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Foundations","chapter":"Set Theory","chapter_id":"sets","section":"The hierarchy of sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000C","source_file":"sets.tex","source_line":129,"source_end_line":132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L129-L132","statement_sha256":"f711d033579785024d091b566f562270badcb8abb05d3bc05a40a579356825e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":0,"rank":0,"depth":0,"x":237.89,"y":220.0,"cluster":"categories-foundations"},{"id":"stacks:05N2","tag":"05N2","title":"Cofinality · Lemma 05N2","summary":"Suppose that T = colim_α < β T_α is a colimit of sets indexed by ordinals less than a given ordinal β. Suppose that φ : S → T is a map of sets. Then φ lifts to a map into T_α for some α < β provided that β is not a limit of ordinals indexed by S, in other words, if β is an ordinal with cf(β) > |S|.","statement_latex":"Suppose that $T = \\colim_{\\alpha < \\beta} T_\\alpha$\nis a colimit of sets indexed by ordinals less than a given ordinal $\\beta$.\nSuppose that $\\varphi : S \\to T$ is a map of sets.\nThen $\\varphi$ lifts to a map into $T_\\alpha$ for some $\\alpha < \\beta$\nprovided that $\\beta$ is not a limit of ordinals indexed by $S$,\nin other words, if $\\beta$ is an ordinal with $\\text{cf}(\\beta) > |S|$.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Cofinality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05N2","source_file":"sets.tex","source_line":192,"source_end_line":200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L192-L200","statement_sha256":"13cfb7cb455fdb111c360aa7a3c4448662debdd28ef78496c7d19e1745187dd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1,"rank":1,"depth":0,"x":219.924,"y":227.754,"cluster":"categories-foundations"},{"id":"stacks:05N3","tag":"05N3","title":"Cofinality · Proposition 05N3","summary":"Let kappa be a cardinal. Then there exists an ordinal whose cofinality is bigger than kappa.","statement_latex":"Let $\\kappa$ be a cardinal. Then there exists an ordinal\nwhose cofinality is bigger than $\\kappa$.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Cofinality","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05N3","source_file":"sets.tex","source_line":216,"source_end_line":220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L216-L220","statement_sha256":"4394e5ab10e6800ebd887a64f5d053a9f64f5f9718ed8d1d9e264732cd4537d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2,"rank":2,"depth":0,"x":231.542,"y":205.238,"cluster":"categories-foundations"},{"id":"stacks:000G","tag":"000G","title":"Reflection principle · Theorem 000G","summary":"Suppose given φ_1(x_1, …, x_n), …, φ_m(x_1, …, x_n) a finite collection of formulas of set theory. Let M_0 be a set. There exists a set M such that M_0 ⊂ M and ∀ x_1, …, x_n ∈ M, we have ∀ i = 1, …, m, φ_i^M(x_1, …, x_n) ⇔ ∀ i = 1, …, m, φ_i(x_1, …, x_n). In fact we may take M = V_α for some limit ordinal α.","statement_latex":"Suppose given $\\phi_1(x_1, \\ldots, x_n), \\ldots, \\phi_m(x_1, \\ldots, x_n)$\na {\\bf finite} collection of\nformulas of set theory. Let $M_0$ be a set.\nThere exists a set $M$ such that\n$M_0 \\subset M$ and\n$\\forall x_1, \\ldots, x_n \\in M$, we have\n$$\n\\forall i = 1, \\ldots, m, \\ \\phi_i^{M}(x_1, \\ldots, x_n)\n\\Leftrightarrow\n\\forall i = 1, \\ldots, m, \\ \\phi_i(x_1, \\ldots, x_n).\n$$\nIn fact we may take $M = V_\\alpha$ for some limit ordinal $\\alpha$.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Reflection principle","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000G","source_file":"sets.tex","source_line":264,"source_end_line":278,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L264-L278","statement_sha256":"f14722a5d0e4336d704b8c85ee5526658d3a424a113d2ad3d17edd22e11771f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3,"rank":3,"depth":0,"x":242.701,"y":233.915,"cluster":"categories-foundations"},{"id":"stacks:000I","tag":"000I","title":"Constructing categories of schemes · Lemma 000I","summary":"For every cardinal kappa, there exists a set A such that every element of A is a scheme and such that for every scheme S with size(S) ≤ kappa, there is an element X ∈ A such that X ≅ S (isomorphism of schemes).","statement_latex":"For every cardinal $\\kappa$, there exists a set $A$ such\nthat every element of $A$ is a scheme and such that for every\nscheme $S$ with $\\text{size}(S) \\leq \\kappa$, there is\nan element $X \\in A$ such that $X \\cong S$ (isomorphism\nof schemes).","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000I","source_file":"sets.tex","source_line":328,"source_end_line":335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L328-L335","statement_sha256":"0e588de4b10b62484d3c336df26eac6cc4ffee64a5ba09598601f930554d63b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4,"rank":4,"depth":0,"x":206.693,"y":216.537,"cluster":"categories-foundations"},{"id":"stacks:000J","tag":"000J","title":"Constructing categories of schemes · Lemma 000J","summary":"With notations size, Bound and Sch_α as above. Let S_0 be a set of schemes. There exists a limit ordinal α with the following properties: • We have S_0 ⊂ V_α; in other words, S_0 ⊂ Ob(Sch_α). • For any S ∈ Ob(Sch_α) and any scheme T with size(T) ≤ Bound(size(S)), there exists a scheme S' ∈ Ob(Sch_α) such that T ≅ S'. • For any countable diagram category I and any functor F : I → Sch_α, the limit lim_I F exists in Sch_α if and only if it exists in Sch and moreover, in this…","statement_latex":"With notations $\\text{size}$, $Bound$ and $\\Sch_\\alpha$ as above.\nLet $S_0$ be a set of schemes. There exists a limit ordinal\n$\\alpha$ with the following properties:\n\\begin{enumerate}\n\\item\n\nWe have $S_0 \\subset V_\\alpha$; in other words,\n$S_0 \\subset \\Ob(\\Sch_\\alpha)$.\n\\item\n\nFor any $S \\in \\Ob(\\Sch_\\alpha)$ and any\nscheme $T$ with $\\text{size}(T) \\leq Bound(\\text{size}(S))$,\nthere exists a scheme $S' \\in \\Ob(\\Sch_\\alpha)$\nsuch that $T \\cong S'$.\n\\item\n\nFor any countable\\footnote{Both the set of objects and\nthe morphism sets are countable. In fact you can prove the lemma with\n$\\aleph_0$ replaced by any cardinal whatsoever in (3) and (4).} diagram\ncategory $\\mathcal{I}$ and\nany functor $F : \\mathcal{I} \\to \\Sch_\\alpha$, the limit\n$\\lim_\\mathcal{I} F$ exists in $\\Sch_\\alpha$ if and\nonly if it exists in $\\Sch$ and moreover, in this case,\nthe natural morphism between them is an isomorphism.\n\\item\n\nFor any countable index category $\\mathcal{I}$ and\nany functor $F : \\mathcal{I} \\to \\Sch_\\alpha$, the colimit\n$\\colim_\\mathcal{I} F$ exists in $\\Sch_\\alpha$ if and\nonly if it exists in $\\Sch$ and moreover, in this case,\nthe natural morphism between them is an isomorphism.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000J","source_file":"sets.tex","source_line":355,"source_end_line":389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L355-L389","statement_sha256":"06cadd4519a6c9eb8484a73d2e13e0f76b9001a22033c97596ea0c86c1b4f43b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5,"rank":5,"depth":1,"x":252.078,"y":208.203,"cluster":"categories-foundations"},{"id":"stacks:000P","tag":"000P","title":"Constructing categories of schemes · Lemma 000P","summary":"Let S be an affine scheme. Let R = Γ(S, O_S). Then the size of S is equal to max( aleph_0, |R|).","statement_latex":"Let $S$ be an affine scheme.\nLet $R = \\Gamma(S, \\mathcal{O}_S)$.\nThen the size of $S$ is equal to $\\max\\{ \\aleph_0, |R|\\}$.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000P","source_file":"sets.tex","source_line":503,"source_end_line":508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L503-L508","statement_sha256":"2b07c9eb63bec4bc7e3a66fef8e2d124b585183d65ea81ea286db30a99e192b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6,"rank":6,"depth":0,"x":222.615,"y":243.076,"cluster":"categories-foundations"},{"id":"stacks:000Q","tag":"000Q","title":"Constructing categories of schemes · Lemma 000Q","summary":"Let S be a scheme. Let S = ⋃_i ∈ I S_i be an open covering. Then size(S) ≤ max(|I|, sup_i(size(S_i))).","statement_latex":"Let $S$ be a scheme. Let $S = \\bigcup_{i \\in I} S_i$ be\nan open covering. Then\n$\\text{size}(S) \\leq \\max\\{|I|, \\sup_i\\{\\text{size}(S_i)\\}\\}$.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000Q","source_file":"sets.tex","source_line":524,"source_end_line":529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L524-L529","statement_sha256":"e8efa96754c926664bf6893ddce3bdeb7322c898df8277b137b9cff92f18a8ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":7,"rank":7,"depth":0,"x":215.916,"y":197.222,"cluster":"categories-foundations"},{"id":"stacks:04T6","tag":"04T6","title":"Constructing categories of schemes · Lemma 04T6","summary":"Let f : X → S, g : Y → S be morphisms of schemes. Then we have size(X ×_S Y) ≤ max(size(X), size(Y)).","statement_latex":"Let $f : X \\to S$, $g : Y \\to S$ be morphisms of schemes.\nThen we have\n$\\text{size}(X \\times_S Y) \\leq \\max\\{\\text{size}(X), \\text{size}(Y)\\}$.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04T6","source_file":"sets.tex","source_line":565,"source_end_line":570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L565-L570","statement_sha256":"2728fde8bdf6aae2ffb15acf6d99ccc85a5fe83ce9f3e54667725a4f8b471936","origin":"The Stacks Project","memory_eligible":false,"source_rank":8,"rank":8,"depth":1,"x":260.556,"y":229.374,"cluster":"categories-foundations"},{"id":"stacks:04T7","tag":"04T7","title":"Constructing categories of schemes · Lemma 04T7","summary":"Let S be a scheme. Let f : X → S be locally of finite type with X quasi-compact. Then size(X) ≤ size(S).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to S$ be locally of finite type with $X$ quasi-compact.\nThen $\\text{size}(X) \\leq \\text{size}(S)$.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04T7","source_file":"sets.tex","source_line":609,"source_end_line":614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L609-L614","statement_sha256":"3adf14d341e74e7af83084c00bbe8a355e156b7e6cdb6b477207184f45abb9ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":9,"rank":9,"depth":1,"x":198.212,"y":231.022,"cluster":"categories-foundations"},{"id":"stacks:04VA","tag":"04VA","title":"Constructing categories of schemes · Lemma 04VA","summary":"Let f : X → Y be a monomorphism of schemes. If at least one of the following properties holds, then size(X) ≤ size(Y): • f is quasi-compact, • f is locally of finite presentation, • add more here as needed. But the bound does not hold for monomorphisms which are locally of finite type.","statement_latex":"Let $f : X \\to Y$ be a monomorphism of schemes.\nIf at least one of the following properties\nholds, then $\\text{size}(X) \\leq \\text{size}(Y)$:\n\\begin{enumerate}\n\\item $f$ is quasi-compact,\n\\item $f$ is locally of finite presentation,\n\\item add more here as needed.\n\\end{enumerate}\nBut the bound does not hold for monomorphisms\nwhich are locally of finite type.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VA","source_file":"sets.tex","source_line":636,"source_end_line":648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L636-L648","statement_sha256":"05ad6c2c0d5b5a51d71f1bf060d1d6b5ce1f842223c21094f6201f5ef9042be8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10,"rank":10,"depth":1,"x":245.324,"y":192.493,"cluster":"categories-foundations"},{"id":"stacks:000R","tag":"000R","title":"Constructing categories of schemes · Lemma 000R","summary":"Let α be an ordinal as in Lemma [Tag 000J] above. The category Sch_α satisfies the following properties: • If X, Y, S ∈ Ob(Sch_α), then for any morphisms f : X → S, g : Y → S the fibre product X ×_S Y in Sch_α exists and is a fibre product in the category of schemes. • Given any at most countable collection S_1, S_2, … of elements of Ob(Sch_α), the coproduct coprod_i S_i exists in Ob(Sch_α) and is a coproduct in the category of schemes. • For any S ∈ Ob(Sch_α) and any…","statement_latex":"Let $\\alpha$ be an ordinal as in Lemma \\ref{lemma-construct-category} above.\nThe category $\\Sch_\\alpha$ satisfies the following\nproperties:\n\\begin{enumerate}\n\\item If $X, Y, S \\in \\Ob(\\Sch_\\alpha)$, then\nfor any morphisms $f : X \\to S$, $g : Y \\to S$ the fibre\nproduct $X \\times_S Y$ in $\\Sch_\\alpha$ exists\nand is a fibre product in the category of schemes.\n\\item Given any at most countable collection $S_1, S_2, \\ldots$\nof elements of $\\Ob(\\Sch_\\alpha)$, the coproduct\n$\\coprod_i S_i$ exists in $\\Ob(\\Sch_\\alpha)$ and\nis a coproduct in the category of schemes.\n\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and\nany open immersion $U \\to S$, there exists a\n$V \\in \\Ob(\\Sch_\\alpha)$ with $V \\cong U$.\n\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and\nany closed immersion $T \\to S$, there exists an\n$S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$.\n\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and\nany finite type morphism $T \\to S$, there exists an\n$S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$.\n\\item Suppose $S$ is a scheme which has an open covering\n$S = \\bigcup_{i \\in I} S_i$ such that there exists\na $T \\in \\Ob(\\Sch_\\alpha)$ with\n(a) $\\text{size}(S_i) \\leq \\text{size}(T)^{\\aleph_0}$ for all\n$i \\in I$, and (b) $|I| \\leq \\text{size}(T)^{\\aleph_0}$.\nThen $S$ is isomorphic to an object of $\\Sch_\\alpha$.\n\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and\nany morphism $f : T \\to S$ locally of finite type such\nthat $T$ can be covered by at most\n$\\text{size}(S)^{\\aleph_0}$ open affines, there exists an\n$S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$.\nFor example this holds if $T$ can be covered by at most\n$|\\mathbf{R}| = 2^{\\aleph_0} = \\aleph_0^{\\aleph_0}$ open affines.\n\\item For any $S \\in \\Ob(\\Sch_\\alpha)$ and\nany monomorphism $T \\to S$ which is either locally of finite presentation\nor quasi-compact, there exists an\n$S' \\in \\Ob(\\Sch_\\alpha)$ with $S' \\cong T$.\n\\item Suppose that $T \\in \\Ob(\\Sch_\\alpha)$ is\naffine. Write $R = \\Gamma(T, \\mathcal{O}_T)$.\nThen any of the following schemes is isomorphic to a scheme\nin $\\Sch_\\alpha$:\n\\begin{enumerate}\n\\item For any ideal $I \\subset R$ with completion\n$R^* = \\lim_n R/I^n$, the scheme $\\Spec(R^*)$.\n\\item For any finite type $R$-algebra $R'$, the\nscheme $\\Spec(R')$.\n\\item For any localization $S^{-1}R$, the scheme $\\Spec(S^{-1}R)$.\n\\item For any prime $\\mathfrak p \\subset R$, the scheme\n$\\Spec(\\overline{\\kappa(\\mathfrak p)})$.\n\\item For any subring $R' \\subset R$, the scheme\n$\\Spec(R')$.\n\\item Any scheme of finite type over a ring of cardinality at most\n$|R|^{\\aleph_0}$.\n\\item And so on.\n\\end{enumerate}\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000R","source_file":"sets.tex","source_line":684,"source_end_line":743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L684-L743","statement_sha256":"1084c915981f72036c9613d82bf3a48b868075712c1101e0992150e7790081cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11,"rank":11,"depth":2,"x":241.324,"y":250.326,"cluster":"categories-foundations"},{"id":"stacks:0AHK","tag":"0AHK","title":"Constructing categories of schemes · Lemma 0AHK","summary":"Let f : X → Y be a morphism of schemes. Assume there exists an fpqc covering (g_j : Y_j → Y)_j ∈ J such that g_j factors through f. Then size(Y) ≤ size(X).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume there exists an\nfpqc covering $\\{g_j : Y_j \\to Y\\}_{j \\in J}$ such that $g_j$ factors\nthrough $f$. Then $\\text{size}(Y) \\leq \\text{size}(X)$.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHK","source_file":"sets.tex","source_line":793,"source_end_line":798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L793-L798","statement_sha256":"a7305ad585cc867038749f3344bd1108b33f3037fbc57fbdef9446fd317f9ac1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12,"rank":12,"depth":0,"x":195.869,"y":203.385,"cluster":"categories-foundations"},{"id":"stacks:0AHL","tag":"0AHL","title":"Constructing categories of schemes · Lemma 0AHL","summary":"Let (f_i : X_i → X)_i ∈ I be an fppf covering of a scheme. There exists an fppf covering (W_j → X)_j ∈ J which is a refinement of (X_i → X)_i ∈ I such that size(coprod W_j) ≤ size(X).","statement_latex":"Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fppf covering of a scheme.\nThere exists an fppf covering $\\{W_j \\to X\\}_{j \\in J}$\nwhich is a refinement of $\\{X_i \\to X\\}_{i \\in I}$ such that\n$\\text{size}(\\coprod W_j) \\leq \\text{size}(X)$.","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Constructing categories of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHL","source_file":"sets.tex","source_line":832,"source_end_line":838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L832-L838","statement_sha256":"3338c44d0912d3113d690a437f519d8d07187b185720ca9fdbca393658868f1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13,"rank":13,"depth":2,"x":270.039,"y":212.606,"cluster":"categories-foundations"},{"id":"stacks:000U","tag":"000U","title":"Sets with group action · Lemma 000U","summary":"With notations G, G-Sets_α, size, and Bound as above. Let S_0 be a set of G-sets. There exists a limit ordinal α with the following properties: • We have S_0 ∪ (_GG) ⊂ Ob(G-Sets_α). • For any S ∈ Ob(G-Sets_α) and any G-set T with size(T) ≤ Bound(size(S)), there exists an S' ∈ Ob(G-Sets_α) that is isomorphic to T. • For any countable index category I and any functor F : I → G-Sets_α, the limit lim_I F and colimit colim_I F exist in G-Sets_α and are the same as in G-Sets.","statement_latex":"With notations $G$, $G\\textit{-Sets}_\\alpha$, $\\text{size}$,\nand $Bound$ as above. Let $S_0$ be a set of $G$-sets.\nThere exists a limit ordinal $\\alpha$ with the following properties:\n\\begin{enumerate}\n\\item We have $S_0 \\cup \\{{}_GG\\} \\subset \\Ob(G\\textit{-Sets}_\\alpha)$.\n\\item For any $S \\in \\Ob(G\\textit{-Sets}_\\alpha)$ and any\n$G$-set $T$ with $\\text{size}(T) \\leq Bound(\\text{size}(S))$,\nthere exists an $S' \\in \\Ob(G\\textit{-Sets}_\\alpha)$\nthat is isomorphic to $T$.\n\\item For any countable index category $\\mathcal{I}$ and\nany functor $F : \\mathcal{I} \\to G\\textit{-Sets}_\\alpha$, the\nlimit $\\lim_\\mathcal{I} F$ and colimit\n$\\colim_\\mathcal{I} F$ exist in $G\\textit{-Sets}_\\alpha$\nand are the same as in $G\\textit{-Sets}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Sets with group action","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000U","source_file":"sets.tex","source_line":868,"source_end_line":885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L868-L885","statement_sha256":"40262f0b81510718ee1dd2ad740925b9c44be42e9466bd25a1180b70f258dac2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14,"rank":14,"depth":2,"x":205.565,"y":249.196,"cluster":"categories-foundations"},{"id":"stacks:000V","tag":"000V","title":"Sets with group action · Lemma 000V","summary":"Let α be an ordinal as in Lemma [Tag 000U] above. The category G-Sets_α satisfies the following properties: • The G-set _GG is an object of G-Sets_α. • (Co)Products, fibre products, and pushouts exist in G-Sets_α and are the same as their counterparts in G-Sets. • Given an object U of G-Sets_α, any G-stable subset O ⊂ U is isomorphic to an object of G-Sets_α.","statement_latex":"Let $\\alpha$ be an ordinal as in Lemma \\ref{lemma-sets-with-group-action}\nabove. The category $G\\textit{-Sets}_\\alpha$ satisfies the following\nproperties:\n\\begin{enumerate}\n\\item The $G$-set ${}_GG$ is an object of $G\\textit{-Sets}_\\alpha$.\n\\item (Co)Products, fibre products, and pushouts\nexist in $G\\textit{-Sets}_\\alpha$\nand are the same as their counterparts in $G\\textit{-Sets}$.\n\\item Given an object $U$ of $G\\textit{-Sets}_\\alpha$,\nany $G$-stable subset $O \\subset U$  is isomorphic to an object\nof $G\\textit{-Sets}_\\alpha$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Sets with group action","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000V","source_file":"sets.tex","source_line":892,"source_end_line":906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L892-L906","statement_sha256":"f6e228773f57426f115b2c61a32025bb59990b31b90dc55209f334f99c75bd8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":15,"rank":15,"depth":3,"x":224.355,"y":183.407,"cluster":"categories-foundations"},{"id":"stacks:000X","tag":"000X","title":"Coverings of a site · Lemma 000X","summary":"With notations as above. Let Cov_0 ⊂ Cov(C) be a set contained in Cov(C). There exist a cardinal kappa and a limit ordinal α with the following properties: • We have Cov_0 ⊂ Cov(C)_kappa, α. • The set of coverings Cov(C)_kappa, α satisfies (1), (2), and (3) of Sites, Definition [Tag 00VH] (see above). In other words (C, Cov(C)_kappa, α) is a site. • Every covering in Cov(C) is combinatorially equivalent to a covering in Cov(C)_kappa, α.","statement_latex":"With notations as above.\nLet $\\text{Cov}_0 \\subset \\text{Cov}(\\mathcal{C})$\nbe a set contained in $\\text{Cov}(\\mathcal{C})$.\nThere exist a cardinal $\\kappa$ and a limit ordinal $\\alpha$\nwith the following properties:\n\\begin{enumerate}\n\\item We have $\\text{Cov}_0 \\subset \\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$.\n\\item The set of coverings\n$\\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$ satisfies\n(1), (2), and (3) of Sites, Definition \\ref{sites-definition-site} (see above).\nIn other words $(\\mathcal{C}, \\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha})$\nis a site.\n\\item Every covering in $\\text{Cov}(\\mathcal{C})$\nis combinatorially equivalent\nto a covering in $\\text{Cov}(\\mathcal{C})_{\\kappa, \\alpha}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Coverings of a site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/000X","source_file":"sets.tex","source_line":952,"source_end_line":970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L952-L970","statement_sha256":"f15e27e82d5a906896d129c06a44216ab101c79f2b5174ed87dddf7a5b86ad07","origin":"The Stacks Project","memory_eligible":false,"source_rank":16,"rank":16,"depth":1,"x":264.656,"y":244.535,"cluster":"categories-foundations"},{"id":"stacks:0010","tag":"0010","title":"Abelian categories and injectives · Lemma 0010","summary":"Suppose given a big category A (see Categories, Remark [Tag 0015]). Assume A is abelian and has enough injectives. See Homology, Definitions [Tag 0109] and [Tag 0138]. Then for any given set of objects (A_s)_s∈ S of A, there is an abelian subcategory A' ⊂ A with the following properties: • the inclusion functor A' → A is exact, • Ob(A') is a set, • Ob(A') contains A_s for each s ∈ S, • A' has enough injectives, and • an object of A' is injective if and only if it is an…","statement_latex":"Suppose given a big category $\\mathcal{A}$ (see\nCategories, Remark \\ref{categories-remark-big-categories}).\nAssume $\\mathcal{A}$ is abelian and has enough injectives.\nSee Homology, Definitions \\ref{homology-definition-abelian-category}\nand \\ref{homology-definition-enough-injectives}.\nThen for any given set of objects $\\{A_s\\}_{s\\in S}$\nof $\\mathcal{A}$, there is an abelian subcategory\n$\\mathcal{A}' \\subset \\mathcal{A}$\nwith the following properties:\n\\begin{enumerate}\n\\item the inclusion functor $\\mathcal{A}' \\to \\mathcal{A}$ is exact,\n\\item $\\Ob(\\mathcal{A}')$ is a set,\n\\item $\\Ob(\\mathcal{A}')$ contains $A_s$ for each $s \\in S$,\n\\item $\\mathcal{A}'$ has enough injectives, and\n\\item an object of $\\mathcal{A}'$ is injective if and only if it\nis an injective object of $\\mathcal{A}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Set Theory","chapter_id":"sets","section":"Abelian categories and injectives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0010","source_file":"sets.tex","source_line":1131,"source_end_line":1150,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sets.tex#L1131-L1150","statement_sha256":"024ad0cb730d1c7f0e6f4fd2461b273bd0ee2174afbd194092f6a25ea322b656","origin":"The Stacks Project","memory_eligible":false,"source_rank":17,"rank":17,"depth":0,"x":183.364,"y":221.62,"cluster":"categories-foundations"},{"id":"stacks:0014","tag":"0014","title":"Definitions · Definition 0014","summary":"A category C consists of the following data: • A set of objects Ob(C). • For each pair x, y ∈ Ob(C) a set of morphisms Mor_C(x, y). • For each triple x, y, z∈ Ob(C) a composition map Mor_C(y, z) × Mor_C(x, y) → Mor_C(x, z) , denoted (φ, ψ) ↦ φ ∘ ψ. These data are to satisfy the following rules: • For every element x∈ Ob(C) there exists a morphism id_x∈ Mor_C(x, x) such that id_x ∘ φ = φ and ψ ∘ id_x = ψ whenever these compositions make sense. • Composition is associative,…","statement_latex":"A {\\it category} $\\mathcal{C}$ consists of the following data:\n\\begin{enumerate}\n\\item A set of objects $\\Ob(\\mathcal{C})$.\n\\item For each pair $x, y \\in \\Ob(\\mathcal{C})$ a set of morphisms\n$\\Mor_\\mathcal{C}(x, y)$.\n\\item For each triple $x, y, z\\in \\Ob(\\mathcal{C})$ a composition\nmap $ \\Mor_\\mathcal{C}(y, z) \\times \\Mor_\\mathcal{C}(x, y)\n\\to \\Mor_\\mathcal{C}(x, z) $, denoted $(\\phi, \\psi) \\mapsto\n\\phi \\circ \\psi$.\n\\end{enumerate}\nThese data are to satisfy the following rules:\n\\begin{enumerate}\n\\item For every element $x\\in \\Ob(\\mathcal{C})$ there exists a\nmorphism $\\text{id}_x\\in \\Mor_\\mathcal{C}(x, x)$ such that\n$\\text{id}_x \\circ \\phi = \\phi$ and $\\psi \\circ \\text{id}_x = \\psi $ whenever\nthese compositions make sense.\n\\item Composition is associative, i.e., $(\\phi \\circ \\psi) \\circ \\chi =\n\\phi \\circ ( \\psi \\circ \\chi)$ whenever these compositions make sense.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0014","source_file":"categories.tex","source_line":37,"source_end_line":58,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L37-L58","statement_sha256":"08e66edab81047f61eac5477b9c67a17aee88837b3de87936afa776e9f2b1c0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":18,"rank":18,"depth":0,"x":264.017,"y":191.565,"cluster":"categories-foundations"},{"id":"stacks:0017","tag":"0017","title":"Definitions · Definition 0017","summary":"A morphism φ : x → y is an isomorphism of the category C if there exists a morphism ψ : y → x such that φ ∘ ψ = id_y and ψ ∘ φ = id_x.","statement_latex":"A morphism $\\phi : x \\to y$ is an {\\it isomorphism} of the category\n$\\mathcal{C}$ if there exists a morphism $\\psi : y \\to x$\nsuch that $\\phi \\circ \\psi = \\text{id}_y$ and\n$\\psi \\circ \\phi = \\text{id}_x$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0017","source_file":"categories.tex","source_line":124,"source_end_line":130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L124-L130","statement_sha256":"ef019f4855c20b8a92a866b36879d83c358981e03e1bceafbd32cf288282bec3","origin":"The Stacks Project","memory_eligible":false,"source_rank":19,"rank":19,"depth":0,"x":227.724,"y":261.343,"cluster":"categories-foundations"},{"id":"stacks:0018","tag":"0018","title":"Definitions · Definition 0018","summary":"A groupoid is a category where every morphism is an isomorphism.","statement_latex":"A {\\it groupoid} is a category where every morphism is an isomorphism.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0018","source_file":"categories.tex","source_line":141,"source_end_line":144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L141-L144","statement_sha256":"9ebec151896fbf8d679e68d73f7b3c86f626e206b20e360a078580ab725e5b84","origin":"The Stacks Project","memory_eligible":false,"source_rank":20,"rank":20,"depth":0,"x":197.633,"y":187.419,"cluster":"categories-foundations"},{"id":"stacks:001B","tag":"001B","title":"Definitions · Definition 001B","summary":"A functor F : A → B between two categories A, B is given by the following data: • A map F : Ob(A) → Ob(B). • For every x, y ∈ Ob(A) a map F : Mor_A(x, y) → Mor_B(F(x), F(y)), denoted φ ↦ F(φ). These data should be compatible with composition and identity morphisms in the following manner: F(φ ∘ ψ) = F(φ) ∘ F(ψ) for a composable pair (φ, ψ) of morphisms of A and F(id_x) = id_F(x).","statement_latex":"A {\\it functor} $F : \\mathcal{A} \\to \\mathcal{B}$\nbetween two categories $\\mathcal{A}, \\mathcal{B}$ is given by the\nfollowing data:\n\\begin{enumerate}\n\\item A map $F : \\Ob(\\mathcal{A}) \\to \\Ob(\\mathcal{B})$.\n\\item For every $x, y \\in \\Ob(\\mathcal{A})$ a map\n$F : \\Mor_\\mathcal{A}(x, y) \\to \\Mor_\\mathcal{B}(F(x), F(y))$,\ndenoted $\\phi \\mapsto F(\\phi)$.\n\\end{enumerate}\nThese data should be compatible with composition and identity morphisms\nin the following manner: $F(\\phi \\circ \\psi) =\nF(\\phi) \\circ F(\\psi)$ for a composable pair $(\\phi, \\psi)$ of\nmorphisms of $\\mathcal{A}$ and $F(\\text{id}_x) = \\text{id}_{F(x)}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001B","source_file":"categories.tex","source_line":162,"source_end_line":177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L162-L177","statement_sha256":"79a37a0070a3cfe6232231f7e6ded8529c1e90dec8ba364704a0ccc3ab2521b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":21,"rank":21,"depth":0,"x":281.274,"y":225.795,"cluster":"categories-foundations"},{"id":"stacks:001C","tag":"001C","title":"Definitions · Definition 001C","summary":"Let F : A → B be a functor. • We say F is faithful if for any objects x, y ∈ Ob(A) the map F : Mor_A(x, y) → Mor_B(F(x), F(y)) is injective. • If these maps are all bijective then F is called fully faithful. • The functor F is called essentially surjective if for any object y ∈ Ob(B) there exists an object x ∈ Ob(A) such that F(x) is isomorphic to y in B.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\n\\begin{enumerate}\n\\item We say $F$ is {\\it faithful} if\nfor any objects $x, y \\in \\Ob(\\mathcal{A})$ the map\n$$\nF : \\Mor_\\mathcal{A}(x, y) \\to \\Mor_\\mathcal{B}(F(x), F(y))\n$$\nis injective.\n\\item If these maps are all bijective then $F$ is called\n{\\it fully faithful}.\n\\item\nThe functor $F$ is called {\\it essentially surjective} if for any\nobject $y \\in \\Ob(\\mathcal{B})$ there exists an object\n$x \\in \\Ob(\\mathcal{A})$ such that $F(x)$ is isomorphic to $y$ in\n$\\mathcal{B}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001C","source_file":"categories.tex","source_line":187,"source_end_line":205,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L187-L205","statement_sha256":"3fb1843e85cf301929f1e26d3731cd872ca5f69dbb5847583e2901fb1754e897","origin":"The Stacks Project","memory_eligible":false,"source_rank":22,"rank":22,"depth":0,"x":186.556,"y":245.391,"cluster":"categories-foundations"},{"id":"stacks:001D","tag":"001D","title":"Definitions · Definition 001D","summary":"A subcategory of a category B is a category A whose objects and arrows form subsets of the objects and arrows of B and such that source, target and composition in A agree with those of B and such that the identity morphism of an object of A matches the one in B. We say A is a full subcategory of B if Mor_A(x, y) = Mor_B(x, y) for all x, y ∈ Ob(A). We say A is a strictly full subcategory of B if it is a full subcategory and given x ∈ Ob(A) any object of B which is…","statement_latex":"A {\\it subcategory} of a category $\\mathcal{B}$ is a category $\\mathcal{A}$\nwhose objects and arrows form subsets of the objects and arrows of $\\mathcal{B}$\nand such that source, target and composition in $\\mathcal{A}$ agree with those\nof $\\mathcal{B}$ and such that the identity morphism of an object of\n$\\mathcal{A}$ matches the one in $\\mathcal{B}$. We say $\\mathcal{A}$ is a\n{\\it full subcategory} of $\\mathcal{B}$ if $\\Mor_\\mathcal{A}(x, y)\n= \\Mor_\\mathcal{B}(x, y)$ for all $x, y \\in \\Ob(\\mathcal{A})$.\nWe say $\\mathcal{A}$ is a {\\it strictly full} subcategory of $\\mathcal{B}$\nif it is a full subcategory and given $x \\in \\Ob(\\mathcal{A})$ any\nobject of $\\mathcal{B}$ which is isomorphic to $x$ is also in $\\mathcal{A}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001D","source_file":"categories.tex","source_line":207,"source_end_line":219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L207-L219","statement_sha256":"13883b85fbb776236dfba40f9b797e40a52b1c800303846173f196f0bd1d9e08","origin":"The Stacks Project","memory_eligible":false,"source_rank":23,"rank":23,"depth":0,"x":241.871,"y":175.674,"cluster":"categories-foundations"},{"id":"stacks:001I","tag":"001I","title":"Definitions · Definition 001I","summary":"Let F, G : A → B be functors. A natural transformation, or a morphism of functors t : F → G, is a collection (t_x)_x∈ Ob(A) such that • t_x : F(x) → G(x) is a morphism in the category B, and • for every morphism φ : x → y of A the following diagram is commutative xymatrix F(x) ar[r]^t_x ar[d]_F(φ) & G(x) ar[d]^G(φ) F(y) ar[r]^t_y & G(y)","statement_latex":"Let $F, G : \\mathcal{A} \\to \\mathcal{B}$ be functors.\nA {\\it natural transformation}, or a {\\it morphism of functors}\n$t : F \\to G$, is a collection $\\{t_x\\}_{x\\in \\Ob(\\mathcal{A})}$\nsuch that\n\\begin{enumerate}\n\\item $t_x : F(x) \\to G(x)$ is a morphism in the category $\\mathcal{B}$, and\n\\item for every morphism $\\phi : x \\to y$ of $\\mathcal{A}$ the following\ndiagram is commutative\n$$\n\\xymatrix{\nF(x) \\ar[r]^{t_x} \\ar[d]_{F(\\phi)} & G(x) \\ar[d]^{G(\\phi)} \\\\\nF(y) \\ar[r]^{t_y} & G(y) }\n$$\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001I","source_file":"categories.tex","source_line":290,"source_end_line":306,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L290-L306","statement_sha256":"f41c24db067fd313435bc1592edd111f3ba0e5b616661935e3b193e52fa92f56","origin":"The Stacks Project","memory_eligible":false,"source_rank":24,"rank":24,"depth":0,"x":257.458,"y":260.251,"cluster":"categories-foundations"},{"id":"stacks:001J","tag":"001J","title":"Definitions · Definition 001J","summary":"An equivalence of categories F : A → B is a functor such that there exists a functor G : B → A such that the compositions F ∘ G and G ∘ F are isomorphic to the identity functors id_B, respectively id_A. In this case we say that G is a quasi-inverse to F.","statement_latex":"An {\\it equivalence of categories}\n$F : \\mathcal{A} \\to \\mathcal{B}$ is a functor such that there\nexists a functor $G : \\mathcal{B} \\to \\mathcal{A}$ such that\nthe compositions $F \\circ G$ and $G \\circ F$ are isomorphic to the\nidentity functors $\\text{id}_\\mathcal{B}$,\nrespectively $\\text{id}_\\mathcal{A}$.\nIn this case we say that $G$ is a {\\it quasi-inverse} to $F$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001J","source_file":"categories.tex","source_line":350,"source_end_line":359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L350-L359","statement_sha256":"12eade92fe27a4eebd3a958d5cb9b035c11f08c41dd1874845db06f658866b33","origin":"The Stacks Project","memory_eligible":false,"source_rank":25,"rank":25,"depth":0,"x":176.322,"y":205.615,"cluster":"categories-foundations"},{"id":"stacks:05SG","tag":"05SG","title":"Definitions · Lemma 05SG","summary":"Let F : A → B be a fully faithful functor. Suppose for every X ∈ Ob(B) we are given an object j(X) of A and an isomorphism i_X : X → F(j(X)). Then there is a unique functor j : B → A such that j extends the rule on objects, and the isomorphisms i_X define an isomorphism of functors id_B → F ∘ j. Moreover, j and F are quasi-inverse equivalences of categories.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a fully faithful functor.\nSuppose for every $X \\in \\Ob(\\mathcal{B})$ we are given an\nobject $j(X)$ of $\\mathcal{A}$ and an isomorphism $i_X : X \\to F(j(X))$.\nThen there is a unique functor $j : \\mathcal{B} \\to \\mathcal{A}$\nsuch that $j$ extends the rule on objects, and the isomorphisms\n$i_X$ define an isomorphism of functors\n$\\text{id}_\\mathcal{B} \\to F \\circ j$. Moreover, $j$ and $F$\nare quasi-inverse equivalences of categories.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SG","source_file":"categories.tex","source_line":361,"source_end_line":371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L361-L371","statement_sha256":"6ac7f260d80db61960911b2a0aa185068a78798dd0edfd25795072725da93d0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":26,"rank":26,"depth":0,"x":282.141,"y":199.764,"cluster":"categories-foundations"},{"id":"stacks:02C3","tag":"02C3","title":"Definitions · Lemma 02C3","summary":"A functor is an equivalence of categories if and only if it is both fully faithful and essentially surjective.","statement_latex":"A functor is an equivalence of categories if and only if it is both fully\nfaithful and essentially surjective.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02C3","source_file":"categories.tex","source_line":402,"source_end_line":406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L402-L406","statement_sha256":"9c9390484bf7dfb4b10986978fa196d58b6634dc41291fc84f358850583a7bff","origin":"The Stacks Project","memory_eligible":false,"source_rank":27,"rank":27,"depth":1,"x":207.411,"y":265.339,"cluster":"categories-foundations"},{"id":"stacks:001K","tag":"001K","title":"Definitions · Definition 001K","summary":"Let A, B be categories. We define the product category A × B to be the category with objects Ob(A × B) = Ob(A) × Ob(B) and Mor_A × B((x, y), (x', y')) := Mor_A(x, x')× Mor_B(y, y'). Composition is defined componentwise.","statement_latex":"Let $\\mathcal{A}$, $\\mathcal{B}$ be categories.\nWe define the {\\it product category}\n$\\mathcal{A} \\times \\mathcal{B}$ to be the category with\nobjects\n$\\Ob(\\mathcal{A} \\times \\mathcal{B}) =\n\\Ob(\\mathcal{A}) \\times \\Ob(\\mathcal{B})$\nand\n$$\n\\Mor_{\\mathcal{A} \\times \\mathcal{B}}((x, y), (x', y'))\n:=\n\\Mor_\\mathcal{A}(x, x')\\times\n\\Mor_\\mathcal{B}(y, y').\n$$\nComposition is defined componentwise.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001K","source_file":"categories.tex","source_line":418,"source_end_line":434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L418-L434","statement_sha256":"a23478913724309b32dc1aa576e544912116b24736b7a76e2eb82bff85d9a4d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":28,"rank":28,"depth":0,"x":209.84,"y":172.918,"cluster":"categories-foundations"},{"id":"stacks:001M","tag":"001M","title":"Opposite Categories and the Yoneda Lemma · Definition 001M","summary":"Given a category C the opposite category C^opp is the category with the same objects as C but all morphisms reversed.","statement_latex":"Given a category $\\mathcal{C}$ the {\\it opposite category}\n$\\mathcal{C}^{opp}$ is the category with the same objects\nas $\\mathcal{C}$ but all morphisms reversed.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Opposite Categories and the Yoneda Lemma","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001M","source_file":"categories.tex","source_line":440,"source_end_line":445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L440-L445","statement_sha256":"a6e80f9d4aca58141050c46cc5cfa6476363ad6ae4d52b97f9deba791fb9cdfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":29,"rank":29,"depth":0,"x":283.644,"y":243.683,"cluster":"categories-foundations"},{"id":"stacks:001N","tag":"001N","title":"Opposite Categories and the Yoneda Lemma · Definition 001N","summary":"Let C, S be categories. A contravariant functor F from C to S is a functor C^opp→ S.","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{S}$ be categories.\nA {\\it contravariant} functor $F$\nfrom $\\mathcal{C}$ to $\\mathcal{S}$\nis a functor $\\mathcal{C}^{opp}\\to \\mathcal{S}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Opposite Categories and the Yoneda Lemma","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001N","source_file":"categories.tex","source_line":458,"source_end_line":464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L458-L464","statement_sha256":"774974ccef0568a65601554183beb368b2f51ec8978c7ebd11bc96ed723264d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":30,"rank":30,"depth":0,"x":170.415,"y":233.193,"cluster":"categories-foundations"},{"id":"stacks:02X6","tag":"02X6","title":"Opposite Categories and the Yoneda Lemma · Definition 02X6","summary":"Let C be a category. • A presheaf of sets on C or simply a presheaf is a contravariant functor F from C to Sets. • The category of presheaves is denoted PSh(C).","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item A {\\it presheaf of sets on $\\mathcal{C}$}\nor simply a {\\it presheaf} is a contravariant functor\n$F$ from $\\mathcal{C}$ to $\\textit{Sets}$.\n\\item The category of presheaves is denoted $\\textit{PSh}(\\mathcal{C})$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Opposite Categories and the Yoneda Lemma","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02X6","source_file":"categories.tex","source_line":477,"source_end_line":486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L477-L486","statement_sha256":"779b312cbe76b3497e1b609c44e18c85d281108b0a53f49882522ebcb4978849","origin":"The Stacks Project","memory_eligible":false,"source_rank":31,"rank":31,"depth":0,"x":263.868,"y":175.755,"cluster":"categories-foundations"},{"id":"stacks:001P","tag":"001P","title":"Yoneda lemma · Lemma 001P","summary":"Appeared in some form in [Yoneda-homology]. Used by Grothendieck in a generalized form in [Gr-II]. Let U, V ∈ Ob(C). Given any morphism of functors s : h_U → h_V there is a unique morphism φ : U → V such that h(φ) = s. In other words the functor h is fully faithful. More generally, given any contravariant functor F and any object U of C we have a natural bijection Mor_PSh(C)(h_U, F) → F(U), s ↦ s_U(id_U).","statement_latex":"\\begin{reference}\nAppeared in some form in \\cite{Yoneda-homology}. Used by Grothendieck in a\ngeneralized form in \\cite{Gr-II}.\n\\end{reference}\nLet $U, V \\in \\Ob(\\mathcal{C})$.\nGiven any morphism of functors $s : h_U \\to h_V$\nthere is a unique morphism $\\phi : U \\to V$\nsuch that $h(\\phi) = s$. In other words the\nfunctor $h$ is fully faithful. More generally,\ngiven any contravariant functor $F$ and any object\n$U$ of $\\mathcal{C}$ we have a natural bijection\n$$\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, F) \\longrightarrow F(U),\n\\quad\ns \\longmapsto s_U(\\text{id}_U).\n$$","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Opposite Categories and the Yoneda Lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001P","source_file":"categories.tex","source_line":540,"source_end_line":558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L540-L558","statement_sha256":"149c7284169afbb11d32e3a06f0e68e47ca7ba4049ab69eb6d940eaa32a8b782","origin":"The Stacks Project","memory_eligible":false,"source_rank":32,"rank":32,"depth":0,"x":240.774,"y":272.659,"cluster":"categories-foundations"},{"id":"stacks:001Q","tag":"001Q","title":"Opposite Categories and the Yoneda Lemma · Definition 001Q","summary":"A contravariant functor F : C→ Sets is said to be representable if it is isomorphic to the functor of points h_U for some object U of C.","statement_latex":"A contravariant functor $F : \\mathcal{C}\\to \\textit{Sets}$ is said\nto be {\\it representable} if it is isomorphic to the functor of\npoints $h_U$ for some object $U$ of $\\mathcal{C}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Opposite Categories and the Yoneda Lemma","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001Q","source_file":"categories.tex","source_line":568,"source_end_line":573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L568-L573","statement_sha256":"c1c745303febff11e67207424d2dc1405223b01c5c836e9cc0592d21fd500af9","origin":"The Stacks Project","memory_eligible":false,"source_rank":33,"rank":33,"depth":0,"x":178.942,"y":186.785,"cluster":"categories-foundations"},{"id":"stacks:001S","tag":"001S","title":"Products of pairs · Definition 001S","summary":"Let x, y∈ Ob(C). A product of x and y is an object x × y ∈ Ob(C) together with morphisms p∈ Mor_ C(x × y, x) and q∈Mor_ C(x × y, y) such that the following universal property holds: for any w∈ Ob(C) and morphisms α ∈ Mor_ C(w, x) and β ∈ Mor_C(w, y) there is a unique γ∈ Mor_ C(w, x × y) making the diagram xymatrix w ar[rrrd]^β ar@-->[rrd]_γ ar[rrdd]_α & & & & x × y ar[d]_p ar[r]_q & y & & x & commute.","statement_latex":"Let $x, y\\in \\Ob(\\mathcal{C})$.\nA {\\it product} of $x$ and $y$ is\nan object $x \\times y \\in \\Ob(\\mathcal{C})$\ntogether with morphisms\n$p\\in \\Mor_{\\mathcal C}(x \\times y, x)$ and\n$q\\in\\Mor_{\\mathcal C}(x \\times y, y)$ such\nthat the following universal property holds: for\nany $w\\in \\Ob(\\mathcal{C})$ and morphisms\n$\\alpha \\in \\Mor_{\\mathcal C}(w, x)$ and\n$\\beta \\in \\Mor_\\mathcal{C}(w, y)$\nthere is a unique\n$\\gamma\\in \\Mor_{\\mathcal C}(w, x \\times y)$ making\nthe diagram\n$$\n\\xymatrix{\nw \\ar[rrrd]^\\beta \\ar@{-->}[rrd]_\\gamma \\ar[rrdd]_\\alpha & & \\\\\n& & x \\times y \\ar[d]_p \\ar[r]_q & y \\\\\n& & x &\n}\n$$\ncommute.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Products of pairs","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001S","source_file":"categories.tex","source_line":601,"source_end_line":624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L601-L624","statement_sha256":"1a7f905ecdc0c806b8fef5c85bc1a508c2515c34eb6e9b7c40467b1d8ae5e572","origin":"The Stacks Project","memory_eligible":false,"source_rank":34,"rank":34,"depth":0,"x":295.312,"y":215.455,"cluster":"categories-foundations"},{"id":"stacks:001T","tag":"001T","title":"Products of pairs · Definition 001T","summary":"We say the category C has products of pairs of objects if a product x × y exists for any x, y ∈ Ob(C).","statement_latex":"We say the category $\\mathcal{C}$ {\\it has products of pairs\nof objects} if a product $x \\times y$\nexists for any $x, y \\in \\Ob(\\mathcal{C})$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Products of pairs","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001T","source_file":"categories.tex","source_line":638,"source_end_line":643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L638-L643","statement_sha256":"cec019d57bed7fa19adf687a9220eed696362b062d3d2b12f330ed8bcffa62bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":35,"rank":35,"depth":0,"x":184.855,"y":260.992,"cluster":"categories-foundations"},{"id":"stacks:04AO","tag":"04AO","title":"Coproducts of pairs · Definition 04AO","summary":"Let x, y ∈ Ob(C). A coproduct, or amalgamated sum of x and y is an object x amalg y ∈ Ob(C) together with morphisms i ∈ Mor_ C(x, x amalg y) and j ∈ Mor_ C(y, x amalg y) such that the following universal property holds: for any w ∈ Ob(C) and morphisms α ∈ Mor_ C(x, w) and β ∈ Mor_C(y, w) there is a unique γ ∈ Mor_ C(x amalg y, w) making the diagram xymatrix & y ar[d]^j ar[rrdd]^β x ar[r]^i ar[rrrd]_α & x amalg y ar@-->[rrd]^γ & & & w commute.","statement_latex":"Let $x, y \\in \\Ob(\\mathcal{C})$.\nA {\\it coproduct}, or {\\it amalgamated sum} of $x$ and $y$ is\nan object $x \\amalg y \\in \\Ob(\\mathcal{C})$\ntogether with morphisms\n$i \\in \\Mor_{\\mathcal C}(x, x \\amalg y)$ and\n$j \\in \\Mor_{\\mathcal C}(y, x \\amalg y)$ such\nthat the following universal property holds: for\nany $w \\in \\Ob(\\mathcal{C})$ and morphisms\n$\\alpha \\in \\Mor_{\\mathcal C}(x, w)$ and\n$\\beta \\in \\Mor_\\mathcal{C}(y, w)$\nthere is a unique\n$\\gamma \\in \\Mor_{\\mathcal C}(x \\amalg y, w)$ making\nthe diagram\n$$\n\\xymatrix{\n& y \\ar[d]^j \\ar[rrdd]^\\beta \\\\\nx \\ar[r]^i \\ar[rrrd]_\\alpha & x \\amalg y \\ar@{-->}[rrd]^\\gamma \\\\\n& & & w\n}\n$$\ncommute.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Coproducts of pairs","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AO","source_file":"categories.tex","source_line":658,"source_end_line":681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L658-L681","statement_sha256":"acc202f79370bfe60d539be1a8bb9943e8546912d954e956e484ca56feb2d2b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":36,"rank":36,"depth":0,"x":230.329,"y":163.377,"cluster":"categories-foundations"},{"id":"stacks:04AP","tag":"04AP","title":"Coproducts of pairs · Definition 04AP","summary":"We say the category C has coproducts of pairs of objects if a coproduct x amalg y exists for any x, y ∈ Ob(C).","statement_latex":"We say the category $\\mathcal{C}$ {\\it has coproducts of pairs\nof objects} if a coproduct $x \\amalg y$\nexists for any $x, y \\in \\Ob(\\mathcal{C})$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Coproducts of pairs","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AP","source_file":"categories.tex","source_line":695,"source_end_line":700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L695-L700","statement_sha256":"ded89c701ef10818368d0ef303aa749c13a17cd27c002159664c58abcaecc873","origin":"The Stacks Project","memory_eligible":false,"source_rank":37,"rank":37,"depth":0,"x":275.907,"y":262.509,"cluster":"categories-foundations"},{"id":"stacks:001V","tag":"001V","title":"Fibre products · Definition 001V","summary":"Let x, y, z∈ Ob(C), f∈ Mor_C(x, y) and g∈ Mor_ C(z, y). A fibre product of f and g is an object x ×_y z∈ Ob(C) together with morphisms p ∈ Mor_ C(x ×_y z, x) and q ∈ Mor_ C(x ×_y z, z) making the diagram xymatrix x ×_y z ar[r]_q ar[d]_p & z ar[d]^g x ar[r]^f & y commute, and such that the following universal property holds: for any w∈ Ob(C) and morphisms α ∈ Mor_ C(w, x) and β ∈ Mor_C(w, z) with f ∘ α = g ∘ β there is a unique γ ∈ Mor_ C(w, x ×_y z) making the diagram…","statement_latex":"Let $x, y, z\\in \\Ob(\\mathcal{C})$,\n$f\\in \\Mor_\\mathcal{C}(x, y)$\nand $g\\in \\Mor_{\\mathcal C}(z, y)$.\nA {\\it fibre product} of $f$ and $g$ is\nan object $x \\times_y z\\in \\Ob(\\mathcal{C})$\ntogether with morphisms\n$p \\in \\Mor_{\\mathcal C}(x \\times_y z, x)$ and\n$q \\in \\Mor_{\\mathcal C}(x \\times_y z, z)$ making the diagram\n$$\n\\xymatrix{\nx \\times_y z \\ar[r]_q \\ar[d]_p & z \\ar[d]^g \\\\\nx \\ar[r]^f & y\n}\n$$\ncommute, and such that the following universal property holds: for\nany $w\\in \\Ob(\\mathcal{C})$ and morphisms\n$\\alpha \\in \\Mor_{\\mathcal C}(w, x)$ and\n$\\beta \\in \\Mor_\\mathcal{C}(w, z)$ with\n$f \\circ \\alpha = g \\circ \\beta$\nthere is a unique\n$\\gamma \\in \\Mor_{\\mathcal C}(w, x \\times_y z)$ making\nthe diagram\n$$\n\\xymatrix{\nw \\ar[rrrd]^\\beta \\ar@{-->}[rrd]_\\gamma \\ar[rrdd]_\\alpha & & \\\\\n& & x \\times_y z \\ar[d]^p \\ar[r]_q & z \\ar[d]^g \\\\\n& & x \\ar[r]^f & y\n}\n$$\ncommute.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001V","source_file":"categories.tex","source_line":715,"source_end_line":747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L715-L747","statement_sha256":"e9aa6ef72d7a5becc263d1e6305144f3051cf2e80f92a36eae58251249dcbab9","origin":"The Stacks Project","memory_eligible":false,"source_rank":38,"rank":38,"depth":0,"x":161.064,"y":214.633,"cluster":"categories-foundations"},{"id":"stacks:08N0","tag":"08N0","title":"Fibre products · Definition 08N0","summary":"We say a commutative diagram xymatrix w ar[r] ar[d] & z ar[d] x ar[r] & y in a category is cartesian if w and the morphisms w → x and w → z form a fibre product of the morphisms x → y and z → y.","statement_latex":"We say a commutative diagram\n$$\n\\xymatrix{\nw \\ar[r] \\ar[d] &\nz \\ar[d] \\\\\nx \\ar[r] &\ny\n}\n$$\nin a category is {\\it cartesian} if $w$ and the morphisms $w \\to x$ and\n$w \\to z$ form a fibre product of the morphisms $x \\to y$ and $z \\to y$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08N0","source_file":"categories.tex","source_line":761,"source_end_line":774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L761-L774","statement_sha256":"89f1eb786e06be6751b4ca1adce2ac42848d8a402218eb320cde10e1b8925031","origin":"The Stacks Project","memory_eligible":false,"source_rank":39,"rank":39,"depth":0,"x":285.858,"y":184.389,"cluster":"categories-foundations"},{"id":"stacks:001W","tag":"001W","title":"Fibre products · Definition 001W","summary":"We say the category C has fibre products if the fibre product exists for any f∈ Mor_ C(x, y) and g∈ Mor_ C(z, y).","statement_latex":"We say the category $\\mathcal{C}$ {\\it has fibre products} if\nthe fibre product exists for any $f\\in \\Mor_{\\mathcal C}(x, y)$\nand $g\\in \\Mor_{\\mathcal C}(z, y)$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001W","source_file":"categories.tex","source_line":776,"source_end_line":781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L776-L781","statement_sha256":"ad3f357130749624237d2fb1831b083f84465a1a1845593a8dc1749e03c4f64a","origin":"The Stacks Project","memory_eligible":false,"source_rank":40,"rank":40,"depth":0,"x":217.291,"y":278.682,"cluster":"categories-foundations"},{"id":"stacks:001X","tag":"001X","title":"Fibre products · Definition 001X","summary":"A morphism f : x → y of a category C is said to be representable if for every morphism z → y in C the fibre product x ×_y z exists.","statement_latex":"A morphism $f : x \\to y$ of a category $\\mathcal{C}$ is said to be\n{\\it representable} if for every morphism $z \\to y$\nin $\\mathcal{C}$ the fibre product $x \\times_y z$ exists.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001X","source_file":"categories.tex","source_line":783,"source_end_line":788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L783-L788","statement_sha256":"29ba8b32c11565d76916d0c3ccdf36b0f6fe25db91e7b2577501ff869fb39a42","origin":"The Stacks Project","memory_eligible":false,"source_rank":41,"rank":41,"depth":0,"x":191.718,"y":168.899,"cluster":"categories-foundations"},{"id":"stacks:001Y","tag":"001Y","title":"Fibre products · Lemma 001Y","summary":"Let C be a category. Let f : x → y, and g : y → z be representable. Then g ∘ f : x → z is representable.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $f : x \\to y$, and $g : y \\to z$ be representable.\nThen $g \\circ f : x \\to z$ is representable.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001Y","source_file":"categories.tex","source_line":790,"source_end_line":795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L790-L795","statement_sha256":"872dd631094b6e5b7ed6399c1373602436c82b40520f678b07f74ab7e0d664fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":42,"rank":42,"depth":1,"x":300.152,"y":236.15,"cluster":"categories-foundations"},{"id":"stacks:001Z","tag":"001Z","title":"Fibre products · Lemma 001Z","summary":"Let C be a category. Let f : x → y be representable. Let y' → y be a morphism of C. Then the morphism x' := x ×_y y' → y' is representable also.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $f : x \\to y$ be representable.\nLet $y' \\to y$ be a morphism of $\\mathcal{C}$.\nThen the morphism $x' := x \\times_y y' \\to y'$ is representable also.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/001Z","source_file":"categories.tex","source_line":866,"source_end_line":872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L866-L872","statement_sha256":"4a0e1c8da50326c0d60cc2fc618715c10aad6d7db303e1bfc4d4759bd7c2a7bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":43,"rank":43,"depth":0,"x":164.529,"y":248.223,"cluster":"categories-foundations"},{"id":"stacks:0022","tag":"0022","title":"Fibre products and representability · Lemma 0022","summary":"Let C be a category. Let F, G, H : C^opp → Sets be functors. Let a : F → G and b : H → G be transformations of functors. Then the fibre product F ×_a, G, b H in the category PSh(C) exists and is given by the formula (F ×_a, G, b H)(X) = F(X) ×_a_X, G(X), b_X H(X) for any object X of C.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $F, G, H : \\mathcal{C}^{opp} \\to \\textit{Sets}$\nbe functors. Let $a : F \\to G$ and $b : H \\to G$ be\ntransformations of functors. Then the fibre product\n$F \\times_{a, G, b} H$ in the category\n$\\textit{PSh}(\\mathcal{C})$\nexists and is given by the formula\n$$\n(F \\times_{a, G, b} H)(X) =\nF(X) \\times_{a_X, G(X), b_X} H(X)\n$$\nfor any object $X$ of $\\mathcal{C}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products and representability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0022","source_file":"categories.tex","source_line":939,"source_end_line":953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L939-L953","statement_sha256":"02102652a80e3d2ea04acd111b4f5c80d395a17d3feb7f4a85085f9990a05ccb","origin":"The Stacks Project","memory_eligible":false,"source_rank":44,"rank":44,"depth":0,"x":255.873,"y":161.377,"cluster":"categories-foundations"},{"id":"stacks:0023","tag":"0023","title":"Fibre products and representability · Definition 0023","summary":"Let C be a category. Let F, G : C^opp → Sets be functors. We say a morphism a : F → G is representable, or that F is relatively representable over G, if for every U ∈ Ob(C) and any xi ∈ G(U) the functor h_U ×_xi, G, a F is representable.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $F, G : \\mathcal{C}^{opp} \\to \\textit{Sets}$\nbe functors. We say a morphism $a : F \\to G$ is\n{\\it representable}, or that {\\it $F$ is relatively representable\nover $G$}, if for every $U \\in \\Ob(\\mathcal{C})$\nand any $\\xi \\in G(U)$ the functor\n$h_U \\times_{\\xi, G, a} F$ is representable.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products and representability","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0023","source_file":"categories.tex","source_line":976,"source_end_line":985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L976-L985","statement_sha256":"91084cdbf056e8f940829f3d297fe6ccdc4bc1bec9591e8cb5a4b1b7c6af28fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":45,"rank":45,"depth":0,"x":258.377,"y":278.554,"cluster":"categories-foundations"},{"id":"stacks:03KC","tag":"03KC","title":"Fibre products and representability · Lemma 03KC","summary":"Let C be a category. Let a : F → G be a morphism of contravariant functors from C to Sets. If a is representable, and G is a representable functor, then F is representable.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $a : F \\to G$ be a morphism of contravariant functors\nfrom $\\mathcal{C}$ to $\\textit{Sets}$. If $a$ is representable,\nand $G$ is a representable functor, then $F$ is representable.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products and representability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KC","source_file":"categories.tex","source_line":987,"source_end_line":993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L987-L993","statement_sha256":"602459957792b07fe5d4b443184e34cfb6c2e727a722080fd41f6750a8be66e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":46,"rank":46,"depth":0,"x":161.246,"y":192.629,"cluster":"categories-foundations"},{"id":"stacks:0024","tag":"0024","title":"Fibre products and representability · Lemma 0024","summary":"Let C be a category. Let F : C^opp → Sets be a functor. Assume C has products of pairs of objects and fibre products. The following are equivalent: • the diagonal Δ : F → F × F is representable, • for every U in C, and any xi ∈ F(U) the map xi : h_U → F is representable, • for every pair U, V in C and any xi ∈ F(U), xi' ∈ F(V) the fibre product h_U ×_xi, F, xi' h_V is representable.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $F : \\mathcal{C}^{opp} \\to \\textit{Sets}$ be a functor.\nAssume $\\mathcal{C}$ has products of pairs of objects and fibre products.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the diagonal $\\Delta : F \\to F \\times F$ is representable,\n\\item for every $U$ in $\\mathcal{C}$,\nand any $\\xi \\in F(U)$ the map $\\xi : h_U \\to F$ is representable,\n\\item for every pair $U, V$ in $\\mathcal{C}$\nand any $\\xi \\in F(U)$, $\\xi' \\in F(V)$ the fibre product\n$h_U \\times_{\\xi, F, \\xi'} h_V$ is representable.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibre products and representability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0024","source_file":"categories.tex","source_line":999,"source_end_line":1013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L999-L1013","statement_sha256":"70bf322885fa28ba29d7049a16bee4f91a244c3d3a45d3968d1c6d7046f8f1c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":47,"rank":47,"depth":1,"x":303.485,"y":200.969,"cluster":"categories-foundations"},{"id":"stacks:0026","tag":"0026","title":"Pushouts · Definition 0026","summary":"Let x, y, z∈ Ob(C), f∈ Mor_C(y, x) and g∈ Mor_ C(y, z). A pushout of f and g is an object xamalg_y z∈ Ob(C) together with morphisms p∈ Mor_ C(x, xamalg_y z) and q∈Mor_ C(z, xamalg_y z) making the diagram xymatrix y ar[r]_g ar[d]_f & z ar[d]^q x ar[r]^p & xamalg_y z commute, and such that the following universal property holds: For any w∈ Ob(C) and morphisms α ∈ Mor_ C(x, w) and β ∈ Mor_C(z, w) with α ∘ f = β ∘ g there is a unique γ∈ Mor_ C(xamalg_y z, w) making the…","statement_latex":"Let $x, y, z\\in \\Ob(\\mathcal{C})$,\n$f\\in \\Mor_\\mathcal{C}(y, x)$\nand $g\\in \\Mor_{\\mathcal C}(y, z)$.\nA {\\it pushout} of $f$ and $g$ is\nan object $x\\amalg_y z\\in \\Ob(\\mathcal{C})$\ntogether with morphisms\n$p\\in \\Mor_{\\mathcal C}(x, x\\amalg_y z)$ and\n$q\\in\\Mor_{\\mathcal C}(z, x\\amalg_y z)$ making the diagram\n$$\n\\xymatrix{\ny \\ar[r]_g \\ar[d]_f & z \\ar[d]^q \\\\\nx \\ar[r]^p & x\\amalg_y z\n}\n$$\ncommute, and such that the following universal property holds:\nFor any $w\\in \\Ob(\\mathcal{C})$ and morphisms\n$\\alpha \\in \\Mor_{\\mathcal C}(x, w)$ and\n$\\beta \\in \\Mor_\\mathcal{C}(z, w)$ with\n$\\alpha \\circ f = \\beta \\circ g$ there is a unique\n$\\gamma\\in \\Mor_{\\mathcal C}(x\\amalg_y z, w)$ making\nthe diagram\n$$\n\\xymatrix{\ny \\ar[r]_g \\ar[d]_f & z \\ar[d]^q \\ar[rrdd]^\\beta & & \\\\\nx \\ar[r]^p \\ar[rrrd]^\\alpha & x \\amalg_y z \\ar@{-->}[rrd]^\\gamma & & \\\\\n& & & w\n}\n$$\ncommute.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Pushouts","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0026","source_file":"categories.tex","source_line":1077,"source_end_line":1108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1077-L1108","statement_sha256":"4174cd87e5e9afaa0043264ff4692a43873dc616cf8da3c7ee0088b497d40558","origin":"The Stacks Project","memory_eligible":false,"source_rank":48,"rank":48,"depth":0,"x":190.711,"y":276.313,"cluster":"categories-foundations"},{"id":"stacks:08N1","tag":"08N1","title":"Pushouts · Definition 08N1","summary":"We say a commutative diagram xymatrix y ar[r] ar[d] & z ar[d] x ar[r] & w in a category is cocartesian if w and the morphisms x → w and z → w form a pushout of the morphisms y → x and y → z.","statement_latex":"We say a commutative diagram\n$$\n\\xymatrix{\ny \\ar[r] \\ar[d] & z \\ar[d] \\\\\nx \\ar[r] & w\n}\n$$\nin a category is {\\it cocartesian} if $w$ and the morphisms $x \\to w$ and\n$z \\to w$ form a pushout of the morphisms $y \\to x$ and $y \\to z$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Pushouts","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08N1","source_file":"categories.tex","source_line":1117,"source_end_line":1128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1117-L1128","statement_sha256":"ac842d5194d32ce6477f6e827e0f2bff99fdb5be2791238b2bc85cdc37be4538","origin":"The Stacks Project","memory_eligible":false,"source_rank":49,"rank":49,"depth":0,"x":213.519,"y":155.529,"cluster":"categories-foundations"},{"id":"stacks:0028","tag":"0028","title":"Equalizers · Definition 0028","summary":"Suppose that X, Y are objects of a category C and that a, b : X → Y are morphisms. We say a morphism e : Z → X is an equalizer for the pair (a, b) if a ∘ e = b ∘ e and if (Z, e) satisfies the following universal property: For every morphism t : W → X in C such that a ∘ t = b ∘ t there exists a unique morphism s : W → Z such that t = e ∘ s.","statement_latex":"Suppose that $X$, $Y$ are objects of a category $\\mathcal{C}$\nand that $a, b : X \\to Y$ are morphisms. We say a morphism\n$e : Z \\to X$ is an {\\it equalizer} for the pair $(a, b)$ if\n$a \\circ e = b \\circ e$ and if $(Z, e)$ satisfies the following\nuniversal property: For every morphism $t : W \\to X$\nin $\\mathcal{C}$ such that $a \\circ t = b \\circ t$ there exists\na unique morphism $s : W \\to Z$ such that $t = e \\circ s$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Equalizers","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0028","source_file":"categories.tex","source_line":1134,"source_end_line":1143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1134-L1143","statement_sha256":"55ee21e86ef6d9e607dd3b8b25d2ce33afc4582457707ae4b1da04ffbe649946","origin":"The Stacks Project","memory_eligible":false,"source_rank":50,"rank":50,"depth":0,"x":294.641,"y":258.571,"cluster":"categories-foundations"},{"id":"stacks:002A","tag":"002A","title":"Coequalizers · Definition 002A","summary":"Suppose that X, Y are objects of a category C and that a, b : X → Y are morphisms. We say a morphism c : Y → Z is a coequalizer for the pair (a, b) if c ∘ a = c ∘ b and if (Z, c) satisfies the following universal property: For every morphism t : Y → W in C such that t ∘ a = t ∘ b there exists a unique morphism s : Z → W such that t = s ∘ c.","statement_latex":"Suppose that $X$, $Y$ are objects of a category $\\mathcal{C}$\nand that $a, b : X \\to Y$ are morphisms. We say a morphism\n$c : Y \\to Z$ is a {\\it coequalizer} for the pair $(a, b)$ if\n$c \\circ a = c \\circ b$ and if $(Z, c)$ satisfies the following\nuniversal property: For every morphism $t : Y \\to W$\nin $\\mathcal{C}$ such that $t \\circ a = t \\circ b$ there exists\na unique morphism $s : Z \\to W$ such that $t = s \\circ c$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Coequalizers","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002A","source_file":"categories.tex","source_line":1154,"source_end_line":1163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1154-L1163","statement_sha256":"b13c9ad9afdbde8cfce1ffc692d1b5260fc213d8c7e401ea8ef3099e45cdd527","origin":"The Stacks Project","memory_eligible":false,"source_rank":51,"rank":51,"depth":0,"x":150.544,"y":228.318,"cluster":"categories-foundations"},{"id":"stacks:002C","tag":"002C","title":"Initial and final objects · Definition 002C","summary":"Let C be a category. • An object x of the category C is called an initial object if for every object y of C there is exactly one morphism x → y. • An object x of the category C is called a final object if for every object y of C there is exactly one morphism y → x.","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item An object $x$ of the category $\\mathcal{C}$ is called\nan {\\it initial} object if for every object $y$ of $\\mathcal{C}$\nthere is exactly one morphism $x \\to y$.\n\\item An object $x$ of the category $\\mathcal{C}$ is called\na {\\it final} object if for every object $y$ of $\\mathcal{C}$\nthere is exactly one morphism $y \\to x$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Initial and final objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002C","source_file":"categories.tex","source_line":1175,"source_end_line":1186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1175-L1186","statement_sha256":"e4fef70a38fe884f9309d2029c45a033be4634db4048c58570a0e36accf40997","origin":"The Stacks Project","memory_eligible":false,"source_rank":52,"rank":52,"depth":0,"x":282.401,"y":168.287,"cluster":"categories-foundations"},{"id":"stacks:003B","tag":"003B","title":"Monomorphisms and Epimorphisms · Definition 003B","summary":"Let C be a category and let f : X → Y be a morphism of C. • We say that f is a monomorphism if for every object W and every pair of morphisms a, b : W → X such that f ∘ a = f ∘ b we have a = b. • We say that f is an epimorphism if for every object W and every pair of morphisms a, b : Y → W such that a ∘ f = b ∘ f we have a = b.","statement_latex":"Let $\\mathcal{C}$ be a category and let $f : X \\to Y$ be\na morphism of $\\mathcal{C}$.\n\\begin{enumerate}\n\\item We say that $f$ is a {\\it monomorphism} if for every object\n$W$ and every pair of morphisms $a, b : W \\to X$ such that\n$f \\circ a = f \\circ b$ we have $a = b$.\n\\item We say that $f$ is an {\\it epimorphism} if for every object\n$W$ and every pair of morphisms $a, b : Y \\to W$ such that\n$a \\circ f = b \\circ f$ we have $a = b$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monomorphisms and Epimorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003B","source_file":"categories.tex","source_line":1200,"source_end_line":1212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1200-L1212","statement_sha256":"daef8698c98d25d19bb81e47272ed6b363f03c2512f74ae99118bedc067e74a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":53,"rank":53,"depth":0,"x":232.974,"y":288.507,"cluster":"categories-foundations"},{"id":"stacks:08LR","tag":"08LR","title":"Monomorphisms and Epimorphisms · Lemma 08LR","summary":"Let C be a category, and let f : X → Y be a morphism of C. Then • f is a monomorphism if and only if X is the fibre product X ×_Y X, and • f is an epimorphism if and only if Y is the pushout Y amalg_X Y.","statement_latex":"Let $\\mathcal{C}$ be a category, and let $f : X \\to Y$ be\na morphism of $\\mathcal{C}$. Then\n\\begin{enumerate}\n\\item $f$ is a monomorphism if and only if $X$ is the fibre\nproduct $X \\times_Y X$, and\n\\item $f$ is an epimorphism if and only if $Y$ is the pushout\n$Y \\amalg_X Y$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monomorphisms and Epimorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LR","source_file":"categories.tex","source_line":1220,"source_end_line":1230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1220-L1230","statement_sha256":"ce1a95fde9a6cbb752c868186591e9fdc896144437f4e90306ad835bbd3f53d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":54,"rank":54,"depth":0,"x":172.183,"y":170.718,"cluster":"categories-foundations"},{"id":"stacks:002E","tag":"002E","title":"Limits and colimits · Definition 002E","summary":"A limit of the I-diagram M in the category C is given by an object lim_I M in C together with morphisms p_i : lim_I M → M_i such that • for φ : i → i' a morphism in I we have p_i' = M(φ) ∘ p_i, and • for any object W in C and any family of morphisms q_i : W → M_i (indexed by i ∈ Ob(I)) such that for all φ : i → i' in I we have q_i' = M(φ) ∘ q_i there exists a unique morphism q : W → lim_I M such that q_i = p_i ∘ q for every object i of I.","statement_latex":"A {\\it limit} of the $\\mathcal{I}$-diagram $M$ in the category\n$\\mathcal{C}$ is given by an object $\\lim_\\mathcal{I} M$ in $\\mathcal{C}$\ntogether with morphisms $p_i : \\lim_\\mathcal{I} M \\to M_i$ such that\n\\begin{enumerate}\n\\item for $\\phi : i \\to i'$ a morphism\nin $\\mathcal{I}$ we have $p_{i'} = M(\\phi) \\circ p_i$, and\n\\item for any object $W$ in $\\mathcal{C}$ and any family of\nmorphisms $q_i : W \\to M_i$ (indexed by $i \\in \\Ob(\\mathcal{I})$)\nsuch that for all $\\phi : i \\to i'$\nin $\\mathcal{I}$ we have $q_{i'} = M(\\phi) \\circ q_i$ there\nexists a unique morphism $q : W \\to \\lim_\\mathcal{I} M$ such that\n$q_i = p_i \\circ q$ for every object $i$ of $\\mathcal{I}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002E","source_file":"categories.tex","source_line":1282,"source_end_line":1297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1282-L1297","statement_sha256":"b9acb77b15796fcc1f14a4d445ee2190c8a946f202c91f517914f32f19653c49","origin":"The Stacks Project","memory_eligible":false,"source_rank":55,"rank":55,"depth":0,"x":313.014,"y":223.565,"cluster":"categories-foundations"},{"id":"stacks:002F","tag":"002F","title":"Limits and colimits · Definition 002F","summary":"A colimit of the I-diagram M in the category C is given by an object colim_I M in C together with morphisms s_i : M_i → colim_I M such that • for φ : i → i' a morphism in I we have s_i = s_i' ∘ M(φ), and • for any object W in C and any family of morphisms t_i : M_i → W (indexed by i ∈ Ob(I)) such that for all φ : i → i' in I we have t_i = t_i' ∘ M(φ) there exists a unique morphism t : colim_I M → W such that t_i = t ∘ s_i for every object i of I.","statement_latex":"A {\\it colimit} of the $\\mathcal{I}$-diagram $M$ in the category\n$\\mathcal{C}$ is given by an object $\\colim_\\mathcal{I} M$ in $\\mathcal{C}$\ntogether with morphisms $s_i : M_i \\to \\colim_\\mathcal{I} M$ such that\n\\begin{enumerate}\n\\item for $\\phi : i \\to i'$ a morphism\nin $\\mathcal{I}$ we have $s_i = s_{i'} \\circ M(\\phi)$, and\n\\item for any object $W$ in $\\mathcal{C}$ and any family of\nmorphisms $t_i : M_i \\to W$ (indexed by $i \\in \\Ob(\\mathcal{I})$)\nsuch that for all $\\phi : i \\to i'$\nin $\\mathcal{I}$ we have $t_i = t_{i'} \\circ M(\\phi)$ there\nexists a unique morphism $t : \\colim_\\mathcal{I} M \\to W$ such that\n$t_i = t \\circ s_i$ for every object $i$ of $\\mathcal{I}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002F","source_file":"categories.tex","source_line":1309,"source_end_line":1324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1309-L1324","statement_sha256":"7b73db691dd9556bc75edfdc8ed654d0983f6801e0ceebddcde4f039df3e01c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":56,"rank":56,"depth":0,"x":165.346,"y":264.873,"cluster":"categories-foundations"},{"id":"stacks:002I","tag":"002I","title":"Limits and colimits · Definition 002I","summary":"Suppose that I is a set, and suppose given for every i ∈ I an object M_i of the category C. A product ∏_i∈ I M_i is by definition lim_I M (if it exists) where I is the category having only identities as morphisms and having the elements of I as objects.","statement_latex":"Suppose that $I$ is a set, and suppose given for every $i \\in I$ an\nobject $M_i$ of the category $\\mathcal{C}$. A {\\it product}\n$\\prod_{i\\in I} M_i$ is by definition $\\lim_\\mathcal{I} M$\n(if it exists)\nwhere $\\mathcal{I}$ is the category having only identities as\nmorphisms and having the elements of $I$ as objects.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002I","source_file":"categories.tex","source_line":1404,"source_end_line":1412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1404-L1412","statement_sha256":"bd6dd80cbea09065f753315a07f355accda524780a22c98378855f0b56f4c908","origin":"The Stacks Project","memory_eligible":false,"source_rank":57,"rank":57,"depth":0,"x":241.691,"y":149.612,"cluster":"categories-foundations"},{"id":"stacks:002J","tag":"002J","title":"Limits and colimits · Definition 002J","summary":"Suppose that I is a set, and suppose given for every i ∈ I an object M_i of the category C. A coproduct coprod_i∈ I M_i is by definition colim_I M (if it exists) where I is the category having only identities as morphisms and having the elements of I as objects.","statement_latex":"Suppose that $I$ is a set, and suppose given for every $i \\in I$ an\nobject $M_i$ of the category $\\mathcal{C}$. A {\\it coproduct}\n$\\coprod_{i\\in I} M_i$ is by definition $\\colim_\\mathcal{I} M$\n(if it exists) where $\\mathcal{I}$ is the category having only\nidentities as morphisms and having the elements of $I$ as objects.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002J","source_file":"categories.tex","source_line":1420,"source_end_line":1427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1420-L1427","statement_sha256":"0d2080b324d97fb50b4f96b2fc40b3192ab435b2ff6de897dd82302ec171bf56","origin":"The Stacks Project","memory_eligible":false,"source_rank":58,"rank":58,"depth":0,"x":278.398,"y":279.042,"cluster":"categories-foundations"},{"id":"stacks:002K","tag":"002K","title":"Limits and colimits · Lemma 002K","summary":"Suppose that M : I → C, and N : J → C are diagrams whose colimits exist. Suppose H : I → J is a functor, and suppose t : M → N ∘ H is a transformation of functors. Then there is a unique morphism theta : colim_I M → colim_J N such that all the diagrams xymatrix M_i ar[d]_t_i ar[r] & colim_I M ar[d]^theta N_H(i) ar[r] & colim_J N commute.","statement_latex":"Suppose that $M : \\mathcal{I} \\to \\mathcal{C}$,\nand $N : \\mathcal{J} \\to \\mathcal{C}$ are diagrams\nwhose colimits exist. Suppose\n$H : \\mathcal{I} \\to \\mathcal{J}$ is\na functor, and suppose $t : M \\to N \\circ H$\nis a transformation of functors.\nThen there is a unique morphism\n$$\n\\theta :\n\\colim_\\mathcal{I} M\n\\longrightarrow\n\\colim_\\mathcal{J} N\n$$\nsuch that all the diagrams\n$$\n\\xymatrix{\nM_i \\ar[d]_{t_i} \\ar[r]\n&\n\\colim_\\mathcal{I} M \\ar[d]^{\\theta}\n\\\\\nN_{H(i)} \\ar[r]\n&\n\\colim_\\mathcal{J} N\n}\n$$\ncommute.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002K","source_file":"categories.tex","source_line":1436,"source_end_line":1464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1436-L1464","statement_sha256":"dea8f09947f73cfcaa03a7f25a01ca7529c40a816aae68628b1de20227140aac","origin":"The Stacks Project","memory_eligible":false,"source_rank":59,"rank":59,"depth":0,"x":146.126,"y":203.789,"cluster":"categories-foundations"},{"id":"stacks:002L","tag":"002L","title":"Limits and colimits · Lemma 002L","summary":"Suppose that M : I → C, and N : J → C are diagrams whose limits exist. Suppose H : I → J is a functor, and suppose t : N ∘ H → M is a transformation of functors. Then there is a unique morphism theta : lim_J N → lim_I M such that all the diagrams xymatrix lim_J N ar[d]^theta ar[r] & N_H(i) ar[d]_t_i lim_I M ar[r] & M_i commute.","statement_latex":"Suppose that $M : \\mathcal{I} \\to \\mathcal{C}$,\nand $N : \\mathcal{J} \\to \\mathcal{C}$ are diagrams\nwhose limits exist. Suppose $H : \\mathcal{I} \\to \\mathcal{J}$ is\na functor, and suppose $t : N \\circ H \\to M$\nis a transformation of functors.\nThen there is a unique morphism\n$$\n\\theta :\n\\lim_\\mathcal{J} N\n\\longrightarrow\n\\lim_\\mathcal{I} M\n$$\nsuch that all the diagrams\n$$\n\\xymatrix{\n\\lim_\\mathcal{J} N \\ar[d]^{\\theta} \\ar[r]\n&\nN_{H(i)} \\ar[d]_{t_i}\n\\\\\n\\lim_\\mathcal{I} M \\ar[r]\n&\nM_i\n}\n$$\ncommute.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002L","source_file":"categories.tex","source_line":1470,"source_end_line":1497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1470-L1497","statement_sha256":"21be8571814c50c62796fb0e72032094e5b1f45cb0143ea1fcf2805f16a22110","origin":"The Stacks Project","memory_eligible":false,"source_rank":60,"rank":60,"depth":0,"x":305.509,"y":184.065,"cluster":"categories-foundations"},{"id":"stacks:002M","tag":"002M","title":"Limits and colimits · Lemma 002M","summary":"Let I, J be index categories. Let M : I × J → C be a functor. Assume that M_i, ∞ = colim_j M_i,j exists for all i. Then the resulting functor M_-, ∞ : I → C has a colimit if and only if M does, and then the colimits coincide. In particular, we have colim_i colim_j M_i, j = colim_i, j M_i, j = colim_j colim_i M_i, j provided all the indicated colimits exist. Similar for limits.","statement_latex":"Let $\\mathcal{I}$, $\\mathcal{J}$ be index categories. Let\n$M : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{C}$ be a functor.\nAssume that $M_{i, \\infty} = \\colim_j M_{i,j}$ \nexists for all $i$. Then the resulting functor\n$M_{-, \\infty} : \\mathcal{I} \\to \\mathcal{C}$\nhas a colimit if and only if $M$ does, and then the colimits coincide.\nIn particular, we have\n$$\n\\colim_i \\colim_j M_{i, j}\n=\n\\colim_{i, j} M_{i, j}\n=\n\\colim_j \\colim_i M_{i, j}\n$$\nprovided all the indicated colimits exist. Similar for limits.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002M","source_file":"categories.tex","source_line":1504,"source_end_line":1521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1504-L1521","statement_sha256":"80f78483a90a201dc8064d6f327a2bc976cca6d9195a9915509c1d7216c7e3b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":61,"rank":61,"depth":0,"x":202.999,"y":289.913,"cluster":"categories-foundations"},{"id":"stacks:002N","tag":"002N","title":"Limits and colimits · Lemma 002N","summary":"Let M : I → C be a diagram. Write I = Ob(I) and A = Arrows(I). Denote s, t : A → I the source and target maps. Suppose that ∏_i ∈ I M_i and ∏_a ∈ A M_t(a) exist. Suppose that the equalizer of xymatrix ∏_i ∈ I M_i ar@<1ex>[r]^φ ar@<-1ex>[r]_ψ & ∏_a ∈ A M_t(a) exists, where the morphisms are determined by their components as follows: p_a ∘ ψ = M(a) ∘ p_s(a) and p_a ∘ φ = p_t(a). Then this equalizer is the limit of the diagram.","statement_latex":"Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram.\nWrite $I = \\Ob(\\mathcal{I})$ and $A = \\text{Arrows}(\\mathcal{I})$.\nDenote $s, t : A \\to I$ the source and target maps.\nSuppose that $\\prod_{i \\in I} M_i$ and $\\prod_{a \\in A} M_{t(a)}$\nexist. Suppose that the equalizer of\n$$\n\\xymatrix{\n\\prod_{i \\in I} M_i\n\\ar@<1ex>[r]^\\phi \\ar@<-1ex>[r]_\\psi\n&\n\\prod_{a \\in A} M_{t(a)}\n}\n$$\nexists, where the morphisms are determined by their components\nas follows: $p_a \\circ \\psi = M(a) \\circ p_{s(a)}$\nand $p_a \\circ \\phi = p_{t(a)}$. Then this equalizer is the\nlimit of the diagram.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002N","source_file":"categories.tex","source_line":1527,"source_end_line":1546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1527-L1546","statement_sha256":"d2e2dbcf70706c909b08743974485944da78dbfbf885b0505f9247a9d6144293","origin":"The Stacks Project","memory_eligible":false,"source_rank":62,"rank":62,"depth":0,"x":193.394,"y":152.586,"cluster":"categories-foundations"},{"id":"stacks:002P","tag":"002P","title":"Limits and colimits · Lemma 002P","summary":"If all coproducts and coequalizers exist, all colimits exist. Let M : I → C be a diagram. Write I = Ob(I) and A = Arrows(I). Denote s, t : A → I the source and target maps. Suppose that coprod_i ∈ I M_i and coprod_a ∈ A M_s(a) exist. Suppose that the coequalizer of xymatrix coprod_a ∈ A M_s(a) ar@<1ex>[r]^φ ar@<-1ex>[r]_ψ & coprod_i ∈ I M_i exists, where the morphisms are determined by their components as follows: The component M_s(a) maps via ψ to the component M_t(a)…","statement_latex":"\\begin{slogan}\nIf all coproducts and coequalizers exist, all colimits exist.\n\\end{slogan}\nLet $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram.\nWrite $I = \\Ob(\\mathcal{I})$ and $A = \\text{Arrows}(\\mathcal{I})$.\nDenote $s, t : A \\to I$ the source and target maps.\nSuppose that $\\coprod_{i \\in I} M_i$ and $\\coprod_{a \\in A} M_{s(a)}$\nexist. Suppose that the coequalizer of\n$$\n\\xymatrix{\n\\coprod_{a \\in A} M_{s(a)}\n\\ar@<1ex>[r]^\\phi \\ar@<-1ex>[r]_\\psi\n&\n\\coprod_{i \\in I} M_i\n}\n$$\nexists, where the morphisms are determined by their components\nas follows: The component $M_{s(a)}$ maps via $\\psi$\nto the component $M_{t(a)}$ via the morphism $M(a)$.\nThe component $M_{s(a)}$ maps via $\\phi$ to the component\n$M_{s(a)}$ by the identity morphism. Then this coequalizer is the\ncolimit of the diagram.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002P","source_file":"categories.tex","source_line":1553,"source_end_line":1577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1553-L1577","statement_sha256":"73aad69c7f96acebba310fe9c344380a0e9f8a892ae207c277a9635ec3587b30","origin":"The Stacks Project","memory_eligible":false,"source_rank":63,"rank":63,"depth":0,"x":311.85,"y":249.169,"cluster":"categories-foundations"},{"id":"stacks:002S","tag":"002S","title":"Connected limits · Definition 002S","summary":"We say that a category I is connected if the equivalence relation generated by x sim y ⇔ Mor_I(x, y) not = ∅ has exactly one equivalence class.","statement_latex":"We say that a category $\\mathcal{I}$ is {\\it connected}\nif the equivalence relation generated by\n$x \\sim y \\Leftrightarrow \\Mor_\\mathcal{I}(x, y) \\not = \\emptyset$\nhas exactly one equivalence class.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Connected limits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002S","source_file":"categories.tex","source_line":1653,"source_end_line":1659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1653-L1659","statement_sha256":"ce3568a59a3cd1448edf84a7a29d12e4c36c5aecda5788bfc252d94ca59df9a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":64,"rank":64,"depth":0,"x":145.533,"y":245.13,"cluster":"categories-foundations"},{"id":"stacks:002T","tag":"002T","title":"Connected limits · Lemma 002T","summary":"Let C be a category. Let X be an object of C. Let M : I → C/X be a diagram in the category of objects over X. If the index category I is connected and the limit of M exists in C/X, then the limit of the composition I → C/X → C exists and is the same.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $M : \\mathcal{I} \\to \\mathcal{C}/X$ be a diagram\nin the category of objects over $X$.\nIf the index category $\\mathcal{I}$ is connected\nand the limit of $M$ exists in $\\mathcal{C}/X$,\nthen the limit of the composition\n$\\mathcal{I} \\to \\mathcal{C}/X \\to \\mathcal{C}$\nexists and is the same.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Connected limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002T","source_file":"categories.tex","source_line":1667,"source_end_line":1678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1667-L1678","statement_sha256":"a26b4248b54b5b14cf296fb5c11420e72005c2b263b86ef5752ae78a70cace10","origin":"The Stacks Project","memory_eligible":false,"source_rank":65,"rank":65,"depth":0,"x":272.4,"y":153.029,"cluster":"categories-foundations"},{"id":"stacks:04AR","tag":"04AR","title":"Connected limits · Lemma 04AR","summary":"Let C be a category. Let X be an object of C. Let M : I → X/C be a diagram in the category of objects under X. If the index category I is connected and the colimit of M exists in X/C, then the colimit of the composition I → X/C → C exists and is the same.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $M : \\mathcal{I} \\to X/\\mathcal{C}$ be a diagram\nin the category of objects under $X$.\nIf the index category $\\mathcal{I}$ is connected\nand the colimit of $M$ exists in $X/\\mathcal{C}$,\nthen the colimit of the composition\n$\\mathcal{I} \\to X/\\mathcal{C} \\to \\mathcal{C}$\nexists and is the same.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Connected limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AR","source_file":"categories.tex","source_line":1698,"source_end_line":1709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1698-L1709","statement_sha256":"84dba22113a6b9cd244cbb283930ec462b52a477494cae7e402f4aae21edfb08","origin":"The Stacks Project","memory_eligible":false,"source_rank":66,"rank":66,"depth":1,"x":252.762,"y":293.999,"cluster":"categories-foundations"},{"id":"stacks:04E6","tag":"04E6","title":"Cofinal and initial categories · Definition 04E6","summary":"Let H : I → J be a functor between categories. We say I is cofinal in J or that H is cofinal if • for all y ∈ Ob(J) there exist an x ∈ Ob(I) and a morphism y → H(x), and • given y ∈ Ob(J), x, x' ∈ Ob(I) and morphisms y → H(x) and y → H(x') there exist a sequence of morphisms x = x_0 ← x_1 → x_2 ← x_3 → … → x_2n = x' in I and morphisms y → H(x_i) in J with y → H(x_0) and y → H(x_2n) the given morphisms such that the diagrams xymatrix & y ar[ld] ar[d] ar[rd] H(x_2k) &…","statement_latex":"Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor between categories.\nWe say {\\it $\\mathcal{I}$ is cofinal in $\\mathcal{J}$} or that\n$H$ is {\\it cofinal} if\n\\begin{enumerate}\n\\item for all $y \\in \\Ob(\\mathcal{J})$ there exist an\n$x \\in \\Ob(\\mathcal{I})$ and a morphism $y \\to H(x)$, and\n\\item given $y \\in \\Ob(\\mathcal{J})$, $x, x' \\in \\Ob(\\mathcal{I})$\nand morphisms $y \\to H(x)$ and $y \\to H(x')$ there exist a sequence\nof morphisms\n$$\nx = x_0 \\leftarrow x_1 \\rightarrow x_2 \\leftarrow x_3 \\rightarrow \\ldots\n\\rightarrow x_{2n} = x'\n$$\nin $\\mathcal{I}$ and morphisms $y \\to H(x_i)$ in $\\mathcal{J}$\nwith $y \\to H(x_0)$ and $y \\to H(x_{2n})$ the given morphisms\nsuch that the diagrams\n$$\n\\xymatrix{\n& y \\ar[ld] \\ar[d] \\ar[rd] \\\\\nH(x_{2k}) & H(x_{2k + 1}) \\ar[l] \\ar[r] & H(x_{2k + 2})\n}\n$$\ncommute for $k = 0, \\ldots, n - 1$.\n\\end{enumerate}\nIn other words, fixing an object $y$ of $\\mathcal{J}$ consider the\nset $S$ of pairs $(x, b)$ where $x$ is an object of $\\mathcal{I}$ and\n$b : y \\to H(x)$ is a morphism. Consider the equivalence relation\non $S$ generated by $(x, b) \\sim (x', b')$ if there exists a\nmorphism $a : x \\to x'$ with $b' = H(a) \\circ b$. Then $S$ should\nconsist of exactly one equivalence class.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Cofinal and initial categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04E6","source_file":"categories.tex","source_line":1725,"source_end_line":1757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1725-L1757","statement_sha256":"3d953c220aa5ccbf96733a842a3ea91be5929aad12a24101e2c14f12867664c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":67,"rank":67,"depth":0,"x":153.138,"y":178.039,"cluster":"categories-foundations"},{"id":"stacks:04E7","tag":"04E7","title":"Cofinal and initial categories · Lemma 04E7","summary":"Let H : I → J be a functor of categories. Assume I is cofinal in J. Then for every diagram M : J → C we have a canonical isomorphism colim_I M ∘ H = colim_J M if either side exists.","statement_latex":"Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor of categories. Assume\n$\\mathcal{I}$ is cofinal in $\\mathcal{J}$. Then for every diagram\n$M : \\mathcal{J} \\to \\mathcal{C}$ we have a canonical isomorphism\n$$\n\\colim_\\mathcal{I} M \\circ H\n=\n\\colim_\\mathcal{J} M\n$$\nif either side exists.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Cofinal and initial categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04E7","source_file":"categories.tex","source_line":1759,"source_end_line":1770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1759-L1770","statement_sha256":"f68a2facac8022103c38ccad37d4076f505d25826a935d749cb8afc624db3b70","origin":"The Stacks Project","memory_eligible":false,"source_rank":68,"rank":68,"depth":0,"x":321.086,"y":207.235,"cluster":"categories-foundations"},{"id":"stacks:09WP","tag":"09WP","title":"Cofinal and initial categories · Definition 09WP","summary":"Let H : I → J be a functor between categories. We say I is initial in J or that H is initial if • for all y ∈ Ob(J) there exist an x ∈ Ob(I) and a morphism H(x) → y, • for any y ∈ Ob(J), x , x' ∈ Ob(I) and morphisms H(x) → y, H(x') → y in J there exist a sequence of morphisms x = x_0 ← x_1 → x_2 ← x_3 → … → x_2n = x' in I and morphisms H(x_i) → y in J such that the diagrams xymatrix H(x_2k) ar[rd] & H(x_2k + 1) ar[l] ar[r] ar[d] & H(x_2k + 2) ar[ld] & y commute for k = 0,…","statement_latex":"Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor between categories.\nWe say {\\it $\\mathcal{I}$ is initial in $\\mathcal{J}$} or that\n$H$ is {\\it initial} if\n\\begin{enumerate}\n\\item for all $y \\in \\Ob(\\mathcal{J})$ there exist an\n$x \\in \\Ob(\\mathcal{I})$ and a morphism $H(x) \\to y$,\n\\item for any $y \\in \\Ob(\\mathcal{J})$, $x , x' \\in \\Ob(\\mathcal{I})$ and\nmorphisms $H(x) \\to y$, $H(x') \\to y$ in $\\mathcal{J}$\nthere exist a sequence of morphisms\n$$\nx = x_0 \\leftarrow x_1 \\rightarrow x_2 \\leftarrow x_3 \\rightarrow \\ldots\n\\rightarrow x_{2n} = x'\n$$\nin $\\mathcal{I}$ and morphisms $H(x_i) \\to y$ in $\\mathcal{J}$\nsuch that the diagrams\n$$\n\\xymatrix{\nH(x_{2k}) \\ar[rd] &\nH(x_{2k + 1}) \\ar[l] \\ar[r] \\ar[d] &\nH(x_{2k + 2}) \\ar[ld] \\\\\n& y\n}\n$$\ncommute for $k = 0, \\ldots, n - 1$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Cofinal and initial categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WP","source_file":"categories.tex","source_line":1776,"source_end_line":1803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1776-L1803","statement_sha256":"707f59841707294d6530cf405e0937578d1167ee091723a60a2cd542ea1d37c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":69,"rank":69,"depth":0,"x":172.687,"y":281.541,"cluster":"categories-foundations"},{"id":"stacks:002R","tag":"002R","title":"Cofinal and initial categories · Lemma 002R","summary":"Let H : I → J be a functor of categories. Assume I is initial in J. Then for every diagram M : J → C we have a canonical isomorphism lim_I M ∘ H = lim_J M if either side exists.","statement_latex":"Let $H : \\mathcal{I} \\to \\mathcal{J}$ be a functor of categories.\nAssume $\\mathcal{I}$ is initial in $\\mathcal{J}$.\nThen for every diagram $M : \\mathcal{J} \\to \\mathcal{C}$ we\nhave a canonical isomorphism\n$$\n\\lim_\\mathcal{I} M \\circ H = \\lim_\\mathcal{J} M\n$$\nif either side exists.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Cofinal and initial categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002R","source_file":"categories.tex","source_line":1808,"source_end_line":1818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1808-L1818","statement_sha256":"95154a10bc955217a3ac3b720a87c0a4b32bee9b3a3eb69179fdf55201582c16","origin":"The Stacks Project","memory_eligible":false,"source_rank":70,"rank":70,"depth":0,"x":222.721,"y":141.543,"cluster":"categories-foundations"},{"id":"stacks:05US","tag":"05US","title":"Cofinal and initial categories · Lemma 05US","summary":"Let F : I → I' be a functor. Assume • the fibre categories (see Definition [Tag 02XH]) of I over I' are all connected, and • for every morphism α' : x' → y' in I' there exists a morphism α : x → y in I such that F(α) = α'. Then for every diagram M : I' → C the colimit colim_I M ∘ F exists if and only if colim_I' M exists and if so these colimits agree.","statement_latex":"Let $F : \\mathcal{I} \\to \\mathcal{I}'$ be a functor.\nAssume\n\\begin{enumerate}\n\\item the fibre categories (see\nDefinition \\ref{definition-fibre-category})\nof $\\mathcal{I}$ over $\\mathcal{I}'$ are all connected, and\n\\item for every morphism $\\alpha' : x' \\to y'$ in $\\mathcal{I}'$ there\nexists a morphism $\\alpha : x \\to y$ in $\\mathcal{I}$ such that\n$F(\\alpha) = \\alpha'$.\n\\end{enumerate}\nThen for every diagram $M : \\mathcal{I}' \\to \\mathcal{C}$\nthe colimit $\\colim_\\mathcal{I} M \\circ F$ exists if and only\nif $\\colim_{\\mathcal{I}'} M$ exists and if so these colimits\nagree.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Cofinal and initial categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05US","source_file":"categories.tex","source_line":1824,"source_end_line":1840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1824-L1840","statement_sha256":"5f63c4d8458cfbdc23f14652ac8141fb674064a10d34594dfd4ffc46673aa2ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":71,"rank":71,"depth":1,"x":298.943,"y":274.101,"cluster":"categories-foundations"},{"id":"stacks:0A2B","tag":"0A2B","title":"Cofinal and initial categories · Lemma 0A2B","summary":"Let I and J be a categories and denote p : I × J → J the projection. If I is connected, then for a diagram M : J → C the colimit colim_J M exists if and only if colim_I × J M ∘ p exists and if so these colimits are equal.","statement_latex":"Let $\\mathcal{I}$ and $\\mathcal{J}$ be a categories and denote\n$p : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{J}$ the projection.\nIf $\\mathcal{I}$ is connected, then for a diagram\n$M : \\mathcal{J} \\to \\mathcal{C}$ the colimit $\\colim_\\mathcal{J} M$ exists\nif and only if $\\colim_{\\mathcal{I} \\times \\mathcal{J}} M \\circ p$ exists and\nif so these colimits are equal.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Cofinal and initial categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2B","source_file":"categories.tex","source_line":1866,"source_end_line":1874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1866-L1874","statement_sha256":"53d38bc1f7a0c15d40115bc2b30fc11c9d6dee351c86ffc2ea185487dc9c92e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":72,"rank":72,"depth":2,"x":135.001,"y":219.221,"cluster":"categories-foundations"},{"id":"stacks:05XU","tag":"05XU","title":"Finite limits and colimits · Lemma 05XU","summary":"Let I be a category with • Ob(I) is finite, and • there exist finitely many morphisms f_1, …, f_m ∈ Arrows(I) such that every morphism of I is a composition f_j_1 ∘ f_j_2 ∘ … ∘ f_j_k. Then there exists a functor F : J → I such that • [(a)] J is a finite category, and • [(b)] for any diagram M : I → C the (co)limit of M over I exists if and only if the (co)limit of M ∘ F over J exists and in this case the (co)limits are canonically isomorphic. Moreover, J is connected…","statement_latex":"Let $\\mathcal{I}$ be a category with\n\\begin{enumerate}\n\\item $\\Ob(\\mathcal{I})$ is finite, and\n\\item there exist finitely many morphisms\n$f_1, \\ldots, f_m \\in \\text{Arrows}(\\mathcal{I})$ such\nthat every morphism of $\\mathcal{I}$ is a composition\n$f_{j_1} \\circ f_{j_2} \\circ \\ldots \\circ f_{j_k}$.\n\\end{enumerate}\nThen there exists a functor $F : \\mathcal{J} \\to \\mathcal{I}$\nsuch that\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{J}$ is a finite category, and\n\\item[(b)] for any diagram $M : \\mathcal{I} \\to \\mathcal{C}$ the\n(co)limit of $M$ over $\\mathcal{I}$ exists if and only if\nthe (co)limit of $M \\circ F$ over $\\mathcal{J}$ exists and in this case\nthe (co)limits are canonically isomorphic.\n\\end{enumerate}\nMoreover, $\\mathcal{J}$ is connected (resp.\\ nonempty) if and only if\n$\\mathcal{I}$ is so.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Finite limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XU","source_file":"categories.tex","source_line":1897,"source_end_line":1918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1897-L1918","statement_sha256":"aef4e8417b1905e045f8ffb34869f7a50569820bbf66fe5e88fc295403c5e985","origin":"The Stacks Project","memory_eligible":false,"source_rank":73,"rank":73,"depth":1,"x":301.161,"y":166.304,"cluster":"categories-foundations"},{"id":"stacks:04AT","tag":"04AT","title":"Finite limits and colimits · Lemma 04AT","summary":"Let C be a category. The following are equivalent: • Connected finite limits exist in C. • Equalizers and fibre products exist in C.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe following are equivalent:\n\\begin{enumerate}\n\\item Connected finite limits exist in $\\mathcal{C}$.\n\\item Equalizers and fibre products exist in $\\mathcal{C}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Finite limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AT","source_file":"categories.tex","source_line":1968,"source_end_line":1976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L1968-L1976","statement_sha256":"d8e914ebd11fb0913b23025700c60efb357aa29920aca82012de046b4bd9c8fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":74,"rank":74,"depth":2,"x":220.645,"y":300.514,"cluster":"categories-foundations"},{"id":"stacks:04AU","tag":"04AU","title":"Finite limits and colimits · Lemma 04AU","summary":"Let C be a category. The following are equivalent: • Nonempty finite limits exist in C. • Products of pairs and equalizers exist in C. • Products of pairs and fibre products exist in C.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe following are equivalent:\n\\begin{enumerate}\n\\item Nonempty finite limits exist in $\\mathcal{C}$.\n\\item Products of pairs and equalizers exist in $\\mathcal{C}$.\n\\item Products of pairs and fibre products exist in $\\mathcal{C}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Finite limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AU","source_file":"categories.tex","source_line":2022,"source_end_line":2031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2022-L2031","statement_sha256":"cfec8d6f1dc2685e9aaefc7a2688f8a319fe3e498edb9870dd5c26586341c5e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":75,"rank":75,"depth":1,"x":171.766,"y":154.891,"cluster":"categories-foundations"},{"id":"stacks:002O","tag":"002O","title":"Finite limits and colimits · Lemma 002O","summary":"Let C be a category. The following are equivalent: • Finite limits exist in C. • Finite products and equalizers exist. • The category has a final object and fibre products exist.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe following are equivalent:\n\\begin{enumerate}\n\\item Finite limits exist in $\\mathcal{C}$.\n\\item Finite products and equalizers exist.\n\\item The category has a final object and fibre products exist.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Finite limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002O","source_file":"categories.tex","source_line":2047,"source_end_line":2056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2047-L2056","statement_sha256":"2ba5d9eca7e2a9c037146926f874980d5acea8631230736797436d4695e98896","origin":"The Stacks Project","memory_eligible":false,"source_rank":76,"rank":76,"depth":1,"x":325.927,"y":235.065,"cluster":"categories-foundations"},{"id":"stacks:04AV","tag":"04AV","title":"Finite limits and colimits · Lemma 04AV","summary":"Let C be a category. The following are equivalent: • Connected finite colimits exist in C. • Coequalizers and pushouts exist in C.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe following are equivalent:\n\\begin{enumerate}\n\\item Connected finite colimits exist in $\\mathcal{C}$.\n\\item Coequalizers and pushouts exist in $\\mathcal{C}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Finite limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AV","source_file":"categories.tex","source_line":2073,"source_end_line":2081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2073-L2081","statement_sha256":"40fe19347f75d9eebbf90657a76a6450f1f859bd2fd1a35ab5d368221cc3d084","origin":"The Stacks Project","memory_eligible":false,"source_rank":77,"rank":77,"depth":3,"x":146.612,"y":263.603,"cluster":"categories-foundations"},{"id":"stacks:04AW","tag":"04AW","title":"Finite limits and colimits · Lemma 04AW","summary":"Let C be a category. The following are equivalent: • Nonempty finite colimits exist in C. • Coproducts of pairs and coequalizers exist in C. • Coproducts of pairs and pushouts exist in C.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe following are equivalent:\n\\begin{enumerate}\n\\item Nonempty finite colimits exist in $\\mathcal{C}$.\n\\item Coproducts of pairs and coequalizers exist in $\\mathcal{C}$.\n\\item Coproducts of pairs and pushouts exist in $\\mathcal{C}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Finite limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AW","source_file":"categories.tex","source_line":2088,"source_end_line":2097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2088-L2097","statement_sha256":"d9d6cf72396fae03c4203134a60238e8dc108376cd0294479bf5b131bd6bc339","origin":"The Stacks Project","memory_eligible":false,"source_rank":78,"rank":78,"depth":2,"x":256.594,"y":140.022,"cluster":"categories-foundations"},{"id":"stacks:002Q","tag":"002Q","title":"Finite limits and colimits · Lemma 002Q","summary":"Let C be a category. The following are equivalent: • Finite colimits exist in C. • Finite coproducts and coequalizers exist in C. • The category has an initial object and pushouts exist.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe following are equivalent:\n\\begin{enumerate}\n\\item Finite colimits exist in $\\mathcal{C}$.\n\\item Finite coproducts and coequalizers exist in $\\mathcal{C}$.\n\\item The category has an initial object and pushouts exist.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Finite limits and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002Q","source_file":"categories.tex","source_line":2104,"source_end_line":2113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2104-L2113","statement_sha256":"7bf1f2e79aa1772d5b75d845daa6d91240db026ea410a83937f65a2c98ab1b36","origin":"The Stacks Project","memory_eligible":false,"source_rank":79,"rank":79,"depth":2,"x":274.989,"y":294.533,"cluster":"categories-foundations"},{"id":"stacks:002V","tag":"002V","title":"Filtered colimits · Definition 002V","summary":"We say that a diagram M : I → C is directed, or filtered if the following conditions hold: • the category I has at least one object, • for every pair of objects x, y of I there exist an object z and morphisms x → z, y → z, and • for every pair of objects x, y of I and every pair of morphisms a, b : x → y of I there exists a morphism c : y → z of I such that M(c ∘ a) = M(c ∘ b) as morphisms in C. We say that an index category I is directed, or filtered if id : I → I is…","statement_latex":"We say that a diagram $M : \\mathcal{I} \\to \\mathcal{C}$ is {\\it directed},\nor {\\it filtered} if the following conditions hold:\n\\begin{enumerate}\n\\item the category $\\mathcal{I}$ has at least one object,\n\\item for every pair of objects $x, y$ of $\\mathcal{I}$\nthere exist an object $z$ and morphisms $x \\to z$,\n$y \\to z$, and\n\\item for every pair of objects $x, y$ of $\\mathcal{I}$\nand every pair of morphisms $a, b : x \\to y$ of $\\mathcal{I}$\nthere exists a morphism $c : y \\to z$ of $\\mathcal{I}$\nsuch that $M(c \\circ a) = M(c \\circ b)$ as morphisms in $\\mathcal{C}$.\n\\end{enumerate}\nWe say that an index category $\\mathcal{I}$ is {\\it directed}, or\n{\\it filtered} if $\\text{id} : \\mathcal{I} \\to \\mathcal{I}$ is filtered\n(in other words you erase the $M$ in part (3) above).","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Filtered colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002V","source_file":"categories.tex","source_line":2130,"source_end_line":2147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2130-L2147","statement_sha256":"46d0835a55098c546d613c0ccfedd229612aad9a71b447fc6028cbc6916b7f19","origin":"The Stacks Project","memory_eligible":false,"source_rank":80,"rank":80,"depth":0,"x":136.306,"y":190.384,"cluster":"categories-foundations"},{"id":"stacks:002W","tag":"002W","title":"Filtered colimits · Lemma 002W","summary":"Let I and J be index categories. Assume that I is filtered and J is finite. Let M : I × J → Sets, (i, j) ↦ M_i, j be a diagram of diagrams of sets. In this case colim_i lim_j M_i, j = lim_j colim_i M_i, j. In particular, colimits over I commute with finite products, fibre products, and equalizers of sets.","statement_latex":"Let $\\mathcal{I}$ and $\\mathcal{J}$ be index categories.\nAssume that $\\mathcal{I}$ is filtered and $\\mathcal{J}$ is finite.\nLet $M : \\mathcal{I} \\times \\mathcal{J} \\to \\textit{Sets}$,\n$(i, j) \\mapsto M_{i, j}$ be a diagram of diagrams of sets.\nIn this case\n$$\n\\colim_i \\lim_j M_{i, j}\n=\n\\lim_j \\colim_i M_{i, j}.\n$$\nIn particular, colimits over $\\mathcal{I}$ commute with finite products,\nfibre products, and equalizers of sets.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Filtered colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002W","source_file":"categories.tex","source_line":2181,"source_end_line":2195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2181-L2195","statement_sha256":"23a3e1a655f27cdde21eac475224965c47466fcee2433c26e1735707e2707023","origin":"The Stacks Project","memory_eligible":false,"source_rank":81,"rank":81,"depth":0,"x":323.478,"y":188.481,"cluster":"categories-foundations"},{"id":"stacks:0BUC","tag":"0BUC","title":"Filtered colimits · Lemma 0BUC","summary":"[KS] Let I be a category. Let J be a full subcategory. Assume that I is filtered. Assume also that for any object i of I, there exists a morphism i → j to some object j of J. Then J is filtered and cofinal in I.","statement_latex":"\\begin{reference}\n\\cite[Proposition 3.2.4]{KS}\n\\end{reference}\nLet $\\mathcal{I}$ be a category. Let $\\mathcal{J}$ be a full subcategory.\nAssume that $\\mathcal{I}$ is filtered. Assume also that for any object\n$i$ of $\\mathcal{I}$, there exists a morphism $i \\to j$\nto some object $j$ of $\\mathcal{J}$. Then $\\mathcal{J}$\nis filtered and cofinal in $\\mathcal{I}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Filtered colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUC","source_file":"categories.tex","source_line":2229,"source_end_line":2239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2229-L2239","statement_sha256":"eb5b2c90f5e8dc35a5246eb5d78eac9066a7e80fb67713ba84ba6819557ee275","origin":"The Stacks Project","memory_eligible":false,"source_rank":82,"rank":82,"depth":0,"x":186.152,"y":296.748,"cluster":"categories-foundations"},{"id":"stacks:09WQ","tag":"09WQ","title":"Filtered colimits · Lemma 09WQ","summary":"Let I be an index category, i.e., a category. Assume that for every pair of objects x, y of I there exist an object z and morphisms x → z and y → z. Then • If M and N are diagrams of sets over I, then colim (M_i × N_i) → colim M_i × colim N_i is surjective, • in general colimits of diagrams of sets over I do not commute with finite nonempty products.","statement_latex":"Let $\\mathcal{I}$ be an index category, i.e., a category. Assume\nthat for every pair of objects $x, y$ of $\\mathcal{I}$\nthere exist an object $z$ and morphisms $x \\to z$ and $y \\to z$.\nThen\n\\begin{enumerate}\n\\item If $M$ and $N$ are diagrams of sets over $\\mathcal{I}$,\nthen $\\colim (M_i \\times N_i) \\to \\colim M_i \\times \\colim N_i$\nis surjective,\n\\item in general colimits of diagrams of sets over $\\mathcal{I}$\ndo not commute with finite nonempty products.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Filtered colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WQ","source_file":"categories.tex","source_line":2250,"source_end_line":2263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2250-L2263","statement_sha256":"04f1d431dee732c6b27e5aa349da0723aad0919c76eda51f56a2ef3cd6394c7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":83,"rank":83,"depth":1,"x":200.437,"y":138.036,"cluster":"categories-foundations"},{"id":"stacks:09WR","tag":"09WR","title":"Filtered colimits · Lemma 09WR","summary":"Let I be an index category, i.e., a category. Assume that for every pair of objects x, y of I there exist an object z and morphisms x → z and y → z. Let M : I → Ab be a diagram of abelian groups over I. Then the colimit of M in the category of sets surjects onto the colimit of M in the category of abelian groups.","statement_latex":"Let $\\mathcal{I}$ be an index category, i.e., a category. Assume\nthat for every pair of objects $x, y$ of $\\mathcal{I}$\nthere exist an object $z$ and morphisms $x \\to z$ and $y \\to z$.\nLet $M : \\mathcal{I} \\to \\textit{Ab}$ be a diagram of abelian\ngroups over $\\mathcal{I}$. Then the colimit of $M$ in the category\nof sets surjects onto the colimit of $M$ in the category of\nabelian groups.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Filtered colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WR","source_file":"categories.tex","source_line":2295,"source_end_line":2304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2295-L2304","statement_sha256":"a044d88b79f01e9015e2b8c316a91ff13e1a9c84bfdb1416c242f0aa5ebdd716","origin":"The Stacks Project","memory_eligible":false,"source_rank":84,"rank":84,"depth":0,"x":318.234,"y":263.924,"cluster":"categories-foundations"},{"id":"stacks:09WS","tag":"09WS","title":"Filtered colimits · Lemma 09WS","summary":"Let I be an index category, i.e., a category. Assume that for every solid diagram xymatrix x ar[d] ar[r] & y ar@..>[d] z ar@..>[r] & w in I there exist an object w and dotted arrows making the diagram commute. Then I is either empty or a nonempty disjoint union of connected categories having the same property.","statement_latex":"Let $\\mathcal{I}$ be an index category, i.e., a category. Assume\nthat for every solid diagram\n$$\n\\xymatrix{\nx \\ar[d] \\ar[r] & y \\ar@{..>}[d] \\\\\nz \\ar@{..>}[r] & w\n}\n$$\nin $\\mathcal{I}$ there exist an object $w$ and dotted arrows\nmaking the diagram commute. Then $\\mathcal{I}$ is either empty\nor a nonempty disjoint union of connected categories having\nthe same property.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Filtered colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WS","source_file":"categories.tex","source_line":2320,"source_end_line":2334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2320-L2334","statement_sha256":"f7f8e3f85040afb886c70e0cb12bcbe96bb0e82587ae4a834b5d657609b91502","origin":"The Stacks Project","memory_eligible":false,"source_rank":85,"rank":85,"depth":0,"x":129.025,"y":237.782,"cluster":"categories-foundations"},{"id":"stacks:09WT","tag":"09WT","title":"Filtered colimits · Lemma 09WT","summary":"Let I be an index category, i.e., a category. Assume that for every solid diagram xymatrix x ar[d] ar[r] & y ar@..>[d] z ar@..>[r] & w in I there exist an object w and dotted arrows making the diagram commute. Then • an injective morphism M → N of diagrams of sets over I gives rise to an injective map colim M_i → colim N_i of sets, • in general the same is not the case for diagrams of abelian groups and their colimits.","statement_latex":"Let $\\mathcal{I}$ be an index category, i.e., a category. Assume\nthat for every solid diagram\n$$\n\\xymatrix{\nx \\ar[d] \\ar[r] & y \\ar@{..>}[d] \\\\\nz \\ar@{..>}[r] & w\n}\n$$\nin $\\mathcal{I}$ there exist an object $w$ and dotted arrows\nmaking the diagram commute. Then\n\\begin{enumerate}\n\\item an injective morphism $M \\to N$ of diagrams of sets over\n$\\mathcal{I}$ gives rise to an injective map $\\colim M_i \\to \\colim N_i$\nof sets,\n\\item in general the same is not the case for diagrams of abelian\ngroups and their colimits.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Filtered colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WT","source_file":"categories.tex","source_line":2348,"source_end_line":2367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2348-L2367","statement_sha256":"4d72df2089fe38f468a345bbcb1100d897aeb9c2e1453cef9752245f9cc8a6af","origin":"The Stacks Project","memory_eligible":false,"source_rank":86,"rank":86,"depth":1,"x":290.508,"y":149.184,"cluster":"categories-foundations"},{"id":"stacks:002X","tag":"002X","title":"Filtered colimits · Lemma 002X","summary":"Let I be an index category, i.e., a category. Assume • for every pair of morphisms a : w → x and b : w → y in I there exist an object z and morphisms c : x → z and d : y → z such that c ∘ a = d ∘ b, and • for every pair of morphisms a, b : x → y there exists a morphism c : y → z such that c ∘ a = c ∘ b. Then I is a (possibly empty) union of disjoint filtered index categories I_j.","statement_latex":"Let $\\mathcal{I}$ be an index category, i.e., a category.\nAssume\n\\begin{enumerate}\n\\item for every pair of morphisms $a : w \\to x$ and $b : w \\to y$\nin $\\mathcal{I}$ there exist an object $z$ and morphisms $c : x \\to z$\nand $d : y \\to z$ such that $c \\circ a = d \\circ b$, and\n\\item for every pair of morphisms $a, b : x \\to y$ there exists\na morphism $c : y \\to z$ such that $c \\circ a = c \\circ b$.\n\\end{enumerate}\nThen $\\mathcal{I}$ is a (possibly empty) union\nof disjoint filtered index categories $\\mathcal{I}_j$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Filtered colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002X","source_file":"categories.tex","source_line":2451,"source_end_line":2464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2451-L2464","statement_sha256":"d7ca0f917e8ce194351579c1d0d5bbfc6d971c02fd657f8ee8c472ee948e8b0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":87,"rank":87,"depth":0,"x":242.402,"y":307.049,"cluster":"categories-foundations"},{"id":"stacks:002Y","tag":"002Y","title":"Filtered colimits · Lemma 002Y","summary":"Let I be an index category satisfying the hypotheses of Lemma [Tag 002X] above. Then colimits over I commute with fibre products and equalizers in sets (and more generally with finite connected limits).","statement_latex":"Let $\\mathcal{I}$ be an index category satisfying the hypotheses of\nLemma \\ref{lemma-split-into-directed} above. Then colimits over $\\mathcal{I}$\ncommute with fibre products and equalizers in sets (and more generally\nwith finite connected limits).","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Filtered colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/002Y","source_file":"categories.tex","source_line":2477,"source_end_line":2483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2477-L2483","statement_sha256":"a81437de905f24348491dcbab4566fa103d7f15d65c8c3496a806f916bfb0c9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":88,"rank":88,"depth":1,"x":150.403,"y":162.524,"cluster":"categories-foundations"},{"id":"stacks:04AZ","tag":"04AZ","title":"Cofiltered limits · Definition 04AZ","summary":"We say that a diagram M : I → C is codirected or cofiltered if the following conditions hold: • the category I has at least one object, • for every pair of objects x, y of I there exist an object z and morphisms z → x, z → y, and • for every pair of objects x, y of I and every pair of morphisms a, b : x → y of I there exists a morphism c : w → x of I such that M(a ∘ c) = M(b ∘ c) as morphisms in C. We say that an index category I is codirected, or cofiltered if id : I → I…","statement_latex":"We say that a diagram $M : \\mathcal{I} \\to \\mathcal{C}$ is {\\it codirected}\nor {\\it cofiltered} if the following conditions hold:\n\\begin{enumerate}\n\\item the category $\\mathcal{I}$ has at least one object,\n\\item for every pair of objects $x, y$ of $\\mathcal{I}$\nthere exist an object $z$ and morphisms $z \\to x$,\n$z \\to y$, and\n\\item for every pair of objects $x, y$ of $\\mathcal{I}$\nand every pair of morphisms $a, b : x \\to y$ of $\\mathcal{I}$\nthere exists a morphism $c : w \\to x$ of $\\mathcal{I}$\nsuch that $M(a \\circ c) = M(b \\circ c)$ as morphisms in $\\mathcal{C}$.\n\\end{enumerate}\nWe say that an index category $\\mathcal{I}$ is {\\it codirected}, or\n{\\it cofiltered} if $\\text{id} : \\mathcal{I} \\to \\mathcal{I}$ is\ncofiltered (in other words you erase the $M$ in part (3) above).","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Cofiltered limits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AZ","source_file":"categories.tex","source_line":2524,"source_end_line":2541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2524-L2541","statement_sha256":"0e1991562fcf060f7a2f5f2d386c9b5eb2b83400f8fedc498740eeb0989bb8c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":89,"rank":89,"depth":0,"x":335.503,"y":217.201,"cluster":"categories-foundations"},{"id":"stacks:00D3","tag":"00D3","title":"Limits and colimits over preordered sets · Definition 00D3","summary":"Let I be a set and let ≤ be a binary relation on I. • We say ≤ is a preorder if it is transitive (if i ≤ j and j ≤ k then i ≤ k) and reflexive (i ≤ i for all i ∈ I). • A preordered set is a set endowed with a preorder. • A directed set is a preordered set (I, ≤) such that I is not empty and such that ∀ i, j ∈ I, there exists k ∈ I with i ≤ k, j ≤ k. • We say ≤ is a partial order if it is a preorder which is antisymmetric (if i ≤ j and j ≤ i, then i = j). • A partially…","statement_latex":"Let $I$ be a set and let $\\leq$ be a binary relation on $I$.\n\\begin{enumerate}\n\\item We say $\\leq$ is a {\\it preorder} if it is\ntransitive (if $i \\leq j$ and $j \\leq k$ then $i \\leq k$) and\nreflexive ($i \\leq i$ for all $i \\in I$).\n\\item A {\\it preordered set} is a set endowed with a preorder.\n\\item A {\\it directed set} is a preordered set $(I, \\leq)$\nsuch that $I$ is not empty and such that $\\forall i, j \\in I$,\nthere exists $k \\in I$ with $i \\leq k, j \\leq k$.\n\\item We say $\\leq$ is a {\\it partial order} if it is a preorder\nwhich is antisymmetric (if $i \\leq j$ and $j \\leq i$, then $i = j$).\n\\item A {\\it partially ordered set} is a set endowed with a partial order.\n\\item A {\\it directed partially ordered set} is a directed set\nwhose ordering is a partial order.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits over preordered sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00D3","source_file":"categories.tex","source_line":2571,"source_end_line":2588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2571-L2588","statement_sha256":"d8575922aa021e4d2ae8ada3c6d8627b4b25146e4542d4a36a816db7595615f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":90,"rank":90,"depth":0,"x":154.035,"y":282.273,"cluster":"categories-foundations"},{"id":"stacks:0030","tag":"0030","title":"Limits and colimits over preordered sets · Definition 0030","summary":"Let (I, ≤) be a preordered set. Let C be a category. • A system over I in C, sometimes called a inductive system over I in C is given by objects M_i of C and for every i ≤ i' a morphism f_ii' : M_i → M_i' such that f_ii = id and such that f_ii\" = f_i'i\" ∘ f_i i' whenever i ≤ i' ≤ i\". • An inverse system over I in C, sometimes called a projective system over I in C is given by objects M_i of C and for every i' ≤ i a morphism f_ii' : M_i → M_i' such that f_ii = id and such…","statement_latex":"Let $(I, \\leq)$ be a preordered set. Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item A {\\it system over $I$ in $\\mathcal{C}$}, sometimes called a\n{\\it inductive system over $I$ in $\\mathcal{C}$} is given by\nobjects $M_i$ of $\\mathcal{C}$ and for every $i \\leq i'$ a\nmorphism $f_{ii'} : M_i \\to M_{i'}$ such that $f_{ii}\n= \\text{id}$ and such that $f_{ii''} = f_{i'i''} \\circ f_{i i'}$\nwhenever $i \\leq i' \\leq i''$.\n\\item An {\\it inverse system over $I$ in $\\mathcal{C}$},\nsometimes called a {\\it projective system over $I$ in $\\mathcal{C}$}\nis given by objects $M_i$ of $\\mathcal{C}$ and for every $i' \\leq i$ a\nmorphism $f_{ii'} : M_i \\to M_{i'}$ such that $f_{ii}\n= \\text{id}$ and such that $f_{ii''} = f_{i'i''} \\circ f_{i i'}$\nwhenever $i'' \\leq i' \\leq i$. (Note reversal of inequalities.)\n\\end{enumerate}\nWe will say $(M_i, f_{ii'})$ is a (inverse) system over $I$ to\ndenote this. The maps $f_{ii'}$ are sometimes\ncalled the {\\it transition maps}.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits over preordered sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0030","source_file":"categories.tex","source_line":2606,"source_end_line":2626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2606-L2626","statement_sha256":"7645c7a770ffeb4fe24cdad39d30e18a4dc4b5da974a6adaf2cd7a7f5925c059","origin":"The Stacks Project","memory_eligible":false,"source_rank":91,"rank":91,"depth":0,"x":235.97,"y":130.488,"cluster":"categories-foundations"},{"id":"stacks:0031","tag":"0031","title":"Limits and colimits over preordered sets · Definition 0031","summary":"Let I be a preordered set. We say a system (resp. inverse system) (M_i, f_ii') is a directed system (resp. directed inverse system) if I is a directed set (Definition [Tag 00D3]): I is nonempty and for all i_1, i_2 ∈ I there exists i∈ I such that i_1 ≤ i and i_2 ≤ i.","statement_latex":"Let $I$ be a preordered set. We say a system (resp.\\ inverse system)\n$(M_i, f_{ii'})$ is a\n{\\it directed system} (resp.\\ {\\it directed inverse system})\nif $I$ is a directed set\n(Definition \\ref{definition-directed-set}): $I$ is nonempty and\nfor all $i_1, i_2 \\in I$ there exists $i\\in I$ such that\n$i_1 \\leq i$ and $i_2 \\leq i$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits over preordered sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0031","source_file":"categories.tex","source_line":2691,"source_end_line":2700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2691-L2700","statement_sha256":"be6dcfb8dae6be14e1fd16e6c0df53b4473cbdd25013e34698dcd7886f61df05","origin":"The Stacks Project","memory_eligible":false,"source_rank":92,"rank":92,"depth":1,"x":297.948,"y":289.769,"cluster":"categories-foundations"},{"id":"stacks:0032","tag":"0032","title":"Limits and colimits over preordered sets · Lemma 0032","summary":"Let I be a filtered index category. There exist a directed set I and a system (x_i, φ_ii') over I in I with the following properties: • For every category C and every diagram M : I → C with values in C, denote (M(x_i), M(φ_ii')) the corresponding system over I. If colim_i ∈ I M(x_i) exists then so does colim_I M and the transformation theta : colim_i ∈ I M(x_i) → colim_I M of Lemma [Tag 002K] is an isomorphism. • For every category C and every diagram M : I^opp → C in C,…","statement_latex":"Let $\\mathcal{I}$ be a filtered index category.\nThere exist a directed set $I$\nand a system $(x_i, \\varphi_{ii'})$ over $I$ in $\\mathcal{I}$\nwith the following properties:\n\\begin{enumerate}\n\\item For every category $\\mathcal{C}$ and every diagram\n$M : \\mathcal{I} \\to \\mathcal{C}$ with values in $\\mathcal{C}$,\ndenote $(M(x_i), M(\\varphi_{ii'}))$\nthe corresponding system over $I$. If\n$\\colim_{i \\in I} M(x_i)$ exists then so does\n$\\colim_\\mathcal{I} M$ and the transformation\n$$\n\\theta :\n\\colim_{i \\in I} M(x_i)\n\\longrightarrow\n\\colim_\\mathcal{I} M\n$$\nof Lemma \\ref{lemma-functorial-colimit} is an isomorphism.\n\\item For every category $\\mathcal{C}$ and every diagram\n$M : \\mathcal{I}^{opp} \\to \\mathcal{C}$ in $\\mathcal{C}$, denote\n$(M(x_i), M(\\varphi_{ii'}))$ the corresponding inverse system\nover $I$. If $\\lim_{i \\in I} M(x_i)$ exists then so does\n$\\lim_{\\mathcal{I}^{opp}} M$ and the transformation\n$$\n\\theta :\n\\lim_{\\mathcal{I}^{opp}} M\n\\longrightarrow\n\\lim_{i \\in I} M(x_i)\n$$\nof Lemma \\ref{lemma-functorial-limit} is an isomorphism.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits over preordered sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0032","source_file":"categories.tex","source_line":2709,"source_end_line":2742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2709-L2742","statement_sha256":"76db28e107987874940961d4ca608225e56a54350982adbf2839633cac2f91ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":93,"rank":93,"depth":1,"x":123.22,"y":207.039,"cluster":"categories-foundations"},{"id":"stacks:086J","tag":"086J","title":"Limits and colimits over preordered sets · Lemma 086J","summary":"If S : I → Sets is a cofiltered diagram of sets and all the S_i are finite nonempty, then lim_i S_i is nonempty. In other words, the limit of a directed inverse system of finite nonempty sets is nonempty.","statement_latex":"If $S : \\mathcal{I} \\to \\textit{Sets}$ is a cofiltered diagram of sets\nand all the $S_i$ are finite nonempty, then $\\lim_i S_i$ is nonempty.\nIn other words, the limit of a directed inverse system of finite nonempty sets\nis nonempty.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Limits and colimits over preordered sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086J","source_file":"categories.tex","source_line":2862,"source_end_line":2868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2862-L2868","statement_sha256":"0f4b8abf7e96069c3b9cc093e66f6088e98851b4a1f55369d3c4a9b964afbd80","origin":"The Stacks Project","memory_eligible":false,"source_rank":94,"rank":94,"depth":2,"x":319.634,"y":168.696,"cluster":"categories-foundations"},{"id":"stacks:05PU","tag":"05PU","title":"Essentially constant systems · Definition 05PU","summary":"Let M : I → C be a diagram in a category C. • Assume the index category I is filtered and let (X, (M_i → X)_i) be a cocone for M, see Remark [Tag 0G2U]. We say M is essentially constant with value X if there exist an i ∈ I and a morphism X → M_i such that • X → M_i → X is id_X, and • for all j there exist k and morphisms i → k and j → k such that the morphism M_j → M_k equals the composition M_j → X → M_i → M_k. • Assume the index category I is cofiltered and let (X, (X →…","statement_latex":"Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram in a category\n$\\mathcal{C}$.\n\\begin{enumerate}\n\\item Assume the index category $\\mathcal{I}$ is filtered and\nlet $(X, \\{M_i \\to X\\}_i)$ be a cocone for $M$, see\nRemark \\ref{remark-cones-and-cocones}. We say $M$ is\n{\\it essentially constant} with {\\it value} $X$ if there exist an\n$i \\in \\mathcal{I}$ and a morphism $X \\to M_i$ such that\n\\begin{enumerate}\n\\item $X \\to M_i \\to X$ is $\\text{id}_X$, and\n\\item for all $j$ there exist $k$ and morphisms $i \\to k$ and $j \\to k$\nsuch that the morphism $M_j \\to M_k$ equals the composition\n$M_j \\to X \\to M_i \\to M_k$.\n\\end{enumerate}\n\\item Assume the index category $\\mathcal{I}$ is cofiltered and let\n$(X, \\{X \\to M_i\\}_i)$ be a cone for $M$, see\nRemark \\ref{remark-cones-and-cocones}. We say\n$M$ is {\\it essentially constant} with {\\it value} $X$ if\nthere exist an $i \\in \\mathcal{I}$\nand a morphism $M_i \\to X$ such that\n\\begin{enumerate}\n\\item $X \\to M_i \\to X$ is $\\text{id}_X$, and\n\\item for all $j$ there exist $k$ and morphisms $k \\to i$ and $k \\to j$\nsuch that the morphism $M_k \\to M_j$ equals the composition\n$M_k \\to M_i \\to X \\to M_j$.\n\\end{enumerate}\n\\end{enumerate}\nPlease keep in mind Lemma \\ref{lemma-essentially-constant-is-limit-colimit}\nwhen using this definition.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Essentially constant systems","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PU","source_file":"categories.tex","source_line":2927,"source_end_line":2958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2927-L2958","statement_sha256":"d21610c2f8f6d13d1e7d7379e0710361565e55a620900d8e9c9673211f9b4736","origin":"The Stacks Project","memory_eligible":false,"source_rank":95,"rank":95,"depth":0,"x":205.032,"y":309.157,"cluster":"categories-foundations"},{"id":"stacks:05PV","tag":"05PV","title":"Essentially constant systems · Definition 05PV","summary":"Let C be a category. A directed system (M_i, f_ii') is an essentially constant system if M viewed as a functor I → C defines an essentially constant diagram. A directed inverse system (M_i, f_ii') is an essentially constant inverse system if M viewed as a functor I^opp → C defines an essentially constant inverse diagram.","statement_latex":"Let $\\mathcal{C}$ be a category. A directed system\n$(M_i, f_{ii'})$ is an {\\it essentially constant system}\nif $M$ viewed as a functor $I \\to \\mathcal{C}$\ndefines an essentially constant diagram. A directed inverse system\n$(M_i, f_{ii'})$ is an {\\it essentially constant inverse system} if\n$M$ viewed as a functor $I^{opp} \\to \\mathcal{C}$ defines an\nessentially constant inverse diagram.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Essentially constant systems","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PV","source_file":"categories.tex","source_line":2970,"source_end_line":2979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L2970-L2979","statement_sha256":"4d5826e28a6f45c23c927824f5acfcfa4aeb659aec4e49bca00dcde5ebdf2857","origin":"The Stacks Project","memory_eligible":false,"source_rank":96,"rank":96,"depth":0,"x":176.436,"y":139.674,"cluster":"categories-foundations"},{"id":"stacks:0G2V","tag":"0G2V","title":"Essentially constant systems · Lemma 0G2V","summary":"Let M : I → C be a diagram. If I is filtered and M is essentially constant as an ind-object, then X = colim M_i exists and M is essentially constant with value X. If I is cofiltered and M is essentially constant as a pro-object, then X = lim M_i exists and M is essentially constant with value X.","statement_latex":"Let $M : \\mathcal{I} \\to \\mathcal{C}$ be a diagram.\nIf $\\mathcal{I}$ is filtered and $M$ is essentially\nconstant as an ind-object, then $X = \\colim M_i$ exists and $M$\nis essentially constant with value $X$.\nIf $\\mathcal{I}$ is cofiltered and $M$ is essentially\nconstant as a pro-object, then $X = \\lim M_i$ exists and $M$ is\nessentially constant with value $X$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2V","source_file":"categories.tex","source_line":3002,"source_end_line":3011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3002-L3011","statement_sha256":"9e8d6cc75ee6c921ef068e474d1b266efd9a434bcdaa56035365fa6b7b753e82","origin":"The Stacks Project","memory_eligible":false,"source_rank":97,"rank":97,"depth":0,"x":334.629,"y":248.986,"cluster":"categories-foundations"},{"id":"stacks:05SH","tag":"05SH","title":"Essentially constant systems · Lemma 05SH","summary":"Let C be a category. Let M : I → C be a diagram with filtered (resp. cofiltered) index category I. Let F : C → D be a functor. If M is essentially constant as an ind-object (resp. pro-object), then so is F ∘ M : I → D.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $M : \\mathcal{I} \\to \\mathcal{C}$\nbe a diagram with filtered (resp.\\ cofiltered) index category $\\mathcal{I}$.\nLet $F : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nIf $M$ is essentially constant as an ind-object (resp.\\ pro-object),\nthen so is $F \\circ M : \\mathcal{I} \\to \\mathcal{D}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SH","source_file":"categories.tex","source_line":3119,"source_end_line":3126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3119-L3126","statement_sha256":"90417219ca8dba37ec50d7608a0a7239db5cd766d2ba3ea5e08b8108585ea387","origin":"The Stacks Project","memory_eligible":false,"source_rank":98,"rank":98,"depth":0,"x":129.027,"y":258.188,"cluster":"categories-foundations"},{"id":"stacks:05PY","tag":"05PY","title":"Essentially constant systems · Lemma 05PY","summary":"Let C be a category. Let M : I → C be a diagram with filtered index category I. The following are equivalent • M is an essentially constant ind-object, • there exists a cocone (X, (M_i → X)_i) such that for any W in C the map colim_i Mor_C(W, M_i) → Mor_C(W, X) is bijective, • X = colim_i M_i exists and for any W in C the map colim_i Mor_C(W, M_i) → Mor_C(W, X) is bijective, and • there exists an i in I and a morphism X → M_i such that for any W in C the map Mor_C(W, X) →…","statement_latex":"Let $\\mathcal{C}$ be a category. Let $M : \\mathcal{I} \\to \\mathcal{C}$\nbe a diagram with filtered index category $\\mathcal{I}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ is an essentially constant ind-object,\n\\item there exists a cocone $(X, \\{M_i \\to X\\}_i)$ such that for any\n$W$ in $\\mathcal{C}$ the map\n$\\colim_i \\Mor_\\mathcal{C}(W, M_i) \\to \\Mor_\\mathcal{C}(W, X)$\nis bijective,\n\\item $X = \\colim_i M_i$ exists and for any $W$ in $\\mathcal{C}$\nthe map\n$\\colim_i \\Mor_\\mathcal{C}(W, M_i) \\to \\Mor_\\mathcal{C}(W, X)$\nis bijective, and\n\\item there exists an $i$ in $\\mathcal{I}$ and a morphism $X \\to M_i$\nsuch that for any $W$ in $\\mathcal{C}$ the map $\\Mor_\\mathcal{C}(W, X) \\to\n\\colim_{j \\in \\mathcal{I}} \\Mor_\\mathcal{C}(W, M_j)$\nis bijective.\n\\end{enumerate}\nIn cases (2), (3), and (4) the value of the essentially constant system is $X$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PY","source_file":"categories.tex","source_line":3133,"source_end_line":3154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3133-L3154","statement_sha256":"0a707aec5d3f5bef40fb5fe2aeeacdd6e8cea03ca9347f9306a8a73c7d413498","origin":"The Stacks Project","memory_eligible":false,"source_rank":99,"rank":99,"depth":0,"x":273.967,"y":134.115,"cluster":"categories-foundations"},{"id":"stacks:05PZ","tag":"05PZ","title":"Essentially constant systems · Lemma 05PZ","summary":"Let C be a category. Let M : I → C be a diagram with cofiltered index category I. The following are equivalent • M is an essentially constant pro-object, • there exists a cone (X, (X → M_i)) such that for any W in C the map colim_i ∈ I^opp Mor_C(M_i, W) → Mor_C(X, W) is bijective, • X = lim_i M_i exists and for any W in C the map colim_i ∈ I^opp Mor_C(M_i, W) → Mor_C(X, W) is bijective, and • there exists an i in I and a morphism M_i → X such that for any W in C the map…","statement_latex":"Let $\\mathcal{C}$ be a category. Let $M : \\mathcal{I} \\to \\mathcal{C}$\nbe a diagram with cofiltered index category $\\mathcal{I}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ is an essentially constant pro-object,\n\\item there exists a cone $(X, \\{X \\to M_i\\})$\nsuch that for any $W$ in $\\mathcal{C}$ the map\n$\\colim_{i \\in \\mathcal{I}^{opp}} \\Mor_\\mathcal{C}(M_i, W) \\to\n\\Mor_\\mathcal{C}(X, W)$ is bijective,\n\\item $X = \\lim_i M_i$ exists and for any $W$ in $\\mathcal{C}$\nthe map\n$\\colim_{i \\in \\mathcal{I}^{opp}} \\Mor_\\mathcal{C}(M_i, W) \\to\n\\Mor_\\mathcal{C}(X, W)$ is bijective, and\n\\item there exists an $i$ in $\\mathcal{I}$ and a morphism $M_i \\to X$\nsuch that for any $W$ in $\\mathcal{C}$ the map $\\Mor_\\mathcal{C}(X, W) \\to\n\\colim_{j \\in \\mathcal{I}^{opp}} \\Mor_\\mathcal{C}(M_j, W)$ is bijective.\n\\end{enumerate}\nIn cases (2), (3), and (4) the value of the essentially constant system is $X$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PZ","source_file":"categories.tex","source_line":3195,"source_end_line":3215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3195-L3215","statement_sha256":"656e8014612ee1ede48f9e7fbf839095481ad6f6919575264d23f867ecf28cd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":100,"rank":100,"depth":1,"x":266.829,"y":308.719,"cluster":"categories-foundations"},{"id":"stacks:0A1S","tag":"0A1S","title":"Essentially constant systems · Lemma 0A1S","summary":"Let C be a category. Let H : I → J be a functor of filtered index categories. If H is cofinal, then any diagram M : J → C is essentially constant if and only if M ∘ H is essentially constant.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $H : \\mathcal{I} \\to \\mathcal{J}$\nbe a functor of filtered index categories. If $H$ is cofinal, then\nany diagram $M : \\mathcal{J} \\to \\mathcal{C}$ is essentially constant\nif and only if $M \\circ H$ is essentially constant.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1S","source_file":"categories.tex","source_line":3221,"source_end_line":3227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3221-L3227","statement_sha256":"09569d81f53d34c5a0161ba4f7195c84465b6a5d79c9a4d760ebc0f776f236aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":101,"rank":101,"depth":1,"x":131.011,"y":175.258,"cluster":"categories-foundations"},{"id":"stacks:0A2C","tag":"0A2C","title":"Essentially constant systems · Lemma 0A2C","summary":"Let I and J be filtered categories and denote p : I × J → J the projection. Then I × J is filtered and a diagram M : J → C is essentially constant if and only if M ∘ p : I × J → C is essentially constant.","statement_latex":"Let $\\mathcal{I}$ and $\\mathcal{J}$ be filtered categories and denote\n$p : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{J}$ the projection.\nThen $\\mathcal{I} \\times \\mathcal{J}$ is filtered and a diagram\n$M : \\mathcal{J} \\to \\mathcal{C}$ is essentially constant if and only\nif $M \\circ p : \\mathcal{I} \\times \\mathcal{J} \\to \\mathcal{C}$\nis essentially constant.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2C","source_file":"categories.tex","source_line":3235,"source_end_line":3243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3235-L3243","statement_sha256":"6e2348c7b4d41859f633be75bf9d958a3d3d2577c0ceabfbfcb6e1c74cd98835","origin":"The Stacks Project","memory_eligible":false,"source_rank":102,"rank":102,"depth":2,"x":339.507,"y":196.71,"cluster":"categories-foundations"},{"id":"stacks:0A1T","tag":"0A1T","title":"Essentially constant systems · Lemma 0A1T","summary":"Let C be a category. Let H : I → J be a functor of cofiltered index categories. If H is initial, then any diagram M : J → C is essentially constant if and only if M ∘ H is essentially constant.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $H : \\mathcal{I} \\to \\mathcal{J}$\nbe a functor of cofiltered index categories. If $H$ is initial, then\nany diagram $M : \\mathcal{J} \\to \\mathcal{C}$ is essentially constant\nif and only if $M \\circ H$ is essentially constant.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1T","source_file":"categories.tex","source_line":3252,"source_end_line":3258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3252-L3258","statement_sha256":"dad42d4136e34ba336ce5d4aac5024a7ba5326e32287dd76d66c0852764c2a89","origin":"The Stacks Project","memory_eligible":false,"source_rank":103,"rank":103,"depth":2,"x":167.68,"y":299.694,"cluster":"categories-foundations"},{"id":"stacks:0034","tag":"0034","title":"Exact functors · Definition 0034","summary":"Let F : A → B be a functor. • Suppose all finite limits exist in A. We say F is left exact if it commutes with all finite limits. • Suppose all finite colimits exist in A. We say F is right exact if it commutes with all finite colimits. • We say F is exact if it is both left and right exact.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\n\\begin{enumerate}\n\\item Suppose all finite limits exist in $\\mathcal{A}$.\nWe say $F$ is {\\it left exact} if it commutes\nwith all finite limits.\n\\item Suppose all finite colimits exist in $\\mathcal{A}$.\nWe say $F$ is {\\it right exact} if it commutes\nwith all finite colimits.\n\\item We say $F$ is {\\it exact} if it is both left and right\nexact.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Exact functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0034","source_file":"categories.tex","source_line":3278,"source_end_line":3291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3278-L3291","statement_sha256":"7261a92609c7b63a5daa840509b517f041a2e827e518099524da06140a63b8b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":104,"rank":104,"depth":0,"x":211.779,"y":125.421,"cluster":"categories-foundations"},{"id":"stacks:0035","tag":"0035","title":"Exact functors · Lemma 0035","summary":"Let F : A → B be a functor. Suppose all finite limits exist in A, see Lemma [Tag 002O]. The following are equivalent: • F is left exact, • F commutes with finite products and equalizers, and • F transforms a final object of A into a final object of B, and commutes with fibre products.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\nSuppose all finite limits exist in $\\mathcal{A}$,\nsee Lemma \\ref{lemma-finite-limits-exist}.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $F$ is left exact,\n\\item $F$ commutes with finite products and equalizers, and\n\\item $F$ transforms a final object of $\\mathcal{A}$\ninto a final object of $\\mathcal{B}$, and commutes with fibre products.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Exact functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0035","source_file":"categories.tex","source_line":3293,"source_end_line":3305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3293-L3305","statement_sha256":"ef3dde668eefb8db9e60b15aa641c07434dbc2355c49b865fd9af1fb4bf5c427","origin":"The Stacks Project","memory_eligible":false,"source_rank":105,"rank":105,"depth":2,"x":319.919,"y":279.684,"cluster":"categories-foundations"},{"id":"stacks:0GMN","tag":"0GMN","title":"Exact functors · Lemma 0GMN","summary":"Let F : A → B be a functor. Suppose all finite colimits exist in A, see Lemma [Tag 002Q]. The following are equivalent: • F is right exact, • F commutes with finite coproducts and coequalizers, and • F transforms an initial object of A into an initial object of B, and commutes with pushouts.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\nSuppose all finite colimits exist in $\\mathcal{A}$,\nsee Lemma \\ref{lemma-colimits-exist}.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $F$ is right exact,\n\\item $F$ commutes with finite coproducts and coequalizers, and\n\\item $F$ transforms an initial object of $\\mathcal{A}$\ninto an initial object of $\\mathcal{B}$, and commutes with pushouts.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Exact functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMN","source_file":"categories.tex","source_line":3319,"source_end_line":3331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3319-L3331","statement_sha256":"95a688bb4b6b18022d9c653a5b9813ffc07941939c52a5b673987d1695dc00d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":106,"rank":106,"depth":3,"x":115.161,"y":227.045,"cluster":"categories-foundations"},{"id":"stacks:0037","tag":"0037","title":"Adjoint functors · Definition 0037","summary":"Let C, D be categories. Let u : C → D and v : D → C be functors. We say that u is a left adjoint of v, or that v is a right adjoint to u if there are bijections Mor_D(u(X), Y) → Mor_C(X, v(Y)) functorial in X ∈ Ob(C), and Y ∈ Ob(D).","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be categories.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ and\n$v : \\mathcal{D} \\to \\mathcal{C}$ be functors.\nWe say that $u$ is a {\\it left adjoint} of $v$, or that\n$v$ is a {\\it right adjoint} to $u$ if there are bijections\n$$\n\\Mor_\\mathcal{D}(u(X), Y)\n\\longrightarrow\n\\Mor_\\mathcal{C}(X, v(Y))\n$$\nfunctorial in $X \\in \\Ob(\\mathcal{C})$, and\n$Y \\in \\Ob(\\mathcal{D})$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Adjoint functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0037","source_file":"categories.tex","source_line":3343,"source_end_line":3357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3343-L3357","statement_sha256":"03693ef3f66db9473fb828224caf0cfb8a8bc21c28a0ef7c380384af856e9f8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":107,"rank":107,"depth":0,"x":309.384,"y":149.315,"cluster":"categories-foundations"},{"id":"stacks:0A8B","tag":"0A8B","title":"Adjoint functors · Lemma 0A8B","summary":"Let u : C → D be a functor between categories. If for each y ∈ Ob(D) the functor x ↦ Mor_D(u(x), y) is representable, then u has a right adjoint.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories.\nIf for each $y \\in \\Ob(\\mathcal{D})$ the functor\n$x \\mapsto \\Mor_\\mathcal{D}(u(x), y)$ is representable, then\n$u$ has a right adjoint.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8B","source_file":"categories.tex","source_line":3385,"source_end_line":3391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3385-L3391","statement_sha256":"45cc945af7266d6317b4d531de88e9a5a5439db65b72784c8d7092765c408309","origin":"The Stacks Project","memory_eligible":false,"source_rank":108,"rank":108,"depth":1,"x":228.299,"y":317.615,"cluster":"categories-foundations"},{"id":"stacks:0FWV","tag":"0FWV","title":"Adjoint functors · Lemma 0FWV","summary":"Bhargav Bhatt, private communication. Let u be a left adjoint to v as in Definition [Tag 0037]. • If v ∘ u is fully faithful, then u is fully faithful. • If u ∘ v is fully faithful, then v is fully faithful.","statement_latex":"\\begin{reference}\nBhargav Bhatt, private communication.\n\\end{reference}\nLet $u$ be a left adjoint to $v$ as in Definition \\ref{definition-adjoint}.\n\\begin{enumerate}\n\\item If $v \\circ u$ is fully faithful, then $u$ is fully faithful.\n\\item If $u \\circ v$ is fully faithful, then $v$ is fully faithful.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWV","source_file":"categories.tex","source_line":3407,"source_end_line":3417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3407-L3417","statement_sha256":"4822bc574bc7da4f6877a9e7000b3a3248a47fd326c6e975b3d75d3913087e1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":109,"rank":109,"depth":1,"x":152.401,"y":146.721,"cluster":"categories-foundations"},{"id":"stacks:07RB","tag":"07RB","title":"Adjoint functors · Lemma 07RB","summary":"Let u be a left adjoint to v as in Definition [Tag 0037]. Then • u is fully faithful ⇔ id ≅ v ∘ u ⇔ eta : id → v ∘ u is an isomorphism, • v is fully faithful ⇔ u ∘ v ≅ id ⇔ ε : u ∘ v → id is an isomorphism.","statement_latex":"Let $u$ be a left adjoint to $v$ as in Definition \\ref{definition-adjoint}.\nThen\n\\begin{enumerate}\n\\item $u$ is fully faithful $\\Leftrightarrow$ $\\text{id} \\cong v \\circ u$\n$\\Leftrightarrow$ $\\eta : \\text{id} \\to v \\circ u$ is an isomorphism,\n\\item $v$ is fully faithful $\\Leftrightarrow$\n$u \\circ v \\cong \\text{id}$ $\\Leftrightarrow$\n$\\epsilon : u \\circ v \\to \\text{id}$ is an isomorphism.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RB","source_file":"categories.tex","source_line":3439,"source_end_line":3450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3439-L3450","statement_sha256":"2a004279faa921f4c2bcd52220f59275ef1e3a3e51d0cf70073a0fea351077ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":110,"rank":110,"depth":2,"x":346.676,"y":230.049,"cluster":"categories-foundations"},{"id":"stacks:0038","tag":"0038","title":"Adjoint functors · Lemma 0038","summary":"Let u be a left adjoint to v as in Definition [Tag 0037]. • Suppose that M : I → C is a diagram, and suppose that colim_I M exists in C. Then u(colim_I M) = colim_I u ∘ M. In other words, u commutes with (representable) colimits. • Suppose that M : I → D is a diagram, and suppose that lim_I M exists in D. Then v(lim_I M) = lim_I v ∘ M. In other words v commutes with representable limits.","statement_latex":"Let $u$ be a left adjoint to $v$ as in Definition \\ref{definition-adjoint}.\n\\begin{enumerate}\n\\item Suppose that $M : \\mathcal{I} \\to \\mathcal{C}$ is a diagram,\nand suppose that $\\colim_\\mathcal{I} M$ exists in\n$\\mathcal{C}$. Then $u(\\colim_\\mathcal{I} M) =\n\\colim_\\mathcal{I} u \\circ M$. In other words,\n$u$ commutes with (representable) colimits.\n\\item Suppose that $M : \\mathcal{I} \\to \\mathcal{D}$ is a diagram,\nand suppose that $\\lim_\\mathcal{I} M$ exists in\n$\\mathcal{D}$. Then $v(\\lim_\\mathcal{I} M) =\n\\lim_\\mathcal{I} v \\circ M$. In other words $v$ commutes\nwith representable limits.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0038","source_file":"categories.tex","source_line":3471,"source_end_line":3486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3471-L3486","statement_sha256":"dad80b2fbe4dcb0cf4b82cfb09e423ae925a25edcc96cb5063564cc008630c13","origin":"The Stacks Project","memory_eligible":false,"source_rank":111,"rank":111,"depth":1,"x":135.461,"y":279.059,"cluster":"categories-foundations"},{"id":"stacks:0039","tag":"0039","title":"Adjoint functors · Lemma 0039","summary":"Let u be a left adjoint of v as in Definition [Tag 0037]. • If C has finite colimits, then u is right exact. • If D has finite limits, then v is left exact.","statement_latex":"Let $u$ be a left adjoint of $v$ as in Definition \\ref{definition-adjoint}.\n\\begin{enumerate}\n\\item If $\\mathcal{C}$ has finite colimits, then $u$ is right exact.\n\\item If $\\mathcal{D}$ has finite limits, then $v$ is left exact.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0039","source_file":"categories.tex","source_line":3509,"source_end_line":3516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3509-L3516","statement_sha256":"166c0adb9b0a0a59756999bce236ea034262d75289cd71ebfa04a4518d26f6e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":112,"rank":112,"depth":2,"x":252.317,"y":122.374,"cluster":"categories-foundations"},{"id":"stacks:0GLL","tag":"0GLL","title":"Adjoint functors · Lemma 0GLL","summary":"Let u : C → D be a left adjoint to the functor v : D → C. Let eta_X : X → v(u(X)) be the unit and ε_Y : u(v(Y)) → Y be the counit. Then u(X) xrightarrowu(eta_X) u(v(u(X)) xrightarrowε_u(X) u(X) and v(Y) xrightarroweta_v(Y) v(u(v(Y))) xrightarrowv(ε_Y) v(Y) are the identity morphisms.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a left adjoint to the functor\n$v : \\mathcal{D} \\to \\mathcal{C}$. Let $\\eta_X : X \\to v(u(X))$ be the unit\nand $\\epsilon_Y : u(v(Y)) \\to Y$ be the counit. Then\n$$\nu(X) \\xrightarrow{u(\\eta_X)} u(v(u(X))\n\\xrightarrow{\\epsilon_{u(X)}} u(X)\n\\quad\\text{and}\\quad\nv(Y) \\xrightarrow{\\eta_{v(Y)}} v(u(v(Y))) \\xrightarrow{v(\\epsilon_Y)}\nv(Y)\n$$\nare the identity morphisms.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLL","source_file":"categories.tex","source_line":3522,"source_end_line":3535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3522-L3535","statement_sha256":"ac0f0e097e5994b2ae695f0418f0adeb70a825774c34128f0e20444a7ea208ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":113,"rank":113,"depth":0,"x":292.326,"y":305.024,"cluster":"categories-foundations"},{"id":"stacks:0B65","tag":"0B65","title":"Adjoint functors · Lemma 0B65","summary":"Let u_1, u_2 : C → D be functors with right adjoints v_1, v_2 : D → C. Let β : u_2 → u_1 be a transformation of functors. Let β^vee : v_1 → v_2 be the corresponding transformation of adjoint functors. Then xymatrix u_2 ∘ v_1 ar[r]_β ar[d]_β^vee & u_1 ∘ v_1 ar[d] u_2 ∘ v_2 ar[r] & id is commutative where the unlabeled arrows are the counit transformations.","statement_latex":"Let $u_1, u_2 : \\mathcal{C} \\to \\mathcal{D}$ be functors with right\nadjoints $v_1, v_2 : \\mathcal{D} \\to \\mathcal{C}$. Let $\\beta : u_2 \\to u_1$\nbe a transformation of functors. Let $\\beta^\\vee : v_1 \\to v_2$ be\nthe corresponding transformation of adjoint functors. Then\n$$\n\\xymatrix{\nu_2 \\circ v_1 \\ar[r]_\\beta \\ar[d]_{\\beta^\\vee} &\nu_1 \\circ v_1 \\ar[d] \\\\\nu_2 \\circ v_2 \\ar[r] & \\text{id}\n}\n$$\nis commutative where the unlabeled arrows are the counit transformations.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B65","source_file":"categories.tex","source_line":3541,"source_end_line":3555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3541-L3555","statement_sha256":"72f46f71bf813ce58685396779956853a3adec81ebd9d5b02c70a0271abfe511","origin":"The Stacks Project","memory_eligible":false,"source_rank":114,"rank":114,"depth":0,"x":115.168,"y":192.55,"cluster":"categories-foundations"},{"id":"stacks:0DV0","tag":"0DV0","title":"Adjoint functors · Lemma 0DV0","summary":"Let A, B, and C be categories. Let v : A → B and v' : B → C be functors with left adjoints u and u' respectively. Then • The functor v\" = v' ∘ v has a left adjoint equal to u\" = u ∘ u'. • Given X in A we have ε_X^v ∘ u(ε^v'_v(X)) = ε^v\"_X : u\"(v\"(X)) → X Where ε is the counit of the adjunctions.","statement_latex":"Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be categories.\nLet $v : \\mathcal{A} \\to \\mathcal{B}$ and\n$v' : \\mathcal{B} \\to \\mathcal{C}$ be functors\nwith left adjoints $u$ and $u'$ respectively. Then\n\\begin{enumerate}\n\\item The functor $v'' = v' \\circ v$ has a left adjoint equal to\n$u'' = u \\circ u'$.\n\\item Given $X$ in $\\mathcal{A}$ we have\n\\begin{equation}\n\n\\epsilon_X^v \\circ u(\\epsilon^{v'}_{v(X)}) = \\epsilon^{v''}_X :\nu''(v''(X)) \\to X\n\\end{equation}\nWhere $\\epsilon$ is the counit of the adjunctions.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DV0","source_file":"categories.tex","source_line":3575,"source_end_line":3592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3575-L3592","statement_sha256":"27c9e28a9c953713da1bc250362d1140569f2f3bd40221b319fca01730262841","origin":"The Stacks Project","memory_eligible":false,"source_rank":115,"rank":115,"depth":1,"x":337.213,"y":174.888,"cluster":"categories-foundations"},{"id":"stacks:0AHN","tag":"0AHN","title":"A criterion for representability · Lemma 0AHN","summary":"Let C be a big category which has limits. Let F : C → Sets be a functor. Assume that • F commutes with limits, • there exist a family (x_i)_i ∈ I of objects of C and for each i ∈ I an element f_i ∈ F(x_i) such that for y ∈ Ob(C) and g ∈ F(y) there exist an i and a morphism φ : x_i → y with F(φ)(f_i) = g. Then F is representable, i.e., there exists an object x of C such that F(y) = Mor_C(x, y) functorially in y.","statement_latex":"Let $\\mathcal{C}$ be a big\\footnote{See Remark \\ref{remark-big-categories}.}\ncategory which has limits. Let $F : \\mathcal{C} \\to \\textit{Sets}$ be a\nfunctor. Assume that\n\\begin{enumerate}\n\\item $F$ commutes with limits,\n\\item there exist a family $\\{x_i\\}_{i \\in I}$ of objects of $\\mathcal{C}$\nand for each $i \\in I$ an element $f_i \\in F(x_i)$\nsuch that for $y \\in \\Ob(\\mathcal{C})$ and $g \\in F(y)$\nthere exist an $i$ and a morphism $\\varphi : x_i \\to y$\nwith $F(\\varphi)(f_i) = g$.\n\\end{enumerate}\nThen $F$ is representable, i.e., there exists an object $x$\nof $\\mathcal{C}$ such that\n$$\nF(y) = \\Mor_\\mathcal{C}(x, y)\n$$\nfunctorially in $y$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"A criterion for representability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHN","source_file":"categories.tex","source_line":3634,"source_end_line":3653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3634-L3653","statement_sha256":"3fcfba11423e0f7fc473926db2d0932f4e9841b94f3e5ca5119accc175163363","origin":"The Stacks Project","memory_eligible":false,"source_rank":116,"rank":116,"depth":0,"x":187.037,"y":314.505,"cluster":"categories-foundations"},{"id":"stacks:0AHQ","tag":"0AHQ","title":"Adjoint functor theorem · Theorem 0AHQ","summary":"Let G : C → D be a functor of big categories. Assume C has limits, G commutes with them, and for every object y of D there exists a set of pairs (x_i, f_i)_i ∈ I with x_i ∈ Ob(C), f_i ∈ Mor_D(y, G(x_i)) such that for any pair (x, f) with x ∈ Ob(C), f ∈ Mor_D(y, G(x)) there are an i and a morphism h : x_i → x such that f = G(h) ∘ f_i. Then G has a left adjoint F.","statement_latex":"Let $G : \\mathcal{C} \\to \\mathcal{D}$ be a functor of big categories.\nAssume $\\mathcal{C}$ has limits, $G$ commutes with them, and for\nevery object $y$ of $\\mathcal{D}$ there exists a set of pairs\n$(x_i, f_i)_{i \\in I}$ with $x_i \\in \\Ob(\\mathcal{C})$,\n$f_i \\in \\Mor_\\mathcal{D}(y, G(x_i))$ such that for any\npair $(x, f)$ with $x \\in \\Ob(\\mathcal{C})$,\n$f \\in \\Mor_\\mathcal{D}(y, G(x))$ there are an $i$ and a morphism\n$h : x_i \\to x$ such that $f = G(h) \\circ f_i$.\nThen $G$ has a left adjoint $F$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"A criterion for representability","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHQ","source_file":"categories.tex","source_line":3746,"source_end_line":3757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3746-L3757","statement_sha256":"a88d2ea590380616f52b9733a0356b27213c29b6fc44a86415e5a6610f3b5577","origin":"The Stacks Project","memory_eligible":false,"source_rank":117,"rank":117,"depth":1,"x":185.493,"y":125.535,"cluster":"categories-foundations"},{"id":"stacks:0FWX","tag":"0FWX","title":"Categorically compact objects · Definition 0FWX","summary":"Let C be a big category. An object X of C is called a categorically compact if we have Mor_C(X, colim_i M_i) = colim_i Mor_C(X, M_i) for every filtered diagram M : I → C such that colim_i M_i exists.","statement_latex":"Let $\\mathcal{C}$ be a big\\footnote{See Remark \\ref{remark-big-categories}.}\ncategory. An object $X$ of $\\mathcal{C}$ is called a {\\it categorically compact}\nif we have\n$$\n\\Mor_\\mathcal{C}(X, \\colim_i M_i) =\n\\colim_i \\Mor_\\mathcal{C}(X, M_i)\n$$\nfor every filtered diagram $M : \\mathcal{I} \\to \\mathcal{C}$ such that\n$\\colim_i M_i$ exists.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categorically compact objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWX","source_file":"categories.tex","source_line":3779,"source_end_line":3790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3779-L3790","statement_sha256":"cdf241bf3f726f44867484cf30b37f42d4d75d6f6115a08c98f94a9867ffef3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":118,"rank":118,"depth":0,"x":339.244,"y":264.591,"cluster":"categories-foundations"},{"id":"stacks:0FWY","tag":"0FWY","title":"Categorically compact objects · Lemma 0FWY","summary":"Let C and D be big categories having filtered colimits. Let C' ⊂ C be a small full subcategory consisting of categorically compact objects of C such that every object of C is a filtered colimit of objects of C'. Then every functor F' : C' → D has a unique extension F : C → D commuting with filtered colimits.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be big categories having filtered\ncolimits. Let $\\mathcal{C}' \\subset \\mathcal{C}$ be a small full subcategory\nconsisting of categorically compact objects of $\\mathcal{C}$ such that every\nobject of $\\mathcal{C}$ is a filtered colimit of objects of $\\mathcal{C}'$.\nThen every functor $F' : \\mathcal{C}' \\to \\mathcal{D}$ has a unique\nextension $F : \\mathcal{C} \\to \\mathcal{D}$ commuting with filtered colimits.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categorically compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWY","source_file":"categories.tex","source_line":3796,"source_end_line":3804,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3796-L3804","statement_sha256":"86a3db928532378a2c53bdc0cc9c92cc4b21a3aa1146e22b3f57d319fcf3226a","origin":"The Stacks Project","memory_eligible":false,"source_rank":119,"rank":119,"depth":0,"x":113.098,"y":249.229,"cluster":"categories-foundations"},{"id":"stacks:04VC","tag":"04VC","title":"Localization in categories · Definition 04VC","summary":"Let C be a category. A set of arrows S of C is called a left multiplicative system if it has the following properties: • [LMS1] The identity of every object of C is in S and the composition of two composable elements of S is in S. • [LMS2] Every solid diagram xymatrix X ar[d]_t ar[r]_g & Y ar@..>[d]^s Z ar@..>[r]^f & W with t ∈ S can be completed to a commutative dotted square with s ∈ S. • [LMS3] For every pair of morphisms f, g : X → Y and t ∈ S with target X such that…","statement_latex":"Let $\\mathcal{C}$ be a category. A set of arrows $S$ of $\\mathcal{C}$ is\ncalled a {\\it left multiplicative system} if it has the following properties:\n\\begin{enumerate}\n\\item[LMS1] The identity of every object of $\\mathcal{C}$ is in $S$ and\nthe composition of two composable elements of $S$ is in $S$.\n\\item[LMS2] Every solid diagram\n$$\n\\xymatrix{\nX \\ar[d]_t \\ar[r]_g & Y \\ar@{..>}[d]^s \\\\\nZ \\ar@{..>}[r]^f & W\n}\n$$\nwith $t \\in S$ can be completed to a commutative dotted square with\n$s \\in S$.\n\\item[LMS3] For every pair of morphisms $f, g : X \\to Y$ and\n$t \\in S$ with target $X$ such that $f \\circ t = g \\circ t$\nthere exists an $s \\in S$ with source $Y$ such that\n$s \\circ f = s \\circ g$.\n\\end{enumerate}\nA set of arrows $S$ of $\\mathcal{C}$ is\ncalled a {\\it right multiplicative system}\nif it has the following properties:\n\\begin{enumerate}\n\\item[RMS1] The identity of every object of $\\mathcal{C}$ is in $S$ and\nthe composition of two composable elements of $S$ is in $S$.\n\\item[RMS2] Every solid diagram\n$$\n\\xymatrix{\nX \\ar@{..>}[d]_t \\ar@{..>}[r]_g & Y \\ar[d]^s \\\\\nZ \\ar[r]^f & W\n}\n$$\nwith $s \\in S$ can be completed to a commutative dotted square with\n$t \\in S$.\n\\item[RMS3] For every pair of morphisms $f, g : X \\to Y$ and\n$s \\in S$ with source $Y$ such that $s \\circ f = s \\circ g$\nthere exists a $t \\in S$ with target $X$ such that\n$f \\circ t = g \\circ t$.\n\\end{enumerate}\nA set of arrows $S$ of $\\mathcal{C}$ is called a {\\it multiplicative system}\nif it is both a left multiplicative system and a right multiplicative system.\nIn other words, this means that MS1, MS2, MS3 hold, where\nMS1 $=$ LMS1 $+$ RMS1, MS2 $=$ LMS2 $+$ RMS2, and\nMS3 $=$ LMS3 $+$ RMS3. (That said, of course LMS1 $=$ RMS1\n$=$ MS1.)","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VC","source_file":"categories.tex","source_line":3905,"source_end_line":3952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L3905-L3952","statement_sha256":"350af5dd3c3ec77ee58b9860c774d8f86ab3125bc9920ab793b11a55e846a241","origin":"The Stacks Project","memory_eligible":false,"source_rank":120,"rank":120,"depth":0,"x":292.956,"y":131.749,"cluster":"categories-foundations"},{"id":"stacks:04VD","tag":"04VD","title":"Localization in categories · Lemma 04VD","summary":"Let C be a category and let S be a left multiplicative system. • The relation on pairs defined above is an equivalence relation. • The composition rule given above is well defined on equivalence classes. • Composition is associative (and the identity morphisms satisfy the identity axioms), and hence S^-1C is a category.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative\nsystem.\n\\begin{enumerate}\n\\item The relation on pairs defined above is an equivalence relation.\n\\item The composition rule given above is well defined on equivalence\nclasses.\n\\item Composition is associative (and the identity morphisms satisfy\nthe identity axioms), and hence $S^{-1}\\mathcal{C}$ is a category.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VD","source_file":"categories.tex","source_line":4001,"source_end_line":4012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4001-L4012","statement_sha256":"d11bdc9ea2e85132521aad247024b7af475b1c4eb3799fcdb6611da065a7eb88","origin":"The Stacks Project","memory_eligible":false,"source_rank":121,"rank":121,"depth":0,"x":254.647,"y":321.213,"cluster":"categories-foundations"},{"id":"stacks:0BM2","tag":"0BM2","title":"Localization in categories · Definition 0BM2","summary":"Let C be a category and let S be a left multiplicative system of morphisms of C. Given any morphism f : X → Y' in C and any morphism s : Y → Y' in S, we denote by s^-1 f the equivalence class of the pair (f : X → Y', s : Y → Y'). This is a morphism from X to Y in S^-1 C.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative\nsystem of morphisms of $\\mathcal{C}$. Given any morphism\n$f : X \\to Y'$ in $\\mathcal{C}$ and any morphism $s : Y \\to Y'$ in\n$S$, we denote by {\\it $s^{-1} f$} the equivalence class of the pair\n$(f : X \\to Y', s : Y \\to Y')$. This is a morphism from $X$ to $Y$\nin $S^{-1} \\mathcal{C}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BM2","source_file":"categories.tex","source_line":4238,"source_end_line":4246,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4238-L4246","statement_sha256":"fbbcc708629ef98643313862cfdf08c7a99139ccc97e62001232db513a372db9","origin":"The Stacks Project","memory_eligible":false,"source_rank":122,"rank":122,"depth":0,"x":130.026,"y":159.104,"cluster":"categories-foundations"},{"id":"stacks:04VE","tag":"04VE","title":"Localization in categories · Lemma 04VE","summary":"Let C be a category and let S be a left multiplicative system of morphisms of C. Given any finite collection g_i : X_i → Y of morphisms of S^-1C (indexed by i), we can find an element s : Y → Y' of S and a family of morphisms f_i : X_i → Y' of C such that each g_i is the equivalence class of the pair (f_i : X_i → Y', s : Y → Y').","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative\nsystem of morphisms of $\\mathcal{C}$. Given any finite collection\n$g_i : X_i \\to Y$ of morphisms of $S^{-1}\\mathcal{C}$\n(indexed by $i$),\nwe can find an element $s : Y \\to Y'$ of $S$ and\na family of morphisms $f_i : X_i \\to Y'$ of $\\mathcal{C}$ such that\neach $g_i$ is the equivalence class of the pair\n$(f_i : X_i \\to Y', s : Y \\to Y')$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VE","source_file":"categories.tex","source_line":4267,"source_end_line":4277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4267-L4277","statement_sha256":"a333c8a618b5f874c5b0c8b3140ad3d2bb1c4316e06ea2dd8021b9d9cdc20c96","origin":"The Stacks Project","memory_eligible":false,"source_rank":123,"rank":123,"depth":1,"x":353.187,"y":208.128,"cluster":"categories-foundations"},{"id":"stacks:04VF","tag":"04VF","title":"Localization in categories · Lemma 04VF","summary":"Let C be a category and let S be a left multiplicative system of morphisms of C. Let A, B : X → Y be morphisms of S^-1C which are the equivalence classes of (f : X → Y', s : Y → Y') and (g : X → Y', s : Y → Y'). The following are equivalent • A = B • there exists a morphism t : Y' → Y\" in S with t ∘ f = t ∘ g, and • there exists a morphism a : Y' → Y\" such that a ∘ f = a ∘ g and a ∘ s ∈ S.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative\nsystem of morphisms of $\\mathcal{C}$. Let $A, B : X \\to Y$ be morphisms\nof $S^{-1}\\mathcal{C}$ which are the equivalence classes of\n$(f : X \\to Y', s : Y \\to Y')$ and $(g : X \\to Y', s : Y \\to Y')$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A = B$\n\\item there exists a morphism $t : Y' \\to Y''$\nin $S$ with $t \\circ f = t \\circ g$, and\n\\item there exists a morphism $a : Y' \\to Y''$\nsuch that $a \\circ f = a \\circ g$ and $a \\circ s \\in S$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VF","source_file":"categories.tex","source_line":4303,"source_end_line":4317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4303-L4317","statement_sha256":"0eddb7a9911af485d35a6ed2a93fd49b40b84f27a0dcb6272322f76a79e1d002","origin":"The Stacks Project","memory_eligible":false,"source_rank":124,"rank":124,"depth":1,"x":148.384,"y":298.97,"cluster":"categories-foundations"},{"id":"stacks:04VG","tag":"04VG","title":"Localization in categories · Lemma 04VG","summary":"Let C be a category and let S be a left multiplicative system of morphisms of C. • The rules X ↦ X and (f : X → Y) ↦ (f : X → Y, id_Y : Y → Y) define a functor Q : C → S^-1C. • For any s ∈ S the morphism Q(s) is an isomorphism in S^-1C. • If G : C → D is any functor such that G(s) is invertible for every s ∈ S, then there exists a unique functor H : S^-1C → D such that H ∘ Q = G.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative\nsystem of morphisms of $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The rules $X \\mapsto X$ and\n$(f : X \\to Y) \\mapsto (f : X \\to Y, \\text{id}_Y : Y \\to Y)$\ndefine a functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$.\n\\item For any $s \\in S$ the morphism $Q(s)$ is an isomorphism in\n$S^{-1}\\mathcal{C}$.\n\\item If $G : \\mathcal{C} \\to \\mathcal{D}$ is any functor such that\n$G(s)$ is invertible for every $s \\in S$, then there exists a\nunique functor $H : S^{-1}\\mathcal{C} \\to \\mathcal{D}$\nsuch that $H \\circ Q = G$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VG","source_file":"categories.tex","source_line":4405,"source_end_line":4420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4405-L4420","statement_sha256":"3447b0a83c5ce4247f518a059506f0d7c54f8d65e36f09208fb798d5611578a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":125,"rank":125,"depth":0,"x":226.664,"y":115.042,"cluster":"categories-foundations"},{"id":"stacks:05Q2","tag":"05Q2","title":"Localization in categories · Lemma 05Q2","summary":"Let C be a category and let S be a left multiplicative system of morphisms of C. The localization functor Q : C → S^-1C commutes with finite colimits.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a left multiplicative\nsystem of morphisms of $\\mathcal{C}$. The localization functor\n$Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ commutes with finite colimits.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Q2","source_file":"categories.tex","source_line":4430,"source_end_line":4435,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4430-L4435","statement_sha256":"419bd2d1f9a1fd911509b16c6181ffa236bb893aa9d42f42e3fe8fc914ac3d7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":126,"rank":126,"depth":1,"x":317.208,"y":295.8,"cluster":"categories-foundations"},{"id":"stacks:05Q3","tag":"05Q3","title":"Localization in categories · Lemma 05Q3","summary":"Let C be a category. Let S be a left multiplicative system. If f : X → Y, f' : X' → Y' are two morphisms of C and if xymatrix Q(X) ar[d]_Q(f) ar[r]_a & Q(X') ar[d]^Q(f') Q(Y) ar[r]^b & Q(Y') is a commutative diagram in S^-1C, then there exist a morphism f\" : X\" → Y\" in C and a commutative diagram xymatrix X ar[d]_f ar[r]_g & X\" ar[d]^f\" & X' ar[d]^f' ar[l]^s Y ar[r]^h & Y\" & Y' ar[l]_t in C with s, t ∈ S and a = s^-1g, b = t^-1h.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $S$ be a left multiplicative\nsystem. If $f : X \\to Y$, $f' : X' \\to Y'$ are two morphisms of\n$\\mathcal{C}$ and if\n$$\n\\xymatrix{\nQ(X) \\ar[d]_{Q(f)} \\ar[r]_a & Q(X') \\ar[d]^{Q(f')} \\\\\nQ(Y) \\ar[r]^b & Q(Y')\n}\n$$\nis a commutative diagram in $S^{-1}\\mathcal{C}$, then there exist\na morphism $f'' : X'' \\to Y''$ in $\\mathcal{C}$ and a commutative\ndiagram\n$$\n\\xymatrix{\nX \\ar[d]_f \\ar[r]_g & X'' \\ar[d]^{f''} & X' \\ar[d]^{f'} \\ar[l]^s \\\\\nY \\ar[r]^h & Y'' & Y' \\ar[l]_t\n}\n$$\nin $\\mathcal{C}$ with $s, t \\in S$ and $a = s^{-1}g$, $b = t^{-1}h$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Q3","source_file":"categories.tex","source_line":4463,"source_end_line":4484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4463-L4484","statement_sha256":"9b4f304791b58cefd15b88e69037ab5385d5e0d89ce2c09851588fc05bb40774","origin":"The Stacks Project","memory_eligible":false,"source_rank":127,"rank":127,"depth":2,"x":104.246,"y":213.565,"cluster":"categories-foundations"},{"id":"stacks:04VH","tag":"04VH","title":"Localization in categories · Lemma 04VH","summary":"Let C be a category and let S be a right multiplicative system. • The relation on pairs defined above is an equivalence relation. • The composition rule given above is well defined on equivalence classes. • Composition is associative (and the identity morphisms satisfy the identity axioms), and hence S^-1C is a category.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative\nsystem.\n\\begin{enumerate}\n\\item The relation on pairs defined above is an equivalence relation.\n\\item The composition rule given above is well defined on equivalence\nclasses.\n\\item Composition is associative (and the identity morphisms satisfy\nthe identity axioms), and hence $S^{-1}\\mathcal{C}$ is a category.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VH","source_file":"categories.tex","source_line":4581,"source_end_line":4592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4581-L4592","statement_sha256":"b7e990fcf5001e757367cf9adbb667432d03936b25e96ae1c1dd77bbb179b6dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":128,"rank":128,"depth":1,"x":328.285,"y":153.13,"cluster":"categories-foundations"},{"id":"stacks:0BM3","tag":"0BM3","title":"Localization in categories · Definition 0BM3","summary":"Let C be a category and let S be a right multiplicative system of morphisms of C. Given any morphism f : X' → Y in C and any morphism s : X' → X in S, we denote by f s^-1 the equivalence class of the pair (f : X' → Y, s : X' → X). This is a morphism from X to Y in S^-1 C.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative\nsystem of morphisms of $\\mathcal{C}$. Given any morphism\n$f : X' \\to Y$ in $\\mathcal{C}$ and any morphism $s : X' \\to X$ in\n$S$, we denote by {\\it $f s^{-1}$} the equivalence class of the pair\n$(f : X' \\to Y, s : X' \\to X)$. This is a morphism from $X$ to $Y$\nin $S^{-1} \\mathcal{C}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BM3","source_file":"categories.tex","source_line":4602,"source_end_line":4610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4602-L4610","statement_sha256":"b1bc16593821e2d285258a9539799299ad88948a5c764ba1101ee193eef40dd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":129,"rank":129,"depth":0,"x":211.229,"y":325.484,"cluster":"categories-foundations"},{"id":"stacks:04VI","tag":"04VI","title":"Localization in categories · Lemma 04VI","summary":"Let C be a category and let S be a right multiplicative system of morphisms of C. Given any finite collection g_i : X → Y_i of morphisms of S^-1C (indexed by i), we can find an element s : X' → X of S and a family of morphisms f_i : X' → Y_i of C such that g_i is the equivalence class of the pair (f_i : X' → Y_i, s : X' → X).","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative\nsystem of morphisms of $\\mathcal{C}$. Given any finite collection\n$g_i : X \\to Y_i$ of morphisms of $S^{-1}\\mathcal{C}$\n(indexed by $i$),\nwe can find an element $s : X' \\to X$ of $S$ and a family\nof morphisms $f_i : X' \\to Y_i$ of $\\mathcal{C}$ such that\n$g_i$ is the equivalence class of the pair\n$(f_i : X' \\to Y_i, s : X' \\to X)$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VI","source_file":"categories.tex","source_line":4618,"source_end_line":4628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4618-L4628","statement_sha256":"8e0a2807d3bcbb782c59310690df1819739944e9ce062e2999499b820c59d4e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":130,"rank":130,"depth":2,"x":158.742,"y":131.228,"cluster":"categories-foundations"},{"id":"stacks:04VJ","tag":"04VJ","title":"Localization in categories · Lemma 04VJ","summary":"Let C be a category and let S be a right multiplicative system of morphisms of C. Let A, B : X → Y be morphisms of S^-1C which are the equivalence classes of (f : X' → Y, s : X' → X) and (g : X' → Y, s : X' → X). The following are equivalent • A = B, • there exists a morphism t : X\" → X' in S with f ∘ t = g ∘ t, and • there exists a morphism a : X\" → X' with f ∘ a = g ∘ a and s ∘ a ∈ S.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative\nsystem of morphisms of $\\mathcal{C}$. Let $A, B : X \\to Y$ be\nmorphisms of $S^{-1}\\mathcal{C}$ which are the equivalence\nclasses of $(f : X' \\to Y, s : X' \\to X)$ and\n$(g : X' \\to Y, s : X' \\to X)$. The following are equivalent\n\\begin{enumerate}\n\\item $A = B$,\n\\item there exists a morphism $t : X'' \\to X'$ in $S$ with\n$f \\circ t = g \\circ t$, and\n\\item there exists a morphism $a : X'' \\to X'$ with\n$f \\circ a = g \\circ a$ and $s \\circ a \\in S$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VJ","source_file":"categories.tex","source_line":4641,"source_end_line":4655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4641-L4655","statement_sha256":"519beda7bbd3318d8cc0bc7ddd884dbeb8aa778e2e316eec22be448bce634af2","origin":"The Stacks Project","memory_eligible":false,"source_rank":131,"rank":131,"depth":2,"x":354.404,"y":245.121,"cluster":"categories-foundations"},{"id":"stacks:04VK","tag":"04VK","title":"Localization in categories · Lemma 04VK","summary":"Let C be a category and let S be a right multiplicative system of morphisms of C. • The rules X ↦ X and (f : X → Y) ↦ (f : X → Y, id_X : X → X) define a functor Q : C → S^-1C. • For any s ∈ S the morphism Q(s) is an isomorphism in S^-1C. • If G : C → D is any functor such that G(s) is invertible for every s ∈ S, then there exists a unique functor H : S^-1C → D such that H ∘ Q = G.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative\nsystem of morphisms of $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The rules $X \\mapsto X$ and\n$(f : X \\to Y) \\mapsto (f : X \\to Y, \\text{id}_X : X \\to X)$\ndefine a functor $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$.\n\\item For any $s \\in S$ the morphism $Q(s)$ is an isomorphism in\n$S^{-1}\\mathcal{C}$.\n\\item If $G : \\mathcal{C} \\to \\mathcal{D}$ is any functor such that\n$G(s)$ is invertible for every $s \\in S$, then there exists a\nunique functor $H : S^{-1}\\mathcal{C} \\to \\mathcal{D}$\nsuch that $H \\circ Q = G$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VK","source_file":"categories.tex","source_line":4692,"source_end_line":4707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4692-L4707","statement_sha256":"c9b369d9fc727d6faed91ffe424ff6550e4a3b4871e4fbcaa75bf2eaa08fed45","origin":"The Stacks Project","memory_eligible":false,"source_rank":132,"rank":132,"depth":1,"x":117.642,"y":272.262,"cluster":"categories-foundations"},{"id":"stacks:05Q6","tag":"05Q6","title":"Localization in categories · Lemma 05Q6","summary":"Let C be a category and let S be a right multiplicative system of morphisms of C. The localization functor Q : C → S^-1C commutes with finite limits.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a right multiplicative\nsystem of morphisms of $\\mathcal{C}$. The localization functor\n$Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ commutes with finite limits.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Q6","source_file":"categories.tex","source_line":4716,"source_end_line":4721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4716-L4721","statement_sha256":"0d0840417c66c6a085f7fcf339ce97b3248883eb05dafef6239181a2ff4d1407","origin":"The Stacks Project","memory_eligible":false,"source_rank":133,"rank":133,"depth":2,"x":270.976,"y":117.325,"cluster":"categories-foundations"},{"id":"stacks:05Q7","tag":"05Q7","title":"Localization in categories · Lemma 05Q7","summary":"Let C be a category. Let S be a right multiplicative system. If f : X → Y, f' : X' → Y' are two morphisms of C and if xymatrix Q(X) ar[d]_Q(f) ar[r]_a & Q(X') ar[d]^Q(f') Q(Y) ar[r]^b & Q(Y') is a commutative diagram in S^-1C, then there exist a morphism f\" : X\" → Y\" in C and a commutative diagram xymatrix X ar[d]_f & X\" ar[l]^s ar[d]^f\" ar[r]_g & X' ar[d]^f' Y & Y\" ar[l]_t ar[r]^h & Y' in C with s, t ∈ S and a = gs^-1, b = ht^-1.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $S$ be a right multiplicative\nsystem. If $f : X \\to Y$, $f' : X' \\to Y'$ are two morphisms of\n$\\mathcal{C}$ and if\n$$\n\\xymatrix{\nQ(X) \\ar[d]_{Q(f)} \\ar[r]_a & Q(X') \\ar[d]^{Q(f')} \\\\\nQ(Y) \\ar[r]^b & Q(Y')\n}\n$$\nis a commutative diagram in $S^{-1}\\mathcal{C}$, then there exist\na morphism $f'' : X'' \\to Y''$ in $\\mathcal{C}$ and a commutative\ndiagram\n$$\n\\xymatrix{\nX \\ar[d]_f & X'' \\ar[l]^s \\ar[d]^{f''} \\ar[r]_g & X' \\ar[d]^{f'} \\\\\nY & Y'' \\ar[l]_t \\ar[r]^h & Y'\n}\n$$\nin $\\mathcal{C}$ with $s, t \\in S$ and $a = gs^{-1}$, $b = ht^{-1}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Q7","source_file":"categories.tex","source_line":4727,"source_end_line":4748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4727-L4748","statement_sha256":"ce4033bd9357c7cfe617bbf6c405fdfe6734bd50f12f5bb4018e125702c3f162","origin":"The Stacks Project","memory_eligible":false,"source_rank":134,"rank":134,"depth":3,"x":282.548,"y":319.329,"cluster":"categories-foundations"},{"id":"stacks:04VL","tag":"04VL","title":"Localization in categories · Lemma 04VL","summary":"Let C be a category and let S be a multiplicative system. The category of left fractions and the category of right fractions S^-1C are canonically isomorphic.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a multiplicative system.\nThe category of left fractions and the category of right fractions\n$S^{-1}\\mathcal{C}$ are canonically isomorphic.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VL","source_file":"categories.tex","source_line":4760,"source_end_line":4765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4760-L4765","statement_sha256":"02d513e3bd3bae7cb6c06bd5cfaa729c74439ecc81154f10645a49a4366e849a","origin":"The Stacks Project","memory_eligible":false,"source_rank":135,"rank":135,"depth":2,"x":110.936,"y":176.413,"cluster":"categories-foundations"},{"id":"stacks:05Q8","tag":"05Q8","title":"Localization in categories · Definition 05Q8","summary":"Let C be a category and let S be a multiplicative system. We say S is saturated if, in addition to MS1, MS2, MS3, we also have • [MS4] Given three composable morphisms f, g, h, if fg, gh ∈ S, then g ∈ S.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a multiplicative system.\nWe say $S$ is {\\it saturated} if, in addition to MS1, MS2, MS3, we\nalso have\n\\begin{enumerate}\n\\item[MS4] Given three composable morphisms $f, g, h$, if\n$fg, gh \\in S$, then $g \\in S$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Q8","source_file":"categories.tex","source_line":4778,"source_end_line":4787,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4778-L4787","statement_sha256":"dd57ff9c29ae9fe6e77417e88997a6d7c71c9b1b2e91491915dd28d555f26b7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":136,"rank":136,"depth":0,"x":353.297,"y":184.451,"cluster":"categories-foundations"},{"id":"stacks:05Q9","tag":"05Q9","title":"Localization in categories · Lemma 05Q9","summary":"Let C be a category and let S be a multiplicative system. Denote Q : C → S^-1C the localization functor. The set hat S = (f ∈ Arrows(C) mid Q(f) is an isomorphism) is equal to S' = (f ∈ Arrows(C) mid there exist g, h such that gf, fh ∈ S) and is the smallest saturated multiplicative system containing S. In particular, if S is saturated, then hat S = S.","statement_latex":"Let $\\mathcal{C}$ be a category and let $S$ be a multiplicative system.\nDenote $Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ the localization functor.\nThe set\n$$\n\\hat S = \\{f \\in \\text{Arrows}(\\mathcal{C}) \\mid\nQ(f) \\text{ is an isomorphism}\\}\n$$\nis equal to\n$$\nS' = \\{f \\in \\text{Arrows}(\\mathcal{C}) \\mid\n\\text{there exist }g, h\\text{ such that }gf, fh \\in S\\}\n$$\nand is the smallest saturated multiplicative system containing $S$.\nIn particular, if $S$ is saturated, then $\\hat S = S$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Localization in categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Q9","source_file":"categories.tex","source_line":4795,"source_end_line":4811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L4795-L4811","statement_sha256":"22994f8cc09a37e522b9758b6737f6d922daaeb128a27271b09a60b8640bc3b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":137,"rank":137,"depth":3,"x":167.443,"y":316.524,"cluster":"categories-foundations"},{"id":"stacks:003E","tag":"003E","title":"Formal properties · Definition 003E","summary":"Given a diagram as in the left hand side of: xymatrix A rtwocell^F_F't & B rtwocell^G_G's & C gives xymatrix A rrtwocell^G ∘ F _G' ∘ F' s star t & & C we define the horizontal composition s star t to be the transformation of functors _G't ∘ s_F = s_F'∘ _Gt.","statement_latex":"Given a diagram as in the left hand side of:\n$$\n\\xymatrix{\n\\mathcal{A}\n\\rtwocell^F_{F'}{t}\n&\n\\mathcal{B}\n\\rtwocell^G_{G'}{s}\n&\n\\mathcal{C}\n}\n\\text{ gives }\n\\xymatrix{\n\\mathcal{A}\n\\rrtwocell^{G \\circ F} _{G' \\circ F'}{\\ \\ s \\star t}\n& &\n\\mathcal{C}\n}\n$$\nwe define the {\\it horizontal} composition $s \\star t$ to be the\ntransformation of functors ${}_{G'}t \\circ s_F = s_{F'}\\circ {}_Gt$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Formal properties","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003E","source_file":"categories.tex","source_line":5011,"source_end_line":5034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5011-L5034","statement_sha256":"64028470226ecff1467c6384ae4cdebe9070b70bca0573008312db421903db39","origin":"The Stacks Project","memory_eligible":false,"source_rank":138,"rank":138,"depth":0,"x":198.392,"y":112.943,"cluster":"categories-foundations"},{"id":"stacks:003F","tag":"003F","title":"Formal properties · Lemma 003F","summary":"The horizontal and vertical compositions have the following properties • ∘ and star are associative, • the identity transformations id_F are units for ∘, • the identity transformations of the identity functors id_id_A are units for star and ∘, and • given a diagram xymatrix A rruppertwocell^Ft ar[rr]_(.3)F' rrlowertwocell_F\"t' & & B rruppertwocell^Gs ar[rr]_(.3)G' rrlowertwocell_G\"s' & & C we have (s' ∘ s) star (t' ∘ t) = (s' star t') ∘ (s star t).","statement_latex":"The horizontal and vertical compositions have the following\nproperties\n\\begin{enumerate}\n\\item $\\circ$ and $\\star$ are associative,\n\\item the identity transformations $\\text{id}_F$\nare units for $\\circ$,\n\\item the identity transformations of the identity functors\n$\\text{id}_{\\text{id}_\\mathcal{A}}$\nare units for $\\star$ and $\\circ$, and\n\\item given a diagram\n$$\n\\xymatrix{\n\\mathcal{A}\n\\rruppertwocell^F{t}\n\\ar[rr]_(.3){F'}\n\\rrlowertwocell_{F''}{t'}\n& &\n\\mathcal{B}\n\\rruppertwocell^G{s}\n\\ar[rr]_(.3){G'}\n\\rrlowertwocell_{G''}{s'}\n& &\n\\mathcal{C}\n}\n$$\nwe have $ (s' \\circ s) \\star (t' \\circ t) = (s' \\star t') \\circ (s \\star t)$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Formal properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003F","source_file":"categories.tex","source_line":5043,"source_end_line":5072,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5043-L5072","statement_sha256":"2a324dc19314ce9ab0894f16cca0f94f3e3d9c14abbf3561076e01a2b020dfd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":139,"rank":139,"depth":0,"x":339.791,"y":281.225,"cluster":"categories-foundations"},{"id":"stacks:003H","tag":"003H","title":"2-categories · Definition 003H","summary":"A (strict) 2-category C consists of the following data • A set of objects Ob(C). • For each pair x, y ∈ Ob(C) a category Mor_C(x, y). The objects of Mor_C(x, y) will be called 1-morphisms and denoted F : x → y. The morphisms between these 1-morphisms will be called 2-morphisms and denoted t : F' → F. The composition of 2-morphisms in Mor_C(x, y) will be called vertical composition and will be denoted t ∘ t' for t : F' → F and t' : F\" → F'. • For each triple x, y, z∈ Ob(C)…","statement_latex":"A (strict) {\\it $2$-category} $\\mathcal{C}$ consists of the following data\n\\begin{enumerate}\n\\item A set of objects $\\Ob(\\mathcal{C})$.\n\\item For each pair $x, y \\in \\Ob(\\mathcal{C})$\na category $\\Mor_\\mathcal{C}(x, y)$. The objects of\n$\\Mor_\\mathcal{C}(x, y)$ will be called {\\it $1$-morphisms}\nand denoted $F : x \\to y$. The morphisms between these $1$-morphisms\nwill be called {\\it $2$-morphisms} and denoted $t : F' \\to F$.\nThe composition of $2$-morphisms in $\\Mor_\\mathcal{C}(x, y)$\nwill be called {\\it vertical} composition and will be\ndenoted $t \\circ t'$ for $t : F' \\to F$ and $t' : F'' \\to F'$.\n\\item For each triple $x, y, z\\in \\Ob(\\mathcal{C})$ a\nfunctor\n$$\n(\\circ, \\star) :\n\\Mor_\\mathcal{C}(y, z) \\times \\Mor_\\mathcal{C}(x, y)\n\\longrightarrow\n\\Mor_\\mathcal{C}(x, z).\n$$\nThe image of the pair of $1$-morphisms $(F, G)$ on the left hand side\nwill be called the {\\it composition} of $F$ and $G$, and denoted\n$F\\circ G$. The image of the pair of $2$-morphisms $(t, s)$ will\nbe called the {\\it horizontal} composition and denoted $t \\star s$.\n\\end{enumerate}\nThese data are to satisfy the following rules:\n\\begin{enumerate}\n\\item The set of objects together with the set of $1$-morphisms endowed\nwith composition of $1$-morphisms forms a category.\n\\item Horizontal composition of $2$-morphisms is associative.\n\\item The identity $2$-morphism $\\text{id}_{\\text{id}_x}$\nof the identity $1$-morphism $\\text{id}_x$ is a unit for\nhorizontal composition.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003H","source_file":"categories.tex","source_line":5131,"source_end_line":5166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5131-L5166","statement_sha256":"3f4df532ca950e22575acc2b9d75487f043b91e063ebdad5849e2fe785d77414","origin":"The Stacks Project","memory_eligible":false,"source_rank":140,"rank":140,"depth":0,"x":99.343,"y":237.212,"cluster":"categories-foundations"},{"id":"stacks:02X7","tag":"02X7","title":"2-categories · Definition 02X7","summary":"Let C be a 2-category. A sub 2-category C' of C, is given by a subset Ob(C') of Ob(C) and sub categories Mor_C'(x, y) of the categories Mor_C(x, y) for all x, y ∈ Ob(C') such that these, together with the operations ∘ (composition 1-morphisms), ∘ (vertical composition 2-morphisms), and star (horizontal composition) form a 2-category.","statement_latex":"Let $\\mathcal{C}$ be a $2$-category.\nA {\\it sub $2$-category} $\\mathcal{C}'$ of $\\mathcal{C}$, is given by a subset\n$\\Ob(\\mathcal{C}')$ of $\\Ob(\\mathcal{C})$\nand sub categories $\\Mor_{\\mathcal{C}'}(x, y)$ of the\ncategories $\\Mor_\\mathcal{C}(x, y)$ for all\n$x, y \\in \\Ob(\\mathcal{C}')$ such that these, together with\nthe operations $\\circ$ (composition $1$-morphisms), $\\circ$ (vertical\ncomposition $2$-morphisms), and $\\star$ (horizontal composition)\nform a $2$-category.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02X7","source_file":"categories.tex","source_line":5180,"source_end_line":5191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5180-L5191","statement_sha256":"6e06a59810814699c81147096c0b89daad6c85b398bde227509bbff7ea2c7870","origin":"The Stacks Project","memory_eligible":false,"source_rank":141,"rank":141,"depth":0,"x":312.794,"y":132.863,"cluster":"categories-foundations"},{"id":"stacks:003L","tag":"003L","title":"2-categories · Definition 003L","summary":"Two objects x, y of a 2-category are equivalent if there exist 1-morphisms F : x → y and G : y → x such that F ∘ G is 2-isomorphic to id_y and G ∘ F is 2-isomorphic to id_x.","statement_latex":"Two objects $x, y$ of a $2$-category are {\\it equivalent} if there exist\n$1$-morphisms $F : x \\to y$ and $G : y \\to x$ such that $F \\circ G$ is\n$2$-isomorphic to $\\text{id}_y$ and $G \\circ F$ is $2$-isomorphic to\n$\\text{id}_x$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003L","source_file":"categories.tex","source_line":5230,"source_end_line":5236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5230-L5236","statement_sha256":"f7b32aaedb746e0cec226db9b251234be18e8a68dfa12ab58ef4ced01d7a0226","origin":"The Stacks Project","memory_eligible":false,"source_rank":142,"rank":142,"depth":0,"x":239.053,"y":331.622,"cluster":"categories-foundations"},{"id":"stacks:003N","tag":"003N","title":"2-categories · Definition 003N","summary":"Let A be a category and let C be a 2-category. • A functor from an ordinary category into a 2-category will ignore the 2-morphisms unless mentioned otherwise. In other words, it will be a \"usual\" functor into the category formed out of 2-category by forgetting all the 2-morphisms. • A weak functor, or a pseudo functor φ from A into the 2-category C is given by the following data • a map φ : Ob(A) → Ob(C), • for every pair x, y∈ Ob(A), and every morphism f : x → y a…","statement_latex":"Let $\\mathcal{A}$ be a category and let $\\mathcal{C}$ be a $2$-category.\n\\begin{enumerate}\n\\item A {\\it functor} from an ordinary category into a $2$-category\nwill ignore the\n$2$-morphisms unless mentioned otherwise. In other words, it will be a\n``usual'' functor into the category formed out of 2-category by forgetting\nall the 2-morphisms.\n\\item A {\\it weak functor}, or\na {\\it pseudo functor} $\\varphi$ from $\\mathcal{A}$ into the 2-category\n$\\mathcal{C}$ is given by the following data\n\\begin{enumerate}\n\\item a map $\\varphi : \\Ob(\\mathcal{A}) \\to \\Ob(\\mathcal{C})$,\n\\item for every pair $x, y\\in \\Ob(\\mathcal{A})$, and every\nmorphism $f : x \\to y$ a $1$-morphism $\\varphi(f) : \\varphi(x) \\to \\varphi(y)$,\n\\item for every $x\\in \\Ob(A)$ a $2$-morphism\n$\\alpha_x : \\text{id}_{\\varphi(x)} \\to \\varphi(\\text{id}_x)$, and\n\\item for every pair of composable morphisms $f : x \\to y$,\n$g : y \\to z$ of $\\mathcal{A}$ a $2$-morphism\n$\\alpha_{g, f} : \\varphi(g \\circ f) \\to \\varphi(g) \\circ \\varphi(f)$.\n\\end{enumerate}\nThese data are subject to the following conditions:\n\\begin{enumerate}\n\\item the $2$-morphisms $\\alpha_x$ and $\\alpha_{g, f}$ are all\nisomorphisms,\n\\item for any morphism $f : x \\to y$ in $\\mathcal{A}$ we have\n$\\alpha_{\\text{id}_y, f} = \\alpha_y \\star \\text{id}_{\\varphi(f)}$:\n$$\n\\xymatrix{\n\\varphi(x)\n\\rrtwocell^{\\varphi(f)}_{\\varphi(f)}{\\ \\ \\ \\ \\text{id}_{\\varphi(f)}}\n& &\n\\varphi(y)\n\\rrtwocell^{\\text{id}_{\\varphi(y)}}_{\\varphi(\\text{id}_y)}{\\alpha_y}\n& &\n\\varphi(y)\n}\n=\n\\xymatrix{\n\\varphi(x)\n\\rrtwocell^{\\varphi(f)}_{\\varphi(\\text{id}_y) \\circ \\varphi(f)}{\\ \\ \\ \\ \\alpha_{\\text{id}_y, f}}\n& &\n\\varphi(y)\n}\n$$\n\\item for any morphism $f : x \\to y$ in $\\mathcal{A}$ we have\n$\\alpha_{f, \\text{id}_x} = \\text{id}_{\\varphi(f)} \\star \\alpha_x$,\n\\item for any triple of composable morphisms\n$f : w \\to x$, $g : x \\to y$, and $h : y \\to z$ of $\\mathcal{A}$\nwe have\n$$\n(\\text{id}_{\\varphi(h)} \\star \\alpha_{g, f})\n\\circ\n\\alpha_{h, g \\circ f}\n=\n(\\alpha_{h, g} \\star \\text{id}_{\\varphi(f)})\n\\circ\n\\alpha_{h \\circ g, f}\n$$\nin other words the following diagram with objects\n$1$-morphisms and arrows $2$-morphisms commutes\n$$\n\\xymatrix{\n\\varphi(h \\circ g \\circ f)\n\\ar[d]_{\\alpha_{h, g \\circ f}}\n\\ar[rr]_{\\alpha_{h \\circ g, f}}\n& &\n\\varphi(h \\circ g) \\circ \\varphi(f)\n\\ar[d]^{\\alpha_{h, g} \\star \\text{id}_{\\varphi(f)}} \\\\\n\\varphi(h) \\circ \\varphi(g \\circ f)\n\\ar[rr]^{\\text{id}_{\\varphi(h)} \\star \\alpha_{g, f}}\n& &\n\\varphi(h) \\circ \\varphi(g) \\circ \\varphi(f)\n}\n$$\n\\end{enumerate}\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003N","source_file":"categories.tex","source_line":5242,"source_end_line":5320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5242-L5320","statement_sha256":"73e70772fa7598ec1b1bc56639427fc961fd29d3be3f206b42809fdc2a71caf8","origin":"The Stacks Project","memory_eligible":false,"source_rank":143,"rank":143,"depth":0,"x":133.225,"y":142.56,"cluster":"categories-foundations"},{"id":"stacks:003I","tag":"003I","title":"(2, 1)-categories · Definition 003I","summary":"A (strict) (2, 1)-category is a 2-category in which all 2-morphisms are isomorphisms.","statement_latex":"A (strict) {\\it $(2, 1)$-category} is a $2$-category in which all\n$2$-morphisms are isomorphisms.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"(2, 1)-categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003I","source_file":"categories.tex","source_line":5339,"source_end_line":5343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5339-L5343","statement_sha256":"ebb28594a9f7713c51b19129335b98a054cfeecf369e7d3a0589f4af1c735df3","origin":"The Stacks Project","memory_eligible":false,"source_rank":144,"rank":144,"depth":0,"x":364.098,"y":222.198,"cluster":"categories-foundations"},{"id":"stacks:003P","tag":"003P","title":"2-fibre products · Definition 003P","summary":"A final object of a (2, 1)-category C is an object x such that • for every y ∈ Ob(C) there is a morphism y → x, and • every two morphisms y → x are isomorphic by a unique 2-morphism.","statement_latex":"A {\\it final object} of a $(2, 1)$-category\n$\\mathcal{C}$ is an object $x$ such that\n\\begin{enumerate}\n\\item for every $y \\in \\Ob(\\mathcal{C})$ there is a morphism $y \\to x$,\nand\n\\item every two morphisms $y \\to x$ are isomorphic by a unique 2-morphism.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003P","source_file":"categories.tex","source_line":5506,"source_end_line":5515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5506-L5515","statement_sha256":"fae90eeb3fab7f9e2231dc09cfddcc11fac2b938aa189984ba978e812efe42d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":145,"rank":145,"depth":0,"x":129.005,"y":294.725,"cluster":"categories-foundations"},{"id":"stacks:003Q","tag":"003Q","title":"2-fibre products · Definition 003Q","summary":"Let C be a (2, 1)-category. Let x, y, z∈ Ob(C) and f∈ Mor_C(x, z) and g∈ Mor_ C(y, z). A 2-fibre product of f and g is a final object in the category of 2-commutative diagrams described above. If a 2-fibre product exists we will denote it x ×_z y∈ Ob(C), and denote the required morphisms p∈ Mor_ C(x ×_z y, x) and q∈ Mor_ C(x ×_z y, y) making the diagram xymatrix & x ×_z y ar[r]^p ar[d]_q & x ar[d]^f & y ar[r]^g & z 2-commute and we will denote the given invertible…","statement_latex":"Let $\\mathcal{C}$ be a $(2, 1)$-category.\nLet $x, y, z\\in \\Ob(\\mathcal{C})$ and\n$f\\in \\Mor_\\mathcal{C}(x, z)$\nand $g\\in \\Mor_{\\mathcal C}(y, z)$. A\n{\\it 2-fibre product of $f$ and $g$} is\na final object in the category of 2-commutative diagrams\ndescribed above. If a 2-fibre product exists we\nwill denote it $x \\times_z y\\in \\Ob(\\mathcal{C})$, and denote the\nrequired morphisms $p\\in \\Mor_{\\mathcal C}(x \\times_z y, x)$ and\n$q\\in \\Mor_{\\mathcal C}(x \\times_z y, y)$ making the diagram\n$$\n\\xymatrix{\n& x \\times_z y \\ar[r]^{p} \\ar[d]_q & x \\ar[d]^{f} \\\\\n& y \\ar[r]^{g} & z }\n$$\n2-commute and we will denote the given invertible\n2-morphism exhibiting this by $\\psi : f \\circ p \\to g \\circ q$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003Q","source_file":"categories.tex","source_line":5522,"source_end_line":5541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5522-L5541","statement_sha256":"eb66e88469327fe7377e16e72b59867fd63785b36d5317e98fa79069c17ba5b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":146,"rank":146,"depth":0,"x":244.429,"y":107.209,"cluster":"categories-foundations"},{"id":"stacks:02X9","tag":"02X9","title":"2-fibre products · Lemma 02X9","summary":"In the (2, 1)-category of categories 2-fibre products exist and are given by the construction of Example [Tag 003R].","statement_latex":"In the $(2, 1)$-category of categories $2$-fibre products exist and\nare given by the construction of\nExample \\ref{example-2-fibre-product-categories}.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02X9","source_file":"categories.tex","source_line":5612,"source_end_line":5617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5612-L5617","statement_sha256":"961b56e6e5890f381499bea682db177644a354371f8547fdff2ee7505cef892e","origin":"The Stacks Project","memory_eligible":false,"source_rank":147,"rank":147,"depth":0,"x":310.335,"y":311.667,"cluster":"categories-foundations"},{"id":"stacks:02XA","tag":"02XA","title":"2-fibre products · Lemma 02XA","summary":"Let xymatrix & Y ar[d]_I ar[rd]^K & X ar[r]^H ar[rd]^L & Z ar[rd]^M & B ar[d]^G & A ar[r]^F & C be a 2-commutative diagram of categories. A choice of isomorphisms α : G ∘ K → M ∘ I and β : M ∘ H → F ∘ L determines a morphism X ×_Z Y → A ×_C B of 2-fibre products associated to this situation.","statement_latex":"Let\n$$\n\\xymatrix{\n& \\mathcal{Y} \\ar[d]_I \\ar[rd]^K & \\\\\n\\mathcal{X} \\ar[r]^H \\ar[rd]^L &\n\\mathcal{Z} \\ar[rd]^M & \\mathcal{B} \\ar[d]^G \\\\\n& \\mathcal{A} \\ar[r]^F & \\mathcal{C}\n}\n$$\nbe a $2$-commutative diagram of categories.\nA choice of isomorphisms\n$\\alpha : G \\circ K \\to M \\circ I$ and\n$\\beta : M \\circ H \\to F \\circ L$\ndetermines a morphism\n$$\n\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y}\n\\longrightarrow\n\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}\n$$\nof $2$-fibre products associated to this situation.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XA","source_file":"categories.tex","source_line":5678,"source_end_line":5700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5678-L5700","statement_sha256":"a677a8fd2cbb0876e08e058d43700644ef3ec0a4933181f0dc7981026682e7cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":148,"rank":148,"depth":0,"x":96.599,"y":197.916,"cluster":"categories-foundations"},{"id":"stacks:02XB","tag":"02XB","title":"2-fibre products · Lemma 02XB","summary":"Assumptions as in Lemma [Tag 02XA]. • If K and L are faithful then the morphism X ×_Z Y → A ×_C B is faithful. • If K and L are fully faithful and M is faithful then the morphism X ×_Z Y → A ×_C B is fully faithful. • If K and L are equivalences and M is fully faithful then the morphism X ×_Z Y → A ×_C B is an equivalence.","statement_latex":"Assumptions as in Lemma \\ref{lemma-functoriality-2-fibre-product}.\n\\begin{enumerate}\n\\item If $K$ and $L$ are faithful\nthen the morphism\n$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to\n\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$\nis faithful.\n\\item If $K$ and $L$ are fully faithful and $M$ is faithful\nthen the morphism\n$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to\n\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$\nis fully faithful.\n\\item If $K$ and $L$ are equivalences and $M$ is fully faithful\nthen the morphism\n$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to\n\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B}$\nis an equivalence.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XB","source_file":"categories.tex","source_line":5715,"source_end_line":5735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5715-L5735","statement_sha256":"942866cfcf2def3b22afc67f7943ff2d1e5cbd841aacbf4da2c5cff3711652a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":149,"rank":149,"depth":1,"x":346.515,"y":160.391,"cluster":"categories-foundations"},{"id":"stacks:02XC","tag":"02XC","title":"2-fibre products · Lemma 02XC","summary":"Let xymatrix A ar[rd] & & C ar[ld] ar[rd] & & E ar[ld] & B & & D be a diagram of categories and functors. Then there is a canonical isomorphism (A ×_B C) ×_D E ≅ A ×_B (C ×_D E) of categories.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{A} \\ar[rd] & & \\mathcal{C} \\ar[ld] \\ar[rd] & & \\mathcal{E} \\ar[ld] \\\\\n& \\mathcal{B} & & \\mathcal{D}\n}\n$$\nbe a diagram of categories and functors.\nThen there is a canonical isomorphism\n$$\n(\\mathcal{A} \\times_\\mathcal{B} \\mathcal{C}) \\times_\\mathcal{D} \\mathcal{E}\n\\cong\n\\mathcal{A} \\times_\\mathcal{B} (\\mathcal{C} \\times_\\mathcal{D} \\mathcal{E})\n$$\nof categories.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XC","source_file":"categories.tex","source_line":5764,"source_end_line":5781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5764-L5781","statement_sha256":"1ede11d51702305ceb98282fd45780d1d2577f2250263e5c2ff526ad067d788e","origin":"The Stacks Project","memory_eligible":false,"source_rank":150,"rank":150,"depth":0,"x":191.893,"y":330.433,"cluster":"categories-foundations"},{"id":"stacks:04S7","tag":"04S7","title":"2-fibre products · Lemma 04S7","summary":"Let xymatrix A ar[rd] & & C ar[ld] ar[rd] & & E ar[ld] & B ar[rd]_F & & D ar[ld]^G & & F & be a commutative diagram of categories and functors. Then there is a canonical functor pr_02 : A ×_B C ×_D E → A ×_F E of categories.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{A} \\ar[rd] & & \\mathcal{C} \\ar[ld] \\ar[rd] & & \\mathcal{E} \\ar[ld] \\\\\n& \\mathcal{B} \\ar[rd]_F & & \\mathcal{D} \\ar[ld]^G \\\\\n& & \\mathcal{F} &\n}\n$$\nbe a commutative diagram of categories and functors.\nThen there is a canonical functor\n$$\n\\text{pr}_{02} :\n\\mathcal{A} \\times_\\mathcal{B} \\mathcal{C} \\times_\\mathcal{D} \\mathcal{E}\n\\longrightarrow\n\\mathcal{A} \\times_\\mathcal{F} \\mathcal{E}\n$$\nof categories.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04S7","source_file":"categories.tex","source_line":5797,"source_end_line":5816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5797-L5816","statement_sha256":"c64d3077a4aefd1241ca51a9d7a85901f2830501056ede6d6a6e7f3d29616b0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":151,"rank":151,"depth":0,"x":169.092,"y":116.606,"cluster":"categories-foundations"},{"id":"stacks:02XD","tag":"02XD","title":"2-fibre products · Lemma 02XD","summary":"Let A → B ← C ← D be a diagram of categories and functors. Then there is a canonical isomorphism A ×_B C ×_C D ≅ A ×_B D of categories.","statement_latex":"Let\n$$\n\\mathcal{A} \\to\n\\mathcal{B} \\leftarrow \\mathcal{C} \\leftarrow \\mathcal{D}\n$$\nbe a diagram of categories and functors.\nThen there is a canonical isomorphism\n$$\n\\mathcal{A} \\times_\\mathcal{B} \\mathcal{C} \\times_\\mathcal{C} \\mathcal{D}\n\\cong\n\\mathcal{A} \\times_\\mathcal{B} \\mathcal{D}\n$$\nof categories.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XD","source_file":"categories.tex","source_line":5834,"source_end_line":5849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5834-L5849","statement_sha256":"693314cf827324934e3fc5c737982d0b44591b9c61052b4f8cd67f31a03bd155","origin":"The Stacks Project","memory_eligible":false,"source_rank":152,"rank":152,"depth":0,"x":358.479,"y":261.817,"cluster":"categories-foundations"},{"id":"stacks:02XE","tag":"02XE","title":"2-fibre products · Lemma 02XE","summary":"Let xymatrix C_3 ar[r] ar[d] & S ar[d]^Δ C_1 × C_2 ar[r]^G_1 × G_2 & S × S be a 2-fibre product of categories. Then there is a canonical isomorphism C_3 ≅ C_1 ×_G_1, S, G_2 C_2.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{C}_3 \\ar[r] \\ar[d] & \\mathcal{S} \\ar[d]^\\Delta \\\\\n\\mathcal{C}_1 \\times \\mathcal{C}_2 \\ar[r]^{G_1 \\times G_2} &\n\\mathcal{S} \\times \\mathcal{S}\n}\n$$\nbe a $2$-fibre product of categories.\nThen there is a canonical isomorphism\n$\\mathcal{C}_3 \\cong\n\\mathcal{C}_1 \\times_{G_1, \\mathcal{S}, G_2} \\mathcal{C}_2$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XE","source_file":"categories.tex","source_line":5860,"source_end_line":5874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5860-L5874","statement_sha256":"b871d6197a3d7a18987d8c50f0e8c935abb95eddceefbbdc57832f966dc44892","origin":"The Stacks Project","memory_eligible":false,"source_rank":153,"rank":153,"depth":0,"x":101.216,"y":262.203,"cluster":"categories-foundations"},{"id":"stacks:02XF","tag":"02XF","title":"2-fibre products · Lemma 02XF","summary":"Let xymatrix C' ar[r] ar[d] & S ar[d]^Δ C ar[r]^(G_1, G_2) & S × S be a 2-fibre product of categories. Then there is a canonical isomorphism C' ≅ (C ×_G_1, S, G_2 C) ×_(p, q), C × C, Δ C.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{C}' \\ar[r] \\ar[d] & \\mathcal{S} \\ar[d]^\\Delta \\\\\n\\mathcal{C} \\ar[r]^{(G_1, G_2)} &\n\\mathcal{S} \\times \\mathcal{S}\n}\n$$\nbe a $2$-fibre product of categories.\nThen there is a canonical isomorphism\n$$\n\\mathcal{C}' \\cong\n(\\mathcal{C} \\times_{G_1, \\mathcal{S}, G_2} \\mathcal{C})\n\\times_{(p, q), \\mathcal{C} \\times \\mathcal{C}, \\Delta}\n\\mathcal{C}.\n$$","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XF","source_file":"categories.tex","source_line":5894,"source_end_line":5912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5894-L5912","statement_sha256":"75f329531d273250f0a56ee0de9c3f6ff116fd74c8709acee07f031eb75811ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":154,"rank":154,"depth":0,"x":291.222,"y":115.468,"cluster":"categories-foundations"},{"id":"stacks:04Z1","tag":"04Z1","title":"2-fibre products · Lemma 04Z1","summary":"Let A → C, B → C and C → D be functors between categories. Then the diagram xymatrix A ×_C B ar[d] ar[r] & A ×_D B ar[d] C ar[r]^-Δ_C/D ar[r] & C ×_D C is a 2-fibre product diagram.","statement_latex":"Let $\\mathcal{A} \\to \\mathcal{C}$, $\\mathcal{B} \\to \\mathcal{C}$\nand $\\mathcal{C} \\to \\mathcal{D}$ be functors between categories.\nThen the diagram\n$$\n\\xymatrix{\n\\mathcal{A} \\times_\\mathcal{C} \\mathcal{B} \\ar[d] \\ar[r] &\n\\mathcal{A} \\times_\\mathcal{D} \\mathcal{B} \\ar[d] \\\\\n\\mathcal{C} \\ar[r]^-{\\Delta_{\\mathcal{C}/\\mathcal{D}}} \\ar[r] &\n\\mathcal{C} \\times_\\mathcal{D} \\mathcal{C}\n}\n$$\nis a $2$-fibre product diagram.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Z1","source_file":"categories.tex","source_line":5925,"source_end_line":5939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5925-L5939","statement_sha256":"2d4e1595edb0e9c28bd8555180e1b257ee7069453c102545fd74f105819dbed6","origin":"The Stacks Project","memory_eligible":false,"source_rank":155,"rank":155,"depth":0,"x":269.043,"y":332.177,"cluster":"categories-foundations"},{"id":"stacks:04YR","tag":"04YR","title":"2-fibre products · Lemma 04YR","summary":"Let xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y be a 2-fibre product of categories. Then the diagram xymatrix U ar[d] ar[r] & U ×_V U ar[d] X ar[r] & X ×_Y X is 2-cartesian.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{U} \\ar[d] \\ar[r] & \\mathcal{V} \\ar[d] \\\\\n\\mathcal{X} \\ar[r] & \\mathcal{Y}\n}\n$$\nbe a $2$-fibre product of categories. Then the diagram\n$$\n\\xymatrix{\n\\mathcal{U} \\ar[d] \\ar[r] &\n\\mathcal{U} \\times_\\mathcal{V} \\mathcal{U} \\ar[d] \\\\\n\\mathcal{X} \\ar[r] &\n\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}\n}\n$$\nis $2$-cartesian.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"2-fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YR","source_file":"categories.tex","source_line":5945,"source_end_line":5964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L5945-L5964","statement_sha256":"2f504b0db317fb37183fa55073b01f62429e7f1cac35b55e9f302873378952ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":156,"rank":156,"depth":1,"x":110.621,"y":159.243,"cluster":"categories-foundations"},{"id":"stacks:003Y","tag":"003Y","title":"Categories over categories · Definition 003Y","summary":"Let C be a category. The 2-category of categories over C is the 2-category defined as follows: • Its objects will be functors p : S → C. • Its 1-morphisms (S, p) → (S', p') will be functors G : S → S' such that p' ∘ G = p. • Its 2-morphisms t : G → H for G, H : (S, p) → (S', p') will be morphisms of functors such that p'(t_x) = id_p(x) for all x ∈ Ob(S). In this situation we will denote Mor_Cat/C(S, S') the category of 1-morphisms between (S, p) and (S', p')","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe {\\it $2$-category of categories over $\\mathcal{C}$}\nis the $2$-category defined as follows:\n\\begin{enumerate}\n\\item Its objects will be functors $p : \\mathcal{S} \\to \\mathcal{C}$.\n\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that\n$p' \\circ G = p$.\n\\item Its $2$-morphisms $t : G \\to H$ for\n$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be morphisms of functors\nsuch that $p'(t_x) = \\text{id}_{p(x)}$\nfor all $x \\in \\Ob(\\mathcal{S})$.\n\\end{enumerate}\nIn this situation we will denote\n$$\n\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{S}')\n$$\nthe category of $1$-morphisms between\n$(\\mathcal{S}, p)$ and $(\\mathcal{S}', p')$","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories over categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003Y","source_file":"categories.tex","source_line":6005,"source_end_line":6027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6005-L6027","statement_sha256":"61161686539dd8ab2006721730609476d58559e287bfc7e35b1f59e2a0061db0","origin":"The Stacks Project","memory_eligible":false,"source_rank":157,"rank":157,"depth":0,"x":367.321,"y":196.991,"cluster":"categories-foundations"},{"id":"stacks:02XH","tag":"02XH","title":"Categories over categories · Definition 02XH","summary":"Let C be a category. Let p : S → C be a category over C. • The fibre category over an object U∈ Ob(C) is the category S_U with objects Ob(S_U) = (x∈ Ob(S) : p(x) = U) and morphisms Mor_S_U(x, y) = ( φ ∈ Mor_S(x, y) : p(φ) = id_U). • A lift of an object U ∈ Ob(C) is an object x∈ Ob(S) such that p(x) = U, i.e., x∈ Ob(S_U). We will also sometime say that x lies over U. • Similarly, a lift of a morphism f : V → U in C is a morphism φ : y → x in S such that p(φ) = f. We…","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The {\\it fibre category} over an object $U\\in \\Ob(\\mathcal{C})$\nis the category $\\mathcal{S}_U$ with objects\n$$\n\\Ob(\\mathcal{S}_U) = \\{x\\in \\Ob(\\mathcal{S}) :\np(x) = U\\}\n$$\nand morphisms\n$$\n\\Mor_{\\mathcal{S}_U}(x, y) = \\{ \\phi \\in \\Mor_\\mathcal{S}(x, y) :\np(\\phi) = \\text{id}_U\\}.\n$$\n\\item A {\\it lift} of an object $U \\in \\Ob(\\mathcal{C})$\nis an object $x\\in \\Ob(\\mathcal{S})$ such that $p(x) = U$, i.e.,\n$x\\in \\Ob(\\mathcal{S}_U)$. We will also sometime say\nthat {\\it $x$ lies over $U$}.\n\\item Similarly, a {\\it lift} of a morphism $f : V \\to U$ in $\\mathcal{C}$\nis a morphism $\\phi : y \\to x$ in $\\mathcal{S}$ such that $p(\\phi) = f$.\nWe sometimes say that {\\it $\\phi$ lies over $f$}.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories over categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XH","source_file":"categories.tex","source_line":6043,"source_end_line":6067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6043-L6067","statement_sha256":"ca2c11bd882d1e7339326232f26e1d60d145240d87349489e89d687a78af9eb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":158,"rank":158,"depth":0,"x":146.984,"y":315.185,"cluster":"categories-foundations"},{"id":"stacks:0040","tag":"0040","title":"Categories over categories · Lemma 0040","summary":"Let C be a category. The (2, 1)-category of categories over C has 2-fibre products. Suppose that F : X → S and G : Y → S are morphisms of categories over C. An explicit 2-fibre product X ×_SY is given by the following description • an object of X ×_S Y is a quadruple (U, x, y, f), where U ∈ Ob(C), x∈ Ob(X_U), y∈ Ob(Y_U), and f : F(x) → G(y) is an isomorphism in S_U, • a morphism (U, x, y, f) → (U', x', y', f') is given by a pair (a, b), where a : x → x' is a morphism in…","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe $(2, 1)$-category of categories\nover $\\mathcal{C}$ has 2-fibre products.\nSuppose that\n$F : \\mathcal{X} \\to \\mathcal{S}$ and\n$G : \\mathcal{Y} \\to \\mathcal{S}$ are morphisms of categories over\n$\\mathcal{C}$.\nAn explicit 2-fibre product\n$\\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y}$ is given by the following\ndescription\n\\begin{enumerate}\n\\item an object of $\\mathcal{X} \\times_\\mathcal{S} \\mathcal{Y}$ is a quadruple\n$(U, x, y, f)$, where $U \\in \\Ob(\\mathcal{C})$,\n$x\\in \\Ob(\\mathcal{X}_U)$, $y\\in \\Ob(\\mathcal{Y}_U)$,\nand $f : F(x) \\to G(y)$ is an isomorphism in $\\mathcal{S}_U$,\n\\item a morphism $(U, x, y, f) \\to (U', x', y', f')$ is given by a pair\n$(a, b)$, where $a : x \\to x'$ is a morphism in $\\mathcal{X}$, and\n$b : y \\to y'$ is a\nmorphism in $\\mathcal{Y}$ such that\n\\begin{enumerate}\n\\item $a$ and $b$ induce the same morphism $U \\to U'$, and\n\\item the diagram\n$$\n\\xymatrix{\nF(x) \\ar[r]^f \\ar[d]^{F(a)} & G(y) \\ar[d]^{G(b)} \\\\\nF(x') \\ar[r]^{f'} & G(y')\n}\n$$\nis commutative.\n\\end{enumerate}\n\\end{enumerate}\nThe functors $p : \\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y} \\to \\mathcal{X}$\nand $q : \\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y} \\to \\mathcal{Y}$ are the\nforgetful functors in this case. The transformation $\\psi : F \\circ p \\to\nG \\circ q$ is given on the object $\\xi = (U, x, y, f)$ by\n$\\psi_\\xi = f : F(p(\\xi)) = F(x) \\to G(y) = G(q(\\xi))$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories over categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0040","source_file":"categories.tex","source_line":6080,"source_end_line":6118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6080-L6118","statement_sha256":"172ab85fbcc9ebbfc29d2b2047b99ec9cf7ce1b6655bb29edecc57786d667ece","origin":"The Stacks Project","memory_eligible":false,"source_rank":159,"rank":159,"depth":1,"x":214.621,"y":102.34,"cluster":"categories-foundations"},{"id":"stacks:02XI","tag":"02XI","title":"Categories over categories · Lemma 02XI","summary":"Let C be a category. Let f : X → S and g : Y → S be morphisms of categories over C. For any object U of C we have the following identity of fibre categories (X ×_SY)_U = X_U ×_S_U Y_U","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $f : \\mathcal{X} \\to \\mathcal{S}$ and\n$g : \\mathcal{Y} \\to \\mathcal{S}$ be morphisms of categories over\n$\\mathcal{C}$. For any object $U$ of $\\mathcal{C}$ we have\nthe following identity of\nfibre categories\n$$\n\\left(\\mathcal{X} \\times_\\mathcal{S}\\mathcal{Y}\\right)_U\n=\n\\mathcal{X}_U \\times_{\\mathcal{S}_U} \\mathcal{Y}_U\n$$","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories over categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XI","source_file":"categories.tex","source_line":6152,"source_end_line":6165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6152-L6165","statement_sha256":"cbecec5ca8a01e11542a1e5dc95c84e1e8472a7a1c5029490306c1f81e24d2ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":160,"rank":160,"depth":0,"x":336.288,"y":298.277,"cluster":"categories-foundations"},{"id":"stacks:02XK","tag":"02XK","title":"Fibred categories · Definition 02XK","summary":"Let C be a category. Let p : S → C be a category over C. A strongly cartesian morphism, or more precisely a strongly C-cartesian morphism is a morphism φ : y → x of S such that for every z ∈ Ob(S) the map Mor_S(z, y) → Mor_S(z, x) ×_Mor_C(p(z), p(x)) Mor_C(p(z), p(y)), given by ψ ↦ (φ ∘ ψ, p(ψ)) is bijective.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.\nA {\\it strongly cartesian morphism}, or more precisely a\n{\\it strongly $\\mathcal{C}$-cartesian morphism} is a\nmorphism $\\varphi : y \\to x$ of $\\mathcal{S}$ such that\nfor every $z \\in \\Ob(\\mathcal{S})$ the map\n$$\n\\Mor_\\mathcal{S}(z, y)\n\\longrightarrow\n\\Mor_\\mathcal{S}(z, x)\n\\times_{\\Mor_\\mathcal{C}(p(z), p(x))}\n\\Mor_\\mathcal{C}(p(z), p(y)),\n$$\ngiven by $\\psi \\longmapsto (\\varphi \\circ \\psi, p(\\psi))$\nis bijective.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XK","source_file":"categories.tex","source_line":6204,"source_end_line":6221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6204-L6221","statement_sha256":"ffc0ea163ce2fcad334405e90a1ef60047d2310df019d11924e31bf10aee256b","origin":"The Stacks Project","memory_eligible":false,"source_rank":161,"rank":161,"depth":0,"x":88.24,"y":222.598,"cluster":"categories-foundations"},{"id":"stacks:02XL","tag":"02XL","title":"Fibred categories · Lemma 02XL","summary":"Let C be a category. Let p : S → C be a category over C. • The composition of two strongly cartesian morphisms is strongly cartesian. • Any isomorphism of S is strongly cartesian. • Any strongly cartesian morphism φ such that p(φ) is an isomorphism, is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The composition of two strongly cartesian morphisms\nis strongly cartesian.\n\\item Any isomorphism of $\\mathcal{S}$ is strongly cartesian.\n\\item Any strongly cartesian morphism $\\varphi$ such that $p(\\varphi)$\nis an isomorphism, is an isomorphism.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XL","source_file":"categories.tex","source_line":6241,"source_end_line":6252,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6241-L6252","statement_sha256":"6a87909670b6571a7eecbe9a75d2a0f74751f10df07c3ed3daaeb44fc371151f","origin":"The Stacks Project","memory_eligible":false,"source_rank":162,"rank":162,"depth":1,"x":332.757,"y":137.393,"cluster":"categories-foundations"},{"id":"stacks:09WU","tag":"09WU","title":"Fibred categories · Lemma 09WU","summary":"Let F : A → B and G : B → C be composable functors between categories. Let x → y be a morphism of A. If x → y is strongly B-cartesian and F(x) → F(y) is strongly C-cartesian, then x → y is strongly C-cartesian.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{C}$\nbe composable functors between categories. Let $x \\to y$ be a morphism of\n$\\mathcal{A}$. If $x \\to y$ is strongly $\\mathcal{B}$-cartesian\nand $F(x) \\to F(y)$ is strongly $\\mathcal{C}$-cartesian, then\n$x \\to y$ is strongly $\\mathcal{C}$-cartesian.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WU","source_file":"categories.tex","source_line":6300,"source_end_line":6307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6300-L6307","statement_sha256":"22030745da75a0b1811d2f10fcf8a2284247818acc7a52020718e34443e2448a","origin":"The Stacks Project","memory_eligible":false,"source_rank":163,"rank":163,"depth":0,"x":220.63,"y":339.583,"cluster":"categories-foundations"},{"id":"stacks:06N4","tag":"06N4","title":"Fibred categories · Lemma 06N4","summary":"Let C be a category. Let p : S → C be a category over C. Let x → y and z → y be morphisms of S. Assume • x → y is strongly cartesian, • p(x) ×_p(y) p(z) exists, and • there exists a strongly cartesian morphism a : w → z in S with p(w) = p(x) ×_p(y) p(z) and p(a) = pr_2 : p(x) ×_p(y) p(z) → p(z). Then the fibre product x ×_y z exists and is isomorphic to w.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.\nLet $x \\to y$ and $z \\to y$ be morphisms of $\\mathcal{S}$.\nAssume\n\\begin{enumerate}\n\\item $x \\to y$ is strongly cartesian,\n\\item $p(x) \\times_{p(y)} p(z)$ exists, and\n\\item there exists a strongly cartesian morphism $a : w \\to z$ in\n$\\mathcal{S}$ with $p(w) = p(x) \\times_{p(y)} p(z)$ and\n$p(a) = \\text{pr}_2 : p(x) \\times_{p(y)} p(z) \\to p(z)$.\n\\end{enumerate}\nThen the fibre product $x \\times_y z$ exists and is isomorphic to $w$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06N4","source_file":"categories.tex","source_line":6313,"source_end_line":6327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6313-L6327","statement_sha256":"bbc0645c0917e67550ef8ec88b1403e01fc9e99812e9c741546d9525e61a0d6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":164,"rank":164,"depth":0,"x":140.473,"y":126.221,"cluster":"categories-foundations"},{"id":"stacks:02XM","tag":"02XM","title":"Fibred categories · Definition 02XM","summary":"Let C be a category. Let p : S → C be a category over C. We say S is a fibred category over C if given any x ∈ Ob(S) lying over U ∈ Ob(C) and any morphism f : V → U of C, there exists a strongly cartesian morphism f^*x → x lying over f.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.\nWe say $\\mathcal{S}$ is a {\\it fibred category over $\\mathcal{C}$}\nif given any $x \\in \\Ob(\\mathcal{S})$ lying over\n$U \\in \\Ob(\\mathcal{C})$ and any morphism $f : V \\to U$ of\n$\\mathcal{C}$, there exists a strongly cartesian morphism $f^*x \\to x$\nlying over $f$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XM","source_file":"categories.tex","source_line":6360,"source_end_line":6369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6360-L6369","statement_sha256":"31e92e43e70b62068a35e9637d2211d948d06002d0e6ef1ed5fb75167f66b30e","origin":"The Stacks Project","memory_eligible":false,"source_rank":165,"rank":165,"depth":0,"x":371.856,"y":238.407,"cluster":"categories-foundations"},{"id":"stacks:02XN","tag":"02XN","title":"Fibred categories · Definition 02XN","summary":"Assume p : S → C is a fibred category. • A choice of pullbacks for p : S → C is given by a choice of a strongly cartesian morphism f^ast x → x lying over f for any morphism f: V → U of C and any x ∈ Ob(S_U). • Given a choice of pullbacks, for any morphism f : V → U of C the functor f^* : S_U → S_V described above is called a pullback functor (associated to the choices f^*x → x made above).","statement_latex":"Assume $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category.\n\\begin{enumerate}\n\\item A {\\it choice of pullbacks}\\footnote{This is probably nonstandard\nterminology. In some texts this is called a ``cleavage'' but it conjures up\nthe wrong image. Maybe a ``cleaving'' would be a better word.\nA related notion is that of a ``splitting'', but in many texts a ``splitting''\nmeans a choice of pullbacks such that $g^*f^* = (f \\circ g)^*$\nfor any composable pair of morphisms. Compare\nalso with Definition \\ref{definition-split-fibred-category}.}\nfor $p : \\mathcal{S} \\to \\mathcal{C}$\nis given by a choice of a strongly cartesian morphism\n$f^\\ast x \\to x$ lying over $f$ for any morphism\n$f: V \\to U$ of $\\mathcal{C}$ and any $x \\in \\Ob(\\mathcal{S}_U)$.\n\\item Given a choice of pullbacks,\nfor any morphism $f : V \\to U$ of $\\mathcal{C}$\nthe functor $f^* : \\mathcal{S}_U \\to \\mathcal{S}_V$ described\nabove is called a {\\it pullback functor} (associated to the choices\n$f^*x \\to x$ made above).\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XN","source_file":"categories.tex","source_line":6391,"source_end_line":6412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6391-L6412","statement_sha256":"15b36fb8b8e7dd994c275150fac77bb9901a84c5c548520558ee4dfe89961987","origin":"The Stacks Project","memory_eligible":false,"source_rank":166,"rank":166,"depth":0,"x":110.238,"y":287.12,"cluster":"categories-foundations"},{"id":"stacks:02XO","tag":"02XO","title":"Fibred categories · Lemma 02XO","summary":"Assume p : S → C is a fibred category. Assume given a choice of pullbacks for p : S → C. • For any pair of composable morphisms f : V → U, g : W → V there is a unique isomorphism α_g, f : (f ∘ g)^ast → g^ast ∘ f^ast as functors S_U → S_W such that for every y∈ Ob(S_U) the following diagram commutes xymatrix g^ast f^ast y ar[r] & f^ast y ar[d] (f ∘ g)^ast y ar[r] ar[u]^(α_g, f)_y & y • If f = id_U, then there is a canonical isomorphism α_U : id → (id_U)^* as functors S_U →…","statement_latex":"Assume $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category.\nAssume given a choice of pullbacks for $p : \\mathcal{S} \\to \\mathcal{C}$.\n\\begin{enumerate}\n\\item For any pair of composable morphisms $f : V \\to U$,\n$g : W \\to V$ there is a unique isomorphism\n$$\n\\alpha_{g, f} :\n(f \\circ g)^\\ast\n\\longrightarrow\ng^\\ast \\circ f^\\ast\n$$\nas functors $\\mathcal{S}_U \\to \\mathcal{S}_W$\nsuch that for every $y\\in \\Ob(\\mathcal{S}_U)$ the following\ndiagram commutes\n$$\n\\xymatrix{\ng^\\ast f^\\ast y \\ar[r]\n&\nf^\\ast y \\ar[d] \\\\\n(f \\circ g)^\\ast y \\ar[r]\n\\ar[u]^{(\\alpha_{g, f})_y}\n&\ny\n}\n$$\n\\item If $f = \\text{id}_U$, then there is a canonical isomorphism\n$\\alpha_U : \\text{id} \\to (\\text{id}_U)^*$ as functors\n$\\mathcal{S}_U \\to \\mathcal{S}_U$.\n\\item The quadruple\n$(U \\mapsto \\mathcal{S}_U, f \\mapsto f^*, \\alpha_{g, f}, \\alpha_U)$\ndefines a pseudo functor from $\\mathcal{C}^{opp}$ to\nthe $(2, 1)$-category of categories, see\nDefinition \\ref{definition-functor-into-2-category}.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XO","source_file":"categories.tex","source_line":6419,"source_end_line":6455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6419-L6455","statement_sha256":"e962215f5593810d4db29462d66d8b9ebd68631a3d2a2d9d2d8ef8ceac5c3265","origin":"The Stacks Project","memory_eligible":false,"source_rank":167,"rank":167,"depth":2,"x":264.437,"y":102.201,"cluster":"categories-foundations"},{"id":"stacks:042G","tag":"042G","title":"Fibred categories · Lemma 042G","summary":"Let C be a category. Let S_1, S_2 be categories over C. Suppose that S_1 and S_2 are equivalent as categories over C. Then S_1 is fibred over C if and only if S_2 is fibred over C.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.\nSuppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent\nas categories over $\\mathcal{C}$.\nThen $\\mathcal{S}_1$ is fibred over $\\mathcal{C}$ if and only if\n$\\mathcal{S}_2$ is fibred over $\\mathcal{C}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042G","source_file":"categories.tex","source_line":6478,"source_end_line":6486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6478-L6486","statement_sha256":"473ac7e19d6c3b5c38306bef5f89f88c1496a92f47ebaff7c7b398d573b388cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":168,"rank":168,"depth":0,"x":299.543,"y":326.719,"cluster":"categories-foundations"},{"id":"stacks:02XP","tag":"02XP","title":"Fibred categories · Definition 02XP","summary":"Let C be a category. The 2-category of fibred categories over C is the sub 2-category of the 2-category of categories over C (see Definition [Tag 003Y]) defined as follows: • Its objects will be fibred categories p : S → C. • Its 1-morphisms (S, p) → (S', p') will be functors G : S → S' such that p' ∘ G = p and such that G maps strongly cartesian morphisms to strongly cartesian morphisms. • Its 2-morphisms t : G → H for G, H : (S, p) → (S', p') will be morphisms of…","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe {\\it $2$-category of fibred categories over $\\mathcal{C}$}\nis the sub $2$-category of the $2$-category of categories\nover $\\mathcal{C}$ (see Definition \\ref{definition-categories-over-C})\ndefined as follows:\n\\begin{enumerate}\n\\item Its objects will be fibred categories\n$p : \\mathcal{S} \\to \\mathcal{C}$.\n\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that\n$p' \\circ G = p$ and such that $G$ maps strongly cartesian\nmorphisms to strongly cartesian morphisms.\n\\item Its $2$-morphisms $t : G \\to H$ for\n$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be morphisms of functors\nsuch that $p'(t_x) = \\text{id}_{p(x)}$\nfor all $x \\in \\Ob(\\mathcal{S})$.\n\\end{enumerate}\nIn this situation we will denote\n$$\n\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{S}')\n$$\nthe category of $1$-morphisms between\n$(\\mathcal{S}, p)$ and $(\\mathcal{S}', p')$","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XP","source_file":"categories.tex","source_line":6523,"source_end_line":6549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6523-L6549","statement_sha256":"043f8c2df652baf5861ba2b7db246fff4406bceec77fb6fedbf1ee484d406db9","origin":"The Stacks Project","memory_eligible":false,"source_rank":169,"rank":169,"depth":1,"x":92.497,"y":180.652,"cluster":"categories-foundations"},{"id":"stacks:02XQ","tag":"02XQ","title":"Fibred categories · Lemma 02XQ","summary":"Let C be a category. The (2, 1)-category of fibred categories over C has 2-fibre products, and they are described as in Lemma [Tag 0040].","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe $(2, 1)$-category of fibred categories\nover $\\mathcal{C}$ has 2-fibre products, and\nthey are described as in\nLemma \\ref{lemma-2-product-categories-over-C}.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XQ","source_file":"categories.tex","source_line":6557,"source_end_line":6564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6557-L6564","statement_sha256":"a2cb16a1e63037d1376a716e62c74f28e6401eed4f7e19c5f1baa61eade6182d","origin":"The Stacks Project","memory_eligible":false,"source_rank":170,"rank":170,"depth":2,"x":363.424,"y":170.848,"cluster":"categories-foundations"},{"id":"stacks:02XR","tag":"02XR","title":"Fibred categories · Lemma 02XR","summary":"Let C be a category. Let U ∈ Ob(C). If p : S → C is a fibred category and p factors through p' : S → C/U then p' : S → C/U is a fibred category.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$.\nIf $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category\nand $p$ factors through $p' : \\mathcal{S} \\to \\mathcal{C}/U$\nthen $p' : \\mathcal{S} \\to \\mathcal{C}/U$ is a fibred category.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XR","source_file":"categories.tex","source_line":6591,"source_end_line":6597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6591-L6597","statement_sha256":"119d65b3f31c0f3e652b20aa928e4f1a3fe52e44b5953e6e9643e93ae0707028","origin":"The Stacks Project","memory_eligible":false,"source_rank":171,"rank":171,"depth":1,"x":170.97,"y":332.277,"cluster":"categories-foundations"},{"id":"stacks:09WV","tag":"09WV","title":"Fibred categories · Lemma 09WV","summary":"Let A → B → C be functors between categories. If A is fibred over B and B is fibred over C, then A is fibred over C.","statement_latex":"Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$ be functors between\ncategories. If $\\mathcal{A}$ is fibred over $\\mathcal{B}$ and\n$\\mathcal{B}$ is fibred over $\\mathcal{C}$, then $\\mathcal{A}$\nis fibred over $\\mathcal{C}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WV","source_file":"categories.tex","source_line":6634,"source_end_line":6640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6634-L6640","statement_sha256":"37d8051d9a5d69d88cb8c631969cbdfa9d5a6e4453c1af0df98641c976bd81ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":172,"rank":172,"depth":1,"x":183.102,"y":103.378,"cluster":"categories-foundations"},{"id":"stacks:06N5","tag":"06N5","title":"Fibred categories · Lemma 06N5","summary":"Let p : S → C be a fibred category. Let x → y and z → y be morphisms of S with x → y strongly cartesian. If p(x) ×_p(y) p(z) exists, then x ×_y z exists, p(x ×_y z) = p(x) ×_p(y) p(z), and x ×_y z → z is strongly cartesian.","statement_latex":"Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.\nLet $x \\to y$ and $z \\to y$ be morphisms of $\\mathcal{S}$\nwith $x \\to y$ strongly cartesian. If $p(x) \\times_{p(y)} p(z)$ exists,\nthen $x \\times_y z$ exists, $p(x \\times_y z) = p(x) \\times_{p(y)} p(z)$,\nand $x \\times_y z \\to z$ is strongly cartesian.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06N5","source_file":"categories.tex","source_line":6647,"source_end_line":6654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6647-L6654","statement_sha256":"0e24286d8ddd6ea0a731d7065cbe9d0b484799ad0fe571d4634ebac32654bc7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":173,"rank":173,"depth":1,"x":358.735,"y":279.555,"cluster":"categories-foundations"},{"id":"stacks:08NF","tag":"08NF","title":"Fibred categories · Lemma 08NF","summary":"Let C be a category. Let F : X → Y be a 1-morphism of fibred categories over C. There exist 1-morphisms of fibred categories over C xymatrix X ar@<1ex>[r]^u & X' ar[r]^v ar@<1ex>[l]^w & Y such that F = v ∘ u and such that • u : X → X' is fully faithful, • w is left adjoint to u, and • v : X' → Y is a fibred category.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of fibred categories over $\\mathcal{C}$.\nThere exist $1$-morphisms of fibred categories over $\\mathcal{C}$\n$$\n\\xymatrix{\n\\mathcal{X} \\ar@<1ex>[r]^u &\n\\mathcal{X}' \\ar[r]^v \\ar@<1ex>[l]^w & \\mathcal{Y}\n}\n$$\nsuch that $F = v \\circ u$ and such that\n\\begin{enumerate}\n\\item $u : \\mathcal{X} \\to \\mathcal{X}'$ is fully faithful,\n\\item $w$ is left adjoint to $u$, and\n\\item $v : \\mathcal{X}' \\to \\mathcal{Y}$ is a fibred category.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NF","source_file":"categories.tex","source_line":6664,"source_end_line":6681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6664-L6681","statement_sha256":"206138f1c444c28471512d8583e644b785d336622fe497011bb518b084a1f515","origin":"The Stacks Project","memory_eligible":false,"source_rank":174,"rank":174,"depth":0,"x":86.772,"y":249.215,"cluster":"categories-foundations"},{"id":"stacks:034H","tag":"034H","title":"Inertia · Lemma 034H","summary":"Let C be a category. Let p : S → C and p' : S' → C be fibred categories. Let F : S → S' be a 1-morphism of fibred categories over C. Consider the category I_S/S' over C whose • objects are pairs (x, α) where x ∈ Ob(S) and α : x → x is an automorphism with F(α) = id, • morphisms (x, α) → (y, β) are given by morphisms φ : x → y such that xymatrix xar[r]_φar[d]_α & yar[d]^β xar[r]^φ & y commutes, and • the functor I_S/S' → C is given by (x, α) ↦ p(x). Then • there is an…","statement_latex":"Let $\\mathcal{C}$ be a category. Let\n$p : \\mathcal{S} \\to \\mathcal{C}$ and\n$p' : \\mathcal{S}' \\to \\mathcal{C}$ be fibred categories.\nLet $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of\nfibred categories over $\\mathcal{C}$. Consider the category\n$\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ over $\\mathcal{C}$ whose\n\\begin{enumerate}\n\\item objects are pairs $(x, \\alpha)$ where $x \\in \\Ob(\\mathcal{S})$\nand $\\alpha : x \\to x$ is an automorphism with $F(\\alpha) = \\text{id}$,\n\\item morphisms $(x, \\alpha) \\to (y, \\beta)$ are given by morphisms\n$\\phi : x \\to y$ such that\n$$\n\\xymatrix{\nx\\ar[r]_\\phi\\ar[d]_\\alpha &\ny\\ar[d]^{\\beta} \\\\\nx\\ar[r]^\\phi &\ny \\\\\n}\n$$\ncommutes, and\n\\item the functor $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\to \\mathcal{C}$\nis given by $(x, \\alpha) \\mapsto p(x)$.\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item there is an equivalence\n$$\n\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\longrightarrow\n\\mathcal{S}\n\\times_{\\Delta, (\\mathcal{S} \\times_{\\mathcal{S}'} \\mathcal{S}), \\Delta}\n\\mathcal{S}\n$$\nin the $(2, 1)$-category of categories over $\\mathcal{C}$, and\n\\item $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ is a fibred category over\n$\\mathcal{C}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Inertia","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034H","source_file":"categories.tex","source_line":6779,"source_end_line":6817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6779-L6817","statement_sha256":"b750e765093860184d597c53715f19839a40c033b278b31eb19a818888b040aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":175,"rank":175,"depth":3,"x":312.353,"y":116.894,"cluster":"categories-foundations"},{"id":"stacks:034I","tag":"034I","title":"Inertia · Definition 034I","summary":"Let C be a category. • Let F : S → S' be a 1-morphism of fibred categories over C. The relative inertia of S over S' is the fibred category I_S/S' → C of Lemma [Tag 034H]. • By the inertia fibred category I_S of S we mean I_S = I_S/C.","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of\nfibred categories over $\\mathcal{C}$. The {\\it relative inertia\nof $\\mathcal{S}$ over $\\mathcal{S}'$} is the fibred category\n$\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'} \\to \\mathcal{C}$ of\nLemma \\ref{lemma-inertia-fibred-category}.\n\\item By the {\\it inertia fibred category $\\mathcal{I}_\\mathcal{S}$\nof $\\mathcal{S}$} we mean\n$\\mathcal{I}_\\mathcal{S} = \\mathcal{I}_{\\mathcal{S}/\\mathcal{C}}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Inertia","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034I","source_file":"categories.tex","source_line":6878,"source_end_line":6891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6878-L6891","statement_sha256":"a6026e2134ed63cb5638e8a34f801178d2c0dae56b2f14df9e9f56653718bc8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":176,"rank":176,"depth":4,"x":252.252,"y":343.104,"cluster":"categories-foundations"},{"id":"stacks:04Z6","tag":"04Z6","title":"Inertia · Lemma 04Z6","summary":"Let F : S → S' be a 1-morphism of categories fibred over a category C. Then the diagram xymatrix I_S/S' ar[d]_F ∘ ([Tag 042H]) ar[rr]_([Tag 04Z5]) & & I_S ar[d]^([Tag 04Z4]) S' ar[rr]^e & & I_S' is a 2-fibre product.","statement_latex":"Let $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of categories\nfibred over a category $\\mathcal{C}$. Then the diagram\n$$\n\\xymatrix{\n\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}\n\\ar[d]_{F \\circ (\\ref{equation-inertia-structure-map})}\n\\ar[rr]_{(\\ref{equation-comparison})} & &\n\\mathcal{I}_\\mathcal{S} \\ar[d]^{(\\ref{equation-functorial})} \\\\\n\\mathcal{S}' \\ar[rr]^e & &\n\\mathcal{I}_{\\mathcal{S}'}\n}\n$$\nis a $2$-fibre product.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Inertia","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Z6","source_file":"categories.tex","source_line":6944,"source_end_line":6959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6944-L6959","statement_sha256":"f088fc26668a575ad16fc032fbca993851bc2a202db5b47335414f3bcb0aba54","origin":"The Stacks Project","memory_eligible":false,"source_rank":177,"rank":177,"depth":0,"x":114.271,"y":141.632,"cluster":"categories-foundations"},{"id":"stacks:003T","tag":"003T","title":"Categories fibred in groupoids · Definition 003T","summary":"Let p : S → C be a functor. We say that S is fibred in groupoids over C if the following two conditions hold: • For every morphism f : V → U in C and every lift x of U there is a lift φ : y → x of f with target x. • For every pair of morphisms φ : y → x and ψ : z → x and any morphism f : p(z) → p(y) such that p(φ) ∘ f = p(ψ) there exists a unique lift chi : z → y of f such that φ ∘ chi = ψ.","statement_latex":"Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a functor.\nWe say that $\\mathcal{S}$ is {\\it fibred in groupoids} over $\\mathcal{C}$ if\nthe following two conditions hold:\n\\begin{enumerate}\n\\item For every morphism $f : V \\to U$ in $\\mathcal{C}$ and every\nlift $x$ of $U$ there is a lift $\\phi : y \\to x$ of $f$ with\ntarget $x$.\n\\item For every pair of morphisms $\\phi : y \\to x$ and $ \\psi : z \\to x$\nand any morphism $f : p(z) \\to p(y)$ such that $p(\\phi) \\circ f = p(\\psi)$\nthere exists a unique lift $\\chi : z \\to y$ of $f$ such that\n$\\phi \\circ \\chi = \\psi$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003T","source_file":"categories.tex","source_line":6977,"source_end_line":6991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L6977-L6991","statement_sha256":"09e3358f5b33f15b4bf00493187b8c436137aefcc3d992f7ad83454c55ffff4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":178,"rank":178,"depth":0,"x":378.773,"y":212.098,"cluster":"categories-foundations"},{"id":"stacks:003V","tag":"003V","title":"Categories fibred in groupoids · Lemma 003V","summary":"Let p : S → C be a functor. The following are equivalent • p : S → C is a category fibred in groupoids, and • all fibre categories are groupoids and S is a fibred category over C. Moreover, in this case every morphism of S is strongly cartesian. In addition, given f^ast x → x lying over f for all f: V → U = p(x) the data (U ↦ S_U, f ↦ f^*, α_f, g, α_U) constructed in Lemma [Tag 02XO] defines a pseudo functor from C^opp in to the (2, 1)-category of groupoids.","statement_latex":"Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a functor.\nThe following are equivalent\n\\begin{enumerate}\n\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a category\nfibred in groupoids, and\n\\item all fibre categories are groupoids and\n$\\mathcal{S}$ is a fibred category over $\\mathcal{C}$.\n\\end{enumerate}\nMoreover, in this case every morphism of $\\mathcal{S}$ is\nstrongly cartesian. In addition, given $f^\\ast x \\to x$\nlying over $f$ for all $f: V \\to U = p(x)$ the data\n$(U \\mapsto \\mathcal{S}_U, f \\mapsto f^*, \\alpha_{f, g}, \\alpha_U)$\nconstructed in Lemma \\ref{lemma-fibred}\ndefines a pseudo functor from $\\mathcal{C}^{opp}$ in to\nthe $(2, 1)$-category of groupoids.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003V","source_file":"categories.tex","source_line":7035,"source_end_line":7052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7035-L7052","statement_sha256":"3f472629914c8f19c04987bbdf8c961399973dfb73a8a600703752d3ce676ec9","origin":"The Stacks Project","memory_eligible":false,"source_rank":179,"rank":179,"depth":3,"x":126.365,"y":310.494,"cluster":"categories-foundations"},{"id":"stacks:03WQ","tag":"03WQ","title":"Categories fibred in groupoids · Lemma 03WQ","summary":"Let C be a category. Let p : S → C be a fibred category. Let S' be the subcategory of S defined as follows • Ob(S') = Ob(S), and • for x, y ∈ Ob(S') the set of morphisms between x and y in S' is the set of strongly cartesian morphisms between x and y in S. Let p' : S' → C be the restriction of p to S'. Then p' : S' → C is fibred in groupoids.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.\nLet $\\mathcal{S}'$ be the subcategory of $\\mathcal{S}$ defined\nas follows\n\\begin{enumerate}\n\\item $\\Ob(\\mathcal{S}') = \\Ob(\\mathcal{S})$, and\n\\item for $x, y \\in \\Ob(\\mathcal{S}')$ the set of morphisms between $x$\nand $y$ in $\\mathcal{S}'$ is the set of strongly cartesian morphisms between\n$x$ and $y$ in $\\mathcal{S}$.\n\\end{enumerate}\nLet $p' : \\mathcal{S}' \\to \\mathcal{C}$ be the restriction of $p$\nto $\\mathcal{S}'$. Then $p' : \\mathcal{S}' \\to \\mathcal{C}$ is fibred\nin groupoids.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WQ","source_file":"categories.tex","source_line":7100,"source_end_line":7115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7100-L7115","statement_sha256":"1a7c5d2e3a9718576afd4ae8ab003aa3fd19a38db63e48791bd5624abda11500","origin":"The Stacks Project","memory_eligible":false,"source_rank":180,"rank":180,"depth":2,"x":233.656,"y":94.119,"cluster":"categories-foundations"},{"id":"stacks:02XS","tag":"02XS","title":"Categories fibred in groupoids · Definition 02XS","summary":"Let C be a category. The 2-category of categories fibred in groupoids over C is the sub 2-category of the 2-category of fibred categories over C (see Definition [Tag 02XP]) defined as follows: • Its objects will be categories p : S → C fibred in groupoids. • Its 1-morphisms (S, p) → (S', p') will be functors G : S → S' such that p' ∘ G = p (since every morphism is strongly cartesian G automatically preserves them). • Its 2-morphisms t : G → H for G, H : (S, p) → (S', p')…","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe {\\it $2$-category of categories fibred in groupoids over $\\mathcal{C}$}\nis the sub $2$-category of the $2$-category of fibred categories\nover $\\mathcal{C}$ (see Definition \\ref{definition-fibred-categories-over-C})\ndefined as follows:\n\\begin{enumerate}\n\\item Its objects will be categories\n$p : \\mathcal{S} \\to \\mathcal{C}$ fibred in groupoids.\n\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that\n$p' \\circ G = p$ (since every morphism is strongly cartesian\n$G$ automatically preserves them).\n\\item Its $2$-morphisms $t : G \\to H$ for\n$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be morphisms of functors\nsuch that $p'(t_x) = \\text{id}_{p(x)}$\nfor all $x \\in \\Ob(\\mathcal{S})$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XS","source_file":"categories.tex","source_line":7201,"source_end_line":7221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7201-L7221","statement_sha256":"2b244d58ee0089d1b38f32c4ebfca9dc5449fb7d80af5789c871136592b8108f","origin":"The Stacks Project","memory_eligible":false,"source_rank":181,"rank":181,"depth":2,"x":328.804,"y":315.157,"cluster":"categories-foundations"},{"id":"stacks:0041","tag":"0041","title":"Categories fibred in groupoids · Lemma 0041","summary":"Let C be a category. The 2-category of categories fibred in groupoids over C has 2-fibre products, and they are described as in Lemma [Tag 0040].","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe $2$-category of categories fibred in groupoids\nover $\\mathcal{C}$ has 2-fibre products, and they are described as in\nLemma \\ref{lemma-2-product-categories-over-C}.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0041","source_file":"categories.tex","source_line":7228,"source_end_line":7234,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7228-L7234","statement_sha256":"89e5f44516c096825bbae1b03dfc9b8b98f2657d7fdddc9014c9837aaa5ab2c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":182,"rank":182,"depth":4,"x":80.212,"y":205.858,"cluster":"categories-foundations"},{"id":"stacks:003Z","tag":"003Z","title":"Categories fibred in groupoids · Lemma 003Z","summary":"Let p : S→ C and p' : S'→ C be categories fibred in groupoids, and suppose that G : S→ S' is a functor over C. • Then G is faithful (resp. fully faithful, resp. an equivalence) if and only if for each U∈Ob(C) the induced functor G_U : S_U→ S'_U is faithful (resp. fully faithful, resp. an equivalence). • If G is an equivalence, then G is an equivalence in the 2-category of categories fibred in groupoids over C.","statement_latex":"Let $p : \\mathcal{S}\\to \\mathcal{C}$ and\n$p' : \\mathcal{S'}\\to \\mathcal{C}$ be categories fibred in groupoids, and\nsuppose that $G : \\mathcal{S}\\to \\mathcal {S}'$ is a functor over\n$\\mathcal{C}$.\n\\begin{enumerate}\n\\item Then $G$ is faithful (resp.\\ fully faithful, resp.\\ an equivalence)\nif and only if for each $U\\in\\Ob(\\mathcal{C})$ the induced functor\n$G_U : \\mathcal{S}_U\\to \\mathcal{S}'_U$ is faithful\n(resp.\\ fully faithful, resp.\\ an equivalence).\n\\item If $G$ is an equivalence, then $G$ is an equivalence in the\n$2$-category of categories fibred in groupoids over $\\mathcal{C}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/003Z","source_file":"categories.tex","source_line":7270,"source_end_line":7284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7270-L7284","statement_sha256":"ef33baa27adfac4f6339bb56491b2f9c8845674cd4f9b5963b20900c26cce2db","origin":"The Stacks Project","memory_eligible":false,"source_rank":183,"rank":183,"depth":2,"x":352.155,"y":145.233,"cluster":"categories-foundations"},{"id":"stacks:04Z7","tag":"04Z7","title":"Categories fibred in groupoids · Lemma 04Z7","summary":"Let C be a category. Let p : S→ C and p' : S'→ C be categories fibred in groupoids. Let G : S→ S' be a functor over C. Then G is fully faithful if and only if the diagonal Δ_G : S → S ×_G, S', G S is an equivalence.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $p : \\mathcal{S}\\to \\mathcal{C}$ and\n$p' : \\mathcal{S'}\\to \\mathcal{C}$ be categories fibred in groupoids.\nLet $G : \\mathcal{S}\\to \\mathcal {S}'$ be a functor over $\\mathcal{C}$.\nThen $G$ is fully faithful if and only if the diagonal\n$$\n\\Delta_G :\n\\mathcal{S}\n\\longrightarrow\n\\mathcal{S} \\times_{G, \\mathcal{S}', G} \\mathcal{S}\n$$\nis an equivalence.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Z7","source_file":"categories.tex","source_line":7364,"source_end_line":7377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7364-L7377","statement_sha256":"c782a643fcd366aa033576c0258582fb8059a794601f5c41d043e9a982be4c7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":184,"rank":184,"depth":3,"x":199.97,"y":344.782,"cluster":"categories-foundations"},{"id":"stacks:03YT","tag":"03YT","title":"Categories fibred in groupoids · Lemma 03YT","summary":"Let C be a category. Let S_i, i = 1, 2, 3, 4 be categories fibred in groupoids over C. Suppose that φ : S_1 → S_2 and ψ : S_3 → S_4 are equivalences over C. Then Mor_Cat/C(S_2, S_3) → Mor_Cat/C(S_1, S_4), α ↦ ψ ∘ α ∘ φ is an equivalence of categories.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{S}_i$, $i = 1, 2, 3, 4$ be categories fibred in\ngroupoids over $\\mathcal{C}$.\nSuppose that $\\varphi : \\mathcal{S}_1 \\to \\mathcal{S}_2$ and\n$\\psi : \\mathcal{S}_3 \\to \\mathcal{S}_4$ are equivalences\nover $\\mathcal{C}$. Then\n$$\n\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}_2, \\mathcal{S}_3)\n\\longrightarrow\n\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}_1, \\mathcal{S}_4),\n\\quad \\alpha \\longmapsto \\psi \\circ \\alpha \\circ \\varphi\n$$\nis an equivalence of categories.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YT","source_file":"categories.tex","source_line":7395,"source_end_line":7410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7395-L7410","statement_sha256":"ccf1308917bff28b0ddfafe53cd46ffa7c72a15ea257dde3dde088f001bcec02","origin":"The Stacks Project","memory_eligible":false,"source_rank":185,"rank":185,"depth":0,"x":151.588,"y":110.655,"cluster":"categories-foundations"},{"id":"stacks:042I","tag":"042I","title":"Categories fibred in groupoids · Lemma 042I","summary":"Let C be a category. If p : S → C is fibred in groupoids, then so is the inertia fibred category I_S → C.","statement_latex":"Let $\\mathcal{C}$ be a category.\nIf $p : \\mathcal{S} \\to \\mathcal{C}$ is fibred in groupoids, then\nso is the inertia fibred category $\\mathcal{I}_\\mathcal{S} \\to \\mathcal{C}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042I","source_file":"categories.tex","source_line":7416,"source_end_line":7421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7416-L7421","statement_sha256":"8c5028369251ad7fb7ecaebe9eb5bb7099f299939718595d6f71e4d089c4ab49","origin":"The Stacks Project","memory_eligible":false,"source_rank":186,"rank":186,"depth":5,"x":376.141,"y":256.233,"cluster":"categories-foundations"},{"id":"stacks:02XT","tag":"02XT","title":"Categories fibred in groupoids · Lemma 02XT","summary":"Let C be a category. Let U ∈ Ob(C). If p : S → C is a category fibred in groupoids and p factors through p' : S → C/U then p' : S → C/U is fibred in groupoids.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$.\nIf $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in groupoids\nand $p$ factors through $p' : \\mathcal{S} \\to \\mathcal{C}/U$\nthen $p' : \\mathcal{S} \\to \\mathcal{C}/U$ is fibred in groupoids.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XT","source_file":"categories.tex","source_line":7433,"source_end_line":7439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7433-L7439","statement_sha256":"23a83f4c69a91972b16b0071bfa2c6adea47e4230f902c0e581f145926083ee5","origin":"The Stacks Project","memory_eligible":false,"source_rank":187,"rank":187,"depth":4,"x":92.737,"y":276.356,"cluster":"categories-foundations"},{"id":"stacks:09WW","tag":"09WW","title":"Categories fibred in groupoids · Lemma 09WW","summary":"Let A → B → C be functors between categories. If A is fibred in groupoids over B and B is fibred in groupoids over C, then A is fibred in groupoids over C.","statement_latex":"Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$ be functors between\ncategories. If $\\mathcal{A}$ is fibred in groupoids over $\\mathcal{B}$\nand $\\mathcal{B}$ is fibred in groupoids over $\\mathcal{C}$, then\n$\\mathcal{A}$ is fibred in groupoids over $\\mathcal{C}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WW","source_file":"categories.tex","source_line":7454,"source_end_line":7460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7454-L7460","statement_sha256":"39f5c7014c8d4de69d2619185c430874890d9f49e8338005b50a1e18d8237242","origin":"The Stacks Project","memory_eligible":false,"source_rank":188,"rank":188,"depth":4,"x":286.043,"y":100.242,"cluster":"categories-foundations"},{"id":"stacks:06N6","tag":"06N6","title":"Categories fibred in groupoids · Lemma 06N6","summary":"Let p : S → C be a category fibred in groupoids. Let x → y and z → y be morphisms of S. If p(x) ×_p(y) p(z) exists, then x ×_y z exists and p(x ×_y z) = p(x) ×_p(y) p(z).","statement_latex":"Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.\nLet $x \\to y$ and $z \\to y$ be morphisms of $\\mathcal{S}$.\nIf $p(x) \\times_{p(y)} p(z)$ exists, then\n$x \\times_y z$ exists and $p(x \\times_y z) = p(x) \\times_{p(y)} p(z)$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06N6","source_file":"categories.tex","source_line":7476,"source_end_line":7482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7476-L7482","statement_sha256":"85c727de172c3c3cb9d566d906adbabf10e381fb6af6839e7e97375164703613","origin":"The Stacks Project","memory_eligible":false,"source_rank":189,"rank":189,"depth":2,"x":285.125,"y":340.423,"cluster":"categories-foundations"},{"id":"stacks:06N7","tag":"06N7","title":"Categories fibred in groupoids · Lemma 06N7","summary":"Let C be a category. Let F : X → Y be a 1-morphism of categories fibred in groupoids over C. There exists a factorization X → X' → Y by 1-morphisms of categories fibred in groupoids over C such that X → X' is an equivalence over C and such that X' is a category fibred in groupoids over Y.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of categories fibred in groupoids over $\\mathcal{C}$.\nThere exists a factorization $\\mathcal{X} \\to \\mathcal{X}' \\to \\mathcal{Y}$\nby $1$-morphisms of categories fibred in groupoids over $\\mathcal{C}$ such\nthat $\\mathcal{X} \\to \\mathcal{X}'$ is an equivalence over $\\mathcal{C}$\nand such that $\\mathcal{X}'$ is a category fibred in groupoids over\n$\\mathcal{Y}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06N7","source_file":"categories.tex","source_line":7489,"source_end_line":7498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7489-L7498","statement_sha256":"a1644a99d31abda8b9c2017b3fa5b6caf8765f5c5d9de8728f25c03377a25ea4","origin":"The Stacks Project","memory_eligible":false,"source_rank":190,"rank":190,"depth":5,"x":92.151,"y":162.33,"cluster":"categories-foundations"},{"id":"stacks:06N8","tag":"06N8","title":"Categories fibred in groupoids · Lemma 06N8","summary":"Let C be a category. Let F : X → Y be a 1-morphism of categories fibred in groupoids over C. Assume we have a 2-commutative diagram xymatrix X' ar[rd]_f & X ar[l]^a ar[d]^F ar[r]_b & X\" ar[ld]^g & Y where a and b are equivalences of categories over C and f and g are categories fibred in groupoids. Then there exists an equivalence h : X\" → X' of categories over Y such that h ∘ b is 2-isomorphic to a as 1-morphisms of categories over C. If the diagram above actually…","statement_latex":"Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of categories fibred in groupoids over $\\mathcal{C}$.\nAssume we have a $2$-commutative diagram\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[rd]_f &\n\\mathcal{X} \\ar[l]^a \\ar[d]^F \\ar[r]_b &\n\\mathcal{X}'' \\ar[ld]^g \\\\\n& \\mathcal{Y}\n}\n$$\nwhere $a$ and $b$ are equivalences of categories over $\\mathcal{C}$\nand $f$ and $g$ are categories fibred in groupoids. Then there exists\nan equivalence $h : \\mathcal{X}'' \\to \\mathcal{X}'$ of categories over\n$\\mathcal{Y}$ such that $h \\circ b$ is $2$-isomorphic to $a$ as $1$-morphisms\nof categories over $\\mathcal{C}$. If the diagram above actually commutes, then\nwe can arrange it so that $h \\circ b$ is $2$-isomorphic to $a$ as\n$1$-morphisms of categories over $\\mathcal{Y}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06N8","source_file":"categories.tex","source_line":7554,"source_end_line":7574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7554-L7574","statement_sha256":"196ffe3838149f3ef9bce1f333e738282c86b8e84886890b3224fe58f7af569f","origin":"The Stacks Project","memory_eligible":false,"source_rank":191,"rank":191,"depth":6,"x":378.409,"y":184.213,"cluster":"categories-foundations"},{"id":"stacks:02XW","tag":"02XW","title":"Presheaves of categories · Definition 02XW","summary":"Let C be a category. Suppose that F : C^opp → Cat is a functor to the 2-category of categories. We will write p_F : S_F → C for the fibred category constructed in Example [Tag 02XV]. A split fibred category is a fibred category isomorphic (!) over C to one of these categories S_F.","statement_latex":"Let $\\mathcal{C}$ be a category.\nSuppose that $F : \\mathcal{C}^{opp} \\to \\textit{Cat}$ is a functor\nto the $2$-category of categories.\nWe will write $p_F : \\mathcal{S}_F \\to \\mathcal{C}$ for the\nfibred category constructed in\nExample \\ref{example-functor-categories}.\nA {\\it split fibred category} is a fibred category isomorphic (!)\nover $\\mathcal{C}$ to one of these categories {\\it $\\mathcal{S}_F$}.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Presheaves of categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XW","source_file":"categories.tex","source_line":7720,"source_end_line":7730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7720-L7730","statement_sha256":"c9868ac5c74343eb02e5773cef10637a96ca96c6bd2b8a0cc5cb94cef8e94f91","origin":"The Stacks Project","memory_eligible":false,"source_rank":192,"rank":192,"depth":0,"x":149.136,"y":330.885,"cluster":"categories-foundations"},{"id":"stacks:02XX","tag":"02XX","title":"Presheaves of categories · Lemma 02XX","summary":"Let C be a category. Let S be a fibred category over C. Then S is split if and only if for some choice of pullbacks (see Definition [Tag 02XN]) the pullback functors (f ∘ g)^* and g^* ∘ f^* are equal.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{S}$ be a fibred category over $\\mathcal{C}$.\nThen $\\mathcal{S}$ is split if and only if for some choice\nof pullbacks (see Definition \\ref{definition-pullback-functor-fibred-category})\nthe pullback functors\n$(f \\circ g)^*$ and $g^* \\circ f^*$ are equal.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Presheaves of categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XX","source_file":"categories.tex","source_line":7732,"source_end_line":7740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7732-L7740","statement_sha256":"4b23fb8626438d08468aa5acf44d9953b5aa45541e2517101620a3ce244b48c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":193,"rank":193,"depth":1,"x":200.381,"y":92.022,"cluster":"categories-foundations"},{"id":"stacks:004A","tag":"004A","title":"Presheaves of categories · Lemma 004A","summary":"Let p : S → C be a fibred category. There exists a contravariant functor F : C → Cat such that S is equivalent to S_F in the 2-category of fibred categories over C. In other words, every fibred category is equivalent to a split one.","statement_latex":"Let $ p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.\nThere exists a contravariant functor $F : \\mathcal{C} \\to \\textit{Cat}$\nsuch that $\\mathcal{S}$ is equivalent to $\\mathcal{S}_F$\nin the $2$-category of fibred categories over $\\mathcal{C}$. In other\nwords, every fibred category is equivalent to a split one.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Presheaves of categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004A","source_file":"categories.tex","source_line":7746,"source_end_line":7753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7746-L7753","statement_sha256":"0455fb9a5658aece245385f059b57cefc67d9b20658e267dff6fd6951b6f678c","origin":"The Stacks Project","memory_eligible":false,"source_rank":194,"rank":194,"depth":3,"x":355.076,"y":297.761,"cluster":"categories-foundations"},{"id":"stacks:04TL","tag":"04TL","title":"Presheaves of groupoids · Definition 04TL","summary":"Let C be a category. Suppose that F : C^opp → Groupoids is a functor to the 2-category of groupoids. We will write p_F : S_F → C for the category fibred in groupoids constructed in Example [Tag 0049]. A split category fibred in groupoids is a category fibred in groupoids isomorphic (!) over C to one of these categories S_F.","statement_latex":"Let $\\mathcal{C}$ be a category.\nSuppose that $F : \\mathcal{C}^{opp} \\to \\textit{Groupoids}$ is a functor\nto the $2$-category of groupoids.\nWe will write $p_F : \\mathcal{S}_F \\to \\mathcal{C}$ for the\ncategory fibred in groupoids constructed in\nExample \\ref{example-functor-groupoids}.\nA {\\it split category fibred in groupoids} is a\ncategory fibred in groupoids isomorphic (!)\nover $\\mathcal{C}$ to one of these categories {\\it $\\mathcal{S}_F$}.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Presheaves of groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TL","source_file":"categories.tex","source_line":7881,"source_end_line":7892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7881-L7892","statement_sha256":"9a9001256d5e39b15c45dddbcc25658690b9608bbcec4b9c28772d4f847184d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":195,"rank":195,"depth":0,"x":74.844,"y":233.666,"cluster":"categories-foundations"},{"id":"stacks:02XY","tag":"02XY","title":"Presheaves of groupoids · Lemma 02XY","summary":"Let p : S → C be a category fibred in groupoids. There exists a contravariant functor F : C → Groupoids such that S is equivalent to S_F over C. In other words, every category fibred in groupoids is equivalent to a split one.","statement_latex":"Let $ p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.\nThere exists a contravariant functor $F : \\mathcal{C} \\to \\textit{Groupoids}$\nsuch that $\\mathcal{S}$ is equivalent to $\\mathcal{S}_F$ over $\\mathcal{C}$.\nIn other words, every category fibred in groupoids is equivalent to a split one.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Presheaves of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XY","source_file":"categories.tex","source_line":7894,"source_end_line":7900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7894-L7900","statement_sha256":"b4e87a27e1732aeeaece3ac0422efaf1241b3fdbadca9008aaef5d2c687b019e","origin":"The Stacks Project","memory_eligible":false,"source_rank":196,"rank":196,"depth":4,"x":333.682,"y":121.635,"cluster":"categories-foundations"},{"id":"stacks:02Y0","tag":"02Y0","title":"Categories fibred in sets · Definition 02Y0","summary":"A category is called discrete if the only morphisms are the identity morphisms.","statement_latex":"A category is called {\\it discrete} if the only morphisms are the identity\nmorphisms.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Y0","source_file":"categories.tex","source_line":7960,"source_end_line":7964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7960-L7964","statement_sha256":"e0d1b1d4a31bc2a292c390b4eff8f7f34cb401d64f5fa162086691c67c49e7e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":197,"rank":197,"depth":0,"x":232.656,"y":351.696,"cluster":"categories-foundations"},{"id":"stacks:0043","tag":"0043","title":"Categories fibred in sets · Definition 0043","summary":"Let C be a category. A category fibred in sets, or a category fibred in discrete categories is a category fibred in groupoids all of whose fibre categories are discrete.","statement_latex":"Let $\\mathcal{C}$ be a category.\nA {\\it category fibred in sets}, or a {\\it category fibred\nin discrete categories} is a category fibred in groupoids all\nof whose fibre categories are discrete.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0043","source_file":"categories.tex","source_line":7971,"source_end_line":7977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7971-L7977","statement_sha256":"b217b2f63871ed6bbb198ab5f8eedd05448589574ab074a0f9a2c7609f0db1c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":198,"rank":198,"depth":0,"x":121.865,"y":124.157,"cluster":"categories-foundations"},{"id":"stacks:04S8","tag":"04S8","title":"Categories fibred in sets · Definition 04S8","summary":"Let C be a category. The 2-category of categories fibred in sets over C is the sub 2-category of the category of categories fibred in groupoids over C (see Definition [Tag 02XS]) defined as follows: • Its objects will be categories p : S → C fibred in sets. • Its 1-morphisms (S, p) → (S', p') will be functors G : S → S' such that p' ∘ G = p (since every morphism is strongly cartesian G automatically preserves them). • Its 2-morphisms t : G → H for G, H : (S, p) → (S', p')…","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe {\\it $2$-category of categories fibred in sets over $\\mathcal{C}$}\nis the sub $2$-category of the category of categories fibred in groupoids\nover $\\mathcal{C}$ (see\nDefinition \\ref{definition-categories-fibred-in-groupoids-over-C})\ndefined as follows:\n\\begin{enumerate}\n\\item Its objects will be categories\n$p : \\mathcal{S} \\to \\mathcal{C}$ fibred in sets.\n\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that\n$p' \\circ G = p$ (since every morphism is strongly cartesian\n$G$ automatically preserves them).\n\\item Its $2$-morphisms $t : G \\to H$ for\n$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be morphisms of functors\nsuch that $p'(t_x) = \\text{id}_{p(x)}$\nfor all $x \\in \\Ob(\\mathcal{S})$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04S8","source_file":"categories.tex","source_line":7984,"source_end_line":8005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L7984-L8005","statement_sha256":"0ae8363cfed60ddd802c3e2be2ef300a4e5de3592f8f2cf8f139c1506026f29c","origin":"The Stacks Project","memory_eligible":false,"source_rank":199,"rank":199,"depth":3,"x":387.202,"y":229.338,"cluster":"categories-foundations"},{"id":"stacks:0047","tag":"0047","title":"Categories fibred in sets · Lemma 0047","summary":"Let C be a category. The 2-category of categories fibred in sets over C has 2-fibre products. More precisely, the 2-fibre product described in Lemma [Tag 0040] returns a category fibred in sets if one starts out with such.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe 2-category of categories fibred in sets over $\\mathcal{C}$\nhas 2-fibre products. More precisely, the 2-fibre product described in\nLemma \\ref{lemma-2-product-categories-over-C}\nreturns a category fibred in sets if one starts out with such.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0047","source_file":"categories.tex","source_line":8012,"source_end_line":8019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8012-L8019","statement_sha256":"0472d3c05bed586116f9eb8468a89960e92c7d5929d8d014fe50c3586efd0a71","origin":"The Stacks Project","memory_eligible":false,"source_rank":200,"rank":200,"depth":2,"x":106.266,"y":302.519,"cluster":"categories-foundations"},{"id":"stacks:02Y2","tag":"02Y2","title":"Categories fibred in sets · Lemma 02Y2","summary":"Categories fibred in sets are precisely presheaves. Let C be a category. The only 2-morphisms between categories fibred in sets are identities. In other words, the 2-category of categories fibred in sets is a category. Moreover, there is an equivalence of categories ( the category of presheaves of sets over C ) ↔ ( the category of categories fibred in sets over C ) The functor from left to right is the construction F → S_F discussed in Example [Tag 04TM]. The functor from…","statement_latex":"\\begin{slogan}\nCategories fibred in sets are precisely presheaves.\n\\end{slogan}\nLet $\\mathcal{C}$ be a category.\nThe only $2$-morphisms between categories fibred in sets are identities.\nIn other words, the $2$-category of categories fibred in sets is a category.\nMoreover, there is an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{the category of presheaves}\\\\\n\\text{of sets over }\\mathcal{C}\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{the category of categories}\\\\\n\\text{fibred in sets over }\\mathcal{C}\n\\end{matrix}\n\\right\\}\n$$\nThe functor from left to right is the construction\n$F \\to \\mathcal{S}_F$ discussed in\nExample \\ref{example-presheaf}.\nThe functor from right to left assigns to $p : \\mathcal{S} \\to \\mathcal{C}$\nthe presheaf of objects $U \\mapsto \\Ob(\\mathcal{S}_U)$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Y2","source_file":"categories.tex","source_line":8059,"source_end_line":8088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8059-L8088","statement_sha256":"655fcddcdbdc7c82f220c425fec81fa56d2257542dafdf5d9578c07344596837","origin":"The Stacks Project","memory_eligible":false,"source_rank":201,"rank":201,"depth":4,"x":254.942,"y":88.618,"cluster":"categories-foundations"},{"id":"stacks:02XZ","tag":"02XZ","title":"Categories fibred in setoids · Definition 02XZ","summary":"Let us call a category a setoid if it is a groupoid where every object has exactly one automorphism: the identity.","statement_latex":"Let us call a category a {\\it setoid}\\footnote{A set on steroids!?}\nif it is a groupoid where every object\nhas exactly one automorphism: the identity.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in setoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02XZ","source_file":"categories.tex","source_line":8149,"source_end_line":8154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8149-L8154","statement_sha256":"4bdf10db49879ac1780a9477d75bb8a869a522029a7f1d05d88c859dd3e3e25e","origin":"The Stacks Project","memory_eligible":false,"source_rank":202,"rank":202,"depth":0,"x":317.476,"y":331.304,"cluster":"categories-foundations"},{"id":"stacks:04SA","tag":"04SA","title":"Categories fibred in setoids · Definition 04SA","summary":"Let C be a category. A category fibred in setoids is a category fibred in groupoids all of whose fibre categories are setoids.","statement_latex":"Let $\\mathcal{C}$ be a category. A {\\it category fibred in setoids}\nis a category fibred in groupoids all of whose fibre categories are\nsetoids.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in setoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SA","source_file":"categories.tex","source_line":8174,"source_end_line":8179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8174-L8179","statement_sha256":"38d50e1d7c756475cf93a6efe1337b71a73a8e59ab754fc4c262d0b0334d93b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":203,"rank":203,"depth":0,"x":75.612,"y":187.483,"cluster":"categories-foundations"},{"id":"stacks:02Y1","tag":"02Y1","title":"Categories fibred in setoids · Definition 02Y1","summary":"Let C be a category. The 2-category of categories fibred in setoids over C is the sub 2-category of the category of categories fibred in groupoids over C (see Definition [Tag 02XS]) defined as follows: • Its objects will be categories p : S → C fibred in setoids. • Its 1-morphisms (S, p) → (S', p') will be functors G : S → S' such that p' ∘ G = p (since every morphism is strongly cartesian G automatically preserves them). • Its 2-morphisms t : G → H for G, H : (S, p) →…","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe {\\it $2$-category of categories fibred in setoids over $\\mathcal{C}$}\nis the sub $2$-category of the category of categories fibred in groupoids\nover $\\mathcal{C}$ (see\nDefinition \\ref{definition-categories-fibred-in-groupoids-over-C})\ndefined as follows:\n\\begin{enumerate}\n\\item Its objects will be categories\n$p : \\mathcal{S} \\to \\mathcal{C}$ fibred in setoids.\n\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that\n$p' \\circ G = p$ (since every morphism is strongly cartesian\n$G$ automatically preserves them).\n\\item Its $2$-morphisms $t : G \\to H$ for\n$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be morphisms of functors\nsuch that $p'(t_x) = \\text{id}_{p(x)}$\nfor all $x \\in \\Ob(\\mathcal{S})$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in setoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Y1","source_file":"categories.tex","source_line":8185,"source_end_line":8206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8185-L8206","statement_sha256":"ed350048bbde30ff7ca9b381bdcd5c4ebadf2d9a64d7cc1c03e2a0920359fb1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":204,"rank":204,"depth":3,"x":370.333,"y":156.22,"cluster":"categories-foundations"},{"id":"stacks:04SB","tag":"04SB","title":"Categories fibred in setoids · Lemma 04SB","summary":"Let C be a category. The 2-category of categories fibred in setoids over C has 2-fibre products. More precisely, the 2-fibre product described in Lemma [Tag 0040] returns a category fibred in setoids if one starts out with such.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe 2-category of categories fibred in setoids over $\\mathcal{C}$\nhas 2-fibre products. More precisely, the 2-fibre product described in\nLemma \\ref{lemma-2-product-categories-over-C} returns a category fibred in\nsetoids if one starts out with such.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SB","source_file":"categories.tex","source_line":8215,"source_end_line":8222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8215-L8222","statement_sha256":"dee52eb6e105bb99fa43d8b24535cb429d93d5d58382e0ab8e712f34a65ac62f","origin":"The Stacks Project","memory_eligible":false,"source_rank":205,"rank":205,"depth":2,"x":177.685,"y":346.966,"cluster":"categories-foundations"},{"id":"stacks:0045","tag":"0045","title":"Categories fibred in setoids · Lemma 0045","summary":"Let C be a category. Let S be a category over C. • If S → S' is an equivalence over C with S' fibred in sets over C, then • S is fibred in setoids over C, and • for each U ∈ Ob(C) the map Ob(S_U) → Ob(S'_U) identifies the target as the set of isomorphism classes of the source. • If p : S → C is a category fibred in setoids, then there exists a category fibred in sets p' : S' → C and an equivalence can : S → S' over C.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $\\mathcal{S}$ be a category\nover $\\mathcal{C}$.\n\\begin{enumerate}\n\\item If $\\mathcal{S} \\to \\mathcal{S}'$ is an equivalence\nover $\\mathcal{C}$ with $\\mathcal{S}'$ fibred in sets over $\\mathcal{C}$,\nthen\n\\begin{enumerate}\n\\item $\\mathcal{S}$ is fibred in setoids over $\\mathcal{C}$, and\n\\item for each $U \\in \\Ob(\\mathcal{C})$ the map\n$\\Ob(\\mathcal{S}_U) \\to \\Ob(\\mathcal{S}'_U)$\nidentifies the target as the set of isomorphism classes of the source.\n\\end{enumerate}\n\\item If $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in setoids,\nthen there exists a category fibred in sets\n$p' : \\mathcal{S}' \\to \\mathcal{C}$ and an equivalence\n$\\text{can} : \\mathcal{S} \\to \\mathcal{S}'$ over $\\mathcal{C}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0045","source_file":"categories.tex","source_line":8228,"source_end_line":8247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8228-L8247","statement_sha256":"be343ce9484faed85980a2f6c72ce7a449a47156f062f536ac479d931fa08aae","origin":"The Stacks Project","memory_eligible":false,"source_rank":206,"rank":206,"depth":0,"x":166.321,"y":96.395,"cluster":"categories-foundations"},{"id":"stacks:04SC","tag":"04SC","title":"Categories fibred in setoids · Lemma 04SC","summary":"Let C be a category. The construction of Lemma [Tag 0045] part (2) gives a functor F : ( the 2-category of categories fibred in setoids over C ) → ( the category of categories fibred in sets over C ) (see Definition [Tag 003N]). This functor is an equivalence in the following sense: • for any two 1-morphisms f, g : S_1 → S_2 with F(f) = F(g) there exists a unique 2-isomorphism f → g, • for any morphism h : F(S_1) → F(S_2) there exists a 1-morphism f : S_1 → S_2 with F(f)…","statement_latex":"Let $\\mathcal{C}$ be a category. The construction of\nLemma \\ref{lemma-setoid-fibres}\npart (2) gives a functor\n$$\nF :\n\\left\\{\n\\begin{matrix}\n\\text{the 2-category of categories}\\\\\n\\text{fibred in setoids over }\\mathcal{C}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{the category of categories}\\\\\n\\text{fibred in sets over }\\mathcal{C}\n\\end{matrix}\n\\right\\}\n$$\n(see\nDefinition \\ref{definition-functor-into-2-category}).\nThis functor is an equivalence in the following sense:\n\\begin{enumerate}\n\\item for any two 1-morphisms $f, g : \\mathcal{S}_1 \\to \\mathcal{S}_2$\nwith $F(f) = F(g)$ there exists a unique 2-isomorphism $f \\to g$,\n\\item for any morphism $h : F(\\mathcal{S}_1) \\to F(\\mathcal{S}_2)$\nthere exists a 1-morphism $f : \\mathcal{S}_1 \\to \\mathcal{S}_2$\nwith $F(f) = h$, and\n\\item any category fibred in sets $\\mathcal{S}$ is equal to $F(\\mathcal{S})$.\n\\end{enumerate}\nIn particular, defining $F_i \\in \\textit{PSh}(\\mathcal{C})$ by the\nrule $F_i(U) = \\Ob(\\mathcal{S}_{i, U})/\\cong$, we have\n$$\n\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{S}_1, \\mathcal{S}_2)\n\\Big/\n2\\text{-isomorphism}\n=\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(F_1, F_2)\n$$\nMore precisely, given any map $\\phi : F_1 \\to F_2$ there exists a\n$1$-morphism $f : \\mathcal{S}_1 \\to \\mathcal{S}_2$ which induces\n$\\phi$ on isomorphism classes of objects and\nwhich is unique up to unique $2$-isomorphism.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SC","source_file":"categories.tex","source_line":8271,"source_end_line":8316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8271-L8316","statement_sha256":"acaa250f2fc5dfe367ce819745e17173c0abf53715bce1925e525a96a0017db3","origin":"The Stacks Project","memory_eligible":false,"source_rank":207,"rank":207,"depth":5,"x":376.706,"y":275.143,"cluster":"categories-foundations"},{"id":"stacks:042J","tag":"042J","title":"Categories fibred in setoids · Lemma 042J","summary":"Let C be a category. Let p : S → C be a category fibred in groupoids. The following are equivalent: • p : S → C is a category fibred in setoids, and • the canonical 1-morphism I_S → S, see ([Tag 042H]), is an equivalence (of categories over C).","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a category fibred in setoids, and\n\\item the canonical $1$-morphism $\\mathcal{I}_\\mathcal{S} \\to \\mathcal{S}$,\nsee (\\ref{equation-inertia-structure-map}), is an equivalence (of categories\nover $\\mathcal{C}$).\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042J","source_file":"categories.tex","source_line":8345,"source_end_line":8356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8345-L8356","statement_sha256":"d78056abe945229f3debdbd4a0feed786ec01bed99f34e6a1c18b5847ef6fdf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":208,"rank":208,"depth":0,"x":77.113,"y":262.684,"cluster":"categories-foundations"},{"id":"stacks:04SD","tag":"04SD","title":"Categories fibred in setoids · Lemma 04SD","summary":"Let C be a category. The construction of Lemma [Tag 04SC] which associates to a category fibred in setoids a presheaf is compatible with products, in the sense that the presheaf associated to a 2-fibre product X ×_Y Z is the fibre product of the presheaves associated to X, Y, Z.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe construction of\nLemma \\ref{lemma-2-category-fibred-setoids}\nwhich associates to a category fibred in setoids a presheaf is\ncompatible with products, in the sense that the presheaf associated\nto a $2$-fibre product $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$\nis the fibre product of the presheaves associated to\n$\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories fibred in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SD","source_file":"categories.tex","source_line":8374,"source_end_line":8384,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8374-L8384","statement_sha256":"16ab4d7da292bf4a32172e0976dd40426790267a13ce3f84007cddf034f1bf19","origin":"The Stacks Project","memory_eligible":false,"source_rank":209,"rank":209,"depth":6,"x":308.597,"y":101.493,"cluster":"categories-foundations"},{"id":"stacks:0046","tag":"0046","title":"Representable categories fibred in groupoids · Definition 0046","summary":"Let C be a category. A category fibred in groupoids p : S → C is called representable if there exist an object X of C and an equivalence j : S → C/X (in the 2-category of categories fibred in groupoids over C).","statement_latex":"Let $\\mathcal{C}$ be a category.\nA category fibred in groupoids $p : \\mathcal{S} \\to \\mathcal{C}$ is\ncalled {\\it representable} if there exist an object\n$X$ of $\\mathcal{C}$ and an equivalence $j : \\mathcal{S} \\to \\mathcal{C}/X$\n(in the $2$-category of categories fibred in groupoids over $\\mathcal{C}$).","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Representable categories fibred in groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0046","source_file":"categories.tex","source_line":8415,"source_end_line":8422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8415-L8422","statement_sha256":"1cfccddde29f5c143ee95d30901db408ebb28042a981e1c9a8217665e5cc3ae6","origin":"The Stacks Project","memory_eligible":false,"source_rank":210,"rank":210,"depth":0,"x":267.432,"y":352.295,"cluster":"categories-foundations"},{"id":"stacks:02Y3","tag":"02Y3","title":"Representable categories fibred in groupoids · Lemma 02Y3","summary":"Let C be a category. Let p : S → C be a category fibred in groupoids. • S is representable if and only if the following conditions are satisfied: • S is fibred in setoids, and • the presheaf U ↦ Ob(S_U)/≅ is representable. • If S is representable the pair (X, j), where j is the equivalence j : S → C/X, is uniquely determined up to isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.\n\\begin{enumerate}\n\\item $\\mathcal{S}$ is representable if and only if\nthe following conditions are satisfied:\n\\begin{enumerate}\n\\item $\\mathcal{S}$ is fibred in setoids, and\n\\item the presheaf $U \\mapsto \\Ob(\\mathcal{S}_U)/\\cong$ is\nrepresentable.\n\\end{enumerate}\n\\item If $\\mathcal{S}$ is representable the pair $(X, j)$, where $j$ is the\nequivalence $j : \\mathcal{S} \\to \\mathcal{C}/X$, is uniquely determined\nup to isomorphism.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Representable categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Y3","source_file":"categories.tex","source_line":8428,"source_end_line":8444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8428-L8444","statement_sha256":"78a2b770636102d788fe9004aaf4bbe07a6c88ba93853e6e7f99c63c3d12fa6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":211,"rank":211,"depth":5,"x":95.695,"y":143.508,"cluster":"categories-foundations"},{"id":"stacks:04SF","tag":"04SF","title":"Representable categories fibred in groupoids · Lemma 04SF","summary":"Let C be a category. Let X, Y be categories fibred in groupoids over C. Assume that X, Y are representable by objects X, Y of C. Then Mor_Cat/C(X, Y) Big/ 2-isomorphism = Mor_C(X, Y) More precisely, given φ : X → Y there exists a 1-morphism f : X → Y which induces φ on isomorphism classes of objects and which is unique up to unique 2-isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids\nover $\\mathcal{C}$. Assume that $\\mathcal{X}$, $\\mathcal{Y}$\nare representable by objects $X$, $Y$ of $\\mathcal{C}$.\nThen\n$$\n\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{X}, \\mathcal{Y})\n\\Big/\n2\\text{-isomorphism}\n=\n\\Mor_\\mathcal{C}(X, Y)\n$$\nMore precisely, given $\\phi : X \\to Y$ there exists a\n$1$-morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ which induces\n$\\phi$ on isomorphism classes of objects and\nwhich is unique up to unique $2$-isomorphism.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Representable categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SF","source_file":"categories.tex","source_line":8461,"source_end_line":8479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8461-L8479","statement_sha256":"d618fb3036be50b2503ea80351ad126f0b926ca1c866da53b46f2326e4f6c244","origin":"The Stacks Project","memory_eligible":false,"source_rank":212,"rank":212,"depth":6,"x":390.923,"y":200.15,"cluster":"categories-foundations"},{"id":"stacks:0GWI","tag":"0GWI","title":"2-Yoneda lemma for fibred categories · Lemma 0GWI","summary":"Let C be a category. Let S → C be a fibred category over C. Let U ∈ Ob(C). The functor Mor_Fib/C(C/U, S) → S_U given by G ↦ G(id_U) is an equivalence.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{S} \\to \\mathcal{C}$ be a fibred category over $\\mathcal{C}$.\nLet $U \\in \\Ob(\\mathcal{C})$.\nThe functor\n$$\n\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{C}/U, \\mathcal{S})\n\\longrightarrow\n\\mathcal{S}_U\n$$\ngiven by $G \\mapsto G(\\text{id}_U)$ is an equivalence.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"The 2-Yoneda lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWI","source_file":"categories.tex","source_line":8525,"source_end_line":8537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8525-L8537","statement_sha256":"ea7c741ccb9997642a291db3a02ee3581c216841024c6d81078231a2b294b27e","origin":"The Stacks Project","memory_eligible":false,"source_rank":213,"rank":213,"depth":3,"x":127.062,"y":326.195,"cluster":"categories-foundations"},{"id":"stacks:004B","tag":"004B","title":"2-Yoneda lemma · Lemma 004B","summary":"Let S→ C be fibred in groupoids. Let U ∈ Ob(C). The functor Mor_Cat/C(C/U, S) → S_U given by G ↦ G(id_U) is an equivalence.","statement_latex":"Let $\\mathcal{S}\\to \\mathcal{C}$ be fibred in groupoids.\nLet $U \\in \\Ob(\\mathcal{C})$.\nThe functor\n$$\n\\Mor_{\\textit{Cat}/\\mathcal{C}}(\\mathcal{C}/U, \\mathcal{S})\n\\longrightarrow\n\\mathcal{S}_U\n$$\ngiven by $G \\mapsto G(\\text{id}_U)$ is an equivalence.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"The 2-Yoneda lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004B","source_file":"categories.tex","source_line":8581,"source_end_line":8592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8581-L8592","statement_sha256":"95110c6db98d03f65b4e7e26c2a2645e6ad61c279e203ae60f28479a64a371fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":214,"rank":214,"depth":4,"x":220.484,"y":82.967,"cluster":"categories-foundations"},{"id":"stacks:02Y5","tag":"02Y5","title":"Representable 1-morphisms · Lemma 02Y5","summary":"In the situation above the fibre category of (C/U) ×_Y X over an object f : V → U of C/U is the category described as follows: • objects are pairs (x, φ), where x ∈ Ob(X_V), and φ : f^*y → F(x) is a morphism in Y_V, • the set of morphisms between (x, φ) and (x', φ') is the set of morphisms ψ : x → x' in X_V such that F(ψ) = φ' ∘ φ^-1.","statement_latex":"In the situation above the fibre category of\n$(\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X}$ over\nan object $f : V \\to U$ of $\\mathcal{C}/U$\nis the category described as follows:\n\\begin{enumerate}\n\\item objects are pairs $(x, \\phi)$,\nwhere $x \\in \\Ob(\\mathcal{X}_V)$, and\n$\\phi : f^*y \\to F(x)$ is a morphism in $\\mathcal{Y}_V$,\n\\item the set of morphisms between $(x, \\phi)$ and $(x', \\phi')$\nis the set of morphisms $\\psi : x \\to x'$ in $\\mathcal{X}_V$\nsuch that $F(\\psi) = \\phi' \\circ \\phi^{-1}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Representable 1-morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Y5","source_file":"categories.tex","source_line":8693,"source_end_line":8707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8693-L8707","statement_sha256":"187511887482683e21c3160cd9c4413a455a2cc44d8ee1c45f282c3db4521a6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":215,"rank":215,"depth":0,"x":347.486,"y":315.867,"cluster":"categories-foundations"},{"id":"stacks:02Y6","tag":"02Y6","title":"Representable 1-morphisms · Lemma 02Y6","summary":"Let C be a category. Let X, Y be categories fibred in groupoids over C. Let F : X → Y be a 1-morphism. Let G : C/U → Y be a 1-morphism. Then (C/U) ×_Y X → C/U is a category fibred in groupoids.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids\nover $\\mathcal{C}$.\nLet $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism.\nLet $G : \\mathcal{C}/U \\to \\mathcal{Y}$ be a $1$-morphism.\nThen\n$$\n(\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X}\n\\longrightarrow\n\\mathcal{C}/U\n$$\nis a category fibred in groupoids.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Representable 1-morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Y6","source_file":"categories.tex","source_line":8713,"source_end_line":8727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8713-L8727","statement_sha256":"d7e889f8f6c5aaaa1c72f4a3dbebcec36a079da552a5eb94cdeacd05db62d9b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":216,"rank":216,"depth":5,"x":65.898,"y":215.964,"cluster":"categories-foundations"},{"id":"stacks:02Y7","tag":"02Y7","title":"Representable 1-morphisms · Definition 02Y7","summary":"Let C be a category. Let X, Y be categories fibred in groupoids over C. Let F : X → Y be a 1-morphism. We say F is representable, or that X is relatively representable over Y, if for every U ∈ Ob(C) and any G : C/U → Y the category fibred in groupoids (C/U) ×_Y X → C/U is representable.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids\nover $\\mathcal{C}$.\nLet $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism.\nWe say $F$ is {\\it representable}, or that\n{\\it $\\mathcal{X}$ is relatively representable over $\\mathcal{Y}$},\nif for every $U \\in \\Ob(\\mathcal{C})$\nand any $G : \\mathcal{C}/U \\to \\mathcal{Y}$\nthe category fibred in groupoids\n$$\n(\\mathcal{C}/U) \\times_\\mathcal{Y} \\mathcal{X}\n\\longrightarrow\n\\mathcal{C}/U\n$$\nis representable.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Representable 1-morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Y7","source_file":"categories.tex","source_line":8743,"source_end_line":8760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8743-L8760","statement_sha256":"8b5b378a2e1dd3e118d8e0e84c486e547e5bb2f5bcb4aa3576103e1889c858f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":217,"rank":217,"depth":0,"x":354.536,"y":129.655,"cluster":"categories-foundations"},{"id":"stacks:02Y8","tag":"02Y8","title":"Representable 1-morphisms · Lemma 02Y8","summary":"Let C be a category. Let X, Y be categories fibred in groupoids over C. Let F : X → Y be a 1-morphism. If F is representable then every one of the functors F_U : X_U → Y_U between fibre categories is faithful.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids\nover $\\mathcal{C}$.\nLet $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism.\nIf $F$ is representable then every one of the functors\n$$\nF_U : \\mathcal{X}_U \\longrightarrow \\mathcal{Y}_U\n$$\nbetween fibre categories is faithful.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Representable 1-morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Y8","source_file":"categories.tex","source_line":8762,"source_end_line":8773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8762-L8773","statement_sha256":"a271a8f3f5356e1c0e10086b93660bd631536a252579853138a4028301e953c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":218,"rank":218,"depth":6,"x":210.779,"y":357.596,"cluster":"categories-foundations"},{"id":"stacks:02Y9","tag":"02Y9","title":"Representable 1-morphisms · Lemma 02Y9","summary":"Let C be a category. Let X, Y be categories fibred in groupoids over C. Let F : X → Y be a 1-morphism. Make a choice of pullbacks for Y. Assume • each functor F_U : X_U → Y_U between fibre categories is faithful, and • for each U and each y ∈ Y_U the presheaf (f : V → U) ↦ ((x, φ) mid x ∈ X_V, φ : f^*y → F(x))/≅ is a representable presheaf on C/U. Then F is representable.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids\nover $\\mathcal{C}$.\nLet $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism.\nMake a choice of pullbacks for $\\mathcal{Y}$.\nAssume\n\\begin{enumerate}\n\\item each functor $F_U : \\mathcal{X}_U \\longrightarrow \\mathcal{Y}_U$\nbetween fibre categories is faithful, and\n\\item for each $U$ and each $y \\in \\mathcal{Y}_U$ the presheaf\n$$\n(f : V \\to U)\n\\longmapsto\n\\{(x, \\phi) \\mid x \\in \\mathcal{X}_V, \\phi : f^*y \\to F(x)\\}/\\cong\n$$\nis a representable presheaf on $\\mathcal{C}/U$.\n\\end{enumerate}\nThen $F$ is representable.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Representable 1-morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Y9","source_file":"categories.tex","source_line":8782,"source_end_line":8802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8782-L8802","statement_sha256":"a52cd292ded73b715b43cbc95fdcf2ccdcf3dc8a0d46cd30cd57fe7b1875e700","origin":"The Stacks Project","memory_eligible":false,"source_rank":219,"rank":219,"depth":6,"x":133.304,"y":107.378,"cluster":"categories-foundations"},{"id":"stacks:02YA","tag":"02YA","title":"Representable 1-morphisms · Lemma 02YA","summary":"Let C be a category. Let S → C be a category fibred in groupoids. Assume C has products of pairs of objects and fibre products. The following are equivalent: • The diagonal S → S × S is representable. • For every U in C, any G : C/U → S is representable.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids.\nAssume $\\mathcal{C}$ has products of pairs of objects and fibre products.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The diagonal $\\mathcal{S} \\to \\mathcal{S} \\times \\mathcal{S}$\nis representable.\n\\item For every $U$ in $\\mathcal{C}$, any $G : \\mathcal{C}/U \\to \\mathcal{S}$\nis representable.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Representable 1-morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YA","source_file":"categories.tex","source_line":8825,"source_end_line":8837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L8825-L8837","statement_sha256":"81d129ed879d57d37b473b4ba5627cea8199e783126ac1f55b56d154addd2cfc","origin":"The Stacks Project","memory_eligible":false,"source_rank":220,"rank":220,"depth":1,"x":392.235,"y":248.242,"cluster":"categories-foundations"},{"id":"stacks:0HAW","tag":"0HAW","title":"Monoidal categories · Lemma 0HAW","summary":"Let (C, ⊗, φ) be as above. There is a 1-to-1 correspondence between units (1, l, r) in C and pairs (1, 1) where 1 is an object of C and 1 : 1 ⊗ 1 → 1 is an isomorphism such that the functors L : X ↦ 1 ⊗ X and R : X ↦ X ⊗ 1 are equivalences.","statement_latex":"Let $(\\mathcal{C}, \\otimes, \\phi)$ be as above. There is a 1-to-1\ncorrespondence between units $(\\mathbf{1}, l, r)$ in $\\mathcal{C}$ and pairs\n$(\\mathbf{1}, 1)$ where $\\mathbf{1}$ is an object of $\\mathcal{C}$ and\n$1 : \\mathbf{1} \\otimes \\mathbf{1} \\to \\mathbf{1}$\nis an isomorphism such that the functors $L : X \\mapsto \\mathbf{1} \\otimes X$\nand $R : X \\mapsto X \\otimes \\mathbf{1}$ are equivalences.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAW","source_file":"categories.tex","source_line":9013,"source_end_line":9021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9013-L9021","statement_sha256":"0b4616437edd2909f0011871da7ce65b856b62a1c0daacda200e240d2ff7e85b","origin":"The Stacks Project","memory_eligible":false,"source_rank":221,"rank":221,"depth":0,"x":87.34,"y":291.391,"cluster":"categories-foundations"},{"id":"stacks:0HAX","tag":"0HAX","title":"Monoidal categories · Lemma 0HAX","summary":"Let (C, ⊗, φ) be as above. Let (1, 1) be a unit (see Lemma [Tag 0HAW]). Then • 1 ⊗ id_1 = id_1 ⊗ 1 • Γ = Mor(1, 1) is a commutative monoid, • a = 1 ∘ (a ⊗ id_1) ∘ 1^-1 = 1 ∘ (id_1 ⊗ a) ∘ 1^-1 for all a ∈ Γ, • any other unit is isomorphic to (1, 1) by a unique isomorphism.","statement_latex":"Let $(\\mathcal{C}, \\otimes, \\phi)$ be as above.\nLet $(\\mathbf{1}, 1)$ be a unit (see Lemma \\ref{lemma-monoidal-unit}).\nThen\n\\begin{enumerate}\n\\item $1 \\otimes \\text{id}_\\mathbf{1} = \\text{id}_\\mathbf{1} \\otimes 1$\n\\item $\\Gamma = \\text{Mor}(\\mathbf{1}, \\mathbf{1})$ is a commutative monoid,\n\\item  $a = 1 \\circ (a \\otimes \\text{id}_\\mathbf{1}) \\circ 1^{-1} =\n1 \\circ (\\text{id}_\\mathbf{1} \\otimes a) \\circ 1^{-1}$ for all $a \\in \\Gamma$,\n\\item any other unit is isomorphic to $(\\mathbf{1}, 1)$\nby a unique isomorphism.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAX","source_file":"categories.tex","source_line":9103,"source_end_line":9116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9103-L9116","statement_sha256":"aa81870d514187cdc61f6bd7f7d084da5f5c6e4c138d639af5cd300cafe7c264","origin":"The Stacks Project","memory_eligible":false,"source_rank":222,"rank":222,"depth":1,"x":277.892,"y":86.111,"cluster":"categories-foundations"},{"id":"stacks:0FFK","tag":"0FFK","title":"Monoidal categories · Definition 0FFK","summary":"A triple (C, ⊗, φ) where C is a category, ⊗ : C × C → C is a functor, and φ is an associativity constraint is called a monoidal category if there exists a unit 1.","statement_latex":"A triple $(\\mathcal{C}, \\otimes, \\phi)$ where $\\mathcal{C}$ is a category,\n$\\otimes : \\mathcal{C} \\times \\mathcal{C} \\to \\mathcal{C}$ is a functor,\nand $\\phi$ is an associativity constraint is called a {\\it monoidal category}\nif there exists a unit $\\mathbf{1}$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFK","source_file":"categories.tex","source_line":9169,"source_end_line":9175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9169-L9175","statement_sha256":"0c42a6785ab137671ff2585f7b73f7af3e3262772c1e233f501d2c1d2feec964","origin":"The Stacks Project","memory_eligible":false,"source_rank":223,"rank":223,"depth":0,"x":302.516,"y":346.183,"cluster":"categories-foundations"},{"id":"stacks:0HAY","tag":"0HAY","title":"Monoidal categories · Lemma 0HAY","summary":"In a monoidal category C, ⊗, φ, 1, 1 and with notation as in the proof of Lemma [Tag 0HAW] we have • the arrows 1, r_1, l_1 : 1 ⊗ 1 → 1 agree, • the arrows l_X ⊗ id_Y, l_X ⊗ Y : 1 ⊗ X ⊗ Y → X ⊗ Y agree, and • the arrows id_X ⊗ r_Y , r_X ⊗ Y : X ⊗ Y ⊗ 1 → X ⊗ Y agree. A monoidal category satisfies the assumptions of [associativity].","statement_latex":"In a monoidal category $\\mathcal{C}, \\otimes, \\phi, \\mathbf{1}, 1$\nand with notation as in the proof of Lemma \\ref{lemma-monoidal-unit}\nwe have\n\\begin{enumerate}\n\\item the arrows $1, r_\\mathbf{1}, l_\\mathbf{1} :\n\\mathbf{1} \\otimes \\mathbf{1} \\to \\mathbf{1}$ agree,\n\\item the arrows\n$l_X \\otimes \\text{id}_Y, l_{X \\otimes Y} :\n\\mathbf{1} \\otimes X \\otimes Y \\to X \\otimes Y$ agree, and\n\\item the arrows\n$\\text{id}_X \\otimes r_Y , r_{X \\otimes Y} :\nX \\otimes Y \\otimes \\mathbf{1} \\to X \\otimes Y$ agree.\n\\end{enumerate}\nA monoidal category satisfies the assumptions of\n\\cite[Theorem 5.2]{associativity}.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAY","source_file":"categories.tex","source_line":9184,"source_end_line":9201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9184-L9201","statement_sha256":"0d1f2ebc89f1e73c1fa8012f10c97d5071558deb6b8d3acb286e79c3d4391db7","origin":"The Stacks Project","memory_eligible":false,"source_rank":224,"rank":224,"depth":1,"x":74.712,"y":167.988,"cluster":"categories-foundations"},{"id":"stacks:0FFL","tag":"0FFL","title":"Monoidal categories · Definition 0FFL","summary":"Let C and C' be monoidal categories. A functor of monoidal categories F : C → C' is given by a functor F as indicated and an isomorphism F(X) ⊗ F(Y) → F(X ⊗ Y) functorial in X and Y such that for all objects X, Y, and Z the diagram xymatrix F(X) ⊗ (F(Y) ⊗ F(Z)) ar[r] ar[d] & F(X) ⊗ F(Y ⊗ Z) ar[r] & F(X ⊗ (Y ⊗ Z)) ar[d] (F(X) ⊗ F(Y)) ⊗ F(Z) ar[r] & F(X ⊗ Y) ⊗ F(Z) ar[r] & F((X ⊗ Y) ⊗ Z) commutes and such that F(1) is a unit in C'.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{C}'$ be monoidal categories.\nA {\\it functor of monoidal categories} $F : \\mathcal{C} \\to \\mathcal{C}'$\nis given by a functor $F$ as indicated and an isomorphism\n$$\nF(X) \\otimes F(Y) \\to F(X \\otimes Y)\n$$\nfunctorial in $X$ and $Y$\nsuch that for all objects $X$, $Y$, and $Z$ the diagram\n$$\n\\xymatrix{\nF(X) \\otimes (F(Y) \\otimes F(Z)) \\ar[r] \\ar[d] &\nF(X) \\otimes F(Y \\otimes Z) \\ar[r] &\nF(X \\otimes (Y \\otimes Z)) \\ar[d] \\\\\n(F(X) \\otimes F(Y)) \\otimes F(Z) \\ar[r] &\nF(X \\otimes Y) \\otimes F(Z) \\ar[r] &\nF((X \\otimes Y) \\otimes Z)\n}\n$$\ncommutes and such that $F(\\mathbf{1})$ is a unit in $\\mathcal{C}'$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFL","source_file":"categories.tex","source_line":9237,"source_end_line":9258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9237-L9258","statement_sha256":"4cd068545899e04806ef24ad18e02dd6f90df91c099fdbbe253ffbc59d65898c","origin":"The Stacks Project","memory_eligible":false,"source_rank":225,"rank":225,"depth":0,"x":386.678,"y":170.129,"cluster":"categories-foundations"},{"id":"stacks:0FFM","tag":"0FFM","title":"Monoidal categories · Lemma 0FFM","summary":"Let C be a monoidal category. Let X be an object of C. The following are equivalent • the functor L : Y ↦ X ⊗ Y is an equivalence, • the functor R : Y ↦ Y ⊗ X is an equivalence, • there exists an object X' such that X ⊗ X' ≅ X' ⊗ X ≅ 1.","statement_latex":"Let $\\mathcal{C}$ be a monoidal category. Let $X$ be an object of\n$\\mathcal{C}$. The following are equivalent\n\\begin{enumerate}\n\\item the functor $L : Y \\mapsto X \\otimes Y$ is an equivalence,\n\\item the functor $R : Y \\mapsto Y \\otimes X$ is an equivalence,\n\\item there exists an object $X'$ such that\n$X \\otimes X' \\cong X' \\otimes X \\cong \\mathbf{1}$.\n\\end{enumerate}","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFM","source_file":"categories.tex","source_line":9267,"source_end_line":9277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9267-L9277","statement_sha256":"9596a8eb0a36320a86fdbd2ba10e23041a4642f007c97aa59fe1e8d4a564b5a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":226,"rank":226,"depth":0,"x":154.407,"y":345.953,"cluster":"categories-foundations"},{"id":"stacks:0FFN","tag":"0FFN","title":"Monoidal categories · Definition 0FFN","summary":"Let C be a monoidal category. An object X of C is called invertible if any (or all) of the equivalent conditions of Lemma [Tag 0FFM] hold.","statement_latex":"Let $\\mathcal{C}$ be a monoidal category. An object $X$ of $\\mathcal{C}$\nis called {\\it invertible} if any (or all) of the equivalent conditions of\nLemma \\ref{lemma-invertible} hold.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFN","source_file":"categories.tex","source_line":9296,"source_end_line":9301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9296-L9301","statement_sha256":"14f5f80ff6bd467d5412026b60ad8759305e7f5d2a1a83252779ad008f3d860c","origin":"The Stacks Project","memory_eligible":false,"source_rank":227,"rank":227,"depth":1,"x":184.354,"y":83.935,"cluster":"categories-foundations"},{"id":"stacks:0FFP","tag":"0FFP","title":"Monoidal categories · Definition 0FFP","summary":"Given a monoidal category (C, ⊗, φ) and an object X a left dual is an object Y together with morphisms eta : 1 → X ⊗ Y and ε : Y ⊗ X → 1 such that the diagrams vcenter xymatrix X ar[rd]_1 ar[r]_-eta ⊗ 1 & X ⊗ Y ⊗ X ar[d]^1 ⊗ ε & X and vcenter xymatrix Y ar[rd]_1 ar[r]_-1 ⊗ eta & Y ⊗ X ⊗ Y ar[d]^ε ⊗ 1 & Y commute. In this situation we say that X is a right dual of Y.","statement_latex":"Given a monoidal category $(\\mathcal{C}, \\otimes, \\phi)$\nand an object $X$ a {\\it left dual} is an object $Y$ together with\nmorphisms $\\eta : \\mathbf{1} \\to X \\otimes Y$ and\n$\\epsilon : Y \\otimes X \\to \\mathbf{1}$\nsuch that the diagrams\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[rd]_1 \\ar[r]_-{\\eta \\otimes 1} &\nX \\otimes Y \\otimes X \\ar[d]^{1 \\otimes \\epsilon} \\\\\n& X\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nY \\ar[rd]_1 \\ar[r]_-{1 \\otimes \\eta} &\nY \\otimes X \\otimes Y \\ar[d]^{\\epsilon \\otimes 1} \\\\\n& Y\n}\n}\n$$\ncommute. In this situation we say that $X$ is a {\\it right dual} of $Y$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFP","source_file":"categories.tex","source_line":9308,"source_end_line":9333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9308-L9333","statement_sha256":"4a8e7606e6080337ef73dde2657bf9c26799b3f4772e9088397b445cda5d62a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":228,"rank":228,"depth":0,"x":373.39,"y":294.594,"cluster":"categories-foundations"},{"id":"stacks:0FFQ","tag":"0FFQ","title":"Monoidal categories · Lemma 0FFQ","summary":"Let C be a monoidal category. If Y is a left dual to X, then Mor(Z' ⊗ X, Z) = Mor(Z', Z ⊗ Y) and Mor(Y ⊗ Z', Z) = Mor(Z', X ⊗ Z) functorially in Z and Z'.","statement_latex":"Let $\\mathcal{C}$ be a monoidal category. If $Y$ is a left dual to $X$,\nthen\n$$\n\\Mor(Z' \\otimes X, Z) = \\Mor(Z', Z \\otimes Y)\n\\quad\\text{and}\\quad\n\\Mor(Y \\otimes Z', Z) = \\Mor(Z', X \\otimes Z)\n$$\nfunctorially in $Z$ and $Z'$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFQ","source_file":"categories.tex","source_line":9340,"source_end_line":9350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9340-L9350","statement_sha256":"3f1230f567dc7f83dd6b3d99d298d2aff195cd883de13c418f0d493054efcb8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":229,"rank":229,"depth":0,"x":63.922,"y":246.415,"cluster":"categories-foundations"},{"id":"stacks:0FFS","tag":"0FFS","title":"Monoidal categories · Lemma 0FFS","summary":"Let C be a monoidal category. If Y_i, i = 1, 2 are left duals of X_i, i = 1, 2, then Y_2 ⊗ Y_1 is a left dual of X_1 ⊗ X_2.","statement_latex":"Let $\\mathcal{C}$ be a monoidal category. If $Y_i$, $i = 1, 2$\nare left duals of $X_i$, $i = 1, 2$, then $Y_2 \\otimes Y_1$ is\na left dual of $X_1 \\otimes X_2$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFS","source_file":"categories.tex","source_line":9440,"source_end_line":9445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9440-L9445","statement_sha256":"a5261f56fe10f465dc05072df3b856d372046d805a9d201e179674c2e88938af","origin":"The Stacks Project","memory_eligible":false,"source_rank":230,"rank":230,"depth":0,"x":331.439,"y":106.04,"cluster":"categories-foundations"},{"id":"stacks:0FFW","tag":"0FFW","title":"Monoidal categories · Definition 0FFW","summary":"A quadruple (C, ⊗, φ, ψ) where C is a category, ⊗ : C ⊗ C → C is a functor, φ is an associativity constraint, and ψ is a commutativity constraint compatible with φ is called a symmetric monoidal category if there exists a unit.","statement_latex":"A quadruple $(\\mathcal{C}, \\otimes, \\phi, \\psi)$ where\n$\\mathcal{C}$ is a category,\n$\\otimes : \\mathcal{C} \\otimes \\mathcal{C} \\to \\mathcal{C}$ is a functor,\n$\\phi$ is an associativity constraint, and\n$\\psi$ is a commutativity constraint compatible with $\\phi$\nis called a {\\it symmetric monoidal category} if there exists\na unit.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFW","source_file":"categories.tex","source_line":9478,"source_end_line":9487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9478-L9487","statement_sha256":"88482df298a8255ab8a34b1047596b9c6a65f2a1fcaf83e0cbad03e2db1c1d24","origin":"The Stacks Project","memory_eligible":false,"source_rank":231,"rank":231,"depth":0,"x":246.88,"y":361.895,"cluster":"categories-foundations"},{"id":"stacks:0HB0","tag":"0HB0","title":"Monoidal categories · Lemma 0HB0","summary":"In a symmetric monoidal category C, ⊗, φ, ψ, 1, 1 we have • the arrows 1 ∘ ψ, 1 : 1 ⊗ 1 → 1 agree, • the arrows id_X ⊗ l_Y, (l_X ⊗ id) ∘ (ψ ⊗ id_Y): X ⊗ 1 ⊗ Y → X ⊗ Y agree, A symmetric monoidal category satisfies the assumptions of [associativity].","statement_latex":"In a symmetric monoidal category\n$\\mathcal{C}, \\otimes, \\phi, \\psi, \\mathbf{1}, 1$\nwe have\n\\begin{enumerate}\n\\item the arrows $1 \\circ \\psi, 1 :\n\\mathbf{1} \\otimes \\mathbf{1} \\to \\mathbf{1}$ agree,\n\\item the arrows $\\text{id}_X \\otimes l_Y,\n(l_X \\otimes \\text{id}) \\circ (\\psi \\otimes \\text{id}_Y):\nX \\otimes \\mathbf{1} \\otimes Y \\to X \\otimes Y$ agree,\n\\end{enumerate}\nA symmetric monoidal category satisfies the assumptions of\n\\cite[Theorem 5.1]{associativity}.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HB0","source_file":"categories.tex","source_line":9495,"source_end_line":9509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9495-L9509","statement_sha256":"be24419fd0e12827fd4046eebb0747709a70996d775b7cdaeed89b0e9a9b51b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":232,"rank":232,"depth":2,"x":103.174,"y":124.744,"cluster":"categories-foundations"},{"id":"stacks:0FN8","tag":"0FN8","title":"Monoidal categories · Lemma 0FN8","summary":"Let (C, ⊗, φ, ψ) be a symmetric monoidal category. Let X be an object of C and let Y, eta : 1 → X ⊗ Y, and ε : Y ⊗ X → 1 be a left dual of X as in Definition [Tag 0FFP]. Then eta' = ψ ∘ eta : 1 → Y ⊗ X and ε' = ε ∘ ψ : X ⊗ Y → 1 makes X into a left dual of Y.","statement_latex":"Let $(\\mathcal{C}, \\otimes, \\phi, \\psi)$ be a symmetric monoidal category.\nLet $X$ be an object of $\\mathcal{C}$ and let $Y$,\n$\\eta : \\mathbf{1} \\to X \\otimes Y$, and\n$\\epsilon : Y \\otimes X \\to \\mathbf{1}$\nbe a left dual of $X$ as in Definition \\ref{definition-dual}.\nThen $\\eta' = \\psi \\circ \\eta : \\mathbf{1} \\to Y \\otimes X$\nand $\\epsilon' = \\epsilon \\circ \\psi : X \\otimes Y \\to \\mathbf{1}$\nmakes $X$ into a left dual of $Y$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FN8","source_file":"categories.tex","source_line":9576,"source_end_line":9586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9576-L9586","statement_sha256":"87967dbd6b292dbd560ba7f75b103e8a86551492b8e1174e28724b2481fe96ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":233,"rank":233,"depth":1,"x":400.484,"y":218.273,"cluster":"categories-foundations"},{"id":"stacks:0FFY","tag":"0FFY","title":"Monoidal categories · Definition 0FFY","summary":"Let C and C' be symmetric monoidal categories. A functor of symmetric monoidal categories F : C → C' is given by a functor F as indicated and an isomorphism F(X) ⊗ F(Y) → F(X ⊗ Y) functorial in X and Y such that F is a functor of monoidal categories and such that for all objects X and Y the diagram xymatrix F(X) ⊗ F(Y) ar[r] ar[d] & F(X ⊗ Y) ar[d] F(Y) ⊗ F(X) ar[r] & F(Y ⊗ X) commutes.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{C}'$ be symmetric monoidal categories.\nA {\\it functor of symmetric monoidal categories}\n$F : \\mathcal{C} \\to \\mathcal{C}'$\nis given by a functor $F$ as indicated and an isomorphism\n$$\nF(X) \\otimes F(Y) \\to F(X \\otimes Y)\n$$\nfunctorial in $X$ and $Y$\nsuch that $F$ is a functor of monoidal categories and such that\nfor all objects $X$ and $Y$ the diagram\n$$\n\\xymatrix{\nF(X) \\otimes F(Y) \\ar[r] \\ar[d] &\nF(X \\otimes Y) \\ar[d] \\\\\nF(Y) \\otimes F(X) \\ar[r] &\nF(Y \\otimes X)\n}\n$$\ncommutes.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Monoidal categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFY","source_file":"categories.tex","source_line":9592,"source_end_line":9613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9592-L9613","statement_sha256":"7d06c37867e5980c4cec717ecd7191f9e58f4d9f052781c041e310c3ef49d94c","origin":"The Stacks Project","memory_eligible":false,"source_rank":234,"rank":234,"depth":0,"x":105.414,"y":318.218,"cluster":"categories-foundations"},{"id":"stacks:0H18","tag":"0H18","title":"Categories of dotted arrows · Definition 0H18","summary":"Let C be a (2,1)-category. Consider a 2-commutative solid diagram vcenter xymatrix S ar[r]_-x ar[d]_j & X ar[d]^f T ar[r]^-y ar@..>[ru] & Y in C. Fix a 2-isomorphism γ : y ∘ j → f ∘ x witnessing the 2-commutativity of the diagram. Given ([Tag 0H19]) and γ, a emphdotted arrow is a triple (a, α, β) consisting of a morphism a colon T → X and 2-isomorphisms α : a ∘ j → x, β : y → f ∘ a such that γ = (id_f star α) ∘ (β star id_j), in other words such that xymatrix & f ∘ a ∘ j…","statement_latex":"Let $\\mathcal{C}$ be a $(2,1)$-category. Consider a $2$-commutative \nsolid diagram\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\nS \\ar[r]_-x \\ar[d]_j & X \\ar[d]^f \\\\\nT \\ar[r]^-y \\ar@{..>}[ru] & Y\n}\n}\n\\end{equation}\nin $\\mathcal{C}$. Fix a $2$-isomorphism \n$$\n\\gamma : y \\circ j \\rightarrow f \\circ x\n$$\nwitnessing the $2$-commutativity of the diagram.\nGiven (\\ref{equation-dotted-arrows}) and $\\gamma$, a \\emph{dotted arrow} \nis a triple $(a, \\alpha, \\beta)$ consisting of a morphism \n$a \\colon T \\to X$ and $2$-isomorphisms\n$\\alpha : a \\circ j \\to x$, $\\beta : y \\to f \\circ a$\nsuch that\n$\\gamma = (\\text{id}_f \\star \\alpha) \\circ (\\beta \\star \\text{id}_j)$,\nin other words such that\n$$\n\\xymatrix{\n& f \\circ a \\circ j \\ar[rd]^{\\text{id}_f \\star \\alpha} \\\\\ny \\circ j \\ar[ru]^{\\beta \\star \\text{id}_j} \\ar[rr]^\\gamma & &\nf \\circ x\n}\n$$\nis commutative. A {\\it morphism of dotted arrows}\n$(a, \\alpha, \\beta) \\to (a', \\alpha', \\beta')$ is a\n$2$-arrow $\\theta : a \\to a'$ such that\n$\\alpha = \\alpha' \\circ (\\theta \\star \\text{id}_j)$ and\n$\\beta' = (\\text{id}_f \\star \\theta) \\circ \\beta$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories of dotted arrows","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H18","source_file":"categories.tex","source_line":9681,"source_end_line":9718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9681-L9718","statement_sha256":"750e7e1184adae45914e255d39e4a2f15a89b64a1c6f273e4b71b5ae4232d3ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":235,"rank":235,"depth":0,"x":242.911,"y":76.581,"cluster":"categories-foundations"},{"id":"stacks:0H1A","tag":"0H1A","title":"Categories of dotted arrows · Lemma 0H1A","summary":"Let C be a (2,1)-category. Assume given a 2-commutative diagram xymatrix S ar[r]_-x' ar[d]_j & X' ar[d]^p ar[r]_q & X ar[d]^f T ar[r]^-y' & Y' ar[r]^g & Y in C, where the right square is 2-cartesian with respect to a 2-isomorphism φ colon g ∘ p → f ∘ q. Choose a 2-arrow γ' : y' ∘ j → p ∘ x'. Set x = q ∘ x', y = g ∘ y' and let γ : y ∘ j → f ∘ x be the 2-isomorphism γ = (φ star id_x') ∘ (id_g star γ'). Then the category D' of dotted arrows for the left square and γ' is…","statement_latex":"Let $\\mathcal{C}$ be a $(2,1)$-category. Assume given a $2$-commutative diagram\n$$\n\\xymatrix{\nS \\ar[r]_-{x'} \\ar[d]_j &\nX' \\ar[d]^p \\ar[r]_q &\nX \\ar[d]^f \\\\\nT \\ar[r]^-{y'} &\nY' \\ar[r]^g &\nY\n}\n$$\nin $\\mathcal{C}$, where the right square is $2$-cartesian \nwith respect to a $2$-isomorphism $\\phi \\colon g \\circ p \\to f \\circ q$. \nChoose a $2$-arrow\n$\\gamma' : y' \\circ j \\to p \\circ x'$. Set\n$x = q \\circ x'$, $y = g \\circ y'$ and let\n$\\gamma : y \\circ j \\to f \\circ x$ be the $2$-isomorphism \n$\\gamma = (\\phi \\star \\text{id}_{x'}) \\circ (\\text{id}_g \\star \\gamma')$. \nThen the category $\\mathcal{D}'$ of dotted arrows\nfor the left square and $\\gamma'$ is equivalent to the category \n$\\mathcal{D}$ of dotted\narrows for the outer rectangle and $\\gamma$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories of dotted arrows","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1A","source_file":"categories.tex","source_line":9727,"source_end_line":9751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9727-L9751","statement_sha256":"11241f74e61847322f36b0f15451cdc7f8818a6ac0226eb4f4e860722f6cd730","origin":"The Stacks Project","memory_eligible":false,"source_rank":236,"rank":236,"depth":0,"x":336.035,"y":333.319,"cluster":"categories-foundations"},{"id":"stacks:0H1B","tag":"0H1B","title":"Categories of dotted arrows · Lemma 0H1B","summary":"Let C be a (2,1)-category. Assume given a solid 2-commutative diagram xymatrix S ar[r]_-x ar[dd]_j & X ar[d]^f & Y ar[d]^g T ar[r]^-z ar@..>[ruu] & Z in C. Choose a 2-isomorphism γ colon z ∘ j → g ∘ f ∘ x. Let D be the category of dotted arrows for the outer rectangle and γ. Let D' be the category of dotted arrows for the solid square xymatrix S ar[r]_-f ∘ x ar[d]_j & Y ar[d]^g T ar[r]^-z ar@..>[ru] & Z and γ. Then D is equivalent to a category D\" which has the following…","statement_latex":"Let $\\mathcal{C}$ be a $(2,1)$-category. Assume given a solid $2$-commutative\ndiagram\n$$\n\\xymatrix{\nS \\ar[r]_-x \\ar[dd]_j & X \\ar[d]^f \\\\\n& Y \\ar[d]^g \\\\\nT \\ar[r]^-z \\ar@{..>}[ruu] & Z\n}\n$$\nin $\\mathcal{C}$. \nChoose a $2$-isomorphism $\\gamma \\colon z \\circ j \\to g \\circ f \\circ x$. \nLet $\\mathcal{D}$ be the category of dotted arrows for \nthe outer rectangle and $\\gamma$. Let $\\mathcal{D}'$ be \nthe category of dotted arrows for the solid square\n$$\n\\xymatrix{\nS \\ar[r]_-{f \\circ x} \\ar[d]_j & Y \\ar[d]^g \\\\\nT \\ar[r]^-z \\ar@{..>}[ru] & Z\n}\n$$\nand $\\gamma$. Then $\\mathcal{D}$ is equivalent to \na category $\\mathcal{D}''$ which has the following property: \nthere is a functor $\\mathcal{D}'' \\to \\mathcal{D}'$ which turns $\\mathcal{D}''$\ninto a category fibred in groupoids over $\\mathcal{D}'$ and whose fibre\ncategories are isomorphic to categories of dotted arrows for certain\nsolid squares of the form\n$$\n\\xymatrix{\nS \\ar[r]_-x \\ar[d]_j & X \\ar[d]^f \\\\\nT \\ar[r]^-y \\ar@{..>}[ru] & Y\n}\n$$\nand some choices of $2$-isomorphism $y \\circ j \\to f \\circ x$.","area":"Categories & Foundations","chapter":"Categories","chapter_id":"categories","section":"Categories of dotted arrows","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1B","source_file":"categories.tex","source_line":9763,"source_end_line":9798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/categories.tex#L9763-L9798","statement_sha256":"c3fb791e4a05ccd9f24ff08b65a21cd7a05abfdc20732ba448fe8c58184805ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":237,"rank":237,"depth":0,"x":60.33,"y":196.559,"cluster":"categories-foundations"},{"id":"stacks:08ZE","tag":"08ZE","title":"Hausdorff spaces · Lemma 08ZE","summary":"Let X be a topological space. The following are equivalent: • X is Hausdorff, • the diagonal Δ(X) ⊂ X × X is closed.","statement_latex":"Let $X$ be a topological space. The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is Hausdorff,\n\\item the diagonal $\\Delta(X) \\subset X \\times X$ is closed.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Hausdorff spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZE","source_file":"topology.tex","source_line":122,"source_end_line":129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L122-L129","statement_sha256":"6c87bc62b77e05ce874518f80d973733d536023cedd218b6c74f5545f18c3b95","origin":"The Stacks Project","memory_eligible":false,"source_rank":238,"rank":238,"depth":0,"x":688.772,"y":220.0,"cluster":"topology"},{"id":"stacks:08ZF","tag":"08ZF","title":"Hausdorff spaces · Lemma 08ZF","summary":"Graphs of maps to Hausdorff spaces are closed. Let f : X → Y be a continuous map of topological spaces. If Y is Hausdorff, then the graph of f is closed in X × Y.","statement_latex":"\\begin{slogan}\nGraphs of maps to Hausdorff spaces are closed.\n\\end{slogan}\nLet $f : X \\to Y$ be a continuous map of topological spaces.\nIf $Y$ is Hausdorff, then the graph of $f$ is closed in $X \\times Y$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Hausdorff spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZF","source_file":"topology.tex","source_line":147,"source_end_line":154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L147-L154","statement_sha256":"bc1ca2a16b7b8c2ad0776ed7f35d965013c498c8e2bf5ae2b412a6c45bc72177","origin":"The Stacks Project","memory_eligible":false,"source_rank":239,"rank":239,"depth":1,"x":668.797,"y":228.621,"cluster":"topology"},{"id":"stacks:08ZG","tag":"08ZG","title":"Hausdorff spaces · Lemma 08ZG","summary":"Let f : X → Y be a continuous map of topological spaces. Let s : Y → X be a continuous map such that f ∘ s = id_Y. If X is Hausdorff, then s(Y) is closed.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $s : Y \\to X$ be a continuous map such that $f \\circ s = \\text{id}_Y$.\nIf $X$ is Hausdorff, then $s(Y)$ is closed.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Hausdorff spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZG","source_file":"topology.tex","source_line":162,"source_end_line":167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L162-L167","statement_sha256":"76479a19e57e9e11b7c79b6f1025045f14e75d061394f476142943834c7f7d63","origin":"The Stacks Project","memory_eligible":false,"source_rank":240,"rank":240,"depth":1,"x":681.715,"y":203.587,"cluster":"topology"},{"id":"stacks:08ZH","tag":"08ZH","title":"Hausdorff spaces · Lemma 08ZH","summary":"Let X → Z and Y → Z be continuous maps of topological spaces. If Z is Hausdorff, then X ×_Z Y is closed in X × Y.","statement_latex":"Let $X \\to Z$ and $Y \\to Z$ be continuous maps of topological spaces.\nIf $Z$ is Hausdorff, then $X \\times_Z Y$ is closed in $X \\times Y$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Hausdorff spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZH","source_file":"topology.tex","source_line":174,"source_end_line":178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L174-L178","statement_sha256":"21142eaee57fdb4585d58bc1c5e124b0a9fa6594ba3f9023605f4e56ae1144d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":241,"rank":241,"depth":1,"x":694.121,"y":235.471,"cluster":"topology"},{"id":"stacks:0CY1","tag":"0CY1","title":"Separated maps · Definition 0CY1","summary":"A continuous map f : X → Y of topological spaces is called separated if and only if the diagonal Δ : X → X ×_Y X is a closed map.","statement_latex":"A continuous map $f : X \\to Y$ of topological spaces is called\n{\\it separated} if and only if the diagonal $\\Delta : X \\to X \\times_Y X$\nis a closed map.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Separated maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CY1","source_file":"topology.tex","source_line":195,"source_end_line":200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L195-L200","statement_sha256":"03bd5b670e1f2adff24f5f9134f1683b22d4934de376ee118c949002bbd86e74","origin":"The Stacks Project","memory_eligible":false,"source_rank":242,"rank":242,"depth":0,"x":654.086,"y":216.15,"cluster":"topology"},{"id":"stacks:0CY2","tag":"0CY2","title":"Separated maps · Lemma 0CY2","summary":"Let f : X → Y be continuous map of topological spaces. The following are equivalent: • f is separated, • Δ(X) ⊂ X ×_Y X is a closed subset, • given distinct points x, x' ∈ X mapping to the same point of Y, there exist disjoint open neighbourhoods of x and x'.","statement_latex":"Let $f : X \\to Y$ be continuous map of topological spaces.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is separated,\n\\item $\\Delta(X) \\subset X \\times_Y X$ is a closed subset,\n\\item given distinct points $x, x' \\in X$ mapping to the same point of\n$Y$, there exist disjoint open neighbourhoods of $x$ and $x'$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Separated maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CY2","source_file":"topology.tex","source_line":202,"source_end_line":212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L202-L212","statement_sha256":"e8f8627ac1ee2fb53437b796b6b8fd3935df32decde263055360160a2698323d","origin":"The Stacks Project","memory_eligible":false,"source_rank":243,"rank":243,"depth":1,"x":704.548,"y":206.883,"cluster":"topology"},{"id":"stacks:0CY3","tag":"0CY3","title":"Separated maps · Lemma 0CY3","summary":"Let f : X → Y be continuous map of topological spaces. If X is Hausdorff, then f is separated.","statement_latex":"Let $f : X \\to Y$ be continuous map of topological spaces.\nIf $X$ is Hausdorff, then $f$ is separated.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Separated maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CY3","source_file":"topology.tex","source_line":247,"source_end_line":251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L247-L251","statement_sha256":"322e18fffad38016d2277b69a7c82e86aa2493d8973268f41cb56a666124df93","origin":"The Stacks Project","memory_eligible":false,"source_rank":244,"rank":244,"depth":2,"x":671.789,"y":245.656,"cluster":"topology"},{"id":"stacks:0CY4","tag":"0CY4","title":"Separated maps · Lemma 0CY4","summary":"Let f : X → Y and Y' → Y be continuous maps of topological spaces. If f is separated, then f' : Y' ×_Y X → Y' is separated.","statement_latex":"Let $f : X \\to Y$ and $Y' \\to Y$ be continuous maps of topological spaces.\nIf $f$ is separated, then $f' : Y' \\times_Y X \\to Y'$ is separated.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Separated maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CY4","source_file":"topology.tex","source_line":259,"source_end_line":263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L259-L263","statement_sha256":"90d9128489bffef404d2f786610bc61cf53e7d1e2b8b97f94f7f9726a5df4eb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":245,"rank":245,"depth":2,"x":664.341,"y":194.674,"cluster":"topology"},{"id":"stacks:004P","tag":"004P","title":"Bases · Definition 004P","summary":"Let X be a topological space. A collection of subsets B of X is called a base for the topology on X or a basis for the topology on X if the following conditions hold: • Every element B ∈ B is open in X. • For every open U ⊂ X and every x ∈ U, there exists an element B ∈ B such that x ∈ B ⊂ U.","statement_latex":"Let $X$ be a topological space. A collection of subsets $\\mathcal{B}$ of $X$\nis called a {\\it base for the topology on $X$} or a {\\it basis for the\ntopology on $X$} if the following conditions hold:\n\\begin{enumerate}\n\\item Every element $B \\in \\mathcal{B}$ is open in $X$.\n\\item For every open $U \\subset X$ and every $x \\in U$,\nthere exists an element $B \\in \\mathcal{B}$ such that\n$x \\in B \\subset U$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Bases","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004P","source_file":"topology.tex","source_line":281,"source_end_line":292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L281-L292","statement_sha256":"2ce248f20997de50e132bbc39ec7dd71791dc3e4bd08212ab4c8d15a99137e78","origin":"The Stacks Project","memory_eligible":false,"source_rank":246,"rank":246,"depth":0,"x":713.973,"y":230.422,"cluster":"topology"},{"id":"stacks:0D5P","tag":"0D5P","title":"Bases · Lemma 0D5P","summary":"Let X be a set and let B be a collection of subsets. Assume that X = ⋃_B ∈ B B and that given x ∈ B_1 ∩ B_2 with B_1, B_2 ∈ B there is a B_3 ∈ B with x ∈ B_3 ⊂ B_1 ∩ B_2. Then there is a unique topology on X such that B is a basis for this topology.","statement_latex":"Let $X$ be a set and let $\\mathcal{B}$ be a collection of subsets.\nAssume that $X = \\bigcup_{B \\in \\mathcal{B}} B$ and that given\n$x \\in B_1 \\cap B_2$ with $B_1, B_2 \\in \\mathcal{B}$ there is a\n$B_3 \\in \\mathcal{B}$ with $x \\in B_3 \\subset B_1 \\cap B_2$.\nThen there is a unique topology on $X$ such that $\\mathcal{B}$\nis a basis for this topology.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Bases","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5P","source_file":"topology.tex","source_line":297,"source_end_line":305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L297-L305","statement_sha256":"bf9642796c006ec8de32b4ec008dacceb858335c32a0897da68cee20859c8807","origin":"The Stacks Project","memory_eligible":false,"source_rank":247,"rank":247,"depth":1,"x":644.657,"y":232.255,"cluster":"topology"},{"id":"stacks:004Q","tag":"004Q","title":"Bases · Lemma 004Q","summary":"Let X be a topological space. Let B be a basis for the topology on X. Let U : U = ⋃_i U_i be an open covering of U ⊂ X. There exists an open covering U = ⋃ V_j which is a refinement of U such that each V_j is an element of the basis B.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{B}$ be a basis for the topology on $X$.\nLet $\\mathcal{U} : U = \\bigcup_i U_i$ be an open covering of\n$U \\subset X$. There exists an open covering $U = \\bigcup V_j$\nwhich is a refinement of $\\mathcal{U}$ such that each\n$V_j$ is an element of the basis $\\mathcal{B}$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Bases","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004Q","source_file":"topology.tex","source_line":334,"source_end_line":342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L334-L342","statement_sha256":"8f266f60d89f29d98cfec9978499afd6fa6516fbec790e1893afffe71fd24749","origin":"The Stacks Project","memory_eligible":false,"source_rank":248,"rank":248,"depth":0,"x":697.038,"y":189.417,"cluster":"topology"},{"id":"stacks:08ZI","tag":"08ZI","title":"Bases · Definition 08ZI","summary":"Let X be a topological space. A collection of subsets B of X is called a subbase for the topology on X or a subbasis for the topology on X if the finite intersections of elements of B form a basis for the topology on X.","statement_latex":"Let $X$ be a topological space. A collection of subsets $\\mathcal{B}$ of $X$\nis called a {\\it subbase for the topology on $X$} or a {\\it subbasis for the\ntopology on $X$} if the finite intersections of\nelements of $\\mathcal{B}$ form a basis for the topology on $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Bases","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZI","source_file":"topology.tex","source_line":352,"source_end_line":358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L352-L358","statement_sha256":"53720a33b2108b867d28a9102dbc1c9bea7ec6455f33189ae516a568972b5641","origin":"The Stacks Project","memory_eligible":false,"source_rank":249,"rank":249,"depth":0,"x":692.591,"y":253.718,"cluster":"topology"},{"id":"stacks:08ZJ","tag":"08ZJ","title":"Bases · Lemma 08ZJ","summary":"Let X be a set. Given any collection B of subsets of X there is a unique topology on X such that B is a subbase for this topology.","statement_latex":"Let $X$ be a set. Given any collection $\\mathcal{B}$ of subsets of $X$\nthere is a unique topology on $X$ such that $\\mathcal{B}$ is a subbase\nfor this topology.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Bases","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZJ","source_file":"topology.tex","source_line":363,"source_end_line":368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L363-L368","statement_sha256":"924fb6279d08402f152b90ad0c4973d1383d4480cf3dfc3d689e1837cd335116","origin":"The Stacks Project","memory_eligible":false,"source_rank":250,"rank":250,"depth":2,"x":642.052,"y":201.527,"cluster":"topology"},{"id":"stacks:0D5Q","tag":"0D5Q","title":"Bases · Lemma 0D5Q","summary":"Let X be a topological space. Let B be a collection of opens of X. Assume X = ⋃_U ∈ B U and for U, V ∈ B we have U ∩ V = ⋃_W ∈ B, W ⊂ U ∩ V W. Then there is a continuous map f : X → Y of topological spaces such that • for U ∈ B the image f(U) is open, • for U ∈ B we have f^-1(f(U)) = U, and • the opens f(U), U ∈ B form a basis for the topology on Y.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{B}$ be a collection\nof opens of $X$. Assume $X = \\bigcup_{U \\in \\mathcal{B}} U$ and\nfor $U, V \\in \\mathcal{B}$ we have\n$U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$.\nThen there is a continuous map $f : X \\to Y$ of topological spaces\nsuch that\n\\begin{enumerate}\n\\item for $U \\in \\mathcal{B}$ the image $f(U)$ is open,\n\\item for $U \\in \\mathcal{B}$ we have $f^{-1}(f(U)) = U$, and\n\\item the opens $f(U)$, $U \\in \\mathcal{B}$\nform a basis for the topology on $Y$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Bases","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5Q","source_file":"topology.tex","source_line":376,"source_end_line":390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L376-L390","statement_sha256":"d4148ab235c0a24b2ad9b3295efd1de7dfc137d94d6242720154d2e758098b86","origin":"The Stacks Project","memory_eligible":false,"source_rank":251,"rank":251,"depth":2,"x":724.517,"y":211.779,"cluster":"topology"},{"id":"stacks:09R8","tag":"09R8","title":"Submersive maps · Lemma 09R8","summary":"Let X be a topological space. Let Y be a set and let f : Y → X be an injective map of sets. The induced topology on Y is the topology characterized by each of the following statements: • it is the weakest topology on Y such that f is continuous, • the open subsets of Y are f^-1(U) for U ⊂ X open, • the closed subsets of Y are the sets f^-1(Z) for Z ⊂ X closed.","statement_latex":"Let $X$ be a topological space. Let $Y$ be a set and let\n$f : Y \\to X$ be an injective map of sets. The induced\ntopology on $Y$ is the topology characterized by\neach of the following statements:\n\\begin{enumerate}\n\\item it is the weakest topology on $Y$ such that $f$ is continuous,\n\\item the open subsets of $Y$ are $f^{-1}(U)$ for $U \\subset X$ open,\n\\item the closed subsets of $Y$ are the sets $f^{-1}(Z)$ for $Z \\subset X$\nclosed.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Submersive maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09R8","source_file":"topology.tex","source_line":420,"source_end_line":432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L420-L432","statement_sha256":"cd03ef398159a3f2dddbcb5e589f2cc6ba9437dc71453e01726de79745d45d7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":252,"rank":252,"depth":0,"x":652.832,"y":252.461,"cluster":"topology"},{"id":"stacks:08ZK","tag":"08ZK","title":"Submersive maps · Lemma 08ZK","summary":"Let X be a topological space. Let Y be a set and let f : X → Y be a surjective map of sets. The quotient topology on Y is the topology characterized by each of the following statements: • it is the strongest topology on Y such that f is continuous, • a subset V of Y is open if and only if f^-1(V) is open, • a subset Z of Y is closed if and only if f^-1(Z) is closed.","statement_latex":"Let $X$ be a topological space. Let $Y$ be a set and let $f : X \\to Y$\nbe a surjective map of sets. The quotient topology on $Y$ is the\ntopology characterized by each of the following statements:\n\\begin{enumerate}\n\\item it is the strongest topology on $Y$ such that $f$ is continuous,\n\\item a subset $V$ of $Y$ is open if and only if $f^{-1}(V)$ is open,\n\\item a subset $Z$ of $Y$ is closed if and only if $f^{-1}(Z)$ is closed.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Submersive maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZK","source_file":"topology.tex","source_line":474,"source_end_line":484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L474-L484","statement_sha256":"e3e4a61f5c3a669f7e73a2d78068218a95f736b7796a9e46bf5ae0619ef1dcc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":253,"rank":253,"depth":0,"x":673.724,"y":179.314,"cluster":"topology"},{"id":"stacks:0406","tag":"0406","title":"Submersive maps · Definition 0406","summary":"Let f : X → Y be a continuous map of topological spaces. • We say f is a strict map of topological spaces if the induced topology and the quotient topology on f(X) agree (see discussion above). • We say f is submersive if f is surjective and strict.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\n\\begin{enumerate}\n\\item We say $f$ is a {\\it strict map of topological spaces}\nif the induced topology and the quotient topology on $f(X)$ agree\n(see discussion above).\n\\item We say $f$ is {\\it submersive}\\footnote{This is very different from\nthe notion of a submersion between differential manifolds! It is probably\na good idea to use ``strict and surjective'' instead of ``submersive''.}\nif $f$ is surjective and strict.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Submersive maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0406","source_file":"topology.tex","source_line":532,"source_end_line":544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L532-L544","statement_sha256":"f4f59529d9e4424b4fbab5668848597816b1902bb9c84006ad31b9dd63eeab84","origin":"The Stacks Project","memory_eligible":false,"source_rank":254,"rank":254,"depth":0,"x":718.531,"y":247.278,"cluster":"topology"},{"id":"stacks:02YB","tag":"02YB","title":"Submersive maps · Lemma 02YB","summary":"Let f : X → Y be surjective, open, continuous map of topological spaces. Let T ⊂ Y be a subset. Then • f^-1(overlineT) = overlinef^-1(T), • T ⊂ Y is closed if and only if f^-1(T) is closed, • T ⊂ Y is open if and only if f^-1(T) is open, and • T ⊂ Y is locally closed if and only if f^-1(T) is locally closed. In particular we see that f is submersive.","statement_latex":"Let $f : X \\to Y$ be surjective, open, continuous map of topological spaces.\nLet $T \\subset Y$ be a subset. Then\n\\begin{enumerate}\n\\item $f^{-1}(\\overline{T}) = \\overline{f^{-1}(T)}$,\n\\item $T \\subset Y$ is closed if and only if $f^{-1}(T)$ is closed,\n\\item $T \\subset Y$ is open if and only if $f^{-1}(T)$ is open, and\n\\item $T \\subset Y$ is locally closed if and only if $f^{-1}(T)$\nis locally closed.\n\\end{enumerate}\nIn particular we see that $f$ is submersive.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Submersive maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YB","source_file":"topology.tex","source_line":553,"source_end_line":565,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L553-L565","statement_sha256":"17cddf0890b9523ab2dbe335d4f8c4565386d9619768ab1da5607af5bc169426","origin":"The Stacks Project","memory_eligible":false,"source_rank":255,"rank":255,"depth":0,"x":628.149,"y":221.801,"cluster":"topology"},{"id":"stacks:0AAU","tag":"0AAU","title":"Submersive maps · Lemma 0AAU","summary":"Let f : X → Y be surjective, closed, continuous map of topological spaces. Let T ⊂ Y be a subset. Then • overlineT = f(overlinef^-1(T)), • T ⊂ Y is closed if and only if f^-1(T) is closed, • T ⊂ Y is open if and only if f^-1(T) is open, and • T ⊂ Y is locally closed if and only if f^-1(T) is locally closed. In particular we see that f is submersive.","statement_latex":"Let $f : X \\to Y$ be surjective, closed, continuous map of topological spaces.\nLet $T \\subset Y$ be a subset. Then\n\\begin{enumerate}\n\\item $\\overline{T} = f(\\overline{f^{-1}(T)})$,\n\\item $T \\subset Y$ is closed if and only if $f^{-1}(T)$ is closed,\n\\item $T \\subset Y$ is open if and only if $f^{-1}(T)$ is open, and\n\\item $T \\subset Y$ is locally closed if and only if\n$f^{-1}(T)$ is locally closed.\n\\end{enumerate}\nIn particular we see that $f$ is submersive.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Submersive maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAU","source_file":"topology.tex","source_line":583,"source_end_line":595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L583-L595","statement_sha256":"4ca1c2e560f16f8b3eed7f9b4a34124e8516734cb3ef12fa9598bc62d8c87090","origin":"The Stacks Project","memory_eligible":false,"source_rank":256,"rank":256,"depth":0,"x":717.822,"y":188.385,"cluster":"topology"},{"id":"stacks:004S","tag":"004S","title":"Connected components · Definition 004S","summary":"Let X be a topological space. • We say X is connected if X is not empty and whenever X = T_1 amalg T_2 with T_i ⊂ X open and closed, then either T_1 = ∅ or T_2 = ∅. • We say T ⊂ X is a connected component of X if T is a maximal connected subset of X.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item We say $X$ is {\\it connected} if $X$ is not empty and whenever\n$X = T_1 \\amalg T_2$ with $T_i \\subset X$ open and closed, then either\n$T_1 = \\emptyset$ or $T_2 = \\emptyset$.\n\\item We say $T \\subset X$ is a {\\it connected component} of $X$ if\n$T$ is a maximal connected subset of $X$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004S","source_file":"topology.tex","source_line":623,"source_end_line":633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L623-L633","statement_sha256":"4f8da2e19bc49c853bd0470b344c88ae53b06a301babcf8f3d15c4c5daac8c75","origin":"The Stacks Project","memory_eligible":false,"source_rank":257,"rank":257,"depth":0,"x":677.47,"y":265.967,"cluster":"topology"},{"id":"stacks:0376","tag":"0376","title":"Connected components · Lemma 0376","summary":"Let f : X → Y be a continuous map of topological spaces. If E ⊂ X is a connected subset, then f(E) ⊂ Y is connected as well.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nIf $E \\subset X$ is a connected subset, then $f(E) \\subset Y$\nis connected as well.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0376","source_file":"topology.tex","source_line":638,"source_end_line":643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L638-L643","statement_sha256":"dddeb780bf887df5e0bdd275b13ea970c73400ce47931a04eb5fe9a8e401d176","origin":"The Stacks Project","memory_eligible":false,"source_rank":258,"rank":258,"depth":0,"x":644.013,"y":183.775,"cluster":"topology"},{"id":"stacks:004T","tag":"004T","title":"Connected components · Lemma 004T","summary":"Let X be a topological space. • If T ⊂ X is connected, then so is its closure. • Any connected component of X is closed (but not necessarily open). • Every connected subset of X is contained in a unique connected component of X. • Every point of X is contained in a unique connected component, in other words, X is the disjoint union of its connected components.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item If $T \\subset X$ is connected, then so is its closure.\n\\item Any connected component of $X$ is closed (but not necessarily open).\n\\item Every connected subset of $X$ is contained in a unique connected\ncomponent of $X$.\n\\item Every point of $X$ is contained in a unique connected component, in other\nwords, $X$ is the disjoint union of its connected components.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004T","source_file":"topology.tex","source_line":657,"source_end_line":668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L657-L668","statement_sha256":"5ee0ab8be8a1e14787fa26eb29b4599fae39f0d72ae86bb18c87522990f5a4df","origin":"The Stacks Project","memory_eligible":false,"source_rank":259,"rank":259,"depth":0,"x":737.008,"y":226.443,"cluster":"topology"},{"id":"stacks:0377","tag":"0377","title":"Connected components · Lemma 0377","summary":"Let f : X → Y be a continuous map of topological spaces. Assume that • all fibres of f are connected, and • a set T ⊂ Y is closed if and only if f^-1(T) is closed. Then f induces a bijection between the sets of connected components of X and Y.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nAssume that\n\\begin{enumerate}\n\\item all fibres of $f$ are connected, and\n\\item a set $T \\subset Y$ is closed if and only if $f^{-1}(T)$ is closed.\n\\end{enumerate}\nThen $f$ induces a bijection between the sets of connected\ncomponents of $X$ and $Y$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0377","source_file":"topology.tex","source_line":720,"source_end_line":730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L720-L730","statement_sha256":"f20f7e99438783cb1c0131203cba563ed6899a3174902eebc19d7ff8b06647fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":260,"rank":260,"depth":1,"x":631.697,"y":248.23,"cluster":"topology"},{"id":"stacks:0378","tag":"0378","title":"Connected components · Lemma 0378","summary":"Let f : X → Y be a continuous map of topological spaces. Assume that (a) f is open, (b) all fibres of f are connected. Then f induces a bijection between the sets of connected components of X and Y.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nAssume that\n(a) $f$ is open,\n(b) all fibres of $f$ are connected.\nThen $f$ induces a bijection between the sets of connected\ncomponents of $X$ and $Y$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0378","source_file":"topology.tex","source_line":748,"source_end_line":756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L748-L756","statement_sha256":"7aef9be71d3ee19a675496879bf594fcfe6fffc3d4529998f4717344b1d6b32b","origin":"The Stacks Project","memory_eligible":false,"source_rank":261,"rank":261,"depth":2,"x":693.199,"y":170.716,"cluster":"topology"},{"id":"stacks:07VB","tag":"07VB","title":"Connected components · Lemma 07VB","summary":"Let f : X → Y be a continuous map of nonempty topological spaces. Assume that (a) Y is connected, (b) f is open and closed, and (c) there is a point y∈ Y such that the fiber f^-1(y) is a finite set. Then X has at most |f^-1(y)| connected components. Hence any connected component T of X is open and closed, and f(T) is a nonempty open and closed subset of Y, which is therefore equal to Y.","statement_latex":"Let $f : X \\to Y$ be a continuous map of nonempty topological spaces. Assume\nthat\n(a) $Y$ is connected,\n(b) $f$ is open and closed, and\n(c) there is a point $y\\in Y$ such that the fiber $f^{-1}(y)$ is a finite set.\nThen $X$ has at most $|f^{-1}(y)|$ connected components. Hence any connected \ncomponent $T$ of $X$ is open and closed, and $f(T)$ is a nonempty open and \nclosed subset of $Y$, which is therefore equal to $Y$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VB","source_file":"topology.tex","source_line":763,"source_end_line":773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L763-L773","statement_sha256":"d4e4223335f44a5cc14cea00afe16a88b05e93aa00743bbee7429cc450b7309b","origin":"The Stacks Project","memory_eligible":false,"source_rank":262,"rank":262,"depth":0,"x":710.529,"y":264.752,"cluster":"topology"},{"id":"stacks:04MC","tag":"04MC","title":"Connected components · Definition 04MC","summary":"A topological space is totally disconnected if the connected components are all singletons.","statement_latex":"A topological space is {\\it totally disconnected} if the connected components\nare all singletons.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MC","source_file":"topology.tex","source_line":785,"source_end_line":789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L785-L789","statement_sha256":"559df05028e72e9a4a389a626400c323ee60159e244385e12d61597075975554","origin":"The Stacks Project","memory_eligible":false,"source_rank":263,"rank":263,"depth":0,"x":620.319,"y":204.007,"cluster":"topology"},{"id":"stacks:08ZL","tag":"08ZL","title":"Connected components · Lemma 08ZL","summary":"Let X be a topological space. Let π_0(X) be the set of connected components of X. Let X → π_0(X) be the map which sends x ∈ X to the connected component of X passing through x. Endow π_0(X) with the quotient topology. Then π_0(X) is a totally disconnected space and any continuous map X → Y from X to a totally disconnected space Y factors through π_0(X).","statement_latex":"Let $X$ be a topological space. Let $\\pi_0(X)$ be the set of connected\ncomponents of $X$. Let $X \\to \\pi_0(X)$ be the map which sends\n$x \\in X$ to the connected component of $X$ passing through $x$.\nEndow $\\pi_0(X)$ with the quotient topology. Then $\\pi_0(X)$ is a\ntotally disconnected space and any continuous map $X \\to Y$\nfrom $X$ to a totally disconnected space $Y$ factors through $\\pi_0(X)$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZL","source_file":"topology.tex","source_line":796,"source_end_line":804,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L796-L804","statement_sha256":"04ce45a7627bac58f2af5e15dab4574f5d6b5ce45bb2a153c46700520bcbe43d","origin":"The Stacks Project","memory_eligible":false,"source_rank":264,"rank":264,"depth":2,"x":737.972,"y":197.501,"cluster":"topology"},{"id":"stacks:04MD","tag":"04MD","title":"Connected components · Definition 04MD","summary":"A topological space X is called locally connected if every point x ∈ X has a fundamental system of connected neighbourhoods.","statement_latex":"A topological space $X$ is called {\\it locally connected} if\nevery point $x \\in X$ has a fundamental system of connected neighbourhoods.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MD","source_file":"topology.tex","source_line":813,"source_end_line":817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L813-L817","statement_sha256":"199602667694bbf4df67b875d2fab250ac39aaffed0e9c5f94cd80b466d5df39","origin":"The Stacks Project","memory_eligible":false,"source_rank":265,"rank":265,"depth":0,"x":654.885,"y":270.409,"cluster":"topology"},{"id":"stacks:04ME","tag":"04ME","title":"Connected components · Lemma 04ME","summary":"Let X be a topological space. If X is locally connected, then • any open subset of X is locally connected, and • the connected components of X are open. So also the connected components of open subsets of X are open. In particular, every point has a fundamental system of open connected neighbourhoods.","statement_latex":"Let $X$ be a topological space. If $X$ is locally connected, then\n\\begin{enumerate}\n\\item any open subset of $X$ is locally connected, and\n\\item the connected components of $X$ are open.\n\\end{enumerate}\nSo also the connected components of open subsets of $X$ are open.\nIn particular, every point has a fundamental system of open connected\nneighbourhoods.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Connected components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ME","source_file":"topology.tex","source_line":819,"source_end_line":829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L819-L829","statement_sha256":"1cf3aa7322551b9c2935972cf24123804a39616ed38a0e9ae2f5d5e7e8426147","origin":"The Stacks Project","memory_eligible":false,"source_rank":266,"rank":266,"depth":0,"x":657.585,"y":167.653,"cluster":"topology"},{"id":"stacks:004V","tag":"004V","title":"Irreducible components · Definition 004V","summary":"Let X be a topological space. • We say X is irreducible, if X is not empty, and whenever X = Z_1 ∪ Z_2 with Z_i closed, we have X = Z_1 or X = Z_2. • We say Z ⊂ X is an irreducible component of X if Z is a maximal irreducible subset of X.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item We say $X$ is {\\it irreducible}, if $X$ is not empty, and whenever\n$X = Z_1 \\cup Z_2$ with $Z_i$ closed, we have $X = Z_1$ or $X = Z_2$.\n\\item We say $Z \\subset X$ is an {\\it irreducible component} of $X$\nif $Z$ is a maximal irreducible subset of $X$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004V","source_file":"topology.tex","source_line":854,"source_end_line":863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L854-L863","statement_sha256":"02064db7a0c47545a7958c3c22f88e5463f31611d3c77e7c9c15f94837a81888","origin":"The Stacks Project","memory_eligible":false,"source_rank":267,"rank":267,"depth":0,"x":739.643,"y":246.331,"cluster":"topology"},{"id":"stacks:0379","tag":"0379","title":"Irreducible components · Lemma 0379","summary":"Let f : X → Y be a continuous map of topological spaces. If E ⊂ X is an irreducible subset, then f(E) ⊂ Y is irreducible as well.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nIf $E \\subset X$ is an irreducible subset, then $f(E) \\subset Y$\nis irreducible as well.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0379","source_file":"topology.tex","source_line":868,"source_end_line":873,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L868-L873","statement_sha256":"2f95121001f1c670e5878589a6db2d484cd2fc3459570633a686a17302e47f40","origin":"The Stacks Project","memory_eligible":false,"source_rank":268,"rank":268,"depth":0,"x":613.752,"y":234.669,"cluster":"topology"},{"id":"stacks:004W","tag":"004W","title":"Irreducible components · Lemma 004W","summary":"Let X be a topological space. • If T ⊂ X is irreducible so is its closure in X. • Any irreducible component of X is closed. • Any irreducible subset of X is contained in an irreducible component of X. • Every point of X is contained in some irreducible component of X, in other words, X is the union of its irreducible components.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item If $T \\subset X$ is irreducible so is its closure in $X$.\n\\item Any irreducible component of $X$ is closed.\n\\item Any irreducible subset of $X$ is contained in an\nirreducible component of $X$.\n\\item Every point of $X$ is contained in some irreducible component\nof $X$, in other words, $X$ is the union of its irreducible components.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004W","source_file":"topology.tex","source_line":885,"source_end_line":896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L885-L896","statement_sha256":"e431d6a4260136615e4a54b06df2aacd1bee81b1656791479bb94ed779e1d5a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":269,"rank":269,"depth":0,"x":717.656,"y":170.807,"cluster":"topology"},{"id":"stacks:0G2Y","tag":"0G2Y","title":"Irreducible components · Lemma 0G2Y","summary":"Let X be a topological space and suppose X = ⋃_i = 1, …, n X_i where each X_i is an irreducible closed subset of X and no X_i is contained in the union of the other members. Then each X_i is an irreducible component of X and each irreducible component of X is one of the X_i.","statement_latex":"Let $X$ be a topological space and suppose $X = \\bigcup_{i = 1, \\ldots, n} X_i$\nwhere each $X_i$ is an irreducible closed subset of $X$ and no $X_i$\nis contained in the union of the other members.  Then each $X_i$ is an\nirreducible component of $X$ and each irreducible component of $X$\nis one of the $X_i$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2Y","source_file":"topology.tex","source_line":924,"source_end_line":931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L924-L931","statement_sha256":"c77436925bc95e872934eae007e51d181d82bf1605d2ac2762055314642f4b58","origin":"The Stacks Project","memory_eligible":false,"source_rank":270,"rank":270,"depth":1,"x":691.979,"y":278.548,"cluster":"topology"},{"id":"stacks:0GM2","tag":"0GM2","title":"Irreducible components · Lemma 0GM2","summary":"Let f : X → Y be a surjective, continuous map of topological spaces. If X has a finite number, say n, of irreducible components, then Y has ≤ n irreducible components.","statement_latex":"Let $f : X \\to Y$ be a surjective, continuous map of topological spaces.\nIf $X$ has a finite number, say $n$, of irreducible components, then\n$Y$ has $\\leq n$ irreducible components.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GM2","source_file":"topology.tex","source_line":948,"source_end_line":953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L948-L953","statement_sha256":"403245817d8583cd48370ab587fe41d01b66a8c58b915fa9748d10cf0e38aecf","origin":"The Stacks Project","memory_eligible":false,"source_rank":271,"rank":271,"depth":2,"x":623.232,"y":183.07,"cluster":"topology"},{"id":"stacks:004X","tag":"004X","title":"Irreducible components · Definition 004X","summary":"Let X be a topological space. • Let Z ⊂ X be an irreducible closed subset. A generic point of Z is a point xi ∈ Z such that Z = overline(xi). • The space X is called Kolmogorov, if for every x, x' ∈ X, x not = x' there exists a closed subset of X which contains exactly one of the two points. • The space X is called quasi-sober if every irreducible closed subset has a generic point. • The space X is called sober if every irreducible closed subset has a unique generic point.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item Let $Z \\subset X$ be an irreducible closed subset.\nA {\\it generic point} of $Z$ is a point $\\xi \\in Z$ such\nthat $Z = \\overline{\\{\\xi\\}}$.\n\\item The space $X$ is called {\\it Kolmogorov}, if for every $x, x' \\in X$,\n$x \\not = x'$ there exists a closed subset of $X$ which contains\nexactly one of the two points.\n\\item The space $X$ is called {\\it quasi-sober} if every\nirreducible closed subset has a generic point.\n\\item The space $X$ is called {\\it sober} if every\nirreducible closed subset has a unique generic point.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004X","source_file":"topology.tex","source_line":971,"source_end_line":986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L971-L986","statement_sha256":"f83c7138fcc4819b72ad3903aa44e17f520c32034a102373c6eadbaae2efe285","origin":"The Stacks Project","memory_eligible":false,"source_rank":272,"rank":272,"depth":0,"x":752.617,"y":214.946,"cluster":"topology"},{"id":"stacks:0B31","tag":"0B31","title":"Irreducible components · Lemma 0B31","summary":"Let X be a topological space and let Y⊂ X. • If X is Kolmogorov then so is Y. • Suppose Y is locally closed in X. If X is quasi-sober then so is Y. • Suppose Y is locally closed in X. If X is sober then so is Y.","statement_latex":"Let $X$ be a topological space and let $Y\\subset X$.\n\\begin{enumerate}\n\\item If $X$ is Kolmogorov then so is $Y$.\n\\item Suppose $Y$ is locally closed in $X$. If $X$ is quasi-sober then\nso is $Y$.\n\\item Suppose $Y$ is locally closed in $X$. If $X$ is sober then so is $Y$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B31","source_file":"topology.tex","source_line":995,"source_end_line":1004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L995-L1004","statement_sha256":"e005c0019906033c5091abae1d37641fda149206cfefa62f4d83b8d71277dcf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":273,"rank":273,"depth":0,"x":629.807,"y":265.576,"cluster":"topology"},{"id":"stacks:06N9","tag":"06N9","title":"Irreducible components · Lemma 06N9","summary":"Let X be a topological space and let (X_i)_i∈ I be a covering of X. • Suppose X_i is locally closed in X for every i∈ I. Then, X is Kolmogorov if and only if X_i is Kolmogorov for every i∈ I. • Suppose X_i is open in X for every i∈ I. Then, X is quasi-sober if and only if X_i is quasi-sober for every i∈ I. • Suppose X_i is open in X for every i∈ I. Then, X is sober if and only if X_i is sober for every i∈ I.","statement_latex":"Let $X$ be a topological space and let $(X_i)_{i\\in I}$ be a covering of $X$.\n\\begin{enumerate}\n\\item Suppose $X_i$ is locally closed in $X$ for every $i\\in I$. Then, $X$ is\nKolmogorov if and only if $X_i$ is Kolmogorov for every $i\\in I$.\n\\item Suppose $X_i$ is open in $X$ for every $i\\in I$. Then, $X$ is\nquasi-sober if and only if $X_i$ is quasi-sober for every $i\\in I$.\n\\item Suppose $X_i$ is open in $X$ for every $i\\in I$. Then, $X$ is sober if\nand  only if $X_i$ is sober for every $i\\in I$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06N9","source_file":"topology.tex","source_line":1031,"source_end_line":1042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1031-L1042","statement_sha256":"72f6a74aa2bfa51f910a56d4d2188e800ee79ce24a725fde3e58e2a8e8735b96","origin":"The Stacks Project","memory_eligible":false,"source_rank":274,"rank":274,"depth":1,"x":680.366,"y":157.045,"cluster":"topology"},{"id":"stacks:004Z","tag":"004Z","title":"Irreducible components · Lemma 004Z","summary":"Let f : X → Y be a continuous map of topological spaces. Assume that (a) Y is irreducible, (b) f is open, and (c) there exists a dense collection of points y ∈ Y such that f^-1(y) is irreducible. Then X is irreducible.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nAssume that\n(a) $Y$ is irreducible,\n(b) $f$ is open, and\n(c) there exists a dense collection of points $y \\in Y$ such\nthat $f^{-1}(y)$ is irreducible.\nThen $X$ is irreducible.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/004Z","source_file":"topology.tex","source_line":1114,"source_end_line":1123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1114-L1123","statement_sha256":"c29e57a0b8ce4243d396e1f479f61ef0407f79f5894d1f662ece25f442a8ac85","origin":"The Stacks Project","memory_eligible":false,"source_rank":275,"rank":275,"depth":0,"x":731.041,"y":267.263,"cluster":"topology"},{"id":"stacks:037A","tag":"037A","title":"Irreducible components · Lemma 037A","summary":"Let f : X → Y be a continuous map of topological spaces. Assume that (a) f is open, and (b) for every y ∈ Y the fibre f^-1(y) is irreducible. Then f induces a bijection between irreducible components.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nAssume that (a) $f$ is open, and\n(b) for every $y \\in Y$ the fibre $f^{-1}(y)$ is irreducible.\nThen $f$ induces a bijection between irreducible components.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037A","source_file":"topology.tex","source_line":1138,"source_end_line":1144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1138-L1144","statement_sha256":"9c97100b22ec16753274916a15437b6acc139359c044285975d30dbf2bc7850f","origin":"The Stacks Project","memory_eligible":false,"source_rank":276,"rank":276,"depth":1,"x":603.355,"y":214.033,"cluster":"topology"},{"id":"stacks:0A2N","tag":"0A2N","title":"Irreducible components · Lemma 0A2N","summary":"[EGA1-second] Let X be a topological space. There is a canonical continuous map c : X → X' from X to a sober topological space X' which is universal among continuous maps from X to sober topological spaces. Moreover, the assignment U' ↦ c^-1(U') is a bijection between opens of X' and X which commutes with finite intersections and arbitrary unions. The image c(X) is a Kolmogorov topological space and the map c : X → c(X) is universal for maps of X into Kolmogorov spaces.","statement_latex":"\\begin{reference}\n\\cite[Page 68 ff]{EGA1-second}\n\\end{reference}\nLet $X$ be a topological space. There is a canonical continuous map\n$$\nc : X \\longrightarrow X'\n$$\nfrom $X$ to a sober topological space $X'$ which is universal\namong continuous maps from $X$ to sober topological spaces.\nMoreover, the assignment $U' \\mapsto c^{-1}(U')$ is a bijection\nbetween opens of $X'$ and $X$ which commutes with finite intersections\nand arbitrary unions.\nThe image $c(X)$ is a Kolmogorov topological space and the\nmap $c : X \\to c(X)$ is universal for maps of $X$ into Kolmogorov spaces.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2N","source_file":"topology.tex","source_line":1162,"source_end_line":1178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1162-L1178","statement_sha256":"66b511bc36226179ffce8ac163e1c2618423a9d0611b79036f67ce0daf051a17","origin":"The Stacks Project","memory_eligible":false,"source_rank":277,"rank":277,"depth":0,"x":742.105,"y":180.406,"cluster":"topology"},{"id":"stacks:0GM3","tag":"0GM3","title":"Irreducible components · Lemma 0GM3","summary":"Let X be a connected topological space with a finite number of irreducible components X_1, …, X_n. If n > 1 there is an 1 ≤ j ≤ n such that X' = ⋃_i not = j X_i is connected.","statement_latex":"Let $X$ be a connected topological space with a finite number of\nirreducible components $X_1, \\ldots, X_n$. If $n > 1$ there is an\n$1 \\leq j \\leq n$ such that $X' = \\bigcup_{i \\not = j} X_i$ is connected.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Irreducible components","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GM3","source_file":"topology.tex","source_line":1222,"source_end_line":1227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1222-L1227","statement_sha256":"ea19a3f202c1e1ee9df7d908f1c98a6a4fe128da9f834593b320ff5ae56526df","origin":"The Stacks Project","memory_eligible":false,"source_rank":278,"rank":278,"depth":0,"x":665.87,"y":285.245,"cluster":"topology"},{"id":"stacks:0051","tag":"0051","title":"Noetherian topological spaces · Definition 0051","summary":"A topological space is called Noetherian if the descending chain condition holds for closed subsets of X. A topological space is called locally Noetherian if every point has a neighbourhood which is Noetherian.","statement_latex":"A topological space is called {\\it Noetherian}\nif the descending chain condition holds for\nclosed subsets of $X$. A topological space is called\n{\\it locally Noetherian} if every point has a neighbourhood\nwhich is Noetherian.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Noetherian topological spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0051","source_file":"topology.tex","source_line":1243,"source_end_line":1250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1243-L1250","statement_sha256":"c679e342d51f74a5e8a8fa8b5798c9b958278d466275432f4558c8c8716af51a","origin":"The Stacks Project","memory_eligible":false,"source_rank":279,"rank":279,"depth":0,"x":637.436,"y":163.184,"cluster":"topology"},{"id":"stacks:0052","tag":"0052","title":"Noetherian topological spaces · Lemma 0052","summary":"Let X be a Noetherian topological space. • Any subset of X with the induced topology is Noetherian. • The space X has finitely many irreducible components. • Each irreducible component of X contains a nonempty open of X.","statement_latex":"Let $X$ be a Noetherian topological space.\n\\begin{enumerate}\n\\item Any subset of $X$ with the induced topology is Noetherian.\n\\item The space $X$ has finitely many irreducible components.\n\\item Each irreducible component of $X$ contains a nonempty open of $X$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Noetherian topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0052","source_file":"topology.tex","source_line":1252,"source_end_line":1260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1252-L1260","statement_sha256":"b85447cdc3e4620c76561b54555c6293b14db00237acf75f6c765d6c6c2b0907","origin":"The Stacks Project","memory_eligible":false,"source_rank":280,"rank":280,"depth":2,"x":757.997,"y":237.956,"cluster":"topology"},{"id":"stacks:04Z8","tag":"04Z8","title":"Noetherian topological spaces · Lemma 04Z8","summary":"Let f : X → Y be a continuous map of topological spaces. • If X is Noetherian, then f(X) is Noetherian. • If X is locally Noetherian and f open, then f(X) is locally Noetherian.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\n\\begin{enumerate}\n\\item If $X$ is Noetherian, then $f(X)$ is Noetherian.\n\\item If $X$ is locally Noetherian and $f$ open, then $f(X)$ is\nlocally Noetherian.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Noetherian topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Z8","source_file":"topology.tex","source_line":1303,"source_end_line":1311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1303-L1311","statement_sha256":"15926a994b8d69d0693753ffbbb97f03912aca2634da69bb55be94cd5f5fe0c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":281,"rank":281,"depth":0,"x":607.206,"y":251.379,"cluster":"topology"},{"id":"stacks:0053","tag":"0053","title":"Noetherian topological spaces · Lemma 0053","summary":"Let X be a topological space. Let X_i ⊂ X, i = 1, …, n be a finite collection of subsets. If each X_i is Noetherian (with the induced topology), then ⋃_i = 1, …, n X_i is Noetherian (with the induced topology).","statement_latex":"Let $X$ be a topological space.\nLet $X_i \\subset X$, $i = 1, \\ldots, n$ be a finite collection of subsets.\nIf each $X_i$ is Noetherian (with the induced topology), then\n$\\bigcup_{i = 1, \\ldots, n}  X_i$ is Noetherian (with the induced topology).","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Noetherian topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0053","source_file":"topology.tex","source_line":1327,"source_end_line":1333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1327-L1333","statement_sha256":"66f602fbbb4910470b87c4431961e577fa6528325fc08516711a0fb914695015","origin":"The Stacks Project","memory_eligible":false,"source_rank":282,"rank":282,"depth":0,"x":708.767,"y":154.821,"cluster":"topology"},{"id":"stacks:04MF","tag":"04MF","title":"Noetherian topological spaces · Lemma 04MF","summary":"Let X be a locally Noetherian topological space. Then X is locally connected.","statement_latex":"Let $X$ be a locally Noetherian topological space.\nThen $X$ is locally connected.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Noetherian topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MF","source_file":"topology.tex","source_line":1365,"source_end_line":1369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1365-L1369","statement_sha256":"c47ef8a7e0d3fab1a8e979c45365682d5689a80e82f3e66799017603031349c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":283,"rank":283,"depth":3,"x":711.551,"y":285.103,"cluster":"topology"},{"id":"stacks:0055","tag":"0055","title":"Krull dimension · Definition 0055","summary":"Let X be a topological space. • A chain of irreducible closed subsets of X is a sequence Z_0 ⊂ Z_1 ⊂ … ⊂ Z_n ⊂ X with Z_i closed irreducible and Z_i not = Z_i + 1 for i = 0, …, n - 1. • The length of a chain Z_0 ⊂ Z_1 ⊂ … ⊂ Z_n ⊂ X of irreducible closed subsets of X is the integer n. • The dimension or more precisely the Krull dimension dim(X) of X is the element of (-∞, 0, 1, 2, 3, …, ∞) defined by the formula: dim(X) = sup (lengths of chains of irreducible closed…","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it chain of irreducible closed subsets} of $X$\nis a sequence $Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_n \\subset X$\nwith $Z_i$ closed irreducible and $Z_i \\not = Z_{i + 1}$ for\n$i = 0, \\ldots, n - 1$.\n\\item The {\\it length} of a chain\n$Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_n \\subset X$\nof irreducible closed subsets of $X$ is the\ninteger $n$.\n\\item The {\\it dimension} or more precisely the {\\it Krull dimension}\n$\\dim(X)$ of $X$ is the element of\n$\\{-\\infty, 0, 1, 2, 3, \\ldots, \\infty\\}$ defined by the formula:\n$$\n\\dim(X) =\n\\sup \\{\\text{lengths of chains of irreducible closed subsets}\\}\n$$\nThus $\\dim(X) = -\\infty$ if and only if $X$ is the empty space.\n\\item Let $x \\in X$.\nThe {\\it Krull dimension of $X$ at $x$} is defined as\n$$\n\\dim_x(X) = \\min \\{\\dim(U), x\\in U\\subset X\\text{ open}\\}\n$$\nthe minimum of $\\dim(U)$ where $U$ runs over the open\nneighbourhoods of $x$ in $X$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Krull dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0055","source_file":"topology.tex","source_line":1391,"source_end_line":1419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1391-L1419","statement_sha256":"88ffd0eb1b8ae9c3bbda8ab531f5690d96f4ce9074bd1b04bff1b955b2bf5c84","origin":"The Stacks Project","memory_eligible":false,"source_rank":284,"rank":284,"depth":0,"x":603.556,"y":189.568,"cluster":"topology"},{"id":"stacks:0B7I","tag":"0B7I","title":"Krull dimension · Lemma 0B7I","summary":"Let X be a topological space. Then dim(X) = sup dim_x(X) where the supremum runs over the points x of X.","statement_latex":"Let $X$ be a topological space. Then $\\dim(X) = \\sup \\dim_x(X)$\nwhere the supremum runs over the points $x$ of $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Krull dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7I","source_file":"topology.tex","source_line":1427,"source_end_line":1431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1427-L1431","statement_sha256":"af88a790df6c9d8a538247e78e5c0d141e35322d9bc7f5b7106ca9cce540e855","origin":"The Stacks Project","memory_eligible":false,"source_rank":285,"rank":285,"depth":1,"x":761.703,"y":198.84,"cluster":"topology"},{"id":"stacks:0058","tag":"0058","title":"Krull dimension · Definition 0058","summary":"Let X be a topological space. We say that X is equidimensional if every irreducible component of X has the same dimension.","statement_latex":"Let $X$ be a topological space.\nWe say that $X$ is {\\it equidimensional} if every irreducible\ncomponent of $X$ has the same dimension.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Krull dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0058","source_file":"topology.tex","source_line":1468,"source_end_line":1473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1468-L1473","statement_sha256":"0201102b1a7f6c54823ca471e4315783575e0a508768dda2e761806ecc25ea66","origin":"The Stacks Project","memory_eligible":false,"source_rank":286,"rank":286,"depth":0,"x":636.317,"y":282.611,"cluster":"topology"},{"id":"stacks:02I3","tag":"02I3","title":"Codimension and catenary spaces · Definition 02I3","summary":"Let X be a topological space. Let Y ⊂ X be an irreducible closed subset. The codimension of Y in X is the supremum of the lengths e of chains Y = Y_0 ⊂ Y_1 ⊂ … ⊂ Y_e ⊂ X of irreducible closed subsets in X starting with Y. We will denote this codim(Y, X).","statement_latex":"Let $X$ be a topological space.\nLet $Y \\subset X$ be an irreducible closed subset.\nThe {\\it codimension} of $Y$ in $X$ is the supremum of\nthe lengths $e$ of chains\n$$\nY = Y_0 \\subset Y_1 \\subset \\ldots \\subset Y_e \\subset X\n$$\nof irreducible closed subsets in $X$ starting with $Y$.\nWe will denote this $\\text{codim}(Y, X)$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Codimension and catenary spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02I3","source_file":"topology.tex","source_line":1485,"source_end_line":1496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1485-L1496","statement_sha256":"884464105e03221346587eb13bf9247ced6e1ff37c64707ae7e3eaada4129cf3","origin":"The Stacks Project","memory_eligible":false,"source_rank":287,"rank":287,"depth":0,"x":661.675,"y":148.319,"cluster":"topology"},{"id":"stacks:02I4","tag":"02I4","title":"Codimension and catenary spaces · Lemma 02I4","summary":"Let X be a topological space. Let Y ⊂ X be an irreducible closed subset. Let U ⊂ X be an open subset such that Y ∩ U is nonempty. Then codim(Y, X) = codim(Y ∩ U, U)","statement_latex":"Let $X$ be a topological space.\nLet $Y \\subset X$ be an irreducible closed subset.\nLet $U \\subset X$ be an open subset such that $Y \\cap U$ is nonempty.\nThen\n$$\n\\text{codim}(Y, X) = \\text{codim}(Y \\cap U, U)\n$$","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Codimension and catenary spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02I4","source_file":"topology.tex","source_line":1503,"source_end_line":1512,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1503-L1512","statement_sha256":"6c73da04b0a52abf2f7d621c83961c1c6f968bce98c35a4bf9da12f6154ad69d","origin":"The Stacks Project","memory_eligible":false,"source_rank":288,"rank":288,"depth":0,"x":751.87,"y":262.884,"cluster":"topology"},{"id":"stacks:02I1","tag":"02I1","title":"Codimension and catenary spaces · Definition 02I1","summary":"Let X be a topological space. We say X is catenary if for every pair of irreducible closed subsets T ⊂ T' we have codim(T, T') < ∞ and every maximal chain of irreducible closed subsets T = T_0 ⊂ T_1 ⊂ … ⊂ T_e = T' has the same length (equal to the codimension).","statement_latex":"Let $X$ be a topological space. We say $X$ is {\\it catenary} if\nfor every pair of irreducible closed subsets $T \\subset T'$\nwe have $\\text{codim}(T, T') < \\infty$ and every maximal chain\nof irreducible closed subsets\n$$\nT = T_0 \\subset T_1 \\subset \\ldots \\subset T_e = T'\n$$\nhas the same length (equal to the codimension).","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Codimension and catenary spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02I1","source_file":"topology.tex","source_line":1534,"source_end_line":1544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1534-L1544","statement_sha256":"c04fdd7b0230cad0d87cabbb19832e003a06aced570c6635e29b88e843f5e06c","origin":"The Stacks Project","memory_eligible":false,"source_rank":289,"rank":289,"depth":0,"x":591.658,"y":229.248,"cluster":"topology"},{"id":"stacks:02I2","tag":"02I2","title":"Codimension and catenary spaces · Lemma 02I2","summary":"Let X be a topological space. The following are equivalent: • X is catenary, • X has an open covering by catenary spaces. Moreover, in this case any locally closed subspace of X is catenary.","statement_latex":"Let $X$ be a topological space.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $X$ is catenary,\n\\item $X$ has an open covering by catenary spaces.\n\\end{enumerate}\nMoreover, in this case any locally closed subspace of $X$ is catenary.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Codimension and catenary spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02I2","source_file":"topology.tex","source_line":1546,"source_end_line":1555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1546-L1555","statement_sha256":"1938c940a87a0682abb1809e7bb962c5aa9e65fde266b3680024cbe9beb41ab6","origin":"The Stacks Project","memory_eligible":false,"source_rank":290,"rank":290,"depth":0,"x":738.261,"y":162.504,"cluster":"topology"},{"id":"stacks:02I6","tag":"02I6","title":"Codimension and catenary spaces · Lemma 02I6","summary":"Let X be a topological space. The following are equivalent: • X is catenary, and • for every pair of irreducible closed subsets Y ⊂ Y' we have codim(Y, Y') < ∞ and for every triple Y ⊂ Y' ⊂ Y\" of irreducible closed subsets we have codim(Y, Y\") = codim(Y, Y') + codim(Y', Y\").","statement_latex":"Let $X$ be a topological space. The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is catenary, and\n\\item for every pair of irreducible closed subsets $Y \\subset Y'$ we have\n$\\text{codim}(Y, Y') < \\infty$ and for every triple\n$Y \\subset Y' \\subset Y''$ of irreducible closed subsets we have\n$$\n\\text{codim}(Y, Y'') = \\text{codim}(Y, Y') + \\text{codim}(Y', Y'').\n$$\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Codimension and catenary spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02I6","source_file":"topology.tex","source_line":1565,"source_end_line":1577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1565-L1577","statement_sha256":"7c560f43a06aaaa0d561e6c29969d16e68ddb4924cdb190e9d96e123e75d6e0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":291,"rank":291,"depth":1,"x":683.307,"y":296.169,"cluster":"topology"},{"id":"stacks:005A","tag":"005A","title":"Quasi-compact spaces and maps · Definition 005A","summary":"Quasi-compactness. • We say that a topological space X is quasi-compact if every open covering of X has a finite subcover. • We say that a continuous map f : X → Y is quasi-compact if the inverse image f^-1(V) of every quasi-compact open V ⊂ Y is quasi-compact. • We say a subset Z ⊂ X is retrocompact if the inclusion map Z → X is quasi-compact.","statement_latex":"Quasi-compactness.\n\\begin{enumerate}\n\\item We say that a topological space $X$ is {\\it quasi-compact}\nif every open covering of $X$ has a finite subcover.\n\\item We say that a continuous map $f : X \\to Y$ is {\\it quasi-compact}\nif the inverse image $f^{-1}(V)$ of every quasi-compact open $V \\subset Y$\nis quasi-compact.\n\\item We say a subset $Z \\subset X$ is {\\it retrocompact}\nif the inclusion map $Z \\to X$ is quasi-compact.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005A","source_file":"topology.tex","source_line":1627,"source_end_line":1639,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1627-L1639","statement_sha256":"639b0a54f986e5de891ff9aeb80e1a0a994b7d2c34c3edecabe6e80a46a55f8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":292,"rank":292,"depth":0,"x":615.717,"y":165.207,"cluster":"topology"},{"id":"stacks:005B","tag":"005B","title":"Quasi-compact spaces and maps · Lemma 005B","summary":"A composition of quasi-compact maps is quasi-compact.","statement_latex":"A composition of quasi-compact maps is quasi-compact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005B","source_file":"topology.tex","source_line":1652,"source_end_line":1655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1652-L1655","statement_sha256":"08147eff06029bd6c156fad221a47abef8133a7501084b5b91286dcecf7a76d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":293,"rank":293,"depth":0,"x":772.298,"y":223.964,"cluster":"topology"},{"id":"stacks:005C","tag":"005C","title":"Quasi-compact spaces and maps · Lemma 005C","summary":"A closed subset of a quasi-compact topological space is quasi-compact.","statement_latex":"A closed subset of a quasi-compact topological space is quasi-compact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005C","source_file":"topology.tex","source_line":1661,"source_end_line":1664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1661-L1664","statement_sha256":"a9a5ffdb57ce7638d3e727d6baeab63419084966430ec5e07b5f1092c141f26c","origin":"The Stacks Project","memory_eligible":false,"source_rank":294,"rank":294,"depth":0,"x":608.116,"y":269.891,"cluster":"topology"},{"id":"stacks:08YB","tag":"08YB","title":"Quasi-compact spaces and maps · Lemma 08YB","summary":"Let X be a Hausdorff topological space. • If E ⊂ X is quasi-compact, then it is closed. • If E_1, E_2 ⊂ X are disjoint quasi-compact subsets then there exists opens E_i ⊂ U_i with U_1 ∩ U_2 = ∅.","statement_latex":"Let $X$ be a Hausdorff topological space.\n\\begin{enumerate}\n\\item If $E \\subset X$ is quasi-compact, then it is closed.\n\\item If $E_1, E_2 \\subset X$ are disjoint quasi-compact subsets\nthen there exists opens $E_i \\subset U_i$ with $U_1 \\cap U_2 = \\emptyset$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YB","source_file":"topology.tex","source_line":1676,"source_end_line":1684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1676-L1684","statement_sha256":"ea015aa3b4cfb709fae4da3b963a317ed7b90353f0c6643cf66c24c1569c9da7","origin":"The Stacks Project","memory_eligible":false,"source_rank":295,"rank":295,"depth":0,"x":692.998,"y":141.74,"cluster":"topology"},{"id":"stacks:08YC","tag":"08YC","title":"Quasi-compact spaces and maps · Lemma 08YC","summary":"Let X be a quasi-compact Hausdorff space. Let E ⊂ X. The following are equivalent: (a) E is closed in X, (b) E is quasi-compact.","statement_latex":"Let $X$ be a quasi-compact Hausdorff space. Let $E \\subset X$.\nThe following are equivalent: (a) $E$ is closed in $X$, (b)\n$E$ is quasi-compact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YC","source_file":"topology.tex","source_line":1705,"source_end_line":1710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1705-L1710","statement_sha256":"b0bd13d0873c14ced7541d5510b787075c42aa8e696b0951f056bf9078d4000c","origin":"The Stacks Project","memory_eligible":false,"source_rank":296,"rank":296,"depth":1,"x":733.81,"y":285.645,"cluster":"topology"},{"id":"stacks:005D","tag":"005D","title":"Quasi-compact spaces and maps · Lemma 005D","summary":"Let X be a quasi-compact topological space. If (Z_α)_α ∈ A is a collection of closed subsets such that the intersection of each finite subcollection is nonempty, then ⋂_α ∈ A Z_α is nonempty.","statement_latex":"Let $X$ be a quasi-compact topological space.\nIf $\\{Z_\\alpha\\}_{\\alpha \\in A}$ is a collection of closed subsets\nsuch that the intersection of each finite subcollection\nis nonempty, then $\\bigcap_{\\alpha \\in A} Z_\\alpha$ is nonempty.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005D","source_file":"topology.tex","source_line":1723,"source_end_line":1729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1723-L1729","statement_sha256":"71d723965799922d4652e1b6f2c53291a93393aee1c7a819c61fcadc3c1b3ec3","origin":"The Stacks Project","memory_eligible":false,"source_rank":297,"rank":297,"depth":0,"x":586.746,"y":201.976,"cluster":"topology"},{"id":"stacks:04Z9","tag":"04Z9","title":"Quasi-compact spaces and maps · Lemma 04Z9","summary":"Let f : X → Y be a continuous map of topological spaces. • If X is quasi-compact, then f(X) is quasi-compact. • If f is quasi-compact, then f(X) is retrocompact.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\n\\begin{enumerate}\n\\item If $X$ is quasi-compact, then $f(X)$ is quasi-compact.\n\\item If $f$ is quasi-compact, then $f(X)$ is retrocompact.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Z9","source_file":"topology.tex","source_line":1746,"source_end_line":1753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1746-L1753","statement_sha256":"ca41cc9f2e3196fe6412beefc72ca0c3e1fc357a51f3eff1ef3290dc053e2b60","origin":"The Stacks Project","memory_eligible":false,"source_rank":298,"rank":298,"depth":0,"x":763.954,"y":180.046,"cluster":"topology"},{"id":"stacks:005E","tag":"005E","title":"Quasi-compact spaces and maps · Lemma 005E","summary":"Let X be a topological space. Assume that • X is nonempty, • X is quasi-compact, and • X is Kolmogorov. Then X has a closed point.","statement_latex":"Let $X$ be a topological space. Assume that\n\\begin{enumerate}\n\\item $X$ is nonempty,\n\\item $X$ is quasi-compact, and\n\\item $X$ is Kolmogorov.\n\\end{enumerate}\nThen $X$ has a closed point.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005E","source_file":"topology.tex","source_line":1767,"source_end_line":1776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1767-L1776","statement_sha256":"4ceec39a0c1b717070c1712c883871486d217801558baf355addcead42a46127","origin":"The Stacks Project","memory_eligible":false,"source_rank":299,"rank":299,"depth":1,"x":649.98,"y":297.732,"cluster":"topology"},{"id":"stacks:08ZM","tag":"08ZM","title":"Quasi-compact spaces and maps · Lemma 08ZM","summary":"Let X be a quasi-compact Kolmogorov space. Then the set X_0 of closed points of X is quasi-compact.","statement_latex":"Let $X$ be a quasi-compact Kolmogorov space. Then the set $X_0$ of\nclosed points of $X$ is quasi-compact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZM","source_file":"topology.tex","source_line":1797,"source_end_line":1801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1797-L1801","statement_sha256":"4d8b76b625049e61f899ef8d4e076ce3d7aef3cc19a13dd52188f8fe045b8d2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":300,"rank":300,"depth":2,"x":639.301,"y":145.047,"cluster":"topology"},{"id":"stacks:005F","tag":"005F","title":"Quasi-compact spaces and maps · Lemma 005F","summary":"Let X be a topological space. Assume • X is quasi-compact, • X has a basis for the topology consisting of quasi-compact opens, and • the intersection of two quasi-compact opens is quasi-compact. For any x ∈ X the connected component of X containing x is the intersection of all open and closed subsets of X containing x.","statement_latex":"Let $X$ be a topological space.\nAssume\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item $X$ has a basis for the topology consisting of quasi-compact opens, and\n\\item the intersection of two quasi-compact opens is quasi-compact.\n\\end{enumerate}\nFor any $x \\in X$ the connected component of $X$ containing\n$x$ is the intersection of all open and closed subsets\nof $X$ containing $x$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005F","source_file":"topology.tex","source_line":1815,"source_end_line":1827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1815-L1827","statement_sha256":"63049e6d18f9d4ff637276314eacf278de06fd659bd382cf93b671f55cb6e21f","origin":"The Stacks Project","memory_eligible":false,"source_rank":301,"rank":301,"depth":1,"x":771.004,"y":252.431,"cluster":"topology"},{"id":"stacks:08ZN","tag":"08ZN","title":"Quasi-compact spaces and maps · Lemma 08ZN","summary":"Let X be a topological space. Assume X is quasi-compact and Hausdorff. For any x ∈ X the connected component of X containing x is the intersection of all open and closed subsets of X containing x.","statement_latex":"Let $X$ be a topological space. Assume $X$ is quasi-compact and Hausdorff.\nFor any $x \\in X$ the connected component of $X$ containing\n$x$ is the intersection of all open and closed subsets\nof $X$ containing $x$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZN","source_file":"topology.tex","source_line":1860,"source_end_line":1866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1860-L1866","statement_sha256":"1eeab1c68d3640f62c957e2129c01b8dc141898524dae23119a95279fc5b8bd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":302,"rank":302,"depth":1,"x":586.086,"y":247.94,"cluster":"topology"},{"id":"stacks:04PL","tag":"04PL","title":"Quasi-compact spaces and maps · Lemma 04PL","summary":"Let X be a topological space. Assume • X is quasi-compact, • X has a basis for the topology consisting of quasi-compact opens, and • the intersection of two quasi-compact opens is quasi-compact. For a subset T ⊂ X the following are equivalent: • [(a)] T is an intersection of open and closed subsets of X, and • [(b)] T is closed in X and is a union of connected components of X.","statement_latex":"Let $X$ be a topological space.\nAssume\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item $X$ has a basis for the topology consisting of quasi-compact opens, and\n\\item the intersection of two quasi-compact opens is quasi-compact.\n\\end{enumerate}\nFor a subset $T \\subset X$ the following are equivalent:\n\\begin{enumerate}\n\\item[(a)] $T$ is an intersection of open and closed subsets of $X$, and\n\\item[(b)] $T$ is closed in $X$ and is a union of connected components of $X$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PL","source_file":"topology.tex","source_line":1894,"source_end_line":1908,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1894-L1908","statement_sha256":"20cff2be09c4dc281b3549a549136559c65efbc7f228137f5bb617faccfe39cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":303,"rank":303,"depth":2,"x":727.142,"y":145.539,"cluster":"topology"},{"id":"stacks:04ZA","tag":"04ZA","title":"Quasi-compact spaces and maps · Lemma 04ZA","summary":"Let X be a Noetherian topological space. • The space X is quasi-compact. • Any subset of X is retrocompact.","statement_latex":"Let $X$ be a Noetherian topological space.\n\\begin{enumerate}\n\\item The space $X$ is quasi-compact.\n\\item Any subset of $X$ is retrocompact.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZA","source_file":"topology.tex","source_line":1931,"source_end_line":1938,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1931-L1938","statement_sha256":"34d5c2371b199503af13bd0b75b9eb299b9ddb156d8cd33bd64d08fd9690c5d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":304,"rank":304,"depth":3,"x":705.308,"y":302.275,"cluster":"topology"},{"id":"stacks:04ZB","tag":"04ZB","title":"Quasi-compact spaces and maps · Lemma 04ZB","summary":"A quasi-compact locally Noetherian space is Noetherian.","statement_latex":"A quasi-compact locally Noetherian space is Noetherian.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZB","source_file":"topology.tex","source_line":1954,"source_end_line":1957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1954-L1957","statement_sha256":"0339b2b771cc57a0a170a20a68e1b6153d6a2903809e694827e5ee308f57fec7","origin":"The Stacks Project","memory_eligible":false,"source_rank":305,"rank":305,"depth":1,"x":594.541,"y":173.346,"cluster":"topology"},{"id":"stacks:08ZP","tag":"08ZP","title":"Alexander subbase theorem · Lemma 08ZP","summary":"Let X be a topological space. Let B be a subbase for X. If every covering of X by elements of B has a finite refinement, then X is quasi-compact.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{B}$ be a subbase for $X$.\nIf every covering of $X$ by elements of $\\mathcal{B}$ has a finite\nrefinement, then $X$ is quasi-compact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Quasi-compact spaces and maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZP","source_file":"topology.tex","source_line":1965,"source_end_line":1970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L1965-L1970","statement_sha256":"62cd5a1ed705e87f1c9a4addcb7cdd757380b1d2534bf7b7000a2ea505f5ba57","origin":"The Stacks Project","memory_eligible":false,"source_rank":306,"rank":306,"depth":0,"x":781.273,"y":205.807,"cluster":"topology"},{"id":"stacks:0068","tag":"0068","title":"Locally quasi-compact spaces · Definition 0068","summary":"A topological space X is called locally quasi-compact if every point has a fundamental system of quasi-compact neighbourhoods.","statement_latex":"A topological space $X$ is called\n{\\it locally quasi-compact}\\footnote{This may not be standard notation.\nAlternative notions used in the literature are: (1) Every point has some\nquasi-compact neighbourhood, and (2) Every point has a closed quasi-compact\nneighbourhood. A scheme has the property that every point has a fundamental\nsystem of open quasi-compact neighbourhoods.} if every\npoint has a fundamental system of quasi-compact neighbourhoods.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Locally quasi-compact spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0068","source_file":"topology.tex","source_line":2006,"source_end_line":2015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2006-L2015","statement_sha256":"8a564e9c4bb10ccbed3a52027993c4079943b56b3ba927e279b4fc3de18e9112","origin":"The Stacks Project","memory_eligible":false,"source_rank":307,"rank":307,"depth":0,"x":616.278,"y":288.423,"cluster":"topology"},{"id":"stacks:08ZR","tag":"08ZR","title":"Locally quasi-compact spaces · Lemma 08ZR","summary":"A Hausdorff space is locally quasi-compact if and only if every point has a quasi-compact neighbourhood.","statement_latex":"A Hausdorff space is locally quasi-compact if and only if every point\nhas a quasi-compact neighbourhood.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZR","source_file":"topology.tex","source_line":2021,"source_end_line":2025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2021-L2025","statement_sha256":"39a9dbd1bb41ef4da147e7a25e1698161ec4adcd728ec00d01a365f788c8088f","origin":"The Stacks Project","memory_eligible":false,"source_rank":308,"rank":308,"depth":1,"x":671.906,"y":132.769,"cluster":"topology"},{"id":"stacks:0CQN","tag":"0CQN","title":"Baire category theorem · Lemma 0CQN","summary":"Let X be a locally quasi-compact Hausdorff space. Let U_n ⊂ X, n ≥ 1 be dense open subsets. Then ⋂_n ≥ 1 U_n is dense in X.","statement_latex":"Let $X$ be a locally quasi-compact Hausdorff space.\nLet $U_n \\subset X$, $n \\geq 1$ be dense open subsets. Then\n$\\bigcap_{n \\geq 1} U_n$ is dense in $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQN","source_file":"topology.tex","source_line":2043,"source_end_line":2048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2043-L2048","statement_sha256":"0a1672ea694615de85fae13793adfe72f8869929e383ef4997a7a1ddeae8bad8","origin":"The Stacks Project","memory_eligible":false,"source_rank":309,"rank":309,"depth":1,"x":756.653,"y":280.151,"cluster":"topology"},{"id":"stacks:09UV","tag":"09UV","title":"Locally quasi-compact spaces · Lemma 09UV","summary":"Let X be a Hausdorff and quasi-compact space. Let X = ⋃_i ∈ I U_i be an open covering. Then there exists an open covering X = ⋃_i ∈ I V_i such that overlineV_i ⊂ U_i for all i.","statement_latex":"Let $X$ be a Hausdorff and quasi-compact space.\nLet $X = \\bigcup_{i \\in I} U_i$ be an open covering.\nThen there exists an open covering $X = \\bigcup_{i \\in I} V_i$\nsuch that $\\overline{V_i} \\subset U_i$ for all $i$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UV","source_file":"topology.tex","source_line":2084,"source_end_line":2090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2084-L2090","statement_sha256":"1749e70a7315ecaf1e776d80d2382283cd3ca9a22817d6285e5eed5c7228a859","origin":"The Stacks Project","memory_eligible":false,"source_rank":310,"rank":310,"depth":1,"x":574.377,"y":219.134,"cluster":"topology"},{"id":"stacks:09UW","tag":"09UW","title":"Locally quasi-compact spaces · Lemma 09UW","summary":"Let X be a Hausdorff and quasi-compact space. Let X = ⋃_i ∈ I U_i be an open covering. Suppose given an integer p ≥ 0 and for every (p + 1)-tuple i_0, …, i_p of I an open covering U_i_0 ∩ … ∩ U_i_p = ⋃ W_i_0 … i_p, k. Then there exists an open covering X = ⋃_j ∈ J V_j and a map α : J → I such that overlineV_j ⊂ U_α(j) and such that each V_j_0 ∩ … ∩ V_j_p is contained in W_α(j_0) … α(j_p), k for some k.","statement_latex":"Let $X$ be a Hausdorff and quasi-compact space.\nLet $X = \\bigcup_{i \\in I} U_i$ be an open covering.\nSuppose given an integer $p \\geq 0$ and for every $(p + 1)$-tuple\n$i_0, \\ldots, i_p$ of $I$ an open covering\n$U_{i_0} \\cap \\ldots \\cap U_{i_p} = \\bigcup W_{i_0 \\ldots i_p, k}$.\nThen there exists an open covering $X = \\bigcup_{j \\in J} V_j$\nand a map $\\alpha : J \\to I$ such that $\\overline{V_j} \\subset U_{\\alpha(j)}$\nand such that each $V_{j_0} \\cap \\ldots \\cap V_{j_p}$\nis contained in $W_{\\alpha(j_0) \\ldots \\alpha(j_p), k}$\nfor some $k$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UW","source_file":"topology.tex","source_line":2105,"source_end_line":2117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2105-L2117","statement_sha256":"80d11ca0dc78b9d62e2f9ad1f0facddeaddb29014b92dd53ef4bf1d979821ac1","origin":"The Stacks Project","memory_eligible":false,"source_rank":311,"rank":311,"depth":2,"x":759.12,"y":160.299,"cluster":"topology"},{"id":"stacks:09UX","tag":"09UX","title":"Locally quasi-compact spaces · Lemma 09UX","summary":"Let X be a Hausdorff and locally quasi-compact space. Let Z ⊂ X be a quasi-compact (hence closed) subset. Suppose given an integer p ≥ 0, a set I, for every i ∈ I an open U_i ⊂ X, and for every (p + 1)-tuple i_0, …, i_p of I an open W_i_0 … i_p ⊂ U_i_0 ∩ … ∩ U_i_p such that • Z ⊂ ⋃ U_i, and • for every i_0, …, i_p we have W_i_0 … i_p ∩ Z = U_i_0 ∩ … ∩ U_i_p ∩ Z. Then there exist opens V_i of X such that we have Z ⊂ ⋃ V_i, for all i we have overlineV_i ⊂ U_i, and we have…","statement_latex":"Let $X$ be a Hausdorff and locally quasi-compact space.\nLet $Z \\subset X$ be a quasi-compact (hence closed) subset.\nSuppose given an integer $p \\geq 0$, a set $I$, for every $i \\in I$\nan open $U_i \\subset X$, and for every $(p + 1)$-tuple\n$i_0, \\ldots, i_p$ of $I$ an open\n$W_{i_0 \\ldots i_p} \\subset U_{i_0} \\cap \\ldots \\cap U_{i_p}$\nsuch that\n\\begin{enumerate}\n\\item $Z \\subset \\bigcup U_i$, and\n\\item for every $i_0, \\ldots, i_p$ we have\n$W_{i_0 \\ldots i_p} \\cap Z = U_{i_0} \\cap \\ldots \\cap U_{i_p} \\cap Z$.\n\\end{enumerate}\nThen there exist opens $V_i$ of $X$ such that\nwe have $Z \\subset \\bigcup V_i$,\nfor all $i$ we have $\\overline{V_i} \\subset U_i$, and\nwe have $V_{i_0} \\cap \\ldots \\cap V_{i_p} \\subset W_{i_0 \\ldots i_p}$\nfor all $(p + 1)$-tuples $i_0, \\ldots, i_p$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UX","source_file":"topology.tex","source_line":2191,"source_end_line":2210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2191-L2210","statement_sha256":"cc83ff79af68872f6761d2f49c3b8170af34d81535341fb149abb2fbff25b878","origin":"The Stacks Project","memory_eligible":false,"source_rank":312,"rank":312,"depth":2,"x":669.599,"y":309.518,"cluster":"topology"},{"id":"stacks:0CY5","tag":"0CY5","title":"Locally quasi-compact spaces · Lemma 0CY5","summary":"Let X be a topological space. Let Z ⊂ X be a quasi-compact subset such that any two points of Z have disjoint open neighbourhoods in X. Suppose given an integer p ≥ 0, a set I, for every i ∈ I an open U_i ⊂ X, and for every (p + 1)-tuple i_0, …, i_p of I an open W_i_0 … i_p ⊂ U_i_0 ∩ … ∩ U_i_p such that • Z ⊂ ⋃ U_i, and • for every i_0, …, i_p we have W_i_0 … i_p ∩ Z = U_i_0 ∩ … ∩ U_i_p ∩ Z. Then there exist opens V_i of X such that • Z ⊂ ⋃ V_i, • V_i ⊂ U_i for all i, •…","statement_latex":"Let $X$ be a topological space. Let $Z \\subset X$ be a quasi-compact subset\nsuch that any two points of $Z$ have disjoint open neighbourhoods in $X$.\nSuppose given an integer $p \\geq 0$, a set $I$, for every $i \\in I$\nan open $U_i \\subset X$, and for every $(p + 1)$-tuple\n$i_0, \\ldots, i_p$ of $I$ an open\n$W_{i_0 \\ldots i_p} \\subset U_{i_0} \\cap \\ldots \\cap U_{i_p}$\nsuch that\n\\begin{enumerate}\n\\item $Z \\subset \\bigcup U_i$, and\n\\item for every $i_0, \\ldots, i_p$ we have\n$W_{i_0 \\ldots i_p} \\cap Z = U_{i_0} \\cap \\ldots \\cap U_{i_p} \\cap Z$.\n\\end{enumerate}\nThen there exist opens $V_i$ of $X$ such that\n\\begin{enumerate}\n\\item $Z \\subset \\bigcup V_i$,\n\\item $V_i \\subset U_i$ for all $i$,\n\\item $\\overline{V_i} \\cap Z \\subset U_i$ for all $i$, and\n\\item $V_{i_0} \\cap \\ldots \\cap V_{i_p} \\subset W_{i_0 \\ldots i_p}$\nfor all $(p + 1)$-tuples $i_0, \\ldots, i_p$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CY5","source_file":"topology.tex","source_line":2259,"source_end_line":2281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2259-L2281","statement_sha256":"742f39e33d5437495a1e7c0a4f315b0116e803f6d97bdc1ce42ae8eace433018","origin":"The Stacks Project","memory_eligible":false,"source_rank":313,"rank":313,"depth":1,"x":615.253,"y":147.609,"cluster":"topology"},{"id":"stacks:08ZT","tag":"08ZT","title":"Limits of spaces · Lemma 08ZT","summary":"The category of topological spaces has limits and the forgetful functor to sets commutes with them.","statement_latex":"The category of topological spaces has limits and the forgetful functor\nto sets commutes with them.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZT","source_file":"topology.tex","source_line":2357,"source_end_line":2361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2357-L2361","statement_sha256":"63995d12f69a101082dc47a3902fe8f12c14487577bc0f852944413efb637fe5","origin":"The Stacks Project","memory_eligible":false,"source_rank":314,"rank":314,"depth":2,"x":786.655,"y":236.75,"cluster":"topology"},{"id":"stacks:0A2P","tag":"0A2P","title":"Limits of spaces · Lemma 0A2P","summary":"Let I be a cofiltered category. Let i ↦ X_i be a diagram of topological spaces over I. Let X = lim X_i be the limit with projection maps f_i : X → X_i. • Any open of X is of the form ⋃_j ∈ J f_j^-1(U_j) for some subset J ⊂ I and opens U_j ⊂ X_j. • Any quasi-compact open of X is of the form f_i^-1(U_i) for some i and some U_i ⊂ X_i open.","statement_latex":"Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram\nof topological spaces over $\\mathcal{I}$. Let $X = \\lim X_i$ be the limit\nwith projection maps $f_i : X \\to X_i$.\n\\begin{enumerate}\n\\item Any open of $X$ is of the form $\\bigcup_{j \\in J} f_j^{-1}(U_j)$\nfor some subset $J \\subset I$ and opens $U_j \\subset X_j$.\n\\item Any quasi-compact open of $X$ is of the form\n$f_i^{-1}(U_i)$ for some $i$ and some $U_i \\subset X_i$ open.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2P","source_file":"topology.tex","source_line":2373,"source_end_line":2384,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2373-L2384","statement_sha256":"1c79f9aebaa3c72cccdc3b8b4b631cea65cb5a7a052cbf61c667e1f9463867b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":315,"rank":315,"depth":0,"x":587.286,"y":268.48,"cluster":"topology"},{"id":"stacks:0A2Q","tag":"0A2Q","title":"Limits of spaces · Lemma 0A2Q","summary":"Let I be a cofiltered category. Let i ↦ X_i be a diagram of topological spaces over I. Let X be a topological space such that • X = lim X_i as a set (denote f_i the projection maps), • the sets f_i^-1(U_i) for i ∈ Ob(I) and U_i ⊂ X_i open form a basis for the topology of X. Then X is the limit of the X_i as a topological space.","statement_latex":"Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram\nof topological spaces over $\\mathcal{I}$. Let $X$ be a topological\nspace such that\n\\begin{enumerate}\n\\item $X = \\lim X_i$ as a set (denote $f_i$ the projection maps),\n\\item the sets $f_i^{-1}(U_i)$ for $i \\in \\Ob(\\mathcal{I})$ and\n$U_i \\subset X_i$ open form a basis for the topology of $X$.\n\\end{enumerate}\nThen $X$ is the limit of the $X_i$ as a topological space.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2Q","source_file":"topology.tex","source_line":2427,"source_end_line":2438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2427-L2438","statement_sha256":"769811d5051ce674f18c502c7405b08e757e91878224d5ad1c5f3b468923b1a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":316,"rank":316,"depth":1,"x":709.568,"y":131.077,"cluster":"topology"},{"id":"stacks:08ZU","tag":"08ZU","title":"Tychonov · Theorem 08ZU","summary":"A product of quasi-compact spaces is quasi-compact.","statement_latex":"A product of quasi-compact spaces is quasi-compact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spaces","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZU","source_file":"topology.tex","source_line":2445,"source_end_line":2448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2445-L2448","statement_sha256":"6c3908eb3961df5d0340e03455387d52413467953cea1eddff26f38582babd62","origin":"The Stacks Project","memory_eligible":false,"source_rank":317,"rank":317,"depth":1,"x":730.021,"y":302.869,"cluster":"topology"},{"id":"stacks:08ZV","tag":"08ZV","title":"Limits of spaces · Lemma 08ZV","summary":"Let I be a category and let i ↦ X_i be a diagram over I in the category of topological spaces. If each X_i is quasi-compact and Hausdorff, then lim X_i is quasi-compact.","statement_latex":"Let $\\mathcal{I}$ be a category and let $i \\mapsto X_i$\nbe a diagram over $\\mathcal{I}$ in the category of topological\nspaces. If each $X_i$ is quasi-compact and Hausdorff, then\n$\\lim X_i$ is quasi-compact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZV","source_file":"topology.tex","source_line":2474,"source_end_line":2480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2474-L2480","statement_sha256":"815e5935969fb4a320952975fabb748bb72bad7a1aac4b3ca7b559c0a348e43c","origin":"The Stacks Project","memory_eligible":false,"source_rank":318,"rank":318,"depth":2,"x":575.828,"y":187.072,"cluster":"topology"},{"id":"stacks:0A2R","tag":"0A2R","title":"Limits of spaces · Lemma 0A2R","summary":"Let I be a cofiltered category and let i ↦ X_i be a diagram over I in the category of topological spaces. If each X_i is quasi-compact, Hausdorff, and nonempty, then lim X_i is nonempty.","statement_latex":"Let $\\mathcal{I}$ be a cofiltered category and let $i \\mapsto X_i$\nbe a diagram over $\\mathcal{I}$ in the category of topological\nspaces. If each $X_i$ is quasi-compact, Hausdorff, and nonempty, then\n$\\lim X_i$ is nonempty.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2R","source_file":"topology.tex","source_line":2506,"source_end_line":2512,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2506-L2512","statement_sha256":"ccc1e81000b2e994afa1b4bfecf26dbc4b583b41b7c13e9a74057b259746bd13","origin":"The Stacks Project","memory_eligible":false,"source_rank":319,"rank":319,"depth":3,"x":783.932,"y":184.956,"cluster":"topology"},{"id":"stacks:005G","tag":"005G","title":"Constructible sets · Definition 005G","summary":"Let X be a topological space. Let E ⊂ X be a subset of X. • We say E is constructible in X if E is a finite union of subsets of the form U ∩ V^c where U, V ⊂ X are open and retrocompact in X. • We say E is locally constructible in X if there exists an open covering X = ⋃ V_i such that each E ∩ V_i is constructible in V_i.","statement_latex":"Let $X$ be a topological space. Let $E \\subset X$ be a subset of $X$.\n\\begin{enumerate}\n\\item We say $E$ is {\\it constructible}\\footnote{In the second edition\nof EGA I \\cite{EGA1-second} this was called a ``globally constructible''\nset and a the terminology ``constructible'' was used for what we call a locally\nconstructible set.}\nin $X$ if $E$ is a finite union\nof subsets of the form $U \\cap V^c$ where $U, V \\subset X$ are open and\nretrocompact in $X$.\n\\item We say $E$ is {\\it locally constructible} in $X$ if there exists an open\ncovering $X = \\bigcup V_i$ such that each $E \\cap V_i$ is constructible\nin $V_i$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005G","source_file":"topology.tex","source_line":2553,"source_end_line":2568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2553-L2568","statement_sha256":"9f7d54d1f5b792eaac80038082e600de92bba842aca7b1178bf76c504f94d8e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":320,"rank":320,"depth":0,"x":631.249,"y":305.332,"cluster":"topology"},{"id":"stacks:005H","tag":"005H","title":"Constructible sets · Lemma 005H","summary":"The collection of constructible sets is closed under finite intersections, finite unions and complements.","statement_latex":"The collection of constructible sets is closed under\nfinite intersections, finite unions and complements.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005H","source_file":"topology.tex","source_line":2570,"source_end_line":2574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2570-L2574","statement_sha256":"7c04336cfd86fd7889f74230b7d667bf9d5ba8245a032bf070df7c2d60f10e0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":321,"rank":321,"depth":0,"x":647.13,"y":128.87,"cluster":"topology"},{"id":"stacks:005I","tag":"005I","title":"Constructible sets · Lemma 005I","summary":"Let f : X → Y be a continuous map of topological spaces. If the inverse image of every retrocompact open subset of Y is retrocompact in X, then inverse images of constructible sets are constructible.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nIf the inverse image of every retrocompact open subset of $Y$\nis retrocompact in $X$, then inverse images of constructible\nsets are constructible.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005I","source_file":"topology.tex","source_line":2590,"source_end_line":2596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2590-L2596","statement_sha256":"b207683d70c221751411b53d609e75b440c898c2d21d024c54f82154a503f146","origin":"The Stacks Project","memory_eligible":false,"source_rank":322,"rank":322,"depth":0,"x":778.102,"y":268.836,"cluster":"topology"},{"id":"stacks:005J","tag":"005J","title":"Constructible sets · Lemma 005J","summary":"Let U ⊂ X be open. For a constructible set E ⊂ X the intersection E ∩ U is constructible in U.","statement_latex":"Let $U \\subset X$ be open. For a constructible set\n$E \\subset X$ the intersection $E \\cap U$ is constructible\nin $U$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005J","source_file":"topology.tex","source_line":2603,"source_end_line":2608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2603-L2608","statement_sha256":"b6c84095f13e45d5c10f61f88195ca540ff532951efc30d736ff2d1c068b92a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":323,"rank":323,"depth":1,"x":567.732,"y":239.77,"cluster":"topology"},{"id":"stacks:09YD","tag":"09YD","title":"Constructible sets · Lemma 09YD","summary":"Let U ⊂ X be a retrocompact open. Let E ⊂ U. If E is constructible in U, then E is constructible in X.","statement_latex":"Let $U \\subset X$ be a retrocompact open. Let $E \\subset U$.\nIf $E$ is constructible in $U$, then $E$ is constructible in $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YD","source_file":"topology.tex","source_line":2619,"source_end_line":2623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2619-L2623","statement_sha256":"c1ae3ed1e0fe7031754a1c51479b4eba672a51afac01f232664a2a22af6bba4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":324,"rank":324,"depth":1,"x":747.274,"y":141.264,"cluster":"topology"},{"id":"stacks:053W","tag":"053W","title":"Constructible sets · Lemma 053W","summary":"Let X be a topological space. Let E ⊂ X be a subset. Let X = V_1 ∪ … ∪ V_m be a finite covering by retrocompact opens. Then E is constructible in X if and only if E ∩ V_j is constructible in V_j for each j = 1, …, m.","statement_latex":"Let $X$ be a topological space. Let $E \\subset X$ be a subset.\nLet $X = V_1 \\cup \\ldots \\cup V_m$ be a finite covering by\nretrocompact opens.\nThen $E$ is constructible in $X$ if and only if $E \\cap V_j$\nis constructible in $V_j$ for each $j = 1, \\ldots, m$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053W","source_file":"topology.tex","source_line":2634,"source_end_line":2641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2634-L2641","statement_sha256":"d7e24e51dfd50f60fbe18255c2f176f93459864b42b0805850f43ce0be6dd643","origin":"The Stacks Project","memory_eligible":false,"source_rank":325,"rank":325,"depth":2,"x":693.789,"y":316.785,"cluster":"topology"},{"id":"stacks:09YE","tag":"09YE","title":"Constructible sets · Lemma 09YE","summary":"Let X be a topological space. Let Z ⊂ X be a closed subset such that X setminus Z is quasi-compact. Then for a constructible set E ⊂ X the intersection E ∩ Z is constructible in Z.","statement_latex":"Let $X$ be a topological space. Let $Z \\subset X$ be a closed\nsubset such that $X \\setminus Z$ is quasi-compact.\nThen for a constructible set $E \\subset X$ the intersection\n$E \\cap Z$ is constructible in $Z$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YE","source_file":"topology.tex","source_line":2655,"source_end_line":2661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2655-L2661","statement_sha256":"a4a34017b2248ab938f5fcb70a863016480fa19169b063ca92548306c769367f","origin":"The Stacks Project","memory_eligible":false,"source_rank":326,"rank":326,"depth":1,"x":591.501,"y":156.096,"cluster":"topology"},{"id":"stacks:09YF","tag":"09YF","title":"Constructible sets · Lemma 09YF","summary":"Let X be a topological space. Let T ⊂ X be a subset. Suppose • T is retrocompact in X, • quasi-compact opens form a basis for the topology on X. Then for a constructible set E ⊂ X the intersection E ∩ T is constructible in T.","statement_latex":"Let $X$ be a topological space. Let $T \\subset X$ be a subset. Suppose\n\\begin{enumerate}\n\\item $T$ is retrocompact in $X$,\n\\item quasi-compact opens form a basis for the topology on $X$.\n\\end{enumerate}\nThen for a constructible set $E \\subset X$ the intersection $E \\cap T$ is\nconstructible in $T$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YF","source_file":"topology.tex","source_line":2675,"source_end_line":2684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2675-L2684","statement_sha256":"e7efa217a9fdad579913cd5c7a611ce583ef78f9304840cfbf54a5c3fd99d318","origin":"The Stacks Project","memory_eligible":false,"source_rank":327,"rank":327,"depth":1,"x":797.302,"y":216.888,"cluster":"topology"},{"id":"stacks:09YG","tag":"09YG","title":"Constructible sets · Lemma 09YG","summary":"Let Z ⊂ X be a closed subset whose complement is retrocompact open. Let E ⊂ Z. If E is constructible in Z, then E is constructible in X.","statement_latex":"Let $Z \\subset X$ be a closed subset whose complement is retrocompact open.\nLet $E \\subset Z$. If $E$ is constructible in $Z$, then $E$ is constructible\nin $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YG","source_file":"topology.tex","source_line":2698,"source_end_line":2703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2698-L2703","statement_sha256":"4ca886e8284eae8684658efe7897d4f83fe1390d8c9b5825c446b40e52c72f13","origin":"The Stacks Project","memory_eligible":false,"source_rank":328,"rank":328,"depth":1,"x":595.54,"y":289.237,"cluster":"topology"},{"id":"stacks:09YH","tag":"09YH","title":"Constructible sets · Lemma 09YH","summary":"Let X be a topological space. Every constructible subset of X is retrocompact.","statement_latex":"Let $X$ be a topological space. Every constructible\nsubset of $X$ is retrocompact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YH","source_file":"topology.tex","source_line":2726,"source_end_line":2730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2726-L2730","statement_sha256":"0923b18933f20587f22cec5250e00262f909c5a05dd10b8cffb15f14375e421b","origin":"The Stacks Project","memory_eligible":false,"source_rank":329,"rank":329,"depth":1,"x":686.638,"y":120.478,"cluster":"topology"},{"id":"stacks:09YI","tag":"09YI","title":"Constructible sets · Lemma 09YI","summary":"Let X be a topological space. Assume X has a basis consisting of quasi-compact opens. For E, E' constructible in X, the intersection E ∩ E' is constructible in E.","statement_latex":"Let $X$ be a topological space. Assume\n$X$ has a basis consisting of quasi-compact opens.\nFor $E, E'$ constructible in $X$, the intersection\n$E \\cap E'$ is constructible in $E$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YI","source_file":"topology.tex","source_line":2749,"source_end_line":2755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2749-L2755","statement_sha256":"7400f2595ef3dd90fc634599fb86d6052cd4484b4978582fe4db93f97b85b711","origin":"The Stacks Project","memory_eligible":false,"source_rank":330,"rank":330,"depth":2,"x":755.547,"y":297.571,"cluster":"topology"},{"id":"stacks:09YJ","tag":"09YJ","title":"Constructible sets · Lemma 09YJ","summary":"Let X be a topological space. Assume X has a basis consisting of quasi-compact opens. Let E be constructible in X and F ⊂ E constructible in E. Then F is constructible in X.","statement_latex":"Let $X$ be a topological space. Assume\n$X$ has a basis consisting of quasi-compact opens.\nLet $E$ be constructible in $X$ and $F \\subset E$ constructible in $E$.\nThen $F$ is constructible in $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YJ","source_file":"topology.tex","source_line":2762,"source_end_line":2768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2762-L2768","statement_sha256":"52b2d4a7ff6dba3a3d35a397a5e1a090b21812278c5b99e2719660c33f609c39","origin":"The Stacks Project","memory_eligible":false,"source_rank":331,"rank":331,"depth":3,"x":561.278,"y":205.59,"cluster":"topology"},{"id":"stacks:0F2K","tag":"0F2K","title":"Constructible sets · Lemma 0F2K","summary":"Let X be a quasi-compact topological space having a basis consisting of quasi-compact opens such that the intersection of any two quasi-compact opens is quasi-compact. Let T ⊂ X be a locally closed subset such that T is quasi-compact and T^c is retrocompact in X. Then T is constructible in X.","statement_latex":"Let $X$ be a quasi-compact topological space having a basis consisting of\nquasi-compact opens such that the intersection of any two\nquasi-compact opens is quasi-compact.\nLet $T \\subset X$ be a locally closed subset\nsuch that $T$ is quasi-compact and $T^c$ is retrocompact in $X$.\nThen $T$ is constructible in $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2K","source_file":"topology.tex","source_line":2793,"source_end_line":2801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2793-L2801","statement_sha256":"954fa7871c093a6dbee74bfab4bb56adf3ff618f5d502169bc552a76533d947a","origin":"The Stacks Project","memory_eligible":false,"source_rank":332,"rank":332,"depth":0,"x":779.658,"y":162.959,"cluster":"topology"},{"id":"stacks:09YK","tag":"09YK","title":"Constructible sets · Lemma 09YK","summary":"Let X be a topological space which has a basis for the topology consisting of quasi-compact opens. Let E ⊂ X be a subset. Let X = E_1 ∪ … ∪ E_m be a finite covering by constructible subsets. Then E is constructible in X if and only if E ∩ E_j is constructible in E_j for each j = 1, …, m.","statement_latex":"Let $X$ be a topological space which has a basis for the topology\nconsisting of quasi-compact opens. Let $E \\subset X$ be a subset.\nLet $X = E_1 \\cup \\ldots \\cup E_m$ be a finite covering by constructible\nsubsets. Then $E$ is constructible in $X$ if and only if $E \\cap E_j$\nis constructible in $E_j$ for each $j = 1, \\ldots, m$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YK","source_file":"topology.tex","source_line":2815,"source_end_line":2822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2815-L2822","statement_sha256":"9366d3d12b94e1c2ba35bf1e1d28ab98d9231680e8014f50ddf3d2cde4f10a78","origin":"The Stacks Project","memory_eligible":false,"source_rank":333,"rank":333,"depth":4,"x":652.239,"y":319.128,"cluster":"topology"},{"id":"stacks:005K","tag":"005K","title":"Constructible sets · Lemma 005K","summary":"Let X be a topological space. Suppose that Z ⊂ X is irreducible. Let E ⊂ X be a finite union of locally closed subsets (e.g. E is constructible). The following are equivalent • The intersection E ∩ Z contains an open dense subset of Z. • The intersection E ∩ Z is dense in Z. If Z has a generic point xi, then this is also equivalent to • [(3)] We have xi ∈ E.","statement_latex":"Let $X$ be a topological space. Suppose that\n$Z \\subset X$ is irreducible. Let $E \\subset X$\nbe a finite union of locally closed subsets (e.g.\\ $E$\nis constructible). The following are equivalent\n\\begin{enumerate}\n\\item The intersection $E \\cap Z$ contains an open\ndense subset of $Z$.\n\\item The intersection $E \\cap Z$ is dense in $Z$.\n\\end{enumerate}\nIf $Z$ has a generic point $\\xi$, then this is\nalso equivalent to\n\\begin{enumerate}\n\\item[(3)] We have $\\xi \\in E$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005K","source_file":"topology.tex","source_line":2830,"source_end_line":2846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2830-L2846","statement_sha256":"2f659ca0dab6a9a927c9fd9367b4902308c1dc3a5484c7bed159d77ca60aa82c","origin":"The Stacks Project","memory_eligible":false,"source_rank":334,"rank":334,"depth":0,"x":620.446,"y":130.69,"cluster":"topology"},{"id":"stacks:005L","tag":"005L","title":"Constructible sets and Noetherian spaces · Lemma 005L","summary":"Let X be a Noetherian topological space. The constructible sets in X are precisely the finite unions of locally closed subsets of X.","statement_latex":"Let $X$ be a Noetherian topological space.\nThe constructible sets in $X$ are precisely the finite unions\nof locally closed subsets of $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets and Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005L","source_file":"topology.tex","source_line":2870,"source_end_line":2875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2870-L2875","statement_sha256":"2ef27e79b6d064066d2205318e805746f3953478cd409efb65d277f169be7068","origin":"The Stacks Project","memory_eligible":false,"source_rank":335,"rank":335,"depth":4,"x":796.33,"y":252.228,"cluster":"topology"},{"id":"stacks:053Y","tag":"053Y","title":"Constructible sets and Noetherian spaces · Lemma 053Y","summary":"Let f : X → Y be a continuous map of Noetherian topological spaces. If E ⊂ Y is constructible in Y, then f^-1(E) is constructible in X.","statement_latex":"Let $f : X \\to Y$ be a continuous map of Noetherian topological spaces.\nIf $E \\subset Y$ is constructible in $Y$, then $f^{-1}(E)$ is constructible\nin $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets and Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053Y","source_file":"topology.tex","source_line":2882,"source_end_line":2887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2882-L2887","statement_sha256":"e6bb26048ecbc10c76de2b40e60a2156315cbfeb2e7603aa81985e44d080de01","origin":"The Stacks Project","memory_eligible":false,"source_rank":336,"rank":336,"depth":5,"x":567.734,"y":262.459,"cluster":"topology"},{"id":"stacks:053Z","tag":"053Z","title":"Constructible sets and Noetherian spaces · Lemma 053Z","summary":"Let X be a Noetherian topological space. Let E ⊂ X be a subset. The following are equivalent: • E is constructible in X, and • for every irreducible closed Z ⊂ X the intersection E ∩ Z either contains a nonempty open of Z or is not dense in Z.","statement_latex":"Let $X$ be a Noetherian topological space.\nLet $E \\subset X$ be a subset.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $E$ is constructible in $X$, and\n\\item for every irreducible closed $Z \\subset X$ the intersection\n$E \\cap Z$ either contains a nonempty open of $Z$ or is not dense in $Z$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets and Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053Z","source_file":"topology.tex","source_line":2895,"source_end_line":2905,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2895-L2905","statement_sha256":"1443a0e1830b6478dcfdbf769b488fedbd0b55d2730b491d5acbd79ab9f41c74","origin":"The Stacks Project","memory_eligible":false,"source_rank":337,"rank":337,"depth":6,"x":728.884,"y":124.51,"cluster":"topology"},{"id":"stacks:0540","tag":"0540","title":"Constructible sets and Noetherian spaces · Lemma 0540","summary":"Let X be a Noetherian topological space. Let x ∈ X. Let E ⊂ X be constructible in X. The following are equivalent: • E is a neighbourhood of x, and • for every irreducible closed subset Y of X which contains x the intersection E ∩ Y is dense in Y.","statement_latex":"Let $X$ be a Noetherian topological space.\nLet $x \\in X$.\nLet $E \\subset X$ be constructible in $X$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $E$ is a neighbourhood of $x$, and\n\\item for every irreducible closed subset $Y$ of $X$ which contains\n$x$ the intersection $E \\cap Y$ is dense in $Y$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets and Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0540","source_file":"topology.tex","source_line":2946,"source_end_line":2957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2946-L2957","statement_sha256":"3b0446a929ab62bde032c8ed82bf756663ea3c908d08bb54639b818970b8dbb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":338,"rank":338,"depth":7,"x":720.948,"y":318.641,"cluster":"topology"},{"id":"stacks:0541","tag":"0541","title":"Constructible sets and Noetherian spaces · Lemma 0541","summary":"Let X be a Noetherian topological space. Let E ⊂ X be a subset. The following are equivalent: • E is open in X, and • for every irreducible closed subset Y of X the intersection E ∩ Y is either empty or contains a nonempty open of Y.","statement_latex":"Let $X$ be a Noetherian topological space.\nLet $E \\subset X$ be a subset.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $E$ is open in $X$, and\n\\item for every irreducible closed subset $Y$ of $X$\nthe intersection $E \\cap Y$ is either empty or\ncontains a nonempty open of $Y$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Constructible sets and Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0541","source_file":"topology.tex","source_line":2982,"source_end_line":2993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L2982-L2993","statement_sha256":"7e631d6ce399723af08ef2c6c9fdae400fddd0f31e870b485f690cc4abfae01c","origin":"The Stacks Project","memory_eligible":false,"source_rank":339,"rank":339,"depth":8,"x":569.941,"y":170.254,"cluster":"topology"},{"id":"stacks:005N","tag":"005N","title":"Tube lemma · Lemma 005N","summary":"Let X and Y be topological spaces. Let A ⊂ X and B ⊂ Y be quasi-compact subsets. Let A × B ⊂ W ⊂ X × Y with W open in X × Y. Then there exists opens A ⊂ U ⊂ X and B ⊂ V ⊂ Y such that U × V ⊂ W.","statement_latex":"Let $X$ and $Y$ be topological spaces.\nLet $A \\subset X$ and $B \\subset Y$ be quasi-compact subsets.\nLet $A \\times B \\subset W \\subset X \\times Y$ with $W$\nopen in $X \\times Y$. Then there exists opens $A \\subset U \\subset X$\nand $B \\subset V \\subset Y$ such that $U \\times V \\subset W$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Characterizing proper maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005N","source_file":"topology.tex","source_line":3042,"source_end_line":3049,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3042-L3049","statement_sha256":"2f82cbdfeb8bbe11ed429f0f5cad03b46cfdd70fa758334064239dec61a06165","origin":"The Stacks Project","memory_eligible":false,"source_rank":340,"rank":340,"depth":0,"x":801.753,"y":194.106,"cluster":"topology"},{"id":"stacks:005O","tag":"005O","title":"Characterizing proper maps · Definition 005O","summary":"Let f : X→ Y be a continuous map between topological spaces. • We say that the map f is closed if the image of every closed subset is closed. • We say that the map f is Bourbaki-proper if the map Z × X→ Z × Y is closed for any topological space Z. • We say that the map f is quasi-proper if the inverse image f^-1(V) of every quasi-compact subset V ⊂ Y is quasi-compact. • We say that f is universally closed if the map f': Z ×_Y X → Z is closed for any continuous map g: Z →…","statement_latex":"Let $f : X\\to Y$ be a continuous map between topological spaces.\n\\begin{enumerate}\n\\item We say that the map $f$ is {\\it closed}\nif the image of every closed subset is closed.\n\\item We say that the map $f$ is {\\it Bourbaki-proper}\\footnote{This is the\nterminology used in \\cite{Bourbaki}. Sometimes this property may be\ncalled ``universally closed'' in the literature.} if the map\n$Z \\times X\\to Z \\times Y$ is closed for any topological space $Z$.\n\\item We say that the map $f$ is {\\it quasi-proper} if\nthe inverse image $f^{-1}(V)$ of every quasi-compact subset $V \\subset Y$\nis quasi-compact.\n\\item We say that $f$ is {\\it universally closed} if\nthe map $f': Z \\times_Y X \\to Z$ is closed for any continuous map $g: Z \\to Y$.\n\\item We say that $f$ is {\\it proper} if $f$ is separated\nand universally closed.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Characterizing proper maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005O","source_file":"topology.tex","source_line":3077,"source_end_line":3095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3077-L3095","statement_sha256":"41f3c60d6194432a988a96608bb3dff31c81b664d7ad9eb70e1a2c57a5501a8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":341,"rank":341,"depth":0,"x":610.71,"y":308.607,"cluster":"topology"},{"id":"stacks:005P","tag":"005P","title":"Characterizing proper maps · Lemma 005P","summary":"Combination of [Bourbaki] and [Bourbaki]. A topological space X is quasi-compact if and only if the projection map Z × X → Z is closed for any topological space Z.","statement_latex":"\\begin{reference}\nCombination of\n\\cite[I, p. 75, Lemme 1]{Bourbaki} and\n\\cite[I, p. 76, Corrolaire 1]{Bourbaki}.\n\\end{reference}\nA topological space $X$ is quasi-compact if and only if the\nprojection map $Z \\times X \\to Z$ is closed for\nany topological space $Z$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Characterizing proper maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005P","source_file":"topology.tex","source_line":3100,"source_end_line":3110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3100-L3110","statement_sha256":"b2a5ba060f62475e49743a425e00ee0161c432193a5d46bf54e4123fc01b86d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":342,"rank":342,"depth":1,"x":659.741,"y":114.844,"cluster":"topology"},{"id":"stacks:005R","tag":"005R","title":"Characterizing proper maps · Theorem 005R","summary":"In [Bourbaki] you can find: (2) ⇔ (4). In [Bourbaki] you can find: (2) ⇒ (1). Let f: X→ Y be a continuous map between topological spaces. The following conditions are equivalent: • The map f is quasi-proper and closed. • The map f is Bourbaki-proper. • The map f is universally closed. • The map f is closed and f^-1(y) is quasi-compact for any y∈ Y.","statement_latex":"\\begin{reference}\nIn \\cite[I, p. 75, Theorem 1]{Bourbaki} you can find:\n(2) $\\Leftrightarrow$ (4).\nIn \\cite[I, p. 77, Proposition 6]{Bourbaki} you can find:\n(2) $\\Rightarrow$ (1).\n\\end{reference}\nLet $f: X\\to Y$ be a continuous map between\ntopological spaces. The following conditions are equivalent:\n\\begin{enumerate}\n\\item The map $f$ is quasi-proper and closed.\n\\item The map $f$ is Bourbaki-proper.\n\\item The map $f$ is universally closed.\n\\item The map $f$ is closed and $f^{-1}(y)$ is quasi-compact for any\n$y\\in Y$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Characterizing proper maps","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005R","source_file":"topology.tex","source_line":3152,"source_end_line":3169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3152-L3169","statement_sha256":"bc1283e42c40e2cfb86d363a885f75be332c5e3b09966686651734ce74afbbbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":343,"rank":343,"depth":2,"x":779.975,"y":286.359,"cluster":"topology"},{"id":"stacks:08YD","tag":"08YD","title":"Characterizing proper maps · Lemma 08YD","summary":"A map from a compact space to a Hausdorff space is universally closed. Let f : X → Y be a continuous map of topological spaces. If X is quasi-compact and Y is Hausdorff, then f is universally closed.","statement_latex":"\\begin{slogan}\nA map from a compact space to a Hausdorff space is universally closed.\n\\end{slogan}\nLet $f : X \\to Y$ be a continuous map of topological spaces.\nIf $X$ is quasi-compact and $Y$ is Hausdorff, then $f$ is\nuniversally closed.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Characterizing proper maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YD","source_file":"topology.tex","source_line":3256,"source_end_line":3264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3256-L3264","statement_sha256":"9bf9ef45ddab446f117d532c7ef0a86832cb6692c90154bd9b09d49174baaf18","origin":"The Stacks Project","memory_eligible":false,"source_rank":344,"rank":344,"depth":3,"x":552.318,"y":227.833,"cluster":"topology"},{"id":"stacks:08YE","tag":"08YE","title":"Characterizing proper maps · Lemma 08YE","summary":"Let f : X → Y be a continuous map of topological spaces. If f is bijective, X is quasi-compact, and Y is Hausdorff, then f is a homeomorphism.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nIf $f$ is bijective, $X$ is quasi-compact, and $Y$ is Hausdorff,\nthen $f$ is a homeomorphism.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Characterizing proper maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YE","source_file":"topology.tex","source_line":3280,"source_end_line":3285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3280-L3285","statement_sha256":"a9a96679d54479ea144fc87c31f77792919a90acefa2b0da113530478e6c0acc","origin":"The Stacks Project","memory_eligible":false,"source_rank":345,"rank":345,"depth":1,"x":768.262,"y":141.409,"cluster":"topology"},{"id":"stacks:005U","tag":"005U","title":"Jacobson spaces · Definition 005U","summary":"Let X be a topological space. Let X_0 be the set of closed points of X. We say that X is Jacobson if every closed subset Z ⊂ X is the closure of Z ∩ X_0.","statement_latex":"Let $X$ be a topological space.\nLet $X_0$ be the set of closed points of $X$.\nWe say that $X$ is {\\it Jacobson} if every\nclosed subset $Z \\subset X$ is the closure\nof $Z \\cap X_0$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Jacobson spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005U","source_file":"topology.tex","source_line":3310,"source_end_line":3317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3310-L3317","statement_sha256":"4455f076a4f56d35897bc7bb1e27837ce28b8ca43302323708d6a718732941f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":346,"rank":346,"depth":0,"x":678.109,"y":328.532,"cluster":"topology"},{"id":"stacks:005V","tag":"005V","title":"Jacobson spaces · Lemma 005V","summary":"Let X be a topological space. Let X_0 be the set of closed points of X. Suppose that for every point x∈ X the intersection X_0 ∩ overline(x) is dense in overline(x). Then X is Jacobson.","statement_latex":"Let $X$ be a topological space. Let $X_0$ be the set\nof closed points of $X$.\nSuppose that for every point $x\\in X$\nthe intersection $X_0 \\cap \\overline{\\{x\\}}$ is dense in $\\overline{\\{x\\}}$.\nThen $X$ is Jacobson.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005V","source_file":"topology.tex","source_line":3334,"source_end_line":3341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3334-L3341","statement_sha256":"ec6898875de173c17dbf62cd657a8ce6de7854a0fdad262eccb1b506264c0e20","origin":"The Stacks Project","memory_eligible":false,"source_rank":347,"rank":347,"depth":0,"x":593.723,"y":138.526,"cluster":"topology"},{"id":"stacks:02I7","tag":"02I7","title":"Jacobson spaces · Lemma 02I7","summary":"Let X be a Kolmogorov topological space with a basis of quasi-compact open sets. If X is not Jacobson, then there exists a non-closed point x ∈ X such that (x) is locally closed.","statement_latex":"Let $X$ be a Kolmogorov topological space with a basis of quasi-compact\nopen sets.\nIf $X$ is not Jacobson, then there exists a non-closed point\n$x \\in X$ such that $\\{x\\}$ is locally closed.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02I7","source_file":"topology.tex","source_line":3352,"source_end_line":3358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3352-L3358","statement_sha256":"33da910fe1503e8e4286afe2190c6cbd7febb0e358276ff602cf13dc5011d7de","origin":"The Stacks Project","memory_eligible":false,"source_rank":348,"rank":348,"depth":2,"x":809.724,"y":231.172,"cluster":"topology"},{"id":"stacks:005W","tag":"005W","title":"Jacobson spaces · Lemma 005W","summary":"Let X be a topological space. Let X = ⋃ U_i be an open covering. Then X is Jacobson if and only if each U_i is Jacobson. Moreover, in this case X_0 = ⋃ U_i, 0.","statement_latex":"Let $X$ be a topological space.\nLet $X = \\bigcup U_i$ be an open covering.\nThen $X$ is Jacobson if and only if each $U_i$ is Jacobson.\nMoreover, in this case $X_0 = \\bigcup U_{i, 0}$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005W","source_file":"topology.tex","source_line":3371,"source_end_line":3377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3371-L3377","statement_sha256":"cd9a525c32f1b864d71cbf9dc9ecf56c212321477250b014d0f2fa33bba8ed3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":349,"rank":349,"depth":0,"x":574.888,"y":285.664,"cluster":"topology"},{"id":"stacks:005X","tag":"005X","title":"Jacobson spaces · Lemma 005X","summary":"Let X be Jacobson. The following types of subsets T ⊂ X are Jacobson: • Open subspaces. • Closed subspaces. • Locally closed subspaces. • Unions of locally closed subspaces. • Constructible sets. • Any subset T ⊂ X which locally on X is a union of locally closed subsets. In each of these cases closed points of T are closed in X.","statement_latex":"Let $X$ be Jacobson. The following types of subsets $T \\subset X$\nare Jacobson:\n\\begin{enumerate}\n\\item Open subspaces.\n\\item Closed subspaces.\n\\item Locally closed subspaces.\n\\item Unions of locally closed subspaces.\n\\item Constructible sets.\n\\item Any subset $T \\subset X$ which locally on $X$\nis a union of locally closed subsets.\n\\end{enumerate}\nIn each of these cases closed points of $T$ are\nclosed in $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005X","source_file":"topology.tex","source_line":3424,"source_end_line":3439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3424-L3439","statement_sha256":"c37be496482d2936cf134296b965167f1f64da6ee18b03b4c434d498d2531a73","origin":"The Stacks Project","memory_eligible":false,"source_rank":350,"rank":350,"depth":0,"x":704.812,"y":111.456,"cluster":"topology"},{"id":"stacks:07JU","tag":"07JU","title":"Jacobson spaces · Lemma 07JU","summary":"A finite Jacobson space is discrete. A Jacobson space with finitely many closed points is discrete.","statement_latex":"A finite Jacobson space is discrete.\nA Jacobson space with finitely many closed points is discrete.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JU","source_file":"topology.tex","source_line":3460,"source_end_line":3464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3460-L3464","statement_sha256":"45c99699bb105fbaf67a4606d2612ae29ef74373cbd2aee0f4ebe6071262da54","origin":"The Stacks Project","memory_eligible":false,"source_rank":351,"rank":351,"depth":0,"x":749.296,"y":314.533,"cluster":"topology"},{"id":"stacks:005Z","tag":"005Z","title":"Jacobson spaces · Lemma 005Z","summary":"For Jacobson spaces, closed points see everything about the topology. Suppose X is a Jacobson topological space. Let X_0 be the set of closed points of X. There is a bijective, inclusion preserving correspondence (finite unions loc. closed subsets of X) ↔ (finite unions loc. closed subsets of X_0) given by E ↦ E ∩ X_0. This correspondence preserves the subsets of locally closed, of open and of closed subsets.","statement_latex":"\\begin{slogan}\nFor Jacobson spaces, closed points see everything about the topology.\n\\end{slogan}\nSuppose $X$ is a Jacobson topological space.\nLet $X_0$ be the set of closed points of $X$.\nThere is a bijective, inclusion preserving correspondence\n$$\n\\{\\text{finite unions loc. closed subsets of } X\\}\n\\leftrightarrow\n\\{\\text{finite unions loc. closed subsets of } X_0\\}\n$$\ngiven by $E \\mapsto E \\cap X_0$. This correspondence preserves\nthe subsets of locally closed, of open and of closed subsets.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005Z","source_file":"topology.tex","source_line":3476,"source_end_line":3491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3476-L3491","statement_sha256":"8931cd62ba3e6fb180b4deb0a412baaf2099695049d1e0188e6ed01c4d42e08c","origin":"The Stacks Project","memory_eligible":false,"source_rank":352,"rank":352,"depth":0,"x":552.325,"y":189.48,"cluster":"topology"},{"id":"stacks:005Y","tag":"005Y","title":"Jacobson spaces · Lemma 005Y","summary":"Suppose X is a Jacobson topological space. Let X_0 be the set of closed points of X. There is a bijective, inclusion preserving correspondence (constructible subsets of X) ↔ (constructible subsets of X_0) given by E ↦ E ∩ X_0. This correspondence preserves the subset of retrocompact open subsets, as well as complements of these.","statement_latex":"Suppose $X$ is a Jacobson topological space.\nLet $X_0$ be the set of closed points of $X$.\nThere is a bijective, inclusion preserving correspondence\n$$\n\\{\\text{constructible subsets of } X\\}\n\\leftrightarrow\n\\{\\text{constructible subsets of } X_0\\}\n$$\ngiven by $E \\mapsto E \\cap X_0$. This correspondence preserves\nthe subset of retrocompact open subsets, as well as complements\nof these.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/005Y","source_file":"topology.tex","source_line":3501,"source_end_line":3514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3501-L3514","statement_sha256":"67b813bdd05406ae248e85b055997f1aaf0c1526d2753d9fabd4d8b4e0fb51cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":353,"rank":353,"depth":1,"x":799.203,"y":169.843,"cluster":"topology"},{"id":"stacks:0061","tag":"0061","title":"Specialization · Definition 0061","summary":"Let X be a topological space. • If x, x' ∈ X then we say x is a specialization of x', or x' is a generalization of x if x ∈ overline(x'). Notation: x' leadsto x. • A subset T ⊂ X is stable under specialization if for all x' ∈ T and every specialization x' leadsto x we have x ∈ T. • A subset T ⊂ X is stable under generalization if for all x ∈ T and every generalization x' leadsto x we have x' ∈ T.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item If $x, x' \\in X$ then we say $x$ is a {\\it specialization} of $x'$,\nor $x'$ is a {\\it generalization} of $x$ if $x \\in \\overline{\\{x'\\}}$.\nNotation: $x' \\leadsto x$.\n\\item A subset $T \\subset X$ is {\\it stable under specialization}\nif for all $x' \\in T$ and every specialization $x' \\leadsto x$ we have\n$x \\in T$.\n\\item A subset $T \\subset X$ is {\\it stable under generalization}\nif for all $x \\in T$ and every generalization $x' \\leadsto x$ we have\n$x' \\in T$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0061","source_file":"topology.tex","source_line":3551,"source_end_line":3565,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3551-L3565","statement_sha256":"ae6deca7e6bb9e4b5033ee3706f32f0098aafa10907ee6c97e9d4fd9f1b483e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":354,"rank":354,"depth":0,"x":632.232,"y":325.074,"cluster":"topology"},{"id":"stacks:0062","tag":"0062","title":"Specialization · Lemma 0062","summary":"Let X be a topological space. • Any closed subset of X is stable under specialization. • Any open subset of X is stable under generalization. • A subset T ⊂ X is stable under specialization if and only if the complement T^c is stable under generalization.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item Any closed subset of $X$ is stable under specialization.\n\\item Any open subset of $X$ is stable under generalization.\n\\item A subset $T \\subset X$ is stable under specialization\nif and only if\nthe complement $T^c$ is stable under generalization.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0062","source_file":"topology.tex","source_line":3567,"source_end_line":3577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3567-L3577","statement_sha256":"1f8eac9ca2033c52766ffd9ac53069cd2b8d501d707ed0c179ef93755393cb7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":355,"rank":355,"depth":0,"x":630.516,"y":114.97,"cluster":"topology"},{"id":"stacks:0EES","tag":"0EES","title":"Specialization · Lemma 0EES","summary":"Let T ⊂ X be a subset of a topological space X. The following are equivalent • T is stable under specialization, and • T is a (directed) union of closed subsets of X.","statement_latex":"Let $T \\subset X$ be a subset of a topological space $X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $T$ is stable under specialization, and\n\\item $T$ is a (directed) union of closed subsets of $X$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EES","source_file":"topology.tex","source_line":3594,"source_end_line":3602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3594-L3602","statement_sha256":"4a64e2132dd96e7735a3089254e0dbd2440d878b086fec9cc095aaea4cf1724c","origin":"The Stacks Project","memory_eligible":false,"source_rank":356,"rank":356,"depth":0,"x":801.462,"y":269.578,"cluster":"topology"},{"id":"stacks:0063","tag":"0063","title":"Specialization · Definition 0063","summary":"Let f : X → Y be a continuous map of topological spaces. • We say that specializations lift along f or that f is specializing if given y' leadsto y in Y and any x'∈ X with f(x') = y' there exists a specialization x' leadsto x of x' in X such that f(x) = y. • We say that generalizations lift along f or that f is generalizing if given y' leadsto y in Y and any x∈ X with f(x) = y there exists a generalization x' leadsto x of x in X such that f(x') = y'.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\n\\begin{enumerate}\n\\item We say that {\\it specializations lift along $f$} or that $f$ is\n{\\it specializing} if given $y' \\leadsto y$ in $Y$ and any $x'\\in X$ with\n$f(x') = y'$ there exists a specialization $x' \\leadsto x$ of $x'$ in $X$ such\nthat $f(x) = y$.\n\\item We say that {\\it generalizations lift along $f$} or that $f$ is\n{\\it generalizing} if given $y' \\leadsto y$ in $Y$ and any $x\\in X$ with\n$f(x) = y$ there exists a generalization $x' \\leadsto x$ of $x$ in $X$ such\nthat $f(x') = y'$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0063","source_file":"topology.tex","source_line":3614,"source_end_line":3627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3614-L3627","statement_sha256":"c5c50d38e14f6a5adb36392972e2a4a0e87b1d9921c8590ab9eeb232e24064c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":357,"rank":357,"depth":0,"x":550.025,"y":252.498,"cluster":"topology"},{"id":"stacks:0064","tag":"0064","title":"Specialization · Lemma 0064","summary":"Suppose f : X → Y and g : Y → Z are continuous maps of topological spaces. If specializations lift along both f and g then specializations lift along g ∘ f. Similarly for \"generalizations lift along\".","statement_latex":"Suppose $f : X \\to Y$ and $g : Y \\to Z$ are continuous maps\nof topological spaces. If specializations lift along both $f$ and $g$\nthen specializations lift along $g \\circ f$. Similarly for\n``generalizations lift along''.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0064","source_file":"topology.tex","source_line":3629,"source_end_line":3635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3629-L3635","statement_sha256":"d661ccf4c062cf101567ceaa2d8f6bf5a8cefaa9e174f6d8c7bda9e8cbb14d1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":358,"rank":358,"depth":0,"x":749.997,"y":121.879,"cluster":"topology"},{"id":"stacks:0065","tag":"0065","title":"Specialization · Lemma 0065","summary":"Let f : X → Y be a continuous map of topological spaces. • If specializations lift along f, and if T ⊂ X is stable under specialization, then f(T) ⊂ Y is stable under specialization. • If generalizations lift along f, and if T ⊂ X is stable under generalization, then f(T) ⊂ Y is stable under generalization.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\n\\begin{enumerate}\n\\item If specializations lift along $f$, and if $T \\subset X$\nis stable under specialization, then $f(T) \\subset Y$ is\nstable under specialization.\n\\item If generalizations lift along $f$, and if $T \\subset X$\nis stable under generalization, then $f(T) \\subset Y$ is\nstable under generalization.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0065","source_file":"topology.tex","source_line":3647,"source_end_line":3658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3647-L3658","statement_sha256":"077bcf0ece86e155f34c34f2f22c0bd616738e86b82e0f74fabfbb4a1fab6fd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":359,"rank":359,"depth":0,"x":707.403,"y":332.533,"cluster":"topology"},{"id":"stacks:0066","tag":"0066","title":"Specialization · Lemma 0066","summary":"Let f : X → Y be a continuous map of topological spaces. • If f is closed then specializations lift along f. • If f is open, X is a Noetherian topological space, each irreducible closed subset of X has a generic point, and Y is Kolmogorov then generalizations lift along f.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\n\\begin{enumerate}\n\\item If $f$ is closed then specializations lift along $f$.\n\\item If $f$ is open, $X$ is a Noetherian topological space,\neach irreducible closed subset of $X$ has a generic point,\nand $Y$ is Kolmogorov then generalizations lift along $f$.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0066","source_file":"topology.tex","source_line":3671,"source_end_line":3680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3671-L3680","statement_sha256":"095269449027c6647553f12e2e62dc1328fb6c322ca00dc2d3aede5504f8642b","origin":"The Stacks Project","memory_eligible":false,"source_rank":360,"rank":360,"depth":3,"x":568.845,"y":152.294,"cluster":"topology"},{"id":"stacks:06NA","tag":"06NA","title":"Specialization · Lemma 06NA","summary":"Suppose that s, t : R → U and π : U → X are continuous maps of topological spaces such that • π is open, • U is sober, • s, t have finite fibres, • generalizations lift along s, t, • (t, s)(R) ⊂ U × U is an equivalence relation on U and X is the quotient of U by this equivalence relation (as a set). Then X is Kolmogorov.","statement_latex":"Suppose that $s, t : R \\to U$ and $\\pi : U \\to X$ are continuous maps\nof topological spaces such that\n\\begin{enumerate}\n\\item $\\pi$ is open,\n\\item $U$ is sober,\n\\item $s, t$ have finite fibres,\n\\item generalizations lift along $s, t$,\n\\item $(t, s)(R) \\subset U \\times U$ is an equivalence relation on $U$ and\n$X$ is the quotient of $U$ by this equivalence relation (as a set).\n\\end{enumerate}\nThen $X$ is Kolmogorov.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NA","source_file":"topology.tex","source_line":3715,"source_end_line":3728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3715-L3728","statement_sha256":"65ba24795f9f3d5bbd599d9f0b0a784e4ab0e41cdb1c07f02fa45de0a6629d5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":361,"rank":361,"depth":0,"x":816.964,"y":206.8,"cluster":"topology"},{"id":"stacks:02JF","tag":"02JF","title":"Specialization · Lemma 02JF","summary":"Let f : X → Y be a morphism of topological spaces. Suppose that Y is a sober topological space, and f is surjective. If either specializations or generalizations lift along f, then dim(X) ≥ dim(Y).","statement_latex":"Let $f : X \\to Y$ be a morphism of topological spaces.\nSuppose that $Y$ is a sober topological space, and $f$ is surjective.\nIf either specializations or generalizations lift along $f$, then\n$\\dim(X) \\geq \\dim(Y)$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JF","source_file":"topology.tex","source_line":3765,"source_end_line":3771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3765-L3771","statement_sha256":"8947e3714a2b1b721e03bfc126cc56d62f3b8581dba68ca95dafca7fac9110c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":362,"rank":362,"depth":0,"x":589.257,"y":307.801,"cluster":"topology"},{"id":"stacks:0542","tag":"0542","title":"Specialization · Lemma 0542","summary":"Let X be a Noetherian sober topological space. Let E ⊂ X be a subset of X. • If E is constructible and stable under specialization, then E is closed. • If E is constructible and stable under generalization, then E is open.","statement_latex":"Let $X$ be a Noetherian sober topological space.\nLet $E \\subset X$ be a subset of $X$.\n\\begin{enumerate}\n\\item If $E$ is constructible and stable under specialization, then\n$E$ is closed.\n\\item If $E$ is constructible and stable under generalization, then\n$E$ is open.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Specialization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0542","source_file":"topology.tex","source_line":3797,"source_end_line":3807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3797-L3807","statement_sha256":"fe3310ce478c191e58e395d15b7e81830f00187186c332a11ba226ad088f9f18","origin":"The Stacks Project","memory_eligible":false,"source_rank":363,"rank":363,"depth":9,"x":676.291,"y":103.303,"cluster":"topology"},{"id":"stacks:02I9","tag":"02I9","title":"Dimension functions · Definition 02I9","summary":"Let X be a topological space. • Let x, y ∈ X, x not = y. Suppose x leadsto y, that is y is a specialization of x. We say y is an immediate specialization of x if there is no z ∈ X setminus (x, y) with x leadsto z and z leadsto y. • A map δ : X → Z is called a dimension function if • whenever x leadsto y and x not = y we have δ(x) > δ(y), and • for every immediate specialization x leadsto y in X we have δ(x) = δ(y) + 1.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item  Let $x, y \\in X$, $x \\not = y$. Suppose $x \\leadsto y$, that\nis $y$ is a specialization of $x$.\nWe say $y$ is an {\\it immediate specialization}\nof $x$ if there is no\n$z \\in X \\setminus \\{x, y\\}$ with $x \\leadsto z$ and $z \\leadsto y$.\n\\item A map $\\delta : X \\to \\mathbf{Z}$ is called a\n{\\it dimension function}\\footnote{This is likely nonstandard\nnotation. This notion is usually introduced only for (locally) Noetherian\nschemes, in which case condition (a) is implied by (b).} if\n\\begin{enumerate}\n\\item whenever $x \\leadsto y$ and $x \\not = y$\nwe have $\\delta(x) > \\delta(y)$, and\n\\item for every immediate specialization $x \\leadsto y$ in $X$\nwe have $\\delta(x) = \\delta(y) + 1$.\n\\end{enumerate}\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Dimension functions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02I9","source_file":"topology.tex","source_line":3839,"source_end_line":3859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3839-L3859","statement_sha256":"e30d21321e9926e92e3a4d041433494a8ff211f8351e8271a87c2d4b84d8e81a","origin":"The Stacks Project","memory_eligible":false,"source_rank":364,"rank":364,"depth":0,"x":776.961,"y":304.277,"cluster":"topology"},{"id":"stacks:02IA","tag":"02IA","title":"Dimension functions · Lemma 02IA","summary":"Let X be a topological space. If X is sober and has a dimension function, then X is catenary. Moreover, for any x leadsto y we have δ(x) - δ(y) = codim(overline(y), overline(x)).","statement_latex":"Let $X$ be a topological space. If $X$ is sober and has a dimension\nfunction, then $X$ is catenary. Moreover, for any $x \\leadsto y$\nwe have\n$$\n\\delta(x) - \\delta(y) =\n\\text{codim}\\left(\\overline{\\{y\\}}, \\ \\overline{\\{x\\}}\\right).\n$$","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IA","source_file":"topology.tex","source_line":3865,"source_end_line":3874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3865-L3874","statement_sha256":"495e83407a3ff9e969f53ebb923bc0db5bcfc8e4d431334349c325ca0e7d03f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":365,"rank":365,"depth":0,"x":540.182,"y":212.845,"cluster":"topology"},{"id":"stacks:02IB","tag":"02IB","title":"Dimension functions · Lemma 02IB","summary":"Let X be a topological space. Let δ, δ' be two dimension functions on X. If X is locally Noetherian and sober then δ - δ' is locally constant on X.","statement_latex":"Let $X$ be a topological space.\nLet $\\delta$, $\\delta'$ be two dimension functions on $X$.\nIf $X$ is locally Noetherian and sober then $\\delta - \\delta'$ is\nlocally constant on $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IB","source_file":"topology.tex","source_line":3892,"source_end_line":3898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3892-L3898","statement_sha256":"24ce0f71a377562d1be00e3f3c28a9194772225d3884f78ac5f4c34e19b59b9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":366,"rank":366,"depth":3,"x":789.277,"y":145.652,"cluster":"topology"},{"id":"stacks:02IC","tag":"02IC","title":"Dimension functions · Lemma 02IC","summary":"Let X be locally Noetherian, sober and catenary. Then any point has an open neighbourhood U ⊂ X which has a dimension function.","statement_latex":"Let $X$ be locally Noetherian, sober and catenary.\nThen any point has an open neighbourhood\n$U \\subset X$ which has a dimension function.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IC","source_file":"topology.tex","source_line":3923,"source_end_line":3928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3923-L3928","statement_sha256":"c5e2bf27d7b1c6df016aae92886e1c9cf03ce8b6f8ed5b06eeb6dc35683bfffb","origin":"The Stacks Project","memory_eligible":false,"source_rank":367,"rank":367,"depth":4,"x":659.13,"y":337.281,"cluster":"topology"},{"id":"stacks:03HN","tag":"03HN","title":"Nowhere dense sets · Definition 03HN","summary":"Let X be a topological space. • Given a subset T ⊂ X the interior of T is the largest open subset of X contained in T. • A subset T ⊂ X is called nowhere dense if the closure of T has empty interior.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item Given a subset $T \\subset X$ the {\\it interior} of $T$ is the\nlargest open subset of $X$ contained in $T$.\n\\item A subset $T \\subset X$ is called {\\it nowhere dense} if the closure of\n$T$ has empty interior.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Nowhere dense sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HN","source_file":"topology.tex","source_line":3993,"source_end_line":4002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L3993-L4002","statement_sha256":"1cf8742bf2b38ca0f0c71b134e82a33d501b23b8defa8ef87816e489a627b91b","origin":"The Stacks Project","memory_eligible":false,"source_rank":368,"rank":368,"depth":0,"x":600.773,"y":121.3,"cluster":"topology"},{"id":"stacks:03HO","tag":"03HO","title":"Nowhere dense sets · Lemma 03HO","summary":"Let X be a topological space. The union of a finite number of nowhere dense sets is a nowhere dense set.","statement_latex":"Let $X$ be a topological space. The union of a finite number of nowhere\ndense sets is a nowhere dense set.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Nowhere dense sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HO","source_file":"topology.tex","source_line":4004,"source_end_line":4008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4004-L4008","statement_sha256":"b4d7107267b6e7c434370f7cba6835ae584aec3b92cbd668d0816f9cac4bb102","origin":"The Stacks Project","memory_eligible":false,"source_rank":369,"rank":369,"depth":0,"x":818.317,"y":247.93,"cluster":"topology"},{"id":"stacks:03J0","tag":"03J0","title":"Nowhere dense sets · Lemma 03J0","summary":"Let X be a topological space. Let U ⊂ X be an open. Let T ⊂ U be a subset. If T is nowhere dense in U, then T is nowhere dense in X.","statement_latex":"Let $X$ be a topological space.\nLet $U \\subset X$ be an open.\nLet $T \\subset U$ be a subset.\nIf $T$ is nowhere dense in $U$, then $T$ is nowhere dense in $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Nowhere dense sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03J0","source_file":"topology.tex","source_line":4021,"source_end_line":4027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4021-L4027","statement_sha256":"0c8b8eb3ca2668c999e1b7c22f108ea251667924b2219ebbbbfd9310e8fd5322","origin":"The Stacks Project","memory_eligible":false,"source_rank":370,"rank":370,"depth":0,"x":555.077,"y":278.107,"cluster":"topology"},{"id":"stacks:03HP","tag":"03HP","title":"Nowhere dense sets · Lemma 03HP","summary":"Let X be a topological space. Let X = ⋃ U_i be an open covering. Let T ⊂ X be a subset. If T ∩ U_i is nowhere dense in U_i for all i, then T is nowhere dense in X.","statement_latex":"Let $X$ be a topological space.\nLet $X = \\bigcup U_i$ be an open covering.\nLet $T \\subset X$ be a subset.\nIf $T \\cap U_i$ is nowhere dense in $U_i$ for all $i$,\nthen $T$ is nowhere dense in $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Nowhere dense sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HP","source_file":"topology.tex","source_line":4039,"source_end_line":4046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4039-L4046","statement_sha256":"8d299c59bc58d2d18a525a9ad6056bcd6ccd571af73c052bb461f6bae3f1c26c","origin":"The Stacks Project","memory_eligible":false,"source_rank":371,"rank":371,"depth":0,"x":725.558,"y":105.842,"cluster":"topology"},{"id":"stacks:03HQ","tag":"03HQ","title":"Nowhere dense sets · Lemma 03HQ","summary":"Let f : X → Y be a continuous map of topological spaces. Let T ⊂ X be a subset. If f is a homeomorphism of X onto a closed subset of Y and T is nowhere dense in X, then also f(T) is nowhere dense in Y.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $T \\subset X$ be a subset.\nIf $f$ is a homeomorphism of $X$ onto a closed subset of $Y$\nand $T$ is nowhere dense in $X$, then also $f(T)$ is nowhere dense in $Y$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Nowhere dense sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HQ","source_file":"topology.tex","source_line":4061,"source_end_line":4067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4061-L4067","statement_sha256":"3509b3482bcc057bcd764fa14210acf67d68b301cf335a0915b043c88703de21","origin":"The Stacks Project","memory_eligible":false,"source_rank":372,"rank":372,"depth":0,"x":738.425,"y":330.438,"cluster":"topology"},{"id":"stacks:03HR","tag":"03HR","title":"Nowhere dense sets · Lemma 03HR","summary":"Let f : X → Y be a continuous map of topological spaces. Let T ⊂ Y be a subset. If f is open and T is a closed nowhere dense subset of Y, then also f^-1(T) is a closed nowhere dense subset of X. If f is surjective and open, then T is closed nowhere dense if and only if f^-1(T) is closed nowhere dense.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $T \\subset Y$ be a subset.\nIf $f$ is open and $T$ is a closed nowhere dense subset of $Y$,\nthen also $f^{-1}(T)$ is a closed nowhere dense subset of $X$.\nIf $f$ is surjective and open, then\n$T$ is closed nowhere dense if and only\nif $f^{-1}(T)$ is closed nowhere dense.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Nowhere dense sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HR","source_file":"topology.tex","source_line":4078,"source_end_line":4087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4078-L4087","statement_sha256":"2e4c1a26e01a17dd2ede7c6d551c62e7f55a5d06bee3afe8b815e0d9731e2240","origin":"The Stacks Project","memory_eligible":false,"source_rank":373,"rank":373,"depth":0,"x":547.621,"y":171.538,"cluster":"topology"},{"id":"stacks:08ZX","tag":"08ZX","title":"Profinite spaces · Definition 08ZX","summary":"A topological space is profinite if it is homeomorphic to a limit of a diagram of finite discrete spaces.","statement_latex":"A topological space is {\\it profinite} if it is homeomorphic to a limit\nof a diagram of finite discrete spaces.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Profinite spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZX","source_file":"topology.tex","source_line":4124,"source_end_line":4128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4124-L4128","statement_sha256":"0eb50b193fd443fec997c829ca47512e70c718edf8557772c71962fd7074d078","origin":"The Stacks Project","memory_eligible":false,"source_rank":374,"rank":374,"depth":0,"x":817.086,"y":180.476,"cluster":"topology"},{"id":"stacks:08ZY","tag":"08ZY","title":"Profinite spaces · Lemma 08ZY","summary":"Let X be a topological space. The following are equivalent • X is a profinite space, and • X is Hausdorff, quasi-compact, and totally disconnected. If this is true, then X is a cofiltered limit of finite discrete spaces.","statement_latex":"Let $X$ be a topological space.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is a profinite space, and\n\\item $X$ is Hausdorff, quasi-compact, and totally disconnected.\n\\end{enumerate}\nIf this is true, then $X$ is a cofiltered limit of finite discrete\nspaces.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Profinite spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZY","source_file":"topology.tex","source_line":4133,"source_end_line":4143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4133-L4143","statement_sha256":"d98dbe0c8ed279a1d490eb2f8070f6da0db64692c47ff292854da3ddc03142a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":375,"rank":375,"depth":3,"x":610.447,"y":327.319,"cluster":"topology"},{"id":"stacks:0ET8","tag":"0ET8","title":"Profinite spaces · Lemma 0ET8","summary":"A limit of profinite spaces is profinite.","statement_latex":"A limit of profinite spaces is profinite.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Profinite spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET8","source_file":"topology.tex","source_line":4181,"source_end_line":4184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4181-L4184","statement_sha256":"53fc83be1a0d8dd059b92dffde8146fc1eadee2bfebc2cb023c3a2f1f7d5980a","origin":"The Stacks Project","memory_eligible":false,"source_rank":376,"rank":376,"depth":4,"x":644.857,"y":100.971,"cluster":"topology"},{"id":"stacks:08ZZ","tag":"08ZZ","title":"Profinite spaces · Lemma 08ZZ","summary":"Let X be a profinite space. Every open covering of X has a refinement by a finite covering X = coprod U_i with U_i open and closed.","statement_latex":"Let $X$ be a profinite space. Every open covering of $X$ has a refinement\nby a finite covering $X = \\coprod U_i$ with $U_i$ open and closed.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Profinite spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ZZ","source_file":"topology.tex","source_line":4206,"source_end_line":4210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4206-L4210","statement_sha256":"9d4636bcfc449de839c7199958ca6fe5e0ed06af5019119ed10e210af8775dbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":377,"rank":377,"depth":4,"x":802.07,"y":288.073,"cluster":"topology"},{"id":"stacks:0900","tag":"0900","title":"Profinite spaces · Lemma 0900","summary":"Let X be a topological space. If X is quasi-compact and every connected component of X is the intersection of the open and closed subsets containing it, then π_0(X) is a profinite space.","statement_latex":"Let $X$ be a topological space. If $X$ is quasi-compact\nand every connected component of $X$ is the intersection\nof the open and closed subsets containing it, then $\\pi_0(X)$\nis a profinite space.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Profinite spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0900","source_file":"topology.tex","source_line":4231,"source_end_line":4237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4231-L4237","statement_sha256":"3389bb9e94dc7dd784556879bf4877682466b39df6ae7a5ac7f6a53ebfa3fe76","origin":"The Stacks Project","memory_eligible":false,"source_rank":378,"rank":378,"depth":4,"x":534.731,"y":239.137,"cluster":"topology"},{"id":"stacks:08YG","tag":"08YG","title":"Spectral spaces · Definition 08YG","summary":"A topological space X is called spectral if it is sober, quasi-compact, the intersection of two quasi-compact opens is quasi-compact, and the collection of quasi-compact opens forms a basis for the topology. A continuous map f : X → Y of spectral spaces is called spectral if the inverse image of a quasi-compact open is quasi-compact.","statement_latex":"A topological space $X$ is called {\\it spectral} if it is sober,\nquasi-compact, the intersection of two quasi-compact opens is\nquasi-compact, and the collection of quasi-compact opens forms a\nbasis for the topology. A continuous map $f : X \\to Y$ of spectral\nspaces is called {\\it spectral} if the inverse image of a quasi-compact\nopen is quasi-compact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YG","source_file":"topology.tex","source_line":4275,"source_end_line":4283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4275-L4283","statement_sha256":"19a71e25122226760041547831d4d5b2cc1f8d8b1476248af43a89c83b4ed534","origin":"The Stacks Project","memory_eligible":false,"source_rank":379,"rank":379,"depth":0,"x":772.054,"y":123.118,"cluster":"topology"},{"id":"stacks:0901","tag":"0901","title":"Spectral spaces · Lemma 0901","summary":"Let X be a spectral space. The constructible topology is Hausdorff, totally disconnected, and quasi-compact.","statement_latex":"Let $X$ be a spectral space. The constructible topology is\nHausdorff, totally disconnected, and quasi-compact.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0901","source_file":"topology.tex","source_line":4304,"source_end_line":4308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4304-L4308","statement_sha256":"2ecc3972cf95d33c27f4be1ab3c877ec3d1d34844e79f352890ed06bd1474eff","origin":"The Stacks Project","memory_eligible":false,"source_rank":380,"rank":380,"depth":1,"x":690.066,"y":344.106,"cluster":"topology"},{"id":"stacks:0A2S","tag":"0A2S","title":"Spectral spaces · Lemma 0A2S","summary":"Let f : X → Y be a spectral map of spectral spaces. Then • f is continuous in the constructible topology, • the fibres of f are quasi-compact, and • the image is closed in the constructible topology.","statement_latex":"Let $f : X \\to Y$ be a spectral map of spectral spaces. Then\n\\begin{enumerate}\n\\item $f$ is continuous in the constructible topology,\n\\item the fibres of $f$ are quasi-compact, and\n\\item the image is closed in the constructible topology.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2S","source_file":"topology.tex","source_line":4371,"source_end_line":4379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4371-L4379","statement_sha256":"1e063ab859df2e580add8697714b85987302a501d9ab55667b78facd8acd6231","origin":"The Stacks Project","memory_eligible":false,"source_rank":381,"rank":381,"depth":2,"x":572.402,"y":133.899,"cluster":"topology"},{"id":"stacks:0G1J","tag":"0G1J","title":"Spectral spaces · Lemma 0G1J","summary":"Let X and Y be spectral spaces. Let f : X → Y be a continuous map. Then f is spectral if and only if f is continuous in the constructible topology.","statement_latex":"Let $X$ and $Y$ be spectral spaces. Let $f : X \\to Y$ be a continuous map.\nThen $f$ is spectral if and only if $f$ is continuous in the constructible\ntopology.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1J","source_file":"topology.tex","source_line":4403,"source_end_line":4408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4403-L4408","statement_sha256":"79e8973452de1d12e54d83e645ac4e62462fcfd516b629f0ced86608df945402","origin":"The Stacks Project","memory_eligible":false,"source_rank":382,"rank":382,"depth":3,"x":829.095,"y":222.444,"cluster":"topology"},{"id":"stacks:0902","tag":"0902","title":"Spectral spaces · Lemma 0902","summary":"Let X be a spectral space. Let E ⊂ X be closed in the constructible topology (for example constructible or closed). Then E with the induced topology is a spectral space.","statement_latex":"Let $X$ be a spectral space. Let $E \\subset X$ be closed in the constructible\ntopology (for example constructible or closed). Then $E$ with the induced\ntopology is a spectral space.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0902","source_file":"topology.tex","source_line":4427,"source_end_line":4432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4427-L4432","statement_sha256":"92ed00bef65a7c673cce99b4c33a18cf3b0412175c7d06622ab7a4a5c14b90a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":383,"rank":383,"depth":2,"x":567.71,"y":303.082,"cluster":"topology"},{"id":"stacks:0903","tag":"0903","title":"Spectral spaces · Lemma 0903","summary":"Let X be a spectral space. Let E ⊂ X be a subset closed in the constructible topology (for example constructible). • If x ∈ overlineE, then x is the specialization of a point of E. • If E is stable under specialization, then E is closed. • If E' ⊂ X is open in the constructible topology (for example constructible) and stable under generalization, then E' is open.","statement_latex":"Let $X$ be a spectral space. Let $E \\subset X$ be a subset closed\nin the constructible topology (for example constructible).\n\\begin{enumerate}\n\\item If $x \\in \\overline{E}$, then $x$ is the specialization of a point of\n$E$.\n\\item If $E$ is stable under specialization, then $E$ is closed.\n\\item If $E' \\subset X$ is open in the constructible topology\n(for example constructible) and stable under generalization, then $E'$ is open.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0903","source_file":"topology.tex","source_line":4460,"source_end_line":4471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4460-L4471","statement_sha256":"a15431c3e885797a81bfbc7f9ae8fd285751d6e583cc51fe9c7cddbae5f74792","origin":"The Stacks Project","memory_eligible":false,"source_rank":384,"rank":384,"depth":2,"x":696.043,"y":94.595,"cluster":"topology"},{"id":"stacks:0904","tag":"0904","title":"Spectral spaces · Lemma 0904","summary":"Let X be a spectral space. Let x, y ∈ X. Then either there exists a third point specializing to both x and y, or there exist disjoint open neighbourhoods containing x and y.","statement_latex":"Let $X$ be a spectral space. Let $x, y \\in X$. Then either there exists\na third point specializing to both $x$ and $y$, or there exist disjoint\nopen neighbourhoods containing $x$ and $y$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0904","source_file":"topology.tex","source_line":4497,"source_end_line":4502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4497-L4502","statement_sha256":"a10af2fb29ee7935c15da8ca93a72d8d1c85f22acfadd0b88ee27d62fd9cfb2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":385,"rank":385,"depth":2,"x":769.319,"y":321.919,"cluster":"topology"},{"id":"stacks:0905","tag":"0905","title":"Spectral spaces · Lemma 0905","summary":"Let X be a spectral space. The following are equivalent: • X is profinite, • X is Hausdorff, • X is totally disconnected, • every quasi-compact open is closed, • there are no nontrivial specializations between points, • every point of X is closed, • every point of X is the generic point of an irreducible component of X, • the constructible topology equals the given topology on X, and • add more here.","statement_latex":"Let $X$ be a spectral space. The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is profinite,\n\\item $X$ is Hausdorff,\n\\item $X$ is totally disconnected,\n\\item every quasi-compact open is closed,\n\\item there are no nontrivial specializations between points,\n\\item every point of $X$ is closed,\n\\item every point of $X$ is the generic point of an irreducible component\nof $X$,\n\\item the constructible topology equals the given topology on $X$, and\n\\item add more here.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0905","source_file":"topology.tex","source_line":4524,"source_end_line":4539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4524-L4539","statement_sha256":"3d157a41c2bf202ca0735a1afb096564e5293af3c460059590a98ed7442d02b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":386,"rank":386,"depth":4,"x":531.68,"y":195.446,"cluster":"topology"},{"id":"stacks:0906","tag":"0906","title":"Spectral spaces · Lemma 0906","summary":"If X is a spectral space, then π_0(X) is a profinite space.","statement_latex":"If $X$ is a spectral space, then $\\pi_0(X)$ is a profinite space.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0906","source_file":"topology.tex","source_line":4560,"source_end_line":4563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4560-L4563","statement_sha256":"19d34c9e4d84c0f95983111ab47b992a4ae6da1ebdeaf9d8047b72cf669ec082","origin":"The Stacks Project","memory_eligible":false,"source_rank":387,"rank":387,"depth":5,"x":809.546,"y":153.725,"cluster":"topology"},{"id":"stacks:0907","tag":"0907","title":"Spectral spaces · Lemma 0907","summary":"The product of two spectral spaces is spectral.","statement_latex":"The product of two spectral spaces is spectral.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0907","source_file":"topology.tex","source_line":4570,"source_end_line":4573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4570-L4573","statement_sha256":"0ab2e83c77ca0793dc23e4148bd7f3f7b59380b9e60570a3d8ba64c0c9181e6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":388,"rank":388,"depth":0,"x":637.631,"y":342.784,"cluster":"topology"},{"id":"stacks:09XU","tag":"09XU","title":"Spectral spaces · Lemma 09XU","summary":"Let f : X → Y be a continuous map of topological spaces. If • X and Y are spectral, • f is spectral and bijective, and • generalizations (resp. specializations) lift along f. Then f is a homeomorphism.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces. If\n\\begin{enumerate}\n\\item $X$ and $Y$ are spectral,\n\\item $f$ is spectral and bijective, and\n\\item generalizations (resp.\\ specializations) lift along $f$.\n\\end{enumerate}\nThen $f$ is a homeomorphism.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XU","source_file":"topology.tex","source_line":4604,"source_end_line":4613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4604-L4613","statement_sha256":"91b8a40b222d295bc084383f9e59d7d818090c67f050a79c37ee05a9fbaffb9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":389,"rank":389,"depth":3,"x":612.28,"y":105.043,"cluster":"topology"},{"id":"stacks:09XV","tag":"09XV","title":"Spectral spaces · Lemma 09XV","summary":"The inverse limit of a directed inverse system of finite sober topological spaces is a spectral topological space.","statement_latex":"The inverse limit of a directed inverse system of finite sober\ntopological spaces is a spectral topological space.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XV","source_file":"topology.tex","source_line":4633,"source_end_line":4637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4633-L4637","statement_sha256":"64f321513cffdc351a9a72087dc5ed9a761f99c255e0a0a726e28ad9cf1ed1f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":390,"rank":390,"depth":3,"x":822.847,"y":266.494,"cluster":"topology"},{"id":"stacks:09XW","tag":"09XW","title":"Spectral spaces · Lemma 09XW","summary":"Let W be the topological space with two points, one closed, the other not. A topological space is spectral if and only if it is homeomorphic to a subspace of a product of copies of W which is closed in the constructible topology.","statement_latex":"Let $W$ be the topological space with two points, one closed,\nthe other not. A topological space is spectral if and only if\nit is homeomorphic to a subspace of a product of\ncopies of $W$ which is closed in the constructible topology.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XW","source_file":"topology.tex","source_line":4659,"source_end_line":4665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4659-L4665","statement_sha256":"c8c573b63b27809beaaae596a1e1937a263eee198d20855c85965f1d7f91740b","origin":"The Stacks Project","memory_eligible":false,"source_rank":391,"rank":391,"depth":4,"x":536.814,"y":266.923,"cluster":"topology"},{"id":"stacks:09XX","tag":"09XX","title":"Spectral spaces · Lemma 09XX","summary":"A topological space is spectral if and only if it is a directed inverse limit of finite sober topological spaces.","statement_latex":"A topological space is spectral if and only if it is a directed\ninverse limit of finite sober topological spaces.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XX","source_file":"topology.tex","source_line":4703,"source_end_line":4707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4703-L4707","statement_sha256":"e046581369c29c49c49450dd57a808b575b034928fc80a02469a295dc8d1298a","origin":"The Stacks Project","memory_eligible":false,"source_rank":392,"rank":392,"depth":5,"x":748.068,"y":103.778,"cluster":"topology"},{"id":"stacks:0A2T","tag":"0A2T","title":"Spectral spaces · Lemma 0A2T","summary":"Let X be a topological space and let c : X → X' be the universal map from X to a sober topological space, see Lemma [Tag 0A2N]. • If X is quasi-compact, so is X'. • If X is quasi-compact, has a basis of quasi-compact opens, and the intersection of two quasi-compact opens is quasi-compact, then X' is spectral. • If X is Noetherian, then X' is a Noetherian spectral space.","statement_latex":"Let $X$ be a topological space and let $c : X \\to X'$ be the universal\nmap from $X$ to a sober topological space, see\nLemma \\ref{lemma-make-sober}.\n\\begin{enumerate}\n\\item If $X$ is quasi-compact, so is $X'$.\n\\item If $X$ is quasi-compact, has a basis of quasi-compact opens,\nand the intersection of two quasi-compact opens is quasi-compact, then\n$X'$ is spectral.\n\\item If $X$ is Noetherian, then $X'$ is a Noetherian spectral space.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2T","source_file":"topology.tex","source_line":4729,"source_end_line":4741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4729-L4741","statement_sha256":"0556870e9af6e91da227f3945e2d62fd71e898b957637fe9fd79002769bb778c","origin":"The Stacks Project","memory_eligible":false,"source_rank":393,"rank":393,"depth":1,"x":723.409,"y":344.723,"cluster":"topology"},{"id":"stacks:0A2V","tag":"0A2V","title":"Limits of spectral spaces · Lemma 0A2V","summary":"Let I be a category. Let i ↦ X_i be a diagram of spectral spaces such that for a : j → i in I the corresponding map f_a : X_j → X_i is spectral. • Given subsets Z_i ⊂ X_i closed in the constructible topology with f_a(Z_j) ⊂ Z_i for all a : j → i in I, then lim Z_i is quasi-compact. • The space X = lim X_i is quasi-compact.","statement_latex":"Let $\\mathcal{I}$ be a category. Let $i \\mapsto X_i$ be a diagram\nof spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$\nthe corresponding map $f_a : X_j \\to X_i$ is spectral.\n\\begin{enumerate}\n\\item Given subsets $Z_i \\subset X_i$ closed in the constructible\ntopology with $f_a(Z_j) \\subset Z_i$ for all $a : j \\to i$ in $\\mathcal{I}$,\nthen $\\lim Z_i$ is quasi-compact.\n\\item The space $X = \\lim X_i$ is quasi-compact.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2V","source_file":"topology.tex","source_line":4778,"source_end_line":4789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4778-L4789","statement_sha256":"4de37b4ae9abd7dadf3fa7b4883c99f572098cfcc9b08641e831a2fd6a67a570","origin":"The Stacks Project","memory_eligible":false,"source_rank":394,"rank":394,"depth":3,"x":547.27,"y":152.448,"cluster":"topology"},{"id":"stacks:0A2W","tag":"0A2W","title":"Limits of spectral spaces · Lemma 0A2W","summary":"Let I be a cofiltered category. Let i ↦ X_i be a diagram of spectral spaces such that for a : j → i in I the corresponding map f_a : X_j → X_i is spectral. • Given nonempty subsets Z_i ⊂ X_i closed in the constructible topology with f_a(Z_j) ⊂ Z_i for all a : j → i in I, then lim Z_i is nonempty. • If each X_i is nonempty, then X = lim X_i is nonempty.","statement_latex":"Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram\nof spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$\nthe corresponding map $f_a : X_j \\to X_i$ is spectral.\n\\begin{enumerate}\n\\item Given nonempty subsets $Z_i \\subset X_i$ closed in the constructible\ntopology with $f_a(Z_j) \\subset Z_i$ for all $a : j \\to i$ in $\\mathcal{I}$,\nthen $\\lim Z_i$ is nonempty.\n\\item If each $X_i$ is nonempty, then $X = \\lim X_i$ is nonempty.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2W","source_file":"topology.tex","source_line":4809,"source_end_line":4820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4809-L4820","statement_sha256":"46feb8e63a7eeb1000f11f92dca20b7a118014209a15b901f426d12ef22861e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":395,"rank":395,"depth":4,"x":832.679,"y":194.417,"cluster":"topology"},{"id":"stacks:0A2X","tag":"0A2X","title":"Limits of spectral spaces · Lemma 0A2X","summary":"Let I be a cofiltered category. Let i ↦ X_i be a diagram of spectral spaces such that for a : j → i in I the corresponding map f_a : X_j → X_i is spectral. Let X = lim X_i with projections p_i : X → X_i. Let i ∈ Ob(I) and let E, F ⊂ X_i be subsets with E closed in the constructible topology and F open in the constructible topology. Then p_i^-1(E) ⊂ p_i^-1(F) if and only if there is a morphism a : j → i in I such that f_a^-1(E) ⊂ f_a^-1(F).","statement_latex":"Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram\nof spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$\nthe corresponding map $f_a : X_j \\to X_i$ is spectral. Let $X = \\lim X_i$\nwith projections $p_i : X \\to X_i$. Let $i \\in \\Ob(\\mathcal{I})$ and let\n$E, F \\subset X_i$ be subsets with $E$ closed in the constructible topology\nand $F$ open in the constructible topology.\nThen $p_i^{-1}(E) \\subset p_i^{-1}(F)$ if and only if there is a morphism\n$a : j \\to i$ in $\\mathcal{I}$ such that $f_a^{-1}(E) \\subset f_a^{-1}(F)$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2X","source_file":"topology.tex","source_line":4835,"source_end_line":4845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4835-L4845","statement_sha256":"721761fe36545eea946ca4cb9724a6e457dc93386ac06b35c6484dd726eee86e","origin":"The Stacks Project","memory_eligible":false,"source_rank":396,"rank":396,"depth":5,"x":587.7,"y":325.83,"cluster":"topology"},{"id":"stacks:0A2Y","tag":"0A2Y","title":"Limits of spectral spaces · Lemma 0A2Y","summary":"Let I be a cofiltered category. Let i ↦ X_i be a diagram of spectral spaces such that for a : j → i in I the corresponding map f_a : X_j → X_i is spectral. Let X = lim X_i with projections p_i : X → X_i. Let E ⊂ X be a constructible subset. Then there exists an i ∈ Ob(I) and a constructible subset E_i ⊂ X_i such that p_i^-1(E_i) = E. If E is open, resp. closed, we may choose E_i open, resp. closed.","statement_latex":"Let $\\mathcal{I}$ be a cofiltered category. Let $i \\mapsto X_i$ be a diagram\nof spectral spaces such that for $a : j \\to i$ in $\\mathcal{I}$\nthe corresponding map $f_a : X_j \\to X_i$ is spectral. Let $X = \\lim X_i$\nwith projections $p_i : X \\to X_i$. Let $E \\subset X$ be a constructible\nsubset. Then there exists an $i \\in \\Ob(\\mathcal{I})$ and a constructible\nsubset $E_i \\subset X_i$ such that $p_i^{-1}(E_i) = E$. If $E$ is open,\nresp.\\ closed, we may choose $E_i$ open, resp.\\ closed.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2Y","source_file":"topology.tex","source_line":4861,"source_end_line":4870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4861-L4870","statement_sha256":"f7aea3899c48b2fcc7b8835ccf10235c995c87be280d13d3e6972105f7765d89","origin":"The Stacks Project","memory_eligible":false,"source_rank":397,"rank":397,"depth":1,"x":662.901,"y":89.181,"cluster":"topology"},{"id":"stacks:0A2Z","tag":"0A2Z","title":"Limits of spectral spaces · Lemma 0A2Z","summary":"Let I be a cofiltered index category. Let i ↦ X_i be a diagram of spectral spaces such that for a : j → i in I the corresponding map f_a : X_j → X_i is spectral. Then the inverse limit X = lim X_i is a spectral topological space and the projection maps p_i : X → X_i are spectral.","statement_latex":"Let $\\mathcal{I}$ be a cofiltered index category.\nLet $i \\mapsto X_i$ be a diagram of spectral spaces such\nthat for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map\n$f_a : X_j \\to X_i$ is spectral. Then\nthe inverse limit $X = \\lim X_i$ is a spectral topological space\nand the projection maps $p_i : X \\to X_i$ are spectral.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2Z","source_file":"topology.tex","source_line":4901,"source_end_line":4909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4901-L4909","statement_sha256":"ba7955bcc13aac69c75e3803cdff015803c3c7f092891a7ccca8c66cada03454","origin":"The Stacks Project","memory_eligible":false,"source_rank":398,"rank":398,"depth":4,"x":798.175,"y":307.031,"cluster":"topology"},{"id":"stacks:0A30","tag":"0A30","title":"Limits of spectral spaces · Lemma 0A30","summary":"Let I be a cofiltered index category. Let i ↦ X_i be a diagram of spectral spaces such that for a : j → i in I the corresponding map f_a : X_j → X_i is spectral. Set X = lim X_i and denote p_i : X → X_i the projection. • Given any quasi-compact open U ⊂ X there exists an i ∈ Ob(I) and a quasi-compact open U_i ⊂ X_i such that p_i^-1(U_i) = U. • Given U_i ⊂ X_i and U_j ⊂ X_j quasi-compact opens such that p_i^-1(U_i) ⊂ p_j^-1(U_j) there exist k ∈ Ob(I) and morphisms a : k →…","statement_latex":"Let $\\mathcal{I}$ be a cofiltered index category.\nLet $i \\mapsto X_i$ be a diagram of spectral spaces such\nthat for $a : j \\to i$ in $\\mathcal{I}$ the corresponding map\n$f_a : X_j \\to X_i$ is spectral. Set $X = \\lim X_i$ and denote\n$p_i : X \\to X_i$ the projection.\n\\begin{enumerate}\n\\item Given any quasi-compact open $U \\subset X$\nthere exists an $i \\in \\Ob(\\mathcal{I})$ and a quasi-compact open\n$U_i \\subset X_i$ such that $p_i^{-1}(U_i) = U$.\n\\item Given $U_i \\subset X_i$ and $U_j \\subset X_j$\nquasi-compact opens such that $p_i^{-1}(U_i) \\subset p_j^{-1}(U_j)$\nthere exist $k \\in \\Ob(\\mathcal{I})$ and morphisms\n$a : k \\to i$ and $b : k \\to j$ such that $f_a^{-1}(U_i) \\subset f_b^{-1}(U_j)$.\n\\item If $U_i, U_{1, i}, \\ldots, U_{n, i} \\subset X_i$ are quasi-compact\nopens and\n$p_i^{-1}(U_i) = p_i^{-1}(U_{1, i}) \\cup \\ldots \\cup p_i^{-1}(U_{n, i})$\nthen\n$f_a^{-1}(U_i) = f_a^{-1}(U_{1, i}) \\cup \\ldots \\cup f_a^{-1}(U_{n, i})$\nfor some morphism $a : j \\to i$ in $\\mathcal{I}$.\n\\item Same statement as in (3) but for intersections.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A30","source_file":"topology.tex","source_line":4968,"source_end_line":4991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L4968-L4991","statement_sha256":"06a151a3a9b52cc941e27dd90ee8fd8fc366a68b5683a49664a51e1d0e4492a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":399,"rank":399,"depth":6,"x":522.386,"y":222.889,"cluster":"topology"},{"id":"stacks:0A31","tag":"0A31","title":"Limits of spectral spaces · Lemma 0A31","summary":"Let W be a subset of a spectral space X. The following are equivalent: • W is an intersection of constructible sets and closed under generalizations, • W is quasi-compact and closed under generalizations, • there exists a quasi-compact subset E ⊂ X such that W is the set of points specializing to E, • W is an intersection of quasi-compact open subsets, • there exists a nonempty set I and quasi-compact opens U_i ⊂ X, i ∈ I such that W = ⋂ U_i and for all i, j ∈ I there…","statement_latex":"Let $W$ be a subset of a spectral space $X$. The following are equivalent:\n\\begin{enumerate}\n\\item $W$ is an intersection of constructible sets and\nclosed under generalizations,\n\\item $W$ is quasi-compact and closed under generalizations,\n\\item there exists a quasi-compact subset $E \\subset X$ such that\n$W$ is the set of points specializing to $E$,\n\\item $W$ is an intersection of quasi-compact open subsets,\n\\item\n\nthere exists a nonempty set $I$ and quasi-compact opens\n$U_i \\subset X$, $i \\in I$\nsuch that $W = \\bigcap U_i$ and for all $i, j \\in I$ there exists a\n$k \\in I$ with $U_k \\subset U_i \\cap U_j$.\n\\end{enumerate}\nIn this case we have (a) $W$ is a spectral space, (b) $W = \\lim U_i$\nas topological spaces, and (c) for any open $U$ containing $W$\nthere exists an $i$ with $U_i \\subset U$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A31","source_file":"topology.tex","source_line":5004,"source_end_line":5024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5004-L5024","statement_sha256":"0ef511236f404d01f32aecb5fb2a5c3648f656213779c950ab6565daf1549839","origin":"The Stacks Project","memory_eligible":false,"source_rank":400,"rank":400,"depth":5,"x":794.249,"y":128.155,"cluster":"topology"},{"id":"stacks:0AP0","tag":"0AP0","title":"Limits of spectral spaces · Lemma 0AP0","summary":"Let X be a spectral space. Let E ⊂ X be a constructible subset. Let W ⊂ X be the set of points of X which specialize to a point of E. Then W setminus E is a spectral space. If W = ⋂ U_i with U_i as in Lemma [Tag 0A31] ([Tag 0ANZ]) then W setminus E = lim (U_i setminus E).","statement_latex":"Let $X$ be a spectral space. Let $E \\subset X$ be a constructible subset.\nLet $W \\subset X$ be the set of points of $X$ which specialize\nto a point of $E$. Then $W \\setminus E$ is a spectral space.\nIf $W = \\bigcap U_i$ with $U_i$ as in\nLemma \\ref{lemma-make-spectral-space}\n(\\ref{item-intersection-quasi-compact-open})\nthen $W \\setminus E = \\lim (U_i \\setminus E)$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Limits of spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AP0","source_file":"topology.tex","source_line":5068,"source_end_line":5077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5068-L5077","statement_sha256":"1a876d7cb5bed7da0bc4a92c7ccfc9f771cbbaa44a586013cac5399474d9e37c","origin":"The Stacks Project","memory_eligible":false,"source_rank":401,"rank":401,"depth":6,"x":669.583,"y":352.957,"cluster":"topology"},{"id":"stacks:0909","tag":"0909","title":"Stone-v Cech compactification · Lemma 0909","summary":"Let f : X → Y be a continuous map of topological spaces. Assume that f(X) is dense in Y and that Y is Hausdorff. Then the cardinality of Y is at most the cardinality of P(P(X)) where P is the power set operation.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces. Assume that\n$f(X)$ is dense in $Y$ and that $Y$ is Hausdorff. Then the cardinality\nof $Y$ is at most the cardinality of $P(P(X))$ where $P$ is the power\nset operation.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Stone-v Cech compactification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0909","source_file":"topology.tex","source_line":5105,"source_end_line":5111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5105-L5111","statement_sha256":"6afa442d1d8444aa8300fb924cc557528da660e056874df817c3d54977e7066e","origin":"The Stacks Project","memory_eligible":false,"source_rank":402,"rank":402,"depth":0,"x":580.46,"y":115.733,"cluster":"topology"},{"id":"stacks:090A","tag":"090A","title":"Stone-v Cech compactification · Lemma 090A","summary":"Let X be a Hausdorff, locally quasi-compact space. There exists a map X → X^* which identifies X as an open subspace of a quasi-compact Hausdorff space X^* such that X^* setminus X is a singleton (one point compactification). In particular, the map X → β(X) identifies X with an open subspace of β(X).","statement_latex":"Let $X$ be a Hausdorff, locally quasi-compact space.\nThere exists a map $X \\to X^*$ which identifies $X$ as an open\nsubspace of a quasi-compact Hausdorff space $X^*$ such that\n$X^* \\setminus X$ is a singleton (one point compactification).\nIn particular, the map $X \\to \\beta(X)$ identifies $X$\nwith an open subspace of $\\beta(X)$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Stone-v Cech compactification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090A","source_file":"topology.tex","source_line":5152,"source_end_line":5160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5152-L5160","statement_sha256":"2aae78d37d88ec70be10308d903efcf16f5ab0787f406a6f92fd5f65355d7bd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":403,"rank":403,"depth":2,"x":837.721,"y":240.465,"cluster":"topology"},{"id":"stacks:08YI","tag":"08YI","title":"Extremally disconnected spaces · Definition 08YI","summary":"A topological space X is called extremally disconnected if the closure of every open subset of X is open.","statement_latex":"A topological space $X$ is called {\\it extremally disconnected}\nif the closure of every open subset of $X$ is open.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Extremally disconnected spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YI","source_file":"topology.tex","source_line":5214,"source_end_line":5218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5214-L5218","statement_sha256":"64a2102785a0fa5e09b8f4d7070bf64f41efecc0cc4401550d29572864c41050","origin":"The Stacks Project","memory_eligible":false,"source_rank":404,"rank":404,"depth":0,"x":546.844,"y":294.627,"cluster":"topology"},{"id":"stacks:08YJ","tag":"08YJ","title":"Extremally disconnected spaces · Lemma 08YJ","summary":"Let f : X → Y be a continuous map of topological spaces. Assume f is surjective and f(E) not = Y for all proper closed subsets E ⊂ X. Then for U ⊂ X open the subset f(U) is contained in the closure of Y setminus f(X setminus U).","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nAssume $f$ is surjective and $f(E) \\not = Y$ for all proper\nclosed subsets $E \\subset X$. Then for $U \\subset X$ open the subset\n$f(U)$ is contained in the closure of $Y \\setminus f(X \\setminus U)$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Extremally disconnected spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YJ","source_file":"topology.tex","source_line":5231,"source_end_line":5237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5231-L5237","statement_sha256":"92dad67413852e66dba2603ee0b9ec540149adaf57774b74c2744e9d84344305","origin":"The Stacks Project","memory_eligible":false,"source_rank":405,"rank":405,"depth":0,"x":718.288,"y":89.026,"cluster":"topology"},{"id":"stacks:08YK","tag":"08YK","title":"Extremally disconnected spaces · Lemma 08YK","summary":"Let X be an extremally disconnected space. If U, V ⊂ X are disjoint open subsets, then overlineU and overlineV are disjoint too.","statement_latex":"Let $X$ be an extremally disconnected space.\nIf $U, V \\subset X$ are disjoint open subsets, then\n$\\overline{U}$ and $\\overline{V}$ are disjoint too.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Extremally disconnected spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YK","source_file":"topology.tex","source_line":5248,"source_end_line":5253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5248-L5253","statement_sha256":"b8093bb8555880ac31c0a3029d6dd59b4ae113fbe9e4bc2ef14c2f106d76dce7","origin":"The Stacks Project","memory_eligible":false,"source_rank":406,"rank":406,"depth":0,"x":757.32,"y":338.654,"cluster":"topology"},{"id":"stacks:08YL","tag":"08YL","title":"Extremally disconnected spaces · Lemma 08YL","summary":"Let f : X → Y be a continuous map of Hausdorff quasi-compact topological spaces. If Y is extremally disconnected, f is surjective, and f(Z) not = Y for every proper closed subset Z of X, then f is a homeomorphism.","statement_latex":"Let $f : X \\to Y$ be a continuous map of Hausdorff quasi-compact\ntopological spaces. If $Y$ is extremally disconnected, $f$ is surjective,\nand $f(Z) \\not = Y$ for every proper closed subset $Z$ of $X$, then\n$f$ is a homeomorphism.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Extremally disconnected spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YL","source_file":"topology.tex","source_line":5263,"source_end_line":5269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5263-L5269","statement_sha256":"608c37f88e8b215e3443fcd885e0d440ef3738d43d67ec879313549b240a5051","origin":"The Stacks Project","memory_eligible":false,"source_rank":407,"rank":407,"depth":4,"x":527.119,"y":176.252,"cluster":"topology"},{"id":"stacks:08YM","tag":"08YM","title":"Extremally disconnected spaces · Lemma 08YM","summary":"Let f : X → Y be a continuous surjective map of Hausdorff quasi-compact topological spaces. There exists a quasi-compact subset E ⊂ X such that f(E) = Y but f(E') not = Y for all proper closed subsets E' ⊂ E.","statement_latex":"Let $f : X \\to Y$ be a continuous surjective map of Hausdorff quasi-compact\ntopological spaces. There exists a quasi-compact subset $E \\subset X$\nsuch that $f(E) = Y$ but $f(E') \\not = Y$ for all proper closed subsets\n$E' \\subset E$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Extremally disconnected spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YM","source_file":"topology.tex","source_line":5285,"source_end_line":5291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5285-L5291","statement_sha256":"af71be3b3664bc33739701e1005f7eacb5e3b67db200539dc0f9c3fca9c1dcaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":408,"rank":408,"depth":2,"x":828.346,"y":165.351,"cluster":"topology"},{"id":"stacks:08YN","tag":"08YN","title":"Extremally disconnected spaces · Proposition 08YN","summary":"Let X be a Hausdorff, quasi-compact topological space. The following are equivalent • X is extremally disconnected, • for any surjective continuous map f : Y → X with Y Hausdorff quasi-compact there exists a continuous section, and • for any solid commutative diagram xymatrix & Y ar[d] X ar@..>[ru] ar[r] & Z of continuous maps of quasi-compact Hausdorff spaces with Y → Z surjective, there is a dotted arrow in the category of topological spaces making the diagram commute.","statement_latex":"Let $X$ be a Hausdorff, quasi-compact topological space.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is extremally disconnected,\n\\item for any surjective continuous map $f : Y \\to X$ with $Y$ Hausdorff\nquasi-compact there exists a continuous section, and\n\\item for any solid commutative diagram\n$$\n\\xymatrix{\n& Y \\ar[d] \\\\\nX \\ar@{..>}[ru] \\ar[r] & Z\n}\n$$\nof continuous maps of quasi-compact Hausdorff spaces with $Y \\to Z$\nsurjective, there is a dotted arrow\nin the category of topological spaces making the diagram commute.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Extremally disconnected spaces","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YN","source_file":"topology.tex","source_line":5312,"source_end_line":5331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5312-L5331","statement_sha256":"c41225f6cb461ea8d17b3c001c35e3b8d88c81043669375defbf24bb914b3d08","origin":"The Stacks Project","memory_eligible":false,"source_rank":409,"rank":409,"depth":5,"x":614.369,"y":344.834,"cluster":"topology"},{"id":"stacks:090B","tag":"090B","title":"Extremally disconnected spaces · Lemma 090B","summary":"Let f : X → X be a surjective continuous selfmap of a Hausdorff topological space. If f is not id_X, then there exists a proper closed subset E ⊂ X such that X = E ∪ f(E).","statement_latex":"Let $f : X \\to X$ be a surjective continuous selfmap of a Hausdorff\ntopological space. If $f$ is not $\\text{id}_X$, then there exists a\nproper closed subset $E \\subset X$ such that $X = E \\cup f(E)$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Extremally disconnected spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090B","source_file":"topology.tex","source_line":5355,"source_end_line":5360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5355-L5360","statement_sha256":"ec19fa2c5543044cd1b9efedf40f549f9167aadfbc1a31efbdfd7581813ecf4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":410,"rank":410,"depth":0,"x":627.857,"y":90.335,"cluster":"topology"},{"id":"stacks:090D","tag":"090D","title":"Extremally disconnected spaces · Lemma 090D","summary":"Every quasi-compact Hausdorff space has a canonical extremally disconnected cover Let X be a quasi-compact Hausdorff space. There exists a continuous surjection X' → X with X' quasi-compact, Hausdorff, and extremally disconnected. If we require that every proper closed subset of X' does not map onto X, then X' is unique up to isomorphism.","statement_latex":"\\begin{slogan}\nEvery quasi-compact Hausdorff space has a canonical\nextremally disconnected cover\n\\end{slogan}\nLet $X$ be a quasi-compact Hausdorff space.\nThere exists a continuous surjection $X' \\to X$ with $X'$\nquasi-compact, Hausdorff, and extremally disconnected.\nIf we require that every proper closed subset of $X'$ does not\nmap onto $X$, then $X'$ is unique up to isomorphism.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Extremally disconnected spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090D","source_file":"topology.tex","source_line":5396,"source_end_line":5407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5396-L5407","statement_sha256":"4801be4c8c654e29d8c3f15049696e457a2ded72b948aa68da14a9644570a117","origin":"The Stacks Project","memory_eligible":false,"source_rank":411,"rank":411,"depth":6,"x":823.132,"y":286.216,"cluster":"topology"},{"id":"stacks:0069","tag":"0069","title":"Miscellany · Lemma 0069","summary":"Let X be a topological space which • has a basis of the topology consisting of quasi-compact opens, and • has the property that the intersection of any two quasi-compact opens is quasi-compact. Then • X is locally quasi-compact, • a quasi-compact open U ⊂ X is retrocompact, • any quasi-compact open U ⊂ X has a cofinal system of open coverings U : U = ⋃_j∈ J U_j with J finite and all U_j and U_j ∩ U_j' quasi-compact, • add more here.","statement_latex":"Let $X$ be a topological space which\n\\begin{enumerate}\n\\item has a basis of the topology consisting of quasi-compact opens, and\n\\item has the property that the intersection of any two quasi-compact\nopens is quasi-compact.\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item $X$ is locally quasi-compact,\n\\item a quasi-compact open $U \\subset X$ is retrocompact,\n\\item any quasi-compact open $U \\subset X$ has a cofinal system of open\ncoverings $\\mathcal{U} : U = \\bigcup_{j\\in J} U_j$ with $J$ finite\nand all $U_j$ and $U_j \\cap U_{j'}$ quasi-compact,\n\\item add more here.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0069","source_file":"topology.tex","source_line":5469,"source_end_line":5486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5469-L5486","statement_sha256":"9a194ae4011de8b2d56aed8231fe4c2ab1216972ce135b43cf0ad137cb5da405","origin":"The Stacks Project","memory_eligible":false,"source_rank":412,"rank":412,"depth":0,"x":520.754,"y":252.483,"cluster":"topology"},{"id":"stacks:06RM","tag":"06RM","title":"Miscellany · Definition 06RM","summary":"Let X be a topological space. We say x ∈ X is an isolated point of X if (x) is open in X.","statement_latex":"Let $X$ be a topological space. We say $x \\in X$ is an\n{\\it isolated point} of $X$ if $\\{x\\}$ is open in $X$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Miscellany","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RM","source_file":"topology.tex","source_line":5492,"source_end_line":5496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5492-L5496","statement_sha256":"a84aa529cc709d0245467ae21cb8ef0311036d6b069cc76983137732a78d1222","origin":"The Stacks Project","memory_eligible":false,"source_rank":413,"rank":413,"depth":0,"x":771.563,"y":105.363,"cluster":"topology"},{"id":"stacks:09XZ","tag":"09XZ","title":"Partitions and stratifications · Definition 09XZ","summary":"Let X be a topological space. A partition of X is a decomposition X = coprod X_i into locally closed subsets X_i. The X_i are called the parts of the partition. Given two partitions of X we say one refines the other if the parts of one are unions of parts of the other.","statement_latex":"Let $X$ be a topological space. A {\\it partition} of $X$ is a\ndecomposition $X = \\coprod X_i$ into locally closed subsets $X_i$.\nThe $X_i$ are called the {\\it parts} of the partition.\nGiven two partitions of $X$ we say one {\\it refines} the other if\nthe parts of one are unions of parts of the other.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Partitions and stratifications","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XZ","source_file":"topology.tex","source_line":5507,"source_end_line":5514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5507-L5514","statement_sha256":"138ca4da5023b4d4c202ce174bc6c6b540aa816e37fdc3aa4dc5948d2c6f0b32","origin":"The Stacks Project","memory_eligible":false,"source_rank":414,"rank":414,"depth":0,"x":704.74,"y":356.872,"cluster":"topology"},{"id":"stacks:09Y0","tag":"09Y0","title":"Partitions and stratifications · Definition 09Y0","summary":"Let X be a topological space. A good stratification of X is a partition X = coprod X_i such that for all i, j ∈ I we have X_i ∩ overlineX_j not = ∅ ⇒ X_i ⊂ overlineX_j.","statement_latex":"Let $X$ be a topological space. A {\\it good stratification}\nof $X$ is a partition $X = \\coprod X_i$ such that for all\n$i, j \\in I$ we have\n$$\nX_i \\cap \\overline{X_j} \\not = \\emptyset\n\\Rightarrow\nX_i \\subset \\overline{X_j}.\n$$","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Partitions and stratifications","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Y0","source_file":"topology.tex","source_line":5524,"source_end_line":5534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5524-L5534","statement_sha256":"356ef2f04f8e86733fd0f5844206342e76de93fa58121298489f23d0c9179c4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":415,"rank":415,"depth":0,"x":551.328,"y":132.867,"cluster":"topology"},{"id":"stacks:09Y1","tag":"09Y1","title":"Partitions and stratifications · Definition 09Y1","summary":"Let X be a topological space. A stratification of X is given by a partition X = coprod_i ∈ I X_i and a partial ordering on I such that for each j ∈ I we have overlineX_j ⊂ ⋃_i ≤ j X_i The parts X_i are called the strata of the stratification.","statement_latex":"Let $X$ be a topological space. A {\\it stratification} of $X$ is\ngiven by a partition $X = \\coprod_{i \\in I} X_i$ and a partial ordering\non $I$ such that for each $j \\in I$ we have\n$$\n\\overline{X_j} \\subset \\bigcup\\nolimits_{i \\leq j} X_i\n$$\nThe parts $X_i$ are called the {\\it strata} of the stratification.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Partitions and stratifications","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Y1","source_file":"topology.tex","source_line":5547,"source_end_line":5556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5547-L5556","statement_sha256":"ac39288c3f458a1c00f5ed698e7a334db5f82617012d1cb53021a1f1440db754","origin":"The Stacks Project","memory_eligible":false,"source_rank":416,"rank":416,"depth":0,"x":845.411,"y":211.214,"cluster":"topology"},{"id":"stacks:0BDS","tag":"0BDS","title":"Partitions and stratifications · Definition 0BDS","summary":"Let X be a topological space. Let I be a set and for i ∈ I let E_i ⊂ X be a subset. We say the collection (E_i)_i ∈ I is locally finite if for all x ∈ X there exists an open neighbourhood U of x such that (i ∈ I | E_i ∩ U not = ∅) is finite.","statement_latex":"Let $X$ be a topological space. Let $I$ be a set and for $i \\in I$\nlet $E_i \\subset X$ be a subset. We say the collection $\\{E_i\\}_{i \\in I}$\nis {\\it locally finite} if for all $x \\in X$ there exists an open\nneighbourhood $U$ of $x$ such that\n$\\{i \\in I | E_i \\cap U \\not = \\emptyset\\}$ is finite.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Partitions and stratifications","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDS","source_file":"topology.tex","source_line":5565,"source_end_line":5572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5565-L5572","statement_sha256":"934d33a2530e8041ecbc18bfe531c71049940909525dbd7db427ac16a398bcd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":417,"rank":417,"depth":0,"x":564.775,"y":320.615,"cluster":"topology"},{"id":"stacks:09Y3","tag":"09Y3","title":"Partitions and stratifications · Lemma 09Y3","summary":"Let X be a topological space. Let X = coprod X_i be a finite partition of X. Then there exists a finite stratification of X refining it.","statement_latex":"Let $X$ be a topological space. Let $X = \\coprod X_i$ be a finite partition\nof $X$. Then there exists a finite stratification of $X$ refining it.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Partitions and stratifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Y3","source_file":"topology.tex","source_line":5590,"source_end_line":5594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5590-L5594","statement_sha256":"5f5c3ef03f918ca56ee28187d7e97b2fa018aef5ed48ca73db01fffbd392dd29","origin":"The Stacks Project","memory_eligible":false,"source_rank":418,"rank":418,"depth":0,"x":684.065,"y":80.041,"cluster":"topology"},{"id":"stacks:09Y4","tag":"09Y4","title":"Partitions and stratifications · Lemma 09Y4","summary":"Let X be a topological space. Suppose X = T_1 ∪ … ∪ T_n is written as a union of constructible subsets. There exists a finite stratification X = coprod X_i with each X_i constructible such that each T_k is a union of strata.","statement_latex":"Let $X$ be a topological space. Suppose $X = T_1 \\cup \\ldots \\cup T_n$\nis written as a union of constructible subsets. There exists a finite\nstratification $X = \\coprod X_i$ with each $X_i$ constructible\nsuch that each $T_k$ is a union of strata.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Partitions and stratifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Y4","source_file":"topology.tex","source_line":5606,"source_end_line":5612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5606-L5612","statement_sha256":"49c8edb6c37c4c042c8e15a24f5ab8743d074413cd41d9f50bdc565e19bf71f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":419,"rank":419,"depth":1,"x":789.854,"y":325.8,"cluster":"topology"},{"id":"stacks:09Y5","tag":"09Y5","title":"Partitions and stratifications · Lemma 09Y5","summary":"Let X be a Noetherian topological space. Any finite partition of X can be refined by a finite good stratification.","statement_latex":"Let $X$ be a Noetherian topological space. Any finite partition\nof $X$ can be refined by a finite good stratification.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Partitions and stratifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Y5","source_file":"topology.tex","source_line":5630,"source_end_line":5634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5630-L5634","statement_sha256":"bf3a0f0a0e8a5834ce5eefaff361957e33173e5223203512b34c02e6fafec603","origin":"The Stacks Project","memory_eligible":false,"source_rank":420,"rank":420,"depth":3,"x":513.461,"y":204.276,"cluster":"topology"},{"id":"stacks:0B1X","tag":"0B1X","title":"Colimits of spaces · Lemma 0B1X","summary":"The category of topological spaces has colimits and the forgetful functor to sets commutes with them.","statement_latex":"The category of topological spaces has colimits and the forgetful functor\nto sets commutes with them.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Colimits of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1X","source_file":"topology.tex","source_line":5684,"source_end_line":5688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5684-L5688","statement_sha256":"5439177acb62ae4901082f0ba30792048690f728e1468ed4a4cd3026ce0cafef","origin":"The Stacks Project","memory_eligible":false,"source_rank":421,"rank":421,"depth":2,"x":815.816,"y":136.871,"cluster":"topology"},{"id":"stacks:0B1Z","tag":"0B1Z","title":"Topological groups, rings, modules · Definition 0B1Z","summary":"A topological group is a group G endowed with a topology such that multiplication G × G → G, (x, y) ↦ xy and inverse G → G, x ↦ x^-1 are continuous. A homomorphism of topological groups is a homomorphism of groups which is continuous.","statement_latex":"A {\\it topological group} is a group $G$ endowed with a topology\nsuch that multiplication $G \\times G \\to G$, $(x, y) \\mapsto xy$ and\ninverse $G \\to G$, $x \\mapsto x^{-1}$ are continuous.\nA {\\it homomorphism of topological groups} is a homomorphism of groups\nwhich is continuous.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1Z","source_file":"topology.tex","source_line":5712,"source_end_line":5719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5712-L5719","statement_sha256":"24f29bf6becd36d0c0b311f1c7e7d8c21b955d7bd2b4e824b37a586b26fdf8ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":422,"rank":422,"depth":0,"x":646.612,"y":358.737,"cluster":"topology"},{"id":"stacks:0B20","tag":"0B20","title":"Topological groups, rings, modules · Lemma 0B20","summary":"The category of topological groups has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of groups.","statement_latex":"The category of topological groups has limits and limits commute\nwith the forgetful functors to (a) the category of topological spaces and\n(b) the category of groups.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B20","source_file":"topology.tex","source_line":5757,"source_end_line":5762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5757-L5762","statement_sha256":"9095da3ce46fdf4f24eaa809171955584914c5470ee2d25e579e08f248238ca0","origin":"The Stacks Project","memory_eligible":false,"source_rank":423,"rank":423,"depth":1,"x":592.818,"y":98.427,"cluster":"topology"},{"id":"stacks:0BR1","tag":"0BR1","title":"Topological groups, rings, modules · Lemma 0BR1","summary":"Let G be a topological group. The following are equivalent • G as a topological space is profinite, • G is a limit of a diagram of finite discrete topological groups, • G is a cofiltered limit of finite discrete topological groups.","statement_latex":"Let $G$ be a topological group. The following are equivalent\n\\begin{enumerate}\n\\item $G$ as a topological space is profinite,\n\\item $G$ is a limit of a diagram of finite discrete topological groups,\n\\item $G$ is a cofiltered limit of finite discrete topological groups.\n\\end{enumerate}","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BR1","source_file":"topology.tex","source_line":5780,"source_end_line":5788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5780-L5788","statement_sha256":"6c116d83d8da778fb3f7ead752b750e608a3022a4b7a333a5adde31d57a55792","origin":"The Stacks Project","memory_eligible":false,"source_rank":424,"rank":424,"depth":4,"x":842.485,"y":260.285,"cluster":"topology"},{"id":"stacks:0BR2","tag":"0BR2","title":"Topological groups, rings, modules · Definition 0BR2","summary":"A topological group is called a profinite group if it satisfies the equivalent conditions of Lemma [Tag 0BR1].","statement_latex":"A topological group is called a {\\it profinite group} if it satisfies\nthe equivalent conditions of Lemma \\ref{lemma-profinite-group}.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BR2","source_file":"topology.tex","source_line":5844,"source_end_line":5848,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5844-L5848","statement_sha256":"beada486912e583b3e97b1af886b276b4f4469abb945895899a570b400e945ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":425,"rank":425,"depth":5,"x":527.386,"y":282.659,"cluster":"topology"},{"id":"stacks:0B21","tag":"0B21","title":"Topological groups, rings, modules · Lemma 0B21","summary":"The category of topological groups has colimits and colimits commute with the forgetful functor to the category of groups.","statement_latex":"The category of topological groups has colimits and colimits commute\nwith the forgetful functor to the category of groups.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B21","source_file":"topology.tex","source_line":5858,"source_end_line":5862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5858-L5862","statement_sha256":"df89ed3b3f30e8812fd42b9a57de547f9f2a27725f27c2325a8c4ddcf5b55733","origin":"The Stacks Project","memory_eligible":false,"source_rank":426,"rank":426,"depth":2,"x":742.311,"y":86.849,"cluster":"topology"},{"id":"stacks:0B22","tag":"0B22","title":"Topological groups, rings, modules · Definition 0B22","summary":"A topological ring is a ring R endowed with a topology such that addition R × R → R, (x, y) ↦ x + y and multiplication R × R → R, (x, y) ↦ xy are continuous. A homomorphism of topological rings is a homomorphism of rings which is continuous.","statement_latex":"A {\\it topological ring} is a ring $R$ endowed with a topology\nsuch that addition $R \\times R \\to R$, $(x, y) \\mapsto x + y$ and\nmultiplication $R \\times R \\to R$, $(x, y) \\mapsto xy$ are continuous.\nA {\\it homomorphism of topological rings} is a homomorphism of rings\nwhich is continuous.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B22","source_file":"topology.tex","source_line":5895,"source_end_line":5902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5895-L5902","statement_sha256":"7b906678eaa115642d94a25d43a636f11aaf9ff5d91c9b1b2cb445554b38f4ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":427,"rank":427,"depth":0,"x":741.29,"y":353.891,"cluster":"topology"},{"id":"stacks:0B23","tag":"0B23","title":"Topological groups, rings, modules · Lemma 0B23","summary":"The category of topological rings has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of rings.","statement_latex":"The category of topological rings has limits and limits commute\nwith the forgetful functors to (a) the category of topological spaces and\n(b) the category of rings.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B23","source_file":"topology.tex","source_line":5914,"source_end_line":5919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5914-L5919","statement_sha256":"5a0ceae625500f66eba01100a505659c4abb1eb6a22aa3f300ac9864f2de0640","origin":"The Stacks Project","memory_eligible":false,"source_rank":428,"rank":428,"depth":1,"x":526.735,"y":155.881,"cluster":"topology"},{"id":"stacks:0B24","tag":"0B24","title":"Topological groups, rings, modules · Lemma 0B24","summary":"The category of topological rings has colimits and colimits commute with the forgetful functor to the category of rings.","statement_latex":"The category of topological rings has colimits and colimits commute\nwith the forgetful functor to the category of rings.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B24","source_file":"topology.tex","source_line":5936,"source_end_line":5940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5936-L5940","statement_sha256":"119f27f9d2ae30c79b10fb32714aecc2b0b8b108564971fd5de64803072eb677","origin":"The Stacks Project","memory_eligible":false,"source_rank":429,"rank":429,"depth":3,"x":845.006,"y":180.211,"cluster":"topology"},{"id":"stacks:0B25","tag":"0B25","title":"Topological groups, rings, modules · Definition 0B25","summary":"Let R be a topological ring. A topological module is an R-module M endowed with a topology such that addition M × M → M and scalar multiplication R × M → M are continuous. A homomorphism of topological modules is a homomorphism of modules which is continuous.","statement_latex":"Let $R$ be a topological ring. A {\\it topological module} is an $R$-module\n$M$ endowed with a topology such that addition $M \\times M \\to M$ and\nscalar multiplication $R \\times M \\to M$ are continuous.\nA {\\it homomorphism of topological modules} is a homomorphism of\nmodules which is continuous.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B25","source_file":"topology.tex","source_line":5951,"source_end_line":5958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5951-L5958","statement_sha256":"f9b0cca5fdd0e40b4f8c4bb86ef6c20e7a9fa6e46cea1551ec930eb57a9e0413","origin":"The Stacks Project","memory_eligible":false,"source_rank":430,"rank":430,"depth":0,"x":590.092,"y":343.286,"cluster":"topology"},{"id":"stacks:0B26","tag":"0B26","title":"Topological groups, rings, modules · Lemma 0B26","summary":"Let R be a topological ring. The category of topological modules over R has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of R-modules.","statement_latex":"Let $R$ be a topological ring. The category of topological modules over $R$\nhas limits and limits commute with the forgetful functors to\n(a) the category of topological spaces and\n(b) the category of $R$-modules.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B26","source_file":"topology.tex","source_line":5970,"source_end_line":5976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5970-L5976","statement_sha256":"f82cbc573faf6a69f0fee05e7817c88362085bc739e79dce33a63f84cb28c2c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":431,"rank":431,"depth":1,"x":647.069,"y":77.71,"cluster":"topology"},{"id":"stacks:0B27","tag":"0B27","title":"Topological groups, rings, modules · Lemma 0B27","summary":"Let R be a topological ring. The category of topological modules over R has colimits and colimits commute with the forgetful functor to the category of modules over R.","statement_latex":"Let $R$ be a topological ring. The category of topological modules over $R$\nhas colimits and colimits commute with the forgetful functor to the category\nof modules over $R$.","area":"Topology","chapter":"Topology","chapter_id":"topology","section":"Topological groups, rings, modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B27","source_file":"topology.tex","source_line":5994,"source_end_line":5999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topology.tex#L5994-L5999","statement_sha256":"92e82801abb37b9c0eb8e7050bd8dda56297c9c31beb975cf341a3b0792b54f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":432,"rank":432,"depth":3,"x":819.064,"y":306.457,"cluster":"topology"},{"id":"stacks:006E","tag":"006E","title":"Presheaves · Definition 006E","summary":"Let X be a topological space. • A presheaf F of sets on X is a rule which assigns to each open U ⊂ X a set F(U) and to each inclusion V ⊂ U a map ρ^U_V : F(U) → F(V) such that ρ^U_U = id_F(U) and whenever W ⊂ V ⊂ U we have ρ^U_W = ρ^V_W ∘ ρ ^U_V. • A morphism φ : F → G of presheaves of sets on X is a rule which assigns to each open U ⊂ X a map of sets φ : F(U) → G(U) compatible with restriction maps, i.e., whenever V ⊂ U ⊂ X are open the diagram xymatrix F(U) ar[r]^φ…","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf $\\mathcal{F}$ of sets on $X$} is a rule which\nassigns to each open $U \\subset X$ a set $\\mathcal{F}(U)$ and\nto each inclusion $V \\subset U$ a map\n$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that\n$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ and\nwhenever $W \\subset V \\subset U$ we have\n$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves of sets on $X$} is a rule which assigns to each\nopen $U \\subset X$ a map of sets $\\varphi : \\mathcal{F}(U)\n\\to \\mathcal{G}(U)$ compatible with restriction maps,\ni.e., whenever $V \\subset U \\subset X$ are open the\ndiagram\n$$\n\\xymatrix{\n\\mathcal{F}(U) \\ar[r]^\\varphi \\ar[d]^{\\rho^U_V} &\n\\mathcal{G}(U) \\ar[d]^{\\rho^U_V} \\\\\n\\mathcal{F}(V) \\ar[r]^\\varphi & \\mathcal{G}(V)\n}\n$$\ncommutes.\n\\item The category of presheaves of sets on $X$ will be denoted\n$\\textit{PSh}(X)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006E","source_file":"sheaves.tex","source_line":73,"source_end_line":101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L73-L101","statement_sha256":"55d3283c661b0273b0efa5425bec754c3c8dce4fba2b1f457b31bb1f665e3cbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":433,"rank":433,"depth":0,"x":1134.434,"y":220.0,"cluster":"sheaves-sites"},{"id":"stacks:006F","tag":"006F","title":"Presheaves · Definition 006F","summary":"Let X be a topological space. Let A be a set. The constant presheaf with value A is the presheaf that assigns the set A to every open U ⊂ X, and such that all restriction mappings are id_A.","statement_latex":"Let $X$ be a topological space. Let $A$ be a set.\nThe {\\it constant presheaf with value $A$} is the\npresheaf that assigns the set $A$ to every open\n$U \\subset X$, and such that all restriction mappings\nare $\\text{id}_A$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006F","source_file":"sheaves.tex","source_line":127,"source_end_line":134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L127-L134","statement_sha256":"564afcc18670c247313630218e35016c7190e0977778639af5bf8e72a4828273","origin":"The Stacks Project","memory_eligible":false,"source_rank":434,"rank":434,"depth":0,"x":1124.338,"y":224.357,"cluster":"sheaves-sites"},{"id":"stacks:006I","tag":"006I","title":"Abelian presheaves · Lemma 006I","summary":"Let X be a topological space. The category of presheaves of sets on X has products (see Categories, Definition [Tag 002I]). Moreover, the set of sections of the product F × G over an open U is the product of the sets of sections of F and G over U.","statement_latex":"Let $X$ be a topological space. The category of presheaves of sets\non $X$ has products (see\nCategories, Definition \\ref{categories-definition-product}).\nMoreover, the set of\nsections of the product $\\mathcal{F} \\times \\mathcal{G}$\nover an open $U$ is the product of the sets of sections of\n$\\mathcal{F}$ and $\\mathcal{G}$ over $U$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Abelian presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006I","source_file":"sheaves.tex","source_line":156,"source_end_line":165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L156-L165","statement_sha256":"0ace905ceb403e0d80b5057d7a754791f69ebe1a6107c1716bb4d147603a213f","origin":"The Stacks Project","memory_eligible":false,"source_rank":435,"rank":435,"depth":1,"x":1130.867,"y":211.704,"cluster":"sheaves-sites"},{"id":"stacks:006J","tag":"006J","title":"Abelian presheaves · Lemma 006J","summary":"Let X be a topological space. Let F be a presheaf of sets. Consider the following types of structure on F: • For every open U the structure of an abelian group on F(U) such that all restriction maps are abelian group homomorphisms. • A map of presheaves + : F × F → F, a map of presheaves - : F → F and a map 0 : * → F (see Example [Tag 006H]) satisfying all the axioms of +, -, 0 in a usual abelian group. • A map of presheaves + : F × F → F, a map of presheaves - : F → F…","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{F}$ be a presheaf of sets.\nConsider the following types of structure on $\\mathcal{F}$:\n\\begin{enumerate}\n\\item For every open $U$ the structure of an abelian group\non $\\mathcal{F}(U)$ such that all restriction maps are\nabelian group homomorphisms.\n\\item A map of presheaves\n$+ : \\mathcal{F} \\times \\mathcal{F} \\to \\mathcal{F}$,\na map of presheaves $- : \\mathcal{F} \\to \\mathcal{F}$\nand a map $0 : * \\to \\mathcal{F}$\n(see Example \\ref{example-singleton-presheaf})\nsatisfying all the axioms of $+, -, 0$ in a usual\nabelian group.\n\\item A map of presheaves\n$+ : \\mathcal{F} \\times \\mathcal{F} \\to \\mathcal{F}$,\na map of presheaves $- : \\mathcal{F} \\to \\mathcal{F}$\nand a map $0 : * \\to \\mathcal{F}$\nsuch that for each open $U \\subset X$ the quadruple\n$(\\mathcal{F}(U), +, -, 0)$ is an abelian group,\n\\item A map of presheaves $+ : \\mathcal{F} \\times \\mathcal{F}\n\\to \\mathcal{F}$ such that for every open $U \\subset X$\nthe map $+ : \\mathcal{F}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$\ndefines the structure of an abelian group.\n\\end{enumerate}\nThere are natural bijections between the collections of\ntypes of data (1) - (4) above.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Abelian presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006J","source_file":"sheaves.tex","source_line":197,"source_end_line":226,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L197-L226","statement_sha256":"fd4474af51e58b189a460acc21c17f68cd20fa70a783de9a0a96ab838f87c711","origin":"The Stacks Project","memory_eligible":false,"source_rank":436,"rank":436,"depth":0,"x":1137.137,"y":227.82,"cluster":"sheaves-sites"},{"id":"stacks:006K","tag":"006K","title":"Abelian presheaves · Definition 006K","summary":"Let X be a topological space. • A presheaf of abelian groups on X or an abelian presheaf over X is a presheaf of sets F such that for each open U ⊂ X the set F(U) is endowed with the structure of an abelian group, and such that all restriction maps ρ^U_V are homomorphisms of abelian groups, see Lemma [Tag 006J] above. • A morphism of abelian presheaves over X φ : F → G is a morphism of presheaves of sets which induces a homomorphism of abelian groups F(U) → G(U) for every…","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf of abelian groups on $X$} or an\n{\\it abelian presheaf over $X$}\nis a presheaf of sets $\\mathcal{F}$ such that for each open\n$U \\subset X$ the set $\\mathcal{F}(U)$ is endowed with\nthe structure of an abelian group, and such that all restriction\nmaps $\\rho^U_V$ are homomorphisms of abelian groups, see\nLemma \\ref{lemma-abelian-presheaves} above.\n\\item A {\\it morphism of abelian presheaves over $X$}\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of presheaves\nof sets which induces\na homomorphism of abelian groups $\\mathcal{F}(U) \\to \\mathcal{G}(U)$\nfor every open $U \\subset X$.\n\\item The category of presheaves of abelian groups on $X$ is denoted\n$\\textit{PAb}(X)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Abelian presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006K","source_file":"sheaves.tex","source_line":244,"source_end_line":263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L244-L263","statement_sha256":"0a738feda8678c7d7ef4c0653d5a284499e9997e27e1b3143d05366c7c24fd2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":437,"rank":437,"depth":1,"x":1116.903,"y":218.054,"cluster":"sheaves-sites"},{"id":"stacks:006N","tag":"006N","title":"Presheaves of algebraic structures · Definition 006N","summary":"Let X be a topological space. Let C be a category. • A presheaf F on X with values in C is given by a rule which assigns to every open U ⊂ X an object F(U) of C and to each inclusion V ⊂ U a morphism ρ_V^U : F(U) → F(V) in C such that whenever W ⊂ V ⊂ U we have ρ_W^U = ρ_W^V ∘ ρ_V^U. • A morphism φ : F → G of presheaves with value in C is given by a morphism φ : F(U) → G(U) in C compatible with restriction morphisms.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item A {\\it presheaf $\\mathcal{F}$ on $X$ with values in $\\mathcal{C}$}\nis given by a rule which assigns to every open $U \\subset X$\nan object $\\mathcal{F}(U)$ of $\\mathcal{C}$\nand to each inclusion $V \\subset U$\na morphism $\\rho_V^U : \\mathcal{F}(U) \\to \\mathcal{F}(V)$\nin $\\mathcal{C}$ such that whenever $W \\subset V \\subset U$\nwe have $\\rho_W^U = \\rho_W^V \\circ \\rho_V^U$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves with value in $\\mathcal{C}$} is given by a\nmorphism $\\varphi : \\mathcal{F}(U) \\to \\mathcal{G}(U)$\nin $\\mathcal{C}$ compatible with restriction morphisms.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Presheaves of algebraic structures","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006N","source_file":"sheaves.tex","source_line":339,"source_end_line":356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L339-L356","statement_sha256":"f15f6d96ad9b6650299e85537d8cea84d6311afd78fa0bdee29efc3c1fc985f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":438,"rank":438,"depth":0,"x":1142.407,"y":213.37,"cluster":"sheaves-sites"},{"id":"stacks:006O","tag":"006O","title":"Presheaves of algebraic structures · Definition 006O","summary":"Let X be a topological space. Let C be a category. Let F : C → Sets be a faithful functor. Let F be a presheaf on X with values in C. The presheaf of sets U ↦ F(F(U)) is called the underlying presheaf of sets of F.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{C}$ be a category.\nLet $F : \\mathcal{C} \\to \\textit{Sets}$ be a faithful functor.\nLet $\\mathcal{F}$ be a presheaf on $X$ with values in $\\mathcal{C}$.\nThe presheaf of sets $U \\mapsto F(\\mathcal{F}(U))$\nis called the {\\it underlying presheaf of sets of $\\mathcal{F}$}.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Presheaves of algebraic structures","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006O","source_file":"sheaves.tex","source_line":358,"source_end_line":365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L358-L365","statement_sha256":"66fdb453b4cedba82d48d332e7af53faa808eb867411dead123c7dc448a90a20","origin":"The Stacks Project","memory_eligible":false,"source_rank":439,"rank":439,"depth":0,"x":1125.85,"y":232.967,"cluster":"sheaves-sites"},{"id":"stacks:006Q","tag":"006Q","title":"Presheaves of modules · Definition 006Q","summary":"Let X be a topological space, and let O be a presheaf of rings on X. • A presheaf of O-modules is given by an abelian presheaf F together with a map of presheaves of sets O × F → F such that for every open U ⊂ X the map O(U) × F(U) → F(U) defines the structure of an O(U)-module structure on the abelian group F(U). • A morphism φ : F → G of presheaves of O-modules is a morphism of abelian presheaves φ : F → G such that the diagram xymatrix O × F ar[r] ar[d]_id × φ & F…","statement_latex":"Let $X$ be a topological space, and let $\\mathcal{O}$ be\na presheaf of rings on $X$.\n\\begin{enumerate}\n\\item A {\\it presheaf of $\\mathcal{O}$-modules}\nis given by an abelian presheaf $\\mathcal{F}$ together with a\nmap of presheaves of sets\n$$\n\\mathcal{O} \\times \\mathcal{F} \\longrightarrow \\mathcal{F}\n$$\nsuch that for every open $U \\subset X$ the map\n$\\mathcal{O}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$\ndefines the structure of an $\\mathcal{O}(U)$-module\nstructure on the abelian group $\\mathcal{F}(U)$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves of $\\mathcal{O}$-modules} is a morphism of abelian presheaves\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ such that\nthe diagram\n$$\n\\xymatrix{\n\\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d]_{\\text{id} \\times \\varphi} &\n\\mathcal{F} \\ar[d]^{\\varphi} \\\\\n\\mathcal{O} \\times \\mathcal{G} \\ar[r] &\n\\mathcal{G}\n}\n$$\ncommutes.\n\\item The set of $\\mathcal{O}$-module morphisms as above is\ndenoted $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$.\n\\item The category of presheaves of $\\mathcal{O}$-modules\nis denoted $\\textit{PMod}(\\mathcal{O})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Presheaves of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006Q","source_file":"sheaves.tex","source_line":396,"source_end_line":429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L396-L429","statement_sha256":"542afb7271aa7f6e3ff3058cbc8b308b875f185c8dada380c3fd4eaef5617c6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":440,"rank":440,"depth":0,"x":1122.086,"y":207.2,"cluster":"sheaves-sites"},{"id":"stacks:006R","tag":"006R","title":"Presheaves of modules · Lemma 006R","summary":"With X, O_1, O_2, F and G as above there exists a canonical bijection Hom_O_1(G, F_O_1) = Hom_O_2( O_2 ⊗_p, O_1 G, F ) In other words, the restriction and change of rings functors are adjoint to each other.","statement_latex":"With $X$, $\\mathcal{O}_1$, $\\mathcal{O}_2$, $\\mathcal{F}$ and\n$\\mathcal{G}$ as above there exists a canonical bijection\n$$\n\\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1})\n=\n\\Hom_{\\mathcal{O}_2}(\n\\mathcal{O}_2 \\otimes_{p, \\mathcal{O}_1} \\mathcal{G},\n\\mathcal{F}\n)\n$$\nIn other words, the restriction and change of rings functors\nare adjoint to each other.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006R","source_file":"sheaves.tex","source_line":474,"source_end_line":488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L474-L488","statement_sha256":"8a4bc7cf890031b1fbb62169bc09b060805b3381a00131a6eebcd276f1496d52","origin":"The Stacks Project","memory_eligible":false,"source_rank":441,"rank":441,"depth":0,"x":1147.171,"y":225.267,"cluster":"sheaves-sites"},{"id":"stacks:006T","tag":"006T","title":"Sheaves · Definition 006T","summary":"Let X be a topological space. • A sheaf F of sets on X is a presheaf of sets which satisfies the following additional property: Given any open covering U = ⋃_i ∈ I U_i and any collection of sections s_i ∈ F(U_i), i ∈ I such that ∀ i, j∈ I s_i|_U_i ∩ U_j = s_j|_U_i ∩ U_j there exists a unique section s ∈ F(U) such that s_i = s|_U_i for all i ∈ I. • A morphism of sheaves of sets is simply a morphism of presheaves of sets. • The category of sheaves of sets on X is denoted Sh(X).","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it sheaf $\\mathcal{F}$ of sets on $X$} is a presheaf\nof sets which satisfies the following additional property: Given\nany open covering $U = \\bigcup_{i \\in I} U_i$ and any collection\nof sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that\n$\\forall i, j\\in I$\n$$\ns_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}\n$$\nthere exists a unique section $s \\in \\mathcal{F}(U)$ such that\n$s_i = s|_{U_i}$ for all $i \\in I$.\n\\item A {\\it morphism of sheaves of sets} is simply a\nmorphism of presheaves of sets.\n\\item The category of sheaves of sets on $X$ is denoted\n$\\Sh(X)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006T","source_file":"sheaves.tex","source_line":506,"source_end_line":525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L506-L525","statement_sha256":"6293937868da3008d8e5d438fb5577aecdeb2b20efcf862c72f1ad6634d1f1fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":442,"rank":442,"depth":0,"x":1112.137,"y":226.194,"cluster":"sheaves-sites"},{"id":"stacks:006W","tag":"006W","title":"Sheaves · Definition 006W","summary":"Let X be a topological space. Let A be a set. The constant sheaf with value A denoted underlineA, or underlineA_X is the sheaf that assigns to an open U ⊂ X the set of all locally constant maps U → A with restriction mappings given by restrictions of functions.","statement_latex":"Let $X$ be a topological space. Let $A$ be a set.\nThe {\\it constant sheaf with value $A$} denoted $\\underline{A}$, or\n$\\underline{A}_X$ is the sheaf that assigns to an open $U \\subset X$\nthe set of all locally constant maps $U \\to A$ with restriction mappings\ngiven by restrictions of functions.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/006W","source_file":"sheaves.tex","source_line":587,"source_end_line":594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L587-L594","statement_sha256":"2fd647e25587358061d1b4242f6979fa00c7bfd16f27edd94e593ac39366f13f","origin":"The Stacks Project","memory_eligible":false,"source_rank":443,"rank":443,"depth":0,"x":1138.611,"y":204.542,"cluster":"sheaves-sites"},{"id":"stacks:0070","tag":"0070","title":"Abelian sheaves · Definition 0070","summary":"Let X be a topological space. • An abelian sheaf on X or sheaf of abelian groups on X is an abelian presheaf on X such that the underlying presheaf of sets is a sheaf. • The category of sheaves of abelian groups is denoted Ab(X).","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item An {\\it abelian sheaf on $X$} or\n{\\it sheaf of abelian groups on $X$}\nis an abelian presheaf on $X$ such that the underlying presheaf of\nsets is a sheaf.\n\\item The category of sheaves of abelian groups\nis denoted $\\textit{Ab}(X)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Abelian sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0070","source_file":"sheaves.tex","source_line":646,"source_end_line":657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L646-L657","statement_sha256":"8a949919344ce3ce8e86ae24bcc1bb77b32b735271a513b7a6c523bdd7adfc5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":444,"rank":444,"depth":0,"x":1136.364,"y":237.042,"cluster":"sheaves-sites"},{"id":"stacks:0072","tag":"0072","title":"Sheaves of algebraic structures · Definition 0072","summary":"Let X be a topological space. Let C be a category with products. A presheaf F with values in C on X is a sheaf if for every open covering the diagram xymatrix F(U) ar[r] & ∏_i∈ I F(U_i) ar@<1ex>[r] ar@<-1ex>[r] & ∏_(i_0, i_1) ∈ I × I F(U_i_0 ∩ U_i_1) is an equalizer diagram in the category C.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{C}$ be\na category with products. A presheaf $\\mathcal{F}$ with\nvalues in $\\mathcal{C}$ on $X$ is a {\\it sheaf}\nif for every open covering the diagram\n$$\n\\xymatrix{\n\\mathcal{F}(U) \\ar[r]\n&\n\\prod\\nolimits_{i\\in I}\n\\mathcal{F}(U_i)\n\\ar@<1ex>[r] \\ar@<-1ex>[r]\n&\n\\prod\\nolimits_{(i_0, i_1) \\in I \\times I}\n\\mathcal{F}(U_{i_0} \\cap U_{i_1})\n}\n$$\nis an equalizer diagram in the category $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheaves of algebraic structures","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0072","source_file":"sheaves.tex","source_line":718,"source_end_line":737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L718-L737","statement_sha256":"e6ea9514fb3557e0277ab63bf7b39faaf1453371b4ad5877e1d93bbdd9ca7655","origin":"The Stacks Project","memory_eligible":false,"source_rank":445,"rank":445,"depth":0,"x":1110.82,"y":210.663,"cluster":"sheaves-sites"},{"id":"stacks:0073","tag":"0073","title":"Sheaves of algebraic structures · Lemma 0073","summary":"Suppose the category C and the functor F : C → Sets have the following properties: • F is faithful, • C has limits and F commutes with them, and • the functor F reflects isomorphisms. Let X be a topological space. Let F be a presheaf with values in C. Then F is a sheaf if and only if the underlying presheaf of sets is a sheaf.","statement_latex":"Suppose the category $\\mathcal{C}$ and\nthe functor $F : \\mathcal{C} \\to \\textit{Sets}$\nhave the following properties:\n\\begin{enumerate}\n\\item $F$ is faithful,\n\\item $\\mathcal{C}$ has limits and $F$ commutes with them, and\n\\item the functor $F$ reflects isomorphisms.\n\\end{enumerate}\nLet $X$ be a topological space. Let $\\mathcal{F}$\nbe a presheaf with values in $\\mathcal{C}$.\nThen $\\mathcal{F}$ is a sheaf if and only if the\nunderlying presheaf of sets is a sheaf.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheaves of algebraic structures","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0073","source_file":"sheaves.tex","source_line":776,"source_end_line":790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L776-L790","statement_sha256":"5a6f84248d60bd8a5c07f5a634d0c54352b642376d62ebaaab0c3a6d0037ebee","origin":"The Stacks Project","memory_eligible":false,"source_rank":446,"rank":446,"depth":1,"x":1152.5,"y":215.845,"cluster":"sheaves-sites"},{"id":"stacks:0077","tag":"0077","title":"Sheaves of modules · Definition 0077","summary":"Let X be a topological space. Let O be a sheaf of rings on X. • A sheaf of O-modules is a presheaf of O-modules F, see Definition [Tag 006Q], such that the underlying presheaf of abelian groups F is a sheaf. • A morphism of sheaves of O-modules is a morphism of presheaves of O-modules. • Given sheaves of O-modules F and G we denote Hom_O(F, G) the set of morphism of sheaves of O-modules. • The category of sheaves of O-modules is denoted Mod(O).","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{O}$ be a sheaf of rings on $X$.\n\\begin{enumerate}\n\\item A {\\it sheaf of $\\mathcal{O}$-modules} is a presheaf\nof $\\mathcal{O}$-modules $\\mathcal{F}$,\nsee Definition \\ref{definition-presheaf-modules},\nsuch that the underlying presheaf of abelian groups $\\mathcal{F}$\nis a sheaf.\n\\item A {\\it morphism of sheaves of $\\mathcal{O}$-modules}\nis a morphism of presheaves of $\\mathcal{O}$-modules.\n\\item Given sheaves of $\\mathcal{O}$-modules\n$\\mathcal{F}$ and $\\mathcal{G}$ we denote\n$\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$\nthe set of morphism of sheaves of $\\mathcal{O}$-modules.\n\\item The category of sheaves of $\\mathcal{O}$-modules\nis denoted $\\textit{Mod}(\\mathcal{O})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheaves of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0077","source_file":"sheaves.tex","source_line":859,"source_end_line":878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L859-L878","statement_sha256":"34225be92db3d41f9e5ae9cb97250c4496a20a2e8f2de9c2d51f6211f70771d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":447,"rank":447,"depth":1,"x":1116.268,"y":236.407,"cluster":"sheaves-sites"},{"id":"stacks:0079","tag":"0079","title":"Stalks · Lemma 0079","summary":"Let F be a sheaf of sets on the topological space X. For every open U ⊂ X the map F(U) → ∏_x ∈ U F_x is injective.","statement_latex":"Let $\\mathcal{F}$ be a sheaf of sets on the topological space $X$.\nFor every open $U \\subset X$ the map\n$$\n\\mathcal{F}(U)\n\\longrightarrow\n\\prod\\nolimits_{x \\in U} \\mathcal{F}_x\n$$\nis injective.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Stalks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0079","source_file":"sheaves.tex","source_line":945,"source_end_line":955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L945-L955","statement_sha256":"d6dc09d2831cec211497b2586fbb8e708b887269eb1958d2bfd54ebbd223ca9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":448,"rank":448,"depth":0,"x":1126.828,"y":199.436,"cluster":"sheaves-sites"},{"id":"stacks:007A","tag":"007A","title":"Stalks · Definition 007A","summary":"Let X be a topological space. A presheaf of sets F on X is separated if for every open U ⊂ X the map F(U) → ∏_x ∈ U F_x is injective.","statement_latex":"Let $X$ be a topological space.\nA presheaf of sets $\\mathcal{F}$ on $X$ is {\\it separated}\nif for every open $U \\subset X$ the map\n$\\mathcal{F}(U) \\to \\prod_{x \\in U} \\mathcal{F}_x$ is\ninjective.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Stalks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007A","source_file":"sheaves.tex","source_line":966,"source_end_line":973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L966-L973","statement_sha256":"602cf934a95d4f22e9f447e7b866b259c640b721be30d0fc232fc4405c5bb306","origin":"The Stacks Project","memory_eligible":false,"source_rank":449,"rank":449,"depth":0,"x":1149.475,"y":233.787,"cluster":"sheaves-sites"},{"id":"stacks:007F","tag":"007F","title":"Stalks of abelian presheaves · Lemma 007F","summary":"Let X be a topological space. Let F be a presheaf of abelian groups on X. There exists a unique structure of an abelian group on F_x such that for every U ⊂ X open, x∈ U the map F(U) → F_x is a group homomorphism. Moreover, F_x = colim_x∈ U F(U) holds in the category of abelian groups.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{F}$ be a presheaf\nof abelian groups on $X$. There exists a unique structure of an\nabelian group on $\\mathcal{F}_x$ such that for every\n$U \\subset X$ open, $x\\in U$ the map $\\mathcal{F}(U) \\to \\mathcal{F}_x$\nis a group homomorphism. Moreover,\n$$\n\\mathcal{F}_x\n=\n\\colim_{x\\in U} \\mathcal{F}(U)\n$$\nholds in the category of abelian groups.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Stalks of abelian presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007F","source_file":"sheaves.tex","source_line":1057,"source_end_line":1070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1057-L1070","statement_sha256":"9038895821320192875a8d8d0de893af72c7e6f200ee5a8bd05e50dd9525c83c","origin":"The Stacks Project","memory_eligible":false,"source_rank":450,"rank":450,"depth":0,"x":1103.793,"y":220.91,"cluster":"sheaves-sites"},{"id":"stacks:007H","tag":"007H","title":"Stalks of presheaves of algebraic structures · Lemma 007H","summary":"Let C be a category. Let F : C → Sets be a functor. Assume that • F is faithful, and • directed colimits exist in C and F commutes with them. Let X be a topological space. Let x ∈ X. Let F be a presheaf with values in C. Then F_x = colim_x∈ U F(U) exists in C. Its underlying set is equal to the stalk of the underlying presheaf of sets of F. Furthermore, the construction F ↦ F_x is a functor from the category of presheaves with values in C to C.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $F : \\mathcal{C} \\to \\textit{Sets}$\nbe a functor. Assume that\n\\begin{enumerate}\n\\item $F$ is faithful, and\n\\item directed colimits exist in $\\mathcal{C}$ and $F$ commutes with\nthem.\n\\end{enumerate}\nLet $X$ be a topological space. Let $x \\in X$. Let $\\mathcal{F}$\nbe a presheaf with values in $\\mathcal{C}$.\nThen\n$$\n\\mathcal{F}_x = \\colim_{x\\in U} \\mathcal{F}(U)\n$$\nexists in $\\mathcal{C}$. Its underlying set is equal to the\nstalk of the underlying presheaf of sets of $\\mathcal{F}$.\nFurthermore, the construction $\\mathcal{F} \\mapsto \\mathcal{F}_x$\nis a functor from the category of presheaves with values in\n$\\mathcal{C}$ to $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Stalks of presheaves of algebraic structures","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007H","source_file":"sheaves.tex","source_line":1097,"source_end_line":1117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1097-L1117","statement_sha256":"bea3fca46712d2b1403e06dfb557ec08a991c94bd2c99b896c642d5e9090b567","origin":"The Stacks Project","memory_eligible":false,"source_rank":451,"rank":451,"depth":0,"x":1149.116,"y":204.021,"cluster":"sheaves-sites"},{"id":"stacks:007J","tag":"007J","title":"Stalks of presheaves of modules · Lemma 007J","summary":"Let X be a topological space. Let O be a presheaf of rings on X. Let F be a presheaf of O-modules. Let x ∈ X. The canonical map O_x × F_x → F_x coming from the multiplication map O × F → F defines a O_x-module structure on the abelian group F_x.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{O}$ be a presheaf of rings on $X$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.\nLet $x \\in X$.\nThe canonical map $\\mathcal{O}_x \\times \\mathcal{F}_x\n\\to \\mathcal{F}_x$ coming from the multiplication map\n$\\mathcal{O} \\times \\mathcal{F} \\to \\mathcal{F}$ defines\na $\\mathcal{O}_x$-module structure on the abelian group\n$\\mathcal{F}_x$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Stalks of presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007J","source_file":"sheaves.tex","source_line":1140,"source_end_line":1151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1140-L1151","statement_sha256":"16431db45550331e9570e556c24ce7e4274c126a1b36561ce9f625804029f20a","origin":"The Stacks Project","memory_eligible":false,"source_rank":452,"rank":452,"depth":0,"x":1128.721,"y":243.233,"cluster":"sheaves-sites"},{"id":"stacks:007K","tag":"007K","title":"Stalks of presheaves of modules · Lemma 007K","summary":"Let X be a topological space. Let O → O' be a morphism of presheaves of rings on X. Let F be a presheaf of O-modules. Let x ∈ X. We have F_x ⊗_O_x O'_x = (F ⊗_p, O O')_x as O'_x-modules.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{O} \\to \\mathcal{O}'$ be a morphism of\npresheaves of rings on $X$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.\nLet $x \\in X$. We have\n$$\n\\mathcal{F}_x \\otimes_{\\mathcal{O}_x} \\mathcal{O}'_x\n=\n(\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{O}')_x\n$$\nas $\\mathcal{O}'_x$-modules.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Stalks of presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007K","source_file":"sheaves.tex","source_line":1157,"source_end_line":1170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1157-L1170","statement_sha256":"54cb72f9f6b7ffa8426a5c466118cb31ccfa93704f2127fe27d9403b8ed582a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":453,"rank":453,"depth":0,"x":1111.811,"y":201.691,"cluster":"sheaves-sites"},{"id":"stacks:007M","tag":"007M","title":"Algebraic structures · Definition 007M","summary":"A type of algebraic structure is given by a category C and a functor F : C → Sets with the following properties • F is faithful, • C has limits and F commutes with limits, • C has filtered colimits and F commutes with them, and • F reflects isomorphisms.","statement_latex":"A {\\it type of algebraic structure} is given by a category $\\mathcal{C}$\nand a functor $F : \\mathcal{C} \\to \\textit{Sets}$ with the\nfollowing properties\n\\begin{enumerate}\n\\item $F$ is faithful,\n\\item $\\mathcal{C}$ has limits and $F$ commutes with limits,\n\\item $\\mathcal{C}$ has filtered colimits and $F$ commutes with them, and\n\\item $F$ reflects isomorphisms.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Algebraic structures","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007M","source_file":"sheaves.tex","source_line":1184,"source_end_line":1195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1184-L1195","statement_sha256":"09e4653411cd5e5a511214f81160ea22f4aab949fccd8998504f02012181f752","origin":"The Stacks Project","memory_eligible":false,"source_rank":454,"rank":454,"depth":0,"x":1158.813,"y":223.256,"cluster":"sheaves-sites"},{"id":"stacks:007N","tag":"007N","title":"Algebraic structures · Lemma 007N","summary":"The following categories, endowed with the obvious forgetful functor, define types of algebraic structures: • The category of pointed sets. • The category of abelian groups. • The category of groups. • The category of monoids. • The category of rings. • The category of R-modules for a fixed ring R. • The category of Lie algebras over a fixed field.","statement_latex":"The following categories, endowed with the obvious forgetful\nfunctor, define types of algebraic structures:\n\\begin{enumerate}\n\\item The category of pointed sets.\n\\item The category of abelian groups.\n\\item The category of groups.\n\\item The category of monoids.\n\\item The category of rings.\n\\item The category of $R$-modules for a fixed ring $R$.\n\\item The category of Lie algebras over a fixed field.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Algebraic structures","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007N","source_file":"sheaves.tex","source_line":1206,"source_end_line":1219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1206-L1219","statement_sha256":"306345f6177afd590919f178aedaf02111353ff390240a15e4a14988833c2b9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":455,"rank":455,"depth":0,"x":1105.587,"y":234.268,"cluster":"sheaves-sites"},{"id":"stacks:007O","tag":"007O","title":"Algebraic structures · Lemma 007O","summary":"Let (C, F) be a type of algebraic structure. • C has a final object 0 and F(0) = ( * ). • C has products and F(∏ A_i) = ∏ F(A_i). • C has fibre products and F(A ×_B C) = F(A)×_F(B)F(C). • C has equalizers, and if E → A is the equalizer of a, b : A → B, then F(E) → F(A) is the equalizer of F(a), F(b) : F(A) → F(B). • A → B is a monomorphism if and only if F(A) → F(B) is injective. • if F(a) : F(A) → F(B) is surjective, then a is an epimorphism. • given A_1 → A_2 → A_3 → …,…","statement_latex":"Let $(\\mathcal{C}, F)$ be a type of algebraic structure.\n\\begin{enumerate}\n\\item $\\mathcal{C}$ has a final object $0$ and $F(0) = \\{ * \\}$.\n\\item $\\mathcal{C}$ has products and $F(\\prod A_i) = \\prod F(A_i)$.\n\\item $\\mathcal{C}$ has fibre products and\n$F(A \\times_B C) = F(A)\\times_{F(B)}F(C)$.\n\\item $\\mathcal{C}$ has equalizers, and if $E \\to A$\nis the equalizer of $a, b : A \\to B$, then\n$F(E) \\to F(A)$ is the equalizer of $F(a), F(b) : F(A) \\to F(B)$.\n\\item $A \\to B$ is a monomorphism if and only if\n$F(A) \\to F(B)$ is injective.\n\\item if $F(a) : F(A) \\to F(B)$ is surjective, then\n$a$ is an epimorphism.\n\\item given $A_1 \\to A_2 \\to A_3 \\to \\ldots$, then\n$\\colim A_i$ exists and $F(\\colim A_i) = \\colim F(A_i)$,\nand more generally for any filtered colimit.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Algebraic structures","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007O","source_file":"sheaves.tex","source_line":1235,"source_end_line":1254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1235-L1254","statement_sha256":"681e64ff357e7a71c49ade59ed306923a103c8e689202899a33433b9b86b15b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":456,"rank":456,"depth":0,"x":1136.671,"y":195.091,"cluster":"sheaves-sites"},{"id":"stacks:007P","tag":"007P","title":"Algebraic structures · Lemma 007P","summary":"Let (C, F) be a type of algebraic structure. Suppose that A, B, C ∈ Ob(C). Let f : A → B and g : C → B be morphisms of C. If F(g) is injective, and Im(F(f)) ⊂ Im(F(g)), then f factors as f = g ∘ t for some morphism t : A → C.","statement_latex":"Let $(\\mathcal{C}, F)$ be a type of algebraic structure.\nSuppose that $A, B, C \\in \\Ob(\\mathcal{C})$.\nLet $f : A \\to B$ and $g : C \\to B$ be morphisms of\n$\\mathcal{C}$. If $F(g)$ is injective, and\n$\\Im(F(f)) \\subset \\Im(F(g))$, then\n$f$ factors as $f = g \\circ t$ for some morphism\n$t : A \\to C$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Algebraic structures","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007P","source_file":"sheaves.tex","source_line":1263,"source_end_line":1272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1263-L1272","statement_sha256":"483e55abf685c0713a813be6f8245580b3715eb0f1a6dca169c322f6bad09cd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":457,"rank":457,"depth":0,"x":1145.43,"y":242.619,"cluster":"sheaves-sites"},{"id":"stacks:007T","tag":"007T","title":"Exactness and points · Lemma 007T","summary":"Let X be a topological space. Let φ : F → G be a morphism of sheaves of sets on X. • The map φ is a monomorphism in the category of sheaves if and only if for all x ∈ X the map φ_x : F_x → G_x is injective. • The map φ is an epimorphism in the category of sheaves if and only if for all x ∈ X the map φ_x : F_x → G_x is surjective. • The map φ is an isomorphism in the category of sheaves if and only if for all x ∈ X the map φ_x : F_x → G_x is bijective.","statement_latex":"Let $X$ be a topological space. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nbe a morphism of sheaves of sets on $X$.\n\\begin{enumerate}\n\\item The map $\\varphi$ is a monomorphism in the category of sheaves\nif and only if for all $x \\in X$ the map\n$\\varphi_x : \\mathcal{F}_x \\to \\mathcal{G}_x$\nis injective.\n\\item The map $\\varphi$ is an epimorphism in the category of sheaves\nif and only if for all $x \\in X$ the map\n$\\varphi_x : \\mathcal{F}_x \\to \\mathcal{G}_x$\nis surjective.\n\\item The map $\\varphi$ is an isomorphism in the category of sheaves\nif and only if for all $x \\in X$ the map\n$\\varphi_x : \\mathcal{F}_x \\to \\mathcal{G}_x$\nis bijective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Exactness and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007T","source_file":"sheaves.tex","source_line":1340,"source_end_line":1358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1340-L1358","statement_sha256":"99a539c0bb93077c6209de505b62421acb89866d7b900ef25cdb1c72572c164a","origin":"The Stacks Project","memory_eligible":false,"source_rank":458,"rank":458,"depth":0,"x":1099.836,"y":211.917,"cluster":"sheaves-sites"},{"id":"stacks:007U","tag":"007U","title":"Exactness and points · Definition 007U","summary":"Let X be a topological space. • A presheaf F is called a subpresheaf of a presheaf G if F(U) ⊂ G(U) for all open U ⊂ X such that the restriction maps of G induce the restriction maps of F. If F and G are sheaves, then F is called a subsheaf of G. We sometimes indicate this by the notation F ⊂ G. • A morphism of presheaves of sets φ : F → G on X is called injective if and only if F(U) → G(U) is injective for all U open in X. • A morphism of presheaves of sets φ : F → G on…","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item A presheaf $\\mathcal{F}$ is called a {\\it subpresheaf} of a presheaf\n$\\mathcal{G}$ if $\\mathcal{F}(U) \\subset \\mathcal{G}(U)$ for all open\n$U \\subset X$ such that the restriction maps of $\\mathcal{G}$ induce the\nrestriction maps of $\\mathcal{F}$. If $\\mathcal{F}$ and\n$\\mathcal{G}$ are sheaves, then $\\mathcal{F}$ is called a {\\it subsheaf}\nof $\\mathcal{G}$. We sometimes indicate this by the notation\n$\\mathcal{F} \\subset \\mathcal{G}$.\n\\item A morphism of presheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\non $X$ is called {\\it injective} if and only if\n$\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is injective for all $U$ open in $X$.\n\\item A morphism of presheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\non $X$ is called {\\it surjective} if and only if\n$\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is surjective for all $U$ open in $X$.\n\\item A morphism of sheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\non $X$ is called {\\it injective} if and only if\n$\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is injective for all $U$ open in $X$.\n\\item A morphism of sheaves of sets $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\non $X$ is called {\\it surjective} if and only if for every open\n$U$ of $X$ and every section $s$ of $\\mathcal{G}(U)$ there exists an\nopen covering $U = \\bigcup U_i$ such that $s|_{U_i}$ is in\nthe image of $\\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)$ for all $i$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Exactness and points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007U","source_file":"sheaves.tex","source_line":1369,"source_end_line":1395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1369-L1395","statement_sha256":"2d4841ee5f4e383ab2718a22b0b38a278a5528c2c05c08aa44fe3c3542c3a73a","origin":"The Stacks Project","memory_eligible":false,"source_rank":459,"rank":459,"depth":0,"x":1159.301,"y":208.628,"cluster":"sheaves-sites"},{"id":"stacks:0H7H","tag":"0H7H","title":"Exactness and points · Lemma 0H7H","summary":"Let X be a topological space. • Epimorphisms (resp. monomorphisms) in the category of presheaves are exactly the surjective (resp. injective) maps of presheaves. • Epimorphisms (resp. monomorphisms) in the category of sheaves are exactly the surjective (resp. injective) maps of sheaves, and are exactly those maps which are surjective (resp. injective) on all the stalks.","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item Epimorphisms (resp.\\ monomorphisms) in the category of\npresheaves are exactly the surjective (resp.\\ injective) maps\nof presheaves.\n\\item Epimorphisms (resp.\\ monomorphisms) in the category of\nsheaves are exactly the surjective (resp.\\ injective) maps\nof sheaves, and are exactly those maps which are surjective\n(resp.\\ injective) on all the stalks.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Exactness and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7H","source_file":"sheaves.tex","source_line":1398,"source_end_line":1410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1398-L1410","statement_sha256":"6cde2593d428d94084378529f24bc42b779990b770733e9f8427fc87a7c39334","origin":"The Stacks Project","memory_eligible":false,"source_rank":460,"rank":460,"depth":0,"x":1117.306,"y":245.478,"cluster":"sheaves-sites"},{"id":"stacks:007W","tag":"007W","title":"Exactness and points · Lemma 007W","summary":"let X be a topological space. Let (C, F) be a type of algebraic structure. Suppose that F, G are sheaves on X with values in C. Let φ : F → G be a map of the underlying sheaves of sets. If for all points x ∈ X the map F_x → G_x is a morphism of algebraic structures, then φ is a morphism of sheaves of algebraic structures.","statement_latex":"let $X$ be a topological space.\nLet $(\\mathcal{C}, F)$ be a type of algebraic structure.\nSuppose that $\\mathcal{F}$, $\\mathcal{G}$ are sheaves on $X$\nwith values in $\\mathcal{C}$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nbe a map of the underlying sheaves of sets.\nIf for all points $x \\in X$ the map\n$\\mathcal{F}_x \\to \\mathcal{G}_x$\nis a morphism of algebraic structures,\nthen $\\varphi$ is a morphism of sheaves of algebraic structures.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Exactness and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007W","source_file":"sheaves.tex","source_line":1417,"source_end_line":1429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1417-L1429","statement_sha256":"9fe18e0db7e426a05358593865078a7ad5d54500d9ea53620af6a3a14e705605","origin":"The Stacks Project","memory_eligible":false,"source_rank":461,"rank":461,"depth":1,"x":1118.671,"y":193.542,"cluster":"sheaves-sites"},{"id":"stacks:007Y","tag":"007Y","title":"Sheafification · Lemma 007Y","summary":"The presheaf F^\\# is a sheaf.","statement_latex":"The presheaf $\\mathcal{F}^{\\#}$ is a sheaf.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007Y","source_file":"sheaves.tex","source_line":1536,"source_end_line":1539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1536-L1539","statement_sha256":"ac4c4171995856615e389c634d0b256f849770ef1668fcfd4715bd2658f8bb4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":462,"rank":462,"depth":0,"x":1160.145,"y":233.308,"cluster":"sheaves-sites"},{"id":"stacks:007Z","tag":"007Z","title":"Sheafification · Lemma 007Z","summary":"Let X be a topological space. Let F be a presheaf of sets on X. Let x ∈ X. Then F_x = F^\\#_x.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{F}$ be a presheaf of sets on $X$.\nLet $x \\in X$. Then $\\mathcal{F}_x = \\mathcal{F}^\\#_x$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/007Z","source_file":"sheaves.tex","source_line":1562,"source_end_line":1567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1562-L1567","statement_sha256":"625a45f287fdc5c4373a52edc27f6caeaee20aa07f43f02d43f2bfa757ef0d96","origin":"The Stacks Project","memory_eligible":false,"source_rank":463,"rank":463,"depth":0,"x":1096.516,"y":227.414,"cluster":"sheaves-sites"},{"id":"stacks:0080","tag":"0080","title":"Sheafification · Lemma 0080","summary":"Let F be a presheaf of sets on X. Any map F → G into a sheaf of sets factors uniquely as F → F^\\# → G.","statement_latex":"Let $\\mathcal{F}$ be a presheaf of sets on $X$.\nAny map $\\mathcal{F} \\to \\mathcal{G}$ into a sheaf of sets\nfactors uniquely as\n$\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0080","source_file":"sheaves.tex","source_line":1588,"source_end_line":1594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1588-L1594","statement_sha256":"d54ddeb6545c9aeebefc0cec60d2f310f1bfcd49241f6b4cbdc9960ae80d4d6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":464,"rank":464,"depth":1,"x":1149.032,"y":195.136,"cluster":"sheaves-sites"},{"id":"stacks:0082","tag":"0082","title":"Sheafification · Lemma 0082","summary":"Let X be a topological space. A presheaf F is separated (see Definition [Tag 007A]) if and only if the canonical map F → F^\\# is injective.","statement_latex":"Let $X$ be a topological space.\nA presheaf $\\mathcal{F}$ is separated (see\nDefinition \\ref{definition-separated}) if and only if\nthe canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ is injective.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0082","source_file":"sheaves.tex","source_line":1639,"source_end_line":1645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1639-L1645","statement_sha256":"465723ab76749b7fb1bbbc5b7fb66fb9c1fb8da268716b3adc208c0e4379fade","origin":"The Stacks Project","memory_eligible":false,"source_rank":465,"rank":465,"depth":1,"x":1136.054,"y":249.592,"cluster":"sheaves-sites"},{"id":"stacks:0H7I","tag":"0H7I","title":"Sheafification · Lemma 0H7I","summary":"Let X be a topological space. The sheafification of a surjective (resp. injective) morphism of presheaves of sets is surjective (resp. injective).","statement_latex":"Let $X$ be a topological space. The sheafification of a surjective\n(resp.\\ injective) morphism of presheaves of sets is surjective\n(resp.\\ injective).","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7I","source_file":"sheaves.tex","source_line":1652,"source_end_line":1657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1652-L1657","statement_sha256":"2b882483ec7615f07f3abb07ecb8c71bf06f6e895e0ab5c4dca77cd1d8c479d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":466,"rank":466,"depth":0,"x":1101.308,"y":201.335,"cluster":"sheaves-sites"},{"id":"stacks:0084","tag":"0084","title":"Sheafification of abelian presheaves · Lemma 0084","summary":"Let X be a topological space. Let F be a presheaf of sets on X. Let U ⊂ X be open. There is a canonical fibre product diagram xymatrix F^\\#(U) ar[d] ar[r] & Pi(F)(U) ar[d] ∏_x ∈ U F_x ar[r] & ∏_x ∈ U Pi(F)_x where the maps are the following: • The left vertical map has components F^\\#(U) → F^\\#_x = F_x where the equality is Lemma [Tag 007Z]. • The top horizontal map comes from the map of presheaves F → Pi(F) described in Section [Tag 007X]. • The right vertical map has…","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{F}$ be\na presheaf of sets on $X$. Let $U \\subset X$ be open.\nThere is a canonical fibre product diagram\n$$\n\\xymatrix{\n\\mathcal{F}^\\#(U) \\ar[d] \\ar[r] &\n\\Pi(\\mathcal{F})(U) \\ar[d] \\\\\n\\prod_{x \\in U} \\mathcal{F}_x\n\\ar[r] &\n\\prod_{x \\in U} \\Pi(\\mathcal{F})_x\n}\n$$\nwhere the maps are the following:\n\\begin{enumerate}\n\\item The left vertical map has components\n$\\mathcal{F}^\\#(U) \\to \\mathcal{F}^\\#_x = \\mathcal{F}_x$\nwhere the equality is Lemma \\ref{lemma-stalk-sheafification}.\n\\item The top horizontal map comes from the\nmap of presheaves $\\mathcal{F} \\to \\Pi(\\mathcal{F})$ described\nin Section \\ref{section-sheafification}.\n\\item The right vertical map has obvious component\nmaps $\\Pi(\\mathcal{F})(U) \\to \\Pi(\\mathcal{F})_x$.\n\\item The bottom horizontal map has components\n$\\mathcal{F}_x \\to \\Pi(\\mathcal{F})_x$\nwhich come from the map of presheaves\n$\\mathcal{F} \\to \\Pi(\\mathcal{F})$ described\nin Section \\ref{section-sheafification}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification of abelian presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0084","source_file":"sheaves.tex","source_line":1672,"source_end_line":1702,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1672-L1702","statement_sha256":"fcf95ceb29cf88280b95793a829d81b0b14f7d82dde10cb6de102a53c7ee88c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":467,"rank":467,"depth":1,"x":1166.702,"y":217.446,"cluster":"sheaves-sites"},{"id":"stacks:0085","tag":"0085","title":"Sheafification of abelian presheaves · Lemma 0085","summary":"Let X be a topological space. Let F be an abelian presheaf on X. Then there exists a unique structure of abelian sheaf on F^\\# such that F → F^\\# is a morphism of abelian presheaves. Moreover, the following adjointness property holds Mor_PAb(X)(F, i(G)) = Mor_Ab(X)(F^\\#, G).","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThen there exists a unique structure of\nabelian sheaf on $\\mathcal{F}^\\#$ such that\n$\\mathcal{F} \\to \\mathcal{F}^\\#$ is a morphism\nof abelian presheaves. Moreover, the following adjointness\nproperty holds\n$$\n\\Mor_{\\textit{PAb}(X)}(\\mathcal{F}, i(\\mathcal{G}))\n=\n\\Mor_{\\textit{Ab}(X)}(\\mathcal{F}^\\#, \\mathcal{G}).\n$$","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification of abelian presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0085","source_file":"sheaves.tex","source_line":1719,"source_end_line":1733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1719-L1733","statement_sha256":"3bcec6f0cc7f358a3526f32861fb4d701f12f4f3958a4a89e305d5e002b39844","origin":"The Stacks Project","memory_eligible":false,"source_rank":468,"rank":468,"depth":2,"x":1104.631,"y":243.035,"cluster":"sheaves-sites"},{"id":"stacks:0087","tag":"0087","title":"Sheafification of presheaves of algebraic structures · Lemma 0087","summary":"Let X be a topological space. Let (C, F) be a type of algebraic structure. Let F be a presheaf with values in C on X. Then there exists a sheaf F^\\# with values in C and a morphism F → F^\\# of presheaves with values in C with the following properties: • The map F → F^\\# identifies the underlying sheaf of sets of F^\\# with the sheafification of the underlying presheaf of sets of F. • For any morphism F → G, where G is a sheaf with values in C there exists a unique…","statement_latex":"Let $X$ be a topological space.\nLet $(\\mathcal{C}, F)$ be a type of algebraic structure.\nLet $\\mathcal{F}$ be a presheaf with values in $\\mathcal{C}$\non $X$. Then there exists a sheaf $\\mathcal{F}^\\#$ with values\nin $\\mathcal{C}$ and a morphism $\\mathcal{F} \\to \\mathcal{F}^\\#$\nof presheaves with values in $\\mathcal{C}$ with the\nfollowing properties:\n\\begin{enumerate}\n\\item The map $\\mathcal{F} \\to \\mathcal{F}^\\#$ identifies\nthe underlying sheaf of sets of $\\mathcal{F}^\\#$ with\nthe sheafification of the underlying presheaf of sets of $\\mathcal{F}$.\n\\item For any morphism $\\mathcal{F} \\to \\mathcal{G}$, where\n$\\mathcal{G}$ is a sheaf with values in $\\mathcal{C}$ there exists\na unique factorization $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification of presheaves of algebraic structures","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0087","source_file":"sheaves.tex","source_line":1782,"source_end_line":1799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1782-L1799","statement_sha256":"9743891bc294206c03bef967987469d96bd20f580ef155026dfd852fa9accfb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":469,"rank":469,"depth":3,"x":1130.185,"y":188.181,"cluster":"sheaves-sites"},{"id":"stacks:0089","tag":"0089","title":"Sheafification of presheaves of modules · Lemma 0089","summary":"Let X be a topological space. Let O be a presheaf of rings on X. Let F be a presheaf of O-modules. Let O^\\# be the sheafification of O. Let F^\\# be the sheafification of F as a presheaf of abelian groups. There exists a map of sheaves of sets O^\\# × F^\\# → F^\\# which makes the diagram xymatrix O × F ar[r] ar[d] & F ar[d] O^\\# × F^\\# ar[r] & F^\\# commute and which makes F^\\# into a sheaf of O^\\#-modules. In addition, if G is a sheaf of O^\\#-modules, then any morphism of…","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{O}$ be a presheaf of rings on $X$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.\nLet $\\mathcal{O}^\\#$ be the sheafification of $\\mathcal{O}$.\nLet $\\mathcal{F}^\\#$ be the sheafification of $\\mathcal{F}$\nas a presheaf of abelian groups. There exists a map of\nsheaves of sets\n$$\n\\mathcal{O}^\\# \\times \\mathcal{F}^\\#\n\\longrightarrow\n\\mathcal{F}^\\#\n$$\nwhich makes the diagram\n$$\n\\xymatrix{\n\\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d] &\n\\mathcal{F} \\ar[d] \\\\\n\\mathcal{O}^\\# \\times \\mathcal{F}^\\# \\ar[r] &\n\\mathcal{F}^\\#\n}\n$$\ncommute and which makes $\\mathcal{F}^\\#$ into a sheaf\nof $\\mathcal{O}^\\#$-modules. In addition, if $\\mathcal{G}$\nis a sheaf of $\\mathcal{O}^\\#$-modules, then any morphism\nof presheaves of $\\mathcal{O}$-modules $\\mathcal{F} \\to \\mathcal{G}$\n(into the restriction of $\\mathcal{G}$ to a $\\mathcal{O}$-module)\nfactors uniquely as $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$\nwhere $\\mathcal{F}^\\# \\to \\mathcal{G}$ is a morphism of\n$\\mathcal{O}^\\#$-modules.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification of presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0089","source_file":"sheaves.tex","source_line":1824,"source_end_line":1855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1824-L1855","statement_sha256":"db354d783fd33a1b8edcd0ea8fd4fcf6933919e4a253e8c488f89742214dabb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":470,"rank":470,"depth":0,"x":1155.798,"y":243.888,"cluster":"sheaves-sites"},{"id":"stacks:008A","tag":"008A","title":"Sheafification of presheaves of modules · Lemma 008A","summary":"With X, O_1, O_2, F and G as above there exists a canonical bijection Hom_O_1(G, F_O_1) = Hom_O_2( O_2 ⊗_O_1 G, F ) In other words, the restriction and change of rings functors are adjoint to each other.","statement_latex":"With $X$, $\\mathcal{O}_1$, $\\mathcal{O}_2$, $\\mathcal{F}$ and\n$\\mathcal{G}$ as above there exists a canonical bijection\n$$\n\\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1})\n=\n\\Hom_{\\mathcal{O}_2}(\n\\mathcal{O}_2 \\otimes_{\\mathcal{O}_1} \\mathcal{G},\n\\mathcal{F}\n)\n$$\nIn other words, the restriction and change of rings functors\nare adjoint to each other.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification of presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008A","source_file":"sheaves.tex","source_line":1915,"source_end_line":1929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1915-L1929","statement_sha256":"44282dddee264d459336630e31e244969bdd97d0c8a18a7b6b1e0b70efd95f92","origin":"The Stacks Project","memory_eligible":false,"source_rank":471,"rank":471,"depth":1,"x":1091.262,"y":216.984,"cluster":"sheaves-sites"},{"id":"stacks:008B","tag":"008B","title":"Sheafification of presheaves of modules · Lemma 008B","summary":"Let X be a topological space. Let O → O' be a morphism of sheaves of rings on X. Let F be a sheaf O-modules. Let x ∈ X. We have F_x ⊗_O_x O'_x = (F ⊗_O O')_x as O'_x-modules.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{O} \\to \\mathcal{O}'$ be a morphism of\nsheaves of rings on $X$.\nLet $\\mathcal{F}$ be a sheaf $\\mathcal{O}$-modules.\nLet $x \\in X$. We have\n$$\n\\mathcal{F}_x \\otimes_{\\mathcal{O}_x} \\mathcal{O}'_x\n=\n(\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{O}')_x\n$$\nas $\\mathcal{O}'_x$-modules.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Sheafification of presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008B","source_file":"sheaves.tex","source_line":1947,"source_end_line":1960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L1947-L1960","statement_sha256":"67a9ce34e1d309c0ae1106d090c3e35d5ac3bfa3321358cf764cf857ac118127","origin":"The Stacks Project","memory_eligible":false,"source_rank":472,"rank":472,"depth":1,"x":1161.39,"y":199.988,"cluster":"sheaves-sites"},{"id":"stacks:008D","tag":"008D","title":"Continuous maps and sheaves · Lemma 008D","summary":"Let f : X → Y be a continuous map. Let F be a sheaf of sets on X. Then f_*F is a sheaf on Y.","statement_latex":"Let $f : X \\to Y$ be a continuous map.\nLet $\\mathcal{F}$ be a sheaf of sets on $X$.\nThen $f_*\\mathcal{F}$ is a sheaf on $Y$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008D","source_file":"sheaves.tex","source_line":2016,"source_end_line":2021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2016-L2021","statement_sha256":"f6bdb281d04ac3dcf4a483ee7fc791362392c093fe947e291b2e8e54a46bccd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":473,"rank":473,"depth":0,"x":1122.858,"y":252.977,"cluster":"sheaves-sites"},{"id":"stacks:008E","tag":"008E","title":"Continuous maps and sheaves · Lemma 008E","summary":"Let f : X → Y and g : Y → Z be continuous maps of topological spaces. The functors (g ∘ f)_* and g_* ∘ f_* are equal (on both presheaves and sheaves of sets).","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be continuous maps\nof topological spaces. The functors $(g \\circ f)_*$\nand $g_* \\circ f_*$ are equal (on both presheaves\nand sheaves of sets).","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008E","source_file":"sheaves.tex","source_line":2037,"source_end_line":2043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2037-L2043","statement_sha256":"b48dc3ea0678ad722477167fa6077882202291a8c48bf767e2d485ee75c1570c","origin":"The Stacks Project","memory_eligible":false,"source_rank":474,"rank":474,"depth":0,"x":1108.487,"y":191.284,"cluster":"sheaves-sites"},{"id":"stacks:008F","tag":"008F","title":"Continuous maps and sheaves · Lemma 008F","summary":"Let f : X → Y be a continuous map. There exists a functor f_p : PSh(Y) → PSh(X) which is left adjoint to f_*. For a presheaf G it is determined by the rule f_pG(U) = colim_f(U) ⊂ V G(V) where the colimit is over the collection of open neighbourhoods V of f(U) in Y. The colimits are over directed partially ordered sets. (The restriction mappings of f_pG are explained in the proof.)","statement_latex":"Let $f : X \\to Y$ be a continuous map.\nThere exists a functor\n$f_p : \\textit{PSh}(Y) \\to \\textit{PSh}(X)$\nwhich is left adjoint to $f_*$. For a presheaf\n$\\mathcal{G}$ it is determined by the rule\n$$\nf_p\\mathcal{G}(U) = \\colim_{f(U) \\subset V} \\mathcal{G}(V)\n$$\nwhere the colimit is over the collection of open neighbourhoods\n$V$ of $f(U)$ in $Y$. The colimits are over\ndirected partially ordered sets.\n(The restriction mappings of $f_p\\mathcal{G}$ are explained in the proof.)","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008F","source_file":"sheaves.tex","source_line":2066,"source_end_line":2080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2066-L2080","statement_sha256":"f602a1dc921c8a09145bf487fe15307fc5554d5e9dc0e6edb3691876dc1a5a50","origin":"The Stacks Project","memory_eligible":false,"source_rank":475,"rank":475,"depth":1,"x":1169.422,"y":229.075,"cluster":"sheaves-sites"},{"id":"stacks:008G","tag":"008G","title":"Continuous maps and sheaves · Lemma 008G","summary":"Let f : X → Y be a continuous map. Let x ∈ X. Let G be a presheaf of sets on Y. There is a canonical bijection of stalks (f_pG)_x = G_f(x).","statement_latex":"Let $f : X \\to Y$ be a continuous map.\nLet $x \\in X$. Let $\\mathcal{G}$ be a presheaf of sets on $Y$.\nThere is a canonical bijection of stalks\n$(f_p\\mathcal{G})_x = \\mathcal{G}_{f(x)}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008G","source_file":"sheaves.tex","source_line":2140,"source_end_line":2146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2140-L2146","statement_sha256":"536544cb6bbfed768fcdd8d8b37042e197c56d39ddcf37ba413a8afda1f686c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":476,"rank":476,"depth":1,"x":1093.208,"y":235.86,"cluster":"sheaves-sites"},{"id":"stacks:008H","tag":"008H","title":"Continuous maps and sheaves · Lemma 008H","summary":"Let x ∈ X. Let G be a sheaf of sets on Y. There is a canonical bijection of stalks (f^-1G)_x = G_f(x).","statement_latex":"Let $x \\in X$. Let $\\mathcal{G}$ be a sheaf of sets on $Y$.\nThere is a canonical bijection of stalks\n$(f^{-1}\\mathcal{G})_x = \\mathcal{G}_{f(x)}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008H","source_file":"sheaves.tex","source_line":2196,"source_end_line":2201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2196-L2201","statement_sha256":"f2fc48e667a7c7768b9d7345b131880cfcbfcff0a2879e58d370aabd57ec3bd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":477,"rank":477,"depth":2,"x":1144.54,"y":187.057,"cluster":"sheaves-sites"},{"id":"stacks:008I","tag":"008I","title":"Continuous maps and sheaves · Lemma 008I","summary":"Let f : X → Y and g : Y → Z be continuous maps of topological spaces. The functors (g ∘ f)^-1 and f^-1 ∘ g^-1 are canonically isomorphic. Similarly (g ∘ f)_p ≅ f_p ∘ g_p on presheaves.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be continuous maps\nof topological spaces. The functors $(g \\circ f)^{-1}$\nand $f^{-1} \\circ g^{-1}$ are canonically isomorphic.\nSimilarly $(g \\circ f)_p \\cong f_p \\circ g_p$ on\npresheaves.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008I","source_file":"sheaves.tex","source_line":2208,"source_end_line":2215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2208-L2215","statement_sha256":"214545227134b2542587a4b526af298e152b4ae88a5a6180cc96e80ae92f22d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":478,"rank":478,"depth":1,"x":1145.947,"y":252.905,"cluster":"sheaves-sites"},{"id":"stacks:008J","tag":"008J","title":"Continuous maps and sheaves · Definition 008J","summary":"Let f : X → Y be a continuous map. Let F be a sheaf of sets on X and let G be a sheaf of sets on Y. An f-map xi : G → F is a collection of maps xi_V : G(V) → F(f^-1(V)) indexed by open subsets V ⊂ Y such that xymatrix G(V) ar[r]_xi_V ar[d]_restriction of G & F(f^-1V) ar[d]^restriction of F G(V') ar[r]^xi_V' & F(f^-1V') commutes for all V' ⊂ V ⊂ Y open.","statement_latex":"Let $f : X \\to Y$ be a continuous map.\nLet $\\mathcal{F}$ be a sheaf of sets on $X$ and\nlet $\\mathcal{G}$ be a sheaf of sets on $Y$.\nAn {\\it $f$-map $\\xi : \\mathcal{G} \\to \\mathcal{F}$}\nis a collection of maps\n$\\xi_V : \\mathcal{G}(V) \\to \\mathcal{F}(f^{-1}(V))$\nindexed by open subsets $V \\subset Y$ such that\n$$\n\\xymatrix{\n\\mathcal{G}(V) \\ar[r]_{\\xi_V} \\ar[d]_{\\text{restriction of }\\mathcal{G}} &\n\\mathcal{F}(f^{-1}V) \\ar[d]^{\\text{restriction of }\\mathcal{F}} \\\\\n\\mathcal{G}(V') \\ar[r]^{\\xi_{V'}} &\n\\mathcal{F}(f^{-1}V')\n}\n$$\ncommutes for all $V' \\subset V \\subset Y$ open.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008J","source_file":"sheaves.tex","source_line":2223,"source_end_line":2241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2223-L2241","statement_sha256":"5753aaac807763139b45e5ae7201169a4e2e209c8411e89789d2342bd99a71e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":479,"rank":479,"depth":0,"x":1091.363,"y":204.619,"cluster":"sheaves-sites"},{"id":"stacks:008K","tag":"008K","title":"Continuous maps and sheaves · Lemma 008K","summary":"Let f : X → Y be a continuous map. There are bijections between the following four sets • the set of maps G → f_*F, • the set of maps f^-1G → F, • the set of f-maps xi : G → F, and • the set of all collections of maps xi_U, V : G(V) → F(U) for all U ⊂ X and V ⊂ Y open such that f(U) ⊂ V compatible with all restriction maps, functorially in F ∈ Sh(X) and G ∈ Sh(Y).","statement_latex":"Let $f : X \\to Y$ be a continuous map.\nThere are bijections between the following four sets\n\\begin{enumerate}\n\\item the set of maps $\\mathcal{G} \\to f_*\\mathcal{F}$,\n\\item the set of maps $f^{-1}\\mathcal{G} \\to \\mathcal{F}$,\n\\item the set of $f$-maps $\\xi : \\mathcal{G} \\to \\mathcal{F}$, and\n\\item the set of all collections of maps\n$\\xi_{U, V} : \\mathcal{G}(V) \\to \\mathcal{F}(U)$ for all\n$U \\subset X$ and $V \\subset Y$ open such that $f(U) \\subset V$\ncompatible with all restriction maps,\n\\end{enumerate}\nfunctorially in $\\mathcal{F} \\in \\Sh(X)$ and $\\mathcal{G} \\in \\Sh(Y)$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008K","source_file":"sheaves.tex","source_line":2248,"source_end_line":2262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2248-L2262","statement_sha256":"a45bd087519dd3f2abd6c214aa151b5600c2809742637e5d55d33e9ffaf6352f","origin":"The Stacks Project","memory_eligible":false,"source_rank":480,"rank":480,"depth":1,"x":1171.295,"y":209.305,"cluster":"sheaves-sites"},{"id":"stacks:008L","tag":"008L","title":"Continuous maps and sheaves · Definition 008L","summary":"Suppose that f : X → Y and g : Y → Z are continuous maps of topological spaces. Suppose that F is a sheaf on X, G is a sheaf on Y, and H is a sheaf on Z. Let φ : G → F be an f-map. Let ψ : H → G be an g-map. The composition of φ and ψ is the (g ∘ f)-map φ ∘ ψ defined by the commutativity of the diagrams xymatrix H(W) ar[rr]_(φ ∘ ψ)_W ar[rd]_ψ_W & & F(f^-1g^-1W) & G(g^-1W) ar[ru]_φ_g^-1W","statement_latex":"Suppose that $f : X \\to Y$ and $g : Y \\to Z$ are continuous\nmaps of topological spaces. Suppose that $\\mathcal{F}$ is\na sheaf on $X$, $\\mathcal{G}$ is a sheaf on $Y$, and\n$\\mathcal{H}$ is a sheaf on $Z$.\nLet $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be an $f$-map.\nLet $\\psi : \\mathcal{H} \\to \\mathcal{G}$ be an $g$-map.\nThe {\\it composition of $\\varphi$ and $\\psi$} is the\n$(g \\circ f)$-map $\\varphi \\circ \\psi$ defined\nby the commutativity of the diagrams\n$$\n\\xymatrix{\n\\mathcal{H}(W) \\ar[rr]_{(\\varphi \\circ \\psi)_W}\n\\ar[rd]_{\\psi_W} & &\n\\mathcal{F}(f^{-1}g^{-1}W) \\\\\n&\n\\mathcal{G}(g^{-1}W)\n\\ar[ru]_{\\varphi_{g^{-1}W}}\n}\n$$","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008L","source_file":"sheaves.tex","source_line":2313,"source_end_line":2334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2313-L2334","statement_sha256":"5467b9ad723539427c61facd8682baaedad4dcfc1c33439e7dc5b5b0256d5308","origin":"The Stacks Project","memory_eligible":false,"source_rank":481,"rank":481,"depth":0,"x":1107.922,"y":251.645,"cluster":"sheaves-sites"},{"id":"stacks:008M","tag":"008M","title":"Continuous maps and sheaves · Lemma 008M","summary":"Suppose that f : X → Y and g : Y → Z are continuous maps of topological spaces. Suppose that F is a sheaf on X, G is a sheaf on Y, and H is a sheaf on Z. Let φ : G → F be an f-map. Let ψ : H → G be an g-map. Let x ∈ X be a point. The map on stalks (φ ∘ ψ)_x : H_g(f(x)) → F_x is the composition H_g(f(x)) xrightarrowψ_f(x) G_f(x) xrightarrowφ_x F_x","statement_latex":"Suppose that $f : X \\to Y$ and $g : Y \\to Z$ are continuous\nmaps of topological spaces. Suppose that $\\mathcal{F}$ is\na sheaf on $X$, $\\mathcal{G}$ is a sheaf on $Y$, and\n$\\mathcal{H}$ is a sheaf on $Z$.\nLet $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be an $f$-map.\nLet $\\psi : \\mathcal{H} \\to \\mathcal{G}$ be an $g$-map.\nLet $x \\in X$ be a point. The map on stalks\n$(\\varphi \\circ \\psi)_x : \\mathcal{H}_{g(f(x))}\n\\to \\mathcal{F}_x$ is the composition\n$$\n\\mathcal{H}_{g(f(x))}\n\\xrightarrow{\\psi_{f(x)}}\n\\mathcal{G}_{f(x)}\n\\xrightarrow{\\varphi_x}\n\\mathcal{F}_x\n$$","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008M","source_file":"sheaves.tex","source_line":2373,"source_end_line":2391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2373-L2391","statement_sha256":"6fe3c2d316aaa03e6335100a63fb3d2f58abc7cd7153a1a7dd0be2d934c391ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":482,"rank":482,"depth":1,"x":1120.738,"y":183.771,"cluster":"sheaves-sites"},{"id":"stacks:008O","tag":"008O","title":"Continuous maps and abelian sheaves · Lemma 008O","summary":"Let f : X → Y be a continuous map. • Let G be an abelian presheaf on Y. Let x ∈ X. The bijection G_f(x) → (f_pG)_x of Lemma [Tag 008G] is an isomorphism of abelian groups. • Let G be an abelian sheaf on Y. Let x ∈ X. The bijection G_f(x) → (f^-1G)_x of Lemma [Tag 008H] is an isomorphism of abelian groups.","statement_latex":"Let $f : X \\to Y$ be a continuous map.\n\\begin{enumerate}\n\\item Let $\\mathcal{G}$ be an abelian presheaf on $Y$.\nLet $x \\in X$. The bijection\n$\\mathcal{G}_{f(x)} \\to (f_p\\mathcal{G})_x$ of\nLemma \\ref{lemma-stalk-pullback-presheaf} is an isomorphism of abelian groups.\n\\item Let $\\mathcal{G}$ be an abelian sheaf on $Y$.\nLet $x \\in X$. The bijection\n$\\mathcal{G}_{f(x)} \\to (f^{-1}\\mathcal{G})_x$ of\nLemma \\ref{lemma-stalk-pullback} is an isomorphism of abelian groups.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008O","source_file":"sheaves.tex","source_line":2499,"source_end_line":2512,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2499-L2512","statement_sha256":"f47c4a2d29c087ad481e3fc81cfce2229748d3b7b2e99f2a5c704237e675136c","origin":"The Stacks Project","memory_eligible":false,"source_rank":483,"rank":483,"depth":3,"x":1166.325,"y":241.675,"cluster":"sheaves-sites"},{"id":"stacks:008Q","tag":"008Q","title":"Continuous maps and sheaves of algebraic structures · Lemma 008Q","summary":"Let f : X → Y be a continuous map of topological spaces. Suppose given sheaves of algebraic structures F on X, G on Y. Let φ : G → F be an f-map of underlying sheaves of sets. If for every V ⊂ Y open the map of sets φ_V : G(V) → F(f^-1V) is the effect of a morphism in C on underlying sets, then φ comes from a unique f-morphism between sheaves of algebraic structures.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nSuppose given sheaves of algebraic structures\n$\\mathcal{F}$ on $X$, $\\mathcal{G}$ on $Y$. Let\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be an $f$-map\nof underlying sheaves of sets. If for every $V \\subset Y$ open the\nmap of sets $\\varphi_V : \\mathcal{G}(V) \\to \\mathcal{F}(f^{-1}V)$\nis the effect of a morphism in $\\mathcal{C}$ on underlying sets,\nthen $\\varphi$ comes from a unique $f$-morphism between\nsheaves of algebraic structures.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves of algebraic structures","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008Q","source_file":"sheaves.tex","source_line":2662,"source_end_line":2673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2662-L2673","statement_sha256":"b93d0a61957b9d656d402b7d8df43f7ea3e9a4ef183e7837e5f01c0852e13b3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":484,"rank":484,"depth":0,"x":1085.35,"y":224.674,"cluster":"sheaves-sites"},{"id":"stacks:008S","tag":"008S","title":"Continuous maps and sheaves of modules · Lemma 008S","summary":"Let f : X → Y be a continuous map of topological spaces. Let O be a presheaf of rings on X. Let F be a presheaf of O-modules. There is a natural map of underlying presheaves of sets f_*O × f_*F → f_*F which turns f_*F into a presheaf of f_*O-modules. This construction is functorial in F.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $\\mathcal{O}$ be a presheaf of rings on $X$. Let\n$\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.\nThere is a natural map of underlying presheaves of sets\n$$\nf_*\\mathcal{O} \\times f_*\\mathcal{F}\n\\longrightarrow\nf_*\\mathcal{F}\n$$\nwhich turns $f_*\\mathcal{F}$ into a presheaf of\n$f_*\\mathcal{O}$-modules. This construction is\nfunctorial in $\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008S","source_file":"sheaves.tex","source_line":2692,"source_end_line":2706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2692-L2706","statement_sha256":"9a82115d7dace05c49ecfa4a5363daf609aa859fd14245bacf8f38dfba5a59be","origin":"The Stacks Project","memory_eligible":false,"source_rank":485,"rank":485,"depth":0,"x":1159.447,"y":190.94,"cluster":"sheaves-sites"},{"id":"stacks:008T","tag":"008T","title":"Continuous maps and sheaves of modules · Lemma 008T","summary":"Let f : X → Y be a continuous map of topological spaces. Let O be a presheaf of rings on Y. Let G be a presheaf of O-modules. There is a natural map of underlying presheaves of sets f_pO × f_pG → f_pG which turns f_pG into a presheaf of f_pO-modules. This construction is functorial in G.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $\\mathcal{O}$ be a presheaf of rings on $Y$. Let\n$\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules.\nThere is a natural map of underlying presheaves of sets\n$$\nf_p\\mathcal{O} \\times f_p\\mathcal{G}\n\\longrightarrow\nf_p\\mathcal{G}\n$$\nwhich turns $f_p\\mathcal{G}$ into a presheaf of $f_p\\mathcal{O}$-modules.\nThis construction is functorial in $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008T","source_file":"sheaves.tex","source_line":2726,"source_end_line":2739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2726-L2739","statement_sha256":"3c5a14f064ebd83daf8276105085e1e250f666168f80942bbf8d7c10ae5b96f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":486,"rank":486,"depth":1,"x":1131.671,"y":258.498,"cluster":"sheaves-sites"},{"id":"stacks:008U","tag":"008U","title":"Continuous maps and sheaves of modules · Lemma 008U","summary":"Let f : X → Y be a continuous map of topological spaces. Let O be a presheaf of rings on Y. Let G be a presheaf of O-modules. Let F be a presheaf of f_pO-modules. Then Mor_PMod(f_pO)(f_pG, F) = Mor_PMod(O)(G, f_*F). Here we use Lemmas [Tag 008T] and [Tag 008S], and we think of f_*F as an O-module via the map i_O : O → f_*f_pO (defined first in the proof of Lemma [Tag 008F]).","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $\\mathcal{O}$ be a presheaf of rings on $Y$.\nLet $\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules.\nLet $\\mathcal{F}$ be a presheaf of $f_p\\mathcal{O}$-modules.\nThen\n$$\n\\Mor_{\\textit{PMod}(f_p\\mathcal{O})}(f_p\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{PMod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).\n$$\nHere we use\nLemmas \\ref{lemma-pullback-presheaf-module}\nand \\ref{lemma-pushforward-presheaf-module}, and we think of\n$f_*\\mathcal{F}$ as an $\\mathcal{O}$-module via the map\n$i_\\mathcal{O} : \\mathcal{O} \\to f_*f_p\\mathcal{O}$\n(defined first in the proof of Lemma \\ref{lemma-pullback-presheaves}).","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008U","source_file":"sheaves.tex","source_line":2781,"source_end_line":2799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2781-L2799","statement_sha256":"ec13999057fc09d333d024291e3514c4496fd3a356d7f4c9eed30f0a51362c00","origin":"The Stacks Project","memory_eligible":false,"source_rank":487,"rank":487,"depth":2,"x":1097.51,"y":192.306,"cluster":"sheaves-sites"},{"id":"stacks:008V","tag":"008V","title":"Continuous maps and sheaves of modules · Lemma 008V","summary":"Let f : X → Y be a continuous map of topological spaces. Let O be a presheaf of rings on X. Let F be a presheaf of O-modules. Let G be a presheaf of f_*O-modules. Then Mor_PMod(O)( O ⊗_p, f_pf_*O f_pG, F) = Mor_PMod(f_*O)(G, f_*F). Here we use Lemmas [Tag 008T] and [Tag 008S], and we use the map c_O : f_pf_*O → O in the definition of the tensor product.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $\\mathcal{O}$ be a presheaf of rings on $X$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.\nLet $\\mathcal{G}$ be a presheaf of $f_*\\mathcal{O}$-modules.\nThen\n$$\n\\Mor_{\\textit{PMod}(\\mathcal{O})}(\n\\mathcal{O} \\otimes_{p, f_pf_*\\mathcal{O}} f_p\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{PMod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).\n$$\nHere we use\nLemmas \\ref{lemma-pullback-presheaf-module}\nand \\ref{lemma-pushforward-presheaf-module}, and we use\nthe map $c_\\mathcal{O} : f_pf_*\\mathcal{O} \\to \\mathcal{O}$\nin the definition of the tensor product.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008V","source_file":"sheaves.tex","source_line":2833,"source_end_line":2851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2833-L2851","statement_sha256":"61df196eeecb6f6881634086ed88e7f713c80e235f67f451b9f412b7f0fe057e","origin":"The Stacks Project","memory_eligible":false,"source_rank":488,"rank":488,"depth":3,"x":1176.65,"y":222.004,"cluster":"sheaves-sites"},{"id":"stacks:008W","tag":"008W","title":"Continuous maps and sheaves of modules · Lemma 008W","summary":"Let f : X → Y be a continuous map of topological spaces. Let O be a sheaf of rings on X. Let F be a sheaf of O-modules. The pushforward f_*F, as defined in Lemma [Tag 008S] is a sheaf of f_*O-modules.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $\\mathcal{O}$ be a sheaf of rings on $X$. Let\n$\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nThe pushforward $f_*\\mathcal{F}$, as defined in\nLemma \\ref{lemma-pushforward-presheaf-module}\nis a sheaf of $f_*\\mathcal{O}$-modules.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008W","source_file":"sheaves.tex","source_line":2881,"source_end_line":2889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2881-L2889","statement_sha256":"d5bf6431ca84877e79c5f7fbe0d3b44998b81f0d23256426e1c6cf6a50115e20","origin":"The Stacks Project","memory_eligible":false,"source_rank":489,"rank":489,"depth":1,"x":1093.668,"y":245.216,"cluster":"sheaves-sites"},{"id":"stacks:008X","tag":"008X","title":"Continuous maps and sheaves of modules · Lemma 008X","summary":"Let f : X → Y be a continuous map of topological spaces. Let O be a sheaf of rings on Y. Let G be a sheaf of O-modules. There is a natural map of underlying presheaves of sets f^-1O × f^-1G → f^-1G which turns f^-1G into a sheaf of f^-1O-modules.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $\\mathcal{O}$ be a sheaf of rings on $Y$. Let\n$\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules.\nThere is a natural map of underlying presheaves of sets\n$$\nf^{-1}\\mathcal{O} \\times f^{-1}\\mathcal{G}\n\\longrightarrow\nf^{-1}\\mathcal{G}\n$$\nwhich turns $f^{-1}\\mathcal{G}$ into a\nsheaf of $f^{-1}\\mathcal{O}$-modules.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008X","source_file":"sheaves.tex","source_line":2895,"source_end_line":2908,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2895-L2908","statement_sha256":"907c1ad2a8b449ee56b086bf68415af17ba168b4ad328375b7173196970c8c48","origin":"The Stacks Project","memory_eligible":false,"source_rank":490,"rank":490,"depth":2,"x":1136.57,"y":180.445,"cluster":"sheaves-sites"},{"id":"stacks:008Y","tag":"008Y","title":"Continuous maps and sheaves of modules · Lemma 008Y","summary":"Let f : X → Y be a continuous map of topological spaces. Let O be a sheaf of rings on Y. Let G be a sheaf of O-modules. Let F be a sheaf of f^-1O-modules. Then Mor_Mod(f^-1O)(f^-1G, F) = Mor_Mod(O)(G, f_*F). Here we use Lemmas [Tag 008X] and [Tag 008W], and we think of f_*F as an O-module by restriction via O → f_*f^-1O.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $\\mathcal{O}$ be a sheaf of rings on $Y$.\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules.\nLet $\\mathcal{F}$ be a sheaf of $f^{-1}\\mathcal{O}$-modules.\nThen\n$$\n\\Mor_{\\textit{Mod}(f^{-1}\\mathcal{O})}(f^{-1}\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).\n$$\nHere we use\nLemmas \\ref{lemma-pullback-module}\nand \\ref{lemma-pushforward-module}, and we think of\n$f_*\\mathcal{F}$ as an $\\mathcal{O}$-module by restriction via\n$\\mathcal{O} \\to f_*f^{-1}\\mathcal{O}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008Y","source_file":"sheaves.tex","source_line":2932,"source_end_line":2949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2932-L2949","statement_sha256":"d8b748a0140152145f2e789171c3355edbd176c38d6a47fca3fcd2246e65e46d","origin":"The Stacks Project","memory_eligible":false,"source_rank":491,"rank":491,"depth":3,"x":1157.197,"y":253.179,"cluster":"sheaves-sites"},{"id":"stacks:008Z","tag":"008Z","title":"Continuous maps and sheaves of modules · Lemma 008Z","summary":"Let f : X → Y be a continuous map of topological spaces. Let O be a sheaf of rings on X. Let F be a sheaf of O-modules. Let G be a sheaf of f_*O-modules. Then Mor_Mod(O)( O ⊗_f^-1f_*O f^-1G, F) = Mor_Mod(f_*O)(G, f_*F). Here we use Lemmas [Tag 008X] and [Tag 008W], and we use the canonical map f^-1f_*O → O in the definition of the tensor product.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $\\mathcal{O}$ be a sheaf of rings on $X$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nLet $\\mathcal{G}$ be a sheaf of $f_*\\mathcal{O}$-modules.\nThen\n$$\n\\Mor_{\\textit{Mod}(\\mathcal{O})}(\n\\mathcal{O} \\otimes_{f^{-1}f_*\\mathcal{O}} f^{-1}\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{Mod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).\n$$\nHere we use\nLemmas \\ref{lemma-pullback-module}\nand \\ref{lemma-pushforward-module}, and we use\nthe canonical map $f^{-1}f_*\\mathcal{O} \\to \\mathcal{O}$\nin the definition of the tensor product.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Continuous maps and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/008Z","source_file":"sheaves.tex","source_line":2965,"source_end_line":2983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L2965-L2983","statement_sha256":"a0a29e6f7c903d72c90d7df29f7b5660bd3b8827c64914b374da58fdb535f5b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":492,"rank":492,"depth":4,"x":1082.867,"y":210.89,"cluster":"sheaves-sites"},{"id":"stacks:0091","tag":"0091","title":"Ringed spaces · Definition 0091","summary":"A ringed space is a pair (X, O_X) consisting of a topological space X and a sheaf of rings O_X on X. A morphism of ringed spaces (X, O_X) → (Y, O_Y) is a pair consisting of a continuous map f : X → Y and an f-map of sheaves of rings f^sharp : O_Y → O_X.","statement_latex":"A {\\it ringed space} is a pair $(X, \\mathcal{O}_X)$ consisting\nof a topological space $X$ and a sheaf of rings $\\mathcal{O}_X$\non $X$. A {\\it morphism of ringed spaces}\n$(X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ is a pair\nconsisting of a continuous map $f : X \\to Y$ and an\n$f$-map of sheaves of rings\n$f^\\sharp : \\mathcal{O}_Y \\to \\mathcal{O}_X$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0091","source_file":"sheaves.tex","source_line":3047,"source_end_line":3056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3047-L3056","statement_sha256":"d17ad0dbb04d7428f4370732fc93749d4834027c7aee3c20284d697a78e08196","origin":"The Stacks Project","memory_eligible":false,"source_rank":493,"rank":493,"depth":0,"x":1172.432,"y":199.806,"cluster":"sheaves-sites"},{"id":"stacks:0093","tag":"0093","title":"Ringed spaces · Definition 0093","summary":"Let (f, f^sharp) : (X, O_X) → (Y, O_Y) and (g, g^sharp) : (Y, O_Y) → (Z, O_Z) be morphisms of ringed spaces. Then we define the composition of morphisms of ringed spaces by the rule (g, g^sharp) ∘ (f, f^sharp) = (g ∘ f, f^sharp ∘ g^sharp). Here we use composition of f-maps defined in Definition [Tag 008L].","statement_latex":"Let\n$(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ and\n$(g, g^\\sharp) : (Y, \\mathcal{O}_Y) \\to (Z, \\mathcal{O}_Z)$\nbe morphisms of ringed spaces. Then we define\nthe {\\it composition of morphisms of ringed spaces}\nby the rule\n$$\n(g, g^\\sharp) \\circ (f, f^\\sharp) = (g \\circ f, f^\\sharp \\circ g^\\sharp).\n$$\nHere we use composition of $f$-maps defined in\nDefinition \\ref{definition-composition-f-maps}.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0093","source_file":"sheaves.tex","source_line":3092,"source_end_line":3105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3092-L3105","statement_sha256":"ba448ae775d79c3db0849d2967d08e5d6435248242b1c9300f44a814e07d9cee","origin":"The Stacks Project","memory_eligible":false,"source_rank":494,"rank":494,"depth":1,"x":1114.827,"y":259.288,"cluster":"sheaves-sites"},{"id":"stacks:0095","tag":"0095","title":"Morphisms of ringed spaces and modules · Definition 0095","summary":"Let (f, f^sharp) : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. • Let F be a sheaf of O_X-modules. We define the pushforward of F as the sheaf of O_Y-modules which as a sheaf of abelian groups equals f_*F and with module structure given by the restriction via f^sharp : O_Y → f_*O_X of the module structure given in Lemma [Tag 008W]. • Let G be a sheaf of O_Y-modules. We define the pullback f^*G to be the sheaf of O_X-modules defined by the formula f^*G = O_X…","statement_latex":"Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nWe define the {\\it pushforward} of $\\mathcal{F}$ as the\nsheaf of $\\mathcal{O}_Y$-modules which as a sheaf\nof abelian groups equals $f_*\\mathcal{F}$ and with\nmodule structure given by the restriction\nvia $f^\\sharp : \\mathcal{O}_Y \\to f_*\\mathcal{O}_X$\nof the module structure given\nin Lemma \\ref{lemma-pushforward-module}.\n\\item Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.\nWe define the {\\it pullback} $f^*\\mathcal{G}$ to be the\nsheaf of $\\mathcal{O}_X$-modules defined by the formula\n$$\nf^*\\mathcal{G}\n=\n\\mathcal{O}_X \\otimes_{f^{-1}\\mathcal{O}_Y} f^{-1}\\mathcal{G}\n$$\nwhere the ring map $f^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X$\nis the map corresponding to $f^\\sharp$, and where the  module\nstructure is given by Lemma \\ref{lemma-pullback-module}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Morphisms of ringed spaces and modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0095","source_file":"sheaves.tex","source_line":3116,"source_end_line":3141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3116-L3141","statement_sha256":"9dc26b4d91a1d339e90c634c77ff93935547bcff6c2d9870984f5efc8e1208eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":495,"rank":495,"depth":3,"x":1109.429,"y":182.117,"cluster":"sheaves-sites"},{"id":"stacks:0096","tag":"0096","title":"Morphisms of ringed spaces and modules · Lemma 0096","summary":"Let (f, f^sharp) : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let F be a sheaf of O_X-modules. Let G be a sheaf of O_Y-modules. There is a canonical bijection Hom_O_X(f^*G, F) = Hom_O_Y(G, f_*F). In other words: the functor f^* is the left adjoint to f_*.","statement_latex":"Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.\nThere is a canonical bijection\n$$\n\\Hom_{\\mathcal{O}_X}(f^*\\mathcal{G}, \\mathcal{F})\n=\n\\Hom_{\\mathcal{O}_Y}(\\mathcal{G}, f_*\\mathcal{F}).\n$$\nIn other words: the functor $f^*$ is the left adjoint to\n$f_*$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Morphisms of ringed spaces and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0096","source_file":"sheaves.tex","source_line":3156,"source_end_line":3170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3156-L3170","statement_sha256":"bf0319fa5949cc1742a20b804724c4135573c669d2f2532e832524f64071fb5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":496,"rank":496,"depth":4,"x":1175.996,"y":236.392,"cluster":"sheaves-sites"},{"id":"stacks:0097","tag":"0097","title":"Morphisms of ringed spaces and modules · Lemma 0097","summary":"Let f : X → Y and g : Y → Z be morphisms of ringed spaces. The functors (g ∘ f)_* and g_* ∘ f_* are equal. There is a canonical isomorphism of functors (g ∘ f)^* ≅ f^* ∘ g^*.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.\nThe functors $(g \\circ f)_*$ and $g_* \\circ f_*$ are equal.\nThere is a canonical isomorphism of functors\n$(g \\circ f)^* \\cong f^* \\circ g^*$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Morphisms of ringed spaces and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0097","source_file":"sheaves.tex","source_line":3189,"source_end_line":3195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3189-L3195","statement_sha256":"a815f2379f0d47458428777d8f41d45741f40101ebfad8a2dd063a8d3396391a","origin":"The Stacks Project","memory_eligible":false,"source_rank":497,"rank":497,"depth":2,"x":1082.533,"y":234.122,"cluster":"sheaves-sites"},{"id":"stacks:0098","tag":"0098","title":"Morphisms of ringed spaces and modules · Lemma 0098","summary":"Let (f, f^sharp) : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let G be a sheaf of O_Y-modules. Let x ∈ X. Then (f^*G)_x = G_f(x) ⊗_O_Y, f(x) O_X, x as O_X, x-modules where the tensor product on the right uses f^sharp_x : O_Y, f(x) → O_X, x.","statement_latex":"Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.\nLet $x \\in X$. Then\n$$\n(f^*\\mathcal{G})_x =\n\\mathcal{G}_{f(x)}\n\\otimes_{\\mathcal{O}_{Y, f(x)}}\n\\mathcal{O}_{X, x}\n$$\nas $\\mathcal{O}_{X, x}$-modules where the tensor product on the right\nuses $f^\\sharp_x : \\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Morphisms of ringed spaces and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0098","source_file":"sheaves.tex","source_line":3243,"source_end_line":3257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3243-L3257","statement_sha256":"a689212f6d5e3df6f6d433395f6ce9bf9244b193c59d889d7d602cf5d81dffcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":498,"rank":498,"depth":2,"x":1153.827,"y":182.366,"cluster":"sheaves-sites"},{"id":"stacks:009A","tag":"009A","title":"Skyscraper sheaves and stalks · Definition 009A","summary":"Let X be a topological space. • Let x ∈ X be a point. Denote i_x : (x) → X the inclusion map. Let A be a set and think of A as a sheaf on the one point space (x). We call i_x, *A the skyscraper sheaf at x with value A. • If in (1) above A is an abelian group then we think of i_x, *A as a sheaf of abelian groups on X. • If in (1) above A is an algebraic structure then we think of i_x, *A as a sheaf of algebraic structures. • If (X, O_X) is a ringed space, then we think of…","statement_latex":"Let $X$ be a topological space.\n\\begin{enumerate}\n\\item Let $x \\in X$ be a point. Denote $i_x : \\{x\\} \\to X$ the inclusion map.\nLet $A$ be a set and think of $A$ as a sheaf on the one point space $\\{x\\}$.\nWe call $i_{x, *}A$ the {\\it skyscraper sheaf at $x$ with value $A$}.\n\\item If in (1) above $A$ is an abelian group then we think of\n$i_{x, *}A$ as a sheaf of abelian groups on $X$.\n\\item If in (1) above $A$ is an algebraic structure then we think\nof $i_{x, *}A$ as a sheaf of algebraic structures.\n\\item If $(X, \\mathcal{O}_X)$ is a ringed space, then we think\nof $i_x : \\{x\\} \\to X$ as a morphism of ringed spaces\n$(\\{x\\}, \\mathcal{O}_{X, x}) \\to (X, \\mathcal{O}_X)$\nand if $A$ is a $\\mathcal{O}_{X, x}$-module, then we think\nof $i_{x, *}A$ as a sheaf of $\\mathcal{O}_X$-modules.\n\\item We say a sheaf of sets $\\mathcal{F}$ is a {\\it skyscraper sheaf}\nif there exists a point $x$ of $X$ and a set $A$ such\nthat $\\mathcal{F} \\cong i_{x, *}A$.\n\\item We say a sheaf of abelian groups $\\mathcal{F}$ is a\n{\\it skyscraper sheaf} if there exists a point $x$ of $X$\nand an abelian group $A$ such that $\\mathcal{F} \\cong i_{x, *}A$\nas sheaves of abelian groups.\n\\item We say a sheaf of algebraic structures $\\mathcal{F}$ is a\n{\\it skyscraper sheaf} if there exists a point $x$ of $X$\nand an algebraic structure $A$ such that $\\mathcal{F} \\cong i_{x, *}A$\nas sheaves of algebraic structures.\n\\item If $(X, \\mathcal{O}_X)$ is a ringed space and\n$\\mathcal{F}$ is a sheaf of $\\mathcal{O}_X$-modules, then\nwe say $\\mathcal{F}$ is a {\\it skyscraper sheaf} if there\nexists a point $x \\in X$ and a $\\mathcal{O}_{X, x}$-module\n$A$ such that $\\mathcal{F} \\cong i_{x, *}A$\nas sheaves of $\\mathcal{O}_X$-modules.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Skyscraper sheaves and stalks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009A","source_file":"sheaves.tex","source_line":3272,"source_end_line":3306,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3272-L3306","statement_sha256":"225d1ee53ff8475840913da05bf350dcf49826f9d40e3e1729a9d112545adf44","origin":"The Stacks Project","memory_eligible":false,"source_rank":499,"rank":499,"depth":0,"x":1142.791,"y":261.584,"cluster":"sheaves-sites"},{"id":"stacks:009B","tag":"009B","title":"Skyscraper sheaves and stalks · Lemma 009B","summary":"Let X be a topological space, x ∈ X a point, and A a set. For any point x' ∈ X the stalk of the skyscraper sheaf at x with value A at x' is (i_x, *A)_x' = ( A & if & x' ∈ overline(x) (*) & if & x' not∈ overline(x) . A similar description holds for the case of abelian groups, algebraic structures and sheaves of modules.","statement_latex":"Let $X$ be a topological space, $x \\in X$ a point, and\n$A$ a set. For any point $x' \\in X$ the stalk of the\nskyscraper sheaf at $x$ with value $A$ at $x'$ is\n$$\n(i_{x, *}A)_{x'} =\n\\left\\{\n\\begin{matrix}\nA & \\text{if} & x' \\in \\overline{\\{x\\}} \\\\\n\\{*\\} & \\text{if} & x' \\not\\in \\overline{\\{x\\}}\n\\end{matrix}\n\\right.\n$$\nA similar description holds for the case of\nabelian groups, algebraic structures and\nsheaves of modules.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Skyscraper sheaves and stalks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009B","source_file":"sheaves.tex","source_line":3308,"source_end_line":3325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3308-L3325","statement_sha256":"b9124d52b2fabf53cc279d7d83f30111430c44dd35f8bb9c346c9568d3f1f462","origin":"The Stacks Project","memory_eligible":false,"source_rank":500,"rank":500,"depth":0,"x":1086.807,"y":196.42,"cluster":"sheaves-sites"},{"id":"stacks:009C","tag":"009C","title":"Skyscraper sheaves and stalks · Lemma 009C","summary":"Let X be a topological space, and let x ∈ X a point. The functors F ↦ F_x and A ↦ i_x, *A are adjoint. In a formula Mor_Sets(F_x, A) = Mor_Sh(X)(F, i_x, *A). A similar statement holds for the case of abelian groups, algebraic structures. In the case of sheaves of modules we have Hom_O_X, x(F_x, A) = Hom_O_X(F, i_x, *A).","statement_latex":"Let $X$ be a topological space, and let $x \\in X$ a point.\nThe functors $\\mathcal{F} \\mapsto \\mathcal{F}_x$ and\n$A \\mapsto i_{x, *}A$ are adjoint. In a formula\n$$\n\\Mor_{\\textit{Sets}}(\\mathcal{F}_x, A)\n=\n\\Mor_{\\Sh(X)}(\\mathcal{F}, i_{x, *}A).\n$$\nA similar statement holds for the case of\nabelian groups, algebraic structures. In the case of\nsheaves of modules we have\n$$\n\\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, A)\n=\n\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, i_{x, *}A).\n$$","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Skyscraper sheaves and stalks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009C","source_file":"sheaves.tex","source_line":3331,"source_end_line":3349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3331-L3349","statement_sha256":"8bd02dfb09b0d3fcd26d9f18bd29b0623cbf5597007bc6c07d90d14cf5a96282","origin":"The Stacks Project","memory_eligible":false,"source_rank":501,"rank":501,"depth":0,"x":1181.186,"y":212.826,"cluster":"sheaves-sites"},{"id":"stacks:009F","tag":"009F","title":"Limits and colimits of sheaves · Lemma 009F","summary":"Let X be a topological space. Let I be a directed set. Let (F_i, φ_ii') be a system of sheaves of sets over I, see Categories, Section [Tag 002Z]. Let U ⊂ X be an open subset. Consider the canonical map Ψ : colim_i F_i(U) → (colim_i F_i)(U) • If all the transition maps are injective then Ψ is injective for any open U. • If U is quasi-compact, then Ψ is injective. • If U is quasi-compact and all the transition maps are injective then Ψ is an isomorphism. • If U has a…","statement_latex":"Let $X$ be a topological space. Let $I$ be a directed set.\nLet $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system of sheaves of sets\nover $I$, see\nCategories, Section \\ref{categories-section-posets-limits}.\nLet $U \\subset X$ be an open subset.\nConsider the canonical map\n$$\n\\Psi :\n\\colim_i \\mathcal{F}_i(U)\n\\longrightarrow\n\\left(\\colim_i \\mathcal{F}_i\\right)(U)\n$$\n\\begin{enumerate}\n\\item If all the transition maps are injective then\n$\\Psi$ is injective for any open $U$.\n\\item If $U$ is quasi-compact, then $\\Psi$ is injective.\n\\item If $U$ is quasi-compact and all the transition maps are injective\nthen $\\Psi$ is an isomorphism.\n\\item If $U$ has a cofinal system of open coverings\n$\\mathcal{U} : U = \\bigcup_{j\\in J} U_j$ with\n$J$ finite and $U_j \\cap U_{j'}$ quasi-compact\nfor all $j, j' \\in J$, then $\\Psi$ is bijective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Limits and colimits of sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009F","source_file":"sheaves.tex","source_line":3443,"source_end_line":3468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3443-L3468","statement_sha256":"c962e1ef19539d82bbf608897581ed71a2664b868977cfb2fabfe72bc577bac6","origin":"The Stacks Project","memory_eligible":false,"source_rank":502,"rank":502,"depth":2,"x":1097.793,"y":254.583,"cluster":"sheaves-sites"},{"id":"stacks:0A32","tag":"0A32","title":"Limits and colimits of sheaves · Lemma 0A32","summary":"In the situation described above, let i ∈ Ob(I) and let G be a sheaf on X_i. For U_i ⊂ X_i quasi-compact open we have p_i^-1G(p_i^-1(U_i)) = colim_a : j → i f_a^-1G(f_a^-1(U_i))","statement_latex":"In the situation described above, let $i \\in \\Ob(\\mathcal{I})$ and let\n$\\mathcal{G}$ be a sheaf on $X_i$. For $U_i \\subset X_i$\nquasi-compact open we have\n$$\np_i^{-1}\\mathcal{G}(p_i^{-1}(U_i)) =\n\\colim_{a : j \\to i} f_a^{-1}\\mathcal{G}(f_a^{-1}(U_i))\n$$","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Limits and colimits of sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A32","source_file":"sheaves.tex","source_line":3564,"source_end_line":3573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3564-L3573","statement_sha256":"4ce1ecbf13ad0c707d569f7c1a07c381a15c402da744e2dcdbe5383958eb6045","origin":"The Stacks Project","memory_eligible":false,"source_rank":503,"rank":503,"depth":7,"x":1125.909,"y":175.911,"cluster":"sheaves-sites"},{"id":"stacks:0A33","tag":"0A33","title":"Limits and colimits of sheaves · Lemma 0A33","summary":"In the situation described above, let i ∈ Ob(I) and let U_i ⊂ X_i be a quasi-compact open. Then colim_a : j → i F_j(f_a^-1(U_i)) = F(p_i^-1(U_i))","statement_latex":"In the situation described above, let $i \\in \\Ob(\\mathcal{I})$ and let\n$U_i \\subset X_i$ be a quasi-compact open. Then\n$$\n\\colim_{a : j \\to i} \\mathcal{F}_j(f_a^{-1}(U_i)) = \\mathcal{F}(p_i^{-1}(U_i))\n$$","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Limits and colimits of sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A33","source_file":"sheaves.tex","source_line":3636,"source_end_line":3643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3636-L3643","statement_sha256":"37e3eae47a1d6e46397152b863c702411b568b734dafd41ae70cb51ae28bd038","origin":"The Stacks Project","memory_eligible":false,"source_rank":504,"rank":504,"depth":8,"x":1168.742,"y":250.402,"cluster":"sheaves-sites"},{"id":"stacks:009I","tag":"009I","title":"Bases and sheaves · Definition 009I","summary":"Let X be a topological space. Let B be a basis for the topology on X. • A presheaf F of sets on B is a rule which assigns to each U ∈ B a set F(U) and to each inclusion V ⊂ U of elements of B a map ρ^U_V : F(U) → F(V) such that ρ^U_U = id_F(U) for all U ∈ B whenever W ⊂ V ⊂ U in B we have ρ^U_W = ρ^V_W ∘ ρ ^U_V. • A morphism φ : F → G of presheaves of sets on B is a rule which assigns to each element U ∈ B a map of sets φ : F(U) → G(U) compatible with restriction maps.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{B}$ be a\nbasis for the topology on $X$.\n\\begin{enumerate}\n\\item A {\\it presheaf $\\mathcal{F}$ of sets on $\\mathcal{B}$}\nis a rule which assigns to each $U \\in \\mathcal{B}$ a set\n$\\mathcal{F}(U)$ and to each inclusion $V \\subset U$\nof elements of $\\mathcal{B}$ a map\n$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that\n$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ for all $U \\in \\mathcal{B}$\nwhenever $W \\subset V \\subset U$ in $\\mathcal{B}$ we have\n$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves of sets on $\\mathcal{B}$} is a rule which assigns to each\nelement $U \\in \\mathcal{B}$ a map of sets $\\varphi : \\mathcal{F}(U)\n\\to \\mathcal{G}(U)$ compatible with restriction maps.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009I","source_file":"sheaves.tex","source_line":3695,"source_end_line":3713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3695-L3713","statement_sha256":"b891763d8e27db9ccc3b4d8f0a5d85fbfbace23f153e6a30f8ac16e7646394cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":505,"rank":505,"depth":0,"x":1076.615,"y":219.562,"cluster":"sheaves-sites"},{"id":"stacks:009J","tag":"009J","title":"Bases and sheaves · Definition 009J","summary":"Let X be a topological space. Let B be a basis for the topology on X. • A sheaf F of sets on B is a presheaf of sets on B which satisfies the following additional property: Given any U ∈ B, and any covering U = ⋃_i ∈ I U_i with U_i ∈ B, and any coverings U_i ∩ U_j = ⋃_k ∈ I_ij U_ijk with U_ijk ∈ B the sheaf condition holds: • [(**)] For any collection of sections s_i ∈ F(U_i), i ∈ I such that ∀ i, j∈ I, ∀ k∈ I_ij s_i|_U_ijk = s_j|_U_ijk there exists a unique section s ∈…","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{B}$ be a\nbasis for the topology on $X$.\n\\begin{enumerate}\n\\item A {\\it sheaf $\\mathcal{F}$ of sets on $\\mathcal{B}$} is a presheaf\nof sets on $\\mathcal{B}$ which satisfies the following additional\nproperty: Given any $U \\in \\mathcal{B}$, and any covering\n$U = \\bigcup_{i \\in I} U_i$ with $U_i \\in \\mathcal{B}$, and\nany coverings $U_i \\cap U_j = \\bigcup_{k \\in I_{ij}} U_{ijk}$ with\n$U_{ijk} \\in \\mathcal{B}$ the sheaf condition holds:\n\\begin{itemize}\n\\item[(**)] For any collection\nof sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that\n$\\forall i, j\\in I$, $\\forall k\\in I_{ij}$\n$$\ns_i|_{U_{ijk}} = s_j|_{U_{ijk}}\n$$\nthere exists a unique section $s \\in \\mathcal{F}(U)$ such that\n$s_i = s|_{U_i}$ for all $i \\in I$.\n\\end{itemize}\n\\item A {\\it morphism of sheaves of sets on $\\mathcal{B}$} is simply a\nmorphism of presheaves of sets.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009J","source_file":"sheaves.tex","source_line":3733,"source_end_line":3757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3733-L3757","statement_sha256":"203ab04786dedfc7d86aff32f8bc588372364d0aa5a20ed0c998c08a20c4626e","origin":"The Stacks Project","memory_eligible":false,"source_rank":506,"rank":506,"depth":0,"x":1169.989,"y":189.825,"cluster":"sheaves-sites"},{"id":"stacks:009K","tag":"009K","title":"Bases and sheaves · Lemma 009K","summary":"With notation as above. For each U ∈ B, let C(U) ⊂ Cov_B(U) be a cofinal system. For each U ∈ B, and each U : U = ⋃ U_i in C(U), let coverings U_ij : U_i ∩ U_j = ⋃ U_ijk, U_ijk ∈ B be given. Let F be a presheaf of sets on B. The following are equivalent • The presheaf F is a sheaf on B. • For every U ∈ B and every covering U : U = ⋃ U_i in C(U) the sheaf condition (**) holds (for the given coverings U_ij).","statement_latex":"With notation as above.\nFor each $U \\in \\mathcal{B}$, let $C(U) \\subset \\text{Cov}_\\mathcal{B}(U)$\nbe a cofinal system. For each $U \\in \\mathcal{B}$, and each\n$\\mathcal{U} : U = \\bigcup U_i$ in $C(U)$, let coverings\n$\\mathcal{U}_{ij} : U_i \\cap U_j = \\bigcup U_{ijk}$,\n$U_{ijk} \\in \\mathcal{B}$ be given.\nLet $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{B}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item The presheaf $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$.\n\\item For every $U \\in \\mathcal{B}$ and every covering\n$\\mathcal{U} : U = \\bigcup U_i$ in $C(U)$ the sheaf condition\n$(**)$ holds (for the given coverings $\\mathcal{U}_{ij}$).\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009K","source_file":"sheaves.tex","source_line":3772,"source_end_line":3788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3772-L3788","statement_sha256":"f952df8e809f030d82ee71d22b87b5866d705ac1c5dab35212457f460d508eb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":507,"rank":507,"depth":0,"x":1124.743,"y":265.245,"cluster":"sheaves-sites"},{"id":"stacks:009L","tag":"009L","title":"Bases and sheaves · Lemma 009L","summary":"Let X be a topological space. Let B be a basis for the topology on X. Assume that for every triple U, U', U\" ∈ B with U' ⊂ U and U\" ⊂ U we have U' ∩ U\" ∈ B. For each U ∈ B, let C(U) ⊂ Cov_B(U) be a cofinal system. Let F be a presheaf of sets on B. The following are equivalent • The presheaf F is a sheaf on B. • For every U ∈ B and every covering U : U = ⋃ U_i in C(U) and for every family of sections s_i ∈ F(U_i) such that s_i|_U_i ∩ U_j = s_j|_U_i ∩ U_j there exists a…","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the\ntopology on $X$. Assume that for every triple $U, U', U'' \\in \\mathcal{B}$\nwith $U' \\subset U$ and $U'' \\subset U$ we have $U' \\cap U'' \\in \\mathcal{B}$.\nFor each $U \\in \\mathcal{B}$, let $C(U) \\subset \\text{Cov}_\\mathcal{B}(U)$\nbe a cofinal system.\nLet $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{B}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item The presheaf $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$.\n\\item For every $U \\in \\mathcal{B}$ and every covering\n$\\mathcal{U} : U = \\bigcup U_i$ in $C(U)$ and for every\nfamily of sections $s_i \\in \\mathcal{F}(U_i)$ such\nthat $s_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}$ there\nexists a unique section $s \\in \\mathcal{F}(U)$ which\nrestricts to $s_i$ on $U_i$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009L","source_file":"sheaves.tex","source_line":3861,"source_end_line":3879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3861-L3879","statement_sha256":"c6d21d459fb922b87a12d1d51fa861aceef40d85ec69a149cf32768aa1a60789","origin":"The Stacks Project","memory_eligible":false,"source_rank":508,"rank":508,"depth":1,"x":1097.275,"y":183.412,"cluster":"sheaves-sites"},{"id":"stacks:009M","tag":"009M","title":"Bases and sheaves · Lemma 009M","summary":"Let X be a topological space. Let B be a basis for the topology on X. Let U ∈ B. Let F be a sheaf of sets on B. The map F(U) → ∏_x ∈ U F_x identifies F(U) with the elements (s_x)_x∈ U with the property • [(*)] For any x ∈ U there exists a V ∈ B, with x ∈ V ⊂ U and a section σ ∈ F(V) such that for all y ∈ V we have s_y = (V, σ) in F_y.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{B}$ be a basis for the topology on $X$.\nLet $U \\in \\mathcal{B}$.\nLet $\\mathcal{F}$ be a sheaf of sets on $\\mathcal{B}$.\nThe map\n$$\n\\mathcal{F}(U) \\to \\prod\\nolimits_{x \\in U} \\mathcal{F}_x\n$$\nidentifies $\\mathcal{F}(U)$ with the elements $(s_x)_{x\\in U}$\nwith the property\n\\begin{itemize}\n\\item[(*)] For any $x \\in U$ there exists a $V \\in \\mathcal{B}$,\nwith $x \\in V \\subset U$ and a section $\\sigma \\in \\mathcal{F}(V)$\nsuch that for all $y \\in V$ we have $s_y = (V, \\sigma)$ in $\\mathcal{F}_y$.\n\\end{itemize}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009M","source_file":"sheaves.tex","source_line":3889,"source_end_line":3906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3889-L3906","statement_sha256":"cb09b63b6c0251d3155a0b8cb63d76d15f03b3008d2b1df70d66a96a940c82df","origin":"The Stacks Project","memory_eligible":false,"source_rank":509,"rank":509,"depth":1,"x":1183.906,"y":228.466,"cluster":"sheaves-sites"},{"id":"stacks:009N","tag":"009N","title":"Bases and sheaves · Lemma 009N","summary":"Let X be a topological space. Let B be a basis for the topology on X. Let F be a sheaf of sets on B. There exists a unique sheaf of sets F^ext on X such that F^ext(U) = F(U) for all U ∈ B compatibly with the restriction mappings.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{B}$ be a basis for the topology on $X$.\nLet $\\mathcal{F}$ be a sheaf of sets on $\\mathcal{B}$.\nThere exists a unique sheaf of sets $\\mathcal{F}^{ext}$\non $X$ such that $\\mathcal{F}^{ext}(U) = \\mathcal{F}(U)$\nfor all $U \\in \\mathcal{B}$ compatibly with the restriction\nmappings.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009N","source_file":"sheaves.tex","source_line":3933,"source_end_line":3942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3933-L3942","statement_sha256":"67a5546914f08f75bc2dc21405f668c86b05bf1d857d6e589b06b6f9cb64f99e","origin":"The Stacks Project","memory_eligible":false,"source_rank":510,"rank":510,"depth":2,"x":1083.14,"y":244.503,"cluster":"sheaves-sites"},{"id":"stacks:009O","tag":"009O","title":"Bases and sheaves · Lemma 009O","summary":"Let X be a topological space. Let B be a basis for the topology on X. Denote Sh(B) the category of sheaves on B. There is an equivalence of categories Sh(X) → Sh(B) which assigns to a sheaf on X its restriction to the members of B.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{B}$ be a basis for the topology on $X$.\nDenote $\\Sh(\\mathcal{B})$ the category of\nsheaves on $\\mathcal{B}$.\nThere is an equivalence of categories\n$$\n\\Sh(X) \\longrightarrow \\Sh(\\mathcal{B})\n$$\nwhich assigns to a sheaf on $X$ its restriction to\nthe members of $\\mathcal{B}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009O","source_file":"sheaves.tex","source_line":3968,"source_end_line":3980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3968-L3980","statement_sha256":"04435efc34dc77521d4b6b3b8e18b33fa5fc97a1aa855a5877b99fd6dc1dc509","origin":"The Stacks Project","memory_eligible":false,"source_rank":511,"rank":511,"depth":3,"x":1144.944,"y":175.056,"cluster":"sheaves-sites"},{"id":"stacks:009P","tag":"009P","title":"Bases and sheaves · Definition 009P","summary":"Let X be a topological space. Let B be a basis for the topology on X. Let (C, F) be a type of algebraic structure. • A presheaf F with values in C on B is a rule which assigns to each U ∈ B an object F(U) of C and to each inclusion V ⊂ U of elements of B a morphism ρ^U_V : F(U) → F(V) in C such that ρ^U_U = id_F(U) for all U ∈ B and whenever W ⊂ V ⊂ U in B we have ρ^U_W = ρ^V_W ∘ ρ ^U_V. • A morphism φ : F → G of presheaves with values in C on B is a rule which assigns to…","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{B}$ be a\nbasis for the topology on $X$. Let $(\\mathcal{C}, F)$ be\na type of algebraic structure.\n\\begin{enumerate}\n\\item A {\\it presheaf $\\mathcal{F}$ with values in $\\mathcal{C}$\non $\\mathcal{B}$} is a rule which assigns to each\n$U \\in \\mathcal{B}$ an object\n$\\mathcal{F}(U)$ of $\\mathcal{C}$ and to each inclusion $V \\subset U$\nof elements of $\\mathcal{B}$ a morphism\n$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ in $\\mathcal{C}$ such that\n$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ for all $U \\in \\mathcal{B}$ and\nwhenever $W \\subset V \\subset U$ in $\\mathcal{B}$ we have\n$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves with values in $\\mathcal{C}$\non $\\mathcal{B}$} is a rule which assigns to each\nelement $U \\in \\mathcal{B}$ a morphism of\nalgebraic structures $\\varphi : \\mathcal{F}(U) \\to \\mathcal{G}(U)$\ncompatible with restriction maps.\n\\item Given a presheaf $\\mathcal{F}$ with values in $\\mathcal{C}$\non $\\mathcal{B}$ we say that $U \\mapsto F(\\mathcal{F}(U))$ is the\nunderlying presheaf of sets.\n\\item A {\\it sheaf $\\mathcal{F}$ with values in $\\mathcal{C}$\non $\\mathcal{B}$} is a presheaf with values in $\\mathcal{C}$\non $\\mathcal{B}$ whose underlying presheaf of sets is a sheaf.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009P","source_file":"sheaves.tex","source_line":3996,"source_end_line":4024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L3996-L4024","statement_sha256":"6a991bb18fabd420d349d05a81ce6583ff09614875f55ce571191bdca922690b","origin":"The Stacks Project","memory_eligible":false,"source_rank":512,"rank":512,"depth":0,"x":1155.282,"y":261.884,"cluster":"sheaves-sites"},{"id":"stacks:009Q","tag":"009Q","title":"Bases and sheaves · Lemma 009Q","summary":"Let X be a topological space. Let (C, F) be a type of algebraic structure. Let B be a basis for the topology on X. Let F be a sheaf with values in C on B. There exists a unique sheaf F^ext with values in C on X such that F^ext(U) = F(U) for all U ∈ B compatibly with the restriction mappings.","statement_latex":"Let $X$ be a topological space. Let $(\\mathcal{C}, F)$ be\na type of algebraic structure.\nLet $\\mathcal{B}$ be a basis for the topology on $X$.\nLet $\\mathcal{F}$ be a sheaf with values in $\\mathcal{C}$\non $\\mathcal{B}$.\nThere exists a unique sheaf $\\mathcal{F}^{ext}$ with values in $\\mathcal{C}$\non $X$ such that $\\mathcal{F}^{ext}(U) = \\mathcal{F}(U)$\nfor all $U \\in \\mathcal{B}$ compatibly with the restriction\nmappings.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009Q","source_file":"sheaves.tex","source_line":4047,"source_end_line":4058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4047-L4058","statement_sha256":"b1881ae552004fdee9935582effa58f3873d2a82ad601ad67d981969f917ff74","origin":"The Stacks Project","memory_eligible":false,"source_rank":513,"rank":513,"depth":2,"x":1077.349,"y":203.357,"cluster":"sheaves-sites"},{"id":"stacks:009R","tag":"009R","title":"Bases and sheaves · Lemma 009R","summary":"Let X be a topological space. Let B be a basis for the topology on X. Let (C, F) be a type of algebraic structure. Denote Sh(B, C) the category of sheaves with values in C on B. There is an equivalence of categories Sh(X, C) → Sh(B, C) which assigns to a sheaf on X its restriction to the members of B.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{B}$ be a basis for the topology on $X$.\nLet $(\\mathcal{C}, F)$ be a type of algebraic structure.\nDenote $\\Sh(\\mathcal{B}, \\mathcal{C})$ the category of\nsheaves with values in $\\mathcal{C}$ on $\\mathcal{B}$.\nThere is an equivalence of categories\n$$\n\\Sh(X, \\mathcal{C})\n\\longrightarrow\n\\Sh(\\mathcal{B}, \\mathcal{C})\n$$\nwhich assigns to a sheaf on $X$ its restriction to\nthe members of $\\mathcal{B}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009R","source_file":"sheaves.tex","source_line":4104,"source_end_line":4119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4104-L4119","statement_sha256":"d0e2d576fbe3640ccbf6c636dd9e059c2f19323ced800cc8e0cc56ee74c0a59b","origin":"The Stacks Project","memory_eligible":false,"source_rank":514,"rank":514,"depth":3,"x":1182.53,"y":202.288,"cluster":"sheaves-sites"},{"id":"stacks:009S","tag":"009S","title":"Bases and sheaves · Definition 009S","summary":"Let X be a topological space. Let B be a basis for the topology on X. Let O be a presheaf of rings on B. • A presheaf of O-modules F on B is a presheaf of abelian groups on B together with a morphism of presheaves of sets O × F → F such that for all U ∈ B the map O(U) × F(U) → F(U) turns the group F(U) into an O(U)-module. • A morphism φ : F → G of presheaves of O-modules on B is a morphism of abelian presheaves on B which induces an O(U)-module homomorphism F(U) → G(U)…","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{B}$ be a\nbasis for the topology on $X$. Let $\\mathcal{O}$ be\na presheaf of rings on $\\mathcal{B}$.\n\\begin{enumerate}\n\\item A {\\it presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$\non $\\mathcal{B}$} is a presheaf of abelian groups on\n$\\mathcal{B}$ together with a morphism of presheaves\nof sets $\\mathcal{O} \\times \\mathcal{F} \\to \\mathcal{F}$\nsuch that for all $U \\in \\mathcal{B}$ the map\n$\\mathcal{O}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$\nturns the group $\\mathcal{F}(U)$ into an $\\mathcal{O}(U)$-module.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves of $\\mathcal{O}$-modules on $\\mathcal{B}$}\nis a morphism of abelian presheaves on $\\mathcal{B}$\nwhich induces an $\\mathcal{O}(U)$-module homomorphism\n$\\mathcal{F}(U) \\to \\mathcal{G}(U)$ for every $U \\in \\mathcal{B}$.\n\\item Suppose that $\\mathcal{O}$ is a sheaf of rings\non $\\mathcal{B}$. A {\\it sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules\non $\\mathcal{B}$} is a presheaf of $\\mathcal{O}$-modules\non $\\mathcal{B}$ whose underlying presheaf of abelian groups\nis a sheaf.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009S","source_file":"sheaves.tex","source_line":4135,"source_end_line":4159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4135-L4159","statement_sha256":"5be38849094a83d65b04b513ab58a69b1347f87bec58d8021220354cd24684d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":515,"rank":515,"depth":0,"x":1105.36,"y":263.129,"cluster":"sheaves-sites"},{"id":"stacks:009T","tag":"009T","title":"Bases and sheaves · Lemma 009T","summary":"Let X be a topological space. Let B be a basis for the topology on X. Let O be a sheaf of rings on B. Let F be a sheaf of O-modules on B. Let O^ext be the sheaf of rings on X extending O and let F^ext be the abelian sheaf on X extending F, see Lemma [Tag 009Q]. There exists a canonical map O^ext × F^ext → F^ext which agrees with the given map over elements of B and which endows F^ext with the structure of an O^ext-module.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{B}$ be a basis for the topology on $X$.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{B}$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules\non $\\mathcal{B}$. Let $\\mathcal{O}^{ext}$ be the sheaf\nof rings on $X$ extending $\\mathcal{O}$ and let\n$\\mathcal{F}^{ext}$ be the abelian sheaf on $X$ extending\n$\\mathcal{F}$, see Lemma \\ref{lemma-extend-off-basis-structures}.\nThere exists a canonical map\n$$\n\\mathcal{O}^{ext} \\times \\mathcal{F}^{ext}\n\\longrightarrow\n\\mathcal{F}^{ext}\n$$\nwhich agrees with the given map over elements of $\\mathcal{B}$\nand which endows $\\mathcal{F}^{ext}$ with the structure\nof an $\\mathcal{O}^{ext}$-module.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009T","source_file":"sheaves.tex","source_line":4179,"source_end_line":4198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4179-L4198","statement_sha256":"171aaa23b8db44e0c42d1a892912badcfc162a020de92c5b64cb60f4b57978fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":516,"rank":516,"depth":3,"x":1113.387,"y":173.94,"cluster":"sheaves-sites"},{"id":"stacks:009U","tag":"009U","title":"Bases and sheaves · Lemma 009U","summary":"Let X be a topological space. Let B be a basis for the topology on X. Let O be a sheaf of rings on X. Denote Mod(O|_B) the category of sheaves of O|_B-modules on B. There is an equivalence of categories Mod(O) → Mod(O|_B) which assigns to a sheaf of O-modules on X its restriction to the members of B.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{B}$ be a basis for the topology on $X$.\nLet $\\mathcal{O}$ be a sheaf of rings on $X$.\nDenote $\\textit{Mod}(\\mathcal{O}|_\\mathcal{B})$ the category of\nsheaves of $\\mathcal{O}|_\\mathcal{B}$-modules on $\\mathcal{B}$.\nThere is an equivalence of categories\n$$\n\\textit{Mod}(\\mathcal{O})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}|_\\mathcal{B})\n$$\nwhich assigns to a sheaf of $\\mathcal{O}$-modules on $X$ its restriction to\nthe members of $\\mathcal{B}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009U","source_file":"sheaves.tex","source_line":4224,"source_end_line":4239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4224-L4239","statement_sha256":"7951d64bdc01c423620fc6e3d0f4242cea4ab7a7f1726e915bfd8381dcd2117a","origin":"The Stacks Project","memory_eligible":false,"source_rank":517,"rank":517,"depth":4,"x":1179.583,"y":244.683,"cluster":"sheaves-sites"},{"id":"stacks:009V","tag":"009V","title":"Bases and sheaves · Lemma 009V","summary":"Let f : X → Y be a continuous map of topological spaces. Let (C, F) be a type of algebraic structures. Let F be a sheaf with values in C on X. Let G be a sheaf with values in C on Y. Let B be a basis for the topology on Y. Suppose given for every V ∈ B a morphism φ_V : G(V) → F(f^-1V) of C compatible with restriction mappings. Then there is a unique f-map (see Definition [Tag 008J] and discussion of f-maps in Section [Tag 008P]) φ : G → F recovering φ_V for V ∈ B.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $(\\mathcal{C}, F)$ be a type of algebraic structures.\nLet $\\mathcal{F}$ be a sheaf with values in $\\mathcal{C}$ on $X$.\nLet $\\mathcal{G}$ be a sheaf with values in $\\mathcal{C}$ on $Y$.\nLet $\\mathcal{B}$ be a basis for the topology on $Y$.\nSuppose given for every $V \\in \\mathcal{B}$ a morphism\n$$\n\\varphi_V :\n\\mathcal{G}(V)\n\\longrightarrow\n\\mathcal{F}(f^{-1}V)\n$$\nof $\\mathcal{C}$ compatible with restriction mappings.\nThen there is a unique $f$-map (see Definition \\ref{definition-f-map}\nand discussion of $f$-maps in\nSection \\ref{section-presheaves-structures-functorial})\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$\nrecovering $\\varphi_V$ for $V \\in \\mathcal{B}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009V","source_file":"sheaves.tex","source_line":4253,"source_end_line":4273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4253-L4273","statement_sha256":"fa50b59470f118567e2cf2e8f70fe170d07b284e5237cf323cab65c8ebe3362b","origin":"The Stacks Project","memory_eligible":false,"source_rank":518,"rank":518,"depth":4,"x":1073.257,"y":229.992,"cluster":"sheaves-sites"},{"id":"stacks:009W","tag":"009W","title":"Bases and sheaves · Lemma 009W","summary":"Let (f, f^sharp) : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let F be a sheaf of O_X-modules. Let G be a sheaf of O_Y-modules. Let B be a basis for the topology on Y. Suppose given for every V ∈ B a O_Y(V)-module map φ_V : G(V) → F(f^-1V) (where F(f^-1V) has a module structure using f^sharp_V : O_Y(V) → O_X(f^-1V)) compatible with restriction mappings. Then there is a unique f-map (see discussion of f-maps in Section [Tag 0094]) φ : G → F recovering φ_V for V ∈ B.","statement_latex":"Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.\nLet $\\mathcal{B}$ be a basis for the topology on $Y$.\nSuppose given for every $V \\in \\mathcal{B}$ a\n$\\mathcal{O}_Y(V)$-module map\n$$\n\\varphi_V :\n\\mathcal{G}(V)\n\\longrightarrow\n\\mathcal{F}(f^{-1}V)\n$$\n(where $\\mathcal{F}(f^{-1}V)$ has a module structure using\n$f^\\sharp_V : \\mathcal{O}_Y(V) \\to \\mathcal{O}_X(f^{-1}V)$)\ncompatible with restriction mappings.\nThen there is a unique $f$-map (see discussion of $f$-maps in\nSection \\ref{section-ringed-spaces-functoriality-modules})\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$\nrecovering $\\varphi_V$ for $V \\in \\mathcal{B}$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009W","source_file":"sheaves.tex","source_line":4291,"source_end_line":4313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4291-L4313","statement_sha256":"3a399e17c42f6a5f0cd8a4ab27855db7edb36c8ffd7571c888be71b0da2af19f","origin":"The Stacks Project","memory_eligible":false,"source_rank":519,"rank":519,"depth":0,"x":1164.002,"y":180.205,"cluster":"sheaves-sites"},{"id":"stacks:009X","tag":"009X","title":"Bases and sheaves · Lemma 009X","summary":"Let f : X → Y be a continuous map of topological spaces. Let (C, F) be a type of algebraic structures. Let F be a sheaf with values in C on X. Let G be a sheaf with values in C on Y. Let B_Y be a basis for the topology on Y. Let B_X be a basis for the topology on X. Suppose given for every V ∈ B_Y, and U ∈ B_X such that f(U) ⊂ V a morphism φ_V^U : G(V) → F(U) of C compatible with restriction mappings. Then there is a unique f-map (see Definition [Tag 008J] and the…","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $(\\mathcal{C}, F)$ be a type of algebraic structures.\nLet $\\mathcal{F}$ be a sheaf with values in $\\mathcal{C}$ on $X$.\nLet $\\mathcal{G}$ be a sheaf with values in $\\mathcal{C}$ on $Y$.\nLet $\\mathcal{B}_Y$ be a basis for the topology on $Y$.\nLet $\\mathcal{B}_X$ be a basis for the topology on $X$.\nSuppose given for every $V \\in \\mathcal{B}_Y$, and\n$U \\in \\mathcal{B}_X$ such that $f(U) \\subset V$ a morphism\n$$\n\\varphi_V^U :\n\\mathcal{G}(V)\n\\longrightarrow\n\\mathcal{F}(U)\n$$\nof $\\mathcal{C}$ compatible with restriction mappings.\nThen there is a unique $f$-map (see\nDefinition \\ref{definition-f-map} and the discussion\nof $f$-maps in Section \\ref{section-presheaves-structures-functorial})\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$\nrecovering $\\varphi_V^U$ as the composition\n$$\n\\mathcal{G}(V) \\xrightarrow{\\varphi_V}\n\\mathcal{F}(f^{-1}(V)) \\xrightarrow{\\text{restr.}}\n\\mathcal{F}(U)\n$$\nfor every pair $(U, V)$ as above.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009X","source_file":"sheaves.tex","source_line":4320,"source_end_line":4348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4320-L4348","statement_sha256":"8b759f025d35a7f6e44895380198df127d842a6889f3ea1bb164c06d89d3b84e","origin":"The Stacks Project","memory_eligible":false,"source_rank":520,"rank":520,"depth":5,"x":1136.969,"y":268.918,"cluster":"sheaves-sites"},{"id":"stacks:009Y","tag":"009Y","title":"Bases and sheaves · Lemma 009Y","summary":"Let (f, f^sharp) : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let F be a sheaf of O_X-modules. Let G be a sheaf of O_Y-modules. Let B_Y be a basis for the topology on Y. Let B_X be a basis for the topology on X. Suppose given for every V ∈ B_Y, and U ∈ B_X such that f(U) ⊂ V a O_Y(V)-module map φ_V^U : G(V) → F(U) compatible with restriction mappings. Here the O_Y(V)-module structure on F(U) comes from the O_X(U)-module structure via the map f^sharp_V : O_Y(V) →…","statement_latex":"Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_Y$-modules.\nLet $\\mathcal{B}_Y$ be a basis for the topology on $Y$.\nLet $\\mathcal{B}_X$ be a basis for the topology on $X$.\nSuppose given for every $V \\in \\mathcal{B}_Y$, and\n$U \\in \\mathcal{B}_X$ such that $f(U) \\subset V$ a\n$\\mathcal{O}_Y(V)$-module map\n$$\n\\varphi_V^U :\n\\mathcal{G}(V)\n\\longrightarrow\n\\mathcal{F}(U)\n$$\ncompatible with restriction mappings. Here the\n$\\mathcal{O}_Y(V)$-module structure on $\\mathcal{F}(U)$\ncomes from the $\\mathcal{O}_X(U)$-module structure\nvia the map $f^\\sharp_V : \\mathcal{O}_Y(V)\n\\to \\mathcal{O}_X(f^{-1}V) \\to \\mathcal{O}_X(U)$.\nThen there is a unique $f$-map of sheaves of modules (see\nDefinition \\ref{definition-f-map} and the discussion\nof $f$-maps in Section \\ref{section-ringed-spaces-functoriality-modules})\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$\nrecovering $\\varphi_V^U$ as the composition\n$$\n\\mathcal{G}(V) \\xrightarrow{\\varphi_V}\n\\mathcal{F}(f^{-1}(V)) \\xrightarrow{\\text{restr.}}\n\\mathcal{F}(U)\n$$\nfor every pair $(U, V)$ as above.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Bases and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/009Y","source_file":"sheaves.tex","source_line":4400,"source_end_line":4433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4400-L4433","statement_sha256":"9a988a94dba802e9d227f63905be346cee6eb34184d3518ae976a1da8e84518d","origin":"The Stacks Project","memory_eligible":false,"source_rank":521,"rank":521,"depth":1,"x":1085.27,"y":187.701,"cluster":"sheaves-sites"},{"id":"stacks:00A0","tag":"00A0","title":"Open immersions and (pre)sheaves · Lemma 00A0","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset U into X. • Let G be a presheaf of sets on X. The presheaf j_pG (see Section [Tag 008C]) is given by the rule V ↦ G(V) for V ⊂ U open. • Let G be a sheaf of sets on X. The sheaf j^-1G is given by the rule V ↦ G(V) for V ⊂ U open. • For any point u ∈ U and any sheaf G on X we have a canonical identification of stalks j^-1G_u = (G|_U)_u = G_u. • On the category of presheaves of U we have j_pj_* =…","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset $U$ into $X$.\n\\begin{enumerate}\n\\item Let $\\mathcal{G}$ be a presheaf of sets on $X$.\nThe presheaf $j_p\\mathcal{G}$\n(see Section \\ref{section-presheaves-functorial}) is given by the rule\n$V \\mapsto \\mathcal{G}(V)$ for $V \\subset U$ open.\n\\item Let $\\mathcal{G}$ be a sheaf of sets on $X$.\nThe sheaf $j^{-1}\\mathcal{G}$ is given by the rule\n$V \\mapsto \\mathcal{G}(V)$ for $V \\subset U$ open.\n\\item For any point $u \\in U$ and any sheaf $\\mathcal{G}$ on $X$\nwe have a canonical identification of stalks\n$$\nj^{-1}\\mathcal{G}_u = (\\mathcal{G}|_U)_u = \\mathcal{G}_u.\n$$\n\\item On the category of presheaves of $U$ we have $j_pj_* = \\text{id}$.\n\\item On the category of sheaves of $U$ we have $j^{-1}j_* = \\text{id}$.\n\\end{enumerate}\nThe same description holds for (pre)sheaves of abelian groups,\n(pre)sheaves of algebraic structures, and (pre)sheaves of modules.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A0","source_file":"sheaves.tex","source_line":4452,"source_end_line":4474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4452-L4474","statement_sha256":"de0bc2540b1c9ffc63a51ecfa5627fc1a14298dc7a1d800e39f6082af1718c02","origin":"The Stacks Project","memory_eligible":false,"source_rank":522,"rank":522,"depth":0,"x":1189.288,"y":218.427,"cluster":"sheaves-sites"},{"id":"stacks:00A1","tag":"00A1","title":"Open immersions and (pre)sheaves · Definition 00A1","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset. • Let G be a presheaf of sets, abelian groups or algebraic structures on X. The presheaf j_pG described in Lemma [Tag 00A0] is called the restriction of G to U and denoted G|_U. • Let G be a sheaf of sets on X, abelian groups or algebraic structures on X. The sheaf j^-1G is called the restriction of G to U and denoted G|_U. • If (X, O) is a ringed space, then the pair (U, O|_U) is called the…","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset.\n\\begin{enumerate}\n\\item Let $\\mathcal{G}$ be a presheaf of sets, abelian groups or\nalgebraic structures on $X$. The presheaf $j_p\\mathcal{G}$ described\nin Lemma \\ref{lemma-j-pullback} is called\nthe {\\it restriction of $\\mathcal{G}$ to $U$} and denoted $\\mathcal{G}|_U$.\n\\item Let $\\mathcal{G}$ be a sheaf of sets on $X$, abelian groups or\nalgebraic structures on $X$. The sheaf $j^{-1}\\mathcal{G}$ is called\nthe {\\it restriction of $\\mathcal{G}$ to $U$} and denoted $\\mathcal{G}|_U$.\n\\item If $(X, \\mathcal{O})$ is a ringed space, then the pair\n$(U, \\mathcal{O}|_U)$ is called the\n{\\it open subspace of $(X, \\mathcal{O})$ associated to $U$}.\n\\item If $\\mathcal{G}$ is a presheaf of $\\mathcal{O}$-modules\nthen $\\mathcal{G}|_U$ together with the multiplication map\n$\\mathcal{O}|_U \\times \\mathcal{G}|_U \\to \\mathcal{G}|_U$\n(see Lemma \\ref{lemma-pullback-module})\nis called the {\\it restriction of $\\mathcal{G}$ to $U$}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A1","source_file":"sheaves.tex","source_line":4494,"source_end_line":4515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4494-L4515","statement_sha256":"8259787049533720539f6fc01a06944d400b5c3f305c9a525c09b35a94db627c","origin":"The Stacks Project","memory_eligible":false,"source_rank":523,"rank":523,"depth":3,"x":1087.311,"y":254.994,"cluster":"sheaves-sites"},{"id":"stacks:00A2","tag":"00A2","title":"Open immersions and (pre)sheaves · Definition 00A2","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset. • Let F be a presheaf of sets on U. We define the extension of F by the empty set j_p!F to be the presheaf of sets on X defined by the rule j_p!F(V) = ( ∅ & if & V not ⊂ U F(V) & if & V ⊂ U . with obvious restriction mappings. • Let F be a sheaf of sets on U. We define the extension of F by the empty set j_!F to be the sheafification of the presheaf j_p!F.","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a presheaf of sets on $U$. We define\nthe {\\it extension of $\\mathcal{F}$ by the empty set $j_{p!}\\mathcal{F}$}\nto be the presheaf of sets on $X$ defined by the rule\n$$\nj_{p!}\\mathcal{F}(V) =\n\\left\\{\n\\begin{matrix}\n\\emptyset & \\text{if} & V \\not \\subset U \\\\\n\\mathcal{F}(V) & \\text{if} & V \\subset U\n\\end{matrix}\n\\right.\n$$\nwith obvious restriction mappings.\n\\item Let $\\mathcal{F}$ be a sheaf of sets on $U$. We define\nthe {\\it extension of $\\mathcal{F}$ by the empty set $j_!\\mathcal{F}$}\nto be the sheafification of the presheaf $j_{p!}\\mathcal{F}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A2","source_file":"sheaves.tex","source_line":4524,"source_end_line":4546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4524-L4546","statement_sha256":"6a4b1736d07f0feba07ce0b51692708c6bf38e4d9f4c831d705634dfebbdcdea","origin":"The Stacks Project","memory_eligible":false,"source_rank":524,"rank":524,"depth":0,"x":1133.355,"y":169.699,"cluster":"sheaves-sites"},{"id":"stacks:00A3","tag":"00A3","title":"Open immersions and (pre)sheaves · Lemma 00A3","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset. • The functor j_p! is a left adjoint to the restriction functor j_p (see Lemma [Tag 00A0]). • The functor j_! is a left adjoint to restriction, in a formula Mor_Sh(X)(j_!F, G) = Mor_Sh(U)(F, j^-1G) = Mor_Sh(U)(F, G|_U) bifunctorially in F and G. • Let F be a sheaf of sets on U. The stalks of the sheaf j_!F are described as follows j_!F_x = ( ∅ & if & x not ∈ U F_x & if & x ∈ U . • On the…","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset.\n\\begin{enumerate}\n\\item The functor $j_{p!}$ is a left adjoint to the\nrestriction functor $j_p$ (see Lemma \\ref{lemma-j-pullback}).\n\\item The functor $j_!$ is a left adjoint to restriction,\nin a formula\n$$\n\\Mor_{\\Sh(X)}(j_!\\mathcal{F}, \\mathcal{G})\n=\n\\Mor_{\\Sh(U)}(\\mathcal{F}, j^{-1}\\mathcal{G})\n=\n\\Mor_{\\Sh(U)}(\\mathcal{F}, \\mathcal{G}|_U)\n$$\nbifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.\n\\item Let $\\mathcal{F}$ be a sheaf of sets on $U$.\nThe stalks of the sheaf $j_!\\mathcal{F}$ are described\nas follows\n$$\nj_{!}\\mathcal{F}_x =\n\\left\\{\n\\begin{matrix}\n\\emptyset & \\text{if} & x \\not \\in U \\\\\n\\mathcal{F}_x & \\text{if} & x \\in U\n\\end{matrix}\n\\right.\n$$\n\\item On the category of presheaves of $U$ we have $j_pj_{p!} = \\text{id}$.\n\\item On the category of sheaves of $U$ we have $j^{-1}j_! = \\text{id}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A3","source_file":"sheaves.tex","source_line":4548,"source_end_line":4580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4548-L4580","statement_sha256":"28d7a2aa564f466113c0ee642d433bf672a6202326f3dd96593eb2aaa06d206d","origin":"The Stacks Project","memory_eligible":false,"source_rank":525,"rank":525,"depth":1,"x":1168.183,"y":259.207,"cluster":"sheaves-sites"},{"id":"stacks:00A4","tag":"00A4","title":"Open immersions and (pre)sheaves · Definition 00A4","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset. • Let F be an abelian presheaf on U. We define the extension j_p!F of F by 0 to be the abelian presheaf on X defined by the rule j_p!F(V) = ( 0 & if & V not ⊂ U F(V) & if & V ⊂ U . with obvious restriction mappings. • Let F be an abelian sheaf on U. We define the extension j_!F of F by 0 to be the sheafification of the abelian presheaf j_p!F. • Let C be a category having an initial object e.…","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be an abelian presheaf on $U$.\nWe define the {\\it extension $j_{p!}\\mathcal{F}$ of $\\mathcal{F}$ by $0$}\nto be the abelian presheaf on $X$ defined by the rule\n$$\nj_{p!}\\mathcal{F}(V) =\n\\left\\{\n\\begin{matrix}\n0 & \\text{if} & V \\not \\subset U \\\\\n\\mathcal{F}(V) & \\text{if} & V \\subset U\n\\end{matrix}\n\\right.\n$$\nwith obvious restriction mappings.\n\\item Let $\\mathcal{F}$ be an abelian sheaf on $U$. We define\nthe {\\it extension $j_!\\mathcal{F}$ of $\\mathcal{F}$ by $0$}\nto be the sheafification of the abelian presheaf $j_{p!}\\mathcal{F}$.\n\\item Let $\\mathcal{C}$ be a category having an initial object $e$.\nLet $\\mathcal{F}$ be a presheaf on $U$ with values in $\\mathcal{C}$.\nWe define the {\\it extension $j_{p!}\\mathcal{F}$ of $\\mathcal{F}$ by $e$}\nto be the presheaf on $X$ with values in $\\mathcal{C}$ defined by the\nrule\n$$\nj_{p!}\\mathcal{F}(V) =\n\\left\\{\n\\begin{matrix}\ne & \\text{if} & V \\not \\subset U \\\\\n\\mathcal{F}(V) & \\text{if} & V \\subset U\n\\end{matrix}\n\\right.\n$$\nwith obvious restriction mappings.\n\\item Let $(\\mathcal{C}, F)$ be a type of algebraic structure\nsuch that $\\mathcal{C}$ has an initial object $e$.\nLet $\\mathcal{F}$ be a sheaf of algebraic structures on $U$\n(of the give type). We define the\n{\\it extension $j_!\\mathcal{F}$ of $\\mathcal{F}$ by $e$}\nto be the sheafification of the presheaf $j_{p!}\\mathcal{F}$\ndefined above.\n\\item Let $\\mathcal{O}$ be a presheaf of rings on $X$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}|_U$-modules.\nIn this case we define the {\\it extension by $0$}\nto be the presheaf of $\\mathcal{O}$-modules which is equal to\n$j_{p!}\\mathcal{F}$ as an abelian presheaf endowed with\nthe multiplication map\n$\\mathcal{O} \\times j_{p!}\\mathcal{F} \\to j_{p!}\\mathcal{F}$.\n\\item Let $\\mathcal{O}$ be a sheaf of rings on $X$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}|_U$-modules.\nIn this case we define the {\\it extension by $0$}\nto be the $\\mathcal{O}$-module which is equal to\n$j_!\\mathcal{F}$ as an abelian sheaf endowed with\nthe multiplication map $\\mathcal{O} \\times j_!\\mathcal{F} \\to j_!\\mathcal{F}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A4","source_file":"sheaves.tex","source_line":4611,"source_end_line":4668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4611-L4668","statement_sha256":"6968fde10259cd930da55862e73c20d33f5d6a3668a150700c923e7e9022b04e","origin":"The Stacks Project","memory_eligible":false,"source_rank":526,"rank":526,"depth":0,"x":1069.995,"y":212.717,"cluster":"sheaves-sites"},{"id":"stacks:00A5","tag":"00A5","title":"Open immersions and (pre)sheaves · Lemma 00A5","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset. Consider the functors of restriction and extension by 0 for abelian (pre)sheaves. • The functor j_p! is a left adjoint to the restriction functor j_p (see Lemma [Tag 00A0]). • The functor j_! is a left adjoint to restriction, in a formula Mor_Ab(X)(j_!F, G) = Mor_Ab(U)(F, j^-1G) = Mor_Ab(U)(F, G|_U) bifunctorially in F and G. • Let F be an abelian sheaf on U. The stalks of the sheaf j_!F are…","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset.\nConsider the functors of restriction and extension\nby $0$ for abelian (pre)sheaves.\n\\begin{enumerate}\n\\item The functor $j_{p!}$ is a left adjoint to the\nrestriction functor $j_p$ (see Lemma \\ref{lemma-j-pullback}).\n\\item The functor $j_!$ is a left adjoint to restriction,\nin a formula\n$$\n\\Mor_{\\textit{Ab}(X)}(j_!\\mathcal{F}, \\mathcal{G})\n=\n\\Mor_{\\textit{Ab}(U)}(\\mathcal{F}, j^{-1}\\mathcal{G})\n=\n\\Mor_{\\textit{Ab}(U)}(\\mathcal{F}, \\mathcal{G}|_U)\n$$\nbifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.\n\\item Let $\\mathcal{F}$ be an abelian sheaf on $U$.\nThe stalks of the sheaf $j_!\\mathcal{F}$ are described\nas follows\n$$\nj_{!}\\mathcal{F}_x =\n\\left\\{\n\\begin{matrix}\n0 & \\text{if} & x \\not \\in U \\\\\n\\mathcal{F}_x & \\text{if} & x \\in U\n\\end{matrix}\n\\right.\n$$\n\\item On the category of abelian presheaves of $U$\nwe have $j_pj_{p!} = \\text{id}$.\n\\item On the category of abelian sheaves of $U$\nwe have $j^{-1}j_! = \\text{id}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A5","source_file":"sheaves.tex","source_line":4684,"source_end_line":4720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4684-L4720","statement_sha256":"c803caabb580c7bab0d50e26c769e4edd593910eccce5a873e84cec2633f31e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":527,"rank":527,"depth":1,"x":1180.37,"y":191.17,"cluster":"sheaves-sites"},{"id":"stacks:00A6","tag":"00A6","title":"Open immersions and (pre)sheaves · Lemma 00A6","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset. Let (C, F) be a type of algebraic structure such that C has an initial object e. Consider the functors of restriction and extension by e for (pre)sheaves of algebraic structure defined above. • The functor j_p! is a left adjoint to the restriction functor j_p (see Lemma [Tag 00A0]). • The functor j_! is a left adjoint to restriction, in a formula Mor_Sh(X, C)(j_!F, G) = Mor_Sh(U, C)(F, j^-1G)…","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset.\nLet $(\\mathcal{C}, F)$ be a type of algebraic structure\nsuch that $\\mathcal{C}$ has an initial object $e$.\nConsider the functors of restriction and extension\nby $e$ for (pre)sheaves of algebraic structure defined above.\n\\begin{enumerate}\n\\item The functor $j_{p!}$ is a left adjoint to the\nrestriction functor $j_p$ (see Lemma \\ref{lemma-j-pullback}).\n\\item The functor $j_!$ is a left adjoint to restriction,\nin a formula\n$$\n\\Mor_{\\Sh(X, \\mathcal{C})}(j_!\\mathcal{F}, \\mathcal{G})\n=\n\\Mor_{\\Sh(U, \\mathcal{C})}(\\mathcal{F}, j^{-1}\\mathcal{G})\n=\n\\Mor_{\\Sh(U, \\mathcal{C})}(\\mathcal{F}, \\mathcal{G}|_U)\n$$\nbifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.\n\\item Let $\\mathcal{F}$ be a sheaf on $U$.\nThe stalks of the sheaf $j_!\\mathcal{F}$ are described\nas follows\n$$\nj_{!}\\mathcal{F}_x =\n\\left\\{\n\\begin{matrix}\ne & \\text{if} & x \\not \\in U \\\\\n\\mathcal{F}_x & \\text{if} & x \\in U\n\\end{matrix}\n\\right.\n$$\n\\item On the category of presheaves of algebraic structures on $U$\nwe have $j_pj_{p!} = \\text{id}$.\n\\item On the category of sheaves of algebraic structures on $U$\nwe have $j^{-1}j_! = \\text{id}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A6","source_file":"sheaves.tex","source_line":4726,"source_end_line":4764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4726-L4764","statement_sha256":"31ec19ee0144b451ed238aa3c345b2cfb866733de6ccaa97fae44900c2dc32d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":528,"rank":528,"depth":1,"x":1115.969,"y":270.102,"cluster":"sheaves-sites"},{"id":"stacks:00A7","tag":"00A7","title":"Open immersions and (pre)sheaves · Lemma 00A7","summary":"Let (X, O) be a ringed space. Let j : (U, O|_U) → (X, O) be an open subspace. Consider the functors of restriction and extension by 0 for (pre)sheaves of modules defined above. • The functor j_p! is a left adjoint to restriction, in a formula Mor_PMod(O)(j_p!F, G) = Mor_PMod(O|_U)(F, G|_U) bifunctorially in F and G. • The functor j_! is a left adjoint to restriction, in a formula Mor_Mod(O)(j_!F, G) = Mor_Mod(O|_U)(F, G|_U) bifunctorially in F and G. • Let F be a sheaf of…","statement_latex":"Let $(X, \\mathcal{O})$ be a ringed space.\nLet $j : (U, \\mathcal{O}|_U) \\to (X, \\mathcal{O})$\nbe an open subspace.\nConsider the functors of restriction and extension\nby $0$ for (pre)sheaves of modules defined above.\n\\begin{enumerate}\n\\item The functor $j_{p!}$ is a left adjoint to restriction,\nin a formula\n$$\n\\Mor_{\\textit{PMod}(\\mathcal{O})}(j_{p!}\\mathcal{F}, \\mathcal{G})\n=\n\\Mor_{\\textit{PMod}(\\mathcal{O}|_U)}(\\mathcal{F}, \\mathcal{G}|_U)\n$$\nbifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.\n\\item The functor $j_!$ is a left adjoint to restriction,\nin a formula\n$$\n\\Mor_{\\textit{Mod}(\\mathcal{O})}(j_!\\mathcal{F}, \\mathcal{G})\n=\n\\Mor_{\\textit{Mod}(\\mathcal{O}|_U)}(\\mathcal{F}, \\mathcal{G}|_U)\n$$\nbifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.\n\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $U$.\nThe stalks of the sheaf $j_!\\mathcal{F}$ are described\nas follows\n$$\nj_{!}\\mathcal{F}_x =\n\\left\\{\n\\begin{matrix}\n0 & \\text{if} & x \\not \\in U \\\\\n\\mathcal{F}_x & \\text{if} & x \\in U\n\\end{matrix}\n\\right.\n$$\n\\item On the category of sheaves of $\\mathcal{O}|_U$-modules on $U$\nwe have $j^{-1}j_! = \\text{id}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A7","source_file":"sheaves.tex","source_line":4770,"source_end_line":4809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4770-L4809","statement_sha256":"57cfc5fdd5fe79c93e0761ceaa19e6a73066b51e3ceb0e40ac9e338d4bb7c9e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":529,"rank":529,"depth":0,"x":1099.9,"y":174.861,"cluster":"sheaves-sites"},{"id":"stacks:00A8","tag":"00A8","title":"Open immersions and (pre)sheaves · Lemma 00A8","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset. The functor j_! : Sh(U) → Sh(X) is fully faithful. Its essential image consists exactly of those sheaves G such that G_x = ∅ for all x ∈ X setminus U.","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset.\nThe functor\n$$\nj_! : \\Sh(U) \\longrightarrow \\Sh(X)\n$$\nis fully faithful. Its essential image consists exactly\nof those sheaves $\\mathcal{G}$ such that\n$\\mathcal{G}_x = \\emptyset$ for all $x \\in X \\setminus U$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A8","source_file":"sheaves.tex","source_line":4823,"source_end_line":4834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4823-L4834","statement_sha256":"1f3bd0192fa829d7b36739366d75da240b486590931e3264b34d9bf79114e552","origin":"The Stacks Project","memory_eligible":false,"source_rank":530,"rank":530,"depth":0,"x":1188.796,"y":236.289,"cluster":"sheaves-sites"},{"id":"stacks:00A9","tag":"00A9","title":"Open immersions and (pre)sheaves · Lemma 00A9","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset. The functor j_! : Ab(U) → Ab(X) is fully faithful. Its essential image consists exactly of those sheaves G such that G_x = 0 for all x ∈ X setminus U.","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset.\nThe functor\n$$\nj_! : \\textit{Ab}(U) \\longrightarrow \\textit{Ab}(X)\n$$\nis fully faithful. Its essential image consists exactly\nof those sheaves $\\mathcal{G}$ such that\n$\\mathcal{G}_x = 0$ for all $x \\in X \\setminus U$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00A9","source_file":"sheaves.tex","source_line":4848,"source_end_line":4859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4848-L4859","statement_sha256":"93a901c095e42395b5541ec2c2f2de1dd0ac6fb1e0a196aed245b0c8eeecc56c","origin":"The Stacks Project","memory_eligible":false,"source_rank":531,"rank":531,"depth":0,"x":1073.258,"y":241.46,"cluster":"sheaves-sites"},{"id":"stacks:00AA","tag":"00AA","title":"Open immersions and (pre)sheaves · Lemma 00AA","summary":"Let X be a topological space. Let j : U → X be the inclusion of an open subset. Let (C, F) be a type of algebraic structure such that C has an initial object e. The functor j_! : Sh(U, C) → Sh(X, C) is fully faithful. Its essential image consists exactly of those sheaves G such that G_x = e for all x ∈ X setminus U.","statement_latex":"Let $X$ be a topological space.\nLet $j : U \\to X$ be the inclusion of an open subset.\nLet $(\\mathcal{C}, F)$ be a type of algebraic structure\nsuch that $\\mathcal{C}$ has an initial object $e$.\nThe functor\n$$\nj_! : \\Sh(U, \\mathcal{C}) \\longrightarrow \\Sh(X, \\mathcal{C})\n$$\nis fully faithful. Its essential image consists exactly\nof those sheaves $\\mathcal{G}$ such that\n$\\mathcal{G}_x = e$ for all $x \\in X \\setminus U$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00AA","source_file":"sheaves.tex","source_line":4865,"source_end_line":4878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4865-L4878","statement_sha256":"98683c6998e3254d1242e9a0e66e915b4fa1f957d2c37928bd82d6fb6d63c0c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":532,"rank":532,"depth":0,"x":1154.707,"y":171.737,"cluster":"sheaves-sites"},{"id":"stacks:00AB","tag":"00AB","title":"Open immersions and (pre)sheaves · Lemma 00AB","summary":"Let (X, O) be a ringed space. Let j : (U, O|_U) → (X, O) be an open subspace. The functor j_! : Mod(O|_U) → Mod(O) is fully faithful. Its essential image consists exactly of those sheaves G such that G_x = 0 for all x ∈ X setminus U.","statement_latex":"Let $(X, \\mathcal{O})$ be a ringed space.\nLet $j : (U, \\mathcal{O}|_U) \\to (X, \\mathcal{O})$\nbe an open subspace.\nThe functor\n$$\nj_! : \\textit{Mod}(\\mathcal{O}|_U) \\longrightarrow \\textit{Mod}(\\mathcal{O})\n$$\nis fully faithful. Its essential image consists exactly\nof those sheaves $\\mathcal{G}$ such that\n$\\mathcal{G}_x = 0$ for all $x \\in X \\setminus U$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Open immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00AB","source_file":"sheaves.tex","source_line":4885,"source_end_line":4897,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4885-L4897","statement_sha256":"65b2f0df2e1ce2c4b44ca6966654c0979c2b50793c33f109716f85f6fcfb7408","origin":"The Stacks Project","memory_eligible":false,"source_rank":533,"rank":533,"depth":0,"x":1150.696,"y":269.856,"cluster":"sheaves-sites"},{"id":"stacks:00AE","tag":"00AE","title":"Closed immersions and (pre)sheaves · Lemma 00AE","summary":"Let X be a topological space. Let i : Z → X be the inclusion of a closed subset Z into X. Let F be a sheaf of sets on Z. The stalks of i_*F are described as follows i_*F_x = ( (*) & if & x not ∈ Z F_x & if & x ∈ Z . where (*) denotes a singleton set. Moreover, i^-1i_* = id on the category of sheaves of sets on Z. Moreover, the same holds for abelian sheaves on Z, resp. sheaves of algebraic structures on Z where (*) has to be replaced by 0, resp. a final object of the…","statement_latex":"Let $X$ be a topological space.\nLet $i : Z \\to X$ be the inclusion of a closed subset $Z$ into $X$.\nLet $\\mathcal{F}$ be a sheaf of sets on $Z$.\nThe stalks of $i_*\\mathcal{F}$ are described as follows\n$$\ni_*\\mathcal{F}_x =\n\\left\\{\n\\begin{matrix}\n\\{*\\} & \\text{if} & x \\not \\in Z \\\\\n\\mathcal{F}_x & \\text{if} & x \\in Z\n\\end{matrix}\n\\right.\n$$\nwhere $\\{*\\}$ denotes a singleton set. Moreover,\n$i^{-1}i_* = \\text{id}$ on the category of sheaves\nof sets on $Z$. Moreover, the same holds for abelian\nsheaves on $Z$, resp.\\ sheaves of algebraic structures on $Z$\nwhere $\\{*\\}$ has to be replaced by $0$, resp.\\ a\nfinal object of the category of algebraic structures.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Closed immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00AE","source_file":"sheaves.tex","source_line":4935,"source_end_line":4956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4935-L4956","statement_sha256":"d272aa9153a0cb4813a3df4a366dfe4878737ce7467231d6013643ab039ce050","origin":"The Stacks Project","memory_eligible":false,"source_rank":534,"rank":534,"depth":0,"x":1074.373,"y":194.857,"cluster":"sheaves-sites"},{"id":"stacks:00AF","tag":"00AF","title":"Closed immersions and (pre)sheaves · Lemma 00AF","summary":"Let X be a topological space. Let i : Z → X be the inclusion of a closed subset. The functor i_* : Sh(Z) → Sh(X) is fully faithful. Its essential image consists exactly of those sheaves G such that G_x = (*) for all x ∈ X setminus Z.","statement_latex":"Let $X$ be a topological space.\nLet $i : Z \\to X$ be the inclusion of a closed subset.\nThe functor\n$$\ni_* : \\Sh(Z) \\longrightarrow \\Sh(X)\n$$\nis fully faithful. Its essential image consists exactly\nof those sheaves $\\mathcal{G}$ such that\n$\\mathcal{G}_x = \\{*\\}$ for all $x \\in X \\setminus Z$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Closed immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00AF","source_file":"sheaves.tex","source_line":4977,"source_end_line":4988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L4977-L4988","statement_sha256":"8f7635e32ca5ffde27467d38a58b9e3a06f71b3fbf482c9e9d6473c76d801c80","origin":"The Stacks Project","memory_eligible":false,"source_rank":535,"rank":535,"depth":0,"x":1191.537,"y":206.912,"cluster":"sheaves-sites"},{"id":"stacks:00AG","tag":"00AG","title":"Closed immersions and (pre)sheaves · Lemma 00AG","summary":"Let X be a topological space. Let i : Z → X be the inclusion of a closed subset. The functor i_* : Ab(Z) → Ab(X) is fully faithful. Its essential image consists exactly of those sheaves G such that G_x = 0 for all x ∈ X setminus Z.","statement_latex":"Let $X$ be a topological space.\nLet $i : Z \\to X$ be the inclusion of a closed subset.\nThe functor\n$$\ni_* : \\textit{Ab}(Z) \\longrightarrow \\textit{Ab}(X)\n$$\nis fully faithful. Its essential image consists exactly\nof those sheaves $\\mathcal{G}$ such that\n$\\mathcal{G}_x = 0$ for all $x \\in X \\setminus Z$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Closed immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00AG","source_file":"sheaves.tex","source_line":5002,"source_end_line":5013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L5002-L5013","statement_sha256":"b895cab0e18ae29de91e6f0733fa686e6438ba0888e51bade9cbdaaf8ae792d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":536,"rank":536,"depth":0,"x":1094.979,"y":264.784,"cluster":"sheaves-sites"},{"id":"stacks:00AH","tag":"00AH","title":"Closed immersions and (pre)sheaves · Lemma 00AH","summary":"Let X be a topological space. Let i : Z → X be the inclusion of a closed subset. Let (C, F) be a type of algebraic structure with final object 0. The functor i_* : Sh(Z, C) → Sh(X, C) is fully faithful. Its essential image consists exactly of those sheaves G such that G_x = 0 for all x ∈ X setminus Z.","statement_latex":"Let $X$ be a topological space.\nLet $i : Z \\to X$ be the inclusion of a closed subset.\nLet $(\\mathcal{C}, F)$ be a type of algebraic structure\nwith final object $0$. The functor\n$$\ni_* : \\Sh(Z, \\mathcal{C}) \\longrightarrow \\Sh(X, \\mathcal{C})\n$$\nis fully faithful. Its essential image consists exactly\nof those sheaves $\\mathcal{G}$ such that\n$\\mathcal{G}_x = 0$ for all $x \\in X \\setminus Z$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Closed immersions and (pre)sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00AH","source_file":"sheaves.tex","source_line":5019,"source_end_line":5031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L5019-L5031","statement_sha256":"d86ba8ec51a297f75509d01420c64a0252a1dac2e9ab0eff04c01a371a960927","origin":"The Stacks Project","memory_eligible":false,"source_rank":537,"rank":537,"depth":0,"x":1119.761,"y":166.851,"cluster":"sheaves-sites"},{"id":"stacks:04TN","tag":"04TN","title":"Glueing sheaves · Lemma 04TN","summary":"Let X be a topological space. Let X = ⋃ U_i be an open covering. Let F, G be sheaves of sets on X. Given a collection φ_i : F|_U_i → G|_U_i of maps of sheaves such that for all i, j ∈ I the maps φ_i, φ_j restrict to the same map F|_U_i ∩ U_j → G|_U_i ∩ U_j then there exists a unique map of sheaves φ : F → G whose restriction to each U_i agrees with φ_i.","statement_latex":"Let $X$ be a topological space.\nLet $X = \\bigcup U_i$ be an open covering.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of sets on $X$.\nGiven a collection\n$$\n\\varphi_i :\n\\mathcal{F}|_{U_i}\n\\longrightarrow\n\\mathcal{G}|_{U_i}\n$$\nof maps of sheaves such that for all $i, j \\in I$ the maps\n$\\varphi_i, \\varphi_j$ restrict to the same map\n$\\mathcal{F}|_{U_i \\cap U_j} \\to \\mathcal{G}|_{U_i \\cap U_j}$\nthen there exists a unique map of sheaves\n$$\n\\varphi :\n\\mathcal{F}\n\\longrightarrow\n\\mathcal{G}\n$$\nwhose restriction to each $U_i$ agrees with $\\varphi_i$.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TN","source_file":"sheaves.tex","source_line":5077,"source_end_line":5100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L5077-L5100","statement_sha256":"155fbbed1fd43a06edc17e14f84a8ecfda62cfd7951e4429929797ba515a2fa1","origin":"The Stacks Project","memory_eligible":false,"source_rank":538,"rank":538,"depth":0,"x":1180.53,"y":253.54,"cluster":"sheaves-sites"},{"id":"stacks:00AL","tag":"00AL","title":"Glueing sheaves · Lemma 00AL","summary":"Let X be a topological space. Let X = ⋃_i∈ I U_i be an open covering. Given any glueing data (F_i, φ_ij) for sheaves of sets with respect to the covering X = ⋃ U_i there exists a sheaf of sets F on X together with isomorphisms φ_i : F|_U_i → F_i such that the diagrams xymatrix F|_U_i ∩ U_j ar[r]_φ_i ar[d]_id & F_i|_U_i ∩ U_j ar[d]^φ_ij F|_U_i ∩ U_j ar[r]^φ_j & F_j|_U_i ∩ U_j are commutative.","statement_latex":"Let $X$ be a topological space.\nLet $X = \\bigcup_{i\\in I} U_i$ be an open covering.\nGiven any glueing data $(\\mathcal{F}_i, \\varphi_{ij})$\nfor sheaves of sets with respect to the covering $X = \\bigcup U_i$\nthere exists a sheaf of sets $\\mathcal{F}$ on $X$\ntogether with isomorphisms\n$$\n\\varphi_i : \\mathcal{F}|_{U_i} \\to \\mathcal{F}_i\n$$\nsuch that the diagrams\n$$\n\\xymatrix{\n\\mathcal{F}|_{U_i \\cap U_j} \\ar[r]_{\\varphi_i} \\ar[d]_{\\text{id}} &\n\\mathcal{F}_i|_{U_i \\cap U_j} \\ar[d]^{\\varphi_{ij}} \\\\\n\\mathcal{F}|_{U_i \\cap U_j} \\ar[r]^{\\varphi_j} &\n\\mathcal{F}_j|_{U_i \\cap U_j}\n}\n$$\nare commutative.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00AL","source_file":"sheaves.tex","source_line":5191,"source_end_line":5212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L5191-L5212","statement_sha256":"cbdce393c6accafa96b1111ee75160edf906a2339ed4557549653872a392ba27","origin":"The Stacks Project","memory_eligible":false,"source_rank":539,"rank":539,"depth":3,"x":1065.466,"y":223.959,"cluster":"sheaves-sites"},{"id":"stacks:00AM","tag":"00AM","title":"Glueing sheaves · Lemma 00AM","summary":"Let X be a topological space. Let X = ⋃ U_i be an open covering. Let (F_i, φ_ij) be a glueing data of sheaves of abelian groups, resp. sheaves of algebraic structures, resp. sheaves of O-modules for some sheaf of rings O on X. Then the construction in the proof of Lemma [Tag 00AL] above leads to a sheaf of abelian groups, resp. sheaf of algebraic structures, resp. sheaf of O-modules.","statement_latex":"Let $X$ be a topological space.\nLet $X = \\bigcup U_i$ be an open covering.\nLet $(\\mathcal{F}_i, \\varphi_{ij})$ be a glueing data\nof sheaves of abelian groups, resp.\\ sheaves of algebraic structures,\nresp.\\ sheaves of $\\mathcal{O}$-modules for some sheaf of rings\n$\\mathcal{O}$ on $X$. Then the construction in the proof of\nLemma \\ref{lemma-glue-sheaves} above leads to a sheaf\nof abelian groups, resp.\\ sheaf of algebraic structures,\nresp.\\ sheaf of $\\mathcal{O}$-modules.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00AM","source_file":"sheaves.tex","source_line":5252,"source_end_line":5263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L5252-L5263","statement_sha256":"2e84c4fc61ca8d18c4fee35ae233ac3d8a637ba518dc41259f438c9399313886","origin":"The Stacks Project","memory_eligible":false,"source_rank":540,"rank":540,"depth":4,"x":1174.61,"y":180.278,"cluster":"sheaves-sites"},{"id":"stacks:00AN","tag":"00AN","title":"Glueing sheaves · Lemma 00AN","summary":"Let X be a topological space. Let X = ⋃_i∈ I U_i be an open covering. The functor which associates to a sheaf of sets F the following collection of glueing data (F|_U_i, (F|_U_i)|_U_i ∩ U_j → (F|_U_j)|_U_i ∩ U_j ) with respect to the covering X = ⋃ U_i defines an equivalence of categories between Sh(X) and the category of glueing data. A similar statement holds for abelian sheaves, resp. sheaves of algebraic structures, resp. sheaves of O-modules.","statement_latex":"Let $X$ be a topological space.\nLet $X = \\bigcup_{i\\in I} U_i$ be an open covering.\nThe functor which associates to a sheaf of\nsets $\\mathcal{F}$ the following collection of\nglueing data\n$$\n(\\mathcal{F}|_{U_i},\n(\\mathcal{F}|_{U_i})|_{U_i \\cap U_j}\n\\to\n(\\mathcal{F}|_{U_j})|_{U_i \\cap U_j}\n)\n$$\nwith respect to the covering $X = \\bigcup U_i$\ndefines an equivalence of categories between\n$\\Sh(X)$ and the category of glueing\ndata. A similar statement holds for\nabelian sheaves, resp.\\ sheaves of algebraic structures,\nresp.\\ sheaves of $\\mathcal{O}$-modules.","area":"Sheaves & Sites","chapter":"Sheaves on Spaces","chapter_id":"sheaves","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00AN","source_file":"sheaves.tex","source_line":5283,"source_end_line":5303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sheaves.tex#L5283-L5303","statement_sha256":"651c040425082c89d5736d9a25c08959e33fded1b5dd77a8da8eb921376ef854","origin":"The Stacks Project","memory_eligible":false,"source_rank":541,"rank":541,"depth":4,"x":1129.044,"y":274.855,"cluster":"sheaves-sites"},{"id":"stacks:00V2","tag":"00V2","title":"Presheaves · Definition 00V2","summary":"A presheaf of sets on C is a contravariant functor from C to Sets. Morphisms of presheaves are transformations of functors. The category of presheaves of sets is denoted PSh(C).","statement_latex":"A {\\it presheaf of sets} on $\\mathcal{C}$ is a contravariant\nfunctor from $\\mathcal{C}$ to $\\textit{Sets}$. {\\it Morphisms\nof presheaves} are transformations of functors. The category\nof presheaves of sets is denoted $\\textit{PSh}(\\mathcal{C})$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00V2","source_file":"sites.tex","source_line":71,"source_end_line":77,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L71-L77","statement_sha256":"db0e2a93d7038ee0e6f20edc5acecfd71156a8cc880e2d303ed84554b22545cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":542,"rank":542,"depth":0,"x":1086.393,"y":178.821,"cluster":"sheaves-sites"},{"id":"stacks:00V3","tag":"00V3","title":"Presheaves · Definition 00V3","summary":"Let C, A be categories. A presheaf F on C with values in A is a contravariant functor from C to A, i.e., F : C^opp → A. A morphism of presheaves F → G on C with values in A is a transformation of functors from F to G.","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{A}$ be categories.\nA {\\it presheaf} $\\mathcal{F}$ on $\\mathcal{C}$\nwith values in $\\mathcal{A}$ is a contravariant\nfunctor from $\\mathcal{C}$ to $\\mathcal{A}$,\ni.e., $\\mathcal{F} : \\mathcal{C}^{opp} \\to \\mathcal{A}$.\nA {\\it morphism} of presheaves $\\mathcal{F} \\to \\mathcal{G}$\non $\\mathcal{C}$ with values in $\\mathcal{A}$ is a transformation\nof functors from $\\mathcal{F}$ to $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00V3","source_file":"sites.tex","source_line":98,"source_end_line":108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L98-L108","statement_sha256":"f1c8d6ddf97e6982fcaa32b6946db9721bdfaa84dddf00b869843bfeadeed4ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":543,"rank":543,"depth":0,"x":1195.566,"y":225.647,"cluster":"sheaves-sites"},{"id":"stacks:00V6","tag":"00V6","title":"Injective and surjective maps of presheaves · Definition 00V6","summary":"Let C be a category, and let φ : F → G be a map of presheaves of sets. • We say that φ is injective if for every object U of C the map φ_U : F(U) → G(U) is injective. • We say that φ is surjective if for every object U of C the map φ_U : F(U) → G(U) is surjective.","statement_latex":"Let $\\mathcal{C}$ be a category, and let $\\varphi : \\mathcal{F}\n\\to \\mathcal{G}$ be a map of presheaves of sets.\n\\begin{enumerate}\n\\item We say that $\\varphi$ is {\\it injective} if for every object\n$U$ of $\\mathcal{C}$ the map $\\varphi_U : \\mathcal{F}(U)\n\\to \\mathcal{G}(U)$ is injective.\n\\item We say that $\\varphi$ is {\\it surjective} if for every object\n$U$ of $\\mathcal{C}$ the map $\\varphi_U : \\mathcal{F}(U)\n\\to \\mathcal{G}(U)$ is surjective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Injective and surjective maps of presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00V6","source_file":"sites.tex","source_line":141,"source_end_line":153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L141-L153","statement_sha256":"05fdbd2d600df0fce087a2bf1d58d20f252d1aa7d92053c34ee77aff0ba2db56","origin":"The Stacks Project","memory_eligible":false,"source_rank":544,"rank":544,"depth":0,"x":1076.874,"y":253.188,"cluster":"sheaves-sites"},{"id":"stacks:00V7","tag":"00V7","title":"Injective and surjective maps of presheaves · Lemma 00V7","summary":"The injective (resp. surjective) maps defined above are exactly the monomorphisms (resp. epimorphisms) of PSh(C). A map is an isomorphism if and only if it is both injective and surjective.","statement_latex":"The injective (resp.\\ surjective) maps defined above\nare exactly the monomorphisms (resp.\\ epimorphisms) of\n$\\textit{PSh}(\\mathcal{C})$. A map is an isomorphism\nif and only if it is both injective and surjective.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Injective and surjective maps of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00V7","source_file":"sites.tex","source_line":155,"source_end_line":161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L155-L161","statement_sha256":"cd4286930c9a77b1003c7c20133d2abf94e9eb5ead0e3ec6122f9d90296376bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":545,"rank":545,"depth":0,"x":1142.541,"y":165.139,"cluster":"sheaves-sites"},{"id":"stacks:00V8","tag":"00V8","title":"Injective and surjective maps of presheaves · Definition 00V8","summary":"We say F is a subpresheaf of G if for every object U ∈ Ob(C) the set F(U) is a subset of G(U), compatibly with the restriction mappings.","statement_latex":"We say $\\mathcal{F}$ is a {\\it subpresheaf} of $\\mathcal{G}$\nif for every object $U \\in \\Ob(\\mathcal{C})$ the set\n$\\mathcal{F}(U)$ is a subset of $\\mathcal{G}(U)$, compatibly\nwith the restriction mappings.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Injective and surjective maps of presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00V8","source_file":"sites.tex","source_line":239,"source_end_line":245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L239-L245","statement_sha256":"82e365fa6f8c6816ce8da7319b7a8edfa1e846df2f01e17d9691264c3c5d9be6","origin":"The Stacks Project","memory_eligible":false,"source_rank":546,"rank":546,"depth":0,"x":1165.024,"y":267.779,"cluster":"sheaves-sites"},{"id":"stacks:00V9","tag":"00V9","title":"Injective and surjective maps of presheaves · Lemma 00V9","summary":"Let C be a category. Suppose that φ : F → G is a morphism of presheaves of sets on C. There exists a unique subpresheaf G' ⊂ G such that φ factors as F → G' → G and such that the first map is surjective.","statement_latex":"Let $\\mathcal{C}$ be a category.\nSuppose that $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a\nmorphism of presheaves of sets on $\\mathcal{C}$.\nThere exists a unique subpresheaf $\\mathcal{G}' \\subset \\mathcal{G}$\nsuch that $\\varphi$ factors as\n$\\mathcal{F} \\to \\mathcal{G}' \\to \\mathcal{G}$\nand such that the first map is surjective.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Injective and surjective maps of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00V9","source_file":"sites.tex","source_line":253,"source_end_line":262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L253-L262","statement_sha256":"8c1e272f56c2754f6a44d543832a7ce37ff1811af860386b09960db926a987ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":547,"rank":547,"depth":0,"x":1065.47,"y":204.574,"cluster":"sheaves-sites"},{"id":"stacks:00VA","tag":"00VA","title":"Injective and surjective maps of presheaves · Definition 00VA","summary":"Notation as in Lemma [Tag 00V9]. We say that G' is the image of φ.","statement_latex":"Notation as in Lemma \\ref{lemma-image}. We\nsay that $\\mathcal{G}'$ is the {\\it image of $\\varphi$}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Injective and surjective maps of presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VA","source_file":"sites.tex","source_line":274,"source_end_line":278,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L274-L278","statement_sha256":"36a54de401326ffaa641ef149ee84848f14fd1acb6e13a6ca3ea52e231171c50","origin":"The Stacks Project","memory_eligible":false,"source_rank":548,"rank":548,"depth":1,"x":1190.248,"y":194.649,"cluster":"sheaves-sites"},{"id":"stacks:00X4","tag":"00X4","title":"Functoriality of categories of presheaves · Lemma 00X4","summary":"Let u : C → D be a functor between categories. Suppose that C has fibre products and equalizers, and that u commutes with them. Then the categories (I_V)^opp satisfy the hypotheses of Categories, Lemma [Tag 002X].","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories.\nSuppose that $\\mathcal{C}$ has fibre products and equalizers, and that\n$u$ commutes with them. Then the categories $(\\mathcal{I}_V)^{opp}$\nsatisfy the hypotheses of\nCategories, Lemma \\ref{categories-lemma-split-into-directed}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Functoriality of categories of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00X4","source_file":"sites.tex","source_line":416,"source_end_line":423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L416-L423","statement_sha256":"e0073723b14a82ee945208652ac309fdf99fe568568ad5510beac8aebc664abb","origin":"The Stacks Project","memory_eligible":false,"source_rank":549,"rank":549,"depth":1,"x":1105.857,"y":273.107,"cluster":"sheaves-sites"},{"id":"stacks:00X3","tag":"00X3","title":"Functoriality of categories of presheaves · Lemma 00X3","summary":"Let u : C → D be a functor between categories. Assume • the category C has a final object X and u(X) is a final object of D , and • the category C has fibre products and u commutes with them. Then the index categories (I^u_V)^opp are filtered (see Categories, Definition [Tag 002V]).","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories.\nAssume\n\\begin{enumerate}\n\\item the category $\\mathcal{C}$ has a final object $X$ and\n$u(X)$ is a final object of $\\mathcal{D}$ , and\n\\item the category $\\mathcal{C}$ has fibre products and\n$u$ commutes with them.\n\\end{enumerate}\nThen the index categories $(\\mathcal{I}^u_V)^{opp}$ are filtered (see\nCategories, Definition \\ref{categories-definition-directed}).","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Functoriality of categories of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00X3","source_file":"sites.tex","source_line":451,"source_end_line":463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L451-L463","statement_sha256":"cd901057733b8bab9271f20d6873af26b7d5946d8960482ee0e16da0306009f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":550,"rank":550,"depth":2,"x":1104.99,"y":166.915,"cluster":"sheaves-sites"},{"id":"stacks:00VD","tag":"00VD","title":"Functoriality of categories of presheaves · Lemma 00VD","summary":"There is a canonical map F(U) → u_pF(u(U)), which is compatible with restriction maps (on F and on u_pF).","statement_latex":"There is a canonical map\n$\\mathcal{F}(U) \\to u_p\\mathcal{F}(u(U))$,\nwhich is compatible with restriction maps\n(on $\\mathcal{F}$ and on $u_p\\mathcal{F}$).","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Functoriality of categories of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VD","source_file":"sites.tex","source_line":541,"source_end_line":547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L541-L547","statement_sha256":"ce77bd0c77ce18432af0945626abd34ce75abff478776e52facbea767f7c130a","origin":"The Stacks Project","memory_eligible":false,"source_rank":551,"rank":551,"depth":0,"x":1191.39,"y":245.058,"cluster":"sheaves-sites"},{"id":"stacks:00VE","tag":"00VE","title":"Functoriality of categories of presheaves · Lemma 00VE","summary":"The functor u_p is a left adjoint to the functor u^p. In other words the formula Mor_PSh(C)(F, u^pG) = Mor_PSh(D)(u_pF, G) holds bifunctorially in F and G.","statement_latex":"The functor $u_p$ is a left adjoint to the functor $u^p$.\nIn other words the formula\n$$\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{F}, u^p\\mathcal{G})\n=\n\\Mor_{\\textit{PSh}(\\mathcal{D})}(u_p\\mathcal{F}, \\mathcal{G})\n$$\nholds bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Functoriality of categories of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VE","source_file":"sites.tex","source_line":566,"source_end_line":576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L566-L576","statement_sha256":"b30283c7979b987838eb967239b8b06b5c8d2f9f1f96e17ce3407a148c6e547a","origin":"The Stacks Project","memory_eligible":false,"source_rank":552,"rank":552,"depth":1,"x":1064.307,"y":236.425,"cluster":"sheaves-sites"},{"id":"stacks:04D2","tag":"04D2","title":"Functoriality of categories of presheaves · Lemma 04D2","summary":"Let u : C → D be a functor between categories. For any object U of C we have u_ph_U = h_u(U).","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor between categories.\nFor any object $U$ of $\\mathcal{C}$ we have $u_ph_U = h_{u(U)}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Functoriality of categories of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04D2","source_file":"sites.tex","source_line":622,"source_end_line":626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L622-L626","statement_sha256":"6e6591abee8da923f6718b01cf5d29170e8a88bda51a57dad1050c821e7f0f2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":553,"rank":553,"depth":0,"x":1165.378,"y":170.407,"cluster":"sheaves-sites"},{"id":"stacks:0396","tag":"0396","title":"Sites · Definition 0396","summary":"Let C be a category, see Conventions, Section [Tag 0005]. A family of morphisms with fixed target in C is given by an object U ∈ Ob(C), a set I and for each i∈ I a morphism U_i → U of C with target U. We use the notation (U_i → U)_i∈ I to indicate this.","statement_latex":"Let $\\mathcal{C}$ be a category, see\nConventions, Section \\ref{conventions-section-categories}.\nA {\\it family of morphisms with fixed target} in $\\mathcal{C}$ is\ngiven by an object $U \\in \\Ob(\\mathcal{C})$, a set $I$ and\nfor each $i\\in I$ a morphism $U_i \\to U$ of $\\mathcal{C}$ with target $U$.\nWe use the notation $\\{U_i \\to U\\}_{i\\in I}$ to indicate this.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0396","source_file":"sites.tex","source_line":664,"source_end_line":672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L664-L672","statement_sha256":"e7abf09c208133176a1594b4741feaee3309123e55cfcf0197fa7cabbc39fa80","origin":"The Stacks Project","memory_eligible":false,"source_rank":554,"rank":554,"depth":0,"x":1143.85,"y":276.877,"cluster":"sheaves-sites"},{"id":"stacks:00VH","tag":"00VH","title":"Sites · Definition 00VH","summary":"A site is given by a category C and a set Cov(C) of families of morphisms with fixed target (U_i → U)_i ∈ I, called coverings of C, satisfying the following axioms • If V → U is an isomorphism then (V → U) ∈ Cov(C). • If (U_i → U)_i∈ I ∈ Cov(C) and for each i we have (V_ij → U_i)_j∈ J_i ∈ Cov(C), then (V_ij → U)_i ∈ I, j∈ J_i ∈ Cov(C). • If (U_i → U)_i∈ I∈ Cov(C) and V → U is a morphism of C then U_i ×_U V exists for all i and (U_i ×_U V → V )_i∈ I ∈ Cov(C).","statement_latex":"A {\\it site}\\footnote{This notation differs from that of \\cite{SGA4}, as\nexplained in the introduction.} is given by a category $\\mathcal{C}$ and a set\n$\\text{Cov}(\\mathcal{C})$ of families of morphisms with fixed target\n$\\{U_i \\to U\\}_{i \\in I}$, called {\\it coverings of $\\mathcal{C}$},\nsatisfying the following axioms\n\\begin{enumerate}\n\\item If $V \\to U$ is an isomorphism then $\\{V \\to U\\} \\in\n\\text{Cov}(\\mathcal{C})$.\n\\item If $\\{U_i \\to U\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$ and for each\n$i$ we have $\\{V_{ij} \\to U_i\\}_{j\\in J_i} \\in \\text{Cov}(\\mathcal{C})$, then\n$\\{V_{ij} \\to U\\}_{i \\in I, j\\in J_i} \\in \\text{Cov}(\\mathcal{C})$.\n\\item If $\\{U_i \\to U\\}_{i\\in I}\\in \\text{Cov}(\\mathcal{C})$\nand $V \\to U$ is a morphism of $\\mathcal{C}$ then $U_i \\times_U V$\nexists for all $i$ and\n$\\{U_i \\times_U V \\to V \\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VH","source_file":"sites.tex","source_line":678,"source_end_line":696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L678-L696","statement_sha256":"b88401f565e8c5a9670f89fb40bbc2e5063333da4cea4a065071ae16573b824b","origin":"The Stacks Project","memory_eligible":false,"source_rank":555,"rank":555,"depth":0,"x":1073.82,"y":185.78,"cluster":"sheaves-sites"},{"id":"stacks:00VM","tag":"00VM","title":"Sheaves · Definition 00VM","summary":"Let C be a site, and let F be a presheaf of sets on C. We say F is a sheaf if for every covering (U_i → U)_i ∈ I ∈ Cov(C) the diagram xymatrix F(U) ar[r] & ∏_i∈ I F(U_i) ar@<1ex>[r]^-pr_0^* ar@<-1ex>[r]_-pr_1^* & ∏_(i_0, i_1) ∈ I × I F(U_i_0 ×_U U_i_1) represents the first arrow as the equalizer of pr_0^* and pr_1^*.","statement_latex":"Let $\\mathcal{C}$ be a site, and let $\\mathcal{F}$ be a presheaf of sets\non $\\mathcal{C}$. We say $\\mathcal{F}$ is a {\\it sheaf} if\nfor every covering $\\{U_i \\to U\\}_{i \\in I} \\in \\text{Cov}(\\mathcal{C})$\nthe diagram\n\\begin{equation}\n\n\\xymatrix{\n\\mathcal{F}(U) \\ar[r]\n&\n\\prod\\nolimits_{i\\in I}\n\\mathcal{F}(U_i)\n\\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*}\n&\n\\prod\\nolimits_{(i_0, i_1) \\in I \\times I}\n\\mathcal{F}(U_{i_0} \\times_U U_{i_1})\n}\n\\end{equation}\nrepresents the first arrow as the equalizer of $\\text{pr}_0^*$\nand $\\text{pr}_1^*$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VM","source_file":"sites.tex","source_line":917,"source_end_line":938,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L917-L938","statement_sha256":"ac35868f14b23129cebb392f0395828741b8af07301ea7f912c8edcfb6d36049","origin":"The Stacks Project","memory_eligible":false,"source_rank":556,"rank":556,"depth":0,"x":1199.225,"y":213.328,"cluster":"sheaves-sites"},{"id":"stacks:00VQ","tag":"00VQ","title":"Sheaves · Definition 00VQ","summary":"The category Sh(C) of sheaves of sets is the full subcategory of the category PSh(C) whose objects are the sheaves of sets.","statement_latex":"The category {\\it $\\Sh(\\mathcal{C})$}\nof sheaves of sets is the full subcategory of the category\n$\\textit{PSh}(\\mathcal{C})$ whose objects are the sheaves of sets.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VQ","source_file":"sites.tex","source_line":997,"source_end_line":1002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L997-L1002","statement_sha256":"7aaa9a0e06d5421b32252f641f0cb815ae7d7460b3ddac26ea37c3fc02331902","origin":"The Stacks Project","memory_eligible":false,"source_rank":557,"rank":557,"depth":0,"x":1084.136,"y":264.377,"cluster":"sheaves-sites"},{"id":"stacks:00VR","tag":"00VR","title":"Sheaves · Definition 00VR","summary":"Let C be a site, let A be a category and let F be a presheaf on C with values in A. We say that F is a sheaf if for all objects X of A the presheaf of sets F_X (defined above) is a sheaf.","statement_latex":"Let $\\mathcal{C}$ be a site, let $\\mathcal{A}$ be a category\nand let $\\mathcal{F}$ be a presheaf on $\\mathcal{C}$ with values in\n$\\mathcal{A}$. We say that $\\mathcal{F}$ is a {\\it sheaf}\nif for all objects $X$ of $\\mathcal{A}$ the presheaf of sets\n$\\mathcal{F}_X$ (defined above) is a sheaf.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VR","source_file":"sites.tex","source_line":1051,"source_end_line":1058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1051-L1058","statement_sha256":"04c078714cbded17e4b826e69889ec3f4ef0d3bc383396009210f8ba19684035","origin":"The Stacks Project","memory_eligible":false,"source_rank":558,"rank":558,"depth":0,"x":1128.125,"y":161.019,"cluster":"sheaves-sites"},{"id":"stacks:00VT","tag":"00VT","title":"Families of morphisms with fixed target · Definition 00VT","summary":"Let C be a category. Let U = (U_i → U)_i∈ I be a family of morphisms of C with fixed target. Let V = (V_j → V)_j∈ J be another. • A morphism of families of maps with fixed target of C from U to V, or simply a morphism from U to V is given by a morphism U → V, a map of sets α : I → J and for each i∈ I a morphism U_i → V_α(i) such that the diagram xymatrix U_i ar[r] ar[d] & V_α(i) ar[d] U ar[r] & V is commutative. • In the special case that U = V and U → V is the identity…","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{U} = \\{U_i \\to U\\}_{i\\in I}$ be a family\nof morphisms of $\\mathcal{C}$ with fixed target.\nLet $\\mathcal{V} = \\{V_j \\to V\\}_{j\\in J}$ be another.\n\\begin{enumerate}\n\\item\nA {\\it morphism of families of maps with fixed target\nof $\\mathcal{C}$ from  $\\mathcal{U}$ to $\\mathcal{V}$},\nor simply a {\\it morphism from $\\mathcal{U}$ to $\\mathcal{V}$}\nis given by a morphism $U \\to V$, a map of sets\n$\\alpha : I \\to J$ and for each $i\\in I$\na morphism $U_i \\to V_{\\alpha(i)}$ such that the diagram\n$$\n\\xymatrix{\nU_i \\ar[r] \\ar[d]\n&\nV_{\\alpha(i)} \\ar[d]\n\\\\\nU \\ar[r]\n&\nV\n}\n$$\nis commutative.\n\\item In the special case that $U = V$ and $U \\to V$ is the identity\nwe call $\\mathcal{U}$ a {\\it refinement} of the family $\\mathcal{V}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Families of morphisms with fixed target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VT","source_file":"sites.tex","source_line":1077,"source_end_line":1106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1077-L1106","statement_sha256":"648f74459ad4614191ad5529fe1ada7a84a2072a7df1312d50171f78d36ec6fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":559,"rank":559,"depth":0,"x":1179.007,"y":262.596,"cluster":"sheaves-sites"},{"id":"stacks:00VU","tag":"00VU","title":"Families of morphisms with fixed target · Definition 00VU","summary":"Let C be a category. Let U = (φ_i : U_i → U)_i∈ I, and V = (ψ_j : V_j → U)_j∈ J be two families of morphisms with fixed target. • We say U and V are combinatorially equivalent if there exist maps α : I → J and β : J→ I such that φ_i = ψ_α(i) and ψ_j = φ_β(j). • We say U and V are tautologically equivalent if there exist maps α : I → J and β : J→ I and for all i∈ I and j ∈ J commutative diagrams xymatrix U_i ar[rd] ar[rr] & & V_α(i) ar[ld] & & V_j ar[rd] ar[rr] & & U_β(j)…","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$, and\n$\\mathcal{V} = \\{\\psi_j : V_j \\to U\\}_{j\\in J}$ be two families of morphisms\nwith fixed target.\n\\begin{enumerate}\n\\item  We say $\\mathcal{U}$ and $\\mathcal{V}$ are\n{\\it combinatorially equivalent}\nif there exist maps\n$\\alpha : I \\to J$ and $\\beta : J\\to I$ such that\n$\\varphi_i = \\psi_{\\alpha(i)}$ and $\\psi_j = \\varphi_{\\beta(j)}$.\n\\item We say $\\mathcal{U}$ and $\\mathcal{V}$ are\n{\\it tautologically equivalent} if there exist maps\n$\\alpha : I \\to J$ and $\\beta : J\\to I$ and\nfor all $i\\in I$ and $j \\in J$ commutative diagrams\n$$\n\\xymatrix{\nU_i \\ar[rd] \\ar[rr] & &\nV_{\\alpha(i)} \\ar[ld] & &\nV_j \\ar[rd] \\ar[rr] & &\nU_{\\beta(j)} \\ar[ld] \\\\\n&\nU & & & &\nU &\n}\n$$\nwith isomorphisms as horizontal arrows.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Families of morphisms with fixed target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VU","source_file":"sites.tex","source_line":1115,"source_end_line":1144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1115-L1144","statement_sha256":"fd45c1b8c104642c6208665d8f9323af0ff5e92d21e89829a99b822344b61d9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":560,"rank":560,"depth":0,"x":1059.332,"y":216.384,"cluster":"sheaves-sites"},{"id":"stacks:00VV","tag":"00VV","title":"Families of morphisms with fixed target · Lemma 00VV","summary":"Let C be a category. Let U = (φ_i : U_i → U)_i∈ I, and V = (ψ_j : V_j → U)_j∈ J be two families of morphisms with the same fixed target. • If U and V are combinatorially equivalent then they are tautologically equivalent. • If U and V are tautologically equivalent then U is a refinement of V and V is a refinement of U. • The relation \"being combinatorially equivalent\" is an equivalence relation on all families of morphisms with fixed target. • The relation \"being…","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$, and\n$\\mathcal{V} = \\{\\psi_j : V_j \\to U\\}_{j\\in J}$ be two families of morphisms\nwith the same fixed target.\n\\begin{enumerate}\n\\item If $\\mathcal{U}$ and $\\mathcal{V}$ are combinatorially equivalent\nthen they are tautologically equivalent.\n\\item If $\\mathcal{U}$ and $\\mathcal{V}$ are tautologically equivalent\nthen $\\mathcal{U}$ is a refinement of $\\mathcal{V}$ and\n$\\mathcal{V}$ is a refinement of $\\mathcal{U}$.\n\\item The relation ``being combinatorially equivalent'' is an\nequivalence relation on all families of morphisms with fixed target.\n\\item The relation ``being tautologically equivalent'' is an\nequivalence relation on all families of morphisms with fixed target.\n\\item The relation ``$\\mathcal{U}$ refines $\\mathcal{V}$ and\n$\\mathcal{V}$ refines $\\mathcal{U}$'' is an equivalence relation on\nall families of morphisms with fixed target.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Families of morphisms with fixed target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VV","source_file":"sites.tex","source_line":1146,"source_end_line":1166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1146-L1166","statement_sha256":"f62c43f52dfed5b6946eaa2fe0c5fe2d00bca315fe1959a0f7f24df769706848","origin":"The Stacks Project","memory_eligible":false,"source_rank":561,"rank":561,"depth":0,"x":1185.231,"y":182.422,"cluster":"sheaves-sites"},{"id":"stacks:00VW","tag":"00VW","title":"Families of morphisms with fixed target · Lemma 00VW","summary":"Let C be a category. Let U = (φ_i : U_i → U)_i∈ I, and V = (ψ_j : V_j → U)_j∈ J be two families of morphisms with the same fixed target. Assume that the fibre products U_i ×_U U_i' and V_j ×_U V_j' exist. If U and V are tautologically equivalent, then for any presheaf F on C the sheaf condition for F with respect to U is equivalent to the sheaf condition for F with respect to V.","statement_latex":"Let $\\mathcal{C}$ be a category. Let\n$\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$, and\n$\\mathcal{V} = \\{\\psi_j : V_j \\to U\\}_{j\\in J}$ be two families of morphisms\nwith the same fixed target. Assume that the fibre products\n$U_i \\times_U U_{i'}$ and $V_j \\times_U V_{j'}$ exist.\nIf $\\mathcal{U}$ and $\\mathcal{V}$ are\ntautologically equivalent, then for any presheaf $\\mathcal{F}$ on\n$\\mathcal{C}$ the sheaf condition for $\\mathcal{F}$ with respect to\n$\\mathcal{U}$ is equivalent to the sheaf condition for $\\mathcal{F}$\nwith respect to $\\mathcal{V}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Families of morphisms with fixed target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VW","source_file":"sites.tex","source_line":1181,"source_end_line":1193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1181-L1193","statement_sha256":"5eae013de28429b88ca81091ff0998e7cdb6960cae69b695a67aa5790286a054","origin":"The Stacks Project","memory_eligible":false,"source_rank":562,"rank":562,"depth":0,"x":1119.452,"y":279.277,"cluster":"sheaves-sites"},{"id":"stacks:0G1K","tag":"0G1K","title":"Families of morphisms with fixed target · Lemma 0G1K","summary":"Let C be a category. Let V = (V_j → U)_j ∈ J → U = (U_i → U)_i ∈ I be a morphism of families of maps with fixed target of C given by id : U → U, α : J → I and f_j : V_j → U_α(j). Let F be a presheaf on C. If F(U) → ∏_j ∈ J F(V_j) is injective then F(U) → ∏_i ∈ I F(U_i) is injective.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J} \\to\n\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a morphism of families of maps\nwith fixed target of $\\mathcal{C}$ given by $\\text{id} : U \\to U$,\n$\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$. Let $\\mathcal{F}$\nbe a presheaf on $\\mathcal{C}$. If\n$\\mathcal{F}(U) \\to \\prod_{j \\in J} \\mathcal{F}(V_j)$ is\ninjective then\n$\\mathcal{F}(U) \\to \\prod_{i \\in I} \\mathcal{F}(U_i)$ is\ninjective.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Families of morphisms with fixed target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1K","source_file":"sites.tex","source_line":1228,"source_end_line":1239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1228-L1239","statement_sha256":"39812a418fda0716c323e8e3493bb85ee6b51db6da9abb23d27da94e9a7f406a","origin":"The Stacks Project","memory_eligible":false,"source_rank":563,"rank":563,"depth":0,"x":1089.957,"y":170.114,"cluster":"sheaves-sites"},{"id":"stacks:0G1L","tag":"0G1L","title":"Families of morphisms with fixed target · Lemma 0G1L","summary":"Let C be a category. Let V = (V_j → U)_j ∈ J → U = (U_i → U)_i ∈ I be a morphism of families of maps with fixed target of C given by id : U → U, α : J → I and f_j : V_j → U_α(j). Let F be a presheaf on C. If • the fibre products U_i ×_U U_i', U_i ×_U V_j, V_j ×_U V_j' exist, • F satisfies the sheaf condition with respect to V, and • for every i ∈ I the map F(U_i) → ∏_j ∈ J F(V_j ×_U U_i) is injective. Then F satisfies the sheaf condition with respect to U.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J} \\to\n\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a morphism of families of maps\nwith fixed target of $\\mathcal{C}$ given by $\\text{id} : U \\to U$,\n$\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$. Let $\\mathcal{F}$\nbe a presheaf on $\\mathcal{C}$. If\n\\begin{enumerate}\n\\item the fibre products $U_i \\times_U U_{i'}$, $U_i \\times_U V_j$,\n$V_j \\times_U V_{j'}$ exist,\n\\item $\\mathcal{F}$ satisfies the sheaf condition with respect to\n$\\mathcal{V}$, and\n\\item for every $i \\in I$ the map\n$\\mathcal{F}(U_i) \\to \\prod_{j \\in J} \\mathcal{F}(V_j \\times_U U_i)$\nis injective.\n\\end{enumerate}\nThen $\\mathcal{F}$ satisfies the sheaf condition with respect to $\\mathcal{U}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Families of morphisms with fixed target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1L","source_file":"sites.tex","source_line":1245,"source_end_line":1262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1245-L1262","statement_sha256":"8a0e363c3c8711ad018bf216fe8e718010bf5abb02d1aeb1f48879d666796131","origin":"The Stacks Project","memory_eligible":false,"source_rank":564,"rank":564,"depth":1,"x":1199.909,"y":234.117,"cluster":"sheaves-sites"},{"id":"stacks:00VX","tag":"00VX","title":"Families of morphisms with fixed target · Lemma 00VX","summary":"Let C be a category. Let Cov_i, i = 1, 2 be two sets of families of morphisms with fixed target which each define the structure of a site on C. • If every U ∈ Cov_1 is tautologically equivalent to some V ∈ Cov_2, then Sh(C, Cov_2) ⊂ Sh(C, Cov_1). If also, every U ∈ Cov_2 is tautologically equivalent to some V ∈ Cov_1 then the category of sheaves are equal. • Suppose that for each U ∈ Cov_1 there exists a V ∈ Cov_2 such that V refines U. In this case Sh(C, Cov_2) ⊂ Sh(C,…","statement_latex":"Let $\\mathcal{C}$ be a category. Let $\\text{Cov}_i$, $i = 1, 2$\nbe two sets of families of morphisms with fixed target which\neach define the structure of a site on $\\mathcal{C}$.\n\\begin{enumerate}\n\\item If every $\\mathcal{U} \\in \\text{Cov}_1$ is tautologically\nequivalent to some $\\mathcal{V} \\in \\text{Cov}_2$, then\n$\\Sh(\\mathcal{C}, \\text{Cov}_2) \\subset\n\\Sh(\\mathcal{C}, \\text{Cov}_1)$.\nIf also, every $\\mathcal{U} \\in \\text{Cov}_2$ is tautologically\nequivalent to some $\\mathcal{V} \\in \\text{Cov}_1$ then\nthe category of sheaves are equal.\n\\item Suppose\nthat for each $\\mathcal{U} \\in \\text{Cov}_1$ there exists a\n$\\mathcal{V} \\in \\text{Cov}_2$ such that $\\mathcal{V}$ refines\n$\\mathcal{U}$. In this case\n$\\Sh(\\mathcal{C}, \\text{Cov}_2) \\subset\n\\Sh(\\mathcal{C}, \\text{Cov}_1)$.\nIf also for every $\\mathcal{U} \\in \\text{Cov}_2$\nthere exists a $\\mathcal{V} \\in \\text{Cov}_1$ such that $\\mathcal{V}$\nrefines $\\mathcal{U}$, then the categories of sheaves\nare equal.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Families of morphisms with fixed target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VX","source_file":"sites.tex","source_line":1298,"source_end_line":1322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1298-L1322","statement_sha256":"70c5b958c32578fc4c0297692b6230ea1996e18f66ee6b7bbfa928f92d86bdf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":565,"rank":565,"depth":2,"x":1066.861,"y":249.369,"cluster":"sheaves-sites"},{"id":"stacks:00VY","tag":"00VY","title":"Families of morphisms with fixed target · Lemma 00VY","summary":"Let C be a category. Let Cov(C) be a proper class of coverings satisfying conditions (1), (2) and (3) of Definition [Tag 00VH]. Let Cov_1, Cov_2 ⊂ Cov(C) be two subsets of Cov(C) which endow C with the structure of a site. If every covering U ∈ Cov(C) is combinatorially equivalent to a covering in Cov_1 and combinatorially equivalent to a covering in Cov_2, then Sh(C, Cov_1) = Sh(C, Cov_2).","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\text{Cov}(\\mathcal{C})$ be a proper class of coverings\nsatisfying conditions (1), (2) and (3) of Definition \\ref{definition-site}.\nLet $\\text{Cov}_1, \\text{Cov}_2 \\subset \\text{Cov}(\\mathcal{C})$\nbe two subsets of $\\text{Cov}(\\mathcal{C})$ which endow\n$\\mathcal{C}$ with the structure of a site. If\nevery covering $\\mathcal{U} \\in \\text{Cov}(\\mathcal{C})$\nis combinatorially equivalent to a covering in\n$\\text{Cov}_1$ and combinatorially equivalent to a\ncovering in $\\text{Cov}_2$, then\n$\\Sh(\\mathcal{C}, \\text{Cov}_1) =\n\\Sh(\\mathcal{C}, \\text{Cov}_2)$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Families of morphisms with fixed target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00VY","source_file":"sites.tex","source_line":1342,"source_end_line":1356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1342-L1356","statement_sha256":"44347560f13547c5033f9b4a2005f8ef128d70e0a73d559ddf62d7134b00873d","origin":"The Stacks Project","memory_eligible":false,"source_rank":566,"rank":566,"depth":3,"x":1153.026,"y":162.302,"cluster":"sheaves-sites"},{"id":"stacks:00W0","tag":"00W0","title":"The example of G-sets · Proposition 00W0","summary":"The functors F ↦ F(_GG) and S ↦ F_S define quasi-inverse equivalences between Sh(T_G) and G-Sets.","statement_latex":"The functors $\\mathcal{F} \\mapsto \\mathcal{F}({}_GG)$\nand $S \\mapsto \\mathcal{F}_S$ define quasi-inverse\nequivalences between $\\Sh(\\mathcal{T}_G)$\nand $G\\textit{-Sets}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"The example of G-sets","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00W0","source_file":"sites.tex","source_line":1527,"source_end_line":1533,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1527-L1533","statement_sha256":"282f34bede978848ef7a0537756845d36c7800f835dd827473b1afd8f4791abd","origin":"The Stacks Project","memory_eligible":false,"source_rank":567,"rank":567,"depth":1,"x":1159.529,"y":275.818,"cluster":"sheaves-sites"},{"id":"stacks:00W2","tag":"00W2","title":"Sheafification · Lemma 00W2","summary":"Limit of diagram of sheaves exists and coincides with limit as presheaves Let F : I → Sh(C) be a diagram. Then lim_I F exists and is equal to the limit in the category of presheaves.","statement_latex":"\\begin{slogan}\nLimit of diagram of sheaves exists and coincides with limit as presheaves\n\\end{slogan}\nLet $\\mathcal{F} : \\mathcal{I} \\to \\Sh(\\mathcal{C})$\nbe a diagram. Then $\\lim_\\mathcal{I} \\mathcal{F}$ exists\nand is equal to the limit in the category of presheaves.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00W2","source_file":"sites.tex","source_line":1679,"source_end_line":1687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1679-L1687","statement_sha256":"047de3a2d67297baa67203ae2cfd186dfa21b075ffc792adc0632cc5a98b3008","origin":"The Stacks Project","memory_eligible":false,"source_rank":568,"rank":568,"depth":0,"x":1063.092,"y":195.506,"cluster":"sheaves-sites"},{"id":"stacks:00W4","tag":"00W4","title":"Sheafification · Lemma 00W4","summary":"The constructions above define a presheaf F^+ together with a canonical map of presheaves F → F^+.","statement_latex":"The constructions above define a presheaf\n$\\mathcal{F}^+$ together with a canonical\nmap of presheaves $\\mathcal{F} \\to \\mathcal{F}^+$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00W4","source_file":"sites.tex","source_line":1775,"source_end_line":1780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1775-L1780","statement_sha256":"1f66fd5f5798e9b7c58bbcefe7c02c990d32a58c4de67ab63eb24c9edca7cc78","origin":"The Stacks Project","memory_eligible":false,"source_rank":569,"rank":569,"depth":0,"x":1199.287,"y":200.023,"cluster":"sheaves-sites"},{"id":"stacks:00W5","tag":"00W5","title":"Sheafification · Lemma 00W5","summary":"The association F ↦ (F → F^+) is a functor.","statement_latex":"The association $\\mathcal{F} \\mapsto\n(\\mathcal{F} \\to \\mathcal{F}^+)$\nis a functor.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00W5","source_file":"sites.tex","source_line":1807,"source_end_line":1812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1807-L1812","statement_sha256":"3ee4bc609d2678db3198d6593a02a98cf8c76de3d9c8624f9d6b44d56752dc33","origin":"The Stacks Project","memory_eligible":false,"source_rank":570,"rank":570,"depth":0,"x":1094.846,"y":274.242,"cluster":"sheaves-sites"},{"id":"stacks:00W6","tag":"00W6","title":"Sheafification · Lemma 00W6","summary":"Given a pair of coverings (U_i → U) and (V_j → U) of a given object U of the site C, there exists a covering which is a common refinement.","statement_latex":"Given a pair of coverings $\\{U_i \\to U\\}$\nand $\\{V_j \\to U\\}$ of a given object $U$ of the site\n$\\mathcal{C}$, there exists a covering which is a\ncommon refinement.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00W6","source_file":"sites.tex","source_line":1835,"source_end_line":1841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1835-L1841","statement_sha256":"88c2f1be54e30cb7e59041c2da7840d60b744cdc21c948b813be7c4232c799a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":571,"rank":571,"depth":0,"x":1112.238,"y":159.839,"cluster":"sheaves-sites"},{"id":"stacks:00W7","tag":"00W7","title":"Sheafification · Lemma 00W7","summary":"Any two morphisms f, g: U → V of coverings inducing the same morphism U → V induce the same map H^0(V, F) → H^0(U, F).","statement_latex":"Any two morphisms $f, g: \\mathcal{U} \\to \\mathcal{V}$ of coverings\ninducing the same morphism $U \\to V$ induce the same\nmap $H^0(\\mathcal{V}, \\mathcal{F}) \\to  H^0(\\mathcal{U}, \\mathcal{F})$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00W7","source_file":"sites.tex","source_line":1851,"source_end_line":1856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1851-L1856","statement_sha256":"7ab74ae4d5b6401f79ff57576408b446ab6d29183a361b7f406d679197622a3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":572,"rank":572,"depth":0,"x":1191.697,"y":254.406,"cluster":"sheaves-sites"},{"id":"stacks:00W9","tag":"00W9","title":"Sheafification · Lemma 00W9","summary":"The map theta : F → F^+ has the following property: For every object U of C and every section s ∈ F^+(U) there exists a covering (U_i → U) such that s|_U_i is in the image of theta : F(U_i) → F^+(U_i).","statement_latex":"The map $\\theta : \\mathcal{F} \\to \\mathcal{F}^+$ has the following\nproperty: For every object $U$ of $\\mathcal{C}$ and every section\n$s \\in \\mathcal{F}^+(U)$ there exists a covering $\\{U_i \\to U\\}$\nsuch that $s|_{U_i}$ is in the image of $\\theta : \\mathcal{F}(U_i)\n\\to \\mathcal{F}^{+}(U_i)$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00W9","source_file":"sites.tex","source_line":1946,"source_end_line":1953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1946-L1953","statement_sha256":"d5bdb114b7aa379939e13040154b4ca342d1a153f8579751ee2245e8cc81dfbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":573,"rank":573,"depth":1,"x":1056.577,"y":229.672,"cluster":"sheaves-sites"},{"id":"stacks:00WA","tag":"00WA","title":"Sheafification · Definition 00WA","summary":"We say that a presheaf of sets F on a site C is separated if, for all coverings of (U_i → U), the map F(U) → ∏ F(U_i) is injective.","statement_latex":"We say that a presheaf of sets $\\mathcal{F}$ on a site\n$\\mathcal{C}$ is {\\it separated} if, for all coverings\nof $\\{U_i \\rightarrow U\\}$, the map\n$\\mathcal{F}(U) \\to \\prod \\mathcal{F}(U_i)$ is injective.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WA","source_file":"sites.tex","source_line":1966,"source_end_line":1972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1966-L1972","statement_sha256":"3877aa9fb70b534ca6dbd8f8a92c842c0c23a990ec5e36af23de9014fad68583","origin":"The Stacks Project","memory_eligible":false,"source_rank":574,"rank":574,"depth":0,"x":1176.526,"y":171.033,"cluster":"sheaves-sites"},{"id":"stacks:00WB","tag":"00WB","title":"Sheafification · Theorem 00WB","summary":"With F as above • The presheaf F^+ is separated. • If F is separated, then F^+ is a sheaf and the map of presheaves F → F^+ is injective. • If F is a sheaf, then F → F^+ is an isomorphism. • The presheaf F^++ is always a sheaf.","statement_latex":"With $\\mathcal{F}$ as above\n\\begin{enumerate}\n\\item\n\nThe presheaf $\\mathcal{F}^+$ is separated.\n\\item\n\nIf $\\mathcal{F}$ is separated, then $\\mathcal{F}^+$ is a sheaf\nand the map of presheaves $\\mathcal{F} \\to \\mathcal{F}^+$ is injective.\n\\item\n\nIf $\\mathcal{F}$ is a sheaf, then $\\mathcal{F} \\to \\mathcal{F}^+$\nis an isomorphism.\n\\item\n\nThe presheaf $\\mathcal{F}^{++}$ is always a sheaf.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WB","source_file":"sites.tex","source_line":1974,"source_end_line":1993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L1974-L1993","statement_sha256":"308ce0bf64d625164842b7c4790266bf6864d41a21833c8364cf78bc43c8ae19","origin":"The Stacks Project","memory_eligible":false,"source_rank":575,"rank":575,"depth":2,"x":1135.088,"y":282.726,"cluster":"sheaves-sites"},{"id":"stacks:00WG","tag":"00WG","title":"Sheafification · Definition 00WG","summary":"Let C be a site and let F be a presheaf of sets on C. The sheaf F^\\# := F^++ together with the canonical map F → F^\\# is called the sheaf associated to F.","statement_latex":"Let $\\mathcal{C}$ be a site and let $\\mathcal{F}$ be a presheaf\nof sets on $\\mathcal{C}$. The sheaf $\\mathcal{F}^\\# := \\mathcal{F}^{++}$\ntogether with the canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$\nis called the {\\it sheaf associated to $\\mathcal{F}$}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WG","source_file":"sites.tex","source_line":2044,"source_end_line":2050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2044-L2050","statement_sha256":"933625fd175b68fdb69bd7de7b4165499928150fb2a532e2ed2525a877ad1614","origin":"The Stacks Project","memory_eligible":false,"source_rank":576,"rank":576,"depth":0,"x":1075.617,"y":176.482,"cluster":"sheaves-sites"},{"id":"stacks:00WH","tag":"00WH","title":"Sheafification · Proposition 00WH","summary":"The canonical map F → F^\\# has the following universal property: For any map F → G, where G is a sheaf of sets, there is a unique map F^\\# → G such that F → F^\\# → G equals the given map.","statement_latex":"The canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ has the\nfollowing universal property: For any map $\\mathcal{F} \\to \\mathcal{G}$,\nwhere $\\mathcal{G}$ is a sheaf of sets, there is a unique map\n$\\mathcal{F}^\\# \\to \\mathcal{G}$ such that $\\mathcal{F} \\to \\mathcal{F}^\\#\n\\to \\mathcal{G}$ equals the given map.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WH","source_file":"sites.tex","source_line":2052,"source_end_line":2059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2052-L2059","statement_sha256":"356e6a32ba52b97f30382b2fa70076a14c83552ded8c3b1fed200b35215b1617","origin":"The Stacks Project","memory_eligible":false,"source_rank":577,"rank":577,"depth":3,"x":1205.356,"y":221.235,"cluster":"sheaves-sites"},{"id":"stacks:00WI","tag":"00WI","title":"Sheafification · Lemma 00WI","summary":"Colimit in category of sheaves equals sheafification of colimit in category of presheaves. Let F : I → Sh(C) be a diagram. Then colim_I F exists and is the sheafification of the colimit in the category of presheaves.","statement_latex":"\\begin{slogan}\nColimit in category of sheaves equals sheafification of colimit in\ncategory of presheaves.\n\\end{slogan}\nLet $\\mathcal{F} : \\mathcal{I} \\to \\Sh(\\mathcal{C})$\nbe a diagram. Then $\\colim_\\mathcal{I} \\mathcal{F}$ exists\nand is the sheafification of the colimit in the category of presheaves.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WI","source_file":"sites.tex","source_line":2113,"source_end_line":2122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2113-L2122","statement_sha256":"df67218cf0045f2225341cdd612f5a79a11da21e92343eff9312377722ec1728","origin":"The Stacks Project","memory_eligible":false,"source_rank":578,"rank":578,"depth":2,"x":1073.246,"y":261.992,"cluster":"sheaves-sites"},{"id":"stacks:00WJ","tag":"00WJ","title":"Sheafification · Lemma 00WJ","summary":"The functor PSh(C) → Sh(C), F ↦ F^\\# is exact.","statement_latex":"The functor $\\textit{PSh}(\\mathcal{C}) \\to \\Sh(\\mathcal{C})$,\n$\\mathcal{F} \\mapsto \\mathcal{F}^\\#$ is exact.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WJ","source_file":"sites.tex","source_line":2133,"source_end_line":2137,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2133-L2137","statement_sha256":"7729a88d1bbb67088b9d5b0f4096bc2feaef054b98005e03cd82d2a5abcbf393","origin":"The Stacks Project","memory_eligible":false,"source_rank":579,"rank":579,"depth":3,"x":1138.109,"y":156.617,"cluster":"sheaves-sites"},{"id":"stacks:00WK","tag":"00WK","title":"Sheafification · Lemma 00WK","summary":"Let C be a site. Let F be a presheaf of sets on C. Denote theta^2 : F → F^\\# the canonical map of F into its sheafification. Let U be an object of C. Let s ∈ F^\\#(U). There exists a covering (U_i → U) and sections s_i ∈ F(U_i) such that • s|_U_i = theta^2(s_i), and • for every i, j there exists a covering (U_ijk → U_i ×_U U_j) of C such that the pullbacks of s_i and s_j to each U_ijk agree. Conversely, given any covering (U_i → U), elements s_i ∈ F(U_i) such that (2)…","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$.\nDenote $\\theta^2 : \\mathcal{F} \\to \\mathcal{F}^\\#$ the canonical\nmap of $\\mathcal{F}$ into its sheafification.\nLet $U$ be an object of $\\mathcal{C}$.\nLet $s \\in \\mathcal{F}^\\#(U)$. There exists\na covering $\\{U_i \\to U\\}$ and sections\n$s_i \\in \\mathcal{F}(U_i)$ such that\n\\begin{enumerate}\n\\item $s|_{U_i} = \\theta^2(s_i)$, and\n\\item for every $i, j$ there exists a covering\n$\\{U_{ijk} \\to U_i \\times_U U_j\\}$ of $\\mathcal{C}$ such that\nthe pullbacks of $s_i$ and $s_j$ to each $U_{ijk}$ agree.\n\\end{enumerate}\nConversely, given any covering $\\{U_i \\to U\\}$, elements\n$s_i \\in \\mathcal{F}(U_i)$ such that (2) holds, then there\nexists a unique section $s \\in \\mathcal{F}^\\#(U)$ such\nthat (1) holds.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WK","source_file":"sites.tex","source_line":2155,"source_end_line":2175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2155-L2175","statement_sha256":"2acac84fd62e6a8c4a61d36cf2e38e8d13737b53803bc8ed1eda5aa15d14893c","origin":"The Stacks Project","memory_eligible":false,"source_rank":580,"rank":580,"depth":0,"x":1175.144,"y":271.513,"cluster":"sheaves-sites"},{"id":"stacks:0H6Y","tag":"0H6Y","title":"Sheafification · Lemma 0H6Y","summary":"Let C be a site. Let F → G be a map of presheaves of sets on C. Denote B the set of U ∈ Ob(C) such that F(U) → G(U) is bijective. If every object of C has a covering by elements of B, then F^\\# → G^\\# is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{F} \\to \\mathcal{G}$ be a map\nof presheaves of sets on $\\mathcal{C}$. Denote $\\mathcal{B}$ the set of\n$U \\in \\Ob(\\mathcal{C})$ such that $\\mathcal{F}(U) \\to \\mathcal{G}(U)$\nis bijective. If every object of $\\mathcal{C}$ has a covering by elements\nof $\\mathcal{B}$, then $\\mathcal{F}^\\# \\to \\mathcal{G}^\\#$ is an isomorphism.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6Y","source_file":"sites.tex","source_line":2181,"source_end_line":2188,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2181-L2188","statement_sha256":"ba64a9505f91dd4face525f89026cd996f920f69d839d0e46277f0f8704d984c","origin":"The Stacks Project","memory_eligible":false,"source_rank":581,"rank":581,"depth":1,"x":1055.035,"y":207.59,"cluster":"sheaves-sites"},{"id":"stacks:00WM","tag":"00WM","title":"Injective and surjective maps of sheaves · Definition 00WM","summary":"Let C be a site, and let φ : F → G be a map of sheaves of sets. • We say that φ is injective if for every object U of C the map φ : F(U) → G(U) is injective. • We say that φ is surjective if for every object U of C and every section s∈ G(U) there exists a covering (U_i → U) such that for all i the restriction s|_U_i is in the image of φ : F(U_i) → G(U_i).","statement_latex":"Let $\\mathcal{C}$ be a site, and let $\\varphi : \\mathcal{F}\n\\to \\mathcal{G}$ be a map of sheaves of sets.\n\\begin{enumerate}\n\\item We say that $\\varphi$ is {\\it injective} if for every object\n$U$ of $\\mathcal{C}$ the map $\\varphi : \\mathcal{F}(U)\n\\to \\mathcal{G}(U)$ is injective.\n\\item We say that $\\varphi$ is {\\it surjective} if for every object\n$U$ of $\\mathcal{C}$ and every section $s\\in \\mathcal{G}(U)$\nthere exists a covering $\\{U_i \\to U\\}$ such that for\nall $i$ the restriction $s|_{U_i}$ is in the image of\n$\\varphi : \\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Injective and surjective maps of sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WM","source_file":"sites.tex","source_line":2239,"source_end_line":2253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2239-L2253","statement_sha256":"3eff28f2c5b9fdc4f3ddef12543572b2618904e970d83d4fc0262699827af21f","origin":"The Stacks Project","memory_eligible":false,"source_rank":582,"rank":582,"depth":0,"x":1195.476,"y":186.503,"cluster":"sheaves-sites"},{"id":"stacks:00WN","tag":"00WN","title":"Injective and surjective maps of sheaves · Lemma 00WN","summary":"The injective (resp. surjective) maps defined above are exactly the monomorphisms (resp. epimorphisms) of the category Sh(C). A map of sheaves is an isomorphism if and only if it is both injective and surjective.","statement_latex":"The injective (resp.\\ surjective) maps defined above\nare exactly the monomorphisms (resp.\\ epimorphisms) of\nthe category $\\Sh(\\mathcal{C})$. A map of sheaves\nis an isomorphism if and only if it is both injective\nand surjective.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Injective and surjective maps of sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WN","source_file":"sites.tex","source_line":2255,"source_end_line":2262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2255-L2262","statement_sha256":"ab1fcc35e385dd4ee8033096cb14e55b6ac8b88adc8d8f4c8c4509b47bfc0211","origin":"The Stacks Project","memory_eligible":false,"source_rank":583,"rank":583,"depth":0,"x":1108.586,"y":282.058,"cluster":"sheaves-sites"},{"id":"stacks:086K","tag":"086K","title":"Injective and surjective maps of sheaves · Lemma 086K","summary":"Let C be a site. Let F → G be a surjection of sheaves of sets. Then the diagram xymatrix F ×_G F ar@<1ex>[r] ar@<-1ex>[r] & F ar[r] & G represents G as a coequalizer.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{F} \\to \\mathcal{G}$\nbe a surjection of sheaves of sets. Then the diagram\n$$\n\\xymatrix{\n\\mathcal{F} \\times_\\mathcal{G} \\mathcal{F}\n\\ar@<1ex>[r] \\ar@<-1ex>[r]\n&\n\\mathcal{F} \\ar[r]\n&\n\\mathcal{G}}\n$$\nrepresents $\\mathcal{G}$ as a coequalizer.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Injective and surjective maps of sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086K","source_file":"sites.tex","source_line":2268,"source_end_line":2282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2268-L2282","statement_sha256":"2d024e0fb68976cb1b26f009adbf42ebe9ff9b26c38598399a739aa4b92768d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":584,"rank":584,"depth":1,"x":1095.773,"y":161.897,"cluster":"sheaves-sites"},{"id":"stacks:00WP","tag":"00WP","title":"Representable sheaves · Definition 00WP","summary":"Let C be a category. We say that a family (U_i → U)_i ∈ I is an effective epimorphism if all the morphisms U_i → U are representable (see Categories, Definition [Tag 001X]), and for any X∈ Ob(C) the sequence xymatrix Mor_C(U, X) ar[r] & ∏_i ∈ I Mor_C(U_i, X) ar@<1ex>[r] ar@<-1ex>[r] & ∏_(i, j) ∈ I^2 Mor_C(U_i ×_U U_j, X) is an equalizer diagram. We say that a family (U_i → U) is a universal effective epimorphism if for any morphism V → U the base change (U_i ×_U V → V) is…","statement_latex":"Let $\\mathcal{C}$ be a category. We say that a family $\\{U_i \\to U\\}_{i \\in I}$\nis an {\\it effective epimorphism} if all the morphisms $U_i \\to U$ are\nrepresentable (see\nCategories, Definition \\ref{categories-definition-representable-morphism}),\nand for any $X\\in \\Ob(\\mathcal{C})$ the sequence\n$$\n\\xymatrix{\n\\Mor_\\mathcal{C}(U, X) \\ar[r]\n&\n\\prod\\nolimits_{i \\in I} \\Mor_\\mathcal{C}(U_i, X)\n\\ar@<1ex>[r] \\ar@<-1ex>[r]\n&\n\\prod\\nolimits_{(i, j) \\in I^2} \\Mor_\\mathcal{C}(U_i \\times_U U_j, X)\n}\n$$\nis an equalizer diagram. We say that a family $\\{U_i \\to U\\}$ is a\n{\\it universal effective epimorphism} if for any morphism $V \\to U$\nthe base change $\\{U_i \\times_U V \\to V\\}$ is an effective epimorphism.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Representable sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WP","source_file":"sites.tex","source_line":2329,"source_end_line":2349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2329-L2349","statement_sha256":"cf8af0ffa96929c84d973a0952c8b07dc2b6ef0e6d1e9f23914448fc39f4720c","origin":"The Stacks Project","memory_eligible":false,"source_rank":585,"rank":585,"depth":1,"x":1202.199,"y":243.499,"cluster":"sheaves-sites"},{"id":"stacks:00WQ","tag":"00WQ","title":"Representable sheaves · Definition 00WQ","summary":"We say that the topology on a site C is weaker than the canonical topology, or that the topology is subcanonical if all the coverings of C are universal effective epimorphisms.","statement_latex":"We say that the topology on a site $\\mathcal{C}$ is\n{\\it weaker than the canonical topology}, or that the topology is\n{\\it subcanonical} if all the coverings\nof $\\mathcal{C}$ are universal effective epimorphisms.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Representable sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WQ","source_file":"sites.tex","source_line":2365,"source_end_line":2371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2365-L2371","statement_sha256":"3e0ec03ffc0bbb34ea1c20f0ae63f88c76aea3bd1f7f9d98937b3dfc3fade138","origin":"The Stacks Project","memory_eligible":false,"source_rank":586,"rank":586,"depth":0,"x":1057.63,"y":243.716,"cluster":"sheaves-sites"},{"id":"stacks:00WR","tag":"00WR","title":"Representable sheaves · Definition 00WR","summary":"Let C be a site whose topology is subcanonical. The Yoneda embedding h (see Categories, Section [Tag 001L]) presents C as a full subcategory of the category of sheaves of C. In this case we call sheaves of the form h_U with U ∈ Ob(C) representable sheaves on C. Notation: Sometimes, the representable sheaf h_U associated to U is denoted underlineU.","statement_latex":"Let $\\mathcal{C}$ be a site whose topology is subcanonical.\nThe Yoneda embedding $h$ (see\nCategories, Section \\ref{categories-section-opposite})\npresents $\\mathcal{C}$ as a full subcategory of the\ncategory of sheaves of $\\mathcal{C}$. In this case\nwe call sheaves of the form $h_U$ with $U \\in \\Ob(\\mathcal{C})$\n{\\it representable sheaves} on $\\mathcal{C}$.\nNotation: Sometimes, the representable sheaf $h_U$ associated to $U$ is\ndenoted {\\it $\\underline{U}$}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Representable sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WR","source_file":"sites.tex","source_line":2378,"source_end_line":2389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2378-L2389","statement_sha256":"8185d0d8899df69c20547faedffda8b9b7e37d9f64f185b2457448b199b1e399","origin":"The Stacks Project","memory_eligible":false,"source_rank":587,"rank":587,"depth":0,"x":1164.404,"y":161.258,"cluster":"sheaves-sites"},{"id":"stacks:00WT","tag":"00WT","title":"Representable sheaves · Lemma 00WT","summary":"Coverings become surjective after sheafification. Let C be a site. If (U_i → U)_i ∈ I is a covering of the site C, then the morphism of presheaves of sets coprod_i ∈ I h_U_i → h_U becomes surjective after sheafification.","statement_latex":"\\begin{slogan}\nCoverings become surjective after sheafification.\n\\end{slogan}\nLet $\\mathcal{C}$ be a site. If\n$\\{U_i \\to U\\}_{i \\in I}$ is a covering of the site\n$\\mathcal{C}$, then the morphism of presheaves of sets\n$$\n\\coprod\\nolimits_{i \\in I} h_{U_i} \\to h_U\n$$\nbecomes surjective after sheafification.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Representable sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WT","source_file":"sites.tex","source_line":2410,"source_end_line":2422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2410-L2422","statement_sha256":"6f0c7b21b83e019018d47ca9465edab2706a462de7d72054cf57068db1a5ea65","origin":"The Stacks Project","memory_eligible":false,"source_rank":588,"rank":588,"depth":1,"x":1151.94,"y":283.038,"cluster":"sheaves-sites"},{"id":"stacks:00WS","tag":"00WS","title":"Representable sheaves · Lemma 00WS","summary":"Let C be a site. Let E ⊂ Ob(C) be a subset such that every object of C has a covering by elements of E. Let F be a sheaf of sets. There exists a diagram of sheaves of sets xymatrix F_1 ar@<1ex>[r] ar@<-1ex>[r] & F_0 ar[r] & F which represents F as a coequalizer, such that F_i, i = 0, 1 are coproducts of sheaves of the form h_U^\\# with U ∈ E.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $E \\subset \\Ob(\\mathcal{C})$ be a\nsubset such that every object of $\\mathcal{C}$ has a covering by\nelements of $E$. Let $\\mathcal{F}$ be a sheaf of sets. There exists a\ndiagram of sheaves of sets\n$$\n\\xymatrix{\n\\mathcal{F}_1 \\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\mathcal{F}_0 \\ar[r] &\n\\mathcal{F}\n}\n$$\nwhich represents $\\mathcal{F}$ as a coequalizer,\nsuch that $\\mathcal{F}_i$, $i = 0, 1$ are coproducts\nof sheaves of the form $h_U^\\#$ with $U \\in E$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Representable sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WS","source_file":"sites.tex","source_line":2440,"source_end_line":2456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2440-L2456","statement_sha256":"ffec4d1c1d9e7d356a27db8bb02d3bed51f8bc3ff67f46b38580e7de50b9687e","origin":"The Stacks Project","memory_eligible":false,"source_rank":589,"rank":589,"depth":2,"x":1062.915,"y":185.857,"cluster":"sheaves-sites"},{"id":"stacks:0GLW","tag":"0GLW","title":"Representable sheaves · Lemma 0GLW","summary":"Let C be a site. Let F be a sheaf of sets on C. Then there exists a diagram I → C, i ↦ U_i such that F = colim_i ∈ I h_U_i^\\# Moreover, if E ⊂ Ob(C) is a subset such that every object of C has a covering by elements of E, then we may assume U_i is an element of E for all i ∈ Ob(I).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf of sets on\n$\\mathcal{C}$. Then there exists a diagram $\\mathcal{I} \\to \\mathcal{C}$,\n$i \\mapsto U_i$ such that\n$$\n\\mathcal{F} = \\colim_{i \\in \\mathcal{I}} h_{U_i}^\\#\n$$\nMoreover, if $E \\subset \\Ob(\\mathcal{C})$ is a subset such that every\nobject of $\\mathcal{C}$ has a covering by elements of $E$, then we may\nassume $U_i$ is an element of $E$ for all $i \\in \\Ob(\\mathcal{I})$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Representable sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLW","source_file":"sites.tex","source_line":2475,"source_end_line":2486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2475-L2486","statement_sha256":"9139e653f5547294f29565a9e991289a61198768f047c187366a910e6dbc8fda","origin":"The Stacks Project","memory_eligible":false,"source_rank":590,"rank":590,"depth":4,"x":1207.168,"y":207.07,"cluster":"sheaves-sites"},{"id":"stacks:00WV","tag":"00WV","title":"Continuous functors · Definition 00WV","summary":"Let C and D be sites. A functor u : C → D is called continuous if for every (V_i → V)_i∈ I ∈ Cov(C) we have the following • (u(V_i) → u(V))_i∈ I is in Cov(D), and • for any morphism T → V in C the morphism u(T ×_V V_i) → u(T) ×_u(V) u(V_i) is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nA functor $u : \\mathcal{C} \\to \\mathcal{D}$ is called\n{\\it continuous} if for every\n$\\{V_i \\to V\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$\nwe have the following\n\\begin{enumerate}\n\\item $\\{u(V_i) \\to u(V)\\}_{i\\in I}$ is in $\\text{Cov}(\\mathcal{D})$, and\n\\item for any morphism $T \\to V$ in $\\mathcal{C}$ the morphism\n$u(T \\times_V V_i) \\to u(T) \\times_{u(V)} u(V_i)$ is an isomorphism.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Continuous functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WV","source_file":"sites.tex","source_line":2523,"source_end_line":2535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2523-L2535","statement_sha256":"3b84361d034d631856513df63d20820c24550133e7862a260dbd85cb9cb222ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":591,"rank":591,"depth":0,"x":1083.349,"y":273.489,"cluster":"sheaves-sites"},{"id":"stacks:00WW","tag":"00WW","title":"Continuous functors · Lemma 00WW","summary":"Let C and D be sites. Let u : C → D be a continuous functor. If F is a sheaf on D then u^pF is a sheaf as well.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor.\nIf $\\mathcal{F}$ is a sheaf on $\\mathcal{D}$ then\n$u^p\\mathcal{F}$ is a sheaf as well.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Continuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WW","source_file":"sites.tex","source_line":2547,"source_end_line":2553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2547-L2553","statement_sha256":"b3c2ffcf2e300a6e77c91451effb8d777da7683c9150c8f5bde04415d8d50e14","origin":"The Stacks Project","memory_eligible":false,"source_rank":592,"rank":592,"depth":0,"x":1121.358,"y":153.881,"cluster":"sheaves-sites"},{"id":"stacks:00WX","tag":"00WX","title":"Continuous functors · Lemma 00WX","summary":"In the situation of Lemma [Tag 00WW]. The functor u_s : G ↦ (u_p G)^\\# is a left adjoint to u^s.","statement_latex":"In the situation of Lemma \\ref{lemma-pushforward-sheaf}.\nThe functor $u_s : \\mathcal{G} \\mapsto (u_p \\mathcal{G})^\\#$\nis a left adjoint to $u^s$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Continuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WX","source_file":"sites.tex","source_line":2578,"source_end_line":2583,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2578-L2583","statement_sha256":"6935cbad6d7c03c8328881998f7ed23578e62cb8d509ebf57b9e28ef0ee9efcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":593,"rank":593,"depth":4,"x":1189.729,"y":263.988,"cluster":"sheaves-sites"},{"id":"stacks:00WY","tag":"00WY","title":"Continuous functors · Lemma 00WY","summary":"In the situation of Lemma [Tag 00WW]. For any presheaf G on C we have (u_pG)^\\# = (u_p(G^\\#))^\\#.","statement_latex":"In the situation of Lemma \\ref{lemma-pushforward-sheaf}.\nFor any presheaf $\\mathcal{G}$ on $\\mathcal{C}$\nwe have $(u_p\\mathcal{G})^\\# = (u_p(\\mathcal{G}^\\#))^\\#$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Continuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00WY","source_file":"sites.tex","source_line":2593,"source_end_line":2598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2593-L2598","statement_sha256":"2aaabb1ca6e5b303c4f61349ba89f5a8bf865586816de16a5eb65f6d3d6599e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":594,"rank":594,"depth":1,"x":1050.338,"y":221.46,"cluster":"sheaves-sites"},{"id":"stacks:04D3","tag":"04D3","title":"Continuous functors · Lemma 04D3","summary":"Let u : C → D be a continuous functor between sites. For any object U of C we have u_sh_U^\\# = h_u(U)^\\#.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor\nbetween sites.\nFor any object $U$ of $\\mathcal{C}$ we have $u_sh_U^\\# = h_{u(U)}^\\#$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Continuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04D3","source_file":"sites.tex","source_line":2618,"source_end_line":2623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2618-L2623","statement_sha256":"a430af55e48c7a25680c185944272af9ae80f859ca1255549525d42fb883991a","origin":"The Stacks Project","memory_eligible":false,"source_rank":595,"rank":595,"depth":2,"x":1187.744,"y":173.579,"cluster":"sheaves-sites"},{"id":"stacks:00X1","tag":"00X1","title":"Morphisms of sites · Definition 00X1","summary":"Let C and D be sites. A morphism of sites f : D → C is given by a continuous functor u : C → D such that the functor u_s is exact.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nA {\\it morphism of sites} $f : \\mathcal{D} \\to \\mathcal{C}$\nis given by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$\nsuch that the functor $u_s$ is exact.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of sites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00X1","source_file":"sites.tex","source_line":2686,"source_end_line":2692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2686-L2692","statement_sha256":"fd7925aa25cae99e78f0b325c958c6336ac142b7ede12c94f5f01eb7267e38a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":596,"rank":596,"depth":0,"x":1124.735,"y":287.2,"cluster":"sheaves-sites"},{"id":"stacks:03CB","tag":"03CB","title":"Morphisms of sites · Lemma 03CB","summary":"Composition of site functors respects continuity of site functors. Let C_i, i = 1, 2, 3 be sites. Let u : C_2 → C_1 and v : C_3 → C_2 be continuous functors which induce morphisms of sites. Then the functor u ∘ v : C_3 → C_1 is continuous and defines a morphism of sites C_1 → C_3.","statement_latex":"\\begin{slogan}\nComposition of site functors respects continuity of site functors.\n\\end{slogan}\nLet $\\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let\n$u : \\mathcal{C}_2 \\to \\mathcal{C}_1$ and\n$v : \\mathcal{C}_3 \\to \\mathcal{C}_2$ be continuous functors\nwhich induce morphisms of sites. Then the functor\n$u \\circ v : \\mathcal{C}_3 \\to \\mathcal{C}_1$ is continuous and\ndefines a morphism of sites $\\mathcal{C}_1 \\to \\mathcal{C}_3$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CB","source_file":"sites.tex","source_line":2763,"source_end_line":2774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2763-L2774","statement_sha256":"07241e899f814695d07f6dc07f1fca50ca998041ea407f2bbbc5fdd70353f283","origin":"The Stacks Project","memory_eligible":false,"source_rank":597,"rank":597,"depth":0,"x":1079.69,"y":167.301,"cluster":"sheaves-sites"},{"id":"stacks:03CC","tag":"03CC","title":"Morphisms of sites · Definition 03CC","summary":"Let C_i, i = 1, 2, 3 be sites. Let f : C_1 → C_2 and g : C_2 → C_3 be morphisms of sites given by continuous functors u : C_2 → C_1 and v : C_3 → C_2. The composition g ∘ f is the morphism of sites corresponding to the functor u ∘ v.","statement_latex":"Let $\\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let\n$f : \\mathcal{C}_1 \\to \\mathcal{C}_2$ and\n$g : \\mathcal{C}_2 \\to \\mathcal{C}_3$ be morphisms of sites\ngiven by continuous functors $u : \\mathcal{C}_2 \\to \\mathcal{C}_1$\nand $v : \\mathcal{C}_3 \\to \\mathcal{C}_2$. The {\\it composition}\n$g \\circ f$ is the morphism of sites corresponding to the\nfunctor $u \\circ v$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of sites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CC","source_file":"sites.tex","source_line":2785,"source_end_line":2794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2785-L2794","statement_sha256":"b7b3e0bb37c9d1f6313ee6031d8cc89df7be1bd34e216f85eba7e9261cc78fa3","origin":"The Stacks Project","memory_eligible":false,"source_rank":598,"rank":598,"depth":0,"x":1209.716,"y":230.344,"cluster":"sheaves-sites"},{"id":"stacks:00X5","tag":"00X5","title":"Morphisms of sites · Lemma 00X5","summary":"Let C and D be sites. Let u : C → D be continuous. Assume all the categories (I_V^u)^opp of Section [Tag 00VC] are filtered. Then u defines a morphism of sites D → C, in other words u_s is exact.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let\n$u : \\mathcal{C} \\to \\mathcal{D}$ be continuous.\nAssume all the categories $(\\mathcal{I}_V^u)^{opp}$ of\nSection \\ref{section-functoriality-PSh}\nare filtered. Then $u$ defines a morphism of sites $\\mathcal{D} \\to\n\\mathcal{C}$, in other words $u_s$ is exact.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00X5","source_file":"sites.tex","source_line":2801,"source_end_line":2809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2801-L2809","statement_sha256":"93f15e48acc8ff39307a3eb79412ea7ad02d4c069e44ec347bb29ba81f111f66","origin":"The Stacks Project","memory_eligible":false,"source_rank":599,"rank":599,"depth":4,"x":1062.699,"y":257.718,"cluster":"sheaves-sites"},{"id":"stacks:00X6","tag":"00X6","title":"Morphisms of sites · Proposition 00X6","summary":"Let C and D be sites. Let u : C → D be continuous. Assume furthermore the following: • the category C has a final object X and u(X) is a final object of D , and • the category C has fibre products and u commutes with them. Then u defines a morphism of sites D → C, in other words u_s is exact.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let\n$u : \\mathcal{C} \\to \\mathcal{D}$ be continuous.\nAssume furthermore the following:\n\\begin{enumerate}\n\\item the category $\\mathcal{C}$ has a final object $X$ and\n$u(X)$ is a final object of $\\mathcal{D}$ , and\n\\item the category $\\mathcal{C}$ has fibre products and\n$u$ commutes with them.\n\\end{enumerate}\nThen $u$ defines a morphism of sites $\\mathcal{D} \\to\n\\mathcal{C}$, in other words $u_s$ is exact.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of sites","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00X6","source_file":"sites.tex","source_line":2826,"source_end_line":2839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2826-L2839","statement_sha256":"27e59d21e475650a332985eeee2597decd28ceb83597b639fd8b0fc421f977d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":600,"rank":600,"depth":5,"x":1149.352,"y":153.802,"cluster":"sheaves-sites"},{"id":"stacks:08H2","tag":"08H2","title":"Morphisms of sites · Lemma 08H2","summary":"Let f : D → C be a morphism of sites given by the functor u : C → D. Given any object V of D there exists a covering (V_j → V) such that for every j there exists a morphism V_j → u(U_j) for some object U_j of C.","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given by the\nfunctor $u : \\mathcal{C} \\to \\mathcal{D}$. Given any object $V$ of\n$\\mathcal{D}$ there exists a covering $\\{V_j \\to V\\}$ such that for every\n$j$ there exists a morphism $V_j \\to u(U_j)$ for some object $U_j$\nof $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08H2","source_file":"sites.tex","source_line":2884,"source_end_line":2891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2884-L2891","statement_sha256":"0fdff8ca75b93153afaf72b1b8b540532be6be2be8d30648689c0690660bdaf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":601,"rank":601,"depth":0,"x":1169.08,"y":279.971,"cluster":"sheaves-sites"},{"id":"stacks:00XA","tag":"00XA","title":"Topoi · Definition 00XA","summary":"A topos is the category Sh(C) of sheaves on a site C. • Let C, D be sites. A morphism of topoi f from Sh(D) to Sh(C) is given by a pair of functors f_* : Sh(D) → Sh(C) and f^-1 : Sh(C) → Sh(D) such that • we have Mor_Sh(D)(f^-1G, F) = Mor_Sh(C)(G, f_*F) bifunctorially, and • the functor f^-1 commutes with finite limits, i.e., is left exact. • Let C, D, E be sites. Given morphisms of topoi f :Sh(D) → Sh(C) and g :Sh(E) → Sh(D) the composition f∘ g is the morphism of topoi…","statement_latex":"A {\\it topos} is the category $\\Sh(\\mathcal{C})$ of sheaves\non a site $\\mathcal{C}$.\n\\begin{enumerate}\n\\item Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nA {\\it morphism of topoi} $f$ from $\\Sh(\\mathcal{D})$\nto $\\Sh(\\mathcal{C})$ is given by a pair of functors\n$f_* : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$\nand\n$f^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nsuch that\n\\begin{enumerate}\n\\item we have\n$$\n\\Mor_{\\Sh(\\mathcal{D})}(f^{-1}\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, f_*\\mathcal{F})\n$$\nbifunctorially, and\n\\item the functor $f^{-1}$ commutes with finite limits, i.e.,\nis left exact.\n\\end{enumerate}\n\\item Let $\\mathcal{C}$, $\\mathcal{D}$, $\\mathcal{E}$ be sites.\nGiven morphisms of topoi\n$f :\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ and\n$g :\\Sh(\\mathcal{E}) \\to \\Sh(\\mathcal{D})$ the\n{\\it composition $f\\circ g$} is the morphism of topoi defined\nby the functors\n$(f \\circ g)_* = f_* \\circ g_*$ and\n$(f \\circ g)^{-1} = g^{-1} \\circ f^{-1}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XA","source_file":"sites.tex","source_line":2941,"source_end_line":2973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L2941-L2973","statement_sha256":"bf369499cbfc93fee160966b4096dbc7142e90574f22d3b292faa17957d56f13","origin":"The Stacks Project","memory_eligible":false,"source_rank":602,"rank":602,"depth":0,"x":1052.73,"y":197.889,"cluster":"sheaves-sites"},{"id":"stacks:00XC","tag":"00XC","title":"Topoi · Lemma 00XC","summary":"Given a morphism of sites f : D → C corresponding to the functor u : C → D the pair of functors (f^-1 = u_s, f_* = u^s) is a morphism of topoi Sh(D) → Sh(C).","statement_latex":"Given a morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$\ncorresponding to the functor $u : \\mathcal{C} \\to \\mathcal{D}$\nthe pair of functors $(f^{-1} = u_s, f_* = u^s)$ is a morphism of topoi\n$\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XC","source_file":"sites.tex","source_line":3015,"source_end_line":3021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3015-L3021","statement_sha256":"849477bfe742754f7af182cfd0bf9c15908ff6f0eb2bb83f60723b17e4447b36","origin":"The Stacks Project","memory_eligible":false,"source_rank":603,"rank":603,"depth":1,"x":1204.978,"y":192.379,"cluster":"sheaves-sites"},{"id":"stacks:090H","tag":"090H","title":"Quasi-compact objects and colimits · Definition 090H","summary":"Let C be a site. An object U of C is quasi-compact if given a covering U = (U_i → U)_i ∈ I in C there exists another covering V = (V_j → U)_j ∈ J and a morphism V → U of families of maps with fixed target given by id : U → U, α : J → I, and V_j → U_α(j) (see Definition [Tag 00VT]) such that the image of α is a finite subset of I.","statement_latex":"Let $\\mathcal{C}$ be a site. An object $U$ of $\\mathcal{C}$ is\n{\\it quasi-compact} if given a covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$\nin $\\mathcal{C}$ there exists another covering\n$\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J}$ and a morphism\n$\\mathcal{V} \\to \\mathcal{U}$ of families of maps with fixed target\ngiven by $\\text{id} : U \\to U$, $\\alpha : J \\to I$, and $V_j \\to U_{\\alpha(j)}$\n(see Definition \\ref{definition-morphism-coverings})\nsuch that the image of $\\alpha$ is a finite subset of $I$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Quasi-compact objects and colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090H","source_file":"sites.tex","source_line":3252,"source_end_line":3262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3252-L3262","statement_sha256":"6d831cf8f267514de7d100940bc46aefc8afb6b29e2774c1def579684de847bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":604,"rank":604,"depth":1,"x":1096.828,"y":283.094,"cluster":"sheaves-sites"},{"id":"stacks:0D05","tag":"0D05","title":"Quasi-compact objects and colimits · Lemma 0D05","summary":"Let C be a site. Let U be an object of C. Consider the following conditions • U is quasi-compact, • for every covering (U_i → U)_i ∈ I in C there exists a finite covering (V_j → U)_j = 1, …, m of C refining U, and • for every covering (U_i → U)_i ∈ I in C there exists a finite subset I' ⊂ I such that (U_i → U)_i ∈ I' is a covering in C. Then we always have (3) ⇒ (2) ⇒ (1) but the reverse implications do not hold in general.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.\nConsider the following conditions\n\\begin{enumerate}\n\\item $U$ is quasi-compact,\n\\item for every covering $\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$\nthere exists a finite covering $\\{V_j \\to U\\}_{j = 1, \\ldots, m}$\nof $\\mathcal{C}$ refining $\\mathcal{U}$, and\n\\item for every covering $\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$\nthere exists a finite subset $I' \\subset I$ such that\n$\\{U_i \\to U\\}_{i \\in I'}$ is a covering in $\\mathcal{C}$.\n\\end{enumerate}\nThen we always have (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1)\nbut the reverse implications do not hold in general.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Quasi-compact objects and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D05","source_file":"sites.tex","source_line":3268,"source_end_line":3283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3268-L3283","statement_sha256":"511e5b77af988906fe668cded5b85e6509f360b6038bcff4470dbee4b25edd77","origin":"The Stacks Project","memory_eligible":false,"source_rank":605,"rank":605,"depth":2,"x":1103.646,"y":154.464,"cluster":"sheaves-sites"},{"id":"stacks:0D06","tag":"0D06","title":"Quasi-compact objects and colimits · Lemma 0D06","summary":"Let C be a site. Let U be an object of C. The following are equivalent • U is quasi-compact, and • for every surjection of sheaves coprod_i ∈ I F_i → h_U^\\# there is a finite subset J ⊂ I such that coprod_i ∈ J F_i → h_U^\\# is surjective.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $U$ is quasi-compact, and\n\\item for every surjection of sheaves\n$\\coprod_{i \\in I} \\mathcal{F}_i \\to h_U^\\#$\nthere is a finite subset $J \\subset I$ such that\n$\\coprod_{i \\in J} \\mathcal{F}_i \\to h_U^\\#$ is surjective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Quasi-compact objects and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D06","source_file":"sites.tex","source_line":3303,"source_end_line":3314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3303-L3314","statement_sha256":"7959e5b158805a397b24c618b3c7823f8dfbaa1b5a9642477f1da0ec576400f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":606,"rank":606,"depth":3,"x":1202.343,"y":253.467,"cluster":"sheaves-sites"},{"id":"stacks:0D07","tag":"0D07","title":"Quasi-compact objects and colimits · Definition 0D07","summary":"An object F of a topos Sh(C) is quasi-compact if for any surjective map coprod_i ∈ I F_i → F of Sh(C) there exists a finite subset J ⊂ I such that coprod_i ∈ J F_i → F is surjective. A topos Sh(C) is said to be quasi-compact if its final object * is a quasi-compact object.","statement_latex":"An object $\\mathcal{F}$ of a topos $\\Sh(\\mathcal{C})$ is {\\it quasi-compact}\nif for any surjective map $\\coprod_{i \\in I} \\mathcal{F}_i \\to \\mathcal{F}$\nof $\\Sh(\\mathcal{C})$ there exists a finite subset $J \\subset I$ such\nthat $\\coprod_{i \\in J} \\mathcal{F}_i \\to \\mathcal{F}$ is surjective.\nA topos $\\Sh(\\mathcal{C})$ is said to be {\\it quasi-compact}\nif its final object $*$ is a quasi-compact object.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Quasi-compact objects and colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D07","source_file":"sites.tex","source_line":3367,"source_end_line":3375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3367-L3375","statement_sha256":"f2f25373979521eac17f01ce4cfc63b42141996cd2cd922a097230c226995d94","origin":"The Stacks Project","memory_eligible":false,"source_rank":607,"rank":607,"depth":0,"x":1049.513,"y":236.418,"cluster":"sheaves-sites"},{"id":"stacks:0GMP","tag":"0GMP","title":"Quasi-compact objects and colimits · Lemma 0GMP","summary":"Sheaf surjections transmit quasi-compactness. Let C be a site. • If U → V is a morphism of C such that h_U^\\# → h_V^\\# is surjective and U is quasi-compact, then V is quasi-compact. • If F → G is a surjection of sheaves of sets and F is quasi-compact, then G is quasi-compact.","statement_latex":"\\begin{slogan}\nSheaf surjections transmit quasi-compactness.\n\\end{slogan}\nLet $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item If $U \\to V$ is a morphism of $\\mathcal{C}$ such that\n$h_U^\\# \\to h_V^\\#$ is surjective and $U$ is quasi-compact, then\n$V$ is quasi-compact.\n\\item If $\\mathcal{F} \\to \\mathcal{G}$ is a surjection of sheaves\nof sets and $\\mathcal{F}$ is quasi-compact, then $\\mathcal{G}$\nis quasi-compact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Quasi-compact objects and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMP","source_file":"sites.tex","source_line":3382,"source_end_line":3396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3382-L3396","statement_sha256":"75e2a52cc8917103e999e651f4204127f468d683e4484973759184756459dd22","origin":"The Stacks Project","memory_eligible":false,"source_rank":608,"rank":608,"depth":0,"x":1176.278,"y":162.06,"cluster":"sheaves-sites"},{"id":"stacks:0GMQ","tag":"0GMQ","title":"Quasi-compact objects and colimits · Lemma 0GMQ","summary":"Finite copoducts of sheaves conserve quasi-compactness. Let C be a site. If n ≥ 1 and F_1, …, F_n are quasi-compact sheaves on C, then coprod_i = 1, …, n F_i is quasi-compact.","statement_latex":"\\begin{slogan}\nFinite copoducts of sheaves conserve quasi-compactness.\n\\end{slogan}\nLet $\\mathcal{C}$ be a site. If $n \\geq 1$ and\n$\\mathcal{F}_1, \\ldots, \\mathcal{F}_n$ are quasi-compact\nsheaves on $\\mathcal{C}$, then $\\coprod_{i = 1, \\ldots, n} \\mathcal{F}_i$\nis quasi-compact.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Quasi-compact objects and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMQ","source_file":"sites.tex","source_line":3402,"source_end_line":3411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3402-L3411","statement_sha256":"9591efba8c6764542a4737e33fe2f4e9c204a7140f6b43a29169c579517f8f91","origin":"The Stacks Project","memory_eligible":false,"source_rank":609,"rank":609,"depth":0,"x":1142.504,"y":289.179,"cluster":"sheaves-sites"},{"id":"stacks:0738","tag":"0738","title":"Quasi-compact objects and colimits · Lemma 0738","summary":"Let C be a site. Let I → Sh(C), i ↦ F_i be a filtered diagram of sheaves of sets. Let U ∈ Ob(C). Consider the canonical map Ψ : colim_i F_i(U) → (colim_i F_i)(U) With the terminology introduced above: • If all the transition maps are injective then Ψ is injective for any U. • If U is quasi-compact, then Ψ is injective. • If U is quasi-compact and all the transition maps are injective then Ψ is an isomorphism. • If U has a cofinal system of coverings (U_j → U)_j ∈ J with J…","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$\\mathcal{I} \\to \\Sh(\\mathcal{C})$, $i \\mapsto \\mathcal{F}_i$\nbe a filtered diagram of sheaves of sets.\nLet $U \\in \\Ob(\\mathcal{C})$.\nConsider the canonical map\n$$\n\\Psi :\n\\colim_i \\mathcal{F}_i(U)\n\\longrightarrow\n\\left(\\colim_i \\mathcal{F}_i\\right)(U)\n$$\nWith the terminology introduced above:\n\\begin{enumerate}\n\\item If all the transition maps are injective then\n$\\Psi$ is injective for any $U$.\n\\item If $U$ is quasi-compact, then $\\Psi$ is injective.\n\\item If $U$ is quasi-compact and all the transition maps are injective\nthen $\\Psi$ is an isomorphism.\n\\item If $U$ has a cofinal system of coverings\n$\\{U_j \\to U\\}_{j \\in J}$ with\n$J$ finite and $U_j \\times_U U_{j'}$ quasi-compact\nfor all $j, j' \\in J$, then $\\Psi$ is bijective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Quasi-compact objects and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0738","source_file":"sites.tex","source_line":3422,"source_end_line":3447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3422-L3447","statement_sha256":"c70a4861bda18df647e029c056a04a7e8d4818af613319a5eb47a3bbbb8727cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":610,"rank":610,"depth":3,"x":1064.966,"y":175.961,"cluster":"sheaves-sites"},{"id":"stacks:0GMR","tag":"0GMR","title":"Quasi-compact objects and colimits · Lemma 0GMR","summary":"Let C be a site. Let I → Sh(C), i ↦ F_i be a filtered diagram of sheaves of sets. Consider the canonical map Ψ : colim_i Γ(C, F_i) → Γ(C, colim_i F_i) We have the following: • If all the transition maps are injective then Ψ is injective. • If Sh(C) is quasi-compact, then Ψ is injective. • If Sh(C) is quasi-compact and all the transition maps are injective then Ψ is an isomorphism. • Assume there exists a set S ⊂ Ob(Sh(C)) with the following properties: • for every…","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$\\mathcal{I} \\to \\Sh(\\mathcal{C})$, $i \\mapsto \\mathcal{F}_i$\nbe a filtered diagram of sheaves of sets.\nConsider the canonical map\n$$\n\\Psi :\n\\colim_i \\Gamma(\\mathcal{C}, \\mathcal{F}_i)\n\\longrightarrow\n\\Gamma(\\mathcal{C}, \\colim_i \\mathcal{F}_i)\n$$\nWe have the following:\n\\begin{enumerate}\n\\item If all the transition maps are injective then $\\Psi$ is injective.\n\\item If $\\Sh(\\mathcal{C})$ is quasi-compact, then $\\Psi$ is injective.\n\\item If $\\Sh(\\mathcal{C})$ is quasi-compact and all the transition maps\nare injective then $\\Psi$ is an isomorphism.\n\\item Assume there exists a set $S \\subset \\Ob(\\Sh(\\mathcal{C}))$\nwith the following properties:\n\\begin{enumerate}\n\\item for every surjection $\\mathcal{F} \\to *$ there exists a\n$\\mathcal{K} \\in S$ and a map $\\mathcal{K} \\to \\mathcal{F}$\nsuch that $\\mathcal{K} \\to *$ is surjective,\n\\item for $\\mathcal{K} \\in S$ the product\n$\\mathcal{K} \\times \\mathcal{K}$ is quasi-compact.\n\\end{enumerate}\nThen $\\Psi$ is bijective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Quasi-compact objects and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMR","source_file":"sites.tex","source_line":3514,"source_end_line":3543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3514-L3543","statement_sha256":"9f12c85a0883a33db48dff971798af22caeecf30020d280162466c7b9770c466","origin":"The Stacks Project","memory_eligible":false,"source_rank":611,"rank":611,"depth":3,"x":1213.603,"y":215.56,"cluster":"sheaves-sites"},{"id":"stacks:0GS0","tag":"0GS0","title":"Quasi-compact objects and colimits · Lemma 0GS0","summary":"Let C be a site. Let β be an ordinal. Let β → Sh(C), α ↦ F_α be a system of sheaves over β. For U ∈ Ob(C) consider the canonical map colim_α < β F_α(U) → (colim_α < β F_α )(U) If the cofinality of β is large enough, then this map is bijective for all U.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\beta$ be an ordinal.\nLet $\\beta \\to \\Sh(\\mathcal{C})$, $\\alpha \\mapsto \\mathcal{F}_\\alpha$\nbe a system of sheaves over $\\beta$. For $U \\in \\Ob(\\mathcal{C})$\nconsider the canonical map\n$$\n\\colim_{\\alpha < \\beta} \\mathcal{F}_\\alpha(U)\n\\longrightarrow\n\\left(\\colim_{\\alpha < \\beta} \\mathcal{F}_\\alpha \\right)(U)\n$$\nIf the cofinality of $\\beta$ is large enough, then this map\nis bijective for all $U$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Quasi-compact objects and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GS0","source_file":"sites.tex","source_line":3703,"source_end_line":3716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3703-L3716","statement_sha256":"e8543748f6abd49d54e7f056d3d1acaea2de0e59c0ceea1699d7c1ca87464903","origin":"The Stacks Project","memory_eligible":false,"source_rank":612,"rank":612,"depth":3,"x":1071.762,"y":270.853,"cluster":"sheaves-sites"},{"id":"stacks:09YL","tag":"09YL","title":"Colimits of sites · Lemma 09YL","summary":"In Situation [Tag 0A34] we can construct a site (C, Cov(C)) as follows • as a category C = colim C_i, and • Cov(C) is the union of the images of Cov(C_i) by u_i : C_i → C.","statement_latex":"In Situation \\ref{situation-inverse-limit-sites} we can construct\na site $(\\mathcal{C}, \\text{Cov}(\\mathcal{C}))$ as follows\n\\begin{enumerate}\n\\item as a category $\\mathcal{C} = \\colim \\mathcal{C}_i$, and\n\\item $\\text{Cov}(\\mathcal{C})$ is the union of the images\nof $\\text{Cov}(\\mathcal{C}_i)$ by $u_i : \\mathcal{C}_i \\to \\mathcal{C}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Colimits of sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YL","source_file":"sites.tex","source_line":3783,"source_end_line":3792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3783-L3792","statement_sha256":"328c0c955ec6becc9c937c1ea2ec277ff5da44a930c049834a267fb3b2429419","origin":"The Stacks Project","memory_eligible":false,"source_rank":613,"rank":613,"depth":1,"x":1132.054,"y":149.261,"cluster":"sheaves-sites"},{"id":"stacks:0A35","tag":"0A35","title":"Colimits of sites · Lemma 0A35","summary":"In Situation [Tag 0A34] let u_i : C_i → C be as constructed in Lemma [Tag 09YL]. Then u_i defines a morphism of sites f_i : C → C_i. For U_i ∈ Ob(C_i) and sheaf F on C_i we have f_i^-1F(u_i(U_i)) = colim_a : j → i f_a^-1F(u_a(U_i))","statement_latex":"In Situation \\ref{situation-inverse-limit-sites} let\n$u_i : \\mathcal{C}_i \\to \\mathcal{C}$ be as constructed in\nLemma \\ref{lemma-colimit-sites}. Then $u_i$ defines a morphism\nof sites $f_i : \\mathcal{C} \\to \\mathcal{C}_i$. For\n$U_i \\in \\Ob(\\mathcal{C}_i)$ and sheaf $\\mathcal{F}$ on $\\mathcal{C}_i$ we have\n\\begin{equation}\n\nf_i^{-1}\\mathcal{F}(u_i(U_i)) =\n\\colim_{a : j \\to i} f_a^{-1}\\mathcal{F}(u_a(U_i))\n\\end{equation}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Colimits of sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A35","source_file":"sites.tex","source_line":3858,"source_end_line":3870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3858-L3870","statement_sha256":"504f69be8672524e3de724c341d194afc819b89ec4ce4a276d2bce3f2abc3021","origin":"The Stacks Project","memory_eligible":false,"source_rank":614,"rank":614,"depth":3,"x":1185.523,"y":273.474,"cluster":"sheaves-sites"},{"id":"stacks:09YN","tag":"09YN","title":"Colimits of sites · Lemma 09YN","summary":"In Situation [Tag 0A34] assume given • a sheaf F_i on C_i for all i ∈ Ob(I), • for a : j → i a map φ_a : f_a^-1F_i → F_j of sheaves on C_j such that φ_c = φ_b ∘ f_b^-1φ_a whenever c = a ∘ b. Set F = colim f_i^-1F_i on the site C of Lemma [Tag 09YL]. Let i ∈ Ob(I) and X_i ∈ Ob(C_i). Then colim_a : j → i F_j(u_a(X_i)) = F(u_i(X_i))","statement_latex":"In Situation \\ref{situation-inverse-limit-sites} assume given\n\\begin{enumerate}\n\\item a sheaf $\\mathcal{F}_i$ on $\\mathcal{C}_i$ for all\n$i \\in \\Ob(\\mathcal{I})$,\n\\item for $a : j \\to i$ a map\n$\\varphi_a : f_a^{-1}\\mathcal{F}_i \\to \\mathcal{F}_j$\nof sheaves on $\\mathcal{C}_j$\n\\end{enumerate}\nsuch that $\\varphi_c = \\varphi_b \\circ f_b^{-1}\\varphi_a$\nwhenever $c = a \\circ b$. Set $\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$\non the site $\\mathcal{C}$ of Lemma \\ref{lemma-colimit-sites}.\nLet $i \\in \\Ob(\\mathcal{I})$ and $X_i \\in \\text{Ob}(\\mathcal{C}_i)$. Then\n$$\n\\colim_{a : j \\to i} \\mathcal{F}_j(u_a(X_i)) = \\mathcal{F}(u_i(X_i))\n$$","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Colimits of sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YN","source_file":"sites.tex","source_line":3965,"source_end_line":3982,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3965-L3982","statement_sha256":"6438f2f78e12e6a5539a32889638c3fb311d4eff7ed81ab6001190e7a0d3aef1","origin":"The Stacks Project","memory_eligible":false,"source_rank":615,"rank":615,"depth":4,"x":1045.827,"y":212.053,"cluster":"sheaves-sites"},{"id":"stacks:0EXJ","tag":"0EXJ","title":"Colimits of sites · Lemma 0EXJ","summary":"In Situation [Tag 0A34] assume we have a sheaf F on C. Then F = colim f_i^-1f_i, *F where the transition maps are f_j^-1φ_a for a : j → i where φ_a : f_a^-1f_i, *F → f_j, *F is a canonical map satisfying a cocycle condition as in Lemma [Tag 09YN].","statement_latex":"In Situation \\ref{situation-inverse-limit-sites} assume\nwe have a sheaf $\\mathcal{F}$ on $\\mathcal{C}$. Then\n$$\n\\mathcal{F} = \\colim f_i^{-1}f_{i, *}\\mathcal{F}\n$$\nwhere the transition maps are $f_j^{-1}\\varphi_a$\nfor $a : j \\to i$ where\n$\\varphi_a : f_a^{-1}f_{i, *}\\mathcal{F} \\to f_{j, *}\\mathcal{F}$\nis a canonical map\nsatisfying a cocycle condition as in Lemma \\ref{lemma-colimit}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Colimits of sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXJ","source_file":"sites.tex","source_line":3997,"source_end_line":4009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L3997-L4009","statement_sha256":"b9142743221f2016113ab6ebf069cb7f3f977c002d300c411b057d4e285e6c65","origin":"The Stacks Project","memory_eligible":false,"source_rank":616,"rank":616,"depth":5,"x":1198.645,"y":177.984,"cluster":"sheaves-sites"},{"id":"stacks:00XG","tag":"00XG","title":"More functoriality of presheaves · Lemma 00XG","summary":"There is a canonical map _puF(u(U)) → F(U), which is compatible with restriction maps.","statement_latex":"There is a canonical map\n${}_pu\\mathcal{F}(u(U)) \\to \\mathcal{F}(U)$,\nwhich is compatible with restriction maps.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"More functoriality of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XG","source_file":"sites.tex","source_line":4118,"source_end_line":4123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4118-L4123","statement_sha256":"07432ee58f5d04283ac1faee37bee15bb76dbb8a2907d7c13fa52af8c81b0f77","origin":"The Stacks Project","memory_eligible":false,"source_rank":617,"rank":617,"depth":0,"x":1113.125,"y":290.121,"cluster":"sheaves-sites"},{"id":"stacks:00XH","tag":"00XH","title":"More functoriality of presheaves · Lemma 00XH","summary":"The functor _pu is a right adjoint to the functor u^p. In other words the formula Mor_PSh(C)(u^pG, F) = Mor_PSh(D)(G, _puF) holds bifunctorially in F and G.","statement_latex":"The functor ${}_pu$ is a right adjoint to the functor $u^p$.\nIn other words the formula\n$$\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(u^p\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{PSh}(\\mathcal{D})}(\\mathcal{G}, {}_pu\\mathcal{F})\n$$\nholds bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"More functoriality of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XH","source_file":"sites.tex","source_line":4142,"source_end_line":4152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4142-L4152","statement_sha256":"ff346d88204fd312066fcf8915e70e02d5d55f98efa9b7329de4e7088ccbe81a","origin":"The Stacks Project","memory_eligible":false,"source_rank":618,"rank":618,"depth":2,"x":1085.936,"y":158.554,"cluster":"sheaves-sites"},{"id":"stacks:09VQ","tag":"09VQ","title":"More functoriality of presheaves · Lemma 09VQ","summary":"Let u : C → D and v : D → C be functors of categories. Assume that v is right adjoint to u. Then we have • u^ph_V = h_v(V) for any V in D, • the category I^v_U has an initial object, • the category _V^uI has a final object, • _pu = v^p, and • u^p = v_p.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ and $v : \\mathcal{D} \\to \\mathcal{C}$\nbe functors of categories. Assume that $v$ is right adjoint to $u$.\nThen we have\n\\begin{enumerate}\n\\item $u^ph_V = h_{v(V)}$ for any $V$ in $\\mathcal{D}$,\n\\item the category $\\mathcal{I}^v_U$ has an initial object,\n\\item the category ${}_V^u\\mathcal{I}$ has a final object,\n\\item ${}_pu = v^p$, and\n\\item $u^p = v_p$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"More functoriality of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VQ","source_file":"sites.tex","source_line":4183,"source_end_line":4195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4183-L4195","statement_sha256":"9f0f5b3c23b76e2ac251c5b4927284cd9072d0f301f6504233f31fd7a81c1646","origin":"The Stacks Project","memory_eligible":false,"source_rank":619,"rank":619,"depth":0,"x":1212.124,"y":240.361,"cluster":"sheaves-sites"},{"id":"stacks:00XJ","tag":"00XJ","title":"Cocontinuous functors · Definition 00XJ","summary":"Let C and D be sites. Let u : C → D be a functor. The functor u is called cocontinuous if for every U ∈ Ob(C) and every covering (V_j → u(U))_j ∈ J of D there exists a covering (U_i → U)_i∈ I of C such that the family of maps (u(U_i) → u(U))_i ∈ I refines the covering (V_j → u(U))_j ∈ J.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nThe functor $u$ is called {\\it cocontinuous}\nif for every $U \\in \\Ob(\\mathcal{C})$\nand every covering $\\{V_j \\to u(U)\\}_{j \\in J}$ of $\\mathcal{D}$\nthere exists a covering\n$\\{U_i \\to U\\}_{i\\in I}$ of $\\mathcal{C}$\nsuch that the family of maps $\\{u(U_i) \\to u(U)\\}_{i \\in I}$\nrefines the covering $\\{V_j \\to u(U)\\}_{j \\in J}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XJ","source_file":"sites.tex","source_line":4250,"source_end_line":4261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4250-L4261","statement_sha256":"10848527dc8eca2ee85bc2414ddedaf2a1f51360e61e9d6b69093115bd398714","origin":"The Stacks Project","memory_eligible":false,"source_rank":620,"rank":620,"depth":0,"x":1052.865,"y":251.669,"cluster":"sheaves-sites"},{"id":"stacks:00XK","tag":"00XK","title":"Cocontinuous functors · Lemma 00XK","summary":"Let C and D be sites. Let u : C → D be cocontinuous. Let F be a sheaf on C. Then _puF is a sheaf on D, which we will denote _suF.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be cocontinuous.\nLet $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.\nThen ${}_pu\\mathcal{F}$ is a sheaf on $\\mathcal{D}$,\nwhich we will denote ${}_su\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XK","source_file":"sites.tex","source_line":4267,"source_end_line":4274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4267-L4274","statement_sha256":"26f0152eda6feb01308cfef891f0931155f4763b17b83f0f1be59ddfc5d50b1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":621,"rank":621,"depth":3,"x":1161.494,"y":152.702,"cluster":"sheaves-sites"},{"id":"stacks:00XL","tag":"00XL","title":"Cocontinuous functors · Lemma 00XL","summary":"Let C and D be sites. Let u : C → D be cocontinuous. The functor Sh(D) → Sh(C), G ↦ (u^pG)^\\# is a left adjoint to the functor _su introduced in Lemma [Tag 00XK] above. Moreover, it is exact.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be cocontinuous.\nThe functor\n$\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$,\n$\\mathcal{G} \\mapsto (u^p\\mathcal{G})^\\#$\nis a left adjoint to the functor ${}_su$ introduced\nin Lemma \\ref{lemma-pu-sheaf} above. Moreover, it\nis exact.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XL","source_file":"sites.tex","source_line":4343,"source_end_line":4353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4343-L4353","statement_sha256":"522e967f031c0d0ccba2ac9658bb52fe483a10669b62d3d3bb723a776151087f","origin":"The Stacks Project","memory_eligible":false,"source_rank":622,"rank":622,"depth":4,"x":1160.977,"y":287.672,"cluster":"sheaves-sites"},{"id":"stacks:00XM","tag":"00XM","title":"Cocontinuous functors · Lemma 00XM","summary":"In the situation of Lemma [Tag 00XL]. For any presheaf G on D we have (u^pG)^\\# = (u^p(G^\\#))^\\#.","statement_latex":"In the situation of Lemma \\ref{lemma-exact-cocontinuous}.\nFor any presheaf $\\mathcal{G}$ on $\\mathcal{D}$\nwe have $(u^p\\mathcal{G})^\\# = (u^p(\\mathcal{G}^\\#))^\\#$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XM","source_file":"sites.tex","source_line":4386,"source_end_line":4391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4386-L4391","statement_sha256":"31b1188793369a11cf9c1592d0920a94531968e24c03d7c2bd4f05c5aac9527b","origin":"The Stacks Project","memory_eligible":false,"source_rank":623,"rank":623,"depth":5,"x":1052.536,"y":187.592,"cluster":"sheaves-sites"},{"id":"stacks:00XO","tag":"00XO","title":"Cocontinuous functors and morphisms of topoi · Lemma 00XO","summary":"Let C and D be sites. Let u : C → D be cocontinuous. The functors g_* = _su and g^-1 = (u^p )^\\# define a morphism of topoi g from Sh(C) to Sh(D).","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be cocontinuous.\nThe functors $g_* = {}_su$ and $g^{-1} = (u^p\\ )^\\#$\ndefine a morphism of topoi\n$g$ from  $\\Sh(\\mathcal{C})$ to $\\Sh(\\mathcal{D})$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XO","source_file":"sites.tex","source_line":4450,"source_end_line":4457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4450-L4457","statement_sha256":"3d64ec05b623b0a9e4a02e433604517b953f5fa5438cc241035bde7700869ce1","origin":"The Stacks Project","memory_eligible":false,"source_rank":624,"rank":624,"depth":5,"x":1213.398,"y":199.89,"cluster":"sheaves-sites"},{"id":"stacks:03L5","tag":"03L5","title":"Cocontinuous functors and morphisms of topoi · Lemma 03L5","summary":"Composition of site functors respects cocontinuity of site functors. Let u : C → D, and v : D → E be cocontinuous functors. Then v ∘ u is cocontinuous and we have h = g ∘ f where f : Sh(C) → Sh(D), resp. g : Sh(D) → Sh(E), resp. h : Sh(C) → Sh(E) is the morphism of topoi associated to u, resp. v, resp. v ∘ u.","statement_latex":"\\begin{slogan}\nComposition of site functors respects cocontinuity of site functors.\n\\end{slogan}\nLet $u : \\mathcal{C} \\to \\mathcal{D}$, and $v : \\mathcal{D} \\to \\mathcal{E}$\nbe cocontinuous functors. Then $v \\circ u$ is cocontinuous and we\nhave $h = g \\circ f$\nwhere $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$,\nresp.\\ $g : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{E})$,\nresp.\\ $h : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{E})$ is the\nmorphism of topoi associated to $u$, resp.\\ $v$, resp.\\ $v \\circ u$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03L5","source_file":"sites.tex","source_line":4463,"source_end_line":4475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4463-L4475","statement_sha256":"3fff1da6ff9d022db56275eb16f0305c2350f8957f270b920f3a3e5195d38ce9","origin":"The Stacks Project","memory_eligible":false,"source_rank":625,"rank":625,"depth":4,"x":1084.558,"y":282.312,"cluster":"sheaves-sites"},{"id":"stacks:00XR","tag":"00XR","title":"Cocontinuous functors and morphisms of topoi · Lemma 00XR","summary":"Sheafification is redundant in topoi morphisms associated to simultaneously continuous and cocontinuous site functors. Let C and D be sites. Let u : C → D be a functor. Assume that • [(a)] u is cocontinuous, and • [(b)] u is continuous. Let g : Sh(C) → Sh(D) be the associated morphism of topoi. Then • sheafification in the formula g^-1 = (u^p )^\\# is unnecessary, in other words g^-1(G)(U) = G(u(U)), • g^-1 has a left adjoint g_! = (u_p )^\\#, and • g^-1 commutes with…","statement_latex":"\\begin{slogan}\nSheafification is redundant in topoi morphisms associated to simultaneously\ncontinuous and cocontinuous site functors.\n\\end{slogan}\nLet $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item[(a)] $u$ is cocontinuous, and\n\\item[(b)] $u$ is continuous.\n\\end{enumerate}\nLet $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe the associated morphism of topoi. Then\n\\begin{enumerate}\n\\item sheafification in the formula $g^{-1} = (u^p\\ )^\\#$ is\nunnecessary, in other words $g^{-1}(\\mathcal{G})(U) = \\mathcal{G}(u(U))$,\n\\item $g^{-1}$ has a left adjoint $g_{!} = (u_p\\ )^\\#$, and\n\\item $g^{-1}$ commutes with arbitrary limits and colimits.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XR","source_file":"sites.tex","source_line":4577,"source_end_line":4598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4577-L4598","statement_sha256":"9239115d044fa48cfdc08902116d3b7cfb4ca3c58bed6289928c18c2d3918483","origin":"The Stacks Project","memory_eligible":false,"source_rank":626,"rank":626,"depth":5,"x":1113.356,"y":148.083,"cluster":"sheaves-sites"},{"id":"stacks:00XS","tag":"00XS","title":"Cocontinuous functors and morphisms of topoi · Lemma 00XS","summary":"Let C and D be sites. Let u : C → D be a functor. Assume that • [(a)] u is cocontinuous, • [(b)] u is continuous, and • [(c)] fibre products and equalizers exist in C and u commutes with them. In this case the functor g_! above commutes with fibre products and equalizers (and more generally with finite connected limits).","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item[(a)] $u$ is cocontinuous,\n\\item[(b)] $u$ is continuous, and\n\\item[(c)] fibre products and equalizers exist in $\\mathcal{C}$ and\n$u$ commutes with them.\n\\end{enumerate}\nIn this case the functor $g_!$ above commutes with fibre products and\nequalizers (and more generally with finite connected limits).","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XS","source_file":"sites.tex","source_line":4626,"source_end_line":4639,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4626-L4639","statement_sha256":"b995c90299de208fe8b24a0f8bb41676e3c6c607599cfde2179e915a1688cc18","origin":"The Stacks Project","memory_eligible":false,"source_rank":627,"rank":627,"depth":4,"x":1200.287,"y":263.698,"cluster":"sheaves-sites"},{"id":"stacks:00XT","tag":"00XT","title":"Cocontinuous functors and morphisms of topoi · Lemma 00XT","summary":"Let C and D be sites. Let u : C → D be a functor. Assume that • [(a)] u is cocontinuous, • [(b)] u is continuous, and • [(c)] u is fully faithful. For g_!, g^-1, g_* as above the canonical maps F → g^-1g_!F and g^-1g_*F → F are isomorphisms for all sheaves F on C.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item[(a)] $u$ is cocontinuous,\n\\item[(b)] $u$ is continuous, and\n\\item[(c)] $u$ is fully faithful.\n\\end{enumerate}\nFor $g_!, g^{-1}, g_*$ as above\nthe canonical maps $\\mathcal{F} \\to g^{-1}g_!\\mathcal{F}$\nand $g^{-1}g_*\\mathcal{F} \\to \\mathcal{F}$ are isomorphisms\nfor all sheaves $\\mathcal{F}$ on $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XT","source_file":"sites.tex","source_line":4662,"source_end_line":4676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4662-L4676","statement_sha256":"1aace4d4b9464e3a00fe535349e3a79d11fd87770d62008c77d63d35ef9bb9b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":628,"rank":628,"depth":6,"x":1042.81,"y":227.68,"cluster":"sheaves-sites"},{"id":"stacks:00XU","tag":"00XU","title":"Cocontinuous functors and morphisms of topoi · Lemma 00XU","summary":"Let C and D be sites. Let u : C → D be a functor. Assume that • [(a)] u is cocontinuous, • [(b)] u is continuous, • [(c)] u is fully faithful, • [(d)] fibre products exist in C and u commutes with them, and • [(e)] there exist final objects e_C ∈ Ob(C), e_D ∈ Ob(D) such that u(e_C) = e_D. Let g_!, g^-1, g_* be as above. Then, u defines a morphism of sites f : D → C with f_* = g^-1, f^-1 = g_!. The composition xymatrix Sh(C) ar[r]^g & Sh(D) ar[r]^f & Sh(C) is isomorphic to…","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item[(a)] $u$ is cocontinuous,\n\\item[(b)] $u$ is continuous,\n\\item[(c)] $u$ is fully faithful,\n\\item[(d)] fibre products exist in $\\mathcal{C}$ and $u$ commutes with them,\nand\n\\item[(e)] there exist final objects\n$e_\\mathcal{C} \\in \\Ob(\\mathcal{C})$,\n$e_\\mathcal{D} \\in \\Ob(\\mathcal{D})$ such that\n$u(e_\\mathcal{C}) = e_\\mathcal{D}$.\n\\end{enumerate}\nLet $g_!, g^{-1}, g_*$ be as above. Then, $u$ defines a morphism of sites\n$f : \\mathcal{D} \\to \\mathcal{C}$ with $f_* = g^{-1}$, $f^{-1} = g_!$.\nThe composition\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C}) \\ar[r]^g &\n\\Sh(\\mathcal{D}) \\ar[r]^f &\n\\Sh(\\mathcal{C})\n}\n$$\nis isomorphic to the identity morphism of the topos\n$\\Sh(\\mathcal{C})$. Moreover, the functor $f^{-1}$ is fully faithful.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XU","source_file":"sites.tex","source_line":4729,"source_end_line":4757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4729-L4757","statement_sha256":"6d7b2c8b888973aeec19da1535a97149b1cc596b78ee9810cd57b893a2c5fb3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":629,"rank":629,"depth":7,"x":1188.264,"y":164.724,"cluster":"sheaves-sites"},{"id":"stacks:00XX","tag":"00XX","title":"Cocontinuous functors which have a right adjoint · Lemma 00XX","summary":"Let C and D be sites. Let u : C → D, and v : D → C be functors. Assume that u is cocontinuous and that v is a right adjoint to u. Let g : Sh(C) → Sh(D) be the morphism of topoi associated to u, see Lemma [Tag 00XO]. Then • for a sheaf F on C the sheaf g_*F is equal to the presheaf v^pF, in other words, (g_*F)(V) = F(v(V)), and • for a sheaf G on D we have g^-1G = (v_pG)^\\#.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let\n$u : \\mathcal{C} \\to \\mathcal{D}$, and $v : \\mathcal{D} \\to \\mathcal{C}$\nbe functors. Assume that $u$ is cocontinuous\nand that $v$ is a right adjoint to $u$.\nLet $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be\nthe morphism of topoi associated to $u$, see\nLemma \\ref{lemma-cocontinuous-morphism-topoi}.\nThen\n\\begin{enumerate}\n\\item for a sheaf $\\mathcal{F}$ on $\\mathcal{C}$ the sheaf\n$g_*\\mathcal{F}$ is equal to the presheaf $v^p\\mathcal{F}$, in other words,\n$(g_*\\mathcal{F})(V) = \\mathcal{F}(v(V))$, and\n\\item for a sheaf $\\mathcal{G}$ on $\\mathcal{D}$ we have\n$g^{-1}\\mathcal{G} = (v_p\\mathcal{G})^\\#$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors which have a right adjoint","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XX","source_file":"sites.tex","source_line":4838,"source_end_line":4855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4838-L4855","statement_sha256":"584a2a29a7c6059f26ccfb8eda65f9054cb95a7db2ea39bf5ccae36d463fc2b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":630,"rank":630,"depth":6,"x":1131.492,"y":294.006,"cluster":"sheaves-sites"},{"id":"stacks:00XY","tag":"00XY","title":"Cocontinuous functors which have a right adjoint · Lemma 00XY","summary":"Notation and assumptions as in Lemma [Tag 00XX]. If in addition v is continuous then v defines a morphism of sites f : C → D whose associated morphism of topoi is equal to g.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-have-functor-other-way}.\nIf in addition $v$ is continuous then $v$ defines a morphism of sites\n$f : \\mathcal{C} \\to \\mathcal{D}$ whose associated morphism\nof topoi is equal to $g$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors which have a right adjoint","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00XY","source_file":"sites.tex","source_line":4879,"source_end_line":4885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4879-L4885","statement_sha256":"c0a57c58e0bed47ff38af99ac67ca8e91b793c91f9def30181bce120c795a964","origin":"The Stacks Project","memory_eligible":false,"source_rank":631,"rank":631,"depth":7,"x":1069.234,"y":166.141,"cluster":"sheaves-sites"},{"id":"stacks:09VR","tag":"09VR","title":"Cocontinuous functors which have a right adjoint · Lemma 09VR","summary":"Let C and D be sites and let v : D → C be a continuous functor. Assume v has a left adjoint u : C → D. Then • u is cocontinuous, • the results of Lemmas [Tag 00XX] and [Tag 00XY] hold. In particular, v defines a morphism of sites f : C → D.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites and let\n$v : \\mathcal{D} \\to \\mathcal{C}$ be a continuous functor.\nAssume $v$ has a left adjoint $u : \\mathcal{C} \\to \\mathcal{D}$. Then\n\\begin{enumerate}\n\\item $u$ is cocontinuous,\n\\item the results of Lemmas \\ref{lemma-have-functor-other-way} and\n\\ref{lemma-have-functor-other-way-morphism} hold.\n\\end{enumerate}\nIn particular, $v$ defines a morphism of sites\n$f : \\mathcal{C} \\to \\mathcal{D}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors which have a right adjoint","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VR","source_file":"sites.tex","source_line":4919,"source_end_line":4931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4919-L4931","statement_sha256":"d9e4fb5bd0f54bfe43c119026410d51d0d72e3c2cd50383e8e0b9d1dfd9dd6d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":632,"rank":632,"depth":8,"x":1218.34,"y":225.248,"cluster":"sheaves-sites"},{"id":"stacks:08NH","tag":"08NH","title":"Cocontinuous functors which have a left adjoint · Lemma 08NH","summary":"Let C and D be sites. Let g : Sh(C) → Sh(D) be the morphism of topoi associated to a continuous and cocontinuous functor u : C → D, see Lemmas [Tag 00XO] and [Tag 00XR]. • If w : D → C is a left adjoint to u, then • g_!F is the sheaf associated to the presheaf w^pF, and • g_! is exact. • if w is a continuous left adjoint, then g_! has a left adjoint. • If w is a cocontinuous left adjoint, then g_! = h^-1 and g^-1 = h_* where h : Sh(D) → Sh(C) is the morphism of topoi…","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let\n$g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be\nthe morphism of topoi associated to a continuous and cocontinuous functor\n$u : \\mathcal{C} \\to \\mathcal{D}$, see\nLemmas \\ref{lemma-cocontinuous-morphism-topoi} and\n\\ref{lemma-when-shriek}.\n\\begin{enumerate}\n\\item If $w : \\mathcal{D} \\to \\mathcal{C}$ is a left adjoint to $u$, then\n\\begin{enumerate}\n\\item $g_!\\mathcal{F}$ is the sheaf associated to the presheaf\n$w^p\\mathcal{F}$, and\n\\item $g_!$ is exact.\n\\end{enumerate}\n\\item if $w$ is a continuous left adjoint, then $g_!$\nhas a left adjoint.\n\\item If $w$ is a cocontinuous left adjoint, then $g_! = h^{-1}$ and\n$g^{-1} = h_*$ where $h : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ is\nthe morphism of topoi associated to $w$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Cocontinuous functors which have a left adjoint","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NH","source_file":"sites.tex","source_line":4982,"source_end_line":5003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L4982-L5003","statement_sha256":"f6a00704b6bd752cbe15cfad0f2258554ad0e785ba22111858eb9df095b2ed71","origin":"The Stacks Project","memory_eligible":false,"source_rank":633,"rank":633,"depth":7,"x":1060.467,"y":266.372,"cluster":"sheaves-sites"},{"id":"stacks:09YX","tag":"09YX","title":"Existence of lower shriek · Lemma 09YX","summary":"Let C, D be two sites. Let f : Sh(D) → Sh(C) be a morphism of topoi. Let E ⊂ Ob(D) be a subset such that • for V ∈ E there exists a sheaf G on C such that f^-1F(V) = Mor_Sh(C)(G, F) functorially for F in Sh(C), • every object of D has a covering by objects of E. Then f^-1 has a left adjoint f_!.","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be two sites.\nLet $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.\nLet $E \\subset \\Ob(\\mathcal{D})$ be a subset such that\n\\begin{enumerate}\n\\item for $V \\in E$ there exists a sheaf $\\mathcal{G}$\non $\\mathcal{C}$ such that $f^{-1}\\mathcal{F}(V) = \n\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, \\mathcal{F})$ functorially\nfor $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$,\n\\item every object of $\\mathcal{D}$ has a covering by objects of $E$.\n\\end{enumerate}\nThen $f^{-1}$ has a left adjoint $f_!$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Existence of lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YX","source_file":"sites.tex","source_line":5049,"source_end_line":5062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5049-L5062","statement_sha256":"01632639c111bb6e8df2db55424951f0c0fa4be70a8e5aace21a4ed81c7ac8d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":634,"rank":634,"depth":3,"x":1144.016,"y":146.17,"cluster":"sheaves-sites"},{"id":"stacks:00Y0","tag":"00Y0","title":"Localization · Definition 00Y0","summary":"Let C be a site. Let U ∈ Ob(C). • The site C/U is called the localization of the site C at the object U. • The morphism of topoi j_U : Sh(C/U) → Sh(C) is called the localization morphism. • The functor j_U* is called the direct image functor. • For a sheaf F on C the sheaf j_U^-1F is called the restriction of F to C/U. • For a sheaf G on C/U the sheaf j_U!G is called the extension of G by the empty set.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $U \\in \\Ob(\\mathcal{C})$.\n\\begin{enumerate}\n\\item The site $\\mathcal{C}/U$ is called the {\\it localization of\nthe site $\\mathcal{C}$ at the object $U$}.\n\\item The morphism of topoi\n$j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$\nis called the {\\it localization morphism}.\n\\item The functor $j_{U*}$ is called the {\\it direct image functor}.\n\\item For a sheaf $\\mathcal{F}$ on $\\mathcal{C}$ the sheaf\n$j_U^{-1}\\mathcal{F}$ is called the {\\it restriction of $\\mathcal{F}$\nto $\\mathcal{C}/U$}.\n\\item For a sheaf $\\mathcal{G}$ on $\\mathcal{C}/U$\nthe sheaf $j_{U!}\\mathcal{G}$ is called the\n{\\it extension of $\\mathcal{G}$ by the empty set}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Y0","source_file":"sites.tex","source_line":5188,"source_end_line":5206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5188-L5206","statement_sha256":"aec80acd55823a762b77994244a6b4b13469966662e8b9feb186984098f11cb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":635,"rank":635,"depth":0,"x":1179.157,"y":282.548,"cluster":"sheaves-sites"},{"id":"stacks:03CD","tag":"03CD","title":"Localization · Lemma 03CD","summary":"Let C be a site. Let U ∈ Ob(C). Let G be a presheaf on C/U. Then j_U!(G^\\#) is the sheaf associated to the presheaf V ↦ coprod_φ ∈ Mor_C(V, U) G(V xrightarrowφ U) with obvious restriction mappings.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $U \\in \\Ob(\\mathcal{C})$.\nLet $\\mathcal{G}$ be a presheaf on $\\mathcal{C}/U$.\nThen $j_{U!}(\\mathcal{G}^\\#)$ is the sheaf associated to the presheaf\n$$\nV\n\\longmapsto\n\\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}\n\\mathcal{G}(V \\xrightarrow{\\varphi} U)\n$$\nwith obvious restriction mappings.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CD","source_file":"sites.tex","source_line":5214,"source_end_line":5227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5214-L5227","statement_sha256":"2a723a8db865890a48af2101114c19940b156f5273789b8595c5e1c1dae37c6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":636,"rank":636,"depth":6,"x":1043.242,"y":201.727,"cluster":"sheaves-sites"},{"id":"stacks:03HU","tag":"03HU","title":"Localization · Lemma 03HU","summary":"Let C be a site. Let U ∈ Ob(C). Let X/U be an object of C/U. Then we have j_U!(h_X/U^\\#) = h_X^\\#.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $U \\in \\Ob(\\mathcal{C})$.\nLet $X/U$ be an object of $\\mathcal{C}/U$.\nThen we have $j_{U!}(h_{X/U}^\\#) = h_X^\\#$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HU","source_file":"sites.tex","source_line":5247,"source_end_line":5253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5247-L5253","statement_sha256":"4668a3595288ae93a72daf58873a42833ed982b8e862b454f0729ec5858ffb59","origin":"The Stacks Project","memory_eligible":false,"source_rank":637,"rank":637,"depth":7,"x":1208.86,"y":184.159,"cluster":"sheaves-sites"},{"id":"stacks:00Y1","tag":"00Y1","title":"Localization · Lemma 00Y1","summary":"Let C be a site. Let U ∈ Ob(C). The functor j_U! gives an equivalence of categories Sh(C/U) → Sh(C)/h_U^\\#","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $U \\in \\Ob(\\mathcal{C})$.\nThe functor $j_{U!}$ gives an equivalence of categories\n$$\n\\Sh(\\mathcal{C}/U)\n\\longrightarrow\n\\Sh(\\mathcal{C})/h_U^\\#\n$$","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Y1","source_file":"sites.tex","source_line":5276,"source_end_line":5286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5276-L5286","statement_sha256":"607c460888950ca6a14ef97982f311bd68e70be3700b613661f6acf03b20bd58","origin":"The Stacks Project","memory_eligible":false,"source_rank":638,"rank":638,"depth":7,"x":1100.601,"y":291.348,"cluster":"sheaves-sites"},{"id":"stacks:04BB","tag":"04BB","title":"Localization · Lemma 04BB","summary":"Let C be a site. Let U ∈ Ob(C). The functor j_U! commutes with fibre products and equalizers (and more generally finite connected limits). In particular, if F ⊂ F' in Sh(C/U), then j_U!F ⊂ j_U!F'.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.\nThe functor $j_{U!}$ commutes with fibre products and equalizers (and\nmore generally finite connected limits). In particular, if\n$\\mathcal{F} \\subset \\mathcal{F}'$ in $\\Sh(\\mathcal{C}/U)$, then\n$j_{U!}\\mathcal{F} \\subset j_{U!}\\mathcal{F}'$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BB","source_file":"sites.tex","source_line":5405,"source_end_line":5412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5405-L5412","statement_sha256":"b718dc42191f2a8021d592ad6c63892817f31c496b7a4ebefbb36a837fb9c3c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":639,"rank":639,"depth":8,"x":1094.216,"y":150.54,"cluster":"sheaves-sites"},{"id":"stacks:0E8E","tag":"0E8E","title":"Localization · Lemma 0E8E","summary":"Let C be a site. Let U ∈ Ob(C). The functor j_U! reflects injections and surjections.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.\nThe functor $j_{U!}$ reflects injections and surjections.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8E","source_file":"sites.tex","source_line":5427,"source_end_line":5431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5427-L5431","statement_sha256":"2da5d8943660cbbe5dc6495b8af7f349364fc150767314107333c77540a18d81","origin":"The Stacks Project","memory_eligible":false,"source_rank":640,"rank":640,"depth":8,"x":1212.442,"y":250.988,"cluster":"sheaves-sites"},{"id":"stacks:03EE","tag":"03EE","title":"Localization · Lemma 03EE","summary":"Let C be a site. Let U ∈ Ob(C). For any sheaf F on C we have j_U!j_U^-1F = F × h_U^\\#.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.\nFor any sheaf $\\mathcal{F}$ on $\\mathcal{C}$ we have\n$j_{U!}j_U^{-1}\\mathcal{F} = \\mathcal{F} \\times h_U^\\#$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EE","source_file":"sites.tex","source_line":5442,"source_end_line":5447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5442-L5447","statement_sha256":"e39862227ed3bd23d84db95e4e8499b7c767ac4cb1224aef7a28e4d2c9fa569c","origin":"The Stacks Project","memory_eligible":false,"source_rank":641,"rank":641,"depth":7,"x":1044.085,"y":243.986,"cluster":"sheaves-sites"},{"id":"stacks:03EH","tag":"03EH","title":"Localization · Lemma 03EH","summary":"Let C be a site. Let f : V → U be a morphism of C. Then there exists a commutative diagram xymatrix C/V ar[rd]_j_V ar[rr]_j & & C/U ar[ld]^j_U & C & of continuous and cocontinuous functors. The functor j : C/V → C/U, (a : W → V) ↦ (f ∘ a : W → U) is identified with the functor j_V/U : (C/U)/(V/U) → C/U via the identification (C/U)/(V/U) = C/V. Moreover we have j_V! = j_U! ∘ j_!, j_V^-1 = j^-1 ∘ j_U^-1, and j_V* = j_U* ∘ j_*.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $f : V \\to U$ be a morphism of $\\mathcal{C}$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n\\mathcal{C}/V \\ar[rd]_{j_V} \\ar[rr]_j & &\n\\mathcal{C}/U \\ar[ld]^{j_U} \\\\\n& \\mathcal{C} &\n}\n$$\nof continuous and cocontinuous functors.\nThe functor $j : \\mathcal{C}/V \\to \\mathcal{C}/U$,\n$(a : W \\to V) \\mapsto (f \\circ a : W \\to U)$\nis identified with the functor\n$j_{V/U} : (\\mathcal{C}/U)/(V/U) \\to \\mathcal{C}/U$\nvia the identification $(\\mathcal{C}/U)/(V/U) = \\mathcal{C}/V$.\nMoreover we have $j_{V!} = j_{U!} \\circ j_!$,\n$j_V^{-1} = j^{-1} \\circ j_U^{-1}$, and\n$j_{V*} = j_{U*} \\circ j_*$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EH","source_file":"sites.tex","source_line":5454,"source_end_line":5475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5454-L5475","statement_sha256":"ca1f8a144c93883dd421fd2d9efd9936e9811a82932099f73ceac31c6a5fabe3","origin":"The Stacks Project","memory_eligible":false,"source_rank":642,"rank":642,"depth":5,"x":1174.168,"y":153.405,"cluster":"sheaves-sites"},{"id":"stacks:04IL","tag":"04IL","title":"Localization · Lemma 04IL","summary":"Notation C, f : V → U, j_U, j_V, and j as in Lemma [Tag 03EH]. Via the identifications Sh(C/V) = Sh(C)/h_V^\\# and Sh(C/U) = Sh(C)/h_U^\\# of Lemma [Tag 00Y1] we have • the functor j^-1 has the following description j^-1(H xrightarrowφ h_U^\\#) = (H ×_φ, h_U^\\#, f h_V^\\# → h_V^\\#). • the functor j_! has the following description j_!(H xrightarrowφ h_V^\\#) = (H xrightarrowh_f ∘ φ h_U^\\#)","statement_latex":"Notation $\\mathcal{C}$, $f : V \\to U$, $j_U$, $j_V$, and $j$ as in\nLemma \\ref{lemma-relocalize}. Via the identifications\n$\\Sh(\\mathcal{C}/V) = \\Sh(\\mathcal{C})/h_V^\\#$\nand\n$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$\nof\nLemma \\ref{lemma-essential-image-j-shriek}\nwe have\n\\begin{enumerate}\n\\item the functor $j^{-1}$ has the following description\n$$\nj^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} h_U^\\#)\n=\n(\\mathcal{H} \\times_{\\varphi, h_U^\\#, f} h_V^\\# \\to h_V^\\#).\n$$\n\\item the functor $j_!$ has the following description\n$$\nj_!(\\mathcal{H} \\xrightarrow{\\varphi} h_V^\\#) =\n(\\mathcal{H} \\xrightarrow{h_f \\circ \\varphi} h_U^\\#)\n$$\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IL","source_file":"sites.tex","source_line":5498,"source_end_line":5521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5498-L5521","statement_sha256":"7a99eead2fda3c4c61ff54d39cb48d8e21d5df51b6042e19d35a1665a9c6b685","origin":"The Stacks Project","memory_eligible":false,"source_rank":643,"rank":643,"depth":8,"x":1151.035,"y":294.343,"cluster":"sheaves-sites"},{"id":"stacks:04TQ","tag":"04TQ","title":"Glueing sheaves · Lemma 04TQ","summary":"Maps of sheaves glue. Let C be a site. Let (U_i → U) be a covering of C. Let F, G be sheaves on C. Given a collection φ_i : F|_C/U_i → G|_C/U_i of maps of sheaves such that for all i, j ∈ I the maps φ_i, φ_j restrict to the same map φ_ij : F|_C/U_i ×_U U_j → G|_C/U_i ×_U U_j then there exists a unique map of sheaves φ : F|_C/U → G|_C/U whose restriction to each C/U_i agrees with φ_i.","statement_latex":"\\begin{slogan}\nMaps of sheaves glue.\n\\end{slogan}\nLet $\\mathcal{C}$ be a site.\nLet $\\{U_i \\to U\\}$ be a covering of $\\mathcal{C}$.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be sheaves on $\\mathcal{C}$.\nGiven a collection\n$$\n\\varphi_i :\n\\mathcal{F}|_{\\mathcal{C}/U_i}\n\\longrightarrow\n\\mathcal{G}|_{\\mathcal{C}/U_i}\n$$\nof maps of sheaves such that for all $i, j \\in I$ the maps\n$\\varphi_i, \\varphi_j$ restrict to the same map\n$\\varphi_{ij} : \\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j} \\to\n\\mathcal{G}|_{\\mathcal{C}/U_i \\times_U U_j}$\nthen there exists a unique map of sheaves\n$$\n\\varphi :\n\\mathcal{F}|_{\\mathcal{C}/U}\n\\longrightarrow\n\\mathcal{G}|_{\\mathcal{C}/U}\n$$\nwhose restriction to each $\\mathcal{C}/U_i$ agrees with $\\varphi_i$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TQ","source_file":"sites.tex","source_line":5637,"source_end_line":5664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5637-L5664","statement_sha256":"2eb11881c25f8d6d1c9f1229d84c017e2a257840bf2b893da082cd8053486f90","origin":"The Stacks Project","memory_eligible":false,"source_rank":644,"rank":644,"depth":6,"x":1054.527,"y":177.015,"cluster":"sheaves-sites"},{"id":"stacks:0BWQ","tag":"0BWQ","title":"Glueing sheaves · Lemma 0BWQ","summary":"The category of sheaves on a site is cartesian closed Let C be a site. Let F, G and H be sheaves on C. There is a canonical bijection Mor_Sh(C)(F×G,H) = Mor_Sh(C)(F,SheafHom(G,H)) which is functorial in all three entries.","statement_latex":"\\begin{slogan}\nThe category of sheaves on a site is cartesian closed\n\\end{slogan}\nLet $\\mathcal{C}$ be a site. Let $\\mathcal{F}$, $\\mathcal{G}$ and\n$\\mathcal{H}$ be sheaves on $\\mathcal{C}$. There is a canonical bijection\n$$\n\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F}\\times\\mathcal{G},\\mathcal{H}) =\n\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{F},\\SheafHom(\\mathcal{G},\\mathcal{H}))\n$$\nwhich is functorial in all three entries.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWQ","source_file":"sites.tex","source_line":5716,"source_end_line":5728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5716-L5728","statement_sha256":"131077a4fc97767762dd7a473205c30aea0ac2f8ed5ae31d13b9e2f942168a94","origin":"The Stacks Project","memory_eligible":false,"source_rank":645,"rank":645,"depth":0,"x":1220.431,"y":208.845,"cluster":"sheaves-sites"},{"id":"stacks:0D7X","tag":"0D7X","title":"Glueing sheaves · Lemma 0D7X","summary":"Let C be a site and U ∈ Ob(C). Then SheafHom(h_U^\\#, F) = j_*(F|_C/U) for F in Sh(C).","statement_latex":"Let $\\mathcal{C}$ be a site and $U \\in \\Ob(\\mathcal{C})$.\nThen $\\SheafHom(h_U^\\#, \\mathcal{F}) = j_*(\\mathcal{F}|_{\\mathcal{C}/U})$\nfor $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7X","source_file":"sites.tex","source_line":5788,"source_end_line":5793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5788-L5793","statement_sha256":"4fe48f75b2d76b101363e3977b97336f31c53d5587491b4dfb15caf0474c45ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":646,"rank":646,"depth":8,"x":1072.154,"y":279.677,"cluster":"sheaves-sites"},{"id":"stacks:04TR","tag":"04TR","title":"Glueing sheaves · Lemma 04TR","summary":"Let C be a site. Let (U_i → U)_i ∈ I be a covering of C. Given any glueing data (F_i, φ_ij) for sheaves of sets with respect to the covering (U_i → U)_i ∈ I there exists a sheaf of sets F on C/U together with isomorphisms φ_i : F|_C/U_i → F_i such that the diagrams xymatrix F|_C/U_i ×_U U_j ar[d]_id ar[r]_φ_i & F_i|_C/U_i ×_U U_j ar[d]^φ_ij F|_C/U_i ×_U U_j ar[r]^φ_j & F_j|_C/U_i ×_U U_j are commutative.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.\nGiven any glueing data $(\\mathcal{F}_i, \\varphi_{ij})$\nfor sheaves of sets with respect to the covering $\\{U_i \\to U\\}_{i \\in I}$\nthere exists a sheaf of sets $\\mathcal{F}$ on $\\mathcal{C}/U$\ntogether with isomorphisms\n$$\n\\varphi_i : \\mathcal{F}|_{\\mathcal{C}/U_i} \\to \\mathcal{F}_i\n$$\nsuch that the diagrams\n$$\n\\xymatrix{\n\\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j}\n\\ar[d]_{\\text{id}} \\ar[r]_{\\varphi_i} &\n\\mathcal{F}_i|_{\\mathcal{C}/U_i \\times_U U_j} \\ar[d]^{\\varphi_{ij}} \\\\\n\\mathcal{F}|_{\\mathcal{C}/U_i \\times_U U_j} \\ar[r]^{\\varphi_j} &\n\\mathcal{F}_j|_{\\mathcal{C}/U_i \\times_U U_j}\n}\n$$\nare commutative.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TR","source_file":"sites.tex","source_line":5847,"source_end_line":5869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5847-L5869","statement_sha256":"ca5d765e1f351f4590c88d7e82423516bb523d9d357c57dc7bc971ac196b8a49","origin":"The Stacks Project","memory_eligible":false,"source_rank":647,"rank":647,"depth":0,"x":1124.652,"y":142.994,"cluster":"sheaves-sites"},{"id":"stacks:04TS","tag":"04TS","title":"Glueing sheaves · Lemma 04TS","summary":"Let C be a site. Let (U_i → U)_i ∈ I be a covering of C. The category Sh(C/U) is equivalent to the category of glueing data via the functor that associates to F on C/U the canonical glueing data.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.\nThe category $\\Sh(\\mathcal{C}/U)$ is equivalent\nto the category of glueing data via the functor that associates\nto $\\mathcal{F}$ on $\\mathcal{C}/U$ the canonical glueing data.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TS","source_file":"sites.tex","source_line":5909,"source_end_line":5916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5909-L5916","statement_sha256":"c3d90f87bfe38fec9c50472f636660883c1c13071cc0d9b429618639fdeab91e","origin":"The Stacks Project","memory_eligible":false,"source_rank":648,"rank":648,"depth":7,"x":1196.021,"y":273.873,"cluster":"sheaves-sites"},{"id":"stacks:0GWK","tag":"0GWK","title":"Glueing sheaves · Lemma 0GWK","summary":"Let C be a site. The category Sh(C) is equivalent to the category of absolute glueing data via the functor that associates to F on C the canonical absolute glueing data.","statement_latex":"Let $\\mathcal{C}$ be a site. The category\n$\\Sh(\\mathcal{C})$ is equivalent to the category\nof absolute glueing data via the functor that\nassociates to $\\mathcal{F}$ on $\\mathcal{C}$ the\ncanonical absolute glueing data.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Glueing sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWK","source_file":"sites.tex","source_line":5970,"source_end_line":5977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L5970-L5977","statement_sha256":"685a96c33bc1d15089961ddf85139f3eb069e0115846f5f9593a86151ba7e1ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":649,"rank":649,"depth":0,"x":1037.783,"y":217.732,"cluster":"sheaves-sites"},{"id":"stacks:03HT","tag":"03HT","title":"More localization · Lemma 03HT","summary":"Let C be a site. Let U ∈ Ob(C). If the topology on C is subcanonical, see Definition [Tag 00WQ], and if G is a sheaf on C/U, then j_U!(G)(V) = coprod_φ ∈ Mor_C(V, U) G(V xrightarrowφ U), in other words sheafification is not necessary in Lemma [Tag 03CD].","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $U \\in \\Ob(\\mathcal{C})$.\nIf the topology on $\\mathcal{C}$ is subcanonical, see\nDefinition \\ref{definition-weaker-than-canonical},\nand if $\\mathcal{G}$ is a sheaf on $\\mathcal{C}/U$, then\n$$\nj_{U!}(\\mathcal{G})(V)\n=\n\\coprod\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}\n\\mathcal{G}(V \\xrightarrow{\\varphi} U),\n$$\nin other words sheafification is not necessary in\nLemma \\ref{lemma-describe-j-shriek}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"More localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HT","source_file":"sites.tex","source_line":6050,"source_end_line":6065,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6050-L6065","statement_sha256":"042e464a14b13615f4730fa17990a04848b67b52f23c5753d31df0eaff65050f","origin":"The Stacks Project","memory_eligible":false,"source_rank":650,"rank":650,"depth":7,"x":1199.983,"y":169.23,"cluster":"sheaves-sites"},{"id":"stacks:03CE","tag":"03CE","title":"More localization · Lemma 03CE","summary":"Let C be a site. Let U ∈ Ob(C). Assume C has products of pairs of objects. Then • the functor j_U has a continuous right adjoint, namely the functor v(X) = X × U / U, • the functor v defines a morphism of sites C/U → C whose associated morphism of topoi equals j_U : Sh(C/U) → Sh(C), and • we have j_U*F(X) = F(X × U/U).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $U \\in \\Ob(\\mathcal{C})$.\nAssume $\\mathcal{C}$ has products of pairs of objects.\nThen\n\\begin{enumerate}\n\\item the functor $j_U$ has a continuous right adjoint,\nnamely the functor $v(X) = X \\times U / U$,\n\\item the functor $v$ defines a morphism of sites\n$\\mathcal{C}/U \\to \\mathcal{C}$ whose associated morphism of topoi equals\n$j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$, and\n\\item we have $j_{U*}\\mathcal{F}(X) = \\mathcal{F}(X \\times U/U)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"More localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CE","source_file":"sites.tex","source_line":6093,"source_end_line":6107,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6093-L6107","statement_sha256":"5b9968d51431c8e6b7d79446a2b9b6f5f056a3f814b3282768d46698e6223e16","origin":"The Stacks Project","memory_eligible":false,"source_rank":651,"rank":651,"depth":8,"x":1119.199,"y":297.322,"cluster":"sheaves-sites"},{"id":"stacks:09W9","tag":"09W9","title":"More localization · Lemma 09W9","summary":"Let C be a site. Let U → V be a morphism of C. Assume C has fibre products. Let j be as in Lemma [Tag 03EH]. Then • the functor j : C/U → C/V has a continuous right adjoint, namely the functor v : (X/V) ↦ (X ×_V U/U), • the functor v defines a morphism of sites C/U → C/V whose associated morphism of topoi equals j, and • we have j_*F(X/V) = F(X ×_V U/U).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U \\to V$ be a morphism of $\\mathcal{C}$.\nAssume $\\mathcal{C}$ has fibre products. Let $j$ be as in\nLemma \\ref{lemma-relocalize}. Then\n\\begin{enumerate}\n\\item the functor $j : \\mathcal{C}/U \\to \\mathcal{C}/V$\nhas a continuous right adjoint, namely the functor\n$v : (X/V) \\mapsto (X \\times_V U/U)$,\n\\item the functor $v$ defines a morphism of sites\n$\\mathcal{C}/U \\to \\mathcal{C}/V$ whose associated morphism of topoi equals\n$j$, and\n\\item we have $j_*\\mathcal{F}(X/V) = \\mathcal{F}(X \\times_V U/U)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"More localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09W9","source_file":"sites.tex","source_line":6131,"source_end_line":6145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6131-L6145","statement_sha256":"965ce6bf5413b87691539b52319bdc2e1771034af393c68f8024e7cab2b8f99a","origin":"The Stacks Project","memory_eligible":false,"source_rank":652,"rank":652,"depth":9,"x":1075.662,"y":156.712,"cluster":"sheaves-sites"},{"id":"stacks:00Y2","tag":"00Y2","title":"More localization · Lemma 00Y2","summary":"Let C be a site. Let U ∈ Ob(C). Assume that every X in C has at most one morphism to U. Let F be a sheaf on C/U. The canonical maps F → j_U^-1j_U!F and j_U^-1j_U*F → F are isomorphisms.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $U \\in \\Ob(\\mathcal{C})$.\nAssume that every $X$ in $\\mathcal{C}$ has at most\none morphism to $U$. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}/U$.\nThe canonical maps $\\mathcal{F} \\to j_U^{-1}j_{U!}\\mathcal{F}$\nand $j_U^{-1}j_{U*}\\mathcal{F} \\to \\mathcal{F}$ are\nisomorphisms.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"More localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Y2","source_file":"sites.tex","source_line":6159,"source_end_line":6168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6159-L6168","statement_sha256":"a5f88e0c5a98b6141d14ac50a1fdd6ee3d447b7651bc05ab81cbc4b89866478a","origin":"The Stacks Project","memory_eligible":false,"source_rank":653,"rank":653,"depth":7,"x":1221.168,"y":235.87,"cluster":"sheaves-sites"},{"id":"stacks:0EYV","tag":"0EYV","title":"More localization · Lemma 0EYV","summary":"Let C be a site. Let xymatrix U' ar[d] ar[r] & U ar[d] V' ar[r] & V be a commutative diagram of C. The morphisms of Lemma [Tag 03EH] produce commutative diagrams vcenter xymatrix C/U' ar[d]_j_U'/V' ar[r]_j_U'/U & C/U ar[d]^j_U/V C/V' ar[r]^j_V'/V & C/V and vcenter xymatrix Sh(C/U') ar[d]_j_U'/V' ar[r]_j_U'/U & Sh(C/U) ar[d]^j_U/V Sh(C/V') ar[r]^j_V'/V & Sh(C/V) of continuous and cocontinuous functors and of topoi. Moreover, if the initial diagram of C is cartesian, then…","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$$\n\\xymatrix{\nU' \\ar[d] \\ar[r] & U \\ar[d] \\\\\nV' \\ar[r] & V\n}\n$$\nbe a commutative diagram of $\\mathcal{C}$. The\nmorphisms of Lemma \\ref{lemma-relocalize}\nproduce commutative diagrams\n$$\n\\vcenter{\n\\xymatrix{\n\\mathcal{C}/U' \\ar[d]_{j_{U'/V'}} \\ar[r]_{j_{U'/U}} &\n\\mathcal{C}/U \\ar[d]^{j_{U/V}} \\\\\n\\mathcal{C}/V' \\ar[r]^{j_{V'/V}} & \\mathcal{C}/V\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\n\\Sh(\\mathcal{C}/U') \\ar[d]_{j_{U'/V'}} \\ar[r]_{j_{U'/U}} &\n\\Sh(\\mathcal{C}/U) \\ar[d]^{j_{U/V}} \\\\\n\\Sh(\\mathcal{C}/V') \\ar[r]^{j_{V'/V}} &\n\\Sh(\\mathcal{C}/V)\n}\n}\n$$\nof continuous and cocontinuous functors and of topoi.\nMoreover, if the initial diagram of $\\mathcal{C}$ is cartesian,\nthen we have\n$j_{V'/V}^{-1} \\circ j_{U/V, *} = j_{U'/V', *} \\circ j_{U'/U}^{-1}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"More localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYV","source_file":"sites.tex","source_line":6176,"source_end_line":6210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6176-L6210","statement_sha256":"89948e86308c855466776943b99968f67cdccce483b4aa4788b0bb89cfbb4d16","origin":"The Stacks Project","memory_eligible":false,"source_rank":654,"rank":654,"depth":9,"x":1049.832,"y":260.118,"cluster":"sheaves-sites"},{"id":"stacks:03CF","tag":"03CF","title":"Localization and morphisms · Lemma 03CF","summary":"Let f : C → D be a morphism of sites corresponding to the continuous functor u : D → C. Let V ∈ Ob(D) and set U = u(V). Then the functor u' : D/V → C/U, V'/V ↦ u(V')/U determines a morphism of sites f' : C/U → D/V. The morphism f' fits into a commutative diagram of topoi xymatrix Sh(C/U) ar[r]_j_U ar[d]_f' & Sh(C) ar[d]^f Sh(D/V) ar[r]^j_V & Sh(D). Using the identifications Sh(C/U) = Sh(C)/h_U^\\# and Sh(D/V) = Sh(D)/h_V^\\# of Lemma [Tag 00Y1] the functor (f')^-1 is…","statement_latex":"Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites\ncorresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nLet $V \\in \\Ob(\\mathcal{D})$ and set $U = u(V)$.\nThen the functor $u' : \\mathcal{D}/V \\to \\mathcal{C}/U$,\n$V'/V \\mapsto u(V')/U$ determines a morphism of sites\n$f' : \\mathcal{C}/U \\to \\mathcal{D}/V$.\nThe morphism $f'$ fits into a commutative diagram of topoi\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f'} &\n\\Sh(\\mathcal{C}) \\ar[d]^f \\\\\n\\Sh(\\mathcal{D}/V) \\ar[r]^{j_V} &\n\\Sh(\\mathcal{D}).\n}\n$$\nUsing the identifications\n$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ and\n$\\Sh(\\mathcal{D}/V) = \\Sh(\\mathcal{D})/h_V^\\#$ of\nLemma \\ref{lemma-essential-image-j-shriek}\nthe functor $(f')^{-1}$ is described by the rule\n$$\n(f')^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} h_V^\\#)\n=\n(f^{-1}\\mathcal{H} \\xrightarrow{f^{-1}\\varphi} h_U^\\#).\n$$\nFinally, we have $f'_*j_U^{-1} = j_V^{-1}f_*$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CF","source_file":"sites.tex","source_line":6264,"source_end_line":6292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6264-L6292","statement_sha256":"653463f5f0cf655f8dd13951f5708d4274570649e4f5a1bb5d75543a35641de7","origin":"The Stacks Project","memory_eligible":false,"source_rank":655,"rank":655,"depth":8,"x":1156.913,"y":144.761,"cluster":"sheaves-sites"},{"id":"stacks:03EF","tag":"03EF","title":"Localization and morphisms · Lemma 03EF","summary":"Let C, D be sites. Let u : D → C be a functor. Let V ∈ Ob(D). Set U = u(V). Assume that • C and D have all finite limits, • u is continuous, and • u commutes with finite limits. There exists a commutative diagram of morphisms of sites xymatrix C/U ar[r]_j_U ar[d]_f' & C ar[d]^f D/V ar[r]^j_V & D where the right vertical arrow corresponds to u, the left vertical arrow corresponds to the functor u' : D/V → C/U, V'/V ↦ u(V')/u(V) and the horizontal arrows correspond to the…","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{D} \\to \\mathcal{C}$ be a functor.\nLet $V \\in \\Ob(\\mathcal{D})$. Set $U = u(V)$.\nAssume that\n\\begin{enumerate}\n\\item $\\mathcal{C}$ and $\\mathcal{D}$ have\nall finite limits,\n\\item $u$ is continuous, and\n\\item $u$ commutes with finite limits.\n\\end{enumerate}\nThere exists a commutative diagram of morphisms of sites\n$$\n\\xymatrix{\n\\mathcal{C}/U \\ar[r]_{j_U} \\ar[d]_{f'} & \\mathcal{C} \\ar[d]^f \\\\\n\\mathcal{D}/V \\ar[r]^{j_V} & \\mathcal{D}\n}\n$$\nwhere the right vertical arrow corresponds to $u$,\nthe left vertical arrow corresponds to the\nfunctor $u' : \\mathcal{D}/V \\to \\mathcal{C}/U$, $V'/V \\mapsto u(V')/u(V)$\nand the horizontal arrows correspond to the functors\n$\\mathcal{C} \\to \\mathcal{C}/U$, $X \\mapsto X \\times U$\nand $\\mathcal{D} \\to \\mathcal{D}/V$, $Y \\mapsto Y \\times V$\nas in Lemma \\ref{lemma-localize-given-products}.\nMoreover, the associated diagram of morphisms of topoi is\nequal to the diagram of\nLemma \\ref{lemma-localize-morphism}.\nIn particular we have $f'_*j_U^{-1} = j_V^{-1}f_*$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EF","source_file":"sites.tex","source_line":6387,"source_end_line":6417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6387-L6417","statement_sha256":"d54a06064fc44de508e259b00ba514c5fded0529e0b6fcb81245836bef3490c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":656,"rank":656,"depth":9,"x":1170.751,"y":290.909,"cluster":"sheaves-sites"},{"id":"stacks:04IN","tag":"04IN","title":"Localization and morphisms · Lemma 04IN","summary":"Let f : C → D be a morphism of sites corresponding to the continuous functor u : D → C. Let V ∈ Ob(D), U ∈ Ob(C) and c : U → u(V) a morphism of C. There exists a commutative diagram of topoi xymatrix Sh(C/U) ar[r]_j_U ar[d]_f_c & Sh(C) ar[d]^f Sh(D/V) ar[r]^j_V & Sh(D). We have f_c = f' ∘ j_U/u(V) where f' : Sh(C/u(V)) → Sh(D/V) is as in Lemma [Tag 03CF] and j_U/u(V) : Sh(C/U) → Sh(C/u(V)) is as in Lemma [Tag 03EH]. Using the identifications Sh(C/U) = Sh(C)/h_U^\\# and…","statement_latex":"Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites\ncorresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nLet $V \\in \\Ob(\\mathcal{D})$, $U \\in \\Ob(\\mathcal{C})$\nand $c : U \\to u(V)$ a morphism of $\\mathcal{C}$.\nThere exists a commutative diagram of topoi\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f_c} &\n\\Sh(\\mathcal{C}) \\ar[d]^f \\\\\n\\Sh(\\mathcal{D}/V) \\ar[r]^{j_V} &\n\\Sh(\\mathcal{D}).\n}\n$$\nWe have $f_c = f' \\circ j_{U/u(V)}$ where\n$f' : \\Sh(\\mathcal{C}/u(V)) \\to \\Sh(\\mathcal{D}/V)$\nis as in\nLemma \\ref{lemma-localize-morphism}\nand\n$j_{U/u(V)} : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C}/u(V))$\nis as in\nLemma \\ref{lemma-relocalize}.\nUsing the identifications\n$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ and\n$\\Sh(\\mathcal{D}/V) = \\Sh(\\mathcal{D})/h_V^\\#$ of\nLemma \\ref{lemma-essential-image-j-shriek}\nthe functor $(f_c)^{-1}$ is described by the rule\n$$\n(f_c)^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} h_V^\\#)\n=\n(f^{-1}\\mathcal{H} \\times_{f^{-1}\\varphi, h_{u(V)}^\\#, c} h_U^\\#\n\\rightarrow h_U^\\#).\n$$\nFinally, given any morphisms $b : V' \\to V$, $a : U' \\to U$ and\n$c' : U' \\to u(V')$ such that\n$$\n\\xymatrix{\nU' \\ar[r]_-{c'} \\ar[d]_a & u(V') \\ar[d]^{u(b)} \\\\\nU \\ar[r]^-c & u(V)\n}\n$$\ncommutes, then the diagram\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C}/U') \\ar[r]_{j_{U'/U}} \\ar[d]_{f_{c'}} &\n\\Sh(\\mathcal{C}/U) \\ar[d]^{f_c} \\\\\n\\Sh(\\mathcal{D}/V') \\ar[r]^{j_{V'/V}} &\n\\Sh(\\mathcal{D}/V).\n}\n$$\ncommutes.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IN","source_file":"sites.tex","source_line":6442,"source_end_line":6494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6442-L6494","statement_sha256":"f6cdd380c70e6eb631f835159e3a93a9ed037d40c78cfc9719d9282681b47565","origin":"The Stacks Project","memory_eligible":false,"source_rank":657,"rank":657,"depth":9,"x":1042.736,"y":190.772,"cluster":"sheaves-sites"},{"id":"stacks:03EG","tag":"03EG","title":"Localization and morphisms · Lemma 03EG","summary":"Let C, D be sites. Let u : C → D be a cocontinuous functor. Let U be an object of C, and set V = u(U). We have a commutative diagram xymatrix C/U ar[r]_j_U ar[d]_u' & C ar[d]^u D/V ar[r]^-j_V & D where the left vertical arrow is u' : C/U → D/V, U'/U ↦ V'/V. Then u' is cocontinuous also and we get a commutative diagram of topoi xymatrix Sh(C/U) ar[r]_j_U ar[d]_f' & Sh(C) ar[d]^f Sh(D/V) ar[r]^-j_V & Sh(D) where f (resp. f') corresponds to u (resp. u').","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a cocontinuous functor.\nLet $U$ be an object of $\\mathcal{C}$, and set $V = u(U)$.\nWe have a commutative diagram\n$$\n\\xymatrix{\n\\mathcal{C}/U \\ar[r]_{j_U} \\ar[d]_{u'} & \\mathcal{C} \\ar[d]^u \\\\\n\\mathcal{D}/V \\ar[r]^-{j_V} & \\mathcal{D}\n}\n$$\nwhere the left vertical arrow is\n$u' : \\mathcal{C}/U \\to \\mathcal{D}/V$, $U'/U \\mapsto V'/V$.\nThen $u'$ is cocontinuous also and we get a commutative diagram of topoi\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d]_{f'} &\n\\Sh(\\mathcal{C}) \\ar[d]^f \\\\\n\\Sh(\\mathcal{D}/V) \\ar[r]^-{j_V} &\n\\Sh(\\mathcal{D})\n}\n$$\nwhere $f$ (resp.\\ $f'$) corresponds to $u$ (resp.\\ $u'$).","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EG","source_file":"sites.tex","source_line":6530,"source_end_line":6554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6530-L6554","statement_sha256":"b08f235b9dbfc32152556d81e7d5a644e371ca9c707e39300ed7d3d8389a57c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":658,"rank":658,"depth":0,"x":1218.046,"y":191.975,"cluster":"sheaves-sites"},{"id":"stacks:0D5R","tag":"0D5R","title":"Localization and morphisms · Lemma 0D5R","summary":"Let C, D be sites. Let u : C → D be a cocontinuous functor. Let V be an object of D. Let ^u_VI be the category introduced in Section [Tag 00XF]. We have a commutative diagram vcenter xymatrix _V^uI ar[r]_j ar[d]_u' & C ar[d]^u D/V ar[r]^-j_V & D where j : (U, ψ) ↦ U u' : (U, ψ) ↦ (ψ : u(U) → V) Declare a family of morphisms ((U_i, ψ_i) → (U, ψ)) of ^u_VI to be a covering if and only if (U_i → U) is a covering in C. Then • ^u_VI is a site, • j is continuous and…","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a cocontinuous functor.\nLet $V$ be an object of $\\mathcal{D}$. Let\n${}^u_V\\mathcal{I}$ be the category introduced in\nSection \\ref{section-more-functoriality-PSh}.\nWe have a commutative diagram\n$$\n\\vcenter{\n\\xymatrix{\n\\,_V^u\\mathcal{I} \\ar[r]_j \\ar[d]_{u'} &\n\\mathcal{C} \\ar[d]^u \\\\\n\\mathcal{D}/V \\ar[r]^-{j_V} &\n\\mathcal{D}\n}\n}\n\\quad\\text{where}\\quad\n\\begin{matrix}\nj : (U, \\psi) \\mapsto U \\\\\nu' : (U, \\psi) \\mapsto (\\psi : u(U) \\to V)\n\\end{matrix}\n$$\nDeclare a family of morphisms $\\{(U_i, \\psi_i) \\to (U, \\psi)\\}$\nof ${}^u_V\\mathcal{I}$ to be a covering if and only if\n$\\{U_i \\to U\\}$ is a covering in $\\mathcal{C}$.\nThen\n\\begin{enumerate}\n\\item ${}^u_V\\mathcal{I}$ is a site,\n\\item $j$ is continuous and cocontinuous,\n\\item $u'$ is cocontinuous,\n\\item we get a commutative diagram of topoi\n$$\n\\xymatrix{\n\\Sh({}^u_V\\mathcal{I}) \\ar[r]_j \\ar[d]_{f'} &\n\\Sh(\\mathcal{C}) \\ar[d]^f \\\\\n\\Sh(\\mathcal{D}/V) \\ar[r]^-{j_V} &\n\\Sh(\\mathcal{D})\n}\n$$\nwhere $f$ (resp.\\ $f'$) corresponds to $u$ (resp.\\ $u'$), and\n\\item we have $f'_*j^{-1} = j_V^{-1}f_*$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5R","source_file":"sites.tex","source_line":6578,"source_end_line":6621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6578-L6621","statement_sha256":"86fa8bd6bb22d7933b51f68261c7d3dfb5fd8c790aec4806406ce2a650b21d20","origin":"The Stacks Project","memory_eligible":false,"source_rank":659,"rank":659,"depth":6,"x":1087.52,"y":290.779,"cluster":"sheaves-sites"},{"id":"stacks:0FN1","tag":"0FN1","title":"Localization and morphisms · Lemma 0FN1","summary":"Assume given sites C', C, D', D and functors xymatrix C' ar[r]_v' ar[d]_u' & C ar[d]^u D' ar[r]^v & D Assume • u, u', v, and v' are cocontinuous giving rise to morphisms of topoi f, f', g, and g' by Lemma [Tag 00XO], • v ∘ u' = u ∘ v', • v and v' are continuous as well as cocontinuous, and • for any object V' of D' the functor ^u'_V'I → ^ u_v(V')I given by v is cofinal. Then f'_* ∘ (g')^-1 = g^-1 ∘ f_* and g'_! ∘ (f')^-1 = f^-1 ∘ g_!.","statement_latex":"Assume given sites $\\mathcal{C}', \\mathcal{C}, \\mathcal{D}', \\mathcal{D}$\nand functors\n$$\n\\xymatrix{\n\\mathcal{C}' \\ar[r]_{v'} \\ar[d]_{u'} &\n\\mathcal{C} \\ar[d]^u \\\\\n\\mathcal{D}' \\ar[r]^v &\n\\mathcal{D}\n}\n$$\nAssume\n\\begin{enumerate}\n\\item $u$, $u'$, $v$, and $v'$ are cocontinuous giving rise to morphisms of\ntopoi $f$, $f'$, $g$, and $g'$ by Lemma \\ref{lemma-cocontinuous-morphism-topoi},\n\\item $v \\circ u' = u \\circ v'$,\n\\item $v$ and $v'$ are continuous as well as cocontinuous, and\n\\item for any object $V'$ of $\\mathcal{D}'$ the functor\n${}^{u'}_{V'}\\mathcal{I} \\to {}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$\ngiven by $v$ is cofinal.\n\\end{enumerate}\nThen $f'_* \\circ (g')^{-1} = g^{-1} \\circ f_*$ and\n$g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FN1","source_file":"sites.tex","source_line":6648,"source_end_line":6672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6648-L6672","statement_sha256":"458cfa6328fa2c4cf021a9f8e60e445711a478b290068f6c2d255e62d20feb6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":660,"rank":660,"depth":6,"x":1104.349,"y":143.538,"cluster":"sheaves-sites"},{"id":"stacks:0FN2","tag":"0FN2","title":"Localization and morphisms · Lemma 0FN2","summary":"Assume given sites C', C, D', D and functors xymatrix C' ar[r]_v' & C D' ar[r]^v ar[u]^u' & D ar[u]_u With notation as in Sections [Tag 00X0] and [Tag 00XN] assume • u and u' are continuous giving rise to morphisms of sites f and f', • v and v' are cocontinuous giving rise to morphisms of topoi g and g', • u ∘ v = v' ∘ u', and • v and v' are continuous as well as cocontinuous. Then f'_* ∘ (g')^-1 = g^-1 ∘ f_* and g'_! ∘ (f')^-1 = f^-1 ∘ g_!.","statement_latex":"Assume given sites $\\mathcal{C}', \\mathcal{C}, \\mathcal{D}', \\mathcal{D}$\nand functors\n$$\n\\xymatrix{\n\\mathcal{C}' \\ar[r]_{v'} &\n\\mathcal{C} \\\\\n\\mathcal{D}' \\ar[r]^v \\ar[u]^{u'} &\n\\mathcal{D} \\ar[u]_u\n}\n$$\nWith notation as in Sections \\ref{section-morphism-sites}\nand \\ref{section-cocontinuous-morphism-topoi} assume\n\\begin{enumerate}\n\\item $u$ and $u'$ are continuous giving rise to morphisms of\nsites $f$ and $f'$,\n\\item $v$ and $v'$ are cocontinuous giving rise to morphisms\nof topoi $g$ and $g'$,\n\\item $u \\circ v = v' \\circ u'$, and\n\\item $v$ and $v'$ are continuous as well as cocontinuous.\n\\end{enumerate}\nThen\\footnote{In this generality\nwe don't know $f \\circ g'$ is equal to $g \\circ f'$\nas morphisms of topoi (there is a canonical $2$-arrow from\nthe first to the second which may not be an isomorphism).}\n$f'_* \\circ (g')^{-1} = g^{-1} \\circ f_*$ and\n$g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FN2","source_file":"sites.tex","source_line":6700,"source_end_line":6728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6700-L6728","statement_sha256":"afde686aa4351f41d26f25c8d609a5b54c57007c2581b0865bad62172373f05c","origin":"The Stacks Project","memory_eligible":false,"source_rank":661,"rank":661,"depth":6,"x":1210.578,"y":261.918,"cluster":"sheaves-sites"},{"id":"stacks:03A0","tag":"03A0","title":"Morphisms of topoi · Lemma 03A0","summary":"Let C, D be sites. Let u : C → D be a functor. Assume that • u is cocontinuous, • u is continuous, • given a, b : U' → U in C such that u(a) = u(b), then there exists a covering (f_i : U'_i → U') in C such that a ∘ f_i = b ∘ f_i, • given U', U ∈ Ob(C) and a morphism c : u(U') → u(U) in D there exists a covering (f_i : U_i' → U') in C and morphisms c_i : U_i' → U such that u(c_i) = c ∘ u(f_i), and • given V ∈ Ob(D) there exists a covering of V in D of the form (u(U_i) →…","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item $u$ is cocontinuous,\n\\item $u$ is continuous,\n\\item given $a, b : U' \\to U$ in $\\mathcal{C}$ such that\n$u(a) = u(b)$, then there exists a covering $\\{f_i : U'_i \\to U'\\}$\nin $\\mathcal{C}$ such that $a \\circ f_i = b \\circ f_i$,\n\\item given $U', U \\in \\Ob(\\mathcal{C})$ and\na morphism $c : u(U') \\to u(U)$ in $\\mathcal{D}$ there exists\na covering $\\{f_i : U_i' \\to U'\\}$ in $\\mathcal{C}$\nand morphisms $c_i : U_i' \\to U$ such that $u(c_i) = c \\circ u(f_i)$, and\n\\item given $V \\in \\Ob(\\mathcal{D})$ there exists a covering\nof $V$ in $\\mathcal{D}$ of the form $\\{u(U_i) \\to V\\}_{i \\in I}$.\n\\end{enumerate}\nThen the morphism of topoi\n$$\ng : \\Sh(\\mathcal{C}) \\longrightarrow \\Sh(\\mathcal{D})\n$$\nassociated to the cocontinuous functor $u$ by\nLemma \\ref{lemma-cocontinuous-morphism-topoi}\nis an equivalence.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03A0","source_file":"sites.tex","source_line":6773,"source_end_line":6798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6773-L6798","statement_sha256":"84132f27da78765d8c4684997e767fb4ad4b372ad5ae3597218fb26695b52849","origin":"The Stacks Project","memory_eligible":false,"source_rank":662,"rank":662,"depth":6,"x":1036.672,"y":234.844,"cluster":"sheaves-sites"},{"id":"stacks:03CG","tag":"03CG","title":"Morphisms of topoi · Definition 03CG","summary":"Let C, D be sites. A special cocontinuous functor u from C to D is a cocontinuous functor u : C → D satisfying the assumptions and conclusions of Lemma [Tag 03A0].","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nA {\\it special cocontinuous functor $u$ from $\\mathcal{C}$ to $\\mathcal{D}$}\nis a cocontinuous functor $u : \\mathcal{C} \\to \\mathcal{D}$ satisfying\nthe assumptions and conclusions of Lemma \\ref{lemma-equivalence}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CG","source_file":"sites.tex","source_line":6893,"source_end_line":6899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6893-L6899","statement_sha256":"128ebc9c922e7d819a5575304a285549a87956b8cb68834dd28e7cbf17901f5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":663,"rank":663,"depth":7,"x":1187.004,"y":155.96,"cluster":"sheaves-sites"},{"id":"stacks:03CH","tag":"03CH","title":"Morphisms of topoi · Lemma 03CH","summary":"Let C, D be sites. Let u : C → D be a special cocontinuous functor. For every object U of C we have a commutative diagram xymatrix C/U ar[r]_j_U ar[d] & C ar[d]^u D/u(U) ar[r]^-j_u(U) & D as in Lemma [Tag 03EG]. The left vertical arrow is a special cocontinuous functor. Hence in the commutative diagram of topoi xymatrix Sh(C/U) ar[r]_j_U ar[d] & Sh(C) ar[d]^u Sh(D/u(U)) ar[r]^-j_u(U) & Sh(D) the vertical arrows are equivalences.","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a special cocontinuous functor.\nFor every object $U$ of $\\mathcal{C}$ we have a commutative diagram\n$$\n\\xymatrix{\n\\mathcal{C}/U \\ar[r]_{j_U} \\ar[d] & \\mathcal{C} \\ar[d]^u \\\\\n\\mathcal{D}/u(U) \\ar[r]^-{j_{u(U)}} & \\mathcal{D}\n}\n$$\nas in Lemma \\ref{lemma-localize-cocontinuous}.\nThe left vertical arrow is a special cocontinuous functor.\nHence in the commutative diagram of topoi\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C}/U) \\ar[r]_{j_U} \\ar[d] &\n\\Sh(\\mathcal{C}) \\ar[d]^u \\\\\n\\Sh(\\mathcal{D}/u(U)) \\ar[r]^-{j_{u(U)}} &\n\\Sh(\\mathcal{D})\n}\n$$\nthe vertical arrows are equivalences.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CH","source_file":"sites.tex","source_line":6901,"source_end_line":6924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6901-L6924","statement_sha256":"9734f20a8d777e2d4c3cec8c935c8f8bbf73dca7d2a621887d249d51081d4fa5","origin":"The Stacks Project","memory_eligible":false,"source_rank":664,"rank":664,"depth":7,"x":1139.486,"y":299.738,"cluster":"sheaves-sites"},{"id":"stacks:03A1","tag":"03A1","title":"Morphisms of topoi · Lemma 03A1","summary":"Let C be a site. Let C' ⊂ Sh(C) be a full subcategory (with a set of objects) such that • h_U^\\# ∈ Ob(C') for all U ∈ Ob(C), and • C' is preserved under fibre products in Sh(C). Declare a covering of C' to be any family (F_i → F)_i ∈ I of maps such that coprod_i ∈ I F_i → F is a surjective map of sheaves. Then • C' is a site (after choosing a set of coverings, see Sets, Lemma [Tag 000X]), • representable presheaves on C' are sheaves (i.e., the topology on C' is…","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$\\mathcal{C}' \\subset \\Sh(\\mathcal{C})$\nbe a full subcategory (with a set of objects) such that\n\\begin{enumerate}\n\\item $h_U^\\# \\in \\Ob(\\mathcal{C}')$ for all\n$U \\in \\Ob(\\mathcal{C})$, and\n\\item $\\mathcal{C}'$ is preserved under fibre products in\n$\\Sh(\\mathcal{C})$.\n\\end{enumerate}\nDeclare a covering of $\\mathcal{C}'$ to be any family\n$\\{\\mathcal{F}_i \\to \\mathcal{F}\\}_{i \\in I}$ of maps such that\n$\\coprod_{i \\in I} \\mathcal{F}_i \\to \\mathcal{F}$ is a surjective\nmap of sheaves. Then\n\\begin{enumerate}\n\\item $\\mathcal{C}'$ is a site (after\nchoosing a set of coverings, see Sets, Lemma \\ref{sets-lemma-coverings-site}),\n\\item representable presheaves on $\\mathcal{C}'$ are sheaves\n(i.e., the topology on $\\mathcal{C}'$ is subcanonical, see\nDefinition \\ref{definition-weaker-than-canonical}),\n\\item the functor $v : \\mathcal{C} \\to \\mathcal{C}'$,\n$U \\mapsto h_U^\\#$ is a special cocontinuous functor, hence induces an\nequivalence $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$,\n\\item for any $\\mathcal{F} \\in \\Ob(\\mathcal{C}')$ we have\n$g^{-1}h_\\mathcal{F} = \\mathcal{F}$, and\n\\item for any $U \\in \\Ob(\\mathcal{C})$ we have\n$g_*h_U^\\# = h_{v(U)} = h_{h_U^\\#}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03A1","source_file":"sites.tex","source_line":6981,"source_end_line":7010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L6981-L7010","statement_sha256":"45f232e8f1f2e634f1d521fc9baf3daa35900916e1f737391ecba5e84b79e8b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":665,"rank":665,"depth":7,"x":1058.73,"y":166.471,"cluster":"sheaves-sites"},{"id":"stacks:03CI","tag":"03CI","title":"Morphisms of topoi · Lemma 03CI","summary":"Let Sh(C) be a topos. Let (F_i)_i ∈ I be a set of sheaves on C. There exists an equivalence of topoi g : Sh(C) → Sh(C') induced by a special cocontinuous functor u : C → C' of sites such that • C' has a subcanonical topology, • a family (V_j → V) of morphisms of C' is (combinatorially equivalent to) a covering of C' if and only if coprod h_V_j → h_V is surjective, • C' has fibre products and a final object (i.e., C' has all finite limits), • every subsheaf of a…","statement_latex":"Let $\\Sh(\\mathcal{C})$ be a topos. Let $\\{\\mathcal{F}_i\\}_{i \\in I}$\nbe a set of sheaves on $\\mathcal{C}$. There exists an equivalence of topoi\n$g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$ induced by a special\ncocontinuous functor $u : \\mathcal{C} \\to \\mathcal{C}'$ of sites\nsuch that\n\\begin{enumerate}\n\\item $\\mathcal{C}'$ has a subcanonical topology,\n\\item a family $\\{V_j \\to V\\}$ of morphisms of $\\mathcal{C}'$\nis (combinatorially equivalent to) a covering of $\\mathcal{C}'$\nif and only if $\\coprod h_{V_j} \\to h_V$ is surjective,\n\\item $\\mathcal{C}'$ has fibre products and a final object\n(i.e., $\\mathcal{C}'$ has all finite limits),\n\\item every subsheaf of a representable sheaf on $\\mathcal{C}'$\nis representable, and\n\\item each $g_*\\mathcal{F}_i$ is a representable sheaf.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CI","source_file":"sites.tex","source_line":7091,"source_end_line":7109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7091-L7109","statement_sha256":"eb09dad4aecc0a5237f184077f77f794a2111b38e6633c4b916f1b133725551d","origin":"The Stacks Project","memory_eligible":false,"source_rank":666,"rank":666,"depth":8,"x":1225.803,"y":219.029,"cluster":"sheaves-sites"},{"id":"stacks:03A2","tag":"03A2","title":"Morphisms of topoi · Lemma 03A2","summary":"This statement is closely related to [SGA4]. In order to get the whole result, one should also use [SGA4]. Let C, D be sites. Let f : Sh(C) → Sh(D) be a morphism of topoi. Then there exists a site C' and a diagram of functors xymatrix C ar[r]_v & C' & D ar[l]^u such that • the functor v is a special cocontinuous functor, • the functor u commutes with fibre products, is continuous and defines a morphism of sites C' → D, and • the morphism of topoi f agrees with the…","statement_latex":"\\begin{reference}\nThis statement is closely related to\n\\cite[Proposition 4.9.4. Expos\\'e IV]{SGA4}.\nIn order to get the whole result, one should also use\n\\cite[Remarque 4.7.4, Expos\\'e IV]{SGA4}.\n\\end{reference}\nLet $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a\nmorphism of topoi.\nThen there exists a site $\\mathcal{C}'$ and a diagram of functors\n$$\n\\xymatrix{\n\\mathcal{C} \\ar[r]_v & \\mathcal{C}' & \\mathcal{D} \\ar[l]^u\n}\n$$\nsuch that\n\\begin{enumerate}\n\\item the functor $v$ is a special cocontinuous functor,\n\\item the functor $u$ commutes with fibre products, is\ncontinuous and defines a morphism of sites\n$\\mathcal{C}' \\to \\mathcal{D}$, and\n\\item the morphism of topoi $f$ agrees with the composition\nof morphisms of topoi\n$$\n\\Sh(\\mathcal{C}) \\longrightarrow\n\\Sh(\\mathcal{C}') \\longrightarrow\n\\Sh(\\mathcal{D})\n$$\nwhere the first arrow comes from $v$ via Lemma \\ref{lemma-equivalence}\nand the second arrow from $u$ via Lemma \\ref{lemma-morphism-sites-topoi}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03A2","source_file":"sites.tex","source_line":7140,"source_end_line":7173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7140-L7173","statement_sha256":"6cd64b5ebddc1fdf5f6c6af0cb66515ec88a3959cd6ebc460646f607276ec817","origin":"The Stacks Project","memory_eligible":false,"source_rank":667,"rank":667,"depth":8,"x":1059.989,"y":275.193,"cluster":"sheaves-sites"},{"id":"stacks:04GZ","tag":"04GZ","title":"Localization of topoi · Lemma 04GZ","summary":"Let C be a site. Let F be a sheaf on C. Then the category Sh(C)/F is a topos. There is a canonical morphism of topoi j_F : Sh(C)/F → Sh(C) which is a localization as in Section [Tag 00XZ] such that • the functor j_F^-1 is the functor H ↦ H × F/F, and • the functor j_F! is the forgetful functor G/F ↦ G.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.\nThen the category $\\Sh(\\mathcal{C})/\\mathcal{F}$\nis a topos. There is a canonical morphism of topoi\n$$\nj_\\mathcal{F} :\n\\Sh(\\mathcal{C})/\\mathcal{F}\n\\longrightarrow\n\\Sh(\\mathcal{C})\n$$\nwhich is a localization as in\nSection \\ref{section-localize}\nsuch that\n\\begin{enumerate}\n\\item the functor $j_\\mathcal{F}^{-1}$ is the functor\n$\\mathcal{H} \\mapsto \\mathcal{H} \\times \\mathcal{F}/\\mathcal{F}$, and\n\\item the functor $j_{\\mathcal{F}!}$ is the forgetful\nfunctor $\\mathcal{G}/\\mathcal{F} \\mapsto \\mathcal{G}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GZ","source_file":"sites.tex","source_line":7324,"source_end_line":7345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7324-L7345","statement_sha256":"e131f3903a032d4179233cae7257ee714c2facb756f2ee6c9bf1dfad25b103f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":668,"rank":668,"depth":9,"x":1137.256,"y":139.405,"cluster":"sheaves-sites"},{"id":"stacks:04H0","tag":"04H0","title":"Localization of topoi · Lemma 04H0","summary":"In the situation of Lemma [Tag 04GZ], the functor j_F, * is the one associates to φ : G → F the sheaf U ↦ (α : F|_U → G|_U such that α is a right inverse to φ|_U ).","statement_latex":"In the situation of Lemma \\ref{lemma-localize-topos}, the functor\n$j_{\\mathcal{F}, *}$ is the one associates to\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ the sheaf\n$$\nU\n\\longmapsto\n\\{\\alpha : \\mathcal{F}|_U \\to \\mathcal{G}|_U\n\\text{ such that } \\alpha \\text{ is a right inverse to }\\varphi|_U \\}.\n$$","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04H0","source_file":"sites.tex","source_line":7376,"source_end_line":7387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7376-L7387","statement_sha256":"9e82cbf98e03470e43d8217792626a60e2fef4f0046a4696013c80b9f00d81bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":669,"rank":669,"depth":10,"x":1189.587,"y":283.68,"cluster":"sheaves-sites"},{"id":"stacks:0791","tag":"0791","title":"Localization of topoi · Lemma 0791","summary":"Let C be a site. Let F be a sheaf on C. Let C/F be the category of pairs (U, s) where U ∈ Ob(C) and s ∈ F(U). Let a covering in C/F be a family ((U_i, s_i) → (U, s)) such that (U_i → U) is a covering of C. Then j : C/F → C is a continuous and cocontinuous functor of sites which induces a morphism of topoi j : Sh(C/F) → Sh(C). In fact, there is an equivalence Sh(C/F) = Sh(C)/F which turns j into j_F.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.\nLet $\\mathcal{C}/\\mathcal{F}$ be the category of pairs $(U, s)$ where\n$U \\in \\Ob(\\mathcal{C})$ and $s \\in \\mathcal{F}(U)$. Let a covering in\n$\\mathcal{C}/\\mathcal{F}$ be a family $\\{(U_i, s_i) \\to (U, s)\\}$\nsuch that $\\{U_i \\to U\\}$ is a covering of $\\mathcal{C}$.\nThen $j : \\mathcal{C}/\\mathcal{F} \\to \\mathcal{C}$ is a continuous\nand cocontinuous functor of sites which induces a morphism of topoi\n$j : \\Sh(\\mathcal{C}/\\mathcal{F}) \\to \\Sh(\\mathcal{C})$. In fact, there\nis an equivalence $\\Sh(\\mathcal{C}/\\mathcal{F}) =\n\\Sh(\\mathcal{C})/\\mathcal{F}$ which turns $j$ into $j_\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0791","source_file":"sites.tex","source_line":7448,"source_end_line":7460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7448-L7460","statement_sha256":"cec94d2f082b86d93664fa2feeabc6f966b2cb0b9310c39f648073776198f7fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":670,"rank":670,"depth":6,"x":1034.653,"y":206.827,"cluster":"sheaves-sites"},{"id":"stacks:04IP","tag":"04IP","title":"Localization of topoi · Definition 04IP","summary":"Let C be a site. Let F be a sheaf on C. • The topos Sh(C)/F is called the localization of the topos Sh(C) at F. • The morphism of topoi j_F : Sh(C)/F → Sh(C) of Lemma [Tag 04GZ] is called the localization morphism.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The topos $\\Sh(\\mathcal{C})/\\mathcal{F}$\nis called the\n{\\it localization of the topos $\\Sh(\\mathcal{C})$ at $\\mathcal{F}$}.\n\\item The morphism of topoi\n$j_\\mathcal{F} :\n\\Sh(\\mathcal{C})/\\mathcal{F}\n\\to\n\\Sh(\\mathcal{C})$ of\nLemma \\ref{lemma-localize-topos}\nis called the {\\it localization morphism}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization of topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IP","source_file":"sites.tex","source_line":7491,"source_end_line":7507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7491-L7507","statement_sha256":"75508a8639208d9ba2198e4d2b6908e21381982779aed67e660000dea62bc5aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":671,"rank":671,"depth":10,"x":1211.069,"y":175.519,"cluster":"sheaves-sites"},{"id":"stacks:04IQ","tag":"04IQ","title":"Localization of topoi · Lemma 04IQ","summary":"Let C be a site. Let F = h_U^\\# for some object U of C. Then j_F : Sh(C)/F → Sh(C) constructed in Lemma [Tag 04GZ] agrees with the morphism of topoi j_U : Sh(C/U) → Sh(C) constructed in Section [Tag 00XZ] via the identification Sh(C/U) = Sh(C)/h_U^\\# of Lemma [Tag 00Y1].","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{F} = h_U^\\#$ for some object $U$\nof $\\mathcal{C}$. Then $j_\\mathcal{F} : \\Sh(\\mathcal{C})/\\mathcal{F}\n\\to \\Sh(\\mathcal{C})$ constructed in\nLemma \\ref{lemma-localize-topos}\nagrees with the morphism of topoi\n$j_U : \\Sh(\\mathcal{C}/U) \\to \\Sh(\\mathcal{C})$\nconstructed in\nSection \\ref{section-localize}\nvia the identification\n$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$\nof\nLemma \\ref{lemma-essential-image-j-shriek}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IQ","source_file":"sites.tex","source_line":7516,"source_end_line":7530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7516-L7530","statement_sha256":"a873e15460e0ac4d914df3d23fe0338d7e034466ead451ce783b88c9ced6ec17","origin":"The Stacks Project","memory_eligible":false,"source_rank":672,"rank":672,"depth":10,"x":1105.942,"y":298.963,"cluster":"sheaves-sites"},{"id":"stacks:04IR","tag":"04IR","title":"Localization of topoi · Lemma 04IR","summary":"Let C be a site. If s : G → F is a morphism of sheaves on C then there exists a natural commutative diagram of morphisms of topoi xymatrix Sh(C)/G ar[rd]_j_G ar[rr]_j & & Sh(C)/F ar[ld]^j_F & Sh(C) & where j = j_G/F is the localization of the topos Sh(C)/F at the object G/F. In particular we have j^-1(H → F) = (H ×_F G → G) and j_!(E xrightarrowe G) = (E xrightarrows ∘ e F).","statement_latex":"Let $\\mathcal{C}$ be a site.\nIf $s : \\mathcal{G} \\to \\mathcal{F}$ is a morphism of sheaves\non $\\mathcal{C}$ then there exists a natural commutative diagram of\nmorphisms of topoi\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C})/\\mathcal{G} \\ar[rd]_{j_\\mathcal{G}} \\ar[rr]_j & &\n\\Sh(\\mathcal{C})/\\mathcal{F} \\ar[ld]^{j_\\mathcal{F}} \\\\\n& \\Sh(\\mathcal{C}) &\n}\n$$\nwhere $j = j_{\\mathcal{G}/\\mathcal{F}}$ is the localization of the\ntopos $\\Sh(\\mathcal{C})/\\mathcal{F}$ at the object\n$\\mathcal{G}/\\mathcal{F}$. In particular we have\n$$\nj^{-1}(\\mathcal{H} \\to \\mathcal{F}) =\n(\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G} \\to \\mathcal{G})\n$$\nand\n$$\nj_!(\\mathcal{E} \\xrightarrow{e} \\mathcal{G}) =\n(\\mathcal{E} \\xrightarrow{s \\circ e} \\mathcal{F}).\n$$","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IR","source_file":"sites.tex","source_line":7547,"source_end_line":7572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7547-L7572","statement_sha256":"846a24ceaa6601692dd1749829e0877761158e6ae7feb45afe5061c973291836","origin":"The Stacks Project","memory_eligible":false,"source_rank":673,"rank":673,"depth":10,"x":1084.146,"y":147.974,"cluster":"sheaves-sites"},{"id":"stacks:04IS","tag":"04IS","title":"Localization of topoi · Lemma 04IS","summary":"Assume C and s : G → F are as in Lemma [Tag 04IR]. If G = h_V^\\# and F = h_U^\\# and s : G → F comes from a morphism V → U of C then the diagram in Lemma [Tag 04IR] is identified with diagram ([Tag 04IK]) via the identifications Sh(C/V) = Sh(C)/h_V^\\# and Sh(C/U) = Sh(C)/h_U^\\# of Lemma [Tag 00Y1].","statement_latex":"Assume $\\mathcal{C}$ and $s : \\mathcal{G} \\to \\mathcal{F}$ are as in\nLemma \\ref{lemma-relocalize-topos}.\nIf $\\mathcal{G} = h_V^\\#$ and $\\mathcal{F} = h_U^\\#$ and\n$s : \\mathcal{G} \\to \\mathcal{F}$ comes from a morphism $V \\to U$\nof $\\mathcal{C}$ then the diagram in\nLemma \\ref{lemma-relocalize-topos}\nis identified with\ndiagram (\\ref{equation-relocalize})\nvia the identifications\n$\\Sh(\\mathcal{C}/V) = \\Sh(\\mathcal{C})/h_V^\\#$\nand\n$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$\nof\nLemma \\ref{lemma-essential-image-j-shriek}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IS","source_file":"sites.tex","source_line":7586,"source_end_line":7602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7586-L7602","statement_sha256":"dddc59c4eaac8c3412544f5beba5ab8996ea5b7e6537fd12f8ab3668ab49c11a","origin":"The Stacks Project","memory_eligible":false,"source_rank":674,"rank":674,"depth":11,"x":1221.922,"y":247.148,"cluster":"sheaves-sites"},{"id":"stacks:04H1","tag":"04H1","title":"Localization and morphisms of topoi · Lemma 04H1","summary":"Let f : Sh(C) → Sh(D) be a morphism of topoi. Let G be a sheaf on D. Set F = f^-1G. Then there exists a commutative diagram of topoi xymatrix Sh(C)/F ar[r]_j_F ar[d]_f' & Sh(C) ar[d]^f Sh(D)/G ar[r]^j_G & Sh(D). The morphism f' is characterized by the property that (f')^-1(H xrightarrowφ G) = (f^-1H xrightarrowf^-1φ F) and we have f'_*j_F^-1 = j_G^-1f_*.","statement_latex":"Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe a morphism of topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$.\nSet $\\mathcal{F} = f^{-1}\\mathcal{G}$. Then there exists\na commutative diagram of topoi\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C})/\\mathcal{F} \\ar[r]_{j_\\mathcal{F}} \\ar[d]_{f'} &\n\\Sh(\\mathcal{C}) \\ar[d]^f \\\\\n\\Sh(\\mathcal{D})/\\mathcal{G} \\ar[r]^{j_\\mathcal{G}} &\n\\Sh(\\mathcal{D}).\n}\n$$\nThe morphism $f'$ is characterized by the property that\n$$\n(f')^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} \\mathcal{G})\n=\n(f^{-1}\\mathcal{H} \\xrightarrow{f^{-1}\\varphi} \\mathcal{F})\n$$\nand we have $f'_*j_\\mathcal{F}^{-1} = j_\\mathcal{G}^{-1}f_*$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04H1","source_file":"sites.tex","source_line":7625,"source_end_line":7646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7625-L7646","statement_sha256":"3ecb25ae0571af1c6486e7bf30a625c198b4dddc685618efd654658552d7f5bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":675,"rank":675,"depth":11,"x":1040.203,"y":252.206,"cluster":"sheaves-sites"},{"id":"stacks:04IU","tag":"04IU","title":"Localization and morphisms of topoi · Lemma 04IU","summary":"Let f : C → D be a morphism of sites given by the continuous functor u : D → C. Let V be an object of D. Set U = u(V). Set G = h_V^\\#, and F = h_U^\\# = f^-1h_V^\\# (see Lemma [Tag 04D3]). Then the diagram of morphisms of topoi of Lemma [Tag 04H1] agrees with the diagram of morphisms of topoi of Lemma [Tag 03CF] via the identifications j_F= j_U and j_G = j_V of Lemma [Tag 04IQ].","statement_latex":"Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites given\nby the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nLet $V$ be an object of $\\mathcal{D}$. Set $U = u(V)$.\nSet $\\mathcal{G} = h_V^\\#$, and\n$\\mathcal{F} = h_U^\\# = f^{-1}h_V^\\#$ (see\nLemma \\ref{lemma-pullback-representable-sheaf}).\nThen the diagram of morphisms of topoi of\nLemma \\ref{lemma-localize-morphism-topoi}\nagrees with the diagram of morphisms of topoi of\nLemma \\ref{lemma-localize-morphism}\nvia the identifications $j_\\mathcal{F}= j_U$\nand $j_\\mathcal{G} = j_V$ of\nLemma \\ref{lemma-localize-compare}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IU","source_file":"sites.tex","source_line":7678,"source_end_line":7693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7678-L7693","statement_sha256":"009a114b59e1d74cc557560f49359954e4fe446dc441b87d60a3d03afd0b1b20","origin":"The Stacks Project","memory_eligible":false,"source_rank":676,"rank":676,"depth":12,"x":1170.398,"y":145.147,"cluster":"sheaves-sites"},{"id":"stacks:04IV","tag":"04IV","title":"Localization and morphisms of topoi · Lemma 04IV","summary":"Let f : Sh(C) → Sh(D) be a morphism of topoi. Let G ∈ Sh(D), F ∈ Sh(C) and s : F → f^-1G a morphism of sheaves. There exists a commutative diagram of topoi xymatrix Sh(C)/F ar[r]_j_F ar[d]_f_s & Sh(C) ar[d]^f Sh(D)/G ar[r]^j_G & Sh(D). We have f_s = f' ∘ j_F/f^-1G where f' : Sh(C)/f^-1G → Sh(D)/F is as in Lemma [Tag 04H1] and j_F/f^-1G : Sh(C)/F → Sh(C)/f^-1G is as in Lemma [Tag 04IR]. The functor (f_s)^-1 is described by the rule (f_s)^-1(H xrightarrowφ G) = (f^-1H…","statement_latex":"Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe a morphism of topoi.\nLet $\\mathcal{G} \\in \\Sh(\\mathcal{D})$,\n$\\mathcal{F} \\in \\Sh(\\mathcal{C})$\nand $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ a morphism of sheaves.\nThere exists a commutative diagram of topoi\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C})/\\mathcal{F} \\ar[r]_{j_\\mathcal{F}} \\ar[d]_{f_s} &\n\\Sh(\\mathcal{C}) \\ar[d]^f \\\\\n\\Sh(\\mathcal{D})/\\mathcal{G} \\ar[r]^{j_\\mathcal{G}} &\n\\Sh(\\mathcal{D}).\n}\n$$\nWe have $f_s = f' \\circ j_{\\mathcal{F}/f^{-1}\\mathcal{G}}$ where\n$f' :\n\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}\n\\to\n\\Sh(\\mathcal{D})/\\mathcal{F}$\nis as in\nLemma \\ref{lemma-localize-morphism-topoi}\nand\n$j_{\\mathcal{F}/f^{-1}\\mathcal{G}} :\n\\Sh(\\mathcal{C})/\\mathcal{F}\n\\to\n\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}$\nis as in\nLemma \\ref{lemma-relocalize-topos}.\nThe functor $(f_s)^{-1}$ is described by the rule\n$$\n(f_s)^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} \\mathcal{G})\n=\n(f^{-1}\\mathcal{H} \\times_{f^{-1}\\varphi, f^{-1}\\mathcal{G}, s} \\mathcal{F}\n\\rightarrow \\mathcal{F}).\n$$\nFinally, given any morphisms $b : \\mathcal{G}' \\to \\mathcal{G}$,\n$a : \\mathcal{F}' \\to \\mathcal{F}$ and\n$s' : \\mathcal{F}' \\to f^{-1}\\mathcal{G}'$ such that\n$$\n\\xymatrix{\n\\mathcal{F}' \\ar[r]_-{s'} \\ar[d]_a & f^{-1}\\mathcal{G}' \\ar[d]^{f^{-1}b} \\\\\n\\mathcal{F} \\ar[r]^-s & f^{-1}\\mathcal{G}\n}\n$$\ncommutes, then the diagram\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C})/\\mathcal{F}'\n\\ar[r]_{j_{\\mathcal{F}'/\\mathcal{F}}} \\ar[d]_{f_{s'}} &\n\\Sh(\\mathcal{C})/\\mathcal{F} \\ar[d]^{f_s} \\\\\n\\Sh(\\mathcal{D})/\\mathcal{G}' \\ar[r]^{j_{\\mathcal{G}'/\\mathcal{G}}} &\n\\Sh(\\mathcal{D})/\\mathcal{G}.\n}\n$$\ncommutes.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IV","source_file":"sites.tex","source_line":7705,"source_end_line":7762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7705-L7762","statement_sha256":"233fd69d092f3c952a7d75dcd49c319c450e17e546fbc9b269cd79c62a001233","origin":"The Stacks Project","memory_eligible":false,"source_rank":677,"rank":677,"depth":12,"x":1160.468,"y":298.277,"cluster":"sheaves-sites"},{"id":"stacks:04IW","tag":"04IW","title":"Localization and morphisms of topoi · Lemma 04IW","summary":"Let f : C → D be a morphism of sites given by the continuous functor u : D → C. Let V be an object of D. Let c : U → u(V) be a morphism. Set G = h_V^\\# and F = h_U^\\# = f^-1h_V^\\#. Let s : F → f^-1G be the map induced by c. Then the diagram of morphisms of topoi of Lemma [Tag 04IN] agrees with the diagram of morphisms of topoi of Lemma [Tag 04IV] via the identifications j_F = j_U and j_G = j_V of Lemma [Tag 04IQ].","statement_latex":"Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites given\nby the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nLet $V$ be an object of $\\mathcal{D}$. Let $c : U \\to u(V)$ be a morphism.\nSet $\\mathcal{G} = h_V^\\#$ and $\\mathcal{F} = h_U^\\# = f^{-1}h_V^\\#$.\nLet $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ be the map induced by $c$.\nThen the diagram of morphisms of topoi of\nLemma \\ref{lemma-relocalize-morphism}\nagrees with the diagram of morphisms of topoi of\nLemma \\ref{lemma-relocalize-morphism-topoi}\nvia the identifications $j_\\mathcal{F} = j_U$\nand $j_\\mathcal{G} = j_V$ of\nLemma \\ref{lemma-localize-compare}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IW","source_file":"sites.tex","source_line":7788,"source_end_line":7802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7788-L7802","statement_sha256":"378b0f3f36c431c23c9ef82d96c37a4a24665c6cb0fe6f86ce8fcdcff358afc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":678,"rank":678,"depth":13,"x":1044.412,"y":179.486,"cluster":"sheaves-sites"},{"id":"stacks:00Y4","tag":"00Y4","title":"Points · Definition 00Y4","summary":"Let C be a site. A point of the topos Sh(C) is a morphism of topoi p from Sh(pt) to Sh(C).","statement_latex":"Let $\\mathcal{C}$ be a site.\nA {\\it point of the topos $\\Sh(\\mathcal{C})$}\nis a morphism of topoi $p$ from $\\Sh(pt)$ to\n$\\Sh(\\mathcal{C})$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Y4","source_file":"sites.tex","source_line":7830,"source_end_line":7836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7830-L7836","statement_sha256":"087f0be1c406be13bf4e0241484799d00cb8e5d2a38a680e4608ecdbf19e29ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":679,"rank":679,"depth":0,"x":1225.883,"y":201.272,"cluster":"sheaves-sites"},{"id":"stacks:00Y5","tag":"00Y5","title":"Points · Definition 00Y5","summary":"Let C be a site. A point p of the site C is given by a functor u : C → Sets such that • For every covering (U_i → U) of C the map coprod u(U_i) → u(U) is surjective. • For every covering (U_i → U) of C and every morphism V → U the maps u(U_i ×_U V) → u(U_i) ×_u(U) u(V) are bijective. • The stalk functor Sh(C) → Sets, F ↦ F_p is left exact.","statement_latex":"Let $\\mathcal{C}$ be a site. A {\\it point $p$ of the site\n$\\mathcal{C}$} is given by a functor $u : \\mathcal{C}\n\\to \\textit{Sets}$ such that\n\\begin{enumerate}\n\\item For every covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$ the map\n$\\coprod u(U_i) \\to u(U)$ is surjective.\n\\item For every covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$ and\nevery morphism $V \\to U$ the maps\n$u(U_i \\times_U V) \\to u(U_i) \\times_{u(U)} u(V)$ are bijective.\n\\item The stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,\n$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is left exact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Y5","source_file":"sites.tex","source_line":7890,"source_end_line":7904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7890-L7904","statement_sha256":"4149284b22f88c338cc7e9b04c74a2ca0600f9c649e9ee525a0e342f236d111c","origin":"The Stacks Project","memory_eligible":false,"source_rank":680,"rank":680,"depth":0,"x":1074.246,"y":288.353,"cluster":"sheaves-sites"},{"id":"stacks:00Y6","tag":"00Y6","title":"Points · Lemma 00Y6","summary":"Let C be a site. Let p = u : C → Sets be a functor. There are functorial isomorphisms (h_U)_p = u(U) for U ∈ Ob(C).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p = u : \\mathcal{C} \\to \\textit{Sets}$ be a functor.\nThere are functorial isomorphisms\n$(h_U)_p = u(U)$ for $U \\in \\Ob(\\mathcal{C})$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Y6","source_file":"sites.tex","source_line":7918,"source_end_line":7924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7918-L7924","statement_sha256":"8b1f3b6e261d76dac28c522568242a53b67d5707cedf2dec594780548e36bd4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":681,"rank":681,"depth":0,"x":1116.117,"y":137.798,"cluster":"sheaves-sites"},{"id":"stacks:00Y7","tag":"00Y7","title":"Points · Lemma 00Y7","summary":"For any functor u : C → Sets. The functor u^p is a right adjoint to the stalk functor on presheaves.","statement_latex":"For any functor $u : \\mathcal{C} \\to \\textit{Sets}$.\nThe functor $u^p$ is a right adjoint to the stalk functor\non presheaves.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Y7","source_file":"sites.tex","source_line":7957,"source_end_line":7962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7957-L7962","statement_sha256":"9356f2aec7474938918760d5793899f73630731d7aba388fb064978d00cc97e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":682,"rank":682,"depth":0,"x":1206.494,"y":272.842,"cluster":"sheaves-sites"},{"id":"stacks:00Y8","tag":"00Y8","title":"Points · Lemma 00Y8","summary":"Let C be a site. Let p = u : C → Sets be a functor. Suppose that for every covering (U_i → U) of C • the map coprod u(U_i) → u(U) is surjective, and • the maps u(U_i ×_U U_j) → u(U_i) ×_u(U) u(U_j) are surjective. Then we have • the presheaf u^pE is a sheaf for all sets E, denote it u^sE, • the stalk functor Sh(C) → Sets and the functor u^s: Sets → Sh(C) are adjoint, and • we have F_p = F^\\#_p for every presheaf of sets F.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p = u : \\mathcal{C} \\to \\textit{Sets}$\nbe a functor. Suppose that for every covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$\n\\begin{enumerate}\n\\item the map $\\coprod u(U_i) \\to u(U)$ is surjective, and\n\\item the maps\n$u(U_i \\times_U U_j) \\to u(U_i) \\times_{u(U)} u(U_j)$ are surjective.\n\\end{enumerate}\nThen we have\n\\begin{enumerate}\n\\item the presheaf $u^pE$ is a sheaf for all sets $E$, denote it $u^sE$,\n\\item the stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$\nand the functor $u^s: \\textit{Sets} \\to \\Sh(\\mathcal{C})$ are\nadjoint, and\n\\item we have $\\mathcal{F}_p = \\mathcal{F}^\\#_p$\nfor every presheaf of sets $\\mathcal{F}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Y8","source_file":"sites.tex","source_line":7982,"source_end_line":8000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L7982-L8000","statement_sha256":"f8f187187916c712f53de8e89a818c16425ac648fd15a097dc0d9fee56ed181e","origin":"The Stacks Project","memory_eligible":false,"source_rank":683,"rank":683,"depth":1,"x":1030.905,"y":224.448,"cluster":"sheaves-sites"},{"id":"stacks:00Y9","tag":"00Y9","title":"Points · Definition 00Y9","summary":"Let p be a point of the site C given by the functor u. For a set E we define p_*E = u^sE the sheaf described in Lemma [Tag 00Y8] above. We sometimes call this a skyscraper sheaf.","statement_latex":"Let $p$ be a point of the site $\\mathcal{C}$ given by the functor $u$.\nFor a set $E$ we define $p_*E = u^sE$ the sheaf\ndescribed in Lemma \\ref{lemma-point-pushforward-sheaf} above.\nWe sometimes call this a {\\it skyscraper sheaf}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Y9","source_file":"sites.tex","source_line":8022,"source_end_line":8028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8022-L8028","statement_sha256":"d57401a52e92cb9e86db688e664224e71c4aaa1fcb761a66ca4eb0c5142e33c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":684,"rank":684,"depth":2,"x":1199.631,"y":160.373,"cluster":"sheaves-sites"},{"id":"stacks:00YA","tag":"00YA","title":"Points · Lemma 00YA","summary":"Let C be a site. • Let p be a point of the site C. Then the pair of functors (p_*, p^-1) introduced above define a morphism of topoi Sh(pt) → Sh(C). • Let p = (p_*, p^-1) be a point of the topos Sh(C). Then the functor u : U ↦ p^-1(h_U^\\#) gives rise to a point p' of the site C whose associated morphism of topoi (p'_*, (p')^-1) is equal to p.","statement_latex":"Let $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item Let $p$ be a point of the site $\\mathcal{C}$.\nThen the pair of functors $(p_*, p^{-1})$ introduced\nabove define a morphism of topoi\n$\\Sh(pt) \\to \\Sh(\\mathcal{C})$.\n\\item Let $p = (p_*, p^{-1})$\nbe a point of the topos $\\Sh(\\mathcal{C})$.\nThen the functor $u : U \\mapsto p^{-1}(h_U^\\#)$ gives\nrise to a point $p'$ of the site $\\mathcal{C}$\nwhose associated morphism of topoi $(p'_*, (p')^{-1})$\nis equal to $p$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YA","source_file":"sites.tex","source_line":8041,"source_end_line":8056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8041-L8056","statement_sha256":"7aa9cd3f36caa1433f3a6f1cb87c31166b26d970133f07bfa6f1bb16931fcb5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":685,"rank":685,"depth":4,"x":1126.598,"y":303.642,"cluster":"sheaves-sites"},{"id":"stacks:04EL","tag":"04EL","title":"Points · Lemma 04EL","summary":"Let C be a site. Let p be a point of C given by u : C → Sets. Let S_0 be an infinite set such that u(U) ⊂ S_0 for all U ∈ Ob(C). Let S be the site constructed out of the powerset S = P(S_0) in Remark [Tag 00XD]. Then • there is an equivalence i : Sh(pt) → Sh(S), • the functor u : C → S induces a morphism of sites f : S → C, and • the composition Sh(pt) → Sh(S) → Sh(C) is the morphism of topoi (p_*, p^-1) of Lemma [Tag 00YA].","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ given by\n$u : \\mathcal{C} \\to \\textit{Sets}$. Let $S_0$ be an infinite set such that\n$u(U) \\subset S_0$ for all $U \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{S}$\nbe the site constructed out of the powerset $S = \\mathcal{P}(S_0)$ in\nRemark \\ref{remark-pt-topos}.\nThen\n\\begin{enumerate}\n\\item there is an equivalence\n$i : \\Sh(pt) \\to \\Sh(\\mathcal{S})$,\n\\item the functor $u : \\mathcal{C} \\to \\mathcal{S}$ induces a morphism of\nsites $f : \\mathcal{S} \\to \\mathcal{C}$, and\n\\item the composition\n$$\n\\Sh(pt) \\to\n\\Sh(\\mathcal{S}) \\to\n\\Sh(\\mathcal{C})\n$$\nis the morphism of topoi $(p_*, p^{-1})$ of\nLemma \\ref{lemma-point-site-topos}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EL","source_file":"sites.tex","source_line":8121,"source_end_line":8143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8121-L8143","statement_sha256":"719a4edcdc71835fd499d7b5940ee377b57e7816d6b3ca814d51fbd8d220718a","origin":"The Stacks Project","memory_eligible":false,"source_rank":686,"rank":686,"depth":5,"x":1065.119,"y":156.269,"cluster":"sheaves-sites"},{"id":"stacks:05UX","tag":"05UX","title":"Points · Lemma 05UX","summary":"Let C be a site. Let p : Sh(pt) → Sh(C) be a point of the topos associated to C. For any set E there are canonical maps E → (p_*E)_p → E whose composition is id_E.","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$p : \\Sh(pt) \\to \\Sh(\\mathcal{C})$ be a point of\nthe topos associated to $\\mathcal{C}$.\nFor any set $E$ there are canonical maps\n$$\nE \\longrightarrow (p_*E)_p \\longrightarrow E\n$$\nwhose composition is $\\text{id}_E$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UX","source_file":"sites.tex","source_line":8169,"source_end_line":8179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8169-L8179","statement_sha256":"eefd10ac8843da949a086e0a34097b3ef8ea5dfb312badf91d3d31b2463935c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":687,"rank":687,"depth":0,"x":1229.286,"y":230.199,"cluster":"sheaves-sites"},{"id":"stacks:05UY","tag":"05UY","title":"Points · Lemma 05UY","summary":"Let C be a site. Let p : Sh(pt) → Sh(C) be a point of the topos associated to C. The functor p_* : Sets → Sh(C) has the following properties: It commutes with arbitrary limits, it is left exact, it is faithful, it transforms surjections into surjections, it commutes with coequalizers, it reflects injections, it reflects surjections, and it reflects isomorphisms.","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$p : \\Sh(pt) \\to \\Sh(\\mathcal{C})$ be a point of\nthe topos associated to $\\mathcal{C}$.\nThe functor $p_* : \\textit{Sets} \\to \\Sh(\\mathcal{C})$\nhas the following properties: It commutes with arbitrary limits,\nit is left exact, it is faithful, it transforms surjections into surjections,\nit commutes with coequalizers, it reflects injections, it reflects\nsurjections, and it reflects isomorphisms.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UY","source_file":"sites.tex","source_line":8196,"source_end_line":8206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8196-L8206","statement_sha256":"4982075452b3acd271de6ac1768a3408160c37d91d25f78a9859f2b04dbbe074","origin":"The Stacks Project","memory_eligible":false,"source_rank":688,"rank":688,"depth":5,"x":1048.428,"y":268.912,"cluster":"sheaves-sites"},{"id":"stacks:0F4E","tag":"0F4E","title":"Constructing points · Lemma 0F4E","summary":"Let C be a site. Let p = u : C → Sets be a functor. If the category of neighbourhoods of p is cofiltered, then the stalk functor ([Tag 04EH]) is left exact.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p = u : \\mathcal{C} \\to \\textit{Sets}$\nbe a functor. If the category of neighbourhoods of $p$ is\ncofiltered, then the stalk functor (\\ref{equation-stalk})\nis left exact.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Constructing points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4E","source_file":"sites.tex","source_line":8237,"source_end_line":8243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8237-L8243","statement_sha256":"5879415d4017e41398e9862c357dc6a51a924e01dcc9a533a5904abe21024a2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":689,"rank":689,"depth":1,"x":1150.857,"y":137.488,"cluster":"sheaves-sites"},{"id":"stacks:00YB","tag":"00YB","title":"Constructing points · Lemma 00YB","summary":"Let C be a site. Assume that C has a final object X and fibred products. Let p = u : C → Sets be a functor such that • u(X) is a singleton set, and • for every pair of morphisms U → W and V → W with the same target the map u(U ×_W V) → u(U) ×_u(W) u(V) is bijective. Then the the category of neighbourhoods of p is cofiltered and consequently the stalk functor Sh(C) → Sets, F → F_p commutes with finite limits.","statement_latex":"Let $\\mathcal{C}$ be a site. Assume that $\\mathcal{C}$ has\na final object $X$ and fibred products.\nLet $p = u : \\mathcal{C} \\to \\textit{Sets}$ be a functor such that\n\\begin{enumerate}\n\\item $u(X)$ is a singleton set, and\n\\item for every pair of morphisms $U \\to W$ and $V \\to W$ with\nthe same target the map\n$u(U \\times_W V) \\to u(U) \\times_{u(W)} u(V)$ is bijective.\n\\end{enumerate}\nThen the the category of neighbourhoods of $p$ is cofiltered\nand consequently the stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,\n$\\mathcal{F} \\to \\mathcal{F}_p$ commutes with finite limits.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Constructing points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YB","source_file":"sites.tex","source_line":8267,"source_end_line":8281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8267-L8281","statement_sha256":"03dfc8a59c00c7a77f51dbb96e6062e6ddcda9214d655e810d73ea0afc7ed8df","origin":"The Stacks Project","memory_eligible":false,"source_rank":690,"rank":690,"depth":3,"x":1181.072,"y":292.818,"cluster":"sheaves-sites"},{"id":"stacks:00YC","tag":"00YC","title":"Constructing points · Proposition 00YC","summary":"Let C be a site. Assume that finite limits exist in C. (I.e., C has fibre products, and a final object.) A point p of such a site C is given by a functor u : C → Sets such that • u commutes with finite limits, and • if (U_i → U) is a covering, then coprod_i u(U_i) → u(U) is surjective.","statement_latex":"Let $\\mathcal{C}$ be a site. Assume that finite limits exist\nin $\\mathcal{C}$. (I.e., $\\mathcal{C}$ has fibre products, and a\nfinal object.) A point $p$ of such a site $\\mathcal{C}$\nis given by a functor $u : \\mathcal{C} \\to \\textit{Sets}$ such that\n\\begin{enumerate}\n\\item $u$ commutes with finite limits, and\n\\item if $\\{U_i \\to U\\}$ is a covering, then\n$\\coprod_i u(U_i) \\to u(U)$ is surjective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Constructing points","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YC","source_file":"sites.tex","source_line":8308,"source_end_line":8319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8308-L8319","statement_sha256":"cca35740ae0bbf843daeb90d9c01a39650094bea0f69e43083825f59ff444d4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":691,"rank":691,"depth":4,"x":1033.597,"y":195.238,"cluster":"sheaves-sites"},{"id":"stacks:0F4F","tag":"0F4F","title":"Points and morphisms of topoi · Lemma 0F4F","summary":"Let u : C → D be a functor. Let v : D → Sets be a functor and set w = v ∘ u. Denote q, resp., p the stalk functor ([Tag 04EH]) associated to v, resp. w. Then (u_pF)_q = F_p functorially in the presheaf F on C.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor. Let\n$v : \\mathcal{D} \\to \\textit{Sets}$ be a functor and set\n$w = v \\circ u$. Denote $q$, resp., $p$ the stalk functor\n(\\ref{equation-stalk}) associated to $v$, resp.\\ $w$.\nThen $(u_p\\mathcal{F})_q = \\mathcal{F}_p$ functorially in the\npresheaf $\\mathcal{F}$ on $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4F","source_file":"sites.tex","source_line":8513,"source_end_line":8521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8513-L8521","statement_sha256":"3af9385423e172ced942d79723e4d4a9549ae81bac097164a25281bcadb12e5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":692,"rank":692,"depth":2,"x":1221.173,"y":183.489,"cluster":"sheaves-sites"},{"id":"stacks:05V1","tag":"05V1","title":"Points and morphisms of topoi · Lemma 05V1","summary":"A map of sites defines a map on points, and pullback respects stalks. Let f : D → C be a morphism of sites given by a continuous functor u : C → D. Let q be a point of D given by the functor v : D → Sets, see Definition [Tag 00Y5]. Then the functor v ∘ u : C → Sets defines a point p of C and moreover there is a canonical identification (f^-1F)_q = F_p for any sheaf F on C.","statement_latex":"\\begin{slogan}\nA map of sites defines a map on points, and pullback respects stalks.\n\\end{slogan}\nLet $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites\ngiven by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.\nLet $q$ be a point of $\\mathcal{D}$ given by the functor\n$v : \\mathcal{D} \\to \\textit{Sets}$, see\nDefinition \\ref{definition-point}.\nThen the functor $v \\circ u : \\mathcal{C} \\to \\textit{Sets}$\ndefines a point $p$ of $\\mathcal{C}$ and moreover there is\na canonical identification\n$$\n(f^{-1}\\mathcal{F})_q = \\mathcal{F}_p\n$$\nfor any sheaf $\\mathcal{F}$ on $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05V1","source_file":"sites.tex","source_line":8557,"source_end_line":8574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8557-L8574","statement_sha256":"3522e9722c5a0171ac034ca9e41f08a9ffbe9a702780d829da4f505c97433fe0","origin":"The Stacks Project","memory_eligible":false,"source_rank":693,"rank":693,"depth":3,"x":1092.06,"y":298.806,"cluster":"sheaves-sites"},{"id":"stacks:05V2","tag":"05V2","title":"Points and morphisms of topoi · Lemma 05V2","summary":"Let f : Sh(D) → Sh(C) be a morphism of topoi. Let q : Sh(pt) → Sh(D) be a point. Then p = f ∘ q is a point of the topos Sh(C) and we have a canonical identification (f^-1F)_q = F_p for any sheaf F on C.","statement_latex":"Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$\nbe a morphism of topoi. Let $q : \\Sh(pt) \\to \\Sh(\\mathcal{D})$\nbe a point. Then $p = f \\circ q$ is a point of the topos\n$\\Sh(\\mathcal{C})$ and we have\na canonical identification\n$$\n(f^{-1}\\mathcal{F})_q = \\mathcal{F}_p\n$$\nfor any sheaf $\\mathcal{F}$ on $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points and morphisms of topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05V2","source_file":"sites.tex","source_line":8638,"source_end_line":8649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8638-L8649","statement_sha256":"a2550a56701b1f3e1105896f7fedc1d2570310cf2ddd6ecb24fdda3b56fc63ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":694,"rank":694,"depth":0,"x":1094.535,"y":140.21,"cluster":"sheaves-sites"},{"id":"stacks:04H2","tag":"04H2","title":"Localization and points · Lemma 04H2","summary":"Let C be a site. Let p be a point of C given by u : C → Sets. Let U be an object of C and let x ∈ u(U). The functor v : C/U → Sets, (φ : V → U) ↦ (y ∈ u(V) mid u(φ)(y) = x) defines a point q of the site C/U such that the diagram xymatrix & Sh(pt) ar[d]^p ar[ld]_q Sh(C/U) ar[r]^j_U & Sh(C) commutes. In other words F_p = (j_U^-1F)_q for any sheaf on C.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ given by\n$u : \\mathcal{C} \\to \\textit{Sets}$. Let $U$ be an object of $\\mathcal{C}$\nand let $x \\in u(U)$. The functor\n$$\nv : \\mathcal{C}/U \\longrightarrow \\textit{Sets}, \\quad\n(\\varphi : V \\to U) \\longmapsto \\{y \\in u(V) \\mid u(\\varphi)(y) = x\\}\n$$\ndefines a point $q$ of the site $\\mathcal{C}/U$ such that the diagram\n$$\n\\xymatrix{\n& \\Sh(pt) \\ar[d]^p \\ar[ld]_q \\\\\n\\Sh(\\mathcal{C}/U) \\ar[r]^{j_U} &\n\\Sh(\\mathcal{C})\n}\n$$\ncommutes. In other words\n$\\mathcal{F}_p = (j_U^{-1}\\mathcal{F})_q$ for any\nsheaf on $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04H2","source_file":"sites.tex","source_line":8667,"source_end_line":8687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8667-L8687","statement_sha256":"6dcf8126b1e3fe0efb078fe2c20c57f424112d3dec4d8ace1c81d5a311a518d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":695,"rank":695,"depth":9,"x":1220.486,"y":258.785,"cluster":"sheaves-sites"},{"id":"stacks:04H3","tag":"04H3","title":"Localization and points · Lemma 04H3","summary":"Let C, p, u, U be as in Lemma [Tag 04H2]. The construction of Lemma [Tag 04H2] gives a one to one correspondence between points q of C/U lying over p and elements x of u(U).","statement_latex":"Let $\\mathcal{C}$, $p$, $u$, $U$ be as in\nLemma \\ref{lemma-point-localize}.\nThe construction of\nLemma \\ref{lemma-point-localize}\ngives a one to one correspondence between points $q$\nof $\\mathcal{C}/U$ lying over $p$ and elements $x$ of $u(U)$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04H3","source_file":"sites.tex","source_line":8749,"source_end_line":8757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8749-L8757","statement_sha256":"161156d0be19108948b401f0aa84e3df0882af023ea9270b4263487e53916b51","origin":"The Stacks Project","memory_eligible":false,"source_rank":696,"rank":696,"depth":10,"x":1031.902,"y":242.787,"cluster":"sheaves-sites"},{"id":"stacks:04H4","tag":"04H4","title":"Localization and points · Lemma 04H4","summary":"Let C be a site. Let p be a point of C given by u : C → Sets. Let U be an object of C. For any sheaf G on C/U we have (j_U!G)_p = coprod_q G_q where the coproduct is over the points q of C/U associated to elements x ∈ u(U) as in Lemma [Tag 04H2].","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p$ be a point of $\\mathcal{C}$ given by\n$u : \\mathcal{C} \\to \\textit{Sets}$. Let $U$ be an object of $\\mathcal{C}$.\nFor any sheaf $\\mathcal{G}$ on $\\mathcal{C}/U$ we have\n$$\n(j_{U!}\\mathcal{G})_p =\n\\coprod\\nolimits_q \\mathcal{G}_q\n$$\nwhere the coproduct is over the points $q$ of $\\mathcal{C}/U$\nassociated to elements $x \\in u(U)$ as in\nLemma \\ref{lemma-point-localize}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Localization and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04H4","source_file":"sites.tex","source_line":8813,"source_end_line":8825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8813-L8825","statement_sha256":"fece7557a357fe59e95bc49816c4915da078235fa9534cd12b9ec1fa4657b017","origin":"The Stacks Project","memory_eligible":false,"source_rank":697,"rank":697,"depth":10,"x":1184.112,"y":147.398,"cluster":"sheaves-sites"},{"id":"stacks:04IA","tag":"04IA","title":"2-morphisms of topoi · Definition 04IA","summary":"Let f, g : Sh(C) → Sh(D) be two morphisms of topoi. A 2-morphism from f to g is given by a transformation of functors t : f_* → g_*.","statement_latex":"Let $f, g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe two morphisms of topoi. A {\\it 2-morphism from $f$ to $g$}\nis given by a transformation of functors $t : f_* \\to g_*$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"2-morphisms of topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IA","source_file":"sites.tex","source_line":8871,"source_end_line":8876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8871-L8876","statement_sha256":"61f356cdc5d6ceedc25f05726c79949655fc95eb0f19d16a12584c1d2d906f74","origin":"The Stacks Project","memory_eligible":false,"source_rank":698,"rank":698,"depth":0,"x":1148.517,"y":304.397,"cluster":"sheaves-sites"},{"id":"stacks:00YH","tag":"00YH","title":"Morphisms between points · Lemma 00YH","summary":"Let C be a site. Let u, u' : C → Sets be two functors, and let t : u' → u be a transformation of functors. Then we obtain a canonical transformation of stalk functors t_stalk : F_p' → F_p which agrees with t via the identifications of Lemma [Tag 00Y6].","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $u, u' : \\mathcal{C} \\to \\textit{Sets}$ be two\nfunctors, and let $t : u' \\to u$ be a transformation of functors.\nThen we obtain a canonical transformation of stalk\nfunctors $t_{stalk} : \\mathcal{F}_{p'} \\to \\mathcal{F}_p$\nwhich agrees with $t$ via the identifications of\nLemma \\ref{lemma-points-recover}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms between points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YH","source_file":"sites.tex","source_line":8962,"source_end_line":8971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8962-L8971","statement_sha256":"e02abb5ec2c1c36b91d6af635e980423c284e64a60eb2ccf75af7c2488973f6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":699,"rank":699,"depth":1,"x":1048.324,"y":168.178,"cluster":"sheaves-sites"},{"id":"stacks:00YI","tag":"00YI","title":"Morphisms between points · Definition 00YI","summary":"Let C be a site. Let p, p' be points of C given by functors u, u' : C → Sets. A morphism f : p → p' is given by a transformation of functors f_u : u' → u.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p, p'$ be points of $\\mathcal{C}$\ngiven by functors $u, u' : \\mathcal{C} \\to \\textit{Sets}$.\nA {\\it morphism $f : p \\to p'$} is given by a transformation of\nfunctors\n$$\nf_u : u' \\to u.\n$$","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Morphisms between points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YI","source_file":"sites.tex","source_line":8977,"source_end_line":8986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L8977-L8986","statement_sha256":"8bf1b869ce6fd8b990514ddf7b0ed823d92ac1cc0b7f33687dcb55331711aaad","origin":"The Stacks Project","memory_eligible":false,"source_rank":700,"rank":700,"depth":0,"x":1232.089,"y":211.853,"cluster":"sheaves-sites"},{"id":"stacks:00YK","tag":"00YK","title":"Sites with enough points · Definition 00YK","summary":"Let C be a site. • A family of points (p_i)_i∈ I is called conservative if every map of sheaves φ : F → G which is an isomorphism on all the fibres F_p_i → G_p_i is an isomorphism. • We say that C has enough points if there exists a conservative family of points.","statement_latex":"Let $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item A family of points $\\{p_i\\}_{i\\in I}$ is called {\\it conservative}\nif every map of sheaves $\\phi : \\mathcal{F} \\to \\mathcal{G}$\nwhich is an isomorphism on all the fibres $\\mathcal{F}_{p_i}\n\\to \\mathcal{G}_{p_i}$ is an isomorphism.\n\\item  We say that $\\mathcal{C}$ {\\it has enough points}\nif there exists a conservative family of points.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sites with enough points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YK","source_file":"sites.tex","source_line":9027,"source_end_line":9038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9027-L9038","statement_sha256":"4d74f1418051a1d266327752120cb3ab28cde131787efaf1fde047d6306ae9c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":701,"rank":701,"depth":0,"x":1061.146,"y":284.053,"cluster":"sheaves-sites"},{"id":"stacks:00YL","tag":"00YL","title":"Sites with enough points · Lemma 00YL","summary":"Let C be a site and let (p_i)_i∈ I be a conservative family of points. Then • Given any map of sheaves φ : F → G we have ∀ i, φ_p_i injective implies φ injective. • Given any map of sheaves φ : F → G we have ∀ i, φ_p_i surjective implies φ surjective. • Given any pair of maps of sheaves φ_1, φ_2 : F → G we have ∀ i, φ_1, p_i = φ_2, p_i implies φ_1 = φ_2. • Given a finite diagram G : J → Sh(C), a sheaf F and morphisms q_j : F → G_j then (F, q_j) is a limit of the diagram…","statement_latex":"Let $\\mathcal{C}$ be a site and let $\\{p_i\\}_{i\\in I}$ be a conservative\nfamily of points. Then\n\\begin{enumerate}\n\\item Given any map of sheaves $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nwe have $\\forall i, \\varphi_{p_i}$ injective implies $\\varphi$ injective.\n\\item Given any map of sheaves $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nwe have $\\forall i, \\varphi_{p_i}$ surjective implies $\\varphi$ surjective.\n\\item Given any pair of maps of sheaves\n$\\varphi_1, \\varphi_2 : \\mathcal{F} \\to \\mathcal{G}$\nwe have $\\forall i, \\varphi_{1, p_i} = \\varphi_{2, p_i}$\nimplies $\\varphi_1 = \\varphi_2$.\n\\item Given a finite diagram $\\mathcal{G} : \\mathcal{J}\n\\to \\Sh(\\mathcal{C})$, a sheaf $\\mathcal{F}$ and morphisms\n$q_j : \\mathcal{F} \\to \\mathcal{G}_j$ then $(\\mathcal{F}, q_j)$\nis a limit of the diagram if and only if for each $i$ the stalk\n$(\\mathcal{F}_{p_i}, (q_j)_{p_i})$ is one.\n\\item Given a finite diagram $\\mathcal{F} : \\mathcal{J}\n\\to \\Sh(\\mathcal{C})$, a sheaf $\\mathcal{G}$ and morphisms\n$e_j : \\mathcal{F}_j \\to \\mathcal{G}$ then $(\\mathcal{G}, e_j)$\nis a colimit of the diagram if and only if for each $i$ the stalk\n$(\\mathcal{G}_{p_i}, (e_j)_{p_i})$ is one.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sites with enough points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YL","source_file":"sites.tex","source_line":9043,"source_end_line":9067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9043-L9067","statement_sha256":"04d174052b6a9089a22ee2f396e1b9d4dc9b1b4cab9d46246478595c7b4afc65","origin":"The Stacks Project","memory_eligible":false,"source_rank":702,"rank":702,"depth":2,"x":1129.261,"y":133.54,"cluster":"sheaves-sites"},{"id":"stacks:00YM","tag":"00YM","title":"Sites with enough points · Lemma 00YM","summary":"Let C be a site and let ((p_i, u_i))_i∈ I be a family of points. The family is conservative if and only if for every sheaf F and every U∈ Ob(C) and every pair of distinct sections s, s' ∈ F(U), s not = s' there exists an i and x∈ u_i(U) such that the triples (U, x, s) and (U, x, s') define distinct elements of F_p_i.","statement_latex":"Let $\\mathcal{C}$ be a site and let $\\{(p_i, u_i)\\}_{i\\in I}$ be a\nfamily of points. The family is conservative if and only if for every\nsheaf $\\mathcal{F}$ and every $U\\in \\Ob(\\mathcal{C})$ and every\npair of distinct sections $s, s' \\in \\mathcal{F}(U)$, $s \\not = s'$ there\nexists an $i$ and $x\\in u_i(U)$ such that the triples\n$(U, x, s)$ and $(U, x, s')$ define distinct elements of\n$\\mathcal{F}_{p_i}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sites with enough points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YM","source_file":"sites.tex","source_line":9092,"source_end_line":9101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9092-L9101","statement_sha256":"96bbba9d315d18b666dbe502f429e3ca80038630a03e27ffbbb1ca9076b93350","origin":"The Stacks Project","memory_eligible":false,"source_rank":703,"rank":703,"depth":3,"x":1200.202,"y":283.451,"cluster":"sheaves-sites"},{"id":"stacks:04H6","tag":"04H6","title":"Sites with enough points · Lemma 04H6","summary":"Let C be a site. Let U be an object of C. let ((p_i, u_i))_i∈ I be a family of points of C. For x ∈ u_i(U) let q_i, x be the point of C/U constructed in Lemma [Tag 04H2]. If (p_i) is a conservative family of points, then (q_i, x)_i ∈ I, x ∈ u_i(U) is a conservative family of points of C/U. In particular, if C has enough points, then so does every localization C/U.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.\nlet $\\{(p_i, u_i)\\}_{i\\in I}$ be a family of points of $\\mathcal{C}$.\nFor $x \\in u_i(U)$ let $q_{i, x}$ be the point of $\\mathcal{C}/U$\nconstructed in\nLemma \\ref{lemma-point-localize}.\nIf $\\{p_i\\}$ is a conservative family of points, then\n$\\{q_{i, x}\\}_{i \\in I, x \\in u_i(U)}$ is a conservative\nfamily of points of $\\mathcal{C}/U$.\nIn particular, if $\\mathcal{C}$ has enough points, then so\ndoes every localization $\\mathcal{C}/U$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sites with enough points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04H6","source_file":"sites.tex","source_line":9137,"source_end_line":9149,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9137-L9149","statement_sha256":"22aa4207246ae8b236448b55b257a7fb006949693fa6f27e3935577ca58452be","origin":"The Stacks Project","memory_eligible":false,"source_rank":704,"rank":704,"depth":11,"x":1027.021,"y":213.034,"cluster":"sheaves-sites"},{"id":"stacks:06UL","tag":"06UL","title":"Sites with enough points · Lemma 06UL","summary":"Let C be a site. Let (U_i)_i ∈ I be a family of objects of C. Assume • coprod h_U_i^\\# → * is a surjective map of sheaves, and • each localization C/U_i has enough points. Then C has enough points.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\{U_i\\}_{i \\in I}$ be a family of\nobjects of $\\mathcal{C}$. Assume\n\\begin{enumerate}\n\\item $\\coprod h_{U_i}^\\# \\to *$ is a surjective map of sheaves, and\n\\item each localization $\\mathcal{C}/U_i$ has enough points.\n\\end{enumerate}\nThen $\\mathcal{C}$ has enough points.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sites with enough points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UL","source_file":"sites.tex","source_line":9165,"source_end_line":9174,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9165-L9174","statement_sha256":"b6e49d97ee89822a9636d56dbdbf16f1a9943d962183d6db205e1f1fad263979","origin":"The Stacks Project","memory_eligible":false,"source_rank":705,"rank":705,"depth":1,"x":1211.686,"y":166.607,"cluster":"sheaves-sites"},{"id":"stacks:0F4G","tag":"0F4G","title":"Sites with enough points · Lemma 0F4G","summary":"Let u : C → D be a continuous functor of sites. Let ((q_i, v_i))_i∈ I be a conservative family of points of D. If each functor u_i = v_i ∘ u defines a point of C, then u defines a morphism of sites f : D → C.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites.\nLet $\\{(q_i, v_i)\\}_{i\\in I}$ be a conservative family of points of\n$\\mathcal{D}$. If each functor $u_i = v_i \\circ u$ defines\na point of $\\mathcal{C}$,\nthen $u$ defines a morphism of sites $f : \\mathcal{D} \\to \\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sites with enough points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4G","source_file":"sites.tex","source_line":9206,"source_end_line":9213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9206-L9213","statement_sha256":"d442ee0453eedfc81a21e60405c424fb388c28e53f82adda7087e629797794e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":706,"rank":706,"depth":3,"x":1112.672,"y":305.877,"cluster":"sheaves-sites"},{"id":"stacks:00YO","tag":"00YO","title":"Criterion for existence of points · Lemma 00YO","summary":"Let C be a site. Let (J, ≥, V_j, g_jj') be a system as above with associated pair of functors (u', p'). Let F be a sheaf on C. Let s, s' ∈ F_p' be distinct elements. Let (W_k → W) be a finite covering of C. Let f ∈ u'(W). There exists a refinement (I, ≥, U_i, f_ii') of (J, ≥, V_j, g_jj') such that s, s' map to distinct elements of F_p and that the image of f in u(W) is in the image of one of the u(W_k).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $(J, \\geq, V_j, g_{jj'})$ be a system as above with associated\npair of functors $(u', p')$.\nLet $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.\nLet $s, s' \\in \\mathcal{F}_{p'}$ be distinct elements.\nLet $\\{W_k \\to W\\}$ be a finite covering of $\\mathcal{C}$.\nLet $f \\in u'(W)$.\nThere exists a refinement $(I, \\geq, U_i, f_{ii'})$\nof $(J, \\geq, V_j, g_{jj'})$ such that $s, s'$ map\nto distinct elements of $\\mathcal{F}_p$ and that\nthe image of $f$ in $u(W)$ is in the image of one of\nthe $u(W_k)$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Criterion for existence of points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YO","source_file":"sites.tex","source_line":9285,"source_end_line":9299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9285-L9299","statement_sha256":"afbbe342cbb9a8ff98afe6844279d4433b24d2173e6a4bf51aeb51a18f7f538c","origin":"The Stacks Project","memory_eligible":false,"source_rank":707,"rank":707,"depth":1,"x":1073.616,"y":146.711,"cluster":"sheaves-sites"},{"id":"stacks:00YP","tag":"00YP","title":"Criterion for existence of points · Lemma 00YP","summary":"Let C be a site. Let (J, ≥, V_j, g_jj') be a system as above with associated pair of functors (u', p'). Let F be a sheaf on C. Let s, s' ∈ F_p' be distinct elements. There exists a refinement (I, ≥, U_i, f_ii') of (J, ≥, V_j, g_jj') such that s, s' map to distinct elements of F_p and such that for every finite covering (W_k → W) of the site C, and any f ∈ u'(W) the image of f in u(W) is in the image of one of the u(W_k).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $(J, \\geq, V_j, g_{jj'})$ be a system as above with associated\npair of functors $(u', p')$.\nLet $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.\nLet $s, s' \\in \\mathcal{F}_{p'}$ be distinct elements.\nThere exists a refinement $(I, \\geq, U_i, f_{ii'})$\nof $(J, \\geq, V_j, g_{jj'})$ such that $s, s'$ map\nto distinct elements of $\\mathcal{F}_p$ and such that\nfor every finite covering $\\{W_k \\to W\\}$ of the site\n$\\mathcal{C}$, and any $f \\in u'(W)$ the image of $f$ in $u(W)$\nis in the image of one of the $u(W_k)$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Criterion for existence of points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YP","source_file":"sites.tex","source_line":9329,"source_end_line":9342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9329-L9342","statement_sha256":"7b455307799b334c11c8fa3105ad337396d5a7c6e6a37e8464f874cff89a9549","origin":"The Stacks Project","memory_eligible":false,"source_rank":708,"rank":708,"depth":2,"x":1230.694,"y":242.088,"cluster":"sheaves-sites"},{"id":"stacks:00YQ","tag":"00YQ","title":"Criterion for existence of points · Proposition 00YQ","summary":"[SGA4] Let C be a site. Assume that • finite limits exist in C, and • every covering (U_i → U)_i ∈ I has a refinement by a finite covering of C. Then C has enough points.","statement_latex":"\\begin{reference}\n\\cite[Expos\\'e VI, Appendix by Deligne, Proposition 9.0]{SGA4}\n\\end{reference}\nLet $\\mathcal{C}$ be a site. Assume that\n\\begin{enumerate}\n\\item finite limits exist in $\\mathcal{C}$, and\n\\item every covering $\\{U_i \\to U\\}_{i \\in I}$\nhas a refinement by a finite covering of $\\mathcal{C}$.\n\\end{enumerate}\nThen $\\mathcal{C}$ has enough points.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Criterion for existence of points","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YQ","source_file":"sites.tex","source_line":9400,"source_end_line":9412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9400-L9412","statement_sha256":"f1b1e64cacb5ff57b9c863ee4c11234dba9af8e586f82c35be1bef12dc938cb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":709,"rank":709,"depth":5,"x":1037.822,"y":260.923,"cluster":"sheaves-sites"},{"id":"stacks:0DW0","tag":"0DW0","title":"Criterion for existence of points · Lemma 0DW0","summary":"Let C be a site. Let I be a set and for i ∈ I let U_i be an object of C such that • coprod h_U_i surjects onto the final object of Sh(C), and • C/U_i satisfies the hypotheses of Proposition [Tag 00YQ]. Then C has enough points.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $I$ be a set and for\n$i \\in I$ let $U_i$ be an object of $\\mathcal{C}$ such that\n\\begin{enumerate}\n\\item $\\coprod h_{U_i}$ surjects onto\nthe final object of $\\Sh(\\mathcal{C})$, and\n\\item $\\mathcal{C}/U_i$ satisfies the hypotheses of\nProposition \\ref{proposition-criterion-points}.\n\\end{enumerate}\nThen $\\mathcal{C}$ has enough points.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Criterion for existence of points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DW0","source_file":"sites.tex","source_line":9438,"source_end_line":9449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9438-L9449","statement_sha256":"3f821beb197d4b1108c0120b4653e9044bdf8c78e5e71c78e6eb55063181471d","origin":"The Stacks Project","memory_eligible":false,"source_rank":710,"rank":710,"depth":6,"x":1165.124,"y":137.373,"cluster":"sheaves-sites"},{"id":"stacks:090K","tag":"090K","title":"Weakly contractible objects · Lemma 090K","summary":"Let C be a site. Let U be an object of C. The following conditions are equivalent • For every covering (U_i → U) there exists a map of sheaves h_U^\\# → coprod h_U_i^\\# right inverse to the sheafification of coprod h_U_i → h_U. • For every surjection of sheaves of sets F → G the map F(U) → G(U) is surjective.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.\nThe following conditions are equivalent\n\\begin{enumerate}\n\\item For every covering $\\{U_i \\to U\\}$ there exists a map of\nsheaves $h_U^\\# \\to \\coprod h_{U_i}^\\#$ right inverse to the sheafification\nof $\\coprod h_{U_i} \\to h_U$.\n\\item For every surjection of sheaves of sets $\\mathcal{F} \\to \\mathcal{G}$\nthe map $\\mathcal{F}(U) \\to \\mathcal{G}(U)$ is surjective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Weakly contractible objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090K","source_file":"sites.tex","source_line":9479,"source_end_line":9490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9479-L9490","statement_sha256":"260d5fcffc414a6b3a1a15b11bed49133b5dc347ee5a98d828a753027b3a8ae4","origin":"The Stacks Project","memory_eligible":false,"source_rank":711,"rank":711,"depth":2,"x":1170.618,"y":301.002,"cluster":"sheaves-sites"},{"id":"stacks:090L","tag":"090L","title":"Weakly contractible objects · Definition 090L","summary":"Let C be a site. • We say an object U of C is weakly contractible if the equivalent conditions of Lemma [Tag 090K] hold. • We say a site has enough weakly contractible objects if every object U of C has a covering (U_i → U) with U_i weakly contractible for all i. • More generally, if P is a property of objects of C we say that C has enough P objects if every object U of C has a covering (U_i → U) such that U_i has P for all i.","statement_latex":"Let $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item We say an object $U$ of $\\mathcal{C}$ is {\\it weakly contractible}\nif the equivalent conditions of Lemma \\ref{lemma-w-contractible} hold.\n\\item We say a site has {\\it enough weakly contractible objects}\nif every object $U$ of $\\mathcal{C}$ has a covering $\\{U_i \\to U\\}$\nwith $U_i$ weakly contractible for all $i$.\n\\item More generally, if $P$ is a property of objects of $\\mathcal{C}$\nwe say that $\\mathcal{C}$ has {\\it enough $P$ objects} if every object $U$ of\n$\\mathcal{C}$ has a covering $\\{U_i \\to U\\}$ such that $U_i$ has $P$\nfor all $i$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Weakly contractible objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090L","source_file":"sites.tex","source_line":9514,"source_end_line":9528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9514-L9528","statement_sha256":"e6b3ab4c460e91aecfc4316e95e381da4c08dfa458ca025f957bda0b3841b2e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":712,"rank":712,"depth":3,"x":1034.741,"y":183.253,"cluster":"sheaves-sites"},{"id":"stacks:04D6","tag":"04D6","title":"Exactness properties of pushforward · Lemma 04D6","summary":"Let f : Sh(C) → Sh(D) be a morphism of topoi. Consider the following properties (on sheaves of sets): • f_* is faithful, • f_* is fully faithful, • f^-1f_*F → F is surjective for all F in Sh(C), • f_* transforms surjections into surjections, • f_* commutes with coequalizers, • f_* commutes with pushouts, • f^-1f_*F → F is an isomorphism for all F in Sh(C), • f_* reflects injections, • f_* reflects surjections, • f_* reflects bijections, and • for any surjection F → f^-1G…","statement_latex":"Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be\na morphism of topoi. Consider the following properties (on sheaves\nof sets):\n\\begin{enumerate}\n\\item $f_*$ is faithful,\n\\item $f_*$ is fully faithful,\n\\item $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is surjective for\nall $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$,\n\\item $f_*$ transforms surjections into surjections,\n\\item $f_*$ commutes with coequalizers,\n\\item $f_*$ commutes with pushouts,\n\\item $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is an isomorphism for\nall $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$,\n\\item $f_*$ reflects injections,\n\\item $f_*$ reflects surjections,\n\\item $f_*$ reflects bijections, and\n\\item for any surjection $\\mathcal{F} \\to f^{-1}\\mathcal{G}$ there\nexists a surjection $\\mathcal{G}' \\to \\mathcal{G}$ such that\n$f^{-1}\\mathcal{G}' \\to f^{-1}\\mathcal{G}$ factors through\n$\\mathcal{F} \\to f^{-1}\\mathcal{G}$.\n\\end{enumerate}\nThen we have the following implications\n\\begin{enumerate}\n\\item[(a)] (2) $\\Rightarrow$ (1),\n\\item[(b)] (3) $\\Rightarrow$ (1),\n\\item[(c)] (7) $\\Rightarrow$ (1), (2), (3), (8), (9), (10).\n\\item[(d)] (3) $\\Leftrightarrow$ (9),\n\\item[(e)] (6) $\\Rightarrow$ (4) and (5) $\\Rightarrow$ (4),\n\\item[(f)] (4) $\\Leftrightarrow$ (11),\n\\item[(g)] (9) $\\Rightarrow$ (8), (10), and\n\\item[(h)] (2) $\\Leftrightarrow$ (7).\n\\end{enumerate}\nPicture\n$$\n\\xymatrix{\n(6) \\ar@{=>}[rd] & & & & & (9) \\ar@{=>}[r] \\ar@{=>}[rd] & (8) \\\\\n& (4) \\ar@{<=>}[r] & (11) &\n(2) \\ar@{<=>}[r] &\n(7) \\ar@{=>}[ru] \\ar@{=>}[rd] & & (10) \\\\\n(5) \\ar@{=>}[ur] & & & & & (3) \\ar@{=>}[r] & (1)\n}\n$$","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Exactness properties of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04D6","source_file":"sites.tex","source_line":9553,"source_end_line":9597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9553-L9597","statement_sha256":"afacdd387c4994e4f937b6cc492bf8d992496c4a722b03975d290f3ae7f768d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":713,"rank":713,"depth":3,"x":1229.97,"y":192.997,"cluster":"sheaves-sites"},{"id":"stacks:04D7","tag":"04D7","title":"Exactness properties of pushforward · Lemma 04D7","summary":"Let f : D → C be a morphism of sites associated to the continuous functor u : C → D. Assume that for any object U of C and any covering (V_j → u(U)) in D there exists a covering (U_i → U) in C such that the map of sheaves coprod h_u(U_i)^\\# → h_u(U)^\\# factors through the map of sheaves coprod h_V_j^\\# → h_u(U)^\\#. Then f_* transforms surjective maps of sheaves into surjective maps of sheaves.","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites associated to\nthe continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.\nAssume that for any object $U$ of $\\mathcal{C}$ and any covering\n$\\{V_j \\to u(U)\\}$ in $\\mathcal{D}$ there exists a covering $\\{U_i \\to U\\}$\nin $\\mathcal{C}$ such that the map of sheaves\n$$\n\\coprod h_{u(U_i)}^\\# \\to h_{u(U)}^\\#\n$$\nfactors through the map of sheaves\n$$\n\\coprod h_{V_j}^\\# \\to h_{u(U)}^\\#.\n$$\nThen $f_*$ transforms surjective maps of sheaves into surjective maps of\nsheaves.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Exactness properties of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04D7","source_file":"sites.tex","source_line":9749,"source_end_line":9765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9749-L9765","statement_sha256":"41d6f80e87f04c0ec1fccb0dcbddbca2e532b70fa5fbffb5dca2796f3e77ff6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":714,"rank":714,"depth":0,"x":1077.908,"y":296.772,"cluster":"sheaves-sites"},{"id":"stacks:04D9","tag":"04D9","title":"Exactness properties of pushforward · Lemma 04D9","summary":"Let f : D → C be a morphism of sites given by the functor u : C → D. Assume that for every object V of D there exist objects U_i of C and morphisms u(U_i) → V such that (u(U_i) → V) is a covering of D. In this case the functor f_* : Sh(D) → Sh(C) reflects injections and surjections.","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites given\nby the functor $u : \\mathcal{C} \\to \\mathcal{D}$.\nAssume that for every object $V$ of $\\mathcal{D}$ there exist objects\n$U_i$ of $\\mathcal{C}$ and morphisms $u(U_i) \\to V$ such that\n$\\{u(U_i) \\to V\\}$ is a covering of $\\mathcal{D}$. In this case the functor\n$f_* : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ reflects\ninjections and surjections.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Exactness properties of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04D9","source_file":"sites.tex","source_line":9829,"source_end_line":9838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9829-L9838","statement_sha256":"72c9fd13977c225fcb99751afa42ad94a110aea7e6f93d3b409e3da6a045aa85","origin":"The Stacks Project","memory_eligible":false,"source_rank":715,"rank":715,"depth":0,"x":1106.633,"y":133.68,"cluster":"sheaves-sites"},{"id":"stacks:04B5","tag":"04B5","title":"Almost cocontinuous functors · Definition 04B5","summary":"Let C be a site. We say an object U of C is sheaf theoretically empty if ∅^\\# → h_U^\\# is an isomorphism of sheaves.","statement_latex":"Let $\\mathcal{C}$ be a site. We say an object $U$ of $\\mathcal{C}$\nis {\\it sheaf theoretically empty} if $\\emptyset^\\# \\to h_U^\\#$\nis an isomorphism of sheaves.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Almost cocontinuous functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04B5","source_file":"sites.tex","source_line":9907,"source_end_line":9912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9907-L9912","statement_sha256":"1db27eee00b7268ca94036fc106ddef02ca939e9429cc6188390538b75797786","origin":"The Stacks Project","memory_eligible":false,"source_rank":716,"rank":716,"depth":0,"x":1216.798,"y":270.48,"cluster":"sheaves-sites"},{"id":"stacks:04B6","tag":"04B6","title":"Almost cocontinuous functors · Lemma 04B6","summary":"Let C be a site. Let U be an object of C. The following are equivalent: • U is sheaf theoretically empty, • F(U) is a singleton for each sheaf F, • ∅^\\#(U) is a singleton, • ∅^\\#(U) is nonempty, and • the empty family is a covering of U in C. Moreover, if U is sheaf theoretically empty, then for any morphism U' → U of C the object U' is sheaf theoretically empty.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U$ be an object of $\\mathcal{C}$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $U$ is sheaf theoretically empty,\n\\item $\\mathcal{F}(U)$ is a singleton for each sheaf $\\mathcal{F}$,\n\\item $\\emptyset^\\#(U)$ is a singleton,\n\\item $\\emptyset^\\#(U)$ is nonempty, and\n\\item the empty family is a covering of $U$ in $\\mathcal{C}$.\n\\end{enumerate}\nMoreover, if $U$ is sheaf theoretically empty, then for any morphism\n$U' \\to U$ of $\\mathcal{C}$ the object $U'$ is sheaf theoretically empty.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Almost cocontinuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04B6","source_file":"sites.tex","source_line":9917,"source_end_line":9930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9917-L9930","statement_sha256":"1da6f775e1da75c4efb8e09605c64c996922858a5184c0aefa8124123d6d4a99","origin":"The Stacks Project","memory_eligible":false,"source_rank":717,"rank":717,"depth":3,"x":1025.22,"y":232.049,"cluster":"sheaves-sites"},{"id":"stacks:04B7","tag":"04B7","title":"Almost cocontinuous functors · Definition 04B7","summary":"Let C, D be sites. Let u : C → D be a functor. We say u is almost cocontinuous if for every object U of C and every covering (V_j → u(U))_j ∈ J there exists a covering (U_i → U)_i ∈ I in C such that for each i in I we have at least one of the following two conditions • u(U_i) is sheaf theoretically empty, or • the morphism u(U_i) → u(U) factors through V_j for some j ∈ J.","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nWe say $u$ is {\\it almost cocontinuous} if for every\nobject $U$ of $\\mathcal{C}$ and every covering\n$\\{V_j \\to u(U)\\}_{j \\in J}$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$ such that\nfor each $i$ in $I$ we have at least one of the following two conditions\n\\begin{enumerate}\n\\item $u(U_i)$ is sheaf theoretically empty, or\n\\item the morphism $u(U_i) \\to u(U)$ factors through $V_j$ for some $j \\in J$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Almost cocontinuous functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04B7","source_file":"sites.tex","source_line":9952,"source_end_line":9965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9952-L9965","statement_sha256":"c4f87556ec63349667ba7a1dec8124a62968c3aa97c1e596a6c4f671102a258f","origin":"The Stacks Project","memory_eligible":false,"source_rank":718,"rank":718,"depth":0,"x":1197.691,"y":151.542,"cluster":"sheaves-sites"},{"id":"stacks:04B8","tag":"04B8","title":"Almost cocontinuous functors · Lemma 04B8","summary":"Let C, D be sites. Let u : C → D be a functor. Assume that u is continuous and almost cocontinuous. Let G be a presheaf on D such that G(V) is a singleton whenever V is sheaf theoretically empty. Then (u^pG)^\\# = u^p(G^\\#).","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume that $u$ is continuous and almost cocontinuous.\nLet $\\mathcal{G}$ be a presheaf on $\\mathcal{D}$ such that $\\mathcal{G}(V)$ is\na singleton whenever $V$ is sheaf theoretically empty.\nThen $(u^p\\mathcal{G})^\\# = u^p(\\mathcal{G}^\\#)$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Almost cocontinuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04B8","source_file":"sites.tex","source_line":9979,"source_end_line":9987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L9979-L9987","statement_sha256":"28e2d612c9281deb1a5e684b10a54e36378775b2f0613af251ba7bc3cc5d08d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":719,"rank":719,"depth":1,"x":1135.146,"y":309.043,"cluster":"sheaves-sites"},{"id":"stacks:04B9","tag":"04B9","title":"Almost cocontinuous functors · Lemma 04B9","summary":"Let C, D be sites. Let u : C → D be a functor. Assume that u is continuous and almost cocontinuous. Then u^s = u^p : Sh(D) → Sh(C) commutes with pushouts and coequalizers (and more generally finite connected colimits).","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume that $u$ is continuous and almost cocontinuous.\nThen $u^s = u^p : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$\ncommutes with pushouts and coequalizers (and more generally\nfinite connected colimits).","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Almost cocontinuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04B9","source_file":"sites.tex","source_line":10029,"source_end_line":10037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10029-L10037","statement_sha256":"7291fd9d2842da0256e804a68d50e6f16e8f1102e314d7273ab6ef07b3d1cb49","origin":"The Stacks Project","memory_eligible":false,"source_rank":720,"rank":720,"depth":3,"x":1054.469,"y":157.153,"cluster":"sheaves-sites"},{"id":"stacks:04BA","tag":"04BA","title":"Almost cocontinuous functors · Lemma 04BA","summary":"Let f : D → C be a morphism of sites associated to the continuous functor u : C → D. If u is almost cocontinuous then f_* commutes with pushouts and coequalizers (and more generally finite connected colimits).","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites\nassociated to the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.\nIf $u$ is almost cocontinuous then $f_*$ commutes with\npushouts and coequalizers (and more generally finite connected colimits).","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Almost cocontinuous functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BA","source_file":"sites.tex","source_line":10063,"source_end_line":10069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10063-L10069","statement_sha256":"16d473c4ff45b070b4b990c0fa3b92d3fcbff4534d8fd0859ca33ed7d5fa9133","origin":"The Stacks Project","memory_eligible":false,"source_rank":721,"rank":721,"depth":4,"x":1236.417,"y":223.49,"cluster":"sheaves-sites"},{"id":"stacks:08LU","tag":"08LU","title":"Subtopoi · Definition 08LU","summary":"Let C and D be sites. A morphism of topoi f : Sh(D) → Sh(C) is called an embedding if f_* is fully faithful.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nA morphism of topoi $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$\nis called an {\\it embedding} if $f_*$ is fully faithful.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Subtopoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LU","source_file":"sites.tex","source_line":10088,"source_end_line":10093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10088-L10093","statement_sha256":"f9f21cfdcfdb5e897d93360aef0192e651eb79913f5eec6004e86ee84877a80f","origin":"The Stacks Project","memory_eligible":false,"source_rank":722,"rank":722,"depth":0,"x":1048.584,"y":277.909,"cluster":"sheaves-sites"},{"id":"stacks:08LV","tag":"08LV","title":"Subtopoi · Definition 08LV","summary":"Let C be a site. A strictly full subcategory E ⊂ Sh(C) is a subtopos if there exists an embedding of topoi f : Sh(D) → Sh(C) such that E is equal to the essential image of the functor f_*.","statement_latex":"Let $\\mathcal{C}$ be a site. A strictly full subcategory\n$E \\subset \\Sh(\\mathcal{C})$ is a {\\it subtopos} if there\nexists an embedding of topoi $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$\nsuch that $E$ is equal to the essential image of the functor $f_*$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Subtopoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LV","source_file":"sites.tex","source_line":10101,"source_end_line":10107,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10101-L10107","statement_sha256":"4d551ee311030647248123b283b8e216dd046302c493b1f443e5f4bd07ca4d45","origin":"The Stacks Project","memory_eligible":false,"source_rank":723,"rank":723,"depth":0,"x":1143.489,"y":130.95,"cluster":"sheaves-sites"},{"id":"stacks:08LW","tag":"08LW","title":"Subtopoi · Lemma 08LW","summary":"Let C be a site. Let F be a sheaf on C. The following are equivalent • F is a subobject of the final object of Sh(C), and • the topos Sh(C)/F is a subtopos of Sh(C).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf on\n$\\mathcal{C}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a subobject of the final object of\n$\\Sh(\\mathcal{C})$, and\n\\item the topos $\\Sh(\\mathcal{C})/\\mathcal{F}$ is a subtopos of\n$\\Sh(\\mathcal{C})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Subtopoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LW","source_file":"sites.tex","source_line":10113,"source_end_line":10123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10113-L10123","statement_sha256":"bfe5d23f2f23127a8c924cdc0d483815fefe78b73a7d5229e12a8f2b2d9e00d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":724,"rank":724,"depth":10,"x":1191.769,"y":293.443,"cluster":"sheaves-sites"},{"id":"stacks:08LX","tag":"08LX","title":"Subtopoi · Definition 08LX","summary":"Let C be a site. A strictly full subcategory E ⊂ Sh(C) is an open subtopos if there exists a subsheaf F of the final object of Sh(C) such that E is the subtopos Sh(C)/F described in Lemma [Tag 08LW].","statement_latex":"Let $\\mathcal{C}$ be a site. A strictly full subcategory\n$E \\subset \\Sh(\\mathcal{C})$ is an {\\it open subtopos}\nif there exists a subsheaf $\\mathcal{F}$ of the final object\nof $\\Sh(\\mathcal{C})$ such that $E$ is the subtopos\n$\\Sh(\\mathcal{C})/\\mathcal{F}$ described in Lemma \\ref{lemma-open-subtopos}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Subtopoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LX","source_file":"sites.tex","source_line":10161,"source_end_line":10168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10161-L10168","statement_sha256":"6e316d5d4c7018b5f2e766005154cadd5a67adc160d033c8723409dcfc69ff14","origin":"The Stacks Project","memory_eligible":false,"source_rank":725,"rank":725,"depth":11,"x":1025.214,"y":200.862,"cluster":"sheaves-sites"},{"id":"stacks:08LY","tag":"08LY","title":"Subtopoi · Lemma 08LY","summary":"Let C be a site. Let F be a subsheaf of the final object * of Sh(C). The full subcategory of sheaves G such that F × G → F is an isomorphism is a subtopos of Sh(C).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a subsheaf of the final\nobject $*$ of $\\Sh(\\mathcal{C})$. The full subcategory of sheaves\n$\\mathcal{G}$ such that $\\mathcal{F} \\times \\mathcal{G} \\to \\mathcal{F}$\nis an isomorphism is a subtopos of $\\Sh(\\mathcal{C})$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Subtopoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LY","source_file":"sites.tex","source_line":10175,"source_end_line":10181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10175-L10181","statement_sha256":"a0740c803417ff2ad2b359c64b1915ba8504f5cd3f478613e2fe0ec96742ff0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":726,"rank":726,"depth":9,"x":1222.814,"y":174.578,"cluster":"sheaves-sites"},{"id":"stacks:08LZ","tag":"08LZ","title":"Subtopoi · Definition 08LZ","summary":"Let C be a site. A strictly full subcategory E ⊂ Sh(C) is an closed subtopos if there exists a subsheaf F of the final object of Sh(C) such that E is the subtopos described in Lemma [Tag 08LY].","statement_latex":"Let $\\mathcal{C}$ be a site. A strictly full subcategory\n$E \\subset \\Sh(\\mathcal{C})$ is an {\\it closed subtopos}\nif there exists a subsheaf $\\mathcal{F}$ of the final object\nof $\\Sh(\\mathcal{C})$ such that $E$ is the subtopos\ndescribed in Lemma \\ref{lemma-closed-subtopos}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Subtopoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LZ","source_file":"sites.tex","source_line":10236,"source_end_line":10243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10236-L10243","statement_sha256":"3cdc7c1725b1186aa6c8ddd5bf9ae4422ca006ded2c06b6375b5930042dbc3c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":727,"rank":727,"depth":10,"x":1098.034,"y":306.303,"cluster":"sheaves-sites"},{"id":"stacks:08M0","tag":"08M0","title":"Subtopoi · Definition 08M0","summary":"Let f : Sh(D) → Sh(C) be a morphism of topoi. • We say f is an open immersion if f is an embedding and the essential image of f_* is an open subtopos. • We say f is a closed immersion if f is an embedding and the essential image of f_* is a closed subtopos.","statement_latex":"Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.\n\\begin{enumerate}\n\\item We say $f$ is an {\\it open immersion} if $f$ is an embedding\nand the essential image of $f_*$ is an open subtopos.\n\\item We say $f$ is a {\\it closed immersion} if $f$ is an embedding\nand the essential image of $f_*$ is a closed subtopos.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Subtopoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08M0","source_file":"sites.tex","source_line":10249,"source_end_line":10258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10249-L10258","statement_sha256":"a61319f8cd1e09abb19f656b59019923d8ed380314b689ab28071e612d336eb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":728,"rank":728,"depth":0,"x":1084.092,"y":138.086,"cluster":"sheaves-sites"},{"id":"stacks:08M1","tag":"08M1","title":"Subtopoi · Lemma 08M1","summary":"Let i : Sh(D) → Sh(C) be a closed immersion of topoi. Then i_* is fully faithful, transforms surjections into surjections, commutes with coequalizers, commutes with pushouts, reflects injections, reflects surjections, and reflects bijections.","statement_latex":"Let $i : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a closed immersion of\ntopoi. Then $i_*$ is fully faithful, transforms surjections into surjections,\ncommutes with coequalizers, commutes with pushouts, reflects injections,\nreflects surjections, and reflects bijections.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Subtopoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08M1","source_file":"sites.tex","source_line":10260,"source_end_line":10266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10260-L10266","statement_sha256":"9c9f8f20d527f00bf15ffa12e24686ccb2aea475aac4b463109affc36824dbd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":729,"rank":729,"depth":10,"x":1229.891,"y":254.41,"cluster":"sheaves-sites"},{"id":"stacks:00YS","tag":"00YS","title":"Sheaves of algebraic structures · Definition 00YS","summary":"Let f : D → C be a morphism of sites given by a functor u : C → D. We define the pushforward functor for presheaves of algebraic structures by the rule u^pF(U) = F(uU), and for sheaves of algebraic structures by the same rule, namely f_*F(U) = F(uU).","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites\ngiven by a functor $u : \\mathcal{C} \\to \\mathcal{D}$.\nWe define the {\\it pushforward} functor for presheaves of algebraic structures\nby the rule $u^p\\mathcal{F}(U) = \\mathcal{F}(uU)$,\nand for sheaves of algebraic structures by the same rule, namely\n$f_*\\mathcal{F}(U) = \\mathcal{F}(uU)$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheaves of algebraic structures","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YS","source_file":"sites.tex","source_line":10373,"source_end_line":10381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10373-L10381","statement_sha256":"6399bb423eb1785491b3e5d1bc4c732ea6a6441961662c2fdc5c1e72f713830b","origin":"The Stacks Project","memory_eligible":false,"source_rank":730,"rank":730,"depth":0,"x":1028.501,"y":251.359,"cluster":"sheaves-sites"},{"id":"stacks:00YT","tag":"00YT","title":"Sheaves of algebraic structures · Lemma 00YT","summary":"Suppose the functor u : C → D satisfies the hypotheses of Proposition [Tag 00X6], and hence gives rise to a morphism of sites f : D → C. In this case the pullback functor f^-1 (resp. u_p) and the pushforward functor f_* (resp. u^p) extend to an adjoint pair of functors on the categories of sheaves (resp. presheaves) of algebraic structures. Moreover, these functors commute with taking the underlying sheaf (resp. presheaf) of sets.","statement_latex":"Suppose the functor $u : \\mathcal{C} \\to \\mathcal{D}$ satisfies\nthe hypotheses of Proposition \\ref{proposition-get-morphism},\nand hence gives rise to a morphism of sites\n$f : \\mathcal{D} \\to \\mathcal{C}$. In this case\nthe pullback functor $f^{-1}$ (resp.\\ $u_p$) and the pushforward\nfunctor $f_*$ (resp. $u^p$) extend to an adjoint pair of functors on\nthe categories of sheaves (resp.\\ presheaves)  of algebraic structures.\nMoreover, these functors commute with taking\nthe underlying sheaf (resp.\\ presheaf) of sets.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheaves of algebraic structures","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YT","source_file":"sites.tex","source_line":10393,"source_end_line":10404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10393-L10404","statement_sha256":"b2318c0d5013896b495eb9d38a9320e97a4202d1f1f3ce2bb24fd726c3d1140c","origin":"The Stacks Project","memory_eligible":false,"source_rank":731,"rank":731,"depth":6,"x":1179.708,"y":139.15,"cluster":"sheaves-sites"},{"id":"stacks:00YV","tag":"00YV","title":"Sheaves of algebraic structures · Proposition 00YV","summary":"Morphisms of topoi preserve algebraic structure. Let C, D be sites. Let f = (f^-1, f_*) be a morphism of topoi from Sh(D) → Sh(C). The method introduced above gives rise to an adjoint pair of functors (f^-1, f_*) on sheaves of algebraic structures compatible with taking the underlying sheaves of sets for the following types of algebraic structures: • pointed sets, • abelian groups, • groups, • monoids, • rings, • modules over a fixed ring, and • lie algebras over a fixed…","statement_latex":"\\begin{slogan}\nMorphisms of topoi preserve algebraic structure.\n\\end{slogan}\nLet $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $f = (f^{-1}, f_*)$ be a morphism of topoi\nfrom $\\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$.\nThe method introduced above gives rise to an adjoint\npair of functors $(f^{-1}, f_*)$ on sheaves of algebraic structures\ncompatible with taking the underlying sheaves of sets\nfor the following types of algebraic structures:\n\\begin{enumerate}\n\\item pointed sets,\n\\item abelian groups,\n\\item groups,\n\\item monoids,\n\\item rings,\n\\item modules over a fixed ring, and\n\\item lie algebras over a fixed field.\n\\end{enumerate}\nMoreover, in each of these cases the results above labeled ($\\alpha$),\n($\\beta$), ($\\gamma$), ($\\delta$), ($\\epsilon$), and ($\\zeta$) hold.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheaves of algebraic structures","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YV","source_file":"sites.tex","source_line":10472,"source_end_line":10495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10472-L10495","statement_sha256":"4294d0e7f743d7def29c7ccc327e4aa90e8f9db0f3a522b0ad0982df354c1dfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":732,"rank":732,"depth":1,"x":1158.411,"y":307.968,"cluster":"sheaves-sites"},{"id":"stacks:06UN","tag":"06UN","title":"Pullback maps · Definition 06UN","summary":"The global sections of a presheaf of sets F over a site C is the set Γ(C, F) = Mor_PSh(C)(*, F) where * is the final object in the category of presheaves on C, i.e., the presheaf which associates to every object a singleton.","statement_latex":"The {\\it global sections} of a presheaf of sets $\\mathcal{F}$ over a\nsite $\\mathcal{C}$ is the set\n$$\n\\Gamma(\\mathcal{C}, \\mathcal{F}) =\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(*, \\mathcal{F})\n$$\nwhere $*$ is the final object in the category of presheaves on\n$\\mathcal{C}$, i.e., the presheaf which associates to every object\na singleton.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Pullback maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UN","source_file":"sites.tex","source_line":10603,"source_end_line":10614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10603-L10614","statement_sha256":"c189626294702663b2a8f6473c8a4b192bc7c837d5bb913e0c4640483b7d70d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":733,"rank":733,"depth":0,"x":1038.158,"y":171.174,"cluster":"sheaves-sites"},{"id":"stacks:0792","tag":"0792","title":"Pullback maps · Lemma 0792","summary":"Let C be a site. Let a, b : V → U be objects of C such that xymatrix h_V^\\# ar@<1ex>[r] ar@<-1ex>[r] & h_U^\\# ar[r] & * is a coequalizer in Sh(C). Then Γ(C, F) is the equalizer of a^*, b^* : F(U) → F(V).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $a, b : V \\to U$ be objects of $\\mathcal{C}$\nsuch that\n$$\n\\xymatrix{\nh_V^\\# \\ar@<1ex>[r] \\ar@<-1ex>[r] & h_U^\\# \\ar[r] & {*}\n}\n$$\nis a coequalizer in $\\Sh(\\mathcal{C})$. Then\n$\\Gamma(\\mathcal{C}, \\mathcal{F})$ is the equalizer of\n$a^*, b^* : \\mathcal{F}(U) \\to \\mathcal{F}(V)$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Pullback maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0792","source_file":"sites.tex","source_line":10620,"source_end_line":10632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10620-L10632","statement_sha256":"bc93cf1ff5296b4d5d9332617f220d2c85d8f6e1d743baf1c45463795c5ebb29","origin":"The Stacks Project","memory_eligible":false,"source_rank":734,"rank":734,"depth":0,"x":1237.163,"y":203.863,"cluster":"sheaves-sites"},{"id":"stacks:00YX","tag":"00YX","title":"Topologies · Definition 00YX","summary":"Let C be a category. Let U ∈ Ob(C). A sieve S on U is a subpresheaf S ⊂ h_U.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$.\nA {\\it sieve $S$ on $U$} is a subpresheaf $S \\subset h_U$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YX","source_file":"sites.tex","source_line":10876,"source_end_line":10880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10876-L10880","statement_sha256":"0987dd40e7a75bf04798de039a220881c8efa1a62861c58634c0bbbca7821e5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":735,"rank":735,"depth":0,"x":1063.848,"y":292.825,"cluster":"sheaves-sites"},{"id":"stacks:00YZ","tag":"00YZ","title":"Topologies · Lemma 00YZ","summary":"Let C be a category. Let U ∈ Ob(C). • The collection of sieves on U is a set. • Inclusion defines a partial ordering on this set. • Unions and intersections of sieves are sieves. • Given a family of morphisms (U_i → U)_i∈ I of C with target U there exists a unique smallest sieve S on U such that each U_i → U belongs to S(U_i). • The sieve S = h_U is the maximal sieve. • The empty subpresheaf is the minimal sieve.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $U \\in \\Ob(\\mathcal{C})$.\n\\begin{enumerate}\n\\item The collection of sieves on $U$ is a set.\n\\item Inclusion defines a partial ordering on this set.\n\\item Unions and intersections of sieves are sieves.\n\\item\n\nGiven a family of morphisms $\\{U_i \\to U\\}_{i\\in I}$\nof $\\mathcal{C}$ with target $U$\nthere exists a unique smallest sieve $S$ on $U$ such that\neach $U_i \\to U$ belongs to $S(U_i)$.\n\\item The sieve $S = h_U$ is the maximal sieve.\n\\item The empty subpresheaf is the minimal sieve.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00YZ","source_file":"sites.tex","source_line":10902,"source_end_line":10918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10902-L10918","statement_sha256":"a384df47ce1f986bb1e6aa1600d4b88e4830da22e5ff1882423c094a9298685a","origin":"The Stacks Project","memory_eligible":false,"source_rank":736,"rank":736,"depth":2,"x":1120.2,"y":128.615,"cluster":"sheaves-sites"},{"id":"stacks:00Z1","tag":"00Z1","title":"Topologies · Definition 00Z1","summary":"Let C be a category. Given a family of morphisms (f_i : U_i → U)_i∈ I of C with target U we say the sieve S on U described in Lemma [Tag 00YZ] part ([Tag 00Z0]) is the sieve on U generated by the morphisms f_i.","statement_latex":"Let $\\mathcal{C}$ be a category.\nGiven a family of morphisms $\\{f_i : U_i \\to U\\}_{i\\in I}$\nof $\\mathcal{C}$ with target $U$ we say the sieve\n$S$ on $U$ described in Lemma \\ref{lemma-sieves-set}\npart (\\ref{item-sieve-generated}) is the {\\it sieve  on $U$\ngenerated by the morphisms $f_i$}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Z1","source_file":"sites.tex","source_line":10949,"source_end_line":10957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10949-L10957","statement_sha256":"b9d2cae71cfe64453a599a6f61b389e78ab9312392de463b942ac38c59e258ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":737,"rank":737,"depth":3,"x":1210.847,"y":281.925,"cluster":"sheaves-sites"},{"id":"stacks:00Z2","tag":"00Z2","title":"Topologies · Definition 00Z2","summary":"Let C be a category. Let f : V → U be a morphism of C. Let S ⊂ h_U be a sieve. We define the pullback of S by f to be the sieve S ×_U V of V defined by the rule (α : T → V) ∈ (S ×_U V)(T) ⇔ (f ∘ α : T → U) ∈ S(T)","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $f : V \\to U$ be a morphism of $\\mathcal{C}$.\nLet $S \\subset h_U$ be a sieve. We define the\n{\\it pullback of $S$ by $f$} to be the sieve\n$S \\times_U V$ of $V$ defined by the rule\n$$\n(\\alpha : T \\to V) \\in (S \\times_U V)(T)\n\\Leftrightarrow\n(f \\circ \\alpha : T \\to U) \\in S(T)\n$$","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Z2","source_file":"sites.tex","source_line":10959,"source_end_line":10971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10959-L10971","statement_sha256":"a43e1d54489bf16fda3aa7100c91ebb7343e2ad7bf32b407cc290556194d5e1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":738,"rank":738,"depth":0,"x":1020.409,"y":220.212,"cluster":"sheaves-sites"},{"id":"stacks:00Z3","tag":"00Z3","title":"Topologies · Lemma 00Z3","summary":"Let C be a category. Let U ∈ Ob(C). Let S be a sieve on U. If f : V → U is in S, then S ×_U V = h_V is maximal.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U \\in \\Ob(\\mathcal{C})$.\nLet $S$ be a sieve on $U$.\nIf $f : V \\to U$ is in $S$, then\n$S \\times_U V = h_V$ is maximal.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Z3","source_file":"sites.tex","source_line":10979,"source_end_line":10986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10979-L10986","statement_sha256":"2529e782b88c600dccf59374ba3439fd470c507511bfb3cbbde4dc71799d28cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":739,"rank":739,"depth":0,"x":1210.77,"y":157.559,"cluster":"sheaves-sites"},{"id":"stacks:00Z4","tag":"00Z4","title":"Topologies · Definition 00Z4","summary":"Let C be a category. A topology on C is given by a rule which assigns to every U ∈ Ob(C) a subset J(U) of the set of all sieves on U satisfying the following conditions • For every morphism f : V → U in C, and every element S ∈ J(U) the pullback S ×_U V is an element of J(V). • If S and S' are sieves on U ∈ Ob(C), if S ∈ J(U), and if for all f ∈ S(V) the pullback S' ×_U V belongs to J(V), then S' belongs to J(U). • For every U ∈ Ob(C) the maximal sieve S = h_U belongs to…","statement_latex":"Let $\\mathcal{C}$ be a category. A {\\it topology on $\\mathcal{C}$} is given\nby a rule which assigns to every $U \\in \\Ob(\\mathcal{C})$\na subset $J(U)$ of the set of all sieves on $U$ satisfying\nthe following conditions\n\\begin{enumerate}\n\\item For every morphism $f : V \\to U$ in $\\mathcal{C}$, and\nevery element $S \\in J(U)$ the pullback $S \\times_U V$\nis an element of $J(V)$.\n\\item If $S$ and $S'$ are sieves on $U \\in \\Ob(\\mathcal{C})$,\nif $S \\in J(U)$, and if for all $f \\in S(V)$ the pullback\n$S' \\times_U V$ belongs to $J(V)$, then $S'$ belongs to $J(U)$.\n\\item For every $U \\in \\Ob(\\mathcal{C})$ the\nmaximal sieve $S = h_U$ belongs to $J(U)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Z4","source_file":"sites.tex","source_line":10992,"source_end_line":11008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L10992-L11008","statement_sha256":"166a25f83c5d1e10f06bcd2195aea8c1d7cabbbb117b519b859a38de9f513ba9","origin":"The Stacks Project","memory_eligible":false,"source_rank":740,"rank":740,"depth":0,"x":1120.64,"y":312.022,"cluster":"sheaves-sites"},{"id":"stacks:00Z5","tag":"00Z5","title":"Topologies · Lemma 00Z5","summary":"Let C be a category. Let J be a topology on C. Let U ∈ Ob(C). • Finite intersections of elements of J(U) are in J(U). • If S ∈ J(U) and S' ⊃ S, then S' ∈ J(U).","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $J$ be a topology on $\\mathcal{C}$.\nLet $U \\in \\Ob(\\mathcal{C})$.\n\\begin{enumerate}\n\\item Finite intersections of elements of $J(U)$ are in $J(U)$.\n\\item If $S \\in J(U)$ and $S' \\supset S$, then $S' \\in J(U)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Z5","source_file":"sites.tex","source_line":11014,"source_end_line":11023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11014-L11023","statement_sha256":"6b82b77cbf80c120b5bd6eb5f1bffd604efdf35e2c0e2e95e0a6fe2bb5fb2553","origin":"The Stacks Project","memory_eligible":false,"source_rank":741,"rank":741,"depth":1,"x":1062.793,"y":146.716,"cluster":"sheaves-sites"},{"id":"stacks:00Z6","tag":"00Z6","title":"Topologies · Definition 00Z6","summary":"Let C be a category. Let J, J' be two topologies on C. We say that J is finer or stronger than J' if and only if for every object U of C we have J'(U) ⊂ J(U). In this case we also say that J' is coarser or weaker than J.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $J$, $J'$ be\ntwo topologies on $\\mathcal{C}$. We say that $J$ is\n{\\it finer} or {\\it stronger} than $J'$ if and only if for every object\n$U$ of $\\mathcal{C}$ we have $J'(U) \\subset J(U)$.\nIn this case we also say that $J'$ is\n{\\it coarser} or {\\it weaker} than $J$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Z6","source_file":"sites.tex","source_line":11044,"source_end_line":11052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11044-L11052","statement_sha256":"7a253a8a78527d829bdaa4f7efe74fcd41afbd25480309b4d019a613294af7d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":742,"rank":742,"depth":0,"x":1238.664,"y":235.929,"cluster":"sheaves-sites"},{"id":"stacks:00Z7","tag":"00Z7","title":"Topologies · Lemma 00Z7","summary":"Let C be a category. Let (J_i)_i∈ I be a set of topologies. • The rule J(U) = ⋂ J_i(U) defines a topology on C. • There is a coarsest topology finer than all of the topologies J_i.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\{J_i\\}_{i\\in I}$ be a set of topologies.\n\\begin{enumerate}\n\\item The rule $J(U) = \\bigcap J_i(U)$ defines\na topology on $\\mathcal{C}$.\n\\item There is a coarsest topology finer than\nall of the topologies $J_i$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Z7","source_file":"sites.tex","source_line":11065,"source_end_line":11075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11065-L11075","statement_sha256":"800196fa018915d79ccfc96a40de090414a80ea4bd4f587f06210c2e8be3d1cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":743,"rank":743,"depth":0,"x":1036.915,"y":269.992,"cluster":"sheaves-sites"},{"id":"stacks:00Z8","tag":"00Z8","title":"Topologies · Definition 00Z8","summary":"Let C be a category endowed with a topology J. Let F be a presheaf of sets on C. We say that F is a sheaf on C if for every U ∈ Ob(C) and for every covering sieve S of U the canonical map Mor_PSh(C)(h_U, F) → Mor_PSh(C)(S, F) is bijective.","statement_latex":"Let $\\mathcal{C}$ be a category endowed with a\ntopology $J$. Let $\\mathcal{F}$ be a presheaf of sets\non $\\mathcal{C}$.\nWe say that $\\mathcal{F}$ is a\n{\\it sheaf} on $\\mathcal{C}$\nif for every $U \\in \\Ob(\\mathcal{C})$ and for\nevery covering sieve $S$ of $U$ the canonical map\n$$\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F})\n\\longrightarrow\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F})\n$$\nis bijective.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Z8","source_file":"sites.tex","source_line":11087,"source_end_line":11102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11087-L11102","statement_sha256":"8785a0a3238cf275aa6dd7ed3468f1ce8a6b5066efeeaa3ff40ffd16b481afdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":744,"rank":744,"depth":0,"x":1158.482,"y":130.176,"cluster":"sheaves-sites"},{"id":"stacks:00Z9","tag":"00Z9","title":"Topologies · Lemma 00Z9","summary":"Let C be a category. Let ( F_i )_i∈ I be a collection of presheaves of sets on C. For each U ∈ Ob(C) denote J(U) the set of sieves S on U with the following property: For every morphism V → U, the maps Mor_PSh(C)(h_V, F_i) → Mor_PSh(C)(S ×_U V, F_i) are bijective for all i ∈ I. Then J defines a topology on C. This topology is the finest topology in which all of the F_i are sheaves.","statement_latex":"Let $\\mathcal{C}$ be a category. Let $\\{ \\mathcal{F}_i \\}_{i\\in I}$ be a\ncollection of presheaves of sets on $\\mathcal{C}$. For each\n$U \\in \\Ob(\\mathcal{C})$ denote\n$J(U)$ the set of sieves $S$ on $U$ with the following property:\nFor every morphism $V \\to U$, the maps\n$$\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_V, \\mathcal{F}_i)\n\\longrightarrow\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(S \\times_U V, \\mathcal{F}_i)\n$$\nare bijective for all $i \\in I$. Then $J$ defines a\ntopology on $\\mathcal{C}$. This topology is the finest\ntopology in which all of the $\\mathcal{F}_i$ are sheaves.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Z9","source_file":"sites.tex","source_line":11113,"source_end_line":11128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11113-L11128","statement_sha256":"f6eaf0606aad84214a19194cf794af8fb5036f09cdccb6fa3fcd693165ef95d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":745,"rank":745,"depth":1,"x":1181.313,"y":302.527,"cluster":"sheaves-sites"},{"id":"stacks:00ZA","tag":"00ZA","title":"Topologies · Definition 00ZA","summary":"Let C be a category. The finest topology on C such that all representable presheaves are sheaves, see Lemma [Tag 00Z9], is called the canonical topology of C.","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe finest topology on $\\mathcal{C}$ such that\nall representable presheaves are sheaves, see\nLemma \\ref{lemma-topology-presheaves-sheaves},\nis called the {\\it canonical topology} of $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZA","source_file":"sites.tex","source_line":11198,"source_end_line":11205,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11198-L11205","statement_sha256":"03a26e0f643e6acbc731feacb98a3980e31ec8cb6e7e6462941b22a069911e0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":746,"rank":746,"depth":2,"x":1025.632,"y":188.212,"cluster":"sheaves-sites"},{"id":"stacks:00ZC","tag":"00ZC","title":"The topology defined by a site · Lemma 00ZC","summary":"Let C be a site with coverings Cov(C). For every object U of C, let J(U) denote the set of sieves S on U with the following property: there exists a covering (f_i : U_i → U)_i∈ I ∈ Cov(C) so that the sieve S' generated by the f_i (see Definition [Tag 00Z1]) is contained in S. • This J is a topology on C. • A presheaf F is a sheaf for this topology (see Definition [Tag 00Z8]) if and only if it is a sheaf on the site (see Definition [Tag 00VM]).","statement_latex":"Let $\\mathcal{C}$ be a site with coverings $\\text{Cov}(\\mathcal{C})$.\nFor every object $U$ of $\\mathcal{C}$, let $J(U)$ denote\nthe set of sieves $S$ on $U$ with the following property:\nthere exists a covering\n$\\{f_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov}(\\mathcal{C})$\nso that the sieve $S'$ generated by the $f_i$ (see Definition\n\\ref{definition-sieve-generated}) is contained in $S$.\n\\begin{enumerate}\n\\item This $J$ is a topology on $\\mathcal{C}$.\n\\item A presheaf $\\mathcal{F}$ is a sheaf for this topology\n(see Definition \\ref{definition-sheaf-sets-topology})\nif and only if it is a sheaf on the site (see\nDefinition \\ref{definition-sheaf-sets}).\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"The topology defined by a site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZC","source_file":"sites.tex","source_line":11243,"source_end_line":11259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11243-L11259","statement_sha256":"b27d9c9f47b70143128fec980579b5a454f5819faac071613a5a6a7b93780768","origin":"The Stacks Project","memory_eligible":false,"source_rank":747,"rank":747,"depth":4,"x":1232.683,"y":184.163,"cluster":"sheaves-sites"},{"id":"stacks:00ZD","tag":"00ZD","title":"The topology defined by a site · Definition 00ZD","summary":"Let C be a site with coverings Cov(C). The topology associated to C is the topology J constructed in Lemma [Tag 00ZC] above.","statement_latex":"Let $\\mathcal{C}$ be a site with coverings $\\text{Cov}(\\mathcal{C})$.\nThe {\\it topology associated to $\\mathcal{C}$} is the topology\n$J$ constructed in Lemma \\ref{lemma-site-gives-topology} above.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"The topology defined by a site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZD","source_file":"sites.tex","source_line":11335,"source_end_line":11340,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11335-L11340","statement_sha256":"7c1f90ec00fedf26bf438fe4fd5361c0b82329443363df129442c9c47a703a70","origin":"The Stacks Project","memory_eligible":false,"source_rank":748,"rank":748,"depth":5,"x":1083.029,"y":304.823,"cluster":"sheaves-sites"},{"id":"stacks:00ZH","tag":"00ZH","title":"Sheafification in a topology · Lemma 00ZH","summary":"In the situation above. • The assignment U ↦ LF(U) combined with the restriction mappings defined above is a presheaf. • The maps ℓ glue to give a morphism of presheaves ℓ : F → LF. • The rule F ↦ (F xrightarrowℓ LF) is a functor. • If F is a subpresheaf of G, then LF is a subpresheaf of LG. • The map ℓ : F → LF has the following property: For every section s ∈ LF(U) there exists a covering sieve S on U and an element φ ∈ Mor_PSh(C)(S, F) such that ℓ(φ) equals the…","statement_latex":"In the situation above.\n\\begin{enumerate}\n\\item The assignment $U \\mapsto L\\mathcal{F}(U)$ combined with the\nrestriction mappings defined above is a presheaf.\n\\item The maps $\\ell$ glue to give a morphism of presheaves\n$\\ell : \\mathcal{F} \\to L\\mathcal{F}$.\n\\item The rule $\\mathcal{F} \\mapsto (\\mathcal{F} \\xrightarrow{\\ell}\nL\\mathcal{F})$ is a functor.\n\\item If $\\mathcal{F}$ is a subpresheaf of $\\mathcal{G}$, then\n$L\\mathcal{F}$ is a subpresheaf of $L\\mathcal{G}$.\n\\item The map $\\ell : \\mathcal{F} \\to L\\mathcal{F}$ has the\nfollowing property: For every section $s \\in L\\mathcal{F}(U)$\nthere exists a covering sieve $S$ on $U$ and an element\n$\\varphi \\in \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F})$\nsuch that $\\ell(\\varphi)$ equals the restriction of\n$s$ to $S$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification in a topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZH","source_file":"sites.tex","source_line":11509,"source_end_line":11528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11509-L11528","statement_sha256":"878c07417abe3e5953c7dd9a7cfdd4818f3f6df2d62ad6f38e0e81717d7d519b","origin":"The Stacks Project","memory_eligible":false,"source_rank":749,"rank":749,"depth":0,"x":1096.371,"y":130.66,"cluster":"sheaves-sites"},{"id":"stacks:00ZI","tag":"00ZI","title":"Sheafification in a topology · Definition 00ZI","summary":"Let C be a category. Let J be a topology on C. We say that a presheaf of sets F is separated if for every object U and every covering sieve S on U the canonical map F(U) → Mor_PSh(C)(S, F) is injective.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $J$ be a topology on $\\mathcal{C}$.\nWe say that a presheaf of sets $\\mathcal{F}$\nis {\\it separated} if for every object $U$ and\nevery covering sieve $S$ on $U$ the canonical map\n$\\mathcal{F}(U) \\to \\Mor_{\\textit{PSh}(\\mathcal{C})}(S, \\mathcal{F})$\nis injective.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification in a topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZI","source_file":"sites.tex","source_line":11534,"source_end_line":11543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11534-L11543","statement_sha256":"05ef69a0e003d7f9c4714844b4dbcd66706695c132c625208730cba9ca0be571","origin":"The Stacks Project","memory_eligible":false,"source_rank":750,"rank":750,"depth":0,"x":1226.792,"y":266.868,"cluster":"sheaves-sites"},{"id":"stacks:00ZJ","tag":"00ZJ","title":"Sheafification in a topology · Theorem 00ZJ","summary":"Let C be a category. Let J be a topology on C. Let F be a presheaf of sets. • The presheaf LF is separated. • If F is separated, then LF is a sheaf and the map of presheaves F → LF is injective. • If F is a sheaf, then F → LF is an isomorphism. • The presheaf LLF is always a sheaf.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $J$ be a topology on $\\mathcal{C}$.\nLet $\\mathcal{F}$ be a presheaf of sets.\n\\begin{enumerate}\n\\item The presheaf $L\\mathcal{F}$ is separated.\n\\item If $\\mathcal{F}$ is separated, then $L\\mathcal{F}$ is a sheaf\nand the map of presheaves $\\mathcal{F} \\to L\\mathcal{F}$ is injective.\n\\item If $\\mathcal{F}$ is a sheaf, then $\\mathcal{F} \\to L\\mathcal{F}$\nis an isomorphism.\n\\item The presheaf $LL\\mathcal{F}$ is always a sheaf.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification in a topology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZJ","source_file":"sites.tex","source_line":11545,"source_end_line":11558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11545-L11558","statement_sha256":"a2df8beadea874183b71d2ee3d531f3a0b29d4b799af2e1c3fb82f19357b5597","origin":"The Stacks Project","memory_eligible":false,"source_rank":751,"rank":751,"depth":3,"x":1020.767,"y":240.394,"cluster":"sheaves-sites"},{"id":"stacks:00ZL","tag":"00ZL","title":"Sheafification in a topology · Definition 00ZL","summary":"Let C be a category endowed with a topology J. Let F be a presheaf of sets on C. The sheaf F^\\# := LLF together with the canonical map F → F^\\# is called the sheaf associated to F.","statement_latex":"Let $\\mathcal{C}$ be a category endowed with a topology $J$.\nLet $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$.\nThe sheaf $\\mathcal{F}^\\# := LL\\mathcal{F}$\ntogether with the canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$\nis called the {\\it sheaf associated to $\\mathcal{F}$}.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification in a topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZL","source_file":"sites.tex","source_line":11644,"source_end_line":11651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11644-L11651","statement_sha256":"94d3b4a2d925b6dec7f4a3d7cc448f036c36bfb18d1dd11f6394bc553132c9db","origin":"The Stacks Project","memory_eligible":false,"source_rank":752,"rank":752,"depth":0,"x":1194.246,"y":142.861,"cluster":"sheaves-sites"},{"id":"stacks:00ZM","tag":"00ZM","title":"Sheafification in a topology · Proposition 00ZM","summary":"Let C be a category endowed with a topology. Let F be a presheaf of sets on C. The canonical map F → F^\\# has the following universal property: For any map F → G, where G is a sheaf of sets, there is a unique map F^\\# → G such that F → F^\\# → G equals the given map.","statement_latex":"Let $\\mathcal{C}$ be a category endowed with a topology.\nLet $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{C}$.\nThe canonical map $\\mathcal{F} \\to \\mathcal{F}^\\#$ has the\nfollowing universal property: For any map\n$\\mathcal{F} \\to \\mathcal{G}$,\nwhere $\\mathcal{G}$ is a sheaf of sets, there is a unique map\n$\\mathcal{F}^\\# \\to \\mathcal{G}$ such that $\\mathcal{F} \\to \\mathcal{F}^\\#\n\\to \\mathcal{G}$ equals the given map.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Sheafification in a topology","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZM","source_file":"sites.tex","source_line":11653,"source_end_line":11663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11653-L11663","statement_sha256":"80bf46ab19b80a62f9bb0edfacf46b69274e7a6ae87f04e8b21d7d48bc301134","origin":"The Stacks Project","memory_eligible":false,"source_rank":753,"rank":753,"depth":4,"x":1144.682,"y":313.479,"cluster":"sheaves-sites"},{"id":"stacks:00ZO","tag":"00ZO","title":"Topologies and sheaves · Lemma 00ZO","summary":"Let C be a category endowed with a topology J. Let U be an object of C. Let S be a sieve on U. The following are equivalent • The sieve S is a covering sieve. • The sheafification S^\\# → h_U^\\# of the map S → h_U is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category endowed with a topology $J$.\nLet $U$ be an object of $\\mathcal{C}$.\nLet $S$ be a sieve on $U$. The following are equivalent\n\\begin{enumerate}\n\\item The sieve $S$ is a covering sieve.\n\\item The sheafification $S^\\# \\to h_U^\\#$\nof the map $S \\to h_U$ is an isomorphism.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZO","source_file":"sites.tex","source_line":11685,"source_end_line":11695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11685-L11695","statement_sha256":"4f2fb3d25d2266ba26c8e9386d0a2b9f9aeae412072e2797f37ead15208d7bc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":754,"rank":754,"depth":1,"x":1043.868,"y":159.307,"cluster":"sheaves-sites"},{"id":"stacks:00ZP","tag":"00ZP","title":"Topologies and sheaves · Theorem 00ZP","summary":"Let C be a category. Let J, J' be topologies on C. The following are equivalent • J = J', • sheaves for the topology J are the same as sheaves for the topology J'.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $J$, $J'$ be topologies on $\\mathcal{C}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $J = J'$,\n\\item sheaves for the topology $J$ are the same as\nsheaves for the topology $J'$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies and sheaves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZP","source_file":"sites.tex","source_line":11734,"source_end_line":11744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11734-L11744","statement_sha256":"52d64fa92c0463867547c593b5236485a389ee81303d801daded275a8d188173","origin":"The Stacks Project","memory_eligible":false,"source_rank":755,"rank":755,"depth":2,"x":1242.492,"y":215.874,"cluster":"sheaves-sites"},{"id":"stacks:00ZQ","tag":"00ZQ","title":"Topologies and sheaves · Lemma 00ZQ","summary":"Assumption and notation as in Theorem [Tag 00ZP]. Then J ⊂ J' if and only if every sheaf for the topology J' is a sheaf for the topology J.","statement_latex":"Assumption and notation as in Theorem \\ref{theorem-topology-and-topos}.\nThen $J \\subset J'$ if and only if every sheaf for the\ntopology $J'$ is a sheaf for the topology $J$.","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Topologies and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZQ","source_file":"sites.tex","source_line":11761,"source_end_line":11766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11761-L11766","statement_sha256":"b27db9019240d5ce0526c84f9dc7fe6569f01f530377ca1dc0290376bccfe7bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":756,"rank":756,"depth":3,"x":1050.246,"y":286.975,"cluster":"sheaves-sites"},{"id":"stacks:00ZT","tag":"00ZT","title":"Points and topologies · Definition 00ZT","summary":"Let C be a category. Let J be a topology on C. A point p of the topology is given by a functor u : C → Sets such that • For every covering sieve S on U the map S_p → (h_U)_p is surjective. • The stalk functor Sh(C) → Sets, F → F_p is exact.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $J$ be a topology on $\\mathcal{C}$.\nA {\\it point $p$} of the topology is given by a functor\n$u : \\mathcal{C} \\to \\textit{Sets}$ such that\n\\begin{enumerate}\n\\item For every covering sieve $S$ on $U$ the map\n$S_p \\to (h_U)_p$ is surjective.\n\\item The stalk functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,\n$\\mathcal{F} \\to \\mathcal{F}_p$ is exact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sites and Sheaves","chapter_id":"sites","section":"Points and topologies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZT","source_file":"sites.tex","source_line":11830,"source_end_line":11842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites.tex#L11830-L11842","statement_sha256":"02afd1034b55a301cd74c7d76ed3230195689ef10a69ee102201d00323118ade","origin":"The Stacks Project","memory_eligible":false,"source_rank":757,"rank":757,"depth":0,"x":1134.957,"y":125.216,"cluster":"sheaves-sites"},{"id":"stacks:026A","tag":"026A","title":"Presheaves of morphisms associated to fibred categories · Lemma 026A","summary":"This actually does give a presheaf.","statement_latex":"This actually does give a presheaf.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Presheaves of morphisms associated to fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/026A","source_file":"stacks.tex","source_line":76,"source_end_line":79,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L76-L79","statement_sha256":"009dbce18190744f2d232b7f252a6c41ed7f580cde5369b94abebf074ac8df59","origin":"The Stacks Project","memory_eligible":false,"source_rank":758,"rank":758,"depth":1,"x":1202.678,"y":292.816,"cluster":"sheaves-sites"},{"id":"stacks:02ZB","tag":"02ZB","title":"Presheaves of morphisms associated to fibred categories · Definition 02ZB","summary":"Let C be a category. Let p : S → C be a fibred category, see Categories, Section [Tag 02XJ]. Given an object U of C and objects x, y of the fibre category, the presheaf of morphisms from x to y is the presheaf (f : V → U) ↦ Mor_S_V(f^*x, f^*y) described above. It is denoted mathitMor(x, y). The subpresheaf mathitIsom(x, y) whose value over V is the set of isomorphisms f^*x → f^*y in the fibre category S_V is called the presheaf of isomorphisms from x to y.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category,\nsee Categories, Section \\ref{categories-section-fibred-categories}.\nGiven an object $U$ of $\\mathcal{C}$ and objects\n$x$, $y$ of the fibre category, the {\\it presheaf\nof morphisms from $x$ to $y$} is the presheaf\n$$\n(f : V \\to U) \\longmapsto \\Mor_{\\mathcal{S}_V}(f^*x, f^*y)\n$$\ndescribed above. It is denoted $\\mathit{Mor}(x, y)$.\nThe subpresheaf $\\mathit{Isom}(x, y)$ whose value\nover $V$ is the set of isomorphisms\n$f^*x \\to f^*y$ in the fibre category $\\mathcal{S}_V$\nis called the {\\it presheaf of isomorphisms from $x$ to $y$}.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Presheaves of morphisms associated to fibred categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZB","source_file":"stacks.tex","source_line":147,"source_end_line":163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L147-L163","statement_sha256":"1a9f508c91d44d875b1e08ed8301a9dbf9049362bb3a6083ff09731754cada62","origin":"The Stacks Project","memory_eligible":false,"source_rank":759,"rank":759,"depth":0,"x":1017.682,"y":207.527,"cluster":"sheaves-sites"},{"id":"stacks:042V","tag":"042V","title":"Presheaves of morphisms associated to fibred categories · Lemma 042V","summary":"Let F : S_1 → S_2 be a 1-morphism of fibred categories over the category C. Let U ∈ Ob(C) and x, y∈ Ob((S_1)_U). Then F defines a canonical morphism of presheaves mathitMor_S_1(x, y) → mathitMor_S_2(F(x), F(y)) on C/U.","statement_latex":"Let $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$ be a $1$-morphism of fibred\ncategories over the category $\\mathcal{C}$. Let $U \\in \\Ob(\\mathcal{C})$\nand $x, y\\in \\Ob((\\mathcal{S}_1)_U)$. Then $F$ defines a canonical\nmorphism of presheaves\n$$\n\\mathit{Mor}_{\\mathcal{S}_1}(x, y)\n\\longrightarrow\n\\mathit{Mor}_{\\mathcal{S}_2}(F(x), F(y))\n$$\non $\\mathcal{C}/U$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Presheaves of morphisms associated to fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042V","source_file":"stacks.tex","source_line":170,"source_end_line":182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L170-L182","statement_sha256":"15eae0aa7474637aecea3675d722687acbc209e32786f067f0ae04362e5f40d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":760,"rank":760,"depth":2,"x":1222.992,"y":165.383,"cluster":"sheaves-sites"},{"id":"stacks:04SI","tag":"04SI","title":"Presheaves of morphisms associated to fibred categories · Lemma 04SI","summary":"Let C be a category. Let p : S → C be a fibred category, see Categories, Section [Tag 02XJ]. Let U ∈ Ob(C) and let x, y ∈ Ob(S_U). Denote x, y : C/U → S also the corresponding 1-morphisms, see Categories, Lemma [Tag 004B]. Then • the 2-fibre product S ×_S × S, (x, y) C/U is fibred in setoids over C/U, and • mathitIsom(x, y) is the presheaf of sets corresponding to this category fibred in setoids, see Categories, Lemma [Tag 04SC].","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category,\nsee Categories, Section \\ref{categories-section-fibred-categories}.\nLet $U \\in \\Ob(\\mathcal{C})$ and let $x, y \\in \\Ob(\\mathcal{S}_U)$.\nDenote $x, y : \\mathcal{C}/U \\to \\mathcal{S}$ also the corresponding\n$1$-morphisms, see\nCategories, Lemma \\ref{categories-lemma-yoneda-2category}.\nThen\n\\begin{enumerate}\n\\item the $2$-fibre product\n$\\mathcal{S} \\times_{\\mathcal{S} \\times \\mathcal{S}, (x, y)} \\mathcal{C}/U$\nis fibred in setoids over $\\mathcal{C}/U$, and\n\\item $\\mathit{Isom}(x, y)$ is the presheaf of sets corresponding\nto this category fibred in setoids, see\nCategories, Lemma \\ref{categories-lemma-2-category-fibred-setoids}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Presheaves of morphisms associated to fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SI","source_file":"stacks.tex","source_line":225,"source_end_line":243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L225-L243","statement_sha256":"e51faf1611ab870836f65cc5551648002e51bfb89a4db554782d0a4bb398c3e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":761,"rank":761,"depth":6,"x":1105.313,"y":313.179,"cluster":"sheaves-sites"},{"id":"stacks:026B","tag":"026B","title":"Descent data in fibred categories · Definition 026B","summary":"Let C be a category. Let p : S → C be a fibred category. Make a choice of pullbacks as in Categories, Definition [Tag 02XN]. Let U = (f_i : U_i → U)_i ∈ I be a family of morphisms of C. Assume all the fibre products U_i ×_U U_j, and U_i ×_U U_j ×_U U_k exist. • A descent datum (X_i, φ_ij) in S relative to the family (f_i : U_i → U) is given by an object X_i of S_U_i for each i ∈ I, an isomorphism φ_ij : pr_0^*X_i → pr_1^*X_j in S_U_i ×_U U_j for each pair (i, j) ∈ I^2…","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.\nMake a choice of pullbacks as in Categories,\nDefinition \\ref{categories-definition-pullback-functor-fibred-category}.\nLet $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$\nbe a family of morphisms of $\\mathcal{C}$. Assume all the fibre products\n$U_i \\times_U U_j$, and $U_i \\times_U U_j \\times_U U_k$ exist.\n\\begin{enumerate}\n\\item A {\\it descent datum $(X_i, \\varphi_{ij})$ in $\\mathcal{S}$\nrelative to the family $\\{f_i : U_i \\to U\\}$} is given by an object $X_i$\nof $\\mathcal{S}_{U_i}$ for each $i \\in I$, an isomorphism\n$\\varphi_{ij} : \\text{pr}_0^*X_i \\to \\text{pr}_1^*X_j$\nin $\\mathcal{S}_{U_i \\times_U U_j}$ for each pair $(i, j) \\in I^2$\nsuch that for every triple of indices $(i, j, k) \\in I^3$ the\ndiagram\n$$\n\\xymatrix{\n\\text{pr}_0^*X_i \\ar[rd]_{\\text{pr}_{01}^*\\varphi_{ij}}\n\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & &\n\\text{pr}_2^*X_k \\\\\n& \\text{pr}_1^*X_j \\ar[ru]_{\\text{pr}_{12}^*\\varphi_{jk}} &\n}\n$$\nin the category $\\mathcal{S}_{U_i \\times_U U_j \\times_U U_k}$\ncommutes. This is called the {\\it cocycle condition}.\n\\item A {\\it morphism $\\psi : (X_i, \\varphi_{ij}) \\to\n(X'_i, \\varphi'_{ij})$ of descent data} is given\nby a family $\\psi = (\\psi_i)_{i\\in I}$ of morphisms\n$\\psi_i : X_i \\to X'_i$ in $\\mathcal{S}_{U_i}$\nsuch that all the diagrams\n$$\n\\xymatrix{\n\\text{pr}_0^*X_i \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\text{pr}_0^*\\psi_i}\n& \\text{pr}_1^*X_j \\ar[d]^{\\text{pr}_1^*\\psi_j} \\\\\n\\text{pr}_0^*X'_i \\ar[r]^{\\varphi'_{ij}} &\n\\text{pr}_1^*X'_j \\\\\n}\n$$\nin the categories $\\mathcal{S}_{U_i \\times_U U_j}$ commute.\n\\item The category of descent data relative to\n$\\mathcal{U}$ is denoted $DD(\\mathcal{U})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Descent data in fibred categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/026B","source_file":"stacks.tex","source_line":280,"source_end_line":324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L280-L324","statement_sha256":"36f36b5da7ec5c6549359a64406e16d4a272d22d0389f52f2fc83c30fff0a482","origin":"The Stacks Project","memory_eligible":false,"source_rank":762,"rank":762,"depth":1,"x":1073.186,"y":137.159,"cluster":"sheaves-sites"},{"id":"stacks:02ZD","tag":"02ZD","title":"Descent data in fibred categories · Lemma 02ZD","summary":"(Pullback of descent data.) Let C be a category. Let p : S → C be a fibred category. Make a choice pullbacks as in Categories, Definition [Tag 02XN]. Let U = (f_i : U_i → U)_i ∈ I, and V = (V_j → V)_j ∈ J be a families of morphisms of C with fixed target. Assume all the fibre products U_i ×_U U_i', U_i ×_U U_i' ×_U U_i\", V_j ×_V V_j', and V_j ×_V V_j' ×_V V_j\" exist. Let α : I → J, h : U → V and g_i : U_i → V_α(i) be a morphism of families of maps with fixed target, see…","statement_latex":"(Pullback of descent data.)\nLet $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.\nMake a choice pullbacks as in Categories,\nDefinition \\ref{categories-definition-pullback-functor-fibred-category}.\nLet $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$, and\n$\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$\nbe a families of morphisms of $\\mathcal{C}$ with fixed target.\nAssume all the fibre products\n$U_i \\times_U U_{i'}$, $U_i \\times_U U_{i'} \\times_U U_{i''}$,\n$V_j \\times_V V_{j'}$, and $V_j \\times_V V_{j'} \\times_V V_{j''}$ exist.\nLet $\\alpha : I \\to J$, $h : U \\to V$ and\n$g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families\nof maps with fixed target, see\nSites, Definition \\ref{sites-definition-morphism-coverings}.\n\\begin{enumerate}\n\\item Let $(Y_j, \\varphi_{jj'})$ be a descent datum relative to the\nfamily $\\{V_j \\to V\\}$. The system\n$$\n\\left(\ng_i^*Y_{\\alpha(i)},\n(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}\n\\right)\n$$\nis a descent datum relative to $\\mathcal{U}$.\n\\item This construction defines a functor between descent data relative\nto $\\mathcal{V}$ and descent data relative to $\\mathcal{U}$.\n\\item Given a second $\\alpha' : I \\to J$, $h' : U \\to V$ and\n$g'_i : U_i \\to V_{\\alpha'(i)}$ morphism of families\nof maps with fixed target, then if $h = h'$ the two resulting functors\nbetween descent data are canonically isomorphic.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Descent data in fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZD","source_file":"stacks.tex","source_line":365,"source_end_line":399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L365-L399","statement_sha256":"c66b6b0a7e9808c4f89f104651c715ac5b27d9281545b3ab395c959552b7c7f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":763,"rank":763,"depth":1,"x":1238.674,"y":248.891,"cluster":"sheaves-sites"},{"id":"stacks:02ZE","tag":"02ZE","title":"Descent data in fibred categories · Definition 02ZE","summary":"With U = (U_i → U)_i ∈ I, V = (V_j → V)_j ∈ J, α : I → J, h : U → V, and g_i : U_i → V_α(i) as in Lemma [Tag 02ZD] the functor (Y_j, φ_jj') ↦ (g_i^*Y_α(i), (g_i × g_i')^*φ_α(i)α(i')) constructed in that lemma is called the pullback functor on descent data.","statement_latex":"With $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$,\n$\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$,\n$\\alpha : I \\to J$, $h : U \\to V$,\nand $g_i : U_i \\to V_{\\alpha(i)}$ as in Lemma \\ref{lemma-pullback}\nthe functor\n$$\n(Y_j, \\varphi_{jj'}) \\longmapsto\n(g_i^*Y_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')})\n$$\nconstructed in that lemma\nis called the {\\it pullback functor} on descent data.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Descent data in fibred categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZE","source_file":"stacks.tex","source_line":405,"source_end_line":418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L405-L418","statement_sha256":"4261b7eeb2fae1cc962eb72f354278dd11952d88396d7e0417d5d7560897d1b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":764,"rank":764,"depth":2,"x":1026.478,"y":260.42,"cluster":"sheaves-sites"},{"id":"stacks:026E","tag":"026E","title":"Descent data in fibred categories · Definition 026E","summary":"Let C be a category. Let p : S → C be a fibred category. Make a choice of pullbacks as in Categories, Definition [Tag 02XN]. Let U = (f_i : U_i → U)_i ∈ I be a family of morphisms with target U. Assume all the fibre products U_i ×_U U_j and U_i ×_U U_j ×_U U_k exist. • Given an object X of S_U the trivial descent datum is the descent datum (X, id_X) with respect to the family (id_U : U → U). • Given an object X of S_U we have a canonical descent datum on the family of…","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category.\nMake a choice of pullbacks as in Categories,\nDefinition \\ref{categories-definition-pullback-functor-fibred-category}.\nLet $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms\nwith target $U$. Assume all the fibre products\n$U_i \\times_U U_j$ and $U_i \\times_U U_j \\times_U U_k$ exist.\n\\begin{enumerate}\n\\item Given an object $X$ of $\\mathcal{S}_U$ the {\\it trivial descent datum}\nis the descent datum $(X, \\text{id}_X)$ with respect to the family\n$\\{\\text{id}_U : U \\to U\\}$.\n\\item Given an object $X$ of $\\mathcal{S}_U$\nwe have a {\\it canonical descent datum} on the family of\nobjects $f_i^*X$ by pulling back the trivial\ndescent datum $(X, \\text{id}_X)$ via the\nobvious map $\\{f_i : U_i \\to U\\} \\to \\{\\text{id}_U : U \\to U\\}$.\nWe denote this descent datum $(f_i^*X, can)$.\n\\item A descent datum $(X_i, \\varphi_{ij})$\nrelative to $\\{f_i : U_i \\to U\\}$ is called {\\it effective}\nif there exists an object $X$ of $\\mathcal{S}_U$ such that\n$(X_i, \\varphi_{ij})$ is isomorphic to $(f_i^*X, can)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Descent data in fibred categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/026E","source_file":"stacks.tex","source_line":428,"source_end_line":452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L428-L452","statement_sha256":"5660b2bfc17c9bd545ea2b696cae8ab74d2bcbd1b5f66578efa248fec611c99b","origin":"The Stacks Project","memory_eligible":false,"source_rank":765,"rank":765,"depth":1,"x":1173.896,"y":131.322,"cluster":"sheaves-sites"},{"id":"stacks:026D","tag":"026D","title":"Descent data in fibred categories · Lemma 026D","summary":"In the situation of Definition [Tag 026E] part (2) the maps can_ij : pr_0^*f_i^*X → pr_1^*f_j^*X are equal to (α_pr_1, f_j)_X ∘ (α_pr_0, f_i)_X^-1 where α_·, · is as in Categories, Lemma [Tag 02XO] and where we use the equality f_i ∘ pr_0 = f_j ∘ pr_1 as maps U_i ×_U U_j → U.","statement_latex":"In the situation of\nDefinition \\ref{definition-effective-descent-datum} part (2) the maps\n$can_{ij} : \\text{pr}_0^*f_i^*X \\to \\text{pr}_1^*f_j^*X$ are equal to\n$(\\alpha_{\\text{pr}_1, f_j})_X \\circ (\\alpha_{\\text{pr}_0, f_i})_X^{-1}$\nwhere $\\alpha_{\\cdot, \\cdot}$ is as in\nCategories, Lemma \\ref{categories-lemma-fibred}\nand where we\nuse the equality $f_i \\circ \\text{pr}_0 = f_j \\circ \\text{pr}_1$\nas maps $U_i \\times_U U_j \\to U$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Descent data in fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/026D","source_file":"stacks.tex","source_line":465,"source_end_line":476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L465-L476","statement_sha256":"5a3bf52448a5c43da35824ab5b42d9e5991bfc49d9220f8eaa354a4ee6d5c091","origin":"The Stacks Project","memory_eligible":false,"source_rank":766,"rank":766,"depth":3,"x":1169.002,"y":310.431,"cluster":"sheaves-sites"},{"id":"stacks:0GEA","tag":"0GEA","title":"Descent data in fibred categories · Lemma 0GEA","summary":"Let C be a category. Let V = (V_j → U)_j ∈ J → U = (U_i → U)_i ∈ I be a morphism of families of maps with fixed target of C given by id : U → U, α : J → I and f_j : V_j → U_α(j). Let p : S → C be a fibred category. If • for 0 ≤ p ≤ 3 and 0 ≤ q ≤ 3 with p + q ≥ 2 and i_1, …, i_p ∈ I and j_1, …, j_q ∈ J the fibre products U_i_1 ×_U … ×_U U_i_p ×_U V_j_1 ×_U … ×_U V_j_q exist, • the functor S_U → DD(V) is an equivalence, • for every i ∈ I the functor S_U_i → DD(V_i) is fully…","statement_latex":"Let $\\mathcal{C}$ be a category. Let\n$\\mathcal{V} = \\{V_j \\to U\\}_{j \\in J} \\to\n\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$\nbe a morphism of families of maps with fixed target of\n$\\mathcal{C}$ given by $\\text{id} : U \\to U$,\n$\\alpha : J \\to I$ and $f_j : V_j \\to U_{\\alpha(j)}$. Let\n$p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category. If\n\\begin{enumerate}\n\\item for $0 \\leq p \\leq 3$ and $0 \\leq q \\leq 3$ with $p + q \\geq 2$\nand $i_1, \\ldots, i_p \\in I$ and $j_1, \\ldots, j_q \\in J$\nthe fibre products $U_{i_1} \\times_U \\ldots \\times_U U_{i_p} \\times_U\nV_{j_1} \\times_U \\ldots \\times_U V_{j_q}$ exist,\n\\item the functor $\\mathcal{S}_U \\to DD(\\mathcal{V})$\nis an equivalence,\n\\item for every $i \\in I$ the functor\n$\\mathcal{S}_{U_i} \\to DD(\\mathcal{V}_i)$\nis fully faithful, and\n\\item for every $i, i' \\in I$ the functor\n$\\mathcal{S}_{U_i \\times_U U_{i'}} \\to DD(\\mathcal{V}_{ii'})$\nis faithful.\n\\end{enumerate}\nHere $\\mathcal{V}_i = \\{U_i \\times_U V_j \\to U_i\\}_{j \\in J}$ and\n$\\mathcal{V}_{ii'} =\n\\{U_i \\times_U U_{i'} \\times_U V_j \\to U_i \\times_U U_{i'}\\}_{j \\in J}$.\nThen $\\mathcal{S}_U \\to DD(\\mathcal{U})$ is an equivalence.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Descent data in fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEA","source_file":"stacks.tex","source_line":482,"source_end_line":509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L482-L509","statement_sha256":"0c6375e3546e96177ad1223a0bd776298c1fcd5b2db6916986bbde0382982531","origin":"The Stacks Project","memory_eligible":false,"source_rank":767,"rank":767,"depth":2,"x":1028.369,"y":175.382,"cluster":"sheaves-sites"},{"id":"stacks:026F","tag":"026F","title":"Stacks · Definition 026F","summary":"Let C be a site. A stack over C is a category p : S → C over C which satisfies the following conditions: • p : S → C is a fibred category, see Categories, Definition [Tag 02XM], • for any U ∈ Ob(C) and any x, y ∈ S_U the presheaf mathitMor(x, y) (see Definition [Tag 02ZB]) is a sheaf on the site C/U, and • for any covering U = (f_i : U_i → U)_i ∈ I of the site C, any descent datum in S relative to U is effective.","statement_latex":"Let $\\mathcal{C}$ be a site. A {\\it stack} over $\\mathcal{C}$\nis a category $p : \\mathcal{S} \\to \\mathcal{C}$ over $\\mathcal{C}$ which\nsatisfies the following conditions:\n\\begin{enumerate}\n\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category, see\nCategories, Definition \\ref{categories-definition-fibred-category},\n\\item for any $U \\in \\Ob(\\mathcal{C})$ and any $x, y \\in \\mathcal{S}_U$\nthe presheaf $\\mathit{Mor}(x, y)$ (see\nDefinition \\ref{definition-mor-presheaf}) is a sheaf on\nthe site $\\mathcal{C}/U$, and\n\\item for any covering $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$\nof the site $\\mathcal{C}$, any descent datum in $\\mathcal{S}$\nrelative to $\\mathcal{U}$ is effective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/026F","source_file":"stacks.tex","source_line":567,"source_end_line":583,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L567-L583","statement_sha256":"eb9efa7379c34e29a709c17eb8979643534ab0b4cf915d6fa049f1ba939565d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":768,"rank":768,"depth":1,"x":1240.985,"y":195.196,"cluster":"sheaves-sites"},{"id":"stacks:02ZF","tag":"02ZF","title":"Stacks · Lemma 02ZF","summary":"Let C be a site. Let p : S → C be a fibred category over C. The following are equivalent • S is a stack over C, and • for any covering U = (f_i : U_i → U)_i ∈ I of the site C the functor S_U → DD(U) which associates to an object its canonical descent datum is an equivalence.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category\nover $\\mathcal{C}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{S}$ is a stack over $\\mathcal{C}$, and\n\\item for any covering $\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$\nof the site $\\mathcal{C}$ the functor\n$$\n\\mathcal{S}_U \\longrightarrow DD(\\mathcal{U})\n$$\nwhich associates to an\nobject its canonical descent datum is an equivalence.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZF","source_file":"stacks.tex","source_line":596,"source_end_line":611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L596-L611","statement_sha256":"8735b3d378e74c8d1a03f4d007dfe21c266316d793dab2bcaddac98e70435447","origin":"The Stacks Project","memory_eligible":false,"source_rank":769,"rank":769,"depth":0,"x":1068.017,"y":301.385,"cluster":"sheaves-sites"},{"id":"stacks:04TU","tag":"04TU","title":"Stacks · Lemma 04TU","summary":"Let p : S → C be a stack over the site C. Let S' be a subcategory of S. Assume • if φ : y → x is a strongly cartesian morphism of S and x is an object of S', then y is isomorphic to an object of S', • S' is a full subcategory of S, and • if (f_i : U_i → U) is a covering of C, and x an object of S over U such that f_i^*x is isomorphic to an object of S' for each i, then x is isomorphic to an object of S'. Then S' → C is a stack.","statement_latex":"Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a stack over the site $\\mathcal{C}$.\nLet $\\mathcal{S}'$ be a subcategory of $\\mathcal{S}$.\nAssume\n\\begin{enumerate}\n\\item if $\\varphi : y \\to x$ is a strongly cartesian\nmorphism of $\\mathcal{S}$ and\n$x$ is an object of $\\mathcal{S}'$, then $y$ is isomorphic to an\nobject of $\\mathcal{S}'$,\n\\item $\\mathcal{S}'$ is a full subcategory of $\\mathcal{S}$, and\n\\item if $\\{f_i : U_i \\to U\\}$ is a covering of $\\mathcal{C}$,\nand $x$ an object of $\\mathcal{S}$ over $U$ such that $f_i^*x$\nis isomorphic to an object of $\\mathcal{S}'$ for each $i$,\nthen $x$ is isomorphic to an object of $\\mathcal{S}'$.\n\\end{enumerate}\nThen $\\mathcal{S}' \\to \\mathcal{C}$ is a stack.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TU","source_file":"stacks.tex","source_line":617,"source_end_line":634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L617-L634","statement_sha256":"fcd0abf1e7fd2724f065cfbc81b1b7c53abf8b3665d471efb5842a5531f2b4e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":770,"rank":770,"depth":0,"x":1110.229,"y":124.678,"cluster":"sheaves-sites"},{"id":"stacks:042W","tag":"042W","title":"Stacks · Lemma 042W","summary":"Let C be a site. Let S_1, S_2 be categories over C. Suppose that S_1 and S_2 are equivalent as categories over C. Then S_1 is a stack over C if and only if S_2 is a stack over C.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.\nSuppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent\nas categories over $\\mathcal{C}$.\nThen $\\mathcal{S}_1$ is a stack over $\\mathcal{C}$ if and only if\n$\\mathcal{S}_2$ is a stack over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042W","source_file":"stacks.tex","source_line":647,"source_end_line":655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L647-L655","statement_sha256":"bae998334cf4d4c4423a300bd4e3f8bf9b0aa7ed580ce1eaeae56b1e20b9f8c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":771,"rank":771,"depth":3,"x":1221.367,"y":279.156,"cluster":"sheaves-sites"},{"id":"stacks:02ZG","tag":"02ZG","title":"Stacks · Definition 02ZG","summary":"Let C be a site. The 2-category of stacks over C is the sub 2-category of the 2-category of fibred categories over C (see Categories, Definition [Tag 02XP]) defined as follows: • Its objects will be stacks p : S → C. • Its 1-morphisms (S, p) → (S', p') will be functors G : S → S' such that p' ∘ G = p and such that G maps strongly cartesian morphisms to strongly cartesian morphisms. • Its 2-morphisms t : G → H for G, H : (S, p) → (S', p') will be morphisms of functors such…","statement_latex":"Let $\\mathcal{C}$ be a site.\nThe {\\it $2$-category of stacks over $\\mathcal{C}$}\nis the sub $2$-category of the $2$-category of fibred categories\nover $\\mathcal{C}$ (see\nCategories, Definition \\ref{categories-definition-fibred-categories-over-C})\ndefined as follows:\n\\begin{enumerate}\n\\item Its objects will be stacks $p : \\mathcal{S} \\to \\mathcal{C}$.\n\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that\n$p' \\circ G = p$ and such that $G$ maps strongly cartesian\nmorphisms to strongly cartesian morphisms.\n\\item Its $2$-morphisms $t : G \\to H$ for\n$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be morphisms of functors\nsuch that $p'(t_x) = \\text{id}_{p(x)}$\nfor all $x \\in \\Ob(\\mathcal{S})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZG","source_file":"stacks.tex","source_line":699,"source_end_line":719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L699-L719","statement_sha256":"f4e21d522631ffcc3f53dcbb05dc00259f5128743e4cad21d640daab2efa2f9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":772,"rank":772,"depth":2,"x":1014.888,"y":228.235,"cluster":"sheaves-sites"},{"id":"stacks:026G","tag":"026G","title":"Stacks · Lemma 026G","summary":"Let C be a site. The (2, 1)-category of stacks over C has 2-fibre products, and they are described as in Categories, Lemma [Tag 0040].","statement_latex":"Let $\\mathcal{C}$ be a site.\nThe $(2, 1)$-category of stacks over $\\mathcal{C}$\nhas 2-fibre products, and they are described as in\nCategories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/026G","source_file":"stacks.tex","source_line":721,"source_end_line":727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L721-L727","statement_sha256":"3c9cfb857a11a8994c304fbf6b5c4443430c9099d051c0bf8a8a79d6a526a268","origin":"The Stacks Project","memory_eligible":false,"source_rank":773,"rank":773,"depth":3,"x":1208.373,"y":148.507,"cluster":"sheaves-sites"},{"id":"stacks:04WQ","tag":"04WQ","title":"Stacks · Lemma 04WQ","summary":"Let C be a site. Let S_1, S_2 be stacks over C. Let F : S_1 → S_2 be a 1-morphism. Then the following are equivalent • F is fully faithful, • for every U ∈ Ob(C) and for every x, y ∈ Ob(S_1, U) the map F : mathitMor_S_1(x, y) → mathitMor_S_2(F(x), F(y)) is an isomorphism of sheaves on C/U.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{S}_1$, $\\mathcal{S}_2$ be stacks over $\\mathcal{C}$.\nLet $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$ be a $1$-morphism.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $F$ is fully faithful,\n\\item for every $U \\in \\Ob(\\mathcal{C})$ and for every\n$x, y \\in \\Ob(\\mathcal{S}_{1, U})$ the map\n$$\nF :\n\\mathit{Mor}_{\\mathcal{S}_1}(x, y)\n\\longrightarrow\n\\mathit{Mor}_{\\mathcal{S}_2}(F(x), F(y))\n$$\nis an isomorphism of sheaves on $\\mathcal{C}/U$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WQ","source_file":"stacks.tex","source_line":791,"source_end_line":809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L791-L809","statement_sha256":"6c63374cef197150fe38eb07b38d180574ed9304628c855209827f237fc7530e","origin":"The Stacks Project","memory_eligible":false,"source_rank":774,"rank":774,"depth":0,"x":1129.702,"y":317.329,"cluster":"sheaves-sites"},{"id":"stacks:046N","tag":"046N","title":"Stacks · Lemma 046N","summary":"Let C be a site. Let S_1, S_2 be stacks over C. Let F : S_1 → S_2 be a 1-morphism which is fully faithful. Then the following are equivalent • F is an equivalence, • for every U ∈ Ob(C) and for every x ∈ Ob(S_2, U) there exists a covering (f_i : U_i → U) such that f_i^*x is in the essential image of the functor F : S_1, U_i → S_2, U_i.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{S}_1$, $\\mathcal{S}_2$ be stacks over $\\mathcal{C}$.\nLet $F : \\mathcal{S}_1 \\to \\mathcal{S}_2$ be a $1$-morphism which is\nfully faithful. Then the following are equivalent\n\\begin{enumerate}\n\\item $F$ is an equivalence,\n\\item for every $U \\in \\Ob(\\mathcal{C})$ and for every\n$x \\in \\Ob(\\mathcal{S}_{2, U})$ there exists a covering\n$\\{f_i : U_i \\to U\\}$ such that $f_i^*x$ is in the essential image\nof the functor $F : \\mathcal{S}_{1, U_i} \\to \\mathcal{S}_{2, U_i}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046N","source_file":"stacks.tex","source_line":844,"source_end_line":857,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L844-L857","statement_sha256":"89fc63ea9574710ec373029681ad714ee29fb05b5586ea03edc3546c6f9236bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":775,"rank":775,"depth":0,"x":1051.838,"y":147.958,"cluster":"sheaves-sites"},{"id":"stacks:02ZI","tag":"02ZI","title":"Stacks in groupoids · Definition 02ZI","summary":"A stack in groupoids over a site C is a category p : S → C over C such that • p : S → C is fibred in groupoids over C (see Categories, Definition [Tag 003T]), • for all U ∈ Ob(C), for all x, y∈ Ob(S_U) the presheaf mathitIsom(x, y) is a sheaf on the site C/U, and • for all coverings U = (U_i → U) in C, all descent data (x_i, φ_ij) for U are effective.","statement_latex":"A {\\it stack in groupoids} over a site $\\mathcal{C}$ is a\ncategory $p : \\mathcal{S} \\to \\mathcal{C}$ over $\\mathcal{C}$\nsuch that\n\\begin{enumerate}\n\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is fibred\nin groupoids over $\\mathcal{C}$ (see\nCategories, Definition \\ref{categories-definition-fibred-groupoids}),\n\\item for all $U \\in \\Ob(\\mathcal{C})$,\nfor all $x, y\\in \\Ob(\\mathcal{S}_U)$ the presheaf\n$\\mathit{Isom}(x, y)$ is a sheaf on the site $\\mathcal{C}/U$, and\n\\item for all coverings $\\mathcal{U} = \\{U_i \\to U\\}$ in $\\mathcal{C}$,\nall descent data $(x_i, \\phi_{ij})$ for $\\mathcal{U}$ are effective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZI","source_file":"stacks.tex","source_line":931,"source_end_line":946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L931-L946","statement_sha256":"d29ea2ee0b85156a2beed5598bb7b054472fea7eade284082d8dbbcd406233fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":776,"rank":776,"depth":1,"x":1245.736,"y":228.784,"cluster":"sheaves-sites"},{"id":"stacks:02ZJ","tag":"02ZJ","title":"Stacks in groupoids · Lemma 02ZJ","summary":"Let C be a site. Let p : S → C be a category over C. The following are equivalent • S is a stack in groupoids over C, • S is a stack over C and all fibre categories are groupoids, and • S is fibred in groupoids over C and is a stack over C.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category over $\\mathcal{C}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{S}$ is a stack in groupoids over $\\mathcal{C}$,\n\\item $\\mathcal{S}$ is a stack over $\\mathcal{C}$ and all\nfibre categories are groupoids, and\n\\item $\\mathcal{S}$ is fibred in groupoids over $\\mathcal{C}$\nand is a stack over $\\mathcal{C}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZJ","source_file":"stacks.tex","source_line":952,"source_end_line":964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L952-L964","statement_sha256":"37f291ec9882bcd08125dd5e6542998fa85d6915646ea61bad57ab1a2a070129","origin":"The Stacks Project","memory_eligible":false,"source_rank":777,"rank":777,"depth":4,"x":1037.462,"y":279.279,"cluster":"sheaves-sites"},{"id":"stacks:03YI","tag":"03YI","title":"Stacks in groupoids · Lemma 03YI","summary":"Let C be a site. Let p : S → C be a stack. Let p' : S' → C be the category fibred in groupoids associated to S constructed in Categories, Lemma [Tag 03WQ]. Then p' : S' → C is a stack in groupoids.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a stack.\nLet $p' : \\mathcal{S}' \\to \\mathcal{C}$\nbe the category fibred in groupoids associated to $\\mathcal{S}$\nconstructed in\nCategories, Lemma \\ref{categories-lemma-fibred-gives-fibred-groupoids}.\nThen $p' : \\mathcal{S}' \\to \\mathcal{C}$ is a stack in groupoids.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YI","source_file":"stacks.tex","source_line":970,"source_end_line":979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L970-L979","statement_sha256":"f8126ac3d5a3cb96cd04c807b4ebb3583f63ec7a4170cf67f989196c77dbd84b","origin":"The Stacks Project","memory_eligible":false,"source_rank":778,"rank":778,"depth":3,"x":1150.595,"y":123.643,"cluster":"sheaves-sites"},{"id":"stacks:042X","tag":"042X","title":"Stacks in groupoids · Lemma 042X","summary":"Let C be a site. Let S_1, S_2 be categories over C. Suppose that S_1 and S_2 are equivalent as categories over C. Then S_1 is a stack in groupoids over C if and only if S_2 is a stack in groupoids over C.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.\nSuppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent\nas categories over $\\mathcal{C}$.\nThen $\\mathcal{S}_1$ is a stack in groupoids over $\\mathcal{C}$ if and only if\n$\\mathcal{S}_2$ is a stack in groupoids over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042X","source_file":"stacks.tex","source_line":989,"source_end_line":997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L989-L997","statement_sha256":"30d1bb3aed6716bd1dafe5e2e4192d6c2c9adbae324eb4926cd818c8bf0f6ea1","origin":"The Stacks Project","memory_eligible":false,"source_rank":779,"rank":779,"depth":5,"x":1192.39,"y":302.856,"cluster":"sheaves-sites"},{"id":"stacks:02ZK","tag":"02ZK","title":"Stacks in groupoids · Definition 02ZK","summary":"Let C be a site. The 2-category of stacks in groupoids over C is the sub 2-category of the 2-category of stacks over C (see Definition [Tag 02ZG]) defined as follows: • Its objects will be stacks in groupoids p : S → C. • Its 1-morphisms (S, p) → (S', p') will be functors G : S → S' such that p' ∘ G = p. (Since every morphism is strongly cartesian every functor preserves them.) • Its 2-morphisms t : G → H for G, H : (S, p) → (S', p') will be morphisms of functors such…","statement_latex":"Let $\\mathcal{C}$ be a site.\nThe {\\it $2$-category of stacks in groupoids over $\\mathcal{C}$}\nis the sub $2$-category of the $2$-category of stacks\nover $\\mathcal{C}$ (see Definition \\ref{definition-stacks-over-C})\ndefined as follows:\n\\begin{enumerate}\n\\item Its objects will be stacks in groupoids\n$p : \\mathcal{S} \\to \\mathcal{C}$.\n\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that\n$p' \\circ G = p$. (Since every morphism is strongly cartesian\nevery functor preserves them.)\n\\item Its $2$-morphisms $t : G \\to H$ for\n$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be morphisms of functors\nsuch that $p'(t_x) = \\text{id}_{p(x)}$\nfor all $x \\in \\Ob(\\mathcal{S})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZK","source_file":"stacks.tex","source_line":1008,"source_end_line":1028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1008-L1028","statement_sha256":"c6a43af9a2b1c33f82cb48c211ab289748feabfdb218d62e8d5404146d1191f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":780,"rank":780,"depth":3,"x":1017.204,"y":194.268,"cluster":"sheaves-sites"},{"id":"stacks:02ZL","tag":"02ZL","title":"Stacks in groupoids · Lemma 02ZL","summary":"Let C be a category. The 2-category of stacks in groupoids over C has 2-fibre products, and they are described as in Categories, Lemma [Tag 0040].","statement_latex":"Let $\\mathcal{C}$ be a category.\nThe $2$-category of stacks in groupoids over $\\mathcal{C}$\nhas 2-fibre products, and they are described as in\nCategories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZL","source_file":"stacks.tex","source_line":1035,"source_end_line":1041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1035-L1041","statement_sha256":"f1eca470a8487b7fdf407f5bf683a70b4d1ac1ca98d0f6e42c5e7ab48ff9c8b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":781,"rank":781,"depth":5,"x":1234.014,"y":174.908,"cluster":"sheaves-sites"},{"id":"stacks:042Z","tag":"042Z","title":"Stacks in setoids · Definition 042Z","summary":"Let C be a site. • A stack in setoids over C is a stack over C all of whose fibre categories are setoids. • A stack in sets, or a stack in discrete categories is a stack over C all of whose fibre categories are discrete.","statement_latex":"Let $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item A {\\it stack in setoids} over $\\mathcal{C}$\nis a stack over $\\mathcal{C}$ all of whose fibre categories are\nsetoids.\n\\item A {\\it stack in sets}, or a {\\it stack in discrete categories}\nis a stack over $\\mathcal{C}$ all of whose fibre categories are discrete.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042Z","source_file":"stacks.tex","source_line":1066,"source_end_line":1076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1066-L1076","statement_sha256":"163c6c31215ba4f61409960b8c61d7be5ecfc6944dccc8aeb1fe11a4d7adaebf","origin":"The Stacks Project","memory_eligible":false,"source_rank":782,"rank":782,"depth":0,"x":1089.506,"y":312.401,"cluster":"sheaves-sites"},{"id":"stacks:0430","tag":"0430","title":"Stacks in setoids · Lemma 0430","summary":"Let C be a site. Under the equivalence ( the category of presheaves of sets over C ) ↔ ( the category of categories fibred in sets over C ) of Categories, Lemma [Tag 02Y2] the stacks in sets correspond precisely to the sheaves.","statement_latex":"Let $\\mathcal{C}$ be a site. Under the equivalence\n$$\n\\left\\{\n\\begin{matrix}\n\\text{the category of presheaves}\\\\\n\\text{of sets over }\\mathcal{C}\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{the category of categories}\\\\\n\\text{fibred in sets over }\\mathcal{C}\n\\end{matrix}\n\\right\\}\n$$\nof\nCategories, Lemma \\ref{categories-lemma-2-category-fibred-sets}\nthe stacks in sets correspond precisely to the sheaves.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0430","source_file":"stacks.tex","source_line":1085,"source_end_line":1106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1085-L1106","statement_sha256":"c141e681511656b7d570981eee176d097e57ed14718e8914a7e5ec19f9f35a58","origin":"The Stacks Project","memory_eligible":false,"source_rank":783,"rank":783,"depth":5,"x":1085.49,"y":128.76,"cluster":"sheaves-sites"},{"id":"stacks:0432","tag":"0432","title":"Stacks in setoids · Lemma 0432","summary":"Let C be a site. Let S be a category fibred in setoids over C. Then S is a stack in setoids if and only if the unique equivalent category S' fibred in sets (see Categories, Lemma [Tag 0045]) is a stack in sets. In other words, if and only if the presheaf U ↦ Ob(S_U)/ ≅ is a sheaf.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{S}$ be a category fibred in setoids over $\\mathcal{C}$.\nThen $\\mathcal{S}$ is a stack in setoids if and only if the unique\nequivalent category $\\mathcal{S}'$ fibred in sets (see\nCategories, Lemma \\ref{categories-lemma-setoid-fibres})\nis a stack in sets. In other words, if and only if the presheaf\n$$\nU \\longmapsto \\Ob(\\mathcal{S}_U)/\\!\\!\\cong\n$$\nis a sheaf.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0432","source_file":"stacks.tex","source_line":1113,"source_end_line":1125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1113-L1125","statement_sha256":"c166dfd8b0c8010860e8a430da775578bf1f5571979a028c39e5085face125ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":784,"rank":784,"depth":1,"x":1236.343,"y":262.082,"cluster":"sheaves-sites"},{"id":"stacks:0431","tag":"0431","title":"Stacks in setoids · Lemma 0431","summary":"Let C be a site. Let S_1, S_2 be categories over C. Suppose that S_1 and S_2 are equivalent as categories over C. Then S_1 is a stack in setoids over C if and only if S_2 is a stack in setoids over C.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.\nSuppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent\nas categories over $\\mathcal{C}$.\nThen $\\mathcal{S}_1$ is a stack in setoids over $\\mathcal{C}$ if and only if\n$\\mathcal{S}_2$ is a stack in setoids over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0431","source_file":"stacks.tex","source_line":1131,"source_end_line":1139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1131-L1139","statement_sha256":"b2fb8dc3271ea4a6135174dc20c112030a70b36e8b0b24a5b5ff7845f9e2591a","origin":"The Stacks Project","memory_eligible":false,"source_rank":785,"rank":785,"depth":2,"x":1017.586,"y":249.351,"cluster":"sheaves-sites"},{"id":"stacks:0433","tag":"0433","title":"Stacks in setoids · Definition 0433","summary":"Let C be a site. The 2-category of stacks in setoids over C is the sub 2-category of the 2-category of stacks over C (see Definition [Tag 02ZG]) defined as follows: • Its objects will be stacks in setoids p : S → C. • Its 1-morphisms (S, p) → (S', p') will be functors G : S → S' such that p' ∘ G = p. (Since every morphism is strongly cartesian every functor preserves them.) • Its 2-morphisms t : G → H for G, H : (S, p) → (S', p') will be morphisms of functors such that…","statement_latex":"Let $\\mathcal{C}$ be a site.\nThe {\\it $2$-category of stacks in setoids over $\\mathcal{C}$}\nis the sub $2$-category of the $2$-category of stacks\nover $\\mathcal{C}$ (see Definition \\ref{definition-stacks-over-C})\ndefined as follows:\n\\begin{enumerate}\n\\item Its objects will be stacks in setoids\n$p : \\mathcal{S} \\to \\mathcal{C}$.\n\\item Its $1$-morphisms $(\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be functors $G : \\mathcal{S} \\to \\mathcal{S}'$ such that\n$p' \\circ G = p$. (Since every morphism is strongly cartesian\nevery functor preserves them.)\n\\item Its $2$-morphisms $t : G \\to H$ for\n$G, H : (\\mathcal{S}, p) \\to (\\mathcal{S}', p')$\nwill be morphisms of functors\nsuch that $p'(t_x) = \\text{id}_{p(x)}$\nfor all $x \\in \\Ob(\\mathcal{S})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0433","source_file":"stacks.tex","source_line":1157,"source_end_line":1177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1157-L1177","statement_sha256":"b7dff206c04a3023591be287e782042df6f9ffe1c881fbb82f76d8d46d801c2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":786,"rank":786,"depth":3,"x":1189.372,"y":134.451,"cluster":"sheaves-sites"},{"id":"stacks:0434","tag":"0434","title":"Stacks in setoids · Lemma 0434","summary":"Let C be a site. The 2-category of stacks in setoids over C has 2-fibre products, and they are described as in Categories, Lemma [Tag 0040].","statement_latex":"Let $\\mathcal{C}$ be a site.\nThe $2$-category of stacks in setoids over $\\mathcal{C}$\nhas 2-fibre products, and they are described as in\nCategories, Lemma \\ref{categories-lemma-2-product-categories-over-C}.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0434","source_file":"stacks.tex","source_line":1184,"source_end_line":1190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1184-L1190","statement_sha256":"97770968eaec1d4b1d1c4ca7bded7c314ec63819808aaf3f9dff7e97c4b0b33f","origin":"The Stacks Project","memory_eligible":false,"source_rank":787,"rank":787,"depth":5,"x":1155.051,"y":316.906,"cluster":"sheaves-sites"},{"id":"stacks:05UI","tag":"05UI","title":"Stacks in setoids · Lemma 05UI","summary":"Let C be a site. Let S, T be stacks in groupoids over C and let R be a stack in setoids over C. Let f : T → S and g : R → S be 1-morphisms. If f is faithful, then the 2-fibre product T ×_f, S, g R is a stack in setoids over C.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{S}, \\mathcal{T}$ be stacks in groupoids over $\\mathcal{C}$\nand let $\\mathcal{R}$ be a stack in setoids over $\\mathcal{C}$.\nLet $f : \\mathcal{T} \\to \\mathcal{S}$ and $g : \\mathcal{R} \\to \\mathcal{S}$\nbe $1$-morphisms. If $f$ is faithful, then the $2$-fibre product\n$$\n\\mathcal{T} \\times_{f, \\mathcal{S}, g} \\mathcal{R}\n$$\nis a stack in setoids over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UI","source_file":"stacks.tex","source_line":1201,"source_end_line":1212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1201-L1212","statement_sha256":"5a3989b4e18f726b32ee327f70abd2abd14574fad2e66a3b6a32bf77b17e8bd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":788,"rank":788,"depth":2,"x":1033.464,"y":162.678,"cluster":"sheaves-sites"},{"id":"stacks:05UJ","tag":"05UJ","title":"Stacks in setoids · Lemma 05UJ","summary":"Let C be a site. Let S be a stack in groupoids over C and let S_i, i = 1, 2 be stacks in setoids over C. Let f_i : S_i → S be 1-morphisms. Then the 2-fibre product S_1 ×_f_1, S, f_2 S_2 is a stack in setoids over C.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{S}$ be a stack in groupoids over $\\mathcal{C}$ and\nlet $\\mathcal{S}_i$, $i = 1, 2$ be stacks in setoids over $\\mathcal{C}$.\nLet $f_i : \\mathcal{S}_i \\to \\mathcal{S}$ be $1$-morphisms.\nThen the $2$-fibre product\n$$\n\\mathcal{S}_1 \\times_{f_1, \\mathcal{S}, f_2} \\mathcal{S}_2\n$$\nis a stack in setoids over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UJ","source_file":"stacks.tex","source_line":1219,"source_end_line":1230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1219-L1230","statement_sha256":"25a7881727fc111977d3801a6a59585553aaa6f759c032585daa1f6dd3125d46","origin":"The Stacks Project","memory_eligible":false,"source_rank":789,"rank":789,"depth":3,"x":1247.443,"y":207.474,"cluster":"sheaves-sites"},{"id":"stacks:06DV","tag":"06DV","title":"Stacks in setoids · Lemma 06DV","summary":"Let C be a site. Let xymatrix T_2 ar[r] ar[d]_G' & T_1 ar[d]^G S_2 ar[r]^F & S_1 be a 2-cartesian diagram of stacks in groupoids over C. Assume • for every U ∈ Ob(C) and x ∈ Ob((S_1)_U) there exists a covering (U_i → U) such that x|_U_i is in the essential image of F : (S_2)_U_i → (S_1)_U_i, and • G' is faithful, then G is faithful.","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$$\n\\xymatrix{\n\\mathcal{T}_2 \\ar[r] \\ar[d]_{G'} & \\mathcal{T}_1 \\ar[d]^G \\\\\n\\mathcal{S}_2 \\ar[r]^F & \\mathcal{S}_1\n}\n$$\nbe a $2$-cartesian diagram of stacks in groupoids over $\\mathcal{C}$.\nAssume\n\\begin{enumerate}\n\\item for every $U \\in \\Ob(\\mathcal{C})$ and\n$x \\in \\Ob((\\mathcal{S}_1)_U)$ there exists a covering\n$\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential\nimage of $F : (\\mathcal{S}_2)_{U_i} \\to (\\mathcal{S}_1)_{U_i}$, and\n\\item $G'$ is faithful,\n\\end{enumerate}\nthen $G$ is faithful.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DV","source_file":"stacks.tex","source_line":1238,"source_end_line":1257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1238-L1257","statement_sha256":"1f70f6f17ec501606690ce2a442bc867b56b593e921fb03e0cb29b29ea4eace5","origin":"The Stacks Project","memory_eligible":false,"source_rank":790,"rank":790,"depth":3,"x":1053.367,"y":295.981,"cluster":"sheaves-sites"},{"id":"stacks:05W9","tag":"05W9","title":"Stacks in setoids · Lemma 05W9","summary":"Let C be a site. Let xymatrix T_2 ar[r] ar[d] & T_1 ar[d]^G S_2 ar[r]^F & S_1 be a 2-cartesian diagram of stacks in groupoids over C. If • F : S_2 → S_1 is fully faithful, • for every U ∈ Ob(C) and x ∈ Ob((S_1)_U) there exists a covering (U_i → U) such that x|_U_i is in the essential image of F : (S_2)_U_i → (S_1)_U_i, and • T_2 is a stack in setoids. then T_1 is a stack in setoids.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet\n$$\n\\xymatrix{\n\\mathcal{T}_2 \\ar[r] \\ar[d] & \\mathcal{T}_1 \\ar[d]^G \\\\\n\\mathcal{S}_2 \\ar[r]^F & \\mathcal{S}_1\n}\n$$\nbe a $2$-cartesian diagram of stacks in groupoids over $\\mathcal{C}$.\nIf\n\\begin{enumerate}\n\\item $F : \\mathcal{S}_2 \\to \\mathcal{S}_1$ is fully faithful,\n\\item for every $U \\in \\Ob(\\mathcal{C})$ and\n$x \\in \\Ob((\\mathcal{S}_1)_U)$ there exists a covering\n$\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential\nimage of $F : (\\mathcal{S}_2)_{U_i} \\to (\\mathcal{S}_1)_{U_i}$, and\n\\item $\\mathcal{T}_2$ is a stack in setoids.\n\\end{enumerate}\nthen $\\mathcal{T}_1$ is a stack in setoids.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05W9","source_file":"stacks.tex","source_line":1288,"source_end_line":1309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1288-L1309","statement_sha256":"d2c5ac0bc3185aaf81adf696805d9d04523a0b6e99b4a38e09f973da34a3b88b","origin":"The Stacks Project","memory_eligible":false,"source_rank":791,"rank":791,"depth":2,"x":1125.4,"y":120.352,"cluster":"sheaves-sites"},{"id":"stacks:0CKJ","tag":"0CKJ","title":"Stacks in setoids · Lemma 0CKJ","summary":"Let C be a site. Let F : S → T be a 1-morphism of categories fibred in groupoids over C. Assume that • T is a stack in groupoids over C, • for every U ∈ Ob(C) the functor S_U → T_U of fibre categories is faithful, • for each U and each y ∈ Ob(T_U) the presheaf (h : V → U) ↦ ((x, f) mid x ∈ Ob(S_V), f : F(x) → f^*y over V)/≅ is a sheaf on C/U. Then S is a stack in groupoids over C.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $F : \\mathcal{S} \\to \\mathcal{T}$\nbe a $1$-morphism of categories fibred in groupoids over $\\mathcal{C}$.\nAssume that\n\\begin{enumerate}\n\\item $\\mathcal{T}$ is a stack in groupoids over $\\mathcal{C}$,\n\\item for every $U \\in \\Ob(\\mathcal{C})$ the functor\n$\\mathcal{S}_U \\to \\mathcal{T}_U$ of fibre categories is faithful,\n\\item for each $U$ and each $y \\in \\Ob(\\mathcal{T}_U)$ the presheaf\n$$\n(h : V \\to U)\n\\longmapsto\n\\{(x, f) \\mid x \\in \\Ob(\\mathcal{S}_V), f : F(x) \\to f^*y\\text{ over }V\\}/\\cong\n$$\nis a sheaf on $\\mathcal{C}/U$.\n\\end{enumerate}\nThen $\\mathcal{S}$ is a stack in groupoids over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks in setoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKJ","source_file":"stacks.tex","source_line":1333,"source_end_line":1351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1333-L1351","statement_sha256":"7bb18ea951ab1c4bcb402955e32813fc791532e211fb9ddd9f7dff70b2ac3481","origin":"The Stacks Project","memory_eligible":false,"source_rank":792,"rank":792,"depth":0,"x":1213.64,"y":290.966,"cluster":"sheaves-sites"},{"id":"stacks:036Y","tag":"036Y","title":"The inertia stack · Lemma 036Y","summary":"Let C be a site. Let p : S → C and p' : S' → C be stacks over the site C. Let F : S → S' be a 1-morphism of stacks over C. • The inertia I_S/S' and I_S are stacks over C. • If S, S' are stacks in groupoids over C, then so are I_S/S' and I_S. • If S, S' are stacks in setoids over C, then so are I_S/S' and I_S.","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$p : \\mathcal{S} \\to \\mathcal{C}$ and\n$p' : \\mathcal{S}' \\to \\mathcal{C}$\nbe stacks over the site $\\mathcal{C}$.\nLet $F : \\mathcal{S} \\to \\mathcal{S}'$ be a $1$-morphism of\nstacks over $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The inertia $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ and\n$\\mathcal{I}_\\mathcal{S}$ are stacks over $\\mathcal{C}$.\n\\item If $\\mathcal{S}, \\mathcal{S}'$ are stacks in groupoids over\n$\\mathcal{C}$, then so are $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ and\n$\\mathcal{I}_\\mathcal{S}$.\n\\item If $\\mathcal{S}, \\mathcal{S}'$ are stacks in setoids over $\\mathcal{C}$,\nthen so are $\\mathcal{I}_{\\mathcal{S}/\\mathcal{S}'}$ and\n$\\mathcal{I}_\\mathcal{S}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"The inertia stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036Y","source_file":"stacks.tex","source_line":1427,"source_end_line":1445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1427-L1445","statement_sha256":"f220f16dcb6205e185b0e0235b4e2eb3bf04636248d89c543a77128481717efa","origin":"The Stacks Project","memory_eligible":false,"source_rank":793,"rank":793,"depth":6,"x":1011.094,"y":215.124,"cluster":"sheaves-sites"},{"id":"stacks:04ZM","tag":"04ZM","title":"The inertia stack · Lemma 04ZM","summary":"Let C be a site. If S is a stack in groupoids, then the canonical 1-morphism I_S → S is an equivalence if and only if S is a stack in setoids.","statement_latex":"Let $\\mathcal{C}$ be a site.\nIf $\\mathcal{S}$ is a stack in groupoids, then the\ncanonical $1$-morphism $\\mathcal{I}_\\mathcal{S} \\to \\mathcal{S}$\nis an equivalence if and only if $\\mathcal{S}$ is a stack in setoids.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"The inertia stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZM","source_file":"stacks.tex","source_line":1456,"source_end_line":1462,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1456-L1462","statement_sha256":"0ad07817c20b53191312be48ff890f2108319749b8e1020506f1dd6da7c125a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":794,"rank":794,"depth":1,"x":1221.726,"y":156.038,"cluster":"sheaves-sites"},{"id":"stacks:02ZN","tag":"02ZN","title":"Stackification of fibred categories · Lemma 02ZN","summary":"Let C be a site. Let p : S → C be a fibred category over C. There exists a stack p' : S' → C and a 1-morphism G : S → S' of fibred categories over C (see Categories, Definition [Tag 02XP]) such that • for every U ∈ Ob(C), and any x, y ∈ Ob(S_U) the map mathitMor(x, y) → mathitMor(G(x), G(y)) induced by G identifies the right hand side with the sheafification of the left hand side, and • for every U ∈ Ob(C), and any x' ∈ Ob(S'_U) there exists a covering (U_i → U)_i ∈ I…","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category over $\\mathcal{C}$.\nThere exists a stack $p' : \\mathcal{S}' \\to \\mathcal{C}$ and a\n$1$-morphism $G : \\mathcal{S} \\to \\mathcal{S}'$\nof fibred categories over $\\mathcal{C}$ (see\nCategories, Definition \\ref{categories-definition-fibred-categories-over-C})\nsuch that\n\\begin{enumerate}\n\\item for every $U \\in \\Ob(\\mathcal{C})$, and any\n$x, y \\in \\Ob(\\mathcal{S}_U)$ the map\n$$\n\\mathit{Mor}(x, y) \\longrightarrow \\mathit{Mor}(G(x), G(y))\n$$\ninduced by $G$ identifies the right hand side with the sheafification\nof the left hand side, and\n\\item for every $U \\in \\Ob(\\mathcal{C})$, and any\n$x' \\in \\Ob(\\mathcal{S}'_U)$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ such that for every $i \\in I$ the\nobject $x'|_{U_i}$ is in the essential image of the\nfunctor $G : \\mathcal{S}_{U_i} \\to \\mathcal{S}'_{U_i}$.\n\\end{enumerate}\nMoreover the stack $\\mathcal{S}'$ is determined up to unique\n$2$-isomorphism by these conditions.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stackification of fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZN","source_file":"stacks.tex","source_line":1479,"source_end_line":1504,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1479-L1504","statement_sha256":"fdd5cb17cb2e1f919e1bab4e196472b017f867a6f06aaae82b6d84879374efc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":795,"rank":795,"depth":2,"x":1113.777,"y":319.347,"cluster":"sheaves-sites"},{"id":"stacks:0435","tag":"0435","title":"Stackification of fibred categories · Lemma 0435","summary":"Let C be a site. Let p : S → C be a fibred category over C. Let p' : S' → C and G : S → S' the stack and 1-morphism constructed in Lemma [Tag 02ZN]. This construction has the following universal property: Given a stack q : X → C and a 1-morphism F : S → X of fibred categories over C there exists a 1-morphism H : S' → X such that the diagram xymatrix S ar[rr]_F ar[rd]_G & & X & S' ar[ru]_H is 2-commutative.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a fibred category over $\\mathcal{C}$.\nLet $p' : \\mathcal{S}' \\to \\mathcal{C}$ and $G : \\mathcal{S} \\to \\mathcal{S}'$\nthe stack and $1$-morphism constructed in Lemma \\ref{lemma-stackify}.\nThis construction has the following universal property: Given a stack\n$q : \\mathcal{X} \\to \\mathcal{C}$ and a $1$-morphism\n$F : \\mathcal{S} \\to \\mathcal{X}$ of fibred categories over $\\mathcal{C}$\nthere exists a $1$-morphism $H : \\mathcal{S}' \\to \\mathcal{X}$\nsuch that the diagram\n$$\n\\xymatrix{\n\\mathcal{S} \\ar[rr]_F \\ar[rd]_G & & \\mathcal{X} \\\\\n& \\mathcal{S}' \\ar[ru]_H\n}\n$$\nis $2$-commutative.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stackification of fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0435","source_file":"stacks.tex","source_line":1730,"source_end_line":1748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1730-L1748","statement_sha256":"2b39eed992ef2551b9018eb94e84bd7957bac4d1c2658a1e4cae47e511a24cd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":796,"rank":796,"depth":3,"x":1061.979,"y":137.426,"cluster":"sheaves-sites"},{"id":"stacks:04W9","tag":"04W9","title":"Stackification of fibred categories · Lemma 04W9","summary":"Notation and assumptions as in Lemma [Tag 0435]. There is a canonical equivalence of categories Mor_Fib/C(S, X) = Mor_Stacks/C(S', X) given by the constructions in the proof of the aforementioned lemma.","statement_latex":"Notation and assumptions as in\nLemma \\ref{lemma-stackify-universal-property}.\nThere is a canonical equivalence of categories\n$$\n\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{X})\n=\n\\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}', \\mathcal{X})\n$$\ngiven by the constructions in the proof of the aforementioned lemma.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stackification of fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04W9","source_file":"stacks.tex","source_line":1770,"source_end_line":1781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1770-L1781","statement_sha256":"8cf1934a5e2f2ebe401b13aaf8912f5937406c208c6ab6691c0e1732001163b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":797,"rank":797,"depth":4,"x":1246.72,"y":242.322,"cluster":"sheaves-sites"},{"id":"stacks:04Y1","tag":"04Y1","title":"Stackification of fibred categories · Lemma 04Y1","summary":"Let C be a site. Let f : X → Y and g : Z → Y be morphisms of fibred categories over C. In this case the stackification of the 2-fibre product is the 2-fibre product of the stackifications.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Z} \\to \\mathcal{Y}$\nbe morphisms of fibred categories over $\\mathcal{C}$.\nIn this case the stackification of the $2$-fibre product is the $2$-fibre\nproduct of the stackifications.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stackification of fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Y1","source_file":"stacks.tex","source_line":1787,"source_end_line":1794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1787-L1794","statement_sha256":"9c1c1e794e71acff0cbea8b34e27dded82e5db48a5d821973b0940d389d32dfc","origin":"The Stacks Project","memory_eligible":false,"source_rank":798,"rank":798,"depth":4,"x":1025.841,"y":269.837,"cluster":"sheaves-sites"},{"id":"stacks:06NS","tag":"06NS","title":"Stackification of fibred categories · Lemma 06NS","summary":"Let C be a site. Let X be a fibred category over C. The stackification of the inertia fibred category I_X is inertia of the stackification of X.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{X}$ be a fibred category over $\\mathcal{C}$.\nThe stackification of the inertia fibred category $\\mathcal{I}_\\mathcal{X}$\nis inertia of the stackification of $\\mathcal{X}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stackification of fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NS","source_file":"stacks.tex","source_line":1866,"source_end_line":1872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1866-L1872","statement_sha256":"ed85c35cb21901a6c12672b87ff201952f922a97fd37a15ff06ebdc39d7c88cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":799,"rank":799,"depth":5,"x":1166.777,"y":124.02,"cluster":"sheaves-sites"},{"id":"stacks:02ZP","tag":"02ZP","title":"Stackification of categories fibred in groupoids · Lemma 02ZP","summary":"Let C be a site. Let p : S → C be a category fibred in groupoids over C. There exists a stack in groupoids p' : S' → C and a 1-morphism G : S → S' of categories fibred in groupoids over C (see Categories, Definition [Tag 02XS]) such that • for every U ∈ Ob(C), and any x, y ∈ Ob(S_U) the map mathitMor(x, y) → mathitMor(G(x), G(y)) induced by G identifies the right hand side with the sheafification of the left hand side, and • for every U ∈ Ob(C), and any x' ∈ Ob(S'_U)…","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category\nfibred in groupoids over $\\mathcal{C}$.\nThere exists a stack in groupoids\n$p' : \\mathcal{S}' \\to \\mathcal{C}$ and a\n$1$-morphism $G : \\mathcal{S} \\to \\mathcal{S}'$\nof categories fibred in groupoids over $\\mathcal{C}$ (see\nCategories, Definition\n\\ref{categories-definition-categories-fibred-in-groupoids-over-C})\nsuch that\n\\begin{enumerate}\n\\item for every $U \\in \\Ob(\\mathcal{C})$, and any\n$x, y \\in \\Ob(\\mathcal{S}_U)$ the map\n$$\n\\mathit{Mor}(x, y) \\longrightarrow \\mathit{Mor}(G(x), G(y))\n$$\ninduced by $G$ identifies the right hand side with the sheafification\nof the left hand side, and\n\\item for every $U \\in \\Ob(\\mathcal{C})$, and any\n$x' \\in \\Ob(\\mathcal{S}'_U)$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ such that for every $i \\in I$ the\nobject $x'|_{U_i}$ is in the essential image of the\nfunctor $G : \\mathcal{S}_{U_i} \\to \\mathcal{S}'_{U_i}$.\n\\end{enumerate}\nMoreover the stack in groupoids $\\mathcal{S}'$ is determined up to unique\n$2$-isomorphism by these conditions.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stackification of categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZP","source_file":"stacks.tex","source_line":1893,"source_end_line":1921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1893-L1921","statement_sha256":"d4fe29192e450025a3f7d36ce9c124293c25999c6dde3e36465d3077043245ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":800,"rank":800,"depth":5,"x":1180.133,"y":311.765,"cluster":"sheaves-sites"},{"id":"stacks:0436","tag":"0436","title":"Stackification of categories fibred in groupoids · Lemma 0436","summary":"Let C be a site. Let p : S → C be a category fibred in groupoids over C. Let p' : S' → C and G : S → S' the stack in groupoids and 1-morphism constructed in Lemma [Tag 02ZP]. This construction has the following universal property: Given a stack in groupoids q : X → C and a 1-morphism F : S → X of categories over C there exists a 1-morphism H : S' → X such that the diagram xymatrix S ar[rr]_F ar[rd]_G & & X & S' ar[ru]_H is 2-commutative.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ be a category fibred in groupoids\nover $\\mathcal{C}$. Let $p' : \\mathcal{S}' \\to \\mathcal{C}$ and\n$G : \\mathcal{S} \\to \\mathcal{S}'$\nthe stack in groupoids and $1$-morphism constructed in\nLemma \\ref{lemma-stackify-groupoids}.\nThis construction has the following universal property: Given a stack\nin groupoids $q : \\mathcal{X} \\to \\mathcal{C}$ and a $1$-morphism\n$F : \\mathcal{S} \\to \\mathcal{X}$ of categories over $\\mathcal{C}$\nthere exists a $1$-morphism $H : \\mathcal{S}' \\to \\mathcal{X}$\nsuch that the diagram\n$$\n\\xymatrix{\n\\mathcal{S} \\ar[rr]_F \\ar[rd]_G & & \\mathcal{X} \\\\\n& \\mathcal{S}' \\ar[ru]_H\n}\n$$\nis $2$-commutative.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stackification of categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0436","source_file":"stacks.tex","source_line":1928,"source_end_line":1948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1928-L1948","statement_sha256":"212e05188491ba91f4965d1f482b1b8aa4440468b86e6ad29806622f754d856e","origin":"The Stacks Project","memory_eligible":false,"source_rank":801,"rank":801,"depth":6,"x":1019.09,"y":180.728,"cluster":"sheaves-sites"},{"id":"stacks:04Y2","tag":"04Y2","title":"Stackification of categories fibred in groupoids · Lemma 04Y2","summary":"Let C be a site. Let f : X → Y and g : Z → Y be morphisms of categories fibred in groupoids over C. In this case the stackification of the 2-fibre product is the 2-fibre product of the stackifications.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Z} \\to \\mathcal{Y}$\nbe morphisms of categories fibred in groupoids over $\\mathcal{C}$.\nIn this case the stackification of the $2$-fibre product is the $2$-fibre\nproduct of the stackifications.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stackification of categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Y2","source_file":"stacks.tex","source_line":1955,"source_end_line":1962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1955-L1962","statement_sha256":"eea3869024d5465bf63cc8565fa2d35c3b84db04c2181da4b01eeccec0840568","origin":"The Stacks Project","memory_eligible":false,"source_rank":802,"rank":802,"depth":5,"x":1243.517,"y":185.98,"cluster":"sheaves-sites"},{"id":"stacks:06NU","tag":"06NU","title":"Inherited topologies · Lemma 06NU","summary":"Let C be a site. Let p : S → C be a fibred category. Let Cov(S) be the set of families (x_i → x)_i ∈ I of morphisms in S with fixed target such that (a) each x_i → x is strongly cartesian, and (b) (p(x_i) → p(x))_i ∈ I is a covering of C. Then (S, Cov(S)) is a site.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$\nbe a fibred category. Let $\\text{Cov}(\\mathcal{S})$\nbe the set of families $\\{x_i \\to x\\}_{i \\in I}$ of morphisms in $\\mathcal{S}$\nwith fixed target such that (a) each $x_i \\to x$ is strongly cartesian,\nand (b) $\\{p(x_i) \\to p(x)\\}_{i \\in I}$ is a covering of $\\mathcal{C}$.\nThen $(\\mathcal{S}, \\text{Cov}(\\mathcal{S}))$ is a site.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Inherited topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NU","source_file":"stacks.tex","source_line":1984,"source_end_line":1992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L1984-L1992","statement_sha256":"6373f97b889cc2b100294ee8cbfdf7da2b346cf55514e91384d758f3c1e35b3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":803,"rank":803,"depth":2,"x":1073.577,"y":309.617,"cluster":"sheaves-sites"},{"id":"stacks:06NV","tag":"06NV","title":"Inherited topologies · Definition 06NV","summary":"Let C be a site. Let p : S → C be a fibred category. We say (S, Cov(S)) as in Lemma [Tag 06NU] is the structure of site on S inherited from C. We sometimes indicate this by saying that S is endowed with the topology inherited from C.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a\nfibred category. We say $(\\mathcal{S}, \\text{Cov}(\\mathcal{S}))$ as in\nLemma \\ref{lemma-topology-inherited}\nis the {\\it structure of site on $\\mathcal{S}$ inherited from $\\mathcal{C}$}.\nWe sometimes indicate this by saying that\n{\\it $\\mathcal{S}$ is endowed with the topology inherited from $\\mathcal{C}$}.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Inherited topologies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NV","source_file":"stacks.tex","source_line":2037,"source_end_line":2045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2037-L2045","statement_sha256":"60f47b2c2f2180644edc12b061a63cdd14cb17d14b3575c148892ffb74d0e1c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":804,"rank":804,"depth":3,"x":1099.497,"y":121.772,"cluster":"sheaves-sites"},{"id":"stacks:06NW","tag":"06NW","title":"Inherited topologies · Lemma 06NW","summary":"Let C be a site. Let F : X → Y be a 1-morphism of fibred categories over C. Then F is a continuous and cocontinuous functor between the structure of sites inherited from C. Hence F induces a morphism of topoi f : Sh(X) → Sh(Y) with f_* = _sF = _pF and f^-1 = F^s = F^p. In particular f^-1(G)(x) = G(F(x)) for a sheaf G on Y and object x of X.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $F : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of fibred categories over $\\mathcal{C}$.\nThen $F$ is a continuous and cocontinuous functor between the structure\nof sites inherited from $\\mathcal{C}$. Hence $F$ induces a morphism of topoi\n$f : \\Sh(\\mathcal{X}) \\to \\Sh(\\mathcal{Y})$ with\n$f_* = {}_sF = {}_pF$ and $f^{-1} = F^s = F^p$. In particular\n$f^{-1}(\\mathcal{G})(x) = \\mathcal{G}(F(x))$\nfor a sheaf $\\mathcal{G}$ on $\\mathcal{Y}$ and object $x$ of $\\mathcal{X}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Inherited topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NW","source_file":"stacks.tex","source_line":2052,"source_end_line":2062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2052-L2062","statement_sha256":"7981d4cb2847e2e61268e1e2d73c5b5f19146d7bdb24e946c41620b9813e5fb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":805,"rank":805,"depth":6,"x":1231.619,"y":275.197,"cluster":"sheaves-sites"},{"id":"stacks:0CN0","tag":"0CN0","title":"Inherited topologies · Lemma 0CN0","summary":"Let C be a site. Let p : X → C be a category fibred in groupoids. Let x ∈ Ob(X) lying over U = p(x). The functor p induces an equivalence of sites X/x → C/U where X is endowed with the topology inherited from C.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$\nbe a category fibred in groupoids. Let $x \\in \\Ob(\\mathcal{X})$\nlying over $U = p(x)$. The functor $p$ induces an equivalence of sites\n$\\mathcal{X}/x \\to \\mathcal{C}/U$ where $\\mathcal{X}$ is endowed with\nthe topology inherited from $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Inherited topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CN0","source_file":"stacks.tex","source_line":2106,"source_end_line":2113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2106-L2113","statement_sha256":"ff938edbe621ff66793c99895be0778c436b93e7db58d7048e331df382b9d085","origin":"The Stacks Project","memory_eligible":false,"source_rank":806,"rank":806,"depth":4,"x":1010.523,"y":236.982,"cluster":"sheaves-sites"},{"id":"stacks:06NX","tag":"06NX","title":"Inherited topologies · Lemma 06NX","summary":"Let C be a site. Let p : X → C and q : Y → C be stacks in groupoids. Let F : X → Y be a 1-morphism of categories over C. If F turns X into a category fibred in groupoids over Y, then X is a stack in groupoids over Y (with topology inherited from C).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$\nand $q : \\mathcal{Y} \\to \\mathcal{C}$\nbe stacks in groupoids. Let $F : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of categories over $\\mathcal{C}$. If $F$ turns\n$\\mathcal{X}$ into a category fibred in groupoids over $\\mathcal{Y}$,\nthen $\\mathcal{X}$ is a stack in groupoids over $\\mathcal{Y}$ (with\ntopology inherited from $\\mathcal{C}$).","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Inherited topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NX","source_file":"stacks.tex","source_line":2127,"source_end_line":2136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2127-L2136","statement_sha256":"d1147db966e75a30b6023df41b1b2548d439d95bceb00bd16841f3f64742b78f","origin":"The Stacks Project","memory_eligible":false,"source_rank":807,"rank":807,"depth":0,"x":1204.542,"y":139.578,"cluster":"sheaves-sites"},{"id":"stacks:09WX","tag":"09WX","title":"Inherited topologies · Lemma 09WX","summary":"Let C be a site. Let p : X → C be a stack. Endow X with the topology inherited from C and let q : Y → X be a stack. Then Y is a stack over C. If p and q define stacks in groupoids, then Y is a stack in groupoids over C.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$\nbe a stack. Endow $\\mathcal{X}$ with the topology inherited from\n$\\mathcal{C}$ and let $q : \\mathcal{Y} \\to \\mathcal{X}$ be a stack.\nThen $\\mathcal{Y}$ is a stack over $\\mathcal{C}$.\nIf $p$ and $q$ define stacks in groupoids, then\n$\\mathcal{Y}$ is a stack in groupoids over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Inherited topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WX","source_file":"stacks.tex","source_line":2163,"source_end_line":2171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2163-L2171","statement_sha256":"570b5f39e7eef0b40e6e529ddc4dab0eb191887085e81d6e2bdd8c2a05f31e2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":808,"rank":808,"depth":5,"x":1139.72,"y":321.732,"cluster":"sheaves-sites"},{"id":"stacks:06NZ","tag":"06NZ","title":"Gerbes · Definition 06NZ","summary":"A gerbe over a site C is a category p : S → C over C such that • p : S → C is a stack in groupoids over C (see Definition [Tag 02ZI]), • for U ∈ Ob(C) there exists a covering (U_i → U) in C such that S_U_i is nonempty, and • for U ∈ Ob(C) and x, y ∈ Ob(S_U) there exists a covering (U_i → U) in C such that x|_U_i ≅ y|_U_i in S_U_i.","statement_latex":"A {\\it gerbe} over a site $\\mathcal{C}$ is a category\n$p : \\mathcal{S} \\to \\mathcal{C}$ over $\\mathcal{C}$ such that\n\\begin{enumerate}\n\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a stack\nin groupoids over $\\mathcal{C}$ (see\nDefinition \\ref{definition-stack-in-groupoids}),\n\\item for $U \\in \\Ob(\\mathcal{C})$ there exists\na covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ such that\n$\\mathcal{S}_{U_i}$ is nonempty, and\n\\item for $U \\in \\Ob(\\mathcal{C})$ and\n$x, y \\in \\Ob(\\mathcal{S}_U)$ there exists\na covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ such that\n$x|_{U_i} \\cong y|_{U_i}$ in $\\mathcal{S}_{U_i}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Gerbes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NZ","source_file":"stacks.tex","source_line":2238,"source_end_line":2254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2238-L2254","statement_sha256":"4038def4873e1b5195bb5f41cf739c81030e54a25a340d2c0190844a2f6040f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":809,"rank":809,"depth":2,"x":1040.906,"y":150.408,"cluster":"sheaves-sites"},{"id":"stacks:06P0","tag":"06P0","title":"Gerbes · Lemma 06P0","summary":"Let C be a site. Let S_1, S_2 be categories over C. Suppose that S_1 and S_2 are equivalent as categories over C. Then S_1 is a gerbe over C if and only if S_2 is a gerbe over C.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{S}_1$, $\\mathcal{S}_2$ be categories over $\\mathcal{C}$.\nSuppose that $\\mathcal{S}_1$ and $\\mathcal{S}_2$ are equivalent\nas categories over $\\mathcal{C}$.\nThen $\\mathcal{S}_1$ is a gerbe over $\\mathcal{C}$ if and only if\n$\\mathcal{S}_2$ is a gerbe over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06P0","source_file":"stacks.tex","source_line":2260,"source_end_line":2268,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2260-L2268","statement_sha256":"e02334606f5dd07599a37dbd56b4c3779cc581c3be6241d60746f27e1ca0afda","origin":"The Stacks Project","memory_eligible":false,"source_rank":810,"rank":810,"depth":6,"x":1251.819,"y":220.763,"cluster":"sheaves-sites"},{"id":"stacks:06P1","tag":"06P1","title":"Gerbes · Lemma 06P1","summary":"Let C be a site. Let p : X → C and q : Y → C be stacks in groupoids. Let F : X → Y be a 1-morphism of categories over C. The following are equivalent • For some (equivalently any) factorization F = F' ∘ a where a : X → X' is an equivalence of categories over C and F' is fibred in groupoids, the map F' : X' → Y is a gerbe (with the topology on Y inherited from C). • The following two conditions are satisfied • for y ∈ Ob(Y) lying over U ∈ Ob(C) there exists a covering (U_i…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{X} \\to \\mathcal{C}$\nand $q : \\mathcal{Y} \\to \\mathcal{C}$ be stacks in groupoids.\nLet $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nover $\\mathcal{C}$. The following are equivalent\n\\begin{enumerate}\n\\item For some (equivalently any) factorization $F = F' \\circ a$ where\n$a : \\mathcal{X} \\to \\mathcal{X}'$ is an equivalence of categories over\n$\\mathcal{C}$ and $F'$ is fibred in groupoids, the map\n$F' : \\mathcal{X}' \\to \\mathcal{Y}$ is a gerbe (with the topology\non $\\mathcal{Y}$ inherited from $\\mathcal{C}$).\n\\item The following two conditions are satisfied\n\\begin{enumerate}\n\\item for $y \\in \\Ob(\\mathcal{Y})$ lying over\n$U \\in \\Ob(\\mathcal{C})$ there exists a covering\n$\\{U_i \\to U\\}$ in $\\mathcal{C}$ and objects $x_i$ of $\\mathcal{X}$\nover $U_i$ such that $F(x_i) \\cong y|_{U_i}$ in $\\mathcal{Y}_{U_i}$, and\n\\item for $U \\in \\Ob(\\mathcal{C})$,\n$x, x' \\in \\Ob(\\mathcal{X}_U)$, and $b : F(x) \\to F(x')$ in\n$\\mathcal{Y}_U$ there exists\na covering $\\{U_i \\to U\\}$ in $\\mathcal{C}$ and morphisms\n$a_i : x|_{U_i} \\to x'|_{U_i}$ in $\\mathcal{X}_{U_i}$ with\n$F(a_i) = b|_{U_i}$.\n\\end{enumerate}\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06P1","source_file":"stacks.tex","source_line":2304,"source_end_line":2330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2304-L2330","statement_sha256":"2d0f00da07401c2f977b931a1d111a195383776ba14c7a89f212cfb17ecac6ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":811,"rank":811,"depth":7,"x":1039.441,"y":288.65,"cluster":"sheaves-sites"},{"id":"stacks:06P2","tag":"06P2","title":"Gerbes · Definition 06P2","summary":"Let C be a site. Let X and Y be stacks in groupoids over C. Let F : X → Y be a 1-morphism of categories over C. We say X is a gerbe over Y if the equivalent conditions of Lemma [Tag 06P1] are satisfied.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{X}$\nand $\\mathcal{Y}$ be stacks in groupoids over $\\mathcal{C}$.\nLet $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nover $\\mathcal{C}$. We say $\\mathcal{X}$ is a {\\it gerbe over} $\\mathcal{Y}$\nif the equivalent conditions of\nLemma \\ref{lemma-when-gerbe}\nare satisfied.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Gerbes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06P2","source_file":"stacks.tex","source_line":2379,"source_end_line":2388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2379-L2388","statement_sha256":"e31dd4d9c880bd26d7b1f8ad11de7971f96ada36d47aeb4347a217ee24298428","origin":"The Stacks Project","memory_eligible":false,"source_rank":812,"rank":812,"depth":8,"x":1141.585,"y":117.861,"cluster":"sheaves-sites"},{"id":"stacks:06P3","tag":"06P3","title":"Gerbes · Lemma 06P3","summary":"Let C be a site. Let xymatrix X' ar[r]_G' ar[d]_F' & X ar[d]^F Y' ar[r]^G & Y be a 2-fibre product of stacks in groupoids over C. If X is a gerbe over Y, then X' is a gerbe over Y'.","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r]_{G'} \\ar[d]_{F'} & \\mathcal{X} \\ar[d]^F \\\\\n\\mathcal{Y}' \\ar[r]^G & \\mathcal{Y}\n}\n$$\nbe a $2$-fibre product of stacks in groupoids over $\\mathcal{C}$.\nIf $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then\n$\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06P3","source_file":"stacks.tex","source_line":2411,"source_end_line":2423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2411-L2423","statement_sha256":"8976c0a0fbf5c03d63e44601698fdb6a2e1a0d82feca7849adbf8fae00c9955f","origin":"The Stacks Project","memory_eligible":false,"source_rank":813,"rank":813,"depth":8,"x":1203.69,"y":301.996,"cluster":"sheaves-sites"},{"id":"stacks:06R3","tag":"06R3","title":"Gerbes · Lemma 06R3","summary":"Let C be a site. Let F : X → Y and G : Y → Z be 1-morphisms of stacks in groupoids over C. If X is a gerbe over Y and Y is a gerbe over Z, then X is a gerbe over Z.","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$F : \\mathcal{X} \\to \\mathcal{Y}$ and $G : \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of stacks in groupoids over $\\mathcal{C}$.\nIf $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ and\n$\\mathcal{Y}$ is a gerbe over $\\mathcal{Z}$, then\n$\\mathcal{X}$ is a gerbe over $\\mathcal{Z}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R3","source_file":"stacks.tex","source_line":2456,"source_end_line":2464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2456-L2464","statement_sha256":"892e6aeeaf1473e655ef4421d8eeac2a4d432d7935b7c5355d40a38d1bfcb6cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":814,"rank":814,"depth":8,"x":1009.568,"y":201.327,"cluster":"sheaves-sites"},{"id":"stacks:06P4","tag":"06P4","title":"Gerbes · Lemma 06P4","summary":"Let C be a site. Let xymatrix X' ar[r]_G' ar[d]_F' & X ar[d]^F Y' ar[r]^G & Y be a 2-cartesian diagram of stacks in groupoids over C. If for every U ∈ Ob(C) and x ∈ Ob(Y_U) there exists a covering (U_i → U) such that x|_U_i is in the essential image of G : Y'_U_i → Y_U_i and X' is a gerbe over Y', then X is a gerbe over Y.","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r]_{G'} \\ar[d]_{F'} & \\mathcal{X} \\ar[d]^F \\\\\n\\mathcal{Y}' \\ar[r]^G & \\mathcal{Y}\n}\n$$\nbe a $2$-cartesian diagram of stacks in groupoids over $\\mathcal{C}$.\nIf for every $U \\in \\Ob(\\mathcal{C})$ and\n$x \\in \\Ob(\\mathcal{Y}_U)$ there exists a covering\n$\\{U_i \\to U\\}$ such that $x|_{U_i}$ is in the essential\nimage of $G : \\mathcal{Y}'_{U_i} \\to \\mathcal{Y}_{U_i}$ and\n$\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$, then\n$\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06P4","source_file":"stacks.tex","source_line":2497,"source_end_line":2513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2497-L2513","statement_sha256":"20c0f3c9f811cd4d4683ea2ccc516161a5ce83875ac1d2907ed7f24b5533b9c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":815,"rank":815,"depth":8,"x":1233.955,"y":165.363,"cluster":"sheaves-sites"},{"id":"stacks:0CJY","tag":"0CJY","title":"Gerbes · Lemma 0CJY","summary":"Let p : S → C be a gerbe over a site C. Assume that for all U ∈ Ob(C) and x ∈ Ob(S_U) the sheaf of groups mathitAut(x) = mathitIsom(x, x) on C/U is abelian. Then there exist • a sheaf G of abelian groups on C, • for every U ∈ Ob(C) and every x ∈ Ob(S_U) an isomorphism G|_U → mathitAut(x) such that for every U and every morphism φ : x → y in S_U the diagram xymatrix G|_U ar[d] ar@=[rr] & & G|_U ar[d] mathitAut(x) ar[rr]^α ↦ φ ∘ α ∘ φ^-1 & & mathitAut(y) is commutative.","statement_latex":"Let $p : \\mathcal{S} \\to \\mathcal{C}$ be a gerbe over a site $\\mathcal{C}$.\nAssume that for all $U \\in \\Ob(\\mathcal{C})$ and $x \\in \\Ob(\\mathcal{S}_U)$\nthe sheaf of groups $\\mathit{Aut}(x) = \\mathit{Isom}(x, x)$ on $\\mathcal{C}/U$\nis abelian. Then there exist\n\\begin{enumerate}\n\\item a sheaf $\\mathcal{G}$ of abelian groups on $\\mathcal{C}$,\n\\item for every $U \\in \\Ob(\\mathcal{C})$ and every $x \\in \\Ob(\\mathcal{S}_U)$\nan isomorphism $\\mathcal{G}|_U \\to \\mathit{Aut}(x)$\n\\end{enumerate}\nsuch that for every $U$ and every morphism $\\varphi : x \\to y$\nin $\\mathcal{S}_U$ the diagram\n$$\n\\xymatrix{\n\\mathcal{G}|_U \\ar[d] \\ar@{=}[rr] & & \\mathcal{G}|_U \\ar[d] \\\\\n\\mathit{Aut}(x)\n\\ar[rr]^{\\alpha \\mapsto \\varphi \\circ \\alpha \\circ \\varphi^{-1}} & &\n\\mathit{Aut}(y)\n}\n$$\nis commutative.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJY","source_file":"stacks.tex","source_line":2567,"source_end_line":2589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2567-L2589","statement_sha256":"b21331900a46804dfc61af833fbd710dcfcb99ea7bb2bc930cd3f372e4e50a28","origin":"The Stacks Project","memory_eligible":false,"source_rank":816,"rank":816,"depth":7,"x":1097.241,"y":319.402,"cluster":"sheaves-sites"},{"id":"stacks:04WB","tag":"04WB","title":"Functoriality for stacks · Lemma 04WB","summary":"In the situation above, if S is a fibred category over D then u^pS is a fibred category over C.","statement_latex":"In the situation above, if $\\mathcal{S}$ is a fibred category over\n$\\mathcal{D}$ then $u^p\\mathcal{S}$ is a fibred category over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WB","source_file":"stacks.tex","source_line":2697,"source_end_line":2701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2697-L2701","statement_sha256":"c01c8a7279742021f981d501cf67bbde6ce6d35d7ef2518c51ea230c7ec9ca54","origin":"The Stacks Project","memory_eligible":false,"source_rank":817,"rank":817,"depth":2,"x":1074.148,"y":127.996,"cluster":"sheaves-sites"},{"id":"stacks:04WC","tag":"04WC","title":"Functoriality for stacks · Lemma 04WC","summary":"Let u : C → D be a continuous functor of sites. Let p : S → D be a stack over D. Then u^pS is a stack over C.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites.\nLet $p : \\mathcal{S} \\to \\mathcal{D}$ be a stack over $\\mathcal{D}$.\nThen $u^p\\mathcal{S}$ is a stack over $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WC","source_file":"stacks.tex","source_line":2758,"source_end_line":2763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2758-L2763","statement_sha256":"720359fbbac7a7ad7551b6334154c5ddeb9d7169268a2f82e10ca84bd2bd90bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":818,"rank":818,"depth":3,"x":1245.319,"y":256.196,"cluster":"sheaves-sites"},{"id":"stacks:04WD","tag":"04WD","title":"Functoriality for stacks · Lemma 04WD","summary":"Let u : C → D be a continuous functor of sites. Let p : S → D be a stack in groupoids over D. Then u^pS is a stack in groupoids over C.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous functor of sites.\nLet $p : \\mathcal{S} \\to \\mathcal{D}$ be a stack in groupoids\nover $\\mathcal{D}$. Then $u^p\\mathcal{S}$ is a stack in groupoids\nover $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WD","source_file":"stacks.tex","source_line":2792,"source_end_line":2798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2792-L2798","statement_sha256":"81510783d77e7300a384b2eb41202cca1f922f609e6537195f87e680e2e4a3b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":819,"rank":819,"depth":4,"x":1015.712,"y":258.793,"cluster":"sheaves-sites"},{"id":"stacks:04WE","tag":"04WE","title":"Functoriality for stacks · Definition 04WE","summary":"Let f : D → C be a morphism of sites given by the continuous functor u : C → D. Let S be a fibred category over D. In this setting we write f_*S for the fibred category u^pS defined above. We say that f_*S is the pushforward of S along f.","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites\ngiven by the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.\nLet $\\mathcal{S}$ be a fibred category over $\\mathcal{D}$.\nIn this setting we write {\\it $f_*\\mathcal{S}$} for the fibred\ncategory $u^p\\mathcal{S}$ defined above. We say that\n$f_*\\mathcal{S}$ is the {\\it pushforward of $\\mathcal{S}$ along $f$}.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WE","source_file":"stacks.tex","source_line":2806,"source_end_line":2814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2806-L2814","statement_sha256":"f8cc6decba4a4e326691cff4d349d9c2bea8c302fe6143c81cdff56d3b52a64a","origin":"The Stacks Project","memory_eligible":false,"source_rank":820,"rank":820,"depth":0,"x":1183.145,"y":126.426,"cluster":"sheaves-sites"},{"id":"stacks:04WF","tag":"04WF","title":"Functoriality for stacks · Lemma 04WF","summary":"In the situation above assume • p : S → C is a fibred category, • C has nonempty finite limits, and • u : C → D commutes with nonempty finite limits. Consider the set R ⊂ Arrows(u_ppS) of morphisms of the form (a, id_V, α) : (U', φ' : V → u(U'), x') → (U, φ : V → u(U), x) with α strongly cartesian. Then R is a right multiplicative system.","statement_latex":"In the situation above assume\n\\begin{enumerate}\n\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category,\n\\item $\\mathcal{C}$ has nonempty finite limits, and\n\\item $u : \\mathcal{C} \\to \\mathcal{D}$ commutes with nonempty finite limits.\n\\end{enumerate}\nConsider the set $R \\subset \\text{Arrows}(u_{pp}\\mathcal{S})$ of morphisms\nof the form\n$$\n(a, \\text{id}_V, \\alpha) :\n(U', \\phi' : V \\to u(U'), x')\n\\longrightarrow\n(U, \\phi : V \\to u(U), x)\n$$\nwith $\\alpha$ strongly cartesian. Then $R$ is a right multiplicative system.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WF","source_file":"stacks.tex","source_line":2860,"source_end_line":2877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2860-L2877","statement_sha256":"ee588a4d400221db1adf03be39287ac6b64d0076705991a52f0751a90fe08eb2","origin":"The Stacks Project","memory_eligible":false,"source_rank":821,"rank":821,"depth":2,"x":1166.107,"y":319.282,"cluster":"sheaves-sites"},{"id":"stacks:04WG","tag":"04WG","title":"Functoriality for stacks · Lemma 04WG","summary":"With notation and assumptions as in Lemma [Tag 04WF]. Set u_pS = R^-1u_ppS, see Categories, Section [Tag 04VB]. Then u_pS is a fibred category over D.","statement_latex":"With notation and assumptions as in\nLemma \\ref{lemma-right-multiplicative-system}.\nSet $u_p\\mathcal{S} = R^{-1}u_{pp}\\mathcal{S}$, see\nCategories, Section \\ref{categories-section-localization}.\nThen $u_p\\mathcal{S}$ is a fibred category over $\\mathcal{D}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WG","source_file":"stacks.tex","source_line":2970,"source_end_line":2977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L2970-L2977","statement_sha256":"40610039e5961df9dae37e6a8526c251d93f2e66a4f97cf9758f86583eea24c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":822,"rank":822,"depth":3,"x":1023.401,"y":167.212,"cluster":"sheaves-sites"},{"id":"stacks:04WH","tag":"04WH","title":"Functoriality for stacks · Lemma 04WH","summary":"With notation and assumptions as in Lemma [Tag 04WG]. If S is fibred in groupoids, then u_pS is fibred in groupoids.","statement_latex":"With notation and assumptions as in\nLemma \\ref{lemma-fibred-category-pullback}.\nIf $\\mathcal{S}$ is fibred in groupoids, then $u_p\\mathcal{S}$ is fibred\nin groupoids.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WH","source_file":"stacks.tex","source_line":3142,"source_end_line":3148,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L3142-L3148","statement_sha256":"d28745ac72090b8acfb1a2f7758f36c5de50e4c323ae44d91d01c519de08c5f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":823,"rank":823,"depth":4,"x":1251.207,"y":198.411,"cluster":"sheaves-sites"},{"id":"stacks:04WI","tag":"04WI","title":"Functoriality for stacks · Lemma 04WI","summary":"Let u : C → D be a functor. Let p : S → C and q : T → D be categories over C and D. Assume that • p : S → C is a fibred category, • q : T → D is a fibred category, • C has nonempty finite limits, and • u : C → D commutes with nonempty finite limits. Then we have a canonical equivalence of categories Mor_Fib/C(S, u^pT) = Mor_Fib/D(u_pS, T) of morphism categories.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ and $q : \\mathcal{T} \\to \\mathcal{D}$\nbe categories over $\\mathcal{C}$ and $\\mathcal{D}$. Assume that\n\\begin{enumerate}\n\\item $p : \\mathcal{S} \\to \\mathcal{C}$ is a fibred category,\n\\item $q : \\mathcal{T} \\to \\mathcal{D}$ is a fibred category,\n\\item $\\mathcal{C}$ has nonempty finite limits, and\n\\item $u : \\mathcal{C} \\to \\mathcal{D}$ commutes with nonempty finite limits.\n\\end{enumerate}\nThen we have a canonical equivalence of categories\n$$\n\\Mor_{\\textit{Fib}/\\mathcal{C}}(\\mathcal{S}, u^p\\mathcal{T})\n=\n\\Mor_{\\textit{Fib}/\\mathcal{D}}(u_p\\mathcal{S}, \\mathcal{T})\n$$\nof morphism categories.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WI","source_file":"stacks.tex","source_line":3166,"source_end_line":3184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L3166-L3184","statement_sha256":"e1fc1d8e70526db712c56f6069c7e66fd98e7682820ef8c8fc2375f7695c6e74","origin":"The Stacks Project","memory_eligible":false,"source_rank":824,"rank":824,"depth":4,"x":1057.894,"y":304.802,"cluster":"sheaves-sites"},{"id":"stacks:04WJ","tag":"04WJ","title":"Functoriality for stacks · Definition 04WJ","summary":"Let f : D → C be a morphism of sites given by a continuous functor u : C → D satisfying the hypotheses and conclusions of Sites, Proposition [Tag 00X6]. Let S be a stack over C. In this setting we write f^-1S for the stackification of the fibred category u_pS over D constructed above. We say that f^-1S is the pullback of S along f.","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites\ngiven by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$\nsatisfying the hypotheses and conclusions of\nSites, Proposition \\ref{sites-proposition-get-morphism}.\nLet $\\mathcal{S}$ be a stack over $\\mathcal{C}$.\nIn this setting we write {\\it $f^{-1}\\mathcal{S}$} for the stackification\nof the fibred category $u_p\\mathcal{S}$ over $\\mathcal{D}$ constructed\nabove. We say that $f^{-1}\\mathcal{S}$ is the\n{\\it pullback of $\\mathcal{S}$ along $f$}.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WJ","source_file":"stacks.tex","source_line":3299,"source_end_line":3310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L3299-L3310","statement_sha256":"3b311b48b5cf60fd1190afae802c0aa33cb02355541e4363eb961c006e9f98fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":825,"rank":825,"depth":6,"x":1114.955,"y":116.424,"cluster":"sheaves-sites"},{"id":"stacks:04WK","tag":"04WK","title":"Functoriality for stacks · Lemma 04WK","summary":"Let f : D → C be a morphism of sites given by a continuous functor u : C → D satisfying the hypotheses and conclusions of Sites, Proposition [Tag 00X6]. Let p : S → C and q : T → D be stacks. Then we have a canonical equivalence of categories Mor_Stacks/C(S, f_*T) = Mor_Stacks/D(f^-1S, T) of morphism categories.","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites\ngiven by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$\nsatisfying the hypotheses and conclusions of\nSites, Proposition \\ref{sites-proposition-get-morphism}.\nLet $p : \\mathcal{S} \\to \\mathcal{C}$ and\n$q : \\mathcal{T} \\to \\mathcal{D}$ be stacks.\nThen we have a canonical equivalence of categories\n$$\n\\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}, f_*\\mathcal{T})\n=\n\\Mor_{\\textit{Stacks}/\\mathcal{D}}(f^{-1}\\mathcal{S}, \\mathcal{T})\n$$\nof morphism categories.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WK","source_file":"stacks.tex","source_line":3318,"source_end_line":3333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L3318-L3333","statement_sha256":"5e7cad97f0f886028b0f6469ba8709f3171df7ecda99c9bd76cee5e4d3b206d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":826,"rank":826,"depth":6,"x":1224.505,"y":287.923,"cluster":"sheaves-sites"},{"id":"stacks:04WR","tag":"04WR","title":"Functoriality for stacks · Lemma 04WR","summary":"Let f : D → C be a morphism of sites given by a continuous functor u : C → D satisfying the hypotheses and conclusions of Sites, Proposition [Tag 00X6]. Let S → C be a fibred category, and let S → S' be the stackification of S. Then f^-1S' is the stackification of u_pS.","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites\ngiven by a continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$\nsatisfying the hypotheses and conclusions of\nSites, Proposition \\ref{sites-proposition-get-morphism}.\nLet $\\mathcal{S} \\to \\mathcal{C}$ be a fibred category, and\nlet $\\mathcal{S} \\to \\mathcal{S}'$ be the stackification of $\\mathcal{S}$.\nThen $f^{-1}\\mathcal{S}'$ is the stackification of\n$u_p\\mathcal{S}$.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WR","source_file":"stacks.tex","source_line":3353,"source_end_line":3363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L3353-L3363","statement_sha256":"c8c988b958b294e884795f4577bfba142dad220fe0b80e9aedfea16b65a58e70","origin":"The Stacks Project","memory_eligible":false,"source_rank":827,"rank":827,"depth":6,"x":1005.536,"y":223.543,"cluster":"sheaves-sites"},{"id":"stacks:04WS","tag":"04WS","title":"Functoriality for stacks · Lemma 04WS","summary":"Let C and D be sites. Let u : C → D be a functor satisfying the assumptions of Sites, Lemma [Tag 00XU]. Let f : D → C be the corresponding morphism of sites. Then • for every stack p : S → C the canonical functor S → f_*f^-1S is an equivalence of stacks, • given stacks S, S' over C the construction f^-1 induces an equivalence Mor_Stacks/C(S, S') → Mor_Stacks/D(f^-1S, f^-1S') of morphism categories.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor satisfying the\nassumptions of\nSites, Lemma \\ref{sites-lemma-bigger-site}.\nLet $f : \\mathcal{D} \\to \\mathcal{C}$ be the corresponding\nmorphism of sites. Then\n\\begin{enumerate}\n\\item for every stack $p : \\mathcal{S} \\to \\mathcal{C}$ the\ncanonical functor $\\mathcal{S} \\to f_*f^{-1}\\mathcal{S}$ is\nan equivalence of stacks,\n\\item given stacks $\\mathcal{S}, \\mathcal{S}'$ over $\\mathcal{C}$\nthe construction $f^{-1}$\ninduces an equivalence\n$$\n\\Mor_{\\textit{Stacks}/\\mathcal{C}}(\\mathcal{S}, \\mathcal{S}')\n\\longrightarrow\n\\Mor_{\\textit{Stacks}/\\mathcal{D}}(f^{-1}\\mathcal{S}, f^{-1}\\mathcal{S}')\n$$\nof morphism categories.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Functoriality for stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WS","source_file":"stacks.tex","source_line":3377,"source_end_line":3399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L3377-L3399","statement_sha256":"c85eff85bc8a6e56a7ba22710f24578492298bd4982820722f5991614daff307","origin":"The Stacks Project","memory_eligible":false,"source_rank":828,"rank":828,"depth":8,"x":1219.039,"y":146.672,"cluster":"sheaves-sites"},{"id":"stacks:04WU","tag":"04WU","title":"Stacks and localization · Lemma 04WU","summary":"Let C be a site. Let U ∈ Ob(C). Then j_U : C/U → C is a stack over C if and only if h_U is a sheaf.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.\nThen $j_U : \\mathcal{C}/U \\to \\mathcal{C}$ is a stack over $\\mathcal{C}$\nif and only if $h_U$ is a sheaf.","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks and localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WU","source_file":"stacks.tex","source_line":3518,"source_end_line":3523,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L3518-L3523","statement_sha256":"fb33208e1549185e37d42ef49136b1a2552e2267091c9dba3ff345cec78a0b42","origin":"The Stacks Project","memory_eligible":false,"source_rank":829,"rank":829,"depth":2,"x":1123.304,"y":324.724,"cluster":"sheaves-sites"},{"id":"stacks:04WV","tag":"04WV","title":"Stacks and localization · Lemma 04WV","summary":"Assume that C is a site, and U is an object of C whose associated representable presheaf is a sheaf. Constructions A and B above define mutually inverse (!) functors of 2-categories ( 2-category of stacks over C/U ) ↔ ( 2-category of pairs (T, p) consisting of a stack T over C and a morphism p : T → C/U of stacks over C )","statement_latex":"Assume that $\\mathcal{C}$ is a site,\nand $U$ is an object of $\\mathcal{C}$ whose associated representable\npresheaf is a sheaf. Constructions A and B above define\nmutually inverse (!) functors of $2$-categories\n$$\n\\left\\{\n\\begin{matrix}\n2\\text{-category of}\\\\\n\\text{stacks over }\\mathcal{C}/U\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n2\\text{-category of pairs }(\\mathcal{T}, p)\n\\text{ consisting} \\\\\n\\text{of a stack }\\mathcal{T}\\text{ over }\\mathcal{C}\\text{ and a morphism} \\\\\np : \\mathcal{T} \\to \\mathcal{C}/U\\text{ of stacks over }\\mathcal{C}\n\\end{matrix}\n\\right\\}\n$$","area":"Sheaves & Sites","chapter":"Stacks","chapter_id":"stacks","section":"Stacks and localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WV","source_file":"stacks.tex","source_line":3563,"source_end_line":3586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks.tex#L3563-L3586","statement_sha256":"fd2f264b24c57bb1862b733a39a52f483ded398328a08faead6137a3222e9913","origin":"The Stacks Project","memory_eligible":false,"source_rank":830,"rank":830,"depth":0,"x":1050.624,"y":138.878,"cluster":"sheaves-sites"},{"id":"stacks:09FD","tag":"09FD","title":"Basic definitions · Definition 09FD","summary":"A field is a nonzero ring where every nonzero element is invertible. Given a field a subfield is a subring that is itself a field.","statement_latex":"A {\\it field} is a nonzero ring where every nonzero element is invertible.\nGiven a field a {\\it subfield} is a subring that is itself a field.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Basic definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09FD","source_file":"fields.tex","source_line":49,"source_end_line":53,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L49-L53","statement_sha256":"25ddbf0b9ddf9aa278fe1f29adbab18933ed14f7c5675ac14d9f8375e6518af7","origin":"The Stacks Project","memory_eligible":false,"source_rank":831,"rank":831,"depth":0,"x":1590.104,"y":220.0,"cluster":"fields-brauer-groups"},{"id":"stacks:09FE","tag":"09FE","title":"Basic definitions · Definition 09FE","summary":"A domain or an integral domain is a nonzero ring where 0 is the only zerodivisor.","statement_latex":"A {\\it domain} or an {\\it integral domain} is a nonzero ring where $0$\nis the only zerodivisor.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Basic definitions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09FE","source_file":"fields.tex","source_line":60,"source_end_line":64,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L60-L64","statement_sha256":"7f5c1a47c3f0f1f493d1cc196d57ad5cc6848281a61fde8be5a346295c2fabe2","origin":"The Stacks Project","memory_eligible":false,"source_rank":832,"rank":832,"depth":0,"x":1567.096,"y":229.93,"cluster":"fields-brauer-groups"},{"id":"stacks:09FN","tag":"09FN","title":"Vector spaces · Lemma 09FN","summary":"If k is a field, then every k-module is free.","statement_latex":"If $k$ is a field, then every $k$-module is free.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Vector spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09FN","source_file":"fields.tex","source_line":154,"source_end_line":157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L154-L157","statement_sha256":"b62c3fa1ae7676f966feca1160ce6ecba40fd7f758439f513ab2220989c01560","origin":"The Stacks Project","memory_eligible":false,"source_rank":833,"rank":833,"depth":0,"x":1581.975,"y":201.095,"cluster":"fields-brauer-groups"},{"id":"stacks:09FP","tag":"09FP","title":"Vector spaces · Lemma 09FP","summary":"Every exact sequence of modules over a field splits.","statement_latex":"Every exact sequence of modules over a field splits.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Vector spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09FP","source_file":"fields.tex","source_line":165,"source_end_line":168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L165-L168","statement_sha256":"edb025688cad6f504b307c6a6becb8dc08cc5c967f1877e08ec09c85f048438c","origin":"The Stacks Project","memory_eligible":false,"source_rank":834,"rank":834,"depth":1,"x":1596.265,"y":237.82,"cluster":"fields-brauer-groups"},{"id":"stacks:09FR","tag":"09FR","title":"The characteristic of a field · Definition 09FR","summary":"The characteristic of a field F is 0 if Z ⊂ F, or is a prime p if p = 0 in F. The prime subfield of F is the smallest subfield of F which is either Q ⊂ F if the characteristic is zero, or F_p ⊂ F if the characteristic is p > 0.","statement_latex":"The {\\it characteristic} of a field $F$ is $0$ if\n$\\mathbf{Z} \\subset F$, or is a prime $p$ if $p = 0$ in $F$.\nThe {\\it prime subfield of $F$} is the smallest subfield of $F$\nwhich is either $\\mathbf{Q} \\subset F$ if the characteristic is zero, or\n$\\mathbf{F}_p \\subset F$ if the characteristic is $p > 0$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"The characteristic of a field","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09FR","source_file":"fields.tex","source_line":224,"source_end_line":231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L224-L231","statement_sha256":"4af76d62089cd1957800658bb63bea58ca2dc90a47347610ed6ad6f7bb84d87a","origin":"The Stacks Project","memory_eligible":false,"source_rank":835,"rank":835,"depth":0,"x":1550.152,"y":215.565,"cluster":"fields-brauer-groups"},{"id":"stacks:09FU","tag":"09FU","title":"Field extensions · Lemma 09FU","summary":"If F is a field and R is a nonzero ring, then any ring homomorphism φ : F → R is injective.","statement_latex":"If $F$ is a field and $R$ is a nonzero ring, then any ring homomorphism\n$\\varphi : F \\to R$ is injective.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Field extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09FU","source_file":"fields.tex","source_line":253,"source_end_line":257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L253-L257","statement_sha256":"f65366561c284828aed737e49564ee2882b5d66c8a8799cbd3e1d6bbea000256","origin":"The Stacks Project","memory_eligible":false,"source_rank":836,"rank":836,"depth":0,"x":1608.274,"y":204.892,"cluster":"fields-brauer-groups"},{"id":"stacks:09FV","tag":"09FV","title":"Field extensions · Definition 09FV","summary":"If F is a field contained in a field E, then E is said to be a field extension of F. We shall write E/F to indicate that E is an extension of F.","statement_latex":"If $F$ is a field contained in a field $E$, then $E$ is said\nto be a {\\it field extension} of $F$. We shall write $E/F$ to indicate\nthat $E$ is an extension of $F$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Field extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09FV","source_file":"fields.tex","source_line":265,"source_end_line":270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L265-L270","statement_sha256":"7e91454526ddc9a329297a3194ebf79b2c2fff4104835be9b9372ec77eb9f824","origin":"The Stacks Project","memory_eligible":false,"source_rank":837,"rank":837,"depth":0,"x":1570.543,"y":249.551,"cluster":"fields-brauer-groups"},{"id":"stacks:09FW","tag":"09FW","title":"Field extensions · Definition 09FW","summary":"A tower of fields E_n/E_n - 1/…/E_0 consists of a sequence of extensions of fields E_n/E_n - 1, E_n - 1/E_n - 2, …, E_1/E_0.","statement_latex":"A {\\it tower} of fields $E_n/E_{n - 1}/\\ldots/E_0$ consists of a sequence of\nextensions of fields\n$E_n/E_{n - 1}$, $E_{n - 1}/E_{n - 2}$, $\\ldots$, $E_1/E_0$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Field extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09FW","source_file":"fields.tex","source_line":298,"source_end_line":303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L298-L303","statement_sha256":"6b3f7fd13b003199cca1e42616109a93a1af365d6cd2ac65a4dd24a7225a0fda","origin":"The Stacks Project","memory_eligible":false,"source_rank":838,"rank":838,"depth":0,"x":1561.964,"y":190.829,"cluster":"fields-brauer-groups"},{"id":"stacks:09FZ","tag":"09FZ","title":"Field extensions · Definition 09FZ","summary":"Let k be a field. If F/k is an extension of fields and S ⊂ F, we write k(S) for the smallest subfield of F containing k and S. We will say that S generates the field extension k(S)/k. If S = (α) is a singleton, then we write k(α) instead of k((α)). We say F/k is a finitely generated field extension if there exists a finite subset S ⊂ F with F = k(S).","statement_latex":"Let $k$ be a field. If $F/k$ is an extension of fields and\n$S \\subset F$, we write $k(S)$ for the smallest subfield of $F$\ncontaining $k$ and $S$. We will say that $S$ {\\it generates the\nfield extension} $k(S)/k$. If $S = \\{\\alpha\\}$ is a singleton, then we\nwrite $k(\\alpha)$ instead of $k(\\{\\alpha\\})$. We say $F/k$ is a\n{\\it finitely generated field extension} if there exists a\nfinite subset $S \\subset F$ with $F = k(S)$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Field extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09FZ","source_file":"fields.tex","source_line":333,"source_end_line":342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L333-L342","statement_sha256":"cf35c69324ef7ee0279fec8a6cc9035e4918a4d6a87035bce7de53bd2072d88c","origin":"The Stacks Project","memory_eligible":false,"source_rank":839,"rank":839,"depth":0,"x":1619.131,"y":232.004,"cluster":"fields-brauer-groups"},{"id":"stacks:09G1","tag":"09G1","title":"Classification of simple extensions · Lemma 09G1","summary":"If a field extension F/k is generated by one element, then it is k-isomorphic either to the rational function field k(t)/k or to one of the extensions k[t]/(P) for P ∈ k[t] irreducible.","statement_latex":"If a field extension $F/k$ is generated by one element, then it is\n$k$-isomorphic either to the rational function field $k(t)/k$ or to one\nof the extensions $k[t]/(P)$ for $P \\in k[t]$ irreducible.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Field extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09G1","source_file":"fields.tex","source_line":356,"source_end_line":361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L356-L361","statement_sha256":"8923fd9ba7782552e5a89fffe2b59f9939cf80274faf64379aa5adcb8f9bd250","origin":"The Stacks Project","memory_eligible":false,"source_rank":840,"rank":840,"depth":0,"x":1539.291,"y":234.115,"cluster":"fields-brauer-groups"},{"id":"stacks:0H7K","tag":"0H7K","title":"Field extensions · Lemma 0H7K","summary":"Let k be a field and let E/k and F/k be field extensions. Then there exists a common field extension M/k, i.e., an extension field such that there exist maps E → M and F → M of extensions of k.","statement_latex":"Let $k$ be a field and let $E/k$ and $F/k$ be field extensions.\nThen there exists a common field extension $M/k$, i.e., an extension\nfield such that there exist maps $E \\to M$ and $F \\to M$ of extensions of $k$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Field extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7K","source_file":"fields.tex","source_line":394,"source_end_line":399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L394-L399","statement_sha256":"4badf9eb885d11ae3a79d197968014081170e65bbde0ccf011ae255b9a1b869d","origin":"The Stacks Project","memory_eligible":false,"source_rank":841,"rank":841,"depth":1,"x":1599.624,"y":184.774,"cluster":"fields-brauer-groups"},{"id":"stacks:09G3","tag":"09G3","title":"Finite extensions · Definition 09G3","summary":"Let F/E be an extension of fields. The dimension of F considered as an E-vector space is called the degree of the extension and is denoted [F : E]. If [F : E] < ∞ then F is said to be a finite extension of E.","statement_latex":"Let $F/E$ be an extension of fields. The dimension of $F$ considered as an\n$E$-vector space is called the {\\it degree} of the extension and is\ndenoted $[F : E]$. If $[F : E] < \\infty$ then $F$ is said to be a\n{\\it finite} extension of $E$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Finite extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09G3","source_file":"fields.tex","source_line":435,"source_end_line":441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L435-L441","statement_sha256":"34d092a0c7dd0c9254704ad7a4ea08a7b2ba636915ef81142ec0ccc721789d26","origin":"The Stacks Project","memory_eligible":false,"source_rank":842,"rank":842,"depth":0,"x":1594.502,"y":258.837,"cluster":"fields-brauer-groups"},{"id":"stacks:09G5","tag":"09G5","title":"Finite extensions · Lemma 09G5","summary":"Let K/E/F be a tower of algebraic field extensions. If K is finite over F, then K is finite over E.","statement_latex":"Let $K/E/F$ be a tower of algebraic field extensions.\nIf $K$ is finite over $F$, then $K$ is finite over $E$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Finite extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09G5","source_file":"fields.tex","source_line":450,"source_end_line":454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L450-L454","statement_sha256":"e7f82dd68e7d25bdb66dc212fe23f3a86f8693d2ceeed08ef63cadae6c75af25","origin":"The Stacks Project","memory_eligible":false,"source_rank":843,"rank":843,"depth":0,"x":1536.291,"y":198.723,"cluster":"fields-brauer-groups"},{"id":"stacks:0BU1","tag":"0BU1","title":"Finite extensions · Lemma 0BU1","summary":"A finite extension of fields is a finitely generated field extension. The converse is not true.","statement_latex":"A finite extension of fields is a finitely generated field extension.\nThe converse is not true.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Finite extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BU1","source_file":"fields.tex","source_line":489,"source_end_line":493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L489-L493","statement_sha256":"31e9deb0238f64b60ece605bcc6cd658ecd1ac4bcc3e2fa6d5898ff63e582629","origin":"The Stacks Project","memory_eligible":false,"source_rank":844,"rank":844,"depth":0,"x":1631.275,"y":210.531,"cluster":"fields-brauer-groups"},{"id":"stacks:09G9","tag":"09G9","title":"Multiplicativity · Lemma 09G9","summary":"Suppose given a tower of fields F/E/k. Then [F:k] = [F:E][E:k]","statement_latex":"Suppose given a tower of fields $F/E/k$. Then\n$$\n[F:k] = [F:E][E:k]\n$$","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Finite extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09G9","source_file":"fields.tex","source_line":541,"source_end_line":547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L541-L547","statement_sha256":"e8cce3c00fc25c109c8f9dc1f7c64c1209ea280f7615130a88c93901e6b7e663","origin":"The Stacks Project","memory_eligible":false,"source_rank":845,"rank":845,"depth":0,"x":1548.707,"y":257.389,"cluster":"fields-brauer-groups"},{"id":"stacks:09GA","tag":"09GA","title":"Finite extensions · Definition 09GA","summary":"A field K is said to be a number field if it has characteristic 0 and the extension K/Q is finite.","statement_latex":"A field $K$ is said to be a {\\it number field} if it has characteristic\n$0$ and the extension $K/\\mathbf{Q}$ is finite.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Finite extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GA","source_file":"fields.tex","source_line":587,"source_end_line":591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L587-L591","statement_sha256":"32fc40f553996d6b46236113988b954cb31b1d8093f3ae0346973e19dddd5d74","origin":"The Stacks Project","memory_eligible":false,"source_rank":846,"rank":846,"depth":0,"x":1572.771,"y":173.138,"cluster":"fields-brauer-groups"},{"id":"stacks:09GC","tag":"09GC","title":"Algebraic extensions · Definition 09GC","summary":"Consider a field extension F/E. An element α ∈ F is said to be algebraic over E if α is the root of some nonzero polynomial with coefficients in E. If all elements of F are algebraic then F is said to be an algebraic extension of E.","statement_latex":"Consider a field extension $F/E$. An element $\\alpha \\in F$ is said to be\n{\\it algebraic} over $E$ if $\\alpha$ is the root of some nonzero polynomial\nwith coefficients in $E$. If all elements of $F$ are algebraic then $F$ is\nsaid to be an {\\it algebraic extension} of $E$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GC","source_file":"fields.tex","source_line":608,"source_end_line":614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L608-L614","statement_sha256":"34d6056351d02c14bf4ba9cc0a69d7a61ce92f8f3eaca9b87d538a8e96a1517c","origin":"The Stacks Project","memory_eligible":false,"source_rank":847,"rank":847,"depth":0,"x":1624.381,"y":251.42,"cluster":"fields-brauer-groups"},{"id":"stacks:09GF","tag":"09GF","title":"Algebraic extensions · Lemma 09GF","summary":"Let K/E/F be a tower of field extensions. • If α ∈ K is algebraic over F, then α is algebraic over E. • If K is algebraic over F, then K is algebraic over E.","statement_latex":"Let $K/E/F$ be a tower of field extensions.\n\\begin{enumerate}\n\\item If $\\alpha \\in K$ is algebraic over $F$, then $\\alpha$ is algebraic\nover $E$.\n\\item If $K$ is algebraic over $F$, then $K$ is algebraic over $E$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GF","source_file":"fields.tex","source_line":642,"source_end_line":650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L642-L650","statement_sha256":"fd2f946239b529bb2674b9799015eed68981a3d32f277633c04468d19edb6626","origin":"The Stacks Project","memory_eligible":false,"source_rank":848,"rank":848,"depth":0,"x":1520.277,"y":222.075,"cluster":"fields-brauer-groups"},{"id":"stacks:09GG","tag":"09GG","title":"Algebraic extensions · Lemma 09GG","summary":"A finite extension is algebraic. In fact, an extension E/k is algebraic if and only if every subextension k(α)/k generated by some α ∈ E is finite.","statement_latex":"A finite extension is algebraic. In fact, an extension $E/k$ is algebraic\nif and only if every subextension $k(\\alpha)/k$ generated by some\n$\\alpha \\in E$ is finite.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GG","source_file":"fields.tex","source_line":660,"source_end_line":665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L660-L665","statement_sha256":"3f7fd3746c57cf3094b657ea809ce4892d74c831c2f0b4bb2745579e1c7a223c","origin":"The Stacks Project","memory_eligible":false,"source_rank":849,"rank":849,"depth":0,"x":1623.563,"y":183.585,"cluster":"fields-brauer-groups"},{"id":"stacks:09GH","tag":"09GH","title":"Algebraic extensions · Lemma 09GH","summary":"A finitely generated algebraic extension is finite. Let k be a field, and let α_1, α_2, …, α_n be elements of some extension field such that each α_i is algebraic over k. Then the extension k(α_1, …, α_n)/k is finite. That is, a finitely generated algebraic extension is finite.","statement_latex":"\\begin{slogan}\nA finitely generated algebraic extension is finite.\n\\end{slogan}\nLet $k$ be a field, and let $\\alpha_1, \\alpha_2, \\ldots, \\alpha_n$ be elements\nof some extension field such that each $\\alpha_i$ is algebraic over $k$. Then\nthe extension $k(\\alpha_1, \\ldots, \\alpha_n)/k$ is finite.\nThat is, a finitely generated algebraic extension is finite.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GH","source_file":"fields.tex","source_line":692,"source_end_line":701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L692-L701","statement_sha256":"626fd04babd387bc309fb202994723539ce8171298e7247340bdbd8aebafa934","origin":"The Stacks Project","memory_eligible":false,"source_rank":850,"rank":850,"depth":1,"x":1577.085,"y":272.945,"cluster":"fields-brauer-groups"},{"id":"stacks:09GI","tag":"09GI","title":"Algebraic extensions · Lemma 09GI","summary":"Let E/k be a field extension. Then the elements of E algebraic over k form a subextension of E/k.","statement_latex":"Let $E/k$ be a field extension. Then the elements of $E$ algebraic over $k$\nform a subextension of $E/k$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GI","source_file":"fields.tex","source_line":720,"source_end_line":724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L720-L724","statement_sha256":"e7e6f743609fe7234ce74a1269057dc40b7eb436e550075047be96e654ef24c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":851,"rank":851,"depth":2,"x":1538.549,"y":178.276,"cluster":"fields-brauer-groups"},{"id":"stacks:09GJ","tag":"09GJ","title":"Algebraic extensions · Lemma 09GJ","summary":"Let E/k and F/E be algebraic extensions of fields. Then F/k is an algebraic extension of fields.","statement_latex":"Let $E/k$ and $F/E$ be algebraic extensions of fields. Then $F/k$ is an\nalgebraic extension of fields.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GJ","source_file":"fields.tex","source_line":739,"source_end_line":743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L739-L743","statement_sha256":"4ba144e0e62e4e2734fa890b974e0aec3c29f3eb35243214cd46ff70c4619de0","origin":"The Stacks Project","memory_eligible":false,"source_rank":852,"rank":852,"depth":2,"x":1645.662,"y":227.421,"cluster":"fields-brauer-groups"},{"id":"stacks:09GK","tag":"09GK","title":"Algebraic extensions · Lemma 09GK","summary":"Let E/F be an algebraic extension of fields. Then the cardinality |E| of E is at most max(aleph_0, |F|).","statement_latex":"Let $E/F$ be an algebraic extension of fields. Then the cardinality $|E|$\nof $E$ is at most $\\max(\\aleph_0, |F|)$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GK","source_file":"fields.tex","source_line":768,"source_end_line":772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L768-L772","statement_sha256":"2eaf94f68b1c4db8239fe4575fd5d3c8f3c579c5a6a1b4642080b404c80b47df","origin":"The Stacks Project","memory_eligible":false,"source_rank":853,"rank":853,"depth":0,"x":1524.365,"y":252.516,"cluster":"fields-brauer-groups"},{"id":"stacks:0BID","tag":"0BID","title":"Algebraic extensions · Lemma 0BID","summary":"Let E/F be a finite or more generally an algebraic extension of fields. Any subring F ⊂ R ⊂ E is a field.","statement_latex":"Let $E/F$ be a finite or more generally an algebraic extension of fields.\nAny subring $F \\subset R \\subset E$ is a field.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BID","source_file":"fields.tex","source_line":785,"source_end_line":789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L785-L789","statement_sha256":"8ffbe3d01c69baf8f78dbde5240ab6d3bd67db0b4c287e82c991a0e45a7f9fe1","origin":"The Stacks Project","memory_eligible":false,"source_rank":854,"rank":854,"depth":1,"x":1595.203,"y":163.234,"cluster":"fields-brauer-groups"},{"id":"stacks:0BMD","tag":"0BMD","title":"Algebraic extensions · Lemma 0BMD","summary":"Let E/F an algebraic extension of fields. Any F-algebra map f : E → E is an automorphism.","statement_latex":"Let $E/F$ an algebraic extension of fields. Any $F$-algebra map\n$f : E \\to E$ is an automorphism.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMD","source_file":"fields.tex","source_line":806,"source_end_line":810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L806-L810","statement_sha256":"be9a7b4fa72fce46968557b19e6d5ecad9e0e7bdede039ef654ffe900118e1dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":855,"rank":855,"depth":2,"x":1615.163,"y":271.546,"cluster":"fields-brauer-groups"},{"id":"stacks:09GM","tag":"09GM","title":"Minimal polynomials · Definition 09GM","summary":"The polynomial P above is called the minimal polynomial of α over k.","statement_latex":"The polynomial $P$ above is called the {\\it minimal polynomial}\nof $\\alpha$ over $k$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Minimal polynomials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GM","source_file":"fields.tex","source_line":850,"source_end_line":854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L850-L854","statement_sha256":"b1f9c4cadb3b1bcf52d5bbb764abd6d638938b2dfcb5d69871e459be47655a8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":856,"rank":856,"depth":0,"x":1511.259,"y":201.579,"cluster":"fields-brauer-groups"},{"id":"stacks:09GN","tag":"09GN","title":"Minimal polynomials · Lemma 09GN","summary":"The degree of the minimal polynomial is [k(α) : k].","statement_latex":"The degree of the minimal polynomial is $[k(\\alpha) : k]$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Minimal polynomials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GN","source_file":"fields.tex","source_line":869,"source_end_line":872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L869-L872","statement_sha256":"09f338aa86e3845082cc7317925b9f03440a85e2e140d5153441bd2f4fdf6479","origin":"The Stacks Project","memory_eligible":false,"source_rank":857,"rank":857,"depth":1,"x":1646.773,"y":194.085,"cluster":"fields-brauer-groups"},{"id":"stacks:09GQ","tag":"09GQ","title":"Algebraic closure · Definition 09GQ","summary":"A field F is said to be algebraically closed if every algebraic extension E/F is trivial, i.e., E = F.","statement_latex":"A field $F$ is said to be {\\it algebraically closed} if every algebraic\nextension $E/F$ is trivial, i.e., $E = F$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic closure","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GQ","source_file":"fields.tex","source_line":900,"source_end_line":904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L900-L904","statement_sha256":"27f035261b5b1e702af4fd7b7014343f675d795daa2a3f0fe832ba96ded3e5fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":858,"rank":858,"depth":0,"x":1551.072,"y":278.062,"cluster":"fields-brauer-groups"},{"id":"stacks:09GR","tag":"09GR","title":"Algebraic closure · Lemma 09GR","summary":"Let F be a field. The following are equivalent • F is algebraically closed, • every irreducible polynomial over F is linear, • every nonconstant polynomial over F has a root, • every nonconstant polynomial over F is a product of linear factors.","statement_latex":"Let $F$ be a field. The following are equivalent\n\\begin{enumerate}\n\\item $F$ is algebraically closed,\n\\item every irreducible polynomial over $F$ is linear,\n\\item every nonconstant polynomial over $F$ has a root,\n\\item every nonconstant polynomial over $F$ is a product of linear factors.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GR","source_file":"fields.tex","source_line":910,"source_end_line":919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L910-L919","statement_sha256":"5331e685f9d236c54e5b817fb976097e0aba98c25997f9ebb228dfa6371f7305","origin":"The Stacks Project","memory_eligible":false,"source_rank":859,"rank":859,"depth":0,"x":1554.183,"y":159.706,"cluster":"fields-brauer-groups"},{"id":"stacks:09GS","tag":"09GS","title":"Algebraic closure · Definition 09GS","summary":"Let F be a field. An algebraic closure of F is a field overlineF containing F such that: • overlineF is algebraic over F. • overlineF is algebraically closed.","statement_latex":"Let $F$ be a field. An {\\it algebraic closure} of $F$ is a field\n$\\overline{F}$ containing $F$ such that:\n\\begin{enumerate}\n\\item $\\overline{F}$ is algebraic over $F$.\n\\item $\\overline{F}$ is algebraically closed.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic closure","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GS","source_file":"fields.tex","source_line":952,"source_end_line":960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L952-L960","statement_sha256":"b43019d62f3eea1a6f3ea7a77e5aecbcdfb537aa452e8281c9cb877bda6b68ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":860,"rank":860,"depth":0,"x":1648.697,"y":250.329,"cluster":"fields-brauer-groups"},{"id":"stacks:09GT","tag":"09GT","title":"Algebraic closure · Theorem 09GT","summary":"Every field has an algebraic closure.","statement_latex":"Every field has an algebraic closure.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic closure","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GT","source_file":"fields.tex","source_line":966,"source_end_line":969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L966-L969","statement_sha256":"ea2d6b34b850a6c3609eb51331a81c6b213c2b93086489a5bd800920be271e09","origin":"The Stacks Project","memory_eligible":false,"source_rank":861,"rank":861,"depth":0,"x":1503.695,"y":236.896,"cluster":"fields-brauer-groups"},{"id":"stacks:09GU","tag":"09GU","title":"Algebraic closure · Lemma 09GU","summary":"Let F be a field. Let overlineF be an algebraic closure of F. Let M/F be an algebraic extension. Then there is a morphism of F-extensions M → overlineF.","statement_latex":"Let $F$ be a field. Let $\\overline{F}$ be an algebraic closure of $F$.\nLet $M/F$ be an algebraic extension. Then there is a morphism of\n$F$-extensions $M \\to \\overline{F}$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GU","source_file":"fields.tex","source_line":1022,"source_end_line":1027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1022-L1027","statement_sha256":"ddc22a407675cfcb0ac84b49275e42ede798f0122c642630e6ab9fdf3e51d072","origin":"The Stacks Project","memory_eligible":false,"source_rank":862,"rank":862,"depth":1,"x":1623.373,"y":163.338,"cluster":"fields-brauer-groups"},{"id":"stacks:09GV","tag":"09GV","title":"Algebraic closure · Lemma 09GV","summary":"Any two algebraic closures of a field are isomorphic.","statement_latex":"Any two algebraic closures of a field are isomorphic.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Algebraic closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GV","source_file":"fields.tex","source_line":1051,"source_end_line":1054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1051-L1054","statement_sha256":"b5fec50ecd7d27724b23c572928c13f2b324b16120e785c6bbc8942bea1bb623","origin":"The Stacks Project","memory_eligible":false,"source_rank":863,"rank":863,"depth":2,"x":1593.797,"y":287.436,"cluster":"fields-brauer-groups"},{"id":"stacks:09GX","tag":"09GX","title":"Relatively prime polynomials · Definition 09GX","summary":"If k is any field, we say that two polynomials in k[x] are relatively prime if they generate the unit ideal in k[x].","statement_latex":"If $k$ is any field, we say that two polynomials in $k[x]$ are\n{\\it relatively prime} if they generate the unit ideal in $k[x]$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Relatively prime polynomials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GX","source_file":"fields.tex","source_line":1086,"source_end_line":1090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1086-L1090","statement_sha256":"bb7b544e037827bfb59ed95456c875015cd47f7a079ad4e6ac4381075966f288","origin":"The Stacks Project","memory_eligible":false,"source_rank":864,"rank":864,"depth":0,"x":1514.614,"y":177.464,"cluster":"fields-brauer-groups"},{"id":"stacks:09GY","tag":"09GY","title":"Relatively prime polynomials · Lemma 09GY","summary":"Two polynomials in k[x] are relatively prime precisely when they have no common roots in an algebraic closure overlinek of k.","statement_latex":"Two polynomials in $k[x]$ are relatively prime precisely when they\nhave no common roots in an algebraic closure $\\overline{k}$ of $k$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Relatively prime polynomials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09GY","source_file":"fields.tex","source_line":1104,"source_end_line":1108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1104-L1108","statement_sha256":"76e6aa2a826e7f5bbcf0a6ee2a49a7306484732cb0d10891a4bb59d5bb7768ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":865,"rank":865,"depth":0,"x":1663.64,"y":214.179,"cluster":"fields-brauer-groups"},{"id":"stacks:09H0","tag":"09H0","title":"Separable algebraic extensions · Lemma 09H0","summary":"Let F be a field. Let P ∈ F[x] be an irreducible polynomial over F. Let P' = dP/dx be the derivative of P with respect to x. Then one of the following two cases happens • P and P' are relatively prime, or • P' is the zero polynomial. The second case can only happen if F has characteristic p > 0. In this case P(x) = Q(x^q) where q = p^f is a power of p and Q ∈ F[x] is an irreducible polynomial such that Q and Q' are relatively prime.","statement_latex":"Let $F$ be a field. Let $P \\in F[x]$ be an irreducible polynomial over $F$.\nLet $P' = \\text{d}P/\\text{d}x$ be the derivative of $P$ with respect\nto $x$. Then one of the following two cases happens\n\\begin{enumerate}\n\\item $P$ and $P'$ are relatively prime, or\n\\item $P'$ is the zero polynomial.\n\\end{enumerate}\nThe second case can only happen if $F$ has characteristic $p > 0$.\nIn this case $P(x) = Q(x^q)$ where $q = p^f$ is a power of $p$ and\n$Q \\in F[x]$ is an irreducible polynomial such that $Q$ and $Q'$\nare relatively prime.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09H0","source_file":"fields.tex","source_line":1130,"source_end_line":1143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1130-L1143","statement_sha256":"6c103c0fdb2da65547db6567b8edce44f0dd2c02c6e052fab837a3efab77d2c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":866,"rank":866,"depth":0,"x":1522.187,"y":272.495,"cluster":"fields-brauer-groups"},{"id":"stacks:09H1","tag":"09H1","title":"Separable algebraic extensions · Definition 09H1","summary":"Let F be a field. Let K/F be an extension of fields. • We say an irreducible polynomial P over F is separable if it is relatively prime to its derivative. • Given α ∈ K algebraic over F we say α is separable over F if its minimal polynomial is separable over F. • If K is an algebraic extension of F, we say K is separable over F if every element of K is separable over F.","statement_latex":"Let $F$ be a field. Let $K/F$ be an extension of fields.\n\\begin{enumerate}\n\\item We say an irreducible polynomial $P$ over $F$ is {\\it separable}\nif it is relatively prime to its derivative.\n\\item Given $\\alpha \\in K$ algebraic over $F$ we say $\\alpha$ is\n{\\it separable} over $F$ if its minimal polynomial is separable over $F$.\n\\item If $K$ is an algebraic extension of $F$, we say $K$ is\n{\\it separable}\\footnote{For nonalgebraic extensions\nthis definition does not make sense and is not the correct one. We refer\nthe reader to Algebra, Sections \\ref{algebra-section-separability} and\n\\ref{algebra-section-separability-continued}.}\nover $F$ if every element of $K$ is separable over $F$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09H1","source_file":"fields.tex","source_line":1164,"source_end_line":1179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1164-L1179","statement_sha256":"ee172808c23b8ed932954b7ebab6256613417b03fdb0b22ff6adef0ff91439b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":867,"rank":867,"depth":0,"x":1580.421,"y":147.487,"cluster":"fields-brauer-groups"},{"id":"stacks:09H2","tag":"09H2","title":"Separable algebraic extensions · Lemma 09H2","summary":"Let K/E/F be a tower of algebraic field extensions. • If α ∈ K is separable over F, then α is separable over E. • if K is separable over F, then K is separable over E.","statement_latex":"Let $K/E/F$ be a tower of algebraic field extensions.\n\\begin{enumerate}\n\\item If $\\alpha \\in K$ is separable over $F$, then $\\alpha$ is separable\nover $E$.\n\\item if $K$ is separable over $F$, then $K$ is separable over $E$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09H2","source_file":"fields.tex","source_line":1186,"source_end_line":1194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1186-L1194","statement_sha256":"692d662ac498fd73e15014fa2f8b0fb68264ec4f7ea4ce9eb0d09985a913bcf5","origin":"The Stacks Project","memory_eligible":false,"source_rank":868,"rank":868,"depth":1,"x":1638.79,"y":274.438,"cluster":"fields-brauer-groups"},{"id":"stacks:09H3","tag":"09H3","title":"Separable algebraic extensions · Lemma 09H3","summary":"Let F be a field. An irreducible polynomial P over F is separable if and only if P has pairwise distinct roots in an algebraic closure of F.","statement_latex":"Let $F$ be a field. An irreducible polynomial $P$ over $F$\nis separable if and only if $P$ has pairwise distinct roots in an\nalgebraic closure of $F$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09H3","source_file":"fields.tex","source_line":1206,"source_end_line":1211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1206-L1211","statement_sha256":"63ae0f5d0e6c5317c98e4d83337935948557e25deee9bd23119f3e06bc376f5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":869,"rank":869,"depth":1,"x":1491.719,"y":213.127,"cluster":"fields-brauer-groups"},{"id":"stacks:09H4","tag":"09H4","title":"Separable algebraic extensions · Lemma 09H4","summary":"Let F be a field and let overlineF be an algebraic closure of F. Let p > 0 be the characteristic of F. Let P be a polynomial over F. Then the set of roots of P and P(x^p) in overlineF have the same cardinality (not counting multiplicity).","statement_latex":"Let $F$ be a field and let $\\overline{F}$ be an algebraic closure of $F$.\nLet $p > 0$ be the characteristic of $F$. Let $P$ be a polynomial\nover $F$. Then the set of roots of $P$ and $P(x^p)$ in $\\overline{F}$\nhave the same cardinality (not counting multiplicity).","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09H4","source_file":"fields.tex","source_line":1224,"source_end_line":1230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1224-L1230","statement_sha256":"07dec6bd844252ab51a6290cf071f64682643d8dfdb717767e0dc94a9d227b85","origin":"The Stacks Project","memory_eligible":false,"source_rank":870,"rank":870,"depth":0,"x":1651.533,"y":174.395,"cluster":"fields-brauer-groups"},{"id":"stacks:09H5","tag":"09H5","title":"Separable algebraic extensions · Definition 09H5","summary":"Let F be a field. Let P be an irreducible polynomial over F. The separable degree of P is the cardinality of the set of roots of P in any algebraic closure of F (see discussion above). Notation deg_s(P).","statement_latex":"Let $F$ be a field. Let $P$ be an irreducible polynomial over $F$.\nThe {\\it separable degree} of $P$ is the cardinality of the\nset of roots of $P$ in any algebraic closure of $F$ (see discussion\nabove). Notation $\\deg_s(P)$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09H5","source_file":"fields.tex","source_line":1255,"source_end_line":1261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1255-L1261","statement_sha256":"79d3a3d709434a57858314dc57d7f2691ddc39590af15d4c415eb8c24a30caf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":871,"rank":871,"depth":0,"x":1563.725,"y":295.15,"cluster":"fields-brauer-groups"},{"id":"stacks:09H7","tag":"09H7","title":"Separable algebraic extensions · Lemma 09H7","summary":"In Situation [Tag 09H6] the correspondence Mor_F(K, overlineF) → ((β_1, …, β_n) as below), φ ↦ (φ(α_1), …, φ(α_n)) is a bijection. Here the right hand side is the set of n-tuples (β_1, …, β_n) of elements of overlineF having the following property: • β_1 ∈ overlineF is a root of P_1; let φ_1 : K_1 → overlineF be the homomorphism over F sending α_1 to β_1, • β_2 ∈ overlineF is a root of P_2^φ_1; let φ_2 : K_2 → overlineF be the homomorphism extending φ_1 sending α_2 to…","statement_latex":"In Situation \\ref{situation-finitely-generated} the correspondence\n$$\n\\Mor_F(K, \\overline{F})\n\\longrightarrow\n\\{(\\beta_1, \\ldots, \\beta_n)\\text{ as below}\\},\n\\quad\n\\varphi \\longmapsto (\\varphi(\\alpha_1), \\ldots, \\varphi(\\alpha_n))\n$$\nis a bijection. Here the right hand side is the set of $n$-tuples\n$(\\beta_1, \\ldots, \\beta_n)$ of elements of $\\overline{F}$ having\nthe following property:\n\\begin{enumerate}\n\\item $\\beta_1 \\in \\overline{F}$ is a root of $P_1$;\nlet $\\varphi_1 : K_1 \\to \\overline{F}$ be the homomorphism\nover $F$ sending $\\alpha_1$ to $\\beta_1$,\n\\item $\\beta_2 \\in \\overline{F}$ is a root of $P_2^{\\varphi_1}$;\nlet $\\varphi_2 : K_2 \\to \\overline{F}$ be the homomorphism\nextending $\\varphi_1$ sending $\\alpha_2$ to $\\beta_2$,\n\\item and so on until,\n\\item $\\beta_n \\in \\overline{F}$ is a root of $P_n^{\\varphi_{n - 1}}$.\n\\end{enumerate}\nIn each step the homorphism $\\varphi_i$ exists and is unique because\n$K_i = K_{i - 1}[x]/(P_i)$ and $\\beta_i$ is a root of $P_i^{\\varphi_{i - 1}}$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09H7","source_file":"fields.tex","source_line":1290,"source_end_line":1315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1290-L1315","statement_sha256":"105b807328b62a166a0d6e9c553d0d2f2bb8aeda47974627b1fdb41c37ff9248","origin":"The Stacks Project","memory_eligible":false,"source_rank":872,"rank":872,"depth":0,"x":1530.974,"y":154.559,"cluster":"fields-brauer-groups"},{"id":"stacks:09H8","tag":"09H8","title":"Separable algebraic extensions · Lemma 09H8","summary":"In Situation [Tag 09H6] we have |Mor_F(K, overlineF)| = ∏_i = 1^n deg_s(P_i).","statement_latex":"In Situation \\ref{situation-finitely-generated} we have\n$|\\Mor_F(K, \\overline{F})| = \\prod_{i = 1}^n \\deg_s(P_i)$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09H8","source_file":"fields.tex","source_line":1325,"source_end_line":1329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1325-L1329","statement_sha256":"e0808a05f5c7ae6163a0ee16eccb2823b83b03219b3662de5eecee45fb636568","origin":"The Stacks Project","memory_eligible":false,"source_rank":873,"rank":873,"depth":1,"x":1669.838,"y":240.682,"cluster":"fields-brauer-groups"},{"id":"stacks:09H9","tag":"09H9","title":"Separable algebraic extensions · Lemma 09H9","summary":"Assumptions and notation as in Situation [Tag 09H6]. If each P_i is separable, i.e., each α_i is separable over K_i - 1, then |Mor_F(K, overlineF)| = [K : F] and the field extension K/F is separable. If one of the α_i is not separable over K_i - 1, then |Mor_F(K, overlineF)| < [K : F].","statement_latex":"Assumptions and notation as in Situation \\ref{situation-finitely-generated}.\nIf each $P_i$ is separable, i.e., each $\\alpha_i$ is separable over\n$K_{i - 1}$, then\n$$\n|\\Mor_F(K, \\overline{F})| = [K : F]\n$$\nand the field extension $K/F$ is separable. If one of the $\\alpha_i$ is\nnot separable over $K_{i - 1}$, then\n$|\\Mor_F(K, \\overline{F})| < [K : F]$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09H9","source_file":"fields.tex","source_line":1342,"source_end_line":1353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1342-L1353","statement_sha256":"5bb7bfe038ccd330b1370a6962cca88b3d238743db7691039fce1792e2414bd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":874,"rank":874,"depth":2,"x":1496.156,"y":256.143,"cluster":"fields-brauer-groups"},{"id":"stacks:09HA","tag":"09HA","title":"Separable algebraic extensions · Lemma 09HA","summary":"Let K/F be a finite extension of fields. Let overlineF be an algebraic closure of F. Then we have |Mor_F(K, overlineF)| ≤ [K : F] with equality if and only if K is separable over F.","statement_latex":"Let $K/F$ be a finite extension of fields. Let $\\overline{F}$ be an\nalgebraic closure of $F$. Then we have\n$$\n|\\Mor_F(K, \\overline{F})| \\leq [K : F]\n$$\nwith equality if and only if $K$ is separable over $F$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HA","source_file":"fields.tex","source_line":1383,"source_end_line":1391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1383-L1391","statement_sha256":"c883706c5e1917c534f132ca6bbe90373d482768299be71d456751d841252d1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":875,"rank":875,"depth":3,"x":1613.134,"y":144.927,"cluster":"fields-brauer-groups"},{"id":"stacks:09HB","tag":"09HB","title":"Separable algebraic extensions · Lemma 09HB","summary":"Let E/k and F/E be separable algebraic extensions of fields. Then F/k is a separable extension of fields.","statement_latex":"Let $E/k$ and $F/E$ be separable algebraic extensions of fields. Then $F/k$\nis a separable extension of fields.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HB","source_file":"fields.tex","source_line":1408,"source_end_line":1412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1408-L1412","statement_sha256":"b6b7e77d49e15b524906ceeb8c240bad9a0d968abfa268a87cef36ab97b617af","origin":"The Stacks Project","memory_eligible":false,"source_rank":876,"rank":876,"depth":3,"x":1616.34,"y":294.986,"cluster":"fields-brauer-groups"},{"id":"stacks:09HC","tag":"09HC","title":"Separable algebraic extensions · Lemma 09HC","summary":"Let E/k be a field extension. Then the elements of E separable over k form a subextension of E/k.","statement_latex":"Let $E/k$ be a field extension. Then the elements of $E$ separable\nover $k$ form a subextension of $E/k$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Separable algebraic extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HC","source_file":"fields.tex","source_line":1434,"source_end_line":1438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1434-L1438","statement_sha256":"9d74d16436cc26919541054f0dd04b9e8349255a5ea54875929f47c72ddf8949","origin":"The Stacks Project","memory_eligible":false,"source_rank":877,"rank":877,"depth":3,"x":1491.951,"y":184.949,"cluster":"fields-brauer-groups"},{"id":"stacks:0CKL","tag":"0CKL","title":"Linear independence of characters · Lemma 0CKL","summary":"Let L be a field. Let G be a monoid, for example a group. Let chi_1, …, chi_n : G → L be pairwise distinct homomorphisms of monoids where L is regarded as a monoid by multiplication. Then chi_1, …, chi_n are L-linearly independent: if λ_1, …, λ_n ∈ L not all zero, then ∑ λ_ichi_i(g) not = 0 for some g ∈ G.","statement_latex":"Let $L$ be a field. Let $G$ be a monoid, for example a group. Let\n$\\chi_1, \\ldots, \\chi_n : G \\to L$ be pairwise distinct\nhomomorphisms of monoids where $L$ is regarded as a monoid\nby multiplication. Then $\\chi_1, \\ldots, \\chi_n$\nare $L$-linearly independent: if $\\lambda_1, \\ldots, \\lambda_n \\in L$\nnot all zero, then $\\sum \\lambda_i\\chi_i(g) \\not = 0$\nfor some $g \\in G$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Linear independence of characters","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKL","source_file":"fields.tex","source_line":1458,"source_end_line":1467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1458-L1467","statement_sha256":"e75ec868fa4f5c9742faa78cc93af6459e307c9bb329b7d77d2cd3ea26898684","origin":"The Stacks Project","memory_eligible":false,"source_rank":878,"rank":878,"depth":0,"x":1674.107,"y":195.628,"cluster":"fields-brauer-groups"},{"id":"stacks:0EM9","tag":"0EM9","title":"Linear independence of characters · Lemma 0EM9","summary":"Let L be a field. Let n ≥ 1 and α_1, …, α_n ∈ L pairwise distinct elements of L. Then there exists an e ≥ 0 such that ∑_i = 1, …, n α_i^e not = 0.","statement_latex":"Let $L$ be a field. Let $n \\geq 1$ and $\\alpha_1, \\ldots, \\alpha_n \\in L$\npairwise distinct elements of $L$. Then there exists an\n$e \\geq 0$ such that $\\sum_{i = 1, \\ldots, n} \\alpha_i^e \\not = 0$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Linear independence of characters","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EM9","source_file":"fields.tex","source_line":1499,"source_end_line":1504,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1499-L1504","statement_sha256":"bde2fa526e0c27dabe86a90922ee5eca60538b1a3d0e61069fd889abbfe5f692","origin":"The Stacks Project","memory_eligible":false,"source_rank":879,"rank":879,"depth":1,"x":1529.686,"y":292.116,"cluster":"fields-brauer-groups"},{"id":"stacks:0CKM","tag":"0CKM","title":"Linear independence of characters · Lemma 0CKM","summary":"Let K/F and L/F be field extensions. Let σ_1, …, σ_n : K → L be pairwise distinct morphisms of F-extensions. Then σ_1, …, σ_n are L-linearly independent: if λ_1, …, λ_n ∈ L not all zero, then ∑ λ_iσ_i(α) not = 0 for some α ∈ K.","statement_latex":"Let $K/F$ and $L/F$ be field extensions. Let\n$\\sigma_1, \\ldots, \\sigma_n : K \\to L$ be pairwise distinct\nmorphisms of $F$-extensions. Then $\\sigma_1, \\ldots, \\sigma_n$\nare $L$-linearly independent: if $\\lambda_1, \\ldots, \\lambda_n \\in L$\nnot all zero, then $\\sum \\lambda_i\\sigma_i(\\alpha) \\not = 0$\nfor some $\\alpha \\in K$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Linear independence of characters","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKM","source_file":"fields.tex","source_line":1513,"source_end_line":1521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1513-L1521","statement_sha256":"73b6396d04ffc339af64fc278cbff0b488c08b7362c80506560acdd9d802d0cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":880,"rank":880,"depth":1,"x":1558.894,"y":137.437,"cluster":"fields-brauer-groups"},{"id":"stacks:0CKN","tag":"0CKN","title":"Linear independence of characters · Lemma 0CKN","summary":"Let K/F and L/F be field extensions with K/F finite separable and L algebraically closed. Then the map K ⊗_F L → ∏_σ ∈ Hom_F(K, L) L, α ⊗ β ↦ (σ(α)β)_σ is an isomorphism of L-algebras.","statement_latex":"Let $K/F$ and $L/F$ be field extensions with\n$K/F$ finite separable and $L$ algebraically closed.\nThen the map\n$$\nK \\otimes_F L\n\\longrightarrow\n\\prod\\nolimits_{\\sigma \\in \\Hom_F(K, L)} L,\\quad\n\\alpha \\otimes \\beta \\mapsto (\\sigma(\\alpha)\\beta)_\\sigma\n$$\nis an isomorphism of $L$-algebras.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Linear independence of characters","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKN","source_file":"fields.tex","source_line":1528,"source_end_line":1540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1528-L1540","statement_sha256":"1e89319021dd315baa057eb23dbee6dfcbb0e6805c89d4d4a13b7fa65a0dc2b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":881,"rank":881,"depth":4,"x":1662.78,"y":269.395,"cluster":"fields-brauer-groups"},{"id":"stacks:09HE","tag":"09HE","title":"Purely inseparable extensions · Definition 09HE","summary":"Let F be a field of characteristic p > 0. Let K/F be an extension. • An element α ∈ K is purely inseparable over F if there exists a power q of p such that α^q ∈ F. • The extension K/F is said to be purely inseparable if and only if every element of K is purely inseparable over F. If we have a field extension L/M (with no condition on the characteristic of M), then we will say the extension is purely inseparable if either L = M or the characteristic of M is a prime number…","statement_latex":"Let $F$ be a field of characteristic $p > 0$. Let $K/F$ be an extension.\n\\begin{enumerate}\n\\item An element $\\alpha \\in K$ is {\\it purely inseparable} over $F$\nif there exists a power $q$ of $p$ such that $\\alpha^q \\in F$.\n\\item The extension $K/F$ is said to be {\\it purely inseparable}\nif and only if every element of $K$ is purely inseparable over $F$.\n\\end{enumerate}\nIf we have a field extension $L/M$ (with no condition on the characteristic\nof $M$), then we will say the extension is {\\it purely inseparable}\nif either $L = M$ or the characteristic of $M$ is a prime number $p$\nand $L/M$ is purely inseparable in the sense defined above.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Purely inseparable extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HE","source_file":"fields.tex","source_line":1580,"source_end_line":1593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1580-L1593","statement_sha256":"e6efe9af38548dc59ce162c27e415e2ef43ca106865806c927c68f4f0add1a11","origin":"The Stacks Project","memory_eligible":false,"source_rank":882,"rank":882,"depth":0,"x":1478.247,"y":230.652,"cluster":"fields-brauer-groups"},{"id":"stacks:09HF","tag":"09HF","title":"Purely inseparable extensions · Lemma 09HF","summary":"Let p be a prime number. Let F be a field of characteristic p. Let t ∈ F be an element which does not have a pth root in F. Then the polynomial x^p - t is irreducible over F.","statement_latex":"Let $p$ be a prime number. Let $F$ be a field of characteristic $p$.\nLet $t \\in F$ be an element which does not have a $p$th root in $F$.\nThen the polynomial $x^p - t$ is irreducible over $F$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Purely inseparable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HF","source_file":"fields.tex","source_line":1616,"source_end_line":1621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1616-L1621","statement_sha256":"082527ad44fa00dfe886227b8a5b226ea7126d9353fae7f09e809dccf654e2f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":883,"rank":883,"depth":0,"x":1647.106,"y":153.775,"cluster":"fields-brauer-groups"},{"id":"stacks:09HG","tag":"09HG","title":"Purely inseparable extensions · Lemma 09HG","summary":"Let E/k and F/E be purely inseparable extensions of fields. Then F/k is a purely inseparable extension of fields.","statement_latex":"Let $E/k$ and $F/E$ be purely inseparable extensions of fields. Then $F/k$\nis a purely inseparable extension of fields.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Purely inseparable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HG","source_file":"fields.tex","source_line":1639,"source_end_line":1643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1639-L1643","statement_sha256":"b4412c6c8a8235a76d08dffe1e3c61f2782eac53290deb49ac54cef770f20123","origin":"The Stacks Project","memory_eligible":false,"source_rank":884,"rank":884,"depth":0,"x":1583.809,"y":307.732,"cluster":"fields-brauer-groups"},{"id":"stacks:09HH","tag":"09HH","title":"Purely inseparable extensions · Lemma 09HH","summary":"Let E/k be a field extension. Then the elements of E purely-inseparable over k form a subextension of E/k.","statement_latex":"Let $E/k$ be a field extension. Then the elements of $E$ purely-inseparable\nover $k$ form a subextension of $E/k$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Purely inseparable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HH","source_file":"fields.tex","source_line":1651,"source_end_line":1655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1651-L1655","statement_sha256":"a775b1863bd8b330c4ba3ad0feae5bf2b35e7973350d4dfb9e876cc11dc8e9f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":885,"rank":885,"depth":0,"x":1505.959,"y":156.888,"cluster":"fields-brauer-groups"},{"id":"stacks:09HI","tag":"09HI","title":"Purely inseparable extensions · Lemma 09HI","summary":"Let E/F be a finite purely inseparable field extension of characteristic p > 0. Then there exists a sequence of elements α_1, …, α_n ∈ E such that we obtain a tower of fields E = F(α_1, …, α_n) ⊃ F(α_1, …, α_n - 1) ⊃ … ⊃ F(α_1) ⊃ F such that each intermediate extension is of degree p and comes from adjoining a pth root. Namely, α_i^p ∈ F(α_1, …, α_i - 1) is an element which does not have a pth root in F(α_1, …, α_i - 1) for i = 1, …, n.","statement_latex":"Let $E/F$ be a finite purely inseparable field extension of\ncharacteristic $p > 0$. Then there exists a sequence of elements\n$\\alpha_1, \\ldots, \\alpha_n \\in E$ such that we obtain a tower\nof fields\n$$\nE = F(\\alpha_1, \\ldots, \\alpha_n) \\supset\nF(\\alpha_1, \\ldots, \\alpha_{n - 1}) \\supset\n\\ldots\n\\supset F(\\alpha_1) \\supset F\n$$\nsuch that each intermediate extension is of degree $p$ and comes\nfrom adjoining a $p$th root. Namely,\n$\\alpha_i^p \\in F(\\alpha_1, \\ldots, \\alpha_{i - 1})$\nis an element which does not have a $p$th root in\n$F(\\alpha_1, \\ldots, \\alpha_{i - 1})$ for $i = 1, \\ldots, n$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Purely inseparable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HI","source_file":"fields.tex","source_line":1668,"source_end_line":1685,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1668-L1685","statement_sha256":"57f90bf2d1e4fdf1655253f6addad9f96cb70918b8a35767bd3fe7ead390d692","origin":"The Stacks Project","memory_eligible":false,"source_rank":886,"rank":886,"depth":1,"x":1686.309,"y":224.566,"cluster":"fields-brauer-groups"},{"id":"stacks:030K","tag":"030K","title":"Purely inseparable extensions · Lemma 030K","summary":"Any algebraic field extension is uniquely a separable field extension followed by a purely inseparable one. Let E/F be an algebraic field extension. There exists a unique subextension E/E_sep/F such that E_sep/F is separable and E/E_sep is purely inseparable.","statement_latex":"\\begin{slogan}\nAny algebraic field extension is uniquely a separable field extension\nfollowed by a purely inseparable one.\n\\end{slogan}\nLet $E/F$ be an algebraic field extension. There exists a unique subextension\n$E/E_{sep}/F$ such that $E_{sep}/F$ is separable and $E/E_{sep}$ is\npurely inseparable.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Purely inseparable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030K","source_file":"fields.tex","source_line":1703,"source_end_line":1712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1703-L1712","statement_sha256":"39249962a6f89c4df2dec6e1903086875d68862cb4101d13bc549ca8e52dfb2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":887,"rank":887,"depth":4,"x":1497.203,"y":277.465,"cluster":"fields-brauer-groups"},{"id":"stacks:030L","tag":"030L","title":"Purely inseparable extensions · Definition 030L","summary":"Let E/F be an algebraic field extension. Let E_sep be the subextension found in Lemma [Tag 030K]. • The integer [E_sep : F] is called the separable degree of the extension. Notation [E : F]_s. • The integer [E : E_sep] is called the inseparable degree, or the degree of inseparability of the extension. Notation [E : F]_i.","statement_latex":"Let $E/F$ be an algebraic field extension. Let $E_{sep}$ be the subextension\nfound in Lemma \\ref{lemma-separable-first}.\n\\begin{enumerate}\n\\item The integer $[E_{sep} : F]$ is called the {\\it separable\ndegree} of the extension. Notation $[E : F]_s$.\n\\item The integer $[E : E_{sep}]$ is called the {\\it inseparable\ndegree}, or the {\\it degree of inseparability} of the extension.\nNotation $[E : F]_i$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Purely inseparable extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030L","source_file":"fields.tex","source_line":1726,"source_end_line":1737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1726-L1737","statement_sha256":"a2e1af56a85b5e9eb47363fb9f3501a146f9adbaabd841591aa90487e1d805d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":888,"rank":888,"depth":5,"x":1594.972,"y":129.86,"cluster":"fields-brauer-groups"},{"id":"stacks:09HJ","tag":"09HJ","title":"Purely inseparable extensions · Lemma 09HJ","summary":"Let K/F be a finite extension. Let overlineF be an algebraic closure of F. Then [K : F]_s = |Mor_F(K, overlineF)|.","statement_latex":"Let $K/F$ be a finite extension. Let $\\overline{F}$ be an algebraic\nclosure of $F$. Then $[K : F]_s = |\\Mor_F(K, \\overline{F})|$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Purely inseparable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HJ","source_file":"fields.tex","source_line":1750,"source_end_line":1754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1750-L1754","statement_sha256":"cda8dd498293c8221102d2f76c57b53f940bd2f3258b532aa7d6d1cb9561b4aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":889,"rank":889,"depth":5,"x":1641.979,"y":295.611,"cluster":"fields-brauer-groups"},{"id":"stacks:09HK","tag":"09HK","title":"Multiplicativity · Lemma 09HK","summary":"Suppose given a tower of algebraic field extensions K/E/F. Then [K : F]_s = [K : E]_s [E : F]_s and [K : F]_i = [K : E]_i [E : F]_i","statement_latex":"Suppose given a tower of algebraic field extensions $K/E/F$. Then\n$$\n[K : F]_s = [K : E]_s [E : F]_s\n\\quad\\text{and}\\quad\n[K : F]_i = [K : E]_i [E : F]_i\n$$","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Purely inseparable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HK","source_file":"fields.tex","source_line":1778,"source_end_line":1786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1778-L1786","statement_sha256":"2070e96fbd5a14b404be4b0cc1c99fd76f6f382ed1d0c78cff320274940318f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":890,"rank":890,"depth":6,"x":1472.589,"y":199.239,"cluster":"fields-brauer-groups"},{"id":"stacks:09HM","tag":"09HM","title":"Normal extensions · Definition 09HM","summary":"Let E/F be an algebraic field extension. We say E is normal over F if for all α ∈ E the minimal polynomial P of α over F splits completely into linear factors over E.","statement_latex":"Let $E/F$ be an algebraic field extension. We say $E$ is {\\it normal}\nover $F$ if for all $\\alpha \\in E$ the minimal polynomial $P$\nof $\\alpha$ over $F$ splits completely into linear factors over $E$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HM","source_file":"fields.tex","source_line":1826,"source_end_line":1831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1826-L1831","statement_sha256":"4073a15d1aa359d67e46d48631fe6f3dc3e0a58fe9fe33de6578996ebe46a07f","origin":"The Stacks Project","memory_eligible":false,"source_rank":891,"rank":891,"depth":0,"x":1676.699,"y":173.98,"cluster":"fields-brauer-groups"},{"id":"stacks:09HN","tag":"09HN","title":"Normal extensions · Lemma 09HN","summary":"Let K/E/F be a tower of algebraic field extensions. If K is normal over F, then K is normal over E.","statement_latex":"Let $K/E/F$ be a tower of algebraic field extensions.\nIf $K$ is normal over $F$, then $K$ is normal over $E$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HN","source_file":"fields.tex","source_line":1837,"source_end_line":1841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1837-L1841","statement_sha256":"ce240ef0d592828724f65a3e9aeab61df8586a6453f577011aea966ef113bc94","origin":"The Stacks Project","memory_eligible":false,"source_rank":892,"rank":892,"depth":0,"x":1545.422,"y":309.532,"cluster":"fields-brauer-groups"},{"id":"stacks:09HP","tag":"09HP","title":"Normal extensions · Lemma 09HP","summary":"Let F be a field. Let M/F be an algebraic extension. Let M/E_i/F, i ∈ I be subextensions with E_i/F normal. Then ⋂ E_i is normal over F.","statement_latex":"Let $F$ be a field. Let $M/F$ be an algebraic extension. Let\n$M/E_i/F$, $i \\in I$ be subextensions with\n$E_i/F$ normal. Then $\\bigcap E_i$ is normal over $F$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HP","source_file":"fields.tex","source_line":1850,"source_end_line":1855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1850-L1855","statement_sha256":"d3cc00210c27b18853aeeadb67787895b1a89b4dac3785676b8e561864f87bff","origin":"The Stacks Project","memory_eligible":false,"source_rank":893,"rank":893,"depth":0,"x":1533.122,"y":133.668,"cluster":"fields-brauer-groups"},{"id":"stacks:0EXK","tag":"0EXK","title":"Normal extensions · Lemma 0EXK","summary":"Let E/F be a normal algebraic field extension. Then the subextension E/E_sep/F of Lemma [Tag 030K] is normal.","statement_latex":"Let $E/F$ be a normal algebraic field extension. Then the subextension\n$E/E_{sep}/F$ of Lemma \\ref{lemma-separable-first} is normal.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXK","source_file":"fields.tex","source_line":1861,"source_end_line":1865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1861-L1865","statement_sha256":"5b8b34ff36c2166d4e0bd403c5704e78c20e4bac9d936c5ae95f6299244c5913","origin":"The Stacks Project","memory_eligible":false,"source_rank":894,"rank":894,"depth":5,"x":1684.819,"y":257.354,"cluster":"fields-brauer-groups"},{"id":"stacks:09HQ","tag":"09HQ","title":"Normal extensions · Lemma 09HQ","summary":"Let E/F be an algebraic extension of fields. Let overlineF be an algebraic closure of F. The following are equivalent • E is normal over F, and • for every pair σ, σ' ∈ Mor_F(E, overlineF) we have σ(E) = σ'(E).","statement_latex":"Let $E/F$ be an algebraic extension of fields. Let $\\overline{F}$ be an\nalgebraic closure of $F$. The following are equivalent\n\\begin{enumerate}\n\\item $E$ is normal over $F$, and\n\\item for every pair $\\sigma, \\sigma' \\in \\Mor_F(E, \\overline{F})$ we\nhave $\\sigma(E) = \\sigma'(E)$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HQ","source_file":"fields.tex","source_line":1878,"source_end_line":1887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1878-L1887","statement_sha256":"6450c147550db6537f72531de7f2fd5521d7d9c876a7d92f23d7e4c5679a2683","origin":"The Stacks Project","memory_eligible":false,"source_rank":895,"rank":895,"depth":2,"x":1471.829,"y":252.182,"cluster":"fields-brauer-groups"},{"id":"stacks:0BR3","tag":"0BR3","title":"Normal extensions · Lemma 0BR3","summary":"Let E/F be an algebraic extension of fields. If E is generated by α_i ∈ E, i ∈ I over F and if for each i the minimal polynomial of α_i over F splits completely in E, then E/F is normal.","statement_latex":"Let $E/F$ be an algebraic extension of fields.\nIf $E$ is generated by $\\alpha_i \\in E$, $i \\in I$\nover $F$ and if for each $i$ the minimal polynomial\nof $\\alpha_i$ over $F$ splits completely in $E$, then\n$E/F$ is normal.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BR3","source_file":"fields.tex","source_line":1921,"source_end_line":1928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1921-L1928","statement_sha256":"73c45adc1e37fcf868092786b4ecece23e1ee68537eec229ecf9f0ccb51f0577","origin":"The Stacks Project","memory_eligible":false,"source_rank":896,"rank":896,"depth":3,"x":1634.299,"y":134.235,"cluster":"fields-brauer-groups"},{"id":"stacks:0BME","tag":"0BME","title":"Normal extensions · Lemma 0BME","summary":"Let L/M/K be a tower of algebraic extensions. • If M/K is normal, then any automorphism τ of L/K induces an automorphism τ|_M : M → M. • If L/K is normal, then any K-algebra map σ : M → L extends to an automorphism of L.","statement_latex":"Let $L/M/K$ be a tower of algebraic extensions.\n\\begin{enumerate}\n\\item If $M/K$ is normal, then any automorphism $\\tau$ of $L/K$\ninduces an automorphism $\\tau|_M : M \\to M$.\n\\item If $L/K$ is normal, then any $K$-algebra map $\\sigma : M \\to L$\nextends to an automorphism of $L$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BME","source_file":"fields.tex","source_line":1945,"source_end_line":1954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1945-L1954","statement_sha256":"d922df0afab5a49db2a49fad118a4b75b5d6854d7565f7752844e4ad069e8050","origin":"The Stacks Project","memory_eligible":false,"source_rank":897,"rank":897,"depth":3,"x":1609.15,"y":314.765,"cluster":"fields-brauer-groups"},{"id":"stacks:09HR","tag":"09HR","title":"Normal extensions · Definition 09HR","summary":"Let E/F be an extension of fields. Then Aut(E/F) or Aut_F(E) denotes the automorphism group of E as an object of the category of F-extensions. Elements of Aut(E/F) are called automorphisms of E over F or automorphisms of E/F.","statement_latex":"Let $E/F$ be an extension of fields. Then $\\text{Aut}(E/F)$ or\n$\\text{Aut}_F(E)$ denotes the automorphism group of $E$ as an object\nof the category of $F$-extensions. Elements of $\\text{Aut}(E/F)$\nare called {\\it automorphisms of $E$ over $F$} or\n{\\it automorphisms of $E/F$}.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HR","source_file":"fields.tex","source_line":1981,"source_end_line":1988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1981-L1988","statement_sha256":"c8e95e4446d55e9623c9a5327683affd94785cc6bdf29be1d0ecccb989ae9a4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":898,"rank":898,"depth":0,"x":1481.568,"y":166.264,"cluster":"fields-brauer-groups"},{"id":"stacks:09HS","tag":"09HS","title":"Normal extensions · Lemma 09HS","summary":"Let E/F be a finite extension. We have |Aut(E/F)| ≤ [E : F]_s with equality if and only if E is normal over F.","statement_latex":"Let $E/F$ be a finite extension. We have\n$$\n|\\text{Aut}(E/F)| \\leq [E : F]_s\n$$\nwith equality if and only if $E$ is normal over $F$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HS","source_file":"fields.tex","source_line":1993,"source_end_line":2000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L1993-L2000","statement_sha256":"a7327498bd74542af147d8a2e6e05d96cc42e20206d7f288c4b8b840135deedf","origin":"The Stacks Project","memory_eligible":false,"source_rank":899,"rank":899,"depth":3,"x":1696.648,"y":203.652,"cluster":"fields-brauer-groups"},{"id":"stacks:0BR4","tag":"0BR4","title":"Normal extensions · Lemma 0BR4","summary":"Let L/K be an algebraic normal extension of fields. Let E/K be an extension of fields. Then either there is no K-embedding from L to E or there is one τ : L → E and every other one is of the form τ ∘ σ where σ ∈ Aut(L/K).","statement_latex":"Let $L/K$ be an algebraic normal extension of fields.\nLet $E/K$ be an extension of fields. Then either\nthere is no $K$-embedding from $L$ to $E$ or\nthere is one $\\tau : L \\to E$ and every other\none is of the form $\\tau \\circ \\sigma$ where $\\sigma \\in \\text{Aut}(L/K)$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Normal extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BR4","source_file":"fields.tex","source_line":2024,"source_end_line":2031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2024-L2031","statement_sha256":"66217897fa53ebf8af245e741c9b78be74f1332651d16762afc23cccf5e97263","origin":"The Stacks Project","memory_eligible":false,"source_rank":900,"rank":900,"depth":4,"x":1506.604,"y":298.811,"cluster":"fields-brauer-groups"},{"id":"stacks:09HU","tag":"09HU","title":"Splitting fields · Lemma 09HU","summary":"Let F be a field. Let P ∈ F[x] be a nonconstant polynomial. There exists a smallest field extension E/F such that P splits completely over E. Moreover, the field extension E/F is normal and unique up to (nonunique) isomorphism.","statement_latex":"Let $F$ be a field. Let $P \\in F[x]$ be a nonconstant polynomial.\nThere exists a smallest field extension $E/F$ such that $P$\nsplits completely over $E$. Moreover, the field extension $E/F$ is normal\nand unique up to (nonunique) isomorphism.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Splitting fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HU","source_file":"fields.tex","source_line":2048,"source_end_line":2054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2048-L2054","statement_sha256":"6ef9e89a252356405827b4fce201576d98ed97678d78695ec710f176997b3712","origin":"The Stacks Project","memory_eligible":false,"source_rank":901,"rank":901,"depth":3,"x":1570.678,"y":119.527,"cluster":"fields-brauer-groups"},{"id":"stacks:09HV","tag":"09HV","title":"Splitting fields · Definition 09HV","summary":"Let F be a field. Let P ∈ F[x] be a nonconstant polynomial. The field extension E/F constructed in Lemma [Tag 09HU] is called the splitting field of P over F.","statement_latex":"Let $F$ be a field. Let $P \\in F[x]$ be a nonconstant polynomial.\nThe field extension $E/F$ constructed in Lemma \\ref{lemma-splitting-field}\nis called the {\\it splitting field of $P$ over $F$}.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Splitting fields","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HV","source_file":"fields.tex","source_line":2078,"source_end_line":2083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2078-L2083","statement_sha256":"c1936e233c5bf17fa91f3b175c6be5514a0ef2f7bd3c735e365b81235c38e731","origin":"The Stacks Project","memory_eligible":false,"source_rank":902,"rank":902,"depth":4,"x":1668.29,"y":289.283,"cluster":"fields-brauer-groups"},{"id":"stacks:09DT","tag":"09DT","title":"Splitting fields · Lemma 09DT","summary":"Existence of normal closure of finite extensions of fields. Let E/F be a finite extension of fields. There exists a unique smallest finite extension K/E such that K is normal over F.","statement_latex":"\\begin{slogan}\nExistence of normal closure of finite extensions of fields.\n\\end{slogan}\nLet $E/F$ be a finite extension of fields. There exists a unique\nsmallest finite extension $K/E$ such that $K$ is normal over $F$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Splitting fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DT","source_file":"fields.tex","source_line":2085,"source_end_line":2092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2085-L2092","statement_sha256":"054dd04034afc6e7414696dbaac18124fc021b3becbee010bcb1aedc758e6ee8","origin":"The Stacks Project","memory_eligible":false,"source_rank":903,"rank":903,"depth":2,"x":1458.342,"y":219.003,"cluster":"fields-brauer-groups"},{"id":"stacks:0BMF","tag":"0BMF","title":"Splitting fields · Definition 0BMF","summary":"Let E/F be a finite extension of fields. The field extension K/E constructed in Lemma [Tag 09DT] is called the normal closure E over F.","statement_latex":"Let $E/F$ be a finite extension of fields. The field extension $K/E$\nconstructed in Lemma \\ref{lemma-normal-closure}\nis called the {\\it normal closure $E$ over $F$}.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Splitting fields","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMF","source_file":"fields.tex","source_line":2123,"source_end_line":2128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2123-L2128","statement_sha256":"f7439ef66e9edebeec062dfd886b74ac3c09b7ac177f5b6ae3f890025e043d6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":904,"rank":904,"depth":3,"x":1671.131,"y":151.236,"cluster":"fields-brauer-groups"},{"id":"stacks:0BMG","tag":"0BMG","title":"Splitting fields · Lemma 0BMG","summary":"Let L/K be an algebraic normal extension. • If L/M/K is a subextension with M/K finite, then there exists a tower L/M'/M/K with M'/K finite and normal. • If L/M'/M/K is a tower with M/K normal and M'/M finite, then there exists a tower L/M\"/M'/M/K with M\"/M finite and M\"/K normal.","statement_latex":"Let $L/K$ be an algebraic normal extension.\n\\begin{enumerate}\n\\item If $L/M/K$ is a subextension with $M/K$ finite, then there exists\na tower $L/M'/M/K$ with $M'/K$ finite and normal.\n\\item If $L/M'/M/K$ is a tower with $M/K$ normal and $M'/M$ finite,\nthen there exists a tower $L/M''/M'/M/K$ with $M''/M$\nfinite and $M''/K$ normal.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Splitting fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMG","source_file":"fields.tex","source_line":2133,"source_end_line":2143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2133-L2143","statement_sha256":"284cc27a5fafab2857e0413c5787d09a43c75ad2c71548f4209a7744af317e3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":905,"rank":905,"depth":4,"x":1568.02,"y":323.108,"cluster":"fields-brauer-groups"},{"id":"stacks:0EXL","tag":"0EXL","title":"Splitting fields · Lemma 0EXL","summary":"Let L/K be a finite extension. Let M/L be the normal closure of L over K. Then there is a surjective map L ⊗_K L ⊗_K … ⊗_K L → M of K-algebras where the number of tensors can be taken [L : K]_s ≤ [L : K].","statement_latex":"Let $L/K$ be a finite extension. Let $M/L$ be the normal\nclosure of $L$ over $K$. Then there is a surjective map\n$$\nL \\otimes_K L \\otimes_K \\ldots \\otimes_K L \\longrightarrow M\n$$\nof $K$-algebras where the number of tensors can be taken\n$[L : K]_s \\leq [L : K]$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Splitting fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXL","source_file":"fields.tex","source_line":2164,"source_end_line":2173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2164-L2173","statement_sha256":"0b75f5b77db2896a254e94b9a187b526b878c82d00bdf81857d9be8bc2d1ff7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":906,"rank":906,"depth":6,"x":1505.424,"y":136.62,"cluster":"fields-brauer-groups"},{"id":"stacks:09HX","tag":"09HX","title":"Roots of unity · Lemma 09HX","summary":"Let A be an abelian group of exponent dividing n such that (x ∈ A mid dx = 0) has cardinality at most d for all d | n. Then A is cyclic of order dividing n.","statement_latex":"Let $A$ be an abelian group of exponent dividing $n$ such that\n$\\{x \\in A \\mid dx = 0\\}$ has cardinality at most $d$ for all $d | n$.\nThen $A$ is cyclic of order dividing $n$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Roots of unity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09HX","source_file":"fields.tex","source_line":2232,"source_end_line":2237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2232-L2237","statement_sha256":"d4d58c74ac14e4aa188dfb7116d9a4ec6f51a4309f44bb62b3056b2a9a1d1c20","origin":"The Stacks Project","memory_eligible":false,"source_rank":907,"rank":907,"depth":0,"x":1702.846,"y":239.293,"cluster":"fields-brauer-groups"},{"id":"stacks:030N","tag":"030N","title":"Primitive element · Lemma 030N","summary":"Let E/F be a finite extension of fields. The following are equivalent • there exists a primitive element for E over F, and • there are finitely many subextensions E/K/F. Moreover, (1) and (2) hold if E/F is separable.","statement_latex":"Let $E/F$ be a finite extension of fields. The following are equivalent\n\\begin{enumerate}\n\\item there exists a primitive element for $E$ over $F$, and\n\\item there are finitely many subextensions $E/K/F$.\n\\end{enumerate}\nMoreover, (1) and (2) hold if $E/F$ is separable.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Primitive elements","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030N","source_file":"fields.tex","source_line":2300,"source_end_line":2308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2300-L2308","statement_sha256":"e92567d9eb0461f000d2f2be81daea6bc54160802d31bb11809ab1f7a526d11b","origin":"The Stacks Project","memory_eligible":false,"source_rank":908,"rank":908,"depth":4,"x":1473.211,"y":275.84,"cluster":"fields-brauer-groups"},{"id":"stacks:0BIF","tag":"0BIF","title":"Trace and norm · Definition 0BIF","summary":"Let L/K be a finite extension of fields. For α ∈ L we define the trace Trace_L/K(α) = Trace_K(α : L → L) and the norm Norm_L/K(α) = det_K(α : L → L).","statement_latex":"Let $L/K$ be a finite extension of fields. For $\\alpha \\in L$ we define\nthe {\\it trace}\n$\\text{Trace}_{L/K}(\\alpha) = \\text{Trace}_K(\\alpha : L \\to L)$\nand the {\\it norm}\n$\\text{Norm}_{L/K}(\\alpha) = \\det_K(\\alpha : L \\to L)$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Trace and norm","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIF","source_file":"fields.tex","source_line":2379,"source_end_line":2386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2379-L2386","statement_sha256":"98c71ef23510f7c3da74cbf575e18c5b7989ee68b0cc5bef038fc7c4eec2c8e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":909,"rank":909,"depth":0,"x":1614.057,"y":117.578,"cluster":"fields-brauer-groups"},{"id":"stacks:0BIG","tag":"0BIG","title":"Trace and norm · Lemma 0BIG","summary":"Let L/K be a finite extension of fields. Let α ∈ L and let P be the minimal polynomial of α over K. Then the characteristic polynomial of the K-linear map α : L → L is equal to P^e with e deg(P) = [L : K].","statement_latex":"Let $L/K$ be a finite extension of fields. Let $\\alpha \\in L$ and let $P$\nbe the minimal polynomial of $\\alpha$ over $K$. Then the characteristic\npolynomial of the $K$-linear map $\\alpha : L \\to L$ is equal to\n$P^e$ with $e \\deg(P) = [L : K]$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Trace and norm","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIG","source_file":"fields.tex","source_line":2398,"source_end_line":2404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2398-L2404","statement_sha256":"b357003216d30bcd3b4a0fcb8f7a3880cf8da5153f0b2a42d2ad2d470f18cb45","origin":"The Stacks Project","memory_eligible":false,"source_rank":910,"rank":910,"depth":2,"x":1637.615,"y":315.449,"cluster":"fields-brauer-groups"},{"id":"stacks:0BIH","tag":"0BIH","title":"Trace and norm · Lemma 0BIH","summary":"Let L/K be a finite extension of fields. Let α ∈ L and let P = x^d + a_1 x^d - 1 + … + a_d be the minimal polynomial of α over K. Then Norm_L/K(α) = (-1)^[L : K] a_d^e and Trace_L/K(α) = - e a_1 where e d = [L : K].","statement_latex":"Let $L/K$ be a finite extension of fields. Let $\\alpha \\in L$ and let\n$P = x^d + a_1 x^{d - 1} + \\ldots + a_d$\nbe the minimal polynomial of $\\alpha$ over $K$. Then\n$$\n\\text{Norm}_{L/K}(\\alpha) = (-1)^{[L : K]} a_d^e\n\\quad\\text{and}\\quad\n\\text{Trace}_{L/K}(\\alpha) = - e a_1\n$$\nwhere $e d = [L : K]$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Trace and norm","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIH","source_file":"fields.tex","source_line":2425,"source_end_line":2436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2425-L2436","statement_sha256":"31eb868755c3290d75243f299890fd7a34ca8edf6245454417277db491f1cd53","origin":"The Stacks Project","memory_eligible":false,"source_rank":911,"rank":911,"depth":3,"x":1460.013,"y":182.074,"cluster":"fields-brauer-groups"},{"id":"stacks:0BII","tag":"0BII","title":"Trace and norm · Lemma 0BII","summary":"Let L/K be a finite extension of fields. Let V be a finite dimensional vector space over L. Let φ : V → V be an L-linear map. Then Trace_K(φ : V → V) = Trace_L/K(Trace_L(φ : V → V)) and det_K(φ : V → V) = Norm_L/K(det_L(φ : V → V))","statement_latex":"Let $L/K$ be a finite extension of fields. Let $V$ be a finite dimensional\nvector space over $L$. Let $\\varphi : V \\to V$ be an $L$-linear map.\nThen\n$$\n\\text{Trace}_K(\\varphi : V \\to V) =\n\\text{Trace}_{L/K}(\\text{Trace}_L(\\varphi : V \\to V))\n$$\nand\n$$\n\\det\\nolimits_K(\\varphi : V \\to V) =\n\\text{Norm}_{L/K}(\\det\\nolimits_L(\\varphi : V \\to V))\n$$","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Trace and norm","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BII","source_file":"fields.tex","source_line":2443,"source_end_line":2457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2443-L2457","statement_sha256":"3169a4a79daf5967568371cca34ad90c2c9a4472e65d2cd759a3d2c370a51852","origin":"The Stacks Project","memory_eligible":false,"source_rank":912,"rank":912,"depth":0,"x":1699.71,"y":179.635,"cluster":"fields-brauer-groups"},{"id":"stacks:0BIJ","tag":"0BIJ","title":"Trace and norm · Lemma 0BIJ","summary":"Let M/L/K be a tower of finite extensions of fields. Then Trace_M/K = Trace_L/K ∘ Trace_M/L and Norm_M/K = Norm_L/K ∘ Norm_M/L","statement_latex":"Let $M/L/K$ be a tower of finite extensions of fields. Then\n$$\n\\text{Trace}_{M/K} = \\text{Trace}_{L/K} \\circ \\text{Trace}_{M/L}\n\\quad\\text{and}\\quad\n\\text{Norm}_{M/K} = \\text{Norm}_{L/K} \\circ \\text{Norm}_{M/L}\n$$","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Trace and norm","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIJ","source_file":"fields.tex","source_line":2495,"source_end_line":2503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2495-L2503","statement_sha256":"608e017188ea8d14349db0824772dfe1ce42fd6babe336dcf67845d724c40a7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":913,"rank":913,"depth":1,"x":1523.848,"y":318.286,"cluster":"fields-brauer-groups"},{"id":"stacks:0BIK","tag":"0BIK","title":"Trace and norm · Definition 0BIK","summary":"Let L/K be a finite extension of fields. The trace pairing for L/K is the symmetric K-bilinear form Q_L/K : L × L → K, (α, β) ↦ Trace_L/K(αβ)","statement_latex":"Let $L/K$ be a finite extension of fields. The {\\it trace pairing}\nfor $L/K$ is the symmetric $K$-bilinear form\n$$\nQ_{L/K} : L \\times L \\longrightarrow K,\\quad\n(\\alpha, \\beta) \\longmapsto \\text{Trace}_{L/K}(\\alpha\\beta)\n$$","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Trace and norm","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIK","source_file":"fields.tex","source_line":2513,"source_end_line":2521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2513-L2521","statement_sha256":"38ee6ad9755bdcebed84fd6ecf96b497692e3ceac8f281b717a9d5baf26d5bef","origin":"The Stacks Project","memory_eligible":false,"source_rank":914,"rank":914,"depth":0,"x":1542.141,"y":115.035,"cluster":"fields-brauer-groups"},{"id":"stacks:0BIL","tag":"0BIL","title":"Trace and norm · Lemma 0BIL","summary":"Let L/K be a finite extension of fields. The following are equivalent: • L/K is separable, • Trace_L/K is not identically zero, and • the trace pairing Q_L/K is nondegenerate.","statement_latex":"Let $L/K$ be a finite extension of fields. The following are equivalent:\n\\begin{enumerate}\n\\item $L/K$ is separable,\n\\item $\\text{Trace}_{L/K}$ is not identically zero, and\n\\item the trace pairing $Q_{L/K}$ is nondegenerate.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Trace and norm","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIL","source_file":"fields.tex","source_line":2527,"source_end_line":2535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2527-L2535","statement_sha256":"6a1511ec3459809c4461eb4cbdb7ecc9dc52ae79829d6162c55498f998935ab6","origin":"The Stacks Project","memory_eligible":false,"source_rank":915,"rank":915,"depth":5,"x":1692.995,"y":276.25,"cluster":"fields-brauer-groups"},{"id":"stacks:0BIM","tag":"0BIM","title":"Trace and norm · Definition 0BIM","summary":"Let L/K be a finite extension of fields. The discriminant of L/K is the discriminant of the trace pairing Q_L/K.","statement_latex":"Let $L/K$ be a finite extension of fields. The\n{\\it discriminant of $L/K$} is the discriminant of\nthe trace pairing $Q_{L/K}$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Trace and norm","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIM","source_file":"fields.tex","source_line":2607,"source_end_line":2612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2607-L2612","statement_sha256":"31e7eb274fa1a4d38c135d34ea166d0b5cceb95a68d0f24230ba3a0273e7c5d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":916,"rank":916,"depth":0,"x":1450.689,"y":242.771,"cluster":"fields-brauer-groups"},{"id":"stacks:09I0","tag":"09I0","title":"Galois theory · Definition 09I0","summary":"A field extension E/F is called Galois if it is algebraic, separable, and normal.","statement_latex":"A field extension $E/F$ is called {\\it Galois} if it is algebraic,\nseparable, and normal.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Galois theory","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09I0","source_file":"fields.tex","source_line":2653,"source_end_line":2657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2653-L2657","statement_sha256":"457219acc9c9e3ff590cb1d5c9dbdcd67226fc48b23cca7ac05be5232c06d2db","origin":"The Stacks Project","memory_eligible":false,"source_rank":917,"rank":917,"depth":0,"x":1657.487,"y":129.311,"cluster":"fields-brauer-groups"},{"id":"stacks:09I1","tag":"09I1","title":"Galois theory · Lemma 09I1","summary":"Let E/F be a finite extension of fields. Then E is Galois over F if and only if |Aut(E/F)| = [E : F].","statement_latex":"Let $E/F$ be a finite extension of fields. Then $E$ is Galois over $F$\nif and only if $|\\text{Aut}(E/F)| = [E : F]$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Galois theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09I1","source_file":"fields.tex","source_line":2663,"source_end_line":2667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2663-L2667","statement_sha256":"d78a2f0e67baf009a1c579f17aa0f44142540888158ee7b5b9b0310e8792a973","origin":"The Stacks Project","memory_eligible":false,"source_rank":918,"rank":918,"depth":4,"x":1595.883,"y":331.478,"cluster":"fields-brauer-groups"},{"id":"stacks:09DV","tag":"09DV","title":"Galois theory · Definition 09DV","summary":"If E/F is a Galois extension, then the group Aut(E/F) is called the Galois group and it is denoted Gal(E/F).","statement_latex":"If $E/F$ is a Galois extension, then the group $\\text{Aut}(E/F)$ is\ncalled the {\\it Galois group} and it is denoted $\\text{Gal}(E/F)$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Galois theory","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DV","source_file":"fields.tex","source_line":2682,"source_end_line":2686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2682-L2686","statement_sha256":"2a35ee729d37fe0c6770a9f821f5b2184eccd8a035757ab55bdb51434d95b6a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":919,"rank":919,"depth":0,"x":1478.066,"y":146.395,"cluster":"fields-brauer-groups"},{"id":"stacks:09I2","tag":"09I2","title":"Galois theory · Lemma 09I2","summary":"Let K/E/F be a tower of algebraic field extensions. If K is Galois over F, then K is Galois over E.","statement_latex":"Let $K/E/F$ be a tower of algebraic field extensions.\nIf $K$ is Galois over $F$, then $K$ is Galois over $E$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Galois theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09I2","source_file":"fields.tex","source_line":2693,"source_end_line":2697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2693-L2697","statement_sha256":"185a47dae571db2576e81114175413fbd93c1118e64792eb552c10f51dbac61b","origin":"The Stacks Project","memory_eligible":false,"source_rank":920,"rank":920,"depth":2,"x":1715.11,"y":216.416,"cluster":"fields-brauer-groups"},{"id":"stacks:0EXM","tag":"0EXM","title":"Galois theory · Lemma 0EXM","summary":"Let L/K be a finite separable extension of fields. Let M be the normal closure of L over K (Definition [Tag 0BMF]). Then M/K is Galois.","statement_latex":"Let $L/K$ be a finite separable extension of fields.\nLet $M$ be the normal closure of $L$ over $K$\n(Definition \\ref{definition-normal-closure}).\nThen $M/K$ is Galois.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Galois theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXM","source_file":"fields.tex","source_line":2703,"source_end_line":2709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2703-L2709","statement_sha256":"1b5d757462c7da855b4c217ffbabb121277e4dd3de1e56269e59902771f07bf9","origin":"The Stacks Project","memory_eligible":false,"source_rank":921,"rank":921,"depth":6,"x":1482.718,"y":299.748,"cluster":"fields-brauer-groups"},{"id":"stacks:09I3","tag":"09I3","title":"Galois theory · Lemma 09I3","summary":"Let K be a field. Let G be a finite group acting faithfully on K. Then the extension K/K^G is Galois, we have [K : K^G] = |G|, and the Galois group of the extension is G.","statement_latex":"Let $K$ be a field. Let $G$ be a finite group acting faithfully on $K$.\nThen the extension $K/K^G$ is Galois, we have $[K : K^G] = |G|$,\nand the Galois group of the extension is $G$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Galois theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09I3","source_file":"fields.tex","source_line":2726,"source_end_line":2731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2726-L2731","statement_sha256":"e936af969c16e4e8414c06e840215cd392d385c4ebdc1d62157c4ab130393606","origin":"The Stacks Project","memory_eligible":false,"source_rank":922,"rank":922,"depth":5,"x":1587.645,"y":105.369,"cluster":"fields-brauer-groups"},{"id":"stacks:09DW","tag":"09DW","title":"Fundamental theorem of Galois theory · Theorem 09DW","summary":"Let L/K be a finite Galois extension with Galois group G. Then we have K = L^G and the map (subgroups of G) → (subextensions L/M/K), H ↦ L^H is a bijection whose inverse maps M to Gal(L/M). The normal subgroups H of G correspond exactly to those subextensions M with M/K Galois.","statement_latex":"Let $L/K$ be a finite Galois extension with Galois group $G$.\nThen we have $K = L^G$ and the map\n$$\n\\{\\text{subgroups of }G\\}\n\\longrightarrow\n\\{\\text{subextensions }L/M/K\\},\\quad\nH \\longmapsto L^H\n$$\nis a bijection whose inverse maps $M$ to $\\text{Gal}(L/M)$.\nThe normal subgroups $H$ of $G$ correspond exactly to those\nsubextensions $M$ with $M/K$ Galois.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Galois theory","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DW","source_file":"fields.tex","source_line":2783,"source_end_line":2796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2783-L2796","statement_sha256":"17af8d1add47a53289685da5c4f7d8ae8d9605f092b271106fcd69b2f3ec36d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":923,"rank":923,"depth":6,"x":1667.015,"y":309.348,"cluster":"fields-brauer-groups"},{"id":"stacks:0BMH","tag":"0BMH","title":"Galois theory · Lemma 0BMH","summary":"Let L/M/K be a tower of fields. Assume L/K and M/K are finite Galois. Then we obtain a short exact sequence 1 → Gal(L/M) → Gal(L/K) → Gal(M/K) → 1 of finite groups.","statement_latex":"Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are finite Galois.\nThen we obtain a short exact sequence\n$$\n1 \\to \\text{Gal}(L/M) \\to \\text{Gal}(L/K) \\to \\text{Gal}(M/K) \\to 1\n$$\nof finite groups.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Galois theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMH","source_file":"fields.tex","source_line":2824,"source_end_line":2832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2824-L2832","statement_sha256":"017ee47c2f8665c9a74c44371468b651f1416c202d4f97a8e85b69c57292d646","origin":"The Stacks Project","memory_eligible":false,"source_rank":924,"rank":924,"depth":4,"x":1443.255,"y":203.402,"cluster":"fields-brauer-groups"},{"id":"stacks:0BMJ","tag":"0BMJ","title":"Infinite Galois theory · Lemma 0BMJ","summary":"Let E/F be a Galois extension. Endow Gal(E/F) with the coarsest topology such that Gal(E/F) × E → E is continuous when E is given the discrete topology. Then • for any topological space X and map X → Gal(E/F) such that the action X × E → E is continuous the induced map X → Gal(E/F) is continuous, • this topology turns Gal(E/F) into a profinite topological group.","statement_latex":"Let $E/F$ be a Galois extension. Endow $\\text{Gal}(E/F)$ with the coarsest\ntopology such that\n$$\n\\text{Gal}(E/F) \\times E \\longrightarrow E\n$$\nis continuous when $E$ is given the discrete topology. Then\n\\begin{enumerate}\n\\item for any topological space $X$ and map $X \\to \\text{Gal}(E/F)$\nsuch that the action $X \\times E \\to E$ is continuous the induced map\n$X \\to \\text{Gal}(E/F)$ is continuous,\n\\item this topology turns $\\text{Gal}(E/F)$ into\na profinite topological group.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Infinite Galois theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMJ","source_file":"fields.tex","source_line":2858,"source_end_line":2873,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2858-L2873","statement_sha256":"19bdf7d4f4fdc845145bd680bd3f2552f37292559f4151dea8a609a306708fc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":925,"rank":925,"depth":5,"x":1694.787,"y":154.299,"cluster":"fields-brauer-groups"},{"id":"stacks:0BMK","tag":"0BMK","title":"Infinite Galois theory · Lemma 0BMK","summary":"Let L/M/K be a tower of fields. Assume both L/K and M/K are Galois. Then there is a canonical surjective continuous homomorphism c : Gal(L/K) → Gal(M/K).","statement_latex":"Let $L/M/K$ be a tower of fields. Assume both $L/K$ and\n$M/K$ are Galois. Then there is a canonical surjective continuous\nhomomorphism $c : \\text{Gal}(L/K) \\to \\text{Gal}(M/K)$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Infinite Galois theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMK","source_file":"fields.tex","source_line":2950,"source_end_line":2955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2950-L2955","statement_sha256":"9bf60dbff66d22d718f6816cab91e422a4d9fad2b09f3de55584777dbb7a64f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":926,"rank":926,"depth":6,"x":1548.025,"y":334.177,"cluster":"fields-brauer-groups"},{"id":"stacks:0BU2","tag":"0BU2","title":"Infinite Galois theory · Lemma 0BU2","summary":"Let L/K be a Galois extension with Galois group G. Let Lambda be the set of finite Galois subextensions, i.e., λ ∈ Lambda corresponds to L/L_λ/K with L_λ/K finite Galois with Galois group G_λ. Define a partial ordering on Lambda by the rule λ ≥ λ' if and only if L_λ ⊃ L_λ'. Then • Lambda is a directed partially ordered set, • L_λ is a system of K-extensions over Lambda and L = colim L_λ, • G_λ is an inverse system of finite groups over Lambda, the transition maps are…","statement_latex":"Let $L/K$ be a Galois extension with Galois group $G$.\nLet $\\Lambda$ be the set of finite Galois subextensions,\ni.e., $\\lambda \\in \\Lambda$ corresponds to $L/L_\\lambda/K$\nwith $L_\\lambda/K$ finite Galois with Galois group $G_\\lambda$.\nDefine a partial ordering on $\\Lambda$ by the rule\n$\\lambda \\geq \\lambda'$ if and only if\n$L_\\lambda \\supset L_{\\lambda'}$. Then\n\\begin{enumerate}\n\\item $\\Lambda$ is a directed partially ordered set,\n\\item $L_\\lambda$ is a system of $K$-extensions over $\\Lambda$\nand $L = \\colim L_\\lambda$,\n\\item $G_\\lambda$ is an inverse system of finite groups over $\\Lambda$,\nthe transition maps are surjective, and\n$$\nG = \\lim_{\\lambda \\in \\Lambda} G_\\lambda\n$$\nas a profinite group, and\n\\item each of the projections $G \\to G_\\lambda$ is continuous and surjective.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Infinite Galois theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BU2","source_file":"fields.tex","source_line":2979,"source_end_line":3000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L2979-L3000","statement_sha256":"6f2d55b3cf3a573b5534b0865a9355a3c16b5b757b0c7925f0e7af917aa8d1dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":927,"rank":927,"depth":7,"x":1511.405,"y":117.132,"cluster":"fields-brauer-groups"},{"id":"stacks:0BML","tag":"0BML","title":"Fundamental theorem of infinite Galois theory · Theorem 0BML","summary":"Let L/K be a Galois extension. Let G = Gal(L/K) be the Galois group viewed as a profinite topological group (Lemma [Tag 0BMJ]). Then we have K = L^G and the map (closed subgroups of G) → (subextensions L/M/K), H ↦ L^H is a bijection whose inverse maps M to Gal(L/M). The finite subextensions M correspond exactly to the open subgroups H ⊂ G. The normal closed subgroups H of G correspond exactly to subextensions M Galois over K.","statement_latex":"Let $L/K$ be a Galois extension. Let $G = \\text{Gal}(L/K)$\nbe the Galois group viewed as a profinite topological group\n(Lemma \\ref{lemma-galois-profinite}). Then we have $K = L^G$ and the map\n$$\n\\{\\text{closed subgroups of }G\\}\n\\longrightarrow\n\\{\\text{subextensions }L/M/K\\},\\quad\nH \\longmapsto L^H\n$$\nis a bijection whose inverse maps $M$ to $\\text{Gal}(L/M)$.\nThe finite subextensions $M$ correspond exactly to the open\nsubgroups $H \\subset G$. The normal closed subgroups $H$ of $G$\ncorrespond exactly to subextensions $M$ Galois over $K$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Infinite Galois theory","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BML","source_file":"fields.tex","source_line":3053,"source_end_line":3068,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3053-L3068","statement_sha256":"62edf6770cda365b1030fb9c1acdb94e4a585397f2ab294944943527b571143f","origin":"The Stacks Project","memory_eligible":false,"source_rank":928,"rank":928,"depth":7,"x":1713.99,"y":257.121,"cluster":"fields-brauer-groups"},{"id":"stacks:0BMM","tag":"0BMM","title":"Infinite Galois theory · Lemma 0BMM","summary":"Let L/M/K be a tower of fields. Assume L/K and M/K are Galois. Then we obtain a short exact sequence 1 → Gal(L/M) → Gal(L/K) → Gal(M/K) → 1 of profinite topological groups.","statement_latex":"Let $L/M/K$ be a tower of fields. Assume $L/K$ and $M/K$ are Galois.\nThen we obtain a short exact sequence\n$$\n1 \\to \\text{Gal}(L/M) \\to \\text{Gal}(L/K) \\to \\text{Gal}(M/K) \\to 1\n$$\nof profinite topological groups.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Infinite Galois theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMM","source_file":"fields.tex","source_line":3136,"source_end_line":3144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3136-L3144","statement_sha256":"aa3cdae8005306d809c32da204b59b714d476b8a2af217ae33da6dc959bf50af","origin":"The Stacks Project","memory_eligible":false,"source_rank":929,"rank":929,"depth":7,"x":1450.691,"y":268.905,"cluster":"fields-brauer-groups"},{"id":"stacks:09I5","tag":"09I5","title":"Fundamental theorem of algebra · Lemma 09I5","summary":"The field C is algebraically closed.","statement_latex":"The field $\\mathbf{C}$ is algebraically closed.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"The complex numbers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09I5","source_file":"fields.tex","source_line":3201,"source_end_line":3204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3201-L3204","statement_sha256":"4b818a8dd1471e51cf4dfd224ea2c304d81c6c196e9a1b39dc849b21f509393f","origin":"The Stacks Project","memory_eligible":false,"source_rank":930,"rank":930,"depth":0,"x":1636.305,"y":110.014,"cluster":"fields-brauer-groups"},{"id":"stacks:09DX","tag":"09DX","title":"Kummer extensions · Lemma 09DX","summary":"Let L/K be a Galois extension of fields whose Galois group is Z/nZ. Assume moreover that the characteristic of K is prime to n and that K contains a primitive nth root of 1. Then L = K[z] with z^n ∈ K.","statement_latex":"Let $L/K$ be a Galois extension of fields whose Galois group is\n$\\mathbf{Z}/n\\mathbf{Z}$. Assume moreover that the characteristic of $K$\nis prime to $n$ and that $K$ contains a primitive $n$th root of $1$.\nThen $L = K[z]$ with $z^n \\in K$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Kummer extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DX","source_file":"fields.tex","source_line":3230,"source_end_line":3236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3230-L3236","statement_sha256":"e192598861428c51b74fe79e81f81e3a2b943c8244ad282667282ee5a7ccdb43","origin":"The Stacks Project","memory_eligible":false,"source_rank":931,"rank":931,"depth":1,"x":1627.164,"y":333.615,"cluster":"fields-brauer-groups"},{"id":"stacks:0EXN","tag":"0EXN","title":"Kummer extensions · Lemma 0EXN","summary":"Let K be a field with algebraic closure overlineK. Let p be a prime different from the characteristic of K. Let zeta ∈ overlineK be a primitive pth root of 1. Then K(zeta)/K is a Galois extension of degree dividing p - 1.","statement_latex":"Let $K$ be a field with algebraic closure $\\overline{K}$.\nLet $p$ be a prime different from the characteristic of $K$.\nLet $\\zeta \\in \\overline{K}$ be a primitive $p$th root\nof $1$. Then $K(\\zeta)/K$ is a Galois extension of degree dividing $p - 1$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Kummer extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXN","source_file":"fields.tex","source_line":3258,"source_end_line":3264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3258-L3264","statement_sha256":"d6c679f4dd787be1eb0f0db38b8427035ca3ad70963c8ea1c441015a7fdf98d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":932,"rank":932,"depth":0,"x":1453.232,"y":162.702,"cluster":"fields-brauer-groups"},{"id":"stacks:0EXP","tag":"0EXP","title":"Kummer extensions · Lemma 0EXP","summary":"[Radical] Let K be a field. Let L/K be a finite extension of degree e which is generated by an element α with a = α^e ∈ K. If every eth root of unity in L is contained in K, then any sub extension L/L'/K is generated by α^d for some d | e.","statement_latex":"\\begin{reference}\n\\cite[Theorem 5.2]{Radical}\n\\end{reference}\nLet $K$ be a field. Let $L/K$ be a finite extension of degree $e$\nwhich is generated by an element $\\alpha$ with $a = \\alpha^e \\in K$.\nIf every $e$th root of unity in $L$ is contained in $K$, then \nany sub extension $L/L'/K$ is generated by $\\alpha^d$ for some $d | e$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Kummer extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXP","source_file":"fields.tex","source_line":3276,"source_end_line":3285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3276-L3285","statement_sha256":"8ea03f842a2bfdc43b1ebf9c09ce19b8069f17482bf7750bcebf2098429f4c8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":933,"rank":933,"depth":0,"x":1720.237,"y":190.175,"cluster":"fields-brauer-groups"},{"id":"stacks:09DY","tag":"09DY","title":"Artin-Schreier extensions · Lemma 09DY","summary":"Let L/K be a Galois extension of fields of characteristic p > 0 with Galois group Z/pZ. Then L = K[z] with z^p - z ∈ K.","statement_latex":"Let $L/K$ be a Galois extension of fields of characteristic $p > 0$\nwith Galois group $\\mathbf{Z}/p\\mathbf{Z}$. Then $L = K[z]$ with\n$z^p - z \\in K$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Artin-Schreier extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DY","source_file":"fields.tex","source_line":3331,"source_end_line":3336,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3331-L3336","statement_sha256":"2c8f886fe08d4bccb3791c828589799dcd9ed97bcfa5a5304eedce25ca738bd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":934,"rank":934,"depth":1,"x":1500.191,"y":322.059,"cluster":"fields-brauer-groups"},{"id":"stacks:030E","tag":"030E","title":"Transcendence · Definition 030E","summary":"Let K/k be a field extension. • A collection of elements (x_i)_i ∈ I of K is called algebraically independent over k if the map k[X_i; i∈ I] → K which maps X_i to x_i is injective. • The field of fractions of a polynomial ring k[x_i; i ∈ I] is denoted k(x_i; i∈ I). • A purely transcendental extension of k is any field extension K/k isomorphic to the field of fractions of a polynomial ring over k. • A transcendence basis of K/k is a collection of elements (x_i)_i ∈ I which…","statement_latex":"Let $K/k$ be a field extension.\n\\begin{enumerate}\n\\item A collection of elements $\\{x_i\\}_{i \\in I}$ of $K$ is called\n{\\it algebraically independent} over $k$ if the map\n$$\nk[X_i; i\\in I] \\longrightarrow K\n$$\nwhich maps $X_i$ to $x_i$ is injective.\n\\item The field of fractions of a polynomial ring\n$k[x_i; i \\in I]$ is denoted $k(x_i; i\\in I)$.\n\\item A {\\it purely transcendental extension} of $k$ is any\nfield extension $K/k$ isomorphic to the field of\nfractions of a polynomial ring over $k$.\n\\item A {\\it transcendence basis} of $K/k$ is a\ncollection of elements $\\{x_i\\}_{i \\in I}$ which are\nalgebraically independent over $k$ and such that\nthe extension $K/k(x_i; i\\in I)$ is algebraic.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Transcendence","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030E","source_file":"fields.tex","source_line":3368,"source_end_line":3388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3368-L3388","statement_sha256":"9e9b3995c69798264fe582c861b8149fa6d56b380012dad91c33401b90e70f54","origin":"The Stacks Project","memory_eligible":false,"source_rank":935,"rank":935,"depth":0,"x":1556.665,"y":98.88,"cluster":"fields-brauer-groups"},{"id":"stacks:030F","tag":"030F","title":"Transcendence · Lemma 030F","summary":"Let E/F be a field extension. A transcendence basis of E over F exists. Any two transcendence bases have the same cardinality.","statement_latex":"Let $E/F$ be a field extension. A transcendence basis of $E$ over $F$ exists.\nAny two transcendence bases have the same cardinality.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Transcendence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030F","source_file":"fields.tex","source_line":3397,"source_end_line":3401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3397-L3401","statement_sha256":"64be86bda5fe3b0b745e124faf980176c38b444be28509f85bcd78b1af450c33","origin":"The Stacks Project","memory_eligible":false,"source_rank":936,"rank":936,"depth":0,"x":1695.153,"y":296.433,"cluster":"fields-brauer-groups"},{"id":"stacks:030G","tag":"030G","title":"Transcendence · Definition 030G","summary":"Let K/k be a field extension. The transcendence degree of K over k is the cardinality of a transcendence basis of K over k. It is denoted trdeg_k(K).","statement_latex":"Let $K/k$ be a field extension.\nThe {\\it transcendence degree} of $K$ over $k$ is\nthe cardinality of a transcendence basis of $K$ over $k$.\nIt is denoted $\\text{trdeg}_k(K)$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Transcendence","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030G","source_file":"fields.tex","source_line":3473,"source_end_line":3479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3473-L3479","statement_sha256":"741804ccab99565059b5e93b9d9c14283bf9058b611649fd72c1616b48e00da4","origin":"The Stacks Project","memory_eligible":false,"source_rank":937,"rank":937,"depth":0,"x":1432.934,"y":229.022,"cluster":"fields-brauer-groups"},{"id":"stacks:030H","tag":"030H","title":"Transcendence · Lemma 030H","summary":"Let L/K/k be field extensions. Then trdeg_k(L) = trdeg_K(L) + trdeg_k(K).","statement_latex":"Let $L/K/k$ be field extensions.\nThen\n$$\n\\text{trdeg}_k(L) =\n\\text{trdeg}_K(L) +\n\\text{trdeg}_k(K).\n$$","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Transcendence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030H","source_file":"fields.tex","source_line":3481,"source_end_line":3490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3481-L3490","statement_sha256":"e4558b832981a737c280d3eb9f2a1f9cbbe247167943b0f151fc4e51866e0037","origin":"The Stacks Project","memory_eligible":false,"source_rank":938,"rank":938,"depth":0,"x":1681.661,"y":129.478,"cluster":"fields-brauer-groups"},{"id":"stacks:037I","tag":"037I","title":"Transcendence · Definition 037I","summary":"Let K/k be a field extension. • The algebraic closure of k in K is the subfield k' of K consisting of elements of K which are algebraic over k. • We say k is algebraically closed in K if every element of K which is algebraic over k is contained in k.","statement_latex":"Let $K/k$ be a field extension.\n\\begin{enumerate}\n\\item The {\\it algebraic closure of $k$ in $K$} is the subfield\n$k'$ of $K$ consisting of elements of $K$ which are algebraic over $k$.\n\\item We say $k$ is {\\it algebraically closed in $K$} if\nevery element of $K$ which is algebraic over $k$ is\ncontained in $k$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Transcendence","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037I","source_file":"fields.tex","source_line":3546,"source_end_line":3556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3546-L3556","statement_sha256":"70e2daf3e6c33850d2ded4a87bb8dfef7672e5a9f95f16824b10cf0b7424139e","origin":"The Stacks Project","memory_eligible":false,"source_rank":939,"rank":939,"depth":0,"x":1577.822,"y":345.009,"cluster":"fields-brauer-groups"},{"id":"stacks:0G1M","tag":"0G1M","title":"Transcendence · Lemma 0G1M","summary":"Let k'/k be a finite extension of fields. Let k'(x_1, …, x_r)/k(x_1, …, x_r) be the induced extension of purely transcendental extensions. Then [k'(x_1, …, x_r) : k(x_1, …, x_r)] = [k' : k] < ∞.","statement_latex":"Let $k'/k$ be a finite extension of fields. Let\n$k'(x_1, \\ldots, x_r)/k(x_1, \\ldots, x_r)$ be the\ninduced extension of purely transcendental extensions.\nThen $[k'(x_1, \\ldots, x_r) : k(x_1, \\ldots, x_r)] = [k' : k] < \\infty$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Transcendence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1M","source_file":"fields.tex","source_line":3558,"source_end_line":3564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3558-L3564","statement_sha256":"e87713c263bb3533022542c82d8c0c0e0727a915e53abb5247b9b82ae5318a75","origin":"The Stacks Project","memory_eligible":false,"source_rank":940,"rank":940,"depth":1,"x":1480.625,"y":126.157,"cluster":"fields-brauer-groups"},{"id":"stacks:037J","tag":"037J","title":"Transcendence · Lemma 037J","summary":"Let K/k be a finitely generated field extension. The algebraic closure of k in K is finite over k.","statement_latex":"Let $K/k$ be a finitely generated field extension.\nThe algebraic closure of $k$ in $K$ is finite over $k$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Transcendence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037J","source_file":"fields.tex","source_line":3583,"source_end_line":3587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3583-L3587","statement_sha256":"7c3dbe9975b1d52bc17246eb4ffa308eefaafb9835599312588f6cb447947304","origin":"The Stacks Project","memory_eligible":false,"source_rank":941,"rank":941,"depth":2,"x":1729.418,"y":232.869,"cluster":"fields-brauer-groups"},{"id":"stacks:09ID","tag":"09ID","title":"Linearly disjoint extensions · Definition 09ID","summary":"Consider a diagram vcenter xymatrix L ar[r] & Ω k ar[r] ar[u] & K ar[u] of field extensions. The compositum of K and L in Ω written KL is the smallest subfield of Ω containing both L and K.","statement_latex":"Consider a diagram\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\nL \\ar[r] & \\Omega \\\\\nk \\ar[r] \\ar[u] & K \\ar[u]\n}\n}\n\\end{equation}\nof field extensions. The {\\it compositum of $K$ and $L$ in $\\Omega$}\nwritten $KL$ is the smallest subfield of $\\Omega$ containing both\n$L$ and $K$.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Linearly disjoint extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ID","source_file":"fields.tex","source_line":3617,"source_end_line":3632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3617-L3632","statement_sha256":"98ed61316f332039295ed09eeeb7c0d65515263f5ab265d6e90ef87096101b7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":942,"rank":942,"depth":0,"x":1458.931,"y":295.633,"cluster":"fields-brauer-groups"},{"id":"stacks:09IF","tag":"09IF","title":"Linearly disjoint extensions · Definition 09IF","summary":"Consider a diagram of fields as in ([Tag 09IE]). We say that K and L are linearly disjoint over k in Ω if the map K ⊗_k L → KL, ∑ x_i ⊗ y_i ↦ ∑ x_i y_i is injective.","statement_latex":"Consider a diagram of fields as in (\\ref{equation-inside-omega}).\nWe say that $K$ and $L$ are {\\it linearly disjoint over $k$ in $\\Omega$}\nif the map\n$$\nK \\otimes_k L \\longrightarrow KL,\\quad\n\\sum x_i \\otimes y_i \\longmapsto \\sum x_i y_i\n$$\nis injective.","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Linearly disjoint extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IF","source_file":"fields.tex","source_line":3650,"source_end_line":3660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3650-L3660","statement_sha256":"e78a72cfa3442f5f86b1f10a85ac6973261ff55d61692db056ced36f4f906e15","origin":"The Stacks Project","memory_eligible":false,"source_rank":943,"rank":943,"depth":1,"x":1608.579,"y":94.978,"cluster":"fields-brauer-groups"},{"id":"stacks:030M","tag":"030M","title":"Linearly disjoint extensions · Lemma 030M","summary":"Let E/F be a normal algebraic field extension. There exist subextensions E / E_sep /F and E / E_insep / F such that • F ⊂ E_sep is Galois and E_sep ⊂ E is purely inseparable, • F ⊂ E_insep is purely inseparable and E_insep ⊂ E is Galois, • E = E_sep ⊗_F E_insep.","statement_latex":"Let $E/F$ be a normal algebraic field extension. There exist subextensions\n$E / E_{sep} /F$ and $E / E_{insep} / F$ such that\n\\begin{enumerate}\n\\item $F \\subset E_{sep}$ is Galois and $E_{sep} \\subset E$\nis purely inseparable,\n\\item $F \\subset E_{insep}$ is purely inseparable and $E_{insep} \\subset E$\nis Galois,\n\\item $E = E_{sep} \\otimes_F E_{insep}$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Linearly disjoint extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030M","source_file":"fields.tex","source_line":3665,"source_end_line":3676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3665-L3676","statement_sha256":"dda692afceb74cb2afcd22cc2215e873fd34b5f7782467bda27cdc96c54f096a","origin":"The Stacks Project","memory_eligible":false,"source_rank":944,"rank":944,"depth":5,"x":1659.816,"y":328.884,"cluster":"fields-brauer-groups"},{"id":"stacks:030J","tag":"030J","title":"Review · Definition 030J","summary":"Algebraic field extensions. • A field extension K/k is called algebraic if every element of K is algebraic over k. • An algebraic extension k'/k is called separable if every α ∈ k' is separable over k. • An algebraic extension k'/k is called purely inseparable if the characteristic of k is p > 0 and for every element α ∈ k' there exists a power q of p such that α^q ∈ k. • An algebraic extension k'/k is called normal if for every α ∈ k' the minimal polynomial P(T) ∈ k[T]…","statement_latex":"Algebraic field extensions.\n\\begin{enumerate}\n\\item A field extension $K/k$ is called {\\it algebraic}\nif every element of $K$ is algebraic over $k$.\n\\item An algebraic extension $k'/k$ is called {\\it separable}\nif every $\\alpha \\in k'$ is separable over $k$.\n\\item An algebraic\nextension $k'/k$ is called {\\it purely inseparable} if\nthe characteristic of $k$ is $p > 0$ and for every element\n$\\alpha \\in k'$ there exists a power $q$ of $p$ such that\n$\\alpha^q \\in k$.\n\\item An algebraic extension $k'/k$ is called {\\it normal}\nif for every $\\alpha \\in k'$ the minimal polynomial $P(T) \\in k[T]$\nof $\\alpha$ over $k$ splits completely into linear factors over $k'$.\n\\item An algebraic extension $k'/k$ is called {\\it Galois}\nif it is separable and normal.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Review","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030J","source_file":"fields.tex","source_line":3722,"source_end_line":3741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3722-L3741","statement_sha256":"3ea33c25b69d3fb9cc681d1cd9531f3d0d19210ec9db4e4baa48fbd80709b56f","origin":"The Stacks Project","memory_eligible":false,"source_rank":945,"rank":945,"depth":0,"x":1432.943,"y":184.847,"cluster":"fields-brauer-groups"},{"id":"stacks:031V","tag":"031V","title":"Review · Lemma 031V","summary":"Let K be a field of characteristic p > 0. Let L/K be a separable algebraic extension. Let α ∈ L. • If the coefficients of the minimal polynomial of α over K are pth powers in K then α is a pth power in L. • More generally, if P ∈ K[T] is a polynomial such that (a) α is a root of P, (b) P has pairwise distinct roots in an algebraic closure, and (c) all coefficients of P are pth powers, then α is a pth power in L.","statement_latex":"Let $K$ be a field of characteristic $p > 0$. Let $L/K$ be a separable\nalgebraic extension. Let $\\alpha \\in L$.\n\\begin{enumerate}\n\\item If the coefficients of the minimal polynomial of $\\alpha$\nover $K$ are $p$th powers in $K$ then $\\alpha$ is a $p$th\npower in $L$.\n\\item More generally, if $P \\in K[T]$ is a polynomial such that (a) $\\alpha$\nis a root of $P$, (b) $P$ has pairwise distinct roots in an algebraic closure,\nand (c) all coefficients of $P$ are $p$th powers, then $\\alpha$ is a\n$p$th power in $L$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Fields","chapter_id":"fields","section":"Review","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031V","source_file":"fields.tex","source_line":3746,"source_end_line":3759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/fields.tex#L3746-L3759","statement_sha256":"62c80f2bc978d6ea0f17eae6aeb283b269da2d1c3ebda58aa92b188f9130a1b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":946,"rank":946,"depth":1,"x":1717.299,"y":162.228,"cluster":"fields-brauer-groups"},{"id":"stacks:07JW","tag":"07JW","title":"Snake lemma · Lemma 07JW","summary":"[Cartan-Eilenberg] Given a commutative diagram xymatrix & X ar[r] ar[d]^α & Y ar[r] ar[d]^β & Z ar[r] ar[d]^γ & 0 0 ar[r] & U ar[r] & V ar[r] & W of abelian groups with exact rows, there is a canonical exact sequence Ker(α) → Ker(β) → Ker(γ) → Coker(α) → Coker(β) → Coker(γ) Moreover: if X → Y is injective, then the first map is injective; if V → W is surjective, then the last map is surjective.","statement_latex":"\\begin{reference}\n\\cite[III, Lemma 3.3]{Cartan-Eilenberg}\n\\end{reference}\nGiven a commutative diagram\n$$\n\\xymatrix{\n& X \\ar[r] \\ar[d]^\\alpha &\nY \\ar[r] \\ar[d]^\\beta &\nZ \\ar[r] \\ar[d]^\\gamma &\n0 \\\\\n0 \\ar[r] & U \\ar[r] & V \\ar[r] & W\n}\n$$\nof abelian groups with exact rows, there is a canonical exact sequence\n$$\n\\Ker(\\alpha) \\to \\Ker(\\beta) \\to \\Ker(\\gamma)\n\\to\n\\Coker(\\alpha) \\to \\Coker(\\beta) \\to \\Coker(\\gamma)\n$$\nMoreover: if $X \\to Y$ is injective, then the first map is\ninjective; if $V \\to W$ is surjective, then the last\nmap is surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Snake lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JW","source_file":"algebra.tex","source_line":303,"source_end_line":327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L303-L327","statement_sha256":"c74b6cc9fdd7737bdc8a6406b15a52520696c3b79c36b267807957579ebbb3d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":947,"rank":947,"depth":0,"x":2033.415,"y":220.0,"cluster":"commutative-algebra"},{"id":"stacks:0518","tag":"0518","title":"Finite modules and finitely presented modules · Definition 0518","summary":"Let R be a ring. Let M be an R-module. • We say M is a finite R-module, or a finitely generated R-module if there exist n ∈ N and x_1, …, x_n ∈ M such that every element of M is an R-linear combination of the x_i. Equivalently, this means there exists a surjection R^⊕ n → M for some n ∈ N. • We say M is a finitely presented R-module or an R-module of finite presentation if there exist integers n, m ∈ N and an exact sequence R^⊕ m → R^⊕ n → M → 0","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item We say $M$ is a {\\it finite $R$-module}, or a {\\it finitely generated\n$R$-module} if there exist $n \\in \\mathbf{N}$ and $x_1, \\ldots, x_n \\in M$\nsuch that every element of $M$ is an $R$-linear combination of the $x_i$.\nEquivalently, this means there exists a surjection\n$R^{\\oplus n} \\to M$ for some $n \\in \\mathbf{N}$.\n\\item We say $M$ is a {\\it finitely presented $R$-module} or an\n{\\it $R$-module of finite presentation} if there exist integers\n$n, m \\in \\mathbf{N}$ and an exact sequence\n$$\nR^{\\oplus m} \\longrightarrow R^{\\oplus n} \\longrightarrow M \\longrightarrow 0\n$$\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite modules and finitely presented modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0518","source_file":"algebra.tex","source_line":349,"source_end_line":365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L349-L365","statement_sha256":"4b975a0d3948b1cb05cb75b5adfe9106f5b56678966b8558db027fa82558cec1","origin":"The Stacks Project","memory_eligible":false,"source_rank":948,"rank":948,"depth":0,"x":2025.638,"y":223.356,"cluster":"commutative-algebra"},{"id":"stacks:07JX","tag":"07JX","title":"Finite modules and finitely presented modules · Lemma 07JX","summary":"Let R be a ring. Let α : R^⊕ n → M and β : N → M be module maps. If Im(α) ⊂ Im(β), then there exists an R-module map γ : R^⊕ n → N such that α = β ∘ γ.","statement_latex":"Let $R$ be a ring. Let $\\alpha : R^{\\oplus n} \\to M$ and $\\beta : N \\to M$ be\nmodule maps. If $\\Im(\\alpha) \\subset \\Im(\\beta)$, then there\nexists an $R$-module map $\\gamma : R^{\\oplus n} \\to N$ such that\n$\\alpha = \\beta \\circ \\gamma$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite modules and finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JX","source_file":"algebra.tex","source_line":374,"source_end_line":380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L374-L380","statement_sha256":"358ac00d8802f4652786e1279565651e01f4630f4c0b81e811b300fcddfa49de","origin":"The Stacks Project","memory_eligible":false,"source_rank":949,"rank":949,"depth":0,"x":2030.668,"y":213.61,"cluster":"commutative-algebra"},{"id":"stacks:0519","tag":"0519","title":"Finite modules and finitely presented modules · Lemma 0519","summary":"Let R be a ring. Let 0 → M_1 → M_2 → M_3 → 0 be a short exact sequence of R-modules. • If M_1 and M_3 are finite R-modules, then M_2 is a finite R-module. • If M_1 and M_3 are finitely presented R-modules, then M_2 is a finitely presented R-module. • If M_2 is a finite R-module, then M_3 is a finite R-module. • If M_2 is a finitely presented R-module and M_1 is a finite R-module, then M_3 is a finitely presented R-module. • If M_3 is a finitely presented R-module and M_2…","statement_latex":"Let $R$ be a ring.\nLet\n$$\n0 \\to M_1 \\to M_2 \\to M_3 \\to 0\n$$\nbe a short exact sequence of $R$-modules.\n\\begin{enumerate}\n\\item If $M_1$ and $M_3$ are finite $R$-modules, then $M_2$ is a finite\n$R$-module.\n\\item If $M_1$ and $M_3$ are finitely presented $R$-modules, then $M_2$\nis a finitely presented $R$-module.\n\\item If $M_2$ is a finite $R$-module, then $M_3$ is a finite $R$-module.\n\\item If $M_2$ is a finitely presented $R$-module and $M_1$ is a\nfinite $R$-module, then $M_3$ is a finitely presented $R$-module.\n\\item If $M_3$ is a finitely presented $R$-module and $M_2$ is a finite\n$R$-module, then $M_1$ is a finite $R$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite modules and finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0519","source_file":"algebra.tex","source_line":390,"source_end_line":409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L390-L409","statement_sha256":"14211ba0e56bccaef97e4d89541ed7d3a0f06321fea6e005fa3bc45b2b28ffdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":950,"rank":950,"depth":1,"x":2035.497,"y":226.023,"cluster":"commutative-algebra"},{"id":"stacks:00KZ","tag":"00KZ","title":"Finite modules and finitely presented modules · Lemma 00KZ","summary":"Finite modules have filtrations such that successive quotients are cyclic modules. Let R be a ring, and let M be a finite R-module. There exists a filtration by finite R-submodules 0 = M_0 ⊂ M_1 ⊂ … ⊂ M_n = M such that each quotient M_i/M_i - 1 is isomorphic to R/I_i for some ideal I_i of R.","statement_latex":"\\begin{slogan}\nFinite modules have filtrations such that successive quotients are\ncyclic modules.\n\\end{slogan}\nLet $R$ be a ring, and let $M$ be a finite $R$-module.\nThere exists a filtration by finite $R$-submodules\n$$\n0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M\n$$\nsuch that each quotient $M_i/M_{i - 1}$ is isomorphic\nto $R/I_i$ for some ideal $I_i$ of $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite modules and finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KZ","source_file":"algebra.tex","source_line":477,"source_end_line":490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L477-L490","statement_sha256":"373e5d2fa68c81de41832a6b2b211eeb4adbdf5978326d41b6491eefc37d13de","origin":"The Stacks Project","memory_eligible":false,"source_rank":951,"rank":951,"depth":0,"x":2019.912,"y":218.501,"cluster":"commutative-algebra"},{"id":"stacks:0560","tag":"0560","title":"Finite modules and finitely presented modules · Lemma 0560","summary":"Let R → S be a ring map. Let M be an S-module. If M is finite as an R-module, then M is finite as an S-module.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nIf $M$ is finite as an $R$-module, then $M$ is finite as an $S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite modules and finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0560","source_file":"algebra.tex","source_line":500,"source_end_line":505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L500-L505","statement_sha256":"7a7b7f907c2b23b98728d78f4f63c362ef63f8771c09a84eface8664873d9155","origin":"The Stacks Project","memory_eligible":false,"source_rank":952,"rank":952,"depth":0,"x":2039.557,"y":214.894,"cluster":"commutative-algebra"},{"id":"stacks:00F3","tag":"00F3","title":"Ring maps of finite type and of finite presentation · Definition 00F3","summary":"Let R → S be a ring map. • We say R → S is of finite type, or that S is a finite type R-algebra if there exist an n ∈ N and an surjection of R-algebras R[x_1, …, x_n] → S. • We say R → S is of finite presentation if there exist integers n, m ∈ N and polynomials f_1, …, f_m ∈ R[x_1, …, x_n] and an isomorphism of R-algebras R[x_1, …, x_n]/(f_1, …, f_m) ≅ S.","statement_latex":"Let $R \\to S$ be a ring map.\n\\begin{enumerate}\n\\item We say $R \\to S$ is of {\\it finite type}, or that {\\it $S$ is a finite\ntype $R$-algebra} if there exist an $n \\in \\mathbf{N}$ and an surjection\nof $R$-algebras $R[x_1, \\ldots, x_n] \\to S$.\n\\item We say $R \\to S$ is of {\\it finite presentation} if there\nexist integers $n, m \\in \\mathbf{N}$ and polynomials\n$f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$\nand an isomorphism of $R$-algebras\n$R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m) \\cong S$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ring maps of finite type and of finite presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00F3","source_file":"algebra.tex","source_line":517,"source_end_line":530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L517-L530","statement_sha256":"fc2fec276961fa6f06310253ba5839b25161a75cc71789932ecb0049adce4fb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":953,"rank":953,"depth":0,"x":2026.804,"y":229.988,"cluster":"commutative-algebra"},{"id":"stacks:00F4","tag":"00F4","title":"Ring maps of finite type and of finite presentation · Lemma 00F4","summary":"The notions finite type and finite presentation have the following permanence properties. • A composition of ring maps of finite type is of finite type. • A composition of ring maps of finite presentation is of finite presentation. • Given R → S' → S with R → S of finite type, then S' → S is of finite type. • Given R → S' → S, with R → S of finite presentation, and R → S' of finite type, then S' → S is of finite presentation.","statement_latex":"The notions finite type and finite presentation have the following\npermanence properties.\n\\begin{enumerate}\n\\item A composition of ring maps of finite type is of finite type.\n\\item A composition of ring maps of finite presentation is of finite\npresentation.\n\\item Given $R \\to S' \\to S$ with $R \\to S$ of finite type,\nthen $S' \\to S$ is of finite type.\n\\item Given $R \\to S' \\to S$, with $R \\to S$ of finite presentation,\nand $R \\to S'$ of finite type, then $S' \\to S$ is of finite presentation.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ring maps of finite type and of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00F4","source_file":"algebra.tex","source_line":539,"source_end_line":552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L539-L552","statement_sha256":"a8608c56e8618c9be744727cadb43baebc37b0be3026465acaadc341abdccb48","origin":"The Stacks Project","memory_eligible":false,"source_rank":954,"rank":954,"depth":0,"x":2023.904,"y":210.14,"cluster":"commutative-algebra"},{"id":"stacks:00R2","tag":"00R2","title":"Ring maps of finite type and of finite presentation · Lemma 00R2","summary":"Let R → S be a ring map of finite presentation. For any surjection α : R[x_1, …, x_n] → S the kernel of α is a finitely generated ideal in R[x_1, …, x_n].","statement_latex":"Let $R \\to S$ be a ring map of finite presentation.\nFor any surjection $\\alpha : R[x_1, \\ldots, x_n] \\to S$ the\nkernel of $\\alpha$ is a finitely generated ideal in $R[x_1, \\ldots, x_n]$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ring maps of finite type and of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00R2","source_file":"algebra.tex","source_line":565,"source_end_line":570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L565-L570","statement_sha256":"6e09c3765de8cf81fa815569341e7602013b000c24e85ceb61639bb9f05de5fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":955,"rank":955,"depth":0,"x":2043.226,"y":224.057,"cluster":"commutative-algebra"},{"id":"stacks:0561","tag":"0561","title":"Ring maps of finite type and of finite presentation · Lemma 0561","summary":"Let R → S be a ring map. Let M be an S-module. Assume R → S is of finite type and M is finitely presented as an R-module. Then M is finitely presented as an S-module.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nAssume $R \\to S$ is of finite type and\n$M$ is finitely presented as an $R$-module.\nThen $M$ is finitely presented as an $S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ring maps of finite type and of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0561","source_file":"algebra.tex","source_line":583,"source_end_line":590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L583-L590","statement_sha256":"9dc366893a009bf78f8cb5aee63530b0362efa6fe7d2fe5546e76ec057db83e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":956,"rank":956,"depth":1,"x":2016.241,"y":224.771,"cluster":"commutative-algebra"},{"id":"stacks:0563","tag":"0563","title":"Finite ring maps · Definition 0563","summary":"Let φ : R → S be a ring map. We say φ : R → S is finite if S is finite as an R-module.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. We say $\\varphi : R \\to S$ is\n{\\it finite} if $S$ is finite as an $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0563","source_file":"algebra.tex","source_line":635,"source_end_line":639,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L635-L639","statement_sha256":"d678f74f979454cf9adf7d7511603935bb87f62d92450d95c1886419de6ce731","origin":"The Stacks Project","memory_eligible":false,"source_rank":957,"rank":957,"depth":0,"x":2036.633,"y":208.094,"cluster":"commutative-algebra"},{"id":"stacks:00GJ","tag":"00GJ","title":"Finite ring maps · Lemma 00GJ","summary":"Let R → S be a finite ring map. Let M be an S-module. Then M is finite as an R-module if and only if M is finite as an S-module.","statement_latex":"Let $R \\to S$ be a finite ring map.\nLet $M$ be an $S$-module.\nThen $M$ is finite as an $R$-module if and only if $M$ is finite\nas an $S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GJ","source_file":"algebra.tex","source_line":641,"source_end_line":647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L641-L647","statement_sha256":"841a9ec84caa52f4fe73aac45aba3f24f0f8afc39266da6d3c4954643a2144da","origin":"The Stacks Project","memory_eligible":false,"source_rank":958,"rank":958,"depth":1,"x":2034.902,"y":233.127,"cluster":"commutative-algebra"},{"id":"stacks:00GL","tag":"00GL","title":"Finite ring maps · Lemma 00GL","summary":"Suppose that R → S and S → T are finite ring maps. Then R → T is finite.","statement_latex":"Suppose that $R \\to S$ and $S \\to T$ are finite ring maps.\nThen $R \\to T$ is finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GL","source_file":"algebra.tex","source_line":658,"source_end_line":662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L658-L662","statement_sha256":"25d4c6668fa518dfbc0bdc1f4dd972f4d91bcb22aac3c770c32104c04a13e532","origin":"The Stacks Project","memory_eligible":false,"source_rank":959,"rank":959,"depth":2,"x":2015.227,"y":212.808,"cluster":"commutative-algebra"},{"id":"stacks:0D46","tag":"0D46","title":"Finite ring maps · Lemma 0D46","summary":"Let φ : R → S be a ring map. • If φ is finite, then φ is of finite type. • If S is of finite presentation as an R-module, then φ is of finite presentation.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\n\\begin{enumerate}\n\\item If $\\varphi$ is finite, then $\\varphi$ is of finite type.\n\\item If $S$ is of finite presentation as an $R$-module, then\n$\\varphi$ is of finite presentation.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D46","source_file":"algebra.tex","source_line":671,"source_end_line":679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L671-L679","statement_sha256":"1c30e79ac0d885a6268f03c273a9c57b58aa3a89e53bf8d75be25e06ad0619e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":960,"rank":960,"depth":0,"x":2047.331,"y":216.799,"cluster":"commutative-algebra"},{"id":"stacks:00D4","tag":"00D4","title":"Colimits · Definition 00D4","summary":"Let (I, ≤) be a preordered set. A system (M_i, μ_ij) of R-modules over I consists of a family of R-modules (M_i)_i∈ I indexed by I and a family of R-module maps (μ_ij : M_i → M_j)_i ≤ j such that for all i ≤ j ≤ k μ_ii = id_M_i μ_ik = μ_jk∘ μ_ij We say (M_i, μ_ij) is a directed system if I is a directed set.","statement_latex":"Let $(I, \\leq)$ be a preordered set.\nA {\\it system $(M_i, \\mu_{ij})$ of $R$-modules over $I$}\nconsists of a family of $R$-modules $\\{M_i\\}_{i\\in I}$ indexed\nby $I$ and a family of $R$-module maps $\\{\\mu_{ij} : M_i \\to M_j\\}_{i \\leq j}$\nsuch that for all $i \\leq j \\leq k$\n$$\n\\mu_{ii} = \\text{id}_{M_i}\\quad\n\\mu_{ik} = \\mu_{jk}\\circ \\mu_{ij}\n$$\nWe say $(M_i, \\mu_{ij})$ is a {\\it directed system} if $I$ is a directed set.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00D4","source_file":"algebra.tex","source_line":714,"source_end_line":726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L714-L726","statement_sha256":"b45ca1be714ae5a6fcffc6872a6e3dc5b6b7382ca5cc9e506fc38c65dfa237f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":961,"rank":961,"depth":0,"x":2019.423,"y":232.637,"cluster":"commutative-algebra"},{"id":"stacks:00D5","tag":"00D5","title":"Colimits · Lemma 00D5","summary":"Let (M_i, μ_ij) be a system of R-modules over the preordered set I. The colimit of the system (M_i, μ_ij) is the quotient R-module (bigoplus_i∈ I M_i) /Q where Q is the R-submodule generated by all elements iota_i(x_i) - iota_j(μ_ij(x_i)) where iota_i : M_i → bigoplus_i∈ I M_i is the natural inclusion. We denote the colimit M = colim_i M_i. We denote π : bigoplus_i∈ I M_i → M the projection map and φ_i = π ∘ iota_i : M_i → M.","statement_latex":"Let $(M_i, \\mu_{ij})$ be a system of $R$-modules over the preordered set $I$.\nThe colimit of the system $(M_i, \\mu_{ij})$ is the quotient $R$-module\n$(\\bigoplus_{i\\in I} M_i) /Q$ where $Q$ is the\n$R$-submodule generated by all elements\n$$\n\\iota_i(x_i) - \\iota_j(\\mu_{ij}(x_i))\n$$\nwhere $\\iota_i : M_i \\to \\bigoplus_{i\\in I} M_i$\nis the natural inclusion. We denote the colimit\n$M = \\colim_i M_i$. We denote\n$\\pi : \\bigoplus_{i\\in I} M_i \\to M$ the\nprojection map and\n$\\phi_i = \\pi \\circ \\iota_i : M_i \\to M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00D5","source_file":"algebra.tex","source_line":736,"source_end_line":751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L736-L751","statement_sha256":"2a424da4c30531136264c28d8fa51cea61559a2f757a69051d37204c2d7f4bde","origin":"The Stacks Project","memory_eligible":false,"source_rank":962,"rank":962,"depth":1,"x":2027.557,"y":204.161,"cluster":"commutative-algebra"},{"id":"stacks:00D6","tag":"00D6","title":"Colimits · Lemma 00D6","summary":"Let (M_i, μ_ij) be a system of R-modules over the preordered set I. Assume that I is directed. The colimit of the system (M_i, μ_ij) is canonically isomorphic to the module M defined as follows: • as a set let M = (coprod_i ∈ I M_i)/sim where for m ∈ M_i and m' ∈ M_i' we have m sim m' ⇔ μ_ij(m) = μ_i'j(m') for some j ≥ i, i' • as an abelian group for m ∈ M_i and m' ∈ M_i' we define the sum of the classes of m and m' in M to be the class of μ_ij(m) + μ_i'j(m') where j ∈ I…","statement_latex":"Let $(M_i, \\mu_{ij})$ be a system of $R$-modules over the\npreordered set $I$. Assume that $I$ is directed.\nThe colimit of the system $(M_i, \\mu_{ij})$ is canonically\nisomorphic to the module $M$ defined as follows:\n\\begin{enumerate}\n\\item as a set let\n$$\nM = \\left(\\coprod\\nolimits_{i \\in I} M_i\\right)/\\sim\n$$\nwhere for $m \\in M_i$ and $m' \\in M_{i'}$ we have\n$$\nm \\sim m' \\Leftrightarrow\n\\mu_{ij}(m) = \\mu_{i'j}(m')\\text{ for some }j \\geq i, i'\n$$\n\\item as an abelian group for $m \\in M_i$ and $m' \\in M_{i'}$\nwe define the sum of the classes of $m$ and $m'$ in $M$\nto be the class of $\\mu_{ij}(m) + \\mu_{i'j}(m')$ where\n$j \\in I$ is any index with $i \\leq j$ and $i' \\leq j$, and\n\\item as an $R$-module define for $m \\in M_i$ and $x \\in R$\nthe product of $x$ and the class of $m$ in $M$ to be the\nclass of $xm$ in $M$.\n\\end{enumerate}\nThe canonical maps $\\phi_i : M_i \\to M$ are induced by the canonical\nmaps $M_i \\to \\coprod_{i \\in I} M_i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00D6","source_file":"algebra.tex","source_line":777,"source_end_line":803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L777-L803","statement_sha256":"9ebebcef0862962e4b8bf8d35826540d8554a0fe31e6820826bbbe523bb3f394","origin":"The Stacks Project","memory_eligible":false,"source_rank":963,"rank":963,"depth":0,"x":2045.001,"y":230.62,"cluster":"commutative-algebra"},{"id":"stacks:00D7","tag":"00D7","title":"Colimits · Lemma 00D7","summary":"Let (M_i, μ_ij) be a directed system. Let M = colim M_i with μ_i : M_i → M. Then, μ_i(x_i) = 0 for x_i ∈ M_i if and only if there exists j ≥ i such that μ_ij(x_i) = 0.","statement_latex":"Let $(M_i, \\mu_{ij})$ be a directed system.\nLet $M = \\colim M_i$ with $\\mu_i : M_i \\to M$.\nThen, $\\mu_i(x_i) = 0$ for $x_i \\in M_i$ if and only if\nthere exists $j \\geq i$ such that $\\mu_{ij}(x_i) = 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00D7","source_file":"algebra.tex","source_line":810,"source_end_line":816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L810-L816","statement_sha256":"c072793e6c324ee472b2fed29ab4e6e4e61c76278f4c372fbd69c39aa7cbb03d","origin":"The Stacks Project","memory_eligible":false,"source_rank":964,"rank":964,"depth":1,"x":2009.814,"y":220.701,"cluster":"commutative-algebra"},{"id":"stacks:00D9","tag":"00D9","title":"Colimits · Definition 00D9","summary":"Let (M_i, μ_ij), (N_i, ν_ij) be systems of R-modules over the same preordered set I. A homomorphism of systems Φ from (M_i, μ_ij) to (N_i, ν_ij) is by definition a family of R-module homomorphisms φ_i : M_i → N_i such that φ_j ∘ μ_ij = ν_ij ∘ φ_i for all i ≤ j.","statement_latex":"Let $(M_i, \\mu_{ij})$, $(N_i, \\nu_{ij})$ be\nsystems of $R$-modules over the same preordered set $I$.\nA {\\it homomorphism of systems} $\\Phi$ from $(M_i, \\mu_{ij})$ to\n$(N_i, \\nu_{ij})$ is by definition a family of $R$-module homomorphisms\n$\\phi_i : M_i \\to N_i$\nsuch that $\\phi_j \\circ \\mu_{ij} = \\nu_{ij} \\circ \\phi_i$\nfor all $i \\leq j$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00D9","source_file":"algebra.tex","source_line":843,"source_end_line":852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L843-L852","statement_sha256":"98482b1b80d5c4b2dc0f355a9701334c3f2904954b5384266772528ba902f759","origin":"The Stacks Project","memory_eligible":false,"source_rank":965,"rank":965,"depth":0,"x":2044.724,"y":207.692,"cluster":"commutative-algebra"},{"id":"stacks:00DA","tag":"00DA","title":"Colimits · Lemma 00DA","summary":"Let (M_i, μ_ij), (N_i, ν_ij) be systems of R-modules over the same preordered set. A morphism of systems Φ = (φ_i) from (M_i, μ_ij) to (N_i, ν_ij) induces a unique homomorphism colim φ_i : colim M_i → colim N_i such that xymatrix M_i ar[r] ar[d]_φ_i & colim M_i ar[d]^colim φ_i N_i ar[r] & colim N_i commutes for all i ∈ I.","statement_latex":"Let $(M_i, \\mu_{ij})$, $(N_i, \\nu_{ij})$ be\nsystems of $R$-modules over the same preordered set.\nA morphism of systems $\\Phi = (\\phi_i)$ from $(M_i, \\mu_{ij})$ to\n$(N_i, \\nu_{ij})$ induces a unique homomorphism\n$$\n\\colim \\phi_i : \\colim M_i \\longrightarrow \\colim N_i\n$$\nsuch that\n$$\n\\xymatrix{\nM_i \\ar[r] \\ar[d]_{\\phi_i} & \\colim M_i \\ar[d]^{\\colim \\phi_i} \\\\\nN_i \\ar[r] & \\colim N_i\n}\n$$\ncommutes for all $i \\in I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DA","source_file":"algebra.tex","source_line":862,"source_end_line":879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L862-L879","statement_sha256":"41d45d4aca7f640f671e95ef7664cfdb0d7639a04536f96b8e732b39c7e91910","origin":"The Stacks Project","memory_eligible":false,"source_rank":966,"rank":966,"depth":2,"x":2029.015,"y":237.895,"cluster":"commutative-algebra"},{"id":"stacks:00DB","tag":"00DB","title":"Colimits · Lemma 00DB","summary":"Filtered colimits are exact. Directed colimits are exact. Let I be a directed set. Let (L_i, λ_ij), (M_i, μ_ij), and (N_i, ν_ij) be systems of R-modules over I. Let φ_i : L_i → M_i and ψ_i : M_i → N_i be morphisms of systems over I. Assume that for all i ∈ I the sequence of R-modules xymatrix L_i ar[r]^φ_i & M_i ar[r]^ψ_i & N_i is a complex with homology H_i. Then the R-modules H_i form a system over I, the sequence of R-modules xymatrix colim_i L_i ar[r]^φ & colim_i M_i…","statement_latex":"\\begin{slogan}\nFiltered colimits are exact. Directed colimits are exact.\n\\end{slogan}\nLet $I$ be a directed set.\nLet $(L_i, \\lambda_{ij})$, $(M_i, \\mu_{ij})$, and\n$(N_i, \\nu_{ij})$ be systems of $R$-modules over $I$.\nLet $\\varphi_i : L_i \\to M_i$ and $\\psi_i : M_i \\to N_i$ be\nmorphisms of systems over $I$. Assume that for all $i \\in I$ the\nsequence of $R$-modules\n$$\n\\xymatrix{\nL_i \\ar[r]^{\\varphi_i} &\nM_i \\ar[r]^{\\psi_i} &\nN_i\n}\n$$\nis a complex with homology $H_i$.\nThen the $R$-modules $H_i$ form a system over $I$,\nthe sequence of $R$-modules\n$$\n\\xymatrix{\n\\colim_i L_i \\ar[r]^\\varphi &\n\\colim_i M_i \\ar[r]^\\psi &\n\\colim_i N_i\n}\n$$\nis a complex as well, and denoting $H$ its homology we have\n$$\nH = \\colim_i H_i.\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DB","source_file":"algebra.tex","source_line":907,"source_end_line":939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L907-L939","statement_sha256":"46f39a0145510e5a5cedf6ca6f2164439af7763d1d0dff73c954e439407e38b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":967,"rank":967,"depth":1,"x":2015.99,"y":205.897,"cluster":"commutative-algebra"},{"id":"stacks:04B0","tag":"04B0","title":"Colimits · Lemma 04B0","summary":"Let I be an index category satisfying the assumptions of Categories, Lemma [Tag 002X]. Then taking colimits of diagrams of abelian groups over I is exact (i.e., the analogue of Lemma [Tag 00DB] holds in this situation).","statement_latex":"Let $\\mathcal{I}$ be an index category satisfying the assumptions of\nCategories, Lemma \\ref{categories-lemma-split-into-directed}.\nThen taking colimits of diagrams of abelian groups over $\\mathcal{I}$\nis exact (i.e., the analogue of\nLemma \\ref{lemma-directed-colimit-exact}\nholds in this situation).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04B0","source_file":"algebra.tex","source_line":1000,"source_end_line":1008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1000-L1008","statement_sha256":"d1b8005ed14474857b6a1c4eb91528e38282ec26c84b05eea74a98b804aa87b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":968,"rank":968,"depth":2,"x":2052.194,"y":222.508,"cluster":"commutative-algebra"},{"id":"stacks:00CN","tag":"00CN","title":"Localization · Definition 00CN","summary":"Let R be a ring, S a subset of R. We say S is a multiplicative subset of R if 1∈ S and S is closed under multiplication, i.e., s, s' ∈ S ⇒ ss' ∈ S.","statement_latex":"Let $R$ be a ring, $S$ a subset of $R$.\nWe say $S$ is a {\\it multiplicative subset of $R$} if\n$1\\in S$ and $S$ is closed\nunder multiplication, i.e., $s, s' \\in S \\Rightarrow ss' \\in S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CN","source_file":"algebra.tex","source_line":1041,"source_end_line":1047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1041-L1047","statement_sha256":"c70f49f048e634fb7c14654c2cabed0b94efbeda058dfe66a7e6cbf2d278c87f","origin":"The Stacks Project","memory_eligible":false,"source_rank":969,"rank":969,"depth":0,"x":2011.195,"y":230.99,"cluster":"commutative-algebra"},{"id":"stacks:00CO","tag":"00CO","title":"Localization · Definition 00CO","summary":"This ring is called the localization of A with respect to S.","statement_latex":"This ring is called the {\\it localization of $A$ with respect to $S$}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CO","source_file":"algebra.tex","source_line":1067,"source_end_line":1070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1067-L1070","statement_sha256":"076c3abe26c5583dcf17929f833c95a6160602e22a616f1e371ec04033d0e5bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":970,"rank":970,"depth":0,"x":2035.139,"y":200.813,"cluster":"commutative-algebra"},{"id":"stacks:00CP","tag":"00CP","title":"Localization · Proposition 00CP","summary":"Let f : A → B be a ring map that sends every element in S to a unit of B. Then there is a unique homomorphism g : S^-1A → B such that the following diagram commutes. xymatrix A ar[rr]^f ar[dr] & & B & S^-1A ar[ur]_g","statement_latex":"Let $f : A \\to B$ be a ring map that sends every element in $S$ to a unit\nof $B$. Then there is a unique homomorphism $g : S^{-1}A \\to B$ such\nthat the following diagram commutes.\n$$\n\\xymatrix{\nA \\ar[rr]^{f} \\ar[dr] & & B \\\\\n& S^{-1}A \\ar[ur]_g\n}\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CP","source_file":"algebra.tex","source_line":1083,"source_end_line":1094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1083-L1094","statement_sha256":"d946d503d0def792120812feb98bb1072dae98a64b47b2db50d843c7ccb4e181","origin":"The Stacks Project","memory_eligible":false,"source_rank":971,"rank":971,"depth":0,"x":2041.885,"y":237.423,"cluster":"commutative-algebra"},{"id":"stacks:00CQ","tag":"00CQ","title":"Localization · Lemma 00CQ","summary":"The localization S^-1A is the zero ring if and only if 0∈ S.","statement_latex":"The localization $S^{-1}A$ is the zero ring if and only if $0\\in S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CQ","source_file":"algebra.tex","source_line":1110,"source_end_line":1113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1110-L1113","statement_sha256":"74bcfa1f84a20e7fcddac2cda6f6e02b8cfffbc84362035d2b1deda9b5822642","origin":"The Stacks Project","memory_eligible":false,"source_rank":972,"rank":972,"depth":0,"x":2006.766,"y":213.774,"cluster":"commutative-algebra"},{"id":"stacks:07JY","tag":"07JY","title":"Localization · Lemma 07JY","summary":"Let R be a ring. Let S ⊂ R be a multiplicative subset. The category of S^-1R-modules is equivalent to the category of R-modules N with the property that every s ∈ S acts as an automorphism on N.","statement_latex":"Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.\nThe category of $S^{-1}R$-modules is equivalent to the category\nof $R$-modules $N$ with the property that every $s \\in S$ acts as\nan automorphism on $N$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JY","source_file":"algebra.tex","source_line":1120,"source_end_line":1126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1120-L1126","statement_sha256":"606e1118302e4959302d89196e42a9c8f11bfb277ab027e1adc2432b3beb290a","origin":"The Stacks Project","memory_eligible":false,"source_rank":973,"rank":973,"depth":0,"x":2052.569,"y":211.241,"cluster":"commutative-algebra"},{"id":"stacks:07JZ","tag":"07JZ","title":"Localization · Definition 07JZ","summary":"The S^-1A-module S^-1M is called the localization of M with respect to S.","statement_latex":"The $S^{-1}A$-module $S^{-1}M$ is called the {\\it localization} of $M$\nwith respect to $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JZ","source_file":"algebra.tex","source_line":1158,"source_end_line":1162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1158-L1162","statement_sha256":"b63a79fdc7cafe2145b3329edabd11d800cdc25758cae002354a0e4704cdf24e","origin":"The Stacks Project","memory_eligible":false,"source_rank":974,"rank":974,"depth":0,"x":2020.222,"y":239.625,"cluster":"commutative-algebra"},{"id":"stacks:07K0","tag":"07K0","title":"Localization · Lemma 07K0","summary":"Let R be a ring. Let S ⊂ R a multiplicative subset. Let M, N be R-modules. Assume all the elements of S act as automorphisms on N. Then the canonical map Hom_R(S^-1M, N) → Hom_R(M, N) induced by the localization map, is an isomorphism.","statement_latex":"Let $R$ be a ring. Let $S \\subset R$ a multiplicative subset. Let $M$, $N$\nbe $R$-modules. Assume all the elements of $S$ act as automorphisms on $N$.\nThen the canonical map\n$$\n\\Hom_R(S^{-1}M, N) \\longrightarrow \\Hom_R(M, N)\n$$\ninduced by the localization map, is an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07K0","source_file":"algebra.tex","source_line":1170,"source_end_line":1179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1170-L1179","statement_sha256":"ec208a9c7dc4e60b9aac26b171d746851b3205023d5dc041ec96345efb340844","origin":"The Stacks Project","memory_eligible":false,"source_rank":975,"rank":975,"depth":0,"x":2021.274,"y":199.621,"cluster":"commutative-algebra"},{"id":"stacks:00CR","tag":"00CR","title":"Localization · Lemma 00CR","summary":"Let R be a ring. Let S ⊂ R be a multiplicative subset. Let M be an R-module. Then S^-1M = colim_f ∈ S M_f where the preorder on S is given by f ≥ f' ⇔ f = f'f\" for some f\" ∈ R in which case the map M_f' → M_f is given by m/(f')^e ↦ m(f\")^e/f^e.","statement_latex":"Let $R$ be a ring.\nLet $S \\subset R$ be a multiplicative subset.\nLet $M$ be an $R$-module.\nThen\n$$\nS^{-1}M = \\colim_{f \\in S} M_f\n$$\nwhere the preorder on $S$ is given by\n$f \\geq f' \\Leftrightarrow f = f'f''$ for some $f'' \\in R$\nin which case the map $M_{f'} \\to M_f$ is given\nby $m/(f')^e \\mapsto m(f'')^e/f^e$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CR","source_file":"algebra.tex","source_line":1242,"source_end_line":1255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1242-L1255","statement_sha256":"3326234499af318e5eeb0d295115e806dd442988e81dcb542df747d2465a2ceb","origin":"The Stacks Project","memory_eligible":false,"source_rank":976,"rank":976,"depth":1,"x":2053.22,"y":230.251,"cluster":"commutative-algebra"},{"id":"stacks:02C6","tag":"02C6","title":"Localization · Proposition 02C6","summary":"Let overlineS be the image of S in S'^-1A, then (SS')^-1A is isomorphic to overlineS^-1(S'^-1A).","statement_latex":"Let $\\overline{S}$ be the image of $S$ in $S'^{-1}A$, then\n$(SS')^{-1}A$ is isomorphic to $\\overline{S}^{-1}(S'^{-1}A)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02C6","source_file":"algebra.tex","source_line":1275,"source_end_line":1279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1275-L1279","statement_sha256":"1feff366d5e90170d928b7d703398b007ce24b7d1363a12f8c51a7256313b254","origin":"The Stacks Project","memory_eligible":false,"source_rank":977,"rank":977,"depth":0,"x":2004.209,"y":225.711,"cluster":"commutative-algebra"},{"id":"stacks:02C7","tag":"02C7","title":"Localization · Proposition 02C7","summary":"View S'^-1M as an A-module, then S^-1(S'^-1M) is isomorphic to (SS')^-1M.","statement_latex":"View $S'^{-1}M$ as an $A$-module, then $S^{-1}(S'^{-1}M)$ is\nisomorphic to $(SS')^{-1}M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02C7","source_file":"algebra.tex","source_line":1301,"source_end_line":1305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1301-L1305","statement_sha256":"fab876441d2488b108ce6e1040a4495e072d9dcee23c8effe4c9f4a8e3d08e2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":978,"rank":978,"depth":0,"x":2044.66,"y":200.849,"cluster":"commutative-algebra"},{"id":"stacks:00CS","tag":"00CS","title":"Localization · Proposition 00CS","summary":"Localization is exact. Let Lxrightarrowu Mxrightarrowv N be an exact sequence of R-modules. Then S^-1L → S^-1M → S^-1N is also exact.","statement_latex":"\\begin{slogan}\nLocalization is exact.\n\\end{slogan}\nLet $L\\xrightarrow{u} M\\xrightarrow{v} N$ be an exact sequence\nof $R$-modules. Then\n$S^{-1}L \\to S^{-1}M \\to S^{-1}N$ is also exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CS","source_file":"algebra.tex","source_line":1335,"source_end_line":1343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1335-L1343","statement_sha256":"118401cd4899a3d8bf7f0d1555d88567eda73edb4e5a58bf6d6031240fdf1747","origin":"The Stacks Project","memory_eligible":false,"source_rank":979,"rank":979,"depth":0,"x":2034.663,"y":242.793,"cluster":"commutative-algebra"},{"id":"stacks:02C8","tag":"02C8","title":"Localization · Lemma 02C8","summary":"Localization respects quotients, i.e. if N is a submodule of M, then S^-1(M/N)≃ (S^-1M)/(S^-1N).","statement_latex":"Localization respects quotients, i.e. if $N$ is a submodule of\n$M$, then $S^{-1}(M/N)\\simeq (S^{-1}M)/(S^{-1}N)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02C8","source_file":"algebra.tex","source_line":1355,"source_end_line":1359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1355-L1359","statement_sha256":"5a03f40eebd9ab9b76207d1ccb6ed935bf1b8c011498ce567c7bdb2b3065a12a","origin":"The Stacks Project","memory_eligible":false,"source_rank":980,"rank":980,"depth":0,"x":2007.9,"y":205.623,"cluster":"commutative-algebra"},{"id":"stacks:00CT","tag":"00CT","title":"Localization · Proposition 00CT","summary":"Let I be an ideal of A, S a multiplicative set of A. Then S^-1I is an ideal of S^-1A and overlineS^-1(A/I) is isomorphic to S^-1A/S^-1I, where overlineS is the image of S in A/I.","statement_latex":"Let $I$ be an ideal of $A$, $S$ a multiplicative set of $A$. Then\n$S^{-1}I$ is an ideal of $S^{-1}A$ and $\\overline{S}^{-1}(A/I)$ is\nisomorphic to $S^{-1}A/S^{-1}I$, where $\\overline{S}$ is\nthe image of $S$ in $A/I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CT","source_file":"algebra.tex","source_line":1379,"source_end_line":1385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1379-L1385","statement_sha256":"c5d9d3bf0e29d70d5e350933a6b61ea9948a31678d2b14ab2caaa0911424e828","origin":"The Stacks Project","memory_eligible":false,"source_rank":981,"rank":981,"depth":0,"x":2058.27,"y":218.033,"cluster":"commutative-algebra"},{"id":"stacks:00CU","tag":"00CU","title":"Localization · Lemma 00CU","summary":"Any submodule N' of S^-1M is of the form S^-1N for some N⊂ M. Indeed one can take N to be the inverse image of N' in M.","statement_latex":"Any submodule $N'$ of $S^{-1}M$ is of the form $S^{-1}N$ for some\n$N\\subset M$. Indeed one can take $N$ to be the inverse image of\n$N'$ in $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CU","source_file":"algebra.tex","source_line":1420,"source_end_line":1425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1420-L1425","statement_sha256":"c299bb857f6686daad39853d08740dfe9dcfd8fca14f17d01d757a902df94b9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":982,"rank":982,"depth":0,"x":2010.459,"y":237.743,"cluster":"commutative-algebra"},{"id":"stacks:02C9","tag":"02C9","title":"Localization · Lemma 02C9","summary":"Ideals in the localization of a ring are localizations of ideals. Each ideal I' of S^-1A takes the form S^-1I, where one can take I to be the inverse image of I' in A.","statement_latex":"\\begin{slogan}\nIdeals in the localization of a ring are localizations of ideals.\n\\end{slogan}\nEach ideal $I'$ of $S^{-1}A$ takes the form $S^{-1}I$, where one can\ntake $I$ to be the inverse image of $I'$ in $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02C9","source_file":"algebra.tex","source_line":1440,"source_end_line":1447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1440-L1447","statement_sha256":"7ec9f05b364ebdb99779fc41f7250a6008ebf224d933b14c236dc5f97c38194a","origin":"The Stacks Project","memory_eligible":false,"source_rank":983,"rank":983,"depth":1,"x":2030.142,"y":195.491,"cluster":"commutative-algebra"},{"id":"stacks:0582","tag":"0582","title":"Internal Hom · Lemma 0582","summary":"Exactness and Hom_R. Let R be a ring. Let M_1, M_2, M_3 be R-modules. Let M_1 → M_2 and M_2 → M_3 be R-module maps. • M_1 → M_2 → M_3 → 0 is exact if and only if 0 → Hom_R(M_3, N) → Hom_R(M_2, N) → Hom_R(M_1, N) is exact for all R-modules N. • 0 → M_1 → M_2 → M_3 is exact if and only if 0 → Hom_R(N, M_1) → Hom_R(N, M_2) → Hom_R(N, M_3) is exact for all R-modules N.","statement_latex":"Exactness and $\\Hom_R$. Let $R$ be a ring. Let $M_1$, $M_2$, $M_3$\nbe $R$-modules. Let $M_1 \\to M_2$ and $M_2 \\to M_3$ be $R$-module maps.\n\\begin{enumerate}\n\\item $M_1 \\to M_2 \\to M_3 \\to 0$ is exact if and only if\n$0 \\to \\Hom_R(M_3, N) \\to \\Hom_R(M_2, N) \\to \\Hom_R(M_1, N)$\nis exact for all $R$-modules $N$.\n\\item $0 \\to M_1 \\to M_2 \\to M_3$ is exact if and only if\n$0 \\to \\Hom_R(N, M_1) \\to \\Hom_R(N, M_2) \\to \\Hom_R(N, M_3)$\nis exact for all $R$-modules $N$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0582","source_file":"algebra.tex","source_line":1495,"source_end_line":1507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1495-L1507","statement_sha256":"1045168e2828b02f8045352e117025ccbae35c7faec5e8207dbb0d665f9e46e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":984,"rank":984,"depth":0,"x":2049.871,"y":238.4,"cluster":"commutative-algebra"},{"id":"stacks:0583","tag":"0583","title":"Internal Hom · Lemma 0583","summary":"Let R be a ring. Let M be a finitely presented R-module. Let N be an R-module. • For f ∈ R we have Hom_R(M, N)_f = Hom_R_f(M_f, N_f) = Hom_R(M_f, N_f), • for a multiplicative subset S of R we have S^-1Hom_R(M, N) = Hom_S^-1R(S^-1M, S^-1N) = Hom_R(S^-1M, S^-1N).","statement_latex":"Let $R$ be a ring. Let $M$ be a finitely presented $R$-module.\nLet $N$ be an $R$-module.\n\\begin{enumerate}\n\\item For $f \\in R$ we have\n$\\Hom_R(M, N)_f = \\Hom_{R_f}(M_f, N_f) = \\Hom_R(M_f, N_f)$,\n\\item for a multiplicative subset $S$ of $R$ we have\n$$\nS^{-1}\\Hom_R(M, N) = \\Hom_{S^{-1}R}(S^{-1}M, S^{-1}N) =\n\\Hom_R(S^{-1}M, S^{-1}N).\n$$\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0583","source_file":"algebra.tex","source_line":1513,"source_end_line":1526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1513-L1526","statement_sha256":"952863e0958a3ee84a652224f93ab5cb636f73779e13b3bf5185f2750e365517","origin":"The Stacks Project","memory_eligible":false,"source_rank":985,"rank":985,"depth":1,"x":2000.161,"y":217.677,"cluster":"commutative-algebra"},{"id":"stacks:0G8N","tag":"0G8N","title":"Characterizing finite and finitely presented modules · Lemma 0G8N","summary":"Let R be a ring. Let N be an R-module. The following are equivalent • N is a finite R-module, • for any filtered colimit M = colim M_i of R-modules the map colim Hom_R(N, M_i) → Hom_R(N, M) is injective.","statement_latex":"Let $R$ be a ring. Let $N$ be an $R$-module. The following are equivalent\n\\begin{enumerate}\n\\item $N$ is a finite $R$-module,\n\\item for any filtered colimit $M = \\colim M_i$ of $R$-modules the map\n$\\colim \\Hom_R(N, M_i) \\to \\Hom_R(N, M)$ is injective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing finite and finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8N","source_file":"algebra.tex","source_line":1588,"source_end_line":1596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1588-L1596","statement_sha256":"adcad16ebb0630f69b73096d74ec275931e1fc0cb7e47e0b4fbb358cd84e262d","origin":"The Stacks Project","memory_eligible":false,"source_rank":986,"rank":986,"depth":0,"x":2054.178,"y":204.586,"cluster":"commutative-algebra"},{"id":"stacks:07N8","tag":"07N8","title":"Characterizing finite and finitely presented modules · Definition 07N8","summary":"Let R be a ring. Let M be an R-module. Let n ≥ 0 and x_i ∈ M for i = 1, …, n. A relation between x_1, …, x_n in M is a sequence of elements f_1, …, f_n ∈ R such that ∑_i = 1, …, n f_i x_i = 0.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $n \\geq 0$ and $x_i \\in M$ for $i = 1, \\ldots, n$.\nA {\\it relation} between $x_1, \\ldots, x_n$ in $M$ is a\nsequence of elements $f_1, \\ldots, f_n \\in R$ such that\n$\\sum_{i = 1, \\ldots, n} f_i x_i = 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing finite and finitely presented modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07N8","source_file":"algebra.tex","source_line":1620,"source_end_line":1627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1620-L1627","statement_sha256":"682889a48b30176daac0f5e494a1d914519d69b0f2facc8c0a5746e972b07751","origin":"The Stacks Project","memory_eligible":false,"source_rank":987,"rank":987,"depth":0,"x":2024.499,"y":245.401,"cluster":"commutative-algebra"},{"id":"stacks:00HA","tag":"00HA","title":"Characterizing finite and finitely presented modules · Lemma 00HA","summary":"Let R be a ring and let M be an R-module. Then M is the colimit of a directed system (M_i, μ_ij) of R-modules with all M_i finitely presented R-modules.","statement_latex":"Let $R$ be a ring and let $M$ be an $R$-module.\nThen $M$ is the colimit of a directed system\n$(M_i, \\mu_{ij})$ of $R$-modules\nwith all $M_i$ finitely presented $R$-modules.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing finite and finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HA","source_file":"algebra.tex","source_line":1629,"source_end_line":1635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1629-L1635","statement_sha256":"8f5ef5d0de32fde894bffcba4b9f0b55797fb45745d774ff80426e2e121c9bdb","origin":"The Stacks Project","memory_eligible":false,"source_rank":988,"rank":988,"depth":0,"x":2013.43,"y":197.881,"cluster":"commutative-algebra"},{"id":"stacks:0G8P","tag":"0G8P","title":"Characterizing finite and finitely presented modules · Lemma 0G8P","summary":"Let R be a ring. Let N be an R-module. The following are equivalent • N is a finitely presented R-module, • for any filtered colimit M = colim M_i of R-modules the map colim Hom_R(N, M_i) → Hom_R(N, M) is bijective.","statement_latex":"Let $R$ be a ring. Let $N$ be an $R$-module. The following are equivalent\n\\begin{enumerate}\n\\item $N$ is a finitely presented $R$-module,\n\\item for any filtered colimit $M = \\colim M_i$ of $R$-modules the map\n$\\colim \\Hom_R(N, M_i) \\to \\Hom_R(N, M)$ is bijective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing finite and finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8P","source_file":"algebra.tex","source_line":1658,"source_end_line":1666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1658-L1666","statement_sha256":"1d3c616682c5f3c9805e931a08b009e888eeaedd2d78ff526bd5af4115c1ef09","origin":"The Stacks Project","memory_eligible":false,"source_rank":989,"rank":989,"depth":2,"x":2060.365,"y":226.99,"cluster":"commutative-algebra"},{"id":"stacks:00CW","tag":"00CW","title":"Tensor products · Definition 00CW","summary":"Let R be a ring, M, N, P be three R-modules. A mapping f : M × N → P (where M × N is viewed only as Cartesian product of two R-modules) is said to be R-bilinear if for each x ∈ M the mapping y↦ f(x, y) of N into P is R-linear, and for each y∈ N the mapping x↦ f(x, y) is also R-linear.","statement_latex":"Let $R$ be a ring, $M, N, P$ be three $R$-modules.\nA mapping $f : M \\times N \\to P$ (where $M \\times N$\nis viewed only as Cartesian product of two $R$-modules) is said to be\n{\\it $R$-bilinear} if for each $x \\in M$\nthe mapping $y\\mapsto f(x, y)$ of $N$ into $P$ is $R$-linear, and for each\n$y\\in N$ the mapping $x\\mapsto f(x, y)$ is also $R$-linear.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CW","source_file":"algebra.tex","source_line":1705,"source_end_line":1713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1705-L1713","statement_sha256":"eafec3147dbd7046faad862cfea57520d10189eb3e38f5ea3adc508c0382305c","origin":"The Stacks Project","memory_eligible":false,"source_rank":990,"rank":990,"depth":0,"x":2001.661,"y":232.216,"cluster":"commutative-algebra"},{"id":"stacks:00CX","tag":"00CX","title":"Tensor products · Lemma 00CX","summary":"Let M, N be R-modules. Then there exists a pair (T, g) where T is an R-module, and g : M × N → T an R-bilinear mapping, with the following universal property: For any R-module P and any R-bilinear mapping f : M × N → P, there exists a unique R-linear mapping tildef : T → P such that f = tildef ∘ g. In other words, the following diagram commutes: xymatrix M × N ar[rr]^f ar[dr]_g & & P & T ar[ur]_tilde f Moreover, if (T, g) and (T', g') are two pairs with this property,…","statement_latex":"Let $M, N$ be $R$-modules. Then there exists a pair $(T, g)$\nwhere $T$ is an $R$-module, and\n$g : M \\times N \\to T$ an $R$-bilinear\nmapping, with the following universal property:\nFor any $R$-module $P$ and any $R$-bilinear mapping\n$f : M \\times N \\to P$, there\nexists a unique $R$-linear\nmapping $\\tilde{f} : T \\to P$ such that $f = \\tilde{f} \\circ g$.\nIn other words, the following diagram commutes:\n$$\n\\xymatrix{\nM \\times N \\ar[rr]^f \\ar[dr]_g & & P\\\\\n& T \\ar[ur]_{\\tilde f}\n}\n$$\nMoreover, if $(T, g)$ and $(T', g')$\nare two pairs with this property, then there\nexists a unique isomorphism\n$j : T \\to T'$ such that $j\\circ g = g'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CX","source_file":"algebra.tex","source_line":1715,"source_end_line":1736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1715-L1736","statement_sha256":"9dc6800e9c4d120b1dfca8ad0cc562654115b6347bb3a41232947aa39ceb384a","origin":"The Stacks Project","memory_eligible":false,"source_rank":991,"rank":991,"depth":0,"x":2041.199,"y":194.625,"cluster":"commutative-algebra"},{"id":"stacks:00CY","tag":"00CY","title":"Tensor products · Lemma 00CY","summary":"Let M, N, P be R-modules, then the bilinear maps (x, y) & ↦ y ⊗ x (x + y, z) & ↦ x ⊗ z + y ⊗ z (r, x) & ↦ rx induce unique isomorphisms M ⊗_R N & → N ⊗_R M, (M⊕ N)⊗_R P & → (M ⊗_R P)⊕(N ⊗_R P), R ⊗_R M & → M","statement_latex":"Let $M, N, P$ be $R$-modules, then the bilinear maps\n\\begin{align*}\n(x, y) & \\mapsto y \\otimes x\\\\\n(x + y, z) & \\mapsto x \\otimes z + y \\otimes z\\\\\n(r, x) & \\mapsto rx\n\\end{align*}\ninduce unique isomorphisms\n\\begin{align*}\nM \\otimes_R N & \\to N \\otimes_R M, \\\\\n(M\\oplus N)\\otimes_R P & \\to (M \\otimes_R P)\\oplus(N \\otimes_R P),  \\\\\nR \\otimes_R M & \\to M\n\\end{align*}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CY","source_file":"algebra.tex","source_line":1778,"source_end_line":1792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1778-L1792","statement_sha256":"669458d32ffe1fe0b96cf2548b0f3968437c15b1521d8a03e27daf91f5feb297","origin":"The Stacks Project","memory_eligible":false,"source_rank":992,"rank":992,"depth":0,"x":2042.283,"y":245.345,"cluster":"commutative-algebra"},{"id":"stacks:00CZ","tag":"00CZ","title":"Tensor products · Lemma 00CZ","summary":"Let M_1, …, M_r be R-modules. Then there exists a pair (T, g) consisting of an R-module T and an R-multilinear mapping g : M_1× … × M_r → T with the universal property: For any R-multilinear mapping f : M_1× … × M_r → P there exists a unique R-module homomorphism f' : T → P such that f'∘ g = f. Such a module T is unique up to unique isomorphism. We denote it M_1⊗_R … ⊗_R M_r and we denote the universal multilinear map (m_1, …, m_r) ↦ m_1 ⊗ … ⊗ m_r.","statement_latex":"Let $M_1, \\ldots, M_r$ be $R$-modules. Then there exists a pair $(T, g)$\nconsisting of an $R$-module T and an $R$-multilinear mapping\n$g : M_1\\times \\ldots \\times M_r \\to T$ with the universal\nproperty: For any $R$-multilinear mapping\n$f : M_1\\times \\ldots \\times M_r \\to P$ there exists a unique $R$-module\nhomomorphism $f' : T \\to P$ such that $f'\\circ g = f$.\nSuch a module $T$ is unique up to unique isomorphism. We denote it\n$M_1\\otimes_R \\ldots \\otimes_R M_r$ and we denote the universal\nmultilinear map $(m_1, \\ldots, m_r) \\mapsto m_1 \\otimes \\ldots \\otimes m_r$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00CZ","source_file":"algebra.tex","source_line":1805,"source_end_line":1816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1805-L1816","statement_sha256":"c7b3d86c98d2ba2067a1758baa04875c587235e4fae9b11f937f443cd4104994","origin":"The Stacks Project","memory_eligible":false,"source_rank":993,"rank":993,"depth":0,"x":2000.24,"y":208.153,"cluster":"commutative-algebra"},{"id":"stacks:00D0","tag":"00D0","title":"Tensor products · Lemma 00D0","summary":"The homomorphisms (M ⊗_R N)⊗_R P → M ⊗_R N ⊗_R P → M ⊗_R (N ⊗_R P) such that f((x ⊗ y)⊗ z) = x ⊗ y ⊗ z and g(x ⊗ y ⊗ z) = x ⊗ (y ⊗ z), x∈ M, y∈ N, z∈ P are well-defined and are isomorphisms.","statement_latex":"The homomorphisms\n$$\n(M \\otimes_R N)\\otimes_R P \\to\nM \\otimes_R N \\otimes_R P \\to\nM \\otimes_R (N \\otimes_R P)\n$$\nsuch that\n$f((x \\otimes y)\\otimes z) = x \\otimes y \\otimes z$\nand $g(x \\otimes y \\otimes z) = x \\otimes (y \\otimes z)$,\n$x\\in M, y\\in N, z\\in P$ are well-defined and are isomorphisms.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00D0","source_file":"algebra.tex","source_line":1822,"source_end_line":1834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1822-L1834","statement_sha256":"307cfbd2fa762f2b50ab966dee9b6fc55f42d89ebed3bb0fcc9bb7b45aaf34e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":994,"rank":994,"depth":0,"x":2061.808,"y":211.762,"cluster":"commutative-algebra"},{"id":"stacks:00D1","tag":"00D1","title":"Tensor products · Definition 00D1","summary":"An abelian group N is called an (A, B)-bimodule if it is both an A-module and a B-module and for all a ∈ A and b ∈ B the multiplication by a and b commute, so b(an) = a(bn) for all n ∈ N. In this situation we usually write the B-action on the right: so for b ∈ B and n ∈ N the result of multiplying n by b is denoted nb. With this convention the compatibility above is that (ax)b = a(xb) for all a∈ A, b∈ B, x∈ N. The shorthand _AN_B is used to denote an (A, B)-bimodule N.","statement_latex":"An abelian group $N$ is called an {\\it $(A, B)$-bimodule} if it is both an\n$A$-module and a $B$-module and for all $a \\in A$ and $b \\in B$ the\nmultiplication by $a$ and $b$ commute, so $b(an) = a(bn)$ for all $n \\in N$.\nIn this situation we usually write the $B$-action on the right: so\nfor $b \\in B$ and $n \\in N$ the result of multiplying $n$ by $b$\nis denoted $nb$. With this convention the compatibility above is\nthat $(ax)b = a(xb)$ for all $a\\in A, b\\in B, x\\in N$.\nThe shorthand $_AN_B$ is used to denote an $(A, B)$-bimodule $N$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00D1","source_file":"algebra.tex","source_line":1868,"source_end_line":1878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1868-L1878","statement_sha256":"1615442066234ebfabf93d483d45fbfe90181dbb3189330b3ee2f07898782a84","origin":"The Stacks Project","memory_eligible":false,"source_rank":995,"rank":995,"depth":0,"x":2012.994,"y":244.375,"cluster":"commutative-algebra"},{"id":"stacks:00D2","tag":"00D2","title":"Tensor products · Lemma 00D2","summary":"For A-module M, B-module P and (A, B)-bimodule N, the modules (M ⊗_A N)⊗_B P and M ⊗_A(N ⊗_B P) can both be given (A, B)-bimodule structure, and moreover (M ⊗_A N)⊗_B P ≅ M ⊗_A(N ⊗_B P).","statement_latex":"For $A$-module $M$, $B$-module $P$ and $(A, B)$-bimodule $N$, the modules\n$(M \\otimes_A N)\\otimes_B P$ and $M \\otimes_A(N \\otimes_B P)$ can both be\ngiven $(A, B)$-bimodule structure,\nand moreover\n$$\n(M \\otimes_A N)\\otimes_B P \\cong M \\otimes_A(N \\otimes_B P).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00D2","source_file":"algebra.tex","source_line":1880,"source_end_line":1889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1880-L1889","statement_sha256":"c985427d3b387c9ab7322419e98648ac3f55c175f56d1e11d34a2dc3aa7cdf97","origin":"The Stacks Project","memory_eligible":false,"source_rank":996,"rank":996,"depth":1,"x":2022.866,"y":192.094,"cluster":"commutative-algebra"},{"id":"stacks:00DE","tag":"00DE","title":"Tensor products · Lemma 00DE","summary":"For any three R-modules M, N, P, Hom_R(M ⊗_R N, P) ≅ Hom_R(M, Hom_R(N, P))","statement_latex":"For any three $R$-modules $M, N, P$,\n$$\n\\Hom_R(M \\otimes_R N, P) \\cong \\Hom_R(M, \\Hom_R(N, P))\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DE","source_file":"algebra.tex","source_line":1905,"source_end_line":1911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1905-L1911","statement_sha256":"0274003e047973bd106fe120920e3a6a18e1ed99f1fec296a8b2c3866e294744","origin":"The Stacks Project","memory_eligible":false,"source_rank":997,"rank":997,"depth":0,"x":2057.98,"y":236.695,"cluster":"commutative-algebra"},{"id":"stacks:00DD","tag":"00DD","title":"Tensor products commute with colimits · Lemma 00DD","summary":"Let (M_i, μ_ij) be a system over the preordered set I. Let N be an R-module. Then colim (M_i ⊗ N) ≅ (colim M_i)⊗ N. Moreover, the isomorphism is induced by the homomorphisms μ_i ⊗ 1: M_i ⊗ N → M ⊗ N where M = colim_i M_i with natural maps μ_i : M_i → M.","statement_latex":"Let $(M_i, \\mu_{ij})$ be a system over the preordered set $I$.\nLet $N$ be an $R$-module. Then\n$$\n\\colim (M_i \\otimes N) \\cong (\\colim M_i)\\otimes N.\n$$\nMoreover, the isomorphism is induced by the homomorphisms\n$\\mu_i \\otimes 1: M_i \\otimes N \\to M \\otimes N$\nwhere $M = \\colim_i M_i$ with natural maps $\\mu_i : M_i \\to M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DD","source_file":"algebra.tex","source_line":1933,"source_end_line":1943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1933-L1943","statement_sha256":"7f0c238a5d797445107d6f38e0a8d839594a0cf3ac4cb45095de751a2e2fe361","origin":"The Stacks Project","memory_eligible":false,"source_rank":998,"rank":998,"depth":3,"x":1995.608,"y":223.6,"cluster":"commutative-algebra"},{"id":"stacks:00DF","tag":"00DF","title":"Tensor products · Lemma 00DF","summary":"Let M_1xrightarrowf M_2xrightarrowg M_3 → 0 be an exact sequence of R-modules and homomorphisms, and let N be any R-module. Then the sequence M_1⊗ Nxrightarrowf ⊗ 1 M_2⊗ N xrightarrowg ⊗ 1 M_3⊗ N → 0 is exact. In other words, the functor - ⊗_R N is right exact, in the sense that tensoring each term in the original right exact sequence preserves the exactness.","statement_latex":"Let\n\\begin{align*}\nM_1\\xrightarrow{f} M_2\\xrightarrow{g} M_3 \\to 0\n\\end{align*}\nbe an exact sequence of $R$-modules and homomorphisms, and let $N$ be any\n$R$-module. Then the sequence\n\\begin{equation}\n\nM_1\\otimes N\\xrightarrow{f \\otimes 1} M_2\\otimes N \\xrightarrow{g \\otimes 1}\nM_3\\otimes N \\to 0\n\\end{equation}\nis exact. In other words, the functor $- \\otimes_R N$ is\n{\\it right exact}, in the sense that tensoring\neach term in the original right exact sequence preserves the exactness.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DF","source_file":"algebra.tex","source_line":1993,"source_end_line":2009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L1993-L2009","statement_sha256":"13cca3ca9027c20ada7ff05852c99431284ae40f223bac4945d4f26e19e969bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":999,"rank":999,"depth":1,"x":2052.682,"y":197.616,"cluster":"commutative-algebra"},{"id":"stacks:05BS","tag":"05BS","title":"Tensor products · Lemma 05BS","summary":"Let R be a ring. Let M and N be R-modules. • If N and M are finite, then so is M ⊗_R N. • If N and M are finitely presented, then so is M ⊗_R N.","statement_latex":"Let $R$ be a ring. Let $M$ and $N$ be $R$-modules.\n\\begin{enumerate}\n\\item If $N$ and $M$ are finite, then so is $M \\otimes_R N$.\n\\item If $N$ and $M$ are finitely presented, then so is $M \\otimes_R N$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BS","source_file":"algebra.tex","source_line":2063,"source_end_line":2070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2063-L2070","statement_sha256":"09c9968d7b8dd823aa8bb4b00dfcb33f6b5d8e9ae9ace196abe4d67392fb8e92","origin":"The Stacks Project","memory_eligible":false,"source_rank":1000,"rank":1000,"depth":2,"x":2031.287,"y":249.653,"cluster":"commutative-algebra"},{"id":"stacks:00DK","tag":"00DK","title":"Tensor products · Lemma 00DK","summary":"Let M be an R-module. Then the S^-1R-modules S^-1M and S^-1R ⊗_R M are canonically isomorphic, and the canonical isomorphism f : S^-1R ⊗_R M → S^-1M is given by f((a/s) ⊗ m) = am/s, ∀ a ∈ R, m ∈ M, s ∈ S","statement_latex":"Let $M$ be an $R$-module. Then the $S^{-1}R$-modules $S^{-1}M$\nand $S^{-1}R \\otimes_R M$ are canonically isomorphic, and the\ncanonical isomorphism $f : S^{-1}R \\otimes_R M \\to S^{-1}M$\nis given by\n$$\nf((a/s) \\otimes m) = am/s, \\forall a \\in R, m \\in M, s \\in S\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DK","source_file":"algebra.tex","source_line":2087,"source_end_line":2096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2087-L2096","statement_sha256":"4baa6765a9335992ffbd0c5eb35bc0b07cc35e6e821b26a22479b2feff429333","origin":"The Stacks Project","memory_eligible":false,"source_rank":1001,"rank":1001,"depth":0,"x":2004.974,"y":198.668,"cluster":"commutative-algebra"},{"id":"stacks:00DL","tag":"00DL","title":"Tensor products · Lemma 00DL","summary":"Let M, N be R-modules, then there is a canonical S^-1R-module isomorphism f : S^-1M ⊗_S^-1RS^-1N → S^-1(M ⊗_R N), given by f((m/s)⊗(n/t)) = (m ⊗ n)/st","statement_latex":"Let $M, N$ be $R$-modules, then there is a canonical\n$S^{-1}R$-module isomorphism\n$f : S^{-1}M \\otimes_{S^{-1}R}S^{-1}N \\to S^{-1}(M \\otimes_R N)$,\ngiven by\n$$\nf((m/s)\\otimes(n/t)) = (m \\otimes n)/st\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DL","source_file":"algebra.tex","source_line":2123,"source_end_line":2132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2123-L2132","statement_sha256":"fb8fc25970d488802bb36d7fcbf52bfad71c0181e60acdfbb96d9d50d360c303","origin":"The Stacks Project","memory_eligible":false,"source_rank":1002,"rank":1002,"depth":2,"x":2065.932,"y":221.543,"cluster":"commutative-algebra"},{"id":"stacks:00DN","tag":"00DN","title":"Tensor algebra · Lemma 00DN","summary":"Let R be a ring. Let M be an R-module. If M is a free R-module, so is each symmetric and exterior power.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nIf $M$ is a free $R$-module, so is each symmetric and exterior power.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DN","source_file":"algebra.tex","source_line":2230,"source_end_line":2234,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2230-L2234","statement_sha256":"1f740842557c93e8ebe44646c11c0f5fabb7b76d824f937c93b27a12f892a8ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":1003,"rank":1003,"depth":0,"x":2002.015,"y":239.423,"cluster":"commutative-algebra"},{"id":"stacks:00DO","tag":"00DO","title":"Tensor algebra · Lemma 00DO","summary":"Let R be a ring. Let M_2 → M_1 → M → 0 be an exact sequence of R-modules. There are exact sequences M_2 ⊗_R Sym^n - 1(M_1) → Sym^n(M_1) → Sym^n(M) → 0 and similarly M_2 ⊗_R wedge^n - 1(M_1) → wedge^n(M_1) → wedge^n(M) → 0","statement_latex":"Let $R$ be a ring.\nLet $M_2 \\to M_1 \\to M \\to 0$ be an exact sequence of $R$-modules.\nThere are exact sequences\n$$\nM_2 \\otimes_R \\text{Sym}^{n - 1}(M_1)\n\\to\n\\text{Sym}^n(M_1)\n\\to\n\\text{Sym}^n(M)\n\\to\n0\n$$\nand similarly\n$$\nM_2 \\otimes_R \\wedge^{n - 1}(M_1)\n\\to\n\\wedge^n(M_1)\n\\to\n\\wedge^n(M)\n\\to\n0\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DO","source_file":"algebra.tex","source_line":2240,"source_end_line":2264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2240-L2264","statement_sha256":"f56755948bba076ee5b770f93c8001b7ef641bb2a513d25d0d306512a052bf68","origin":"The Stacks Project","memory_eligible":false,"source_rank":1004,"rank":1004,"depth":0,"x":2035.06,"y":189.533,"cluster":"commutative-algebra"},{"id":"stacks:00DP","tag":"00DP","title":"Tensor algebra · Lemma 00DP","summary":"Let R be a ring. Let M be an R-module. Let x_i, i ∈ I be a given system of generators of M as an R-module. Let n ≥ 2. There exists a canonical exact sequence bigoplus_1 ≤ j_1 < j_2 ≤ n bigoplus_i_1, i_2 ∈ I T^n - 2(M) ⊕ bigoplus_1 ≤ j_1 < j_2 ≤ n bigoplus_i ∈ I T^n - 2(M) → T^n(M) → wedge^n(M) → 0 where the pure tensor m_1 ⊗ … ⊗ m_n - 2 in the first summand maps to underbrace m_1 ⊗ … ⊗ x_i_1 ⊗ … ⊗ x_i_2 ⊗ … ⊗ m_n - 2 _with x_i_1 and x_i_2 occupying slots j_1 and j_2 in…","statement_latex":"Let $R$ be a ring.\nLet $M$ be an $R$-module.\nLet $x_i$, $i \\in I$ be a given system of generators of\n$M$ as an $R$-module. Let $n \\geq 2$.\nThere exists a canonical exact sequence\n$$\n\\bigoplus_{1 \\leq j_1 < j_2 \\leq n}\n\\bigoplus_{i_1, i_2 \\in I}\n\\text{T}^{n - 2}(M)\n\\oplus\n\\bigoplus_{1 \\leq j_1 < j_2 \\leq n}\n\\bigoplus_{i \\in I}\n\\text{T}^{n - 2}(M)\n\\to\n\\text{T}^n(M)\n\\to\n\\wedge^n(M)\n\\to\n0\n$$\nwhere the pure tensor $m_1 \\otimes \\ldots \\otimes m_{n - 2}$ in the first\nsummand maps to\n\\begin{align*}\n\\underbrace{\nm_1 \\otimes \\ldots \\otimes x_{i_1} \\otimes \\ldots\n\\otimes x_{i_2} \\otimes \\ldots \\otimes m_{n - 2}\n}_{\\text{with } x_{i_1} \\text{ and } x_{i_2}\n\\text{ occupying slots } j_1 \\text{ and } j_2\n\\text{ in the tensor}} \\\\\n+\n\\underbrace{\nm_1 \\otimes \\ldots \\otimes x_{i_2} \\otimes \\ldots\n\\otimes x_{i_1} \\otimes \\ldots \\otimes m_{n - 2}\n}_{\\text{with } x_{i_2} \\text{ and } x_{i_1}\n\\text{ occupying slots } j_1 \\text{ and } j_2\n\\text{ in the tensor}}\n\\end{align*}\nand $m_1 \\otimes \\ldots \\otimes m_{n - 2}$ in the second\nsummand maps to\n$$\n\\underbrace{\nm_1 \\otimes \\ldots \\otimes x_i \\otimes \\ldots\n\\otimes x_i \\otimes \\ldots \\otimes m_{n - 2}\n}_{\\text{with } x_{i} \\text{ and } x_{i}\n\\text{ occupying slots } j_1 \\text{ and } j_2\n\\text{ in the tensor}}\n$$\nThere is also a canonical exact sequence\n$$\n\\bigoplus_{1 \\leq j_1 < j_2 \\leq n}\n\\bigoplus_{i_1, i_2 \\in I}\n\\text{T}^{n - 2}(M)\n\\to\n\\text{T}^n(M)\n\\to\n\\text{Sym}^n(M)\n\\to\n0\n$$\nwhere the pure tensor $m_1 \\otimes \\ldots \\otimes m_{n - 2}$ maps to\n\\begin{align*}\n\\underbrace{\nm_1 \\otimes \\ldots \\otimes x_{i_1} \\otimes \\ldots\n\\otimes x_{i_2} \\otimes \\ldots \\otimes m_{n - 2}\n}_{\\text{with } x_{i_1} \\text{ and } x_{i_2}\n\\text{ occupying slots } j_1 \\text{ and } j_2\n\\text{ in the tensor}} \\\\\n-\n\\underbrace{\nm_1 \\otimes \\ldots \\otimes x_{i_2} \\otimes \\ldots\n\\otimes x_{i_1} \\otimes \\ldots \\otimes m_{n - 2}\n}_{\\text{with } x_{i_2} \\text{ and } x_{i_1}\n\\text{ occupying slots } j_1 \\text{ and } j_2\n\\text{ in the tensor}}\n\\end{align*}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DP","source_file":"algebra.tex","source_line":2270,"source_end_line":2347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2270-L2347","statement_sha256":"fd9482911b06cd6810454ff02115e7353793db75c18f2ab4437aca1de203245d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1005,"rank":1005,"depth":0,"x":2050.949,"y":245.556,"cluster":"commutative-algebra"},{"id":"stacks:0H1C","tag":"0H1C","title":"Tensor algebra · Lemma 0H1C","summary":"Let A → B be a ring map. Let M be a B-module. Let n > 1. The kernel of the A-linear map M ⊗_A … ⊗_A M → wedge^n_B(M) is generated as an A-module by the elements m_1 ⊗ … ⊗ m_n with m_i = m_j for i not = j, m_1, …, m_n ∈ M and the elements m_1 ⊗ … ⊗ bm_i ⊗ … ⊗ m_n - m_1 ⊗ … ⊗ bm_j ⊗ … ⊗ m_n for i not = j, m_1, …, m_n ∈ M, and b ∈ B.","statement_latex":"Let $A \\to B$ be a ring map. Let $M$ be a $B$-module.\nLet $n > 1$. The kernel of the $A$-linear map\n$M \\otimes_A \\ldots \\otimes_A M \\to \\wedge^n_B(M)$\nis generated as an $A$-module by the elements\n$m_1 \\otimes \\ldots \\otimes m_n$ with $m_i = m_j$\nfor $i \\not = j$, $m_1, \\ldots, m_n \\in M$ and the elements\n$m_1 \\otimes \\ldots \\otimes bm_i \\otimes \\ldots \\otimes m_n -\nm_1 \\otimes \\ldots \\otimes bm_j \\otimes \\ldots \\otimes m_n$\nfor $i \\not = j$, $m_1, \\ldots, m_n \\in M$, and $b \\in B$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1C","source_file":"algebra.tex","source_line":2353,"source_end_line":2364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2353-L2364","statement_sha256":"ae8696461b65de54fc57e06297a4c42f50304f59b65b7f076b5013ae0f305605","origin":"The Stacks Project","memory_eligible":false,"source_rank":1006,"rank":1006,"depth":0,"x":1993.695,"y":212.983,"cluster":"commutative-algebra"},{"id":"stacks:00DQ","tag":"00DQ","title":"Tensor algebra · Lemma 00DQ","summary":"Taking tensor algebras commutes with filtered colimits. Let R be a ring. Let M_i be a directed system of R-modules. Then colim_i T(M_i) = T(colim_i M_i) and similarly for the symmetric and exterior algebras.","statement_latex":"\\begin{slogan}\nTaking tensor algebras commutes with filtered colimits.\n\\end{slogan}\nLet $R$ be a ring. Let $M_i$ be a directed system of\n$R$-modules. Then\n$\\colim_i \\text{T}(M_i) = \\text{T}(\\colim_i M_i)$\nand similarly for the symmetric and exterior algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DQ","source_file":"algebra.tex","source_line":2370,"source_end_line":2379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2370-L2379","statement_sha256":"1a694e7fdf7bd2a74cd31f48f9cfe8a6ac0dbb7fb9b736b6950be5b40fbdfd7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1007,"rank":1007,"depth":4,"x":2062.684,"y":204.445,"cluster":"commutative-algebra"},{"id":"stacks:0C6F","tag":"0C6F","title":"Tensor algebra · Lemma 0C6F","summary":"Let R be a ring and let S ⊂ R be a multiplicative subset. Then S^-1T_R(M) = T_S^-1R(S^-1M) for any R-module M. Similar for symmetric and exterior algebras.","statement_latex":"Let $R$ be a ring and let $S \\subset R$ be a multiplicative subset.\nThen $S^{-1}T_R(M) = T_{S^{-1}R}(S^{-1}M)$ for any $R$-module $M$.\nSimilar for symmetric and exterior algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tensor algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6F","source_file":"algebra.tex","source_line":2385,"source_end_line":2390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2385-L2390","statement_sha256":"036776f440236c0692887d9fdbcc69309b319327ac276edb2aba5360bd15a366","origin":"The Stacks Project","memory_eligible":false,"source_rank":1008,"rank":1008,"depth":3,"x":2018.313,"y":250.262,"cluster":"commutative-algebra"},{"id":"stacks:05G4","tag":"05G4","title":"Base change · Definition 05G4","summary":"Let φ : R → S be a ring map. Let M be an S-module. Let R → R' be any ring map. The base change of φ by R → R' is the ring map R' → S ⊗_R R'. In this situation we often write S' = S ⊗_R R'. The base change of the S-module M is the S'-module M ⊗_R R'.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module.\nLet $R \\to R'$ be any ring map. The {\\it base change} of $\\varphi$\nby $R \\to R'$ is the ring map $R' \\to S \\otimes_R R'$. In this situation\nwe often write $S' = S \\otimes_R R'$.\nThe {\\it base change} of the $S$-module $M$ is the $S'$-module\n$M \\otimes_R R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Base change","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05G4","source_file":"algebra.tex","source_line":2410,"source_end_line":2418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2410-L2418","statement_sha256":"d11b518d665a82ea82695fdae494379347705050808848e71793df16fe97b0a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1009,"rank":1009,"depth":0,"x":2014.155,"y":190.82,"cluster":"commutative-algebra"},{"id":"stacks:05G5","tag":"05G5","title":"Base change · Lemma 05G5","summary":"Let R → S be a ring map. Let M be an S-module. Let R → R' be a ring map and let S' = S ⊗_R R' and M' = M ⊗_R R' be the base changes. • If M is a finite S-module, then the base change M' is a finite S'-module. • If M is an S-module of finite presentation, then the base change M' is an S'-module of finite presentation. • If R → S is of finite type, then the base change R' → S' is of finite type. • If R → S is of finite presentation, then the base change R' → S' is of finite…","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.\nLet $R \\to R'$ be a ring map and let $S' = S \\otimes_R R'$ and\n$M' = M \\otimes_R R'$ be the base changes.\n\\begin{enumerate}\n\\item If $M$ is a finite $S$-module, then the base change\n$M'$ is a finite $S'$-module.\n\\item If $M$ is an $S$-module of finite presentation, then\nthe base change $M'$ is an $S'$-module of finite presentation.\n\\item If $R \\to S$ is of finite type, then the base change\n$R' \\to S'$ is of finite type.\n\\item If $R \\to S$ is of finite presentation, then\nthe base change $R' \\to S'$ is of finite presentation.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05G5","source_file":"algebra.tex","source_line":2427,"source_end_line":2442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2427-L2442","statement_sha256":"12b9358cc4855253e094d1022e972cb4e7b7f2a39351f0570f052b1a6ba13cdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1010,"rank":1010,"depth":2,"x":2065.429,"y":232.626,"cluster":"commutative-algebra"},{"id":"stacks:05DQ","tag":"05DQ","title":"Base change · Lemma 05DQ","summary":"Let R → S be a ring map. The functors Mod_S → Mod_R, N ↦ N_R (restriction) and Mod_R → Mod_S, M ↦ M ⊗_R S (base change) are adjoint functors. In a formula Hom_R(M, N_R) = Hom_S(M ⊗_R S, N)","statement_latex":"Let $R \\to S$ be a ring map. The functors\n$\\text{Mod}_S \\to \\text{Mod}_R$, $N \\mapsto N_R$ (restriction)\nand $\\text{Mod}_R \\to \\text{Mod}_S$, $M \\mapsto M \\otimes_R S$\n(base change) are adjoint functors. In a formula\n$$\n\\Hom_R(M, N_R) = \\Hom_S(M \\otimes_R S, N)\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DQ","source_file":"algebra.tex","source_line":2467,"source_end_line":2476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2467-L2476","statement_sha256":"6cd343226767d82d1eedd7d3911b2338918b60c4a52c43fa88ea195e05656e5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1011,"rank":1011,"depth":0,"x":1993.439,"y":230.877,"cluster":"commutative-algebra"},{"id":"stacks:08YP","tag":"08YP","title":"Base change · Lemma 08YP","summary":"Let R → S be a ring map. The functors Mod_S → Mod_R, N ↦ N_R (restriction) and Mod_R → Mod_S, M ↦ Hom_R(S, M) are adjoint functors. In a formula Hom_R(N_R, M) = Hom_S(N, Hom_R(S, M))","statement_latex":"Let $R \\to S$ be a ring map. The functors\n$\\text{Mod}_S \\to \\text{Mod}_R$, $N \\mapsto N_R$ (restriction)\nand $\\text{Mod}_R \\to \\text{Mod}_S$, $M \\mapsto \\Hom_R(S, M)$\nare adjoint functors. In a formula\n$$\n\\Hom_R(N_R, M) = \\Hom_S(N, \\Hom_R(S, M))\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YP","source_file":"algebra.tex","source_line":2491,"source_end_line":2500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2491-L2500","statement_sha256":"a19faac838863468e8333a947e082319d85b024d78f64ea4fe2e8cda0d29c2ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":1012,"rank":1012,"depth":0,"x":2048.353,"y":191.012,"cluster":"commutative-algebra"},{"id":"stacks:08YQ","tag":"08YQ","title":"Base change · Lemma 08YQ","summary":"Let R → S be a ring map. Given S-modules M, N and an R-module P we have Hom_R(M ⊗_S N, P) = Hom_S(M, Hom_R(N, P))","statement_latex":"Let $R \\to S$ be a ring map. Given $S$-modules $M, N$ and an $R$-module $P$\nwe have\n$$\n\\Hom_R(M \\otimes_S N, P) = \\Hom_S(M, \\Hom_R(N, P))\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YQ","source_file":"algebra.tex","source_line":2511,"source_end_line":2518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2511-L2518","statement_sha256":"9949eb619f28ca586051f941b58fcf88120b6f9538e47dcb161f14cd4eea906e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1013,"rank":1013,"depth":1,"x":2039.853,"y":252.03,"cluster":"commutative-algebra"},{"id":"stacks:07K1","tag":"07K1","title":"Miscellany · Lemma 07K1","summary":"Let R be a ring, I and J two ideals and p a prime ideal containing the product IJ. Then p contains I or J.","statement_latex":"Let $R$ be a ring, $I$ and $J$ two ideals and $\\mathfrak p$ a prime ideal\ncontaining the product $IJ$. Then $\\mathfrak{p}$ contains $I$ or $J$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07K1","source_file":"algebra.tex","source_line":2550,"source_end_line":2554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2550-L2554","statement_sha256":"7b395954eef1bd83189bc940061e863e04754d4aff95ff8d635ea7d836d48c7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1014,"rank":1014,"depth":0,"x":1996.73,"y":201.837,"cluster":"commutative-algebra"},{"id":"stacks:00DS","tag":"00DS","title":"Prime avoidance · Lemma 00DS","summary":"1. In an affine scheme if a finite number of points are contained in an open subset then they are contained in a smaller principal open subset. 2. Affine opens are cofinal among the neighborhoods of a given finite set of an affine scheme Let R be a ring. Let I_i ⊂ R, i = 1, …, r, and J ⊂ R be ideals. Assume • J not⊂ I_i for i = 1, …, r, and • all but two of I_i are prime ideals. Then there exists an x ∈ J, xnot∈ I_i for all i.","statement_latex":"\\begin{slogan}\n1. In an affine scheme if a finite number of points are contained in an\nopen subset then they are contained in a smaller principal open subset.\n2. Affine opens are cofinal among the neighborhoods of a given finite set\nof an affine scheme\n\\end{slogan}\nLet $R$ be a ring. Let $I_i \\subset R$, $i = 1, \\ldots, r$,\nand $J \\subset R$ be ideals. Assume\n\\begin{enumerate}\n\\item $J \\not\\subset I_i$ for $i = 1, \\ldots, r$, and\n\\item all but two of $I_i$ are prime ideals.\n\\end{enumerate}\nThen there exists an $x \\in J$, $x\\not\\in I_i$ for all $i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DS","source_file":"algebra.tex","source_line":2563,"source_end_line":2578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2563-L2578","statement_sha256":"20bea3e2a76047c225b3e7be0df4b76d812057d8c5e52b2b9cec9c4d0cd4a052","origin":"The Stacks Project","memory_eligible":false,"source_rank":1015,"rank":1015,"depth":1,"x":2069.427,"y":214.475,"cluster":"commutative-algebra"},{"id":"stacks:0EHL","tag":"0EHL","title":"Miscellany · Lemma 0EHL","summary":"Let R be a ring. Let x ∈ R, I ⊂ R an ideal, and p_i, i = 1, …, r be prime ideals. Suppose that x + I not ⊂ p_i for i = 1, …, r. Then there exists a y ∈ I such that x + y not ∈ p_i for all i.","statement_latex":"Let $R$ be a ring. Let $x \\in R$, $I \\subset R$ an ideal, and\n$\\mathfrak p_i$, $i = 1, \\ldots, r$ be prime ideals.\nSuppose that $x + I \\not \\subset \\mathfrak p_i$ for\n$i = 1, \\ldots, r$. Then there exists a $y \\in I$\nsuch that $x + y \\not \\in \\mathfrak p_i$ for all $i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHL","source_file":"algebra.tex","source_line":2596,"source_end_line":2603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2596-L2603","statement_sha256":"e796e8f2bee674fb944d6f690aea6bf2fce6a9c7d9b9ce134a7923acb0a293fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1016,"rank":1016,"depth":0,"x":2005.192,"y":246.638,"cluster":"commutative-algebra"},{"id":"stacks:00DT","tag":"00DT","title":"Chinese remainder · Lemma 00DT","summary":"Let R be a ring. • If I_1, …, I_r are ideals such that I_a + I_b = R when a not = b, then I_1 ∩ … ∩ I_r = I_1I_2… I_r and R/(I_1I_2… I_r) ≅ R/I_1 × … × R/I_r. • If m_1, …, m_r are pairwise distinct maximal ideals then m_a + m_b = R for a not = b and the above applies.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item If $I_1, \\ldots, I_r$ are ideals such that $I_a + I_b = R$\nwhen $a \\not = b$, then $I_1 \\cap \\ldots \\cap I_r =\nI_1I_2\\ldots I_r$ and $R/(I_1I_2\\ldots I_r)\n\\cong R/I_1 \\times \\ldots \\times R/I_r$.\n\\item If $\\mathfrak m_1, \\ldots, \\mathfrak m_r$ are pairwise distinct maximal\nideals then $\\mathfrak m_a + \\mathfrak m_b = R$ for $a \\not = b$ and the\nabove applies.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DT","source_file":"algebra.tex","source_line":2615,"source_end_line":2627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2615-L2627","statement_sha256":"3bcec0b9a3b11b1ff4ae7a82aed27efa2dc1c271e0591561ae23ab0a776412d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1017,"rank":1017,"depth":0,"x":2026.849,"y":186.04,"cluster":"commutative-algebra"},{"id":"stacks:07DQ","tag":"07DQ","title":"Miscellany · Lemma 07DQ","summary":"Let R be a ring. Let n ≥ m. Let A be an n × m matrix with coefficients in R. Let J ⊂ R be the ideal generated by the m × m minors of A. • For any f ∈ J there exists a m × n matrix B such that BA = f 1_m × m. • If f ∈ R and BA = f 1_m × m for some m × n matrix B, then f^m ∈ J.","statement_latex":"Let $R$ be a ring. Let $n \\geq m$. Let $A$ be an\n$n \\times m$ matrix with coefficients in $R$. Let $J \\subset R$\nbe the ideal generated by the $m \\times m$ minors of $A$.\n\\begin{enumerate}\n\\item For any $f \\in J$ there exists a $m \\times n$ matrix $B$\nsuch that $BA = f 1_{m \\times m}$.\n\\item If $f \\in R$ and $BA = f 1_{m \\times m}$ for some $m \\times n$ matrix\n$B$, then $f^m \\in J$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DQ","source_file":"algebra.tex","source_line":2667,"source_end_line":2678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2667-L2678","statement_sha256":"a79cb155f82136781cbb03955136c856d6a44d5b47b6ee452c1abda3acc3e37e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1018,"rank":1018,"depth":0,"x":2059.842,"y":243.417,"cluster":"commutative-algebra"},{"id":"stacks:080R","tag":"080R","title":"Miscellany · Lemma 080R","summary":"Let R be a ring. Let n ≥ m. Let A = (a_ij) be an n × m matrix with coefficients in R, written in block form as A = ( A_1 A_2 ) where A_1 has size m × m. Let B be the adjugate (transpose of cofactor) matrix to A_1. Then AB = ( f 1_m × m C ) where f = det(A_1) and c_ij is (up to sign) the determinant of the m × m minor of A corresponding to the rows 1, …, hat j, …, m, i.","statement_latex":"Let $R$ be a ring. Let $n \\geq m$. Let $A = (a_{ij})$ be an\n$n \\times m$ matrix with coefficients in $R$, written in block form\nas\n$$\nA =\n\\left(\n\\begin{matrix}\nA_1 \\\\\nA_2\n\\end{matrix}\n\\right)\n$$\nwhere $A_1$ has size $m \\times m$. Let $B$ be the adjugate (transpose of\ncofactor) matrix to $A_1$. Then\n$$\nAB = \n\\left(\n\\begin{matrix}\nf 1_{m \\times m} \\\\\nC\n\\end{matrix}\n\\right)\n$$\nwhere $f = \\det(A_1)$ and $c_{ij}$ is (up to sign) the determinant of the\n$m \\times m$ minor of $A$ corresponding to the rows\n$1, \\ldots, \\hat j, \\ldots, m, i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080R","source_file":"algebra.tex","source_line":2705,"source_end_line":2733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2705-L2733","statement_sha256":"37e48e54342759ba5dfb7d03d97408bf37d9f7b3f341b43b8b68c858c771be0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1019,"rank":1019,"depth":0,"x":1988.88,"y":219.663,"cluster":"commutative-algebra"},{"id":"stacks:05WI","tag":"05WI","title":"Miscellany · Lemma 05WI","summary":"A map of finite free modules cannot be injective if the source has rank bigger than the target. Let R be a nonzero ring. Let n ≥ 1. Let M be an R-module generated by < n elements. Then any R-module map f : R^⊕ n → M has a nonzero kernel.","statement_latex":"\\begin{slogan}\nA map of finite free modules cannot be injective if the source has\nrank bigger than the target.\n\\end{slogan}\nLet $R$ be a nonzero ring. Let $n \\geq 1$. Let $M$ be an $R$-module generated\nby $< n$ elements. Then any $R$-module map $f : R^{\\oplus n} \\to M$ has a\nnonzero kernel.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WI","source_file":"algebra.tex","source_line":2746,"source_end_line":2755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2746-L2755","statement_sha256":"613b80c6ee837b5d3be5e4e18f540212c7e2c73b139c6e135db18202b319c698","origin":"The Stacks Project","memory_eligible":false,"source_rank":1020,"rank":1020,"depth":1,"x":2060.802,"y":196.758,"cluster":"commutative-algebra"},{"id":"stacks:0FJ7","tag":"0FJ7","title":"Miscellany · Lemma 0FJ7","summary":"The rank of a finite free module is well defined. Let R be a nonzero ring. Let n, m ≥ 0 be integers. If R^⊕ n is isomorphic to R^⊕ m as R-modules, then n = m.","statement_latex":"\\begin{slogan}\nThe rank of a finite free module is well defined.\n\\end{slogan}\nLet $R$ be a nonzero ring. Let $n, m \\geq 0$ be integers.\nIf $R^{\\oplus n}$ is isomorphic to $R^{\\oplus m}$ as\n$R$-modules, then $n = m$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJ7","source_file":"algebra.tex","source_line":2788,"source_end_line":2796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2788-L2796","statement_sha256":"8cd1b131293f6a73f708bd8a6e2808074eca22da788844ae926b1001f7bbc7a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1021,"rank":1021,"depth":2,"x":2025.951,"y":254.85,"cluster":"commutative-algebra"},{"id":"stacks:00DX","tag":"00DX","title":"Cayley-Hamilton · Lemma 00DX","summary":"Let R be a ring. Let A = (a_ij) be an n × n matrix with coefficients in R. Let P(x) ∈ R[x] be the characteristic polynomial of A (defined as det(xid_n × n - A)). Then P(A) = 0 in Mat(n × n, R).","statement_latex":"Let $R$ be a ring. Let $A = (a_{ij})$ be an $n \\times n$\nmatrix with coefficients in $R$. Let $P(x) \\in R[x]$\nbe the characteristic polynomial of $A$ (defined\nas $\\det(x\\text{id}_{n \\times n} - A)$).\nThen $P(A) = 0$ in $\\text{Mat}(n \\times n, R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cayley-Hamilton","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DX","source_file":"algebra.tex","source_line":2809,"source_end_line":2816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2809-L2816","statement_sha256":"0908e1d53057f1854809dc7804cae1b4dad22e478540e131bc93b63b878478c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1022,"rank":1022,"depth":0,"x":2004.793,"y":191.818,"cluster":"commutative-algebra"},{"id":"stacks:05BT","tag":"05BT","title":"Cayley-Hamilton · Lemma 05BT","summary":"Let R be a ring. Let M be a finite R-module. Let φ : M → M be an endomorphism. Then there exists a monic polynomial P ∈ R[T] such that P(φ) = 0 as an endomorphism of M.","statement_latex":"Let $R$ be a ring.\nLet $M$ be a finite $R$-module.\nLet $\\varphi : M \\to M$ be an endomorphism.\nThen there exists a monic polynomial $P \\in R[T]$ such that\n$P(\\varphi) = 0$ as an endomorphism of $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cayley-Hamilton","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BT","source_file":"algebra.tex","source_line":2834,"source_end_line":2841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2834-L2841","statement_sha256":"49a3ed8658b805612271f5c27052578499ca014b273ea595db2cbb6a9c8596ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":1023,"rank":1023,"depth":1,"x":2071.522,"y":226.521,"cluster":"commutative-algebra"},{"id":"stacks:05G7","tag":"05G7","title":"Cayley-Hamilton · Lemma 05G7","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be a finite R-module. Let φ : M → M be an endomorphism such that φ(M) ⊂ IM. Then there exists a monic polynomial P = t^n + a_1 t^n - 1 + … + a_n ∈ R[T] such that a_j ∈ I^j and P(φ) = 0 as an endomorphism of M.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nLet $M$ be a finite $R$-module.\nLet $\\varphi : M \\to M$ be an endomorphism such\nthat $\\varphi(M) \\subset IM$.\nThen there exists a monic polynomial\n$P = t^n + a_1 t^{n - 1} + \\ldots + a_n \\in R[T]$\nsuch that $a_j \\in I^j$ and $P(\\varphi) = 0$ as an endomorphism of $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cayley-Hamilton","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05G7","source_file":"algebra.tex","source_line":2860,"source_end_line":2869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2860-L2869","statement_sha256":"45369ee3abe469e58078b8a34ac602debb04c49ee4cfbb01a25002af0eb137c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1024,"rank":1024,"depth":1,"x":1993.906,"y":238.874,"cluster":"commutative-algebra"},{"id":"stacks:05G8","tag":"05G8","title":"Cayley-Hamilton · Lemma 05G8","summary":"Let R be a ring. Let M be a finite R-module. Let φ : M → M be a surjective R-module map. Then φ is an isomorphism.","statement_latex":"Let $R$ be a ring.\nLet $M$ be a finite $R$-module.\nLet $\\varphi : M \\to M$ be a surjective $R$-module map.\nThen $\\varphi$ is an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cayley-Hamilton","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05G8","source_file":"algebra.tex","source_line":2892,"source_end_line":2898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2892-L2898","statement_sha256":"893dd2b6c5e35cdc2562225c66ecd7fb0da0454fea984ed11f653c867d2dbad3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1025,"rank":1025,"depth":2,"x":2041.511,"y":185.382,"cluster":"commutative-algebra"},{"id":"stacks:00DZ","tag":"00DZ","title":"The spectrum of a ring · Definition 00DZ","summary":"Let R be a ring. • The spectrum of R is the set of prime ideals of R. It is usually denoted Spec(R). • Given a subset T ⊂ R we let V(T) ⊂ Spec(R) be the set of primes containing T, i.e., V(T) = ( p ∈ Spec(R) mid ∀ f∈ T, f∈ p). • Given an element f ∈ R we let D(f) ⊂ Spec(R) be the set of primes not containing f.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item The {\\it spectrum} of $R$ is the set of prime ideals of $R$.\nIt is usually denoted $\\Spec(R)$.\n\\item Given a subset $T \\subset R$ we let $V(T) \\subset \\Spec(R)$\nbe the set of primes containing $T$, i.e., $V(T) = \\{ \\mathfrak p \\in\n\\Spec(R) \\mid \\forall f\\in T, f\\in \\mathfrak p\\}$.\n\\item Given an element $f \\in R$ we let $D(f) \\subset \\Spec(R)$\nbe the set of primes not containing $f$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DZ","source_file":"algebra.tex","source_line":2972,"source_end_line":2984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2972-L2984","statement_sha256":"d93af62f2d21c38dc5031a5854801efc36d808e3d856c6a4a5db665d1fa2277a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1026,"rank":1026,"depth":0,"x":2049.474,"y":252.262,"cluster":"commutative-algebra"},{"id":"stacks:00E0","tag":"00E0","title":"The spectrum of a ring · Lemma 00E0","summary":"Let R be a ring. • The spectrum of a ring R is empty if and only if R is the zero ring. • Every nonzero ring has a maximal ideal. • Every nonzero ring has a minimal prime ideal. • Given an ideal I ⊂ R and a prime ideal I ⊂ p there exists a prime I ⊂ q ⊂ p such that q is minimal over I. • If T ⊂ R, and if (T) is the ideal generated by T in R, then V((T)) = V(T). • If I is an ideal and sqrtI is its radical, see basic notion ([Tag 00BI]), then V(I) = V(sqrtI). • Given an…","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item The spectrum of a ring $R$ is empty if and only if $R$\nis the zero ring.\n\\item Every nonzero ring has a maximal ideal.\n\\item Every nonzero ring has a minimal prime ideal.\n\\item Given an ideal $I \\subset R$ and a prime ideal\n$I \\subset \\mathfrak p$ there exists a prime\n$I \\subset \\mathfrak q \\subset \\mathfrak p$ such\nthat $\\mathfrak q$ is minimal over $I$.\n\\item If $T \\subset R$, and if $(T)$ is the ideal generated by\n$T$ in $R$, then $V((T)) = V(T)$.\n\\item If $I$ is an ideal and $\\sqrt{I}$ is its radical,\nsee basic notion (\\ref{item-radical-ideal}), then $V(I) = V(\\sqrt{I})$.\n\\item Given an ideal $I$ of $R$ we have $\\sqrt{I} =\n\\bigcap_{I \\subset \\mathfrak p} \\mathfrak p$.\n\\item If $I$ is an ideal then $V(I) = \\emptyset$ if and only\nif $I$ is the unit ideal.\n\\item If $I$, $J$ are ideals of $R$ then $V(I) \\cup V(J) =\nV(I \\cap J)$.\n\\item If $(I_a)_{a\\in A}$ is a set of ideals of $R$ then\n$\\bigcap_{a\\in A} V(I_a) = V(\\bigcup_{a\\in A} I_a)$.\n\\item If $f \\in R$, then $D(f) \\amalg V(f) = \\Spec(R)$.\n\\item If $f \\in R$ then $D(f) = \\emptyset$ if and only if $f$\nis nilpotent.\n\\item If $f = u f'$ for some unit $u \\in R$, then $D(f) = D(f')$.\n\\item If $I \\subset R$ is an ideal, and $\\mathfrak p$ is a prime of\n$R$ with $\\mathfrak p \\not\\in V(I)$, then there exists an $f \\in R$\nsuch that $\\mathfrak p \\in D(f)$, and $D(f) \\cap V(I) = \\emptyset$.\n\\item If $f, g \\in R$, then $D(fg) = D(f) \\cap D(g)$.\n\\item If $f_i \\in R$ for $i \\in I$, then\n$\\bigcup_{i\\in I} D(f_i)$ is the complement of $V(\\{f_i \\}_{i\\in I})$\nin $\\Spec(R)$.\n\\item If $f \\in R$ and $D(f) = \\Spec(R)$, then $f$ is a unit.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The spectrum of a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00E0","source_file":"algebra.tex","source_line":2986,"source_end_line":3023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L2986-L3023","statement_sha256":"606d6149f84fb2e73b902dff29eaa863cfe3e81d00146e98d413a672472f219a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1027,"rank":1027,"depth":0,"x":1989.445,"y":207.181,"cluster":"commutative-algebra"},{"id":"stacks:00E1","tag":"00E1","title":"The spectrum of a ring · Definition 00E1","summary":"Let R be a ring. The topology on Spec(R) whose closed sets are the sets V(T) is called the Zariski topology. The open subsets D(f) are called the standard opens of Spec(R).","statement_latex":"Let $R$ be a ring.\nThe topology on $\\Spec(R)$ whose closed sets are the\nsets $V(T)$ is called the {\\it Zariski} topology. The open\nsubsets $D(f)$ are called the {\\it standard opens} of $\\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00E1","source_file":"algebra.tex","source_line":3111,"source_end_line":3117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3111-L3117","statement_sha256":"35f04b86acc17e4da9fbdd7c1a83cf6111cd66babfce591df6897d1b4dd34e1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1028,"rank":1028,"depth":0,"x":2070.462,"y":206.357,"cluster":"commutative-algebra"},{"id":"stacks:00E2","tag":"00E2","title":"The spectrum of a ring · Lemma 00E2","summary":"Functoriality of the spectrum Suppose that φ : R → R' is a ring homomorphism. The induced map Spec(φ) : Spec(R') → Spec(R), p' ↦ φ^-1( p') is continuous for the Zariski topologies. In fact, for any element f ∈ R we have Spec(φ)^-1(D(f)) = D(φ(f)).","statement_latex":"\\begin{slogan}\nFunctoriality of the spectrum\n\\end{slogan}\nSuppose that $\\varphi : R \\to R'$ is a ring homomorphism.\nThe induced map\n$$\n\\Spec(\\varphi) : \\Spec(R') \\longrightarrow \\Spec(R),\n\\quad\n\\mathfrak p' \\longmapsto \\varphi^{-1}(\\mathfrak p')\n$$\nis continuous for the Zariski topologies. In fact, for any\nelement $f \\in R$ we have\n$\\Spec(\\varphi)^{-1}(D(f)) = D(\\varphi(f))$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The spectrum of a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00E2","source_file":"algebra.tex","source_line":3123,"source_end_line":3138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3123-L3138","statement_sha256":"3d5052d345f341802219d5a32a0e4738b0e4f196ecb17a62f90e53bd3037a605","origin":"The Stacks Project","memory_eligible":false,"source_rank":1029,"rank":1029,"depth":0,"x":2011.021,"y":253.22,"cluster":"commutative-algebra"},{"id":"stacks:00E3","tag":"00E3","title":"The spectrum of a ring · Lemma 00E3","summary":"Let R be a ring. Let S ⊂ R be a multiplicative subset. The map R → S^-1R induces via the functoriality of Spec a homeomorphism Spec(S^-1R) → ( p ∈ Spec(R) mid S ∩ p = ∅ ) where the topology on the right hand side is that induced from the Zariski topology on Spec(R). The inverse map is given by p ↦ S^-1 p = p(S^-1R).","statement_latex":"Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.\nThe map $R \\to S^{-1}R$ induces via the functoriality of $\\Spec$\na homeomorphism\n$$\n\\Spec(S^{-1}R)\n\\longrightarrow\n\\{\\mathfrak p \\in \\Spec(R) \\mid S \\cap \\mathfrak p = \\emptyset \\}\n$$\nwhere the topology on the right hand side is that induced from the\nZariski topology on $\\Spec(R)$. The inverse map is given\nby $\\mathfrak p \\mapsto S^{-1}\\mathfrak p = \\mathfrak p(S^{-1}R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The spectrum of a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00E3","source_file":"algebra.tex","source_line":3162,"source_end_line":3175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3162-L3175","statement_sha256":"5d92e604b13d3e11b5fd2bf7b3e62daf19bca7b1637d394fcb573cfc0e47c007","origin":"The Stacks Project","memory_eligible":false,"source_rank":1030,"rank":1030,"depth":1,"x":2017.204,"y":184.522,"cluster":"commutative-algebra"},{"id":"stacks:00E4","tag":"00E4","title":"The spectrum of a ring · Lemma 00E4","summary":"Let R be a ring. Let f ∈ R. The map R → R_f induces via the functoriality of Spec a homeomorphism Spec(R_f) → D(f) ⊂ Spec(R). The inverse is given by p ↦ p · R_f.","statement_latex":"Let $R$ be a ring. Let $f \\in R$.\nThe map $R \\to R_f$ induces via the functoriality of\n$\\Spec$ a homeomorphism\n$$\n\\Spec(R_f) \\longrightarrow D(f) \\subset \\Spec(R).\n$$\nThe inverse is given by $\\mathfrak p \\mapsto \\mathfrak p \\cdot R_f$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The spectrum of a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00E4","source_file":"algebra.tex","source_line":3206,"source_end_line":3215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3206-L3215","statement_sha256":"6ee1d4aacb83016600d3aa20bdcf01e563ff7753813b8896aafe1a7df990f403","origin":"The Stacks Project","memory_eligible":false,"source_rank":1031,"rank":1031,"depth":2,"x":2068.192,"y":239.012,"cluster":"commutative-algebra"},{"id":"stacks:00E5","tag":"00E5","title":"The spectrum of a ring · Lemma 00E5","summary":"Let R be a ring. Let I ⊂ R be an ideal. The map R → R/I induces via the functoriality of Spec a homeomorphism Spec(R/I) → V(I) ⊂ Spec(R). The inverse is given by p ↦ p / I.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nThe map $R \\to R/I$ induces via the functoriality of\n$\\Spec$ a homeomorphism\n$$\n\\Spec(R/I) \\longrightarrow V(I) \\subset \\Spec(R).\n$$\nThe inverse is given by $\\mathfrak p \\mapsto \\mathfrak p / I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The spectrum of a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00E5","source_file":"algebra.tex","source_line":3226,"source_end_line":3235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3226-L3235","statement_sha256":"b9cd4d1cad28b8995a155f1dc31059b35d386387c36888ad3c9f123005320457","origin":"The Stacks Project","memory_eligible":false,"source_rank":1032,"rank":1032,"depth":0,"x":1986.293,"y":227.697,"cluster":"commutative-algebra"},{"id":"stacks:00E8","tag":"00E8","title":"The spectrum of a ring · Lemma 00E8","summary":"The spectrum of a ring is quasi-compact Let R be a ring. The space Spec(R) is quasi-compact.","statement_latex":"\\begin{slogan}\nThe spectrum of a ring is quasi-compact\n\\end{slogan}\nLet $R$ be a ring. The space $\\Spec(R)$ is quasi-compact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The spectrum of a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00E8","source_file":"algebra.tex","source_line":3251,"source_end_line":3257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3251-L3257","statement_sha256":"cba890e31ed07f61f572cdedf2c161d93ddcc11e644d47f4f32175a295bb7b9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1033,"rank":1033,"depth":1,"x":2056.191,"y":189.347,"cluster":"commutative-algebra"},{"id":"stacks:04PM","tag":"04PM","title":"The spectrum of a ring · Lemma 04PM","summary":"Let R be a ring. The topology on X = Spec(R) has the following properties: • X is quasi-compact, • X has a basis for the topology consisting of quasi-compact opens, and • the intersection of any two quasi-compact opens is quasi-compact.","statement_latex":"Let $R$ be a ring.\nThe topology on $X = \\Spec(R)$ has the following properties:\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item $X$ has a basis for the topology consisting of quasi-compact opens, and\n\\item the intersection of any two quasi-compact opens is quasi-compact.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The spectrum of a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PM","source_file":"algebra.tex","source_line":3275,"source_end_line":3284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3275-L3284","statement_sha256":"f29a016b82eec8b0b9773679de9642bdac6aeb7428dc00e8ca89fbdfa0c6eab2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1034,"rank":1034,"depth":3,"x":2035.368,"y":257.679,"cluster":"commutative-algebra"},{"id":"stacks:07BI","tag":"07BI","title":"Local rings · Definition 07BI","summary":"A local ring is a ring with exactly one maximal ideal. If R is a local ring, then the maximal ideal is often denoted m_R and the field R/ m_R is called the residue field of the local ring R. We often say \"let (R, m) be a local ring\" or \"let (R, m, kappa) be a local ring\" to indicate that R is local, m is its unique maximal ideal and kappa = R/ m is its residue field. A local homomorphism of local rings is a ring map φ : R → S such that R and S are local rings and such…","statement_latex":"A {\\it local ring} is a ring with exactly one maximal ideal.\nIf $R$ is a local ring, then the maximal ideal is often denoted\n$\\mathfrak m_R$ and the field $R/\\mathfrak m_R$ is called the\n{\\it residue field} of the local ring $R$.\nWe often say ``let $(R, \\mathfrak m)$ be a local ring''\nor ``let $(R, \\mathfrak m, \\kappa)$ be a local ring''\nto indicate that $R$ is local, $\\mathfrak m$ is its unique\nmaximal ideal and $\\kappa = R/\\mathfrak m$ is its residue field.\nA {\\it local homomorphism of local rings} is a ring map\n$\\varphi : R \\to S$ such that $R$ and $S$ are local rings and such\nthat $\\varphi(\\mathfrak m_R) \\subset \\mathfrak m_S$.\nIf it is given that $R$ and $S$ are local rings, then the phrase\n``{\\it local ring map $\\varphi : R \\to S$}'' means that $\\varphi$\nis a local homomorphism of local rings.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BI","source_file":"algebra.tex","source_line":3311,"source_end_line":3327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3311-L3327","statement_sha256":"8f9fd2af4e0f32a3fe2d851f6fcdac5e363b4db231b843205f28c9b35ac8cbed","origin":"The Stacks Project","memory_eligible":false,"source_rank":1035,"rank":1035,"depth":0,"x":1995.547,"y":195.122,"cluster":"commutative-algebra"},{"id":"stacks:00E9","tag":"00E9","title":"Local rings · Lemma 00E9","summary":"Let R be a ring. The following are equivalent: • R is a local ring, • Spec(R) has exactly one closed point, • R has a maximal ideal m and every element of R setminus m is a unit, and • R is not the zero ring and for every x ∈ R either x or 1 - x is invertible or both.","statement_latex":"Let $R$ be a ring. The following are equivalent:\n\\begin{enumerate}\n\\item $R$ is a local ring,\n\\item $\\Spec(R)$ has exactly one closed point,\n\\item $R$ has a maximal ideal $\\mathfrak m$\nand every element of $R \\setminus \\mathfrak m$\nis a unit, and\n\\item $R$ is not the zero ring and for every $x \\in R$ either $x$\nor $1 - x$ is invertible or both.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00E9","source_file":"algebra.tex","source_line":3366,"source_end_line":3378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3366-L3378","statement_sha256":"f4f1646189bdb386d42b9c8530fe2b31425305e81383913dd96f9be6b81e471e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1036,"rank":1036,"depth":1,"x":2075.667,"y":218.788,"cluster":"commutative-algebra"},{"id":"stacks:07BJ","tag":"07BJ","title":"Local rings · Lemma 07BJ","summary":"Let φ : R → S be a ring map. Assume R and S are local rings. The following are equivalent: • φ is a local ring map, • φ( m_R) ⊂ m_S, • φ^-1( m_S) = m_R, and • for any x ∈ R, if φ(x) is invertible in S, then x is invertible in R.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume $R$ and $S$ are local rings.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\varphi$ is a local ring map,\n\\item $\\varphi(\\mathfrak m_R) \\subset \\mathfrak m_S$,\n\\item $\\varphi^{-1}(\\mathfrak m_S) = \\mathfrak m_R$, and\n\\item for any $x \\in R$, if $\\varphi(x)$ is invertible in $S$, then $x$\nis invertible in $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BJ","source_file":"algebra.tex","source_line":3399,"source_end_line":3410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3399-L3410","statement_sha256":"57eb8a8c3552a140d7269d0a41bca1cfbbcfbdbd63227be07a443f8c7a993ada","origin":"The Stacks Project","memory_eligible":false,"source_rank":1037,"rank":1037,"depth":2,"x":1997.119,"y":246.955,"cluster":"commutative-algebra"},{"id":"stacks:00E7","tag":"00E7","title":"Local rings · Lemma 00E7","summary":"Let φ : R → S be a ring map. Let p be a prime of R. The following are equivalent • p is in the image of Spec(S) → Spec(R), • S ⊗_R kappa( p) not = 0, • S_ p/ p S_ p not = 0, • (S/ pS)_ p not = 0, and • p = φ^-1( pS).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Let $\\mathfrak p$\nbe a prime of $R$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathfrak p$ is in the image of\n$\\Spec(S) \\to \\Spec(R)$,\n\\item $S \\otimes_R \\kappa(\\mathfrak p) \\not = 0$,\n\\item $S_{\\mathfrak p}/\\mathfrak p S_{\\mathfrak p} \\not = 0$,\n\\item $(S/\\mathfrak pS)_{\\mathfrak p} \\not = 0$, and\n\\item $\\mathfrak p = \\varphi^{-1}(\\mathfrak pS)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00E7","source_file":"algebra.tex","source_line":3499,"source_end_line":3511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3499-L3511","statement_sha256":"72a3bfbb221b88f352f3a21d8dcac87fe6bfdd5172deb4bde0a9fcbadf1c7692","origin":"The Stacks Project","memory_eligible":false,"source_rank":1038,"rank":1038,"depth":0,"x":2032.584,"y":181.255,"cluster":"commutative-algebra"},{"id":"stacks:0AME","tag":"0AME","title":"The Jacobson radical of a ring · Lemma 0AME","summary":"Let R be a ring with Jacobson radical rad(R). Let I ⊂ R be an ideal. The following are equivalent • I ⊂ rad(R), and • every element of 1 + I is a unit in R. In this case every element of R which maps to a unit of R/I is a unit.","statement_latex":"Let $R$ be a ring with Jacobson radical $\\text{rad}(R)$.\nLet $I \\subset R$ be an ideal. The following are\nequivalent\n\\begin{enumerate}\n\\item $I \\subset \\text{rad}(R)$, and\n\\item every element of $1 + I$ is a unit in $R$.\n\\end{enumerate}\nIn this case every element of $R$ which maps to a unit of $R/I$ is a unit.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Jacobson radical of a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AME","source_file":"algebra.tex","source_line":3554,"source_end_line":3564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3554-L3564","statement_sha256":"813c3bce72153fdfcc9d1a67da39663bd7977c76bab92362b4f5007dd4decc25","origin":"The Stacks Project","memory_eligible":false,"source_rank":1039,"rank":1039,"depth":1,"x":2059.411,"y":250.199,"cluster":"commutative-algebra"},{"id":"stacks:0B7C","tag":"0B7C","title":"The Jacobson radical of a ring · Lemma 0B7C","summary":"Let φ : R → S be a ring map such that the induced map Spec(S) → Spec(R) is surjective. Then an element x ∈ R is a unit if and only if φ(x) ∈ S is a unit.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map such that the induced map\n$\\Spec(S) \\to \\Spec(R)$ is surjective. Then an element $x \\in R$\nis a unit if and only if $\\varphi(x) \\in S$ is a unit.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Jacobson radical of a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7C","source_file":"algebra.tex","source_line":3586,"source_end_line":3591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3586-L3591","statement_sha256":"eaeb98f0088a896c4cb505189792480e5112e2bdeb4d6c0b818d8d0bb984c3ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":1040,"rank":1040,"depth":1,"x":1983.781,"y":214.39,"cluster":"commutative-algebra"},{"id":"stacks:00DV","tag":"00DV","title":"Nakayama's lemma · Lemma 00DV","summary":"[MatCA] We quote from [MatCA]: \"This simple but important lemma is due to T. Nakayama, G. Azumaya and W. Krull. Priority is obscure, and although it is usually called the Lemma of Nakayama, late Prof. Nakayama did not like the name.\" Let R be a ring with Jacobson radical rad(R). Let M be an R-module. Let I ⊂ R be an ideal. • If IM = M and M is finite, then there exists an f ∈ 1 + I such that fM = 0. • If IM = M, M is finite, and I ⊂ rad(R), then M = 0. • If N, N' ⊂ M, M =…","statement_latex":"\\begin{reference}\n\\cite[1.M Lemma (NAK) page 11]{MatCA}\n\\end{reference}\n\\begin{history}\nWe quote from \\cite{MatCA}: ``This simple but\nimportant lemma is due to T.~Nakayama, G.~Azumaya and W.~Krull. Priority\nis obscure, and although it is usually called the Lemma of Nakayama, late\nProf.~Nakayama did not like the name.''\n\\end{history}\nLet $R$ be a ring with Jacobson radical $\\text{rad}(R)$.\nLet $M$ be an $R$-module. Let $I \\subset R$\nbe an ideal.\n\\begin{enumerate}\n\\item\n\nIf $IM = M$ and $M$ is finite, then there exists an $f \\in 1 + I$ such that\n$fM = 0$.\n\\item If $IM = M$, $M$ is finite, and $I \\subset \\text{rad}(R)$, then $M = 0$.\n\\item If $N, N' \\subset M$, $M = N + IN'$, and $N'$ is finite,\nthen there exists an $f \\in 1 + I$ such that $fM \\subset N$ and $M_f = N_f$.\n\\item If $N, N' \\subset M$, $M = N + IN'$, $N'$ is finite, and\n$I \\subset \\text{rad}(R)$, then $M = N$.\n\\item If $N \\to M$ is a module map, $N/IN \\to M/IM$ is\nsurjective, and $M$ is finite, then there exists an $f \\in 1 + I$\nsuch that $N_f \\to M_f$ is surjective.\n\\item If $N \\to M$ is a module map, $N/IN \\to M/IM$ is\nsurjective, $M$ is finite, and $I \\subset \\text{rad}(R)$,\nthen $N \\to M$ is surjective.\n\\item If $x_1, \\ldots, x_n \\in M$ generate $M/IM$ and $M$ is finite,\nthen there exists an $f \\in 1 + I$ such that $x_1, \\ldots, x_n$\ngenerate $M_f$ over $R_f$.\n\\item If $x_1, \\ldots, x_n \\in M$ generate $M/IM$, $M$ is finite, and\n$I \\subset \\text{rad}(R)$, then $M$ is generated by $x_1, \\ldots, x_n$.\n\\item If $IM = M$, $I$ is nilpotent, then $M = 0$.\n\\item If $N, N' \\subset M$, $M = N + IN'$, and $I$ is nilpotent then $M = N$.\n\\item If $N \\to M$ is a module map, $I$ is nilpotent, and $N/IN \\to M/IM$\nis surjective, then $N \\to M$ is surjective.\n\\item If $\\{x_\\alpha\\}_{\\alpha \\in A}$ is a set of elements of $M$\nwhich generate $M/IM$ and $I$ is nilpotent, then $M$ is generated\nby the $x_\\alpha$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nakayama's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00DV","source_file":"algebra.tex","source_line":3615,"source_end_line":3658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3615-L3658","statement_sha256":"eb5dba3e62ebf23e73f5a8679c3a4e5997d7c5c8ac580964e0c9f5bd34550b18","origin":"The Stacks Project","memory_eligible":false,"source_rank":1041,"rank":1041,"depth":2,"x":2068.798,"y":197.793,"cluster":"commutative-algebra"},{"id":"stacks:0GLX","tag":"0GLX","title":"Nakayama's lemma · Lemma 0GLX","summary":"Let R be a ring, let S ⊂ R be a multiplicative subset, let I ⊂ R be an ideal, and let M be a finite R-module. If x_1, …, x_r ∈ M generate S^-1(M/IM) as an S^-1(R/I)-module, then there exists an f ∈ S + I such that x_1, …, x_r generate M_f as an R_f-module.M, then x_1, …, x_r generate M_f for some f ∈ S. (II) I = p is a prime ideal and S = R setminus p. The lemma says if x_1, …, x_r generate M ⊗_R kappa( p) then x_1, …, x_r generate M_f for some f ∈ R, f not ∈ p.","statement_latex":"Let $R$ be a ring, let $S \\subset R$ be a multiplicative subset,\nlet $I \\subset R$ be an ideal, and let $M$ be a finite $R$-module.\nIf $x_1, \\ldots, x_r \\in M$ generate $S^{-1}(M/IM)$\nas an $S^{-1}(R/I)$-module, then there exists an $f \\in S + I$\nsuch that $x_1, \\ldots, x_r$ generate $M_f$ as an\n$R_f$-module.\\footnote{Special cases: (I) $I = 0$. The lemma says\nif $x_1, \\ldots, x_r$ generate $S^{-1}M$, then $x_1, \\ldots, x_r$\ngenerate $M_f$ for some $f \\in S$. (II) $I = \\mathfrak p$ is\na prime ideal and $S = R \\setminus \\mathfrak p$. The lemma says if\n$x_1, \\ldots, x_r$ generate $M \\otimes_R \\kappa(\\mathfrak p)$\nthen $x_1, \\ldots, x_r$ generate $M_f$ for some\n$f \\in R$, $f \\not \\in \\mathfrak p$.}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nakayama's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLX","source_file":"algebra.tex","source_line":3698,"source_end_line":3712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3698-L3712","statement_sha256":"9d59537675d6366de489f253958cd1b4815ebfaa7e49cda40a322f2db2b4a513","origin":"The Stacks Project","memory_eligible":false,"source_rank":1042,"rank":1042,"depth":3,"x":2019.193,"y":258.591,"cluster":"commutative-algebra"},{"id":"stacks:0E8M","tag":"0E8M","title":"Nakayama's lemma · Lemma 0E8M","summary":"Let A → B be a local homomorphism of local rings. Assume • B is finite as an A-module, • m_B is a finitely generated ideal, • A → B induces an isomorphism on residue fields, and • m_A/ m_A^2 → m_B/ m_B^2 is surjective. Then A → B is surjective.","statement_latex":"Let $A \\to B$ be a local homomorphism of local rings.\nAssume\n\\begin{enumerate}\n\\item $B$ is finite as an $A$-module,\n\\item $\\mathfrak m_B$ is a finitely generated ideal,\n\\item $A \\to B$ induces an isomorphism on residue fields, and\n\\item $\\mathfrak m_A/\\mathfrak m_A^2 \\to \\mathfrak m_B/\\mathfrak m_B^2$\nis surjective.\n\\end{enumerate}\nThen $A \\to B$ is surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nakayama's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8M","source_file":"algebra.tex","source_line":3735,"source_end_line":3747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3735-L3747","statement_sha256":"36605d88b9de5eb1b607608075ad99909707233a79b8729598acfaa249170317","origin":"The Stacks Project","memory_eligible":false,"source_rank":1043,"rank":1043,"depth":3,"x":2006.815,"y":185.231,"cluster":"commutative-algebra"},{"id":"stacks:00EC","tag":"00EC","title":"Open and closed subsets of spectra · Lemma 00EC","summary":"Let R be a ring. Let e ∈ R be an idempotent. In this case Spec(R) = D(e) amalg D(1-e).","statement_latex":"Let $R$ be a ring. Let $e \\in R$ be an idempotent.\nIn this case\n$$\n\\Spec(R) = D(e) \\amalg D(1-e).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Open and closed subsets of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EC","source_file":"algebra.tex","source_line":3774,"source_end_line":3781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3774-L3781","statement_sha256":"24c60021b7dbcc3d199960db8529c409c058f671377d2b6c4581a70e1c79cf92","origin":"The Stacks Project","memory_eligible":false,"source_rank":1044,"rank":1044,"depth":0,"x":2075.288,"y":232.547,"cluster":"commutative-algebra"},{"id":"stacks:00ED","tag":"00ED","title":"Open and closed subsets of spectra · Lemma 00ED","summary":"Let R_1 and R_2 be rings. Let R = R_1 × R_2. The maps R → R_1, (x, y) ↦ x and R → R_2, (x, y) ↦ y induce continuous maps Spec(R_1) → Spec(R) and Spec(R_2) → Spec(R). The induced map Spec(R_1) amalg Spec(R_2) → Spec(R) is a homeomorphism. In other words, the spectrum of R = R_1× R_2 is the disjoint union of the spectrum of R_1 and the spectrum of R_2.","statement_latex":"Let $R_1$ and $R_2$ be rings.\nLet $R = R_1 \\times R_2$.\nThe maps $R \\to R_1$, $(x, y) \\mapsto x$ and $R \\to R_2$,\n$(x, y) \\mapsto y$\ninduce continuous maps $\\Spec(R_1) \\to \\Spec(R)$ and\n$\\Spec(R_2) \\to \\Spec(R)$.\nThe induced map\n$$\n\\Spec(R_1) \\amalg \\Spec(R_2)\n\\longrightarrow\n\\Spec(R)\n$$\nis a homeomorphism. In other words,\nthe spectrum of $R = R_1\\times R_2$ is the\ndisjoint union of the spectrum of $R_1$ and the\nspectrum of $R_2$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Open and closed subsets of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ED","source_file":"algebra.tex","source_line":3821,"source_end_line":3839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3821-L3839","statement_sha256":"14e0c5b4fb538203e5fe5378516a883f66da53d63686d675f48d510b2ea2388c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1045,"rank":1045,"depth":3,"x":1986.294,"y":236.53,"cluster":"commutative-algebra"},{"id":"stacks:00EE","tag":"00EE","title":"Open and closed subsets of spectra · Lemma 00EE","summary":"Let R be a ring. For each U ⊂ Spec(R) which is open and closed there exists a unique idempotent e ∈ R such that U = D(e). This induces a 1-1 correspondence between open and closed subsets U ⊂ Spec(R) and idempotents e ∈ R.","statement_latex":"Let $R$ be a ring. For each $U \\subset \\Spec(R)$\nwhich is open and closed\nthere exists a unique idempotent $e \\in R$ such that\n$U = D(e)$. This induces a 1-1 correspondence between\nopen and closed subsets $U \\subset \\Spec(R)$ and\nidempotents $e \\in R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Open and closed subsets of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EE","source_file":"algebra.tex","source_line":3857,"source_end_line":3865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3857-L3865","statement_sha256":"336aa78c75e52adeba8085c02963f65e9b84358c69db897bc0252a74dcecd738","origin":"The Stacks Project","memory_eligible":false,"source_rank":1046,"rank":1046,"depth":2,"x":2049.031,"y":182.825,"cluster":"commutative-algebra"},{"id":"stacks:00EF","tag":"00EF","title":"Open and closed subsets of spectra · Lemma 00EF","summary":"Let R be a nonzero ring. Then Spec(R) is connected if and only if R has no nontrivial idempotents.","statement_latex":"Let $R$ be a nonzero ring. Then $\\Spec(R)$ is\nconnected if and only if $R$ has no nontrivial\nidempotents.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Open and closed subsets of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EF","source_file":"algebra.tex","source_line":3909,"source_end_line":3914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3909-L3914","statement_sha256":"b23e8c026af7a0e2cbcf4f432ca7abda8b43d7415636caae25a38d28594cd272","origin":"The Stacks Project","memory_eligible":false,"source_rank":1047,"rank":1047,"depth":3,"x":2045.941,"y":258.402,"cluster":"commutative-algebra"},{"id":"stacks:00EH","tag":"00EH","title":"Open and closed subsets of spectra · Lemma 00EH","summary":"Let I ⊂ R be a finitely generated ideal of a ring R such that I = I^2. Then • there exists an idempotent e ∈ R such that I = (e), • R/I ≅ R_e' for the idempotent e' = 1 - e ∈ R, and • V(I) is open and closed in Spec(R).","statement_latex":"Let $I \\subset R$ be a finitely generated ideal of a ring $R$\nsuch that $I = I^2$. Then\n\\begin{enumerate}\n\\item there exists an idempotent $e \\in R$ such that $I = (e)$,\n\\item $R/I \\cong R_{e'}$ for the idempotent $e' = 1 - e \\in R$, and\n\\item $V(I)$ is open and closed in $\\Spec(R)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Open and closed subsets of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EH","source_file":"algebra.tex","source_line":3921,"source_end_line":3930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3921-L3930","statement_sha256":"d075b821daaf3db37550b73f726e7f68e92ba7bf88d297bd9b5d2be372527f6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1048,"rank":1048,"depth":3,"x":1987.153,"y":200.633,"cluster":"commutative-algebra"},{"id":"stacks:04PP","tag":"04PP","title":"Connected components of spectra · Lemma 04PP","summary":"Let R be a ring. Let T ⊂ Spec(R) be a subset of the spectrum. The following are equivalent • T is closed and is a union of connected components of Spec(R), • T is an intersection of open and closed subsets of Spec(R), and • T = V(I) where I ⊂ R is an ideal generated by idempotents. Moreover, the ideal in (3) if it exists is unique.","statement_latex":"Let $R$ be a ring. Let $T \\subset \\Spec(R)$ be a subset of the spectrum.\nThe following are equivalent\n\\begin{enumerate}\n\\item $T$ is closed and is a union of connected components of\n$\\Spec(R)$,\n\\item $T$ is an intersection of open and closed subsets of\n$\\Spec(R)$, and\n\\item $T = V(I)$ where $I \\subset R$ is an ideal generated by idempotents.\n\\end{enumerate}\nMoreover, the ideal in (3) if it exists is unique.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Connected components of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PP","source_file":"algebra.tex","source_line":3967,"source_end_line":3979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L3967-L3979","statement_sha256":"cc1f5633667df1810fee29277d9b7f3ebce34245210b79a24cc84c44fe7b2ec6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1049,"rank":1049,"depth":4,"x":2077.4,"y":209.919,"cluster":"commutative-algebra"},{"id":"stacks:00EG","tag":"00EG","title":"Connected components of spectra · Lemma 00EG","summary":"Let R be a ring. A connected component of Spec(R) is of the form V(I), where I is an ideal generated by idempotents such that every idempotent of R either maps to 0 or 1 in R/I.","statement_latex":"Let $R$ be a ring.\nA connected component of\n$\\Spec(R)$ is of the form $V(I)$,\nwhere $I$ is an ideal generated by idempotents\nsuch that every idempotent of $R$ either maps to\n$0$ or $1$ in $R/I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Connected components of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EG","source_file":"algebra.tex","source_line":4007,"source_end_line":4015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4007-L4015","statement_sha256":"867f12553e38094bee3875b212606516d58aa5873b73dd00d854843c2ed50cf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1050,"rank":1050,"depth":4,"x":2003.025,"y":254.496,"cluster":"commutative-algebra"},{"id":"stacks:00HN","tag":"00HN","title":"Glueing properties · Lemma 00HN","summary":"Let R be a ring. • For an element x of an R-module M the following are equivalent • x = 0, • x maps to zero in M_ p for all p ∈ Spec(R), • x maps to zero in M_ m for all maximal ideals m of R. In other words, the map M → ∏_ m M_ m is injective. • Given an R-module M the following are equivalent • M is zero, • M_ p is zero for all p ∈ Spec(R), • M_ m is zero for all maximal ideals m of R. • Given a complex M_1 → M_2 → M_3 of R-modules the following are equivalent • M_1 →…","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item For an element $x$ of an $R$-module $M$ the following are equivalent\n\\begin{enumerate}\n\\item $x = 0$,\n\\item $x$ maps to zero in $M_\\mathfrak p$ for all $\\mathfrak p \\in \\Spec(R)$,\n\\item $x$ maps to zero in $M_{\\mathfrak m}$ for all maximal ideals\n$\\mathfrak m$ of $R$.\n\\end{enumerate}\nIn other words, the map $M \\to \\prod_{\\mathfrak m} M_{\\mathfrak m}$\nis injective.\n\\item Given an $R$-module $M$ the following are equivalent\n\\begin{enumerate}\n\\item $M$ is zero,\n\\item $M_{\\mathfrak p}$ is zero for all $\\mathfrak p \\in \\Spec(R)$,\n\\item $M_{\\mathfrak m}$ is zero for all maximal ideals $\\mathfrak m$ of $R$.\n\\end{enumerate}\n\\item Given a complex $M_1 \\to M_2 \\to M_3$\nof $R$-modules the following are equivalent\n\\begin{enumerate}\n\\item $M_1 \\to M_2 \\to M_3$ is exact,\n\\item for every prime $\\mathfrak p$ of $R$ the localization\n$M_{1, \\mathfrak p} \\to M_{2, \\mathfrak p} \\to M_{3, \\mathfrak p}$\nis exact,\n\\item for every maximal ideal $\\mathfrak m$ of $R$ the localization\n$M_{1, \\mathfrak m} \\to M_{2, \\mathfrak m} \\to M_{3, \\mathfrak m}$\nis exact.\n\\end{enumerate}\n\\item Given a map $f : M \\to M'$ of $R$-modules the following are equivalent\n\\begin{enumerate}\n\\item $f$ is injective,\n\\item $f_{\\mathfrak p} : M_\\mathfrak p \\to M'_\\mathfrak p$ is injective\nfor all primes $\\mathfrak p$ of $R$,\n\\item $f_{\\mathfrak m} : M_\\mathfrak m \\to M'_\\mathfrak m$ is injective\nfor all maximal ideals $\\mathfrak m$ of $R$.\n\\end{enumerate}\n\\item Given a map $f : M \\to M'$ of $R$-modules the following are equivalent\n\\begin{enumerate}\n\\item $f$ is surjective,\n\\item $f_{\\mathfrak p} : M_\\mathfrak p \\to M'_\\mathfrak p$ is surjective\nfor all primes $\\mathfrak p$ of $R$,\n\\item $f_{\\mathfrak m} : M_\\mathfrak m \\to M'_\\mathfrak m$ is surjective\nfor all maximal ideals $\\mathfrak m$ of $R$.\n\\end{enumerate}\n\\item Given a map $f : M \\to M'$ of $R$-modules the following are equivalent\n\\begin{enumerate}\n\\item $f$ is bijective,\n\\item $f_{\\mathfrak p} : M_\\mathfrak p \\to M'_\\mathfrak p$ is bijective\nfor all primes $\\mathfrak p$ of $R$,\n\\item $f_{\\mathfrak m} : M_\\mathfrak m \\to M'_\\mathfrak m$ is bijective\nfor all maximal ideals $\\mathfrak m$ of $R$.\n\\end{enumerate}\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Glueing properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HN","source_file":"algebra.tex","source_line":4050,"source_end_line":4105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4050-L4105","statement_sha256":"44d9644f5fb5dd9a6f76be1d05465d4e9dac940ad8b5d9d1f76bc950a89f9a40","origin":"The Stacks Project","memory_eligible":false,"source_rank":1051,"rank":1051,"depth":1,"x":2022.113,"y":179.062,"cluster":"commutative-algebra"},{"id":"stacks:00EO","tag":"00EO","title":"Glueing properties · Lemma 00EO","summary":"Zariski-local properties of modules and algebras Let R be a ring. Let M be an R-module. Let S be an R-algebra. Suppose that f_1, …, f_n is a finite list of elements of R such that ⋃ D(f_i) = Spec(R), in other words (f_1, …, f_n) = R. • If each M_f_i = 0 then M = 0. • If each M_f_i is a finite R_f_i-module, then M is a finite R-module. • If each M_f_i is a finitely presented R_f_i-module, then M is a finitely presented R-module. • Let M → N be a map of R-modules. If M_f_i…","statement_latex":"\\begin{slogan}\nZariski-local properties of modules and algebras\n\\end{slogan}\nLet $R$ be a ring. Let $M$ be an $R$-module. Let $S$ be an $R$-algebra.\nSuppose that $f_1, \\ldots, f_n$ is a finite list of\nelements of $R$ such that $\\bigcup D(f_i) = \\Spec(R)$,\nin other words $(f_1, \\ldots, f_n) = R$.\n\\begin{enumerate}\n\\item If each $M_{f_i} = 0$ then $M = 0$.\n\\item If each $M_{f_i}$ is a finite $R_{f_i}$-module,\nthen $M$ is a finite $R$-module.\n\\item If each $M_{f_i}$ is a finitely presented $R_{f_i}$-module,\nthen $M$ is a finitely presented $R$-module.\n\\item Let $M \\to N$ be a map of $R$-modules. If $M_{f_i} \\to N_{f_i}$\nis an isomorphism for each $i$ then $M \\to N$ is an isomorphism.\n\\item Let $0 \\to M'' \\to M \\to M' \\to 0$ be a complex of $R$-modules.\nIf $0 \\to M''_{f_i} \\to M_{f_i} \\to M'_{f_i} \\to 0$ is exact for each $i$,\nthen $0 \\to M'' \\to M \\to M' \\to 0$ is exact.\n\\item If each $R_{f_i}$ is Noetherian, then $R$ is Noetherian.\n\\item If each $S_{f_i}$ is a finite type $R_{f_i}$-algebra, then\n$S$ is a finite type $R$-algebra.\n\\item If each $S_{f_i}$ is of finite presentation over $R_{f_i}$, then\n$S$ is a finitely presented $R$-algebra.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Glueing properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EO","source_file":"algebra.tex","source_line":4133,"source_end_line":4159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4133-L4159","statement_sha256":"a50f3806b77f7165cb46f36ec9694eff70b1e7ab80a44230f973a8458536649f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1052,"rank":1052,"depth":2,"x":2068.921,"y":245.834,"cluster":"commutative-algebra"},{"id":"stacks:00EP","tag":"00EP","title":"Glueing properties · Lemma 00EP","summary":"Let R → S be a ring map. Suppose that g_1, …, g_n is a finite list of elements of S such that ⋃ D(g_i) = Spec(S) in other words (g_1, …, g_n) = S. • If each S_g_i is of finite type over R, then S is of finite type over R. • If each S_g_i is of finite presentation over R, then S is of finite presentation over R.","statement_latex":"Let $R \\to S$ be a ring map.\nSuppose that $g_1, \\ldots, g_n$ is a finite list of\nelements of $S$ such that $\\bigcup D(g_i) = \\Spec(S)$\nin other words $(g_1, \\ldots, g_n) = S$.\n\\begin{enumerate}\n\\item If each $S_{g_i}$ is of finite type over $R$, then $S$ is\nof finite type over $R$.\n\\item If each $S_{g_i}$ is of finite presentation over $R$,\nthen $S$ is of finite presentation over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Glueing properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EP","source_file":"algebra.tex","source_line":4219,"source_end_line":4231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4219-L4231","statement_sha256":"046aa124fd0f9ec590047cba6b7eb2fb08473f39b155791fbc0c03c7bef42582","origin":"The Stacks Project","memory_eligible":false,"source_rank":1053,"rank":1053,"depth":3,"x":1980.292,"y":223.049,"cluster":"commutative-algebra"},{"id":"stacks:00EK","tag":"00EK","title":"Glueing functions · Lemma 00EK","summary":"Let R be a ring. Let f_1, …, f_n be elements of R generating the unit ideal. Let M be an R-module. The sequence 0 → M xrightarrowα bigoplus_i = 1^n M_f_i xrightarrowβ bigoplus_i, j = 1^n M_f_i f_j is exact, where α(m) = (m/1, …, m/1) and β(m_1/f_1^e_1, …, m_n/f_n^e_n) = (m_i/f_i^e_i - m_j/f_j^e_j)_(i, j).","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_n$ be elements of $R$\ngenerating the unit ideal. Let $M$ be an $R$-module.\nThe sequence\n$$\n0 \\to\nM \\xrightarrow{\\alpha}\n\\bigoplus\\nolimits_{i = 1}^n M_{f_i} \\xrightarrow{\\beta}\n\\bigoplus\\nolimits_{i, j = 1}^n M_{f_i f_j}\n$$\nis exact, where $\\alpha(m) = (m/1, \\ldots, m/1)$\nand $\\beta(m_1/f_1^{e_1}, \\ldots, m_n/f_n^{e_n})\n= (m_i/f_i^{e_i} - m_j/f_j^{e_j})_{(i, j)}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Glueing functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EK","source_file":"algebra.tex","source_line":4306,"source_end_line":4320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4306-L4320","statement_sha256":"96b92b0942076f381a71d07095e0a4987848a22a1526ca8ab49fa513e596d8cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1054,"rank":1054,"depth":2,"x":2064.361,"y":189.404,"cluster":"commutative-algebra"},{"id":"stacks:00EJ","tag":"00EJ","title":"Glueing functions · Lemma 00EJ","summary":"Let R be a ring, and let f_1, f_2, … f_n∈ R generate the unit ideal in R. Then the following sequence is exact: 0 → R → bigoplus_i R_f_i → bigoplus_i, jR_f_if_j where the maps α : R → bigoplus_i R_f_i and β : bigoplus_i R_f_i → bigoplus_i, j R_f_if_j are defined as α(x) = (fracx1, …, fracx1) and β(fracx_1f_1^r_1, …, fracx_nf_n^r_n) = (fracx_if_i^r_i-fracx_jf_j^r_j in R_f_if_j).","statement_latex":"Let $R$ be a ring, and let $f_1, f_2, \\ldots f_n\\in R$ generate\nthe unit ideal in $R$.\nThen the following sequence is exact:\n$$\n0 \\longrightarrow\nR \\longrightarrow\n\\bigoplus\\nolimits_i R_{f_i} \\longrightarrow\n\\bigoplus\\nolimits_{i, j}R_{f_if_j}\n$$\nwhere the maps $\\alpha : R \\longrightarrow \\bigoplus_i R_{f_i}$\nand $\\beta : \\bigoplus_i R_{f_i} \\longrightarrow \\bigoplus_{i, j} R_{f_if_j}$\nare defined as\n$$\n\\alpha(x) = \\left(\\frac{x}{1}, \\ldots, \\frac{x}{1}\\right)\n\\text{ and }\n\\beta\\left(\\frac{x_1}{f_1^{r_1}}, \\ldots, \\frac{x_n}{f_n^{r_n}}\\right)\n=\n\\left(\\frac{x_i}{f_i^{r_i}}-\\frac{x_j}{f_j^{r_j}}~\\text{in}~R_{f_if_j}\\right).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Glueing functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EJ","source_file":"algebra.tex","source_line":4353,"source_end_line":4374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4353-L4374","statement_sha256":"301af7b9fe1a1eff8e7a47284159d6733b4c470ee417ea38b267c05c68620ffe","origin":"The Stacks Project","memory_eligible":false,"source_rank":1055,"rank":1055,"depth":3,"x":2029.264,"y":262.253,"cluster":"commutative-algebra"},{"id":"stacks:00EM","tag":"00EM","title":"Glueing functions · Lemma 00EM","summary":"Let R be a ring. If Spec(R) = U amalg V with both U and V open then R ≅ R_1 × R_2 with U ≅ Spec(R_1) and V ≅ Spec(R_2) via the maps in Lemma [Tag 00ED]. Moreover, both R_1 and R_2 are localizations as well as quotients of the ring R.","statement_latex":"Let $R$ be a ring.\nIf $\\Spec(R) = U \\amalg V$ with both $U$ and $V$ open\nthen $R \\cong R_1 \\times R_2$ with $U \\cong \\Spec(R_1)$\nand $V \\cong \\Spec(R_2)$ via the maps in Lemma \\ref{lemma-spec-product}.\nMoreover, both $R_1$ and $R_2$ are localizations as well as quotients\nof the ring $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Glueing functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EM","source_file":"algebra.tex","source_line":4384,"source_end_line":4392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4384-L4392","statement_sha256":"9d9b0bc8db077cf7cc9f8c94f5cc4ff06e693eedddf9ba207170726f8d1a8557","origin":"The Stacks Project","memory_eligible":false,"source_rank":1056,"rank":1056,"depth":4,"x":1996.411,"y":188.281,"cluster":"commutative-algebra"},{"id":"stacks:0565","tag":"0565","title":"Glueing functions · Lemma 0565","summary":"Let R be a ring. Let f_1, …, f_n ∈ R. Let M be an R-module. Then M → bigoplus M_f_i is injective if and only if M → bigoplus_i = 1, …, n M, m ↦ (f_1m, …, f_nm) is injective.","statement_latex":"Let $R$ be a ring.\nLet $f_1, \\ldots, f_n \\in R$.\nLet $M$ be an $R$-module.\nThen $M \\to \\bigoplus M_{f_i}$ is injective if and only if\n$$\nM \\longrightarrow \\bigoplus\\nolimits_{i = 1, \\ldots, n} M, \\quad\nm \\longmapsto (f_1m, \\ldots, f_nm)\n$$\nis injective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Glueing functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0565","source_file":"algebra.tex","source_line":4404,"source_end_line":4415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4404-L4415","statement_sha256":"40a26817ed4e73f979225701e4236c9c72a746f8c18b3a86071a244ec9f33a41","origin":"The Stacks Project","memory_eligible":false,"source_rank":1057,"rank":1057,"depth":0,"x":2080.503,"y":224.35,"cluster":"commutative-algebra"},{"id":"stacks:00EQ","tag":"00EQ","title":"Glueing functions · Lemma 00EQ","summary":"Let R be a ring. Let f_1, …, f_n ∈ R. Suppose we are given the following data: • For each i an R_f_i-module M_i. • For each pair i, j an R_f_if_j-module isomorphism ψ_ij : (M_i)_f_j → (M_j)_f_i. which satisfy the \"cocycle condition\" that all the diagrams xymatrix (M_i)_f_jf_k ar[rd]_ψ_ij ar[rr]^ψ_ik & & (M_k)_f_if_j & (M_j)_f_if_k ar[ru]_ψ_jk commute (for all triples i, j, k). Given this data define M = Ker( bigoplus_1 ≤ i ≤ n M_i → bigoplus_1 ≤ i, j ≤ n (M_i)_f_j ) where…","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_n \\in R$. Suppose we are given\nthe following data:\n\\begin{enumerate}\n\\item For each $i$ an $R_{f_i}$-module $M_i$.\n\\item For each pair $i, j$ an $R_{f_if_j}$-module isomorphism\n$\\psi_{ij} : (M_i)_{f_j} \\to (M_j)_{f_i}$.\n\\end{enumerate}\nwhich satisfy the ``cocycle condition'' that all the diagrams\n$$\n\\xymatrix{\n(M_i)_{f_jf_k}\n\\ar[rd]_{\\psi_{ij}}\n\\ar[rr]^{\\psi_{ik}}\n& &\n(M_k)_{f_if_j} \\\\\n&\n(M_j)_{f_if_k} \\ar[ru]_{\\psi_{jk}}\n}\n$$\ncommute (for all triples $i, j, k$). Given this data define\n$$\nM = \\Ker\\left(\n\\bigoplus\\nolimits_{1 \\leq i \\leq n} M_i\n\\longrightarrow\n\\bigoplus\\nolimits_{1 \\leq i, j \\leq n} (M_i)_{f_j}\n\\right)\n$$\nwhere $(m_1, \\ldots, m_n)$ maps to the element whose\n$(i, j)$th entry is $m_i/1 - \\psi_{ji}(m_j/1)$.\nThen the natural map $M \\to M_i$ induces an isomorphism\n$M_{f_i} \\to M_i$. Moreover $\\psi_{ij}(m/1) = m/1$\nfor all $m \\in M$ (with obvious notation).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Glueing functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EQ","source_file":"algebra.tex","source_line":4438,"source_end_line":4472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4438-L4472","statement_sha256":"36b496d86edf9a4d664370acd22879e8ffc22dd5283f9d284f11289e434040bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1058,"rank":1058,"depth":3,"x":1989.079,"y":245.564,"cluster":"commutative-algebra"},{"id":"stacks:00EU","tag":"00EU","title":"Zerodivisors and total rings of fractions · Lemma 00EU","summary":"Let p be a minimal prime of a ring R. Every element of the maximal ideal of R_ p is nilpotent. If R is reduced then R_ p is a field.","statement_latex":"Let $\\mathfrak p$ be a minimal prime of a ring $R$.\nEvery element of the maximal ideal of $R_{\\mathfrak p}$\nis nilpotent. If $R$ is reduced then $R_{\\mathfrak p}$\nis a field.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zerodivisors and total rings of fractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EU","source_file":"algebra.tex","source_line":4542,"source_end_line":4548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4542-L4548","statement_sha256":"aa8a29486aa9bc549df21ebebae2ad7b5cdf585b37b420c6936ac67271089585","origin":"The Stacks Project","memory_eligible":false,"source_rank":1059,"rank":1059,"depth":1,"x":2039.66,"y":177.743,"cluster":"commutative-algebra"},{"id":"stacks:00EW","tag":"00EW","title":"Zerodivisors and total rings of fractions · Lemma 00EW","summary":"Let R be a reduced ring. Then • R is a subring of a product of fields, • R → ∏_ p minimal R_ p is an embedding into a product of fields, • ⋃_ p minimal p is the set of zerodivisors of R.","statement_latex":"Let $R$ be a reduced ring. Then\n\\begin{enumerate}\n\\item $R$ is a subring of a product of fields,\n\\item $R \\to \\prod_{\\mathfrak p\\text{ minimal}} R_{\\mathfrak p}$\nis an embedding into a product of fields,\n\\item $\\bigcup_{\\mathfrak p\\text{ minimal}} \\mathfrak p$ is the set\nof zerodivisors of $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zerodivisors and total rings of fractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EW","source_file":"algebra.tex","source_line":4559,"source_end_line":4569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4559-L4569","statement_sha256":"06fd331d94a0176ac85e4002686f72f138a588fe8329d5ac165026c5bc6c3de0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1060,"rank":1060,"depth":2,"x":2056.978,"y":256.803,"cluster":"commutative-algebra"},{"id":"stacks:02LW","tag":"02LW","title":"Zerodivisors and total rings of fractions · Lemma 02LW","summary":"Let R be a ring. Let S ⊂ R be a multiplicative subset consisting of nonzerodivisors. Then Q(R) ≅ Q(S^-1R). In particular Q(R) ≅ Q(Q(R)).","statement_latex":"Let $R$ be a ring.\nLet $S \\subset R$ be a multiplicative subset consisting of nonzerodivisors.\nThen $Q(R) \\cong Q(S^{-1}R)$.\nIn particular $Q(R) \\cong Q(Q(R))$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zerodivisors and total rings of fractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LW","source_file":"algebra.tex","source_line":4592,"source_end_line":4598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4592-L4598","statement_sha256":"f44bac57670758d9668a6bf40999c1004e151b631237b7b4c50c04c0a9d3d412","origin":"The Stacks Project","memory_eligible":false,"source_rank":1061,"rank":1061,"depth":0,"x":1980.295,"y":208.118,"cluster":"commutative-algebra"},{"id":"stacks:02LX","tag":"02LX","title":"Zerodivisors and total rings of fractions · Lemma 02LX","summary":"Let R be a ring. Assume that R has finitely many minimal primes q_1, …, q_t, and that q_1 ∪ … ∪ q_t is the set of zerodivisors of R. Then the total ring of fractions Q(R) is equal to R_ q_1 × … × R_ q_t.","statement_latex":"Let $R$ be a ring.\nAssume that $R$ has finitely many minimal primes\n$\\mathfrak q_1, \\ldots, \\mathfrak q_t$, and that\n$\\mathfrak q_1 \\cup \\ldots \\cup \\mathfrak q_t$ is the set\nof zerodivisors of $R$.\nThen the total ring of fractions $Q(R)$ is equal to\n$R_{\\mathfrak q_1} \\times \\ldots \\times R_{\\mathfrak q_t}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zerodivisors and total rings of fractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LX","source_file":"algebra.tex","source_line":4611,"source_end_line":4620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4611-L4620","statement_sha256":"b4ce7483224ca837722fcbb074504d999c08b83c8af3cec10f7f06216000e18c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1062,"rank":1062,"depth":5,"x":2076.407,"y":200.473,"cluster":"commutative-algebra"},{"id":"stacks:00ES","tag":"00ES","title":"Irreducible components of spectra · Lemma 00ES","summary":"Let R be a ring. • For a prime p ⊂ R the closure of ( p) in the Zariski topology is V( p). In a formula overline( p) = V( p). • The irreducible closed subsets of Spec(R) are exactly the subsets V( p), with p ⊂ R a prime. • The irreducible components (see Topology, Definition [Tag 004V]) of Spec(R) are exactly the subsets V( p), with p ⊂ R a minimal prime.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item For a prime $\\mathfrak p \\subset R$ the closure\nof $\\{\\mathfrak p\\}$ in the Zariski topology is $V(\\mathfrak p)$.\nIn a formula $\\overline{\\{\\mathfrak p\\}} = V(\\mathfrak p)$.\n\\item The irreducible closed subsets of $\\Spec(R)$ are\nexactly the subsets $V(\\mathfrak p)$, with $\\mathfrak p \\subset R$\na prime.\n\\item The irreducible components (see Topology,\nDefinition \\ref{topology-definition-irreducible-components})\nof $\\Spec(R)$ are  exactly the subsets $V(\\mathfrak p)$,\nwith $\\mathfrak p \\subset R$ a minimal prime.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Irreducible components of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ES","source_file":"algebra.tex","source_line":4664,"source_end_line":4679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4664-L4679","statement_sha256":"47c94726f06e53599047dbffb5caa8c95361caffbfc2fef4dc4b10a62ae22c30","origin":"The Stacks Project","memory_eligible":false,"source_rank":1063,"rank":1063,"depth":1,"x":2011.404,"y":260.906,"cluster":"commutative-algebra"},{"id":"stacks:090M","tag":"090M","title":"Irreducible components of spectra · Lemma 090M","summary":"The spectrum of a ring is a spectral space, see Topology, Definition [Tag 08YG].","statement_latex":"The spectrum of a ring is a spectral space, see Topology, Definition\n\\ref{topology-definition-spectral-space}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Irreducible components of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090M","source_file":"algebra.tex","source_line":4706,"source_end_line":4710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4706-L4710","statement_sha256":"054ebfdad705ddf7481e720bfd2e15bac856d770ba90932cecb7ce0bd1f1dad9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1064,"rank":1064,"depth":4,"x":2010.735,"y":179.111,"cluster":"commutative-algebra"},{"id":"stacks:00ET","tag":"00ET","title":"Irreducible components of spectra · Lemma 00ET","summary":"Let R be a ring. Let p ⊂ R be a prime. • the set of irreducible closed subsets of Spec(R) passing through p is in one-to-one correspondence with primes q ⊂ R_ p. • The set of irreducible components of Spec(R) passing through p is in one-to-one correspondence with minimal primes q ⊂ R_ p.","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime.\n\\begin{enumerate}\n\\item the set of irreducible closed subsets of $\\Spec(R)$\npassing through $\\mathfrak p$ is in one-to-one correspondence with\nprimes $\\mathfrak q \\subset R_{\\mathfrak p}$.\n\\item The set of irreducible components of $\\Spec(R)$ passing through\n$\\mathfrak p$ is in one-to-one correspondence with minimal\nprimes $\\mathfrak q \\subset R_{\\mathfrak p}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Irreducible components of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ET","source_file":"algebra.tex","source_line":4717,"source_end_line":4728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4717-L4728","statement_sha256":"ff3ede2293f66e584a3a95ae8569ca140445abdf1d3a43fb5498d5c638d1bc01","origin":"The Stacks Project","memory_eligible":false,"source_rank":1065,"rank":1065,"depth":2,"x":2077.286,"y":239.301,"cluster":"commutative-algebra"},{"id":"stacks:00EV","tag":"00EV","title":"Irreducible components of spectra · Lemma 00EV","summary":"Let R be a ring. Let p be a minimal prime of R. Let W ⊂ Spec(R) be a quasi-compact open not containing the point p. Then there exists an f ∈ R, f not ∈ p such that D(f) ∩ W = ∅.","statement_latex":"Let $R$ be a ring.\nLet $\\mathfrak p$ be a minimal prime of $R$.\nLet $W \\subset \\Spec(R)$ be a quasi-compact open\nnot containing the point $\\mathfrak p$. Then there\nexists an $f \\in R$, $f \\not \\in \\mathfrak p$ such\nthat $D(f) \\cap W = \\emptyset$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Irreducible components of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EV","source_file":"algebra.tex","source_line":4738,"source_end_line":4746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4738-L4746","statement_sha256":"48a76b101e18924fa3d362afdef4f96cfffe61321fe178f6f018c034e3c97074","origin":"The Stacks Project","memory_eligible":false,"source_rank":1066,"rank":1066,"depth":2,"x":1979.399,"y":232.652,"cluster":"commutative-algebra"},{"id":"stacks:04MG","tag":"04MG","title":"Irreducible components of spectra · Lemma 04MG","summary":"Let R be a ring. Let X = Spec(R) as a topological space. The following are equivalent • X is profinite, • X is Hausdorff, • X is totally disconnected. • every quasi-compact open of X is closed, • there are no nontrivial inclusions between its prime ideals, • every prime ideal is a maximal ideal, • every prime ideal is minimal, • every standard open D(f) ⊂ X is closed, and • add more here.","statement_latex":"Let $R$ be a ring. Let $X = \\Spec(R)$ as a topological space.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is profinite,\n\\item $X$ is Hausdorff,\n\\item $X$ is totally disconnected.\n\\item every quasi-compact open of $X$ is closed,\n\\item there are no nontrivial inclusions between its prime ideals,\n\\item every prime ideal is a maximal ideal,\n\\item every prime ideal is minimal,\n\\item every standard open $D(f) \\subset X$ is closed, and\n\\item add more here.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Irreducible components of spectra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MG","source_file":"algebra.tex","source_line":4758,"source_end_line":4773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L4758-L4773","statement_sha256":"90da4637d2ddfaeb7f01c90f0e82bd1f17697448f4fb28e59bf2f469fda70a09","origin":"The Stacks Project","memory_eligible":false,"source_rank":1067,"rank":1067,"depth":5,"x":2057.251,"y":181.801,"cluster":"commutative-algebra"},{"id":"stacks:05K8","tag":"05K8","title":"A meta-observation about prime ideals · Lemma 05K8","summary":"Let R be a ring. For a principal ideal J ⊂ R, and for any ideal I ⊂ J we have I = J (I : J).","statement_latex":"Let $R$ be a ring. For a principal ideal $J \\subset R$, and for any ideal\n$I \\subset J$ we have $I = J (I : J)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"A meta-observation about prime ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05K8","source_file":"algebra.tex","source_line":5115,"source_end_line":5119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5115-L5119","statement_sha256":"fe0516fff152349f7358901a545821feeb3e39f91774bb01c3a2809e74340ed4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1068,"rank":1068,"depth":0,"x":2040.668,"y":263.81,"cluster":"commutative-algebra"},{"id":"stacks:05K9","tag":"05K9","title":"A meta-observation about prime ideals · Definition 05K9","summary":"Let R be a ring. Let F be a set of ideals of R. We say F is an Oka family if R ∈ F and whenever I ⊂ R is an ideal and (I : a), (I, a) ∈ F for some a ∈ R, then I ∈ F.","statement_latex":"Let $R$ be a ring. Let $\\mathcal{F}$ be a set of ideals of $R$. We say\n$\\mathcal{F}$ is an {\\it Oka family} if $R \\in \\mathcal{F}$ and\nwhenever $I \\subset R$ is an ideal and $(I : a), (I, a) \\in \\mathcal{F}$\nfor some $a \\in R$, then $I \\in \\mathcal{F}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"A meta-observation about prime ideals","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05K9","source_file":"algebra.tex","source_line":5134,"source_end_line":5140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5134-L5140","statement_sha256":"d98cad136b31b8e39e718529d6325ca351ef4980c98e221e7f8307e6b624d3fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1069,"rank":1069,"depth":0,"x":1986.727,"y":193.642,"cluster":"commutative-algebra"},{"id":"stacks:05KE","tag":"05KE","title":"A meta-observation about prime ideals · Proposition 05KE","summary":"If F is an Oka family of ideals, then any maximal element of the complement of F is prime.","statement_latex":"If $\\mathcal{F}$ is an Oka family of ideals, then any maximal element of\nthe complement of $\\mathcal{F}$ is prime.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"A meta-observation about prime ideals","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KE","source_file":"algebra.tex","source_line":5202,"source_end_line":5206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5202-L5206","statement_sha256":"8833a41c6d3196125e5a7f3e50309ec3656213cb258c86b9019c5893f4df83fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1070,"rank":1070,"depth":0,"x":2083.321,"y":214.861,"cluster":"commutative-algebra"},{"id":"stacks:05KF","tag":"05KF","title":"A meta-observation about prime ideals · Lemma 05KF","summary":"Let R be a ring. Let S be a multiplicative subset of R. An ideal I ⊂ R which is maximal with respect to the property that I ∩ S = ∅ is prime.","statement_latex":"Let $R$ be a ring. Let $S$ be a multiplicative subset of $R$.\nAn ideal $I \\subset R$ which is maximal with respect to the property\nthat $I \\cap S = \\emptyset$ is prime.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"A meta-observation about prime ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KF","source_file":"algebra.tex","source_line":5222,"source_end_line":5227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5222-L5227","statement_sha256":"832550ec2ff2860228561c1cff133c684dc509426c8d027754408b1a6b3164b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1071,"rank":1071,"depth":1,"x":1994.673,"y":254.182,"cluster":"commutative-algebra"},{"id":"stacks:05KG","tag":"05KG","title":"A meta-observation about prime ideals · Lemma 05KG","summary":"Let R be a ring. • An ideal I ⊂ R maximal with respect to not being finitely generated is prime. • If every prime ideal of R is finitely generated, then every ideal of R is finitely generated.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item An ideal $I \\subset R$ maximal with respect to not being\nfinitely generated is prime.\n\\item If every prime ideal of $R$ is\nfinitely generated, then\nevery ideal of $R$ is finitely generated\\footnote{Later we will say\nthat $R$ is Noetherian.}.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"A meta-observation about prime ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KG","source_file":"algebra.tex","source_line":5237,"source_end_line":5248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5237-L5248","statement_sha256":"7ddee8132f6c4f9553b61e147f50e11bef23a927f84a90780a49047d05d31571","origin":"The Stacks Project","memory_eligible":false,"source_rank":1072,"rank":1072,"depth":1,"x":2028.556,"y":174.569,"cluster":"commutative-algebra"},{"id":"stacks:05KH","tag":"05KH","title":"A meta-observation about prime ideals · Lemma 05KH","summary":"Let R be a ring. • An ideal I ⊂ R maximal with respect to not being principal is prime. • If every prime ideal of R is principal, then every ideal of R is principal.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item An ideal $I \\subset R$ maximal with respect to not being\nprincipal is prime.\n\\item If every prime ideal of $R$ is principal, then\nevery ideal of $R$ is principal.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"A meta-observation about prime ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KH","source_file":"algebra.tex","source_line":5264,"source_end_line":5273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5264-L5273","statement_sha256":"6d9dea3a0dcb904c5e3681539b48fb91b32f14c244f829e4a9f4a5070e2cda96","origin":"The Stacks Project","memory_eligible":false,"source_rank":1073,"rank":1073,"depth":1,"x":2067.748,"y":252.81,"cluster":"commutative-algebra"},{"id":"stacks:05KI","tag":"05KI","title":"A meta-observation about prime ideals · Lemma 05KI","summary":"Let R be a ring. • An ideal maximal among the ideals which do not contain a nonzerodivisor is prime. • If R is nonzero and every nonzero prime ideal in R contains a nonzerodivisor, then R is a domain.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item An ideal maximal among the ideals which do not contain a\nnonzerodivisor is prime.\n\\item If $R$ is nonzero and every nonzero prime ideal in $R$\ncontains a nonzerodivisor, then $R$ is a domain.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"A meta-observation about prime ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KI","source_file":"algebra.tex","source_line":5288,"source_end_line":5297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5288-L5297","statement_sha256":"4fc0efa49438a990714d6f937ad38c876f2fc03ab4d206c4ae46d79647406d58","origin":"The Stacks Project","memory_eligible":false,"source_rank":1074,"rank":1074,"depth":2,"x":1975.568,"y":217.215,"cluster":"commutative-algebra"},{"id":"stacks:00F6","tag":"00F6","title":"Images of ring maps of finite presentation · Lemma 00F6","summary":"Let U ⊂ Spec(R) be open. The following are equivalent: • U is retrocompact in Spec(R), • U is quasi-compact, • U is a finite union of standard opens, and • there exists a finitely generated ideal I ⊂ R such that X setminus V(I) = U.","statement_latex":"Let $U \\subset \\Spec(R)$ be open. The following\nare equivalent:\n\\begin{enumerate}\n\\item $U$ is retrocompact in $\\Spec(R)$,\n\\item $U$ is quasi-compact,\n\\item $U$ is a finite union of standard opens, and\n\\item there exists a finitely generated ideal $I \\subset R$ such\nthat $X \\setminus V(I) = U$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00F6","source_file":"algebra.tex","source_line":5382,"source_end_line":5393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5382-L5393","statement_sha256":"63258705a189ae51f39a6cf66b73abdbfb8f555ab523667509e97c3f317fe43f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1075,"rank":1075,"depth":3,"x":2072.542,"y":191.055,"cluster":"commutative-algebra"},{"id":"stacks:00F7","tag":"00F7","title":"Images of ring maps of finite presentation · Lemma 00F7","summary":"Let φ : R → S be a ring map. The induced continuous map f : Spec(S) → Spec(R) is quasi-compact. For any constructible set E ⊂ Spec(R) the inverse image f^-1(E) is constructible in Spec(S).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nThe induced continuous map $f : \\Spec(S) \\to \\Spec(R)$\nis quasi-compact. For any constructible set $E \\subset \\Spec(R)$\nthe inverse image $f^{-1}(E)$ is constructible in $\\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00F7","source_file":"algebra.tex","source_line":5414,"source_end_line":5420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5414-L5420","statement_sha256":"acb6869d3a4b3dda7c50ef4d879144f5536eefe297360512f8aeafa066622d27","origin":"The Stacks Project","memory_eligible":false,"source_rank":1076,"rank":1076,"depth":4,"x":2021.875,"y":265.658,"cluster":"commutative-algebra"},{"id":"stacks:0G1P","tag":"0G1P","title":"Images of ring maps of finite presentation · Lemma 0G1P","summary":"Let R be a ring. A subset of Spec(R) is constructible if and only if it can be written as a finite union of subsets of the form D(f) ∩ V(g_1, …, g_m) for f, g_1, …, g_m ∈ R.","statement_latex":"Let $R$ be a ring. A subset of $\\Spec(R)$ is constructible if and only\nif it can be written as a finite union of subsets of the form\n$D(f) \\cap V(g_1, \\ldots, g_m)$ for $f, g_1, \\ldots, g_m \\in R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1P","source_file":"algebra.tex","source_line":5433,"source_end_line":5438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5433-L5438","statement_sha256":"ba16ca5ecd978cd9b8859337d2419442d71404c63d86171c22e30f1a32370548","origin":"The Stacks Project","memory_eligible":false,"source_rank":1077,"rank":1077,"depth":4,"x":1999.156,"y":181.575,"cluster":"commutative-algebra"},{"id":"stacks:00F8","tag":"00F8","title":"Images of ring maps of finite presentation · Lemma 00F8","summary":"Let R be a ring and let T ⊂ Spec(R) be constructible. Then there exists a ring map R → S of finite presentation such that T is the image of Spec(S) in Spec(R).","statement_latex":"Let $R$ be a ring and let $T \\subset \\Spec(R)$\nbe constructible. Then there exists a ring map $R \\to S$ of\nfinite presentation such that $T$ is the image of\n$\\Spec(S)$ in $\\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00F8","source_file":"algebra.tex","source_line":5453,"source_end_line":5459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5453-L5459","statement_sha256":"c4f63e0ecaeb472d2042c08868f8e029dd1a5457c7e43908249290f86fae528f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1078,"rank":1078,"depth":5,"x":2083.848,"y":230.874,"cluster":"commutative-algebra"},{"id":"stacks:00F9","tag":"00F9","title":"Images of ring maps of finite presentation · Lemma 00F9","summary":"Let R be a ring. Let f be an element of R. Let S = R_f. Then the image of a constructible subset of Spec(S) is constructible in Spec(R).","statement_latex":"Let $R$ be a ring.\nLet $f$ be an element of $R$.\nLet $S = R_f$.\nThen the image of a constructible subset of $\\Spec(S)$\nis constructible in $\\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00F9","source_file":"algebra.tex","source_line":5474,"source_end_line":5481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5474-L5481","statement_sha256":"9efb2505ca75d7ea441279159584cfc1dc37f0c2e1207eedd515f902581d070e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1079,"rank":1079,"depth":4,"x":1981.366,"y":242.622,"cluster":"commutative-algebra"},{"id":"stacks:00FA","tag":"00FA","title":"Images of ring maps of finite presentation · Lemma 00FA","summary":"Let R be a ring. Let I be a finitely generated ideal of R. Let S = R/I. Then the image of a constructible subset of Spec(S) is constructible in Spec(R).","statement_latex":"Let $R$ be a ring.\nLet $I$ be a finitely generated ideal of $R$.\nLet $S = R/I$.\nThen the image of a constructible subset of $\\Spec(S)$\nis constructible in $\\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FA","source_file":"algebra.tex","source_line":5495,"source_end_line":5502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5495-L5502","statement_sha256":"efdaba6911ffc06fbaece113baf7ee7e8c58537b3192bd5852633399b5fd6a50","origin":"The Stacks Project","memory_eligible":false,"source_rank":1080,"rank":1080,"depth":4,"x":2047.736,"y":175.557,"cluster":"commutative-algebra"},{"id":"stacks:00FB","tag":"00FB","title":"Images of ring maps of finite presentation · Lemma 00FB","summary":"Let R be a ring. The map Spec(R[x]) → Spec(R) is open, and the image of any standard open is a quasi-compact open.","statement_latex":"Let $R$ be a ring. The map $\\Spec(R[x]) \\to \\Spec(R)$\nis open, and the image of any standard open is a quasi-compact\nopen.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FB","source_file":"algebra.tex","source_line":5524,"source_end_line":5529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5524-L5529","statement_sha256":"296a311cede5e161262ed74d44267a79bdcec76fd4b49bd2f81a85bded989324","origin":"The Stacks Project","memory_eligible":false,"source_rank":1081,"rank":1081,"depth":1,"x":2052.745,"y":262.994,"cluster":"commutative-algebra"},{"id":"stacks:00FC","tag":"00FC","title":"Images of ring maps of finite presentation · Lemma 00FC","summary":"Let R → A be a ring homomorphism. Assume A ≅ R^⊕ n as an R-module. Let f ∈ A. The multiplication map m_f: A → A is R-linear and hence has a characteristic polynomial P(T) = T^n + r_n-1T^n-1 + … + r_0 ∈ R[T]. For any prime p ∈ Spec(R), f acts nilpotently on A ⊗_R kappa(p) if and only if p ∈ V(r_0, …, r_n-1).","statement_latex":"Let $R \\to A$ be a ring homomorphism.\nAssume $A \\cong R^{\\oplus n}$ as an $R$-module.\nLet $f \\in A$. The multiplication map $m_f: A\n\\to A$ is $R$-linear and hence\nhas a characteristic polynomial\n$P(T) = T^n + r_{n-1}T^{n-1} + \\ldots + r_0 \\in R[T]$.\nFor any prime\n$\\mathfrak{p} \\in \\Spec(R)$, $f$ acts nilpotently on $A\n\\otimes_R \\kappa(\\mathfrak{p})$ if and only if $\\mathfrak p \\in\nV(r_0, \\ldots, r_{n-1})$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FC","source_file":"algebra.tex","source_line":5554,"source_end_line":5566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5554-L5566","statement_sha256":"16818eac3828f91d52cfdf51ad6b7470f13fc061caa68f4be75b6f7253b33292","origin":"The Stacks Project","memory_eligible":false,"source_rank":1082,"rank":1082,"depth":0,"x":1978.464,"y":201.133,"cluster":"commutative-algebra"},{"id":"stacks:00FD","tag":"00FD","title":"Images of ring maps of finite presentation · Lemma 00FD","summary":"Let R be a ring. Let f, g ∈ R[x] be polynomials. Assume the leading coefficient of g is a unit of R. There exists elements r_i∈ R, i = 1…, n such that the image of D(f) ∩ V(g) in Spec(R) is ⋃_i = 1, …, n D(r_i).","statement_latex":"Let $R$ be a ring. Let $f, g \\in R[x]$ be polynomials.\nAssume the leading coefficient of $g$ is a unit of $R$.\nThere exists elements $r_i\\in R$, $i = 1\\ldots, n$ such that\nthe image of $D(f) \\cap V(g)$ in $\\Spec(R)$ is\n$\\bigcup_{i = 1, \\ldots, n} D(r_i)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FD","source_file":"algebra.tex","source_line":5597,"source_end_line":5604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5597-L5604","statement_sha256":"5820402bd32bc30831fbf2581af834e8e639f8dd2b348dbf56c94bc118c49965","origin":"The Stacks Project","memory_eligible":false,"source_rank":1083,"rank":1083,"depth":1,"x":2083.369,"y":204.613,"cluster":"commutative-algebra"},{"id":"stacks:00FE","tag":"00FE","title":"Chevalley's Theorem · Theorem 00FE","summary":"Suppose that R → S is of finite presentation. The image of a constructible subset of Spec(S) in Spec(R) is constructible.","statement_latex":"Suppose that $R \\to S$ is of finite presentation.\nThe image of a constructible subset of\n$\\Spec(S)$ in $\\Spec(R)$ is constructible.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Images of ring maps of finite presentation","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FE","source_file":"algebra.tex","source_line":5645,"source_end_line":5650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5645-L5650","statement_sha256":"e95ea3867f6dd7d524d265d4a400be3769e16a8c65ea5e9250174ff431881be0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1084,"rank":1084,"depth":5,"x":2002.923,"y":261.78,"cluster":"commutative-algebra"},{"id":"stacks:00FG","tag":"00FG","title":"More on images · Lemma 00FG","summary":"Let R ⊂ S be an inclusion of domains. Assume that R → S is of finite type. There exists a nonzero f ∈ R, and a nonzero g ∈ S such that R_f → S_fg is of finite presentation.","statement_latex":"Let $R \\subset S$ be an inclusion of domains.\nAssume that $R \\to S$ is of finite type.\nThere exists a nonzero $f \\in R$, and a nonzero $g \\in S$\nsuch that $R_f \\to S_{fg}$ is of finite presentation.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More on images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FG","source_file":"algebra.tex","source_line":5729,"source_end_line":5735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5729-L5735","statement_sha256":"b16fab6e3ae98a70785dd1eeb43780780cd955ab0dc0d965519f124f9f9508d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1085,"rank":1085,"depth":0,"x":2016.319,"y":173.661,"cluster":"commutative-algebra"},{"id":"stacks:00FH","tag":"00FH","title":"More on images · Lemma 00FH","summary":"Let R → S be a finite type ring map. Denote X = Spec(R) and Y = Spec(S). Write f : Y → X the induced map of spectra. Let E ⊂ Y = Spec(S) be a constructible set. If a point xi ∈ X is in f(E), then overline(xi) ∩ f(E) contains an open dense subset of overline(xi).","statement_latex":"Let $R \\to S$ be a finite type ring map.\nDenote $X = \\Spec(R)$ and $Y = \\Spec(S)$.\nWrite $f : Y \\to X$ the induced\nmap of spectra. Let $E \\subset Y = \\Spec(S)$ be a\nconstructible set.\nIf a point $\\xi \\in X$ is in $f(E)$, then\n$\\overline{\\{\\xi\\}} \\cap f(E)$ contains an open\ndense subset of $\\overline{\\{\\xi\\}}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More on images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FH","source_file":"algebra.tex","source_line":5769,"source_end_line":5779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5769-L5779","statement_sha256":"64066b8e7abd88454a24209126ff8663f9ce5e9959e8cbfcdb05184a663289c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1086,"rank":1086,"depth":6,"x":2077.523,"y":246.501,"cluster":"commutative-algebra"},{"id":"stacks:00FI","tag":"00FI","title":"More on images · Lemma 00FI","summary":"Let φ : R → S be a ring map. The following are equivalent: • The map Spec(S) → Spec(R) is surjective. • For any ideal I ⊂ R the inverse image of sqrtIS in R is equal to sqrtI. • For any radical ideal I ⊂ R the inverse image of IS in R is equal to I. • For every prime p of R the inverse image of p S in R is p. In this case the same is true after any base change: Given a ring map R → R' the ring map R' → R' ⊗_R S has the equivalent properties (1), (2), (3) as well.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The map $\\Spec(S) \\to \\Spec(R)$ is surjective.\n\\item For any ideal $I \\subset R$\nthe inverse image of $\\sqrt{IS}$ in $R$ is equal to $\\sqrt{I}$.\n\\item For any radical ideal $I \\subset R$ the inverse image\nof $IS$ in $R$ is equal to $I$.\n\\item For every prime $\\mathfrak p$ of $R$ the inverse\nimage of $\\mathfrak p S$ in $R$ is $\\mathfrak p$.\n\\end{enumerate}\nIn this case the same is true after any base change: Given a ring map\n$R \\to R'$ the ring map $R' \\to R' \\otimes_R S$ has the equivalent\nproperties (1), (2), (3) as well.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More on images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FI","source_file":"algebra.tex","source_line":5821,"source_end_line":5837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5821-L5837","statement_sha256":"c0d2a1dfc7198e1f820b14643b149478de35884ba9961ff71bb6236560dda152","origin":"The Stacks Project","memory_eligible":false,"source_rank":1087,"rank":1087,"depth":1,"x":1973.445,"y":227.45,"cluster":"commutative-algebra"},{"id":"stacks:00FJ","tag":"00FJ","title":"More on images · Lemma 00FJ","summary":"Let R be a domain. Let φ : R → S be a ring map. The following are equivalent: • The ring map R → S is injective. • The image Spec(S) → Spec(R) contains a dense set of points. • There exists a prime ideal q ⊂ S whose inverse image in R is (0).","statement_latex":"Let $R$ be a domain. Let $\\varphi : R \\to S$ be a ring map.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The ring map $R \\to S$ is injective.\n\\item The image $\\Spec(S) \\to \\Spec(R)$\ncontains a dense set of points.\n\\item There exists a prime ideal $\\mathfrak q \\subset S$\nwhose inverse image in $R$ is $(0)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More on images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FJ","source_file":"algebra.tex","source_line":5871,"source_end_line":5882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5871-L5882","statement_sha256":"2d4d4a25c51c7d411da6de9d5e46a8e34ab0991ce50c3d24a2ab267c61774485","origin":"The Stacks Project","memory_eligible":false,"source_rank":1088,"rank":1088,"depth":1,"x":2065.837,"y":182.283,"cluster":"commutative-algebra"},{"id":"stacks:00FK","tag":"00FK","title":"More on images · Lemma 00FK","summary":"Let R ⊂ S be an injective ring map. Then Spec(S) → Spec(R) hits all the minimal primes.","statement_latex":"Let $R \\subset S$ be an injective ring map.\nThen $\\Spec(S) \\to \\Spec(R)$\nhits all the minimal primes.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More on images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FK","source_file":"algebra.tex","source_line":5905,"source_end_line":5910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5905-L5910","statement_sha256":"c745eb007c41747eaf339b607a81acb58bffb14683051e6ac8d556179ac3ca65","origin":"The Stacks Project","memory_eligible":false,"source_rank":1089,"rank":1089,"depth":1,"x":2033.919,"y":268.316,"cluster":"commutative-algebra"},{"id":"stacks:00FL","tag":"00FL","title":"More on images · Lemma 00FL","summary":"Let R → S be a ring map. The following are equivalent: • The kernel of R → S consists of nilpotent elements. • The minimal primes of R are in the image of Spec(S) → Spec(R). • The image of Spec(S) → Spec(R) is dense in Spec(R).","statement_latex":"Let $R \\to S$ be a ring map. The following are equivalent:\n\\begin{enumerate}\n\\item The kernel of $R \\to S$ consists of nilpotent elements.\n\\item The minimal primes of $R$ are in the image of\n$\\Spec(S) \\to \\Spec(R)$.\n\\item The image of $\\Spec(S) \\to \\Spec(R)$ is dense\nin $\\Spec(R)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More on images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FL","source_file":"algebra.tex","source_line":5920,"source_end_line":5930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5920-L5930","statement_sha256":"46ffb771c2753fdf9d23c5dbc2aa4f7b105611bd2d942b3a0cf3760d04f7c130","origin":"The Stacks Project","memory_eligible":false,"source_rank":1090,"rank":1090,"depth":2,"x":1988.111,"y":186.48,"cluster":"commutative-algebra"},{"id":"stacks:0CAN","tag":"0CAN","title":"More on images · Lemma 0CAN","summary":"Let R → S be a ring map. If a minimal prime p ⊂ R is in the image of Spec(S) → Spec(R), then it is the image of a minimal prime.","statement_latex":"Let $R \\to S$ be a ring map. If a minimal prime $\\mathfrak p \\subset R$\nis in the image of $\\Spec(S) \\to \\Spec(R)$, then it is the image\nof a minimal prime.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More on images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAN","source_file":"algebra.tex","source_line":5946,"source_end_line":5951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5946-L5951","statement_sha256":"d22ec99307173b65373aff9b7065dec7b166c80b568d3e8ab01f5f6079f09e35","origin":"The Stacks Project","memory_eligible":false,"source_rank":1091,"rank":1091,"depth":1,"x":2088.044,"y":220.952,"cluster":"commutative-algebra"},{"id":"stacks:0H7L","tag":"0H7L","title":"More on images · Lemma 0H7L","summary":"Let A ⊂ B be an inclusion of domains inducing an algebraic extension of fraction fields. If J ⊂ B is a nonzero ideal, then A ∩ J is nonzero too. Thus the image of a proper closed subset of Spec(B) is not dense in Spec(A).","statement_latex":"Let $A \\subset B$ be an inclusion of domains inducing an algebraic extension\nof fraction fields. If $J \\subset B$ is a nonzero ideal, then $A \\cap J$\nis nonzero too. Thus the image of a proper closed subset of $\\Spec(B)$\nis not dense in $\\Spec(A)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More on images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7L","source_file":"algebra.tex","source_line":5961,"source_end_line":5967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5961-L5967","statement_sha256":"8f4384598c0cb17adabd0d9843a5cdf574777f0109226f90535f2931f5e4b613","origin":"The Stacks Project","memory_eligible":false,"source_rank":1092,"rank":1092,"depth":0,"x":1986.285,"y":252.345,"cluster":"commutative-algebra"},{"id":"stacks:00FN","tag":"00FN","title":"Noetherian rings · Lemma 00FN","summary":"Noetherian property is stable by passage to finite type extension and localization. Any finitely generated ring over a Noetherian ring is Noetherian. Any localization of a Noetherian ring is Noetherian.","statement_latex":"\\begin{slogan}\nNoetherian property is stable by passage to finite type extension\nand localization.\n\\end{slogan}\nAny finitely generated ring over a Noetherian ring\nis Noetherian. Any localization of a Noetherian ring\nis Noetherian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FN","source_file":"algebra.tex","source_line":5990,"source_end_line":5999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L5990-L5999","statement_sha256":"613474a81c5a3982a1f126afb13032dc0d13d7d1e7fabb85337420e0a4b75ad5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1093,"rank":1093,"depth":0,"x":2036.246,"y":171.179,"cluster":"commutative-algebra"},{"id":"stacks:0306","tag":"0306","title":"Noetherian rings · Lemma 0306","summary":"If R is a Noetherian ring, then so is the formal power series ring R[[x_1, …, x_n]].","statement_latex":"If $R$ is a Noetherian ring, then so is the formal power\nseries ring $R[[x_1, \\ldots, x_n]]$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0306","source_file":"algebra.tex","source_line":6027,"source_end_line":6031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6027-L6031","statement_sha256":"b394b51c7305cf51264623a76f0b753f14ffe6df2541b4b5edbad6939dbcc3f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1094,"rank":1094,"depth":0,"x":2064.773,"y":259.678,"cluster":"commutative-algebra"},{"id":"stacks:00FO","tag":"00FO","title":"Noetherian rings · Lemma 00FO","summary":"Any finite type algebra over a field is Noetherian. Any finite type algebra over Z is Noetherian.","statement_latex":"Any finite type algebra over a field is Noetherian.\nAny finite type algebra over $\\mathbf{Z}$ is Noetherian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FO","source_file":"algebra.tex","source_line":6074,"source_end_line":6078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6074-L6078","statement_sha256":"1a0f8d87337a8a87ad034dae56949c8bfd6fbe44fe7e0061ddd45695a9c0483f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1095,"rank":1095,"depth":1,"x":1972.258,"y":210.441,"cluster":"commutative-algebra"},{"id":"stacks:00FP","tag":"00FP","title":"Noetherian rings · Lemma 00FP","summary":"Let R be a Noetherian ring. • Any finite R-module is of finite presentation. • Any submodule of a finite R-module is finite. • Any finite type R-algebra is of finite presentation over R.","statement_latex":"Let $R$ be a Noetherian ring.\n\\begin{enumerate}\n\\item Any finite $R$-module is of finite presentation.\n\\item Any submodule of a finite $R$-module is finite.\n\\item Any finite type $R$-algebra is of finite presentation over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FP","source_file":"algebra.tex","source_line":6087,"source_end_line":6095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6087-L6095","statement_sha256":"a72a4fbee4da5e08069fc0efd49786489eeeaa95a1209f372215015466a42807","origin":"The Stacks Project","memory_eligible":false,"source_rank":1096,"rank":1096,"depth":2,"x":2080.433,"y":194.199,"cluster":"commutative-algebra"},{"id":"stacks:00FQ","tag":"00FQ","title":"Noetherian rings · Lemma 00FQ","summary":"If R is a Noetherian ring then Spec(R) is a Noetherian topological space, see Topology, Definition [Tag 0051].","statement_latex":"If $R$ is a Noetherian ring then $\\Spec(R)$\nis a Noetherian topological space, see Topology,\nDefinition \\ref{topology-definition-noetherian}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FQ","source_file":"algebra.tex","source_line":6119,"source_end_line":6124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6119-L6124","statement_sha256":"351ea928f78cd8a06de430f76c80d3890ec5240ce9fc1e2e5f56d80fd0c6e131","origin":"The Stacks Project","memory_eligible":false,"source_rank":1097,"rank":1097,"depth":1,"x":2013.505,"y":267.801,"cluster":"commutative-algebra"},{"id":"stacks:00FR","tag":"00FR","title":"Noetherian rings · Lemma 00FR","summary":"A Noetherian affine scheme has finitely many generic points. If R is a Noetherian ring then Spec(R) has finitely many irreducible components. In other words R has finitely many minimal primes.","statement_latex":"\\begin{slogan}\nA Noetherian affine scheme has finitely many generic points.\n\\end{slogan}\nIf $R$ is a Noetherian ring then $\\Spec(R)$\nhas finitely many irreducible components. In other words\n$R$ has finitely many minimal primes.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FR","source_file":"algebra.tex","source_line":6134,"source_end_line":6142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6134-L6142","statement_sha256":"767d6d8300a9df1535d8f5fc9c635338b1aa100e23f83c60f9d4547671893bd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1098,"rank":1098,"depth":3,"x":2003.636,"y":175.246,"cluster":"commutative-algebra"},{"id":"stacks:0CY6","tag":"0CY6","title":"Noetherian rings · Lemma 0CY6","summary":"Let R → S be a ring map. Let R → R' be of finite type. If S is Noetherian, then the base change S' = R' ⊗_R S is Noetherian.","statement_latex":"Let $R \\to S$ be a ring map. Let $R \\to R'$ be of finite type.\nIf $S$ is Noetherian, then the base change $S' = R' \\otimes_R S$\nis Noetherian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CY6","source_file":"algebra.tex","source_line":6152,"source_end_line":6157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6152-L6157","statement_sha256":"501cee71e84db1f7c7d8e855fd1879ac2893161f02d715571749e3678b370094","origin":"The Stacks Project","memory_eligible":false,"source_rank":1099,"rank":1099,"depth":3,"x":2085.612,"y":238.1,"cluster":"commutative-algebra"},{"id":"stacks:045I","tag":"045I","title":"Noetherian rings · Lemma 045I","summary":"Let k be a field and let R be a Noetherian k-algebra. If K/k is a finitely generated field extension then K ⊗_k R is Noetherian.","statement_latex":"Let $k$ be a field and let $R$ be a Noetherian $k$-algebra.\nIf $K/k$ is a finitely generated field extension then\n$K \\otimes_k R$ is Noetherian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045I","source_file":"algebra.tex","source_line":6165,"source_end_line":6170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6165-L6170","statement_sha256":"2be0e6efa95634afb69cf0d0200bd68334633e3192e89ff267d0339467baa0b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1100,"rank":1100,"depth":4,"x":1974.256,"y":238.268,"cluster":"commutative-algebra"},{"id":"stacks:0BX1","tag":"0BX1","title":"Noetherian rings · Lemma 0BX1","summary":"Let R be a ring and p ⊂ R be a prime. There exists an f ∈ R, f not ∈ p such that R_f → R_ p is injective in each of the following cases • R is a domain, • R is Noetherian, or • R is reduced and has finitely many minimal primes.","statement_latex":"Let $R$ be a ring and $\\mathfrak p \\subset R$ be a prime.\nThere exists an $f \\in R$, $f \\not \\in \\mathfrak p$ such\nthat $R_f \\to R_\\mathfrak p$ is injective in each of the\nfollowing cases\n\\begin{enumerate}\n\\item $R$ is a domain,\n\\item $R$ is Noetherian, or\n\\item $R$ is reduced and has finitely many minimal primes.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BX1","source_file":"algebra.tex","source_line":6186,"source_end_line":6197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6186-L6197","statement_sha256":"7e1a0e6d119e9ec0870403ff724fe9cd78aed2e39c7d54e6516b127867ba50f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1101,"rank":1101,"depth":3,"x":2056.5,"y":174.754,"cluster":"commutative-algebra"},{"id":"stacks:06RN","tag":"06RN","title":"Noetherian rings · Lemma 06RN","summary":"Any surjective endomorphism of a Noetherian ring is an isomorphism.","statement_latex":"Any surjective endomorphism of a Noetherian ring is an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RN","source_file":"algebra.tex","source_line":6218,"source_end_line":6221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6218-L6221","statement_sha256":"6259eea56ef87aac351299c688288a9ade81581f69133ed627eb6f751e4b9812","origin":"The Stacks Project","memory_eligible":false,"source_rank":1102,"rank":1102,"depth":0,"x":2046.899,"y":268.556,"cluster":"commutative-algebra"},{"id":"stacks:00IL","tag":"00IL","title":"Locally nilpotent ideals · Definition 00IL","summary":"Let R be a ring. Let I ⊂ R be an ideal. We say I is locally nilpotent if for every x ∈ I there exists an n ∈ N such that x^n = 0. We say I is nilpotent if there exists an n ∈ N such that I^n = 0.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nWe say $I$ is {\\it locally nilpotent} if for every\n$x \\in I$ there exists an $n \\in \\mathbf{N}$ such\nthat $x^n = 0$. We say $I$ is {\\it nilpotent} if\nthere exists an $n \\in \\mathbf{N}$ such that $I^n = 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Locally nilpotent ideals","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IL","source_file":"algebra.tex","source_line":6244,"source_end_line":6251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6244-L6251","statement_sha256":"29f779a4cb6d2efad3f04a63816d2e33a2f77b34e2b6cd549bbd65dba68a992b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1103,"rank":1103,"depth":0,"x":1978.327,"y":193.701,"cluster":"commutative-algebra"},{"id":"stacks:0544","tag":"0544","title":"Locally nilpotent ideals · Lemma 0544","summary":"Let R → R' be a ring map and let I ⊂ R be a locally nilpotent ideal. Then IR' is a locally nilpotent ideal of R'.","statement_latex":"Let $R \\to R'$ be a ring map and let $I \\subset R$ be a locally nilpotent\nideal. Then $IR'$ is a locally nilpotent ideal of $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Locally nilpotent ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0544","source_file":"algebra.tex","source_line":6267,"source_end_line":6271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6267-L6271","statement_sha256":"c4ade75e0e5e861aaa7e6b47c0704b08f42f8bf01b082146d90779073e271a68","origin":"The Stacks Project","memory_eligible":false,"source_rank":1104,"rank":1104,"depth":0,"x":2089.439,"y":210.04,"cluster":"commutative-algebra"},{"id":"stacks:0AMG","tag":"0AMG","title":"Locally nilpotent ideals · Lemma 0AMG","summary":"Let R be a ring and let I ⊂ R be a locally nilpotent ideal. An element x of R is a unit if and only if the image of x in R/I is a unit.","statement_latex":"Let $R$ be a ring and let $I \\subset R$ be a locally nilpotent\nideal.\nAn element $x$ of $R$ is a unit if and only if the image of $x$\nin $R/I$ is a unit.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Locally nilpotent ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMG","source_file":"algebra.tex","source_line":6279,"source_end_line":6285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6279-L6285","statement_sha256":"a7d72b30c68b7b2aaac531db381300aae7fad99263b5994c9ea656506edbfc79","origin":"The Stacks Project","memory_eligible":false,"source_rank":1105,"rank":1105,"depth":0,"x":1994.067,"y":261.201,"cluster":"commutative-algebra"},{"id":"stacks:00IM","tag":"00IM","title":"Locally nilpotent ideals · Lemma 00IM","summary":"An ideal in a Noetherian ring is nilpotent if each element of the ideal is nilpotent. Let R be a Noetherian ring. Let I, J be ideals of R. Suppose J ⊂ sqrtI. Then J^n ⊂ I for some n. In particular, in a Noetherian ring the notions of \"locally nilpotent ideal\" and \"nilpotent ideal\" coincide.","statement_latex":"\\begin{slogan}\nAn ideal in a Noetherian ring is nilpotent if each element\nof the ideal is nilpotent.\n\\end{slogan}\nLet $R$ be a Noetherian ring. Let $I, J$ be ideals of $R$.\nSuppose $J \\subset \\sqrt{I}$. Then $J^n \\subset I$ for some $n$.\nIn particular, in a Noetherian ring the notions of\n``locally nilpotent ideal''\nand ``nilpotent ideal'' coincide.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Locally nilpotent ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IM","source_file":"algebra.tex","source_line":6299,"source_end_line":6310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6299-L6310","statement_sha256":"e68f52f0f1931dc10b1cf41681f6ef71df1cf8aefc6af67f91287be31b1f9a94","origin":"The Stacks Project","memory_eligible":false,"source_rank":1106,"rank":1106,"depth":0,"x":2023.343,"y":169.071,"cluster":"commutative-algebra"},{"id":"stacks:00J9","tag":"00J9","title":"Locally nilpotent ideals · Lemma 00J9","summary":"Let R be a ring. Let I ⊂ R be a locally nilpotent ideal. Then R → R/I induces a bijection on idempotents.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal.\nThen $R \\to R/I$ induces a bijection on idempotents.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Locally nilpotent ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00J9","source_file":"algebra.tex","source_line":6318,"source_end_line":6322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6318-L6322","statement_sha256":"ef85dfb6d09c7e30adf8548feba7316ff3fa200489e79cf29cd484a10700d29b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1107,"rank":1107,"depth":3,"x":2076.007,"y":253.882,"cluster":"commutative-algebra"},{"id":"stacks:05BU","tag":"05BU","title":"Locally nilpotent ideals · Lemma 05BU","summary":"Let A be a possibly noncommutative algebra. Let e ∈ A be an element such that x = e^2 - e is nilpotent. Then there exists an idempotent of the form e' = e + x(∑ a_i, je^ix^j) ∈ A with a_i, j ∈ Z.","statement_latex":"Let $A$ be a possibly noncommutative algebra.\nLet $e \\in A$ be an element such that $x = e^2 - e$ is nilpotent.\nThen there exists an idempotent of the form\n$e' = e + x(\\sum a_{i, j}e^ix^j) \\in A$\nwith $a_{i, j} \\in \\mathbf{Z}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Locally nilpotent ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BU","source_file":"algebra.tex","source_line":6385,"source_end_line":6392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6385-L6392","statement_sha256":"32ec7c5eace74e578c72be64e92ae15934c7c97e5acf40a7762bc145e9c1651a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1108,"rank":1108,"depth":4,"x":1968.64,"y":221.125,"cluster":"commutative-algebra"},{"id":"stacks:0CAP","tag":"0CAP","title":"Locally nilpotent ideals · Lemma 0CAP","summary":"Let R be a ring. Let I ⊂ R be a locally nilpotent ideal. Let n ≥ 1 be an integer which is invertible in R/I. Then • the nth power map 1 + I → 1 + I, 1 + x ↦ (1 + x)^n is a bijection, • a unit of R is a nth power if and only if its image in R/I is an nth power.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal.\nLet $n \\geq 1$ be an integer which is invertible in $R/I$. Then\n\\begin{enumerate}\n\\item the $n$th power map $1 + I \\to 1 + I$, $1 + x \\mapsto (1 + x)^n$\nis a bijection,\n\\item a unit of $R$ is a $n$th power if and only if its image in $R/I$\nis an $n$th power.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Locally nilpotent ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAP","source_file":"algebra.tex","source_line":6401,"source_end_line":6411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6401-L6411","statement_sha256":"c17a78e87ca0ab812364284060c5379c7a546b439231509729e2502104748f24","origin":"The Stacks Project","memory_eligible":false,"source_rank":1109,"rank":1109,"depth":1,"x":2074.478,"y":184.244,"cluster":"commutative-algebra"},{"id":"stacks:02JH","tag":"02JH","title":"Curiosity · Lemma 02JH","summary":"Let R be a ring. Let S ⊂ R be a multiplicative subset. Assume the image of the map Spec(S^-1R) → Spec(R) is closed. Then S^-1R ≅ R/I for some ideal I ⊂ R.","statement_latex":"Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.\nAssume the image of the map $\\Spec(S^{-1}R) \\to \\Spec(R)$\nis closed. Then $S^{-1}R \\cong R/I$ for some ideal $I \\subset R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Curiosity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JH","source_file":"algebra.tex","source_line":6455,"source_end_line":6460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6455-L6460","statement_sha256":"f25f15c215d1e4028a17a11ac88cb08c079643c458e435d6f371fe2b2656a462","origin":"The Stacks Project","memory_eligible":false,"source_rank":1110,"rank":1110,"depth":1,"x":2025.944,"y":271.761,"cluster":"commutative-algebra"},{"id":"stacks:02JI","tag":"02JI","title":"Curiosity · Lemma 02JI","summary":"Let R be a ring. Let S ⊂ R be a multiplicative subset. Assume the image of the map Spec(S^-1R) → Spec(R) is closed. If R is Noetherian, or Spec(R) is a Noetherian topological space, or S is finitely generated as a monoid, then R ≅ S^-1R × R' for some ring R'.","statement_latex":"Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.\nAssume the image of the map $\\Spec(S^{-1}R) \\to \\Spec(R)$\nis closed. If $R$ is Noetherian, or $\\Spec(R)$ is a\nNoetherian topological space, or $S$ is finitely generated as a monoid,\nthen $R \\cong S^{-1}R \\times R'$ for some ring $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Curiosity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JI","source_file":"algebra.tex","source_line":6479,"source_end_line":6486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6479-L6486","statement_sha256":"7226cc2977593aefc0ed2539045c67cf14060f5b06f57c725c276647dc2e385f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1111,"rank":1111,"depth":5,"x":1991.248,"y":179.408,"cluster":"commutative-algebra"},{"id":"stacks:00FV","tag":"00FV","title":"Hilbert Nullstellensatz · Theorem 00FV","summary":"Let k be a field. • For any maximal ideal m ⊂ k[x_1, …, x_n] the field extension kappa( m)/k is finite. • Any radical ideal I ⊂ k[x_1, …, x_n] is the intersection of maximal ideals containing it. The same is true in any finite type k-algebra.","statement_latex":"Let $k$ be a field.\n\\begin{enumerate}\n\\item\n\nFor any maximal ideal $\\mathfrak m \\subset k[x_1, \\ldots, x_n]$\nthe field extension $\\kappa(\\mathfrak m)/k$ is finite.\n\\item\n\nAny radical ideal $I \\subset k[x_1, \\ldots, x_n]$\nis the intersection of maximal ideals containing it.\n\\end{enumerate}\nThe same is true in any finite type $k$-algebra.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Hilbert Nullstellensatz","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FV","source_file":"algebra.tex","source_line":6523,"source_end_line":6537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6523-L6537","statement_sha256":"ad2497a3bf054d6f5ea7b30d3e6cf8ce90d602f02450833eb8d5be9167a3ebbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1112,"rank":1112,"depth":6,"x":2091.402,"y":227.967,"cluster":"commutative-algebra"},{"id":"stacks:00FY","tag":"00FY","title":"Hilbert Nullstellensatz · Lemma 00FY","summary":"Let R be a ring. Let K be a field. If R ⊂ K and K is of finite type over R, then there exists an f ∈ R such that R_f is a field, and K/R_f is a finite field extension.","statement_latex":"Let $R$ be a ring. Let $K$ be a field.\nIf $R \\subset K$ and $K$ is of finite type over $R$,\nthen there exists an $f \\in R$ such that $R_f$ is a field,\nand $K/R_f$ is a finite field extension.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Hilbert Nullstellensatz","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FY","source_file":"algebra.tex","source_line":6610,"source_end_line":6616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6610-L6616","statement_sha256":"0e1e3f97995f60c463af8c7b1b97a74f87cfe346eeb4c42324a7122816001f4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1113,"rank":1113,"depth":7,"x":1978.161,"y":249.053,"cluster":"commutative-algebra"},{"id":"stacks:00G0","tag":"00G0","title":"Jacobson rings · Definition 00G0","summary":"Let R be a ring. We say that R is a Jacobson ring if every radical ideal I is the intersection of the maximal ideals containing it.","statement_latex":"Let $R$ be a ring. We say that $R$ is a\n{\\it Jacobson ring} if every radical\nideal $I$ is the intersection of the\nmaximal ideals containing it.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00G0","source_file":"algebra.tex","source_line":6659,"source_end_line":6665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6659-L6665","statement_sha256":"03556743ac1b75010a24bd8dfb38e7f4d7f0bfe81f9b35db08190414ab877e67","origin":"The Stacks Project","memory_eligible":false,"source_rank":1114,"rank":1114,"depth":0,"x":2044.906,"y":169.011,"cluster":"commutative-algebra"},{"id":"stacks:00G1","tag":"00G1","title":"Jacobson rings · Lemma 00G1","summary":"Any algebra of finite type over a field is Jacobson.","statement_latex":"Any algebra of finite type over a field is Jacobson.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00G1","source_file":"algebra.tex","source_line":6667,"source_end_line":6670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6667-L6670","statement_sha256":"07b8c1566c8695d426ccca9f914f8d39941c19b8a437bcc6cf26988e99674c8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1115,"rank":1115,"depth":7,"x":2060.102,"y":266.193,"cluster":"commutative-algebra"},{"id":"stacks:00G2","tag":"00G2","title":"Jacobson rings · Lemma 00G2","summary":"Let R be a ring. If every prime ideal of R is the intersection of the maximal ideals containing it, then R is Jacobson.","statement_latex":"Let $R$ be a ring. If every prime ideal of $R$ is the\nintersection of the maximal ideals containing it,\nthen $R$ is Jacobson.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00G2","source_file":"algebra.tex","source_line":6677,"source_end_line":6682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6677-L6682","statement_sha256":"9c1294809b4571a2dd6ea87943ff1244f5d43f2df3e1ca38e356d95a0552d887","origin":"The Stacks Project","memory_eligible":false,"source_rank":1116,"rank":1116,"depth":1,"x":1970.482,"y":202.968,"cluster":"commutative-algebra"},{"id":"stacks:00G3","tag":"00G3","title":"Jacobson rings · Lemma 00G3","summary":"A ring R is Jacobson if and only if Spec(R) is Jacobson, see Topology, Definition [Tag 005U].","statement_latex":"A ring $R$ is Jacobson if and only if $\\Spec(R)$\nis Jacobson, see Topology,\nDefinition \\ref{topology-definition-space-jacobson}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00G3","source_file":"algebra.tex","source_line":6691,"source_end_line":6696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6691-L6696","statement_sha256":"d9ae5fdf452a30a67ae35cb399083b230a33c7f1ad607c356d280ca92259cff6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1117,"rank":1117,"depth":1,"x":2087.752,"y":198.725,"cluster":"commutative-algebra"},{"id":"stacks:034J","tag":"034J","title":"Jacobson rings · Lemma 034J","summary":"Let R be a ring. If R is not Jacobson there exist a prime p ⊂ R, an element f ∈ R such that the following hold • p is not a maximal ideal, • f not ∈ p, • V( p) ∩ D(f) = ( p), and • (R/ p)_f is a field. On the other hand, if R is Jacobson, then for any pair ( p, f) such that (1) and (2) hold the set V( p) ∩ D(f) is infinite.","statement_latex":"Let $R$ be a ring. If $R$ is not Jacobson there exist\na prime $\\mathfrak p \\subset R$, an element $f \\in R$\nsuch that the following hold\n\\begin{enumerate}\n\\item $\\mathfrak p$ is not a maximal ideal,\n\\item $f \\not \\in \\mathfrak p$,\n\\item $V(\\mathfrak p) \\cap D(f) = \\{\\mathfrak p\\}$, and\n\\item $(R/\\mathfrak p)_f$ is a field.\n\\end{enumerate}\nOn the other hand, if $R$ is Jacobson, then for any pair $(\\mathfrak p, f)$\nsuch that (1) and (2) hold the set $V(\\mathfrak p) \\cap D(f)$ is\ninfinite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034J","source_file":"algebra.tex","source_line":6727,"source_end_line":6741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6727-L6741","statement_sha256":"1919361870f5b80658ab35241ed69a48284dba5aa4d5109eed270d3473b46b42","origin":"The Stacks Project","memory_eligible":false,"source_rank":1118,"rank":1118,"depth":3,"x":2004.449,"y":268.599,"cluster":"commutative-algebra"},{"id":"stacks:00G4","tag":"00G4","title":"Jacobson rings · Lemma 00G4","summary":"The ring Z is a Jacobson ring. More generally, let R be a ring such that • R is a domain, • R is Noetherian, • any nonzero prime ideal is a maximal ideal, and • R has infinitely many maximal ideals. Then R is a Jacobson ring.","statement_latex":"The ring $\\mathbf{Z}$ is a Jacobson ring.\nMore generally, let $R$ be a ring such that\n\\begin{enumerate}\n\\item $R$ is a domain,\n\\item $R$ is Noetherian,\n\\item any nonzero prime ideal is a maximal ideal, and\n\\item $R$ has infinitely many maximal ideals.\n\\end{enumerate}\nThen $R$ is a Jacobson ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00G4","source_file":"algebra.tex","source_line":6778,"source_end_line":6789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6778-L6789","statement_sha256":"adf2832e273d9ba2588845b9aca915bf9aae9895d37da23fb45a2e8b65dea0d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1119,"rank":1119,"depth":3,"x":2009.7,"y":169.52,"cluster":"commutative-algebra"},{"id":"stacks:00GA","tag":"00GA","title":"Jacobson rings · Lemma 00GA","summary":"Let R → S be a ring map. Let m ⊂ R be a maximal ideal. Let q ⊂ S be a prime ideal lying over m such that kappa( q)/kappa( m) is an algebraic field extension. Then q is a maximal ideal of S.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak m \\subset R$ be a maximal ideal.\nLet $\\mathfrak q \\subset S$ be a prime ideal\nlying over $\\mathfrak m$ such that $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak m)$\nis an algebraic field extension.\nThen $\\mathfrak q$ is a maximal ideal of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GA","source_file":"algebra.tex","source_line":6844,"source_end_line":6852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6844-L6852","statement_sha256":"32effca5044dc5121c2fb02e1f4b2901fdd39f8d95f10920abd4eb8fb766aba3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1120,"rank":1120,"depth":2,"x":2085.723,"y":245.778,"cluster":"commutative-algebra"},{"id":"stacks:00FT","tag":"00FT","title":"Jacobson rings · Lemma 00FT","summary":"Suppose that k is a field and suppose that V is a nonzero vector space over k. Assume the dimension of V (which is a cardinal number) is smaller than the cardinality of k. Then for any linear operator T : V → V there exists some monic polynomial P(t) ∈ k[t] such that P(T) is not invertible.","statement_latex":"Suppose that $k$ is a field and suppose that $V$ is a nonzero vector\nspace over $k$. Assume the dimension of $V$ (which is a cardinal number)\nis smaller than the cardinality of $k$. Then for any linear operator\n$T : V \\to V$ there exists some monic polynomial $P(t) \\in k[t]$ such that\n$P(T)$ is not invertible.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FT","source_file":"algebra.tex","source_line":6872,"source_end_line":6879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6872-L6879","statement_sha256":"13218aeb39c4cf2c6caf4b65a6f3a48d0a723d8138327af06d86d0d393eb32c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1121,"rank":1121,"depth":0,"x":1968.004,"y":232.646,"cluster":"commutative-algebra"},{"id":"stacks:00FU","tag":"00FU","title":"Jacobson rings · Theorem 00FU","summary":"Let k be a field. Let S be a k-algebra generated over k by the elements (x_i)_i ∈ I. Assume the cardinality of I is smaller than the cardinality of k. Then • for all maximal ideals m ⊂ S the field extension kappa( m)/k is algebraic, and • S is a Jacobson ring.","statement_latex":"Let $k$ be a field. Let $S$ be a $k$-algebra generated over $k$\nby the elements $\\{x_i\\}_{i \\in I}$. Assume the cardinality of $I$\nis smaller than the cardinality of $k$. Then\n\\begin{enumerate}\n\\item for all maximal ideals $\\mathfrak m \\subset S$ the field\nextension $\\kappa(\\mathfrak m)/k$\nis algebraic, and\n\\item $S$ is a Jacobson ring.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00FU","source_file":"algebra.tex","source_line":6891,"source_end_line":6902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6891-L6902","statement_sha256":"e9282ca1e6f1c1bb16417573e58d02cf29bbf49a2e0b9abbe4beb2a85776be99","origin":"The Stacks Project","memory_eligible":false,"source_rank":1122,"rank":1122,"depth":7,"x":2065.646,"y":175.371,"cluster":"commutative-algebra"},{"id":"stacks:046V","tag":"046V","title":"Jacobson rings · Lemma 046V","summary":"Let k be a field. Let S be a k-algebra. For any field extension K/k whose cardinality is larger than the cardinality of S we have • for every maximal ideal m of S_K the field kappa( m) is algebraic over K, and • S_K is a Jacobson ring.","statement_latex":"Let $k$ be a field. Let $S$ be a $k$-algebra.\nFor any field extension $K/k$ whose cardinality is larger\nthan the cardinality of $S$ we have\n\\begin{enumerate}\n\\item for every maximal ideal $\\mathfrak m$ of $S_K$ the field\n$\\kappa(\\mathfrak m)$ is algebraic over $K$, and\n\\item $S_K$ is a Jacobson ring.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046V","source_file":"algebra.tex","source_line":6945,"source_end_line":6955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6945-L6955","statement_sha256":"cdb4e88101abe3830911c43783ebaa092f3039559be50bd49f0be2684991d8d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1123,"rank":1123,"depth":8,"x":2039.632,"y":273.285,"cluster":"commutative-algebra"},{"id":"stacks:00G6","tag":"00G6","title":"Jacobson rings · Lemma 00G6","summary":"Let R be a Jacobson ring. Let f ∈ R. The ring R_f is Jacobson and maximal ideals of R_f correspond to maximal ideals of R not containing f.","statement_latex":"Let $R$ be a Jacobson ring. Let $f \\in R$. The ring $R_f$ is Jacobson and\nmaximal ideals of $R_f$ correspond to maximal ideals of $R$ not containing $f$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00G6","source_file":"algebra.tex","source_line":6980,"source_end_line":6984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L6980-L6984","statement_sha256":"3a7eb35cd802a0fc5ab1cf53ef5d9641409f800152188f030f10c9aed377edd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1124,"rank":1124,"depth":2,"x":1979.907,"y":186.078,"cluster":"commutative-algebra"},{"id":"stacks:00G9","tag":"00G9","title":"Jacobson rings · Lemma 00G9","summary":"Let R be a Jacobson ring. Let I ⊂ R be an ideal. The ring R/I is Jacobson and maximal ideals of R/I correspond to maximal ideals of R containing I.","statement_latex":"Let $R$ be a Jacobson ring. Let $I \\subset R$ be an ideal.\nThe ring $R/I$ is Jacobson and maximal ideals\nof $R/I$ correspond to maximal ideals of $R$ containing $I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00G9","source_file":"algebra.tex","source_line":7021,"source_end_line":7026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7021-L7026","statement_sha256":"bc76425bab0e22767b5d0fea6c8d0350a1334f0a8736fa7b3fd08450efe40678","origin":"The Stacks Project","memory_eligible":false,"source_rank":1125,"rank":1125,"depth":3,"x":2094.396,"y":216.58,"cluster":"commutative-algebra"},{"id":"stacks:0CY7","tag":"0CY7","title":"Jacobson rings · Lemma 0CY7","summary":"Let R be a Jacobson ring. Let K be a field. Let R ⊂ K and K is of finite type over R. Then R is a field and K/R is a finite field extension.","statement_latex":"Let $R$ be a Jacobson ring. Let $K$ be a field. Let $R \\subset K$ and\n$K$ is of finite type over $R$. Then $R$ is a field and $K/R$\nis a finite field extension.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CY7","source_file":"algebra.tex","source_line":7033,"source_end_line":7038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7033-L7038","statement_sha256":"3ae64f207365f406c05e8aa2d8090b188eccec6eefa2b3e79cab663f58ad1b5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1126,"rank":1126,"depth":8,"x":1985.142,"y":259.17,"cluster":"commutative-algebra"},{"id":"stacks:00GB","tag":"00GB","title":"Jacobson rings · Proposition 00GB","summary":"Let R be a Jacobson ring. Let R → S be a ring map of finite type. Then • The ring S is Jacobson. • The map Spec(S) → Spec(R) transforms closed points to closed points. • For m' ⊂ S maximal lying over m ⊂ R the field extension kappa( m')/kappa( m) is finite.","statement_latex":"Let $R$ be a Jacobson ring. Let $R \\to S$ be a\nring map of finite type. Then\n\\begin{enumerate}\n\\item The ring $S$ is Jacobson.\n\\item The map $\\Spec(S) \\to \\Spec(R)$ transforms\nclosed points to closed points.\n\\item For $\\mathfrak m' \\subset S$ maximal lying over $\\mathfrak m \\subset R$\nthe field extension $\\kappa(\\mathfrak m')/\\kappa(\\mathfrak m)$\nis finite.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GB","source_file":"algebra.tex","source_line":7050,"source_end_line":7062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7050-L7062","statement_sha256":"e5a0a5fc8bff25cb2d47a11c9811ea2d7543fba8322c26a80dbdeda2d295c299","origin":"The Stacks Project","memory_eligible":false,"source_rank":1127,"rank":1127,"depth":9,"x":2031.582,"y":165.513,"cluster":"commutative-algebra"},{"id":"stacks:00GC","tag":"00GC","title":"Jacobson rings · Lemma 00GC","summary":"Any finite type algebra over Z is Jacobson.","statement_latex":"Any finite type algebra over $\\mathbf{Z}$ is Jacobson.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GC","source_file":"algebra.tex","source_line":7089,"source_end_line":7092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7089-L7092","statement_sha256":"b61bb9cc058e37072ad157337d768f17f8f52e4411c7362160d41c5d83501b9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1128,"rank":1128,"depth":10,"x":2072.767,"y":261.189,"cluster":"commutative-algebra"},{"id":"stacks:00GD","tag":"00GD","title":"Jacobson rings · Lemma 00GD","summary":"Let R → S be a finite type ring map of Jacobson rings. Denote X = Spec(R) and Y = Spec(S). Write f : Y → X the induced map of spectra. Let E ⊂ Y = Spec(S) be a constructible set. Denote with a subscript _0 the set of closed points of a topological space. • We have f(E)_0 = f(E_0) = X_0 ∩ f(E). • A point xi ∈ X is in f(E) if and only if overline(xi) ∩ f(E_0) is dense in overline(xi).","statement_latex":"Let $R \\to S$ be a finite type ring map of Jacobson rings.\nDenote $X = \\Spec(R)$ and $Y = \\Spec(S)$.\nWrite $f : Y \\to X$ the induced\nmap of spectra. Let $E \\subset Y = \\Spec(S)$ be a\nconstructible set. Denote with a subscript ${}_0$ the set\nof closed points of a topological space.\n\\begin{enumerate}\n\\item We have $f(E)_0 = f(E_0) = X_0 \\cap f(E)$.\n\\item A point $\\xi \\in X$ is in $f(E)$ if and only if\n$\\overline{\\{\\xi\\}} \\cap f(E_0)$ is dense in $\\overline{\\{\\xi\\}}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GD","source_file":"algebra.tex","source_line":7099,"source_end_line":7112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7099-L7112","statement_sha256":"a23786d7829b8182eed454ec57bc06deb8ae7ac36a2d4aeac435c5e381307939","origin":"The Stacks Project","memory_eligible":false,"source_rank":1129,"rank":1129,"depth":10,"x":1965.165,"y":213.879,"cluster":"commutative-algebra"},{"id":"stacks:00GE","tag":"00GE","title":"Jacobson rings · Lemma 00GE","summary":"With notation as above. Assume that R is a Noetherian Jacobson ring. Further assume R → S is of finite type. There is a commutative diagram xymatrix Constr(Y) ar[r]^E ↦ E_0 ar[d]^E ↦ f(E) & Constr(Y_0) ar[d]^E ↦ f(E) Constr(X) ar[r]^E ↦ E_0 & Constr(X_0) where the horizontal arrows are the bijections from Topology, Lemma [Tag 005Y].","statement_latex":"With notation as above. Assume that $R$ is a Noetherian Jacobson ring.\nFurther assume $R \\to S$ is of finite type.\nThere is a commutative diagram\n$$\n\\xymatrix{\n\\text{Constr}(Y) \\ar[r]^{E \\mapsto E_0} \\ar[d]^{E \\mapsto f(E)} &\n\\text{Constr}(Y_0) \\ar[d]^{E \\mapsto f(E)} \\\\\n\\text{Constr}(X) \\ar[r]^{E \\mapsto E_0} &\n\\text{Constr}(X_0)\n}\n$$\nwhere the horizontal arrows are the bijections from\nTopology, Lemma \\ref{topology-lemma-jacobson-equivalent-constructible}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GE","source_file":"algebra.tex","source_line":7189,"source_end_line":7204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7189-L7204","statement_sha256":"467956e958c9ddbe380102cf95c9f4cc64e4f4bda68dcdf94bd1055c708302b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1130,"rank":1130,"depth":11,"x":2082.874,"y":187.637,"cluster":"commutative-algebra"},{"id":"stacks:00GI","tag":"00GI","title":"Finite and integral ring extensions · Definition 00GI","summary":"Let φ : R → S be a ring map. • An element s ∈ S is integral over R if there exists a monic polynomial P(x) ∈ R[x] such that P^φ(s) = 0, where P^φ(x) ∈ S[x] is the image of P under φ : R[x] → S[x]. • The ring map φ is integral if every s ∈ S is integral over R.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\n\\begin{enumerate}\n\\item An element $s \\in S$\nis {\\it integral over $R$} if there exists a monic\npolynomial $P(x) \\in R[x]$ such that\n$P^\\varphi(s) = 0$, where $P^\\varphi(x) \\in S[x]$\nis the image of $P$ under $\\varphi : R[x] \\to S[x]$.\n\\item  The ring map $\\varphi$ is {\\it integral}\nif every $s \\in S$ is integral over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GI","source_file":"algebra.tex","source_line":7441,"source_end_line":7453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7441-L7453","statement_sha256":"fc5ee0542e403bf7a01cf4991dff538f49f927ec916ffd01f8d34395c97e27f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1131,"rank":1131,"depth":0,"x":2017.002,"y":274.011,"cluster":"commutative-algebra"},{"id":"stacks:052I","tag":"052I","title":"Finite and integral ring extensions · Lemma 052I","summary":"Let φ : R → S be a ring map. Let y ∈ S. If there exists a finite R-submodule M of S such that 1 ∈ M and yM ⊂ M, then y is integral over R.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Let $y \\in S$. If there exists a\nfinite $R$-submodule $M$ of $S$ such that $1 \\in M$ and $yM \\subset M$,\nthen $y$ is integral over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052I","source_file":"algebra.tex","source_line":7455,"source_end_line":7460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7455-L7460","statement_sha256":"24dcf3a382f08041cc04526e0cae7e4337422929c80968569c477da24690f5ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":1132,"rank":1132,"depth":2,"x":1996.059,"y":172.67,"cluster":"commutative-algebra"},{"id":"stacks:00GK","tag":"00GK","title":"Finite and integral ring extensions · Lemma 00GK","summary":"A finite ring map is integral.","statement_latex":"A finite ring map is integral.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GK","source_file":"algebra.tex","source_line":7469,"source_end_line":7472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7469-L7472","statement_sha256":"7f238df0b44871f1b38d0558e90a9fb001bdce497474802a9d3fe1170839366d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1133,"rank":1133,"depth":3,"x":2093.257,"y":235.683,"cluster":"commutative-algebra"},{"id":"stacks:00GM","tag":"00GM","title":"Finite and integral ring extensions · Lemma 00GM","summary":"Let φ : R → S be a ring map. Let s_1, …, s_n be a finite set of elements of S. In this case s_i is integral over R for all i = 1, …, n if and only if there exists an R-subalgebra S' ⊂ S finite over R containing all of the s_i.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Let $s_1, \\ldots, s_n$\nbe a finite set of elements of $S$.\nIn this case $s_i$ is integral over $R$ for all $i = 1, \\ldots, n$\nif and only if\nthere exists an $R$-subalgebra $S' \\subset S$ finite over $R$\ncontaining all of the $s_i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GM","source_file":"algebra.tex","source_line":7480,"source_end_line":7488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7480-L7488","statement_sha256":"15805fdd5329573473a85b998373be572bccd1aceb231648cf500c3fefd7d636","origin":"The Stacks Project","memory_eligible":false,"source_rank":1134,"rank":1134,"depth":4,"x":1970.586,"y":244.394,"cluster":"commutative-algebra"},{"id":"stacks:02JJ","tag":"02JJ","title":"Finite and integral ring extensions · Lemma 02JJ","summary":"Let R → S be a ring map. The following are equivalent • R → S is finite, • R → S is integral and of finite type, and • there exist x_1, …, x_n ∈ S which generate S as an algebra over R such that each x_i is integral over R.","statement_latex":"Let $R \\to S$ be a ring map. The following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is finite,\n\\item $R \\to S$ is integral and of finite type, and\n\\item there exist $x_1, \\ldots, x_n \\in S$ which generate $S$ as an\nalgebra over $R$ such that each $x_i$ is integral over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JJ","source_file":"algebra.tex","source_line":7502,"source_end_line":7511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7502-L7511","statement_sha256":"bb7bae68435cf25d2c1527b9a05d0f7ce8c1d048582e1ae5700f4ab10319483f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1135,"rank":1135,"depth":5,"x":2054.258,"y":168.163,"cluster":"commutative-algebra"},{"id":"stacks:00GN","tag":"00GN","title":"Finite and integral ring extensions · Lemma 00GN","summary":"A composition of integral ring maps is integral Suppose that R → S and S → T are integral ring maps. Then R → T is integral.","statement_latex":"\\begin{slogan}\nA composition of integral ring maps is integral\n\\end{slogan}\nSuppose that $R \\to S$ and $S \\to T$ are integral\nring maps. Then $R \\to T$ is integral.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GN","source_file":"algebra.tex","source_line":7517,"source_end_line":7524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7517-L7524","statement_sha256":"96ad3f093fe337671adeb14745908d1a37d47c507a47abcfa67ef3a9b9f32f36","origin":"The Stacks Project","memory_eligible":false,"source_rank":1136,"rank":1136,"depth":5,"x":2053.861,"y":272.125,"cluster":"commutative-algebra"},{"id":"stacks:00GO","tag":"00GO","title":"Finite and integral ring extensions · Lemma 00GO","summary":"Let R → S be a ring homomorphism. The set S' = (s ∈ S mid s is integral over R) is an R-subalgebra of S.","statement_latex":"Let $R \\to S$ be a ring homomorphism.\nThe set\n$$\nS' = \\{s \\in S \\mid s\\text{ is integral over }R\\}\n$$\nis an $R$-subalgebra of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GO","source_file":"algebra.tex","source_line":7541,"source_end_line":7549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7541-L7549","statement_sha256":"81d9efe098f97faf6edb9e80e77133aa9622e8d8cd443d425b2adc164c5fc380","origin":"The Stacks Project","memory_eligible":false,"source_rank":1137,"rank":1137,"depth":5,"x":1970.332,"y":195.038,"cluster":"commutative-algebra"},{"id":"stacks:0CY8","tag":"0CY8","title":"Finite and integral ring extensions · Lemma 0CY8","summary":"Let R_i→ S_i be ring maps i = 1, …, n. Let R and S denote the product of the R_i and S_i respectively. Then an element s = (s_1, …, s_n) ∈ S is integral over R if and only if each s_i is integral over R_i.","statement_latex":"Let $R_i\\to S_i$ be ring maps $i = 1, \\ldots, n$.\nLet $R$ and $S$ denote the product of the $R_i$ and $S_i$ respectively.\nThen an element $s = (s_1, \\ldots, s_n) \\in S$ is integral over $R$\nif and only if each $s_i$ is integral over $R_i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CY8","source_file":"algebra.tex","source_line":7556,"source_end_line":7562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7556-L7562","statement_sha256":"67f5f76d0742b7186b0734f2cff06e8c90530e7054579995a6540a38bc976121","origin":"The Stacks Project","memory_eligible":false,"source_rank":1138,"rank":1138,"depth":0,"x":2094.238,"y":204.51,"cluster":"commutative-algebra"},{"id":"stacks:00GP","tag":"00GP","title":"Finite and integral ring extensions · Definition 00GP","summary":"Let R → S be a ring map. The ring S' ⊂ S of elements integral over R, see Lemma [Tag 00GO], is called the integral closure of R in S. If R ⊂ S we say that R is integrally closed in S if R = S'.","statement_latex":"Let $R \\to S$ be a ring map.\nThe ring $S' \\subset S$ of elements integral over\n$R$, see Lemma \\ref{lemma-integral-closure-is-ring},\nis called the {\\it integral closure} of $R$\nin $S$. If $R \\subset S$ we say that $R$ is\n{\\it integrally closed} in $S$ if $R = S'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GP","source_file":"algebra.tex","source_line":7568,"source_end_line":7576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7568-L7576","statement_sha256":"8e7f632f71c62ea1059a4434429528dd7cdab7bb9f2f7f3866e3bc19af69c09d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1139,"rank":1139,"depth":6,"x":1994.998,"y":267.996,"cluster":"commutative-algebra"},{"id":"stacks:0CY9","tag":"0CY9","title":"Finite and integral ring extensions · Lemma 0CY9","summary":"Let R_i→ S_i be ring maps i = 1, …, n. Denote the integral closure of R_i in S_i by S'_i. Further let R and S denote the product of the R_i and S_i respectively. Then the integral closure of R in S is the product of the S'_i. In particular R → S is integrally closed if and only if each R_i → S_i is integrally closed.","statement_latex":"Let $R_i\\to S_i$ be ring maps $i = 1, \\ldots, n$.\nDenote the integral closure of $R_i$ in $S_i$ by $S'_i$.\nFurther let $R$ and $S$ denote the product of the $R_i$ and $S_i$ respectively.\nThen the integral closure of $R$ in $S$\nis the product of the $S'_i$. In particular $R \\to S$ is\nintegrally closed if and only if each $R_i \\to S_i$ is integrally closed.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CY9","source_file":"algebra.tex","source_line":7582,"source_end_line":7590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7582-L7590","statement_sha256":"06cc5b15cf5b0fb98ff56cb90fe3b6324a7684033823e70eccc5fd6e2cd5a60d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1140,"rank":1140,"depth":1,"x":2017.179,"y":164.605,"cluster":"commutative-algebra"},{"id":"stacks:0307","tag":"0307","title":"Finite and integral ring extensions · Lemma 0307","summary":"Integral closure commutes with localization: If A → B is a ring map, and S ⊂ A is a multiplicative subset, then the integral closure of S^-1A in S^-1B is S^-1B', where B' ⊂ B is the integral closure of A in B.","statement_latex":"Integral closure commutes with localization: If $A \\to B$ is a ring\nmap, and $S \\subset A$ is a multiplicative subset, then the integral\nclosure of $S^{-1}A$ in $S^{-1}B$ is $S^{-1}B'$, where $B' \\subset B$\nis the integral closure of $A$ in $B$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0307","source_file":"algebra.tex","source_line":7596,"source_end_line":7602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7596-L7602","statement_sha256":"6f10f0def6de9cbd30566f9c3439b478bb8a6cf018fd9881bcd62092caff293d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1141,"rank":1141,"depth":0,"x":2084.139,"y":253.659,"cluster":"commutative-algebra"},{"id":"stacks:034K","tag":"034K","title":"Finite and integral ring extensions · Lemma 034K","summary":"An element of an algebra over a ring is integral over the ring if and only if it is locally integral at every prime ideal of the ring. Let φ : R → S be a ring map. Let x ∈ S. The following are equivalent: • x is integral over R, and • for every prime ideal p ⊂ R the element x ∈ S_ p is integral over R_ p.","statement_latex":"\\begin{slogan}\nAn element of an algebra over a ring is integral over the ring\nif and only if it is locally integral at every prime ideal of the ring.\n\\end{slogan}\nLet $\\varphi : R \\to S$ be a ring map.\nLet $x \\in S$. The following are equivalent:\n\\begin{enumerate}\n\\item $x$ is integral over $R$, and\n\\item for every prime ideal $\\mathfrak p \\subset R$ the element\n$x \\in S_{\\mathfrak p}$ is integral over $R_{\\mathfrak p}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034K","source_file":"algebra.tex","source_line":7629,"source_end_line":7642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7629-L7642","statement_sha256":"7e8667737118838e8a81df4f8b83b001cf680211f0f800176b925d2613117c1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1142,"rank":1142,"depth":5,"x":1962.841,"y":225.915,"cluster":"commutative-algebra"},{"id":"stacks:02JK","tag":"02JK","title":"Finite and integral ring extensions · Lemma 02JK","summary":"Integrality and finiteness are preserved under base change. Let R → S and R → R' be ring maps. Set S' = R' ⊗_R S. • If R → S is integral so is R' → S'. • If R → S is finite so is R' → S'.","statement_latex":"\\begin{slogan}\nIntegrality and finiteness are preserved under base change.\n\\end{slogan}\nLet $R \\to S$ and $R \\to R'$ be ring maps.\nSet $S' = R' \\otimes_R S$.\n\\begin{enumerate}\n\\item If $R \\to S$ is integral so is $R' \\to S'$.\n\\item If $R \\to S$ is finite so is $R' \\to S'$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JK","source_file":"algebra.tex","source_line":7664,"source_end_line":7675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7664-L7675","statement_sha256":"c40fe03eecd0d3a35b3141748c2d6127f38c9e8af46edf03dcd5343dfa95acf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1143,"rank":1143,"depth":0,"x":2074.878,"y":177.423,"cluster":"commutative-algebra"},{"id":"stacks:02JL","tag":"02JL","title":"Finite and integral ring extensions · Lemma 02JL","summary":"Let R → S be a ring map. Let f_1, …, f_n ∈ R generate the unit ideal. • If each R_f_i → S_f_i is integral, so is R → S. • If each R_f_i → S_f_i is finite, so is R → S.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $f_1, \\ldots, f_n \\in R$ generate the unit ideal.\n\\begin{enumerate}\n\\item If each $R_{f_i} \\to S_{f_i}$ is integral, so is $R \\to S$.\n\\item If each $R_{f_i} \\to S_{f_i}$ is finite, so is $R \\to S$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JL","source_file":"algebra.tex","source_line":7688,"source_end_line":7696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7688-L7696","statement_sha256":"e2b42ae604e9f62bcf20885e2eb4f59fb0834205d72d65465494a1010980b640","origin":"The Stacks Project","memory_eligible":false,"source_rank":1144,"rank":1144,"depth":0,"x":2031.15,"y":277.004,"cluster":"commutative-algebra"},{"id":"stacks:02JM","tag":"02JM","title":"Finite and integral ring extensions · Lemma 02JM","summary":"Let A → B → C be ring maps. • If A → C is integral so is B → C. • If A → C is finite so is B → C.","statement_latex":"Let $A \\to B \\to C$ be ring maps.\n\\begin{enumerate}\n\\item If $A \\to C$ is integral so is $B \\to C$.\n\\item If $A \\to C$ is finite so is $B \\to C$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JM","source_file":"algebra.tex","source_line":7709,"source_end_line":7716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7709-L7716","statement_sha256":"1f0d5c44d4833cf64212572ccbb0d108e9ce9ad8219cb6d196ae1b5eba7e7262","origin":"The Stacks Project","memory_eligible":false,"source_rank":1145,"rank":1145,"depth":0,"x":1983.194,"y":178.515,"cluster":"commutative-algebra"},{"id":"stacks:0308","tag":"0308","title":"Finite and integral ring extensions · Lemma 0308","summary":"Let A → B → C be ring maps. Let B' be the integral closure of A in B, let C' be the integral closure of B' in C. Then C' is the integral closure of A in C.","statement_latex":"Let $A \\to B \\to C$ be ring maps.\nLet $B'$ be the integral closure of $A$ in $B$,\nlet $C'$ be the integral closure of $B'$ in $C$. Then\n$C'$ is the integral closure of $A$ in $C$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0308","source_file":"algebra.tex","source_line":7722,"source_end_line":7728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7722-L7728","statement_sha256":"58e4005d562474ae3a09ac0ee21b9ba965cd88b51fc87cc8fb291e7830b072cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1146,"rank":1146,"depth":0,"x":2098.045,"y":224.042,"cluster":"commutative-algebra"},{"id":"stacks:00GQ","tag":"00GQ","title":"Finite and integral ring extensions · Lemma 00GQ","summary":"Suppose that R → S is an integral ring extension with R ⊂ S. Then φ : Spec(S) → Spec(R) is surjective.","statement_latex":"Suppose that $R \\to S$ is an integral\nring extension with $R \\subset S$.\nThen $\\varphi : \\Spec(S) \\to \\Spec(R)$\nis surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GQ","source_file":"algebra.tex","source_line":7734,"source_end_line":7740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7734-L7740","statement_sha256":"86ab93eaafd1aa3774dd845ea20a1d39451989835894caf27f34d8d3b5ca74ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":1147,"rank":1147,"depth":5,"x":1976.442,"y":255.718,"cluster":"commutative-algebra"},{"id":"stacks:00GR","tag":"00GR","title":"Finite and integral ring extensions · Lemma 00GR","summary":"Let R be a ring. Let K be a field. If R ⊂ K and K is integral over R, then R is a field and K is an algebraic extension. If R ⊂ K and K is finite over R, then R is a field and K is a finite algebraic extension.","statement_latex":"Let $R$ be a ring. Let $K$ be a field.\nIf $R \\subset K$ and $K$ is integral over $R$,\nthen $R$ is a field and $K$ is an algebraic extension.\nIf $R \\subset K$ and $K$ is finite over $R$,\nthen $R$ is a field and $K$ is a finite algebraic extension.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GR","source_file":"algebra.tex","source_line":7764,"source_end_line":7771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7764-L7771","statement_sha256":"b6395038e7cb6f5f37513b4058a1761f6172c0386a94483686bb51f40f138ece","origin":"The Stacks Project","memory_eligible":false,"source_rank":1148,"rank":1148,"depth":6,"x":2040.796,"y":163.132,"cluster":"commutative-algebra"},{"id":"stacks:00GS","tag":"00GS","title":"Finite and integral ring extensions · Lemma 00GS","summary":"Let k be a field. Let S be a k-algebra over k. • If S is a domain and finite dimensional over k, then S is a field. • If S is integral over k and a domain, then S is a field. • If S is integral over k then every prime of S is a maximal ideal (see Lemma [Tag 04MG] for more consequences).","statement_latex":"Let $k$ be a field. Let $S$ be a $k$-algebra over $k$.\n\\begin{enumerate}\n\\item If $S$ is a domain and finite dimensional over $k$,\nthen $S$ is a field.\n\\item If $S$ is integral over $k$ and a domain,\nthen $S$ is a field.\n\\item If $S$ is integral over $k$ then every prime of\n$S$ is a maximal ideal (see\nLemma \\ref{lemma-ring-with-only-minimal-primes}\nfor more consequences).\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GS","source_file":"algebra.tex","source_line":7781,"source_end_line":7794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7781-L7794","statement_sha256":"e47d6b26c848dfd739dea01d8b519f7991d9b1ad890b806846de63ff6faf09cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1149,"rank":1149,"depth":6,"x":2067.864,"y":268.178,"cluster":"commutative-algebra"},{"id":"stacks:00GT","tag":"00GT","title":"Finite and integral ring extensions · Lemma 00GT","summary":"Suppose R → S is integral. Let q, q' ∈ Spec(S) be distinct primes having the same image in Spec(R). Then neither q ⊂ q' nor q' ⊂ q.","statement_latex":"Suppose $R \\to S$ is integral.\nLet $\\mathfrak q, \\mathfrak q' \\in \\Spec(S)$\nbe distinct primes\nhaving the same image in $\\Spec(R)$.\nThen neither $\\mathfrak q \\subset \\mathfrak q'$\nnor $\\mathfrak q' \\subset \\mathfrak q$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GT","source_file":"algebra.tex","source_line":7812,"source_end_line":7820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7812-L7820","statement_sha256":"353e64ef2906858bf6c995e1abcb54a531ecf6a0c7ba84187fc3f8c5f08523f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1150,"rank":1150,"depth":7,"x":1963.174,"y":205.925,"cluster":"commutative-algebra"},{"id":"stacks:05DR","tag":"05DR","title":"Finite and integral ring extensions · Lemma 05DR","summary":"Suppose R → S is finite. Then the fibres of Spec(S) → Spec(R) are finite.","statement_latex":"Suppose $R \\to S$ is finite.\nThen the fibres of $\\Spec(S) \\to \\Spec(R)$ are finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DR","source_file":"algebra.tex","source_line":7831,"source_end_line":7835,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7831-L7835","statement_sha256":"37534724c3f96c1e57de243900570dfca4c46b700005eb51b0156fc20f9d90a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1151,"rank":1151,"depth":8,"x":2090.743,"y":192.393,"cluster":"commutative-algebra"},{"id":"stacks:00GU","tag":"00GU","title":"Finite and integral ring extensions · Lemma 00GU","summary":"Let R → S be a ring map such that S is integral over R. Let p ⊂ p' ⊂ R be primes. Let q be a prime of S mapping to p. Then there exists a prime q' with q ⊂ q' mapping to p'.","statement_latex":"Let $R \\to S$ be a ring map such that\n$S$ is integral over $R$.\nLet $\\mathfrak p \\subset \\mathfrak p' \\subset R$\nbe primes. Let $\\mathfrak q$ be a prime of $S$ mapping\nto $\\mathfrak p$. Then there exists a prime $\\mathfrak q'$\nwith $\\mathfrak q \\subset \\mathfrak q'$\nmapping to $\\mathfrak p'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GU","source_file":"algebra.tex","source_line":7852,"source_end_line":7861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7852-L7861","statement_sha256":"9fc293474d9d8cb960d0ce1225c39e1b4b68346ca6a50000216af6dc85270d47","origin":"The Stacks Project","memory_eligible":false,"source_rank":1152,"rank":1152,"depth":6,"x":2007.355,"y":274.957,"cluster":"commutative-algebra"},{"id":"stacks:0564","tag":"0564","title":"Finite and integral ring extensions · Lemma 0564","summary":"Let R → S be a finite and finitely presented ring map. Let M be an S-module. Then M is finitely presented as an R-module if and only if M is finitely presented as an S-module.","statement_latex":"Let $R \\to S$ be a finite and finitely presented ring map.\nLet $M$ be an $S$-module.\nThen $M$ is finitely presented as an $R$-module if and only if\n$M$ is finitely presented as an $S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0564","source_file":"algebra.tex","source_line":7875,"source_end_line":7881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7875-L7881","statement_sha256":"c99700e14e9cb2670f8d358daa307f499a8fafc7b5915fb6bb1691709306605c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1153,"rank":1153,"depth":3,"x":2002.437,"y":166.498,"cluster":"commutative-algebra"},{"id":"stacks:052J","tag":"052J","title":"Finite and integral ring extensions · Lemma 052J","summary":"Let R be a ring. Let x, y ∈ R be nonzerodivisors. Let R[x/y] ⊂ R_xy be the R-subalgebra generated by x/y, and similarly for the subalgebras R[y/x] and R[x/y, y/x]. If R is integrally closed in R_x or R_y, then the sequence 0 → R xrightarrow(-1, 1) R[x/y] ⊕ R[y/x] xrightarrow(1, 1) R[x/y, y/x] → 0 is a short exact sequence of R-modules.","statement_latex":"Let $R$ be a ring. Let $x, y \\in R$ be nonzerodivisors.\nLet $R[x/y] \\subset R_{xy}$ be the $R$-subalgebra generated\nby $x/y$, and similarly for the subalgebras $R[y/x]$ and $R[x/y, y/x]$.\nIf $R$ is integrally closed in $R_x$ or $R_y$, then the sequence\n$$\n0 \\to R \\xrightarrow{(-1, 1)} R[x/y] \\oplus R[y/x] \\xrightarrow{(1, 1)}\nR[x/y, y/x] \\to 0\n$$\nis a short exact sequence of $R$-modules.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite and integral ring extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052J","source_file":"algebra.tex","source_line":7927,"source_end_line":7938,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7927-L7938","statement_sha256":"fc04438a5e73250d93a1ba79cb088b72800d93d7a3b58ef5c35a42d119ce139a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1154,"rank":1154,"depth":3,"x":2093.501,"y":243.869,"cluster":"commutative-algebra"},{"id":"stacks:0309","tag":"0309","title":"Normal rings · Definition 0309","summary":"A domain R is called normal if it is integrally closed in its field of fractions.","statement_latex":"A domain $R$ is called {\\it normal} if it is integrally\nclosed in its field of fractions.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0309","source_file":"algebra.tex","source_line":7978,"source_end_line":7982,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7978-L7982","statement_sha256":"10f4f2267c7b55aec445a00aeadb82dd71389dd6120a6ad9733110aba64c687c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1155,"rank":1155,"depth":0,"x":1963.823,"y":238.476,"cluster":"commutative-algebra"},{"id":"stacks:034L","tag":"034L","title":"Normal rings · Lemma 034L","summary":"Let R → S be a ring map. If S is a normal domain, then the integral closure of R in S is a normal domain.","statement_latex":"Let $R \\to S$ be a ring map.\nIf $S$ is a normal domain, then the integral closure of $R$\nin $S$ is a normal domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034L","source_file":"algebra.tex","source_line":7984,"source_end_line":7989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7984-L7989","statement_sha256":"3ecc05e441ad6bc034f54913779990c73bcc082b258351a662f01b37d192041f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1156,"rank":1156,"depth":0,"x":2064.021,"y":168.704,"cluster":"commutative-algebra"},{"id":"stacks:00GW","tag":"00GW","title":"Normal rings · Definition 00GW","summary":"Let R be a domain. • An element g of the fraction field of R is called almost integral over R if there exists an element r ∈ R, rnot = 0 such that rg^n ∈ R for all n ≥ 0. • The domain R is called completely normal if every almost integral element of the fraction field of R is contained in R.","statement_latex":"Let $R$ be a domain.\n\\begin{enumerate}\n\\item An element $g$ of the fraction\nfield of $R$ is called {\\it almost integral over $R$}\nif there exists an element $r \\in R$, $r\\not = 0$\nsuch that $rg^n \\in R$ for all $n \\geq 0$.\n\\item The domain $R$ is called {\\it completely normal} if every\nalmost integral element of the fraction field of $R$ is\ncontained in $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GW","source_file":"algebra.tex","source_line":7999,"source_end_line":8011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L7999-L8011","statement_sha256":"fc6977e09200ad6bf9a01ec14cc981da3e11ce40f5c2d06c2ba2f53a9a48e5c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1157,"rank":1157,"depth":0,"x":2046.202,"y":277.264,"cluster":"commutative-algebra"},{"id":"stacks:00GX","tag":"00GX","title":"Normal rings · Lemma 00GX","summary":"Let R be a domain with fraction field K. If u, v ∈ K are almost integral over R, then so are u + v and uv. Any element g ∈ K which is integral over R is almost integral over R. If R is Noetherian then the converse holds as well.","statement_latex":"Let $R$ be a domain with fraction field $K$.\nIf $u, v \\in K$ are almost integral over $R$, then so are\n$u + v$ and $uv$. Any element $g \\in K$ which is integral over $R$\nis almost integral over $R$. If $R$ is Noetherian\nthen the converse holds as well.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GX","source_file":"algebra.tex","source_line":8017,"source_end_line":8024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8017-L8024","statement_sha256":"3c279a1ba415f9f12feea6e20f5fdfa1d5f97b486881a6d4c04556e30a07eb13","origin":"The Stacks Project","memory_eligible":false,"source_rank":1158,"rank":1158,"depth":4,"x":1971.866,"y":186.891,"cluster":"commutative-algebra"},{"id":"stacks:00GY","tag":"00GY","title":"Normal rings · Lemma 00GY","summary":"Any localization of a normal domain is normal.","statement_latex":"Any localization of a normal domain is normal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GY","source_file":"algebra.tex","source_line":8047,"source_end_line":8050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8047-L8050","statement_sha256":"dd0024c4966e1c677ccc1e64ead579dea0f76fa540b1fbb23cb76e4da94c8eba","origin":"The Stacks Project","memory_eligible":false,"source_rank":1159,"rank":1159,"depth":0,"x":2099.655,"y":211.408,"cluster":"commutative-algebra"},{"id":"stacks:00GZ","tag":"00GZ","title":"Normal rings · Lemma 00GZ","summary":"A principal ideal domain is normal.","statement_latex":"A principal ideal domain is normal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GZ","source_file":"algebra.tex","source_line":8065,"source_end_line":8068,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8065-L8068","statement_sha256":"62005dbfae87c8027a2a746277492af602be686bb8a09dd90a8ba1abe2393a40","origin":"The Stacks Project","memory_eligible":false,"source_rank":1160,"rank":1160,"depth":0,"x":1985.443,"y":265.966,"cluster":"commutative-algebra"},{"id":"stacks:00H0","tag":"00H0","title":"Normal rings · Lemma 00H0","summary":"Let R be a domain with fraction field K. Suppose f = ∑ α_i x^i is an element of K[x]. • If f is integral over R[x] then all α_i are integral over R, and • If f is almost integral over R[x] then all α_i are almost integral over R.","statement_latex":"Let $R$ be a domain with fraction field $K$.\nSuppose $f = \\sum \\alpha_i x^i$ is an\nelement of $K[x]$.\n\\begin{enumerate}\n\\item If $f$ is integral over $R[x]$\nthen all $\\alpha_i$ are integral over $R$, and\n\\item If $f$ is almost integral over $R[x]$\nthen all $\\alpha_i$ are almost integral over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00H0","source_file":"algebra.tex","source_line":8081,"source_end_line":8092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8081-L8092","statement_sha256":"d5e4300497666fec2e7cac5fb428af101a0953784ab2dc0a30a32d37f21cd28e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1161,"rank":1161,"depth":5,"x":2025.881,"y":160.685,"cluster":"commutative-algebra"},{"id":"stacks:030A","tag":"030A","title":"Normal rings · Lemma 030A","summary":"Let R be a normal domain. Then R[x] is a normal domain.","statement_latex":"Let $R$ be a normal domain.\nThen $R[x]$ is a normal domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030A","source_file":"algebra.tex","source_line":8129,"source_end_line":8133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8129-L8133","statement_sha256":"71a7343544fce9f76145e093ef27829e888ec041ac31f45b81a39985a82edbf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1162,"rank":1162,"depth":6,"x":2080.853,"y":261.496,"cluster":"commutative-algebra"},{"id":"stacks:0BI0","tag":"0BI0","title":"Normal rings · Lemma 0BI0","summary":"Let R be a Noetherian normal domain. Then R[[x]] is a Noetherian normal domain.","statement_latex":"Let $R$ be a Noetherian normal domain. Then $R[[x]]$ is\na Noetherian normal domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BI0","source_file":"algebra.tex","source_line":8149,"source_end_line":8153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8149-L8153","statement_sha256":"c6c1a1558da236a7cd926252a0337888eed37d1fea87d4923cfa7ae9de16934c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1163,"rank":1163,"depth":5,"x":1958.969,"y":218.253,"cluster":"commutative-algebra"},{"id":"stacks:030B","tag":"030B","title":"Normal rings · Lemma 030B","summary":"Let R be a domain. The following are equivalent: • The domain R is a normal domain, • for every prime p ⊂ R the local ring R_ p is a normal domain, and • for every maximal ideal m the ring R_ m is a normal domain.","statement_latex":"Let $R$ be a domain. The following are equivalent:\n\\begin{enumerate}\n\\item The domain $R$ is a normal domain,\n\\item for every prime $\\mathfrak p \\subset R$ the local ring\n$R_{\\mathfrak p}$ is a normal domain, and\n\\item for every maximal ideal $\\mathfrak m$ the ring $R_{\\mathfrak m}$\nis a normal domain.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030B","source_file":"algebra.tex","source_line":8176,"source_end_line":8186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8176-L8186","statement_sha256":"188087a35c01b40544c99b6fdccef8579e6f05a24f66b6b48817ce7d65587b1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1164,"rank":1164,"depth":1,"x":2083.905,"y":180.894,"cluster":"commutative-algebra"},{"id":"stacks:00GV","tag":"00GV","title":"Normal rings · Definition 00GV","summary":"A ring R is called normal if for every prime p ⊂ R the localization R_ p is a normal domain (see Definition [Tag 0309]).","statement_latex":"A ring $R$ is called {\\it normal} if for every prime\n$\\mathfrak p \\subset R$ the localization $R_{\\mathfrak p}$ is\na normal domain (see Definition \\ref{definition-domain-normal}).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00GV","source_file":"algebra.tex","source_line":8205,"source_end_line":8210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8205-L8210","statement_sha256":"334d37182ff40d2ea404aa1130a9f260404ee642d1dcb998db3f58d40298bc9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1165,"rank":1165,"depth":1,"x":2021.68,"y":279.558,"cluster":"commutative-algebra"},{"id":"stacks:034M","tag":"034M","title":"Normal rings · Lemma 034M","summary":"A normal ring is integrally closed in its total ring of fractions.","statement_latex":"A normal ring is integrally closed in its total ring of fractions.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034M","source_file":"algebra.tex","source_line":8217,"source_end_line":8220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8217-L8220","statement_sha256":"714961f8a09c4928a86e587f20cf6f0865c7a7cec1687868762fef515fff6453","origin":"The Stacks Project","memory_eligible":false,"source_rank":1166,"rank":1166,"depth":0,"x":1988.145,"y":171.252,"cluster":"commutative-algebra"},{"id":"stacks:037C","tag":"037C","title":"Normal rings · Lemma 037C","summary":"A localization of a normal ring is a normal ring.","statement_latex":"A localization of a normal ring is a normal ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037C","source_file":"algebra.tex","source_line":8237,"source_end_line":8240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8237-L8240","statement_sha256":"746f774e74a617da196dcf660f17bd53dff42297954af0e048045867918e344a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1167,"rank":1167,"depth":0,"x":2100.223,"y":232.224,"cluster":"commutative-algebra"},{"id":"stacks:00H1","tag":"00H1","title":"Normal rings · Lemma 00H1","summary":"Let R be a normal ring. Then R[x] is a normal ring.","statement_latex":"Let $R$ be a normal ring. Then $R[x]$ is a normal ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00H1","source_file":"algebra.tex","source_line":8246,"source_end_line":8249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8246-L8249","statement_sha256":"f14c5e8ff1a02abe5cab6aadf00cc1b1fc6b08b7df232702cf55cdd7e7172355","origin":"The Stacks Project","memory_eligible":false,"source_rank":1168,"rank":1168,"depth":7,"x":1968.25,"y":250.901,"cluster":"commutative-algebra"},{"id":"stacks:0CYA","tag":"0CYA","title":"Normal rings · Lemma 0CYA","summary":"A finite product of normal rings is normal.","statement_latex":"A finite product of normal rings is normal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYA","source_file":"algebra.tex","source_line":8259,"source_end_line":8262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8259-L8262","statement_sha256":"b19ce9f55fd6e079d83c3eeb87a0838be3caa293eac611743dcfa150f1109726","origin":"The Stacks Project","memory_eligible":false,"source_rank":1169,"rank":1169,"depth":3,"x":2050.73,"y":162.046,"cluster":"commutative-algebra"},{"id":"stacks:030C","tag":"030C","title":"Normal rings · Lemma 030C","summary":"Let R be a ring. Assume R is reduced and has finitely many minimal primes. Then the following are equivalent: • R is a normal ring, • R is integrally closed in its total ring of fractions, and • R is a finite product of normal domains.","statement_latex":"Let $R$ be a ring. Assume $R$ is reduced and has finitely many\nminimal primes. Then the following are equivalent:\n\\begin{enumerate}\n\\item $R$ is a normal ring,\n\\item $R$ is integrally closed in its total ring of fractions, and\n\\item $R$ is a finite product of normal domains.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030C","source_file":"algebra.tex","source_line":8274,"source_end_line":8283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8274-L8283","statement_sha256":"40bcb8dbb8f03722a74c87b180661a6d5e6671327f641b1a37712988b191ebbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1170,"rank":1170,"depth":6,"x":2061.389,"y":274.618,"cluster":"commutative-algebra"},{"id":"stacks:037D","tag":"037D","title":"Normal rings · Lemma 037D","summary":"Let (R_i, φ_ii') be a directed system (Categories, Definition [Tag 00D4]) of rings. If each R_i is a normal ring so is R = colim_i R_i.","statement_latex":"Let $(R_i, \\varphi_{ii'})$ be a directed system\n(Categories, Definition \\ref{definition-directed-system})\nof rings. If each $R_i$ is a normal ring so is\n$R = \\colim_i R_i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Normal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037D","source_file":"algebra.tex","source_line":8311,"source_end_line":8317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8311-L8317","statement_sha256":"c0f74f060de69461ccf4ed1db93c99771e1051a36abcb66323f08485822e1dbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1171,"rank":1171,"depth":1,"x":1962.784,"y":197.487,"cluster":"commutative-algebra"},{"id":"stacks:00H2","tag":"00H2","title":"Going down for integral over normal · Definition 00H2","summary":"Let φ : R → S be a ring map. Let I ⊂ R be an ideal. We say an element g ∈ S is integral over I if there exists a monic polynomial P = x^d + ∑_j < d a_j x^j with coefficients a_j ∈ I^d-j such that P^φ(g) = 0 in S.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nLet $I \\subset R$ be an ideal.\nWe say an element $g \\in S$ is\n{\\it integral over $I$} if\nthere exists a monic\npolynomial $P = x^d + \\sum_{j < d} a_j x^j$\nwith coefficients $a_j \\in I^{d-j}$ such\nthat $P^\\varphi(g) = 0$ in $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going down for integral over normal","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00H2","source_file":"algebra.tex","source_line":8347,"source_end_line":8357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8347-L8357","statement_sha256":"7e1db534223b2cd22134d92f0b1c6e5aea18cc81ac2a023f3056eaf2f917033a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1172,"rank":1172,"depth":0,"x":2097.818,"y":198.414,"cluster":"commutative-algebra"},{"id":"stacks:00H3","tag":"00H3","title":"Going down for integral over normal · Lemma 00H3","summary":"Let φ : R → S be a ring map. Let I ⊂ R be an ideal. Let A = ∑ I^nt^n ⊂ R[t] be the subring of the polynomial ring generated by R ⊕ It ⊂ R[t]. An element s ∈ S is integral over I if and only if the element st ∈ S[t] is integral over A.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nLet $I \\subset R$ be an ideal.\nLet $A = \\sum I^nt^n \\subset R[t]$ be the\nsubring of the polynomial ring\ngenerated by $R \\oplus It \\subset R[t]$.\nAn element $s \\in S$ is integral over $I$ if\nand only if the element $st \\in S[t]$\nis integral over $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going down for integral over normal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00H3","source_file":"algebra.tex","source_line":8365,"source_end_line":8375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8365-L8375","statement_sha256":"eba30dbd8edb4855b73ccef5649f8b00bb4c16c809e78a0b685f0dbcf4b1b7f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1173,"rank":1173,"depth":0,"x":1997.28,"y":274.518,"cluster":"commutative-algebra"},{"id":"stacks:00H4","tag":"00H4","title":"Going down for integral over normal · Lemma 00H4","summary":"Let φ : R → S be a ring map. Let I ⊂ R be an ideal. The set of elements of S which are integral over I form a R-submodule of S. Furthermore, if s ∈ S is integral over R, and s' is integral over I, then ss' is integral over I.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nLet $I \\subset R$ be an ideal.\nThe set of elements of $S$ which are integral\nover $I$ form a $R$-submodule of $S$.\nFurthermore, if $s \\in S$ is integral over\n$R$, and $s'$ is integral over $I$, then\n$ss'$ is integral over $I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going down for integral over normal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00H4","source_file":"algebra.tex","source_line":8398,"source_end_line":8407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8398-L8407","statement_sha256":"5032b8ceb05443294cb07afad4e9f6b5b6ad0e20c88573ef9f85a5252ba817f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1174,"rank":1174,"depth":6,"x":2010.242,"y":161.104,"cluster":"commutative-algebra"},{"id":"stacks:00H5","tag":"00H5","title":"Going down for integral over normal · Lemma 00H5","summary":"Suppose φ : R → S is integral. Suppose I ⊂ R is an ideal. Then every element of IS is integral over I.","statement_latex":"Suppose $\\varphi : R \\to S$ is integral.\nSuppose $I \\subset R$ is an ideal.\nThen every element of $IS$ is integral over $I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going down for integral over normal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00H5","source_file":"algebra.tex","source_line":8421,"source_end_line":8426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8421-L8426","statement_sha256":"a7464e8f4154bb96260a462993edb2fd0afb2a1861f32a1795d2c4b437f21ef1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1175,"rank":1175,"depth":7,"x":2092.066,"y":252.288,"cluster":"commutative-algebra"},{"id":"stacks:00H6","tag":"00H6","title":"Going down for integral over normal · Lemma 00H6","summary":"Let K be a field. Let n, m ∈ N and a_0, …, a_n - 1, b_0, …, b_m - 1 ∈ K. If the polynomial x^n + a_n - 1x^n - 1 + … + a_0 divides the polynomial x^m + b_m - 1 x^m - 1 + … + b_0 in K[x] then • a_0, …, a_n - 1 are integral over any subring R_0 of K containing the elements b_0, …, b_m - 1, and • each a_i lies in sqrt(b_0, …, b_m-1)R for any subring R ⊂ K containing the elements a_0, …, a_n - 1, b_0, …, b_m - 1.","statement_latex":"Let $K$ be a field. Let $n, m \\in \\mathbf{N}$ and\n$a_0, \\ldots, a_{n - 1}, b_0, \\ldots, b_{m - 1} \\in K$.\nIf the polynomial $x^n + a_{n - 1}x^{n - 1} + \\ldots + a_0$\ndivides the polynomial $x^m + b_{m - 1} x^{m - 1} + \\ldots + b_0$\nin $K[x]$ then\n\\begin{enumerate}\n\\item $a_0, \\ldots, a_{n - 1}$ are integral over any subring\n$R_0$ of $K$ containing the elements $b_0, \\ldots, b_{m - 1}$, and\n\\item each $a_i$ lies in $\\sqrt{(b_0, \\ldots, b_{m-1})R}$\nfor any subring $R \\subset K$ containing the elements\n$a_0, \\ldots, a_{n - 1}, b_0, \\ldots, b_{m - 1}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going down for integral over normal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00H6","source_file":"algebra.tex","source_line":8432,"source_end_line":8446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8432-L8446","statement_sha256":"80f5ff804244ef4841c40ec589d736922820b6adf6fdcb97df7f689f6979f54f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1176,"rank":1176,"depth":6,"x":1958.113,"y":231.434,"cluster":"commutative-algebra"},{"id":"stacks:00H7","tag":"00H7","title":"Going down for integral over normal · Lemma 00H7","summary":"Let R ⊂ S be an inclusion of domains. Assume R is normal. Let g ∈ S be integral over R. Then the minimal polynomial of g has coefficients in R.","statement_latex":"Let $R \\subset S$ be an inclusion of domains.\nAssume $R$ is normal. Let $g \\in S$ be integral\nover $R$. Then the minimal polynomial of $g$\nhas coefficients in $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going down for integral over normal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00H7","source_file":"algebra.tex","source_line":8498,"source_end_line":8504,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8498-L8504","statement_sha256":"884c227034457e92ee2e8d694f1f8add1fda185a7872e3b38fa545b9a8986479","origin":"The Stacks Project","memory_eligible":false,"source_rank":1177,"rank":1177,"depth":7,"x":2073.908,"y":170.673,"cluster":"commutative-algebra"},{"id":"stacks:00H8","tag":"00H8","title":"Going down for integral over normal · Proposition 00H8","summary":"Let R ⊂ S be an inclusion of domains. Assume R is normal and S integral over R. Let p ⊂ p' ⊂ R be primes. Let q' be a prime of S with p' = R ∩ q'. Then there exists a prime q with q ⊂ q' such that p = R ∩ q. In other words: the going down property holds for R → S, see Definition [Tag 00HV].","statement_latex":"Let $R \\subset S$ be an inclusion of domains.\nAssume $R$ is normal and $S$ integral over $R$.\nLet $\\mathfrak p \\subset \\mathfrak p' \\subset R$\nbe primes. Let $\\mathfrak q'$ be a prime of $S$\nwith $\\mathfrak p' = R \\cap \\mathfrak q'$.\nThen there exists a prime $\\mathfrak q$\nwith $\\mathfrak q \\subset \\mathfrak q'$\nsuch that $\\mathfrak p = R \\cap \\mathfrak q$. In other words:\nthe going down property holds for $R \\to S$, see\nDefinition \\ref{definition-going-up-down}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going down for integral over normal","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00H8","source_file":"algebra.tex","source_line":8517,"source_end_line":8529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8517-L8529","statement_sha256":"4b8fe1a6bea6d4f24f361446fcc7c2a1c0122f47cc75296560b01da09b723736","origin":"The Stacks Project","memory_eligible":false,"source_rank":1178,"rank":1178,"depth":8,"x":2037.307,"y":281.419,"cluster":"commutative-algebra"},{"id":"stacks:00HB","tag":"00HB","title":"Flat modules and flat ring maps · Definition 00HB","summary":"Let R be a ring. • An R-module M is called flat if whenever N_1 → N_2 → N_3 is an exact sequence of R-modules the sequence M ⊗_R N_1 → M ⊗_R N_2 → M ⊗_R N_3 is exact as well. • An R-module M is called faithfully flat if the complex of R-modules N_1 → N_2 → N_3 is exact if and only if the sequence M ⊗_R N_1 → M ⊗_R N_2 → M ⊗_R N_3 is exact. • A ring map R → S is called flat if S is flat as an R-module. • A ring map R → S is called faithfully flat if S is faithfully flat as…","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item An $R$-module $M$ is called {\\it flat} if whenever\n$N_1 \\to N_2 \\to N_3$ is an exact sequence of $R$-modules\nthe sequence $M \\otimes_R N_1 \\to M \\otimes_R N_2 \\to M \\otimes_R N_3$\nis exact as well.\n\\item An $R$-module $M$ is called {\\it faithfully flat} if the\ncomplex of $R$-modules\n$N_1 \\to N_2 \\to N_3$ is exact if and only if\nthe sequence $M \\otimes_R N_1 \\to M \\otimes_R N_2 \\to M \\otimes_R N_3$\nis exact.\n\\item A ring map $R \\to S$ is called {\\it flat} if\n$S$ is flat as an $R$-module.\n\\item A ring map $R \\to S$ is called {\\it faithfully flat} if\n$S$ is faithfully flat as an $R$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HB","source_file":"algebra.tex","source_line":8620,"source_end_line":8638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8620-L8638","statement_sha256":"842ff5d1c1ab9a4c1e24a551ddd700df78e5a0228a2021b7396db7f4507380da","origin":"The Stacks Project","memory_eligible":false,"source_rank":1179,"rank":1179,"depth":0,"x":1975.104,"y":178.769,"cluster":"commutative-algebra"},{"id":"stacks:0BBY","tag":"0BBY","title":"Flat modules and flat ring maps · Lemma 0BBY","summary":"Let R be a ring. Let I, J ⊂ R be ideals. Let M be a flat R-module. Then IM ∩ JM = (I ∩ J)M.","statement_latex":"Let $R$ be a ring. Let $I, J \\subset R$ be ideals. Let $M$ be a flat\n$R$-module. Then $IM \\cap JM = (I \\cap J)M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBY","source_file":"algebra.tex","source_line":8643,"source_end_line":8647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8643-L8647","statement_sha256":"2a91b6ec363a14abd46d31871607f3336f353b10d0533487d187c572ac1376a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1180,"rank":1180,"depth":0,"x":2103.793,"y":219.252,"cluster":"commutative-algebra"},{"id":"stacks:05UT","tag":"05UT","title":"Flat modules and flat ring maps · Lemma 05UT","summary":"Let R be a ring. Let (M_i, φ_ii') be a directed system of flat R-modules. Then colim_i M_i is a flat R-module.","statement_latex":"Let $R$ be a ring. Let $\\{M_i, \\varphi_{ii'}\\}$ be a directed system of\nflat $R$-modules. Then $\\colim_i M_i$ is a flat $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UT","source_file":"algebra.tex","source_line":8659,"source_end_line":8663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8659-L8663","statement_sha256":"3f1cabb286ed64824b75da18b65bc5cff7e38ad08825ddaa63afeab3bbe85ed5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1181,"rank":1181,"depth":2,"x":1976.073,"y":262.513,"cluster":"commutative-algebra"},{"id":"stacks:00HC","tag":"00HC","title":"Flat modules and flat ring maps · Lemma 00HC","summary":"A composition of (faithfully) flat ring maps is (faithfully) flat. If R → R' is (faithfully) flat, and M' is a (faithfully) flat R'-module, then M' is a (faithfully) flat R-module.","statement_latex":"A composition of (faithfully) flat ring maps is\n(faithfully) flat.\nIf $R \\to R'$ is (faithfully) flat, and $M'$ is a\n(faithfully) flat $R'$-module, then $M'$ is a\n(faithfully) flat $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HC","source_file":"algebra.tex","source_line":8671,"source_end_line":8678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8671-L8678","statement_sha256":"3f37b4beae9eb18c566c86005728ea9788a7ca5682f88534dbebec3a7653329f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1182,"rank":1182,"depth":2,"x":2035.589,"y":157.921,"cluster":"commutative-algebra"},{"id":"stacks:00HD","tag":"00HD","title":"Flat modules and flat ring maps · Lemma 00HD","summary":"Let M be an R-module. The following are equivalent: • M is flat over R. • for every injection of R-modules N ⊂ N' the map N ⊗_R M → N'⊗_R M is injective. • for every ideal I ⊂ R the map I ⊗_R M → R ⊗_R M = M is injective. • for every finitely generated ideal I ⊂ R the map I ⊗_R M → R ⊗_R M = M is injective.","statement_latex":"Let $M$ be an $R$-module. The following are equivalent:\n\\begin{enumerate}\n\\item\n\n$M$ is flat over $R$.\n\\item\n\nfor every injection of $R$-modules $N \\subset N'$\nthe map $N \\otimes_R M \\to N'\\otimes_R M$ is injective.\n\\item\n\nfor every ideal $I \\subset R$ the map\n$I \\otimes_R M \\to R \\otimes_R M = M$ is injective.\n\\item\n\nfor every finitely generated ideal $I \\subset R$\nthe map $I \\otimes_R M \\to R \\otimes_R M = M$ is injective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HD","source_file":"algebra.tex","source_line":8707,"source_end_line":8727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8707-L8727","statement_sha256":"3e104f7985435fe9c6489c04c81ede0e110228424a1f401cbe83fac85f842b33","origin":"The Stacks Project","memory_eligible":false,"source_rank":1183,"rank":1183,"depth":2,"x":2075.897,"y":269.05,"cluster":"commutative-algebra"},{"id":"stacks:05UU","tag":"05UU","title":"Flat modules and flat ring maps · Lemma 05UU","summary":"Let (R_i, φ_ii') be a system of rings over the directed set I. Let R = colim_i R_i. • If M is an R-module such that M is flat as an R_i-module for all i, then M is flat as an R-module. • For i ∈ I let M_i be a flat R_i-module and for i' ≥ i let f_ii' : M_i → M_i' be a φ_ii'-linear map such that f_i' i\" ∘ f_i i' = f_i i\". Then M = colim_i ∈ I M_i is a flat R-module.","statement_latex":"Let $\\{R_i, \\varphi_{ii'}\\}$ be a system of rings over the directed set $I$.\nLet $R = \\colim_i R_i$.\n\\begin{enumerate}\n\\item If $M$ is an $R$-module such that $M$ is flat as an $R_i$-module\nfor all $i$, then $M$ is flat as an $R$-module.\n\\item For $i \\in I$ let $M_i$ be a flat $R_i$-module and\nfor $i' \\geq i$ let $f_{ii'} : M_i \\to M_{i'}$ be a $\\varphi_{ii'}$-linear\nmap such that $f_{i' i''} \\circ f_{i i'} = f_{i i''}$. Then\n$M = \\colim_{i \\in I} M_i$ is a flat $R$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UU","source_file":"algebra.tex","source_line":8856,"source_end_line":8868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8856-L8868","statement_sha256":"a8799b3b5b80038a1bbffdf37710302f4ecb3c14d14ba2286bba47d69e69fa92","origin":"The Stacks Project","memory_eligible":false,"source_rank":1184,"rank":1184,"depth":3,"x":1956.559,"y":209.853,"cluster":"commutative-algebra"},{"id":"stacks:00HI","tag":"00HI","title":"Flat modules and flat ring maps · Lemma 00HI","summary":"Suppose that M is (faithfully) flat over R, and that R → R' is a ring map. Then M ⊗_R R' is (faithfully) flat over R'.","statement_latex":"Suppose that $M$ is (faithfully) flat over $R$, and that $R \\to R'$\nis a ring map. Then $M \\otimes_R R'$ is (faithfully) flat over $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HI","source_file":"algebra.tex","source_line":8888,"source_end_line":8892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8888-L8892","statement_sha256":"e0bf24bd803666842df6c2343b23992b74e77fe01689b748da2dc4b1d93818f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1185,"rank":1185,"depth":0,"x":2092.444,"y":185.738,"cluster":"commutative-algebra"},{"id":"stacks:00HJ","tag":"00HJ","title":"Flat modules and flat ring maps · Lemma 00HJ","summary":"Let R → R' be a faithfully flat ring map. Let M be a module over R, and set M' = R' ⊗_R M. Then M is flat over R if and only if M' is flat over R'.","statement_latex":"Let $R \\to R'$ be a faithfully flat ring map.\nLet $M$ be a module over $R$, and set $M' = R' \\otimes_R M$.\nThen $M$ is flat over $R$ if and only if $M'$ is flat over $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HJ","source_file":"algebra.tex","source_line":8902,"source_end_line":8907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8902-L8907","statement_sha256":"a8bf34777420c9fbedd9c03c2503e2e7ef97befc6d17a7b2aac491c12f0ee23d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1186,"rank":1186,"depth":1,"x":2011.469,"y":280.822,"cluster":"commutative-algebra"},{"id":"stacks:0584","tag":"0584","title":"Flat modules and flat ring maps · Lemma 0584","summary":"Let R be a ring. Let S → S' be a flat map of R-algebras. Let M be a module over S, and set M' = S' ⊗_S M. • If M is flat over R, then M' is flat over R. • If S → S' is faithfully flat, then M is flat over R if and only if M' is flat over R.","statement_latex":"Let $R$ be a ring. Let $S \\to S'$ be a flat map of $R$-algebras.\nLet $M$ be a module over $S$, and set $M' = S' \\otimes_S M$.\n\\begin{enumerate}\n\\item If $M$ is flat over $R$, then $M'$ is flat over $R$.\n\\item If $S \\to S'$ is faithfully flat, then $M$ is flat\nover $R$ if and only if $M'$ is flat over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0584","source_file":"algebra.tex","source_line":8929,"source_end_line":8938,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8929-L8938","statement_sha256":"d56c244b5e70d560bd65b1559de3e980842589e820de4c9be1f198c1202a11a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1187,"rank":1187,"depth":3,"x":1994.68,"y":164.521,"cluster":"commutative-algebra"},{"id":"stacks:039V","tag":"039V","title":"Flat modules and flat ring maps · Lemma 039V","summary":"Let R → S be a ring map. Let M be an S-module. If M is flat as an R-module and faithfully flat as an S-module, then R → S is flat.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.\nIf $M$ is flat as an $R$-module and faithfully flat as an $S$-module,\nthen $R \\to S$ is flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039V","source_file":"algebra.tex","source_line":8957,"source_end_line":8962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8957-L8962","statement_sha256":"b98ef14f1caf23bba1a6f041b3f22332b861a1a252ce677af0c85b8a749023c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1188,"rank":1188,"depth":0,"x":2100.804,"y":240.911,"cluster":"commutative-algebra"},{"id":"stacks:00HK","tag":"00HK","title":"Equational criterion of flatness · Lemma 00HK","summary":"A module M over R is flat if and only if every relation in M is trivial.","statement_latex":"A module $M$ over $R$ is flat if and only if\nevery relation in $M$ is trivial.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HK","source_file":"algebra.tex","source_line":8994,"source_end_line":8998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L8994-L8998","statement_sha256":"b6a1772f501a577948d231927b162d7fb27a6ba0332f2257a1c38a59b2cac758","origin":"The Stacks Project","memory_eligible":false,"source_rank":1189,"rank":1189,"depth":0,"x":1960.833,"y":244.807,"cluster":"commutative-algebra"},{"id":"stacks:00HL","tag":"00HL","title":"Flat modules and flat ring maps · Lemma 00HL","summary":"Suppose that R is a ring, 0 → M\" → M' → M → 0 a short exact sequence, and N an R-module. If M is flat then N ⊗_R M\" → N ⊗_R M' is injective, i.e., the sequence 0 → N ⊗_R M\" → N ⊗_R M' → N ⊗_R M → 0 is a short exact sequence.","statement_latex":"Suppose that $R$ is a ring, $0 \\to M'' \\to M' \\to M \\to 0$\na short exact sequence, and $N$ an $R$-module. If $M$ is flat\nthen $N \\otimes_R M'' \\to N \\otimes_R M'$ is injective, i.e., the\nsequence\n$$\n0 \\to N \\otimes_R M'' \\to N \\otimes_R M' \\to N \\otimes_R M \\to 0\n$$\nis a short exact sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HL","source_file":"algebra.tex","source_line":9036,"source_end_line":9046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9036-L9046","statement_sha256":"39e8b7b77552fe81d1adf0c24f3f551a68e46b7dc1088137ed05261af7b463dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1190,"rank":1190,"depth":0,"x":2061.117,"y":162.344,"cluster":"commutative-algebra"},{"id":"stacks:00HM","tag":"00HM","title":"Flat modules and flat ring maps · Lemma 00HM","summary":"Suppose that 0 → M' → M → M\" → 0 is a short exact sequence of R-modules. If M' and M\" are flat so is M. If M and M\" are flat so is M'.","statement_latex":"Suppose that $0 \\to M' \\to M \\to M'' \\to 0$ is\na short exact sequence of $R$-modules.\nIf $M'$ and $M''$ are flat so is $M$.\nIf $M$ and $M''$ are flat so is $M'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HM","source_file":"algebra.tex","source_line":9070,"source_end_line":9076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9070-L9076","statement_sha256":"d4b5f760d6a5668b675beaec30353ef6197fc1ee6cdd488cc8b1c7ed3276c740","origin":"The Stacks Project","memory_eligible":false,"source_rank":1191,"rank":1191,"depth":3,"x":2053.468,"y":280.293,"cluster":"commutative-algebra"},{"id":"stacks:00HO","tag":"00HO","title":"Flat modules and flat ring maps · Lemma 00HO","summary":"Let R be a ring. Let M be an R-module. The following are equivalent • M is faithfully flat, and • M is flat and for all R-module homomorphisms α : N → N' we have α = 0 if and only if α ⊗ id_M = 0.","statement_latex":"Let $R$ be a ring.\nLet $M$ be an $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ is faithfully flat, and\n\\item $M$ is flat and for all $R$-module homomorphisms $\\alpha : N \\to N'$\nwe have $\\alpha = 0$ if and only if $\\alpha \\otimes \\text{id}_M = 0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HO","source_file":"algebra.tex","source_line":9096,"source_end_line":9106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9096-L9106","statement_sha256":"90a6221de72e18f03c7260d14d5a599d2679b41c531f79d957efd61319e6527c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1192,"rank":1192,"depth":0,"x":1964.076,"y":188.794,"cluster":"commutative-algebra"},{"id":"stacks:00HP","tag":"00HP","title":"Flat modules and flat ring maps · Lemma 00HP","summary":"A flat module is faithfully flat if and only if it has nonzero fibers. Let M be a flat R-module. The following are equivalent: • M is faithfully flat, • for every nonzero R-module N, then tensor product M ⊗_R N is nonzero, • for all p ∈ Spec(R) the tensor product M ⊗_R kappa( p) is nonzero, and • for all maximal ideals m of R the tensor product M ⊗_R kappa( m) = M/ mM is nonzero.","statement_latex":"\\begin{slogan}\nA flat module is faithfully flat if and only if it has nonzero fibers.\n\\end{slogan}\nLet $M$ be a flat $R$-module.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $M$ is faithfully flat,\n\\item for every nonzero $R$-module $N$, then tensor product $M \\otimes_R N$\nis nonzero,\n\\item for all $\\mathfrak p \\in \\Spec(R)$\nthe tensor product $M \\otimes_R \\kappa(\\mathfrak p)$ is nonzero, and\n\\item for all maximal ideals $\\mathfrak m$ of $R$\nthe tensor product $M \\otimes_R \\kappa(\\mathfrak m) = M/{\\mathfrak m}M$\nis nonzero.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HP","source_file":"algebra.tex","source_line":9123,"source_end_line":9140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9123-L9140","statement_sha256":"31f253cee1fcfc065c381de934a461b477ba708961e0ccc9454fb2eb2cd89f4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1193,"rank":1193,"depth":1,"x":2103.855,"y":205.575,"cluster":"commutative-algebra"},{"id":"stacks:00HQ","tag":"00HQ","title":"Flat modules and flat ring maps · Lemma 00HQ","summary":"Let R → S be a flat ring map. The following are equivalent: • R → S is faithfully flat, • the induced map on Spec is surjective, and • any closed point x ∈ Spec(R) is in the image of the map Spec(S) → Spec(R).","statement_latex":"Let $R \\to S$ be a flat ring map.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $R \\to S$ is faithfully flat,\n\\item the induced map on $\\Spec$ is surjective, and\n\\item any closed point $x \\in \\Spec(R)$ is\nin the image of the map $\\Spec(S) \\to \\Spec(R)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HQ","source_file":"algebra.tex","source_line":9163,"source_end_line":9173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9163-L9173","statement_sha256":"d90e483b9b448812652fab1468999b423f552732e01407a1c09bd56e755dc30e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1194,"rank":1194,"depth":2,"x":1987.055,"y":272.65,"cluster":"commutative-algebra"},{"id":"stacks:00HR","tag":"00HR","title":"Flat modules and flat ring maps · Lemma 00HR","summary":"A flat local ring homomorphism of local rings is faithfully flat.","statement_latex":"A flat local ring homomorphism of local rings is faithfully flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HR","source_file":"algebra.tex","source_line":9183,"source_end_line":9186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9183-L9186","statement_sha256":"bb397da02b2d41aa3fb82bd316d8f0a703f3447fd98928dff0ed73e41370717c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1195,"rank":1195,"depth":3,"x":2019.306,"y":156.683,"cluster":"commutative-algebra"},{"id":"stacks:00HT","tag":"00HT","title":"Flat modules and flat ring maps · Lemma 00HT","summary":"Let R be a ring. Let S ⊂ R be a multiplicative subset. • The localization S^-1R is a flat R-algebra. • If M is an S^-1R-module, then M is a flat R-module if and only if M is a flat S^-1R-module. • Suppose M is an R-module. Then M is a flat R-module if and only if M_ p is a flat R_ p-module for all primes p of R. • Suppose M is an R-module. Then M is a flat R-module if and only if M_ m is a flat R_ m-module for all maximal ideals m of R. • Suppose R → A is a ring map, M is…","statement_latex":"Let $R$ be a ring. Let $S \\subset R$ be a multiplicative subset.\n\\begin{enumerate}\n\\item The localization $S^{-1}R$ is a flat $R$-algebra.\n\\item If $M$ is an $S^{-1}R$-module, then $M$ is a flat $R$-module\nif and only if $M$ is a flat $S^{-1}R$-module.\n\\item Suppose $M$ is an $R$-module. Then\n$M$ is a flat $R$-module if and only if $M_{\\mathfrak p}$ is a flat\n$R_{\\mathfrak p}$-module for all primes $\\mathfrak p$ of $R$.\n\\item Suppose $M$ is an $R$-module. Then $M$ is a flat $R$-module if\nand only if $M_{\\mathfrak m}$ is a flat\n$R_{\\mathfrak m}$-module for all maximal ideals $\\mathfrak m$ of $R$.\n\\item Suppose $R \\to A$ is a ring map, $M$ is an $A$-module,\nand $g_1, \\ldots, g_m \\in A$ are elements generating the unit\nideal of $A$. Then $M$ is flat over $R$ if and only if each localization\n$M_{g_i}$ is flat over $R$.\n\\item Suppose $R \\to A$ is a ring map, and $M$ is an $A$-module.\nThen $M$ is a flat $R$-module if and only if the localization\n$M_{\\mathfrak q}$ is a flat $R_{\\mathfrak p}$-module\n(with $\\mathfrak p$ the prime of $R$ lying under $\\mathfrak q$)\nfor all primes $\\mathfrak q$ of $A$.\n\\item Suppose $R \\to A$ is a ring map, and $M$ is an $A$-module.\nThen $M$ is a flat $R$-module if and only if the localization\n$M_{\\mathfrak m}$ is a flat $R_{\\mathfrak p}$-module\n(with $\\mathfrak p = R \\cap \\mathfrak m$)\nfor all maximal ideals $\\mathfrak m$ of $A$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HT","source_file":"algebra.tex","source_line":9195,"source_end_line":9223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9195-L9223","statement_sha256":"e9cbaaf5ce74cdfa4e57721c2d7b56c6f11441d3a8a598574998eefd09ee96c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1196,"rank":1196,"depth":2,"x":2088.92,"y":260.702,"cluster":"commutative-algebra"},{"id":"stacks:00HS","tag":"00HS","title":"Flat modules and flat ring maps · Lemma 00HS","summary":"Let R → S be flat. Let p ⊂ p' be primes of R. Let q' ⊂ S be a prime of S mapping to p'. Then there exists a prime q ⊂ q' mapping to p.","statement_latex":"Let $R \\to S$ be flat. Let $\\mathfrak p \\subset \\mathfrak p'$\nbe primes of $R$. Let $\\mathfrak q' \\subset S$ be a prime of $S$\nmapping to $\\mathfrak p'$. Then there exists a prime\n$\\mathfrak q \\subset \\mathfrak q'$ mapping to $\\mathfrak p$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HS","source_file":"algebra.tex","source_line":9264,"source_end_line":9270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9264-L9270","statement_sha256":"9b276dc79cdf000b80889d95168461d33495454535f42e3af875cbd24dde3af7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1197,"rank":1197,"depth":4,"x":1953.671,"y":223.426,"cluster":"commutative-algebra"},{"id":"stacks:090N","tag":"090N","title":"Flat modules and flat ring maps · Lemma 090N","summary":"Let R be a ring. Let (S_i, φ_ii') be a directed system of faithfully flat R-algebras. Then S = colim_i S_i is a faithfully flat R-algebra.","statement_latex":"Let $R$ be a ring. Let $\\{S_i, \\varphi_{ii'}\\}$ be a directed system of\nfaithfully flat $R$-algebras. Then $S = \\colim_i S_i$ is a faithfully flat\n$R$-algebra.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flat modules and flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090N","source_file":"algebra.tex","source_line":9285,"source_end_line":9290,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9285-L9290","statement_sha256":"090a7268de01e0204f25eec80f1c8ca7cb87cb8d2cf448962d64348128b4d91f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1198,"rank":1198,"depth":3,"x":2083.634,"y":174.072,"cluster":"commutative-algebra"},{"id":"stacks:00L1","tag":"00L1","title":"Supports and annihilators · Definition 00L1","summary":"Let R be a ring and let M be an R-module. The support of M is the set Supp(M) = ( p ∈ Spec(R) mid M_ p not = 0 )","statement_latex":"Let $R$ be a ring and let $M$ be an $R$-module.\nThe {\\it support of $M$} is the set\n$$\n\\text{Supp}(M)\n=\n\\{\n\\mathfrak p \\in \\Spec(R)\n\\mid\nM_{\\mathfrak p} \\not = 0\n\\}\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Supports and annihilators","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00L1","source_file":"algebra.tex","source_line":9313,"source_end_line":9326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9313-L9326","statement_sha256":"626176c62d30a0457d103d22e3bc38a78d59274d0101434752df50007fa3a590","origin":"The Stacks Project","memory_eligible":false,"source_rank":1199,"rank":1199,"depth":0,"x":2027.38,"y":284.426,"cluster":"commutative-algebra"},{"id":"stacks:0585","tag":"0585","title":"Supports and annihilators · Lemma 0585","summary":"A module over a ring has empty support if and only if it is the trivial module. Let R be a ring. Let M be an R-module. Then M = (0) ⇔ Supp(M) = ∅.","statement_latex":"\\begin{slogan}\nA module over a ring has empty support if and only if it is the trivial module.\n\\end{slogan}\nLet $R$ be a ring. Let $M$ be an $R$-module. Then\n$$\nM = (0) \\Leftrightarrow \\text{Supp}(M) = \\emptyset.\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Supports and annihilators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0585","source_file":"algebra.tex","source_line":9328,"source_end_line":9337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9328-L9337","statement_sha256":"d752b1c288b395564f0a905a9a5ea5fbd577f6ce05c3a75bb6a8c7eee9678029","origin":"The Stacks Project","memory_eligible":false,"source_rank":1200,"rank":1200,"depth":2,"x":1980.025,"y":170.911,"cluster":"commutative-algebra"},{"id":"stacks:07T7","tag":"07T7","title":"Supports and annihilators · Definition 07T7","summary":"Let R be a ring. Let M be an R-module. • Given an element m ∈ M the annihilator of m is the ideal Ann_R(m) = Ann(m) = (f ∈ R mid fm = 0). • The annihilator of M is the ideal Ann_R(M) = Ann(M) = (f ∈ R mid fm = 0 ∀ m ∈ M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item Given an element $m \\in M$ the {\\it annihilator of $m$}\nis the ideal\n$$\n\\text{Ann}_R(m) = \\text{Ann}(m) = \\{f \\in R \\mid fm = 0\\}.\n$$\n\\item The {\\it annihilator of $M$}\nis the ideal\n$$\n\\text{Ann}_R(M) = \\text{Ann}(M) = \\{f \\in R \\mid fm = 0\\ \\forall m \\in M\\}.\n$$\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Supports and annihilators","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07T7","source_file":"algebra.tex","source_line":9346,"source_end_line":9361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9346-L9361","statement_sha256":"848130559f440a50fa3a3e30e5840ce9d255f0cff8cb743cdf4f122bce8f1769","origin":"The Stacks Project","memory_eligible":false,"source_rank":1201,"rank":1201,"depth":0,"x":2106.476,"y":227.856,"cluster":"commutative-algebra"},{"id":"stacks:07T8","tag":"07T8","title":"Supports and annihilators · Lemma 07T8","summary":"Let R → S be a flat ring map. Let M be an R-module and m ∈ M. Then Ann_R(m) S = Ann_S(m ⊗ 1). If M is a finite R-module, then Ann_R(M) S = Ann_S(M ⊗_R S).","statement_latex":"Let $R \\to S$ be a flat ring map. Let $M$ be an $R$-module and\n$m \\in M$. Then $\\text{Ann}_R(m) S = \\text{Ann}_S(m \\otimes 1)$.\nIf $M$ is a finite $R$-module, then\n$\\text{Ann}_R(M) S = \\text{Ann}_S(M \\otimes_R S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Supports and annihilators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07T8","source_file":"algebra.tex","source_line":9363,"source_end_line":9369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9363-L9369","statement_sha256":"dbab668306b97bc51671e12195e86b8208fc498074a8f29ec7156dc9192eeffd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1202,"rank":1202,"depth":1,"x":1967.168,"y":257.675,"cluster":"commutative-algebra"},{"id":"stacks:00L2","tag":"00L2","title":"Supports and annihilators · Lemma 00L2","summary":"Let R be a ring and let M be an R-module. If M is finite, then Supp(M) is closed. More precisely, if I = Ann(M) is the annihilator of M, then V(I) = Supp(M).","statement_latex":"Let $R$ be a ring and let $M$ be an $R$-module.\nIf $M$ is finite, then $\\text{Supp}(M)$ is closed.\nMore precisely, if $I = \\text{Ann}(M)$ is the annihilator of $M$, then\n$V(I) = \\text{Supp}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Supports and annihilators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00L2","source_file":"algebra.tex","source_line":9384,"source_end_line":9390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9384-L9390","statement_sha256":"7f4a1c7151e632a6fb1a5a00203a995d5b845d6614585267aad77a6a2e05fdc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1203,"rank":1203,"depth":2,"x":2046.065,"y":156.445,"cluster":"commutative-algebra"},{"id":"stacks:0BUR","tag":"0BUR","title":"Supports and annihilators · Lemma 0BUR","summary":"Let R → R' be a ring map and let M be a finite R-module. Then Supp(M ⊗_R R') is the inverse image of Supp(M).","statement_latex":"Let $R \\to R'$ be a ring map and let $M$ be a finite $R$-module.\nThen $\\text{Supp}(M \\otimes_R R')$ is the inverse image of\n$\\text{Supp}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Supports and annihilators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUR","source_file":"algebra.tex","source_line":9412,"source_end_line":9417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9412-L9417","statement_sha256":"c5a02c5ef78917140edb2710695bd396cc1128bb27e3ca8b52065d927582192c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1204,"rank":1204,"depth":3,"x":2069.339,"y":276.088,"cluster":"commutative-algebra"},{"id":"stacks:07Z5","tag":"07Z5","title":"Supports and annihilators · Lemma 07Z5","summary":"Let R be a ring, let M be an R-module, and let m ∈ M. Then p ∈ V(Ann(m)) if and only if m does not map to zero in M_ p.","statement_latex":"Let $R$ be a ring, let $M$ be an $R$-module, and let $m \\in M$.\nThen $\\mathfrak p \\in V(\\text{Ann}(m))$ if and only if\n$m$ does not map to zero in $M_\\mathfrak p$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Supports and annihilators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Z5","source_file":"algebra.tex","source_line":9446,"source_end_line":9451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9446-L9451","statement_sha256":"8c1f288f155a309151e196e297df98c806af1e78abe42e1a1f3e8f605b025b32","origin":"The Stacks Project","memory_eligible":false,"source_rank":1205,"rank":1205,"depth":3,"x":1955.745,"y":200.927,"cluster":"commutative-algebra"},{"id":"stacks:051B","tag":"051B","title":"Supports and annihilators · Lemma 051B","summary":"Let R be a ring and let M be an R-module. If M is a finitely presented R-module, then Supp(M) is a closed subset of Spec(R) whose complement is quasi-compact.","statement_latex":"Let $R$ be a ring and let $M$ be an $R$-module.\nIf $M$ is a finitely presented $R$-module, then $\\text{Supp}(M)$ is a\nclosed subset of $\\Spec(R)$ whose complement is quasi-compact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Supports and annihilators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051B","source_file":"algebra.tex","source_line":9460,"source_end_line":9465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9460-L9465","statement_sha256":"d972a0e75685a4ecf4053730ac56919dc358e79d2794d3d91d5e6869797ba336","origin":"The Stacks Project","memory_eligible":false,"source_rank":1206,"rank":1206,"depth":3,"x":2100.227,"y":191.877,"cluster":"commutative-algebra"},{"id":"stacks:00L3","tag":"00L3","title":"Supports and annihilators · Lemma 00L3","summary":"Let R be a ring and let M be an R-module. • If M is finite then the support of M/IM is Supp(M) ∩ V(I). • If N ⊂ M, then Supp(N) ⊂ Supp(M). • If Q is a quotient module of M then Supp(Q) ⊂ Supp(M). • If 0 → N → M → Q → 0 is a short exact sequence then Supp(M) = Supp(Q) ∪ Supp(N).","statement_latex":"Let $R$ be a ring and let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item If $M$ is finite then the support\nof $M/IM$ is $\\text{Supp}(M) \\cap V(I)$.\n\\item If $N \\subset M$, then $\\text{Supp}(N) \\subset\n\\text{Supp}(M)$.\n\\item If $Q$ is a quotient module of $M$ then $\\text{Supp}(Q) \\subset\n\\text{Supp}(M)$.\n\\item If $0 \\to N \\to M \\to Q \\to 0$ is a short exact sequence\nthen $\\text{Supp}(M) = \\text{Supp}(Q) \\cup \\text{Supp}(N)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Supports and annihilators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00L3","source_file":"algebra.tex","source_line":9486,"source_end_line":9499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9486-L9499","statement_sha256":"12dadce2ad491da5416213f7d1f3449ef56d972a379554dd291297c584d332a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1207,"rank":1207,"depth":3,"x":2000.776,"y":280.701,"cluster":"commutative-algebra"},{"id":"stacks:00HV","tag":"00HV","title":"Going up and going down · Definition 00HV","summary":"Let φ : R → S be a ring map. • We say a φ : R → S satisfies going up if given primes p ⊂ p' in R and a prime q in S lying over p there exists a prime q' of S such that (a) q ⊂ q', and (b) q' lies over p'. • We say a φ : R → S satisfies going down if given primes p ⊂ p' in R and a prime q' in S lying over p' there exists a prime q of S such that (a) q ⊂ q', and (b) q lies over p.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\n\\begin{enumerate}\n\\item We say a $\\varphi : R \\to S$ satisfies {\\it going up} if\ngiven primes $\\mathfrak p \\subset \\mathfrak p'$ in $R$\nand a prime $\\mathfrak q$ in $S$ lying over $\\mathfrak p$\nthere exists a prime $\\mathfrak q'$ of $S$ such that\n(a) $\\mathfrak q \\subset \\mathfrak q'$, and (b)\n$\\mathfrak q'$ lies over $\\mathfrak p'$.\n\\item We say a $\\varphi : R \\to S$ satisfies {\\it going down} if\ngiven primes $\\mathfrak p \\subset \\mathfrak p'$ in $R$\nand a prime $\\mathfrak q'$ in $S$ lying over $\\mathfrak p'$\nthere exists a prime $\\mathfrak q$ of $S$ such that\n(a) $\\mathfrak q \\subset \\mathfrak q'$, and (b)\n$\\mathfrak q$ lies over $\\mathfrak p$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HV","source_file":"algebra.tex","source_line":9531,"source_end_line":9548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9531-L9548","statement_sha256":"f74b2a5e4e50dbcb8756caff0cd0feac11d410ed2ffa83d08ff664b3de53614d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1208,"rank":1208,"depth":0,"x":2002.683,"y":158.541,"cluster":"commutative-algebra"},{"id":"stacks:0407","tag":"0407","title":"Going up and going down · Lemma 0407","summary":"Let R → S be a ring map. If the induced map φ : Spec(S) → Spec(R) is open, then R → S satisfies going down.","statement_latex":"Let $R \\to S$ be a ring map. If the induced map\n$\\varphi : \\Spec(S) \\to \\Spec(R)$ is open, then\n$R \\to S$ satisfies going down.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0407","source_file":"algebra.tex","source_line":9566,"source_end_line":9571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9566-L9571","statement_sha256":"f9b292fadccf76fbd26dfda7cffe6628034c3f04ed27f55629fa4f192f55995f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1209,"rank":1209,"depth":4,"x":2099.698,"y":249.875,"cluster":"commutative-algebra"},{"id":"stacks:00HW","tag":"00HW","title":"Going up and going down · Lemma 00HW","summary":"Let R → S be a ring map. • R → S satisfies going down if and only if generalizations lift along the map Spec(S) → Spec(R), see Topology, Definition [Tag 0063]. • R → S satisfies going up if and only if specializations lift along the map Spec(S) → Spec(R), see Topology, Definition [Tag 0063].","statement_latex":"Let $R \\to S$ be a ring map.\n\\begin{enumerate}\n\\item $R \\to S$ satisfies going down if and only if\ngeneralizations lift along the map $\\Spec(S) \\to \\Spec(R)$,\nsee Topology, Definition \\ref{topology-definition-lift-specializations}.\n\\item $R \\to S$ satisfies going up if and only if\nspecializations lift along the map $\\Spec(S) \\to \\Spec(R)$,\nsee Topology, Definition \\ref{topology-definition-lift-specializations}.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HW","source_file":"algebra.tex","source_line":9588,"source_end_line":9599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9588-L9599","statement_sha256":"f8f144d0470968f5bd6f20af329f2732095fea7f2e9dc285f8c297c2d2202818","origin":"The Stacks Project","memory_eligible":false,"source_rank":1210,"rank":1210,"depth":1,"x":1954.44,"y":237.552,"cluster":"commutative-algebra"},{"id":"stacks:00HX","tag":"00HX","title":"Going up and going down · Lemma 00HX","summary":"Suppose R → S and S → T are ring maps satisfying going down. Then so does R → T. Similarly for going up.","statement_latex":"Suppose $R \\to S$ and $S \\to T$ are ring maps satisfying\ngoing down. Then so does $R \\to T$. Similarly for going up.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HX","source_file":"algebra.tex","source_line":9605,"source_end_line":9609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9605-L9609","statement_sha256":"117efcfe49f8f59969daaf6c40576b7a2a913121809e2c5e9eda9f6746053b78","origin":"The Stacks Project","memory_eligible":false,"source_rank":1211,"rank":1211,"depth":2,"x":2071.68,"y":164.078,"cluster":"commutative-algebra"},{"id":"stacks:00HY","tag":"00HY","title":"Going up and going down · Lemma 00HY","summary":"Let R → S be a ring map. Let T ⊂ Spec(R) be the image of Spec(S). If T is stable under specialization, then T is closed.","statement_latex":"Let $R \\to S$ be a ring map. Let $T \\subset \\Spec(R)$\nbe the image of $\\Spec(S)$. If $T$ is stable under specialization,\nthen $T$ is closed.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HY","source_file":"algebra.tex","source_line":9617,"source_end_line":9622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9617-L9622","statement_sha256":"cc9b42711d72c48baa943377a04651ffccf5c3fa58625a8fe3a0f9b2a4d6c966","origin":"The Stacks Project","memory_eligible":false,"source_rank":1212,"rank":1212,"depth":2,"x":2044.263,"y":285.008,"cluster":"commutative-algebra"},{"id":"stacks:00HZ","tag":"00HZ","title":"Going up and going down · Lemma 00HZ","summary":"Let R → S be a ring map. The following are equivalent: • Going up holds for R → S, and • the map Spec(S) → Spec(R) is closed.","statement_latex":"Let $R \\to S$ be a ring map. The following are equivalent:\n\\begin{enumerate}\n\\item Going up holds for $R \\to S$, and\n\\item the map $\\Spec(S) \\to \\Spec(R)$ is closed.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00HZ","source_file":"algebra.tex","source_line":9651,"source_end_line":9658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9651-L9658","statement_sha256":"f45b9c038fe1893453c3e0bbdd9c354417f8960f66c19f8027cb498ba056b388","origin":"The Stacks Project","memory_eligible":false,"source_rank":1213,"rank":1213,"depth":4,"x":1967.089,"y":180.084,"cluster":"commutative-algebra"},{"id":"stacks:00I0","tag":"00I0","title":"Going up and going down · Lemma 00I0","summary":"Let R be a ring. Let E ⊂ Spec(R) be a constructible subset. • If E is stable under specialization, then E is closed. • If E is stable under generalization, then E is open.","statement_latex":"Let $R$ be a ring. Let $E \\subset \\Spec(R)$ be a constructible subset.\n\\begin{enumerate}\n\\item If $E$ is stable under specialization, then $E$ is closed.\n\\item If $E$ is stable under generalization, then $E$ is open.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00I0","source_file":"algebra.tex","source_line":9681,"source_end_line":9688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9681-L9688","statement_sha256":"e267a9b09eff75560735a4bfb5b6a134d95ec3eaae817ca2d714751328b79020","origin":"The Stacks Project","memory_eligible":false,"source_rank":1214,"rank":1214,"depth":6,"x":2108.635,"y":213.725,"cluster":"commutative-algebra"},{"id":"stacks:00I1","tag":"00I1","title":"Going up and going down · Proposition 00I1","summary":"Let R → S be flat and of finite presentation. Then Spec(S) → Spec(R) is open. More generally this holds for any ring map R → S of finite presentation which satisfies going down.","statement_latex":"Let $R \\to S$ be flat and of finite presentation.\nThen $\\Spec(S) \\to \\Spec(R)$ is open.\nMore generally this holds for any ring map $R \\to S$ of\nfinite presentation which satisfies going down.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00I1","source_file":"algebra.tex","source_line":9707,"source_end_line":9713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9707-L9713","statement_sha256":"66a12e92732548ff39d31004206e615d647f49fefd53159e76a9dd8995ba9c98","origin":"The Stacks Project","memory_eligible":false,"source_rank":1215,"rank":1215,"depth":7,"x":1976.964,"y":269.338,"cluster":"commutative-algebra"},{"id":"stacks:037F","tag":"037F","title":"Going up and going down · Lemma 037F","summary":"Let k be a field, and let R, S be k-algebras. Let S' ⊂ S be a sub k-algebra, and let f ∈ S' ⊗_k R. In the commutative diagram xymatrix Spec((S ⊗_k R)_f) ar[rd] ar[rr] & & Spec((S' ⊗_k R)_f) ar[ld] & Spec(R) & the images of the diagonal arrows are the same.","statement_latex":"Let $k$ be a field, and let $R$, $S$ be $k$-algebras.\nLet $S' \\subset S$ be a sub $k$-algebra, and let $f \\in S' \\otimes_k R$.\nIn the commutative diagram\n$$\n\\xymatrix{\n\\Spec((S \\otimes_k R)_f) \\ar[rd] \\ar[rr] & &\n\\Spec((S' \\otimes_k R)_f) \\ar[ld] \\\\\n& \\Spec(R) &\n}\n$$\nthe images of the diagonal arrows are the same.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037F","source_file":"algebra.tex","source_line":9739,"source_end_line":9752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9739-L9752","statement_sha256":"c027da58f70c01481c93baf0cf4a03dd869ba2e60157ac848ee71396279704e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1216,"rank":1216,"depth":1,"x":2029.431,"y":153.403,"cluster":"commutative-algebra"},{"id":"stacks:037G","tag":"037G","title":"Going up and going down · Lemma 037G","summary":"Let k be a field. Let R and S be k-algebras. The map Spec(S ⊗_k R) → Spec(R) is open.","statement_latex":"Let $k$ be a field.\nLet $R$ and $S$ be $k$-algebras.\nThe map $\\Spec(S \\otimes_k R) \\to \\Spec(R)$\nis open.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037G","source_file":"algebra.tex","source_line":9774,"source_end_line":9780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9774-L9780","statement_sha256":"4f887a3f478dee2e8f801705c7226fb39371e0c8c0bc8b4a057f54db6f54dc8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1217,"rank":1217,"depth":8,"x":2084.074,"y":268.874,"cluster":"commutative-algebra"},{"id":"stacks:00EA","tag":"00EA","title":"Going up and going down · Lemma 00EA","summary":"Let R → S be a ring map. Let p ⊂ R be a prime. Assume that • there exists a unique prime q ⊂ S lying over p, and • either • going up holds for R → S, or • going down holds for R → S and there is at most one prime of S above every prime of R. Then S_ p = S_ q.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak p \\subset R$ be a prime.\nAssume that\n\\begin{enumerate}\n\\item there exists a unique prime $\\mathfrak q \\subset S$ lying over\n$\\mathfrak p$, and\n\\item either\n\\begin{enumerate}\n\\item going up holds for $R \\to S$, or\n\\item going down holds for $R \\to S$ and there is at most one prime\nof $S$ above every prime of $R$.\n\\end{enumerate}\n\\end{enumerate}\nThen $S_{\\mathfrak p} = S_{\\mathfrak q}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00EA","source_file":"algebra.tex","source_line":9796,"source_end_line":9812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9796-L9812","statement_sha256":"8cc0414313e55a3c425a82c738f77d2317e9ed8fbcc56c8518fa37ca65de40bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1218,"rank":1218,"depth":0,"x":1950.679,"y":214.634,"cluster":"commutative-algebra"},{"id":"stacks:080T","tag":"080T","title":"Going up and going down · Lemma 080T","summary":"Let R → S be a ring map. Let N be a finite S-module flat over R. Endow Supp(N) ⊂ Spec(S) with the induced topology. Then generalizations lift along Supp(N) → Spec(R).","statement_latex":"Let $R \\to S$ be a ring map. Let $N$ be a finite $S$-module flat over $R$.\nEndow $\\text{Supp}(N) \\subset \\Spec(S)$ with the induced topology.\nThen generalizations lift along $\\text{Supp}(N) \\to \\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Going up and going down","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080T","source_file":"algebra.tex","source_line":9851,"source_end_line":9856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9851-L9856","statement_sha256":"4068aff1ccb360214d7f73b9fdd805401265a69c090d15a0a2ab3159905a0f6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1219,"rank":1219,"depth":3,"x":2092.919,"y":178.873,"cluster":"commutative-algebra"},{"id":"stacks:030O","tag":"030O","title":"Separable extensions · Definition 030O","summary":"Let K/k be a field extension. • We say K is separably generated over k if there exists a transcendence basis (x_i; i ∈ I) of K/k such that the extension K/k(x_i; i ∈ I) is a separable algebraic extension. • We say K is separable over k if for every subextension k ⊂ K' ⊂ K with K' finitely generated over k, the extension K'/k is separably generated.","statement_latex":"Let $K/k$ be a field extension.\n\\begin{enumerate}\n\\item We say $K$ is {\\it separably generated over $k$} if there exists\na transcendence basis $\\{x_i; i \\in I\\}$ of $K/k$ such that the extension\n$K/k(x_i; i \\in I)$ is a separable algebraic extension.\n\\item We say $K$ is {\\it separable over $k$} if for every subextension\n$k \\subset K' \\subset K$ with $K'$ finitely generated\nover $k$, the extension $K'/k$ is separably generated.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Separable extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030O","source_file":"algebra.tex","source_line":9914,"source_end_line":9925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9914-L9925","statement_sha256":"1d4feffcd79482017341865ed1afdd823330798577178549ca86e26e5e8b4f58","origin":"The Stacks Project","memory_eligible":false,"source_rank":1220,"rank":1220,"depth":0,"x":2016.653,"y":286.148,"cluster":"commutative-algebra"},{"id":"stacks:030P","tag":"030P","title":"Separable extensions · Lemma 030P","summary":"Let K/k be a separable field extension. For any subextension K/K'/k the field extension K'/k is separable.","statement_latex":"Let $K/k$ be a separable field extension.\nFor any subextension $K/K'/k$ the field\nextension $K'/k$ is separable.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Separable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030P","source_file":"algebra.tex","source_line":9933,"source_end_line":9938,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9933-L9938","statement_sha256":"166d5ca594259d4c564b166449f9329bf8859a3051f1c79bc756041f54951b79","origin":"The Stacks Project","memory_eligible":false,"source_rank":1221,"rank":1221,"depth":0,"x":1986.569,"y":163.549,"cluster":"commutative-algebra"},{"id":"stacks:030Q","tag":"030Q","title":"Separable extensions · Lemma 030Q","summary":"Let K/k be a separably generated, and finitely generated field extension. Set r = trdeg_k(K). Then there exist elements x_1, …, x_r + 1 of K such that • x_1, …, x_r is a transcendence basis of K over k, • K = k(x_1, …, x_r + 1), and • x_r + 1 is separable over k(x_1, …, x_r).","statement_latex":"Let $K/k$ be a separably generated, and finitely generated\nfield extension.\nSet $r = \\text{trdeg}_k(K)$. Then there exist elements\n$x_1, \\ldots, x_{r + 1}$ of $K$ such that\n\\begin{enumerate}\n\\item $x_1, \\ldots, x_r$ is a transcendence basis of $K$ over $k$,\n\\item $K = k(x_1, \\ldots, x_{r + 1})$, and\n\\item $x_{r + 1}$ is separable over $k(x_1, \\ldots, x_r)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Separable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030Q","source_file":"algebra.tex","source_line":9944,"source_end_line":9955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9944-L9955","statement_sha256":"5521372547ea573d5e0b70109519dddb8309e9709b9cf7899a396a8129dca78e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1222,"rank":1222,"depth":5,"x":2107.561,"y":237.013,"cluster":"commutative-algebra"},{"id":"stacks:04KM","tag":"04KM","title":"Separable extensions · Lemma 04KM","summary":"Let K/k be a finitely generated field extension. There exists a diagram xymatrix K ar[r] & K' k ar[u] ar[r] & k' ar[u] where k'/k, K'/K are finite purely inseparable field extensions such that K'/k' is a separably generated field extension.","statement_latex":"Let $K/k$ be a finitely generated field extension.\nThere exists a diagram\n$$\n\\xymatrix{\nK \\ar[r] & K' \\\\\nk \\ar[u] \\ar[r] & k' \\ar[u]\n}\n$$\nwhere $k'/k$, $K'/K$ are finite purely inseparable field\nextensions such that $K'/k'$ is a separably generated field extension.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Separable extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KM","source_file":"algebra.tex","source_line":9961,"source_end_line":9973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L9961-L9973","statement_sha256":"963c195c40ee1c1f19ceb5dd9594d280212a71a7fb7fd14462938af6c63be845","origin":"The Stacks Project","memory_eligible":false,"source_rank":1223,"rank":1223,"depth":5,"x":1958.999,"y":251.521,"cluster":"commutative-algebra"},{"id":"stacks:030S","tag":"030S","title":"Geometrically reduced algebras · Definition 030S","summary":"Let k be a field. Let S be a k-algebra. We say S is geometrically reduced over k if for every field extension K/k the K-algebra K ⊗_k S is reduced.","statement_latex":"Let $k$ be a field. Let $S$ be a $k$-algebra.\nWe say $S$ is {\\it geometrically reduced over $k$}\nif for every field extension $K/k$ the\n$K$-algebra $K \\otimes_k S$ is reduced.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically reduced algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030S","source_file":"algebra.tex","source_line":10045,"source_end_line":10051,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10045-L10051","statement_sha256":"30fea62b0b27e5604b8dc20e22780f28d16f4e37aca7c844cf0b8bb8ed809ea4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1224,"rank":1224,"depth":0,"x":2057.055,"y":156.356,"cluster":"commutative-algebra"},{"id":"stacks:030T","tag":"030T","title":"Geometrically reduced algebras · Lemma 030T","summary":"Elementary properties of geometrically reduced algebras. Let k be a field. Let S be a k-algebra. • If S is geometrically reduced over k so is every k-subalgebra. • If all finitely generated k-subalgebras of S are geometrically reduced, then S is geometrically reduced. • A directed colimit of geometrically reduced k-algebras is geometrically reduced. • If S is geometrically reduced over k, then any localization of S is geometrically reduced over k.","statement_latex":"Elementary properties of geometrically reduced algebras.\nLet $k$ be a field. Let $S$ be a $k$-algebra.\n\\begin{enumerate}\n\\item If $S$ is geometrically reduced over $k$ so is every\n$k$-subalgebra.\n\\item If all finitely generated $k$-subalgebras of $S$ are\ngeometrically reduced, then $S$ is geometrically reduced.\n\\item A directed colimit of geometrically reduced $k$-algebras\nis geometrically reduced.\n\\item If $S$ is geometrically reduced over $k$, then any localization\nof $S$ is geometrically reduced over $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically reduced algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030T","source_file":"algebra.tex","source_line":10062,"source_end_line":10076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10062-L10076","statement_sha256":"9e9f6982a845d658e868d5b00b3277545b2c2cea7f6f47ddc21f8b8355b1b4ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":1225,"rank":1225,"depth":0,"x":2061.286,"y":282.392,"cluster":"commutative-algebra"},{"id":"stacks:04KN","tag":"04KN","title":"Geometrically reduced algebras · Lemma 04KN","summary":"Let k be a field. If R is geometrically reduced over k, and S ⊂ R is a multiplicative subset, then the localization S^-1R is geometrically reduced over k. If R is geometrically reduced over k, then R[x] is geometrically reduced over k.","statement_latex":"Let $k$ be a field.\nIf $R$ is geometrically reduced over $k$,\nand $S \\subset R$ is a multiplicative subset, then the localization\n$S^{-1}R$ is geometrically reduced over $k$.\nIf $R$ is geometrically reduced over $k$, then $R[x]$ is geometrically\nreduced over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically reduced algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KN","source_file":"algebra.tex","source_line":10083,"source_end_line":10091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10083-L10091","statement_sha256":"f9dc887d78d21ff2ce0b56c5c63139945486762a35d911e0a475040bda135bf9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1226,"rank":1226,"depth":0,"x":1956.626,"y":191.695,"cluster":"commutative-algebra"},{"id":"stacks:00I3","tag":"00I3","title":"Geometrically reduced algebras · Lemma 00I3","summary":"Let k be a field. Let R, S be k-algebras. • If R ⊗_k S is nonreduced, then there exist finitely generated subalgebras R' ⊂ R, S' ⊂ S such that R' ⊗_k S' is not reduced. • If R ⊗_k S contains a nonzero zerodivisor, then there exist finitely generated subalgebras R' ⊂ R, S' ⊂ S such that R' ⊗_k S' contains a nonzero zerodivisor. • If R ⊗_k S contains a nontrivial idempotent, then there exist finitely generated subalgebras R' ⊂ R, S' ⊂ S such that R' ⊗_k S' contains a…","statement_latex":"Let $k$ be a field. Let $R$, $S$ be $k$-algebras.\n\\begin{enumerate}\n\\item If $R \\otimes_k S$ is nonreduced, then there exist\nfinitely generated subalgebras $R' \\subset R$,\n$S' \\subset S$ such that $R' \\otimes_k S'$ is not reduced.\n\\item If $R \\otimes_k S$ contains a nonzero zerodivisor, then there exist\nfinitely generated subalgebras $R' \\subset R$,\n$S' \\subset S$ such that $R' \\otimes_k S'$ contains a nonzero zerodivisor.\n\\item If $R \\otimes_k S$ contains a nontrivial idempotent, then there exist\nfinitely generated subalgebras $R' \\subset R$,\n$S' \\subset S$ such that $R' \\otimes_k S'$ contains a nontrivial idempotent.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically reduced algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00I3","source_file":"algebra.tex","source_line":10105,"source_end_line":10119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10105-L10119","statement_sha256":"532e8dc4d319da70d99600b9e9093d5b1b575e41cabcbcb2eb2b2c821ff239b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1227,"rank":1227,"depth":0,"x":2107.002,"y":199.2,"cluster":"commutative-algebra"},{"id":"stacks:034N","tag":"034N","title":"Geometrically reduced algebras · Lemma 034N","summary":"Let k be a field. Let S be a geometrically reduced k-algebra. Let R be any reduced k-algebra. Then R ⊗_k S is reduced.","statement_latex":"Let $k$ be a field.\nLet $S$ be a geometrically reduced $k$-algebra.\nLet $R$ be any reduced $k$-algebra.\nThen $R \\otimes_k S$ is reduced.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically reduced algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034N","source_file":"algebra.tex","source_line":10129,"source_end_line":10135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10129-L10135","statement_sha256":"7776622ca6133c542889a7c5859c70824ddf7e139e7983fe60dc0a0a112115aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":1228,"rank":1228,"depth":6,"x":1989.876,"y":279.134,"cluster":"commutative-algebra"},{"id":"stacks:030U","tag":"030U","title":"Geometrically reduced algebras · Lemma 030U","summary":"Let k be a field. Let S be a reduced k-algebra. Let K/k be either a separable field extension, or a separably generated field extension. Then K ⊗_k S is reduced.","statement_latex":"Let $k$ be a field.\nLet $S$ be a reduced $k$-algebra.\nLet $K/k$ be either a separable field extension,\nor a separably generated field extension.\nThen $K \\otimes_k S$ is reduced.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically reduced algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030U","source_file":"algebra.tex","source_line":10151,"source_end_line":10158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10151-L10158","statement_sha256":"46f15384328e69c7fe3c31d2d5428cb66a23cd291fac895e309a2da7e62af798","origin":"The Stacks Project","memory_eligible":false,"source_rank":1229,"rank":1229,"depth":6,"x":2012.002,"y":153.511,"cluster":"commutative-algebra"},{"id":"stacks:07K2","tag":"07K2","title":"Geometrically reduced algebras · Lemma 07K2","summary":"Let k be a field and let S be a k-algebra. Assume that S is reduced and that S_ p is geometrically reduced for every minimal prime p of S. Then S is geometrically reduced.","statement_latex":"Let $k$ be a field and let $S$ be a $k$-algebra. Assume that\n$S$ is reduced and that $S_{\\mathfrak p}$ is geometrically\nreduced for every minimal prime $\\mathfrak p$ of $S$.\nThen $S$ is geometrically reduced.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically reduced algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07K2","source_file":"algebra.tex","source_line":10196,"source_end_line":10202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10196-L10202","statement_sha256":"873c693cedca14c28d5db6e9dc1f69676977949ed9012a039f1583120f07fec0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1230,"rank":1230,"depth":3,"x":2096.857,"y":258.883,"cluster":"commutative-algebra"},{"id":"stacks:0C2X","tag":"0C2X","title":"Geometrically reduced algebras · Lemma 0C2X","summary":"Let k'/k be a separable algebraic extension. Then there exists a multiplicative subset S ⊂ k' ⊗_k k' such that the multiplication map k' ⊗_k k' → k' is identified with k' ⊗_k k' → S^-1(k' ⊗_k k').","statement_latex":"Let $k'/k$ be a separable algebraic extension.\nThen there exists a multiplicative subset $S \\subset k' \\otimes_k k'$\nsuch that the multiplication map $k' \\otimes_k k' \\to k'$\nis identified with $k' \\otimes_k k' \\to S^{-1}(k' \\otimes_k k')$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically reduced algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2X","source_file":"algebra.tex","source_line":10220,"source_end_line":10226,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10220-L10226","statement_sha256":"59f89d1e60ea59c86126f7a3aa0de4154513b5dc2809fd2c6a932c78adb1beaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1231,"rank":1231,"depth":5,"x":1949.292,"y":229.281,"cluster":"commutative-algebra"},{"id":"stacks:0C2Y","tag":"0C2Y","title":"Geometrically reduced algebras · Lemma 0C2Y","summary":"Let k'/k be a separable algebraic field extension. Let A be an algebra over k'. Then A is geometrically reduced over k if and only if it is geometrically reduced over k'.","statement_latex":"Let $k'/k$ be a separable algebraic field extension.\nLet $A$ be an algebra over $k'$. Then $A$ is geometrically\nreduced over $k$ if and only if it is geometrically reduced over $k'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically reduced algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2Y","source_file":"algebra.tex","source_line":10257,"source_end_line":10262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10257-L10262","statement_sha256":"cd589eb89502fdef4a4b7e83760c1fe9c06537b41a4e1ffea2b09008004b37d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1232,"rank":1232,"depth":7,"x":2082.14,"y":167.27,"cluster":"commutative-algebra"},{"id":"stacks:0H71","tag":"0H71","title":"Separable extensions, continued · Lemma 0H71","summary":"Let k be a field of characteristic p > 1. Let K/k be a field extension generated by x_1, …, x_n + 1 ∈ K such that • (x_1, …, x_n) is a transcendence base of K/k, • for every k-linearly independent subset (a_1, …, a_m) of K the set (a^p_1, …, a_m^p) is k-linearly independent. Then there is 1 ≤ j ≤ n+1 such that ( x_1, …, widehatx_j, …, x_n+1) is a separating transcendence base for K / k.","statement_latex":"Let $k$ be a field of characteristic $p > 1$. Let $K/k$\nbe a field extension generated by $x_1, \\ldots, x_{n + 1} \\in K$ such that\n\\begin{enumerate}\n\\item $\\{x_1, \\ldots, x_n\\}$ is a transcendence base of $K/k$,\n\\item for every $k$-linearly independent subset\n$\\{a_1, \\ldots, a_m\\}$ of $K$ the set\n$\\{a^p_1, \\ldots, a_m^p\\}$ is $k$-linearly independent.\n\\end{enumerate}\nThen there is $1 \\leq j \\leq n+1$ such that\n$\\{ x_1, \\ldots, \\widehat{x}_j, \\ldots, x_{n+1}\\}$\nis a separating transcendence base for $K / k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Separable extensions, continued","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H71","source_file":"algebra.tex","source_line":10295,"source_end_line":10308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10295-L10308","statement_sha256":"a0f346ce7b575132fea0487d56b50cdd6c1d4d0d12a89ae177f8f0ff1aceb1dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1233,"rank":1233,"depth":1,"x":2033.964,"y":288.586,"cluster":"commutative-algebra"},{"id":"stacks:030W","tag":"030W","title":"Separable extensions, continued · Lemma 030W","summary":"Let k be a field of characteristic p > 0. Let K/k be a field extension. The following are equivalent: • K is separable over k, • for every k-linearly independent subset (a_1, …, a_m) of K the set (a^p_1, …, a_m^p) is k-linearly independent, • the ring K ⊗_k k^1/p is reduced, and • K is geometrically reduced over k.","statement_latex":"Let $k$ be a field of characteristic $p > 0$.\nLet $K/k$ be a field extension.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $K$ is separable over $k$,\n\\item for every $k$-linearly independent subset $\\{a_1, \\ldots, a_m\\}$\nof $K$ the set $\\{a^p_1, \\ldots, a_m^p\\}$ is $k$-linearly independent,\n\\item the ring $K \\otimes_k k^{1/p}$ is reduced, and\n\\item $K$ is geometrically reduced over $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Separable extensions, continued","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030W","source_file":"algebra.tex","source_line":10359,"source_end_line":10371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10359-L10371","statement_sha256":"70fdf23b97d4cc61a95b20aee1325d33e69a399bfaced362e02c392dab357316","origin":"The Stacks Project","memory_eligible":false,"source_rank":1234,"rank":1234,"depth":7,"x":1971.822,"y":171.592,"cluster":"commutative-algebra"},{"id":"stacks:030X","tag":"030X","title":"Separable extensions, continued · Lemma 030X","summary":"A separably generated field extension is separable.","statement_latex":"A separably generated field extension is separable.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Separable extensions, continued","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030X","source_file":"algebra.tex","source_line":10409,"source_end_line":10412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10409-L10412","statement_sha256":"716c208ef3cf38b0ddcc28010b0b5edeadf2eae7cc1d77ca2b86b941a77c9214","origin":"The Stacks Project","memory_eligible":false,"source_rank":1235,"rank":1235,"depth":8,"x":2111.969,"y":222.688,"cluster":"commutative-algebra"},{"id":"stacks:030V","tag":"030V","title":"Separable extensions, continued · Lemma 030V","summary":"Let k be a field. Let S be a k-algebra. The following are equivalent: • k' ⊗_k S is reduced for every finite purely inseparable extension k' of k, • k^1/p ⊗_k S is reduced, • k^perf ⊗_k S is reduced, where k^perf is the perfect closure of k, • overlinek ⊗_k S is reduced, where overlinek is the algebraic closure of k, and • S is geometrically reduced over k.","statement_latex":"Let $k$ be a field. Let $S$ be a $k$-algebra.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $k' \\otimes_k S$ is reduced for every finite\npurely inseparable extension $k'$ of $k$,\n\\item $k^{1/p} \\otimes_k S$ is reduced,\n\\item $k^{perf} \\otimes_k S$ is reduced, where $k^{perf}$ is the\nperfect closure of $k$,\n\\item $\\overline{k} \\otimes_k S$ is reduced, where $\\overline{k}$ is the\nalgebraic closure of $k$, and\n\\item $S$ is geometrically reduced over $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Separable extensions, continued","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030V","source_file":"algebra.tex","source_line":10424,"source_end_line":10438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10424-L10438","statement_sha256":"2acc60a8be212f07f95695b5dfa6c72f1648729b35d063ddf2fd6ef443741877","origin":"The Stacks Project","memory_eligible":false,"source_rank":1236,"rank":1236,"depth":8,"x":1967.289,"y":264.605,"cluster":"commutative-algebra"},{"id":"stacks:030Y","tag":"030Y","title":"Perfect fields · Definition 030Y","summary":"Let k be a field. We say k is perfect if every field extension of k is separable over k.","statement_latex":"Let $k$ be a field. We say $k$ is {\\it perfect}\nif every field extension of $k$ is separable over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Perfect fields","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030Y","source_file":"algebra.tex","source_line":10488,"source_end_line":10492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10488-L10492","statement_sha256":"e190026d30a711ffdcdf06c416eb0acf8e1ef429e788d321b6e6c4685744d142","origin":"The Stacks Project","memory_eligible":false,"source_rank":1237,"rank":1237,"depth":0,"x":2040.39,"y":151.408,"cluster":"commutative-algebra"},{"id":"stacks:030Z","tag":"030Z","title":"Perfect fields · Lemma 030Z","summary":"A field k is perfect if and only if it is a field of characteristic 0 or a field of characteristic p > 0 such that every element has a pth root.","statement_latex":"A field $k$ is perfect if and only if it is a field of characteristic $0$\nor a field of characteristic $p > 0$ such that every element has a $p$th\nroot.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Perfect fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030Z","source_file":"algebra.tex","source_line":10494,"source_end_line":10499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10494-L10499","statement_sha256":"e7cbdc5a89a4597248e67148c67b7f2385d6bfb0a2fd22eddd90caf60c824bce","origin":"The Stacks Project","memory_eligible":false,"source_rank":1238,"rank":1238,"depth":8,"x":2077.578,"y":276.57,"cluster":"commutative-algebra"},{"id":"stacks:030R","tag":"030R","title":"Perfect fields · Lemma 030R","summary":"Let K/k be a finitely generated field extension. There exists a diagram xymatrix K ar[r] & K' k ar[u] ar[r] & k' ar[u] where k'/k, K'/K are finite purely inseparable field extensions such that K'/k' is a separable field extension. In this situation we can assume that K' = k'K is the compositum, and also that K' = (k' ⊗_k K)_red.","statement_latex":"Let $K/k$ be a finitely generated field extension.\nThere exists a diagram\n$$\n\\xymatrix{\nK \\ar[r] & K' \\\\\nk \\ar[u] \\ar[r] & k' \\ar[u]\n}\n$$\nwhere $k'/k$, $K'/K$ are finite purely inseparable field\nextensions such that $K'/k'$ is a separable field extension.\nIn this situation we can assume that $K' = k'K$ is the compositum,\nand also that $K' = (k' \\otimes_k K)_{red}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Perfect fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/030R","source_file":"algebra.tex","source_line":10512,"source_end_line":10526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10512-L10526","statement_sha256":"9dd64b63e83d83716f9c5c234f4e05771821bdb497874affdf30e423352f7343","origin":"The Stacks Project","memory_eligible":false,"source_rank":1239,"rank":1239,"depth":9,"x":1949.288,"y":205.258,"cluster":"commutative-algebra"},{"id":"stacks:046W","tag":"046W","title":"Perfect fields · Lemma 046W","summary":"Every field has a unique perfect closure. For every field k there exists a purely inseparable extension k'/k such that k' is perfect. The field extension k'/k is unique up to unique isomorphism.","statement_latex":"\\begin{slogan}\nEvery field has a unique perfect closure.\n\\end{slogan}\nFor every field $k$ there exists a purely inseparable extension\n$k'/k$ such that $k'$ is perfect. The field extension\n$k'/k$ is unique up to unique isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Perfect fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046W","source_file":"algebra.tex","source_line":10544,"source_end_line":10552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10544-L10552","statement_sha256":"914ff45adc6414427b5074df68af2da544c64a9fb1d0ce2974651c55e8468faf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1240,"rank":1240,"depth":0,"x":2101.491,"y":185.013,"cluster":"commutative-algebra"},{"id":"stacks:046X","tag":"046X","title":"Perfect fields · Definition 046X","summary":"Let k be a field. The field extension k'/k of Lemma [Tag 046W] is called the perfect closure of k. Notation k^perf/k.","statement_latex":"Let $k$ be a field. The field extension $k'/k$ of Lemma \\ref{lemma-perfection}\nis called the {\\it perfect closure} of $k$. Notation $k^{perf}/k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Perfect fields","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046X","source_file":"algebra.tex","source_line":10568,"source_end_line":10572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10568-L10572","statement_sha256":"510c433cf1e0f0cd2f7807b93218cfdca1d76c60371f1125ae6507b4ab4aa02b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1241,"rank":1241,"depth":1,"x":2005.378,"y":286.476,"cluster":"commutative-algebra"},{"id":"stacks:00I4","tag":"00I4","title":"Perfect fields · Lemma 00I4","summary":"Let k be a perfect field. Any reduced k algebra is geometrically reduced over k. Let R, S be k-algebras. Assume both R and S are reduced. Then the k-algebra R ⊗_k S is reduced.","statement_latex":"Let $k$ be a perfect field.\nAny reduced $k$ algebra is geometrically reduced over $k$.\nLet $R$, $S$ be $k$-algebras.\nAssume both $R$ and $S$ are reduced.\nThen the $k$-algebra $R \\otimes_k S$ is reduced.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Perfect fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00I4","source_file":"algebra.tex","source_line":10580,"source_end_line":10587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10580-L10587","statement_sha256":"dcc2a0496b7e5ba025f4c061695a16a02db914c11c7b791345c9f417c7f86dfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1242,"rank":1242,"depth":9,"x":1994.639,"y":156.905,"cluster":"commutative-algebra"},{"id":"stacks:0BR6","tag":"0BR6","title":"Universal homeomorphisms · Lemma 0BR6","summary":"Let φ : R → S be a surjective map with locally nilpotent kernel. Then φ induces a homeomorphism of spectra and isomorphisms on residue fields. For any ring map R → R' the ring map R' → R' ⊗_R S is surjective with locally nilpotent kernel.","statement_latex":"Let $\\varphi : R \\to S$ be a surjective map with locally nilpotent kernel.\nThen $\\varphi$ induces a homeomorphism of spectra and isomorphisms\non residue fields. For any ring map $R \\to R'$ the ring map\n$R' \\to R' \\otimes_R S$ is surjective with locally nilpotent kernel.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BR6","source_file":"algebra.tex","source_line":10614,"source_end_line":10620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10614-L10620","statement_sha256":"cfc7c1c1ab2eebd3ffc24aab9fc182d981221fea73086277bb924aaf9c95eef9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1243,"rank":1243,"depth":1,"x":2106.942,"y":246.505,"cluster":"commutative-algebra"},{"id":"stacks:0BR7","tag":"0BR7","title":"Universal homeomorphisms · Lemma 0BR7","summary":"[Alper-adequate] Let k'/k be a field extension. The following are equivalent • for each x ∈ k' there exists an n > 0 such that x^n ∈ k, and • k' = k or k and k' have characteristic p > 0 and either k'/k is a purely inseparable extension or k and k' are algebraic extensions of F_p.","statement_latex":"\\begin{reference}\n\\cite[Lemma 3.1.6]{Alper-adequate}\n\\end{reference}\nLet $k'/k$ be a field extension. The following are equivalent\n\\begin{enumerate}\n\\item for each $x \\in k'$ there exists an $n > 0$ such that $x^n \\in k$, and\n\\item $k' = k$ or $k$ and $k'$ have characteristic $p > 0$ and\neither $k'/k$ is a purely inseparable extension or\n$k$ and $k'$ are algebraic extensions of $\\mathbf{F}_p$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BR7","source_file":"algebra.tex","source_line":10632,"source_end_line":10644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10632-L10644","statement_sha256":"e7c4715174ec6f650ace4ae800263aeada20967660ba9924059dfe085359444f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1244,"rank":1244,"depth":5,"x":1951.82,"y":244.155,"cluster":"commutative-algebra"},{"id":"stacks:0BR8","tag":"0BR8","title":"Universal homeomorphisms · Lemma 0BR8","summary":"Let φ : R → S be a ring map. If • for any x ∈ S there exists n > 0 such that x^n is in the image of φ, and • Ker(φ) is locally nilpotent, then φ induces a homeomorphism on spectra and induces residue field extensions satisfying the equivalent conditions of Lemma [Tag 0BR7].","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. If\n\\begin{enumerate}\n\\item for any $x \\in S$ there exists $n > 0$ such that\n$x^n$ is in the image of $\\varphi$, and\n\\item $\\Ker(\\varphi)$ is locally nilpotent,\n\\end{enumerate}\nthen $\\varphi$ induces a homeomorphism on spectra and induces residue\nfield extensions satisfying the equivalent conditions of\nLemma \\ref{lemma-powers-field}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BR8","source_file":"algebra.tex","source_line":10717,"source_end_line":10728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10717-L10728","statement_sha256":"e0810321feb3c538e0667c91feb1fe1fe867afce149e15609e385739c58c1aac","origin":"The Stacks Project","memory_eligible":false,"source_rank":1245,"rank":1245,"depth":6,"x":2068.288,"y":157.724,"cluster":"commutative-algebra"},{"id":"stacks:0EUH","tag":"0EUH","title":"Universal homeomorphisms · Lemma 0EUH","summary":"Let φ : R → S be a ring map. Assume • [(a)] S is generated as an R-algebra by elements x such that x^2, x^3 ∈ φ(R), and • [(b)] Ker(φ) is locally nilpotent, Then φ induces isomorphisms on residue fields and a homeomorphism of spectra. For any ring map R → R' the ring map R' → R' ⊗_R S also satisfies (a) and (b).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such\nthat $x^2, x^3 \\in \\varphi(R)$, and\n\\item[(b)] $\\Ker(\\varphi)$ is locally nilpotent,\n\\end{enumerate}\nThen $\\varphi$ induces isomorphisms on residue fields and\na homeomorphism of spectra. For any ring map $R \\to R'$\nthe ring map $R' \\to R' \\otimes_R S$ also satisfies (a) and (b).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUH","source_file":"algebra.tex","source_line":10769,"source_end_line":10780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10769-L10780","statement_sha256":"6576ecaad177f202c73b377732b61ce1a540435fb51806d5e4eefc9546fb39e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1246,"rank":1246,"depth":7,"x":2051.884,"y":287.758,"cluster":"commutative-algebra"},{"id":"stacks:0545","tag":"0545","title":"Universal homeomorphisms · Lemma 0545","summary":"Let p be a prime number. Let n, m > 0 be two integers. There exists an integer a such that (x + y)^p^a, p^a(x + y) ∈ Z[x^p^n, p^nx, y^p^m, p^my].","statement_latex":"Let $p$ be a prime number. Let $n, m > 0$ be two integers. There exists\nan integer $a$ such that\n$(x + y)^{p^a}, p^a(x + y) \\in \\mathbf{Z}[x^{p^n}, p^nx, y^{p^m}, p^my]$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0545","source_file":"algebra.tex","source_line":10813,"source_end_line":10818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10813-L10818","statement_sha256":"e44a7f96c3908490cee4f6c7a41cccddd8c9878af797ef850f1a98e944d7088a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1247,"rank":1247,"depth":0,"x":1959.258,"y":182.392,"cluster":"commutative-algebra"},{"id":"stacks:0BR9","tag":"0BR9","title":"Universal homeomorphisms · Lemma 0BR9","summary":"Let k'/k be a field extension. Let p be a prime number. The following are equivalent • k' is generated as a field extension of k by elements x such that there exists an n > 0 with x^p^n ∈ k and p^nx ∈ k, and • k = k' or the characteristic of k and k' is p and k'/k is purely inseparable.","statement_latex":"Let $k'/k$ be a field extension. Let $p$ be a prime number.\nThe following are equivalent\n\\begin{enumerate}\n\\item $k'$ is generated as a field extension of $k$ by elements\n$x$ such that there exists an $n > 0$ with $x^{p^n} \\in k$ and\n$p^nx \\in k$, and\n\\item $k = k'$ or the characteristic of $k$\nand $k'$ is $p$ and $k'/k$ is purely inseparable.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BR9","source_file":"algebra.tex","source_line":10847,"source_end_line":10858,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10847-L10858","statement_sha256":"e313060cdb8d6ac702bdb4ac34081547f95ab88a6f77d70b08ec275e91bdad1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1248,"rank":1248,"depth":0,"x":2112.543,"y":207.571,"cluster":"commutative-algebra"},{"id":"stacks:0BRA","tag":"0BRA","title":"Universal homeomorphisms · Lemma 0BRA","summary":"Let φ : R → S be a ring map. Let p be a prime number. Assume • [(a)] S is generated as an R-algebra by elements x such that there exists an n > 0 with x^p^n ∈ φ(R) and p^nx ∈ φ(R), and • [(b)] Ker(φ) is locally nilpotent, Then φ induces a homeomorphism of spectra and induces residue field extensions satisfying the equivalent conditions of Lemma [Tag 0BR9]. For any ring map R → R' the ring map R' → R' ⊗_R S also satisfies (a) and (b).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Let $p$ be a prime number. Assume\n\\begin{enumerate}\n\\item[(a)] $S$ is generated as an $R$-algebra by elements $x$ such\nthat there exists an $n > 0$ with $x^{p^n} \\in \\varphi(R)$ and\n$p^nx \\in \\varphi(R)$, and\n\\item[(b)] $\\Ker(\\varphi)$ is locally nilpotent,\n\\end{enumerate}\nThen $\\varphi$ induces a homeomorphism of spectra and induces\nresidue field extensions satisfying the equivalent conditions\nof Lemma \\ref{lemma-p-ring-map-field}. For any ring map $R \\to R'$\nthe ring map $R' \\to R' \\otimes_R S$ also satisfies (a) and (b).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRA","source_file":"algebra.tex","source_line":10867,"source_end_line":10880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10867-L10880","statement_sha256":"d3ec4df6e580aae6c34bcca431e18a8ad8150c495dd7c01a15f2023f2ad2d7aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":1249,"rank":1249,"depth":7,"x":1979.046,"y":276.094,"cluster":"commutative-algebra"},{"id":"stacks:0BRB","tag":"0BRB","title":"Universal homeomorphisms · Lemma 0BRB","summary":"Let φ : R → S be a ring map. Assume • φ induces an injective map of spectra, • φ induces purely inseparable residue field extensions. Then for any ring map R → R' properties (1) and (2) are true for R' → R' ⊗_R S.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ induces an injective map of spectra,\n\\item $\\varphi$ induces purely inseparable residue field extensions.\n\\end{enumerate}\nThen for any ring map $R \\to R'$ properties (1) and (2) are true for\n$R' \\to R' \\otimes_R S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRB","source_file":"algebra.tex","source_line":10912,"source_end_line":10921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10912-L10921","statement_sha256":"2a22505993d1cbd3ca82d850cf4551c957000fb7fc3723ba4a2ad504201cdf0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1250,"rank":1250,"depth":8,"x":2022.451,"y":149.61,"cluster":"commutative-algebra"},{"id":"stacks:0BRC","tag":"0BRC","title":"Universal homeomorphisms · Lemma 0BRC","summary":"Let φ : R → S be a ring map. Assume • φ is integral, • φ induces an injective map of spectra, • φ induces purely inseparable residue field extensions. Then φ induces a homeomorphism from Spec(S) onto a closed subset of Spec(R) and for any ring map R → R' properties (1), (2), (3) are true for R' → R' ⊗_R S.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is integral,\n\\item $\\varphi$ induces an injective map of spectra,\n\\item $\\varphi$ induces purely inseparable residue field extensions.\n\\end{enumerate}\nThen $\\varphi$ induces a homeomorphism from $\\Spec(S)$ onto a closed\nsubset of $\\Spec(R)$ and for any ring map\n$R \\to R'$ properties (1), (2), (3) are true for $R' \\to R' \\otimes_R S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRC","source_file":"algebra.tex","source_line":10968,"source_end_line":10979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10968-L10979","statement_sha256":"828e27bdcc89706caa7397ae131f8186d9aaa1e03b65fa30dcc0d577225d8ed5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1251,"rank":1251,"depth":9,"x":2092.273,"y":267.698,"cluster":"commutative-algebra"},{"id":"stacks:0BRD","tag":"0BRD","title":"Universal homeomorphisms · Lemma 0BRD","summary":"Let φ : R → S be a ring map. Assume • φ is integral, • φ induces an bijective map of spectra, • φ induces purely inseparable residue field extensions. Then φ induces a homeomorphism on spectra and for any ring map R → R' properties (1), (2), (3) are true for R' → R' ⊗_R S.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is integral,\n\\item $\\varphi$ induces an bijective map of spectra,\n\\item $\\varphi$ induces purely inseparable residue field extensions.\n\\end{enumerate}\nThen $\\varphi$ induces a homeomorphism on spectra and for any ring map\n$R \\to R'$ properties (1), (2), (3) are true for $R' \\to R' \\otimes_R S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRD","source_file":"algebra.tex","source_line":10988,"source_end_line":10998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L10988-L10998","statement_sha256":"30e95c0bd9040c0656392cfdf7159fd7f86103e86d3339b2cfcf4b526df90ea1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1252,"rank":1252,"depth":10,"x":1945.587,"y":220.163,"cluster":"commutative-algebra"},{"id":"stacks:09EF","tag":"09EF","title":"Universal homeomorphisms · Lemma 09EF","summary":"Let φ : R → S be a ring map such that • the kernel of φ is locally nilpotent, and • S is generated as an R-algebra by elements x such that there exist n > 0 and a polynomial P(T) ∈ R[T] whose image in S[T] is (T - x)^n. Then Spec(S) → Spec(R) is a homeomorphism and R → S induces purely inseparable extensions of residue fields. Moreover, conditions (1) and (2) remain true on arbitrary base change.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map such that\n\\begin{enumerate}\n\\item the kernel of $\\varphi$ is locally nilpotent, and\n\\item $S$ is generated as an $R$-algebra by elements $x$\nsuch that there exist $n > 0$ and a polynomial $P(T) \\in R[T]$\nwhose image in $S[T]$ is $(T - x)^n$.\n\\end{enumerate}\nThen $\\Spec(S) \\to \\Spec(R)$ is a homeomorphism and $R \\to S$\ninduces purely inseparable extensions of residue fields.\nMoreover, conditions (1) and (2) remain true on arbitrary base change.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EF","source_file":"algebra.tex","source_line":11005,"source_end_line":11017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11005-L11017","statement_sha256":"0b51089e35acd2c9992b224a49c19d9bb92ace3f4517d0fab3f5d41c494fac08","origin":"The Stacks Project","memory_eligible":false,"source_rank":1253,"rank":1253,"depth":8,"x":2092.214,"y":171.904,"cluster":"commutative-algebra"},{"id":"stacks:00I6","tag":"00I6","title":"Geometrically irreducible algebras · Lemma 00I6","summary":"Let R → S be a ring map. Assume • [(a)] Spec(R) is irreducible, • [(b)] R → S is flat, • [(c)] R → S is of finite presentation, • [(d)] the fibre rings S ⊗_R kappa( p) have irreducible spectra for a dense collection of primes p of R. Then Spec(S) is irreducible. This is true more generally with (b) + (c) replaced by \"the map Spec(S) → Spec(R) is open\".","statement_latex":"Let $R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item[(a)] $\\Spec(R)$ is irreducible,\n\\item[(b)] $R \\to S$ is flat,\n\\item[(c)] $R \\to S$ is of finite presentation,\n\\item[(d)] the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$\nhave irreducible spectra for a dense collection of primes $\\mathfrak p$ of $R$.\n\\end{enumerate}\nThen $\\Spec(S)$ is irreducible.\nThis is true more generally with (b) $+$ (c)\nreplaced by ``the map $\\Spec(S) \\to \\Spec(R)$ is open''.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00I6","source_file":"algebra.tex","source_line":11082,"source_end_line":11095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11082-L11095","statement_sha256":"1f3d4002220d013e866ab15d579ad6eb1a2547985a1ef73c8bd5782bf0dd7979","origin":"The Stacks Project","memory_eligible":false,"source_rank":1254,"rank":1254,"depth":8,"x":2022.79,"y":290.881,"cluster":"commutative-algebra"},{"id":"stacks:00I7","tag":"00I7","title":"Geometrically irreducible algebras · Lemma 00I7","summary":"Let k be a separably closed field. Let R, S be k-algebras. If R, S have a unique minimal prime, so does R ⊗_k S.","statement_latex":"Let $k$ be a separably closed field.\nLet $R$, $S$ be $k$-algebras. If $R$, $S$ have a unique\nminimal prime, so does $R \\otimes_k S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00I7","source_file":"algebra.tex","source_line":11105,"source_end_line":11110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11105-L11110","statement_sha256":"a1d884974cc38f449e786ace1a18e48cdb0facdf32c03ddc2fe6c3abbe895896","origin":"The Stacks Project","memory_eligible":false,"source_rank":1255,"rank":1255,"depth":10,"x":1978.233,"y":163.552,"cluster":"commutative-algebra"},{"id":"stacks:037K","tag":"037K","title":"Geometrically irreducible algebras · Lemma 037K","summary":"Let k be a field. Let R be a k-algebra. The following are equivalent • for every field extension k'/k the spectrum of R ⊗_k k' is irreducible, • for every finite separable field extension k'/k the spectrum of R ⊗_k k' is irreducible, • the spectrum of R ⊗_k overlinek is irreducible where overlinek is the separable algebraic closure of k, and • the spectrum of R ⊗_k overlinek is irreducible where overlinek is the algebraic closure of k.","statement_latex":"Let $k$ be a field.\nLet $R$ be a $k$-algebra.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every field extension $k'/k$ the\nspectrum of $R \\otimes_k k'$ is irreducible,\n\\item for every finite separable field extension $k'/k$ the\nspectrum of $R \\otimes_k k'$ is irreducible,\n\\item the spectrum of $R \\otimes_k \\overline{k}$ is irreducible\nwhere $\\overline{k}$ is the separable algebraic closure of $k$, and\n\\item the spectrum of $R \\otimes_k \\overline{k}$ is irreducible\nwhere $\\overline{k}$ is the algebraic closure of $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037K","source_file":"algebra.tex","source_line":11145,"source_end_line":11160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11145-L11160","statement_sha256":"9b97a056f228e923d4a238a02993b7b3c916c5819ed9bba74ada6e598c937358","origin":"The Stacks Project","memory_eligible":false,"source_rank":1256,"rank":1256,"depth":11,"x":2113.699,"y":232.269,"cluster":"commutative-algebra"},{"id":"stacks:037L","tag":"037L","title":"Geometrically irreducible algebras · Definition 037L","summary":"Let k be a field. Let S be a k-algebra. We say S is geometrically irreducible over k if for every field extension k'/k the spectrum of S ⊗_k k' is irreducible.","statement_latex":"Let $k$ be a field.\nLet $S$ be a $k$-algebra.\nWe say $S$ is {\\it geometrically irreducible over $k$}\nif for every field extension $k'/k$ the spectrum of\n$S \\otimes_k k'$ is irreducible\\footnote{An irreducible space is nonempty.}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037L","source_file":"algebra.tex","source_line":11206,"source_end_line":11213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11206-L11213","statement_sha256":"28e430648b75b1db023d24ca3a354989b0b1913f466b06962ece0080577c86a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1257,"rank":1257,"depth":0,"x":1958.301,"y":258.507,"cluster":"commutative-algebra"},{"id":"stacks:037M","tag":"037M","title":"Geometrically irreducible algebras · Lemma 037M","summary":"Let k be a field. Let R be a k-algebra. If k is separably algebraically closed then R is geometrically irreducible over k if and only if the spectrum of R is irreducible.","statement_latex":"Let $k$ be a field.\nLet $R$ be a $k$-algebra.\nIf $k$ is separably algebraically closed then $R$ is\ngeometrically irreducible over $k$ if and only if the\nspectrum of $R$ is irreducible.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037M","source_file":"algebra.tex","source_line":11220,"source_end_line":11227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11220-L11227","statement_sha256":"23c44e748a9d59357b76ca2b1ec621941c1af5a6c3e2411ccac1969eefd3e7e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1258,"rank":1258,"depth":1,"x":2051.939,"y":150.812,"cluster":"commutative-algebra"},{"id":"stacks:037N","tag":"037N","title":"Geometrically irreducible algebras · Lemma 037N","summary":"Let k be a field. Let S be a k-algebra. • If S is geometrically irreducible over k so is every k-subalgebra. • If all finitely generated k-subalgebras of S are geometrically irreducible, then S is geometrically irreducible. • A directed colimit of geometrically irreducible k-algebras is geometrically irreducible.","statement_latex":"Let $k$ be a field. Let $S$ be a $k$-algebra.\n\\begin{enumerate}\n\\item If $S$ is geometrically irreducible over $k$ so is every\n$k$-subalgebra.\n\\item If all finitely generated $k$-subalgebras of $S$ are\ngeometrically irreducible, then $S$ is geometrically irreducible.\n\\item A directed colimit of geometrically irreducible $k$-algebras\nis geometrically irreducible.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037N","source_file":"algebra.tex","source_line":11234,"source_end_line":11245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11234-L11245","statement_sha256":"4d45a8cc509209d1ab40d2d0b0fa9e4f3e01c6a40aeec46188330eec208d07f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1259,"rank":1259,"depth":2,"x":2069.524,"y":283.567,"cluster":"commutative-algebra"},{"id":"stacks:037O","tag":"037O","title":"Geometrically irreducible algebras · Lemma 037O","summary":"Let k be a field. Let S be a geometrically irreducible k-algebra. Let R be any k-algebra. The map Spec(R ⊗_k S) → Spec(R) induces a bijection on irreducible components.","statement_latex":"Let $k$ be a field.\nLet $S$ be a geometrically irreducible $k$-algebra.\nLet $R$ be any $k$-algebra.\nThe map\n$$\n\\Spec(R \\otimes_k S) \\longrightarrow \\Spec(R)\n$$\ninduces a bijection on irreducible components.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037O","source_file":"algebra.tex","source_line":11256,"source_end_line":11266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11256-L11266","statement_sha256":"7b8cb1ba499a77b8c64ba96cea18be545d7cd12d834c318f165bd8f0b0a30c1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1260,"rank":1260,"depth":5,"x":1949.61,"y":195.515,"cluster":"commutative-algebra"},{"id":"stacks:037P","tag":"037P","title":"Geometrically irreducible algebras · Lemma 037P","summary":"Let K/k be a field extension. If k is algebraically closed in K, then K is geometrically irreducible over k.","statement_latex":"Let $K/k$ be a field extension. If $k$ is algebraically closed in $K$, then\n$K$ is geometrically irreducible over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037P","source_file":"algebra.tex","source_line":11296,"source_end_line":11300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11296-L11300","statement_sha256":"e6cb1b06892218c7c9ac2fc33bcd027f3ac0d833a5ea444868f5d2cf0b6766ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":1261,"rank":1261,"depth":12,"x":2109.093,"y":192.396,"cluster":"commutative-algebra"},{"id":"stacks:0G30","tag":"0G30","title":"Geometrically irreducible algebras · Lemma 0G30","summary":"Let K/k be a geometrically irreducible field extension. Let S be a geometrically irreducible K-algebra. Then S is geometrically irreducible over k.","statement_latex":"Let $K/k$ be a geometrically irreducible field extension.\nLet $S$ be a geometrically irreducible $K$-algebra.\nThen $S$ is geometrically irreducible over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G30","source_file":"algebra.tex","source_line":11322,"source_end_line":11327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11322-L11327","statement_sha256":"e344c48ec18b10be49a25726ce0704a03bacda8e07e1db3a118732167580f0f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1262,"rank":1262,"depth":12,"x":1993.82,"y":285.336,"cluster":"commutative-algebra"},{"id":"stacks:0G31","tag":"0G31","title":"Geometrically irreducible algebras · Lemma 0G31","summary":"Let K/k be a field extension. The following are equivalent • K is geometrically irreducible over k, and • the induced extension K(t)/k(t) of purely transcendental extensions is geometrically irreducible.","statement_latex":"Let $K/k$ be a field extension. The following are equivalent\n\\begin{enumerate}\n\\item $K$ is geometrically irreducible over $k$, and\n\\item the induced extension $K(t)/k(t)$ of purely transcendental extensions\nis geometrically irreducible.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G31","source_file":"algebra.tex","source_line":11340,"source_end_line":11348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11340-L11348","statement_sha256":"a30a3881f3068ee130dd40ab4f42e5cde13fef8ec84b7a2a685713cd70752a36","origin":"The Stacks Project","memory_eligible":false,"source_rank":1263,"rank":1263,"depth":12,"x":2004.097,"y":151.186,"cluster":"commutative-algebra"},{"id":"stacks:0G32","tag":"0G32","title":"Geometrically irreducible algebras · Lemma 0G32","summary":"Let K/L/M be a tower of fields with L/M geometrically irreducible. Let x ∈ K be transcendental over L. Then L(x)/M(x) is geometrically irreducible.","statement_latex":"Let $K/L/M$ be a tower of fields with $L/M$ geometrically irreducible.\nLet $x \\in K$ be transcendental over $L$. Then $L(x)/M(x)$ is geometrically\nirreducible.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G32","source_file":"algebra.tex","source_line":11377,"source_end_line":11382,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11377-L11382","statement_sha256":"2a29d19e8ce04d44cf9133e0818915eedba89b11dfc74a4dd985330c3ec99137","origin":"The Stacks Project","memory_eligible":false,"source_rank":1264,"rank":1264,"depth":13,"x":2104.555,"y":256.101,"cluster":"commutative-algebra"},{"id":"stacks:0G33","tag":"0G33","title":"Geometrically irreducible algebras · Lemma 0G33","summary":"Let K/k be a field extension. The following are equivalent • K/k is geometrically irreducible, and • every element α ∈ K separably algebraic over k is in k.","statement_latex":"Let $K/k$ be a field extension. The following are equivalent\n\\begin{enumerate}\n\\item $K/k$ is geometrically irreducible, and\n\\item every element $\\alpha \\in K$ separably algebraic over $k$ is\nin $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G33","source_file":"algebra.tex","source_line":11391,"source_end_line":11399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11391-L11399","statement_sha256":"b8a10fbcd44e44b5b2c61fa29ea0b1373dbe0b3565f875f1dde4bdfe95a6f43d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1265,"rank":1265,"depth":13,"x":1945.862,"y":235.709,"cluster":"commutative-algebra"},{"id":"stacks:037Q","tag":"037Q","title":"Geometrically irreducible algebras · Lemma 037Q","summary":"Let K/k be a field extension. Consider the subextension K/k'/k consisting of elements separably algebraic over k. Then K is geometrically irreducible over k'. If K/k is a finitely generated field extension, then [k' : k] < ∞.","statement_latex":"Let $K/k$ be a field extension. Consider the subextension $K/k'/k$ consisting\nof elements separably algebraic over $k$. Then $K$ is geometrically irreducible\nover $k'$. If $K/k$ is a finitely generated field extension, then\n$[k' : k] < \\infty$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037Q","source_file":"algebra.tex","source_line":11427,"source_end_line":11433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11427-L11433","statement_sha256":"7b165a97633c5efb293e710525f2bedecdf76d51315915b37c8d4864ea2d1380","origin":"The Stacks Project","memory_eligible":false,"source_rank":1266,"rank":1266,"depth":14,"x":2079.486,"y":160.583,"cluster":"commutative-algebra"},{"id":"stacks:04KP","tag":"04KP","title":"Geometrically irreducible algebras · Lemma 04KP","summary":"Let K/k be an extension of fields. Let overlinek/k be a separable algebraic closure. Then Gal(overlinek/k) acts transitively on the primes of overlinek ⊗_k K.","statement_latex":"Let $K/k$ be an extension of fields.\nLet $\\overline{k}/k$ be a separable algebraic closure.\nThen $\\text{Gal}(\\overline{k}/k)$ acts transitively on the\nprimes of $\\overline{k} \\otimes_k K$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically irreducible algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KP","source_file":"algebra.tex","source_line":11445,"source_end_line":11451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11445-L11451","statement_sha256":"02629c2dbe56fcb61889de37ba9c41d91871bdf581895d5f525c309efd07114c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1267,"rank":1267,"depth":15,"x":2041.309,"y":292.003,"cluster":"commutative-algebra"},{"id":"stacks:037R","tag":"037R","title":"Geometrically connected algebras · Lemma 037R","summary":"Let k be a separably algebraically closed field. Let R, S be k-algebras. If Spec(R), and Spec(S) are connected, then so is Spec(R ⊗_k S).","statement_latex":"Let $k$ be a separably algebraically closed field.\nLet $R$, $S$ be $k$-algebras. If $\\Spec(R)$, and\n$\\Spec(S)$ are connected, then so is\n$\\Spec(R \\otimes_k S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically connected algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037R","source_file":"algebra.tex","source_line":11487,"source_end_line":11493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11487-L11493","statement_sha256":"31b7b200364f80e5cf8454fdca78b6cb09ed5697bbdfc6d8886f4e66e499ef64","origin":"The Stacks Project","memory_eligible":false,"source_rank":1268,"rank":1268,"depth":11,"x":1963.656,"y":173.251,"cluster":"commutative-algebra"},{"id":"stacks:037S","tag":"037S","title":"Geometrically connected algebras · Lemma 037S","summary":"Let k be a field. Let R be a k-algebra. The following are equivalent • for every field extension k'/k the spectrum of R ⊗_k k' is connected, and • for every finite separable field extension k'/k the spectrum of R ⊗_k k' is connected.","statement_latex":"Let $k$ be a field.\nLet $R$ be a $k$-algebra.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every field extension $k'/k$ the\nspectrum of $R \\otimes_k k'$ is connected, and\n\\item for every finite separable field extension $k'/k$ the\nspectrum of $R \\otimes_k k'$ is connected.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically connected algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037S","source_file":"algebra.tex","source_line":11530,"source_end_line":11541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11530-L11541","statement_sha256":"1e2369297c81f8055507ecad0dae6b38eb4cbd2e2ba8512471cf992f7804f1da","origin":"The Stacks Project","memory_eligible":false,"source_rank":1269,"rank":1269,"depth":12,"x":2116.648,"y":216.822,"cluster":"commutative-algebra"},{"id":"stacks:037T","tag":"037T","title":"Geometrically connected algebras · Definition 037T","summary":"Let k be a field. Let S be a k-algebra. We say S is geometrically connected over k if for every field extension k'/k the spectrum of S ⊗_k k' is connected.","statement_latex":"Let $k$ be a field.\nLet $S$ be a $k$-algebra.\nWe say $S$ is {\\it geometrically connected over $k$}\nif for every field extension $k'/k$ the spectrum\nof $S \\otimes_k k'$ is connected.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically connected algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037T","source_file":"algebra.tex","source_line":11560,"source_end_line":11567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11560-L11567","statement_sha256":"f5c3e740705267f11426172c5eebc95ce10b9ffe9001582ff4f6655330fa803e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1270,"rank":1270,"depth":0,"x":1968.569,"y":271.588,"cluster":"commutative-algebra"},{"id":"stacks:037U","tag":"037U","title":"Geometrically connected algebras · Lemma 037U","summary":"Let k be a field. Let R be a k-algebra. If k is separably algebraically closed then R is geometrically connected over k if and only if the spectrum of R is connected.","statement_latex":"Let $k$ be a field.\nLet $R$ be a $k$-algebra.\nIf $k$ is separably algebraically closed then $R$ is\ngeometrically connected over $k$ if and only if the\nspectrum of $R$ is connected.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically connected algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037U","source_file":"algebra.tex","source_line":11573,"source_end_line":11580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11573-L11580","statement_sha256":"2fec46a1cf68f467cd15fe989635ba3d9445009aa5543ca81d901b828c35bdd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1271,"rank":1271,"depth":1,"x":2033.818,"y":146.991,"cluster":"commutative-algebra"},{"id":"stacks:037V","tag":"037V","title":"Geometrically connected algebras · Lemma 037V","summary":"Let k be a field. Let S be a k-algebra. • If S is geometrically connected over k so is every k-subalgebra. • If all finitely generated k-subalgebras of S are geometrically connected, then S is geometrically connected. • A directed colimit of geometrically connected k-algebras is geometrically connected.","statement_latex":"Let $k$ be a field. Let $S$ be a $k$-algebra.\n\\begin{enumerate}\n\\item If $S$ is geometrically connected over $k$ so is every\n$k$-subalgebra.\n\\item If all finitely generated $k$-subalgebras of $S$ are\ngeometrically connected, then $S$ is geometrically connected.\n\\item A directed colimit of geometrically connected $k$-algebras\nis geometrically connected.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically connected algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037V","source_file":"algebra.tex","source_line":11587,"source_end_line":11598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11587-L11598","statement_sha256":"e3bcbf7a1f83701af5dbdf529b33003aea38f46cec64d6c5749f3e290c6a511d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1272,"rank":1272,"depth":0,"x":2085.981,"y":276.087,"cluster":"commutative-algebra"},{"id":"stacks:037W","tag":"037W","title":"Geometrically connected algebras · Lemma 037W","summary":"Let k be a field. Let S be a geometrically connected k-algebra. Let R be any k-algebra. The map R → R ⊗_k S induces a bijection on idempotents, and the map Spec(R ⊗_k S) → Spec(R) induces a bijection on connected components.","statement_latex":"Let $k$ be a field.\nLet $S$ be a geometrically connected $k$-algebra.\nLet $R$ be any $k$-algebra.\nThe map\n$$\nR \\longrightarrow R \\otimes_k S\n$$\ninduces a bijection on idempotents, and the map\n$$\n\\Spec(R \\otimes_k S) \\longrightarrow \\Spec(R)\n$$\ninduces a bijection on connected components.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically connected algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037W","source_file":"algebra.tex","source_line":11610,"source_end_line":11624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11610-L11624","statement_sha256":"4d6ce6ab5115b3787a51c5601c655fa16c76bc8e5d686dc7633c3d982421b5ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":1273,"rank":1273,"depth":8,"x":1943.486,"y":210.393,"cluster":"commutative-algebra"},{"id":"stacks:05DX","tag":"05DX","title":"Geometrically integral algebras · Definition 05DX","summary":"Let k be a field. Let S be a k-algebra. We say S is geometrically integral over k if for every field extension k'/k the ring of S ⊗_k k' is a domain.","statement_latex":"Let $k$ be a field.\nLet $S$ be a $k$-algebra.\nWe say $S$ is {\\it geometrically integral over $k$}\nif for every field extension $k'/k$ the ring\nof $S \\otimes_k k'$ is a domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically integral algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DX","source_file":"algebra.tex","source_line":11660,"source_end_line":11667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11660-L11667","statement_sha256":"9552f89f547c45f685840d6c0b376d3ea0b85b0a024d5a1b45bfafef55e746ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":1274,"rank":1274,"depth":0,"x":2101.628,"y":177.931,"cluster":"commutative-algebra"},{"id":"stacks:05DY","tag":"05DY","title":"Geometrically integral algebras · Lemma 05DY","summary":"Let k be a field. Let S be a k-algebra. In this case S is geometrically integral over k if and only if S is geometrically irreducible as well as geometrically reduced over k.","statement_latex":"Let $k$ be a field.\nLet $S$ be a $k$-algebra.\nIn this case $S$ is geometrically integral over $k$ if and only if\n$S$ is geometrically irreducible as well as geometrically reduced over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically integral algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DY","source_file":"algebra.tex","source_line":11673,"source_end_line":11679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11673-L11679","statement_sha256":"4c50ee7e830138fe87ce8da68fd58ee0b29907db36d05fb57ddcd8b917105c79","origin":"The Stacks Project","memory_eligible":false,"source_rank":1275,"rank":1275,"depth":0,"x":2010.985,"y":291.772,"cluster":"commutative-algebra"},{"id":"stacks:0FWF","tag":"0FWF","title":"Geometrically integral algebras · Lemma 0FWF","summary":"Let k be a field. Let S be a k-algebra. The following are equivalent • S is geometrically integral over k, • for every finite extension k'/k of fields the ring S ⊗_k k' is a domain, • S ⊗_k overlinek is a domain where overlinek is the algebraic closure of k.","statement_latex":"Let $k$ be a field. Let $S$ be a $k$-algebra.\nThe following are equivalent\n\\begin{enumerate}\n\\item $S$ is geometrically integral over $k$,\n\\item for every finite extension $k'/k$ of fields\nthe ring $S \\otimes_k k'$ is a domain,\n\\item $S \\otimes_k \\overline{k}$ is a domain\nwhere $\\overline{k}$ is the algebraic closure of $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically integral algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWF","source_file":"algebra.tex","source_line":11685,"source_end_line":11696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11685-L11696","statement_sha256":"f7cf920e02e052e0ffc720ad96c26fbb7cfebfeea0972e8dfcc24c55e2a860f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1276,"rank":1276,"depth":12,"x":1986.239,"y":156.191,"cluster":"commutative-algebra"},{"id":"stacks:09P9","tag":"09P9","title":"Geometrically integral algebras · Lemma 09P9","summary":"Let k be a field. Let S be a geometrically integral k-algebra. Let R be a k-algebra and an integral domain. Then R ⊗_k S is an integral domain.","statement_latex":"Let $k$ be a field. Let $S$ be a geometrically integral $k$-algebra.\nLet $R$ be a $k$-algebra and an integral domain. Then $R \\otimes_k S$\nis an integral domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically integral algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09P9","source_file":"algebra.tex","source_line":11704,"source_end_line":11709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11704-L11709","statement_sha256":"70eb8641cd1a205caf4103877bc29a40170b26e5d5b4c717a4d0708e53f699c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1277,"rank":1277,"depth":7,"x":2113.707,"y":242.254,"cluster":"commutative-algebra"},{"id":"stacks:00I9","tag":"00I9","title":"Valuation rings · Definition 00I9","summary":"Valuation rings. • Let K be a field. Let A, B be local rings contained in K. We say that B dominates A if A ⊂ B and m_A = A ∩ m_B. • Let A be a ring. We say A is a valuation ring if A is a local domain and if A is maximal for the relation of domination among local rings contained in the fraction field of A. • Let A be a valuation ring with fraction field K. If R ⊂ K is a subring of K, then we say A is centered on R if R ⊂ A.","statement_latex":"Valuation rings.\n\\begin{enumerate}\n\\item Let $K$ be a field. Let $A$, $B$ be local rings contained\nin $K$. We say that $B$ {\\it dominates} $A$ if $A \\subset B$\nand $\\mathfrak m_A = A \\cap \\mathfrak m_B$.\n\\item Let $A$ be a ring. We say $A$ is a {\\it valuation ring}\nif $A$ is a local domain and if $A$ is maximal\nfor the relation of domination among local rings contained in\nthe fraction field of $A$.\n\\item Let $A$ be a valuation ring with fraction field $K$.\nIf $R \\subset K$ is a subring of $K$, then we say $A$\nis {\\it centered} on $R$ if $R \\subset A$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00I9","source_file":"algebra.tex","source_line":11729,"source_end_line":11744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11729-L11744","statement_sha256":"594ac7f00e8ddb2689f0c58810a2e67e75df09f024d0b9c6556bcae3bbfc5a41","origin":"The Stacks Project","memory_eligible":false,"source_rank":1278,"rank":1278,"depth":0,"x":1950.261,"y":251.134,"cluster":"commutative-algebra"},{"id":"stacks:00IA","tag":"00IA","title":"Valuation rings · Lemma 00IA","summary":"Let K be a field. Let A ⊂ K be a local subring. Then there exists a valuation ring with fraction field K dominating A.","statement_latex":"Let $K$ be a field. Let $A \\subset K$ be a local subring.\nThen there exists a valuation ring with fraction field $K$\ndominating $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IA","source_file":"algebra.tex","source_line":11749,"source_end_line":11754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11749-L11754","statement_sha256":"aaa9bc98c9923ea6bc781a8982d2d9314d5188994af2a62f0bd9857e540576bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1279,"rank":1279,"depth":6,"x":2063.811,"y":151.695,"cluster":"commutative-algebra"},{"id":"stacks:00IC","tag":"00IC","title":"Valuation rings · Lemma 00IC","summary":"Let A be a valuation ring. Then A is a normal domain.","statement_latex":"Let $A$ be a valuation ring.\nThen $A$ is a normal domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IC","source_file":"algebra.tex","source_line":11782,"source_end_line":11786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11782-L11786","statement_sha256":"82d155b0559344231a5026be66eb349f36b697722c9f0fd27b001ac570281597","origin":"The Stacks Project","memory_eligible":false,"source_rank":1280,"rank":1280,"depth":6,"x":2060.042,"y":289.655,"cluster":"commutative-algebra"},{"id":"stacks:00IB","tag":"00IB","title":"Valuation rings · Lemma 00IB","summary":"Let A be a valuation ring with maximal ideal m and fraction field K. Let x ∈ K. Then either x ∈ A or x^-1 ∈ A or both.","statement_latex":"Let $A$ be a valuation ring with maximal ideal $\\mathfrak m$ and\nfraction field $K$.\nLet $x \\in K$. Then either $x \\in A$ or $x^{-1} \\in A$ or both.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IB","source_file":"algebra.tex","source_line":11800,"source_end_line":11805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11800-L11805","statement_sha256":"1b049f403ab576f87743ae0b592ec8a29f1a32d38ec9f1613a6eeeb023a1f334","origin":"The Stacks Project","memory_eligible":false,"source_rank":1281,"rank":1281,"depth":7,"x":1951.718,"y":185.633,"cluster":"commutative-algebra"},{"id":"stacks:052K","tag":"052K","title":"Valuation rings · Lemma 052K","summary":"Let A ⊂ K be a subring of a field K such that for all x ∈ K either x ∈ A or x^-1 ∈ A or both. Then A is a valuation ring with fraction field K.","statement_latex":"Let $A \\subset K$ be a subring of a field $K$ such that for all\n$x \\in K$ either $x \\in A$ or $x^{-1} \\in A$ or both.\nThen $A$ is a valuation ring with fraction field $K$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052K","source_file":"algebra.tex","source_line":11822,"source_end_line":11827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11822-L11827","statement_sha256":"2ce7b5882f51a61b5e0d69c0886cf18a9443474eee66530d87d28999b8c597cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1282,"rank":1282,"depth":0,"x":2115.487,"y":200.894,"cluster":"commutative-algebra"},{"id":"stacks:0AS4","tag":"0AS4","title":"Valuation rings · Lemma 0AS4","summary":"Valuation rings are stable under filtered direct limits Let I be a directed set. Let (A_i, φ_ij) be a system of valuation rings over I. Then A = colim A_i is a valuation ring.","statement_latex":"\\begin{slogan}\nValuation rings are stable under filtered direct limits\n\\end{slogan}\nLet $I$ be a directed set. Let $(A_i, \\varphi_{ij})$\nbe a system of valuation rings over $I$.\nThen $A = \\colim A_i$ is a valuation ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AS4","source_file":"algebra.tex","source_line":11843,"source_end_line":11851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11843-L11851","statement_sha256":"42a387e1b164c92a5c26e8e8c0ed1186c6acec7849770825d1646894e77f62eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1283,"rank":1283,"depth":8,"x":1982.257,"y":282.687,"cluster":"commutative-algebra"},{"id":"stacks:052L","tag":"052L","title":"Valuation rings · Lemma 052L","summary":"Let L/K be an extension of fields. If B ⊂ L is a valuation ring, then A = K ∩ B is a valuation ring.","statement_latex":"Let $L/K$ be an extension of fields. If $B \\subset L$\nis a valuation ring, then $A = K \\cap B$ is a valuation ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052L","source_file":"algebra.tex","source_line":11862,"source_end_line":11866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11862-L11866","statement_sha256":"0d8ce24f35a80526fbff9890108574938ab17b493e86154ae5aebf38c2b66bc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1284,"rank":1284,"depth":8,"x":2014.771,"y":146.578,"cluster":"commutative-algebra"},{"id":"stacks:0AAV","tag":"0AAV","title":"Valuation rings · Lemma 0AAV","summary":"Let L/K be an algebraic extension of fields. If B ⊂ L is a valuation ring with fraction field L and not a field, then A = K ∩ B is a valuation ring and not a field.","statement_latex":"Let $L/K$ be an algebraic extension of fields. If $B \\subset L$\nis a valuation ring with fraction field $L$ and not a field, then\n$A = K \\cap B$ is a valuation ring and not a field.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAV","source_file":"algebra.tex","source_line":11875,"source_end_line":11880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11875-L11880","statement_sha256":"05b670fa5040e948c1bc865816ffd5f9594e66b1ad5505bee14d4475bfbee090","origin":"The Stacks Project","memory_eligible":false,"source_rank":1285,"rank":1285,"depth":9,"x":2100.376,"y":265.565,"cluster":"commutative-algebra"},{"id":"stacks:088Y","tag":"088Y","title":"Valuation rings · Lemma 088Y","summary":"Let A be a valuation ring. For any prime ideal p ⊂ A the quotient A/ p is a valuation ring. The same is true for the localization A_ p and in fact any localization of A.","statement_latex":"Let $A$ be a valuation ring. For any prime ideal $\\mathfrak p \\subset A$ the\nquotient $A/\\mathfrak p$ is a valuation ring. The same is true for the\nlocalization $A_\\mathfrak p$ and in fact any localization of $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088Y","source_file":"algebra.tex","source_line":11890,"source_end_line":11895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11890-L11895","statement_sha256":"0fd37b4613a91c189727c9f19f9fcef7fa5d7ad6c39a1fe7a8bcae3fef7d28fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":1286,"rank":1286,"depth":1,"x":1941.334,"y":226.343,"cluster":"commutative-algebra"},{"id":"stacks:088Z","tag":"088Z","title":"Valuation rings · Lemma 088Z","summary":"Let A' be a valuation ring with residue field K. Let A be a valuation ring with fraction field K. Then C = (λ ∈ A' mid λ bmod m_A' ∈ A) is a valuation ring.","statement_latex":"Let $A'$ be a valuation ring with residue field $K$.\nLet $A$ be a valuation ring with fraction field $K$.\nThen\n$C = \\{\\lambda \\in A' \\mid \\lambda \\bmod \\mathfrak m_{A'} \\in A\\}$\nis a valuation ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088Z","source_file":"algebra.tex","source_line":11902,"source_end_line":11909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11902-L11909","statement_sha256":"1192f4378c2b677378e2f98e896bd7f4852d24225a964cd1ff3823d6d88cc6e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1287,"rank":1287,"depth":1,"x":2090.367,"y":164.932,"cluster":"commutative-algebra"},{"id":"stacks:090P","tag":"090P","title":"Valuation rings · Lemma 090P","summary":"Let A be a normal domain with fraction field K. • For every x ∈ K, x not ∈ A there exists a valuation ring A ⊂ V ⊂ K with fraction field K such that x not ∈ V. • If A is local, we can moreover choose V which dominates A. In other words, A is the intersection of all valuation rings in K containing A and if A is local, then A is the intersection of all valuation rings in K dominating A.","statement_latex":"Let $A$ be a normal domain with fraction field $K$.\n\\begin{enumerate}\n\\item For every $x \\in K$, $x \\not \\in A$ there exists a valuation ring\n$A \\subset V \\subset K$ with fraction field $K$ such that $x \\not \\in V$.\n\\item If $A$ is local, we can moreover choose $V$ which dominates $A$.\n\\end{enumerate}\nIn other words, $A$ is the intersection of all valuation rings in $K$\ncontaining $A$ and if $A$ is local, then $A$ is the intersection of\nall valuation rings in $K$ dominating $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090P","source_file":"algebra.tex","source_line":11923,"source_end_line":11934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11923-L11934","statement_sha256":"427670f2afe77826057b43d0caf73b3f6741fef29217bd1da8bfc0ed7f523827","origin":"The Stacks Project","memory_eligible":false,"source_rank":1288,"rank":1288,"depth":7,"x":2029.77,"y":294.969,"cluster":"commutative-algebra"},{"id":"stacks:00ID","tag":"00ID","title":"Valuation rings · Lemma 00ID","summary":"Let A be a valuation ring with field of fractions K. Set Γ = K^*/A^* (with group law written additively). For γ, γ' ∈ Γ define γ ≥ γ' if and only if γ - γ' is in the image of A - (0) → Γ. Then (Γ, ≥) is a totally ordered abelian group.","statement_latex":"Let $A$ be a valuation ring with field of fractions $K$.\nSet $\\Gamma = K^*/A^*$ (with group law written additively).\nFor $\\gamma, \\gamma' \\in \\Gamma$\ndefine $\\gamma \\geq \\gamma'$ if and only if\n$\\gamma - \\gamma'$ is in the image of $A - \\{0\\} \\to \\Gamma$.\nThen $(\\Gamma, \\geq)$ is a totally ordered abelian group.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ID","source_file":"algebra.tex","source_line":11969,"source_end_line":11977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11969-L11977","statement_sha256":"9dafef28975fe7690d32866d9c7feafc8e1d594c4ea176a9a4ca9c665f1c8684","origin":"The Stacks Project","memory_eligible":false,"source_rank":1289,"rank":1289,"depth":8,"x":1969.795,"y":164.509,"cluster":"commutative-algebra"},{"id":"stacks:00IE","tag":"00IE","title":"Valuation rings · Definition 00IE","summary":"Let A be a valuation ring. • The totally ordered abelian group (Γ, ≥) of Lemma [Tag 00ID] is called the value group of the valuation ring A. • The map v : A - (0) → Γ and also v : K^* → Γ is called the valuation associated to A. • The valuation ring A is called a discrete valuation ring if Γ ≅ Z.","statement_latex":"Let $A$ be a valuation ring.\n\\begin{enumerate}\n\\item The totally ordered abelian group $(\\Gamma, \\geq)$ of\nLemma \\ref{lemma-valuation-group} is called the\n{\\it value group} of the valuation ring $A$.\n\\item The map $v : A - \\{0\\} \\to \\Gamma$ and also $v : K^* \\to \\Gamma$ is\ncalled the {\\it valuation} associated to $A$.\n\\item The valuation ring $A$ is called a {\\it discrete valuation ring}\nif $\\Gamma \\cong \\mathbf{Z}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IE","source_file":"algebra.tex","source_line":11986,"source_end_line":11998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L11986-L11998","statement_sha256":"4714707b3b1ba35cc0d55fbb3e2a891124e3493a90799c6721a7c558a0e7285f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1290,"rank":1290,"depth":9,"x":2119.146,"y":226.766,"cluster":"commutative-algebra"},{"id":"stacks:00IF","tag":"00IF","title":"Valuation rings · Lemma 00IF","summary":"Let A be a valuation ring. The valuation v : A -(0) → Γ_≥ 0 has the following properties: • v(a) = 0 ⇔ a ∈ A^*, • v(ab) = v(a) + v(b), • v(a + b) ≥ min(v(a), v(b)) provided a + b not = 0.","statement_latex":"Let $A$ be a valuation ring. The valuation $v : A -\\{0\\} \\to \\Gamma_{\\geq 0}$\nhas the following properties:\n\\begin{enumerate}\n\\item $v(a) = 0 \\Leftrightarrow a \\in A^*$,\n\\item $v(ab) = v(a) + v(b)$,\n\\item $v(a + b) \\geq \\min(v(a), v(b))$ provided $a + b \\not = 0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IF","source_file":"algebra.tex","source_line":12005,"source_end_line":12014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12005-L12014","statement_sha256":"e0fb837c1e113877fab88e5df65633ea73cfff893a667248f95b7919101c1970","origin":"The Stacks Project","memory_eligible":false,"source_rank":1291,"rank":1291,"depth":0,"x":1958.722,"y":265.66,"cluster":"commutative-algebra"},{"id":"stacks:090Q","tag":"090Q","title":"Valuation rings · Lemma 090Q","summary":"Let A be a ring. The following are equivalent • A is a valuation ring, • A is a local domain and every finitely generated ideal of A is principal.","statement_latex":"Let $A$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $A$ is a valuation ring,\n\\item $A$ is a local domain and every finitely generated\nideal of $A$ is principal.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090Q","source_file":"algebra.tex","source_line":12020,"source_end_line":12028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12020-L12028","statement_sha256":"3240b7319cf6c13801ffdbe46b6d081f6759434f1ae4c8d4459eb2a5df16da12","origin":"The Stacks Project","memory_eligible":false,"source_rank":1292,"rank":1292,"depth":1,"x":2045.864,"y":145.78,"cluster":"commutative-algebra"},{"id":"stacks:00IG","tag":"00IG","title":"Valuation rings · Lemma 00IG","summary":"Let (Γ, ≥) be a totally ordered abelian group. Let K be a field. Let v : K^* → Γ be a homomorphism of abelian groups such that v(a + b) ≥ min(v(a), v(b)) for a, b ∈ K with a, b, a + b not zero. Then A = ( x ∈ K mid x = 0 or v(x) ≥ 0 ) is a valuation ring with value group Im(v) ⊂ Γ, with maximal ideal m = ( x ∈ K mid x = 0 or v(x) > 0 ) and with group of units A^* = ( x ∈ K^* mid v(x) = 0 ).","statement_latex":"Let $(\\Gamma, \\geq)$ be a totally ordered abelian group.\nLet $K$ be a field. Let $v : K^* \\to \\Gamma$ be a homomorphism\nof abelian groups such that $v(a + b) \\geq \\min(v(a), v(b))$ for\n$a, b \\in K$ with $a, b, a + b$ not zero. Then\n$$\nA =\n\\{\nx \\in K \\mid x = 0 \\text{ or } v(x) \\geq 0\n\\}\n$$\nis a valuation ring with value group $\\Im(v) \\subset \\Gamma$,\nwith maximal ideal\n$$\n\\mathfrak m =\n\\{\nx \\in K \\mid x = 0 \\text{ or } v(x) > 0\n\\}\n$$\nand with group of units\n$$\nA^* =\n\\{\nx \\in K^* \\mid v(x) = 0\n\\}.\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IG","source_file":"algebra.tex","source_line":12042,"source_end_line":12069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12042-L12069","statement_sha256":"7f24939d2193f983aade69bc23f137874d86a9925cb76b5e89ab1b0ebb205929","origin":"The Stacks Project","memory_eligible":false,"source_rank":1293,"rank":1293,"depth":0,"x":2078.056,"y":283.821,"cluster":"commutative-algebra"},{"id":"stacks:00IH","tag":"00IH","title":"Valuation rings · Lemma 00IH","summary":"Let A be a valuation ring. Ideals in A correspond 1 - 1 with ideals of Γ. This bijection is inclusion preserving, and maps prime ideals to prime ideals.","statement_latex":"Let $A$ be a valuation ring.\nIdeals in $A$ correspond $1 - 1$ with ideals of $\\Gamma$.\nThis bijection is inclusion preserving, and maps prime\nideals to prime ideals.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IH","source_file":"algebra.tex","source_line":12084,"source_end_line":12090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12084-L12090","statement_sha256":"cd9b2c328d4dde8e2be9d436ac0e33e02038a2b2565cb9350fb80fa4f78b6a72","origin":"The Stacks Project","memory_eligible":false,"source_rank":1294,"rank":1294,"depth":0,"x":1943.118,"y":200.18,"cluster":"commutative-algebra"},{"id":"stacks:00II","tag":"00II","title":"Valuation rings · Lemma 00II","summary":"A valuation ring is Noetherian if and only if it is a discrete valuation ring or a field.","statement_latex":"A valuation ring is Noetherian if and only if it is\na discrete valuation ring or a field.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00II","source_file":"algebra.tex","source_line":12096,"source_end_line":12100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12096-L12100","statement_sha256":"12ca7c1e343cd67684b588a03e46bc257a74c1e5756e33d59a213d029f125edc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1295,"rank":1295,"depth":1,"x":2110.118,"y":185.267,"cluster":"commutative-algebra"},{"id":"stacks:00IK","tag":"00IK","title":"More Noetherian rings · Lemma 00IK","summary":"Let R be a Noetherian ring. Any finite R-module is of finite presentation. Any submodule of a finite R-module is finite. The ascending chain condition holds for R-submodules of a finite R-module.","statement_latex":"Let $R$ be a Noetherian ring.\nAny finite $R$-module is of finite presentation.\nAny submodule of a finite $R$-module is finite.\nThe ascending chain condition holds for $R$-submodules\nof a finite $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IK","source_file":"algebra.tex","source_line":12153,"source_end_line":12160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12153-L12160","statement_sha256":"e625680a28368fd27174142f20f8be466e41aa0889d6225829f8a9442b035dd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1296,"rank":1296,"depth":0,"x":1998.809,"y":291.173,"cluster":"commutative-algebra"},{"id":"stacks:00IN","tag":"00IN","title":"Artin-Rees · Lemma 00IN","summary":"Suppose that R is Noetherian, I ⊂ R an ideal. Let N ⊂ M be finite R-modules. There exists a constant c > 0 such that I^n M ∩ N = I^n-c(I^cM ∩ N) for all n ≥ c.","statement_latex":"Suppose that $R$ is Noetherian, $I \\subset R$ an ideal.\nLet $N \\subset M$ be finite $R$-modules.\nThere exists a constant $c > 0$ such that\n$I^n M \\cap N  =  I^{n-c}(I^cM \\cap N)$ for all $n \\geq c$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IN","source_file":"algebra.tex","source_line":12186,"source_end_line":12192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12186-L12192","statement_sha256":"fa6296da560608d743590656afdac4e21786c2e32c97222374c4b0bc9d88fa60","origin":"The Stacks Project","memory_eligible":false,"source_rank":1297,"rank":1297,"depth":1,"x":1995.716,"y":149.721,"cluster":"commutative-algebra"},{"id":"stacks:00IO","tag":"00IO","title":"More Noetherian rings · Lemma 00IO","summary":"Suppose that 0 → K → M xrightarrowf N is an exact sequence of finitely generated modules over a Noetherian ring R. Let I ⊂ R be an ideal. Then there exists a c such that f^-1(I^nN) = K + I^n-cf^-1(I^cN) and f(M) ∩ I^nN ⊂ f(I^n - cM) for all n ≥ c.","statement_latex":"Suppose that $0 \\to K \\to M \\xrightarrow{f} N$ is an\nexact sequence of finitely generated modules\nover a Noetherian ring $R$. Let $I \\subset R$ be an ideal.\nThen there exists a $c$ such that\n$$\nf^{-1}(I^nN) = K + I^{n-c}f^{-1}(I^cN)\n\\quad\\text{and}\\quad\nf(M) \\cap I^nN \\subset f(I^{n - c}M)\n$$\nfor all $n \\geq c$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IO","source_file":"algebra.tex","source_line":12220,"source_end_line":12232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12220-L12232","statement_sha256":"de142271892c319123c077345cff11b7039d84d5affdccf0f77438198bc03513","origin":"The Stacks Project","memory_eligible":false,"source_rank":1298,"rank":1298,"depth":2,"x":2111.911,"y":252.414,"cluster":"commutative-algebra"},{"id":"stacks:00IP","tag":"00IP","title":"Krull's intersection theorem · Lemma 00IP","summary":"Let R be a Noetherian local ring. Let I ⊂ R be a proper ideal. Let M be a finite R-module. Then ⋂_n ≥ 0 I^nM = 0.","statement_latex":"Let $R$ be a Noetherian local ring. Let $I \\subset R$ be\na proper ideal. Let $M$ be a finite $R$-module.\nThen $\\bigcap_{n \\geq 0} I^nM = 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IP","source_file":"algebra.tex","source_line":12240,"source_end_line":12245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12240-L12245","statement_sha256":"a14ec52c326702ac9e2d6340d5d95797be2073670f6c0e0642ae6f007e689caf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1299,"rank":1299,"depth":3,"x":1943.412,"y":242.608,"cluster":"commutative-algebra"},{"id":"stacks:00IQ","tag":"00IQ","title":"More Noetherian rings · Lemma 00IQ","summary":"Let R be a Noetherian ring. Let I ⊂ R be an ideal. Let M be a finite R-module. Let N = ⋂_n I^n M. • For every prime p, I ⊂ p there exists a f ∈ R, f not ∈ p such that N_f = 0. • If I is contained in the Jacobson radical of R, then N = 0.","statement_latex":"Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal.\nLet $M$ be a finite $R$-module. Let $N = \\bigcap_n I^n M$.\n\\begin{enumerate}\n\\item For every prime $\\mathfrak p$, $I \\subset \\mathfrak p$ there\nexists a $f \\in R$, $f \\not \\in \\mathfrak p$ such that $N_f = 0$.\n\\item If $I$ is contained in the Jacobson radical\nof $R$, then $N = 0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IQ","source_file":"algebra.tex","source_line":12256,"source_end_line":12266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12256-L12266","statement_sha256":"fcc06d4a3f320c4ab0f40c5177acd28b6498107a8bf432ee31d72fe33484cb43","origin":"The Stacks Project","memory_eligible":false,"source_rank":1300,"rank":1300,"depth":4,"x":2075.731,"y":154.105,"cluster":"commutative-algebra"},{"id":"stacks:00IS","tag":"00IS","title":"Artin-Tate · Lemma 00IS","summary":"Let R be a Noetherian ring. Let S be a finitely generated R-algebra. If T ⊂ S is an R-subalgebra such that S is finitely generated as a T-module, then T is of finite type over R.","statement_latex":"Let $R$ be a Noetherian ring. Let $S$ be a finitely\ngenerated $R$-algebra. If $T \\subset S$ is an $R$-subalgebra such\nthat $S$ is finitely generated as a $T$-module, then $T$ is of\nfinite type over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IS","source_file":"algebra.tex","source_line":12296,"source_end_line":12302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12296-L12302","statement_sha256":"0bb35727ee5e22f09e76974c86327181dd1ee18f092c8c39be122247c7053132","origin":"The Stacks Project","memory_eligible":false,"source_rank":1301,"rank":1301,"depth":0,"x":2049.296,"y":294.643,"cluster":"commutative-algebra"},{"id":"stacks:02LY","tag":"02LY","title":"Length · Definition 02LY","summary":"Let R be a ring. For any R-module M we define the length of M over R by the formula length_R(M) = sup ( n mid ∃ 0 = M_0 ⊂ M_1 ⊂ … ⊂ M_n = M, M_i not = M_i + 1 ).","statement_latex":"Let $R$ be a ring. For any $R$-module $M$\nwe define the {\\it length} of $M$ over $R$ by the\nformula\n$$\n\\text{length}_R(M)\n=\n\\sup\n\\{\nn\n\\mid\n\\exists\\ 0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M,\n\\text{ }M_i \\not = M_{i + 1}\n\\}.\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LY","source_file":"algebra.tex","source_line":12349,"source_end_line":12365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12349-L12365","statement_sha256":"24f59d35d292fe25ab4cb3ef28bc1ece1fcf16c64e32dc6b71057d21a5035a59","origin":"The Stacks Project","memory_eligible":false,"source_rank":1302,"rank":1302,"depth":0,"x":1955.643,"y":175.847,"cluster":"commutative-algebra"},{"id":"stacks:02LZ","tag":"02LZ","title":"Length · Lemma 02LZ","summary":"Modules of finite length are finite. Let R be a ring. Let M be an R-module. If length_R(M) < ∞ then M is a finite R-module.","statement_latex":"\\begin{slogan}\nModules of finite length are finite.\n\\end{slogan}\nLet $R$ be a ring.\nLet $M$ be an $R$-module.\nIf $\\text{length}_R(M) < \\infty$ then $M$ is a finite $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LZ","source_file":"algebra.tex","source_line":12379,"source_end_line":12387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12379-L12387","statement_sha256":"3185fa28b8b0e114d02fa6441dec7fb4635593ee3ee2708082be84477982944f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1303,"rank":1303,"depth":0,"x":2120.461,"y":210.352,"cluster":"commutative-algebra"},{"id":"stacks:00IV","tag":"00IV","title":"Length · Lemma 00IV","summary":"Length is additive in short exact sequences. If 0 → M' → M → M\" → 0 is a short exact sequence of modules over R then the length of M is the sum of the lengths of M' and M\".","statement_latex":"\\begin{slogan}\nLength is additive in short exact sequences.\n\\end{slogan}\nIf $0 \\to M' \\to M \\to M'' \\to 0$\nis a short exact sequence of modules over $R$ then\nthe length of $M$ is the sum of the\nlengths of $M'$ and $M''$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IV","source_file":"algebra.tex","source_line":12393,"source_end_line":12402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12393-L12402","statement_sha256":"85675c952ff5686f20ddc64a52a77c9650bd6b3eade302140cf62a8449e355b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1304,"rank":1304,"depth":0,"x":1970.973,"y":278.525,"cluster":"commutative-algebra"},{"id":"stacks:00IW","tag":"00IW","title":"Length · Lemma 00IW","summary":"Let R be a local ring with maximal ideal m. If M is an R-module and m^n M not = 0 for all n ≥ 0, then length_R(M) = ∞. In other words, if M has finite length then m^nM = 0 for some n.","statement_latex":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$.\nIf $M$ is an $R$-module and $\\mathfrak m^n M \\not = 0$ for all $n \\geq 0$,\nthen $\\text{length}_R(M) = \\infty$. In other words, if $M$\nhas finite length then $\\mathfrak m^nM = 0$ for some $n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IW","source_file":"algebra.tex","source_line":12420,"source_end_line":12426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12420-L12426","statement_sha256":"59ee3b9f9f859804b16ed982a5db028d161ab1753fb9458fd651896a195a2898","origin":"The Stacks Project","memory_eligible":false,"source_rank":1305,"rank":1305,"depth":0,"x":2026.457,"y":143.246,"cluster":"commutative-algebra"},{"id":"stacks:00IX","tag":"00IX","title":"Length · Lemma 00IX","summary":"Let R → S be a ring map. Let M be an S-module. We always have length_R(M) ≥ length_S(M). If R → S is surjective then equality holds.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.\nWe always have $\\text{length}_R(M) \\geq \\text{length}_S(M)$.\nIf $R \\to S$ is surjective then equality holds.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IX","source_file":"algebra.tex","source_line":12444,"source_end_line":12449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12444-L12449","statement_sha256":"c58d2f3bf8535e8760370ff665b611801b34ae58f02edfd18f48e8b7dd71783d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1306,"rank":1306,"depth":0,"x":2094.425,"y":274.662,"cluster":"commutative-algebra"},{"id":"stacks:00IY","tag":"00IY","title":"Length · Lemma 00IY","summary":"Let R be a ring with maximal ideal m. Suppose that M is an R-module with m M = 0. Then the length of M as an R-module agrees with the dimension of M as a R/ m vector space. The length is finite if and only if M is a finite R-module.","statement_latex":"Let $R$ be a ring with maximal ideal $\\mathfrak m$.\nSuppose that $M$ is an $R$-module with\n$\\mathfrak m M  =  0$. Then the length of $M$ as\nan $R$-module agrees with the dimension of $M$ as\na $R/\\mathfrak m$ vector space.\nThe length is finite if and only if $M$ is a finite $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IY","source_file":"algebra.tex","source_line":12459,"source_end_line":12467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12459-L12467","statement_sha256":"7f7c31cf61ea93c855a007f59c2b64e4c943bdfdd4f2b6b1328a6a3566b8ce7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1307,"rank":1307,"depth":1,"x":1938.411,"y":216.244,"cluster":"commutative-algebra"},{"id":"stacks:00IZ","tag":"00IZ","title":"Length · Lemma 00IZ","summary":"Let R be a ring. Let M be an R-module. Let S ⊂ R be a multiplicative subset. Then length_R(M) ≥ length_S^-1R(S^-1M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \\subset R$ be\na multiplicative subset. Then\n$\\text{length}_R(M) \\geq \\text{length}_{S^{-1}R}(S^{-1}M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IZ","source_file":"algebra.tex","source_line":12476,"source_end_line":12481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12476-L12481","statement_sha256":"7025c141fdf2af912065ef34adcde3e57d43e68b970c3a7bb75568365dc25f8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1308,"rank":1308,"depth":1,"x":2100.653,"y":170.733,"cluster":"commutative-algebra"},{"id":"stacks:00J0","tag":"00J0","title":"Length · Lemma 00J0","summary":"Let R be a ring with finitely generated maximal ideal m. (For example R Noetherian.) Suppose that M is a finite R-module with m^n M = 0 for some n. Then length_R(M) < ∞.","statement_latex":"Let $R$ be a ring with finitely generated\nmaximal ideal $\\mathfrak m$. (For example $R$ Noetherian.)\nSuppose that $M$ is a finite $R$-module with\n$\\mathfrak m^n M  =  0$ for some $n$.\nThen $\\text{length}_R(M) < \\infty$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00J0","source_file":"algebra.tex","source_line":12489,"source_end_line":12496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12489-L12496","statement_sha256":"3ab7e21e339db0a1648e2b8127e8a1d481d50af241b2586261f87aac6ef80c72","origin":"The Stacks Project","memory_eligible":false,"source_rank":1309,"rank":1309,"depth":2,"x":2017.504,"y":296.523,"cluster":"commutative-algebra"},{"id":"stacks:00J1","tag":"00J1","title":"Length · Definition 00J1","summary":"Let R be a ring. Let M be an R-module. We say M is simple if M not = 0 and every submodule of M is either equal to M or to 0.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nWe say $M$ is {\\it simple} if $M \\not = 0$ and\nevery submodule of $M$ is either equal to $M$ or\nto $0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00J1","source_file":"algebra.tex","source_line":12508,"source_end_line":12514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12508-L12514","statement_sha256":"7512af636a5fc2eec931a27878baf709574e67d61218bb333c48b606efd093fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1310,"rank":1310,"depth":0,"x":1977.606,"y":156.396,"cluster":"commutative-algebra"},{"id":"stacks:00J2","tag":"00J2","title":"Length · Lemma 00J2","summary":"Let R be a ring. Let M be an R-module. The following are equivalent: • M is simple, • length_R(M) = 1, and • M ≅ R/ m for some maximal ideal m ⊂ R.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $M$ is simple,\n\\item $\\text{length}_R(M) = 1$, and\n\\item $M \\cong R/\\mathfrak m$ for some maximal ideal\n$\\mathfrak m \\subset R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00J2","source_file":"algebra.tex","source_line":12516,"source_end_line":12526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12516-L12526","statement_sha256":"600bef8ab2cfbf8f6c71e1acfb39be36981836dcaad34bebe6fb078b980725ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":1311,"rank":1311,"depth":2,"x":2119.904,"y":237.194,"cluster":"commutative-algebra"},{"id":"stacks:00J3","tag":"00J3","title":"Length · Lemma 00J3","summary":"Let R be a ring. Let M be a finite length R-module. Choose any maximal chain of submodules 0 = M_0 ⊂ M_1 ⊂ M_2 ⊂ … ⊂ M_n = M with M_i not = M_i-1, i = 1, …, n. Then • n = length_R(M), • each M_i/M_i-1 is simple, • each M_i/M_i-1 is of the form R/ m_i for some maximal ideal m_i, • given a maximal ideal m ⊂ R we have \\# (i mid m_i = m) = length_R_ m (M_ m).","statement_latex":"Let $R$ be a ring. Let $M$ be a finite length $R$-module.\nChoose any maximal chain of submodules\n$$\n0 = M_0 \\subset M_1 \\subset M_2 \\subset \\ldots \\subset M_n = M\n$$\nwith $M_i \\not = M_{i-1}$, $i = 1, \\ldots, n$. Then\n\\begin{enumerate}\n\\item $n = \\text{length}_R(M)$,\n\\item each $M_i/M_{i-1}$ is simple,\n\\item each $M_i/M_{i-1}$ is of the form\n$R/\\mathfrak m_i$ for some maximal ideal $\\mathfrak m_i$,\n\\item given a maximal ideal $\\mathfrak m \\subset R$\nwe have\n$$\n\\# \\{i \\mid \\mathfrak m_i = \\mathfrak m\\}\n=\n\\text{length}_{R_{\\mathfrak m}} (M_{\\mathfrak m}).\n$$\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00J3","source_file":"algebra.tex","source_line":12545,"source_end_line":12566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12545-L12566","statement_sha256":"087e2f134fab79b5c9164faad2281611cc3477b61ec1d7e03383db18acbf445c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1312,"rank":1312,"depth":3,"x":1949.771,"y":258.387,"cluster":"commutative-algebra"},{"id":"stacks:02M0","tag":"02M0","title":"Length · Lemma 02M0","summary":"Let A be a local ring with maximal ideal m. Let B be a semi-local ring with maximal ideals m_i, i = 1, …, n. Suppose that A → B is a homomorphism such that each m_i lies over m and such that [kappa( m_i) : kappa( m)] < ∞. Let M be a B-module of finite length. Then length_A(M) = ∑_i = 1, …, n [kappa( m_i) : kappa( m)] length_B_ m_i(M_ m_i), in particular length_A(M) < ∞.","statement_latex":"Let $A$ be a local ring with maximal ideal $\\mathfrak m$.\nLet $B$ be a semi-local ring with maximal ideals $\\mathfrak m_i$,\n$i = 1, \\ldots, n$.\nSuppose that $A \\to B$ is a homomorphism such that each $\\mathfrak m_i$\nlies over $\\mathfrak m$ and such that\n$$\n[\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m)] < \\infty.\n$$\nLet $M$ be a $B$-module of finite length.\nThen\n$$\n\\text{length}_A(M) = \\sum\\nolimits_{i = 1, \\ldots, n}\n[\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m)]\n\\text{length}_{B_{\\mathfrak m_i}}(M_{\\mathfrak m_i}),\n$$\nin particular $\\text{length}_A(M) < \\infty$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02M0","source_file":"algebra.tex","source_line":12602,"source_end_line":12620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12602-L12620","statement_sha256":"b7447f03f138b79807624b92cd8fdba75ff67f9ac284ba14746b561fc41d44b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1313,"rank":1313,"depth":4,"x":2058.328,"y":146.071,"cluster":"commutative-algebra"},{"id":"stacks:02M1","tag":"02M1","title":"Length · Lemma 02M1","summary":"Let A → B be a flat local homomorphism of local rings. Then for any A-module M we have length_A(M) length_B(B/ m_AB) = length_B(M ⊗_A B). In particular, if length_B(B/ m_AB) < ∞ then M has finite length if and only if M ⊗_A B has finite length.","statement_latex":"Let $A \\to B$ be a flat local homomorphism of local rings.\nThen for any $A$-module $M$ we have\n$$\n\\text{length}_A(M) \\text{length}_B(B/\\mathfrak m_AB)\n=\n\\text{length}_B(M \\otimes_A B).\n$$\nIn particular, if $\\text{length}_B(B/\\mathfrak m_AB) < \\infty$\nthen $M$ has finite length if and only if $M \\otimes_A B$ has finite length.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02M1","source_file":"algebra.tex","source_line":12641,"source_end_line":12652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12641-L12652","statement_sha256":"d7824ffa3646c383bcc5511df38cf68610b44dca81f5da4b9378b88ee1a1b7d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1314,"rank":1314,"depth":4,"x":2068.615,"y":290.683,"cluster":"commutative-algebra"},{"id":"stacks:02M2","tag":"02M2","title":"Length · Lemma 02M2","summary":"Let A → B → C be flat local homomorphisms of local rings. Then length_B(B/ m_A B) length_C(C/ m_B C) = length_C(C/ m_A C)","statement_latex":"Let $A \\to B \\to C$ be flat local homomorphisms of local rings. Then\n$$\n\\text{length}_B(B/\\mathfrak m_A B)\n\\text{length}_C(C/\\mathfrak m_B C)\n=\n\\text{length}_C(C/\\mathfrak m_A C)\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Length","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02M2","source_file":"algebra.tex","source_line":12673,"source_end_line":12682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12673-L12682","statement_sha256":"711a2cb8dd601bc6f59959edd678ff117c0e9d3e458fbe6e27dcf1cbb9759a46","origin":"The Stacks Project","memory_eligible":false,"source_rank":1315,"rank":1315,"depth":5,"x":1944.57,"y":189.75,"cluster":"commutative-algebra"},{"id":"stacks:00J5","tag":"00J5","title":"Artinian rings · Definition 00J5","summary":"A ring R is Artinian if it satisfies the descending chain condition for ideals.","statement_latex":"A ring $R$ is {\\it Artinian} if it satisfies the\ndescending chain condition for ideals.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Artinian rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00J5","source_file":"algebra.tex","source_line":12707,"source_end_line":12711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12707-L12711","statement_sha256":"4ed418632ba8e80388701b88351efbe3c29f9e3ec6015d46c43e0eafb60b1e62","origin":"The Stacks Project","memory_eligible":false,"source_rank":1316,"rank":1316,"depth":0,"x":2117.437,"y":193.796,"cluster":"commutative-algebra"},{"id":"stacks:00J6","tag":"00J6","title":"Artinian rings · Lemma 00J6","summary":"Suppose R is a finite dimensional algebra over a field. Then R is Artinian.","statement_latex":"Suppose $R$ is a finite dimensional algebra over a field.\nThen $R$ is Artinian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00J6","source_file":"algebra.tex","source_line":12713,"source_end_line":12717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12713-L12717","statement_sha256":"23e7a57a59362da0f8ed2f5be5f44c1aae33bc9355f5e964373fd5bcd88c7708","origin":"The Stacks Project","memory_eligible":false,"source_rank":1317,"rank":1317,"depth":0,"x":1986.54,"y":289.028,"cluster":"commutative-algebra"},{"id":"stacks:00J7","tag":"00J7","title":"Artinian rings · Lemma 00J7","summary":"If R is Artinian then R has only finitely many maximal ideals.","statement_latex":"If $R$ is Artinian then $R$ has only finitely many maximal ideals.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00J7","source_file":"algebra.tex","source_line":12723,"source_end_line":12726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12723-L12726","statement_sha256":"fc66f82d1c2aeae34647766dec9bebdcab7bd5703f62c9304dd2e41f9451c7ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":1318,"rank":1318,"depth":0,"x":2006.505,"y":144.339,"cluster":"commutative-algebra"},{"id":"stacks:00J8","tag":"00J8","title":"Artinian rings · Lemma 00J8","summary":"Let R be Artinian. The Jacobson radical of R is a nilpotent ideal.","statement_latex":"Let $R$ be Artinian. The Jacobson radical of $R$ is a nilpotent ideal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00J8","source_file":"algebra.tex","source_line":12738,"source_end_line":12741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12738-L12741","statement_sha256":"b8a8ccbe79f857abac5a7b88e6b91d1ece126d805d2d9cd6082f74fb5f52d2d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1319,"rank":1319,"depth":3,"x":2108.273,"y":262.516,"cluster":"commutative-algebra"},{"id":"stacks:00JA","tag":"00JA","title":"Artinian rings · Lemma 00JA","summary":"Any ring with finitely many maximal ideals and locally nilpotent Jacobson radical is the product of its localizations at its maximal ideals. Also, all primes are maximal.","statement_latex":"Any ring with finitely many maximal ideals and\nlocally nilpotent Jacobson radical is the product of its localizations\nat its maximal ideals. Also, all primes are maximal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JA","source_file":"algebra.tex","source_line":12757,"source_end_line":12762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12757-L12762","statement_sha256":"b55c6ea7c8b12ffbad0cf80f4e63e9651ed1eff4c2b9cbc0fc4d50330b59eed9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1320,"rank":1320,"depth":5,"x":1937.972,"y":233.081,"cluster":"commutative-algebra"},{"id":"stacks:00JB","tag":"00JB","title":"Artinian rings · Lemma 00JB","summary":"A ring R is Artinian if and only if it has finite length as a module over itself. Any such ring R is both Artinian and Noetherian, any prime ideal of R is a maximal ideal, and R is equal to the (finite) product of its localizations at its maximal ideals.","statement_latex":"A ring $R$ is Artinian if and only if it has finite length\nas a module over itself. Any such ring $R$ is both Artinian and\nNoetherian, any prime ideal of $R$ is a maximal ideal, and $R$ is equal\nto the (finite) product of its localizations at its maximal ideals.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JB","source_file":"algebra.tex","source_line":12786,"source_end_line":12792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12786-L12792","statement_sha256":"79112bf8352d7f4d8bd81e2449d516f754c4a5fb7e7d6109dcf7d5fbefcd5dd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1321,"rank":1321,"depth":6,"x":2087.417,"y":158.054,"cluster":"commutative-algebra"},{"id":"stacks:00QM","tag":"00QM","title":"Homomorphisms essentially of finite type · Definition 00QM","summary":"Let R → S be a ring map. • We say that R → S is essentially of finite type if S is the localization of an R-algebra of finite type. • We say that R → S is essentially of finite presentation if S is the localization of an R-algebra of finite presentation.","statement_latex":"Let $R \\to S$ be a ring map.\n\\begin{enumerate}\n\\item We say that $R \\to S$ is {\\it essentially of finite type} if\n$S$ is the localization of an $R$-algebra of finite type.\n\\item We say that $R \\to S$ is {\\it essentially of finite presentation} if\n$S$ is the localization of an $R$-algebra of finite presentation.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms essentially of finite type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QM","source_file":"algebra.tex","source_line":12835,"source_end_line":12844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12835-L12844","statement_sha256":"0f7f13d38dabbc9216461600cd2d4c950a4bed38df13912dce1f8c555cfcc6f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1322,"rank":1322,"depth":0,"x":2037.487,"y":298.36,"cluster":"commutative-algebra"},{"id":"stacks:07DS","tag":"07DS","title":"Homomorphisms essentially of finite type · Lemma 07DS","summary":"The class of ring maps which are essentially of finite type is preserved under composition. Similarly for essentially of finite presentation.","statement_latex":"The class of ring maps which are essentially of finite type is\npreserved under composition. Similarly for essentially of finite\npresentation.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms essentially of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DS","source_file":"algebra.tex","source_line":12846,"source_end_line":12851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12846-L12851","statement_sha256":"2103b0d7a8c39fe0334cf52a907123895d3a6b8b0d101a63e0004803b98a3430","origin":"The Stacks Project","memory_eligible":false,"source_rank":1323,"rank":1323,"depth":0,"x":1961.375,"y":166.396,"cluster":"commutative-algebra"},{"id":"stacks:0AUF","tag":"0AUF","title":"Homomorphisms essentially of finite type · Lemma 0AUF","summary":"The class of ring maps which are essentially of finite type is preserved by base change. Similarly for essentially of finite presentation.","statement_latex":"The class of ring maps which are essentially of finite type is\npreserved by base change. Similarly for essentially of finite\npresentation.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms essentially of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUF","source_file":"algebra.tex","source_line":12857,"source_end_line":12862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12857-L12862","statement_sha256":"001223868bc7e9ed86db70a5f6e7a6866f5f3cc6653afb34b40f0ce56ba2678e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1324,"rank":1324,"depth":0,"x":2123.832,"y":220.587,"cluster":"commutative-algebra"},{"id":"stacks:07DT","tag":"07DT","title":"Homomorphisms essentially of finite type · Lemma 07DT","summary":"Let R → S be a ring map. Assume S is an Artinian local ring with maximal ideal m. Then • R → S is finite if and only if R → S/ m is finite, • R → S is of finite type if and only if R → S/ m is of finite type. • R → S is essentially of finite type if and only if the composition R → S/ m is essentially of finite type.","statement_latex":"Let $R \\to S$ be a ring map. Assume $S$ is an Artinian local ring with\nmaximal ideal $\\mathfrak m$. Then\n\\begin{enumerate}\n\\item $R \\to S$ is finite if and only if $R \\to S/\\mathfrak m$ is finite,\n\\item $R \\to S$ is of finite type if and only if $R \\to S/\\mathfrak m$\nis of finite type.\n\\item $R \\to S$ is essentially of finite type if and\nonly if the composition $R \\to S/\\mathfrak m$ is essentially\nof finite type.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms essentially of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DT","source_file":"algebra.tex","source_line":12868,"source_end_line":12880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12868-L12880","statement_sha256":"ba98aa9b5e4979fe4b0a1c6af1d244cda159d6cc43241aca217b832b01ad2bc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1325,"rank":1325,"depth":7,"x":1960.246,"y":272.878,"cluster":"commutative-algebra"},{"id":"stacks:0AUG","tag":"0AUG","title":"Homomorphisms essentially of finite type · Lemma 0AUG","summary":"Let φ : R → S be essentially of finite type with R and S local (but not necessarily φ local). Then there exists an n and a maximal ideal m ⊂ R[x_1, …, x_n] lying over m_R such that S is a localization of a quotient of R[x_1, …, x_n]_ m.","statement_latex":"Let $\\varphi : R \\to S$ be essentially of finite type with $R$ and $S$\nlocal (but not necessarily $\\varphi$ local). Then there exists\nan $n$ and a maximal ideal $\\mathfrak m \\subset R[x_1, \\ldots, x_n]$\nlying over $\\mathfrak m_R$ such that $S$ is a localization of a\nquotient of $R[x_1, \\ldots, x_n]_\\mathfrak m$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms essentially of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUG","source_file":"algebra.tex","source_line":12930,"source_end_line":12937,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L12930-L12937","statement_sha256":"f4a1b4cb7e5eccbf72d77e6ed1fb3efbf8be41041011c464751754c43947bdd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1326,"rank":1326,"depth":9,"x":2038.924,"y":141.326,"cluster":"commutative-algebra"},{"id":"stacks:00JD","tag":"00JD","title":"K-groups · Lemma 00JD","summary":"If R is an Artinian local ring then the length function defines a natural abelian group homomorphism length_R : K'_0(R) → Z.","statement_latex":"If $R$ is an Artinian local ring then the length function\ndefines a natural abelian group homomorphism\n$\\text{length}_R : K'_0(R) \\to \\mathbf{Z}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JD","source_file":"algebra.tex","source_line":13024,"source_end_line":13029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13024-L13029","statement_sha256":"914bbdaf13e9f9b9644b986df6ee697b635788f4e77b610dd83a4a15e632b23e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1327,"rank":1327,"depth":7,"x":2086.76,"y":283.158,"cluster":"commutative-algebra"},{"id":"stacks:00JH","tag":"00JH","title":"K-groups · Lemma 00JH","summary":"Let R = R_1 × R_2. Then K_0(R) = K_0(R_1) × K_0(R_2) and K'_0(R) = K'_0(R_1) × K'_0(R_2)","statement_latex":"Let $R = R_1 \\times R_2$. Then $K_0(R) = K_0(R_1) \\times K_0(R_2)$\nand $K'_0(R) = K'_0(R_1) \\times K'_0(R_2)$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JH","source_file":"algebra.tex","source_line":13108,"source_end_line":13112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13108-L13112","statement_sha256":"3202461077a4f6fa491972824787a230f6e1043e97089669a569abeecf340fac","origin":"The Stacks Project","memory_eligible":false,"source_rank":1328,"rank":1328,"depth":0,"x":1937.236,"y":205.617,"cluster":"commutative-algebra"},{"id":"stacks:00JI","tag":"00JI","title":"K-groups · Lemma 00JI","summary":"Let R be an Artinian local ring. The map length_R : K'_0(R) → Z of Lemma [Tag 00JD] is an isomorphism.","statement_latex":"Let $R$ be an Artinian local ring.\nThe map $\\text{length}_R : K'_0(R) \\to \\mathbf{Z}$\nof Lemma \\ref{lemma-length-K0} is an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JI","source_file":"algebra.tex","source_line":13118,"source_end_line":13123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13118-L13123","statement_sha256":"1140fe92903960be37783e3d79a985aabf0cda89526982685309c6bc6bfa39ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":1329,"rank":1329,"depth":8,"x":2110.072,"y":177.915,"cluster":"commutative-algebra"},{"id":"stacks:00JJ","tag":"00JJ","title":"K-groups · Lemma 00JJ","summary":"Let (R, m) be a local ring. Every finite projective R-module is finite free. The map rank_R : K_0(R) → Z defined by [M] → rank_R(M) is well defined and an isomorphism.","statement_latex":"Let $(R, \\mathfrak m)$ be a local ring. Every finite projective $R$-module\nis finite free. The map $\\text{rank}_R : K_0(R) \\to \\mathbf{Z}$\ndefined by $[M] \\to \\text{rank}_R(M)$ is well defined\nand an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JJ","source_file":"algebra.tex","source_line":13129,"source_end_line":13135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13129-L13135","statement_sha256":"ed8e31b7394f8574ee790c95777896329adf7ecec9fecb392dd73d9da81720f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1330,"rank":1330,"depth":3,"x":2004.767,"y":296.566,"cluster":"commutative-algebra"},{"id":"stacks:00JK","tag":"00JK","title":"K-groups · Lemma 00JK","summary":"Let R be a local Artinian ring. There is a commutative diagram xymatrix K_0(R) ar[rr] ar[d]_rank_R & & K'_0(R) ar[d]^length_R Z ar[rr]^length_R(R) & & Z where the vertical maps are isomorphisms by Lemmas [Tag 00JI] and [Tag 00JJ].","statement_latex":"Let $R$ be a local Artinian ring. There is a commutative\ndiagram\n$$\n\\xymatrix{\nK_0(R) \\ar[rr] \\ar[d]_{\\text{rank}_R} & &\nK'_0(R) \\ar[d]^{\\text{length}_R} \\\\\n\\mathbf{Z} \\ar[rr]^{\\text{length}_R(R)} & &\n\\mathbf{Z}\n}\n$$\nwhere the vertical maps are isomorphisms by\nLemmas \\ref{lemma-K0prime-Artinian} and \\ref{lemma-K0-local}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JK","source_file":"algebra.tex","source_line":13157,"source_end_line":13171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13157-L13171","statement_sha256":"e175c0691617da9ceda3d077af513a0ab3daf52322c5d5c04ca63f244e82de07","origin":"The Stacks Project","memory_eligible":false,"source_rank":1331,"rank":1331,"depth":9,"x":1986.979,"y":149.133,"cluster":"commutative-algebra"},{"id":"stacks:0EKB","tag":"0EKB","title":"Graded rings · Lemma 0EKB","summary":"Let S be a graded ring. Let M be a graded S-module. • If S_+M = M and M is finite, then M = 0. • If N, N' ⊂ M are graded submodules, M = N + S_+N', and N' is finite, then M = N. • If N → M is a map of graded modules, N/S_+N → M/S_+M is surjective, and M is finite, then N → M is surjective. • If x_1, …, x_n ∈ M are homogeneous and generate M/S_+M and M is finite, then x_1, …, x_n generate M.","statement_latex":"Let $S$ be a graded ring. Let $M$ be a graded $S$-module.\n\\begin{enumerate}\n\\item If $S_+M = M$ and $M$ is finite, then $M = 0$.\n\\item If $N, N' \\subset M$ are graded submodules,\n$M = N + S_+N'$, and $N'$ is finite, then $M = N$.\n\\item If $N \\to M$ is a map of graded modules, $N/S_+N \\to M/S_+M$\nis surjective, and $M$ is finite, then $N \\to M$ is surjective.\n\\item If $x_1, \\ldots, x_n \\in M$ are homogeneous and generate $M/S_+M$\nand $M$ is finite, then $x_1, \\ldots, x_n$ generate $M$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Graded rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKB","source_file":"algebra.tex","source_line":13249,"source_end_line":13261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13249-L13261","statement_sha256":"9244eb9d34fd3f21d4e4e9ada5a6e59aab874f1e672c734ac89a0a19c37391f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1332,"rank":1332,"depth":0,"x":2118.825,"y":247.881,"cluster":"commutative-algebra"},{"id":"stacks:0EGH","tag":"0EGH","title":"Graded rings · Lemma 0EGH","summary":"Let S be a graded ring, which is finitely generated over S_0. Then for all sufficiently divisible d the algebra S^(d) is generated in degree 1 over S_0.","statement_latex":"Let $S$ be a graded ring, which is finitely generated over $S_0$.\nThen for all sufficiently divisible $d$ the algebra\n$S^{(d)}$ is generated in degree $1$ over $S_0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Graded rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EGH","source_file":"algebra.tex","source_line":13282,"source_end_line":13287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13282-L13287","statement_sha256":"23c980ab1ce2b05d3a3dfb077e340e1c9cb263da310614e2e7cbaa8b582daa30","origin":"The Stacks Project","memory_eligible":false,"source_rank":1333,"rank":1333,"depth":0,"x":1941.969,"y":249.881,"cluster":"commutative-algebra"},{"id":"stacks:077G","tag":"077G","title":"Graded rings · Lemma 077G","summary":"[Huneke-Swanson] Let R → S be a homomorphism of graded rings. Let S' ⊂ S be the integral closure of R in S. Then S' = bigoplus_d ≥ 0 S' ∩ S_d, i.e., S' is a graded R-subalgebra of S.","statement_latex":"\\begin{reference}\n\\cite[Theorem 2.3.2]{Huneke-Swanson}\n\\end{reference}\nLet $R \\to S$ be a homomorphism of graded rings.\nLet $S' \\subset S$ be the integral closure of $R$ in $S$.\nThen\n$$\nS' = \\bigoplus\\nolimits_{d \\geq 0} S' \\cap S_d,\n$$\ni.e., $S'$ is a graded $R$-subalgebra of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Graded rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077G","source_file":"algebra.tex","source_line":13307,"source_end_line":13319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13307-L13319","statement_sha256":"66b52cbd17c2201b17df04a408fb275ebd935e63e1ee9a920e3170bf38eeff5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1334,"rank":1334,"depth":6,"x":2070.936,"y":147.924,"cluster":"commutative-algebra"},{"id":"stacks:00JN","tag":"00JN","title":"Proj of a graded ring · Definition 00JN","summary":"Let S be a graded ring. We define Proj(S) to be the set of homogeneous prime ideals p of S such that S_+ not ⊂ p. The set Proj(S) is a subset of Spec(S) and we endow it with the induced topology. The topological space Proj(S) is called the homogeneous spectrum of the graded ring S.","statement_latex":"Let $S$ be a graded ring.\nWe define $\\text{Proj}(S)$ to be the set of homogeneous\nprime ideals $\\mathfrak p$ of $S$ such that\n$S_{+} \\not \\subset \\mathfrak p$.\nThe set $\\text{Proj}(S)$ is a subset of $\\Spec(S)$\nand we endow it with the induced topology.\nThe topological space $\\text{Proj}(S)$ is called the\n{\\it homogeneous spectrum} of the graded ring $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Proj of a graded ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JN","source_file":"algebra.tex","source_line":13382,"source_end_line":13392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13382-L13392","statement_sha256":"f431427d59f749cbb640a3715a0cbea16b6f8076630a1c351ede2c2ced97d70d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1335,"rank":1335,"depth":0,"x":2057.812,"y":296.473,"cluster":"commutative-algebra"},{"id":"stacks:00JO","tag":"00JO","title":"Proj of a graded ring · Lemma 00JO","summary":"Let S be a Z-graded ring containing a homogeneous invertible element of positive degree. Then the set G ⊂ Spec(S) of Z-graded primes of S (with induced topology) maps homeomorphically to Spec(S_0).","statement_latex":"Let $S$ be a $\\mathbf{Z}$-graded ring containing a homogeneous\ninvertible element of positive degree. Then the set\n$G \\subset \\Spec(S)$ of $\\mathbf{Z}$-graded primes of $S$\n(with induced topology) maps homeomorphically to $\\Spec(S_0)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JO","source_file":"algebra.tex","source_line":13410,"source_end_line":13416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13410-L13416","statement_sha256":"b6f3ae0ae466ebf3cfe7d59b696ff624f760761f752d9848f72dce45498eb9b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1336,"rank":1336,"depth":0,"x":1947.891,"y":179.34,"cluster":"commutative-algebra"},{"id":"stacks:00JP","tag":"00JP","title":"Topology on Proj · Lemma 00JP","summary":"Let S = ⊕_d ≥ 0 S_d be a graded ring. • The sets D_+(f) are open in Proj(S). • We have D_+(ff') = D_+(f) ∩ D_+(f'). • Let g = g_0 + … + g_m be an element of S with g_i ∈ S_i. Then D(g) ∩ Proj(S) = (D(g_0) ∩ Proj(S)) ∪ ⋃_i ≥ 1 D_+(g_i). • Let g_0∈ S_0 be a homogeneous element of degree 0. Then D(g_0) ∩ Proj(S) = ⋃_f ∈ S_d, d≥ 1 D_+(g_0 f). • The open sets D_+(f) form a basis for the topology of Proj(S). • Let f ∈ S be homogeneous of positive degree. The ring S_f has a…","statement_latex":"Let $S = \\oplus_{d \\geq 0} S_d$ be a graded ring.\n\\begin{enumerate}\n\\item The sets $D_{+}(f)$ are open in $\\text{Proj}(S)$.\n\\item We have $D_{+}(ff') = D_{+}(f) \\cap D_{+}(f')$.\n\\item Let $g = g_0 + \\ldots + g_m$ be an element\nof $S$ with $g_i \\in S_i$. Then\n$$\nD(g) \\cap \\text{Proj}(S) =\n(D(g_0) \\cap \\text{Proj}(S))\n\\cup\n\\bigcup\\nolimits_{i \\geq 1} D_{+}(g_i).\n$$\n\\item\nLet $g_0\\in S_0$ be a homogeneous element of degree $0$. Then\n$$\nD(g_0) \\cap \\text{Proj}(S)\n=\n\\bigcup\\nolimits_{f \\in S_d, \\ d\\geq 1} D_{+}(g_0 f).\n$$\n\\item The open sets $D_{+}(f)$ form a\nbasis for the topology of $\\text{Proj}(S)$.\n\\item Let $f \\in S$ be homogeneous of positive degree.\nThe ring $S_f$ has a natural $\\mathbf{Z}$-grading.\nThe ring maps $S \\to S_f \\leftarrow S_{(f)}$ induce\nhomeomorphisms\n$$\nD_{+}(f)\n\\leftarrow\n\\{\\mathbf{Z}\\text{-graded primes of }S_f\\}\n\\to\n\\Spec(S_{(f)}).\n$$\n\\item There exists an $S$ such that $\\text{Proj}(S)$ is not\nquasi-compact.\n\\item The sets $V_{+}(I)$ are closed.\n\\item Any closed subset $T \\subset \\text{Proj}(S)$ is of\nthe form $V_{+}(I)$ for some homogeneous ideal $I \\subset S$.\n\\item For any graded ideal $I \\subset S$ we have\n$V_{+}(I) = \\emptyset$ if and only if $S_{+} \\subset \\sqrt{I}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JP","source_file":"algebra.tex","source_line":13451,"source_end_line":13493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13451-L13493","statement_sha256":"642853358734820aa1724ac2cca67e1376c8481d5aca3224daf6901124808af7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1337,"rank":1337,"depth":3,"x":2123.361,"y":203.371,"cluster":"commutative-algebra"},{"id":"stacks:00JR","tag":"00JR","title":"Proj of a graded ring · Lemma 00JR","summary":"Let S be a graded ring. Let M be a graded S-module. Let p be an element of Proj(S). Let f ∈ S be a homogeneous element of positive degree such that f not ∈ p, i.e., p ∈ D_+(f). Let p' ⊂ S_(f) be the element of Spec(S_(f)) corresponding to p as in Lemma [Tag 00JP]. Then S_( p) = (S_(f))_ p' and compatibly M_( p) = (M_(f))_ p'.","statement_latex":"Let $S$ be a graded ring. Let $M$ be a graded $S$-module.\nLet $\\mathfrak p$ be an element of $\\text{Proj}(S)$.\nLet $f \\in S$ be a homogeneous element of positive degree\nsuch that $f \\not \\in \\mathfrak p$, i.e., $\\mathfrak p \\in D_{+}(f)$.\nLet $\\mathfrak p' \\subset S_{(f)}$ be the element of\n$\\Spec(S_{(f)})$ corresponding to $\\mathfrak p$ as in\nLemma \\ref{lemma-topology-proj}. Then\n$S_{(\\mathfrak p)} = (S_{(f)})_{\\mathfrak p'}$\nand compatibly\n$M_{(\\mathfrak p)} = (M_{(f)})_{\\mathfrak p'}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JR","source_file":"algebra.tex","source_line":13599,"source_end_line":13611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13599-L13611","statement_sha256":"281b4909b0faa0b1e1c224b87e95b6403fbb394670d0e057c4889ccb0455c2fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":1338,"rank":1338,"depth":4,"x":1974.46,"y":285.32,"cluster":"commutative-algebra"},{"id":"stacks:00JS","tag":"00JS","title":"Proj of a graded ring · Lemma 00JS","summary":"Suppose S is a graded ring, p_i, i = 1, …, r homogeneous prime ideals and I ⊂ S_+ a graded ideal. Assume I not⊂ p_i for all i. Then there exists a homogeneous element x∈ I of positive degree such that xnot∈ p_i for all i.","statement_latex":"Suppose $S$ is a graded ring, $\\mathfrak p_i$, $i = 1, \\ldots, r$\nhomogeneous prime ideals and $I \\subset S_{+}$ a graded ideal.\nAssume $I \\not\\subset \\mathfrak p_i$ for all $i$. Then there\nexists a homogeneous element $x\\in I$ of positive degree such\nthat $x\\not\\in \\mathfrak p_i$ for all $i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JS","source_file":"algebra.tex","source_line":13629,"source_end_line":13636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13629-L13636","statement_sha256":"a8f633ff9be06901a9f3fab0c79a366c26975eaf9ab299f94979103e3452a74f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1339,"rank":1339,"depth":0,"x":2018.412,"y":140.22,"cluster":"commutative-algebra"},{"id":"stacks:00JT","tag":"00JT","title":"Proj of a graded ring · Lemma 00JT","summary":"Let S be a graded ring. Let p ⊂ S be a prime. Let q be the homogeneous ideal of S generated by the homogeneous elements of p. Then q is a prime ideal of S.","statement_latex":"Let $S$ be a graded ring.\nLet $\\mathfrak p \\subset S$ be a prime.\nLet $\\mathfrak q$ be the homogeneous ideal of $S$ generated by the\nhomogeneous elements of $\\mathfrak p$. Then $\\mathfrak q$ is a\nprime ideal of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JT","source_file":"algebra.tex","source_line":13650,"source_end_line":13657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13650-L13657","statement_sha256":"cb5e0d411b0f008efe5fb4ca35021cf578ce03c17517270e6713ee369a75e145","origin":"The Stacks Project","memory_eligible":false,"source_rank":1340,"rank":1340,"depth":0,"x":2102.793,"y":272.319,"cluster":"commutative-algebra"},{"id":"stacks:00JU","tag":"00JU","title":"Proj of a graded ring · Lemma 00JU","summary":"Let S be a graded ring. • Any minimal prime of S is a homogeneous ideal of S. • Given a homogeneous ideal I ⊂ S any minimal prime over I is homogeneous.","statement_latex":"Let $S$ be a graded ring.\n\\begin{enumerate}\n\\item Any minimal prime of $S$ is a homogeneous ideal of $S$.\n\\item Given a homogeneous ideal $I \\subset S$ any minimal\nprime over $I$ is homogeneous.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JU","source_file":"algebra.tex","source_line":13669,"source_end_line":13677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13669-L13677","statement_sha256":"03f32cd526d52032cbf72812f41da706e04b2ec81fc25c0afdfa8339bfd837fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1341,"rank":1341,"depth":1,"x":1934.131,"y":222.729,"cluster":"commutative-algebra"},{"id":"stacks:07Z2","tag":"07Z2","title":"Proj of a graded ring · Lemma 07Z2","summary":"Let R be a ring. Let S be a graded R-algebra. Let f ∈ S_+ be homogeneous. Assume that S is of finite type over R. Then • the ring S_(f) is of finite type over R, and • for any finite graded S-module M the module M_(f) is a finite S_(f)-module.","statement_latex":"Let $R$ be a ring. Let $S$ be a graded $R$-algebra. Let $f \\in S_{+}$\nbe homogeneous. Assume that $S$ is of finite type over $R$. Then\n\\begin{enumerate}\n\\item the ring $S_{(f)}$ is of finite type over $R$, and\n\\item for any finite graded $S$-module $M$ the module $M_{(f)}$\nis a finite $S_{(f)}$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Z2","source_file":"algebra.tex","source_line":13685,"source_end_line":13694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13685-L13694","statement_sha256":"cd55b664b5932837acff4bfd9c79abc3d834ad2b490fedb56ab16449665d6a97","origin":"The Stacks Project","memory_eligible":false,"source_rank":1342,"rank":1342,"depth":0,"x":2098.583,"y":163.519,"cluster":"commutative-algebra"},{"id":"stacks:052N","tag":"052N","title":"Proj of a graded ring · Lemma 052N","summary":"Let R be a ring. Let R' be a finite type R-algebra, and let M be a finite R'-module. There exists a graded R-algebra S, a graded S-module N and an element f ∈ S homogeneous of degree 1 such that • R' ≅ S_(f) and M ≅ N_(f) (as modules), • S_0 = R and S is generated by finitely many elements of degree 1 over R, and • N is a finite S-module.","statement_latex":"Let $R$ be a ring.\nLet $R'$ be a finite type $R$-algebra, and let $M$ be a finite $R'$-module.\nThere exists a graded $R$-algebra $S$, a graded $S$-module $N$ and\nan element $f \\in S$ homogeneous of degree $1$ such that\n\\begin{enumerate}\n\\item $R' \\cong S_{(f)}$ and $M \\cong N_{(f)}$ (as modules),\n\\item $S_0 = R$ and $S$ is generated by finitely many elements\nof degree $1$ over $R$, and\n\\item $N$ is a finite $S$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052N","source_file":"algebra.tex","source_line":13729,"source_end_line":13741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13729-L13741","statement_sha256":"3a7d11012991facefce4334b988a8ccf247ec61f454bbd8c59d3f161353ef435","origin":"The Stacks Project","memory_eligible":false,"source_rank":1343,"rank":1343,"depth":0,"x":2024.842,"y":300.664,"cluster":"commutative-algebra"},{"id":"stacks:07Z4","tag":"07Z4","title":"Noetherian graded rings · Lemma 07Z4","summary":"Let S be a graded ring. A set of homogeneous elements f_i ∈ S_+ generates S as an algebra over S_0 if and only if they generate S_+ as an ideal of S.","statement_latex":"Let $S$ be a graded ring. A set of homogeneous elements\n$f_i \\in S_{+}$ generates $S$ as an algebra over $S_0$ if\nand only if they generate $S_{+}$ as an ideal of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian graded rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Z4","source_file":"algebra.tex","source_line":13781,"source_end_line":13786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13781-L13786","statement_sha256":"bda9d0a48fd14578b68afcc88ff5e511508e8ac6fe462a7f4372ae1b3d446e72","origin":"The Stacks Project","memory_eligible":false,"source_rank":1344,"rank":1344,"depth":0,"x":1968.86,"y":157.515,"cluster":"commutative-algebra"},{"id":"stacks:00JW","tag":"00JW","title":"Noetherian graded rings · Lemma 00JW","summary":"A graded ring S is Noetherian if and only if S_0 is Noetherian and S_+ is finitely generated as an ideal of S.","statement_latex":"A graded ring $S$ is Noetherian if and only if $S_0$ is\nNoetherian and $S_{+}$ is finitely generated as an ideal of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian graded rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JW","source_file":"algebra.tex","source_line":13802,"source_end_line":13806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13802-L13806","statement_sha256":"65fee4c2a34cbf3aa6d16ed70452e96c5f5cb55be78d3236d4cb545fd0be817f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1345,"rank":1345,"depth":1,"x":2125.45,"y":231.397,"cluster":"commutative-algebra"},{"id":"stacks:00JX","tag":"00JX","title":"Noetherian graded rings · Definition 00JX","summary":"Let A be an abelian group. We say that a function f : n ↦ f(n) ∈ A defined for all sufficient large integers n is a numerical polynomial if there exists r ≥ 0, elements a_0, …, a_r∈ A such that f(n) = ∑_i = 0^r binomni a_i for all n gg 0.","statement_latex":"Let $A$ be an abelian group.\nWe say that a function $f : n \\mapsto f(n) \\in A$\ndefined for all sufficient large integers $n$ is a\n{\\it numerical polynomial} if there exists $r \\geq 0$,\nelements $a_0, \\ldots, a_r\\in A$ such that\n$$\nf(n) = \\sum\\nolimits_{i = 0}^r \\binom{n}{i} a_i\n$$\nfor all $n \\gg 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian graded rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JX","source_file":"algebra.tex","source_line":13820,"source_end_line":13831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13820-L13831","statement_sha256":"7d27745d4b43d7b98ce08a0853b982b15ec2c23338c6e497a08f679bf4fa67c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1346,"rank":1346,"depth":0,"x":1950.354,"y":265.813,"cluster":"commutative-algebra"},{"id":"stacks:00JY","tag":"00JY","title":"Noetherian graded rings · Lemma 00JY","summary":"If A → A' is a homomorphism of abelian groups and if f : n ↦ f(n) ∈ A is a numerical polynomial, then so is the composition.","statement_latex":"If $A \\to A'$ is a homomorphism of abelian groups and if\n$f : n \\mapsto f(n) \\in A$ is a numerical polynomial,\nthen so is the composition.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian graded rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JY","source_file":"algebra.tex","source_line":13841,"source_end_line":13846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13841-L13846","statement_sha256":"e24228ceb057d28d7f11b0b59b771ae90ae0de284b4524fef568a5c40ba4fc9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1347,"rank":1347,"depth":0,"x":2051.915,"y":140.928,"cluster":"commutative-algebra"},{"id":"stacks:00JZ","tag":"00JZ","title":"Noetherian graded rings · Lemma 00JZ","summary":"Suppose that f: n ↦ f(n) ∈ A is defined for all n sufficiently large and suppose that n ↦ f(n) - f(n-1) is a numerical polynomial. Then f is a numerical polynomial.","statement_latex":"Suppose that $f: n \\mapsto f(n) \\in A$\nis defined for all $n$ sufficiently large\nand suppose that $n \\mapsto f(n) - f(n-1)$\nis a numerical polynomial. Then $f$ is a\nnumerical polynomial.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian graded rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00JZ","source_file":"algebra.tex","source_line":13852,"source_end_line":13859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13852-L13859","statement_sha256":"685d1bc0ef4060cc2505663bb9f704f5c548dadf8b511748765d1b5356745776","origin":"The Stacks Project","memory_eligible":false,"source_rank":1348,"rank":1348,"depth":0,"x":2077.486,"y":290.829,"cluster":"commutative-algebra"},{"id":"stacks:00K0","tag":"00K0","title":"Noetherian graded rings · Lemma 00K0","summary":"If M is a finitely generated graded S-module, and if S is finitely generated over S_0, then each M_n is a finite S_0-module.","statement_latex":"If $M$ is a finitely generated graded $S$-module,\nand if $S$ is finitely generated over $S_0$, then\neach $M_n$ is a finite $S_0$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian graded rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00K0","source_file":"algebra.tex","source_line":13872,"source_end_line":13877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13872-L13877","statement_sha256":"c00e0077c54b98e7efbc441870a90dd85739772ee4affe4c7b99b78d8d1c0992","origin":"The Stacks Project","memory_eligible":false,"source_rank":1349,"rank":1349,"depth":0,"x":1937.914,"y":194.686,"cluster":"commutative-algebra"},{"id":"stacks:00K1","tag":"00K1","title":"Noetherian graded rings · Proposition 00K1","summary":"Suppose that S is a Noetherian graded ring and M a finite graded S-module. Consider the function Z → K'_0(S_0), n ↦ [M_n] see Lemma [Tag 00K0]. If S_+ is generated by elements of degree 1, then this function is a numerical polynomial.","statement_latex":"Suppose that $S$ is a Noetherian graded ring\nand $M$ a finite graded $S$-module. Consider the\nfunction\n$$\n\\mathbf{Z} \\longrightarrow K'_0(S_0), \\quad\nn \\longmapsto [M_n]\n$$\nsee Lemma \\ref{lemma-graded-module-fg}.\nIf $S_{+}$ is generated by elements of degree $1$,\nthen this function is a numerical polynomial.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian graded rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00K1","source_file":"algebra.tex","source_line":13889,"source_end_line":13901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13889-L13901","statement_sha256":"3bf7ee1889462f7f99c8e2d06b87eefdf8970de093b85917a8ad0a89f7327e5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1350,"rank":1350,"depth":1,"x":2118.368,"y":186.373,"cluster":"commutative-algebra"},{"id":"stacks:00K3","tag":"00K3","title":"Noetherian graded rings · Lemma 00K3","summary":"Let k be a field. Suppose that I ⊂ k[X_1, …, X_d] is a nonzero graded ideal. Let M = k[X_1, …, X_d]/I. Then the numerical polynomial n ↦ dim_k(M_n) (see Example [Tag 00K2]) has degree < d - 1 (or is zero if d = 1).","statement_latex":"Let $k$ be a field. Suppose that $I \\subset k[X_1, \\ldots, X_d]$\nis a nonzero graded ideal. Let $M = k[X_1, \\ldots, X_d]/I$.\nThen the numerical polynomial $n \\mapsto \\dim_k(M_n)$ (see\nExample \\ref{example-hilbert-function})\nhas degree $ < d - 1$ (or is zero if $d = 1$).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian graded rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00K3","source_file":"algebra.tex","source_line":13964,"source_end_line":13971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L13964-L13971","statement_sha256":"06d374ddd221057b472f22d546b3b4f4ee53e074e631228020decfc7cf946ceb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1351,"rank":1351,"depth":0,"x":1991.834,"y":295.029,"cluster":"commutative-algebra"},{"id":"stacks:07DU","tag":"07DU","title":"Noetherian local rings · Definition 07DU","summary":"Let (R, m) be a local Noetherian ring. An ideal I ⊂ R such that sqrtI = m is called an ideal of definition of R.","statement_latex":"Let $(R, \\mathfrak m)$ be a local Noetherian ring.\nAn ideal $I \\subset R$ such that $\\sqrt{I} = \\mathfrak m$ is called\n{\\it an ideal of definition of $R$}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DU","source_file":"algebra.tex","source_line":14019,"source_end_line":14024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14019-L14024","statement_sha256":"d99fbf1b8ccb705e12e121f07fd715a04fd3713a629b7515bccfaa3ce3b4556e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1352,"rank":1352,"depth":0,"x":1997.768,"y":142.925,"cluster":"commutative-algebra"},{"id":"stacks:00K5","tag":"00K5","title":"Noetherian local rings · Lemma 00K5","summary":"Suppose that M' ⊂ M are finite R-modules with finite length quotient. Then there exists a constants c_1, c_2 such that for all n ≥ c_2 we have c_1 + chi_I, M'(n - c_2) ≤ chi_I, M(n) ≤ c_1 + chi_I, M'(n)","statement_latex":"Suppose that $M' \\subset M$ are finite $R$-modules\nwith finite length quotient. Then there exists a\nconstants $c_1, c_2$ such that for all $n \\geq c_2$ we have\n$$\nc_1 + \\chi_{I, M'}(n - c_2) \\leq \\chi_{I, M}(n) \\leq\nc_1 + \\chi_{I, M'}(n)\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00K5","source_file":"algebra.tex","source_line":14044,"source_end_line":14053,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14044-L14053","statement_sha256":"6dbf1c8fce5bb1a61a2a654d352624fab77c2ffe6014c513d09cb95867bfe335","origin":"The Stacks Project","memory_eligible":false,"source_rank":1353,"rank":1353,"depth":0,"x":2115.853,"y":258.591,"cluster":"commutative-algebra"},{"id":"stacks:00K6","tag":"00K6","title":"Noetherian local rings · Lemma 00K6","summary":"Suppose that 0 → M' → M → M\" → 0 is a short exact sequence of finite R-modules. Then there exists a submodule N ⊂ M' with finite colength l and c ≥ 0 such that chi_I, M(n) = chi_I, M\"(n) + chi_I, N(n - c) + l and φ_I, M(n) = φ_I, M\"(n) + φ_I, N(n - c) for all n ≥ c.","statement_latex":"Suppose that $0 \\to M' \\to M \\to M'' \\to 0$\nis a short exact sequence of finite $R$-modules.\nThen there exists a submodule $N \\subset M'$ with\nfinite colength $l$ and $c \\geq 0$ such that\n$$\n\\chi_{I, M}(n) = \\chi_{I, M''}(n) + \\chi_{I, N}(n - c) + l\n$$\nand\n$$\n\\varphi_{I, M}(n) = \\varphi_{I, M''}(n) + \\varphi_{I, N}(n - c)\n$$\nfor all $n \\geq c$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00K6","source_file":"algebra.tex","source_line":14087,"source_end_line":14101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14087-L14101","statement_sha256":"58f05b00995f9dfb43ef0ee3335796b742ff6169eca4c038555abcdf981bc932","origin":"The Stacks Project","memory_eligible":false,"source_rank":1354,"rank":1354,"depth":2,"x":1935.545,"y":240.283,"cluster":"commutative-algebra"},{"id":"stacks:00K7","tag":"00K7","title":"Noetherian local rings · Lemma 00K7","summary":"Suppose that I, I' are two ideals of definition for the Noetherian local ring R. Let M be a finite R-module. There exists a constant a such that chi_I, M(n) ≤ chi_I', M(an) for n ≥ 1.","statement_latex":"Suppose that $I$, $I'$ are two ideals of definition\nfor the Noetherian local ring $R$. Let $M$ be a\nfinite $R$-module. There exists a constant $a$ such that\n$\\chi_{I, M}(n) \\leq \\chi_{I', M}(an)$ for $n \\geq 1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00K7","source_file":"algebra.tex","source_line":14134,"source_end_line":14140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14134-L14140","statement_sha256":"6c3e54cc67a0ce0125f3545a3fdc4e793e925929c5866964ac3377900a74b1de","origin":"The Stacks Project","memory_eligible":false,"source_rank":1355,"rank":1355,"depth":0,"x":2083.403,"y":151.365,"cluster":"commutative-algebra"},{"id":"stacks:00K8","tag":"00K8","title":"Noetherian local rings · Proposition 00K8","summary":"Let R be a Noetherian local ring. Let M be a finite R-module. Let I ⊂ R be an ideal of definition. The Hilbert function φ_I, M and the function chi_I, M are numerical polynomials.","statement_latex":"Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module.\nLet $I \\subset R$ be an ideal of definition.\nThe Hilbert function $\\varphi_{I, M}$ and the function\n$\\chi_{I, M}$ are numerical polynomials.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00K8","source_file":"algebra.tex","source_line":14148,"source_end_line":14154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14148-L14154","statement_sha256":"403f749cce1da65ebc6aeb482d72eece2c2f0ad29904fd558315fe6586ebbed5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1356,"rank":1356,"depth":8,"x":2045.835,"y":301.009,"cluster":"commutative-algebra"},{"id":"stacks:09CA","tag":"09CA","title":"Noetherian local rings · Definition 09CA","summary":"Let R be a Noetherian local ring. Let M be a finite R-module. The Hilbert polynomial of M over R is the element P(t) ∈ Q[t] such that P(n) = φ_M(n) for n gg 0.","statement_latex":"Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module.\nThe {\\it Hilbert polynomial} of $M$ over $R$ is the element\n$P(t) \\in \\mathbf{Q}[t]$ such that $P(n) = \\varphi_M(n)$ for $n \\gg 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CA","source_file":"algebra.tex","source_line":14167,"source_end_line":14172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14167-L14172","statement_sha256":"56811f6c5caf59579c05050423b808660b82c840d44fe3663bd50f401cb73c7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1357,"rank":1357,"depth":0,"x":1953.086,"y":169.189,"cluster":"commutative-algebra"},{"id":"stacks:00K9","tag":"00K9","title":"Noetherian local rings · Lemma 00K9","summary":"Let R be a Noetherian local ring. Let M be a finite R-module. • The degree of the numerical polynomial φ_I, M is independent of the ideal of definition I. • The degree of the numerical polynomial chi_I, M is independent of the ideal of definition I.","statement_latex":"Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module.\n\\begin{enumerate}\n\\item The degree of the numerical polynomial $\\varphi_{I, M}$ is independent\nof the ideal of definition $I$.\n\\item The degree of the numerical polynomial $\\chi_{I, M}$ is independent\nof the ideal of definition $I$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00K9","source_file":"algebra.tex","source_line":14178,"source_end_line":14187,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14178-L14187","statement_sha256":"09fbc2fd4ee8e08ac8277581dff9d3396b8fdeaf0391a1de2c800aa50ac36ec2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1358,"rank":1358,"depth":1,"x":2127.693,"y":213.817,"cluster":"commutative-algebra"},{"id":"stacks:00KA","tag":"00KA","title":"Noetherian local rings · Definition 00KA","summary":"Let R be a local Noetherian ring and M a finite R-module. We denote d(M) the element of (-∞, 0, 1, 2, … ) defined as follows: • If M = 0 we set d(M) = -∞, • if M not = 0 then d(M) is the degree of the numerical polynomial chi_M.","statement_latex":"Let $R$ be a local Noetherian ring and $M$ a finite $R$-module.\nWe denote {\\it $d(M)$} the element of $\\{-\\infty, 0, 1, 2, \\ldots \\}$\ndefined as follows:\n\\begin{enumerate}\n\\item If $M = 0$ we set $d(M) = -\\infty$,\n\\item if $M \\not = 0$ then $d(M)$ is the degree of the numerical\npolynomial $\\chi_M$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KA","source_file":"algebra.tex","source_line":14196,"source_end_line":14206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14196-L14206","statement_sha256":"db1777a14b571303fdf14f36d661b13ee905950fb491d5bb97ce7dba5c7a61f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1359,"rank":1359,"depth":0,"x":1962.855,"y":280.064,"cluster":"commutative-algebra"},{"id":"stacks:00KB","tag":"00KB","title":"Noetherian local rings · Lemma 00KB","summary":"Let R be a Noetherian local ring. Let I ⊂ R be an ideal of definition. Let M be a finite R-module which does not have finite length. If M' ⊂ M is a submodule with finite colength, then chi_I, M - chi_I, M' is a polynomial of degree < degree of either polynomial.","statement_latex":"Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal\nof definition. Let $M$ be a finite $R$-module\nwhich does not have finite length. If $M' \\subset M$ is a submodule\nwith finite colength, then $\\chi_{I, M} - \\chi_{I, M'}$\nis a polynomial of degree $<$ degree of either polynomial.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KB","source_file":"algebra.tex","source_line":14212,"source_end_line":14219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14212-L14219","statement_sha256":"102f88d7a27a3c9a6d6addb0c9f9309d67e56933c94ebd5679f851ef7170dbd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1360,"rank":1360,"depth":1,"x":2031.211,"y":137.512,"cluster":"commutative-algebra"},{"id":"stacks:00KC","tag":"00KC","title":"Noetherian local rings · Lemma 00KC","summary":"Let R be a Noetherian local ring. Let I ⊂ R be an ideal of definition. Let 0 → M' → M → M\" → 0 be a short exact sequence of finite R-modules. Then • if M' does not have finite length, then chi_I, M - chi_I, M\" - chi_I, M' is a numerical polynomial of degree < the degree of chi_I, M', • max( deg(chi_I, M'), deg(chi_I, M\") ) = deg(chi_I, M), and • max(d(M'), d(M\")) = d(M),","statement_latex":"Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal of\ndefinition. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence\nof finite $R$-modules. Then\n\\begin{enumerate}\n\\item if $M'$ does not have finite length, then\n$\\chi_{I, M} - \\chi_{I, M''} - \\chi_{I, M'}$\nis a numerical polynomial of degree $<$ the degree of\n$\\chi_{I, M'}$,\n\\item $\\max\\{ \\deg(\\chi_{I, M'}), \\deg(\\chi_{I, M''}) \\} = \\deg(\\chi_{I, M})$,\nand\n\\item $\\max\\{d(M'), d(M'')\\} = d(M)$,\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noetherian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KC","source_file":"algebra.tex","source_line":14225,"source_end_line":14239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14225-L14239","statement_sha256":"ef77473ffbae66142240086d0e33d579164ead09d85aa3ce4c3d59ef40f85beb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1361,"rank":1361,"depth":3,"x":2095.519,"y":281.585,"cluster":"commutative-algebra"},{"id":"stacks:0GIE","tag":"0GIE","title":"Dimension · Definition 0GIE","summary":"Let R be a ring. A chain of prime ideals is a sequence p_0 ⊂ p_1 ⊂ … ⊂ p_n of prime ideals of R such that p_i not = p_i + 1 for i = 0, …, n - 1. The length of this chain of prime ideals is n.","statement_latex":"Let $R$ be a ring. A {\\it chain of prime ideals} is a sequence\n$\\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_n$\nof prime ideals of $R$ such that $\\mathfrak p_i \\not = \\mathfrak p_{i + 1}$\nfor $i = 0, \\ldots, n - 1$. The {\\it length} of this chain of prime\nideals is $n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIE","source_file":"algebra.tex","source_line":14312,"source_end_line":14319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14312-L14319","statement_sha256":"fb230a2917970ce41dc869a793a2bcc768ff00d9249a3ed1386dfa92cee5195a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1362,"rank":1362,"depth":0,"x":1932.046,"y":211.755,"cluster":"commutative-algebra"},{"id":"stacks:00KE","tag":"00KE","title":"Dimension · Definition 00KE","summary":"The Krull dimension of the ring R is the Krull dimension of the topological space Spec(R), see Topology, Definition [Tag 0055]. In other words it is the supremum of the integers n≥ 0 such that R has a chain of prime ideals p_0 ⊂ p_1 ⊂ … ⊂ p_n, p_i not = p_i + 1. of length n.","statement_latex":"The {\\it Krull dimension} of the ring $R$ is the\nKrull dimension of the topological space $\\Spec(R)$, see\nTopology, Definition \\ref{topology-definition-Krull}.\nIn other words it is the supremum of the integers $n\\geq 0$\nsuch that $R$ has a chain of prime ideals\n$$\n\\mathfrak p_0\n\\subset\n\\mathfrak p_1\n\\subset\n\\ldots\n\\subset\n\\mathfrak p_n, \\quad\n\\mathfrak p_i \\not = \\mathfrak p_{i + 1}.\n$$\nof length $n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KE","source_file":"algebra.tex","source_line":14326,"source_end_line":14344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14326-L14344","statement_sha256":"e2e250e003e86aa4565ae98141b585b926a17a372e5d7a8987710d57eba71f19","origin":"The Stacks Project","memory_eligible":false,"source_rank":1363,"rank":1363,"depth":1,"x":2108.953,"y":170.44,"cluster":"commutative-algebra"},{"id":"stacks:00KF","tag":"00KF","title":"Dimension · Definition 00KF","summary":"The height of a prime ideal p of a ring R is the dimension of the local ring R_ p.","statement_latex":"The {\\it height} of a prime ideal $\\mathfrak p$ of\na ring $R$ is the dimension of the local ring $R_{\\mathfrak p}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KF","source_file":"algebra.tex","source_line":14346,"source_end_line":14350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14346-L14350","statement_sha256":"23b508b33aafbf5764b421ce46cd6d74371a1a255aae0adbe558314a53b37598","origin":"The Stacks Project","memory_eligible":false,"source_rank":1364,"rank":1364,"depth":0,"x":2011.615,"y":301.441,"cluster":"commutative-algebra"},{"id":"stacks:00KG","tag":"00KG","title":"Dimension · Lemma 00KG","summary":"The Krull dimension of R is the supremum of the heights of its (maximal) primes.","statement_latex":"The Krull dimension of $R$ is the supremum of the\nheights of its (maximal) primes.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KG","source_file":"algebra.tex","source_line":14352,"source_end_line":14356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14352-L14356","statement_sha256":"9ab56660790b2d8da20db4c0f16eb4b0641e150b53d3e8a9ba63fe9b1ba5fa33","origin":"The Stacks Project","memory_eligible":false,"source_rank":1365,"rank":1365,"depth":0,"x":1978.004,"y":149.432,"cluster":"commutative-algebra"},{"id":"stacks:00KH","tag":"00KH","title":"Dimension · Lemma 00KH","summary":"A Noetherian ring of dimension 0 is Artinian. Conversely, any Artinian ring is Noetherian of dimension zero.","statement_latex":"A Noetherian ring of dimension $0$ is Artinian.\nConversely, any Artinian ring is Noetherian of dimension zero.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KH","source_file":"algebra.tex","source_line":14363,"source_end_line":14367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14363-L14367","statement_sha256":"881eb1278dc738b859b5860987bab0cc52356b011f0174e40248497e5607d59a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1366,"rank":1366,"depth":7,"x":2125.202,"y":242.558,"cluster":"commutative-algebra"},{"id":"stacks:00KI","tag":"00KI","title":"Dimension · Lemma 00KI","summary":"Let R be a Noetherian local ring. Then dim(R) = 0 ⇔ d(R) = 0.","statement_latex":"Let $R$ be a Noetherian local ring.\nThen $\\dim(R) = 0 \\Leftrightarrow d(R) = 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KI","source_file":"algebra.tex","source_line":14403,"source_end_line":14407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14403-L14407","statement_sha256":"5db9a1e19227d07261bf89672b6064a0cc5cdea008a4b7a1da4840e219dfaa87","origin":"The Stacks Project","memory_eligible":false,"source_rank":1367,"rank":1367,"depth":7,"x":1941.555,"y":257.429,"cluster":"commutative-algebra"},{"id":"stacks:00KJ","tag":"00KJ","title":"Dimension · Proposition 00KJ","summary":"Let R be a ring. The following are equivalent: • R is Artinian, • R is Noetherian and dim(R) = 0, • R has finite length as a module over itself, • R is a finite product of Artinian local rings, • R is Noetherian and Spec(R) is a finite discrete topological space, • R is a finite product of Noetherian local rings of dimension 0, • R is a finite product of Noetherian local rings R_i with d(R_i) = 0, • R is a finite product of Noetherian local rings R_i whose maximal ideals…","statement_latex":"Let $R$ be a ring. The following are equivalent:\n\\begin{enumerate}\n\\item $R$ is Artinian,\n\\item $R$ is Noetherian and $\\dim(R) = 0$,\n\\item $R$ has finite length as a module over itself,\n\\item $R$ is a finite product of Artinian local rings,\n\\item $R$ is Noetherian and $\\Spec(R)$ is a\nfinite discrete topological space,\n\\item $R$ is a finite product of Noetherian local rings\nof dimension $0$,\n\\item $R$ is a finite product of Noetherian local rings\n$R_i$ with $d(R_i) = 0$,\n\\item $R$ is a finite product of Noetherian local rings\n$R_i$ whose maximal ideals are nilpotent,\n\\item $R$ is Noetherian, has finitely many maximal\nideals and its Jacobson radical ideal is nilpotent, and\n\\item $R$ is Noetherian and there are no strict inclusions\namong its primes.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KJ","source_file":"algebra.tex","source_line":14414,"source_end_line":14435,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14414-L14435","statement_sha256":"aeda6e9dd463d2137f84205c1afae58883cc2d3a994b1dfb33508209854de669","origin":"The Stacks Project","memory_eligible":false,"source_rank":1368,"rank":1368,"depth":8,"x":2065.159,"y":142.124,"cluster":"commutative-algebra"},{"id":"stacks:00KK","tag":"00KK","title":"Dimension · Lemma 00KK","summary":"Let R be a local Noetherian ring. The following are equivalent: • dim(R) = 1, • d(R) = 1, • there exists an x ∈ m, x not nilpotent such that V(x) = ( m), • there exists an x ∈ m, x not nilpotent such that m = sqrt(x), and • there exists an ideal of definition generated by 1 element, and no ideal of definition is generated by 0 elements.","statement_latex":"Let $R$ be a local Noetherian ring.\nThe following are equivalent:\n\\begin{enumerate}\n\\item\n\n$\\dim(R) = 1$,\n\\item\n\n$d(R) = 1$,\n\\item\n\nthere exists an $x \\in \\mathfrak m$, $x$ not nilpotent\nsuch that $V(x) = \\{\\mathfrak m\\}$,\n\\item\n\nthere exists an $x \\in \\mathfrak m$, $x$ not nilpotent\nsuch that $\\mathfrak m = \\sqrt{(x)}$, and\n\\item\n\nthere exists an ideal of definition generated by $1$ element,\nand no ideal of definition is generated by $0$ elements.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KK","source_file":"algebra.tex","source_line":14445,"source_end_line":14469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14445-L14469","statement_sha256":"d1ada7e86dadc3852b8bc0076e33eb4e0aba19689597a3f967de000d3b21e612","origin":"The Stacks Project","memory_eligible":false,"source_rank":1369,"rank":1369,"depth":8,"x":2066.743,"y":297.465,"cluster":"commutative-algebra"},{"id":"stacks:00KQ","tag":"00KQ","title":"Dimension · Proposition 00KQ","summary":"Let R be a local Noetherian ring. Let d ≥ 0 be an integer. The following are equivalent: • dim(R) = d, • d(R) = d, • there exists an ideal of definition generated by d elements, and no ideal of definition is generated by fewer than d elements.","statement_latex":"Let $R$ be a local Noetherian ring. Let $d \\geq 0$ be an integer.\nThe following are equivalent:\n\\begin{enumerate}\n\\item\n\n$\\dim(R) = d$,\n\\item\n\n$d(R) = d$,\n\\item\n\nthere exists an ideal of definition generated by $d$ elements,\nand no ideal of definition is generated by fewer than $d$ elements.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KQ","source_file":"algebra.tex","source_line":14520,"source_end_line":14536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14520-L14536","statement_sha256":"2852627771d18eb64fa6959ad8c30619ab5c24e353379f273cf6c3e1a5d7b317","origin":"The Stacks Project","memory_eligible":false,"source_rank":1370,"rank":1370,"depth":9,"x":1940.508,"y":183.685,"cluster":"commutative-algebra"},{"id":"stacks:00KU","tag":"00KU","title":"Dimension · Definition 00KU","summary":"Let (R, m) be a Noetherian local ring of dimension d. • A system of parameters of R is a sequence of elements x_1, …, x_d ∈ m which generates an ideal of definition of R, • if there exist x_1, …, x_d ∈ m such that m = (x_1, …, x_d) then we call R a regular local ring and x_1, …, x_d a regular system of parameters.","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$.\n\\begin{enumerate}\n\\item A {\\it system of parameters of $R$} is a sequence of elements\n$x_1, \\ldots, x_d \\in \\mathfrak m$ which generates an ideal of\ndefinition of $R$,\n\\item if there exist $x_1, \\ldots, x_d \\in \\mathfrak m$\nsuch that $\\mathfrak m = (x_1, \\ldots, x_d)$ then we call\n$R$ a {\\it regular local ring} and $x_1, \\ldots, x_d$ a {\\it regular\nsystem of parameters}.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KU","source_file":"algebra.tex","source_line":14614,"source_end_line":14626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14614-L14626","statement_sha256":"11932307e947122816a69914ff1a9be2523470229dcd007085d7e3b7289a1d36","origin":"The Stacks Project","memory_eligible":false,"source_rank":1371,"rank":1371,"depth":0,"x":2125.304,"y":195.97,"cluster":"commutative-algebra"},{"id":"stacks:00KV","tag":"00KV","title":"Dimension · Lemma 00KV","summary":"Let R be a Noetherian ring. Let x ∈ R. • If p is minimal over (x) then the height of p is 0 or 1. • If p, q ∈ Spec(R) and q is minimal over ( p, x), then there is no prime strictly between p and q.","statement_latex":"Let $R$ be a Noetherian ring. Let $x \\in R$.\n\\begin{enumerate}\n\\item If $\\mathfrak p$ is minimal over $(x)$\nthen the height of $\\mathfrak p$ is $0$ or $1$.\n\\item If $\\mathfrak p, \\mathfrak q \\in \\Spec(R)$ and $\\mathfrak q$\nis minimal over $(\\mathfrak p, x)$, then there is no prime strictly\nbetween $\\mathfrak p$ and $\\mathfrak q$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KV","source_file":"algebra.tex","source_line":14633,"source_end_line":14643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14633-L14643","statement_sha256":"723768c237a810dd267049f75d2992766c5731644397fcb413f779a6458fd8ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":1372,"rank":1372,"depth":9,"x":1978.99,"y":291.88,"cluster":"commutative-algebra"},{"id":"stacks:0BBZ","tag":"0BBZ","title":"Dimension · Lemma 0BBZ","summary":"Let R be a Noetherian ring. Let f_1, …, f_r ∈ R. • If p is minimal over (f_1, …, f_r) then the height of p is ≤ r. • If p, q ∈ Spec(R) and q is minimal over ( p, f_1, …, f_r), then every chain of primes between p and q has length at most r.","statement_latex":"Let $R$ be a Noetherian ring. Let $f_1, \\ldots, f_r \\in R$.\n\\begin{enumerate}\n\\item If $\\mathfrak p$ is minimal over $(f_1, \\ldots, f_r)$\nthen the height of $\\mathfrak p$ is $\\leq r$.\n\\item If $\\mathfrak p, \\mathfrak q \\in \\Spec(R)$ and\n$\\mathfrak q$ is minimal over $(\\mathfrak p, f_1, \\ldots, f_r)$,\nthen every chain of primes between $\\mathfrak p$ and $\\mathfrak q$\nhas length at most $r$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBZ","source_file":"algebra.tex","source_line":14664,"source_end_line":14675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14664-L14675","statement_sha256":"ed9c0d0624758ba61fb1b7017ad725de7fa091697dc4f40f94408c17e8b3ebbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1373,"rank":1373,"depth":10,"x":2009.787,"y":137.958,"cluster":"commutative-algebra"},{"id":"stacks:00KW","tag":"00KW","title":"Dimension · Lemma 00KW","summary":"Suppose that R is a Noetherian local ring and x∈ m an element of its maximal ideal. Then dim R ≤ dim R/xR + 1. If x is not contained in any of the minimal primes of R then equality holds. (For example if x is a nonzerodivisor.)","statement_latex":"Suppose that $R$ is a Noetherian local ring and $x\\in \\mathfrak m$ an\nelement of its maximal ideal. Then $\\dim R \\leq \\dim R/xR + 1$.\nIf $x$ is not contained in any of the minimal primes of $R$\nthen equality holds. (For example if $x$ is a nonzerodivisor.)","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KW","source_file":"algebra.tex","source_line":14693,"source_end_line":14699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14693-L14699","statement_sha256":"bf8bb62428c68bc18525adbe4f626d19dd06a00f3f7d5943cf87c014e4f16540","origin":"The Stacks Project","memory_eligible":false,"source_rank":1374,"rank":1374,"depth":10,"x":2110.974,"y":269.084,"cluster":"commutative-algebra"},{"id":"stacks:02IE","tag":"02IE","title":"Dimension · Lemma 02IE","summary":"Let (R, m) be a Noetherian local ring. Suppose x_1, …, x_d ∈ m generate an ideal of definition and d = dim(R). Then dim(R/(x_1, …, x_i)) = d - i for all i = 1, …, d.","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring.\nSuppose $x_1, \\ldots, x_d \\in \\mathfrak m$ generate an\nideal of definition and $d = \\dim(R)$. Then\n$\\dim(R/(x_1, \\ldots, x_i)) = d - i$ for all $i = 1, \\ldots, d$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IE","source_file":"algebra.tex","source_line":14712,"source_end_line":14718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14712-L14718","statement_sha256":"3d76e061e74e35427fb9b553e46325c8adaa7d413f0ecf04bd539345dd0b75a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1375,"rank":1375,"depth":11,"x":1930.705,"y":229.764,"cluster":"commutative-algebra"},{"id":"stacks:02IG","tag":"02IG","title":"Applications of dimension theory · Lemma 02IG","summary":"Let R be a Noetherian local domain of dimension ≥ 2. A nonempty open subset U ⊂ Spec(R) is infinite.","statement_latex":"Let $R$ be a Noetherian local domain of dimension $\\geq 2$.\nA nonempty open subset $U \\subset \\Spec(R)$ is\ninfinite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Applications of dimension theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IG","source_file":"algebra.tex","source_line":14738,"source_end_line":14743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14738-L14743","statement_sha256":"8e568068538687ac99bd6d667a964a3321326ffe1bced1e59725c7d250b3ee69","origin":"The Stacks Project","memory_eligible":false,"source_rank":1376,"rank":1376,"depth":12,"x":2095.441,"y":156.385,"cluster":"commutative-algebra"},{"id":"stacks:0ALV","tag":"0ALV","title":"Applications of dimension theory · Lemma 0ALV","summary":"A Noetherian ring with finitely many primes has dimension ≤ 1.","statement_latex":"A Noetherian ring with finitely many primes has dimension $\\leq 1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Applications of dimension theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALV","source_file":"algebra.tex","source_line":14772,"source_end_line":14775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14772-L14775","statement_sha256":"4766b54fec64d1681b4ceb1185c7bc8d2791497cd68aec293681bafc0f767a69","origin":"The Stacks Project","memory_eligible":false,"source_rank":1377,"rank":1377,"depth":13,"x":2032.905,"y":304.137,"cluster":"commutative-algebra"},{"id":"stacks:0ALW","tag":"0ALW","title":"Applications of dimension theory · Lemma 0ALW","summary":"Let S be a nonzero finite type algebra over a field k. The following are equivalent • dim(S) = 0, • S has finitely many primes, • S has finitely many maximal ideals, • Spec(S) satisfies one of the equivalent conditions of Lemma [Tag 04MG], • dim_k(S) < ∞, • S is Artinian, • Spec(S) is a discrete topological space, • add more here.","statement_latex":"Let $S$ be a nonzero finite type algebra over a field $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\dim(S) = 0$,\n\\item $S$ has finitely many primes,\n\\item $S$ has finitely many maximal ideals,\n\\item $\\Spec(S)$ satisfies one of the equivalent conditions of\nLemma \\ref{lemma-ring-with-only-minimal-primes},\n\\item $\\dim_k(S) < \\infty$,\n\\item $S$ is Artinian,\n\\item $\\Spec(S)$ is a discrete topological space,\n\\item add more here.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Applications of dimension theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALW","source_file":"algebra.tex","source_line":14790,"source_end_line":14805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14790-L14805","statement_sha256":"e9b89315b8c4d5ea2ad7b76be99ef9fbd1456c2f0559ce9b46a3fd3753b87ed0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1378,"rank":1378,"depth":9,"x":1960.117,"y":159.538,"cluster":"commutative-algebra"},{"id":"stacks:00KX","tag":"00KX","title":"Applications of dimension theory · Lemma 00KX","summary":"Noetherian Jacobson rings. • Any Noetherian domain R of dimension 1 with infinitely many primes is Jacobson. • Any Noetherian ring such that every prime p is either maximal or contained in infinitely many prime ideals is Jacobson.","statement_latex":"Noetherian Jacobson rings.\n\\begin{enumerate}\n\\item Any Noetherian domain $R$ of dimension $1$\nwith infinitely many primes is Jacobson.\n\\item Any Noetherian ring such that every prime\n$\\mathfrak p$ is either maximal or contained in\ninfinitely many prime ideals is Jacobson.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Applications of dimension theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00KX","source_file":"algebra.tex","source_line":14841,"source_end_line":14851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14841-L14851","statement_sha256":"04f053a905a50a3ef11800a849d09975832be5ec98dc233548774d68bb959eed","origin":"The Stacks Project","memory_eligible":false,"source_rank":1379,"rank":1379,"depth":13,"x":2130.266,"y":224.936,"cluster":"commutative-algebra"},{"id":"stacks:00L0","tag":"00L0","title":"Support and dimension of modules · Lemma 00L0","summary":"Let R be a Noetherian ring, and let M be a finite R-module. There exists a filtration by R-submodules 0 = M_0 ⊂ M_1 ⊂ … ⊂ M_n = M such that each quotient M_i/M_i-1 is isomorphic to R/ p_i for some prime ideal p_i of R.","statement_latex":"Let $R$ be a Noetherian ring, and let $M$ be a finite $R$-module.\nThere exists a filtration by $R$-submodules\n$$\n0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M\n$$\nsuch that each quotient $M_i/M_{i-1}$ is isomorphic\nto $R/\\mathfrak p_i$ for some prime ideal $\\mathfrak p_i$\nof $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Support and dimension of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00L0","source_file":"algebra.tex","source_line":14915,"source_end_line":14925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14915-L14925","statement_sha256":"6e988b218eef374b513626c472a425a96977b598b4bfbd52e1c650c7ad68ca7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1380,"rank":1380,"depth":1,"x":1952.008,"y":273.314,"cluster":"commutative-algebra"},{"id":"stacks:00L4","tag":"00L4","title":"Support and dimension of modules · Lemma 00L4","summary":"Let R, M, M_i, p_i as in Lemma [Tag 00L0]. Then Supp(M) = ⋃ V( p_i) and in particular p_i ∈ Supp(M).","statement_latex":"Let $R$, $M$, $M_i$, $\\mathfrak p_i$ as in\nLemma \\ref{lemma-filter-Noetherian-module}.\nThen $\\text{Supp}(M) = \\bigcup V(\\mathfrak p_i)$\nand in particular $\\mathfrak p_i \\in \\text{Supp}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Support and dimension of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00L4","source_file":"algebra.tex","source_line":14960,"source_end_line":14966,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14960-L14966","statement_sha256":"99aa1b69e3ff2f95614ae0aa21950c447155790eb1794cdb1401e1c3366c71dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1381,"rank":1381,"depth":4,"x":2044.653,"y":136.338,"cluster":"commutative-algebra"},{"id":"stacks:00L5","tag":"00L5","title":"Support and dimension of modules · Lemma 00L5","summary":"Suppose that R is a Noetherian local ring with maximal ideal m. Let M be a nonzero finite R-module. Then Supp(M) = ( m) if and only if M has finite length over R.","statement_latex":"Suppose that $R$ is a Noetherian local ring with\nmaximal ideal $\\mathfrak m$. Let $M$ be a nonzero finite\n$R$-module. Then $\\text{Supp}(M) = \\{ \\mathfrak m\\}$\nif and only if $M$ has finite length over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Support and dimension of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00L5","source_file":"algebra.tex","source_line":14973,"source_end_line":14979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14973-L14979","statement_sha256":"4893bce1343e5b31ae3c62179755b4338a80109f382f2b196d855bca2ed614e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1382,"rank":1382,"depth":5,"x":2086.537,"y":290.085,"cluster":"commutative-algebra"},{"id":"stacks:00L6","tag":"00L6","title":"Support and dimension of modules · Lemma 00L6","summary":"Let R be a Noetherian ring. Let I ⊂ R be an ideal. Let M be a finite R-module. Then I^nM = 0 for some n ≥ 0 if and only if Supp(M) ⊂ V(I).","statement_latex":"Let $R$ be a Noetherian ring.\nLet $I \\subset R$ be an ideal.\nLet $M$ be a finite $R$-module.\nThen $I^nM = 0$ for some $n \\geq 0$ if and only if\n$\\text{Supp}(M) \\subset V(I)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Support and dimension of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00L6","source_file":"algebra.tex","source_line":14996,"source_end_line":15003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L14996-L15003","statement_sha256":"6c621efdd93469fb9a148459bc742edc0c3929259a907c850c0ceaec69d658bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1383,"rank":1383,"depth":3,"x":1931.839,"y":200.379,"cluster":"commutative-algebra"},{"id":"stacks:00L7","tag":"00L7","title":"Support and dimension of modules · Lemma 00L7","summary":"Let R, M, M_i, p_i as in Lemma [Tag 00L0]. The minimal elements of the set ( p_i) are the minimal elements of Supp(M). The number of times a minimal prime p occurs is \\#(i mid p_i = p) = length_R_ p M_ p.","statement_latex":"Let $R$, $M$, $M_i$, $\\mathfrak p_i$ as in\nLemma \\ref{lemma-filter-Noetherian-module}.\nThe minimal elements of the set $\\{\\mathfrak p_i\\}$\nare the minimal elements of $\\text{Supp}(M)$.\nThe number of times a minimal prime $\\mathfrak p$\noccurs is\n$$\n\\#\\{i \\mid \\mathfrak p_i = \\mathfrak p\\}\n=\n\\text{length}_{R_\\mathfrak p} M_{\\mathfrak p}.\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Support and dimension of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00L7","source_file":"algebra.tex","source_line":15016,"source_end_line":15029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15016-L15029","statement_sha256":"2b32c03bb82135fb5e68aef0324ae462fa421cfe2677a5e9d395e22d87b08624","origin":"The Stacks Project","memory_eligible":false,"source_rank":1384,"rank":1384,"depth":6,"x":2118.26,"y":178.723,"cluster":"commutative-algebra"},{"id":"stacks:00L8","tag":"00L8","title":"Support and dimension of modules · Lemma 00L8","summary":"Let R be a Noetherian local ring. Let M be a finite R-module. Then d(M) = dim(Supp(M)) where d(M) is as in Definition [Tag 00KA].","statement_latex":"Let $R$ be a Noetherian local ring.\nLet $M$ be a finite $R$-module.\nThen $d(M) = \\dim(\\text{Supp}(M))$ where $d(M)$ is as in\nDefinition \\ref{definition-d}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Support and dimension of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00L8","source_file":"algebra.tex","source_line":15053,"source_end_line":15059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15053-L15059","statement_sha256":"6ceabe82d330f324f96f2ec51f75fab45eea41c617d641f67395a6e25d9b4b32","origin":"The Stacks Project","memory_eligible":false,"source_rank":1385,"rank":1385,"depth":10,"x":1998.077,"y":300.608,"cluster":"commutative-algebra"},{"id":"stacks:0B51","tag":"0B51","title":"Support and dimension of modules · Lemma 0B51","summary":"Let R be a Noetherian ring. Let 0 → M' → M → M\" → 0 be a short exact sequence of finite R-modules. Then max(dim(Supp(M')), dim(Supp(M\"))) = dim(Supp(M)).","statement_latex":"Let $R$ be a Noetherian ring. Let $0 \\to M' \\to M \\to M'' \\to 0$\nbe a short exact sequence of finite $R$-modules. Then\n$\\max\\{\\dim(\\text{Supp}(M')), \\dim(\\text{Supp}(M''))\\} =\n\\dim(\\text{Supp}(M))$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Support and dimension of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B51","source_file":"algebra.tex","source_line":15072,"source_end_line":15078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15072-L15078","statement_sha256":"d8f6c58dae7d6ba70e236a716b0e8d5447ace21f4af940f6e388a4282bdec165","origin":"The Stacks Project","memory_eligible":false,"source_rank":1386,"rank":1386,"depth":11,"x":1988.671,"y":142.36,"cluster":"commutative-algebra"},{"id":"stacks:00LA","tag":"00LA","title":"Associated primes · Definition 00LA","summary":"Let R be a ring. Let M be an R-module. A prime p of R is associated to M if there exists an element m ∈ M whose annihilator is p. The set of all such primes is denoted Ass_R(M) or Ass(M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nA prime $\\mathfrak p$ of $R$ is {\\it associated} to $M$\nif there exists an element $m \\in M$ whose annihilator\nis $\\mathfrak p$.\nThe set of all such primes is denoted $\\text{Ass}_R(M)$\nor $\\text{Ass}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LA","source_file":"algebra.tex","source_line":15107,"source_end_line":15115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15107-L15115","statement_sha256":"6613a024e60754773ac5d946398eb7a306614fe1d7181cba5be3e6f694243a0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1387,"rank":1387,"depth":0,"x":2123.015,"y":253.837,"cluster":"commutative-algebra"},{"id":"stacks:0586","tag":"0586","title":"Associated primes · Lemma 0586","summary":"Let R be a ring. Let M be an R-module. Then Ass(M) ⊂ Supp(M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nThen $\\text{Ass}(M) \\subset \\text{Supp}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0586","source_file":"algebra.tex","source_line":15117,"source_end_line":15121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15117-L15121","statement_sha256":"617b5bd5f46f9bb48592875579b1422d21d7afdc044d66d6eb4c5e9c22c78c95","origin":"The Stacks Project","memory_eligible":false,"source_rank":1388,"rank":1388,"depth":0,"x":1934.095,"y":247.859,"cluster":"commutative-algebra"},{"id":"stacks:02M3","tag":"02M3","title":"Associated primes · Lemma 02M3","summary":"Let R be a ring. Let 0 → M' → M → M\" → 0 be a short exact sequence of R-modules. Then Ass(M') ⊂ Ass(M) and Ass(M) ⊂ Ass(M') ∪ Ass(M\"). Also Ass(M' ⊕ M\") = Ass(M') ∪ Ass(M\").","statement_latex":"Let $R$ be a ring. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence\nof $R$-modules. Then $\\text{Ass}(M') \\subset \\text{Ass}(M)$ and\n$\\text{Ass}(M) \\subset \\text{Ass}(M') \\cup \\text{Ass}(M'')$.\nAlso $\\text{Ass}(M' \\oplus M'') = \\text{Ass}(M') \\cup \\text{Ass}(M'')$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02M3","source_file":"algebra.tex","source_line":15130,"source_end_line":15136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15130-L15136","statement_sha256":"29c1f4168f23fd82807211a98261789f6aa47bad8e0751cf6240b99c9a2214e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1389,"rank":1389,"depth":0,"x":2078.369,"y":144.955,"cluster":"commutative-algebra"},{"id":"stacks:00LB","tag":"00LB","title":"Associated primes · Lemma 00LB","summary":"Let R be a ring, and M an R-module. Suppose there exists a filtration by R-submodules 0 = M_0 ⊂ M_1 ⊂ … ⊂ M_n = M such that each quotient M_i/M_i-1 is isomorphic to R/ p_i for some prime ideal p_i of R. Then Ass(M) ⊂ ( p_1, …, p_n).","statement_latex":"Let $R$ be a ring, and $M$ an $R$-module.\nSuppose there exists a filtration by $R$-submodules\n$$\n0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M\n$$\nsuch that each quotient $M_i/M_{i-1}$ is isomorphic to $R/\\mathfrak p_i$\nfor some prime ideal $\\mathfrak p_i$ of $R$.\nThen $\\text{Ass}(M) \\subset \\{\\mathfrak p_1, \\ldots, \\mathfrak p_n\\}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LB","source_file":"algebra.tex","source_line":15152,"source_end_line":15162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15152-L15162","statement_sha256":"f63c18ffdc404dfa9fecb84ab3197c8d94b4db9e06fe45e07501dfb164e8f5f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1390,"rank":1390,"depth":0,"x":2054.709,"y":302.875,"cluster":"commutative-algebra"},{"id":"stacks:00LC","tag":"00LC","title":"Associated primes · Lemma 00LC","summary":"Let R be a Noetherian ring. Let M be a finite R-module. Then Ass(M) is finite.","statement_latex":"Let $R$ be a Noetherian ring.\nLet $M$ be a finite $R$-module.\nThen $\\text{Ass}(M)$ is finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LC","source_file":"algebra.tex","source_line":15175,"source_end_line":15180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15175-L15180","statement_sha256":"768ba5e1190b0431e7852d84875b7897df6a4af8fec1b521386f93a751555138","origin":"The Stacks Project","memory_eligible":false,"source_rank":1391,"rank":1391,"depth":2,"x":1945.04,"y":172.858,"cluster":"commutative-algebra"},{"id":"stacks:02CE","tag":"02CE","title":"Associated primes · Proposition 02CE","summary":"Let R be a Noetherian ring. Let M be a finite R-module. The following sets of primes are the same: • The minimal primes in the support of M. • The minimal primes in Ass(M). • For any filtration 0 = M_0 ⊂ M_1 ⊂ … ⊂ M_n-1 ⊂ M_n = M with M_i/M_i-1 ≅ R/ p_i the minimal primes of the set ( p_i).","statement_latex":"Let $R$ be a Noetherian ring.\nLet $M$ be a finite $R$-module.\nThe following sets of primes are the same:\n\\begin{enumerate}\n\\item The minimal primes in the support of $M$.\n\\item The minimal primes in $\\text{Ass}(M)$.\n\\item For any filtration $0 = M_0 \\subset M_1 \\subset \\ldots\n\\subset M_{n-1} \\subset M_n = M$ with $M_i/M_{i-1} \\cong R/\\mathfrak p_i$\nthe minimal primes of the set $\\{\\mathfrak p_i\\}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02CE","source_file":"algebra.tex","source_line":15187,"source_end_line":15199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15187-L15199","statement_sha256":"ca877bfddd0037ea9bc8f77b806e82d19dca74f416b3e4214fd1f7fca7cd652b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1392,"rank":1392,"depth":7,"x":2130.669,"y":206.539,"cluster":"commutative-algebra"},{"id":"stacks:0587","tag":"0587","title":"Associated primes · Lemma 0587","summary":"Over a Noetherian ring each nonzero module has an associated prime. Let R be a Noetherian ring. Let M be an R-module. Then M = (0) ⇔ Ass(M) = ∅.","statement_latex":"\\begin{slogan}\nOver a Noetherian ring each nonzero module has an associated prime.\n\\end{slogan}\nLet $R$ be a Noetherian ring. Let $M$ be an $R$-module.\nThen\n$$\nM = (0) \\Leftrightarrow \\text{Ass}(M) = \\emptyset.\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0587","source_file":"algebra.tex","source_line":15233,"source_end_line":15243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15233-L15243","statement_sha256":"9dbb4cadd5f59a480b36ccdee356823e6049843bea7591804eb792d253bd53a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1393,"rank":1393,"depth":8,"x":1966.524,"y":287.122,"cluster":"commutative-algebra"},{"id":"stacks:05BV","tag":"05BV","title":"Associated primes · Lemma 05BV","summary":"Let R be a Noetherian ring. Let M be an R-module. Any p ∈ Supp(M) which is minimal among the elements of Supp(M) is an element of Ass(M).","statement_latex":"Let $R$ be a Noetherian ring.\nLet $M$ be an $R$-module.\nAny $\\mathfrak p \\in \\text{Supp}(M)$ which is minimal among the elements\nof $\\text{Supp}(M)$ is an element of $\\text{Ass}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BV","source_file":"algebra.tex","source_line":15259,"source_end_line":15265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15259-L15265","statement_sha256":"c9abc33316b4f09a191c00a086795d57ab1df2f51f6cc156207194f699790583","origin":"The Stacks Project","memory_eligible":false,"source_rank":1394,"rank":1394,"depth":8,"x":2022.821,"y":134.394,"cluster":"commutative-algebra"},{"id":"stacks:00LD","tag":"00LD","title":"Associated primes · Lemma 00LD","summary":"Let R be a Noetherian ring. Let M be an R-module. The union ⋃_ q ∈ Ass(M) q is the set of elements of R which are zerodivisors on M.","statement_latex":"Let $R$ be a Noetherian ring.\nLet $M$ be an $R$-module.\nThe union $\\bigcup_{\\mathfrak q \\in \\text{Ass}(M)} \\mathfrak q$\nis the set of elements of $R$ which are zerodivisors on $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LD","source_file":"algebra.tex","source_line":15277,"source_end_line":15283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15277-L15283","statement_sha256":"b893913692b5d0e9736598e7a6524e2fdc2b248654e866dd690cee6a7760ffc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1395,"rank":1395,"depth":9,"x":2104.217,"y":279.116,"cluster":"commutative-algebra"},{"id":"stacks:0B52","tag":"0B52","title":"Associated primes · Lemma 0B52","summary":"Let R is a Noetherian local ring, M a finite R-module, and f ∈ m an element of the maximal ideal of R. Then dim(Supp(M/fM)) ≤ dim(Supp(M)) ≤ dim(Supp(M/fM)) + 1 If f is not in any of the minimal primes of the support of M (for example if f is a nonzerodivisor on M), then equality holds for the right inequality.","statement_latex":"Let $R$ is a Noetherian local ring, $M$ a finite $R$-module, and\n$f \\in \\mathfrak m$ an element of the maximal ideal of $R$. Then\n$$\n\\dim(\\text{Supp}(M/fM)) \\leq\n\\dim(\\text{Supp}(M)) \\leq\n\\dim(\\text{Supp}(M/fM)) + 1\n$$\nIf $f$ is not in any of the minimal primes of the support of $M$\n(for example if $f$ is a nonzerodivisor on $M$), then equality\nholds for the right inequality.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B52","source_file":"algebra.tex","source_line":15296,"source_end_line":15308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15296-L15308","statement_sha256":"71c4c9095da6b5b97eae1b497c64c4bbe370729db0104c03dd7973f4474cdee2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1396,"rank":1396,"depth":11,"x":1927.622,"y":218.52,"cluster":"commutative-algebra"},{"id":"stacks:05BW","tag":"05BW","title":"Associated primes · Lemma 05BW","summary":"Let φ : R → S be a ring map. Let M be an S-module. Then Spec(φ)(Ass_S(M)) ⊂ Ass_R(M).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nThen $\\Spec(\\varphi)(\\text{Ass}_S(M)) \\subset \\text{Ass}_R(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BW","source_file":"algebra.tex","source_line":15328,"source_end_line":15333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15328-L15333","statement_sha256":"5be9fdc23d4c38e4ddc3e3cb834cfb25967ecc826d4f32c2eef0eab26f77de21","origin":"The Stacks Project","memory_eligible":false,"source_rank":1397,"rank":1397,"depth":0,"x":2106.766,"y":162.938,"cluster":"commutative-algebra"},{"id":"stacks:05DZ","tag":"05DZ","title":"Associated primes · Lemma 05DZ","summary":"Let φ : R → S be a ring map. Let M be an S-module. If S is Noetherian, then Spec(φ)(Ass_S(M)) = Ass_R(M).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nLet $M$ be an $S$-module. If $S$ is Noetherian, then\n$\\Spec(\\varphi)(\\text{Ass}_S(M)) = \\text{Ass}_R(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DZ","source_file":"algebra.tex","source_line":15352,"source_end_line":15357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15352-L15357","statement_sha256":"00a1781b8c9b2b17cda49162955e8e9bf57abd85090e5daf446d15316430c334","origin":"The Stacks Project","memory_eligible":false,"source_rank":1398,"rank":1398,"depth":8,"x":2019.27,"y":305.729,"cluster":"commutative-algebra"},{"id":"stacks:05BY","tag":"05BY","title":"Associated primes · Lemma 05BY","summary":"Let R be a ring. Let I be an ideal. Let M be an R/I-module. Via the canonical injection Spec(R/I) → Spec(R) we have Ass_R/I(M) = Ass_R(M).","statement_latex":"Let $R$ be a ring.\nLet $I$ be an ideal.\nLet $M$ be an $R/I$-module.\nVia the canonical injection\n$\\Spec(R/I) \\to \\Spec(R)$\nwe have $\\text{Ass}_{R/I}(M) = \\text{Ass}_R(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BY","source_file":"algebra.tex","source_line":15379,"source_end_line":15387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15379-L15387","statement_sha256":"0976622cde33fd9eee406414358b02112e6a0a396370a836269b9726dc45b252","origin":"The Stacks Project","memory_eligible":false,"source_rank":1399,"rank":1399,"depth":0,"x":1968.905,"y":150.621,"cluster":"commutative-algebra"},{"id":"stacks:0310","tag":"0310","title":"Associated primes · Lemma 0310","summary":"Let R be a ring. Let M be an R-module. Let p ⊂ R be a prime. • If p ∈ Ass(M) then pR_ p ∈ Ass(M_ p). • If p is finitely generated then the converse holds as well.","statement_latex":"Let $R$ be a ring.\nLet $M$ be an $R$-module.\nLet $\\mathfrak p \\subset R$ be a prime.\n\\begin{enumerate}\n\\item If $\\mathfrak p \\in \\text{Ass}(M)$ then\n$\\mathfrak pR_{\\mathfrak p} \\in \\text{Ass}(M_{\\mathfrak p})$.\n\\item If $\\mathfrak p$ is finitely generated then the converse holds\nas well.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0310","source_file":"algebra.tex","source_line":15393,"source_end_line":15404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15393-L15404","statement_sha256":"c526170c272f810172907798d8fbbdaf1c9ea6a9447d353862561e05080c0b65","origin":"The Stacks Project","memory_eligible":false,"source_rank":1400,"rank":1400,"depth":1,"x":2130.952,"y":236.51,"cluster":"commutative-algebra"},{"id":"stacks:05BZ","tag":"05BZ","title":"Associated primes · Lemma 05BZ","summary":"Let R be a ring. Let M be an R-module. Let S ⊂ R be a multiplicative subset. Via the canonical injection Spec(S^-1R) → Spec(R) we have • Ass_R(S^-1M) = Ass_S^-1R(S^-1M), • Ass_R(M) ∩ Spec(S^-1R) ⊂ Ass_R(S^-1M), and • if R is Noetherian this inclusion is an equality.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $S \\subset R$ be a multiplicative subset.\nVia the canonical injection $\\Spec(S^{-1}R) \\to \\Spec(R)$\nwe have\n\\begin{enumerate}\n\\item $\\text{Ass}_R(S^{-1}M) = \\text{Ass}_{S^{-1}R}(S^{-1}M)$,\n\\item\n$\\text{Ass}_R(M) \\cap \\Spec(S^{-1}R) \\subset \\text{Ass}_R(S^{-1}M)$, and\n\\item if $R$ is Noetherian this inclusion is an equality.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BZ","source_file":"algebra.tex","source_line":15422,"source_end_line":15434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15422-L15434","statement_sha256":"f621ee718058822b0071b14cb2324ea687c4997fe870a0eb9b216d5af07aa1ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":1401,"rank":1401,"depth":2,"x":1942.188,"y":265.157,"cluster":"commutative-algebra"},{"id":"stacks:05C0","tag":"05C0","title":"Associated primes · Lemma 05C0","summary":"Let R be a ring. Let M be an R-module. Let S ⊂ R be a multiplicative subset. Assume that every s ∈ S is a nonzerodivisor on M. Then Ass_R(M) = Ass_R(S^-1M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $S \\subset R$ be a multiplicative subset.\nAssume that every $s \\in S$ is a nonzerodivisor on $M$.\nThen\n$$\n\\text{Ass}_R(M) = \\text{Ass}_R(S^{-1}M).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05C0","source_file":"algebra.tex","source_line":15450,"source_end_line":15459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15450-L15459","statement_sha256":"a78b4c3a30232b1fbad8225907d475a04423b6f402f0f29eb109bdb13a86e8a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1402,"rank":1402,"depth":1,"x":2058.468,"y":136.785,"cluster":"commutative-algebra"},{"id":"stacks:00LL","tag":"00LL","title":"Associated primes · Lemma 00LL","summary":"Let R be a Noetherian local ring with maximal ideal m. Let I ⊂ m be an ideal. Let M be a finite R-module. The following are equivalent: • There exists an x ∈ I which is not a zerodivisor on M. • We have I not ⊂ q for all q ∈ Ass(M).","statement_latex":"Let $R$ be a Noetherian local ring with\nmaximal ideal $\\mathfrak m$. Let $I \\subset \\mathfrak m$\nbe an ideal. Let $M$ be a finite $R$-module.\nThe following are equivalent:\n\\begin{enumerate}\n\\item There exists an $x \\in I$ which is not a zerodivisor on $M$.\n\\item We have $I \\not \\subset \\mathfrak q$ for all\n$\\mathfrak q \\in \\text{Ass}(M)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LL","source_file":"algebra.tex","source_line":15470,"source_end_line":15481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15470-L15481","statement_sha256":"96e63b4449686445b203486b92c51068eb2633e85f181b7a25af82ea070f7b94","origin":"The Stacks Project","memory_eligible":false,"source_rank":1403,"rank":1403,"depth":10,"x":2075.977,"y":297.598,"cluster":"commutative-algebra"},{"id":"stacks:0311","tag":"0311","title":"Associated primes · Lemma 0311","summary":"Let R be a ring. Let M be an R-module. If R is Noetherian the map M → ∏_ p ∈ Ass(M) M_ p is injective.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. If $R$ is Noetherian\nthe map\n$$\nM\n\\longrightarrow\n\\prod\\nolimits_{\\mathfrak p \\in \\text{Ass}(M)} M_{\\mathfrak p}\n$$\nis injective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0311","source_file":"algebra.tex","source_line":15495,"source_end_line":15505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15495-L15505","statement_sha256":"a6a7e9295c5a819e0718e5bf72d23edb49b2a18bf98b419400c2b7e7833d9162","origin":"The Stacks Project","memory_eligible":false,"source_rank":1404,"rank":1404,"depth":9,"x":1933.592,"y":188.836,"cluster":"commutative-algebra"},{"id":"stacks:0GEC","tag":"0GEC","title":"Associated primes · Lemma 0GEC","summary":"Let k be a field. Let S be a finite type k algebra. If dim(S) > 0, then there exists an element f ∈ S which is a nonzerodivisor and a nonunit.","statement_latex":"Let $k$ be a field. Let $S$ be a finite type $k$ algebra.\nIf $\\dim(S) > 0$, then there exists an element $f \\in S$\nwhich is a nonzerodivisor and a nonunit.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEC","source_file":"algebra.tex","source_line":15521,"source_end_line":15526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15521-L15526","statement_sha256":"ceec2adf9c8c2b22aceffc4e8623f069b5aefcb5a4b63b4ad2571a4a35143fec","origin":"The Stacks Project","memory_eligible":false,"source_rank":1405,"rank":1405,"depth":10,"x":2126.254,"y":188.242,"cluster":"commutative-algebra"},{"id":"stacks:0313","tag":"0313","title":"Symbolic powers · Definition 0313","summary":"Let R be a ring. Let p be a prime ideal. For n ≥ 0 the nth symbolic power of p is the ideal p^(n) = Ker(R → R_ p/ p^nR_ p).","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p$ be a prime ideal. For $n \\geq 0$ the\n$n$th {\\it symbolic power} of $\\mathfrak p$ is the ideal\n$\\mathfrak p^{(n)} = \\Ker(R \\to R_\\mathfrak p/\\mathfrak p^nR_\\mathfrak p)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Symbolic powers","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0313","source_file":"algebra.tex","source_line":15548,"source_end_line":15553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15548-L15553","statement_sha256":"7af260039e2f565447235f27cd93925ebd3b3a2f4134c308d8819835f47362e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1406,"rank":1406,"depth":0,"x":1984.514,"y":298.118,"cluster":"commutative-algebra"},{"id":"stacks:0314","tag":"0314","title":"Symbolic powers · Lemma 0314","summary":"Let R be a Noetherian ring. Let p be a prime ideal. Let n > 0. Then Ass(R/ p^(n)) = ( p).","statement_latex":"Let $R$ be a Noetherian ring.\nLet $\\mathfrak p$ be a prime ideal.\nLet $n > 0$. Then $\\text{Ass}(R/\\mathfrak p^{(n)}) = \\{\\mathfrak p\\}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Symbolic powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0314","source_file":"algebra.tex","source_line":15559,"source_end_line":15564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15559-L15564","statement_sha256":"8cc2f9b9c9dd52ae3b86d036390bcdaa28d839820374c7b88a57a7993915d8c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1407,"rank":1407,"depth":0,"x":2000.689,"y":136.498,"cluster":"commutative-algebra"},{"id":"stacks:0BC0","tag":"0BC0","title":"Symbolic powers · Lemma 0BC0","summary":"Let R → S be flat ring map. Let p ⊂ R be a prime such that q = p S is a prime of S. Then p^(n) S = q^(n).","statement_latex":"Let $R \\to S$ be flat ring map. Let $\\mathfrak p \\subset R$ be a prime\nsuch that $\\mathfrak q = \\mathfrak p S$ is a prime of $S$.\nThen $\\mathfrak p^{(n)} S = \\mathfrak q^{(n)}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Symbolic powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BC0","source_file":"algebra.tex","source_line":15577,"source_end_line":15582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15577-L15582","statement_sha256":"692c319c1459263be0057f7d353d009d5eee774e457d094373b61c393dd8dcdb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1408,"rank":1408,"depth":0,"x":2118.858,"y":264.989,"cluster":"commutative-algebra"},{"id":"stacks:05GB","tag":"05GB","title":"Relative assassin · Lemma 05GB","summary":"Let R → S be a ring map. Let N be an S-module. Let A, A', A_fin, B, and B_fin be the subsets of Spec(S) introduced above. • We always have A = A'. • We always have A_fin ⊂ A, B_fin ⊂ B, A_fin ⊂ A'_fin ⊂ B_fin and A ⊂ B. • If S is Noetherian, then A = A_fin and B = B_fin. • If N is flat over R, then A = A_fin = A'_fin and B = B_fin. • If R is Noetherian and N is flat over R, then all of the sets are equal, i.e., A = A' = A_fin = A'_fin = B = B_fin.","statement_latex":"Let $R \\to S$ be a ring map. Let $N$ be an $S$-module.\nLet $A$, $A'$, $A_{fin}$, $B$, and $B_{fin}$ be the subsets of\n$\\Spec(S)$ introduced above.\n\\begin{enumerate}\n\\item We always have $A = A'$.\n\\item We always have $A_{fin} \\subset A$,\n$B_{fin} \\subset B$, $A_{fin} \\subset A'_{fin} \\subset B_{fin}$\nand $A \\subset B$.\n\\item If $S$ is Noetherian, then $A = A_{fin}$ and $B = B_{fin}$.\n\\item If $N$ is flat over $R$, then $A = A_{fin} = A'_{fin}$ and $B = B_{fin}$.\n\\item If $R$ is Noetherian and $N$ is flat over $R$, then all of the sets\nare equal, i.e., $A = A' = A_{fin} = A'_{fin} = B = B_{fin}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Relative assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GB","source_file":"algebra.tex","source_line":15642,"source_end_line":15657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15642-L15657","statement_sha256":"a8f25a74def3477394446f110b697c51d045e64c019ae28ff5d3753d2044bd6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1409,"rank":1409,"depth":3,"x":1928.191,"y":237.264,"cluster":"commutative-algebra"},{"id":"stacks:05GC","tag":"05GC","title":"Relative assassin · Definition 05GC","summary":"Let R → S be a ring map. Let N be an S-module. The relative assassin of N over S/R is the set Ass_S/R(N) = ( q ⊂ S mid q ∈ Ass_S(N ⊗_R kappa( p)) with p = R ∩ q). This is the set named A in Lemma [Tag 05GB].","statement_latex":"Let $R \\to S$ be a ring map. Let $N$ be an $S$-module.\nThe {\\it relative assassin of $N$ over $S/R$} is the set\n$$\n\\text{Ass}_{S/R}(N)\n=\n\\{ \\mathfrak q \\subset S \\mid\n\\mathfrak q \\in \\text{Ass}_S(N \\otimes_R \\kappa(\\mathfrak p))\n\\text{ with }\\mathfrak p = R \\cap \\mathfrak q\\}.\n$$\nThis is the set named $A$ in\nLemma \\ref{lemma-compare-relative-assassins}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Relative assassin","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GC","source_file":"algebra.tex","source_line":15754,"source_end_line":15767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15754-L15767","statement_sha256":"a24ed124e489a0283335a7625305227ee76a691d71afa0735e91d3b758a6f54c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1410,"rank":1410,"depth":4,"x":2091.254,"y":149.426,"cluster":"commutative-algebra"},{"id":"stacks:0312","tag":"0312","title":"Relative assassin · Lemma 0312","summary":"Let R → S be a ring map. Let M be an R-module, and let N be an S-module. If N is flat as R-module, then Ass_S(M ⊗_R N) ⊃ ⋃_ p ∈ Ass_R(M) Ass_S(N/ pN) and if R is Noetherian then we have equality.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $R$-module, and let $N$ be an $S$-module.\nIf $N$ is flat as $R$-module, then\n$$\n\\text{Ass}_S(M \\otimes_R N)\n\\supset\n\\bigcup\\nolimits_{\\mathfrak p \\in \\text{Ass}_R(M)} \\text{Ass}_S(N/\\mathfrak pN)\n$$\nand if $R$ is Noetherian then we have equality.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Relative assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0312","source_file":"algebra.tex","source_line":15773,"source_end_line":15784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15773-L15784","statement_sha256":"8d6022454d125fb8af83f3266278c97f4bd8352aaab2fd0d5b27c11c4f99bd33","origin":"The Stacks Project","memory_eligible":false,"source_rank":1411,"rank":1411,"depth":11,"x":2041.598,"y":306.889,"cluster":"commutative-algebra"},{"id":"stacks:05C1","tag":"05C1","title":"Relative assassin · Lemma 05C1","summary":"Let R → S be a ring map. Let N be an S-module. Assume N is flat as an R-module and R is a domain with fraction field K. Then Ass_S(N) = Ass_S(N ⊗_R K) = Ass_S ⊗_R K(N ⊗_R K) via the canonical inclusion Spec(S ⊗_R K) ⊂ Spec(S).","statement_latex":"Let $R \\to S$ be a ring map.\nLet $N$ be an $S$-module.\nAssume $N$ is flat as an $R$-module and\n$R$ is a domain with fraction field $K$.\nThen\n$$\n\\text{Ass}_S(N) =\n\\text{Ass}_S(N \\otimes_R K) =\n\\text{Ass}_{S \\otimes_R K}(N \\otimes_R K)\n$$\nvia the canonical inclusion\n$\\Spec(S \\otimes_R K) \\subset \\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Relative assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05C1","source_file":"algebra.tex","source_line":15853,"source_end_line":15867,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15853-L15867","statement_sha256":"2f637b117e8c077d3450b106ea4e1599164ea2d0239dbe0b18e2564b9f6b1534","origin":"The Stacks Project","memory_eligible":false,"source_rank":1412,"rank":1412,"depth":3,"x":1951.491,"y":162.45,"cluster":"commutative-algebra"},{"id":"stacks:05C2","tag":"05C2","title":"Relative assassin · Lemma 05C2","summary":"Let R → S be a ring map. Let M be an R-module, and let N be an S-module. Assume N is flat as R-module. Then Ass_S(M ⊗_R N) ⊃ ⋃_ p ∈ Ass_R(M) Ass_S ⊗_R kappa( p)(N ⊗_R kappa( p)) where we use Remark [Tag 00E6] to think of the spectra of fibre rings as subsets of Spec(S). If R is Noetherian then this inclusion is an equality.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $R$-module, and let $N$ be an $S$-module.\nAssume $N$ is flat as $R$-module. Then\n$$\n\\text{Ass}_S(M \\otimes_R N)\n\\supset\n\\bigcup\\nolimits_{\\mathfrak p \\in \\text{Ass}_R(M)}\n\\text{Ass}_{S \\otimes_R \\kappa(\\mathfrak p)}(N \\otimes_R \\kappa(\\mathfrak p))\n$$\nwhere we use\nRemark \\ref{remark-fundamental-diagram}\nto think of the spectra of fibre rings as subsets of $\\Spec(S)$.\nIf $R$ is Noetherian then this inclusion is an equality.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Relative assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05C2","source_file":"algebra.tex","source_line":15879,"source_end_line":15894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15879-L15894","statement_sha256":"c9ef0e5720329c84f07454562cd303290a8748983b683854a374323ee812be19","origin":"The Stacks Project","memory_eligible":false,"source_rank":1413,"rank":1413,"depth":12,"x":2134.281,"y":217.887,"cluster":"commutative-algebra"},{"id":"stacks:0547","tag":"0547","title":"Weakly associated primes · Definition 0547","summary":"Let R be a ring. Let M be an R-module. A prime p of R is weakly associated to M if there exists an element m ∈ M such that p is minimal among the prime ideals containing the annihilator Ann(m) = (f ∈ R mid fm = 0). The set of all such primes is denoted WeakAss_R(M) or WeakAss(M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nA prime $\\mathfrak p$ of $R$ is {\\it weakly associated} to $M$\nif there exists an element $m \\in M$ such that $\\mathfrak p$ is minimal\namong the prime ideals containing the annihilator\n$\\text{Ann}(m) = \\{f \\in R \\mid fm = 0\\}$.\nThe set of all such primes is denoted $\\text{WeakAss}_R(M)$\nor $\\text{WeakAss}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0547","source_file":"algebra.tex","source_line":15932,"source_end_line":15941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15932-L15941","statement_sha256":"726fb31e87ac9b02c87968b1f30175afc8f6095d32c6d52c434edcf63615c26d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1414,"rank":1414,"depth":0,"x":1954.725,"y":280.794,"cluster":"commutative-algebra"},{"id":"stacks:0566","tag":"0566","title":"Weakly associated primes · Lemma 0566","summary":"Let R be a ring. Let M be an R-module. Let p be a prime of R. The following are equivalent: • p is weakly associated to M, • pR_ p is weakly associated to M_ p, and • M_ p contains an element whose annihilator has radical equal to pR_ p.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $\\mathfrak p$ be a prime of $R$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\mathfrak p$ is weakly associated to $M$,\n\\item $\\mathfrak pR_{\\mathfrak p}$ is weakly associated to $M_{\\mathfrak p}$,\nand\n\\item $M_{\\mathfrak p}$ contains an element whose\nannihilator has radical equal to $\\mathfrak pR_{\\mathfrak p}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0566","source_file":"algebra.tex","source_line":15947,"source_end_line":15959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15947-L15959","statement_sha256":"05f0af0df310887c70aaf3228a80269761e857c0419fd46d0e68777639e3ac77","origin":"The Stacks Project","memory_eligible":false,"source_rank":1415,"rank":1415,"depth":0,"x":2036.625,"y":132.367,"cluster":"commutative-algebra"},{"id":"stacks:0EMA","tag":"0EMA","title":"Weakly associated primes · Lemma 0EMA","summary":"For a reduced ring the weakly associated primes of the ring are the minimal primes.","statement_latex":"For a reduced ring the weakly associated primes of the ring are\nthe minimal primes.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMA","source_file":"algebra.tex","source_line":15985,"source_end_line":15989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L15985-L15989","statement_sha256":"139bade7fbff721043cd66fe36590fa8f550570abbf7e86db00edf02c7e8f78f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1416,"rank":1416,"depth":1,"x":2095.655,"y":288.45,"cluster":"commutative-algebra"},{"id":"stacks:0548","tag":"0548","title":"Weakly associated primes · Lemma 0548","summary":"Let R be a ring. Let 0 → M' → M → M\" → 0 be a short exact sequence of R-modules. Then WeakAss(M') ⊂ WeakAss(M) and WeakAss(M) ⊂ WeakAss(M') ∪ WeakAss(M\").","statement_latex":"Let $R$ be a ring.\nLet $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence\nof $R$-modules.\nThen $\\text{WeakAss}(M') \\subset \\text{WeakAss}(M)$ and\n$\\text{WeakAss}(M) \\subset \\text{WeakAss}(M') \\cup \\text{WeakAss}(M'')$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0548","source_file":"algebra.tex","source_line":16003,"source_end_line":16010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16003-L16010","statement_sha256":"c35c48c55bf95ef24c9fa50256133c6e8885cc85ab58d3e08ac22b085000c5a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1417,"rank":1417,"depth":1,"x":1926.434,"y":206.767,"cluster":"commutative-algebra"},{"id":"stacks:0588","tag":"0588","title":"Weakly associated primes · Lemma 0588","summary":"Every nonzero module has a weakly associated prime. Let R be a ring. Let M be an R-module. Then M = (0) ⇔ WeakAss(M) = ∅","statement_latex":"\\begin{slogan}\nEvery nonzero module has a weakly associated prime.\n\\end{slogan}\nLet $R$ be a ring. Let $M$ be an $R$-module. Then\n$$\nM = (0) \\Leftrightarrow \\text{WeakAss}(M) = \\emptyset\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0588","source_file":"algebra.tex","source_line":16028,"source_end_line":16037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16028-L16037","statement_sha256":"9c9a842e624c53762a4729544b82197ea97d470f6a0370d9858f4b4c046d63c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1418,"rank":1418,"depth":2,"x":2117.1,"y":170.941,"cluster":"commutative-algebra"},{"id":"stacks:0589","tag":"0589","title":"Weakly associated primes · Lemma 0589","summary":"Let R be a ring. Let M be an R-module. Then Ass(M) ⊂ WeakAss(M) ⊂ Supp(M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. Then\n$$\n\\text{Ass}(M) \\subset \\text{WeakAss}(M) \\subset \\text{Supp}(M).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0589","source_file":"algebra.tex","source_line":16052,"source_end_line":16058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16052-L16058","statement_sha256":"6c2805d70bd9b9ca4990c0967f4f2b7913f4512e16b4a63e586e6a1dc9c0d8ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":1419,"rank":1419,"depth":1,"x":2005.2,"y":305.687,"cluster":"commutative-algebra"},{"id":"stacks:05C3","tag":"05C3","title":"Weakly associated primes · Lemma 05C3","summary":"Let R be a ring. Let M be an R-module. The union ⋃_ q ∈ WeakAss(M) q is the set of elements of R which are zerodivisors on M.","statement_latex":"Let $R$ be a ring.\nLet $M$ be an $R$-module.\nThe union $\\bigcup_{\\mathfrak q \\in \\text{WeakAss}(M)} \\mathfrak q$\nis the set of elements of $R$ which are zerodivisors on $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05C3","source_file":"algebra.tex","source_line":16067,"source_end_line":16073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16067-L16073","statement_sha256":"818f373d6c4e7bac2f529255176daf969ee26b960d664b39a510a74204f4b380","origin":"The Stacks Project","memory_eligible":false,"source_rank":1420,"rank":1420,"depth":3,"x":1979.328,"y":142.663,"cluster":"commutative-algebra"},{"id":"stacks:05C4","tag":"05C4","title":"Weakly associated primes · Lemma 05C4","summary":"Let R be a ring. Let M be an R-module. Any p ∈ Supp(M) which is minimal among the elements of Supp(M) is an element of WeakAss(M).","statement_latex":"Let $R$ be a ring.\nLet $M$ be an $R$-module.\nAny $\\mathfrak p \\in \\text{Supp}(M)$ which is minimal among the elements\nof $\\text{Supp}(M)$ is an element of $\\text{WeakAss}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05C4","source_file":"algebra.tex","source_line":16092,"source_end_line":16098,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16092-L16098","statement_sha256":"499e4d6ec083fe73958655d40523cb7314097d422c5ee172aa78fdab9c559ac0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1421,"rank":1421,"depth":3,"x":2129.66,"y":248.303,"cluster":"commutative-algebra"},{"id":"stacks:058A","tag":"058A","title":"Weakly associated primes · Lemma 058A","summary":"Let R be a ring. Let M be an R-module. Let p be a prime ideal of R which is finitely generated. Then p ∈ Ass(M) ⇔ p ∈ WeakAss(M). In particular, if R is Noetherian, then Ass(M) = WeakAss(M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $\\mathfrak p$ be a prime ideal of $R$ which is finitely generated.\nThen\n$$\n\\mathfrak p \\in \\text{Ass}(M) \\Leftrightarrow\n\\mathfrak p \\in \\text{WeakAss}(M).\n$$\nIn particular, if $R$ is Noetherian, then $\\text{Ass}(M) = \\text{WeakAss}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058A","source_file":"algebra.tex","source_line":16114,"source_end_line":16124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16114-L16124","statement_sha256":"caccd2b813ecfb924bbff3fce93b3ec83d718762243c770e6100cbea0c82a4e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1422,"rank":1422,"depth":2,"x":1933.652,"y":255.716,"cluster":"commutative-algebra"},{"id":"stacks:05C6","tag":"05C6","title":"Weakly associated primes · Lemma 05C6","summary":"Let φ : R → S be a ring map. Let M be an S-module. Then we have Spec(φ)(WeakAss_S(M)) ⊃ WeakAss_R(M).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module.\nThen we have\n$\\Spec(\\varphi)(\\text{WeakAss}_S(M)) \\supset \\text{WeakAss}_R(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05C6","source_file":"algebra.tex","source_line":16168,"source_end_line":16173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16168-L16173","statement_sha256":"7b9724663ae3cc2ba152ae43cd114c7c1a430671fa651d3bce0565413959ef14","origin":"The Stacks Project","memory_eligible":false,"source_rank":1423,"rank":1423,"depth":1,"x":2072.367,"y":138.91,"cluster":"commutative-algebra"},{"id":"stacks:05E1","tag":"05E1","title":"Weakly associated primes · Lemma 05E1","summary":"Let φ : R → S be a ring map. Let M be an S-module. Denote f : Spec(S) → Spec(R) the associated map on spectra. If φ is a finite ring map, then WeakAss_R(M) = f(WeakAss_S(M)).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Let $M$ be an $S$-module.\nDenote $f : \\Spec(S) \\to \\Spec(R)$ the associated map on spectra.\nIf $\\varphi$ is a finite ring map, then\n$$\n\\text{WeakAss}_R(M) = f(\\text{WeakAss}_S(M)).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05E1","source_file":"algebra.tex","source_line":16211,"source_end_line":16219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16211-L16219","statement_sha256":"862e402ae779bc7fd7f9c1825695c2f6eab263a596c7bebebda9beea9c652121","origin":"The Stacks Project","memory_eligible":false,"source_rank":1424,"rank":1424,"depth":9,"x":2064.004,"y":303.921,"cluster":"commutative-algebra"},{"id":"stacks:05C8","tag":"05C8","title":"Weakly associated primes · Lemma 05C8","summary":"Let R be a ring. Let I be an ideal. Let M be an R/I-module. Via the canonical injection Spec(R/I) → Spec(R) we have WeakAss_R/I(M) = WeakAss_R(M).","statement_latex":"Let $R$ be a ring.\nLet $I$ be an ideal.\nLet $M$ be an $R/I$-module.\nVia the canonical injection\n$\\Spec(R/I) \\to \\Spec(R)$\nwe have $\\text{WeakAss}_{R/I}(M) = \\text{WeakAss}_R(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05C8","source_file":"algebra.tex","source_line":16256,"source_end_line":16264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16256-L16264","statement_sha256":"641c5b34472a01d200c6d1cee90a15745b1219044605c1cfb43c241dfdda4af1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1425,"rank":1425,"depth":10,"x":1937.344,"y":177.369,"cluster":"commutative-algebra"},{"id":"stacks:05C9","tag":"05C9","title":"Weakly associated primes · Lemma 05C9","summary":"Let R be a ring. Let M be an R-module. Let S ⊂ R be a multiplicative subset. Via the canonical injection Spec(S^-1R) → Spec(R) we have WeakAss_R(S^-1M) = WeakAss_S^-1R(S^-1M) and WeakAss(M) ∩ Spec(S^-1R) = WeakAss(S^-1M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $S \\subset R$ be a multiplicative subset.\nVia the canonical injection $\\Spec(S^{-1}R) \\to \\Spec(R)$\nwe have $\\text{WeakAss}_R(S^{-1}M) = \\text{WeakAss}_{S^{-1}R}(S^{-1}M)$\nand\n$$\n\\text{WeakAss}(M) \\cap \\Spec(S^{-1}R) = \\text{WeakAss}(S^{-1}M).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05C9","source_file":"algebra.tex","source_line":16270,"source_end_line":16280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16270-L16280","statement_sha256":"a342d2e68e820396dec499ee9fcf9f5aee4a456a63e0a8cc215d1299f30fa487","origin":"The Stacks Project","memory_eligible":false,"source_rank":1426,"rank":1426,"depth":1,"x":2132.71,"y":198.839,"cluster":"commutative-algebra"},{"id":"stacks:05CA","tag":"05CA","title":"Weakly associated primes · Lemma 05CA","summary":"Let R be a ring. Let M be an R-module. Let S ⊂ R be a multiplicative subset. Assume that every s ∈ S is a nonzerodivisor on M. Then WeakAss(M) = WeakAss(S^-1M).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $S \\subset R$ be a multiplicative subset.\nAssume that every $s \\in S$ is a nonzerodivisor on $M$.\nThen\n$$\n\\text{WeakAss}(M) = \\text{WeakAss}(S^{-1}M).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CA","source_file":"algebra.tex","source_line":16295,"source_end_line":16304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16295-L16304","statement_sha256":"56bb43e452044cf327da0d90ce794408e9b4c1b4f53dd8bff4451a78bcc92712","origin":"The Stacks Project","memory_eligible":false,"source_rank":1427,"rank":1427,"depth":2,"x":1971.22,"y":293.96,"cluster":"commutative-algebra"},{"id":"stacks:05CB","tag":"05CB","title":"Weakly associated primes · Lemma 05CB","summary":"Let R be a ring. Let M be an R-module. The map M → ∏_ p ∈ WeakAss(M) M_ p is injective.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. The map\n$$\nM\n\\longrightarrow\n\\prod\\nolimits_{\\mathfrak p \\in \\text{WeakAss}(M)} M_{\\mathfrak p}\n$$\nis injective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CB","source_file":"algebra.tex","source_line":16316,"source_end_line":16325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16316-L16325","statement_sha256":"09d626d396e56cd7c2623c2ae4e6ccb58d4fdb39097d77dc73f0759aaab25817","origin":"The Stacks Project","memory_eligible":false,"source_rank":1428,"rank":1428,"depth":3,"x":2013.85,"y":132.021,"cluster":"commutative-algebra"},{"id":"stacks:05CC","tag":"05CC","title":"Weakly associated primes · Lemma 05CC","summary":"Let R → S be a ring map. Let N be an S-module. Assume N is flat as an R-module and R is a domain with fraction field K. Then WeakAss_S(N) = WeakAss_S ⊗_R K(N ⊗_R K) via the canonical inclusion Spec(S ⊗_R K) ⊂ Spec(S).","statement_latex":"Let $R \\to S$ be a ring map.\nLet $N$ be an $S$-module.\nAssume $N$ is flat as an $R$-module and\n$R$ is a domain with fraction field $K$.\nThen\n$$\n\\text{WeakAss}_S(N) = \\text{WeakAss}_{S \\otimes_R K}(N \\otimes_R K)\n$$\nvia the canonical inclusion\n$\\Spec(S \\otimes_R K) \\subset \\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CC","source_file":"algebra.tex","source_line":16338,"source_end_line":16350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16338-L16350","statement_sha256":"4214c7ac4db06491a6fde0ed1e62cd56dd14ca8b252761939534aa09a9970b0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1429,"rank":1429,"depth":3,"x":2112.743,"y":275.768,"cluster":"commutative-algebra"},{"id":"stacks:0CUB","tag":"0CUB","title":"Weakly associated primes · Lemma 0CUB","summary":"Let K/k be a field extension. Let R be a k-algebra. Let M be an R-module. Let q ⊂ R ⊗_k K be a prime lying over p ⊂ R. If q is weakly associated to M ⊗_k K, then p is weakly associated to M.","statement_latex":"Let $K/k$ be a field extension. Let $R$ be a $k$-algebra.\nLet $M$ be an $R$-module. Let $\\mathfrak q \\subset R \\otimes_k K$\nbe a prime lying over $\\mathfrak p \\subset R$. If\n$\\mathfrak q$ is weakly associated to $M \\otimes_k K$,\nthen $\\mathfrak p$ is weakly associated to $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Weakly associated primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUB","source_file":"algebra.tex","source_line":16360,"source_end_line":16367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16360-L16367","statement_sha256":"c973c47543d704c6af72d552249596f77001b04ac00f34998343088d72d12ed3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1430,"rank":1430,"depth":10,"x":1924.032,"y":225.834,"cluster":"commutative-algebra"},{"id":"stacks:02M5","tag":"02M5","title":"Embedded primes · Definition 02M5","summary":"Let R be a ring. Let M be an R-module. • The associated primes of M which are not minimal among the associated primes of M are called the embedded associated primes of M. • The embedded primes of R are the embedded associated primes of R as an R-module.","statement_latex":"Let $R$ be a ring.\nLet $M$ be an $R$-module.\n\\begin{enumerate}\n\\item  The associated primes of $M$ which are\nnot minimal among the associated primes of $M$ are called the\n{\\it embedded associated primes} of $M$.\n\\item The {\\it embedded primes of $R$}\nare the embedded associated primes of $R$ as an $R$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Embedded primes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02M5","source_file":"algebra.tex","source_line":16451,"source_end_line":16462,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16451-L16462","statement_sha256":"16c160a0a209203bc97d40ede353fa54edb73881779f48244f6736ffbacf4fc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1431,"rank":1431,"depth":0,"x":2103.522,"y":155.504,"cluster":"commutative-algebra"},{"id":"stacks:02M6","tag":"02M6","title":"Embedded primes · Lemma 02M6","summary":"Let R be a Noetherian ring. Let M be a finite R-module. Consider the set of R-submodules ( K ⊂ M mid Supp(K) nowhere dense in Supp(M) ). This set has a maximal element K and the quotient M' = M/K has the following properties • Supp(M) = Supp(M'), • M' has no embedded associated primes, • for any f ∈ R which is contained in all embedded associated primes of M we have M_f ≅ M'_f.","statement_latex":"Let $R$ be a Noetherian ring.\nLet $M$ be a finite $R$-module.\nConsider the set of $R$-submodules\n$$\n\\{\nK \\subset M\n\\mid\n\\text{Supp}(K)\n\\text{ nowhere dense in }\n\\text{Supp}(M)\n\\}.\n$$\nThis set has a maximal element $K$ and the quotient\n$M' = M/K$ has the following properties\n\\begin{enumerate}\n\\item $\\text{Supp}(M) = \\text{Supp}(M')$,\n\\item $M'$ has no embedded associated primes,\n\\item for any $f \\in R$ which is contained in all\nembedded associated primes of $M$ we have $M_f \\cong M'_f$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Embedded primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02M6","source_file":"algebra.tex","source_line":16467,"source_end_line":16489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16467-L16489","statement_sha256":"7162d9e6b5fb1bc3f599f975d1eefb533961dff8186a2f616f5d358b97625191","origin":"The Stacks Project","memory_eligible":false,"source_rank":1432,"rank":1432,"depth":8,"x":2027.65,"y":309.366,"cluster":"commutative-algebra"},{"id":"stacks:02M7","tag":"02M7","title":"Embedded primes · Lemma 02M7","summary":"Let R be a Noetherian ring. Let M be a finite R-module. For any f ∈ R we have (M')_f = (M_f)' where M → M' and M_f → (M_f)' are the quotients constructed in Lemma [Tag 02M6].","statement_latex":"Let $R$ be a Noetherian ring.\nLet $M$ be a finite $R$-module.\nFor any $f \\in R$ we have $(M')_f = (M_f)'$ where\n$M \\to M'$ and $M_f \\to (M_f)'$ are the quotients\nconstructed in Lemma \\ref{lemma-remove-embedded-primes}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Embedded primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02M7","source_file":"algebra.tex","source_line":16535,"source_end_line":16542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16535-L16542","statement_sha256":"a968500fbc0ea9b9f7583006f953210fd83cbba3eca991d1f890c1d75d0530bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1433,"rank":1433,"depth":9,"x":1959.797,"y":152.701,"cluster":"commutative-algebra"},{"id":"stacks:02M8","tag":"02M8","title":"Embedded primes · Lemma 02M8","summary":"Let R be a Noetherian ring. Let M be a finite R-module without embedded associated primes. Let I = (x ∈ R mid xM = 0). Then the ring R/I has no embedded primes.","statement_latex":"Let $R$ be a Noetherian ring.\nLet $M$ be a finite $R$-module without embedded associated primes.\nLet $I = \\{x \\in R \\mid xM = 0\\}$. Then the ring $R/I$ has no\nembedded primes.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Embedded primes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02M8","source_file":"algebra.tex","source_line":16548,"source_end_line":16554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16548-L16554","statement_sha256":"0133cb560b310cb9597bee3c8aa9a9776de4ec68a6fb6c7b0bae337bde2f586e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1434,"rank":1434,"depth":3,"x":2135.993,"y":229.8,"cluster":"commutative-algebra"},{"id":"stacks:00LF","tag":"00LF","title":"Regular sequences · Definition 00LF","summary":"Let R be a ring. Let M be an R-module. A sequence of elements f_1, …, f_r of R is called an M-regular sequence if the following conditions hold: • f_i is a nonzerodivisor on M/(f_1, …, f_i - 1)M for each i = 1, …, r, and • the module M/(f_1, …, f_r)M is not zero. If I is an ideal of R and f_1, …, f_r ∈ I then we call f_1, …, f_r an M-regular sequence in I. If M = R, we call f_1, …, f_r simply a regular sequence (in I).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. A sequence of elements\n$f_1, \\ldots, f_r$ of $R$ is called an {\\it $M$-regular sequence}\nif the following conditions hold:\n\\begin{enumerate}\n\\item $f_i$ is a nonzerodivisor on\n$M/(f_1, \\ldots, f_{i - 1})M$\nfor each $i = 1, \\ldots, r$, and\n\\item the module $M/(f_1, \\ldots, f_r)M$ is not zero.\n\\end{enumerate}\nIf $I$ is an ideal of $R$ and $f_1, \\ldots, f_r \\in I$\nthen we call $f_1, \\ldots, f_r$ an {\\it $M$-regular sequence\nin $I$}. If $M = R$, we call $f_1, \\ldots, f_r$ simply a\n{\\it regular sequence} (in $I$).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular sequences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LF","source_file":"algebra.tex","source_line":16593,"source_end_line":16608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16593-L16608","statement_sha256":"5b667d64d07555fcb9710cabcf4e9be5b3e7d5d3b3e767f20c1e1a911e94b254","origin":"The Stacks Project","memory_eligible":false,"source_rank":1435,"rank":1435,"depth":0,"x":1943.875,"y":272.97,"cluster":"commutative-algebra"},{"id":"stacks:00LJ","tag":"00LJ","title":"Regular sequences · Lemma 00LJ","summary":"Let R be a local Noetherian ring. Let M be a finite R-module. Let x_1, …, x_c be an M-regular sequence. Then any permutation of the x_i is a regular sequence as well.","statement_latex":"Let $R$ be a local Noetherian ring.\nLet $M$ be a finite $R$-module.\nLet $x_1, \\ldots, x_c$ be an $M$-regular sequence.\nThen any permutation of the $x_i$ is a regular\nsequence as well.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LJ","source_file":"algebra.tex","source_line":16638,"source_end_line":16645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16638-L16645","statement_sha256":"c52a7c21a429b5da1413bf770ed44a496892b56af679a1bdc1aa2632de21dcc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1436,"rank":1436,"depth":3,"x":2050.931,"y":131.983,"cluster":"commutative-algebra"},{"id":"stacks:00LM","tag":"00LM","title":"Regular sequences · Lemma 00LM","summary":"Flat local ring homomorphisms preserve and reflect regular sequences. Let R, S be local rings. Let R → S be a flat local ring homomorphism. Let x_1, …, x_r be a sequence in R. Let M be an R-module. The following are equivalent • x_1, …, x_r is an M-regular sequence in R, and • the images of x_1, …, x_r in S form a M ⊗_R S-regular sequence.","statement_latex":"\\begin{slogan}\nFlat local ring homomorphisms preserve and reflect regular sequences.\n\\end{slogan}\nLet $R, S$ be local rings. Let $R \\to S$ be a flat local ring homomorphism.\nLet $x_1, \\ldots, x_r$ be a sequence in $R$. Let $M$ be an $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $x_1, \\ldots, x_r$ is an $M$-regular sequence in $R$, and\n\\item the images of $x_1, \\ldots, x_r$ in $S$ form a $M \\otimes_R S$-regular\nsequence.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LM","source_file":"algebra.tex","source_line":16675,"source_end_line":16688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16675-L16688","statement_sha256":"fe21ea4a88ef9cea5080b98197fe1a5f0d8a940314da76be7e5f4481566f9e47","origin":"The Stacks Project","memory_eligible":false,"source_rank":1437,"rank":1437,"depth":4,"x":2085.402,"y":296.856,"cluster":"commutative-algebra"},{"id":"stacks:061L","tag":"061L","title":"Regular sequences · Lemma 061L","summary":"Let R be a Noetherian ring. Let M be a finite R-module. Let p be a prime. Let x_1, …, x_r be a sequence in R whose image in R_ p forms an M_ p-regular sequence. Then there exists a g ∈ R, g not ∈ p such that the image of x_1, …, x_r in R_g forms an M_g-regular sequence.","statement_latex":"Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module.\nLet $\\mathfrak p$ be a prime. Let $x_1, \\ldots, x_r$ be a sequence\nin $R$ whose image in $R_{\\mathfrak p}$ forms an $M_{\\mathfrak p}$-regular\nsequence. Then there exists a $g \\in R$, $g \\not \\in \\mathfrak p$\nsuch that the image of $x_1, \\ldots, x_r$ in $R_g$ forms\nan $M_g$-regular sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061L","source_file":"algebra.tex","source_line":16695,"source_end_line":16703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16695-L16703","statement_sha256":"aba2f1893acdd281ec3ceebf52bcbb9d3f1a3e1a718d408c5c32b23b060c00a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1438,"rank":1438,"depth":0,"x":1927.24,"y":194.739,"cluster":"commutative-algebra"},{"id":"stacks:065K","tag":"065K","title":"Regular sequences · Lemma 065K","summary":"Let A be a ring. Let I be an ideal generated by a regular sequence f_1, …, f_n in A. Let g_1, …, g_m ∈ A be elements whose images overlineg_1, …, overlineg_m form a regular sequence in A/I. Then f_1, …, f_n, g_1, …, g_m is a regular sequence in A.","statement_latex":"Let $A$ be a ring. Let $I$ be an ideal generated by a regular\nsequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be\nelements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form a\nregular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$\nis a regular sequence in $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065K","source_file":"algebra.tex","source_line":16716,"source_end_line":16723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16716-L16723","statement_sha256":"71d861e909c263d2c1293c7d4d7a64f53df5cf46736adf871652c60455d078de","origin":"The Stacks Project","memory_eligible":false,"source_rank":1439,"rank":1439,"depth":0,"x":2126.183,"y":180.279,"cluster":"commutative-algebra"},{"id":"stacks:0F1T","tag":"0F1T","title":"Regular sequences · Lemma 0F1T","summary":"Let R be a ring. Let 0 → M_1 → M_2 → M_3 → 0 be a short exact sequence of R-modules. Let f_1, …, f_r ∈ R. If f_1, …, f_r is M_1-regular and M_3-regular, then f_1, …, f_r is M_2-regular.","statement_latex":"Let $R$ be a ring. Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$\nbe a short exact sequence of $R$-modules. Let $f_1, \\ldots, f_r \\in R$.\nIf $f_1, \\ldots, f_r$ is $M_1$-regular and $M_3$-regular, then\n$f_1, \\ldots, f_r$ is $M_2$-regular.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1T","source_file":"algebra.tex","source_line":16729,"source_end_line":16735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16729-L16735","statement_sha256":"95100c218b79b7a268f154ff84c842d98bc7588955d3c442fa43fde9164ca88d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1440,"rank":1440,"depth":1,"x":1990.979,"y":303.95,"cluster":"commutative-algebra"},{"id":"stacks:07DV","tag":"07DV","title":"Regular sequences · Lemma 07DV","summary":"Let R be a ring. Let M be an R-module. Let f_1, …, f_r ∈ R and e_1, …, e_r > 0 integers. Then f_1, …, f_r is an M-regular sequence if and only if f_1^e_1, …, f_r^e_r is an M-regular sequence.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $f_1, \\ldots, f_r \\in R$ and $e_1, \\ldots, e_r > 0$ integers.\nThen $f_1, \\ldots, f_r$ is an $M$-regular sequence\nif and only if $f_1^{e_1}, \\ldots, f_r^{e_r}$\nis an $M$-regular sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DV","source_file":"algebra.tex","source_line":16747,"source_end_line":16754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16747-L16754","statement_sha256":"1a4caa4f1657b26ba7f6fdd65c6893e558b6a75f823c6e8efe53b47b4d0755f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1441,"rank":1441,"depth":2,"x":1991.225,"y":135.872,"cluster":"commutative-algebra"},{"id":"stacks:07DW","tag":"07DW","title":"Regular sequences · Lemma 07DW","summary":"Let R be a ring. Let f_1, …, f_r ∈ R which do not generate the unit ideal. The following are equivalent: • any permutation of f_1, …, f_r is a regular sequence, • any subsequence of f_1, …, f_r (in the given order) is a regular sequence, and • f_1x_1, …, f_rx_r is a regular sequence in the polynomial ring R[x_1, …, x_r].","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ which do not generate\nthe unit ideal. The following are equivalent:\n\\begin{enumerate}\n\\item any permutation of $f_1, \\ldots, f_r$ is a regular sequence,\n\\item any subsequence of $f_1, \\ldots, f_r$ (in the given order) is\na regular sequence, and\n\\item $f_1x_1, \\ldots, f_rx_r$ is a regular sequence in the polynomial\nring $R[x_1, \\ldots, x_r]$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DW","source_file":"algebra.tex","source_line":16796,"source_end_line":16807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16796-L16807","statement_sha256":"38e76c6f53c86b219d65c19f92c89d81c489f086f43ffd8d205858770ac34cfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1442,"rank":1442,"depth":0,"x":2126.341,"y":260.072,"cluster":"commutative-algebra"},{"id":"stacks:061P","tag":"061P","title":"Quasi-regular sequences · Definition 061P","summary":"Let R be a ring. Let M be an R-module. A sequence of elements f_1, …, f_c of R is called M-quasi-regular if ([Tag 061N]) is an isomorphism. If M = R, we call f_1, …, f_c simply a quasi-regular sequence.","statement_latex":"Let $R$ be a ring.\nLet $M$ be an $R$-module.\nA sequence of elements $f_1, \\ldots, f_c$ of $R$ is called\n{\\it $M$-quasi-regular} if (\\ref{equation-quasi-regular})\nis an isomorphism. If $M = R$, we call $f_1, \\ldots, f_c$ simply a\n{\\it quasi-regular sequence}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-regular sequences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061P","source_file":"algebra.tex","source_line":16877,"source_end_line":16885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16877-L16885","statement_sha256":"db3806b439e3ced3289b94a6ba5cacd44afbef89c887e1dbe7701580fdc7797a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1443,"rank":1443,"depth":0,"x":1926.633,"y":245.142,"cluster":"commutative-algebra"},{"id":"stacks:00LN","tag":"00LN","title":"Quasi-regular sequences · Lemma 00LN","summary":"Let R be a ring. • A regular sequence f_1, …, f_c of R is a quasi-regular sequence. • Suppose that M is an R-module and that f_1, …, f_c is an M-regular sequence. Then f_1, …, f_c is an M-quasi-regular sequence.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item A regular sequence $f_1, \\ldots, f_c$ of $R$ is a quasi-regular\nsequence.\n\\item Suppose that $M$ is an $R$-module and that $f_1, \\ldots, f_c$\nis an $M$-regular sequence. Then $f_1, \\ldots, f_c$ is an\n$M$-quasi-regular sequence.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LN","source_file":"algebra.tex","source_line":16895,"source_end_line":16905,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16895-L16905","statement_sha256":"324bcaf7166334d25a957e0224d359a5998a10b63389e1bda84618af53eb5f68","origin":"The Stacks Project","memory_eligible":false,"source_rank":1444,"rank":1444,"depth":0,"x":2086.058,"y":142.732,"cluster":"commutative-algebra"},{"id":"stacks:065L","tag":"065L","title":"Quasi-regular sequences · Lemma 065L","summary":"Let R → R' be a flat ring map. Let M be an R-module. Suppose that f_1, …, f_r ∈ R form an M-quasi-regular sequence. Then the images of f_1, …, f_r in R' form a M ⊗_R R'-quasi-regular sequence.","statement_latex":"Let $R \\to R'$ be a flat ring map. Let $M$ be an $R$-module.\nSuppose that $f_1, \\ldots, f_r \\in R$ form an $M$-quasi-regular sequence.\nThen the images of $f_1, \\ldots, f_r$ in\n$R'$ form a $M \\otimes_R R'$-quasi-regular sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065L","source_file":"algebra.tex","source_line":16980,"source_end_line":16986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L16980-L16986","statement_sha256":"e5fb0fbcc9fae91a6d21457bd4f513e908f39cbe7473064a37136b60797a2172","origin":"The Stacks Project","memory_eligible":false,"source_rank":1445,"rank":1445,"depth":0,"x":2050.821,"y":308.872,"cluster":"commutative-algebra"},{"id":"stacks:061Q","tag":"061Q","title":"Quasi-regular sequences · Lemma 061Q","summary":"Let R be a Noetherian ring. Let M be a finite R-module. Let p be a prime. Let x_1, …, x_c be a sequence in R whose image in R_ p forms an M_ p-quasi-regular sequence. Then there exists a g ∈ R, g not ∈ p such that the image of x_1, …, x_c in R_g forms an M_g-quasi-regular sequence.","statement_latex":"Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module.\nLet $\\mathfrak p$ be a prime. Let $x_1, \\ldots, x_c$ be a sequence\nin $R$ whose image in $R_{\\mathfrak p}$ forms an\n$M_{\\mathfrak p}$-quasi-regular sequence. Then there exists a\n$g \\in R$, $g \\not \\in \\mathfrak p$\nsuch that the image of $x_1, \\ldots, x_c$ in $R_g$ forms\nan $M_g$-quasi-regular sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061Q","source_file":"algebra.tex","source_line":17004,"source_end_line":17013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17004-L17013","statement_sha256":"68f92a3b883b080a717206beca00cc8e196ebc8a42299cceb415474a631c0508","origin":"The Stacks Project","memory_eligible":false,"source_rank":1446,"rank":1446,"depth":0,"x":1943.093,"y":166.229,"cluster":"commutative-algebra"},{"id":"stacks:061R","tag":"061R","title":"Quasi-regular sequences · Lemma 061R","summary":"Let R be a ring. Let M be an R-module. Let f_1, …, f_c ∈ R be an M-quasi-regular sequence. For any i the sequence overlinef_i + 1, …, overlinef_c of overlineR = R/(f_1, …, f_i) is an overlineM = M/(f_1, …, f_i)M-quasi-regular sequence.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $f_1, \\ldots, f_c \\in R$ be an $M$-quasi-regular sequence.\nFor any $i$ the sequence\n$\\overline{f}_{i + 1}, \\ldots, \\overline{f}_c$\nof $\\overline{R} = R/(f_1, \\ldots, f_i)$ is an\n$\\overline{M} = M/(f_1, \\ldots, f_i)M$-quasi-regular sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061R","source_file":"algebra.tex","source_line":17026,"source_end_line":17034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17026-L17034","statement_sha256":"80688b0867c4f8dd977dfeb2ed0823fdac90b22656551bc0939a068d22764e38","origin":"The Stacks Project","memory_eligible":false,"source_rank":1447,"rank":1447,"depth":0,"x":2137.43,"y":210.327,"cluster":"commutative-algebra"},{"id":"stacks:061S","tag":"061S","title":"Quasi-regular sequences · Lemma 061S","summary":"Let (R, m) be a local Noetherian ring. Let M be a nonzero finite R-module. Let f_1, …, f_c ∈ m be an M-quasi-regular sequence. Then f_1, …, f_c is an M-regular sequence.","statement_latex":"Let $(R, \\mathfrak m)$ be a local Noetherian ring.\nLet $M$ be a nonzero finite $R$-module.\nLet $f_1, \\ldots, f_c \\in \\mathfrak m$ be an $M$-quasi-regular sequence.\nThen $f_1, \\ldots, f_c$ is an $M$-regular sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061S","source_file":"algebra.tex","source_line":17059,"source_end_line":17065,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17059-L17065","statement_sha256":"7b5539f09db8bd0f2bc277228b289318b0df596d50f9473569807d37701a447f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1448,"rank":1448,"depth":4,"x":1958.491,"y":288.158,"cluster":"commutative-algebra"},{"id":"stacks:065N","tag":"065N","title":"Quasi-regular sequences · Lemma 065N","summary":"Let R be a ring. Let J = (f_1, …, f_r) be an ideal of R. Let M be an R-module. Set overlineR = R/⋂_n ≥ 0 J^n, overlineM = M/⋂_n ≥ 0 J^nM, and denote overlinef_i the image of f_i in overlineR. Then f_1, …, f_r is M-quasi-regular if and only if overlinef_1, …, overlinef_r is overlineM-quasi-regular.","statement_latex":"Let $R$ be a ring. Let $J = (f_1, \\ldots, f_r)$ be an ideal of $R$.\nLet $M$ be an $R$-module. Set $\\overline{R} = R/\\bigcap_{n \\geq 0} J^n$,\n$\\overline{M} = M/\\bigcap_{n \\geq 0} J^nM$, and denote\n$\\overline{f}_i$ the image of $f_i$ in $\\overline{R}$.\nThen $f_1, \\ldots, f_r$ is $M$-quasi-regular if and only if\n$\\overline{f}_1, \\ldots, \\overline{f}_r$ is $\\overline{M}$-quasi-regular.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065N","source_file":"algebra.tex","source_line":17116,"source_end_line":17124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17116-L17124","statement_sha256":"0a116ea4af742020a923edbd4d09e130a211630d0a53b410439236595846814e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1449,"rank":1449,"depth":0,"x":2027.917,"y":129.077,"cluster":"commutative-algebra"},{"id":"stacks:052Q","tag":"052Q","title":"Blow up algebras · Definition 052Q","summary":"Let R be a ring. Let I ⊂ R be an ideal. • The blowup algebra, or the Rees algebra, associated to the pair (R, I) is the graded R-algebra Bl_I(R) = bigoplus_n ≥ 0 I^n = R ⊕ I ⊕ I^2 ⊕ … where the summand I^n is placed in degree n. • Let a ∈ I be an element. Denote a^(1) the element a seen as an element of degree 1 in the Rees algebra. Then the affine blowup algebra R[fracIa] is the algebra (Bl_I(R))_(a^(1)) constructed in Section [Tag 00JM].","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\n\\begin{enumerate}\n\\item The {\\it blowup algebra}, or the {\\it Rees algebra}, associated to\nthe pair $(R, I)$ is the graded $R$-algebra\n$$\n\\text{Bl}_I(R) =\n\\bigoplus\\nolimits_{n \\geq 0} I^n =\nR \\oplus I \\oplus I^2 \\oplus \\ldots\n$$\nwhere the summand $I^n$ is placed in degree $n$.\n\\item Let $a \\in I$ be an element. Denote $a^{(1)}$ the element $a$\nseen as an element of degree $1$ in the Rees algebra. Then the\n{\\it affine blowup algebra} $R[\\frac{I}{a}]$ is the algebra\n$(\\text{Bl}_I(R))_{(a^{(1)})}$ constructed in Section \\ref{section-proj}.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052Q","source_file":"algebra.tex","source_line":17142,"source_end_line":17160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17142-L17160","statement_sha256":"797e5ecdbb78f42f7713bb50604f5a1ce9188c7dc797f03c2f6c983bcfc23c75","origin":"The Stacks Project","memory_eligible":false,"source_rank":1450,"rank":1450,"depth":0,"x":2104.726,"y":285.927,"cluster":"commutative-algebra"},{"id":"stacks:07Z3","tag":"07Z3","title":"Blow up algebras · Lemma 07Z3","summary":"Let R be a ring, I ⊂ R an ideal, and a ∈ I. Let R' = R[fracIa] be the affine blowup algebra. Then • the image of a in R' is a nonzerodivisor, • IR' = aR', and • (R')_a = R_a.","statement_latex":"Let $R$ be a ring, $I \\subset R$ an ideal, and $a \\in I$.\nLet $R' = R[\\frac{I}{a}]$ be the affine blowup algebra. Then\n\\begin{enumerate}\n\\item the image of $a$ in $R'$ is a nonzerodivisor,\n\\item $IR' = aR'$, and\n\\item $(R')_a = R_a$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Z3","source_file":"algebra.tex","source_line":17168,"source_end_line":17177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17168-L17177","statement_sha256":"50f2ba56b963a43305ce27d9a8b24648bc436a8048ea27f02603c416676caa15","origin":"The Stacks Project","memory_eligible":false,"source_rank":1451,"rank":1451,"depth":0,"x":1921.776,"y":213.782,"cluster":"commutative-algebra"},{"id":"stacks:0BIP","tag":"0BIP","title":"Blow up algebras · Lemma 0BIP","summary":"Let R → S be a ring map. Let I ⊂ R be an ideal and a ∈ I. Set J = IS and let b ∈ J be the image of a. Then S[fracJb] is the quotient of S ⊗_R R[fracIa] by the ideal of elements annihilated by some power of b.","statement_latex":"Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal\nand $a \\in I$. Set $J = IS$ and let $b \\in J$ be the image of $a$.\nThen $S[\\frac{J}{b}]$ is the quotient of $S \\otimes_R R[\\frac{I}{a}]$\nby the ideal of elements annihilated by some power of $b$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIP","source_file":"algebra.tex","source_line":17183,"source_end_line":17189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17183-L17189","statement_sha256":"accd3761e917cefc5f9e72f16f5875728c70169837f843466ec0aa26af672c45","origin":"The Stacks Project","memory_eligible":false,"source_rank":1452,"rank":1452,"depth":0,"x":2114.885,"y":163.121,"cluster":"commutative-algebra"},{"id":"stacks:0G8S","tag":"0G8S","title":"Blow up algebras · Lemma 0G8S","summary":"Let R be a ring. Let I = (a_1, …, a_n) be an ideal of R. Let a = a_1. Then there is a surjection R[x_2, …, x_n]/(a x_2 - a_2, …, a x_n - a_n) → textstyleR[fracIa] whose kernel is the a-power torsion in the source.","statement_latex":"Let $R$ be a ring. Let $I = (a_1, \\ldots, a_n)$ be an ideal of $R$.\nLet $a = a_1$. Then there is a surjection\n$$\nR[x_2, \\ldots, x_n]/(a x_2 - a_2, \\ldots, a x_n - a_n)\n\\longrightarrow\n\\textstyle{R[\\frac{I}{a}]}\n$$\nwhose kernel is the $a$-power torsion in the source.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8S","source_file":"algebra.tex","source_line":17249,"source_end_line":17259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17249-L17259","statement_sha256":"ca84887e770dbaf51367621ef67220c3023b7059d2bee1d0368830aa8ba3907e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1453,"rank":1453,"depth":1,"x":2013.131,"y":310.194,"cluster":"commutative-algebra"},{"id":"stacks:080U","tag":"080U","title":"Blow up algebras · Lemma 080U","summary":"Let R be a ring, I ⊂ R an ideal, and a ∈ I. Set R' = R[fracIa]. If f ∈ R is such that V(f) = V(I), then f maps to a nonzerodivisor in R' and R'_f = R'_a = R_a.","statement_latex":"Let $R$ be a ring, $I \\subset R$ an ideal, and $a \\in I$.\nSet $R' = R[\\frac{I}{a}]$. If $f \\in R$ is such that $V(f) = V(I)$,\nthen $f$ maps to a nonzerodivisor in $R'$ and $R'_f = R'_a = R_a$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080U","source_file":"algebra.tex","source_line":17270,"source_end_line":17275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17270-L17275","statement_sha256":"f6f7b31e093a6882ebc4e06f5853104d3e75254773c5a93c42aba0ae800fc012","origin":"The Stacks Project","memory_eligible":false,"source_rank":1454,"rank":1454,"depth":1,"x":1969.849,"y":143.846,"cluster":"commutative-algebra"},{"id":"stacks:0BBI","tag":"0BBI","title":"Blow up algebras · Lemma 0BBI","summary":"Let R be a ring, I ⊂ R an ideal, a ∈ I, and f ∈ R. Set R' = R[fracIa] and R\" = R[fracfIfa]. Then there is a surjective R-algebra map R' → R\" whose kernel is the set of f-power torsion elements of R'.","statement_latex":"Let $R$ be a ring, $I \\subset R$ an ideal, $a \\in I$, and $f \\in R$.\nSet $R' = R[\\frac{I}{a}]$ and $R'' = R[\\frac{fI}{fa}]$. Then\nthere is a surjective $R$-algebra map $R' \\to R''$ whose kernel\nis the set of $f$-power torsion elements of $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBI","source_file":"algebra.tex","source_line":17285,"source_end_line":17291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17285-L17291","statement_sha256":"878051fb38ffcf14813ffdf26e24bb16268d0e5fc7a977899255bb77fc07cf6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1455,"rank":1455,"depth":1,"x":2135.697,"y":242.045,"cluster":"commutative-algebra"},{"id":"stacks:052S","tag":"052S","title":"Blow up algebras · Lemma 052S","summary":"Being reduced is invariant under blowup If R is reduced then every (affine) blowup algebra of R is reduced.","statement_latex":"\\begin{slogan}\nBeing reduced is invariant under blowup\n\\end{slogan}\nIf $R$ is reduced then every (affine) blowup algebra of $R$ is reduced.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052S","source_file":"algebra.tex","source_line":17305,"source_end_line":17311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17305-L17311","statement_sha256":"093ca3fc3a725a7874961917691f6bf21c0543abd7d8e52c9f012d0a2f3b5cd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1456,"rank":1456,"depth":0,"x":1934.241,"y":263.761,"cluster":"commutative-algebra"},{"id":"stacks:052R","tag":"052R","title":"Blow up algebras · Lemma 052R","summary":"Let R be a domain, I ⊂ R an ideal, and a ∈ I a nonzero element. Then the affine blowup algebra R[fracIa] is a domain.","statement_latex":"Let $R$ be a domain, $I \\subset R$ an ideal, and $a \\in I$ a nonzero\nelement. Then the affine blowup algebra $R[\\frac{I}{a}]$ is a domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052R","source_file":"algebra.tex","source_line":17322,"source_end_line":17326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17322-L17326","statement_sha256":"36117ec87d1edfbf8676208c9c11c5253cbff58eb6ebcd484265609fe3e4011d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1457,"rank":1457,"depth":0,"x":2065.453,"y":133.312,"cluster":"commutative-algebra"},{"id":"stacks:052T","tag":"052T","title":"Blow up algebras · Lemma 052T","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let a ∈ I. If a is not contained in any minimal prime of R, then Spec(R[fracIa]) → Spec(R) has dense image.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $a \\in I$.\nIf $a$ is not contained in any minimal prime of $R$, then\n$\\Spec(R[\\frac{I}{a}]) \\to \\Spec(R)$ has dense image.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052T","source_file":"algebra.tex","source_line":17335,"source_end_line":17340,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17335-L17340","statement_sha256":"e4a8a100d65b8f307e4f85d44d389d4594121e5a7039cd2fc92c644943a71431","origin":"The Stacks Project","memory_eligible":false,"source_rank":1458,"rank":1458,"depth":3,"x":2073.611,"y":304.12,"cluster":"commutative-algebra"},{"id":"stacks:052M","tag":"052M","title":"Blow up algebras · Lemma 052M","summary":"Let (R, m) be a local domain with fraction field K. Let R ⊂ A ⊂ K be a valuation ring which dominates R. Then A = colim R[textstylefracIa] is a directed colimit of affine blowups R → R[fracIa] with the following properties • a ∈ I ⊂ m, • I is finitely generated, and • the fibre ring of R → R[fracIa] at m is not zero.","statement_latex":"Let $(R, \\mathfrak m)$ be a local domain with fraction field $K$.\nLet $R \\subset A \\subset K$ be a valuation ring which dominates $R$.\nThen\n$$\nA = \\colim R[\\textstyle{\\frac{I}{a}}]\n$$\nis a directed colimit of affine blowups $R \\to R[\\frac{I}{a}]$ with\nthe following properties\n\\begin{enumerate}\n\\item $a \\in I \\subset \\mathfrak m$,\n\\item $I$ is finitely generated, and\n\\item the fibre ring of $R \\to R[\\frac{I}{a}]$ at $\\mathfrak m$\nis not zero.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Blow up algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052M","source_file":"algebra.tex","source_line":17351,"source_end_line":17367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17351-L17367","statement_sha256":"d15470d817daee78bb8815ed9ac7d0e9d12790a36e04a4fe3cd0720c55767760","origin":"The Stacks Project","memory_eligible":false,"source_rank":1459,"rank":1459,"depth":1,"x":1930.1,"y":182.682,"cluster":"commutative-algebra"},{"id":"stacks:00LP","tag":"00LP","title":"Ext groups · Lemma 00LP","summary":"Let R be a ring. Let M be an R-module. • There exists an exact complex … → F_2 → F_1 → F_0 → M → 0. with F_i free R-modules. • If R is Noetherian and M finite over R, then we can choose the complex such that F_i is finite free. In other words, we can find an exact complex … → R^⊕ n_2 → R^⊕ n_1 → R^⊕ n_0 → M → 0.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item There exists an exact complex\n$$\n\\ldots \\to F_2 \\to F_1 \\to F_0 \\to M \\to 0.\n$$\nwith $F_i$ free $R$-modules.\n\\item If $R$ is Noetherian and $M$ finite over $R$, then we\ncan choose the complex such that $F_i$ is finite free.\nIn other words, we can find an exact complex\n$$\n\\ldots \\to R^{\\oplus n_2} \\to R^{\\oplus n_1} \\to R^{\\oplus n_0} \\to M \\to 0.\n$$\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LP","source_file":"algebra.tex","source_line":17410,"source_end_line":17426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17410-L17426","statement_sha256":"dfd10c60d309a2bb22832271b3eec1749f93905d046d39eae9b3fc23ea2a568b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1460,"rank":1460,"depth":0,"x":2133.774,"y":190.804,"cluster":"commutative-algebra"},{"id":"stacks:00LQ","tag":"00LQ","title":"Ext groups · Definition 00LQ","summary":"Let R be a ring. Let M be an R-module. • A (left) resolution F_bullet → M of M is an exact complex … → F_2 → F_1 → F_0 → M → 0 of R-modules. • A resolution of M by free R-modules is a resolution F_bullet → M where each F_i is a free R-module. • A resolution of M by finite free R-modules is a resolution F_bullet → M where each F_i is a finite free R-module.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item A (left) {\\it resolution} $F_\\bullet \\to M$ of $M$ is an exact complex\n$$\n\\ldots \\to F_2 \\to F_1 \\to F_0 \\to M \\to 0\n$$\nof $R$-modules.\n\\item A {\\it resolution of $M$ by free $R$-modules} is a resolution\n$F_\\bullet \\to M$ where each $F_i$ is a free $R$-module.\n\\item A {\\it resolution of $M$ by finite free $R$-modules} is a resolution\n$F_\\bullet \\to M$ where each $F_i$ is a finite free $R$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ext groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LQ","source_file":"algebra.tex","source_line":17437,"source_end_line":17451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17437-L17451","statement_sha256":"a2870c97cef8690a1221ff8c6dda429dd153d50e9782cdbdc7b2d42aa4e7fb12","origin":"The Stacks Project","memory_eligible":false,"source_rank":1461,"rank":1461,"depth":0,"x":1976.906,"y":300.489,"cluster":"commutative-algebra"},{"id":"stacks:00LR","tag":"00LR","title":"Ext groups · Lemma 00LR","summary":"Any two homotopic maps of complexes induce the same maps on (co)homology groups.","statement_latex":"Any two homotopic maps of complexes induce the same maps on\n(co)homology groups.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LR","source_file":"algebra.tex","source_line":17489,"source_end_line":17493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17489-L17493","statement_sha256":"30d656b4149827366d7c919caa39093635814a2a28496fb10bfbf3c1f9f5540b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1462,"rank":1462,"depth":0,"x":2004.399,"y":130.437,"cluster":"commutative-algebra"},{"id":"stacks:00LS","tag":"00LS","title":"Ext groups · Lemma 00LS","summary":"Let R be a ring. Let M → N be a map of R-modules. Let N_bullet → N be an arbitrary resolution. Let … → F_2 → F_1 → F_0 → M be a complex of R-modules where each F_i is a free R-module. Then • there exists a map of complexes F_bullet → N_bullet such that xymatrix F_0 ar[r] ar[d] & M ar[d] N_0 ar[r] & N is commutative, and • any two maps α, β : F_bullet → N_bullet as in (1) are homotopic.","statement_latex":"Let $R$ be a ring. Let $M \\to N$ be a map of $R$-modules.\nLet $N_\\bullet \\to N$ be an arbitrary resolution.\nLet\n$$\n\\ldots \\to F_2 \\to F_1 \\to F_0 \\to M\n$$\nbe a complex of $R$-modules where each $F_i$ is a free $R$-module. Then\n\\begin{enumerate}\n\\item there exists a map of complexes $F_\\bullet \\to N_\\bullet$ such that\n$$\n\\xymatrix{\nF_0 \\ar[r] \\ar[d] & M \\ar[d] \\\\\nN_0 \\ar[r] & N\n}\n$$\nis commutative, and\n\\item any two maps $\\alpha, \\beta : F_\\bullet \\to N_\\bullet$ as in (1)\nare homotopic.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LS","source_file":"algebra.tex","source_line":17499,"source_end_line":17520,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17499-L17520","statement_sha256":"2db43538ec8b8c9e5ca18a9d142614c957b0eeed57eb41b3bac4d69445222ac3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1463,"rank":1463,"depth":0,"x":2120.988,"y":271.565,"cluster":"commutative-algebra"},{"id":"stacks:00LT","tag":"00LT","title":"Ext groups · Lemma 00LT","summary":"Let R be a ring. Let M_1, M_2, N be R-modules. Suppose that F_bullet is a free resolution of the module M_1, and G_bullet is a free resolution of the module M_2. Let φ : M_1 → M_2 be a module map. Let α : F_bullet → G_bullet be a map of complexes inducing φ on M_1 = Coker(d_F, 1) → M_2 = Coker(d_G, 1), see Lemma [Tag 00LS]. Then the induced maps H^i(α) : H^i(Hom_R(F_bullet, N)) → H^i(Hom_R(G_bullet, N)) are independent of the choice of α. If φ is an isomorphism, so are…","statement_latex":"Let $R$ be a ring. Let $M_1, M_2, N$ be $R$-modules.\nSuppose that $F_{\\bullet}$ is a free resolution of the module $M_1$,\nand $G_{\\bullet}$ is a free resolution of the module $M_2$.\nLet $\\varphi : M_1 \\to M_2$ be a module map.\nLet $\\alpha : F_{\\bullet} \\to G_{\\bullet}$ be\na map of complexes inducing $\\varphi$ on\n$M_1 = \\Coker(d_{F, 1}) \\to M_2 = \\Coker(d_{G, 1})$,\nsee Lemma \\ref{lemma-compare-resolutions}.\nThen the induced maps\n$$\nH^i(\\alpha) :\nH^i(\\Hom_R(F_{\\bullet}, N))\n\\longrightarrow\nH^i(\\Hom_R(G_{\\bullet}, N))\n$$\nare independent of the choice of $\\alpha$.\nIf $\\varphi$ is an isomorphism, so are all the maps\n$H^i(\\alpha)$. If $M_1 = M_2$, $F_\\bullet = G_\\bullet$, and\n$\\varphi$ is the identity, so are all the maps $H_i(\\alpha)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LT","source_file":"algebra.tex","source_line":17574,"source_end_line":17595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17574-L17595","statement_sha256":"7b129520a36f193f3e6e75fa312cab12530e9ae617962979e12e341470a2da79","origin":"The Stacks Project","memory_eligible":false,"source_rank":1464,"rank":1464,"depth":1,"x":1921.337,"y":233.619,"cluster":"commutative-algebra"},{"id":"stacks:00LU","tag":"00LU","title":"Ext groups · Lemma 00LU","summary":"Let R be a ring. Let M be an R-module. Let 0 → N' → N → N\" → 0 be a short exact sequence. Then we get a long exact sequence 0 → Hom_R(M, N') → Hom_R(M, N) → Hom_R(M, N\") phantom0 → Ext^1_R(M, N') → Ext^1_R(M, N) → Ext^1_R(M, N\") → …","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $0 \\to N' \\to N \\to N'' \\to 0$ be a\nshort exact sequence. Then we get a long exact\nsequence\n$$\n\\begin{matrix}\n0\n\\to \\Hom_R(M, N')\n\\to \\Hom_R(M, N)\n\\to \\Hom_R(M, N'')\n\\\\\n\\phantom{0\\ }\n\\to \\Ext^1_R(M, N')\n\\to \\Ext^1_R(M, N)\n\\to \\Ext^1_R(M, N'')\n\\to \\ldots\n\\end{matrix}\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LU","source_file":"algebra.tex","source_line":17622,"source_end_line":17642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17622-L17642","statement_sha256":"879c9d2367730e3a67f827eddf69d109ee7816abc8c9ace8d251ed99f333c2be","origin":"The Stacks Project","memory_eligible":false,"source_rank":1465,"rank":1465,"depth":0,"x":2099.24,"y":148.232,"cluster":"commutative-algebra"},{"id":"stacks:065P","tag":"065P","title":"Ext groups · Lemma 065P","summary":"Let R be a ring. Let N be an R-module. Let 0 → M' → M → M\" → 0 be a short exact sequence. Then we get a long exact sequence 0 → Hom_R(M\", N) → Hom_R(M, N) → Hom_R(M', N) phantom0 → Ext^1_R(M\", N) → Ext^1_R(M, N) → Ext^1_R(M', N) → …","statement_latex":"Let $R$ be a ring. Let $N$ be an $R$-module.\nLet $0 \\to M' \\to M \\to M'' \\to 0$ be a\nshort exact sequence. Then we get a long exact\nsequence\n$$\n\\begin{matrix}\n0\n\\to \\Hom_R(M'', N)\n\\to \\Hom_R(M, N)\n\\to \\Hom_R(M', N)\n\\\\\n\\phantom{0\\ }\n\\to \\Ext^1_R(M'', N)\n\\to \\Ext^1_R(M, N)\n\\to \\Ext^1_R(M', N)\n\\to \\ldots\n\\end{matrix}\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065P","source_file":"algebra.tex","source_line":17659,"source_end_line":17679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17659-L17679","statement_sha256":"200f69ba3239de637f9161d4e89d0f6010f34897018a029ee6b405cfa54177c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1466,"rank":1466,"depth":0,"x":2036.663,"y":312.296,"cluster":"commutative-algebra"},{"id":"stacks:00LV","tag":"00LV","title":"Ext groups · Lemma 00LV","summary":"Let R be a ring. Let M, N be R-modules. Any x∈ R such that either xN = 0, or xM = 0 annihilates each of the modules Ext^i_R(M, N).","statement_latex":"Let $R$ be a ring. Let $M$, $N$ be $R$-modules.\nAny $x\\in R$ such that either $xN = 0$, or $xM = 0$\nannihilates each of the modules $\\Ext^i_R(M, N)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LV","source_file":"algebra.tex","source_line":17723,"source_end_line":17728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17723-L17728","statement_sha256":"0f85b1bb2d46bb7a64e19ed6185978a79a1f6bb4526250e9156e8aa7a22135a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1467,"rank":1467,"depth":2,"x":1950.791,"y":155.663,"cluster":"commutative-algebra"},{"id":"stacks:08YR","tag":"08YR","title":"Ext groups · Lemma 08YR","summary":"Let R be a Noetherian ring. Let M, N be finite R-modules. Then Ext^i_R(M, N) is a finite R-module for all i.","statement_latex":"Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules.\nThen $\\Ext^i_R(M, N)$ is a finite $R$-module for all $i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YR","source_file":"algebra.tex","source_line":17743,"source_end_line":17747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17743-L17747","statement_sha256":"6d5c9fdfcc1e19243802f742635cf677df261db7037a83a2d6b09cfb59a0dcf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1468,"rank":1468,"depth":1,"x":2140.249,"y":222.498,"cluster":"commutative-algebra"},{"id":"stacks:00LI","tag":"00LI","title":"Depth · Definition 00LI","summary":"Let R be a ring, and I ⊂ R an ideal. Let M be a finite R-module. The I-depth of M, denoted depth_I(M), is defined as follows: • if IM not = M, then depth_I(M) is the supremum in (0, 1, 2, …, ∞) of the lengths of M-regular sequences in I, • if IM = M we set depth_I(M) = ∞. If (R, m) is local we call depth_ m(M) simply the depth of M.","statement_latex":"Let $R$ be a ring, and $I \\subset R$ an ideal. Let $M$ be a finite $R$-module.\nThe {\\it $I$-depth} of $M$, denoted $\\text{depth}_I(M)$, is defined as follows:\n\\begin{enumerate}\n\\item if $IM \\not = M$, then $\\text{depth}_I(M)$ is the supremum in\n$\\{0, 1, 2, \\ldots, \\infty\\}$ of the lengths of $M$-regular sequences in $I$,\n\\item if $IM = M$ we set $\\text{depth}_I(M) = \\infty$.\n\\end{enumerate}\nIf $(R, \\mathfrak m)$ is local we call $\\text{depth}_{\\mathfrak m}(M)$ simply\nthe {\\it depth} of $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LI","source_file":"algebra.tex","source_line":17765,"source_end_line":17776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17765-L17776","statement_sha256":"652a93ce833a8e8cdd79557cbeb2a724d50cc8a77ffe878b0fc5627bae67afa1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1469,"rank":1469,"depth":0,"x":1946.617,"y":280.773,"cluster":"commutative-algebra"},{"id":"stacks:0AUI","tag":"0AUI","title":"Depth · Lemma 0AUI","summary":"Let R be a ring, I ⊂ R an ideal, and M a finite R-module. Then depth_I(M) is equal to the supremum of the lengths of sequences f_1, …, f_r ∈ I such that f_i is a nonzerodivisor on M/(f_1, …, f_i - 1)M.","statement_latex":"Let $R$ be a ring, $I \\subset R$ an ideal, and $M$ a finite $R$-module.\nThen $\\text{depth}_I(M)$ is equal to the supremum of the lengths of\nsequences $f_1, \\ldots, f_r \\in I$ such that $f_i$ is a nonzerodivisor\non $M/(f_1, \\ldots, f_{i - 1})M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUI","source_file":"algebra.tex","source_line":17796,"source_end_line":17802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17796-L17802","statement_sha256":"cffdf3f90a2ba47d6ed42f390139b0fd4f52f6f485d146d8e57cbace60de628c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1470,"rank":1470,"depth":3,"x":2042.625,"y":127.787,"cluster":"commutative-algebra"},{"id":"stacks:00LK","tag":"00LK","title":"Depth · Lemma 00LK","summary":"Let (R, m) be a Noetherian local ring. Let M be a nonzero finite R-module. Then dim(Supp(M)) ≥ depth(M).","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring.\nLet $M$ be a nonzero finite $R$-module.\nThen $\\dim(\\text{Supp}(M)) \\geq \\text{depth}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LK","source_file":"algebra.tex","source_line":17813,"source_end_line":17818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17813-L17818","statement_sha256":"13ae462f5f4175a8d66bca4ec631dad62072e4c4cae949d1999454f09e773154","origin":"The Stacks Project","memory_eligible":false,"source_rank":1471,"rank":1471,"depth":12,"x":2094.906,"y":295.23,"cluster":"commutative-algebra"},{"id":"stacks:0AUJ","tag":"0AUJ","title":"Depth · Lemma 0AUJ","summary":"Let R be a Noetherian ring, I ⊂ R an ideal, and M a finite nonzero R-module such that IM not = M. Then depth_I(M) < ∞.","statement_latex":"Let $R$ be a Noetherian ring, $I \\subset R$ an ideal, and $M$ a\nfinite nonzero $R$-module such that $IM \\not = M$. Then\n$\\text{depth}_I(M) < \\infty$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUJ","source_file":"algebra.tex","source_line":17839,"source_end_line":17844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17839-L17844","statement_sha256":"b038e9a65df4b54dbf28d3a274b4c7bb30634741be4c32b7f0a93bf485820e1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1472,"rank":1472,"depth":13,"x":1921.541,"y":201.338,"cluster":"commutative-algebra"},{"id":"stacks:00LW","tag":"00LW","title":"Depth · Lemma 00LW","summary":"Let R be a Noetherian local ring with maximal ideal m. Let M be a nonzero finite R-module. Then depth(M) is equal to the smallest integer i such that Ext^i_R(R/ m, M) is nonzero.","statement_latex":"Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$.\nLet $M$ be a nonzero finite $R$-module. Then $\\text{depth}(M)$\nis equal to the smallest integer $i$ such that\n$\\Ext^i_R(R/\\mathfrak m, M)$ is nonzero.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LW","source_file":"algebra.tex","source_line":17862,"source_end_line":17868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17862-L17868","statement_sha256":"3e927e5222b7bb1552fd011d0148102db31bf3581f44902afa0f19e4c2ac5d1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1473,"rank":1473,"depth":11,"x":2125.072,"y":172.174,"cluster":"commutative-algebra"},{"id":"stacks:00LX","tag":"00LX","title":"Depth · Lemma 00LX","summary":"Let R be a local Noetherian ring. Let 0 → N' → N → N\" → 0 be a short exact sequence of nonzero finite R-modules. • depth(N) ≥ min(depth(N'), depth(N\")) • depth(N\") ≥ min(depth(N), depth(N') - 1) • depth(N') ≥ min(depth(N), depth(N\") + 1)","statement_latex":"Let $R$ be a local Noetherian ring. Let $0 \\to N' \\to N \\to N'' \\to 0$\nbe a short exact sequence of nonzero finite $R$-modules.\n\\begin{enumerate}\n\\item\n$\\text{depth}(N) \\geq \\min\\{\\text{depth}(N'), \\text{depth}(N'')\\}$\n\\item\n$\\text{depth}(N'') \\geq \\min\\{\\text{depth}(N), \\text{depth}(N') - 1\\}$\n\\item\n$\\text{depth}(N') \\geq \\min\\{\\text{depth}(N), \\text{depth}(N'') + 1\\}$\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LX","source_file":"algebra.tex","source_line":17904,"source_end_line":17916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17904-L17916","statement_sha256":"da32605eaec5642832d1d427a2dab9c88225c8e8b9f151c17ef082ab6c5b9ece","origin":"The Stacks Project","memory_eligible":false,"source_rank":1474,"rank":1474,"depth":12,"x":1998.326,"y":309.295,"cluster":"commutative-algebra"},{"id":"stacks:090R","tag":"090R","title":"Depth · Lemma 090R","summary":"Let R be a local Noetherian ring and M a nonzero finite R-module. • If x ∈ m is a nonzerodivisor on M, then depth(M/xM) = depth(M) - 1. • Any M-regular sequence x_1, …, x_r can be extended to an M-regular sequence of length depth(M).","statement_latex":"Let $R$ be a local Noetherian ring and $M$ a nonzero finite $R$-module.\n\\begin{enumerate}\n\\item If $x \\in \\mathfrak m$ is a nonzerodivisor on $M$, then\n$\\text{depth}(M/xM) = \\text{depth}(M) - 1$.\n\\item Any $M$-regular sequence $x_1, \\ldots, x_r$ can be extended to an\n$M$-regular sequence of length $\\text{depth}(M)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090R","source_file":"algebra.tex","source_line":17939,"source_end_line":17948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17939-L17948","statement_sha256":"f37230a79ffc1ba91f9f1d7c9b6c9a6394dccff18ef4110d79d576f22d520de9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1475,"rank":1475,"depth":13,"x":1981.502,"y":136.105,"cluster":"commutative-algebra"},{"id":"stacks:0CN5","tag":"0CN5","title":"Depth · Lemma 0CN5","summary":"Let (R, m) be a local Noetherian ring and M a finite R-module. Let x ∈ m, p ∈ Ass(M), and q minimal over p + (x). Then q ∈ Ass(M/x^nM) for some n ≥ 1.","statement_latex":"Let $(R, \\mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module.\nLet $x \\in \\mathfrak m$, $\\mathfrak p \\in \\text{Ass}(M)$, and $\\mathfrak q$\nminimal over $\\mathfrak p + (x)$. Then $\\mathfrak q \\in \\text{Ass}(M/x^nM)$\nfor some $n \\geq 1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CN5","source_file":"algebra.tex","source_line":17960,"source_end_line":17966,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17960-L17966","statement_sha256":"a8dbc49ae782cf98466e9f6d84a809cb72c885ea797b00a5c58ec85dd83554ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":1476,"rank":1476,"depth":9,"x":2133.323,"y":254.376,"cluster":"commutative-algebra"},{"id":"stacks:0BK4","tag":"0BK4","title":"Depth · Lemma 0BK4","summary":"Let (R, m) be a local Noetherian ring and M a finite R-module. For p ∈ Ass(M) we have dim(R/ p) ≥ depth(M).","statement_latex":"Let $(R, \\mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module.\nFor $\\mathfrak p \\in \\text{Ass}(M)$ we have\n$\\dim(R/\\mathfrak p) \\geq \\text{depth}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BK4","source_file":"algebra.tex","source_line":17984,"source_end_line":17989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L17984-L17989","statement_sha256":"7654ddebc7dfbd85f1235bd63692fe5dab8826ea2c8251fa3727db6becb1fe03","origin":"The Stacks Project","memory_eligible":false,"source_rank":1477,"rank":1477,"depth":14,"x":1926.072,"y":253.31,"cluster":"commutative-algebra"},{"id":"stacks:0FCC","tag":"0FCC","title":"Depth · Lemma 0FCC","summary":"Let R be a local Noetherian ring and M a finite R-module. For a prime ideal p ⊂ R we have depth(M_ p) + dim(R/ p) ≥ depth(M).","statement_latex":"Let $R$ be a local Noetherian ring and $M$ a finite $R$-module.\nFor a prime ideal $\\mathfrak p \\subset R$ we have\n$\\text{depth}(M_\\mathfrak p) + \\dim(R/\\mathfrak p) \\geq \\text{depth}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCC","source_file":"algebra.tex","source_line":18022,"source_end_line":18027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18022-L18027","statement_sha256":"eea625d36f6f4eb125387e28805ba5644977f8e16d4b53e6411e94e022e3efd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1478,"rank":1478,"depth":15,"x":2079.894,"y":136.389,"cluster":"commutative-algebra"},{"id":"stacks:0AUK","tag":"0AUK","title":"Depth · Lemma 0AUK","summary":"Let (R, m) be a Noetherian local ring. Let R → S be a finite ring map. Let m_1, …, m_n be the maximal ideals of S. Let N be a finite S-module. Then min_i = 1, …, n depth(N_ m_i) = depth_ m(N)","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let $R \\to S$\nbe a finite ring map. Let $\\mathfrak m_1, \\ldots, \\mathfrak m_n$\nbe the maximal ideals of $S$. Let $N$ be a finite $S$-module.\nThen\n$$\n\\min\\nolimits_{i = 1, \\ldots, n} \\text{depth}(N_{\\mathfrak m_i}) =\n\\text{depth}_\\mathfrak m(N)\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUK","source_file":"algebra.tex","source_line":18047,"source_end_line":18057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18047-L18057","statement_sha256":"52b57e3abea80d9eab6a21f080a8c3382b70d4c37d28c903bf3bafbea0d3feda","origin":"The Stacks Project","memory_eligible":false,"source_rank":1479,"rank":1479,"depth":14,"x":2060.475,"y":310.047,"cluster":"commutative-algebra"},{"id":"stacks:087N","tag":"087N","title":"Functorialities for Ext · Lemma 087N","summary":"Given a flat ring map R → R', an R-module M, and an R'-module N' the natural map Ext^i_R'(M ⊗_R R', N') → Ext^i_R(M, N') is an isomorphism for i ≥ 0.","statement_latex":"Given a flat ring map $R \\to R'$, an $R$-module $M$, and an\n$R'$-module $N'$ the natural map\n$$\n\\Ext^i_{R'}(M \\otimes_R R', N') \\to \\text{Ext}^i_R(M, N')\n$$\nis an isomorphism for $i \\geq 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Functorialities for Ext","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087N","source_file":"algebra.tex","source_line":18106,"source_end_line":18114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18106-L18114","statement_sha256":"5fffc34f90a2e2e0e22a485c7e9cdba3cdefc18ef6d8566ac2beb30709e3e86e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1480,"rank":1480,"depth":1,"x":1935.028,"y":170.848,"cluster":"commutative-algebra"},{"id":"stacks:02HO","tag":"02HO","title":"An application of Ext groups · Lemma 02HO","summary":"Let R be a Noetherian ring. Let I ⊂ R be an ideal contained in the Jacobson radical of R. Let N → M be a homomorphism of finite R-modules. Suppose that there exists arbitrarily large n such that N/I^nN → M/I^nM is a split injection. Then N → M is a split injection.","statement_latex":"Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal\ncontained in the Jacobson radical of $R$.\nLet $N \\to M$ be a homomorphism of finite $R$-modules.\nSuppose that there exists arbitrarily large $n$ such that\n$N/I^nN \\to M/I^nM$ is a split injection.\nThen $N \\to M$ is a split injection.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"An application of Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HO","source_file":"algebra.tex","source_line":18141,"source_end_line":18149,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18141-L18149","statement_sha256":"da6a89a2eee957fb1b1b9c6d6671ee5057c56f74c8448d2acbd835fe7d5af39c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1481,"rank":1481,"depth":5,"x":2139.658,"y":202.339,"cluster":"commutative-algebra"},{"id":"stacks:00LZ","tag":"00LZ","title":"Tor groups and flatness · Lemma 00LZ","summary":"Let R be a ring. Let M_1, M_2, N be R-modules. Suppose that F_bullet is a free resolution of the module M_1 and that G_bullet is a free resolution of the module M_2. Let φ : M_1 → M_2 be a module map. Let α : F_bullet → G_bullet be a map of complexes inducing φ on M_1 = Coker(d_F, 1) → M_2 = Coker(d_G, 1), see Lemma [Tag 00LS]. Then the induced maps H_i(α) : H_i(F_bullet ⊗_R N) → H_i(G_bullet ⊗_R N) are independent of the choice of α. If φ is an isomorphism, so are all…","statement_latex":"Let $R$ be a ring. Let $M_1, M_2, N$ be $R$-modules.\nSuppose that $F_\\bullet$ is a free resolution of\nthe module $M_1$ and that $G_\\bullet$ is a free\nresolution of the module $M_2$. Let $\\varphi : M_1 \\to M_2$\nbe a module map. Let $\\alpha : F_\\bullet \\to G_\\bullet$\nbe a map of complexes inducing $\\varphi$ on\n$M_1 = \\Coker(d_{F, 1}) \\to M_2 = \\Coker(d_{G, 1})$,\nsee Lemma \\ref{lemma-compare-resolutions}.\nThen the induced maps\n$$\nH_i(\\alpha) :\nH_i(F_\\bullet \\otimes_R N)\n\\longrightarrow\nH_i(G_\\bullet \\otimes_R N)\n$$\nare independent of the choice of $\\alpha$. If $\\varphi$\nis an isomorphism, so are all the maps $H_i(\\alpha)$.\nIf $M_1 = M_2$, $F_\\bullet = G_\\bullet$, and\n$\\varphi$ is the identity, so are all the maps $H_i(\\alpha)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tor groups and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00LZ","source_file":"algebra.tex","source_line":18238,"source_end_line":18259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18238-L18259","statement_sha256":"751eccf48d254ab0f67c0d2a62dcfbd16d5a280ce76fea4b647a9ab4147a07fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1482,"rank":1482,"depth":2,"x":1963.282,"y":295.314,"cluster":"commutative-algebra"},{"id":"stacks:00M0","tag":"00M0","title":"Tor groups and flatness · Lemma 00M0","summary":"Let R be a ring and let M be an R-module. Suppose that 0 → N' → N → N\" → 0 is a short exact sequence of R-modules. There exists a long exact sequence Tor_1^R(M, N') → Tor_1^R(M, N) → Tor_1^R(M, N\") → M ⊗_R N' → M ⊗_R N → M ⊗_R N\" → 0","statement_latex":"Let $R$ be a ring and let $M$ be an $R$-module.\nSuppose that $0 \\to N' \\to N \\to N'' \\to 0$ is a short\nexact sequence of $R$-modules. There exists a long\nexact sequence\n$$\n\\text{Tor}_1^R(M, N')\n\\to \\text{Tor}_1^R(M, N)\n\\to \\text{Tor}_1^R(M, N'')\n\\to\nM \\otimes_R N'\n\\to M \\otimes_R N\n\\to M \\otimes_R N''\n\\to 0\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tor groups and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00M0","source_file":"algebra.tex","source_line":18290,"source_end_line":18306,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18290-L18306","statement_sha256":"ee81151e3ef1bc301c3395a4e82b3b0ce2132be9532ae718e2492ed584a335e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1483,"rank":1483,"depth":1,"x":2018.621,"y":126.522,"cluster":"commutative-algebra"},{"id":"stacks:00M1","tag":"00M1","title":"Tor groups and flatness · Lemma 00M1","summary":"Let (A_bullet, bullet, d, δ) be a double complex such that • Each row A_bullet, j is a resolution of R(A)_j. • Each column A_i, bullet is a resolution of U(A)_i. Then there are canonical isomorphisms H_i(R(A)_bullet) ≅ H_i(U(A)_bullet). The isomorphisms are functorial with respect to morphisms of double complexes with the properties above.","statement_latex":"Let $(A_{\\bullet, \\bullet}, d, \\delta)$ be a double complex such\nthat\n\\begin{enumerate}\n\\item Each row $A_{\\bullet, j}$ is a resolution of $R(A)_j$.\n\\item Each column $A_{i, \\bullet}$ is a resolution of $U(A)_i$.\n\\end{enumerate}\nThen there are canonical isomorphisms\n$$\nH_i(R(A)_\\bullet)\n\\cong\nH_i(U(A)_\\bullet).\n$$\nThe isomorphisms are functorial with respect to morphisms\nof double complexes with the properties above.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tor groups and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00M1","source_file":"algebra.tex","source_line":18397,"source_end_line":18413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18397-L18413","statement_sha256":"d64d083bb0e18e2f0044e84a37ce38d833e0bd018610088e8545bddfa395ac25","origin":"The Stacks Project","memory_eligible":false,"source_rank":1484,"rank":1484,"depth":0,"x":2113.64,"y":282.529,"cluster":"commutative-algebra"},{"id":"stacks:00M3","tag":"00M3","title":"Tor groups and flatness · Lemma 00M3","summary":"Let R be a ring. For any i ≥ 0 the functors Mod_R × Mod_R → Mod_R, (M, N) ↦ Tor_i^R(M, N) and (M, N) ↦ Tor_i^R(N, M) are canonically isomorphic.","statement_latex":"Let $R$ be a ring. For any $i \\geq 0$ the functors\n$\\text{Mod}_R \\times \\text{Mod}_R \\to \\text{Mod}_R$,\n$(M, N) \\mapsto \\text{Tor}_i^R(M, N)$ and\n$(M, N) \\mapsto \\text{Tor}_i^R(N, M)$ are\ncanonically isomorphic.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tor groups and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00M3","source_file":"algebra.tex","source_line":18533,"source_end_line":18540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18533-L18540","statement_sha256":"856d1c50a4538527020809a4ca41465ae0a162938836d18b124f7c680cb82dca","origin":"The Stacks Project","memory_eligible":false,"source_rank":1485,"rank":1485,"depth":1,"x":1917.939,"y":221.352,"cluster":"commutative-algebra"},{"id":"stacks:0AZ4","tag":"0AZ4","title":"Tor groups and flatness · Lemma 0AZ4","summary":"Let R be a Noetherian ring. Let M, N be finite R-modules. Then Tor_p^R(M, N) is a finite R-module for all p.","statement_latex":"Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules.\nThen $\\text{Tor}_p^R(M, N)$ is a finite $R$-module for all $p$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tor groups and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZ4","source_file":"algebra.tex","source_line":18636,"source_end_line":18640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18636-L18640","statement_sha256":"e5df313dd04f98e8f825990d1f244c31539334d60d93c19373a6a8b5403c7478","origin":"The Stacks Project","memory_eligible":false,"source_rank":1486,"rank":1486,"depth":1,"x":2111.619,"y":155.358,"cluster":"commutative-algebra"},{"id":"stacks:00M5","tag":"00M5","title":"Tor groups and flatness · Lemma 00M5","summary":"Let R be a ring. Let M be an R-module. The following are equivalent: • The module M is flat over R. • For all i > 0 the functor Tor_i^R(M, -) is zero. • The functor Tor_1^R(M, -) is zero. • For all ideals I ⊂ R we have Tor_1^R(M, R/I) = 0. • For all finitely generated ideals I ⊂ R we have Tor_1^R(M, R/I) = 0.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The module $M$ is flat over $R$.\n\\item For all $i > 0$ the functor $\\text{Tor}_i^R(M, -)$ is zero.\n\\item The functor $\\text{Tor}_1^R(M, -)$ is zero.\n\\item For all ideals $I \\subset R$ we have $\\text{Tor}_1^R(M, R/I) = 0$.\n\\item For all finitely generated ideals $I \\subset R$ we have\n$\\text{Tor}_1^R(M, R/I) = 0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Tor groups and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00M5","source_file":"algebra.tex","source_line":18649,"source_end_line":18661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18649-L18661","statement_sha256":"2072af4f2490105d38e00c44565d3d71606ae9c4beaba128f7e8b0d42ea03e2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1487,"rank":1487,"depth":3,"x":2021.791,"y":314.063,"cluster":"commutative-algebra"},{"id":"stacks:00M8","tag":"00M8","title":"Functorialities for Tor · Lemma 00M8","summary":"Given a flat ring map R → R' and R-modules M, N the natural R-module map Tor_i^R(M, N)⊗_R R' → Tor_i^R'(M ⊗_R R', N ⊗_R R') is an isomorphism for all i.","statement_latex":"Given a flat ring map $R \\to R'$ and $R$-modules\n$M$, $N$ the natural $R$-module map\n$\\text{Tor}_i^R(M, N)\\otimes_R R'\n\\to \\text{Tor}_i^{R'}(M \\otimes_R R', N \\otimes_R R')$\nis an isomorphism for all $i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Functorialities for Tor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00M8","source_file":"algebra.tex","source_line":18734,"source_end_line":18741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18734-L18741","statement_sha256":"69d16a2bdb7eba282637ca075a8f551642ae970a7e1d9778d29f04bebe24657e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1488,"rank":1488,"depth":0,"x":1960.347,"y":145.915,"cluster":"commutative-algebra"},{"id":"stacks:0BNF","tag":"0BNF","title":"Functorialities for Tor · Lemma 0BNF","summary":"Let R be a ring. Let M = colim M_i be a filtered colimit of R-modules. Let N be an R-module. Then Tor_n^R(M, N) = colim Tor_n^R(M_i, N) for all n.","statement_latex":"Let $R$ be a ring. Let $M = \\colim M_i$ be a filtered colimit of\n$R$-modules. Let $N$ be an $R$-module. Then\n$\\text{Tor}_n^R(M, N) = \\colim \\text{Tor}_n^R(M_i, N)$ for all $n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Functorialities for Tor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNF","source_file":"algebra.tex","source_line":18753,"source_end_line":18758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18753-L18758","statement_sha256":"be7fe913c7c012a6ffe57d1041449cec85ccf02a90f8d599f88eeeb172323095","origin":"The Stacks Project","memory_eligible":false,"source_rank":1489,"rank":1489,"depth":4,"x":2141.038,"y":235.12,"cluster":"commutative-algebra"},{"id":"stacks:05CE","tag":"05CE","title":"Projective modules · Definition 05CE","summary":"Let R be a ring. An R-module P is projective if and only if the functor Hom_R(P, -) : Mod_R → Mod_R is an exact functor.","statement_latex":"Let $R$ be a ring. An $R$-module $P$ is {\\it projective} if and only if\nthe functor $\\Hom_R(P, -) : \\text{Mod}_R \\to \\text{Mod}_R$ is\nan exact functor.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CE","source_file":"algebra.tex","source_line":18780,"source_end_line":18785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18780-L18785","statement_sha256":"9f5cd4d7956f826b1b8509037a1db287933f7bfaad89fcbd44cfe27c985ffaef","origin":"The Stacks Project","memory_eligible":false,"source_rank":1490,"rank":1490,"depth":0,"x":1935.878,"y":271.903,"cluster":"commutative-algebra"},{"id":"stacks:05CF","tag":"05CF","title":"Projective modules · Lemma 05CF","summary":"Let R be a ring. Let P be an R-module. The following are equivalent • P is projective, • P is a direct summand of a free R-module, and • Ext^1_R(P, M) = 0 for every R-module M.","statement_latex":"Let $R$ be a ring. Let $P$ be an $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $P$ is projective,\n\\item $P$ is a direct summand of a free $R$-module, and\n\\item $\\Ext^1_R(P, M) = 0$ for every $R$-module $M$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CF","source_file":"algebra.tex","source_line":18794,"source_end_line":18803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18794-L18803","statement_sha256":"048f9878bdd73fc8f70446c4d1da37b44afd8b55f53742108aa060d2f386c602","origin":"The Stacks Project","memory_eligible":false,"source_rank":1491,"rank":1491,"depth":0,"x":2057.69,"y":128.238,"cluster":"commutative-algebra"},{"id":"stacks:0G8T","tag":"0G8T","title":"Projective modules · Lemma 0G8T","summary":"Let R be a Noetherian ring. Let P be a finite R-module. If Ext^1_R(P, M) = 0 for every finite R-module M, then P is projective.","statement_latex":"Let $R$ be a Noetherian ring. Let $P$ be a finite $R$-module.\nIf $\\Ext^1_R(P, M) = 0$ for every finite $R$-module $M$, then\n$P$ is projective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8T","source_file":"algebra.tex","source_line":18845,"source_end_line":18850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18845-L18850","statement_sha256":"3717c1413081989a2c0424d10b5b65c795064c334e7c9b2ad981311e23645f63","origin":"The Stacks Project","memory_eligible":false,"source_rank":1492,"rank":1492,"depth":0,"x":2083.422,"y":303.45,"cluster":"commutative-algebra"},{"id":"stacks:065Q","tag":"065Q","title":"Projective modules · Lemma 065Q","summary":"A direct sum of projective modules is projective.","statement_latex":"A direct sum of projective modules is projective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065Q","source_file":"algebra.tex","source_line":18863,"source_end_line":18866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18863-L18866","statement_sha256":"e186acd58a0dd01f9b6e1c88a39009238d48bf6dbfcf5d13401de3940cc8b52e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1493,"rank":1493,"depth":1,"x":1923.404,"y":188.75,"cluster":"commutative-algebra"},{"id":"stacks:07LV","tag":"07LV","title":"Projective modules · Lemma 07LV","summary":"Let R be a ring. Let I ⊂ R be a nilpotent ideal. Let overlineP be a projective R/I-module. Then there exists a projective R-module P such that P/IP ≅ overlineP.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be a nilpotent ideal. Let\n$\\overline{P}$ be a projective $R/I$-module. Then there exists a\nprojective $R$-module $P$ such that $P/IP \\cong \\overline{P}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LV","source_file":"algebra.tex","source_line":18874,"source_end_line":18879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18874-L18879","statement_sha256":"04f8aa5b7aea98cb7fe6a941f185fb2fd8c559283ca52a39990591071a1c529d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1494,"rank":1494,"depth":5,"x":2133.826,"y":182.525,"cluster":"commutative-algebra"},{"id":"stacks:0D47","tag":"0D47","title":"Projective modules · Lemma 0D47","summary":"Let R be a ring. Let I ⊂ R be a locally nilpotent ideal. Let overlineP be a finite projective R/I-module. Then there exists a finite projective R-module P such that P/IP ≅ overlineP.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal. Let\n$\\overline{P}$ be a finite projective $R/I$-module. Then there exists a\nfinite projective $R$-module $P$ such that $P/IP \\cong \\overline{P}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D47","source_file":"algebra.tex","source_line":18897,"source_end_line":18902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18897-L18902","statement_sha256":"ea41a590307987d26b4700e32de64dc7765e3c79e48af6f2b0f400607928e13e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1495,"rank":1495,"depth":5,"x":1983.536,"y":306.624,"cluster":"commutative-algebra"},{"id":"stacks:05CG","tag":"05CG","title":"Projective modules · Lemma 05CG","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. Assume • I is nilpotent, • M/IM is a projective R/I-module, • M is a flat R-module. Then M is a projective R-module.","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\nLet $M$ be an $R$-module.\nAssume\n\\begin{enumerate}\n\\item $I$ is nilpotent,\n\\item $M/IM$ is a projective $R/I$-module,\n\\item $M$ is a flat $R$-module.\n\\end{enumerate}\nThen $M$ is a projective $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CG","source_file":"algebra.tex","source_line":18923,"source_end_line":18935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18923-L18935","statement_sha256":"e5a5cdc13d7af06bf7743e9d2b6c92fc48e777eff7ced8921cdf5cac5f7cfc43","origin":"The Stacks Project","memory_eligible":false,"source_rank":1496,"rank":1496,"depth":6,"x":1994.57,"y":129.68,"cluster":"commutative-algebra"},{"id":"stacks:0H7M","tag":"0H7M","title":"Projective modules · Lemma 0H7M","summary":"Let R be a ring. Let I, J ⊂ R be ideals such that I ∩ J = 0. Let P be an R-module such that P/IP is a projective R/I-module and P/JP is a projective R/J-module. Then P is a projective R-module.","statement_latex":"Let $R$ be a ring. Let $I, J \\subset R$ be ideals such that $I \\cap J = 0$.\nLet $P$ be an $R$-module such that $P/IP$ is a projective $R/I$-module and\n$P/JP$ is a projective $R/J$-module. Then $P$ is a projective $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7M","source_file":"algebra.tex","source_line":18960,"source_end_line":18965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L18960-L18965","statement_sha256":"a7b31c2c69a6a76ad907a38f6183d0f968759008b21312a51203b2def0336ad2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1497,"rank":1497,"depth":0,"x":2128.847,"y":266.538,"cluster":"commutative-algebra"},{"id":"stacks:00NW","tag":"00NW","title":"Finite projective modules · Definition 00NW","summary":"Let R be a ring and M an R-module. • We say that M is locally free if we can cover Spec(R) by standard opens D(f_i), i ∈ I such that M_f_i is a free R_f_i-module for all i ∈ I. • We say that M is finite locally free if we can choose the covering such that each M_f_i is finite free. • We say that M is finite locally free of rank r if we can choose the covering such that each M_f_i is isomorphic to R_f_i^⊕ r.","statement_latex":"Let $R$ be a ring and $M$ an $R$-module.\n\\begin{enumerate}\n\\item We say that $M$ is {\\it locally free} if we can cover $\\Spec(R)$ by\nstandard opens $D(f_i)$, $i \\in I$ such that $M_{f_i}$ is a free\n$R_{f_i}$-module for all $i \\in I$.\n\\item We say that $M$ is {\\it finite locally free} if we can choose\nthe covering such that each $M_{f_i}$ is finite free.\n\\item We say that $M$ is {\\it finite locally free of rank $r$}\nif we can choose the covering such that each $M_{f_i}$ is isomorphic\nto $R_{f_i}^{\\oplus r}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite projective modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NW","source_file":"algebra.tex","source_line":19003,"source_end_line":19016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19003-L19016","statement_sha256":"582c5139202e3d9b109fc55c39bce1ed73e3e370fbdf07f64c5087fd2d64f921","origin":"The Stacks Project","memory_eligible":false,"source_rank":1498,"rank":1498,"depth":0,"x":1919.589,"y":241.791,"cluster":"commutative-algebra"},{"id":"stacks:00NX","tag":"00NX","title":"Finite projective modules · Lemma 00NX","summary":"Let R be a ring and let M be an R-module. The following are equivalent • M is finitely presented and R-flat, • M is finite projective, • M is a direct summand of a finite free R-module, • M is finitely presented and for all p ∈ Spec(R) the localization M_ p is free, • M is finitely presented and for all maximal ideals m ⊂ R the localization M_ m is free, • M is finite and locally free, • M is finite locally free, and • M is finite, for every prime p the module M_ p is…","statement_latex":"Let $R$ be a ring and let $M$ be an $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ is finitely presented and $R$-flat,\n\\item $M$ is finite projective,\n\\item $M$ is a direct summand of a finite free $R$-module,\n\\item $M$ is finitely presented and\nfor all $\\mathfrak p \\in \\Spec(R)$ the\nlocalization $M_{\\mathfrak p}$ is free,\n\\item $M$ is finitely presented and\nfor all maximal ideals $\\mathfrak m \\subset R$ the\nlocalization $M_{\\mathfrak m}$ is free,\n\\item $M$ is finite and locally free,\n\\item $M$ is finite locally free, and\n\\item $M$ is finite, for every prime $\\mathfrak p$ the module\n$M_{\\mathfrak p}$ is free, and the function\n$$\n\\rho_M : \\Spec(R) \\to \\mathbf{Z}, \\quad\n\\mathfrak p\n\\longmapsto\n\\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p)\n$$\nis locally constant in the Zariski topology.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NX","source_file":"algebra.tex","source_line":19026,"source_end_line":19052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19026-L19052","statement_sha256":"0a8969676b3496361a56cf53f7a23e6465b66f605331ce756c1003717faeee25","origin":"The Stacks Project","memory_eligible":false,"source_rank":1499,"rank":1499,"depth":4,"x":2093.948,"y":141.213,"cluster":"commutative-algebra"},{"id":"stacks:0FWG","tag":"0FWG","title":"Finite projective modules · Lemma 0FWG","summary":"Let R be a reduced ring and let M be an R-module. Then the equivalent conditions of Lemma [Tag 00NX] are also equivalent to • [(9)] M is finite and the function ρ_M : Spec(R) → Z, p ↦ dim_kappa( p) M ⊗_R kappa( p) is locally constant in the Zariski topology.","statement_latex":"Let $R$ be a reduced ring and let $M$ be an $R$-module. Then the\nequivalent conditions of Lemma \\ref{lemma-finite-projective}\nare also equivalent to\n\\begin{enumerate}\n\\item[(9)] $M$ is finite and the function\n$\\rho_M : \\Spec(R) \\to \\mathbf{Z}$, $\\mathfrak p \\mapsto\n\\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p)$\nis locally constant in the Zariski topology.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWG","source_file":"algebra.tex","source_line":19169,"source_end_line":19180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19169-L19180","statement_sha256":"5c56c96d933aa2ac1d113d6ea7b52d53dfa0e49ab330b39caf91615f6fb42d38","origin":"The Stacks Project","memory_eligible":false,"source_rank":1500,"rank":1500,"depth":5,"x":2046.219,"y":314.466,"cluster":"commutative-algebra"},{"id":"stacks:00NZ","tag":"00NZ","title":"Finite projective modules · Lemma 00NZ","summary":"(Warning: see Remark [Tag 00NY].) Suppose R is a local ring, and M is a finite flat R-module. Then M is finite free.","statement_latex":"(Warning: see Remark \\ref{remark-warning}.)\nSuppose $R$ is a local ring, and $M$ is a finite\nflat $R$-module. Then $M$ is finite free.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NZ","source_file":"algebra.tex","source_line":19222,"source_end_line":19227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19222-L19227","statement_sha256":"8c7f5f53c4ef75ae284408b9bed67b516128ba14a857034a9c8dff47e6f6d9c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1501,"rank":1501,"depth":3,"x":1941.997,"y":159.492,"cluster":"commutative-algebra"},{"id":"stacks:00O1","tag":"00O1","title":"Finite projective modules · Lemma 00O1","summary":"Let R → S be a flat local homomorphism of local rings. Let M be a finite R-module. Then M is finite projective over R if and only if M ⊗_R S is finite projective over S.","statement_latex":"Let $R \\to S$ be a flat local homomorphism of local rings.\nLet $M$ be a finite $R$-module. Then $M$ is finite projective\nover $R$ if and only if $M \\otimes_R S$ is finite projective\nover $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00O1","source_file":"algebra.tex","source_line":19268,"source_end_line":19274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19268-L19274","statement_sha256":"c790d38e2eabc208a5eacb47cb8ae78fca1a2f265142b697037f0bea0e0fff2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1502,"rank":1502,"depth":5,"x":2143.651,"y":214.678,"cluster":"commutative-algebra"},{"id":"stacks:02M9","tag":"02M9","title":"Finite projective modules · Lemma 02M9","summary":"Let R be a semi-local ring. Let M be a finite locally free module. If M has constant rank, then M is free. In particular, if R has connected spectrum, then M is free.","statement_latex":"Let $R$ be a semi-local ring.\nLet $M$ be a finite locally free module.\nIf $M$ has constant rank, then $M$ is free. In particular, if $R$ has\nconnected spectrum, then $M$ is free.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02M9","source_file":"algebra.tex","source_line":19290,"source_end_line":19296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19290-L19296","statement_sha256":"5e7b63d0b95ce39e243d03a9badcbc49e3a27789ba3d9babed40518f8fc4874c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1503,"rank":1503,"depth":0,"x":1950.406,"y":288.473,"cluster":"commutative-algebra"},{"id":"stacks:03C1","tag":"03C1","title":"Finite projective modules · Lemma 03C1","summary":"Let R be a local ring with maximal ideal m and infinite residue field. Let R → S be a ring map. Let M be an S-module and let N ⊂ M be an R-submodule. Assume • S is semi-local and mS is contained in the Jacobson radical of S, • M is a finite free S-module, and • N generates M as an S-module. Then N contains an S-basis of M.","statement_latex":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and\ninfinite residue field.\nLet $R \\to S$ be a ring map.\nLet $M$ be an $S$-module and let $N \\subset M$ be an $R$-submodule.\nAssume\n\\begin{enumerate}\n\\item $S$ is semi-local and $\\mathfrak mS$ is contained in the\nJacobson radical of $S$,\n\\item $M$ is a finite free $S$-module, and\n\\item $N$ generates $M$ as an $S$-module.\n\\end{enumerate}\nThen $N$ contains an $S$-basis of $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03C1","source_file":"algebra.tex","source_line":19310,"source_end_line":19324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19310-L19324","statement_sha256":"08ffd93d2243ff218bbcb84a6c12db9b5839ce67d7f3103ac46fabbcec2d3bda","origin":"The Stacks Project","memory_eligible":false,"source_rank":1504,"rank":1504,"depth":3,"x":2033.631,"y":124.261,"cluster":"commutative-algebra"},{"id":"stacks:0DVB","tag":"0DVB","title":"Finite projective modules · Lemma 0DVB","summary":"Let R be ring. Let L, M, N be R-modules. The canonical map Hom_R(M, N) ⊗_R L → Hom_R(M, N ⊗_R L) is an isomorphism if M is finite projective.","statement_latex":"Let $R$ be ring. Let $L$, $M$, $N$ be $R$-modules.\nThe canonical map\n$$\n\\Hom_R(M, N) \\otimes_R L \\to \\Hom_R(M, N \\otimes_R L)\n$$\nis an isomorphism if $M$ is finite projective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVB","source_file":"algebra.tex","source_line":19349,"source_end_line":19357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19349-L19357","statement_sha256":"0574c725c1736d44105db6fd00f0e00bf8e7f30c60552d183b610f58093d6d4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1505,"rank":1505,"depth":5,"x":2104.378,"y":292.72,"cluster":"commutative-algebra"},{"id":"stacks:05GE","tag":"05GE","title":"Open loci defined by module maps · Lemma 05GE","summary":"Let R be a ring. Let φ : M → N be a map of R-modules with N a finite R-module. Then we have the equality U & = ( p ⊂ R mid φ_ p : M_ p → N_ p is surjective) & = ( p ⊂ R mid φ ⊗ kappa( p) : M ⊗ kappa( p) → N ⊗ kappa( p) is surjective) and U is an open subset of Spec(R). Moreover, for any f ∈ R such that D(f) ⊂ U the map M_f → N_f is surjective.","statement_latex":"Let $R$ be a ring. Let $\\varphi : M \\to N$ be a map of $R$-modules\nwith $N$ a finite $R$-module. Then we have the equality\n\\begin{align*}\nU & = \\{\\mathfrak p \\subset R \\mid\n\\varphi_{\\mathfrak p} : M_{\\mathfrak p} \\to N_{\\mathfrak p}\n\\text{ is surjective}\\} \\\\\n& = \\{\\mathfrak p \\subset R \\mid\n\\varphi \\otimes \\kappa(\\mathfrak p) :\nM \\otimes \\kappa(\\mathfrak p) \\to N \\otimes \\kappa(\\mathfrak p)\n\\text{ is surjective}\\}\n\\end{align*}\nand $U$ is an open subset of $\\Spec(R)$. Moreover, for any $f \\in R$\nsuch that $D(f) \\subset U$ the map $M_f \\to N_f$ is surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Open loci defined by module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GE","source_file":"algebra.tex","source_line":19383,"source_end_line":19398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19383-L19398","statement_sha256":"9f1879482f5a2225cffa2f794af0778172c65559acb819d41cde52ba85d5809b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1506,"rank":1506,"depth":3,"x":1916.576,"y":208.571,"cluster":"commutative-algebra"},{"id":"stacks:05GF","tag":"05GF","title":"Open loci defined by module maps · Lemma 05GF","summary":"Let R be a ring. Let φ : M → N be a map of R-modules with M finite and N finitely presented. Then U = ( p ⊂ R mid φ_ p : M_ p → N_ p is an isomorphism) is an open subset of Spec(R).","statement_latex":"Let $R$ be a ring. Let $\\varphi : M \\to N$ be a map of $R$-modules\nwith $M$ finite and $N$ finitely presented. Then\n$$\nU = \\{\\mathfrak p \\subset R \\mid\n\\varphi_{\\mathfrak p} : M_{\\mathfrak p} \\to N_{\\mathfrak p}\n\\text{ is an isomorphism}\\}\n$$\nis an open subset of $\\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Open loci defined by module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GF","source_file":"algebra.tex","source_line":19408,"source_end_line":19418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19408-L19418","statement_sha256":"6edac35115884e280a957ce4915c8357fe096053d30c580d463b2f36c354522c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1507,"rank":1507,"depth":4,"x":2122.908,"y":164.019,"cluster":"commutative-algebra"},{"id":"stacks:0GWM","tag":"0GWM","title":"Open loci defined by module maps · Lemma 0GWM","summary":"Let R be a ring. Let p ⊂ R be a prime. Let M be a finitely presented R-module. If M_ p is free, then there is an f ∈ R, f not ∈ p such that M_f is a free R_f-module.","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime.\nLet $M$ be a finitely presented $R$-module. If $M_\\mathfrak p$\nis free, then there is an $f \\in R$, $f \\not \\in \\mathfrak p$\nsuch that $M_f$ is a free $R_f$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Open loci defined by module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWM","source_file":"algebra.tex","source_line":19446,"source_end_line":19452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19446-L19452","statement_sha256":"10f643e930348e077a5611fdb8dc689eecfecff3ead4978df5c92a34d9d5bf9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1508,"rank":1508,"depth":5,"x":2006.489,"y":314.08,"cluster":"commutative-algebra"},{"id":"stacks:00O0","tag":"00O0","title":"Open loci defined by module maps · Lemma 00O0","summary":"Let R be a ring. Let φ : P_1 → P_2 be a map of finite projective modules. Then • The set U of primes p ∈ Spec(R) such that φ ⊗ kappa( p) is injective is open and for any f∈ R such that D(f) ⊂ U we have • P_1, f → P_2, f is injective, and • the module Coker(φ)_f is finite projective over R_f. • The set W of primes p ∈ Spec(R) such that φ ⊗ kappa( p) is surjective is open and for any f∈ R such that D(f) ⊂ W we have • P_1, f → P_2, f is surjective, and • the module Ker(φ)_f…","statement_latex":"Let $R$ be a ring. Let $\\varphi : P_1 \\to P_2$ be a map of\nfinite projective modules. Then\n\\begin{enumerate}\n\\item The set $U$ of primes $\\mathfrak p \\in \\Spec(R)$ such that\n$\\varphi \\otimes \\kappa(\\mathfrak p)$ is injective is open and\nfor any $f\\in R$ such that $D(f) \\subset U$ we have\n\\begin{enumerate}\n\\item $P_{1, f} \\to P_{2, f}$ is injective, and\n\\item the module $\\Coker(\\varphi)_f$ is finite projective over $R_f$.\n\\end{enumerate}\n\\item The set $W$ of primes $\\mathfrak p \\in \\Spec(R)$ such that\n$\\varphi \\otimes \\kappa(\\mathfrak p)$ is surjective is open and\nfor any $f\\in R$ such that $D(f) \\subset W$ we have\n\\begin{enumerate}\n\\item $P_{1, f} \\to P_{2, f}$ is surjective, and\n\\item the module $\\Ker(\\varphi)_f$ is finite projective over $R_f$.\n\\end{enumerate}\n\\item The set $V$ of primes $\\mathfrak p \\in \\Spec(R)$ such that\n$\\varphi \\otimes \\kappa(\\mathfrak p)$ is an isomorphism is open and\nfor any $f\\in R$ such that $D(f) \\subset V$ the map\n$\\varphi : P_{1, f} \\to P_{2, f}$ is an isomorphism of modules over $R_f$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Open loci defined by module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00O0","source_file":"algebra.tex","source_line":19471,"source_end_line":19495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19471-L19495","statement_sha256":"596b646ded9a0cc29fcf2596742adf6d1bc921a9a16ef5978f88420c0ed31e6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1509,"rank":1509,"depth":5,"x":1971.63,"y":137.214,"cluster":"commutative-algebra"},{"id":"stacks:058D","tag":"058D","title":"Characterizing flatness · Lemma 058D","summary":"Let M be an R-module. The following are equivalent: • M is flat. • If f: R^n → M is a module map and x ∈ Ker(f), then there are module maps h: R^n → R^m and g: R^m → M such that f = g ∘ h and x ∈ Ker(h). • Suppose f: R^n → M is a module map, N ⊂ Ker(f) any submodule, and h: R^n → R^m a map such that N ⊂ Ker(h) and f factors through h. Then given any x ∈ Ker(f) we can find a map h': R^n → R^m' such that N + Rx ⊂ Ker(h') and f factors through h'. • If f: R^n → M is a module…","statement_latex":"Let $M$ be an $R$-module.  The following are equivalent:\n\\begin{enumerate}\n\\item $M$ is flat.\n\\item If $f: R^n \\to M$ is a module map and $x \\in \\Ker(f)$, then there\nare module maps $h: R^n \\to R^m$ and $g: R^m \\to M$ such that\n$f = g \\circ h$ and $x \\in \\Ker(h)$.\n\\item Suppose $f: R^n \\to M$ is a module map, $N \\subset \\Ker(f)$ any\nsubmodule, and $h: R^n \\to R^{m}$ a map such that $N \\subset \\Ker(h)$\nand $f$ factors through $h$.  Then given any $x \\in \\Ker(f)$ we can find a map\n$h': R^n \\to R^{m'}$ such that $N + Rx \\subset \\Ker(h')$ and $f$\nfactors through $h'$.\n\\item If $f: R^n \\to M$ is a module map and $N \\subset \\Ker(f)$ is a\nfinitely generated submodule, then there are module maps $h: R^n \\to\nR^m$ and $g: R^m \\to M$ such that $f = g \\circ h$ and $N \\subset\n\\Ker(h)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058D","source_file":"algebra.tex","source_line":19595,"source_end_line":19613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19595-L19613","statement_sha256":"ad4dcfcd2ca3e45d41411d423e0e2ab7286197556a1bdd48c3ecb8332a65ef6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1510,"rank":1510,"depth":1,"x":2139.71,"y":247.949,"cluster":"commutative-algebra"},{"id":"stacks:058E","tag":"058E","title":"Characterizing flatness · Lemma 058E","summary":"Let M be an R-module. Then M is flat if and only if the following condition holds: if P is a finitely presented R-module and f: P → M a module map, then there is a free finite R-module F and module maps h: P → F and g: F → M such that f = g ∘ h.","statement_latex":"Let $M$ be an $R$-module.  Then $M$ is flat if and only if the following\ncondition holds: if $P$ is a finitely presented $R$-module and $f: P\n\\to M$ a module map, then there is a free finite $R$-module $F$ and\nmodule maps $h: P \\to F$ and $g: F \\to M$ such that $f = g\n\\circ h$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058E","source_file":"algebra.tex","source_line":19635,"source_end_line":19642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19635-L19642","statement_sha256":"55c6c04994b940f942ea7fedc525427b8f5f7231ed7de0421878c0ba53343d99","origin":"The Stacks Project","memory_eligible":false,"source_rank":1511,"rank":1511,"depth":2,"x":1926.537,"y":261.679,"cluster":"commutative-algebra"},{"id":"stacks:058F","tag":"058F","title":"Characterizing flatness · Lemma 058F","summary":"Let M be an R-module. Then M is flat if and only if the following condition holds: for every finitely presented R-module P, if N → M is a surjective R-module map, then the induced map Hom_R(P, N) → Hom_R(P, M) is surjective.","statement_latex":"Let $M$ be an $R$-module.  Then $M$ is flat if and only if the following\ncondition holds: for every finitely presented $R$-module $P$, if $N \\to\nM$ is a surjective $R$-module map, then the induced map $\\Hom_R(P, N)\n\\to \\Hom_R(P, M)$ is surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058F","source_file":"algebra.tex","source_line":19649,"source_end_line":19655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19649-L19655","statement_sha256":"2fb834d0bbe4dbcd42c9895e8eddb65f7e635387385c45505e31215b83dfebd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1512,"rank":1512,"depth":3,"x":2072.812,"y":130.481,"cluster":"commutative-algebra"},{"id":"stacks:058G","tag":"058G","title":"Lazard's theorem · Theorem 058G","summary":"Let M be an R-module. Then M is flat if and only if it is the colimit of a directed system of free finite R-modules.","statement_latex":"Let $M$ be an $R$-module.  Then $M$ is flat if and only if it is the colimit of\na directed system of free finite $R$-modules.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing flatness","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058G","source_file":"algebra.tex","source_line":19675,"source_end_line":19679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19675-L19679","statement_sha256":"3a1b9e1274a0e85ff143b3614afe9009b731567ae2e2d6b5b9752e1f6e71f7f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1513,"rank":1513,"depth":3,"x":2070.454,"y":310.38,"cluster":"commutative-algebra"},{"id":"stacks:058I","tag":"058I","title":"Universally injective module maps · Definition 058I","summary":"Let f: M → N be a map of R-modules. Then f is called universally injective if for every R-module Q, the map f ⊗_R id_Q: M ⊗_R Q → N ⊗_R Q is injective. A sequence 0 → M_1 → M_2 → M_3 → 0 of R-modules is called universally exact if it is exact and M_1 → M_2 is universally injective.","statement_latex":"Let $f: M \\to N$ be a map of $R$-modules.  Then $f$ is called\n{\\it universally injective} if for every $R$-module $Q$, the map $f\n\\otimes_R \\text{id}_Q: M \\otimes_R Q \\to N \\otimes_R Q$\nis injective.  A sequence $0 \\to M_1 \\to M_2 \\to M_3\n\\to 0$ of $R$-modules is called {\\it universally exact} if it is exact\nand $M_1 \\to M_2$ is universally injective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058I","source_file":"algebra.tex","source_line":19751,"source_end_line":19759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19751-L19759","statement_sha256":"df6dc735bf3daea26758ca0fc9992d850294665915add652459e502e2c7466b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1514,"rank":1514,"depth":0,"x":1927.4,"y":176.272,"cluster":"commutative-algebra"},{"id":"stacks:058K","tag":"058K","title":"Universally injective module maps · Theorem 058K","summary":"Let 0 → M_1 xrightarrowf_1 M_2 xrightarrowf_2 M_3 → 0 be an exact sequence of R-modules. The following are equivalent: • The sequence 0 → M_1 → M_2 → M_3 → 0 is universally exact. • For every finitely presented R-module Q, the sequence 0 → M_1 ⊗_R Q → M_2 ⊗_R Q → M_3 ⊗_R Q → 0 is exact. • Given elements x_i ∈ M_1 (i = 1, …, n), y_j ∈ M_2 (j = 1, …, m), and a_ij ∈ R (i = 1, …, n, j = 1, …, m) such that for all i f_1(x_i) = ∑_j a_ij y_j, there exists z_j ∈ M_1 (j =1, …, m)…","statement_latex":"Let\n$$\n0 \\to M_1 \\xrightarrow{f_1} M_2 \\xrightarrow{f_2} M_3 \\to 0\n$$\nbe an exact sequence of $R$-modules.  The following are equivalent:\n\\begin{enumerate}\n\\item The sequence $0 \\to M_1 \\to M_2 \\to M_3\n\\to 0$ is universally exact.\n\\item For every finitely presented $R$-module $Q$, the sequence\n$$\n0 \\to M_1 \\otimes_R Q \\to M_2 \\otimes_R Q \\to\nM_3 \\otimes_R Q \\to 0\n$$\nis exact.\n\\item Given elements $x_i \\in M_1$ $(i = 1, \\ldots, n)$, $y_j \\in M_2$ $(j = 1,\n\\ldots, m)$, and $a_{ij} \\in R$ $(i = 1, \\ldots, n, j = 1, \\ldots, m)$ such that\nfor all $i$\n$$\nf_1(x_i) = \\sum\\nolimits_j a_{ij} y_j,\n$$\nthere exists $z_j \\in M_1$ $(j =1, \\ldots, m)$ such that for all $i$,\n$$\nx_i = \\sum\\nolimits_j a_{ij} z_j .\n$$\n\\item Given a commutative diagram of $R$-module maps\n$$\n\\xymatrix{\nR^n \\ar[r] \\ar[d] &  R^m \\ar[d] \\\\\nM_1 \\ar[r]^{f_1}        &  M_2\n}\n$$\nwhere $m$ and $n$ are integers, there exists a map $R^m \\to M_1$ making\nthe top triangle commute.\n\\item For every finitely presented $R$-module $P$, the $R$-module\nmap $\\Hom_R(P, M_2) \\to \\Hom_R(P, M_3)$ is surjective.\n\\item The sequence $0 \\to M_1 \\to M_2 \\to M_3\n\\to 0$ is the colimit of a directed system of split exact sequences of\nthe form\n$$\n0 \\to M_{1} \\to M_{2, i} \\to M_{3, i} \\to 0\n$$\nwhere the $M_{3, i}$ are finitely presented.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058K","source_file":"algebra.tex","source_line":19784,"source_end_line":19829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19784-L19829","statement_sha256":"576b787c106a4f590c4321e355397ef4e68e070319e28c94a9300f696cd76ae0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1515,"rank":1515,"depth":1,"x":2140.915,"y":194.004,"cluster":"commutative-algebra"},{"id":"stacks:058L","tag":"058L","title":"Universally injective module maps · Lemma 058L","summary":"Let 0 → M_1 → M_2 → M_3 → 0 be an exact sequence of R-modules. Suppose M_3 is of finite presentation. Then 0 → M_1 → M_2 → M_3 → 0 is universally exact if and only if it is split.","statement_latex":"Let\n$$\n0 \\to M_1 \\to M_2 \\to M_3 \\to 0\n$$\nbe an exact sequence of $R$-modules.  Suppose $M_3$ is of finite presentation.\nThen\n$$\n0 \\to M_1 \\to M_2 \\to M_3 \\to 0\n$$\nis universally exact if and only if it is split.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058L","source_file":"algebra.tex","source_line":19954,"source_end_line":19966,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19954-L19966","statement_sha256":"b6deadd57102ba267a5b3b5d1109a495a9eb0c7dd7af105f68450657310c7773","origin":"The Stacks Project","memory_eligible":false,"source_rank":1516,"rank":1516,"depth":2,"x":1969.066,"y":302.175,"cluster":"commutative-algebra"},{"id":"stacks:058M","tag":"058M","title":"Universally injective module maps · Lemma 058M","summary":"Let M be an R-module. Then M is flat if and only if any exact sequence of R-modules 0 → M_1 → M_2 → M → 0 is universally exact.","statement_latex":"Let $M$ be an $R$-module.  Then $M$ is flat if and only if any exact sequence\nof $R$-modules\n$$\n0 \\to M_1 \\to M_2 \\to M \\to 0\n$$\nis universally exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058M","source_file":"algebra.tex","source_line":19980,"source_end_line":19988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L19980-L19988","statement_sha256":"d72cf37f608571cfdf16cf2f4aae1b5a0fb807ed6b57f1f87b6d95881a5ef61b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1517,"rank":1517,"depth":4,"x":2008.831,"y":124.748,"cluster":"commutative-algebra"},{"id":"stacks:058P","tag":"058P","title":"Universally injective module maps · Lemma 058P","summary":"Let 0 → M_1 → M_2 → M_3 → 0 be a universally exact sequence of R-modules, and suppose M_2 is flat. Then M_1 and M_3 are flat.","statement_latex":"Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a universally exact sequence\nof $R$-modules, and suppose $M_2$ is flat.\nThen $M_1$ and $M_3$ are flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058P","source_file":"algebra.tex","source_line":20040,"source_end_line":20045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20040-L20045","statement_sha256":"9996e8d157620f7c8d1cf6c0d09118dbc236d0cb909a7cd07c39325be6f92200","origin":"The Stacks Project","memory_eligible":false,"source_rank":1518,"rank":1518,"depth":0,"x":2122.288,"y":278.275,"cluster":"commutative-algebra"},{"id":"stacks:05CH","tag":"05CH","title":"Universally injective module maps · Lemma 05CH","summary":"Let R be a ring. Let M → M' be a universally injective R-module map. Then for any R-module N the map M ⊗_R N → M' ⊗_R N is universally injective.","statement_latex":"Let $R$ be a ring.\nLet $M \\to M'$ be a universally injective $R$-module map.\nThen for any $R$-module $N$ the map $M \\otimes_R N \\to M' \\otimes_R N$\nis universally injective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CH","source_file":"algebra.tex","source_line":20072,"source_end_line":20078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20072-L20078","statement_sha256":"d59421276f3832508b131e2e927b56ea433dc423a671bc34a15ce7a274dd91bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1519,"rank":1519,"depth":0,"x":1914.987,"y":229.403,"cluster":"commutative-algebra"},{"id":"stacks:05CI","tag":"05CI","title":"Universally injective module maps · Lemma 05CI","summary":"Let R be a ring. A composition of universally injective R-module maps is universally injective.","statement_latex":"Let $R$ be a ring. A composition of universally injective\n$R$-module maps is universally injective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CI","source_file":"algebra.tex","source_line":20084,"source_end_line":20088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20084-L20088","statement_sha256":"f0bcce74e4219ae336c95e869b73079c87d3958efcc0fee5d7fa1ed98de03a12","origin":"The Stacks Project","memory_eligible":false,"source_rank":1520,"rank":1520,"depth":0,"x":2107.313,"y":147.744,"cluster":"commutative-algebra"},{"id":"stacks:05CJ","tag":"05CJ","title":"Universally injective module maps · Lemma 05CJ","summary":"Let R be a ring. Let M → M' and M' → M\" be R-module maps. If their composition M → M\" is universally injective, then M → M' is universally injective.","statement_latex":"Let $R$ be a ring. Let $M \\to M'$ and $M' \\to M''$ be $R$-module maps.\nIf their composition $M \\to M''$ is universally injective, then\n$M \\to M'$ is universally injective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CJ","source_file":"algebra.tex","source_line":20094,"source_end_line":20099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20094-L20099","statement_sha256":"b9dcebd5bb110d86b8ca3c30bc76862d637bbf3d1f42a372bcd1dfb5f6d8b825","origin":"The Stacks Project","memory_eligible":false,"source_rank":1521,"rank":1521,"depth":0,"x":2031.098,"y":317.232,"cluster":"commutative-algebra"},{"id":"stacks:05CL","tag":"05CL","title":"Universally injective module maps · Lemma 05CL","summary":"Let R → S be a ring map. Let M → M' be a map of S-modules. The following are equivalent • M → M' is universally injective as a map of R-modules, • for each prime q of S the map M_ q → M'_ q is universally injective as a map of R-modules, • for each maximal ideal m of S the map M_ m → M'_ m is universally injective as a map of R-modules, • for each prime q of S the map M_ q → M'_ q is universally injective as a map of R_ p-modules, where p is the inverse image of q in R,…","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M \\to M'$ be a map of $S$-modules.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M \\to M'$ is universally injective as a map of $R$-modules,\n\\item for each prime $\\mathfrak q$ of $S$ the map\n$M_{\\mathfrak q} \\to M'_{\\mathfrak q}$ is universally injective\nas a map of $R$-modules,\n\\item for each maximal ideal $\\mathfrak m$ of $S$ the map\n$M_{\\mathfrak m} \\to M'_{\\mathfrak m}$ is universally injective\nas a map of $R$-modules,\n\\item for each prime $\\mathfrak q$ of $S$ the map\n$M_{\\mathfrak q} \\to M'_{\\mathfrak q}$ is universally injective\nas a map of $R_{\\mathfrak p}$-modules, where $\\mathfrak p$ is the\ninverse image of $\\mathfrak q$ in $R$, and\n\\item for each maximal ideal $\\mathfrak m$ of $S$ the map\n$M_{\\mathfrak m} \\to M'_{\\mathfrak m}$ is universally injective\nas a map of $R_{\\mathfrak p}$-modules, where $\\mathfrak p$ is the\ninverse image of $\\mathfrak m$ in $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CL","source_file":"algebra.tex","source_line":20105,"source_end_line":20127,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20105-L20127","statement_sha256":"0e8de2c97ccf35bcd9e20373349f2a1d1cc83b96294e0ed7df72a16acf544eee","origin":"The Stacks Project","memory_eligible":false,"source_rank":1522,"rank":1522,"depth":2,"x":1950.932,"y":148.865,"cluster":"commutative-algebra"},{"id":"stacks:05CM","tag":"05CM","title":"Universally injective module maps · Lemma 05CM","summary":"Let φ : A → B be a ring map. Let S ⊂ A and S' ⊂ B be multiplicative subsets such that φ(S) ⊂ S'. Let M → M' be a map of B-modules. • If M → M' is universally injective as a map of A-modules, then (S')^-1M → (S')^-1M' is universally injective as a map of A-modules and as a map of S^-1A-modules. • If M and M' are (S')^-1B-modules, then M → M' is universally injective as a map of A-modules if and only if it is universally injective as a map of S^-1A-modules.","statement_latex":"Let $\\varphi : A \\to B$ be a ring map. Let $S \\subset A$ and\n$S' \\subset B$ be multiplicative subsets such that $\\varphi(S) \\subset S'$.\nLet $M \\to M'$ be a map of $B$-modules.\n\\begin{enumerate}\n\\item If $M \\to M'$ is universally injective as a map of $A$-modules,\nthen $(S')^{-1}M \\to (S')^{-1}M'$ is universally injective as a map of\n$A$-modules and as a map of $S^{-1}A$-modules.\n\\item If $M$ and $M'$ are $(S')^{-1}B$-modules, then $M \\to M'$\nis universally injective as a map of $A$-modules if and only if\nit is universally injective as a map of $S^{-1}A$-modules.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CM","source_file":"algebra.tex","source_line":20160,"source_end_line":20173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20160-L20173","statement_sha256":"3205f2dafdef81e1392d18a33478d5ac65fec0e1adfe68cdb5eaf20ae0649ce9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1523,"rank":1523,"depth":3,"x":2145.606,"y":227.595,"cluster":"commutative-algebra"},{"id":"stacks:0AS5","tag":"0AS5","title":"Universally injective module maps · Lemma 0AS5","summary":"Let R be a ring and let M → M' be a map of R-modules. If M' is flat, then M → M' is universally injective if and only if M/IM → M'/IM' is injective for every finitely generated ideal I of R.","statement_latex":"Let $R$ be a ring and let $M \\to M'$ be a map of $R$-modules.\nIf $M'$ is flat, then $M \\to M'$ is universally injective if\nand only if $M/IM \\to M'/IM'$ is injective for every finitely\ngenerated ideal $I$ of $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AS5","source_file":"algebra.tex","source_line":20196,"source_end_line":20202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20196-L20202","statement_sha256":"f483dfc0fa0d1835ecc7913bc3c6c0220b45c257b80f153d280a0ba351a8f250","origin":"The Stacks Project","memory_eligible":false,"source_rank":1524,"rank":1524,"depth":2,"x":1938.569,"y":280.048,"cluster":"commutative-algebra"},{"id":"stacks:0H9I","tag":"0H9I","title":"Universally injective module maps · Lemma 0H9I","summary":"Let R → S be a ring map which is universally injective as a map of R-modules. Then the functor M ↦ M ⊗_R S on R-modules reflects injections, surjections, and isomorphisms.","statement_latex":"Let $R \\to S$ be a ring map which is universally injective\nas a map of $R$-modules. Then the functor $M \\mapsto M \\otimes_R S$\non $R$-modules reflects injections, surjections, and isomorphisms.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9I","source_file":"algebra.tex","source_line":20236,"source_end_line":20241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20236-L20241","statement_sha256":"ed590fa123370d1c41f5ec1fe154ebb72e0507d0bee10438aaef586d101512d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1525,"rank":1525,"depth":0,"x":2049.147,"y":123.76,"cluster":"commutative-algebra"},{"id":"stacks:05CK","tag":"05CK","title":"Universally injective module maps · Lemma 05CK","summary":"Let R → S be a faithfully flat ring map. Then R → S is universally injective as a map of R-modules. In particular R ∩ IS = I for any ideal I ⊂ R.","statement_latex":"Let $R \\to S$ be a faithfully flat ring map.\nThen $R \\to S$ is universally injective as a map of $R$-modules.\nIn particular $R \\cap IS = I$ for any ideal $I \\subset R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Universally injective module maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CK","source_file":"algebra.tex","source_line":20254,"source_end_line":20259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20254-L20259","statement_sha256":"c4ee2e00ba2f24f0986eb2ce3b19ecca7db7b839b1a26a396fc2fde2ea5bfc26","origin":"The Stacks Project","memory_eligible":false,"source_rank":1526,"rank":1526,"depth":0,"x":2093.328,"y":301.899,"cluster":"commutative-algebra"},{"id":"stacks:058R","tag":"058R","title":"Descent for finite projective modules · Lemma 058R","summary":"Let M be an R-module. Then M is finite projective if and only if M is finitely presented and flat.","statement_latex":"Let $M$ be an $R$-module.  Then $M$ is finite projective if and only if $M$ is\nfinitely presented and flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descent for finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058R","source_file":"algebra.tex","source_line":20285,"source_end_line":20289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20285-L20289","statement_sha256":"5c4e58c0b38e17ccbdb6594af47c5b62d2eb612d4d4fc40338f71180248b1462","origin":"The Stacks Project","memory_eligible":false,"source_rank":1527,"rank":1527,"depth":5,"x":1917.347,"y":195.522,"cluster":"commutative-algebra"},{"id":"stacks:03C4","tag":"03C4","title":"Descent for finite projective modules · Lemma 03C4","summary":"Let R → S be a faithfully flat ring map. Let M be an R-module. Then • if the S-module M ⊗_R S is of finite type, then M is of finite type, • if the S-module M ⊗_R S is of finite presentation, then M is of finite presentation, • if the S-module M ⊗_R S is flat, then M is flat, and • add more here as needed.","statement_latex":"Let $R \\to S$ be a faithfully flat ring map.\nLet $M$ be an $R$-module. Then\n\\begin{enumerate}\n\\item if the $S$-module $M \\otimes_R S$ is of finite type, then\n$M$ is of finite type,\n\\item if the $S$-module $M \\otimes_R S$ is of finite presentation, then\n$M$ is of finite presentation,\n\\item if the $S$-module $M \\otimes_R S$ is flat, then\n$M$ is flat, and\n\\item add more here as needed.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descent for finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03C4","source_file":"algebra.tex","source_line":20306,"source_end_line":20319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20306-L20319","statement_sha256":"11e79bb72f72272b0c4b2da408122149b3e7db7cf2fc10f2a5f791ed891f9567","origin":"The Stacks Project","memory_eligible":false,"source_rank":1528,"rank":1528,"depth":2,"x":2132.839,"y":174.089,"cluster":"commutative-algebra"},{"id":"stacks:058S","tag":"058S","title":"Descent for finite projective modules · Proposition 058S","summary":"Let R → S be a faithfully flat ring map. Let M be an R-module. If the S-module M ⊗_R S is finite projective, then M is finite projective.","statement_latex":"Let $R \\to S$ be a faithfully flat ring map.  Let $M$ be an $R$-module.\nIf the $S$-module $M \\otimes_R S$ is finite projective, then $M$ is finite\nprojective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descent for finite projective modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058S","source_file":"algebra.tex","source_line":20345,"source_end_line":20350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20345-L20350","statement_sha256":"db6de7215ea25b4ffd275610b065fce3f4947a8a85ddad504e0a5807a4882ae9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1529,"rank":1529,"depth":6,"x":1991.055,"y":312.285,"cluster":"commutative-algebra"},{"id":"stacks:058U","tag":"058U","title":"Transfinite dévissage of modules · Definition 058U","summary":"Let M be an R-module. A direct sum dévissage of M is a family of submodules (M_α)_α ∈ S, indexed by an ordinal S and increasing (with respect to inclusion), such that: • [(0)] M_0 = 0; • [(1)] M = ⋃_α M_α; • [(2)] if α ∈ S is a limit ordinal, then M_α = ⋃_β < α M_β; • [(3)] if α + 1 ∈ S, then M_α is a direct summand of M_α + 1. If moreover • [(4)] M_α + 1/M_α is countably generated for α + 1 ∈ S, then (M_α)_α ∈ S is called a Kaplansky dévissage of M.","statement_latex":"Let $M$ be an $R$-module.  A {\\it direct sum d\\'evissage} of $M$ is a family\nof submodules $(M_{\\alpha})_{\\alpha \\in S}$, indexed by an ordinal $S$ and\nincreasing (with respect to inclusion), such that:\n\\begin{enumerate}\n\\item[(0)] $M_0 = 0$;\n\\item[(1)] $M = \\bigcup_{\\alpha} M_{\\alpha}$;\n\\item[(2)] if $\\alpha \\in S$ is a limit ordinal, then $M_{\\alpha} =\n\\bigcup_{\\beta < \\alpha} M_{\\beta}$;\n\\item[(3)] if $\\alpha + 1 \\in S$, then $M_{\\alpha}$ is a direct summand of\n$M_{\\alpha + 1}$.\n\\end{enumerate}\nIf moreover\n\\begin{enumerate}\n\\item[(4)] $M_{\\alpha + 1}/M_{\\alpha}$ is countably generated for\n$\\alpha + 1 \\in S$,\n\\end{enumerate}\nthen $(M_{\\alpha})_{\\alpha \\in S}$ is called a {\\it Kaplansky d\\'evissage}\nof $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Transfinite dévissage of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058U","source_file":"algebra.tex","source_line":20384,"source_end_line":20404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20384-L20404","statement_sha256":"dfa1f3bb487dd404f1016221bee7ffc07e1a641d96782cc071400757f18a4d4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1530,"rank":1530,"depth":0,"x":1984.466,"y":129.777,"cluster":"commutative-algebra"},{"id":"stacks:058V","tag":"058V","title":"Transfinite dévissage of modules · Lemma 058V","summary":"Let M be an R-module. If (M_α)_α ∈ S is a direct sum dévissage of M, then M ≅ bigoplus_α + 1 ∈ S M_α + 1/M_α.","statement_latex":"Let $M$ be an $R$-module.  If $(M_{\\alpha})_{\\alpha \\in S}$ is a direct sum\nd\\'evissage of $M$, then\n$M \\cong \\bigoplus_{\\alpha + 1 \\in S} M_{\\alpha + 1}/M_{\\alpha}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Transfinite dévissage of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058V","source_file":"algebra.tex","source_line":20409,"source_end_line":20414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20409-L20414","statement_sha256":"932cdcd38220ff7a636452eea2f687cf19040ac84f07fe715609d00aa7416e2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1531,"rank":1531,"depth":0,"x":2136.219,"y":260.726,"cluster":"commutative-algebra"},{"id":"stacks:058W","tag":"058W","title":"Transfinite dévissage of modules · Lemma 058W","summary":"Let M be an R-module. Then M is a direct sum of countably generated R-modules if and only if it admits a Kaplansky dévissage.","statement_latex":"Let $M$ be an $R$-module.  Then $M$ is a direct sum of countably generated\n$R$-modules if and only if it admits a Kaplansky d\\'evissage.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Transfinite dévissage of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058W","source_file":"algebra.tex","source_line":20455,"source_end_line":20459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20455-L20459","statement_sha256":"e52dbd561649f9ccac44abed655d2154cab8ae5799fe1f99fab1993759263620","origin":"The Stacks Project","memory_eligible":false,"source_rank":1532,"rank":1532,"depth":0,"x":1918.832,"y":250.266,"cluster":"commutative-algebra"},{"id":"stacks:058X","tag":"058X","title":"Transfinite dévissage of modules · Theorem 058X","summary":"Suppose M is a direct sum of countably generated R-modules. If P is a direct summand of M, then P is also a direct sum of countably generated R-modules.","statement_latex":"Suppose $M$ is a direct sum of countably generated $R$-modules.  If $P$ is a\ndirect summand of $M$, then $P$ is also a direct sum of countably generated\n$R$-modules.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Transfinite dévissage of modules","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058X","source_file":"algebra.tex","source_line":20469,"source_end_line":20474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20469-L20474","statement_sha256":"9df7cb82b3633fe73cdbd6e0d894e56755a46230d3eb8440e55469c4b72c0c78","origin":"The Stacks Project","memory_eligible":false,"source_rank":1533,"rank":1533,"depth":0,"x":2087.683,"y":134.532,"cluster":"commutative-algebra"},{"id":"stacks:058Y","tag":"058Y","title":"Transfinite dévissage of modules · Theorem 058Y","summary":"Any projective module is a direct sum of countably generated projective modules. If P is a projective R-module, then P is a direct sum of countably generated projective R-modules.","statement_latex":"\\begin{slogan}\nAny projective module is a direct sum of countably generated\nprojective modules.\n\\end{slogan}\nIf $P$ is a projective $R$-module, then $P$ is a direct sum of countably\ngenerated projective $R$-modules.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Transfinite dévissage of modules","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/058Y","source_file":"algebra.tex","source_line":20572,"source_end_line":20580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20572-L20580","statement_sha256":"f7461aff68d2634530db608fcd46b8103a5f836f0d9b55dec102ae16b5458825","origin":"The Stacks Project","memory_eligible":false,"source_rank":1534,"rank":1534,"depth":1,"x":2056.218,"y":315.833,"cluster":"commutative-algebra"},{"id":"stacks:0590","tag":"0590","title":"Projective modules over a local ring · Lemma 0590","summary":"Let R be a ring. Then every projective R-module is free if and only if every countably generated projective R-module is free.","statement_latex":"Let $R$ be a ring.  Then every projective $R$-module is free if and only if\nevery countably generated projective $R$-module is free.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules over a local ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0590","source_file":"algebra.tex","source_line":20600,"source_end_line":20604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20600-L20604","statement_sha256":"9afef1f573a5e009633f31b87779470f9db4d3b035fa45e88f92319aa04d4412","origin":"The Stacks Project","memory_eligible":false,"source_rank":1535,"rank":1535,"depth":2,"x":1933.521,"y":164.165,"cluster":"commutative-algebra"},{"id":"stacks:0591","tag":"0591","title":"Projective modules over a local ring · Lemma 0591","summary":"Let M be a countably generated R-module with the following property: if M = N ⊕ N' with N' a finite free R-module, then any element of N is contained in a free direct summand of N. Then M is free.","statement_latex":"Let $M$ be a countably generated $R$-module with the following property:\nif $M = N \\oplus N'$ with $N'$ a finite free $R$-module, then\nany element of $N$ is contained in a free direct summand of $N$.\nThen $M$ is free.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules over a local ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0591","source_file":"algebra.tex","source_line":20614,"source_end_line":20620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20614-L20620","statement_sha256":"600e841ca08363ef4cd456a2a206c7838c8d6e88fd4ddd824257ff006376ba27","origin":"The Stacks Project","memory_eligible":false,"source_rank":1536,"rank":1536,"depth":0,"x":2146.139,"y":206.416,"cluster":"commutative-algebra"},{"id":"stacks:0592","tag":"0592","title":"Projective modules over a local ring · Lemma 0592","summary":"Let P be a projective module over a local ring R. Then any element of P is contained in a free direct summand of P.","statement_latex":"Let $P$ be a projective module over a local ring $R$.  Then any element of $P$\nis contained in a free direct summand of $P$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules over a local ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0592","source_file":"algebra.tex","source_line":20640,"source_end_line":20644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20640-L20644","statement_sha256":"2a8200bde8a55d5cb0623c303439a1308ec60f8ba178b85cf2e499dd3a6ba251","origin":"The Stacks Project","memory_eligible":false,"source_rank":1537,"rank":1537,"depth":0,"x":1955.223,"y":295.979,"cluster":"commutative-algebra"},{"id":"stacks:0593","tag":"0593","title":"Projective modules over a local ring · Theorem 0593","summary":"Projective modules over local rings are free. If P is a projective module over a local ring R, then P is free.","statement_latex":"\\begin{slogan}\nProjective modules over local rings are free.\n\\end{slogan}\nIf $P$ is a projective module over a local ring $R$, then $P$ is free.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Projective modules over a local ring","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0593","source_file":"algebra.tex","source_line":20682,"source_end_line":20688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20682-L20688","statement_sha256":"99f7322b81ebe7ee8b604fda723f08549235ff1220e151546d634c72bcf32f8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1538,"rank":1538,"depth":3,"x":2024.034,"y":121.463,"cluster":"commutative-algebra"},{"id":"stacks:0595","tag":"0595","title":"Mittag-Leffler systems · Definition 0595","summary":"Let (A_i, φ_ji) be a directed inverse system of sets over I. Then we say (A_i, φ_ji) is Mittag-Leffler if for each i ∈ I, the family φ_ji(A_j) ⊂ A_i for j ≥ i stabilizes. Explicitly, this means that for each i ∈ I, there exists j ≥ i such that for k ≥ j we have φ_ki(A_k) = φ_ji( A_j). If (A_i, φ_ji) is a directed inverse system of modules over a ring R, we say that it is Mittag-Leffler if the underlying inverse system of sets is Mittag-Leffler.","statement_latex":"Let $(A_i, \\varphi_{ji})$ be a directed inverse system of sets over $I$.  Then\nwe say  $(A_i, \\varphi_{ji})$ is {\\it Mittag-Leffler} if for\neach $i \\in I$, the family $\\varphi_{ji}(A_j) \\subset A_i$ for\n$j \\geq i$ stabilizes.  Explicitly, this means that for each $i \\in I$, there\nexists $j \\geq i$ such that for $k \\geq j$ we have $\\varphi_{ki}(A_k) =\n\\varphi_{ji}( A_j)$.  If $(A_i, \\varphi_{ji})$ is a directed inverse system\nof modules over a ring $R$, we say that it is Mittag-Leffler if the underlying\ninverse system of sets is Mittag-Leffler.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler systems","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0595","source_file":"algebra.tex","source_line":20723,"source_end_line":20733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20723-L20733","statement_sha256":"ebdfa3dcb6c08fafd18c551305d33310212a914fa7b6b41304018cc244ccf159","origin":"The Stacks Project","memory_eligible":false,"source_rank":1539,"rank":1539,"depth":0,"x":2113.709,"y":289.332,"cluster":"commutative-algebra"},{"id":"stacks:0597","tag":"0597","title":"Mittag-Leffler systems · Lemma 0597","summary":"Let (A_i, φ_ji) be a directed inverse system over I. Suppose I is countable. If (A_i, φ_ji) is Mittag-Leffler and the A_i are nonempty, then lim A_i is nonempty.","statement_latex":"Let $(A_i, \\varphi_{ji})$ be a directed inverse system over $I$.  Suppose $I$\nis countable.  If $(A_i, \\varphi_{ji})$ is Mittag-Leffler and the $A_i$ are\nnonempty, then $\\lim A_i$ is nonempty.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0597","source_file":"algebra.tex","source_line":20747,"source_end_line":20752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20747-L20752","statement_sha256":"df43cbee70a1a3c8dbc055476842c00aaea85d98b777c461f6d7c51ff3156cb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1540,"rank":1540,"depth":0,"x":1912.423,"y":216.371,"cluster":"commutative-algebra"},{"id":"stacks:0598","tag":"0598","title":"Mittag-Leffler systems · Lemma 0598","summary":"Let 0 → A_i xrightarrowf_i B_i xrightarrowg_i C_i → 0 be an exact sequence of directed inverse systems of abelian groups over I. Suppose I is countable. If (A_i) is Mittag-Leffler, then 0 → lim A_i → lim B_i → lim C_i→ 0 is exact.","statement_latex":"Let\n$$\n0 \\to A_i \\xrightarrow{f_i} B_i \\xrightarrow{g_i} C_i \\to 0\n$$\nbe an exact sequence of directed inverse systems of abelian groups over $I$.\nSuppose $I$ is countable.  If $(A_i)$ is Mittag-Leffler, then\n$$\n0 \\to \\lim A_i \\to \\lim B_i \\to \\lim C_i\\to 0\n$$\nis exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0598","source_file":"algebra.tex","source_line":20769,"source_end_line":20781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20769-L20781","statement_sha256":"dd3ec2c210886acd9e526160c51a96da0c5a417ee8bf3c95b03f3a2fab4fa637","origin":"The Stacks Project","memory_eligible":false,"source_rank":1541,"rank":1541,"depth":1,"x":2119.691,"y":155.907,"cluster":"commutative-algebra"},{"id":"stacks:03CA","tag":"03CA","title":"Inverse systems · Lemma 03CA","summary":"Let R be a ring. Let 0 → K_i → L_i → M_i → 0 be short exact sequences of R-modules, i ≥ 1 which fit into maps of short exact sequences xymatrix 0 ar[r] & K_i ar[r] & L_i ar[r] & M_i ar[r] & 0 0 ar[r] & K_i + 1 ar[r] ar[u] & L_i + 1 ar[r] ar[u] & M_i + 1 ar[r] ar[u] & 0 If for every i there exists a c = c(i) ≥ i such that Im(K_c → K_i) = Im(K_j → K_i) for all j ≥ c, then the sequence 0 → lim K_i → lim L_i → lim M_i → 0 is exact.","statement_latex":"Let $R$ be a ring.\nLet $0 \\to K_i \\to L_i \\to M_i \\to 0$ be short exact sequences of\n$R$-modules, $i \\geq 1$ which fit into maps of short exact sequences\n$$\n\\xymatrix{\n0 \\ar[r] &\nK_i \\ar[r] &\nL_i \\ar[r] &\nM_i \\ar[r] &\n0 \\\\\n0 \\ar[r] &\nK_{i + 1} \\ar[r] \\ar[u] &\nL_{i + 1} \\ar[r] \\ar[u] &\nM_{i + 1} \\ar[r] \\ar[u] &\n0}\n$$\nIf for every $i$ there exists a $c = c(i) \\geq i$ such that\n$\\Im(K_c \\to K_i) = \\Im(K_j \\to K_i)$\nfor all $j \\geq c$, then the sequence\n$$\n0 \\to \\lim K_i \\to \\lim L_i \\to \\lim M_i \\to 0\n$$\nis exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Inverse systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CA","source_file":"algebra.tex","source_line":20847,"source_end_line":20872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20847-L20872","statement_sha256":"42778875abddfc2a81609061bf0604ac95c904aa7a207c5995226a3db9541ac1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1542,"rank":1542,"depth":2,"x":2015.393,"y":318.235,"cluster":"commutative-algebra"},{"id":"stacks:059A","tag":"059A","title":"Mittag-Leffler modules · Definition 059A","summary":"Let (M_i, f_ij) be a directed system of R-modules. We say that (M_i, f_ij) is a Mittag-Leffler directed system of modules if each M_i is an R-module of finite presentation and if for every R-module N, the inverse system (Hom_R(M_i, N), Hom_R(f_ij, N)) is Mittag-Leffler.","statement_latex":"Let $(M_i, f_{ij})$ be a directed system of $R$-modules.  We say that\n$(M_i, f_{ij})$ is a {\\it Mittag-Leffler directed system of modules} if each\n$M_i$ is an $R$-module of finite presentation and if for every $R$-module $N$,\nthe inverse system\n$$\n(\\Hom_R(M_i, N), \\Hom_R(f_{ij}, N))\n$$\nis Mittag-Leffler.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059A","source_file":"algebra.tex","source_line":20889,"source_end_line":20899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20889-L20899","statement_sha256":"2ec2bb3f4bc7456f84790db12cb3294d0941c2d20efa86a9c770e86478c35c55","origin":"The Stacks Project","memory_eligible":false,"source_rank":1543,"rank":1543,"depth":0,"x":1961.718,"y":139.209,"cluster":"commutative-algebra"},{"id":"stacks:059B","tag":"059B","title":"Mittag-Leffler modules · Definition 059B","summary":"Let f: M → N and g: M → M' be maps of R-modules. Then we say g dominates f if for any R-module Q, we have Ker(f ⊗_R id_Q) ⊂ Ker(g ⊗_R id_Q).","statement_latex":"Let $f: M \\to N$ and $g: M \\to M'$ be maps of $R$-modules.\nThen we say $g$ {\\it dominates} $f$ if for any $R$-module $Q$, we have $\\Ker(f\n\\otimes_R \\text{id}_Q) \\subset \\Ker(g \\otimes_R \\text{id}_Q)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059B","source_file":"algebra.tex","source_line":20905,"source_end_line":20910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20905-L20910","statement_sha256":"ea4c6405bcd8fd9ab165e188f686b1301a54249769cec3052423c0a2caa59d48","origin":"The Stacks Project","memory_eligible":false,"source_rank":1544,"rank":1544,"depth":0,"x":2145.414,"y":240.846,"cluster":"commutative-algebra"},{"id":"stacks:059C","tag":"059C","title":"Mittag-Leffler modules · Lemma 059C","summary":"Let f: M → N and g: M → M' be maps of R-modules. Then g dominates f if and only if for any finitely presented R-module Q, we have Ker(f ⊗_R id_Q) ⊂ Ker(g ⊗_R id_Q).","statement_latex":"Let $f: M \\to N$ and $g: M \\to M'$ be maps of $R$-modules.\nThen $g$ dominates $f$ if and only if for any finitely presented $R$-module\n$Q$, we have $\\Ker(f \\otimes_R \\text{id}_Q) \\subset \\Ker(g \\otimes_R\n\\text{id}_Q)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059C","source_file":"algebra.tex","source_line":20915,"source_end_line":20921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20915-L20921","statement_sha256":"410cbef6d924a1cb11e9e5bdd56e89d30b6a16522deccb00c7d553f5956528f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1545,"rank":1545,"depth":0,"x":1928.048,"y":270.158,"cluster":"commutative-algebra"},{"id":"stacks:0AUM","tag":"0AUM","title":"Mittag-Leffler modules · Lemma 0AUM","summary":"Let f : M → N and g : M → M' be maps of R-modules. Consider the pushout of f and g, xymatrix M ar[r]_f ar[d]_g & N ar[d]^g' M' ar[r]^f' & N' Then g dominates f if and only if f' is universally injective.","statement_latex":"Let $f : M \\to N$ and $g : M \\to M'$ be maps of $R$-modules.\nConsider the pushout of $f$ and $g$,\n$$\n\\xymatrix{\nM  \\ar[r]_f \\ar[d]_g & N \\ar[d]^{g'} \\\\\nM' \\ar[r]^{f'} & N'\n}\n$$\nThen $g$ dominates $f$ if and only if $f'$ is universally injective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUM","source_file":"algebra.tex","source_line":20935,"source_end_line":20946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20935-L20946","statement_sha256":"31b92ca3aa25fd1535b158b5cd8070f32e1886473d408ca5b761067c0bbc0d6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1546,"rank":1546,"depth":0,"x":2064.87,"y":125.087,"cluster":"commutative-algebra"},{"id":"stacks:059D","tag":"059D","title":"Mittag-Leffler modules · Lemma 059D","summary":"Let f: M → N and g: M → M' be maps of R-modules. Suppose Coker(f) is of finite presentation. Then g dominates f if and only if g factors through f, i.e. there exists a module map h: N → M' such that g = h ∘ f.","statement_latex":"Let $f: M \\to N$ and $g: M \\to M'$ be maps of $R$-modules.\nSuppose $\\Coker(f)$ is of finite presentation.  Then $g$ dominates $f$ if\nand\nonly if $g$ factors through $f$, i.e.\\ there exists a module map $h: N\n\\to M'$ such that $g = h \\circ f$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059D","source_file":"algebra.tex","source_line":20972,"source_end_line":20979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20972-L20979","statement_sha256":"43e3199a549cd56c8825c15d365999cf1574bfc647875059ce6cddd57b0669fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1547,"rank":1547,"depth":3,"x":2080.655,"y":309.846,"cluster":"commutative-algebra"},{"id":"stacks:059E","tag":"059E","title":"Mittag-Leffler modules · Proposition 059E","summary":"Let M be an R-module. Let (M_i, f_ij) be a directed system of finitely presented R-modules, indexed by I, such that M = colim M_i. Let f_i: M_i → M be the canonical map. The following are equivalent: • For every finitely presented R-module P and module map f: P → M, there exists a finitely presented R-module Q and a module map g: P → Q such that g and f dominate each other, i.e., Ker(f ⊗_R id_N) = Ker(g ⊗_R id_N) for every R-module N. • For each i ∈ I, there exists j ≥ i…","statement_latex":"Let $M$ be an $R$-module.  Let $(M_i, f_{ij})$ be a directed system of finitely\npresented $R$-modules, indexed by $I$, such that $M = \\colim M_i$.  Let\n$f_i:\nM_i \\to M$ be the canonical map.  The following are equivalent:\n\\begin{enumerate}\n\\item For every finitely presented $R$-module $P$ and module map $f: P\n\\to M$, there exists a finitely presented $R$-module $Q$ and a module\nmap $g: P \\to Q$ such that $g$ and $f$ dominate each other, i.e.,\n$\\Ker(f \\otimes_R \\text{id}_N) = \\Ker(g \\otimes_R \\text{id}_N)$\nfor every $R$-module $N$.\n\\item For each $i \\in I$, there exists $j \\geq i$ such that $f_{ij}: M_i\n\\to M_j$ dominates $f_i: M_i \\to M$.\n\\item For each $i \\in I$, there exists $j \\geq i$ such that $f_{ij}: M_i\n\\to M_j$ factors through $f_{ik}: M_i \\to M_k$ for all $k \\geq\ni$.\n\\item For every $R$-module $N$, the inverse system\n$(\\Hom_R(M_i, N), \\Hom_R(f_{ij}, N))$ is Mittag-Leffler.\n\\item For $N = \\prod_{s \\in I} M_s$, the inverse system\n$(\\Hom_R(M_i, N), \\Hom_R(f_{ij}, N))$ is Mittag-Leffler.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059E","source_file":"algebra.tex","source_line":20999,"source_end_line":21021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L20999-L21021","statement_sha256":"d621fd554db1091f150ea3579c9bc2b3dfde1821199f6a33bce5098d0c6f701b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1548,"rank":1548,"depth":4,"x":1920.307,"y":182.461,"cluster":"commutative-algebra"},{"id":"stacks:059F","tag":"059F","title":"Mittag-Leffler modules · Definition 059F","summary":"Let M be an R-module. We say that M is Mittag-Leffler if the equivalent conditions of Proposition [Tag 059E] hold.","statement_latex":"Let $M$ be an $R$-module.  We say that $M$ is {\\it Mittag-Leffler} if the\nequivalent conditions of\nProposition \\ref{proposition-ML-characterization}\nhold.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059F","source_file":"algebra.tex","source_line":21105,"source_end_line":21111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21105-L21111","statement_sha256":"48d5bf21b0eff403934a55c0122b6a2e9e914b71d64b6357a32c8597677deea0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1549,"rank":1549,"depth":5,"x":2141.164,"y":185.41,"cluster":"commutative-algebra"},{"id":"stacks:05CN","tag":"05CN","title":"Mittag-Leffler modules · Lemma 05CN","summary":"If R is a ring and M, N are Mittag-Leffler modules over R, then M ⊗_R N is a Mittag-Leffler module.","statement_latex":"If $R$ is a ring and $M$, $N$ are Mittag-Leffler modules over $R$,\nthen $M \\otimes_R N$ is a Mittag-Leffler module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CN","source_file":"algebra.tex","source_line":21132,"source_end_line":21136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21132-L21136","statement_sha256":"8104d29d4e09e60a3bfc6c1dfc001622533d7a0d8128967e7ae1046068278bae","origin":"The Stacks Project","memory_eligible":false,"source_rank":1550,"rank":1550,"depth":5,"x":1975.802,"y":308.654,"cluster":"commutative-algebra"},{"id":"stacks:05CP","tag":"05CP","title":"Mittag-Leffler modules · Lemma 05CP","summary":"Let R be a ring and M an R-module. Then M is Mittag-Leffler if and only if for every finite free R-module F and module map f: F → M, there exists a finitely presented R-module Q and a module map g : F → Q such that g and f dominate each other, i.e., Ker(f ⊗_R id_N) = Ker(g ⊗_R id_N) for every R-module N.","statement_latex":"Let $R$ be a ring and $M$ an $R$-module. Then $M$ is Mittag-Leffler if and\nonly if for every finite free $R$-module $F$ and module map\n$f: F \\to M$, there exists a finitely presented $R$-module $Q$\nand a module map $g : F \\to Q$ such that $g$ and $f$ dominate each other, i.e.,\n$\\Ker(f \\otimes_R \\text{id}_N) = \\Ker(g \\otimes_R \\text{id}_N)$\nfor every $R$-module $N$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CP","source_file":"algebra.tex","source_line":21162,"source_end_line":21170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21162-L21170","statement_sha256":"fac8c1b38d2672c6d1f051d97f646e6ad6ec9d3474d0ecc2de0f5a4d9d1fcd9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1551,"rank":1551,"depth":5,"x":1998.646,"y":123.797,"cluster":"commutative-algebra"},{"id":"stacks:05CQ","tag":"05CQ","title":"Mittag-Leffler modules · Lemma 05CQ","summary":"Let R → S be a finite and finitely presented ring map. Let M be an S-module. If M is a Mittag-Leffler module over S then M is a Mittag-Leffler module over R.","statement_latex":"Let $R \\to S$ be a finite and finitely presented ring map.\nLet $M$ be an $S$-module.\nIf $M$ is a Mittag-Leffler module over $S$ then\n$M$ is a Mittag-Leffler module over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CQ","source_file":"algebra.tex","source_line":21199,"source_end_line":21205,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21199-L21205","statement_sha256":"d0ca28f9864400043aeaa6cc7017da3f52c09b379054371b6d5915bd135d6a17","origin":"The Stacks Project","memory_eligible":false,"source_rank":1552,"rank":1552,"depth":4,"x":2130.565,"y":273.191,"cluster":"commutative-algebra"},{"id":"stacks:05CR","tag":"05CR","title":"Mittag-Leffler modules · Lemma 05CR","summary":"Let R be a ring. Let S = R/I for some finitely generated ideal I. Let M be an S-module. Then M is a Mittag-Leffler module over R if and only if M is a Mittag-Leffler module over S.","statement_latex":"Let $R$ be a ring.\nLet $S = R/I$ for some finitely generated ideal $I$.\nLet $M$ be an $S$-module.\nThen $M$ is a Mittag-Leffler module over $R$ if and only if\n$M$ is a Mittag-Leffler module over $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CR","source_file":"algebra.tex","source_line":21221,"source_end_line":21228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21221-L21228","statement_sha256":"c44436a471b1d9159ae970eac40a5e512f8735ee923d30444767d9b16c148a52","origin":"The Stacks Project","memory_eligible":false,"source_rank":1553,"rank":1553,"depth":5,"x":1912.977,"y":237.855,"cluster":"commutative-algebra"},{"id":"stacks:059J","tag":"059J","title":"Interchanging direct products with tensor · Proposition 059J","summary":"Let M be an R-module. The following are equivalent: • M is finitely generated. • For every family (Q_α)_α ∈ A of R-modules, the canonical map M ⊗_R ( ∏_α Q_α ) → ∏_α (M ⊗_R Q_α) is surjective. • For every R-module Q and every set A, the canonical map M ⊗_R Q^A → (M ⊗_R Q)^A is surjective. • For every set A, the canonical map M ⊗_R R^A → M^A is surjective.","statement_latex":"Let $M$ be an $R$-module.  The following are equivalent:\n\\begin{enumerate}\n\\item $M$ is finitely generated.\n\\item For every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the\ncanonical map $M \\otimes_R \\left( \\prod_{\\alpha} Q_{\\alpha} \\right)\n\\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$ is surjective.\n\\item For every $R$-module $Q$ and every set $A$, the canonical map $M\n\\otimes_R Q^{A} \\to (M \\otimes_R Q)^{A}$ is surjective.\n\\item For every set $A$, the canonical map $M \\otimes_R R^{A} \\to\nM^{A}$ is surjective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059J","source_file":"algebra.tex","source_line":21315,"source_end_line":21328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21315-L21328","statement_sha256":"e43a5025b46fe2892a5af41cbdfcae56ded1c761a6e7d6baffeb811bd76d9795","origin":"The Stacks Project","memory_eligible":false,"source_rank":1554,"rank":1554,"depth":0,"x":2101.99,"y":140.368,"cluster":"commutative-algebra"},{"id":"stacks:059K","tag":"059K","title":"Interchanging direct products with tensor · Proposition 059K","summary":"Let M be an R-module. The following are equivalent: • M is finitely presented. • For every family (Q_α)_α ∈ A of R-modules, the canonical map M ⊗_R ( ∏_α Q_α ) → ∏_α (M ⊗_R Q_α) is bijective. • For every R-module Q and every set A, the canonical map M ⊗_R Q^A → (M ⊗_R Q)^A is bijective. • For every set A, the canonical map M ⊗_R R^A → M^A is bijective.","statement_latex":"Let $M$ be an $R$-module.  The following are equivalent:\n\\begin{enumerate}\n\\item $M$ is finitely presented.\n\\item For every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the\ncanonical map $M \\otimes_R \\left( \\prod_{\\alpha} Q_{\\alpha} \\right)\n\\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$ is bijective.\n\\item For every $R$-module $Q$ and every set $A$, the canonical map $M\n\\otimes_R Q^{A} \\to (M \\otimes_R Q)^{A}$ is bijective.\n\\item For every set $A$, the canonical map $M \\otimes_R R^{A} \\to\nM^{A}$ is bijective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059K","source_file":"algebra.tex","source_line":21358,"source_end_line":21371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21358-L21371","statement_sha256":"cb93f0a31ed4aa012b0c51c6030954106beaf40352c4029eab8825ad5a81c926","origin":"The Stacks Project","memory_eligible":false,"source_rank":1555,"rank":1555,"depth":1,"x":2040.962,"y":319.648,"cluster":"commutative-algebra"},{"id":"stacks:059L","tag":"059L","title":"Interchanging direct products with tensor · Lemma 059L","summary":"Let M be an R-module, P a finitely presented R-module, and f: P → M a map. Let Q be an R-module and suppose x ∈ Ker(P ⊗ Q → M ⊗ Q). Then there exists a finitely presented R-module P' and a map f': P → P' such that f factors through f' and x ∈ Ker(P ⊗ Q → P' ⊗ Q).","statement_latex":"Let $M$ be an $R$-module, $P$ a finitely presented $R$-module, and $f: P\n\\to M$ a map.  Let $Q$ be an $R$-module and suppose $x \\in \\Ker(P\n\\otimes Q \\to M \\otimes Q)$.  Then there exists a finitely presented\n$R$-module $P'$ and a map $f': P \\to P'$ such that $f$ factors through\n$f'$ and $x \\in \\Ker(P \\otimes Q \\to P' \\otimes Q)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059L","source_file":"algebra.tex","source_line":21414,"source_end_line":21421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21414-L21421","statement_sha256":"3e42056e1adb425442174876149e4b15ad8cfdf4f9580fcf41c44f2c1262be81","origin":"The Stacks Project","memory_eligible":false,"source_rank":1556,"rank":1556,"depth":0,"x":1941.712,"y":152.687,"cluster":"commutative-algebra"},{"id":"stacks:059M","tag":"059M","title":"Interchanging direct products with tensor · Proposition 059M","summary":"Let M be an R-module. The following are equivalent: • M is Mittag-Leffler. • For every family (Q_α)_α ∈ A of R-modules, the canonical map M ⊗_R ( ∏_α Q_α ) → ∏_α (M ⊗_R Q_α) is injective.","statement_latex":"Let $M$ be an $R$-module.  The following are equivalent:\n\\begin{enumerate}\n\\item $M$ is Mittag-Leffler.\n\\item For every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the\ncanonical map $M \\otimes_R \\left( \\prod_{\\alpha} Q_{\\alpha} \\right)\n\\to \\prod_{\\alpha} (M \\otimes_R Q_{\\alpha})$ is injective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059M","source_file":"algebra.tex","source_line":21442,"source_end_line":21451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21442-L21451","statement_sha256":"017c35c91e91e0cf0c37b09a4674254dd59c59a1a45bbad6c9380354144f1c91","origin":"The Stacks Project","memory_eligible":false,"source_rank":1557,"rank":1557,"depth":5,"x":2149.328,"y":219.538,"cluster":"commutative-algebra"},{"id":"stacks:0AS6","tag":"0AS6","title":"Interchanging direct products with tensor · Lemma 0AS6","summary":"Let M be a flat Mittag-Leffler module over R. Let F be an R-module and let x ∈ F ⊗_R M. Then there exists a smallest submodule F' ⊂ F such that x ∈ F' ⊗_R M. Also, F' is a finite R-module.","statement_latex":"Let $M$ be a flat Mittag-Leffler module over $R$. Let $F$ be an $R$-module\nand let $x \\in F \\otimes_R M$. Then there exists a smallest submodule\n$F' \\subset F$ such that $x \\in F' \\otimes_R M$.\nAlso, $F'$ is a finite $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AS6","source_file":"algebra.tex","source_line":21533,"source_end_line":21539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21533-L21539","statement_sha256":"863fb8232b5e27b210ce16712af126601c394f47346690af9e2475e3a843cd02","origin":"The Stacks Project","memory_eligible":false,"source_rank":1558,"rank":1558,"depth":6,"x":1942.311,"y":288.105,"cluster":"commutative-algebra"},{"id":"stacks:059N","tag":"059N","title":"Interchanging direct products with tensor · Lemma 059N","summary":"Let 0 → M_1 → M_2 → M_3 → 0 be a universally exact sequence of R-modules. Then: • If M_2 is Mittag-Leffler, then M_1 is Mittag-Leffler. • If M_1 and M_3 are Mittag-Leffler, then M_2 is Mittag-Leffler.","statement_latex":"Let $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a\nuniversally exact sequence of $R$-modules.  Then:\n\\begin{enumerate}\n\\item If $M_2$ is Mittag-Leffler, then $M_1$ is Mittag-Leffler.\n\\item If $M_1$ and $M_3$ are Mittag-Leffler, then $M_2$ is Mittag-Leffler.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059N","source_file":"algebra.tex","source_line":21564,"source_end_line":21572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21564-L21572","statement_sha256":"37a03af7a6b4fd9aba70abe41735ff561a7f315ec7b580ed4e23ebe99e0bbd92","origin":"The Stacks Project","memory_eligible":false,"source_rank":1559,"rank":1559,"depth":6,"x":2039.901,"y":119.944,"cluster":"commutative-algebra"},{"id":"stacks:0EGI","tag":"0EGI","title":"Interchanging direct products with tensor · Lemma 0EGI","summary":"Let M_1 → M_2 → M_3 → 0 be an exact sequence of R-modules. If M_1 is finitely generated and M_2 is Mittag-Leffler, then M_3 is Mittag-Leffler.","statement_latex":"Let $M_1 \\to M_2 \\to M_3 \\to 0$ be an exact sequence of $R$-modules.\nIf $M_1$ is finitely generated and $M_2$ is Mittag-Leffler, then $M_3$\nis Mittag-Leffler.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EGI","source_file":"algebra.tex","source_line":21590,"source_end_line":21595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21590-L21595","statement_sha256":"c58b960da0a25aa300eae126b944aac814673546676663c49f97bf0025c776df","origin":"The Stacks Project","memory_eligible":false,"source_rank":1560,"rank":1560,"depth":6,"x":2103.22,"y":299.461,"cluster":"commutative-algebra"},{"id":"stacks:0AS7","tag":"0AS7","title":"Interchanging direct products with tensor · Lemma 0AS7","summary":"If M = colim M_i is the colimit of a directed system of Mittag-Leffler R-modules M_i with universally injective transition maps, then M is Mittag-Leffler.","statement_latex":"If $M = \\colim M_i$ is the colimit of a directed system of Mittag-Leffler\n$R$-modules $M_i$ with universally injective transition maps, then $M$ is\nMittag-Leffler.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AS7","source_file":"algebra.tex","source_line":21618,"source_end_line":21623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21618-L21623","statement_sha256":"917ac06c26cd831b38f4bd190616129e9dc23f37d0578ec3118e3376960a0969","origin":"The Stacks Project","memory_eligible":false,"source_rank":1561,"rank":1561,"depth":6,"x":1912.015,"y":202.94,"cluster":"commutative-algebra"},{"id":"stacks:059P","tag":"059P","title":"Interchanging direct products with tensor · Lemma 059P","summary":"If M = bigoplus_i ∈ I M_i is a direct sum of R-modules, then M is Mittag-Leffler if and only if each M_i is Mittag-Leffler.","statement_latex":"If $M = \\bigoplus_{i \\in I} M_i$ is a direct sum of $R$-modules, then $M$ is\nMittag-Leffler if and only if each $M_i$ is Mittag-Leffler.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059P","source_file":"algebra.tex","source_line":21638,"source_end_line":21642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21638-L21642","statement_sha256":"10b70424239a215fbf12a5b2711ec7af8c2ba4259188fb7b0515ec772541b5da","origin":"The Stacks Project","memory_eligible":false,"source_rank":1562,"rank":1562,"depth":7,"x":2130.799,"y":165.59,"cluster":"commutative-algebra"},{"id":"stacks:05CT","tag":"05CT","title":"Interchanging direct products with tensor · Lemma 05CT","summary":"Let R → S be a ring map. Let M be an S-module. If S is Mittag-Leffler as an R-module, and M is flat and Mittag-Leffler as an S-module, then M is Mittag-Leffler as an R-module.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.\nIf $S$ is Mittag-Leffler as an $R$-module, and $M$ is flat and Mittag-Leffler\nas an $S$-module, then $M$ is Mittag-Leffler as an $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Interchanging direct products with tensor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CT","source_file":"algebra.tex","source_line":21663,"source_end_line":21668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21663-L21668","statement_sha256":"105c397f26445d3d10dcf299e99ef4b02426c13c06c17dbf0d592e8915a880ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":1563,"rank":1563,"depth":6,"x":1999.403,"y":317.394,"cluster":"commutative-algebra"},{"id":"stacks:05CV","tag":"05CV","title":"Coherent rings · Definition 05CV","summary":"Let R be a ring. Let M be an R-module. • We say M is a coherent module if it is finitely generated and every finitely generated submodule of M is finitely presented over R. • We say R is a coherent ring if it is coherent as a module over itself.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item We say $M$ is a {\\it coherent module} if it is finitely generated\nand every finitely generated submodule of $M$ is finitely presented over\n$R$.\n\\item We say $R$ is a {\\it coherent ring} if it is coherent as a module\nover itself.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Coherent rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CV","source_file":"algebra.tex","source_line":21700,"source_end_line":21710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21700-L21710","statement_sha256":"9f6fb35dc966ae5d49c319c8fb218ae1decc239648636ba7fbaf420b2bd63ef0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1564,"rank":1564,"depth":0,"x":1974.196,"y":130.751,"cluster":"commutative-algebra"},{"id":"stacks:05CW","tag":"05CW","title":"Coherent rings · Lemma 05CW","summary":"Let R be a ring. • A finite submodule of a coherent module is coherent. • Let φ : N → M be a homomorphism from a finite module to a coherent module. Then Ker(φ) is finite, Im(φ) is coherent, and Coker(φ) is coherent. • Let φ : N → M be a homomorphism of coherent modules. Then Ker(φ) and Coker(φ) are coherent modules. • Given a short exact sequence of R-modules 0 → M_1 → M_2 → M_3 → 0 if two out of three are coherent so is the third.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item A finite submodule of a coherent module is coherent.\n\\item Let $\\varphi : N \\to M$ be a homomorphism from a finite\nmodule to a coherent module. Then $\\Ker(\\varphi)$ is finite,\n$\\Im(\\varphi)$ is\ncoherent, and $\\Coker(\\varphi)$ is coherent.\n\\item Let $\\varphi : N \\to M$ be a homomorphism of coherent modules.\nThen $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are coherent\nmodules.\n\\item Given a short exact sequence of $R$-modules\n$0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ if two out of three are coherent\nso is the third.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Coherent rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CW","source_file":"algebra.tex","source_line":21726,"source_end_line":21742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21726-L21742","statement_sha256":"a1e3eea34663043858ab1abf064abbe2143cce04cc329c354f1b4e73b544f788","origin":"The Stacks Project","memory_eligible":false,"source_rank":1565,"rank":1565,"depth":2,"x":2143.01,"y":254.173,"cluster":"commutative-algebra"},{"id":"stacks:05CX","tag":"05CX","title":"Coherent rings · Lemma 05CX","summary":"Let R be a ring. If R is coherent, then a module is coherent if and only if it is finitely presented.","statement_latex":"Let $R$ be a ring. If $R$ is coherent, then a module is coherent\nif and only if it is finitely presented.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Coherent rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CX","source_file":"algebra.tex","source_line":21782,"source_end_line":21786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21782-L21786","statement_sha256":"4c5e342ada055b3faf9df1d462e3da1d96475546ec53507d2f6c1a9668f4b11e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1566,"rank":1566,"depth":3,"x":1919.101,"y":258.956,"cluster":"commutative-algebra"},{"id":"stacks:05CY","tag":"05CY","title":"Coherent rings · Lemma 05CY","summary":"A Noetherian ring is a coherent ring.","statement_latex":"A Noetherian ring is a coherent ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Coherent rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CY","source_file":"algebra.tex","source_line":21795,"source_end_line":21798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21795-L21798","statement_sha256":"7816638b08e2dc5cff671b2c8e4c8948ef01ef43703f85d50e5d61d7400b5092","origin":"The Stacks Project","memory_eligible":false,"source_rank":1567,"rank":1567,"depth":3,"x":2080.487,"y":128.275,"cluster":"commutative-algebra"},{"id":"stacks:05CZ","tag":"05CZ","title":"Coherent rings · Proposition 05CZ","summary":"This is [Chase]. Let R be a ring. The following are equivalent • R is coherent, • any product of flat R-modules is flat, and • for every set A the module R^A is flat.","statement_latex":"\\begin{reference}\nThis is \\cite[Theorem 2.1]{Chase}.\n\\end{reference}\nLet $R$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $R$ is coherent,\n\\item any product of flat $R$-modules is flat, and\n\\item for every set $A$ the module $R^A$ is flat.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Coherent rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05CZ","source_file":"algebra.tex","source_line":21807,"source_end_line":21818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21807-L21818","statement_sha256":"2e4219e7561dbfb954214e535988fc522d415d32578926be49d2a78bba86cdb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1568,"rank":1568,"depth":3,"x":2066.563,"y":316.36,"cluster":"commutative-algebra"},{"id":"stacks:059S","tag":"059S","title":"Examples and non-examples of Mittag-Leffler modules · Lemma 059S","summary":"Let M be a flat R-module. The following are equivalent • M is Mittag-Leffler, and • if F is a finite free R-module and x ∈ F ⊗_R M, then there exists a smallest submodule F' of F such that x ∈ F' ⊗_R M.","statement_latex":"Let $M$ be a flat $R$-module. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is Mittag-Leffler, and\n\\item if $F$ is a finite free $R$-module and\n$x \\in F \\otimes_R M$, then there exists a smallest submodule $F'$ of $F$\nsuch that $x \\in F' \\otimes_R M$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Examples and non-examples of Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059S","source_file":"algebra.tex","source_line":21880,"source_end_line":21889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21880-L21889","statement_sha256":"ceed14dc0bf0b15ebd499bcc3a3a88caf87229b27b4248d09285e7132fa66fba","origin":"The Stacks Project","memory_eligible":false,"source_rank":1569,"rank":1569,"depth":7,"x":1925.468,"y":169.653,"cluster":"commutative-algebra"},{"id":"stacks:05D0","tag":"05D0","title":"Examples and non-examples of Mittag-Leffler modules · Lemma 05D0","summary":"Let R be a Noetherian ring and A a set. Then M = R^A is a flat and Mittag-Leffler R-module.","statement_latex":"Let $R$ be a Noetherian ring and $A$ a set.\nThen $M = R^A$ is a flat and Mittag-Leffler $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Examples and non-examples of Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05D0","source_file":"algebra.tex","source_line":21954,"source_end_line":21958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21954-L21958","statement_sha256":"44fd8d5c2fa9cc5c8682477d5be60360a82dc8697a513ff9f649ca470db29295","origin":"The Stacks Project","memory_eligible":false,"source_rank":1570,"rank":1570,"depth":8,"x":2147.66,"y":197.794,"cluster":"commutative-algebra"},{"id":"stacks:059T","tag":"059T","title":"Examples and non-examples of Mittag-Leffler modules · Lemma 059T","summary":"Let R be a Noetherian ring and n a positive integer. Then the R-module M = R[[t_1, …, t_n]] is flat and Mittag-Leffler.","statement_latex":"Let $R$ be a Noetherian ring and $n$ a positive integer.  Then the $R$-module\n$M = R[[t_1, \\ldots, t_n]]$ is flat and Mittag-Leffler.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Examples and non-examples of Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059T","source_file":"algebra.tex","source_line":21983,"source_end_line":21987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L21983-L21987","statement_sha256":"f6fbf7587e9c3e51eef8c3b13b8439ddb7ac2734db16ecc1f8eaa4d4e03fcf2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1571,"rank":1571,"depth":9,"x":1961.043,"y":303.203,"cluster":"commutative-algebra"},{"id":"stacks:059W","tag":"059W","title":"Countably generated Mittag-Leffler modules · Lemma 059W","summary":"Let M be an R-module. Write M = colim_i ∈ I M_i where (M_i, f_ij) is a directed system of finitely presented R-modules. If M is Mittag-Leffler and countably generated, then there is a directed countable subset I' ⊂ I such that M ≅ colim_i ∈ I' M_i.","statement_latex":"Let $M$ be an $R$-module.  Write $M = \\colim_{i \\in I} M_i$ where $(M_i,\nf_{ij})$ is a directed system of finitely presented $R$-modules.  If $M$ is\nMittag-Leffler and countably generated, then there is a directed countable\nsubset $I' \\subset I$ such that $M \\cong \\colim_{i \\in I'} M_i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Countably generated Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059W","source_file":"algebra.tex","source_line":22052,"source_end_line":22058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22052-L22058","statement_sha256":"c935fc501bb79909d2f1ee86c9db280e15c20597a33ab8e5df2f3f0083d016d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1572,"rank":1572,"depth":5,"x":2013.926,"y":119.442,"cluster":"commutative-algebra"},{"id":"stacks:05D2","tag":"05D2","title":"Countably generated Mittag-Leffler modules · Lemma 05D2","summary":"Let R be a ring. Let M be an R-module. Assume M is Mittag-Leffler and countably generated. For any R-module map f : P → M with P finitely generated there exists an endomorphism α : M → M such that • α : M → M factors through a finitely presented R-module, and • α ∘ f = f.","statement_latex":"Let $R$ be a ring.\nLet $M$ be an $R$-module.\nAssume $M$ is Mittag-Leffler and countably generated.\nFor any $R$-module map $f : P \\to M$ with $P$ finitely generated there\nexists an endomorphism $\\alpha : M \\to M$ such that\n\\begin{enumerate}\n\\item $\\alpha : M \\to M$ factors through a finitely presented $R$-module, and\n\\item $\\alpha \\circ f = f$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Countably generated Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05D2","source_file":"algebra.tex","source_line":22116,"source_end_line":22127,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22116-L22127","statement_sha256":"253bc6a1dd539e4b853512b8d4de663b7cf10fc62a5e5c98072b91b3aed970b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1573,"rank":1573,"depth":6,"x":2122.791,"y":285.08,"cluster":"commutative-algebra"},{"id":"stacks:059X","tag":"059X","title":"Characterizing projective modules · Lemma 059X","summary":"Let M be an R-module. If M is flat, Mittag-Leffler, and countably generated, then M is projective.","statement_latex":"Let $M$ be an $R$-module.  If $M$ is flat, Mittag-Leffler, and countably\ngenerated, then $M$ is projective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059X","source_file":"algebra.tex","source_line":22192,"source_end_line":22196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22192-L22196","statement_sha256":"298c12f552126340ea47ac9a38122ee14ee2801f32497e0e379f45efef684e71","origin":"The Stacks Project","memory_eligible":false,"source_rank":1574,"rank":1574,"depth":6,"x":1909.148,"y":224.667,"cluster":"commutative-algebra"},{"id":"stacks:059Z","tag":"059Z","title":"Characterizing projective modules · Theorem 059Z","summary":"Let M be an R-module. Then M is projective if and only if • M is flat, • M is Mittag-Leffler, • M is a direct sum of countably generated R-modules.","statement_latex":"Let $M$ be an $R$-module. Then $M$ is projective if and only if\n\\begin{enumerate}\n\\item $M$ is flat,\n\\item $M$ is Mittag-Leffler,\n\\item $M$ is a direct sum of countably generated $R$-modules.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing projective modules","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/059Z","source_file":"algebra.tex","source_line":22248,"source_end_line":22256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22248-L22256","statement_sha256":"236f36828832d70ff9bb1cada6da5ab6e7c560e2524852bd4719e8a12d38143c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1575,"rank":1575,"depth":7,"x":2115.428,"y":147.929,"cluster":"commutative-algebra"},{"id":"stacks:05A0","tag":"05A0","title":"Characterizing projective modules · Lemma 05A0","summary":"Let f: M → N be universally injective map of R-modules. Suppose M is a direct sum of countably generated R-modules, and suppose N is flat and Mittag-Leffler. Then M is projective.","statement_latex":"Let $f: M \\to N$ be universally injective map of $R$-modules.  Suppose\n$M$ is a direct sum of countably generated $R$-modules, and suppose $N$ is flat\nand Mittag-Leffler.  Then $M$ is projective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05A0","source_file":"algebra.tex","source_line":22272,"source_end_line":22277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22272-L22277","statement_sha256":"1b413f1a52222f563c8ec3bfc6ee4aaa6d0187ca6e3a1e0743f2c91e30a6f2ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":1576,"rank":1576,"depth":8,"x":2024.961,"y":321.697,"cluster":"commutative-algebra"},{"id":"stacks:05A1","tag":"05A1","title":"Characterizing projective modules · Lemma 05A1","summary":"Let R be a Noetherian ring and let M be a R-module. Suppose M is a direct sum of countably generated R-modules, and suppose there is a universally injective map M → R[[t_1, …, t_n]] for some n. Then M is projective.","statement_latex":"Let $R$ be a Noetherian ring and let $M$ be a $R$-module.  Suppose $M$ is a\ndirect sum of countably generated $R$-modules, and suppose there is a\nuniversally injective map $M \\to R[[t_1, \\ldots, t_n]]$ for some $n$.\nThen $M$ is projective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Characterizing projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05A1","source_file":"algebra.tex","source_line":22287,"source_end_line":22293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22287-L22293","statement_sha256":"a2c89a7076d331ef421e5965406b280cca6d8f1a580d83a068c5fe23e3cf3977","origin":"The Stacks Project","memory_eligible":false,"source_rank":1577,"rank":1577,"depth":10,"x":1951.874,"y":142.091,"cluster":"commutative-algebra"},{"id":"stacks:05A3","tag":"05A3","title":"Ascending properties of modules · Lemma 05A3","summary":"Let R → S be a ring map. Let M be an R-module. Then: • If M is flat, then the S-module M ⊗_R S is flat. • If M is Mittag-Leffler, then the S-module M ⊗_R S is Mittag-Leffler. • If M is a direct sum of countably generated R-modules, then the S-module M ⊗_R S is a direct sum of countably generated S-modules. • If M is projective, then the S-module M ⊗_R S is projective.","statement_latex":"Let $R \\to S$ be a ring map.  Let $M$ be an $R$-module.  Then:\n\\begin{enumerate}\n\\item If $M$ is flat, then the $S$-module $M \\otimes_R S$ is flat.\n\\item If $M$ is Mittag-Leffler, then the $S$-module $M \\otimes_R S$ is\nMittag-Leffler.\n\\item If $M$ is a direct sum of countably generated $R$-modules, then the\n$S$-module $M \\otimes_R S$ is a direct sum of countably generated $S$-modules.\n\\item If $M$ is projective, then the $S$-module $M \\otimes_R S$ is projective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05A3","source_file":"algebra.tex","source_line":22310,"source_end_line":22321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22310-L22321","statement_sha256":"b65d6250bb1f675cc6157ccf7f71f2746caf83321bac70c64f2e29c40fb6cc7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1578,"rank":1578,"depth":6,"x":2150.354,"y":233.129,"cluster":"commutative-algebra"},{"id":"stacks:05A5","tag":"05A5","title":"Descending properties of modules · Lemma 05A5","summary":"Email from Juan Pablo Acosta Lopez dated 12/20/14. Let R → S be a faithfully flat ring map. Let M be an R-module. If the S-module M ⊗_R S is Mittag-Leffler, then M is Mittag-Leffler.","statement_latex":"\\begin{reference}\nEmail from Juan Pablo Acosta Lopez dated 12/20/14.\n\\end{reference}\nLet $R \\to S$ be a faithfully flat ring map. Let $M$ be an $R$-module. If the\n$S$-module $M \\otimes_R S$ is Mittag-Leffler, then $M$ is Mittag-Leffler.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05A5","source_file":"algebra.tex","source_line":22341,"source_end_line":22348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22341-L22348","statement_sha256":"00724a390e5852e071c751276dbad77825cd1107df85a010eebb132269a8f5bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1579,"rank":1579,"depth":5,"x":1930.619,"y":278.656,"cluster":"commutative-algebra"},{"id":"stacks:0GVD","tag":"0GVD","title":"Descending properties of modules · Lemma 0GVD","summary":"Let R → S be a faithfully flat ring map. Let M be an R-module. If the S-module M ⊗_R S is countably generated, then M is countably generated.","statement_latex":"Let $R \\to S$ be a faithfully flat ring map. Let $M$ be an $R$-module. If\nthe $S$-module $M \\otimes_R S$ is countably generated, then $M$\nis countably generated.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVD","source_file":"algebra.tex","source_line":22384,"source_end_line":22389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22384-L22389","statement_sha256":"6b99b0a0da34c4bdb4ea90765718cf962a40983a03ffe66673b04fadf8a864fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1580,"rank":1580,"depth":0,"x":2056.133,"y":120.28,"cluster":"commutative-algebra"},{"id":"stacks:05A6","tag":"05A6","title":"Descending properties of modules · Lemma 05A6","summary":"Let R → S be a faithfully flat ring map. Let M be an R-module. If the S-module M ⊗_R S is countably generated and projective, then M is countably generated and projective.","statement_latex":"Let $R \\to S$ be a faithfully flat ring map.  Let $M$ be an $R$-module.\n If the $S$-module $M \\otimes_R S$ is countably generated and projective,\nthen $M$ is countably generated and projective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05A6","source_file":"algebra.tex","source_line":22406,"source_end_line":22411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22406-L22411","statement_sha256":"bc6d2c20ff798cb868d32983bb8ab81bc7c27d5754b160c5453282c7dc6efd17","origin":"The Stacks Project","memory_eligible":false,"source_rank":1581,"rank":1581,"depth":8,"x":2090.969,"y":308.428,"cluster":"commutative-algebra"},{"id":"stacks:05A7","tag":"05A7","title":"Descending properties of modules · Lemma 05A7","summary":"Let R → S be a ring map, let M be an R-module, and let Q be a countably generated S-submodule of M ⊗_R S. Then there exists a countably generated R-submodule P of M such that Im(P ⊗_R S → M ⊗_R S) contains Q.","statement_latex":"Let $R \\to S$ be a ring map, let $M$ be an $R$-module, and let $Q$ be a\ncountably generated $S$-submodule of $M \\otimes_R S$.  Then there exists a\ncountably generated $R$-submodule $P$ of $M$ such that\n$\\Im(P \\otimes_R S \\to M \\otimes_R S)$ contains $Q$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05A7","source_file":"algebra.tex","source_line":22423,"source_end_line":22429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22423-L22429","statement_sha256":"42fc90877a728ebb0352550d06a8ceba9727b2ea8825b3507d486f102e14d7b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1582,"rank":1582,"depth":0,"x":1913.842,"y":189.366,"cluster":"commutative-algebra"},{"id":"stacks:05A8","tag":"05A8","title":"Descending properties of modules · Lemma 05A8","summary":"Let R → S be a ring map, and let M be an R-module. Suppose M ⊗_R S = bigoplus_i ∈ I Q_i is a direct sum of countably generated S-modules Q_i. If N is a countably generated submodule of M, then there is a countably generated submodule N' of M such that N' ⊃ N and Im(N' ⊗_R S → M ⊗_R S) = bigoplus_i ∈ I' Q_i for some subset I' ⊂ I.","statement_latex":"Let $R \\to S$ be a ring map, and let $M$ be an $R$-module.  Suppose $M\n\\otimes_R S = \\bigoplus_{i \\in I} Q_i$ is a direct sum of countably generated\n$S$-modules $Q_i$.  If $N$ is a countably generated submodule of $M$, then\nthere is a countably generated submodule $N'$ of $M$ such that $N' \\supset N$\nand $\\Im(N' \\otimes_R S \\to M \\otimes_R S) =\n\\bigoplus_{i \\in I'} Q_i$ for some subset $I' \\subset I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05A8","source_file":"algebra.tex","source_line":22437,"source_end_line":22445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22437-L22445","statement_sha256":"f95b228baf8d11e5044085c65806e6ddf4830903c1b926238fe119176f84cfd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1583,"rank":1583,"depth":1,"x":2140.373,"y":176.645,"cluster":"commutative-algebra"},{"id":"stacks:05A9","tag":"05A9","title":"Descending properties of modules · Theorem 05A9","summary":"Let R → S be a faithfully flat ring map. Let M be an R-module. If the S-module M ⊗_R S is projective, then M is projective.","statement_latex":"Let $R \\to S$ be a faithfully flat ring map.  Let $M$ be an $R$-module.\n If the $S$-module $M \\otimes_R S$ is projective, then $M$ is projective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties of modules","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05A9","source_file":"algebra.tex","source_line":22461,"source_end_line":22465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22461-L22465","statement_sha256":"54e2e9734d157c3214b97b6a76cb1e5432dd3262d176609fd98f7f48787b7959","origin":"The Stacks Project","memory_eligible":false,"source_rank":1584,"rank":1584,"depth":9,"x":1983.442,"y":314.67,"cluster":"commutative-algebra"},{"id":"stacks:0315","tag":"0315","title":"Completion · Lemma 0315","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let φ : M → N be a map of R-modules. • If M/IM → N/IN is surjective, then M^wedge → N^wedge is surjective. • If M → N is surjective, then M^wedge → N^wedge is surjective. • If 0 → K → M → N → 0 is a short exact sequence of R-modules and N is flat, then 0 → K^wedge → M^wedge → N^wedge → 0 is a short exact sequence. • The map M ⊗_R R^wedge → M^wedge is surjective for any finite R-module M.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nLet $\\varphi : M \\to N$ be a map of $R$-modules.\n\\begin{enumerate}\n\\item If $M/IM \\to N/IN$ is surjective, then $M^\\wedge \\to N^\\wedge$\nis surjective.\n\\item If $M \\to N$ is surjective, then $M^\\wedge \\to N^\\wedge$ is surjective.\n\\item If $0 \\to K \\to M \\to N \\to 0$ is a short exact sequence of\n$R$-modules and $N$ is flat, then\n$0 \\to K^\\wedge \\to M^\\wedge \\to N^\\wedge \\to 0$ is a short exact sequence.\n\\item The map $M \\otimes_R R^\\wedge \\to M^\\wedge$ is\nsurjective for any finite $R$-module $M$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0315","source_file":"algebra.tex","source_line":22570,"source_end_line":22584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22570-L22584","statement_sha256":"2f1c89d0942a4498f544750fb8ff8433b0f9b118a8fdc9ccfb3be0eeb50a2d21","origin":"The Stacks Project","memory_eligible":false,"source_rank":1585,"rank":1585,"depth":3,"x":1988.168,"y":123.701,"cluster":"commutative-algebra"},{"id":"stacks:0317","tag":"0317","title":"Completion · Definition 0317","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. We say M is I-adically complete if the map M → M^wedge = lim_n M/I^nM is an isomorphism. We say R is I-adically complete if R is I-adically complete as an R-module.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nLet $M$ be an $R$-module. We say $M$ is {\\it $I$-adically complete}\nif the map\n$$\nM \\longrightarrow M^\\wedge = \\lim_n M/I^nM\n$$\nis an isomorphism\\footnote{This includes the condition that\n$\\bigcap I^nM = 0$.}. We say $R$ is {\\it $I$-adically complete}\nif $R$ is $I$-adically complete as an $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0317","source_file":"algebra.tex","source_line":22625,"source_end_line":22636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22625-L22636","statement_sha256":"5e418b99f584826b44432aef9ad5158ed0c1e778fd22c37b3dd45b93eec7bc2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1586,"rank":1586,"depth":0,"x":2138.37,"y":267.309,"cluster":"commutative-algebra"},{"id":"stacks:05GG","tag":"05GG","title":"Completion · Lemma 05GG","summary":"[Matlis]. The slick proof given here is from an email of Bjorn Poonen dated Nov 5, 2016. Let R be a ring. Let I be a finitely generated ideal of R. Let M be an R-module. Then • the completion M^wedge is I-adically complete, and • I^nM^wedge = Ker(M^wedge → M/I^nM) = (I^nM)^wedge for all n ≥ 1. In particular R^wedge is I-adically complete, I^nR^wedge = (I^n)^wedge, and R^wedge/I^nR^wedge = R/I^n.","statement_latex":"\\begin{reference}\n\\cite[Theorem 15]{Matlis}. The slick proof given here is from\nan email of Bjorn Poonen dated Nov 5, 2016.\n\\end{reference}\nLet $R$ be a ring. Let $I$ be a finitely generated ideal of $R$.\nLet $M$ be an $R$-module. Then\n\\begin{enumerate}\n\\item the completion $M^\\wedge$ is $I$-adically complete, and\n\\item $I^nM^\\wedge = \\Ker(M^\\wedge \\to M/I^nM) = (I^nM)^\\wedge$ for all\n$n \\geq 1$.\n\\end{enumerate}\nIn particular $R^\\wedge$ is $I$-adically complete,\n$I^nR^\\wedge = (I^n)^\\wedge$, and\n$R^\\wedge/I^nR^\\wedge = R/I^n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GG","source_file":"algebra.tex","source_line":22644,"source_end_line":22660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22644-L22660","statement_sha256":"b189303d0c4099e6c0f4de19ef7d1f8d59818d8c1beb0e0dc02fe28a097e3783","origin":"The Stacks Project","memory_eligible":false,"source_rank":1587,"rank":1587,"depth":4,"x":1911.955,"y":246.627,"cluster":"commutative-algebra"},{"id":"stacks:0BNG","tag":"0BNG","title":"Completion · Lemma 0BNG","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let 0 → M → N → Q → 0 be an exact sequence of R-modules such that Q is annihilated by a power of I. Then completion produces an exact sequence 0 → M^wedge → N^wedge → Q → 0.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let\n$0 \\to M \\to N \\to Q \\to 0$ be an exact sequence of\n$R$-modules such that $Q$ is annihilated by a power of $I$.\nThen completion produces an exact sequence\n$0 \\to M^\\wedge \\to N^\\wedge \\to Q \\to 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNG","source_file":"algebra.tex","source_line":22680,"source_end_line":22687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22680-L22687","statement_sha256":"15eb40b6b037ebf16defdc28e77739c867196f50f296ebfc6b92d23b6da906bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1588,"rank":1588,"depth":3,"x":2095.682,"y":133.319,"cluster":"commutative-algebra"},{"id":"stacks:0318","tag":"0318","title":"Completion · Lemma 0318","summary":"Taken from an unpublished note of Lenstra and de Smit. Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. Denote K_n = Ker(M^wedge → M/I^nM). Then M^wedge is I-adically complete if and only if K_n is equal to I^nM^wedge for all n ≥ 1.","statement_latex":"\\begin{reference}\nTaken from an unpublished note of Lenstra and de Smit.\n\\end{reference}\nLet $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.\nDenote $K_n = \\Ker(M^\\wedge \\to M/I^nM)$. Then $M^\\wedge$ is $I$-adically\ncomplete if and only if $K_n$ is equal to $I^nM^\\wedge$ for all $n \\geq 1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0318","source_file":"algebra.tex","source_line":22701,"source_end_line":22709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22701-L22709","statement_sha256":"2b91a08e9d148d9aadd0af4102e501be0d59dccee9bab71ab70055d688e26165","origin":"The Stacks Project","memory_eligible":false,"source_rank":1589,"rank":1589,"depth":3,"x":2051.29,"y":321.263,"cluster":"commutative-algebra"},{"id":"stacks:05GI","tag":"05GI","title":"Completion · Lemma 05GI","summary":"Let R be a ring, let I ⊂ R be an ideal, and let R^wedge = lim R/I^n. • any element of R^wedge which maps to a unit of R/I is a unit, • any element of 1 + I maps to an invertible element of R^wedge, • any element of 1 + IR^wedge is invertible in R^wedge, and • the ideals IR^wedge and Ker(R^wedge → R/I) are contained in the Jacobson radical of R^wedge.","statement_latex":"Let $R$ be a ring, let $I \\subset R$ be an ideal, and let\n$R^\\wedge = \\lim R/I^n$.\n\\begin{enumerate}\n\\item any element of $R^\\wedge$ which maps to a unit of $R/I$ is a unit,\n\\item any element of $1 + I$ maps to an invertible element of $R^\\wedge$,\n\\item any element of $1 + IR^\\wedge$ is invertible in $R^\\wedge$, and\n\\item the ideals $IR^\\wedge$ and $\\Ker(R^\\wedge \\to R/I)$ are contained\nin the Jacobson radical of $R^\\wedge$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GI","source_file":"algebra.tex","source_line":22733,"source_end_line":22744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22733-L22744","statement_sha256":"632e8aec038d50617eca23b4f87d3f1b4351a2792bc8d9e2464f161ce7164915","origin":"The Stacks Project","memory_eligible":false,"source_rank":1590,"rank":1590,"depth":2,"x":1932.794,"y":157.363,"cluster":"commutative-algebra"},{"id":"stacks:090S","tag":"090S","title":"Completion · Lemma 090S","summary":"Let A be a ring. Let I = (f_1, …, f_r) be a finitely generated ideal. If M → lim M/f_i^nM is surjective for each i, then M → lim M/I^nM is surjective.","statement_latex":"Let $A$ be a ring. Let $I = (f_1, \\ldots, f_r)$ be a finitely\ngenerated ideal. If $M \\to \\lim M/f_i^nM$ is surjective for\neach $i$, then $M \\to \\lim M/I^nM$ is surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090S","source_file":"algebra.tex","source_line":22755,"source_end_line":22760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22755-L22760","statement_sha256":"5104096acc41269d7b01823e4b53329f323ad12ff349498d2a1c71e4968eeb43","origin":"The Stacks Project","memory_eligible":false,"source_rank":1591,"rank":1591,"depth":0,"x":2152.141,"y":211.024,"cluster":"commutative-algebra"},{"id":"stacks:090T","tag":"090T","title":"Completion · Lemma 090T","summary":"Let A be a ring. Let I ⊂ J ⊂ A be ideals. If M is J-adically complete and I is finitely generated, then M is I-adically complete.","statement_latex":"Let $A$ be a ring. Let $I \\subset J \\subset A$ be ideals.\nIf $M$ is $J$-adically complete and $I$ is finitely generated, then\n$M$ is $I$-adically complete.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090T","source_file":"algebra.tex","source_line":22775,"source_end_line":22780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22775-L22780","statement_sha256":"20b3dce9416e804d620f11e0a9f594f10cc914adf1c3ff7eff1125ccb93be693","origin":"The Stacks Project","memory_eligible":false,"source_rank":1592,"rank":1592,"depth":1,"x":1947.091,"y":295.982,"cluster":"commutative-algebra"},{"id":"stacks:0319","tag":"0319","title":"Completion · Lemma 0319","summary":"Let R be a ring. Let I, J be ideals of R. Assume there exist integers c, d > 0 such that I^c ⊂ J and J^d ⊂ I. Then completion with respect to I agrees with completion with respect to J for any R-module. In particular an R-module M is I-adically complete if and only if it is J-adically complete.","statement_latex":"Let $R$ be a ring.\nLet $I$, $J$ be ideals of $R$.\nAssume there exist integers $c, d > 0$ such that\n$I^c \\subset J$ and $J^d \\subset I$.\nThen completion with respect to $I$ agrees with completion\nwith respect to $J$ for any $R$-module.\nIn particular an $R$-module $M$ is $I$-adically complete\nif and only if it is $J$-adically complete.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0319","source_file":"algebra.tex","source_line":22805,"source_end_line":22815,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22805-L22815","statement_sha256":"d39d4a8bd16053f6284b75dc30a57cf3a5b0f312fc3de40b751d8228ed6ec6cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1593,"rank":1593,"depth":0,"x":2030.033,"y":116.85,"cluster":"commutative-algebra"},{"id":"stacks:031A","tag":"031A","title":"Completion · Lemma 031A","summary":"Let R be a ring. Let I be an ideal of R. Let M be an I-adically complete R-module, and let K ⊂ M be an R-submodule. The following are equivalent • K = ⋂ (K + I^nM) and • M/K is I-adically complete.","statement_latex":"Let $R$ be a ring. Let $I$ be an ideal of $R$.\nLet $M$ be an $I$-adically complete $R$-module,\nand let $K \\subset M$ be an $R$-submodule.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K = \\bigcap (K + I^nM)$ and\n\\item $M/K$ is $I$-adically complete.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031A","source_file":"algebra.tex","source_line":22824,"source_end_line":22834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22824-L22834","statement_sha256":"e22bfbaea32ab95484ca95a14b9c371567197bba48745cee8c88c85846c82f9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1594,"rank":1594,"depth":4,"x":2112.988,"y":296.137,"cluster":"commutative-algebra"},{"id":"stacks:031B","tag":"031B","title":"Completion · Lemma 031B","summary":"Let R be a ring. Let I be an ideal of R. Let M be an R-module. If (a) R is I-adically complete, (b) M is a finite R-module, and (c) ⋂ I^nM = (0), then M is I-adically complete.","statement_latex":"Let $R$ be a ring. Let $I$ be an ideal of $R$.\nLet $M$ be an $R$-module.\nIf (a) $R$ is $I$-adically complete, (b) $M$ is a finite $R$-module,\nand (c) $\\bigcap I^nM = (0)$, then $M$ is $I$-adically complete.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031B","source_file":"algebra.tex","source_line":22844,"source_end_line":22850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22844-L22850","statement_sha256":"2dd95703d48ac344a27b02490d8e18f783d494f637b0ce0390a20be787aefe85","origin":"The Stacks Project","memory_eligible":false,"source_rank":1595,"rank":1595,"depth":4,"x":1907.486,"y":210.94,"cluster":"commutative-algebra"},{"id":"stacks:031D","tag":"031D","title":"Completion · Lemma 031D","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. Assume • R is I-adically complete, • ⋂_n ≥ 1 I^nM = (0), and • M/IM is a finite R/I-module. Then M is a finite R-module.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is $I$-adically complete,\n\\item $\\bigcap_{n \\geq 1} I^nM = (0)$, and\n\\item $M/IM$ is a finite $R/I$-module.\n\\end{enumerate}\nThen $M$ is a finite $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031D","source_file":"algebra.tex","source_line":22859,"source_end_line":22869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22859-L22869","statement_sha256":"bd261e76cc956a8b0d9050afb7dea47b9e85b3a5eead899dc8cce2ed206d2922","origin":"The Stacks Project","memory_eligible":false,"source_rank":1596,"rank":1596,"depth":5,"x":2127.698,"y":157.116,"cluster":"commutative-algebra"},{"id":"stacks:00MA","tag":"00MA","title":"Completion for Noetherian rings · Lemma 00MA","summary":"Let I be an ideal of a Noetherian ring R. Denote ^wedge completion with respect to I. • If K → N is an injective map of finite R-modules, then the map on completions K^wedge → N^wedge is injective. • If 0 → K → N → M → 0 is a short exact sequence of finite R-modules, then 0 → K^wedge → N^wedge → M^wedge → 0 is a short exact sequence. • If M is a finite R-module, then M^wedge = M ⊗_R R^wedge.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $R$.\nDenote ${}^\\wedge$ completion with respect to $I$.\n\\begin{enumerate}\n\\item If $K \\to N$ is an injective map of finite $R$-modules,\nthen the map on completions $K^\\wedge \\to N^\\wedge$ is injective.\n\\item If $0 \\to K \\to N \\to M \\to 0$ is a short exact sequence\nof finite $R$-modules, then $0 \\to K^\\wedge \\to N^\\wedge \\to M^\\wedge \\to 0$\nis a short exact sequence.\n\\item If $M$ is a finite $R$-module, then $M^\\wedge = M \\otimes_R R^\\wedge$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MA","source_file":"algebra.tex","source_line":22900,"source_end_line":22912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22900-L22912","statement_sha256":"a9908afa6adab1b8eb8cdf83ec78569cd6faf68aa5773e9c4b949646a41f4e04","origin":"The Stacks Project","memory_eligible":false,"source_rank":1597,"rank":1597,"depth":4,"x":2008.512,"y":321.882,"cluster":"commutative-algebra"},{"id":"stacks:00MB","tag":"00MB","title":"Completion for Noetherian rings · Lemma 00MB","summary":"Let I be an ideal of a Noetherian ring R. Denote ^wedge completion with respect to I. • The ring map R → R^wedge is flat. • The functor M ↦ M^wedge is exact on the category of finitely generated R-modules.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $R$.\nDenote ${}^\\wedge$ completion with respect to $I$.\n\\begin{enumerate}\n\\item The ring map $R \\to R^\\wedge$ is flat.\n\\item The functor $M \\mapsto M^\\wedge$ is exact on the category of\nfinitely generated $R$-modules.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MB","source_file":"algebra.tex","source_line":22955,"source_end_line":22964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22955-L22964","statement_sha256":"163c7f518df2a460d9cc05efa810b09121da320721c3c8f0973a8ca27fd913b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1598,"rank":1598,"depth":5,"x":1963.865,"y":132.616,"cluster":"commutative-algebra"},{"id":"stacks:00MC","tag":"00MC","title":"Completion for Noetherian rings · Lemma 00MC","summary":"Let I be an ideal of a Noetherian ring R. Denote R^wedge the completion of R with respect to I. If I is contained in the Jacobson radical of R, then the ring map R → R^wedge is faithfully flat. In particular, if (R, m) is a Noetherian local ring, then the completion lim_n R/ m^n is faithfully flat.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $R$. Denote $R^\\wedge$\nthe completion of $R$ with respect to $I$. If $I$ is contained\nin the Jacobson radical of $R$, then the ring map $R \\to R^\\wedge$\nis faithfully flat. In particular, if $(R, \\mathfrak m)$ is a Noetherian\nlocal ring, then the completion $\\lim_n R/\\mathfrak m^n$ is faithfully flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MC","source_file":"algebra.tex","source_line":22976,"source_end_line":22983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22976-L22983","statement_sha256":"70f4720afe190753b36327d008436c3f31e644353edd92553066bc5a4cb17a14","origin":"The Stacks Project","memory_eligible":false,"source_rank":1599,"rank":1599,"depth":6,"x":2149.128,"y":246.929,"cluster":"commutative-algebra"},{"id":"stacks:031C","tag":"031C","title":"Completion for Noetherian rings · Lemma 031C","summary":"Let R be a Noetherian ring. Let I be an ideal of R. Let M be an R-module. Then the completion M^wedge of M with respect to I is I-adically complete, I^n M^wedge = (I^nM)^wedge, and M^wedge/I^nM^wedge = M/I^nM.","statement_latex":"Let $R$ be a Noetherian ring.\nLet $I$ be an ideal of $R$.\nLet $M$ be an $R$-module.\nThen the completion $M^\\wedge$\nof $M$ with respect to $I$ is $I$-adically complete,\n$I^n M^\\wedge = (I^nM)^\\wedge$, and $M^\\wedge/I^nM^\\wedge = M/I^nM$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031C","source_file":"algebra.tex","source_line":22994,"source_end_line":23002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L22994-L23002","statement_sha256":"2e7abf1f604ddd04969d0a8a9affae5abaa08c3e7a5f8076ea97e4cb4bebdf55","origin":"The Stacks Project","memory_eligible":false,"source_rank":1600,"rank":1600,"depth":5,"x":1920.42,"y":267.774,"cluster":"commutative-algebra"},{"id":"stacks:05GH","tag":"05GH","title":"Completion for Noetherian rings · Lemma 05GH","summary":"Let I be an ideal of a ring R. Assume • R/I is a Noetherian ring, • I is finitely generated. Then the completion R^wedge of R with respect to I is a Noetherian ring complete with respect to IR^wedge.","statement_latex":"Let $I$ be an ideal of a ring $R$. Assume\n\\begin{enumerate}\n\\item $R/I$ is a Noetherian ring,\n\\item $I$ is finitely generated.\n\\end{enumerate}\nThen the completion $R^\\wedge$ of $R$ with respect to $I$\nis a Noetherian ring complete with respect to $IR^\\wedge$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GH","source_file":"algebra.tex","source_line":23010,"source_end_line":23019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23010-L23019","statement_sha256":"8216bc05e490a883f99da0c443433bccb5bcf635ef38e6cd881e0b905f4a785c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1601,"rank":1601,"depth":5,"x":2072.416,"y":122.521,"cluster":"commutative-algebra"},{"id":"stacks:0316","tag":"0316","title":"Completion for Noetherian rings · Lemma 0316","summary":"Let R be a Noetherian ring. Let I be an ideal of R. The completion R^wedge of R with respect to I is Noetherian.","statement_latex":"Let $R$ be a Noetherian ring.\nLet $I$ be an ideal of $R$.\nThe completion $R^\\wedge$ of $R$ with respect to $I$ is\nNoetherian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0316","source_file":"algebra.tex","source_line":23064,"source_end_line":23070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23064-L23070","statement_sha256":"4e5d589d017dc38cbd15c5d6b28481457e9d011879d355c3fd7b1efbdec6e639","origin":"The Stacks Project","memory_eligible":false,"source_rank":1602,"rank":1602,"depth":6,"x":2077.148,"y":316.018,"cluster":"commutative-algebra"},{"id":"stacks:0394","tag":"0394","title":"Completion for Noetherian rings · Lemma 0394","summary":"Let R → S be a local homomorphism of local rings (R, m) and (S, n). Let R^wedge, resp. S^wedge be the completion of R, resp. S with respect to m, resp. n. If m and n are finitely generated and dim_kappa( m) S/ mS < ∞, then • S^wedge is equal to the m-adic completion of S, and • S^wedge is a finite R^wedge-module.","statement_latex":"Let $R \\to S$ be a local homomorphism of local rings $(R, \\mathfrak m)$\nand $(S, \\mathfrak n)$. Let $R^\\wedge$, resp.\\ $S^\\wedge$ be the completion\nof $R$, resp.\\ $S$ with respect to $\\mathfrak m$, resp.\\ $\\mathfrak n$.\nIf $\\mathfrak m$ and $\\mathfrak n$ are finitely generated and\n$\\dim_{\\kappa(\\mathfrak m)} S/\\mathfrak mS < \\infty$, then\n\\begin{enumerate}\n\\item $S^\\wedge$ is equal to the $\\mathfrak m$-adic completion of $S$, and\n\\item $S^\\wedge$ is a finite $R^\\wedge$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0394","source_file":"algebra.tex","source_line":23099,"source_end_line":23110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23099-L23110","statement_sha256":"9cc88667261a3b4871bbf401aeda0df95c5d00bd23fad712022d2b792179d364","origin":"The Stacks Project","memory_eligible":false,"source_rank":1603,"rank":1603,"depth":8,"x":1917.935,"y":175.918,"cluster":"commutative-algebra"},{"id":"stacks:07N9","tag":"07N9","title":"Completion for Noetherian rings · Lemma 07N9","summary":"Let R be a Noetherian ring. Let R → S be a finite ring map. Let p ⊂ R be a prime and let q_1, …, q_m be the primes of S lying over p (Lemma [Tag 05DR]). Then R_ p^wedge ⊗_R S = (S_ p)^wedge = S_ q_1^wedge × … × S_ q_m^wedge where the (S_ p)^wedge is the completion with respect to p and the local rings R_ p and S_ q_i are completed with respect to their maximal ideals.","statement_latex":"Let $R$ be a Noetherian ring. Let $R \\to S$ be a finite ring map.\nLet $\\mathfrak p \\subset R$ be a prime and let\n$\\mathfrak q_1, \\ldots, \\mathfrak q_m$ be the primes of $S$\nlying over $\\mathfrak p$\n(Lemma \\ref{lemma-finite-finite-fibres}).\nThen\n$$\nR_\\mathfrak p^\\wedge \\otimes_R S =\n(S_\\mathfrak p)^\\wedge =\nS_{\\mathfrak q_1}^\\wedge \\times \\ldots \\times S_{\\mathfrak q_m}^\\wedge\n$$\nwhere the $(S_\\mathfrak p)^\\wedge$ is the completion with respect to\n$\\mathfrak p$ and the local rings $R_\\mathfrak p$ and\n$S_{\\mathfrak q_i}$ are completed with respect to their maximal ideals.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07N9","source_file":"algebra.tex","source_line":23134,"source_end_line":23150,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23134-L23150","statement_sha256":"8a54533c73352f2d3446a01d1e087cb93d4e6ae18133c42d193166333fab9786","origin":"The Stacks Project","memory_eligible":false,"source_rank":1604,"rank":1604,"depth":9,"x":2148.172,"y":188.894,"cluster":"commutative-algebra"},{"id":"stacks:05D3","tag":"05D3","title":"Completion for Noetherian rings · Lemma 05D3","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let 0 → K → P → M → 0 be a short exact sequence of R-modules. If M is flat over R and M/IM is a projective R/I-module, then the sequence of I-adic completions 0 → K^wedge → P^wedge → M^wedge → 0 is a split exact sequence.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nLet $0 \\to K \\to P \\to M \\to 0$ be a short exact sequence of\n$R$-modules. If $M$ is flat over $R$ and $M/IM$ is a projective\n$R/I$-module, then the sequence of $I$-adic completions\n$$\n0 \\to K^\\wedge \\to P^\\wedge \\to M^\\wedge \\to 0\n$$\nis a split exact sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05D3","source_file":"algebra.tex","source_line":23176,"source_end_line":23186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23176-L23186","statement_sha256":"e1b77d0732c675a3a773f84c9ea81f299afebca0450bca06040b2984b015c0aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":1605,"rank":1605,"depth":7,"x":1967.83,"y":310.057,"cluster":"commutative-algebra"},{"id":"stacks:0DYC","tag":"0DYC","title":"Completion for Noetherian rings · Lemma 0DYC","summary":"Let A be a Noetherian ring. Let I, J ⊂ A be ideals. If A is I-adically complete and A/I is J-adically complete, then A is J-adically complete.","statement_latex":"Let $A$ be a Noetherian ring. Let $I, J \\subset A$ be ideals.\nIf $A$ is $I$-adically complete and $A/I$ is $J$-adically complete,\nthen $A$ is $J$-adically complete.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYC","source_file":"algebra.tex","source_line":23215,"source_end_line":23220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23215-L23220","statement_sha256":"591ebd1f48cf7f2e6be3c57b18f1e50a58ad95059bf8c255f51e6ce7f0bc05de","origin":"The Stacks Project","memory_eligible":false,"source_rank":1606,"rank":1606,"depth":6,"x":2003.402,"y":118.242,"cluster":"commutative-algebra"},{"id":"stacks:0G1Q","tag":"0G1Q","title":"Taking limits of modules · Lemma 0G1Q","summary":"Let I ⊂ A be a finitely generated ideal of a ring. Let (M_n) be an inverse system of A-modules with I^n M_n = 0. Then M = lim M_n is I-adically complete.","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring.\nLet $(M_n)$ be an inverse system of $A$-modules with $I^n M_n = 0$.\nThen $M = \\lim M_n$ is $I$-adically complete.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Taking limits of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1Q","source_file":"algebra.tex","source_line":23253,"source_end_line":23258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23253-L23258","statement_sha256":"dfcf508f155bf89230e7927c071fa6906438b1962b949c5b4e314fe48a0901e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1607,"rank":1607,"depth":5,"x":2131.519,"y":279.987,"cluster":"commutative-algebra"},{"id":"stacks:09B8","tag":"09B8","title":"Taking limits of modules · Lemma 09B8","summary":"Let I ⊂ A be a finitely generated ideal of a ring. Let (M_n) be an inverse system of A-modules with M_n = M_n + 1/I^nM_n + 1. Set M = lim M_n. Then M/I^nM = M_n and M is I-adically complete.","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring.\nLet $(M_n)$ be an inverse system of $A$-modules with\n$M_n = M_{n + 1}/I^nM_{n + 1}$. Set $M = \\lim M_n$.\nThen $M/I^nM = M_n$ and $M$ is $I$-adically complete.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Taking limits of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09B8","source_file":"algebra.tex","source_line":23267,"source_end_line":23273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23267-L23273","statement_sha256":"a43fea63eaff40450a9ae361ff0b758829487a7e8e91b16031cdaa39e7fe5d99","origin":"The Stacks Project","memory_eligible":false,"source_rank":1608,"rank":1608,"depth":6,"x":1906.812,"y":233.381,"cluster":"commutative-algebra"},{"id":"stacks:0EKC","tag":"0EKC","title":"Taking limits of modules · Lemma 0EKC","summary":"Let A be a Noetherian graded ring. Let I ⊂ A_+ be a homogeneous ideal. Let (N_n) be an inverse system of finite graded A-modules with N_n = N_n + 1/I^n N_n + 1. Then there is a finite graded A-module N such that N_n = N/I^nN as graded modules for all n.","statement_latex":"Let $A$ be a Noetherian graded ring. Let $I \\subset A_+$ be a homogeneous\nideal. Let $(N_n)$ be an inverse system of finite graded $A$-modules with\n$N_n = N_{n + 1}/I^n N_{n + 1}$. Then there is a finite graded $A$-module\n$N$ such that $N_n = N/I^nN$ as graded modules for all $n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Taking limits of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKC","source_file":"algebra.tex","source_line":23299,"source_end_line":23305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23299-L23305","statement_sha256":"696fd30407d66f28c5ab6960ff771cfec9a5cbca6b295cc5220804931eb11878","origin":"The Stacks Project","memory_eligible":false,"source_rank":1609,"rank":1609,"depth":1,"x":2110.136,"y":140.175,"cluster":"commutative-algebra"},{"id":"stacks:0EKD","tag":"0EKD","title":"Taking limits of modules · Lemma 0EKD","summary":"Let A be a graded ring. Let I ⊂ A_+ be a homogeneous ideal. Denote A' = lim A/I^n. Let (G_n) be an inverse system of graded A-modules with G_n annihilated by I^n. Let M be a graded A-module and let φ_n : M → G_n be a compatible system of graded A-module maps. If the induced map φ : M ⊗_A A' → lim G_n is an isomorphism, then M_d → lim G_n, d is an isomorphism for all d ∈ Z.","statement_latex":"Let $A$ be a graded ring. Let $I \\subset A_+$ be a homogeneous ideal.\nDenote $A' = \\lim A/I^n$. Let $(G_n)$ be an inverse system\nof graded $A$-modules with $G_n$ annihilated by $I^n$.\nLet $M$ be a graded $A$-module and let\n$\\varphi_n : M \\to G_n$ be a compatible system of graded\n$A$-module maps. If the induced map\n$$\n\\varphi : M \\otimes_A A' \\longrightarrow \\lim G_n\n$$\nis an isomorphism, then $M_d \\to \\lim G_{n, d}$\nis an isomorphism for all $d \\in \\mathbf{Z}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Taking limits of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKD","source_file":"algebra.tex","source_line":23335,"source_end_line":23348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23335-L23348","statement_sha256":"166f37e0ab66c220dae7a037b3c45d7dcfabbaffc73aa1a5feaab95682d96ef6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1610,"rank":1610,"depth":0,"x":2035.106,"y":324.409,"cluster":"commutative-algebra"},{"id":"stacks:00ME","tag":"00ME","title":"Criteria for flatness · Lemma 00ME","summary":"Suppose that R → S is a local homomorphism of local rings with S Noetherian. Denote m the maximal ideal of R. Let M be a flat R-module and N a finite S-module. Let u : N → M be a map of R-modules. If overlineu : N/ m N → M/ m M is injective then u is injective. In this case M/u(N) is flat over R.","statement_latex":"Suppose that $R \\to S$ is a local homomorphism of local rings with $S$\nNoetherian.\nDenote $\\mathfrak m$ the maximal ideal of $R$. Let $M$ be a flat $R$-module\nand $N$ a finite $S$-module. Let $u : N \\to M$ be a map of $R$-modules.\nIf $\\overline{u} : N/\\mathfrak m N \\to M/\\mathfrak m M$\nis injective then $u$ is injective.\nIn this case $M/u(N)$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ME","source_file":"algebra.tex","source_line":23423,"source_end_line":23432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23423-L23432","statement_sha256":"0c5fcf87339b23cdd19aeb4f94064b860a3dc89f3990e234f2ac7b68521b7b3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1611,"rank":1611,"depth":4,"x":1942.207,"y":145.853,"cluster":"commutative-algebra"},{"id":"stacks:00MF","tag":"00MF","title":"Criteria for flatness · Lemma 00MF","summary":"Suppose that R → S is a flat and local ring homomorphism of Noetherian local rings. Denote m the maximal ideal of R. Suppose f ∈ S is a nonzerodivisor in S/ mS. Then S/fS is flat over R, and f is a nonzerodivisor in S.","statement_latex":"Suppose that $R \\to S$ is a flat and local ring homomorphism of Noetherian\nlocal rings. Denote $\\mathfrak m$ the maximal ideal of $R$.\nSuppose $f \\in S$ is a nonzerodivisor in $S/{\\mathfrak m}S$.\nThen $S/fS$ is flat over $R$, and $f$ is a nonzerodivisor in $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MF","source_file":"algebra.tex","source_line":23479,"source_end_line":23485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23479-L23485","statement_sha256":"08608b7733a3ebf53cdb1ece7d8d20ef94661441d8e57792ea0e9b065274c6e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1612,"rank":1612,"depth":5,"x":2154.454,"y":224.863,"cluster":"commutative-algebra"},{"id":"stacks:00MG","tag":"00MG","title":"Criteria for flatness · Lemma 00MG","summary":"Suppose that R → S is a flat and local ring homomorphism of Noetherian local rings. Denote m the maximal ideal of R. Suppose f_1, …, f_c is a sequence of elements of S such that the images overlinef_1, …, overlinef_c form a regular sequence in S/ mS. Then f_1, …, f_c is a regular sequence in S and each of the quotients S/(f_1, …, f_i) is flat over R.","statement_latex":"Suppose that $R \\to S$ is a flat and local ring homomorphism of Noetherian\nlocal rings. Denote $\\mathfrak m$ the maximal ideal of $R$.\nSuppose $f_1, \\ldots, f_c$ is a sequence of elements of\n$S$ such that the images $\\overline{f}_1, \\ldots, \\overline{f}_c$\nform a regular sequence in $S/{\\mathfrak m}S$.\nThen $f_1, \\ldots, f_c$ is a regular sequence in $S$ and each\nof the quotients $S/(f_1, \\ldots, f_i)$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MG","source_file":"algebra.tex","source_line":23491,"source_end_line":23500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23491-L23500","statement_sha256":"ba3605693e32ba1d9a79a3c6270c534aaf41db0d041d56a0798279198b27cee8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1613,"rank":1613,"depth":6,"x":1934.249,"y":287.082,"cluster":"commutative-algebra"},{"id":"stacks:00MH","tag":"00MH","title":"Criteria for flatness · Lemma 00MH","summary":"Let R → S be a local homomorphism of Noetherian local rings. Let m be the maximal ideal of R. Let M be a nonzero finite S-module. Suppose that (a) M/ mM is a free S/ mS-module, and (b) M is flat over R. Then M is free and S is flat over R.","statement_latex":"Let $R \\to S$ be a local homomorphism of Noetherian\nlocal rings. Let $\\mathfrak m$ be the maximal\nideal of $R$. Let $M$ be a nonzero finite $S$-module.\nSuppose that (a) $M/\\mathfrak mM$\nis a free $S/\\mathfrak mS$-module, and (b) $M$ is flat over $R$.\nThen $M$ is free and $S$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MH","source_file":"algebra.tex","source_line":23506,"source_end_line":23514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23506-L23514","statement_sha256":"3d8e7f53ce3dee60d7b7541d881d6f9ef90534f4cf7463b9d56b7936d5fa5ce0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1614,"rank":1614,"depth":5,"x":2046.672,"y":116.128,"cluster":"commutative-algebra"},{"id":"stacks:00MI","tag":"00MI","title":"Criteria for flatness · Lemma 00MI","summary":"Let R → S be a local homomorphism of local Noetherian rings. Let m be the maximal ideal of R. Let 0 → F_e → F_e-1 → … → F_0 be a finite complex of finite S-modules. Assume that each F_i is R-flat, and that the complex 0 → F_e/ m F_e → F_e-1/ m F_e-1 → … → F_0 / m F_0 is exact. Then 0 → F_e → F_e-1 → … → F_0 is exact, and moreover the module Coker(F_1 → F_0) is R-flat.","statement_latex":"Let $R \\to S$ be a local homomorphism of local Noetherian\nrings. Let $\\mathfrak m$ be the maximal ideal of $R$.\nLet $0 \\to F_e \\to F_{e-1} \\to \\ldots \\to F_0$\nbe a finite complex of finite $S$-modules. Assume that\neach $F_i$ is $R$-flat, and that the complex\n$0 \\to F_e/\\mathfrak m F_e \\to F_{e-1}/\\mathfrak m F_{e-1}\n\\to \\ldots \\to F_0 / \\mathfrak m F_0$ is exact.\nThen $0 \\to F_e \\to F_{e-1} \\to \\ldots \\to F_0$\nis exact, and moreover the module\n$\\Coker(F_1 \\to F_0)$ is $R$-flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MI","source_file":"algebra.tex","source_line":23527,"source_end_line":23539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23527-L23539","statement_sha256":"9bbd24b63fac8c6690a97f643e0723142bd211dfe0afb821ccdae72881c1e825","origin":"The Stacks Project","memory_eligible":false,"source_rank":1615,"rank":1615,"depth":5,"x":2101.289,"y":306.116,"cluster":"commutative-algebra"},{"id":"stacks:00MJ","tag":"00MJ","title":"Criteria for flatness · Lemma 00MJ","summary":"Let R be a local ring with maximal ideal m and residue field kappa = R/ m. Let M be an R-module. If Tor_1^R(kappa, M) = 0, then for every finite length R-module N we have Tor_1^R(N, M) = 0.","statement_latex":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$\nand residue field $\\kappa = R/\\mathfrak m$.\nLet $M$ be an $R$-module. If $\\text{Tor}_1^R(\\kappa, M) = 0$,\nthen for every finite length $R$-module $N$ we have\n$\\text{Tor}_1^R(N, M) = 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MJ","source_file":"algebra.tex","source_line":23559,"source_end_line":23566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23559-L23566","statement_sha256":"f7ee16100d8fb600dc3123596d1371d105f781a97369664a8ff1b146b1f8d11c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1616,"rank":1616,"depth":2,"x":1908.092,"y":196.933,"cluster":"commutative-algebra"},{"id":"stacks:00MK","tag":"00MK","title":"Local criterion for flatness · Lemma 00MK","summary":"Let R → S be a local homomorphism of local Noetherian rings. Let m be the maximal ideal of R, and let kappa = R/ m. Let M be a finite S-module. If Tor_1^R(kappa, M) = 0, then M is flat over R.","statement_latex":"Let $R \\to S$ be a local homomorphism of local Noetherian\nrings. Let $\\mathfrak m$ be the maximal ideal of $R$,\nand let $\\kappa = R/\\mathfrak m$.\nLet $M$ be a finite $S$-module. If $\\text{Tor}_1^R(\\kappa, M) = 0$,\nthen $M$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MK","source_file":"algebra.tex","source_line":23582,"source_end_line":23589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23582-L23589","statement_sha256":"2729809b33586af5bae30a3ad243e4e898efc56605bf2d289aab6ab1ea6b4095","origin":"The Stacks Project","memory_eligible":false,"source_rank":1617,"rank":1617,"depth":7,"x":2138.521,"y":167.797,"cluster":"commutative-algebra"},{"id":"stacks:051C","tag":"051C","title":"Criteria for flatness · Lemma 051C","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. If M/IM is flat over R/I and Tor_1^R(R/I, M) = 0 then • M/I^nM is flat over R/I^n for all n ≥ 1, and • for any module N which is annihilated by I^m for some m ≥ 0 we have Tor_1^R(N, M) = 0. In particular, if I is nilpotent, then M is flat over R.","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\nLet $M$ be an $R$-module.\nIf $M/IM$ is flat over $R/I$ and $\\text{Tor}_1^R(R/I, M) = 0$ then\n\\begin{enumerate}\n\\item $M/I^nM$ is flat over $R/I^n$ for all $n \\geq 1$, and\n\\item for any module $N$ which is annihilated by $I^m$ for some $m \\geq 0$\nwe have $\\text{Tor}_1^R(N, M) = 0$.\n\\end{enumerate}\nIn particular, if $I$ is nilpotent, then $M$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051C","source_file":"algebra.tex","source_line":23650,"source_end_line":23662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23650-L23662","statement_sha256":"fd4b217ce8ff8cabe98701ebd31fdcb2e371d5628fbb6333e231ab9268b8ecbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1618,"rank":1618,"depth":0,"x":1991.93,"y":320.144,"cluster":"commutative-algebra"},{"id":"stacks:0AS8","tag":"0AS8","title":"Criteria for flatness · Lemma 0AS8","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. • If M/IM is flat over R/I and M ⊗_R I/I^2 → IM/I^2M is injective, then M/I^2M is flat over R/I^2. • If M/IM is flat over R/I and M ⊗_R I^n/I^n + 1 → I^nM/I^n + 1M is injective for n = 1, …, k, then M/I^k + 1M is flat over R/I^k + 1.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nLet $M$ be an $R$-module.\n\\begin{enumerate}\n\\item If $M/IM$ is flat over $R/I$ and $M \\otimes_R I/I^2 \\to IM/I^2M$\nis injective, then $M/I^2M$ is flat over $R/I^2$.\n\\item If $M/IM$ is flat over $R/I$ and $M \\otimes_R I^n/I^{n + 1}\n\\to I^nM/I^{n + 1}M$ is injective for $n = 1, \\ldots, k$,\nthen $M/I^{k + 1}M$ is flat over $R/I^{k + 1}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AS8","source_file":"algebra.tex","source_line":23718,"source_end_line":23729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23718-L23729","statement_sha256":"4bb0379f00eb576eda5d9ea0a77bded17150d303e6667cb31a387b08b41c3e6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1619,"rank":1619,"depth":1,"x":1977.501,"y":124.485,"cluster":"commutative-algebra"},{"id":"stacks:00ML","tag":"00ML","title":"Variant of the local criterion · Lemma 00ML","summary":"Let R → S be a local homomorphism of Noetherian local rings. Let I not = R be an ideal in R. Let M be a finite S-module. If Tor_1^R(M, R/I) = 0 and M/IM is flat over R/I, then M is flat over R.","statement_latex":"Let $R \\to S$ be a local homomorphism of Noetherian\nlocal rings. Let $I \\not = R$ be an ideal in $R$.\nLet $M$ be a finite $S$-module. If $\\text{Tor}_1^R(M, R/I) = 0$\nand $M/IM$ is flat over $R/I$, then $M$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ML","source_file":"algebra.tex","source_line":23750,"source_end_line":23756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23750-L23756","statement_sha256":"f8487cc74b131d8c1365118b268dd689e0e475b5e4f26878f9f9c670c516559e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1620,"rank":1620,"depth":8,"x":2145.606,"y":260.672,"cluster":"commutative-algebra"},{"id":"stacks:0523","tag":"0523","title":"Criteria for flatness · Lemma 0523","summary":"Let R → S be a ring map. Let I ⊂ R be an ideal. Let M be an S-module. Assume • R is a Noetherian ring, • S is a Noetherian ring, • M is a finite S-module, and • for each n ≥ 1 the module M/I^n M is flat over R/I^n. Then for every q ∈ V(IS) the localization M_ q is flat over R. In particular, if S is local and IS is contained in its maximal ideal, then M is flat over R.","statement_latex":"Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal.\nLet $M$ be an $S$-module. Assume\n\\begin{enumerate}\n\\item $R$ is a Noetherian ring,\n\\item $S$ is a Noetherian ring,\n\\item $M$ is a finite $S$-module, and\n\\item for each $n \\geq 1$ the module $M/I^n M$ is flat over\n$R/I^n$.\n\\end{enumerate}\nThen for every $\\mathfrak q \\in V(IS)$\nthe localization $M_{\\mathfrak q}$ is flat over $R$.\nIn particular, if $S$ is local and $IS$ is contained\nin its maximal ideal, then $M$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0523","source_file":"algebra.tex","source_line":23807,"source_end_line":23822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23807-L23822","statement_sha256":"3eb7cd2f6590482763641f13d998f23a5c2a846ce2b601b17b556da7d499644e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1621,"rank":1621,"depth":9,"x":1911.962,"y":255.632,"cluster":"commutative-algebra"},{"id":"stacks:00MM","tag":"00MM","title":"Criteria for flatness · Lemma 00MM","summary":"Let R → R' → R\" be ring maps. Let M be an R-module. Suppose that M ⊗_R R' is flat over R'. Then the natural map Tor_1^R(M, R') ⊗_R' R\" → Tor_1^R(M, R\") is onto.","statement_latex":"Let $R \\to R' \\to R''$ be ring maps.\nLet $M$ be an $R$-module. Suppose that $M \\otimes_R R'$\nis flat over $R'$. Then the natural map\n$\\text{Tor}_1^R(M, R') \\otimes_{R'} R'' \\to\n\\text{Tor}_1^R(M, R'')$ is onto.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MM","source_file":"algebra.tex","source_line":23845,"source_end_line":23852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23845-L23852","statement_sha256":"a284345e400f6eff45b024fb72a2c029cb9eef56b3004ac6e75586d2a2fbc14d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1622,"rank":1622,"depth":0,"x":2088.427,"y":126.681,"cluster":"commutative-algebra"},{"id":"stacks:00MN","tag":"00MN","title":"Criteria for flatness · Lemma 00MN","summary":"Let R → R' be a ring map. Let I ⊂ R be an ideal and I' = IR'. Let M be an R-module and set M' = M ⊗_R R'. The natural map Tor_1^R(R'/I', M) → Tor_1^R'(R'/I', M') is surjective.","statement_latex":"Let $R \\to R'$ be a ring map. Let $I \\subset R$ be\nan ideal and $I' = IR'$. Let $M$ be an $R$-module\nand set $M' = M \\otimes_R R'$. The natural map\n$\\text{Tor}_1^R(R'/I', M) \\to \\text{Tor}_1^{R'}(R'/I', M')$\nis surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MN","source_file":"algebra.tex","source_line":23880,"source_end_line":23887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23880-L23887","statement_sha256":"a486033050892af5bef936e28d69eb733e8ade9af5b8aed80e6e4c7059f7f005","origin":"The Stacks Project","memory_eligible":false,"source_rank":1623,"rank":1623,"depth":0,"x":2061.985,"y":322.038,"cluster":"commutative-algebra"},{"id":"stacks:00MO","tag":"00MO","title":"Criteria for flatness · Lemma 00MO","summary":"Let xymatrix S ar[r] & S' R ar[r] ar[u] & R' ar[u] be a commutative diagram of local homomorphisms of local Noetherian rings. Let I ⊂ R be a proper ideal. Let M be a finite S-module. Denote I' = IR' and M' = M ⊗_S S'. Assume that • S' is a localization of the tensor product S ⊗_R R', • M/IM is flat over R/I, • Tor_1^R(M, R/I) → Tor_1^R'(M', R'/I') is zero. Then M' is flat over R'.","statement_latex":"Let\n$$\n\\xymatrix{\nS \\ar[r] & S' \\\\\nR \\ar[r] \\ar[u] & R' \\ar[u]\n}\n$$\nbe a commutative diagram of local homomorphisms of local Noetherian rings.\nLet $I \\subset R$ be a proper ideal.\nLet $M$ be a finite $S$-module.\nDenote $I' = IR'$ and $M' = M \\otimes_S S'$.\nAssume that\n\\begin{enumerate}\n\\item $S'$ is a localization of the tensor product\n$S \\otimes_R R'$,\n\\item $M/IM$ is flat over $R/I$,\n\\item $\\text{Tor}_1^R(M, R/I) \\to \\text{Tor}_1^{R'}(M', R'/I')$\nis zero.\n\\end{enumerate}\nThen $M'$ is flat over $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MO","source_file":"algebra.tex","source_line":23920,"source_end_line":23942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23920-L23942","statement_sha256":"2c080151c69d330b49b1f7712bddc0edb4e3961420dba502bab0c72d8b203017","origin":"The Stacks Project","memory_eligible":false,"source_rank":1624,"rank":1624,"depth":9,"x":1924.283,"y":162.867,"cluster":"commutative-algebra"},{"id":"stacks:00MP","tag":"00MP","title":"Crit\\`ere de platitude par fibres; Noetherian case · Lemma 00MP","summary":"Let R, S, S' be Noetherian local rings and let R → S → S' be local ring homomorphisms. Let m ⊂ R be the maximal ideal. Let M be an S'-module. Assume • The module M is finite over S'. • The module M is not zero. • The module M/ m M is a flat S/ m S-module. • The module M is a flat R-module. Then S is flat over R and M is a flat S-module.","statement_latex":"Let $R$, $S$, $S'$ be Noetherian local rings and let $R \\to S \\to S'$\nbe local ring homomorphisms. Let $\\mathfrak m \\subset R$ be the\nmaximal ideal. Let $M$ be an $S'$-module. Assume\n\\begin{enumerate}\n\\item The module $M$ is finite over $S'$.\n\\item The module $M$ is not zero.\n\\item The module $M/\\mathfrak m M$\nis a flat $S/\\mathfrak m S$-module.\n\\item The module $M$ is a flat $R$-module.\n\\end{enumerate}\nThen $S$ is flat over $R$ and $M$ is a flat $S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MP","source_file":"algebra.tex","source_line":23977,"source_end_line":23990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L23977-L23990","statement_sha256":"d27597b4ff030a7bb274ab82357a28b77701c87725aba827686421a70138e770","origin":"The Stacks Project","memory_eligible":false,"source_rank":1625,"rank":1625,"depth":9,"x":2153.988,"y":202.13,"cluster":"commutative-algebra"},{"id":"stacks:0H7N","tag":"0H7N","title":"Criteria for flatness · Lemma 0H7N","summary":"Let A be a ring, let M be an A-module, and let f ∈ A. If • f is a nonzerodivisor on A and M, • M_f is a flat A_f-module, and • M/fM is a flat A/fA-module, Then M is a flat A-module. Same with \"flat\" replaced by \"faithfully flat\".","statement_latex":"Let $A$ be a ring, let $M$ be an $A$-module, and let $f \\in A$. If\n\\begin{enumerate}\n\\item $f$ is a nonzerodivisor on $A$ and $M$,\n\\item $M_f$ is a flat $A_f$-module, and\n\\item $M/fM$ is a flat $A/fA$-module,\n\\end{enumerate}\nThen $M$ is a flat $A$-module. Same with ``flat'' replaced by\n``faithfully flat''.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7N","source_file":"algebra.tex","source_line":24025,"source_end_line":24035,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24025-L24035","statement_sha256":"a0d5d5bfa1fc7e6969178e9a08b33f30af315fef9ca27d29bc58d377fc5ca7b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1626,"rank":1626,"depth":4,"x":1952.888,"y":303.591,"cluster":"commutative-algebra"},{"id":"stacks:0H7P","tag":"0H7P","title":"Criteria for flatness · Lemma 0H7P","summary":"Let A be a ring, let M be an A-module. Let I = (f_1, …, f_r) be an ideal of A generated by r ≥ 1 elements. If • M_f_i is a flat A_f_i-module for i = 1, …, r, • M/IM is a flat A/I-module, • Tor_i^A(M, A/I) = 0 for i = 1, …, r + 1. Then M is a flat A-module. Same with \"flat\" replaced by \"faithfully flat\".","statement_latex":"Let $A$ be a ring, let $M$ be an $A$-module. Let $I = (f_1, \\ldots, f_r)$\nbe an ideal of $A$ generated by $r \\geq 1$ elements. If\n\\begin{enumerate}\n\\item $M_{f_i}$ is a flat $A_{f_i}$-module for $i = 1, \\ldots, r$,\n\\item $M/IM$ is a flat $A/I$-module,\n\\item $\\text{Tor}_i^A(M, A/I) = 0$ for $i = 1, \\ldots, r + 1$.\n\\end{enumerate}\nThen $M$ is a flat $A$-module. Same with ``flat'' replaced by\n``faithfully flat''.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Criteria for flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7P","source_file":"algebra.tex","source_line":24092,"source_end_line":24103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24092-L24103","statement_sha256":"5059014724a8a5235d7f0499d8f8c7416eace067775c72785c0e28c39cd2db9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1627,"rank":1627,"depth":4,"x":2019.632,"y":114.531,"cluster":"commutative-algebra"},{"id":"stacks:00MQ","tag":"00MQ","title":"Base change and flatness · Lemma 00MQ","summary":"Let xymatrix S ar[r] & S' R ar[r] ar[u] & R' ar[u] be a commutative diagram of local homomorphisms of local rings. Assume that S' is a localization of the tensor product S ⊗_R R'. Let M be an S-module and set M' = S' ⊗_S M. • If M is flat over R then M' is flat over R'. • If M' is flat over R' and R → R' is flat then M is flat over R. In particular we have • [(3)] If S is flat over R then S' is flat over R'. • [(4)] If R' → S' and R → R' are flat then S is flat over R.","statement_latex":"Let\n$$\n\\xymatrix{\nS \\ar[r] & S' \\\\\nR \\ar[r] \\ar[u] & R' \\ar[u]\n}\n$$\nbe a commutative diagram of local homomorphisms of local rings.\nAssume that $S'$ is a localization of the tensor product $S \\otimes_R R'$.\nLet $M$ be an $S$-module and set $M' = S' \\otimes_S M$.\n\\begin{enumerate}\n\\item If $M$ is flat over $R$ then $M'$ is flat over $R'$.\n\\item If $M'$ is flat over $R'$ and $R \\to R'$ is flat then\n$M$ is flat over $R$.\n\\end{enumerate}\nIn particular we have\n\\begin{enumerate}\n\\item[(3)] If $S$ is flat over $R$ then $S'$ is flat over $R'$.\n\\item[(4)] If $R' \\to S'$ and $R \\to R'$ are flat then $S$ is flat over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Base change and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MQ","source_file":"algebra.tex","source_line":24184,"source_end_line":24206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24184-L24206","statement_sha256":"8946e391efff0d2dc159ebc7eedff163299615fdd2d2d14b3e610702128f95ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":1628,"rank":1628,"depth":4,"x":2122.526,"y":291.939,"cluster":"commutative-algebra"},{"id":"stacks:0GEB","tag":"0GEB","title":"Base change and flatness · Lemma 0GEB","summary":"Consider a commutative diagram of local rings and local homomorphisms xymatrix S ar[r] & S' R ar[r] ar[u] & R' ar[u] Let M be a finite S-module. Assume that • the horizontal arrows are flat ring maps • M is flat over R, • m_R R' = m_R', • R' and S' are Noetherian. Then M' = M ⊗_S S' is flat over R'.","statement_latex":"Consider a commutative diagram of local rings and local homomorphisms\n$$\n\\xymatrix{\nS \\ar[r] & S' \\\\\nR \\ar[r] \\ar[u] & R' \\ar[u]\n}\n$$\nLet $M$ be a finite $S$-module. Assume that\n\\begin{enumerate}\n\\item the horizontal arrows are flat ring maps\n\\item $M$ is flat over $R$,\n\\item $\\mathfrak m_R R' = \\mathfrak m_{R'}$,\n\\item $R'$ and $S'$ are Noetherian.\n\\end{enumerate}\nThen $M' = M \\otimes_S S'$ is flat over $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Base change and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEB","source_file":"algebra.tex","source_line":24242,"source_end_line":24259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24242-L24259","statement_sha256":"7745db680c02fa7e50de6e499176aa68a247e3b05915f8516ba0eb5629a115ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":1629,"rank":1629,"depth":8,"x":1903.831,"y":219.454,"cluster":"commutative-algebra"},{"id":"stacks:051F","tag":"051F","title":"Flatness criteria over Artinian rings · Lemma 051F","summary":"Let (R, m) be a local ring with nilpotent maximal ideal m. Let M be a flat R-module. If A is a set and x_α ∈ M, α ∈ A is a collection of elements of M, then the following are equivalent: • (overlinex_α)_α ∈ A forms a basis for the vector space M/ mM over R/ m, and • (x_α)_α ∈ A forms a basis for M over R.","statement_latex":"Let $(R, \\mathfrak m)$ be a local ring with nilpotent maximal ideal\n$\\mathfrak m$. Let $M$ be a flat $R$-module.\nIf $A$ is a set and $x_\\alpha \\in M$, $\\alpha \\in A$ is a collection\nof elements of $M$, then the following are equivalent:\n\\begin{enumerate}\n\\item $\\{\\overline{x}_\\alpha\\}_{\\alpha \\in A}$ forms a basis\nfor the vector space $M/\\mathfrak mM$ over $R/\\mathfrak m$, and\n\\item $\\{x_\\alpha\\}_{\\alpha \\in A}$ forms a basis for $M$ over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flatness criteria over Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051F","source_file":"algebra.tex","source_line":24294,"source_end_line":24305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24294-L24305","statement_sha256":"3b7d6a975177792d67da17d84da0f29c721c62941bb2c8f15bcd8f2342c77513","origin":"The Stacks Project","memory_eligible":false,"source_rank":1630,"rank":1630,"depth":3,"x":2123.54,"y":148.761,"cluster":"commutative-algebra"},{"id":"stacks:051G","tag":"051G","title":"Flatness criteria over Artinian rings · Lemma 051G","summary":"Let R be a local ring with nilpotent maximal ideal. Let M be an R-module. The following are equivalent • M is flat over R, • M is a free R-module, and • M is a projective R-module.","statement_latex":"Let $R$ be a local ring with nilpotent maximal ideal. Let $M$ be an $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ is flat over $R$,\n\\item $M$ is a free $R$-module, and\n\\item $M$ is a projective $R$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flatness criteria over Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051G","source_file":"algebra.tex","source_line":24320,"source_end_line":24329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24320-L24329","statement_sha256":"e77b53a7566eb9dd3f556badff5b5b3e4f3ff3d75a6adea86c33d018dc69297a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1631,"rank":1631,"depth":4,"x":2018.305,"y":325.683,"cluster":"commutative-algebra"},{"id":"stacks:051H","tag":"051H","title":"Flatness criteria over Artinian rings · Lemma 051H","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. Let A be a set and let x_α ∈ M, α ∈ A be a collection of elements of M. Assume • I is nilpotent, • (overlinex_α)_α ∈ A forms a basis for M/IM over R/I, and • Tor_1^R(R/I, M) = 0. Then M is free on (x_α)_α ∈ A over R.","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\nLet $M$ be an $R$-module.\nLet $A$ be a set and let $x_\\alpha \\in M$, $\\alpha \\in A$ be a collection\nof elements of $M$.\nAssume\n\\begin{enumerate}\n\\item $I$ is nilpotent,\n\\item $\\{\\overline{x}_\\alpha\\}_{\\alpha \\in A}$ forms a basis for $M/IM$ over\n$R/I$, and\n\\item $\\text{Tor}_1^R(R/I, M) = 0$.\n\\end{enumerate}\nThen $M$ is free on $\\{x_\\alpha\\}_{\\alpha \\in A}$ over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flatness criteria over Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051H","source_file":"algebra.tex","source_line":24342,"source_end_line":24357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24342-L24357","statement_sha256":"43377c086b9b128b85809c1198293a19e6b8cff8fed7689c1028e63d17973944","origin":"The Stacks Project","memory_eligible":false,"source_rank":1632,"rank":1632,"depth":3,"x":1953.582,"y":135.375,"cluster":"commutative-algebra"},{"id":"stacks:051I","tag":"051I","title":"Flatness criteria over Artinian rings · Lemma 051I","summary":"Let φ : R → R' be a ring map. Let I ⊂ R be an ideal. Let M be an R-module. Assume • M/IM is flat over R/I, and • R' ⊗_R M is flat over R'. Set I_2 = φ^-1(φ(I^2)R'). Then M/I_2M is flat over R/I_2.","statement_latex":"Let $\\varphi : R \\to R'$ be a ring map.\nLet $I \\subset R$ be an ideal.\nLet $M$ be an $R$-module.\nAssume\n\\begin{enumerate}\n\\item $M/IM$ is flat over $R/I$, and\n\\item $R' \\otimes_R M$ is flat over $R'$.\n\\end{enumerate}\nSet $I_2 = \\varphi^{-1}(\\varphi(I^2)R')$.\nThen $M/I_2M$ is flat over $R/I_2$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flatness criteria over Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051I","source_file":"algebra.tex","source_line":24383,"source_end_line":24395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24383-L24395","statement_sha256":"bd6c49a17ca00bc891139643f6744595315db621b345276c6bfbcb2536243d84","origin":"The Stacks Project","memory_eligible":false,"source_rank":1633,"rank":1633,"depth":1,"x":2154.49,"y":239.053,"cluster":"commutative-algebra"},{"id":"stacks:051J","tag":"051J","title":"Flatness criteria over Artinian rings · Lemma 051J","summary":"Let φ : R → R' be a ring map. Let I ⊂ R be an ideal. Let M be an R-module. Assume • I is nilpotent, • R → R' is injective, • M/IM is flat over R/I, and • R' ⊗_R M is flat over R'. Then M is flat over R.","statement_latex":"Let $\\varphi : R \\to R'$ be a ring map.\nLet $I \\subset R$ be an ideal.\nLet $M$ be an $R$-module.\nAssume\n\\begin{enumerate}\n\\item $I$ is nilpotent,\n\\item $R \\to R'$ is injective,\n\\item $M/IM$ is flat over $R/I$, and\n\\item $R' \\otimes_R M$ is flat over $R'$.\n\\end{enumerate}\nThen $M$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flatness criteria over Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051J","source_file":"algebra.tex","source_line":24418,"source_end_line":24431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24418-L24431","statement_sha256":"1e6b85cdf581e6ebc25175410f00e49abc1c57b35d3abc8ded11b5a237a6c8f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1634,"rank":1634,"depth":2,"x":1922.805,"y":276.629,"cluster":"commutative-algebra"},{"id":"stacks:051K","tag":"051K","title":"Flatness criteria over Artinian rings · Lemma 051K","summary":"Let R be an Artinian local ring. Let M be an R-module. Let I ⊂ R be a proper ideal. The following are equivalent • M is flat over R, and • M/IM is flat over R/I and Tor_1^R(R/I, M) = 0.","statement_latex":"Let $R$ be an Artinian local ring. Let $M$ be an $R$-module.\nLet $I \\subset R$ be a proper ideal. The following are\nequivalent\n\\begin{enumerate}\n\\item $M$ is flat over $R$, and\n\\item $M/IM$ is flat over $R/I$ and $\\text{Tor}_1^R(R/I, M) = 0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flatness criteria over Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051K","source_file":"algebra.tex","source_line":24447,"source_end_line":24456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24447-L24456","statement_sha256":"5ea55cb0ea487758baaa1fa50b9b5aa797f263692aa2d52e1ef6fe8c9ec66f93","origin":"The Stacks Project","memory_eligible":false,"source_rank":1635,"rank":1635,"depth":5,"x":2063.528,"y":117.345,"cluster":"commutative-algebra"},{"id":"stacks:051L","tag":"051L","title":"Flatness criteria over Artinian rings · Lemma 051L","summary":"Let R → S be a ring map. Let M be an R-module. Assume • R is Artinian • R → S is injective, and • M ⊗_R S is a flat S-module. Then M is a flat R-module.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $R$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is Artinian\n\\item $R \\to S$ is injective, and\n\\item $M \\otimes_R S$ is a flat $S$-module.\n\\end{enumerate}\nThen $M$ is a flat $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flatness criteria over Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051L","source_file":"algebra.tex","source_line":24474,"source_end_line":24484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24474-L24484","statement_sha256":"e287c4fe6df8f25a06711a03d7bb28d7cf51a9e340e8f89b244f780d5aa61b35","origin":"The Stacks Project","memory_eligible":false,"source_rank":1636,"rank":1636,"depth":7,"x":2087.87,"y":314.787,"cluster":"commutative-algebra"},{"id":"stacks:06A5","tag":"06A5","title":"Crit\\`ere de platitude par fibres: Nilpotent case · Lemma 06A5","summary":"Let xymatrix S ar[rr] & & S' & R ar[lu] ar[ru] be a commutative diagram in the category of rings. Let I ⊂ R be a nilpotent ideal and M an S'-module. Assume • The module M/IM is a flat S/IS-module. • The module M is a flat R-module. Then M is a flat S-module and S_ q is flat over R for every q ⊂ S such that M ⊗_S kappa( q) is nonzero.","statement_latex":"Let\n$$\n\\xymatrix{\nS \\ar[rr] & & S' \\\\\n& R \\ar[lu] \\ar[ru]\n}\n$$\nbe a commutative diagram in the category of rings.\nLet $I \\subset R$ be a nilpotent ideal and $M$ an $S'$-module. Assume\n\\begin{enumerate}\n\\item The module $M/IM$ is a flat $S/IS$-module.\n\\item The module $M$ is a flat $R$-module.\n\\end{enumerate}\nThen $M$ is a flat $S$-module and $S_{\\mathfrak q}$ is flat over $R$\nfor every $\\mathfrak q \\subset S$ such that $M \\otimes_S \\kappa(\\mathfrak q)$\nis nonzero.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Flatness criteria over Artinian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06A5","source_file":"algebra.tex","source_line":24528,"source_end_line":24546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24528-L24546","statement_sha256":"718d87af21e12b6dc182fd04b387b61e7099b346e4a447da021164aa5cd109e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1637,"rank":1637,"depth":2,"x":1911.019,"y":182.916,"cluster":"commutative-algebra"},{"id":"stacks:00MT","tag":"00MT","title":"What makes a complex exact? · Lemma 00MT","summary":"Suppose R is a ring. Let … xrightarrowφ_i + 1 R^n_i xrightarrowφ_i R^n_i-1 xrightarrowφ_i-1 … be a complex of finite free R-modules. Suppose that for some i some matrix coefficient of the map φ_i is invertible. Then the displayed complex is isomorphic to the direct sum of a complex … → R^n_i + 2 xrightarrowφ_i + 2 R^n_i + 1 → R^n_i - 1 → R^n_i - 1 - 1 → R^n_i - 2 xrightarrowφ_i - 2 R^n_i - 3 → … and the complex … → 0 → R → R → 0 → … where the map R → R is the identity map.","statement_latex":"Suppose $R$ is a ring. Let\n$$\n\\ldots\n\\xrightarrow{\\varphi_{i + 1}}\nR^{n_i}\n\\xrightarrow{\\varphi_i}\nR^{n_{i-1}}\n\\xrightarrow{\\varphi_{i-1}}\n\\ldots\n$$\nbe a complex of finite free $R$-modules. Suppose that for some $i$\nsome matrix coefficient of the map $\\varphi_i$ is invertible.\nThen the displayed complex is isomorphic to the direct sum of a complex\n$$\n\\ldots \\to\nR^{n_{i + 2}} \\xrightarrow{\\varphi_{i + 2}}\nR^{n_{i + 1}} \\to\nR^{n_i - 1} \\to\nR^{n_{i - 1} - 1} \\to\nR^{n_{i - 2}} \\xrightarrow{\\varphi_{i - 2}}\nR^{n_{i - 3}} \\to\n\\ldots\n$$\nand the complex $\\ldots \\to 0 \\to R \\to R \\to 0 \\to \\ldots$\nwhere the map $R \\to R$ is the identity map.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"What makes a complex exact?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MT","source_file":"algebra.tex","source_line":24607,"source_end_line":24634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24607-L24634","statement_sha256":"394896b17ccabed68646dd1ffd6eee53c04d4593314915f695e9b6141de42225","origin":"The Stacks Project","memory_eligible":false,"source_rank":1638,"rank":1638,"depth":0,"x":2147.639,"y":179.804,"cluster":"commutative-algebra"},{"id":"stacks:00MY","tag":"00MY","title":"What makes a complex exact? · Lemma 00MY","summary":"In Situation [Tag 00MS]. Suppose R is a local Noetherian ring with maximal ideal m. Assume m ∈ Ass(R), in other words R has depth 0. Suppose that 0 → R^n_e → R^n_e-1 → … → R^n_0 is exact at R^n_e, …, R^n_1. Then the complex is isomorphic to a direct sum of trivial complexes.","statement_latex":"In Situation \\ref{situation-complex}. Suppose $R$ is\na local Noetherian ring with maximal ideal $\\mathfrak m$.\nAssume $\\mathfrak m \\in \\text{Ass}(R)$, in other words\n$R$ has depth $0$. Suppose that\n$0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$\nis exact at $R^{n_e}, \\ldots, R^{n_1}$.\nThen the complex is isomorphic to a direct sum of trivial\ncomplexes.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"What makes a complex exact?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MY","source_file":"algebra.tex","source_line":24677,"source_end_line":24687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24677-L24687","statement_sha256":"9e849b4feac613c058b03eb21eb75a1e1166275c33dc2355bf521f6a13717da8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1639,"rank":1639,"depth":1,"x":1975.541,"y":316.459,"cluster":"commutative-algebra"},{"id":"stacks:00MU","tag":"00MU","title":"What makes a complex exact? · Lemma 00MU","summary":"In Situation [Tag 00MS]. Let R be a Artinian local ring. Suppose that 0 → R^n_e → R^n_e-1 → … → R^n_0 is exact at R^n_e, …, R^n_1. Then the complex is isomorphic to a direct sum of trivial complexes.","statement_latex":"In Situation \\ref{situation-complex}. Let $R$ be a Artinian local ring.\nSuppose that $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$\nis exact at $R^{n_e}, \\ldots, R^{n_1}$. Then the complex is isomorphic\nto a direct sum of trivial complexes.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"What makes a complex exact?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MU","source_file":"algebra.tex","source_line":24702,"source_end_line":24708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24702-L24708","statement_sha256":"3ec173c87071314fd36773a752ec4a79986f5b52d686ba0e192d2d9a3e1ea7d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1640,"rank":1640,"depth":2,"x":1992.562,"y":117.9,"cluster":"commutative-algebra"},{"id":"stacks:00MV","tag":"00MV","title":"What makes a complex exact? · Definition 00MV","summary":"Let R be a ring. Suppose that φ : R^m → R^n is a map of finite free modules. • The rank of φ is the maximal r such that wedge^r φ : wedge^r R^m → wedge^r R^n is nonzero. • We let I(φ) ⊂ R be the ideal generated by the r × r minors of the matrix of φ, where r is the rank as defined above.","statement_latex":"Let $R$ be a ring. Suppose that $\\varphi : R^m \\to R^n$ is a map\nof finite free modules.\n\\begin{enumerate}\n\\item The {\\it rank} of $\\varphi$ is the maximal $r$ such that\n$\\wedge^r \\varphi : \\wedge^r R^m \\to \\wedge^r R^n$ is nonzero.\n\\item We let $I(\\varphi) \\subset R$ be the ideal generated by\nthe $r \\times r$ minors of the matrix of $\\varphi$, where $r$\nis the rank as defined above.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"What makes a complex exact?","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MV","source_file":"algebra.tex","source_line":24721,"source_end_line":24732,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24721-L24732","statement_sha256":"c4c91dc263e46df2b716e369802094bc2109f774a901a591aae8ed0d0c2276ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":1641,"rank":1641,"depth":0,"x":2139.789,"y":274.082,"cluster":"commutative-algebra"},{"id":"stacks:00MW","tag":"00MW","title":"What makes a complex exact? · Lemma 00MW","summary":"In Situation [Tag 00MS], suppose the complex is isomorphic to a direct sum of trivial complexes. Then we have • the maps φ_i have rank r_i = n_i - n_i + 1 + … + (-1)^e-i-1 n_e-1 + (-1)^e-i n_e, • for all i, 1 ≤ i ≤ e - 1 we have rank(φ_i + 1) + rank(φ_i) = n_i, • each I(φ_i) = R.","statement_latex":"In Situation \\ref{situation-complex}, suppose the complex is\nisomorphic to a direct sum of trivial complexes. Then\nwe have\n\\begin{enumerate}\n\\item the maps $\\varphi_i$ have rank\n$r_i = n_i - n_{i + 1} + \\ldots + (-1)^{e-i-1} n_{e-1} + (-1)^{e-i} n_e$,\n\\item for all $i$, $1 \\leq i \\leq e - 1$ we have\n$\\text{rank}(\\varphi_{i + 1}) + \\text{rank}(\\varphi_i) = n_i$,\n\\item each $I(\\varphi_i) = R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"What makes a complex exact?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MW","source_file":"algebra.tex","source_line":24738,"source_end_line":24750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24738-L24750","statement_sha256":"b0b133df87f17c09d99f7ea7f6afd2ff3b5bb6c6d39edee426fbbe5bca7d76b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1642,"rank":1642,"depth":0,"x":1905.465,"y":242.434,"cluster":"commutative-algebra"},{"id":"stacks:00MZ","tag":"00MZ","title":"What makes a complex exact? · Lemma 00MZ","summary":"In Situation [Tag 00MS]. Suppose R is a local ring with maximal ideal m. Suppose that 0 → R^n_e → R^n_e-1 → … → R^n_0 is exact at R^n_e, …, R^n_1. Let x ∈ m be a nonzerodivisor. The complex 0 → (R/xR)^n_e → … → (R/xR)^n_1 is exact at (R/xR)^n_e, …, (R/xR)^n_2.","statement_latex":"In Situation \\ref{situation-complex}. Suppose $R$ is\na local ring with maximal ideal $\\mathfrak m$.\nSuppose that $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$\nis exact at $R^{n_e}, \\ldots, R^{n_1}$.\nLet $x \\in \\mathfrak m$ be a nonzerodivisor. The complex\n$0 \\to (R/xR)^{n_e} \\to \\ldots \\to (R/xR)^{n_1}$\nis exact at $(R/xR)^{n_e}, \\ldots, (R/xR)^{n_2}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"What makes a complex exact?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MZ","source_file":"algebra.tex","source_line":24765,"source_end_line":24774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24765-L24774","statement_sha256":"4e0bc8354aec5e8af3e7ebd32f84021bcee4f97b4c0150adf715d08a1fd91838","origin":"The Stacks Project","memory_eligible":false,"source_rank":1643,"rank":1643,"depth":0,"x":2103.841,"y":132.733,"cluster":"commutative-algebra"},{"id":"stacks:00N0","tag":"00N0","title":"Acyclicity lemma · Lemma 00N0","summary":"[Peskine-Szpiro] Let R be a local Noetherian ring. Let 0 → M_e → M_e-1 → … → M_0 be a complex of finite R-modules. Assume depth(M_i) ≥ i. Let i be the largest index such that the complex is not exact at M_i. If i > 0 then Ker(M_i → M_i-1)/Im(M_i + 1 → M_i) has depth ≥ 1.","statement_latex":"\\begin{reference}\n\\cite[Lemma 1.8]{Peskine-Szpiro}\n\\end{reference}\nLet $R$ be a local Noetherian ring.\nLet $0 \\to M_e \\to M_{e-1} \\to \\ldots \\to M_0$\nbe a complex of finite $R$-modules.\nAssume $\\text{depth}(M_i) \\geq i$.\nLet $i$ be the largest index such that the complex is\nnot exact at $M_i$. If $i > 0$ then\n$\\Ker(M_i \\to M_{i-1})/\\Im(M_{i + 1} \\to M_i)$\nhas depth $\\geq 1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"What makes a complex exact?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00N0","source_file":"algebra.tex","source_line":24792,"source_end_line":24805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24792-L24805","statement_sha256":"bf13771f0353dcfb3b95ced820325164425185bcf035585630d676c07ea7dd78","origin":"The Stacks Project","memory_eligible":false,"source_rank":1644,"rank":1644,"depth":13,"x":2045.74,"y":326.322,"cluster":"commutative-algebra"},{"id":"stacks:00N1","tag":"00N1","title":"What makes a complex exact? · Proposition 00N1","summary":"[WhatExact] In Situation [Tag 00MS], suppose R is a local Noetherian ring. The following are equivalent • 0 → R^n_e → R^n_e-1 → … → R^n_0 is exact at R^n_e, …, R^n_1, and • for all i, 1 ≤ i ≤ e the following two conditions are satisfied: • rank(φ_i) = r_i where r_i = n_i - n_i + 1 + … + (-1)^e-i-1 n_e-1 + (-1)^e-i n_e, • I(φ_i) = R, or I(φ_i) contains a regular sequence of length i.","statement_latex":"\\begin{reference}\n\\cite[Corollary 1]{WhatExact}\n\\end{reference}\nIn Situation \\ref{situation-complex}, suppose $R$ is\na local Noetherian ring. The following are equivalent\n\\begin{enumerate}\n\\item $0 \\to R^{n_e} \\to R^{n_{e-1}} \\to \\ldots \\to R^{n_0}$\nis exact at $R^{n_e}, \\ldots, R^{n_1}$, and\n\\item for all $i$, $1 \\leq i \\leq e$\nthe following two conditions are satisfied:\n\\begin{enumerate}\n\\item $\\text{rank}(\\varphi_i) = r_i$ where\n$r_i = n_i - n_{i + 1} + \\ldots + (-1)^{e-i-1} n_{e-1} + (-1)^{e-i} n_e$,\n\\item $I(\\varphi_i) = R$, or $I(\\varphi_i)$ contains a\nregular sequence of length $i$.\n\\end{enumerate}\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"What makes a complex exact?","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00N1","source_file":"algebra.tex","source_line":24833,"source_end_line":24852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24833-L24852","statement_sha256":"dc3cae847dff71171e3d752420c52c840e0ecfdd9bbdf2b787c4e3834967af35","origin":"The Stacks Project","memory_eligible":false,"source_rank":1645,"rank":1645,"depth":14,"x":1932.825,"y":150.482,"cluster":"commutative-algebra"},{"id":"stacks:00N3","tag":"00N3","title":"Cohen-Macaulay modules · Definition 00N3","summary":"Let R be a Noetherian local ring. Let M be a finite R-module. We say M is Cohen-Macaulay if dim(Supp(M)) = depth(M).","statement_latex":"Let $R$ be a Noetherian local ring.\nLet $M$ be a finite $R$-module.\nWe say $M$ is {\\it Cohen-Macaulay}\nif $\\dim(\\text{Supp}(M)) = \\text{depth}(M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00N3","source_file":"algebra.tex","source_line":24959,"source_end_line":24965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24959-L24965","statement_sha256":"bf842e3b80343ffe68f2e9ee9eaa9622d8c31b7340ab55d743242a777e51a4ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":1646,"rank":1646,"depth":0,"x":2157.648,"y":216.119,"cluster":"commutative-algebra"},{"id":"stacks:00N4","tag":"00N4","title":"Cohen-Macaulay modules · Lemma 00N4","summary":"Notation and assumptions as above. If g is good with respect to (M, f_1, …, f_d), then (a) g is a nonzerodivisor on M, and (b) M/gM is Cohen-Macaulay with maximal regular sequence f_1, …, f_d - 1.","statement_latex":"Notation and assumptions as above. If $g$ is good with respect to\n$(M, f_1, \\ldots, f_d)$, then (a) $g$ is a nonzerodivisor on $M$,\nand (b) $M/gM$ is Cohen-Macaulay with maximal regular\nsequence $f_1, \\ldots, f_{d - 1}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00N4","source_file":"algebra.tex","source_line":24984,"source_end_line":24990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L24984-L24990","statement_sha256":"bbc0d9511d69c2fbf3cf924549d63b07fc567d684926b809b829743644dde489","origin":"The Stacks Project","memory_eligible":false,"source_rank":1647,"rank":1647,"depth":4,"x":1938.932,"y":295.345,"cluster":"commutative-algebra"},{"id":"stacks:00N5","tag":"00N5","title":"Cohen-Macaulay modules · Lemma 00N5","summary":"Let R be a Noetherian local ring. Let M be a Cohen-Macaulay module over R. Suppose g ∈ m is such that dim(Supp(M) ∩ V(g)) = dim(Supp(M)) - 1. Then (a) g is a nonzerodivisor on M, and (b) M/gM is Cohen-Macaulay of depth one less.","statement_latex":"Let $R$ be a Noetherian local ring.\nLet $M$ be a Cohen-Macaulay module over $R$.\nSuppose $g \\in \\mathfrak m$ is such that $\\dim(\\text{Supp}(M) \\cap V(g))\n= \\dim(\\text{Supp}(M)) - 1$. Then (a) $g$ is a nonzerodivisor on $M$,\nand (b) $M/gM$ is Cohen-Macaulay of depth one less.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00N5","source_file":"algebra.tex","source_line":25017,"source_end_line":25024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25017-L25024","statement_sha256":"3dbb025695253b8a3a1f06c8bb6322e5e159222c4956627d90fcb1d76e5a7ce1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1648,"rank":1648,"depth":5,"x":2036.567,"y":112.694,"cluster":"commutative-algebra"},{"id":"stacks:00N6","tag":"00N6","title":"Cohen-Macaulay modules · Proposition 00N6","summary":"Let R be a Noetherian local ring, with maximal ideal m. Let M be a Cohen-Macaulay module over R whose support has dimension d. Suppose that g_1, …, g_c are elements of m such that dim(Supp(M/(g_1, …, g_c)M)) = d - c. Then g_1, …, g_c is an M-regular sequence, and can be extended to a maximal M-regular sequence.","statement_latex":"Let $R$ be a Noetherian local ring, with maximal ideal $\\mathfrak m$.\nLet $M$ be a Cohen-Macaulay module over $R$ whose support has dimension $d$.\nSuppose that $g_1, \\ldots, g_c$ are elements of\n$\\mathfrak m$ such that $\\dim(\\text{Supp}(M/(g_1, \\ldots, g_c)M))\n= d - c$. Then $g_1, \\ldots, g_c$ is an $M$-regular sequence,\nand can be extended to a maximal $M$-regular sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00N6","source_file":"algebra.tex","source_line":25051,"source_end_line":25059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25051-L25059","statement_sha256":"c810100ca8ddcf057a4191de22469806d83fb70cf5373e3a444b9821184d1495","origin":"The Stacks Project","memory_eligible":false,"source_rank":1649,"rank":1649,"depth":11,"x":2111.507,"y":302.909,"cluster":"commutative-algebra"},{"id":"stacks:0C6G","tag":"0C6G","title":"Cohen-Macaulay modules · Lemma 0C6G","summary":"Let R be a Noetherian local ring with maximal ideal m. Let M be a finite R-module. Let x ∈ m be a nonzerodivisor on M. Then M is Cohen-Macaulay if and only if M/xM is Cohen-Macaulay.","statement_latex":"Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$.\nLet $M$ be a finite $R$-module. Let $x \\in \\mathfrak m$ be a\nnonzerodivisor on $M$. Then $M$ is Cohen-Macaulay if and only\nif $M/xM$ is Cohen-Macaulay.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6G","source_file":"algebra.tex","source_line":25081,"source_end_line":25087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25081-L25087","statement_sha256":"c683adbb95bf62b9cce8afa6d8dbc3643feca8f2934878dc2bbf53e528d2a528","origin":"The Stacks Project","memory_eligible":false,"source_rank":1650,"rank":1650,"depth":14,"x":1903.137,"y":205.103,"cluster":"commutative-algebra"},{"id":"stacks:0AAD","tag":"0AAD","title":"Cohen-Macaulay modules · Lemma 0AAD","summary":"Let R → S be a surjective homomorphism of Noetherian local rings. Let N be a finite S-module. Then N is Cohen-Macaulay as an S-module if and only if N is Cohen-Macaulay as an R-module.","statement_latex":"Let $R \\to S$ be a surjective homomorphism of Noetherian local rings.\nLet $N$ be a finite $S$-module. Then $N$ is Cohen-Macaulay as an $S$-module\nif and only if $N$ is Cohen-Macaulay as an $R$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAD","source_file":"algebra.tex","source_line":25096,"source_end_line":25101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25096-L25101","statement_sha256":"4b355f9f02181d44427e9363e2f35404073654e5a19bcd52b1e728756797c27a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1651,"rank":1651,"depth":0,"x":2135.6,"y":158.958,"cluster":"commutative-algebra"},{"id":"stacks:0BUS","tag":"0BUS","title":"Cohen-Macaulay modules · Lemma 0BUS","summary":"[EGA] Let R be a Noetherian local ring. Let M be a finite Cohen-Macaulay R-module. If p ∈ Ass(M), then dim(R/ p) = dim(Supp(M)) and p is a minimal prime in the support of M. In particular, M has no embedded associated primes.","statement_latex":"\\begin{reference}\n\\cite[Chapter 0, Proposition 16.5.4]{EGA}\n\\end{reference}\nLet $R$ be a Noetherian local ring. Let $M$ be a finite Cohen-Macaulay\n$R$-module. If $\\mathfrak p \\in \\text{Ass}(M)$, then\n$\\dim(R/\\mathfrak p) = \\dim(\\text{Supp}(M))$ and $\\mathfrak p$\nis a minimal prime in the support of $M$.\nIn particular, $M$ has no embedded associated primes.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUS","source_file":"algebra.tex","source_line":25107,"source_end_line":25117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25107-L25117","statement_sha256":"5ea72828cd7aae4e5245cdfe66b372be5a8ca9ceba8db29fed4276b773878078","origin":"The Stacks Project","memory_eligible":false,"source_rank":1652,"rank":1652,"depth":15,"x":2001.201,"y":325.003,"cluster":"commutative-algebra"},{"id":"stacks:00NF","tag":"00NF","title":"Cohen-Macaulay modules · Definition 00NF","summary":"Let R be a Noetherian local ring. A finite module M over R is called a maximal Cohen-Macaulay module if depth(M) = dim(R).","statement_latex":"Let $R$ be a Noetherian local ring.\nA finite module $M$ over $R$ is called a {\\it maximal Cohen-Macaulay}\nmodule if $\\text{depth}(M) = \\dim(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NF","source_file":"algebra.tex","source_line":25134,"source_end_line":25139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25134-L25139","statement_sha256":"3af44425c0431f1e2fa43101790830407820b72262e611127775f869fc656c14","origin":"The Stacks Project","memory_eligible":false,"source_rank":1653,"rank":1653,"depth":0,"x":1966.752,"y":126.166,"cluster":"commutative-algebra"},{"id":"stacks:0AAE","tag":"0AAE","title":"Cohen-Macaulay modules · Lemma 0AAE","summary":"In a local Cohen-Macaulay ring, any maximal chain of prime ideals has length equal to the dimension. Let R be a Noetherian local ring. Assume there exists a Cohen-Macaulay module M with Spec(R) = Supp(M). Then any maximal chain of prime ideals p_0 ⊂ p_1 ⊂ … ⊂ p_n has length n = dim(R).","statement_latex":"\\begin{slogan}\nIn a local Cohen-Macaulay ring, any maximal chain of prime ideals has\nlength equal to the dimension.\n\\end{slogan}\nLet $R$ be a Noetherian local ring. Assume there exists a\nCohen-Macaulay module $M$ with $\\Spec(R) = \\text{Supp}(M)$.\nThen any maximal chain of prime ideals $\\mathfrak p_0 \\subset\n\\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_n$\nhas length $n = \\dim(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAE","source_file":"algebra.tex","source_line":25150,"source_end_line":25161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25150-L25161","statement_sha256":"31b28622ceb65f7338c70bc3b756ee2a2aec6b35dc57feebd7ff71522f4095a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1654,"rank":1654,"depth":16,"x":2152.18,"y":253.326,"cluster":"commutative-algebra"},{"id":"stacks:0AAF","tag":"0AAF","title":"Cohen-Macaulay modules · Lemma 0AAF","summary":"Suppose R is a Noetherian local ring. Assume there exists a Cohen-Macaulay module M with Spec(R) = Supp(M). Then for a prime p ⊂ R we have dim(R) = dim(R_ p) + dim(R/ p).","statement_latex":"Suppose $R$ is a Noetherian local ring. Assume there exists a\nCohen-Macaulay module $M$ with $\\Spec(R) = \\text{Supp}(M)$. Then for\na prime $\\mathfrak p \\subset R$ we have\n$$\n\\dim(R) = \\dim(R_{\\mathfrak p}) + \\dim(R/\\mathfrak p).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAF","source_file":"algebra.tex","source_line":25187,"source_end_line":25195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25187-L25195","statement_sha256":"14d8a5e712ff0182bfccb639fc868325154ba64240d08e61b6b8911bbe1d4b3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1655,"rank":1655,"depth":17,"x":1913.026,"y":264.785,"cluster":"commutative-algebra"},{"id":"stacks:0AAG","tag":"0AAG","title":"Cohen-Macaulay modules · Lemma 0AAG","summary":"Suppose R is a Noetherian local ring. Let M be a Cohen-Macaulay module over R. For any prime p ⊂ R the module M_ p is Cohen-Macaulay over R_ p.","statement_latex":"Suppose $R$ is a Noetherian local ring. Let $M$ be a Cohen-Macaulay\nmodule over $R$. For any prime $\\mathfrak p \\subset R$ the\nmodule $M_{\\mathfrak p}$ is Cohen-Macaulay over $R_\\mathfrak p$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAG","source_file":"algebra.tex","source_line":25201,"source_end_line":25206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25201-L25206","statement_sha256":"3c5c8150e0e07a54497e89dc730d77c9a39c6ff0622f6302d035b9a2ba89f7c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1656,"rank":1656,"depth":16,"x":2080.274,"y":120.535,"cluster":"commutative-algebra"},{"id":"stacks:0AAH","tag":"0AAH","title":"Cohen-Macaulay modules · Definition 0AAH","summary":"Let R be a Noetherian ring. Let M be a finite R-module. We say M is Cohen-Macaulay if M_ p is a Cohen-Macaulay module over R_ p for all primes p of R.","statement_latex":"Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module.\nWe say $M$ is {\\it Cohen-Macaulay} if $M_\\mathfrak p$ is a Cohen-Macaulay\nmodule over $R_\\mathfrak p$ for all primes $\\mathfrak p$ of $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAH","source_file":"algebra.tex","source_line":25227,"source_end_line":25232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25227-L25232","statement_sha256":"8c988aaac7ed7230c434bbb7fc59a26b764ca80e9092a38c54a107adc3d609c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1657,"rank":1657,"depth":0,"x":2072.945,"y":321.941,"cluster":"commutative-algebra"},{"id":"stacks:0AAI","tag":"0AAI","title":"Cohen-Macaulay modules · Lemma 0AAI","summary":"Let R be a Noetherian ring. Let M be a Cohen-Macaulay module over R. Then M ⊗_R R[x_1, …, x_n] is a Cohen-Macaulay module over R[x_1, …, x_n].","statement_latex":"Let $R$ be a Noetherian ring. Let $M$ be a Cohen-Macaulay module\nover $R$. Then $M \\otimes_R R[x_1, \\ldots, x_n]$ is a\nCohen-Macaulay module over $R[x_1, \\ldots, x_n]$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAI","source_file":"algebra.tex","source_line":25238,"source_end_line":25243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25238-L25243","statement_sha256":"fd621858f5056669e12e63e91c233da156faa481831a58cb5745e804ff0139f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1658,"rank":1658,"depth":7,"x":1916.277,"y":169.164,"cluster":"commutative-algebra"},{"id":"stacks:00N8","tag":"00N8","title":"Cohen-Macaulay rings · Definition 00N8","summary":"A Noetherian local ring R is called Cohen-Macaulay if it is Cohen-Macaulay as a module over itself.","statement_latex":"A Noetherian local ring $R$ is called {\\it Cohen-Macaulay}\nif it is Cohen-Macaulay as a module over itself.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00N8","source_file":"algebra.tex","source_line":25305,"source_end_line":25309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25305-L25309","statement_sha256":"bad487f6b470197fdb4b426b0dda73af66618c58f01c0edc79c7346c39becaf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1659,"rank":1659,"depth":0,"x":2154.823,"y":192.938,"cluster":"commutative-algebra"},{"id":"stacks:02JN","tag":"02JN","title":"Cohen-Macaulay rings · Lemma 02JN","summary":"Regular sequences in Cohen-Macaulay local rings are characterized by cutting out something of the correct dimension. Let R be a Noetherian local Cohen-Macaulay ring with maximal ideal m . Let x_1, …, x_c ∈ m be elements. Then x_1, …, x_c is a regular sequence ⇔ dim(R/(x_1, …, x_c)) = dim(R) - c If so x_1, …, x_c can be extended to a regular sequence of length dim(R) and each quotient R/(x_1, …, x_i) is a Cohen-Macaulay ring of dimension dim(R) - i.","statement_latex":"\\begin{slogan}\nRegular sequences in Cohen-Macaulay local rings are characterized\nby cutting out something of the correct dimension.\n\\end{slogan}\nLet $R$ be a Noetherian local Cohen-Macaulay ring with maximal\nideal $\\mathfrak m $. Let $x_1, \\ldots, x_c \\in \\mathfrak m$ be\nelements. Then\n$$\nx_1, \\ldots, x_c\n\\text{ is a regular sequence }\n\\Leftrightarrow\n\\dim(R/(x_1, \\ldots, x_c)) = \\dim(R) - c\n$$\nIf so\n$x_1, \\ldots, x_c$ can be extended to\na regular sequence of length $\\dim(R)$ and each quotient\n$R/(x_1, \\ldots, x_i)$ is a Cohen-Macaulay ring of dimension\n$\\dim(R) - i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JN","source_file":"algebra.tex","source_line":25318,"source_end_line":25338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25318-L25338","statement_sha256":"f8e9ea0487c0bbf5035792ed2aba221255ba9f1f6680648b2899c71fb9882cf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1660,"rank":1660,"depth":12,"x":1959.672,"y":310.844,"cluster":"commutative-algebra"},{"id":"stacks:00N9","tag":"00N9","title":"Cohen-Macaulay rings · Lemma 00N9","summary":"Let R be Noetherian local. Suppose R is Cohen-Macaulay of dimension d. Any maximal chain of ideals p_0 ⊂ p_1 ⊂ … ⊂ p_n has length n = d.","statement_latex":"Let $R$ be Noetherian local.\nSuppose $R$ is Cohen-Macaulay of dimension $d$.\nAny maximal chain of ideals $\\mathfrak p_0 \\subset\n\\mathfrak p_1 \\subset \\ldots \\subset \\mathfrak p_n$\nhas length $n = d$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00N9","source_file":"algebra.tex","source_line":25344,"source_end_line":25351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25344-L25351","statement_sha256":"fc859af546b82b1b413823cab8a3d1a644ffa9d78fa29ee007657c5998251562","origin":"The Stacks Project","memory_eligible":false,"source_rank":1661,"rank":1661,"depth":17,"x":2008.79,"y":113.034,"cluster":"commutative-algebra"},{"id":"stacks:00NA","tag":"00NA","title":"Cohen-Macaulay rings · Lemma 00NA","summary":"Suppose R is a Noetherian local Cohen-Macaulay ring of dimension d. For any prime p ⊂ R we have dim(R) = dim(R_ p) + dim(R/ p).","statement_latex":"Suppose $R$ is a Noetherian local Cohen-Macaulay ring of dimension $d$.\nFor any prime $\\mathfrak p \\subset R$ we have\n$$\n\\dim(R) = \\dim(R_{\\mathfrak p}) + \\dim(R/\\mathfrak p).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NA","source_file":"algebra.tex","source_line":25357,"source_end_line":25364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25357-L25364","statement_sha256":"a7c8be1a15aef4a2ed89d7727e48503fc72cbad7e2f5b618ea2eed9aa39ce82d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1662,"rank":1662,"depth":18,"x":2131.728,"y":286.885,"cluster":"commutative-algebra"},{"id":"stacks:00NB","tag":"00NB","title":"Cohen-Macaulay rings · Lemma 00NB","summary":"Suppose R is a Cohen-Macaulay local ring. For any prime p ⊂ R the ring R_ p is Cohen-Macaulay as well.","statement_latex":"Suppose $R$ is a Cohen-Macaulay local ring.\nFor any prime $\\mathfrak p \\subset R$ the\nring $R_{\\mathfrak p}$ is Cohen-Macaulay as well.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NB","source_file":"algebra.tex","source_line":25371,"source_end_line":25376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25371-L25376","statement_sha256":"a2b810a42e46d176fccd97f80d96bf71324207b90780051a62b34096e8d91f0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1663,"rank":1663,"depth":17,"x":1901.113,"y":228.408,"cluster":"commutative-algebra"},{"id":"stacks:00NC","tag":"00NC","title":"Cohen-Macaulay rings · Definition 00NC","summary":"A Noetherian ring R is called Cohen-Macaulay if all its local rings are Cohen-Macaulay.","statement_latex":"A Noetherian ring $R$ is called {\\it Cohen-Macaulay} if all\nits local rings are Cohen-Macaulay.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NC","source_file":"algebra.tex","source_line":25382,"source_end_line":25386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25382-L25386","statement_sha256":"63142cfe8d591294aaed80abb90fb689ce272884020f1dfecc5893254175636b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1664,"rank":1664,"depth":0,"x":2118.337,"y":140.613,"cluster":"commutative-algebra"},{"id":"stacks:00ND","tag":"00ND","title":"Cohen-Macaulay rings · Lemma 00ND","summary":"Suppose R is a Noetherian Cohen-Macaulay ring. Any polynomial algebra over R is Cohen-Macaulay.","statement_latex":"Suppose $R$ is a Noetherian Cohen-Macaulay ring.\nAny polynomial algebra over $R$ is Cohen-Macaulay.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ND","source_file":"algebra.tex","source_line":25388,"source_end_line":25392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25388-L25392","statement_sha256":"e496c2a79568cc47336b85ad7e03a8fc8be6bf747527b6a2e1af7cb230442c72","origin":"The Stacks Project","memory_eligible":false,"source_rank":1665,"rank":1665,"depth":8,"x":2028.702,"y":328.737,"cluster":"commutative-algebra"},{"id":"stacks:00NE","tag":"00NE","title":"Cohen-Macaulay rings · Lemma 00NE","summary":"Let R be a Noetherian local Cohen-Macaulay ring of dimension d. Let 0 → K → R^⊕ n → M → 0 be an exact sequence of R-modules. Then either M = 0, or depth(K) > depth(M), or depth(K) = depth(M) = d.","statement_latex":"Let $R$ be a Noetherian local Cohen-Macaulay ring of dimension $d$.\nLet $0 \\to K \\to R^{\\oplus n} \\to M \\to 0$ be an exact sequence of\n$R$-modules. Then either $M = 0$, or $\\text{depth}(K) > \\text{depth}(M)$, or\n$\\text{depth}(K) = \\text{depth}(M) = d$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NE","source_file":"algebra.tex","source_line":25398,"source_end_line":25404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25398-L25404","statement_sha256":"f708c6317a220e08542b6a90a772b5a53b9d33413fef14668cbe4631065b3610","origin":"The Stacks Project","memory_eligible":false,"source_rank":1666,"rank":1666,"depth":13,"x":1943.456,"y":139.028,"cluster":"commutative-algebra"},{"id":"stacks:00NG","tag":"00NG","title":"Cohen-Macaulay rings · Lemma 00NG","summary":"Let R be a local Noetherian Cohen-Macaulay ring of dimension d. Let M be a finite R-module of depth e. There exists an exact complex 0 → K → F_d-e-1 → … → F_0 → M → 0 with each F_i finite free and K maximal Cohen-Macaulay.","statement_latex":"Let $R$ be a local Noetherian Cohen-Macaulay ring of dimension $d$.\nLet $M$ be a finite $R$-module of depth $e$.\nThere exists an exact complex\n$$\n0 \\to K \\to F_{d-e-1} \\to \\ldots \\to F_0 \\to M \\to 0\n$$\nwith each $F_i$ finite free and $K$ maximal Cohen-Macaulay.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NG","source_file":"algebra.tex","source_line":25410,"source_end_line":25419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25410-L25419","statement_sha256":"5abe8557229c7a376f6fb1efd8027c546af8a094e307eafa8a950599e17d5708","origin":"The Stacks Project","memory_eligible":false,"source_rank":1667,"rank":1667,"depth":14,"x":2159.019,"y":230.607,"cluster":"commutative-algebra"},{"id":"stacks:06LC","tag":"06LC","title":"Cohen-Macaulay rings · Lemma 06LC","summary":"Let φ : A → B be a map of local rings. Assume that B is Noetherian and Cohen-Macaulay and that m_B = sqrtφ( m_A) B. Then there exists a sequence of elements f_1, …, f_dim(B) in A such that φ(f_1), …, φ(f_dim(B)) is a regular sequence in B.","statement_latex":"Let $\\varphi : A \\to B$ be a map of local rings.\nAssume that $B$ is Noetherian and Cohen-Macaulay and that\n$\\mathfrak m_B = \\sqrt{\\varphi(\\mathfrak m_A) B}$. Then there exists\na sequence of elements $f_1, \\ldots, f_{\\dim(B)}$ in $A$\nsuch that $\\varphi(f_1), \\ldots, \\varphi(f_{\\dim(B)})$ is a\nregular sequence in $B$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LC","source_file":"algebra.tex","source_line":25425,"source_end_line":25433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25425-L25433","statement_sha256":"d6192d69977662564546f08018b77c387446c67d0d727d3fb1e062223786f4d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1668,"rank":1668,"depth":13,"x":1926.264,"y":285.43,"cluster":"commutative-algebra"},{"id":"stacks:00NI","tag":"00NI","title":"Catenary rings · Definition 00NI","summary":"A ring R is said to be catenary if for any pair of prime ideals p ⊂ q, there exists an integer bounding the lengths of all finite chains of prime ideals p = p_0 ⊂ p_1 ⊂ … ⊂ p_e = q and all maximal such chains have the same length.","statement_latex":"A ring $R$ is said to be {\\it catenary} if for any pair of prime ideals\n$\\mathfrak p \\subset \\mathfrak q$, there exists an integer bounding the\nlengths of all finite chains of prime ideals\n$\\mathfrak p = \\mathfrak p_0 \\subset \\mathfrak p_1 \\subset \\ldots \\subset\n\\mathfrak p_e = \\mathfrak q$ and all maximal such chains have the same length.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NI","source_file":"algebra.tex","source_line":25462,"source_end_line":25469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25462-L25469","statement_sha256":"95d44d4581a69b5b360e1af451ea6d60e1fe2f0b10db0698053a6d73e73fdbd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1669,"rank":1669,"depth":0,"x":2053.892,"y":112.818,"cluster":"commutative-algebra"},{"id":"stacks:02IH","tag":"02IH","title":"Catenary rings · Lemma 02IH","summary":"A ring R is catenary if and only if the topological space Spec(R) is catenary (see Topology, Definition [Tag 02I1]).","statement_latex":"A ring $R$ is catenary if and only if the topological space\n$\\Spec(R)$ is catenary (see\nTopology, Definition \\ref{topology-definition-catenary}).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IH","source_file":"algebra.tex","source_line":25471,"source_end_line":25476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25471-L25476","statement_sha256":"97ea148248bb439e9693cc5bc48068976278fda1f970fc350db5b57003fc295e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1670,"rank":1670,"depth":2,"x":2098.621,"y":312.653,"cluster":"commutative-algebra"},{"id":"stacks:00NL","tag":"00NL","title":"Catenary rings · Definition 00NL","summary":"A Noetherian ring R is said to be universally catenary if every R-algebra of finite type is catenary.","statement_latex":"A Noetherian ring $R$ is said to be {\\it universally catenary}\nif every $R$-algebra of finite type is catenary.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NL","source_file":"algebra.tex","source_line":25488,"source_end_line":25492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25488-L25492","statement_sha256":"a843938e7e065e979c061c8d1aba80b2d83382af8bb14a7d40a180479ecff612","origin":"The Stacks Project","memory_eligible":false,"source_rank":1671,"rank":1671,"depth":0,"x":1904.807,"y":190.596,"cluster":"commutative-algebra"},{"id":"stacks:00NJ","tag":"00NJ","title":"Catenary rings · Lemma 00NJ","summary":"Any localization of a catenary ring is catenary. Any localization of a Noetherian universally catenary ring is universally catenary.","statement_latex":"Any localization of a catenary ring is catenary.\nAny localization of a Noetherian universally catenary\nring is universally catenary.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NJ","source_file":"algebra.tex","source_line":25501,"source_end_line":25506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25501-L25506","statement_sha256":"40c260a0b6bfc1aebb9de8494f8d131ce2115f1b2f0a6d401f945fdf85ebd8c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1672,"rank":1672,"depth":2,"x":2146.038,"y":170.611,"cluster":"commutative-algebra"},{"id":"stacks:0ECE","tag":"0ECE","title":"Catenary rings · Lemma 0ECE","summary":"Let A be a Noetherian universally catenary ring. Any A-algebra essentially of finite type over A is universally catenary.","statement_latex":"Let $A$ be a Noetherian universally catenary ring.\nAny $A$-algebra essentially of finite type over $A$\nis universally catenary.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECE","source_file":"algebra.tex","source_line":25517,"source_end_line":25522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25517-L25522","statement_sha256":"66c6b94b7b406ed61f0e29d428e3490dbfa2930af40f8e33b105b702ccba8c19","origin":"The Stacks Project","memory_eligible":false,"source_rank":1673,"rank":1673,"depth":3,"x":1984.121,"y":322.33,"cluster":"commutative-algebra"},{"id":"stacks:0AUN","tag":"0AUN","title":"Catenary rings · Lemma 0AUN","summary":"Let R be a ring. The following are equivalent • R is catenary, • R_ p is catenary for all prime ideals p, • R_ m is catenary for all maximal ideals m. Assume R is Noetherian. The following are equivalent • R is universally catenary, • R_ p is universally catenary for all prime ideals p, • R_ m is universally catenary for all maximal ideals m.","statement_latex":"Let $R$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $R$ is catenary,\n\\item $R_\\mathfrak p$ is catenary for all prime ideals $\\mathfrak p$,\n\\item $R_\\mathfrak m$ is catenary for all maximal ideals $\\mathfrak m$.\n\\end{enumerate}\nAssume $R$ is Noetherian. The following are equivalent\n\\begin{enumerate}\n\\item $R$ is universally catenary,\n\\item $R_\\mathfrak p$ is universally catenary for all prime ideals\n$\\mathfrak p$,\n\\item $R_\\mathfrak m$ is universally catenary for all maximal ideals\n$\\mathfrak m$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUN","source_file":"algebra.tex","source_line":25534,"source_end_line":25550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25534-L25550","statement_sha256":"918923958d5d220d50f5724cca7e6dae562220e12759964d41d969d8e1240885","origin":"The Stacks Project","memory_eligible":false,"source_rank":1674,"rank":1674,"depth":3,"x":1981.507,"y":118.443,"cluster":"commutative-algebra"},{"id":"stacks:00NK","tag":"00NK","title":"Catenary rings · Lemma 00NK","summary":"Any quotient of a catenary ring is catenary. Any quotient of a Noetherian universally catenary ring is universally catenary.","statement_latex":"Any quotient of a catenary ring is catenary.\nAny quotient of a Noetherian universally catenary ring is\nuniversally catenary.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NK","source_file":"algebra.tex","source_line":25574,"source_end_line":25579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25574-L25579","statement_sha256":"65c51f872602958316d5abcd16c9104ff290b490cecef2b3cd69bfa81ca4be51","origin":"The Stacks Project","memory_eligible":false,"source_rank":1675,"rank":1675,"depth":4,"x":2147.505,"y":267.402,"cluster":"commutative-algebra"},{"id":"stacks:0AUP","tag":"0AUP","title":"Catenary rings · Lemma 0AUP","summary":"Let R be a Noetherian ring. • R is catenary if and only if R/ p is catenary for every minimal prime p. • R is universally catenary if and only if R/ p is universally catenary for every minimal prime p.","statement_latex":"Let $R$ be a Noetherian ring.\n\\begin{enumerate}\n\\item $R$ is catenary if and only if $R/\\mathfrak p$ is catenary\nfor every minimal prime $\\mathfrak p$.\n\\item $R$ is universally catenary if and only if $R/\\mathfrak p$ is\nuniversally catenary for every minimal prime $\\mathfrak p$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUP","source_file":"algebra.tex","source_line":25589,"source_end_line":25598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25589-L25598","statement_sha256":"718bb34cfc08062fbd325c350577ec7b0d929a57e50333849f904a4d4928f045","origin":"The Stacks Project","memory_eligible":false,"source_rank":1676,"rank":1676,"depth":0,"x":1905.151,"y":251.743,"cluster":"commutative-algebra"},{"id":"stacks:00NM","tag":"00NM","title":"Catenary rings · Lemma 00NM","summary":"A Noetherian Cohen-Macaulay ring is universally catenary. More generally, if R is a Noetherian ring and M is a Cohen-Macaulay R-module with Supp(M) = Spec(R), then R is universally catenary.","statement_latex":"A Noetherian Cohen-Macaulay ring is universally catenary.\nMore generally, if $R$ is a Noetherian ring and $M$ is\na Cohen-Macaulay $R$-module with $\\text{Supp}(M) = \\Spec(R)$,\nthen $R$ is universally catenary.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NM","source_file":"algebra.tex","source_line":25606,"source_end_line":25612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25606-L25612","statement_sha256":"cc803898bf3e45e6b4d9e4c95462e1601848412ae46e7b2b0265c2e3f84fc224","origin":"The Stacks Project","memory_eligible":false,"source_rank":1677,"rank":1677,"depth":18,"x":2096.579,"y":125.689,"cluster":"commutative-algebra"},{"id":"stacks:0ECF","tag":"0ECF","title":"Catenary rings · Lemma 0ECF","summary":"Let (A, m) be a Noetherian local ring. The following are equivalent • A is catenary, and • p ↦ dim(A/ p) is a dimension function on Spec(A).","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. The following are equivalent\n\\begin{enumerate}\n\\item $A$ is catenary, and\n\\item $\\mathfrak p \\mapsto \\dim(A/\\mathfrak p)$ is a dimension function\non $\\Spec(A)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECF","source_file":"algebra.tex","source_line":25640,"source_end_line":25648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25640-L25648","statement_sha256":"d6417232fd6f8465a2a39b4fa3a5717ac2d065f1946260757bf23b1780bf7b81","origin":"The Stacks Project","memory_eligible":false,"source_rank":1678,"rank":1678,"depth":5,"x":2056.766,"y":327.393,"cluster":"commutative-algebra"},{"id":"stacks:00NO","tag":"00NO","title":"Regular local rings · Lemma 00NO","summary":"Let (R, m, kappa) be a regular local ring of dimension d. The graded ring bigoplus m^n / m^n + 1 is isomorphic to the graded polynomial algebra kappa[X_1, …, X_d].","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $d$.\nThe graded ring $\\bigoplus \\mathfrak m^n / \\mathfrak m^{n + 1}$\nis isomorphic to the graded polynomial algebra\n$\\kappa[X_1, \\ldots, X_d]$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NO","source_file":"algebra.tex","source_line":25690,"source_end_line":25696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25690-L25696","statement_sha256":"8cf6a448d347712412bcb353cf4b413f7d5b934f68627bf8467bbbc048d75339","origin":"The Stacks Project","memory_eligible":false,"source_rank":1679,"rank":1679,"depth":10,"x":1923.83,"y":155.955,"cluster":"commutative-algebra"},{"id":"stacks:00NP","tag":"00NP","title":"Regular local rings · Lemma 00NP","summary":"Any regular local ring is a domain.","statement_latex":"Any regular local ring is a domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NP","source_file":"algebra.tex","source_line":25714,"source_end_line":25717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25714-L25717","statement_sha256":"b44375d3cf7d89ed976cfd5b580ace3b15b1bb56ec4f34b97fa431a1607d3835","origin":"The Stacks Project","memory_eligible":false,"source_rank":1680,"rank":1680,"depth":11,"x":2159.877,"y":206.974,"cluster":"commutative-algebra"},{"id":"stacks:00NQ","tag":"00NQ","title":"Regular local rings · Lemma 00NQ","summary":"Let R be a regular local ring and let x_1, …, x_d be a minimal set of generators for the maximal ideal m. Then x_1, …, x_d is a regular sequence, and each R/(x_1, …, x_c) is a regular local ring of dimension d - c. In particular R is Cohen-Macaulay.","statement_latex":"Let $R$ be a regular local ring and let\n$x_1, \\ldots, x_d$ be a minimal set of generators\nfor the maximal ideal $\\mathfrak m$. Then\n$x_1, \\ldots, x_d$ is a regular sequence, and\neach $R/(x_1, \\ldots, x_c)$ is a regular local ring\nof dimension $d - c$. In particular $R$ is Cohen-Macaulay.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NQ","source_file":"algebra.tex","source_line":25731,"source_end_line":25739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25731-L25739","statement_sha256":"14ba61036c18568b3bd53a7b1f5015b8229cbddc2c6aa2df5ca61db5867b9ed3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1681,"rank":1681,"depth":12,"x":1944.65,"y":303.356,"cluster":"commutative-algebra"},{"id":"stacks:00NR","tag":"00NR","title":"Regular local rings · Lemma 00NR","summary":"Let R be a regular local ring. Let I ⊂ R be an ideal such that R/I is a regular local ring as well. Then there exists a minimal set of generators x_1, …, x_d for the maximal ideal m of R such that I = (x_1, …, x_c) for some 0 ≤ c ≤ d.","statement_latex":"Let $R$ be a regular local ring. Let $I \\subset R$ be an ideal\nsuch that $R/I$ is a regular local ring as well. Then\nthere exists a minimal set of generators $x_1, \\ldots, x_d$\nfor the maximal ideal $\\mathfrak m$ of $R$ such that\n$I = (x_1, \\ldots, x_c)$ for some $0 \\leq c \\leq d$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NR","source_file":"algebra.tex","source_line":25749,"source_end_line":25756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25749-L25756","statement_sha256":"9a9919cba1efb82f48a09242399b9c414e508952b36450666203f820e82b0239","origin":"The Stacks Project","memory_eligible":false,"source_rank":1682,"rank":1682,"depth":13,"x":2025.901,"y":110.033,"cluster":"commutative-algebra"},{"id":"stacks:00NS","tag":"00NS","title":"Regular local rings · Lemma 00NS","summary":"Let R be a Noetherian local ring. Let x ∈ m. Let M be a finite R-module such that x is a nonzerodivisor on M and M/xM is free over R/xR. Then M is free over R.","statement_latex":"Let $R$ be a Noetherian local ring.\nLet $x \\in \\mathfrak m$.\nLet $M$ be a finite $R$-module such that\n$x$ is a nonzerodivisor on $M$ and\n$M/xM$ is free over $R/xR$.\nThen $M$ is free over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NS","source_file":"algebra.tex","source_line":25779,"source_end_line":25787,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25779-L25787","statement_sha256":"644a4713ada3db35b0203aa4c5bf82ed1449340a358cffa15a9ea536e81a6684","origin":"The Stacks Project","memory_eligible":false,"source_rank":1683,"rank":1683,"depth":3,"x":2121.515,"y":298.814,"cluster":"commutative-algebra"},{"id":"stacks:00NT","tag":"00NT","title":"Regular local rings · Lemma 00NT","summary":"Let R be a regular local ring. Any maximal Cohen-Macaulay module over R is free.","statement_latex":"Let $R$ be a regular local ring.\nAny maximal Cohen-Macaulay module over $R$ is free.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NT","source_file":"algebra.tex","source_line":25799,"source_end_line":25803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25799-L25803","statement_sha256":"5f2532717d9417bc406dec541373d53015690a4f17c41a5bbc6c1d93db9e27ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":1684,"rank":1684,"depth":12,"x":1899.052,"y":213.808,"cluster":"commutative-algebra"},{"id":"stacks:00NU","tag":"00NU","title":"Regular local rings · Lemma 00NU","summary":"Suppose R is a Noetherian local ring. Let x ∈ m be a nonzerodivisor such that R/xR is a regular local ring. Then R is a regular local ring. More generally, if x_1, …, x_r is a regular sequence in R such that R/(x_1, …, x_r) is a regular local ring, then R is a regular local ring.","statement_latex":"Suppose $R$ is a Noetherian local ring.\nLet $x \\in \\mathfrak m$ be a nonzerodivisor\nsuch that $R/xR$ is a regular local ring. Then $R$ is a regular local ring.\nMore generally, if $x_1, \\ldots, x_r$ is a regular sequence in $R$\nsuch that $R/(x_1, \\ldots, x_r)$ is a regular local ring, then\n$R$ is a regular local ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00NU","source_file":"algebra.tex","source_line":25815,"source_end_line":25823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25815-L25823","statement_sha256":"4f85e2b9e94cc85d69db9de024983fdb6ebf5f2a619730eab016f18609b57b6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1685,"rank":1685,"depth":11,"x":2131.606,"y":150.217,"cluster":"commutative-algebra"},{"id":"stacks:07DX","tag":"07DX","title":"Regular local rings · Lemma 07DX","summary":"Let (R_i, φ_ii') be a directed system of local rings whose transition maps are local ring maps. If each R_i is a regular local ring and R = colim R_i is Noetherian, then R is a regular local ring.","statement_latex":"Let $(R_i, \\varphi_{ii'})$ be a directed system of local rings whose\ntransition maps are local ring maps. If each $R_i$ is a regular local ring and\n$R = \\colim R_i$ is Noetherian, then $R$ is a regular local ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DX","source_file":"algebra.tex","source_line":25833,"source_end_line":25838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25833-L25838","statement_sha256":"983358aee5623d014fe345560341b0a550ae64a7cabe875b9062f65e3f4ba42d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1686,"rank":1686,"depth":13,"x":2011.183,"y":329.182,"cluster":"commutative-algebra"},{"id":"stacks:04VN","tag":"04VN","title":"Epimorphisms of rings · Lemma 04VN","summary":"Let R → S be a ring map. The following are equivalent • R → S is an epimorphism, • the two ring maps S → S ⊗_R S are equal, • either of the ring maps S → S ⊗_R S is an isomorphism, and • the ring map S ⊗_R S → S is an isomorphism.","statement_latex":"Let $R \\to S$ be a ring map. The following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is an epimorphism,\n\\item the two ring maps $S \\to S \\otimes_R S$ are equal,\n\\item either of the ring maps $S \\to S \\otimes_R S$ is an isomorphism, and\n\\item the ring map $S \\otimes_R S \\to S$ is an isomorphism.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VN","source_file":"algebra.tex","source_line":25869,"source_end_line":25878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25869-L25878","statement_sha256":"ab1a748c754e701cdd8b74276a2c7be529f9e14bee93084874a270e49b50a672","origin":"The Stacks Project","memory_eligible":false,"source_rank":1687,"rank":1687,"depth":0,"x":1956.026,"y":128.754,"cluster":"commutative-algebra"},{"id":"stacks:04VP","tag":"04VP","title":"Epimorphisms of rings · Lemma 04VP","summary":"The composition of two epimorphisms of rings is an epimorphism.","statement_latex":"The composition of two epimorphisms of rings is an epimorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VP","source_file":"algebra.tex","source_line":25884,"source_end_line":25887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25884-L25887","statement_sha256":"4da470c40f3fd9cbb663b5e32f1eaedd38c985744f32885633a3bd2f687ae667","origin":"The Stacks Project","memory_eligible":false,"source_rank":1688,"rank":1688,"depth":0,"x":2158.008,"y":245.325,"cluster":"commutative-algebra"},{"id":"stacks:04VQ","tag":"04VQ","title":"Epimorphisms of rings · Lemma 04VQ","summary":"If R → S is an epimorphism of rings and R → R' is any ring map, then R' → R' ⊗_R S is an epimorphism.","statement_latex":"If $R \\to S$ is an epimorphism of rings and $R \\to R'$ is any ring map,\nthen $R' \\to R' \\otimes_R S$ is an epimorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VQ","source_file":"algebra.tex","source_line":25893,"source_end_line":25897,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25893-L25897","statement_sha256":"f56f0a88465bcf8166e09fe8da72d39798090bee1eab03a15f859aaf129ee58b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1689,"rank":1689,"depth":0,"x":1915.168,"y":273.996,"cluster":"commutative-algebra"},{"id":"stacks:04VR","tag":"04VR","title":"Epimorphisms of rings · Lemma 04VR","summary":"If A → B → C are ring maps and A → C is an epimorphism, so is B → C.","statement_latex":"If $A \\to B \\to C$ are ring maps and $A \\to C$ is an epimorphism, so is\n$B \\to C$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VR","source_file":"algebra.tex","source_line":25903,"source_end_line":25907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25903-L25907","statement_sha256":"84cc07ba85d9ccb6258203cf23cafe518dcfd54eb2776a07c05eeef0739b82fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":1690,"rank":1690,"depth":0,"x":2071.281,"y":114.958,"cluster":"commutative-algebra"},{"id":"stacks:04VS","tag":"04VS","title":"Epimorphisms of rings · Lemma 04VS","summary":"Let R → S be a ring map. The following are equivalent: • R → S is an epimorphism, and • R_ p → S_ p is an epimorphism for each prime p of R.","statement_latex":"Let $R \\to S$ be a ring map. The following are equivalent:\n\\begin{enumerate}\n\\item $R \\to S$ is an epimorphism, and\n\\item $R_{\\mathfrak p} \\to S_{\\mathfrak p}$ is an epimorphism for\neach prime $\\mathfrak p$ of $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VS","source_file":"algebra.tex","source_line":25920,"source_end_line":25928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25920-L25928","statement_sha256":"d12e1ee688b27ce06ab1f17e78119cc52cbcbfefda309a283890ea2ee62faf5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1691,"rank":1691,"depth":2,"x":2084.068,"y":320.946,"cluster":"commutative-algebra"},{"id":"stacks:04VT","tag":"04VT","title":"Epimorphisms of rings · Lemma 04VT","summary":"A ring map is surjective if and only if it is a finite epimorphism. Let R → S be a ring map. The following are equivalent • R → S is an epimorphism and finite, and • R → S is surjective.","statement_latex":"\\begin{slogan}\nA ring map is surjective if and only if it is a finite epimorphism.\n\\end{slogan}\nLet $R \\to S$ be a ring map. The following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is an epimorphism and finite, and\n\\item $R \\to S$ is surjective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VT","source_file":"algebra.tex","source_line":25943,"source_end_line":25953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25943-L25953","statement_sha256":"1313bfe5c81e188b2a31b397722a41ae201f1dd09877e439edaf63bb55e1f5fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1692,"rank":1692,"depth":1,"x":1908.875,"y":176.215,"cluster":"commutative-algebra"},{"id":"stacks:04VU","tag":"04VU","title":"Epimorphisms of rings · Lemma 04VU","summary":"A faithfully flat epimorphism is an isomorphism.","statement_latex":"A faithfully flat epimorphism is an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VU","source_file":"algebra.tex","source_line":25976,"source_end_line":25979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25976-L25979","statement_sha256":"e9c6f107509415557e8364b2e81e1a0142898e4087524dafb2c5ee802afb6990","origin":"The Stacks Project","memory_eligible":false,"source_rank":1693,"rank":1693,"depth":1,"x":2154.608,"y":183.533,"cluster":"commutative-algebra"},{"id":"stacks:04VV","tag":"04VV","title":"Epimorphisms of rings · Lemma 04VV","summary":"If k → S is an epimorphism and k is a field, then S = k or S = 0.","statement_latex":"If $k \\to S$ is an epimorphism and $k$ is a field, then $S = k$ or $S = 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VV","source_file":"algebra.tex","source_line":25987,"source_end_line":25990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L25987-L25990","statement_sha256":"91e1a0380f0b46bfc218b4e5cc8fe7673e2554e46246bc41671998d4e848d22b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1694,"rank":1694,"depth":2,"x":1967.401,"y":317.658,"cluster":"commutative-algebra"},{"id":"stacks:04VW","tag":"04VW","title":"Epimorphisms of rings · Lemma 04VW","summary":"Let R → S be an epimorphism of rings. Then • Spec(S) → Spec(R) is injective, and • for q ⊂ S lying over p ⊂ R we have kappa( p) = kappa( q).","statement_latex":"Let $R \\to S$ be an epimorphism of rings. Then\n\\begin{enumerate}\n\\item $\\Spec(S) \\to \\Spec(R)$ is injective, and\n\\item for $\\mathfrak q \\subset S$ lying over $\\mathfrak p \\subset R$\nwe have $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VW","source_file":"algebra.tex","source_line":26000,"source_end_line":26008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26000-L26008","statement_sha256":"65bb07ad5db700bf1416471bfe12eb2e6489b26478529b73951721baaf03beee","origin":"The Stacks Project","memory_eligible":false,"source_rank":1695,"rank":1695,"depth":3,"x":1997.604,"y":112.398,"cluster":"commutative-algebra"},{"id":"stacks:04VX","tag":"04VX","title":"Epimorphisms of rings · Lemma 04VX","summary":"Let R be a ring. Let M, N be R-modules. Let (x_i)_i ∈ I be a set of generators of M. Let (y_j)_j ∈ J be a set of generators of N. Let (m_j)_j ∈ J be a family of elements of M with m_j = 0 for all but finitely many j. Then ∑_j ∈ J m_j ⊗ y_j = 0 in M ⊗_R N is equivalent to the following: There exist a_i, j ∈ R with a_i, j = 0 for all but finitely many pairs (i, j) such that m_j & = ∑_i ∈ I a_i, j x_i for all j ∈ J, 0 & = ∑_j ∈ J a_i, j y_j for all i ∈ I.","statement_latex":"Let $R$ be a ring.\nLet $M$, $N$ be $R$-modules.\nLet $\\{x_i\\}_{i \\in I}$ be a set of generators of $M$.\nLet $\\{y_j\\}_{j \\in J}$ be a set of generators of $N$.\nLet $\\{m_j\\}_{j \\in J}$ be a family of elements of $M$ with $m_j = 0$\nfor all but finitely many $j$.\nThen\n$$\n\\sum\\nolimits_{j \\in J} m_j \\otimes y_j = 0 \\text{ in } M \\otimes_R N\n$$\nis equivalent to the following:\nThere exist $a_{i, j} \\in R$ with $a_{i, j} = 0$ for all but finitely many\npairs $(i, j)$ such that\n\\begin{align*}\nm_j & = \\sum\\nolimits_{i \\in I} a_{i, j} x_i \\quad\\text{for all } j \\in J, \\\\\n0 & = \\sum\\nolimits_{j \\in J} a_{i, j} y_j \\quad\\text{for all } i \\in I.\n\\end{align*}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VX","source_file":"algebra.tex","source_line":26022,"source_end_line":26041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26022-L26041","statement_sha256":"ecdec990cf552f48d0a5a6a462c0e3cecd3b39abdb0fba2b2579e5d9852cee8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1696,"rank":1696,"depth":0,"x":2140.49,"y":281.001,"cluster":"commutative-algebra"},{"id":"stacks:04VY","tag":"04VY","title":"Epimorphisms of rings · Lemma 04VY","summary":"Let φ : R → S be a ring map. Let g ∈ S. The following are equivalent: • g ⊗ 1 = 1 ⊗ g in S ⊗_R S, and • there exist n ≥ 0 and elements y_i, z_j ∈ S and x_i, j ∈ R for 1 ≤ i, j ≤ n such that • g = ∑_i, j ≤ n x_i, j y_i z_j, • for each j we have ∑ x_i, jy_i ∈ φ(R), and • for each i we have ∑ x_i, jz_j ∈ φ(R).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nLet $g \\in S$. The following are equivalent:\n\\begin{enumerate}\n\\item $g \\otimes 1 = 1 \\otimes g$ in $S \\otimes_R S$, and\n\\item there exist $n \\geq 0$ and elements $y_i, z_j \\in S$\nand $x_{i, j} \\in R$ for $1 \\leq i, j \\leq n$ such that\n\\begin{enumerate}\n\\item $g = \\sum_{i, j \\leq n} x_{i, j} y_i z_j$,\n\\item for each $j$ we have $\\sum x_{i, j}y_i \\in \\varphi(R)$, and\n\\item for each $i$ we have $\\sum x_{i, j}z_j \\in \\varphi(R)$.\n\\end{enumerate}\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04VY","source_file":"algebra.tex","source_line":26060,"source_end_line":26074,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26060-L26074","statement_sha256":"ab4cff869bc2a5104b24f1d2fd4268eef738fdb759e48d3d62887b96952fc0cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1697,"rank":1697,"depth":1,"x":1899.387,"y":237.725,"cluster":"commutative-algebra"},{"id":"stacks:04W0","tag":"04W0","title":"Epimorphisms of rings · Lemma 04W0","summary":"Let R → S be an epimorphism of rings. Then the cardinality of S is at most the cardinality of R. In a formula: |S| ≤ |R|.","statement_latex":"Let $R \\to S$ be an epimorphism of rings.\nThen the cardinality of $S$ is at most the cardinality of $R$.\nIn a formula: $|S| \\leq |R|$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04W0","source_file":"algebra.tex","source_line":26125,"source_end_line":26130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26125-L26130","statement_sha256":"e0ccc07e092ea706e29ec5ce18c0a61a88177cb6537f7c799aadff53701d076d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1698,"rank":1698,"depth":1,"x":2112.111,"y":132.761,"cluster":"commutative-algebra"},{"id":"stacks:08YS","tag":"08YS","title":"Epimorphisms of rings · Lemma 08YS","summary":"For a ring homomorphism R → S the following are equivalent • R → S is an epimorphism of rings, • for any S-modules N_1, N_2 we have Hom_S(N_1, N_2) = Hom_R(N_1, N_2), and • the restriction functor Mod_S → Mod_R is fully faithful.","statement_latex":"For a ring homomorphism $R \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is an epimorphism of rings,\n\\item for any $S$-modules $N_1, N_2$ we have\n$\\Hom_S(N_1, N_2) = \\Hom_R(N_1, N_2)$, and\n\\item the restriction functor $\\text{Mod}_S \\to \\text{Mod}_R$\nis fully faithful.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Epimorphisms of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YS","source_file":"algebra.tex","source_line":26158,"source_end_line":26168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26158-L26168","statement_sha256":"0428657045aef56ec847667fb3587e8357e72956ed83cb2220951ec1e999e81d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1699,"rank":1699,"depth":1,"x":2039.614,"y":330.992,"cluster":"commutative-algebra"},{"id":"stacks:04PR","tag":"04PR","title":"Pure ideals · Definition 04PR","summary":"Let R be a ring. We say that I ⊂ R is pure if the quotient ring R/I is flat over R.","statement_latex":"Let $R$ be a ring. We say that $I \\subset R$ is {\\it pure}\nif the quotient ring $R/I$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Pure ideals","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PR","source_file":"algebra.tex","source_line":26199,"source_end_line":26203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26199-L26203","statement_sha256":"aec76c9e897d9c7509d33e1dacc47e65a67a01b7f98dd4c214ab347777c0ed10","origin":"The Stacks Project","memory_eligible":false,"source_rank":1700,"rank":1700,"depth":0,"x":1933.592,"y":143.562,"cluster":"commutative-algebra"},{"id":"stacks:04PS","tag":"04PS","title":"Pure ideals · Lemma 04PS","summary":"Let R be a ring. Let I ⊂ R be an ideal. The following are equivalent: • I is pure, • for every ideal J ⊂ R we have J ∩ I = IJ, • for every finitely generated ideal J ⊂ R we have J ∩ I = JI, • for every x ∈ R we have (x) ∩ I = xI, • for every x ∈ I we have x = yx for some y ∈ I, • for every x_1, …, x_n ∈ I there exists a y ∈ I such that x_i = yx_i for all i = 1, …, n, • for every prime p of R we have IR_ p = 0 or IR_ p = R_ p, • Supp(I) = Spec(R) setminus V(I), • I is the…","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $I$ is pure,\n\\item for every ideal $J \\subset R$ we have $J \\cap I = IJ$,\n\\item for every finitely generated ideal $J \\subset R$ we have\n$J \\cap I = JI$,\n\\item for every $x \\in R$ we have $(x) \\cap I = xI$,\n\\item for every $x \\in I$ we have $x = yx$ for some $y \\in I$,\n\\item for every $x_1, \\ldots, x_n \\in I$ there exists a\n$y \\in I$ such that $x_i = yx_i$ for all $i = 1, \\ldots, n$,\n\\item for every prime $\\mathfrak p$ of $R$ we have\n$IR_{\\mathfrak p} = 0$ or $IR_{\\mathfrak p} = R_{\\mathfrak p}$,\n\\item $\\text{Supp}(I) = \\Spec(R) \\setminus V(I)$,\n\\item $I$ is the kernel of the map $R \\to (1 + I)^{-1}R$,\n\\item $R/I \\cong S^{-1}R$ as $R$-algebras for some multiplicative\nsubset $S$ of $R$, and\n\\item $R/I \\cong (1 + I)^{-1}R$ as $R$-algebras.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Pure ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PS","source_file":"algebra.tex","source_line":26205,"source_end_line":26227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26205-L26227","statement_sha256":"959898f6bcbc140f24af34406d435f53c6f1da996c5719fd4aabf9c0de70d7bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1701,"rank":1701,"depth":3,"x":2162.644,"y":221.661,"cluster":"commutative-algebra"},{"id":"stacks:04PT","tag":"04PT","title":"Pure ideals · Lemma 04PT","summary":"Pure ideals are determined by their vanishing locus. Let R be a ring. If I, J ⊂ R are pure ideals, then V(I) = V(J) implies I = J.","statement_latex":"\\begin{slogan}\nPure ideals are determined by their vanishing locus.\n\\end{slogan}\nLet $R$ be a ring.\nIf $I, J \\subset R$ are pure ideals, then $V(I) = V(J)$\nimplies $I = J$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Pure ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PT","source_file":"algebra.tex","source_line":26288,"source_end_line":26296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26288-L26296","statement_sha256":"6a6128ed374e8a423412a659dfd9bdb8cadbd79290f6ec03b744a9135719f110","origin":"The Stacks Project","memory_eligible":false,"source_rank":1702,"rank":1702,"depth":4,"x":1930.791,"y":294.088,"cluster":"commutative-algebra"},{"id":"stacks:04PU","tag":"04PU","title":"Pure ideals · Lemma 04PU","summary":"Let R be a ring. The rule I ↦ V(I) determines a bijection (I ⊂ R pure) ↔ (Z ⊂ Spec(R) closed and closed under generalizations)","statement_latex":"Let $R$ be a ring. The rule\n$I \\mapsto V(I)$\ndetermines a bijection\n$$\n\\{I \\subset R \\text{ pure}\\}\n\\leftrightarrow\n\\{Z \\subset \\Spec(R)\\text{ closed and closed under generalizations}\\}\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Pure ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PU","source_file":"algebra.tex","source_line":26306,"source_end_line":26316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26306-L26316","statement_sha256":"690550570cf9e26a2fad426816f3a84c27b30613adfd1f6424336c989abf8ce7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1703,"rank":1703,"depth":5,"x":2043.584,"y":109.004,"cluster":"commutative-algebra"},{"id":"stacks:05KK","tag":"05KK","title":"Pure ideals · Lemma 05KK","summary":"Let R be a ring. Let I ⊂ R be an ideal. The following are equivalent • I is pure and finitely generated, • I is generated by an idempotent, • I is pure and V(I) is open, and • R/I is a projective R-module.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nThe following are equivalent\n\\begin{enumerate}\n\\item $I$ is pure and finitely generated,\n\\item $I$ is generated by an idempotent,\n\\item $I$ is pure and $V(I)$ is open, and\n\\item $R/I$ is a projective $R$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Pure ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KK","source_file":"algebra.tex","source_line":26366,"source_end_line":26376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26366-L26376","statement_sha256":"c4285f5dc8704555d54f58f5fe8751cdbee948f2526f255bbf24a4f4ab8eef42","origin":"The Stacks Project","memory_eligible":false,"source_rank":1704,"rank":1704,"depth":5,"x":2109.294,"y":309.612,"cluster":"commutative-algebra"},{"id":"stacks:052U","tag":"052U","title":"Pure ideals · Lemma 052U","summary":"Let R be a ring. The following are equivalent: • every Z ⊂ Spec(R) which is closed and closed under generalizations is also open, and • any finite flat R-module is finite locally free.","statement_latex":"Let $R$ be a ring. The following are equivalent:\n\\begin{enumerate}\n\\item every $Z \\subset \\Spec(R)$ which is closed and closed under\ngeneralizations is also open, and\n\\item any finite flat $R$-module is finite locally free.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Pure ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052U","source_file":"algebra.tex","source_line":26405,"source_end_line":26413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26405-L26413","statement_sha256":"fde3188b1d036d7c53a97e3480e0a1a05d6ce8008f7b4ab226af07a71b7fa21b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1705,"rank":1705,"depth":5,"x":1899.383,"y":198.901,"cluster":"commutative-algebra"},{"id":"stacks:00O3","tag":"00O3","title":"Schanuel's lemma · Lemma 00O3","summary":"Let R be a ring. Let M be an R-module. Suppose that 0 → K xrightarrowc_1 P_1 xrightarrowp_1 M → 0 and 0 → L xrightarrowc_2 P_2 xrightarrowp_2 M → 0 are two short exact sequences, with P_i projective. Then K ⊕ P_2 ≅ L ⊕ P_1. More precisely, there exist a commutative diagram xymatrix 0 ar[r] & K ⊕ P_2 ar[r]_(c_1, id) ar[d] & P_1 ⊕ P_2 ar[r]_(p_1, 0) ar[d] & M ar[r] ar@=[d] & 0 0 ar[r] & P_1 ⊕ L ar[r]^(id, c_2) & P_1 ⊕ P_2 ar[r]^(0, p_2) & M ar[r] & 0 whose vertical arrows…","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nSuppose that\n$$\n0 \\to K \\xrightarrow{c_1} P_1 \\xrightarrow{p_1} M \\to 0\n\\quad\\text{and}\\quad\n0 \\to L \\xrightarrow{c_2} P_2 \\xrightarrow{p_2} M \\to 0\n$$\nare two short exact sequences, with $P_i$ projective.\nThen $K \\oplus P_2 \\cong L \\oplus P_1$. More precisely,\nthere exist a commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\nK \\oplus P_2 \\ar[r]_{(c_1, \\text{id})} \\ar[d] &\nP_1 \\oplus P_2 \\ar[r]_{(p_1, 0)} \\ar[d] &\nM \\ar[r] \\ar@{=}[d] &\n0 \\\\\n0 \\ar[r] &\nP_1 \\oplus L \\ar[r]^{(\\text{id}, c_2)} &\nP_1 \\oplus P_2 \\ar[r]^{(0, p_2)} &\nM \\ar[r] &\n0\n}\n$$\nwhose vertical arrows are isomorphisms.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00O3","source_file":"algebra.tex","source_line":26484,"source_end_line":26511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26484-L26511","statement_sha256":"68d3ed8f312cf161dcd1043a9a39c5260087600fd4df97012e9195aa34dba45a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1706,"rank":1706,"depth":0,"x":2143.354,"y":161.406,"cluster":"commutative-algebra"},{"id":"stacks:00O4","tag":"00O4","title":"Rings of finite global dimension · Definition 00O4","summary":"Let R be a ring. Let M be an R-module. We say M has finite projective dimension if it has a finite length resolution by projective R-modules. The minimal length of such a resolution is called the projective dimension of M.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. We say $M$ has\n{\\it finite projective dimension} if it has a finite length\nresolution by projective $R$-modules. The minimal length of such a\nresolution is called the {\\it projective dimension}\nof $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00O4","source_file":"algebra.tex","source_line":26568,"source_end_line":26575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26568-L26575","statement_sha256":"ce7e1f86262679567ee38cac92bd8a967aa40ad5b50c16a8e4e8aa55b203988f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1707,"rank":1707,"depth":0,"x":1993.511,"y":327.595,"cluster":"commutative-algebra"},{"id":"stacks:00O5","tag":"00O5","title":"Rings of finite global dimension · Lemma 00O5","summary":"Let R be a ring. Suppose that M is an R-module of projective dimension d. Suppose that F_e → F_e-1 → … → F_0 → M → 0 is exact with F_i projective and e ≥ d - 1. Then the kernel of F_e → F_e-1 is projective (or the kernel of F_0 → M is projective in case e = 0).","statement_latex":"Let $R$ be a ring. Suppose that $M$ is an $R$-module of projective\ndimension $d$. Suppose that $F_e \\to F_{e-1} \\to \\ldots \\to F_0 \\to M \\to 0$\nis exact with $F_i$ projective and $e \\geq d - 1$.\nThen the kernel of $F_e \\to F_{e-1}$ is projective\n(or the kernel of $F_0 \\to M$ is projective in case\n$e = 0$).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00O5","source_file":"algebra.tex","source_line":26584,"source_end_line":26592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26584-L26592","statement_sha256":"c6ab523177de49e06a1db8b108ec6e2de4a5a9812d342d13c0bba0b5dcd8519f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1708,"rank":1708,"depth":1,"x":1970.344,"y":119.893,"cluster":"commutative-algebra"},{"id":"stacks:0CXC","tag":"0CXC","title":"Rings of finite global dimension · Lemma 0CXC","summary":"Let R be a ring. Let M be an R-module. Let d ≥ 0. The following are equivalent • M has projective dimension ≤ d, • there exists a resolution 0 → P_d → P_d - 1 → … → P_0 → M → 0 with P_i projective, • for some resolution … → P_2 → P_1 → P_0 → M → 0 with P_i projective we have Ker(P_d - 1 → P_d - 2) is projective if d ≥ 2, or Ker(P_0 → M) is projective if d = 1, or M is projective if d = 0, • for any resolution … → P_2 → P_1 → P_0 → M → 0 with P_i projective we have Ker(P_d…","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. Let $d \\geq 0$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ has projective dimension $\\leq d$,\n\\item there exists a resolution\n$0 \\to P_d \\to P_{d - 1} \\to \\ldots \\to P_0 \\to M \\to 0$\nwith $P_i$ projective,\n\\item for some resolution\n$\\ldots \\to P_2 \\to P_1 \\to P_0 \\to M \\to 0$ with\n$P_i$ projective we have $\\Ker(P_{d - 1} \\to P_{d - 2})$\nis projective if $d \\geq 2$, or $\\Ker(P_0 \\to M)$ is projective if\n$d = 1$, or $M$ is projective if $d = 0$,\n\\item for any resolution\n$\\ldots \\to P_2 \\to P_1 \\to P_0 \\to M \\to 0$ with\n$P_i$ projective we have $\\Ker(P_{d - 1} \\to P_{d - 2})$\nis projective if $d \\geq 2$, or $\\Ker(P_0 \\to M)$ is projective if\n$d = 1$, or $M$ is projective if $d = 0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXC","source_file":"algebra.tex","source_line":26629,"source_end_line":26649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26629-L26649","statement_sha256":"41ed7168dd604d05e5dbf09fdcf70d1d6b505e53809ab8fea01f5e6f7f95c24d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1709,"rank":1709,"depth":2,"x":2154.572,"y":259.992,"cluster":"commutative-algebra"},{"id":"stacks:0CXD","tag":"0CXD","title":"Rings of finite global dimension · Lemma 0CXD","summary":"Let R be a local ring. Let M be an R-module. Let d ≥ 0. The equivalent conditions (1) -- (4) of Lemma [Tag 0CXC] are also equivalent to • [(5)] there exists a resolution 0 → P_d → P_d - 1 → … → P_0 → M → 0 with P_i free.","statement_latex":"Let $R$ be a local ring. Let $M$ be an $R$-module. Let $d \\geq 0$.\nThe equivalent conditions (1) -- (4) of\nLemma \\ref{lemma-what-kind-of-resolutions}\nare also equivalent to\n\\begin{enumerate}\n\\item[(5)] there exists a resolution\n$0 \\to P_d \\to P_{d - 1} \\to \\ldots \\to P_0 \\to M \\to 0$\nwith $P_i$ free.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXD","source_file":"algebra.tex","source_line":26659,"source_end_line":26670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26659-L26670","statement_sha256":"b5156e56d20adee1dd6d9d9be26484ac5f5d4130f9e64c40500e28e53e20e1b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1710,"rank":1710,"depth":4,"x":1905.903,"y":261.222,"cluster":"commutative-algebra"},{"id":"stacks:0CXE","tag":"0CXE","title":"Rings of finite global dimension · Lemma 0CXE","summary":"Let R be a Noetherian ring. Let M be a finite R-module. Let d ≥ 0. The equivalent conditions (1) -- (4) of Lemma [Tag 0CXC] are also equivalent to • [(6)] there exists a resolution 0 → P_d → P_d - 1 → … → P_0 → M → 0 with P_i finite projective.","statement_latex":"Let $R$ be a Noetherian ring. Let $M$ be a finite $R$-module.\nLet $d \\geq 0$. The equivalent conditions (1) -- (4) of\nLemma \\ref{lemma-what-kind-of-resolutions}\nare also equivalent to\n\\begin{enumerate}\n\\item[(6)] there exists a resolution\n$0 \\to P_d \\to P_{d - 1} \\to \\ldots \\to P_0 \\to M \\to 0$\nwith $P_i$ finite projective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXE","source_file":"algebra.tex","source_line":26677,"source_end_line":26688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26677-L26688","statement_sha256":"49de50bba38622a8de78ee01ce65cc0706fb9692fdb14d84d439fa9e13448883","origin":"The Stacks Project","memory_eligible":false,"source_rank":1711,"rank":1711,"depth":3,"x":2088.395,"y":119.124,"cluster":"commutative-algebra"},{"id":"stacks:0CXF","tag":"0CXF","title":"Rings of finite global dimension · Lemma 0CXF","summary":"Let R be a local Noetherian ring. Let M be a finite R-module. Let d ≥ 0. The equivalent conditions (1) -- (4) of Lemma [Tag 0CXC], condition (5) of Lemma [Tag 0CXD], and condition (6) of Lemma [Tag 0CXE] are also equivalent to • [(7)] there exists a resolution 0 → F_d → F_d - 1 → … → F_0 → M → 0 with F_i finite free.","statement_latex":"Let $R$ be a local Noetherian ring. Let $M$ be a finite $R$-module.\nLet $d \\geq 0$. The equivalent conditions (1) -- (4) of\nLemma \\ref{lemma-what-kind-of-resolutions},\ncondition (5) of Lemma \\ref{lemma-what-kind-of-resolutions-local},\nand condition (6) of Lemma \\ref{lemma-what-kind-of-resolutions-Noetherian}\nare also equivalent to\n\\begin{enumerate}\n\\item[(7)] there exists a resolution\n$0 \\to F_d \\to F_{d - 1} \\to \\ldots \\to F_0 \\to M \\to 0$\nwith $F_i$ finite free.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXF","source_file":"algebra.tex","source_line":26701,"source_end_line":26714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26701-L26714","statement_sha256":"829c343009135fdbc4e3c29508c3e0e7d3e57f92cf99b9dcc5a82b85b17f1999","origin":"The Stacks Project","memory_eligible":false,"source_rank":1712,"rank":1712,"depth":5,"x":2068.086,"y":327.587,"cluster":"commutative-algebra"},{"id":"stacks:065R","tag":"065R","title":"Rings of finite global dimension · Lemma 065R","summary":"Let R be a ring. Let M be an R-module. Let n ≥ 0. The following are equivalent • M has projective dimension ≤ n, • Ext^i_R(M, N) = 0 for all R-modules N and all i ≥ n + 1, and • Ext^n + 1_R(M, N) = 0 for all R-modules N.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. Let $n \\geq 0$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ has projective dimension $\\leq n$,\n\\item $\\Ext^i_R(M, N) = 0$ for all $R$-modules $N$ and all\n$i \\geq n + 1$, and\n\\item $\\Ext^{n + 1}_R(M, N) = 0$ for all $R$-modules $N$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065R","source_file":"algebra.tex","source_line":26724,"source_end_line":26734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26724-L26734","statement_sha256":"3286f9200355587a2bd5df78d344e497b670324e43b1017deb6e21fd648f45d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1713,"rank":1713,"depth":2,"x":1915.325,"y":162.242,"cluster":"commutative-algebra"},{"id":"stacks:065S","tag":"065S","title":"Rings of finite global dimension · Lemma 065S","summary":"Let R be a ring. Let 0 → M' → M → M\" → 0 be a short exact sequence of R-modules. • If M has projective dimension ≤ n and M\" has projective dimension ≤ n + 1, then M' has projective dimension ≤ n. • If M' and M\" have projective dimension ≤ n then M has projective dimension ≤ n. • If M' has projective dimension ≤ n and M has projective dimension ≤ n + 1 then M\" has projective dimension ≤ n + 1.","statement_latex":"Let $R$ be a ring. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short\nexact sequence of $R$-modules.\n\\begin{enumerate}\n\\item If $M$ has projective dimension $\\leq n$ and $M''$\nhas projective dimension $\\leq n + 1$, then $M'$ has projective\ndimension $\\leq n$.\n\\item If $M'$ and $M''$ have projective dimension\n$\\leq n$ then $M$ has projective dimension $\\leq n$.\n\\item If $M'$ has projective dimension $\\leq n$ and\n$M$ has projective dimension $\\leq n + 1$ then\n$M''$ has projective dimension $\\leq n + 1$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065S","source_file":"algebra.tex","source_line":26761,"source_end_line":26775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26761-L26775","statement_sha256":"b644ad9476ac9061170e3396fc706a95e276893d3c0010e5bc9619d850a48ada","origin":"The Stacks Project","memory_eligible":false,"source_rank":1714,"rank":1714,"depth":3,"x":2161.09,"y":197.507,"cluster":"commutative-algebra"},{"id":"stacks:00O6","tag":"00O6","title":"Rings of finite global dimension · Definition 00O6","summary":"Let R be a ring. The ring R is said to have finite global dimension if there exists an integer n such that every R-module has a resolution by projective R-modules of length at most n. The minimal such n is then called the global dimension of R.","statement_latex":"Let $R$ be a ring. The ring\n$R$ is said to have {\\it finite global dimension}\nif there exists an integer $n$ such that\nevery $R$-module has a resolution by\nprojective $R$-modules of length at most $n$.\nThe minimal such $n$ is then called the {\\it global dimension}\nof $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00O6","source_file":"algebra.tex","source_line":26784,"source_end_line":26793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26784-L26793","statement_sha256":"a4dda8b5a217d2e88516db886c2361928c501cf1b6ae95979cf737bb3aaba2f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1715,"rank":1715,"depth":0,"x":1951.376,"y":311.027,"cluster":"commutative-algebra"},{"id":"stacks:0D1U","tag":"0D1U","title":"Rings of finite global dimension · Lemma 0D1U","summary":"Let R be a ring. Suppose we have a module M = ⋃_e ∈ E M_e where the M_e are submodules well-ordered by inclusion. Assume the quotients M_e/⋃_e' < e M_e' have projective dimension ≤ n. Then M has projective dimension ≤ n.","statement_latex":"Let $R$ be a ring. Suppose we have a module $M = \\bigcup_{e \\in E} M_e$\nwhere the $M_e$ are submodules well-ordered by inclusion. Assume the quotients\n$M_e/\\bigcup\\nolimits_{e' < e} M_{e'}$ have projective dimension $\\leq n$.\nThen $M$ has projective dimension $\\leq n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1U","source_file":"algebra.tex","source_line":26799,"source_end_line":26805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26799-L26805","statement_sha256":"b781502454219e4722d8dbfe521077a51ab20b973c9871653323bf2de5908a76","origin":"The Stacks Project","memory_eligible":false,"source_rank":1716,"rank":1716,"depth":4,"x":2014.766,"y":108.195,"cluster":"commutative-algebra"},{"id":"stacks:065T","tag":"065T","title":"Rings of finite global dimension · Lemma 065T","summary":"Let R be a ring. The following are equivalent • R has finite global dimension ≤ n, • every finite R-module has projective dimension ≤ n, and • every cyclic R-module R/I has projective dimension ≤ n.","statement_latex":"Let $R$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $R$ has finite global dimension $\\leq n$,\n\\item every finite $R$-module has projective dimension $\\leq n$, and\n\\item every cyclic $R$-module $R/I$ has projective dimension $\\leq n$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065T","source_file":"algebra.tex","source_line":26855,"source_end_line":26863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26855-L26863","statement_sha256":"8144c87b78c9490c49784679df8d9c66ffadbfcc298df1ee4ff16041a1c5a6dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1717,"rank":1717,"depth":5,"x":2131.208,"y":293.846,"cluster":"commutative-algebra"},{"id":"stacks:00O8","tag":"00O8","title":"Rings of finite global dimension · Lemma 00O8","summary":"Let R be a ring. Let M be an R-module. Let S ⊂ R be a multiplicative subset. • If M has projective dimension ≤ n, then S^-1M has projective dimension ≤ n over S^-1R. • If R has finite global dimension ≤ n, then S^-1R has finite global dimension ≤ n.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $S \\subset R$ be a multiplicative subset.\n\\begin{enumerate}\n\\item If $M$ has projective dimension $\\leq n$, then $S^{-1}M$ has\nprojective dimension $\\leq n$ over $S^{-1}R$.\n\\item If $R$ has finite global dimension $\\leq n$, then\n$S^{-1}R$ has finite global dimension $\\leq n$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Rings of finite global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00O8","source_file":"algebra.tex","source_line":26880,"source_end_line":26890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26880-L26890","statement_sha256":"da7947a92d195e2bf5479972d81d64b887c7e6d129b47c167130489ad3e3006f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1718,"rank":1718,"depth":7,"x":1895.902,"y":222.977,"cluster":"commutative-algebra"},{"id":"stacks:00O7","tag":"00O7","title":"Regular rings and global dimension · Proposition 00O7","summary":"Let R be a regular local ring of dimension d. Every finite R-module M of depth e has a finite free resolution 0 → F_d-e → … → F_0 → M → 0. In particular a regular local ring has global dimension ≤ d.","statement_latex":"Let $R$ be a regular local ring of dimension $d$.\nEvery finite $R$-module $M$ of depth $e$ has a finite free\nresolution\n$$\n0 \\to F_{d-e} \\to \\ldots \\to F_0 \\to M \\to 0.\n$$\nIn particular a regular local ring has global dimension $\\leq d$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular rings and global dimension","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00O7","source_file":"algebra.tex","source_line":26921,"source_end_line":26930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26921-L26930","statement_sha256":"f105145b19346dacd9bd4dc4586fac9eeeb27a74640fd041f5570e9a18576fac","origin":"The Stacks Project","memory_eligible":false,"source_rank":1719,"rank":1719,"depth":15,"x":2126.548,"y":141.665,"cluster":"commutative-algebra"},{"id":"stacks:00O9","tag":"00O9","title":"Regular rings and global dimension · Lemma 00O9","summary":"Let R be a Noetherian ring. Let n ≥ 0 be an integer. Then R has finite global dimension ≤ n if and only if for all maximal ideals m of R the ring R_ m has global dimension ≤ n.","statement_latex":"Let $R$ be a Noetherian ring. Let $n \\geq 0$ be an integer.\nThen $R$ has finite global dimension $\\leq n$ if and\nonly if for all maximal ideals $\\mathfrak m$ of $R$\nthe ring $R_{\\mathfrak m}$ has global dimension $\\leq n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular rings and global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00O9","source_file":"algebra.tex","source_line":26938,"source_end_line":26944,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26938-L26944","statement_sha256":"c7697401ff48b3ca93a659080b2ffec4cd4282c22e6e17fef45732c84f086c92","origin":"The Stacks Project","memory_eligible":false,"source_rank":1720,"rank":1720,"depth":8,"x":2021.796,"y":332.617,"cluster":"commutative-algebra"},{"id":"stacks:00OA","tag":"00OA","title":"Regular rings and global dimension · Lemma 00OA","summary":"Suppose that R is a Noetherian local ring with maximal ideal m and residue field kappa. In this case the projective dimension of kappa is ≥ dim_kappa m / m^2.","statement_latex":"Suppose that $R$ is a Noetherian local ring\nwith maximal ideal $\\mathfrak m$ and\nresidue field $\\kappa$. In this case\nthe projective dimension of $\\kappa$ is\n$\\geq \\dim_\\kappa \\mathfrak m / \\mathfrak m^2$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular rings and global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OA","source_file":"algebra.tex","source_line":26965,"source_end_line":26972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L26965-L26972","statement_sha256":"bc0771df7d09c1453d1e34a8318130e81d7a5f9e934ffc18ae14bd502be02c95","origin":"The Stacks Project","memory_eligible":false,"source_rank":1721,"rank":1721,"depth":6,"x":1945.433,"y":132.248,"cluster":"commutative-algebra"},{"id":"stacks:00OB","tag":"00OB","title":"Regular rings and global dimension · Lemma 00OB","summary":"Let R be a Noetherian local ring. Suppose that the residue field kappa has finite projective dimension n over R. In this case dim(R) ≥ n.","statement_latex":"Let $R$ be a Noetherian local ring.\nSuppose that the residue field $\\kappa$ has finite\nprojective dimension $n$ over $R$.\nIn this case $\\dim(R) \\geq n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular rings and global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OB","source_file":"algebra.tex","source_line":27071,"source_end_line":27077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27071-L27077","statement_sha256":"e6cd43bf8ae71bd319c89a12c82bfc0919eb759ec232dcbfd2a69dcdfb81e123","origin":"The Stacks Project","memory_eligible":false,"source_rank":1722,"rank":1722,"depth":15,"x":2163.009,"y":236.732,"cluster":"commutative-algebra"},{"id":"stacks:00OC","tag":"00OC","title":"Regular rings and global dimension · Proposition 00OC","summary":"Let (R, m, kappa) be a Noetherian local ring. The following are equivalent • kappa has finite projective dimension as an R-module, • R has finite global dimension, • R is a regular local ring. Moreover, in this case the global dimension of R equals dim(R) = dim_kappa( m/ m^2).","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\kappa$ has finite projective dimension as an $R$-module,\n\\item $R$ has finite global dimension,\n\\item $R$ is a regular local ring.\n\\end{enumerate}\nMoreover, in this case the global dimension of $R$ equals\n$\\dim(R) = \\dim_\\kappa(\\mathfrak m/\\mathfrak m^2)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular rings and global dimension","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OC","source_file":"algebra.tex","source_line":27095,"source_end_line":27106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27095-L27106","statement_sha256":"24dafc314a81e40b55c9468692b815b343a31c29241bb451b6b4854b2a2aceaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1723,"rank":1723,"depth":16,"x":1918.396,"y":283.174,"cluster":"commutative-algebra"},{"id":"stacks:0AFS","tag":"0AFS","title":"Regular rings and global dimension · Lemma 0AFS","summary":"A Noetherian local ring R is a regular local ring if and only if it has finite global dimension. In this case R_ p is a regular local ring for all primes p.","statement_latex":"A Noetherian local ring $R$ is a regular local ring if and only if\nit has finite global dimension. In this case\n$R_{\\mathfrak p}$ is a regular local ring for all primes $\\mathfrak p$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular rings and global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFS","source_file":"algebra.tex","source_line":27125,"source_end_line":27130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27125-L27130","statement_sha256":"9099cbc6ff895b35692e4c51ad997bdb9cd153c6ba478b915259d4dc31e1e35f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1724,"rank":1724,"depth":17,"x":2061.512,"y":110.022,"cluster":"commutative-algebra"},{"id":"stacks:00OD","tag":"00OD","title":"Regular rings and global dimension · Definition 00OD","summary":"A Noetherian ring R is said to be regular if all the localizations R_ p at primes are regular local rings.","statement_latex":"A Noetherian ring $R$ is said to be {\\it regular}\nif all the localizations $R_{\\mathfrak p}$ at primes are\nregular local rings.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular rings and global dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OD","source_file":"algebra.tex","source_line":27148,"source_end_line":27153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27148-L27153","statement_sha256":"33ec19309ca7b6d4d1da93e569a2912f91e050d09cb844c9a4487c92125a5e56","origin":"The Stacks Project","memory_eligible":false,"source_rank":1725,"rank":1725,"depth":0,"x":2095.246,"y":319.038,"cluster":"commutative-algebra"},{"id":"stacks:00OE","tag":"00OE","title":"Regular rings and global dimension · Lemma 00OE","summary":"Let R be a Noetherian ring. The following are equivalent: • R has finite global dimension n, • R is a regular ring of dimension n, • there exists an integer n such that all the localizations R_ m at maximal ideals are regular of dimension ≤ n with equality for at least one m, and • there exists an integer n such that all the localizations R_ p at prime ideals are regular of dimension ≤ n with equality for at least one p.","statement_latex":"Let $R$ be a Noetherian ring.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $R$ has finite global dimension $n$,\n\\item $R$ is a regular ring of dimension $n$,\n\\item there exists an integer $n$ such that\nall the localizations $R_{\\mathfrak m}$ at maximal ideals\nare regular of dimension $\\leq n$ with equality for at least\none $\\mathfrak m$, and\n\\item there exists an integer $n$ such that\nall the localizations $R_{\\mathfrak p}$ at prime ideals\nare regular of dimension $\\leq n$ with equality for at least\none $\\mathfrak p$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular rings and global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OE","source_file":"algebra.tex","source_line":27163,"source_end_line":27179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27163-L27179","statement_sha256":"11afc78784017a8e342a735e6bca0f0c8f36ed85a6adc8013811ac527bce5968","origin":"The Stacks Project","memory_eligible":false,"source_rank":1726,"rank":1726,"depth":17,"x":1902.166,"y":183.97,"cluster":"commutative-algebra"},{"id":"stacks:00OF","tag":"00OF","title":"Regular rings and global dimension · Lemma 00OF","summary":"Let R → S be a local homomorphism of local Noetherian rings. Assume that R → S is flat and that S is regular. Then R is regular.","statement_latex":"Let $R \\to S$ be a local homomorphism of local Noetherian rings.\nAssume that $R \\to S$ is flat and that $S$ is regular.\nThen $R$ is regular.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Regular rings and global dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OF","source_file":"algebra.tex","source_line":27188,"source_end_line":27193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27188-L27193","statement_sha256":"8482ac849ed8de58e1b4dd2ee413ed2f2a828bca8ea831274e370de0afb01b0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1727,"rank":1727,"depth":17,"x":2153.314,"y":174.003,"cluster":"commutative-algebra"},{"id":"stacks:090V","tag":"090V","title":"Auslander-Buchsbaum · Proposition 090V","summary":"Let R be a Noetherian local ring. Let M be a nonzero finite R-module which has finite projective dimension pd_R(M). Then we have depth(R) = pd_R(M) + depth(M)","statement_latex":"Let $R$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module\nwhich has finite projective dimension $\\text{pd}_R(M)$. Then we have\n$$\n\\text{depth}(R) = \\text{pd}_R(M) + \\text{depth}(M)\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Auslander-Buchsbaum","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090V","source_file":"algebra.tex","source_line":27228,"source_end_line":27235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27228-L27235","statement_sha256":"fe12b6652ebc30776b596062c2fde2cffeb521c0e61073f1ab3910c235028277","origin":"The Stacks Project","memory_eligible":false,"source_rank":1728,"rank":1728,"depth":15,"x":1976.027,"y":323.953,"cluster":"commutative-algebra"},{"id":"stacks:00OH","tag":"00OH","title":"Homomorphisms and dimension · Lemma 00OH","summary":"Suppose R → S is a ring map satisfying either going up, see Definition [Tag 00HV], or going down see Definition [Tag 00HV]. Assume in addition that Spec(S) → Spec(R) is surjective. Then dim(R) ≤ dim(S).","statement_latex":"Suppose $R \\to S$ is a ring map satisfying either going up, see\nDefinition \\ref{definition-going-up-down}, or going down\nsee Definition \\ref{definition-going-up-down}.\nAssume in addition that $\\Spec(S) \\to \\Spec(R)$\nis surjective. Then $\\dim(R) \\leq \\dim(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OH","source_file":"algebra.tex","source_line":27312,"source_end_line":27319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27312-L27319","statement_sha256":"281e8b5bdb252ef6a77d92d369e5b8bd398bdf8b974f980a29e5ea93f1d3710b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1729,"rank":1729,"depth":1,"x":1986.176,"y":112.655,"cluster":"commutative-algebra"},{"id":"stacks:00OI","tag":"00OI","title":"Homomorphisms and dimension · Lemma 00OI","summary":"Suppose that R → S is a ring map with the going up property, see Definition [Tag 00HV]. If q ⊂ S is a maximal ideal. Then the inverse image of q in R is a maximal ideal too.","statement_latex":"Suppose that $R \\to S$ is a ring map with the going up property,\nsee Definition \\ref{definition-going-up-down}. If\n$\\mathfrak q \\subset S$ is a maximal ideal.\nThen the inverse image of $\\mathfrak q$ in $R$\nis a maximal ideal too.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OI","source_file":"algebra.tex","source_line":27334,"source_end_line":27341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27334-L27341","statement_sha256":"4d58cad348859e9f391d32929cd6aa0f74d526d4b69708faa8c7b7c11b871f6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1730,"rank":1730,"depth":1,"x":2148.713,"y":274.322,"cluster":"commutative-algebra"},{"id":"stacks:00OJ","tag":"00OJ","title":"Homomorphisms and dimension · Lemma 00OJ","summary":"Suppose that R → S is a ring map such that S is integral over R. Then dim (R) ≥ dim(S), and every closed point of Spec(S) maps to a closed point of Spec(R).","statement_latex":"Suppose that $R \\to S$ is a ring map such that $S$ is integral over $R$.\nThen $\\dim (R) \\geq \\dim(S)$, and every closed point of $\\Spec(S)$\nmaps to a closed point of $\\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OJ","source_file":"algebra.tex","source_line":27347,"source_end_line":27352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27347-L27352","statement_sha256":"ae642b51c8eebf5ae8b94afbce5877511aff941455e285d8d6ceba407f6c3ee1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1731,"rank":1731,"depth":8,"x":1898.698,"y":247.321,"cluster":"commutative-algebra"},{"id":"stacks:00OK","tag":"00OK","title":"Homomorphisms and dimension · Lemma 00OK","summary":"Suppose R ⊂ S and S integral over R. Then dim(R) = dim(S).","statement_latex":"Suppose $R \\subset S$ and $S$ integral over $R$.\nThen $\\dim(R) = \\dim(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OK","source_file":"algebra.tex","source_line":27360,"source_end_line":27364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27360-L27364","statement_sha256":"fcf83946802ddb6ddaf98d2d470a78d5462b8771e7cb72c2e9bebe441edfa749","origin":"The Stacks Project","memory_eligible":false,"source_rank":1732,"rank":1732,"depth":9,"x":2104.895,"y":125.292,"cluster":"commutative-algebra"},{"id":"stacks:00OL","tag":"00OL","title":"Homomorphisms and dimension · Definition 00OL","summary":"Suppose that R → S is a ring map. Let q ⊂ S be a prime lying over the prime p of R. The local ring of the fibre at q is the local ring S_ q/ pS_ q = (S/ pS)_ q = (S ⊗_R kappa( p))_ q","statement_latex":"Suppose that $R \\to S$ is a ring map.\nLet $\\mathfrak q \\subset S$ be a prime lying\nover the prime $\\mathfrak p$ of $R$.\nThe {\\it local ring of the fibre at $\\mathfrak q$}\nis the local ring\n$$\nS_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}\n=\n(S/\\mathfrak pS)_{\\mathfrak q}\n=\n(S \\otimes_R \\kappa(\\mathfrak p))_{\\mathfrak q}\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms and dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OL","source_file":"algebra.tex","source_line":27374,"source_end_line":27388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27374-L27388","statement_sha256":"2937c452510f3c68b5aa799e07c00fd439030ba7ef11866b6832508a2725d5fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":1733,"rank":1733,"depth":0,"x":2050.948,"y":332.403,"cluster":"commutative-algebra"},{"id":"stacks:00OM","tag":"00OM","title":"Homomorphisms and dimension · Lemma 00OM","summary":"Let R → S be a homomorphism of Noetherian rings. Let q ⊂ S be a prime lying over the prime p. Then dim(S_ q) ≤ dim(R_ p) + dim(S_ q/ pS_ q).","statement_latex":"Let $R \\to S$ be a homomorphism of Noetherian rings.\nLet $\\mathfrak q \\subset S$ be a prime lying\nover the prime $\\mathfrak p$. Then\n$$\n\\dim(S_{\\mathfrak q})\n\\leq\n\\dim(R_{\\mathfrak p})\n+\n\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OM","source_file":"algebra.tex","source_line":27390,"source_end_line":27402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27390-L27402","statement_sha256":"1ee7bf4c7b5391b074ae050ded8f287ae8f19fc7cbd1f38c8c0566d3527e15cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1734,"rank":1734,"depth":10,"x":1924.097,"y":148.959,"cluster":"commutative-algebra"},{"id":"stacks:00ON","tag":"00ON","title":"Homomorphisms and dimension · Lemma 00ON","summary":"Let R → S be a homomorphism of Noetherian rings. Let q ⊂ S be a prime lying over the prime p. Assume the going down property holds for R → S (for example if R → S is flat, see Lemma [Tag 00HS]). Then dim(S_ q) = dim(R_ p) + dim(S_ q/ pS_ q).","statement_latex":"Let $R \\to S$ be a homomorphism of Noetherian rings.\nLet $\\mathfrak q \\subset S$ be a prime lying\nover the prime $\\mathfrak p$. Assume the going down property holds\nfor $R \\to S$ (for example if $R \\to S$ is flat, see\nLemma \\ref{lemma-flat-going-down}). Then\n$$\n\\dim(S_{\\mathfrak q})\n=\n\\dim(R_{\\mathfrak p})\n+\n\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ON","source_file":"algebra.tex","source_line":27420,"source_end_line":27434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27420-L27434","statement_sha256":"44023383b2dbfca12c8912ab561d45260f0954b34879f13928e62306ab676fca","origin":"The Stacks Project","memory_eligible":false,"source_rank":1735,"rank":1735,"depth":11,"x":2165.303,"y":212.288,"cluster":"commutative-algebra"},{"id":"stacks:031E","tag":"031E","title":"Homomorphisms and dimension · Lemma 031E","summary":"Let R → S be a local homomorphism of local Noetherian rings. Assume • R is regular, • S/ m_RS is regular, and • R → S is flat. Then S is regular.","statement_latex":"Let $R \\to S$ be a local homomorphism of local Noetherian rings.\nAssume\n\\begin{enumerate}\n\\item $R$ is regular,\n\\item $S/\\mathfrak m_RS$ is regular, and\n\\item $R \\to S$ is flat.\n\\end{enumerate}\nThen $S$ is regular.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031E","source_file":"algebra.tex","source_line":27458,"source_end_line":27468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27458-L27468","statement_sha256":"a45e3c51b9921583566352c63e8760fa7a7c1f2023c0625dda6d45180ee38dfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":1736,"rank":1736,"depth":12,"x":1936.374,"y":302.512,"cluster":"commutative-algebra"},{"id":"stacks:00R5","tag":"00R5","title":"Homomorphisms and dimension · Lemma 00R5","summary":"Let R → S be a local homomorphism of Noetherian local rings. Assume R Cohen-Macaulay. If S is finite flat over R, or if S is flat over R and dim(S) ≤ dim(R), then S is Cohen-Macaulay and dim(R) = dim(S).","statement_latex":"Let $R \\to S$ be a local homomorphism of Noetherian local rings.\nAssume $R$ Cohen-Macaulay.\nIf $S$ is finite flat over $R$, or if $S$ is flat over $R$ and\n$\\dim(S) \\leq \\dim(R)$, then $S$ is Cohen-Macaulay and $\\dim(R) = \\dim(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Homomorphisms and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00R5","source_file":"algebra.tex","source_line":27488,"source_end_line":27494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27488-L27494","statement_sha256":"acc37abb48abc5817ffa16150347ec032585cbc4024161ba44e0c803afe89a25","origin":"The Stacks Project","memory_eligible":false,"source_rank":1737,"rank":1737,"depth":10,"x":2032.686,"y":105.962,"cluster":"commutative-algebra"},{"id":"stacks:02IJ","tag":"02IJ","title":"The dimension formula · Lemma 02IJ","summary":"Let R → S be a ring map. Let q be a prime of S lying over the prime p of R. Assume that • R is Noetherian, • R → S is of finite type, • R, S are domains, and • R ⊂ S. Then we have height( q) ≤ height( p) + trdeg_R(S) - trdeg_kappa( p) kappa( q) with equality if R is universally catenary.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q$ be a prime of $S$ lying over the prime $\\mathfrak p$ of $R$.\nAssume that\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $R \\to S$ is of finite type,\n\\item $R$, $S$ are domains, and\n\\item $R \\subset S$.\n\\end{enumerate}\nThen we have\n$$\n\\text{height}(\\mathfrak q)\n\\leq\n\\text{height}(\\mathfrak p) + \\text{trdeg}_R(S)\n- \\text{trdeg}_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak q)\n$$\nwith equality if $R$ is universally catenary.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The dimension formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IJ","source_file":"algebra.tex","source_line":27519,"source_end_line":27538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27519-L27538","statement_sha256":"bd953e7be9d33b2c5dfc6e5148dc2529b4bc7a558f98daad9677d0e9ac227329","origin":"The Stacks Project","memory_eligible":false,"source_rank":1738,"rank":1738,"depth":12,"x":2119.78,"y":305.667,"cluster":"commutative-algebra"},{"id":"stacks:02MA","tag":"02MA","title":"The dimension formula · Lemma 02MA","summary":"Let A → B be a ring map. Assume • A ⊂ B is an extension of domains, • the induced extension of fraction fields is finite, • A is Noetherian, and • A → B is of finite type. Let p ⊂ A be a prime of height 1. Then there are at most finitely many primes of B lying over p and they all have height 1.","statement_latex":"Let $A \\to B$ be a ring map.\nAssume\n\\begin{enumerate}\n\\item $A \\subset B$ is an extension of domains,\n\\item the induced extension of fraction fields is finite,\n\\item $A$ is Noetherian, and\n\\item $A \\to B$ is of finite type.\n\\end{enumerate}\nLet $\\mathfrak p \\subset A$ be a prime of height $1$.\nThen there are at most finitely many primes of $B$\nlying over $\\mathfrak p$ and they all have height $1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The dimension formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MA","source_file":"algebra.tex","source_line":27602,"source_end_line":27615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27602-L27615","statement_sha256":"0d782bdf2ebcf3b4b184737f38e951b5e0330d8373dc6801e8542b110a7d82cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1739,"rank":1739,"depth":13,"x":1894.824,"y":207.767,"cluster":"commutative-algebra"},{"id":"stacks:00OP","tag":"00OP","title":"Dimension of finite type algebras over fields · Lemma 00OP","summary":"Let m be a maximal ideal in k[x_1, …, x_n]. The ideal m is generated by n elements. The dimension of k[x_1, …, x_n]_ m is n. Hence k[x_1, …, x_n]_ m is a regular local ring of dimension n.","statement_latex":"Let $\\mathfrak m$ be a maximal ideal in $k[x_1, \\ldots, x_n]$.\nThe ideal $\\mathfrak m$ is generated by $n$ elements.\nThe dimension of $k[x_1, \\ldots, x_n]_{\\mathfrak m}$ is $n$.\nHence $k[x_1, \\ldots, x_n]_{\\mathfrak m}$ is a regular local\nring of dimension $n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OP","source_file":"algebra.tex","source_line":27659,"source_end_line":27666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27659-L27666","statement_sha256":"42a21075bb9d8acb04ebbb3d65498833e6d59a44d1b6d92affea6f47c93b6af4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1740,"rank":1740,"depth":7,"x":2139.581,"y":152.277,"cluster":"commutative-algebra"},{"id":"stacks:00OQ","tag":"00OQ","title":"Dimension of finite type algebras over fields · Proposition 00OQ","summary":"A polynomial algebra in n variables over a field is a regular ring. It has global dimension n. All localizations at maximal ideals are regular local rings of dimension n.","statement_latex":"A polynomial algebra in $n$ variables over a field is a regular ring.\nIt has global dimension $n$. All localizations at maximal ideals\nare regular local rings of dimension $n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OQ","source_file":"algebra.tex","source_line":27722,"source_end_line":27727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27722-L27727","statement_sha256":"028b36e3e422f679857ce5d870c132b15a25a4191b0113768cba3969b4477d83","origin":"The Stacks Project","memory_eligible":false,"source_rank":1741,"rank":1741,"depth":18,"x":2003.641,"y":332.185,"cluster":"commutative-algebra"},{"id":"stacks:00OR","tag":"00OR","title":"Dimension of finite type algebras over fields · Lemma 00OR","summary":"Let k be a field. Let p ⊂ q ⊂ k[x_1, …, x_n] be a pair of primes. Any maximal chain of primes between p and q has length height( q) - height( p).","statement_latex":"Let $k$ be a field.\nLet $\\mathfrak p \\subset \\mathfrak q \\subset k[x_1, \\ldots, x_n]$\nbe a pair of primes.\nAny maximal chain of primes between $\\mathfrak p$ and $\\mathfrak q$\nhas length $\\text{height}(\\mathfrak q) - \\text{height}(\\mathfrak p)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OR","source_file":"algebra.tex","source_line":27736,"source_end_line":27743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27736-L27743","statement_sha256":"1562018a9c75b9441d99c6a90ab1c373dc72018b2c0a2d254deb2c9186c972b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1742,"rank":1742,"depth":19,"x":1959.178,"y":122.261,"cluster":"commutative-algebra"},{"id":"stacks:00OS","tag":"00OS","title":"Dimension of finite type algebras over fields · Lemma 00OS","summary":"Let k be a field. Let S be a finite type k-algebra which is an integral domain. Then dim(S) = dim(S_ m) for any maximal ideal m of S. In words: every maximal chain of primes has length equal to the dimension of S.","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra which is an integral domain.\nThen $\\dim(S) = \\dim(S_{\\mathfrak m})$ for any maximal\nideal $\\mathfrak m$ of $S$. In words: every maximal chain\nof primes has length equal to the dimension of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OS","source_file":"algebra.tex","source_line":27762,"source_end_line":27769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27762-L27769","statement_sha256":"a34fc2d15e12b550e380844f9af0d68801449481587018ea4158af0c14066590","origin":"The Stacks Project","memory_eligible":false,"source_rank":1743,"rank":1743,"depth":20,"x":2160.902,"y":251.904,"cluster":"commutative-algebra"},{"id":"stacks:00OT","tag":"00OT","title":"Dimension of finite type algebras over fields · Lemma 00OT","summary":"Let k be a field. Let S be a finite type k-algebra. Let X = Spec(S). Let p ⊂ S be a prime ideal and let x ∈ X be the corresponding point. The following numbers are equal • dim_x(X), • max dim(Z) where the maximum is over those irreducible components Z of X passing through x, and • min dim(S_ m) where the minimum is over maximal ideals m with p ⊂ m.","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra.\nLet $X = \\Spec(S)$.\nLet $\\mathfrak p \\subset S$ be a prime ideal and let\n$x \\in X$ be the corresponding point.\nThe following numbers are equal\n\\begin{enumerate}\n\\item $\\dim_x(X)$,\n\\item $\\max \\dim(Z)$ where the maximum is over those\nirreducible components $Z$ of $X$ passing through $x$, and\n\\item $\\min \\dim(S_{\\mathfrak m})$ where the minimum\nis over maximal ideals $\\mathfrak m$ with\n$\\mathfrak p \\subset \\mathfrak m$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OT","source_file":"algebra.tex","source_line":27786,"source_end_line":27802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27786-L27802","statement_sha256":"63536b0aa2f44a36cc1a073f9d068dc19f202de8d8642cf6c1216052db73f55c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1744,"rank":1744,"depth":21,"x":1907.744,"y":270.782,"cluster":"commutative-algebra"},{"id":"stacks:00OU","tag":"00OU","title":"Dimension of finite type algebras over fields · Lemma 00OU","summary":"Let k be a field. Let S be a finite type k-algebra. Let X = Spec(S). Let m ⊂ S be a maximal ideal and let x ∈ X be the associated closed point. Then dim_x(X) = dim(S_ m).","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra.\nLet $X = \\Spec(S)$.\nLet $\\mathfrak m \\subset S$ be a maximal ideal and let\n$x \\in X$ be the associated closed point.\nThen $\\dim_x(X) = \\dim(S_{\\mathfrak m})$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OU","source_file":"algebra.tex","source_line":27854,"source_end_line":27862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27854-L27862","statement_sha256":"75c46c09e9411da18a4b7b86653a45158346323256c5442d7b4a66990cef32f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1745,"rank":1745,"depth":22,"x":2079.342,"y":113.119,"cluster":"commutative-algebra"},{"id":"stacks:00OV","tag":"00OV","title":"Dimension of finite type algebras over fields · Lemma 00OV","summary":"Let k be a field. Let S be a finite type k algebra. Assume that S is Cohen-Macaulay. Then Spec(S) = coprod T_d is a finite disjoint union of open and closed subsets T_d with T_d equidimensional (see Topology, Definition [Tag 0058]) of dimension d. Equivalently, S is a product of rings S_d, d = 0, …, dim(S) such that every maximal ideal m of S_d has height d.","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$ algebra.\nAssume that $S$ is Cohen-Macaulay.\nThen $\\Spec(S) = \\coprod T_d$ is a finite disjoint union of\nopen and closed subsets $T_d$ with $T_d$ equidimensional\n(see Topology, Definition \\ref{topology-definition-equidimensional})\nof dimension $d$. Equivalently, $S$ is a product of rings\n$S_d$, $d = 0, \\ldots, \\dim(S)$ such that every maximal ideal\n$\\mathfrak m$ of $S_d$ has height $d$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OV","source_file":"algebra.tex","source_line":27869,"source_end_line":27880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27869-L27880","statement_sha256":"4275079adce6263c9e52a3be43b502d8f155ad15604bffd0e1acfb1e2fb49f00","origin":"The Stacks Project","memory_eligible":false,"source_rank":1746,"rank":1746,"depth":21,"x":2079.597,"y":326.875,"cluster":"commutative-algebra"},{"id":"stacks:051M","tag":"051M","title":"Noether normalization · Lemma 051M","summary":"Let n ∈ N. Let N be a finite nonempty set of multi-indices ν = (ν_1, …, ν_n). Given e = (e_1, …, e_n) we set e · ν = ∑ e_iν_i. Then for e_1 gg e_2 gg … gg e_n-1 gg e_n we have: If ν, ν' ∈ N then (e · ν = e · ν') ⇔ (ν = ν')","statement_latex":"Let $n \\in \\mathbf{N}$.\nLet $N$ be a finite nonempty\nset of multi-indices $\\nu = (\\nu_1, \\ldots, \\nu_n)$.\nGiven $e = (e_1, \\ldots, e_n)$ we set $e \\cdot \\nu = \\sum e_i\\nu_i$.\nThen for $e_1 \\gg e_2 \\gg \\ldots \\gg e_{n-1} \\gg e_n$ we have:\nIf $\\nu, \\nu' \\in N$ then\n$$\n(e \\cdot \\nu = e \\cdot \\nu')\n\\Leftrightarrow\n(\\nu = \\nu')\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051M","source_file":"algebra.tex","source_line":27927,"source_end_line":27940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27927-L27940","statement_sha256":"385ffa7106fbd47e2dede1a0fa5e3289943cd401b958ffb46a633228494d5b9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1747,"rank":1747,"depth":0,"x":1907.408,"y":169.304,"cluster":"commutative-algebra"},{"id":"stacks:051N","tag":"051N","title":"Noether normalization · Lemma 051N","summary":"Let R be a ring. Let g ∈ R[x_1, …, x_n] be an element which is nonconstant, i.e., g not ∈ R. For e_1 gg e_2 gg … gg e_n-1 gg e_n = 1 the polynomial g(x_1 + x_n^e_1, x_2 + x_n^e_2, …, x_n - 1 + x_n^e_n - 1, x_n) = ax_n^d + lower order terms in x_n where d > 0 and a ∈ R is one of the nonzero coefficients of g.","statement_latex":"Let $R$ be a ring. Let $g \\in R[x_1, \\ldots, x_n]$ be an element\nwhich is nonconstant, i.e., $g \\not \\in R$.\nFor $e_1 \\gg e_2 \\gg \\ldots \\gg e_{n-1} \\gg e_n = 1$ the polynomial\n$$\ng(x_1 + x_n^{e_1}, x_2 + x_n^{e_2}, \\ldots, x_{n - 1} + x_n^{e_{n - 1}}, x_n)\n=\nax_n^d + \\text{lower order terms in }x_n\n$$\nwhere $d > 0$ and $a \\in R$ is one of the nonzero coefficients of $g$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051N","source_file":"algebra.tex","source_line":27962,"source_end_line":27973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27962-L27973","statement_sha256":"bf888a9dc8414a82223d29d65a57bafac8e94a4cca9e335f3ea25991118a5318","origin":"The Stacks Project","memory_eligible":false,"source_rank":1748,"rank":1748,"depth":1,"x":2161.245,"y":187.802,"cluster":"commutative-algebra"},{"id":"stacks:00OX","tag":"00OX","title":"Noether normalization · Lemma 00OX","summary":"Let k be a field. Let S = k[x_1, …, x_n]/I for some proper ideal I. If I not = 0, then there exist y_1, …, y_n-1 ∈ k[x_1, …, x_n] such that S is finite over k[y_1, …, y_n-1]. Moreover we may choose y_i to be in the Z-subalgebra of k[x_1, …, x_n] generated by x_1, …, x_n.","statement_latex":"Let $k$ be a field.\nLet $S = k[x_1, \\ldots, x_n]/I$ for some proper ideal $I$.\nIf $I \\not = 0$, then there exist $y_1, \\ldots, y_{n-1} \\in k[x_1, \\ldots, x_n]$\nsuch that $S$ is finite over $k[y_1, \\ldots, y_{n-1}]$. Moreover we may\nchoose $y_i$ to be in the $\\mathbf{Z}$-subalgebra of $k[x_1, \\ldots, x_n]$\ngenerated by $x_1, \\ldots, x_n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OX","source_file":"algebra.tex","source_line":27996,"source_end_line":28004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L27996-L28004","statement_sha256":"d3dc113caae051b5eb39492d639a85a33a786739465933cb870772d6e16af026","origin":"The Stacks Project","memory_eligible":false,"source_rank":1749,"rank":1749,"depth":2,"x":1959.072,"y":318.273,"cluster":"commutative-algebra"},{"id":"stacks:00OY","tag":"00OY","title":"Noether normalization · Lemma 00OY","summary":"Noether normalization Let k be a field. Let S = k[x_1, …, x_n]/I for some ideal I. If I ≠ (1), there exist r≥ 0, and y_1, …, y_r ∈ k[x_1, …, x_n] such that (a) the map k[y_1, …, y_r] → S is injective where the source is the polynomial ring on y_1, …, y_r, and (b) the map k[y_1, …, y_r] → S is finite. In this case the integer r is the dimension of S. Moreover we may choose y_i to be in the Z-subalgebra of k[x_1, …, x_n] generated by x_1, …, x_n.","statement_latex":"\\begin{slogan}\nNoether normalization\n\\end{slogan}\nLet $k$ be a field. Let $S = k[x_1, \\ldots, x_n]/I$ for some ideal $I$.\nIf $I \\neq (1)$, there exist $r\\geq 0$, and\n$y_1, \\ldots, y_r \\in k[x_1, \\ldots, x_n]$\nsuch that (a) the map $k[y_1, \\ldots, y_r] \\to S$ is injective\nwhere the source is the polynomial ring on $y_1, \\ldots, y_r$,\nand (b) the map $k[y_1, \\ldots, y_r] \\to S$ is finite.\nIn this case the integer $r$ is the dimension of $S$.\nMoreover we may choose $y_i$ to be in the\n$\\mathbf{Z}$-subalgebra of $k[x_1, \\ldots, x_n]$\ngenerated by $x_1, \\ldots, x_n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OY","source_file":"algebra.tex","source_line":28024,"source_end_line":28039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28024-L28039","statement_sha256":"db0c49022b68489b5a1e0819d409220b4da591fe5ef49ddfbdbe61664f9817a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1750,"rank":1750,"depth":10,"x":2003.256,"y":107.221,"cluster":"commutative-algebra"},{"id":"stacks:00OZ","tag":"00OZ","title":"Noether normalization · Lemma 00OZ","summary":"Let k be a field. Let S be a finite type k algebra and denote X = Spec(S). Let q be a prime of S, and let x ∈ X be the corresponding point. There exists a g ∈ S, g not ∈ q such that dim(S_g) = dim_x(X) =: d and such that there exists a finite injective map k[y_1, …, y_d] → S_g.","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$ algebra and denote $X = \\Spec(S)$.\nLet $\\mathfrak q$ be a prime of $S$, and let $x \\in X$ be the\ncorresponding point. There exists a $g \\in S$, $g \\not \\in \\mathfrak q$\nsuch that $\\dim(S_g) = \\dim_x(X) =: d$ and such that\nthere exists a finite injective map $k[y_1, \\ldots, y_d] \\to S_g$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00OZ","source_file":"algebra.tex","source_line":28056,"source_end_line":28064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28056-L28064","statement_sha256":"d1b390c29a405b7b925909af2543906d5191199474f0817e78d744c364049178","origin":"The Stacks Project","memory_eligible":false,"source_rank":1751,"rank":1751,"depth":11,"x":2140.481,"y":288.027,"cluster":"commutative-algebra"},{"id":"stacks:051P","tag":"051P","title":"Noether normalization · Lemma 051P","summary":"Let k be a field. Let q ⊂ k[x_1, …, x_n] be a prime ideal. Set r = trdeg_k kappa( q). Then there exists a finite ring map φ : k[y_1, …, y_n] → k[x_1, …, x_n] such that φ^-1( q) = (y_r + 1, …, y_n).","statement_latex":"Let $k$ be a field. Let $\\mathfrak q \\subset k[x_1, \\ldots, x_n]$\nbe a prime ideal. Set $r = \\text{trdeg}_k\\ \\kappa(\\mathfrak q)$.\nThen there exists a finite ring map\n$\\varphi : k[y_1, \\ldots, y_n] \\to k[x_1, \\ldots, x_n]$ such\nthat $\\varphi^{-1}(\\mathfrak q) = (y_{r + 1}, \\ldots, y_n)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051P","source_file":"algebra.tex","source_line":28073,"source_end_line":28080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28073-L28080","statement_sha256":"4cca919295d2787a6aa0533474ef8ba19ec2be1101c6d9e0132efcb3b1d8a4c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1752,"rank":1752,"depth":2,"x":1893.746,"y":232.535,"cluster":"commutative-algebra"},{"id":"stacks:07NA","tag":"07NA","title":"Noether normalization · Lemma 07NA","summary":"Let R → S be an injective finite type ring map. Assume R is a domain. Then there exists an integer d and a factorization R → R[y_1, …, y_d] → S' → S by injective maps such that S' is finite over R[y_1, …, y_d] and such that S'_f ≅ S_f for some nonzero f ∈ R.","statement_latex":"Let $R \\to S$ be an injective finite type ring map. Assume $R$ is a domain.\nThen there exists an integer $d$ and a factorization\n$$\nR \\to R[y_1, \\ldots, y_d] \\to S' \\to S\n$$\nby injective maps such that $S'$ is finite over $R[y_1, \\ldots, y_d]$\nand such that $S'_f \\cong S_f$ for some nonzero $f \\in R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NA","source_file":"algebra.tex","source_line":28125,"source_end_line":28134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28125-L28134","statement_sha256":"7672f2c03f40e490029240eba973ba35b9109f6d92164dc8bb3d8e988a612338","origin":"The Stacks Project","memory_eligible":false,"source_rank":1753,"rank":1753,"depth":11,"x":2120.446,"y":133.392,"cluster":"commutative-algebra"},{"id":"stacks:00P0","tag":"00P0","title":"Dimension of finite type algebras over fields, reprise · Lemma 00P0","summary":"Let k be a field. Let S be a finite type k algebra which is an integral domain. Let K be the field of fractions of S. Let r = trdeg(K/k) be the transcendence degree of K over k. Then dim(S) = r. Moreover, the local ring of S at every maximal ideal has dimension r.","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$ algebra which is an integral domain.\nLet $K$ be the field of fractions of $S$.\nLet $r = \\text{trdeg}(K/k)$ be the transcendence degree of $K$ over $k$.\nThen $\\dim(S) = r$. Moreover, the local ring of $S$ at every maximal\nideal has dimension $r$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00P0","source_file":"algebra.tex","source_line":28177,"source_end_line":28185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28177-L28185","statement_sha256":"e9cff500622543af4a12c49e92c5b482cbb70357acc38412de9111660a9bb0db","origin":"The Stacks Project","memory_eligible":false,"source_rank":1754,"rank":1754,"depth":20,"x":2032.957,"y":335.254,"cluster":"commutative-algebra"},{"id":"stacks:06RP","tag":"06RP","title":"Dimension of finite type algebras over fields, reprise · Lemma 06RP","summary":"Let k be a field. Let S be a finite type k-algebra. Let q ⊂ q' ⊂ S be distinct prime ideals. Then trdeg_k kappa( q') < trdeg_k kappa( q).","statement_latex":"Let $k$ be a field. Let $S$ be a finite type $k$-algebra.\nLet $\\mathfrak q \\subset \\mathfrak q' \\subset S$ be distinct\nprime ideals. Then\n$\\text{trdeg}_k\\ \\kappa(\\mathfrak q') < \\text{trdeg}_k\\ \\kappa(\\mathfrak q)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RP","source_file":"algebra.tex","source_line":28200,"source_end_line":28206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28200-L28206","statement_sha256":"15e2a9a1d770ede346fb393172f4798cadc1ebd07407edcc9bb81c8d599432f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1755,"rank":1755,"depth":21,"x":1935.079,"y":136.641,"cluster":"commutative-algebra"},{"id":"stacks:00P1","tag":"00P1","title":"Dimension of finite type algebras over fields, reprise · Lemma 00P1","summary":"Let k be a field. Let S be a finite type k algebra. Let X = Spec(S). Let p ⊂ S be a prime ideal, and let x ∈ X be the corresponding point. Then we have dim_x(X) = dim(S_ p) + trdeg_k kappa( p).","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$ algebra.\nLet $X = \\Spec(S)$.\nLet $\\mathfrak p \\subset S$ be a prime ideal,\nand let $x \\in X$ be the corresponding point.\nThen we have\n$$\n\\dim_x(X) = \\dim(S_{\\mathfrak p}) + \\text{trdeg}_k\\ \\kappa(\\mathfrak p).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00P1","source_file":"algebra.tex","source_line":28221,"source_end_line":28232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28221-L28232","statement_sha256":"09bf3d11d2ffb57471a4717e38839de8234c8da97c6e09206bcc359ba54f466e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1756,"rank":1756,"depth":22,"x":2167.11,"y":227.612,"cluster":"commutative-algebra"},{"id":"stacks:00P2","tag":"00P2","title":"Dimension of finite type algebras over fields, reprise · Lemma 00P2","summary":"Let k be a field. Let S' → S be a surjection of finite type k algebras. Let p ⊂ S be a prime ideal, and let p' be the corresponding prime ideal of S'. Let X = Spec(S), resp. X' = Spec(S'), and let x ∈ X, resp. x'∈ X' be the point corresponding to p, resp. p'. Then dim_x' X' - dim_x X = height( p') - height( p).","statement_latex":"Let $k$ be a field.\nLet $S' \\to S$ be a surjection of finite type $k$ algebras.\nLet $\\mathfrak p \\subset S$ be a prime ideal,\nand let $\\mathfrak p'$ be the corresponding prime ideal of $S'$.\nLet $X = \\Spec(S)$, resp.\\ $X' = \\Spec(S')$,\nand let $x \\in X$, resp. $x'\\in X'$ be the point corresponding\nto $\\mathfrak p$, resp.\\ $\\mathfrak p'$.\nThen\n$$\n\\dim_{x'} X' - \\dim_x X =\n\\text{height}(\\mathfrak p') - \\text{height}(\\mathfrak p).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00P2","source_file":"algebra.tex","source_line":28269,"source_end_line":28283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28269-L28283","statement_sha256":"570adbdeb42ae23a605743cd6f0b4837f5595e42fff67c64985052887646c31f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1757,"rank":1757,"depth":23,"x":1922.713,"y":292.23,"cluster":"commutative-algebra"},{"id":"stacks:00P3","tag":"00P3","title":"Dimension of finite type algebras over fields, reprise · Lemma 00P3","summary":"Let k be a field. Let S be a finite type k-algebra. Let K/k be a field extension. Then dim(S) = dim(K ⊗_k S).","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra.\nLet $K/k$ be a field extension.\nThen $\\dim(S) = \\dim(K \\otimes_k S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00P3","source_file":"algebra.tex","source_line":28289,"source_end_line":28295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28289-L28295","statement_sha256":"4d7808dab50980b720f371bd215c50a4e159de605dfed56135bb8242b1ddf2ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":1758,"rank":1758,"depth":11,"x":2051.04,"y":105.794,"cluster":"commutative-algebra"},{"id":"stacks:00P4","tag":"00P4","title":"Dimension of finite type algebras over fields, reprise · Lemma 00P4","summary":"Let k be a field. Let S be a finite type k-algebra. Set X = Spec(S). Let K/k be a field extension. Set S_K = K ⊗_k S, and X_K = Spec(S_K). Let q ⊂ S be a prime corresponding to x ∈ X and let q_K ⊂ S_K be a prime corresponding to x_K ∈ X_K lying over q. Then dim_x X = dim_x_K X_K.","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra.\nSet $X = \\Spec(S)$.\nLet $K/k$ be a field extension.\nSet $S_K = K \\otimes_k S$, and $X_K = \\Spec(S_K)$.\nLet $\\mathfrak q \\subset S$ be a prime corresponding to $x \\in X$\nand let $\\mathfrak q_K \\subset S_K$ be a prime corresponding\nto $x_K \\in X_K$ lying over $\\mathfrak q$.\nThen $\\dim_x X = \\dim_{x_K} X_K$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00P4","source_file":"algebra.tex","source_line":28306,"source_end_line":28317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28306-L28317","statement_sha256":"c28bd75878fcc6e3cc6b01e30142fb088c57a73d8c088a6268b825aac67bff51","origin":"The Stacks Project","memory_eligible":false,"source_rank":1759,"rank":1759,"depth":24,"x":2106.373,"y":316.21,"cluster":"commutative-algebra"},{"id":"stacks:0CWE","tag":"0CWE","title":"Dimension of finite type algebras over fields, reprise · Lemma 0CWE","summary":"Let k be a field. Let S be a finite type k-algebra. Let K/k be a field extension. Set S_K = K ⊗_k S. Let q ⊂ S be a prime and let q_K ⊂ S_K be a prime lying over q. Then dim (S_K ⊗_S kappa( q))_ q_K = dim (S_K)_ q_K - dim S_ q = trdeg_k kappa( q) - trdeg_K kappa( q_K) Moreover, given q we can always choose q_K such that the number above is zero.","statement_latex":"Let $k$ be a field. Let $S$ be a finite type $k$-algebra.\nLet $K/k$ be a field extension. Set $S_K = K \\otimes_k S$.\nLet $\\mathfrak q \\subset S$ be a prime and let $\\mathfrak q_K \\subset S_K$\nbe a prime lying over $\\mathfrak q$. Then\n$$\n\\dim (S_K \\otimes_S \\kappa(\\mathfrak q))_{\\mathfrak q_K} =\n\\dim (S_K)_{\\mathfrak q_K} - \\dim S_\\mathfrak q =\n\\text{trdeg}_k \\kappa(\\mathfrak q) - \\text{trdeg}_K \\kappa(\\mathfrak q_K)\n$$\nMoreover, given $\\mathfrak q$ we can always choose $\\mathfrak q_K$ such\nthat the number above is zero.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of finite type algebras over fields, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWE","source_file":"algebra.tex","source_line":28358,"source_end_line":28371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28358-L28371","statement_sha256":"e29d110ccdbbd8ed7f49a6528ed6e08ef450ec4eb00ac2f9011eb92ace2125c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1760,"rank":1760,"depth":25,"x":1896.235,"y":192.376,"cluster":"commutative-algebra"},{"id":"stacks:00P6","tag":"00P6","title":"Dimension of graded algebras over a field · Lemma 00P6","summary":"Let k be a field. Let S be a graded k-algebra generated over k by finitely many elements of degree 1. Assume S_0 = k. Let P(T) ∈ Q[T] be the polynomial such that dim(S_d) = P(d) for all d gg 0. See Proposition [Tag 00K1]. Then • The irrelevant ideal S_+ is a maximal ideal m. • Any minimal prime of S is a homogeneous ideal and is contained in S_+ = m. • We have dim(S) = deg(P) + 1 = dim_xSpec(S) (with the convention that deg(0) = -1) where x is the point corresponding to…","statement_latex":"Let $k$ be a field. Let $S$ be a graded $k$-algebra generated over $k$\nby finitely many elements of degree $1$.\nAssume $S_0 = k$. Let $P(T) \\in \\mathbf{Q}[T]$ be the polynomial\nsuch that $\\dim(S_d) = P(d)$ for all $d \\gg 0$. See\nProposition \\ref{proposition-graded-hilbert-polynomial}.\nThen\n\\begin{enumerate}\n\\item The irrelevant ideal $S_{+}$ is a maximal ideal $\\mathfrak m$.\n\\item Any minimal prime of $S$ is a homogeneous ideal and is contained\nin $S_{+} = \\mathfrak m$.\n\\item We have $\\dim(S) = \\deg(P) + 1 = \\dim_x\\Spec(S)$\n(with the convention that $\\deg(0) = -1$)\nwhere $x$ is the point corresponding to the maximal ideal\n$S_{+} = \\mathfrak m$.\n\\item The Hilbert function of the local ring $R = S_{\\mathfrak m}$\nis equal to the Hilbert function of $S$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of graded algebras over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00P6","source_file":"algebra.tex","source_line":28401,"source_end_line":28420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28401-L28420","statement_sha256":"98b6fbcf9d81e9236997289ad24f80c5517b2f60cd31d7c7b698ef31b24b0b56","origin":"The Stacks Project","memory_eligible":false,"source_rank":1761,"rank":1761,"depth":22,"x":2150.922,"y":164.435,"cluster":"commutative-algebra"},{"id":"stacks:051R","tag":"051R","title":"Generic flatness · Lemma 051R","summary":"Let R → S be a ring map. Let M be an S-module. Assume • R is Noetherian, • R is a domain, • R → S is of finite type, and • M is a finite type S-module. Then there exists a nonzero f ∈ R such that M_f is a free R_f-module.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $R$ is a domain,\n\\item $R \\to S$ is of finite type, and\n\\item $M$ is a finite type $S$-module.\n\\end{enumerate}\nThen there exists a nonzero $f \\in R$ such that\n$M_f$ is a free $R_f$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Generic flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051R","source_file":"algebra.tex","source_line":28457,"source_end_line":28470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28457-L28470","statement_sha256":"6d42a81ddeeab1221e745645b8069ab561b0fd56034ee8e55929cf60e795d69b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1762,"rank":1762,"depth":12,"x":1985.491,"y":329.651,"cluster":"commutative-algebra"},{"id":"stacks:051S","tag":"051S","title":"Generic flatness · Lemma 051S","summary":"Generic freeness. Let R → S be a ring map. Let M be an S-module. Assume • R is a domain, • R → S is of finite presentation, and • M is an S-module of finite presentation. Then there exists a nonzero f ∈ R such that M_f is a free R_f-module.","statement_latex":"\\begin{slogan}\nGeneric freeness.\n\\end{slogan}\nLet $R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is a domain,\n\\item $R \\to S$ is of finite presentation, and\n\\item $M$ is an $S$-module of finite presentation.\n\\end{enumerate}\nThen there exists a nonzero $f \\in R$ such that\n$M_f$ is a free $R_f$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Generic flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051S","source_file":"algebra.tex","source_line":28526,"source_end_line":28541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28526-L28541","statement_sha256":"ae2653a6b6700887c5005ba2a3ae9239651a09a3de40d407141e3631d6aabc84","origin":"The Stacks Project","memory_eligible":false,"source_rank":1763,"rank":1763,"depth":13,"x":1974.609,"y":113.826,"cluster":"commutative-algebra"},{"id":"stacks:051T","tag":"051T","title":"Generic flatness · Lemma 051T","summary":"Let R → S be a ring map. Let M be an S-module. Assume • R is a domain, • R → S is of finite type, and • M is a finite type S-module. Then there exists a nonzero f ∈ R such that • [(a)] M_f and S_f are free as R_f-modules, and • [(b)] S_f is a finitely presented R_f-algebra and M_f is a finitely presented S_f-module.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is a domain,\n\\item $R \\to S$ is of finite type, and\n\\item $M$ is a finite type $S$-module.\n\\end{enumerate}\nThen there exists a nonzero $f \\in R$ such that\n\\begin{enumerate}\n\\item[(a)] $M_f$ and $S_f$ are free as $R_f$-modules, and\n\\item[(b)] $S_f$ is a finitely presented $R_f$-algebra and $M_f$ is a\nfinitely presented $S_f$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Generic flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051T","source_file":"algebra.tex","source_line":28564,"source_end_line":28580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28564-L28580","statement_sha256":"6698292411c51ba8bb38c1c31dff378bdc78ec84b7d984ec726428b439e47779","origin":"The Stacks Project","memory_eligible":false,"source_rank":1764,"rank":1764,"depth":14,"x":2156.301,"y":266.888,"cluster":"commutative-algebra"},{"id":"stacks:051W","tag":"051W","title":"Generic flatness · Lemma 051W","summary":"Let R → S be a ring map. Let 0 → M_1 → M_2 → M_3 → 0 be a short exact sequence of S-modules. Then U(R → S, M_1) ∩ U(R → S, M_3) ⊂ U(R → S, M_2).","statement_latex":"Let $R \\to S$ be a ring map.\nLet $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a short exact sequence\nof $S$-modules.\nThen\n$$\nU(R \\to S, M_1) \\cap U(R \\to S, M_3) \\subset U(R \\to S, M_2).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Generic flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051W","source_file":"algebra.tex","source_line":28651,"source_end_line":28660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28651-L28660","statement_sha256":"e93063b926dfcf84c427c9a5a2ee0632ba11c14b686a5c0fa6cc9931207aeb77","origin":"The Stacks Project","memory_eligible":false,"source_rank":1765,"rank":1765,"depth":2,"x":1899.084,"y":257.113,"cluster":"commutative-algebra"},{"id":"stacks:051X","tag":"051X","title":"Generic flatness · Lemma 051X","summary":"Let R → S be a ring map. Let M be an S-module. Let f ∈ R. Using the identification Spec(R_f) = D(f) we have U(R_f → S_f, M_f) = D(f) ∩ U(R → S, M).","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nLet $f \\in R$.\nUsing the identification $\\Spec(R_f) = D(f)$ we have\n$U(R_f \\to S_f, M_f) = D(f) \\cap U(R \\to S, M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Generic flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051X","source_file":"algebra.tex","source_line":28675,"source_end_line":28682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28675-L28682","statement_sha256":"7463412610c42076463c8dd81665570bce6936d323e671a92bfb9b7a18f4e768","origin":"The Stacks Project","memory_eligible":false,"source_rank":1766,"rank":1766,"depth":0,"x":2096.729,"y":118.288,"cluster":"commutative-algebra"},{"id":"stacks:051Y","tag":"051Y","title":"Generic flatness · Lemma 051Y","summary":"Let R → S be a ring map. Let M be an S-module. Let U ⊂ Spec(R) be a dense open. Assume there is a covering U = ⋃_i ∈ I D(f_i) of opens such that U(R_f_i → S_f_i, M_f_i) is dense in D(f_i) for each i ∈ I. Then U(R → S, M) is dense in Spec(R).","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nLet $U \\subset \\Spec(R)$ be a dense open.\nAssume there is a covering $U = \\bigcup_{i \\in I} D(f_i)$ of\nopens such that $U(R_{f_i} \\to S_{f_i}, M_{f_i})$ is dense in\n$D(f_i)$ for each $i \\in I$. Then $U(R \\to S, M)$ is dense in\n$\\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Generic flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051Y","source_file":"algebra.tex","source_line":28702,"source_end_line":28711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28702-L28711","statement_sha256":"10be768eeeb1520b06fe559ef5aa46d9548a49f73b0ef56f09618f4a4826b5da","origin":"The Stacks Project","memory_eligible":false,"source_rank":1767,"rank":1767,"depth":1,"x":2062.608,"y":332.931,"cluster":"commutative-algebra"},{"id":"stacks:051Z","tag":"051Z","title":"Generic flatness · Lemma 051Z","summary":"Let R → S be a ring map. Let M be an S-module. Assume • R → S is of finite type, • M is a finite S-module, and • R is reduced. Then there exists a subset U ⊂ Spec(R) such that • U is open and dense in Spec(R), • for every u ∈ U there exists an f ∈ R such that u ∈ D(f) ⊂ U and such that we have • M_f and S_f are free over R_f, • S_f is a finitely presented R_f-algebra, and • M_f is a finitely presented S_f-module.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.\nAssume\n\\begin{enumerate}\n\\item $R \\to S$ is of finite type,\n\\item $M$ is a finite $S$-module, and\n\\item $R$ is reduced.\n\\end{enumerate}\nThen there exists a subset $U \\subset \\Spec(R)$ such that\n\\begin{enumerate}\n\\item $U$ is open and dense in $\\Spec(R)$,\n\\item for every $u \\in U$ there exists an $f \\in R$ such\nthat $u \\in D(f) \\subset U$ and such that we have\n\\begin{enumerate}\n\\item $M_f$ and $S_f$ are free over $R_f$,\n\\item $S_f$ is a finitely presented $R_f$-algebra, and\n\\item $M_f$ is a finitely presented $S_f$-module.\n\\end{enumerate}\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Generic flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/051Z","source_file":"algebra.tex","source_line":28725,"source_end_line":28745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28725-L28745","statement_sha256":"9c98e6eaf26a5834e18753ec7f6ddb544c09757f7a3a2fc5252166178607254f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1768,"rank":1768,"depth":3,"x":1915.072,"y":155.191,"cluster":"commutative-algebra"},{"id":"stacks:00P8","tag":"00P8","title":"Around Krull-Akizuki · Lemma 00P8","summary":"Let R be a local Noetherian domain with fraction field K. Assume R is not a field. Then there exist R ⊂ R' ⊂ K with • R' local Noetherian of dimension 1, • R → R' a local ring map, i.e., R' dominates R, and • R → R' essentially of finite type.","statement_latex":"Let $R$ be a local Noetherian domain with fraction field $K$.\nAssume $R$ is not a field.\nThen there exist $R \\subset R' \\subset K$ with\n\\begin{enumerate}\n\\item $R'$ local Noetherian of dimension $1$,\n\\item $R \\to R'$ a local ring map, i.e., $R'$ dominates $R$, and\n\\item $R \\to R'$ essentially of finite type.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00P8","source_file":"algebra.tex","source_line":28891,"source_end_line":28901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28891-L28901","statement_sha256":"98180bffc664cdeb35712a390f474e7e369dc4d26f50ec859b748e7088a6e0dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1769,"rank":1769,"depth":10,"x":2166.944,"y":202.566,"cluster":"commutative-algebra"},{"id":"stacks:0BHZ","tag":"0BHZ","title":"Koll\\'ar · Lemma 0BHZ","summary":"This is taken from a forthcoming paper by J\\'anos Koll\\'ar entitled \"Variants of normality for Noetherian schemes\". Let (R, m) be a local Noetherian ring. Then exactly one of the following holds: • (R, m) is Artinian, • (R, m) is regular of dimension 1, • depth(R) ≥ 2, or • there exists a finite ring map R → R' which is not an isomorphism whose kernel and cokernel are annihilated by a power of m such that m is not an associated prime of R' and R' not = 0.","statement_latex":"\\begin{reference}\nThis is taken from a forthcoming paper by\nJ\\'anos Koll\\'ar entitled ``Variants of normality for Noetherian schemes''.\n\\end{reference}\nLet $(R, \\mathfrak m)$ be a local Noetherian ring.\nThen exactly one of the following holds:\n\\begin{enumerate}\n\\item $(R, \\mathfrak m)$ is Artinian,\n\\item $(R, \\mathfrak m)$ is regular of dimension $1$,\n\\item $\\text{depth}(R) \\geq 2$, or\n\\item there exists a finite ring map $R \\to R'$ which is not\nan isomorphism whose kernel and cokernel are annihilated by a power\nof $\\mathfrak m$ such that $\\mathfrak m$ is not an associated\nprime of $R'$ and $R' \\not = 0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHZ","source_file":"algebra.tex","source_line":28922,"source_end_line":28939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L28922-L28939","statement_sha256":"54c7d37089d6452cbf04c28637a500a313f7ff8891cf28363281987a4ab5e6bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1770,"rank":1770,"depth":13,"x":1942.989,"y":310.614,"cluster":"commutative-algebra"},{"id":"stacks:00P9","tag":"00P9","title":"Around Krull-Akizuki · Lemma 00P9","summary":"Let R be a local ring with maximal ideal m. Assume R is Noetherian, has dimension 1, and that dim( m/ m^2) > 1. Then there exists a ring map R → R' such that • R → R' is finite, • R → R' is not an isomorphism, • the kernel and cokernel of R → R' are annihilated by a power of m, and • m is not an associated prime of R'.","statement_latex":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$.\nAssume $R$ is Noetherian, has dimension $1$, and that\n$\\dim(\\mathfrak m/\\mathfrak m^2) > 1$. Then there exists\na ring map $R \\to R'$ such that\n\\begin{enumerate}\n\\item $R \\to R'$ is finite,\n\\item $R \\to R'$ is not an isomorphism,\n\\item the kernel and cokernel of $R \\to R'$ are annihilated by\na power of $\\mathfrak m$, and\n\\item $\\mathfrak m$ is not an associated prime of $R'$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00P9","source_file":"algebra.tex","source_line":29008,"source_end_line":29021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29008-L29021","statement_sha256":"d697cd51ba709d119b067f82fe33c57265b73a0d0c0b7d29e130ea9033cc31af","origin":"The Stacks Project","memory_eligible":false,"source_rank":1771,"rank":1771,"depth":14,"x":2021.287,"y":103.742,"cluster":"commutative-algebra"},{"id":"stacks:00PD","tag":"00PD","title":"Around Krull-Akizuki · Lemma 00PD","summary":"Let A be a ring. The following are equivalent. • The ring A is a discrete valuation ring. • The ring A is a valuation ring and Noetherian but not a field. • The ring A is a regular local ring of dimension 1. • The ring A is a Noetherian local domain with maximal ideal m generated by a single nonzero element. • The ring A is a Noetherian local normal domain of dimension 1. In this case if π is a generator of the maximal ideal of A, then every nonzero element of A can be…","statement_latex":"Let $A$ be a ring. The following are equivalent.\n\\begin{enumerate}\n\\item The ring $A$ is a discrete valuation ring.\n\\item The ring $A$ is a valuation ring and Noetherian but not a field.\n\\item The ring $A$ is a regular local ring of dimension $1$.\n\\item The ring $A$ is a Noetherian local domain with maximal ideal\n$\\mathfrak m$ generated by a single nonzero element.\n\\item The ring $A$ is a Noetherian local normal domain of dimension $1$.\n\\end{enumerate}\nIn this case if $\\pi$ is a generator of the maximal ideal of\n$A$, then every nonzero element of $A$ can be uniquely written as\n$u\\pi^n$, where $u \\in A$ is a unit.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PD","source_file":"algebra.tex","source_line":29103,"source_end_line":29117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29103-L29117","statement_sha256":"5a8510a2176ad16f09747bf1d697872c29e0dee3f6d45ec79ee947430d952606","origin":"The Stacks Project","memory_eligible":false,"source_rank":1772,"rank":1772,"depth":15,"x":2129.975,"y":300.83,"cluster":"commutative-algebra"},{"id":"stacks:09DZ","tag":"09DZ","title":"Around Krull-Akizuki · Definition 09DZ","summary":"Let A be a discrete valuation ring. A uniformizer is an element π ∈ A which generates the maximal ideal of A.","statement_latex":"Let $A$ be a discrete valuation ring. A {\\it uniformizer} is an element\n$\\pi \\in A$ which generates the maximal ideal of $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DZ","source_file":"algebra.tex","source_line":29160,"source_end_line":29164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29160-L29164","statement_sha256":"f6a72084ab3920fb0e33e9a560f08944f8dd5141b14c5292c482463ce86f671e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1773,"rank":1773,"depth":0,"x":1891.198,"y":217.124,"cluster":"commutative-algebra"},{"id":"stacks:00PE","tag":"00PE","title":"Around Krull-Akizuki · Lemma 00PE","summary":"Let R be a domain with fraction field K. Let M be an R-submodule of K^⊕ r. Assume R is local Noetherian of dimension 1. For any nonzero x ∈ R we have length_R(R/xR) < ∞ and length_R(M/xM) ≤ r · length_R(R/xR).","statement_latex":"Let $R$ be a domain with fraction field $K$.\nLet $M$ be an $R$-submodule of $K^{\\oplus r}$.\nAssume $R$ is local Noetherian of dimension $1$.\nFor any nonzero $x \\in R$ we have $\\text{length}_R(R/xR) < \\infty$\nand\n$$\n\\text{length}_R(M/xM) \\leq r \\cdot \\text{length}_R(R/xR).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PE","source_file":"algebra.tex","source_line":29170,"source_end_line":29180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29170-L29180","statement_sha256":"db08c945d9202fbcd76b0c3cdd92ed52187aa23748a2c91b164189471bde1c45","origin":"The Stacks Project","memory_eligible":false,"source_rank":1774,"rank":1774,"depth":0,"x":2134.724,"y":143.317,"cluster":"commutative-algebra"},{"id":"stacks:031F","tag":"031F","title":"Around Krull-Akizuki · Lemma 031F","summary":"Let R → S be a homomorphism of domains inducing an injection of fraction fields K ⊂ L. If R is Noetherian local of dimension 1 and [L : K] < ∞ then • each prime ideal n_i of S lying over the maximal ideal m of R is maximal, • there are finitely many of these, and • [kappa( n_i) : kappa( m)] < ∞ for each i.","statement_latex":"Let $R \\to S$ be a homomorphism of domains inducing an\ninjection of fraction fields $K \\subset L$. If $R$ is Noetherian\nlocal of dimension $1$ and $[L : K] < \\infty$ then\n\\begin{enumerate}\n\\item each prime ideal $\\mathfrak n_i$ of $S$ lying over\nthe maximal ideal $\\mathfrak m$ of $R$ is maximal,\n\\item there are finitely many of these, and\n\\item $[\\kappa(\\mathfrak n_i) : \\kappa(\\mathfrak m)] < \\infty$ for each $i$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031F","source_file":"algebra.tex","source_line":29240,"source_end_line":29251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29240-L29251","statement_sha256":"9fe73c175a77a479f941d6153b4f4f8c57a6aed7c6a6dccbb96aa97c4aa593d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1775,"rank":1775,"depth":7,"x":2014.435,"y":336.036,"cluster":"commutative-algebra"},{"id":"stacks:00PF","tag":"00PF","title":"Around Krull-Akizuki · Lemma 00PF","summary":"Let R be a domain with fraction field K. Let M be an R-submodule of K^⊕ r. Assume R is Noetherian of dimension 1. For any nonzero x ∈ R we have length_R(M/xM) < ∞.","statement_latex":"Let $R$ be a domain with fraction field $K$.\nLet $M$ be an $R$-submodule of $K^{\\oplus r}$.\nAssume $R$ is Noetherian of dimension $1$.\nFor any nonzero $x \\in R$ we have\n$\\text{length}_R(M/xM) < \\infty$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PF","source_file":"algebra.tex","source_line":29266,"source_end_line":29273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29266-L29273","statement_sha256":"075c16a5eb8691d9e912b0792b9411222d02d2dd91e0aee5ac04987d761e5dc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1776,"rank":1776,"depth":9,"x":1948.117,"y":125.55,"cluster":"commutative-algebra"},{"id":"stacks:00PG","tag":"00PG","title":"Krull-Akizuki · Lemma 00PG","summary":"Let R be a domain with fraction field K. Let L/K be a finite extension of fields. Assume R is Noetherian and dim(R) = 1. In this case any ring A with R ⊂ A ⊂ L is Noetherian.","statement_latex":"Let $R$ be a domain with fraction field $K$.\nLet $L/K$ be a finite extension of fields.\nAssume $R$ is Noetherian and $\\dim(R) = 1$.\nIn this case any ring $A$ with $R \\subset A \\subset L$ is\nNoetherian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PG","source_file":"algebra.tex","source_line":29292,"source_end_line":29299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29292-L29299","statement_sha256":"a75fa4cdf5861115d07a2bd60453f69952de60edb1841722d9ddc2025fd91cd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1777,"rank":1777,"depth":10,"x":2166.412,"y":243.197,"cluster":"commutative-algebra"},{"id":"stacks:00PH","tag":"00PH","title":"Around Krull-Akizuki · Lemma 00PH","summary":"Let R be a Noetherian local domain with fraction field K. Assume that R is not a field. Let L/K be a finitely generated field extension. Then there exists discrete valuation ring A with fraction field L which dominates R.","statement_latex":"Let $R$ be a Noetherian local domain with fraction field $K$.\nAssume that $R$ is not a field.\nLet $L/K$ be a finitely generated field extension.\nThen there exists discrete valuation ring $A$ with fraction field\n$L$ which dominates $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Around Krull-Akizuki","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PH","source_file":"algebra.tex","source_line":29311,"source_end_line":29318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29311-L29318","statement_sha256":"ea1250194c0d0990e357bdab416cb032dad623c9d3836636e7c77dad76f85d79","origin":"The Stacks Project","memory_eligible":false,"source_rank":1778,"rank":1778,"depth":16,"x":1910.688,"y":280.334,"cluster":"commutative-algebra"},{"id":"stacks:034P","tag":"034P","title":"Factorization · Definition 034P","summary":"Let R be a domain. • Elements x, y ∈ R are called associates if there exists a unit u ∈ R^* such that x = uy. • An element x ∈ R is called irreducible if it is nonzero, not a unit and whenever x = yz, y, z ∈ R, then y is either a unit or an associate of x. • An element x ∈ R is called prime if the ideal generated by x is a prime ideal.","statement_latex":"Let $R$ be a domain.\n\\begin{enumerate}\n\\item Elements $x, y \\in R$ are called {\\it associates} if\nthere exists a unit $u \\in R^*$ such that $x = uy$.\n\\item An element $x \\in R$ is called {\\it irreducible}\nif it is nonzero, not a unit and whenever $x = yz$, $y, z \\in R$,\nthen $y$ is either a unit or an associate of $x$.\n\\item An element $x \\in R$ is called {\\it prime} if the ideal\ngenerated by $x$ is a prime ideal.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034P","source_file":"algebra.tex","source_line":29354,"source_end_line":29366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29354-L29366","statement_sha256":"4407fca7506d6983baf1792e88c46db76979cb1f95f04aeab74b552ee416da7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1779,"rank":1779,"depth":0,"x":2069.483,"y":107.745,"cluster":"commutative-algebra"},{"id":"stacks:034Q","tag":"034Q","title":"Factorization · Lemma 034Q","summary":"Let R be a domain. Let x, y ∈ R. Then x, y are associates if and only if (x) = (y).","statement_latex":"Let $R$ be a domain. Let $x, y \\in R$.\nThen $x$, $y$ are associates if and only if $(x) = (y)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034Q","source_file":"algebra.tex","source_line":29368,"source_end_line":29372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29368-L29372","statement_sha256":"8015e08879cd646ba0b1f4edc7ff7194ff3b567b1d5581619c20f6201219380f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1780,"rank":1780,"depth":0,"x":2091.194,"y":325.239,"cluster":"commutative-algebra"},{"id":"stacks:034R","tag":"034R","title":"Factorization · Lemma 034R","summary":"Let R be a domain. Consider the following conditions: • The ring R satisfies the ascending chain condition for principal ideals. • Every nonzero, nonunit element a ∈ R has a factorization a = b_1 … b_k with each b_i an irreducible element of R. Then (1) implies (2).","statement_latex":"Let $R$ be a domain. Consider the following conditions:\n\\begin{enumerate}\n\\item The ring $R$ satisfies the ascending chain condition for\nprincipal ideals.\n\\item Every nonzero, nonunit element $a \\in R$\nhas a factorization $a = b_1 \\ldots b_k$\nwith each $b_i$ an irreducible element of $R$.\n\\end{enumerate}\nThen (1) implies (2).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034R","source_file":"algebra.tex","source_line":29382,"source_end_line":29393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29382-L29393","statement_sha256":"c25c25e6f3b8ae9a815a9f5e2cb2f7e9e7620507737b94f53d685d7a54e1134d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1781,"rank":1781,"depth":0,"x":1900.171,"y":177.096,"cluster":"commutative-algebra"},{"id":"stacks:034S","tag":"034S","title":"Factorization · Definition 034S","summary":"A unique factorization domain, abbreviated UFD, is a domain R such that if x ∈ R is a nonzero, nonunit, then x has a factorization into irreducibles, and if x = a_1 … a_m = b_1 … b_n are factorizations into irreducibles then n = m and there exists a permutation σ : (1, …, n) → (1, …, n) such that a_i and b_σ(i) are associates.","statement_latex":"A {\\it unique factorization domain}, abbreviated {\\it UFD},\nis a domain $R$ such that\nif $x \\in R$ is a nonzero, nonunit, then $x$ has a factorization\ninto irreducibles, and if\n$$\nx = a_1 \\ldots a_m = b_1 \\ldots b_n\n$$\nare factorizations into irreducibles then $n = m$ and\nthere exists a permutation $\\sigma : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, n\\}$\nsuch that $a_i$ and $b_{\\sigma(i)}$ are associates.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034S","source_file":"algebra.tex","source_line":29412,"source_end_line":29424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29412-L29424","statement_sha256":"4daebc07accca57854474f53344d2bcf88fbc754ef7fe877b3571e7d4ce4f789","origin":"The Stacks Project","memory_eligible":false,"source_rank":1782,"rank":1782,"depth":0,"x":2160.311,"y":177.944,"cluster":"commutative-algebra"},{"id":"stacks:034T","tag":"034T","title":"Factorization · Lemma 034T","summary":"Let R be a domain. Assume every nonzero, nonunit factors into irreducibles. Then R is a UFD if and only if every irreducible element is prime.","statement_latex":"Let $R$ be a domain. Assume every nonzero, nonunit factors into\nirreducibles. Then $R$ is a UFD if and only if every irreducible\nelement is prime.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034T","source_file":"algebra.tex","source_line":29426,"source_end_line":29431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29426-L29431","statement_sha256":"582a72ea3f98694ba2ae46020de3fffc1363837ad21e05ccbd320f3609ad2b72","origin":"The Stacks Project","memory_eligible":false,"source_rank":1783,"rank":1783,"depth":0,"x":1967.695,"y":325.013,"cluster":"commutative-algebra"},{"id":"stacks:0AFT","tag":"0AFT","title":"Factorization · Lemma 0AFT","summary":"Let R be a Noetherian domain. Then R is a UFD if and only if every height 1 prime ideal is principal.","statement_latex":"Let $R$ be a Noetherian domain. Then $R$ is a UFD if and only if every\nheight $1$ prime ideal is principal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFT","source_file":"algebra.tex","source_line":29458,"source_end_line":29462,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29458-L29462","statement_sha256":"dc7d28d770c4273b0bfe5a3030a80d8ae558082b6717d835ae93d1c7529cdcbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1784,"rank":1784,"depth":10,"x":1991.472,"y":107.146,"cluster":"commutative-algebra"},{"id":"stacks:0AFU","tag":"0AFU","title":"Nagata's criterion for factoriality · Lemma 0AFU","summary":"[Nagata-UFD] Let A be a domain. Let S ⊂ A be a multiplicative subset generated by prime elements. Let x ∈ A be irreducible. Then • the image of x in S^-1A is irreducible or a unit, and • x is prime if and only if the image of x in S^-1A is a prime element or a unit in S^-1A. Moreover, then A is a UFD if and only if every nonzero nonunit element of A has a factorization into irreducibles and S^-1A is a UFD.","statement_latex":"\\begin{reference}\n\\cite[Lemma 2]{Nagata-UFD}\n\\end{reference}\nLet $A$ be a domain. Let $S \\subset A$ be a multiplicative subset\ngenerated by prime elements. Let $x \\in A$ be irreducible. Then\n\\begin{enumerate}\n\\item the image of $x$ in $S^{-1}A$ is irreducible or a unit, and\n\\item $x$ is prime if and only if the image of $x$ in $S^{-1}A$ is\na prime element or a unit in $S^{-1}A$.\n\\end{enumerate}\nMoreover, then $A$ is a UFD if and only if every nonzero nonunit\nelement of $A$ has a factorization into irreducibles and $S^{-1}A$ is a UFD.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFU","source_file":"algebra.tex","source_line":29484,"source_end_line":29498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29484-L29498","statement_sha256":"506d782a399709e609a9aade167da98be5ebe723d91d0f6d800b63fe839d601f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1785,"rank":1785,"depth":1,"x":2149.232,"y":281.39,"cluster":"commutative-algebra"},{"id":"stacks:0BUD","tag":"0BUD","title":"Factorization · Lemma 0BUD","summary":"A UFD satisfies the ascending chain condition for principal ideals.","statement_latex":"A UFD satisfies the ascending chain condition for principal\nideals.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUD","source_file":"algebra.tex","source_line":29542,"source_end_line":29546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29542-L29546","statement_sha256":"fb99343142034b9d8e4e110df4d0ceaae37574a0d6f0f12ec3c69d7e04132117","origin":"The Stacks Project","memory_eligible":false,"source_rank":1786,"rank":1786,"depth":0,"x":1892.633,"y":242.4,"cluster":"commutative-algebra"},{"id":"stacks:0BUE","tag":"0BUE","title":"Factorization · Lemma 0BUE","summary":"Let R be a domain. Assume R has the ascending chain condition for principal ideals. Then the same property holds for a polynomial ring over R.","statement_latex":"Let $R$ be a domain. Assume $R$ has the ascending chain condition\nfor principal ideals. Then the same property holds for a polynomial\nring over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUE","source_file":"algebra.tex","source_line":29556,"source_end_line":29561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29556-L29561","statement_sha256":"f05dc9b4a28adf4a5a538c7bc06dc1bef80c7bd1df7bf0e3951fe92a51ff851f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1787,"rank":1787,"depth":1,"x":2113.327,"y":125.483,"cluster":"commutative-algebra"},{"id":"stacks:0BC1","tag":"0BC1","title":"Factorization · Lemma 0BC1","summary":"A polynomial ring over a UFD is a UFD. In particular, if k is a field, then k[x_1, …, x_n] is a UFD.","statement_latex":"A polynomial ring over a UFD is a UFD. In particular, if $k$ is a field,\nthen $k[x_1, \\ldots, x_n]$ is a UFD.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BC1","source_file":"algebra.tex","source_line":29577,"source_end_line":29581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29577-L29581","statement_sha256":"687c898bc0f3a4d5f9af5ca21d955226d746e1fdd300814a9524158ed56af2d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1788,"rank":1788,"depth":2,"x":2044.572,"y":337.044,"cluster":"commutative-algebra"},{"id":"stacks:0AFV","tag":"0AFV","title":"Factorization · Lemma 0AFV","summary":"A unique factorization domain is normal.","statement_latex":"A unique factorization domain is normal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFV","source_file":"algebra.tex","source_line":29600,"source_end_line":29603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29600-L29603","statement_sha256":"7e0f9b0d100ccd87485b642f6af46b77ce31b42682008cff17a24be2a0d9a8bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1789,"rank":1789,"depth":0,"x":1925.071,"y":141.917,"cluster":"commutative-algebra"},{"id":"stacks:034U","tag":"034U","title":"Factorization · Definition 034U","summary":"A principal ideal domain, abbreviated PID, is a domain R such that every ideal is a principal ideal.","statement_latex":"A {\\it principal ideal domain}, abbreviated {\\it PID},\nis a domain $R$ such that every ideal is a principal ideal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034U","source_file":"algebra.tex","source_line":29621,"source_end_line":29625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29621-L29625","statement_sha256":"eea3eee79d628b3f2d236b2700d3260e776f5c907ee17cb5d4786afdf27468c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1790,"rank":1790,"depth":0,"x":2170.245,"y":218.036,"cluster":"commutative-algebra"},{"id":"stacks:034V","tag":"034V","title":"Factorization · Lemma 034V","summary":"A principal ideal domain is a unique factorization domain.","statement_latex":"A principal ideal domain is a unique factorization domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034V","source_file":"algebra.tex","source_line":29627,"source_end_line":29630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29627-L29630","statement_sha256":"fb907aaf84833f1d078907a9f87a438587a26a69ccad2feb6376b3b604e71d8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1791,"rank":1791,"depth":11,"x":1928.106,"y":301.073,"cluster":"commutative-algebra"},{"id":"stacks:034W","tag":"034W","title":"Factorization · Definition 034W","summary":"A Dedekind domain is a domain R such that every nonzero ideal I ⊂ R can be written as a product I = p_1 … p_r of nonzero prime ideals uniquely up to permutation of the p_i.","statement_latex":"A {\\it Dedekind domain} is a domain $R$ such that every\nnonzero ideal $I \\subset R$ can be written as a product\n$$\nI = \\mathfrak p_1 \\ldots \\mathfrak p_r\n$$\nof nonzero prime ideals uniquely up to permutation of the $\\mathfrak p_i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034W","source_file":"algebra.tex","source_line":29637,"source_end_line":29645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29637-L29645","statement_sha256":"f15acd5ba969d66a8c7676102e9bcebe1667cb0cac18d5661a57a7c7201edab8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1792,"rank":1792,"depth":0,"x":2039.944,"y":102.334,"cluster":"commutative-algebra"},{"id":"stacks:0AUQ","tag":"0AUQ","title":"Factorization · Lemma 0AUQ","summary":"A PID is a Dedekind domain.","statement_latex":"A PID is a Dedekind domain.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUQ","source_file":"algebra.tex","source_line":29647,"source_end_line":29650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29647-L29650","statement_sha256":"268415252e618c19d498d78733164542edf80557fbc07b3825ce223794e9decb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1793,"rank":1793,"depth":12,"x":2117.341,"y":312.46,"cluster":"commutative-algebra"},{"id":"stacks:09ME","tag":"09ME","title":"Factorization · Lemma 09ME","summary":"A product of ideals is an invertible module iff both factors are. Let A be a ring. Let I and J be nonzero ideals of A such that IJ = (f) for some nonzerodivisor f ∈ A. Then I and J are finitely generated ideals and finitely locally free of rank 1 as A-modules.","statement_latex":"\\begin{slogan}\nA product of ideals is an invertible module iff both factors are.\n\\end{slogan}\nLet $A$ be a ring. Let $I$ and $J$ be nonzero ideals of $A$\nsuch that $IJ = (f)$ for some nonzerodivisor $f \\in A$. Then $I$ and $J$ are\nfinitely generated ideals and finitely locally free of rank $1$ as $A$-modules.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ME","source_file":"algebra.tex","source_line":29660,"source_end_line":29668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29660-L29668","statement_sha256":"73012239cb1ad8355d70928a4581d5493090b882540cf3d05a4e174e5cde2d5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1794,"rank":1794,"depth":5,"x":1891.164,"y":201.37,"cluster":"commutative-algebra"},{"id":"stacks:034X","tag":"034X","title":"Factorization · Lemma 034X","summary":"Let R be a ring. The following are equivalent • R is a Dedekind domain, • R is a Noetherian domain and for every nonzero maximal ideal m the local ring R_ m is a discrete valuation ring, and • R is a Noetherian, normal domain, and dim(R) ≤ 1.","statement_latex":"Let $R$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $R$ is a Dedekind domain,\n\\item $R$ is a Noetherian domain and for every nonzero maximal ideal\n$\\mathfrak m$ the local ring $R_{\\mathfrak m}$ is a discrete valuation ring, and\n\\item $R$ is a Noetherian, normal domain, and $\\dim(R) \\leq 1$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034X","source_file":"algebra.tex","source_line":29685,"source_end_line":29694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29685-L29694","statement_sha256":"48673f0af11417ae3fcb668b80d0f6a2b3fbb436a67f4b6382f83b0bcf453235","origin":"The Stacks Project","memory_eligible":false,"source_rank":1795,"rank":1795,"depth":16,"x":2147.424,"y":154.921,"cluster":"commutative-algebra"},{"id":"stacks:09IG","tag":"09IG","title":"Factorization · Lemma 09IG","summary":"Let A be a Noetherian domain of dimension 1 with fraction field K. Let L/K be a finite extension. Let B be the integral closure of A in L. Then B is a Dedekind domain and Spec(B) → Spec(A) is surjective, has finite fibres, and induces finite residue field extensions.","statement_latex":"Let $A$ be a Noetherian domain of dimension $1$ with fraction field $K$.\nLet $L/K$ be a finite extension. Let $B$ be the\nintegral closure of $A$ in $L$. Then $B$ is a Dedekind domain and\n$\\Spec(B) \\to \\Spec(A)$ is surjective, has finite fibres, and\ninduces finite residue field extensions.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IG","source_file":"algebra.tex","source_line":29753,"source_end_line":29760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29753-L29760","statement_sha256":"0a13d58a13fce2a7355630c5d149f8506840f9f4cd7dd6a9f87d1ad0a33d877d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1796,"rank":1796,"depth":17,"x":1995.728,"y":334.682,"cluster":"commutative-algebra"},{"id":"stacks:02MC","tag":"02MC","title":"Orders of vanishing · Lemma 02MC","summary":"Let R be a semi-local Noetherian ring of dimension 1. If a, b ∈ R are nonzerodivisors then length_R(R/(ab)) = length_R(R/(a)) + length_R(R/(b)) and these lengths are finite.","statement_latex":"Let $R$ be a semi-local Noetherian ring of dimension $1$.\nIf $a, b \\in R$ are nonzerodivisors then\n$$\n\\text{length}_R(R/(ab)) =\n\\text{length}_R(R/(a)) +\n\\text{length}_R(R/(b))\n$$\nand these lengths are finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Orders of vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MC","source_file":"algebra.tex","source_line":29782,"source_end_line":29792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29782-L29792","statement_sha256":"5b5d51f1045f9a742fc5a01ddd31ae4bf7b4be0730240c8631afd69762d37669","origin":"The Stacks Project","memory_eligible":false,"source_rank":1797,"rank":1797,"depth":10,"x":1963.009,"y":115.929,"cluster":"commutative-algebra"},{"id":"stacks:02MD","tag":"02MD","title":"Orders of vanishing · Definition 02MD","summary":"Suppose that K is a field, and R ⊂ K is a local Noetherian subring of dimension 1 with fraction field K. In this case we define the order of vanishing along R ord_R : K^* → Z by the rule ord_R(x) = length_R(R/(x)) if x ∈ R and we set ord_R(x/y) = ord_R(x) - ord_R(y) for x, y ∈ R both nonzero.","statement_latex":"Suppose that $K$ is a field, and $R \\subset K$ is a\nlocal\\footnote{We could also define this when $R$ is only\nsemi-local but this is probably never really what you want!}\nNoetherian subring of dimension $1$ with fraction field $K$.\nIn this case we define the {\\it order of vanishing along $R$}\n$$\n\\text{ord}_R : K^* \\longrightarrow \\mathbf{Z}\n$$\nby the rule\n$$\n\\text{ord}_R(x) = \\text{length}_R(R/(x))\n$$\nif $x \\in R$ and we set\n$\\text{ord}_R(x/y) = \\text{ord}_R(x) - \\text{ord}_R(y)$\nfor $x, y \\in R$ both nonzero.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Orders of vanishing","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MD","source_file":"algebra.tex","source_line":29802,"source_end_line":29819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29802-L29819","statement_sha256":"844aec15dc6334ebb91fea22eb98bc6d7c0ca27b3056afd77e0c793e5bc0fa62","origin":"The Stacks Project","memory_eligible":false,"source_rank":1798,"rank":1798,"depth":0,"x":2163.164,"y":258.75,"cluster":"commutative-algebra"},{"id":"stacks:02ME","tag":"02ME","title":"Orders of vanishing · Definition 02ME","summary":"Let R be a Noetherian local domain of dimension 1 with fraction field K. Let V be a finite dimensional K-vector space. A lattice in V is a finite R-submodule M ⊂ V such that V = K ⊗_R M.","statement_latex":"Let $R$ be a Noetherian local domain of dimension $1$ with\nfraction field $K$. Let $V$ be a finite dimensional $K$-vector space.\nA {\\it lattice in $V$} is a finite $R$-submodule $M \\subset V$ such\nthat $V = K \\otimes_R M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Orders of vanishing","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ME","source_file":"algebra.tex","source_line":29825,"source_end_line":29831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29825-L29831","statement_sha256":"24dcedc4e1e8c15449449d12b49b94a3f89bb5f6c2a316a5f235f5ce94d61c8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1799,"rank":1799,"depth":0,"x":1900.572,"y":267.013,"cluster":"commutative-algebra"},{"id":"stacks:02MF","tag":"02MF","title":"Orders of vanishing · Lemma 02MF","summary":"Let R be a Noetherian local domain of dimension 1 with fraction field K. Let V be a finite dimensional K-vector space. • If M is a lattice in V and M ⊂ M' ⊂ V is an R-submodule of V containing M then the following are equivalent • M' is a lattice, • length_R(M'/M) is finite, and • M' is finitely generated. • If M is a lattice in V and M' ⊂ M is an R-submodule of M then M' is a lattice if and only if length_R(M/M') is finite. • If M, M' are lattices in V, then so are M ∩…","statement_latex":"Let $R$ be a Noetherian local domain of dimension $1$ with\nfraction field $K$. Let $V$ be a finite dimensional $K$-vector space.\n\\begin{enumerate}\n\\item If $M$ is a lattice in $V$ and $M \\subset M' \\subset V$\nis an $R$-submodule of $V$ containing $M$\nthen the following are equivalent\n\\begin{enumerate}\n\\item $M'$ is a lattice,\n\\item $\\text{length}_R(M'/M)$ is finite, and\n\\item $M'$ is finitely generated.\n\\end{enumerate}\n\\item If $M$ is a lattice in $V$ and $M' \\subset M$ is an $R$-submodule\nof $M$ then $M'$ is a lattice if and only if\n$\\text{length}_R(M/M')$ is finite.\n\\item If $M$, $M'$ are lattices in $V$, then so are\n$M \\cap M'$ and $M + M'$.\n\\item If $M \\subset M' \\subset M'' \\subset V$ are lattices in $V$\nthen\n$$\n\\text{length}_R(M''/M) =\n\\text{length}_R(M'/M) +\n\\text{length}_R(M''/M').\n$$\n\\item If $M$, $M'$, $N$, $N'$ are lattices in $V$ and\n$N \\subset M \\cap M'$, $M + M' \\subset N'$, then we have\n\\begin{eqnarray*}\n& & \\text{length}_R(M/M \\cap M') - \\text{length}_R(M'/M \\cap M')\\\\\n& = &\n\\text{length}_R(M/N) - \\text{length}_R(M'/N) \\\\\n& = &\n\\text{length}_R(M + M' / M') - \\text{length}_R(M + M'/M) \\\\\n& = &\n\\text{length}_R(N' / M') - \\text{length}_R(N'/M)\n\\end{eqnarray*}\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Orders of vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MF","source_file":"algebra.tex","source_line":29841,"source_end_line":29878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29841-L29878","statement_sha256":"f6d9cec576a43e441a751f7b37348e6c01db3ab2161cd63c3b2d8e40720d2f20","origin":"The Stacks Project","memory_eligible":false,"source_rank":1800,"rank":1800,"depth":11,"x":2087.664,"y":111.831,"cluster":"commutative-algebra"},{"id":"stacks:02MG","tag":"02MG","title":"Orders of vanishing · Definition 02MG","summary":"Let R be a Noetherian local domain of dimension 1 with fraction field K. Let V be a finite dimensional K-vector space. Let M, M' be two lattices in V. The distance between M and M' is the integer d(M, M') = length_R(M/M ∩ M') - length_R(M'/M ∩ M') of Lemma [Tag 02MF] part (5).","statement_latex":"Let $R$ be a Noetherian local domain of dimension $1$ with\nfraction field $K$. Let $V$ be a finite dimensional $K$-vector space.\nLet $M$, $M'$ be two lattices in $V$. The {\\it distance between\n$M$ and $M'$} is the integer\n$$\nd(M, M') = \\text{length}_R(M/M \\cap M') - \\text{length}_R(M'/M \\cap M')\n$$\nof Lemma \\ref{lemma-compare-lattices} part (5).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Orders of vanishing","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MG","source_file":"algebra.tex","source_line":29933,"source_end_line":29943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29933-L29943","statement_sha256":"4a9ae7f6c9b351e3a62592a8a195bba7be2cee98338e08dc0a571bcbad91b155","origin":"The Stacks Project","memory_eligible":false,"source_rank":1801,"rank":1801,"depth":12,"x":2074.491,"y":332.545,"cluster":"commutative-algebra"},{"id":"stacks:02MH","tag":"02MH","title":"Orders of vanishing · Lemma 02MH","summary":"Let R be a Noetherian local domain of dimension 1 with fraction field K. Let V be a finite dimensional K-vector space. This distance function has the property that d(M, M\") = d(M, M') + d(M', M\") whenever given three lattices M, M', M\" of V. In particular we have d(M, M') = - d(M', M).","statement_latex":"Let $R$ be a Noetherian local domain of dimension $1$ with\nfraction field $K$. Let $V$ be a finite dimensional $K$-vector space.\nThis distance function has the property that\n$$\nd(M, M'') = d(M, M') + d(M', M'')\n$$\nwhenever given three lattices $M$, $M'$, $M''$ of $V$.\nIn particular we have $d(M, M') = - d(M', M)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Orders of vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MH","source_file":"algebra.tex","source_line":29949,"source_end_line":29959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29949-L29959","statement_sha256":"024d13ed326605de127e46ee70eee9482243bacd36f7b1d21dcd33b81540c5f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1802,"rank":1802,"depth":0,"x":1906.618,"y":162.223,"cluster":"commutative-algebra"},{"id":"stacks:02MI","tag":"02MI","title":"Orders of vanishing · Lemma 02MI","summary":"Let R be a Noetherian local domain of dimension 1 with fraction field K. Let V be a finite dimensional K-vector space. Let φ : V → V be a K-linear isomorphism. For any lattice M ⊂ V we have d(M, φ(M)) = ord_R(det(φ))","statement_latex":"Let $R$ be a Noetherian local domain of dimension $1$ with\nfraction field $K$. Let $V$ be a finite dimensional $K$-vector space.\nLet $\\varphi : V \\to V$ be a $K$-linear isomorphism.\nFor any lattice $M \\subset V$ we have\n$$\nd(M, \\varphi(M)) = \\text{ord}_R(\\det(\\varphi))\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Orders of vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MI","source_file":"algebra.tex","source_line":29965,"source_end_line":29974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L29965-L29974","statement_sha256":"8b69f2741448872556a76463406876c33330567224092d930ae2a10436ec65cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1803,"rank":1803,"depth":0,"x":2167.52,"y":192.578,"cluster":"commutative-algebra"},{"id":"stacks:02MJ","tag":"02MJ","title":"Orders of vanishing · Lemma 02MJ","summary":"Let A → B be a ring map. Assume • A is a Noetherian local domain of dimension 1, • A ⊂ B is a finite extension of domains. Let L/K be the corresponding finite extension of fraction fields. Let y ∈ L^* and x = Nm_L/K(y). In this situation B is semi-local. Let m_i, i = 1, …, n be the maximal ideals of B. Then ord_A(x) = ∑_i [kappa( m_i) : kappa( m_A)] ord_B_ m_i(y) where ord is defined as in Definition [Tag 02MD].","statement_latex":"Let $A \\to B$ be a ring map. Assume\n\\begin{enumerate}\n\\item $A$ is a Noetherian local domain of dimension $1$,\n\\item $A \\subset B$ is a finite extension of domains.\n\\end{enumerate}\nLet $L/K$ be the corresponding finite extension of fraction fields.\nLet $y \\in L^*$ and $x = \\text{Nm}_{L/K}(y)$.\nIn this situation $B$ is semi-local.\nLet $\\mathfrak m_i$, $i = 1, \\ldots, n$ be the maximal ideals of $B$.\nThen\n$$\n\\text{ord}_A(x) =\n\\sum\\nolimits_i\n[\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m_A)]\n\\text{ord}_{B_{\\mathfrak m_i}}(y)\n$$\nwhere $\\text{ord}$ is defined as in Definition \\ref{definition-ord}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Orders of vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MJ","source_file":"algebra.tex","source_line":30037,"source_end_line":30056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30037-L30056","statement_sha256":"a5d00d1cf209fbf60de1c90392f67a6f262f6a4e5581a7fe7f0697e503bcc40a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1804,"rank":1804,"depth":14,"x":1950.602,"y":318.308,"cluster":"commutative-algebra"},{"id":"stacks:00PJ","tag":"00PJ","title":"Quasi-finite maps · Lemma 00PJ","summary":"Let k be a field. Let S be a finite type k-algebra. Let q be a prime of S. The following are equivalent: • q is an isolated point of Spec(S), • S_ q is finite over k, • there exists a g ∈ S, g not∈ q such that D(g) = ( q ), • dim_ q Spec(S) = 0, • q is a closed point of Spec(S) and dim(S_ q) = 0, and • the field extension kappa( q)/k is finite and dim(S_ q) = 0. In this case S = S_ q × S' for some finite type k-algebra S'. Also, the element g as in (3) has the property S_…","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra.\nLet $\\mathfrak q$ be a prime of $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\mathfrak q$ is an isolated point of $\\Spec(S)$,\n\\item $S_{\\mathfrak q}$ is finite over $k$,\n\\item there exists a $g \\in S$, $g \\not\\in \\mathfrak q$ such that\n$D(g) = \\{ \\mathfrak q \\}$,\n\\item $\\dim_{\\mathfrak q} \\Spec(S) = 0$,\n\\item $\\mathfrak q$ is a closed point of $\\Spec(S)$ and\n$\\dim(S_{\\mathfrak q}) = 0$, and\n\\item the field extension $\\kappa(\\mathfrak q)/k$ is finite\nand $\\dim(S_{\\mathfrak q}) = 0$.\n\\end{enumerate}\nIn this case $S = S_{\\mathfrak q} \\times S'$ for some\nfinite type $k$-algebra $S'$. Also, the element $g$\nas in (3) has the property $S_{\\mathfrak q} = S_g$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PJ","source_file":"algebra.tex","source_line":30105,"source_end_line":30125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30105-L30125","statement_sha256":"bcc3054762c1d93e7000a0d869591a8091ef8c3a47abd4197f6c1fab2aad481d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1805,"rank":1805,"depth":23,"x":2009.479,"y":102.391,"cluster":"commutative-algebra"},{"id":"stacks:00PK","tag":"00PK","title":"Quasi-finite maps · Lemma 00PK","summary":"Equivalent conditions for isolated points in fibres Let R → S be a ring map of finite type. Let q ⊂ S be a prime lying over p ⊂ R. Let F = Spec(S ⊗_R kappa( p)) be the fibre of Spec(S) → Spec(R), see Remark [Tag 00E6]. Denote overline q ∈ F the point corresponding to q. The following are equivalent • overline q is an isolated point of F, • S_ q/ pS_ q is finite over kappa( p), • there exists a g ∈ S, g not ∈ q such that the only prime of D(g) mapping to p is q, •…","statement_latex":"\\begin{slogan}\nEquivalent conditions for isolated points in fibres\n\\end{slogan}\nLet $R \\to S$ be a ring map of finite type.\nLet $\\mathfrak q \\subset S$ be a prime lying over\n$\\mathfrak p \\subset R$. Let $F = \\Spec(S \\otimes_R \\kappa(\\mathfrak p))$\nbe the fibre of $\\Spec(S) \\to \\Spec(R)$, see\nRemark \\ref{remark-fundamental-diagram}.\nDenote $\\overline{\\mathfrak q} \\in F$ the point corresponding to\n$\\mathfrak q$. The following are equivalent\n\\begin{enumerate}\n\\item $\\overline{\\mathfrak q}$ is an isolated point of $F$,\n\\item $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$ is finite over\n$\\kappa(\\mathfrak p)$,\n\\item there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that\nthe only prime of $D(g)$ mapping to $\\mathfrak p$ is $\\mathfrak q$,\n\\item $\\dim_{\\overline{\\mathfrak q}}(F) = 0$,\n\\item $\\overline{\\mathfrak q}$ is a closed point of $F$ and\n$\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) = 0$, and\n\\item the field extension $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$\nis finite and $\\dim(S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}) = 0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PK","source_file":"algebra.tex","source_line":30195,"source_end_line":30219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30195-L30219","statement_sha256":"03e31c720df397b06f5ff38eb14fe41ef79216b65ec3ee697cff0c60385c1d12","origin":"The Stacks Project","memory_eligible":false,"source_rank":1806,"rank":1806,"depth":24,"x":2139.771,"y":295.121,"cluster":"commutative-algebra"},{"id":"stacks:00PL","tag":"00PL","title":"Quasi-finite maps · Definition 00PL","summary":"Let R → S be a finite type ring map. Let q ⊂ S be a prime. • If the equivalent conditions of Lemma [Tag 00PK] are satisfied then we say R → S is quasi-finite at q. • We say a ring map A → B is quasi-finite if it is of finite type and quasi-finite at all primes of B.","statement_latex":"Let $R \\to S$ be a finite type ring map.\nLet $\\mathfrak q \\subset S$ be a prime.\n\\begin{enumerate}\n\\item If the equivalent conditions of Lemma \\ref{lemma-isolated-point-fibre}\nare satisfied then we say $R \\to S$ is {\\it quasi-finite at $\\mathfrak q$}.\n\\item We say a ring map $A \\to B$ is {\\it quasi-finite}\nif it is of finite type and quasi-finite at all primes of $B$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PL","source_file":"algebra.tex","source_line":30232,"source_end_line":30242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30232-L30242","statement_sha256":"b819175af04ed1c4141f08a9ddf5169b366f2f9afdd02465b8d478783d69858f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1807,"rank":1807,"depth":25,"x":1888.567,"y":226.898,"cluster":"commutative-algebra"},{"id":"stacks:0H8X","tag":"0H8X","title":"Quasi-finite maps · Lemma 0H8X","summary":"Let R → S be a finite type ring map and p be a prime ideal of R. Then the following are equivalent: • R → S is quasi-finite at all primes of S lying over p, • S ⊗_R kappa( p) is a finite kappa( p)-algebra, and • Spec(S ⊗_R kappa( p)) is a finite set.","statement_latex":"Let $R \\to S$ be a finite type ring map\nand $\\mathfrak p$ be a prime ideal of $R$.\nThen the following are equivalent:\n\\begin{enumerate}\n\\item $R \\to S$ is quasi-finite at all primes of $S$ lying over $\\mathfrak p$,\n\\item $S \\otimes_R \\kappa(\\mathfrak p)$ is a finite\n$\\kappa(\\mathfrak p)$-algebra, and\n\\item $\\Spec(S \\otimes_R \\kappa(\\mathfrak p))$ is a finite set.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8X","source_file":"algebra.tex","source_line":30244,"source_end_line":30255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30244-L30255","statement_sha256":"e7ca50ec64db600ffd2b9f468269c21695831c1d03cdb74b2461cde1921447c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1808,"rank":1808,"depth":10,"x":2128.799,"y":134.613,"cluster":"commutative-algebra"},{"id":"stacks:00PM","tag":"00PM","title":"Quasi-finite maps · Lemma 00PM","summary":"Let R → S be a finite type ring map. Then R → S is quasi-finite if and only if for all primes p ⊂ R the ring S ⊗_R kappa( p) is finite over kappa( p).","statement_latex":"Let $R \\to S$ be a finite type ring map. Then $R \\to S$ is quasi-finite\nif and only if for all primes $\\mathfrak p \\subset R$ the ring\n$S \\otimes_R \\kappa(\\mathfrak p)$ is finite over $\\kappa(\\mathfrak p)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PM","source_file":"algebra.tex","source_line":30264,"source_end_line":30269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30264-L30269","statement_sha256":"5e542d32414da2ba1f0627153d9e6d468d53f60768532e73cd4f8925d91340c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1809,"rank":1809,"depth":11,"x":2025.811,"y":339.09,"cluster":"commutative-algebra"},{"id":"stacks:077H","tag":"077H","title":"Quasi-finite maps · Lemma 077H","summary":"Let R → S be a finite type ring map. Let q ⊂ S be a prime lying over p ⊂ R. Let f ∈ R, f not ∈ p and g ∈ S, g not ∈ q. Then R → S is quasi-finite at q if and only if R_f → S_fg is quasi-finite at qS_fg.","statement_latex":"Let $R \\to S$ be a finite type ring map. Let $\\mathfrak q \\subset S$\nbe a prime lying over $\\mathfrak p \\subset R$. Let\n$f \\in R$, $f \\not \\in \\mathfrak p$ and $g \\in S$, $g \\not \\in \\mathfrak q$.\nThen $R \\to S$ is quasi-finite at $\\mathfrak q$ if and only if\n$R_f \\to S_{fg}$ is quasi-finite at $\\mathfrak qS_{fg}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077H","source_file":"algebra.tex","source_line":30276,"source_end_line":30283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30276-L30283","statement_sha256":"39ce8244a747bdfccd4e6f433a666f393cf28b00d981821918a51e2d7df3399a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1810,"rank":1810,"depth":25,"x":1937.269,"y":129.757,"cluster":"commutative-algebra"},{"id":"stacks:00PN","tag":"00PN","title":"Quasi-finite maps · Lemma 00PN","summary":"Let xymatrix S ar[r] & S' & & q ar@-[r] & q' R ar[u] ar[r] & R' ar[u] & & p ar@-[r] ar@-[u] & p' ar@-[u] be a commutative diagram of rings with primes as indicated. Assume R → S of finite type, and S ⊗_R R' → S' surjective. If R → S is quasi-finite at q, then R' → S' is quasi-finite at q'.","statement_latex":"Let\n$$\n\\xymatrix{\nS \\ar[r] & S' & &\n\\mathfrak q \\ar@{-}[r] & \\mathfrak q' \\\\\nR \\ar[u] \\ar[r] &  R' \\ar[u] & &\n\\mathfrak p \\ar@{-}[r] \\ar@{-}[u] & \\mathfrak p' \\ar@{-}[u]\n}\n$$\nbe a commutative diagram of rings with primes as indicated.\nAssume $R \\to S$ of finite type, and $S \\otimes_R R' \\to S'$ surjective.\nIf $R \\to S$ is quasi-finite at $\\mathfrak q$, then\n$R' \\to S'$ is quasi-finite at $\\mathfrak q'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PN","source_file":"algebra.tex","source_line":30292,"source_end_line":30307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30292-L30307","statement_sha256":"93b170fb195a7650e61373eb331ccb576088f8679990a3ce35ae36f3f021098f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1811,"rank":1811,"depth":24,"x":2171.028,"y":233.933,"cluster":"commutative-algebra"},{"id":"stacks:00PO","tag":"00PO","title":"Quasi-finite maps · Lemma 00PO","summary":"A composition of quasi-finite ring maps is quasi-finite.","statement_latex":"A composition of quasi-finite ring maps is quasi-finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PO","source_file":"algebra.tex","source_line":30341,"source_end_line":30344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30341-L30344","statement_sha256":"45430a77d114942adac59aec32da2e2f2fca8f552de39ae061ced35f2ed75680","origin":"The Stacks Project","memory_eligible":false,"source_rank":1812,"rank":1812,"depth":1,"x":1914.739,"y":289.787,"cluster":"commutative-algebra"},{"id":"stacks:00PP","tag":"00PP","title":"Quasi-finite maps · Lemma 00PP","summary":"Let R → S be a ring map of finite type. Let R → R' be any ring map. Set S' = R' ⊗_R S. • The set ( q' mid R' → S' quasi-finite at q') is the inverse image of the corresponding set of Spec(S) under the canonical map Spec(S') → Spec(S). • If Spec(R') → Spec(R) is surjective, then R → S is quasi-finite if and only if R' → S' is quasi-finite. • Any base change of a quasi-finite ring map is quasi-finite.","statement_latex":"Let $R \\to S$ be a ring map of finite type.\nLet $R \\to R'$ be any ring map. Set $S' = R' \\otimes_R S$.\n\\begin{enumerate}\n\\item The set\n$\\{\\mathfrak q' \\mid R' \\to S' \\text{ quasi-finite at }\\mathfrak q'\\}$\nis the inverse image of the corresponding set of $\\Spec(S)$\nunder the canonical map $\\Spec(S') \\to \\Spec(S)$.\n\\item If $\\Spec(R') \\to \\Spec(R)$ is surjective,\nthen $R \\to S$ is quasi-finite if and only if $R' \\to S'$ is quasi-finite.\n\\item Any base change of a quasi-finite ring map is quasi-finite.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PP","source_file":"algebra.tex","source_line":30361,"source_end_line":30374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30361-L30374","statement_sha256":"054bc9c352ae35684877fe2e68df0d03f2ade71f40a236774154235c9a2cd306","origin":"The Stacks Project","memory_eligible":false,"source_rank":1813,"rank":1813,"depth":25,"x":2058.887,"y":103.073,"cluster":"commutative-algebra"},{"id":"stacks:0C6H","tag":"0C6H","title":"Quasi-finite maps · Lemma 0C6H","summary":"Let A → B and B → C be ring homomorphisms such that A → C is of finite type. Let r be a prime of C lying over q ⊂ B and p ⊂ A. If A → C is quasi-finite at r, then B → C is quasi-finite at r.","statement_latex":"Let $A \\to B$ and $B \\to C$ be ring homomorphisms such that $A \\to C$\nis of finite type. Let $\\mathfrak r$ be a prime of $C$ lying over\n$\\mathfrak q \\subset B$ and $\\mathfrak p \\subset A$.\nIf $A \\to C$ is quasi-finite at $\\mathfrak r$, then\n$B \\to C$ is quasi-finite at $\\mathfrak r$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6H","source_file":"algebra.tex","source_line":30392,"source_end_line":30399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30392-L30399","statement_sha256":"3872ca1cd3e482ceb124e4b9afdbf7f8e3e33d34f5bf4e504e392f757f998668","origin":"The Stacks Project","memory_eligible":false,"source_rank":1814,"rank":1814,"depth":25,"x":2102.769,"y":322.668,"cluster":"commutative-algebra"},{"id":"stacks:02ML","tag":"02ML","title":"Quasi-finite maps · Lemma 02ML","summary":"Let R → S be a ring map of finite type. Let p ⊂ R be a minimal prime. Assume that there are at most finitely many primes of S lying over p. Then there exists a g ∈ R, g not ∈ p such that the ring map R_g → S_g is finite.","statement_latex":"Let $R \\to S$ be a ring map of finite type.\nLet $\\mathfrak p \\subset R$ be a minimal prime.\nAssume that there are at most finitely many primes of $S$\nlying over $\\mathfrak p$. Then there exists a\n$g \\in R$, $g \\not \\in \\mathfrak p$ such that the\nring map $R_g \\to S_g$ is finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Quasi-finite maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ML","source_file":"algebra.tex","source_line":30417,"source_end_line":30425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30417-L30425","statement_sha256":"fff7fa7fc6598440afa52e041da5f23e20b6b47c6f590792b105586f8840574f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1815,"rank":1815,"depth":11,"x":1893.703,"y":185.566,"cluster":"commutative-algebra"},{"id":"stacks:00PQ","tag":"00PQ","title":"Zariski's Main Theorem · Lemma 00PQ","summary":"Let φ : R → S be a ring map. Suppose t ∈ S satisfies the relation φ(a_0) + φ(a_1)t + … + φ(a_n) t^n = 0. Then φ(a_n)t is integral over R.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nSuppose $t \\in S$ satisfies the\nrelation $\\varphi(a_0) + \\varphi(a_1)t + \\ldots + \\varphi(a_n) t^n = 0$.\nThen $\\varphi(a_n)t$ is integral over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PQ","source_file":"algebra.tex","source_line":30480,"source_end_line":30486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30480-L30486","statement_sha256":"eb01c2f002dbc5d4315af7967bbbb52e61ca8853bdd9fecea52f46ba2ab25fd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1816,"rank":1816,"depth":0,"x":2158.265,"y":168.024,"cluster":"commutative-algebra"},{"id":"stacks:00PT","tag":"00PT","title":"Zariski's Main Theorem · Lemma 00PT","summary":"Let R be a ring. Let φ : R[x] → S be a ring map. Let t ∈ S. Assume that (a) t is integral over R[x], and (b) there exists a monic p ∈ R[x] such that t φ(p) ∈ Im(φ). Then there exists a q ∈ R[x] such that t - φ(q) is integral over R.","statement_latex":"Let $R$ be a ring. Let $\\varphi : R[x] \\to S$ be\na ring map. Let $t \\in S$.\nAssume that (a) $t$ is integral over $R[x]$,\nand (b) there exists a monic $p \\in R[x]$ such that\n$t \\varphi(p) \\in \\Im(\\varphi)$. Then there\nexists a $q \\in R[x]$ such that $t - \\varphi(q)$\nis integral over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PT","source_file":"algebra.tex","source_line":30502,"source_end_line":30511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30502-L30511","statement_sha256":"54b53152c2b43fcfa63034f652315612714bbc53cb8416544a175929e0896e8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1817,"rank":1817,"depth":6,"x":1977.188,"y":331.169,"cluster":"commutative-algebra"},{"id":"stacks:00PV","tag":"00PV","title":"Zariski's Main Theorem · Lemma 00PV","summary":"Let R be a ring. Let φ : R[x] → S be a ring map. Let t ∈ S. Assume t is integral over R[x]. Let p ∈ R[x], p = a_0 + a_1x + … + a_k x^k such that t φ(p) ∈ Im(φ). Then there exists a q ∈ R[x] and n ≥ 0 such that φ(a_k)^n t - φ(q) is integral over R.","statement_latex":"Let $R$ be a ring. Let $\\varphi : R[x] \\to S$ be\na ring map. Let $t \\in S$. Assume $t$ is integral\nover $R[x]$. Let $p \\in R[x]$, $p = a_0 + a_1x + \\ldots +\na_k x^k$ such that $t \\varphi(p) \\in \\Im(\\varphi)$.\nThen there exists a $q \\in R[x]$ and $n \\geq 0$\nsuch that $\\varphi(a_k)^n t - \\varphi(q)$ is integral\nover $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PV","source_file":"algebra.tex","source_line":30540,"source_end_line":30549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30540-L30549","statement_sha256":"89ec5bec26ba5b172959ce269e6486e29fcec00f9979e679b083c8572bdf24e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1818,"rank":1818,"depth":7,"x":1979.516,"y":107.997,"cluster":"commutative-algebra"},{"id":"stacks:00PX","tag":"00PX","title":"Zariski's Main Theorem · Lemma 00PX","summary":"In Situation [Tag 00PW]. Suppose u ∈ S, a_0, …, a_k ∈ R, u φ(a_0 + a_1x + … + a_k x^k) ∈ J. Then there exists an m ≥ 0 such that u φ(a_k)^m ∈ J.","statement_latex":"In Situation \\ref{situation-one-transcendental-element}.\nSuppose $u \\in S$, $a_0, \\ldots, a_k \\in R$,\n$u \\varphi(a_0 + a_1x + \\ldots + a_k x^k) \\in J$.\nThen there exists an $m \\geq 0$ such that\n$u \\varphi(a_k)^m \\in J$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PX","source_file":"algebra.tex","source_line":30582,"source_end_line":30589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30582-L30589","statement_sha256":"54f7bf4a48f01a9f33dc2f13d4a500064a79f2d74d0d509e4937b7a2371bcdae","origin":"The Stacks Project","memory_eligible":false,"source_rank":1819,"rank":1819,"depth":8,"x":2157.366,"y":273.973,"cluster":"commutative-algebra"},{"id":"stacks:00PY","tag":"00PY","title":"Zariski's Main Theorem · Lemma 00PY","summary":"In Situation [Tag 00PW]. Suppose u ∈ S, a_0, …, a_k ∈ R, u φ(a_0 + a_1x + … + a_k x^k) ∈ sqrtJ. Then u φ(a_i) ∈ sqrtJ for all i.","statement_latex":"In Situation \\ref{situation-one-transcendental-element}.\nSuppose $u \\in S$, $a_0, \\ldots, a_k \\in R$,\n$u \\varphi(a_0 + a_1x + \\ldots + a_k x^k) \\in \\sqrt{J}$.\nThen $u \\varphi(a_i) \\in \\sqrt{J}$ for all $i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PY","source_file":"algebra.tex","source_line":30606,"source_end_line":30612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30606-L30612","statement_sha256":"e909f711b76a3eb588c57525117abaca13b99ccfbf7823c304264756aa1fa346","origin":"The Stacks Project","memory_eligible":false,"source_rank":1820,"rank":1820,"depth":9,"x":1892.603,"y":252.489,"cluster":"commutative-algebra"},{"id":"stacks:00PZ","tag":"00PZ","title":"Zariski's Main Theorem · Definition 00PZ","summary":"Given an inclusion of rings R ⊂ S and an element x ∈ S we say that x is strongly transcendental over R if whenever u(a_0 + a_1 x + … + a_k x^k) = 0 with u ∈ S and a_i ∈ R, then we have ua_i = 0 for all i.","statement_latex":"Given an inclusion of rings $R \\subset S$ and\nan element $x \\in S$ we say that $x$ is\n{\\it strongly transcendental over $R$} if\nwhenever $u(a_0 + a_1 x + \\ldots + a_k x^k) = 0$\nwith $u \\in S$ and $a_i \\in R$, then\nwe have $ua_i = 0$ for all $i$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00PZ","source_file":"algebra.tex","source_line":30628,"source_end_line":30636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30628-L30636","statement_sha256":"64cb630d701dc37a1bb493054d167ef6dfa97eabe34c054e9930ce7bec68b1c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1821,"rank":1821,"depth":0,"x":2105.229,"y":118.025,"cluster":"commutative-algebra"},{"id":"stacks:00Q0","tag":"00Q0","title":"Zariski's Main Theorem · Lemma 00Q0","summary":"Suppose R ⊂ S is an inclusion of reduced rings and suppose that x ∈ S is strongly transcendental over R. Let q ⊂ S be a minimal prime and let p = R ∩ q. Then the image of x in S/ q is strongly transcendental over the subring R/ p.","statement_latex":"Suppose $R \\subset S$ is an inclusion of reduced rings\nand suppose that $x \\in S$ is strongly transcendental over $R$.\nLet $\\mathfrak q \\subset S$ be a minimal prime\nand let $\\mathfrak p = R \\cap \\mathfrak q$.\nThen the image of $x$ in $S/\\mathfrak q$ is strongly\ntranscendental over the subring $R/\\mathfrak p$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Q0","source_file":"algebra.tex","source_line":30643,"source_end_line":30651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30643-L30651","statement_sha256":"8f1b20ac4615901be55125c271bcc970d4b6ac72ce9094d92feff12aad4a5b4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1822,"rank":1822,"depth":2,"x":2056.548,"y":337.947,"cluster":"commutative-algebra"},{"id":"stacks:00Q1","tag":"00Q1","title":"Zariski's Main Theorem · Lemma 00Q1","summary":"Suppose R⊂ S is an inclusion of domains and let x ∈ S. Assume x is (strongly) transcendental over R and that S is finite over R[x]. Then R → S is not quasi-finite at any prime of S.","statement_latex":"Suppose $R\\subset S$ is an inclusion of domains and\nlet $x \\in S$. Assume $x$ is (strongly) transcendental over $R$\nand that $S$ is finite over $R[x]$. Then $R \\to S$ is not\nquasi-finite at any prime of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Q1","source_file":"algebra.tex","source_line":30665,"source_end_line":30671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30665-L30671","statement_sha256":"0faf5b06bdbfce1942974bce9d9bf56419f63cbe6018555f1a3190c4d829f063","origin":"The Stacks Project","memory_eligible":false,"source_rank":1823,"rank":1823,"depth":25,"x":1915.512,"y":148.052,"cluster":"commutative-algebra"},{"id":"stacks:00Q2","tag":"00Q2","title":"Zariski's Main Theorem · Lemma 00Q2","summary":"Suppose R ⊂ S is an inclusion of reduced rings. Assume x ∈ S be strongly transcendental over R, and S finite over R[x]. Then R → S is not quasi-finite at any prime of S.","statement_latex":"Suppose $R \\subset S$ is an inclusion of reduced rings.\nAssume $x \\in S$ be strongly transcendental over $R$,\nand $S$ finite over $R[x]$. Then $R \\to S$ is not\nquasi-finite at any prime of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Q2","source_file":"algebra.tex","source_line":30713,"source_end_line":30719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30713-L30719","statement_sha256":"40862f29a1dc37b576dbb188d39fb9cc4b821f86804caa7000aee9fa0bfecff2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1824,"rank":1824,"depth":26,"x":2172.359,"y":208.083,"cluster":"commutative-algebra"},{"id":"stacks:00Q8","tag":"00Q8","title":"Zariski's Main Theorem · Lemma 00Q8","summary":"Let R be a ring. Let S = R[x]/I. Let q ⊂ S be a prime. Assume R → S is quasi-finite at q. Let S' ⊂ S be the integral closure of R in S. Then there exists an element g ∈ S', g not∈ q such that S'_g ≅ S_g.","statement_latex":"Let $R$ be a ring. Let $S = R[x]/I$.\nLet $\\mathfrak q \\subset S$ be a prime.\nAssume $R \\to S$ is quasi-finite at $\\mathfrak q$.\nLet $S' \\subset S$ be the integral closure of $R$ in $S$.\nThen there exists an element\n$g \\in S'$, $g \\not\\in \\mathfrak q$ such that\n$S'_g \\cong S_g$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Q8","source_file":"algebra.tex","source_line":30734,"source_end_line":30743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30734-L30743","statement_sha256":"b632e1a8883241e2338d7efb2afd2738037f97ecb99bd533a679b5ffcb0573af","origin":"The Stacks Project","memory_eligible":false,"source_rank":1825,"rank":1825,"depth":1,"x":1934.558,"y":309.614,"cluster":"commutative-algebra"},{"id":"stacks:00Q9","tag":"00Q9","title":"Zariski's Main Theorem · Theorem 00Q9","summary":"Let R be a ring. Let R → S be a finite type R-algebra. Let S' ⊂ S be the integral closure of R in S. Let q ⊂ S be a prime of S. If R → S is quasi-finite at q then there exists a g ∈ S', g not ∈ q such that S'_g ≅ S_g.","statement_latex":"Let $R$ be a ring. Let $R \\to S$ be a finite type $R$-algebra.\nLet $S' \\subset S$ be the integral closure of $R$ in $S$.\nLet $\\mathfrak q \\subset S$ be a prime of $S$.\nIf $R \\to S$ is quasi-finite at $\\mathfrak q$ then\nthere exists a $g \\in S'$, $g \\not \\in \\mathfrak q$\nsuch that $S'_g \\cong S_g$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00Q9","source_file":"algebra.tex","source_line":30778,"source_end_line":30786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30778-L30786","statement_sha256":"d41d2989f86c79c594a4423d47efd0957a8eb413f6bb5e9580f59a0f8dc6b5ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":1826,"rank":1826,"depth":27,"x":2028.312,"y":99.698,"cluster":"commutative-algebra"},{"id":"stacks:00QA","tag":"00QA","title":"Zariski's Main Theorem · Lemma 00QA","summary":"Let R → S be a finite type ring map. The set of points q of Spec(S) at which S/R is quasi-finite is open in Spec(S).","statement_latex":"Let $R \\to S$ be a finite type ring map.\nThe set of points $\\mathfrak q$ of $\\Spec(S)$ at which\n$S/R$ is quasi-finite is open in $\\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QA","source_file":"algebra.tex","source_line":30879,"source_end_line":30884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30879-L30884","statement_sha256":"1f328217387d28d393661985dd21ffbdd12ff36dad55eaea7c25c2164463ff1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1827,"rank":1827,"depth":28,"x":2128.042,"y":307.799,"cluster":"commutative-algebra"},{"id":"stacks:00QB","tag":"00QB","title":"Zariski's Main Theorem · Lemma 00QB","summary":"Let R → S be a finite type ring map. Suppose that S is quasi-finite over R. Let S' ⊂ S be the integral closure of R in S. Then • Spec(S) → Spec(S') is a homeomorphism onto an open subset, • if g ∈ S' and D(g) is contained in the image of the map, then S'_g ≅ S_g, and • there exists a finite R-algebra S\" ⊂ S' such that (1) and (2) hold for the ring map S\" → S.","statement_latex":"Let $R \\to S$ be a finite type ring map.\nSuppose that $S$ is quasi-finite over $R$.\nLet $S' \\subset S$ be the integral closure of $R$ in $S$. Then\n\\begin{enumerate}\n\\item $\\Spec(S) \\to \\Spec(S')$ is a homeomorphism\nonto an open subset,\n\\item if $g \\in S'$ and $D(g)$ is contained in the image\nof the map, then $S'_g \\cong S_g$, and\n\\item there exists a finite $R$-algebra $S'' \\subset S'$\nsuch that (1) and (2) hold for the ring map\n$S'' \\to S$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QB","source_file":"algebra.tex","source_line":30905,"source_end_line":30919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30905-L30919","statement_sha256":"f48b2e9124eec0f43210cba0f1e9a1105f4ba37925e2f579eac08558290341d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1828,"rank":1828,"depth":28,"x":1887.022,"y":210.885,"cluster":"commutative-algebra"},{"id":"stacks:02MM","tag":"02MM","title":"Applications of Zariski's Main Theorem · Lemma 02MM","summary":"Let A ⊂ B be an extension of domains. Assume • A is a local Noetherian ring of dimension 1, • A → B is of finite type, and • the induced extension L/K of fraction fields is finite. Then B is semi-local. Let x ∈ m_A, x not = 0. Let m_i, i = 1, …, n be the maximal ideals of B. Then [L : K]ord_A(x) ≥ ∑_i [kappa( m_i) : kappa( m_A)] ord_B_ m_i(x) where ord is defined as in Definition [Tag 02MD]. We have equality if and only if A → B is finite.","statement_latex":"Let $A \\subset B$ be an extension of domains. Assume\n\\begin{enumerate}\n\\item $A$ is a local Noetherian ring of dimension $1$,\n\\item $A \\to B$ is of finite type, and\n\\item the induced extension $L/K$ of fraction fields is finite.\n\\end{enumerate}\nThen $B$ is semi-local.\nLet $x \\in \\mathfrak m_A$, $x \\not = 0$.\nLet $\\mathfrak m_i$, $i = 1, \\ldots, n$\nbe the maximal ideals of $B$.\nThen\n$$\n[L : K]\\text{ord}_A(x)\n\\geq\n\\sum\\nolimits_i\n[\\kappa(\\mathfrak m_i) : \\kappa(\\mathfrak m_A)]\n\\text{ord}_{B_{\\mathfrak m_i}}(x)\n$$\nwhere $\\text{ord}$ is defined as in Definition \\ref{definition-ord}.\nWe have equality if and only if $A \\to B$ is finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Applications of Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MM","source_file":"algebra.tex","source_line":30963,"source_end_line":30985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L30963-L30985","statement_sha256":"3600f0d7c1297189e01f62d95d60cf1fd7b8eaeadb4e4af481202ee95e5bf95d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1829,"rank":1829,"depth":29,"x":2142.822,"y":145.551,"cluster":"commutative-algebra"},{"id":"stacks:052V","tag":"052V","title":"Applications of Zariski's Main Theorem · Lemma 052V","summary":"Let (R, m_R) → (S, m_S) be a local homomorphism of local rings. Assume • R → S is essentially of finite type, • kappa( m_R) ⊂ kappa( m_S) is finite, and • dim(S/ m_RS) = 0. Then S is the localization of a finite R-algebra.","statement_latex":"Let $(R, \\mathfrak m_R) \\to (S, \\mathfrak m_S)$ be a local homomorphism\nof local rings. Assume\n\\begin{enumerate}\n\\item $R \\to S$ is essentially of finite type,\n\\item $\\kappa(\\mathfrak m_R) \\subset \\kappa(\\mathfrak m_S)$ is finite, and\n\\item $\\dim(S/\\mathfrak m_RS) = 0$.\n\\end{enumerate}\nThen $S$ is the localization of a finite $R$-algebra.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Applications of Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052V","source_file":"algebra.tex","source_line":31022,"source_end_line":31032,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31022-L31032","statement_sha256":"f7fa895c7b35745f796f0573f1cb5302f0340d39e572f6eb2ebd9698797766bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1830,"rank":1830,"depth":29,"x":2006.664,"y":338.98,"cluster":"commutative-algebra"},{"id":"stacks:07NC","tag":"07NC","title":"Applications of Zariski's Main Theorem · Lemma 07NC","summary":"Let R → S be a ring map, q a prime of S lying over p in R. If • R is Noetherian, • R → S is of finite type, and • R → S is quasi-finite at q, then R_ p^wedge ⊗_R S = S_ q^wedge × B for some R_ p^wedge-algebra B.","statement_latex":"Let $R \\to S$ be a ring map, $\\mathfrak q$ a prime of $S$\nlying over $\\mathfrak p$ in $R$. If\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $R \\to S$ is of finite type, and\n\\item $R \\to S$ is quasi-finite at $\\mathfrak q$,\n\\end{enumerate}\nthen $R_\\mathfrak p^\\wedge \\otimes_R S = S_\\mathfrak q^\\wedge \\times B$\nfor some $R_\\mathfrak p^\\wedge$-algebra $B$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Applications of Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NC","source_file":"algebra.tex","source_line":31051,"source_end_line":31062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31051-L31062","statement_sha256":"192d33408dfbbc8786c8218b38c8d313d00806715583f4fd41f0cfa63e9128f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1831,"rank":1831,"depth":29,"x":1951.485,"y":118.97,"cluster":"commutative-algebra"},{"id":"stacks:00QD","tag":"00QD","title":"Dimension of fibres · Definition 00QD","summary":"Suppose that R → S is of finite type, and let q ⊂ S be a prime lying over a prime p of R. We define the relative dimension of S/R at q, denoted dim_ q(S/R), to be the dimension of Spec(S ⊗_R kappa( p)) at the point corresponding to q. We let dim(S/R) be the supremum of dim_ q(S/R) over all q. This is called the relative dimension of S/R.","statement_latex":"Suppose that $R \\to S$ is of finite type, and let\n$\\mathfrak q \\subset S$ be a prime lying over a prime\n$\\mathfrak p$ of $R$.\nWe define the {\\it relative dimension\nof $S/R$ at $\\mathfrak q$}, denoted\n$\\dim_{\\mathfrak q}(S/R)$, to be the dimension\nof $\\Spec(S \\otimes_R \\kappa(\\mathfrak p))$\nat the point corresponding to $\\mathfrak q$. We let\n$\\dim(S/R)$ be the supremum of $\\dim_{\\mathfrak q}(S/R)$\nover all $\\mathfrak q$. This is called the\n{\\it relative dimension of} $S/R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of fibres","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QD","source_file":"algebra.tex","source_line":31149,"source_end_line":31162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31149-L31162","statement_sha256":"dc18c6a17b6b41d647d88c0490e146a1246fdd966e96f99f0a934130adfde431","origin":"The Stacks Project","memory_eligible":false,"source_rank":1832,"rank":1832,"depth":0,"x":2169.217,"y":249.963,"cluster":"commutative-algebra"},{"id":"stacks:00QE","tag":"00QE","title":"Dimension of fibres · Lemma 00QE","summary":"Let R → S be a finite type ring map. Let q ⊂ S be a prime. Suppose that dim_ q(S/R) = n. There exists a g ∈ S, g not∈ q such that S_g is quasi-finite over a polynomial algebra R[t_1, …, t_n].","statement_latex":"Let $R \\to S$ be a finite type ring map.\nLet $\\mathfrak q \\subset S$ be a prime.\nSuppose that $\\dim_{\\mathfrak q}(S/R) = n$.\nThere exists a $g \\in S$, $g \\not\\in \\mathfrak q$\nsuch that $S_g$ is quasi-finite over a\npolynomial algebra $R[t_1, \\ldots, t_n]$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QE","source_file":"algebra.tex","source_line":31169,"source_end_line":31177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31169-L31177","statement_sha256":"5bb40c54586af8dc82a3be0015e091032592b752e8fc4676985cbebd8dbfca03","origin":"The Stacks Project","memory_eligible":false,"source_rank":1833,"rank":1833,"depth":29,"x":1903.179,"y":276.932,"cluster":"commutative-algebra"},{"id":"stacks:0520","tag":"0520","title":"Dimension of fibres · Lemma 0520","summary":"Let R → S be a ring map. Let q ⊂ S be a prime lying over the prime p of R. Assume • R → S is of finite type, • dim_ q(S/R) = n, and • trdeg_kappa( p)kappa( q) = r. Then there exist f ∈ R, f not ∈ p, g ∈ S, g not∈ q and a quasi-finite ring map φ : R_f[x_1, …, x_n] → S_g such that φ^-1( qS_g) = ( p, x_r + 1, …, x_n)R_f[x_1, …, x_n]","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$\nbe a prime lying over the prime $\\mathfrak p$ of $R$.\nAssume\n\\begin{enumerate}\n\\item $R \\to S$ is of finite type,\n\\item $\\dim_{\\mathfrak q}(S/R) = n$, and\n\\item $\\text{trdeg}_{\\kappa(\\mathfrak p)}\\kappa(\\mathfrak q) = r$.\n\\end{enumerate}\nThen there exist $f \\in R$, $f \\not \\in \\mathfrak p$,\n$g \\in S$, $g \\not\\in \\mathfrak q$ and a quasi-finite ring map\n$$\n\\varphi : R_f[x_1, \\ldots, x_n] \\longrightarrow S_g\n$$\nsuch that $\\varphi^{-1}(\\mathfrak qS_g) =\n(\\mathfrak p, x_{r + 1}, \\ldots, x_n)R_f[x_1, \\ldots, x_n]$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0520","source_file":"algebra.tex","source_line":31210,"source_end_line":31227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31210-L31227","statement_sha256":"4776085103195d123c4d05569438f524482337c54dff73acc15c7ce38fa76deb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1834,"rank":1834,"depth":30,"x":2077.759,"y":105.996,"cluster":"commutative-algebra"},{"id":"stacks:00QF","tag":"00QF","title":"Dimension of fibres · Lemma 00QF","summary":"Let R → S be a finite type ring map. Let q ⊂ S be a prime lying over p ⊂ R. If R → S is quasi-finite at q, then dim(S_ q) ≤ dim(R_ p).","statement_latex":"Let $R \\to S$ be a finite type ring map.\nLet $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p \\subset R$.\nIf $R \\to S$ is quasi-finite at $\\mathfrak q$, then\n$\\dim(S_{\\mathfrak q}) \\leq \\dim(R_{\\mathfrak p})$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QF","source_file":"algebra.tex","source_line":31266,"source_end_line":31272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31266-L31272","statement_sha256":"f3723bb82186182e4ff018839d826245ad5df4584316e9fa630776e948d67c07","origin":"The Stacks Project","memory_eligible":false,"source_rank":1835,"rank":1835,"depth":28,"x":2086.493,"y":331.224,"cluster":"commutative-algebra"},{"id":"stacks:00QG","tag":"00QG","title":"Dimension of fibres · Lemma 00QG","summary":"A quasi-finite cover of affine n-space has dimension at most n. Let k be a field. Let S be a finite type k-algebra. Suppose there is a quasi-finite k-algebra map k[t_1, …, t_n] ⊂ S. Then dim(S) ≤ n.","statement_latex":"\\begin{slogan}\nA quasi-finite cover of affine n-space has dimension at most n.\n\\end{slogan}\nLet $k$ be a field. Let $S$ be a finite type $k$-algebra.\nSuppose there is a quasi-finite $k$-algebra map\n$k[t_1, \\ldots, t_n] \\subset S$. Then $\\dim(S) \\leq n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QG","source_file":"algebra.tex","source_line":31290,"source_end_line":31298,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31290-L31298","statement_sha256":"a3bd03b9438e2619d6b296c199d238994e39e29513d368d28c5a5fa2b3d7ef8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1836,"rank":1836,"depth":29,"x":1898.829,"y":170.013,"cluster":"commutative-algebra"},{"id":"stacks:00QH","tag":"00QH","title":"Dimension of fibres · Lemma 00QH","summary":"Let R → S be a finite type ring map. Let q ⊂ S be a prime. Suppose that dim_ q(S/R) = n. There exists an open neighbourhood V of q in Spec(S) such that dim_ q'(S/R) ≤ n for all q' ∈ V.","statement_latex":"Let $R \\to S$ be a finite type ring map.\nLet $\\mathfrak q \\subset S$ be a prime.\nSuppose that $\\dim_{\\mathfrak q}(S/R) = n$.\nThere exists an open neighbourhood $V$ of $\\mathfrak q$\nin $\\Spec(S)$ such that\n$\\dim_{\\mathfrak q'}(S/R) \\leq n$ for all $\\mathfrak q' \\in V$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QH","source_file":"algebra.tex","source_line":31307,"source_end_line":31315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31307-L31315","statement_sha256":"db2993311d4a290854bba9ad41df948298e67f2272ac497ff8f39f53a7641f25","origin":"The Stacks Project","memory_eligible":false,"source_rank":1837,"rank":1837,"depth":30,"x":2166.996,"y":182.409,"cluster":"commutative-algebra"},{"id":"stacks:00QI","tag":"00QI","title":"Dimension of fibres · Lemma 00QI","summary":"Let R → S be a finite type ring map. Let R → R' be any ring map. Set S' = R' ⊗_R S and denote f : Spec(S') → Spec(S) the associated map on spectra. Let n ≥ 0. The inverse image f^-1(( q ∈ Spec(S) mid dim_ q(S/R) ≤ n)) is equal to ( q' ∈ Spec(S') mid dim_ q'(S'/R') ≤ n).","statement_latex":"Let $R \\to S$ be a finite type ring map.\nLet $R \\to R'$ be any ring map.\nSet $S' = R' \\otimes_R S$ and denote $f : \\Spec(S') \\to \\Spec(S)$\nthe associated map on spectra.\nLet $n \\geq 0$.\nThe inverse image\n$f^{-1}(\\{\\mathfrak q \\in \\Spec(S) \\mid\n\\dim_{\\mathfrak q}(S/R) \\leq n\\})$\nis equal to\n$\\{\\mathfrak q' \\in \\Spec(S') \\mid\n\\dim_{\\mathfrak q'}(S'/R') \\leq n\\}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QI","source_file":"algebra.tex","source_line":31333,"source_end_line":31346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31333-L31346","statement_sha256":"7072aa77617ec48b9afcef57d8070576e4c8edaa9a03361d6b79fe1b030b036f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1838,"rank":1838,"depth":25,"x":1959.173,"y":325.51,"cluster":"commutative-algebra"},{"id":"stacks:00QJ","tag":"00QJ","title":"Dimension of fibres · Lemma 00QJ","summary":"Let R → S be a ring homomorphism of finite presentation. Let n ≥ 0. The set V_n = ( q ∈ Spec(S) mid dim_ q(S/R) ≤ n) is a quasi-compact open subset of Spec(S).","statement_latex":"Let $R \\to S$ be a ring homomorphism of finite presentation.\nLet $n \\geq 0$. The set\n$$\nV_n = \\{\\mathfrak q \\in \\Spec(S) \\mid \\dim_{\\mathfrak q}(S/R) \\leq n\\}\n$$\nis a quasi-compact open subset of $\\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QJ","source_file":"algebra.tex","source_line":31356,"source_end_line":31364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31356-L31364","statement_sha256":"9b586cefe4bc80ec0ca9e68f7c923140442463c178f6ff1e59bbc79ede601524","origin":"The Stacks Project","memory_eligible":false,"source_rank":1839,"rank":1839,"depth":31,"x":1997.361,"y":101.946,"cluster":"commutative-algebra"},{"id":"stacks:00QK","tag":"00QK","title":"Dimension of fibres · Lemma 00QK","summary":"Let R be a valuation ring with residue field k and field of fractions K. Let S be a domain containing R such that S is of finite type over R. If S ⊗_R k is not the zero ring then dim(S ⊗_R k) = dim(S ⊗_R K) In fact, Spec(S ⊗_R k) is equidimensional.","statement_latex":"Let $R$ be a valuation ring with residue field $k$ and field\nof fractions $K$. Let $S$ be a domain containing $R$ such that\n$S$ is of finite type over $R$. If $S \\otimes_R k$ is not the\nzero ring then\n$$\n\\dim(S \\otimes_R k) = \\dim(S \\otimes_R K)\n$$\nIn fact, $\\Spec(S \\otimes_R k)$ is equidimensional.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QK","source_file":"algebra.tex","source_line":31379,"source_end_line":31389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31379-L31389","statement_sha256":"a37c0918320b6d91236614b2b80583592a56809f2bd2e31f4504926682ed04bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":1840,"rank":1840,"depth":31,"x":2149.068,"y":288.568,"cluster":"commutative-algebra"},{"id":"stacks:00QP","tag":"00QP","title":"Algebras and modules of finite presentation · Lemma 00QP","summary":"Let R → S be a ring map. Let R → R' be a faithfully flat ring map. Set S' = R'⊗_R S. Then R → S is of finite type if and only if R' → S' is of finite type.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $R \\to R'$ be a faithfully flat ring map.\nSet $S' = R'\\otimes_R S$.\nThen $R \\to S$ is of finite type if and only if $R' \\to S'$\nis of finite type.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QP","source_file":"algebra.tex","source_line":31442,"source_end_line":31449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31442-L31449","statement_sha256":"cc5d0de9b834a6aa70ea984946f303f91115b0018fd58ea58980412f56a545cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1841,"rank":1841,"depth":0,"x":1886.984,"y":237.01,"cluster":"commutative-algebra"},{"id":"stacks:00QQ","tag":"00QQ","title":"Algebras and modules of finite presentation · Lemma 00QQ","summary":"Let R → S be a ring map. Let R → R' be a faithfully flat ring map. Set S' = R'⊗_R S. Then R → S is of finite presentation if and only if R' → S' is of finite presentation.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $R \\to R'$ be a faithfully flat ring map.\nSet $S' = R'\\otimes_R S$.\nThen $R \\to S$ is of finite presentation if and only if $R' \\to S'$\nis of finite presentation.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QQ","source_file":"algebra.tex","source_line":31463,"source_end_line":31470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31463-L31470","statement_sha256":"dfc52aac1022d68e238eae37d733775b4d92b9fc56d476344505f6bf5c6e0118","origin":"The Stacks Project","memory_eligible":false,"source_rank":1842,"rank":1842,"depth":1,"x":2121.829,"y":126.256,"cluster":"commutative-algebra"},{"id":"stacks:05N5","tag":"05N5","title":"Algebras and modules of finite presentation · Lemma 05N5","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let S ⊂ R be a multiplicative subset. Set R' = S^-1(R/I) = S^-1R/S^-1I. • For any finite R'-module M' there exists a finite R-module M such that S^-1(M/IM) ≅ M'. • For any finitely presented R'-module M' there exists a finitely presented R-module M such that S^-1(M/IM) ≅ M'.","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\nLet $S \\subset R$ be a multiplicative subset.\nSet $R' = S^{-1}(R/I) = S^{-1}R/S^{-1}I$.\n\\begin{enumerate}\n\\item For any finite $R'$-module $M'$ there exists a\nfinite $R$-module $M$ such that $S^{-1}(M/IM) \\cong M'$.\n\\item For any finitely presented $R'$-module $M'$ there exists a\nfinitely presented $R$-module $M$ such that $S^{-1}(M/IM) \\cong M'$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05N5","source_file":"algebra.tex","source_line":31489,"source_end_line":31501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31489-L31501","statement_sha256":"c5f504799633c01322148abe1c59a1060e9d720e5a60be2595b02a5e7ffd1f3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1843,"rank":1843,"depth":0,"x":2037.677,"y":341.296,"cluster":"commutative-algebra"},{"id":"stacks:05N6","tag":"05N6","title":"Algebras and modules of finite presentation · Lemma 05N6","summary":"Let R be a ring. Let S ⊂ R be a multiplicative subset. Let M be an R-module. • If S^-1M is a finite S^-1R-module then there exists a finite R-module M' and a map M' → M which induces an isomorphism S^-1M' → S^-1M. • If S^-1M is a finitely presented S^-1R-module then there exists an R-module M' of finite presentation and a map M' → M which induces an isomorphism S^-1M' → S^-1M.","statement_latex":"Let $R$ be a ring.\nLet $S \\subset R$ be a multiplicative subset.\nLet $M$ be an $R$-module.\n\\begin{enumerate}\n\\item If $S^{-1}M$ is a finite $S^{-1}R$-module then there\nexists a finite $R$-module $M'$ and a map $M' \\to M$ which induces an\nisomorphism $S^{-1}M' \\to S^{-1}M$.\n\\item If $S^{-1}M$ is a finitely presented $S^{-1}R$-module\nthen there exists an $R$-module $M'$ of finite presentation\nand a map $M' \\to M$ which induces an isomorphism\n$S^{-1}M' \\to S^{-1}M$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05N6","source_file":"algebra.tex","source_line":31526,"source_end_line":31540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31526-L31540","statement_sha256":"a1681613fea14b828126b36df78ef29b9804cc2a241610cb4a264db5ecaf0a41","origin":"The Stacks Project","memory_eligible":false,"source_rank":1844,"rank":1844,"depth":2,"x":1926.74,"y":134.869,"cluster":"commutative-algebra"},{"id":"stacks:05GJ","tag":"05GJ","title":"Algebras and modules of finite presentation · Lemma 05GJ","summary":"Let R be a ring. Let p ⊂ R be a prime ideal. Let M be an R-module. • If M_ p is a finite R_ p-module then there exists a finite R-module M' and a map M' → M which induces an isomorphism M'_ p → M_ p. • If M_ p is a finitely presented R_ p-module then there exists an R-module M' of finite presentation and a map M' → M which induces an isomorphism M'_ p → M_ p.","statement_latex":"Let $R$ be a ring.\nLet $\\mathfrak p \\subset R$ be a prime ideal.\nLet $M$ be an $R$-module.\n\\begin{enumerate}\n\\item If $M_{\\mathfrak p}$ is a finite $R_{\\mathfrak p}$-module then there\nexists a finite $R$-module $M'$ and a map $M' \\to M$ which induces an\nisomorphism $M'_{\\mathfrak p} \\to M_{\\mathfrak p}$.\n\\item If $M_{\\mathfrak p}$ is a finitely presented $R_{\\mathfrak p}$-module\nthen there exists an $R$-module $M'$ of finite presentation\nand a map $M' \\to M$ which induces an isomorphism\n$M'_{\\mathfrak p} \\to M_{\\mathfrak p}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GJ","source_file":"algebra.tex","source_line":31559,"source_end_line":31573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31559-L31573","statement_sha256":"d9b2ff3cf58a53dc3eebfc375c70c62924a3938ae4eb62ff807b8b260b13f14b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1845,"rank":1845,"depth":3,"x":2174.68,"y":224.185,"cluster":"commutative-algebra"},{"id":"stacks:00QR","tag":"00QR","title":"Algebras and modules of finite presentation · Lemma 00QR","summary":"Let φ : R → S be a ring map. Let q ⊂ S be a prime lying over p ⊂ R. Assume • S is of finite presentation over R, • φ induces an isomorphism R_ p ≅ S_ q. Then there exist f ∈ R, f not ∈ p and an R_f-algebra C such that S_f ≅ R_f × C as R_f-algebras.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$\nbe a prime lying over $\\mathfrak p \\subset R$. Assume\n\\begin{enumerate}\n\\item $S$ is of finite presentation over $R$,\n\\item $\\varphi$ induces an isomorphism $R_\\mathfrak p \\cong S_\\mathfrak q$.\n\\end{enumerate}\nThen there exist $f \\in R$, $f \\not \\in \\mathfrak p$ and an\n$R_f$-algebra $C$ such that $S_f \\cong R_f \\times C$ as $R_f$-algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QR","source_file":"algebra.tex","source_line":31580,"source_end_line":31590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31580-L31590","statement_sha256":"166de7ab2feca3ddd4715ae55bed2320fb420a2a6ecb99e7a94cf3a3c5ef7801","origin":"The Stacks Project","memory_eligible":false,"source_rank":1846,"rank":1846,"depth":6,"x":1919.891,"y":299.051,"cluster":"commutative-algebra"},{"id":"stacks:00QS","tag":"00QS","title":"Algebras and modules of finite presentation · Lemma 00QS","summary":"Let R be a ring. Let S, S' be of finite presentation over R. Let q ⊂ S and q' ⊂ S' be primes. If S_ q ≅ S'_ q' as R-algebras, then there exist g ∈ S, g not ∈ q and g' ∈ S', g' not ∈ q' such that S_g ≅ S'_g' as R-algebras.","statement_latex":"Let $R$ be a ring.\nLet $S$, $S'$ be of finite presentation over $R$.\nLet $\\mathfrak q \\subset S$ and $\\mathfrak q' \\subset S'$\nbe primes. If $S_{\\mathfrak q} \\cong S'_{\\mathfrak q'}$ as\n$R$-algebras, then there exist $g \\in S$, $g \\not \\in \\mathfrak q$\nand $g' \\in S'$, $g' \\not \\in \\mathfrak q'$ such that\n$S_g \\cong S'_{g'}$ as $R$-algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QS","source_file":"algebra.tex","source_line":31623,"source_end_line":31632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31623-L31632","statement_sha256":"c0fcc01be7ac6072ad3785a33e980a9a3737d32e85bcfba563d875f971db3d91","origin":"The Stacks Project","memory_eligible":false,"source_rank":1847,"rank":1847,"depth":7,"x":2047.631,"y":99.166,"cluster":"commutative-algebra"},{"id":"stacks:0G8U","tag":"0G8U","title":"Algebras and modules of finite presentation · Lemma 0G8U","summary":"Let R be a ring. Let I ⊂ R be a nilpotent ideal. Let S be an R-algebra such that R/I → S/IS is of finite type. Then R → S is of finite type.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be a nilpotent ideal.\nLet $S$ be an $R$-algebra such that $R/I \\to S/IS$\nis of finite type. Then $R \\to S$ is of finite type.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8U","source_file":"algebra.tex","source_line":31662,"source_end_line":31667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31662-L31667","statement_sha256":"f573de4521dcf077854e31e41b0c8eafb78a63eab7b6316ee31ee0ee803f9104","origin":"The Stacks Project","memory_eligible":false,"source_rank":1848,"rank":1848,"depth":3,"x":2114.215,"y":319.158,"cluster":"commutative-algebra"},{"id":"stacks:07RD","tag":"07RD","title":"Algebras and modules of finite presentation · Lemma 07RD","summary":"Let R be a ring. Let I ⊂ R be a locally nilpotent ideal. Let S → S' be an R-algebra map such that S → S'/IS' is surjective and such that S' is of finite type over R. Then S → S' is surjective.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be a locally nilpotent ideal.\nLet $S \\to S'$ be an $R$-algebra map such that $S \\to S'/IS'$ is surjective\nand such that $S'$ is of finite type over $R$. Then $S \\to S'$ is surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RD","source_file":"algebra.tex","source_line":31676,"source_end_line":31681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31676-L31681","statement_sha256":"b6cd549dfe24a7881b20386dd9fddfd052dfc1272c5f368e586d28e0e7da3e47","origin":"The Stacks Project","memory_eligible":false,"source_rank":1849,"rank":1849,"depth":1,"x":1888.085,"y":194.655,"cluster":"commutative-algebra"},{"id":"stacks:087P","tag":"087P","title":"Algebras and modules of finite presentation · Lemma 087P","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let S → S' be an R-algebra map. Let IS ⊂ q ⊂ S be a prime ideal. Assume that • S → S' is surjective, • S_ q/IS_ q → S'_ q/IS'_ q is an isomorphism, • S is of finite type over R, • S' of finite presentation over R, and • S'_ q is flat over R. Then S_g → S'_g is an isomorphism for some g ∈ S, g not ∈ q.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $S \\to S'$\nbe an $R$-algebra map. Let $IS \\subset \\mathfrak q \\subset S$\nbe a prime\nideal. Assume that\n\\begin{enumerate}\n\\item $S \\to S'$ is surjective,\n\\item $S_\\mathfrak q/IS_\\mathfrak q \\to S'_\\mathfrak q/IS'_\\mathfrak q$\nis an isomorphism,\n\\item $S$ is of finite type over $R$,\n\\item $S'$ of finite presentation over $R$, and\n\\item $S'_\\mathfrak q$ is flat over $R$.\n\\end{enumerate}\nThen $S_g \\to S'_g$ is an isomorphism for some\n$g \\in S$, $g \\not \\in \\mathfrak q$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087P","source_file":"algebra.tex","source_line":31697,"source_end_line":31713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31697-L31713","statement_sha256":"29a003efd147b113711182b7eea7de2ac25495e7b6f155494c95d13f5d3d2881","origin":"The Stacks Project","memory_eligible":false,"source_rank":1850,"rank":1850,"depth":3,"x":2155.095,"y":158.13,"cluster":"commutative-algebra"},{"id":"stacks:07RE","tag":"07RE","title":"Algebras and modules of finite presentation · Lemma 07RE","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let S → S' be an R-algebra map. Assume that • I is locally nilpotent, • S/IS → S'/IS' is an isomorphism, • S is of finite type over R, • S' of finite presentation over R, and • S' is flat over R. Then S → S' is an isomorphism.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $S \\to S'$\nbe an $R$-algebra map. Assume that\n\\begin{enumerate}\n\\item $I$ is locally nilpotent,\n\\item $S/IS \\to S'/IS'$ is an isomorphism,\n\\item $S$ is of finite type over $R$,\n\\item $S'$ of finite presentation over $R$, and\n\\item $S'$ is flat over $R$.\n\\end{enumerate}\nThen $S \\to S'$ is an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Algebras and modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RE","source_file":"algebra.tex","source_line":31728,"source_end_line":31740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31728-L31740","statement_sha256":"04cf365618a348837eb04b83851825d14db56e920b8a1daa573fa7596c8152cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":1851,"rank":1851,"depth":4,"x":1987.488,"y":336.665,"cluster":"commutative-algebra"},{"id":"stacks:0BUF","tag":"0BUF","title":"Colimits and maps of finite presentation · Lemma 0BUF","summary":"Let R → A be a ring map. Consider the category I of all diagrams of R-algebra maps A' → A with A' finitely presented over R. Then I is filtered, and the colimit of the A' over I is isomorphic to A.","statement_latex":"Let $R \\to A$ be a ring map. Consider the category $\\mathcal{I}$ of all\ndiagrams of $R$-algebra maps $A' \\to A$ with $A'$ finitely presented over\n$R$. Then $\\mathcal{I}$ is filtered, and the colimit of the $A'$ over\n$\\mathcal{I}$ is isomorphic to $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUF","source_file":"algebra.tex","source_line":31770,"source_end_line":31776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31770-L31776","statement_sha256":"7a93f50d6fd4affdae5dc68ee3a8fd371e3b28186615639f0b5d73980e500c48","origin":"The Stacks Project","memory_eligible":false,"source_rank":1852,"rank":1852,"depth":3,"x":1967.495,"y":109.792,"cluster":"commutative-algebra"},{"id":"stacks:00QN","tag":"00QN","title":"Colimits and maps of finite presentation · Lemma 00QN","summary":"Let R → A be a ring map. There exists a directed system A_λ of R-algebras of finite presentation such that A = colim_λ A_λ. If A is of finite type over R we may arrange it so that all the transition maps in the system of A_λ are surjective.","statement_latex":"Let $R \\to A$ be a ring map. There exists a directed system $A_\\lambda$ of\n$R$-algebras of finite presentation such that $A = \\colim_\\lambda A_\\lambda$.\nIf $A$ is of finite type over $R$ we may arrange it so that all the\ntransition maps in the system of $A_\\lambda$ are surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QN","source_file":"algebra.tex","source_line":31837,"source_end_line":31843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31837-L31843","statement_sha256":"a169b9656bc25d5bb3cf0b3222be66117fd6b79d34724e0175bd345c1432399e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1853,"rank":1853,"depth":4,"x":2164.788,"y":265.823,"cluster":"commutative-algebra"},{"id":"stacks:00QO","tag":"00QO","title":"Colimits and maps of finite presentation · Lemma 00QO","summary":"Let φ : R → S be a ring map. The following are equivalent • φ is of finite presentation, • for every directed system A_λ of R-algebras the map colim_λ Hom_R(S, A_λ) → Hom_R(S, colim_λ A_λ) is bijective, and • for every directed system A_λ of R-algebras the map colim_λ Hom_R(S, A_λ) → Hom_R(S, colim_λ A_λ) is surjective.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. The following are equivalent\n\\begin{enumerate}\n\\item $\\varphi$ is of finite presentation,\n\\item for every directed system $A_\\lambda$ of $R$-algebras\nthe map\n$$\n\\colim_\\lambda \\Hom_R(S, A_\\lambda) \\longrightarrow\n\\Hom_R(S, \\colim_\\lambda A_\\lambda)\n$$\nis bijective, and\n\\item for every directed system $A_\\lambda$ of $R$-algebras\nthe map\n$$\n\\colim_\\lambda \\Hom_R(S, A_\\lambda) \\longrightarrow\n\\Hom_R(S, \\colim_\\lambda A_\\lambda)\n$$\nis surjective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QO","source_file":"algebra.tex","source_line":31885,"source_end_line":31905,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31885-L31905","statement_sha256":"7bd32c4f82a4ec6d048f9febc4f62c10f20ab4494f799819b7e37f3067b90fde","origin":"The Stacks Project","memory_eligible":false,"source_rank":1854,"rank":1854,"depth":5,"x":1893.688,"y":262.715,"cluster":"commutative-algebra"},{"id":"stacks:07C3","tag":"07C3","title":"Colimits and maps of finite presentation · Lemma 07C3","summary":"Let R → Lambda be a ring map. Let E be a set of R-algebras such that each A ∈ E is of finite presentation over R. Then the following two statements are equivalent • Lambda is a filtered colimit of elements of E, and • for any R algebra map A → Lambda with A of finite presentation over R we can find a factorization A → B → Lambda with B ∈ E.","statement_latex":"Let $R \\to \\Lambda$ be a ring map. Let $\\mathcal{E}$ be a set of $R$-algebras\nsuch that each $A \\in \\mathcal{E}$ is of finite presentation over $R$.\nThen the following two statements are equivalent\n\\begin{enumerate}\n\\item $\\Lambda$ is a filtered colimit of elements of $\\mathcal{E}$, and\n\\item for any $R$ algebra map $A \\to \\Lambda$ with $A$ of finite\npresentation over $R$ we can find a factorization $A \\to B \\to \\Lambda$\nwith $B \\in \\mathcal{E}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07C3","source_file":"algebra.tex","source_line":31951,"source_end_line":31962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31951-L31962","statement_sha256":"7e8034469f565187fd51f795ffa3164849074bfdb8dd10b7c42f24c7bd03fe97","origin":"The Stacks Project","memory_eligible":false,"source_rank":1855,"rank":1855,"depth":6,"x":2096.199,"y":111.098,"cluster":"commutative-algebra"},{"id":"stacks:05LI","tag":"05LI","title":"Colimits and maps of finite presentation · Lemma 05LI","summary":"Let A be a ring and let M, N be A-modules. Suppose that R = colim_i ∈ I R_i is a directed colimit of A-algebras. • If M is a finite A-module, and u, u' : M → N are A-module maps such that u ⊗ 1 = u' ⊗ 1 : M ⊗_A R → N ⊗_A R then for some i we have u ⊗ 1 = u' ⊗ 1 : M ⊗_A R_i → N ⊗_A R_i. • If N is a finite A-module and u : M → N is an A-module map such that u ⊗ 1 : M ⊗_A R → N ⊗_A R is surjective, then for some i the map u ⊗ 1 : M ⊗_A R_i → N ⊗_A R_i is surjective. • If N…","statement_latex":"Let $A$ be a ring and let $M, N$ be $A$-modules.\nSuppose that $R = \\colim_{i \\in I} R_i$ is a directed colimit\nof $A$-algebras.\n\\begin{enumerate}\n\\item If $M$ is a finite $A$-module, and $u, u' : M \\to N$ are\n$A$-module maps such that\n$u \\otimes 1 = u' \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$\nthen for some $i$ we have\n$u \\otimes 1 = u' \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$.\n\\item If $N$ is a finite $A$-module and $u : M \\to N$ is an $A$-module\nmap such that $u \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ is surjective,\nthen for some $i$ the map $u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$\nis surjective.\n\\item If $N$ is a finitely presented $A$-module, and\n$v : N \\otimes_A R \\to M \\otimes_A R$ is an $R$-module\nmap, then there exists an $i$ and an $R_i$-module map\n$v_i : N \\otimes_A R_i \\to M \\otimes_A R_i$ such that $v = v_i \\otimes 1$.\n\\item If $M$ is a finite $A$-module, $N$ is a finitely presented $A$-module,\nand $u : M \\to N$ is an $A$-module map such that\n$u \\otimes 1 : M \\otimes_A R \\to N \\otimes_A R$ is an isomorphism, then\nfor some $i$ the map $u \\otimes 1 : M \\otimes_A R_i \\to N \\otimes_A R_i$\nis an isomorphism.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LI","source_file":"algebra.tex","source_line":31988,"source_end_line":32013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L31988-L32013","statement_sha256":"7dc0da521c7b586524502378cf3650b7557003249e97870e10c0dca7c6a33207","origin":"The Stacks Project","memory_eligible":false,"source_rank":1856,"rank":1856,"depth":0,"x":2068.782,"y":337.928,"cluster":"commutative-algebra"},{"id":"stacks:05N7","tag":"05N7","title":"Colimits and maps of finite presentation · Lemma 05N7","summary":"Suppose that R = colim_λ ∈ Lambda R_λ is a directed colimit of rings. Then the category of finitely presented R-modules is the colimit of the categories of finitely presented R_λ-modules. More precisely • Given a finitely presented R-module M there exists a λ ∈ Lambda and a finitely presented R_λ-module M_λ such that M ≅ M_λ ⊗_R_λ R. • Given a λ ∈ Lambda, finitely presented R_λ-modules M_λ, N_λ, and an R-module map φ : M_λ ⊗_R_λ R → N_λ ⊗_R_λ R, then there exists a μ ≥ λ…","statement_latex":"Suppose that $R = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ is a directed colimit\nof rings. Then the category of finitely presented $R$-modules is\nthe colimit of the categories of finitely presented $R_\\lambda$-modules.\nMore precisely\n\\begin{enumerate}\n\\item Given a finitely presented $R$-module $M$ there exists a\n$\\lambda \\in \\Lambda$ and a finitely presented $R_\\lambda$-module\n$M_\\lambda$ such that $M \\cong M_\\lambda \\otimes_{R_\\lambda} R$.\n\\item Given a $\\lambda \\in \\Lambda$, finitely presented\n$R_\\lambda$-modules $M_\\lambda, N_\\lambda$, and an $R$-module map\n$\\varphi : M_\\lambda \\otimes_{R_\\lambda} R \\to N_\\lambda \\otimes_{R_\\lambda} R$,\nthen there exists a $\\mu \\geq \\lambda$ and an $R_\\mu$-module map\n$\\varphi_\\mu : M_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to\nN_\\lambda \\otimes_{R_\\lambda} R_\\mu$\nsuch that $\\varphi = \\varphi_\\mu \\otimes 1_R$.\n\\item Given a $\\lambda \\in \\Lambda$, finitely presented\n$R_\\lambda$-modules $M_\\lambda, N_\\lambda$, and $R_\\lambda$-module maps\n$\\varphi, \\psi : M_\\lambda \\to N_\\lambda$\nsuch that $\\varphi \\otimes 1_R = \\psi \\otimes 1_R$, then\n$\\varphi \\otimes 1_{R_\\mu} = \\psi \\otimes 1_{R_\\mu}$ for some\n$\\mu \\geq \\lambda$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05N7","source_file":"algebra.tex","source_line":32060,"source_end_line":32084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32060-L32084","statement_sha256":"1c08aafea64ab8d88b1b31f954256d8a9e33dac1fddf39b3962671c9554a4f1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1857,"rank":1857,"depth":1,"x":1906.503,"y":155.013,"cluster":"commutative-algebra"},{"id":"stacks:05N8","tag":"05N8","title":"Colimits and maps of finite presentation · Lemma 05N8","summary":"Let A be a ring and let B, C be A-algebras. Suppose that R = colim_i ∈ I R_i is a directed colimit of A-algebras. • If B is a finite type A-algebra, and u, u' : B → C are A-algebra maps such that u ⊗ 1 = u' ⊗ 1 : B ⊗_A R → C ⊗_A R then for some i we have u ⊗ 1 = u' ⊗ 1 : B ⊗_A R_i → C ⊗_A R_i. • If C is a finite type A-algebra and u : B → C is an A-algebra map such that u ⊗ 1 : B ⊗_A R → C ⊗_A R is surjective, then for some i the map u ⊗ 1 : B ⊗_A R_i → C ⊗_A R_i is…","statement_latex":"Let $A$ be a ring and let $B, C$ be $A$-algebras.\nSuppose that $R = \\colim_{i \\in I} R_i$ is a directed colimit\nof $A$-algebras.\n\\begin{enumerate}\n\\item If $B$ is a finite type $A$-algebra, and $u, u' : B \\to C$ are\n$A$-algebra maps such that\n$u \\otimes 1 = u' \\otimes 1 : B \\otimes_A R \\to C \\otimes_A R$\nthen for some $i$ we have\n$u \\otimes 1 = u' \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$.\n\\item If $C$ is a finite type $A$-algebra and $u : B \\to C$ is an\n$A$-algebra map such that\n$u \\otimes 1 : B \\otimes_A R \\to C \\otimes_A R$ is surjective, then\nfor some $i$ the map $u \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$\nis surjective.\n\\item If $C$ is of finite presentation over $A$ and\n$v : C \\otimes_A R \\to B \\otimes_A R$ is an $R$-algebra map, then there\nexists an $i$ and an $R_i$-algebra map\n$v_i : C \\otimes_A R_i \\to B \\otimes_A R_i$ such that\n$v = v_i \\otimes 1$.\n\\item If $B$ is a finite type $A$-algebra, $C$ is a finitely presented\n$A$-algebra, and\n$u \\otimes 1 : B \\otimes_A R \\to C \\otimes_A R$ is an isomorphism, then\nfor some $i$ the map $u \\otimes 1 : B \\otimes_A R_i \\to C \\otimes_A R_i$\nis an isomorphism.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05N8","source_file":"algebra.tex","source_line":32102,"source_end_line":32129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32102-L32129","statement_sha256":"d68c896ccd3bc7b0228829f42caa9739b880ea78151af10c3669ff0bfaa202f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1858,"rank":1858,"depth":0,"x":2173.401,"y":197.833,"cluster":"commutative-algebra"},{"id":"stacks:05N9","tag":"05N9","title":"Colimits and maps of finite presentation · Lemma 05N9","summary":"Suppose that R = colim_λ ∈ Lambda R_λ is a directed colimit of rings. Then the category of finitely presented R-algebras is the colimit of the categories of finitely presented R_λ-algebras. More precisely • Given a finitely presented R-algebra A there exists a λ ∈ Lambda and a finitely presented R_λ-algebra A_λ such that A ≅ A_λ ⊗_R_λ R. • Given a λ ∈ Lambda, finitely presented R_λ-algebras A_λ, B_λ, and an R-algebra map φ : A_λ ⊗_R_λ R → B_λ ⊗_R_λ R, then there exists a…","statement_latex":"Suppose that $R = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ is a directed colimit\nof rings. Then the category of finitely presented $R$-algebras is\nthe colimit of the categories of finitely presented $R_\\lambda$-algebras.\nMore precisely\n\\begin{enumerate}\n\\item Given a finitely presented $R$-algebra $A$ there exists a\n$\\lambda \\in \\Lambda$ and a finitely presented $R_\\lambda$-algebra\n$A_\\lambda$ such that $A \\cong A_\\lambda \\otimes_{R_\\lambda} R$.\n\\item Given a $\\lambda \\in \\Lambda$, finitely presented\n$R_\\lambda$-algebras $A_\\lambda, B_\\lambda$, and an $R$-algebra map\n$\\varphi : A_\\lambda \\otimes_{R_\\lambda} R \\to B_\\lambda \\otimes_{R_\\lambda} R$,\nthen there exists a $\\mu \\geq \\lambda$ and an $R_\\mu$-algebra map\n$\\varphi_\\mu : A_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to\nB_\\lambda \\otimes_{R_\\lambda} R_\\mu$\nsuch that $\\varphi = \\varphi_\\mu \\otimes 1_R$.\n\\item Given a $\\lambda \\in \\Lambda$, finitely presented\n$R_\\lambda$-algebras $A_\\lambda, B_\\lambda$, and $R_\\lambda$-algebra maps\n$\\varphi_\\lambda, \\psi_\\lambda : A_\\lambda \\to B_\\lambda$\nsuch that $\\varphi \\otimes 1_R = \\psi \\otimes 1_R$, then\n$\\varphi \\otimes 1_{R_\\mu} = \\psi \\otimes 1_{R_\\mu}$ for some\n$\\mu \\geq \\lambda$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05N9","source_file":"algebra.tex","source_line":32177,"source_end_line":32201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32177-L32201","statement_sha256":"70755bcf73d7ef51ec4f1585d504b66fccc316c849000bcc58798c59c664762e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1859,"rank":1859,"depth":1,"x":1942.038,"y":317.766,"cluster":"commutative-algebra"},{"id":"stacks:00QT","tag":"00QT","title":"Colimits and maps of finite presentation · Lemma 00QT","summary":"Suppose R → S is a local homomorphism of local rings. There exists a directed set (Lambda, ≤), and a system of local homomorphisms R_λ → S_λ of local rings such that • The colimit of the system R_λ → S_λ is equal to R → S. • Each R_λ is essentially of finite type over Z. • Each S_λ is essentially of finite type over R_λ.","statement_latex":"Suppose $R \\to S$ is a local homomorphism of local rings.\nThere exists a directed set $(\\Lambda, \\leq)$, and\na system of local homomorphisms $R_\\lambda \\to S_\\lambda$\nof local rings such that\n\\begin{enumerate}\n\\item The colimit of the system $R_\\lambda \\to S_\\lambda$\nis equal to $R \\to S$.\n\\item Each $R_\\lambda$ is essentially of finite type\nover $\\mathbf{Z}$.\n\\item Each $S_\\lambda$ is essentially of finite type\nover $R_\\lambda$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QT","source_file":"algebra.tex","source_line":32218,"source_end_line":32232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32218-L32232","statement_sha256":"e1c3514776e9062291412a8afbb20d4d4fe7aaf3c8358de79fcaa6385c64aab7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1860,"rank":1860,"depth":0,"x":2016.234,"y":97.933,"cluster":"commutative-algebra"},{"id":"stacks:00QU","tag":"00QU","title":"Colimits and maps of finite presentation · Lemma 00QU","summary":"Suppose R → S is a local homomorphism of local rings. Assume that S is essentially of finite type over R. Then there exists a directed set (Lambda, ≤), and a system of local homomorphisms R_λ → S_λ of local rings such that • The colimit of the system R_λ → S_λ is equal to R → S. • Each R_λ is essentially of finite type over Z. • Each S_λ is essentially of finite type over R_λ. • For each λ ≤ μ the map S_λ ⊗_R_λ R_μ → S_μ presents S_μ as the localization of a quotient of…","statement_latex":"Suppose $R \\to S$ is a local homomorphism of local rings.\nAssume that $S$ is essentially of finite type over $R$.\nThen there exists a directed set $(\\Lambda, \\leq)$, and\na system of local homomorphisms $R_\\lambda \\to S_\\lambda$\nof local rings such that\n\\begin{enumerate}\n\\item The colimit of the system $R_\\lambda \\to S_\\lambda$\nis equal to $R \\to S$.\n\\item Each $R_\\lambda$ is essentially of finite type\nover $\\mathbf{Z}$.\n\\item Each $S_\\lambda$ is essentially of finite type\nover $R_\\lambda$.\n\\item For each $\\lambda \\leq \\mu$ the map\n$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$\npresents $S_\\mu$ as the localization of a quotient\nof $S_\\lambda \\otimes_{R_\\lambda} R_\\mu$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QU","source_file":"algebra.tex","source_line":32272,"source_end_line":32291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32272-L32291","statement_sha256":"d518b3f794e3c2b1ae953d89354f5805d7e2ea2ff8cc41867152b83652425869","origin":"The Stacks Project","memory_eligible":false,"source_rank":1861,"rank":1861,"depth":0,"x":2138.371,"y":302.243,"cluster":"commutative-algebra"},{"id":"stacks:00QV","tag":"00QV","title":"Colimits and maps of finite presentation · Lemma 00QV","summary":"Suppose R → S is a local homomorphism of local rings. Assume that S is essentially of finite presentation over R. Then there exists a directed set (Lambda, ≤), and a system of local homomorphism R_λ → S_λ of local rings such that • The colimit of the system R_λ → S_λ is equal to R → S. • Each R_λ is essentially of finite type over Z. • Each S_λ is essentially of finite type over R_λ. • For each λ ≤ μ the map S_λ ⊗_R_λ R_μ → S_μ presents S_μ as the localization of S_λ…","statement_latex":"Suppose $R \\to S$ is a local homomorphism of local rings.\nAssume that $S$ is essentially of finite presentation over $R$.\nThen there exists a directed set $(\\Lambda, \\leq)$, and\na system of local homomorphism $R_\\lambda \\to S_\\lambda$\nof local rings such that\n\\begin{enumerate}\n\\item The colimit of the system $R_\\lambda \\to S_\\lambda$\nis equal to $R \\to S$.\n\\item Each $R_\\lambda$ is essentially of finite type\nover $\\mathbf{Z}$.\n\\item Each $S_\\lambda$ is essentially of finite type\nover $R_\\lambda$.\n\\item For each $\\lambda \\leq \\mu$ the map\n$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$\npresents $S_\\mu$ as the localization of\n$S_\\lambda \\otimes_{R_\\lambda} R_\\mu$\nat a prime ideal.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QV","source_file":"algebra.tex","source_line":32345,"source_end_line":32365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32345-L32365","statement_sha256":"6c823af92fc1503a4eee56cc8aecdd3be3ce07d048b5b50c3316e6a85563a874","origin":"The Stacks Project","memory_eligible":false,"source_rank":1862,"rank":1862,"depth":0,"x":1883.875,"y":220.848,"cluster":"commutative-algebra"},{"id":"stacks:00QX","tag":"00QX","title":"Colimits and maps of finite presentation · Lemma 00QX","summary":"Suppose R → S is a local homomorphism of local rings. Assume that S is essentially of finite presentation over R. Let M be a finitely presented S-module. Then there exists a directed set (Lambda, ≤), and a system of local homomorphisms R_λ → S_λ of local rings together with S_λ-modules M_λ, such that • The colimit of the system R_λ → S_λ is equal to R → S. The colimit of the system M_λ is M. • Each R_λ is essentially of finite type over Z. • Each S_λ is essentially of…","statement_latex":"Suppose $R \\to S$ is a local homomorphism of local rings.\nAssume that $S$ is essentially of finite presentation over $R$.\nLet $M$ be a finitely presented $S$-module.\nThen there exists a directed set $(\\Lambda, \\leq)$, and\na system of local homomorphisms $R_\\lambda \\to S_\\lambda$\nof local rings together with $S_\\lambda$-modules $M_\\lambda$,\nsuch that\n\\begin{enumerate}\n\\item The colimit of the system $R_\\lambda \\to S_\\lambda$\nis equal to $R \\to S$. The colimit of the system $M_\\lambda$\nis $M$.\n\\item Each $R_\\lambda$ is essentially of finite type\nover $\\mathbf{Z}$.\n\\item Each $S_\\lambda$ is essentially of finite type\nover $R_\\lambda$.\n\\item Each $M_\\lambda$ is finite over $S_\\lambda$.\n\\item For each $\\lambda \\leq \\mu$ the map\n$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$\npresents $S_\\mu$ as the localization of\n$S_\\lambda \\otimes_{R_\\lambda} R_\\mu$\nat a prime ideal.\n\\item For each $\\lambda \\leq \\mu$ the map\n$M_\\lambda \\otimes_{S_\\lambda} S_\\mu \\to M_\\mu$\nis an isomorphism.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QX","source_file":"algebra.tex","source_line":32426,"source_end_line":32453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32426-L32453","statement_sha256":"24cf2d0a12697e28efda580cdd6ff89cb49832d50a14a0898a3755a6fee83650","origin":"The Stacks Project","memory_eligible":false,"source_rank":1863,"rank":1863,"depth":1,"x":2137.124,"y":136.416,"cluster":"commutative-algebra"},{"id":"stacks:00QY","tag":"00QY","title":"Colimits and maps of finite presentation · Lemma 00QY","summary":"Suppose R → S is a ring map. Then there exists a directed set (Lambda, ≤), and a system of ring maps R_λ → S_λ such that • The colimit of the system R_λ → S_λ is equal to R → S. • Each R_λ is of finite type over Z. • Each S_λ is of finite type over R_λ.","statement_latex":"Suppose $R \\to S$ is a ring map.\nThen there exists a directed set $(\\Lambda, \\leq)$, and\na system of ring maps $R_\\lambda \\to S_\\lambda$\nsuch that\n\\begin{enumerate}\n\\item The colimit of the system $R_\\lambda \\to S_\\lambda$\nis equal to $R \\to S$.\n\\item Each $R_\\lambda$ is of finite type\nover $\\mathbf{Z}$.\n\\item Each $S_\\lambda$ is of finite type\nover $R_\\lambda$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QY","source_file":"algebra.tex","source_line":32483,"source_end_line":32497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32483-L32497","statement_sha256":"8aa27a77f0e69509fcb428e8a9fc749f161ccc0608d4a9881405db8f46a6b5cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1864,"rank":1864,"depth":1,"x":2018.218,"y":342.483,"cluster":"commutative-algebra"},{"id":"stacks:0BTG","tag":"0BTG","title":"Colimits and maps of finite presentation · Lemma 0BTG","summary":"Suppose R → S is a ring map. Assume that S is integral over R. Then there exists a directed set (Lambda, ≤), and a system of ring maps R_λ → S_λ such that • The colimit of the system R_λ → S_λ is equal to R → S. • Each R_λ is of finite type over Z. • Each S_λ is finite over R_λ.","statement_latex":"Suppose $R \\to S$ is a ring map.\nAssume that $S$ is integral over $R$.\nThen there exists a directed set $(\\Lambda, \\leq)$, and\na system of ring maps $R_\\lambda \\to S_\\lambda$\nsuch that\n\\begin{enumerate}\n\\item The colimit of the system $R_\\lambda \\to S_\\lambda$\nis equal to $R \\to S$.\n\\item Each $R_\\lambda$ is of finite type\nover $\\mathbf{Z}$.\n\\item Each $S_\\lambda$ is finite over $R_\\lambda$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTG","source_file":"algebra.tex","source_line":32505,"source_end_line":32519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32505-L32519","statement_sha256":"083bd1713a793ae145710a7c2fa98a54caeeba371a940ddb0bc3157d07b0e91b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1865,"rank":1865,"depth":6,"x":1940.144,"y":122.947,"cluster":"commutative-algebra"},{"id":"stacks:00QZ","tag":"00QZ","title":"Colimits and maps of finite presentation · Lemma 00QZ","summary":"Suppose R → S is a ring map. Assume that S is of finite type over R. Then there exists a directed set (Lambda, ≤), and a system of ring maps R_λ → S_λ such that • The colimit of the system R_λ → S_λ is equal to R → S. • Each R_λ is of finite type over Z. • Each S_λ is of finite type over R_λ. • For each λ ≤ μ the map S_λ ⊗_R_λ R_μ → S_μ presents S_μ as a quotient of S_λ ⊗_R_λ R_μ.","statement_latex":"Suppose $R \\to S$ is a ring map.\nAssume that $S$ is of finite type over $R$.\nThen there exists a directed set $(\\Lambda, \\leq)$, and\na system of ring maps $R_\\lambda \\to S_\\lambda$\nsuch that\n\\begin{enumerate}\n\\item The colimit of the system $R_\\lambda \\to S_\\lambda$\nis equal to $R \\to S$.\n\\item Each $R_\\lambda$ is of finite type\nover $\\mathbf{Z}$.\n\\item Each $S_\\lambda$ is of finite type\nover $R_\\lambda$.\n\\item For each $\\lambda \\leq \\mu$ the map\n$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$\npresents $S_\\mu$ as a quotient\nof $S_\\lambda \\otimes_{R_\\lambda} R_\\mu$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00QZ","source_file":"algebra.tex","source_line":32539,"source_end_line":32558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32539-L32558","statement_sha256":"bf24137441cf57ff3d3a2c313066fa063f624f7645c92e05402e3a61a8baf9c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1866,"rank":1866,"depth":1,"x":2174.382,"y":240.59,"cluster":"commutative-algebra"},{"id":"stacks:00R0","tag":"00R0","title":"Colimits and maps of finite presentation · Lemma 00R0","summary":"Suppose R → S is a ring map. Assume that S is of finite presentation over R. Then there exists a directed set (Lambda, ≤), and a system of ring maps R_λ → S_λ such that • The colimit of the system R_λ → S_λ is equal to R → S. • Each R_λ is of finite type over Z. • Each S_λ is of finite type over R_λ. • For each λ ≤ μ the map S_λ ⊗_R_λ R_μ → S_μ is an isomorphism.","statement_latex":"Suppose $R \\to S$ is a ring map.\nAssume that $S$ is of finite presentation over $R$.\nThen there exists a directed set $(\\Lambda, \\leq)$, and\na system of ring maps $R_\\lambda \\to S_\\lambda$\nsuch that\n\\begin{enumerate}\n\\item The colimit of the system $R_\\lambda \\to S_\\lambda$\nis equal to $R \\to S$.\n\\item Each $R_\\lambda$ is of finite type\nover $\\mathbf{Z}$.\n\\item Each $S_\\lambda$ is of finite type\nover $R_\\lambda$.\n\\item For each $\\lambda \\leq \\mu$ the map\n$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$\nis an isomorphism.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00R0","source_file":"algebra.tex","source_line":32566,"source_end_line":32584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32566-L32584","statement_sha256":"8bad793397e3aba21bb671ba38e6576af88b701ad1872e76ea8ecf54b4d2f34a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1867,"rank":1867,"depth":1,"x":1906.913,"y":286.778,"cluster":"commutative-algebra"},{"id":"stacks:00R1","tag":"00R1","title":"Colimits and maps of finite presentation · Lemma 00R1","summary":"Suppose R → S is a ring map. Assume that S is of finite presentation over R. Let M be a finitely presented S-module. Then there exists a directed set (Lambda, ≤), and a system of ring maps R_λ → S_λ together with S_λ-modules M_λ, such that • The colimit of the system R_λ → S_λ is equal to R → S. The colimit of the system M_λ is M. • Each R_λ is of finite type over Z. • Each S_λ is of finite type over R_λ. • Each M_λ is finite over S_λ. • For each λ ≤ μ the map S_λ ⊗_R_λ…","statement_latex":"Suppose $R \\to S$ is a ring map.\nAssume that $S$ is of finite presentation over $R$.\nLet $M$ be a finitely presented $S$-module.\nThen there exists a directed set $(\\Lambda, \\leq)$, and\na system of ring maps $R_\\lambda \\to S_\\lambda$\ntogether with $S_\\lambda$-modules $M_\\lambda$,\nsuch that\n\\begin{enumerate}\n\\item The colimit of the system $R_\\lambda \\to S_\\lambda$\nis equal to $R \\to S$. The colimit of the system $M_\\lambda$\nis $M$.\n\\item Each $R_\\lambda$ is of finite type\nover $\\mathbf{Z}$.\n\\item Each $S_\\lambda$ is of finite type\nover $R_\\lambda$.\n\\item Each $M_\\lambda$ is finite over $S_\\lambda$.\n\\item For each $\\lambda \\leq \\mu$ the map\n$S_\\lambda \\otimes_{R_\\lambda} R_\\mu \\to S_\\mu$\nis an isomorphism.\n\\item For each $\\lambda \\leq \\mu$ the map\n$M_\\lambda \\otimes_{S_\\lambda} S_\\mu \\to M_\\mu$\nis an isomorphism.\n\\end{enumerate}\nIn particular, for every $\\lambda \\in \\Lambda$ we have\n$$\nM = M_\\lambda \\otimes_{S_\\lambda} S\n= M_\\lambda \\otimes_{R_\\lambda} R.\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00R1","source_file":"algebra.tex","source_line":32592,"source_end_line":32622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32592-L32622","statement_sha256":"2287a51d64810ebe1839187e7854591df6d99de7526ad08e9cc562e855bb4b8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1868,"rank":1868,"depth":2,"x":2067.081,"y":100.855,"cluster":"commutative-algebra"},{"id":"stacks:00R4","tag":"00R4","title":"More flatness criteria · Lemma 00R4","summary":"Miracle flatness Let R → S be a local homomorphism of Noetherian local rings. Assume • R is regular, • S Cohen-Macaulay, • dim(S) = dim(R) + dim(S/ m_R S). Then R → S is flat.","statement_latex":"\\begin{slogan}\nMiracle flatness\n\\end{slogan}\nLet $R \\to S$ be a local homomorphism of Noetherian local\nrings. Assume\n\\begin{enumerate}\n\\item $R$ is regular,\n\\item $S$ Cohen-Macaulay,\n\\item $\\dim(S) = \\dim(R) + \\dim(S/\\mathfrak m_R S)$.\n\\end{enumerate}\nThen $R \\to S$ is flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00R4","source_file":"algebra.tex","source_line":32653,"source_end_line":32666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32653-L32666","statement_sha256":"87628bfbd5af6df43c45735a2eefa5db4d44e85903950d440fd5bd81f2e6b9d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1869,"rank":1869,"depth":18,"x":2098.506,"y":328.953,"cluster":"commutative-algebra"},{"id":"stacks:07DY","tag":"07DY","title":"More flatness criteria · Lemma 07DY","summary":"Let R → S be a homomorphism of Noetherian local rings. Assume that R is a regular local ring and that a regular system of parameters maps to a regular sequence in S. Then R → S is flat.","statement_latex":"Let $R \\to S$ be a homomorphism of Noetherian local rings.\nAssume that $R$ is a regular local ring and that a regular system\nof parameters maps to a regular sequence in $S$. Then $R \\to S$\nis flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DY","source_file":"algebra.tex","source_line":32693,"source_end_line":32699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32693-L32699","statement_sha256":"f9b38049c89df6241e33e16c4af20841e30ba226318a3e47b8627ee09855bf19","origin":"The Stacks Project","memory_eligible":false,"source_rank":1870,"rank":1870,"depth":9,"x":1891.795,"y":178.51,"cluster":"commutative-algebra"},{"id":"stacks:00R6","tag":"00R6","title":"More flatness criteria · Lemma 00R6","summary":"Let R → S, M, Lambda, R_λ → S_λ, M_λ be as in Lemma [Tag 00QX]. Assume that M is flat over R. Then for some λ ∈ Lambda the module M_λ is flat over R_λ.","statement_latex":"Let $R \\to S$, $M$, $\\Lambda$, $R_\\lambda \\to S_\\lambda$, $M_\\lambda$\nbe as in Lemma \\ref{lemma-limit-module-essentially-finite-presentation}.\nAssume that $M$ is flat over $R$.\nThen for some $\\lambda \\in \\Lambda$ the module\n$M_\\lambda$ is flat over $R_\\lambda$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00R6","source_file":"algebra.tex","source_line":32722,"source_end_line":32729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32722-L32729","statement_sha256":"12d445a349473a35b200d0778b74cea756995cc6a6e39d79a6127265183d47f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1871,"rank":1871,"depth":10,"x":2165.345,"y":172.148,"cluster":"commutative-algebra"},{"id":"stacks:046Y","tag":"046Y","title":"More flatness criteria · Lemma 046Y","summary":"Suppose that R → S is a local homomorphism of local rings. Denote m the maximal ideal of R. Let u : M → N be a map of S-modules. Assume • S is essentially of finite presentation over R, • M, N are finitely presented over S, • N is flat over R, and • overlineu : M/ mM → N/ mN is injective. Then u is injective, and N/u(M) is flat over R.","statement_latex":"Suppose that $R \\to S$ is a local homomorphism of local rings.\nDenote $\\mathfrak m$ the maximal ideal of $R$.\nLet $u : M \\to N$ be a map of $S$-modules.\nAssume\n\\begin{enumerate}\n\\item $S$ is essentially of finite presentation over $R$,\n\\item $M$, $N$ are finitely presented over $S$,\n\\item $N$ is flat over $R$, and\n\\item $\\overline{u} : M/\\mathfrak mM \\to N/\\mathfrak mN$ is injective.\n\\end{enumerate}\nThen $u$ is injective, and $N/u(M)$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046Y","source_file":"algebra.tex","source_line":32769,"source_end_line":32782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32769-L32782","statement_sha256":"7d35317c849aa992ebe94ba1d5928951bd85abf28eab830a17310306b0302161","origin":"The Stacks Project","memory_eligible":false,"source_rank":1872,"rank":1872,"depth":11,"x":1968.648,"y":332.141,"cluster":"commutative-algebra"},{"id":"stacks:046Z","tag":"046Z","title":"More flatness criteria · Lemma 046Z","summary":"Suppose that R → S is a local ring homomorphism of local rings. Denote m the maximal ideal of R. Suppose • S is essentially of finite presentation over R, • S is flat over R, and • f ∈ S is a nonzerodivisor in S/ mS. Then S/fS is flat over R, and f is a nonzerodivisor in S.","statement_latex":"Suppose that $R \\to S$ is a local ring homomorphism of local rings.\nDenote $\\mathfrak m$ the maximal ideal of $R$.\nSuppose\n\\begin{enumerate}\n\\item $S$ is essentially of finite presentation over $R$,\n\\item $S$ is flat over $R$, and\n\\item $f \\in S$ is a nonzerodivisor in $S/{\\mathfrak m}S$.\n\\end{enumerate}\nThen $S/fS$ is flat over $R$, and $f$ is a nonzerodivisor in $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046Z","source_file":"algebra.tex","source_line":32868,"source_end_line":32879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32868-L32879","statement_sha256":"dc4faf584e7f21d89d6ce34a257d4d3dbc5db7125b533532743adee2680e161d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1873,"rank":1873,"depth":12,"x":1985.036,"y":102.435,"cluster":"commutative-algebra"},{"id":"stacks:0470","tag":"0470","title":"More flatness criteria · Lemma 0470","summary":"Suppose that R → S is a local ring homomorphism of local rings. Denote m the maximal ideal of R. Suppose • R → S is essentially of finite presentation, • R → S is flat, and • f_1, …, f_c is a sequence of elements of S such that the images overlinef_1, …, overlinef_c form a regular sequence in S/ mS. Then f_1, …, f_c is a regular sequence in S and each of the quotients S/(f_1, …, f_i) is flat over R.","statement_latex":"Suppose that $R \\to S$ is a local ring homomorphism of local rings.\nDenote $\\mathfrak m$ the maximal ideal of $R$.\nSuppose\n\\begin{enumerate}\n\\item $R \\to S$ is essentially of finite presentation,\n\\item $R \\to S$ is flat, and\n\\item $f_1, \\ldots, f_c$ is a sequence of elements of\n$S$ such that the images $\\overline{f}_1, \\ldots, \\overline{f}_c$\nform a regular sequence in $S/{\\mathfrak m}S$.\n\\end{enumerate}\nThen $f_1, \\ldots, f_c$ is a regular sequence in $S$ and each\nof the quotients $S/(f_1, \\ldots, f_i)$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0470","source_file":"algebra.tex","source_line":32885,"source_end_line":32899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32885-L32899","statement_sha256":"3c581156affa5a0aa7f9bd89dbf86d25c0dcffaff672e061c23b405e6821bd0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1874,"rank":1874,"depth":13,"x":2157.764,"y":281.209,"cluster":"commutative-algebra"},{"id":"stacks:0471","tag":"0471","title":"More flatness criteria · Lemma 0471","summary":"Let R → S be a local homomorphism of local rings. Let I not = R be an ideal in R. Let M be an S-module. Assume • S is essentially of finite presentation over R, • M is of finite presentation over S, • Tor_1^R(M, R/I) = 0, and • M/IM is flat over R/I. Then M is flat over R.","statement_latex":"Let $R \\to S$ be a local homomorphism of local rings.\nLet $I \\not = R$ be an ideal in $R$. Let $M$ be an $S$-module. Assume\n\\begin{enumerate}\n\\item $S$ is essentially of finite presentation over $R$,\n\\item $M$ is of finite presentation over $S$,\n\\item $\\text{Tor}_1^R(M, R/I) = 0$, and\n\\item $M/IM$ is flat over $R/I$.\n\\end{enumerate}\nThen $M$ is flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0471","source_file":"algebra.tex","source_line":32909,"source_end_line":32920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32909-L32920","statement_sha256":"dc14216dabd74144b0c883dcacf53f0688ebe2fd5033c5078a64d32dd9395ba4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1875,"rank":1875,"depth":11,"x":1886.492,"y":247.376,"cluster":"commutative-algebra"},{"id":"stacks:00R7","tag":"00R7","title":"Crit\\`ere de platitude par fibres · Lemma 00R7","summary":"Let R, S, S' be local rings and let R → S → S' be local ring homomorphisms. Let M be an S'-module. Let m ⊂ R be the maximal ideal. Assume • The ring maps R → S and R → S' are essentially of finite presentation. • The module M is of finite presentation over S'. • The module M is not zero. • The module M/ mM is a flat S/ mS-module. • The module M is a flat R-module. Then S is flat over R and M is a flat S-module.","statement_latex":"Let $R$, $S$, $S'$ be local rings and let $R \\to S \\to S'$ be local ring\nhomomorphisms. Let $M$ be an $S'$-module. Let $\\mathfrak m \\subset R$\nbe the maximal ideal. Assume\n\\begin{enumerate}\n\\item The ring maps $R \\to S$ and $R \\to S'$ are essentially\nof finite presentation.\n\\item The module $M$ is of finite presentation over $S'$.\n\\item The module $M$ is not zero.\n\\item The module $M/\\mathfrak mM$ is a flat $S/\\mathfrak mS$-module.\n\\item The module $M$ is a flat $R$-module.\n\\end{enumerate}\nThen $S$ is flat over $R$ and $M$ is a flat $S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00R7","source_file":"algebra.tex","source_line":32991,"source_end_line":33005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L32991-L33005","statement_sha256":"3f9bc5eff12981613e3df99fbfcc634998788137acc391567112150cf82cf49d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1876,"rank":1876,"depth":11,"x":2113.849,"y":118.331,"cluster":"commutative-algebra"},{"id":"stacks:05UV","tag":"05UV","title":"More flatness criteria · Lemma 05UV","summary":"Let R, S, S' be local rings and let R → S → S' be local ring homomorphisms. Let M be an S'-module. Let m ⊂ R be the maximal ideal. Assume • R → S' is essentially of finite presentation, • R → S is essentially of finite type, • M is of finite presentation over S', • M is not zero, • M/ mM is a flat S/ mS-module, and • M is a flat R-module. Then S is essentially of finite presentation and flat over R and M is a flat S-module.","statement_latex":"Let $R$, $S$, $S'$ be local rings and let $R \\to S \\to S'$ be local ring\nhomomorphisms. Let $M$ be an $S'$-module. Let $\\mathfrak m \\subset R$\nbe the maximal ideal. Assume\n\\begin{enumerate}\n\\item $R \\to S'$ is essentially of finite presentation,\n\\item $R \\to S$ is essentially of finite type,\n\\item $M$ is of finite presentation over $S'$,\n\\item $M$ is not zero,\n\\item $M/\\mathfrak mM$ is a flat $S/\\mathfrak mS$-module, and\n\\item $M$ is a flat $R$-module.\n\\end{enumerate}\nThen $S$ is essentially of finite presentation and flat over $R$\nand $M$ is a flat $S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UV","source_file":"algebra.tex","source_line":33118,"source_end_line":33133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33118-L33133","statement_sha256":"06f65a84326b5b2d214b2019aa3191ea300ff68a0166ff177fc50dd985c46204","origin":"The Stacks Project","memory_eligible":false,"source_rank":1877,"rank":1877,"depth":12,"x":2049.941,"y":342.61,"cluster":"commutative-algebra"},{"id":"stacks:0CEL","tag":"0CEL","title":"Crit\\`ere de platitude par fibres: locally nilpotent case · Lemma 0CEL","summary":"Let xymatrix S ar[rr] & & S' & R ar[lu] ar[ru] be a commutative diagram in the category of rings. Let I ⊂ R be a locally nilpotent ideal and M an S'-module. Assume • R → S is of finite type, • R → S' is of finite presentation, • M is a finitely presented S'-module, • M/IM is flat as a S/IS-module, and • M is flat as an R-module. Then M is a flat S-module and S_ q is flat and essentially of finite presentation over R for every q ⊂ S such that M ⊗_S kappa( q) is nonzero.","statement_latex":"Let\n$$\n\\xymatrix{\nS \\ar[rr] & & S' \\\\\n& R \\ar[lu] \\ar[ru]\n}\n$$\nbe a commutative diagram in the category of rings.\nLet $I \\subset R$ be a locally nilpotent ideal and\n$M$ an $S'$-module. Assume\n\\begin{enumerate}\n\\item $R \\to S$ is of finite type,\n\\item $R \\to S'$ is of finite presentation,\n\\item $M$ is a finitely presented $S'$-module,\n\\item $M/IM$ is flat as a $S/IS$-module, and\n\\item $M$ is flat as an $R$-module.\n\\end{enumerate}\nThen $M$ is a flat $S$-module and $S_\\mathfrak q$ is flat\nand essentially of finite presentation over $R$\nfor every $\\mathfrak q \\subset S$ such that\n$M \\otimes_S \\kappa(\\mathfrak q)$ is nonzero.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"More flatness criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEL","source_file":"algebra.tex","source_line":33171,"source_end_line":33194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33171-L33194","statement_sha256":"248eb3a4184d225dddba4e73fec76df65ac18caf176fe45f553174b8c8fc20b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1878,"rank":1878,"depth":13,"x":1916.637,"y":140.863,"cluster":"commutative-algebra"},{"id":"stacks:00R9","tag":"00R9","title":"Openness of the flat locus · Lemma 00R9","summary":"Let k be a field. Let S be a finite type k-algebra. Let f_1, …, f_i be elements of S. Assume that S is Cohen-Macaulay and equidimensional of dimension d, and that dim V(f_1, …, f_i) ≤ d - i. Then equality holds and f_1, …, f_i forms a regular sequence in S_ q for every prime q of V(f_1, …, f_i).","statement_latex":"Let $k$ be a field. Let $S$ be a finite type\n$k$-algebra. Let $f_1, \\ldots, f_i$ be elements\nof $S$. Assume that $S$ is Cohen-Macaulay and\nequidimensional of dimension $d$, and that\n$\\dim V(f_1, \\ldots, f_i) \\leq d - i$. Then equality\nholds and $f_1, \\ldots, f_i$ forms a regular\nsequence in $S_{\\mathfrak q}$ for every prime $\\mathfrak q$\nof $V(f_1, \\ldots, f_i)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of the flat locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00R9","source_file":"algebra.tex","source_line":33235,"source_end_line":33245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33235-L33245","statement_sha256":"8f69636c3f6ad3c11aad438bfe0a193548a8d9fc7ccfd74b56f6a927829c5ec8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1879,"rank":1879,"depth":22,"x":2177.307,"y":214.026,"cluster":"commutative-algebra"},{"id":"stacks:00RA","tag":"00RA","title":"Openness of the flat locus · Lemma 00RA","summary":"Let R → S be a finite type ring map. Let d be an integer such that all fibres S ⊗_R kappa( p) are Cohen-Macaulay and equidimensional of dimension d. Let f_1, …, f_i be elements of S. The set ( q ∈ V(f_1, …, f_i) mid f_1, …, f_i are a regular sequence in S_ q/ p S_ q where p = R ∩ q ) is open in V(f_1, …, f_i).","statement_latex":"Let $R \\to S$ be a finite type ring map. Let $d$ be an integer such that all\nfibres $S \\otimes_R \\kappa(\\mathfrak p)$ are Cohen-Macaulay and equidimensional\nof dimension $d$. Let $f_1, \\ldots, f_i$ be elements of $S$. The set\n$$\n\\{ \\mathfrak q \\in V(f_1, \\ldots, f_i)\n\\mid f_1, \\ldots, f_i\n\\text{ are a regular sequence in }\nS_{\\mathfrak q}/\\mathfrak p S_{\\mathfrak q}\n\\text{ where }\\mathfrak p = R \\cap \\mathfrak q\n\\}\n$$\nis open in $V(f_1, \\ldots, f_i)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of the flat locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RA","source_file":"algebra.tex","source_line":33261,"source_end_line":33275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33261-L33275","statement_sha256":"432640874317fa038e69f679d53b7de0734c9454ea36d06488ae3ca5d11d707b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1880,"rank":1880,"depth":31,"x":1926.128,"y":308.036,"cluster":"commutative-algebra"},{"id":"stacks:00RB","tag":"00RB","title":"Openness of the flat locus · Lemma 00RB","summary":"Let R → S be a ring map. Consider a finite homological complex of finite free S-modules: F_bullet : 0 → S^n_e xrightarrowφ_e S^n_e-1 xrightarrowφ_e-1 … xrightarrowφ_i + 1 S^n_i xrightarrowφ_i S^n_i-1 xrightarrowφ_i-1 … xrightarrowφ_1 S^n_0 For every prime q of S consider the complex overlineF_bullet, q = F_bullet, q ⊗_R kappa( p) where p is inverse image of q in R. Assume R is Noetherian and there exists an integer d such that R → S is finite type, flat with fibres S ⊗_R…","statement_latex":"Let $R \\to S$ be a ring map. Consider a finite homological complex of\nfinite free $S$-modules:\n$$\nF_{\\bullet} :\n0\n\\to\nS^{n_e}\n\\xrightarrow{\\varphi_e}\nS^{n_{e-1}}\n\\xrightarrow{\\varphi_{e-1}}\n\\ldots\n\\xrightarrow{\\varphi_{i + 1}}\nS^{n_i}\n\\xrightarrow{\\varphi_i}\nS^{n_{i-1}}\n\\xrightarrow{\\varphi_{i-1}}\n\\ldots\n\\xrightarrow{\\varphi_1}\nS^{n_0}\n$$\nFor every prime $\\mathfrak q$ of $S$ consider the\ncomplex $\\overline{F}_{\\bullet, \\mathfrak q} =\nF_{\\bullet, \\mathfrak q} \\otimes_R \\kappa(\\mathfrak p)$\nwhere $\\mathfrak p$ is inverse image of $\\mathfrak q$ in $R$.\nAssume $R$ is Noetherian and there exists an integer $d$ such\nthat $R \\to S$ is finite type, flat\nwith fibres $S \\otimes_R \\kappa(\\mathfrak p)$\nCohen-Macaulay of dimension $d$.\nThe set\n$$\n\\{\\mathfrak q \\in \\Spec(S) \\mid\n\\overline{F}_{\\bullet, \\mathfrak q}\\text{ is exact}\\}\n$$\nis open in $\\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of the flat locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RB","source_file":"algebra.tex","source_line":33306,"source_end_line":33342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33306-L33342","statement_sha256":"35c3e7df11e6611a07f99ad2a144237b82dd32e365b16bf658061b7f95098640","origin":"The Stacks Project","memory_eligible":false,"source_rank":1881,"rank":1881,"depth":32,"x":2035.8,"y":96.081,"cluster":"commutative-algebra"},{"id":"stacks:00RC","tag":"00RC","title":"Openness of the flat locus · Theorem 00RC","summary":"Let R be a ring. Let R → S be a ring map of finite presentation. Let M be a finitely presented S-module. The set ( q ∈ Spec(S) mid M_ q is flat over R) is open in Spec(S).","statement_latex":"Let $R$ be a ring. Let $R \\to S$ be a ring map of finite\npresentation. Let $M$ be a finitely presented $S$-module.\nThe set\n$$\n\\{ \\mathfrak q \\in \\Spec(S) \\mid\nM_{\\mathfrak q}\\text{ is flat over }R\\}\n$$\nis open in $\\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of the flat locus","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RC","source_file":"algebra.tex","source_line":33406,"source_end_line":33416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33406-L33416","statement_sha256":"614f3b4f6edc46660f1cc3935a048b2467d0b16265e4e5ac6ff61f77335c4cf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1882,"rank":1882,"depth":33,"x":2125.424,"y":314.716,"cluster":"commutative-algebra"},{"id":"stacks:00RE","tag":"00RE","title":"Openness of Cohen-Macaulay loci · Lemma 00RE","summary":"Let S be a finite type algebra over a field k. Let φ : k[y_1, …, y_d] → S be a quasi-finite ring map. As subsets of Spec(S) we have ( q mid S_ q flat over k[y_1, …, y_d]) = ( q mid S_ q CM and dim_ q(S/k) = d) For notation see Definition [Tag 00QD].","statement_latex":"Let $S$ be a finite type algebra over a field $k$.\nLet $\\varphi : k[y_1, \\ldots, y_d] \\to S$ be a quasi-finite ring map.\nAs subsets of $\\Spec(S)$ we have\n$$\n\\{ \\mathfrak q \\mid\nS_{\\mathfrak q} \\text{ flat over }k[y_1, \\ldots, y_d]\\}\n=\n\\{ \\mathfrak q \\mid\nS_{\\mathfrak q} \\text{ CM and }\\dim_{\\mathfrak q}(S/k) = d\\}\n$$\nFor notation see Definition \\ref{definition-relative-dimension}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of Cohen-Macaulay loci","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RE","source_file":"algebra.tex","source_line":33524,"source_end_line":33537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33524-L33537","statement_sha256":"685f2dadc172034336850f047ba63a589e56342f06e3622431c90011af04e086","origin":"The Stacks Project","memory_eligible":false,"source_rank":1883,"rank":1883,"depth":29,"x":1883.392,"y":204.296,"cluster":"commutative-algebra"},{"id":"stacks:00RF","tag":"00RF","title":"Openness of Cohen-Macaulay loci · Lemma 00RF","summary":"Let S be a finite type algebra over a field k. The set of primes q such that S_ q is Cohen-Macaulay is open in S.","statement_latex":"Let $S$ be a finite type algebra over a field $k$.\nThe set of primes $\\mathfrak q$ such that $S_{\\mathfrak q}$ is\nCohen-Macaulay is open in $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of Cohen-Macaulay loci","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RF","source_file":"algebra.tex","source_line":33571,"source_end_line":33576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33571-L33576","statement_sha256":"90fc48ac04aa934b63606105522e22800a8601699e9266440c65ea6c5abbf5d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1884,"rank":1884,"depth":0,"x":2150.797,"y":148.355,"cluster":"commutative-algebra"},{"id":"stacks:00RG","tag":"00RG","title":"Openness of Cohen-Macaulay loci · Lemma 00RG","summary":"Let k be a field. Let S be a finite type k algebra. The set of Cohen-Macaulay primes forms a dense open U ⊂ Spec(S).","statement_latex":"Let $k$ be a field. Let $S$ be a finite type $k$ algebra.\nThe set of Cohen-Macaulay primes forms a dense open\n$U \\subset \\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of Cohen-Macaulay loci","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RG","source_file":"algebra.tex","source_line":33608,"source_end_line":33613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33608-L33613","statement_sha256":"a99631e7e54da3a895d5a13158486cfb4dbc9ed2cb58fb71a872b43dc4bf346e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1885,"rank":1885,"depth":1,"x":1998.525,"y":341.435,"cluster":"commutative-algebra"},{"id":"stacks:00RH","tag":"00RH","title":"Openness of Cohen-Macaulay loci · Lemma 00RH","summary":"Let R be a ring. Let R → S be of finite presentation and flat. For any d ≥ 0 the set ( q ∈ Spec(S) such that setting p = R ∩ q the fibre ring S_ q/ pS_ q is Cohen-Macaulay and dim_ q(S/R) = d ) is open in Spec(S).","statement_latex":"Let $R$ be a ring. Let $R \\to S$ be of finite presentation\nand flat. For any $d \\geq 0$ the set\n$$\n\\left\\{\n\\begin{matrix}\n\\mathfrak q \\in \\Spec(S)\n\\text{ such that setting }\\mathfrak p = R \\cap \\mathfrak q\n\\text{ the fibre ring}\\\\\nS_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}\n\\text{ is Cohen-Macaulay}\n\\text{ and } \\dim_{\\mathfrak q}(S/R) = d\n\\end{matrix}\n\\right\\}\n$$\nis open in $\\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of Cohen-Macaulay loci","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RH","source_file":"algebra.tex","source_line":33622,"source_end_line":33639,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33622-L33639","statement_sha256":"66aceaa83c186a44bdb176fc44d3592fdbe7014dce09507e0b06c182334c4543","origin":"The Stacks Project","memory_eligible":false,"source_rank":1886,"rank":1886,"depth":34,"x":1955.516,"y":112.541,"cluster":"commutative-algebra"},{"id":"stacks:00RI","tag":"00RI","title":"Openness of Cohen-Macaulay loci · Lemma 00RI","summary":"Let R be a ring. Let R → S be flat of finite presentation. The set of primes q such that the fibre ring S_ q ⊗_R kappa( p), with p = R ∩ q is Cohen-Macaulay is open and dense in every fibre of Spec(S) → Spec(R).","statement_latex":"Let $R$ be a ring. Let $R \\to S$ be flat of finite presentation.\nThe set of primes $\\mathfrak q$ such that the fibre ring\n$S_{\\mathfrak q} \\otimes_R \\kappa(\\mathfrak p)$,\nwith $\\mathfrak p = R \\cap \\mathfrak q$ is Cohen-Macaulay\nis open and dense in every fibre of $\\Spec(S) \\to \\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of Cohen-Macaulay loci","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RI","source_file":"algebra.tex","source_line":33683,"source_end_line":33690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33683-L33690","statement_sha256":"f733f1f87059135df7e3d9f27b7d75b0e77bce881ac07560d32bc565999fdc16","origin":"The Stacks Project","memory_eligible":false,"source_rank":1887,"rank":1887,"depth":35,"x":2171.411,"y":256.993,"cluster":"commutative-algebra"},{"id":"stacks:00RJ","tag":"00RJ","title":"Openness of Cohen-Macaulay loci · Lemma 00RJ","summary":"Let k be a field. Let S be a finite type k-algebra. Let K/k be a field extension, and set S_K = K ⊗_k S. Let q ⊂ S be a prime of S. Let q_K ⊂ S_K be a prime of S_K lying over q. Then S_ q is Cohen-Macaulay if and only if (S_K)_ q_K is Cohen-Macaulay.","statement_latex":"Let $k$ be a field. Let $S$ be a finite type $k$-algebra.\nLet $K/k$ be a field extension, and set $S_K = K \\otimes_k S$.\nLet $\\mathfrak q \\subset S$ be a prime of $S$.\nLet $\\mathfrak q_K \\subset S_K$ be a prime of $S_K$ lying\nover $\\mathfrak q$. Then $S_{\\mathfrak q}$ is Cohen-Macaulay\nif and only if $(S_K)_{\\mathfrak q_K}$ is Cohen-Macaulay.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of Cohen-Macaulay loci","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RJ","source_file":"algebra.tex","source_line":33700,"source_end_line":33708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33700-L33708","statement_sha256":"01658dff400f9d1cc38629da87cf97ebbc459d277ea65e015923f13ad049a94f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1888,"rank":1888,"depth":30,"x":1895.908,"y":272.989,"cluster":"commutative-algebra"},{"id":"stacks:00RK","tag":"00RK","title":"Openness of Cohen-Macaulay loci · Lemma 00RK","summary":"Let R be a ring. Let R → S be of finite type. Let R → R' be any ring map. Set S' = R' ⊗_R S. Denote f : Spec(S') → Spec(S) the map associated to the ring map S → S'. Set W equal to the set of primes q such that the fibre ring S_ q ⊗_R kappa( p), p = R ∩ q is Cohen-Macaulay, and let W' denote the analogue for S'/R'. Then W' = f^-1(W).","statement_latex":"Let $R$ be a ring. Let $R \\to S$ be of finite type.\nLet $R \\to R'$ be any ring map. Set $S' = R' \\otimes_R S$.\nDenote $f : \\Spec(S') \\to \\Spec(S)$ the map\nassociated to the ring map $S \\to S'$.\nSet $W$ equal to the\nset of primes $\\mathfrak q$ such that the fibre ring\n$S_{\\mathfrak q} \\otimes_R \\kappa(\\mathfrak p)$,\n$\\mathfrak p = R \\cap \\mathfrak q$ is Cohen-Macaulay,\nand let $W'$ denote the analogue for $S'/R'$. Then\n$W' = f^{-1}(W)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of Cohen-Macaulay loci","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RK","source_file":"algebra.tex","source_line":33738,"source_end_line":33750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33738-L33750","statement_sha256":"fc8803f4dd4f5b282148b635deaee875441aaef58455a9510508aa67a97d1a8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1889,"rank":1889,"depth":31,"x":2086.294,"y":104.781,"cluster":"commutative-algebra"},{"id":"stacks:00RL","tag":"00RL","title":"Openness of Cohen-Macaulay loci · Lemma 00RL","summary":"Let R be a ring. Let R → S be a ring map which is (a) flat, (b) of finite presentation, (c) has Cohen-Macaulay fibres. Then we can write S = S_0 × … × S_n as a product of R-algebras S_d such that each S_d satisfies (a), (b), (c) and has all fibres equidimensional of dimension d.","statement_latex":"Let $R$ be a ring. Let $R \\to S$ be a ring map which is (a) flat,\n(b) of finite presentation, (c) has Cohen-Macaulay fibres. Then we can write\n$S = S_0 \\times \\ldots \\times S_n$ as a product of $R$-algebras $S_d$\nsuch that each $S_d$ satisfies\n(a), (b), (c) and has all fibres equidimensional of dimension $d$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Openness of Cohen-Macaulay loci","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RL","source_file":"algebra.tex","source_line":33756,"source_end_line":33763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33756-L33763","statement_sha256":"1e66f2d15c536bf11e5458cbfb32cd30728a5aa1457109e3e97f27d8328be677","origin":"The Stacks Project","memory_eligible":false,"source_rank":1890,"rank":1890,"depth":35,"x":2081.171,"y":336.963,"cluster":"commutative-algebra"},{"id":"stacks:00RN","tag":"00RN","title":"Differentials · Definition 00RN","summary":"Let φ : R → S be a ring map and let M be an S-module. A derivation, or more precisely an R-derivation into M is a map D : S → M which is additive, annihilates elements of φ(R), and satisfies the Leibniz rule: D(ab) = aD(b) + bD(a).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map and let $M$ be an $S$-module.\nA {\\it derivation}, or more precisely an\n{\\it $R$-derivation} into $M$ is a map $D : S \\to M$\nwhich is additive, annihilates elements of $\\varphi(R)$,\nand satisfies the {\\it Leibniz rule}: $D(ab) = aD(b) + bD(a)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RN","source_file":"algebra.tex","source_line":33794,"source_end_line":33801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33794-L33801","statement_sha256":"b0ba44cf3ec3bd8e874b47079de20c998e7c0d1363fdf8406113ac94b94f796d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1891,"rank":1891,"depth":0,"x":1898.142,"y":162.76,"cluster":"commutative-algebra"},{"id":"stacks:07BK","tag":"07BK","title":"Differentials · Definition 07BK","summary":"The pair (Ω_S/R, d) is called the module of K\\\"ahler differentials or the module of differentials of S over R.","statement_latex":"The pair $(\\Omega_{S/R}, \\text{d})$ is called the {\\it module\nof K\\\"ahler differentials} or the {\\it module of differentials}\nof $S$ over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BK","source_file":"algebra.tex","source_line":33848,"source_end_line":33853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33848-L33853","statement_sha256":"dd5e36485cd27113e4c59a2f56ebf22cb37f8ce0f062be75e4af7f4a8dd30a37","origin":"The Stacks Project","memory_eligible":false,"source_rank":1892,"rank":1892,"depth":0,"x":2173.333,"y":187.372,"cluster":"commutative-algebra"},{"id":"stacks:00RO","tag":"00RO","title":"Differentials · Lemma 00RO","summary":"Maps out of the module of differentials are the same as derivations. The module of differentials of S over R has the following universal property. The map Hom_S(Ω_S/R, M) → Der_R(S, M), α ↦ α ∘ d is an isomorphism of functors.","statement_latex":"\\begin{slogan}\nMaps out of the module of differentials are the same as derivations.\n\\end{slogan}\nThe module of differentials of $S$ over $R$ has the following\nuniversal property. The map\n$$\n\\Hom_S(\\Omega_{S/R}, M)\n\\longrightarrow\n\\text{Der}_R(S, M), \\quad\n\\alpha\n\\longmapsto\n\\alpha \\circ \\text{d}\n$$\nis an isomorphism of functors.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RO","source_file":"algebra.tex","source_line":33855,"source_end_line":33871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33855-L33871","statement_sha256":"bce9208494be91ae51c6f14e1cd07e0404191d328e041130e67ed77d6c8970ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":1893,"rank":1893,"depth":0,"x":1950.507,"y":325.444,"cluster":"commutative-algebra"},{"id":"stacks:00RP","tag":"00RP","title":"Differentials · Lemma 00RP","summary":"Suppose that R → S is surjective. Then Ω_S/R = 0.","statement_latex":"Suppose that $R \\to S$ is surjective.\nThen $\\Omega_{S/R} = 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RP","source_file":"algebra.tex","source_line":33882,"source_end_line":33886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33882-L33886","statement_sha256":"89a0e191f1accd41e42d48dfb95e38fbab8b7b3fc7404eaa086f709a433e7ed4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1894,"rank":1894,"depth":0,"x":2003.809,"y":97.079,"cluster":"commutative-algebra"},{"id":"stacks:031G","tag":"031G","title":"Differentials · Lemma 031G","summary":"Let I be a directed set. Let (R_i → S_i, φ_ii') be a system of ring maps over I, see Categories, Section [Tag 002Z]. Then we have Ω_S/R = colim_i Ω_S_i/R_i. where R → S = colim (R_i → S_i).","statement_latex":"Let $I$ be a directed set.\nLet $(R_i \\to S_i, \\varphi_{ii'})$ be a system of\nring maps over $I$, see\nCategories, Section \\ref{categories-section-posets-limits}.\nThen we have\n$$\n\\Omega_{S/R} =\n\\colim_i \\Omega_{S_i/R_i}.\n$$\nwhere $R \\to S = \\colim (R_i \\to S_i)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031G","source_file":"algebra.tex","source_line":33958,"source_end_line":33970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33958-L33970","statement_sha256":"d7fff0c3e92b7f9dbae4c647309451be52080cab2b33132861fac1d07c68f261","origin":"The Stacks Project","memory_eligible":false,"source_rank":1895,"rank":1895,"depth":0,"x":2148.223,"y":295.817,"cluster":"commutative-algebra"},{"id":"stacks:00RR","tag":"00RR","title":"Differentials · Lemma 00RR","summary":"In diagram ([Tag 00RQ]), suppose that S → S' is surjective with kernel I ⊂ S. Then Ω_S/R → Ω_S'/R' is surjective with kernel generated as an S-module by the elements da, where a ∈ S is such that φ(a) ∈ β(R'). (This includes in particular the elements d(i), i ∈ I.)","statement_latex":"In diagram (\\ref{equation-functorial-omega}), suppose\nthat $S \\to S'$ is surjective with kernel $I \\subset S$.\nThen $\\Omega_{S/R} \\to \\Omega_{S'/R'}$ is surjective with\nkernel generated as an $S$-module by the elements\n$\\text{d}a$, where $a \\in S$ is such that $\\varphi(a) \\in \\beta(R')$.\n(This includes in particular the elements $\\text{d}(i)$, $i \\in I$.)","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RR","source_file":"algebra.tex","source_line":33977,"source_end_line":33985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L33977-L33985","statement_sha256":"44368074cc4cdb04cdee1a71dedc8e3d4d95e518930b35c3fd6e8c18a950138f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1896,"rank":1896,"depth":0,"x":1881.779,"y":231.182,"cluster":"commutative-algebra"},{"id":"stacks:00RS","tag":"00RS","title":"Differentials · Lemma 00RS","summary":"Let A → B → C be ring maps. Then there is a canonical exact sequence C ⊗_B Ω_B/A → Ω_C/A → Ω_C/B → 0 of C-modules.","statement_latex":"Let $A \\to B \\to C$ be ring maps.\nThen there is a canonical exact sequence\n$$\nC \\otimes_B \\Omega_{B/A} \\to\n\\Omega_{C/A} \\to\n\\Omega_{C/B} \\to 0\n$$\nof $C$-modules.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RS","source_file":"algebra.tex","source_line":34056,"source_end_line":34066,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34056-L34066","statement_sha256":"930a9e3b8a8c9f314a7234781e6072bc7fd6fbeb3bb377a5d6385331dc0b0c52","origin":"The Stacks Project","memory_eligible":false,"source_rank":1897,"rank":1897,"depth":1,"x":2130.354,"y":127.604,"cluster":"commutative-algebra"},{"id":"stacks:00RT","tag":"00RT","title":"Differentials · Lemma 00RT","summary":"Let φ : A → B be a ring map. • If S ⊂ A is a multiplicative subset mapping to invertible elements of B, then Ω_B/A = Ω_B/S^-1A. • If S ⊂ B is a multiplicative subset then S^-1Ω_B/A = Ω_S^-1B/A.","statement_latex":"Let $\\varphi : A \\to B$ be a ring map.\n\\begin{enumerate}\n\\item If $S \\subset A$ is a multiplicative subset mapping to\ninvertible elements of $B$, then $\\Omega_{B/A} = \\Omega_{B/S^{-1}A}$.\n\\item If $S \\subset B$ is a multiplicative subset then\n$S^{-1}\\Omega_{B/A} = \\Omega_{S^{-1}B/A}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RT","source_file":"algebra.tex","source_line":34077,"source_end_line":34086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34077-L34086","statement_sha256":"c4c319b4b8038d5f9179952f625bcb8defe060f44151c45118848e82b5825e73","origin":"The Stacks Project","memory_eligible":false,"source_rank":1898,"rank":1898,"depth":0,"x":2030.303,"y":345.138,"cluster":"commutative-algebra"},{"id":"stacks:00RU","tag":"00RU","title":"Differentials · Lemma 00RU","summary":"In diagram ([Tag 00RQ]), suppose that S → S' is surjective with kernel I ⊂ S, and assume that R' = R. Then there is a canonical exact sequence of S'-modules I/I^2 → Ω_S/R ⊗_S S' → Ω_S'/R → 0 The leftmost map is characterized by the rule that f ∈ I maps to df ⊗ 1.","statement_latex":"In diagram (\\ref{equation-functorial-omega}),\nsuppose that $S \\to S'$ is surjective with kernel $I \\subset S$,\nand assume that $R' = R$.\nThen there is a canonical exact sequence of $S'$-modules\n$$\nI/I^2\n\\longrightarrow\n\\Omega_{S/R} \\otimes_S S'\n\\longrightarrow\n\\Omega_{S'/R}\n\\longrightarrow\n0\n$$\nThe leftmost map is characterized by the rule that\n$f \\in I$ maps to $\\text{d}f \\otimes 1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RU","source_file":"algebra.tex","source_line":34102,"source_end_line":34119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34102-L34119","statement_sha256":"b9a1400ba088e62cfa2a179688eb3c7552181c57a59dcc1f9261c0a337d04799","origin":"The Stacks Project","memory_eligible":false,"source_rank":1899,"rank":1899,"depth":1,"x":1929.094,"y":127.851,"cluster":"commutative-algebra"},{"id":"stacks:02HP","tag":"02HP","title":"Differentials · Lemma 02HP","summary":"In diagram ([Tag 00RQ]), suppose that S → S' is surjective with kernel I ⊂ S, and assume that R' = R. Moreover, assume that there exists an R-algebra map S' → S which is a right inverse to S → S'. Then the exact sequence of S'-modules of Lemma [Tag 00RU] turns into a short exact sequence 0 → I/I^2 → Ω_S/R ⊗_S S' → Ω_S'/R → 0 which is even a split short exact sequence.","statement_latex":"In diagram (\\ref{equation-functorial-omega}),\nsuppose that $S \\to S'$ is surjective with kernel $I \\subset S$,\nand assume that $R' = R$. Moreover, assume that there exists\nan $R$-algebra map $S' \\to S$ which is a right inverse to\n$S \\to S'$. Then the exact sequence of $S'$-modules\nof Lemma \\ref{lemma-differential-seq} turns into a short exact sequence\n$$\n0 \\longrightarrow\nI/I^2\n\\longrightarrow\n\\Omega_{S/R} \\otimes_S S'\n\\longrightarrow\n\\Omega_{S'/R}\n\\longrightarrow\n0\n$$\nwhich is even a split short exact sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HP","source_file":"algebra.tex","source_line":34157,"source_end_line":34176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34157-L34176","statement_sha256":"dd24f5795e6cf9286e88c58e128c5241c275a8794c80f290213021e89c20a755","origin":"The Stacks Project","memory_eligible":false,"source_rank":1900,"rank":1900,"depth":2,"x":2178.586,"y":230.698,"cluster":"commutative-algebra"},{"id":"stacks:02HQ","tag":"02HQ","title":"Differentials · Lemma 02HQ","summary":"Let R → S be a ring map. Let I ⊂ S be an ideal. Let n ≥ 1 be an integer. Set S' = S/I^n + 1. The map Ω_S/R → Ω_S'/R induces an isomorphism Ω_S/R ⊗_S S/I^n → Ω_S'/R ⊗_S' S/I^n.","statement_latex":"Let $R \\to S$ be a ring map. Let $I \\subset S$ be an ideal.\nLet $n \\geq 1$ be an integer. Set $S' = S/I^{n + 1}$.\nThe map $\\Omega_{S/R} \\to \\Omega_{S'/R}$ induces an\nisomorphism\n$$\n\\Omega_{S/R} \\otimes_S S/I^n\n\\longrightarrow\n\\Omega_{S'/R} \\otimes_{S'} S/I^n.\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HQ","source_file":"algebra.tex","source_line":34193,"source_end_line":34204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34193-L34204","statement_sha256":"563e506e4b6b1e9fc91e56d49aa97af3b5fc89edf6442793cc94ddafe46bbc76","origin":"The Stacks Project","memory_eligible":false,"source_rank":1901,"rank":1901,"depth":2,"x":1911.773,"y":296.461,"cluster":"commutative-algebra"},{"id":"stacks:00RV","tag":"00RV","title":"Differentials · Lemma 00RV","summary":"Suppose that we have ring maps R → R' and R → S. Set S' = S ⊗_R R', so that we obtain a diagram ([Tag 00RQ]). Then the canonical map defined above induces an isomorphism Ω_S/R ⊗_R R' = Ω_S'/R'.","statement_latex":"Suppose that we have ring maps $R \\to R'$ and $R \\to S$.\nSet $S' = S \\otimes_R R'$, so that we obtain a diagram\n(\\ref{equation-functorial-omega}). Then the canonical map defined above\ninduces an isomorphism $\\Omega_{S/R} \\otimes_R R' = \\Omega_{S'/R'}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RV","source_file":"algebra.tex","source_line":34212,"source_end_line":34218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34212-L34218","statement_sha256":"dbfe1ab68c97240d85d7e85f73be4b8c14cc415f780e2aab4274a22728cfe456","origin":"The Stacks Project","memory_eligible":false,"source_rank":1902,"rank":1902,"depth":0,"x":2055.704,"y":96.472,"cluster":"commutative-algebra"},{"id":"stacks:00RW","tag":"00RW","title":"Differentials · Lemma 00RW","summary":"Let R → S be a ring map. Let J = Ker(S ⊗_R S → S) be the kernel of the multiplication map. There is a canonical isomorphism of S-modules Ω_S/R → J/J^2, a d b ↦ a ⊗ b - ab ⊗ 1.","statement_latex":"Let $R \\to S$ be a ring map. Let $J = \\Ker(S \\otimes_R S \\to S)$\nbe the kernel of the multiplication map. There is a canonical\nisomorphism of $S$-modules $\\Omega_{S/R} \\to J/J^2$,\n$a \\text{d} b \\mapsto a \\otimes b - ab \\otimes 1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RW","source_file":"algebra.tex","source_line":34241,"source_end_line":34247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34241-L34247","statement_sha256":"355e3dd41a9ecae2362b3194ea1f4df95f999d26de5ccdfcfb37bd6ed70b3653","origin":"The Stacks Project","memory_eligible":false,"source_rank":1903,"rank":1903,"depth":3,"x":2110.424,"y":325.726,"cluster":"commutative-algebra"},{"id":"stacks:00RX","tag":"00RX","title":"Differentials · Lemma 00RX","summary":"If S = R[x_1, …, x_n], then Ω_S/R is a finite free S-module with basis dx_1, …, dx_n.","statement_latex":"If $S = R[x_1, \\ldots, x_n]$, then\n$\\Omega_{S/R}$ is a finite free $S$-module with\nbasis $\\text{d}x_1, \\ldots, \\text{d}x_n$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RX","source_file":"algebra.tex","source_line":34309,"source_end_line":34314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34309-L34314","statement_sha256":"70c991834ac901d7323b5cabf5e5132b380153a25ca3c195d888f7b61de4841e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1904,"rank":1904,"depth":0,"x":1885.602,"y":187.658,"cluster":"commutative-algebra"},{"id":"stacks:00RY","tag":"00RY","title":"Differentials · Lemma 00RY","summary":"Suppose R → S is of finite presentation. Then Ω_S/R is a finitely presented S-module.","statement_latex":"Suppose $R \\to S$ is of finite presentation.\nThen $\\Omega_{S/R}$ is a finitely presented\n$S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RY","source_file":"algebra.tex","source_line":34341,"source_end_line":34346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34341-L34346","statement_sha256":"1d06756a9a934a1fddb44d0998178ccb020220f39cd640dffe8e0b9967dff2c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1905,"rank":1905,"depth":2,"x":2162.552,"y":161.885,"cluster":"commutative-algebra"},{"id":"stacks:00RZ","tag":"00RZ","title":"Differentials · Lemma 00RZ","summary":"Suppose R → S is of finite type. Then Ω_S/R is finitely generated S-module.","statement_latex":"Suppose $R \\to S$ is of finite type.\nThen $\\Omega_{S/R}$ is finitely generated\n$S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00RZ","source_file":"algebra.tex","source_line":34368,"source_end_line":34373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34368-L34373","statement_sha256":"e83aac1cef603088ac5e95f0bc0e434efcdcb259c5d035ecfb6442b11d7c898d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1906,"rank":1906,"depth":3,"x":1978.967,"y":338.125,"cluster":"commutative-algebra"},{"id":"stacks:0H8Y","tag":"0H8Y","title":"The de Rham complex · Lemma 0H8Y","summary":"Suppose that we have ring maps A → A' and A → B. Set B' = B ⊗_A A', so that we obtain a diagram as above. Then the canonical map defined above induces an isomorphism Ω^bullet_B/A ⊗_A A' = Ω^bullet_B'/A' of complexes.","statement_latex":"Suppose that we have ring maps $A \\to A'$ and $A \\to B$.\nSet $B' = B \\otimes_A A'$, so that we obtain a diagram as above.\nThen the canonical map defined above\ninduces an isomorphism\n$\\Omega^\\bullet_{B/A} \\otimes_A A' = \\Omega^\\bullet_{B'/A'}$\nof complexes.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The de Rham complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8Y","source_file":"algebra.tex","source_line":34559,"source_end_line":34567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34559-L34567","statement_sha256":"830729f07cd863b12573c3a3693236d07cade56893af523ef09960aebd883a45","origin":"The Stacks Project","memory_eligible":false,"source_rank":1907,"rank":1907,"depth":1,"x":1972.609,"y":103.881,"cluster":"commutative-algebra"},{"id":"stacks:07HY","tag":"07HY","title":"The de Rham complex · Lemma 07HY","summary":"Let A → B be a ring map. Let π : Ω_B/A → Ω be a surjective B-module map. Denote d : B → Ω the composition of π with the universal derivation d_B/A : B → Ω_B/A. Set Ω^i = wedge_B^i(Ω). Assume that the kernel of π is generated, as a B-module, by elements ω ∈ Ω_B/A such that d_B/A(ω) ∈ Ω_B/A^2 maps to zero in Ω^2. Then there is a de Rham complex Ω^0 → Ω^1 → Ω^2 → … whose differential is defined by the rule d : Ω^p → Ω^p + 1, d(f_0df_1 wedge … wedge df_p) = df_0 wedge df_1…","statement_latex":"Let $A \\to B$ be a ring map. Let $\\pi : \\Omega_{B/A} \\to \\Omega$\nbe a surjective $B$-module map. Denote $\\text{d} : B \\to \\Omega$\nthe composition of $\\pi$ with the universal derivation\n$\\text{d}_{B/A} : B \\to \\Omega_{B/A}$. Set $\\Omega^i = \\wedge_B^i(\\Omega)$.\nAssume that the kernel of $\\pi$ is generated, as a $B$-module,\nby elements $\\omega \\in \\Omega_{B/A}$ such that\n$\\text{d}_{B/A}(\\omega) \\in \\Omega_{B/A}^2$ maps to zero in $\\Omega^2$.\nThen there is a de Rham complex\n$$\n\\Omega^0 \\to \\Omega^1 \\to \\Omega^2 \\to \\ldots\n$$\nwhose differential is defined by the rule\n$$\n\\text{d} : \\Omega^p \\to \\Omega^{p + 1},\\quad\n\\text{d}\\left(f_0\\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_p\\right) =\n\\text{d}f_0 \\wedge \\text{d}f_1 \\wedge \\ldots \\wedge \\text{d}f_p\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The de Rham complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HY","source_file":"algebra.tex","source_line":34574,"source_end_line":34593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34574-L34593","statement_sha256":"e95794f439596e590ddb927aed9abbe02e855960307b6997b6e0756f759f8930","origin":"The Stacks Project","memory_eligible":false,"source_rank":1908,"rank":1908,"depth":0,"x":2165.767,"y":273.086,"cluster":"commutative-algebra"},{"id":"stacks:09CI","tag":"09CI","title":"Finite order differential operators · Definition 09CI","summary":"Let R → S be a ring map. Let M, N be S-modules. Let k ≥ 0 be an integer. We inductively define a differential operator D : M → N of order k to be an R-linear map such that for all g ∈ S the map m ↦ D(gm) - gD(m) is a differential operator of order k - 1. For the base case k = 0 we define a differential operator of order 0 to be an S-linear map.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$, $N$ be $S$-modules.\nLet $k \\geq 0$ be an integer. We inductively define a\n{\\it differential operator $D : M \\to N$ of order $k$}\nto be an $R$-linear map such that for all $g \\in S$ the map\n$m \\mapsto D(gm) - gD(m)$ is a differential operator of\norder $k - 1$. For the base case $k = 0$ we define a\ndifferential operator of order $0$ to be an $S$-linear map.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CI","source_file":"algebra.tex","source_line":34657,"source_end_line":34666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34657-L34666","statement_sha256":"15713e7f1a6455d53bc6822e95ddace2111599286203b3064ba2aa5023a13ebb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1909,"rank":1909,"depth":0,"x":1887.127,"y":257.912,"cluster":"commutative-algebra"},{"id":"stacks:09CJ","tag":"09CJ","title":"Finite order differential operators · Lemma 09CJ","summary":"Let R → S be a ring map. Let L, M, N be S-modules. If D : L → M and D' : M → N are differential operators of order k and k', then D' ∘ D is a differential operator of order k + k'.","statement_latex":"Let $R \\to S$ be a ring map. Let $L, M, N$ be $S$-modules.\nIf $D : L \\to M$ and $D' : M \\to N$ are differential\noperators of order $k$ and $k'$, then $D' \\circ D$ is a\ndifferential operator of order $k + k'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CJ","source_file":"algebra.tex","source_line":34683,"source_end_line":34689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34683-L34689","statement_sha256":"d69121c20751cf2d99f707994db81c67cb1add56fca181307ecdc991866e344a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1910,"rank":1910,"depth":0,"x":2104.903,"y":110.921,"cluster":"commutative-algebra"},{"id":"stacks:09CK","tag":"09CK","title":"Finite order differential operators · Lemma 09CK","summary":"Let R → S be a ring map. Let M be an S-module. Let k ≥ 0. There exists an S-module P^k_S/R(M) and a canonical isomorphism Diff^k_S/R(M, N) = Hom_S(P^k_S/R(M), N) functorial in the S-module N.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.\nLet $k \\geq 0$. There exists an $S$-module $P^k_{S/R}(M)$\nand a canonical isomorphism\n$$\n\\text{Diff}^k_{S/R}(M, N) = \\Hom_S(P^k_{S/R}(M), N)\n$$\nfunctorial in the $S$-module $N$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CK","source_file":"algebra.tex","source_line":34700,"source_end_line":34709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34700-L34709","statement_sha256":"af3ad9c8d9f027e78530d14ce3b603ed40f68acd0ef77266c4c924e98b415ff9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1911,"rank":1911,"depth":0,"x":2062.503,"y":342.996,"cluster":"commutative-algebra"},{"id":"stacks:09CL","tag":"09CL","title":"Finite order differential operators · Definition 09CL","summary":"Let R → S be a ring map. Let M be an S-module. The module P^k_S/R(M) constructed in Lemma [Tag 09CK] is called the module of principal parts of order k of M.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. The module\n$P^k_{S/R}(M)$ constructed in Lemma \\ref{lemma-module-principal-parts}\nis called the {\\it module of principal parts of order $k$} of $M$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CL","source_file":"algebra.tex","source_line":34753,"source_end_line":34758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34753-L34758","statement_sha256":"3ac55da607105d5961a47537114980ce2e7019ddb0f37e56e2ff4565b7dce0a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":1912,"rank":1912,"depth":1,"x":1907.062,"y":147.712,"cluster":"commutative-algebra"},{"id":"stacks:09CN","tag":"09CN","title":"Finite order differential operators · Lemma 09CN","summary":"Let R → S be a ring map. Let M be an S-module. There is a canonical short exact sequence 0 → Ω_S/R ⊗_S M → P^1_S/R(M) → M → 0 functorial in M called the sequence of principal parts.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $S$-module. There is a\ncanonical short exact sequence\n$$\n0 \\to \\Omega_{S/R} \\otimes_S M \\to P^1_{S/R}(M) \\to M \\to 0\n$$\nfunctorial in $M$ called the {\\it sequence of principal parts}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CN","source_file":"algebra.tex","source_line":34786,"source_end_line":34794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34786-L34794","statement_sha256":"0ccfbd2f021a0470e6c0e571d35e22e0b445474d76d380a93beb6e374c61685e","origin":"The Stacks Project","memory_eligible":false,"source_rank":1913,"rank":1913,"depth":1,"x":2178.859,"y":203.538,"cluster":"commutative-algebra"},{"id":"stacks:0H8Z","tag":"0H8Z","title":"Finite order differential operators · Lemma 0H8Z","summary":"Suppose that we have ring maps A → A' and A → B and a B-module M. Set B' = B ⊗_A A' and view M' = M ⊗_A A' as a B'-module. The map of Remark [Tag 09CP] induces an isomorphism P^k_B/A(M) ⊗_A A' = P^k_B'/A'(M').","statement_latex":"Suppose that we have ring maps $A \\to A'$ and $A \\to B$ and a\n$B$-module $M$. Set $B' = B \\otimes_A A'$ and view $M' = M \\otimes_A A'$\nas a $B'$-module.\nThe map of Remark \\ref{remark-functoriality-principal-parts}\ninduces an isomorphism $P^k_{B/A}(M) \\otimes_A A' = P^k_{B'/A'}(M')$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8Z","source_file":"algebra.tex","source_line":34904,"source_end_line":34911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34904-L34911","statement_sha256":"46fbcc71990bb682d5f2e39d0451c51d7a3723c8a2db7f0aea4112a912880128","origin":"The Stacks Project","memory_eligible":false,"source_rank":1914,"rank":1914,"depth":0,"x":1933.424,"y":316.653,"cluster":"commutative-algebra"},{"id":"stacks:0H90","tag":"0H90","title":"Finite order differential operators · Lemma 0H90","summary":"Let R → S be a ring map. Let M be an S-module. Let J = Ker(S ⊗_R S → S) be the kernel of the multiplication map. There is a canonical isomorphism of S-modules P^k_S/R(M) → (S ⊗_R M)/J^k + 1(S ⊗_R M) where s ∈ S acts on the target via multiplication by s ⊗ 1 and such that the universal differential operators of order k to the map given by m ↦ class of 1 ⊗ m.","statement_latex":"Let $R \\to S$ be a ring map. Let $M$ be an $S$-module.\nLet $J = \\Ker(S \\otimes_R S \\to S)$ be the kernel of the multiplication map. \nThere is a canonical isomorphism of $S$-modules\n$$\nP^k_{S/R}(M) \\longrightarrow (S \\otimes_R M)/J^{k + 1}(S \\otimes_R M)\n$$\nwhere $s \\in S$ acts on the target via multiplication by $s \\otimes 1$\nand such that the universal differential operators of order $k$\nto the map given by $m \\mapsto \\text{class of }1 \\otimes m$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H90","source_file":"algebra.tex","source_line":34929,"source_end_line":34940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34929-L34940","statement_sha256":"c08c8181e76a76ff110230532452699842dbadb0f7c2439bcb0b2af09d208f84","origin":"The Stacks Project","memory_eligible":false,"source_rank":1915,"rank":1915,"depth":1,"x":2023.485,"y":93.868,"cluster":"commutative-algebra"},{"id":"stacks:0G34","tag":"0G34","title":"Finite order differential operators · Lemma 0G34","summary":"Let A → B be a ring map. The differentials d : Ω^i_B/A → Ω^i + 1_B/A are differential operators of order 1.","statement_latex":"Let $A \\to B$ be a ring map. The differentials\n$\\text{d} : \\Omega^i_{B/A}  \\to \\Omega^{i + 1}_{B/A}$\nare differential operators of order $1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G34","source_file":"algebra.tex","source_line":34972,"source_end_line":34977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34972-L34977","statement_sha256":"521fa2715a373e4eda5f4ed3046f385596b5d124ade2e0f8ce9b5972617616cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1916,"rank":1916,"depth":0,"x":2136.289,"y":309.355,"cluster":"commutative-algebra"},{"id":"stacks:0G35","tag":"0G35","title":"Finite order differential operators · Lemma 0G35","summary":"Let A → B be a ring map. Let g_i ∈ B, i ∈ I be a set of generators for B as an A-algebra. Let M, N be B-modules. Let D : M → N be an A-linear map. In order to show that D is a differential operator of order k it suffices to show that D ∘ g_i - g_i ∘ D is a differential operator of order k - 1 for i ∈ I.","statement_latex":"Let $A \\to B$ be a ring map. Let $g_i \\in B$, $i \\in I$ be a set of generators\nfor $B$ as an $A$-algebra. Let $M, N$ be $B$-modules.\nLet $D : M \\to N$ be an $A$-linear map. In order to show\nthat $D$ is a differential operator of order $k$ it suffices\nto show that $D \\circ g_i - g_i \\circ D$ is a differential\noperator of order $k - 1$ for $i \\in I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G35","source_file":"algebra.tex","source_line":34998,"source_end_line":35006,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L34998-L35006","statement_sha256":"cf61bb8c1461d1294394f09b98914b5d830d4073f075f52c4f28092f72e55ac4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1917,"rank":1917,"depth":1,"x":1879.693,"y":214.419,"cluster":"commutative-algebra"},{"id":"stacks:0G36","tag":"0G36","title":"Finite order differential operators · Lemma 0G36","summary":"Let A → B be a ring map. Let M, N be B-modules. Let S ⊂ B be a multiplicative subset. Any differential operator D : M → N of order k extends uniquely to a differential operator E : S^-1M → S^-1N of order k.","statement_latex":"Let $A \\to B$ be a ring map. Let $M, N$ be $B$-modules.\nLet $S \\subset B$ be a multiplicative subset. Any differential operator\n$D : M \\to N$ of order $k$ extends uniquely to a differential operator\n$E : S^{-1}M \\to S^{-1}N$ of order $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G36","source_file":"algebra.tex","source_line":35025,"source_end_line":35031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35025-L35031","statement_sha256":"bddee8a076027fbd4a09fcbdac9e6e0bb4476920a931d521184f6d4605cdf0bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1918,"rank":1918,"depth":2,"x":2145.379,"y":138.788,"cluster":"commutative-algebra"},{"id":"stacks:0G37","tag":"0G37","title":"Finite order differential operators · Lemma 0G37","summary":"Let R → A and R → B be ring maps. Let M and M' be A-modules. Let D : M → M' be a differential operator of order k with respect to R → A. Let N be any B-module. Then the map D ⊗ id_N : M ⊗_R N → M' ⊗_R N is a differential operator of order k with respect to B → A ⊗_R B.","statement_latex":"Let $R \\to A$ and $R \\to B$ be ring maps. Let $M$ and $M'$ be $A$-modules.\nLet $D : M \\to M'$ be a differential operator of order $k$ with respect to\n$R \\to A$. Let $N$ be any $B$-module. Then the map\n$$\nD \\otimes \\text{id}_N : M \\otimes_R N \\to M' \\otimes_R N\n$$\nis a differential operator of order $k$ with respect to $B \\to A \\otimes_R B$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G37","source_file":"algebra.tex","source_line":35088,"source_end_line":35097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35088-L35097","statement_sha256":"41f7a2df0efa12e4bf4872f59b01b53d4148b754fe14a604d95d2cc77f0eafca","origin":"The Stacks Project","memory_eligible":false,"source_rank":1919,"rank":1919,"depth":2,"x":2010.22,"y":345.416,"cluster":"commutative-algebra"},{"id":"stacks:07BN","tag":"07BN","title":"The naive cotangent complex · Definition 07BN","summary":"Let R → S be a ring map. The naive cotangent complex NL_S/R is the chain complex ([Tag 07BM]) NL_S/R = (I/I^2 → Ω_R[S]/R ⊗_R[S] S) with I/I^2 placed in (homological) degree 1 and Ω_R[S]/R ⊗_R[S] S placed in degree 0. We will denote H_1(L_S/R) = H_1(NL_S/R) in the literature. the homology in degree 1.","statement_latex":"Let $R \\to S$ be a ring map. The {\\it naive cotangent complex}\n$\\NL_{S/R}$ is the chain complex (\\ref{equation-naive-cotangent-complex})\n$$\n\\NL_{S/R} = \\left(I/I^2 \\longrightarrow \\Omega_{R[S]/R} \\otimes_{R[S]} S\\right)\n$$\nwith $I/I^2$ placed in (homological) degree $1$ and\n$\\Omega_{R[S]/R} \\otimes_{R[S]} S$ placed in degree $0$. We will denote\n$H_1(L_{S/R}) = H_1(\\NL_{S/R})$\\footnote{This module is sometimes\ndenoted $\\Gamma_{S/R}$ in the literature.} the homology in degree $1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BN","source_file":"algebra.tex","source_line":35143,"source_end_line":35154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35143-L35154","statement_sha256":"9d8fe3202f67a6da4eec696f2f7c2e3192776a0c8104277e572652c4a3ed0869","origin":"The Stacks Project","memory_eligible":false,"source_rank":1920,"rank":1920,"depth":0,"x":1943.687,"y":116.246,"cluster":"commutative-algebra"},{"id":"stacks:00S1","tag":"00S1","title":"The naive cotangent complex · Lemma 00S1","summary":"Suppose given a diagram ([Tag 06RQ]). Let α : P → S and α' : P' → S' be presentations. • There exists a morphism of presentations from α to α'. • Any two morphisms of presentations induce homotopic morphisms of complexes NL(α) → NL(α'). • The construction is compatible with compositions of morphisms of presentations (see proof for exact statement). • If R → R' and S → S' are isomorphisms, then for any map φ of presentations from α to α' the induced map NL(α) → NL(α') is a…","statement_latex":"Suppose given a diagram (\\ref{equation-functoriality-NL}).\nLet $\\alpha : P \\to S$ and $\\alpha' : P' \\to S'$ be presentations.\n\\begin{enumerate}\n\\item There exists a morphism of presentations from $\\alpha$ to $\\alpha'$.\n\\item Any two morphisms of presentations induce homotopic\nmorphisms of complexes $\\NL(\\alpha) \\to \\NL(\\alpha')$.\n\\item The construction is compatible with compositions of morphisms\nof presentations (see proof for exact statement).\n\\item If $R \\to R'$ and $S \\to S'$ are isomorphisms, then\nfor any map $\\varphi$ of presentations from $\\alpha$ to $\\alpha'$\nthe induced map $\\NL(\\alpha) \\to \\NL(\\alpha')$ is a homotopy equivalence\nand a quasi-isomorphism.\n\\end{enumerate}\nIn particular, comparing $\\alpha$ to the canonical presentation\n(\\ref{equation-canonical-presentation}) we conclude there is a\nquasi-isomorphism $\\NL(\\alpha) \\to \\NL_{S/R}$ well defined\nup to homotopy and compatible with all functorialities (up to homotopy).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00S1","source_file":"algebra.tex","source_line":35273,"source_end_line":35292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35273-L35292","statement_sha256":"7f1458280dfe335b9fefca39b4322a00c7b806dfa40198e5383998a05e5d44d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1921,"rank":1921,"depth":0,"x":2177.154,"y":247.544,"cluster":"commutative-algebra"},{"id":"stacks:08Q1","tag":"08Q1","title":"The naive cotangent complex · Lemma 08Q1","summary":"Let A → B be a polynomial algebra. Then NL_B/A is homotopy equivalent to the chain complex (0 → Ω_B/A) with Ω_B/A in degree 0.","statement_latex":"Let $A \\to B$ be a polynomial algebra. Then $\\NL_{B/A}$ is homotopy equivalent\nto the chain complex $(0 \\to \\Omega_{B/A})$ with $\\Omega_{B/A}$\nin degree $0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Q1","source_file":"algebra.tex","source_line":35390,"source_end_line":35395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35390-L35395","statement_sha256":"d4ee625c87d5525dbaf4aa642c4eb7d21d9b6a1d25861d34a6cce0ac09f7833c","origin":"The Stacks Project","memory_eligible":false,"source_rank":1922,"rank":1922,"depth":1,"x":1899.276,"y":283.219,"cluster":"commutative-algebra"},{"id":"stacks:00S2","tag":"00S2","title":"Jacobi-Zariski sequence · Lemma 00S2","summary":"Let A → B → C be ring maps. Choose a presentation α : A[x_s, s ∈ S] → B with kernel I. Choose a presentation β : B[y_t, t ∈ T] → C with kernel J. Let γ : A[x_s, y_t] → C be the induced presentation of C with kernel K. Then we get a canonical commutative diagram xymatrix 0 ar[r] & Ω_A[x_s]/A ⊗ C ar[r] & Ω_A[x_s, y_t]/A ⊗ C ar[r] & Ω_B[y_t]/B ⊗ C ar[r] & 0 & I/I^2 ⊗ C ar[r] ar[u] & K/K^2 ar[r] ar[u] & J/J^2 ar[r] ar[u] & 0 with exact rows. We get the following exact…","statement_latex":"Let $A \\to B \\to C$ be ring maps. Choose a presentation\n$\\alpha : A[x_s, s \\in S] \\to B$ with kernel $I$. Choose a presentation\n$\\beta : B[y_t, t \\in T] \\to C$ with kernel $J$. Let\n$\\gamma : A[x_s, y_t] \\to C$ be the induced presentation of $C$ with kernel\n$K$. Then we get a canonical commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\n\\Omega_{A[x_s]/A} \\otimes C \\ar[r] &\n\\Omega_{A[x_s, y_t]/A} \\otimes C \\ar[r] &\n\\Omega_{B[y_t]/B} \\otimes C \\ar[r] &\n0 \\\\\n&\nI/I^2 \\otimes C \\ar[r] \\ar[u] &\nK/K^2 \\ar[r] \\ar[u] &\nJ/J^2 \\ar[r] \\ar[u] &\n0\n}\n$$\nwith exact rows. We get the following exact sequence\nof homology groups\n$$\nH_1(\\NL_{B/A} \\otimes_B C) \\to\nH_1(L_{C/A}) \\to\nH_1(L_{C/B}) \\to\nC \\otimes_B \\Omega_{B/A} \\to\n\\Omega_{C/A} \\to\n\\Omega_{C/B} \\to 0\n$$\nof $C$-modules extending the sequence of\nLemma \\ref{lemma-exact-sequence-differentials}.\nIf $\\text{Tor}_1^B(\\Omega_{B/A}, C) = 0$ and\n$\\text{Tor}_2^B(\\Omega_{B/A}, C) = 0$, then\n$H_1(\\NL_{B/A} \\otimes_B C) = H_1(L_{B/A}) \\otimes_B C$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00S2","source_file":"algebra.tex","source_line":35411,"source_end_line":35447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35411-L35447","statement_sha256":"eee0cba84ebc19b88892fa415a97498a182c37923cf8593debe27a36b40086f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1923,"rank":1923,"depth":2,"x":2075.577,"y":99.148,"cluster":"commutative-algebra"},{"id":"stacks:07BP","tag":"07BP","title":"The naive cotangent complex · Lemma 07BP","summary":"Let A → B be a surjective ring map with kernel I. Then NL_B/A is homotopy equivalent to the chain complex (I/I^2 → 0) with I/I^2 in degree 1. In particular H_1(L_B/A) = I/I^2.","statement_latex":"Let $A \\to B$ be a surjective ring map with kernel $I$.\nThen $\\NL_{B/A}$ is homotopy equivalent to the chain complex\n$(I/I^2 \\to 0)$ with $I/I^2$ in degree $1$. In particular\n$H_1(L_{B/A}) = I/I^2$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BP","source_file":"algebra.tex","source_line":35505,"source_end_line":35511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35505-L35511","statement_sha256":"56073b0c0741157cb0b6f7df1c24908a6fd1177d8cd26991431af54d28a6007f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1924,"rank":1924,"depth":1,"x":2093.609,"y":335.033,"cluster":"commutative-algebra"},{"id":"stacks:065V","tag":"065V","title":"The naive cotangent complex · Lemma 065V","summary":"Let A → B → C be ring maps. Assume A → C is surjective (so also B → C is). Denote I = Ker(A → C) and J = Ker(B → C). Then the sequence I/I^2 → J/J^2 → Ω_B/A ⊗_B B/J → 0 is exact.","statement_latex":"Let $A \\to B \\to C$ be ring maps. Assume $A \\to C$ is surjective (so\nalso $B \\to C$ is). Denote $I = \\Ker(A \\to C)$ and\n$J = \\Ker(B \\to C)$. Then the sequence\n$$\nI/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0\n$$\nis exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065V","source_file":"algebra.tex","source_line":35518,"source_end_line":35527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35518-L35527","statement_sha256":"ba822db42861c5afef242ade71553d5b41c8d15b52c9f10bdbf27623077114ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":1925,"rank":1925,"depth":3,"x":1890.521,"y":171.246,"cluster":"commutative-algebra"},{"id":"stacks:00S4","tag":"00S4","title":"Flat base change · Lemma 00S4","summary":"Let R → S be a ring map. Let α : P → S be a presentation. Let R → R' be a flat ring map. Let α' : P ⊗_R R' → S' = S ⊗_R R' be the induced presentation. Then NL(α) ⊗_R R' = NL(α) ⊗_S S' = NL(α'). In particular, the canonical map NL_S/R ⊗_S S' → NL_S ⊗_R R'/R' is a homotopy equivalence if R → R' is flat.","statement_latex":"Let $R \\to S$ be a ring map. Let $\\alpha : P \\to S$ be a presentation.\nLet $R \\to R'$ be a flat ring map.\nLet $\\alpha' : P \\otimes_R R' \\to S' = S \\otimes_R R'$\nbe the induced presentation.\nThen $\\NL(\\alpha) \\otimes_R R' = \\NL(\\alpha) \\otimes_S S' = \\NL(\\alpha')$.\nIn particular, the canonical map\n$$\n\\NL_{S/R} \\otimes_S S' \\longrightarrow \\NL_{S \\otimes_R R'/R'}\n$$\nis a homotopy equivalence if $R \\to R'$ is flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00S4","source_file":"algebra.tex","source_line":35536,"source_end_line":35548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35536-L35548","statement_sha256":"be260820b1f149542707abceb0b21cb0eee2f716366d65eed6b85bdd51c842b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1926,"rank":1926,"depth":0,"x":2172.125,"y":176.785,"cluster":"commutative-algebra"},{"id":"stacks:07BQ","tag":"07BQ","title":"The naive cotangent complex · Lemma 07BQ","summary":"Let R_λ → S_λ be a system of ring maps over the directed set Lambda. Set R = colim R_λ and S = colim S_λ. Then NL_S/R = colim NL_S_λ/R_λ.","statement_latex":"Let $R_\\lambda \\to S_\\lambda$ be a system of ring maps over\nthe directed set $\\Lambda$.\nSet $R = \\colim R_\\lambda$ and $S = \\colim S_\\lambda$.\nThen $\\NL_{S/R} = \\colim \\NL_{S_\\lambda/R_\\lambda}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BQ","source_file":"algebra.tex","source_line":35556,"source_end_line":35562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35556-L35562","statement_sha256":"adf83069908fc8a20537d5b35c73f2f20535cff7138cce5ad4f2c7396d52a636","origin":"The Stacks Project","memory_eligible":false,"source_rank":1927,"rank":1927,"depth":0,"x":1959.917,"y":332.566,"cluster":"commutative-algebra"},{"id":"stacks:07BR","tag":"07BR","title":"The naive cotangent complex · Lemma 07BR","summary":"If S ⊂ A is a multiplicative subset of A, then NL_S^-1A/A is homotopy equivalent to the zero complex.","statement_latex":"If $S \\subset A$ is a multiplicative subset of $A$, then\n$\\NL_{S^{-1}A/A}$ is homotopy equivalent to the zero complex.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BR","source_file":"algebra.tex","source_line":35574,"source_end_line":35578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35574-L35578","statement_sha256":"a5d25f547fb41bcb4bee8f23a35461b8092baa45cb75726e497ccd36ac1c42b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1928,"rank":1928,"depth":2,"x":1991.137,"y":97.169,"cluster":"commutative-algebra"},{"id":"stacks:07BS","tag":"07BS","title":"The naive cotangent complex · Lemma 07BS","summary":"Let S ⊂ A is a multiplicative subset of A. Let S^-1A → B be a ring map. Then NL_B/A → NL_B/S^-1A is a homotopy equivalence.","statement_latex":"Let $S \\subset A$ is a multiplicative subset of $A$.\nLet $S^{-1}A \\to B$ be a ring map.\nThen $\\NL_{B/A} \\to \\NL_{B/S^{-1}A}$ is a homotopy equivalence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BS","source_file":"algebra.tex","source_line":35589,"source_end_line":35594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35589-L35594","statement_sha256":"43cb48aaa030128bc80f6ee70d0548e364dcfa621547382d3749113d9fcaea8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1929,"rank":1929,"depth":1,"x":2157.497,"y":288.556,"cluster":"commutative-algebra"},{"id":"stacks:08JZ","tag":"08JZ","title":"The naive cotangent complex · Lemma 08JZ","summary":"The formation of the naive cotangent complex commutes with localization at an element. Let A → B be a ring map. Let g ∈ B. Suppose α : P → B is a presentation with kernel I. Then a presentation of B_g over A is the map β : P[x] → B_g extending α and sending x to 1/g. The kernel J of β is generated by I and the element f x - 1 where f ∈ P is an element mapped to g ∈ B by α. In this situation we have • J/J^2 = (I/I^2)_g ⊕ B_g (f x - 1), • Ω_P[x]/A ⊗_P[x] B_g = Ω_P/A ⊗_P B_g…","statement_latex":"\\begin{slogan}\nThe formation of the naive cotangent complex commutes with localization\nat an element.\n\\end{slogan}\nLet $A \\to B$ be a ring map. Let $g \\in B$. Suppose $\\alpha : P \\to B$\nis a presentation with kernel $I$. Then a presentation of $B_g$ over $A$ is\nthe map\n$$\n\\beta : P[x] \\longrightarrow B_g\n$$\nextending $\\alpha$ and sending $x$ to $1/g$.\nThe kernel $J$ of $\\beta$ is generated by $I$ and the element $f x - 1$\nwhere $f \\in P$ is an element mapped to $g \\in B$ by $\\alpha$. In this\nsituation we have\n\\begin{enumerate}\n\\item $J/J^2 = (I/I^2)_g \\oplus B_g (f x - 1)$,\n\\item $\\Omega_{P[x]/A} \\otimes_{P[x]} B_g =\n\\Omega_{P/A} \\otimes_P B_g \\oplus B_g \\text{d}x$,\n\\item $\\NL(\\beta) \\cong\n\\NL(\\alpha) \\otimes_B B_g \\oplus (B_g \\xrightarrow{g} B_g)$\n\\end{enumerate}\nHence the canonical map $\\NL_{B/A} \\otimes_B B_g \\to \\NL_{B_g/A}$\nis a homotopy equivalence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JZ","source_file":"algebra.tex","source_line":35604,"source_end_line":35629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35604-L35629","statement_sha256":"a3926be371ef56fdca681a0897f9080d90bce4d8052970cb1d8f4e1ee7a29cc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1930,"rank":1930,"depth":3,"x":1880.783,"y":241.804,"cluster":"commutative-algebra"},{"id":"stacks:00S7","tag":"00S7","title":"The naive cotangent complex · Lemma 00S7","summary":"Let A → B be a ring map. Let S ⊂ B be a multiplicative subset. The canonical map NL_B/A ⊗_B S^-1B → NL_S^-1B/A is a quasi-isomorphism.","statement_latex":"Let $A \\to B$ be a ring map. Let $S \\subset B$ be a multiplicative subset.\nThe canonical map $\\NL_{B/A} \\otimes_B S^{-1}B \\to \\NL_{S^{-1}B/A}$\nis a quasi-isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00S7","source_file":"algebra.tex","source_line":35691,"source_end_line":35696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35691-L35696","statement_sha256":"1d6245d5b625a29add3416693bc1ed75fffbc065f6e63fba52114cba4d3b60b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1931,"rank":1931,"depth":4,"x":2122.542,"y":119.204,"cluster":"commutative-algebra"},{"id":"stacks:00S3","tag":"00S3","title":"The naive cotangent complex · Lemma 00S3","summary":"Let R be a ring. Let A_1 → A_0, and B_1 → B_0 be two term complexes. Suppose that there exist morphisms of complexes φ : A_bullet → B_bullet and ψ : B_bullet → A_bullet such that φ ∘ ψ and ψ ∘ φ are homotopic to the identity maps. Then A_1 ⊕ B_0 ≅ B_1 ⊕ A_0 as R-modules.","statement_latex":"Let $R$ be a ring.\nLet $A_1 \\to A_0$, and $B_1 \\to B_0$ be\ntwo term complexes. Suppose that there exist\nmorphisms of complexes $\\varphi : A_\\bullet \\to B_\\bullet$\nand $\\psi : B_\\bullet \\to A_\\bullet$ such that\n$\\varphi \\circ \\psi$ and $\\psi \\circ \\varphi$ are\nhomotopic to the identity maps.\nThen $A_1 \\oplus B_0 \\cong B_1 \\oplus A_0$ as\n$R$-modules.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00S3","source_file":"algebra.tex","source_line":35708,"source_end_line":35719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35708-L35719","statement_sha256":"dc019cb4d91e93bdca1193942264ad5c1777204fcfa1f8d1628395a11e6c8c2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":1932,"rank":1932,"depth":0,"x":2042.825,"y":346.898,"cluster":"commutative-algebra"},{"id":"stacks:00S5","tag":"00S5","title":"The naive cotangent complex · Lemma 00S5","summary":"Let R → S be a ring map of finite type. For any presentations α : R[x_1, …, x_n] → S, and β : R[y_1, …, y_m] → S we have I/I^2 ⊕ S^⊕ m ≅ J/J^2 ⊕ S^⊕ n as S-modules where I = Ker(α) and J = Ker(β).","statement_latex":"Let $R \\to S$ be a ring map of finite type.\nFor any presentations $\\alpha : R[x_1, \\ldots, x_n] \\to S$, and\n$\\beta : R[y_1, \\ldots, y_m] \\to S$ we have\n$$\nI/I^2 \\oplus S^{\\oplus m} \\cong J/J^2 \\oplus S^{\\oplus n}\n$$\nas $S$-modules where $I = \\Ker(\\alpha)$ and $J = \\Ker(\\beta)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00S5","source_file":"algebra.tex","source_line":35760,"source_end_line":35769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35760-L35769","statement_sha256":"fdbec440c26f3172bafc8bb4676fcd8a7a58601f2971b133f9dd4262b19a835f","origin":"The Stacks Project","memory_eligible":false,"source_rank":1933,"rank":1933,"depth":1,"x":1918.441,"y":133.663,"cluster":"commutative-algebra"},{"id":"stacks:00S6","tag":"00S6","title":"The naive cotangent complex · Lemma 00S6","summary":"Let R → S be a ring map of finite type. Let g ∈ S. For any presentations α : R[x_1, …, x_n] → S, and β : R[y_1, …, y_m] → S_g we have (I/I^2)_g ⊕ S^⊕ m_g ≅ J/J^2 ⊕ S_g^⊕ n as S_g-modules where I = Ker(α) and J = Ker(β).","statement_latex":"Let $R \\to S$ be a ring map of finite type.\nLet $g \\in S$. For any presentations\n$\\alpha : R[x_1, \\ldots, x_n] \\to S$, and\n$\\beta : R[y_1, \\ldots, y_m] \\to S_g$ we have\n$$\n(I/I^2)_g \\oplus S^{\\oplus m}_g \\cong J/J^2 \\oplus S_g^{\\oplus n}\n$$\nas $S_g$-modules where\n$I = \\Ker(\\alpha)$ and $J = \\Ker(\\beta)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00S6","source_file":"algebra.tex","source_line":35775,"source_end_line":35786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35775-L35786","statement_sha256":"dddb787742899356f3689616d46b589d10ed6d72d51f3c3cdf3dc4fc5f4512eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1934,"rank":1934,"depth":4,"x":2181.765,"y":220.363,"cluster":"commutative-algebra"},{"id":"stacks:00S9","tag":"00S9","title":"Local complete intersections · Definition 00S9","summary":"Let k be a field. Let S be a finite type k-algebra. • We say that S is a global complete intersection over k if there exists a presentation S = k[x_1, …, x_n]/(f_1, …, f_c) such that dim(S) = n - c. • We say that S is a local complete intersection over k if there exists a covering Spec(S) = ⋃ D(g_i) such that each of the rings S_g_i is a global complete intersection over k. We will also use the convention that the zero ring is a global complete intersection over k.","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra.\n\\begin{enumerate}\n\\item We say that $S$ is a {\\it global complete intersection over $k$}\nif there exists a presentation $S = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$\nsuch that $\\dim(S) = n - c$.\n\\item We say that $S$ is a {\\it local complete intersection over $k$}\nif there exists a covering $\\Spec(S) = \\bigcup D(g_i)$ such\nthat each of the rings $S_{g_i}$ is a global complete intersection\nover $k$.\n\\end{enumerate}\nWe will also use the convention that the zero ring is a global\ncomplete intersection over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00S9","source_file":"algebra.tex","source_line":35816,"source_end_line":35831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35816-L35831","statement_sha256":"aa2f36d6aff96cb89439c9cf4d676b247177cd742f64337cfe0a12efa3f3b3a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1935,"rank":1935,"depth":0,"x":1917.744,"y":305.889,"cluster":"commutative-algebra"},{"id":"stacks:00SA","tag":"00SA","title":"Local complete intersections · Lemma 00SA","summary":"Let k be a field. Let S be a finite type k-algebra. Let g ∈ S. • If S is a global complete intersection so is S_g. • If S is a local complete intersection so is S_g.","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra.\nLet $g \\in S$.\n\\begin{enumerate}\n\\item If $S$ is a global complete intersection so is $S_g$.\n\\item If $S$ is a local complete intersection so is $S_g$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SA","source_file":"algebra.tex","source_line":35851,"source_end_line":35860,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35851-L35860","statement_sha256":"b462df4235c75c008b85685f8f455e9bea9041432634e277958c8cbecd66af21","origin":"The Stacks Project","memory_eligible":false,"source_rank":1936,"rank":1936,"depth":1,"x":2043.712,"y":92.909,"cluster":"commutative-algebra"},{"id":"stacks:00SB","tag":"00SB","title":"Local complete intersections · Lemma 00SB","summary":"Let k be a field. Let S be a finite type k-algebra. If S is a local complete intersection, then S is a Cohen-Macaulay ring.","statement_latex":"Let $k$ be a field. Let $S$ be a finite type $k$-algebra.\nIf $S$ is a local complete intersection, then\n$S$ is a Cohen-Macaulay ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SB","source_file":"algebra.tex","source_line":35875,"source_end_line":35880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35875-L35880","statement_sha256":"9516d40f258e9bd10f0a1200ba8bf5ecfdb0ae582d986455f55243836b174e75","origin":"The Stacks Project","memory_eligible":false,"source_rank":1937,"rank":1937,"depth":19,"x":2122.137,"y":321.545,"cluster":"commutative-algebra"},{"id":"stacks:00SC","tag":"00SC","title":"Local complete intersections · Lemma 00SC","summary":"Let k be a field. Let S be a finite type k-algebra. Let q be a prime of S. Choose any presentation S = k[x_1, …, x_n]/I. Let q' be the prime of k[x_1, …, x_n] corresponding to q. Set c = height( q') - height( q), in other words dim_ q(S) = n - c (see Lemma [Tag 00P2]). The following are equivalent • There exists a g ∈ S, g not ∈ q such that S_g is a global complete intersection over k. • The ideal I_ q' ⊂ k[x_1, …, x_n]_ q' can be generated by c elements. • The conormal…","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra.\nLet $\\mathfrak q$ be a prime of $S$.\nChoose any presentation $S = k[x_1, \\ldots, x_n]/I$.\nLet $\\mathfrak q'$ be the prime of $k[x_1, \\ldots, x_n]$ corresponding\nto $\\mathfrak q$. Set\n$c = \\text{height}(\\mathfrak q') - \\text{height}(\\mathfrak q)$,\nin other words $\\dim_{\\mathfrak q}(S) = n - c$\n(see Lemma \\ref{lemma-codimension}). The following are equivalent\n\\begin{enumerate}\n\\item There exists a $g \\in S$, $g \\not \\in \\mathfrak q$\nsuch that $S_g$ is a global complete intersection over $k$.\n\\item The ideal $I_{\\mathfrak q'} \\subset k[x_1, \\ldots, x_n]_{\\mathfrak q'}$\ncan be generated by $c$ elements.\n\\item The conormal module $(I/I^2)_{\\mathfrak q}$ can be generated by\n$c$ elements over $S_{\\mathfrak q}$.\n\\item The conormal module $(I/I^2)_{\\mathfrak q}$ is a free\n$S_{\\mathfrak q}$-module of rank $c$.\n\\item The ideal $I_{\\mathfrak q'}$ can be generated by a regular sequence\nin the regular local ring $k[x_1, \\ldots, x_n]_{\\mathfrak q'}$.\n\\end{enumerate}\nIn this case any $c$ elements of $I_{\\mathfrak q'}$\nwhich generate $I_{\\mathfrak q'}/\\mathfrak q'I_{\\mathfrak q'}$\nform a regular sequence in the local\nring $k[x_1, \\ldots, x_n]_{\\mathfrak q'}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SC","source_file":"algebra.tex","source_line":35908,"source_end_line":35935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L35908-L35935","statement_sha256":"83b0b3a5cfd4b7cd27b8a67855e579b853eefd64425569afbb242a7fb5612f15","origin":"The Stacks Project","memory_eligible":false,"source_rank":1938,"rank":1938,"depth":24,"x":1880.328,"y":197.392,"cluster":"commutative-algebra"},{"id":"stacks:00SD","tag":"00SD","title":"Local complete intersections · Definition 00SD","summary":"Let k be a field. Let S be a local k-algebra essentially of finite type over k. We say S is a complete intersection (over k) if there exists a local k-algebra R and elements f_1, …, f_c ∈ m_R such that • R is essentially of finite type over k, • R is a regular local ring, • f_1, …, f_c form a regular sequence in R, and • S ≅ R/(f_1, …, f_c) as k-algebras.","statement_latex":"Let $k$ be a field. Let $S$ be a local $k$-algebra essentially of finite type\nover $k$. We say $S$ is a {\\it complete intersection (over $k$)}\nif there exists a local $k$-algebra $R$ and elements\n$f_1, \\ldots, f_c \\in \\mathfrak m_R$ such that\n\\begin{enumerate}\n\\item $R$ is essentially of finite type over $k$,\n\\item $R$ is a regular local ring,\n\\item $f_1, \\ldots, f_c$ form a regular sequence in $R$, and\n\\item $S \\cong R/(f_1, \\ldots, f_c)$ as $k$-algebras.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SD","source_file":"algebra.tex","source_line":36013,"source_end_line":36025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36013-L36025","statement_sha256":"84e713b1d57dd9c63a8921224fca64c518a19295f0aa8f30a5542ce702ee78e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1939,"rank":1939,"depth":0,"x":2158.609,"y":151.71,"cluster":"commutative-algebra"},{"id":"stacks:00SE","tag":"00SE","title":"Local complete intersections · Lemma 00SE","summary":"Let A → B → C be surjective local ring homomorphisms. Assume A and B are regular local rings. The following are equivalent • Ker(A → C) is generated by a regular sequence, • Ker(A → C) is generated by dim(A) - dim(C) elements, • Ker(B → C) is generated by a regular sequence, and • Ker(B → C) is generated by dim(B) - dim(C) elements.","statement_latex":"Let $A \\to B \\to C$ be surjective local ring homomorphisms.\nAssume $A$ and $B$ are regular local rings. The following are equivalent\n\\begin{enumerate}\n\\item $\\Ker(A \\to C)$ is generated by a regular sequence,\n\\item $\\Ker(A \\to C)$ is generated by $\\dim(A) - \\dim(C)$ elements,\n\\item $\\Ker(B \\to C)$ is generated by a regular sequence, and\n\\item $\\Ker(B \\to C)$ is generated by $\\dim(B) - \\dim(C)$ elements.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SE","source_file":"algebra.tex","source_line":36037,"source_end_line":36047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36037-L36047","statement_sha256":"207d67cf444794358fb88bc93126554a40a4efa111659bba9d9ccb01e1d5f473","origin":"The Stacks Project","memory_eligible":false,"source_rank":1940,"rank":1940,"depth":14,"x":1990.063,"y":343.391,"cluster":"commutative-algebra"},{"id":"stacks:00SF","tag":"00SF","title":"Local complete intersections · Lemma 00SF","summary":"Let k be a field. Let S be a local k-algebra essentially of finite type over k. The following are equivalent: • S is a complete intersection over k, • for any surjection R → S with R a regular local ring essentially of finite presentation over k the ideal Ker(R → S) can be generated by a regular sequence, • for some surjection R → S with R a regular local ring essentially of finite presentation over k the ideal Ker(R → S) can be generated by dim(R) - dim(S) elements, •…","statement_latex":"Let $k$ be a field. Let $S$ be a local $k$-algebra essentially of finite\ntype over $k$. The following are equivalent:\n\\begin{enumerate}\n\\item $S$ is a complete intersection over $k$,\n\\item for any surjection $R \\to S$ with $R$ a regular local ring\nessentially of finite presentation over $k$ the ideal\n$\\Ker(R \\to S)$ can be generated by a regular sequence,\n\\item for some surjection $R \\to S$ with $R$ a regular local ring\nessentially of finite presentation over $k$ the ideal\n$\\Ker(R \\to S)$ can be generated by\n$\\dim(R) - \\dim(S)$ elements,\n\\item there exists a global complete intersection\n$A$ over $k$ and a prime $\\mathfrak a$ of $A$ such\nthat $S \\cong A_{\\mathfrak a}$, and\n\\item there exists a local complete intersection\n$A$ over $k$ and a prime $\\mathfrak a$ of $A$ such\nthat $S \\cong A_{\\mathfrak a}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SF","source_file":"algebra.tex","source_line":36079,"source_end_line":36099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36079-L36099","statement_sha256":"976ad3ba334e43e437d781c7586d5d20593ec7bc123a26e2b77efcc2db4e9ade","origin":"The Stacks Project","memory_eligible":false,"source_rank":1941,"rank":1941,"depth":25,"x":1960.188,"y":106.297,"cluster":"commutative-algebra"},{"id":"stacks:00SG","tag":"00SG","title":"Local complete intersections · Lemma 00SG","summary":"Let k be a field. Let S be a finite type k-algebra. Let q be a prime of S. The following are equivalent: • The local ring S_ q is a complete intersection ring (Definition [Tag 00SD]). • There exists a g ∈ S, g not ∈ q such that S_g is a local complete intersection over k. • There exists a g ∈ S, g not ∈ q such that S_g is a global complete intersection over k. • For any presentation S = k[x_1, …, x_n]/I with q' ⊂ k[x_1, …, x_n] corresponding to q any of the equivalent…","statement_latex":"Let $k$ be a field. Let $S$ be a finite type $k$-algebra.\nLet $\\mathfrak q$ be a prime of $S$. The following are\nequivalent:\n\\begin{enumerate}\n\\item The local ring $S_{\\mathfrak q}$ is a complete intersection\nring (Definition \\ref{definition-lci-local-ring}).\n\\item There exists a $g \\in S$, $g \\not \\in \\mathfrak q$\nsuch that $S_g$ is a local complete intersection over $k$.\n\\item There exists a $g \\in S$, $g \\not \\in \\mathfrak q$\nsuch that $S_g$ is a global complete intersection over $k$.\n\\item For any presentation $S = k[x_1, \\ldots, x_n]/I$ with\n$\\mathfrak q' \\subset k[x_1, \\ldots, x_n]$ corresponding to $\\mathfrak q$\nany of the equivalent conditions (1) -- (5) of Lemma \\ref{lemma-lci} hold.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SG","source_file":"algebra.tex","source_line":36159,"source_end_line":36175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36159-L36175","statement_sha256":"aeae4492cbf89bea887b9eb506200b94863d9ddc3f4d65ab3943563074672f3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1942,"rank":1942,"depth":26,"x":2172.984,"y":264.251,"cluster":"commutative-algebra"},{"id":"stacks:00SH","tag":"00SH","title":"Local complete intersections · Lemma 00SH","summary":"Let k be a field. Let S be a finite type k-algebra. The following are equivalent: • The ring S is a local complete intersection over k. • All local rings of S are complete intersection rings over k. • All localizations of S at maximal ideals are complete intersection rings over k.","statement_latex":"Let $k$ be a field. Let $S$ be a finite type $k$-algebra.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The ring $S$ is a local complete intersection over $k$.\n\\item All local rings of $S$ are complete intersection rings over $k$.\n\\item All localizations of $S$\nat maximal ideals are complete intersection rings over $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SH","source_file":"algebra.tex","source_line":36182,"source_end_line":36192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36182-L36192","statement_sha256":"3ca7a540098486ab831d4552339b76fa5ed96c99930bcc15625e555d2d711793","origin":"The Stacks Project","memory_eligible":false,"source_rank":1943,"rank":1943,"depth":27,"x":1888.913,"y":268.526,"cluster":"commutative-algebra"},{"id":"stacks:00SI","tag":"00SI","title":"Local complete intersections · Lemma 00SI","summary":"Let K/k be a field extension. Let S be a finite type algebra over k. Let q_K be a prime of S_K = K ⊗_k S and let q be the corresponding prime of S. Then S_ q is a complete intersection over k (Definition [Tag 00SD]) if and only if (S_K)_ q_K is a complete intersection over K.","statement_latex":"Let $K/k$ be a field extension.\nLet $S$ be a finite type algebra over $k$.\nLet $\\mathfrak q_K$ be a prime of $S_K = K \\otimes_k S$\nand let $\\mathfrak q$ be the corresponding prime of $S$.\nThen $S_{\\mathfrak q}$ is a complete intersection\nover $k$ (Definition \\ref{definition-lci-local-ring})\nif and only if $(S_K)_{\\mathfrak q_K}$ is a complete\nintersection over $K$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SI","source_file":"algebra.tex","source_line":36203,"source_end_line":36213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36203-L36213","statement_sha256":"740f28ee265457cbf6d565846b24062604dccc1f402a6dd155a686826ca2d7e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1944,"rank":1944,"depth":27,"x":2095.043,"y":104.106,"cluster":"commutative-algebra"},{"id":"stacks:00SJ","tag":"00SJ","title":"Local complete intersections · Lemma 00SJ","summary":"Let k → K be a field extension. Let S be a finite type k-algebra. Then S is a local complete intersection over k if and only if S ⊗_k K is a local complete intersection over K.","statement_latex":"Let $k \\to K$ be a field extension.\nLet $S$ be a finite type $k$-algebra.\nThen $S$ is a local complete intersection over $k$ if and\nonly if $S \\otimes_k K$ is a local complete intersection over $K$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SJ","source_file":"algebra.tex","source_line":36261,"source_end_line":36267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36261-L36267","statement_sha256":"a6b4a72a39406e6f89d9ca956649ab8cf19fac7f118bc3b92f2b3d7ccf20d595","origin":"The Stacks Project","memory_eligible":false,"source_rank":1945,"rank":1945,"depth":28,"x":2075.259,"y":342.424,"cluster":"commutative-algebra"},{"id":"stacks:02JP","tag":"02JP","title":"Local complete intersections · Lemma 02JP","summary":"Let xymatrix B & S ar[l] A ar[u] & R ar[l] ar[u] be a commutative square of local rings. Assume • R and overlineS = S/ m_R S are regular local rings, • A = R/I and B = S/J for some ideals I, J, • J ⊂ S and overlineJ = J/ m_R ∩ J ⊂ overlineS are generated by regular sequences, and • A → B and R → S are flat. Then I is generated by a regular sequence.","statement_latex":"Let\n$$\n\\xymatrix{\nB & S \\ar[l] \\\\\nA \\ar[u] & R \\ar[l] \\ar[u]\n}\n$$\nbe a commutative square of local rings. Assume\n\\begin{enumerate}\n\\item $R$ and $\\overline{S} = S/\\mathfrak m_R S$ are regular local rings,\n\\item $A = R/I$ and $B = S/J$ for some ideals $I$, $J$,\n\\item $J \\subset S$ and\n$\\overline{J} = J/\\mathfrak m_R \\cap J \\subset \\overline{S}$\nare generated by regular sequences, and\n\\item $A \\to B$ and $R \\to S$ are flat.\n\\end{enumerate}\nThen $I$ is generated by a regular sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JP","source_file":"algebra.tex","source_line":36360,"source_end_line":36379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36360-L36379","statement_sha256":"06786953c609e9ca29523b9d1ea85d39eb1db3d81eefa0f5e1d65ec6e8ff4be2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1946,"rank":1946,"depth":15,"x":1898.113,"y":155.376,"cluster":"commutative-algebra"},{"id":"stacks:00SL","tag":"00SL","title":"Syntomic morphisms · Definition 00SL","summary":"A ring map R → S is called syntomic, or we say S is a flat local complete intersection over R if it is flat, of finite presentation, and if all of its fibre rings S ⊗_R kappa( p) are local complete intersections, see Definition [Tag 00S9].","statement_latex":"A ring map $R \\to S$ is called {\\it syntomic}, or we say $S$ is a\n{\\it flat local complete intersection over $R$}\nif it is flat, of finite presentation, and if all of its fibre rings\n$S \\otimes_R \\kappa(\\mathfrak p)$ are local complete intersections,\nsee Definition \\ref{definition-lci-field}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SL","source_file":"algebra.tex","source_line":36436,"source_end_line":36443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36436-L36443","statement_sha256":"f741cc188d7983b94af81ffd35b576165bfaba292c3ca976803c2fa0e5da456b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1947,"rank":1947,"depth":1,"x":2179.291,"y":192.804,"cluster":"commutative-algebra"},{"id":"stacks:00SM","tag":"00SM","title":"Syntomic morphisms · Lemma 00SM","summary":"Being syntomic is fpqc local on the base. Let R → S be a ring map. Let R → R' be a faithfully flat ring map. Set S' = R'⊗_R S. Then R → S is syntomic if and only if R' → S' is syntomic.","statement_latex":"\\begin{slogan}\nBeing syntomic is fpqc local on the base.\n\\end{slogan}\nLet $R \\to S$ be a ring map.\nLet $R \\to R'$ be a faithfully flat ring map.\nSet $S' = R'\\otimes_R S$.\nThen $R \\to S$ is syntomic if and only if $R' \\to S'$ is syntomic.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SM","source_file":"algebra.tex","source_line":36450,"source_end_line":36459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36450-L36459","statement_sha256":"985d2a0cf29d63a8e650bb2c6694c59cdd067af63f69d510f3087bbf504a5222","origin":"The Stacks Project","memory_eligible":false,"source_rank":1948,"rank":1948,"depth":29,"x":1941.743,"y":324.815,"cluster":"commutative-algebra"},{"id":"stacks:00SN","tag":"00SN","title":"Syntomic morphisms · Lemma 00SN","summary":"Any base change of a syntomic map is syntomic.","statement_latex":"Any base change of a syntomic map is syntomic.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SN","source_file":"algebra.tex","source_line":36477,"source_end_line":36480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36477-L36480","statement_sha256":"ed23b076bf86ca60f00c4264586394f0d8b70beefd49e4abcfec4ada25dd19c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1949,"rank":1949,"depth":29,"x":2010.781,"y":92.571,"cluster":"commutative-algebra"},{"id":"stacks:00SO","tag":"00SO","title":"Syntomic morphisms · Lemma 00SO","summary":"Let R → S be a ring map. Suppose we have g_1, … g_m ∈ S which generate the unit ideal such that each R → S_g_i is syntomic. Then R → S is syntomic.","statement_latex":"Let $R \\to S$ be a ring map.\nSuppose we have $g_1, \\ldots g_m \\in S$ which generate the\nunit ideal such that each $R \\to S_{g_i}$ is syntomic.\nThen $R \\to S$ is syntomic.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SO","source_file":"algebra.tex","source_line":36489,"source_end_line":36495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36489-L36495","statement_sha256":"e7ca7e5612db039c7c34acbf7ba7da50d600553e3278535668729e0eaa36eaa5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1950,"rank":1950,"depth":4,"x":2146.703,"y":303.099,"cluster":"commutative-algebra"},{"id":"stacks:00SP","tag":"00SP","title":"Syntomic morphisms · Definition 00SP","summary":"Let R → S be a ring map. We say that R → S is a relative global complete intersection if there exists a presentation S = R[x_1, …, x_n]/(f_1, …, f_c) and every nonempty fibre of Spec(S) → Spec(R) has dimension n - c. We will say \"let S = R[x_1, …, x_n]/(f_1, …, f_c) be a relative global complete intersection\" to indicate this situation.","statement_latex":"Let $R \\to S$ be a ring map. We say that $R \\to S$ is\na {\\it relative global complete intersection} if there exists\na presentation $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ and\nevery nonempty fibre of $\\Spec(S) \\to \\Spec(R)$ has dimension $n - c$.\nWe will say ``let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative\nglobal complete intersection'' to indicate this situation.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SP","source_file":"algebra.tex","source_line":36505,"source_end_line":36513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36505-L36513","statement_sha256":"e1b626885da9d5778907cdae5dce947c6762fc71d30adb5765b89227a41ce228","origin":"The Stacks Project","memory_eligible":false,"source_rank":1951,"rank":1951,"depth":0,"x":1877.047,"y":224.947,"cluster":"commutative-algebra"},{"id":"stacks:07CF","tag":"07CF","title":"Syntomic morphisms · Lemma 07CF","summary":"Let S be a finitely presented R-algebra which has a presentation S = R[x_1, …, x_n]/I such that I/I^2 is free over S. Then S has a presentation S = R[y_1, …, y_m]/(f_1, …, f_c) such that (f_1, …, f_c)/(f_1, …, f_c)^2 is free with basis given by the classes of f_1, …, f_c.","statement_latex":"Let $S$ be a finitely presented $R$-algebra which has a presentation\n$S = R[x_1, \\ldots, x_n]/I$ such that $I/I^2$ is free over $S$. Then\n$S$ has a presentation $S = R[y_1, \\ldots, y_m]/(f_1, \\ldots, f_c)$\nsuch that $(f_1, \\ldots, f_c)/(f_1, \\ldots, f_c)^2$ is free with\nbasis given by the classes of $f_1, \\ldots, f_c$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CF","source_file":"algebra.tex","source_line":36519,"source_end_line":36526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36519-L36526","statement_sha256":"ce7527ccf9840072ccc64015bc42c3e80dddb949bf08ddafd1f3179654bf1169","origin":"The Stacks Project","memory_eligible":false,"source_rank":1952,"rank":1952,"depth":4,"x":2138.859,"y":129.52,"cluster":"commutative-algebra"},{"id":"stacks:00SS","tag":"00SS","title":"Syntomic morphisms · Lemma 00SS","summary":"Let S = R[x_1, …, x_n]/(f_1, …, f_c) be a relative global complete intersection (Definition [Tag 00SP]) • For any R → R' the base change R' ⊗_R S = R'[x_1, …, x_n]/(f_1, …, f_c) is a relative global complete intersection. • For any g ∈ S which is the image of h ∈ R[x_1, …, x_n] the ring S_g = R[x_1, …, x_n, x_n + 1]/(f_1, …, f_c, hx_n + 1 - 1) is a relative global complete intersection. • If R → S factors as R → R_f → S for some f ∈ R. Then the ring S = R_f[x_1, …,…","statement_latex":"Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a\nrelative global complete intersection\n(Definition \\ref{definition-relative-global-complete-intersection})\n\\begin{enumerate}\n\\item For any $R \\to R'$ the base change\n$R' \\otimes_R S = R'[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a relative\nglobal complete intersection.\n\\item For any $g \\in S$ which is the image of $h \\in R[x_1, \\ldots, x_n]$\nthe ring\n$S_g = R[x_1, \\ldots, x_n, x_{n + 1}]/(f_1, \\ldots, f_c, hx_{n + 1} - 1)$\nis a relative global complete intersection.\n\\item If $R \\to S$ factors as $R \\to R_f \\to S$ for some $f \\in R$.\nThen the ring $S = R_f[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$\nis a relative global complete intersection over $R_f$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SS","source_file":"algebra.tex","source_line":36638,"source_end_line":36655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36638-L36655","statement_sha256":"448979e323d144bd3db272f118cb3e0d33d0e00ce92b0af5138cf02acccc07f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":1953,"rank":1953,"depth":12,"x":2022.487,"y":348.549,"cluster":"commutative-algebra"},{"id":"stacks:00ST","tag":"00ST","title":"Syntomic morphisms · Lemma 00ST","summary":"Let R be a ring. Let S = R[x_1, …, x_n]/(f_1, …, f_c). We will find h ∈ R[x_1, …, x_n] which maps to g ∈ S such that S_g = R[x_1, …, x_n, x_n + 1]/(f_1, …, f_c, hx_n + 1 - 1) is a relative global complete intersection with a presentation as in Definition [Tag 00SP] in each of the following cases: • Let I ⊂ R be an ideal. If the fibres of Spec(S/IS) → Spec(R/I) have dimension n - c, then we can find (h, g) as above such that g maps to 1 ∈ S/IS. • Let p ⊂ R be a prime. If…","statement_latex":"Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$.\nWe will find $h \\in R[x_1, \\ldots, x_n]$ which maps to $g \\in S$\nsuch that\n$$\nS_g = R[x_1, \\ldots, x_n, x_{n + 1}]/(f_1, \\ldots, f_c, hx_{n + 1} - 1)\n$$\nis a relative global complete intersection with a presentation as in\nDefinition \\ref{definition-relative-global-complete-intersection}\nin each of the following cases:\n\\begin{enumerate}\n\\item Let $I \\subset R$ be an ideal. If the fibres of\n$\\Spec(S/IS) \\to \\Spec(R/I)$ have dimension $n - c$, then we can\nfind $(h, g)$ as above such that $g$ maps to $1 \\in S/IS$.\n\\item Let $\\mathfrak p \\subset R$ be a prime. If\n$\\dim(S \\otimes_R \\kappa(\\mathfrak p)) = n - c$, then we can\nfind $(h, g)$ as above such that $g$ maps to a unit of\n$S \\otimes_R \\kappa(\\mathfrak p)$.\n\\item Let $\\mathfrak q \\subset S$ be a prime lying over\n$\\mathfrak p \\subset R$. If $\\dim_{\\mathfrak q}(S/R) = n - c$, then we can\nfind $(h, g)$ as above such that $g \\not \\in \\mathfrak q$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ST","source_file":"algebra.tex","source_line":36670,"source_end_line":36693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36670-L36693","statement_sha256":"6277efc28fd8c459d88dbdf9fe6cf900a2717d6a97a9b92589af8a705720e9e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1954,"rank":1954,"depth":31,"x":1932.118,"y":120.9,"cluster":"commutative-algebra"},{"id":"stacks:00SU","tag":"00SU","title":"Syntomic morphisms · Lemma 00SU","summary":"Let R be a ring. Let S = R[x_1, …, x_n]/(f_1, …, f_c) be a relative global complete intersection (Definition [Tag 00SP]). There exist a finite type Z-subalgebra R_0 ⊂ R such that f_i ∈ R_0[x_1, …, x_n] and such that S_0 = R_0[x_1, …, x_n]/(f_1, …, f_c) is a relative global complete intersection.","statement_latex":"Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$\nbe a relative global complete intersection\n(Definition \\ref{definition-relative-global-complete-intersection}).\nThere exist a finite type $\\mathbf{Z}$-subalgebra $R_0 \\subset R$\nsuch that $f_i \\in R_0[x_1, \\ldots, x_n]$ and such that\n$$\nS_0 = R_0[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)\n$$\nis a relative global complete intersection.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SU","source_file":"algebra.tex","source_line":36726,"source_end_line":36737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36726-L36737","statement_sha256":"ba0fc8648039d460a4e01d74146e519bf1ef3c1e4cc364d8d4735cb59330ccf9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1955,"rank":1955,"depth":31,"x":2181.943,"y":237.543,"cluster":"commutative-algebra"},{"id":"stacks:00SV","tag":"00SV","title":"Syntomic morphisms · Lemma 00SV","summary":"Let R be a ring. Let S = R[x_1, …, x_n]/(f_1, …, f_c) be a relative global complete intersection (Definition [Tag 00SP]). For every prime q of S, let q' denote the corresponding prime of R[x_1, …, x_n]. Then • f_1, …, f_c is a regular sequence in the local ring R[x_1, …, x_n]_ q', • each of the rings R[x_1, …, x_n]_ q'/(f_1, …, f_i) is flat over R, and • the S-module (f_1, …, f_c)/(f_1, …, f_c)^2 is free with basis given by the elements f_i bmod (f_1, …, f_c)^2.","statement_latex":"Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$\nbe a relative global complete intersection (Definition\n\\ref{definition-relative-global-complete-intersection}). For every prime\n$\\mathfrak q$ of $S$, let $\\mathfrak q'$ denote the corresponding\nprime of $R[x_1, \\ldots, x_n]$. Then\n\\begin{enumerate}\n\\item $f_1, \\ldots, f_c$ is a regular sequence in the local ring\n$R[x_1, \\ldots, x_n]_{\\mathfrak q'}$,\n\\item each of the rings\n$R[x_1, \\ldots, x_n]_{\\mathfrak q'}/(f_1, \\ldots, f_i)$ is flat over $R$, and\n\\item the $S$-module $(f_1, \\ldots, f_c)/(f_1, \\ldots, f_c)^2$\nis free with basis given by the elements $f_i \\bmod (f_1, \\ldots, f_c)^2$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SV","source_file":"algebra.tex","source_line":36766,"source_end_line":36781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36766-L36781","statement_sha256":"2f6bb5c500be5775ccc542071727c91fd66f3edd3128cdbe463014fb44ce7a26","origin":"The Stacks Project","memory_eligible":false,"source_rank":1956,"rank":1956,"depth":32,"x":1903.793,"y":293.315,"cluster":"commutative-algebra"},{"id":"stacks:00SW","tag":"00SW","title":"Syntomic morphisms · Lemma 00SW","summary":"A relative global complete intersection is syntomic, i.e., flat.","statement_latex":"A relative global complete intersection is syntomic, i.e., flat.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SW","source_file":"algebra.tex","source_line":36842,"source_end_line":36845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36842-L36845","statement_sha256":"811a103283c041d0e0a98b49dded4bab5ba56fd92eb816c80ba5b483a75019c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1957,"rank":1957,"depth":33,"x":2064.122,"y":94.266,"cluster":"commutative-algebra"},{"id":"stacks:03HS","tag":"03HS","title":"Syntomic morphisms · Lemma 03HS","summary":"Suppose that A is a ring, and P(x) = x^n + b_1 x^n-1 + … + b_n ∈ A[x] is a monic polynomial over A. Then there exists a syntomic, finite free, faithfully flat ring extension A ⊂ A' such that P(x) = ∏_i = 1, …, n (x - β_i) for certain β_i ∈ A'.","statement_latex":"Suppose that $A$ is a ring, and\n$P(x) = x^n + b_1 x^{n-1} + \\ldots + b_n \\in A[x]$ is\na monic polynomial over $A$. Then there exists a\nsyntomic, finite free, faithfully flat ring extension\n$A \\subset A'$ such that $P(x) = \\prod_{i = 1, \\ldots, n} (x - \\beta_i)$\nfor certain $\\beta_i \\in A'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HS","source_file":"algebra.tex","source_line":36856,"source_end_line":36864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36856-L36864","statement_sha256":"4cb95698ce204fa12b3b7beb5f149c0c972dd8a2badc21a705033e9f1b257705","origin":"The Stacks Project","memory_eligible":false,"source_rank":1958,"rank":1958,"depth":34,"x":2105.987,"y":332.129,"cluster":"commutative-algebra"},{"id":"stacks:00SY","tag":"00SY","title":"Syntomic morphisms · Lemma 00SY","summary":"Let R → S be a ring map. Let q ⊂ S be a prime lying over the prime p of R. The following are equivalent: • There exists an element g ∈ S, g not ∈ q such that R → S_g is syntomic. • There exists an element g ∈ S, g not ∈ q such that S_g is a relative global complete intersection over R. • There exists an element g ∈ S, g not ∈ q, such that R → S_g is of finite presentation, the local ring map R_ p → S_ q is flat, and the local ring S_ q/ pS_ q is a complete intersection…","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q \\subset S$ be a prime lying over\nthe prime $\\mathfrak p$ of $R$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item There exists an element $g \\in S$, $g \\not \\in \\mathfrak q$ such that\n$R \\to S_g$ is syntomic.\n\\item There exists an element $g \\in S$, $g \\not \\in \\mathfrak q$\nsuch that $S_g$ is a relative global complete intersection over $R$.\n\\item There exists an element $g \\in S$, $g \\not \\in \\mathfrak q$,\nsuch that $R \\to S_g$ is of finite presentation,\nthe local ring map $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is flat, and\nthe local ring $S_{\\mathfrak q}/\\mathfrak pS_{\\mathfrak q}$ is\na complete intersection ring over $\\kappa(\\mathfrak p)$ (see\nDefinition \\ref{definition-lci-local-ring}).\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SY","source_file":"algebra.tex","source_line":36877,"source_end_line":36895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36877-L36895","statement_sha256":"241e8b98b94080e2781aaa7e2c70cfbc9ce0a66120ff11e8be48dec33ca73860","origin":"The Stacks Project","memory_eligible":false,"source_rank":1959,"rank":1959,"depth":34,"x":1883.729,"y":180.416,"cluster":"commutative-algebra"},{"id":"stacks:07BT","tag":"07BT","title":"Syntomic morphisms · Lemma 07BT","summary":"Let R be a ring. Let S = R[x_1, …, x_n]/I for some finitely generated ideal I. If g ∈ S is such that S_g is syntomic over R, then (I/I^2)_g is a finite projective S_g-module.","statement_latex":"Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/I$ for some\nfinitely generated ideal $I$. If $g \\in S$ is such that\n$S_g$ is syntomic over $R$, then $(I/I^2)_g$ is a finite projective\n$S_g$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BT","source_file":"algebra.tex","source_line":36945,"source_end_line":36951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36945-L36951","statement_sha256":"b072406c169a37534ec91472a2c11259e4dcffeb3181a6ff152a0d0714c421a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":1960,"rank":1960,"depth":35,"x":2169.757,"y":166.165,"cluster":"commutative-algebra"},{"id":"stacks:00SZ","tag":"00SZ","title":"Syntomic morphisms · Lemma 00SZ","summary":"Let R → S, S → S' be ring maps. • If R → S and S → S' are syntomic, then R → S' is syntomic. • If R → S and S → S' are relative global complete intersections, then R → S' is a relative global complete intersection.","statement_latex":"Let $R \\to S$, $S \\to S'$ be ring maps.\n\\begin{enumerate}\n\\item If $R \\to S$ and $S \\to S'$ are syntomic, then $R \\to S'$\nis syntomic.\n\\item If $R \\to S$ and $S \\to S'$ are relative global complete intersections,\nthen $R \\to S'$ is a relative global complete intersection.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00SZ","source_file":"algebra.tex","source_line":36966,"source_end_line":36975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L36966-L36975","statement_sha256":"72d7b67a8dd9372eedd5ca461e10ad4b98d0c73b5059a54d9c475afe94138617","origin":"The Stacks Project","memory_eligible":false,"source_rank":1961,"rank":1961,"depth":35,"x":1970.21,"y":339.054,"cluster":"commutative-algebra"},{"id":"stacks:00T0","tag":"00T0","title":"Syntomic morphisms · Lemma 00T0","summary":"Let R be a ring and let I ⊂ R be an ideal. Let R/I → overlineS be a syntomic map. Then there exists elements overlineg_i ∈ overlineS which generate the unit ideal of overlineS such that each overlineS_g_i ≅ S_i/IS_i for some relative global complete intersection S_i over R.","statement_latex":"Let $R$ be a ring and let $I \\subset R$ be an ideal.\nLet $R/I \\to \\overline{S}$ be a syntomic map.\nThen there exists elements $\\overline{g}_i \\in \\overline{S}$\nwhich generate the unit ideal of $\\overline{S}$\nsuch that each $\\overline{S}_{g_i} \\cong S_i/IS_i$\nfor some relative global complete intersection $S_i$\nover $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00T0","source_file":"algebra.tex","source_line":37021,"source_end_line":37030,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37021-L37030","statement_sha256":"bc2ccfad587ed5bccf89fac606ecb0513142e4e0dfd8c2b022efa79b3f1e4653","origin":"The Stacks Project","memory_eligible":false,"source_rank":1962,"rank":1962,"depth":35,"x":1978.324,"y":98.227,"cluster":"commutative-algebra"},{"id":"stacks:00T2","tag":"00T2","title":"Smooth ring maps · Definition 00T2","summary":"A ring map R → S is smooth if it is of finite presentation and the naive cotangent complex NL_S/R is quasi-isomorphic to a finite projective S-module placed in degree 0: this means that H_1(NL_S/R) = 0 and that Ω_S/R is a finite projective S-module.","statement_latex":"A ring map $R \\to S$ is {\\it smooth} if it is of finite presentation\nand the naive cotangent complex $\\NL_{S/R}$ is quasi-isomorphic to a\nfinite projective $S$-module placed in degree $0$: this means\nthat $H_1(\\NL_{S/R}) = 0$ and that $\\Omega_{S/R}$ is a finite projective\n$S$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00T2","source_file":"algebra.tex","source_line":37083,"source_end_line":37090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37083-L37090","statement_sha256":"e729dfae9eed41e0e617e0b8f7557ff283733f13eb4b5c02af3673bef9844643","origin":"The Stacks Project","memory_eligible":false,"source_rank":1963,"rank":1963,"depth":0,"x":2166.095,"y":280.499,"cluster":"commutative-algebra"},{"id":"stacks:00T3","tag":"00T3","title":"Smooth ring maps · Lemma 00T3","summary":"Let R → S be a smooth ring map. Any localization S_g is smooth over R. If f ∈ R maps to an invertible element of S, then R_f → S is smooth.","statement_latex":"Let $R \\to S$ be a smooth ring map.\nAny localization $S_g$ is smooth over $R$.\nIf $f \\in R$ maps to an invertible element of $S$,\nthen $R_f \\to S$ is smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00T3","source_file":"algebra.tex","source_line":37112,"source_end_line":37118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37112-L37118","statement_sha256":"345d0b4f803f7e2593ceed81e2f54c0ce8e4949342997a0d8ffcfae9621bcedb","origin":"The Stacks Project","memory_eligible":false,"source_rank":1964,"rank":1964,"depth":5,"x":1880.923,"y":252.628,"cluster":"commutative-algebra"},{"id":"stacks:00T4","tag":"00T4","title":"Smooth ring maps · Lemma 00T4","summary":"Smoothness is preserved under base change Let R → S be a smooth ring map. Let R → R' be any ring map. Then the base change R' → S' = R' ⊗_R S is smooth.","statement_latex":"\\begin{slogan}\nSmoothness is preserved under base change\n\\end{slogan}\nLet $R \\to S$ be a smooth ring map.\nLet $R \\to R'$ be any ring map.\nThen the base change $R' \\to S' = R' \\otimes_R S$ is smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00T4","source_file":"algebra.tex","source_line":37132,"source_end_line":37140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37132-L37140","statement_sha256":"59453bf1398851e40ad80345761fc1c94fc7a651ad0afdf1da9e2d166d47913b","origin":"The Stacks Project","memory_eligible":false,"source_rank":1965,"rank":1965,"depth":1,"x":2113.728,"y":111.3,"cluster":"commutative-algebra"},{"id":"stacks:00T5","tag":"00T5","title":"Smooth ring maps · Lemma 00T5","summary":"Let k be a field. Let S be a smooth k-algebra. Then S is a local complete intersection.","statement_latex":"Let $k$ be a field.\nLet $S$ be a smooth $k$-algebra.\nThen $S$ is a local complete intersection.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00T5","source_file":"algebra.tex","source_line":37179,"source_end_line":37184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37179-L37184","statement_sha256":"239cf829844ee95b87ddff5e3542a88b1ca87fe108901f5da15e06037e870855","origin":"The Stacks Project","memory_eligible":false,"source_rank":1966,"rank":1966,"depth":29,"x":2055.686,"y":347.723,"cluster":"commutative-algebra"},{"id":"stacks:00T6","tag":"00T6","title":"Smooth ring maps · Definition 00T6","summary":"Let R be a ring. Given integers n ≥ c ≥ 0 and f_1, …, f_c ∈ R[x_1, …, x_n] we say R[x_1, …, x_n]/(f_1, …, f_c) is a standard smooth algebra over R if the polynomial g = det ( ∂ f_1/∂ x_1 & ∂ f_2/∂ x_1 & … & ∂ f_c/∂ x_1 ∂ f_1/∂ x_2 & ∂ f_2/∂ x_2 & … & ∂ f_c/∂ x_2 … & … & … & … ∂ f_1/∂ x_c & ∂ f_2/∂ x_c & … & ∂ f_c/∂ x_c ) maps to an invertible element in R[x_1, …, x_n]/(f_1, …, f_c). We say an R-algebra S is standard smooth or that the ring map R → S is standard smooth if…","statement_latex":"Let $R$ be a ring. Given integers $n \\geq c \\geq 0$ and\n$f_1, \\ldots, f_c \\in R[x_1, \\ldots, x_n]$ we say\n$$\nR[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)\n$$\nis a {\\it standard smooth algebra over $R$} if the polynomial\n$$\ng =\n\\det\n\\left(\n\\begin{matrix}\n\\partial f_1/\\partial x_1 &\n\\partial f_2/\\partial x_1 &\n\\ldots &\n\\partial f_c/\\partial x_1 \\\\\n\\partial f_1/\\partial x_2 &\n\\partial f_2/\\partial x_2 &\n\\ldots &\n\\partial f_c/\\partial x_2 \\\\\n\\ldots & \\ldots & \\ldots & \\ldots \\\\\n\\partial f_1/\\partial x_c &\n\\partial f_2/\\partial x_c &\n\\ldots &\n\\partial f_c/\\partial x_c\n\\end{matrix}\n\\right)\n$$\nmaps to an invertible element in $R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$.\nWe say an $R$-algebra $S$ is {\\it standard smooth} or that\nthe ring map $R \\to S$ is {\\it standard smooth} if there exist\n$n \\geq c \\geq 0$ and $f_1, \\ldots, f_c \\in R[x_1, \\ldots, x_n]$\nsuch that $R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ is a standard smooth\nalgebra over $R$ and $S$ is isomorphic to\n$R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ as an $R$-algebra.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00T6","source_file":"algebra.tex","source_line":37235,"source_end_line":37271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37235-L37271","statement_sha256":"e64dd766a4f75bb98d872b08284a4fa99157f65eeec362dde6a9579a892f2655","origin":"The Stacks Project","memory_eligible":false,"source_rank":1967,"rank":1967,"depth":0,"x":1908.291,"y":140.357,"cluster":"commutative-algebra"},{"id":"stacks:00T7","tag":"00T7","title":"Smooth ring maps · Lemma 00T7","summary":"Let S = R[x_1, …, x_n]/(f_1, …, f_c) = R[x_1, …, x_n]/I be a standard smooth algebra. Then • the ring map R → S is smooth, • the S-module Ω_S/R is free on dx_c + 1, …, dx_n, • the S-module I/I^2 is free on the classes of f_1, …, f_c, • for any g ∈ S the ring map R → S_g is standard smooth, • for any ring map R → R' the base change R' → R'⊗_R S is standard smooth, • if f ∈ R maps to an invertible element in S, then R_f → S is standard smooth, and • the ring S is a relative…","statement_latex":"Let\n$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c) = R[x_1, \\ldots, x_n]/I$\nbe a standard smooth algebra. Then\n\\begin{enumerate}\n\\item the ring map $R \\to S$ is smooth,\n\\item the $S$-module $\\Omega_{S/R}$ is free on\n$\\text{d}x_{c + 1}, \\ldots, \\text{d}x_n$,\n\\item the $S$-module $I/I^2$ is free on the classes of $f_1, \\ldots, f_c$,\n\\item for any $g \\in S$ the ring map $R \\to S_g$ is standard smooth,\n\\item for any ring map $R \\to R'$ the base change\n$R' \\to R'\\otimes_R S$ is standard smooth,\n\\item if $f \\in R$ maps to an invertible element in $S$, then\n$R_f \\to S$ is standard smooth, and\n\\item the ring $S$ is a relative global complete intersection over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00T7","source_file":"algebra.tex","source_line":37273,"source_end_line":37290,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37273-L37290","statement_sha256":"bb6296d2fd538961d673a15f0190d5500837ca84009aef1d9d33a1d8888b4f02","origin":"The Stacks Project","memory_eligible":false,"source_rank":1968,"rank":1968,"depth":30,"x":2183.865,"y":209.663,"cluster":"commutative-algebra"},{"id":"stacks:00T9","tag":"00T9","title":"Smooth ring maps · Lemma 00T9","summary":"A composition of standard smooth ring maps is standard smooth.","statement_latex":"A composition of standard smooth ring maps is standard smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00T9","source_file":"algebra.tex","source_line":37372,"source_end_line":37375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37372-L37375","statement_sha256":"e2f8839b69b242cad20bb34299065a1afe81d30a4fa1cc663b0931b5b56afd73","origin":"The Stacks Project","memory_eligible":false,"source_rank":1969,"rank":1969,"depth":0,"x":1924.806,"y":314.974,"cluster":"commutative-algebra"},{"id":"stacks:00TA","tag":"00TA","title":"Smooth ring maps · Lemma 00TA","summary":"Let R → S be a smooth ring map. There exists an open covering of Spec(S) by standard opens D(g) such that each S_g is standard smooth over R. In particular R → S is syntomic.","statement_latex":"Let $R \\to S$ be a smooth ring map.\nThere exists an open covering of $\\Spec(S)$ by\nstandard opens $D(g)$ such that each $S_g$ is standard smooth\nover $R$. In particular $R \\to S$ is syntomic.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TA","source_file":"algebra.tex","source_line":37437,"source_end_line":37443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37437-L37443","statement_sha256":"c8b971943b4c030dbd00782149c705a91d7b03b45323edc2fb1933fb5d96534d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1970,"rank":1970,"depth":34,"x":2031.194,"y":90.218,"cluster":"commutative-algebra"},{"id":"stacks:00TB","tag":"00TB","title":"Smooth ring maps · Definition 00TB","summary":"Let R → S be a ring map. Let q be a prime of S. We say R → S is smooth at q if there exists a g ∈ S, g not ∈ q such that R → S_g is smooth.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q$ be a prime of $S$.\nWe say $R \\to S$ is {\\it smooth at $\\mathfrak q$} if there\nexists a $g \\in S$, $g \\not \\in \\mathfrak q$ such\nthat $R \\to S_g$ is smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TB","source_file":"algebra.tex","source_line":37510,"source_end_line":37517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37510-L37517","statement_sha256":"b231741e013802e048fd7736d925661e61e5775c21e8ab181364b0ba7aac04c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1971,"rank":1971,"depth":0,"x":2133.536,"y":316.422,"cluster":"commutative-algebra"},{"id":"stacks:07BU","tag":"07BU","title":"Smooth ring maps · Lemma 07BU","summary":"Let R → S be of finite presentation. Let q be a prime of S. The following are equivalent • R → S is smooth at q, • H_1(L_S/R)_ q = 0 and Ω_S/R, q is a finite free S_ q-module, • H_1(L_S/R)_ q = 0 and Ω_S/R, q is a projective S_ q-module, and • H_1(L_S/R)_ q = 0 and Ω_S/R, q is a flat S_ q-module.","statement_latex":"Let $R \\to S$ be of finite presentation. Let $\\mathfrak q$ be a\nprime of $S$. The following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is smooth at $\\mathfrak q$,\n\\item $H_1(L_{S/R})_\\mathfrak q = 0$ and\n$\\Omega_{S/R, \\mathfrak q}$ is a finite free $S_\\mathfrak q$-module,\n\\item $H_1(L_{S/R})_\\mathfrak q = 0$ and\n$\\Omega_{S/R, \\mathfrak q}$ is a projective $S_\\mathfrak q$-module, and\n\\item $H_1(L_{S/R})_\\mathfrak q = 0$ and\n$\\Omega_{S/R, \\mathfrak q}$ is a flat $S_\\mathfrak q$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BU","source_file":"algebra.tex","source_line":37522,"source_end_line":37535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37522-L37535","statement_sha256":"fd70021289f9e4aa0de402795952336860cf862c52afe03cdcf31798aafda0d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1972,"rank":1972,"depth":5,"x":1876.043,"y":207.643,"cluster":"commutative-algebra"},{"id":"stacks:00TC","tag":"00TC","title":"Smooth ring maps · Lemma 00TC","summary":"A ring map is smooth if and only if it is smooth at all primes of the target Let R → S be a ring map. Then R → S is smooth if and only if R → S is smooth at every prime q of S.","statement_latex":"\\begin{slogan}\nA ring map is smooth if and only if it is smooth at all primes of the target\n\\end{slogan}\nLet $R \\to S$ be a ring map.\nThen $R \\to S$ is smooth if and only if $R \\to S$ is smooth\nat every prime $\\mathfrak q$ of $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TC","source_file":"algebra.tex","source_line":37571,"source_end_line":37579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37571-L37579","statement_sha256":"6435dc4cecf271846027db1fb08e86673879e3219416649f854f6f03502c350d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1973,"rank":1973,"depth":5,"x":2153.52,"y":141.717,"cluster":"commutative-algebra"},{"id":"stacks:00TD","tag":"00TD","title":"Smooth ring maps · Lemma 00TD","summary":"A composition of smooth ring maps is smooth.","statement_latex":"A composition of smooth ring maps is smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TD","source_file":"algebra.tex","source_line":37596,"source_end_line":37599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37596-L37599","statement_sha256":"54d4fef1fb4a78010fd7b64e8d8ddf690f84262cf250b8d2be7008933b809a15","origin":"The Stacks Project","memory_eligible":false,"source_rank":1974,"rank":1974,"depth":35,"x":2001.858,"y":347.873,"cluster":"commutative-algebra"},{"id":"stacks:0GIF","tag":"0GIF","title":"Smooth ring maps · Lemma 0GIF","summary":"Let R be a ring. Let S = S' × S\" be a product of R-algebras. Then S is smooth over R if and only if both S' and S\" are smooth over R.","statement_latex":"Let $R$ be a ring. Let $S = S' \\times S''$ be a product of $R$-algebras.\nThen $S$ is smooth over $R$ if and only if both $S'$ and $S''$ are\nsmooth over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIF","source_file":"algebra.tex","source_line":37612,"source_end_line":37617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37612-L37617","statement_sha256":"37ac0cd984d0171a72f39889824b5a97f16a0080ba6c19132d38648fc2f5cf82","origin":"The Stacks Project","memory_eligible":false,"source_rank":1975,"rank":1975,"depth":6,"x":1947.881,"y":109.689,"cluster":"commutative-algebra"},{"id":"stacks:00TE","tag":"00TE","title":"Smooth ring maps · Lemma 00TE","summary":"Let R be a ring. Let S = R[x_1, …, x_n]/(f_1, …, f_c) be a relative global complete intersection. Let q ⊂ S be a prime. Then R → S is smooth at q if and only if there exists a subset I ⊂ (1, …, n) of cardinality c such that the polynomial g_I = det (∂ f_j/∂ x_i)_j = 1, …, c, i ∈ I. does not map to an element of q.","statement_latex":"Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$\nbe a relative global complete intersection.\nLet $\\mathfrak q \\subset S$ be a prime. Then $R \\to S$\nis smooth at $\\mathfrak q$ if and only if there exists a\nsubset $I \\subset \\{1, \\ldots, n\\}$ of cardinality $c$\nsuch that the polynomial\n$$\ng_I = \\det (\\partial f_j/\\partial x_i)_{j = 1, \\ldots, c, \\ i \\in I}.\n$$\ndoes not map to an element of $\\mathfrak q$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TE","source_file":"algebra.tex","source_line":37628,"source_end_line":37640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37628-L37640","statement_sha256":"429cbb278b91779bca57b2342929cf12b2cd90bbfd7b5abd2d4228a0b7302be6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1976,"rank":1976,"depth":33,"x":2179.332,"y":254.762,"cluster":"commutative-algebra"},{"id":"stacks:00TF","tag":"00TF","title":"Smooth ring maps · Lemma 00TF","summary":"Let R → S be a ring map. Let q ⊂ S be a prime lying over the prime p of R. Assume • there exists a g ∈ S, g not∈ q such that R → S_g is of finite presentation, • the local ring homomorphism R_ p → S_ q is flat, • the fibre S ⊗_R kappa( p) is smooth over kappa( p) at the prime corresponding to q. Then R → S is smooth at q.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q \\subset S$ be a prime lying over the\nprime $\\mathfrak p$ of $R$. Assume\n\\begin{enumerate}\n\\item there exists a $g \\in S$, $g \\not\\in \\mathfrak q$\nsuch that $R \\to S_g$ is of finite presentation,\n\\item the local ring homomorphism\n$R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is flat,\n\\item the fibre $S \\otimes_R \\kappa(\\mathfrak p)$ is smooth\nover $\\kappa(\\mathfrak p)$ at the prime corresponding\nto $\\mathfrak q$.\n\\end{enumerate}\nThen $R \\to S$ is smooth at $\\mathfrak q$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TF","source_file":"algebra.tex","source_line":37672,"source_end_line":37687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37672-L37687","statement_sha256":"51a0d97fe6c40a4faf2e51d855bd48f8ad4423a32dc2d20003c4d93421906d08","origin":"The Stacks Project","memory_eligible":false,"source_rank":1977,"rank":1977,"depth":35,"x":1891.867,"y":279.129,"cluster":"commutative-algebra"},{"id":"stacks:00TG","tag":"00TG","title":"Smooth ring maps · Lemma 00TG","summary":"Let R → S be a ring map of finite presentation. Let R → R' be a flat ring map. Denote S' = R' ⊗_R S the base change. Let U ⊂ Spec(S) be the set of primes at which R → S is smooth. Let V ⊂ Spec(S') the set of primes at which R' → S' is smooth. Then V is the inverse image of U under the map f : Spec(S') → Spec(S).","statement_latex":"Let $R \\to S$ be a ring map of finite presentation.\nLet $R \\to R'$ be a flat ring map.\nDenote $S' = R' \\otimes_R S$ the base change.\nLet $U \\subset \\Spec(S)$ be the set of primes at\nwhich $R \\to S$ is smooth.\nLet $V \\subset \\Spec(S')$ the set of primes at\nwhich $R' \\to S'$ is smooth.\nThen $V$ is the inverse image of $U$ under the\nmap $f : \\Spec(S') \\to \\Spec(S)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TG","source_file":"algebra.tex","source_line":37712,"source_end_line":37723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37712-L37723","statement_sha256":"99a64505ada59349cfd1bdb2b48a3689d35197e55a4c99119a760af58bde5e06","origin":"The Stacks Project","memory_eligible":false,"source_rank":1978,"rank":1978,"depth":6,"x":2084.333,"y":97.962,"cluster":"commutative-algebra"},{"id":"stacks:02UQ","tag":"02UQ","title":"Smooth ring maps · Lemma 02UQ","summary":"Let K/k be a field extension. Let S be a finite type algebra over k. Let q_K be a prime of S_K = K ⊗_k S and let q be the corresponding prime of S. Then S is smooth over k at q if and only if S_K is smooth at q_K over K.","statement_latex":"Let $K/k$ be a field extension.\nLet $S$ be a finite type algebra over $k$.\nLet $\\mathfrak q_K$ be a prime of $S_K = K \\otimes_k S$\nand let $\\mathfrak q$ be the corresponding prime of $S$.\nThen $S$ is smooth over $k$ at $\\mathfrak q$ if and only if\n$S_K$ is smooth at $\\mathfrak q_K$ over $K$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UQ","source_file":"algebra.tex","source_line":37746,"source_end_line":37754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37746-L37754","statement_sha256":"2cab46d8add47a3f0b61468ae77670eb7dd16a6676da1aea10aa314991b4ffdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1979,"rank":1979,"depth":7,"x":2088.102,"y":340.874,"cluster":"commutative-algebra"},{"id":"stacks:04B1","tag":"04B1","title":"Smooth ring maps · Lemma 04B1","summary":"Let R be a ring and let I ⊂ R be an ideal. Let R/I → overlineS be a smooth ring map. Then there exists elements overlineg_i ∈ overlineS which generate the unit ideal of overlineS such that each overlineS_g_i ≅ S_i/IS_i for some (standard) smooth ring S_i over R.","statement_latex":"Let $R$ be a ring and let $I \\subset R$ be an ideal.\nLet $R/I \\to \\overline{S}$ be a smooth ring map.\nThen there exists elements $\\overline{g}_i \\in \\overline{S}$\nwhich generate the unit ideal of $\\overline{S}$\nsuch that each $\\overline{S}_{g_i} \\cong S_i/IS_i$\nfor some (standard) smooth ring $S_i$ over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04B1","source_file":"algebra.tex","source_line":37760,"source_end_line":37768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37760-L37768","statement_sha256":"480130d0292b7dd5edebfb989445922a8a5e5e66843ea7cd32d0bef0d71ef7a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1980,"rank":1980,"depth":35,"x":1889.888,"y":163.812,"cluster":"commutative-algebra"},{"id":"stacks:00TI","tag":"00TI","title":"Formally smooth maps · Definition 00TI","summary":"Let R → S be a ring map. We say S is formally smooth over R if for every commutative solid diagram xymatrix S ar[r] ar@-->[rd] & A/I R ar[r] ar[u] & A ar[u] where I ⊂ A is an ideal of square zero, a dotted arrow exists which makes the diagram commute.","statement_latex":"Let $R \\to S$ be a ring map.\nWe say $S$ is {\\it formally smooth over $R$} if for every\ncommutative solid diagram\n$$\n\\xymatrix{\nS \\ar[r] \\ar@{-->}[rd] & A/I \\\\\nR \\ar[r] \\ar[u] & A \\ar[u]\n}\n$$\nwhere $I \\subset A$ is an ideal of square zero, a dotted\narrow exists which makes the diagram commute.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TI","source_file":"algebra.tex","source_line":37804,"source_end_line":37817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37804-L37817","statement_sha256":"05a8a5fbed446c8813d68d6a537276dc9f05d33b2877b96c3c8bc31f375728d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":1981,"rank":1981,"depth":0,"x":2178.57,"y":181.912,"cluster":"commutative-algebra"},{"id":"stacks:00TJ","tag":"00TJ","title":"Formally smooth maps · Lemma 00TJ","summary":"Let R → S be a formally smooth ring map. Let R → R' be any ring map. Then the base change S' = R' ⊗_R S is formally smooth over R'.","statement_latex":"Let $R \\to S$ be a formally smooth ring map.\nLet $R \\to R'$ be any ring map.\nThen the base change $S' = R' \\otimes_R S$ is formally smooth over $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TJ","source_file":"algebra.tex","source_line":37819,"source_end_line":37824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37819-L37824","statement_sha256":"13a5ba3424300212835e6cb8ff463f5cae20f52fc1fa8fbb81d179e38bea3bb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":1982,"rank":1982,"depth":1,"x":1951.04,"y":332.44,"cluster":"commutative-algebra"},{"id":"stacks:031H","tag":"031H","title":"Formally smooth maps · Lemma 031H","summary":"A composition of formally smooth ring maps is formally smooth.","statement_latex":"A composition of formally smooth ring maps is formally smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031H","source_file":"algebra.tex","source_line":37839,"source_end_line":37842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37839-L37842","statement_sha256":"21d011d4b1e1fc32f3a6f5c92c4e659776325d1f545cd78f3d47e01fa53eb423","origin":"The Stacks Project","memory_eligible":false,"source_rank":1983,"rank":1983,"depth":0,"x":1997.788,"y":92.226,"cluster":"commutative-algebra"},{"id":"stacks:00TK","tag":"00TK","title":"Formally smooth maps · Lemma 00TK","summary":"A polynomial ring over R is formally smooth over R.","statement_latex":"A polynomial ring over $R$ is formally smooth over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TK","source_file":"algebra.tex","source_line":37849,"source_end_line":37852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37849-L37852","statement_sha256":"2c78813b8ea12fc476ba8ab98c72877daef27bf6d9eb3387d4af30292cc39fd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1984,"rank":1984,"depth":1,"x":2156.563,"y":295.976,"cluster":"commutative-algebra"},{"id":"stacks:00TL","tag":"00TL","title":"Formally smooth maps · Lemma 00TL","summary":"Let R → S be a ring map. Let P → S be a surjective R-algebra map from a polynomial ring P onto S. Denote J ⊂ P the kernel. Then R → S is formally smooth if and only if there exists an R-algebra map σ : S → P/J^2 which is a right inverse to the surjection P/J^2 → S.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $P \\to S$ be a surjective $R$-algebra map from a\npolynomial ring $P$ onto $S$. Denote $J \\subset P$ the\nkernel. Then $R \\to S$ is formally smooth if and only\nif there exists an $R$-algebra map $\\sigma : S \\to P/J^2$\nwhich is a right inverse to the surjection\n$P/J^2 \\to S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TL","source_file":"algebra.tex","source_line":37861,"source_end_line":37870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37861-L37870","statement_sha256":"bd17b7af3b142cf438eeb0326b2e0e65d6b5113810d9e58975cff8c2721eaf64","origin":"The Stacks Project","memory_eligible":false,"source_rank":1985,"rank":1985,"depth":2,"x":1875.505,"y":235.799,"cluster":"commutative-algebra"},{"id":"stacks:031I","tag":"031I","title":"Formally smooth maps · Lemma 031I","summary":"Let R → S be a ring map. Let P → S be a surjective R-algebra map from a polynomial ring P onto S. Denote J ⊂ P the kernel. Then R → S is formally smooth if and only if the sequence 0 → J/J^2 → Ω_P/R ⊗_P S → Ω_S/R → 0 of Lemma [Tag 00RU] is a split exact sequence.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $P \\to S$ be a surjective $R$-algebra map from a\npolynomial ring $P$ onto $S$. Denote $J \\subset P$ the\nkernel. Then $R \\to S$ is formally smooth if and only\nif the sequence\n$$\n0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes_P S \\to \\Omega_{S/R} \\to 0\n$$\nof Lemma \\ref{lemma-differential-seq} is a split exact sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031I","source_file":"algebra.tex","source_line":37910,"source_end_line":37921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37910-L37921","statement_sha256":"0feb7926e5b0cc34641ff6786f0a846ca0c97166fb252506f13c49087eca04a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":1986,"rank":1986,"depth":3,"x":2131.264,"y":120.64,"cluster":"commutative-algebra"},{"id":"stacks:031J","tag":"031J","title":"Formally smooth maps · Proposition 031J","summary":"Let R → S be a ring map. Consider a formally smooth R-algebra P and a surjection P → S with kernel J. The following are equivalent • S is formally smooth over R, • for some P → S as above there exists a section to P/J^2 → S, • for all P → S as above there exists a section to P/J^2 → S, • for some P → S as above the sequence 0 → J/J^2 → Ω_P/R ⊗ S → Ω_S/R → 0 is split exact, • for all P → S as above the sequence 0 → J/J^2 → Ω_P/R ⊗ S → Ω_S/R → 0 is split exact, and • the…","statement_latex":"Let $R \\to S$ be a ring map. Consider a formally smooth $R$-algebra $P$ and\na surjection $P \\to S$ with kernel $J$. The following are equivalent\n\\begin{enumerate}\n\\item $S$ is formally smooth over $R$,\n\\item for some $P \\to S$ as above there exists a\nsection to $P/J^2 \\to S$,\n\\item for all $P \\to S$ as above there exists a\nsection to $P/J^2 \\to S$,\n\\item for some $P \\to S$ as above the sequence\n$0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes S \\to \\Omega_{S/R} \\to 0$ is split exact,\n\\item for all $P \\to S$ as above the sequence\n$0 \\to J/J^2 \\to \\Omega_{P/R} \\otimes S \\to \\Omega_{S/R} \\to 0$ is split exact,\nand\n\\item the naive cotangent complex $\\NL_{S/R}$ is quasi-isomorphic to a\nprojective $S$-module placed in degree $0$: this means that\n$H_1(\\NL_{S/R}) = 0$ and that $\\Omega_{S/R}$ is a projective $S$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031J","source_file":"algebra.tex","source_line":37985,"source_end_line":38004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L37985-L38004","statement_sha256":"28aaffec48b42f4ff1602eeb2b0da08e78358f383661b8949df0080cb65b3536","origin":"The Stacks Project","memory_eligible":false,"source_rank":1987,"rank":1987,"depth":4,"x":2035.234,"y":350.786,"cluster":"commutative-algebra"},{"id":"stacks:031K","tag":"031K","title":"Formally smooth maps · Lemma 031K","summary":"Let A → B → C be ring maps. Assume B → C is formally smooth. Then the sequence 0 → Ω_B/A ⊗_B C → Ω_C/A → Ω_C/B → 0 of Lemma [Tag 00RS] is a split short exact sequence.","statement_latex":"Let $A \\to B \\to C$ be ring maps. Assume $B \\to C$ is formally smooth.\nThen the sequence\n$$\n0 \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0\n$$\nof\nLemma \\ref{lemma-exact-sequence-differentials}\nis a split short exact sequence.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031K","source_file":"algebra.tex","source_line":38039,"source_end_line":38049,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38039-L38049","statement_sha256":"3d8fb23c90200da3dcd188b1b07a0127b0d9de73fa9a477334748a27652893c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":1988,"rank":1988,"depth":5,"x":1920.916,"y":126.488,"cluster":"commutative-algebra"},{"id":"stacks:06A6","tag":"06A6","title":"Formally smooth maps · Lemma 06A6","summary":"Let A → B → C be ring maps with A → C formally smooth and B → C surjective with kernel J ⊂ B. Then the exact sequence 0 → J/J^2 → Ω_B/A ⊗_B C → Ω_C/A → 0 of Lemma [Tag 00RU] is split exact.","statement_latex":"Let $A \\to B \\to C$ be ring maps with $A \\to C$ formally smooth\nand $B \\to C$ surjective with kernel $J \\subset B$.\nThen the exact sequence\n$$\n0 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to 0\n$$\nof\nLemma \\ref{lemma-differential-seq}\nis split exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06A6","source_file":"algebra.tex","source_line":38058,"source_end_line":38069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38058-L38069","statement_sha256":"b5b94217529b104c04ee1679cf7f6e9dad41d312a97281b79096281bdfd8da30","origin":"The Stacks Project","memory_eligible":false,"source_rank":1989,"rank":1989,"depth":5,"x":2185.708,"y":227.061,"cluster":"commutative-algebra"},{"id":"stacks:06A7","tag":"06A7","title":"Formally smooth maps · Lemma 06A7","summary":"Let A → B → C be ring maps. Assume A → C is surjective (so also B → C is) and A → B formally smooth. Denote I = Ker(A → C) and J = Ker(B → C). Then the sequence 0 → I/I^2 → J/J^2 → Ω_B/A ⊗_B B/J → 0 of Lemma [Tag 065V] is split exact.","statement_latex":"Let $A \\to B \\to C$ be ring maps. Assume $A \\to C$ is surjective (so\nalso $B \\to C$ is) and $A \\to B$ formally smooth.\nDenote $I = \\Ker(A \\to C)$ and $J = \\Ker(B \\to C)$.\nThen the sequence\n$$\n0 \\to I/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0\n$$\nof\nLemma \\ref{lemma-application-NL}\nis split exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06A7","source_file":"algebra.tex","source_line":38078,"source_end_line":38090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38078-L38090","statement_sha256":"92f5174d8d7bf04d7b6b9e437828984875e9107fab6c757ebd3450f724fabbf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":1990,"rank":1990,"depth":4,"x":1909.45,"y":303.184,"cluster":"commutative-algebra"},{"id":"stacks:031L","tag":"031L","title":"Formally smooth maps · Lemma 031L","summary":"Let R → S be a ring map. Let I ⊂ R be an ideal. Assume • I^2 = 0, • R → S is flat, and • R/I → S/IS is formally smooth. Then R → S is formally smooth.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $I \\subset R$ be an ideal. Assume\n\\begin{enumerate}\n\\item $I^2 = 0$,\n\\item $R \\to S$ is flat, and\n\\item $R/I \\to S/IS$ is formally smooth.\n\\end{enumerate}\nThen $R \\to S$ is formally smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031L","source_file":"algebra.tex","source_line":38099,"source_end_line":38109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38099-L38109","statement_sha256":"62836343d5e7b487d73cbcb28e7f6fcd4a2b21b3122118b7755f002f90dcbf9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":1991,"rank":1991,"depth":4,"x":2052.008,"y":90.199,"cluster":"commutative-algebra"},{"id":"stacks:00TN","tag":"00TN","title":"Formally smooth maps · Proposition 00TN","summary":"Let R → S be a ring map. The following are equivalent • R → S is of finite presentation and formally smooth, • R → S is smooth.","statement_latex":"Let $R \\to S$ be a ring map. The following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is of finite presentation and formally smooth,\n\\item $R \\to S$ is smooth.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TN","source_file":"algebra.tex","source_line":38168,"source_end_line":38175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38168-L38175","statement_sha256":"a959654b0ed4b6475df07fb9a6488d3d1ffbdc941bc9f59cabddc67ff7d40c24","origin":"The Stacks Project","memory_eligible":false,"source_rank":1992,"rank":1992,"depth":5,"x":2118.195,"y":328.251,"cluster":"commutative-algebra"},{"id":"stacks:00TP","tag":"00TP","title":"Formally smooth maps · Lemma 00TP","summary":"Let R → S be a smooth ring map. Then there exists a subring R_0 ⊂ R of finite type over Z and a smooth ring map R_0 → S_0 such that S ≅ R ⊗_R_0 S_0.","statement_latex":"Let $R \\to S$ be a smooth ring map. Then there exists a subring\n$R_0 \\subset R$ of finite type over $\\mathbf{Z}$ and a smooth\nring map $R_0 \\to S_0$ such that $S \\cong R \\otimes_{R_0} S_0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TP","source_file":"algebra.tex","source_line":38186,"source_end_line":38191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38186-L38191","statement_sha256":"0c11806437b09e70d068592300deb300c6b939577e8eb6edf8adbbb9f1f531ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":1993,"rank":1993,"depth":6,"x":1877.845,"y":190.208,"cluster":"commutative-algebra"},{"id":"stacks:0CAQ","tag":"0CAQ","title":"Formally smooth maps · Lemma 0CAQ","summary":"Let A = colim A_i be a filtered colimit of rings. Let A → B be a smooth ring map. There exists an i and a smooth ring map A_i → B_i such that B = B_i ⊗_A_i A.","statement_latex":"Let $A = \\colim A_i$ be a filtered colimit of rings. Let\n$A \\to B$ be a smooth ring map. There exists an $i$ and\na smooth ring map $A_i \\to B_i$ such that $B = B_i \\otimes_{A_i} A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAQ","source_file":"algebra.tex","source_line":38229,"source_end_line":38234,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38229-L38234","statement_sha256":"8035cc9c292532da96b735f30b7dd0a038c8be9943e14eb110363601d692afa9","origin":"The Stacks Project","memory_eligible":false,"source_rank":1994,"rank":1994,"depth":7,"x":2166.217,"y":155.603,"cluster":"commutative-algebra"},{"id":"stacks:06CM","tag":"06CM","title":"Formally smooth maps · Lemma 06CM","summary":"Let R → S be a ring map. Let R → R' be a faithfully flat ring map. Set S' = S ⊗_R R'. Then R → S is formally smooth if and only if R' → S' is formally smooth.","statement_latex":"Let $R \\to S$ be a ring map. Let $R \\to R'$ be a faithfully flat ring map.\nSet $S' = S \\otimes_R R'$. Then $R \\to S$ is formally smooth if and only\nif $R' \\to S'$ is formally smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CM","source_file":"algebra.tex","source_line":38242,"source_end_line":38247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38242-L38247","statement_sha256":"fdacf704a1e34cd69d88c669b52aca91d820cf7199fed0ff49d84cd870bd9be6","origin":"The Stacks Project","memory_eligible":false,"source_rank":1995,"rank":1995,"depth":10,"x":1981.32,"y":344.835,"cluster":"commutative-algebra"},{"id":"stacks:07K4","tag":"07K4","title":"Formally smooth maps · Lemma 07K4","summary":"Let R → S be a smooth ring map. Given a commutative solid diagram xymatrix S ar[r] ar@-->[rd] & A/I R ar[r] ar[u] & A ar[u] where I ⊂ A is a locally nilpotent ideal, a dotted arrow exists which makes the diagram commute.","statement_latex":"Let $R \\to S$ be a smooth ring map. Given a commutative solid diagram\n$$\n\\xymatrix{\nS \\ar[r] \\ar@{-->}[rd] & A/I \\\\\nR \\ar[r] \\ar[u] & A \\ar[u]\n}\n$$\nwhere $I \\subset A$ is a locally nilpotent ideal, a dotted\narrow exists which makes the diagram commute.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally smooth maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07K4","source_file":"algebra.tex","source_line":38290,"source_end_line":38301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38290-L38301","statement_sha256":"ec559918b1bbac5a5c5126e05542c7ebbbeb32c7bfd9f20fd45e8075d75b2afd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1996,"rank":1996,"depth":7,"x":1965.477,"y":100.272,"cluster":"commutative-algebra"},{"id":"stacks:04B2","tag":"04B2","title":"Smoothness and differentials · Lemma 04B2","summary":"Given ring maps A → B → C with B → C smooth, then the sequence 0 → C ⊗_B Ω_B/A → Ω_C/A → Ω_C/B → 0 of Lemma [Tag 00RS] is exact.","statement_latex":"Given ring maps $A \\to B \\to C$ with $B \\to C$ smooth, then the sequence\n$$\n0 \\to C \\otimes_B \\Omega_{B/A} \\to \\Omega_{C/A} \\to \\Omega_{C/B} \\to 0\n$$\nof Lemma \\ref{lemma-exact-sequence-differentials} is exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smoothness and differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04B2","source_file":"algebra.tex","source_line":38346,"source_end_line":38353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38346-L38353","statement_sha256":"83a93a3794b1e89b2203bb0cc1eaf83641c9296b6eaf6e3be2da53312566eb33","origin":"The Stacks Project","memory_eligible":false,"source_rank":1997,"rank":1997,"depth":6,"x":2173.926,"y":271.698,"cluster":"commutative-algebra"},{"id":"stacks:06A8","tag":"06A8","title":"Smoothness and differentials · Lemma 06A8","summary":"Let A → B → C be ring maps with A → C smooth and B → C surjective with kernel J ⊂ B. Then the exact sequence 0 → J/J^2 → Ω_B/A ⊗_B C → Ω_C/A → 0 of Lemma [Tag 00RU] is split exact.","statement_latex":"Let $A \\to B \\to C$ be ring maps with $A \\to C$ smooth\nand $B \\to C$ surjective with kernel $J \\subset B$.\nThen the exact sequence\n$$\n0 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B C \\to \\Omega_{C/A} \\to 0\n$$\nof\nLemma \\ref{lemma-differential-seq}\nis split exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smoothness and differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06A8","source_file":"algebra.tex","source_line":38365,"source_end_line":38376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38365-L38376","statement_sha256":"6cab53a0e8ccb8276e29ff35fca2cb597cd2ec4bed694aef29d53e91e9a6e9bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":1998,"rank":1998,"depth":6,"x":1882.231,"y":263.566,"cluster":"commutative-algebra"},{"id":"stacks:06A9","tag":"06A9","title":"Smoothness and differentials · Lemma 06A9","summary":"Let A → B → C be ring maps. Assume A → C is surjective (so also B → C is) and A → B smooth. Denote I = Ker(A → C) and J = Ker(B → C). Then the sequence 0 → I/I^2 → J/J^2 → Ω_B/A ⊗_B B/J → 0 of Lemma [Tag 065V] is exact.","statement_latex":"Let $A \\to B \\to C$ be ring maps. Assume $A \\to C$ is surjective (so\nalso $B \\to C$ is) and $A \\to B$ smooth.\nDenote $I = \\Ker(A \\to C)$ and $J = \\Ker(B \\to C)$.\nThen the sequence\n$$\n0 \\to I/I^2 \\to J/J^2 \\to \\Omega_{B/A} \\otimes_B B/J \\to 0\n$$\nof\nLemma \\ref{lemma-application-NL}\nis exact.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smoothness and differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06A9","source_file":"algebra.tex","source_line":38385,"source_end_line":38397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38385-L38397","statement_sha256":"1a727f83237ca68144cb4dd3444297cfc0894de4fabd3c93a7a047e86766aa49","origin":"The Stacks Project","memory_eligible":false,"source_rank":1999,"rank":1999,"depth":6,"x":2103.962,"y":103.975,"cluster":"commutative-algebra"},{"id":"stacks:05D5","tag":"05D5","title":"Smoothness and differentials · Lemma 05D5","summary":"If R is a summand of S and S is smooth over R, then the I-adic completion of S is often a power series over R where I is the kernel of the projection map from S to R. Let φ : R → S be a smooth ring map. Let σ : S → R be a left inverse to φ. Set I = Ker(σ). Then • I/I^2 is a finite locally free R-module, and • if I/I^2 is free, then S^wedge ≅ R[[t_1, …, t_d]] as R-algebras, where S^wedge is the I-adic completion of S.","statement_latex":"\\begin{slogan}\nIf $R$ is a summand of $S$ and $S$ is smooth over $R$, then the\n$I$-adic completion of $S$ is often a power series over $R$\nwhere $I$ is the kernel of the projection map from $S$ to $R$.\n\\end{slogan}\nLet $\\varphi : R \\to S$ be a smooth ring map.\nLet $\\sigma : S \\to R$ be a left inverse to $\\varphi$.\nSet $I = \\Ker(\\sigma)$. Then\n\\begin{enumerate}\n\\item $I/I^2$ is a finite locally free $R$-module, and\n\\item if $I/I^2$ is free, then $S^\\wedge \\cong R[[t_1, \\ldots, t_d]]$\nas $R$-algebras, where $S^\\wedge$ is the $I$-adic completion of $S$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smoothness and differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05D5","source_file":"algebra.tex","source_line":38406,"source_end_line":38421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38406-L38421","statement_sha256":"bed5461a4a73e956d8fe07c2896f460c105b45e0933a6e380d6f7a0a12505a95","origin":"The Stacks Project","memory_eligible":false,"source_rank":2000,"rank":2000,"depth":6,"x":2068.783,"y":347.581,"cluster":"commutative-algebra"},{"id":"stacks:00TR","tag":"00TR","title":"Smooth algebras over fields · Lemma 00TR","summary":"Let k be an algebraically closed field. Let S be a finite type k-algebra. Let m ⊂ S be a maximal ideal. Then dim_kappa( m) Ω_S/k ⊗_S kappa( m) = dim_kappa( m) m/ m^2.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $S$ be a finite type $k$-algebra.\nLet $\\mathfrak m \\subset S$ be a maximal ideal.\nThen\n$$\n\\dim_{\\kappa(\\mathfrak m)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m)\n=\n\\dim_{\\kappa(\\mathfrak m)} \\mathfrak m/\\mathfrak m^2.\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TR","source_file":"algebra.tex","source_line":38474,"source_end_line":38485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38474-L38485","statement_sha256":"3a4ab4ebd433b825fe252036c7d11d97ac09d4570f9055418f88f278532bba0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2001,"rank":2001,"depth":7,"x":1898.745,"y":147.898,"cluster":"commutative-algebra"},{"id":"stacks:00TS","tag":"00TS","title":"Smooth algebras over fields · Lemma 00TS","summary":"Let k be an algebraically closed field. Let S be a finite type k-algebra. Let m ⊂ S be a maximal ideal. The following are equivalent: • The ring S_ m is a regular local ring. • We have dim_kappa( m) Ω_S/k ⊗_S kappa( m) ≤ dim(S_ m). • We have dim_kappa( m) Ω_S/k ⊗_S kappa( m) = dim(S_ m). • There exists a g ∈ S, g not ∈ m such that S_g is smooth over k. In other words S/k is smooth at m.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $S$ be a finite type $k$-algebra.\nLet $\\mathfrak m \\subset S$ be a maximal ideal.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The ring $S_{\\mathfrak m}$ is a regular local ring.\n\\item We have\n$\\dim_{\\kappa(\\mathfrak m)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m)\n\\leq \\dim(S_{\\mathfrak m})$.\n\\item We have\n$\\dim_{\\kappa(\\mathfrak m)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak m)\n= \\dim(S_{\\mathfrak m})$.\n\\item There exists a $g \\in S$, $g \\not \\in \\mathfrak m$\nsuch that $S_g$ is smooth over $k$. In other words $S/k$\nis smooth at $\\mathfrak m$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TS","source_file":"algebra.tex","source_line":38504,"source_end_line":38522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38504-L38522","statement_sha256":"c120dc2ef3b9c281a2affd0109daecd86d795cf4e9811fa26cb4db2764bc9e01","origin":"The Stacks Project","memory_eligible":false,"source_rank":2002,"rank":2002,"depth":35,"x":2184.838,"y":198.68,"cluster":"commutative-algebra"},{"id":"stacks:00TT","tag":"00TT","title":"Smooth algebras over fields · Lemma 00TT","summary":"Let k be any field. Let S be a finite type k-algebra. Let X = Spec(S). Let q ⊂ S be a prime corresponding to x ∈ X. The following are equivalent: • The k-algebra S is smooth at q over k. • We have dim_kappa( q) Ω_S/k ⊗_S kappa( q) ≤ dim_x X. • We have dim_kappa( q) Ω_S/k ⊗_S kappa( q) = dim_x X. Moreover, in this case the local ring S_ q is regular.","statement_latex":"Let $k$ be any field.\nLet $S$ be a finite type $k$-algebra.\nLet $X = \\Spec(S)$.\nLet $\\mathfrak q \\subset S$ be a prime\ncorresponding to $x \\in X$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The $k$-algebra $S$ is smooth at $\\mathfrak q$ over $k$.\n\\item We have\n$\\dim_{\\kappa(\\mathfrak q)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak q)\n\\leq \\dim_x X$.\n\\item We have\n$\\dim_{\\kappa(\\mathfrak q)} \\Omega_{S/k} \\otimes_S \\kappa(\\mathfrak q)\n= \\dim_x X$.\n\\end{enumerate}\nMoreover, in this case the local ring $S_{\\mathfrak q}$ is regular.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TT","source_file":"algebra.tex","source_line":38608,"source_end_line":38626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38608-L38626","statement_sha256":"4b8be8d79522d1d0323235bbcacc23b40dab4656feaf5844450af8a7982e9186","origin":"The Stacks Project","memory_eligible":false,"source_rank":2003,"rank":2003,"depth":36,"x":1932.926,"y":323.626,"cluster":"commutative-algebra"},{"id":"stacks:00TU","tag":"00TU","title":"Smooth algebras over fields · Lemma 00TU","summary":"Let k be a field. Let R be a Noetherian local ring containing k. Assume that the residue field kappa = R/ m is a finitely generated separable extension of k. Then the map d : m/ m^2 → Ω_R/k ⊗_R kappa( m) is injective.","statement_latex":"Let $k$ be a field.\nLet $R$ be a Noetherian local ring containing $k$.\nAssume that the residue field $\\kappa = R/\\mathfrak m$\nis a finitely generated separable extension of $k$.\nThen the map\n$$\n\\text{d} :\n\\mathfrak m/\\mathfrak m^2\n\\longrightarrow\n\\Omega_{R/k} \\otimes_R \\kappa(\\mathfrak m)\n$$\nis injective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TU","source_file":"algebra.tex","source_line":38686,"source_end_line":38700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38686-L38700","statement_sha256":"1e4fe12fd47696fb39df78a27c086bf709627ac2f8603c6d14477d423e2ee466","origin":"The Stacks Project","memory_eligible":false,"source_rank":2004,"rank":2004,"depth":3,"x":2018.242,"y":88.446,"cluster":"commutative-algebra"},{"id":"stacks:00TV","tag":"00TV","title":"Smooth algebras over fields · Lemma 00TV","summary":"Let k be a field. Let S be a finite type k-algebra. Let q ⊂ S be a prime. Assume kappa( q) is separable over k. The following are equivalent: • The algebra S is smooth at q over k. • The ring S_ q is regular.","statement_latex":"Let $k$ be a field.\nLet $S$ be a finite type $k$-algebra.\nLet $\\mathfrak q \\subset S$ be a prime.\nAssume $\\kappa(\\mathfrak q)$ is separable over $k$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The algebra $S$ is smooth at $\\mathfrak q$ over $k$.\n\\item The ring $S_{\\mathfrak q}$ is regular.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TV","source_file":"algebra.tex","source_line":38739,"source_end_line":38750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38739-L38750","statement_sha256":"5b8de7ae55ad045b3365fc314873fcdf5a8d87bc22fdf7a9041f1137de7ce56c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2005,"rank":2005,"depth":37,"x":2144.514,"y":310.375,"cluster":"commutative-algebra"},{"id":"stacks:00TW","tag":"00TW","title":"Smooth algebras over fields · Lemma 00TW","summary":"Let R → S be a Q-algebra map. Let f ∈ S be such that Ω_S/R = S df ⊕ C for some S-submodule C. Then • f is not nilpotent, and • if S is a Noetherian local ring, then f is a nonzerodivisor in S.","statement_latex":"Let $R \\to S$ be a $\\mathbf{Q}$-algebra map.\nLet $f \\in S$ be such that $\\Omega_{S/R} = S \\text{d}f \\oplus C$\nfor some $S$-submodule $C$. Then\n\\begin{enumerate}\n\\item $f$ is not nilpotent, and\n\\item if $S$ is a Noetherian local ring, then $f$ is a nonzerodivisor in $S$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TW","source_file":"algebra.tex","source_line":38783,"source_end_line":38792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38783-L38792","statement_sha256":"3cfca402c29106f19231a49e6e0dbc61ef71ea38b8547f345f3d399ee419850c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2006,"rank":2006,"depth":4,"x":1872.811,"y":218.336,"cluster":"commutative-algebra"},{"id":"stacks:00TX","tag":"00TX","title":"Smooth algebras over fields · Lemma 00TX","summary":"Let k be a field of characteristic 0. Let S be a finite type k-algebra. Let q ⊂ S be a prime. The following are equivalent: • The algebra S is smooth at q over k. • The S_ q-module Ω_S/k, q is (finite) free. • The ring S_ q is regular.","statement_latex":"Let $k$ be a field of characteristic $0$.\nLet $S$ be a finite type $k$-algebra.\nLet $\\mathfrak q \\subset S$ be a prime.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The algebra $S$ is smooth at $\\mathfrak q$ over $k$.\n\\item The $S_{\\mathfrak q}$-module $\\Omega_{S/k, \\mathfrak q}$\nis (finite) free.\n\\item The ring $S_{\\mathfrak q}$ is regular.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00TX","source_file":"algebra.tex","source_line":38820,"source_end_line":38832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38820-L38832","statement_sha256":"b10cb4ca598555481e09c9ccccebac8d76bab4dec4794468e70c301f3a249c0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2007,"rank":2007,"depth":38,"x":2147.3,"y":131.995,"cluster":"commutative-algebra"},{"id":"stacks:07ND","tag":"07ND","title":"Smooth algebras over fields · Lemma 07ND","summary":"Let R → S be an injective finite type ring map with R and S domains. Then R → S is smooth at q = (0) if and only if the induced extension L/K of fraction fields is separable.","statement_latex":"Let $R \\to S$ be an injective finite type ring map with $R$ and $S$ domains.\nThen $R \\to S$ is smooth at $\\mathfrak q = (0)$ if and only if\nthe induced extension $L/K$ of fraction fields is separable.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ND","source_file":"algebra.tex","source_line":38890,"source_end_line":38895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38890-L38895","statement_sha256":"83b581f8cf0547dcbba13fcb8be06cf3f88a220283b445d3ecd80b98baa69155","origin":"The Stacks Project","memory_eligible":false,"source_rank":2008,"rank":2008,"depth":38,"x":2014.269,"y":351.511,"cluster":"commutative-algebra"},{"id":"stacks:02HS","tag":"02HS","title":"Smooth ring maps in the Noetherian case · Definition 02HS","summary":"Let φ : B' → B be a ring map. We say φ is a small extension if B' and B are local Artinian rings, φ is surjective and I = Ker(φ) has length 1 as a B'-module.","statement_latex":"Let $\\varphi : B' \\to B$ be a ring map.\nWe say $\\varphi$ is a {\\it small extension} if\n$B'$ and $B$ are local Artinian rings, $\\varphi$ is surjective\nand $I = \\Ker(\\varphi)$ has length $1$ as a $B'$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps in the Noetherian case","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HS","source_file":"algebra.tex","source_line":38936,"source_end_line":38942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38936-L38942","statement_sha256":"9e72816678fc122db454aa54ba8d4055657d23712a13373c964b1c2e3d1bf1f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2009,"rank":2009,"depth":0,"x":1935.799,"y":114.052,"cluster":"commutative-algebra"},{"id":"stacks:02HT","tag":"02HT","title":"Smooth ring maps in the Noetherian case · Lemma 02HT","summary":"Let R → S be a ring map. Let q be a prime ideal of S lying over p ⊂ R. Assume R is Noetherian and R → S of finite type. The following are equivalent: • R → S is smooth at q, • for every surjection of local R-algebras (B', m') → (B, m) with Ker(B' → B) having square zero and every solid commutative diagram xymatrix S ar[r] ar@-->[rd] & B R ar[r] ar[u] & B' ar[u] such that q = S ∩ m there exists a dotted arrow making the diagram commute, • same as in (2) but with B' → B…","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime ideal of\n$S$ lying over $\\mathfrak p \\subset R$. Assume $R$ is Noetherian\nand $R \\to S$ of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $R \\to S$ is smooth at $\\mathfrak q$,\n\\item for every surjection of local $R$-algebras\n$(B', \\mathfrak m') \\to (B, \\mathfrak m)$\nwith $\\Ker(B' \\to B)$ having square zero\nand every solid commutative diagram\n$$\n\\xymatrix{\nS \\ar[r] \\ar@{-->}[rd] & B \\\\\nR \\ar[r] \\ar[u] & B' \\ar[u]\n}\n$$\nsuch that $\\mathfrak q = S \\cap \\mathfrak m$ there exists a dotted\narrow making the diagram commute,\n\\item same as in (2) but with $B' \\to B$ ranging over small extensions, and\n\\item same as in (2) but with $B' \\to B$ ranging over small extensions\nsuch that in addition $S \\to B$ induces an isomorphism\n$\\kappa(\\mathfrak q) \\cong \\kappa(\\mathfrak m)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Smooth ring maps in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HT","source_file":"algebra.tex","source_line":38949,"source_end_line":38974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L38949-L38974","statement_sha256":"648b5e05a67106cc1674c5a276c31d36fb6f85fe216c28c68b66b07f8f615694","origin":"The Stacks Project","memory_eligible":false,"source_rank":2010,"rank":2010,"depth":6,"x":2184.732,"y":244.684,"cluster":"commutative-algebra"},{"id":"stacks:00U1","tag":"00U1","title":"Étale ring maps · Definition 00U1","summary":"Let R → S be a ring map. We say R → S is étale if it is of finite presentation and the naive cotangent complex NL_S/R is quasi-isomorphic to zero: this means that H_1(NL_S/R) = 0 and Ω_S/R = 0. Given a prime q of S we say that R → S is étale at q if there exists a g ∈ S, g not ∈ q such that R → S_g is étale.","statement_latex":"Let $R \\to S$ be a ring map. We say $R \\to S$ is {\\it \\'etale} if it is\nof finite presentation and the naive cotangent complex\n$\\NL_{S/R}$ is quasi-isomorphic to zero: this means that $H_1(\\NL_{S/R}) = 0$\nand $\\Omega_{S/R} = 0$. Given a prime $\\mathfrak q$\nof $S$ we say that $R \\to S$ is {\\it \\'etale at $\\mathfrak q$}\nif there exists a $g \\in S$, $g \\not \\in \\mathfrak q$ such that\n$R \\to S_g$ is \\'etale.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00U1","source_file":"algebra.tex","source_line":39244,"source_end_line":39253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39244-L39253","statement_sha256":"757352ca9dfe0e2bb37da2386dff9daee21ca00ab16b661281e465ac864d3dbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2011,"rank":2011,"depth":0,"x":1895.993,"y":289.628,"cluster":"commutative-algebra"},{"id":"stacks:00U9","tag":"00U9","title":"Étale ring maps · Lemma 00U9","summary":"Any étale ring map is standard smooth. More precisely, if R → S is étale, then there exists a presentation S = R[x_1, …, x_n]/(f_1, …, f_n) such that the image of det(∂ f_j/∂ x_i) is invertible in S.","statement_latex":"Any \\'etale ring map is standard smooth. More precisely, if\n$R \\to S$ is \\'etale, then there exists a presentation\n$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$ such that\nthe image of $\\det(\\partial f_j/\\partial x_i)$ is invertible in $S$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00U9","source_file":"algebra.tex","source_line":39264,"source_end_line":39270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39264-L39270","statement_sha256":"8c6028b8d27216fd625016c968f1a85ce4756d9b9a14f1ee5bb579a56210a455","origin":"The Stacks Project","memory_eligible":false,"source_rank":2012,"rank":2012,"depth":5,"x":2072.841,"y":92.561,"cluster":"commutative-algebra"},{"id":"stacks:00U2","tag":"00U2","title":"Étale ring maps · Lemma 00U2","summary":"Results on étale ring maps. • The ring map R → R_f is étale for any ring R and any f ∈ R. • Compositions of étale ring maps are étale. • A base change of an étale ring map is étale. • The property of being étale is local: Given a ring map R → S and elements g_1, …, g_m ∈ S which generate the unit ideal such that R → S_g_j is étale for j = 1, …, m then R → S is étale. • Given R → S of finite presentation, and a flat ring map R → R', set S' = R' ⊗_R S. The set of primes…","statement_latex":"Results on \\'etale ring maps.\n\\begin{enumerate}\n\\item The ring map $R \\to R_f$ is \\'etale for any ring $R$ and any $f \\in R$.\n\\item Compositions of \\'etale ring maps are \\'etale.\n\\item A base change of an \\'etale ring map is \\'etale.\n\\item The property of being \\'etale is local: Given a ring map\n$R \\to S$ and elements $g_1, \\ldots, g_m \\in S$ which generate the unit ideal\nsuch that $R \\to S_{g_j}$ is \\'etale for $j = 1, \\ldots, m$ then\n$R \\to S$ is \\'etale.\n\\item Given $R \\to S$ of finite presentation, and a flat ring map\n$R \\to R'$, set $S' = R' \\otimes_R S$. The set of primes where $R' \\to S'$\nis \\'etale is the inverse image via $\\Spec(S') \\to \\Spec(S)$\nof the set of primes where $R \\to S$ is \\'etale.\n\\item An \\'etale ring map is syntomic, in particular flat.\n\\item If $S$ is finite type over a field $k$, then $S$ is \\'etale over\n$k$ if and only if $\\Omega_{S/k} = 0$.\n\\item Any \\'etale ring map $R \\to S$ is the base change of an \\'etale\nring map $R_0 \\to S_0$ with $R_0$ of finite type over $\\mathbf{Z}$.\n\\item Let $A = \\colim A_i$ be a filtered colimit of rings.\nLet $A \\to B$ be an \\'etale ring map. Then there exists an \\'etale ring\nmap $A_i \\to B_i$ for some $i$ such that $B \\cong A \\otimes_{A_i} B_i$.\n\\item Let $A$ be a ring. Let $S$ be a multiplicative subset of $A$.\nLet $S^{-1}A \\to B'$ be \\'etale. Then there exists an \\'etale ring map\n$A \\to B$ such that $B' \\cong S^{-1}B$.\n\\item Let $A$ be a ring. Let $B = B' \\times B''$ be a product of $A$-algebras.\nThen $B$ is \\'etale over $A$ if and only if both $B'$ and $B''$ are\n\\'etale over $A$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00U2","source_file":"algebra.tex","source_line":39289,"source_end_line":39319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39289-L39319","statement_sha256":"97dc584597fa855c8c32a7e8052a8ec09f101c442e2378823774ecc9858e155a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2013,"rank":2013,"depth":37,"x":2100.924,"y":338.334,"cluster":"commutative-algebra"},{"id":"stacks:00U3","tag":"00U3","title":"Étale ring maps · Lemma 00U3","summary":"Let k be a field. A ring map k → S is étale if and only if S is isomorphic as a k-algebra to a finite product of finite separable extensions of k.","statement_latex":"Let $k$ be a field. A ring map $k \\to S$ is \\'etale if and only if $S$\nis isomorphic as a $k$-algebra to a finite product of finite separable\nextensions of $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00U3","source_file":"algebra.tex","source_line":39393,"source_end_line":39398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39393-L39398","statement_sha256":"1171b17a408c81b4b00171dbcf2ce94f33d1fda1412b8f227010230ed146f8d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2014,"rank":2014,"depth":39,"x":1882.475,"y":172.966,"cluster":"commutative-algebra"},{"id":"stacks:00U4","tag":"00U4","title":"Étale ring maps · Lemma 00U4","summary":"Let R → S be a ring map. Let q ⊂ S be a prime lying over p in R. If S/R is étale at q then • we have p S_ q = qS_ q is the maximal ideal of the local ring S_ q, and • the field extension kappa( q)/kappa( p) is finite separable.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q \\subset S$ be a prime lying over $\\mathfrak p$ in $R$.\nIf $S/R$ is \\'etale at $\\mathfrak q$ then\n\\begin{enumerate}\n\\item we have $\\mathfrak p S_{\\mathfrak q} = \\mathfrak qS_{\\mathfrak q}$\nis the maximal ideal of the local ring $S_{\\mathfrak q}$, and\n\\item the field extension $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$\nis finite separable.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00U4","source_file":"algebra.tex","source_line":39434,"source_end_line":39445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39434-L39445","statement_sha256":"70f13817baf74c7cf1f28a5e45dcaa1f6f34e06b554a5739a8bee7ca6eb73157","origin":"The Stacks Project","memory_eligible":false,"source_rank":2015,"rank":2015,"depth":40,"x":2176.672,"y":170.951,"cluster":"commutative-algebra"},{"id":"stacks:00U5","tag":"00U5","title":"Étale ring maps · Lemma 00U5","summary":"An étale ring map is quasi-finite.","statement_latex":"An \\'etale ring map is quasi-finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00U5","source_file":"algebra.tex","source_line":39455,"source_end_line":39458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39455-L39458","statement_sha256":"7047d96b634d0cf8afacbb244749f70c2a5f91b2a2c4639ed51f31d368deefb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2016,"rank":2016,"depth":12,"x":1961.26,"y":339.446,"cluster":"commutative-algebra"},{"id":"stacks:00U6","tag":"00U6","title":"Étale ring maps · Lemma 00U6","summary":"Let R → S be a ring map. Let q be a prime of S lying over a prime p of R. If • R → S is of finite presentation, • R_ p → S_ q is flat • p S_ q is the maximal ideal of the local ring S_ q, and • the field extension kappa( q)/kappa( p) is finite separable, then R → S is étale at q.","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$\nlying over a prime $\\mathfrak p$ of $R$. If\n\\begin{enumerate}\n\\item $R \\to S$ is of finite presentation,\n\\item $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is flat\n\\item $\\mathfrak p S_{\\mathfrak q}$ is the maximal ideal\nof the local ring $S_{\\mathfrak q}$, and\n\\item the field extension $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$\nis finite separable,\n\\end{enumerate}\nthen $R \\to S$ is \\'etale at $\\mathfrak q$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00U6","source_file":"algebra.tex","source_line":39469,"source_end_line":39482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39469-L39482","statement_sha256":"e837f249c59b2cabbc25228e9ba20eb11db3514b723dd79f9e70d01f0a3c9da6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2017,"rank":2017,"depth":36,"x":1984.613,"y":92.861,"cluster":"commutative-algebra"},{"id":"stacks:00U7","tag":"00U7","title":"Étale ring maps · Lemma 00U7","summary":"Let R → S and R → S' be étale. Then any R-algebra map S' → S is étale.","statement_latex":"Let $R \\to S$ and $R \\to S'$ be \\'etale.\nThen any $R$-algebra map $S' \\to S$ is \\'etale.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00U7","source_file":"algebra.tex","source_line":39507,"source_end_line":39511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39507-L39511","statement_sha256":"5688b38d52d379e7618a3467b25ae2d5e6775cfdb7455d1194f6e825f2bd4e6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2018,"rank":2018,"depth":41,"x":2165.771,"y":288.027,"cluster":"commutative-algebra"},{"id":"stacks:00U8","tag":"00U8","title":"Étale ring maps · Lemma 00U8","summary":"Let φ : R → S be a ring map. If R → S is surjective, flat and finitely presented then there exist an idempotent e ∈ R such that S = R_e.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. If $R \\to S$ is surjective, flat and\nfinitely presented then there exist an idempotent $e \\in R$ such that\n$S = R_e$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00U8","source_file":"algebra.tex","source_line":39533,"source_end_line":39538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39533-L39538","statement_sha256":"0e3880620a1e4847231ad666e4fe4361e8c8853d881c49ea90f86a92566b0262","origin":"The Stacks Project","memory_eligible":false,"source_rank":2019,"rank":2019,"depth":4,"x":1875.11,"y":246.889,"cluster":"commutative-algebra"},{"id":"stacks:04D1","tag":"04D1","title":"Étale ring maps · Lemma 04D1","summary":"Étale ring maps lift along surjections of rings Let R be a ring and let I ⊂ R be an ideal. Let R/I → overlineS be an étale ring map. Then there exists an étale ring map R → S such that overlineS ≅ S/IS as R/I-algebras.","statement_latex":"\\begin{slogan}\n\\'Etale ring maps lift along surjections of rings\n\\end{slogan}\nLet $R$ be a ring and let $I \\subset R$ be an ideal.\nLet $R/I \\to \\overline{S}$ be an \\'etale ring map.\nThen there exists an \\'etale ring map\n$R \\to S$ such that $\\overline{S} \\cong S/IS$ as $R/I$-algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04D1","source_file":"algebra.tex","source_line":39564,"source_end_line":39573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39564-L39573","statement_sha256":"3059047afa3b6418c97a06ea0e41f12c67be0adfad3bb65bcb711c0eb410a3e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2020,"rank":2020,"depth":6,"x":2122.631,"y":112.236,"cluster":"commutative-algebra"},{"id":"stacks:05YT","tag":"05YT","title":"Étale ring maps · Lemma 05YT","summary":"Consider a commutative diagram xymatrix 0 ar[r] & J ar[r] & B' ar[r] & B ar[r] & 0 0 ar[r] & I ar[r] ar[u] & A' ar[r] ar[u] & A ar[r] ar[u] & 0 with exact rows where B' → B and A' → A are surjective ring maps whose kernels are ideals of square zero. If A → B is étale, and J = I ⊗_A B, then A' → B' is étale.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\nJ \\ar[r] &\nB' \\ar[r] &\nB \\ar[r] & 0 \\\\\n0 \\ar[r] &\nI \\ar[r] \\ar[u] &\nA' \\ar[r] \\ar[u] &\nA \\ar[r] \\ar[u] & 0\n}\n$$\nwith exact rows where $B' \\to B$ and $A' \\to A$ are surjective ring maps\nwhose kernels are ideals of square zero. If $A \\to B$ is \\'etale,\nand $J = I \\otimes_A B$, then $A' \\to B'$ is \\'etale.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YT","source_file":"algebra.tex","source_line":39589,"source_end_line":39607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39589-L39607","statement_sha256":"e1e0997a4e7df139ea50e916bbbf034c9806b9a03632f129f9dc21f7a817e173","origin":"The Stacks Project","memory_eligible":false,"source_rank":2021,"rank":2021,"depth":7,"x":2048.364,"y":352.083,"cluster":"commutative-algebra"},{"id":"stacks:00UH","tag":"00UH","title":"Étale ring maps · Lemma 00UH","summary":"Let R be a ring. Let f ∈ R[x] be a monic polynomial. Let p be a prime of R. Let f bmod p = overlineg overlineh be a factorization of the image of f in kappa( p)[x]. If gcd(overlineg, overlineh) = 1, then there exist • an étale ring map R → R', • a prime p' ⊂ R' lying over p, and • a factorization f = g h in R'[x] such that • kappa( p) = kappa( p'), • overlineg = g bmod p', overlineh = h bmod p', and • the polynomials g, h generate the unit ideal in R'[x].","statement_latex":"Let $R$ be a ring. Let $f \\in R[x]$ be a monic polynomial. Let $\\mathfrak p$\nbe a prime of $R$. Let $f \\bmod \\mathfrak p = \\overline{g} \\overline{h}$\nbe a factorization of the image of $f$ in $\\kappa(\\mathfrak p)[x]$.\nIf $\\gcd(\\overline{g}, \\overline{h}) = 1$, then there exist\n\\begin{enumerate}\n\\item an \\'etale ring map $R \\to R'$,\n\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$, and\n\\item a factorization $f = g h$ in $R'[x]$\n\\end{enumerate}\nsuch that\n\\begin{enumerate}\n\\item $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,\n\\item $\\overline{g} = g \\bmod \\mathfrak p'$,\n$\\overline{h} = h \\bmod \\mathfrak p'$, and\n\\item the polynomials $g, h$ generate the unit ideal in $R'[x]$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UH","source_file":"algebra.tex","source_line":39684,"source_end_line":39702,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39684-L39702","statement_sha256":"0aecb0a2dab865e1d45a752f3a78742ad82c355d96c5b1f0b169bf6098a29bd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2022,"rank":2022,"depth":0,"x":1910.188,"y":132.986,"cluster":"commutative-algebra"},{"id":"stacks:00UB","tag":"00UB","title":"Local structure of étale ring maps · Definition 00UB","summary":"Let R be a ring. Let g , f ∈ R[x]. Assume that f is monic and the derivative f' is invertible in the localization R[x]_g/(f). In this case the ring map R → R[x]_g/(f) is said to be standard étale.","statement_latex":"Let $R$ be a ring. Let $g , f  \\in R[x]$.\nAssume that $f$ is monic and the derivative $f'$ is invertible in\nthe localization $R[x]_g/(f)$.\nIn this case the ring map $R \\to R[x]_g/(f)$ is said to be\n{\\it standard \\'etale}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local structure of étale ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UB","source_file":"algebra.tex","source_line":39768,"source_end_line":39775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39768-L39775","statement_sha256":"2872814aaf870ffb1503299ea6ad82cdcff99130b799f8fc6c2f8c7c63a1ef2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2023,"rank":2023,"depth":0,"x":2188.392,"y":216.177,"cluster":"commutative-algebra"},{"id":"stacks:00UC","tag":"00UC","title":"Local structure of étale ring maps · Lemma 00UC","summary":"Let R → R[x]_g/(f) be standard étale. • The ring map R → R[x]_g/(f) is étale. • For any ring map R → R' the base change R' → R'[x]_g/(f) of the standard étale ring map R → R[x]_g/(f) is standard étale. • Any principal localization of R[x]_g/(f) is standard étale over R. • A composition of standard étale maps is not standard étale in general.","statement_latex":"Let $R \\to R[x]_g/(f)$ be standard \\'etale.\n\\begin{enumerate}\n\\item The ring map $R \\to R[x]_g/(f)$ is \\'etale.\n\\item For any ring map $R \\to R'$ the base change $R' \\to R'[x]_g/(f)$\nof the standard \\'etale ring map $R \\to R[x]_g/(f)$ is standard \\'etale.\n\\item Any principal localization of $R[x]_g/(f)$ is standard \\'etale over $R$.\n\\item A composition of standard \\'etale maps is {\\bf not} standard \\'etale\nin general.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local structure of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UC","source_file":"algebra.tex","source_line":39781,"source_end_line":39792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39781-L39792","statement_sha256":"5e2cd03b657b73331b9e0502bb7dcf404925e931800b7e6a625caba081a6d20b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2024,"rank":2024,"depth":0,"x":1916.228,"y":312.736,"cluster":"commutative-algebra"},{"id":"stacks:00UD","tag":"00UD","title":"Local structure of étale ring maps · Lemma 00UD","summary":"Let R be a ring. Let p be a prime of R. Let L/kappa( p) be a finite separable field extension. There exists an étale ring map R → R' together with a prime p' lying over p such that the field extension kappa( p')/kappa( p) is isomorphic to kappa( p) ⊂ L.","statement_latex":"Let $R$ be a ring.\nLet $\\mathfrak p$ be a prime of $R$.\nLet $L/\\kappa(\\mathfrak p)$ be a finite separable field extension.\nThere exists an \\'etale ring map $R \\to R'$ together with a prime $\\mathfrak p'$\nlying over $\\mathfrak p$ such that the field extension\n$\\kappa(\\mathfrak p')/\\kappa(\\mathfrak p)$ is isomorphic\nto $\\kappa(\\mathfrak p) \\subset L$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local structure of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UD","source_file":"algebra.tex","source_line":39809,"source_end_line":39818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39809-L39818","statement_sha256":"ffcf1672fa2025ce9c87b7068d9953e96c126c2db70a22a841d847ae13aae902","origin":"The Stacks Project","memory_eligible":false,"source_rank":2025,"rank":2025,"depth":0,"x":2039.322,"y":87.002,"cluster":"commutative-algebra"},{"id":"stacks:00UE","tag":"00UE","title":"Local structure of étale ring maps · Proposition 00UE","summary":"Let R → S be a ring map. Let q ⊂ S be a prime. If R → S is étale at q, then there exists a g ∈ S, g not ∈ q such that R → S_g is standard étale.","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime.\nIf $R \\to S$ is \\'etale at $\\mathfrak q$, then there exists\na $g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$\nis standard \\'etale.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local structure of étale ring maps","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UE","source_file":"algebra.tex","source_line":39843,"source_end_line":39849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L39843-L39849","statement_sha256":"dfe26521de7af37666fad167f81f978bccaea2572d8aa59adaf1ab8a83e13969","origin":"The Stacks Project","memory_eligible":false,"source_rank":2026,"rank":2026,"depth":42,"x":2130.123,"y":323.406,"cluster":"commutative-algebra"},{"id":"stacks:00UF","tag":"00UF","title":"Local structure of étale ring maps · Lemma 00UF","summary":"Let R → S be a standard étale morphism. There exists a ring map R → S' with the following properties • R → S' is finite, finitely presented, and flat (in other words S' is finite projective as an R-module), • Spec(S') → Spec(R) is surjective, • for every prime q ⊂ S, lying over p ⊂ R and every prime q' ⊂ S' lying over p there exists a g' ∈ S', g' not ∈ q' such that the ring map R → S'_g' factors through a map φ : S → S'_g' with φ^-1( q'S'_g') = q.","statement_latex":"Let $R \\to S$ be a standard \\'etale morphism.\nThere exists a ring map $R \\to S'$ with the following properties\n\\begin{enumerate}\n\\item $R \\to S'$ is finite, finitely presented, and flat\n(in other words $S'$ is finite projective as an $R$-module),\n\\item $\\Spec(S') \\to \\Spec(R)$ is surjective,\n\\item for every prime $\\mathfrak q \\subset S$, lying over\n$\\mathfrak p \\subset R$ and every prime\n$\\mathfrak q' \\subset S'$ lying over $\\mathfrak p$ there exists\na $g' \\in S'$, $g' \\not \\in \\mathfrak q'$\nsuch that the ring map $R \\to S'_{g'}$ factors\nthrough a map $\\varphi : S \\to S'_{g'}$ with\n$\\varphi^{-1}(\\mathfrak q'S'_{g'}) = \\mathfrak q$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local structure of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UF","source_file":"algebra.tex","source_line":40069,"source_end_line":40085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40069-L40085","statement_sha256":"0cc5cbbd7f417e3002d07f007d555d6b598386a6ce34dd78b40de71bbc41ce79","origin":"The Stacks Project","memory_eligible":false,"source_rank":2027,"rank":2027,"depth":35,"x":1872.945,"y":200.554,"cluster":"commutative-algebra"},{"id":"stacks:00UG","tag":"00UG","title":"Local structure of étale ring maps · Lemma 00UG","summary":"Let R → S be a ring map. Assume that • R → S is étale, and • Spec(S) → Spec(R) is surjective. Then there exists a ring map R → S' such that • R → S' is finite, finitely presented, and flat (in other words it is finite projective as an R-module), • Spec(S') → Spec(R) is surjective, • for every prime q' ⊂ S' there exists a g' ∈ S', g' not ∈ q' such that the ring map R → S'_g' factors as R → S → S'_g'.","statement_latex":"Let $R \\to S$ be a ring map.\nAssume that\n\\begin{enumerate}\n\\item $R \\to S$ is \\'etale, and\n\\item $\\Spec(S) \\to \\Spec(R)$ is surjective.\n\\end{enumerate}\nThen there exists a ring map $R \\to S'$ such that\n\\begin{enumerate}\n\\item $R \\to S'$ is finite, finitely presented, and flat\n(in other words it is finite projective as an $R$-module),\n\\item $\\Spec(S') \\to \\Spec(R)$ is surjective,\n\\item for every prime $\\mathfrak q' \\subset S'$ there exists a\n$g' \\in S'$, $g' \\not \\in \\mathfrak q'$ such that\nthe ring map $R \\to S'_{g'}$ factors as $R \\to S \\to S'_{g'}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local structure of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UG","source_file":"algebra.tex","source_line":40130,"source_end_line":40147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40130-L40147","statement_sha256":"9b34909882b2b51b4678c402d6fc4588c21c32632b9c60d2b8d7bb5f95dae1fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2028,"rank":2028,"depth":43,"x":2161.506,"y":145.19,"cluster":"commutative-algebra"},{"id":"stacks:00UI","tag":"00UI","title":"Étale local structure of quasi-finite ring maps · Lemma 00UI","summary":"Let R → S' → S be ring maps. Let p ⊂ R be a prime. Let g ∈ S' be an element. Assume • R → S' is integral, • R → S is finite type, • S'_g ≅ S_g, and • g invertible in S' ⊗_R kappa( p). Then there exists a f ∈ R, f not ∈ p such that R_f → S_f is finite.","statement_latex":"Let $R \\to S' \\to S$ be ring maps.\nLet $\\mathfrak p \\subset R$ be a prime.\nLet $g \\in S'$ be an element.\nAssume\n\\begin{enumerate}\n\\item $R \\to S'$ is integral,\n\\item $R \\to S$ is finite type,\n\\item $S'_g \\cong S_g$, and\n\\item $g$ invertible in $S' \\otimes_R \\kappa(\\mathfrak p)$.\n\\end{enumerate}\nThen there exists a $f \\in R$, $f \\not \\in \\mathfrak p$ such\nthat $R_f \\to S_f$ is finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale local structure of quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UI","source_file":"algebra.tex","source_line":40200,"source_end_line":40214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40200-L40214","statement_sha256":"9ab3bcfd91ea01db166926697991561708e99bcb7242df9505007ae2290a083c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2029,"rank":2029,"depth":5,"x":1993.174,"y":349.841,"cluster":"commutative-algebra"},{"id":"stacks:00UJ","tag":"00UJ","title":"Étale local structure of quasi-finite ring maps · Lemma 00UJ","summary":"Let R → S be a ring map. Let q ⊂ S be a prime lying over the prime p ⊂ R. Assume R → S finite type and quasi-finite at q. Then there exists • an étale ring map R → R', • a prime p' ⊂ R' lying over p, • a product decomposition R' ⊗_R S = A × B with the following properties • kappa( p) = kappa( p'), • R' → A is finite, • A has exactly one prime r lying over p', • r lies over q, and • B does not have a prime lying over q and p'.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q \\subset S$ be a prime lying over\nthe prime $\\mathfrak p \\subset R$.\nAssume $R \\to S$ finite type and quasi-finite at $\\mathfrak q$.\nThen there exists\n\\begin{enumerate}\n\\item an \\'etale ring map $R \\to R'$,\n\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$,\n\\item a product decomposition\n$$\nR' \\otimes_R S = A \\times B\n$$\n\\end{enumerate}\nwith the following properties\n\\begin{enumerate}\n\\item $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,\n\\item $R' \\to A$ is finite,\n\\item $A$ has exactly one prime $\\mathfrak r$ lying over $\\mathfrak p'$,\n\\item $\\mathfrak r$ lies over $\\mathfrak q$, and\n\\item $B$ does not have a prime lying over $\\mathfrak q$ and $\\mathfrak p'$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale local structure of quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UJ","source_file":"algebra.tex","source_line":40227,"source_end_line":40250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40227-L40250","statement_sha256":"c7cef85fe681952da981e99d262d397946a7f611afac1b94f1d8190a051e7fba","origin":"The Stacks Project","memory_eligible":false,"source_rank":2030,"rank":2030,"depth":28,"x":1952.707,"y":103.31,"cluster":"commutative-algebra"},{"id":"stacks:00UK","tag":"00UK","title":"Étale local structure of quasi-finite ring maps · Lemma 00UK","summary":"Let R → S be a ring map. Let p ⊂ R be a prime. Assume R → S finite type. Then there exists • an étale ring map R → R', • a prime p' ⊂ R' lying over p, • a product decomposition R' ⊗_R S = A_1 × … × A_n × B with the following properties • we have kappa( p) = kappa( p'), • each A_i is finite over R', • each A_i has exactly one prime r_i lying over p', and • R' → B not quasi-finite at any prime lying over p'.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak p \\subset R$ be a prime.\nAssume $R \\to S$ finite type.\nThen there exists\n\\begin{enumerate}\n\\item an \\'etale ring map $R \\to R'$,\n\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$,\n\\item a product decomposition\n$$\nR' \\otimes_R S = A_1 \\times \\ldots \\times A_n \\times B\n$$\n\\end{enumerate}\nwith the following properties\n\\begin{enumerate}\n\\item we have $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,\n\\item each $A_i$ is finite over $R'$,\n\\item each $A_i$ has exactly one prime $\\mathfrak r_i$ lying over\n$\\mathfrak p'$, and\n\\item $R' \\to B$ not quasi-finite at any prime lying over $\\mathfrak p'$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale local structure of quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UK","source_file":"algebra.tex","source_line":40361,"source_end_line":40383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40361-L40383","statement_sha256":"032bcbf47dc0764fa8384eb3b6406e828cc8a70e4de640e3bf1c19e3099aa307","origin":"The Stacks Project","memory_eligible":false,"source_rank":2031,"rank":2031,"depth":29,"x":2180.9,"y":262.206,"cluster":"commutative-algebra"},{"id":"stacks:00UL","tag":"00UL","title":"Étale local structure of quasi-finite ring maps · Lemma 00UL","summary":"Let R → S be a ring map. Let p ⊂ R be a prime. Assume R → S finite type. Then there exists • an étale ring map R → R', • a prime p' ⊂ R' lying over p, • a product decomposition R' ⊗_R S = A_1 × … × A_n × B with the following properties • each A_i is finite over R', • each A_i has exactly one prime r_i lying over p', • the finite field extensions kappa( r_i)/kappa( p') are purely inseparable, and • R' → B not quasi-finite at any prime lying over p'.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak p \\subset R$ be a prime.\nAssume $R \\to S$ finite type.\nThen there exists\n\\begin{enumerate}\n\\item an \\'etale ring map $R \\to R'$,\n\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$,\n\\item a product decomposition\n$$\nR' \\otimes_R S = A_1 \\times \\ldots \\times A_n \\times B\n$$\n\\end{enumerate}\nwith the following properties\n\\begin{enumerate}\n\\item each $A_i$ is finite over $R'$,\n\\item each $A_i$ has exactly one prime $\\mathfrak r_i$ lying over\n$\\mathfrak p'$,\n\\item the finite field extensions\n$\\kappa(\\mathfrak r_i)/\\kappa(\\mathfrak p')$\nare purely inseparable, and\n\\item $R' \\to B$ not quasi-finite at any prime lying over $\\mathfrak p'$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Étale local structure of quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UL","source_file":"algebra.tex","source_line":40420,"source_end_line":40444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40420-L40444","statement_sha256":"4d8a7147e493ffe4c4637f0fb51731ca57924f5b7f77169635b7f563e5608f72","origin":"The Stacks Project","memory_eligible":false,"source_rank":2032,"rank":2032,"depth":30,"x":1884.724,"y":274.526,"cluster":"commutative-algebra"},{"id":"stacks:0GSD","tag":"0GSD","title":"Local homomorphisms · Lemma 0GSD","summary":"[Lindel], [KC] Let (R, m_R) → (S, m_S) be a local homomorphism of local rings. Assume S is the localization of an étale ring extension of R and that kappa( m_R) → kappa( m_S) is an isomorphism. Then there exists an t ∈ m_R such that R/t^nR → S/t^nS is an isomorphsm for all n ≥ 1.","statement_latex":"\\begin{reference}\n\\cite[Lemma on page 321]{Lindel}, \\cite[Lemma 4.1.5]{KC}\n\\end{reference}\nLet $(R, \\mathfrak m_R) \\to (S, \\mathfrak m_S)$ be a local homomorphism\nof local rings. Assume $S$ is the localization of an \\'etale ring extension\nof $R$ and that $\\kappa(\\mathfrak m_R) \\to \\kappa(\\mathfrak m_S)$\nis an isomorphism. Then there exists an $t \\in \\mathfrak m_R$ such that\n$R/t^nR \\to S/t^nS$ is an isomorphsm for all $n \\geq 1$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSD","source_file":"algebra.tex","source_line":40506,"source_end_line":40516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40506-L40516","statement_sha256":"58e2521dfff3bdedaeb26cf0d4a1f4f0377260fd9c456d7b9ecebad185700534","origin":"The Stacks Project","memory_eligible":false,"source_rank":2033,"rank":2033,"depth":43,"x":2093.304,"y":97.306,"cluster":"commutative-algebra"},{"id":"stacks:053K","tag":"053K","title":"Local homomorphisms · Lemma 053K","summary":"Let (R, m_R) → (S, m_S) be a local homomorphism of local rings. Assume S is the localization of an étale ring extension of R. Then there exists a finite, finitely presented, faithfully flat ring map R → S' such that for every maximal ideal m' of S' there is a factorization R → S → S'_ m'. of the ring map R → S'_ m'.","statement_latex":"Let $(R, \\mathfrak m_R) \\to (S, \\mathfrak m_S)$ be a local homomorphism\nof local rings. Assume $S$ is the localization of an \\'etale ring extension\nof $R$. Then there exists a finite, finitely presented, faithfully flat\nring map $R \\to S'$ such that for every maximal ideal $\\mathfrak m'$ of $S'$\nthere is a factorization\n$$\nR \\to S \\to S'_{\\mathfrak m'}.\n$$\nof the ring map $R \\to S'_{\\mathfrak m'}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053K","source_file":"algebra.tex","source_line":40556,"source_end_line":40567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40556-L40567","statement_sha256":"103f473e8296bd589b74d4355b096df663b8680fa5fc218500940cf9882bc58f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2034,"rank":2034,"depth":43,"x":2082.011,"y":346.448,"cluster":"commutative-algebra"},{"id":"stacks:03GD","tag":"03GD","title":"Integral closure and smooth base change · Lemma 03GD","summary":"Let R be a ring. Let f ∈ R[x] be a monic polynomial. Let R → B be a ring map. If h ∈ B[x]/(f) is integral over R, then the element f' h can be written as f'h = ∑_i b_i x^i with b_i ∈ B integral over R.","statement_latex":"Let $R$ be a ring.\nLet $f \\in R[x]$ be a monic polynomial.\nLet $R \\to B$ be a ring map.\nIf $h \\in B[x]/(f)$ is integral over $R$, then the element\n$f' h$ can be written as $f'h = \\sum_i b_i x^i$ with $b_i \\in B$\nintegral over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Integral closure and smooth base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GD","source_file":"algebra.tex","source_line":40589,"source_end_line":40597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40589-L40597","statement_sha256":"72f7385a20e1450f18f7c5db226d2ecbb69f83843f65141fedf5cff31bc374db","origin":"The Stacks Project","memory_eligible":false,"source_rank":2035,"rank":2035,"depth":35,"x":1889.901,"y":156.243,"cluster":"commutative-algebra"},{"id":"stacks:03GE","tag":"03GE","title":"Integral closure and smooth base change · Lemma 03GE","summary":"Let R → S be an étale ring map. Let R → B be any ring map. Let A ⊂ B be the integral closure of R in B. Let A' ⊂ S ⊗_R B be the integral closure of S in S ⊗_R B. Then the canonical map S ⊗_R A → A' is an isomorphism.","statement_latex":"Let $R \\to S$ be an \\'etale ring map.\nLet $R \\to B$ be any ring map.\nLet $A \\subset B$ be the integral closure of $R$ in $B$.\nLet $A' \\subset S \\otimes_R B$ be the integral closure of $S$ in\n$S \\otimes_R B$. Then the canonical map $S \\otimes_R A \\to A'$ is\nan isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Integral closure and smooth base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GE","source_file":"algebra.tex","source_line":40630,"source_end_line":40638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40630-L40638","statement_sha256":"062ac71e1da6056719796bcee73e94a67bbf80ec29f3443c564d10e594e8901e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2036,"rank":2036,"depth":43,"x":2184.646,"y":187.503,"cluster":"commutative-algebra"},{"id":"stacks:03GG","tag":"03GG","title":"Integral closure and smooth base change · Lemma 03GG","summary":"Let R → S be a smooth ring map. Let R → B be any ring map. Let A ⊂ B be the integral closure of R in B. Let A' ⊂ S ⊗_R B be the integral closure of S in S ⊗_R B. Then the canonical map S ⊗_R A → A' is an isomorphism.","statement_latex":"Let $R \\to S$ be a smooth ring map.\nLet $R \\to B$ be any ring map.\nLet $A \\subset B$ be the integral closure of $R$ in $B$.\nLet $A' \\subset S \\otimes_R B$ be the integral closure of $S$ in\n$S \\otimes_R B$. Then the canonical map $S \\otimes_R A \\to A'$ is\nan isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Integral closure and smooth base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GG","source_file":"algebra.tex","source_line":40691,"source_end_line":40699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40691-L40699","statement_sha256":"497c5688772bef37db93a4a6c70723fd6681a0479759e96473a2b3e9c393d002","origin":"The Stacks Project","memory_eligible":false,"source_rank":2037,"rank":2037,"depth":44,"x":1942.061,"y":331.761,"cluster":"commutative-algebra"},{"id":"stacks:0CBF","tag":"0CBF","title":"Integral closure and smooth base change · Lemma 0CBF","summary":"Let R → S and R → B be ring maps. Let A ⊂ B be the integral closure of R in B. Let A' ⊂ S ⊗_R B be the integral closure of S in S ⊗_R B. If S is a filtered colimit of smooth R-algebras, then the canonical map S ⊗_R A → A' is an isomorphism.","statement_latex":"Let $R \\to S$ and $R \\to B$ be ring maps.\nLet $A \\subset B$ be the integral closure of $R$ in $B$.\nLet $A' \\subset S \\otimes_R B$ be the integral closure of $S$ in\n$S \\otimes_R B$. If $S$ is a filtered colimit of smooth $R$-algebras,\nthen the canonical map $S \\otimes_R A \\to A'$ is an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Integral closure and smooth base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBF","source_file":"algebra.tex","source_line":40748,"source_end_line":40755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40748-L40755","statement_sha256":"52427cfa21a0db7368972ef06919505662d4d6bf5785978775529483decec9d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2038,"rank":2038,"depth":45,"x":2004.959,"y":87.632,"cluster":"commutative-algebra"},{"id":"stacks:00UN","tag":"00UN","title":"Formally unramified maps · Definition 00UN","summary":"Let R → S be a ring map. We say S is formally unramified over R if for every commutative solid diagram xymatrix S ar[r] ar@-->[rd] & A/I R ar[r] ar[u] & A ar[u] where I ⊂ A is an ideal of square zero, there exists at most one dotted arrow making the diagram commute.","statement_latex":"Let $R \\to S$ be a ring map.\nWe say $S$ is {\\it formally unramified over $R$} if for every\ncommutative solid diagram\n$$\n\\xymatrix{\nS \\ar[r] \\ar@{-->}[rd] & A/I \\\\\nR \\ar[r] \\ar[u] & A \\ar[u]\n}\n$$\nwhere $I \\subset A$ is an ideal of square zero, there exists\nat most one dotted arrow making the diagram commute.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally unramified maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UN","source_file":"algebra.tex","source_line":40777,"source_end_line":40790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40777-L40790","statement_sha256":"447b001e8b38b15a6ec8b3e24a0fc61cab14241d2965eab07ef68a88f7ba873b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2039,"rank":2039,"depth":0,"x":2154.966,"y":303.433,"cluster":"commutative-algebra"},{"id":"stacks:0H91","tag":"0H91","title":"Formally unramified maps · Lemma 0H91","summary":"Let R → S be a formally unramified map. Let R → R' be any ring map. Then the base change S' = R' ⊗_R S is formally unramified over R'.","statement_latex":"Let $R \\to S$ be a formally unramified map.\nLet $R \\to R'$ be any ring map.\nThen the base change $S' = R' \\otimes_R S$ is formally unramified over $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally unramified maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H91","source_file":"algebra.tex","source_line":40792,"source_end_line":40797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40792-L40797","statement_sha256":"c3c8456c5bfe2df98c2d3a42b4bcf605eb6308a075d848eb2fb0d96122c3f3f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2040,"rank":2040,"depth":1,"x":1870.688,"y":229.39,"cluster":"commutative-algebra"},{"id":"stacks:00UO","tag":"00UO","title":"Formally unramified maps · Lemma 00UO","summary":"Let R → S be a ring map. The following are equivalent: • R → S is formally unramified, • the module of differentials Ω_S/R is zero.","statement_latex":"Let $R \\to S$ be a ring map.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $R \\to S$ is formally unramified,\n\\item the module of differentials $\\Omega_{S/R}$ is zero.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally unramified maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UO","source_file":"algebra.tex","source_line":40813,"source_end_line":40821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40813-L40821","statement_sha256":"46c6f67861a74deaa33a5b0ff3aad141ada8b231be5781e4bf54d9ade2631f9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2041,"rank":2041,"depth":4,"x":2139.971,"y":122.636,"cluster":"commutative-algebra"},{"id":"stacks:04E8","tag":"04E8","title":"Formally unramified maps · Lemma 04E8","summary":"Let R → S be a ring map. The following are equivalent: • R → S is formally unramified, • R → S_ q is formally unramified for all primes q of S, and • R_ p → S_ q is formally unramified for all primes q of S with p = R ∩ q.","statement_latex":"Let $R \\to S$ be a ring map.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $R \\to S$ is formally unramified,\n\\item $R \\to S_{\\mathfrak q}$ is formally unramified for all\nprimes $\\mathfrak q$ of $S$, and\n\\item $R_{\\mathfrak p} \\to S_{\\mathfrak q}$ is formally unramified\nfor all primes $\\mathfrak q$ of $S$ with $\\mathfrak p = R \\cap \\mathfrak q$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally unramified maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04E8","source_file":"algebra.tex","source_line":40848,"source_end_line":40859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40848-L40859","statement_sha256":"c4f088472ddb1f19fcf809c903c9909c2e1da5737ba297207eded7d30d5eae8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2042,"rank":2042,"depth":5,"x":2027.206,"y":354.253,"cluster":"commutative-algebra"},{"id":"stacks:04E9","tag":"04E9","title":"Formally unramified maps · Lemma 04E9","summary":"Let A → B be a formally unramified ring map. • For S ⊂ A a multiplicative subset, S^-1A → S^-1B is formally unramified. • For S ⊂ B a multiplicative subset, A → S^-1B is formally unramified.","statement_latex":"Let $A \\to B$ be a formally unramified ring map.\n\\begin{enumerate}\n\\item For $S \\subset A$ a multiplicative subset,\n$S^{-1}A \\to S^{-1}B$ is formally unramified.\n\\item For $S \\subset B$ a multiplicative subset,\n$A \\to S^{-1}B$ is formally unramified.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally unramified maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04E9","source_file":"algebra.tex","source_line":40873,"source_end_line":40882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40873-L40882","statement_sha256":"a7fba15d0a1c4d4723223053350d7f8d5a890a07f19a6ac55035c484decc3a2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2043,"rank":2043,"depth":6,"x":1924.052,"y":119.374,"cluster":"commutative-algebra"},{"id":"stacks:07QE","tag":"07QE","title":"Formally unramified maps · Lemma 07QE","summary":"Let R be a ring. Let I be a directed set. Let (S_i, φ_ii') be a system of R-algebras over I. If each R → S_i is formally unramified, then S = colim_i ∈ I S_i is formally unramified over R","statement_latex":"Let $R$ be a ring. Let $I$ be a directed set.\nLet $(S_i, \\varphi_{ii'})$ be a system of $R$-algebras\nover $I$. If each $R \\to S_i$ is formally unramified, then\n$S = \\colim_{i \\in I} S_i$ is formally unramified over $R$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally unramified maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QE","source_file":"algebra.tex","source_line":40893,"source_end_line":40899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40893-L40899","statement_sha256":"e394f84e2384b30194dd25644b205e293f8483745a2a54253c516bd532497d34","origin":"The Stacks Project","memory_eligible":false,"source_rank":2044,"rank":2044,"depth":1,"x":2189.114,"y":234.088,"cluster":"commutative-algebra"},{"id":"stacks:04EB","tag":"04EB","title":"Conormal modules and universal thickenings · Lemma 04EB","summary":"Let R → S be a formally unramified ring map. There exists a surjection of R-algebras S' → S whose kernel is an ideal of square zero with the following universal property: Given any commutative diagram xymatrix S ar[r]_a & A/I R ar[r]^b ar[u] & A ar[u] where I ⊂ A is an ideal of square zero, there is a unique R-algebra map a' : S' → A such that S' → A → A/I is equal to S' → S → A/I.","statement_latex":"Let $R \\to S$ be a formally unramified ring map. There exists a surjection of\n$R$-algebras $S' \\to S$ whose kernel is an ideal of square zero with the\nfollowing universal property: Given any commutative diagram\n$$\n\\xymatrix{\nS \\ar[r]_a & A/I \\\\\nR \\ar[r]^b \\ar[u] & A \\ar[u]\n}\n$$\nwhere $I \\subset A$ is an ideal of square zero, there is a unique $R$-algebra\nmap $a' : S' \\to A$ such that $S' \\to A \\to A/I$ is equal to $S' \\to S \\to A/I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Conormal modules and universal thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EB","source_file":"algebra.tex","source_line":40919,"source_end_line":40932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L40919-L40932","statement_sha256":"e556f0a779112f8db96c061e1e0830b89c31b43f196d89bb018173c744fa714d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2045,"rank":2045,"depth":5,"x":1901.286,"y":299.931,"cluster":"commutative-algebra"},{"id":"stacks:04EC","tag":"04EC","title":"Conormal modules and universal thickenings · Definition 04EC","summary":"Let R → S be a formally unramified ring map. • The universal first order thickening of S over R is the surjection of R-algebras S' → S of Lemma [Tag 04EB]. • The conormal module of R → S is the kernel I of the universal first order thickening S' → S, seen as an S-module. We often denote the conormal module C_S/R in this situation.","statement_latex":"Let $R \\to S$ be a formally unramified ring map.\n\\begin{enumerate}\n\\item The {\\it universal first order thickening} of $S$ over $R$ is\nthe surjection of $R$-algebras $S' \\to S$ of\nLemma \\ref{lemma-universal-thickening}.\n\\item The {\\it conormal module} of $R \\to S$ is the kernel $I$ of the\nuniversal first order thickening $S' \\to S$, seen as an $S$-module.\n\\end{enumerate}\nWe often denote the conormal module {\\it $C_{S/R}$} in this situation.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Conormal modules and universal thickenings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EC","source_file":"algebra.tex","source_line":41004,"source_end_line":41015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41004-L41015","statement_sha256":"3b1e6c080d5f81f2ec2f0f7499d9e78e500753a851cdda7b10f06e89515a5ea7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2046,"rank":2046,"depth":6,"x":2060.646,"y":87.967,"cluster":"commutative-algebra"},{"id":"stacks:04ED","tag":"04ED","title":"Conormal modules and universal thickenings · Lemma 04ED","summary":"Let I ⊂ R be an ideal of a ring. The universal first order thickening of R/I over R is the surjection R/I^2 → R/I. The conormal module of R/I over R is C_(R/I)/R = I/I^2.","statement_latex":"Let $I \\subset R$ be an ideal of a ring.\nThe universal first order thickening of $R/I$ over $R$\nis the surjection $R/I^2 \\to R/I$. The conormal module\nof $R/I$ over $R$ is $C_{(R/I)/R} = I/I^2$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Conormal modules and universal thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ED","source_file":"algebra.tex","source_line":41017,"source_end_line":41023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41017-L41023","statement_sha256":"1775e2b36e92b531e6d97da5b135c057cc7d8b53ff9011b7b9b74891f53f749c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2047,"rank":2047,"depth":0,"x":2113.615,"y":334.798,"cluster":"commutative-algebra"},{"id":"stacks:04EE","tag":"04EE","title":"Conormal modules and universal thickenings · Lemma 04EE","summary":"Let A → B be a formally unramified ring map. Let φ : B' → B be the universal first order thickening of B over A. • Let S ⊂ A be a multiplicative subset. Then S^-1B' → S^-1B is the universal first order thickening of S^-1B over S^-1A. In particular S^-1C_B/A = C_S^-1B/S^-1A. • Let S ⊂ B be a multiplicative subset. Then S' = φ^-1(S) is a multiplicative subset in B' and (S')^-1B' → S^-1B is the universal first order thickening of S^-1B over A. In particular S^-1C_B/A =…","statement_latex":"Let $A \\to B$ be a formally unramified ring map.\nLet $\\varphi : B' \\to B$ be the universal first order thickening of\n$B$ over $A$.\n\\begin{enumerate}\n\\item Let $S \\subset A$ be a multiplicative subset.\nThen $S^{-1}B' \\to S^{-1}B$ is the universal first order thickening of\n$S^{-1}B$ over $S^{-1}A$. In particular $S^{-1}C_{B/A} = C_{S^{-1}B/S^{-1}A}$.\n\\item Let $S \\subset B$ be a multiplicative subset.\nThen $S' = \\varphi^{-1}(S)$ is a multiplicative subset in $B'$\nand $(S')^{-1}B' \\to S^{-1}B$ is the universal first order thickening\nof $S^{-1}B$ over $A$. In particular $S^{-1}C_{B/A} = C_{S^{-1}B/A}$.\n\\end{enumerate}\nNote that the lemma makes sense by\nLemma \\ref{lemma-formally-unramified-localize}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Conormal modules and universal thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EE","source_file":"algebra.tex","source_line":41029,"source_end_line":41045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41029-L41045","statement_sha256":"48e1cd38c2895ece0584c2c5ad9e4646d97474cb71a215e166601787196bee22","origin":"The Stacks Project","memory_eligible":false,"source_rank":2048,"rank":2048,"depth":7,"x":1875.96,"y":182.779,"cluster":"commutative-algebra"},{"id":"stacks:04EF","tag":"04EF","title":"Conormal modules and universal thickenings · Lemma 04EF","summary":"Let R → A → B be ring maps. Assume A → B formally unramified. Let B' → B be the universal first order thickening of B over A. Then B' is formally unramified over A, and the canonical map Ω_A/R ⊗_A B → Ω_B'/R ⊗_B' B is an isomorphism.","statement_latex":"Let $R \\to A  \\to B$ be ring maps. Assume $A \\to B$ formally unramified.\nLet $B' \\to B$ be the universal first order thickening of $B$ over $A$.\nThen $B'$ is formally unramified over $A$, and the canonical map\n$\\Omega_{A/R} \\otimes_A B \\to \\Omega_{B'/R} \\otimes_{B'} B$ is an\nisomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Conormal modules and universal thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EF","source_file":"algebra.tex","source_line":41090,"source_end_line":41097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41090-L41097","statement_sha256":"5c35b12d6c4a7fee83377bdbcb89229887cbb35b52cb08f9e40a2b2662611e13","origin":"The Stacks Project","memory_eligible":false,"source_rank":2049,"rank":2049,"depth":6,"x":2173.581,"y":160.014,"cluster":"commutative-algebra"},{"id":"stacks:00UQ","tag":"00UQ","title":"Formally étale maps · Definition 00UQ","summary":"Let R → S be a ring map. We say S is formally étale over R if for every commutative solid diagram xymatrix S ar[r] ar@-->[rd] & A/I R ar[r] ar[u] & A ar[u] where I ⊂ A is an ideal of square zero, there exists a unique dotted arrow making the diagram commute.","statement_latex":"Let $R \\to S$ be a ring map.\nWe say $S$ is {\\it formally \\'etale over $R$} if for every\ncommutative solid diagram\n$$\n\\xymatrix{\nS \\ar[r] \\ar@{-->}[rd] & A/I \\\\\nR \\ar[r] \\ar[u] & A \\ar[u]\n}\n$$\nwhere $I \\subset A$ is an ideal of square zero, there exists\na unique dotted arrow making the diagram commute.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally étale maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UQ","source_file":"algebra.tex","source_line":41167,"source_end_line":41180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41167-L41180","statement_sha256":"2f7228efddceaabe9f5b08db5ea790acb5253036f40c6bd5a4597f22ec1625ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":2050,"rank":2050,"depth":0,"x":1972.34,"y":345.758,"cluster":"commutative-algebra"},{"id":"stacks:0H92","tag":"0H92","title":"Formally étale maps · Lemma 0H92","summary":"Let R → S be a formally étale map. Let R → R' be any ring map. Then the base change S' = R' ⊗_R S is formally étale over R'.","statement_latex":"Let $R \\to S$ be a formally \\'etale map.\nLet $R \\to R'$ be any ring map.\nThen the base change $S' = R' \\otimes_R S$ is formally \\'etale over $R'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H92","source_file":"algebra.tex","source_line":41186,"source_end_line":41191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41186-L41191","statement_sha256":"0c7dbbfda76f8deef45e50fc7f0f63b6f60a8df8ed179eb1ff47af75df2dd3da","origin":"The Stacks Project","memory_eligible":false,"source_rank":2051,"rank":2051,"depth":2,"x":1971.361,"y":94.496,"cluster":"commutative-algebra"},{"id":"stacks:00UR","tag":"00UR","title":"Formally étale maps · Lemma 00UR","summary":"Let R → S be a ring map of finite presentation. The following are equivalent: • R → S is formally étale, • R → S is étale.","statement_latex":"Let $R \\to S$ be a ring map of finite presentation.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $R \\to S$ is formally \\'etale,\n\\item $R \\to S$ is \\'etale.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UR","source_file":"algebra.tex","source_line":41198,"source_end_line":41206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41198-L41206","statement_sha256":"96df30736d9359973c3fbcd37eb10eaa1f722336daeef66c5d8379e3f8af8630","origin":"The Stacks Project","memory_eligible":false,"source_rank":2052,"rank":2052,"depth":6,"x":2174.229,"y":279.297,"cluster":"commutative-algebra"},{"id":"stacks:031N","tag":"031N","title":"Formally étale maps · Lemma 031N","summary":"Let R be a ring. Let I be a directed set. Let (S_i, φ_ii') be a system of R-algebras over I. If each R → S_i is formally étale, then S = colim_i ∈ I S_i is formally étale over R","statement_latex":"Let $R$ be a ring. Let $I$ be a directed set.\nLet $(S_i, \\varphi_{ii'})$ be a system of $R$-algebras\nover $I$. If each $R \\to S_i$ is formally \\'etale, then\n$S = \\colim_{i \\in I} S_i$ is formally \\'etale over $R$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031N","source_file":"algebra.tex","source_line":41224,"source_end_line":41230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41224-L41230","statement_sha256":"5d6b7d152078e36f80d3b7796b4fddcedf409acfbae617bef58a3ae96908e014","origin":"The Stacks Project","memory_eligible":false,"source_rank":2053,"rank":2053,"depth":1,"x":1875.896,"y":258.13,"cluster":"commutative-algebra"},{"id":"stacks:04EG","tag":"04EG","title":"Formally étale maps · Lemma 04EG","summary":"Let R be a ring. Let S ⊂ R be any multiplicative subset. Then the ring map R → S^-1R is formally étale.","statement_latex":"Let $R$ be a ring. Let $S \\subset R$ be any multiplicative subset.\nThen the ring map $R \\to S^{-1}R$ is formally \\'etale.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EG","source_file":"algebra.tex","source_line":41241,"source_end_line":41245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41241-L41245","statement_sha256":"65c6985fec3fcf363a2d3a6d053614dca353a0f185c88bff0ab2976ef9274553","origin":"The Stacks Project","memory_eligible":false,"source_rank":2054,"rank":2054,"depth":0,"x":2113.006,"y":104.391,"cluster":"commutative-algebra"},{"id":"stacks:0H1D","tag":"0H1D","title":"Formally étale maps · Lemma 0H1D","summary":"Let R → S be a ring map. Let J ⊂ S be an ideal such that R → S/J is surjective; let I ⊂ R be the kernel. If R → S is formally étale, then R/I^n → S/J^n is an isomorphism for all n and bigoplus I^n/I^n + 1 → bigoplus J^n/J^n + 1 is an isomorphism of graded rings.","statement_latex":"Let $R \\to S$ be a ring map. Let $J \\subset S$ be an ideal such\nthat $R \\to S/J$ is surjective; let $I \\subset R$ be the kernel.\nIf $R \\to S$ is formally \\'etale, then $R/I^n \\to S/J^n$ is an\nisomorphism for all $n$ and\n$\\bigoplus I^n/I^{n + 1} \\to \\bigoplus J^n/J^{n + 1}$\nis an isomorphism of graded rings.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1D","source_file":"algebra.tex","source_line":41256,"source_end_line":41264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41256-L41264","statement_sha256":"42e8d4447aec6e9d49dbfedfff4a5e41c7e14d275529e0e74911bf32f76dd222","origin":"The Stacks Project","memory_eligible":false,"source_rank":2055,"rank":2055,"depth":0,"x":2061.775,"y":352.405,"cluster":"commutative-algebra"},{"id":"stacks:0H93","tag":"0H93","title":"Formally étale maps · Lemma 0H93","summary":"Let R → S → S' be ring maps. Let J, resp. J' be the kernel of the multiplication map S ⊗_R S → S, resp. S' ⊗_R' S' → S'. If S → S' is formally étale, then the map S' ⊗_S ((S ⊗_R S)/J^k + 1) → (S' ⊗_R S')/(J')^k + 1 is an isomorphism for all k ≥ 0. In particular, the map S' ⊗_S Ω_S/R → Ω_S'/R is an isomorphism.","statement_latex":"Let $R \\to S \\to S'$ be ring maps. Let $J$, resp.\\ $J'$ be the\nkernel of the multiplication map\n$S \\otimes_R S \\to S$, resp.\\ $S' \\otimes_{R'} S' \\to S'$.\nIf $S \\to S'$ is formally \\'etale, then the map\n$$\nS' \\otimes_S \\left((S \\otimes_R S)/J^{k + 1}\\right)\n\\longrightarrow\n(S' \\otimes_R S')/(J')^{k + 1}\n$$\nis an isomorphism for all $k \\geq 0$. In particular, the map\n$S' \\otimes_S \\Omega_{S/R} \\to \\Omega_{S'/R}$ is an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H93","source_file":"algebra.tex","source_line":41290,"source_end_line":41303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41290-L41303","statement_sha256":"9a9d79dfb37889306acb92747e3912dcf1dceaadcddaf7424780902f6f11dc54","origin":"The Stacks Project","memory_eligible":false,"source_rank":2056,"rank":2056,"depth":4,"x":1900.037,"y":140.363,"cluster":"commutative-algebra"},{"id":"stacks:0H94","tag":"0H94","title":"Formally étale maps · Lemma 0H94","summary":"Let R → S → S' be ring maps with S → S' formally étale (for example étale). Let M be an S-module. Set M' = S' ⊗_R M. Then we have S' ⊗_S P^k_S/R(M) = P^k_S'/R(M') It follows that for any S-module N and any finite order differential operator D : M → N there exists a unique extension D' : M' → S' ⊗_S N of D to a differential operator (of the same or lesser order).","statement_latex":"Let $R \\to S \\to S'$ be ring maps with $S \\to S'$ formally \\'etale\n(for example \\'etale).\nLet $M$ be an $S$-module. Set $M' = S' \\otimes_R M$. Then we have\n$$\nS' \\otimes_S P^k_{S/R}(M) = P^k_{S'/R}(M')\n$$\nIt follows that for any $S$-module $N$ and any\nfinite order differential operator $D : M \\to N$\nthere exists a unique extension $D' : M' \\to S' \\otimes_S N$\nof $D$ to a differential operator (of the same or lesser order).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formally étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H94","source_file":"algebra.tex","source_line":41319,"source_end_line":41331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41319-L41331","statement_sha256":"9ef4efe285ee02a7d0f7cec6cba66b7c2eccc96889dc99f8b6cd4b62a889e7ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":2057,"rank":2057,"depth":5,"x":2189.943,"y":204.973,"cluster":"commutative-algebra"},{"id":"stacks:00UT","tag":"00UT","title":"Unramified ring maps · Definition 00UT","summary":"Let R → S be a ring map. • We say R → S is unramified if R → S is of finite type and Ω_S/R = 0. • We say R → S is G-unramified if R → S is of finite presentation and Ω_S/R = 0. • Given a prime q of S we say that S is unramified at q if there exists a g ∈ S, g not ∈ q such that R → S_g is unramified. • Given a prime q of S we say that S is G-unramified at q if there exists a g ∈ S, g not ∈ q such that R → S_g is G-unramified.","statement_latex":"Let $R \\to S$ be a ring map.\n\\begin{enumerate}\n\\item We say $R \\to S$ is {\\it unramified} if $R \\to S$ is of\nfinite type and $\\Omega_{S/R} = 0$.\n\\item We say $R \\to S$ is {\\it G-unramified} if $R \\to S$ is of finite\npresentation and $\\Omega_{S/R} = 0$.\n\\item Given a prime $\\mathfrak q$ of $S$ we say that $S$ is\n{\\it unramified at $\\mathfrak q$} if there exists a\n$g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is unramified.\n\\item Given a prime $\\mathfrak q$ of $S$ we say that $S$ is\n{\\it G-unramified at $\\mathfrak q$} if there exists a\n$g \\in S$, $g \\not \\in \\mathfrak q$ such that $R \\to S_g$ is G-unramified.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Unramified ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UT","source_file":"algebra.tex","source_line":41384,"source_end_line":41399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41384-L41399","statement_sha256":"9fc2bd38a592a9c48507eae3d9990efef4903f55d830c31e10a144476c1470d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2058,"rank":2058,"depth":0,"x":1924.099,"y":321.88,"cluster":"commutative-algebra"},{"id":"stacks:00UU","tag":"00UU","title":"Unramified ring maps · Lemma 00UU","summary":"Let R → S be a ring map. The following are equivalent • R → S is formally unramified and of finite type, and • R → S is unramified. Moreover, also the following are equivalent • R → S is formally unramified and of finite presentation, and • R → S is G-unramified.","statement_latex":"Let $R \\to S$ be a ring map. The following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is formally unramified and of finite type, and\n\\item $R \\to S$ is unramified.\n\\end{enumerate}\nMoreover, also the following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is formally unramified and of finite presentation, and\n\\item $R \\to S$ is G-unramified.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UU","source_file":"algebra.tex","source_line":41404,"source_end_line":41416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41404-L41416","statement_sha256":"76bb9a6ca4ef1172a9ba9ad20737a70cbd5d0ecf59c8222b74107cd382797099","origin":"The Stacks Project","memory_eligible":false,"source_rank":2059,"rank":2059,"depth":5,"x":2026.159,"y":84.727,"cluster":"commutative-algebra"},{"id":"stacks:00UV","tag":"00UV","title":"Unramified ring maps · Lemma 00UV","summary":"Properties of unramified and G-unramified ring maps. • The base change of an unramified ring map is unramified. The base change of a G-unramified ring map is G-unramified. • The composition of unramified ring maps is unramified. The composition of G-unramified ring maps is G-unramified. • Any principal localization R → R_f is G-unramified and unramified. • If I ⊂ R is an ideal, then R → R/I is unramified. If I ⊂ R is a finitely generated ideal, then R → R/I is…","statement_latex":"Properties of unramified and G-unramified ring maps.\n\\begin{enumerate}\n\\item The base change of an unramified ring map is unramified.\nThe base change of a G-unramified ring map is G-unramified.\n\\item The composition of unramified ring maps is unramified.\nThe composition of G-unramified ring maps is G-unramified.\n\\item Any principal localization $R \\to R_f$ is G-unramified and\nunramified.\n\\item If $I \\subset R$ is an ideal, then $R \\to R/I$ is unramified.\nIf $I \\subset R$ is a finitely generated ideal, then $R \\to R/I$ is\nG-unramified.\n\\item An \\'etale ring map is G-unramified and unramified.\n\\item If $R \\to S$ is of finite type (resp.\\ finite presentation),\n$\\mathfrak q \\subset S$ is a prime and $(\\Omega_{S/R})_{\\mathfrak q} = 0$,\nthen $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$.\n\\item If $R \\to S$ is of finite type (resp.\\ finite presentation),\n$\\mathfrak q \\subset S$ is a prime and\n$\\Omega_{S/R} \\otimes_S \\kappa(\\mathfrak q) = 0$, then\n$R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$.\n\\item If $R \\to S$ is of finite type (resp.\\ finite presentation),\n$\\mathfrak q \\subset S$ is a prime lying over $\\mathfrak p \\subset R$ and\n$(\\Omega_{S \\otimes_R \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)})_{\\mathfrak q}\n= 0$, then $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$.\n\\item If $R \\to S$ is of finite type (resp.\\ presentation),\n$\\mathfrak q \\subset S$ is a prime lying over $\\mathfrak p \\subset R$ and\n$(\\Omega_{S \\otimes_R \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)})\n\\otimes_{S \\otimes_R \\kappa(\\mathfrak p)} \\kappa(\\mathfrak q) = 0$,\nthen $R \\to S$ is unramified (resp.\\ G-unramified) at $\\mathfrak q$.\n\\item If $R \\to S$ is a ring map, $g_1, \\ldots, g_m \\in S$ generate\nthe unit ideal and $R \\to S_{g_j}$ is unramified (resp.\\ G-unramified) for\n$j = 1, \\ldots, m$, then $R \\to S$ is unramified (resp.\\ G-unramified).\n\\item If $R \\to S$ is a ring map which is unramified (resp.\\ G-unramified)\nat every prime of $S$, then $R \\to S$ is unramified (resp.\\ G-unramified).\n\\item If $R \\to S$ is G-unramified, then there exists a finite type\n$\\mathbf{Z}$-algebra $R_0$ and a G-unramified ring map $R_0 \\to S_0$\nand a ring map $R_0 \\to R$ such that $S = R \\otimes_{R_0} S_0$.\n\\item If $R \\to S$ is unramified, then there exists a finite type\n$\\mathbf{Z}$-algebra $R_0$ and an unramified ring map $R_0 \\to S_0$\nand a ring map $R_0 \\to R$ such that $S$ is a quotient of\n$R \\otimes_{R_0} S_0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UV","source_file":"algebra.tex","source_line":41423,"source_end_line":41466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41423-L41466","statement_sha256":"a73b92acad112126d60f0a15201e67b3d079afd70704a62195f89f0406a9d583","origin":"The Stacks Project","memory_eligible":false,"source_rank":2060,"rank":2060,"depth":4,"x":2141.663,"y":317.611,"cluster":"commutative-algebra"},{"id":"stacks:02FL","tag":"02FL","title":"Unramified ring maps · Lemma 02FL","summary":"Let R → S be a ring map. If R → S is unramified, then there exists an idempotent e ∈ S ⊗_R S such that S ⊗_R S → S is isomorphic to S ⊗_R S → (S ⊗_R S)_e.","statement_latex":"Let $R \\to S$ be a ring map.\nIf $R \\to S$ is unramified, then there exists an idempotent\n$e \\in S \\otimes_R S$ such that $S \\otimes_R S \\to S$ is isomorphic\nto $S \\otimes_R S \\to (S \\otimes_R S)_e$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FL","source_file":"algebra.tex","source_line":41549,"source_end_line":41555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41549-L41555","statement_sha256":"dbf4d385ccc23024eaaa2c223c5748fc813358d3ac25f5e8e297ae24d9111848","origin":"The Stacks Project","memory_eligible":false,"source_rank":2061,"rank":2061,"depth":4,"x":1869.097,"y":211.38,"cluster":"commutative-algebra"},{"id":"stacks:00UW","tag":"00UW","title":"Unramified ring maps · Lemma 00UW","summary":"Let R → S be a ring map. Let q ⊂ S be a prime lying over p in R. If S/R is unramified at q then • we have p S_ q = qS_ q is the maximal ideal of the local ring S_ q, and • the field extension kappa( q)/kappa( p) is finite separable.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q \\subset S$ be\na prime lying over $\\mathfrak p$ in $R$.\nIf $S/R$ is unramified at $\\mathfrak q$ then\n\\begin{enumerate}\n\\item we have $\\mathfrak p S_{\\mathfrak q} = \\mathfrak qS_{\\mathfrak q}$\nis the maximal ideal of the local ring $S_{\\mathfrak q}$, and\n\\item the field extension $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$\nis finite separable.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UW","source_file":"algebra.tex","source_line":41567,"source_end_line":41579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41567-L41579","statement_sha256":"43da188817b7923bd4e43f6fd03cc818ecec766d8c055dafb89e2fa916bb85e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2062,"rank":2062,"depth":40,"x":2155.633,"y":135.02,"cluster":"commutative-algebra"},{"id":"stacks:02UR","tag":"02UR","title":"Unramified ring maps · Lemma 02UR","summary":"Let R → S be a finite type ring map. Let q be a prime of S. If R → S is unramified at q then R → S is quasi-finite at q. In particular, an unramified ring map is quasi-finite.","statement_latex":"Let $R \\to S$ be a finite type ring map.\nLet $\\mathfrak q$ be a prime of $S$.\nIf $R \\to S$ is unramified at $\\mathfrak q$ then\n$R \\to S$ is quasi-finite at $\\mathfrak q$.\nIn particular, an unramified ring map is quasi-finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UR","source_file":"algebra.tex","source_line":41598,"source_end_line":41605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41598-L41605","statement_sha256":"4c401b2d3c7fcb4ef6d163e6b090a25d0a3eeab8c80ea12f2c8ed2875b55eeaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":2063,"rank":2063,"depth":41,"x":2005.689,"y":354.008,"cluster":"commutative-algebra"},{"id":"stacks:02FM","tag":"02FM","title":"Unramified ring maps · Lemma 02FM","summary":"Let R → S be a ring map. Let q be a prime of S lying over a prime p of R. If • R → S is of finite type, • p S_ q is the maximal ideal of the local ring S_ q, and • the field extension kappa( q)/kappa( p) is finite separable, then R → S is unramified at q.","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$\nlying over a prime $\\mathfrak p$ of $R$. If\n\\begin{enumerate}\n\\item $R \\to S$ is of finite type,\n\\item $\\mathfrak p S_{\\mathfrak q}$ is the maximal ideal\nof the local ring $S_{\\mathfrak q}$, and\n\\item the field extension $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$\nis finite separable,\n\\end{enumerate}\nthen $R \\to S$ is unramified at $\\mathfrak q$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FM","source_file":"algebra.tex","source_line":41615,"source_end_line":41627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41615-L41627","statement_sha256":"e34b41b6ebe3d26212a06b72181d9c82dcb24528c8351c5dfdc9d3ad0c26b38f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2064,"rank":2064,"depth":5,"x":1940.123,"y":107.342,"cluster":"commutative-algebra"},{"id":"stacks:08WD","tag":"08WD","title":"Unramified ring maps · Lemma 08WD","summary":"Let R → S be a ring map. The following are equivalent • R → S is étale, • R → S is flat and G-unramified, and • R → S is flat, unramified, and of finite presentation.","statement_latex":"Let $R \\to S$ be a ring map. The following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is \\'etale,\n\\item $R \\to S$ is flat and G-unramified, and\n\\item $R \\to S$ is flat, unramified, and of finite presentation.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WD","source_file":"algebra.tex","source_line":41643,"source_end_line":41651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41643-L41651","statement_sha256":"83e8343c5c9e225f25999ea905a3e637a41a5b85bb161b78eb672b66293e9e11","origin":"The Stacks Project","memory_eligible":false,"source_rank":2065,"rank":2065,"depth":41,"x":2186.937,"y":252.088,"cluster":"commutative-algebra"},{"id":"stacks:0G1C","tag":"0G1C","title":"Unramified ring maps · Lemma 0G1C","summary":"Let k be a field. Let φ : k[x_1, …, x_n] → A, x_i ↦ a_i be a finite type ring map. Then φ is étale if and only if we have the following two conditions: (a) the local rings of A at maximal ideals have dimension n, and (b) the elements d(a_1), …, d(a_n) generate Ω_A/k as an A-module.","statement_latex":"Let $k$ be a field. Let\n$$\n\\varphi : k[x_1, \\ldots, x_n] \\to A, \\quad x_i \\longmapsto a_i\n$$\nbe a finite type ring map. Then $\\varphi$ is \\'etale if and only if we\nhave the following two conditions: (a) the local rings of $A$ at maximal ideals\nhave dimension $n$, and (b) the elements $\\text{d}(a_1), \\ldots, \\text{d}(a_n)$\ngenerate $\\Omega_{A/k}$ as an $A$-module.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1C","source_file":"algebra.tex","source_line":41668,"source_end_line":41678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41668-L41678","statement_sha256":"7c24831954c70bed171d2ead6cad1093974169f766c2c1837a6110cb2b83afd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2066,"rank":2066,"depth":42,"x":1888.413,"y":285.417,"cluster":"commutative-algebra"},{"id":"stacks:0395","tag":"0395","title":"Local structure of unramified ring maps · Proposition 0395","summary":"Let R → S be a ring map. Let q ⊂ S be a prime. If R → S is unramified at q, then there exist • a g ∈ S, g not ∈ q, • a standard étale ring map R → S', and • a surjective R-algebra map S' → S_g.","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime.\nIf $R \\to S$ is unramified at $\\mathfrak q$, then there exist\n\\begin{enumerate}\n\\item a $g \\in S$, $g \\not \\in \\mathfrak q$,\n\\item a standard \\'etale ring map $R \\to S'$, and\n\\item a surjective $R$-algebra map $S' \\to S_g$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local structure of unramified ring maps","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0395","source_file":"algebra.tex","source_line":41720,"source_end_line":41729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41720-L41729","statement_sha256":"66f85091467e21a3f23c4395a79ca785b8da3beb4f6b0cf718855b5f5047fc9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2067,"rank":2067,"depth":43,"x":2081.82,"y":91.368,"cluster":"commutative-algebra"},{"id":"stacks:00UX","tag":"00UX","title":"Local structure of unramified ring maps · Lemma 00UX","summary":"Let R → S be a ring map. Let q be a prime of S lying over p ⊂ R. Assume that R → S is of finite type and unramified at q. Then there exist • an étale ring map R → R', • a prime p' ⊂ R' lying over p. • a product decomposition R' ⊗_R S = A × B with the following properties • R' → A is surjective, and • p'A is a prime of A lying over p' and over q.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p \\subset R$.\nAssume that $R \\to S$ is of finite type and unramified at $\\mathfrak q$.\nThen there exist\n\\begin{enumerate}\n\\item an \\'etale ring map $R \\to R'$,\n\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$.\n\\item a product decomposition\n$$\nR' \\otimes_R S = A \\times B\n$$\n\\end{enumerate}\nwith the following properties\n\\begin{enumerate}\n\\item $R' \\to A$ is surjective, and\n\\item $\\mathfrak p'A$ is a prime of $A$ lying over $\\mathfrak p'$ and\nover $\\mathfrak q$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local structure of unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UX","source_file":"algebra.tex","source_line":41942,"source_end_line":41962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L41942-L41962","statement_sha256":"a6e574cdb73edd9116724bf02e839d9a36becafaee29b10e0a458453b70799a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2068,"rank":2068,"depth":42,"x":2095.259,"y":344.308,"cluster":"commutative-algebra"},{"id":"stacks:00UY","tag":"00UY","title":"Local structure of unramified ring maps · Lemma 00UY","summary":"In an unramified ring map, one can separate the points in a fiber by passing to an étale neighbourhood. Let R → S be a ring map. Let p be a prime of R. If R → S is unramified then there exist • an étale ring map R → R', • a prime p' ⊂ R' lying over p. • a product decomposition R' ⊗_R S = A_1 × … × A_n × B with the following properties • R' → A_i is surjective, • p'A_i is a prime of A_i lying over p', and • there is no prime of B lying over p'.","statement_latex":"\\begin{slogan}\nIn an unramified ring map, one can separate the points in a fiber\nby passing to an \\'etale neighbourhood.\n\\end{slogan}\nLet $R \\to S$ be a ring map.\nLet $\\mathfrak p$ be a prime of $R$.\nIf $R \\to S$ is unramified then there exist\n\\begin{enumerate}\n\\item an \\'etale ring map $R \\to R'$,\n\\item a prime $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$.\n\\item a product decomposition\n$$\nR' \\otimes_R S = A_1 \\times \\ldots \\times A_n \\times B\n$$\n\\end{enumerate}\nwith the following properties\n\\begin{enumerate}\n\\item $R' \\to A_i$ is surjective,\n\\item $\\mathfrak p'A_i$ is a prime of $A_i$ lying over $\\mathfrak p'$, and\n\\item there is no prime of $B$ lying over $\\mathfrak p'$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Local structure of unramified ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00UY","source_file":"algebra.tex","source_line":42001,"source_end_line":42024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42001-L42024","statement_sha256":"40bab7a7d7a756d9583a90e4b6fd410fbe1da9aaefc7d03e2c050b38ac47fef2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2069,"rank":2069,"depth":42,"x":1881.852,"y":165.343,"cluster":"commutative-algebra"},{"id":"stacks:04GF","tag":"04GF","title":"Henselian local rings · Definition 04GF","summary":"Let (R, m, kappa) be a local ring. • We say R is henselian if for every monic f ∈ R[T] and every root a_0 ∈ kappa of overlinef such that overlinef'(a_0) not = 0 there exists an a ∈ R such that f(a) = 0 and a_0 = overlinea. • We say R is strictly henselian if R is henselian and its residue field is separably algebraically closed.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring.\n\\begin{enumerate}\n\\item We say $R$ is {\\it henselian} if for every monic $f \\in R[T]$ and\nevery root $a_0 \\in \\kappa$ of $\\overline{f}$ such that\n$\\overline{f'}(a_0) \\not = 0$\nthere exists an $a \\in R$ such that $f(a) = 0$ and\n$a_0 = \\overline{a}$.\n\\item We say $R$ is {\\it strictly henselian} if $R$ is henselian\nand its residue field is separably algebraically closed.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GF","source_file":"algebra.tex","source_line":42074,"source_end_line":42086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42074-L42086","statement_sha256":"ae2f1f9984a700fe0e2f0cc9ea8b91ca36009d60f1a116cbbcaabfe4dc6c9e58","origin":"The Stacks Project","memory_eligible":false,"source_rank":2070,"rank":2070,"depth":0,"x":2183.261,"y":176.221,"cluster":"commutative-algebra"},{"id":"stacks:06RR","tag":"06RR","title":"Henselian local rings · Lemma 06RR","summary":"Let (R, m, kappa) be a local ring. Let f ∈ R[T]. Let a, b ∈ R such that f(a) = f(b) = 0, a = b bmod m, and f'(a) not ∈ m. Then a = b.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring.\nLet $f \\in R[T]$. Let $a, b \\in R$ such that $f(a) = f(b) = 0$,\n$a = b \\bmod \\mathfrak m$, and $f'(a) \\not \\in \\mathfrak m$.\nThen $a = b$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RR","source_file":"algebra.tex","source_line":42093,"source_end_line":42099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42093-L42099","statement_sha256":"57c98362ccab4cad8b8d367894f8218fdced9afe5dd92ead2dcbf898ce942437","origin":"The Stacks Project","memory_eligible":false,"source_rank":2071,"rank":2071,"depth":0,"x":1952.161,"y":339.296,"cluster":"commutative-algebra"},{"id":"stacks:04GG","tag":"04GG","title":"Henselian local rings · Lemma 04GG","summary":"Characterizations of henselian local rings Let (R, m, kappa) be a local ring. The following are equivalent • R is henselian, • for every f ∈ R[T] and every root a_0 ∈ kappa of overlinef such that overlinef'(a_0) not = 0 there exists an a ∈ R such that f(a) = 0 and a_0 = overlinea, • for any monic f ∈ R[T] and any factorization overlinef = g_0 h_0 with gcd(g_0, h_0) = 1 there exists a factorization f = gh in R[T] such that g_0 = overlineg and h_0 = overlineh, • for any…","statement_latex":"\\begin{slogan}\nCharacterizations of henselian local rings\n\\end{slogan}\nLet $(R, \\mathfrak m, \\kappa)$ be a local ring.\nThe following are equivalent\n\\begin{enumerate}\n\\item $R$ is henselian,\n\\item for every $f \\in R[T]$ and every root $a_0 \\in \\kappa$\nof $\\overline{f}$ such that $\\overline{f'}(a_0) \\not = 0$\nthere exists an $a \\in R$ such that $f(a) = 0$ and\n$a_0 = \\overline{a}$,\n\\item for any monic $f \\in R[T]$ and any factorization\n$\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there\nexists a factorization $f = gh$ in $R[T]$ such that\n$g_0 = \\overline{g}$ and $h_0 = \\overline{h}$,\n\\item for any monic $f \\in R[T]$ and any factorization\n$\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there\nexists a factorization $f = gh$ in $R[T]$ such that\n$g_0 = \\overline{g}$ and $h_0 = \\overline{h}$ and moreover\n$\\deg_T(g) = \\deg_T(g_0)$,\n\\item for any $f \\in R[T]$ and any factorization\n$\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there\nexists a factorization $f = gh$ in $R[T]$ such that\n$g_0 = \\overline{g}$ and $h_0 = \\overline{h}$,\n\\item for any $f \\in R[T]$ and any factorization\n$\\overline{f} = g_0 h_0$ with $\\gcd(g_0, h_0) = 1$ there\nexists a factorization $f = gh$ in $R[T]$ such that\n$g_0 = \\overline{g}$ and $h_0 = \\overline{h}$ and\nmoreover $\\deg_T(g) = \\deg_T(g_0)$,\n\\item for any \\'etale ring map $R \\to S$ and prime $\\mathfrak q$ of $S$\nlying over $\\mathfrak m$ with $\\kappa = \\kappa(\\mathfrak q)$\nthere exists a retraction $\\tau : S \\to R$ of $R \\to S$,\n\\item for any \\'etale ring map $R \\to S$ and prime $\\mathfrak q$ of $S$\nlying over $\\mathfrak m$ with $\\kappa = \\kappa(\\mathfrak q)$\nthere exists a unique retraction $\\tau : S \\to R$ of $R \\to S$ such that\n$\\mathfrak q = \\tau^{-1}(\\mathfrak m)$,\n\\item any finite $R$-algebra is a product of local rings,\n\\item any finite $R$-algebra is a finite product of local rings,\n\\item any finite type $R$-algebra $S$ can be written as\n$A \\times B$ with $R \\to A$ finite\nand $R \\to B$ not quasi-finite at any prime lying over $\\mathfrak m$,\n\\item any finite type $R$-algebra $S$ can be written as\n$A \\times B$ with $R \\to A$ finite\nsuch that each irreducible component of $\\Spec(B \\otimes_R \\kappa)$\nhas dimension $\\geq 1$, and\n\\item any quasi-finite $R$-algebra $S$ can be written as\n$S = A \\times B$ with $R \\to A$ finite such that $B \\otimes_R \\kappa = 0$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GG","source_file":"algebra.tex","source_line":42114,"source_end_line":42164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42114-L42164","statement_sha256":"c4b63bddf02e385a098694cdbf765b638387550434b71b084573f89ca658779b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2072,"rank":2072,"depth":43,"x":1991.447,"y":87.809,"cluster":"commutative-algebra"},{"id":"stacks:04GH","tag":"04GH","title":"Henselian local rings · Lemma 04GH","summary":"Let (R, m, kappa) be a henselian local ring. • If R → S is a finite ring map then S is a finite product of henselian local rings each finite over R. • If R → S is a finite ring map and S is local, then S is a henselian local ring and R → S is a (finite) local ring map. • If R → S is a finite type ring map, and q is a prime of S lying over m at which R → S is quasi-finite, then S_ q is henselian and finite over R. • If R → S is quasi-finite then S_ q is henselian and…","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a henselian local ring.\n\\begin{enumerate}\n\\item If $R \\to S$ is a finite ring map then $S$ is\na finite product of henselian local rings each finite over $R$.\n\\item If $R \\to S$ is a finite ring map and $S$ is local, then\n$S$ is a henselian local ring and $R \\to S$ is a (finite) local ring map.\n\\item If $R \\to S$ is a finite type ring map, and $\\mathfrak q$ is\na prime of $S$ lying over $\\mathfrak m$ at which $R \\to S$ is quasi-finite,\nthen $S_{\\mathfrak q}$ is henselian and finite over $R$.\n\\item If $R \\to S$ is quasi-finite then $S_{\\mathfrak q}$ is henselian\nand finite over $R$ for every prime $\\mathfrak q$ lying over $\\mathfrak m$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GH","source_file":"algebra.tex","source_line":42384,"source_end_line":42398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42384-L42398","statement_sha256":"f1a3fad1fb46d9fbc72ceaf69cbd7434eb0c003442f451b73fb84530b5d6656d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2073,"rank":2073,"depth":44,"x":2164.79,"y":295.631,"cluster":"commutative-algebra"},{"id":"stacks:04GJ","tag":"04GJ","title":"Henselian local rings · Lemma 04GJ","summary":"Let (R, m, kappa) be a henselian local ring. Any finite type R-algebra S can be written as S = A_1 × … × A_n × B with A_i local and finite over R and R → B not quasi-finite at any prime of B lying over m.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a henselian local ring.\nAny finite type $R$-algebra $S$ can be written as\n$S = A_1 \\times \\ldots \\times A_n \\times B$ with $A_i$ local\nand finite over $R$ and $R \\to B$ not quasi-finite at any\nprime of $B$ lying over $\\mathfrak m$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GJ","source_file":"algebra.tex","source_line":42417,"source_end_line":42424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42417-L42424","statement_sha256":"74ee89c9a5feee6dfde379de8f9323fbf72839f7ff1b36df24747b8dba41597f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2074,"rank":2074,"depth":44,"x":1869.72,"y":240.722,"cluster":"commutative-algebra"},{"id":"stacks:06DD","tag":"06DD","title":"Henselian local rings · Lemma 06DD","summary":"Let (R, m, kappa) be a strictly henselian local ring. Any finite type R-algebra S can be written as S = A_1 × … × A_n × B with A_i local and finite over R and kappa ⊂ kappa( m_A_i) finite purely inseparable and R → B not quasi-finite at any prime of B lying over m.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a strictly henselian local ring.\nAny finite type $R$-algebra $S$ can be written as\n$S = A_1 \\times \\ldots \\times A_n \\times B$ with $A_i$ local\nand finite over $R$ and $\\kappa \\subset \\kappa(\\mathfrak m_{A_i})$\nfinite purely inseparable and $R \\to B$ not quasi-finite\nat any prime of $B$ lying over $\\mathfrak m$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DD","source_file":"algebra.tex","source_line":42431,"source_end_line":42439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42431-L42439","statement_sha256":"4430b807aea9484b4aaf88eecc1b047336218f7ed13f7815fae0be1c68d48be3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2075,"rank":2075,"depth":45,"x":2131.567,"y":113.728,"cluster":"commutative-algebra"},{"id":"stacks:04GK","tag":"04GK","title":"Henselian local rings · Lemma 04GK","summary":"Let (R, m, kappa) be a henselian local ring. The category of finite étale ring extensions R → S is equivalent to the category of finite étale algebras kappa → overlineS via the functor S ↦ S/ mS.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a henselian local ring.\nThe category of finite \\'etale ring extensions $R \\to S$ is\nequivalent to the category of finite \\'etale algebras\n$\\kappa \\to \\overline{S}$ via the functor $S \\mapsto S/\\mathfrak mS$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GK","source_file":"algebra.tex","source_line":42449,"source_end_line":42455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42449-L42455","statement_sha256":"e49e3a55c3932e89e65c63ced9b529a2bb213104be3d7d6dae33c55640e5b43f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2076,"rank":2076,"depth":45,"x":2040.572,"y":356.052,"cluster":"commutative-algebra"},{"id":"stacks:04GL","tag":"04GL","title":"Henselian local rings · Lemma 04GL","summary":"Let (R, m, kappa) be a strictly henselian local ring. Let R → S be an unramified ring map. Then S = A_1 × … × A_n × B with each R → A_i surjective and no prime of B lying over m.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a strictly henselian local ring.\nLet $R \\to S$ be an unramified ring map. Then\n$$\nS = A_1 \\times \\ldots \\times A_n \\times B\n$$\nwith each $R \\to A_i$ surjective and no prime of $B$ lying\nover $\\mathfrak m$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GL","source_file":"algebra.tex","source_line":42507,"source_end_line":42516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42507-L42516","statement_sha256":"b405badcbfc3407dc721458a71f70ea7e2668c9846176c121722dba81e2b884e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2077,"rank":2077,"depth":45,"x":1912.746,"y":125.636,"cluster":"commutative-algebra"},{"id":"stacks:04GM","tag":"04GM","title":"Henselian local rings · Lemma 04GM","summary":"Complete local rings are Henselian by Newton's method Let (R, m, kappa) be a complete local ring, see Definition [Tag 0324]. Then R is henselian.","statement_latex":"\\begin{slogan}\nComplete local rings are Henselian by Newton's method\n\\end{slogan}\nLet $(R, \\mathfrak m, \\kappa)$ be a complete local ring, see\nDefinition \\ref{definition-complete-local-ring}.\nThen $R$ is henselian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GM","source_file":"algebra.tex","source_line":42533,"source_end_line":42541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42533-L42541","statement_sha256":"a8105f45a1ce1f08a3c7067aaf732f73d8582494c901be4af25d2f3b8d22bf2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2078,"rank":2078,"depth":0,"x":2192.414,"y":223.051,"cluster":"commutative-algebra"},{"id":"stacks:06RS","tag":"06RS","title":"Henselian local rings · Lemma 06RS","summary":"Local rings of dimension zero are henselian. Let (R, m) be a local ring of dimension 0. Then R is henselian.","statement_latex":"\\begin{slogan}\nLocal rings of dimension zero are henselian.\n\\end{slogan}\nLet $(R, \\mathfrak m)$ be a local ring of dimension $0$.\nThen $R$ is henselian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RS","source_file":"algebra.tex","source_line":42570,"source_end_line":42577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42570-L42577","statement_sha256":"ea818d2c370bb5340d0a152e3fe900c3f45bda601864d9e17602abdf72624ce8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2079,"rank":2079,"depth":44,"x":1907.733,"y":309.946,"cluster":"commutative-algebra"},{"id":"stacks:08HQ","tag":"08HQ","title":"Henselian local rings · Lemma 08HQ","summary":"Let R → S be a ring map with S henselian local. Given • an étale ring map R → A, • a prime q of A lying over p = R ∩ m_S, • a kappa( p)-algebra map τ : kappa( q) → S/ m_S, then there exists a unique homomorphism of R-algebras f : A → S such that q = f^-1( m_S) and f induces the map τ on residue fields.","statement_latex":"Let $R \\to S$ be a ring map with $S$ henselian local.\nGiven\n\\begin{enumerate}\n\\item an \\'etale ring map $R \\to A$,\n\\item a prime $\\mathfrak q$ of $A$ lying over\n$\\mathfrak p = R \\cap \\mathfrak m_S$,\n\\item a $\\kappa(\\mathfrak p)$-algebra map\n$\\tau : \\kappa(\\mathfrak q) \\to S/\\mathfrak m_S$,\n\\end{enumerate}\nthen there exists a unique homomorphism of $R$-algebras $f : A \\to S$\nsuch that $\\mathfrak q = f^{-1}(\\mathfrak m_S)$ and\n$f$ induces the  map $\\tau$ on residue fields.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HQ","source_file":"algebra.tex","source_line":42600,"source_end_line":42614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42600-L42614","statement_sha256":"952c53662fc57c382b84297bf63648fce6168d4cfaa9b7cbc27dc81546e7c21e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2080,"rank":2080,"depth":44,"x":2047.833,"y":84.241,"cluster":"commutative-algebra"},{"id":"stacks:04GX","tag":"04GX","title":"Henselian local rings · Lemma 04GX","summary":"Let φ : R → S be a local homomorphism of strictly henselian local rings. Let P_1, …, P_n ∈ R[x_1, …, x_n] be polynomials such that R[x_1, …, x_n]/(P_1, …, P_n) is étale over R. Then the map R^n → S^n, (h_1, …, h_n) ↦ (φ(h_1), …, φ(h_n)) induces a bijection between ( (r_1, …, r_n) ∈ R^n mid P_i(r_1, …, r_n) = 0, i = 1, …, n ) and ( (s_1, …, s_n) ∈ S^n mid P^φ_i(s_1, …, s_n) = 0, i = 1, …, n ) where P^φ_i ∈ S[x_1, …, x_n] are the images of the P_i under φ.","statement_latex":"Let $\\varphi : R \\to S$ be a local homomorphism\nof strictly henselian local rings.\nLet $P_1, \\ldots, P_n \\in R[x_1, \\ldots, x_n]$ be polynomials such that\n$R[x_1, \\ldots, x_n]/(P_1, \\ldots, P_n)$ is \\'etale over $R$.\nThen the map\n$$\nR^n \\longrightarrow S^n, \\quad\n(h_1, \\ldots, h_n) \\longmapsto (\\varphi(h_1), \\ldots, \\varphi(h_n))\n$$\ninduces a bijection between\n$$\n\\{\n(r_1, \\ldots, r_n) \\in R^n\n\\mid\nP_i(r_1, \\ldots, r_n) = 0, \\ i = 1, \\ldots, n\n\\}\n$$\nand\n$$\n\\{\n(s_1, \\ldots, s_n) \\in S^n\n\\mid\nP^\\varphi_i(s_1, \\ldots, s_n) = 0, \\ i = 1, \\ldots, n\n\\}\n$$\nwhere $P^\\varphi_i \\in S[x_1, \\ldots, x_n]$ are the images of the $P_i$\nunder $\\varphi$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GX","source_file":"algebra.tex","source_line":42634,"source_end_line":42663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42634-L42663","statement_sha256":"2f848afa661a5355e73e9f3039169cabaebb950a46c37cc0f5b4c2eef45d5373","origin":"The Stacks Project","memory_eligible":false,"source_rank":2081,"rank":2081,"depth":44,"x":2126.064,"y":330.272,"cluster":"commutative-algebra"},{"id":"stacks:05D6","tag":"05D6","title":"Henselian local rings · Lemma 05D6","summary":"Let R be a henselian local ring. Any countably generated Mittag-Leffler module over R is a direct sum of finitely presented R-modules.","statement_latex":"Let $R$ be a henselian local ring.\nAny countably generated Mittag-Leffler module over $R$ is a direct\nsum of finitely presented $R$-modules.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05D6","source_file":"algebra.tex","source_line":42692,"source_end_line":42697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42692-L42697","statement_sha256":"f50c1907d847158bb3f991c5c1fc5f0d74d120faed4284e428a9589f4d7367a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2082,"rank":2082,"depth":44,"x":1870.42,"y":193.185,"cluster":"commutative-algebra"},{"id":"stacks:0BSH","tag":"0BSH","title":"Filtered colimits of étale ring maps · Lemma 0BSH","summary":"Let R → A and R → R' be ring maps. If A is a filtered colimit of étale R-algebras, then so is R' ⊗_R A is a filtered colimit of étale R'-algebras.","statement_latex":"Let $R \\to A$ and $R \\to R'$ be ring maps. If $A$ is a filtered\ncolimit of \\'etale $R$-algebras, then so is $R' \\otimes_R A$\nis a filtered colimit of \\'etale $R'$-algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Filtered colimits of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSH","source_file":"algebra.tex","source_line":42770,"source_end_line":42775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42770-L42775","statement_sha256":"14dcb173ce4859390523678906e95806905d173f11121867597242a1c062a32c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2083,"rank":2083,"depth":38,"x":2169.294,"y":149.194,"cluster":"commutative-algebra"},{"id":"stacks:0BSI","tag":"0BSI","title":"Filtered colimits of étale ring maps · Lemma 0BSI","summary":"Let A → B → C be ring maps. If B is a filtered colimit of étale A-algebras and C is a filtered colimit of étale B-algebras, then C is a filtered colimit of étale A-algebras.","statement_latex":"Let $A \\to B \\to C$ be ring maps. If $B$ is a filtered\ncolimit of \\'etale $A$-algebras and $C$ is a filtered colimit\nof \\'etale $B$-algebras, then $C$ is a filtered colimit of\n\\'etale $A$-algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Filtered colimits of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSI","source_file":"algebra.tex","source_line":42783,"source_end_line":42789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42783-L42789","statement_sha256":"8ef3e7b3f3a6567d2e63c8f8d4b5f8fbd1bbd12dd1505e0d1dd7f5eae567a661","origin":"The Stacks Project","memory_eligible":false,"source_rank":2084,"rank":2084,"depth":38,"x":1984.208,"y":351.305,"cluster":"commutative-algebra"},{"id":"stacks:0BSJ","tag":"0BSJ","title":"Filtered colimits of étale ring maps · Lemma 0BSJ","summary":"Let R be a ring. Let A = colim A_i be a filtered colimit of R-algebras such that each A_i is a filtered colimit of étale R-algebras. Then A is a filtered colimit of étale R-algebras.","statement_latex":"Let $R$ be a ring. Let $A = \\colim A_i$ be a filtered colimit\nof $R$-algebras such that each $A_i$ is a filtered colimit of\n\\'etale $R$-algebras. Then $A$ is a filtered colimit of \\'etale\n$R$-algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Filtered colimits of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSJ","source_file":"algebra.tex","source_line":42814,"source_end_line":42820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42814-L42820","statement_sha256":"926a5790c98c6f1db10741394c2d455a9f86eca84059ef343fd19894bb0b47b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2085,"rank":2085,"depth":6,"x":1958.144,"y":97.143,"cluster":"commutative-algebra"},{"id":"stacks:0GIM","tag":"0GIM","title":"Filtered colimits of étale ring maps · Lemma 0GIM","summary":"Let I be a directed set. Let i ↦ (R_i → A_i) be a system of arrows of rings over I. Set R = colim R_i and A = colim A_i. If each A_i is a filtered colimit of étale R_i-algebras, then A is a filtered colimit of étale R-algebras.","statement_latex":"Let $I$ be a directed set. Let $i \\mapsto (R_i \\to A_i)$ be a\nsystem of arrows of rings over $I$. Set $R = \\colim R_i$\nand $A = \\colim A_i$. If each $A_i$ is a filtered colimit of\n\\'etale $R_i$-algebras, then $A$ is a filtered colimit of \\'etale\n$R$-algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Filtered colimits of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIM","source_file":"algebra.tex","source_line":42842,"source_end_line":42849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42842-L42849","statement_sha256":"db4e78f6478bc75466fc07fc2947c0deec0e8eabff9014892ed589d4c2e937aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":2086,"rank":2086,"depth":39,"x":2181.847,"y":269.841,"cluster":"commutative-algebra"},{"id":"stacks:08HS","tag":"08HS","title":"Filtered colimits of étale ring maps · Lemma 08HS","summary":"Let R be a ring. Let A → B be an R-algebra homomorphism. If A and B are filtered colimits of étale R-algebras, then B is a filtered colimit of étale A-algebras.","statement_latex":"Let $R$ be a ring. Let $A \\to B$ be an $R$-algebra homomorphism.\nIf $A$ and $B$ are filtered colimits of \\'etale $R$-algebras, then\n$B$ is a filtered colimit of \\'etale $A$-algebras.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Filtered colimits of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HS","source_file":"algebra.tex","source_line":42858,"source_end_line":42863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42858-L42863","statement_sha256":"baa9a44a004b3443c19398c822c129f8a6df01e8f6596cc5d0e6be8d6584c00a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2087,"rank":2087,"depth":42,"x":1877.886,"y":269.43,"cluster":"commutative-algebra"},{"id":"stacks:08HR","tag":"08HR","title":"Filtered colimits of étale ring maps · Lemma 08HR","summary":"Let R → S be a ring map with S henselian local. Given • an R-algebra A which is a filtered colimit of étale R-algebras, • a prime q of A lying over p = R ∩ m_S, • a kappa( p)-algebra map τ : kappa( q) → S/ m_S, then there exists a unique homomorphism of R-algebras f : A → S such that q = f^-1( m_S) and f bmod q = τ.","statement_latex":"Let $R \\to S$ be a ring map with $S$ henselian local. Given\n\\begin{enumerate}\n\\item an $R$-algebra $A$ which is a filtered colimit of \\'etale $R$-algebras,\n\\item a prime $\\mathfrak q$ of $A$ lying over\n$\\mathfrak p = R \\cap \\mathfrak m_S$,\n\\item a $\\kappa(\\mathfrak p)$-algebra map\n$\\tau : \\kappa(\\mathfrak q) \\to S/\\mathfrak m_S$,\n\\end{enumerate}\nthen there exists a unique homomorphism of $R$-algebras $f : A \\to S$\nsuch that $\\mathfrak q = f^{-1}(\\mathfrak m_S)$ and\n$f \\bmod \\mathfrak q = \\tau$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Filtered colimits of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HR","source_file":"algebra.tex","source_line":42880,"source_end_line":42893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42880-L42893","statement_sha256":"3f9d400d63aaddcd550bc90c137a514e78d5483364bc3b2db4eba94401072d33","origin":"The Stacks Project","memory_eligible":false,"source_rank":2088,"rank":2088,"depth":45,"x":2102.447,"y":97.186,"cluster":"commutative-algebra"},{"id":"stacks:08HT","tag":"08HT","title":"Filtered colimits of étale ring maps · Lemma 08HT","summary":"Let R be a ring. Given a commutative diagram of ring maps xymatrix S ar[r] & K R ar[u] ar[r] & S' ar[u] where S, S' are henselian local, S, S' are filtered colimits of étale R-algebras, K is a field and the arrows S → K and S' → K identify K with the residue field of both S and S'. Then there exists a unique R-algebra isomorphism S → S' compatible with the maps to K.","statement_latex":"Let $R$ be a ring. Given a commutative diagram of ring maps\n$$\n\\xymatrix{\nS \\ar[r] & K \\\\\nR \\ar[u] \\ar[r] & S' \\ar[u]\n}\n$$\nwhere $S$, $S'$ are henselian local, $S$, $S'$ are filtered colimits\nof \\'etale $R$-algebras, $K$ is a field and the arrows $S \\to K$ and \n$S' \\to K$ identify $K$ with the residue field of both $S$ and $S'$.\nThen there exists a unique $R$-algebra isomorphism $S \\to S'$\ncompatible with the maps to $K$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Filtered colimits of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HT","source_file":"algebra.tex","source_line":42901,"source_end_line":42915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42901-L42915","statement_sha256":"d515ba9b66553db53e05f5634320fa3f1a70577cf800d42fbfe2516f4c4e397c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2089,"rank":2089,"depth":46,"x":2075.361,"y":351.724,"cluster":"commutative-algebra"},{"id":"stacks:04GI","tag":"04GI","title":"Filtered colimits of étale ring maps · Lemma 04GI","summary":"A filtered colimit of (strictly) henselian local rings along local homomorphisms is (strictly) henselian.","statement_latex":"A filtered colimit of (strictly) henselian local rings along local\nhomomorphisms is (strictly) henselian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Filtered colimits of étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GI","source_file":"algebra.tex","source_line":42925,"source_end_line":42929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42925-L42929","statement_sha256":"c2848b527a1c08c2b091d3e942ac20954e38fad54fb2870d888597f7a53a4e5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2090,"rank":2090,"depth":2,"x":1890.565,"y":148.578,"cluster":"commutative-algebra"},{"id":"stacks:04GN","tag":"04GN","title":"Henselization and strict henselization · Lemma 04GN","summary":"Let (R, m, kappa) be a local ring. There exists a local ring map R → R^h with the following properties • R^h is henselian, • R^h is a filtered colimit of étale R-algebras, • m R^h is the maximal ideal of R^h, and • kappa = R^h/ m R^h.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring. There exists a\nlocal ring map $R \\to R^h$ with the following properties\n\\begin{enumerate}\n\\item $R^h$ is henselian,\n\\item $R^h$ is a filtered colimit of \\'etale $R$-algebras,\n\\item $\\mathfrak m R^h$ is the\nmaximal ideal of $R^h$, and\n\\item $\\kappa = R^h/\\mathfrak m R^h$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GN","source_file":"algebra.tex","source_line":42970,"source_end_line":42981,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L42970-L42981","statement_sha256":"15072f48c45e9e98120d8f4876550313f46ffdf63a654a0707e8f0c55daf5575","origin":"The Stacks Project","memory_eligible":false,"source_rank":2091,"rank":2091,"depth":42,"x":2190.319,"y":193.535,"cluster":"commutative-algebra"},{"id":"stacks:04GP","tag":"04GP","title":"Henselization and strict henselization · Lemma 04GP","summary":"Let (R, m, kappa) be a local ring. Let kappa ⊂ kappa^sep be a separable algebraic closure. There exists a commutative diagram xymatrix kappa ar[r] & kappa ar[r] & kappa^sep R ar[r] ar[u] & R^h ar[r] ar[u] & R^sh ar[u] with the following properties • the map R^h → R^sh is local • R^sh is strictly henselian, • R^sh is a filtered colimit of étale R-algebras, • m R^sh is the maximal ideal of R^sh, and • kappa^sep = R^sh/ m R^sh.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring.\nLet $\\kappa \\subset \\kappa^{sep}$ be a separable algebraic closure.\nThere exists a commutative diagram\n$$\n\\xymatrix{\n\\kappa \\ar[r] & \\kappa \\ar[r] & \\kappa^{sep} \\\\\nR \\ar[r] \\ar[u] & R^h \\ar[r] \\ar[u] & R^{sh} \\ar[u]\n}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item the map $R^h \\to R^{sh}$ is local\n\\item $R^{sh}$ is strictly henselian,\n\\item $R^{sh}$ is a filtered colimit of \\'etale $R$-algebras,\n\\item $\\mathfrak m R^{sh}$ is the\nmaximal ideal of $R^{sh}$, and\n\\item $\\kappa^{sep} = R^{sh}/\\mathfrak m R^{sh}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GP","source_file":"algebra.tex","source_line":43092,"source_end_line":43112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43092-L43112","statement_sha256":"7e175dc5bb47792f50f5250b88c2b781fe51441da26dae86e1d8fdd83124bc93","origin":"The Stacks Project","memory_eligible":false,"source_rank":2092,"rank":2092,"depth":43,"x":1933.025,"y":330.529,"cluster":"commutative-algebra"},{"id":"stacks:04GQ","tag":"04GQ","title":"Henselization and strict henselization · Definition 04GQ","summary":"Let (R, m, kappa) be a local ring. • The local ring map R → R^h constructed in Lemma [Tag 04GN] is called the henselization of R. • Given a separable algebraic closure kappa ⊂ kappa^sep the local ring map R → R^sh constructed in Lemma [Tag 04GP] is called the strict henselization of R with respect to kappa ⊂ kappa^sep. • A local ring map R → R^sh is called a strict henselization of R if it is isomorphic to one of the local ring maps constructed in Lemma [Tag 04GP]","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring.\n\\begin{enumerate}\n\\item The local ring map $R \\to R^h$ constructed in\nLemma \\ref{lemma-henselization}\nis called the {\\it henselization} of $R$.\n\\item Given a separable algebraic closure $\\kappa \\subset \\kappa^{sep}$\nthe local ring map $R \\to R^{sh}$ constructed in\nLemma \\ref{lemma-strict-henselization}\nis called the\n{\\it strict henselization of $R$ with respect to\n$\\kappa \\subset \\kappa^{sep}$}.\n\\item A local ring map $R \\to R^{sh}$ is called a {\\it strict henselization}\nof $R$ if it is isomorphic to one of the local ring maps constructed in\nLemma \\ref{lemma-strict-henselization}\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GQ","source_file":"algebra.tex","source_line":43124,"source_end_line":43141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43124-L43141","statement_sha256":"4ef11c81a8e989b2dcb22cd834822c6255ba8118408fd0463bb1a5bd0e6e7d2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2093,"rank":2093,"depth":44,"x":2012.616,"y":83.415,"cluster":"commutative-algebra"},{"id":"stacks:04GR","tag":"04GR","title":"Henselization and strict henselization · Lemma 04GR","summary":"Let R → S be a local map of local rings. Let S → S^h be the henselization. Let R → A be an étale ring map and let q be a prime of A lying over m_R such that R/ m_R ≅ kappa( q). Then there exists a unique morphism of rings f : A → S^h fitting into the commutative diagram xymatrix A ar[r]_f & S^h R ar[u] ar[r] & S ar[u] such that f^-1( m_S^h) = q.","statement_latex":"Let $R \\to S$ be a local map of local rings.\nLet $S \\to S^h$ be the henselization.\nLet $R \\to A$ be an \\'etale ring map and let $\\mathfrak q$\nbe a prime of $A$ lying over $\\mathfrak m_R$\nsuch that $R/\\mathfrak m_R \\cong \\kappa(\\mathfrak q)$.\nThen there exists a unique morphism of rings\n$f : A \\to S^h$ fitting into the commutative diagram\n$$\n\\xymatrix{\nA \\ar[r]_f & S^h \\\\\nR \\ar[u] \\ar[r] & S \\ar[u]\n}\n$$\nsuch that $f^{-1}(\\mathfrak m_{S^h}) = \\mathfrak q$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GR","source_file":"algebra.tex","source_line":43186,"source_end_line":43202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43186-L43202","statement_sha256":"10c44c90731e887417268e603c477b2c2de2c543d027244f08396f3ba0fda650","origin":"The Stacks Project","memory_eligible":false,"source_rank":2094,"rank":2094,"depth":45,"x":2152.708,"y":310.89,"cluster":"commutative-algebra"},{"id":"stacks:04GS","tag":"04GS","title":"Henselization and strict henselization · Lemma 04GS","summary":"Let R → S be a local map of local rings. Let R → R^h and S → S^h be the henselizations. There exists a unique local ring map R^h → S^h fitting into the commutative diagram xymatrix R^h ar[r]_f & S^h R ar[u] ar[r] & S ar[u]","statement_latex":"Let $R \\to S$ be a local map of local rings.\nLet $R \\to R^h$ and $S \\to S^h$ be the henselizations.\nThere exists a unique local ring map $R^h \\to S^h$ fitting\ninto the commutative diagram\n$$\n\\xymatrix{\nR^h \\ar[r]_f & S^h \\\\\nR \\ar[u] \\ar[r] & S \\ar[u]\n}\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GS","source_file":"algebra.tex","source_line":43208,"source_end_line":43220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43208-L43220","statement_sha256":"19ee59d04b062f0d819b2a3dc0ce1be152ee42ae07658e09f683aba15582541e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2095,"rank":2095,"depth":46,"x":1866.359,"y":222.608,"cluster":"commutative-algebra"},{"id":"stacks:04GV","tag":"04GV","title":"Henselization and strict henselization · Lemma 04GV","summary":"Let R be a ring. Let p ⊂ R be a prime ideal. Consider the category of pairs (S, q) where R → S is étale and q is a prime lying over p such that kappa( p) = kappa( q). This category is filtered and (R_ p)^h = colim_(S, q) S = colim_(S, q) S_ q canonically.","statement_latex":"Let $R$ be a ring.\nLet $\\mathfrak p \\subset R$ be a prime ideal.\nConsider the category of pairs $(S, \\mathfrak q)$ where\n$R \\to S$ is \\'etale and $\\mathfrak q$ is a prime lying over $\\mathfrak p$\nsuch that $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$.\nThis category is filtered and\n$$\n(R_{\\mathfrak p})^h = \\colim_{(S, \\mathfrak q)} S\n= \\colim_{(S, \\mathfrak q)} S_{\\mathfrak q}\n$$\ncanonically.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GV","source_file":"algebra.tex","source_line":43229,"source_end_line":43242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43229-L43242","statement_sha256":"bce60b686cfce5951f90db85993c7a1a32f18f2a5a8d672335f07bbfb9cf22fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":2096,"rank":2096,"depth":43,"x":2148.619,"y":125.184,"cluster":"commutative-algebra"},{"id":"stacks:08HU","tag":"08HU","title":"Henselization and strict henselization · Lemma 08HU","summary":"Let R → S be a ring map. Let q ⊂ S be a prime lying over p ⊂ R. Let R → R^h and S → S^h be the henselizations of R_ p and S_ q. The local ring map R^h → S^h of Lemma [Tag 04GS] identifies S^h with the henselization of R^h ⊗_R S at the unique prime lying over m^h and q.","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying\nover $\\mathfrak p \\subset R$. Let $R \\to R^h$ and $S \\to S^h$ be the\nhenselizations of $R_\\mathfrak p$ and $S_\\mathfrak q$. The local ring map\n$R^h \\to S^h$ of Lemma \\ref{lemma-henselian-functorial} identifies $S^h$\nwith the henselization of $R^h \\otimes_R S$ at the unique prime\nlying over $\\mathfrak m^h$ and $\\mathfrak q$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HU","source_file":"algebra.tex","source_line":43315,"source_end_line":43323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43315-L43323","statement_sha256":"c42125e70f7dc8f4fd01eaf9eb4acdafae02a1fb6094f1f79105fb40894a4910","origin":"The Stacks Project","memory_eligible":false,"source_rank":2097,"rank":2097,"depth":47,"x":2018.776,"y":357.28,"cluster":"commutative-algebra"},{"id":"stacks:04GT","tag":"04GT","title":"Henselization and strict henselization · Lemma 04GT","summary":"Let φ : R → S be a local map of local rings. Let S/ m_S ⊂ kappa^sep be a separable algebraic closure. Let S → S^sh be the strict henselization of S with respect to S/ m_S ⊂ kappa^sep. Let R → A be an étale ring map and let q be a prime of A lying over m_R. Given any commutative diagram xymatrix kappa( q) ar[r]_φ & kappa^sep R/ m_R ar[r]^φ ar[u] & S/ m_S ar[u] there exists a unique morphism of rings f : A → S^sh fitting into the commutative diagram xymatrix A ar[r]_f &…","statement_latex":"Let $\\varphi : R \\to S$ be a local map of local rings.\nLet $S/\\mathfrak m_S \\subset \\kappa^{sep}$ be a separable algebraic closure.\nLet $S \\to S^{sh}$ be the strict henselization of $S$\nwith respect to $S/\\mathfrak m_S \\subset \\kappa^{sep}$.\nLet $R \\to A$ be an \\'etale ring map and let $\\mathfrak q$\nbe a prime of $A$ lying over $\\mathfrak m_R$.\nGiven any commutative diagram\n$$\n\\xymatrix{\n\\kappa(\\mathfrak q) \\ar[r]_{\\phi} & \\kappa^{sep} \\\\\nR/\\mathfrak m_R \\ar[r]^{\\varphi} \\ar[u] & S/\\mathfrak m_S \\ar[u]\n}\n$$\nthere exists a unique morphism of rings\n$f : A \\to S^{sh}$ fitting into the commutative diagram\n$$\n\\xymatrix{\nA \\ar[r]_f & S^{sh} \\\\\nR \\ar[u] \\ar[r]^{\\varphi} & S \\ar[u]\n}\n$$\nsuch that $f^{-1}(\\mathfrak m_{S^h}) = \\mathfrak q$ and the induced\nmap $\\kappa(\\mathfrak q) \\to \\kappa^{sep}$ is the given one.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GT","source_file":"algebra.tex","source_line":43337,"source_end_line":43362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43337-L43362","statement_sha256":"44c48b9c902e677616140b116c50c080f1425ca4046ac642c4d309655e9eb3a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2098,"rank":2098,"depth":45,"x":1927.837,"y":112.359,"cluster":"commutative-algebra"},{"id":"stacks:04GU","tag":"04GU","title":"Henselization and strict henselization · Lemma 04GU","summary":"Let R → S be a local map of local rings. Choose separable algebraic closures R/ m_R ⊂ kappa_1^sep and S/ m_S ⊂ kappa_2^sep. Let R → R^sh and S → S^sh be the corresponding strict henselizations. Given any commutative diagram xymatrix kappa_1^sep ar[r]_φ & kappa_2^sep R/ m_R ar[r]^φ ar[u] & S/ m_S ar[u] There exists a unique local ring map R^sh → S^sh fitting into the commutative diagram xymatrix R^sh ar[r]_f & S^sh R ar[u] ar[r] & S ar[u] and inducing φ on the residue…","statement_latex":"Let $R \\to S$ be a local map of local rings.\nChoose separable algebraic closures\n$R/\\mathfrak m_R \\subset \\kappa_1^{sep}$\nand\n$S/\\mathfrak m_S \\subset \\kappa_2^{sep}$.\nLet $R \\to R^{sh}$ and $S \\to S^{sh}$ be the corresponding strict\nhenselizations. Given any commutative diagram\n$$\n\\xymatrix{\n\\kappa_1^{sep} \\ar[r]_{\\phi} & \\kappa_2^{sep} \\\\\nR/\\mathfrak m_R \\ar[r]^{\\varphi} \\ar[u] & S/\\mathfrak m_S \\ar[u]\n}\n$$\nThere exists a unique local ring map $R^{sh} \\to S^{sh}$ fitting\ninto the commutative diagram\n$$\n\\xymatrix{\nR^{sh} \\ar[r]_f & S^{sh} \\\\\nR \\ar[u] \\ar[r] & S \\ar[u]\n}\n$$\nand inducing $\\phi$ on the residue fields of\n$R^{sh}$ and $S^{sh}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GU","source_file":"algebra.tex","source_line":43368,"source_end_line":43393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43368-L43393","statement_sha256":"4b257bf8a8702ad6a3b5da16ba25918905c006ed8bca3ffacb8d223649faf6b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2099,"rank":2099,"depth":46,"x":2191.962,"y":241.412,"cluster":"commutative-algebra"},{"id":"stacks:04GW","tag":"04GW","title":"Henselization and strict henselization · Lemma 04GW","summary":"Let R be a ring. Let p ⊂ R be a prime ideal. Let kappa( p) ⊂ kappa^sep be a separable algebraic closure. Consider the category of triples (S, q, φ) where R → S is étale, q is a prime lying over p, and φ : kappa( q) → kappa^sep is a kappa( p)-algebra map. This category is filtered and (R_ p)^sh = colim_(S, q, φ) S = colim_(S, q, φ) S_ q canonically.","statement_latex":"Let $R$ be a ring.\nLet $\\mathfrak p \\subset R$ be a prime ideal.\nLet $\\kappa(\\mathfrak p) \\subset \\kappa^{sep}$ be a\nseparable algebraic closure.\nConsider the category of triples $(S, \\mathfrak q, \\phi)$\nwhere $R \\to S$ is \\'etale, $\\mathfrak q$ is a prime lying over $\\mathfrak p$,\nand $\\phi : \\kappa(\\mathfrak q) \\to \\kappa^{sep}$ is a\n$\\kappa(\\mathfrak p)$-algebra map. This category is filtered and\n$$\n(R_{\\mathfrak p})^{sh} =\n\\colim_{(S, \\mathfrak q, \\phi)} S =\n\\colim_{(S, \\mathfrak q, \\phi)} S_{\\mathfrak q}\n$$\ncanonically.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GW","source_file":"algebra.tex","source_line":43399,"source_end_line":43415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43399-L43415","statement_sha256":"ff91129e4b0898645e4277e29b7045c59b47c6f3870ed1e88cd4f58259b6a0fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2100,"rank":2100,"depth":45,"x":1893.296,"y":296.144,"cluster":"commutative-algebra"},{"id":"stacks:08HV","tag":"08HV","title":"Henselization and strict henselization · Lemma 08HV","summary":"Let R → S be a ring map. Let q ⊂ S be a prime lying over p ⊂ R. Choose separable algebraic closures kappa( p) ⊂ kappa_1^sep and kappa( q) ⊂ kappa_2^sep. Let R^sh and S^sh be the corresponding strict henselizations of R_ p and S_ q. Given any commutative diagram xymatrix kappa_1^sep ar[r]_φ & kappa_2^sep kappa( p) ar[r]^φ ar[u] & kappa( q) ar[u] The local ring map R^sh → S^sh of Lemma [Tag 04GU] identifies S^sh with the strict henselization of R^sh ⊗_R S at a prime lying…","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime lying\nover $\\mathfrak p \\subset R$. Choose separable algebraic closures\n$\\kappa(\\mathfrak p) \\subset \\kappa_1^{sep}$\nand\n$\\kappa(\\mathfrak q) \\subset \\kappa_2^{sep}$.\nLet $R^{sh}$ and $S^{sh}$ be the corresponding strict\nhenselizations of $R_\\mathfrak p$ and $S_\\mathfrak q$.\nGiven any commutative diagram\n$$\n\\xymatrix{\n\\kappa_1^{sep} \\ar[r]_{\\phi} & \\kappa_2^{sep} \\\\\n\\kappa(\\mathfrak p) \\ar[r]^{\\varphi} \\ar[u] & \\kappa(\\mathfrak q) \\ar[u]\n}\n$$\nThe local ring map $R^{sh} \\to S^{sh}$ of\nLemma \\ref{lemma-strictly-henselian-functorial} identifies $S^{sh}$\nwith the strict henselization of $R^{sh} \\otimes_R S$ at a prime\nlying over $\\mathfrak q$ and the maximal ideal\n$\\mathfrak m^{sh} \\subset R^{sh}$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HV","source_file":"algebra.tex","source_line":43508,"source_end_line":43529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43508-L43529","statement_sha256":"a5e0b69b0e8f665656734380b7a3f497e1c9e6a8c877e8b40a26999ff7cd2575","origin":"The Stacks Project","memory_eligible":false,"source_rank":2101,"rank":2101,"depth":48,"x":2069.587,"y":86.229,"cluster":"commutative-algebra"},{"id":"stacks:0C2Z","tag":"0C2Z","title":"Henselization and strict henselization · Lemma 0C2Z","summary":"Let R → S be a ring map. Let q ⊂ S be a prime lying over p ⊂ R such that kappa( p) → kappa( q) is an isomorphism. Choose a separable algebraic closure kappa^sep of kappa( p) = kappa( q). Then (S_ q)^sh = (S_ q)^h ⊗_(R_ p)^h (R_ p)^sh","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q \\subset S$ be a prime\nlying over $\\mathfrak p \\subset R$ such that\n$\\kappa(\\mathfrak p) \\to \\kappa(\\mathfrak q)$ is an isomorphism.\nChoose a separable algebraic closure $\\kappa^{sep}$ of\n$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak q)$.\nThen\n$$\n(S_\\mathfrak q)^{sh} =\n(S_\\mathfrak q)^h \\otimes_{(R_\\mathfrak p)^h} (R_\\mathfrak p)^{sh}\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and strict henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2Z","source_file":"algebra.tex","source_line":43540,"source_end_line":43552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43540-L43552","statement_sha256":"d5aa5424cb26b368ea2f7793a95345754dd5ed77b40e9d0413bf2901d15d264c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2102,"rank":2102,"depth":0,"x":2108.416,"y":341.153,"cluster":"commutative-algebra"},{"id":"stacks:05WP","tag":"05WP","title":"Henselization and quasi-finite ring maps · Lemma 05WP","summary":"Let R → S be a ring map. Let q be a prime of S lying over p in R. Assume R → S is quasi-finite at q. The commutative diagram xymatrix R_ p^h ar[r] & S_ q^h R_ p ar[u] ar[r] & S_ q ar[u] of Lemma [Tag 04GS] identifies S_ q^h with the localization of R_ p^h ⊗_R_ p S_ q at the prime generated by q. Moreover, the ring map R_ p^h → S_ q^h is finite.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p$ in $R$.\nAssume $R \\to S$ is quasi-finite at $\\mathfrak q$.\nThe commutative diagram\n$$\n\\xymatrix{\nR_{\\mathfrak p}^h \\ar[r] & S_{\\mathfrak q}^h \\\\\nR_{\\mathfrak p} \\ar[u] \\ar[r] & S_{\\mathfrak q} \\ar[u]\n}\n$$\nof\nLemma \\ref{lemma-henselian-functorial}\nidentifies $S_{\\mathfrak q}^h$ with the localization of\n$R_{\\mathfrak p}^h \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$\nat the prime generated by $\\mathfrak q$. Moreover, the ring\nmap $R_{\\mathfrak p}^h \\to S_{\\mathfrak q}^h$ is finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WP","source_file":"algebra.tex","source_line":43580,"source_end_line":43598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43580-L43598","statement_sha256":"1f994c90fae9b7df6482d7df4465f9bb57c3646b94a8f6018f3dd640e826a41d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2103,"rank":2103,"depth":48,"x":1874.685,"y":175.141,"cluster":"commutative-algebra"},{"id":"stacks:05WQ","tag":"05WQ","title":"Henselization and quasi-finite ring maps · Lemma 05WQ","summary":"Henselization is compatible with quotients. Let R be a local ring with henselization R^h. Let I ⊂ m_R. Then R^h/IR^h is the henselization of R/I.","statement_latex":"\\begin{slogan}\nHenselization is compatible with quotients.\n\\end{slogan}\nLet $R$ be a local ring with henselization $R^h$.\nLet $I \\subset \\mathfrak m_R$.\nThen $R^h/IR^h$ is the henselization of $R/I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WQ","source_file":"algebra.tex","source_line":43616,"source_end_line":43624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43616-L43624","statement_sha256":"0a4a3c3dc38412c64541bd2aedbbda6630ef67a27f0030a830cfda812cc261af","origin":"The Stacks Project","memory_eligible":false,"source_rank":2104,"rank":2104,"depth":49,"x":2180.663,"y":164.927,"cluster":"commutative-algebra"},{"id":"stacks:05WR","tag":"05WR","title":"Henselization and quasi-finite ring maps · Lemma 05WR","summary":"Let R → S be a ring map. Let q be a prime of S lying over p in R. Assume R → S is quasi-finite at q. Let kappa_2^sep/kappa( q) be a separable algebraic closure and denote kappa_1^sep ⊂ kappa_2^sep the subfield of elements separable algebraic over kappa( p) (Fields, Lemma [Tag 030K]). The commutative diagram xymatrix R_ p^sh ar[r] & S_ q^sh R_ p ar[u] ar[r] & S_ q ar[u] of Lemma [Tag 04GU] identifies S_ q^sh with the localization of R_ p^sh ⊗_R_ p S_ q at the prime ideal…","statement_latex":"Let $R \\to S$ be a ring map.\nLet $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p$ in $R$.\nAssume $R \\to S$ is quasi-finite at $\\mathfrak q$.\nLet $\\kappa_2^{sep}/\\kappa(\\mathfrak q)$ be a separable algebraic closure\nand denote $\\kappa_1^{sep} \\subset \\kappa_2^{sep}$ the subfield\nof elements separable algebraic over $\\kappa(\\mathfrak p)$\n(Fields, Lemma \\ref{fields-lemma-separable-first}).\nThe commutative diagram\n$$\n\\xymatrix{\nR_{\\mathfrak p}^{sh} \\ar[r] & S_{\\mathfrak q}^{sh} \\\\\nR_{\\mathfrak p} \\ar[u] \\ar[r] & S_{\\mathfrak q} \\ar[u]\n}\n$$\nof Lemma \\ref{lemma-strictly-henselian-functorial}\nidentifies $S_{\\mathfrak q}^{sh}$ with the localization of\n$R_{\\mathfrak p}^{sh} \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}$\nat the prime ideal which is the kernel of the map\n$$\nR_{\\mathfrak p}^{sh} \\otimes_{R_{\\mathfrak p}} S_{\\mathfrak q}\n\\longrightarrow\n\\kappa_1^{sep} \\otimes_{\\kappa(\\mathfrak p)} \\kappa(\\mathfrak q)\n\\longrightarrow\n\\kappa_2^{sep}\n$$\nMoreover, the ring map $R_{\\mathfrak p}^{sh} \\to S_{\\mathfrak q}^{sh}$\nis a finite local homomorphism of local rings whose residue field extension\nis the extension $\\kappa_2^{sep}/\\kappa_1^{sep}$ which is both finite and\npurely inseparable.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WR","source_file":"algebra.tex","source_line":43631,"source_end_line":43662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43631-L43662","statement_sha256":"bfbe1db09214b723d8e3fe0d803673f06a23f1aa504530be23dc0ba07a6f32f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2105,"rank":2105,"depth":49,"x":1963.164,"y":346.152,"cluster":"commutative-algebra"},{"id":"stacks:05WS","tag":"05WS","title":"Henselization and quasi-finite ring maps · Lemma 05WS","summary":"Let R be a local ring with strict henselization R^sh. Let I ⊂ m_R. Then R^sh/IR^sh is a strict henselization of R/I.","statement_latex":"Let $R$ be a local ring with strict henselization $R^{sh}$.\nLet $I \\subset \\mathfrak m_R$.\nThen $R^{sh}/IR^{sh}$ is a strict henselization of $R/I$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WS","source_file":"algebra.tex","source_line":43702,"source_end_line":43707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43702-L43707","statement_sha256":"31645a5ea6d7b1b1c32b05fbd42147449f2a0313ed05ae332a8b8d513bcd147e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2106,"rank":2106,"depth":50,"x":1977.814,"y":88.999,"cluster":"commutative-algebra"},{"id":"stacks:092Y","tag":"092Y","title":"Henselization and quasi-finite ring maps · Lemma 092Y","summary":"Let A → B and A → C be local homomorphisms of local rings. If A → C is integral and either kappa( m_C)/kappa( m_A) or kappa( m_B)/kappa( m_A) is purely inseparable, then D = B ⊗_A C is a local ring and B → D and C → D are local.","statement_latex":"Let $A \\to B$ and $A \\to C$ be local homomorphisms of local rings.\nIf $A \\to C$ is integral and either\n$\\kappa(\\mathfrak m_C)/\\kappa(\\mathfrak m_A)$ or\n$\\kappa(\\mathfrak m_B)/\\kappa(\\mathfrak m_A)$ is purely\ninseparable, then $D = B \\otimes_A C$ is a local ring and\n$B \\to D$ and $C \\to D$ are local.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092Y","source_file":"algebra.tex","source_line":43714,"source_end_line":43722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43714-L43722","statement_sha256":"6dd9f45f29936c549aa23bdef65d72f9af891c10436514f5981812ff29b34dc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2107,"rank":2107,"depth":7,"x":2173.887,"y":287.014,"cluster":"commutative-algebra"},{"id":"stacks:0GIP","tag":"0GIP","title":"Henselization and quasi-finite ring maps · Lemma 0GIP","summary":"Let A → B and A → C be ring maps. Let kappa be a separably algebraically closed field and let B ⊗_A C → kappa be a ring homomorphism. Denote xymatrix B^sh ar[r] & (B ⊗_A C)^sh A^sh ar[u] ar[r] & C^sh ar[u] the corresponding maps of strict henselizations (see proof). If • A → B is quasi-finite at the prime p_B = Ker(B → kappa), or • B is a filtered colimit of quasi-finite A-algebras, or • B_ p_B is a filtered colimit of quasi-finite algebras over A_ p_A, or • B is integral…","statement_latex":"Let $A \\to B$ and $A \\to C$ be ring maps. Let $\\kappa$ be a separably\nalgebraically closed field and let $B \\otimes_A C \\to \\kappa$\nbe a ring homomorphism. Denote\n$$\n\\xymatrix{\nB^{sh} \\ar[r] & (B \\otimes_A C)^{sh} \\\\\nA^{sh} \\ar[u] \\ar[r] & C^{sh} \\ar[u]\n}\n$$\nthe corresponding maps of strict henselizations (see proof). If\n\\begin{enumerate}\n\\item $A \\to B$ is quasi-finite at the prime\n$\\mathfrak p_B = \\Ker(B \\to \\kappa)$, or\n\\item $B$ is a filtered colimit of quasi-finite $A$-algebras, or\n\\item $B_{\\mathfrak p_B}$ is a filtered colimit of quasi-finite\nalgebras over $A_{\\mathfrak p_A}$, or\n\\item $B$ is integral over $A$,\n\\end{enumerate}\nthen $B^{sh} \\otimes_{A^{sh}} C^{sh} \\to (B \\otimes_A C)^{sh}$\nis an isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Henselization and quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIP","source_file":"algebra.tex","source_line":43741,"source_end_line":43763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43741-L43763","statement_sha256":"3dfcde68ef5332a3ba55a0d3be12f3fd7fd620ea173d7c7b87365db4e356a443","origin":"The Stacks Project","memory_eligible":false,"source_rank":2108,"rank":2108,"depth":50,"x":1869.944,"y":252.243,"cluster":"commutative-algebra"},{"id":"stacks:031P","tag":"031P","title":"Serre's criterion for normality · Definition 031P","summary":"Let R be a Noetherian ring. Let k ≥ 0 be an integer. • We say R has property (R_k) if for every prime p of height ≤ k the local ring R_ p is regular. We also say that R is regular in codimension ≤ k. • We say R has property (S_k) if for every prime p the local ring R_ p has depth at least min(k, dim(R_ p)). • Let M be a finite R-module. We say M has property (S_k) if for every prime p the module M_ p has depth at least min(k, dim(Supp(M_ p))).","statement_latex":"Let $R$ be a Noetherian ring.\nLet $k \\geq 0$ be an integer.\n\\begin{enumerate}\n\\item We say $R$ has property {\\it $(R_k)$} if for every prime $\\mathfrak p$\nof height $\\leq k$ the local ring $R_{\\mathfrak p}$ is regular.\nWe also say that $R$ is {\\it regular in codimension $\\leq k$}.\n\\item We say $R$ has property {\\it $(S_k)$} if for every prime $\\mathfrak p$\nthe local ring $R_{\\mathfrak p}$ has depth at least\n$\\min\\{k, \\dim(R_{\\mathfrak p})\\}$.\n\\item Let $M$ be a finite $R$-module. We say $M$ has property $(S_k)$\nif for every prime $\\mathfrak p$ the module\n$M_{\\mathfrak p}$ has depth at least\n$\\min\\{k, \\dim(\\text{Supp}(M_{\\mathfrak p}))\\}$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Serre's criterion for normality","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031P","source_file":"algebra.tex","source_line":43871,"source_end_line":43887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43871-L43887","statement_sha256":"f21d8dd5ffcf679d6feb208baa0316304c2dcc6a4645b532ccbee354eaa0b46b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2109,"rank":2109,"depth":0,"x":2122.132,"y":105.358,"cluster":"commutative-algebra"},{"id":"stacks:031Q","tag":"031Q","title":"Serre's criterion for normality · Lemma 031Q","summary":"Let R be a Noetherian ring. Let M be a finite R-module. The following are equivalent: • M has no embedded associated prime, and • M has property (S_1).","statement_latex":"Let $R$ be a Noetherian ring.\nLet $M$ be a finite $R$-module.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $M$ has no embedded associated prime, and\n\\item $M$ has property $(S_1)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Serre's criterion for normality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031Q","source_file":"algebra.tex","source_line":43896,"source_end_line":43905,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43896-L43905","statement_sha256":"0d1a3f63fd1502559ca066e6e9bce3ffedcdf374b8507faf40e59a1880594821","origin":"The Stacks Project","memory_eligible":false,"source_rank":2110,"rank":2110,"depth":11,"x":2054.265,"y":356.869,"cluster":"commutative-algebra"},{"id":"stacks:031R","tag":"031R","title":"Serre's criterion for normality · Lemma 031R","summary":"Reduced equals R0 plus S1. Let R be a Noetherian ring. The following are equivalent: • R is reduced, and • R has properties (R_0) and (S_1).","statement_latex":"\\begin{slogan}\nReduced equals R0 plus S1.\n\\end{slogan}\nLet $R$ be a Noetherian ring.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $R$ is reduced, and\n\\item $R$ has properties $(R_0)$ and $(S_1)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Serre's criterion for normality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031R","source_file":"algebra.tex","source_line":43934,"source_end_line":43945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43934-L43945","statement_sha256":"50f1f50d1fcd6c75d0003b75c218482144e95b076c1908b4a4fff14c47623d88","origin":"The Stacks Project","memory_eligible":false,"source_rank":2111,"rank":2111,"depth":10,"x":1901.989,"y":132.808,"cluster":"commutative-algebra"},{"id":"stacks:031S","tag":"031S","title":"Serre's criterion for normality · Lemma 031S","summary":"[EGA] Normal equals R1 plus S2. Let R be a Noetherian ring. The following are equivalent: • R is a normal ring, and • R has properties (R_1) and (S_2).","statement_latex":"\\begin{reference}\n\\cite[IV, Theorem 5.8.6]{EGA}\n\\end{reference}\n\\begin{slogan}\nNormal equals R1 plus S2.\n\\end{slogan}\nLet $R$ be a Noetherian ring.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $R$ is a normal ring, and\n\\item $R$ has properties $(R_1)$ and $(S_2)$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Serre's criterion for normality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031S","source_file":"algebra.tex","source_line":43976,"source_end_line":43990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L43976-L43990","statement_sha256":"e5e0085fad54adcc75968d468045cfebbb037f7ca0c9d6211f8ca7a35096117f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2112,"rank":2112,"depth":16,"x":2194.578,"y":211.654,"cluster":"commutative-algebra"},{"id":"stacks:0567","tag":"0567","title":"Serre's criterion for normality · Lemma 0567","summary":"A regular ring is normal.","statement_latex":"A regular ring is normal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Serre's criterion for normality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0567","source_file":"algebra.tex","source_line":44045,"source_end_line":44048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44045-L44048","statement_sha256":"52c094fa17f634344f7a76386248ec69ad2604a7f14873ac9e365a1975170e35","origin":"The Stacks Project","memory_eligible":false,"source_rank":2113,"rank":2113,"depth":17,"x":1915.308,"y":319.58,"cluster":"commutative-algebra"},{"id":"stacks:031T","tag":"031T","title":"Serre's criterion for normality · Lemma 031T","summary":"Let R be a Noetherian normal domain with fraction field K. Then • for any nonzero a ∈ R the quotient R/aR has no embedded primes, and all its associated primes have height 1 • R = ⋂_height( p) = 1 R_ p • For any nonzero x ∈ K the quotient R/(R ∩ xR) has no embedded primes, and all its associates primes have height 1.","statement_latex":"Let $R$ be a Noetherian normal domain with fraction field $K$. Then\n\\begin{enumerate}\n\\item for any nonzero $a \\in R$ the quotient $R/aR$ has no embedded primes,\nand all its associated primes have height $1$\n\\item\n$$\nR = \\bigcap\\nolimits_{\\text{height}(\\mathfrak p) = 1} R_{\\mathfrak p}\n$$\n\\item For any nonzero $x \\in K$ the quotient $R/(R \\cap xR)$\nhas no embedded primes, and all its associates primes have height $1$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Serre's criterion for normality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031T","source_file":"algebra.tex","source_line":44060,"source_end_line":44073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44060-L44073","statement_sha256":"a49c925bb6e28c9163f1aa5757df973707bd13339f1b693297d141f225ee7fcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2114,"rank":2114,"depth":17,"x":2034.494,"y":81.435,"cluster":"commutative-algebra"},{"id":"stacks:090W","tag":"090W","title":"Formal smoothness of fields · Lemma 090W","summary":"Let K/k be a finitely generated field extension. The following are equivalent • K is a finite separable field extension of k, • Ω_K/k = 0, • K is formally unramified over k, • K is unramified over k, • K is formally étale over k, • K is étale over k.","statement_latex":"Let $K/k$ be a finitely generated field extension.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ is a finite separable field extension of $k$,\n\\item $\\Omega_{K/k} = 0$,\n\\item $K$ is formally unramified over $k$,\n\\item $K$ is unramified over $k$,\n\\item $K$ is formally \\'etale over $k$,\n\\item $K$ is \\'etale over $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090W","source_file":"algebra.tex","source_line":44136,"source_end_line":44148,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44136-L44148","statement_sha256":"43ffebb56f443fcbc7f0a0db691cdfbae13baee95a1244647c7fb166d6233fc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2115,"rank":2115,"depth":41,"x":2138.16,"y":324.768,"cluster":"commutative-algebra"},{"id":"stacks:031W","tag":"031W","title":"Formal smoothness of fields · Lemma 031W","summary":"Let k be a perfect field of characteristic p > 0. Let K/k be an extension. Let a ∈ K. Then da = 0 in Ω_K/k if and only if a is a pth power.","statement_latex":"Let $k$ be a perfect field of characteristic $p > 0$.\nLet $K/k$ be an extension.\nLet $a \\in K$. Then $\\text{d}a = 0$ in $\\Omega_{K/k}$\nif and only if $a$ is a $p$th power.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031W","source_file":"algebra.tex","source_line":44173,"source_end_line":44179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44173-L44179","statement_sha256":"c936641bd4e9124c65108e707c99d8bc128e4e68242814c216587269bc860dcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2116,"rank":2116,"depth":2,"x":1865.926,"y":204.112,"cluster":"commutative-algebra"},{"id":"stacks:07DZ","tag":"07DZ","title":"Formal smoothness of fields · Lemma 07DZ","summary":"Let k be a field of characteristic p > 0. Let a_1, …, a_n ∈ k be elements such that da_1, …, da_n are linearly independent in Ω_k/F_p. Then the field extension k(a_1^1/p, …, a_n^1/p) has degree p^n over k.","statement_latex":"Let $k$ be a field of characteristic $p > 0$.\nLet $a_1, \\ldots, a_n \\in k$ be elements such that\n$\\text{d}a_1, \\ldots, \\text{d}a_n$ are linearly independent in\n$\\Omega_{k/\\mathbf{F}_p}$. Then the field extension\n$k(a_1^{1/p}, \\ldots, a_n^{1/p})$ has degree $p^n$ over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DZ","source_file":"algebra.tex","source_line":44228,"source_end_line":44235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44228-L44235","statement_sha256":"ff4471170e1e7b841848204e179ce4aad27fdbec07649ea89ebc950b3e7e949d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2117,"rank":2117,"depth":3,"x":2163.817,"y":138.584,"cluster":"commutative-algebra"},{"id":"stacks:031X","tag":"031X","title":"Formal smoothness of fields · Lemma 031X","summary":"Let k be a field of characteristic p > 0. The following are equivalent: • the field extension K/k is separable (see Definition [Tag 030O]), and • the map K ⊗_k Ω_k/F_p → Ω_K/F_p is injective.","statement_latex":"Let $k$ be a field of characteristic $p > 0$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the field extension $K/k$ is separable\n(see Definition \\ref{definition-separable-field-extension}), and\n\\item the map\n$K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K/\\mathbf{F}_p}$\nis injective.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031X","source_file":"algebra.tex","source_line":44255,"source_end_line":44266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44255-L44266","statement_sha256":"1a4e5f78d82c4ac0fde41d57a9459fff4717051c9d62b1cd9179862b80497f0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2118,"rank":2118,"depth":8,"x":1996.785,"y":356.021,"cluster":"commutative-algebra"},{"id":"stacks:031Y","tag":"031Y","title":"Formal smoothness of fields · Lemma 031Y","summary":"Let K/k be an extension of fields. If K is formally smooth over k, then K is a separable extension of k.","statement_latex":"Let $K/k$ be an extension of fields.\nIf $K$ is formally smooth over $k$, then $K$ is\na separable extension of $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031Y","source_file":"algebra.tex","source_line":44381,"source_end_line":44386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44381-L44386","statement_sha256":"dfa09011bf9e9aa8ea8b18fa60e4a109a815ba9048215cd43b5b358c51d3d8d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2119,"rank":2119,"depth":9,"x":1945.073,"y":100.805,"cluster":"commutative-algebra"},{"id":"stacks:031Z","tag":"031Z","title":"Formal smoothness of fields · Lemma 031Z","summary":"Let K/k be an extension of fields. Then K is formally smooth over k if and only if H_1(L_K/k) = 0.","statement_latex":"Let $K/k$ be an extension of fields.\nThen $K$ is formally smooth over $k$ if and only if\n$H_1(L_{K/k}) = 0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/031Z","source_file":"algebra.tex","source_line":44396,"source_end_line":44401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44396-L44401","statement_sha256":"9a59db4d6d8224742e161e5112ee1050b775ec3d0e8353bdb65ebad7386b190c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2120,"rank":2120,"depth":5,"x":2188.541,"y":259.719,"cluster":"commutative-algebra"},{"id":"stacks:0320","tag":"0320","title":"Formal smoothness of fields · Lemma 0320","summary":"Let K/k be an extension of fields. • If K is purely transcendental over k, then K is formally smooth over k. • If K is separable algebraic over k, then K is formally smooth over k. • If K is separable over k, then K is formally smooth over k.","statement_latex":"Let $K/k$ be an extension of fields.\n\\begin{enumerate}\n\\item If $K$ is purely transcendental over $k$, then\n$K$ is formally smooth over $k$.\n\\item If $K$ is separable algebraic over $k$, then $K$ is\nformally smooth over $k$.\n\\item If $K$ is separable over $k$, then $K$ is formally smooth\nover $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0320","source_file":"algebra.tex","source_line":44408,"source_end_line":44419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44408-L44419","statement_sha256":"c4689400b59a05f7d95871c97e24f7b54ac0d51ade63adb344f6855cc982e46e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2121,"rank":2121,"depth":7,"x":1881.093,"y":280.696,"cluster":"commutative-algebra"},{"id":"stacks:0321","tag":"0321","title":"Formal smoothness of fields · Lemma 0321","summary":"Formally smooth equals separable for field extensions. Let k be a field. • If the characteristic of k is zero, then any extension field of k is formally smooth over k. • If the characteristic of k is p > 0, then K/k is formally smooth if and only if it is a separable field extension.","statement_latex":"\\begin{slogan}\nFormally smooth equals separable for field extensions.\n\\end{slogan}\nLet $k$ be a field.\n\\begin{enumerate}\n\\item If the characteristic of $k$ is zero, then any extension field\nof $k$ is formally smooth over $k$.\n\\item If the characteristic of $k$ is $p > 0$, then $K/k$ is\nformally smooth if and only if it is a separable field extension.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0321","source_file":"algebra.tex","source_line":44451,"source_end_line":44463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44451-L44463","statement_sha256":"9e5e6db7f6aa799bd9bfbc886d8e352d578e88aee13aa27bc0c84102fa0e8a66","origin":"The Stacks Project","memory_eligible":false,"source_rank":2122,"rank":2122,"depth":10,"x":2091.016,"y":90.698,"cluster":"commutative-algebra"},{"id":"stacks:0322","tag":"0322","title":"Formal smoothness of fields · Proposition 0322","summary":"Let K/k be a field extension. If the characteristic of k is zero then • K is separable over k, • K is geometrically reduced over k, • K is formally smooth over k, • H_1(L_K/k) = 0, and • the map K ⊗_k Ω_k/Z → Ω_K/Z is injective. If the characteristic of k is p > 0, then the following are equivalent: • K is separable over k, • the ring K ⊗_k k^1/p is reduced, • K is geometrically reduced over k, • the map K ⊗_k Ω_k/F_p → Ω_K/F_p is injective, • H_1(L_K/k) = 0, and • K is…","statement_latex":"Let $K/k$ be a field extension.\nIf the characteristic of $k$ is zero then\n\\begin{enumerate}\n\\item $K$ is separable over $k$,\n\\item $K$ is geometrically reduced over $k$,\n\\item $K$ is formally smooth over $k$,\n\\item $H_1(L_{K/k}) = 0$, and\n\\item the map $K \\otimes_k \\Omega_{k/\\mathbf{Z}} \\to \\Omega_{K/\\mathbf{Z}}$\nis injective.\n\\end{enumerate}\nIf the characteristic of $k$ is $p > 0$, then the following are\nequivalent:\n\\begin{enumerate}\n\\item $K$ is separable over $k$,\n\\item the ring $K \\otimes_k k^{1/p}$ is reduced,\n\\item $K$ is geometrically reduced over $k$,\n\\item the map $K \\otimes_k \\Omega_{k/\\mathbf{F}_p} \\to \\Omega_{K/\\mathbf{F}_p}$\nis injective,\n\\item $H_1(L_{K/k}) = 0$, and\n\\item $K$ is formally smooth over $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0322","source_file":"algebra.tex","source_line":44474,"source_end_line":44497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44474-L44497","statement_sha256":"97f38ef7ab5bd3f8ca40db19ee060e06e9a22387e23f0fc6b4a6d3eb5edde0bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2123,"rank":2123,"depth":11,"x":2089.013,"y":350.02,"cluster":"commutative-algebra"},{"id":"stacks:037X","tag":"037X","title":"Formal smoothness of fields · Lemma 037X","summary":"Let K/k be a finitely generated field extension. Then K is separable over k if and only if K is the localization of a smooth k-algebra.","statement_latex":"Let $K/k$ be a finitely generated field extension.\nThen $K$ is separable over $k$ if and only if $K$ is\nthe localization of a smooth $k$-algebra.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037X","source_file":"algebra.tex","source_line":44511,"source_end_line":44516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44511-L44516","statement_sha256":"e6053868ce9e39314594bfae980931d50481fe8b2a48dd6b54614f09e2d7da8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2124,"rank":2124,"depth":39,"x":1881.867,"y":157.585,"cluster":"commutative-algebra"},{"id":"stacks:07BV","tag":"07BV","title":"Formal smoothness of fields · Lemma 07BV","summary":"Let K/k be a field extension. Then K is a filtered colimit of global complete intersection algebras over k. If K/k is separable, then K is a filtered colimit of smooth algebras over k.","statement_latex":"Let $K/k$ be a field extension.\nThen $K$ is a filtered colimit of global complete intersection\nalgebras over $k$. If $K/k$ is separable, then $K$ is a filtered\ncolimit of smooth algebras over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Formal smoothness of fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BV","source_file":"algebra.tex","source_line":44526,"source_end_line":44532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44526-L44532","statement_sha256":"5d7b7abf24a415308d5aced1c8539096c7c9f3d89384d156083d247df2a4ce4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2125,"rank":2125,"depth":40,"x":2189.488,"y":181.954,"cluster":"commutative-algebra"},{"id":"stacks:03C3","tag":"03C3","title":"Constructing flat ring maps · Lemma 03C3","summary":"Let (R, m, k) be a local ring. Let K/k be a field extension. There exists a local ring (R', m', k'), a flat local ring map R → R' such that m' = mR' and such that k' is isomorphic to K as an extension of k.","statement_latex":"Let $(R, \\mathfrak m, k)$ be a local ring. Let $K/k$ be a field\nextension. There exists a local ring $(R', \\mathfrak m', k')$, a flat local\nring map $R \\to R'$ such that $\\mathfrak m' = \\mathfrak mR'$ and such that\n$k'$ is isomorphic to $K$ as an extension of $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Constructing flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03C3","source_file":"algebra.tex","source_line":44572,"source_end_line":44578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44572-L44578","statement_sha256":"647069e832f717dd3916d28dea30c54b28cb6a4b584ee435d46a9c5a727b4565","origin":"The Stacks Project","memory_eligible":false,"source_rank":2126,"rank":2126,"depth":0,"x":1942.956,"y":338.6,"cluster":"commutative-algebra"},{"id":"stacks:09E0","tag":"09E0","title":"Constructing flat ring maps · Lemma 09E0","summary":"Let (R, m, k) be a local ring. If k ⊂ K is a separable algebraic extension, then there exists a directed set I and a system of finite étale extensions R ⊂ R_i, i ∈ I of local rings such that R' = colim R_i has residue field K (as extension of k).","statement_latex":"Let $(R, \\mathfrak m, k)$ be a local ring. If $k \\subset K$ is a\nseparable algebraic extension, then there exists a directed set $I$ and\na system of finite \\'etale extensions $R \\subset R_i$, $i \\in I$\nof local rings such that $R' = \\colim R_i$ has residue field\n$K$ (as extension of $k$).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Constructing flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09E0","source_file":"algebra.tex","source_line":44642,"source_end_line":44649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44642-L44649","statement_sha256":"edcf7ae95f33224a48c541eb130d218200c3d64919bf3cda2d36772801efbaaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":2127,"rank":2127,"depth":1,"x":1998.798,"y":83.101,"cluster":"commutative-algebra"},{"id":"stacks:07NE","tag":"07NE","title":"Constructing flat ring maps · Lemma 07NE","summary":"Let R be a ring. Let p ⊂ R be a prime and let L/kappa( p) be a finite extension of fields. Then there exists a finite free ring map R → S such that q = pS is prime and kappa( q)/kappa( p) is isomorphic to the given extension L/kappa( p).","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime and\nlet $L/\\kappa(\\mathfrak p)$ be a finite extension of fields.\nThen there exists a finite free ring map $R \\to S$ such that\n$\\mathfrak q = \\mathfrak pS$ is prime and\n$\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$ is isomorphic to the given\nextension $L/\\kappa(\\mathfrak p)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Constructing flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NE","source_file":"algebra.tex","source_line":44674,"source_end_line":44682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44674-L44682","statement_sha256":"a757c5450ea685a81c56342ca862f811ab886beaa3597dff97d2587af6650a4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2128,"rank":2128,"depth":0,"x":2163.152,"y":303.276,"cluster":"commutative-algebra"},{"id":"stacks:0GIL","tag":"0GIL","title":"Constructing flat ring maps · Lemma 0GIL","summary":"Let A be a ring. Let kappa = max(|A|, aleph_0). Then every flat A-algebra B is the filtered colimit of its flat A-subalgebras B' ⊂ B of cardinality |B'| ≤ kappa. (Observe that B' is faithfully flat over A if B is faithfully flat over A.)","statement_latex":"Let $A$ be a ring. Let $\\kappa = \\max(|A|, \\aleph_0)$. Then every flat\n$A$-algebra $B$ is the filtered colimit of its flat $A$-subalgebras\n$B' \\subset B$ of cardinality $|B'| \\leq \\kappa$. (Observe that $B'$\nis faithfully flat over $A$ if $B$ is faithfully flat over $A$.)","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Constructing flat ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIL","source_file":"algebra.tex","source_line":44703,"source_end_line":44709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44703-L44709","statement_sha256":"a1bcca19e0a78a7340d4d487d59a5723063585d7b0ba77503d732cf56e10fa0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2129,"rank":2129,"depth":1,"x":1864.781,"y":234.153,"cluster":"commutative-algebra"},{"id":"stacks:0324","tag":"0324","title":"The Cohen structure theorem · Definition 0324","summary":"Let (R, m) be a local ring. We say R is a complete local ring if the canonical map R → lim_n R/ m^n to the completion of R with respect to m is an isomorphism.","statement_latex":"Let $(R, \\mathfrak m)$ be a local ring. We say $R$ is a\n{\\it complete local ring} if the canonical map\n$$\nR \\longrightarrow \\lim_n R/\\mathfrak m^n\n$$\nto the completion of $R$ with respect to $\\mathfrak m$ is an\nisomorphism\\footnote{This includes the condition\nthat $\\bigcap \\mathfrak m^n = (0)$; in some texts this may be indicated\nby saying that $R$ is complete and separated. Warning: It can happen\nthat the completion $\\lim_n R/\\mathfrak m^n$ of a local ring is\nnon-complete, see\nExamples, Lemma \\ref{examples-lemma-noncomplete-completion}.\nThis does not happen when $\\mathfrak m$ is finitely generated, see\nLemma \\ref{lemma-hathat-finitely-generated} in which\ncase the completion is Noetherian, see\nLemma \\ref{lemma-completion-Noetherian}.}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0324","source_file":"algebra.tex","source_line":44785,"source_end_line":44803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44785-L44803","statement_sha256":"488b534c91d1e182333924eaeecae3eb88d0d649dd0fe068d2f2d5bd203aa25f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2130,"rank":2130,"depth":6,"x":2140.493,"y":115.773,"cluster":"commutative-algebra"},{"id":"stacks:0325","tag":"0325","title":"The Cohen structure theorem · Lemma 0325","summary":"Let R be a Noetherian complete local ring. Any quotient of R is also a Noetherian complete local ring. Given a finite ring map R → S, then S is a product of Noetherian complete local rings.","statement_latex":"Let $R$ be a Noetherian complete local ring.\nAny quotient of $R$ is also a Noetherian complete local ring.\nGiven a finite ring map $R \\to S$, then $S$ is a product of\nNoetherian complete local rings.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0325","source_file":"algebra.tex","source_line":44810,"source_end_line":44816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44810-L44816","statement_sha256":"10152915fddb15c1e70207705ab33b3c8771ad5a3560a36760c348d488327378","origin":"The Stacks Project","memory_eligible":false,"source_rank":2131,"rank":2131,"depth":10,"x":2032.342,"y":359.608,"cluster":"commutative-algebra"},{"id":"stacks:032B","tag":"032B","title":"The Cohen structure theorem · Lemma 032B","summary":"Let (R, m) be a complete local ring. If m is a finitely generated ideal then R is Noetherian.","statement_latex":"Let $(R, \\mathfrak m)$ be a complete local ring.\nIf $\\mathfrak m$ is a finitely generated ideal then\n$R$ is Noetherian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032B","source_file":"algebra.tex","source_line":44825,"source_end_line":44830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44825-L44830","statement_sha256":"b468606fcdebfcb31588c130958f1c835505839d238148720b24cd4c5053dacd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2132,"rank":2132,"depth":6,"x":1915.959,"y":118.344,"cluster":"commutative-algebra"},{"id":"stacks:0326","tag":"0326","title":"The Cohen structure theorem · Definition 0326","summary":"Let (R, m) be a complete local ring. A subring Lambda ⊂ R is called a coefficient ring if the following conditions hold: • Lambda is a complete local ring with maximal ideal Lambda ∩ m, • the residue field of Lambda maps isomorphically to the residue field of R, and • Lambda ∩ m = pLambda, where p is the characteristic of the residue field of R.","statement_latex":"Let $(R, \\mathfrak m)$ be a complete local ring.\nA subring $\\Lambda \\subset R$ is\ncalled a {\\it coefficient ring} if the following conditions hold:\n\\begin{enumerate}\n\\item $\\Lambda$ is a complete local ring with maximal ideal\n$\\Lambda \\cap \\mathfrak m$,\n\\item the residue field of $\\Lambda$ maps isomorphically to the\nresidue field of $R$, and\n\\item $\\Lambda \\cap \\mathfrak m = p\\Lambda$, where $p$ is the characteristic\nof the residue field of $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0326","source_file":"algebra.tex","source_line":44836,"source_end_line":44849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44836-L44849","statement_sha256":"1924e51a5952b900e4a17ebc54662f168a2ce06409c7f469aabe55919d4a6027","origin":"The Stacks Project","memory_eligible":false,"source_rank":2133,"rank":2133,"depth":0,"x":2195.908,"y":230.254,"cluster":"commutative-algebra"},{"id":"stacks:0327","tag":"0327","title":"The Cohen structure theorem · Definition 0327","summary":"A Cohen ring is a complete discrete valuation ring with uniformizer p a prime number.","statement_latex":"A {\\it Cohen ring} is a complete discrete valuation ring with\nuniformizer $p$ a prime number.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0327","source_file":"algebra.tex","source_line":44871,"source_end_line":44875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44871-L44875","statement_sha256":"a829145bfaa6765529521d65b3d5fab98576880593982ee43a2b9cb5880ce529","origin":"The Stacks Project","memory_eligible":false,"source_rank":2134,"rank":2134,"depth":0,"x":1899.364,"y":306.614,"cluster":"commutative-algebra"},{"id":"stacks:0328","tag":"0328","title":"The Cohen structure theorem · Lemma 0328","summary":"Let p be a prime number. Let k be a field of characteristic p. There exists a Cohen ring Lambda with Lambda/pLambda ≅ k.","statement_latex":"Let $p$ be a prime number.\nLet $k$ be a field of characteristic $p$.\nThere exists a Cohen ring $\\Lambda$ with $\\Lambda/p\\Lambda \\cong k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0328","source_file":"algebra.tex","source_line":44877,"source_end_line":44882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44877-L44882","statement_sha256":"a25abfd1695dea10c32fa849730d4b10591227353876d6ef12a76f13ed8cc606","origin":"The Stacks Project","memory_eligible":false,"source_rank":2135,"rank":2135,"depth":16,"x":2056.688,"y":81.951,"cluster":"commutative-algebra"},{"id":"stacks:0329","tag":"0329","title":"The Cohen structure theorem · Lemma 0329","summary":"Let p > 0 be a prime. Let Lambda be a Cohen ring with residue field of characteristic p. For every n ≥ 1 the ring map Z/p^nZ → Lambda/p^nLambda is formally smooth.","statement_latex":"Let $p > 0$ be a prime.\nLet $\\Lambda$ be a Cohen ring with residue field of characteristic $p$.\nFor every $n \\geq 1$ the ring map\n$$\n\\mathbf{Z}/p^n\\mathbf{Z} \\to \\Lambda/p^n\\Lambda\n$$\nis formally smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0329","source_file":"algebra.tex","source_line":44906,"source_end_line":44915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44906-L44915","statement_sha256":"a3c867428ed5ac01e347d132d64e1a26fd501d0b7c60234a7a735c7c5f5398c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2136,"rank":2136,"depth":12,"x":2121.372,"y":336.985,"cluster":"commutative-algebra"},{"id":"stacks:032A","tag":"032A","title":"Cohen structure theorem · Theorem 032A","summary":"Let (R, m) be a complete local ring. • R has a coefficient ring (see Definition [Tag 0326]), • if m is a finitely generated ideal, then R is isomorphic to a quotient Lambda[[x_1, …, x_n]]/I where Lambda is either a field or a Cohen ring.","statement_latex":"Let $(R, \\mathfrak m)$ be a complete local ring.\n\\begin{enumerate}\n\\item $R$ has a coefficient ring (see\nDefinition \\ref{definition-coefficient-ring}),\n\\item if $\\mathfrak m$ is a finitely generated ideal, then\n$R$ is isomorphic to a quotient\n$$\n\\Lambda[[x_1, \\ldots, x_n]]/I\n$$\nwhere $\\Lambda$ is either a field or a Cohen ring.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032A","source_file":"algebra.tex","source_line":44928,"source_end_line":44941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L44928-L44941","statement_sha256":"d401fc70b9085a174e730ab91f14d7d5ef85293b5a9f2ee3d8a90c874c694edb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2137,"rank":2137,"depth":17,"x":1868.483,"y":185.57,"cluster":"commutative-algebra"},{"id":"stacks:0C0S","tag":"0C0S","title":"The Cohen structure theorem · Lemma 0C0S","summary":"Let (R, m) be a Noetherian complete local ring. Assume R is regular. • If R contains either F_p or Q, then R is isomorphic to a power series ring over its residue field. • If k is a field and k → R is a ring map inducing an isomorphism k → R/ m, then R is isomorphic as a k-algebra to a power series ring over k.","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian complete local ring.\nAssume $R$ is regular.\n\\begin{enumerate}\n\\item If $R$ contains either $\\mathbf{F}_p$ or $\\mathbf{Q}$, then $R$\nis isomorphic to a power series ring over its residue field.\n\\item If $k$ is a field and $k \\to R$ is a ring map inducing\nan isomorphism $k \\to R/\\mathfrak m$, then $R$ is isomorphic\nas a $k$-algebra to a power series ring over $k$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0S","source_file":"algebra.tex","source_line":45035,"source_end_line":45046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45035-L45046","statement_sha256":"5d2a1d194350fcd733c99d7c93626f3f8ae485a2a38b2ce7a1567224d7eed175","origin":"The Stacks Project","memory_eligible":false,"source_rank":2138,"rank":2138,"depth":18,"x":2176.846,"y":153.713,"cluster":"commutative-algebra"},{"id":"stacks:032D","tag":"032D","title":"The Cohen structure theorem · Lemma 032D","summary":"Let (R, m) be a Noetherian complete local domain. Then there exists a R_0 ⊂ R with the following properties • R_0 is a regular complete local ring, • R_0 ⊂ R is finite and induces an isomorphism on residue fields, • R_0 is either isomorphic to k[[X_1, …, X_d]] where k is a field or Lambda[[X_1, …, X_d]] where Lambda is a Cohen ring.","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian complete local domain.\nThen there exists a $R_0 \\subset R$ with the following properties\n\\begin{enumerate}\n\\item $R_0$ is a regular complete local ring,\n\\item $R_0 \\subset R$ is finite and induces an isomorphism on\nresidue fields,\n\\item $R_0$ is either isomorphic to $k[[X_1, \\ldots, X_d]]$ where $k$\nis a field or $\\Lambda[[X_1, \\ldots, X_d]]$ where $\\Lambda$ is a Cohen ring.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"The Cohen structure theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032D","source_file":"algebra.tex","source_line":45064,"source_end_line":45075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45064-L45075","statement_sha256":"9de16fc3a644f61536778da9c4ce9392615d99ebc3b71ad8941a95e0f11f388c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2139,"rank":2139,"depth":11,"x":1975.002,"y":352.256,"cluster":"commutative-algebra"},{"id":"stacks:032F","tag":"032F","title":"Japanese rings · Definition 032F","summary":"[EGA] Let R be a domain with field of fractions K. • We say R is N-1 if the integral closure of R in K is a finite R-module. • We say R is N-2 or Japanese if for any finite extension L/K of fields the integral closure of R in L is finite over R.","statement_latex":"\\begin{reference}\n\\cite[Chapter 0, Definition 23.1.1]{EGA}\n\\end{reference}\nLet $R$ be a domain with field of fractions $K$.\n\\begin{enumerate}\n\\item We say $R$ is {\\it N-1} if the integral closure of $R$ in $K$\nis a finite $R$-module.\n\\item We say $R$ is {\\it N-2} or {\\it Japanese} if for any finite\nextension $L/K$ of fields the integral closure of $R$ in $L$\nis finite over $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032F","source_file":"algebra.tex","source_line":45134,"source_end_line":45147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45134-L45147","statement_sha256":"4bbd8d7856e7ddfd52ba73ac753e181ff452d10002e4c063fa57225eae3a4fb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2140,"rank":2140,"depth":0,"x":1964.172,"y":91.218,"cluster":"commutative-algebra"},{"id":"stacks:032G","tag":"032G","title":"Japanese rings · Lemma 032G","summary":"Let R be a domain. If R is N-1 then so is any localization of R. Same for N-2.","statement_latex":"Let $R$ be a domain.\nIf $R$ is N-1 then so is any localization of $R$.\nSame for N-2.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032G","source_file":"algebra.tex","source_line":45172,"source_end_line":45177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45172-L45177","statement_sha256":"e5c5ad52abba50d08ad72fbfd876d28ef757698ebbdcd1dfe08d7abf49e15ae7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2141,"rank":2141,"depth":1,"x":2182.164,"y":277.632,"cluster":"commutative-algebra"},{"id":"stacks:032H","tag":"032H","title":"Japanese rings · Lemma 032H","summary":"Let R be a domain. Let f_1, …, f_n ∈ R generate the unit ideal. If each domain R_f_i is N-1 then so is R. Same for N-2.","statement_latex":"Let $R$ be a domain. Let $f_1, \\ldots, f_n \\in R$ generate the\nunit ideal. If each domain $R_{f_i}$ is N-1 then so is $R$.\nSame for N-2.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032H","source_file":"algebra.tex","source_line":45184,"source_end_line":45189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45184-L45189","statement_sha256":"e6ef5ab3027998c8e512e5e1fd56bc0c7d0f89c7590e701655e68e21b4604b12","origin":"The Stacks Project","memory_eligible":false,"source_rank":2142,"rank":2142,"depth":3,"x":1871.388,"y":263.862,"cluster":"commutative-algebra"},{"id":"stacks:032I","tag":"032I","title":"Japanese rings · Lemma 032I","summary":"Let R be a domain. Let R ⊂ S be a quasi-finite extension of domains (for example finite). Assume R is N-2 and Noetherian. Then S is N-2.","statement_latex":"Let $R$ be a domain. Let $R \\subset S$ be a quasi-finite extension of domains\n(for example finite). Assume $R$ is N-2 and Noetherian. Then $S$ is N-2.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032I","source_file":"algebra.tex","source_line":45201,"source_end_line":45205,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45201-L45205","statement_sha256":"ba3f73eab88bf70bbe437064f6f0b2efaa5e7cc82235073025152a6f8b242ff8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2143,"rank":2143,"depth":29,"x":2111.718,"y":97.608,"cluster":"commutative-algebra"},{"id":"stacks:032J","tag":"032J","title":"Japanese rings · Lemma 032J","summary":"Let R be a Noetherian domain. If R[z, z^-1] is N-1, then so is R.","statement_latex":"Let $R$ be a Noetherian domain.\nIf $R[z, z^{-1}]$ is N-1, then so is $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032J","source_file":"algebra.tex","source_line":45234,"source_end_line":45238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45234-L45238","statement_sha256":"054b830a0aad49d8746cf4569032d0235f8e3d6d2feea37e449c7738e523ce75","origin":"The Stacks Project","memory_eligible":false,"source_rank":2144,"rank":2144,"depth":0,"x":2068.182,"y":356.673,"cluster":"commutative-algebra"},{"id":"stacks:032K","tag":"032K","title":"Japanese rings · Lemma 032K","summary":"Let R be a Noetherian domain, and let R ⊂ S be a finite extension of domains. If S is N-1, then so is R. If S is N-2, then so is R.","statement_latex":"Let $R$ be a Noetherian domain, and let $R \\subset S$ be a\nfinite extension of domains. If $S$ is N-1, then so is $R$.\nIf $S$ is N-2, then so is $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032K","source_file":"algebra.tex","source_line":45256,"source_end_line":45261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45256-L45261","statement_sha256":"f25f61c848a74bd8fc6c545045a010a50e42999ceb0e6bb8c59c8b68ad588c57","origin":"The Stacks Project","memory_eligible":false,"source_rank":2145,"rank":2145,"depth":0,"x":1891.882,"y":140.853,"cluster":"commutative-algebra"},{"id":"stacks:032L","tag":"032L","title":"Japanese rings · Lemma 032L","summary":"Let R be a Noetherian normal domain with fraction field K. Let L/K be a finite separable field extension. Then the integral closure of R in L is finite over R.","statement_latex":"Let $R$ be a Noetherian normal domain with fraction field $K$.\nLet $L/K$ be a finite separable field extension.\nThen the integral closure of $R$ in $L$ is finite over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032L","source_file":"algebra.tex","source_line":45268,"source_end_line":45273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45268-L45273","statement_sha256":"fb6dd0db2c6243c6feb2b58dfaaf4c0c2c1d5302df758f5c99be7559cd4d73fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2146,"rank":2146,"depth":8,"x":2195.559,"y":199.982,"cluster":"commutative-algebra"},{"id":"stacks:0AE0","tag":"0AE0","title":"Japanese rings · Lemma 0AE0","summary":"Let R be a Noetherian normal domain with fraction field K of characteristic p > 0. Let a ∈ K be an element such that there exists a derivation D : R → R with D(a) not = 0. Then the integral closure of R in L = K[x]/(x^p - a) is finite over R.","statement_latex":"Let $R$ be a Noetherian normal domain with fraction field $K$\nof characteristic $p > 0$.\nLet $a \\in K$ be an element such that there exists a derivation\n$D : R \\to R$ with $D(a) \\not = 0$. Then the integral closure\nof $R$ in $L = K[x]/(x^p - a)$ is finite over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AE0","source_file":"algebra.tex","source_line":45321,"source_end_line":45328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45321-L45328","statement_sha256":"daa926e29d94dfb56185d2e94c1ddd151d35df9396bf88c3b756a945c216b0fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2147,"rank":2147,"depth":0,"x":1923.975,"y":328.746,"cluster":"commutative-algebra"},{"id":"stacks:032M","tag":"032M","title":"Japanese rings · Lemma 032M","summary":"A Noetherian domain whose fraction field has characteristic zero is N-1 if and only if it is N-2 (i.e., Japanese).","statement_latex":"A Noetherian domain whose fraction field has characteristic zero is N-1\nif and only if it is N-2 (i.e., Japanese).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032M","source_file":"algebra.tex","source_line":45369,"source_end_line":45373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45369-L45373","statement_sha256":"9f9ab31597395aa2ff0d9de7e56193a93eb4b50ac19f7e09c97bb7cef6b81363","origin":"The Stacks Project","memory_eligible":false,"source_rank":2148,"rank":2148,"depth":9,"x":2020.727,"y":79.596,"cluster":"commutative-algebra"},{"id":"stacks:032N","tag":"032N","title":"Japanese rings · Lemma 032N","summary":"Let R be a Noetherian domain with fraction field K of characteristic p > 0. Then R is N-2 if and only if for every finite purely inseparable extension L/K the integral closure of R in L is finite over R.","statement_latex":"Let $R$ be a Noetherian domain with fraction field $K$ of\ncharacteristic $p > 0$. Then $R$ is N-2 if and only if\nfor every finite purely inseparable extension $L/K$ the integral\nclosure of $R$ in $L$ is finite over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032N","source_file":"algebra.tex","source_line":45381,"source_end_line":45387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45381-L45387","statement_sha256":"7a28150f38e28c7d8717528edb88f71cfcf6ee50e6eae0f1f2b45c9c0e95814d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2149,"rank":2149,"depth":9,"x":2149.794,"y":318.309,"cluster":"commutative-algebra"},{"id":"stacks:032O","tag":"032O","title":"Japanese rings · Lemma 032O","summary":"Let R be a Noetherian domain. If R is N-1 then R[x] is N-1. If R is N-2 then R[x] is N-2.","statement_latex":"Let $R$ be a Noetherian domain.\nIf $R$ is N-1 then $R[x]$ is N-1.\nIf $R$ is N-2 then $R[x]$ is N-2.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032O","source_file":"algebra.tex","source_line":45410,"source_end_line":45415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45410-L45415","statement_sha256":"fdef4df47bc7fd02f44e3f761e2683277f0437ed9ed340296729d4c30f911874","origin":"The Stacks Project","memory_eligible":false,"source_rank":2150,"rank":2150,"depth":10,"x":1862.542,"y":215.481,"cluster":"commutative-algebra"},{"id":"stacks:0332","tag":"0332","title":"Japanese rings · Lemma 0332","summary":"Let R be a Noetherian domain. If there exists a nonzero f ∈ R such that R_f is normal then U = ( p ∈ Spec(R) mid R_ p is normal) is open in Spec(R).","statement_latex":"Let $R$ be a Noetherian domain.\nIf there exists a nonzero $f \\in R$ such that $R_f$ is normal\nthen\n$$\nU = \\{\\mathfrak p \\in \\Spec(R) \\mid R_{\\mathfrak p} \\text{ is normal}\\}\n$$\nis open in $\\Spec(R)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0332","source_file":"algebra.tex","source_line":45442,"source_end_line":45451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45442-L45451","statement_sha256":"ab03a3288b8b7ce77c03ebded0c37a5da4b758c1c7069f527b0da20a96e4227f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2151,"rank":2151,"depth":17,"x":2157.165,"y":128.277,"cluster":"commutative-algebra"},{"id":"stacks:0333","tag":"0333","title":"Japanese rings · Lemma 0333","summary":"Let R be a Noetherian domain. Then R is N-1 if and only if the following two conditions hold • there exists a nonzero f ∈ R such that R_f is normal, and • for every maximal ideal m ⊂ R the local ring R_ m is N-1.","statement_latex":"Let $R$ be a Noetherian domain. Then $R$ is N-1 if and only if the following\ntwo conditions hold\n\\begin{enumerate}\n\\item there exists a nonzero $f \\in R$ such that $R_f$ is normal, and\n\\item for every maximal ideal $\\mathfrak m \\subset R$\nthe local ring $R_{\\mathfrak m}$ is N-1.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0333","source_file":"algebra.tex","source_line":45481,"source_end_line":45490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45481-L45490","statement_sha256":"0d7feafcf8eb740017402cb161dc78a0012159e93bc5df97f220421c6b2abbc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2152,"rank":2152,"depth":18,"x":2009.984,"y":359.847,"cluster":"commutative-algebra"},{"id":"stacks:032P","tag":"032P","title":"Tate · Lemma 032P","summary":"[EGA] Let R be a ring. Let x ∈ R. Assume • R is a normal Noetherian domain, • R/xR is a domain and N-2, • R ≅ lim_n R/x^nR is complete with respect to x. Then R is N-2.","statement_latex":"\\begin{reference}\n\\cite[Theorem 23.1.3]{EGA}\n\\end{reference}\nLet $R$ be a ring.\nLet $x \\in R$.\nAssume\n\\begin{enumerate}\n\\item $R$ is a normal Noetherian domain,\n\\item $R/xR$ is a domain and N-2,\n\\item $R \\cong \\lim_n R/x^nR$ is complete with respect to $x$.\n\\end{enumerate}\nThen $R$ is N-2.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032P","source_file":"algebra.tex","source_line":45534,"source_end_line":45548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45534-L45548","statement_sha256":"3a45d3d3bb5a75d99b6cc54257b43d91be8af496448a5b8a94b9536038e14a8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2153,"rank":2153,"depth":16,"x":1932.26,"y":105.477,"cluster":"commutative-algebra"},{"id":"stacks:032Q","tag":"032Q","title":"Japanese rings · Lemma 032Q","summary":"Let R be a ring. If R is Noetherian, a domain, and N-2, then so is R[[x]].","statement_latex":"Let $R$ be a ring.\nIf $R$ is Noetherian, a domain, and N-2, then so is $R[[x]]$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Japanese rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032Q","source_file":"algebra.tex","source_line":45592,"source_end_line":45596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45592-L45596","statement_sha256":"e9a8fd2f2e639f713756510ab40be5776352a3bb6b061021d5b24a099fd119cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":2154,"rank":2154,"depth":17,"x":2194.233,"y":248.999,"cluster":"commutative-algebra"},{"id":"stacks:032R","tag":"032R","title":"Nagata rings · Definition 032R","summary":"Let R be a ring. • We say R is universally Japanese if for any finite type ring map R → S with S a domain we have that S is N-2 (i.e., Japanese). • We say that R is a Nagata ring if R is Noetherian and for every prime ideal p the ring R/ p is N-2.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item We say $R$ is {\\it universally Japanese} if for any finite\ntype ring map $R \\to S$ with $S$ a domain we have that $S$ is N-2\n(i.e., Japanese).\n\\item We say that $R$ is a {\\it Nagata ring} if $R$ is Noetherian and\nfor every prime ideal $\\mathfrak p$ the ring $R/\\mathfrak p$ is N-2.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032R","source_file":"algebra.tex","source_line":45621,"source_end_line":45631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45621-L45631","statement_sha256":"52630a904f1077adf9f3ae55d6d746f3b3db8d863c39cacef394650ea3a2274f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2155,"rank":2155,"depth":0,"x":1885.52,"y":291.834,"cluster":"commutative-algebra"},{"id":"stacks:03GH","tag":"03GH","title":"Nagata rings · Lemma 03GH","summary":"Let R be a Nagata ring. Let R → S be essentially of finite type with S reduced. Then the integral closure A of R in S is finite over R.","statement_latex":"Let $R$ be a Nagata ring.\nLet $R \\to S$ be essentially of finite type with $S$ reduced.\nThen the integral closure $A$ of $R$ in $S$ is finite over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GH","source_file":"algebra.tex","source_line":45639,"source_end_line":45644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45639-L45644","statement_sha256":"fee3a4f2d7c55e75477740c70dea436e258f9823402bbaafeeffdd32d2462432","origin":"The Stacks Project","memory_eligible":false,"source_rank":2156,"rank":2156,"depth":6,"x":2078.789,"y":84.996,"cluster":"commutative-algebra"},{"id":"stacks:0351","tag":"0351","title":"Nagata rings · Lemma 0351","summary":"Let R be a ring. To check that R is universally Japanese it suffices to show: If R → S is of finite type, and S a domain then S is N-1.","statement_latex":"Let $R$ be a ring.\nTo check that $R$ is universally Japanese it suffices to show:\nIf $R \\to S$ is of finite type, and $S$ a domain then $S$ is N-1.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0351","source_file":"algebra.tex","source_line":45669,"source_end_line":45674,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45669-L45674","statement_sha256":"6a80ee091638596c31ca73dd0d845798c70a1b714e8c30f7425955317e88c307","origin":"The Stacks Project","memory_eligible":false,"source_rank":2157,"rank":2157,"depth":3,"x":2102.618,"y":347.284,"cluster":"commutative-algebra"},{"id":"stacks:032S","tag":"032S","title":"Nagata rings · Lemma 032S","summary":"If R is universally Japanese then any algebra essentially of finite type over R is universally Japanese.","statement_latex":"If $R$ is universally Japanese then any algebra essentially of finite type\nover $R$ is universally Japanese.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032S","source_file":"algebra.tex","source_line":45690,"source_end_line":45694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45690-L45694","statement_sha256":"5d8adbecfb411eeead44bdf224dcec3740b6df4413d699a845806bc98262ba05","origin":"The Stacks Project","memory_eligible":false,"source_rank":2158,"rank":2158,"depth":2,"x":1874.033,"y":167.327,"cluster":"commutative-algebra"},{"id":"stacks:032T","tag":"032T","title":"Nagata rings · Lemma 032T","summary":"Let R be a Nagata ring. If R → S is a quasi-finite ring map (for example finite) then S is a Nagata ring also.","statement_latex":"Let $R$ be a Nagata ring.\nIf $R \\to S$ is a quasi-finite ring map (for example finite)\nthen $S$ is a Nagata ring also.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032T","source_file":"algebra.tex","source_line":45702,"source_end_line":45707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45702-L45707","statement_sha256":"e3db4987d79f8dbf8e3edae2633407fc827ddea60fafd744e819942664ffaef8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2159,"rank":2159,"depth":30,"x":2187.427,"y":170.321,"cluster":"commutative-algebra"},{"id":"stacks:032U","tag":"032U","title":"Nagata rings · Lemma 032U","summary":"A localization of a Nagata ring is a Nagata ring.","statement_latex":"A localization of a Nagata ring is a Nagata ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032U","source_file":"algebra.tex","source_line":45720,"source_end_line":45723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45720-L45723","statement_sha256":"ce058c1c52d8ffda2d56d5fee77cfd800dfd237857c823e43980c3e3115f65bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2160,"rank":2160,"depth":2,"x":1953.836,"y":346.009,"cluster":"commutative-algebra"},{"id":"stacks:032V","tag":"032V","title":"Nagata rings · Lemma 032V","summary":"Let R be a ring. Let f_1, …, f_n ∈ R generate the unit ideal. • If each R_f_i is universally Japanese then so is R. • If each R_f_i is Nagata then so is R.","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_n \\in R$ generate the\nunit ideal.\n\\begin{enumerate}\n\\item  If each $R_{f_i}$ is universally Japanese then so is $R$.\n\\item  If each $R_{f_i}$ is Nagata then so is $R$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032V","source_file":"algebra.tex","source_line":45729,"source_end_line":45737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45729-L45737","statement_sha256":"b92e9b17b840cea187e55cadd1cba30a435d67e93d1101755bea7003fa10093e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2161,"rank":2161,"depth":4,"x":1984.811,"y":83.812,"cluster":"commutative-algebra"},{"id":"stacks:032W","tag":"032W","title":"Nagata rings · Lemma 032W","summary":"A Noetherian complete local ring is a Nagata ring.","statement_latex":"A Noetherian complete local ring is a Nagata ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032W","source_file":"algebra.tex","source_line":45754,"source_end_line":45757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45754-L45757","statement_sha256":"7be0ea1d15c5c098aa6c2986cb30e6e23af7bb28eb0184757cf6054376e50b81","origin":"The Stacks Project","memory_eligible":false,"source_rank":2162,"rank":2162,"depth":30,"x":2172.896,"y":294.81,"cluster":"commutative-algebra"},{"id":"stacks:032X","tag":"032X","title":"Nagata rings · Definition 032X","summary":"Let (R, m) be a Noetherian local ring. We say R is analytically unramified if its completion R^wedge = lim_n R/ m^n is reduced. A prime ideal p ⊂ R is said to be analytically unramified if R/ p is analytically unramified.","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring.\nWe say $R$ is {\\it analytically unramified} if its completion\n$R^\\wedge = \\lim_n R/\\mathfrak m^n$ is reduced.\nA prime ideal $\\mathfrak p \\subset R$ is said to be\n{\\it analytically unramified} if $R/\\mathfrak p$ is analytically\nunramified.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032X","source_file":"algebra.tex","source_line":45780,"source_end_line":45788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45780-L45788","statement_sha256":"bf867d2500c429dac824732417830c19bade859f7bc222efc867de25ec6a42c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2163,"rank":2163,"depth":0,"x":1864.406,"y":245.928,"cluster":"commutative-algebra"},{"id":"stacks:032Y","tag":"032Y","title":"Nagata rings · Lemma 032Y","summary":"Let (R, m) be a Noetherian local ring. • If R is analytically unramified, then R is reduced. • If R is analytically unramified, then each minimal prime of R is analytically unramified. • If R is reduced with minimal primes q_1, …, q_t, and each q_i is analytically unramified, then R is analytically unramified. • If R is analytically unramified, then the integral closure of R in its total ring of fractions Q(R) is finite over R. • If R is a domain and analytically…","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring.\n\\begin{enumerate}\n\\item If $R$ is analytically unramified, then $R$ is reduced.\n\\item If $R$ is analytically unramified, then each minimal prime of\n$R$ is analytically unramified.\n\\item If $R$ is reduced with minimal primes\n$\\mathfrak q_1, \\ldots, \\mathfrak q_t$, and each $\\mathfrak q_i$\nis analytically unramified, then $R$ is analytically unramified.\n\\item If $R$ is analytically unramified, then the integral closure\nof $R$ in its total ring of fractions $Q(R)$ is finite over $R$.\n\\item If $R$ is a domain and analytically unramified, then $R$ is N-1.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032Y","source_file":"algebra.tex","source_line":45807,"source_end_line":45821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45807-L45821","statement_sha256":"d2c0572da0b55a4e9d66790a52d083515d653ba04dc559ad48644dc9fee036c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2164,"rank":2164,"depth":31,"x":2131.295,"y":106.875,"cluster":"commutative-algebra"},{"id":"stacks:032Z","tag":"032Z","title":"Nagata rings · Lemma 032Z","summary":"Let R be a Noetherian local ring. Let p ⊂ R be a prime. Assume • R_ p is a discrete valuation ring, and • p is analytically unramified. Then for any associated prime q of R^wedge/ pR^wedge the local ring (R^wedge)_ q is a discrete valuation ring.","statement_latex":"Let $R$ be a Noetherian local ring.\nLet $\\mathfrak p \\subset R$ be a prime.\nAssume\n\\begin{enumerate}\n\\item $R_{\\mathfrak p}$ is a discrete valuation ring, and\n\\item $\\mathfrak p$ is analytically unramified.\n\\end{enumerate}\nThen for any associated prime $\\mathfrak q$ of $R^\\wedge/\\mathfrak pR^\\wedge$\nthe local ring $(R^\\wedge)_{\\mathfrak q}$ is a discrete valuation ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/032Z","source_file":"algebra.tex","source_line":45889,"source_end_line":45900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45889-L45900","statement_sha256":"ee201bd960eb1146b50ea04ea9db565ed5096bed1950f1e8b8edb89280387af4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2165,"rank":2165,"depth":0,"x":2046.285,"y":360.949,"cluster":"commutative-algebra"},{"id":"stacks:0330","tag":"0330","title":"Nagata rings · Lemma 0330","summary":"Let (R, m) be a Noetherian local domain. Let x ∈ m. Assume • x not = 0, • R/xR has no embedded primes, and • for each associated prime p ⊂ R of R/xR we have • the local ring R_ p is regular, and • p is analytically unramified. Then R is analytically unramified.","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local domain.\nLet $x \\in \\mathfrak m$. Assume\n\\begin{enumerate}\n\\item $x \\not = 0$,\n\\item $R/xR$ has no embedded primes, and\n\\item for each associated prime $\\mathfrak p \\subset R$\nof $R/xR$ we have\n\\begin{enumerate}\n\\item the local ring $R_{\\mathfrak p}$ is regular, and\n\\item $\\mathfrak p$ is analytically unramified.\n\\end{enumerate}\n\\end{enumerate}\nThen $R$ is analytically unramified.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0330","source_file":"algebra.tex","source_line":45916,"source_end_line":45931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45916-L45931","statement_sha256":"4a37a2e09fc455cab892a09b6b56c49017a979f4e57758e15b09a4e8c38027fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2166,"rank":2166,"depth":12,"x":1904.596,"y":125.27,"cluster":"commutative-algebra"},{"id":"stacks:0331","tag":"0331","title":"Nagata rings · Lemma 0331","summary":"Let (R, m) be a local ring. If R is Noetherian, a domain, and Nagata, then R is analytically unramified.","statement_latex":"Let $(R, \\mathfrak m)$ be a local ring.\nIf $R$ is Noetherian, a domain, and Nagata, then $R$ is\nanalytically unramified.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0331","source_file":"algebra.tex","source_line":45964,"source_end_line":45969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L45964-L45969","statement_sha256":"91f0f6c53ba851e8fff2040212eb5e387fe5518ee768a79538d1536cea5f4e6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2167,"rank":2167,"depth":31,"x":2198.716,"y":218.694,"cluster":"commutative-algebra"},{"id":"stacks:0BI2","tag":"0BI2","title":"Nagata rings · Lemma 0BI2","summary":"Let (R, m) be a Noetherian local ring. The following are equivalent • R is Nagata, • for R → S finite with S a domain and m' ⊂ S maximal the local ring S_ m' is analytically unramified, • for (R, m) → (S, m') finite local homomorphism with S a domain, then S is analytically unramified.","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring. The following\nare equivalent\n\\begin{enumerate}\n\\item $R$ is Nagata,\n\\item for $R \\to S$ finite with $S$ a domain and $\\mathfrak m' \\subset S$\nmaximal the local ring $S_{\\mathfrak m'}$ is analytically unramified,\n\\item for $(R, \\mathfrak m) \\to (S, \\mathfrak m')$ finite\nlocal homomorphism with $S$ a domain, then $S$ is analytically\nunramified.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BI2","source_file":"algebra.tex","source_line":46017,"source_end_line":46029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46017-L46029","statement_sha256":"35364a2d52a5e942cebe7f9b4fa2f86e502ee0f7ac180ee41b8d528cabf65bf9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2168,"rank":2168,"depth":32,"x":1906.594,"y":316.734,"cluster":"commutative-algebra"},{"id":"stacks:0334","tag":"0334","title":"Nagata · Proposition 0334","summary":"Let R be a ring. The following are equivalent: • R is a Nagata ring, • any finite type R-algebra is Nagata, and • R is universally Japanese and Noetherian.","statement_latex":"Let $R$ be a ring. The following are equivalent:\n\\begin{enumerate}\n\\item $R$ is a Nagata ring,\n\\item any finite type $R$-algebra is Nagata, and\n\\item $R$ is universally Japanese and Noetherian.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0334","source_file":"algebra.tex","source_line":46066,"source_end_line":46074,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46066-L46074","statement_sha256":"6c000d3a966debdf50569635f05f6b2272a5a1be8fc77668627a65fbdaa9e8d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2169,"rank":2169,"depth":32,"x":2043.212,"y":78.591,"cluster":"commutative-algebra"},{"id":"stacks:0335","tag":"0335","title":"Nagata rings · Proposition 0335","summary":"The following types of rings are Nagata and in particular universally Japanese: • fields, • Noetherian complete local rings, • Z, • Dedekind domains with fraction field of characteristic zero, • finite type ring extensions of any of the above.","statement_latex":"The following types of rings are Nagata and in particular universally Japanese:\n\\begin{enumerate}\n\\item fields,\n\\item Noetherian complete local rings,\n\\item $\\mathbf{Z}$,\n\\item Dedekind domains with fraction field of characteristic zero,\n\\item finite type ring extensions of any of the above.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0335","source_file":"algebra.tex","source_line":46224,"source_end_line":46234,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46224-L46234","statement_sha256":"78918cc971b36c874347be923566f066ab3abfd3509023c905af4b8837004185","origin":"The Stacks Project","memory_eligible":false,"source_rank":2170,"rank":2170,"depth":31,"x":2134.015,"y":331.813,"cluster":"commutative-algebra"},{"id":"stacks:09E2","tag":"09E2","title":"Nagata rings · Lemma 09E2","summary":"Let (A, m) be a Noetherian local domain which is Nagata and has fraction field of characteristic p. If a ∈ A has a pth root in A^wedge, then a has a pth root in A.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local domain which is Nagata\nand has fraction field of characteristic $p$. If $a \\in A$ has a\n$p$th root in $A^\\wedge$, then $a$ has a $p$th root in $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Nagata rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09E2","source_file":"algebra.tex","source_line":46264,"source_end_line":46269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46264-L46269","statement_sha256":"f541faa23a69787c49d8a41df210fa7c8179d428c0e3e86e374f1f270b3083b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2171,"rank":2171,"depth":33,"x":1863.32,"y":196.562,"cluster":"commutative-algebra"},{"id":"stacks:0338","tag":"0338","title":"Ascending properties · Lemma 0338","summary":"[EGA] We have depth(M ⊗_R N) = depth(M) + depth(N/ m_RN) where R → S is a local homomorphism of local Noetherian rings, M is a finite R-module, and N is a finite S-module flat over R.","statement_latex":"\\begin{reference}\n\\cite[IV, Proposition 6.3.1]{EGA}\n\\end{reference}\nWe have\n$$\n\\text{depth}(M \\otimes_R N)\n=\n\\text{depth}(M) + \\text{depth}(N/\\mathfrak m_RN)\n$$\nwhere $R \\to S$ is a local homomorphism of local Noetherian rings,\n$M$ is a finite $R$-module, and $N$ is a finite $S$-module flat over $R$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0338","source_file":"algebra.tex","source_line":46302,"source_end_line":46315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46302-L46315","statement_sha256":"9e07c7ad08665b86038168bb314b74a268db0a1517c8ed528d9d3c2b8b5ff48a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2172,"rank":2172,"depth":14,"x":2171.81,"y":142.675,"cluster":"commutative-algebra"},{"id":"stacks:0337","tag":"0337","title":"Ascending properties · Lemma 0337","summary":"Suppose that R → S is a flat and local ring homomorphism of Noetherian local rings. Then depth(S) = depth(R) + depth(S/ m_RS).","statement_latex":"Suppose that $R \\to S$ is a flat and local ring homomorphism of Noetherian\nlocal rings. Then\n$$\n\\text{depth}(S) = \\text{depth}(R) + \\text{depth}(S/\\mathfrak m_RS).\n$$","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0337","source_file":"algebra.tex","source_line":46373,"source_end_line":46380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46373-L46380","statement_sha256":"c88ffd5e0ecd823755ce3bb25eaebc8f98302daf307017255d74909e3033253b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2173,"rank":2173,"depth":15,"x":1987.598,"y":357.538,"cluster":"commutative-algebra"},{"id":"stacks:045J","tag":"045J","title":"Ascending properties · Lemma 045J","summary":"Let R → S be a flat local homomorphism of local Noetherian rings. Then the following are equivalent • S is Cohen-Macaulay, and • R and S/ m_RS are Cohen-Macaulay.","statement_latex":"Let $R \\to S$ be a flat local homomorphism of local Noetherian rings.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $S$ is Cohen-Macaulay, and\n\\item $R$ and $S/\\mathfrak m_RS$ are Cohen-Macaulay.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045J","source_file":"algebra.tex","source_line":46386,"source_end_line":46394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46386-L46394","statement_sha256":"3140e166fc1e5e7099ae0d165c56cdf8c223c449a864b4fcff5e87ce3e19204f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2174,"rank":2174,"depth":16,"x":1950.632,"y":94.473,"cluster":"commutative-algebra"},{"id":"stacks:0339","tag":"0339","title":"Ascending properties · Lemma 0339","summary":"Let φ : R → S be a ring map. Assume • R is Noetherian, • S is Noetherian, • φ is flat, • the fibre rings S ⊗_R kappa( p) are (S_k), and • R has property (S_k). Then S has property (S_k).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $S$ is Noetherian,\n\\item $\\varphi$ is flat,\n\\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are $(S_k)$, and\n\\item $R$ has property $(S_k)$.\n\\end{enumerate}\nThen $S$ has property $(S_k)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0339","source_file":"algebra.tex","source_line":46402,"source_end_line":46413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46402-L46413","statement_sha256":"8c3920f8bb5738d7b1b3d88dc341b4ce0c01c0aa32472792a854002d95e418ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":2175,"rank":2175,"depth":16,"x":2189.532,"y":267.545,"cluster":"commutative-algebra"},{"id":"stacks:033A","tag":"033A","title":"Ascending properties · Lemma 033A","summary":"Let φ : R → S be a ring map. Assume • R is Noetherian, • S is Noetherian • φ is flat, • the fibre rings S ⊗_R kappa( p) have property (R_k), and • R has property (R_k). Then S has property (R_k).","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $S$ is Noetherian\n\\item $\\varphi$ is flat,\n\\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$\nhave property $(R_k)$, and\n\\item $R$ has property $(R_k)$.\n\\end{enumerate}\nThen $S$ has property $(R_k)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033A","source_file":"algebra.tex","source_line":46459,"source_end_line":46471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46459-L46471","statement_sha256":"aee513222bb57cee9f226f2d21ecd728a41c6478a82a1bb3606ab38aa16ab6c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2176,"rank":2176,"depth":13,"x":1874.069,"y":275.485,"cluster":"commutative-algebra"},{"id":"stacks:0C21","tag":"0C21","title":"Ascending properties · Lemma 0C21","summary":"Let φ : R → S be a ring map. Assume • R is Noetherian, • S is Noetherian • φ is flat, • the fibre rings S ⊗_R kappa( p) are reduced, • R is reduced. Then S is reduced.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $S$ is Noetherian\n\\item $\\varphi$ is flat,\n\\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are reduced,\n\\item $R$ is reduced.\n\\end{enumerate}\nThen $S$ is reduced.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C21","source_file":"algebra.tex","source_line":46488,"source_end_line":46499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46488-L46499","statement_sha256":"ea863e47b765a91d235eb3636dda11626b0d0092f80d1846a1280a4702f20fb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2177,"rank":2177,"depth":17,"x":2100.388,"y":90.558,"cluster":"commutative-algebra"},{"id":"stacks:033B","tag":"033B","title":"Ascending properties · Lemma 033B","summary":"Let φ : R → S be a ring map. Assume • φ is smooth, • R is reduced. Then S is reduced.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is smooth,\n\\item $R$ is reduced.\n\\end{enumerate}\nThen $S$ is reduced.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033B","source_file":"algebra.tex","source_line":46510,"source_end_line":46518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46510-L46518","statement_sha256":"53baeedb902ea8fb7eec3c40139c6f52e995d69a5317058ce33310b9437653f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2178,"rank":2178,"depth":18,"x":2082.211,"y":355.441,"cluster":"commutative-algebra"},{"id":"stacks:0C22","tag":"0C22","title":"Ascending properties · Lemma 0C22","summary":"Let φ : R → S be a ring map. Assume • R is Noetherian, • S is Noetherian, • φ is flat, • the fibre rings S ⊗_R kappa( p) are normal, and • R is normal. Then S is normal.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $S$ is Noetherian,\n\\item $\\varphi$ is flat,\n\\item the fibre rings $S \\otimes_R \\kappa(\\mathfrak p)$ are normal, and\n\\item $R$ is normal.\n\\end{enumerate}\nThen $S$ is normal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C22","source_file":"algebra.tex","source_line":46540,"source_end_line":46551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46540-L46551","statement_sha256":"db1b40f6398391931205c4a2d761867b6e7a38ddb115fc757cb3abb10e3ead86","origin":"The Stacks Project","memory_eligible":false,"source_rank":2179,"rank":2179,"depth":17,"x":1882.526,"y":149.727,"cluster":"commutative-algebra"},{"id":"stacks:033C","tag":"033C","title":"Ascending properties · Lemma 033C","summary":"Let φ : R → S be a ring map. Assume • φ is smooth, • R is normal. Then S is normal.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is smooth,\n\\item $R$ is normal.\n\\end{enumerate}\nThen $S$ is normal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033C","source_file":"algebra.tex","source_line":46562,"source_end_line":46570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46562-L46570","statement_sha256":"08071e57fb77e348318d723767a658e943e25cf79f5018a2a5103e3745820fd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2180,"rank":2180,"depth":32,"x":2195.32,"y":188.126,"cluster":"commutative-algebra"},{"id":"stacks:07NF","tag":"07NF","title":"Ascending properties · Lemma 07NF","summary":"Regularity ascends along smooth maps of rings. Let φ : R → S be a ring map. Assume • φ is smooth, • R is a regular ring. Then S is regular.","statement_latex":"\\begin{slogan}\nRegularity ascends along smooth maps of rings.\n\\end{slogan}\nLet $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is smooth,\n\\item $R$ is a regular ring.\n\\end{enumerate}\nThen $S$ is regular.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Ascending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NF","source_file":"algebra.tex","source_line":46609,"source_end_line":46620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46609-L46620","statement_sha256":"977b379501efee7601ce2b5202260de0e0eb41d953e86b61b97913f1ec769a14","origin":"The Stacks Project","memory_eligible":false,"source_rank":2181,"rank":2181,"depth":37,"x":1933.69,"y":337.355,"cluster":"commutative-algebra"},{"id":"stacks:033E","tag":"033E","title":"Descending properties · Lemma 033E","summary":"Let R → S be a ring map. Assume that • R → S is faithfully flat, and • S is Noetherian. Then R is Noetherian.","statement_latex":"Let $R \\to S$ be a ring map.\nAssume that\n\\begin{enumerate}\n\\item $R \\to S$ is faithfully flat, and\n\\item $S$ is Noetherian.\n\\end{enumerate}\nThen $R$ is Noetherian.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033E","source_file":"algebra.tex","source_line":46646,"source_end_line":46655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46646-L46655","statement_sha256":"e5f923df6926afcd735f1e17d2b7dc5627abbce52d3cda4a6fd3fb4270bfa62c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2182,"rank":2182,"depth":1,"x":2006.634,"y":78.762,"cluster":"commutative-algebra"},{"id":"stacks:033F","tag":"033F","title":"Descending properties · Lemma 033F","summary":"Let R → S be a ring map. Assume that • R → S is faithfully flat, and • S is reduced. Then R is reduced.","statement_latex":"Let $R \\to S$ be a ring map.\nAssume that\n\\begin{enumerate}\n\\item $R \\to S$ is faithfully flat, and\n\\item $S$ is reduced.\n\\end{enumerate}\nThen $R$ is reduced.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033F","source_file":"algebra.tex","source_line":46667,"source_end_line":46676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46667-L46676","statement_sha256":"2fb5993f1c99387d93af27e6c6162eb7ad10113b85ab0ead0546869287bcdf3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2183,"rank":2183,"depth":1,"x":2160.86,"y":310.924,"cluster":"commutative-algebra"},{"id":"stacks:033G","tag":"033G","title":"Descending properties · Lemma 033G","summary":"Let R → S be a ring map. Assume that • R → S is faithfully flat, and • S is a normal ring. Then R is a normal ring.","statement_latex":"Let $R \\to S$ be a ring map.\nAssume that\n\\begin{enumerate}\n\\item $R \\to S$ is faithfully flat, and\n\\item $S$ is a normal ring.\n\\end{enumerate}\nThen $R$ is a normal ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033G","source_file":"algebra.tex","source_line":46683,"source_end_line":46692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46683-L46692","statement_sha256":"048c656cf83160db76f8dbf27b34e8003c95f675fbd9a467da0a07c5267d5264","origin":"The Stacks Project","memory_eligible":false,"source_rank":2184,"rank":2184,"depth":4,"x":1860.323,"y":227.21,"cluster":"commutative-algebra"},{"id":"stacks:07NG","tag":"07NG","title":"Descending properties · Lemma 07NG","summary":"Let R → S be a ring map. Assume that • R → S is faithfully flat, and • S is a regular ring. Then R is a regular ring.","statement_latex":"Let $R \\to S$ be a ring map.\nAssume that\n\\begin{enumerate}\n\\item $R \\to S$ is faithfully flat, and\n\\item $S$ is a regular ring.\n\\end{enumerate}\nThen $R$ is a regular ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NG","source_file":"algebra.tex","source_line":46714,"source_end_line":46723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46714-L46723","statement_sha256":"a5d0273461b1eaf8216fae15c85c78afd1e7adada09e33b8a8a3f5f04b03ea9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2185,"rank":2185,"depth":18,"x":2149.365,"y":118.366,"cluster":"commutative-algebra"},{"id":"stacks:0352","tag":"0352","title":"Descending properties · Lemma 0352","summary":"Let R → S be a ring map. Assume that • R → S is faithfully flat, and • S is Noetherian and has property (S_k). Then R is Noetherian and has property (S_k).","statement_latex":"Let $R \\to S$ be a ring map.\nAssume that\n\\begin{enumerate}\n\\item $R \\to S$ is faithfully flat, and\n\\item $S$ is Noetherian and has property $(S_k)$.\n\\end{enumerate}\nThen $R$ is Noetherian and has property $(S_k)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0352","source_file":"algebra.tex","source_line":46734,"source_end_line":46743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46734-L46743","statement_sha256":"5de0699a784df6fca5513f3c13c2d1f4fc4343229a88050a6f2a8d5402dbda0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2186,"rank":2186,"depth":16,"x":2023.711,"y":362.729,"cluster":"commutative-algebra"},{"id":"stacks:0353","tag":"0353","title":"Descending properties · Lemma 0353","summary":"Let R → S be a ring map. Assume that • R → S is faithfully flat, and • S is Noetherian and has property (R_k). Then R is Noetherian and has property (R_k).","statement_latex":"Let $R \\to S$ be a ring map. Assume that\n\\begin{enumerate}\n\\item $R \\to S$ is faithfully flat, and\n\\item $S$ is Noetherian and has property $(R_k)$.\n\\end{enumerate}\nThen $R$ is Noetherian and has property $(R_k)$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0353","source_file":"algebra.tex","source_line":46761,"source_end_line":46769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46761-L46769","statement_sha256":"149e711545bff4e5b9498368b14e70899a2f4f42aab3c506c476e49c2daee0ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":2187,"rank":2187,"depth":18,"x":1919.817,"y":111.144,"cluster":"commutative-algebra"},{"id":"stacks:0354","tag":"0354","title":"Descending properties · Lemma 0354","summary":"Let R → S be a ring map. Assume that • R → S is smooth and surjective on spectra, and • S is a Nagata ring. Then R is a Nagata ring.","statement_latex":"Let $R \\to S$ be a ring map. Assume that\n\\begin{enumerate}\n\\item $R \\to S$ is smooth and surjective on spectra, and\n\\item $S$ is a Nagata ring.\n\\end{enumerate}\nThen $R$ is a Nagata ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0354","source_file":"algebra.tex","source_line":46787,"source_end_line":46795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46787-L46795","statement_sha256":"d8d4fe3d06c8f2cd6a26522b7bcb2803761f3a57460efc2311ef568d76cc964f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2188,"rank":2188,"depth":33,"x":2198.851,"y":237.755,"cluster":"commutative-algebra"},{"id":"stacks:037Z","tag":"037Z","title":"Geometrically normal algebras · Lemma 037Z","summary":"Let k be a field. Let A be a k-algebra. The following properties of A are equivalent: • k' ⊗_k A is a normal ring for every field extension k'/k, • k' ⊗_k A is a normal ring for every finitely generated field extension k'/k, • k' ⊗_k A is a normal ring for every finite purely inseparable extension k'/k, • k^perf ⊗_k A is a normal ring. Here normal ring is defined in Definition [Tag 00GV].","statement_latex":"Let $k$ be a field. Let $A$ be a $k$-algebra.\nThe following properties of $A$ are equivalent:\n\\begin{enumerate}\n\\item $k' \\otimes_k A$ is a normal ring\nfor every field extension $k'/k$,\n\\item $k' \\otimes_k A$ is a normal ring\nfor every finitely generated field extension $k'/k$,\n\\item $k' \\otimes_k A$ is a normal ring\nfor every finite purely inseparable extension $k'/k$,\n\\item $k^{perf} \\otimes_k A$ is a normal ring.\n\\end{enumerate}\nHere normal ring is defined in Definition \\ref{definition-ring-normal}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically normal algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/037Z","source_file":"algebra.tex","source_line":46885,"source_end_line":46899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46885-L46899","statement_sha256":"af4688499df79b565f01937fa6e9807a8db26ec4e23970c3adf310663107a83a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2189,"rank":2189,"depth":40,"x":1891.161,"y":302.749,"cluster":"commutative-algebra"},{"id":"stacks:0380","tag":"0380","title":"Geometrically normal algebras · Definition 0380","summary":"Let k be a field. A k-algebra R is called geometrically normal over k if the equivalent conditions of Lemma [Tag 037Z] hold.","statement_latex":"Let $k$ be a field.\nA $k$-algebra $R$ is called {\\it geometrically normal} over $k$ if\nthe equivalent conditions of Lemma \\ref{lemma-geometrically-normal} hold.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically normal algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0380","source_file":"algebra.tex","source_line":46953,"source_end_line":46958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46953-L46958","statement_sha256":"5ed60374a38b8c8e7c241749d1ddb1b048f82cf37d11b8e6d4ee137475857fa3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2190,"rank":2190,"depth":41,"x":2065.847,"y":80.148,"cluster":"commutative-algebra"},{"id":"stacks:06DE","tag":"06DE","title":"Geometrically normal algebras · Lemma 06DE","summary":"Localization preserves geometric normality. Let k be a field. A localization of a geometrically normal k-algebra is geometrically normal.","statement_latex":"\\begin{slogan}\nLocalization preserves geometric normality.\n\\end{slogan}\nLet $k$ be a field. A localization of a geometrically normal $k$-algebra\nis geometrically normal.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically normal algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DE","source_file":"algebra.tex","source_line":46960,"source_end_line":46967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46960-L46967","statement_sha256":"7aca13af03b98851b363373d193b40bca8a1e2f1cf5c832c46074d5e51c9d155","origin":"The Stacks Project","memory_eligible":false,"source_rank":2191,"rank":2191,"depth":0,"x":2116.065,"y":343.512,"cluster":"commutative-algebra"},{"id":"stacks:0C30","tag":"0C30","title":"Geometrically normal algebras · Lemma 0C30","summary":"Let k be a field. Let K/k be a separable field extension. Then K is geometrically normal over k.","statement_latex":"Let $k$ be a field. Let $K/k$ be a separable field extension.\nThen $K$ is geometrically normal over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically normal algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C30","source_file":"algebra.tex","source_line":46974,"source_end_line":46978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46974-L46978","statement_sha256":"94c81206441ae6855fcce590f669f17400d03d8cb02bbc7f8382f42f71bb48ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":2192,"rank":2192,"depth":11,"x":1867.15,"y":177.744,"cluster":"commutative-algebra"},{"id":"stacks:06DF","tag":"06DF","title":"Geometrically normal algebras · Lemma 06DF","summary":"Let k be a field. Let A, B be k-algebras. Assume A is geometrically normal over k and B is a normal ring. Then A ⊗_k B is a normal ring.","statement_latex":"Let $k$ be a field. Let $A, B$ be $k$-algebras. Assume $A$ is geometrically\nnormal over $k$ and $B$ is a normal ring. Then $A \\otimes_k B$ is a normal\nring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically normal algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DF","source_file":"algebra.tex","source_line":46992,"source_end_line":46997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L46992-L46997","statement_sha256":"58fc8606b92a82624e60214043fa4bd1e09034b262c77a11e8a2c7c97a0950a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2193,"rank":2193,"depth":40,"x":2184.123,"y":158.731,"cluster":"commutative-algebra"},{"id":"stacks:0C31","tag":"0C31","title":"Geometrically normal algebras · Lemma 0C31","summary":"Let k'/k be a separable algebraic field extension. Let A be an algebra over k'. Then A is geometrically normal over k if and only if it is geometrically normal over k'.","statement_latex":"Let $k'/k$ be a separable algebraic field extension.\nLet $A$ be an algebra over $k'$. Then $A$ is geometrically normal\nover $k$ if and only if it is geometrically normal over $k'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically normal algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C31","source_file":"algebra.tex","source_line":47062,"source_end_line":47067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47062-L47067","statement_sha256":"5260891cad6e84f1b9c570258aa8faf0182225f7f6f212d8430f30707c2a2b44","origin":"The Stacks Project","memory_eligible":false,"source_rank":2194,"rank":2194,"depth":6,"x":1965.598,"y":352.682,"cluster":"commutative-algebra"},{"id":"stacks:0381","tag":"0381","title":"Geometrically regular algebras · Lemma 0381","summary":"Let k be a field. Let A be a k-algebra. Assume A is Noetherian. The following properties of A are equivalent: • k' ⊗_k A is regular for every finitely generated field extension k'/k, and • k' ⊗_k A is regular for every finite purely inseparable extension k'/k. Here regular ring is as in Definition [Tag 00OD].","statement_latex":"Let $k$ be a field. Let $A$ be a $k$-algebra.\nAssume $A$ is Noetherian.\nThe following properties of $A$ are equivalent:\n\\begin{enumerate}\n\\item $k' \\otimes_k A$ is regular for every finitely generated field\nextension $k'/k$, and\n\\item $k' \\otimes_k A$ is regular for every finite purely inseparable\nextension $k'/k$.\n\\end{enumerate}\nHere regular ring is as in Definition \\ref{definition-regular}.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically regular algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0381","source_file":"algebra.tex","source_line":47103,"source_end_line":47115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47103-L47115","statement_sha256":"c2ebbd62a06aa14f82ff9a26f1ae50347281235b7fac01b5ae28227d2c3ff255","origin":"The Stacks Project","memory_eligible":false,"source_rank":2195,"rank":2195,"depth":40,"x":1970.767,"y":85.568,"cluster":"commutative-algebra"},{"id":"stacks:0382","tag":"0382","title":"Geometrically regular algebras · Definition 0382","summary":"Let k be a field. Let R be a Noetherian k-algebra. The k-algebra R is called geometrically regular over k if the equivalent conditions of Lemma [Tag 0381] hold.","statement_latex":"Let $k$ be a field. Let $R$ be a Noetherian $k$-algebra.\nThe $k$-algebra $R$ is called {\\it geometrically regular} over $k$ if\nthe equivalent conditions of Lemma \\ref{lemma-geometrically-regular} hold.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically regular algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0382","source_file":"algebra.tex","source_line":47150,"source_end_line":47155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47150-L47155","statement_sha256":"ed3aaacb51f00c79e51e8e539aff871f12691bb08b5bcd260125dc26b8136302","origin":"The Stacks Project","memory_eligible":false,"source_rank":2196,"rank":2196,"depth":41,"x":2181.841,"y":285.542,"cluster":"commutative-algebra"},{"id":"stacks:07NH","tag":"07NH","title":"Geometrically regular algebras · Lemma 07NH","summary":"Geometric regularity descends through faithfully flat maps of algebras Let k be a field. Let A → B be a faithfully flat k-algebra map. If B is geometrically regular over k, so is A.","statement_latex":"\\begin{slogan}\nGeometric regularity descends through faithfully flat maps of algebras\n\\end{slogan}\nLet $k$ be a field. Let $A \\to B$ be a faithfully flat $k$-algebra\nmap. If $B$ is geometrically regular over $k$, so is $A$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically regular algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NH","source_file":"algebra.tex","source_line":47165,"source_end_line":47172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47165-L47172","statement_sha256":"e3e66fc51d9a20400e79d75cc07eac5d5acb05eb197f35b80c5b08ad168c20f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2197,"rank":2197,"depth":19,"x":1865.265,"y":257.843,"cluster":"commutative-algebra"},{"id":"stacks:07QF","tag":"07QF","title":"Geometrically regular algebras · Lemma 07QF","summary":"Let k be a field. Let A → B be a smooth ring map of k-algebras. If A is geometrically regular over k, then B is geometrically regular over k.","statement_latex":"Let $k$ be a field. Let $A \\to B$ be a smooth ring map\nof $k$-algebras. If $A$ is geometrically regular over $k$,\nthen $B$ is geometrically regular over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically regular algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QF","source_file":"algebra.tex","source_line":47186,"source_end_line":47191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47186-L47191","statement_sha256":"72a7000143e038d8afd0468021dfa127f172545acef2a8af0580f9fd5830dd53","origin":"The Stacks Project","memory_eligible":false,"source_rank":2198,"rank":2198,"depth":38,"x":2121.076,"y":98.575,"cluster":"commutative-algebra"},{"id":"stacks:07QG","tag":"07QG","title":"Geometrically regular algebras · Lemma 07QG","summary":"Let k be a field. Let A be an algebra over k. Let k = colim k_i be a directed colimit of subfields. If A is geometrically regular over each k_i, then A is geometrically regular over k.","statement_latex":"Let $k$ be a field. Let $A$ be an algebra over $k$.\nLet $k = \\colim k_i$ be a directed colimit of subfields.\nIf $A$ is geometrically regular over each $k_i$, then\n$A$ is geometrically regular over $k$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically regular algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QG","source_file":"algebra.tex","source_line":47201,"source_end_line":47207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47201-L47207","statement_sha256":"4dc151615e94049882200cd2a38c7b077170b300916ef6669be64bfb24c41675","origin":"The Stacks Project","memory_eligible":false,"source_rank":2199,"rank":2199,"depth":0,"x":2060.501,"y":361.269,"cluster":"commutative-algebra"},{"id":"stacks:07QH","tag":"07QH","title":"Geometrically regular algebras · Lemma 07QH","summary":"Let k'/k be a separable algebraic field extension. Let A be an algebra over k'. Then A is geometrically regular over k if and only if it is geometrically regular over k'.","statement_latex":"Let $k'/k$ be a separable algebraic field extension.\nLet $A$ be an algebra over $k'$. Then $A$ is geometrically\nregular over $k$ if and only if it is geometrically regular over $k'$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically regular algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QH","source_file":"algebra.tex","source_line":47218,"source_end_line":47223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47218-L47223","statement_sha256":"0cfe594bd1d51885be42d52735fae7c28cdf44c554ade914173f665912e8b7a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2200,"rank":2200,"depth":39,"x":1893.853,"y":133.104,"cluster":"commutative-algebra"},{"id":"stacks:045M","tag":"045M","title":"Geometrically Cohen-Macaulay algebras · Lemma 045M","summary":"Let k be a field and let K/k and L/k be two field extensions such that one of them is a field extension of finite type. Then K ⊗_k L is a Noetherian Cohen-Macaulay ring.","statement_latex":"Let $k$ be a field and let $K/k$ and $L/k$ be\ntwo field extensions such that one of them is a field extension of finite type.\nThen $K \\otimes_k L$ is a Noetherian Cohen-Macaulay ring.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically Cohen-Macaulay algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045M","source_file":"algebra.tex","source_line":47266,"source_end_line":47271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47266-L47271","statement_sha256":"fd8dc679b8b4d3b66120d1c1ec7fcfa20c3e962caca72ecd2e38047d3df64590","origin":"The Stacks Project","memory_eligible":false,"source_rank":2201,"rank":2201,"depth":11,"x":2200.336,"y":206.818,"cluster":"commutative-algebra"},{"id":"stacks:045N","tag":"045N","title":"Geometrically Cohen-Macaulay algebras · Lemma 045N","summary":"Let k be a field. Let S be a Noetherian k-algebra. Let K/k be a finitely generated field extension, and set S_K = K ⊗_k S. Let q ⊂ S be a prime of S. Let q_K ⊂ S_K be a prime of S_K lying over q. Then S_ q is Cohen-Macaulay if and only if (S_K)_ q_K is Cohen-Macaulay.","statement_latex":"Let $k$ be a field. Let $S$ be a Noetherian $k$-algebra.\nLet $K/k$ be a finitely generated field extension,\nand set $S_K = K \\otimes_k S$. Let $\\mathfrak q \\subset S$\nbe a prime of $S$. Let $\\mathfrak q_K \\subset S_K$ be a prime\nof $S_K$ lying over $\\mathfrak q$. Then $S_{\\mathfrak q}$ is Cohen-Macaulay\nif and only if $(S_K)_{\\mathfrak q_K}$ is Cohen-Macaulay.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Geometrically Cohen-Macaulay algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045N","source_file":"algebra.tex","source_line":47288,"source_end_line":47296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47288-L47296","statement_sha256":"16ada8c5c5dea8fb0421fbfbd611687a3f1bc6978bc8471bced31a6515c7e40d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2202,"rank":2202,"depth":17,"x":1914.955,"y":326.413,"cluster":"commutative-algebra"},{"id":"stacks:02JO","tag":"02JO","title":"Colimits and maps of finite presentation, II · Lemma 02JO","summary":"Let R → S be a ring map. Let M be an S-module. Assume that • R → S is of finite presentation, • M is a finitely presented S-module, and • M is flat over R. In this case we have the following: • There exists a finite type Z-algebra R_0 and a finite type ring map R_0 → S_0 and a finite S_0-module M_0 such that M_0 is flat over R_0, together with a ring maps R_0 → R and S_0 → S and an S_0-module map M_0 → M such that S ≅ R ⊗_R_0 S_0 and M = S ⊗_S_0 M_0. • If R = colim_λ ∈…","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nAssume that\n\\begin{enumerate}\n\\item $R \\to S$ is of finite presentation,\n\\item $M$ is a finitely presented $S$-module, and\n\\item $M$ is flat over $R$.\n\\end{enumerate}\nIn this case we have the following:\n\\begin{enumerate}\n\\item There exists a finite type $\\mathbf{Z}$-algebra $R_0$ and\na finite type ring map $R_0 \\to S_0$ and a finite $S_0$-module $M_0$\nsuch that $M_0$ is flat over $R_0$, together with a ring maps\n$R_0 \\to R$ and $S_0 \\to S$ and an $S_0$-module map $M_0 \\to M$\nsuch that $S \\cong R \\otimes_{R_0} S_0$ and $M = S \\otimes_{S_0} M_0$.\n\\item If $R = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$ is written\nas a directed colimit, then there exists a $\\lambda$ and a ring map\n$R_\\lambda \\to S_\\lambda$ of finite presentation, and an $S_\\lambda$-module\n$M_\\lambda$ of finite presentation such that $M_\\lambda$ is flat over\n$R_\\lambda$ and such that $S = R \\otimes_{R_\\lambda} S_\\lambda$ and\n$M = S \\otimes_{S_{\\lambda}} M_\\lambda$.\n\\item If\n$$\n(R \\to S, M) =\n\\colim_{\\lambda \\in \\Lambda}\n(R_\\lambda \\to S_\\lambda, M_\\lambda)\n$$\nis written as a directed colimit such that\n\\begin{enumerate}\n\\item $R_\\mu \\otimes_{R_\\lambda} S_\\lambda \\to S_\\mu$ and\n$S_\\mu \\otimes_{S_\\lambda} M_\\lambda \\to M_\\mu$ are isomorphisms\nfor $\\mu \\geq \\lambda$,\n\\item $R_\\lambda \\to S_\\lambda$ is of finite presentation,\n\\item $M_\\lambda$ is a finitely presented $S_\\lambda$-module,\n\\end{enumerate}\nthen for all sufficiently large $\\lambda$ the module $M_\\lambda$\nis flat over $R_\\lambda$.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JO","source_file":"algebra.tex","source_line":47330,"source_end_line":47370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47330-L47370","statement_sha256":"b7479b7e0030d19b688caa48119d99b7f7560469e760e196c659b6dd81daeb83","origin":"The Stacks Project","memory_eligible":false,"source_rank":2203,"rank":2203,"depth":34,"x":2029.258,"y":76.199,"cluster":"commutative-algebra"},{"id":"stacks:034Y","tag":"034Y","title":"Colimits and maps of finite presentation, II · Lemma 034Y","summary":"Let R → A → B be ring maps. Assume A → B faithfully flat of finite presentation. Then there exists a commutative diagram xymatrix R ar[r] ar@=[d] & A_0 ar[d] ar[r] & B_0 ar[d] R ar[r] & A ar[r] & B with R → A_0 of finite presentation, A_0 → B_0 faithfully flat of finite presentation and B = A ⊗_A_0 B_0.","statement_latex":"Let $R \\to A \\to B$ be ring maps.\nAssume $A \\to B$ faithfully flat of finite presentation.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\nR \\ar[r] \\ar@{=}[d] &\nA_0 \\ar[d] \\ar[r] &\nB_0 \\ar[d] \\\\\nR \\ar[r] & A \\ar[r] & B\n}\n$$\nwith $R \\to A_0$ of finite presentation,\n$A_0 \\to B_0$ faithfully flat of finite presentation\nand $B = A \\otimes_{A_0} B_0$.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034Y","source_file":"algebra.tex","source_line":47469,"source_end_line":47485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47469-L47485","statement_sha256":"e218ddde651de0034e67b8e7ec4febeb46250fd0fe7e3183703d1a77dd0e8779","origin":"The Stacks Project","memory_eligible":false,"source_rank":2204,"rank":2204,"depth":35,"x":2146.232,"y":325.655,"cluster":"commutative-algebra"},{"id":"stacks:07RG","tag":"07RG","title":"Colimits and maps of finite presentation, II · Lemma 07RG","summary":"Let A = colim_i ∈ I A_i be a directed colimit of rings. Let 0 ∈ I and φ_0 : B_0 → C_0 a map of A_0-algebras. Assume • A ⊗_A_0 B_0 → A ⊗_A_0 C_0 is finite, • C_0 is of finite type over B_0. Then there exists an i ≥ 0 such that the map A_i ⊗_A_0 B_0 → A_i ⊗_A_0 C_0 is finite.","statement_latex":"Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.\nLet $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.\nAssume\n\\begin{enumerate}\n\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is finite,\n\\item $C_0$ is of finite type over $B_0$.\n\\end{enumerate}\nThen there exists an $i \\geq 0$ such that the map\n$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$\nis finite.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RG","source_file":"algebra.tex","source_line":47521,"source_end_line":47533,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47521-L47533","statement_sha256":"49e6655d110a8ddc7e39675b796bf30650af9a3c891304788d76630d0cb0595a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2205,"rank":2205,"depth":0,"x":1859.263,"y":208.04,"cluster":"commutative-algebra"},{"id":"stacks:07RH","tag":"07RH","title":"Colimits and maps of finite presentation, II · Lemma 07RH","summary":"Let A = colim_i ∈ I A_i be a directed colimit of rings. Let 0 ∈ I and φ_0 : B_0 → C_0 a map of A_0-algebras. Assume • A ⊗_A_0 B_0 → A ⊗_A_0 C_0 is surjective, • C_0 is of finite type over B_0. Then for some i ≥ 0 the map A_i ⊗_A_0 B_0 → A_i ⊗_A_0 C_0 is surjective.","statement_latex":"Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.\nLet $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.\nAssume\n\\begin{enumerate}\n\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is surjective,\n\\item $C_0$ is of finite type over $B_0$.\n\\end{enumerate}\nThen for some $i \\geq 0$ the map\n$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$\nis surjective.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RH","source_file":"algebra.tex","source_line":47546,"source_end_line":47558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47546-L47558","statement_sha256":"9e088617d26909a674b0a2fdea622192a0dd5398f8be8c8ad9d539a19bcb2ae3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2206,"rank":2206,"depth":0,"x":2165.568,"y":131.906,"cluster":"commutative-algebra"},{"id":"stacks:0C4F","tag":"0C4F","title":"Colimits and maps of finite presentation, II · Lemma 0C4F","summary":"Let A = colim_i ∈ I A_i be a directed colimit of rings. Let 0 ∈ I and φ_0 : B_0 → C_0 a map of A_0-algebras. Assume • A ⊗_A_0 B_0 → A ⊗_A_0 C_0 is unramified, • C_0 is of finite type over B_0. Then for some i ≥ 0 the map A_i ⊗_A_0 B_0 → A_i ⊗_A_0 C_0 is unramified.","statement_latex":"Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.\nLet $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.\nAssume\n\\begin{enumerate}\n\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is unramified,\n\\item $C_0$ is of finite type over $B_0$.\n\\end{enumerate}\nThen for some $i \\geq 0$ the map\n$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$\nis unramified.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4F","source_file":"algebra.tex","source_line":47570,"source_end_line":47582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47570-L47582","statement_sha256":"52edf926951755ac72520ece9ebc494c20121d56a6db26ffb1baf49444bdad1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2207,"rank":2207,"depth":2,"x":2000.866,"y":361.937,"cluster":"commutative-algebra"},{"id":"stacks:0C32","tag":"0C32","title":"Colimits and maps of finite presentation, II · Lemma 0C32","summary":"Let A = colim_i ∈ I A_i be a directed colimit of rings. Let 0 ∈ I and φ_0 : B_0 → C_0 a map of A_0-algebras. Assume • A ⊗_A_0 B_0 → A ⊗_A_0 C_0 is an isomorphism, • B_0 → C_0 is of finite presentation. Then for some i ≥ 0 the map A_i ⊗_A_0 B_0 → A_i ⊗_A_0 C_0 is an isomorphism.","statement_latex":"Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.\nLet $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.\nAssume\n\\begin{enumerate}\n\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is\nan isomorphism,\n\\item $B_0 \\to C_0$ is of finite presentation.\n\\end{enumerate}\nThen for some $i \\geq 0$ the map\n$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is\nan isomorphism.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C32","source_file":"algebra.tex","source_line":47598,"source_end_line":47611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47598-L47611","statement_sha256":"623c778502f80dabc626d0ac7959c9bb3861c044502622f4020b7501f7c30175","origin":"The Stacks Project","memory_eligible":false,"source_rank":2208,"rank":2208,"depth":1,"x":1937.307,"y":98.761,"cluster":"commutative-algebra"},{"id":"stacks:07RI","tag":"07RI","title":"Colimits and maps of finite presentation, II · Lemma 07RI","summary":"Let A = colim_i ∈ I A_i be a directed colimit of rings. Let 0 ∈ I and φ_0 : B_0 → C_0 a map of A_0-algebras. Assume • A ⊗_A_0 B_0 → A ⊗_A_0 C_0 is étale, • B_0 → C_0 is of finite presentation. Then for some i ≥ 0 the map A_i ⊗_A_0 B_0 → A_i ⊗_A_0 C_0 is étale.","statement_latex":"Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.\nLet $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.\nAssume\n\\begin{enumerate}\n\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is \\'etale,\n\\item $B_0 \\to C_0$ is of finite presentation.\n\\end{enumerate}\nThen for some $i \\geq 0$ the map\n$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$\nis \\'etale.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RI","source_file":"algebra.tex","source_line":47625,"source_end_line":47637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47625-L47637","statement_sha256":"2284ff03601db297aac5980c28fd2e7fc73f41b1f95add7ee1c4ffd22a82c8d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2209,"rank":2209,"depth":0,"x":2195.91,"y":256.817,"cluster":"commutative-algebra"},{"id":"stacks:0C0B","tag":"0C0B","title":"Colimits and maps of finite presentation, II · Lemma 0C0B","summary":"Let A = colim_i ∈ I A_i be a directed colimit of rings. Let 0 ∈ I and φ_0 : B_0 → C_0 a map of A_0-algebras. Assume • A ⊗_A_0 B_0 → A ⊗_A_0 C_0 is smooth, • B_0 → C_0 is of finite presentation. Then for some i ≥ 0 the map A_i ⊗_A_0 B_0 → A_i ⊗_A_0 C_0 is smooth.","statement_latex":"Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.\nLet $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.\nAssume\n\\begin{enumerate}\n\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is smooth,\n\\item $B_0 \\to C_0$ is of finite presentation.\n\\end{enumerate}\nThen for some $i \\geq 0$ the map\n$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$ is smooth.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0B","source_file":"algebra.tex","source_line":47665,"source_end_line":47676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47665-L47676","statement_sha256":"49633b1d8fe7e9b0b6b621fc1a083e0a6de6e7b70014df74339f34690102c675","origin":"The Stacks Project","memory_eligible":false,"source_rank":2210,"rank":2210,"depth":0,"x":1877.997,"y":287.018,"cluster":"commutative-algebra"},{"id":"stacks:0C33","tag":"0C33","title":"Colimits and maps of finite presentation, II · Lemma 0C33","summary":"Let A = colim_i ∈ I A_i be a directed colimit of rings. Let 0 ∈ I and φ_0 : B_0 → C_0 a map of A_0-algebras. Assume • A ⊗_A_0 B_0 → A ⊗_A_0 C_0 is syntomic (resp. a relative global complete intersection), • C_0 is of finite presentation over B_0. Then there exists an i ≥ 0 such that the map A_i ⊗_A_0 B_0 → A_i ⊗_A_0 C_0 is syntomic (resp. a relative global complete intersection).","statement_latex":"Let $A = \\colim_{i \\in I} A_i$ be a directed colimit of rings.\nLet $0 \\in I$ and $\\varphi_0 : B_0 \\to C_0$ a map of $A_0$-algebras.\nAssume\n\\begin{enumerate}\n\\item $A \\otimes_{A_0} B_0 \\to A \\otimes_{A_0} C_0$ is\nsyntomic (resp.\\ a relative global complete intersection),\n\\item $C_0$ is of finite presentation over $B_0$.\n\\end{enumerate}\nThen there exists an $i \\geq 0$ such that the map\n$A_i \\otimes_{A_0} B_0 \\to A_i \\otimes_{A_0} C_0$\nis syntomic (resp.\\ a relative global complete intersection).","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C33","source_file":"algebra.tex","source_line":47705,"source_end_line":47718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47705-L47718","statement_sha256":"e341df13ff40b869331853f5f038a02cfb3633eee99f9d23a89b454f24b914d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2211,"rank":2211,"depth":35,"x":2088.213,"y":84.281,"cluster":"commutative-algebra"},{"id":"stacks:034Z","tag":"034Z","title":"Colimits and maps of finite presentation, II · Lemma 034Z","summary":"Let R → S be a faithfully flat ring map of finite presentation. Then there exists a commutative diagram xymatrix S ar[rr] & & S' & R ar[lu] ar[ru] where R → S' is quasi-finite, faithfully flat and of finite presentation.","statement_latex":"Let $R \\to S$ be a faithfully flat ring map of finite presentation.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\nS \\ar[rr] & & S' \\\\\n& R \\ar[lu] \\ar[ru]\n}\n$$\nwhere $R \\to S'$ is quasi-finite, faithfully flat and of finite presentation.","area":"Commutative Algebra","chapter":"Commutative Algebra","chapter_id":"algebra","section":"Colimits and maps of finite presentation, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034Z","source_file":"algebra.tex","source_line":47788,"source_end_line":47799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebra.tex#L47788-L47799","statement_sha256":"38ae39cbb252d06e49c215fffe3e9b119464cbd164d7c526549f2e2bc0a6df40","origin":"The Stacks Project","memory_eligible":false,"source_rank":2212,"rank":2212,"depth":36,"x":2096.241,"y":353.158,"cluster":"commutative-algebra"},{"id":"stacks:073Z","tag":"073Z","title":"Noncommutative algebras · Definition 073Z","summary":"Let A be a k-algebra. We say A is finite if dim_k(A) < ∞. In this case we write [A : k] = dim_k(A).","statement_latex":"Let $A$ be a $k$-algebra. We say $A$ is {\\it finite} if $\\dim_k(A) < \\infty$.\nIn this case we write $[A : k] = \\dim_k(A)$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Noncommutative algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/073Z","source_file":"brauer.tex","source_line":39,"source_end_line":43,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L39-L43","statement_sha256":"13871a9f1d1d29b7430e010b901a293e48fae1aea945e0f548b29d183526d55a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2213,"rank":2213,"depth":0,"x":1524.98,"y":341.025,"cluster":"fields-brauer-groups"},{"id":"stacks:0740","tag":"0740","title":"Noncommutative algebras · Definition 0740","summary":"A skew field is a possibly noncommutative ring with an identity element 1, with 1 not = 0, in which every nonzero element has a multiplicative inverse.","statement_latex":"A {\\it skew field} is a possibly noncommutative ring with an identity\nelement $1$, with $1 \\not = 0$, in which every nonzero element\nhas a multiplicative inverse.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Noncommutative algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0740","source_file":"brauer.tex","source_line":45,"source_end_line":50,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L45-L50","statement_sha256":"e32e16bc2872d6069a4c6e3357bae778320d1a43d64406ad6b974447855822ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":2214,"rank":2214,"depth":0,"x":1523.004,"y":99.025,"cluster":"fields-brauer-groups"},{"id":"stacks:0741","tag":"0741","title":"Noncommutative algebras · Definition 0741","summary":"Let A be a k-algebra. We say an A-module M is simple if it is nonzero and the only A-submodules are 0 and M. We say A is simple if the only two-sided ideals of A are 0 and A.","statement_latex":"Let $A$ be a $k$-algebra.\nWe say an $A$-module $M$ is {\\it simple} if it is nonzero and\nthe only $A$-submodules are $0$ and $M$.\nWe say $A$ is {\\it simple} if the only two-sided ideals of $A$ are\n$0$ and $A$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Noncommutative algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0741","source_file":"brauer.tex","source_line":58,"source_end_line":65,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L58-L65","statement_sha256":"c71405e514927426ca2d3efe85a4f9d39cd414bb964f6ca1585714acf83ca125","origin":"The Stacks Project","memory_eligible":false,"source_rank":2215,"rank":2215,"depth":0,"x":1719.901,"y":277.104,"cluster":"fields-brauer-groups"},{"id":"stacks:0742","tag":"0742","title":"Noncommutative algebras · Definition 0742","summary":"A k-algebra A is central if the center of A is the image of k → A.","statement_latex":"A $k$-algebra $A$ is {\\it central} if the center of $A$ is the image of\n$k \\to A$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Noncommutative algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0742","source_file":"brauer.tex","source_line":67,"source_end_line":71,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L67-L71","statement_sha256":"38f1b85d060ef748afc3d7165288048a0f1692b18de4716a0232e137933af143","origin":"The Stacks Project","memory_eligible":false,"source_rank":2216,"rank":2216,"depth":0,"x":1430.293,"y":257.432,"cluster":"fields-brauer-groups"},{"id":"stacks:0743","tag":"0743","title":"Noncommutative algebras · Definition 0743","summary":"Given a k-algebra A we denote A^op the k-algebra we get by reversing the order of multiplication in A. This is called the opposite algebra.","statement_latex":"Given a $k$-algebra $A$ we denote $A^{op}$ the $k$-algebra we get by\nreversing the order of multiplication in $A$. This is called the\n{\\it opposite algebra}.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Noncommutative algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0743","source_file":"brauer.tex","source_line":73,"source_end_line":78,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L73-L78","statement_sha256":"1060dcace79e6574296b033f6900cbcb8db46c6a575a862f9be8d73d4cf9eae5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2217,"rank":2217,"depth":0,"x":1660.624,"y":106.984,"cluster":"fields-brauer-groups"},{"id":"stacks:0745","tag":"0745","title":"Wedderburn's theorem · Lemma 0745","summary":"Let A be a possibly noncommutative ring with 1 which contains no nontrivial two-sided ideal. Let M be a nonzero right ideal in A, and view M as a right A-module. Then A coincides with the bicommutant of M.","statement_latex":"Let $A$ be a possibly noncommutative ring with $1$ which contains no\nnontrivial two-sided ideal. Let $M$ be a nonzero right ideal in $A$,\nand view $M$ as a right $A$-module. Then $A$ coincides with the\nbicommutant of $M$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Wedderburn's theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0745","source_file":"brauer.tex","source_line":91,"source_end_line":97,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L91-L97","statement_sha256":"af4176b8107cacdb3ec97ab8767e90bf0cdaf276d4fd7baee8feb85fec19cf37","origin":"The Stacks Project","memory_eligible":false,"source_rank":2218,"rank":2218,"depth":0,"x":1611.564,"y":349.616,"cluster":"fields-brauer-groups"},{"id":"stacks:0746","tag":"0746","title":"Wedderburn's theorem · Lemma 0746","summary":"Let A be a k-algebra. If A is finite, then • A has a simple module, • any nonzero module contains a simple submodule, • a simple module over A has finite dimension over k, and • if M is a simple A-module, then End_A(M) is a skew field.","statement_latex":"Let $A$ be a $k$-algebra. If $A$ is finite, then\n\\begin{enumerate}\n\\item $A$ has a simple module,\n\\item any nonzero module contains a simple submodule,\n\\item a simple module over $A$ has finite dimension over $k$, and\n\\item if $M$ is a simple $A$-module, then $\\text{End}_A(M)$ is a\nskew field.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Wedderburn's theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0746","source_file":"brauer.tex","source_line":119,"source_end_line":129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L119-L129","statement_sha256":"2eebb248102c70844652021937826599b9a59e2d3bf0a39da7ab7625d97e4e09","origin":"The Stacks Project","memory_eligible":false,"source_rank":2219,"rank":2219,"depth":0,"x":1451.971,"y":142.016,"cluster":"fields-brauer-groups"},{"id":"stacks:0747","tag":"0747","title":"Wedderburn's theorem · Theorem 0747","summary":"Simple finite algebras over a field are matrix algebras over a skew field. Let A be a simple finite k-algebra. Then A is a matrix algebra over a finite k-algebra K which is a skew field.","statement_latex":"\\begin{slogan}\nSimple finite algebras over a field are matrix algebras over a skew field.\n\\end{slogan}\nLet $A$ be a simple finite $k$-algebra. Then $A$ is a matrix algebra over\na finite $k$-algebra $K$ which is a skew field.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Wedderburn's theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0747","source_file":"brauer.tex","source_line":140,"source_end_line":147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L140-L147","statement_sha256":"bf8f374d4eb31c7d06623480d8e33b93f2aab250f494dbcf9a439a9f0b32a848","origin":"The Stacks Project","memory_eligible":false,"source_rank":2220,"rank":2220,"depth":1,"x":1737.756,"y":204.796,"cluster":"fields-brauer-groups"},{"id":"stacks:0749","tag":"0749","title":"Lemmas on algebras · Lemma 0749","summary":"Let A, A' be k-algebras. Let B ⊂ A, B' ⊂ A' be subalgebras with centralizers C, C'. Then the centralizer of B ⊗_k B' in A ⊗_k A' is C ⊗_k C'.","statement_latex":"Let $A$, $A'$ be $k$-algebras. Let $B \\subset A$, $B' \\subset A'$ be\nsubalgebras with centralizers $C$, $C'$. Then the centralizer of\n$B \\otimes_k B'$ in $A \\otimes_k A'$ is $C \\otimes_k C'$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0749","source_file":"brauer.tex","source_line":177,"source_end_line":182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L177-L182","statement_sha256":"5393b54a3ec6a5ea2c273a52cc74568a8ef142d5afbb0e30bda8f95107ab56c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2221,"rank":2221,"depth":0,"x":1475.481,"y":321.13,"cluster":"fields-brauer-groups"},{"id":"stacks:074A","tag":"074A","title":"Lemmas on algebras · Lemma 074A","summary":"Let A be a finite simple k-algebra. Then the center k' of A is a finite field extension of k.","statement_latex":"Let $A$ be a finite simple $k$-algebra. Then the center $k'$ of $A$\nis a finite field extension of $k$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074A","source_file":"brauer.tex","source_line":192,"source_end_line":196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L192-L196","statement_sha256":"185900a189c03cadd46b193283620fc1edb7aa5a51d17b073a014f85c5464030","origin":"The Stacks Project","memory_eligible":false,"source_rank":2222,"rank":2222,"depth":2,"x":1575.727,"y":85.588,"cluster":"fields-brauer-groups"},{"id":"stacks:074B","tag":"074B","title":"Lemmas on algebras · Lemma 074B","summary":"Let V be a k vector space. Let K be a central k-algebra which is a skew field. Let W ⊂ V ⊗_k K be a two-sided K-sub vector space. Then W is generated as a left K-vector space by W ∩ (V ⊗ 1).","statement_latex":"Let $V$ be a $k$ vector space. Let $K$ be a central $k$-algebra\nwhich is a skew field. Let $W \\subset V \\otimes_k K$ be a two-sided\n$K$-sub vector space. Then $W$ is generated as a left $K$-vector\nspace by $W \\cap (V \\otimes 1)$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074B","source_file":"brauer.tex","source_line":208,"source_end_line":214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L208-L214","statement_sha256":"3542b91c86c38251e43c3a5c6362742288ab27cfdca18c47a95827c6e54cf613","origin":"The Stacks Project","memory_eligible":false,"source_rank":2223,"rank":2223,"depth":0,"x":1691.681,"y":317.072,"cluster":"fields-brauer-groups"},{"id":"stacks:074C","tag":"074C","title":"Lemmas on algebras · Lemma 074C","summary":"Let A be a k-algebra. Let K be a central k-algebra which is a skew field. Then any two-sided ideal I ⊂ A ⊗_k K is of the form J ⊗_k K for some two-sided ideal J ⊂ A. In particular, if A is simple, then so is A ⊗_k K.","statement_latex":"Let $A$ be a $k$-algebra. Let $K$ be a central $k$-algebra\nwhich is a skew field. Then any two-sided ideal $I \\subset A \\otimes_k K$\nis of the form $J \\otimes_k K$ for some two-sided ideal $J \\subset A$.\nIn particular, if $A$ is simple, then so is $A \\otimes_k K$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074C","source_file":"brauer.tex","source_line":245,"source_end_line":251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L245-L251","statement_sha256":"2a6c6c7fff375e74f3828d27fb4723c2f25ddb5dae189487357f839b21e9304f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2224,"rank":2224,"depth":1,"x":1418.957,"y":211.759,"cluster":"fields-brauer-groups"},{"id":"stacks:074D","tag":"074D","title":"Lemmas on algebras · Lemma 074D","summary":"Let R be a possibly noncommutative ring. Let n ≥ 1 be an integer. Let R_n = Mat(n × n, R). • The functors M ↦ M^⊕ n and N ↦ Ne_11 define quasi-inverse equivalences of categories Mod_R ↔ Mod_R_n. • A two-sided ideal of R_n is of the form IR_n for some two-sided ideal I of R. • The center of R_n is equal to the center of R.","statement_latex":"Let $R$ be a possibly noncommutative ring. Let $n \\geq 1$ be an integer.\nLet $R_n = \\text{Mat}(n \\times n, R)$.\n\\begin{enumerate}\n\\item The functors $M \\mapsto M^{\\oplus n}$ and\n$N \\mapsto Ne_{11}$ define quasi-inverse equivalences of categories\n$\\text{Mod}_R \\leftrightarrow \\text{Mod}_{R_n}$.\n\\item A two-sided ideal of $R_n$ is of the form $IR_n$ for some\ntwo-sided ideal $I$ of $R$.\n\\item The center of $R_n$ is equal to the center of $R$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074D","source_file":"brauer.tex","source_line":259,"source_end_line":271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L259-L271","statement_sha256":"45db48dd926494d890d1f1dd0d45fc4dcb47c17069737f265242cb90fc9eeb57","origin":"The Stacks Project","memory_eligible":false,"source_rank":2225,"rank":2225,"depth":0,"x":1705.866,"y":134.365,"cluster":"fields-brauer-groups"},{"id":"stacks:074E","tag":"074E","title":"Lemmas on algebras · Lemma 074E","summary":"Let A be a finite simple k-algebra. • There exists exactly one simple A-module M up to isomorphism. • Any finite A-module is a direct sum of copies of a simple module. • Two finite A-modules are isomorphic if and only if they have the same dimension over k. • If A = Mat(n × n, K) with K a finite skew field extension of k, then M = K^⊕ n is a simple A-module and End_A(M) = K^op. • If M is a simple A-module, then L = End_A(M) is a skew field finite over k acting on the left…","statement_latex":"Let $A$ be a finite simple $k$-algebra.\n\\begin{enumerate}\n\\item There exists exactly one simple $A$-module $M$ up to isomorphism.\n\\item Any finite $A$-module is a direct sum of copies of a simple module.\n\\item Two finite $A$-modules are isomorphic if and only if they\nhave the same dimension over $k$.\n\\item If $A = \\text{Mat}(n \\times n, K)$ with $K$ a finite skew field\nextension of $k$, then $M = K^{\\oplus n}$ is a simple $A$-module and\n$\\text{End}_A(M) = K^{op}$.\n\\item If $M$ is a simple $A$-module, then $L = \\text{End}_A(M)$\nis a skew field finite over $k$ acting on the left on $M$, we have\n$A = \\text{End}_L(M)$, and the centers of $A$ and $L$ agree.\nAlso $[A : k] [L : k] = \\dim_k(M)^2$.\n\\item For a finite $A$-module $N$ the algebra $B = \\text{End}_A(N)$ is a\nmatrix algebra over the skew field $L$ of (5). Moreover $\\text{End}_B(N) = A$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074E","source_file":"brauer.tex","source_line":280,"source_end_line":298,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L280-L298","statement_sha256":"09c01213beb2727c5e52ea2ce56a99eb42cd8e72c7e2c2736058b2302203773d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2226,"rank":2226,"depth":2,"x":1555.961,"y":355.085,"cluster":"fields-brauer-groups"},{"id":"stacks:074F","tag":"074F","title":"Lemmas on algebras · Lemma 074F","summary":"Let A, A' be two simple k-algebras one of which is finite and central over k. Then A ⊗_k A' is simple.","statement_latex":"Let $A$, $A'$ be two simple $k$-algebras one of which is finite and central\nover $k$. Then $A \\otimes_k A'$ is simple.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074F","source_file":"brauer.tex","source_line":320,"source_end_line":324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L320-L324","statement_sha256":"2b7f073e4656693cbc9e22f3a308c12f5281722b740ceda763ad6cbb6e75d776","origin":"The Stacks Project","memory_eligible":false,"source_rank":2227,"rank":2227,"depth":2,"x":1488.745,"y":106.316,"cluster":"fields-brauer-groups"},{"id":"stacks:074G","tag":"074G","title":"Lemmas on algebras · Lemma 074G","summary":"The tensor product of finite central simple algebras over k is finite, central, and simple.","statement_latex":"The tensor product of finite central simple algebras over $k$ is finite,\ncentral, and simple.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074G","source_file":"brauer.tex","source_line":337,"source_end_line":341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L337-L341","statement_sha256":"85ecb01175082017e17b71baa6741496db4eb0bb0a72d8d3e17d852b77a711c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2228,"rank":2228,"depth":3,"x":1739.315,"y":252.171,"cluster":"fields-brauer-groups"},{"id":"stacks:074H","tag":"074H","title":"Lemmas on algebras · Lemma 074H","summary":"Let A be a finite central simple algebra over k. Let k'/k be a field extension. Then A' = A ⊗_k k' is a finite central simple algebra over k'.","statement_latex":"Let $A$ be a finite central simple algebra over $k$.\nLet $k'/k$ be a field extension. Then $A' = A \\otimes_k k'$ is\na finite central simple algebra over $k'$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074H","source_file":"brauer.tex","source_line":347,"source_end_line":352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L347-L352","statement_sha256":"32352e96667ac1fd1ff00a4bd233b87c34421bda9091044ef68c2e40f6bbf435","origin":"The Stacks Project","memory_eligible":false,"source_rank":2229,"rank":2229,"depth":3,"x":1436.112,"y":286.928,"cluster":"fields-brauer-groups"},{"id":"stacks:074I","tag":"074I","title":"Lemmas on algebras · Lemma 074I","summary":"Let A be a finite central simple algebra over k. Then A ⊗_k A^op ≅ Mat(n × n, k) where n = [A : k].","statement_latex":"Let $A$ be a finite central simple algebra over $k$.\nThen $A \\otimes_k A^{op} \\cong \\text{Mat}(n \\times n, k)$\nwhere $n = [A : k]$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Lemmas on algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074I","source_file":"brauer.tex","source_line":358,"source_end_line":363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L358-L363","statement_sha256":"7b056caeceb7aba5356de5cb3fff76f315fb43d1b28a9ddb42fef8625330321b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2230,"rank":2230,"depth":4,"x":1632.474,"y":88.512,"cluster":"fields-brauer-groups"},{"id":"stacks:074K","tag":"074K","title":"The Brauer group of a field · Lemma 074K","summary":"Similarity. • Similarity defines an equivalence relation on the set of isomorphism classes of finite central simple algebras over k. • Every similarity class contains a unique (up to isomorphism) finite central skew field extension of k. • If A = Mat(n × n, K) and B = Mat(m × m, K') for some finite central skew fields K, K' over k then A and B are similar if and only if K ≅ K' as k-algebras.","statement_latex":"Similarity.\n\\begin{enumerate}\n\\item Similarity defines an equivalence relation on the set of isomorphism\nclasses of finite central simple algebras over $k$.\n\\item Every similarity class contains a unique (up to isomorphism)\nfinite central skew field extension of $k$.\n\\item If $A = \\text{Mat}(n \\times n, K)$ and $B = \\text{Mat}(m \\times m, K')$\nfor some finite central skew fields $K$, $K'$ over $k$\nthen $A$ and $B$ are similar if and only if $K \\cong K'$ as $k$-algebras.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"The Brauer group of a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074K","source_file":"brauer.tex","source_line":390,"source_end_line":402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L390-L402","statement_sha256":"2ae7c63e515e8d193cf916f23a66a6ce040f5c8ed7ca1da811016397894069f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2231,"rank":2231,"depth":3,"x":1647.294,"y":347.204,"cluster":"fields-brauer-groups"},{"id":"stacks:074L","tag":"074L","title":"The Brauer group of a field · Definition 074L","summary":"Let k be a field. The Brauer group of k is the abelian group of similarity classes of finite central simple k-algebras defined above. Notation Br(k).","statement_latex":"Let $k$ be a field. The {\\it Brauer group} of $k$ is the abelian group\nof similarity classes of finite central simple $k$-algebras defined\nabove. Notation $\\text{Br}(k)$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"The Brauer group of a field","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074L","source_file":"brauer.tex","source_line":430,"source_end_line":435,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L430-L435","statement_sha256":"eee5ed319ef1101e09919ee9c07912e537a0c1077ec82cf632373db7c88a272b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2232,"rank":2232,"depth":0,"x":1427.524,"y":164.181,"cluster":"fields-brauer-groups"},{"id":"stacks:074M","tag":"074M","title":"The Brauer group of a field · Lemma 074M","summary":"The Brauer group of an algebraically closed field is zero.","statement_latex":"The Brauer group of an algebraically closed field is zero.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"The Brauer group of a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074M","source_file":"brauer.tex","source_line":449,"source_end_line":452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L449-L452","statement_sha256":"b7fafdb752ee378c6b26901696d36439772246c42874ad8242cefff7d257277a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2233,"rank":2233,"depth":7,"x":1737.897,"y":174.475,"cluster":"fields-brauer-groups"},{"id":"stacks:074N","tag":"074N","title":"The Brauer group of a field · Lemma 074N","summary":"Let A be a finite central simple algebra over a field k. Then [A : k] is a square.","statement_latex":"Let $A$ be a finite central simple algebra over a field $k$.\nThen $[A : k]$ is a square.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"The Brauer group of a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074N","source_file":"brauer.tex","source_line":463,"source_end_line":467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L463-L467","statement_sha256":"503b87cc1758652e8f46df4760466d00bca3a79667d2a8f00a74a6571c95ca82","origin":"The Stacks Project","memory_eligible":false,"source_rank":2234,"rank":2234,"depth":8,"x":1499.889,"y":343.612,"cluster":"fields-brauer-groups"},{"id":"stacks:074Q","tag":"074Q","title":"Skolem-Noether · Theorem 074Q","summary":"Let A be a finite central simple k-algebra. Let B be a simple k-algebra. Let f, g : B → A be two k-algebra homomorphisms. Then there exists an invertible element x ∈ A such that f(b) = xg(b)x^-1 for all b ∈ B.","statement_latex":"Let $A$ be a finite central simple $k$-algebra. Let $B$ be a simple\n$k$-algebra. Let $f, g : B \\to A$ be two $k$-algebra homomorphisms.\nThen there exists an invertible element $x \\in A$ such that\n$f(b) = xg(b)x^{-1}$ for all $b \\in B$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Skolem-Noether","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074Q","source_file":"brauer.tex","source_line":483,"source_end_line":489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L483-L489","statement_sha256":"5150de7b46f4ede0b30ddda4e1153d13fffc18e2d93440175fd0af282c01070b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2235,"rank":2235,"depth":3,"x":1539.522,"y":82.901,"cluster":"fields-brauer-groups"},{"id":"stacks:074R","tag":"074R","title":"Skolem-Noether · Lemma 074R","summary":"Let A be a finite central simple k-algebra. Any automorphism of A is inner. In particular, any automorphism of Mat(n × n, k) is inner.","statement_latex":"Let $A$ be a finite central simple $k$-algebra. Any automorphism of $A$ is\ninner. In particular, any automorphism of $\\text{Mat}(n \\times n, k)$\nis inner.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Skolem-Noether","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074R","source_file":"brauer.tex","source_line":510,"source_end_line":515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L510-L515","statement_sha256":"a8073a915a4f0dcb12f748df440a596f805e561c26166d2a70e8bafe3bec8e5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2236,"rank":2236,"depth":4,"x":1720.601,"y":298.407,"cluster":"fields-brauer-groups"},{"id":"stacks:074T","tag":"074T","title":"The centralizer theorem · Theorem 074T","summary":"Let A be a finite central simple algebra over k, and let B be a simple subalgebra of A. Then • the centralizer C of B in A is simple, • [A : k] = [B : k][C : k], and • the centralizer of C in A is B.","statement_latex":"Let $A$ be a finite central simple algebra over $k$, and let\n$B$ be a simple subalgebra of $A$. Then\n\\begin{enumerate}\n\\item the centralizer $C$ of $B$ in $A$ is simple,\n\\item $[A : k] = [B : k][C : k]$, and\n\\item the centralizer of $C$ in $A$ is $B$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"The centralizer theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074T","source_file":"brauer.tex","source_line":528,"source_end_line":537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L528-L537","statement_sha256":"8b5d8e520fa35dafdf26b9023e0a7ea40249abb4f656c754d9cae93195b61300","origin":"The Stacks Project","memory_eligible":false,"source_rank":2237,"rank":2237,"depth":3,"x":1412.677,"y":242.042,"cluster":"fields-brauer-groups"},{"id":"stacks:074U","tag":"074U","title":"The centralizer theorem · Lemma 074U","summary":"Let A be a finite central simple algebra over k, and let B be a simple subalgebra of A. If B is a central k-algebra, then A = B ⊗_k C where C is the (central simple) centralizer of B in A.","statement_latex":"Let $A$ be a finite central simple algebra over $k$, and let\n$B$ be a simple subalgebra of $A$. If $B$ is a central\n$k$-algebra, then $A = B \\otimes_k C$ where $C$ is the (central simple)\ncentralizer of $B$ in $A$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"The centralizer theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074U","source_file":"brauer.tex","source_line":566,"source_end_line":572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L566-L572","statement_sha256":"84b14ca714bb0f90fb3e2f4c8ba5d40a736cc4ba2fc51cc2f87967c7cbc65112","origin":"The Stacks Project","memory_eligible":false,"source_rank":2238,"rank":2238,"depth":4,"x":1686.028,"y":108.411,"cluster":"fields-brauer-groups"},{"id":"stacks:074V","tag":"074V","title":"The centralizer theorem · Lemma 074V","summary":"Let A be a finite central simple algebra over k. If K ⊂ A is a subfield, then the following are equivalent • [A : k] = [K : k]^2, • K is its own centralizer, and • K is a maximal commutative subring.","statement_latex":"Let $A$ be a finite central simple algebra over $k$.\nIf $K \\subset A$ is a subfield, then the following are equivalent\n\\begin{enumerate}\n\\item $[A : k] = [K : k]^2$,\n\\item $K$ is its own centralizer, and\n\\item $K$ is a maximal commutative subring.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"The centralizer theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074V","source_file":"brauer.tex","source_line":582,"source_end_line":591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L582-L591","statement_sha256":"4ae8511badb1077462e2d712a999bd47e5c49486134a3a81994866fccf1f24ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":2239,"rank":2239,"depth":4,"x":1591.594,"y":362.947,"cluster":"fields-brauer-groups"},{"id":"stacks:074W","tag":"074W","title":"The centralizer theorem · Lemma 074W","summary":"The dimension of a finite central skew field is the square of the dimension of any maximal subfield. Let A be a finite central skew field over k. Then every maximal subfield K ⊂ A satisfies [A : k] = [K : k]^2.","statement_latex":"\\begin{slogan}\nThe dimension of a finite central skew field is the square of the dimension\nof any maximal subfield.\n\\end{slogan}\nLet $A$ be a finite central skew field over $k$.\nThen every maximal subfield $K \\subset A$ satisfies\n$[A : k] = [K : k]^2$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"The centralizer theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074W","source_file":"brauer.tex","source_line":599,"source_end_line":608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L599-L608","statement_sha256":"62d0d70432a7ea12a13186c95efa1dd91be615188e1cb15ba847b8ab32822868","origin":"The Stacks Project","memory_eligible":false,"source_rank":2240,"rank":2240,"depth":5,"x":1456.067,"y":120.828,"cluster":"fields-brauer-groups"},{"id":"stacks:074Y","tag":"074Y","title":"Splitting fields · Definition 074Y","summary":"Let A be a finite central simple k-algebra. We say a field extension k'/k splits A, or k' is a splitting field for A if A ⊗_k k' is a matrix algebra over k'.","statement_latex":"Let $A$ be a finite central simple $k$-algebra.\nWe say a field extension $k'/k$ {\\it splits} $A$, or\n$k'$ is a {\\it splitting field} for $A$ if $A \\otimes_k k'$ is\na matrix algebra over $k'$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Splitting fields","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074Y","source_file":"brauer.tex","source_line":622,"source_end_line":628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L622-L628","statement_sha256":"b56c21f27d5cc1886b6132a26467278df50974d3a3307fa6aec8a43e93beb19d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2241,"rank":2241,"depth":0,"x":1751.729,"y":222.815,"cluster":"fields-brauer-groups"},{"id":"stacks:074Z","tag":"074Z","title":"Splitting fields · Theorem 074Z","summary":"Let A be a finite central simple k-algebra. Let k'/k be a finite field extension. The following are equivalent • k' splits A, and • there exists a finite central simple algebra B similar to A such that k' ⊂ B and [B : k] = [k' : k]^2.","statement_latex":"Let $A$ be a finite central simple $k$-algebra.\nLet $k'/k$ be a finite field extension.\nThe following are equivalent\n\\begin{enumerate}\n\\item $k'$ splits $A$, and\n\\item there exists a finite central simple algebra $B$ similar to $A$\nsuch that $k' \\subset B$ and $[B : k] = [k' : k]^2$.\n\\end{enumerate}","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Splitting fields","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/074Z","source_file":"brauer.tex","source_line":634,"source_end_line":644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L634-L644","statement_sha256":"82f0d725560d224c3850b0dbd683f47b3a4aaf799c44f4dec0b222fdf9dfc874","origin":"The Stacks Project","memory_eligible":false,"source_rank":2242,"rank":2242,"depth":5,"x":1450.663,"y":315.695,"cluster":"fields-brauer-groups"},{"id":"stacks:0750","tag":"0750","title":"Splitting fields · Lemma 0750","summary":"A maximal subfield of a finite central skew field K over k is a splitting field for K.","statement_latex":"A maximal subfield of a finite central skew field $K$ over $k$ is\na splitting field for $K$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Splitting fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0750","source_file":"brauer.tex","source_line":678,"source_end_line":682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L678-L682","statement_sha256":"7b61776a7bbf1c3e99b4407b478771f7f0650d4ecacba448606157f322177387","origin":"The Stacks Project","memory_eligible":false,"source_rank":2243,"rank":2243,"depth":6,"x":1598.479,"y":75.557,"cluster":"fields-brauer-groups"},{"id":"stacks:0751","tag":"0751","title":"Splitting fields · Lemma 0751","summary":"Consider a finite central skew field K over k. Let d^2 = [K : k]. For any finite splitting field k' for K the degree [k' : k] is divisible by d.","statement_latex":"Consider a finite central skew field $K$ over $k$. Let $d^2 = [K : k]$.\nFor any finite splitting field $k'$ for $K$ the degree $[k' : k]$ is\ndivisible by $d$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Splitting fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0751","source_file":"brauer.tex","source_line":689,"source_end_line":694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L689-L694","statement_sha256":"2153b4d79fad823845cad879950f1bad9c6df2dd483030b5ed0db55c858f4ba2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2244,"rank":2244,"depth":6,"x":1682.879,"y":337.392,"cluster":"fields-brauer-groups"},{"id":"stacks:0752","tag":"0752","title":"Splitting fields · Proposition 0752","summary":"Consider a finite central skew field K over k. There exists a maximal subfield k ⊂ k' ⊂ K which is separable over k. In particular, every Brauer class has a finite separable spitting field.","statement_latex":"Consider a finite central skew field $K$ over $k$.\nThere exists a maximal subfield $k \\subset k' \\subset K$ which\nis separable over $k$.\nIn particular, every Brauer class has a finite separable\nspitting field.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Splitting fields","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0752","source_file":"brauer.tex","source_line":705,"source_end_line":712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L705-L712","statement_sha256":"58a18b5c3e5f8b5c17291edade1a53ce2adc2f530a1bdebfd5f016c0626eed1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2245,"rank":2245,"depth":8,"x":1409.163,"y":191.718,"cluster":"fields-brauer-groups"},{"id":"stacks:0753","tag":"0753","title":"Splitting fields · Lemma 0753","summary":"Let k be a field. For a k-algebra A the following are equivalent • A is finite central simple k-algebra, • A is a finite dimensional k-vector space, k is the center of A, and A has no nontrivial two-sided ideal, • there exists d ≥ 1 such that A ⊗_k bar k ≅ Mat(d × d, bar k), • there exists d ≥ 1 such that A ⊗_k k^sep ≅ Mat(d × d, k^sep), • there exist d ≥ 1 and a finite Galois extension k'/k such that A ⊗_k k' ≅ Mat(d × d, k'), • there exist n ≥ 1 and a finite central…","statement_latex":"Let $k$ be a field. For a $k$-algebra $A$ the following are equivalent\n\\begin{enumerate}\n\\item $A$ is finite central simple $k$-algebra,\n\\item $A$ is a finite dimensional $k$-vector space, $k$ is the center of $A$,\nand $A$ has no nontrivial two-sided ideal,\n\\item there exists $d \\geq 1$ such that\n$A \\otimes_k \\bar k \\cong \\text{Mat}(d \\times d, \\bar k)$,\n\\item there exists $d \\geq 1$ such that\n$A \\otimes_k k^{sep} \\cong \\text{Mat}(d \\times d, k^{sep})$,\n\\item there exist $d \\geq 1$ and a finite Galois extension $k'/k$\nsuch that\n$A \\otimes_k k' \\cong \\text{Mat}(d \\times d, k')$,\n\\item there exist $n \\geq 1$ and a finite central skew field $K$\nover $k$ such that $A \\cong \\text{Mat}(n \\times n, K)$.\n\\end{enumerate}\nThe integer $d$ is called the {\\it degree} of $A$.","area":"Fields & Brauer Groups","chapter":"Brauer Groups","chapter_id":"brauer","section":"Splitting fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0753","source_file":"brauer.tex","source_line":767,"source_end_line":785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/brauer.tex#L767-L785","statement_sha256":"479f05a816fefed3ca5847d4f069d8a10fa45ae550dde12a95840f6e7711e945","origin":"The Stacks Project","memory_eligible":false,"source_rank":2246,"rank":2246,"depth":9,"x":1729.212,"y":143.664,"cluster":"fields-brauer-groups"},{"id":"stacks:00ZY","tag":"00ZY","title":"Preadditive and additive categories · Definition 00ZY","summary":"A category A is called preadditive if each morphism set Mor_A(x, y) is endowed with the structure of an abelian group such that the compositions Mor(x, y) × Mor(y, z) → Mor(x, z) are bilinear. A functor F : A → B of preadditive categories is called additive if and only if F : Mor(x, y) → Mor(F(x), F(y)) is a homomorphism of abelian groups for all x, y ∈ Ob(A).","statement_latex":"A category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZY","source_file":"homology.tex","source_line":47,"source_end_line":63,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L47-L63","statement_sha256":"1b3061e87fef829f7b82866b7a7b8ae366c4314dce78ab3d7815ce92f7cdf14c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2247,"rank":2247,"depth":0,"x":2485.134,"y":220.0,"cluster":"homological-algebra"},{"id":"stacks:00ZZ","tag":"00ZZ","title":"Preadditive and additive categories · Lemma 00ZZ","summary":"Let A be a preadditive category. Let x be an object of A. The following are equivalent • x is an initial object, • x is a final object, and • id_x = 0 in Mor_A(x, x). Furthermore, if such an object 0 exists, then a morphism α : y → z factors through 0 if and only if α = 0.","statement_latex":"Let $\\mathcal{A}$ be a preadditive category.\nLet $x$ be an object of $\\mathcal{A}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $x$ is an initial object,\n\\item $x$ is a final object, and\n\\item $\\text{id}_x = 0$ in $\\Mor_\\mathcal{A}(x, x)$.\n\\end{enumerate}\nFurthermore, if such an object $0$ exists, then a morphism\n$\\alpha : y \\to z$ factors through $0$ if and only if $\\alpha = 0$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00ZZ","source_file":"homology.tex","source_line":69,"source_end_line":81,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L69-L81","statement_sha256":"181492b4552ea9f87324cee575714ca699acb33015c2e3f9d7f50716a3eafacf","origin":"The Stacks Project","memory_eligible":false,"source_rank":2248,"rank":2248,"depth":0,"x":2473.443,"y":225.045,"cluster":"homological-algebra"},{"id":"stacks:0100","tag":"0100","title":"Preadditive and additive categories · Definition 0100","summary":"In a preadditive category A, we call an object that is both final and initial (as in Lemma [Tag 00ZZ]) a zero object and denote it 0.","statement_latex":"In a preadditive category $\\mathcal{A}$, we call an object that is both\nfinal and initial (as in Lemma \\ref{lemma-preadditive-zero}) a\n{\\it zero object} and denote it $0$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0100","source_file":"homology.tex","source_line":99,"source_end_line":104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L99-L104","statement_sha256":"fb8a31214201a8c96b7983209bdf43726febb3d3050dcfaba5c3b27abaa34f2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2249,"rank":2249,"depth":1,"x":2481.004,"y":210.394,"cluster":"homological-algebra"},{"id":"stacks:0101","tag":"0101","title":"Preadditive and additive categories · Lemma 0101","summary":"Let A be a preadditive category. Let x, y ∈ Ob(A). If the product x × y exists, then so does the coproduct x amalg y. If the coproduct x amalg y exists, then so does the product x × y. In this case also x amalg y ≅ x × y.","statement_latex":"Let $\\mathcal{A}$ be a preadditive category.\nLet $x, y \\in \\Ob(\\mathcal{A})$.\nIf the product $x \\times y$ exists, then so does\nthe coproduct $x \\amalg y$.\nIf the coproduct $x \\amalg y$ exists, then so does\nthe product $x \\times y$. In this case\nalso $x \\amalg y \\cong x \\times y$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0101","source_file":"homology.tex","source_line":106,"source_end_line":115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L106-L115","statement_sha256":"ee543ff670f7a9f99e2dd4c8c6f973711fede8fd0fd793adeaf7a5d9af3ff361","origin":"The Stacks Project","memory_eligible":false,"source_rank":2250,"rank":2250,"depth":0,"x":2488.264,"y":229.055,"cluster":"homological-algebra"},{"id":"stacks:0102","tag":"0102","title":"Preadditive and additive categories · Definition 0102","summary":"Given a pair of objects x, y in a preadditive category A, the direct sum x ⊕ y of x and y is the direct product x × y endowed with the morphisms i, j, p, q as in Lemma [Tag 0101] above.","statement_latex":"Given a pair of objects $x, y$ in a preadditive category $\\mathcal{A}$,\nthe {\\it direct sum} $x \\oplus y$ of $x$ and $y$ is the direct\nproduct $x \\times y$ endowed with the morphisms\n$i, j, p, q$ as in Lemma \\ref{lemma-preadditive-direct-sum} above.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0102","source_file":"homology.tex","source_line":146,"source_end_line":152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L146-L152","statement_sha256":"a02e18e0a6ecb1ddc90185d0ea6b23d25cfd1c784e99d73c6670de96f633a204","origin":"The Stacks Project","memory_eligible":false,"source_rank":2251,"rank":2251,"depth":1,"x":2464.834,"y":217.747,"cluster":"homological-algebra"},{"id":"stacks:0105","tag":"0105","title":"Preadditive and additive categories · Lemma 0105","summary":"Let A, B be preadditive categories. Let F : A → B be an additive functor. Then F transforms direct sums to direct sums and zero to zero.","statement_latex":"Let $\\mathcal{A}$, $\\mathcal{B}$ be preadditive categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor.\nThen $F$ transforms direct sums to direct sums and zero to zero.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0105","source_file":"homology.tex","source_line":172,"source_end_line":177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L172-L177","statement_sha256":"0589379b7f3cbc46e4a26d60d8dd3c3ee012edc8aceee8120019ab18b81c76c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2252,"rank":2252,"depth":1,"x":2494.366,"y":212.323,"cluster":"homological-algebra"},{"id":"stacks:0104","tag":"0104","title":"Preadditive and additive categories · Definition 0104","summary":"A category A is called additive if it is preadditive and finite products exist, in other words it has a zero object and direct sums.","statement_latex":"A category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0104","source_file":"homology.tex","source_line":198,"source_end_line":203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L198-L203","statement_sha256":"7c29eea2f36719f191ff1d8b7cec77ea9a0cecd4b4796e603c4c4bd81a90a51c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2253,"rank":2253,"depth":0,"x":2475.195,"y":235.015,"cluster":"homological-algebra"},{"id":"stacks:0106","tag":"0106","title":"Preadditive and additive categories · Definition 0106","summary":"Let A be a preadditive category. Let f : x → y be a morphism. • A kernel of f is a morphism i : z → x such that (a) f ∘ i = 0 and (b) for any i' : z' → x such that f ∘ i' = 0 there exists a unique morphism g : z' → z such that i' = i ∘ g. • If the kernel of f exists, then we denote this Ker(f) → x. • A cokernel of f is a morphism p : y → z such that (a) p ∘ f = 0 and (b) for any p' : y → z' such that p' ∘ f = 0 there exists a unique morphism g : z → z' such that p' = g ∘…","statement_latex":"Let $\\mathcal{A}$ be a preadditive category.\nLet $f : x \\to y$ be a morphism.\n\\begin{enumerate}\n\\item A {\\it kernel} of $f$ is a morphism\n$i : z \\to x$ such that (a) $f \\circ i = 0$ and (b)\nfor any $i' : z' \\to x$ such that $f \\circ i' = 0$ there\nexists a unique morphism $g : z' \\to z$ such that\n$i' = i \\circ g$.\n\\item If the kernel of $f$ exists, then we denote\nthis $\\Ker(f) \\to x$.\n\\item A {\\it cokernel} of $f$ is a morphism\n$p : y \\to z$ such that (a) $p \\circ f = 0$ and (b)\nfor any $p' : y \\to z'$ such that $p' \\circ f = 0$ there\nexists a unique morphism $g : z \\to z'$ such that\n$p' = g \\circ p$.\n\\item If a cokernel of $f$ exists we denote this\n$y \\to \\Coker(f)$.\n\\item If a kernel of $f$ exists, then a {\\it coimage\nof $f$} is a cokernel for the morphism $\\Ker(f) \\to x$.\n\\item If a kernel and coimage exist then we denote this\n$x \\to \\Coim(f)$.\n\\item If a cokernel of $f$ exists, then the {\\it image of\n$f$} is a kernel of the morphism $y \\to \\Coker(f)$.\n\\item If a cokernel and image of $f$ exist then we denote\nthis $\\Im(f) \\to y$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0106","source_file":"homology.tex","source_line":209,"source_end_line":237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L209-L237","statement_sha256":"2169a06f33527f57f362c9325ab172780de8d3ca2c4c5bbb5a19cbe5ae10a824","origin":"The Stacks Project","memory_eligible":false,"source_rank":2254,"rank":2254,"depth":0,"x":2470.836,"y":205.178,"cluster":"homological-algebra"},{"id":"stacks:09QG","tag":"09QG","title":"Preadditive and additive categories · Lemma 09QG","summary":"Let C be a preadditive category. Let x ⊕ y with morphisms i, j, p, q as in Lemma [Tag 0101] be a direct sum in C. Then i : x → x ⊕ y is a kernel of q : x ⊕ y → y. Dually, p is a cokernel for j.","statement_latex":"Let $\\mathcal{C}$ be a preadditive category.\nLet $x \\oplus y$ with morphisms $i, j, p, q$ as in\nLemma \\ref{lemma-preadditive-direct-sum}\nbe a direct sum in $\\mathcal{C}$. Then $i : x \\to x \\oplus y$\nis a kernel of $q : x \\oplus y \\rightarrow y$. Dually, $p$ is\na cokernel for $j$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QG","source_file":"homology.tex","source_line":252,"source_end_line":260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L252-L260","statement_sha256":"7e089c7a3c4e04e7897e36e40b1ffe0fbefdfd12f431c7a86508d4b603df559c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2255,"rank":2255,"depth":1,"x":2499.883,"y":226.099,"cluster":"homological-algebra"},{"id":"stacks:0E43","tag":"0E43","title":"Preadditive and additive categories · Lemma 0E43","summary":"Let C be a preadditive category. Let f : x → y be a morphism in C. • If a kernel of f exists, then this kernel is a monomorphism. • If a cokernel of f exists, then this cokernel is an epimorphism. • If a kernel and coimage of f exist, then the coimage is an epimorphism. • If a cokernel and image of f exist, then the image is a monomorphism.","statement_latex":"Let $\\mathcal{C}$ be a preadditive category.\nLet $f : x \\to y$ be a morphism in $\\mathcal{C}$.\n\\begin{enumerate}\n\\item If a kernel of $f$ exists, then\nthis kernel is a monomorphism.\n\\item If a cokernel of $f$ exists, then\nthis cokernel is an epimorphism.\n\\item If a kernel and coimage of $f$ exist, then\nthe coimage is an epimorphism.\n\\item If a cokernel and image of $f$ exist, then\nthe image is a monomorphism.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E43","source_file":"homology.tex","source_line":274,"source_end_line":288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L274-L288","statement_sha256":"1ed66a467934086922281b546e90bfa5ccbc7d5fef06b8868262826698da975a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2256,"rank":2256,"depth":0,"x":2459.315,"y":227.172,"cluster":"homological-algebra"},{"id":"stacks:0107","tag":"0107","title":"Preadditive and additive categories · Lemma 0107","summary":"Let f : x → y be a morphism in a preadditive category such that the kernel, cokernel, image and coimage all exist. Then f can be factored uniquely as x → Coim(f) → Im(f) → y.","statement_latex":"Let $f : x \\to y$ be a morphism in a preadditive category\nsuch that the kernel, cokernel, image and coimage all exist.\nThen $f$ can be factored uniquely as\n$x \\to \\Coim(f) \\to \\Im(f) \\to y$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0107","source_file":"homology.tex","source_line":297,"source_end_line":303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L297-L303","statement_sha256":"9951fdc940dc2eb476d54b58b3bddaffd414694f8ad76141dfbbd7a0b1b5e217","origin":"The Stacks Project","memory_eligible":false,"source_rank":2257,"rank":2257,"depth":1,"x":2489.971,"y":202.101,"cluster":"homological-algebra"},{"id":"stacks:0H9P","tag":"0H9P","title":"Preadditive and additive categories · Lemma 0H9P","summary":"Let A be a preadditive category, x be an object of A, and f : x → x be idempotent. Write g = id _ x - f and recall by Remark [Tag 0H9N] that f ∘ g = g ∘ f = 0. • If f has kernel i : Ker (f) → x and p is the unique morphism such that g = i ∘ p, then p ∘ i = id _ Ker (f) and p is the cokernel of f. • If f has cokernel p : x → Coker (f) and i is the unique morphism such that g = i ∘ p, then p ∘ i = id _ Coker (f) and i is the kernel of f. • If g has kernel j : Ker (g) → x…","statement_latex":"Let $\\mathcal{A}$ be a preadditive category,\n$x$ be an object of $\\mathcal{A}$,\nand $f : x \\to x$ be idempotent.\nWrite $g = \\text{id} _ x - f$\nand recall by Remark \\ref{remark-idempotent-symmetry}\nthat $f \\circ g = g \\circ f = 0$.\n\\begin{enumerate}\n\\item If $f$ has kernel $i : \\Ker (f) \\to x$\nand $p$ is the unique morphism such that $g = i \\circ p$,\nthen $p \\circ i = \\text{id} _ {\\Ker (f)}$\nand $p$ is the cokernel of $f$.\n\\item If $f$ has cokernel $p : x \\to \\Coker (f)$\nand $i$ is the unique morphism such that $g = i \\circ p$,\nthen $p \\circ i = \\text{id} _ {\\Coker (f)}$\nand $i$ is the kernel of $f$.\n\\item If $g$ has kernel $j : \\Ker (g) \\to x$\nand $q$ is the unique morphism such that $f = j \\circ q$,\nthen $q \\circ j = \\text{id} _ {\\Ker (g)}$\nand $q$ is the cokernel of $g$.\n\\item If $g$ has cokernel $q : x \\to \\Coker (g)$\nand $j$ is the unique morphism such that $f = j \\circ q$,\nthen $q \\circ j = \\text{id} _ {\\Coker (g)}$\nand $j$ is the kernel of $g$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9P","source_file":"homology.tex","source_line":366,"source_end_line":392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L366-L392","statement_sha256":"819be83ad71afa01e818e1ef0f008ae46d28ea958d2045b321365b2e2cac5ceb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2258,"rank":2258,"depth":1,"x":2487.369,"y":239.733,"cluster":"homological-algebra"},{"id":"stacks:0H9Q","tag":"0H9Q","title":"Preadditive and additive categories · Lemma 0H9Q","summary":"Let A be a preadditive category, x be an object of A, and f : x → x be idempotent. Write g = id _ x - f. If each of f and g have at least one of a kernel or cokernel and in turn morphisms i , j , p , q are constructed as in the statement of Lemma [Tag 0H9P], then all four (co)kernels exist and x & ≃ Ker (f) ⊕ Ker (g) & ≃ Coker (f) ⊕ Ker (g) & ≃ Ker (f) ⊕ Coker (g) & ≃ Coker (f) ⊕ Coker (g) with the structure of each direct product given by i , j , p , q.","statement_latex":"Let $\\mathcal{A}$ be a preadditive category,\n$x$ be an object of $\\mathcal{A}$,\nand $f : x \\to x$ be idempotent.\nWrite $g = \\text{id} _ x - f$.\nIf each of $f$ and $g$ have at least one of a kernel or cokernel\nand in turn morphisms $i , j , p , q$ are constructed\nas in the statement of Lemma \\ref{lemma-idempotent-kernel-cokernel},\nthen all four (co)kernels exist and\n\\begin{align*}\nx\n& \\simeq \\Ker (f) \\oplus \\Ker (g) \\\\\n& \\simeq \\Coker (f) \\oplus \\Ker (g) \\\\\n& \\simeq \\Ker (f) \\oplus \\Coker (g) \\\\\n& \\simeq \\Coker (f) \\oplus \\Coker (g)\n\\end{align*}\nwith the structure of each direct product given by $i , j , p , q$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9Q","source_file":"homology.tex","source_line":421,"source_end_line":439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L421-L439","statement_sha256":"821b8a728a6cce410cc0b22bbb366b9c695bb35dc115506ab0b783a265b53d8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2259,"rank":2259,"depth":2,"x":2457.791,"y":209.189,"cluster":"homological-algebra"},{"id":"stacks:0H9R","tag":"0H9R","title":"Preadditive and additive categories · Lemma 0H9R","summary":"Let A be a preadditive category, x and y be objects of A, and j : y → x and q : x → y be morphisms satisfying q ∘ j = id _ y. Then • id _ x - j ∘ q has kernel j. • id _ x - j ∘ q has cokernel q.","statement_latex":"Let $\\mathcal{A}$ be a preadditive category,\n$x$ and $y$ be objects of $\\mathcal{A}$,\nand $j : y \\to x$ and $q : x \\to y$ be morphisms\nsatisfying $q \\circ j = \\text{id} _ y$.\nThen\n\\begin{enumerate}\n\\item $\\text{id} _ x - j \\circ q$ has kernel $j$.\n\\item $\\text{id} _ x - j \\circ q$ has cokernel $q$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9R","source_file":"homology.tex","source_line":449,"source_end_line":460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L449-L460","statement_sha256":"22ffab2c8e9aeeb8582495bd0543842266269d29f47bde3732a3107697198629","origin":"The Stacks Project","memory_eligible":false,"source_rank":2260,"rank":2260,"depth":0,"x":2506.054,"y":215.189,"cluster":"homological-algebra"},{"id":"stacks:0H9S","tag":"0H9S","title":"Preadditive and additive categories · Lemma 0H9S","summary":"Let A be a preadditive category, x and y be objects of A, and j : y → x and q : x → y be morphisms satisfying q ∘ j = id _ y. If j ∘ q has at least one of a kernel or cokernel, then both (co)kernels exist and x & ≃ Ker (j ∘ q) ⊕ y & ≃ Coker (j ∘ q) ⊕ y with the structure of either direct product including j , q, and the canonical (co)kernel morphism, and the remaining map uniquely determined.","statement_latex":"Let $\\mathcal{A}$ be a preadditive category,\n$x$ and $y$ be objects of $\\mathcal{A}$,\nand $j : y \\to x$ and $q : x \\to y$ be morphisms\nsatisfying $q \\circ j = \\text{id} _ y$.\nIf $j \\circ q$ has at least one of a kernel or cokernel,\nthen both (co)kernels exist and\n\\begin{align*}\nx\n& \\simeq \\Ker (j \\circ q) \\oplus y \\\\\n& \\simeq \\Coker (j \\circ q) \\oplus y\n\\end{align*}\nwith the structure of either direct product including $j , q$,\nand the canonical (co)kernel morphism,\nand the remaining map uniquely determined.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Preadditive and additive categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9S","source_file":"homology.tex","source_line":479,"source_end_line":495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L479-L495","statement_sha256":"ebddccf2bc7e60b96ddd49f0ce2362325823fb5c8988aeca11b2301ca222ea3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2261,"rank":2261,"depth":3,"x":2464.1,"y":238.998,"cluster":"homological-algebra"},{"id":"stacks:09SG","tag":"09SG","title":"Karoubian categories · Definition 09SG","summary":"Let C be a preadditive category. We say C is Karoubian if every idempotent endomorphism of an object of C has a kernel.","statement_latex":"Let $\\mathcal{C}$ be a preadditive category. We say $\\mathcal{C}$\nis {\\it Karoubian} if every idempotent endomorphism of an object\nof $\\mathcal{C}$ has a kernel.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Karoubian categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SG","source_file":"homology.tex","source_line":513,"source_end_line":518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L513-L518","statement_sha256":"4a2011ccc2644d96d5904e5f11b2b9d200d5b0244c8a2fcc2cb01ba01e7c4936","origin":"The Stacks Project","memory_eligible":false,"source_rank":2262,"rank":2262,"depth":0,"x":2476.327,"y":196.189,"cluster":"homological-algebra"},{"id":"stacks:09SH","tag":"09SH","title":"Karoubian categories · Lemma 09SH","summary":"Let C be a preadditive category. The following are equivalent • C is Karoubian, • every idempotent endomorphism of an object of C has a cokernel, and • given an idempotent endomorphism p : z → z of C there exists a direct sum decomposition z = x ⊕ y such that p corresponds to the projection onto y.","statement_latex":"Let $\\mathcal{C}$ be a preadditive category. The following\nare equivalent\n\\begin{enumerate}\n\\item $\\mathcal{C}$ is Karoubian,\n\\item every idempotent endomorphism of an object of $\\mathcal{C}$ has a\ncokernel, and\n\\item given an idempotent endomorphism $p : z \\to z$ of $\\mathcal{C}$\nthere exists a direct sum decomposition $z = x \\oplus y$ such\nthat $p$ corresponds to the projection onto $y$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Karoubian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SH","source_file":"homology.tex","source_line":525,"source_end_line":537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L525-L537","statement_sha256":"c839022e49384c095822d85bb94e6db009eaab68191ab42b4bd19bd54c923cd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2263,"rank":2263,"depth":3,"x":2502.55,"y":235.965,"cluster":"homological-algebra"},{"id":"stacks:05QV","tag":"05QV","title":"Karoubian categories · Lemma 05QV","summary":"Let D be a preadditive category. • If D has countable products and kernels of maps which have a right inverse, then D is Karoubian. • If D has countable coproducts and cokernels of maps which have a left inverse, then D is Karoubian.","statement_latex":"Let $\\mathcal{D}$ be a preadditive category.\n\\begin{enumerate}\n\\item If $\\mathcal{D}$ has countable products and kernels of maps which\nhave a right inverse, then $\\mathcal{D}$ is Karoubian.\n\\item If $\\mathcal{D}$ has countable coproducts and cokernels of\nmaps which have a left inverse, then $\\mathcal{D}$ is Karoubian.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Karoubian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QV","source_file":"homology.tex","source_line":544,"source_end_line":553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L544-L553","statement_sha256":"e187fdef9eaef9fa38f52d0a1f9f02006b16e3403209513b95cf1e52cb9bfdb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2264,"rank":2264,"depth":4,"x":2449.654,"y":221.054,"cluster":"homological-algebra"},{"id":"stacks:0109","tag":"0109","title":"Abelian categories · Definition 0109","summary":"A category A is abelian if it is additive, if all kernels and cokernels exist, and if the natural map Coim(f) → Im(f) is an isomorphism for all morphisms f of A.","statement_latex":"A category $\\mathcal{A}$ is {\\it abelian} if\nit is additive, if all kernels and cokernels exist,\nand if the natural map $\\Coim(f) \\to \\Im(f)$\nis an isomorphism for all morphisms $f$ of\n$\\mathcal{A}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0109","source_file":"homology.tex","source_line":607,"source_end_line":614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L607-L614","statement_sha256":"87c03c2a93b71bb25fb273eef0989c7dcb8d181fbec297dcf3969bd3ddc861ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":2265,"rank":2265,"depth":0,"x":2502.135,"y":201.497,"cluster":"homological-algebra"},{"id":"stacks:010A","tag":"010A","title":"Abelian categories · Lemma 010A","summary":"Let A be a preadditive category. The additions on sets of morphisms make A^opp into a preadditive category. Furthermore, A is additive if and only if A^opp is additive, and A is abelian if and only if A^opp is abelian.","statement_latex":"Let $\\mathcal{A}$ be a preadditive category.\nThe additions on sets of morphisms make\n$\\mathcal{A}^{opp}$ into a preadditive category.\nFurthermore, $\\mathcal{A}$ is additive if and only if $\\mathcal{A}^{opp}$\nis additive, and\n$\\mathcal{A}$ is abelian if and only if $\\mathcal{A}^{opp}$ is abelian.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010A","source_file":"homology.tex","source_line":616,"source_end_line":624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L616-L624","statement_sha256":"b4e59d8e51a22b2bf263957e722fdb77318ce7d913fc4c1f782ccf63691876ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":2266,"rank":2266,"depth":0,"x":2478.519,"y":246.902,"cluster":"homological-algebra"},{"id":"stacks:010B","tag":"010B","title":"Abelian categories · Definition 010B","summary":"Let f : x → y be a morphism in an abelian category. • We say f is injective if Ker(f) = 0. • We say f is surjective if Coker(f) = 0. If x → y is injective, then we say that x is a subobject of y and we use the notation x ⊂ y. If x → y is surjective, then we say that y is a quotient of x.","statement_latex":"Let $f : x \\to y$ be a morphism in an abelian category.\n\\begin{enumerate}\n\\item We say $f$ is {\\it injective} if $\\Ker(f) = 0$.\n\\item We say $f$ is {\\it surjective} if $\\Coker(f) = 0$.\n\\end{enumerate}\nIf $x \\to y$ is injective, then we say that $x$ is a {\\it subobject}\nof $y$ and we use the notation $x \\subset y$. If $x \\to y$ is\nsurjective, then we say that $y$ is a {\\it quotient} of $x$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010B","source_file":"homology.tex","source_line":640,"source_end_line":650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L640-L650","statement_sha256":"ba02ceb2b7f118805eb34ba23058983a5ff45c04ba8628fed4e30119dc2e069c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2267,"rank":2267,"depth":0,"x":2458.939,"y":198.8,"cluster":"homological-algebra"},{"id":"stacks:010C","tag":"010C","title":"Abelian categories · Lemma 010C","summary":"Let f : x → y be a morphism in an abelian category A. Then • f is injective if and only if f is a monomorphism, • f is surjective if and only if f is an epimorphism, and • f is an isomorphism if and only if f is injective and surjective.","statement_latex":"Let $f : x \\to y$ be a morphism in an abelian category $\\mathcal{A}$. Then\n\\begin{enumerate}\n\\item $f$ is injective if and only if $f$ is a monomorphism,\n\\item $f$ is surjective if and only if $f$ is an epimorphism, and\n\\item $f$ is an isomorphism if and only if $f$ is injective and surjective.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010C","source_file":"homology.tex","source_line":652,"source_end_line":660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L652-L660","statement_sha256":"a9d687e4662567caf35bd220b50944fe0088d050338639db9caa864cfc015ab8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2268,"rank":2268,"depth":1,"x":2513.364,"y":223.771,"cluster":"homological-algebra"},{"id":"stacks:010D","tag":"010D","title":"Abelian categories · Lemma 010D","summary":"Let A be an abelian category. All finite limits and finite colimits exist in A.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAll finite limits and finite colimits exist in $\\mathcal{A}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010D","source_file":"homology.tex","source_line":684,"source_end_line":688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L684-L688","statement_sha256":"1bc966eac64b6eb04be36e89c88434b6a712b267b37ea32f9cf5b60cc66aa2e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2269,"rank":2269,"depth":2,"x":2451.731,"y":236.522,"cluster":"homological-algebra"},{"id":"stacks:010E","tag":"010E","title":"Abelian categories · Definition 010E","summary":"Let A be an additive category. Consider a sequence of morphisms … → x → y → z → … or x_1 → x_2 → … → x_n in A. We say such a sequence is a complex if the composition of any two consecutive (drawn) arrows is zero. If A is abelian then we say a complex of the first type above is exact at y if Im(x → y) = Ker(y → z) and we say a complex of the second kind is exact at x_i where 1 < i < n if Im(x_i - 1 → x_i) = Ker(x_i → x_i + 1). We a sequence as above is exact or is an exact…","statement_latex":"Let $\\mathcal{A}$ be an additive category. Consider a sequence of morphisms\n$$\n\\ldots \\to x \\to y \\to z \\to \\ldots\n\\quad\\text{or}\\quad\nx_1 \\to x_2 \\to \\ldots \\to x_n\n$$\nin $\\mathcal{A}$. We say such a sequence is a {\\it complex} if the\ncomposition of any two consecutive (drawn) arrows is zero.\nIf $\\mathcal{A}$ is abelian then we say a complex of the first\ntype above is {\\it exact at $y$} if $\\Im(x \\to y) = \\Ker(y \\to z)$\nand we say a complex of the second kind is {\\it exact at $x_i$}\nwhere $1 < i < n$ if\n$\\Im(x_{i - 1} \\to x_i) = \\Ker(x_i \\to x_{i + 1})$. We a\nsequence as above is {\\it exact} or is an {\\it exact sequence} or is an\n{\\it exact complex} if it is a complex and exact at every object (in\nthe first case) or exact at $x_i$ for all $1 < i < n$ (in the second case).\nThere are variants of these notions for sequences of the form\n$$\n\\ldots \\to x_{-3} \\to x_{-2} \\to x_{-1}\n\\quad\\text{and}\\quad\nx_1 \\to x_2 \\to x_3 \\to \\ldots\n$$\nA {\\it short exact sequence} is an exact complex of the form\n$$\n0 \\to A  \\to B \\to C \\to 0.\n$$","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010E","source_file":"homology.tex","source_line":715,"source_end_line":743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L715-L743","statement_sha256":"1001e66cbf9609223251321461e8564af8ed8ea2ede21dfc8b0318b34441a561","origin":"The Stacks Project","memory_eligible":false,"source_rank":2270,"rank":2270,"depth":0,"x":2487.725,"y":191.157,"cluster":"homological-algebra"},{"id":"stacks:05AA","tag":"05AA","title":"Abelian categories · Lemma 05AA","summary":"Let A be an abelian category. Let 0 → M_1 → M_2 → M_3 → 0 be a complex of A. • M_1 → M_2 → M_3 → 0 is exact if and only if 0 → Hom_A(M_3, N) → Hom_A(M_2, N) → Hom_A(M_1, N) is an exact sequence of abelian groups for all objects N of A, and • 0 → M_1 → M_2 → M_3 is exact if and only if 0 → Hom_A(N, M_1) → Hom_A(N, M_2) → Hom_A(N, M_3) is an exact sequence of abelian groups for all objects N of A.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a complex of $\\mathcal{A}$.\n\\begin{enumerate}\n\\item $M_1 \\to M_2 \\to M_3 \\to 0$ is exact if and only if\n$$\n0 \\to \\Hom_\\mathcal{A}(M_3, N) \\to\n\\Hom_\\mathcal{A}(M_2, N) \\to \\Hom_\\mathcal{A}(M_1, N)\n$$\nis an exact sequence of abelian groups for all objects $N$ of\n$\\mathcal{A}$, and\n\\item $0 \\to M_1 \\to M_2 \\to M_3$ is exact if and only if\n$$\n0 \\to \\Hom_\\mathcal{A}(N, M_1) \\to \\Hom_\\mathcal{A}(N, M_2) \\to\n\\Hom_\\mathcal{A}(N, M_3)\n$$\nis an exact sequence of abelian groups for all objects $N$ of $\\mathcal{A}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AA","source_file":"homology.tex","source_line":749,"source_end_line":768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L749-L768","statement_sha256":"e8d7b0917375984ce8587c4b3cfe7281a15dedca183bda13ece1cb9570327b34","origin":"The Stacks Project","memory_eligible":false,"source_rank":2271,"rank":2271,"depth":1,"x":2497.867,"y":246.191,"cluster":"homological-algebra"},{"id":"stacks:010F","tag":"010F","title":"Abelian categories · Definition 010F","summary":"Let A be an abelian category. Let i : A → B and q : B → C be morphisms of A such that 0 → A → B → C → 0 is a short exact sequence. We say the short exact sequence is split if there exist morphisms j : C → B and p : B → A such that (B, i, j, p, q) is the direct sum of A and C.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $i : A \\to B$ and $q : B \\to C$ be morphisms\nof $\\mathcal{A}$ such that\n$0 \\to A \\to B \\to C \\to 0$ is a short\nexact sequence. We say the short exact\nsequence is {\\it split} if there exist\nmorphisms $j : C \\to B$ and $p : B \\to A$ such\nthat $(B, i, j, p, q)$ is the direct sum of $A$ and $C$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010F","source_file":"homology.tex","source_line":775,"source_end_line":785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L775-L785","statement_sha256":"bca60a1d9ee2a3980d9811514dba3a5840a827a389b37e5334554b1ab76416bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2272,"rank":2272,"depth":0,"x":2445.072,"y":210.64,"cluster":"homological-algebra"},{"id":"stacks:010G","tag":"010G","title":"Abelian categories · Lemma 010G","summary":"Let A be an abelian category. Let 0 → A → B → C → 0 be a short exact sequence. • Given a morphism s : C → B right inverse to B → C, there exists a unique π : B → A such that (s, π) splits the short exact sequence as in Definition [Tag 010F]. • Given a morphism π : B → A left inverse to A → B, there exists a unique s : C → B such that (s, π) splits the short exact sequence as in Definition [Tag 010F].","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $0 \\to A \\to B \\to C \\to 0$\nbe a short exact sequence.\n\\begin{enumerate}\n\\item Given a morphism $s : C \\to B$ right inverse to\n$B \\to C$, there exists a unique $\\pi : B \\to A$\nsuch that $(s, \\pi)$ splits the short exact sequence\nas in Definition \\ref{definition-ses-split}.\n\\item Given a morphism $\\pi : B \\to A$ left inverse to\n$A \\to B$, there exists a unique $s : C \\to B$\nsuch that $(s, \\pi)$ splits the short exact sequence\nas in Definition \\ref{definition-ses-split}.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010G","source_file":"homology.tex","source_line":787,"source_end_line":802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L787-L802","statement_sha256":"36092dbf1bfcb692596fd2b7d04bba9a2ae9164b9d53976f715f20a9faaf0d17","origin":"The Stacks Project","memory_eligible":false,"source_rank":2273,"rank":2273,"depth":4,"x":2513.928,"y":206.832,"cluster":"homological-algebra"},{"id":"stacks:08N2","tag":"08N2","title":"Abelian categories · Lemma 08N2","summary":"Let A be an abelian category. Let xymatrix war[r]^far[d]_g & yar[d]^h xar[r]^k & z be a commutative diagram. • The diagram is cartesian if and only if 0 → w xrightarrow(g, f) x ⊕ y xrightarrow(k, -h) z is exact. • The diagram is cocartesian if and only if w xrightarrow(g, -f) x ⊕ y xrightarrow(k, h) z → 0 is exact.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$$\n\\xymatrix{\nw\\ar[r]^f\\ar[d]_g\n& y\\ar[d]^h\\\\\nx\\ar[r]^k\n& z\n}\n$$\nbe a commutative diagram. \n\\begin{enumerate}\n\\item The diagram is cartesian if and only if \n$$\n0 \\to w \\xrightarrow{(g, f)} x \\oplus y \\xrightarrow{(k, -h)} z\n$$\nis exact.\n\\item The diagram is cocartesian if and only if \n$$\nw \\xrightarrow{(g, -f)} x \\oplus y \\xrightarrow{(k, h)} z \\to 0\n$$\nis exact.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08N2","source_file":"homology.tex","source_line":808,"source_end_line":832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L808-L832","statement_sha256":"f88a5acf725a92c41d4fd046f9e6459bb6a625285b3c40e1debeef58d347b326","origin":"The Stacks Project","memory_eligible":false,"source_rank":2274,"rank":2274,"depth":0,"x":2465.302,"y":249.502,"cluster":"homological-algebra"},{"id":"stacks:08N3","tag":"08N3","title":"Abelian categories · Lemma 08N3","summary":"Let A be an abelian category. Let xymatrix war[r]^far[d]_g & yar[d]^h xar[r]^k & z be a commutative diagram. • If the diagram is cartesian, then the morphism Ker(f)→Ker(k) induced by g is an isomorphism. • If the diagram is cocartesian, then the morphism Coker(f)→Coker(k) induced by h is an isomorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$$\n\\xymatrix{\nw\\ar[r]^f\\ar[d]_g\n& y\\ar[d]^h\\\\\nx\\ar[r]^k\n& z\n}\n$$\nbe a commutative diagram.\n\\begin{enumerate}\n\\item If the diagram is cartesian, then the morphism \n$\\Ker(f)\\to\\Ker(k)$ induced by $g$ is an isomorphism.\n\\item If the diagram is cocartesian, then the morphism \n$\\Coker(f)\\to\\Coker(k)$ induced by $h$ is an isomorphism.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08N3","source_file":"homology.tex","source_line":851,"source_end_line":869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L851-L869","statement_sha256":"28f7c3046a5e57b43f3e58a284ab4aba3c5e91b5ef126b074b1a9eae7da5eaff","origin":"The Stacks Project","memory_eligible":false,"source_rank":2275,"rank":2275,"depth":0,"x":2466.882,"y":189.364,"cluster":"homological-algebra"},{"id":"stacks:08N4","tag":"08N4","title":"Abelian categories · Lemma 08N4","summary":"Let A be an abelian category. Let xymatrix war[r]^far[d]_g & yar[d]^h xar[r]^k & z be a commutative diagram. • If the diagram is cartesian and k is an epimorphism, then the diagram is cocartesian and f is an epimorphism. • If the diagram is cocartesian and g is a monomorphism, then the diagram is cartesian and h is a monomorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$$\n\\xymatrix{\nw\\ar[r]^f\\ar[d]_g\n& y\\ar[d]^h\\\\\nx\\ar[r]^k\n& z\n}\n$$\nbe a commutative diagram.\n\\begin{enumerate}\n\\item If the diagram is cartesian and $k$ is an epimorphism, \nthen the diagram is cocartesian and $f$ is an epimorphism.\n\\item If the diagram is cocartesian and $g$ is a monomorphism, \nthen the diagram is cartesian and $h$ is a monomorphism.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08N4","source_file":"homology.tex","source_line":886,"source_end_line":904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L886-L904","statement_sha256":"40fa305ea72e85bd3080a8c97066cfb1accb02881ea2202d95e4df1209c98646","origin":"The Stacks Project","memory_eligible":false,"source_rank":2276,"rank":2276,"depth":2,"x":2514.906,"y":235.41,"cluster":"homological-algebra"},{"id":"stacks:05PK","tag":"05PK","title":"Abelian categories · Lemma 05PK","summary":"Let A be an abelian category. • If x → y is surjective, then for every z → y the projection x ×_y z → z is surjective. • If x → y is injective, then for every x → z the morphism z → z amalg_x y is injective.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item If $x \\to y$ is surjective, then for every $z \\to y$ the\nprojection $x \\times_y z \\to z$ is surjective.\n\\item If $x \\to y$ is injective, then for every $x \\to z$ the\nmorphism $z \\to z \\amalg_x y$ is injective.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PK","source_file":"homology.tex","source_line":918,"source_end_line":927,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L918-L927","statement_sha256":"bb67c15fcf80f055620b779671718721500796dc131c0e132bd9fd88c2026a10","origin":"The Stacks Project","memory_eligible":false,"source_rank":2277,"rank":2277,"depth":3,"x":2441.228,"y":228.585,"cluster":"homological-algebra"},{"id":"stacks:08N5","tag":"08N5","title":"Abelian categories · Lemma 08N5","summary":"Let A be an abelian category. Let f:x→ y and g:y→ z be morphisms with g∘ f=0. Then, the following statements are equivalent: • The sequence xoversetf→ yoversetg→ z is exact. • For every h:w→ y with g∘ h=0 there exist an object v, an epimorphism k:v→ w and a morphism l:v→ x with h∘ k=f∘ l.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $f:x\\to y$ and $g:y\\to z$ \nbe morphisms with $g\\circ f=0$. Then, the following statements are equivalent:\n\\begin{enumerate}\n\\item The sequence $x\\overset{f}\\to y\\overset{g}\\to z$ is exact.\n\\item For every $h:w\\to y$ with $g\\circ h=0$ there exist an object $v$, \nan epimorphism $k:v\\to w$ and a morphism $l:v\\to x$ with $h\\circ k=f\\circ l$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08N5","source_file":"homology.tex","source_line":934,"source_end_line":943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L934-L943","statement_sha256":"8129ce99babe890d0a6481f6ac72804fd04ef927aa5b820fd9133d9a9e2ad44a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2278,"rank":2278,"depth":3,"x":2502.038,"y":191.21,"cluster":"homological-algebra"},{"id":"stacks:08N6","tag":"08N6","title":"Abelian categories · Lemma 08N6","summary":"Let A be an abelian category. Let xymatrix x ar[r]^f ar[d]^α & y ar[r]^g ar[d]^β & z ar[d]^γ u ar[r]^k & v ar[r]^l & w be a commutative diagram. • If the first row is exact and k is a monomorphism, then the induced sequence Ker(α) → Ker(β) → Ker(γ) is exact. • If the second row is exact and g is an epimorphism, then the induced sequence Coker(α) → Coker(β) → Coker(γ) is exact.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$$\n\\xymatrix{\nx \\ar[r]^f \\ar[d]^\\alpha &\ny \\ar[r]^g \\ar[d]^\\beta &\nz \\ar[d]^\\gamma\\\\\nu \\ar[r]^k & v \\ar[r]^l & w\n}\n$$\nbe a commutative diagram.\n\\begin{enumerate}\n\\item If the first row is exact and $k$ is a monomorphism, then the induced \nsequence $\\Ker(\\alpha) \\to \\Ker(\\beta) \\to \\Ker(\\gamma)$ \nis exact.\n\\item If the second row is exact and $g$ is an epimorphism, then the induced \nsequence\n$\\Coker(\\alpha) \\to \\Coker(\\beta) \\to \\Coker(\\gamma)$ \nis exact.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08N6","source_file":"homology.tex","source_line":968,"source_end_line":989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L968-L989","statement_sha256":"13e4e28c3f1fc577a72d0fe17ffb6d16043522f4ccadcdc83f52245bd2c95723","origin":"The Stacks Project","memory_eligible":false,"source_rank":2279,"rank":2279,"depth":4,"x":2487.01,"y":254.265,"cluster":"homological-algebra"},{"id":"stacks:010H","tag":"010H","title":"Abelian categories · Lemma 010H","summary":"Let A be an abelian category. Let xymatrix & x ar[r]^f ar[d]^α & y ar[r]^g ar[d]^β & z ar[r] ar[d]^γ & 0 0 ar[r] & u ar[r]^k & v ar[r]^l & w be a commutative diagram with exact rows. • There exists a unique morphism δ : Ker(γ) → Coker(α) such that the diagram xymatrix y ar[d]_β & y ×_z Ker(γ) ar[l]_π' ar[r]^π & Ker(γ) ar[d]^δ v ar[r]^iota' & Coker(α) amalg_u v & Coker(α) ar[l]_iota commutes, where π and π' are the canonical projections and iota and iota' are the canonical…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$$\n\\xymatrix{\n& x \\ar[r]^f \\ar[d]^\\alpha &\ny \\ar[r]^g \\ar[d]^\\beta &\nz \\ar[r] \\ar[d]^\\gamma &\n0 \\\\\n0 \\ar[r] & u \\ar[r]^k & v \\ar[r]^l & w\n}\n$$\nbe a commutative diagram with exact rows.\n\\begin{enumerate}\n\\item There exists a unique morphism\n$\\delta : \\Ker(\\gamma) \\to \\Coker(\\alpha)$\nsuch that the diagram\n$$\n\\xymatrix{\ny \\ar[d]_\\beta &\ny \\times_z \\Ker(\\gamma) \\ar[l]_{\\pi'} \\ar[r]^{\\pi} &\n\\Ker(\\gamma) \\ar[d]^\\delta \\\\\nv \\ar[r]^{\\iota'} & \\Coker(\\alpha) \\amalg_u v &\n\\Coker(\\alpha) \\ar[l]_\\iota\n}\n$$\ncommutes, where $\\pi$ and $\\pi'$ are the canonical projections\nand $\\iota$ and $\\iota'$ are the canonical coprojections.\n\\item The induced sequence\n$$\n\\Ker(\\alpha) \\xrightarrow{f'} \\Ker(\\beta) \\xrightarrow{g'}\n\\Ker(\\gamma) \\xrightarrow{\\delta} \\Coker(\\alpha) \\xrightarrow{k'}\n\\Coker(\\beta) \\xrightarrow{l'} \\Coker(\\gamma)\n$$\nis exact. If $f$ is injective then so is $f'$, and if $l$ is\nsurjective then so is $l'$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010H","source_file":"homology.tex","source_line":1011,"source_end_line":1048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1011-L1048","statement_sha256":"5b6d26e328c54657f4b41219665071544daa5a2c033bbd5b04ceb7c3c40be044","origin":"The Stacks Project","memory_eligible":false,"source_rank":2280,"rank":2280,"depth":5,"x":2446.777,"y":198.387,"cluster":"homological-algebra"},{"id":"stacks:08N7","tag":"08N7","title":"Abelian categories · Lemma 08N7","summary":"Let A be an abelian category. Let xymatrix & & & xar[ld]ar[rr]ar[dd]^(.4)α & & yar[ld]ar[rr]ar[dd]^(.4)β & & zar[ld]ar[rr]ar[dd]^(.4)γ & & 0 & & x'ar[rr]ar[dd]^(.4)α' & & y'ar[rr]ar[dd]^(.4)β' & & z'ar[rr]ar[dd]^(.4)γ' & & 0 & & 0ar[rr] & & uar[ld]ar[rr] & & var[ld]ar[rr] & & war[ld] & & 0ar[rr] & & u'ar[rr] & & v'ar[rr] & & w' & & & be a commutative diagram with exact rows. Then, the induced diagram xymatrix@C=15pt Ker(α) ar[r] ar[d] & Ker(β) ar[r] ar[d] & Ker(γ)…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let \n$$\n\\xymatrix{\n& & & x\\ar[ld]\\ar[rr]\\ar[dd]^(.4)\\alpha\n& & y\\ar[ld]\\ar[rr]\\ar[dd]^(.4)\\beta\n& & z\\ar[ld]\\ar[rr]\\ar[dd]^(.4)\\gamma\n& & 0\\\\\n& & x'\\ar[rr]\\ar[dd]^(.4){\\alpha'}\n& & y'\\ar[rr]\\ar[dd]^(.4){\\beta'}\n& & z'\\ar[rr]\\ar[dd]^(.4){\\gamma'}\n& & 0\n& \\\\\n& 0\\ar[rr]\n& & u\\ar[ld]\\ar[rr]\n& & v\\ar[ld]\\ar[rr]\n& & w\\ar[ld]\n& & \\\\\n0\\ar[rr]\n& & u'\\ar[rr]\n& & v'\\ar[rr]\n& & w'\n& & &\n}\n$$\nbe a commutative diagram with exact rows. Then, the induced diagram\n$$\n\\xymatrix@C=15pt{\n\\Ker(\\alpha) \\ar[r] \\ar[d] &\n\\Ker(\\beta) \\ar[r] \\ar[d] &\n\\Ker(\\gamma) \\ar[r]^(.45){\\delta} \\ar[d] &\n\\Coker(\\alpha) \\ar[r] \\ar[d] &\n\\Coker(\\beta) \\ar[r] \\ar[d] &\n\\Coker(\\gamma) \\ar[d] \\\\\n\\Ker(\\alpha') \\ar[r] &\n\\Ker(\\beta') \\ar[r] &\n\\Ker(\\gamma') \\ar[r]^(.45){\\delta'} &\n\\Coker(\\alpha') \\ar[r] &\n\\Coker(\\beta') \\ar[r] &\n\\Coker(\\gamma')\n}\n$$\ncommutes.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08N7","source_file":"homology.tex","source_line":1129,"source_end_line":1173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1129-L1173","statement_sha256":"82da081353ac2e5523ba230ceb7d6bbdcc1bde5a103420c1f5f7e66e19f6d7b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2281,"rank":2281,"depth":0,"x":2522.499,"y":217.042,"cluster":"homological-algebra"},{"id":"stacks:05QA","tag":"05QA","title":"Abelian categories · Lemma 05QA","summary":"Let A be an abelian category. Let xymatrix w ar[r] ar[d]^α & x ar[r] ar[d]^β & y ar[r] ar[d]^γ & z ar[d]^δ w' ar[r] & x' ar[r] & y' ar[r] & z' be a commutative diagram with exact rows. • If α, γ are surjective and δ is injective, then β is surjective. • If β, δ are injective and α is surjective, then γ is injective.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$$\n\\xymatrix{\nw \\ar[r] \\ar[d]^\\alpha & x \\ar[r] \\ar[d]^\\beta & y \\ar[r] \\ar[d]^\\gamma &\nz \\ar[d]^\\delta \\\\\nw' \\ar[r] & x' \\ar[r] & y' \\ar[r] & z'\n}\n$$\nbe a commutative diagram with exact rows.\n\\begin{enumerate}\n\\item If $\\alpha, \\gamma$ are surjective and $\\delta$ is injective, then\n$\\beta$ is surjective.\n\\item If $\\beta, \\delta$ are injective and $\\alpha$ is surjective, then\n$\\gamma$ is injective.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QA","source_file":"homology.tex","source_line":1179,"source_end_line":1196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1179-L1196","statement_sha256":"508618b288f0ffe8d6121d70e4536df63047e412a22e68193401ce48c7351716","origin":"The Stacks Project","memory_eligible":false,"source_rank":2282,"rank":2282,"depth":6,"x":2450.625,"y":246.673,"cluster":"homological-algebra"},{"id":"stacks:05QB","tag":"05QB","title":"Abelian categories · Lemma 05QB","summary":"[Eilenberg-Steenrod] Let A be an abelian category. Let xymatrix v ar[r] ar[d]^α & w ar[r] ar[d]^β & x ar[r] ar[d]^γ & y ar[r] ar[d]^δ & z ar[d]^ε v' ar[r] & w' ar[r] & x' ar[r] & y' ar[r] & z' be a commutative diagram with exact rows. If β, δ are isomorphisms, ε is injective, and α is surjective then γ is an isomorphism.","statement_latex":"\\begin{reference}\n\\cite[Lemma 4.5 page 16]{Eilenberg-Steenrod}\n\\end{reference}\nLet $\\mathcal{A}$ be an abelian category. Let\n$$\n\\xymatrix{\nv \\ar[r] \\ar[d]^\\alpha &\nw \\ar[r] \\ar[d]^\\beta &\nx \\ar[r] \\ar[d]^\\gamma &\ny \\ar[r] \\ar[d]^\\delta &\nz \\ar[d]^\\epsilon \\\\\nv' \\ar[r] & w' \\ar[r] & x' \\ar[r] & y' \\ar[r] & z'\n}\n$$\nbe a commutative diagram with exact rows. If $\\beta, \\delta$\nare isomorphisms, $\\epsilon$ is injective, and $\\alpha$ is surjective\nthen $\\gamma$ is an isomorphism.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QB","source_file":"homology.tex","source_line":1226,"source_end_line":1245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1226-L1245","statement_sha256":"8322f5167c537a992f4f0136a4103c033166b26ea14641e869c1a33d4cb0b700","origin":"The Stacks Project","memory_eligible":false,"source_rank":2283,"rank":2283,"depth":7,"x":2480.214,"y":183.156,"cluster":"homological-algebra"},{"id":"stacks:010J","tag":"010J","title":"Extensions · Definition 010J","summary":"Let A be an abelian category. Let A, B ∈ Ob(A). An extension E of B by A is a short exact sequence 0 → A → E → B → 0. A morphism of extensions between two extensions 0 → A → E → B → 0 and 0 → A → F → B → 0 means a morphism f : E → F in A making the diagram xymatrix 0 ar[r] & A ar[r] ar[d]^id & E ar[r] ar[d]^f & B ar[r] ar[d]^id & 0 0 ar[r] & A ar[r] & F ar[r] & B ar[r] & 0 commutative. Thus, the extensions of B by A form a category.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A, B \\in \\Ob(\\mathcal{A})$.\nAn {\\it extension $E$ of $B$ by $A$} is a short\nexact sequence\n$$\n0 \\to A \\to E \\to B \\to 0.\n$$\nA {\\it morphism of extensions} between two\nextensions $0 \\to A \\to E \\to B \\to 0$ and\n$0 \\to A \\to F \\to B \\to 0$ means a morphism\n$f : E \\to F$ in $\\mathcal{A}$ making the diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\nA \\ar[r] \\ar[d]^{\\text{id}} &\nE \\ar[r] \\ar[d]^f &\nB \\ar[r] \\ar[d]^{\\text{id}} &\n0 \\\\\n0 \\ar[r] &\nA \\ar[r] &\nF \\ar[r] &\nB \\ar[r] &\n0\n}\n$$\ncommutative.\nThus, the extensions of $B$ by $A$ form a category.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010J","source_file":"homology.tex","source_line":1262,"source_end_line":1291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1262-L1291","statement_sha256":"37a417aa456718ae8c730fd2a263232c26e1bc7e46bb9d62ba2b8f8f74cfc94d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2284,"rank":2284,"depth":0,"x":2509.872,"y":247.661,"cluster":"homological-algebra"},{"id":"stacks:010K","tag":"010K","title":"Extensions · Definition 010K","summary":"Let A be an abelian category. Let A, B ∈ Ob(A). The set of isomorphism classes of extensions of B by A is denoted Ext_A(B, A). This is called the Ext-group.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A, B \\in \\Ob(\\mathcal{A})$.\nThe set of isomorphism classes of extensions\nof $B$ by $A$ is denoted\n$$\n\\Ext_\\mathcal{A}(B, A).\n$$\nThis is called the {\\it $\\Ext$-group}.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010K","source_file":"homology.tex","source_line":1298,"source_end_line":1308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1298-L1308","statement_sha256":"899cbf9de73420c2adbedc5a3e33eaa693783d91e9d10c3cbc471b1ca0463cd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2285,"rank":2285,"depth":0,"x":2435.144,"y":216.508,"cluster":"homological-algebra"},{"id":"stacks:010L","tag":"010L","title":"Extensions · Lemma 010L","summary":"The construction (E_1, E_2) ↦ E_1 + E_2 above defines a commutative group law on Ext_A(B, A) which is functorial in both variables.","statement_latex":"The construction $(E_1, E_2) \\mapsto E_1 + E_2$\nabove defines a commutative group\nlaw on $\\Ext_\\mathcal{A}(B, A)$ which is\nfunctorial in both variables.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010L","source_file":"homology.tex","source_line":1405,"source_end_line":1411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1405-L1411","statement_sha256":"04c3dabde99a56cf4e1d648426d0d9d941ff590b0d77443bb604b91178a41858","origin":"The Stacks Project","memory_eligible":false,"source_rank":2286,"rank":2286,"depth":0,"x":2516.347,"y":196.828,"cluster":"homological-algebra"},{"id":"stacks:05E2","tag":"05E2","title":"Extensions · Lemma 05E2","summary":"Let A be an abelian category. Let 0 → M_1 → M_2 → M_3 → 0 be a short exact sequence in A. • There is a canonical six term exact sequence of abelian groups xymatrix 0 ar[r] & Hom_A(M_3, N) ar[r] & Hom_A(M_2, N) ar[r] & Hom_A(M_1, N) ar[lld] & Ext_A(M_3, N) ar[r] & Ext_A(M_2, N) ar[r] & Ext_A(M_1, N) for all objects N of A, and • there is a canonical six term exact sequence of abelian groups xymatrix 0 ar[r] & Hom_A(N, M_1) ar[r] & Hom_A(N, M_2) ar[r] & Hom_A(N, M_3)…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $0 \\to M_1 \\to M_2 \\to M_3 \\to 0$ be a short exact sequence\nin $\\mathcal{A}$.\n\\begin{enumerate}\n\\item There is a canonical six term exact sequence of abelian groups\n$$\n\\xymatrix{\n0 \\ar[r] &\n\\Hom_\\mathcal{A}(M_3, N) \\ar[r] &\n\\Hom_\\mathcal{A}(M_2, N) \\ar[r] &\n\\Hom_\\mathcal{A}(M_1, N) \\ar[lld] \\\\\n& \\Ext_\\mathcal{A}(M_3, N) \\ar[r] &\n\\Ext_\\mathcal{A}(M_2, N) \\ar[r] &\n\\Ext_\\mathcal{A}(M_1, N)\n}\n$$\nfor all objects $N$ of $\\mathcal{A}$, and\n\\item there is a canonical six term exact sequence of abelian groups\n$$\n\\xymatrix{\n0 \\ar[r] &\n\\Hom_\\mathcal{A}(N, M_1) \\ar[r] &\n\\Hom_\\mathcal{A}(N, M_2) \\ar[r] &\n\\Hom_\\mathcal{A}(N, M_3) \\ar[lld] \\\\\n& \\Ext_\\mathcal{A}(N, M_1) \\ar[r] &\n\\Ext_\\mathcal{A}(N, M_2) \\ar[r] &\n\\Ext_\\mathcal{A}(N, M_3)\n}\n$$\nfor all objects $N$ of $\\mathcal{A}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05E2","source_file":"homology.tex","source_line":1417,"source_end_line":1450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1417-L1450","statement_sha256":"118c019be88fedc7b030b7d8883488415f36ac4401b7c8c47866fe458470b39a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2287,"rank":2287,"depth":0,"x":2471.73,"y":258.184,"cluster":"homological-algebra"},{"id":"stacks:0DLP","tag":"0DLP","title":"Additive functors · Lemma 0DLP","summary":"Let A and B be additive categories. Let F : A → B be a functor. The following are equivalent • F is additive, • F(A) ⊕ F(B) → F(A ⊕ B) is an isomorphism for all A, B ∈ A, and • F(A ⊕ B) → F(A) ⊕ F(B) is an isomorphism for all A, B ∈ A.","statement_latex":"Let $\\mathcal{A}$ and $\\mathcal{B}$ be additive categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\nThe following are equivalent\n\\begin{enumerate}\n\\item $F$ is additive,\n\\item $F(A) \\oplus F(B) \\to F(A \\oplus B)$ is an isomorphism for\nall $A, B \\in \\mathcal{A}$, and\n\\item $F(A \\oplus B) \\to F(A) \\oplus F(B)$  is an isomorphism for\nall $A, B \\in \\mathcal{A}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Additive functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLP","source_file":"homology.tex","source_line":1469,"source_end_line":1481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1469-L1481","statement_sha256":"6b5dc4f167646b963baa01d1536c0630c0ac6b91a6628a1e3158bf13e5617d0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2288,"rank":2288,"depth":2,"x":2455.09,"y":186.749,"cluster":"homological-algebra"},{"id":"stacks:010N","tag":"010N","title":"Additive functors · Lemma 010N","summary":"Let A and B be abelian categories. Let F : A → B be a functor. • If F is either left or right exact, then it is additive. • F is left exact if and only if for every short exact sequence 0 → A → B → C → 0 the sequence 0 → F(A) → F(B) → F(C) is exact. • F is right exact if and only if for every short exact sequence 0 → A → B → C → 0 the sequence F(A) → F(B) → F(C) → 0 is exact. • F is exact if and only if for every short exact sequence 0 → A → B → C → 0 the sequence 0 →…","statement_latex":"Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\n\\begin{enumerate}\n\\item If $F$ is either left or right exact, then it is additive.\n\\item $F$ is left exact if and only if\nfor every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $0 \\to F(A) \\to F(B) \\to F(C)$\nis exact.\n\\item $F$ is right exact if and only if for every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $F(A) \\to F(B) \\to F(C) \\to 0$\nis exact.\n\\item $F$ is exact if and only if for every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $0 \\to F(A) \\to F(B) \\to F(C) \\to 0$\nis exact.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Additive functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010N","source_file":"homology.tex","source_line":1544,"source_end_line":1564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1544-L1564","statement_sha256":"7931d7bdbec1bb923c5575979dec2913769abc84af8e56388cd369c1611f9300","origin":"The Stacks Project","memory_eligible":false,"source_rank":2289,"rank":2289,"depth":6,"x":2525.648,"y":230.509,"cluster":"homological-algebra"},{"id":"stacks:010O","tag":"010O","title":"Additive functors · Lemma 010O","summary":"Let A and B be abelian categories. Let F : A → B be an exact functor. For every pair of objects A, B of A the functor F induces an abelian group homomorphism Ext_A(B, A) → Ext_B(F(B), F(A)) which maps the extension E to F(E).","statement_latex":"Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor.\nFor every pair of objects $A, B$ of $\\mathcal{A}$ the\nfunctor $F$ induces an abelian group homomorphism\n$$\n\\Ext_\\mathcal{A}(B, A)\n\\longrightarrow\n\\Ext_\\mathcal{B}(F(B), F(A))\n$$\nwhich maps the extension $E$ to $F(E)$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Additive functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010O","source_file":"homology.tex","source_line":1639,"source_end_line":1651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1639-L1651","statement_sha256":"27088e38ec1e67638704ea596ef1fcec35f64e478790cbdd8f8135880da466e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2290,"rank":2290,"depth":0,"x":2437.398,"y":238.365,"cluster":"homological-algebra"},{"id":"stacks:03A3","tag":"03A3","title":"Additive functors · Lemma 03A3","summary":"Let a : A → B and b : B → A be functors. Assume that • A, B are additive categories, a, b are additive functors, and a is right adjoint to b, • B is abelian and b is left exact, and • ba ≅ id_A. Then A is abelian.","statement_latex":"Let $a : \\mathcal{A} \\to \\mathcal{B}$ and $b : \\mathcal{B} \\to \\mathcal{A}$\nbe functors. Assume that\n\\begin{enumerate}\n\\item $\\mathcal{A}$, $\\mathcal{B}$ are additive categories,\n$a$, $b$ are additive functors, and $a$ is right adjoint to $b$,\n\\item $\\mathcal{B}$ is abelian and $b$ is left exact, and\n\\item $ba \\cong \\text{id}_\\mathcal{A}$.\n\\end{enumerate}\nThen $\\mathcal{A}$ is abelian.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Additive functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03A3","source_file":"homology.tex","source_line":1661,"source_end_line":1672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1661-L1672","statement_sha256":"3049129958f69a627ec3464e308a3c8775f9f2ddf47b3b9d9ccdca88ac9ea5de","origin":"The Stacks Project","memory_eligible":false,"source_rank":2291,"rank":2291,"depth":3,"x":2496.836,"y":181.854,"cluster":"homological-algebra"},{"id":"stacks:05QD","tag":"05QD","title":"Localization · Lemma 05QD","summary":"Let C be a preadditive category. Let S be a left or right multiplicative system. There exists a unique preadditive structure on S^-1C such that the localization functor Q : C → S^-1C is additive.","statement_latex":"Let $\\mathcal{C}$ be a preadditive category.\nLet $S$ be a left or right multiplicative system.\nThere exists a unique preadditive structure on\n$S^{-1}\\mathcal{C}$ such that the localization functor\n$Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ is additive.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QD","source_file":"homology.tex","source_line":1753,"source_end_line":1760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1753-L1760","statement_sha256":"59dfbc92d8a756fbc096f42516fc07de58725c393e64106078c81cdd5acd0de0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2292,"rank":2292,"depth":2,"x":2498.465,"y":258.101,"cluster":"homological-algebra"},{"id":"stacks:05QE","tag":"05QE","title":"Localization · Lemma 05QE","summary":"Let C be an additive category. Let S be a left or right multiplicative system. Then S^-1C is an additive category and the localization functor Q : C → S^-1C is additive.","statement_latex":"Let $\\mathcal{C}$ be an additive category.\nLet $S$ be a left or right multiplicative system.\nThen $S^{-1}\\mathcal{C}$ is an additive category and the localization functor\n$Q : \\mathcal{C} \\to S^{-1}\\mathcal{C}$ is additive.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QE","source_file":"homology.tex","source_line":1865,"source_end_line":1871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1865-L1871","statement_sha256":"b0f27cdbcebceba6ced78db4557f6e59091e0bf2678c7cca43b70e3cbc987684","origin":"The Stacks Project","memory_eligible":false,"source_rank":2293,"rank":2293,"depth":3,"x":2435.262,"y":202.19,"cluster":"homological-algebra"},{"id":"stacks:05QF","tag":"05QF","title":"Localization · Lemma 05QF","summary":"Let C be an additive category. Let S be a multiplicative system. Let X be an object of C. The following are equivalent • Q(X) = 0 in S^-1C, • there exists Y ∈ Ob(C) such that 0 : X → Y is an element of S, and • there exists Z ∈ Ob(C) such that 0 : Z → X is an element of S.","statement_latex":"Let $\\mathcal{C}$ be an additive category. Let $S$ be a multiplicative\nsystem. Let $X$ be an object\nof $\\mathcal{C}$. The following are equivalent\n\\begin{enumerate}\n\\item $Q(X) = 0$ in $S^{-1}\\mathcal{C}$,\n\\item there exists $Y \\in \\Ob(\\mathcal{C})$ such that\n$0 : X \\to Y$ is an element of $S$, and\n\\item there exists $Z \\in \\Ob(\\mathcal{C})$ such that\n$0 : Z \\to X$ is an element of $S$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QF","source_file":"homology.tex","source_line":1884,"source_end_line":1896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1884-L1896","statement_sha256":"f83ce2c475fd445ff06ec1f955295de810bee4012aa2acfdd78cd46eb52391c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2294,"rank":2294,"depth":4,"x":2527.817,"y":207.616,"cluster":"homological-algebra"},{"id":"stacks:05QG","tag":"05QG","title":"Localization · Lemma 05QG","summary":"Let A be an abelian category. • If S is a left multiplicative system, then the category S^-1A has cokernels and the functor Q : A → S^-1A commutes with them. • If S is a right multiplicative system, then the category S^-1A has kernels and the functor Q : A → S^-1A commutes with them. • If S is a multiplicative system, then the category S^-1A is abelian and the functor Q : A → S^-1A is exact.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item If $S$ is a left multiplicative system, then\nthe category $S^{-1}\\mathcal{A}$ has cokernels and the functor\n$Q : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ commutes with them.\n\\item If $S$ is a right multiplicative system, then\nthe category $S^{-1}\\mathcal{A}$ has kernels and the functor\n$Q : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ commutes with them.\n\\item If $S$ is a multiplicative system, then the category\n$S^{-1}\\mathcal{A}$ is abelian and the functor\n$Q : \\mathcal{A} \\to S^{-1}\\mathcal{A}$ is exact.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QG","source_file":"homology.tex","source_line":1910,"source_end_line":1924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1910-L1924","statement_sha256":"edd0ecc9f219123f4e06ad44c1446796ec0e22e0a99ca34f0d9e2ef176a4ba54","origin":"The Stacks Project","memory_eligible":false,"source_rank":2295,"rank":2295,"depth":2,"x":2454.435,"y":256.643,"cluster":"homological-algebra"},{"id":"stacks:0FCE","tag":"0FCE","title":"Jordan-Hölder · Definition 0FCE","summary":"Let A be an abelian category. An object A of A is said to be simple if it is nonzero and the only subobjects of A are 0 and A.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. An object $A$ of $\\mathcal{A}$\nis said to be {\\it simple} if it is nonzero and the only subobjects\nof $A$ are $0$ and $A$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Jordan-Hölder","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCE","source_file":"homology.tex","source_line":1960,"source_end_line":1965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1960-L1965","statement_sha256":"02bb860c62550232d6f5e6ca129a5162c5c3a59b20de46775d0cf29a71e59bd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2296,"rank":2296,"depth":0,"x":2469.276,"y":178.049,"cluster":"homological-algebra"},{"id":"stacks:0FCF","tag":"0FCF","title":"Jordan-Hölder · Definition 0FCF","summary":"Let A be an abelian category. • We say an object A of A is Artinian if and only if it satisfies the descending chain condition for subobjects. • We say A is Artinian if every object of A is Artinian.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item We say an object $A$ of $\\mathcal{A}$ is {\\it Artinian} if and only if\nit satisfies the descending chain condition for subobjects.\n\\item We say $\\mathcal{A}$ is {\\it Artinian} if every object of\n$\\mathcal{A}$ is Artinian.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Jordan-Hölder","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCF","source_file":"homology.tex","source_line":1967,"source_end_line":1976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1967-L1976","statement_sha256":"c3ffea441156228c6694c4092a35fee460fd2c2ad052d60fbaa16ca7ffc5ffbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2297,"rank":2297,"depth":0,"x":2522.062,"y":245.098,"cluster":"homological-algebra"},{"id":"stacks:0FCG","tag":"0FCG","title":"Jordan-Hölder · Definition 0FCG","summary":"Let A be an abelian category. • We say an object A of A is Noetherian if and only if it satisfies the ascending chain condition for subobjects. • We say A is Noetherian if every object of A is Noetherian.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item We say an object $A$ of $\\mathcal{A}$ is {\\it Noetherian} if and only if\nit satisfies the ascending chain condition for subobjects.\n\\item We say $\\mathcal{A}$ is {\\it Noetherian} if every object of\n$\\mathcal{A}$ is Noetherian.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Jordan-Hölder","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCG","source_file":"homology.tex","source_line":1978,"source_end_line":1987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1978-L1987","statement_sha256":"1e77afffc611616700b8e301a143ab8205acd9e2d66cc5aca1464864bbb5384e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2298,"rank":2298,"depth":0,"x":2428.298,"y":225.413,"cluster":"homological-algebra"},{"id":"stacks:0FCH","tag":"0FCH","title":"Jordan-Hölder · Lemma 0FCH","summary":"Let A be an abelian category. Let 0 → A_1 → A_2 → A_3 → 0 be a short exact sequence of A. Then A_2 is Artinian if and only if A_1 and A_3 are Artinian.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $0 \\to A_1 \\to A_2 \\to A_3 \\to 0$\nbe a short exact sequence of $\\mathcal{A}$. Then $A_2$ is Artinian\nif and only if $A_1$ and $A_3$ are Artinian.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Jordan-Hölder","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCH","source_file":"homology.tex","source_line":1989,"source_end_line":1994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L1989-L1994","statement_sha256":"81bd2b5923dc948fb39885134df84a0931d0cc944a51498ca9703bf8922e6d66","origin":"The Stacks Project","memory_eligible":false,"source_rank":2299,"rank":2299,"depth":0,"x":2514.097,"y":186.351,"cluster":"homological-algebra"},{"id":"stacks:0FCI","tag":"0FCI","title":"Jordan-Hölder · Lemma 0FCI","summary":"Let A be an abelian category. Let 0 → A_1 → A_2 → A_3 → 0 be a short exact sequence of A. Then A_2 is Noetherian if and only if A_1 and A_3 are Noetherian.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $0 \\to A_1 \\to A_2 \\to A_3 \\to 0$\nbe a short exact sequence of $\\mathcal{A}$. Then $A_2$ is Noetherian\nif and only if $A_1$ and $A_3$ are Noetherian.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Jordan-Hölder","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCI","source_file":"homology.tex","source_line":2000,"source_end_line":2005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2000-L2005","statement_sha256":"90bd21549bff6b339a5d9df63edb066fac61b6bec6a5becfad5c89709c196cca","origin":"The Stacks Project","memory_eligible":false,"source_rank":2300,"rank":2300,"depth":0,"x":2481.935,"y":264.578,"cluster":"homological-algebra"},{"id":"stacks:0FCJ","tag":"0FCJ","title":"Jordan-Hölder · Lemma 0FCJ","summary":"Let A be an abelian category. Let A be an object of A. The following are equivalent • A is Artinian and Noetherian, and • there exists a filtration 0 ⊂ A_1 ⊂ A_2 ⊂ … ⊂ A_n = A by subobjects such that A_i/A_i - 1 is simple for i = 1, …, n.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object\nof $\\mathcal{A}$. The following are equivalent\n\\begin{enumerate}\n\\item $A$ is Artinian and Noetherian, and\n\\item there exists a filtration\n$0 \\subset A_1 \\subset A_2 \\subset \\ldots \\subset A_n = A$\nby subobjects such that $A_i/A_{i - 1}$ is simple for $i = 1, \\ldots, n$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Jordan-Hölder","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCJ","source_file":"homology.tex","source_line":2011,"source_end_line":2021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2011-L2021","statement_sha256":"4b3028e96af232f8297c35022441a5d306c5a6a9a7c63a995cacbae8044014b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2301,"rank":2301,"depth":1,"x":2442.379,"y":187.932,"cluster":"homological-algebra"},{"id":"stacks:0FCK","tag":"0FCK","title":"Jordan-Hölder · Lemma 0FCK","summary":"Let A be an abelian category. Let A be an object of A satisfying the equivalent conditions of Lemma [Tag 0FCJ]. Given two filtrations 0 ⊂ A_1 ⊂ A_2 ⊂ … ⊂ A_n = A and 0 ⊂ B_1 ⊂ B_2 ⊂ … ⊂ B_m = A with S_i = A_i/A_i - 1 and T_j = B_j/B_j - 1 simple objects we have n = m and there exists a permutation σ of (1, …, n) such that S_i ≅ T_σ(i) for all i ∈ (1, …, n).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $A$ be an object\nof $\\mathcal{A}$ satisfying the equivalent conditions of\nLemma \\ref{lemma-finite-length}. Given two filtrations\n$$\n0 \\subset A_1 \\subset A_2 \\subset \\ldots \\subset A_n = A\n\\quad\\text{and}\\quad\n0 \\subset B_1 \\subset B_2 \\subset \\ldots \\subset B_m = A\n$$\nwith $S_i = A_i/A_{i - 1}$ and $T_j = B_j/B_{j - 1}$ simple objects we have\n$n = m$ and there exists a permutation $\\sigma$ of $\\{1, \\ldots, n\\}$\nsuch that $S_i \\cong T_{\\sigma(i)}$ for all $i \\in \\{1, \\ldots, n\\}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Jordan-Hölder","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCK","source_file":"homology.tex","source_line":2042,"source_end_line":2055,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2042-L2055","statement_sha256":"f4da6a64b3076d719c3157ac55cf1e25172b43a13514bdb4c10bb98d42354608","origin":"The Stacks Project","memory_eligible":false,"source_rank":2302,"rank":2302,"depth":2,"x":2534.017,"y":222.32,"cluster":"homological-algebra"},{"id":"stacks:02MO","tag":"02MO","title":"Serre subcategories · Definition 02MO","summary":"[Serre_homotopie_classes] Let A be an abelian category. • A Serre subcategory of A is a nonempty full subcategory C of A such that given an exact sequence A → B → C with A, C ∈ Ob(C), then also B ∈ Ob(C). • A weak Serre subcategory of A is a nonempty full subcategory C of A such that given an exact sequence A_0 → A_1 → A_2 → A_3 → A_4 with A_0, A_1, A_3, A_4 in C, then also A_2 in C.","statement_latex":"\\begin{reference}\n\\cite[Condition (I) on page 259]{Serre_homotopie_classes}\n\\end{reference}\nLet $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item A {\\it Serre subcategory} of $\\mathcal{A}$ is a\nnonempty full subcategory $\\mathcal{C}$ of $\\mathcal{A}$\nsuch that given an exact sequence\\footnote{By\nDefinition \\ref{definition-exact} this means $\\Im(A \\to B) = \\Ker(B \\to C)$.}\n$$\nA \\to B \\to C\n$$\nwith $A, C \\in \\Ob(\\mathcal{C})$, then also\n$B \\in \\Ob(\\mathcal{C})$.\n\\item A {\\it weak Serre subcategory} of $\\mathcal{A}$ is a nonempty\nfull subcategory $\\mathcal{C}$ of $\\mathcal{A}$ such that given an\nexact sequence\n$$\nA_0 \\to A_1 \\to A_2 \\to A_3 \\to A_4\n$$\nwith $A_0, A_1, A_3, A_4$ in $\\mathcal{C}$, then also $A_2$ in $\\mathcal{C}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Serre subcategories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MO","source_file":"homology.tex","source_line":2091,"source_end_line":2115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2091-L2115","statement_sha256":"6fd4f0328e7c9ec3dbd04a1a9e1f2ff704be9f5569155745caab5ec7792023cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2303,"rank":2303,"depth":1,"x":2437.93,"y":249.199,"cluster":"homological-algebra"},{"id":"stacks:02MP","tag":"02MP","title":"Serre subcategories · Lemma 02MP","summary":"Let A be an abelian category. Let C be a subcategory of A. Then C is a Serre subcategory if and only if the following conditions are satisfied: • 0 ∈ Ob(C), • C is a strictly full subcategory of A, • any subobject or quotient of an object of C is an object of C, • if A ∈ Ob(A) is an extension of objects of C then also A ∈ Ob(C). Moreover, a Serre subcategory is an abelian category and the inclusion functor is exact.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\mathcal{C}$ be a subcategory of $\\mathcal{A}$.\nThen $\\mathcal{C}$ is a Serre subcategory if and only if\nthe following conditions are satisfied:\n\\begin{enumerate}\n\\item $0 \\in \\Ob(\\mathcal{C})$,\n\\item $\\mathcal{C}$ is a strictly full subcategory of $\\mathcal{A}$,\n\\item any subobject or quotient of an object of $\\mathcal{C}$ is an object\nof $\\mathcal{C}$,\n\\item if $A \\in \\Ob(\\mathcal{A})$ is an extension of objects of $\\mathcal{C}$\nthen also $A \\in \\Ob(\\mathcal{C})$.\n\\end{enumerate}\nMoreover, a Serre subcategory is an abelian category and\nthe inclusion functor is exact.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Serre subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MP","source_file":"homology.tex","source_line":2126,"source_end_line":2142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2126-L2142","statement_sha256":"12c78c9ed2d3f7e72d8c14a509762a11d3ab33d041a6a1b7a84192b540724577","origin":"The Stacks Project","memory_eligible":false,"source_rank":2304,"rank":2304,"depth":0,"x":2487.607,"y":174.199,"cluster":"homological-algebra"},{"id":"stacks:0754","tag":"0754","title":"Serre subcategories · Lemma 0754","summary":"Let A be an abelian category. Let C be a subcategory of A. Then C is a weak Serre subcategory if and only if the following conditions are satisfied: • 0 ∈ Ob(C), • C is a strictly full subcategory of A, • kernels and cokernels in A of morphisms between objects of C are in C, • if A ∈ Ob(A) is an extension of objects of C then also A ∈ Ob(C). Moreover, a weak Serre subcategory is an abelian category and the inclusion functor is exact.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\mathcal{C}$ be a subcategory of $\\mathcal{A}$.\nThen $\\mathcal{C}$ is a weak Serre subcategory if and only if\nthe following conditions are satisfied:\n\\begin{enumerate}\n\\item $0 \\in \\Ob(\\mathcal{C})$,\n\\item $\\mathcal{C}$ is a strictly full subcategory of $\\mathcal{A}$,\n\\item kernels and cokernels in $\\mathcal{A}$ of morphisms\nbetween objects of $\\mathcal{C}$ are in $\\mathcal{C}$,\n\\item if $A \\in \\Ob(\\mathcal{A})$ is an extension of objects of $\\mathcal{C}$\nthen also $A \\in \\Ob(\\mathcal{C})$.\n\\end{enumerate}\nMoreover, a weak Serre subcategory is an abelian category and\nthe inclusion functor is exact.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Serre subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0754","source_file":"homology.tex","source_line":2148,"source_end_line":2164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2148-L2164","statement_sha256":"889d478151a71c82ba6a4daf9784a19fd3dbdd6608b82a443bee9701a990b1dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2305,"rank":2305,"depth":0,"x":2511.492,"y":258.419,"cluster":"homological-algebra"},{"id":"stacks:02MQ","tag":"02MQ","title":"Serre subcategories · Lemma 02MQ","summary":"Let A, B be abelian categories. Let F : A → B be an exact functor. Then the full subcategory of objects C of A such that F(C) = 0 forms a Serre subcategory of A.","statement_latex":"Let $\\mathcal{A}$, $\\mathcal{B}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor.\nThen the full subcategory of objects $C$ of $\\mathcal{A}$\nsuch that $F(C) = 0$ forms a Serre subcategory of $\\mathcal{A}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Serre subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MQ","source_file":"homology.tex","source_line":2170,"source_end_line":2176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2170-L2176","statement_sha256":"43bbb9b24386a0d8804539de31dedb719d56224f1297f65faaba46ebb72a1018","origin":"The Stacks Project","memory_eligible":false,"source_rank":2306,"rank":2306,"depth":0,"x":2425.424,"y":209.451,"cluster":"homological-algebra"},{"id":"stacks:02MR","tag":"02MR","title":"Serre subcategories · Definition 02MR","summary":"Let A, B be abelian categories. Let F : A → B be an exact functor. Then the full subcategory of objects C of A such that F(C) = 0 is called the kernel of the functor F, and is sometimes denoted Ker(F).","statement_latex":"Let $\\mathcal{A}$, $\\mathcal{B}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor.\nThen the full subcategory of objects $C$ of $\\mathcal{A}$\nsuch that $F(C) = 0$ is called the {\\it kernel of the functor $F$},\nand is sometimes denoted $\\Ker(F)$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Serre subcategories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MR","source_file":"homology.tex","source_line":2182,"source_end_line":2189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2182-L2189","statement_sha256":"a4b233d8ae20dd393468a31d46c3e874743c5f7784286bedc36feee7f7f2dcc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2307,"rank":2307,"depth":0,"x":2529.134,"y":196.617,"cluster":"homological-algebra"},{"id":"stacks:02MS","tag":"02MS","title":"Serre subcategories · Lemma 02MS","summary":"Let A be an abelian category. Let C ⊂ A be a Serre subcategory. There exists an abelian category A/C and an exact functor F : A → A/C which is essentially surjective and whose kernel is C. The category A/C and the functor F are characterized by the following universal property: For any exact functor G : A → B such that C ⊂ Ker(G) there exists a factorization G = H ∘ F for a unique exact functor H : A/C → B.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\mathcal{C} \\subset \\mathcal{A}$ be a Serre subcategory.\nThere exists an abelian category $\\mathcal{A}/\\mathcal{C}$\nand an exact functor\n$$\nF : \\mathcal{A} \\longrightarrow \\mathcal{A}/\\mathcal{C}\n$$\nwhich is essentially surjective and whose kernel is $\\mathcal{C}$.\nThe category $\\mathcal{A}/\\mathcal{C}$ and the functor $F$ are\ncharacterized by the following universal property: For any exact\nfunctor $G : \\mathcal{A} \\to \\mathcal{B}$ such that\n$\\mathcal{C} \\subset \\Ker(G)$ there exists a factorization\n$G = H \\circ F$ for a unique exact functor\n$H : \\mathcal{A}/\\mathcal{C} \\to \\mathcal{B}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Serre subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MS","source_file":"homology.tex","source_line":2197,"source_end_line":2213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2197-L2213","statement_sha256":"ff02b6ddd1d0c641c620dd18b381cfc969f1566fabef1c0e2d260d5ee7e3a9c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2308,"rank":2308,"depth":7,"x":2462.431,"y":265.492,"cluster":"homological-algebra"},{"id":"stacks:06XK","tag":"06XK","title":"Serre subcategories · Lemma 06XK","summary":"Let A, B be abelian categories. Let F : A → B be an exact functor. Let C ⊂ A be a Serre subcategory contained in the kernel of F. Then C = Ker(F) if and only if the induced functor overlineF : A/C → B (Lemma [Tag 02MS]) is faithful.","statement_latex":"Let $\\mathcal{A}$, $\\mathcal{B}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor.\nLet $\\mathcal{C} \\subset \\mathcal{A}$ be a Serre subcategory\ncontained in the kernel of $F$.\nThen $\\mathcal{C} = \\Ker(F)$ if and only if the induced functor\n$\\overline{F} : \\mathcal{A}/\\mathcal{C} \\to \\mathcal{B}$\n(Lemma \\ref{lemma-serre-subcategory-is-kernel}) is faithful.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Serre subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XK","source_file":"homology.tex","source_line":2294,"source_end_line":2303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2294-L2303","statement_sha256":"94370ee7c95d7159069f72fd9fe875a23856a403cf58f0f341d8328e6cb020b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2309,"rank":2309,"depth":8,"x":2456.181,"y":176.134,"cluster":"homological-algebra"},{"id":"stacks:02MU","tag":"02MU","title":"K-groups · Definition 02MU","summary":"Let A be an abelian category. We denote K_0(A) the zeroth K-group of A. It is the abelian group constructed as follows. Take the free abelian group on the objects of A and for every short exact sequence 0 → A → B → C → 0 impose the relation [B] - [A] - [C] = 0.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nWe denote $K_0(\\mathcal{A})$ the\n{\\it zeroth $K$-group of $\\mathcal{A}$}.\nIt is the abelian group constructed as follows.\nTake the free abelian group\non the objects of $\\mathcal{A}$\nand for every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nimpose the relation $[B] - [A] - [C] = 0$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"K-groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MU","source_file":"homology.tex","source_line":2333,"source_end_line":2344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2333-L2344","statement_sha256":"f80bd04ccac2f359e95d8149452751b58f113fcc85ef7183a465062d0449be7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2310,"rank":2310,"depth":0,"x":2533.26,"y":238.98,"cluster":"homological-algebra"},{"id":"stacks:02MV","tag":"02MV","title":"K-groups · Lemma 02MV","summary":"Let F : A → B be an exact functor between abelian categories. Then F induces a homomorphism of K-groups K_0(F) : K_0(A) → K_0(B) by simply setting K_0(F)([A]) = [F(A)].","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor between\nabelian categories. Then $F$ induces a homomorphism of $K$-groups\n$K_0(F) : K_0(\\mathcal{A}) \\to K_0(\\mathcal{B})$ by simply setting\n$K_0(F)([A]) = [F(A)]$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MV","source_file":"homology.tex","source_line":2368,"source_end_line":2374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2368-L2374","statement_sha256":"a4c8975d37d7fd88f6c8f1522bd32bba4e5bd246c42a37b4e72c5d8aac8a50df","origin":"The Stacks Project","memory_eligible":false,"source_rank":2311,"rank":2311,"depth":0,"x":2425.037,"y":236.352,"cluster":"homological-algebra"},{"id":"stacks:02MX","tag":"02MX","title":"K-groups · Lemma 02MX","summary":"Let A be an abelian category. Let C ⊂ A be a Serre subcategory and set B = A/C. • The exact functors C → A and A → B induce an exact sequence K_0(C) → K_0(A) → K_0(B) → 0 of K-groups, and • the kernel of K_0(C) → K_0(A) is equal to the collection of elements of the form [H^0(M, φ, ψ)] - [H^1(M, φ, ψ)] where (M, φ, ψ) is a complex as in ([Tag 02MW]) with the property that it becomes exact in B; in other words that H^0(M, φ, ψ) and H^1(M, φ, ψ) are objects of C.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\mathcal{C} \\subset \\mathcal{A}$ be a Serre subcategory and\nset $\\mathcal{B} = \\mathcal{A}/\\mathcal{C}$.\n\\begin{enumerate}\n\\item The exact functors $\\mathcal{C} \\to \\mathcal{A}$ and\n$\\mathcal{A} \\to \\mathcal{B}$ induce an exact sequence\n$$\nK_0(\\mathcal{C}) \\to\nK_0(\\mathcal{A}) \\to\nK_0(\\mathcal{B}) \\to\n0\n$$\nof $K$-groups, and\n\\item the kernel of $K_0(\\mathcal{C}) \\to K_0(\\mathcal{A})$ is equal\nto the collection of elements of the form\n$$\n[H^0(M, \\varphi, \\psi)] - [H^1(M, \\varphi, \\psi)]\n$$\nwhere $(M, \\varphi, \\psi)$ is a complex as in (\\ref{equation-cyclic-complex})\nwith the property that it becomes exact in $\\mathcal{B}$; in other words\nthat $H^0(M, \\varphi, \\psi)$ and $H^1(M, \\varphi, \\psi)$ are\nobjects of $\\mathcal{C}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MX","source_file":"homology.tex","source_line":2400,"source_end_line":2425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2400-L2425","statement_sha256":"516eae119b5b28c9a6cd87cc8fc9eddd516f012e1ec98e16d0b62d02286e781d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2312,"rank":2312,"depth":8,"x":2507.59,"y":176.422,"cluster":"homological-algebra"},{"id":"stacks:010Q","tag":"010Q","title":"Cohomological delta-functors · Definition 010Q","summary":"Let A, B be abelian categories. A cohomological δ-functor or simply a δ-functor from A to B is given by the following data: • a collection F^n : A → B, n ≥ 0 of additive functors, and • for every short exact sequence 0 → A → B → C → 0 of A a collection δ_A → B → C : F^n(C) → F^n + 1(A), n ≥ 0 of morphisms of B. These data are assumed to satisfy the following axioms • for every short exact sequence as above the sequence xymatrix 0 ar[r] & F^0(A) ar[r] & F^0(B) ar[r] &…","statement_latex":"Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.\nA {\\it cohomological $\\delta$-functor} or simply a\n{\\it $\\delta$-functor} from $\\mathcal{A}$\nto $\\mathcal{B}$ is given by the following data:\n\\begin{enumerate}\n\\item a collection $F^n : \\mathcal{A} \\to \\mathcal{B}$, $n \\geq 0$ of additive\nfunctors, and\n\\item for every short exact sequence $0 \\to A \\to B \\to C \\to 0$\nof $\\mathcal{A}$\na collection $\\delta_{A \\to B \\to C} : F^n(C) \\to F^{n + 1}(A)$, $n \\geq 0$\nof morphisms of $\\mathcal{B}$.\n\\end{enumerate}\nThese data are assumed to satisfy the following axioms\n\\begin{enumerate}\n\\item for every short exact sequence as above the sequence\n$$\n\\xymatrix{\n0 \\ar[r] &\nF^0(A) \\ar[r] &\nF^0(B) \\ar[r] &\nF^0(C) \\ar[lld]^{\\delta_{A \\to B \\to C}} \\\\\n &\nF^1(A) \\ar[r] &\nF^1(B) \\ar[r] &\nF^1(C) \\ar[lld]^{\\delta_{A \\to B \\to C}} \\\\\n &\nF^2(A) \\ar[r] &\nF^2(B) \\ar[r] &\n\\ldots\n}\n$$\nis exact, and\n\\item for every morphism $(A \\to B \\to C) \\to (A' \\to B' \\to C')$\nof short exact sequences of $\\mathcal{A}$ the diagrams\n$$\n\\xymatrix{\nF^n(C) \\ar[d] \\ar[rr]_{\\delta_{A \\to B \\to C}} & & F^{n + 1}(A) \\ar[d] \\\\\nF^n(C') \\ar[rr]^{\\delta_{A' \\to B' \\to C'}} & & F^{n + 1}(A')\n}\n$$\nare commutative.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Cohomological delta-functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010Q","source_file":"homology.tex","source_line":2624,"source_end_line":2668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2624-L2668","statement_sha256":"0fbdc6eae3ffd43dc443224091e3b194ea54b8e7057436110926f08921ae7a75","origin":"The Stacks Project","memory_eligible":false,"source_rank":2313,"rank":2313,"depth":0,"x":2494.811,"y":268.151,"cluster":"homological-algebra"},{"id":"stacks:010R","tag":"010R","title":"Cohomological delta-functors · Definition 010R","summary":"Let A, B be abelian categories. Let (F^n, δ_F) and (G^n, δ_G) be δ-functors from A to B. A morphism of δ-functors from F to G is a collection of transformation of functors t^n : F^n → G^n, n ≥ 0 such that for every short exact sequence 0 → A → B → C → 0 of A the diagrams xymatrix F^n(C) ar[d]_t^n ar[rr]_δ_F, A → B → C & & F^n + 1(A) ar[d]^t^n + 1 G^n(C) ar[rr]^δ_G, A → B → C & & G^n + 1(A) are commutative.","statement_latex":"Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.\nLet $(F^n, \\delta_F)$ and $(G^n, \\delta_G)$ be $\\delta$-functors\nfrom $\\mathcal{A}$ to $\\mathcal{B}$. A {\\it morphism of $\\delta$-functors\nfrom $F$ to $G$} is a collection of\ntransformation of functors $t^n : F^n \\to G^n$, $n \\geq 0$ such\nthat for every short exact sequence $0 \\to A \\to B \\to C \\to 0$\nof $\\mathcal{A}$ the diagrams\n$$\n\\xymatrix{\nF^n(C) \\ar[d]_{t^n} \\ar[rr]_{\\delta_{F, A \\to B \\to C}} &\n& F^{n + 1}(A) \\ar[d]^{t^{n + 1}} \\\\\nG^n(C) \\ar[rr]^{\\delta_{G, A \\to B \\to C}} & & G^{n + 1}(A)\n}\n$$\nare commutative.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Cohomological delta-functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010R","source_file":"homology.tex","source_line":2673,"source_end_line":2690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2673-L2690","statement_sha256":"e5d47dc4b474aafc28bf7755c78d8b7fca3c750dbae4d8b1006898533472aa00","origin":"The Stacks Project","memory_eligible":false,"source_rank":2314,"rank":2314,"depth":0,"x":2429.986,"y":192.696,"cluster":"homological-algebra"},{"id":"stacks:010S","tag":"010S","title":"Cohomological delta-functors · Definition 010S","summary":"Let A, B be abelian categories. Let F = (F^n, δ_F) be a δ-functor from A to B. We say F is a universal δ-functor if and only if for every δ-functor G = (G^n, δ_G) and any morphism of functors t : F^0 → G^0 there exists a unique morphism of δ-functors (t^n)_n ≥ 0 : F → G such that t = t^0.","statement_latex":"Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.\nLet $F = (F^n, \\delta_F)$ be a $\\delta$-functor\nfrom $\\mathcal{A}$ to $\\mathcal{B}$.\nWe say $F$ is a {\\it universal $\\delta$-functor} if and only\nif for every $\\delta$-functor $G = (G^n, \\delta_G)$ and any\nmorphism of functors $t : F^0 \\to G^0$ there exists\na unique morphism of $\\delta$-functors $\\{t^n\\}_{n \\geq 0} : F \\to G$\nsuch that $t = t^0$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Cohomological delta-functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010S","source_file":"homology.tex","source_line":2692,"source_end_line":2702,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2692-L2702","statement_sha256":"b5ae21444c8b35ccc898113490068a35b0ad4fbbe60d8abc075f8897ff00cb14","origin":"The Stacks Project","memory_eligible":false,"source_rank":2315,"rank":2315,"depth":0,"x":2539.27,"y":211.694,"cluster":"homological-algebra"},{"id":"stacks:010T","tag":"010T","title":"Cohomological delta-functors · Lemma 010T","summary":"Let A, B be abelian categories. Let F = (F^n, δ_F) be a δ-functor from A to B. Suppose that for every n > 0 and any A ∈ Ob(A) there exists an injective morphism u : A → B (depending on A and n) such that F^n(u) : F^n(A) → F^n(B) is zero. Then F is a universal δ-functor.","statement_latex":"Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.\nLet $F = (F^n, \\delta_F)$ be a $\\delta$-functor\nfrom $\\mathcal{A}$ to $\\mathcal{B}$.\nSuppose that for every $n > 0$ and any $A \\in \\Ob(\\mathcal{A})$\nthere exists an injective morphism $u : A \\to B$ (depending on $A$ and $n$)\nsuch that $F^n(u) : F^n(A) \\to F^n(B)$ is zero. Then $F$ is a universal\n$\\delta$-functor.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Cohomological delta-functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010T","source_file":"homology.tex","source_line":2704,"source_end_line":2713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2704-L2713","statement_sha256":"e30c11fc88edf03530de95fb5939616e8b791c862e67bcf9b7c91f93189f8416","origin":"The Stacks Project","memory_eligible":false,"source_rank":2316,"rank":2316,"depth":0,"x":2442.707,"y":260.044,"cluster":"homological-algebra"},{"id":"stacks:010U","tag":"010U","title":"Cohomological delta-functors · Lemma 010U","summary":"Let A, B be abelian categories. Let F : A → B be a functor. If there exists a universal δ-functor (F^n, δ_F) from A to B with F^0 = F, then it is determined up to unique isomorphism of δ-functors.","statement_latex":"Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\nIf there exists a universal $\\delta$-functor\n$(F^n, \\delta_F)$ from $\\mathcal{A}$ to $\\mathcal{B}$\nwith $F^0 = F$, then it is determined up to unique isomorphism\nof $\\delta$-functors.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Cohomological delta-functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010U","source_file":"homology.tex","source_line":2753,"source_end_line":2761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2753-L2761","statement_sha256":"eedc4b618dfa0e7792fab0399e88c0e40921a96fb5775832426899c3239e0627","origin":"The Stacks Project","memory_eligible":false,"source_rank":2317,"rank":2317,"depth":0,"x":2475.263,"y":168.948,"cluster":"homological-algebra"},{"id":"stacks:010W","tag":"010W","title":"Complexes · Lemma 010W","summary":"Hom functors of Ch(A) respect the homotopy relation. Let A be an additive category. Let f, g : B_bullet → C_bullet be morphisms of chain complexes. Suppose given morphisms of chain complexes a : A_bullet → B_bullet, and c : C_bullet → D_bullet. If (h_i : B_i → C_i + 1) defines a homotopy between f and g, then (c_i + 1 ∘ h_i ∘ a_i) defines a homotopy between c ∘ f ∘ a and c ∘ g ∘ a.","statement_latex":"\\begin{slogan}\nHom functors of $\\text{Ch}(\\mathcal{A})$ respect the homotopy relation.\n\\end{slogan}\nLet $\\mathcal{A}$ be an additive category.\nLet $f, g : B_\\bullet \\to C_\\bullet$ be morphisms\nof chain complexes. Suppose given morphisms of chain\ncomplexes $a : A_\\bullet \\to B_\\bullet$, and\n$c : C_\\bullet \\to D_\\bullet$.\nIf $\\{h_i : B_i \\to C_{i + 1}\\}$ defines a homotopy\nbetween $f$ and $g$, then $\\{c_{i + 1} \\circ h_i \\circ a_i\\}$\ndefines a homotopy between $c \\circ f \\circ a$ and\n$c \\circ g \\circ a$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010W","source_file":"homology.tex","source_line":2847,"source_end_line":2861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2847-L2861","statement_sha256":"a70f427fd28b878c0253d34385c4bb44692a4e11bca2ea9983055af4e57cf273","origin":"The Stacks Project","memory_eligible":false,"source_rank":2318,"rank":2318,"depth":0,"x":2524.861,"y":255.203,"cluster":"homological-algebra"},{"id":"stacks:010X","tag":"010X","title":"Complexes · Definition 010X","summary":"Let A be an additive category. We say a morphism a : A_bullet → B_bullet is a homotopy equivalence if there exists a morphism b : B_bullet → A_bullet such that there exists a homotopy between b ∘ a and id_A and there exists a homotopy between a ∘ b and id_B. If there exists such a morphism between A_bullet and B_bullet, then we say that A_bullet and B_bullet are homotopy equivalent.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nWe say a morphism $a : A_\\bullet \\to B_\\bullet$\nis a {\\it homotopy equivalence} if there exists\na morphism $b : B_\\bullet \\to A_\\bullet$\nsuch that there exists a homotopy between\n$b \\circ a$ and $\\text{id}_A$\nand there exists a homotopy between $a \\circ b$ and $\\text{id}_B$.\nIf there exists such a morphism between $A_\\bullet$ and $B_\\bullet$, then\nwe say that $A_\\bullet$ and $B_\\bullet$ are {\\it homotopy equivalent}.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010X","source_file":"homology.tex","source_line":2872,"source_end_line":2883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2872-L2883","statement_sha256":"18752994ea7ddea623b6e85dfa4f22a254eac6cd9b12c401232b0a49de95ea32","origin":"The Stacks Project","memory_eligible":false,"source_rank":2319,"rank":2319,"depth":0,"x":2418.184,"y":219.493,"cluster":"homological-algebra"},{"id":"stacks:010Y","tag":"010Y","title":"Complexes · Lemma 010Y","summary":"Let A be an abelian category. • The category of chain complexes in A is abelian. • A morphism of complexes f : A_bullet → B_bullet is injective if and only if each f_n : A_n → B_n is injective. • A morphism of complexes f : A_bullet → B_bullet is surjective if and only if each f_n : A_n → B_n is surjective. • A sequence of chain complexes A_bullet xrightarrowf B_bullet xrightarrowg C_bullet is exact at B_bullet if and only if each sequence A_i xrightarrowf_i B_i…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item The category of chain complexes in $\\mathcal{A}$ is\nabelian.\n\\item A morphism of complexes\n$f : A_\\bullet \\to B_\\bullet$ is injective\nif and only if each $f_n : A_n \\to B_n$ is injective.\n\\item A morphism of complexes\n$f : A_\\bullet \\to B_\\bullet$ is surjective\nif and only if each $f_n : A_n \\to B_n$ is surjective.\n\\item A sequence of chain complexes\n$$\nA_\\bullet \\xrightarrow{f} B_\\bullet \\xrightarrow{g} C_\\bullet\n$$\nis exact at $B_\\bullet$ if and only if each sequence\n$$\nA_i \\xrightarrow{f_i} B_i \\xrightarrow{g_i} C_i\n$$\nis exact at $B_i$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010Y","source_file":"homology.tex","source_line":2889,"source_end_line":2911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2889-L2911","statement_sha256":"af5cba2587cb0a784238bbb1ccf67a1d0dfebbaa260e77418adcabc92d3edbbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2320,"rank":2320,"depth":0,"x":2526.304,"y":185.06,"cluster":"homological-algebra"},{"id":"stacks:010Z","tag":"010Z","title":"Complexes · Definition 010Z","summary":"Let A be an abelian category. • A morphism of chain complexes f : A_bullet → B_bullet is called a quasi-isomorphism if the induced map H_i(f) : H_i(A_bullet) → H_i(B_bullet) is an isomorphism for all i ∈ Z. • A chain complex A_bullet is called acyclic if all of its homology objects H_i(A_bullet) are zero.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item A morphism of chain complexes $f : A_\\bullet \\to B_\\bullet$\nis called a {\\it quasi-isomorphism} if the induced\nmap $H_i(f) : H_i(A_\\bullet) \\to H_i(B_\\bullet)$\nis an isomorphism for all $i \\in \\mathbf{Z}$.\n\\item A chain complex $A_\\bullet$ is called\n{\\it acyclic} if all of its homology objects\n$H_i(A_\\bullet)$ are zero.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/010Z","source_file":"homology.tex","source_line":2936,"source_end_line":2948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2936-L2948","statement_sha256":"642c0c26fb10216424711dc65d433d7d094ab21f665a0977ff31c3a2a3cc615f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2321,"rank":2321,"depth":0,"x":2473.913,"y":272.39,"cluster":"homological-algebra"},{"id":"stacks:0110","tag":"0110","title":"Complexes · Lemma 0110","summary":"Let A be an abelian category. • If the maps f, g : A_bullet → B_bullet are homotopic, then the induced maps H_i(f) and H_i(g) are equal. • If the map f : A_bullet → B_bullet is a homotopy equivalence, then f is a quasi-isomorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item If the maps $f, g : A_\\bullet \\to B_\\bullet$ are\nhomotopic, then the induced maps $H_i(f)$ and $H_i(g)$\nare equal.\n\\item If the map $f : A_\\bullet \\to B_\\bullet$ is a homotopy\nequivalence, then $f$ is a quasi-isomorphism.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0110","source_file":"homology.tex","source_line":2951,"source_end_line":2961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2951-L2961","statement_sha256":"7a4089ef1481346c0e9c3488131e37985f683ae14f75617647db1f74198153a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2322,"rank":2322,"depth":0,"x":2442.107,"y":177.634,"cluster":"homological-algebra"},{"id":"stacks:0111","tag":"0111","title":"Complexes · Lemma 0111","summary":"Let A be an abelian category. Suppose that 0 → A_bullet → B_bullet → C_bullet → 0 is a short exact sequence of chain complexes of A. Then there is a canonical long exact homology sequence xymatrix … & … & … ar[lld] H_i(A_bullet) ar[r] & H_i(B_bullet) ar[r] & H_i(C_bullet) ar[lld] H_i - 1(A_bullet) ar[r] & H_i - 1(B_bullet) ar[r] & H_i - 1(C_bullet) ar[lld] … & … & …","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nSuppose that\n$$\n0 \\to\nA_\\bullet \\to\nB_\\bullet \\to\nC_\\bullet \\to\n0\n$$\nis a short exact sequence of chain complexes of $\\mathcal{A}$.\nThen there is a canonical long exact homology sequence\n$$\n\\xymatrix{\n\\ldots & \\ldots & \\ldots \\ar[lld] \\\\\nH_i(A_\\bullet) \\ar[r] & H_i(B_\\bullet) \\ar[r] & H_i(C_\\bullet) \\ar[lld] \\\\\nH_{i - 1}(A_\\bullet) \\ar[r] &\nH_{i - 1}(B_\\bullet) \\ar[r] &\nH_{i - 1}(C_\\bullet) \\ar[lld] \\\\\n\\ldots & \\ldots & \\ldots \\\\\n}\n$$","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0111","source_file":"homology.tex","source_line":2967,"source_end_line":2990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L2967-L2990","statement_sha256":"8e78e56e0ee8d4337baf57c9f195d5a5ff1fda0383ee785ab46d9fe4a59eed66","origin":"The Stacks Project","memory_eligible":false,"source_rank":2323,"rank":2323,"depth":6,"x":2542.419,"y":229.803,"cluster":"homological-algebra"},{"id":"stacks:0112","tag":"0112","title":"Complexes · Lemma 0112","summary":"Let A be an additive category. Let f, g : B^bullet → C^bullet be morphisms of cochain complexes. Suppose given morphisms of cochain complexes a : A^bullet → B^bullet, and c : C^bullet → D^bullet. If (h^i : B^i → C^i - 1) defines a homotopy between f and g, then (c^i - 1 ∘ h^i ∘ a^i) defines a homotopy between c ∘ f ∘ a and c ∘ g ∘ a.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $f, g : B^\\bullet \\to C^\\bullet$ be morphisms\nof cochain complexes. Suppose given morphisms of cochain\ncomplexes $a : A^\\bullet \\to B^\\bullet$, and\n$c : C^\\bullet \\to D^\\bullet$.\nIf $\\{h^i : B^i \\to C^{i - 1}\\}$ defines a homotopy\nbetween $f$ and $g$, then $\\{c^{i - 1} \\circ h^i \\circ a^i\\}$\ndefines a homotopy between $c \\circ f \\circ a$ and\n$c \\circ g \\circ a$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0112","source_file":"homology.tex","source_line":3073,"source_end_line":3084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3073-L3084","statement_sha256":"838c4fefd275cf4f832dc5a44620bcc30632d91410d41bbaaaee42baa67caac2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2324,"rank":2324,"depth":0,"x":2425.739,"y":248.373,"cluster":"homological-algebra"},{"id":"stacks:0113","tag":"0113","title":"Complexes · Definition 0113","summary":"Let A be an additive category. We say a morphism a : A^bullet → B^bullet is a homotopy equivalence if there exists a morphism b : B^bullet → A^bullet such that there exists a homotopy between b ∘ a and id_A and there exists a homotopy between a ∘ b and id_B. If there exists such a morphism between A^bullet and B^bullet, then we say that A^bullet and B^bullet are homotopy equivalent.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nWe say a morphism $a : A^\\bullet \\to B^\\bullet$\nis a {\\it homotopy equivalence} if there exists\na morphism $b : B^\\bullet \\to A^\\bullet$\nsuch that there exists a homotopy between\n$b \\circ a$ and $\\text{id}_A$\nand there exists a homotopy between $a \\circ b$ and $\\text{id}_B$.\nIf there exists such a morphism between $A^\\bullet$ and $B^\\bullet$, then\nwe say that $A^\\bullet$ and $B^\\bullet$ are {\\it homotopy equivalent}.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0113","source_file":"homology.tex","source_line":3095,"source_end_line":3106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3095-L3106","statement_sha256":"020f1bd4026a0c68ba3ff7e6fe0cffbf82273fab8a12c4b290138c605eb62f4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2325,"rank":2325,"depth":0,"x":2497.305,"y":167.958,"cluster":"homological-algebra"},{"id":"stacks:0114","tag":"0114","title":"Complexes · Lemma 0114","summary":"Let A be an abelian category. • The category of cochain complexes in A is abelian. • A morphism of cochain complexes f : A^bullet → B^bullet is injective if and only if each f^n : A^n → B^n is injective. • A morphism of cochain complexes f : A^bullet → B^bullet is surjective if and only if each f^n : A^n → B^n is surjective. • A sequence of cochain complexes A^bullet xrightarrowf B^bullet xrightarrowg C^bullet is exact at B^bullet if and only if each sequence A^i…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item The category of cochain complexes in $\\mathcal{A}$ is\nabelian.\n\\item A morphism of cochain complexes\n$f : A^\\bullet \\to B^\\bullet$ is injective\nif and only if each $f^n : A^n \\to B^n$ is injective.\n\\item A morphism of cochain complexes\n$f : A^\\bullet \\to B^\\bullet$ is surjective\nif and only if each $f^n : A^n \\to B^n$ is surjective.\n\\item A sequence of cochain complexes\n$$\nA^\\bullet \\xrightarrow{f} B^\\bullet \\xrightarrow{g} C^\\bullet\n$$\nis exact at $B^\\bullet$ if and only if each sequence\n$$\nA^i \\xrightarrow{f^i} B^i \\xrightarrow{g^i} C^i\n$$\nis exact at $B^i$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0114","source_file":"homology.tex","source_line":3112,"source_end_line":3134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3112-L3134","statement_sha256":"589145eabb8dcf4d57c2e1ca0306f26871f608316fc7bf1e4912f10413ab71ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":2326,"rank":2326,"depth":0,"x":2509.275,"y":268.499,"cluster":"homological-algebra"},{"id":"stacks:0115","tag":"0115","title":"Complexes · Definition 0115","summary":"Let A be an abelian category. • A morphism of cochain complexes f : A^bullet → B^bullet of A is called a quasi-isomorphism if the induced maps H^i(f) : H^i(A^bullet) → H^i(B^bullet) is an isomorphism for all i ∈ Z. • A cochain complex A^bullet is called acyclic if all of its cohomology objects H^i(A^bullet) are zero.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item A morphism of cochain complexes $f : A^\\bullet \\to B^\\bullet$\nof $\\mathcal{A}$ is called a {\\it quasi-isomorphism} if the induced\nmaps $H^i(f) : H^i(A^\\bullet) \\to H^i(B^\\bullet)$\nis an isomorphism for all $i \\in \\mathbf{Z}$.\n\\item A cochain complex $A^\\bullet$ is called\n{\\it acyclic} if all of its cohomology objects\n$H^i(A^\\bullet)$ are zero.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0115","source_file":"homology.tex","source_line":3159,"source_end_line":3171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3159-L3171","statement_sha256":"9b36d10cdeed6663edcb100ce7dd7afce5e6f5f0f6579a355a753df50c52bf7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2327,"rank":2327,"depth":0,"x":2419.034,"y":200.729,"cluster":"homological-algebra"},{"id":"stacks:0116","tag":"0116","title":"Complexes · Lemma 0116","summary":"Let A be an abelian category. • If the maps f, g : A^bullet → B^bullet are homotopic, then the induced maps H^i(f) and H^i(g) are equal. • If f : A^bullet → B^bullet is a homotopy equivalence, then f is a quasi-isomorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item If the maps $f, g : A^\\bullet \\to B^\\bullet$ are\nhomotopic, then the induced maps $H^i(f)$ and $H^i(g)$\nare equal.\n\\item If $f : A^\\bullet \\to B^\\bullet$ is a homotopy equivalence,\nthen $f$ is a quasi-isomorphism.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0116","source_file":"homology.tex","source_line":3173,"source_end_line":3183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3173-L3183","statement_sha256":"82969b04f363d06eb49375dfbe292a411216a2643537c264fe52222b21f09327","origin":"The Stacks Project","memory_eligible":false,"source_rank":2328,"rank":2328,"depth":0,"x":2540.826,"y":199.49,"cluster":"homological-algebra"},{"id":"stacks:0117","tag":"0117","title":"Complexes · Lemma 0117","summary":"Short exact sequences of complexes give rise to long exact sequences of (co)homology. Let A be an abelian category. Suppose that 0 → A^bullet → B^bullet → C^bullet → 0 is a short exact sequence of cochain complexes of A. Then there is a long exact cohomology sequence xymatrix … & … & … ar[lld] H^i(A^bullet) ar[r] & H^i(B^bullet) ar[r] & H^i(C^bullet) ar[lld] H^i + 1(A^bullet) ar[r] & H^i + 1(B^bullet) ar[r] & H^i + 1(C^bullet) ar[lld] … & … & … The construction produces…","statement_latex":"\\begin{slogan}\nShort exact sequences of complexes give rise to long exact sequences\nof (co)homology.\n\\end{slogan}\nLet $\\mathcal{A}$ be an abelian category. Suppose that\n$$\n0 \\to\nA^\\bullet \\to\nB^\\bullet \\to\nC^\\bullet \\to\n0\n$$\nis a short exact sequence of cochain complexes of $\\mathcal{A}$.\nThen there is a long exact cohomology sequence\n$$\n\\xymatrix{\n\\ldots & \\ldots & \\ldots \\ar[lld] \\\\\nH^i(A^\\bullet) \\ar[r] &\nH^i(B^\\bullet) \\ar[r] &\nH^i(C^\\bullet) \\ar[lld] \\\\\nH^{i + 1}(A^\\bullet) \\ar[r] &\nH^{i + 1}(B^\\bullet) \\ar[r] &\nH^{i + 1}(C^\\bullet) \\ar[lld] \\\\\n\\ldots & \\ldots & \\ldots \\\\\n}\n$$\nThe construction produces long exact cohomology sequences\nwhich are functorial in the short exact\nsequence and compatible with shifts as in\nDefinition \\ref{definition-cohomology-shift}.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0117","source_file":"homology.tex","source_line":3189,"source_end_line":3221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3189-L3221","statement_sha256":"f38ed70f481f5b99d1c2d20880701be75e2d7630ea9e67be57b3860112cea4c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2329,"rank":2329,"depth":6,"x":2451.468,"y":269.94,"cluster":"homological-algebra"},{"id":"stacks:011A","tag":"011A","title":"Homotopy and the shift functor · Definition 011A","summary":"Let A be an additive category. Let A_bullet be a chain complex with boundary maps d_A, n : A_n → A_n - 1. For any k ∈ Z we define the k-shifted chain complex A[k]_bullet as follows: • we set A[k]_n = A_n + k, and • we set d_A[k], n : A[k]_n → A[k]_n - 1 equal to d_A[k], n = (-1)^k d_A, n + k. If f : A_bullet → B_bullet is a morphism of chain complexes, then we let f[k] : A[k]_bullet → B[k]_bullet be the morphism of chain complexes with f[k]_n = f_k + n.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $A_\\bullet$ be a chain complex\nwith boundary maps $d_{A, n} : A_n \\to A_{n - 1}$.\nFor any $k \\in \\mathbf{Z}$ we define the\n{\\it $k$-shifted chain complex $A[k]_\\bullet$}\nas follows:\n\\begin{enumerate}\n\\item we set $A[k]_n = A_{n + k}$, and\n\\item we set $d_{A[k], n} : A[k]_n \\to A[k]_{n - 1}$\nequal to $d_{A[k], n} = (-1)^k d_{A, n + k}$.\n\\end{enumerate}\nIf $f : A_\\bullet \\to B_\\bullet$ is a morphism of\nchain complexes, then we let\n$f[k] : A[k]_\\bullet \\to B[k]_\\bullet$ be the\nmorphism of chain complexes with\n$f[k]_n = f_{k + n}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011A","source_file":"homology.tex","source_line":3267,"source_end_line":3285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3267-L3285","statement_sha256":"63e3100b47da7f2f833044e47329a3d988d4a8baec120bdfa7d5fe5dd56c3101","origin":"The Stacks Project","memory_eligible":false,"source_rank":2330,"rank":2330,"depth":0,"x":2460.763,"y":166.666,"cluster":"homological-algebra"},{"id":"stacks:011B","tag":"011B","title":"Homotopy and the shift functor · Definition 011B","summary":"Let A be an abelian category. Let A_bullet be a chain complex with boundary maps d_A, n : A_n → A_n - 1. For any k ∈ Z we identify H_i + k(A_bullet) → H_i(A[k]_bullet) via the identification A_i + k = A[k]_i.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A_\\bullet$ be a chain complex\nwith boundary maps $d_{A, n} : A_n \\to A_{n - 1}$.\nFor any $k \\in \\mathbf{Z}$ we identify\n{\\it $H_{i + k}(A_\\bullet) \\rightarrow H_i(A[k]_\\bullet)$}\nvia the identification\n$A_{i + k} = A[k]_i$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011B","source_file":"homology.tex","source_line":3302,"source_end_line":3311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3302-L3311","statement_sha256":"9b39bceefce4423fff09cd331f7ac9e214b99ba1a3775f436bded467456a9b5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2331,"rank":2331,"depth":0,"x":2537.414,"y":248.581,"cluster":"homological-algebra"},{"id":"stacks:011C","tag":"011C","title":"Homotopy and the shift functor · Lemma 011C","summary":"Let A be an additive category. Suppose that A_bullet and B_bullet are chain complexes. Given any morphism of chain complexes a : A_bullet → B_bullet there is a bijection between the set of homotopies from a to a and Mor_Ch(A)(A_bullet, B[1]_bullet). More generally, the set of homotopies between a and b is either empty or a principal homogeneous space under the group Mor_Ch(A)(A_bullet, B[1]_bullet).","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nSuppose that $A_\\bullet$ and $B_\\bullet$ are\nchain complexes. Given any morphism of chain\ncomplexes $a : A_\\bullet \\to B_\\bullet$ there\nis a bijection between the set of homotopies\nfrom $a$ to $a$ and\n$\\Mor_{\\text{Ch}(\\mathcal{A})}(A_\\bullet, B[1]_\\bullet)$.\nMore generally, the set of homotopies between\n$a$ and $b$ is either empty or a principal homogeneous\nspace under the group\n$\\Mor_{\\text{Ch}(\\mathcal{A})}(A_\\bullet, B[1]_\\bullet)$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011C","source_file":"homology.tex","source_line":3342,"source_end_line":3355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3342-L3355","statement_sha256":"4cdebd79515df269f5778b01b8823ecbeee6da6323a5f2cc831ed80a75546bc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2332,"rank":2332,"depth":0,"x":2414.296,"y":231.57,"cluster":"homological-algebra"},{"id":"stacks:011D","tag":"011D","title":"Homotopy and the shift functor · Lemma 011D","summary":"Let A be an abelian category. Let 0 → A_bullet → B_bullet → C_bullet → 0 be a short exact sequence of complexes. Suppose that (s_n : C_n → B_n) is a family of morphisms which split the short exact sequences 0 → A_n → B_n → C_n → 0. Let π_n : B_n → A_n be the associated projections, see Lemma [Tag 010G]. Then the family of morphisms π_n - 1 ∘ d_B, n ∘ s_n : C_n → A_n - 1 define a morphism of complexes δ(s) : C_bullet → A[-1]_bullet.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet\n$$\n0 \\to A_\\bullet \\to B_\\bullet \\to C_\\bullet \\to 0\n$$\nbe a short exact sequence of complexes.\nSuppose that $\\{s_n : C_n \\to B_n\\}$ is a family\nof morphisms which split the short exact sequences\n$0 \\to A_n \\to B_n \\to C_n \\to 0$. Let\n$\\pi_n : B_n \\to A_n$ be the associated\nprojections, see Lemma \\ref{lemma-ses-split}.\nThen the family of morphisms\n$$\n\\pi_{n - 1} \\circ d_{B, n} \\circ s_n\n:\nC_n \\to A_{n - 1}\n$$\ndefine a morphism of complexes $\\delta(s) : C_\\bullet \\to A[-1]_\\bullet$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011D","source_file":"homology.tex","source_line":3361,"source_end_line":3381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3361-L3381","statement_sha256":"1b7afc771ef9525795d9e756bae2c2d83cb594bba544302e6873af0a21ca5a47","origin":"The Stacks Project","memory_eligible":false,"source_rank":2333,"rank":2333,"depth":5,"x":2519.372,"y":173.92,"cluster":"homological-algebra"},{"id":"stacks:011E","tag":"011E","title":"Homotopy and the shift functor · Lemma 011E","summary":"Notation and assumptions as in Lemma [Tag 011D] above. The morphism of complexes δ(s) : C_bullet → A[-1]_bullet induces the maps H_i(δ(s)) : H_i(C_bullet) → H_i(A[-1]_bullet) = H_i - 1(A_bullet) which occur in the long exact homology sequence associated to the short exact sequence of chain complexes by Lemma [Tag 0111].","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-ses-termwise-split} above.\nThe morphism of complexes $\\delta(s) : C_\\bullet \\to A[-1]_\\bullet$\ninduces the maps\n$$\nH_i(\\delta(s)) :\nH_i(C_\\bullet) \\longrightarrow H_i(A[-1]_\\bullet) = H_{i - 1}(A_\\bullet)\n$$\nwhich occur in the long exact homology sequence associated\nto the short exact sequence of chain complexes by\nLemma \\ref{lemma-long-exact-sequence-chain}.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011E","source_file":"homology.tex","source_line":3393,"source_end_line":3405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3393-L3405","statement_sha256":"0d1d4f8fa4429c0cb0eee25eb30e237869c24be0b35b8a459ab947599fb03a35","origin":"The Stacks Project","memory_eligible":false,"source_rank":2334,"rank":2334,"depth":7,"x":2488.07,"y":276.643,"cluster":"homological-algebra"},{"id":"stacks:011F","tag":"011F","title":"Homotopy and the shift functor · Lemma 011F","summary":"Notation and assumptions as in Lemma [Tag 011D] above. Suppose (s'_n : C_n → B_n) is a second choice of splittings. Write s'_n = s_n + i_n ∘ h_n for some unique morphisms h_n : C_n → A_n. The family of maps (h_n : C_n → A[-1]_n + 1) is a homotopy between the associated morphisms δ(s), δ(s') : C_bullet → A[-1]_bullet.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-ses-termwise-split} above.\nSuppose $\\{s'_n : C_n \\to B_n\\}$ is a second choice of splittings.\nWrite $s'_n = s_n + i_n \\circ h_n$ for some unique\nmorphisms $h_n : C_n \\to A_n$. The family of maps\n$\\{h_n : C_n \\to A[-1]_{n + 1}\\}$ is a homotopy between\nthe associated morphisms\n$\\delta(s), \\delta(s') : C_\\bullet \\to A[-1]_\\bullet$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011F","source_file":"homology.tex","source_line":3411,"source_end_line":3420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3411-L3420","statement_sha256":"40a1b12810c954aaba431c0ceb835b72d1b6efb9d6a4cd3de09f815157847491","origin":"The Stacks Project","memory_eligible":false,"source_rank":2335,"rank":2335,"depth":6,"x":2428.206,"y":182.6,"cluster":"homological-algebra"},{"id":"stacks:011G","tag":"011G","title":"Homotopy and the shift functor · Definition 011G","summary":"Let A be an additive category. Let A^bullet be a cochain complex with boundary maps d_A^n : A^n → A^n + 1. For any k ∈ Z we define the k-shifted cochain complex A[k]^bullet as follows: • we set A[k]^n = A^n + k, and • we set d_A[k]^n : A[k]^n → A[k]^n + 1 equal to d_A[k]^n = (-1)^k d_A^n + k. If f : A^bullet → B^bullet is a morphism of cochain complexes, then we let f[k] : A[k]^bullet → B[k]^bullet be the morphism of cochain complexes with f[k]^n = f^k + n.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $A^\\bullet$ be a cochain complex\nwith boundary maps $d_A^n : A^n \\to A^{n + 1}$.\nFor any $k \\in \\mathbf{Z}$ we define the\n{\\it $k$-shifted cochain complex $A[k]^\\bullet$}\nas follows:\n\\begin{enumerate}\n\\item we set $A[k]^n = A^{n + k}$, and\n\\item we set $d_{A[k]}^n : A[k]^n \\to A[k]^{n + 1}$\nequal to $d_{A[k]}^n = (-1)^k d_A^{n + k}$.\n\\end{enumerate}\nIf $f : A^\\bullet \\to B^\\bullet$ is a morphism of\ncochain complexes, then we let\n$f[k] : A[k]^\\bullet \\to B[k]^\\bullet$ be the\nmorphism of cochain complexes with\n$f[k]^n = f^{k + n}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011G","source_file":"homology.tex","source_line":3426,"source_end_line":3444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3426-L3444","statement_sha256":"f6ef3f5473ee7cf0f8a95145fb3fb2d55e70c4a987ddbf167c16ebffbe7dd834","origin":"The Stacks Project","memory_eligible":false,"source_rank":2336,"rank":2336,"depth":0,"x":2548.651,"y":218.179,"cluster":"homological-algebra"},{"id":"stacks:011H","tag":"011H","title":"Homotopy and the shift functor · Definition 011H","summary":"Let A be an abelian category. Let A^bullet be a cochain complex with boundary maps d_A^n : A^n → A^n + 1. For any k ∈ Z we identify H^i + k(A^bullet) → H^i(A[k]^bullet) via the identification A^i + k = A[k]^i.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A^\\bullet$ be a cochain complex\nwith boundary maps $d_A^n : A^n \\to A^{n + 1}$.\nFor any $k \\in \\mathbf{Z}$ we identify\n{\\it $H^{i + k}(A^\\bullet) \\longrightarrow H^i(A[k]^\\bullet)$}\nvia the identification $A^{i + k} = A[k]^i$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011H","source_file":"homology.tex","source_line":3461,"source_end_line":3469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3461-L3469","statement_sha256":"461f06388197e748e0aee79ddeedbb5a54f0d637eacf5278f1414c2699d27716","origin":"The Stacks Project","memory_eligible":false,"source_rank":2337,"rank":2337,"depth":0,"x":2430.57,"y":260.521,"cluster":"homological-algebra"},{"id":"stacks:011I","tag":"011I","title":"Homotopy and the shift functor · Lemma 011I","summary":"Let A be an additive category. Suppose that A^bullet and B^bullet are cochain complexes. Given any morphism of cochain complexes a : A^bullet → B^bullet there is a bijection between the set of homotopies from a to a and Mor_CoCh(A)(A^bullet, B[-1]^bullet). More generally, the set of homotopies between a and b is either empty or a principal homogeneous space under the group Mor_CoCh(A)(A^bullet, B[-1]^bullet).","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nSuppose that $A^\\bullet$ and $B^\\bullet$ are\ncochain complexes. Given any morphism of cochain\ncomplexes $a : A^\\bullet \\to B^\\bullet$ there\nis a bijection between the set of homotopies\nfrom $a$ to $a$ and\n$\\Mor_{\\text{CoCh}(\\mathcal{A})}(A^\\bullet, B[-1]^\\bullet)$.\nMore generally, the set of homotopies between\n$a$ and $b$ is either empty or a principal homogeneous\nspace under the group\n$\\Mor_{\\text{CoCh}(\\mathcal{A})}(A^\\bullet, B[-1]^\\bullet)$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011I","source_file":"homology.tex","source_line":3500,"source_end_line":3513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3500-L3513","statement_sha256":"cb2d25f7449a9a61e313b59d1d7207be7c496058d94c889517b624901445059c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2338,"rank":2338,"depth":0,"x":2483.885,"y":161.755,"cluster":"homological-algebra"},{"id":"stacks:011J","tag":"011J","title":"Homotopy and the shift functor · Lemma 011J","summary":"Let A be an additive category. Let 0 → A^bullet → B^bullet → C^bullet → 0 be a complex (!) of complexes. Suppose that we are given splittings B^n = A^n ⊕ C^n compatible with the maps in the displayed sequence. Let s^n : C^n → B^n and π^n : B^n → A^n be the corresponding maps. Then the family of morphisms π^n + 1 ∘ d_B^n ∘ s^n : C^n → A^n + 1 define a morphism of complexes δ : C^bullet → A[1]^bullet.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet\n$$\n0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0\n$$\nbe a complex (!) of complexes.\nSuppose that we are given splittings $B^n = A^n \\oplus C^n$\ncompatible with the maps in the displayed sequence.\nLet $s^n : C^n \\to B^n$ and $\\pi^n : B^n \\to A^n$ be the\ncorresponding maps. Then the family of morphisms\n$$\n\\pi^{n + 1} \\circ d_B^n \\circ s^n\n:\nC^n \\to A^{n + 1}\n$$\ndefine a morphism of complexes $\\delta : C^\\bullet \\to A[1]^\\bullet$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011J","source_file":"homology.tex","source_line":3519,"source_end_line":3537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3519-L3537","statement_sha256":"4b35506e98da3c8ee6ed5f31892a4bb5dcb8a1c5aaa2905011e071c86e5a0623","origin":"The Stacks Project","memory_eligible":false,"source_rank":2339,"rank":2339,"depth":0,"x":2524.213,"y":265.398,"cluster":"homological-algebra"},{"id":"stacks:011K","tag":"011K","title":"Homotopy and the shift functor · Lemma 011K","summary":"Notation and assumptions as in Lemma [Tag 011J] above. Assume in addition that A is abelian. The morphism of complexes δ : C^bullet → A[1]^bullet induces the maps H^i(δ) : H^i(C^bullet) → H^i(A[1]^bullet) = H^i + 1(A^bullet) which occur in the long exact homology sequence associated to the short exact sequence of cochain complexes by Lemma [Tag 0117].","statement_latex":"Notation and assumptions as in\nLemma \\ref{lemma-ses-termwise-split-cochain} above.\nAssume in addition that $\\mathcal{A}$ is abelian.\nThe morphism of complexes $\\delta : C^\\bullet \\to A[1]^\\bullet$\ninduces the maps\n$$\nH^i(\\delta) :\nH^i(C^\\bullet) \\longrightarrow H^i(A[1]^\\bullet) = H^{i + 1}(A^\\bullet)\n$$\nwhich occur in the long exact homology sequence associated\nto the short exact sequence of cochain complexes by\nLemma \\ref{lemma-long-exact-sequence-cochain}.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011K","source_file":"homology.tex","source_line":3549,"source_end_line":3563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3549-L3563","statement_sha256":"f3e53729d68c48a0a035707c729a69f39a8e02c7be0a86a52492c959e6ad1c7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2340,"rank":2340,"depth":7,"x":2410.519,"y":211.567,"cluster":"homological-algebra"},{"id":"stacks:011L","tag":"011L","title":"Homotopy and the shift functor · Lemma 011L","summary":"Notation and assumptions as in Lemma [Tag 011J]. Let α : A^bullet → B^bullet, β : B^bullet → C^bullet be the given morphisms of complexes. Suppose (s')^n : C^n → B^n and (π')^n : B^n → A^n is a second choice of splittings. Write (s')^n = s^n + α^n ∘ h^n and (π')^n = π^n + g^n ∘ β^n for some unique morphisms h^n : C^n → A^n and g^n : C^n → A^n. Then • g^n = - h^n, and • the family of maps (g^n : C^n → A[1]^n - 1) is a homotopy between δ, δ' : C^bullet → A[1]^bullet, more…","statement_latex":"Notation and assumptions as in\nLemma \\ref{lemma-ses-termwise-split-cochain}.\nLet $\\alpha : A^\\bullet \\to B^\\bullet$,\n$\\beta : B^\\bullet \\to C^\\bullet$ be the given\nmorphisms of complexes.\nSuppose $(s')^n : C^n \\to B^n$ and $(\\pi')^n : B^n \\to A^n$\nis a second choice of splittings.\nWrite $(s')^n = s^n + \\alpha^n \\circ h^n$ and\n$(\\pi')^n = \\pi^n + g^n \\circ \\beta^n$ for some unique\nmorphisms $h^n : C^n \\to A^n$ and $g^n : C^n \\to A^n$. Then\n\\begin{enumerate}\n\\item $g^n = - h^n$, and\n\\item the family of maps $\\{g^n : C^n \\to A[1]^{n - 1}\\}$ is a homotopy\nbetween $\\delta, \\delta' : C^\\bullet \\to A[1]^\\bullet$, more precisely\n$(\\delta')^n = \\delta^n + g^{n + 1} \\circ d_C^n + d_{A[1]}^{n - 1} \\circ g^n$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Homotopy and the shift functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011L","source_file":"homology.tex","source_line":3569,"source_end_line":3587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3569-L3587","statement_sha256":"e3c775bd93f294fb971185ab7e3aac5303d6e4438db00e16495fe51d858c4091","origin":"The Stacks Project","memory_eligible":false,"source_rank":2341,"rank":2341,"depth":1,"x":2538.325,"y":186.617,"cluster":"homological-algebra"},{"id":"stacks:0125","tag":"0125","title":"Graded objects · Definition 0125","summary":"Let A be an additive category. The category of graded objects of A, denoted Gr(A), is the category with • objects A = (A^i) are families of objects A^i, i ∈ Z of objects of A, and • morphisms f : A = (A^i) → B = (B^i) are families of morphisms f^i : A^i → B^i of A.","statement_latex":"Let $\\mathcal{A}$ be an additive category. The {\\it category of graded\nobjects of $\\mathcal{A}$}, denoted $\\text{Gr}(\\mathcal{A})$, is\nthe category with\n\\begin{enumerate}\n\\item objects $A = (A^i)$ are families of objects $A^i$, $i \\in \\mathbf{Z}$\nof objects of $\\mathcal{A}$, and\n\\item morphisms $f : A = (A^i) \\to B = (B^i)$ are families of\nmorphisms $f^i : A^i \\to B^i$ of $\\mathcal{A}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Graded objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0125","source_file":"homology.tex","source_line":3858,"source_end_line":3869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3858-L3869","statement_sha256":"666aa88fdeeb88e0d3c06b1b50e2fb192e8117b128d8320e1d82f914cb0f5121","origin":"The Stacks Project","memory_eligible":false,"source_rank":2342,"rank":2342,"depth":0,"x":2463.753,"y":278.014,"cluster":"homological-algebra"},{"id":"stacks:0126","tag":"0126","title":"Graded objects · Lemma 0126","summary":"Let A be an abelian category. The category of graded objects Gr(A) is abelian.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. The category of graded objects\n$\\text{Gr}(\\mathcal{A})$ is abelian.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Graded objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0126","source_file":"homology.tex","source_line":3899,"source_end_line":3903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3899-L3903","statement_sha256":"5f3ee5786f84cb4a103494701d9ab3bdc44af7b73352c5cbb60627af34196d42","origin":"The Stacks Project","memory_eligible":false,"source_rank":2343,"rank":2343,"depth":0,"x":2445.146,"y":167.732,"cluster":"homological-algebra"},{"id":"stacks:09MG","tag":"09MG","title":"Graded objects · Definition 09MG","summary":"Let A be an additive category. If A = (A^i) is a graded object, then the kth shift A[k] is the graded object with A[k]^i = A^k + i.","statement_latex":"Let $\\mathcal{A}$ be an additive category. If $A = (A^i)$ is a graded object,\nthen the $k$th {\\it shift} $A[k]$ is the graded object with\n$A[k]^i = A^{k + i}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Graded objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MG","source_file":"homology.tex","source_line":3932,"source_end_line":3937,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3932-L3937","statement_sha256":"dcf84d899378b72f7faa755eff4fcfcdaca18cfee0fb61a005359c0d09745610","origin":"The Stacks Project","memory_eligible":false,"source_rank":2344,"rank":2344,"depth":0,"x":2548.082,"y":238.861,"cluster":"homological-algebra"},{"id":"stacks:0FNA","tag":"0FNA","title":"Additive monoidal categories · Definition 0FNA","summary":"An additive monoidal category is an additive category A endowed with a monoidal structure ⊗, φ (Categories, Definition [Tag 0FFK]) such that ⊗ is an additive functor in each variable.","statement_latex":"An {\\it additive monoidal category} is an additive category $\\mathcal{A}$\nendowed with a monoidal structure $\\otimes, \\phi$\n(Categories, Definition \\ref{categories-definition-monoidal-category})\nsuch that $\\otimes$ is an additive functor in each variable.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Additive monoidal categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNA","source_file":"homology.tex","source_line":3979,"source_end_line":3985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3979-L3985","statement_sha256":"d6f2a49d938b74cbf4431782b76cc739e709e1ae796f0f45140dc97f27fab6ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":2345,"rank":2345,"depth":1,"x":2414.297,"y":244.849,"cluster":"homological-algebra"},{"id":"stacks:0FFT","tag":"0FFT","title":"Additive monoidal categories · Lemma 0FFT","summary":"Let A be an additive monoidal category. If Y_i, i = 1, 2 are left duals of X_i, i = 1, 2, then Y_1 ⊕ Y_2 is a left dual of X_1 ⊕ X_2.","statement_latex":"Let $\\mathcal{A}$ be an additive monoidal category.\nIf $Y_i$, $i = 1, 2$ are left duals of $X_i$, $i = 1, 2$, then\n$Y_1 \\oplus Y_2$ is a left dual of $X_1 \\oplus X_2$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Additive monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFT","source_file":"homology.tex","source_line":3987,"source_end_line":3992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3987-L3992","statement_sha256":"3310bd35fa2284d2e1c3f18b09ca2347d17a797628f1687a0fd4b1029f854616","origin":"The Stacks Project","memory_eligible":false,"source_rank":2346,"rank":2346,"depth":0,"x":2508.609,"y":164.115,"cluster":"homological-algebra"},{"id":"stacks:0FFU","tag":"0FFU","title":"Additive monoidal categories · Lemma 0FFU","summary":"In a Karoubian additive monoidal category every summand of an object which has a left dual has a left dual.","statement_latex":"In a Karoubian additive monoidal category every summand\nof an object which has a left dual has a left dual.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Additive monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFU","source_file":"homology.tex","source_line":3999,"source_end_line":4003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L3999-L4003","statement_sha256":"913785c2fb043cb626799b9f9a810dd35d74cf1e8a51dec8205300758079ee28","origin":"The Stacks Project","memory_eligible":false,"source_rank":2347,"rank":2347,"depth":1,"x":2503.964,"y":277.729,"cluster":"homological-algebra"},{"id":"stacks:0FFV","tag":"0FFV","title":"Additive monoidal categories · Lemma 0FFV","summary":"Let F be a field. Let C be the category of graded F-vector spaces viewed as a monoidal category as in Example [Tag 0FFX]. If V in C has a left dual W, then ∑_n dim_F V^n < ∞ and the map ε defines nondegenerate pairings W^-n × V^n → F.","statement_latex":"Let $F$ be a field. Let $\\mathcal{C}$ be the category of graded\n$F$-vector spaces viewed as a monoidal category as in\nExample \\ref{example-graded-vector-spaces}. If $V$ in $\\mathcal{C}$\nhas a left dual $W$, then $\\sum_n \\dim_F V^n < \\infty$\nand the map $\\epsilon$ defines nondegenerate pairings\n$W^{-n} \\times V^n \\to F$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Additive monoidal categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFV","source_file":"homology.tex","source_line":4063,"source_end_line":4071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4063-L4071","statement_sha256":"573d46a7d34dcb3e3a8a6aa82c918a69154c40b3ed88c1d2808e12ad190551a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2348,"rank":2348,"depth":1,"x":2415.588,"y":190.886,"cluster":"homological-algebra"},{"id":"stacks:012Y","tag":"012Y","title":"Double complexes and associated total complexes · Definition 012Y","summary":"Let A be an additive category. A double complex in A is given by a system ((A^p, q, d_1^p, q, d_2^p, q)_p, q∈ Z), where each A^p, q is an object of A and d_1^p, q : A^p, q → A^p + 1, q and d_2^p, q : A^p, q → A^p, q + 1 are morphisms of A such that the following rules hold: • d_1^p + 1, q ∘ d_1^p, q = 0 • d_2^p, q + 1 ∘ d_2^p, q = 0 • d_1^p, q + 1 ∘ d_2^p, q = d_2^p + 1, q ∘ d_1^p, q for all p, q ∈ Z.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nA {\\it double complex} in $\\mathcal{A}$ is given\nby a system $(\\{A^{p, q}, d_1^{p, q}, d_2^{p, q}\\}_{p, q\\in \\mathbf{Z}})$,\nwhere each $A^{p, q}$ is an object of $\\mathcal{A}$ and\n$d_1^{p, q} : A^{p, q} \\to A^{p + 1, q}$ and\n$d_2^{p, q} : A^{p, q} \\to A^{p, q + 1}$ are morphisms of $\\mathcal{A}$\nsuch that the following rules hold:\n\\begin{enumerate}\n\\item $d_1^{p + 1, q} \\circ d_1^{p, q} = 0$\n\\item $d_2^{p, q + 1} \\circ d_2^{p, q} = 0$\n\\item $d_1^{p, q + 1} \\circ d_2^{p, q} = d_2^{p + 1, q} \\circ d_1^{p, q}$\n\\end{enumerate}\nfor all $p, q \\in \\mathbf{Z}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Double complexes and associated total complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012Y","source_file":"homology.tex","source_line":4120,"source_end_line":4135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4120-L4135","statement_sha256":"12bdc049103c35755ff31aa6cdcdb988fa95514b7c924247ce5829a0372b81fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2349,"rank":2349,"depth":0,"x":2551.256,"y":204.845,"cluster":"homological-algebra"},{"id":"stacks:012Z","tag":"012Z","title":"Double complexes and associated total complexes · Definition 012Z","summary":"Let A be an additive category. Let A^bullet, bullet be a double complex. The associated simple complex, denoted sA^bullet, also often called the associated total complex, denoted Tot(A^bullet, bullet), is given by sA^n = Tot^n(A^bullet, bullet) = bigoplus_n = p + q A^p, q (if it exists) with differential d_sA^bullet^n = d_Tot(A^bullet, bullet)^n = ∑_n = p + q (d_1^p, q + (-1)^p d_2^p, q)","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $A^{\\bullet, \\bullet}$ be a double complex.\nThe {\\it associated simple complex}, denoted $sA^\\bullet$, also\noften called the {\\it associated total complex}, denoted\n$\\text{Tot}(A^{\\bullet, \\bullet})$, is\ngiven by\n$$\nsA^n = \\text{Tot}^n(A^{\\bullet, \\bullet}) =\n\\bigoplus\\nolimits_{n = p + q} A^{p, q}\n$$\n(if it exists) with differential\n$$\nd_{sA^\\bullet}^n = d_{\\text{Tot}(A^{\\bullet, \\bullet})}^n =\n\\sum\\nolimits_{n = p + q} (d_1^{p, q} + (-1)^p d_2^{p, q})\n$$","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Double complexes and associated total complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012Z","source_file":"homology.tex","source_line":4194,"source_end_line":4211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4194-L4211","statement_sha256":"d0d412c0d28851678b8e6b46b7e3f5846457fe7360ee1ad573e841cc3f82123a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2350,"rank":2350,"depth":0,"x":2439.448,"y":271.857,"cluster":"homological-algebra"},{"id":"stacks:0121","tag":"0121","title":"Filtrations · Definition 0121","summary":"Let A be an abelian category. • A decreasing filtration F on an object A is a family (F^nA)_n ∈ Z of subobjects of A such that A ⊃ … ⊃ F^nA ⊃ F^n + 1A ⊃ … ⊃ 0 • A filtered object of A is pair (A, F) consisting of an object A of A and a decreasing filtration F on A. • A morphism (A, F) → (B, F) of filtered objects is given by a morphism φ : A → B of A such that φ(F^iA) ⊂ F^iB for all i ∈ Z. • The category of filtered objects is denoted Fil(A). • Given a filtered object (A,…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item A {\\it decreasing filtration} $F$ on an object $A$\nis a family $(F^nA)_{n \\in \\mathbf{Z}}$ of subobjects of $A$ such that\n$$\nA \\supset \\ldots \\supset F^nA \\supset F^{n + 1}A \\supset \\ldots \\supset 0\n$$\n\\item A {\\it filtered object of $\\mathcal{A}$} is\npair $(A, F)$ consisting of an object $A$ of $\\mathcal{A}$\nand a decreasing filtration $F$ on $A$.\n\\item A {\\it morphism $(A, F) \\to (B, F)$ of filtered objects}\nis given by a morphism $\\varphi : A \\to B$ of $\\mathcal{A}$\nsuch that $\\varphi(F^iA) \\subset F^iB$ for all $i \\in \\mathbf{Z}$.\n\\item The category of filtered objects is denoted $\\text{Fil}(\\mathcal{A})$.\n\\item Given a filtered object $(A, F)$ and a subobject $X \\subset A$ the\n{\\it induced filtration} on $X$ is the filtration with $F^nX = X \\cap F^nA$.\n\\item Given a filtered object $(A, F)$ and a surjection\n$\\pi : A \\to Y$ the {\\it quotient filtration} is the filtration with\n$F^nY = \\pi(F^nA)$.\n\\item A filtration $F$ on an object $A$ is said to be {\\it finite}\nif there exist $n, m$ such that $F^nA = A$ and $F^mA = 0$.\n\\item Given a filtered object $(A, F)$ we say $\\bigcap F^iA$ exists\nif there exists a biggest subobject of $A$ contained in all $F^iA$.\nWe say $\\bigcup F^iA$ exists if there exists a smallest subobject\nof $A$ containing all $F^iA$.\n\\item The filtration on a filtered object $(A, F)$ is said to be\n{\\it separated} if $\\bigcap F^iA = 0$ and\n{\\it exhaustive} if $\\bigcup F^iA = A$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0121","source_file":"homology.tex","source_line":4666,"source_end_line":4697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4666-L4697","statement_sha256":"d55688f58d7dedc83ce8973cb290babee2dd4e5940fe6124d94b0025ff382d13","origin":"The Stacks Project","memory_eligible":false,"source_rank":2351,"rank":2351,"depth":0,"x":2468.144,"y":158.458,"cluster":"homological-algebra"},{"id":"stacks:0122","tag":"0122","title":"Filtrations · Lemma 0122","summary":"Let A be an abelian category. The category of filtered objects Fil(A) has the following properties: • It is an additive category. • It has a zero object. • It has kernels and cokernels, images and coimages. • In general it is not an abelian category.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nThe category of filtered objects $\\text{Fil}(\\mathcal{A})$\nhas the following properties:\n\\begin{enumerate}\n\\item It is an additive category.\n\\item It has a zero object.\n\\item It has kernels and cokernels, images and coimages.\n\\item In general it is not an abelian category.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0122","source_file":"homology.tex","source_line":4710,"source_end_line":4721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4710-L4721","statement_sha256":"f8e5aef5727d843d8f9a7c369c6ff8423eb90f3541f8cccf1c4f9d47901fc214","origin":"The Stacks Project","memory_eligible":false,"source_rank":2352,"rank":2352,"depth":0,"x":2538.51,"y":258.836,"cluster":"homological-algebra"},{"id":"stacks:0123","tag":"0123","title":"Filtrations · Definition 0123","summary":"Let A be an abelian category. A morphism f : A → B of filtered objects of A is said to be strict if f(F^iA) = f(A) ∩ F^iB for all i ∈ Z.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nA morphism $f : A \\to B$ of filtered objects of $\\mathcal{A}$ is\nsaid to be {\\it strict} if $f(F^iA) = f(A) \\cap F^iB$ for\nall $i \\in \\mathbf{Z}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0123","source_file":"homology.tex","source_line":4738,"source_end_line":4744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4738-L4744","statement_sha256":"7e571b0df57ce02c3784d8491742f2879d6e77af129db784a41316053622cde2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2353,"rank":2353,"depth":0,"x":2405.274,"y":224.584,"cluster":"homological-algebra"},{"id":"stacks:05SI","tag":"05SI","title":"Filtrations · Lemma 05SI","summary":"Let A be an abelian category. Let f : A → B be a morphism of filtered objects of A. The following are equivalent • f is strict, • the morphism Coim(f) → Im(f) of Lemma [Tag 0107] is an isomorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $f : A \\to B$ be a morphism of filtered objects of $\\mathcal{A}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is strict,\n\\item the morphism $\\Coim(f) \\to \\Im(f)$ of\nLemma \\ref{lemma-coim-im-map}\nis an isomorphism.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SI","source_file":"homology.tex","source_line":4751,"source_end_line":4762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4751-L4762","statement_sha256":"eb0d08899f96d5b02ef379c6963926fe35c1669b0ed863fc69af9f94bdcc64a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2354,"rank":2354,"depth":2,"x":2531.655,"y":174.005,"cluster":"homological-algebra"},{"id":"stacks:05SK","tag":"05SK","title":"Filtrations · Lemma 05SK","summary":"Let A be an abelian category. Let f : A → B be a strict monomorphism of filtered objects. Let g : A → C be a morphism of filtered objects. Then f ⊕ g : A → B ⊕ C is a strict monomorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $f : A \\to B$ be a strict monomorphism of filtered objects.\nLet $g : A \\to C$ be a morphism of filtered objects.\nThen $f \\oplus g : A \\to B \\oplus C$ is a strict monomorphism.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SK","source_file":"homology.tex","source_line":4778,"source_end_line":4784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4778-L4784","statement_sha256":"6e62553afffd7bdc8d274e2087caa9ca5fe5000e4bef4d4afdf5cfbf003c3438","origin":"The Stacks Project","memory_eligible":false,"source_rank":2355,"rank":2355,"depth":0,"x":2478.893,"y":283.518,"cluster":"homological-algebra"},{"id":"stacks:05SL","tag":"05SL","title":"Filtrations · Lemma 05SL","summary":"Let A be an abelian category. Let f : B → A be a strict epimorphism of filtered objects. Let g : C → A be a morphism of filtered objects. Then f ⊕ g : B ⊕ C → A is a strict epimorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $f : B \\to A$ be a strict epimorphism of filtered objects.\nLet $g : C \\to A$ be a morphism of filtered objects.\nThen $f \\oplus g : B \\oplus C \\to A$ is a strict epimorphism.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SL","source_file":"homology.tex","source_line":4790,"source_end_line":4796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4790-L4796","statement_sha256":"cc97e5efc898235c5695f896b9b2f9cbdcae2911a8a63721e8831ef4f5e0a1fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":2356,"rank":2356,"depth":0,"x":2429.507,"y":172.318,"cluster":"homological-algebra"},{"id":"stacks:0124","tag":"0124","title":"Filtrations · Lemma 0124","summary":"Let A be an abelian category. Let (A, F), (B, F) be filtered objects. Let u : A → B be a morphism of filtered objects. If u is injective then u is strict if and only if the filtration on A is the induced filtration. If u is surjective then u is strict if and only if the filtration on B is the quotient filtration.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(A, F)$, $(B, F)$ be filtered objects.\nLet $u : A \\to B$ be a morphism of filtered objects.\nIf $u$ is injective then $u$ is strict if and only if the filtration\non $A$ is the induced filtration.\nIf $u$ is surjective then $u$ is strict if and only if the filtration\non $B$ is the quotient filtration.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0124","source_file":"homology.tex","source_line":4802,"source_end_line":4811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4802-L4811","statement_sha256":"62ea4677a51cc584dfdf2a8a1ad99ed7a59c4cd2ceb34a764d2057c067aa2a73","origin":"The Stacks Project","memory_eligible":false,"source_rank":2357,"rank":2357,"depth":0,"x":2555.921,"y":226.539,"cluster":"homological-algebra"},{"id":"stacks:05SJ","tag":"05SJ","title":"Filtrations · Lemma 05SJ","summary":"Let A be an abelian category. Let f : A → B, g : B → C be strict morphisms of filtered objects. • In general the composition g ∘ f is not strict. • If g is injective, then g ∘ f is strict. • If f is surjective, then g ∘ f is strict.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $f : A \\to B$, $g : B \\to C$\nbe strict morphisms of filtered objects.\n\\begin{enumerate}\n\\item In general the composition $g \\circ f$ is not strict.\n\\item If $g$ is injective, then $g \\circ f$ is strict.\n\\item If $f$ is surjective, then $g \\circ f$ is strict.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SJ","source_file":"homology.tex","source_line":4817,"source_end_line":4826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4817-L4826","statement_sha256":"4f9945e1a7da816e13c1dbaa1943a5c6625f40271a64662a05eee19af1d5d15d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2358,"rank":2358,"depth":1,"x":2418.484,"y":258.43,"cluster":"homological-algebra"},{"id":"stacks:0129","tag":"0129","title":"Filtrations · Lemma 0129","summary":"Let A be an abelian category. Let (A, F) be a filtered object of A. Let X ⊂ Y ⊂ A be subobjects of A. On the object Y/X = Ker(A/X → A/Y) the quotient filtration coming from the induced filtration on Y and the induced filtration coming from the quotient filtration on A/X agree. Any of the morphisms X → Y, X → A, Y → A, Y → A/X, Y → Y/X, Y/X → A/X are strict (with induced/quotient filtrations).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(A, F)$ be a filtered object of $\\mathcal{A}$.\nLet $X \\subset Y \\subset A$ be subobjects of $A$.\nOn the object\n$$\nY/X = \\Ker(A/X \\to A/Y)\n$$\nthe quotient filtration coming from the induced filtration on $Y$ and the\ninduced filtration coming from the quotient filtration on $A/X$ agree.\nAny of the morphisms $X \\to Y$, $X \\to A$, $Y \\to A$, $Y \\to A/X$,\n$Y \\to Y/X$, $Y/X \\to A/X$ are strict (with induced/quotient filtrations).","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0129","source_file":"homology.tex","source_line":4868,"source_end_line":4881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4868-L4881","statement_sha256":"223fa58447b1b91d1dcf125bc42ead0d19d07eab51c29a89217fbfd6007b48c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2359,"rank":2359,"depth":0,"x":2494.521,"y":156.475,"cluster":"homological-algebra"},{"id":"stacks:05SM","tag":"05SM","title":"Filtrations · Lemma 05SM","summary":"Let A be an abelian category. Let A, B, C ∈ Fil(A). Let f : A → B and g : A → C be morphisms. Then there exists a pushout xymatrix A ar[r]_f ar[d]_g & B ar[d]^g' C ar[r]^f' & C amalg_A B in Fil(A). If f is strict, so is f'.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A, B, C \\in \\text{Fil}(\\mathcal{A})$.\nLet $f : A \\to B$ and $g : A \\to C$ be morphisms.\nThen there exists a pushout\n$$\n\\xymatrix{\nA \\ar[r]_f \\ar[d]_g & B \\ar[d]^{g'} \\\\\nC \\ar[r]^{f'} & C \\amalg_A B\n}\n$$\nin $\\text{Fil}(\\mathcal{A})$. If $f$ is strict, so is $f'$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SM","source_file":"homology.tex","source_line":4898,"source_end_line":4911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4898-L4911","statement_sha256":"9997b092032c2a539a452a35750c29d770d52fe53e1deeabbb853b8b4aede1a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2360,"rank":2360,"depth":1,"x":2520.555,"y":275.325,"cluster":"homological-algebra"},{"id":"stacks:05SN","tag":"05SN","title":"Filtrations · Lemma 05SN","summary":"Let A be an abelian category. Let A, B, C ∈ Fil(A). Let f : B → A and g : C → A be morphisms. Then there exists a fibre product xymatrix B ×_A C ar[r]_g' ar[d]_f' & B ar[d]^f C ar[r]^g & A in Fil(A). If f is strict, so is f'.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A, B, C \\in \\text{Fil}(\\mathcal{A})$.\nLet $f : B \\to A$ and $g : C \\to A$ be morphisms.\nThen there exists a fibre product\n$$\n\\xymatrix{\nB \\times_A C \\ar[r]_{g'} \\ar[d]_{f'} & B \\ar[d]^f \\\\\nC \\ar[r]^g & A\n}\n$$\nin $\\text{Fil}(\\mathcal{A})$. If $f$ is strict, so is $f'$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SN","source_file":"homology.tex","source_line":4927,"source_end_line":4940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4927-L4940","statement_sha256":"d2eb23b52d7bc5f7a5a283e3ec772883ef5e291f417f18fd71f55c6db0ea8c21","origin":"The Stacks Project","memory_eligible":false,"source_rank":2361,"rank":2361,"depth":2,"x":2405.279,"y":202.138,"cluster":"homological-algebra"},{"id":"stacks:05SP","tag":"05SP","title":"Filtrations · Lemma 05SP","summary":"Let A be an abelian category. • Let A be a filtered object and X ⊂ A. Then for each p the sequence 0 → gr^p(X) → gr^p(A) → gr^p(A/X) → 0 is exact (with induced filtration on X and quotient filtration on A/X). • Let f : A → B be a morphism of filtered objects of A. Then for each p the sequences 0 → gr^p(Ker(f)) → gr^p(A) → gr^p(Coim(f)) → 0 and 0 → gr^p(Im(f)) → gr^p(B) → gr^p(Coker(f)) → 0 are exact.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item Let $A$ be a filtered object and $X \\subset A$. Then for each $p$\nthe sequence\n$$\n0 \\to \\text{gr}^p(X) \\to \\text{gr}^p(A) \\to \\text{gr}^p(A/X) \\to 0\n$$\nis exact (with induced filtration on $X$ and quotient filtration on $A/X$).\n\\item Let $f : A \\to B$ be a morphism of filtered objects of $\\mathcal{A}$.\nThen for each $p$ the sequences\n$$\n0 \\to \\text{gr}^p(\\Ker(f)) \\to \\text{gr}^p(A) \\to\n\\text{gr}^p(\\Coim(f)) \\to 0\n$$\nand\n$$\n0 \\to \\text{gr}^p(\\Im(f)) \\to \\text{gr}^p(B) \\to\n\\text{gr}^p(\\Coker(f)) \\to 0\n$$\nare exact.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SP","source_file":"homology.tex","source_line":4981,"source_end_line":5004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L4981-L5004","statement_sha256":"be929a36ae2bbfc3c86df2edce0c644c38be86e83c56b64614be17486d8f4ae7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2362,"rank":2362,"depth":0,"x":2549.763,"y":190.646,"cluster":"homological-algebra"},{"id":"stacks:0127","tag":"0127","title":"Filtrations · Lemma 0127","summary":"Let A be an abelian category. Let f : A → B be a morphism of finite filtered objects of A. The following are equivalent • f is strict, • the morphism Coim(f) → Im(f) is an isomorphism, • gr(Coim(f)) → gr(Im(f)) is an isomorphism, • the sequence gr(Ker(f)) → gr(A) → gr(B) is exact, • the sequence gr(A) → gr(B) → gr(Coker(f)) is exact, and • the sequence 0 → gr(Ker(f)) → gr(A) → gr(B) → gr(Coker(f)) → 0 is exact.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $f : A \\to B$ be a morphism of finite\nfiltered objects of $\\mathcal{A}$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is strict,\n\\item the morphism $\\Coim(f) \\to \\Im(f)$ is an isomorphism,\n\\item $\\text{gr}(\\Coim(f)) \\to \\text{gr}(\\Im(f))$ is an\nisomorphism,\n\\item the sequence\n$\\text{gr}(\\Ker(f)) \\to \\text{gr}(A) \\to \\text{gr}(B)$\nis exact,\n\\item the sequence $\\text{gr}(A) \\to \\text{gr}(B) \\to\n\\text{gr}(\\Coker(f))$ is exact, and\n\\item the sequence\n$$\n0 \\to\n\\text{gr}(\\Ker(f)) \\to\n\\text{gr}(A) \\to\n\\text{gr}(B) \\to\n\\text{gr}(\\Coker(f)) \\to 0\n$$\nis exact.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0127","source_file":"homology.tex","source_line":5017,"source_end_line":5042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5017-L5042","statement_sha256":"e0aa73489e09f9d680d630ab95c03cbc6068145f1cd5ac28c76a74d45fc49327","origin":"The Stacks Project","memory_eligible":false,"source_rank":2363,"rank":2363,"depth":3,"x":2452.044,"y":281.494,"cluster":"homological-algebra"},{"id":"stacks:0128","tag":"0128","title":"Filtrations · Lemma 0128","summary":"Let A be an abelian category. Let A → B → C be a complex of filtered objects of A. Assume α : A → B and β : B → C are strict morphisms of filtered objects. Then gr(Ker(β)/Im(α)) = Ker(gr(β))/Im(gr(α))).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $A \\to B \\to C$ be a complex\nof filtered objects of $\\mathcal{A}$. Assume $\\alpha : A \\to B$ and\n$\\beta : B \\to C$ are strict morphisms of filtered objects. Then\n$\\text{gr}(\\Ker(\\beta)/\\Im(\\alpha)) =\n\\Ker(\\text{gr}(\\beta))/\\Im(\\text{gr}(\\alpha)))$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0128","source_file":"homology.tex","source_line":5076,"source_end_line":5083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5076-L5083","statement_sha256":"3ae84651fb7ea182a3881256a0e4ac84f6090e8648e7d62a0ba71a322fcf20b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2364,"rank":2364,"depth":3,"x":2451.04,"y":158.532,"cluster":"homological-algebra"},{"id":"stacks:05QH","tag":"05QH","title":"Filtrations · Lemma 05QH","summary":"Let A be an abelian category. Let A → B → C be a complex of filtered objects of A. Assume A, B, C have finite filtrations and that gr(A) → gr(B) → gr(C) is exact. Then • for each p ∈ Z the sequence gr^p(A) → gr^p(B) → gr^p(C) is exact, • for each p ∈ Z the sequence F^p(A) → F^p(B) → F^p(C) is exact, • for each p ∈ Z the sequence A/F^p(A) → B/F^p(B) → C/F^p(C) is exact, • the maps A → B and B → C are strict, and • A → B → C is exact (as a sequence in A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A \\to B \\to C$ be a complex of filtered objects of $\\mathcal{A}$.\nAssume $A, B, C$ have finite filtrations and that\n$\\text{gr}(A) \\to \\text{gr}(B) \\to \\text{gr}(C)$ is exact.\nThen\n\\begin{enumerate}\n\\item for each $p \\in \\mathbf{Z}$ the sequence\n$\\text{gr}^p(A) \\to \\text{gr}^p(B) \\to \\text{gr}^p(C)$ is exact,\n\\item for each $p \\in \\mathbf{Z}$ the sequence\n$F^p(A) \\to F^p(B) \\to F^p(C)$ is exact,\n\\item for each $p \\in \\mathbf{Z}$ the sequence\n$A/F^p(A) \\to B/F^p(B) \\to C/F^p(C)$ is exact,\n\\item the maps $A \\to B$ and $B \\to C$ are strict, and\n\\item $A \\to B \\to C$ is exact (as a sequence in $\\mathcal{A}$).\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Filtrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QH","source_file":"homology.tex","source_line":5094,"source_end_line":5111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5094-L5111","statement_sha256":"84d233616ffd48706f23f371076064a93490b88f2903915a0a8e23a2f922e449","origin":"The Stacks Project","memory_eligible":false,"source_rank":2365,"rank":2365,"depth":0,"x":2551.085,"y":249.015,"cluster":"homological-algebra"},{"id":"stacks:011N","tag":"011N","title":"Spectral sequences · Definition 011N","summary":"Let A be an abelian category. • A spectral sequence in A is given by a system (E_r, d_r)_r ≥ 1 where each E_r is an object of A, each d_r : E_r → E_r is a morphism such that d_r ∘ d_r = 0 and E_r + 1 = Ker(d_r)/Im(d_r) for r ≥ 1. • A morphism of spectral sequences f : (E_r, d_r)_r ≥ 1 → (E'_r, d'_r)_r ≥ 1 is given by a family of morphisms f_r : E_r → E'_r such that f_r ∘ d_r = d'_r ∘ f_r and such that f_r + 1 is the morphism induced by f_r via the identifications E_r + 1…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item A {\\it spectral sequence in $\\mathcal{A}$} is given by a\nsystem $(E_r, d_r)_{r \\geq 1}$ where each $E_r$ is an object\nof $\\mathcal{A}$, each $d_r : E_r \\to E_r$ is a morphism such\nthat $d_r \\circ d_r = 0$ and $E_{r + 1} = \\Ker(d_r)/\\Im(d_r)$\nfor $r \\geq 1$.\n\\item A {\\it morphism of spectral sequences}\n$f : (E_r, d_r)_{r \\geq 1} \\to (E'_r, d'_r)_{r \\geq 1}$ is\ngiven by a family of morphisms $f_r : E_r \\to E'_r$ such that\n$f_r \\circ d_r = d'_r \\circ f_r$ and such that $f_{r + 1}$\nis the morphism induced by $f_r$ via the identifications\n$E_{r + 1} = \\Ker(d_r)/\\Im(d_r)$\nand\n$E'_{r + 1} = \\Ker(d'_r)/\\Im(d'_r)$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011N","source_file":"homology.tex","source_line":5163,"source_end_line":5181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5163-L5181","statement_sha256":"eba520b052bdbf58c11afd1d98e8d710ddf87dd4a940fd95344019fa0d989673","origin":"The Stacks Project","memory_eligible":false,"source_rank":2366,"rank":2366,"depth":0,"x":2403.932,"y":239.019,"cluster":"homological-algebra"},{"id":"stacks:011O","tag":"011O","title":"Spectral sequences · Definition 011O","summary":"Let A be an abelian category. Let (E_r, d_r)_r ≥ 1 be a spectral sequence. • If the subobjects Z_∞ = ⋂ Z_r and B_∞ = ⋃ B_r of E_1 exist then we define the limit of the spectral sequence to be the object E_∞ = Z_∞/B_∞. • We say that the spectral sequence degenerates at E_r if the differentials d_r, d_r + 1, … are all zero.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(E_r, d_r)_{r \\geq 1}$ be a spectral sequence.\n\\begin{enumerate}\n\\item If the subobjects $Z_{\\infty} = \\bigcap Z_r$\nand $B_{\\infty} = \\bigcup B_r$ of $E_1$ exist then we define\nthe {\\it limit}\\footnote{This notation is not universally accepted. In some\nreferences an additional pair of subobjects\n$Z_\\infty$ and $B_\\infty$ of $E_1$ such that\n$0 = B_1 \\subset B_2 \\subset \\ldots \\subset B_\\infty \\subset Z_\\infty\n\\subset \\ldots \\subset Z_2 \\subset Z_1 = E_1$\nis part of the data comprising a spectral sequence!}\nof the spectral sequence to be the object\n$E_{\\infty} = Z_{\\infty}/B_{\\infty}$.\n\\item We say that the spectral sequence {\\it degenerates at $E_r$}\nif the differentials $d_r, d_{r + 1}, \\ldots$ are all zero.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011O","source_file":"homology.tex","source_line":5218,"source_end_line":5236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5218-L5236","statement_sha256":"252ab9c8b5b316b0f478461d1a73b2e0a3fcbd6f8917fa22ecd87c94231519b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2367,"rank":2367,"depth":0,"x":2520.966,"y":162.575,"cluster":"homological-algebra"},{"id":"stacks:011Q","tag":"011Q","title":"Spectral sequences: exact couples · Definition 011Q","summary":"Let A be an abelian category. • An exact couple is a datum (A, E, α, f, g) where A, E are objects of A and α, f, g are morphisms as in the following diagram xymatrix A ar[rr]_α & & A ar[ld]^g & E ar[lu]^f & with the property that the kernel of each arrow is the image of its predecessor. So Ker(α) = Im(f), Ker(f) = Im(g), and Ker(g) = Im(α). • A morphism of exact couples t : (A, E, α, f, g) → (A', E', α', f', g') is given by morphisms t_A : A → A' and t_E : E → E' such…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item An {\\it exact couple} is a datum $(A, E, \\alpha, f, g)$ where\n$A$, $E$ are objects of $\\mathcal{A}$ and $\\alpha$, $f$, $g$\nare morphisms as in the following diagram\n$$\n\\xymatrix{\nA \\ar[rr]_{\\alpha} & & A \\ar[ld]^g \\\\\n& E \\ar[lu]^f &\n}\n$$\nwith the property that the kernel of each arrow is the image\nof its predecessor. So $\\Ker(\\alpha) = \\Im(f)$,\n$\\Ker(f) = \\Im(g)$, and $\\Ker(g) = \\Im(\\alpha)$.\n\\item A {\\it morphism of exact couples}\n$t : (A, E, \\alpha, f, g) \\to (A', E', \\alpha', f', g')$\nis given by morphisms $t_A : A \\to A'$ and\n$t_E : E \\to E'$ such that\n$\\alpha' \\circ t_A = t_A \\circ \\alpha$,\n$f' \\circ t_E = t_A \\circ f$, and\n$g' \\circ t_A = t_E \\circ g$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: exact couples","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011Q","source_file":"homology.tex","source_line":5288,"source_end_line":5312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5288-L5312","statement_sha256":"bc70ad19cc25f4d9286fdfd7dae27800d01b08b8617f497554605db5ba664dd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2368,"rank":2368,"depth":0,"x":2496.038,"y":285.859,"cluster":"homological-algebra"},{"id":"stacks:011R","tag":"011R","title":"Spectral sequences: exact couples · Lemma 011R","summary":"Let (A, E, α, f, g) be an exact couple in an abelian category A. Set • d = g ∘ f : E → E so that d ∘ d = 0, • E' = Ker(d)/Im(d), • A' = Im(α), • α' : A' → A' induced by α, • f' : E' → A' induced by f, • g' : A' → E' induced by \"g ∘ α^-1\". Then we have • Ker(d) = f^-1(Ker(g)) = f^-1(Im(α)), • Im(d) = g(Im(f)) = g(Ker(α)), • (A', E', α', f', g') is an exact couple.","statement_latex":"Let $(A, E, \\alpha, f, g)$ be an exact couple in an abelian category\n$\\mathcal{A}$. Set\n\\begin{enumerate}\n\\item $d = g \\circ f : E \\to E$ so that $d \\circ d = 0$,\n\\item $E' = \\Ker(d)/\\Im(d)$,\n\\item $A' = \\Im(\\alpha)$,\n\\item $\\alpha' : A' \\to A'$ induced by $\\alpha$,\n\\item $f' : E' \\to A'$ induced by $f$,\n\\item $g' : A' \\to E'$ induced by ``$g \\circ \\alpha^{-1}$''.\n\\end{enumerate}\nThen we have\n\\begin{enumerate}\n\\item $\\Ker(d) = f^{-1}(\\Ker(g)) = f^{-1}(\\Im(\\alpha))$,\n\\item $\\Im(d) = g(\\Im(f)) = g(\\Ker(\\alpha))$,\n\\item $(A', E', \\alpha', f', g')$ is an exact couple.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: exact couples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011R","source_file":"homology.tex","source_line":5314,"source_end_line":5332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5314-L5332","statement_sha256":"a283d6bccc77b1b4f4284a5c32f16ee130e2fef23ef88c88c6fe404ec60026c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2369,"rank":2369,"depth":0,"x":2414.947,"y":180.375,"cluster":"homological-algebra"},{"id":"stacks:011S","tag":"011S","title":"Spectral sequences: exact couples · Definition 011S","summary":"Let A be an abelian category. Let (A, E, α, f, g) be an exact couple. The spectral sequence associated to the exact couple is the spectral sequence (E_r, d_r)_r ≥ 1 with E_1 = E, d_1 = d, E_2 = E', d_2 = d' = g' ∘ f', E_3 = E\", d_3 = d\" = g\" ∘ f\", and so on.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(A, E, \\alpha, f, g)$ be an exact couple.\nThe {\\it spectral sequence associated to the exact couple}\nis the spectral sequence $(E_r, d_r)_{r \\geq 1}$ with\n$E_1 = E$, $d_1 = d$, $E_2 = E'$, $d_2 = d' = g' \\circ f'$,\n$E_3 = E''$, $d_3 = d'' = g'' \\circ f''$,\nand so on.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: exact couples","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011S","source_file":"homology.tex","source_line":5344,"source_end_line":5353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5344-L5353","statement_sha256":"6a5e3c8f8fb7503be66a506ead30642a8af12df620205bad957427da6163ab6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2370,"rank":2370,"depth":0,"x":2560.157,"y":212.275,"cluster":"homological-algebra"},{"id":"stacks:011T","tag":"011T","title":"Spectral sequences: exact couples · Lemma 011T","summary":"Let A be an abelian category. Let (A, E, α, f, g) be an exact couple. Let (E_r, d_r)_r ≥ 1 be the spectral sequence associated to the exact couple. In this case we have 0 = B_1 ⊂ … ⊂ B_r + 1 = g(Ker(α^r)) ⊂ … ⊂ Z_r + 1 = f^-1(Im(α^r)) ⊂ … ⊂ Z_1 = E and the map d_r + 1 : E_r + 1 → E_r + 1 is described by the following rule: For any (test) object T of A and any elements x : T → Z_r + 1 and y : T → A such that f ∘ x = α^r ∘ y we have d_r + 1 ∘ overlinex = overlineg ∘ y where…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(A, E, \\alpha, f, g)$ be an exact couple.\nLet $(E_r, d_r)_{r \\geq 1}$ be the spectral sequence\nassociated to the exact couple.\nIn this case we have\n$$\n0 = B_1 \\subset \\ldots \\subset\nB_{r + 1} = g(\\Ker(\\alpha^r))\n\\subset \\ldots \\subset\nZ_{r + 1} = f^{-1}(\\Im(\\alpha^r))\n\\subset \\ldots \\subset Z_1 = E\n$$\nand the map $d_{r + 1} : E_{r + 1} \\to E_{r + 1}$\nis described by the following rule:\nFor any (test) object $T$ of $\\mathcal{A}$ and any elements\n$x : T \\to Z_{r + 1}$ and $y : T \\to A$ such that\n$f \\circ x = \\alpha^r \\circ y$ we have\n$$\nd_{r + 1} \\circ \\overline{x} = \\overline{g \\circ y}\n$$\nwhere $\\overline{x} : T \\to E_{r + 1}$ is the\ninduced morphism.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: exact couples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011T","source_file":"homology.tex","source_line":5355,"source_end_line":5379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5355-L5379","statement_sha256":"7ce41dd726df82b573bffdeeb143030c6f039cf8d363311a6f0a5d821ffa8d08","origin":"The Stacks Project","memory_eligible":false,"source_rank":2371,"rank":2371,"depth":0,"x":2426.893,"y":271.385,"cluster":"homological-algebra"},{"id":"stacks:011V","tag":"011V","title":"Spectral sequences: differential objects · Definition 011V","summary":"Let A be an abelian category. A differential object of A is a pair (A, d) consisting of an object A of A endowed with a selfmap d such that d ∘ d = 0. A morphism of differential objects (A, d) → (B, d) is given by a morphism α : A → B such that d ∘ α = α ∘ d.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nA {\\it differential object} of $\\mathcal{A}$\nis a pair $(A, d)$ consisting of an\nobject $A$ of $\\mathcal{A}$\nendowed with a selfmap $d$ such that $d \\circ d = 0$.\nA {\\it morphism of differential objects} $(A, d) \\to (B, d)$\nis given by a morphism $\\alpha : A \\to B$ such that\n$d \\circ \\alpha = \\alpha \\circ d$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: differential objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011V","source_file":"homology.tex","source_line":5471,"source_end_line":5481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5471-L5481","statement_sha256":"ff967f22c77ae16b0c3cf0dd96645303ec0e768b20b1e9b3c69be0467e0d3bdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2372,"rank":2372,"depth":0,"x":2477.829,"y":151.704,"cluster":"homological-algebra"},{"id":"stacks:011W","tag":"011W","title":"Spectral sequences: differential objects · Lemma 011W","summary":"The category of differential objects of an abelian category is itself an abelian category. Let A be an abelian category. The category of differential objects of A is abelian.","statement_latex":"\\begin{slogan}\nThe category of differential objects of an abelian category is itself\nan abelian category.\n\\end{slogan}\nLet $\\mathcal{A}$ be an abelian category.\nThe category of differential objects of $\\mathcal{A}$ is abelian.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: differential objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011W","source_file":"homology.tex","source_line":5483,"source_end_line":5491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5483-L5491","statement_sha256":"27b31427ad306c46ac9c34a541cdedf4389a4749b312c95001edebaed712b6b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2373,"rank":2373,"depth":0,"x":2536.746,"y":269.323,"cluster":"homological-algebra"},{"id":"stacks:011X","tag":"011X","title":"Spectral sequences: differential objects · Definition 011X","summary":"For a differential object (A, d) we denote H(A, d) = Ker(d)/Im(d) its homology.","statement_latex":"For a differential object $(A, d)$ we denote\n$$\nH(A, d) = \\Ker(d)/\\Im(d)\n$$\nits {\\it homology}.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: differential objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011X","source_file":"homology.tex","source_line":5497,"source_end_line":5504,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5497-L5504","statement_sha256":"01621c97318e6af9a72791472e8ef7304293009065c593d12de46efcf7dbf1fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":2374,"rank":2374,"depth":0,"x":2398.172,"y":215.813,"cluster":"homological-algebra"},{"id":"stacks:011Y","tag":"011Y","title":"Spectral sequences: differential objects · Lemma 011Y","summary":"Let A be an abelian category. Let 0 → (A, d) → (B, d) → (C, d) → 0 be a short exact sequence of differential objects. Then we get an exact homology sequence … → H(C, d) → H(A, d) → H(B, d) → H(C, d) → …","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $0 \\to (A, d) \\to (B, d) \\to (C, d) \\to 0$ be a short exact sequence\nof differential objects. Then we get an exact homology sequence\n$$\n\\ldots \\to H(C, d) \\to H(A, d) \\to H(B, d) \\to H(C, d) \\to \\ldots\n$$","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: differential objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011Y","source_file":"homology.tex","source_line":5506,"source_end_line":5514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5506-L5514","statement_sha256":"b97230399ad4649a7d29d03f3ce81a68d5cba4b0d5fedeea83f5d2417a8ca19a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2375,"rank":2375,"depth":7,"x":2543.954,"y":176.488,"cluster":"homological-algebra"},{"id":"stacks:011Z","tag":"011Z","title":"Spectral sequences: differential objects · Definition 011Z","summary":"Let A be an abelian category. Let (A, d) be a differential object of A. Let α : A → A be an injective selfmap of A which commutes with d. The spectral sequence associated to (A, d, α) is the spectral sequence (E_r, d_r)_r ≥ 0 described above.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(A, d)$ be a differential object of $\\mathcal{A}$.\nLet $\\alpha : A \\to A$ be an injective selfmap of $A$ which\ncommutes with $d$. The {\\it spectral sequence associated to\n$(A, d, \\alpha)$} is the spectral sequence\n$(E_r, d_r)_{r \\geq 0}$ described above.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: differential objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/011Z","source_file":"homology.tex","source_line":5583,"source_end_line":5591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5583-L5591","statement_sha256":"1fd1bc2b0b52be7505a36ba8dc0f96d7b9cbfe387f408e2a47e4e7d5d4dfd453","origin":"The Stacks Project","memory_eligible":false,"source_rank":2376,"rank":2376,"depth":0,"x":2467.786,"y":288.638,"cluster":"homological-algebra"},{"id":"stacks:012B","tag":"012B","title":"Spectral sequences: filtered differential objects · Definition 012B","summary":"Let A be an abelian category. A filtered differential object (K, F, d) is a filtered object (K, F) of A endowed with an endomorphism d : (K, F) → (K, F) whose square is zero: d ∘ d = 0.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nA {\\it filtered differential object} $(K, F, d)$ is a filtered object\n$(K, F)$ of $\\mathcal{A}$ endowed with an endomorphism\n$d : (K, F) \\to (K, F)$ whose square is zero: $d \\circ d = 0$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered differential objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012B","source_file":"homology.tex","source_line":5669,"source_end_line":5675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5669-L5675","statement_sha256":"eb5e1c00aacce03f00482508eaa071d32d6fcc923e8498f38faea32de3d6418d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2377,"rank":2377,"depth":0,"x":2433.633,"y":162.236,"cluster":"homological-algebra"},{"id":"stacks:012C","tag":"012C","title":"Spectral sequences: filtered differential objects · Lemma 012C","summary":"Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. There is a spectral sequence (E_r, d_r)_r ≥ 0 in Gr(A) associated to (K, F, d) such that d_r : E_r → E_r[r] for all r and such that the graded pieces E_r^p and maps d_r^p : E_r^p → E_r^p + r are as given above. Furthermore, E_0^p = gr^p K, d_0^p = gr^p(d), and E_1^p = H(gr^pK, gr^p(d)).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a\nfiltered differential object of $\\mathcal{A}$. There is a\nspectral sequence $(E_r, d_r)_{r \\geq 0}$ in $\\text{Gr}(\\mathcal{A})$\nassociated to $(K, F, d)$ such that $d_r : E_r \\to E_r[r]$\nfor all $r$ and such that the graded pieces\n$E_r^p$ and maps $d_r^p : E_r^p \\to E_r^{p + r}$\nare as given above. Furthermore, $E_0^p = \\text{gr}^p K$,\n$d_0^p = \\text{gr}^p(d)$, and $E_1^p = H(\\text{gr}^pK, \\text{gr}^p(d))$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered differential objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012C","source_file":"homology.tex","source_line":5748,"source_end_line":5758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5748-L5758","statement_sha256":"f599e8ebc385edc06a60d35a2b9d441764e66711b0ae9424cb69e45d7f140f51","origin":"The Stacks Project","memory_eligible":false,"source_rank":2378,"rank":2378,"depth":0,"x":2560.949,"y":236.346,"cluster":"homological-algebra"},{"id":"stacks:012D","tag":"012D","title":"Spectral sequences: filtered differential objects · Lemma 012D","summary":"Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. The spectral sequence (E_r, d_r)_r ≥ 0 associated to (K, F, d) has d_1^p : E_1^p = H(gr^pK) → H(gr^p + 1K) = E_1^p + 1 equal to the boundary map in homology associated to the short exact sequence of differential objects 0 → gr^p + 1K → F^pK/F^p + 2K → gr^pK → 0.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered\ndifferential object of $\\mathcal{A}$. The spectral sequence\n$(E_r, d_r)_{r \\geq 0}$ associated to $(K, F, d)$ has\n$$\nd_1^p :\nE_1^p = H(\\text{gr}^pK)\n\\longrightarrow\nH(\\text{gr}^{p + 1}K) = E_1^{p + 1}\n$$\nequal to the boundary map in homology associated to the short\nexact sequence of differential objects\n$$\n0 \\to \\text{gr}^{p + 1}K \\to F^pK/F^{p + 2}K \\to \\text{gr}^pK \\to 0.\n$$","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered differential objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012D","source_file":"homology.tex","source_line":5797,"source_end_line":5813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5797-L5813","statement_sha256":"89405a3db916576dabf77e3a2d3c8e19b11ccc69a3a506d76b380b9ee9bd6433","origin":"The Stacks Project","memory_eligible":false,"source_rank":2379,"rank":2379,"depth":1,"x":2406.889,"y":254.007,"cluster":"homological-algebra"},{"id":"stacks:012E","tag":"012E","title":"Spectral sequences: filtered differential objects · Definition 012E","summary":"Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. The induced filtration on H(K, d) is the filtration defined by F^pH(K, d) = Im(H(F^pK, d) → H(K, d)).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$.\nThe {\\it induced filtration} on $H(K, d)$ is the filtration defined\nby $F^pH(K, d) = \\Im(H(F^pK, d) \\to H(K, d))$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered differential objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012E","source_file":"homology.tex","source_line":5820,"source_end_line":5826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5820-L5826","statement_sha256":"d7a1c40aa7d90752a6fb14057ad4ff4e762a411ea784a10a1a2cdfbb93020eac","origin":"The Stacks Project","memory_eligible":false,"source_rank":2380,"rank":2380,"depth":0,"x":2506.663,"y":153.19,"cluster":"homological-algebra"},{"id":"stacks:012F","tag":"012F","title":"Spectral sequences: filtered differential objects · Lemma 012F","summary":"Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. If Z_∞^p and B_∞^p exist (see proof), then • the limit E_∞ exists and is graded having E_∞^p = Z_∞^p/B_∞^p in degree p, and • the associated graded gr(H(K)) of the cohomology of K is a graded subquotient of the graded limit object E_∞.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered\ndifferential object of $\\mathcal{A}$. If $Z_\\infty^p$ and $B_\\infty^p$\nexist (see proof), then\n\\begin{enumerate}\n\\item the limit $E_\\infty$ exists and is graded having\n$E_\\infty^p = Z_\\infty^p/B_\\infty^p$ in degree $p$, and\n\\item the associated graded $\\text{gr}(H(K))$ of the cohomology of $K$\nis a graded subquotient of the graded limit object $E_\\infty$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered differential objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012F","source_file":"homology.tex","source_line":5841,"source_end_line":5852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5841-L5852","statement_sha256":"76bb0b6c69466ba2f5d0c4323066c9d8c6c14a192bc38be8f390d8be2a9732c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2381,"rank":2381,"depth":1,"x":2514.193,"y":284.633,"cluster":"homological-algebra"},{"id":"stacks:012I","tag":"012I","title":"Spectral sequences: filtered differential objects · Definition 012I","summary":"Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. We say the spectral sequence associated to (K, F, d) • weakly converges to H(K) if grH(K) = E_∞ via Lemma [Tag 012F], • abuts to H(K) if it weakly converges to H(K) and we have ⋂ F^pH(K) = 0 and ⋃ F^pH(K) = H(K),","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$.\nWe say the spectral sequence associated to $(K, F, d)$\n\\begin{enumerate}\n\\item {\\it weakly converges to $H(K)$} if $\\text{gr}H(K) = E_{\\infty}$\nvia Lemma \\ref{lemma-compute-filtered-cohomology},\n\\item {\\it abuts to $H(K)$} if it weakly converges to $H(K)$ and\nwe have $\\bigcap F^pH(K) = 0$ and $\\bigcup F^pH(K) = H(K)$,\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered differential objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012I","source_file":"homology.tex","source_line":5905,"source_end_line":5916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5905-L5916","statement_sha256":"84c5a921f166d481d94465c920144c37cd4420f8228de02edc2b14d0070d3320","origin":"The Stacks Project","memory_eligible":false,"source_rank":2382,"rank":2382,"depth":2,"x":2402.526,"y":191.638,"cluster":"homological-algebra"},{"id":"stacks:012J","tag":"012J","title":"Spectral sequences: filtered differential objects · Lemma 012J","summary":"Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. The associated spectral sequence • weakly converges to H(K) if and only if for every p ∈ Z we have equality in equations ([Tag 012H]) and ([Tag 012G]), • abuts to H(K) if and only if it weakly converges to H(K) and ⋂_p (Ker(d) ∩ F^pK + Im(d)) = Im(d) and ⋃_p (Ker(d) ∩ F^pK + Im(d)) = Ker(d).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(K, F, d)$ be a filtered differential object of $\\mathcal{A}$.\nThe associated spectral sequence\n\\begin{enumerate}\n\\item weakly converges to $H(K)$ if and only if for every\n$p \\in \\mathbf{Z}$ we have equality in equations\n(\\ref{equation-at-bottom}) and (\\ref{equation-on-top}),\n\\item abuts to $H(K)$ if and only if it weakly converges to $H(K)$ and\n$\\bigcap_p (\\Ker(d) \\cap F^pK + \\Im(d)) = \\Im(d)$\nand $\\bigcup_p (\\Ker(d) \\cap F^pK + \\Im(d)) = \\Ker(d)$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered differential objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012J","source_file":"homology.tex","source_line":5922,"source_end_line":5935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5922-L5935","statement_sha256":"44d32d4ca790d8c9b8856ba8eddc1124f5592c65a8e799e4921fb93c704790c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2383,"rank":2383,"depth":2,"x":2560.229,"y":196.868,"cluster":"homological-algebra"},{"id":"stacks:012L","tag":"012L","title":"Spectral sequences: filtered complexes · Definition 012L","summary":"Let A be an abelian category. A filtered complex K^bullet of A is a complex of Fil(A) (see Definition [Tag 0121]).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nA {\\it filtered complex $K^\\bullet$ of $\\mathcal{A}$}\nis a complex of $\\text{Fil}(\\mathcal{A})$ (see\nDefinition \\ref{definition-filtered}).","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012L","source_file":"homology.tex","source_line":5957,"source_end_line":5963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L5957-L5963","statement_sha256":"5148476c692af1d50394be255f76c7e469d4a00561a84837d3af745df600ddb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2384,"rank":2384,"depth":1,"x":2439.295,"y":282.808,"cluster":"homological-algebra"},{"id":"stacks:012M","tag":"012M","title":"Spectral sequences: filtered complexes · Lemma 012M","summary":"Let A be an abelian category. Let (K^bullet, F) be a filtered complex of A. There is a spectral sequence (E_r, d_r)_r ≥ 0 in the category of bigraded objects of A associated to (K^bullet, F) such that d_r has bidegree (r, - r + 1) and such that E_r has bigraded pieces E_r^p, q and maps d_r^p, q : E_r^p, q → E_r^p + r, q - r + 1 as given above. Furthermore, we have E_0^p, q = gr^p(K^p + q), d_0^p, q = gr^p(d^p + q), and E_1^p, q = H^p + q(gr^p(K^bullet)).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a\nfiltered complex of $\\mathcal{A}$. There is a spectral sequence\n$(E_r, d_r)_{r \\geq 0}$ in the category of bigraded objects of $\\mathcal{A}$\nassociated to $(K^\\bullet, F)$ such that $d_r$ has bidegree $(r, - r + 1)$\nand such that $E_r$ has bigraded pieces $E_r^{p, q}$ and maps\n$d_r^{p, q} : E_r^{p, q} \\to E_r^{p + r, q - r + 1}$ as given above.\nFurthermore, we have $E_0^{p, q} = \\text{gr}^p(K^{p + q})$,\n$d_0^{p, q} = \\text{gr}^p(d^{p + q})$,\nand $E_1^{p, q} = H^{p + q}(\\text{gr}^p(K^\\bullet))$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012M","source_file":"homology.tex","source_line":6048,"source_end_line":6059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6048-L6059","statement_sha256":"8dec21538300fe8e030c843756d0c2dcea207608938960b646dce5ac1fa25672","origin":"The Stacks Project","memory_eligible":false,"source_rank":2385,"rank":2385,"depth":0,"x":2459.433,"y":150.338,"cluster":"homological-algebra"},{"id":"stacks:012N","tag":"012N","title":"Spectral sequences: filtered complexes · Lemma 012N","summary":"Let A be an abelian category. Let (K^bullet, F) be a filtered complex of A. Assume A has countable direct sums. Let (E_r, d_r)_r ≥ 0 be the spectral sequence associated to (K^bullet, F). • The map d_1^p, q : E_1^p, q = H^p + q(gr^p(K^bullet)) → E_1^p + 1, q = H^p + q + 1(gr^p + 1(K^bullet)) is equal to the boundary map in cohomology associated to the short exact sequence of complexes 0 → gr^p + 1(K^bullet) → F^pK^bullet/F^p + 2K^bullet → gr^p(K^bullet) → 0. • Assume that…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$.\nAssume $\\mathcal{A}$ has countable direct sums.\nLet $(E_r, d_r)_{r \\geq 0}$ be the spectral sequence\nassociated to $(K^\\bullet, F)$.\n\\begin{enumerate}\n\\item The map\n$$\nd_1^{p, q} :\nE_1^{p, q} = H^{p + q}(\\text{gr}^p(K^\\bullet))\n\\longrightarrow\nE_1^{p + 1, q} = H^{p + q + 1}(\\text{gr}^{p + 1}(K^\\bullet))\n$$\nis equal to the boundary map in cohomology associated to the short\nexact sequence of complexes\n$$\n0 \\to \\text{gr}^{p + 1}(K^\\bullet) \\to\nF^pK^\\bullet/F^{p + 2}K^\\bullet \\to \\text{gr}^p(K^\\bullet) \\to 0.\n$$\n\\item Assume that $d(F^pK) \\subset F^{p + 1}K$ for all $p \\in \\mathbf{Z}$.\nThen $d$ induces the zero differential on $\\text{gr}^p(K^\\bullet)$\nand hence\n$E_1^{p, q} = \\text{gr}^p(K^\\bullet)^{p + q}$.\nFurthermore, in this case\n$$\nd_1^{p, q} :\nE_1^{p, q} = \\text{gr}^p(K^\\bullet)^{p + q}\n\\longrightarrow\nE_1^{p + 1, q} = \\text{gr}^{p + 1}(K^\\bullet)^{p + q + 1}\n$$\nis the morphism induced by $d$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012N","source_file":"homology.tex","source_line":6106,"source_end_line":6140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6106-L6140","statement_sha256":"4c24f408fa8d951812f99b636261ab55cb7812377893fb1c130aed940b7b2e2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2386,"rank":2386,"depth":1,"x":2551.441,"y":259.839,"cluster":"homological-algebra"},{"id":"stacks:012O","tag":"012O","title":"Spectral sequences: filtered complexes · Lemma 012O","summary":"Let A be an abelian category. Let α : (K^bullet, F) → (L^bullet, F) be a morphism of filtered complexes of A. Let (E_r(K), d_r)_r ≥ 0, resp. (E_r(L), d_r)_r ≥ 0 be the spectral sequence associated to (K^bullet, F), resp. (L^bullet, F). The morphism α induces a canonical morphism of spectral sequences (α_r : E_r(K) → E_r(L))_r ≥ 0 compatible with the bigradings.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\alpha : (K^\\bullet, F) \\to (L^\\bullet, F)$ be a morphism of\nfiltered complexes of $\\mathcal{A}$. Let $(E_r(K), d_r)_{r \\geq 0}$,\nresp.\\ $(E_r(L), d_r)_{r \\geq 0}$ be the spectral sequence associated\nto $(K^\\bullet, F)$, resp.\\ $(L^\\bullet, F)$.\nThe morphism $\\alpha$ induces a canonical morphism of spectral\nsequences $\\{\\alpha_r : E_r(K) \\to E_r(L)\\}_{r \\geq 0}$ compatible\nwith the bigradings.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012O","source_file":"homology.tex","source_line":6147,"source_end_line":6157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6147-L6157","statement_sha256":"ac5c0c0df9af2e1584ee055ecdba977f514c1b08ce8c4ee1d2436bad25dec92d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2387,"rank":2387,"depth":0,"x":2394.982,"y":231.2,"cluster":"homological-algebra"},{"id":"stacks:012P","tag":"012P","title":"Spectral sequences: filtered complexes · Definition 012P","summary":"Let A be an abelian category. Let (K^bullet, F) be a filtered complex of A. The induced filtration on H^n(K^bullet) is the filtration defined by F^pH^n(K^bullet) = Im(H^n(F^pK^bullet) → H^n(K^bullet)).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(K^\\bullet, F)$ be a filtered complex of $\\mathcal{A}$.\nThe {\\it induced filtration} on $H^n(K^\\bullet)$ is the filtration defined\nby $F^pH^n(K^\\bullet) = \\Im(H^n(F^pK^\\bullet) \\to H^n(K^\\bullet))$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012P","source_file":"homology.tex","source_line":6164,"source_end_line":6170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6164-L6170","statement_sha256":"72328f3fef44b435afc08ce81f095ba4a5ca0c351c521bbb2465e08603d0eab0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2388,"rank":2388,"depth":0,"x":2533.874,"y":163.3,"cluster":"homological-algebra"},{"id":"stacks:012Q","tag":"012Q","title":"Spectral sequences: filtered complexes · Lemma 012Q","summary":"Let A be an abelian category. Let (K^bullet, F) be a filtered complex of A. If Z_∞^p, q and B_∞^p, q exist (see proof), then • the limit E_∞ exists and is a bigraded object having E_∞^p, q = Z_∞^p, q/B_∞^p, q in bidegree (p, q), • the pth graded part gr^pH^n(K^bullet) of the nth cohomology object of K^bullet is a subquotient of E_∞^p, n - p.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered\ncomplex of $\\mathcal{A}$. If $Z_\\infty^{p, q}$ and $B_\\infty^{p, q}$ exist\n(see proof), then\n\\begin{enumerate}\n\\item the limit $E_\\infty$ exists and is a bigraded object having\n$E_\\infty^{p, q} = Z_\\infty^{p, q}/B_\\infty^{p, q}$ in bidegree $(p, q)$,\n\\item the $p$th graded part $\\text{gr}^pH^n(K^\\bullet)$ of the\n$n$th cohomology object of $K^\\bullet$ is a subquotient of\n$E_\\infty^{p, n - p}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012Q","source_file":"homology.tex","source_line":6187,"source_end_line":6199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6187-L6199","statement_sha256":"8718e23bfa7d17cd3d6eb36c207fd42890e4860f8984da3ddf5813288b00a5bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2389,"rank":2389,"depth":1,"x":2485.891,"y":292.633,"cluster":"homological-algebra"},{"id":"stacks:0BDU","tag":"0BDU","title":"Spectral sequences: filtered complexes · Definition 0BDU","summary":"Let A be an abelian category. Let (E_r, d_r)_r ≥ r_0 be a spectral sequence of bigraded objects of A with d_r of bidegree (r, -r + 1). We say such a spectral sequence is • regular if for all p, q ∈ Z there is a b = b(p, q) such that the maps d_r^p, q : E_r^p, q → E_r^p + r, q - r + 1 are zero for r ≥ b, • coregular if for all p, q ∈ Z there is a b = b(p, q) such that the maps d_r^p - r, q + r - 1 : E_r^p - r, q + r - 1 → E_r^p, q are zero for r ≥ b, • bounded if for all n…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(E_r, d_r)_{r \\geq r_0}$\nbe a spectral sequence of bigraded objects of\n$\\mathcal{A}$ with $d_r$ of bidegree $(r, -r + 1)$.\nWe say such a spectral sequence is\n\\begin{enumerate}\n\\item {\\it regular} if for all $p, q \\in \\mathbf{Z}$ there is\na $b = b(p, q)$ such that the maps\n$d_r^{p, q} : E_r^{p, q} \\to E_r^{p + r, q - r + 1}$ are zero for $r \\geq b$,\n\\item {\\it coregular} if for all $p, q \\in \\mathbf{Z}$ there is a\n$b = b(p, q)$ such that the maps\n$d_r^{p - r, q + r - 1} : E_r^{p - r, q + r - 1} \\to E_r^{p, q}$\nare zero for $r \\geq b$,\n\\item {\\it bounded} if for all $n$\nthere are only a finite number of nonzero $E_{r_0}^{p, n - p}$,\n\\item {\\it bounded below} if for all $n$ there is a $b = b(n)$ such that\n$E_{r_0}^{p, n - p} = 0$ for $p \\geq b$.\n\\item {\\it bounded above} if for all $n$ there is a $b = b(n)$ such that\n$E_{r_0}^{p, n - p} = 0$ for $p \\leq b$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDU","source_file":"homology.tex","source_line":6261,"source_end_line":6282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6261-L6282","statement_sha256":"2cd72a0ac9e7a70df5d840d5bf8f79455081e7a4a12333fca6513661b3f677c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2390,"rank":2390,"depth":0,"x":2417.029,"y":169.61,"cluster":"homological-algebra"},{"id":"stacks:0BDV","tag":"0BDV","title":"Spectral sequences: filtered complexes · Lemma 0BDV","summary":"In the situation of Definition [Tag 0BDU]. Let Z_r^p, q, B_r^p, q ⊂ E_r_0^p, q be the (p, q)-graded parts of Z_r, B_r defined as in Section [Tag 011M]. • The spectral sequence is regular if and only if for all p, q there exists an r = r(p, q) such that Z_r^p, q = Z_r + 1^p, q = … • The spectral sequence is coregular if and only if for all p, q there exists an r = r(p, q) such that B_r^p, q = B_r + 1^p, q = … • The spectral sequence is bounded if and only if it is both…","statement_latex":"In the situation of Definition \\ref{definition-bounded-ss}.\nLet $Z_r^{p, q}, B_r^{p, q} \\subset E_{r_0}^{p, q}$ be the\n$(p, q)$-graded parts of $Z_r, B_r$ defined as in\nSection \\ref{section-spectral-sequence}.\n\\begin{enumerate}\n\\item The spectral sequence is regular if and only if for all $p, q$\nthere exists an $r = r(p, q)$ such that\n$Z_r^{p, q} = Z_{r + 1}^{p, q} = \\ldots$\n\\item The spectral sequence is coregular if and only if for all $p, q$\nthere exists an $r = r(p, q)$ such that\n$B_r^{p, q} = B_{r + 1}^{p, q} = \\ldots$\n\\item The spectral sequence is bounded if and only if it is both\nbounded below and bounded above.\n\\item If the spectral sequence is bounded below, then it is regular.\n\\item If the spectral sequence is bounded above, then it is coregular.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDV","source_file":"homology.tex","source_line":6291,"source_end_line":6309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6291-L6309","statement_sha256":"7ab57dd568f6edbd5ad89b25197fc6326420f1aad4860173f634ea7f60df57d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2391,"rank":2391,"depth":1,"x":2567.257,"y":221.43,"cluster":"homological-algebra"},{"id":"stacks:012U","tag":"012U","title":"Spectral sequences: filtered complexes · Definition 012U","summary":"Let A be an abelian category. Let (K^bullet, F) be a filtered complex of A. We say the spectral sequence associated to (K^bullet, F) • weakly converges to H^*(K^bullet) if gr^pH^n(K^bullet) = E_∞^p, n - p via Lemma [Tag 012Q] for all p, n ∈ Z, • abuts to H^*(K^bullet) if it weakly converges to H^*(K^bullet) and ⋂_p F^pH^n(K^bullet) = 0 and ⋃_p F^p H^n(K^bullet) = H^n(K^bullet) for all n, • converges to H^*(K^bullet) if it is regular, abuts to H^*(K^bullet), and…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a\nfiltered complex of $\\mathcal{A}$. We say the spectral sequence\nassociated to $(K^\\bullet, F)$\n\\begin{enumerate}\n\\item {\\it weakly converges to $H^*(K^\\bullet)$} if\n$\\text{gr}^pH^n(K^\\bullet) = E_{\\infty}^{p, n - p}$\nvia Lemma \\ref{lemma-compute-cohomology-filtered-complex}\nfor all $p, n \\in \\mathbf{Z}$,\n\\item {\\it abuts to $H^*(K^\\bullet)$} if it weakly converges to\n$H^*(K^\\bullet)$ and $\\bigcap_p F^pH^n(K^\\bullet) = 0$ and\n$\\bigcup_p F^p H^n(K^\\bullet) = H^n(K^\\bullet)$ for all $n$,\n\\item {\\it converges to $H^*(K^\\bullet)$} if it is regular,\nabuts to $H^*(K^\\bullet)$, and\n$H^n(K^\\bullet) = \\lim_p H^n(K^\\bullet)/F^pH^n(K^\\bullet)$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012U","source_file":"homology.tex","source_line":6316,"source_end_line":6333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6316-L6333","statement_sha256":"5137511bc555faa2eeaf6f493452274ac4e381377aad23fb17ea0e898d8764d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2392,"rank":2392,"depth":2,"x":2414.283,"y":268.623,"cluster":"homological-algebra"},{"id":"stacks:012V","tag":"012V","title":"Spectral sequences: filtered complexes · Lemma 012V","summary":"Let A be an abelian category. Let (K^bullet, F) be a filtered complex of A. The associated spectral sequence • weakly converges to H^*(K^bullet) if and only if for every p, q ∈ Z we have equality in equations ([Tag 012T]) and ([Tag 012S]), • abuts to H^*(K) if and only if it weakly converges to H^*(K^bullet) and we have ⋂_p (Ker(d) ∩ F^pK^n + Im(d) ∩ K^n) = Im(d) ∩ K^n and ⋃_p (Ker(d) ∩ F^pK^n + Im(d) ∩ K^n) = Ker(d) ∩ K^n.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a filtered\ncomplex of $\\mathcal{A}$. The associated spectral sequence\n\\begin{enumerate}\n\\item weakly converges to $H^*(K^\\bullet)$ if and only if for every\n$p, q \\in \\mathbf{Z}$ we have equality in equations\n(\\ref{equation-at-bottom-bigraded}) and (\\ref{equation-on-top-bigraded}),\n\\item abuts to $H^*(K)$ if and only if it weakly converges to $H^*(K^\\bullet)$\nand we have\n$\\bigcap_p (\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n) = \\Im(d) \\cap K^n$\nand\n$\\bigcup_p (\\Ker(d) \\cap F^pK^n + \\Im(d) \\cap K^n) = \\Ker(d) \\cap K^n$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012V","source_file":"homology.tex","source_line":6341,"source_end_line":6355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6341-L6355","statement_sha256":"3e8b8c8789a35bfa187803dc26965721a762935d93a72a135d8a433f5b0851a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2393,"rank":2393,"depth":2,"x":2489.389,"y":146.607,"cluster":"homological-algebra"},{"id":"stacks:012W","tag":"012W","title":"Spectral sequences: filtered complexes · Lemma 012W","summary":"Let A be an abelian category. Let (K^bullet, F) be a filtered complex of A. Assume that the filtration on each K^n is finite (see Definition [Tag 0121]). Then • the spectral sequence associated to (K^bullet, F) is bounded, • the filtration on each H^n(K^bullet) is finite, • the spectral sequence associated to (K^bullet, F) converges to H^*(K^bullet), • if C ⊂ A is a weak Serre subcategory and for some r we have E_r^p, q ∈ C for all p, q ∈ Z, then H^n(K^bullet) is in C.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a\nfiltered complex of $\\mathcal{A}$. Assume that the filtration on each $K^n$\nis finite (see Definition \\ref{definition-filtered}). Then\n\\begin{enumerate}\n\\item the spectral sequence associated to $(K^\\bullet, F)$ is bounded,\n\\item the filtration on each $H^n(K^\\bullet)$ is finite,\n\\item the spectral sequence associated to $(K^\\bullet, F)$ converges\nto $H^*(K^\\bullet)$,\n\\item if $\\mathcal{C} \\subset \\mathcal{A}$ is a weak Serre subcategory\nand for some $r$ we have $E_r^{p, q} \\in \\mathcal{C}$ for all\n$p, q \\in \\mathbf{Z}$, then $H^n(K^\\bullet)$ is in $\\mathcal{C}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/012W","source_file":"homology.tex","source_line":6361,"source_end_line":6375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6361-L6375","statement_sha256":"5f50d15b57366ab2aaaedc12d433a2c3f27f29412a1bb7d995fdaffccc636afc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2394,"rank":2394,"depth":3,"x":2532.274,"y":279.648,"cluster":"homological-algebra"},{"id":"stacks:0BDW","tag":"0BDW","title":"Spectral sequences: filtered complexes · Lemma 0BDW","summary":"Let A be an abelian category. Let (K^bullet, F) be a filtered complex of A. Assume that the filtration on each K^n is finite (see Definition [Tag 0121]) and that for some r we have only a finite number of nonzero E_r^p, q. Then only a finite number of H^n(K^bullet) are nonzero and we have ∑ (-1)^n[H^n(K^bullet)] = ∑ (-1)^p + q [E_r^p, q] in K_0(A') where A' is the smallest weak Serre subcategory of A containing the objects E_r^p, q.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a\nfiltered complex of $\\mathcal{A}$. Assume that the filtration on each $K^n$\nis finite (see Definition \\ref{definition-filtered}) and that for some\n$r$ we have only a finite number of nonzero $E_r^{p, q}$. Then\nonly a finite number of $H^n(K^\\bullet)$ are nonzero and we have\n$$\n\\sum (-1)^n[H^n(K^\\bullet)] = \\sum (-1)^{p + q} [E_r^{p, q}]\n$$\nin $K_0(\\mathcal{A}')$ where $\\mathcal{A}'$ is the smallest weak\nSerre subcategory of $\\mathcal{A}$ containing the objects\n$E_r^{p, q}$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDW","source_file":"homology.tex","source_line":6412,"source_end_line":6425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6412-L6425","statement_sha256":"84e030db4b0c87e8e9dee79d8ce7324299986f710ee5ed83dd01869c8a07ab47","origin":"The Stacks Project","memory_eligible":false,"source_rank":2395,"rank":2395,"depth":4,"x":2393.196,"y":205.63,"cluster":"homological-algebra"},{"id":"stacks:0BK5","tag":"0BK5","title":"Spectral sequences: filtered complexes · Lemma 0BK5","summary":"Let A be an abelian category. Let (K^bullet, F) be a filtered complex of A. Assume • for every n there exist p_0(n) such that H^n(F^pK^bullet) = 0 for p ≥ p_0(n), • for every n there exist p_1(n) such that H^n(F^pK^bullet) → H^n(K^bullet) is an isomorphism for p ≤ p_1(n). Then • the spectral sequence associated to (K^bullet, F) is bounded, • the filtration on each H^n(K^bullet) is finite, • the spectral sequence associated to (K^bullet, F) converges to H^*(K^bullet).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K^\\bullet, F)$ be a\nfiltered complex of $\\mathcal{A}$. Assume\n\\begin{enumerate}\n\\item for every $n$ there exist $p_0(n)$ such that\n$H^n(F^pK^\\bullet) = 0$ for $p \\geq p_0(n)$,\n\\item for every $n$ there exist $p_1(n)$ such that\n$H^n(F^pK^\\bullet) \\to H^n(K^\\bullet)$ is an isomorphism\nfor $p \\leq p_1(n)$.\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item the spectral sequence associated to $(K^\\bullet, F)$ is bounded,\n\\item the filtration on each $H^n(K^\\bullet)$ is finite,\n\\item the spectral sequence associated to $(K^\\bullet, F)$ converges\nto $H^*(K^\\bullet)$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BK5","source_file":"homology.tex","source_line":6463,"source_end_line":6481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6463-L6481","statement_sha256":"9b787b2ca1f3382a7b5af3d756f6f526f3479c3fb8c6f4a1ef0027df7b5e4694","origin":"The Stacks Project","memory_eligible":false,"source_rank":2396,"rank":2396,"depth":3,"x":2555.816,"y":181.213,"cluster":"homological-algebra"},{"id":"stacks:0130","tag":"0130","title":"Spectral sequences: double complexes · Lemma 0130","summary":"Let A be an abelian category. Let K^bullet, bullet be a double complex. The spectral sequences associated to K^bullet, bullet have the following terms: • 'E_0^p, q = K^p, q with 'd_0^p, q = (-1)^p d_2^p, q : K^p, q → K^p, q + 1, • \"E_0^p, q = K^q, p with \"d_0^p, q = d_1^q, p : K^q, p → K^q + 1, p, • 'E_1^p, q = H^q(K^p, bullet) with 'd_1^p, q = H^q(d_1^p, bullet), • \"E_1^p, q = H^q(K^bullet, p) with \"d_1^p, q = (-1)^q H^q(d_2^bullet, p), • 'E_2^p, q =…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^{\\bullet, \\bullet}$ be a double complex.\nThe spectral sequences associated to $K^{\\bullet, \\bullet}$\nhave the following terms:\n\\begin{enumerate}\n\\item ${}'E_0^{p, q} = K^{p, q}$ with\n${}'d_0^{p, q} = (-1)^p d_2^{p, q} : K^{p, q} \\to K^{p, q + 1}$,\n\\item ${}''E_0^{p, q} = K^{q, p}$ with\n${}''d_0^{p, q} = d_1^{q, p} : K^{q, p} \\to K^{q + 1, p}$,\n\\item ${}'E_1^{p, q} = H^q(K^{p, \\bullet})$ with\n${}'d_1^{p, q} = H^q(d_1^{p, \\bullet})$,\n\\item ${}''E_1^{p, q} = H^q(K^{\\bullet, p})$ with\n${}''d_1^{p, q} = (-1)^q H^q(d_2^{\\bullet, p})$,\n\\item ${}'E_2^{p, q} = H^p_I(H^q_{II}(K^{\\bullet, \\bullet}))$,\n\\item ${}''E_2^{p, q} = H^p_{II}(H^q_I(K^{\\bullet, \\bullet}))$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: double complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0130","source_file":"homology.tex","source_line":6607,"source_end_line":6625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6607-L6625","statement_sha256":"f763896febc6a2a81b31a7a168915f2012b06551261f4e5d74f37ad7d73743d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2397,"rank":2397,"depth":0,"x":2455.204,"y":291.859,"cluster":"homological-algebra"},{"id":"stacks:0131","tag":"0131","title":"Spectral sequences: double complexes · Definition 0131","summary":"Let A be an abelian category. Let K^bullet, bullet be a double complex. We say the spectral sequence ('E_r, 'd_r)_r ≥ 0 weakly converges to H^n(Tot(K^bullet, bullet)), abuts to H^n(Tot(K^bullet, bullet)), or converges to H^n(Tot(K^bullet, bullet)) if Definition [Tag 012U] applies. Similarly we say the spectral sequence (\"E_r, \"d_r)_r ≥ 0 weakly converges to H^n(Tot(K^bullet, bullet)), abuts to H^n(Tot(K^bullet, bullet)), or converges to H^n(Tot(K^bullet, bullet)) if…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^{\\bullet, \\bullet}$ be a double complex.\nWe say the spectral sequence $({}'E_r, {}'d_r)_{r \\geq 0}$\n{\\it weakly converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$},\n{\\it abuts to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}, or\n{\\it converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}\nif Definition \\ref{definition-filtered-complex-ss-converges} applies.\nSimilarly we say the spectral sequence $({}''E_r, {}''d_r)_{r \\geq 0}$\n{\\it weakly converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$},\n{\\it abuts to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}, or\n{\\it converges to $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$}\nif Definition \\ref{definition-filtered-complex-ss-converges} applies.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: double complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0131","source_file":"homology.tex","source_line":6636,"source_end_line":6650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6636-L6650","statement_sha256":"34a455b7b2859f990d7a60fc61efc51410c7d063dfd2208341d21134c36b992c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2398,"rank":2398,"depth":3,"x":2440.367,"y":152.722,"cluster":"homological-algebra"},{"id":"stacks:0132","tag":"0132","title":"Spectral sequences: double complexes · Lemma 0132","summary":"Let A be an abelian category. Let K^bullet, bullet be a double complex. Assume that for every n ∈ Z there are only finitely many nonzero K^p, q with p + q = n. Then • the two spectral sequences associated to K^bullet, bullet are bounded, • the filtrations F_I, F_II on each H^n(Tot(K^bullet, bullet)) are finite, • the spectral sequences ('E_r, 'd_r)_r ≥ 0 and (\"E_r, \"d_r)_r ≥ 0 converge to H^*(Tot(K^bullet, bullet)), • if C ⊂ A is a weak Serre subcategory and for some r we…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $K^{\\bullet, \\bullet}$\nbe a double complex. Assume that for every $n \\in \\mathbf{Z}$ there are\nonly finitely many nonzero $K^{p, q}$ with $p + q = n$. Then\n\\begin{enumerate}\n\\item the two spectral sequences associated to $K^{\\bullet, \\bullet}$\nare bounded,\n\\item the filtrations $F_I$, $F_{II}$ on each\n$H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$ are finite,\n\\item the spectral sequences $({}'E_r, {}'d_r)_{r \\geq 0}$ and\n$({}''E_r, {}''d_r)_{r \\geq 0}$ converge to\n$H^*(\\text{Tot}(K^{\\bullet, \\bullet}))$,\n\\item if $\\mathcal{C} \\subset \\mathcal{A}$ is a weak Serre subcategory\nand for some $r$ we have ${}'E_r^{p, q} \\in \\mathcal{C}$ for all\n$p, q \\in \\mathbf{Z}$, then $H^n(\\text{Tot}(K^{\\bullet, \\bullet}))$\nis in $\\mathcal{C}$. Similarly for $({}''E_r, {}''d_r)_{r \\geq 0}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: double complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0132","source_file":"homology.tex","source_line":6671,"source_end_line":6689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6671-L6689","statement_sha256":"6d06b303fa76471c7a6e34a096b42169e18520b9e6f6a7850f0ea86445d2e9ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":2399,"rank":2399,"depth":4,"x":2563.601,"y":247.21,"cluster":"homological-algebra"},{"id":"stacks:0133","tag":"0133","title":"Spectral sequences: double complexes · Lemma 0133","summary":"Let A be an abelian category. Let K^bullet be a complex. Let A^bullet, bullet be a double complex. Let α^p : K^p → A^p, 0 be morphisms. Assume that • For every n ∈ Z there are only finitely many nonzero A^p, q with p + q = n. • We have A^p, q = 0 if q < 0. • The morphisms α^p give rise to a morphism of complexes α : K^bullet → A^bullet, 0. • The complex A^p, bullet is exact in all degrees q not = 0 and the morphism K^p → A^p, 0 induces an isomorphism K^p → Ker(d_2^p, 0).…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet$ be a complex.\nLet $A^{\\bullet, \\bullet}$ be a double complex.\nLet $\\alpha^p : K^p \\to A^{p, 0}$ be morphisms.\nAssume that\n\\begin{enumerate}\n\\item For every $n \\in \\mathbf{Z}$ there are only finitely many nonzero\n$A^{p, q}$ with $p + q = n$.\n\\item We have $A^{p, q} = 0$ if $q < 0$.\n\\item The morphisms $\\alpha^p$ give rise to a morphism\nof complexes $\\alpha : K^\\bullet \\to A^{\\bullet, 0}$.\n\\item The complex $A^{p, \\bullet}$ is exact in all degrees\n$q \\not = 0$ and the morphism $K^p \\to A^{p, 0}$ induces\nan isomorphism $K^p \\to \\Ker(d_2^{p, 0})$.\n\\end{enumerate}\nThen $\\alpha$ induces a quasi-isomorphism\n$$\nK^\\bullet \\longrightarrow \\text{Tot}(A^{\\bullet, \\bullet})\n$$\nof complexes.\nMoreover, there is a variant of this lemma involving the second\nvariable $q$ instead of $p$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: double complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0133","source_file":"homology.tex","source_line":6698,"source_end_line":6722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6698-L6722","statement_sha256":"65b342c1d4f558de98b4ffb84af12838b786aa084341aefaaf56d1cbd17ff21f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2400,"rank":2400,"depth":5,"x":2396.201,"y":247.462,"cluster":"homological-algebra"},{"id":"stacks:0FKH","tag":"0FKH","title":"Spectral sequences: double complexes · Lemma 0FKH","summary":"Let A be an abelian category. Let M^bullet be a complex of A. Let a : M^bullet[0] → (A^0, bullet → A^1, bullet → A^2, bullet → … ) be a homotopy equivalence in the category of complexes of complexes of A. Then the map α : M^bullet → Tot(A^bullet, bullet) induced by M^bullet → A^0, bullet is a homotopy equivalence.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $M^\\bullet$ be a complex of $\\mathcal{A}$. Let\n$$\na :\nM^\\bullet[0]\n\\longrightarrow\n\\left(A^{0, \\bullet} \\to A^{1, \\bullet} \\to A^{2, \\bullet} \\to \\ldots \\right)\n$$\nbe a homotopy equivalence in the category of complexes of complexes\nof $\\mathcal{A}$. Then the map\n$\\alpha : M^\\bullet \\to \\text{Tot}(A^{\\bullet, \\bullet})$\ninduced by $M^\\bullet \\to A^{0, \\bullet}$ is a homotopy equivalence.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Spectral sequences: double complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKH","source_file":"homology.tex","source_line":6745,"source_end_line":6759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6745-L6759","statement_sha256":"d4566aa6882e4cfe1e3d5d26691a1d816c505740d6b21cf718b59ea11feab810","origin":"The Stacks Project","memory_eligible":false,"source_rank":2401,"rank":2401,"depth":0,"x":2519.837,"y":151.981,"cluster":"homological-algebra"},{"id":"stacks:0E1Q","tag":"0E1Q","title":"Double complexes of abelian groups · Lemma 0E1Q","summary":"Let M^bullet be a complex of abelian groups. Let 0 → M^bullet → A_0^bullet → A_1^bullet → A_2^bullet → … be an exact complex of complexes of abelian groups. Set A^p, q = A_p^q to obtain a double complex. Then the map M^bullet → Tot(A^bullet, bullet) induced by M^bullet → A_0^bullet is a quasi-isomorphism.","statement_latex":"Let $M^\\bullet$ be a complex of abelian groups. Let\n$$\n0 \\to M^\\bullet \\to A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots\n$$\nbe an exact complex of complexes of abelian groups. Set\n$A^{p, q} = A_p^q$ to obtain a double complex.\nThen the map $M^\\bullet \\to \\text{Tot}(A^{\\bullet, \\bullet})$\ninduced by $M^\\bullet \\to A_0^\\bullet$ is a quasi-isomorphism.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Double complexes of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1Q","source_file":"homology.tex","source_line":6836,"source_end_line":6846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6836-L6846","statement_sha256":"59879f138629378aff032c73f265ea31a1fe5a91748386eda1867215f966624e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2402,"rank":2402,"depth":6,"x":2505.405,"y":292.994,"cluster":"homological-algebra"},{"id":"stacks:09IZ","tag":"09IZ","title":"Double complexes of abelian groups · Lemma 09IZ","summary":"Let M^bullet be a complex of abelian groups. Let … → A_2^bullet → A_1^bullet → A_0^bullet → M^bullet → 0 be an exact complex of complexes of abelian groups such that for all p ∈ Z the complexes … → Ker(d_A_2^bullet^p) → Ker(d_A_1^bullet^p) → Ker(d_A_0^bullet^p) → Ker(d_M^bullet^p) → 0 are exact as well. Set A^p, q = A_-p^q to obtain a double complex. Then Tot(A^bullet, bullet) → M^bullet induced by A_0^bullet → M^bullet is a quasi-isomorphism.","statement_latex":"Let $M^\\bullet$ be a complex of abelian groups. Let\n$$\n\\ldots \\to A_2^\\bullet \\to A_1^\\bullet \\to A_0^\\bullet \\to M^\\bullet \\to 0\n$$\nbe an exact complex of complexes of abelian groups such that for all\n$p \\in \\mathbf{Z}$ the complexes\n$$\n\\ldots \\to \\Ker(d_{A_2^\\bullet}^p) \\to \\Ker(d_{A_1^\\bullet}^p)\n\\to \\Ker(d_{A_0^\\bullet}^p) \\to \\Ker(d_{M^\\bullet}^p) \\to 0\n$$\nare exact as well. Set $A^{p, q} = A_{-p}^q$ to obtain a double\ncomplex. Then $\\text{Tot}(A^{\\bullet, \\bullet}) \\to M^\\bullet$\ninduced by $A_0^\\bullet \\to M^\\bullet$ is a quasi-isomorphism.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Double complexes of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IZ","source_file":"homology.tex","source_line":6872,"source_end_line":6887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6872-L6887","statement_sha256":"a66f711dd66058ecb78dbc8aba5b725063a7e629fc194eb5f5648e7fd749170e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2403,"rank":2403,"depth":0,"x":2402.32,"y":180.465,"cluster":"homological-algebra"},{"id":"stacks:09J0","tag":"09J0","title":"Double complexes of abelian groups · Lemma 09J0","summary":"Let M^bullet be a complex of abelian groups. Let 0 → M^bullet → A_0^bullet → A_1^bullet → A_2^bullet → … be an exact complex of complexes of abelian groups such that for all p ∈ Z the complexes 0 → Coker(d_M^bullet^p) → Coker(d_A_0^bullet^p) → Coker(d_A_1^bullet^p) → Coker(d_A_2^bullet^p) → … are exact as well. Set A^p, q = A_p^q to obtain a double complex. Let Tot_π(A^bullet, bullet) be the product total complex associated to the double complex (see proof). Then the map…","statement_latex":"Let $M^\\bullet$ be a complex of abelian groups. Let\n$$\n0 \\to M^\\bullet \\to A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots\n$$\nbe an exact complex of complexes of abelian groups\nsuch that for all $p \\in \\mathbf{Z}$ the complexes\n$$\n0 \\to\n\\Coker(d_{M^\\bullet}^p) \\to\n\\Coker(d_{A_0^\\bullet}^p) \\to\n\\Coker(d_{A_1^\\bullet}^p) \\to\n\\Coker(d_{A_2^\\bullet}^p) \\to \\ldots\n$$\nare exact as well. Set $A^{p, q} = A_p^q$ to obtain a double\ncomplex. Let $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$ be the\nproduct total complex associated to the double complex\n(see proof). Then the map\n$M^\\bullet \\to \\text{Tot}_\\pi(A^{\\bullet, \\bullet})$\ninduced by $M^\\bullet \\to A_0^\\bullet$ is a quasi-isomorphism.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Double complexes of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09J0","source_file":"homology.tex","source_line":6927,"source_end_line":6948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L6927-L6948","statement_sha256":"f96ca0d7b0e64e3b813d76bde91b56c38f382c91528981bde6213c6fc1b7950e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2404,"rank":2404,"depth":1,"x":2569.355,"y":205.028,"cluster":"homological-algebra"},{"id":"stacks:0E1R","tag":"0E1R","title":"Double complexes of abelian groups · Lemma 0E1R","summary":"Let M^bullet be a complex of abelian groups. Let … → A_2^bullet → A_1^bullet → A_0^bullet → M^bullet → 0 be an exact complex of complexes of abelian groups. Set A^p, q = A_-p^q to obtain a double complex. Let Tot_π(A^bullet, bullet) be the product total complex associated to the double complex (see proof). Then the map Tot_π(A^bullet, bullet) → M^bullet induced by A_0^bullet → M^bullet is a quasi-isomorphism.","statement_latex":"Let $M^\\bullet$ be a complex of abelian groups. Let\n$$\n\\ldots \\to A_2^\\bullet \\to A_1^\\bullet \\to A_0^\\bullet \\to M^\\bullet \\to 0\n$$\nbe an exact complex of complexes of abelian groups. Set $A^{p, q} = A_{-p}^q$\nto obtain a double complex. Let $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$\nbe the product total complex associated to the double complex (see proof).\nThen the map $\\text{Tot}_\\pi(A^{\\bullet, \\bullet}) \\to M^\\bullet$\ninduced by $A_0^\\bullet \\to M^\\bullet$ is a quasi-isomorphism.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Double complexes of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1R","source_file":"homology.tex","source_line":7016,"source_end_line":7027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7016-L7027","statement_sha256":"0a0c7a3095e2a2bf09561916abb9067dd1dabb8c5d1ca0765d563749d198f781","origin":"The Stacks Project","memory_eligible":false,"source_rank":2405,"rank":2405,"depth":0,"x":2425.982,"y":281.937,"cluster":"homological-algebra"},{"id":"stacks:0135","tag":"0135","title":"Injectives · Definition 0135","summary":"Let A be an abelian category. An object J ∈ Ob(A) is called injective if for every injection A hookrightarrow B and every morphism A → J there exists a morphism B → J making the following diagram commute xymatrix A ar[r] ar[d] & B ar@-->[ld] J &","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAn object $J \\in \\Ob(\\mathcal{A})$ is\ncalled {\\it injective} if for every injection\n$A \\hookrightarrow B$ and every morphism\n$A \\to J$ there exists a morphism $B \\to J$ making\nthe following diagram commute\n$$\n\\xymatrix{\nA \\ar[r] \\ar[d] & B \\ar@{-->}[ld] \\\\\nJ &\n}\n$$","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Injectives","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0135","source_file":"homology.tex","source_line":7075,"source_end_line":7089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7075-L7089","statement_sha256":"30fcee6304e7dd17807a8f48e8784deef16426c733e609dafd5d921f30d362cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2406,"rank":2406,"depth":0,"x":2469.993,"y":143.439,"cluster":"homological-algebra"},{"id":"stacks:0136","tag":"0136","title":"Injectives · Lemma 0136","summary":"Let A be an abelian category. Let I be an object of A. The following are equivalent: • The object I is injective. • The functor B ↦ Hom_A(B, I) is exact. • Any short exact sequence 0 → I → A → B → 0 in A is split. • We have Ext_A(B, I) = 0 for all B ∈ Ob(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $I$ be an object of $\\mathcal{A}$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The object $I$ is injective.\n\\item The functor $B \\mapsto \\Hom_\\mathcal{A}(B, I)$\nis exact.\n\\item Any short exact sequence\n$$\n0 \\to I \\to A \\to B \\to 0\n$$\nin $\\mathcal{A}$ is split.\n\\item We have $\\Ext_\\mathcal{A}(B, I) = 0$ for\nall $B \\in \\Ob(\\mathcal{A})$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Injectives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0136","source_file":"homology.tex","source_line":7094,"source_end_line":7111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7094-L7111","statement_sha256":"04189921d9215e840b3a31a5a895a86fb4cc51e2d7e26f15be42a5f9e25ee76b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2407,"rank":2407,"depth":0,"x":2549.161,"y":270.935,"cluster":"homological-algebra"},{"id":"stacks:0137","tag":"0137","title":"Injectives · Lemma 0137","summary":"Let A be an abelian category. Suppose I_ω, ω ∈ Ω is a set of injective objects of A. If ∏_ω ∈ Ω I_ω exists then it is injective.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nSuppose $I_\\omega$, $\\omega \\in \\Omega$ is a set of injective\nobjects of $\\mathcal{A}$. If $\\prod_{\\omega \\in \\Omega} I_\\omega$\nexists then it is injective.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Injectives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0137","source_file":"homology.tex","source_line":7117,"source_end_line":7123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7117-L7123","statement_sha256":"0268533c457f01a21852bae25a565b451bfb2eedf7611828329b8d1be869e181","origin":"The Stacks Project","memory_eligible":false,"source_rank":2408,"rank":2408,"depth":0,"x":2387.757,"y":221.691,"cluster":"homological-algebra"},{"id":"stacks:0138","tag":"0138","title":"Injectives · Definition 0138","summary":"Let A be an abelian category. We say A has enough injectives if every object A has an injective morphism A → J into an injective object J.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nWe say $\\mathcal{A}$ has {\\it enough injectives}\nif every object $A$ has an injective morphism\n$A \\to J$ into an injective object $J$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Injectives","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0138","source_file":"homology.tex","source_line":7129,"source_end_line":7135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7129-L7135","statement_sha256":"9654ce74991ba848e2986e7f126301411a0a3397883ba0082a72149bb87e7ca3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2409,"rank":2409,"depth":0,"x":2546.864,"y":166.248,"cluster":"homological-algebra"},{"id":"stacks:0139","tag":"0139","title":"Injectives · Definition 0139","summary":"Let A be an abelian category. We say that A has functorial injective embeddings if there exists a functor J : A → Arrows(A) such that • s ∘ J = id_A, • for any object A ∈ Ob(A) the morphism J(A) is injective, and • for any object A ∈ Ob(A) the object t(J(A)) is an injective object of A. We will denote such a functor by A ↦ (A → J(A)).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nWe say that $\\mathcal{A}$ has {\\it functorial injective embeddings}\nif there exists a functor\n$$\nJ : \\mathcal{A} \\longrightarrow \\text{Arrows}(\\mathcal{A})\n$$\nsuch that\n\\begin{enumerate}\n\\item $s \\circ J = \\text{id}_\\mathcal{A}$,\n\\item for any object $A \\in \\Ob(\\mathcal{A})$\nthe morphism $J(A)$ is injective, and\n\\item for any object $A \\in \\Ob(\\mathcal{A})$\nthe object $t(J(A))$ is an injective object of $\\mathcal{A}$.\n\\end{enumerate}\nWe will denote such a functor by\n$A \\mapsto (A \\to J(A))$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Injectives","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0139","source_file":"homology.tex","source_line":7137,"source_end_line":7155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7137-L7155","statement_sha256":"73e8383ca2dfcfc7ca23b93b10fd81ebcfd87100bf57c08cba8e8e9b07f79346","origin":"The Stacks Project","memory_eligible":false,"source_rank":2410,"rank":2410,"depth":0,"x":2473.903,"y":297.813,"cluster":"homological-algebra"},{"id":"stacks:013B","tag":"013B","title":"Projectives · Definition 013B","summary":"Let A be an abelian category. An object P ∈ Ob(A) is called projective if for every surjection A → B and every morphism P → B there exists a morphism P → A making the following diagram commute xymatrix A ar[r] & B P ar@-->[u] ar[ru] &","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAn object $P \\in \\Ob(\\mathcal{A})$ is\ncalled {\\it projective} if for every surjection\n$A \\rightarrow B$ and every morphism\n$P \\to B$ there exists a morphism $P \\to A$ making\nthe following diagram commute\n$$\n\\xymatrix{\nA \\ar[r] & B \\\\\nP \\ar@{-->}[u] \\ar[ru] &\n}\n$$","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Projectives","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013B","source_file":"homology.tex","source_line":7164,"source_end_line":7178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7164-L7178","statement_sha256":"54d8dd055807c4c625c567ee239eca4adc1cc570c18d486224c2ac2c3ca7fcd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2411,"rank":2411,"depth":0,"x":2421.745,"y":158.978,"cluster":"homological-algebra"},{"id":"stacks:013C","tag":"013C","title":"Projectives · Lemma 013C","summary":"Let A be an abelian category. Let P be an object of A. The following are equivalent: • The object P is projective. • The functor B ↦ Hom_A(P, B) is exact. • Any short exact sequence 0 → A → B → P → 0 in A is split. • We have Ext_A(P, A) = 0 for all A ∈ Ob(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $P$ be an object of $\\mathcal{A}$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The object $P$ is projective.\n\\item The functor $B \\mapsto \\Hom_\\mathcal{A}(P, B)$\nis exact.\n\\item Any short exact sequence\n$$\n0 \\to A \\to B \\to P \\to 0\n$$\nin $\\mathcal{A}$ is split.\n\\item We have $\\Ext_\\mathcal{A}(P, A) = 0$ for\nall $A \\in \\Ob(\\mathcal{A})$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Projectives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013C","source_file":"homology.tex","source_line":7183,"source_end_line":7200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7183-L7200","statement_sha256":"a59cb1293035dafe5c3fb235f3ed47f9ed8c3f7e9a3ef209ba7c9c1df3b7b640","origin":"The Stacks Project","memory_eligible":false,"source_rank":2412,"rank":2412,"depth":0,"x":2572.306,"y":231.977,"cluster":"homological-algebra"},{"id":"stacks:013D","tag":"013D","title":"Projectives · Lemma 013D","summary":"Let A be an abelian category. Suppose P_ω, ω ∈ Ω is a set of projective objects of A. If coprod_ω ∈ Ω P_ω exists then it is projective.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nSuppose $P_\\omega$, $\\omega \\in \\Omega$ is a set of projective\nobjects of $\\mathcal{A}$. If $\\coprod_{\\omega \\in \\Omega} P_\\omega$\nexists then it is projective.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Projectives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013D","source_file":"homology.tex","source_line":7206,"source_end_line":7212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7206-L7212","statement_sha256":"974b78406bb5ca4bce319f54b063cdc0361a15fb8e42abf772dbac1afe6b4b58","origin":"The Stacks Project","memory_eligible":false,"source_rank":2413,"rank":2413,"depth":0,"x":2402.071,"y":263.675,"cluster":"homological-algebra"},{"id":"stacks:013E","tag":"013E","title":"Projectives · Definition 013E","summary":"Let A be an abelian category. We say A has enough projectives if every object A has an surjective morphism P → A from an projective object P onto it.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nWe say $\\mathcal{A}$ has {\\it enough projectives}\nif every object $A$ has an surjective morphism\n$P \\to A$ from an projective object $P$ onto it.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Projectives","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013E","source_file":"homology.tex","source_line":7218,"source_end_line":7224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7218-L7224","statement_sha256":"11f55004b7e5f975c4c4b8b1825b71a30a921e2f4618dc2dcf1680b4a2d5e793","origin":"The Stacks Project","memory_eligible":false,"source_rank":2414,"rank":2414,"depth":0,"x":2502.408,"y":143.348,"cluster":"homological-algebra"},{"id":"stacks:013F","tag":"013F","title":"Projectives · Definition 013F","summary":"Let A be an abelian category. We say that A has functorial projective surjections if there exists a functor P : A → Arrows(A) such that • t ∘ P = id_A, • for any object A ∈ Ob(A) the morphism P(A) is surjective, and • for any object A ∈ Ob(A) the object s(P(A)) is an projective object of A. We will denote such a functor by A ↦ (P(A) → A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nWe say that $\\mathcal{A}$ has {\\it functorial projective surjections}\nif there exists a functor\n$$\nP : \\mathcal{A} \\longrightarrow \\text{Arrows}(\\mathcal{A})\n$$\nsuch that\n\\begin{enumerate}\n\\item $t \\circ P = \\text{id}_\\mathcal{A}$,\n\\item for any object $A \\in \\Ob(\\mathcal{A})$\nthe morphism $P(A)$ is surjective, and\n\\item for any object $A \\in \\Ob(\\mathcal{A})$\nthe object $s(P(A))$ is an projective object of $\\mathcal{A}$.\n\\end{enumerate}\nWe will denote such a functor by\n$A \\mapsto (P(A) \\to A)$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Projectives","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013F","source_file":"homology.tex","source_line":7226,"source_end_line":7244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7226-L7244","statement_sha256":"cb57033c183445d8aa27eb5114ee197b96406c3e92d6122fbbf8412646654ef9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2415,"rank":2415,"depth":0,"x":2525.251,"y":289.442,"cluster":"homological-algebra"},{"id":"stacks:015Z","tag":"015Z","title":"Injectives and adjoint functors · Lemma 015Z","summary":"A functor with an exact left adjoint preserves injectives Let A and B be abelian categories. Let u : A → B and v : B → A be additive functors with u right adjoint to v. Consider the following conditions: • [(a)] v transforms injective maps into injective maps, • [(b)] v is exact, and • [(c)] u transforms injectives into injectives. Then (a) ⇔ (b) ⇒ (c). If A has enough injectives, then all three conditions are equivalent.","statement_latex":"\\begin{slogan}\nA functor with an exact left adjoint preserves injectives\n\\end{slogan}\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $u : \\mathcal{A} \\to \\mathcal{B}$ and\n$v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors with\n$u$ right adjoint to $v$. Consider the following conditions:\n\\begin{enumerate}\n\\item[(a)] $v$ transforms injective maps into injective maps,\n\\item[(b)] $v$ is exact, and\n\\item[(c)] $u$ transforms injectives into injectives.\n\\end{enumerate}\nThen (a) $\\Leftrightarrow$ (b) $\\Rightarrow$ (c). If $\\mathcal{A}$\nhas enough injectives, then all three conditions are equivalent.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Injectives and adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015Z","source_file":"homology.tex","source_line":7269,"source_end_line":7285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7269-L7285","statement_sha256":"8e3d617d4280adb697bd50e24f371523148703f61acc5079f5ab6732c0e1dba3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2416,"rank":2416,"depth":7,"x":2390.527,"y":194.397,"cluster":"homological-algebra"},{"id":"stacks:0160","tag":"0160","title":"Injectives and adjoint functors · Lemma 0160","summary":"Let A and B be abelian categories. Let u : A → B and v : B → A be additive functors. Assume • u is right adjoint to v, • v transforms injective maps into injective maps, • A has enough injectives, and • vB = 0 implies B = 0 for any B ∈ Ob(B). Then B has enough injectives.","statement_latex":"Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $u : \\mathcal{A} \\to \\mathcal{B}$ and\n$v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors.\nAssume\n\\begin{enumerate}\n\\item $u$ is right adjoint to $v$,\n\\item $v$ transforms injective maps into injective maps,\n\\item $\\mathcal{A}$ has enough injectives, and\n\\item $vB = 0$ implies $B = 0$ for any $B \\in \\Ob(\\mathcal{B})$.\n\\end{enumerate}\nThen $\\mathcal{B}$ has enough injectives.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Injectives and adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0160","source_file":"homology.tex","source_line":7328,"source_end_line":7341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7328-L7341","statement_sha256":"9264903e7033be40a407906e95a29f999f30fd75a65c2c9bd9c088a82d235791","origin":"The Stacks Project","memory_eligible":false,"source_rank":2417,"rank":2417,"depth":8,"x":2566.819,"y":188.017,"cluster":"homological-algebra"},{"id":"stacks:0161","tag":"0161","title":"Injectives and adjoint functors · Lemma 0161","summary":"Let A and B be abelian categories. Let u : A → B and v : B → A be additive functors. Assume • u is right adjoint to v, • v transforms injective maps into injective maps, • A has enough injectives, • vB = 0 implies B = 0 for any B ∈ Ob(B), and • A has functorial injective embeddings. Then B has functorial injective embeddings.","statement_latex":"Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $u : \\mathcal{A} \\to \\mathcal{B}$ and\n$v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors.\nAssume\n\\begin{enumerate}\n\\item $u$ is right adjoint to $v$,\n\\item $v$ transforms injective maps into injective maps,\n\\item $\\mathcal{A}$ has enough injectives,\n\\item $vB = 0$ implies $B = 0$ for any $B \\in \\Ob(\\mathcal{B})$, and\n\\item $\\mathcal{A}$ has functorial injective embeddings.\n\\end{enumerate}\nThen $\\mathcal{B}$ has functorial injective embeddings.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Injectives and adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0161","source_file":"homology.tex","source_line":7363,"source_end_line":7377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7363-L7377","statement_sha256":"0f569c98426ccdeea479fa34a32270cf2d7be19baa00e8c7ab8bcd74b3eb784c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2418,"rank":2418,"depth":9,"x":2441.59,"y":293.059,"cluster":"homological-algebra"},{"id":"stacks:0793","tag":"0793","title":"Injectives and adjoint functors · Lemma 0793","summary":"Let A and B be abelian categories. Let u : A → B be a functor. If there exists a subset P ⊂ Ob(B) such that • every object of B is a quotient of an element of P, and • for every P ∈ P there exists an object Q of A such that Hom_A(Q, A) = Hom_B(P, u(A)) functorially in A, then there exists a left adjoint v of u.","statement_latex":"Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $u : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\nIf there exists a subset $\\mathcal{P} \\subset \\Ob(\\mathcal{B})$\nsuch that\n\\begin{enumerate}\n\\item every object of $\\mathcal{B}$ is a quotient of an element\nof $\\mathcal{P}$, and\n\\item for every $P \\in \\mathcal{P}$ there exists an object\n$Q$ of $\\mathcal{A}$ such that\n$\\Hom_\\mathcal{A}(Q, A) = \\Hom_\\mathcal{B}(P, u(A))$ functorially\nin $A$,\n\\end{enumerate}\nthen there exists a left adjoint $v$ of $u$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Injectives and adjoint functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0793","source_file":"homology.tex","source_line":7387,"source_end_line":7402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7387-L7402","statement_sha256":"6d11333b62584bb6cb699c73fd4a3b081d5d04b47d02e0899d6eb6fdbc6b7dae","origin":"The Stacks Project","memory_eligible":false,"source_rank":2419,"rank":2419,"depth":1,"x":2449.484,"y":144.114,"cluster":"homological-algebra"},{"id":"stacks:0A2E","tag":"0A2E","title":"Essentially constant systems · Lemma 0A2E","summary":"Let I be a category, let A be a pre-additive Karoubian category, and let M : I → A be a diagram. • Assume I is filtered. The following are equivalent • M is essentially constant, • X = colim M exists and there exists a cofinal filtered subcategory I' ⊂ I and for i' ∈ Ob(I') a direct sum decomposition M_i' = X_i' ⊕ Z_i' such that X_i' maps isomorphically to X and Z_i' to zero in M_i\" for some i' → i\" in I'. • Assume I is cofiltered. The following are equivalent • M is…","statement_latex":"Let $\\mathcal{I}$ be a category, let $\\mathcal{A}$ be a pre-additive\nKaroubian category, and let $M : \\mathcal{I} \\to \\mathcal{A}$ be a diagram.\n\\begin{enumerate}\n\\item Assume $\\mathcal{I}$ is filtered. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is essentially constant,\n\\item $X = \\colim M$ exists and there exists a cofinal filtered subcategory\n$\\mathcal{I}' \\subset \\mathcal{I}$ and for $i' \\in \\Ob(\\mathcal{I}')$\na direct sum decomposition $M_{i'} = X_{i'} \\oplus Z_{i'}$ such that\n$X_{i'}$ maps isomorphically to $X$ and $Z_{i'}$ to zero in $M_{i''}$\nfor some $i' \\to i''$ in $\\mathcal{I}'$.\n\\end{enumerate}\n\\item Assume $\\mathcal{I}$ is cofiltered. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is essentially constant,\n\\item $X = \\lim M$ exists and there exists an initial cofiltered subcategory\n$\\mathcal{I}' \\subset \\mathcal{I}$ and for $i' \\in \\Ob(\\mathcal{I}')$\na direct sum decomposition $M_{i'} = X_{i'} \\oplus Z_{i'}$\nsuch that $X$ maps isomorphically to $X_{i'}$ and $M_{i''} \\to Z_{i'}$\nis zero for some $i'' \\to i'$ in $\\mathcal{I}'$.\n\\end{enumerate}\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2E","source_file":"homology.tex","source_line":7466,"source_end_line":7490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7466-L7490","statement_sha256":"7e67dda43812509946280de452b3b2d77d095fbfa3a6bd78d1d0670070e531c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2420,"rank":2420,"depth":2,"x":2563.768,"y":258.752,"cluster":"homological-algebra"},{"id":"stacks:0A2F","tag":"0A2F","title":"Essentially constant systems · Lemma 0A2F","summary":"Let I be a category. Let A be an additive, Karoubian category. Let F : I → A and G : I → A be functors. The following are equivalent • colim_I F ⊕ G exists, and • colim_I F and colim_I G exist. In this case colim_I F ⊕ G = colim_I F ⊕ colim_I G.","statement_latex":"Let $\\mathcal{I}$ be a category. Let $\\mathcal{A}$ be an additive, Karoubian\ncategory. Let $F : \\mathcal{I} \\to \\mathcal{A}$ and\n$G : \\mathcal{I} \\to \\mathcal{A}$ be functors. The following are equivalent\n\\begin{enumerate}\n\\item $\\colim_\\mathcal{I} F \\oplus G$ exists, and\n\\item $\\colim_\\mathcal{I} F$ and $\\colim_\\mathcal{I} G$ exist.\n\\end{enumerate}\nIn this case $\\colim_\\mathcal{I} F \\oplus G =\n\\colim_\\mathcal{I} F \\oplus \\colim_\\mathcal{I} G$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2F","source_file":"homology.tex","source_line":7532,"source_end_line":7543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7532-L7543","statement_sha256":"7ced134f18fc02e38da7f7a3043b8a55f58dd7f05f3296bd0ca86a84777e5b51","origin":"The Stacks Project","memory_eligible":false,"source_rank":2421,"rank":2421,"depth":4,"x":2386.802,"y":239.01,"cluster":"homological-algebra"},{"id":"stacks:0A2G","tag":"0A2G","title":"Essentially constant systems · Lemma 0A2G","summary":"Let I be a filtered category. Let A be an additive, Karoubian category. Let F : I → A and G : I → A be functors. The following are equivalent • F ⊕ G : I → A is essentially constant, and • F and G are essentially constant.","statement_latex":"Let $\\mathcal{I}$ be a filtered category. Let $\\mathcal{A}$\nbe an additive, Karoubian category. Let $F : \\mathcal{I} \\to \\mathcal{A}$ and\n$G : \\mathcal{I} \\to \\mathcal{A}$ be functors. The following are equivalent\n\\begin{enumerate}\n\\item $F \\oplus G : \\mathcal{I} \\to \\mathcal{A}$\nis essentially constant, and\n\\item $F$ and $G$ are essentially constant.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2G","source_file":"homology.tex","source_line":7558,"source_end_line":7568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7558-L7568","statement_sha256":"4a478f404e51e75a84cb69d6396ddeb3cd79e4f282f101482798b832d07933f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2422,"rank":2422,"depth":5,"x":2533.587,"y":152.909,"cluster":"homological-algebra"},{"id":"stacks:02MZ","tag":"02MZ","title":"Inverse systems · Lemma 02MZ","summary":"Let C be a category. • If C is an additive category, then the category of inverse systems with values in C is an additive category. • If C is an abelian category, then the category of inverse systems with values in C is an abelian category. A sequence (K_i) → (L_i) → (M_i) of inverse systems is exact if and only if each K_i → L_i → N_i is exact.","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item If $\\mathcal{C}$ is an additive category, then the category\nof inverse systems with values in $\\mathcal{C}$ is an additive category.\n\\item If $\\mathcal{C}$ is an abelian category, then the category\nof inverse systems with values in $\\mathcal{C}$ is an abelian category.\nA sequence $(K_i) \\to (L_i) \\to (M_i)$ of inverse systems\nis exact if and only if each $K_i \\to L_i \\to N_i$ is exact.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Inverse systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02MZ","source_file":"homology.tex","source_line":7615,"source_end_line":7626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7615-L7626","statement_sha256":"d6db8b8eda9643fb1dd3fcf243de0e1adb16885747f5dd54e231d200026e3e8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2423,"rank":2423,"depth":0,"x":2494.479,"y":300.104,"cluster":"homological-algebra"},{"id":"stacks:02N0","tag":"02N0","title":"Inverse systems · Definition 02N0","summary":"Let C be an abelian category. We say the inverse system (A_i) satisfies the Mittag-Leffler condition, or for short is ML, if for every i there exists a c = c(i) ≥ i such that Im(A_k → A_i) = Im(A_c → A_i) for all k ≥ c.","statement_latex":"Let $\\mathcal{C}$ be an abelian category.\nWe say the inverse system $(A_i)$\nsatisfies the {\\it Mittag-Leffler condition}, or for short\nis {\\it ML}, if for every $i$ there exists a $c = c(i) \\geq i$\nsuch that\n$$\n\\Im(A_k \\to A_i) = \\Im(A_c \\to A_i)\n$$\nfor all $k \\geq c$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Inverse systems","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02N0","source_file":"homology.tex","source_line":7651,"source_end_line":7662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7651-L7662","statement_sha256":"4e364c70d2a255dac2f1397015d6e9fc9f5356698d6735f4ce79303f89f00289","origin":"The Stacks Project","memory_eligible":false,"source_rank":2424,"rank":2424,"depth":0,"x":2404.695,"y":169.006,"cluster":"homological-algebra"},{"id":"stacks:02N1","tag":"02N1","title":"Inverse systems · Lemma 02N1","summary":"Let 0 → (A_i) → (B_i) → (C_i) → 0 be a short exact sequence of inverse systems of abelian groups. • In any case the sequence 0 → lim_i A_i → lim_i B_i → lim_i C_i is exact. • If (B_i) is ML, then also (C_i) is ML. • If (A_i) is ML, then 0 → lim_i A_i → lim_i B_i → lim_i C_i → 0 is exact.","statement_latex":"Let\n$$\n0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0\n$$\nbe a short exact sequence of inverse systems of abelian groups.\n\\begin{enumerate}\n\\item In any case the sequence\n$$\n0 \\to \\lim_i A_i \\to \\lim_i B_i \\to \\lim_i C_i\n$$\nis exact.\n\\item If $(B_i)$ is ML, then also $(C_i)$ is ML.\n\\item If $(A_i)$ is ML, then\n$$\n0 \\to \\lim_i A_i \\to \\lim_i B_i \\to \\lim_i C_i \\to 0\n$$\nis exact.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Inverse systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02N1","source_file":"homology.tex","source_line":7672,"source_end_line":7692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7672-L7692","statement_sha256":"b87054bc8e5f5c9577465fee7ffc9c4c440ee90175b252a47622c0e1b58719fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2425,"rank":2425,"depth":3,"x":2576.806,"y":214.858,"cluster":"homological-algebra"},{"id":"stacks:070B","tag":"070B","title":"Inverse systems · Lemma 070B","summary":"Let (A_i) → (B_i) → (C_i) → (D_i) be an exact sequence of inverse systems of abelian groups. If the system (A_i) is ML, then the sequence lim_i B_i → lim_i C_i → lim_i D_i is exact.","statement_latex":"Let\n$$\n(A_i) \\to (B_i) \\to (C_i) \\to (D_i)\n$$\nbe an exact sequence of inverse systems of abelian groups. If the\nsystem $(A_i)$ is ML, then the sequence\n$$\n\\lim_i B_i \\to \\lim_i C_i \\to \\lim_i D_i\n$$\nis exact.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Inverse systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070B","source_file":"homology.tex","source_line":7699,"source_end_line":7711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7699-L7711","statement_sha256":"db695621f03bd4af535f34969a53c27f044f9c64f35fabfca15482d5db7201f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2426,"rank":2426,"depth":4,"x":2412.565,"y":278.884,"cluster":"homological-algebra"},{"id":"stacks:070C","tag":"070C","title":"Inverse systems · Lemma 070C","summary":"Let A be an abelian category. Let (A_i) be an inverse system in A with limit A = lim A_i. Then (A_i) is essentially constant (see Categories, Definition [Tag 05PU]) if and only if there exists an i and for all j ≥ i a direct sum decomposition A_j = A ⊕ Z_j such that (a) the maps A_j' → A_j are compatible with the direct sum decompositions, (b) for all j there exists some j' ≥ j such that Z_j' → Z_j is zero.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $(A_i)$ be an inverse system in $\\mathcal{A}$ with limit $A = \\lim A_i$.\nThen $(A_i)$ is essentially constant (see\nCategories, Definition\n\\ref{categories-definition-essentially-constant-diagram})\nif and only if there exists an $i$ and for all $j \\geq i$ a direct sum\ndecomposition $A_j = A \\oplus Z_j$ such that\n(a) the maps $A_{j'} \\to A_j$ are compatible with the direct sum\ndecompositions, (b) for all $j$ there exists some $j' \\geq j$ such that\n$Z_{j'} \\to Z_j$ is zero.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Inverse systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070C","source_file":"homology.tex","source_line":7732,"source_end_line":7744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7732-L7744","statement_sha256":"84a35a6b6f2ead3a081b68fe8313fdcb119eb4e106b034973ae47cae069f6532","origin":"The Stacks Project","memory_eligible":false,"source_rank":2427,"rank":2427,"depth":1,"x":2482.379,"y":138.09,"cluster":"homological-algebra"},{"id":"stacks:070D","tag":"070D","title":"Inverse systems · Lemma 070D","summary":"Let 0 → (A_i) → (B_i) → (C_i) → 0 be an exact sequence of inverse systems of abelian groups. If (C_i) is essentially constant, then (A_i) has ML if and only if (B_i) has ML.","statement_latex":"Let\n$$\n0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0\n$$\nbe an exact sequence of inverse systems of abelian groups.\nIf $(C_i)$ is essentially constant, then $(A_i)$ has ML\nif and only if $(B_i)$ has ML.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Inverse systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070D","source_file":"homology.tex","source_line":7760,"source_end_line":7769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7760-L7769","statement_sha256":"09d8aa05ec4c2e7b3ee286cc79c8fa5e4a1b75dda7c8313eddb0e27b7f66e33c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2428,"rank":2428,"depth":2,"x":2544.292,"y":281.919,"cluster":"homological-algebra"},{"id":"stacks:070E","tag":"070E","title":"Inverse systems · Lemma 070E","summary":"Let (A^-2_i → A^-1_i → A^0_i → A^1_i) be an inverse system of complexes of abelian groups and denote A^-2 → A^-1 → A^0 → A^1 its limit. Denote (H_i^-1), (H_i^0) the inverse systems of cohomologies, and denote H^-1, H^0 the cohomologies of A^-2 → A^-1 → A^0 → A^1. If (A^-2_i) and (A^-1_i) are ML and (H^-1_i) is essentially constant, then H^0 = lim H_i^0.","statement_latex":"Let\n$$\n(A^{-2}_i \\to A^{-1}_i \\to A^0_i \\to A^1_i)\n$$\nbe an inverse system of complexes of abelian groups and denote\n$A^{-2} \\to A^{-1} \\to A^0 \\to A^1$ its limit. Denote\n$(H_i^{-1})$, $(H_i^0)$ the inverse systems of cohomologies, and\ndenote $H^{-1}$, $H^0$ the cohomologies of $A^{-2} \\to A^{-1} \\to A^0 \\to A^1$.\nIf $(A^{-2}_i)$ and $(A^{-1}_i)$ are ML and\n$(H^{-1}_i)$ is essentially constant, then\n$H^0 = \\lim H_i^0$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Inverse systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070E","source_file":"homology.tex","source_line":7809,"source_end_line":7822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7809-L7822","statement_sha256":"f3bb7e15b6d9fbdfd2e76da4679e381d468511a66b9515f6f5223a110fc8231f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2429,"rank":2429,"depth":4,"x":2382.534,"y":210.798,"cluster":"homological-algebra"},{"id":"stacks:0AAT","tag":"0AAT","title":"Inverse systems · Lemma 0AAT","summary":"Let α be an ordinal. Let K_β^bullet, β < α be an inverse system of complexes of abelian groups over α. If for all β < α the complex K_β^bullet is acyclic and the map K^n_β → lim_γ < β K^n_γ is surjective, then the complex lim_β < α K_β^bullet is acyclic.","statement_latex":"Let $\\alpha$ be an ordinal. Let $K_\\beta^\\bullet$, $\\beta < \\alpha$\nbe an inverse system of complexes of abelian groups over $\\alpha$. If\nfor all $\\beta < \\alpha$ the complex $K_\\beta^\\bullet$ is acyclic and\nthe map\n$$\nK^n_\\beta \\longrightarrow \\lim_{\\gamma < \\beta} K^n_\\gamma\n$$\nis surjective, then the complex\n$\\lim_{\\beta < \\alpha} K_\\beta^\\bullet$ is acyclic.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Inverse systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAT","source_file":"homology.tex","source_line":7861,"source_end_line":7872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7861-L7872","statement_sha256":"ed937c7ac2cb5fb78c09d2237f148b6b61ccdd2f969bb8ce0ef9888ecd2b1fb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2430,"rank":2430,"depth":0,"x":2559.486,"y":171.349,"cluster":"homological-algebra"},{"id":"stacks:060K","tag":"060K","title":"Exactness of products · Lemma 060K","summary":"Let I be a set. For i ∈ I let L_i → M_i → N_i be a complex of abelian groups. Let H_i = Ker(M_i → N_i)/Im(L_i → M_i) be the cohomology. Then ∏ L_i → ∏ M_i → ∏ N_i is a complex of abelian groups with homology ∏ H_i.","statement_latex":"Let $I$ be a set. For $i \\in I$ let $L_i \\to M_i \\to N_i$ be a complex\nof abelian groups. Let $H_i = \\Ker(M_i \\to N_i)/\\Im(L_i \\to M_i)$\nbe the cohomology. Then\n$$\n\\prod L_i \\to \\prod M_i \\to \\prod N_i\n$$\nis a complex of abelian groups with homology $\\prod H_i$.","area":"Homological Algebra","chapter":"Homological Algebra","chapter_id":"homology","section":"Exactness of products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060K","source_file":"homology.tex","source_line":7920,"source_end_line":7929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/homology.tex#L7920-L7929","statement_sha256":"8a72a052573d346595c2f02d8b5fada1db742df680e833d9d81e65d9000ce636","origin":"The Stacks Project","memory_eligible":false,"source_rank":2431,"rank":2431,"depth":0,"x":2460.46,"y":301.195,"cluster":"homological-algebra"},{"id":"stacks:0144","tag":"0144","title":"The definition of a triangulated category · Definition 0144","summary":"Let D be an additive category. Let [1] : D → D, E ↦ E[1] be an additive functor which is an auto-equivalence of D. • A triangle is a sextuple (X, Y, Z, f, g, h) where X, Y, Z ∈ Ob(D) and f : X → Y, g : Y → Z and h : Z → X[1] are morphisms of D. • A morphism of triangles (X, Y, Z, f, g, h) → (X', Y', Z', f', g', h') is given by morphisms a : X → X', b : Y → Y' and c : Z → Z' of D such that b ∘ f = f' ∘ a, c ∘ g = g' ∘ b and a[1] ∘ h = h' ∘ c.","statement_latex":"Let $\\mathcal{D}$ be an additive category. Let\n$[1] : \\mathcal{D} \\to \\mathcal{D}$, $E \\mapsto E[1]$\nbe an additive functor which is an auto-equivalence of $\\mathcal{D}$.\n\\begin{enumerate}\n\\item A {\\it triangle} is a sextuple\n$(X, Y, Z, f, g, h)$ where $X, Y, Z \\in \\Ob(\\mathcal{D})$ and\n$f : X \\to Y$, $g : Y \\to Z$ and $h : Z \\to X[1]$ are morphisms\nof $\\mathcal{D}$.\n\\item A {\\it morphism of triangles}\n$(X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')$\nis given by morphisms $a : X \\to X'$, $b : Y \\to Y'$ and $c : Z \\to Z'$\nof $\\mathcal{D}$ such that\n$b \\circ f = f' \\circ a$, $c  \\circ g = g' \\circ b$ and\n$a[1] \\circ h = h' \\circ c$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The definition of a triangulated category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0144","source_file":"derived.tex","source_line":48,"source_end_line":65,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L48-L65","statement_sha256":"a8762deed6a45c01b872ca52b6491ab6679e1626d0f95665da0bd3d48eb65fe0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2432,"rank":2432,"depth":0,"x":234.749,"y":680.0,"cluster":"derived-categories"},{"id":"stacks:0145","tag":"0145","title":"The definition of a triangulated category · Definition 0145","summary":"A triangulated category consists of a triple (D, ([n])_n∈ Z, T) where • D is an additive category, • [1] : D → D, E ↦ E[1] is an additive auto-equivalence and [n] for n ∈ Z is as discussed above, and • T is a set of triangles (Definition [Tag 0144]) called the distinguished triangles subject to the following conditions • [TR1] Any triangle isomorphic to a distinguished triangle is a distinguished triangle. Any triangle of the form (X, X, 0, id, 0, 0) is distinguished. For…","statement_latex":"A {\\it triangulated category} consists of a triple\n$(\\mathcal{D}, \\{[n]\\}_{n\\in \\mathbf{Z}}, \\mathcal{T})$\nwhere\n\\begin{enumerate}\n\\item $\\mathcal{D}$ is an additive category,\n\\item $[1] : \\mathcal{D} \\to \\mathcal{D}$, $E \\mapsto E[1]$\nis an additive auto-equivalence and $[n]$ for $n \\in \\mathbf{Z}$ is\nas discussed above, and\n\\item $\\mathcal{T}$ is a set of triangles (Definition \\ref{definition-triangle})\ncalled the {\\it distinguished triangles}\n\\end{enumerate}\nsubject to the following conditions\n\\begin{enumerate}\n\\item[TR1] Any triangle isomorphic to a distinguished triangle is\na distinguished triangle. Any triangle of the form\n$(X, X, 0, \\text{id}, 0, 0)$ is distinguished.\nFor any morphism $f : X \\to Y$ of $\\mathcal{D}$ there exists a\ndistinguished triangle of the form $(X, Y, Z, f, g, h)$.\n\\item[TR2] The triangle $(X, Y, Z, f, g, h)$ is distinguished\nif and only if the triangle $(Y, Z, X[1], g, h, -f[1])$ is.\n\\item[TR3] Given a solid diagram\n$$\n\\xymatrix{\nX \\ar[r]^f \\ar[d]^a &\nY \\ar[r]^g \\ar[d]^b &\nZ \\ar[r]^h \\ar@{-->}[d] &\nX[1] \\ar[d]^{a[1]} \\\\\nX' \\ar[r]^{f'} &\nY' \\ar[r]^{g'} &\nZ' \\ar[r]^{h'} &\nX'[1]\n}\n$$\nwhose rows are distinguished triangles and which satisfies\n$b \\circ f = f' \\circ a$, there exists a morphism\n$c : Z \\to Z'$ such that $(a, b, c)$ is a morphism of triangles.\n\\item[TR4] Given objects $X$, $Y$, $Z$ of $\\mathcal{D}$, and morphisms\n$f : X \\to Y$, $g : Y \\to Z$, and distinguished triangles\n$(X, Y, Q_1, f, p_1, d_1)$,\n$(X, Z, Q_2, g \\circ f, p_2, d_2)$,\nand\n$(Y, Z, Q_3, g, p_3, d_3)$,\nthere exist\nmorphisms $a : Q_1 \\to Q_2$ and $b : Q_2 \\to Q_3$ such\nthat\n\\begin{enumerate}\n\\item $(Q_1, Q_2, Q_3, a, b, p_1[1] \\circ d_3)$ is a\ndistinguished triangle,\n\\item the triple $(\\text{id}_X, g, a)$ is\na morphism of triangles\n$(X, Y, Q_1, f, p_1, d_1) \\to (X, Z, Q_2, g \\circ f, p_2, d_2)$, and\n\\item the triple $(f, \\text{id}_Z, b)$ is a morphism of triangles\n$(X, Z, Q_2, g \\circ f, p_2, d_2) \\to (Y, Z, Q_3, g, p_3, d_3)$.\n\\end{enumerate}\n\\end{enumerate}\nWe will call $(\\mathcal{D}, [\\ ], \\mathcal{T})$ a\n{\\it pre-triangulated category} if TR1, TR2 and TR3\nhold.\\footnote{We use $[\\ ]$ as an abbreviation for the\nfamily $\\{[n]\\}_{n\\in \\mathbf{Z}}$.}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The definition of a triangulated category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0145","source_file":"derived.tex","source_line":94,"source_end_line":155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L94-L155","statement_sha256":"9bd9dcd120134dfc47bd5be986e2bd99004726a9dd1d4f12b2f1e37afc10a660","origin":"The Stacks Project","memory_eligible":false,"source_rank":2433,"rank":2433,"depth":1,"x":223.935,"y":684.667,"cluster":"derived-categories"},{"id":"stacks:014V","tag":"014V","title":"The definition of a triangulated category · Definition 014V","summary":"Let D, D' be pre-triangulated categories. An exact functor, or a triangulated functor from D to D' is a functor F : D → D' together with given functorial isomorphisms xi_X : F(X[1]) → F(X)[1] such that for every distinguished triangle (X, Y, Z, f, g, h) of D the triangle (F(X), F(Y), F(Z), F(f), F(g), xi_X ∘ F(h)) is a distinguished triangle of D'.","statement_latex":"Let $\\mathcal{D}$, $\\mathcal{D}'$ be pre-triangulated\ncategories. An {\\it exact functor}, or a {\\it triangulated functor}\nfrom $\\mathcal{D}$ to $\\mathcal{D}'$ is a functor\n$F : \\mathcal{D} \\to \\mathcal{D}'$ together\nwith given functorial isomorphisms $\\xi_X : F(X[1]) \\to F(X)[1]$\nsuch that for every distinguished triangle\n$(X, Y, Z, f, g, h)$ of $\\mathcal{D}$ the triangle\n$(F(X), F(Y), F(Z), F(f), F(g), \\xi_X \\circ F(h))$\nis a distinguished triangle of $\\mathcal{D}'$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The definition of a triangulated category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014V","source_file":"derived.tex","source_line":190,"source_end_line":201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L190-L201","statement_sha256":"5852689407017bfda11b4a818e12733ab1ff67e02418bab779b100e736e0dbff","origin":"The Stacks Project","memory_eligible":false,"source_rank":2434,"rank":2434,"depth":0,"x":230.928,"y":671.114,"cluster":"derived-categories"},{"id":"stacks:05QM","tag":"05QM","title":"The definition of a triangulated category · Definition 05QM","summary":"Let (D, [ ], T) be a pre-triangulated category. A pre-triangulated subcategory' is a full subcategory then T' is the intersection of the set of triangles in D' with T, see Lemma [Tag 05QX]. In this case we drop T' from the notation. is a pair (D', T') such that • D' is an additive subcategory of D which is preserved under [1] and such that [1] : D' → D' is an auto-equivalence, • T' ⊂ T is a subset such that for every (X, Y, Z, f, g, h) ∈ T' we have X, Y, Z ∈ Ob(D') and f,…","statement_latex":"Let $(\\mathcal{D}, [\\ ], \\mathcal{T})$ be a pre-triangulated category.\nA {\\it pre-triangulated subcategory}\\footnote{This definition may be\nnonstandard. If $\\mathcal{D}'$ is a full subcategory then $\\mathcal{T}'$\nis the intersection of the set of triangles in $\\mathcal{D}'$ with\n$\\mathcal{T}$, see\nLemma \\ref{lemma-triangulated-subcategory}.\nIn this case we drop $\\mathcal{T}'$ from the notation.}\nis a pair $(\\mathcal{D}', \\mathcal{T}')$ such that\n\\begin{enumerate}\n\\item $\\mathcal{D}'$ is an additive subcategory of $\\mathcal{D}$\nwhich is preserved under $[1]$ and such that\n$[1] : \\mathcal{D}' \\to \\mathcal{D}'$ is an auto-equivalence,\n\\item $\\mathcal{T}' \\subset \\mathcal{T}$ is a subset such that for every\n$(X, Y, Z, f, g, h) \\in \\mathcal{T}'$ we have\n$X, Y, Z \\in \\Ob(\\mathcal{D}')$ and\n$f, g, h \\in \\text{Arrows}(\\mathcal{D}')$, and\n\\item $(\\mathcal{D}', [\\ ], \\mathcal{T}')$ is a pre-triangulated\ncategory.\n\\end{enumerate}\nIf $\\mathcal{D}$ is a triangulated category, then we say\n$(\\mathcal{D}', \\mathcal{T}')$ is a {\\it triangulated subcategory} if\nit is a pre-triangulated subcategory and\n$(\\mathcal{D}', [\\ ], \\mathcal{T}')$ is a triangulated category.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The definition of a triangulated category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QM","source_file":"derived.tex","source_line":219,"source_end_line":244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L219-L244","statement_sha256":"d5d6e335778a7ffa1b89952eb59c0280b80f9b361b856487bfbea243f6fd63f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2435,"rank":2435,"depth":0,"x":237.645,"y":688.376,"cluster":"derived-categories"},{"id":"stacks:0147","tag":"0147","title":"The definition of a triangulated category · Definition 0147","summary":"Let D be a pre-triangulated category. Let A be an abelian category. An additive functor H : D → A is called homological if for every distinguished triangle (X, Y, Z, f, g, h) the sequence H(X) → H(Y) → H(Z) is exact in the abelian category A. An additive functor H : D^opp → A is called cohomological if the corresponding functor D → A^opp is homological.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $\\mathcal{A}$ be an abelian category.\nAn additive functor $H : \\mathcal{D} \\to \\mathcal{A}$ is called\n{\\it homological} if for every distinguished triangle\n$(X, Y, Z, f, g, h)$ the sequence\n$$\nH(X) \\to H(Y) \\to H(Z)\n$$\nis exact in the abelian category $\\mathcal{A}$. An additive functor\n$H : \\mathcal{D}^{opp} \\to \\mathcal{A}$ is called {\\it cohomological}\nif the corresponding functor $\\mathcal{D} \\to \\mathcal{A}^{opp}$ is\nhomological.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The definition of a triangulated category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0147","source_file":"derived.tex","source_line":260,"source_end_line":274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L260-L274","statement_sha256":"6c41a07894a2afd9be84f2bac0d07cc6b54dc09bc0b2a40b0701fa296d900968","origin":"The Stacks Project","memory_eligible":false,"source_rank":2436,"rank":2436,"depth":0,"x":215.971,"y":677.916,"cluster":"derived-categories"},{"id":"stacks:0150","tag":"0150","title":"The definition of a triangulated category · Definition 0150","summary":"Let A be an abelian category. Let D be a triangulated category. A δ-functor from A to D is given by a functor G : A → D and a rule which assigns to every short exact sequence 0 → A xrightarrowa B xrightarrowb C → 0 a morphism δ = δ_A → B → C : G(C) → G(A)[1] such that • the triangle (G(A), G(B), G(C), G(a), G(b), δ_A → B → C) is a distinguished triangle of D for any short exact sequence as above, and • for every morphism (A → B → C) → (A' → B' → C') of short exact…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\mathcal{D}$ be a triangulated category.\nA {\\it $\\delta$-functor from $\\mathcal{A}$ to $\\mathcal{D}$} is\ngiven by a functor $G : \\mathcal{A} \\to \\mathcal{D}$ and\na rule which assigns to every short exact sequence\n$$\n0 \\to A \\xrightarrow{a} B \\xrightarrow{b} C \\to 0\n$$\na morphism $\\delta = \\delta_{A \\to B \\to C} : G(C) \\to G(A)[1]$\nsuch that\n\\begin{enumerate}\n\\item the triangle\n$(G(A), G(B), G(C), G(a), G(b), \\delta_{A \\to B \\to C})$\nis a distinguished triangle of $\\mathcal{D}$\nfor any short exact sequence as above, and\n\\item for every morphism $(A \\to B \\to C) \\to (A' \\to B' \\to C')$\nof short exact sequences the diagram\n$$\n\\xymatrix{\nG(C) \\ar[d] \\ar[rr]_{\\delta_{A \\to B \\to C}} & &\nG(A)[1] \\ar[d] \\\\\nG(C') \\ar[rr]^{\\delta_{A' \\to B' \\to C'}} & &\nG(A')[1]\n}\n$$\nis commutative.\n\\end{enumerate}\nIn this situation we call\n$(G(A), G(B), G(C), G(a), G(b), \\delta_{A \\to B \\to C})$\nthe {\\it image of the short exact sequence under the\ngiven $\\delta$-functor}.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The definition of a triangulated category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0150","source_file":"derived.tex","source_line":298,"source_end_line":331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L298-L331","statement_sha256":"c0040c376548934db0419eb15d1dcebb0c6e5ee7446ccc6dd3caab3be669dac7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2437,"rank":2437,"depth":0,"x":243.289,"y":672.899,"cluster":"derived-categories"},{"id":"stacks:0146","tag":"0146","title":"Elementary results on triangulated categories · Lemma 0146","summary":"Let D be a pre-triangulated category. Let (X, Y, Z, f, g, h) be a distinguished triangle. Then g ∘ f = 0, h ∘ g = 0 and f[1] ∘ h = 0.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $(X, Y, Z, f, g, h)$ be a distinguished triangle.\nThen $g \\circ f = 0$,\n$h \\circ g = 0$ and $f[1] \\circ h = 0$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0146","source_file":"derived.tex","source_line":357,"source_end_line":363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L357-L363","statement_sha256":"b8940f6311b754926dea11372accca6a772cb63e452f84304cdf9667c99310d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2438,"rank":2438,"depth":0,"x":225.555,"y":693.89,"cluster":"derived-categories"},{"id":"stacks:0149","tag":"0149","title":"Elementary results on triangulated categories · Lemma 0149","summary":"Let D be a pre-triangulated category. For any object W of D the functor Hom_D(W, -) is homological, and the functor Hom_D(-, W) is cohomological.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nFor any object $W$ of $\\mathcal{D}$ the functor\n$\\Hom_\\mathcal{D}(W, -)$ is homological, and the functor\n$\\Hom_\\mathcal{D}(-, W)$ is cohomological.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0149","source_file":"derived.tex","source_line":385,"source_end_line":391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L385-L391","statement_sha256":"db404965f54cf7c41ab2d48c689aa10d194632f3c33d13a57e946ed89c1a3e1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2439,"rank":2439,"depth":1,"x":221.523,"y":666.289,"cluster":"derived-categories"},{"id":"stacks:014A","tag":"014A","title":"Elementary results on triangulated categories · Lemma 014A","summary":"Let D be a pre-triangulated category. Let (a, b, c) : (X, Y, Z, f, g, h) → (X', Y', Z', f', g', h') be a morphism of distinguished triangles. If two among a, b, c are isomorphisms so is the third.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet\n$$\n(a, b, c) : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')\n$$\nbe a morphism of distinguished triangles. If two among $a, b, c$\nare isomorphisms so is the third.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014A","source_file":"derived.tex","source_line":413,"source_end_line":422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L413-L422","statement_sha256":"c868953310f0a4bc34d35919000887b942e347b7842644d6ab61139715677238","origin":"The Stacks Project","memory_eligible":false,"source_rank":2440,"rank":2440,"depth":8,"x":248.392,"y":685.642,"cluster":"derived-categories"},{"id":"stacks:05QP","tag":"05QP","title":"Elementary results on triangulated categories · Lemma 05QP","summary":"Let D be a pre-triangulated category. Let (0, b, 0), (0, b', 0) : (X, Y, Z, f, g, h) → (X, Y, Z, f, g, h) be endomorphisms of a distinguished triangle. Then bb' = 0.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet\n$$\n(0, b, 0), (0, b', 0) : (X, Y, Z, f, g, h) \\to (X, Y, Z, f, g, h)\n$$\nbe endomorphisms of a distinguished triangle. Then $bb' = 0$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QP","source_file":"derived.tex","source_line":483,"source_end_line":491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L483-L491","statement_sha256":"01dcb6eeec4e6fe2572794c9931d7fe9f27e359d4adb71e3c47feca66223fec1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2441,"rank":2441,"depth":2,"x":210.866,"y":686.634,"cluster":"derived-categories"},{"id":"stacks:05QQ","tag":"05QQ","title":"Elementary results on triangulated categories · Lemma 05QQ","summary":"Let D be a pre-triangulated category. Let (X, Y, Z, f, g, h) be a distinguished triangle. If xymatrix Z ar[r]_h ar[d]_c & X[1] ar[d]^a[1] Z ar[r]^h & X[1] is commutative and a^2 = a, c^2 = c, then there exists a morphism b : Y → Y with b^2 = b such that (a, b, c) is an endomorphism of the triangle (X, Y, Z, f, g, h).","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $(X, Y, Z, f, g, h)$ be a distinguished triangle.\nIf\n$$\n\\xymatrix{\nZ \\ar[r]_h \\ar[d]_c & X[1] \\ar[d]^{a[1]} \\\\\nZ \\ar[r]^h & X[1]\n}\n$$\nis commutative and $a^2 = a$, $c^2 = c$, then there exists a\nmorphism $b : Y \\to Y$ with $b^2 = b$ such that\n$(a, b, c)$ is an endomorphism of the triangle $(X, Y, Z, f, g, h)$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QQ","source_file":"derived.tex","source_line":513,"source_end_line":527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L513-L527","statement_sha256":"eae26ab6a356ee0b3d641d656cdbf679d79b6aac96493925df8a0c8eb04d0812","origin":"The Stacks Project","memory_eligible":false,"source_rank":2442,"rank":2442,"depth":3,"x":239.224,"y":663.443,"cluster":"derived-categories"},{"id":"stacks:014B","tag":"014B","title":"Elementary results on triangulated categories · Lemma 014B","summary":"Let D be a pre-triangulated category. Let f : X → Y be a morphism of D. There exists a distinguished triangle (X, Y, Z, f, g, h) which is unique up to (nonunique) isomorphism of triangles. More precisely, given a second such distinguished triangle (X, Y, Z', f, g', h') there exists an isomorphism (1, 1, c) : (X, Y, Z, f, g, h) → (X, Y, Z', f, g', h')","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $f : X \\to Y$ be a morphism of $\\mathcal{D}$.\nThere exists a distinguished triangle $(X, Y, Z, f, g, h)$ which\nis unique up to (nonunique) isomorphism of triangles.\nMore precisely, given a second such distinguished triangle\n$(X, Y, Z', f, g', h')$ there exists an isomorphism\n$$\n(1, 1, c) : (X, Y, Z, f, g, h) \\longrightarrow (X, Y, Z', f, g', h')\n$$","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014B","source_file":"derived.tex","source_line":540,"source_end_line":551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L540-L551","statement_sha256":"5ad8f5daae4bd01560bc526247160b30eb1b26c8be784b6d0010d58ec43ccdba","origin":"The Stacks Project","memory_eligible":false,"source_rank":2443,"rank":2443,"depth":9,"x":236.816,"y":698.254,"cluster":"derived-categories"},{"id":"stacks:0FWZ","tag":"0FWZ","title":"Elementary results on triangulated categories · Lemma 0FWZ","summary":"Let D be a pre-triangulated category. Let (a, b, c) : (X, Y, Z, f, g, h) → (X', Y', Z', f', g', h') be a morphism of distinguished triangles. If one of the following conditions holds • Hom(Y, X') = 0, • Hom(Z, Y') = 0, • Hom(X, X') = Hom(Z, X') = 0, • Hom(Z, X') = Hom(Z, Z') = 0, or • Hom(X[1], Z') = Hom(Z, X') = 0 then b is the unique morphism from Y → Y' such that (a, b, c) is a morphism of triangles.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category. Let\n$$\n(a, b, c) : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')\n$$\nbe a morphism of distinguished triangles. If one of the following\nconditions holds\n\\begin{enumerate}\n\\item $\\Hom(Y, X') = 0$,\n\\item $\\Hom(Z, Y') = 0$,\n\\item $\\Hom(X, X') = \\Hom(Z, X') = 0$,\n\\item $\\Hom(Z, X') = \\Hom(Z, Z') = 0$, or\n\\item $\\Hom(X[1], Z') = \\Hom(Z, X') = 0$\n\\end{enumerate}\nthen $b$ is the unique morphism from $Y \\to Y'$ such that\n$(a, b, c)$ is a morphism of triangles.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWZ","source_file":"derived.tex","source_line":558,"source_end_line":575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L558-L575","statement_sha256":"55be782fcd33c4860e5becc325ff2f14ece36249aae8b6606a2d31254da02b70","origin":"The Stacks Project","memory_eligible":false,"source_rank":2444,"rank":2444,"depth":2,"x":209.456,"y":669.999,"cluster":"derived-categories"},{"id":"stacks:05QR","tag":"05QR","title":"Elementary results on triangulated categories · Lemma 05QR","summary":"Let D be a pre-triangulated category. Let f : X → Y be a morphism of D. The following are equivalent • f is an isomorphism, • (X, Y, 0, f, 0, 0) is a distinguished triangle, and • for any distinguished triangle (X, Y, Z, f, g, h) we have Z = 0.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $f : X \\to Y$ be a morphism of $\\mathcal{D}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is an isomorphism,\n\\item $(X, Y, 0, f, 0, 0)$ is a distinguished triangle, and\n\\item for any distinguished triangle $(X, Y, Z, f, g, h)$ we have $Z = 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QR","source_file":"derived.tex","source_line":615,"source_end_line":625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L615-L625","statement_sha256":"c6fc9fe474f14f4f64faea48f7af0aad2619a012b16c583b1ae191c5a4def747","origin":"The Stacks Project","memory_eligible":false,"source_rank":2445,"rank":2445,"depth":9,"x":254.1,"y":675.549,"cluster":"derived-categories"},{"id":"stacks:05QS","tag":"05QS","title":"Elementary results on triangulated categories · Lemma 05QS","summary":"Let D be a pre-triangulated category. Let (X, Y, Z, f, g, h) and (X', Y', Z', f', g', h') be triangles. The following are equivalent • (X ⊕ X', Y ⊕ Y', Z ⊕ Z', f ⊕ f', g ⊕ g', h ⊕ h') is a distinguished triangle, • both (X, Y, Z, f, g, h) and (X', Y', Z', f', g', h') are distinguished triangles.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ be triangles.\nThe following are equivalent\n\\begin{enumerate}\n\\item $(X \\oplus X', Y \\oplus Y', Z \\oplus Z',\nf \\oplus f', g \\oplus g', h \\oplus h')$\nis a distinguished triangle,\n\\item both $(X, Y, Z, f, g, h)$ and $(X', Y', Z', f', g', h')$ are\ndistinguished triangles.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QS","source_file":"derived.tex","source_line":638,"source_end_line":650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L638-L650","statement_sha256":"f79b1ee30340c2d21db44a5eeb0985c1f7ca5498aaac5d9b6bf382abec748121","origin":"The Stacks Project","memory_eligible":false,"source_rank":2446,"rank":2446,"depth":0,"x":215.292,"y":697.573,"cluster":"derived-categories"},{"id":"stacks:05QT","tag":"05QT","title":"Elementary results on triangulated categories · Lemma 05QT","summary":"Let D be a pre-triangulated category. Let (X, Y, Z, f, g, h) be a distinguished triangle. • If h = 0, then there exists a right inverse s : Z → Y to g. • For any right inverse s : Z → Y of g the map f ⊕ s : X ⊕ Z → Y is an isomorphism. • For any objects X', Z' of D the triangle (X', X' ⊕ Z', Z', (1, 0), (0, 1), 0) is distinguished.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $(X, Y, Z, f, g, h)$ be a distinguished triangle.\n\\begin{enumerate}\n\\item If $h = 0$, then there exists a right inverse $s : Z \\to Y$ to $g$.\n\\item For any right inverse $s : Z \\to Y$ of $g$ the map\n$f \\oplus s : X \\oplus Z \\to Y$ is an isomorphism.\n\\item For any objects $X', Z'$ of $\\mathcal{D}$ the triangle\n$(X', X' \\oplus Z', Z', (1, 0), (0, 1), 0)$ is distinguished.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QT","source_file":"derived.tex","source_line":699,"source_end_line":710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L699-L710","statement_sha256":"62d58c6deb3ec29c8c2252db1431e2e62a8c81d91a64070c97fb2dfcd727368a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2447,"rank":2447,"depth":2,"x":226.602,"y":657.974,"cluster":"derived-categories"},{"id":"stacks:05QU","tag":"05QU","title":"Elementary results on triangulated categories · Lemma 05QU","summary":"Let D be a pre-triangulated category. Let f : X → Y be a morphism of D. The following are equivalent • f has a kernel, • f has a cokernel, • f is the isomorphic to a composition K ⊕ Z → Z → Z ⊕ Q of a projection and coprojection for some objects K, Z, Q of D.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $f : X \\to Y$ be a morphism of $\\mathcal{D}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ has a kernel,\n\\item $f$ has a cokernel,\n\\item $f$ is the isomorphic to a composition\n$K \\oplus Z \\to Z \\to Z \\oplus Q$ of a projection and coprojection\nfor some objects $K, Z, Q$ of $\\mathcal{D}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QU","source_file":"derived.tex","source_line":729,"source_end_line":741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L729-L741","statement_sha256":"afd02dadad87acd05ea5f8d1f070abc1ec6496953154d6e60df7e794de9a7f85","origin":"The Stacks Project","memory_eligible":false,"source_rank":2448,"rank":2448,"depth":3,"x":250.86,"y":694.768,"cluster":"derived-categories"},{"id":"stacks:0CRG","tag":"0CRG","title":"Elementary results on triangulated categories · Lemma 0CRG","summary":"Let D be a pre-triangulated category. Let I be a set. • Let X_i, i ∈ I be a family of objects of D. • If ∏ X_i exists, then (∏ X_i)[1] = ∏ X_i[1]. • If bigoplus X_i exists, then (bigoplus X_i)[1] = bigoplus X_i[1]. • Let X_i → Y_i → Z_i → X_i[1] be a family of distinguished triangles of D. • If ∏ X_i, ∏ Y_i, ∏ Z_i exist, then ∏ X_i → ∏ Y_i → ∏ Z_i → ∏ X_i[1] is a distinguished triangle. • If bigoplus X_i, bigoplus Y_i, bigoplus Z_i exist, then bigoplus X_i → bigoplus Y_i…","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $I$ be a set.\n\\begin{enumerate}\n\\item Let $X_i$, $i \\in I$ be a family of objects of $\\mathcal{D}$.\n\\begin{enumerate}\n\\item If $\\prod X_i$ exists, then $(\\prod X_i)[1] = \\prod X_i[1]$.\n\\item If $\\bigoplus X_i$ exists, then $(\\bigoplus X_i)[1] = \\bigoplus X_i[1]$.\n\\end{enumerate}\n\\item Let $X_i \\to Y_i \\to Z_i \\to X_i[1]$ be a family of distinguished\ntriangles of $\\mathcal{D}$.\n\\begin{enumerate}\n\\item If $\\prod X_i$, $\\prod Y_i$, $\\prod Z_i$ exist, then\n$\\prod X_i \\to \\prod Y_i \\to \\prod Z_i \\to \\prod X_i[1]$\nis a distinguished triangle.\n\\item If $\\bigoplus X_i$, $\\bigoplus Y_i$,\n$\\bigoplus Z_i$ exist, then\n$\\bigoplus X_i \\to \\bigoplus Y_i \\to \\bigoplus Z_i \\to \\bigoplus X_i[1]$\nis a distinguished triangle.\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRG","source_file":"derived.tex","source_line":761,"source_end_line":783,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L761-L783","statement_sha256":"bbcd8470bfe17ef480d7fe8331a6dafc956fa3bbafc27a4760c87761dc1dcf49","origin":"The Stacks Project","memory_eligible":false,"source_rank":2449,"rank":2449,"depth":0,"x":201.929,"y":680.975,"cluster":"derived-categories"},{"id":"stacks:05QW","tag":"05QW","title":"Elementary results on triangulated categories · Lemma 05QW","summary":"Let D be a pre-triangulated category. If D has countable products, then D is Karoubian. If D has countable coproducts, then D is Karoubian.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nIf $\\mathcal{D}$ has countable products, then $\\mathcal{D}$\nis Karoubian.\nIf $\\mathcal{D}$ has countable coproducts, then $\\mathcal{D}$\nis Karoubian.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QW","source_file":"derived.tex","source_line":805,"source_end_line":812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L805-L812","statement_sha256":"bd77a57ebc59bbb7982b7a33ae2bf4c6b31b9d4a36bc4dca60abfec67e912907","origin":"The Stacks Project","memory_eligible":false,"source_rank":2450,"rank":2450,"depth":5,"x":250.475,"y":662.884,"cluster":"derived-categories"},{"id":"stacks:014C","tag":"014C","title":"Elementary results on triangulated categories · Lemma 014C","summary":"Let D be a pre-triangulated category. In order to prove TR4 it suffices to show that given any pair of composable morphisms f : X → Y and g : Y → Z there exist • isomorphisms i : X' → X, j : Y' → Y and k : Z' → Z, and then setting f' = j^-1fi : X' → Y' and g' = k^-1gj : Y' → Z' there exist • distinguished triangles (X', Y', Q_1, f', p_1, d_1), (X', Z', Q_2, g' ∘ f', p_2, d_2) and (Y', Z', Q_3, g', p_3, d_3), such that the assertion of TR4 holds.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nIn order to prove TR4 it suffices to show that given\nany pair of composable morphisms\n$f : X \\to Y$ and $g : Y \\to Z$ there exist\n\\begin{enumerate}\n\\item isomorphisms $i : X' \\to X$, $j : Y' \\to Y$ and\n$k : Z' \\to Z$, and then setting $f' = j^{-1}fi : X' \\to Y'$ and\n$g' = k^{-1}gj : Y' \\to Z'$ there exist\n\\item distinguished triangles\n$(X', Y', Q_1, f', p_1, d_1)$,\n$(X', Z', Q_2, g' \\circ f', p_2, d_2)$\nand\n$(Y', Z', Q_3, g', p_3, d_3)$,\nsuch that the assertion of TR4 holds.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014C","source_file":"derived.tex","source_line":828,"source_end_line":845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L828-L845","statement_sha256":"a26550fe0137fdadf96512a1c2b6f528b27112dd1767e979fec089539c428a93","origin":"The Stacks Project","memory_eligible":false,"source_rank":2451,"rank":2451,"depth":10,"x":228.63,"y":704.885,"cluster":"derived-categories"},{"id":"stacks:05QX","tag":"05QX","title":"Elementary results on triangulated categories · Lemma 05QX","summary":"Let D be a pre-triangulated category. Assume that D' is an additive full subcategory of D. The following are equivalent • there exists a set of triangles T' such that (D', T') is a pre-triangulated subcategory of D, • D' is preserved under [1] and [1] : D' → D' is an auto-equivalence and given any morphism f : X → Y in D' there exists a distinguished triangle (X, Y, Z, f, g, h) in D such that Z is isomorphic to an object of D'. In this case T' as in (1) is the set of…","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nAssume that $\\mathcal{D}'$ is an additive full subcategory of $\\mathcal{D}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists a set of triangles $\\mathcal{T}'$ such that\n$(\\mathcal{D}', \\mathcal{T}')$ is a pre-triangulated subcategory\nof $\\mathcal{D}$,\n\\item $\\mathcal{D}'$ is preserved under $[1]$ and\n$[1] : \\mathcal{D}' \\to \\mathcal{D}'$ is an auto-equivalence and\ngiven any morphism $f : X \\to Y$ in $\\mathcal{D}'$ there exists\na distinguished triangle $(X, Y, Z, f, g, h)$ in $\\mathcal{D}$\nsuch that $Z$ is isomorphic to an object of $\\mathcal{D}'$.\n\\end{enumerate}\nIn this case $\\mathcal{T}'$ as in (1) is the set of distinguished triangles\n$(X, Y, Z, f, g, h)$ of $\\mathcal{D}$ such that\n$X, Y, Z \\in \\Ob(\\mathcal{D}')$. Finally, if $\\mathcal{D}$\nis a triangulated category, then (1) and (2) are also equivalent to\n\\begin{enumerate}\n\\item[(3)] $\\mathcal{D}'$ is a triangulated subcategory.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QX","source_file":"derived.tex","source_line":859,"source_end_line":881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L859-L881","statement_sha256":"57dc1100e05a2a8c3d5f26d95203ce44e83f4abc8c273288ef7930db54a514d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2452,"rank":2452,"depth":0,"x":210.518,"y":660.389,"cluster":"derived-categories"},{"id":"stacks:05QY","tag":"05QY","title":"Elementary results on triangulated categories · Lemma 05QY","summary":"An exact functor of pre-triangulated categories is additive.","statement_latex":"An exact functor of pre-triangulated categories is additive.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QY","source_file":"derived.tex","source_line":887,"source_end_line":890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L887-L890","statement_sha256":"779884aa1642041018a2ea9361093e4dea193e2c63ada7e0f592e6d2aa23c32e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2453,"rank":2453,"depth":3,"x":260.862,"y":683.488,"cluster":"derived-categories"},{"id":"stacks:05SQ","tag":"05SQ","title":"Elementary results on triangulated categories · Lemma 05SQ","summary":"Let F : D → D' be a fully faithful exact functor of pre-triangulated categories. Then a triangle (X, Y, Z, f, g, h) of D is distinguished if and only if (F(X), F(Y), F(Z), F(f), F(g), F(h)) is distinguished in D'.","statement_latex":"Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be a fully faithful exact functor\nof pre-triangulated categories. Then a triangle $(X, Y, Z, f, g, h)$\nof $\\mathcal{D}$ is distinguished if and only if\n$(F(X), F(Y), F(Z), F(f), F(g), F(h))$ is distinguished in $\\mathcal{D}'$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SQ","source_file":"derived.tex","source_line":915,"source_end_line":921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L915-L921","statement_sha256":"56f185a1d16080f1a4600019c74a70dca2575683832e3f624d0666b5f6162547","origin":"The Stacks Project","memory_eligible":false,"source_rank":2454,"rank":2454,"depth":10,"x":203.851,"y":695.283,"cluster":"derived-categories"},{"id":"stacks:014Y","tag":"014Y","title":"Elementary results on triangulated categories · Lemma 014Y","summary":"Let D, D', D\" be pre-triangulated categories. Let F : D → D' and F' : D' → D\" be exact functors. Then F' ∘ F is an exact functor.","statement_latex":"Let $\\mathcal{D}, \\mathcal{D}', \\mathcal{D}''$ be pre-triangulated categories.\nLet $F : \\mathcal{D} \\to \\mathcal{D}'$ and\n$F' : \\mathcal{D}' \\to \\mathcal{D}''$ be exact functors.\nThen $F' \\circ F$ is an exact functor.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014Y","source_file":"derived.tex","source_line":938,"source_end_line":944,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L938-L944","statement_sha256":"53afe7d1923931bc09263531d868be1a8269568eba67f3cf75e9ea66d8be076f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2455,"rank":2455,"depth":0,"x":237.146,"y":653.319,"cluster":"derived-categories"},{"id":"stacks:05QZ","tag":"05QZ","title":"Elementary results on triangulated categories · Lemma 05QZ","summary":"Let D be a pre-triangulated category. Let A be an abelian category. Let H : D → A be a homological functor. • Let D' be a pre-triangulated category. Let F : D' → D be an exact functor. Then the composition H ∘ F is a homological functor as well. • Let A' be an abelian category. Let G : A → A' be an exact functor. Then G ∘ H is a homological functor as well.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $\\mathcal{A}$ be an abelian category.\nLet $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor.\n\\begin{enumerate}\n\\item Let $\\mathcal{D}'$ be a pre-triangulated category.\nLet $F : \\mathcal{D}' \\to \\mathcal{D}$ be an exact functor.\nThen the composition $H \\circ F$ is a homological functor as well.\n\\item Let $\\mathcal{A}'$ be an abelian category. Let\n$G : \\mathcal{A} \\to \\mathcal{A}'$ be an exact functor.\nThen $G \\circ H$ is a homological functor as well.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05QZ","source_file":"derived.tex","source_line":950,"source_end_line":963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L950-L963","statement_sha256":"3e7d22f5dc80d5732946374c5e295a5195976e4bfaa705357cd3425bec5fb846","origin":"The Stacks Project","memory_eligible":false,"source_rank":2456,"rank":2456,"depth":0,"x":246.527,"y":704.227,"cluster":"derived-categories"},{"id":"stacks:0151","tag":"0151","title":"Elementary results on triangulated categories · Lemma 0151","summary":"Let D be a triangulated category. Let A be an abelian category. Let G : A → D be a δ-functor. • Let D' be a triangulated category. Let F : D → D' be an exact functor. Then the composition F ∘ G is a δ-functor as well. • Let A' be an abelian category. Let H : A' → A be an exact functor. Then G ∘ H is a δ-functor as well.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $\\mathcal{A}$ be an abelian category.\nLet $G : \\mathcal{A} \\to \\mathcal{D}$ be a $\\delta$-functor.\n\\begin{enumerate}\n\\item Let $\\mathcal{D}'$ be a triangulated category.\nLet $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor.\nThen the composition $F \\circ G$ is a $\\delta$-functor as well.\n\\item Let $\\mathcal{A}'$ be an abelian category. Let\n$H : \\mathcal{A}' \\to \\mathcal{A}$ be an exact functor.\nThen $G \\circ H$ is a $\\delta$-functor as well.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0151","source_file":"derived.tex","source_line":969,"source_end_line":982,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L969-L982","statement_sha256":"2d4924d2cb9978030ea2f7965f98113af1586013af72961c8c87a12262377c50","origin":"The Stacks Project","memory_eligible":false,"source_rank":2457,"rank":2457,"depth":0,"x":197.691,"y":671.342,"cluster":"derived-categories"},{"id":"stacks:05SR","tag":"05SR","title":"Elementary results on triangulated categories · Lemma 05SR","summary":"Let D be a triangulated category. Let A and B be abelian categories. Let G : A → D be a δ-functor. Let H : D → B be a homological functor. Assume that H^-1(G(A)) = 0 for all A in A. Then the collection (H^n ∘ G, H^n(δ_A → B → C))_n ≥ 0 is a δ-functor from A → B, see Homology, Definition [Tag 010Q].","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $G : \\mathcal{A} \\to \\mathcal{D}$ be a $\\delta$-functor.\nLet $H : \\mathcal{D} \\to \\mathcal{B}$ be a homological functor.\nAssume that $H^{-1}(G(A)) = 0$ for all $A$ in $\\mathcal{A}$.\nThen the collection\n$$\n\\{H^n \\circ G, H^n(\\delta_{A \\to B \\to C})\\}_{n \\geq 0}\n$$\nis a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SR","source_file":"derived.tex","source_line":988,"source_end_line":1001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L988-L1001","statement_sha256":"39e58de6d79d6c9e037a2e51bb85d3fb77569edf9dd31d5cf256dd040ac73833","origin":"The Stacks Project","memory_eligible":false,"source_rank":2458,"rank":2458,"depth":1,"x":261.384,"y":667.82,"cluster":"derived-categories"},{"id":"stacks:05R0","tag":"05R0","title":"Elementary results on triangulated categories · Proposition 05R0","summary":"Let D be a triangulated category. Any commutative diagram xymatrix X ar[r] ar[d] & Y ar[d] X' ar[r] & Y' can be extended to a diagram xymatrix X ar[r] ar[d] & Y ar[r] ar[d] & Z ar[r] ar[d] & X[1] ar[d] X' ar[r] ar[d] & Y' ar[r] ar[d] & Z' ar[r] ar[d] & X'[1] ar[d] X\" ar[r] ar[d] & Y\" ar[r] ar[d] & Z\" ar[r] ar[d] & X\"[1] ar[d] X[1] ar[r] & Y[1] ar[r] & Z[1] ar[r] & X[2] where all the squares are commutative, except for the lower right square which is anticommutative.…","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Any commutative diagram\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d] & Y \\ar[d] \\\\\nX' \\ar[r] & Y'\n}\n$$\ncan be extended to a diagram\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d] & Y \\ar[r] \\ar[d] & Z \\ar[r] \\ar[d] & X[1] \\ar[d] \\\\\nX' \\ar[r] \\ar[d] & Y' \\ar[r] \\ar[d] & Z' \\ar[r] \\ar[d] & X'[1] \\ar[d] \\\\\nX'' \\ar[r] \\ar[d] & Y'' \\ar[r] \\ar[d] & Z'' \\ar[r] \\ar[d] & X''[1] \\ar[d] \\\\\nX[1] \\ar[r] & Y[1] \\ar[r] & Z[1] \\ar[r] & X[2]\n}\n$$\nwhere all the squares are commutative, except for the lower right square\nwhich is anticommutative. Moreover, each of the rows and columns are\ndistinguished triangles. Finally, the morphisms on the bottom row\n(resp.\\ right column) are obtained from the morphisms of the top row\n(resp.\\ left column) by applying $[1]$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Elementary results on triangulated categories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05R0","source_file":"derived.tex","source_line":1030,"source_end_line":1053,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1030-L1053","statement_sha256":"17fcef96ec0bfd3956124e0e8f1a4ea1fa26b47cc7406522fa4b5aabf178c175","origin":"The Stacks Project","memory_eligible":false,"source_rank":2459,"rank":2459,"depth":0,"x":216.404,"y":707.29,"cluster":"derived-categories"},{"id":"stacks:05R2","tag":"05R2","title":"Localization of triangulated categories · Definition 05R2","summary":"Let D be a pre-triangulated category. We say a multiplicative system S is compatible with the triangulated structure if the following two conditions hold: • [MS5] For a morphism f of D we have f ∈ S ⇔ f[1] ∈ S. • [MS6] Given a solid commutative square xymatrix X ar[r] ar[d]^s & Y ar[r] ar[d]^s' & Z ar[r] ar@-->[d] & X[1] ar[d]^s[1] X' ar[r] & Y' ar[r] & Z' ar[r] & X'[1] whose rows are distinguished triangles with s, s' ∈ S there exists a morphism s\" : Z → Z' in S such…","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category. We say a multiplicative\nsystem $S$ is {\\it compatible with the triangulated structure} if\nthe following two conditions hold:\n\\begin{enumerate}\n\\item[MS5] For a morphism $f$ of $\\mathcal{D}$ we have\n$f \\in S \\Leftrightarrow f[1] \\in S$\\footnote{See Remark \\ref{remark-MS5}.}.\n\\item[MS6] Given a solid commutative square\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d]^s &\nY \\ar[r] \\ar[d]^{s'} &\nZ \\ar[r] \\ar@{-->}[d] &\nX[1] \\ar[d]^{s[1]} \\\\\nX' \\ar[r] &\nY' \\ar[r] &\nZ' \\ar[r] &\nX'[1]\n}\n$$\nwhose rows are distinguished triangles with $s, s' \\in S$\nthere exists a morphism $s'' : Z \\to Z'$ in $S$ such that\n$(s, s', s'')$ is a morphism of triangles.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Localization of triangulated categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05R2","source_file":"derived.tex","source_line":1156,"source_end_line":1181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1156-L1181","statement_sha256":"67a8d317791bae39f23b7ae31789799ecb4138d81fb49b594adb5ef379b2349d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2460,"rank":2460,"depth":0,"x":217.865,"y":651.661,"cluster":"derived-categories"},{"id":"stacks:05R3","tag":"05R3","title":"Localization of triangulated categories · Lemma 05R3","summary":"Let D be a pre-triangulated category. Let S ⊂ Arrows(D). • If S contains all identities and MS6 holds (Definition [Tag 05R2]), then every isomorphism of D is in S. • If MS1, MS5 (Categories, Definition [Tag 04VC]) and MS6 hold, then MS2 holds.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $S \\subset \\text{Arrows}(\\mathcal{D})$.\n\\begin{enumerate}\n\\item If $S$ contains all identities and MS6 holds\n(Definition \\ref{definition-localization}),\nthen every isomorphism of $\\mathcal{D}$ is in $S$.\n\\item If MS1, MS5 (Categories, Definition\n\\ref{categories-definition-multiplicative-system}) and MS6 hold,\nthen MS2 holds.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Localization of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05R3","source_file":"derived.tex","source_line":1187,"source_end_line":1199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1187-L1199","statement_sha256":"a85db111f0e9c3eb5a7794cef457cd205b250250fcb9b4fc7bc14b180520ec06","origin":"The Stacks Project","memory_eligible":false,"source_rank":2461,"rank":2461,"depth":10,"x":262.289,"y":694.255,"cluster":"derived-categories"},{"id":"stacks:05R4","tag":"05R4","title":"Localization of triangulated categories · Lemma 05R4","summary":"Let F : D → D' be an exact functor of pre-triangulated categories. Let S = (f ∈ Arrows(D) mid F(f) is an isomorphism) Then S is a saturated (see Categories, Definition [Tag 05Q8]) multiplicative system compatible with the triangulated structure on D.","statement_latex":"Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of\npre-triangulated categories.  Let\n$$\nS = \\{f \\in \\text{Arrows}(\\mathcal{D})\n\\mid F(f)\\text{ is an isomorphism}\\}\n$$\nThen $S$ is a saturated (see\nCategories,\nDefinition \\ref{categories-definition-saturated-multiplicative-system})\nmultiplicative system compatible with the\ntriangulated structure on $\\mathcal{D}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Localization of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05R4","source_file":"derived.tex","source_line":1256,"source_end_line":1269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1256-L1269","statement_sha256":"d718a328da78f8b431725fcb8b6990628e2993c9f5a907c2296ce6d6929550bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2462,"rank":2462,"depth":11,"x":194.135,"y":687.941,"cluster":"derived-categories"},{"id":"stacks:05R5","tag":"05R5","title":"Localization of triangulated categories · Lemma 05R5","summary":"Let H : D → A be a homological functor between a pre-triangulated category and an abelian category. Let S = (f ∈ Arrows(D) mid H^i(f) is an isomorphism for all i ∈ Z) Then S is a saturated (see Categories, Definition [Tag 05Q8]) multiplicative system compatible with the triangulated structure on D.","statement_latex":"Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor between a\npre-triangulated category and an abelian category. Let\n$$\nS = \\{f \\in \\text{Arrows}(\\mathcal{D})\n\\mid H^i(f)\\text{ is an isomorphism for all }i \\in \\mathbf{Z}\\}\n$$\nThen $S$ is a saturated (see\nCategories,\nDefinition \\ref{categories-definition-saturated-multiplicative-system})\nmultiplicative system compatible with the\ntriangulated structure on $\\mathcal{D}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Localization of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05R5","source_file":"derived.tex","source_line":1307,"source_end_line":1320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1307-L1320","statement_sha256":"be2b88c2097db8fb85b29787af04b83901098ed869b0cad101e39ad41a6c541d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2463,"rank":2463,"depth":11,"x":250.386,"y":653.368,"cluster":"derived-categories"},{"id":"stacks:05R6","tag":"05R6","title":"Localization of triangulated categories · Proposition 05R6","summary":"Let D be a pre-triangulated category. Let S be a multiplicative system compatible with the triangulated structure. Then there exists a unique structure of a pre-triangulated category on S^-1D such that [1] ∘ Q = Q ∘ [1] and the localization functor Q : D → S^-1D is exact. Moreover, if D is a triangulated category, so is S^-1D.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative\nsystem compatible with the triangulated structure.\nThen there exists a unique structure of a pre-triangulated category on\n$S^{-1}\\mathcal{D}$ such that $[1] \\circ Q = Q \\circ [1]$ and\nthe localization functor $Q : \\mathcal{D} \\to S^{-1}\\mathcal{D}$ is exact.\nMoreover, if $\\mathcal{D}$ is a triangulated category, so is\n$S^{-1}\\mathcal{D}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Localization of triangulated categories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05R6","source_file":"derived.tex","source_line":1359,"source_end_line":1368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1359-L1368","statement_sha256":"26bd7fff2e619b1f939ba12e7288f4d3fe20f410239599e2a8927444956db514","origin":"The Stacks Project","memory_eligible":false,"source_rank":2464,"rank":2464,"depth":11,"x":236.485,"y":711.696,"cluster":"derived-categories"},{"id":"stacks:05R7","tag":"05R7","title":"Localization of triangulated categories · Lemma 05R7","summary":"Let D be a pre-triangulated category. Let S be a multiplicative system compatible with the triangulated structure. Let Q : D → S^-1D be the localization functor, see Proposition [Tag 05R6]. • If H : D → A is a homological functor into an abelian category A such that H(s) is an isomorphism for all s ∈ S, then the unique factorization H' : S^-1D → A such that H = H' ∘ Q (see Categories, Lemma [Tag 04VG]) is a homological functor too. • If F : D → D' is an exact functor into…","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative\nsystem compatible with the triangulated structure. Let\n$Q : \\mathcal{D} \\to S^{-1}\\mathcal{D}$ be the localization functor, see\nProposition \\ref{proposition-construct-localization}.\n\\begin{enumerate}\n\\item If $H : \\mathcal{D} \\to \\mathcal{A}$ is a homological functor into\nan abelian category $\\mathcal{A}$ such that $H(s)$ is an isomorphism for\nall $s \\in S$, then the unique factorization\n$H' : S^{-1}\\mathcal{D} \\to \\mathcal{A}$ such that $H = H' \\circ Q$ (see\nCategories, Lemma \\ref{categories-lemma-properties-left-localization})\nis a homological functor too.\n\\item If $F : \\mathcal{D} \\to \\mathcal{D}'$ is an exact functor into\na pre-triangulated category $\\mathcal{D}'$ such that $F(s)$ is an isomorphism\nfor all $s \\in S$, then the unique factorization\n$F' : S^{-1}\\mathcal{D} \\to \\mathcal{D}'$ such that $F = F' \\circ Q$ (see\nCategories, Lemma \\ref{categories-lemma-properties-left-localization})\nis an exact functor too.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Localization of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05R7","source_file":"derived.tex","source_line":1470,"source_end_line":1490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1470-L1490","statement_sha256":"983f5ac8332b20d896be69aa7e8c4fbd1d2428d5d60f890c91ee05919fff1c3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2465,"rank":2465,"depth":12,"x":199.268,"y":660.007,"cluster":"derived-categories"},{"id":"stacks:0GSL","tag":"0GSL","title":"Localization of triangulated categories · Lemma 0GSL","summary":"Let D be a pre-triangulated category and let D' ⊂ D be a full, pre-triangulated subcategory. Let S be a saturated multiplicative system of D compatible with the triangulated structure. Assume that for each X in D there exists an s : X' → X in S such that X' is an object of D'. Then S' = S ∩ Arrows(D') is a saturated multiplicative system compatible with the triangulated structure and the functor (S')^-1D' → S^-1D is an equivalence of pre-triangulated categories.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category and let\n$\\mathcal{D}' \\subset \\mathcal{D}$ be a full, pre-triangulated subcategory.\nLet $S$ be a saturated multiplicative system of $\\mathcal{D}$\ncompatible with the triangulated structure. Assume that for each $X$\nin $\\mathcal{D}$ there exists an $s : X' \\to X$ in $S$ such that $X'$ is\nan object of $\\mathcal{D}'$. Then\n$S' = S \\cap \\text{Arrows}(\\mathcal{D}')$ is a saturated\nmultiplicative system compatible with the triangulated structure and\nthe functor\n$$\n(S')^{-1}\\mathcal{D}' \\longrightarrow S^{-1}\\mathcal{D}\n$$\nis an equivalence of pre-triangulated categories.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Localization of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSL","source_file":"derived.tex","source_line":1496,"source_end_line":1511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1496-L1511","statement_sha256":"a3717b14872968736bc19fb11fb0d10a6de66de2655e1fd8f9740e16041a9a15","origin":"The Stacks Project","memory_eligible":false,"source_rank":2466,"rank":2466,"depth":13,"x":269.312,"y":677.264,"cluster":"derived-categories"},{"id":"stacks:05R8","tag":"05R8","title":"Localization of triangulated categories · Lemma 05R8","summary":"Let D be a pre-triangulated category. Let S be a multiplicative system compatible with the triangulated structure. Let Z be an object of D. The following are equivalent • Q(Z) = 0 in S^-1D, • there exists Z' ∈ Ob(D) such that 0 : Z → Z' is an element of S, • there exists Z' ∈ Ob(D) such that 0 : Z' → Z is an element of S, and • there exists an object Z' and a distinguished triangle (X, Y, Z ⊕ Z', f, g, h) such that f ∈ S. If S is saturated, then these are also equivalent…","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category. Let $S$ be a multiplicative\nsystem compatible with the triangulated structure. Let $Z$ be an object\nof $\\mathcal{D}$. The following are equivalent\n\\begin{enumerate}\n\\item $Q(Z) = 0$ in $S^{-1}\\mathcal{D}$,\n\\item there exists $Z' \\in \\Ob(\\mathcal{D})$ such that\n$0 : Z \\to Z'$ is an element of $S$,\n\\item there exists $Z' \\in \\Ob(\\mathcal{D})$ such that\n$0 : Z' \\to Z$ is an element of $S$, and\n\\item there exists an object $Z'$ and a distinguished triangle\n$(X, Y, Z \\oplus Z', f, g, h)$ such that $f \\in S$.\n\\end{enumerate}\nIf $S$ is saturated, then these are also equivalent to\n\\begin{enumerate}\n\\item[(5)] the morphism $0 \\to Z$ is an element of $S$,\n\\item[(6)] the morphism $Z \\to 0$ is an element of $S$,\n\\item[(7)] there exists a distinguished triangle $(X, Y, Z, f, g, h)$\nsuch that $f \\in S$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Localization of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05R8","source_file":"derived.tex","source_line":1548,"source_end_line":1569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1548-L1569","statement_sha256":"f6cfd29d494a3311c95bae1c556171f6cdbe61c7315aa0449a2d9741a6b2630d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2467,"rank":2467,"depth":5,"x":202.827,"y":704.674,"cluster":"derived-categories"},{"id":"stacks:05R9","tag":"05R9","title":"Localization of triangulated categories · Lemma 05R9","summary":"Let D be a pre-triangulated category. Let S be a saturated multiplicative system in D that is compatible with the triangulated structure. Let (X, Y, Z, f, g, h) be a distinguished triangle in D. Consider the category of morphisms of triangles I = ((s, s', s\") : (X, Y, Z, f, g, h) → (X', Y', Z', f', g', h') mid s, s', s\" ∈ S) Then I is a filtered category and the functors I → X/S, I → Y/S, and I → Z/S are cofinal.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category.\nLet $S$ be a saturated multiplicative system in $\\mathcal{D}$\nthat is compatible with the triangulated structure.\nLet $(X, Y, Z, f, g, h)$ be a distinguished triangle in $\\mathcal{D}$.\nConsider the category of morphisms of triangles\n$$\n\\mathcal{I} =\n\\{(s, s', s'') : (X, Y, Z, f, g, h) \\to (X', Y', Z', f', g', h')\n\\mid s, s', s'' \\in S\\}\n$$\nThen $\\mathcal{I}$ is a filtered category and the functors\n$\\mathcal{I} \\to X/S$, $\\mathcal{I} \\to Y/S$, and $\\mathcal{I} \\to Z/S$\nare cofinal.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Localization of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05R9","source_file":"derived.tex","source_line":1595,"source_end_line":1610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1595-L1610","statement_sha256":"e0f2c7f7d43eb5390c32ec31fc57a77683a98493df9fc600d5f444997666794a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2468,"rank":2468,"depth":4,"x":230.198,"y":645.918,"cluster":"derived-categories"},{"id":"stacks:05RB","tag":"05RB","title":"Quotients of triangulated categories · Definition 05RB","summary":"Let D be a pre-triangulated category. We say a full pre-triangulated subcategory D' of D is saturated if whenever X ⊕ Y is isomorphic to an object of D' then both X and Y are isomorphic to objects of D'.","statement_latex":"Let $\\mathcal{D}$ be a pre-triangulated category. We say a full\npre-triangulated subcategory $\\mathcal{D}'$ of $\\mathcal{D}$ is\n{\\it saturated} if whenever $X \\oplus Y$ is isomorphic to an object\nof $\\mathcal{D}'$ then both $X$ and $Y$ are isomorphic to objects\nof $\\mathcal{D}'$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RB","source_file":"derived.tex","source_line":1757,"source_end_line":1764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1757-L1764","statement_sha256":"18e03c8f850a653f5d6476d86d3fa05f9546572be2f84d1133c357df23489be2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2469,"rank":2469,"depth":0,"x":257.632,"y":705.587,"cluster":"derived-categories"},{"id":"stacks:05RC","tag":"05RC","title":"Quotients of triangulated categories · Lemma 05RC","summary":"Let F : D → D' be an exact functor of pre-triangulated categories. Let D\" be the full subcategory of D with objects Ob(D\") = (X ∈ Ob(D) mid F(X) = 0) Then D\" is a strictly full saturated pre-triangulated subcategory of D. If D is a triangulated category, then D\" is a triangulated subcategory.","statement_latex":"Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of\npre-triangulated categories. Let $\\mathcal{D}''$ be the full subcategory\nof $\\mathcal{D}$ with objects\n$$\n\\Ob(\\mathcal{D}'') =\n\\{X \\in \\Ob(\\mathcal{D}) \\mid F(X) = 0\\}\n$$\nThen $\\mathcal{D}''$ is a strictly full saturated pre-triangulated\nsubcategory of $\\mathcal{D}$. If $\\mathcal{D}$ is a triangulated category,\nthen $\\mathcal{D}''$ is a triangulated subcategory.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RC","source_file":"derived.tex","source_line":1785,"source_end_line":1797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1785-L1797","statement_sha256":"a70b26906368176d7ab6352da49ae7055a580741fa26099f86ed7a2abab52232","origin":"The Stacks Project","memory_eligible":false,"source_rank":2470,"rank":2470,"depth":1,"x":188.507,"y":676.77,"cluster":"derived-categories"},{"id":"stacks:05RD","tag":"05RD","title":"Quotients of triangulated categories · Lemma 05RD","summary":"Let H : D → A be a homological functor of a pre-triangulated category into an abelian category. Let D' be the full subcategory of D with objects Ob(D') = (X ∈ Ob(D) mid H(X[n]) = 0 for all n ∈ Z) Then D' is a strictly full saturated pre-triangulated subcategory of D. If D is a triangulated category, then D' is a triangulated subcategory.","statement_latex":"Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor of\na pre-triangulated category into an abelian category.\nLet $\\mathcal{D}'$ be the full subcategory of $\\mathcal{D}$ with objects\n$$\n\\Ob(\\mathcal{D}') =\n\\{X \\in \\Ob(\\mathcal{D}) \\mid\nH(X[n]) = 0\\text{ for all }n \\in \\mathbf{Z}\\}\n$$\nThen $\\mathcal{D}'$ is a strictly full saturated pre-triangulated subcategory\nof $\\mathcal{D}$. If $\\mathcal{D}$ is a triangulated category, then\n$\\mathcal{D}'$ is a triangulated subcategory.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RD","source_file":"derived.tex","source_line":1812,"source_end_line":1825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1812-L1825","statement_sha256":"c63fd5f9e3a39a0de6eced7d1b3f19eb541d83df6b2235807cc343967f711eb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2471,"rank":2471,"depth":1,"x":263.622,"y":658.565,"cluster":"derived-categories"},{"id":"stacks:05RE","tag":"05RE","title":"Quotients of triangulated categories · Lemma 05RE","summary":"Let H : D → A be a homological functor of a pre-triangulated category into an abelian category. Let D_H^+, D_H^-, D_H^b be the full subcategory of D with objects Ob(D_H^+) = (X ∈ Ob(D) mid H(X[n]) = 0 for all n ll 0) Ob(D_H^-) = (X ∈ Ob(D) mid H(X[n]) = 0 for all n gg 0) Ob(D_H^b) = (X ∈ Ob(D) mid H(X[n]) = 0 for all |n| gg 0) Each of these is a strictly full saturated pre-triangulated subcategory of D. If D is a triangulated category, then each is a triangulated subcategory.","statement_latex":"Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor of\na pre-triangulated category into an abelian category.\nLet $\\mathcal{D}_H^{+}, \\mathcal{D}_H^{-}, \\mathcal{D}_H^b$\nbe the full subcategory of $\\mathcal{D}$ with objects\n$$\n\\begin{matrix}\n\\Ob(\\mathcal{D}_H^{+}) =\n\\{X \\in \\Ob(\\mathcal{D}) \\mid\nH(X[n]) = 0\\text{ for all }n \\ll 0\\} \\\\\n\\Ob(\\mathcal{D}_H^{-}) =\n\\{X \\in \\Ob(\\mathcal{D}) \\mid\nH(X[n]) = 0\\text{ for all }n \\gg 0\\} \\\\\n\\Ob(\\mathcal{D}_H^b) =\n\\{X \\in \\Ob(\\mathcal{D}) \\mid\nH(X[n]) = 0\\text{ for all }|n| \\gg 0\\}\n\\end{matrix}\n$$\nEach of these is a strictly full saturated pre-triangulated subcategory\nof $\\mathcal{D}$. If $\\mathcal{D}$ is a triangulated category, then\neach is a triangulated subcategory.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RE","source_file":"derived.tex","source_line":1845,"source_end_line":1867,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1845-L1867","statement_sha256":"f72b41a46b9d7fbf947d8c0c3b0d2a3bb9a1b771f725720cff636bef7ee0129e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2472,"rank":2472,"depth":1,"x":222.35,"y":715.322,"cluster":"derived-categories"},{"id":"stacks:05RF","tag":"05RF","title":"Quotients of triangulated categories · Definition 05RF","summary":"Let D be a (pre-)triangulated category. • Let F : D → D' be an exact functor. The kernel of F is the strictly full saturated (pre-)triangulated subcategory described in Lemma [Tag 05RC]. • Let H : D → A be a homological functor. The kernel of H is the strictly full saturated (pre-)triangulated subcategory described in Lemma [Tag 05RD]. These are sometimes denoted Ker(F) or Ker(H).","statement_latex":"Let $\\mathcal{D}$ be a (pre-)triangulated category.\n\\begin{enumerate}\n\\item Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor.\nThe {\\it kernel of $F$} is the strictly full saturated\n(pre-)triangulated subcategory described in\nLemma \\ref{lemma-triangle-functor-kernel}.\n\\item Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor.\nThe {\\it kernel of $H$} is the strictly full saturated\n(pre-)triangulated subcategory described in\nLemma \\ref{lemma-homological-functor-kernel}.\n\\end{enumerate}\nThese are sometimes denoted $\\Ker(F)$ or $\\Ker(H)$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RF","source_file":"derived.tex","source_line":1889,"source_end_line":1903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1889-L1903","statement_sha256":"da823c5dca98bec3e3cb54c517fab621b50e42943aae41a567abf23991df0aa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2473,"rank":2473,"depth":2,"x":206.957,"y":649.242,"cluster":"derived-categories"},{"id":"stacks:05RG","tag":"05RG","title":"Quotients of triangulated categories · Lemma 05RG","summary":"Let D be a triangulated category. Let D' ⊂ D be a full triangulated subcategory. Set S = ( f ∈ Arrows(D) such that there exists a distinguished triangle (X, Y, Z, f, g, h) of D with Z isomorphic to an object of D' ) Then S is a multiplicative system compatible with the triangulated structure on D. In this situation the following are equivalent • S is a saturated multiplicative system, • D' is a saturated triangulated subcategory.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $\\mathcal{D}' \\subset \\mathcal{D}$ be a full triangulated\nsubcategory. Set\n\\begin{equation}\n\nS =\n\\left\\{\n\\begin{matrix}\nf \\in \\text{Arrows}(\\mathcal{D})\n\\text{ such that there exists a distinguished triangle }\\\\\n(X, Y, Z, f, g, h) \\text{ of }\\mathcal{D}\\text{ with }\nZ\\text{ isomorphic to an object of }\\mathcal{D}'\n\\end{matrix}\n\\right\\}\n\\end{equation}\nThen $S$ is a multiplicative system compatible with the triangulated\nstructure on $\\mathcal{D}$. In this situation the following are equivalent\n\\begin{enumerate}\n\\item $S$ is a saturated multiplicative system,\n\\item $\\mathcal{D}'$ is a saturated triangulated subcategory.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RG","source_file":"derived.tex","source_line":1908,"source_end_line":1931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L1908-L1931","statement_sha256":"211ebc3e7fb80d4441a765f48dd4ef4c645d80abd6f53dbdb80e5aa8bb9673da","origin":"The Stacks Project","memory_eligible":false,"source_rank":2474,"rank":2474,"depth":12,"x":272.225,"y":689.721,"cluster":"derived-categories"},{"id":"stacks:05RI","tag":"05RI","title":"Quotients of triangulated categories · Definition 05RI","summary":"Let D be a triangulated category. Let B be a full triangulated subcategory. We define the quotient category D/B by the formula D/B = S^-1D, where S is the multiplicative system of D associated to B via Lemma [Tag 05RG]. The localization functor Q : D → D/B is called the quotient functor in this case.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $\\mathcal{B}$ be a full triangulated subcategory.\nWe define the {\\it quotient category $\\mathcal{D}/\\mathcal{B}$}\nby the formula $\\mathcal{D}/\\mathcal{B} = S^{-1}\\mathcal{D}$, where\n$S$ is the multiplicative system of $\\mathcal{D}$ associated to\n$\\mathcal{B}$ via\nLemma \\ref{lemma-construct-multiplicative-system}.\nThe localization functor $Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B}$\nis called the {\\it quotient functor} in this case.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RI","source_file":"derived.tex","source_line":2065,"source_end_line":2076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2065-L2076","statement_sha256":"bb972a0de0896374457161f9783edae42cf85401ced96b5ba56aa8474d144b28","origin":"The Stacks Project","memory_eligible":false,"source_rank":2475,"rank":2475,"depth":13,"x":190.592,"y":696.988,"cluster":"derived-categories"},{"id":"stacks:05RJ","tag":"05RJ","title":"Quotients of triangulated categories · Lemma 05RJ","summary":"The universal property of the Verdier quotient. Let D be a triangulated category. Let B be a full triangulated subcategory of D. Let Q : D → D/B be the quotient functor. • If H : D → A is a homological functor into an abelian category A such that B ⊂ Ker(H) then there exists a unique factorization H' : D/B → A such that H = H' ∘ Q and H' is a homological functor too. • If F : D → D' is an exact functor into a pre-triangulated category D' such that B ⊂ Ker(F) then there…","statement_latex":"\\begin{slogan}\nThe universal property of the Verdier quotient.\n\\end{slogan}\nLet $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{B}$\nbe a full triangulated subcategory of $\\mathcal{D}$. Let\n$Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B}$ be the quotient functor.\n\\begin{enumerate}\n\\item If $H : \\mathcal{D} \\to \\mathcal{A}$ is a homological functor into\nan abelian category $\\mathcal{A}$ such that\n$\\mathcal{B} \\subset \\Ker(H)$ then there exists a unique factorization\n$H' : \\mathcal{D}/\\mathcal{B} \\to \\mathcal{A}$ such that $H = H' \\circ Q$\nand $H'$ is a homological functor too.\n\\item If $F : \\mathcal{D} \\to \\mathcal{D}'$ is an exact functor into\na pre-triangulated category $\\mathcal{D}'$ such that\n$\\mathcal{B} \\subset \\Ker(F)$ then there exists a unique factorization\n$F' : \\mathcal{D}/\\mathcal{B} \\to \\mathcal{D}'$ such that $F = F' \\circ Q$\nand $F'$ is an exact functor too.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RJ","source_file":"derived.tex","source_line":2085,"source_end_line":2105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2085-L2105","statement_sha256":"27ff99cec89ad8b84a5ed88776767ba08f03a20b65b12cc8c41d32d0e50b468f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2476,"rank":2476,"depth":13,"x":245.573,"y":644.714,"cluster":"derived-categories"},{"id":"stacks:05RK","tag":"05RK","title":"Quotients of triangulated categories · Lemma 05RK","summary":"Let D be a triangulated category. Let B be a full triangulated subcategory. The kernel of the quotient functor Q : D → D/B is the strictly full subcategory of D whose objects are Ob(Ker(Q)) = ( Z ∈ Ob(D) such that there exists a Z' ∈ Ob(D) such that Z ⊕ Z' is isomorphic to an object of B ) In other words it is the smallest strictly full saturated triangulated subcategory of D containing B.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $\\mathcal{B}$ be a full triangulated subcategory.\nThe kernel of the quotient functor\n$Q : \\mathcal{D} \\to \\mathcal{D}/\\mathcal{B}$\nis the strictly full subcategory of $\\mathcal{D}$ whose objects are\n$$\n\\Ob(\\Ker(Q)) =\n\\left\\{\n\\begin{matrix}\nZ \\in \\Ob(\\mathcal{D})\n\\text{ such that there exists a }Z' \\in \\Ob(\\mathcal{D}) \\\\\n\\text{ such that }Z \\oplus Z'\\text{ is isomorphic to an object of }\\mathcal{B}\n\\end{matrix}\n\\right\\}\n$$\nIn other words it is the smallest strictly full saturated triangulated\nsubcategory of $\\mathcal{D}$ containing $\\mathcal{B}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RK","source_file":"derived.tex","source_line":2120,"source_end_line":2139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2120-L2139","statement_sha256":"703fe0b7fb74d95f8dce22657a08aca4598f5067648405c0ba2a0034cc6af79e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2477,"rank":2477,"depth":6,"x":247.08,"y":715.244,"cluster":"derived-categories"},{"id":"stacks:05RL","tag":"05RL","title":"Quotients of triangulated categories · Lemma 05RL","summary":"Let D be a triangulated category. The operations described above have the following properties • S(B(S)) is the \"saturation\" of S, i.e., it is the smallest saturated multiplicative system in D containing S, and • B(S(B)) is the \"saturation\" of B, i.e., it is the smallest strictly full saturated triangulated subcategory of D containing B. In particular, the constructions define mutually inverse maps between the (partially ordered) set of saturated multiplicative systems in…","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. The operations described above\nhave the following properties\n\\begin{enumerate}\n\\item $S(\\mathcal{B}(S))$ is the ``saturation'' of $S$, i.e., it is the\nsmallest saturated multiplicative system in $\\mathcal{D}$ containing $S$, and\n\\item $\\mathcal{B}(S(\\mathcal{B}))$ is the ``saturation'' of $\\mathcal{B}$,\ni.e., it is the smallest strictly full saturated triangulated subcategory of\n$\\mathcal{D}$ containing $\\mathcal{B}$.\n\\end{enumerate}\nIn particular, the constructions define mutually inverse maps between\nthe (partially ordered) set of saturated multiplicative systems in\n$\\mathcal{D}$ compatible with the triangulated structure on $\\mathcal{D}$\nand\nthe (partially ordered) set of strictly full saturated triangulated\nsubcategories of $\\mathcal{D}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RL","source_file":"derived.tex","source_line":2177,"source_end_line":2194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2177-L2194","statement_sha256":"0ad57eb63a8f72e473c0b9b8540c6ea8326ecae097bb7ca16233eb3079af698d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2478,"rank":2478,"depth":10,"x":188.616,"y":663.525,"cluster":"derived-categories"},{"id":"stacks:05RM","tag":"05RM","title":"Quotients of triangulated categories · Lemma 05RM","summary":"Let H : D → A be a homological functor from a triangulated category D to an abelian category A, see Definition [Tag 0147]. The subcategory Ker(H) of D is a strictly full saturated triangulated subcategory of D whose corresponding saturated multiplicative system (see Lemma [Tag 05RL]) is the set S = (f ∈ Arrows(D) mid H^i(f) is an isomorphism for all i ∈ Z). The functor H factors through the quotient functor Q : D → D/Ker(H).","statement_latex":"Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor from a\ntriangulated category $\\mathcal{D}$ to an abelian category $\\mathcal{A}$, see\nDefinition \\ref{definition-homological}.\nThe subcategory $\\Ker(H)$ of $\\mathcal{D}$ is a strictly full\nsaturated triangulated subcategory of $\\mathcal{D}$ whose corresponding\nsaturated multiplicative system (see\nLemma \\ref{lemma-operations})\nis the set\n$$\nS = \\{f \\in \\text{Arrows}(\\mathcal{D}) \\mid\nH^i(f)\\text{ is an isomorphism for all }i \\in \\mathbf{Z}\\}.\n$$\nThe functor $H$ factors through the quotient functor\n$Q : \\mathcal{D} \\to \\mathcal{D}/\\Ker(H)$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Quotients of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RM","source_file":"derived.tex","source_line":2230,"source_end_line":2246,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2230-L2246","statement_sha256":"709fe20f2866c9c95e38dc5f3105a960c7d8c28b8a4d6ec3c2e3dc2d3ad8ad40","origin":"The Stacks Project","memory_eligible":false,"source_rank":2479,"rank":2479,"depth":14,"x":274.232,"y":668.545,"cluster":"derived-categories"},{"id":"stacks:0A8D","tag":"0A8D","title":"Adjoints for exact functors · Lemma 0A8D","summary":"Let F : D → D' be an exact functor between triangulated categories. If F admits a right adjoint G: D' → D, then G is also an exact functor.","statement_latex":"Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor between\ntriangulated categories. If $F$ admits a right adjoint\n$G: \\mathcal{D'} \\to \\mathcal{D}$, then $G$ is also an exact functor.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Adjoints for exact functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8D","source_file":"derived.tex","source_line":2289,"source_end_line":2294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2289-L2294","statement_sha256":"caaa02b6161c2a1b88ff0113a2e9b15b828d1f1f21a196ac29d47915ac9aa6d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2480,"rank":2480,"depth":1,"x":206.352,"y":713.895,"cluster":"derived-categories"},{"id":"stacks:09J1","tag":"09J1","title":"Adjoints for exact functors · Lemma 09J1","summary":"Let D, D' be triangulated categories. Let F : D → D' and G : D' → D be functors. Assume that • F and G are exact functors, • F is fully faithful, • G is a right adjoint to F, and • the kernel of G is zero. Then F is an equivalence of categories.","statement_latex":"Let $\\mathcal{D}$, $\\mathcal{D}'$ be triangulated categories.\nLet $F : \\mathcal{D} \\to \\mathcal{D}'$ and\n$G : \\mathcal{D}' \\to \\mathcal{D}$ be functors. Assume that\n\\begin{enumerate}\n\\item $F$ and $G$ are exact functors,\n\\item $F$ is fully faithful,\n\\item $G$ is a right adjoint to $F$, and\n\\item the kernel of $G$ is zero.\n\\end{enumerate}\nThen $F$ is an equivalence of categories.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Adjoints for exact functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09J1","source_file":"derived.tex","source_line":2372,"source_end_line":2384,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2372-L2384","statement_sha256":"049d1a3059331928ea9a90b6b9cb07fabc7112e9ce3bd3f064087703b00cac71","origin":"The Stacks Project","memory_eligible":false,"source_rank":2481,"rank":2481,"depth":3,"x":220.08,"y":641.194,"cluster":"derived-categories"},{"id":"stacks:013H","tag":"013H","title":"The homotopy category · Definition 013H","summary":"Let A be an additive category. • We set Comp(A) = CoCh(A) be the category of (cochain) complexes. • A complex K^bullet is said to be bounded below if K^n = 0 for all n ll 0. • A complex K^bullet is said to be bounded above if K^n = 0 for all n gg 0. • A complex K^bullet is said to be bounded if K^n = 0 for all |n| gg 0. • We let Comp^+(A), Comp^-(A), resp. Comp^b(A) be the full subcategory of Comp(A) whose objects are the complexes which are bounded below, bounded above,…","statement_latex":"Let $\\mathcal{A}$ be an additive category.\n\\begin{enumerate}\n\\item We set $\\text{Comp}(\\mathcal{A}) = \\text{CoCh}(\\mathcal{A})$\nbe the {\\it category of (cochain) complexes}.\n\\item A complex $K^\\bullet$ is said to be\n{\\it bounded below} if $K^n = 0$ for all $n \\ll 0$.\n\\item A complex $K^\\bullet$ is said to be\n{\\it bounded above} if $K^n = 0$ for all $n \\gg 0$.\n\\item A complex $K^\\bullet$ is said to be\n{\\it bounded} if $K^n = 0$ for all $|n| \\gg 0$.\n\\item We let\n$\\text{Comp}^{+}(\\mathcal{A})$, $\\text{Comp}^{-}(\\mathcal{A})$,\nresp.\\ $\\text{Comp}^b(\\mathcal{A})$ be the full subcategory\nof $\\text{Comp}(\\mathcal{A})$ whose objects are the complexes\nwhich are bounded below, bounded above, resp.\\ bounded.\n\\item We let $K(\\mathcal{A})$ be the category with the same objects\nas $\\text{Comp}(\\mathcal{A})$ but as morphisms homotopy classes of\nmaps of complexes (see\nHomology, Lemma \\ref{homology-lemma-compose-homotopy-cochain}).\n\\item We let $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$,\nresp.\\ $K^b(\\mathcal{A})$ be the full subcategory of $K(\\mathcal{A})$\nwhose objects are bounded below, bounded above, resp.\\ bounded\ncomplexes of $\\mathcal{A}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The homotopy category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013H","source_file":"derived.tex","source_line":2420,"source_end_line":2446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2420-L2446","statement_sha256":"0270c7a61bc8b7609a1e99ed4bc3cba1886ef2d2436c28e05069c3af9751e687","origin":"The Stacks Project","memory_eligible":false,"source_rank":2482,"rank":2482,"depth":1,"x":268.908,"y":703.216,"cluster":"derived-categories"},{"id":"stacks:014E","tag":"014E","title":"Cones and termwise split sequences · Definition 014E","summary":"Let A be an additive category. Let f : K^bullet → L^bullet be a morphism of complexes of A. The cone of f is the complex C(f)^bullet given by C(f)^n = L^n ⊕ K^n + 1 and differential d_C(f)^n = ( d^n_L & f^n + 1 0 & -d_K^n + 1 ) It comes equipped with canonical morphisms of complexes i : L^bullet → C(f)^bullet and p : C(f)^bullet → K^bullet[1] induced by the obvious maps L^n → C(f)^n → K^n + 1.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $f : K^\\bullet \\to L^\\bullet$ be a morphism of\ncomplexes of $\\mathcal{A}$. The {\\it cone} of $f$\nis the complex $C(f)^\\bullet$ given by\n$C(f)^n = L^n \\oplus K^{n + 1}$ and\ndifferential\n$$\nd_{C(f)}^n =\n\\left(\n\\begin{matrix}\nd^n_L & f^{n + 1} \\\\\n0 & -d_K^{n + 1}\n\\end{matrix}\n\\right)\n$$\nIt comes equipped with canonical morphisms of complexes\n$i : L^\\bullet \\to C(f)^\\bullet$ and $p : C(f)^\\bullet \\to K^\\bullet[1]$\ninduced by the obvious maps $L^n \\to C(f)^n \\to K^{n + 1}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014E","source_file":"derived.tex","source_line":2470,"source_end_line":2490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2470-L2490","statement_sha256":"541b0d1a6257b3ec0b62161bba003af7d78ac2fa3a9971e20fecf5c06b4763fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":2483,"rank":2483,"depth":0,"x":182.174,"y":685.007,"cluster":"derived-categories"},{"id":"stacks:014F","tag":"014F","title":"Cones and termwise split sequences · Lemma 014F","summary":"Suppose that xymatrix K_1^bullet ar[r]_f_1 ar[d]_a & L_1^bullet ar[d]^b K_2^bullet ar[r]^f_2 & L_2^bullet is a diagram of morphisms of complexes which is commutative up to homotopy. Then there exists a morphism c : C(f_1)^bullet → C(f_2)^bullet which gives rise to a morphism of triangles (a, b, c) : (K_1^bullet, L_1^bullet, C(f_1)^bullet, f_1, i_1, p_1) → (K_2^bullet, L_2^bullet, C(f_2)^bullet, f_2, i_2, p_2) of K(A).","statement_latex":"Suppose that\n$$\n\\xymatrix{\nK_1^\\bullet \\ar[r]_{f_1} \\ar[d]_a & L_1^\\bullet \\ar[d]^b \\\\\nK_2^\\bullet \\ar[r]^{f_2} & L_2^\\bullet\n}\n$$\nis a diagram of morphisms of complexes which is commutative\nup to homotopy. Then there exists a morphism\n$c : C(f_1)^\\bullet \\to C(f_2)^\\bullet$ which gives rise to\na morphism of triangles\n$(a, b, c) : (K_1^\\bullet, L_1^\\bullet, C(f_1)^\\bullet, f_1, i_1, p_1)\n\\to\n(K_2^\\bullet, L_2^\\bullet, C(f_2)^\\bullet, f_2, i_2, p_2)$\nof $K(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014F","source_file":"derived.tex","source_line":2500,"source_end_line":2517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2500-L2517","statement_sha256":"b6ebb90ac47500910269459f95628505416a671658bb0e520e9a056678054224","origin":"The Stacks Project","memory_eligible":false,"source_rank":2484,"rank":2484,"depth":0,"x":261.541,"y":648.874,"cluster":"derived-categories"},{"id":"stacks:08RI","tag":"08RI","title":"Cones and termwise split sequences · Lemma 08RI","summary":"Suppose that f: K^bullet → L^bullet and g : L^bullet → M^bullet are morphisms of complexes such that g ∘ f is homotopic to zero. Then • g factors through a morphism C(f)^bullet → M^bullet, and • f factors through a morphism K^bullet → C(g)^bullet[-1].","statement_latex":"Suppose that $f: K^\\bullet \\to L^\\bullet$ and $g : L^\\bullet \\to M^\\bullet$\nare morphisms of complexes such that $g \\circ f$ is homotopic to zero.\nThen\n\\begin{enumerate}\n\\item $g$ factors through a morphism $C(f)^\\bullet \\to M^\\bullet$, and\n\\item $f$ factors through a morphism $K^\\bullet \\to C(g)^\\bullet[-1]$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RI","source_file":"derived.tex","source_line":2544,"source_end_line":2553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2544-L2553","statement_sha256":"c0ca61102900baee94f69e7a0ef658af617e60c0e8e356ca51184528842cdf0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2485,"rank":2485,"depth":1,"x":231.79,"y":721.235,"cluster":"derived-categories"},{"id":"stacks:014G","tag":"014G","title":"Cones and termwise split sequences · Definition 014G","summary":"Let A be an additive category. A termwise split injection α : A^bullet → B^bullet is a morphism of complexes such that each A^n → B^n is isomorphic to the inclusion of a direct summand. A termwise split surjection β : B^bullet → C^bullet is a morphism of complexes such that each B^n → C^n is isomorphic to the projection onto a direct summand.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nA {\\it termwise split injection $\\alpha : A^\\bullet \\to B^\\bullet$}\nis a morphism of complexes such that each $A^n \\to B^n$\nis isomorphic to the inclusion of a direct summand.\nA {\\it termwise split surjection $\\beta : B^\\bullet \\to C^\\bullet$}\nis a morphism of complexes such that each $B^n \\to C^n$\nis isomorphic to the projection onto a direct summand.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014G","source_file":"derived.tex","source_line":2579,"source_end_line":2588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2579-L2588","statement_sha256":"3a743164626f26e1c5d10891cf8f40ccddd1b33cbe5564b4c888a1a2de4dd785","origin":"The Stacks Project","memory_eligible":false,"source_rank":2486,"rank":2486,"depth":0,"x":195.2,"y":650.337,"cluster":"derived-categories"},{"id":"stacks:014H","tag":"014H","title":"Cones and termwise split sequences · Lemma 014H","summary":"Let A be an additive category. Let xymatrix A^bullet ar[r]_f ar[d]_a & B^bullet ar[d]^b C^bullet ar[r]^g & D^bullet be a diagram of morphisms of complexes commuting up to homotopy. If f is a termwise split injection, then b is homotopic to a morphism which makes the diagram commute. If g is a termwise split surjection, then a is homotopic to a morphism which makes the diagram commute.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet\n$$\n\\xymatrix{\nA^\\bullet \\ar[r]_f \\ar[d]_a & B^\\bullet \\ar[d]^b \\\\\nC^\\bullet \\ar[r]^g & D^\\bullet\n}\n$$\nbe a diagram of morphisms of complexes commuting up to homotopy.\nIf $f$ is a termwise split injection, then $b$ is homotopic to a\nmorphism which makes the diagram commute.\nIf $g$ is a termwise split surjection, then $a$ is homotopic to a\nmorphism which makes the diagram commute.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014H","source_file":"derived.tex","source_line":2590,"source_end_line":2605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2590-L2605","statement_sha256":"f108c31505e6ecfb627028c17e6fda3ec4764f47db7d8552d80798d0e21e9c50","origin":"The Stacks Project","memory_eligible":false,"source_rank":2487,"rank":2487,"depth":0,"x":279.967,"y":682.146,"cluster":"derived-categories"},{"id":"stacks:013N","tag":"013N","title":"Cones and termwise split sequences · Lemma 013N","summary":"Let A be an additive category. Let α : K^bullet → L^bullet be a morphism of complexes of A. There exists a factorization xymatrix K^bullet ar[r]^tilde α ar@/_1pc/[rr]_α & tilde L^bullet ar[r]^π & L^bullet such that • tilde α is a termwise split injection (see Definition [Tag 014G]), • there is a map of complexes s : L^bullet → tilde L^bullet such that π ∘ s = id_L^bullet and such that s ∘ π is homotopic to id_tilde L^bullet. Moreover, if both K^bullet and L^bullet are in…","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism\nof complexes of $\\mathcal{A}$.\nThere exists a factorization\n$$\n\\xymatrix{\nK^\\bullet \\ar[r]^{\\tilde \\alpha} \\ar@/_1pc/[rr]_\\alpha &\n\\tilde L^\\bullet \\ar[r]^\\pi &\nL^\\bullet\n}\n$$\nsuch that\n\\begin{enumerate}\n\\item $\\tilde \\alpha$ is a termwise split injection (see\nDefinition \\ref{definition-termwise-split-map}),\n\\item there is a map of complexes $s : L^\\bullet \\to \\tilde L^\\bullet$\nsuch that $\\pi \\circ s = \\text{id}_{L^\\bullet}$ and such that\n$s \\circ \\pi$ is homotopic to $\\text{id}_{\\tilde L^\\bullet}$.\n\\end{enumerate}\nMoreover, if both $K^\\bullet$ and $L^\\bullet$ are in\n$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$,\nthen so is $\\tilde L^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013N","source_file":"derived.tex","source_line":2622,"source_end_line":2646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2622-L2646","statement_sha256":"3acdbbca87b268ba39e696f1715ad6e2fb08cbcaac7c119ae4bf811f49e779a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2488,"rank":2488,"depth":1,"x":191.084,"y":707.009,"cluster":"derived-categories"},{"id":"stacks:0642","tag":"0642","title":"Cones and termwise split sequences · Lemma 0642","summary":"Let A be an additive category. Let α : K^bullet → L^bullet be a morphism of complexes of A. There exists a factorization xymatrix K^bullet ar[r]^i ar@/_1pc/[rr]_α & tilde K^bullet ar[r]^tilde α & L^bullet such that • tilde α is a termwise split surjection (see Definition [Tag 014G]), • there is a map of complexes s : tilde K^bullet → K^bullet such that s ∘ i = id_K^bullet and such that i ∘ s is homotopic to id_tilde K^bullet. Moreover, if both K^bullet and L^bullet are in…","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism\nof complexes of $\\mathcal{A}$.\nThere exists a factorization\n$$\n\\xymatrix{\nK^\\bullet \\ar[r]^i \\ar@/_1pc/[rr]_\\alpha &\n\\tilde K^\\bullet \\ar[r]^{\\tilde \\alpha} &\nL^\\bullet\n}\n$$\nsuch that\n\\begin{enumerate}\n\\item $\\tilde \\alpha$ is a termwise split surjection (see\nDefinition \\ref{definition-termwise-split-map}),\n\\item there is a map of complexes $s : \\tilde K^\\bullet \\to K^\\bullet$\nsuch that $s \\circ i = \\text{id}_{K^\\bullet}$ and such that\n$i \\circ s$ is homotopic to $\\text{id}_{\\tilde K^\\bullet}$.\n\\end{enumerate}\nMoreover, if both $K^\\bullet$ and $L^\\bullet$ are in\n$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$,\nthen so is $\\tilde K^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0642","source_file":"derived.tex","source_line":2727,"source_end_line":2751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2727-L2751","statement_sha256":"564dff6184c480b9022470015c59fd181f93e348de01e8562703bc935e4f9e14","origin":"The Stacks Project","memory_eligible":false,"source_rank":2489,"rank":2489,"depth":2,"x":237.037,"y":637.633,"cluster":"derived-categories"},{"id":"stacks:014I","tag":"014I","title":"Cones and termwise split sequences · Definition 014I","summary":"Let A be an additive category. A termwise split exact sequence of complexes of A is a complex of complexes 0 → A^bullet xrightarrowα B^bullet xrightarrowβ C^bullet → 0 together with given direct sum decompositions B^n = A^n ⊕ C^n compatible with α^n and β^n. We often write s^n : C^n → B^n and π^n : B^n → A^n for the maps induced by the direct sum decompositions. According to Homology, Lemma [Tag 011J] we get an associated morphism of complexes δ : C^bullet → A^bullet[1]…","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nA {\\it termwise split exact sequence of complexes of $\\mathcal{A}$}\nis a complex of complexes\n$$\n0 \\to\nA^\\bullet \\xrightarrow{\\alpha}\nB^\\bullet \\xrightarrow{\\beta}\nC^\\bullet \\to 0\n$$\ntogether with given direct sum decompositions\n$B^n = A^n \\oplus C^n$\ncompatible with $\\alpha^n$ and $\\beta^n$.\nWe often write $s^n : C^n \\to B^n$ and $\\pi^n : B^n \\to A^n$\nfor the maps induced by the direct sum decompositions.\nAccording to\nHomology, Lemma \\ref{homology-lemma-ses-termwise-split-cochain}\nwe get an associated morphism of complexes\n$$\n\\delta : C^\\bullet \\longrightarrow A^\\bullet[1]\n$$\nwhich in degree $n$ is the map $\\pi^{n + 1} \\circ d_B^n \\circ s^n$.\nIn other words\n$(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$\nforms a triangle\n$$\nA^\\bullet \\to B^\\bullet \\to C^\\bullet \\to A^\\bullet[1]\n$$\nThis will be the {\\it triangle associated to the termwise\nsplit sequence of complexes}.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014I","source_file":"derived.tex","source_line":2818,"source_end_line":2849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2818-L2849","statement_sha256":"6f31c8611e56c2518745095f76f868e291b3291b5d2fd10023c2670365364834","origin":"The Stacks Project","memory_eligible":false,"source_rank":2490,"rank":2490,"depth":1,"x":259.131,"y":715.538,"cluster":"derived-categories"},{"id":"stacks:05SS","tag":"05SS","title":"Cones and termwise split sequences · Lemma 05SS","summary":"Let A be an additive category. Let 0 → A^bullet → B^bullet → C^bullet → 0 be termwise split exact sequences as in Definition [Tag 014I]. Let (π')^n, (s')^n be a second collection of splittings. Denote δ' : C^bullet → A^bullet[1] the morphism associated to this second set of splittings. Then (1, 1, 1) : (A^bullet, B^bullet, C^bullet, α, β, δ) → (A^bullet, B^bullet, C^bullet, α, β, δ') is an isomorphism of triangles in K(A).","statement_latex":"Let $\\mathcal{A}$ be an additive category. Let\n$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$\nbe termwise split exact sequences as in\nDefinition \\ref{definition-split-ses}.\nLet $(\\pi')^n$, $(s')^n$ be a second collection of splittings.\nDenote $\\delta' : C^\\bullet \\longrightarrow A^\\bullet[1]$ the\nmorphism associated to this second set of splittings.\nThen\n$$\n(1, 1, 1) :\n(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)\n\\longrightarrow\n(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta')\n$$\nis an isomorphism of triangles in $K(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SS","source_file":"derived.tex","source_line":2851,"source_end_line":2868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2851-L2868","statement_sha256":"8d8e734d5471fb954cb5417591e41d18f197b9495106592fc8cae4bd1788dd49","origin":"The Stacks Project","memory_eligible":false,"source_rank":2491,"rank":2491,"depth":2,"x":179.515,"y":670.242,"cluster":"derived-categories"},{"id":"stacks:086L","tag":"086L","title":"Cones and termwise split sequences · Lemma 086L","summary":"Let A be an additive category. Let 0 → A_i^bullet → B_i^bullet → C_i^bullet → 0, i = 1, 2, 3 be termwise split exact sequences of complexes. Let b : B_1^bullet → B_2^bullet and b' : B_2^bullet → B_3^bullet be morphisms of complexes such that vcenter xymatrix A_1^bullet ar[d]_0 ar[r] & B_1^bullet ar[r] ar[d]_b & C_1^bullet ar[d]_0 A_2^bullet ar[r] & B_2^bullet ar[r] & C_2^bullet and vcenter xymatrix A_2^bullet ar[d]^0 ar[r] & B_2^bullet ar[r] ar[d]^b' & C_2^bullet ar[d]^0…","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $0 \\to A_i^\\bullet \\to B_i^\\bullet \\to C_i^\\bullet \\to 0$, $i = 1, 2, 3$\nbe termwise split exact sequences of complexes. Let\n$b : B_1^\\bullet \\to B_2^\\bullet$ and $b' : B_2^\\bullet \\to B_3^\\bullet$\nbe morphisms of complexes such that\n$$\n\\vcenter{\n\\xymatrix{\nA_1^\\bullet \\ar[d]_0 \\ar[r] &\nB_1^\\bullet \\ar[r] \\ar[d]_b &\nC_1^\\bullet \\ar[d]_0 \\\\\nA_2^\\bullet \\ar[r] & B_2^\\bullet \\ar[r] & C_2^\\bullet\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nA_2^\\bullet \\ar[d]^0 \\ar[r] &\nB_2^\\bullet \\ar[r] \\ar[d]^{b'} &\nC_2^\\bullet \\ar[d]^0 \\\\\nA_3^\\bullet \\ar[r] & B_3^\\bullet \\ar[r] & C_3^\\bullet\n}\n}\n$$\ncommute in $K(\\mathcal{A})$. Then $b' \\circ b = 0$ in $K(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086L","source_file":"derived.tex","source_line":2902,"source_end_line":2929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2902-L2929","statement_sha256":"d712ffeb3441469d9a159ad0def64329ab55e6b0b544efe86ff55b8a6eb58993","origin":"The Stacks Project","memory_eligible":false,"source_rank":2492,"rank":2492,"depth":1,"x":275.45,"y":658.37,"cluster":"derived-categories"},{"id":"stacks:014K","tag":"014K","title":"Cones and termwise split sequences · Lemma 014K","summary":"Let A be an additive category. Let f_1 : K_1^bullet → L_1^bullet and f_2 : K_2^bullet → L_2^bullet be morphisms of complexes. Let (a, b, c) : (K_1^bullet, L_1^bullet, C(f_1)^bullet, f_1, i_1, p_1) → (K_2^bullet, L_2^bullet, C(f_2)^bullet, f_2, i_2, p_2) be any morphism of triangles of K(A). If a and b are homotopy equivalences then so is c.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $f_1 : K_1^\\bullet \\to L_1^\\bullet$ and\n$f_2 : K_2^\\bullet \\to L_2^\\bullet$ be morphisms of complexes.\nLet\n$$\n(a, b, c) :\n(K_1^\\bullet, L_1^\\bullet, C(f_1)^\\bullet, f_1, i_1, p_1)\n\\longrightarrow\n(K_2^\\bullet, L_2^\\bullet, C(f_2)^\\bullet, f_2, i_2, p_2)\n$$\nbe any morphism of triangles of $K(\\mathcal{A})$.\nIf $a$ and $b$ are homotopy equivalences then so is $c$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014K","source_file":"derived.tex","source_line":2940,"source_end_line":2954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2940-L2954","statement_sha256":"3248a3c7196561360d79d9fd7e06686c1571fdab0f8696f12e66cde637f71e73","origin":"The Stacks Project","memory_eligible":false,"source_rank":2493,"rank":2493,"depth":2,"x":213.748,"y":722.082,"cluster":"derived-categories"},{"id":"stacks:014L","tag":"014L","title":"Cones and termwise split sequences · Lemma 014L","summary":"Let A be an additive category. • Given a termwise split sequence of complexes (α : A^bullet → B^bullet, β : B^bullet → C^bullet, s^n, π^n) there exists a homotopy equivalence C(α)^bullet → C^bullet such that the diagram xymatrix A^bullet ar[r] ar[d] & B^bullet ar[d] ar[r] & C(α)^bullet ar[r]_-p ar[d] & A^bullet[1] ar[d] A^bullet ar[r] & B^bullet ar[r] & C^bullet ar[r]^δ & A^bullet[1] defines an isomorphism of triangles in K(A). • Given a morphism of complexes f : K^bullet…","statement_latex":"Let $\\mathcal{A}$ be an additive category.\n\\begin{enumerate}\n\\item Given a termwise split sequence of complexes\n$(\\alpha : A^\\bullet \\to B^\\bullet,\n\\beta : B^\\bullet \\to C^\\bullet, s^n, \\pi^n)$\nthere exists a homotopy equivalence $C(\\alpha)^\\bullet \\to C^\\bullet$\nsuch that the diagram\n$$\n\\xymatrix{\nA^\\bullet \\ar[r] \\ar[d] & B^\\bullet \\ar[d] \\ar[r] &\nC(\\alpha)^\\bullet \\ar[r]_{-p} \\ar[d] & A^\\bullet[1] \\ar[d] \\\\\nA^\\bullet \\ar[r] & B^\\bullet \\ar[r] &\nC^\\bullet \\ar[r]^\\delta & A^\\bullet[1]\n}\n$$\ndefines an isomorphism of triangles in $K(\\mathcal{A})$.\n\\item Given a morphism of complexes $f : K^\\bullet \\to L^\\bullet$\nthere exists an isomorphism of triangles\n$$\n\\xymatrix{\nK^\\bullet \\ar[r] \\ar[d] & \\tilde L^\\bullet \\ar[d] \\ar[r] &\nM^\\bullet \\ar[r]_{\\delta} \\ar[d] & K^\\bullet[1] \\ar[d] \\\\\nK^\\bullet \\ar[r] & L^\\bullet \\ar[r] &\nC(f)^\\bullet \\ar[r]^{-p} & K^\\bullet[1]\n}\n$$\nwhere the upper triangle is the triangle associated to a\ntermwise split exact sequence $K^\\bullet \\to \\tilde L^\\bullet \\to M^\\bullet$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014L","source_file":"derived.tex","source_line":2995,"source_end_line":3026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L2995-L3026","statement_sha256":"b904ed48a8f074e5def1e2ce4bc4546018479b2721fa79a8bc4c6bb7660360d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2494,"rank":2494,"depth":3,"x":207.967,"y":639.423,"cluster":"derived-categories"},{"id":"stacks:014M","tag":"014M","title":"Cones and termwise split sequences · Lemma 014M","summary":"Let A be an additive category. Let A_1^bullet → A_2^bullet → … → A_n^bullet be a sequence of composable morphisms of complexes. There exists a commutative diagram xymatrix A_1^bullet ar[r] & A_2^bullet ar[r] & … ar[r] & A_n^bullet B_1^bullet ar[r] ar[u] & B_2^bullet ar[r] ar[u] & … ar[r] & B_n^bullet ar[u] such that each morphism B_i^bullet → B_i + 1^bullet is a termwise split injection and each B_i^bullet → A_i^bullet is a homotopy equivalence. Moreover, if all…","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $A_1^\\bullet \\to A_2^\\bullet \\to \\ldots \\to A_n^\\bullet$\nbe a sequence of composable morphisms of complexes.\nThere exists a commutative diagram\n$$\n\\xymatrix{\nA_1^\\bullet \\ar[r] &\nA_2^\\bullet \\ar[r] &\n\\ldots \\ar[r] &\nA_n^\\bullet \\\\\nB_1^\\bullet \\ar[r] \\ar[u] &\nB_2^\\bullet \\ar[r] \\ar[u] &\n\\ldots \\ar[r] &\nB_n^\\bullet \\ar[u]\n}\n$$\nsuch that each morphism $B_i^\\bullet \\to B_{i + 1}^\\bullet$\nis a termwise split injection and each $B_i^\\bullet \\to A_i^\\bullet$\nis a homotopy equivalence. Moreover, if all $A_i^\\bullet$ are in\n$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, or $K^b(\\mathcal{A})$,\nthen so are the $B_i^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014M","source_file":"derived.tex","source_line":3139,"source_end_line":3162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3139-L3162","statement_sha256":"18e7554b6f88a1372970578f1a3371f3f9f67452a951a9fae0f0255c24068586","origin":"The Stacks Project","memory_eligible":false,"source_rank":2495,"rank":2495,"depth":2,"x":279.266,"y":697.557,"cluster":"derived-categories"},{"id":"stacks:014N","tag":"014N","title":"Cones and termwise split sequences · Lemma 014N","summary":"Let A be an additive category. Let (α : A^bullet → B^bullet, β : B^bullet → C^bullet, s^n, π^n) be a termwise split sequence of complexes. Let (A^bullet, B^bullet, C^bullet, α, β, δ) be the associated triangle. Then the triangle (C^bullet[-1], A^bullet, B^bullet, δ[-1], α, β) is isomorphic to the triangle (C^bullet[-1], A^bullet, C(δ[-1])^bullet, δ[-1], i, p).","statement_latex":"Let $\\mathcal{A}$ be an additive category. Let\n$(\\alpha : A^\\bullet \\to B^\\bullet, \\beta : B^\\bullet \\to C^\\bullet, s^n,\n\\pi^n)$ be a termwise split sequence of complexes.\nLet $(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$\nbe the associated triangle.\nThen the triangle\n$(C^\\bullet[-1], A^\\bullet, B^\\bullet, \\delta[-1], \\alpha, \\beta)$\nis isomorphic to the triangle\n$(C^\\bullet[-1], A^\\bullet, C(\\delta[-1])^\\bullet, \\delta[-1], i, p)$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014N","source_file":"derived.tex","source_line":3175,"source_end_line":3186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3175-L3186","statement_sha256":"39fa45820d3288a1b2ca8b79ef566b0a15219b79e779773a79b73575c3622d47","origin":"The Stacks Project","memory_eligible":false,"source_rank":2496,"rank":2496,"depth":0,"x":179.158,"y":695.126,"cluster":"derived-categories"},{"id":"stacks:014O","tag":"014O","title":"Cones and termwise split sequences · Lemma 014O","summary":"Let A be an additive category. Let f : K^bullet → L^bullet be a morphism of complexes. The triangle (L^bullet, C(f)^bullet, K^bullet[1], i, p, f[1]) is the triangle associated to the termwise split sequence 0 → L^bullet → C(f)^bullet → K^bullet[1] → 0 coming from the definition of the cone of f.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nLet $f : K^\\bullet \\to L^\\bullet$ be a morphism of complexes.\nThe triangle $(L^\\bullet, C(f)^\\bullet, K^\\bullet[1], i, p, f[1])$ is\nthe triangle associated to the termwise split sequence\n$$\n0 \\to L^\\bullet \\to C(f)^\\bullet \\to K^\\bullet[1] \\to 0\n$$\ncoming from the definition of the cone of $f$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cones and termwise split sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014O","source_file":"derived.tex","source_line":3206,"source_end_line":3216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3206-L3216","statement_sha256":"a3b88728478f41c35eb3bceb4fc52eb2e88b37fb4498c4cd40e9c67c630df895","origin":"The Stacks Project","memory_eligible":false,"source_rank":2497,"rank":2497,"depth":0,"x":255.521,"y":639.689,"cluster":"derived-categories"},{"id":"stacks:014Q","tag":"014Q","title":"Distinguished triangles in the homotopy category · Definition 014Q","summary":"Let A be an additive category. A triangle (X, Y, Z, f, g, h) of K(A) is called a distinguished triangle of K(A) if it is isomorphic to the triangle associated to a termwise split exact sequence of complexes, see Definition [Tag 014I]. Same definition for K^+(A), K^-(A), and K^b(A).","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nA triangle $(X, Y, Z, f, g, h)$ of $K(\\mathcal{A})$ is\ncalled a {\\it distinguished triangle of $K(\\mathcal{A})$}\nif it is isomorphic to the triangle associated to\na termwise split exact sequence of complexes, see Definition\n\\ref{definition-split-ses}.\nSame definition for $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, and\n$K^b(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Distinguished triangles in the homotopy category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014Q","source_file":"derived.tex","source_line":3235,"source_end_line":3245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3235-L3245","statement_sha256":"536a1bc282a6599c9154614ad17f7c23abce875c8b8ce116e9844ec3b9a2af9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2498,"rank":2498,"depth":2,"x":243.701,"y":724.541,"cluster":"derived-categories"},{"id":"stacks:014R","tag":"014R","title":"Distinguished triangles in the homotopy category · Lemma 014R","summary":"Let A be an additive category. Suppose that α : A^bullet → B^bullet and β : B^bullet → C^bullet are split injections of complexes. Then there exist distinguished triangles (A^bullet, B^bullet, Q_1^bullet, α, p_1, d_1), (A^bullet, C^bullet, Q_2^bullet, β ∘ α, p_2, d_2) and (B^bullet, C^bullet, Q_3^bullet, β, p_3, d_3) for which TR4 holds.","statement_latex":"Let $\\mathcal{A}$ be an additive category. Suppose that\n$\\alpha : A^\\bullet \\to B^\\bullet$ and $\\beta : B^\\bullet \\to C^\\bullet$\nare split injections of complexes. Then there exist distinguished triangles\n$(A^\\bullet, B^\\bullet, Q_1^\\bullet, \\alpha, p_1, d_1)$,\n$(A^\\bullet, C^\\bullet, Q_2^\\bullet, \\beta \\circ \\alpha, p_2, d_2)$\nand\n$(B^\\bullet, C^\\bullet, Q_3^\\bullet, \\beta, p_3, d_3)$\nfor which TR4 holds.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Distinguished triangles in the homotopy category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014R","source_file":"derived.tex","source_line":3256,"source_end_line":3266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3256-L3266","statement_sha256":"c67c61ecd13e5663682bc1cfcd740c45aafdec0a08d5c7eaaab11265e2b9f0af","origin":"The Stacks Project","memory_eligible":false,"source_rank":2499,"rank":2499,"depth":0,"x":183.736,"y":654.743,"cluster":"derived-categories"},{"id":"stacks:014S","tag":"014S","title":"Distinguished triangles in the homotopy category · Proposition 014S","summary":"Let A be an additive category. The category K(A) of complexes up to homotopy with its natural translation functors and distinguished triangles as defined above is a triangulated category.","statement_latex":"Let $\\mathcal{A}$ be an additive category.\nThe category $K(\\mathcal{A})$ of complexes up to\nhomotopy with its natural translation functors\nand distinguished triangles as defined above\nis a triangulated category.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Distinguished triangles in the homotopy category","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014S","source_file":"derived.tex","source_line":3319,"source_end_line":3326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3319-L3326","statement_sha256":"bbcb665105f4700e7a98acdc211a1fa39cce2896e75f3fa4c554b09b57494a25","origin":"The Stacks Project","memory_eligible":false,"source_rank":2500,"rank":2500,"depth":11,"x":284.826,"y":672.316,"cluster":"derived-categories"},{"id":"stacks:05RQ","tag":"05RQ","title":"Distinguished triangles in the homotopy category · Lemma 05RQ","summary":"Let A be an additive category. The categories K^+(A), K^-(A), and K^b(A) are full triangulated subcategories of K(A).","statement_latex":"Let $\\mathcal{A}$ be an additive category. The categories\n$K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, and $K^b(\\mathcal{A})$\nare full triangulated subcategories of $K(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Distinguished triangles in the homotopy category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RQ","source_file":"derived.tex","source_line":3401,"source_end_line":3406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3401-L3406","statement_sha256":"474a66e2be01e59daa3ead26f1d21365550200dc3bbeca1bc2aa1344fb933cb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2501,"rank":2501,"depth":2,"x":195.503,"y":717.042,"cluster":"derived-categories"},{"id":"stacks:014X","tag":"014X","title":"Distinguished triangles in the homotopy category · Lemma 014X","summary":"Let A, B be additive categories. Let F : A → B be an additive functor. The induced functors F : K(A) → K(B) F : K^+(A) → K^+(B) F : K^-(A) → K^-(B) F : K^b(A) → K^b(B) are exact functors of triangulated categories.","statement_latex":"Let $\\mathcal{A}$, $\\mathcal{B}$ be additive categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor.\nThe induced functors\n$$\n\\begin{matrix}\nF : K(\\mathcal{A}) \\longrightarrow K(\\mathcal{B}) \\\\\nF : K^{+}(\\mathcal{A}) \\longrightarrow K^{+}(\\mathcal{B}) \\\\\nF : K^{-}(\\mathcal{A}) \\longrightarrow K^{-}(\\mathcal{B}) \\\\\nF : K^b(\\mathcal{A}) \\longrightarrow K^b(\\mathcal{B})\n\\end{matrix}\n$$\nare exact functors of triangulated categories.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Distinguished triangles in the homotopy category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014X","source_file":"derived.tex","source_line":3430,"source_end_line":3444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3430-L3444","statement_sha256":"1fc013189a9434f1e692609d634e793f71811413617099e6e95cb7130ab1ae29","origin":"The Stacks Project","memory_eligible":false,"source_rank":2502,"rank":2502,"depth":2,"x":225.618,"y":632.776,"cluster":"derived-categories"},{"id":"stacks:0G6C","tag":"0G6C","title":"Distinguished triangles in the homotopy category · Lemma 0G6C","summary":"Let A be an additive category. Let (A^bullet, B^bullet, C^bullet, a, b, c) be a distinguished triangle in K(A). Then there exists an isomorphic distinguished triangle (A^bullet, (B')^bullet, C^bullet, a', b', c) such that 0 → A^n → (B')^n → C^n → 0 is a split short exact sequence for all n.","statement_latex":"Let $\\mathcal{A}$ be an additive category. Let\n$(A^\\bullet, B^\\bullet, C^\\bullet, a, b, c)$ be a distinguished triangle in\n$K(\\mathcal{A})$. Then there exists an isomorphic distinguished triangle\n$(A^\\bullet, (B')^\\bullet, C^\\bullet, a', b', c)$ such that\n$0 \\to A^n \\to (B')^n \\to C^n \\to 0$ is a split short exact sequence\nfor all $n$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Distinguished triangles in the homotopy category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6C","source_file":"derived.tex","source_line":3458,"source_end_line":3466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3458-L3466","statement_sha256":"5fe4e29368d85ec34c4feeed90d9828232f091be8e0a5eb218a999fb3d40521e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2503,"rank":2503,"depth":12,"x":271.497,"y":712.564,"cluster":"derived-categories"},{"id":"stacks:05RS","tag":"05RS","title":"Derived categories · Lemma 05RS","summary":"Let A be an abelian category. The functor H^0 : K(A) → A is homological.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. The functor\n$$\nH^0 : K(\\mathcal{A}) \\longrightarrow \\mathcal{A}\n$$\nis homological.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RS","source_file":"derived.tex","source_line":3617,"source_end_line":3624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3617-L3624","statement_sha256":"5fdac1a654c26c157c05f0cf09e428a42e102a9cc865ee939d674d186e6ffdd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2504,"rank":2504,"depth":7,"x":172.819,"y":679.531,"cluster":"derived-categories"},{"id":"stacks:05RT","tag":"05RT","title":"Derived categories · Lemma 05RT","summary":"Let A be an abelian category. The full subcategory Ac(A) of K(A) consisting of acyclic complexes is a strictly full saturated triangulated subcategory of K(A). The corresponding saturated multiplicative system (see Lemma [Tag 05RL]) of K(A) is the set Qis(A) of quasi-isomorphisms. In particular, the kernel of the localization functor Q : K(A) → Qis(A)^-1K(A) is Ac(A) and the functor H^0 factors through Q.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. The full subcategory\n$\\text{Ac}(\\mathcal{A})$ of $K(\\mathcal{A})$ consisting of acyclic complexes\nis a strictly full saturated triangulated subcategory of $K(\\mathcal{A})$.\nThe corresponding saturated multiplicative system (see\nLemma \\ref{lemma-operations})\nof $K(\\mathcal{A})$ is the set $\\text{Qis}(\\mathcal{A})$\nof quasi-isomorphisms. In particular, the kernel of the localization\nfunctor $Q : K(\\mathcal{A}) \\to \\text{Qis}(\\mathcal{A})^{-1}K(\\mathcal{A})$\nis $\\text{Ac}(\\mathcal{A})$ and the functor $H^0$ factors through $Q$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RT","source_file":"derived.tex","source_line":3669,"source_end_line":3680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3669-L3680","statement_sha256":"7f6129fc7e01c8ccac26d2a51ab3e8e91c078ce45d452e3f9f5fed56c8b4cc38","origin":"The Stacks Project","memory_eligible":false,"source_rank":2505,"rank":2505,"depth":15,"x":272.833,"y":647.68,"cluster":"derived-categories"},{"id":"stacks:05RU","tag":"05RU","title":"Derived categories · Definition 05RU","summary":"Let A be an abelian category. Let Ac(A) and Qis(A) be as in Lemma [Tag 05RT]. The derived category of A is the triangulated category D(A) = K(A)/Ac(A) = Qis(A)^-1 K(A). We denote H^0 : D(A) → A the unique functor whose composition with the quotient functor gives back the functor H^0 defined above. Using Lemma [Tag 05RE] we introduce the strictly full saturated triangulated subcategories D^+(A), D^-(A), D^b(A) whose sets of objects are Ob(D^+(A)) = (X ∈ Ob(D(A)) mid H^n(X)…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\text{Ac}(\\mathcal{A})$ and $\\text{Qis}(\\mathcal{A})$\nbe as in\nLemma \\ref{lemma-acyclic}.\nThe {\\it derived category of $\\mathcal{A}$} is the triangulated\ncategory\n$$\nD(\\mathcal{A}) =\nK(\\mathcal{A})/\\text{Ac}(\\mathcal{A}) =\n\\text{Qis}(\\mathcal{A})^{-1} K(\\mathcal{A}).\n$$\nWe denote $H^0 : D(\\mathcal{A}) \\to \\mathcal{A}$ the unique functor\nwhose composition with the quotient functor gives back the functor\n$H^0$ defined above. Using\nLemma \\ref{lemma-homological-functor-bounded}\nwe introduce the strictly full saturated triangulated subcategories\n$D^{+}(\\mathcal{A}), D^{-}(\\mathcal{A}), D^b(\\mathcal{A})$\nwhose sets of objects are\n$$\n\\begin{matrix}\n\\Ob(D^{+}(\\mathcal{A})) =\n\\{X \\in \\Ob(D(\\mathcal{A})) \\mid\nH^n(X) = 0\\text{ for all }n \\ll 0\\} \\\\\n\\Ob(D^{-}(\\mathcal{A})) =\n\\{X \\in \\Ob(D(\\mathcal{A})) \\mid\nH^n(X) = 0\\text{ for all }n \\gg 0\\} \\\\\n\\Ob(D^b(\\mathcal{A})) =\n\\{X \\in \\Ob(D(\\mathcal{A})) \\mid\nH^n(X) = 0\\text{ for all }|n| \\gg 0\\}\n\\end{matrix}\n$$\nThe category $D^b(\\mathcal{A})$ is called the {\\it bounded derived\ncategory} of $\\mathcal{A}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RU","source_file":"derived.tex","source_line":3689,"source_end_line":3724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3689-L3724","statement_sha256":"5129e698e4526a4628b353afccec84abab1de487e263107892743d20500883eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2506,"rank":2506,"depth":16,"x":224.369,"y":728.462,"cluster":"derived-categories"},{"id":"stacks:05RV","tag":"05RV","title":"Derived categories · Lemma 05RV","summary":"Let A be an abelian category. Let K^bullet be a complex. • If H^n(K^bullet) = 0 for all n ll 0, then there exists a quasi-isomorphism K^bullet → L^bullet with L^bullet bounded below. • If H^n(K^bullet) = 0 for all n gg 0, then there exists a quasi-isomorphism M^bullet → K^bullet with M^bullet bounded above. • If H^n(K^bullet) = 0 for all |n| gg 0, then there exists a commutative diagram of morphisms of complexes xymatrix K^bullet ar[r] & L^bullet M^bullet ar[u] ar[r] &…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet$ be a complex.\n\\begin{enumerate}\n\\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$, then there exists\na quasi-isomorphism $K^\\bullet \\to L^\\bullet$ with $L^\\bullet$\nbounded below.\n\\item If $H^n(K^\\bullet) = 0$ for all $n \\gg 0$, then there exists\na quasi-isomorphism $M^\\bullet \\to K^\\bullet$ with $M^\\bullet$\nbounded above.\n\\item If $H^n(K^\\bullet) = 0$ for all $|n| \\gg 0$, then there exists\na commutative diagram of morphisms of complexes\n$$\n\\xymatrix{\nK^\\bullet \\ar[r] & L^\\bullet \\\\\nM^\\bullet \\ar[u] \\ar[r] & N^\\bullet \\ar[u]\n}\n$$\nwhere all the arrows are quasi-isomorphisms, $L^\\bullet$\nbounded below, $M^\\bullet$ bounded above, and $N^\\bullet$ a bounded\ncomplex.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RV","source_file":"derived.tex","source_line":3759,"source_end_line":3782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3759-L3782","statement_sha256":"10859c52c6e706fbce94b0cfcd997b071e0d8e2803198288b8dd81eb0774bcf9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2507,"rank":2507,"depth":0,"x":194.948,"y":640.81,"cluster":"derived-categories"},{"id":"stacks:05RW","tag":"05RW","title":"Derived categories · Lemma 05RW","summary":"Let A be an abelian category. The subcategories Ac^+(A), Ac^-(A), resp. Ac^b(A) are strictly full saturated triangulated subcategories of K^+(A), K^-(A), resp. K^b(A). The corresponding saturated multiplicative systems (see Lemma [Tag 05RL]) are the sets Qis^+(A), Qis^-(A), resp. Qis^b(A). • The kernel of the functor K^+(A) → D^+(A) is Ac^+(A) and this induces an equivalence of triangulated categories K^+(A)/Ac^+(A) = Qis^+(A)^-1K^+(A) → D^+(A) • The kernel of the functor…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. The subcategories\n$\\text{Ac}^{+}(\\mathcal{A})$, $\\text{Ac}^{-}(\\mathcal{A})$,\nresp.\\ $\\text{Ac}^b(\\mathcal{A})$\nare strictly full saturated triangulated subcategories\nof $K^{+}(\\mathcal{A})$, $K^{-}(\\mathcal{A})$, resp.\\ $K^b(\\mathcal{A})$.\nThe corresponding saturated multiplicative systems (see\nLemma \\ref{lemma-operations})\nare the sets $\\text{Qis}^{+}(\\mathcal{A})$, $\\text{Qis}^{-}(\\mathcal{A})$,\nresp.\\ $\\text{Qis}^b(\\mathcal{A})$.\n\\begin{enumerate}\n\\item The kernel of the functor $K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$\nis $\\text{Ac}^{+}(\\mathcal{A})$ and this induces an equivalence\nof triangulated categories\n$$\nK^{+}(\\mathcal{A})/\\text{Ac}^{+}(\\mathcal{A}) =\n\\text{Qis}^{+}(\\mathcal{A})^{-1}K^{+}(\\mathcal{A})\n\\longrightarrow\nD^{+}(\\mathcal{A})\n$$\n\\item The kernel of the functor $K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{A})$\nis $\\text{Ac}^{-}(\\mathcal{A})$ and this induces an equivalence\nof triangulated categories\n$$\nK^{-}(\\mathcal{A})/\\text{Ac}^{-}(\\mathcal{A}) =\n\\text{Qis}^{-}(\\mathcal{A})^{-1}K^{-}(\\mathcal{A})\n\\longrightarrow\nD^{-}(\\mathcal{A})\n$$\n\\item The kernel of the functor $K^b(\\mathcal{A}) \\to D^b(\\mathcal{A})$\nis $\\text{Ac}^b(\\mathcal{A})$ and this induces an equivalence\nof triangulated categories\n$$\nK^b(\\mathcal{A})/\\text{Ac}^b(\\mathcal{A}) =\n\\text{Qis}^b(\\mathcal{A})^{-1}K^b(\\mathcal{A})\n\\longrightarrow\nD^b(\\mathcal{A})\n$$\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RW","source_file":"derived.tex","source_line":3804,"source_end_line":3844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3804-L3844","statement_sha256":"66d444c9313822b2b04d43388da65566bcbbdd741fef74df6a7d88b33230f5e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2508,"rank":2508,"depth":15,"x":287.739,"y":689.068,"cluster":"derived-categories"},{"id":"stacks:0152","tag":"0152","title":"The canonical delta-functor · Lemma 0152","summary":"Let A be an abelian category. The functor Comp(A) → D(A) defined has the natural structure of a δ-functor, with δ_A^bullet → B^bullet → C^bullet = - p ∘ q^-1 with p and q as explained above. The same construction turns the functors Comp^+(A) → D^+(A), Comp^-(A) → D^-(A), and Comp^b(A) → D^b(A) into δ-functors.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. The functor\n$\\text{Comp}(\\mathcal{A}) \\to D(\\mathcal{A})$\ndefined has the natural structure of a $\\delta$-functor,\nwith\n$$\n\\delta_{A^\\bullet \\to B^\\bullet \\to C^\\bullet} = - p \\circ q^{-1}\n$$\nwith $p$ and $q$ as explained above. The same construction turns the\nfunctors\n$\\text{Comp}^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$,\n$\\text{Comp}^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{A})$, and\n$\\text{Comp}^b(\\mathcal{A}) \\to D^b(\\mathcal{A})$\ninto $\\delta$-functors.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The canonical delta-functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0152","source_file":"derived.tex","source_line":3964,"source_end_line":3979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L3964-L3979","statement_sha256":"a3f2edacab48195719c7b551d397f0daa46f04417e281843d6a89d945875222a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2509,"rank":2509,"depth":1,"x":179.808,"y":706.245,"cluster":"derived-categories"},{"id":"stacks:0153","tag":"0153","title":"The canonical delta-functor · Lemma 0153","summary":"Let A be an abelian category. Let xymatrix 0 ar[r] & A^bullet ar[r] ar[d] & B^bullet ar[r] ar[d] & C^bullet ar[r] ar[d] & 0 0 ar[r] & D^bullet ar[r] & E^bullet ar[r] & F^bullet ar[r] & 0 be a commutative diagram of morphisms of complexes such that the rows are short exact sequences of complexes, and the vertical arrows are quasi-isomorphisms. The δ-functor of Lemma [Tag 0152] above maps the short exact sequences 0 → A^bullet → B^bullet → C^bullet → 0 and 0 → D^bullet →…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet\n$$\n\\xymatrix{\n0 \\ar[r] &\nA^\\bullet \\ar[r] \\ar[d] &\nB^\\bullet \\ar[r] \\ar[d] &\nC^\\bullet \\ar[r] \\ar[d] &\n0 \\\\\n0 \\ar[r] &\nD^\\bullet \\ar[r] &\nE^\\bullet \\ar[r] &\nF^\\bullet \\ar[r] &\n0\n}\n$$\nbe a commutative diagram of morphisms of complexes\nsuch that the rows are short exact sequences of complexes, and\nthe vertical arrows are quasi-isomorphisms.\nThe $\\delta$-functor of\nLemma \\ref{lemma-derived-canonical-delta-functor}\nabove maps the short exact sequences\n$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$\nand\n$0 \\to D^\\bullet \\to E^\\bullet \\to F^\\bullet \\to 0$\nto isomorphic distinguished triangles.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The canonical delta-functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0153","source_file":"derived.tex","source_line":4008,"source_end_line":4036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4008-L4036","statement_sha256":"b48d452a1fcd6eb12d5e7b35558e4c4d32f42dad022fedd12b12f74c5478a634","origin":"The Stacks Project","memory_eligible":false,"source_rank":2510,"rank":2510,"depth":2,"x":246.007,"y":631.86,"cluster":"derived-categories"},{"id":"stacks:0154","tag":"0154","title":"The canonical delta-functor · Lemma 0154","summary":"Let A be an abelian category. Let xymatrix 0 ar[r] & A^bullet ar[r] & B^bullet ar[r] & C^bullet ar[r] & 0 be a short exact sequences of complexes. Assume this short exact sequence is termwise split. Let (A^bullet, B^bullet, C^bullet, α, β, δ) be the distinguished triangle of K(A) associated to the sequence. The δ-functor of Lemma [Tag 0152] above maps the short exact sequences 0 → A^bullet → B^bullet → C^bullet → 0 to a triangle isomorphic to the distinguished triangle…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$$\n\\xymatrix{\n0 \\ar[r] &\nA^\\bullet \\ar[r] &\nB^\\bullet \\ar[r] &\nC^\\bullet \\ar[r] &\n0\n}\n$$\nbe a short exact sequences of complexes.\nAssume this short exact sequence is termwise split. Let\n$(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta)$\nbe the distinguished triangle of $K(\\mathcal{A})$\nassociated to the sequence. The $\\delta$-functor of\nLemma \\ref{lemma-derived-canonical-delta-functor}\nabove maps the short exact sequences\n$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$\nto a triangle isomorphic to the distinguished triangle\n$$\n(A^\\bullet, B^\\bullet, C^\\bullet, \\alpha, \\beta, \\delta).\n$$","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The canonical delta-functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0154","source_file":"derived.tex","source_line":4044,"source_end_line":4068,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4044-L4068","statement_sha256":"c6247befb00cec2994ccd38d86fff3578c07a7655de3054835c6dca516308ea2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2511,"rank":2511,"depth":4,"x":257.08,"y":724.863,"cluster":"derived-categories"},{"id":"stacks:08Q2","tag":"08Q2","title":"The canonical delta-functor · Lemma 08Q2","summary":"Let A be an abelian category. Let K_0^bullet → K_1^bullet → … → K_n^bullet be maps of complexes such that • H^i(K_0^bullet) = 0 for i > 0, • H^-j(K_j^bullet) → H^-j(K_j + 1^bullet) is zero. Then the composition K_0^bullet → K_n^bullet factors through τ_≤ -nK_n^bullet → K_n^bullet in D(A). Dually, given maps of complexes K_n^bullet → K_n - 1^bullet → … → K_0^bullet such that • H^i(K_0^bullet) = 0 for i < 0, • H^j(K_j + 1^bullet) → H^j(K_j^bullet) is zero, then the…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$$\nK_0^\\bullet \\to K_1^\\bullet \\to \\ldots \\to K_n^\\bullet\n$$\nbe maps of complexes such that\n\\begin{enumerate}\n\\item $H^i(K_0^\\bullet) = 0$ for $i > 0$,\n\\item $H^{-j}(K_j^\\bullet) \\to H^{-j}(K_{j + 1}^\\bullet)$ is zero.\n\\end{enumerate}\nThen the composition $K_0^\\bullet \\to K_n^\\bullet$ factors through\n$\\tau_{\\leq -n}K_n^\\bullet \\to K_n^\\bullet$ in $D(\\mathcal{A})$.\nDually, given maps of complexes\n$$\nK_n^\\bullet \\to K_{n - 1}^\\bullet \\to \\ldots \\to K_0^\\bullet\n$$\nsuch that\n\\begin{enumerate}\n\\item $H^i(K_0^\\bullet) = 0$ for $i < 0$,\n\\item $H^j(K_{j + 1}^\\bullet) \\to H^j(K_j^\\bullet)$ is zero,\n\\end{enumerate}\nthen the composition $K_n^\\bullet \\to K_0^\\bullet$ factors through\n$K_n^\\bullet \\to \\tau_{\\geq n}K_n^\\bullet$ in $D(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"The canonical delta-functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Q2","source_file":"derived.tex","source_line":4111,"source_end_line":4135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4111-L4135","statement_sha256":"16cb4661a78509330849e6c7043f536fd583b0d1a0b4a5bad677d4e81d9b5039","origin":"The Stacks Project","memory_eligible":false,"source_rank":2512,"rank":2512,"depth":2,"x":173.605,"y":662.174,"cluster":"derived-categories"},{"id":"stacks:05RY","tag":"05RY","title":"Filtered derived categories · Definition 05RY","summary":"Let A be an abelian category. The category of finite filtered objects of A is the category of filtered objects (A, F) of A whose filtration F is finite. We denote it Fil^f(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. The\n{\\it category of finite filtered objects of $\\mathcal{A}$}\nis the category of filtered objects\n$(A, F)$ of $\\mathcal{A}$ whose filtration $F$ is finite.\nWe denote it $\\text{Fil}^f(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RY","source_file":"derived.tex","source_line":4193,"source_end_line":4200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4193-L4200","statement_sha256":"5c27369caa5c56f67388372bf48b230492ffdb52c9dbb9af7e3a2a773c6b5f85","origin":"The Stacks Project","memory_eligible":false,"source_rank":2513,"rank":2513,"depth":0,"x":286.265,"y":661.028,"cluster":"derived-categories"},{"id":"stacks:05RZ","tag":"05RZ","title":"Filtered derived categories · Definition 05RZ","summary":"Let A be an abelian category. • Let α : K^bullet → L^bullet be a morphism of K(Fil^f(A)). We say that α is a filtered quasi-isomorphism if the morphism gr(α) is a quasi-isomorphism. • Let K^bullet be an object of K(Fil^f(A)). We say that K^bullet is filtered acyclic if the complex gr(K^bullet) is acyclic.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item Let $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism of\n$K(\\text{Fil}^f(\\mathcal{A}))$. We say that\n$\\alpha$ is a {\\it filtered quasi-isomorphism} if\nthe morphism $\\text{gr}(\\alpha)$ is a quasi-isomorphism.\n\\item Let $K^\\bullet$ be an object of $K(\\text{Fil}^f(\\mathcal{A}))$.\nWe say that $K^\\bullet$ is {\\it filtered acyclic} if\nthe complex $\\text{gr}(K^\\bullet)$ is acyclic.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05RZ","source_file":"derived.tex","source_line":4243,"source_end_line":4255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4243-L4255","statement_sha256":"53eb912fa011eb2f383022302178bf600ab660a74c5ac6fecf906b2f4985bef9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2514,"rank":2514,"depth":0,"x":203.608,"y":726.196,"cluster":"derived-categories"},{"id":"stacks:05S0","tag":"05S0","title":"Filtered derived categories · Lemma 05S0","summary":"Let A be an abelian category. • The functor K(Fil^f(A)) → Gr(A), K^bullet ↦ H^0(gr(K^bullet)) is homological. • The functor K(Fil^f(A)) → A, K^bullet ↦ H^0(gr^p(K^bullet)) is homological. • The functor K(Fil^f(A)) → A, K^bullet ↦ H^0((forget F)K^bullet) is homological.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item The functor\n$K(\\text{Fil}^f(\\mathcal{A})) \\longrightarrow \\text{Gr}(\\mathcal{A})$,\n$K^\\bullet \\longmapsto H^0(\\text{gr}(K^\\bullet))$\nis homological.\n\\item The functor\n$K(\\text{Fil}^f(\\mathcal{A})) \\rightarrow \\mathcal{A}$,\n$K^\\bullet \\longmapsto H^0(\\text{gr}^p(K^\\bullet))$\nis homological.\n\\item The functor\n$K(\\text{Fil}^f(\\mathcal{A})) \\longrightarrow \\mathcal{A}$,\n$K^\\bullet \\longmapsto H^0((\\text{forget }F)K^\\bullet)$\nis homological.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05S0","source_file":"derived.tex","source_line":4263,"source_end_line":4280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4263-L4280","statement_sha256":"dacc062049b8a911b0ca51f49a9be95014ec44e91aaae8469ef31bfbff00985d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2515,"rank":2515,"depth":8,"x":212.206,"y":630.665,"cluster":"derived-categories"},{"id":"stacks:05S1","tag":"05S1","title":"Filtered derived categories · Lemma 05S1","summary":"Let A be an abelian category. The full subcategory FAc(A) of K(Fil^f(A)) consisting of filtered acyclic complexes is a strictly full saturated triangulated subcategory of K(Fil^f(A)). The corresponding saturated multiplicative system (see Lemma [Tag 05RL]) of K(Fil^f(A)) is the set FQis(A) of filtered quasi-isomorphisms. In particular, the kernel of the localization functor Q : K(Fil^f(A)) → FQis(A)^-1K(Fil^f(A)) is FAc(A) and the functor H^0 ∘ gr factors through Q.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. The full subcategory\n$\\text{FAc}(\\mathcal{A})$ of $K(\\text{Fil}^f(\\mathcal{A}))$\nconsisting of filtered acyclic complexes is a strictly full saturated\ntriangulated subcategory of $K(\\text{Fil}^f(\\mathcal{A}))$.\nThe corresponding saturated multiplicative system (see\nLemma \\ref{lemma-operations})\nof $K(\\text{Fil}^f(\\mathcal{A}))$ is the set\n$\\text{FQis}(\\mathcal{A})$ of filtered quasi-isomorphisms.\nIn particular, the kernel of the localization\nfunctor\n$$\nQ :\nK(\\text{Fil}^f(\\mathcal{A}))\n\\longrightarrow\n\\text{FQis}(\\mathcal{A})^{-1}K(\\text{Fil}^f(\\mathcal{A}))\n$$\nis $\\text{FAc}(\\mathcal{A})$ and the functor $H^0 \\circ \\text{gr}$\nfactors through $Q$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05S1","source_file":"derived.tex","source_line":4291,"source_end_line":4311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4291-L4311","statement_sha256":"752d0ece1222d9dc405a3cdee68c2ccf067cc68143c9f4d3f6dd6a461d84dbf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2516,"rank":2516,"depth":15,"x":283.109,"y":706.438,"cluster":"derived-categories"},{"id":"stacks:05S2","tag":"05S2","title":"Filtered derived categories · Definition 05S2","summary":"Let A be an abelian category. Let FAc(A) and FQis(A) be as in Lemma [Tag 05S1]. The filtered derived category of A is the triangulated category DF(A) = K(Fil^f(A))/FAc(A) = FQis(A)^-1 K(Fil^f(A)).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\text{FAc}(\\mathcal{A})$ and $\\text{FQis}(\\mathcal{A})$\nbe as in\nLemma \\ref{lemma-filtered-acyclic}.\nThe {\\it filtered derived category of $\\mathcal{A}$}\nis the triangulated category\n$$\nDF(\\mathcal{A}) =\nK(\\text{Fil}^f(\\mathcal{A}))/\\text{FAc}(\\mathcal{A}) =\n\\text{FQis}(\\mathcal{A})^{-1} K(\\text{Fil}^f(\\mathcal{A})).\n$$","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05S2","source_file":"derived.tex","source_line":4320,"source_end_line":4333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4320-L4333","statement_sha256":"2005ddd5ec40b9c3d3af2c3f0fca7ccb31952e59c6e63b1e8375da3a678765cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2517,"rank":2517,"depth":16,"x":169.222,"y":690.703,"cluster":"derived-categories"},{"id":"stacks:05S3","tag":"05S3","title":"Filtered derived categories · Lemma 05S3","summary":"The functors gr^p, gr, (forget F) induce canonical exact functors gr^p, (forget F): DF(A) → D(A) and gr: DF(A) → D(Gr(A)) which commute with the localization functors.","statement_latex":"The functors $\\text{gr}^p, \\text{gr}, (\\text{forget }F)$ induce\ncanonical exact functors\n$$\n\\text{gr}^p, (\\text{forget }F):\nDF(\\mathcal{A})\n\\longrightarrow\nD(\\mathcal{A})\n$$\nand\n$$\n\\text{gr}:\nDF(\\mathcal{A})\n\\longrightarrow\nD(\\text{Gr}(\\mathcal{A}))\n$$\nwhich commute with the localization functors.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05S3","source_file":"derived.tex","source_line":4335,"source_end_line":4353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4335-L4353","statement_sha256":"2cb8d075798702248b65bad51335a4700a94754555d136897793d55c030da90c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2518,"rank":2518,"depth":16,"x":266.42,"y":637.375,"cluster":"derived-categories"},{"id":"stacks:05S4","tag":"05S4","title":"Filtered derived categories · Definition 05S4","summary":"Let A be an abelian category. The bounded filtered derived category DF^b(A) is the full subcategory of DF(A) with objects those X such that gr(X) ∈ D^b(A). Similarly for the bounded below filtered derived category DF^+(A) and the bounded above filtered derived category DF^-(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nThe {\\it bounded filtered derived category} $DF^b(\\mathcal{A})$ is\nthe full subcategory of $DF(\\mathcal{A})$ with objects those $X$\nsuch that $\\text{gr}(X) \\in D^b(\\mathcal{A})$.\nSimilarly for the bounded below filtered derived category\n$DF^{+}(\\mathcal{A})$ and the bounded above filtered derived category\n$DF^{-}(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05S4","source_file":"derived.tex","source_line":4374,"source_end_line":4383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4374-L4383","statement_sha256":"0a8e8cd91e1bb32bc5ba598e3ac652c3f0c8ff19fa073184bb75964f8c8b4e76","origin":"The Stacks Project","memory_eligible":false,"source_rank":2519,"rank":2519,"depth":0,"x":237.465,"y":732.396,"cluster":"derived-categories"},{"id":"stacks:05S5","tag":"05S5","title":"Filtered derived categories · Lemma 05S5","summary":"Let A be an abelian category. Let K^bullet ∈ K(Fil^f(A)). • If H^n(gr(K^bullet)) = 0 for all n < a, then there exists a filtered quasi-isomorphism K^bullet → L^bullet with L^n = 0 for all n < a. • If H^n(gr(K^bullet)) = 0 for all n > b, then there exists a filtered quasi-isomorphism M^bullet → K^bullet with M^n = 0 for all n > b. • If H^n(gr(K^bullet)) = 0 for all |n| gg 0, then there exists a commutative diagram of morphisms of complexes xymatrix K^bullet ar[r] &…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$.\n\\begin{enumerate}\n\\item If $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $n < a$, then there exists\na filtered quasi-isomorphism $K^\\bullet \\to L^\\bullet$ with\n$L^n = 0$ for all $n < a$.\n\\item If $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $n > b$, then there exists\na filtered quasi-isomorphism $M^\\bullet \\to K^\\bullet$ with\n$M^n = 0$ for all $n > b$.\n\\item If $H^n(\\text{gr}(K^\\bullet)) = 0$ for all $|n| \\gg 0$, then there\nexists a commutative diagram of morphisms of complexes\n$$\n\\xymatrix{\nK^\\bullet \\ar[r] & L^\\bullet \\\\\nM^\\bullet \\ar[u] \\ar[r] & N^\\bullet \\ar[u]\n}\n$$\nwhere all the arrows are filtered quasi-isomorphisms, $L^\\bullet$\nbounded below, $M^\\bullet$ bounded above, and $N^\\bullet$ a bounded\ncomplex.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05S5","source_file":"derived.tex","source_line":4385,"source_end_line":4408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4385-L4408","statement_sha256":"f26e638995f193df5f2f47e9128a94e866472d27564bbeebb4d5ea5f46826995","origin":"The Stacks Project","memory_eligible":false,"source_rank":2520,"rank":2520,"depth":4,"x":182.09,"y":645.405,"cluster":"derived-categories"},{"id":"stacks:05S6","tag":"05S6","title":"Filtered derived categories · Lemma 05S6","summary":"Let A be an abelian category. The subcategories FAc^+(A), FAc^-(A), resp. FAc^b(A) are strictly full saturated triangulated subcategories of K^+(Fil^fA), K^-(Fil^fA), resp. K^b(Fil^fA). The corresponding saturated multiplicative systems (see Lemma [Tag 05RL]) are the sets FQis^+(A), FQis^-(A), resp. FQis^b(A). • The kernel of the functor K^+(Fil^fA) → DF^+(A) is FAc^+(A) and this induces an equivalence of triangulated categories K^+(Fil^fA)/FAc^+(A) =…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. The subcategories\n$\\text{FAc}^{+}(\\mathcal{A})$, $\\text{FAc}^{-}(\\mathcal{A})$,\nresp.\\ $\\text{FAc}^b(\\mathcal{A})$\nare strictly full saturated triangulated subcategories\nof $K^{+}(\\text{Fil}^f\\mathcal{A})$, $K^{-}(\\text{Fil}^f\\mathcal{A})$,\nresp.\\ $K^b(\\text{Fil}^f\\mathcal{A})$.\nThe corresponding saturated multiplicative systems (see\nLemma \\ref{lemma-operations})\nare the sets $\\text{FQis}^{+}(\\mathcal{A})$, $\\text{FQis}^{-}(\\mathcal{A})$,\nresp.\\ $\\text{FQis}^b(\\mathcal{A})$.\n\\begin{enumerate}\n\\item The kernel of the functor\n$K^{+}(\\text{Fil}^f\\mathcal{A}) \\to DF^{+}(\\mathcal{A})$\nis $\\text{FAc}^{+}(\\mathcal{A})$ and this induces an equivalence\nof triangulated categories\n$$\nK^{+}(\\text{Fil}^f\\mathcal{A})/\\text{FAc}^{+}(\\mathcal{A}) =\n\\text{FQis}^{+}(\\mathcal{A})^{-1}K^{+}(\\text{Fil}^f\\mathcal{A})\n\\longrightarrow\nDF^{+}(\\mathcal{A})\n$$\n\\item The kernel of the functor\n$K^{-}(\\text{Fil}^f\\mathcal{A}) \\to DF^{-}(\\mathcal{A})$\nis $\\text{FAc}^{-}(\\mathcal{A})$ and this induces an equivalence\nof triangulated categories\n$$\nK^{-}(\\text{Fil}^f\\mathcal{A})/\\text{FAc}^{-}(\\mathcal{A}) =\n\\text{FQis}^{-}(\\mathcal{A})^{-1}K^{-}(\\text{Fil}^f\\mathcal{A})\n\\longrightarrow\nDF^{-}(\\mathcal{A})\n$$\n\\item The kernel of the functor\n$K^b(\\text{Fil}^f\\mathcal{A}) \\to DF^b(\\mathcal{A})$\nis $\\text{FAc}^b(\\mathcal{A})$ and this induces an equivalence\nof triangulated categories\n$$\nK^b(\\text{Fil}^f\\mathcal{A})/\\text{FAc}^b(\\mathcal{A}) =\n\\text{FQis}^b(\\mathcal{A})^{-1}K^b(\\text{Fil}^f\\mathcal{A})\n\\longrightarrow\nDF^b(\\mathcal{A})\n$$\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05S6","source_file":"derived.tex","source_line":4445,"source_end_line":4489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4445-L4489","statement_sha256":"926bd57f6cabf0690911862b4a68ccbf089d49463c66331b33c5cc94b855f096","origin":"The Stacks Project","memory_eligible":false,"source_rank":2521,"rank":2521,"depth":16,"x":293.504,"y":678.315,"cluster":"derived-categories"},{"id":"stacks:05S9","tag":"05S9","title":"Derived functors in general · Definition 05S9","summary":"Assumptions and notation as in Situation [Tag 05S8]. Let X ∈ Ob(D). • we say the right derived functor RF is defined at X if the ind-object (X/S) → D', (s : X → X') ↦ F(X') is essentially constant; in this case the value Y in D' is called the value of RF at X. • we say the left derived functor LF is defined at X if the pro-object (S/X) → D', (s: X' → X) ↦ F(X') is essentially constant; in this case the value Y in D' is called the value of LF at X. By abuse of notation we…","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-derived-functor}.\nLet $X \\in \\Ob(\\mathcal{D})$.\n\\begin{enumerate}\n\\item we say the {\\it right derived functor $RF$ is defined at}\n$X$ if the ind-object\n$$\n(X/S) \\longrightarrow \\mathcal{D}', \\quad\n(s : X \\to X') \\longmapsto F(X')\n$$\nis essentially constant\\footnote{For a discussion of when an ind-object\nor pro-object of a category is essentially constant we refer to\nCategories, Section \\ref{categories-section-essentially-constant}.};\nin this case the value\n$Y$ in $\\mathcal{D}'$ is called the {\\it value of $RF$ at $X$}.\n\\item we say the {\\it left derived functor $LF$ is defined at} $X$\nif the pro-object\n$$\n(S/X) \\longrightarrow \\mathcal{D}', \\quad\n(s: X' \\to X) \\longmapsto F(X')\n$$\nis essentially constant; in this case the value $Y$ in $\\mathcal{D}'$\nis called the {\\it value of $LF$ at $X$}.\n\\end{enumerate}\nBy abuse of notation we often denote the values simply\n$RF(X)$ or $LF(X)$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05S9","source_file":"derived.tex","source_line":4535,"source_end_line":4563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4535-L4563","statement_sha256":"a03fca3056b157a3f3c2906f63b62b33642c380e93578220139e13e5601e1993","origin":"The Stacks Project","memory_eligible":false,"source_rank":2522,"rank":2522,"depth":0,"x":184.276,"y":717.483,"cluster":"derived-categories"},{"id":"stacks:05SA","tag":"05SA","title":"Derived functors in general · Lemma 05SA","summary":"Assumptions and notation as in Situation [Tag 05S8]. Let f : X → Y be a morphism of D. • If RF is defined at X and Y then there exists a unique morphism RF(f) : RF(X) → RF(Y) between the values such that for any commutative diagram xymatrix X ar[d]_f ar[r]_s & X' ar[d]^f' Y ar[r]^s' & Y' with s, s' ∈ S the diagram xymatrix F(X) ar[d] ar[r] & F(X') ar[d] ar[r] & RF(X) ar[d] F(Y) ar[r] & F(Y') ar[r] & RF(Y) commutes. • If LF is defined at X and Y then there exists a unique…","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-derived-functor}.\nLet $f : X \\to Y$ be a morphism of $\\mathcal{D}$.\n\\begin{enumerate}\n\\item If $RF$ is defined at $X$ and $Y$ then there exists a unique\nmorphism $RF(f) : RF(X) \\to RF(Y)$ between the values such that\nfor any commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f \\ar[r]_s & X' \\ar[d]^{f'} \\\\\nY \\ar[r]^{s'} & Y'\n}\n$$\nwith $s, s' \\in S$ the diagram\n$$\n\\xymatrix{\nF(X) \\ar[d] \\ar[r] & F(X') \\ar[d] \\ar[r] & RF(X) \\ar[d] \\\\\nF(Y) \\ar[r] & F(Y') \\ar[r] & RF(Y)\n}\n$$\ncommutes.\n\\item If $LF$ is defined at $X$ and $Y$ then there exists a unique\nmorphism $LF(f) : LF(X) \\to LF(Y)$ between the values such that\nfor any commutative diagram\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_s & X \\ar[d]^f \\\\\nY' \\ar[r]^{s'} & Y\n}\n$$\nwith $s, s'$ in $S$ the diagram\n$$\n\\xymatrix{\nLF(X) \\ar[d] \\ar[r] & F(X') \\ar[d] \\ar[r] & F(X) \\ar[d] \\\\\nLF(Y) \\ar[r] & F(Y') \\ar[r] & F(Y)\n}\n$$\ncommutes.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SA","source_file":"derived.tex","source_line":4571,"source_end_line":4612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4571-L4612","statement_sha256":"25d26ef092a72320f639e10594ca7e34c100fc04356b62f649e6cad7f341262f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2523,"rank":2523,"depth":0,"x":233.593,"y":626.122,"cluster":"derived-categories"},{"id":"stacks:05SB","tag":"05SB","title":"Derived functors in general · Lemma 05SB","summary":"Assumptions and notation as in Situation [Tag 05S8]. Let s : X → Y be an element of S. • RF is defined at X if and only if it is defined at Y. In this case the map RF(s) : RF(X) → RF(Y) between values is an isomorphism. • LF is defined at X if and only if it is defined at Y. In this case the map LF(s) : LF(X) → LF(Y) between values is an isomorphism.","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-derived-functor}.\nLet $s : X \\to Y$ be an element of $S$.\n\\begin{enumerate}\n\\item $RF$ is defined at $X$ if and only if it is defined at $Y$.\nIn this case the map $RF(s) : RF(X) \\to RF(Y)$ between values\nis an isomorphism.\n\\item $LF$ is defined at $X$ if and only if it is defined at $Y$.\nIn this case the map $LF(s) : LF(X) \\to LF(Y)$ between values\nis an isomorphism.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SB","source_file":"derived.tex","source_line":4637,"source_end_line":4650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4637-L4650","statement_sha256":"969be9999c2a9e16fe0642997f5fb0d1ba69cdac59eddcbd61b9092983f487ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":2524,"rank":2524,"depth":0,"x":270.898,"y":721.995,"cluster":"derived-categories"},{"id":"stacks:05SU","tag":"05SU","title":"Derived functors in general · Lemma 05SU","summary":"Derived functors are compatible with shifts Assumptions and notation as in Situation [Tag 05S8]. Let X be an object of D and n ∈ Z. • RF is defined at X if and only if it is defined at X[n]. In this case there is a canonical isomorphism RF(X)[n]= RF(X[n]) between values. • LF is defined at X if and only if it is defined at X[n]. In this case there is a canonical isomorphism LF(X)[n] → LF(X[n]) between values.","statement_latex":"\\begin{slogan}\nDerived functors are compatible with shifts\n\\end{slogan}\nAssumptions and notation as in\nSituation \\ref{situation-derived-functor}.\nLet $X$ be an object of $\\mathcal{D}$ and $n \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item $RF$ is defined at $X$ if and only if it is defined at $X[n]$.\nIn this case there is a canonical isomorphism\n$RF(X)[n]= RF(X[n])$ between values.\n\\item $LF$ is defined at $X$ if and only if it is defined at $X[n]$.\nIn this case there is a canonical isomorphism\n$LF(X)[n] \\to LF(X[n])$ between values.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SU","source_file":"derived.tex","source_line":4656,"source_end_line":4672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4656-L4672","statement_sha256":"843f6175068a66ba07c01d1aa1163b08f48b537204e78ac6250d0e6f9ec9f10f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2525,"rank":2525,"depth":0,"x":165.728,"y":672.199,"cluster":"derived-categories"},{"id":"stacks:05SC","tag":"05SC","title":"Derived functors in general · Lemma 05SC","summary":"Assumptions and notation as in Situation [Tag 05S8]. Let (X, Y, Z, f, g, h) be a distinguished triangle of D. If RF is defined at two out of three of X, Y, Z, then it is defined at the third. Moreover, in this case (RF(X), RF(Y), RF(Z), RF(f), RF(g), RF(h)) is a distinguished triangle in D'. Similarly for LF.","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-derived-functor}.\nLet $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\\mathcal{D}$.\nIf $RF$ is defined at two out of three of $X, Y, Z$, then it is defined\nat the third. Moreover, in this case\n$$\n(RF(X), RF(Y), RF(Z), RF(f), RF(g), RF(h))\n$$\nis a distinguished triangle in $\\mathcal{D}'$. Similarly for $LF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SC","source_file":"derived.tex","source_line":4678,"source_end_line":4689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4678-L4689","statement_sha256":"fbb8b38d21958a2896032417321e9d13a7fad9ca99971f895a8dad404e06b7ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":2526,"rank":2526,"depth":8,"x":283.952,"y":649.12,"cluster":"derived-categories"},{"id":"stacks:05SD","tag":"05SD","title":"Derived functors in general · Lemma 05SD","summary":"Assumptions and notation as in Situation [Tag 05S8]. Let X, Y be objects of D. • If RF is defined at X and Y, then RF is defined at X ⊕ Y. • If D' is Karoubian and RF is defined at X ⊕ Y, then RF is defined at both X and Y. In either case we have RF(X ⊕ Y) = RF(X) ⊕ RF(Y). Similarly for LF.","statement_latex":"Assumptions and notation as in Situation \\ref{situation-derived-functor}.\nLet $X, Y$ be objects of $\\mathcal{D}$.\n\\begin{enumerate}\n\\item If $RF$ is defined at $X$ and $Y$, then $RF$ is defined at $X \\oplus Y$.\n\\item If $\\mathcal{D}'$ is Karoubian and $RF$ is defined at $X \\oplus Y$,\nthen $RF$ is defined at both $X$ and $Y$.\n\\end{enumerate}\nIn either case we have $RF(X \\oplus Y) = RF(X) \\oplus RF(Y)$.\nSimilarly for $LF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SD","source_file":"derived.tex","source_line":4801,"source_end_line":4812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4801-L4812","statement_sha256":"2dc96504471ab7aa89dd82b4b2d159b323e20ca60fe58235d833ab986661e37a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2527,"rank":2527,"depth":9,"x":214.971,"y":733.665,"cluster":"derived-categories"},{"id":"stacks:05SE","tag":"05SE","title":"Derived functors in general · Proposition 05SE","summary":"Assumptions and notation as in Situation [Tag 05S8]. • The full subcategory E of D consisting of objects at which RF is defined is a strictly full triangulated subcategory of D. • We obtain an exact functor RF : E → D' of triangulated categories. • Elements of S with either source or target in E are morphisms of E. • Any element of S_E = Arrows(E) ∩ S is mapped to an isomorphism by RF. • The set S_E is a saturated multiplicative system in E compatible with the…","statement_latex":"Assumptions and notation as in Situation \\ref{situation-derived-functor}.\n\\begin{enumerate}\n\\item The full subcategory $\\mathcal{E}$ of $\\mathcal{D}$ consisting of\nobjects at which $RF$ is defined is a strictly full triangulated\nsubcategory of $\\mathcal{D}$.\n\\item We obtain an exact functor\n$RF : \\mathcal{E} \\longrightarrow \\mathcal{D}'$\nof triangulated categories.\n\\item Elements of $S$ with either source or target\nin $\\mathcal{E}$ are morphisms of $\\mathcal{E}$.\n\\item Any element of $S_\\mathcal{E} = \\text{Arrows}(\\mathcal{E}) \\cap S$\nis mapped to an isomorphism by $RF$.\n\\item The set $S_\\mathcal{E}$ is a saturated multiplicative system in\n$\\mathcal{E}$ compatible with the triangulated structure.\n\\item The functor $S_\\mathcal{E}^{-1}\\mathcal{E} \\to S^{-1}\\mathcal{D}$\nis a fully faithful exact functor of triangulated categories.\n\\item We obtain an exact functor\n$$\nRF : S_\\mathcal{E}^{-1}\\mathcal{E} \\longrightarrow \\mathcal{D}'.\n$$\n\\item If $\\mathcal{D}'$ is Karoubian, then $\\mathcal{E}$ is a saturated\ntriangulated subcategory of $\\mathcal{D}$.\n\\end{enumerate}\nA similar result holds for $LF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SE","source_file":"derived.tex","source_line":4894,"source_end_line":4920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4894-L4920","statement_sha256":"a8d363813b44377e23a69305e7a3ae0a6d6c352f11ff7c7dc47b0f67c1a5265d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2528,"rank":2528,"depth":13,"x":197.759,"y":631.651,"cluster":"derived-categories"},{"id":"stacks:05SV","tag":"05SV","title":"Derived functors in general · Definition 05SV","summary":"In Situation [Tag 05S8]. We say F is right derivable, or that RF everywhere defined if RF is defined at every object of D. We say F is left derivable, or that LF everywhere defined if LF is defined at every object of D.","statement_latex":"In\nSituation \\ref{situation-derived-functor}.\nWe say $F$ is {\\it right derivable}, or that $RF$ {\\it everywhere defined}\nif $RF$ is defined at every object of $\\mathcal{D}$.\nWe say $F$ is {\\it left derivable}, or that $LF$ {\\it everywhere defined}\nif $LF$ is defined at every object of $\\mathcal{D}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SV","source_file":"derived.tex","source_line":4961,"source_end_line":4969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4961-L4969","statement_sha256":"ddbea60e5c5a492f7c4220a44f89a446ec0a3efa8c099163830975c741648ccc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2529,"rank":2529,"depth":0,"x":292.977,"y":697.447,"cluster":"derived-categories"},{"id":"stacks:05SX","tag":"05SX","title":"Derived functors in general · Definition 05SX","summary":"In Situation [Tag 05S8]. • An object X of D computes RF if RF is defined at X and the canonical map F(X) → RF(X) is an isomorphism. • An object X of D computes LF if LF is defined at X and the canonical map LF(X) → F(X) is an isomorphism.","statement_latex":"In\nSituation \\ref{situation-derived-functor}.\n\\begin{enumerate}\n\\item An object $X$ of $\\mathcal{D}$ {\\it computes} $RF$ if $RF$ is defined\nat $X$ and the canonical map $F(X) \\to RF(X)$ is an isomorphism.\n\\item An object $X$ of $\\mathcal{D}$ {\\it computes} $LF$ if $LF$ is defined\nat $X$ and the canonical map $LF(X) \\to F(X)$ is an isomorphism.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SX","source_file":"derived.tex","source_line":4989,"source_end_line":4999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L4989-L4999","statement_sha256":"8354cb9b033840afbd1fc9fc51313ed961b923c8635bd43f30dfc79e8d182d89","origin":"The Stacks Project","memory_eligible":false,"source_rank":2530,"rank":2530,"depth":0,"x":169.223,"y":702.986,"cluster":"derived-categories"},{"id":"stacks:05SY","tag":"05SY","title":"Derived functors in general · Lemma 05SY","summary":"Assumptions and notation as in Situation [Tag 05S8]. Let X be an object of D and n ∈ Z. • X computes RF if and only if X[n] computes RF. • X computes LF if and only if X[n] computes LF.","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-derived-functor}.\nLet $X$ be an object of $\\mathcal{D}$ and $n \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item $X$ computes $RF$ if and only if $X[n]$ computes $RF$.\n\\item $X$ computes $LF$ if and only if $X[n]$ computes $LF$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SY","source_file":"derived.tex","source_line":5001,"source_end_line":5010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5001-L5010","statement_sha256":"00b395c910413cb1378d98b62e39d14c3b8e39383eb6a11843b529dfb551cbda","origin":"The Stacks Project","memory_eligible":false,"source_rank":2531,"rank":2531,"depth":0,"x":256.464,"y":628.305,"cluster":"derived-categories"},{"id":"stacks:05SZ","tag":"05SZ","title":"Derived functors in general · Lemma 05SZ","summary":"Assumptions and notation as in Situation [Tag 05S8]. Let (X, Y, Z, f, g, h) be a distinguished triangle of D. If X, Y compute RF then so does Z. Similar for LF.","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-derived-functor}.\nLet $(X, Y, Z, f, g, h)$ be a distinguished triangle of $\\mathcal{D}$.\nIf $X, Y$ compute $RF$ then so does $Z$. Similar for $LF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05SZ","source_file":"derived.tex","source_line":5016,"source_end_line":5022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5016-L5022","statement_sha256":"ad91b5d64d15ada4e0107f8a0b4840fa736bbcfd14b9019aa899304a8b8da7be","origin":"The Stacks Project","memory_eligible":false,"source_rank":2532,"rank":2532,"depth":9,"x":252.168,"y":733.401,"cluster":"derived-categories"},{"id":"stacks:05T0","tag":"05T0","title":"Derived functors in general · Lemma 05T0","summary":"Assumptions and notation as in Situation [Tag 05S8]. Let X, Y be objects of D. If X ⊕ Y computes RF, then X and Y compute RF. Similarly for LF.","statement_latex":"Assumptions and notation as in Situation \\ref{situation-derived-functor}.\nLet $X, Y$ be objects of $\\mathcal{D}$. If $X \\oplus Y$ computes $RF$, then\n$X$ and $Y$ compute $RF$. Similarly for $LF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05T0","source_file":"derived.tex","source_line":5039,"source_end_line":5044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5039-L5044","statement_sha256":"d89a6562b63bbe73c38c91fa2df8c31d8a581557c55cde0f10b2e60c7fa248a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2533,"rank":2533,"depth":10,"x":170.417,"y":653.069,"cluster":"derived-categories"},{"id":"stacks:05T1","tag":"05T1","title":"Derived functors in general · Lemma 05T1","summary":"Assumptions and notation as in Situation [Tag 05S8]. • If for every object X ∈ Ob(D) there exists an arrow s : X → X' in S such that X' computes RF, then RF is everywhere defined. • If for every object X ∈ Ob(D) there exists an arrow s : X' → X in S such that X' computes LF, then LF is everywhere defined.","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-derived-functor}.\n\\begin{enumerate}\n\\item If for every object $X \\in \\Ob(\\mathcal{D})$\nthere exists an arrow $s : X \\to X'$ in $S$ such that $X'$ computes\n$RF$, then $RF$ is everywhere defined.\n\\item If for every object $X \\in \\Ob(\\mathcal{D})$\nthere exists an arrow $s : X' \\to X$ in $S$ such that $X'$ computes\n$LF$, then $LF$ is everywhere defined.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05T1","source_file":"derived.tex","source_line":5073,"source_end_line":5085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5073-L5085","statement_sha256":"a9d3f4f292b2343784b7513d85586d0ba32acfa61a448facc08d944a4ee991db","origin":"The Stacks Project","memory_eligible":false,"source_rank":2534,"rank":2534,"depth":0,"x":295.913,"y":665.982,"cluster":"derived-categories"},{"id":"stacks:06XN","tag":"06XN","title":"Derived functors in general · Lemma 06XN","summary":"Assumptions and notation as in Situation [Tag 05S8]. If there exists a subset I ⊂ Ob(D) such that • for all X ∈ Ob(D) there exists s : X → X' in S with X' ∈ I, and • for every arrow s : X → X' in S with X, X' ∈ I the map F(s) : F(X) → F(X') is an isomorphism, then RF is everywhere defined and every X ∈ I computes RF. Dually, if there exists a subset P ⊂ Ob(D) such that • for all X ∈ Ob(D) there exists s : X' → X in S with X' ∈ P, and • for every arrow s : X → X' in S with…","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-derived-functor}.\nIf there exists a subset $\\mathcal{I} \\subset \\Ob(\\mathcal{D})$\nsuch that\n\\begin{enumerate}\n\\item for all $X \\in \\Ob(\\mathcal{D})$\nthere exists $s : X \\to X'$ in $S$ with $X' \\in \\mathcal{I}$,\nand\n\\item for every arrow $s : X \\to X'$ in $S$ with $X, X' \\in \\mathcal{I}$\nthe map $F(s) : F(X) \\to F(X')$ is an isomorphism,\n\\end{enumerate}\nthen $RF$ is everywhere defined and every $X \\in \\mathcal{I}$\ncomputes $RF$. Dually, if there exists a subset\n$\\mathcal{P} \\subset \\Ob(\\mathcal{D})$\nsuch that\n\\begin{enumerate}\n\\item for all $X \\in \\Ob(\\mathcal{D})$\nthere exists $s : X' \\to X$ in $S$ with $X' \\in \\mathcal{P}$,\nand\n\\item for every arrow $s : X \\to X'$ in $S$ with $X, X' \\in \\mathcal{P}$\nthe map $F(s) : F(X) \\to F(X')$ is an isomorphism,\n\\end{enumerate}\nthen $LF$ is everywhere defined and every $X \\in \\mathcal{P}$\ncomputes $LF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XN","source_file":"derived.tex","source_line":5091,"source_end_line":5117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5091-L5117","statement_sha256":"ecbbfe3b60a58461996e3841ea486537271d5ee45ed5fc7f211dd79ca22bc433","origin":"The Stacks Project","memory_eligible":false,"source_rank":2535,"rank":2535,"depth":0,"x":192.489,"y":727.969,"cluster":"derived-categories"},{"id":"stacks:05T2","tag":"05T2","title":"Derived functors in general · Lemma 05T2","summary":"Let A, B, C be triangulated categories. Let S, resp. S' be a saturated multiplicative system in A, resp. B compatible with the triangulated structure. Let F : A → B and G : B → C be exact functors. Denote F' : A → (S')^-1B the composition of F with the localization functor. • If RF', RG, R(G ∘ F) are everywhere defined, then there is a canonical transformation of functors t : R(G ∘ F) → RG ∘ RF'. • If LF', LG, L(G ∘ F) are everywhere defined, then there is a canonical…","statement_latex":"Let $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}$ be triangulated categories.\nLet $S$, resp.\\ $S'$ be a saturated multiplicative system in\n$\\mathcal{A}$, resp.\\ $\\mathcal{B}$ compatible with the triangulated structure.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{C}$\nbe exact functors. Denote $F' : \\mathcal{A} \\to (S')^{-1}\\mathcal{B}$ the\ncomposition of $F$ with the localization functor.\n\\begin{enumerate}\n\\item If $RF'$, $RG$, $R(G \\circ F)$ are everywhere defined, then there\nis a canonical transformation of functors\n$t : R(G \\circ F) \\longrightarrow RG \\circ RF'$.\n\\item If $LF'$, $LG$, $L(G \\circ F)$ are everywhere defined, then there\nis a canonical transformation of functors\n$t : LG \\circ LF' \\to L(G \\circ F)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors in general","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05T2","source_file":"derived.tex","source_line":5129,"source_end_line":5145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5129-L5145","statement_sha256":"c0698c0062aaad54ce0d116688154b82fd25c039e33839fc510e6918f6704919","origin":"The Stacks Project","memory_eligible":false,"source_rank":2536,"rank":2536,"depth":1,"x":219.032,"y":623.072,"cluster":"derived-categories"},{"id":"stacks:05T5","tag":"05T5","title":"Derived functors on derived categories · Lemma 05T5","summary":"In Situation [Tag 05T4]. • Let X be an object of K^+(A). The right derived functor of K(A) → D(B) is defined at X if and only if the right derived functor of K^+(A) → D^+(B) is defined at X. Moreover, the values are canonically isomorphic. • Let X be an object of K^+(A). Then X computes the right derived functor of K(A) → D(B) if and only if X computes the right derived functor of K^+(A) → D^+(B). • Let X be an object of K^-(A). The left derived functor of K(A) → D(B) is…","statement_latex":"In\nSituation \\ref{situation-classical}.\n\\begin{enumerate}\n\\item Let $X$ be an object of $K^{+}(\\mathcal{A})$.\nThe right derived functor of $K(\\mathcal{A}) \\to D(\\mathcal{B})$\nis defined at $X$ if and only if the right derived functor of\n$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is defined at $X$.\nMoreover, the values are canonically isomorphic.\n\\item Let $X$ be an object of $K^{+}(\\mathcal{A})$.\nThen $X$ computes the right derived functor of\n$K(\\mathcal{A}) \\to D(\\mathcal{B})$\nif and only if $X$ computes the right derived functor of\n$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$.\n\\item Let $X$ be an object of $K^{-}(\\mathcal{A})$.\nThe left derived functor of $K(\\mathcal{A}) \\to D(\\mathcal{B})$\nis defined at $X$ if and only if the left derived functor of\n$K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$ is defined at $X$.\nMoreover, the values are canonically isomorphic.\n\\item Let $X$ be an object of $K^{-}(\\mathcal{A})$.\nThen $X$ computes the left derived functor of\n$K(\\mathcal{A}) \\to D(\\mathcal{B})$ if and only if $X$ computes\nthe left derived functor of $K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors on derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05T5","source_file":"derived.tex","source_line":5228,"source_end_line":5253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5228-L5253","statement_sha256":"cab3fd7487fdd2337884d8b779848ea5df3a330a669d9dc3d8fa66f576980c59","origin":"The Stacks Project","memory_eligible":false,"source_rank":2537,"rank":2537,"depth":1,"x":284.123,"y":715.925,"cluster":"derived-categories"},{"id":"stacks:0157","tag":"0157","title":"Derived functors on derived categories · Definition 0157","summary":"In Situation [Tag 05T4]. • The right derived functors of F are the partial functors RF associated to cases (1) and (2) of Situation [Tag 05T4]. • The left derived functors of F are the partial functors LF associated to cases (3) and (4) of Situation [Tag 05T4]. • An object A of A is said to be right acyclic for F, or acyclic for RF if A[0] computes RF. • An object A of A is said to be left acyclic for F, or acyclic for LF if A[0] computes LF.","statement_latex":"In\nSituation \\ref{situation-classical}.\n\\begin{enumerate}\n\\item The {\\it right derived functors of $F$} are the partial functors\n$RF$ associated to cases (1) and (2) of\nSituation \\ref{situation-classical}.\n\\item The {\\it left derived functors of $F$} are the partial functors\n$LF$ associated to cases (3) and (4) of\nSituation \\ref{situation-classical}.\n\\item An object $A$ of $\\mathcal{A}$ is said to be\n{\\it right acyclic for $F$}, or {\\it acyclic for $RF$}\nif $A[0]$ computes $RF$.\n\\item An object $A$ of $\\mathcal{A}$ is said to be\n{\\it left acyclic for $F$}, or {\\it acyclic for $LF$}\nif $A[0]$ computes $LF$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors on derived categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0157","source_file":"derived.tex","source_line":5277,"source_end_line":5295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5277-L5295","statement_sha256":"f4b4d4b2912c8cfc207962264da11a14b66555e6f3fcff01b6e03d2400b29b9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2538,"rank":2538,"depth":0,"x":160.877,"y":684.24,"cluster":"derived-categories"},{"id":"stacks:05T7","tag":"05T7","title":"Derived functors on derived categories · Lemma 05T7","summary":"Let A be an abelian category. Let P ⊂ Ob(A) be a subset containing 0 such that every object of A is a quotient of an element of P. Let a ∈ Z. • Given K^bullet with K^n = 0 for n > a there exists a quasi-isomorphism P^bullet → K^bullet with P^n ∈ P and P^n → K^n surjective for all n and P^n = 0 for n > a. • Given K^bullet with H^n(K^bullet) = 0 for n > a there exists a quasi-isomorphism P^bullet → K^bullet with P^n ∈ P for all n and P^n = 0 for n > a.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset containing $0$\nsuch that every object of $\\mathcal{A}$ is a quotient of an element of\n$\\mathcal{P}$. Let $a \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item Given $K^\\bullet$ with $K^n = 0$ for $n > a$\nthere exists a quasi-isomorphism $P^\\bullet \\to K^\\bullet$\nwith $P^n \\in \\mathcal{P}$ and $P^n \\to K^n$ surjective\nfor all $n$ and $P^n = 0$ for $n > a$.\n\\item Given $K^\\bullet$ with $H^n(K^\\bullet) = 0$ for $n > a$\nthere exists a quasi-isomorphism $P^\\bullet \\to K^\\bullet$\nwith $P^n \\in \\mathcal{P}$ for all $n$ and $P^n = 0$ for $n > a$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors on derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05T7","source_file":"derived.tex","source_line":5301,"source_end_line":5316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5301-L5316","statement_sha256":"c5f8d9d82682d283e163190e096e394b788d3f9b3c70c52b2b833aaba49766e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2539,"rank":2539,"depth":0,"x":277.782,"y":637.454,"cluster":"derived-categories"},{"id":"stacks:05T6","tag":"05T6","title":"Derived functors on derived categories · Lemma 05T6","summary":"Let A be an abelian category. Let I ⊂ Ob(A) be a subset containing 0 such that every object of A is a subobject of an element of I. Let a ∈ Z. • Given K^bullet with K^n = 0 for n < a there exists a quasi-isomorphism K^bullet → I^bullet with K^n → I^n injective and I^n ∈ I for all n and I^n = 0 for n < a, • Given K^bullet with H^n(K^bullet) = 0 for n < a there exists a quasi-isomorphism K^bullet → I^bullet with I^n ∈ I and I^n = 0 for n < a.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$\\mathcal{I} \\subset \\Ob(\\mathcal{A})$ be a subset containing $0$\nsuch that every object of $\\mathcal{A}$ is a subobject of an element of\n$\\mathcal{I}$. Let $a \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item Given $K^\\bullet$ with $K^n = 0$ for $n < a$\nthere exists a quasi-isomorphism $K^\\bullet \\to I^\\bullet$\nwith $K^n \\to I^n$ injective and $I^n \\in \\mathcal{I}$ for all $n$\nand $I^n = 0$ for $n < a$,\n\\item Given $K^\\bullet$ with $H^n(K^\\bullet) = 0$\nfor $n < a$ there exists a quasi-isomorphism $K^\\bullet \\to I^\\bullet$\nwith $I^n \\in \\mathcal{I}$ and $I^n = 0$ for $n < a$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors on derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05T6","source_file":"derived.tex","source_line":5386,"source_end_line":5401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5386-L5401","statement_sha256":"33d2dc490bfeb7b0f7259db970d20d6d55a711cbd57f36f82f32398025248e15","origin":"The Stacks Project","memory_eligible":false,"source_rank":2540,"rank":2540,"depth":1,"x":228.976,"y":738.756,"cluster":"derived-categories"},{"id":"stacks:05T8","tag":"05T8","title":"Derived functors on derived categories · Lemma 05T8","summary":"In Situation [Tag 05T4]. Let I ⊂ Ob(A) be a subset with the following properties: • every object of A is a subobject of an element of I, • for any short exact sequence 0 → P → Q → R → 0 of A with P, Q ∈ I, then R ∈ I, and 0 → F(P) → F(Q) → F(R) → 0 is exact. Then every object of I is acyclic for RF.","statement_latex":"In\nSituation \\ref{situation-classical}.\nLet $\\mathcal{I} \\subset \\Ob(\\mathcal{A})$ be a subset with the\nfollowing properties:\n\\begin{enumerate}\n\\item every object of $\\mathcal{A}$ is a subobject of an element of\n$\\mathcal{I}$,\n\\item for any short exact sequence $0 \\to P \\to Q \\to R \\to 0$ of\n$\\mathcal{A}$ with $P, Q \\in \\mathcal{I}$, then $R \\in \\mathcal{I}$,\nand $0 \\to F(P) \\to F(Q) \\to F(R) \\to 0$ is exact.\n\\end{enumerate}\nThen every object of $\\mathcal{I}$ is acyclic for $RF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors on derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05T8","source_file":"derived.tex","source_line":5407,"source_end_line":5421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5407-L5421","statement_sha256":"2c10e12b10b6e0528d0e6bbcc401ff280d086fbc66f02e0ab26ebb2a03157c77","origin":"The Stacks Project","memory_eligible":false,"source_rank":2541,"rank":2541,"depth":2,"x":183.292,"y":635.893,"cluster":"derived-categories"},{"id":"stacks:05T9","tag":"05T9","title":"Derived functors on derived categories · Lemma 05T9","summary":"In Situation [Tag 05T4]. Let P ⊂ Ob(A) be a subset with the following properties: • every object of A is a quotient of an element of P, • for any short exact sequence 0 → P → Q → R → 0 of A with Q, R ∈ P, then P ∈ P, and 0 → F(P) → F(Q) → F(R) → 0 is exact. Then every object of P is acyclic for LF.","statement_latex":"In\nSituation \\ref{situation-classical}.\nLet $\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset with the\nfollowing properties:\n\\begin{enumerate}\n\\item every object of $\\mathcal{A}$ is a quotient of an element of\n$\\mathcal{P}$,\n\\item for any short exact sequence $0 \\to P \\to Q \\to R \\to 0$ of\n$\\mathcal{A}$ with $Q, R \\in \\mathcal{P}$, then $P \\in \\mathcal{P}$,\nand $0 \\to F(P) \\to F(Q) \\to F(R) \\to 0$ is exact.\n\\end{enumerate}\nThen every object of $\\mathcal{P}$ is acyclic for $LF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived functors on derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05T9","source_file":"derived.tex","source_line":5462,"source_end_line":5476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5462-L5476","statement_sha256":"872c169f1e6814f578b2a9600cde922b844af05d484248cccab459c43af218a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2542,"rank":2542,"depth":3,"x":300.229,"y":686.048,"cluster":"derived-categories"},{"id":"stacks:05TC","tag":"05TC","title":"Higher derived functors · Lemma 05TC","summary":"Let F : A → B be an additive functor between abelian categories. Let K^bullet be a complex of A and a ∈ Z. • If H^i(K^bullet) = 0 for all i < a and RF is defined at K^bullet, then H^i(RF(K^bullet)) = 0 for all i < a. • If RF is defined at K^bullet and τ_≤ aK^bullet, then H^i(RF(τ_≤ aK^bullet)) = H^i(RF(K^bullet)) for all i ≤ a.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories. Let $K^\\bullet$ be a complex of $\\mathcal{A}$\nand $a \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $H^i(K^\\bullet) = 0$ for all $i < a$ and $RF$ is defined at\n$K^\\bullet$, then $H^i(RF(K^\\bullet)) = 0$ for all $i < a$.\n\\item If $RF$ is defined at $K^\\bullet$ and $\\tau_{\\leq a}K^\\bullet$,\nthen $H^i(RF(\\tau_{\\leq a}K^\\bullet)) = H^i(RF(K^\\bullet))$\nfor all $i \\leq a$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Higher derived functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TC","source_file":"derived.tex","source_line":5496,"source_end_line":5508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5496-L5508","statement_sha256":"a0e899c1012cef7d57366bf34e3310b2e96fd4e0f124403e724801c539041b0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2543,"rank":2543,"depth":9,"x":173.096,"y":715.548,"cluster":"derived-categories"},{"id":"stacks:015A","tag":"015A","title":"Higher derived functors · Definition 015A","summary":"Let F : A → B be an additive functor between abelian categories. Assume RF : D^+(A) → D^+(B) is everywhere defined. Let i ∈ Z. The ith right derived functor R^iF of F is the functor R^iF = H^i ∘ RF : A → B","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories. Assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined. Let $i \\in \\mathbf{Z}$.\nThe {\\it $i$th right derived functor $R^iF$ of $F$} is the functor\n$$\nR^iF = H^i \\circ RF :\n\\mathcal{A}\n\\longrightarrow\n\\mathcal{B}\n$$","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Higher derived functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015A","source_file":"derived.tex","source_line":5544,"source_end_line":5557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5544-L5557","statement_sha256":"7d180dafd5c4a7126f6dce101e7dc39991d763f97c895ebd04c850facc4e26c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2544,"rank":2544,"depth":0,"x":243.433,"y":621.238,"cluster":"derived-categories"},{"id":"stacks:05TD","tag":"05TD","title":"Higher derived functors · Lemma 05TD","summary":"Let F : A → B be an additive functor between abelian categories and assume RF : D^+(A) → D^+(B) is everywhere defined. • We have R^iF = 0 for i < 0, • R^0F is left exact, • the map F → R^0F is an isomorphism if and only if F is left exact.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined.\n\\begin{enumerate}\n\\item We have $R^iF = 0$ for $i < 0$,\n\\item $R^0F$ is left exact,\n\\item the map $F \\to R^0F$ is an isomorphism if and\nonly if $F$ is left exact.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Higher derived functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TD","source_file":"derived.tex","source_line":5564,"source_end_line":5576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5564-L5576","statement_sha256":"ef68dd2acee3656d53d7b9b9bf8b4ab6a3fb0da17a96ec442f4b4da354deb6a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2545,"rank":2545,"depth":10,"x":267.515,"y":731.177,"cluster":"derived-categories"},{"id":"stacks:015C","tag":"015C","title":"Higher derived functors · Lemma 015C","summary":"Let F : A → B be an additive functor between abelian categories and assume RF : D^+(A) → D^+(B) is everywhere defined. Let A be an object of A. • A is right acyclic for F if and only if F(A) → R^0F(A) is an isomorphism and R^iF(A) = 0 for all i > 0, • if F is left exact, then A is right acyclic for F if and only if R^iF(A) = 0 for all i > 0.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined. Let $A$ be an object of $\\mathcal{A}$.\n\\begin{enumerate}\n\\item $A$ is right acyclic for $F$ if and only if\n$F(A) \\to R^0F(A)$ is an isomorphism and $R^iF(A) = 0$ for all $i > 0$,\n\\item if $F$ is left exact, then $A$ is right acyclic for $F$\nif and only if $R^iF(A) = 0$ for all $i > 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Higher derived functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015C","source_file":"derived.tex","source_line":5618,"source_end_line":5630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5618-L5630","statement_sha256":"e96af1a481992f75233cf09bc4f0c7a1b09ea101ece19f9aa11a33ebe46ffe40","origin":"The Stacks Project","memory_eligible":false,"source_rank":2546,"rank":2546,"depth":11,"x":160.881,"y":663.478,"cluster":"derived-categories"},{"id":"stacks:015D","tag":"015D","title":"Higher derived functors · Lemma 015D","summary":"Let F : A → B be a left exact functor between abelian categories and assume RF : D^+(A) → D^+(B) is everywhere defined. Let 0 → A → B → C → 0 be a short exact sequence of A. • If A and C are right acyclic for F then so is B. • If A and B are right acyclic for F then so is C. • If B and C are right acyclic for F and F(B) → F(C) is surjective then A is right acyclic for F. In each of the three cases 0 → F(A) → F(B) → F(C) → 0 is a short exact sequence of B.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined. Let $0 \\to A \\to B \\to C \\to 0$ be a short exact sequence\nof $\\mathcal{A}$.\n\\begin{enumerate}\n\\item If $A$ and $C$ are right acyclic for $F$ then so is $B$.\n\\item If $A$ and $B$ are right acyclic for $F$ then so is $C$.\n\\item If $B$ and $C$ are right acyclic for $F$ and $F(B) \\to F(C)$ is\nsurjective then $A$ is right acyclic for $F$.\n\\end{enumerate}\nIn each of the three cases\n$$\n0 \\to F(A) \\to F(B) \\to F(C) \\to 0\n$$\nis a short exact sequence of $\\mathcal{B}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Higher derived functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015D","source_file":"derived.tex","source_line":5643,"source_end_line":5661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5643-L5661","statement_sha256":"380a9e2a8aa02364a27adf2ef21162fa7d0dfacfe356d85a90979fcd645fe7e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2547,"rank":2547,"depth":12,"x":294.533,"y":652.846,"cluster":"derived-categories"},{"id":"stacks:05TE","tag":"05TE","title":"Higher derived functors · Lemma 05TE","summary":"Let F : A → B be an additive functor between abelian categories and assume RF : D^+(A) → D^+(B) is everywhere defined. • The functors R^iF, i ≥ 0 come equipped with a canonical structure of a δ-functor from A → B, see Homology, Definition [Tag 010Q]. • If every object of A is a subobject of a right acyclic object for F, then (R^iF, δ)_i ≥ 0 is a universal δ-functor, see Homology, Definition [Tag 010S].","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.\n\\begin{enumerate}\n\\item The functors $R^iF$, $i \\geq 0$ come equipped with a canonical\nstructure of a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}.\n\\item If every object of $\\mathcal{A}$ is a subobject of a right\nacyclic object for $F$, then $\\{R^iF, \\delta\\}_{i \\geq 0}$ is a\nuniversal $\\delta$-functor, see\nHomology, Definition \\ref{homology-definition-universal-delta-functor}.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Higher derived functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TE","source_file":"derived.tex","source_line":5680,"source_end_line":5694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5680-L5694","statement_sha256":"84a0c5e11b9680700519fe08ac562739676d588876d4e4fc247810a33a3ae2bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2548,"rank":2548,"depth":17,"x":204.14,"y":736.883,"cluster":"derived-categories"},{"id":"stacks:015E","tag":"015E","title":"Leray's acyclicity lemma · Lemma 015E","summary":"Let F : A → B be an additive functor between abelian categories. Let A^bullet be a bounded below complex of right F-acyclic objects such that RF is defined at A^bullet(A) → D^+(B) is everywhere defined.. The canonical map F(A^bullet) → RF(A^bullet) is an isomorphism in D^+(B), i.e., A^bullet computes RF.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories. Let $A^\\bullet$ be a bounded below complex\nof right $F$-acyclic objects such that $RF$ is defined at\n$A^\\bullet$\\footnote{For example this holds if\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.}.\nThe canonical map\n$$\nF(A^\\bullet) \\longrightarrow RF(A^\\bullet)\n$$\nis an isomorphism in $D^{+}(\\mathcal{B})$, i.e., $A^\\bullet$ computes\n$RF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Higher derived functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015E","source_file":"derived.tex","source_line":5716,"source_end_line":5729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5716-L5729","statement_sha256":"80b88ecac34bf226c7865fbcfb4d28c9d0fb32689ec16f73396dba0dbe03cdc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2549,"rank":2549,"depth":10,"x":203.211,"y":623.14,"cluster":"derived-categories"},{"id":"stacks:05TA","tag":"05TA","title":"Higher derived functors · Proposition 05TA","summary":"A functor on Abelian categories is extended to the (bounded below or above) derived category by resolving with a complex that is acyclic for that functor. Let F : A → B be an additive functor of abelian categories. • If every object of A injects into an object acyclic for RF, then RF is defined on all of K^+(A) and we obtain an exact functor RF : D^+(A) → D^+(B) see ([Tag 05SW]). Moreover, any bounded below complex A^bullet whose terms are acyclic for RF computes RF. • If…","statement_latex":"\\begin{slogan}\nA functor on Abelian categories is extended to the (bounded below or above)\nderived category by resolving with a complex that is acyclic for that functor.\n\\end{slogan}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor of\nabelian categories.\n\\begin{enumerate}\n\\item If every object of $\\mathcal{A}$ injects into an object acyclic\nfor $RF$, then $RF$ is defined on all of $K^{+}(\\mathcal{A})$\nand we obtain an exact functor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})\n$$\nsee (\\ref{equation-everywhere}). Moreover, any bounded below complex\n$A^\\bullet$ whose terms are acyclic for $RF$ computes $RF$.\n\\item If every object of $\\mathcal{A}$ is quotient of\nan object acyclic for $LF$, then $LF$ is defined on all of\n$K^{-}(\\mathcal{A})$ and we obtain an exact functor\n$$\nLF : D^{-}(\\mathcal{A}) \\longrightarrow D^{-}(\\mathcal{B})\n$$\nsee (\\ref{equation-everywhere}). Moreover, any bounded above complex\n$A^\\bullet$ whose terms are acyclic for $LF$ computes $LF$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Higher derived functors","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TA","source_file":"derived.tex","source_line":5809,"source_end_line":5835,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5809-L5835","statement_sha256":"2a01933e89812008cdf8e44c2026e67b6a9d89b4b533ed3e5e30f2659e10d72c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2550,"rank":2550,"depth":14,"x":295.755,"y":706.84,"cluster":"derived-categories"},{"id":"stacks:015F","tag":"015F","title":"Higher derived functors · Lemma 015F","summary":"Let F : A → B be an exact functor of abelian categories. Then • every object of A is right acyclic for F, • RF : D^+(A) → D^+(B) is everywhere defined, • RF : D(A) → D(B) is everywhere defined, • every complex computes RF, in other words, the canonical map F(K^bullet) → RF(K^bullet) is an isomorphism for all complexes, and • R^iF = 0 for i not = 0.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an exact functor of\nabelian categories. Then\n\\begin{enumerate}\n\\item every object of $\\mathcal{A}$ is right acyclic for $F$,\n\\item $RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined,\n\\item $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ is everywhere defined,\n\\item every complex computes $RF$, in other words, the canonical\nmap $F(K^\\bullet) \\to RF(K^\\bullet)$ is an isomorphism for all complexes, and\n\\item $R^iF = 0$ for $i \\not = 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Higher derived functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015F","source_file":"derived.tex","source_line":5883,"source_end_line":5895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5883-L5895","statement_sha256":"8eeda8c141c973495dfa8c608a21f3bd967d31dc46656d5ab86b06b33adb4a30","origin":"The Stacks Project","memory_eligible":false,"source_rank":2551,"rank":2551,"depth":0,"x":159.636,"y":697.593,"cluster":"derived-categories"},{"id":"stacks:06UQ","tag":"06UQ","title":"Triangulated subcategories of the derived category · Lemma 06UQ","summary":"Let A be an abelian category. Let B ⊂ A be a weak Serre subcategory. The category D_B(A) is a strictly full saturated triangulated subcategory of D(A). Similarly for the bounded versions.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\mathcal{B} \\subset \\mathcal{A}$ be a weak Serre subcategory.\nThe category $D_\\mathcal{B}(\\mathcal{A})$ is a strictly full\nsaturated triangulated subcategory of $D(\\mathcal{A})$.\nSimilarly for the bounded versions.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Triangulated subcategories of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UQ","source_file":"derived.tex","source_line":5935,"source_end_line":5942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L5935-L5942","statement_sha256":"1e377a5de16738a7fbc4a121ea1f540f6a19eb346981dc8fba433868186ac6ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":2552,"rank":2552,"depth":2,"x":267.894,"y":626.881,"cluster":"derived-categories"},{"id":"stacks:06XL","tag":"06XL","title":"Triangulated subcategories of the derived category · Lemma 06XL","summary":"Let A be an abelian category. Let B ⊂ A be a Serre subcategory. Then D(A) → D(A/B) is essentially surjective.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\mathcal{B} \\subset \\mathcal{A}$ be a Serre subcategory.\nThen $D(\\mathcal{A}) \\to D(\\mathcal{A}/\\mathcal{B})$\nis essentially surjective.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Triangulated subcategories of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XL","source_file":"derived.tex","source_line":6003,"source_end_line":6009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6003-L6009","statement_sha256":"ec120c8b805cb34121a69cf6d20f22be869c8356b2dfc96f1478936062401f7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2553,"rank":2553,"depth":8,"x":244.835,"y":740.921,"cluster":"derived-categories"},{"id":"stacks:06XM","tag":"06XM","title":"Triangulated subcategories of the derived category · Lemma 06XM","summary":"Let A be an abelian category. Let B ⊂ A be a Serre subcategory. Suppose that the functor v : A → A/B has a left adjoint u : A/B → A such that vu ≅ id. Then D(A)/D_B(A) = D(A/B) and similarly for the bounded versions.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\mathcal{B} \\subset \\mathcal{A}$ be a Serre subcategory.\nSuppose that the functor $v : \\mathcal{A} \\to \\mathcal{A}/\\mathcal{B}$\nhas a left adjoint $u : \\mathcal{A}/\\mathcal{B} \\to \\mathcal{A}$\nsuch that $vu \\cong \\text{id}$. Then\n$$\nD(\\mathcal{A})/D_\\mathcal{B}(\\mathcal{A}) = D(\\mathcal{A}/\\mathcal{B})\n$$\nand similarly for the bounded versions.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Triangulated subcategories of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XM","source_file":"derived.tex","source_line":6102,"source_end_line":6113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6102-L6113","statement_sha256":"581345ccb37fc1c57b7cbcaf29550c654c88793a5b43f49af381a34ae1041323","origin":"The Stacks Project","memory_eligible":false,"source_rank":2554,"rank":2554,"depth":9,"x":169.825,"y":643.346,"cluster":"derived-categories"},{"id":"stacks:0FCL","tag":"0FCL","title":"Triangulated subcategories of the derived category · Lemma 0FCL","summary":"Let A be an abelian category. Let B ⊂ A be a Serre subcategory. Assume that for every surjection X → Y with X ∈ Ob(A) and Y ∈ Ob(B) there exists X' ⊂ X, X' ∈ Ob(B) which surjects onto Y. Then the functor D^-(B) → D^-_B(A) of ([Tag 06UR]) is an equivalence.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{B} \\subset \\mathcal{A}$\nbe a Serre subcategory. Assume that for every surjection $X \\to Y$\nwith $X \\in \\Ob(\\mathcal{A})$ and $Y \\in \\Ob(\\mathcal{B})$ there exists\n$X' \\subset X$, $X' \\in \\Ob(\\mathcal{B})$ which surjects onto $Y$.\nThen the functor $D^-(\\mathcal{B}) \\to D^-_\\mathcal{B}(\\mathcal{A})$ of\n(\\ref{equation-compare}) is an equivalence.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Triangulated subcategories of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCL","source_file":"derived.tex","source_line":6143,"source_end_line":6151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6143-L6151","statement_sha256":"350ac7495fecca79621e8fb34b0b4c8328187e8aa1003dc8e58d64ec3acb3946","origin":"The Stacks Project","memory_eligible":false,"source_rank":2555,"rank":2555,"depth":0,"x":304.148,"y":672.854,"cluster":"derived-categories"},{"id":"stacks:013I","tag":"013I","title":"Injective resolutions · Definition 013I","summary":"Let A be an abelian category. Let A ∈ Ob(A). An injective resolution of A is a complex I^bullet together with a map A → I^0 such that: • We have I^n = 0 for n < 0. • Each I^n is an injective object of A. • The map A → I^0 is an isomorphism onto Ker(d^0). • We have H^i(I^bullet) = 0 for i > 0. Hence A[0] → I^bullet is a quasi-isomorphism. In other words the complex … → 0 → A → I^0 → I^1 → … is acyclic. Let K^bullet be a complex in A. An injective resolution of K^bullet is…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A \\in \\Ob(\\mathcal{A})$.\nAn {\\it injective resolution of $A$} is a complex\n$I^\\bullet$ together with a map $A \\to I^0$ such\nthat:\n\\begin{enumerate}\n\\item We have $I^n = 0$ for $n < 0$.\n\\item Each $I^n$ is an injective object of $\\mathcal{A}$.\n\\item The map $A \\to I^0$ is an isomorphism onto $\\Ker(d^0)$.\n\\item We have $H^i(I^\\bullet) = 0$ for $i > 0$.\n\\end{enumerate}\nHence $A[0] \\to I^\\bullet$ is a quasi-isomorphism.\nIn other words the complex\n$$\n\\ldots \\to 0 \\to A \\to I^0 \\to I^1 \\to \\ldots\n$$\nis acyclic.\nLet $K^\\bullet$ be a complex in $\\mathcal{A}$.\nAn {\\it injective resolution of $K^\\bullet$} is a complex\n$I^\\bullet$ together with a map $\\alpha : K^\\bullet \\to I^\\bullet$\nof complexes such that\n\\begin{enumerate}\n\\item We have $I^n = 0$ for $n \\ll 0$, i.e., $I^\\bullet$ is bounded below.\n\\item Each $I^n$ is an injective object of $\\mathcal{A}$.\n\\item The map $\\alpha : K^\\bullet \\to I^\\bullet$ is a\nquasi-isomorphism.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Injective resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013I","source_file":"derived.tex","source_line":6212,"source_end_line":6241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6212-L6241","statement_sha256":"ecc429f68b9f7f0817f17406ea90d8373750cc5dfc8802f73bc6ed382f8a4f33","origin":"The Stacks Project","memory_eligible":false,"source_rank":2556,"rank":2556,"depth":0,"x":180.875,"y":727.533,"cluster":"derived-categories"},{"id":"stacks:013J","tag":"013J","title":"Injective resolutions · Lemma 013J","summary":"Let A be an abelian category. Let K^bullet be a complex of A. • If K^bullet has an injective resolution then H^n(K^bullet) = 0 for n ll 0. • If H^n(K^bullet) = 0 for all n ll 0 then there exists a quasi-isomorphism K^bullet → L^bullet with L^bullet bounded below.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet$ be a complex of $\\mathcal{A}$.\n\\begin{enumerate}\n\\item If $K^\\bullet$ has an injective resolution then\n$H^n(K^\\bullet) = 0$ for $n \\ll 0$.\n\\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$ then there\nexists a quasi-isomorphism $K^\\bullet \\to L^\\bullet$\nwith $L^\\bullet$ bounded below.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013J","source_file":"derived.tex","source_line":6258,"source_end_line":6269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6258-L6269","statement_sha256":"f57f52c62534883fac9e13b6976dd3f816ed52331e17ae2bc7574ad1bccacf2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2557,"rank":2557,"depth":0,"x":227.992,"y":616.824,"cluster":"derived-categories"},{"id":"stacks:013K","tag":"013K","title":"Injective resolutions · Lemma 013K","summary":"Let A be an abelian category. Assume A has enough injectives. • Any object of A has an injective resolution. • If H^n(K^bullet) = 0 for all n ll 0 then K^bullet has an injective resolution. • If K^bullet is a complex with K^n = 0 for n < a, then there exists an injective resolution α : K^bullet → I^bullet with I^n = 0 for n < a such that each α^n : K^n → I^n is injective.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has enough injectives.\n\\begin{enumerate}\n\\item Any object of $\\mathcal{A}$ has an injective resolution.\n\\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$ then\n$K^\\bullet$ has an injective resolution.\n\\item If $K^\\bullet$ is a complex with $K^n = 0$ for $n < a$, then\nthere exists an injective resolution $\\alpha : K^\\bullet \\to I^\\bullet$\nwith $I^n = 0$ for $n < a$ such that each $\\alpha^n : K^n \\to I^n$ is\ninjective.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013K","source_file":"derived.tex","source_line":6279,"source_end_line":6292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6279-L6292","statement_sha256":"a6ea949526a907f129f9b129486113b5516cedf2b173f2e27bd3da3eb21b117b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2558,"rank":2558,"depth":2,"x":282.492,"y":725.625,"cluster":"derived-categories"},{"id":"stacks:013R","tag":"013R","title":"Injective resolutions · Lemma 013R","summary":"Let A be an abelian category. Let K^bullet be an acyclic complex. Let I^bullet be bounded below and consisting of injective objects. Any morphism K^bullet → I^bullet is homotopic to zero.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet$ be an acyclic complex.\nLet $I^\\bullet$ be bounded below and consisting of injective objects.\nAny morphism $K^\\bullet \\to I^\\bullet$ is homotopic to zero.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013R","source_file":"derived.tex","source_line":6307,"source_end_line":6313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6307-L6313","statement_sha256":"9b82b8555bce8c2dcd2b9ede68716d020089bfeb96c3394be4975a8d86980176","origin":"The Stacks Project","memory_eligible":false,"source_rank":2559,"rank":2559,"depth":0,"x":154.307,"y":676.127,"cluster":"derived-categories"},{"id":"stacks:013P","tag":"013P","title":"Injective resolutions · Lemma 013P","summary":"Let A be an abelian category. Consider a solid diagram xymatrix K^bullet ar[r]_α ar[d]_γ & L^bullet ar@-->[dl]^β I^bullet where I^bullet is bounded below and consists of injective objects, and α is a quasi-isomorphism. • There exists a map of complexes β making the diagram commute up to homotopy. • If α is injective in every degree then we can find a β which makes the diagram commute.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nConsider a solid diagram\n$$\n\\xymatrix{\nK^\\bullet \\ar[r]_\\alpha \\ar[d]_\\gamma & L^\\bullet \\ar@{-->}[dl]^\\beta \\\\\nI^\\bullet\n}\n$$\nwhere $I^\\bullet$ is bounded below and consists of injective\nobjects, and $\\alpha$ is a quasi-isomorphism.\n\\begin{enumerate}\n\\item There exists a map of complexes $\\beta$ making the diagram\ncommute up to homotopy.\n\\item If $\\alpha$ is injective in every degree\nthen we can find a $\\beta$ which makes the diagram commute.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013P","source_file":"derived.tex","source_line":6385,"source_end_line":6403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6385-L6403","statement_sha256":"c3ee9570fe22b83ca09ec7a35def5558174526b68257ca56418d896e957eb62f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2560,"rank":2560,"depth":2,"x":289.159,"y":639.75,"cluster":"derived-categories"},{"id":"stacks:013S","tag":"013S","title":"Injective resolutions · Lemma 013S","summary":"Let A be an abelian category. Consider a solid diagram xymatrix K^bullet ar[r]_α ar[d]_γ & L^bullet ar@-->[dl]^β_i I^bullet where I^bullet is bounded below and consists of injective objects, and α is a quasi-isomorphism. Any two morphisms β_1, β_2 making the diagram commute up to homotopy are homotopic.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nConsider a solid diagram\n$$\n\\xymatrix{\nK^\\bullet \\ar[r]_\\alpha \\ar[d]_\\gamma & L^\\bullet \\ar@{-->}[dl]^{\\beta_i} \\\\\nI^\\bullet\n}\n$$\nwhere $I^\\bullet$ is bounded below and consists of injective\nobjects, and $\\alpha$ is a quasi-isomorphism.\nAny two morphisms $\\beta_1, \\beta_2$ making the diagram commute\nup to homotopy are homotopic.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013S","source_file":"derived.tex","source_line":6480,"source_end_line":6494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6480-L6494","statement_sha256":"c2b3ee51ecf0fe1a3d799b2dc78a31e08a348c33344dcbacd677b5f30c5c116e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2561,"rank":2561,"depth":2,"x":218.702,"y":743.492,"cluster":"derived-categories"},{"id":"stacks:05TG","tag":"05TG","title":"Injective resolutions · Lemma 05TG","summary":"Let A be an abelian category. Let I^bullet be bounded below complex consisting of injective objects. Let L^bullet ∈ K(A). Then Mor_K(A)(L^bullet, I^bullet) = Mor_D(A)(L^bullet, I^bullet).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $I^\\bullet$ be bounded below complex consisting of injective\nobjects. Let $L^\\bullet \\in K(\\mathcal{A})$. Then\n$$\n\\Mor_{K(\\mathcal{A})}(L^\\bullet, I^\\bullet)\n=\n\\Mor_{D(\\mathcal{A})}(L^\\bullet, I^\\bullet).\n$$","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TG","source_file":"derived.tex","source_line":6540,"source_end_line":6550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6540-L6550","statement_sha256":"6f67f51e7a58b30f8358c99062df6220dd3788aacf5c0baa80f9e37a6747cef7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2562,"rank":2562,"depth":3,"x":187.109,"y":626.567,"cluster":"derived-categories"},{"id":"stacks:013T","tag":"013T","title":"Injective resolutions · Lemma 013T","summary":"Let A be an abelian category. Assume A has enough injectives. For any short exact sequence 0 → A^bullet → B^bullet → C^bullet → 0 of Comp^+(A) there exists a commutative diagram in Comp^+(A) xymatrix 0 ar[r] & A^bullet ar[r] ar[d] & B^bullet ar[r] ar[d] & C^bullet ar[r] ar[d] & 0 0 ar[r] & I_1^bullet ar[r] & I_2^bullet ar[r] & I_3^bullet ar[r] & 0 where the vertical arrows are injective resolutions and the rows are short exact sequences of complexes. Additionally, • given…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has enough injectives.\nFor any short exact sequence\n$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$\nof $\\text{Comp}^{+}(\\mathcal{A})$ there exists a\ncommutative diagram in $\\text{Comp}^{+}(\\mathcal{A})$\n$$\n\\xymatrix{\n0 \\ar[r] &\nA^\\bullet \\ar[r] \\ar[d] &\nB^\\bullet \\ar[r] \\ar[d] &\nC^\\bullet \\ar[r] \\ar[d] &\n0 \\\\\n0 \\ar[r] &\nI_1^\\bullet \\ar[r] &\nI_2^\\bullet \\ar[r] &\nI_3^\\bullet \\ar[r] &\n0\n}\n$$\nwhere the vertical arrows are injective resolutions and\nthe rows are short exact sequences of complexes. Additionally,\n\\begin{enumerate}\n\\item given any injective resolution $A^\\bullet \\to I^\\bullet$\nwe may assume $I_1^\\bullet = I^\\bullet$,\n\\item if $A^n = B^n = C^n = 0$ for $n < 0$, then we may\nassume $I_j^n = 0$ for $n < 0$,\n\\item we can combine (1) and (2) if also $I^n = 0$ for $n < 0$, and\n\\item add more here.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013T","source_file":"derived.tex","source_line":6570,"source_end_line":6602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6570-L6602","statement_sha256":"c8ac874ac03de51a4523b533ce07eb302cc66fc2aad496d8d08b0fd7e425324c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2563,"rank":2563,"depth":7,"x":304.88,"y":695.121,"cluster":"derived-categories"},{"id":"stacks:0644","tag":"0644","title":"Projective resolutions · Definition 0644","summary":"Let A be an abelian category. Let A ∈ Ob(A). A projective resolution of A is a complex P^bullet together with a map P^0 → A such that: • We have P^n = 0 for n > 0. • Each P^n is a projective object of A. • The map P^0 → A induces an isomorphism Coker(d^-1) → A. • We have H^i(P^bullet) = 0 for i < 0. Hence P^bullet → A[0] is a quasi-isomorphism. In other words the complex … → P^-1 → P^0 → A → 0 → … is acyclic. Let K^bullet be a complex in A. A projective resolution of…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A \\in \\Ob(\\mathcal{A})$.\nA {\\it projective resolution of $A$} is a complex\n$P^\\bullet$ together with a map $P^0 \\to A$ such\nthat:\n\\begin{enumerate}\n\\item We have $P^n = 0$ for $n > 0$.\n\\item Each $P^n$ is a projective object of $\\mathcal{A}$.\n\\item The map $P^0 \\to A$ induces an isomorphism $\\Coker(d^{-1}) \\to A$.\n\\item We have $H^i(P^\\bullet) = 0$ for $i < 0$.\n\\end{enumerate}\nHence $P^\\bullet \\to A[0]$ is a quasi-isomorphism.\nIn other words the complex\n$$\n\\ldots \\to P^{-1} \\to P^0 \\to A \\to 0 \\to \\ldots\n$$\nis acyclic. Let $K^\\bullet$ be a complex in $\\mathcal{A}$.\nA {\\it projective resolution of $K^\\bullet$} is a complex\n$P^\\bullet$ together with a map $\\alpha : P^\\bullet \\to K^\\bullet$\nof complexes such that\n\\begin{enumerate}\n\\item We have $P^n = 0$ for $n \\gg 0$, i.e., $P^\\bullet$ is bounded above.\n\\item Each $P^n$ is a projective object of $\\mathcal{A}$.\n\\item The map $\\alpha : P^\\bullet \\to K^\\bullet$ is a\nquasi-isomorphism.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0644","source_file":"derived.tex","source_line":6680,"source_end_line":6708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6680-L6708","statement_sha256":"d72fa2fa83e4bcca87c14aa51afdbc7b2882d8ad74795854bda01dd9c0ca0fce","origin":"The Stacks Project","memory_eligible":false,"source_rank":2564,"rank":2564,"depth":0,"x":162.371,"y":711.457,"cluster":"derived-categories"},{"id":"stacks:0645","tag":"0645","title":"Projective resolutions · Lemma 0645","summary":"Let A be an abelian category. Let K^bullet be a complex of A. • If K^bullet has a projective resolution then H^n(K^bullet) = 0 for n gg 0. • If H^n(K^bullet) = 0 for n gg 0 then there exists a quasi-isomorphism L^bullet → K^bullet with L^bullet bounded above.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet$ be a complex of $\\mathcal{A}$.\n\\begin{enumerate}\n\\item If $K^\\bullet$ has a projective resolution then\n$H^n(K^\\bullet) = 0$ for $n \\gg 0$.\n\\item If $H^n(K^\\bullet) = 0$ for $n \\gg 0$ then there\nexists a quasi-isomorphism $L^\\bullet \\to K^\\bullet$\nwith $L^\\bullet$ bounded above.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0645","source_file":"derived.tex","source_line":6710,"source_end_line":6721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6710-L6721","statement_sha256":"24ead2d301f38edd339c72af578e8f9d6e3d7c252250efc63b22d2b7685abb02","origin":"The Stacks Project","memory_eligible":false,"source_rank":2565,"rank":2565,"depth":1,"x":254.664,"y":618.199,"cluster":"derived-categories"},{"id":"stacks:0646","tag":"0646","title":"Projective resolutions · Lemma 0646","summary":"Let A be an abelian category. Assume A has enough projectives. • Any object of A has a projective resolution. • If H^n(K^bullet) = 0 for all n gg 0 then K^bullet has a projective resolution. • If K^bullet is a complex with K^n = 0 for n > a, then there exists a projective resolution α : P^bullet → K^bullet with P^n = 0 for n > a such that each α^n : P^n → K^n is surjective.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has enough projectives.\n\\begin{enumerate}\n\\item Any object of $\\mathcal{A}$ has a projective resolution.\n\\item If $H^n(K^\\bullet) = 0$ for all $n \\gg 0$ then\n$K^\\bullet$ has a projective resolution.\n\\item If $K^\\bullet$ is a complex with $K^n = 0$ for $n > a$, then\nthere exists a projective resolution $\\alpha : P^\\bullet \\to K^\\bullet$\nwith $P^n = 0$ for $n > a$ such that each $\\alpha^n : P^n \\to K^n$ is\nsurjective.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0646","source_file":"derived.tex","source_line":6728,"source_end_line":6741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6728-L6741","statement_sha256":"d033e95699650c5f687c5b0ee9566ed73e5ffd8e7de917a0290f593851025310","origin":"The Stacks Project","memory_eligible":false,"source_rank":2566,"rank":2566,"depth":3,"x":261.629,"y":739.787,"cluster":"derived-categories"},{"id":"stacks:0647","tag":"0647","title":"Projective resolutions · Lemma 0647","summary":"Let A be an abelian category. Let K^bullet be an acyclic complex. Let P^bullet be bounded above and consisting of projective objects. Any morphism P^bullet → K^bullet is homotopic to zero.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet$ be an acyclic complex.\nLet $P^\\bullet$ be bounded above and consisting of projective objects.\nAny morphism $P^\\bullet \\to K^\\bullet$ is homotopic to zero.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0647","source_file":"derived.tex","source_line":6748,"source_end_line":6754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6748-L6754","statement_sha256":"b29f95b36d135b169bf2c68a9fca401b0f5817ae54f8b2a241fed81ff3cc8be8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2567,"rank":2567,"depth":1,"x":158.334,"y":653.764,"cluster":"derived-categories"},{"id":"stacks:0649","tag":"0649","title":"Projective resolutions · Lemma 0649","summary":"Let A be an abelian category. Consider a solid diagram xymatrix K^bullet & L^bullet ar[l]^α P^bullet ar[u] ar@-->[ru]_β where P^bullet is bounded above and consists of projective objects, and α is a quasi-isomorphism. • There exists a map of complexes β making the diagram commute up to homotopy. • If α is surjective in every degree then we can find a β which makes the diagram commute.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nConsider a solid diagram\n$$\n\\xymatrix{\nK^\\bullet & L^\\bullet \\ar[l]^\\alpha \\\\\nP^\\bullet \\ar[u] \\ar@{-->}[ru]_\\beta\n}\n$$\nwhere $P^\\bullet$ is bounded above and consists of projective\nobjects, and $\\alpha$ is a quasi-isomorphism.\n\\begin{enumerate}\n\\item There exists a map of complexes $\\beta$ making the diagram\ncommute up to homotopy.\n\\item If $\\alpha$ is surjective in every degree\nthen we can find a $\\beta$ which makes the diagram commute.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0649","source_file":"derived.tex","source_line":6776,"source_end_line":6794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6776-L6794","statement_sha256":"cc3a370b2823c6598e62a6263387986b35410993f7fd9ff2db3110ab3a10a5f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2568,"rank":2568,"depth":3,"x":304.214,"y":658.603,"cluster":"derived-categories"},{"id":"stacks:064A","tag":"064A","title":"Projective resolutions · Lemma 064A","summary":"Let A be an abelian category. Consider a solid diagram xymatrix K^bullet & L^bullet ar[l]^α P^bullet ar[u] ar@-->[ru]_β_i where P^bullet is bounded above and consists of projective objects, and α is a quasi-isomorphism. Any two morphisms β_1, β_2 making the diagram commute up to homotopy are homotopic.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Consider a solid diagram\n$$\n\\xymatrix{\nK^\\bullet & L^\\bullet \\ar[l]^\\alpha \\\\\nP^\\bullet \\ar[u] \\ar@{-->}[ru]_{\\beta_i}\n}\n$$\nwhere $P^\\bullet$ is bounded above and consists of projective\nobjects, and $\\alpha$ is a quasi-isomorphism.\nAny two morphisms $\\beta_1, \\beta_2$ making the diagram commute\nup to homotopy are homotopic.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064A","source_file":"derived.tex","source_line":6801,"source_end_line":6814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6801-L6814","statement_sha256":"502bec27ac463be2bda999fe181a85a8fb754787422b3950929dd567afe1a237","origin":"The Stacks Project","memory_eligible":false,"source_rank":2569,"rank":2569,"depth":3,"x":192.347,"y":738.099,"cluster":"derived-categories"},{"id":"stacks:064B","tag":"064B","title":"Projective resolutions · Lemma 064B","summary":"Let A be an abelian category. Let P^bullet be bounded above complex consisting of projective objects. Let L^bullet ∈ K(A). Then Mor_K(A)(P^bullet, L^bullet) = Mor_D(A)(P^bullet, L^bullet).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $P^\\bullet$ be bounded above complex consisting of projective\nobjects. Let $L^\\bullet \\in K(\\mathcal{A})$. Then\n$$\n\\Mor_{K(\\mathcal{A})}(P^\\bullet, L^\\bullet)\n=\n\\Mor_{D(\\mathcal{A})}(P^\\bullet, L^\\bullet).\n$$","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064B","source_file":"derived.tex","source_line":6821,"source_end_line":6831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6821-L6831","statement_sha256":"6058236294123046f31783a51d83b0a1d0a1bc3285f3a8d79e79af174d4d2075","origin":"The Stacks Project","memory_eligible":false,"source_rank":2570,"rank":2570,"depth":4,"x":210.975,"y":615.561,"cluster":"derived-categories"},{"id":"stacks:064C","tag":"064C","title":"Projective resolutions · Lemma 064C","summary":"Let A be an abelian category. Assume A has enough projectives. For any short exact sequence 0 → A^bullet → B^bullet → C^bullet → 0 of Comp^+(A) there exists a commutative diagram in Comp^+(A) xymatrix 0 ar[r] & P_1^bullet ar[r] ar[d] & P_2^bullet ar[r] ar[d] & P_3^bullet ar[r] ar[d] & 0 0 ar[r] & A^bullet ar[r] & B^bullet ar[r] & C^bullet ar[r] & 0 where the vertical arrows are projective resolutions and the rows are short exact sequences of complexes. In fact, given any…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has enough projectives.\nFor any short exact sequence\n$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$\nof $\\text{Comp}^{+}(\\mathcal{A})$ there exists a\ncommutative diagram in $\\text{Comp}^{+}(\\mathcal{A})$\n$$\n\\xymatrix{\n0 \\ar[r] &\nP_1^\\bullet \\ar[r] \\ar[d] &\nP_2^\\bullet \\ar[r] \\ar[d] &\nP_3^\\bullet \\ar[r] \\ar[d] &\n0 \\\\\n0 \\ar[r] &\nA^\\bullet \\ar[r] &\nB^\\bullet \\ar[r] &\nC^\\bullet \\ar[r] &\n0\n}\n$$\nwhere the vertical arrows are projective resolutions and\nthe rows are short exact sequences of complexes.\nIn fact, given any projective resolution $P^\\bullet \\to C^\\bullet$\nwe may assume $P_3^\\bullet = P^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064C","source_file":"derived.tex","source_line":6838,"source_end_line":6864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6838-L6864","statement_sha256":"1f141e13a9ee78273ba95bacd9799e4f2ea7a3a19e549c6799e0b274354afd9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2571,"rank":2571,"depth":8,"x":296.084,"y":716.852,"cluster":"derived-categories"},{"id":"stacks:064D","tag":"064D","title":"Projective resolutions · Lemma 064D","summary":"Let A be an abelian category. Let P^bullet, K^bullet be complexes. Let n ∈ Z. Assume that • P^bullet is a bounded complex consisting of projective objects, • P^i = 0 for i < n, and • H^i(K^bullet) = 0 for i ≥ n. Then Hom_K(A)(P^bullet, K^bullet) = Hom_D(A)(P^bullet, K^bullet) = 0.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $P^\\bullet$, $K^\\bullet$ be complexes.\nLet $n \\in \\mathbf{Z}$. Assume that\n\\begin{enumerate}\n\\item $P^\\bullet$ is a bounded complex consisting of projective\nobjects,\n\\item $P^i = 0$ for $i < n$, and\n\\item $H^i(K^\\bullet) = 0$ for $i \\geq n$.\n\\end{enumerate}\nThen\n$\\Hom_{K(\\mathcal{A})}(P^\\bullet, K^\\bullet) =\n\\Hom_{D(\\mathcal{A})}(P^\\bullet, K^\\bullet) = 0$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064D","source_file":"derived.tex","source_line":6871,"source_end_line":6885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6871-L6885","statement_sha256":"1a83ed8fb8e1b98bec30be34289b75ddc8886cb5ee465aa442cbbd16bb209d58","origin":"The Stacks Project","memory_eligible":false,"source_rank":2572,"rank":2572,"depth":5,"x":151.356,"y":690.36,"cluster":"derived-categories"},{"id":"stacks:064E","tag":"064E","title":"Projective resolutions · Lemma 064E","summary":"Let A be an abelian category. Let β : P^bullet → L^bullet and α : E^bullet → L^bullet be maps of complexes. Let n ∈ Z. Assume • P^bullet is a bounded complex of projectives and P^i = 0 for i < n, • H^i(α) is an isomorphism for i > n and surjective for i = n. Then there exists a map of complexes γ : P^bullet → E^bullet such that α ∘ γ and β are homotopic.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\beta : P^\\bullet \\to L^\\bullet$ and\n$\\alpha : E^\\bullet \\to L^\\bullet$ be\nmaps of complexes. Let $n \\in \\mathbf{Z}$. Assume\n\\begin{enumerate}\n\\item $P^\\bullet$ is a bounded complex of projectives and\n$P^i = 0$ for $i < n$,\n\\item $H^i(\\alpha)$ is an isomorphism for $i > n$ and surjective\nfor $i = n$.\n\\end{enumerate}\nThen there exists a map of complexes $\\gamma : P^\\bullet \\to E^\\bullet$\nsuch that $\\alpha \\circ \\gamma$ and $\\beta$ are homotopic.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Projective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064E","source_file":"derived.tex","source_line":6903,"source_end_line":6917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6903-L6917","statement_sha256":"39fc13033e8976383ea9603b63c6d992b113b96fcfe388c4fcf5705b403920eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2573,"rank":2573,"depth":6,"x":279.835,"y":627.552,"cluster":"derived-categories"},{"id":"stacks:05TH","tag":"05TH","title":"Right derived functors and injective resolutions · Lemma 05TH","summary":"Let A be an abelian category. Let I ∈ Ob(A) be an injective object. Let I^bullet be a bounded below complex of injectives in A. • I^bullet computes RF relative to Qis^+(A) for any exact functor F : K^+(A) → D into any triangulated category D. • I is right acyclic for any additive functor F : A → B into any abelian category B.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $I \\in \\Ob(\\mathcal{A})$ be an injective object.\nLet $I^\\bullet$ be a bounded below complex of injectives in $\\mathcal{A}$.\n\\begin{enumerate}\n\\item $I^\\bullet$ computes $RF$ relative to $\\text{Qis}^{+}(\\mathcal{A})$\nfor any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto any triangulated category $\\mathcal{D}$.\n\\item $I$ is right acyclic for any additive functor\n$F : \\mathcal{A} \\to \\mathcal{B}$ into any abelian category $\\mathcal{B}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Right derived functors and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TH","source_file":"derived.tex","source_line":6947,"source_end_line":6959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6947-L6959","statement_sha256":"90d93a2f4db18b4df529d7675b4ca7d8a5526ee202587ee837226df2e1aa2751","origin":"The Stacks Project","memory_eligible":false,"source_rank":2574,"rank":2574,"depth":3,"x":235.449,"y":747.187,"cluster":"derived-categories"},{"id":"stacks:05TI","tag":"05TI","title":"Right derived functors and injective resolutions · Lemma 05TI","summary":"Let A be an abelian category with enough injectives. • For any exact functor F : K^+(A) → D into a triangulated category D the right derived functor RF : D^+(A) → D is everywhere defined. • For any additive functor F : A → B into an abelian category B the right derived functor RF : D^+(A) → D^+(B) is everywhere defined.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\n\\begin{enumerate}\n\\item For any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto a triangulated category $\\mathcal{D}$ the right derived\nfunctor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D}\n$$\nis everywhere defined.\n\\item For any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into an\nabelian category $\\mathcal{B}$ the right derived functor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})\n$$\nis everywhere defined.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Right derived functors and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TI","source_file":"derived.tex","source_line":6979,"source_end_line":6997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L6979-L6997","statement_sha256":"5c403810f7214c439ac83bba7fe680a14cdc81dfcecd3619b31617a642c3aba6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2575,"rank":2575,"depth":15,"x":171.75,"y":633.388,"cluster":"derived-categories"},{"id":"stacks:0159","tag":"0159","title":"Right derived functors and injective resolutions · Lemma 0159","summary":"Let A be an abelian category with enough injectives. Let F : A → B be an additive functor. • The functor RF is an exact functor D^+(A) → D^+(B). • The functor RF induces an exact functor K^+(A) → D^+(B). • The functor RF induces a δ-functor Comp^+(A) → D^+(B). • The functor RF induces a δ-functor A → D^+(B).","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor.\n\\begin{enumerate}\n\\item The functor $RF$ is an exact functor\n$D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$.\n\\item The functor $RF$ induces an exact functor\n$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$.\n\\item The functor $RF$ induces a $\\delta$-functor\n$\\text{Comp}^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$.\n\\item The functor $RF$ induces a $\\delta$-functor\n$\\mathcal{A} \\to D^{+}(\\mathcal{B})$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Right derived functors and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0159","source_file":"derived.tex","source_line":7011,"source_end_line":7025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7011-L7025","statement_sha256":"15ad0181d83a1dc3681ba5b884f77bead041673d84a7f8c688f8ab85848cb397","origin":"The Stacks Project","memory_eligible":false,"source_rank":2576,"rank":2576,"depth":16,"x":310.715,"y":681.323,"cluster":"derived-categories"},{"id":"stacks:015B","tag":"015B","title":"Right derived functors and injective resolutions · Lemma 015B","summary":"Let A be an abelian category with enough injectives. Let F : A → B be a left exact functor. • For any short exact sequence 0 → A^bullet → B^bullet → C^bullet → 0 of complexes in Comp^+(A) there is an associated long exact sequence … → H^i(RF(A^bullet)) → H^i(RF(B^bullet)) → H^i(RF(C^bullet)) → H^i + 1(RF(A^bullet)) → … • The functors R^iF : A → B are zero for i < 0. Also R^0F = F : A → B. • We have R^iF(I) = 0 for i > 0 and I injective. • The sequence (R^iF, δ) forms a…","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor.\n\\begin{enumerate}\n\\item For any short exact sequence\n$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$\nof complexes in $\\text{Comp}^{+}(\\mathcal{A})$ there\nis an associated long exact sequence\n$$\n\\ldots \\to\nH^i(RF(A^\\bullet)) \\to\nH^i(RF(B^\\bullet)) \\to\nH^i(RF(C^\\bullet)) \\to\nH^{i + 1}(RF(A^\\bullet)) \\to \\ldots\n$$\n\\item The functors $R^iF : \\mathcal{A} \\to \\mathcal{B}$\nare zero for $i < 0$. Also $R^0F = F : \\mathcal{A} \\to \\mathcal{B}$.\n\\item We have $R^iF(I) = 0$ for $i > 0$ and $I$ injective.\n\\item The sequence $(R^iF, \\delta)$ forms a universal $\\delta$-functor (see\nHomology, Definition \\ref{homology-definition-universal-delta-functor})\nfrom $\\mathcal{A}$ to $\\mathcal{B}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Right derived functors and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015B","source_file":"derived.tex","source_line":7047,"source_end_line":7070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7047-L7070","statement_sha256":"e027fc1fa3991ed1812b1b2ea4cbe295bc07c9c610b240d4deaec0840e9be18e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2577,"rank":2577,"depth":18,"x":169.21,"y":724.978,"cluster":"derived-categories"},{"id":"stacks:015H","tag":"015H","title":"Cartan-Eilenberg resolutions · Definition 015H","summary":"Let A be an abelian category. Let K^bullet be a bounded below complex. A Cartan-Eilenberg resolution of K^bullet is given by a double complex I^bullet, bullet and a morphism of complexes ε : K^bullet → I^bullet, 0 with the following properties: • There exists a i ll 0 such that I^p, q = 0 for all p < i and all q. • We have I^p, q = 0 if q < 0. • The complex I^p, bullet is an injective resolution of K^p. • The complex Ker(d_1^p, bullet) is an injective resolution of…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet$ be a bounded below complex.\nA {\\it Cartan-Eilenberg resolution} of $K^\\bullet$\nis given by a double complex $I^{\\bullet, \\bullet}$\nand a morphism of complexes $\\epsilon : K^\\bullet \\to I^{\\bullet, 0}$\nwith the following properties:\n\\begin{enumerate}\n\\item There exists a $i \\ll 0$ such that $I^{p, q} = 0$ for all $p < i$\nand all $q$.\n\\item We have $I^{p, q} = 0$ if $q < 0$.\n\\item The complex $I^{p, \\bullet}$ is an injective resolution of $K^p$.\n\\item The complex $\\Ker(d_1^{p, \\bullet})$ is an injective resolution\nof $\\Ker(d_K^p)$.\n\\item The complex $\\Im(d_1^{p, \\bullet})$ is an injective resolution\nof $\\Im(d_K^p)$.\n\\item The complex $H^p_I(I^{\\bullet, \\bullet})$ is an injective resolution\nof $H^p(K^\\bullet)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cartan-Eilenberg resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015H","source_file":"derived.tex","source_line":7104,"source_end_line":7124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7104-L7124","statement_sha256":"a3cb525f7713d18b4ea90910a00ad067f23aaa50ec45cb99d77d84146e7f1388","origin":"The Stacks Project","memory_eligible":false,"source_rank":2578,"rank":2578,"depth":0,"x":238.685,"y":612.11,"cluster":"derived-categories"},{"id":"stacks:015I","tag":"015I","title":"Cartan-Eilenberg resolutions · Lemma 015I","summary":"Let A be an abelian category with enough injectives. Let K^bullet be a bounded below complex. There exists a Cartan-Eilenberg resolution of K^bullet.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nLet $K^\\bullet$ be a bounded below complex.\nThere exists a Cartan-Eilenberg resolution of $K^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cartan-Eilenberg resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015I","source_file":"derived.tex","source_line":7126,"source_end_line":7131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7126-L7131","statement_sha256":"5c2c8b4de05f1c3257e1a9dd11d4d647d233c90ca55ca310cb446862731cae41","origin":"The Stacks Project","memory_eligible":false,"source_rank":2579,"rank":2579,"depth":8,"x":278.354,"y":735.176,"cluster":"derived-categories"},{"id":"stacks:015J","tag":"015J","title":"Cartan-Eilenberg resolutions · Lemma 015J","summary":"Let F : A → B be a left exact functor of abelian categories. Let K^bullet be a bounded below complex of A. Let I^bullet, bullet be a Cartan-Eilenberg resolution for K^bullet. The spectral sequences ('E_r, 'd_r)_r ≥ 0 and (\"E_r, \"d_r)_r ≥ 0 associated to the double complex F(I^bullet, bullet) satisfy the relations 'E_1^p, q = R^qF(K^p) and \"E_2^p, q = R^pF(H^q(K^bullet)) Moreover, these spectral sequences are bounded, converge to H^*(RF(K^bullet)), and the associated…","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor of\nabelian categories.\nLet $K^\\bullet$ be a bounded below complex of $\\mathcal{A}$.\nLet $I^{\\bullet, \\bullet}$ be a Cartan-Eilenberg resolution\nfor $K^\\bullet$. The spectral sequences\n$({}'E_r, {}'d_r)_{r \\geq 0}$ and $({}''E_r, {}''d_r)_{r \\geq 0}$\nassociated to the double complex $F(I^{\\bullet, \\bullet})$\nsatisfy the relations\n$$\n{}'E_1^{p, q} = R^qF(K^p)\n\\quad\n\\text{and}\n\\quad\n{}''E_2^{p, q} = R^pF(H^q(K^\\bullet))\n$$\nMoreover, these spectral sequences are bounded, converge to\n$H^*(RF(K^\\bullet))$, and the associated induced filtrations on\n$H^n(RF(K^\\bullet))$ are finite.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Cartan-Eilenberg resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015J","source_file":"derived.tex","source_line":7176,"source_end_line":7196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7176-L7196","statement_sha256":"b0698d22eba0d4c62ef9bc7be4562958846a27e6f73b15b48e01e03c3177798b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2580,"rank":2580,"depth":6,"x":149.704,"y":666.707,"cluster":"derived-categories"},{"id":"stacks:015M","tag":"015M","title":"Composition of right derived functors · Lemma 015M","summary":"Let A, B, C be abelian categories. Let F : A → B and G : B → C be left exact functors. Assume A, B have enough injectives. The following are equivalent • F(I) is right acyclic for G for each injective object I of A, and • the canonical map t : R(G ∘ F) → RG ∘ RF. is isomorphism of functors from D^+(A) to D^+(C).","statement_latex":"Let $\\mathcal{A}, \\mathcal{B}, \\mathcal{C}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{C}$\nbe left exact functors. Assume $\\mathcal{A}$, $\\mathcal{B}$ have\nenough injectives. The following are equivalent\n\\begin{enumerate}\n\\item $F(I)$ is right acyclic for $G$ for each injective object $I$\nof $\\mathcal{A}$, and\n\\item the canonical map\n$$\nt : R(G \\circ F) \\longrightarrow RG \\circ RF.\n$$\nis isomorphism of functors from $D^{+}(\\mathcal{A})$ to $D^{+}(\\mathcal{C})$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Composition of right derived functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015M","source_file":"derived.tex","source_line":7292,"source_end_line":7307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7292-L7307","statement_sha256":"f1e2073b75345f0b7554a01d6f46ce23d7ba91dc28d6c9a3472e71fededf2a6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2581,"rank":2581,"depth":11,"x":300.132,"y":644.121,"cluster":"derived-categories"},{"id":"stacks:015N","tag":"015N","title":"Grothendieck spectral sequence · Lemma 015N","summary":"With assumptions as in Lemma [Tag 015M] and assuming the equivalent conditions (1) and (2) hold. Let X be an object of D^+(A). There exists a spectral sequence (E_r, d_r)_r ≥ 0 consisting of bigraded objects E_r of C with d_r of bidegree (r, - r + 1) and with E_2^p, q = R^pG(H^q(RF(X))) Moreover, this spectral sequence is bounded, converges to H^*(R(G ∘ F)(X)), and induces a finite filtration on each H^n(R(G ∘ F)(X)).","statement_latex":"With assumptions as in Lemma \\ref{lemma-compose-derived-functors}\nand assuming the equivalent conditions (1) and (2) hold.\nLet $X$ be an object of $D^{+}(\\mathcal{A})$.\nThere exists a spectral sequence $(E_r, d_r)_{r \\geq 0}$\nconsisting of bigraded objects $E_r$ of $\\mathcal{C}$ with\n$d_r$ of bidegree $(r, - r + 1)$ and with\n$$\nE_2^{p, q} = R^pG(H^q(RF(X)))\n$$\nMoreover, this spectral sequence is bounded, converges to\n$H^*(R(G \\circ F)(X))$, and induces a finite filtration\non each $H^n(R(G \\circ F)(X))$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Composition of right derived functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015N","source_file":"derived.tex","source_line":7328,"source_end_line":7342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7328-L7342","statement_sha256":"d0da29a722ba08c6338d2b765daf1ffc88bb8cb1b51fdc9f294f3fdb35ed7f77","origin":"The Stacks Project","memory_eligible":false,"source_rank":2582,"rank":2582,"depth":12,"x":207.063,"y":746.471,"cluster":"derived-categories"},{"id":"stacks:013V","tag":"013V","title":"Resolution functors · Proposition 013V","summary":"Let A be an abelian category. Assume A has enough injectives. Denote I ⊂ A the strictly full additive subcategory whose objects are the injective objects of A. The functor K^+(I) → D^+(A) is exact, fully faithful and essentially surjective, i.e., an equivalence of triangulated categories.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has enough injectives.\nDenote $\\mathcal{I} \\subset \\mathcal{A}$ the strictly full\nadditive subcategory whose objects are the injective objects of\n$\\mathcal{A}$.\nThe functor\n$$\nK^{+}(\\mathcal{I}) \\longrightarrow D^{+}(\\mathcal{A})\n$$\nis exact, fully faithful and essentially surjective, i.e.,\nan equivalence of triangulated categories.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Resolution functors","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013V","source_file":"derived.tex","source_line":7377,"source_end_line":7390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7377-L7390","statement_sha256":"7513201cbcd50e2d922f3469c6732a1306a8b02003789033f83a99d704813a62","origin":"The Stacks Project","memory_eligible":false,"source_rank":2583,"rank":2583,"depth":4,"x":193.339,"y":617.766,"cluster":"derived-categories"},{"id":"stacks:013W","tag":"013W","title":"Resolution functors · Definition 013W","summary":"Let A be an abelian category with enough injectives. A resolution functor for A is given by the following data: • for all K^bullet ∈ Ob(K^+(A)) a bounded below complex of injectives j(K^bullet), and • for all K^bullet ∈ Ob(K^+(A)) a quasi-isomorphism i_K^bullet : K^bullet → j(K^bullet).","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nA {\\it resolution functor}\\footnote{This is likely nonstandard terminology.}\nfor $\\mathcal{A}$ is given by the following data:\n\\begin{enumerate}\n\\item for all $K^\\bullet \\in \\Ob(K^{+}(\\mathcal{A}))$ a\nbounded below complex of injectives $j(K^\\bullet)$, and\n\\item for all $K^\\bullet \\in \\Ob(K^{+}(\\mathcal{A}))$ a\nquasi-isomorphism $i_{K^\\bullet} : K^\\bullet \\to j(K^\\bullet)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Resolution functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013W","source_file":"derived.tex","source_line":7407,"source_end_line":7418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7407-L7418","statement_sha256":"eedd3f01879818d842638b8dfe3fb5686e6748f6f9fbb82bdab0ad105c761b50","origin":"The Stacks Project","memory_eligible":false,"source_rank":2584,"rank":2584,"depth":0,"x":307.333,"y":705.17,"cluster":"derived-categories"},{"id":"stacks:05TJ","tag":"05TJ","title":"Resolution functors · Lemma 05TJ","summary":"Let A be an abelian category with enough injectives. Given a resolution functor (j, i) there is a unique way to turn j into a functor and i into a 2-isomorphism producing a 2-commutative diagram xymatrix K^+(A) ar[rd] ar[rr]_j & & K^+(I) ar[ld] & D^+(A) where I is the full additive subcategory of A consisting of injective objects.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nGiven a resolution functor $(j, i)$ there is a unique way to\nturn $j$ into a functor and $i$ into a $2$-isomorphism\nproducing a $2$-commutative diagram\n$$\n\\xymatrix{\nK^{+}(\\mathcal{A}) \\ar[rd] \\ar[rr]_j & & K^{+}(\\mathcal{I}) \\ar[ld] \\\\\n& D^{+}(\\mathcal{A})\n}\n$$\nwhere $\\mathcal{I}$ is the full additive subcategory of $\\mathcal{A}$\nconsisting of injective objects.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Resolution functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TJ","source_file":"derived.tex","source_line":7420,"source_end_line":7434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7420-L7434","statement_sha256":"07e53032e784b9d687f5b7ea59bfa273d9046d66088a93404a9fbbb7fa4e4e5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2585,"rank":2585,"depth":4,"x":152.484,"y":705.403,"cluster":"derived-categories"},{"id":"stacks:013X","tag":"013X","title":"Resolution functors · Lemma 013X","summary":"Let A be an abelian category. Assume A has enough injectives. Then a resolution functor j exists and is unique up to unique isomorphism of functors.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has enough injectives.\nThen a resolution functor $j$ exists and is\nunique up to unique isomorphism of functors.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Resolution functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/013X","source_file":"derived.tex","source_line":7459,"source_end_line":7465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7459-L7465","statement_sha256":"e65d6f32c0804f633b8d3627e54c60a9f2f62cb5dd2d5464fc8db743956ef3e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2586,"rank":2586,"depth":3,"x":266.85,"y":617.081,"cluster":"derived-categories"},{"id":"stacks:014W","tag":"014W","title":"Resolution functors · Lemma 014W","summary":"Let A be an abelian category with enough injectives. Any resolution functor j : K^+(A) → K^+(I) is exact.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nAny resolution functor\n$j : K^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{I})$\nis exact.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Resolution functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/014W","source_file":"derived.tex","source_line":7477,"source_end_line":7483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7477-L7483","statement_sha256":"e3d57d08ca86037e304b8a916f518ff255173daaef5458295bdb2bc5aab1816d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2587,"rank":2587,"depth":11,"x":253.5,"y":747.521,"cluster":"derived-categories"},{"id":"stacks:05TK","tag":"05TK","title":"Resolution functors · Lemma 05TK","summary":"Let A be an abelian category which has enough injectives. Let j be a resolution functor. Write Q : K^+(A) → D^+(A) for the natural functor. Then j = j' ∘ Q for a unique functor j' : D^+(A) → K^+(I) which is quasi-inverse to the canonical functor K^+(I) → D^+(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category which has enough injectives.\nLet $j$ be a resolution functor. Write\n$Q : K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{A})$ for the natural functor.\nThen $j = j' \\circ Q$ for a unique\nfunctor $j' : D^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{I})$ which\nis quasi-inverse to the canonical functor\n$K^{+}(\\mathcal{I}) \\to D^{+}(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Resolution functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TK","source_file":"derived.tex","source_line":7537,"source_end_line":7546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7537-L7546","statement_sha256":"d609ac45145bc95769006c15945707ea89e41b8ad0c60d8b1041bb0505be1d16","origin":"The Stacks Project","memory_eligible":false,"source_rank":2588,"rank":2588,"depth":16,"x":158.144,"y":643.429,"cluster":"derived-categories"},{"id":"stacks:0141","tag":"0141","title":"Functorial injective embeddings and resolution functors · Lemma 0141","summary":"Let A be an abelian category. Assume A has functorial injective embeddings, see Homology, Definition [Tag 0139]. • There exists a functor inj : Comp^+(A) → InjRes(A) such that s ∘ inj = id. • For any functor inj : Comp^+(A) → InjRes(A) such that s ∘ inj = id we obtain a resolution functor, see Definition [Tag 013W].","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has functorial injective embeddings, see\nHomology, Definition \\ref{homology-definition-functorial-injective-embedding}.\n\\begin{enumerate}\n\\item There exists a functor\n$inj : \\text{Comp}^{+}(\\mathcal{A}) \\to \\text{InjRes}(\\mathcal{A})$\nsuch that $s \\circ inj = \\text{id}$.\n\\item For any functor\n$inj : \\text{Comp}^{+}(\\mathcal{A}) \\to \\text{InjRes}(\\mathcal{A})$\nsuch that $s \\circ inj = \\text{id}$ we obtain a resolution functor, see\nDefinition \\ref{definition-localization-functor}.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Functorial injective embeddings and resolution functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0141","source_file":"derived.tex","source_line":7622,"source_end_line":7636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7622-L7636","statement_sha256":"5ecdece0bb767d256ca0f2e3cf07b8f24628064ed8cf82e8c7ad184f087eb575","origin":"The Stacks Project","memory_eligible":false,"source_rank":2589,"rank":2589,"depth":6,"x":312.655,"y":666.15,"cluster":"derived-categories"},{"id":"stacks:05TN","tag":"05TN","title":"Right derived functors via resolution functors · Lemma 05TN","summary":"Let A be an abelian category with enough injectives Let F : A → B be an additive functor into an abelian category. Let (i, j) be a resolution functor, see Definition [Tag 013W]. The right derived functor RF of F fits into the following 2-commutative diagram xymatrix D^+(A) ar[rd]_RF ar[rr]^j' & & K^+(I) ar[ld]^F & D^+(B) where j' is the functor from Lemma [Tag 05TK].","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor into\nan abelian category. Let $(i, j)$ be a resolution functor, see\nDefinition \\ref{definition-localization-functor}.\nThe right derived functor $RF$ of $F$ fits into the following\n$2$-commutative diagram\n$$\n\\xymatrix{\nD^{+}(\\mathcal{A}) \\ar[rd]_{RF} \\ar[rr]^{j'} & &\nK^{+}(\\mathcal{I}) \\ar[ld]^F \\\\\n& D^{+}(\\mathcal{B})\n}\n$$\nwhere $j'$ is the functor from\nLemma \\ref{lemma-resolution-functor-quasi-inverse}.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Right derived functors via resolution functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TN","source_file":"derived.tex","source_line":7778,"source_end_line":7795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7778-L7795","statement_sha256":"42da19302d6acea3128d057c0ea075ab1ae4c8b738bf8f32c1b6c0e351609926","origin":"The Stacks Project","memory_eligible":false,"source_rank":2590,"rank":2590,"depth":17,"x":180.032,"y":737.293,"cluster":"derived-categories"},{"id":"stacks:015P","tag":"015P","title":"Filtered derived category and injective resolutions · Definition 015P","summary":"Let A be an abelian category. We say an object I of Fil^f(A) is filtered injective if each gr^p(I) is an injective object of A.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nWe say an object $I$ of $\\text{Fil}^f(\\mathcal{A})$\nis {\\it filtered injective} if each $\\text{gr}^p(I)$ is\nan injective object of $\\mathcal{A}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015P","source_file":"derived.tex","source_line":7843,"source_end_line":7849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7843-L7849","statement_sha256":"27799adafe9797c470260c8accd900b45ee9135dab70512d39ccfb624e27e185","origin":"The Stacks Project","memory_eligible":false,"source_rank":2591,"rank":2591,"depth":0,"x":220.743,"y":609.179,"cluster":"derived-categories"},{"id":"stacks:05TP","tag":"05TP","title":"Filtered derived category and injective resolutions · Lemma 05TP","summary":"Let A be an abelian category. An object I of Fil^f(A) is filtered injective if and only if there exist a ≤ b, injective objects I_n, a ≤ n ≤ b of A and an isomorphism I ≅ bigoplus_a ≤ n ≤ b I_n such that F^pI = bigoplus_n ≥ p I_n.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAn object $I$ of $\\text{Fil}^f(\\mathcal{A})$ is filtered injective\nif and only if\nthere exist $a \\leq b$, injective objects $I_n$, $a \\leq n \\leq b$\nof $\\mathcal{A}$ and an isomorphism $I \\cong \\bigoplus_{a \\leq n \\leq b} I_n$\nsuch that $F^pI = \\bigoplus_{n \\geq p} I_n$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TP","source_file":"derived.tex","source_line":7851,"source_end_line":7859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7851-L7859","statement_sha256":"0b9ba4ed90a9a7a992c8c3105eb346e1c6643a8c894248913beb8afd69d0d80a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2592,"rank":2592,"depth":0,"x":293.976,"y":727.116,"cluster":"derived-categories"},{"id":"stacks:05TQ","tag":"05TQ","title":"Filtered derived category and injective resolutions · Lemma 05TQ","summary":"Let A be an abelian category. Any strict monomorphism u : I → A of Fil^f(A) where I is a filtered injective object is a split injection.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAny strict monomorphism $u : I \\to A$ of $\\text{Fil}^f(\\mathcal{A})$\nwhere $I$ is a filtered injective object is a split injection.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TQ","source_file":"derived.tex","source_line":7866,"source_end_line":7871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7866-L7871","statement_sha256":"4aceba803a75a0e207de45c9a0f62b04aefbb9066652618b920ffd661985717f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2593,"rank":2593,"depth":4,"x":144.673,"y":681.564,"cluster":"derived-categories"},{"id":"stacks:05TR","tag":"05TR","title":"Filtered derived category and injective resolutions · Lemma 05TR","summary":"Let A be an abelian category. Let u : A → B be a strict monomorphism of Fil^f(A) and f : A → I a morphism from A into a filtered injective object in Fil^f(A). Then there exists a morphism g : B → I such that f = g ∘ u.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $u : A \\to B$ be a strict monomorphism\nof $\\text{Fil}^f(\\mathcal{A})$\nand $f : A \\to I$ a morphism from $A$ into a filtered injective object\nin $\\text{Fil}^f(\\mathcal{A})$.\nThen there exists a morphism $g : B \\to I$ such that $f = g \\circ u$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TR","source_file":"derived.tex","source_line":7895,"source_end_line":7903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7895-L7903","statement_sha256":"21ffb3132edd8af3760e0459bef3df82215de38993eac02b1192c452c2dbdf44","origin":"The Stacks Project","memory_eligible":false,"source_rank":2594,"rank":2594,"depth":5,"x":291.85,"y":630.278,"cluster":"derived-categories"},{"id":"stacks:05TS","tag":"05TS","title":"Filtered derived category and injective resolutions · Lemma 05TS","summary":"Let A be an abelian category with enough injectives. For any object A of Fil^f(A) there exists a strict monomorphism A → I where I is a filtered injective object.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nFor any object $A$ of $\\text{Fil}^f(\\mathcal{A})$ there exists\na strict monomorphism $A \\to I$\nwhere $I$ is a filtered injective object.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TS","source_file":"derived.tex","source_line":7913,"source_end_line":7919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7913-L7919","statement_sha256":"d6ccc935168c4a773b4756198f56ad56b0cd732e6031f88eb7887ac080ebe445","origin":"The Stacks Project","memory_eligible":false,"source_rank":2595,"rank":2595,"depth":0,"x":224.36,"y":751.978,"cluster":"derived-categories"},{"id":"stacks:05TT","tag":"05TT","title":"Filtered derived category and injective resolutions · Lemma 05TT","summary":"Let A be an abelian category with enough injectives. For any object A of Fil^f(A) there exists a filtered quasi-isomorphism A[0] → I^bullet where I^bullet is a complex of filtered injective objects with I^n = 0 for n < 0.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nFor any object $A$ of $\\text{Fil}^f(\\mathcal{A})$ there exists\na filtered quasi-isomorphism $A[0] \\to I^\\bullet$\nwhere $I^\\bullet$ is a complex of filtered injective objects\nwith $I^n = 0$ for $n < 0$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TT","source_file":"derived.tex","source_line":7931,"source_end_line":7938,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7931-L7938","statement_sha256":"cbff5b01e760765b2080bf4877178d47460833941171abb68037b5359006de09","origin":"The Stacks Project","memory_eligible":false,"source_rank":2596,"rank":2596,"depth":4,"x":176.113,"y":623.553,"cluster":"derived-categories"},{"id":"stacks:05TU","tag":"05TU","title":"Filtered derived category and injective resolutions · Lemma 05TU","summary":"Let A be an abelian category with enough injectives. Let f : A → B be a morphism of Fil^f(A). Given filtered quasi-isomorphisms A[0] → I^bullet and B[0] → J^bullet where I^bullet, J^bullet are complexes of filtered injective objects with I^n = J^n = 0 for n < 0, then there exists a commutative diagram xymatrix A[0] ar[r] ar[d] & B[0] ar[d] I^bullet ar[r] & J^bullet","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nLet $f : A \\to B$ be a morphism of $\\text{Fil}^f(\\mathcal{A})$.\nGiven filtered quasi-isomorphisms $A[0] \\to I^\\bullet$ and\n$B[0] \\to J^\\bullet$ where $I^\\bullet, J^\\bullet$ are complexes of\nfiltered injective objects with $I^n = J^n = 0$ for $n < 0$, then\nthere exists a commutative diagram\n$$\n\\xymatrix{\nA[0] \\ar[r] \\ar[d] &\nB[0] \\ar[d] \\\\\nI^\\bullet \\ar[r] &\nJ^\\bullet\n}\n$$","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TU","source_file":"derived.tex","source_line":7959,"source_end_line":7975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L7959-L7975","statement_sha256":"ba78b1a062f7ee9533c38b590b98feeaa559d3d5f899641ff10abc5f93641f49","origin":"The Stacks Project","memory_eligible":false,"source_rank":2597,"rank":2597,"depth":6,"x":315.385,"y":691.079,"cluster":"derived-categories"},{"id":"stacks:05TV","tag":"05TV","title":"Filtered derived category and injective resolutions · Lemma 05TV","summary":"Let A be an abelian category with enough injectives. Let 0 → A → B → C → 0 be a short exact sequence in Fil^f(A). Given filtered quasi-isomorphisms A[0] → I^bullet and C[0] → J^bullet where I^bullet, J^bullet are complexes of filtered injective objects with I^n = J^n = 0 for n < 0, then there exists a commutative diagram xymatrix 0 ar[r] & A[0] ar[r] ar[d] & B[0] ar[r] ar[d] & C[0] ar[r] ar[d] & 0 0 ar[r] & I^bullet ar[r] & M^bullet ar[r] & J^bullet ar[r] & 0 where the…","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nLet $0 \\to A \\to B \\to C \\to 0$ be a short exact sequence in\n$\\text{Fil}^f(\\mathcal{A})$.\nGiven filtered quasi-isomorphisms $A[0] \\to I^\\bullet$ and\n$C[0] \\to J^\\bullet$ where $I^\\bullet, J^\\bullet$ are complexes of\nfiltered injective objects with $I^n = J^n = 0$ for $n < 0$, then\nthere exists a commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\nA[0] \\ar[r] \\ar[d] &\nB[0] \\ar[r] \\ar[d] &\nC[0] \\ar[r] \\ar[d] &\n0 \\\\\n0 \\ar[r] &\nI^\\bullet \\ar[r] &\nM^\\bullet \\ar[r] &\nJ^\\bullet \\ar[r] &\n0\n}\n$$\nwhere the lower row is a termwise split sequence of complexes.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TV","source_file":"derived.tex","source_line":8014,"source_end_line":8038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8014-L8038","statement_sha256":"e93f63c1cee3cc6f0a7e493ee9fb3fd78f53fbc7088782b044303380a1f22d90","origin":"The Stacks Project","memory_eligible":false,"source_rank":2598,"rank":2598,"depth":16,"x":157.914,"y":720.401,"cluster":"derived-categories"},{"id":"stacks:05TW","tag":"05TW","title":"Filtered derived category and injective resolutions · Lemma 05TW","summary":"Let A be an abelian category with enough injectives. For every K^bullet ∈ K^+(Fil^f(A)) there exists a filtered quasi-isomorphism K^bullet → I^bullet with I^bullet bounded below, each I^n a filtered injective object, and each K^n → I^n a strict monomorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nFor every $K^\\bullet \\in K^{+}(\\text{Fil}^f(\\mathcal{A}))$\nthere exists a filtered quasi-isomorphism $K^\\bullet \\to I^\\bullet$\nwith $I^\\bullet$ bounded below,\neach $I^n$ a filtered injective object, and\neach $K^n \\to I^n$ a strict monomorphism.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TW","source_file":"derived.tex","source_line":8095,"source_end_line":8103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8095-L8103","statement_sha256":"90e390a27c85adf708c31a0703459a29c5e1bbca027610f905e56c818cab7f1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2599,"rank":2599,"depth":17,"x":250.728,"y":609.095,"cluster":"derived-categories"},{"id":"stacks:05TX","tag":"05TX","title":"Filtered derived category and injective resolutions · Lemma 05TX","summary":"Let A be an abelian category. Let K^bullet, I^bullet ∈ K(Fil^f(A)). Assume K^bullet is filtered acyclic and I^bullet bounded below and consisting of filtered injective objects. Any morphism K^bullet → I^bullet is homotopic to zero: Hom_K(Fil^f(A))(K^bullet, I^bullet) = 0.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet, I^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$.\nAssume $K^\\bullet$ is filtered acyclic and\n$I^\\bullet$ bounded below and consisting of filtered injective objects.\nAny morphism $K^\\bullet \\to I^\\bullet$ is homotopic to zero:\n$\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(K^\\bullet, I^\\bullet) = 0$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TX","source_file":"derived.tex","source_line":8196,"source_end_line":8204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8196-L8204","statement_sha256":"12298178ada797e04206fa606f158e83db8b2e9fc5f1599752bd3233d6890b29","origin":"The Stacks Project","memory_eligible":false,"source_rank":2600,"rank":2600,"depth":6,"x":271.859,"y":744.235,"cluster":"derived-categories"},{"id":"stacks:05TY","tag":"05TY","title":"Filtered derived category and injective resolutions · Lemma 05TY","summary":"Let A be an abelian category. Let I^bullet ∈ K(Fil^f(A)) be a bounded below complex consisting of filtered injective objects. • Let α : K^bullet → L^bullet in K(Fil^f(A)) be a filtered quasi-isomorphism. Then the map Hom_K(Fil^f(A))(L^bullet, I^bullet) → Hom_K(Fil^f(A))(K^bullet, I^bullet) is bijective. • Let L^bullet ∈ K(Fil^f(A)). Then Hom_K(Fil^f(A))(L^bullet, I^bullet) = Hom_DF(A)(L^bullet, I^bullet).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $I^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$\nbe a bounded below complex consisting of\nfiltered injective objects.\n\\begin{enumerate}\n\\item Let $\\alpha : K^\\bullet \\to L^\\bullet$ in $K(\\text{Fil}^f(\\mathcal{A}))$\nbe a filtered quasi-isomorphism.\nThen the map\n$$\n\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(L^\\bullet, I^\\bullet)\n\\to\n\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(K^\\bullet, I^\\bullet)\n$$\nis bijective.\n\\item Let $L^\\bullet \\in K(\\text{Fil}^f(\\mathcal{A}))$. Then\n$$\n\\Hom_{K(\\text{Fil}^f(\\mathcal{A}))}(L^\\bullet, I^\\bullet)\n=\n\\Hom_{DF(\\mathcal{A})}(L^\\bullet, I^\\bullet).\n$$\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05TY","source_file":"derived.tex","source_line":8240,"source_end_line":8263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8240-L8263","statement_sha256":"b0270326a2c9d4921294940a745b7e25b6bcd611a458f5bd37af903f97dd4e7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2601,"rank":2601,"depth":16,"x":147.235,"y":656.316,"cluster":"derived-categories"},{"id":"stacks:015Q","tag":"015Q","title":"Filtered derived category and injective resolutions · Lemma 015Q","summary":"Let A be an abelian category with enough injectives. Let I^f ⊂ Fil^f(A) denote the strictly full additive subcategory whose objects are the filtered injective objects. The canonical functor K^+(I^f) → DF^+(A) is exact, fully faithful and essentially surjective, i.e., an equivalence of triangulated categories. Furthermore the diagrams xymatrix K^+(I^f) ar[d]_gr^p ar[r] & DF^+(A) ar[d]_gr^p K^+(I) ar[r] & D^+(A) xymatrix K^+(I^f) ar[d]^forget F ar[r] & DF^+(A) ar[d]^forget…","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough injectives.\nLet $\\mathcal{I}^f \\subset \\text{Fil}^f(\\mathcal{A})$\ndenote the strictly full additive subcategory whose objects are\nthe filtered injective objects. The canonical functor\n$$\nK^{+}(\\mathcal{I}^f)\n\\longrightarrow\nDF^{+}(\\mathcal{A})\n$$\nis exact, fully faithful and essentially surjective, i.e., an\nequivalence of triangulated categories. Furthermore the diagrams\n$$\n\\xymatrix{\nK^{+}(\\mathcal{I}^f) \\ar[d]_{\\text{gr}^p} \\ar[r] &\nDF^{+}(\\mathcal{A}) \\ar[d]_{\\text{gr}^p} \\\\\nK^{+}(\\mathcal{I}) \\ar[r] &\nD^{+}(\\mathcal{A})\n}\n\\quad\n\\xymatrix{\nK^{+}(\\mathcal{I}^f) \\ar[d]^{\\text{forget }F} \\ar[r] &\nDF^{+}(\\mathcal{A}) \\ar[d]^{\\text{forget }F} \\\\\nK^{+}(\\mathcal{I}) \\ar[r] &\nD^{+}(\\mathcal{A})\n}\n$$\nare commutative, where $\\mathcal{I} \\subset \\mathcal{A}$ is the\nstrictly full additive subcategory whose objects are\nthe injective objects.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015Q","source_file":"derived.tex","source_line":8308,"source_end_line":8339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8308-L8339","statement_sha256":"536c1c3952955f7230d2bad229f6896ef29e2aa7e4c269fe91a6801370e53678","origin":"The Stacks Project","memory_eligible":false,"source_rank":2602,"rank":2602,"depth":18,"x":310.31,"y":650.415,"cluster":"derived-categories"},{"id":"stacks:015W","tag":"015W","title":"Filtered derived category and injective resolutions · Lemma 015W","summary":"Let A, B be abelian categories. Let T : A → B be a left exact functor. Assume A has enough injectives. Let (K^bullet, F) be an object of Comp^+(Fil^f(A)). There exists a spectral sequence (E_r, d_r)_r≥ 0 consisting of bigraded objects E_r of B and d_r of bidegree (r, - r + 1) and with E_1^p, q = R^p + qT(gr^p(K^bullet)) Moreover, this spectral sequence is bounded, converges to R^*T(K^bullet), and induces a finite filtration on each R^nT(K^bullet). The construction of this…","statement_latex":"Let $\\mathcal{A}, \\mathcal{B}$ be abelian categories. Let\n$T : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor.\nAssume $\\mathcal{A}$ has enough injectives.\nLet $(K^\\bullet, F)$ be an object of\n$\\text{Comp}^{+}(\\text{Fil}^f(\\mathcal{A}))$.\nThere exists a spectral sequence $(E_r, d_r)_{r\\geq 0}$\nconsisting of bigraded objects $E_r$ of $\\mathcal{B}$\nand $d_r$ of bidegree $(r, - r + 1)$ and with\n$$\nE_1^{p, q} = R^{p + q}T(\\text{gr}^p(K^\\bullet))\n$$\nMoreover, this spectral sequence is bounded, converges\nto $R^*T(K^\\bullet)$, and induces a finite\nfiltration on each $R^nT(K^\\bullet)$. The construction\nof this spectral sequence is functorial in the object\n$K^\\bullet$ of $\\text{Comp}^{+}(\\text{Fil}^f(\\mathcal{A}))$\nand the terms $(E_r, d_r)$ for $r \\geq 1$ do not depend\non any choices.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Filtered derived category and injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/015W","source_file":"derived.tex","source_line":8483,"source_end_line":8503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8483-L8503","statement_sha256":"858308066c0aacd6c2897bb3c0f68040c950dd54bb7da26d5284ecf4c2087f3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2603,"rank":2603,"depth":18,"x":194.469,"y":747.581,"cluster":"derived-categories"},{"id":"stacks:06XQ","tag":"06XQ","title":"Ext groups · Definition 06XQ","summary":"Let A be an abelian category. Let i ∈ Z. Let X, Y be objects of D(A). The ith extension group of X by Y is the group Ext^i_A(X, Y) = Hom_D(A)(X, Y[i]) = Hom_D(A)(X[-i], Y). If A, B ∈ Ob(A) we set Ext^i_A(A, B) = Ext^i_A(A[0], B[0]).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $i \\in \\mathbf{Z}$. Let\n$X, Y$ be objects of $D(\\mathcal{A})$. The {\\it $i$th extension group}\nof $X$ by $Y$ is the group\n$$\n\\Ext^i_\\mathcal{A}(X, Y) =\n\\Hom_{D(\\mathcal{A})}(X, Y[i]) =\n\\Hom_{D(\\mathcal{A})}(X[-i], Y).\n$$\nIf $A, B \\in \\Ob(\\mathcal{A})$ we set\n$\\Ext^i_\\mathcal{A}(A, B) = \\text{Ext}^i_\\mathcal{A}(A[0], B[0])$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XQ","source_file":"derived.tex","source_line":8632,"source_end_line":8644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8632-L8644","statement_sha256":"361033f6bdb0181a35f7d6890803b1c347f9000e5382e40402732b117ed2e1d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2604,"rank":2604,"depth":0,"x":201.772,"y":609.804,"cluster":"derived-categories"},{"id":"stacks:06XR","tag":"06XR","title":"Ext groups · Lemma 06XR","summary":"Let A be an abelian category. Let X^bullet, Y^bullet ∈ Ob(K(A)). • Let Y^bullet → I^bullet be an injective resolution (Definition [Tag 013I]). Then Ext^i_A(X^bullet, Y^bullet) = Hom_K(A)(X^bullet, I^bullet[i]). • Let P^bullet → X^bullet be a projective resolution (Definition [Tag 0644]). Then Ext^i_A(X^bullet, Y^bullet) = Hom_K(A)(P^bullet[-i], Y^bullet).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $X^\\bullet, Y^\\bullet \\in \\Ob(K(\\mathcal{A}))$.\n\\begin{enumerate}\n\\item Let $Y^\\bullet \\to I^\\bullet$ be an injective resolution\n(Definition \\ref{definition-injective-resolution}). Then\n$$\n\\Ext^i_\\mathcal{A}(X^\\bullet, Y^\\bullet) =\n\\Hom_{K(\\mathcal{A})}(X^\\bullet, I^\\bullet[i]).\n$$\n\\item Let $P^\\bullet \\to X^\\bullet$ be a projective resolution\n(Definition \\ref{definition-projective-resolution}). Then\n$$\n\\Ext^i_\\mathcal{A}(X^\\bullet, Y^\\bullet) =\n\\Hom_{K(\\mathcal{A})}(P^\\bullet[-i], Y^\\bullet).\n$$\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XR","source_file":"derived.tex","source_line":8681,"source_end_line":8699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8681-L8699","statement_sha256":"e5f8e2917814affc164e38dd01dc5bba5aa61b396001b7609bfd9a16b2c370db","origin":"The Stacks Project","memory_eligible":false,"source_rank":2605,"rank":2605,"depth":5,"x":307.487,"y":715.847,"cluster":"derived-categories"},{"id":"stacks:06XS","tag":"06XS","title":"Ext groups · Lemma 06XS","summary":"Let A be an abelian category. • Let X, Y be objects of D(A). Given a, b ∈ Z such that H^i(X) = 0 for i > a and H^j(Y) = 0 for j < b, we have Ext^n_A(X, Y) = 0 for n < b - a and Ext^b - a_A(X, Y) = Hom_A(H^a(X), H^b(Y)) • Let A, B ∈ Ob(A). For i < 0 we have Ext^i_A(B, A) = 0. We have Ext^0_A(B, A) = Hom_A(B, A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item Let $X$, $Y$ be objects of $D(\\mathcal{A})$. Given $a, b \\in \\mathbf{Z}$\nsuch that $H^i(X) = 0$ for $i > a$ and $H^j(Y) = 0$\nfor $j < b$, we have $\\Ext^n_\\mathcal{A}(X, Y) = 0$ for\n$n < b - a$ and\n$$\n\\Ext^{b - a}_\\mathcal{A}(X, Y) = \\Hom_\\mathcal{A}(H^a(X), H^b(Y))\n$$\n\\item Let $A, B \\in \\Ob(\\mathcal{A})$.\nFor $i < 0$ we have $\\Ext^i_\\mathcal{A}(B, A) = 0$.\nWe have $\\Ext^0_\\mathcal{A}(B, A) = \\Hom_\\mathcal{A}(B, A)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XS","source_file":"derived.tex","source_line":8712,"source_end_line":8727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8712-L8727","statement_sha256":"3d18c450dbc2cdc08d41a699f2c14411cde43a3c9611930933bdc15609034477","origin":"The Stacks Project","memory_eligible":false,"source_rank":2606,"rank":2606,"depth":0,"x":143.79,"y":697.585,"cluster":"derived-categories"},{"id":"stacks:06XT","tag":"06XT","title":"Ext groups · Definition 06XT","summary":"Let A be an abelian category. Let A, B ∈ Ob(A). For i ≥ 1 a degree i Yoneda extension of B by A is an exact sequence E : 0 → A → Z_i - 1 → Z_i - 2 → … → Z_0 → B → 0 in A. We say two Yoneda extensions E and E' of the same degree are equivalent if there exists a commutative diagram xymatrix 0 ar[r] & A ar[r] & Z_i - 1 ar[r] & … ar[r] & Z_0 ar[r] & B ar[r] & 0 0 ar[r] & A ar[r] ar[u]^id ar[d]_id & Z\"_i - 1 ar[r] ar[u] ar[d] & … ar[r] & Z\"_0 ar[r] ar[u] ar[d] & B ar[r]…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $A, B \\in \\Ob(\\mathcal{A})$.\nFor $i \\geq 1$ a degree $i$ {\\it Yoneda extension} of $B$ by $A$ is an\nexact sequence\n$$\nE : 0 \\to A \\to Z_{i - 1} \\to Z_{i - 2} \\to \\ldots \\to Z_0 \\to B \\to 0\n$$\nin $\\mathcal{A}$. We say two Yoneda extensions $E$ and $E'$ of the same degree\nare {\\it equivalent} if there exists a commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] & A \\ar[r] & Z_{i - 1} \\ar[r] & \\ldots \\ar[r] &\nZ_0 \\ar[r] & B \\ar[r] & 0 \\\\\n0 \\ar[r] &\nA \\ar[r] \\ar[u]^{\\text{id}} \\ar[d]_{\\text{id}} &\nZ''_{i - 1} \\ar[r] \\ar[u] \\ar[d] &\n\\ldots \\ar[r] &\nZ''_0 \\ar[r] \\ar[u] \\ar[d] &\nB \\ar[r] \\ar[u]_{\\text{id}} \\ar[d]^{\\text{id}} & 0 \\\\\n0 \\ar[r] & A \\ar[r] & Z'_{i - 1} \\ar[r] & \\ldots \\ar[r] &\nZ'_0 \\ar[r] & B \\ar[r] & 0\n}\n$$\nwhere the middle row is a Yoneda extension as well.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XT","source_file":"derived.tex","source_line":8774,"source_end_line":8799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8774-L8799","statement_sha256":"3c0a716e4a789588048159c356c52c3f1cd85889edede8bd68b78343fb9a4fae","origin":"The Stacks Project","memory_eligible":false,"source_rank":2607,"rank":2607,"depth":0,"x":279.569,"y":617.939,"cluster":"derived-categories"},{"id":"stacks:06XU","tag":"06XU","title":"Ext groups · Lemma 06XU","summary":"Let A be an abelian category with objects A, B and let i ≥ 1. Any element in Ext^i_A(B, A) is δ(E) for some degree i Yoneda extension of B by A. Given two Yoneda extensions E, E' of the same degree then E is equivalent to E' if and only if δ(E) = δ(E').","statement_latex":"Let $\\mathcal{A}$ be an abelian category with objects $A$, $B$ and\nlet $i \\geq 1$.\nAny element in $\\Ext^i_\\mathcal{A}(B, A)$ is $\\delta(E)$\nfor some degree $i$ Yoneda extension of $B$ by $A$.\nGiven two Yoneda extensions $E$, $E'$ of the same degree\nthen $E$ is equivalent to $E'$ if and only if $\\delta(E) = \\delta(E')$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XU","source_file":"derived.tex","source_line":8830,"source_end_line":8838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8830-L8838","statement_sha256":"e7cbad3c6e8e68d3d81af86fa29fbfa2ac1093bbcfaaca14eca4662f57c089fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2608,"rank":2608,"depth":0,"x":243.394,"y":754.098,"cluster":"derived-categories"},{"id":"stacks:06XV","tag":"06XV","title":"Ext groups · Lemma 06XV","summary":"Let A be an abelian category. Let A, B be objects of A. Then Ext^1_A(B, A) is the group Ext_A(B, A) constructed in Homology, Definition [Tag 010K].","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $A$, $B$ be objects\nof $\\mathcal{A}$. Then $\\Ext^1_\\mathcal{A}(B, A)$ is\nthe group $\\Ext_\\mathcal{A}(B, A)$ constructed in\nHomology, Definition \\ref{homology-definition-ext-group}.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XV","source_file":"derived.tex","source_line":8886,"source_end_line":8892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8886-L8892","statement_sha256":"e2a2d2594ccbb66b44fd2f5f88185d9625bc03fe067ff0eda007859c68043bf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2609,"rank":2609,"depth":1,"x":160.341,"y":632.829,"cluster":"derived-categories"},{"id":"stacks:0GSM","tag":"0GSM","title":"Ext groups · Lemma 0GSM","summary":"Let A be an abelian category. Let 0 → A → Z → B → 0 and 0 → B → Z' → C → 0 be short exact sequences in A. Denote [Z] ∈ Ext^1_A(B, A) and [Z'] ∈ Ext^1_A(C, B) their classes. Then [Z] ∘ [Z'] ∈ Ext^2_A(C, A) is 0 if and only if there exists a commutative diagram xymatrix & & 0 ar[d] & 0 ar[d] 0 ar[r] & A ar[r] ar[d]^1 & Z ar[r] ar[d] & B ar[r] ar[d] & 0 0 ar[r] & A ar[r] & W ar[r] ar[d] & Z' ar[r] ar[d] & 0 & & C ar[r]^1 ar[d]& C ar[d] & & 0 & 0 with exact rows and columns in A.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$0 \\to A \\to Z \\to B \\to 0$ and\n$0 \\to B \\to Z' \\to C \\to 0$ be short exact sequences in $\\mathcal{A}$.\nDenote $[Z] \\in \\Ext^1_\\mathcal{A}(B, A)$ and\n$[Z'] \\in \\Ext^1_\\mathcal{A}(C, B)$ their classes.\nThen $[Z] \\circ [Z'] \\in \\Ext^2_\\mathcal{A}(C, A)$ is $0$ if and\nonly if there exists a commutative diagram\n$$\n\\xymatrix{\n&\n&\n0 \\ar[d] &\n0 \\ar[d]\n\\\\\n0 \\ar[r] &\nA \\ar[r] \\ar[d]^1 &\nZ \\ar[r] \\ar[d] &\nB \\ar[r] \\ar[d] &\n0 \\\\\n0 \\ar[r] &\nA \\ar[r]  &\nW \\ar[r] \\ar[d] &\nZ' \\ar[r] \\ar[d] &\n0 \\\\\n&\n&\nC \\ar[r]^1  \\ar[d]&\nC \\ar[d]\\\\\n&\n&\n0 &\n0 \n}\n$$\nwith exact rows and columns in $\\mathcal{A}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSM","source_file":"derived.tex","source_line":8932,"source_end_line":8969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8932-L8969","statement_sha256":"ff4202b11ad94a403bcc9b5907878d9897ef3d199db317d422bc053a92091651","origin":"The Stacks Project","memory_eligible":false,"source_rank":2610,"rank":2610,"depth":1,"x":319.548,"y":675.244,"cluster":"derived-categories"},{"id":"stacks:0EWW","tag":"0EWW","title":"Ext groups · Lemma 0EWW","summary":"Let A be an abelian category and let p ≥ 0. If Ext^p_A(B, A) = 0 for any pair of objects A, B of A, then Ext^i_A(B, A) = 0 for i ≥ p and any pair of objects A, B of A.","statement_latex":"Let $\\mathcal{A}$ be an abelian category and let $p \\geq 0$.\nIf $\\Ext^p_\\mathcal{A}(B, A) = 0$ for any pair of objects $A$, $B$\nof $\\mathcal{A}$, then $\\Ext^i_\\mathcal{A}(B, A) = 0$ for\n$i \\geq p$ and any pair of objects $A$, $B$ of $\\mathcal{A}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWW","source_file":"derived.tex","source_line":8981,"source_end_line":8987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L8981-L8987","statement_sha256":"cce5dc71f8b275bebe66f90acd3e60dd4c7338d91fde3a0cc4c73b8e15f9e38d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2611,"rank":2611,"depth":1,"x":167.621,"y":734.47,"cluster":"derived-categories"},{"id":"stacks:0GM4","tag":"0GM4","title":"Ext groups · Lemma 0GM4","summary":"Let A be an abelian category. Let K be an object of D^b(A) such that Ext^p_A(H^i(K), H^j(K)) = 0 for all p ≥ 2 and i > j. Then K is isomorphic to the direct sum of its cohomologies: K ≅ bigoplus H^i(K)[-i].","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $K$ be an object of\n$D^b(\\mathcal{A})$ such that $\\Ext^p_\\mathcal{A}(H^i(K), H^j(K)) = 0$\nfor all $p \\geq 2$ and $i > j$. Then $K$ is isomorphic to the direct\nsum of its cohomologies: $K \\cong \\bigoplus H^i(K)[-i]$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GM4","source_file":"derived.tex","source_line":9010,"source_end_line":9016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9010-L9016","statement_sha256":"1432333173c183d495d9cb49ccd82cb2b8ef1b2845f0e5a06798e71efe46ed4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2612,"rank":2612,"depth":3,"x":232.201,"y":604.231,"cluster":"derived-categories"},{"id":"stacks:0EWX","tag":"0EWX","title":"Ext groups · Lemma 0EWX","summary":"Let A be an abelian category. Assume Ext^2_A(B, A) = 0 for any pair of objects A, B of A. Then any object K of D^b(A) is isomorphic to the direct sum of its cohomologies: K ≅ bigoplus H^i(K)[-i].","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Assume $\\Ext^2_\\mathcal{A}(B, A) = 0$\nfor any pair of objects $A$, $B$ of $\\mathcal{A}$.\nThen any object $K$ of $D^b(\\mathcal{A})$ is isomorphic to the direct\nsum of its cohomologies: $K \\cong \\bigoplus H^i(K)[-i]$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Ext groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWX","source_file":"derived.tex","source_line":9037,"source_end_line":9043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9037-L9043","statement_sha256":"567c0286150bd2900f8746933eef38ff54f5b5680e8e8a0a7accad48068a9b0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2613,"rank":2613,"depth":4,"x":289.471,"y":737.276,"cluster":"derived-categories"},{"id":"stacks:0FCN","tag":"0FCN","title":"K-groups · Definition 0FCN","summary":"Let D be a triangulated category. We denote K_0(D) the zeroth K-group of D. It is the abelian group constructed as follows. Take the free abelian group on the objects on D and for every distinguished triangle X → Y → Z impose the relation [Y] - [X] - [Z] = 0.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. We denote $K_0(\\mathcal{D})$ the\n{\\it zeroth $K$-group of $\\mathcal{D}$}. It is the abelian group constructed\nas follows. Take the free abelian group on the objects on $\\mathcal{D}$\nand for every distinguished triangle $X \\to Y \\to Z$\nimpose the relation $[Y] - [X] - [Z] = 0$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCN","source_file":"derived.tex","source_line":9063,"source_end_line":9070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9063-L9070","statement_sha256":"8b8b5b0a1056077d102c2c72f70fe1fe19f0a24e6f187ab5426d75eed46a6f52","origin":"The Stacks Project","memory_eligible":false,"source_rank":2614,"rank":2614,"depth":0,"x":139.841,"y":671.488,"cluster":"derived-categories"},{"id":"stacks:0FCP","tag":"0FCP","title":"K-groups · Lemma 0FCP","summary":"Let A be an abelian category. Then there is a canonical identification K_0(D^b(A)) = K_0(A) of zeroth K-groups.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Then there is a canonical\nidentification $K_0(D^b(\\mathcal{A})) = K_0(\\mathcal{A})$\nof zeroth $K$-groups.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCP","source_file":"derived.tex","source_line":9076,"source_end_line":9081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9076-L9081","statement_sha256":"897b83b1b1390988cfa3c8bcb4fd83c8f098a1ee6b7e751b76bb853c6c305df9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2615,"rank":2615,"depth":0,"x":303.526,"y":634.997,"cluster":"derived-categories"},{"id":"stacks:0FCQ","tag":"0FCQ","title":"K-groups · Lemma 0FCQ","summary":"Let F : D → D' be an exact functor of triangulated categories. Then F induces a group homomorphism K_0(D) → K_0(D').","statement_latex":"Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of triangulated\ncategories. Then $F$ induces a group homomorphism\n$K_0(\\mathcal{D}) \\to K_0(\\mathcal{D}')$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCQ","source_file":"derived.tex","source_line":9126,"source_end_line":9131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9126-L9131","statement_sha256":"fc5e4e69d70ecc45d34141a1c9a99d353842ddca2f5b9900dc404b2767bceb22","origin":"The Stacks Project","memory_eligible":false,"source_rank":2616,"rank":2616,"depth":0,"x":211.925,"y":755.107,"cluster":"derived-categories"},{"id":"stacks:0FCR","tag":"0FCR","title":"K-groups · Lemma 0FCR","summary":"Let H : D → A be a homological functor from a triangulated category to an abelian category. Assume that for any X in D only a finite number of the objects H(X[i]) are nonzero in A. Then H induces a group homomorphism K_0(D) → K_0(A) sending [X] to ∑ (-1)^i[H(X[i])].","statement_latex":"Let $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor\nfrom a triangulated category to an abelian category. Assume that\nfor any $X$ in $\\mathcal{D}$ only a finite number of the objects\n$H(X[i])$ are nonzero in $\\mathcal{A}$. Then $H$ induces a group homomorphism\n$K_0(\\mathcal{D}) \\to K_0(\\mathcal{A})$ sending $[X]$ to\n$\\sum (-1)^i[H(X[i])]$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCR","source_file":"derived.tex","source_line":9137,"source_end_line":9145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9137-L9145","statement_sha256":"4d6b92ae3da948a53eccc016358797a98c414145b3fd10da0cf1ed919fb7f1a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2617,"rank":2617,"depth":0,"x":182.803,"y":614.184,"cluster":"derived-categories"},{"id":"stacks:0FCS","tag":"0FCS","title":"K-groups · Lemma 0FCS","summary":"Let B be a weak Serre subcategory of the abelian category A. There is a canonical isomorphism K_0(B) → K_0(D^b_B(A)), [B] ↦ [B[0]] The inverse sends the class [X] of X to the element ∑ (-1)^i[H^i(X)].","statement_latex":"Let $\\mathcal{B}$ be a weak Serre subcategory of the abelian category\n$\\mathcal{A}$. There is a canonical isomorphism\n$$\nK_0(\\mathcal{B}) \\longrightarrow\nK_0(D^b_\\mathcal{B}(\\mathcal{A})),\\quad\n[B] \\longmapsto [B[0]]\n$$\nThe inverse sends the class $[X]$ of $X$\nto the element $\\sum (-1)^i[H^i(X)]$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCS","source_file":"derived.tex","source_line":9151,"source_end_line":9162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9151-L9162","statement_sha256":"1785983ff0b6be41c4fbaae45794082ba49ee94a346b3599ccdb4160b9d089a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2618,"rank":2618,"depth":0,"x":317.964,"y":701.809,"cluster":"derived-categories"},{"id":"stacks:0FCT","tag":"0FCT","title":"K-groups · Lemma 0FCT","summary":"Let D, D', D\" be triangulated categories. Let ⊗ : D × D' → D\" be a functor such that for fixed X in D the functor X ⊗ - : D' → D\" is an exact functor and for fixed X' in D' the functor - ⊗ X' : D → D\" is an exact functor. Then ⊗ induces a bilinear map K_0(D) × K_0(D') → K_0(D\") which sends ([X], [X']) to [X ⊗ X'].","statement_latex":"Let $\\mathcal{D}$, $\\mathcal{D}'$, $\\mathcal{D}''$ be triangulated categories.\nLet\n$$\n\\otimes : \\mathcal{D} \\times \\mathcal{D}' \\longrightarrow \\mathcal{D}''\n$$\nbe a functor such that for fixed $X$ in $\\mathcal{D}$ the functor\n$X \\otimes - : \\mathcal{D}' \\to \\mathcal{D}''$ is an exact functor and\nfor fixed $X'$ in $\\mathcal{D}'$ the functor\n$- \\otimes X' : \\mathcal{D} \\to \\mathcal{D}''$ is an exact functor. Then\n$\\otimes$ induces a bilinear map\n$K_0(\\mathcal{D}) \\times K_0(\\mathcal{D}') \\to K_0(\\mathcal{D}'')$\nwhich sends $([X], [X'])$ to $[X \\otimes X']$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCT","source_file":"derived.tex","source_line":9178,"source_end_line":9192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9178-L9192","statement_sha256":"cf5b5203a2fb0a974e81b5bdf4f944863b23eb635e4d3adbd17b0505a28ea765","origin":"The Stacks Project","memory_eligible":false,"source_rank":2619,"rank":2619,"depth":0,"x":147.38,"y":713.921,"cluster":"derived-categories"},{"id":"stacks:06XX","tag":"06XX","title":"Unbounded complexes · Lemma 06XX","summary":"Let A be an abelian category. Let P ⊂ Ob(A) be a subset. Assume P contains 0, is closed under (finite) direct sums, and every object of A is a quotient of an element of P. Let K^bullet be a complex. There exists a commutative diagram xymatrix P_1^bullet ar[d] ar[r] & P_2^bullet ar[d] ar[r] & … τ_≤ 1K^bullet ar[r] & τ_≤ 2K^bullet ar[r] & … in the category of complexes such that • the vertical arrows are quasi-isomorphisms and termwise surjective, • P_n^bullet is a bounded…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a subset.\nAssume $\\mathcal{P}$ contains $0$, is closed under (finite) direct sums,\nand every object of $\\mathcal{A}$ is a quotient of an\nelement of $\\mathcal{P}$. Let $K^\\bullet$ be a complex.\nThere exists a commutative diagram\n$$\n\\xymatrix{\nP_1^\\bullet \\ar[d] \\ar[r] & P_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\\n\\tau_{\\leq 1}K^\\bullet \\ar[r] & \\tau_{\\leq 2}K^\\bullet \\ar[r] & \\ldots\n}\n$$\nin the category of complexes such that\n\\begin{enumerate}\n\\item the vertical arrows are quasi-isomorphisms and termwise surjective,\n\\item $P_n^\\bullet$ is a bounded above complex with terms in\n$\\mathcal{P}$,\n\\item the arrows $P_n^\\bullet \\to P_{n + 1}^\\bullet$\nare termwise split injections and each cokernel\n$P^i_{n + 1}/P^i_n$ is an element of $\\mathcal{P}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XX","source_file":"derived.tex","source_line":9214,"source_end_line":9237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9214-L9237","statement_sha256":"8d074ddbe8dc8740edd8b864b9275917a0f93e6a5a5c8fd2e0a8e5175feb96e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2620,"rank":2620,"depth":13,"x":263.733,"y":607.916,"cluster":"derived-categories"},{"id":"stacks:0794","tag":"0794","title":"Unbounded complexes · Proposition 0794","summary":"Let F : A → B be a right exact functor of abelian categories. Let P ⊂ Ob(A) be a subset. Assume • P contains 0, is closed under (finite) direct sums, and every object of A is a quotient of an element of P, • for any bounded above acyclic complex P^bullet of A with P^n ∈ P for all n the complex F(P^bullet) is exact, • A and B have colimits of systems over N, • colimits over N are exact in both A and B, and • F commutes with colimits over N. Then LF is defined on all of D(A).","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a right exact functor\nof abelian categories. Let $\\mathcal{P} \\subset \\Ob(\\mathcal{A})$ be a\nsubset. Assume\n\\begin{enumerate}\n\\item $\\mathcal{P}$ contains $0$, is closed under (finite) direct sums,\nand every object of $\\mathcal{A}$ is a quotient of an\nelement of $\\mathcal{P}$,\n\\item for any bounded above acyclic complex $P^\\bullet$ of\n$\\mathcal{A}$ with $P^n \\in \\mathcal{P}$ for all $n$ the\ncomplex $F(P^\\bullet)$ is exact,\n\\item $\\mathcal{A}$ and $\\mathcal{B}$ have colimits\nof systems over $\\mathbf{N}$,\n\\item colimits over $\\mathbf{N}$ are exact in both\n$\\mathcal{A}$ and $\\mathcal{B}$, and\n\\item $F$ commutes with colimits over $\\mathbf{N}$.\n\\end{enumerate}\nThen $LF$ is defined on all of $D(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Unbounded complexes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0794","source_file":"derived.tex","source_line":9332,"source_end_line":9351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9332-L9351","statement_sha256":"2f216acbe944b64808087717cd4c5b8bf44a7361ac2bf7a78b26080ac45bf4e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2621,"rank":2621,"depth":16,"x":263.18,"y":752.484,"cluster":"derived-categories"},{"id":"stacks:070F","tag":"070F","title":"Unbounded complexes · Lemma 070F","summary":"Let A be an abelian category. Let I ⊂ Ob(A) be a subset. Assume I contains 0, is closed under (finite) products, and every object of A is a subobject of an element of I. Let K^bullet be a complex. There exists a commutative diagram xymatrix … ar[r] & τ_≥ -2K^bullet ar[r] ar[d] & τ_≥ -1K^bullet ar[d] … ar[r] & I_2^bullet ar[r] & I_1^bullet in the category of complexes such that • the vertical arrows are quasi-isomorphisms and termwise injective, • I_n^bullet is a bounded…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$\\mathcal{I} \\subset \\Ob(\\mathcal{A})$ be a subset.\nAssume $\\mathcal{I}$ contains $0$, is closed under (finite) products,\nand every object of $\\mathcal{A}$ is a subobject of an\nelement of $\\mathcal{I}$. Let $K^\\bullet$ be a complex.\nThere exists a commutative diagram\n$$\n\\xymatrix{\n\\ldots \\ar[r] &\n\\tau_{\\geq -2}K^\\bullet \\ar[r] \\ar[d] &\n\\tau_{\\geq -1}K^\\bullet \\ar[d] \\\\\n\\ldots \\ar[r] & I_2^\\bullet \\ar[r] & I_1^\\bullet\n}\n$$\nin the category of complexes such that\n\\begin{enumerate}\n\\item the vertical arrows are quasi-isomorphisms and termwise injective,\n\\item $I_n^\\bullet$ is a bounded below complex with terms in $\\mathcal{I}$,\n\\item the arrows $I_{n + 1}^\\bullet \\to I_n^\\bullet$ are termwise split\nsurjections and $\\Ker(I^i_{n + 1} \\to I^i_n)$ is an element of $\\mathcal{I}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070F","source_file":"derived.tex","source_line":9444,"source_end_line":9467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9444-L9467","statement_sha256":"b7e7eb2a3836e5046ce69c86b0c66584d045d2d6350fa4556eb71cf97ab04878","origin":"The Stacks Project","memory_eligible":false,"source_rank":2622,"rank":2622,"depth":14,"x":147.027,"y":645.288,"cluster":"derived-categories"},{"id":"stacks:0FND","tag":"0FND","title":"Deriving adjoints · Lemma 0FND","summary":"In the situation above assume F is right adjoint to G. Let K ∈ Ob(D) and M ∈ Ob(D'). If RF is defined at K and LG is defined at M, then there is a canonical isomorphism Hom_(S')^-1D'(M, RF(K)) = Hom_S^-1D(LG(M), K) This isomorphism is functorial in both variables on the triangulated subcategories of S^-1D and (S')^-1D' where RF and LG are defined.","statement_latex":"In the situation above assume $F$ is right adjoint\nto $G$. Let $K \\in \\Ob(\\mathcal{D})$ and\n$M \\in \\Ob(\\mathcal{D}')$. If $RF$ is defined at $K$\nand $LG$ is defined at $M$, then there is a canonical isomorphism\n$$\n\\Hom_{(S')^{-1}\\mathcal{D}'}(M, RF(K)) =\n\\Hom_{S^{-1}\\mathcal{D}}(LG(M), K)\n$$\nThis isomorphism is functorial in both variables on the triangulated\nsubcategories of $S^{-1}\\mathcal{D}$ and $(S')^{-1}\\mathcal{D}'$\nwhere $RF$ and $LG$ are defined.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Deriving adjoints","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FND","source_file":"derived.tex","source_line":9516,"source_end_line":9529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9516-L9529","statement_sha256":"2f2beef884e0840b56cfa93c8bbecb10abdad422fdb53429256a934c18879064","origin":"The Stacks Project","memory_eligible":false,"source_rank":2623,"rank":2623,"depth":2,"x":319.329,"y":658.459,"cluster":"derived-categories"},{"id":"stacks:0DVC","tag":"0DVC","title":"Deriving adjoints · Lemma 0DVC","summary":"Let F : A → B and G : B → A be functors of abelian categories such that F is a right adjoint to G. Let K^bullet be a complex of A and let M^bullet be a complex of B. If RF is defined at K^bullet and LG is defined at M^bullet, then there is a canonical isomorphism Hom_D(B)(M^bullet, RF(K^bullet)) = Hom_D(A)(LG(M^bullet), K^bullet) This isomorphism is functorial in both variables on the triangulated subcategories of D(A) and D(B) where RF and LG are defined.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{A}$\nbe functors of abelian categories such that $F$ is a right adjoint to $G$.\nLet $K^\\bullet$ be a complex of $\\mathcal{A}$ and let $M^\\bullet$ be\na complex of $\\mathcal{B}$. If $RF$ is defined at $K^\\bullet$\nand $LG$ is defined at $M^\\bullet$, then there is a canonical isomorphism\n$$\n\\Hom_{D(\\mathcal{B})}(M^\\bullet, RF(K^\\bullet)) =\n\\Hom_{D(\\mathcal{A})}(LG(M^\\bullet), K^\\bullet)\n$$\nThis isomorphism is functorial in both variables on the triangulated\nsubcategories of $D(\\mathcal{A})$ and $D(\\mathcal{B})$\nwhere $RF$ and $LG$ are defined.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Deriving adjoints","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVC","source_file":"derived.tex","source_line":9569,"source_end_line":9583,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9569-L9583","statement_sha256":"1bfed0b210c1df6dd13c4f6cf19827ce2d40a7bdac37018c04731e159340715f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2624,"rank":2624,"depth":3,"x":181.327,"y":746.743,"cluster":"derived-categories"},{"id":"stacks:09T5","tag":"09T5","title":"Deriving adjoints · Lemma 09T5","summary":"Let F : A → B and G : B → A be functors of abelian categories such that F is a right adjoint to G. If the derived functors RF : D(A) → D(B) and LG : D(B) → D(A) exist, then RF is a right adjoint to LG.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ and $G : \\mathcal{B} \\to \\mathcal{A}$\nbe functors of abelian categories such that $F$ is a right adjoint to $G$.\nIf the derived functors $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ and\n$LG : D(\\mathcal{B}) \\to D(\\mathcal{A})$ exist, then\n$RF$ is a right adjoint to $LG$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Deriving adjoints","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09T5","source_file":"derived.tex","source_line":9595,"source_end_line":9602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9595-L9602","statement_sha256":"b4c4a74f011c9b6033876c2a8fa2bff3d0a0bf8cdc5ac819e4bfe2e16e678bf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2625,"rank":2625,"depth":4,"x":212.172,"y":602.969,"cluster":"derived-categories"},{"id":"stacks:070H","tag":"070H","title":"K-injective complexes · Definition 070H","summary":"Let A be an abelian category. A complex I^bullet is K-injective if for every acyclic complex M^bullet we have Hom_K(A)(M^bullet, I^bullet) = 0.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. A complex $I^\\bullet$\nis {\\it K-injective} if for every acyclic complex $M^\\bullet$ we\nhave $\\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet) = 0$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-injective complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070H","source_file":"derived.tex","source_line":9619,"source_end_line":9624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9619-L9624","statement_sha256":"e4778ba813a5ea18ae796132baa612418f621a69d679b2a3cd7e54611ee644fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":2626,"rank":2626,"depth":0,"x":305.285,"y":726.805,"cluster":"derived-categories"},{"id":"stacks:070I","tag":"070I","title":"K-injective complexes · Lemma 070I","summary":"Let A be an abelian category. Let I^bullet be a complex. The following are equivalent • I^bullet is K-injective, • for every quasi-isomorphism M^bullet → N^bullet the map Hom_K(A)(N^bullet, I^bullet) → Hom_K(A)(M^bullet, I^bullet) is bijective, and • for every complex N^bullet the map Hom_K(A)(N^bullet, I^bullet) → Hom_D(A)(N^bullet, I^bullet) is an isomorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $I^\\bullet$ be a complex. The following are equivalent\n\\begin{enumerate}\n\\item $I^\\bullet$ is K-injective,\n\\item for every quasi-isomorphism $M^\\bullet \\to N^\\bullet$ the map\n$$\n\\Hom_{K(\\mathcal{A})}(N^\\bullet, I^\\bullet)\n\\to \\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet)\n$$\nis bijective, and\n\\item for every complex $N^\\bullet$ the map\n$$\n\\Hom_{K(\\mathcal{A})}(N^\\bullet, I^\\bullet)\n\\to \\Hom_{D(\\mathcal{A})}(N^\\bullet, I^\\bullet)\n$$\nis an isomorphism.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070I","source_file":"derived.tex","source_line":9631,"source_end_line":9650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9631-L9650","statement_sha256":"bcdb07131a5eccf11b4f2a92c65d99380d30436cfda8c68b75a8b7626dcc3b8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2627,"rank":2627,"depth":0,"x":136.61,"y":688.226,"cluster":"derived-categories"},{"id":"stacks:090X","tag":"090X","title":"K-injective complexes · Lemma 090X","summary":"Let A be an abelian category. Let (K, L, M, f, g, h) be a distinguished triangle of K(A). If two out of K, L, M are K-injective complexes, then the third is too.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $(K, L, M, f, g, h)$\nbe a distinguished triangle of $K(\\mathcal{A})$. If two out of\n$K$, $L$, $M$ are K-injective complexes, then the third is too.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090X","source_file":"derived.tex","source_line":9673,"source_end_line":9678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9673-L9678","statement_sha256":"9885a63b3076bc4fff4eaa6f6d57a20ca77ecfdb425597e1faeae62fa45a61bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2628,"rank":2628,"depth":12,"x":292.407,"y":620.793,"cluster":"derived-categories"},{"id":"stacks:070J","tag":"070J","title":"K-injective complexes · Lemma 070J","summary":"Let A be an abelian category. A bounded below complex of injectives is K-injective.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. A bounded below complex of\ninjectives is K-injective.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070J","source_file":"derived.tex","source_line":9687,"source_end_line":9691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9687-L9691","statement_sha256":"92162c473bf9514bdd0f93f878e0b05eab30ea0c61ee95bd316febc00bf22174","origin":"The Stacks Project","memory_eligible":false,"source_rank":2629,"rank":2629,"depth":4,"x":231.599,"y":759.269,"cluster":"derived-categories"},{"id":"stacks:0BK6","tag":"0BK6","title":"K-injective complexes · Lemma 0BK6","summary":"Let A be an abelian category. Let T be a set and for each t ∈ T let I_t^bullet be a K-injective complex. If I^n = ∏_t I_t^n exists for all n, then I^bullet is a K-injective complex. Moreover, I^bullet represents the product of the objects I_t^bullet in D(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $T$ be a set and for\neach $t \\in T$ let $I_t^\\bullet$ be a K-injective complex. If\n$I^n = \\prod_t I_t^n$ exists for all $n$, then $I^\\bullet$ is\na K-injective complex. Moreover, $I^\\bullet$ represents the\nproduct of the objects $I_t^\\bullet$ in $D(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BK6","source_file":"derived.tex","source_line":9699,"source_end_line":9706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9699-L9706","statement_sha256":"4296cd14737a70861c9f0f06eb74a15bce321b02a6543fcb4ed74e990b31fbaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":2630,"rank":2630,"depth":1,"x":164.912,"y":622.311,"cluster":"derived-categories"},{"id":"stacks:070Y","tag":"070Y","title":"K-injective complexes · Lemma 070Y","summary":"Let A be an abelian category. Let F : K(A) → D' be an exact functor of triangulated categories. Then RF is defined at every complex in K(A) which is quasi-isomorphic to a K-injective complex. In fact, every K-injective complex computes RF.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $F : K(\\mathcal{A}) \\to \\mathcal{D}'$ be an exact functor\nof triangulated categories. Then $RF$ is defined at every complex\nin $K(\\mathcal{A})$ which is quasi-isomorphic to a\nK-injective complex. In fact, every K-injective complex computes $RF$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070Y","source_file":"derived.tex","source_line":9746,"source_end_line":9753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9746-L9753","statement_sha256":"83999da297a81be2f98223b2a1bcff9ce2276def8757fb7a5d7e9c4dc8c0888f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2631,"rank":2631,"depth":1,"x":324.622,"y":685.621,"cluster":"derived-categories"},{"id":"stacks:070K","tag":"070K","title":"K-injective complexes · Lemma 070K","summary":"Let A be an abelian category. Assume every complex has a quasi-isomorphism towards a K-injective complex. Then any exact functor F : K(A) → D' of triangulated categories has a right derived functor RF : D(A) → D' and RF(I^bullet) = F(I^bullet) for K-injective complexes I^bullet.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nAssume every complex has a quasi-isomorphism towards a K-injective complex.\nThen any exact functor $F : K(\\mathcal{A}) \\to \\mathcal{D}'$ of triangulated\ncategories has a right derived functor\n$$\nRF : D(\\mathcal{A}) \\longrightarrow \\mathcal{D}'\n$$\nand $RF(I^\\bullet) = F(I^\\bullet)$ for K-injective complexes $I^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070K","source_file":"derived.tex","source_line":9766,"source_end_line":9776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9766-L9776","statement_sha256":"ac74cac2463eb3dd0bffdfbc1b5364dc86b932ddf1f9cb60e0555ea64e810e37","origin":"The Stacks Project","memory_eligible":false,"source_rank":2632,"rank":2632,"depth":1,"x":155.523,"y":729.669,"cluster":"derived-categories"},{"id":"stacks:070L","tag":"070L","title":"K-injective complexes · Lemma 070L","summary":"The limit of a \"split\" tower of K-injective complexes is K-injective. Let A be an abelian category. Let … → I_3^bullet → I_2^bullet → I_1^bullet be an inverse system of complexes. Assume • each I_n^bullet is K-injective, • each map I_n + 1^m → I_n^m is a split surjection, • the limits I^m = lim I_n^m exist. Then the complex I^bullet is K-injective.","statement_latex":"\\begin{slogan}\nThe limit of a ``split'' tower of K-injective complexes is K-injective.\n\\end{slogan}\nLet $\\mathcal{A}$ be an abelian category. Let\n$$\n\\ldots \\to I_3^\\bullet \\to I_2^\\bullet \\to I_1^\\bullet\n$$\nbe an inverse system of complexes. Assume\n\\begin{enumerate}\n\\item each $I_n^\\bullet$ is $K$-injective,\n\\item each map $I_{n + 1}^m \\to I_n^m$ is a split surjection,\n\\item the limits $I^m = \\lim I_n^m$ exist.\n\\end{enumerate}\nThen the complex $I^\\bullet$ is K-injective.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070L","source_file":"derived.tex","source_line":9793,"source_end_line":9809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9793-L9809","statement_sha256":"83aee9a698b18f9b3454cc030d8ca2a5775d949a92477941b7f6e168b01b94b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2633,"rank":2633,"depth":5,"x":245.013,"y":600.92,"cluster":"derived-categories"},{"id":"stacks:08BJ","tag":"08BJ","title":"K-injective complexes · Lemma 08BJ","summary":"Let A and B be abelian categories. Let u : A → B and v : B → A be additive functors. Assume • u is right adjoint to v, and • v is exact. Then u transforms K-injective complexes into K-injective complexes.","statement_latex":"Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $u : \\mathcal{A} \\to \\mathcal{B}$ and\n$v : \\mathcal{B} \\to \\mathcal{A}$ be additive functors. Assume\n\\begin{enumerate}\n\\item $u$ is right adjoint to $v$, and\n\\item $v$ is exact.\n\\end{enumerate}\nThen $u$ transforms K-injective complexes into K-injective complexes.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BJ","source_file":"derived.tex","source_line":9856,"source_end_line":9866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9856-L9866","statement_sha256":"62595f8fdc1d2f26b0f5c8e804614d244c13bcaa54b5de19e155e048daddf4d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2634,"rank":2634,"depth":0,"x":282.653,"y":746.995,"cluster":"derived-categories"},{"id":"stacks:07K6","tag":"07K6","title":"Bounded cohomological dimension · Lemma 07K6","summary":"Let A be an abelian category. Let d : Ob(A) → (0, 1, 2, …, ∞) be a function. Assume that • every object of A is a subobject of an object A with d(A) = 0, • d(A ⊕ B) ≤ max (d(A), d(B)) for A, B ∈ A, and • if 0 → A → B → C → 0 is short exact, then d(C) ≤ max(d(A) - 1, d(B)). Let K^bullet be a complex such that n + d(K^n) tends to -∞ as n → -∞. Then there exists a quasi-isomorphism K^bullet → L^bullet with d(L^n) = 0 for all n ∈ Z.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let\n$d : \\Ob(\\mathcal{A}) \\to \\{0, 1, 2, \\ldots, \\infty\\}$ be a function.\nAssume that\n\\begin{enumerate}\n\\item every object of $\\mathcal{A}$ is a subobject of an\nobject $A$ with $d(A) = 0$,\n\\item $d(A \\oplus B) \\leq \\max \\{d(A), d(B)\\}$ for $A, B \\in \\mathcal{A}$, and\n\\item if $0 \\to A \\to B \\to C \\to 0$ is short exact, then\n$d(C) \\leq \\max\\{d(A) - 1, d(B)\\}$.\n\\end{enumerate}\nLet $K^\\bullet$ be a complex such that $n + d(K^n)$ tends to $-\\infty$\nas $n \\to -\\infty$. Then there exists a quasi-isomorphism\n$K^\\bullet \\to L^\\bullet$ with $d(L^n) = 0$ for all $n \\in \\mathbf{Z}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Bounded cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07K6","source_file":"derived.tex","source_line":9891,"source_end_line":9906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9891-L9906","statement_sha256":"20bfb21b1d81399ecbcfcd9821d70f42bb328e9caaa4200b93ef5066b9b2eb65","origin":"The Stacks Project","memory_eligible":false,"source_rank":2635,"rank":2635,"depth":2,"x":137.072,"y":660.427,"cluster":"derived-categories"},{"id":"stacks:07K7","tag":"07K7","title":"Bounded cohomological dimension · Lemma 07K7","summary":"Let F : A → B be a left exact functor of abelian categories. Assume • every object of A is a subobject of an object which is right acyclic for F, • there exists an integer n ≥ 0 such that R^nF = 0, Then • RF : D(A) → D(B) exists, • any complex consisting of right acyclic objects for F computes RF, • any complex is the source of a quasi-isomorphism into a complex consisting of right acyclic objects for F, • for E ∈ D(A) • H^i(RF(τ_≤ aE) → H^i(RF(E)) is an isomorphism for i…","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor of\nabelian categories. Assume\n\\begin{enumerate}\n\\item every object of $\\mathcal{A}$ is a subobject of an object\nwhich is right acyclic for $F$,\n\\item there exists an integer $n \\geq 0$ such that $R^nF = 0$,\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item $RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ exists,\n\\item any complex consisting of right acyclic objects for $F$ computes $RF$,\n\\item any complex is the source of a quasi-isomorphism into a complex\nconsisting of right acyclic objects for $F$,\n\\item for $E \\in D(\\mathcal{A})$\n\\begin{enumerate}\n\\item $H^i(RF(\\tau_{\\leq a}E) \\to H^i(RF(E))$ is an isomorphism\nfor $i \\leq a$,\n\\item $H^i(RF(E)) \\to H^i(RF(\\tau_{\\geq b - n + 1}E))$ is an isomorphism\nfor $i \\geq b$,\n\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some\n$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(RF(E)) = 0$\nfor $i \\not \\in [a, b + n - 1]$.\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Bounded cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07K7","source_file":"derived.tex","source_line":9976,"source_end_line":10002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L9976-L10002","statement_sha256":"609d1242406d9e03b24128980d108c959d554f6d6d50bdc03a78784a74160ccb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2636,"rank":2636,"depth":15,"x":314.468,"y":641.61,"cluster":"derived-categories"},{"id":"stacks:07K8","tag":"07K8","title":"Bounded cohomological dimension · Lemma 07K8","summary":"Let F : A → B be a right exact functor of abelian categories. If • every object of A is a quotient of an object which is left acyclic for F, • there exists an integer n ≥ 0 such that L^nF = 0, Then • LF : D(A) → D(B) exists, • any complex consisting of left acyclic objects for F computes LF, • any complex is the target of a quasi-isomorphism from a complex consisting of left acyclic objects for F, • for E ∈ D(A) • H^i(LF(τ_≤ a + n - 1E) → H^i(LF(E)) is an isomorphism for…","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a right exact functor of\nabelian categories. If\n\\begin{enumerate}\n\\item every object of $\\mathcal{A}$ is a quotient of an object\nwhich is left acyclic for $F$,\n\\item there exists an integer $n \\geq 0$ such that $L^nF = 0$,\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item $LF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ exists,\n\\item any complex consisting of left acyclic objects for $F$ computes $LF$,\n\\item any complex is the target of a quasi-isomorphism from a complex\nconsisting of left acyclic objects for $F$,\n\\item for $E \\in D(\\mathcal{A})$\n\\begin{enumerate}\n\\item $H^i(LF(\\tau_{\\leq a + n - 1}E) \\to H^i(LF(E))$ is an isomorphism\nfor $i \\leq a$,\n\\item $H^i(LF(E)) \\to H^i(LF(\\tau_{\\geq b}E))$ is an isomorphism\nfor $i \\geq b$,\n\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some\n$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(LF(E)) = 0$\nfor $i \\not \\in [a - n + 1, b]$.\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Bounded cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07K8","source_file":"derived.tex","source_line":10102,"source_end_line":10128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10102-L10128","statement_sha256":"b767e6c9b1fbbe1e69d339bb1dcb6bb67623da200b82fb565805f0cdc94b66f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2637,"rank":2637,"depth":16,"x":198.511,"y":756.422,"cluster":"derived-categories"},{"id":"stacks:090Z","tag":"090Z","title":"Derived colimits · Definition 090Z","summary":"Let D be a triangulated category. Let (K_n, f_n) be a system of objects of D. We say an object K is a derived colimit, or a homotopy colimit of the system (K_n) if the direct sum bigoplus K_n exists and there is a distinguished triangle bigoplus K_n → bigoplus K_n → K → bigoplus K_n[1] where the map bigoplus K_n → bigoplus K_n is given by 1 - f_n in degree n. If this is the case, then we sometimes indicate this by the notation K = hocolim K_n.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $(K_n, f_n)$ be a system of objects of $\\mathcal{D}$.\nWe say an object $K$ is a {\\it derived colimit}, or a\n{\\it homotopy colimit} of the system $(K_n)$ if\nthe direct sum $\\bigoplus K_n$ exists and there is a distinguished triangle\n$$\n\\bigoplus K_n \\to \\bigoplus K_n \\to K \\to \\bigoplus K_n[1]\n$$\nwhere the map $\\bigoplus K_n \\to \\bigoplus K_n$ is given\nby $1 - f_n$ in degree $n$. If this is the\ncase, then we sometimes indicate this by the notation\n$K = \\text{hocolim} K_n$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090Z","source_file":"derived.tex","source_line":10145,"source_end_line":10159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10145-L10159","statement_sha256":"701380e5e340005d544e586f11938bfa8c166cf3bdfa892974d722af110656ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":2638,"rank":2638,"depth":0,"x":191.671,"y":605.601,"cluster":"derived-categories"},{"id":"stacks:0CRJ","tag":"0CRJ","title":"Derived colimits · Lemma 0CRJ","summary":"Let D be a triangulated category. Let (K_n, f_n) be a system of objects of D. Let n_1 < n_2 < n_3 < … be a sequence of integers. Assume bigoplus K_n and bigoplus K_n_i exist. Then there exists an isomorphism hocolim K_n_i → hocolim K_n such that xymatrix K_n_i ar[r] ar[d]_id & hocolim K_n_i ar[d] K_n_i ar[r] & hocolim K_n commutes for all i.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $(K_n, f_n)$ be a system of objects of $\\mathcal{D}$.\nLet $n_1 < n_2 < n_3 < \\ldots$ be a sequence of integers.\nAssume $\\bigoplus K_n$ and $\\bigoplus K_{n_i}$ exist.\nThen there exists an isomorphism\n$\\text{hocolim} K_{n_i} \\to \\text{hocolim} K_n$\nsuch that\n$$\n\\xymatrix{\nK_{n_i} \\ar[r] \\ar[d]_{\\text{id}} & \\text{hocolim} K_{n_i} \\ar[d] \\\\\nK_{n_i} \\ar[r] & \\text{hocolim} K_n\n}\n$$\ncommutes for all $i$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRJ","source_file":"derived.tex","source_line":10213,"source_end_line":10229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10213-L10229","statement_sha256":"bfaa4742799c62a6f89948ce4921da7722a34d0914262dcb64ae0907306f7ae0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2639,"rank":2639,"depth":9,"x":318.304,"y":713.191,"cluster":"derived-categories"},{"id":"stacks:0A5L","tag":"0A5L","title":"Derived colimits · Lemma 0A5L","summary":"Let A be an abelian category. If A has exact countable direct sums, then D(A) has countable direct sums. In fact given a collection of complexes K_i^bullet indexed by a countable index set I the termwise direct sum bigoplus K_i^bullet is the direct sum of K_i^bullet in D(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nIf $\\mathcal{A}$ has exact countable direct sums, then\n$D(\\mathcal{A})$ has countable direct sums. In fact given\na collection of complexes $K_i^\\bullet$ indexed by a countable\nindex set $I$ the termwise direct sum $\\bigoplus K_i^\\bullet$\nis the direct sum of $K_i^\\bullet$ in $D(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5L","source_file":"derived.tex","source_line":10299,"source_end_line":10307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10299-L10307","statement_sha256":"a5e2a9936b47dec499b6dbf3269eae63be72dfaa037baab8d1bfe0713d1e8699","origin":"The Stacks Project","memory_eligible":false,"source_rank":2640,"rank":2640,"depth":0,"x":137.976,"y":705.692,"cluster":"derived-categories"},{"id":"stacks:093W","tag":"093W","title":"Derived colimits · Lemma 093W","summary":"Let A be an abelian category. Assume colimits over N exist and are exact. Then countable direct sums exists and are exact. Moreover, if (A_n, f_n) is a system over N, then there is a short exact sequence 0 → bigoplus A_n → bigoplus A_n → colim A_n → 0 where the first map in degree n is given by 1 - f_n.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Assume colimits over $\\mathbf{N}$\nexist and are exact. Then countable direct sums exists and are exact.\nMoreover, if $(A_n, f_n)$ is a system over $\\mathbf{N}$, then there is\na short exact sequence\n$$\n0 \\to \\bigoplus A_n \\to \\bigoplus A_n \\to \\colim A_n \\to 0\n$$\nwhere the first map in degree $n$ is given by $1 - f_n$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093W","source_file":"derived.tex","source_line":10325,"source_end_line":10335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10325-L10335","statement_sha256":"55daffef9d7a33feb503c7ac4bf70a92da1f5c9dce61d0df74ffe672ebf4a37b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2641,"rank":2641,"depth":0,"x":277.309,"y":608.669,"cluster":"derived-categories"},{"id":"stacks:0949","tag":"0949","title":"Derived colimits · Lemma 0949","summary":"Let A be an abelian category. Let L_n^bullet be a system of complexes of A. Assume colimits over N exist and are exact in A. Then the termwise colimit L^bullet = colim L_n^bullet is a homotopy colimit of the system in D(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $L_n^\\bullet$\nbe a system of complexes of $\\mathcal{A}$. Assume\ncolimits over $\\mathbf{N}$ exist and are exact in $\\mathcal{A}$.\nThen the termwise\ncolimit $L^\\bullet = \\colim L_n^\\bullet$ is a homotopy colimit of the\nsystem in $D(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0949","source_file":"derived.tex","source_line":10351,"source_end_line":10359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10351-L10359","statement_sha256":"a5269a74de00c62bbc77fd22945c99612c4d763aa5a301cd4e8f2be8a1a7e459","origin":"The Stacks Project","memory_eligible":false,"source_rank":2642,"rank":2642,"depth":2,"x":252.531,"y":759.63,"cluster":"derived-categories"},{"id":"stacks:0CRK","tag":"0CRK","title":"Derived colimits · Lemma 0CRK","summary":"Let D be a triangulated category having countable direct sums. Let A be an abelian category with exact colimits over N. Let H : D → A be a homological functor commuting with countable direct sums. Then H(hocolim K_n) = colim H(K_n) for any system of objects of D.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category having countable\ndirect sums. Let $\\mathcal{A}$ be an abelian category with exact\ncolimits over $\\mathbf{N}$.\nLet $H : \\mathcal{D} \\to \\mathcal{A}$ be a homological functor\ncommuting with countable direct sums.\nThen $H(\\text{hocolim} K_n) = \\colim H(K_n)$\nfor any system of objects of $\\mathcal{D}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRK","source_file":"derived.tex","source_line":10377,"source_end_line":10386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10377-L10386","statement_sha256":"b14c835fe90d6539de89698d1a4c288f3b9172c005dd9cc181a71d8418ffb8d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2643,"rank":2643,"depth":1,"x":149.16,"y":633.959,"cluster":"derived-categories"},{"id":"stacks:094A","tag":"094A","title":"Derived colimits · Lemma 094A","summary":"Let D be a triangulated category with countable direct sums. Let K ∈ D be an object such that for every countable set of objects E_n ∈ D the canonical map bigoplus Hom_D(K, E_n) → Hom_D(K, bigoplus E_n) is a bijection. Then, given any system L_n of D over N whose derived colimit L = hocolim L_n exists we have that colim Hom_D(K, L_n) → Hom_D(K, L) is a bijection.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category with countable direct sums.\nLet $K \\in \\mathcal{D}$ be an object such that for every\ncountable set of objects $E_n \\in \\mathcal{D}$ the canonical map\n$$\n\\bigoplus \\Hom_\\mathcal{D}(K, E_n)\n\\longrightarrow\n\\Hom_\\mathcal{D}(K, \\bigoplus E_n)\n$$\nis a bijection. Then, given any system $L_n$ of $\\mathcal{D}$ over\n$\\mathbf{N}$ whose derived colimit $L = \\text{hocolim} L_n$\nexists we have that\n$$\n\\colim \\Hom_\\mathcal{D}(K, L_n) \\longrightarrow \\Hom_\\mathcal{D}(K, L)\n$$\nis a bijection.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094A","source_file":"derived.tex","source_line":10405,"source_end_line":10422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10405-L10422","statement_sha256":"4a180ff0025b6c3b2db3ad042b0d7da8a6f6f007c228eae4c10d5b2aed4c05e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2644,"rank":2644,"depth":2,"x":326.861,"y":668.052,"cluster":"derived-categories"},{"id":"stacks:08TC","tag":"08TC","title":"Derived limits · Definition 08TC","summary":"Let D be a triangulated category. Let (K_n, f_n) be an inverse system of objects of D. We say an object K is a derived limit, or a homotopy limit of the system (K_n) if the product ∏ K_n exists and there is a distinguished triangle K → ∏ K_n → ∏ K_n → K[1] where the map ∏ K_n → ∏ K_n is given by (k_n) ↦ (k_n - f_n + 1(k_n + 1)). If this is the case, then we sometimes indicate this by the notation K = Rlim K_n.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $(K_n, f_n)$ be an inverse system of objects of $\\mathcal{D}$.\nWe say an object $K$ is a {\\it derived limit}, or a\n{\\it homotopy limit} of the system $(K_n)$ if\nthe product $\\prod K_n$ exists and there is a distinguished triangle\n$$\nK \\to \\prod K_n \\to \\prod K_n \\to K[1]\n$$\nwhere the map $\\prod K_n \\to \\prod K_n$ is given\nby $(k_n) \\mapsto (k_n - f_{n + 1}(k_{n + 1}))$. If this is the\ncase, then we sometimes indicate this by the notation $K = R\\lim K_n$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived limits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TC","source_file":"derived.tex","source_line":10441,"source_end_line":10454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10441-L10454","statement_sha256":"43d93f55e70e20f0ecf2979670affa8952a72ff2b0eea4d4a9ec8ac0f3e3f574","origin":"The Stacks Project","memory_eligible":false,"source_rank":2645,"rank":2645,"depth":0,"x":168.04,"y":743.92,"cluster":"derived-categories"},{"id":"stacks:07KC","tag":"07KC","title":"Derived limits · Lemma 07KC","summary":"Let A be an abelian category with exact countable products. Then • D(A) has countable products, • countable products ∏ K_i in D(A) are obtained by taking termwise products of any complexes representing the K_i, and • H^p(∏ K_i) = ∏ H^p(K_i).","statement_latex":"Let $\\mathcal{A}$ be an abelian category with exact\ncountable products. Then\n\\begin{enumerate}\n\\item $D(\\mathcal{A})$ has countable products,\n\\item countable products $\\prod K_i$ in $D(\\mathcal{A})$ are obtained by\ntaking termwise products of any complexes representing the $K_i$, and\n\\item $H^p(\\prod K_i) = \\prod H^p(K_i)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KC","source_file":"derived.tex","source_line":10474,"source_end_line":10484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10474-L10484","statement_sha256":"5f22f6421f4b89ab2d94465f3fcdc3e2ea27e0fdc7bc7049d7e7aa2fe2c89185","origin":"The Stacks Project","memory_eligible":false,"source_rank":2646,"rank":2646,"depth":0,"x":224.272,"y":597.518,"cluster":"derived-categories"},{"id":"stacks:0BK7","tag":"0BK7","title":"Derived limits · Lemma 0BK7","summary":"Let A be an abelian category with countable products and enough injectives. Let (K_n) be an inverse system of D^+(A). Then Rlim K_n exists.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with countable products and\nenough injectives. Let $(K_n)$ be an inverse system of $D^+(\\mathcal{A})$.\nThen $R\\lim K_n$ exists.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BK7","source_file":"derived.tex","source_line":10509,"source_end_line":10514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10509-L10514","statement_sha256":"8ccb6e935f88e87cbbdfdd394f3e4c2ef88dbdb13e908983fdcf3c7576c9ad90","origin":"The Stacks Project","memory_eligible":false,"source_rank":2647,"rank":2647,"depth":3,"x":300.716,"y":737.703,"cluster":"derived-categories"},{"id":"stacks:070M","tag":"070M","title":"Derived limits · Lemma 070M","summary":"Let A be an abelian category with countable products and enough injectives. Let K^bullet be a complex. Let I_n^bullet be the inverse system of bounded below complexes of injectives produced by Lemma [Tag 070F]. Then I^bullet = lim I_n^bullet exists, is K-injective, represents Rlim τ_≥ -nK^bullet in D(A), and the following are equivalent • the map K^bullet → I^bullet (see proof) is a quasi-isomorphism, • the map K^bullet → Rlim τ_≥ -nK^bullet of Remark [Tag 0H72] is an…","statement_latex":"Let $\\mathcal{A}$ be an abelian category with countable products and\nenough injectives. Let $K^\\bullet$ be a complex. Let $I_n^\\bullet$ be\nthe inverse system of bounded below complexes of injectives produced by\nLemma \\ref{lemma-special-inverse-system}. Then\n$I^\\bullet = \\lim I_n^\\bullet$ exists, is K-injective, represents\n$R\\lim \\tau_{\\geq -n}K^\\bullet$ in $D(\\mathcal{A})$,\nand the following are equivalent\n\\begin{enumerate}\n\\item the map $K^\\bullet \\to I^\\bullet$ (see proof) is a quasi-isomorphism,\n\\item the map $K^\\bullet \\to R\\lim \\tau_{\\geq -n}K^\\bullet$\nof Remark \\ref{remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{A})$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070M","source_file":"derived.tex","source_line":10588,"source_end_line":10603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10588-L10603","statement_sha256":"a8d7c807123707652910a110026299fc9d663562765ee6da1d0f648b5c30a744","origin":"The Stacks Project","memory_eligible":false,"source_rank":2648,"rank":2648,"depth":15,"x":131.225,"y":677.571,"cluster":"derived-categories"},{"id":"stacks:090Y","tag":"090Y","title":"Derived limits · Lemma 090Y","summary":"Let A be an abelian category having enough injectives and exact countable products. Then for every complex there is a quasi-isomorphism to a K-injective complex.","statement_latex":"Let $\\mathcal{A}$ be an abelian category having enough injectives\nand exact countable products. Then for every complex\nthere is a quasi-isomorphism to a K-injective complex.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/090Y","source_file":"derived.tex","source_line":10662,"source_end_line":10667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10662-L10667","statement_sha256":"5e4008b5f271e4e124fc8d4459e7864f789427a7a2ded9df4f183c71ca0fe5fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2649,"rank":2649,"depth":16,"x":304.959,"y":625.62,"cluster":"derived-categories"},{"id":"stacks:0FX1","tag":"0FX1","title":"Operations on full subcategories · Lemma 0FX1","summary":"Let T be a triangulated category. Given full subcategories A, B, C we have (A star B) star C = A star (B star C).","statement_latex":"Let $\\mathcal{T}$ be a triangulated category.\nGiven full subcategories $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$\nwe have $(\\mathcal{A} \\star \\mathcal{B}) \\star \\mathcal{C} =\n\\mathcal{A} \\star (\\mathcal{B} \\star \\mathcal{C})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Operations on full subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FX1","source_file":"derived.tex","source_line":10724,"source_end_line":10730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10724-L10730","statement_sha256":"1db1fa3113690a05c25b8115dd0278c6ae64271c27d401b1cee465484a05b4d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2650,"rank":2650,"depth":0,"x":218.431,"y":762.821,"cluster":"derived-categories"},{"id":"stacks:0FX2","tag":"0FX2","title":"Operations on full subcategories · Lemma 0FX2","summary":"Let T be a triangulated category. Given full subcategories A, B we have smd(A) star smd(B) ⊂ smd(A star B) and smd(smd(A) star smd(B)) = smd(A star B).","statement_latex":"Let $\\mathcal{T}$ be a triangulated category.\nGiven full subcategories $\\mathcal{A}$, $\\mathcal{B}$\nwe have\n$smd(\\mathcal{A}) \\star smd(\\mathcal{B}) \\subset\nsmd(\\mathcal{A} \\star \\mathcal{B})$ and\n$smd(smd(\\mathcal{A}) \\star smd(\\mathcal{B})) =\nsmd(\\mathcal{A} \\star \\mathcal{B})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Operations on full subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FX2","source_file":"derived.tex","source_line":10738,"source_end_line":10747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10738-L10747","statement_sha256":"260d89bfb7fa5007ea15a031d12bb9bdde4bdea5d0919cb865f613ef8efccd76","origin":"The Stacks Project","memory_eligible":false,"source_rank":2651,"rank":2651,"depth":0,"x":171.798,"y":612.211,"cluster":"derived-categories"},{"id":"stacks:0FX3","tag":"0FX3","title":"Operations on full subcategories · Lemma 0FX3","summary":"Let T be a triangulated category. Given full subcategories A, B the full subcategories add(A) star add(B) and smd(add(A)) are closed under direct sums.","statement_latex":"Let $\\mathcal{T}$ be a triangulated category. Given full subcategories\n$\\mathcal{A}$, $\\mathcal{B}$ the full subcategories\n$add(\\mathcal{A}) \\star add(\\mathcal{B})$ and\n$smd(add(\\mathcal{A}))$ are closed under direct sums.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Operations on full subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FX3","source_file":"derived.tex","source_line":10758,"source_end_line":10764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10758-L10764","statement_sha256":"7c4c1eb6a40f00408fea4033bb8ca8c748e275c95687770446e90174cd6c78cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2652,"rank":2652,"depth":0,"x":327.651,"y":696.999,"cluster":"derived-categories"},{"id":"stacks:0FX4","tag":"0FX4","title":"Operations on full subcategories · Lemma 0FX4","summary":"Let T be a triangulated category. Given a full subcategory A for n ≥ 1 the subcategory C_n = smd(add(A)^star n) = smd(add(A) star … star add(A)) defined above is a strictly full subcategory of T closed under direct sums and direct summands and C_n + m = smd(C_n star C_m) for all n, m ≥ 1.","statement_latex":"Let $\\mathcal{T}$ be a triangulated category. Given a full subcategory\n$\\mathcal{A}$ for $n \\geq 1$ the subcategory\n$$\n\\mathcal{C}_n = smd(add(\\mathcal{A})^{\\star n}) =\nsmd(add(\\mathcal{A}) \\star \\ldots \\star add(\\mathcal{A}))\n$$\ndefined above is a strictly full subcategory of $\\mathcal{T}$\nclosed under direct sums and direct summands and\n$\\mathcal{C}_{n + m} = smd(\\mathcal{C}_n \\star \\mathcal{C}_m)$\nfor all $n, m \\geq 1$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Operations on full subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FX4","source_file":"derived.tex","source_line":10774,"source_end_line":10786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10774-L10786","statement_sha256":"d9b12734e0d8152ba5db218c12220d007dda3cad88fd91b7fbf2795e7bdd48f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2653,"rank":2653,"depth":1,"x":144.131,"y":722.971,"cluster":"derived-categories"},{"id":"stacks:0FX7","tag":"0FX7","title":"Operations on full subcategories · Lemma 0FX7","summary":"Let A be an abelian category. Let D = D(A). Let E ⊂ Ob(A) be a subset which we view as a subset of Ob(D) also. Let K be an object of D. • Let b ≥ a and assume H^i(K) is zero for i not ∈ [a, b] and H^i(K) ∈ E if i ∈ [a, b]. Then K is in smd(add(E[a, b])^star (b - a + 1)). • Let b ≥ a and assume H^i(K) is zero for i not ∈ [a, b] and H^i(K) ∈ smd(add(E)) if i ∈ [a, b]. Then K is in smd(add(E[a, b])^star (b - a + 1)). • Let b ≥ a and assume K can be represented by a complex…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D} = D(\\mathcal{A})$.\nLet $\\mathcal{E} \\subset \\Ob(\\mathcal{A})$ be a subset which we view as\na subset of $\\Ob(\\mathcal{D})$ also. Let $K$ be an object of $\\mathcal{D}$.\n\\begin{enumerate}\n\\item Let $b \\geq a$ and assume $H^i(K)$ is zero for $i \\not \\in [a, b]$\nand $H^i(K) \\in \\mathcal{E}$ if $i \\in [a, b]$. Then $K$ is in\n$smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$.\n\\item Let $b \\geq a$ and assume $H^i(K)$ is zero for $i \\not \\in [a, b]$\nand $H^i(K) \\in smd(add(\\mathcal{E}))$ if $i \\in [a, b]$. Then $K$ is in\n$smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$.\n\\item Let $b \\geq a$ and assume $K$ can be represented by a complex $K^\\bullet$\nwith $K^i = 0$ for $i \\not \\in [a, b]$ and $K^i \\in \\mathcal{E}$ for\n$i \\in [a, b]$. Then $K$ is in\n$smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$.\n\\item Let $b \\geq a$ and assume $K$ can be represented by a complex $K^\\bullet$\nwith $K^i = 0$ for $i \\not \\in [a, b]$ and $K^i \\in smd(add(\\mathcal{E}))$ for\n$i \\in [a, b]$. Then $K$ is in\n$smd(add(\\mathcal{E}[a, b])^{\\star (b - a + 1)})$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Operations on full subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FX7","source_file":"derived.tex","source_line":10846,"source_end_line":10867,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10846-L10867","statement_sha256":"1789b799245db7a66b49399f06003a522378eb458bed071d82f142adf410c4b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2654,"rank":2654,"depth":2,"x":258.826,"y":599.41,"cluster":"derived-categories"},{"id":"stacks:0FX8","tag":"0FX8","title":"Operations on full subcategories · Lemma 0FX8","summary":"Let T be a triangulated category. Let H : T → A be a homological functor to an abelian category A. Let a ≤ b and E ⊂ Ob(T) be a subset such that H^i(E) = 0 for E ∈ E and i not ∈ [a, b]. Then for X ∈ smd(add(E[-m, m])^star n) we have H^i(X) = 0 for i not ∈ [-m + na, m + nb].","statement_latex":"Let $\\mathcal{T}$ be a triangulated category. Let\n$H : \\mathcal{T} \\to \\mathcal{A}$ be a homological functor\nto an abelian category $\\mathcal{A}$.\nLet $a \\leq b$ and $\\mathcal{E} \\subset \\Ob(\\mathcal{T})$\nbe a subset such that $H^i(E) = 0$ for $E \\in \\mathcal{E}$\nand $i \\not \\in [a, b]$.\nThen for $X \\in smd(add(\\mathcal{E}[-m, m])^{\\star n})$\nwe have $H^i(X) = 0$ for $i \\not \\in [-m + na, m + nb]$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Operations on full subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FX8","source_file":"derived.tex","source_line":10881,"source_end_line":10891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10881-L10891","statement_sha256":"36c1d2ea9ae46d01c0315ab7f7ad7a678578ec74eb8c8d27c6ac39693edb1b93","origin":"The Stacks Project","memory_eligible":false,"source_rank":2655,"rank":2655,"depth":0,"x":273.649,"y":755.951,"cluster":"derived-categories"},{"id":"stacks:0FX9","tag":"0FX9","title":"Generators of triangulated categories · Lemma 0FX9","summary":"Let T be a triangulated category. Let E be an object of T. For n ≥ 1 we have langle E rangle_n = smd(langle E rangle_1 star … star langle E rangle_1) = smd(langle E rangle_1^star n) = ⋃_m ≥ 1 smd(add(E[-m, m])^star n) For n, n' ≥ 1 we have langle E rangle_n + n' = smd(langle E rangle_n star langle E rangle_n').","statement_latex":"Let $\\mathcal{T}$ be a triangulated category. Let $E$ be an object\nof $\\mathcal{T}$. For $n \\geq 1$ we have\n$$\n\\langle E \\rangle_n =\nsmd(\\langle E \\rangle_1 \\star \\ldots \\star \\langle E \\rangle_1) =\nsmd({\\langle E \\rangle_1}^{\\star n}) =\n\\bigcup\\nolimits_{m \\geq 1} smd(add(E[-m, m])^{\\star n})\n$$\nFor $n, n' \\geq 1$ we have $\\langle E \\rangle_{n + n'} =\nsmd(\\langle E \\rangle_n \\star \\langle E \\rangle_{n'})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Generators of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FX9","source_file":"derived.tex","source_line":10948,"source_end_line":10960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10948-L10960","statement_sha256":"74bfea583e4d34e289525e42525f0ac4db386456260d3bfc58b094f7b8cba2af","origin":"The Stacks Project","memory_eligible":false,"source_rank":2656,"rank":2656,"depth":2,"x":136.53,"y":648.693,"cluster":"derived-categories"},{"id":"stacks:0ATG","tag":"0ATG","title":"Generators of triangulated categories · Lemma 0ATG","summary":"Let D be a triangulated category. Let E be an object of D. The subcategory langle E rangle = ⋃_n langle E rangle_n = ⋃_n, m ≥ 1 smd(add(E[-m, m])^star n) is a strictly full, saturated, triangulated subcategory of D and it is the smallest such subcategory of D containing the object E.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object\nof $\\mathcal{D}$. The subcategory\n$$\n\\langle E \\rangle = \\bigcup\\nolimits_n \\langle E \\rangle_n\n= \\bigcup\\nolimits_{n, m \\geq 1} smd(add(E[-m, m])^{\\star n})\n$$\nis a strictly full, saturated, triangulated subcategory of $\\mathcal{D}$\nand it is the smallest such subcategory of $\\mathcal{D}$ containing\nthe object $E$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Generators of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATG","source_file":"derived.tex","source_line":10974,"source_end_line":10985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L10974-L10985","statement_sha256":"cc8d44d0906f7c2dd18b115cacd5d258465bbcb70bccba0e8c69ea185f883db6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2657,"rank":2657,"depth":3,"x":324.307,"y":649.982,"cluster":"derived-categories"},{"id":"stacks:09SJ","tag":"09SJ","title":"Generators of triangulated categories · Definition 09SJ","summary":"Let D be a triangulated category. Let E be an object of D. • We say E is a classical generator of D if the smallest strictly full, saturated, triangulated subcategory of D containing E is equal to D, in other words, if langle E rangle = D. • We say E is a strong generator of D if langle E rangle_n = D for some n ≥ 1. • We say E is a weak generator or a generator of D if for any nonzero object K of D there exists an integer n and a nonzero map E → K[n].","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object\nof $\\mathcal{D}$.\n\\begin{enumerate}\n\\item We say $E$ is a {\\it classical generator} of $\\mathcal{D}$\nif the smallest strictly full, saturated, triangulated subcategory\nof $\\mathcal{D}$ containing $E$ is equal to $\\mathcal{D}$, in\nother words, if $\\langle E \\rangle = \\mathcal{D}$.\n\\item We say $E$ is a {\\it strong generator} of $\\mathcal{D}$\nif $\\langle E \\rangle_n = \\mathcal{D}$ for some $n \\geq 1$.\n\\item We say $E$ is a {\\it weak generator} or a {\\it generator}\nof $\\mathcal{D}$\nif for any nonzero object $K$ of $\\mathcal{D}$ there exists\nan integer $n$ and a nonzero map $E \\to K[n]$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Generators of triangulated categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SJ","source_file":"derived.tex","source_line":11012,"source_end_line":11028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11012-L11028","statement_sha256":"1b64ba8c962f670351ad0bed73f62097fda39c8e0bf1c44f4a77b316c33f6aa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2658,"rank":2658,"depth":0,"x":184.5,"y":755.812,"cluster":"derived-categories"},{"id":"stacks:09SK","tag":"09SK","title":"Generators of triangulated categories · Lemma 09SK","summary":"Let D be a triangulated category. Let E, K be objects of D. The following are equivalent • Hom(E, K[i]) = 0 for all i ∈ Z, • Hom(E', K) = 0 for all E' ∈ langle E rangle.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let $E, K$ be objects\nof $\\mathcal{D}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\Hom(E, K[i]) = 0$ for all $i \\in \\mathbf{Z}$,\n\\item $\\Hom(E', K) = 0$ for all $E' \\in \\langle E \\rangle$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Generators of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SK","source_file":"derived.tex","source_line":11033,"source_end_line":11041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11033-L11041","statement_sha256":"0a89e152c31d0b4233975aa60226f9f58ccd833df42db4b120efe8c5f4133829","origin":"The Stacks Project","memory_eligible":false,"source_rank":2659,"rank":2659,"depth":2,"x":202.525,"y":598.101,"cluster":"derived-categories"},{"id":"stacks:09SL","tag":"09SL","title":"Generators of triangulated categories · Lemma 09SL","summary":"Let D be a triangulated category. Let E be an object of D. If E is a classical generator of D, then E is a generator.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let $E$ be an object\nof $\\mathcal{D}$. If $E$ is a classical generator of $\\mathcal{D}$,\nthen $E$ is a generator.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Generators of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SL","source_file":"derived.tex","source_line":11052,"source_end_line":11057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11052-L11057","statement_sha256":"42fa7218772e8e4a8a4af4dddba1276d4d405e8fa9e2320e4075ad5992110730","origin":"The Stacks Project","memory_eligible":false,"source_rank":2660,"rank":2660,"depth":3,"x":316.308,"y":724.899,"cluster":"derived-categories"},{"id":"stacks:0FXA","tag":"0FXA","title":"Generators of triangulated categories · Lemma 0FXA","summary":"Let D be a triangulated category which has a strong generator. Let E be an object of D. If E is a classical generator of D, then E is a strong generator.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category which has a strong generator.\nLet $E$ be an object of $\\mathcal{D}$. If $E$ is a classical generator of\n$\\mathcal{D}$, then $E$ is a strong generator.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Generators of triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXA","source_file":"derived.tex","source_line":11068,"source_end_line":11073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11068-L11073","statement_sha256":"c2970bbd5ab73ec1fc8ca5778c01c6acecd28be611c2be8cb4956a5e20cc5f2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2661,"rank":2661,"depth":4,"x":130.036,"y":695.9,"cluster":"derived-categories"},{"id":"stacks:07LS","tag":"07LS","title":"Compact objects · Definition 07LS","summary":"Let D be an additive category with arbitrary direct sums. A compact object of D is an object K such that the map bigoplus_i ∈ I Hom_D(K, E_i) → Hom_D(K, bigoplus_i ∈ I E_i) is bijective for any set I and objects E_i ∈ Ob(D) parametrized by i ∈ I.","statement_latex":"Let $\\mathcal{D}$ be an additive category with arbitrary direct\nsums. A {\\it compact object} of $\\mathcal{D}$ is an object $K$\nsuch that the map\n$$\n\\bigoplus\\nolimits_{i \\in I} \\Hom_{\\mathcal{D}}(K, E_i)\n\\longrightarrow\n\\Hom_{\\mathcal{D}}(K, \\bigoplus\\nolimits_{i \\in I} E_i)\n$$\nis bijective for any set $I$ and objects\n$E_i \\in \\Ob(\\mathcal{D})$ parametrized by $i \\in I$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Compact objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LS","source_file":"derived.tex","source_line":11127,"source_end_line":11139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11127-L11139","statement_sha256":"7ad3fa26cb9737798d69d065e008a6cbeeeb01b9f18e5d3b05cac2fbaf1cd0e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2662,"rank":2662,"depth":0,"x":291.057,"y":611.406,"cluster":"derived-categories"},{"id":"stacks:09QH","tag":"09QH","title":"Compact objects · Lemma 09QH","summary":"Let D be a (pre-)triangulated category with direct sums. Then the compact objects of D form the objects of a Karoubian, saturated, strictly full, (pre-)triangulated subcategory D_c of D.","statement_latex":"Let $\\mathcal{D}$ be a (pre-)triangulated category with direct sums.\nThen the compact objects of $\\mathcal{D}$ form the objects of a\nKaroubian, saturated, strictly full, (pre-)triangulated subcategory\n$\\mathcal{D}_c$ of $\\mathcal{D}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QH","source_file":"derived.tex","source_line":11146,"source_end_line":11152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11146-L11152","statement_sha256":"3c80e58a6152c7cc5bd11c32fac31bbcd752e658c44d0fb2ec62bbd1fa24f2e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2663,"rank":2663,"depth":8,"x":240.16,"y":765.408,"cluster":"derived-categories"},{"id":"stacks:09SN","tag":"09SN","title":"Compact objects · Lemma 09SN","summary":"Let D be a triangulated category with direct sums. Let E_i, i ∈ I be a family of compact objects of D such that bigoplus E_i generates D. Then every object X of D can be written as X = hocolim X_n where X_1 is a direct sum of shifts of the E_i and each transition morphism fits into a distinguished triangle Y_n → X_n → X_n + 1 → Y_n[1] where Y_n is a direct sum of shifts of the E_i.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category with direct sums.\nLet $E_i$, $i \\in I$ be a family of compact objects of $\\mathcal{D}$\nsuch that $\\bigoplus E_i$ generates $\\mathcal{D}$.\nThen every object $X$ of $\\mathcal{D}$ can be written as\n$$\nX = \\text{hocolim} X_n\n$$\nwhere $X_1$ is a direct sum of shifts of the $E_i$ and each transition\nmorphism fits into a distinguished triangle\n$Y_n \\to X_n \\to X_{n + 1} \\to Y_n[1]$\nwhere $Y_n$ is a direct sum of shifts of the $E_i$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SN","source_file":"derived.tex","source_line":11168,"source_end_line":11181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11168-L11181","statement_sha256":"a8747817d1858e0c41104a3cba9af168e7423c416b5ddd5358e2a246ca5cedc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2664,"rank":2664,"depth":3,"x":153.662,"y":622.664,"cluster":"derived-categories"},{"id":"stacks:09SP","tag":"09SP","title":"Compact objects · Lemma 09SP","summary":"With assumptions and notation as in Lemma [Tag 09SN]. If C is a compact object and C → X_n is a morphism, then there is a factorization C → E → X_n where E is an object of langle E_i_1 ⊕ … ⊕ E_i_t rangle for some i_1, …, i_t ∈ I.","statement_latex":"With assumptions and notation as in Lemma \\ref{lemma-write-as-colimit}.\nIf $C$ is a compact object and $C \\to X_n$ is a morphism, then\nthere is a factorization $C \\to E \\to X_n$ where\n$E$ is an object of $\\langle E_{i_1} \\oplus \\ldots \\oplus E_{i_t} \\rangle$\nfor some $i_1, \\ldots, i_t \\in I$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SP","source_file":"derived.tex","source_line":11214,"source_end_line":11221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11214-L11221","statement_sha256":"82ca4de07ad7be0f4c90c1dbaf8971f84bfb264f6cdd4100e4c72231d0925cda","origin":"The Stacks Project","memory_eligible":false,"source_rank":2665,"rank":2665,"depth":4,"x":332.616,"y":678.96,"cluster":"derived-categories"},{"id":"stacks:09SQ","tag":"09SQ","title":"Compact objects · Definition 09SQ","summary":"Let D be a triangulated category with arbitrary direct sums. We say D is compactly generated if there exists a set E_i, i ∈ I of compact objects such that bigoplus E_i generates D.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category with arbitrary direct\nsums. We say $\\mathcal{D}$ is {\\it compactly generated} if\nthere exists a set $E_i$, $i \\in I$ of compact objects such that\n$\\bigoplus E_i$ generates $\\mathcal{D}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Compact objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SQ","source_file":"derived.tex","source_line":11265,"source_end_line":11271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11265-L11271","statement_sha256":"ab9ba19c54bdf2851af13a0a0898c693ea3927e70e91df34b8b640a5aa15b3d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2666,"rank":2666,"depth":0,"x":155.01,"y":739.118,"cluster":"derived-categories"},{"id":"stacks:09SR","tag":"09SR","title":"Compact objects · Proposition 09SR","summary":"Let D be a triangulated category with direct sums. Let E be a compact object of D. The following are equivalent • E is a classical generator for D_c and D is compactly generated, and • E is a generator for D.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category with direct sums.\nLet $E$ be a compact object of $\\mathcal{D}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $E$ is a classical generator for $\\mathcal{D}_c$ and\n$\\mathcal{D}$ is compactly generated, and\n\\item $E$ is a generator for $\\mathcal{D}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Compact objects","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SR","source_file":"derived.tex","source_line":11277,"source_end_line":11287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11277-L11287","statement_sha256":"48febd1176921e981ec476eb115b2fcf8d02da2bf2875943e3698712506cb2d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2667,"rank":2667,"depth":5,"x":237.771,"y":593.674,"cluster":"derived-categories"},{"id":"stacks:0A8F","tag":"0A8F","title":"Brown representability · Lemma 0A8F","summary":"[Neeman-Grothendieck]. Let D be a triangulated category with direct sums which is compactly generated. Let H : D → Ab be a contravariant cohomological functor which transforms direct sums into products. Then H is representable.","statement_latex":"\\begin{reference}\n\\cite[Theorem 3.1]{Neeman-Grothendieck}.\n\\end{reference}\nLet $\\mathcal{D}$ be a triangulated category with direct sums which is\ncompactly generated. Let $H : \\mathcal{D} \\to \\textit{Ab}$ be a contravariant\ncohomological functor which transforms direct sums into products.\nThen $H$ is representable.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Brown representability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8F","source_file":"derived.tex","source_line":11328,"source_end_line":11337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11328-L11337","statement_sha256":"1aec8048c3812d6793de378695921b83a4cdb033d058fa0ee43b5f5d656862a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2668,"rank":2668,"depth":4,"x":293.824,"y":748.208,"cluster":"derived-categories"},{"id":"stacks:0A8G","tag":"0A8G","title":"Brown representability · Proposition 0A8G","summary":"[Neeman-Grothendieck]. Let D be a triangulated category with direct sums which is compactly generated. Let F : D → D' be an exact functor of triangulated categories which transforms direct sums into direct sums. Then F has an exact right adjoint.","statement_latex":"\\begin{reference}\n\\cite[Theorem 4.1]{Neeman-Grothendieck}.\n\\end{reference}\nLet $\\mathcal{D}$ be a triangulated category with direct sums which is\ncompactly generated. Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be an\nexact functor of triangulated categories which transforms direct sums\ninto direct sums. Then $F$ has an exact right adjoint.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Brown representability","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8G","source_file":"derived.tex","source_line":11427,"source_end_line":11436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11427-L11436","statement_sha256":"bbc69cddc6237bbc8f65967507c8c189abcb22e3fa2b8cae7ea2e242a7a354f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2669,"rank":2669,"depth":5,"x":127.873,"y":665.89,"cluster":"derived-categories"},{"id":"stacks:0GYG","tag":"0GYG","title":"Brown representability, bis · Lemma 0GYG","summary":"Weak version of [Krause] Let D be a triangulated category with direct sums. Suppose given a set E of objects of D such that • if X is a nonzero object of D, then there exists an E ∈ E and a nonzero map E → X, and • given objects X_n, n ∈ N of D, E ∈ E, and α : E → bigoplus X_n, there exist E_n ∈ E and β_n : E_n → X_n and a morphism γ : E → bigoplus E_n such that α = (bigoplus β_n) ∘ γ. Let H : D → Ab be a contravariant cohomological functor which transforms direct sums…","statement_latex":"\\begin{reference}\nWeak version of \\cite[Theorem A]{Krause}\n\\end{reference}\nLet $\\mathcal{D}$ be a triangulated category with direct sums.\nSuppose given a set $\\mathcal{E}$ of objects of $\\mathcal{D}$ such that\n\\begin{enumerate}\n\\item if $X$ is a nonzero object of $\\mathcal{D}$, then there exists\nan $E \\in \\mathcal{E}$ and a nonzero map $E \\to X$, and\n\\item given objects $X_n$, $n \\in \\mathbf{N}$ of $\\mathcal{D}$,\n$E \\in \\mathcal{E}$, and $\\alpha : E \\to \\bigoplus X_n$,\nthere exist $E_n \\in \\mathcal{E}$ and $\\beta_n : E_n \\to X_n$ and a morphism\n$\\gamma : E \\to \\bigoplus E_n$ such that\n$\\alpha = (\\bigoplus \\beta_n) \\circ \\gamma$.\n\\end{enumerate}\nLet $H : \\mathcal{D} \\to \\textit{Ab}$ be a contravariant cohomological functor\nwhich transforms direct sums into products. Then $H$ is representable.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Brown representability, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYG","source_file":"derived.tex","source_line":11468,"source_end_line":11486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11468-L11486","statement_sha256":"cd816e0f890f06e57c55af45df1ce8d7caf869fee19e2b52f54df7b6dc3eef1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2670,"rank":2670,"depth":5,"x":316.834,"y":632.356,"cluster":"derived-categories"},{"id":"stacks:0GYH","tag":"0GYH","title":"Brown representability, bis · Proposition 0GYH","summary":"Let D be a triangulated category with direct sums. Assume there exists a set E of objects of D satisfying conditions (1) and (2) of Lemma [Tag 0GYG]. Let F : D → D' be an exact functor of triangulated categories which transforms direct sums into direct sums. Then F has an exact right adjoint.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category with direct sums.\nAssume there exists a set $\\mathcal{E}$ of objects of $\\mathcal{D}$\nsatisfying conditions (1) and (2) of Lemma \\ref{lemma-brown-bis}.\nLet $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of triangulated\ncategories which transforms direct sums into direct sums.\nThen $F$ has an exact right adjoint.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Brown representability, bis","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYH","source_file":"derived.tex","source_line":11633,"source_end_line":11641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11633-L11641","statement_sha256":"1004c77f1e2226bae8e6c08b166371188761528ba1821c701b0f6f8c437b21a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2671,"rank":2671,"depth":6,"x":204.231,"y":764.578,"cluster":"derived-categories"},{"id":"stacks:0FXB","tag":"0FXB","title":"Admissible subcategories · Definition 0FXB","summary":"Let D be an additive category. Let A ⊂ D be a full subcategory. The right orthogonal A^perp of A is the full subcategory consisting of the objects X of D such that Hom(A, X) = 0 for all A ∈ Ob(A). The left orthogonal ^perpA of A is the full subcategory consisting of the objects X of D such that Hom(X, A) = 0 for all A ∈ Ob(A).","statement_latex":"Let $\\mathcal{D}$ be an additive category. Let $\\mathcal{A} \\subset \\mathcal{D}$\nbe a full subcategory. The {\\it right orthogonal} $\\mathcal{A}^\\perp$ of\n$\\mathcal{A}$ is the full subcategory consisting of the objects $X$ of\n$\\mathcal{D}$ such that $\\Hom(A, X) = 0$ for all $A \\in \\Ob(\\mathcal{A})$.\nThe {\\it left orthogonal} ${}^\\perp\\mathcal{A}$ of\n$\\mathcal{A}$ is the full subcategory consisting of the objects $X$ of\n$\\mathcal{D}$ such that $\\Hom(X, A) = 0$ for all $A \\in \\Ob(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXB","source_file":"derived.tex","source_line":11671,"source_end_line":11680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11671-L11680","statement_sha256":"c855b441897987e9a4b9e028ae524e5df5220b0e1d22fe0afd0508ec939f199a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2672,"rank":2672,"depth":0,"x":180.885,"y":602.852,"cluster":"derived-categories"},{"id":"stacks:0CQQ","tag":"0CQQ","title":"Admissible subcategories · Lemma 0CQQ","summary":"Let D be a triangulated category. Let A ⊂ D be a full subcategory invariant under all shifts. Consider a distinguished triangle X → Y → Z → X[1] of D. The following are equivalent • Z is in A^perp, and • Hom(A, X) = Hom(A, Y) for all A ∈ Ob(A).","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $\\mathcal{A} \\subset \\mathcal{D}$\nbe a full subcategory invariant under all shifts.\nConsider a distinguished triangle\n$$\nX \\to Y \\to Z \\to X[1]\n$$\nof $\\mathcal{D}$. The following are equivalent\n\\begin{enumerate}\n\\item $Z$ is in $\\mathcal{A}^\\perp$, and\n\\item $\\Hom(A, X) = \\Hom(A, Y)$ for all $A \\in \\Ob(\\mathcal{A})$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQQ","source_file":"derived.tex","source_line":11682,"source_end_line":11696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11682-L11696","statement_sha256":"387909b3331912d58ae4ec5ed2beb84b86ab28569ac1656091d491a979d73690","origin":"The Stacks Project","memory_eligible":false,"source_rank":2673,"rank":2673,"depth":2,"x":328.458,"y":709.078,"cluster":"derived-categories"},{"id":"stacks:0H0M","tag":"0H0M","title":"Admissible subcategories · Lemma 0H0M","summary":"Let D be a triangulated category. Let B ⊂ D be a full subcategory invariant under all shifts. Consider a distinguished triangle X → Y → Z → X[1] of D. The following are equivalent • X is in ^perpB, and • Hom(Y, B) = Hom(Z, B) for all B ∈ Ob(B).","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $\\mathcal{B} \\subset \\mathcal{D}$\nbe a full subcategory invariant under all shifts.\nConsider a distinguished triangle\n$$\nX \\to Y \\to Z \\to X[1]\n$$\nof $\\mathcal{D}$. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is in ${}^\\perp\\mathcal{B}$, and\n\\item $\\Hom(Y, B) = \\Hom(Z, B)$ for all $B \\in \\Ob(\\mathcal{B})$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0M","source_file":"derived.tex","source_line":11717,"source_end_line":11731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11717-L11731","statement_sha256":"22d8989fee49c1712497a984a572285e5196b9def493e9b26bf1576564d0040a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2674,"rank":2674,"depth":3,"x":133.818,"y":714.496,"cluster":"derived-categories"},{"id":"stacks:0FXC","tag":"0FXC","title":"Admissible subcategories · Lemma 0FXC","summary":"Let D be a triangulated category. Let A ⊂ D be a full subcategory invariant under all shifts. Then both the right orthogonal A^perp and the left orthogonal ^perpA of A are strictly full, saturated, triangulated subcagories of D.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$\\mathcal{A} \\subset \\mathcal{D}$ be a full subcategory invariant\nunder all shifts. Then both the right orthogonal $\\mathcal{A}^\\perp$ and\nthe left orthogonal ${}^\\perp\\mathcal{A}$ of $\\mathcal{A}$\nare strictly full, saturated\\footnote{Definition \\ref{definition-saturated}.},\ntriangulated subcagories of $\\mathcal{D}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXC","source_file":"derived.tex","source_line":11737,"source_end_line":11745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11737-L11745","statement_sha256":"a22b7b09821a47e7fcc6be3aab0ee33d96aba55a9b1767e7a49a9bc6a14e576d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2675,"rank":2675,"depth":3,"x":273.27,"y":599.824,"cluster":"derived-categories"},{"id":"stacks:0CQR","tag":"0CQR","title":"Admissible subcategories · Lemma 0CQR","summary":"Let D be a triangulated category. Let A be a full triangulated subcategory of D. For an object X of D consider the property P(X): there exists a distinguished triangle A → X → B → A[1] in D with A in A and B in A^perp. • If X_1 → X_2 → X_3 → X_1[1] is a distinguished triangle and P holds for two out of three, then it holds for the third. • If P holds for X_1 and X_2, then it holds for X_1 ⊕ X_2.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{A}$\nbe a full triangulated subcategory of $\\mathcal{D}$. For an object $X$\nof $\\mathcal{D}$ consider the property $P(X)$: there exists a\ndistinguished triangle $A \\to X \\to B \\to A[1]$\nin $\\mathcal{D}$ with $A$ in $\\mathcal{A}$ and $B$ in $\\mathcal{A}^\\perp$.\n\\begin{enumerate}\n\\item If $X_1 \\to X_2 \\to X_3 \\to X_1[1]$ is a distinguished triangle\nand $P$ holds for two out of three, then it holds for the third.\n\\item If $P$ holds for $X_1$ and $X_2$, then it holds for $X_1 \\oplus X_2$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQR","source_file":"derived.tex","source_line":11760,"source_end_line":11772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11760-L11772","statement_sha256":"97ca3903eaa67d46291b8053fce40cc36c239237927b4f2fc07753fa40f9d9a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2676,"rank":2676,"depth":4,"x":262.635,"y":763.843,"cluster":"derived-categories"},{"id":"stacks:0H0N","tag":"0H0N","title":"Admissible subcategories · Lemma 0H0N","summary":"Let D be a triangulated category. Let B be a full triangulated subcategory of D. For an object X of D consider the property P(X): there exists a distinguished triangle A → X → B → A[1] in D with B in B and A in ^perpB. • If X_1 → X_2 → X_3 → X_1[1] is a distinguished triangle and P holds for two out of three, then it holds for the third. • If P holds for X_1 and X_2, then it holds for X_1 ⊕ X_2.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let $\\mathcal{B}$\nbe a full triangulated subcategory of $\\mathcal{D}$. For an object $X$\nof $\\mathcal{D}$ consider the property $P(X)$: there exists a\ndistinguished triangle $A \\to X \\to B \\to A[1]$\nin $\\mathcal{D}$ with $B$ in $\\mathcal{B}$ and $A$ in ${}^\\perp\\mathcal{B}$.\n\\begin{enumerate}\n\\item If $X_1 \\to X_2 \\to X_3 \\to X_1[1]$ is a distinguished triangle\nand $P$ holds for two out of three, then it holds for the third.\n\\item If $P$ holds for $X_1$ and $X_2$, then it holds for $X_1 \\oplus X_2$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0N","source_file":"derived.tex","source_line":11813,"source_end_line":11825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11813-L11825","statement_sha256":"f8fdf30653a9f29a82fc5c5b74037ae89cd0ba4501e6489b1aaa8d39fac057a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2677,"rank":2677,"depth":5,"x":138.326,"y":636.606,"cluster":"derived-categories"},{"id":"stacks:0CQS","tag":"0CQS","title":"Admissible subcategories · Lemma 0CQS","summary":"Let D be a triangulated category. Let A ⊂ D be a full triangulated subcategory. The following are equivalent • the inclusion functor A → D has a right adjoint, and • for every X in D there exists a distinguished triangle A → X → B → A[1] in D with A ∈ Ob(A) and B ∈ Ob(A^perp). If this holds, then A is saturated (Definition [Tag 05RB]) and if A is strictly full in D, then A = ^perp(A^perp).","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$\\mathcal{A} \\subset \\mathcal{D}$ be a full triangulated subcategory.\nThe following are equivalent\n\\begin{enumerate}\n\\item the inclusion functor $\\mathcal{A} \\to \\mathcal{D}$\nhas a right adjoint, and\n\\item for every $X$ in $\\mathcal{D}$ there exists a distinguished\ntriangle\n$$\nA \\to X \\to B \\to A[1]\n$$\nin $\\mathcal{D}$ with $A \\in \\Ob(\\mathcal{A})$ and\n$B \\in \\Ob(\\mathcal{A}^\\perp)$.\n\\end{enumerate}\nIf this holds, then $\\mathcal{A}$ is saturated\n(Definition \\ref{definition-saturated}) and if $\\mathcal{A}$\nis strictly full in $\\mathcal{D}$, then\n$\\mathcal{A} = {}^\\perp(\\mathcal{A}^\\perp)$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQS","source_file":"derived.tex","source_line":11831,"source_end_line":11851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11831-L11851","statement_sha256":"c8073c7fecb8d4a6c55950c2481820ea4e284f19ec27834ff6fe7721cef7cae2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2678,"rank":2678,"depth":4,"x":332.702,"y":659.94,"cluster":"derived-categories"},{"id":"stacks:0CQT","tag":"0CQT","title":"Admissible subcategories · Lemma 0CQT","summary":"Let D be a triangulated category. Let B ⊂ D be a full triangulated subcategory. The following are equivalent • the inclusion functor B → D has a left adjoint, and • for every X in D there exists a distinguished triangle A → X → B → A[1] in D with B ∈ Ob(B) and A ∈ Ob(^perpB). If this holds, then B is saturated (Definition [Tag 05RB]) and if B is strictly full in D, then B = (^perpB)^perp.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$\\mathcal{B} \\subset \\mathcal{D}$ be a full triangulated subcategory.\nThe following are equivalent\n\\begin{enumerate}\n\\item the inclusion functor $\\mathcal{B} \\to \\mathcal{D}$\nhas a left adjoint, and\n\\item for every $X$ in $\\mathcal{D}$ there exists a distinguished\ntriangle\n$$\nA \\to X \\to B \\to A[1]\n$$\nin $\\mathcal{D}$ with $B \\in \\Ob(\\mathcal{B})$ and\n$A \\in \\Ob({}^\\perp\\mathcal{B})$.\n\\end{enumerate}\nIf this holds, then $\\mathcal{B}$ is saturated\n(Definition \\ref{definition-saturated}) and if $\\mathcal{B}$\nis strictly full in $\\mathcal{D}$, then\n$\\mathcal{B} = ({}^\\perp\\mathcal{B})^\\perp$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQT","source_file":"derived.tex","source_line":11900,"source_end_line":11920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11900-L11920","statement_sha256":"38067bf124cea2b824ff5e8f94a427eec4e3e31b7f265f4d2a373a48788b1c14","origin":"The Stacks Project","memory_eligible":false,"source_rank":2679,"rank":2679,"depth":5,"x":170.281,"y":753.214,"cluster":"derived-categories"},{"id":"stacks:0FXD","tag":"0FXD","title":"Admissible subcategories · Definition 0FXD","summary":"Let D be a triangulated category. A right admissible subcategory of D is a strictly full triangulated subcategory satisfying the equivalent conditions of Lemma [Tag 0CQS]. A left admissible subcategory of D is a strictly full triangulated subcategory satisfying the equivalent conditions of Lemma [Tag 0CQT]. A two-sided admissible subcategory is one which is both right and left admissible.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. A {\\it right admissible}\nsubcategory of $\\mathcal{D}$ is a strictly full triangulated subcategory\nsatisfying the equivalent conditions of Lemma \\ref{lemma-right-adjoint}.\nA {\\it left admissible}\nsubcategory of $\\mathcal{D}$ is a strictly full triangulated subcategory\nsatisfying the equivalent conditions of Lemma \\ref{lemma-left-adjoint}.\nA {\\it two-sided admissible} subcategory is one which is both\nright and left admissible.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXD","source_file":"derived.tex","source_line":11926,"source_end_line":11936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11926-L11936","statement_sha256":"e90b30aedeeee88b317917eb84a6fb414064e2d4972e9e21e81e7c3666f1e790","origin":"The Stacks Project","memory_eligible":false,"source_rank":2680,"rank":2680,"depth":6,"x":215.13,"y":591.952,"cluster":"derived-categories"},{"id":"stacks:0H0P","tag":"0H0P","title":"Admissible subcategories · Proposition 0H0P","summary":"Let D be a triangulated category. Let A ⊂ D and B ⊂ D be subcategories. The following are equivalent • A is right admissible and B = A^perp, • B is left admissible and A = ^perpB, • Hom(A, B) = 0 for all A ∈ A and B ∈ B and for every X in D there exists a distinguished triangle A → X → B → A[1] in D with A ∈ A and B ∈ B. If this is true, then A → D/B and B → D/A are equivalences of triangulated categories, the right adjoint to the inclusion functor A → D is D → D/B → A,…","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$\\mathcal{A} \\subset \\mathcal{D}$ and $\\mathcal{B} \\subset \\mathcal{D}$\nbe subcategories. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{A}$ is right admissible and $\\mathcal{B} = \\mathcal{A}^\\perp$,\n\\item $\\mathcal{B}$ is left admissible and $\\mathcal{A} = {}^\\perp\\mathcal{B}$,\n\\item $\\Hom(A, B) = 0$ for all $A \\in \\mathcal{A}$ and $B \\in \\mathcal{B}$\nand for every $X$ in $\\mathcal{D}$ there exists a distinguished triangle\n$A \\to X \\to B \\to A[1]$ in $\\mathcal{D}$ with $A \\in \\mathcal{A}$ and\n$B \\in \\mathcal{B}$.\n\\end{enumerate}\nIf this is true, then\n$\\mathcal{A} \\to \\mathcal{D}/\\mathcal{B}$ and\n$\\mathcal{B} \\to \\mathcal{D}/\\mathcal{A}$ are equivalences\nof triangulated categories,\nthe right adjoint to the inclusion functor $\\mathcal{A} \\to \\mathcal{D}$\nis $\\mathcal{D} \\to \\mathcal{D}/\\mathcal{B} \\to \\mathcal{A}$, and\nthe left adjoint to the inclusion functor $\\mathcal{B} \\to \\mathcal{D}$\nis $\\mathcal{D} \\to \\mathcal{D}/\\mathcal{A} \\to \\mathcal{B}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Admissible subcategories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0P","source_file":"derived.tex","source_line":11952,"source_end_line":11973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L11952-L11973","statement_sha256":"2287de421a90ade6da66c96dc38bea0c468554137ed983b1cdbf6a39133acd58","origin":"The Stacks Project","memory_eligible":false,"source_rank":2681,"rank":2681,"depth":6,"x":311.933,"y":736.6,"cluster":"derived-categories"},{"id":"stacks:0D7Z","tag":"0D7Z","title":"Postnikov systems · Definition 0D7Z","summary":"Let D be a triangulated category. Let X_n → X_n - 1 → … → X_0 be a complex in D. A Postnikov system is defined inductively as follows. • If n = 0, then it is an isomorphism Y_0 → X_0. • If n = 1, then it is a choice of an isomorphism Y_0 → X_0 and a choice of a distinguished triangle Y_1 → X_1 → Y_0 → Y_1[1] where X_1 → Y_0 composed with Y_0 → X_0 is the given morphism X_1 → X_0. • If n > 1, then it is a choice of a Postnikov system for X_n - 1 → … → X_0 and a choice of a…","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_0\n$$\nbe a complex in $\\mathcal{D}$. A {\\it Postnikov system} is defined\ninductively as follows.\n\\begin{enumerate}\n\\item If $n = 0$, then it is an isomorphism $Y_0 \\to X_0$.\n\\item If $n = 1$, then it is a choice of an isomorphism $Y_0 \\to X_0$ and\na choice of a distinguished triangle\n$$\nY_1 \\to X_1 \\to Y_0 \\to Y_1[1]\n$$\nwhere $X_1 \\to Y_0$ composed with $Y_0 \\to X_0$ is the given morphism\n$X_1 \\to X_0$.\n\\item If $n > 1$, then it is a choice of a Postnikov system\nfor $X_{n - 1} \\to \\ldots \\to X_0$ and a choice of a distinguished\ntriangle\n$$\nY_n \\to X_n \\to Y_{n - 1} \\to Y_n[1]\n$$\nwhere the morphism $X_n \\to Y_{n - 1}$ composed with\n$Y_{n - 1} \\to X_{n - 1}$ is the given morphism $X_n \\to X_{n - 1}$.\n\\end{enumerate}\nGiven a morphism\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\nX_n \\ar[r] \\ar[d] &\nX_{n - 1} \\ar[r] \\ar[d] &\n\\ldots \\ar[r] &\nX_0 \\ar[d] \\\\\nX'_n \\ar[r] &\nX'_{n - 1} \\ar[r] &\n\\ldots \\ar[r] &\nX'_0\n}\n}\n\\end{equation}\nbetween complexes of the same length in $\\mathcal{D}$\nthere is an obvious notion of a {\\it morphism of Postnikov systems}.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Postnikov systems","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7Z","source_file":"derived.tex","source_line":12020,"source_end_line":12064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12020-L12064","statement_sha256":"12c1d776edaed984008794d3414609af03b1cdb17322531d7e0425cd301ce7aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":2682,"rank":2682,"depth":0,"x":123.858,"y":684.765,"cluster":"derived-categories"},{"id":"stacks:0D81","tag":"0D81","title":"Postnikov systems · Lemma 0D81","summary":"Let D be a triangulated category. Consider Postnikov systems for complexes of length n. • For n = 0 Postnikov systems always exist and any morphism ([Tag 0D80]) of complexes extends to a unique morphism of Postnikov systems. • For n = 1 Postnikov systems always exist and any morphism ([Tag 0D80]) of complexes extends to a (nonunique) morphism of Postnikov systems. • For n = 2 Postnikov systems always exist but morphisms ([Tag 0D80]) of complexes in general do not extend…","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Consider\nPostnikov systems for complexes of length $n$.\n\\begin{enumerate}\n\\item For $n = 0$ Postnikov systems always exist and\nany morphism (\\ref{equation-map-complexes}) of complexes\nextends to a unique morphism of Postnikov systems.\n\\item For $n = 1$ Postnikov systems always exist and\nany morphism (\\ref{equation-map-complexes}) of complexes\nextends to a (nonunique) morphism of Postnikov systems.\n\\item For $n = 2$ Postnikov systems always exist but\nmorphisms (\\ref{equation-map-complexes}) of complexes\nin general do not extend to morphisms of Postnikov systems.\n\\item For $n > 2$ Postnikov systems do not always exist.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Postnikov systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D81","source_file":"derived.tex","source_line":12122,"source_end_line":12138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12122-L12138","statement_sha256":"9e8d0513b9eadca4f96dad51f93b086f20663ea1429bc9e1d961f2ff7fbdb615","origin":"The Stacks Project","memory_eligible":false,"source_rank":2683,"rank":2683,"depth":2,"x":304.583,"y":616.133,"cluster":"derived-categories"},{"id":"stacks:0D82","tag":"0D82","title":"Postnikov systems · Lemma 0D82","summary":"Let D be a triangulated category. Given a map ([Tag 0D80]) consider the condition Hom(X_i[i - j - 1], X'_j) = 0 for i > j + 1 Then • If we have a Postnikov system for X'_n → X'_n - 1 → … → X'_0 then property ([Tag 0DW1]) implies that Hom(X_i[i - j - 1], Y'_j) = 0 for i > j + 1 • If we are given Postnikov systems for both complexes and we have ([Tag 0DW1]), then the map extends to a (nonunique) map of Postnikov systems.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Given a map\n(\\ref{equation-map-complexes}) consider the condition\n\\begin{equation}\n\n\\Hom(X_i[i - j - 1], X'_j) = 0 \\text{ for }i > j + 1\n\\end{equation}\nThen\n\\begin{enumerate}\n\\item If we have a Postnikov system for\n$X'_n \\to X'_{n - 1} \\to \\ldots \\to X'_0$ then\nproperty (\\ref{equation-P}) implies that\n$$\n\\Hom(X_i[i - j - 1], Y'_j) = 0 \\text{ for }i > j + 1\n$$\n\\item If we are given Postnikov systems for both complexes and\nwe have (\\ref{equation-P}), then the map extends to a (nonunique) map\nof Postnikov systems.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Postnikov systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D82","source_file":"derived.tex","source_line":12164,"source_end_line":12184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12164-L12184","statement_sha256":"619ddb4f7fcc1825da83aa0d07a3b77377315b057e6275133f572608eb7a0417","origin":"The Stacks Project","memory_eligible":false,"source_rank":2684,"rank":2684,"depth":3,"x":226.357,"y":769.59,"cluster":"derived-categories"},{"id":"stacks:0FXE","tag":"0FXE","title":"Postnikov systems · Lemma 0FXE","summary":"Let D be a triangulated category. Given a map ([Tag 0D80]) assume we are given Postnikov systems for both complexes. If • Hom(X_i[i], Y'_n[n]) = 0 for i = 1, …, n, or • Hom(Y_n[n], X'_n - i[n - i]) = 0 for i = 1, …, n, or • Hom(X_j - i[-i + 1], X'_j) = 0 and Hom(X_j, X'_j - i[-i]) = 0 for j ≥ i > 0, then there exists at most one morphism between these Postnikov systems.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Given a map\n(\\ref{equation-map-complexes}) assume we are given\nPostnikov systems for both complexes. If\n\\begin{enumerate}\n\\item $\\Hom(X_i[i], Y'_n[n]) = 0$ for $i = 1, \\ldots, n$, or\n\\item $\\Hom(Y_n[n], X'_{n - i}[n - i]) = 0$ for $i = 1, \\ldots, n$, or\n\\item $\\Hom(X_{j - i}[-i + 1], X'_j) = 0$ and\n$\\Hom(X_j, X'_{j - i}[-i]) = 0$ for $j \\geq i > 0$,\n\\end{enumerate}\nthen there exists at most one morphism between these Postnikov systems.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Postnikov systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXE","source_file":"derived.tex","source_line":12242,"source_end_line":12254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12242-L12254","statement_sha256":"7bed6bf31f7aecfcd82d536c7dc9d2d2967aeb23da9ee4a629c77fcd4e0a6168","origin":"The Stacks Project","memory_eligible":false,"source_rank":2685,"rank":2685,"depth":4,"x":160.505,"y":611.737,"cluster":"derived-categories"},{"id":"stacks:0D83","tag":"0D83","title":"Postnikov systems · Lemma 0D83","summary":"Let D be a triangulated category. Let X_n → X_n - 1 → … → X_0 be a complex in D. If Hom(X_i[i - j - 2], X_j) = 0 for i > j + 2 then there exists a Postnikov system. If we have Hom(X_i[i - j - 1], X_j) = 0 for i > j + 1 then any two Postnikov systems are isomorphic.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category.\nLet $X_n \\to X_{n - 1} \\to \\ldots \\to X_0$ be\na complex in $\\mathcal{D}$. If\n$$\n\\Hom(X_i[i - j - 2], X_j) = 0 \\text{ for }i > j + 2\n$$\nthen there exists a Postnikov system. If we have\n$$\n\\Hom(X_i[i - j - 1], X_j) = 0 \\text{ for }i > j + 1\n$$\nthen any two Postnikov systems are isomorphic.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Postnikov systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D83","source_file":"derived.tex","source_line":12319,"source_end_line":12332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12319-L12332","statement_sha256":"3e448ad864373ca0d592a2025d95be75e246c3328d4930170aa167a20982514c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2686,"rank":2686,"depth":9,"x":336.347,"y":690.924,"cluster":"derived-categories"},{"id":"stacks:0G39","tag":"0G39","title":"Essentially constant systems · Lemma 0G39","summary":"Let D be a triangulated category. Let (A_i) be an inverse system in D. Then (A_i) is essentially constant (see Categories, Definition [Tag 05PU]) if and only if there exists an i and for all j ≥ i a direct sum decomposition A_j = A ⊕ Z_j such that (a) the maps A_j' → A_j are compatible with the direct sum decompositions and identity on A, (b) for all j ≥ i there exists some j' ≥ j such that Z_j' → Z_j is zero.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let $(A_i)$ be an inverse system\nin $\\mathcal{D}$. Then $(A_i)$ is essentially constant (see\nCategories, Definition\n\\ref{categories-definition-essentially-constant-diagram})\nif and only if there exists an $i$ and for all $j \\geq i$ a direct sum\ndecomposition $A_j = A \\oplus Z_j$ such that\n(a) the maps $A_{j'} \\to A_j$ are compatible with the direct sum\ndecompositions and identity on $A$, (b) for all $j \\geq i$ there exists some\n$j' \\geq j$ such that $Z_{j'} \\to Z_j$ is zero.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G39","source_file":"derived.tex","source_line":12387,"source_end_line":12398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12387-L12398","statement_sha256":"93b2e8dcbfdec1768c34f296e2bbca15acee81a08f9cfaedc771394f06d16d2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2687,"rank":2687,"depth":4,"x":142.627,"y":732.39,"cluster":"derived-categories"},{"id":"stacks:0G3A","tag":"0G3A","title":"Essentially constant systems · Lemma 0G3A","summary":"Let D be a triangulated category. Let A_n → B_n → C_n → A_n[1] be an inverse system of distinguished triangles in D. If (A_n) and (C_n) are essentially constant, then (B_n) is essentially constant and their values fit into a distinguished triangle A → B → C → A[1] such that for some n ≥ 1 there is a map xymatrix A_n ar[d] ar[r] & B_n ar[d] ar[r] & C_n ar[d] ar[r] & A_n[1] ar[d] A ar[r] & B ar[r] & C ar[r] & A[1] of distinguished triangles which induces an isomorphism…","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$$\nA_n \\to B_n \\to C_n \\to A_n[1]\n$$\nbe an inverse system of distinguished triangles in $\\mathcal{D}$.\nIf $(A_n)$ and $(C_n)$ are essentially constant, then\n$(B_n)$ is essentially constant and their values fit into\na distinguished triangle $A \\to B \\to C \\to A[1]$ such that for\nsome $n \\geq 1$ there is a map\n$$\n\\xymatrix{\nA_n \\ar[d] \\ar[r] &\nB_n \\ar[d] \\ar[r] &\nC_n \\ar[d] \\ar[r] &\nA_n[1] \\ar[d] \\\\\nA \\ar[r] &\nB \\ar[r] &\nC \\ar[r] &\nA[1]\n}\n$$\nof distinguished triangles which induces an isomorphism\n$\\lim_{n' \\geq n} A_{n'} \\to A$ and similarly for $B$ and $C$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3A","source_file":"derived.tex","source_line":12415,"source_end_line":12440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12415-L12440","statement_sha256":"2446d1086f4a3f2c3d87377939286b15b51fe7f47b64d6a234c480481557972d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2688,"rank":2688,"depth":5,"x":252.34,"y":591.621,"cluster":"derived-categories"},{"id":"stacks:0G3B","tag":"0G3B","title":"Essentially constant systems · Lemma 0G3B","summary":"Let A be an abelian category. Let A_n be an inverse system of objects of D(A). Assume • there exist integers a ≤ b such that H^i(A_n) = 0 for i not ∈ [a, b], and • the inverse systems H^i(A_n) of A are essentially constant for all i ∈ Z. Then A_n is an essentially constant system of D(A) whose value A satisfies that H^i(A) is the value of the constant system H^i(A_n) for each i ∈ Z.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $A_n$ be an inverse\nsystem of objects of $D(\\mathcal{A})$. Assume\n\\begin{enumerate}\n\\item there exist integers $a \\leq b$ such that $H^i(A_n) = 0$\nfor $i \\not \\in [a, b]$, and\n\\item the inverse systems $H^i(A_n)$ of $\\mathcal{A}$ are essentially constant\nfor all $i \\in \\mathbf{Z}$.\n\\end{enumerate}\nThen $A_n$ is an essentially constant system of $D(\\mathcal{A})$ whose\nvalue $A$ satisfies that $H^i(A)$ is the value of the constant system\n$H^i(A_n)$ for each $i \\in \\mathbf{Z}$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3B","source_file":"derived.tex","source_line":12489,"source_end_line":12502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12489-L12502","statement_sha256":"9bc455a5f959a6ab5750b43f5bd031589376709605fbd7bb9559e944cd3ee866","origin":"The Stacks Project","memory_eligible":false,"source_rank":2689,"rank":2689,"depth":6,"x":284.704,"y":757.996,"cluster":"derived-categories"},{"id":"stacks:0G3C","tag":"0G3C","title":"Essentially constant systems · Lemma 0G3C","summary":"Let D be a triangulated category. Let A_n → B_n → C_n → A_n[1] be an inverse system of distinguished triangles. If the system C_n is pro-zero (essentially constant with value 0), then the maps A_n → B_n determine a pro-isomorphism between the pro-object (A_n) and the pro-object (B_n).","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$$\nA_n \\to B_n \\to C_n \\to A_n[1]\n$$\nbe an inverse system of distinguished triangles. If the system $C_n$\nis pro-zero (essentially constant with value $0$), then the maps\n$A_n \\to B_n$ determine a pro-isomorphism between the pro-object $(A_n)$\nand the pro-object $(B_n)$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3C","source_file":"derived.tex","source_line":12520,"source_end_line":12530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12520-L12530","statement_sha256":"5f96ea8135da954ca05a6502d2b289455805abfac15505d4abca7676639a9ade","origin":"The Stacks Project","memory_eligible":false,"source_rank":2690,"rank":2690,"depth":2,"x":126.742,"y":653.477,"cluster":"derived-categories"},{"id":"stacks:0G3D","tag":"0G3D","title":"Essentially constant systems · Lemma 0G3D","summary":"Let A be an abelian category. A_n → B_n be an inverse system of maps of D(A). Assume • there exist integers a ≤ b such that H^i(A_n) = 0 and H^i(B_n) = 0 for i not ∈ [a, b], and • the inverse system of maps H^i(A_n) → H^i(B_n) of A define an isomorphism of pro-objects of A for all i ∈ Z. Then the maps A_n → B_n determine a pro-isomorphism between the pro-object (A_n) and the pro-object (B_n).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n$$\nA_n \\to B_n\n$$\nbe an inverse system of maps of $D(\\mathcal{A})$. Assume\n\\begin{enumerate}\n\\item there exist integers $a \\leq b$ such that $H^i(A_n) = 0$\nand $H^i(B_n) = 0$ for $i \\not \\in [a, b]$, and\n\\item the inverse system of maps $H^i(A_n) \\to H^i(B_n)$ of $\\mathcal{A}$\ndefine an isomorphism of pro-objects of $\\mathcal{A}$\nfor all $i \\in \\mathbf{Z}$.\n\\end{enumerate}\nThen the maps $A_n \\to B_n$\ndetermine a pro-isomorphism between the pro-object $(A_n)$\nand the pro-object $(B_n)$.","area":"Derived Categories","chapter":"Derived Categories","chapter_id":"derived","section":"Essentially constant systems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3D","source_file":"derived.tex","source_line":12549,"source_end_line":12566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derived.tex#L12549-L12566","statement_sha256":"542ea2f173e52d2c13454c4b8ae0e3b29713781b5aa9dccd9ef6503ed8496e9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2691,"rank":2691,"depth":7,"x":327.656,"y":640.892,"cluster":"derived-categories"},{"id":"stacks:0165","tag":"0165","title":"The category of finite ordered sets · Definition 0165","summary":"For any integer n≥ 1, and any 0≤ j ≤ n we let δ^n_j : [n-1] → [n] denote the injective order preserving map skipping j. For any integer n≥ 0, and any 0≤ j ≤ n we denote σ^n_j : [n + 1] → [n] the surjective order preserving map with (σ^n_j)^-1((j)) = (j, j + 1).","statement_latex":"For any integer $n\\geq 1$, and any $0\\leq j \\leq n$ we let\n{\\it $\\delta^n_j : [n-1] \\to [n]$}\ndenote the injective order preserving map skipping $j$. For any\ninteger $n\\geq 0$, and any $0\\leq j \\leq n$ we denote\n{\\it $\\sigma^n_j : [n + 1]  \\to [n]$}\nthe surjective order preserving map with\n$(\\sigma^n_j)^{-1}(\\{j\\}) = \\{j, j + 1\\}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"The category of finite ordered sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0165","source_file":"simplicial.tex","source_line":49,"source_end_line":58,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L49-L58","statement_sha256":"7667ee3f8b6d8c53d0b853d2a626735314e8b49d51dc694edbd88bb6c4c05cd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2692,"rank":2692,"depth":0,"x":2428.977,"y":148.85,"cluster":"homological-algebra"},{"id":"stacks:0166","tag":"0166","title":"The category of finite ordered sets · Lemma 0166","summary":"Any morphism in Δ can be written as a composition of the morphisms δ^n_j and σ^n_j.","statement_latex":"Any morphism in $\\Delta$ can be written as a composition\nof the morphisms $\\delta^n_j$ and $\\sigma^n_j$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"The category of finite ordered sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0166","source_file":"simplicial.tex","source_line":60,"source_end_line":64,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L60-L64","statement_sha256":"a2bc665bbc71d6f22b2e213a9082d77e2718b647e4e37ce55fc5a6983f5ea9ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":2693,"rank":2693,"depth":0,"x":2575.094,"y":243.576,"cluster":"homological-algebra"},{"id":"stacks:0167","tag":"0167","title":"The category of finite ordered sets · Lemma 0167","summary":"The morphisms δ^n_j and σ^n_j satisfy the following relations. • If 0 ≤ i < j ≤ n + 1, then δ^n + 1_j ∘ δ^n_i = δ^n + 1_i ∘ δ^n_j - 1. In other words the diagram xymatrix & [n] ar[rd]^δ^n + 1_j & [n - 1] ar[ru]^δ^n_i ar[rd]_δ^n_j - 1 & & [n + 1] & [n] ar[ru]_δ^n + 1_i & commutes. • If 0 ≤ i < j ≤ n - 1, then σ^n - 1_j ∘ δ^n_i = δ^n - 1_i ∘ σ^n - 2_j - 1. In other words the diagram xymatrix & [n] ar[rd]^σ^n - 1_j & [n - 1] ar[ru]^δ^n_i ar[rd]_σ^n - 2_j - 1 & & [n - 1] & [n…","statement_latex":"The morphisms $\\delta^n_j$ and $\\sigma^n_j$ satisfy the following relations.\n\\begin{enumerate}\n\\item If $0 \\leq i < j \\leq n + 1$, then\n$\\delta^{n + 1}_j \\circ \\delta^n_i =\n\\delta^{n + 1}_i \\circ \\delta^n_{j - 1}$.\nIn other words the diagram\n$$\n\\xymatrix{\n& [n] \\ar[rd]^{\\delta^{n + 1}_j} & \\\\\n[n - 1] \\ar[ru]^{\\delta^n_i} \\ar[rd]_{\\delta^n_{j - 1}} & &\n[n + 1] \\\\\n& [n] \\ar[ru]_{\\delta^{n + 1}_i} &\n}\n$$\ncommutes.\n\\item If $0 \\leq i < j \\leq n - 1$, then\n$\\sigma^{n - 1}_j \\circ \\delta^n_i =\n\\delta^{n - 1}_i \\circ \\sigma^{n - 2}_{j - 1}$.\nIn other words the diagram\n$$\n\\xymatrix{\n& [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\\n[n - 1] \\ar[ru]^{\\delta^n_i} \\ar[rd]_{\\sigma^{n - 2}_{j - 1}} & &\n[n - 1] \\\\\n& [n - 2] \\ar[ru]_{\\delta^{n - 1}_i} &\n}\n$$\ncommutes.\n\\item If $0 \\leq j \\leq n - 1$, then\n$\\sigma^{n - 1}_j \\circ \\delta^n_j = \\text{id}_{[n - 1]}$\nand\n$\\sigma^{n - 1}_j \\circ \\delta^n_{j + 1} = \\text{id}_{[n - 1]}$.\nIn other words the diagram\n$$\n\\xymatrix{\n& [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\\n[n - 1]\n\\ar[ru]^{\\delta^n_j}\n\\ar[rd]_{\\delta^n_{j + 1}}\n\\ar[rr]^{\\text{id}_{[n - 1]}} & & [n - 1] \\\\\n& [n] \\ar[ru]_{\\sigma^{n - 1}_j} &\n}\n$$\ncommutes.\n\\item If $0 < j + 1 < i \\leq n$, then\n$\\sigma^{n - 1}_j \\circ \\delta^n_i =\n\\delta^{n - 1}_{i - 1} \\circ \\sigma^{n - 2}_j$.\nIn other words the diagram\n$$\n\\xymatrix{\n& [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\\n[n - 1] \\ar[ru]^{\\delta^n_i} \\ar[rd]_{\\sigma^{n - 2}_j} & &\n[n - 1] \\\\\n& [n - 2] \\ar[ru]_{\\delta^{n - 1}_{i - 1}} &\n}\n$$\ncommutes.\n\\item If $0 \\leq i \\leq j \\leq n - 1$, then\n$\\sigma^{n - 1}_j \\circ \\sigma^n_i =\n\\sigma^{n - 1}_i \\circ \\sigma^n_{j + 1}$.\nIn other words the diagram\n$$\n\\xymatrix{\n& [n] \\ar[rd]^{\\sigma^{n - 1}_j} & \\\\\n[n + 1] \\ar[ru]^{\\sigma^n_i} \\ar[rd]_{\\sigma^n_{j + 1}} & &\n[n - 1] \\\\\n& [n] \\ar[ru]_{\\sigma^{n - 1}_i} &\n}\n$$\ncommutes.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"The category of finite ordered sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0167","source_file":"simplicial.tex","source_line":79,"source_end_line":152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L79-L152","statement_sha256":"0667e9cd096aeda49cf5b7e30ec4f27ed1c8c8c50e9947bd8fb15146b8966eb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2694,"rank":2694,"depth":0,"x":2390.683,"y":256.671,"cluster":"homological-algebra"},{"id":"stacks:0168","tag":"0168","title":"The category of finite ordered sets · Lemma 0168","summary":"The category Δ is the universal category with objects [n], n ≥ 0 and morphisms δ^n_j and σ^n_j such that (a) every morphism is a composition of these morphisms, (b) the relations listed in Lemma [Tag 0167] are satisfied, and (c) any relation among the morphisms is a consequence of those relations.","statement_latex":"The category $\\Delta$ is the universal category\nwith objects $[n]$, $n \\geq 0$ and morphisms\n$\\delta^n_j$ and $\\sigma^n_j$ such that (a) every morphism is\na composition of these morphisms, (b) the relations\nlisted in Lemma \\ref{lemma-relations-face-degeneracy} are satisfied,\nand (c) any relation among the morphisms is a consequence of\nthose relations.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"The category of finite ordered sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0168","source_file":"simplicial.tex","source_line":158,"source_end_line":167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L158-L167","statement_sha256":"929285257855fd90d48b817fd6bc81766c6ae5ffe1c94d5e9905b3f592976432","origin":"The Stacks Project","memory_eligible":false,"source_rank":2695,"rank":2695,"depth":1,"x":2516.467,"y":142.074,"cluster":"homological-algebra"},{"id":"stacks:016A","tag":"016A","title":"Simplicial objects · Definition 016A","summary":"Let C be a category. • A simplicial object U of C is a contravariant functor U from Δ to C, in a formula: U : Δ^opp → C • If C is the category of sets, then we call U a simplicial set. • If C is the category of abelian groups, then we call U a simplicial abelian group. • A morphism of simplicial objects U → U' is a transformation of functors. • The category of simplicial objects of C is denoted Simp(C).","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item A {\\it simplicial object $U$ of $\\mathcal{C}$}\nis a contravariant functor $U$ from $\\Delta$ to\n$\\mathcal{C}$, in a formula:\n$$\nU : \\Delta^{opp} \\longrightarrow \\mathcal{C}\n$$\n\\item If $\\mathcal{C}$ is the category of sets, then we call\n$U$ a {\\it simplicial set}.\n\\item If $\\mathcal{C}$ is the category of abelian groups,\nthen we call $U$ a {\\it simplicial abelian group}.\n\\item A {\\it morphism of simplicial objects $U \\to U'$}\nis a transformation of functors.\n\\item The {\\it category of simplicial objects of $\\mathcal{C}$}\nis denoted $\\text{Simp}(\\mathcal{C})$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016A","source_file":"simplicial.tex","source_line":182,"source_end_line":201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L182-L201","statement_sha256":"1d282bd65417d1cc7e8feb2eaba637dd7774f006338d4e0298f94da026f68c46","origin":"The Stacks Project","memory_eligible":false,"source_rank":2696,"rank":2696,"depth":0,"x":2515.87,"y":298.359,"cluster":"homological-algebra"},{"id":"stacks:016B","tag":"016B","title":"Simplicial objects · Lemma 016B","summary":"Let C be a category. • Given a simplicial object U in C we obtain a sequence of objects U_n = U([n]) endowed with the morphisms d^n_j = U(δ^n_j) : U_n → U_n-1 and s^n_j = U(σ^n_j) : U_n → U_n + 1. These morphisms satisfy the opposites of the relations displayed in Lemma [Tag 0167], namely • If 0 ≤ i < j ≤ n + 1, then d^n_i ∘ d^n + 1_j = d^n_j - 1 ∘ d^n + 1_i. • If 0 ≤ i < j ≤ n - 1, then d^n_i ∘ s^n - 1_j = s^n - 2_j - 1 ∘ d^n - 1_i. • If 0 ≤ j ≤ n - 1, then id = d^n_j ∘…","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item Given a simplicial object $U$ in $\\mathcal{C}$\nwe obtain a sequence of objects $U_n = U([n])$ endowed\nwith the morphisms $d^n_j = U(\\delta^n_j) : U_n \\to U_{n-1}$ and\n$s^n_j = U(\\sigma^n_j) : U_n \\to U_{n + 1}$. These morphisms\nsatisfy the opposites of the relations displayed in\nLemma \\ref{lemma-relations-face-degeneracy}, namely\n\\begin{enumerate}\n\\item If $0 \\leq i < j \\leq n + 1$, then\n$d^n_i \\circ d^{n + 1}_j = d^n_{j - 1} \\circ d^{n + 1}_i$.\n\\item If $0 \\leq i < j \\leq n - 1$, then\n$d^n_i \\circ s^{n - 1}_j = s^{n - 2}_{j - 1} \\circ d^{n - 1}_i$.\n\\item If $0 \\leq j \\leq n - 1$, then\n$\\text{id} = d^n_j \\circ s^{n - 1}_j = d^n_{j + 1} \\circ s^{n - 1}_j$.\n\\item If $0 < j + 1 < i \\leq n$, then\n$d^n_i \\circ s^{n - 1}_j = s^{n - 2}_j \\circ d^{n - 1}_{i - 1}$.\n\\item If $0 \\leq i \\leq j \\leq n - 1$, then\n$s^n_i \\circ s^{n - 1}_j = s^n_{j + 1} \\circ s^{n - 1}_i$.\n\\end{enumerate}\n\\item Conversely, given a sequence of objects $U_n$ and morphisms\n$d^n_j$, $s^n_j$ satisfying (1)(a) -- (e) there exists a unique\nsimplicial object $U$ in $\\mathcal{C}$ such that $U_n = U([n])$,\n$d^n_j = U(\\delta^n_j)$, and $s^n_j = U(\\sigma^n_j)$.\n\\item A morphism between simplicial objects $U$ and $U'$\nis given by a family of morphisms $U_n \\to U'_n$ commuting\nwith the morphisms $d^n_j$ and $s^n_j$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016B","source_file":"simplicial.tex","source_line":217,"source_end_line":247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L217-L247","statement_sha256":"789a5f44a054575e7ce04c0d70795611fad858285957a8b025bcd40378341154","origin":"The Stacks Project","memory_eligible":false,"source_rank":2697,"rank":2697,"depth":2,"x":2390.302,"y":182.474,"cluster":"homological-algebra"},{"id":"stacks:016F","tag":"016F","title":"Simplicial objects · Lemma 016F","summary":"Let C be a category. Let U be a simplicial object of C. Each of the morphisms s^n_i : U_n → U_n + 1 has a left inverse. In particular s^n_i is a monomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U$ be a simplicial object of $\\mathcal{C}$.\nEach of the morphisms $s^n_i : U_n \\to U_{n + 1}$\nhas a left inverse. In particular $s^n_i$ is a monomorphism.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016F","source_file":"simplicial.tex","source_line":330,"source_end_line":336,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L330-L336","statement_sha256":"95e9d7f089210134c00890e0d9e7540aa50e7c34e6f1526e222671b5db43ed99","origin":"The Stacks Project","memory_eligible":false,"source_rank":2698,"rank":2698,"depth":0,"x":2576.569,"y":196.714,"cluster":"homological-algebra"},{"id":"stacks:016J","tag":"016J","title":"Cosimplicial objects · Definition 016J","summary":"Let C be a category. • A cosimplicial object U of C is a covariant functor U from Δ to C, in a formula: U : Δ → C • If C is the category of sets, then we call U a cosimplicial set. • If C is the category of abelian groups, then we call U a cosimplicial abelian group. • A morphism of cosimplicial objects U → U' is a transformation of functors. • The category of cosimplicial objects of C is denoted CoSimp(C).","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item A {\\it cosimplicial object $U$ of $\\mathcal{C}$}\nis a covariant functor $U$ from $\\Delta$ to\n$\\mathcal{C}$, in a formula:\n$$\nU : \\Delta \\longrightarrow \\mathcal{C}\n$$\n\\item If $\\mathcal{C}$ is the category of sets, then we call\n$U$ a {\\it cosimplicial set}.\n\\item If $\\mathcal{C}$ is the category of abelian groups,\nthen we call $U$ a {\\it cosimplicial abelian group}.\n\\item A {\\it morphism of cosimplicial objects $U \\to U'$}\nis a transformation of functors.\n\\item The {\\it category of cosimplicial objects of $\\mathcal{C}$}\nis denoted $\\text{CoSimp}(\\mathcal{C})$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Cosimplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016J","source_file":"simplicial.tex","source_line":387,"source_end_line":406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L387-L406","statement_sha256":"9936fbd3e7319f6db55ceded4759747fc05877c882faebfcaced3d605c5f9541","origin":"The Stacks Project","memory_eligible":false,"source_rank":2699,"rank":2699,"depth":0,"x":2427.382,"y":292.153,"cluster":"homological-algebra"},{"id":"stacks:016K","tag":"016K","title":"Cosimplicial objects · Lemma 016K","summary":"Let C be a category. • Given a cosimplicial object U in C we obtain a sequence of objects U_n = U([n]) endowed with the morphisms δ^n_j = U(δ^n_j) : U_n - 1 → U_n and σ^n_j = U(σ^n_j) : U_n + 1 → U_n. These morphisms satisfy the relations displayed in Lemma [Tag 0167]. • Conversely, given a sequence of objects U_n and morphisms δ^n_j, σ^n_j satisfying these relations there exists a unique cosimplicial object U in C such that U_n = U([n]), δ^n_j = U(δ^n_j), and σ^n_j =…","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item Given a cosimplicial object $U$ in $\\mathcal{C}$\nwe obtain a sequence of objects $U_n = U([n])$ endowed\nwith the morphisms $\\delta^n_j = U(\\delta^n_j) : U_{n - 1} \\to U_n$ and\n$\\sigma^n_j = U(\\sigma^n_j) : U_{n + 1} \\to U_n$. These morphisms\nsatisfy the relations displayed in\nLemma \\ref{lemma-relations-face-degeneracy}.\n\\item Conversely, given a sequence of objects $U_n$ and morphisms\n$\\delta^n_j$, $\\sigma^n_j$ satisfying these relations there exists a unique\ncosimplicial object $U$ in $\\mathcal{C}$ such that $U_n = U([n])$,\n$\\delta^n_j = U(\\delta^n_j)$, and $\\sigma^n_j = U(\\sigma^n_j)$.\n\\item A morphism between cosimplicial objects $U$ and $U'$\nis given by a family of morphisms $U_n \\to U'_n$ commuting\nwith the morphisms $\\delta^n_j$ and $\\sigma^n_j$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016K","source_file":"simplicial.tex","source_line":422,"source_end_line":440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L422-L440","statement_sha256":"146c073e4db817f52c0904443fa017da66d27d04545e081935f30095cdcdba95","origin":"The Stacks Project","memory_eligible":false,"source_rank":2700,"rank":2700,"depth":2,"x":2460.727,"y":136.725,"cluster":"homological-algebra"},{"id":"stacks:016O","tag":"016O","title":"Cosimplicial objects · Lemma 016O","summary":"Let C be a category. Let U be a cosimplicial object of C. Each of the morphisms δ^n_i : U_n - 1 → U_n has a left inverse. In particular δ^n_i is a monomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U$ be a cosimplicial object of $\\mathcal{C}$.\nEach of the morphisms $\\delta^n_i : U_{n - 1} \\to U_n$\nhas a left inverse. In particular $\\delta^n_i$ is a monomorphism.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016O","source_file":"simplicial.tex","source_line":535,"source_end_line":541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L535-L541","statement_sha256":"47dee7bb63be3086ceb380d7d548e947b04ddaa7254977f62f55fcb8329874e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2701,"rank":2701,"depth":0,"x":2561.387,"y":270.599,"cluster":"homological-algebra"},{"id":"stacks:016Q","tag":"016Q","title":"Products of simplicial objects · Definition 016Q","summary":"Let C be a category. Let U and V be simplicial objects of C. Assume the products U_n × V_n exist in C. The product of U and V is the simplicial object U × V defined as follows: • (U × V)_n = U_n × V_n, • d^n_i = (d^n_i, d^n_i), and • s^n_i = (s^n_i, s^n_i). In other words, U × V is the product of the presheaves U and V on Δ.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U$ and $V$ be simplicial objects of $\\mathcal{C}$.\nAssume the products $U_n \\times V_n$ exist in $\\mathcal{C}$.\nThe {\\it product of $U$ and $V$} is the simplicial object\n$U \\times V$ defined as follows:\n\\begin{enumerate}\n\\item $(U \\times V)_n = U_n \\times V_n$,\n\\item $d^n_i = (d^n_i, d^n_i)$, and\n\\item $s^n_i = (s^n_i, s^n_i)$.\n\\end{enumerate}\nIn other words, $U \\times V$ is the product of the presheaves\n$U$ and $V$ on $\\Delta$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Products of simplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016Q","source_file":"simplicial.tex","source_line":584,"source_end_line":598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L584-L598","statement_sha256":"39a82c35cebd1a8046ccb00a8d22f5eca311859730fbcb67461bee692fdd1275","origin":"The Stacks Project","memory_eligible":false,"source_rank":2702,"rank":2702,"depth":0,"x":2379.04,"y":228.893,"cluster":"homological-algebra"},{"id":"stacks:016R","tag":"016R","title":"Products of simplicial objects · Lemma 016R","summary":"If U and V are simplicial objects in the category C, and if U × V exists, then we have Mor(W, U × V) = Mor(W, U) × Mor(W, V) for any third simplicial object W of C.","statement_latex":"If $U$ and $V$ are simplicial objects in the category $\\mathcal{C}$,\nand if $U \\times V$ exists, then we have\n$$\n\\Mor(W, U \\times V) =\n\\Mor(W, U) \\times\n\\Mor(W, V)\n$$\nfor any third simplicial object $W$ of $\\mathcal{C}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Products of simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016R","source_file":"simplicial.tex","source_line":600,"source_end_line":610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L600-L610","statement_sha256":"8ce1e1ed3a59d65147ac45d6e52fa3d95f2c5fbbc47d56cafc27bb031379ab18","origin":"The Stacks Project","memory_eligible":false,"source_rank":2703,"rank":2703,"depth":0,"x":2547.466,"y":155.994,"cluster":"homological-algebra"},{"id":"stacks:016T","tag":"016T","title":"Fibre products of simplicial objects · Definition 016T","summary":"Let C be a category. Let U, V, W be simplicial objects of C. Let a : V → U, b : W → U be morphisms. Assume the fibre products V_n ×_U_n W_n exist in C. The fibre product of V and W over U is the simplicial object V ×_U W defined as follows: • (V ×_U W)_n = V_n ×_U_n W_n, • d^n_i = (d^n_i, d^n_i), and • s^n_i = (s^n_i, s^n_i). In other words, V ×_U W is the fibre product of the presheaves V and W over the presheaf U on Δ.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U, V, W$ be simplicial objects of $\\mathcal{C}$.\nLet $a : V \\to U$, $b : W \\to U$ be morphisms.\nAssume the fibre products $V_n \\times_{U_n} W_n$ exist in $\\mathcal{C}$.\nThe {\\it fibre product of $V$ and $W$ over $U$} is the simplicial object\n$V \\times_U W$ defined as follows:\n\\begin{enumerate}\n\\item $(V \\times_U W)_n = V_n \\times_{U_n} W_n$,\n\\item $d^n_i = (d^n_i, d^n_i)$, and\n\\item $s^n_i = (s^n_i, s^n_i)$.\n\\end{enumerate}\nIn other words, $V \\times_U W$ is the fibre product of the presheaves\n$V$ and $W$ over the presheaf $U$ on $\\Delta$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Fibre products of simplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016T","source_file":"simplicial.tex","source_line":627,"source_end_line":642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L627-L642","statement_sha256":"5b99ebdfafe5c8371757738a651c954e73e662200f4700f0e5218e1c9bf9eb95","origin":"The Stacks Project","memory_eligible":false,"source_rank":2704,"rank":2704,"depth":0,"x":2481.728,"y":305.694,"cluster":"homological-algebra"},{"id":"stacks:016U","tag":"016U","title":"Fibre products of simplicial objects · Lemma 016U","summary":"If U, V, W are simplicial objects in the category C, and if a : V → U, b : W → U are morphisms and if V ×_U W exists, then we have Mor(T, V ×_U W) = Mor(T, V) ×_Mor(T, U) Mor(T, W) for any fourth simplicial object T of C.","statement_latex":"If $U, V, W$ are simplicial objects in the category $\\mathcal{C}$,\nand if $a : V \\to U$, $b : W \\to U$ are morphisms\nand if $V \\times_U W$ exists, then we have\n$$\n\\Mor(T, V \\times_U W) =\n\\Mor(T, V) \\times_{\\Mor(T, U)}\n\\Mor(T, W)\n$$\nfor any fourth simplicial object $T$ of $\\mathcal{C}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Fibre products of simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016U","source_file":"simplicial.tex","source_line":644,"source_end_line":655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L644-L655","statement_sha256":"86b34f3a082bea6d015b46709199b50c2193aef4f4e72ffc1e51bd4f79d6dbd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2705,"rank":2705,"depth":0,"x":2409.637,"y":157.635,"cluster":"homological-algebra"},{"id":"stacks:016W","tag":"016W","title":"Pushouts of simplicial objects · Definition 016W","summary":"Let C be a category. Let U, V, W be simplicial objects of C. Let a : U → V, b : U → W be morphisms. Assume the pushouts V_n amalg_U_n W_n exist in C. The pushout of V and W over U is the simplicial object Vamalg_U W defined as follows: • (V amalg_U W)_n = V_n amalg_U_n W_n, • d^n_i = (d^n_i, d^n_i), and • s^n_i = (s^n_i, s^n_i). In other words, Vamalg_U W is the pushout of the presheaves V and W over the presheaf U on Δ.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U, V, W$ be simplicial objects of $\\mathcal{C}$.\nLet $a : U \\to V$, $b : U \\to W$ be morphisms.\nAssume the pushouts $V_n \\amalg_{U_n} W_n$ exist in $\\mathcal{C}$.\nThe {\\it pushout of $V$ and $W$ over $U$} is the simplicial object\n$V\\amalg_U W$ defined as follows:\n\\begin{enumerate}\n\\item $(V \\amalg_U W)_n = V_n \\amalg_{U_n} W_n$,\n\\item $d^n_i = (d^n_i, d^n_i)$, and\n\\item $s^n_i = (s^n_i, s^n_i)$.\n\\end{enumerate}\nIn other words, $V\\amalg_U W$ is the pushout of the presheaves\n$V$ and $W$ over the presheaf $U$ on $\\Delta$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Pushouts of simplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016W","source_file":"simplicial.tex","source_line":671,"source_end_line":686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L671-L686","statement_sha256":"03beacb2a15fb1dc0409fdd78dfdaf97c25d8155809857bdde650c5d72fa6a24","origin":"The Stacks Project","memory_eligible":false,"source_rank":2706,"rank":2706,"depth":0,"x":2582.291,"y":226.076,"cluster":"homological-algebra"},{"id":"stacks:016X","tag":"016X","title":"Pushouts of simplicial objects · Lemma 016X","summary":"If U, V, W are simplicial objects in the category C, and if a : U → V, b : U → W are morphisms and if Vamalg_U W exists, then we have Mor(Vamalg_U W, T) = Mor(V, T) ×_Mor(U, T) Mor(W, T) for any fourth simplicial object T of C.","statement_latex":"If $U, V, W$ are simplicial objects in the category $\\mathcal{C}$,\nand if $a : U \\to V$, $b : U \\to W$ are morphisms\nand if $V\\amalg_U W$ exists, then we have\n$$\n\\Mor(V\\amalg_U W, T) =\n\\Mor(V, T) \\times_{\\Mor(U, T)}\n\\Mor(W, T)\n$$\nfor any fourth simplicial object $T$ of $\\mathcal{C}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Pushouts of simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016X","source_file":"simplicial.tex","source_line":688,"source_end_line":699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L688-L699","statement_sha256":"ee2006032079369d208e74c555ed150e5210d67381448f9711fb81e44cb44361","origin":"The Stacks Project","memory_eligible":false,"source_rank":2707,"rank":2707,"depth":0,"x":2399.486,"y":273.695,"cluster":"homological-algebra"},{"id":"stacks:016Z","tag":"016Z","title":"Products of cosimplicial objects · Definition 016Z","summary":"Let C be a category. Let U and V be cosimplicial objects of C. Assume the products U_n × V_n exist in C. The product of U and V is the cosimplicial object U × V defined as follows: • (U × V)_n = U_n × V_n, • for any φ : [n] → [m] the map (U × V)(φ) : U_n × V_n → U_m × V_m is the product U(φ) × V(φ).","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U$ and $V$ be cosimplicial objects of $\\mathcal{C}$.\nAssume the products $U_n \\times V_n$ exist in $\\mathcal{C}$.\nThe {\\it product of $U$ and $V$} is the cosimplicial object\n$U \\times V$ defined as follows:\n\\begin{enumerate}\n\\item $(U \\times V)_n = U_n \\times V_n$,\n\\item for any $\\varphi : [n] \\to [m]$ the map\n$(U \\times V)(\\varphi) : U_n \\times V_n \\to U_m \\times V_m$\nis the product $U(\\varphi) \\times V(\\varphi)$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Products of cosimplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/016Z","source_file":"simplicial.tex","source_line":725,"source_end_line":738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L725-L738","statement_sha256":"db221c2b795f788694d8ca93f7f5a4253607e2715d762afb0daf8893b4df924f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2708,"rank":2708,"depth":0,"x":2496.23,"y":134.51,"cluster":"homological-algebra"},{"id":"stacks:0170","tag":"0170","title":"Products of cosimplicial objects · Lemma 0170","summary":"If U and V are cosimplicial objects in the category C, and if U × V exists, then we have Mor(W, U × V) = Mor(W, U) × Mor(W, V) for any third cosimplicial object W of C.","statement_latex":"If $U$ and $V$ are cosimplicial objects in the category $\\mathcal{C}$,\nand if $U \\times V$ exists, then we have\n$$\n\\Mor(W, U \\times V) =\n\\Mor(W, U) \\times\n\\Mor(W, V)\n$$\nfor any third cosimplicial object $W$ of $\\mathcal{C}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Products of cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0170","source_file":"simplicial.tex","source_line":740,"source_end_line":750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L740-L750","statement_sha256":"3e3ea9dd5afd31b1d230209f02ff7ebc258917824bc7b3f5fa9249fb6bc2cc82","origin":"The Stacks Project","memory_eligible":false,"source_rank":2709,"rank":2709,"depth":0,"x":2536.92,"y":292.425,"cluster":"homological-algebra"},{"id":"stacks:0172","tag":"0172","title":"Fibre products of cosimplicial objects · Definition 0172","summary":"Let C be a category. Let U, V, W be cosimplicial objects of C. Let a : V → U and b : W → U be morphisms. Assume the fibre products V_n ×_U_n W_n exist in C. The fibre product of V and W over U is the cosimplicial object V ×_U W defined as follows: • (V ×_U W)_n = V_n ×_U_n W_n, • for any φ : [n] → [m] the map (V ×_U W)(φ) : V_n ×_U_n W_n → V_m ×_U_m W_m is the product V(φ) ×_U(φ) W(φ).","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U, V, W$ be cosimplicial objects of $\\mathcal{C}$.\nLet $a : V \\to U$ and $b : W \\to U$ be morphisms.\nAssume the fibre products $V_n \\times_{U_n} W_n$ exist in $\\mathcal{C}$.\nThe {\\it fibre product of $V$ and $W$ over $U$} is the cosimplicial object\n$V \\times_U W$ defined as follows:\n\\begin{enumerate}\n\\item $(V \\times_U W)_n = V_n \\times_{U_n} W_n$,\n\\item for any $\\varphi : [n] \\to [m]$ the map\n$(V \\times_U W)(\\varphi) : V_n \\times_{U_n} W_n \\to V_m \\times_{U_m} W_m$\nis the product $V(\\varphi) \\times_{U(\\varphi)} W(\\varphi)$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Fibre products of cosimplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0172","source_file":"simplicial.tex","source_line":766,"source_end_line":780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L766-L780","statement_sha256":"4bc2b512d2cdd301f6a2feda155867b84591dfb3a15eef21c28cc59b3d778798","origin":"The Stacks Project","memory_eligible":false,"source_rank":2710,"rank":2710,"depth":0,"x":2379.54,"y":198.841,"cluster":"homological-algebra"},{"id":"stacks:0173","tag":"0173","title":"Fibre products of cosimplicial objects · Lemma 0173","summary":"If U, V, W are cosimplicial objects in the category C, and if a : V → U, b : W → U are morphisms and if V ×_U W exists, then we have Mor(T, V ×_U W) = Mor(T, V) ×_Mor(T, U) Mor(T, W) for any fourth cosimplicial object T of C.","statement_latex":"If $U, V, W$ are cosimplicial objects in the category $\\mathcal{C}$,\nand if $a : V \\to U$, $b : W \\to U$ are morphisms\nand if $V \\times_U W$ exists, then we have\n$$\n\\Mor(T, V \\times_U W) =\n\\Mor(T, V) \\times_{\\Mor(T, U)}\n\\Mor(T, W)\n$$\nfor any fourth cosimplicial object $T$ of $\\mathcal{C}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Fibre products of cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0173","source_file":"simplicial.tex","source_line":782,"source_end_line":793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L782-L793","statement_sha256":"8319bead5e41aaaa2d08789f62516d22afdd7c9b59f358426bbce05b9d8fea98","origin":"The Stacks Project","memory_eligible":false,"source_rank":2711,"rank":2711,"depth":0,"x":2571.314,"y":178.498,"cluster":"homological-algebra"},{"id":"stacks:0175","tag":"0175","title":"Simplicial sets · Definition 0175","summary":"Let U be a simplicial set. We say x is an n-simplex of U to signify that x is an element of U_n. We say that y is the jth face of x to signify that d^n_jx = y. We say that z is the jth degeneracy of x if z = s^n_jx. A simplex is called degenerate if it is the degeneracy of another simplex.","statement_latex":"Let $U$ be a simplicial set.\nWe say $x$ is an {\\it $n$-simplex of $U$} to signify that\n$x$ is an element of $U_n$. We say that $y$ is the $j$th\n{\\it face of $x$} to signify that $d^n_jx = y$. We say that\n$z$ is the $j$th {\\it degeneracy of $x$} if $z = s^n_jx$.\nA simplex is called {\\it degenerate} if it is the degeneracy\nof another simplex.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0175","source_file":"simplicial.tex","source_line":831,"source_end_line":840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L831-L840","statement_sha256":"a422f96c1a56207b3aa1e478fbc3a33369333a05794a202e614fe5b43b5777d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2712,"rank":2712,"depth":0,"x":2445.958,"y":302.616,"cluster":"homological-algebra"},{"id":"stacks:0177","tag":"0177","title":"Simplicial sets · Lemma 0177","summary":"Let U be a simplicial set. Let n ≥ 0 be an integer. There is a canonical bijection Mor(Δ[n], U) → U_n which maps a morphism φ to the value of φ on the unique nondegenerate n-simplex of Δ[n].","statement_latex":"Let $U$ be a simplicial set. Let $n \\geq 0$ be an integer.\nThere is a canonical bijection\n$$\n\\Mor(\\Delta[n], U)\n\\longrightarrow\nU_n\n$$\nwhich maps a morphism $\\varphi$ to the value of $\\varphi$\non the unique nondegenerate $n$-simplex of $\\Delta[n]$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0177","source_file":"simplicial.tex","source_line":858,"source_end_line":869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L858-L869","statement_sha256":"bcab418d9b4eaaadc3564e988e893208412bc48421f15d7bc743266f90f8ab43","origin":"The Stacks Project","memory_eligible":false,"source_rank":2713,"rank":2713,"depth":0,"x":2438.564,"y":139.571,"cluster":"homological-algebra"},{"id":"stacks:0179","tag":"0179","title":"Simplicial sets · Lemma 0179","summary":"Let U, V be simplicial sets. Let a, b ≥ 0 be integers. Assume every n-simplex of U is degenerate if n > a. Assume every n-simplex of V is degenerate if n > b. Then every n-simplex of U × V is degenerate if n > a + b.","statement_latex":"Let $U$, $V$ be simplicial sets.\nLet $a, b \\geq 0$ be integers.\nAssume every $n$-simplex of $U$ is degenerate if $n > a$.\nAssume every $n$-simplex of $V$ is degenerate if $n > b$.\nThen every $n$-simplex of $U \\times V$ is degenerate\nif $n > a + b$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0179","source_file":"simplicial.tex","source_line":905,"source_end_line":913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L905-L913","statement_sha256":"75a6a38a34b47dabdf01c4d87b4804bccf9a822cdf08abfc815aa22fb490fdb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2714,"rank":2714,"depth":0,"x":2575.461,"y":255.882,"cluster":"homological-algebra"},{"id":"stacks:0180","tag":"0180","title":"Truncated simplicial objects and skeleton functors · Definition 0180","summary":"An n-truncated simplicial object of C is a contravariant functor from Δ_≤ n to C. A morphism of n-truncated simplicial objects is a transformation of functors. We denote the category of n-truncated simplicial objects of C by the symbol Simp_n(C).","statement_latex":"An {\\it $n$-truncated simplicial object of $\\mathcal{C}$}\nis a contravariant functor from $\\Delta_{\\leq n}$ to\n$\\mathcal{C}$. A {\\it morphism of $n$-truncated\nsimplicial objects} is a transformation of functors.\nWe denote the category of $n$-truncated\nsimplicial objects of $\\mathcal{C}$ by\nthe symbol $\\text{Simp}_n(\\mathcal{C})$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Truncated simplicial objects and skeleton functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0180","source_file":"simplicial.tex","source_line":936,"source_end_line":945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L936-L945","statement_sha256":"0c24a1aecef0a7bcfb35412d4b609bfb411b9ca03f599987595992358a80e5bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2715,"rank":2715,"depth":0,"x":2380.517,"y":247.775,"cluster":"homological-algebra"},{"id":"stacks:017C","tag":"017C","title":"Products with simplicial sets · Definition 017C","summary":"Let C be a category such that the coproduct of any two objects of C exists. Let U be a simplicial set. Let V be a simplicial object of C. Assume that each U_n is finite nonempty. In this case we define the product U × V of U and V to be the simplicial object of C whose nth term is the object (U × V)_n = coprod_u∈ U_n V_n with maps for φ : [m] → [n] given by the morphism coprod_u∈ U_n V_n → coprod_u'∈ U_m V_m which maps the component V_n corresponding to u to the component…","statement_latex":"Let $\\mathcal{C}$ be a category such that the coproduct of\nany two objects of $\\mathcal{C}$ exists. Let\n$U$ be a simplicial set. Let $V$ be a simplicial\nobject of $\\mathcal{C}$. Assume that each $U_n$ is\nfinite nonempty. In this case we define\nthe {\\it product $U \\times V$ of $U$ and $V$}\nto be the simplicial object of $\\mathcal{C}$ whose\n$n$th term is the object\n$$\n(U \\times V)_n = \\coprod\\nolimits_{u\\in U_n} V_n\n$$\nwith maps for $\\varphi : [m] \\to [n]$ given by the\nmorphism\n$$\n\\coprod\\nolimits_{u\\in U_n} V_n\n\\longrightarrow\n\\coprod\\nolimits_{u'\\in U_m} V_m\n$$\nwhich maps the component $V_n$ corresponding to $u$ to the\ncomponent $V_m$ corresponding to $u' = U(\\varphi)(u)$\nvia the morphism $V(\\varphi)$.\nMore loosely, if all of the coproducts displayed above\nexist (without assuming anything about $\\mathcal{C}$)\nwe will say that the {\\it product $U \\times V$ exists}.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Products with simplicial sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017C","source_file":"simplicial.tex","source_line":991,"source_end_line":1017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L991-L1017","statement_sha256":"f54758501074dae3c0fb8171f4366f62d08f1bf96fdb32e2007935d5cc5a3e5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2716,"rank":2716,"depth":0,"x":2531.143,"y":142.888,"cluster":"homological-algebra"},{"id":"stacks:017D","tag":"017D","title":"Products with simplicial sets · Lemma 017D","summary":"Let C be a category such that the coproduct of any two objects of C exists. Let U be a simplicial set. Let V be a simplicial object of C. Assume that each U_n is finite nonempty. The functor W ↦ Mor_Simp(C)(U × V, W) is canonically isomorphic to the functor which maps W to the set in Equation ([Tag 017B]).","statement_latex":"Let $\\mathcal{C}$ be a category such that the coproduct of\nany two objects of $\\mathcal{C}$ exists. Let\n$U$ be a simplicial set. Let $V$ be a simplicial\nobject of $\\mathcal{C}$. Assume that each $U_n$ is\nfinite nonempty. The functor\n$W \\mapsto \\Mor_{\\text{Simp}(\\mathcal{C})}(U \\times V, W)$\nis canonically isomorphic to the functor which\nmaps $W$ to the set in\nEquation (\\ref{equation-functor-product-with-simplicial-set}).","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Products with simplicial sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017D","source_file":"simplicial.tex","source_line":1019,"source_end_line":1030,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1019-L1030","statement_sha256":"f3d49c2823fa9faf9db8809d577d581d5b4394a120bbb1c4b31823faa14e0733","origin":"The Stacks Project","memory_eligible":false,"source_rank":2717,"rank":2717,"depth":0,"x":2504.357,"y":306.084,"cluster":"homological-algebra"},{"id":"stacks:017E","tag":"017E","title":"Products with simplicial sets · Lemma 017E","summary":"Let C be a category such that the coproduct of any two objects of C exists. Let us temporarily denote FSSets the category of simplicial sets all of whose components are finite nonempty. • The rule (U, V) ↦ U × V defines a functor FSSets × Simp(C) → Simp(C). • For every U, V as above there is a canonical map of simplicial objects U × V → V defined by taking the identity on each component of (U × V)_n = coprod_u V_n.","statement_latex":"Let $\\mathcal{C}$ be a category such that the coproduct of\nany two objects of $\\mathcal{C}$ exists. Let us temporarily\ndenote $\\textit{FSSets}$ the category of simplicial sets\nall of whose components are finite nonempty.\n\\begin{enumerate}\n\\item The rule $(U, V) \\mapsto U \\times V$\ndefines a functor\n$\\textit{FSSets} \\times \\text{Simp}(\\mathcal{C})\n\\to \\text{Simp}(\\mathcal{C})$.\n\\item For every $U$, $V$ as above\nthere is a canonical map of simplicial objects\n$$\nU \\times V \\longrightarrow V\n$$\ndefined by taking the identity on each component of\n$(U \\times V)_n = \\coprod_u V_n$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Products with simplicial sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017E","source_file":"simplicial.tex","source_line":1036,"source_end_line":1055,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1036-L1055","statement_sha256":"ac9ec3aa504ebd27eb4b85291bd936ff049551c30c1d12687a86387c93596c9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2718,"rank":2718,"depth":0,"x":2392.608,"y":170.227,"cluster":"homological-algebra"},{"id":"stacks:017F","tag":"017F","title":"Products with simplicial sets · Lemma 017F","summary":"With X and k as above. For any simplicial object V of C we have the following canonical bijection Mor_Simp(C)(X × Δ[k], V) → Mor_C(X, V_k). which maps γ to the restriction of the morphism γ_k to the component corresponding to id_[k]. Similarly, for any n ≥ k, if W is an n-truncated simplicial object of C, then we have Mor_Simp_n(C)(sk_n(X × Δ[k]), W) = Mor_C(X, W_k).","statement_latex":"With $X$ and $k$ as above.\nFor any simplicial object $V$ of\n$\\mathcal{C}$ we have the following\ncanonical bijection\n$$\n\\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times \\Delta[k], V)\n\\longrightarrow\n\\Mor_\\mathcal{C}(X, V_k).\n$$\nwhich maps $\\gamma$ to the restriction of the\nmorphism $\\gamma_k$ to the component corresponding\nto $\\text{id}_{[k]}$.\nSimilarly, for any $n \\geq k$, if $W$ is an\n$n$-truncated simplicial object\nof $\\mathcal{C}$, then we have\n$$\n\\Mor_{\\text{Simp}_n(\\mathcal{C})}(\\text{sk}_n(X \\times \\Delta[k]), W)\n=\n\\Mor_\\mathcal{C}(X, W_k).\n$$","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Products with simplicial sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017F","source_file":"simplicial.tex","source_line":1074,"source_end_line":1096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1074-L1096","statement_sha256":"3f035765e2948af77d2b5ae8e5da6ad4b51081cd8ce2ac6ff68c34072c86e65e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2719,"rank":2719,"depth":0,"x":2584.712,"y":207.084,"cluster":"homological-algebra"},{"id":"stacks:019V","tag":"019V","title":"Hom from simplicial sets into cosimplicial objects · Definition 019V","summary":"Let C be a category with finite products. Let V be a cosimplicial object of C. Let U be a simplicial set such that each U_n is finite nonempty. We define Hom(U, V) to be the cosimplicial object of C defined as follows: • we set Hom(U, V)_n = ∏_u ∈ U_n V_n, in other words the unique object of C such that its X-valued points satisfy Mor_C(X, Hom(U, V)_n) = Map(U_n, Mor_C(X, V_n)) and • for φ : [m] → [n] we take the map Hom(U, V)_m → Hom(U, V)_n given by f ↦ V(φ) ∘ f ∘ U(φ)…","statement_latex":"Let $\\mathcal{C}$ be a category with finite products.\nLet $V$ be a cosimplicial object of $\\mathcal{C}$.\nLet $U$ be a simplicial set such that each\n$U_n$ is finite nonempty.\nWe define {\\it $\\Hom(U, V)$} to be\nthe cosimplicial object of $\\mathcal{C}$ defined\nas follows:\n\\begin{enumerate}\n\\item we set $\\Hom(U, V)_n = \\prod_{u \\in U_n} V_n$,\nin other words the unique object of $\\mathcal{C}$ such\nthat its $X$-valued points satisfy\n$$\n\\Mor_\\mathcal{C}(X, \\Hom(U, V)_n)\n=\n\\text{Map}(U_n, \\Mor_\\mathcal{C}(X, V_n))\n$$\nand\n\\item for $\\varphi : [m] \\to [n]$ we take the map\n$\\Hom(U, V)_m \\to \\Hom(U, V)_n$\ngiven by $f \\mapsto V(\\varphi) \\circ f \\circ U(\\varphi)$\non $X$-valued points as above.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Hom from simplicial sets into cosimplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019V","source_file":"simplicial.tex","source_line":1157,"source_end_line":1181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1157-L1181","statement_sha256":"40c87a52b46c8445465af37adfc78825bb58276d9413b28c0dd04dc05098305d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2720,"rank":2720,"depth":0,"x":2413.018,"y":289.101,"cluster":"homological-algebra"},{"id":"stacks:0B15","tag":"0B15","title":"Hom from cosimplicial sets into simplicial objects · Definition 0B15","summary":"Let C be a category with finite products. Let V be a simplicial object of C. Let U be a cosimplicial set such that each U_n is finite nonempty. We define Hom(U, V) to be the simplicial object of C defined as follows: • we set Hom(U, V)_n = ∏_u ∈ U_n V_n, in other words the unique object of C such that its X-valued points satisfy Mor_C(X, Hom(U, V)_n) = Map(U_n, Mor_C(X, V_n)) and • for φ : [m] → [n] we take the map Hom(U, V)_n → Hom(U, V)_m given by f ↦ V(φ) ∘ f ∘ U(φ) on…","statement_latex":"Let $\\mathcal{C}$ be a category with finite products.\nLet $V$ be a simplicial object of $\\mathcal{C}$.\nLet $U$ be a cosimplicial set such that each $U_n$ is finite nonempty.\nWe define {\\it $\\Hom(U, V)$} to be\nthe simplicial object of $\\mathcal{C}$ defined\nas follows:\n\\begin{enumerate}\n\\item we set $\\Hom(U, V)_n = \\prod_{u \\in U_n} V_n$,\nin other words the unique object of $\\mathcal{C}$ such\nthat its $X$-valued points satisfy\n$$\n\\Mor_\\mathcal{C}(X, \\Hom(U, V)_n)\n=\n\\text{Map}(U_n, \\Mor_\\mathcal{C}(X, V_n))\n$$\nand\n\\item for $\\varphi : [m] \\to [n]$ we take the map\n$\\Hom(U, V)_n \\to \\Hom(U, V)_m$\ngiven by $f \\mapsto V(\\varphi) \\circ f \\circ U(\\varphi)$\non $X$-valued points as above.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Hom from cosimplicial sets into simplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B15","source_file":"simplicial.tex","source_line":1210,"source_end_line":1233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1210-L1233","statement_sha256":"57dd161329e576c1c22bdaae206342c7386011b40a6f0ce6fddb005c12f27c5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2721,"rank":2721,"depth":0,"x":2473.808,"y":130.833,"cluster":"homological-algebra"},{"id":"stacks:017M","tag":"017M","title":"Hom from cosimplicial sets into simplicial objects · Lemma 017M","summary":"With X, k and U as above. • For any simplicial object V of C we have the following canonical bijection Mor_Simp(C)(V, U) → Mor_C(V_k, X). which maps γ to the morphism γ_k composed with the projection onto the factor corresponding to id_[k]. • Similarly, if W is an k-truncated simplicial object of C, then we have Mor_Simp_k(C)(W, sk_k U) = Mor_C(W_k, X). • The object U constructed above is an incarnation of Hom(C[k], X) where C[k] is the cosimplicial set from Example [Tag…","statement_latex":"With $X$, $k$ and $U$ as above.\n\\begin{enumerate}\n\\item For any simplicial object $V$ of\n$\\mathcal{C}$ we have the following\ncanonical bijection\n$$\n\\Mor_{\\text{Simp}(\\mathcal{C})}(V, U)\n\\longrightarrow\n\\Mor_\\mathcal{C}(V_k, X).\n$$\nwhich maps $\\gamma$ to the morphism $\\gamma_k$ composed with\nthe projection onto the factor corresponding to $\\text{id}_{[k]}$.\n\\item Similarly, if $W$ is an $k$-truncated simplicial object\nof $\\mathcal{C}$, then we have\n$$\n\\Mor_{\\text{Simp}_k(\\mathcal{C})}(W, \\text{sk}_k U)\n=\n\\Mor_\\mathcal{C}(W_k, X).\n$$\n\\item The object $U$ constructed above is an\nincarnation of $\\Hom(C[k], X)$ where $C[k]$ is the cosimplicial set from\nExample \\ref{example-simplex-cosimplicial-set}.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Hom from cosimplicial sets into simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017M","source_file":"simplicial.tex","source_line":1268,"source_end_line":1293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1268-L1293","statement_sha256":"d0f9518af88c80bec00677516d69c83fda3c8b5757ae2390c3a0238a451b0e80","origin":"The Stacks Project","memory_eligible":false,"source_rank":2722,"rank":2722,"depth":0,"x":2556.448,"y":282.381,"cluster":"homological-algebra"},{"id":"stacks:017I","tag":"017I","title":"Hom from simplicial sets into simplicial objects · Definition 017I","summary":"Let C be a category such that the coproduct of any two objects exists. Let U be a simplicial set, with U_n finite nonempty for all n ≥ 0. Let V be a simplicial object of C. We denote Hom(U, V) any simplicial object of C such that Mor_Simp(C)(W, Hom(U, V)) = Mor_Simp(C)(W × U, V) functorially in the simplicial object W of C.","statement_latex":"Let $\\mathcal{C}$ be a category such that the coproduct\nof any two objects exists.\nLet $U$ be a simplicial set, with $U_n$ finite nonempty\nfor all $n \\geq 0$.\nLet $V$ be a simplicial object of $\\mathcal{C}$.\nWe denote {\\it $\\Hom(U, V)$} any simplicial object of\n$\\mathcal{C}$ such that\n$$\n\\Mor_{\\text{Simp}(\\mathcal{C})}(W, \\Hom(U, V))\n=\n\\Mor_{\\text{Simp}(\\mathcal{C})}(W \\times U, V)\n$$\nfunctorially in the simplicial object $W$ of $\\mathcal{C}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Hom from simplicial sets into simplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017I","source_file":"simplicial.tex","source_line":1384,"source_end_line":1399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1384-L1399","statement_sha256":"eb5ba9ed62275e27de76e0c4aea0ddfe0461acb4936dcff2cb9e5d8ee2f249a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2723,"rank":2723,"depth":0,"x":2373.219,"y":217.374,"cluster":"homological-algebra"},{"id":"stacks:017J","tag":"017J","title":"Hom from simplicial sets into simplicial objects · Lemma 017J","summary":"Assume the category C has coproducts of any two objects and countable limits. Let U be a simplicial set, with U_n finite nonempty for all n ≥ 0. Let V be a simplicial object of C. Then the functor C^opp & → & Sets X & ↦ & Mor_Simp(C)(X × U, V) is representable.","statement_latex":"Assume the category $\\mathcal{C}$\nhas coproducts of any two objects and countable\nlimits. Let $U$ be a simplicial set, with $U_n$ finite nonempty\nfor all $n \\geq 0$.\nLet $V$ be a simplicial object of $\\mathcal{C}$.\nThen the functor\n\\begin{eqnarray*}\n\\mathcal{C}^{opp} & \\longrightarrow & \\textit{Sets} \\\\\nX\n& \\longmapsto &\n\\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times U, V)\n\\end{eqnarray*}\nis representable.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Hom from simplicial sets into simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017J","source_file":"simplicial.tex","source_line":1409,"source_end_line":1424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1409-L1424","statement_sha256":"2f4932bbed78d7c1b53310fbdee6e6d225ec8b68fbac1abf01bfe6bc37fcc380","origin":"The Stacks Project","memory_eligible":false,"source_rank":2724,"rank":2724,"depth":0,"x":2561.035,"y":161.212,"cluster":"homological-algebra"},{"id":"stacks:017K","tag":"017K","title":"Hom from simplicial sets into simplicial objects · Lemma 017K","summary":"Assume the category C has coproducts of any two objects and finite limits. Let U be a simplicial set, with U_n finite nonempty for all n ≥ 0. Assume that all n-simplices of U are degenerate for all n gg 0. Let V be a simplicial object of C. Then the functor C^opp & → & Sets X & ↦ & Mor_Simp(C)(X × U, V) is representable.","statement_latex":"Assume the category $\\mathcal{C}$\nhas coproducts of any two objects and finite\nlimits. Let $U$ be a simplicial set, with $U_n$ finite nonempty\nfor all $n \\geq 0$. Assume that all $n$-simplices\nof $U$ are degenerate for all $n \\gg 0$.\nLet $V$ be a simplicial object of $\\mathcal{C}$.\nThen the functor\n\\begin{eqnarray*}\n\\mathcal{C}^{opp} & \\longrightarrow & \\textit{Sets} \\\\\nX\n& \\longmapsto &\n\\Mor_{\\text{Simp}(\\mathcal{C})}(X \\times U, V)\n\\end{eqnarray*}\nis representable.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Hom from simplicial sets into simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017K","source_file":"simplicial.tex","source_line":1460,"source_end_line":1476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1460-L1476","statement_sha256":"12335db57ce63a373a504d1b34fa896d4b6c9191d2492a830c90c9058173d3c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2725,"rank":2725,"depth":1,"x":2467.493,"y":309.533,"cluster":"homological-algebra"},{"id":"stacks:017L","tag":"017L","title":"Hom from simplicial sets into simplicial objects · Lemma 017L","summary":"Assume the category C has coproducts of any two objects and finite limits. Let U be a simplicial set, with U_n finite nonempty for all n ≥ 0. Assume that all n-simplices of U are degenerate for all n gg 0. Let V be a simplicial object of C. Then Hom(U, V) exists, moreover we have the expected equalities Hom(U, V)_n = Hom(U × Δ[n], V)_0.","statement_latex":"Assume the category $\\mathcal{C}$\nhas coproducts of any two objects and finite\nlimits. Let $U$ be a simplicial set, with $U_n$ finite nonempty\nfor all $n \\geq 0$. Assume that all $n$-simplices\nof $U$ are degenerate for all $n \\gg 0$.\nLet $V$ be a simplicial object of $\\mathcal{C}$.\nThen $\\Hom(U, V)$ exists, moreover\nwe have the expected equalities\n$$\n\\Hom(U, V)_n = \\Hom(U \\times \\Delta[n], V)_0.\n$$","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Hom from simplicial sets into simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017L","source_file":"simplicial.tex","source_line":1545,"source_end_line":1558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1545-L1558","statement_sha256":"103fd89a22c26ad54bd1dda36c0f8bbf3c41740574065b0ba53d0a10f3019eed","origin":"The Stacks Project","memory_eligible":false,"source_rank":2726,"rank":2726,"depth":2,"x":2417.08,"y":146.717,"cluster":"homological-algebra"},{"id":"stacks:017N","tag":"017N","title":"Hom from simplicial sets into simplicial objects · Lemma 017N","summary":"Assume the category C has coproducts of any two objects and finite limits. Let a : U → V, b : U → W be morphisms of simplicial sets. Assume U_n, V_n, W_n finite nonempty for all n ≥ 0. Assume that all n-simplices of U, V, W are degenerate for all n gg 0. Let T be a simplicial object of C. Then Hom(V, T) ×_Hom(U, T) Hom(W, T) = Hom(V amalg_U W, T) In other words, the fibre product on the left hand side is represented by the Hom object on the right hand side.","statement_latex":"Assume the category $\\mathcal{C}$\nhas coproducts of any two objects and finite\nlimits. Let $a : U \\to V$, $b : U \\to W$\nbe morphisms of simplicial sets.\nAssume $U_n, V_n, W_n$ finite nonempty for all $n \\geq 0$.\nAssume that all $n$-simplices of $U, V, W$\nare degenerate for all $n \\gg 0$.\nLet $T$ be a simplicial object of $\\mathcal{C}$.\nThen\n$$\n\\Hom(V, T) \\times_{\\Hom(U, T)} \\Hom(W, T)\n=\n\\Hom(V \\amalg_U W, T)\n$$\nIn other words, the fibre product on the left hand\nside is represented by the Hom object on the right hand side.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Hom from simplicial sets into simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017N","source_file":"simplicial.tex","source_line":1685,"source_end_line":1703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1685-L1703","statement_sha256":"a7b1a23372f71bb0b06c7a8a4d7cfcde65a908da33efecf31233f33d0a596ac6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2727,"rank":2727,"depth":3,"x":2585.566,"y":238.377,"cluster":"homological-algebra"},{"id":"stacks:017P","tag":"017P","title":"Splitting simplicial objects · Definition 017P","summary":"Let C be a category which admits finite nonempty coproducts. We say a simplicial object U of C is split if there exist subobjects N(U_m) of U_m, m ≥ 0 with the property that coprod_φ : [n] → [m] surjective N(U_m) → U_n is an isomorphism for all n ≥ 0. If U is an r-truncated simplicial object of C then we say U is split if there exist subobjects N(U_m) of U_m, r ≥ m ≥ 0 with the property that ([Tag 017Q]) is an isomorphism for r ≥ n ≥ 0.","statement_latex":"Let $\\mathcal{C}$ be a category which admits finite nonempty coproducts.\nWe say a simplicial object $U$ of $\\mathcal{C}$ is {\\it split}\nif there exist subobjects $N(U_m)$ of $U_m$, $m \\geq 0$\nwith the property that\n\\begin{equation}\n\n\\coprod\\nolimits_{\\varphi : [n] \\to [m]\\text{ surjective}}\nN(U_m)\n\\longrightarrow\nU_n\n\\end{equation}\nis an isomorphism for all $n \\geq 0$. If $U$ is an $r$-truncated\nsimplicial object of $\\mathcal{C}$ then we say $U$ is {\\it split}\nif there exist subobjects $N(U_m)$ of $U_m$, $r \\geq m \\geq 0$\nwith the property that (\\ref{equation-splitting})\nis an isomorphism for $r \\geq n \\geq 0$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Splitting simplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017P","source_file":"simplicial.tex","source_line":1766,"source_end_line":1784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1766-L1784","statement_sha256":"0c94309d1881122184c326cb123aca6e8a0232df258d5e3c59fff8e4fc2f4ff0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2728,"rank":2728,"depth":0,"x":2387.171,"y":266.454,"cluster":"homological-algebra"},{"id":"stacks:017R","tag":"017R","title":"Splitting simplicial objects · Lemma 017R","summary":"Let U be a simplicial set. Then U has a unique splitting with N(U_m) equal to the set of nondegenerate m-simplices.","statement_latex":"Let $U$ be a simplicial set. Then $U$ has a unique splitting\nwith $N(U_m)$ equal to the set of nondegenerate $m$-simplices.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Splitting simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017R","source_file":"simplicial.tex","source_line":1795,"source_end_line":1799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1795-L1799","statement_sha256":"eb761da70c750270dec055baa1db5baa0c99aa2e45faabc61936b1d7dd7cf703","origin":"The Stacks Project","memory_eligible":false,"source_rank":2729,"rank":2729,"depth":0,"x":2511.163,"y":132.878,"cluster":"homological-algebra"},{"id":"stacks:017S","tag":"017S","title":"Splitting simplicial objects · Lemma 017S","summary":"Let f : U → V be a morphism of simplicial sets. Suppose that (a) the image of every nondegenerate simplex of U is a nondegenerate simplex of V and (b) the restriction of f to a map from the set of nondegenerate simplices of U to the set of nondegenerate simplices of V is injective. Then f_n is injective for all n. Same holds with \"injective\" replaced by \"surjective\" or \"bijective\".","statement_latex":"Let $f : U \\to V$ be a morphism of simplicial sets.\nSuppose that (a) the image of every nondegenerate simplex of\n$U$ is a nondegenerate simplex of $V$ and (b) the restriction\nof $f$ to a map from the set of nondegenerate simplices of $U$\nto the set of nondegenerate simplices of $V$ is injective.\nThen $f_n$ is injective for all $n$.\nSame holds with ``injective'' replaced by\n``surjective'' or ``bijective''.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Splitting simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017S","source_file":"simplicial.tex","source_line":1823,"source_end_line":1833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1823-L1833","statement_sha256":"27c9bbb1270135c55f54e846ebab21301719b24940c1e166820b5be754751d6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2730,"rank":2730,"depth":1,"x":2527.186,"y":302.107,"cluster":"homological-algebra"},{"id":"stacks:017T","tag":"017T","title":"Splitting simplicial objects · Lemma 017T","summary":"Let U be a simplicial set. Let n ≥ 0 be an integer. The rule U'_m = ⋃_φ : [m] → [i], i≤ n Im(U(φ)) defines a sub simplicial set U' ⊂ U with U'_i = U_i for i ≤ n. Moreover, all m-simplices of U' are degenerate for all m > n.","statement_latex":"Let $U$ be a simplicial set.\nLet $n \\geq 0$ be an integer.\nThe rule\n$$\nU'_m = \\bigcup\\nolimits_{\\varphi : [m] \\to [i], \\ i\\leq n} \\Im(U(\\varphi))\n$$\ndefines a sub simplicial set $U' \\subset U$ with\n$U'_i = U_i$ for $i \\leq n$.\nMoreover, all $m$-simplices of $U'$ are degenerate for\nall $m > n$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Splitting simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017T","source_file":"simplicial.tex","source_line":1856,"source_end_line":1868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1856-L1868","statement_sha256":"2a3db828710a07a3e9396a70ca510c9568c299cc6e9b26597269764ab2dcd4b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2731,"rank":2731,"depth":0,"x":2378.954,"y":186.156,"cluster":"homological-algebra"},{"id":"stacks:017U","tag":"017U","title":"Splitting simplicial objects · Lemma 017U","summary":"Let U be a simplicial abelian group. Then U has a splitting obtained by taking N(U_0) = U_0 and for m ≥ 1 taking N(U_m) = ⋂_i = 0^m - 1 Ker(d^m_i). Moreover, this splitting is functorial on the category of simplicial abelian groups.","statement_latex":"Let $U$ be a simplicial abelian group.\nThen $U$ has a splitting obtained by taking $N(U_0) = U_0$ and\nfor $m \\geq 1$ taking\n$$\nN(U_m) = \\bigcap\\nolimits_{i = 0}^{m - 1} \\Ker(d^m_i).\n$$\nMoreover, this splitting is functorial on the category\nof simplicial abelian groups.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Splitting simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017U","source_file":"simplicial.tex","source_line":1879,"source_end_line":1889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L1879-L1889","statement_sha256":"0b656616c445ba6bd11f7bb1b9876b9aab7ae282bd1547a4b3e572257b9ddec7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2732,"rank":2732,"depth":1,"x":2581.95,"y":187.549,"cluster":"homological-algebra"},{"id":"stacks:017V","tag":"017V","title":"Splitting simplicial objects · Lemma 017V","summary":"Let A be an abelian category. Let U be a simplicial object in A. Then U has a splitting obtained by taking N(U_0) = U_0 and for m ≥ 1 taking N(U_m) = ⋂_i = 0^m - 1 Ker(d^m_i). Moreover, this splitting is functorial on the category of simplicial objects of A.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $U$ be a simplicial object in $\\mathcal{A}$.\nThen $U$ has a splitting obtained by taking $N(U_0) = U_0$ and\nfor $m \\geq 1$ taking\n$$\nN(U_m) = \\bigcap\\nolimits_{i = 0}^{m - 1} \\Ker(d^m_i).\n$$\nMoreover, this splitting is functorial on the category of\nsimplicial objects of $\\mathcal{A}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Splitting simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017V","source_file":"simplicial.tex","source_line":2027,"source_end_line":2038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2027-L2038","statement_sha256":"94cc87ae088c2e1fd540a7d36f76ca713731c1949abec27e2e283db32b66605d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2733,"rank":2733,"depth":2,"x":2430.812,"y":301.957,"cluster":"homological-algebra"},{"id":"stacks:017W","tag":"017W","title":"Splitting simplicial objects · Lemma 017W","summary":"The Dold-Kan normalization functor reflects injectivity, surjectivity, and isomorphy. Let A be an abelian category. Let f : U → V be a morphism of simplicial objects of A. If the induced morphisms N(f)_i : N(U)_i → N(V)_i are injective for all i, then f_i is injective for all i. Same holds with \"injective\" replaced with \"surjective\", or \"isomorphism\".","statement_latex":"\\begin{slogan}\nThe Dold-Kan normalization functor reflects\ninjectivity, surjectivity, and isomorphy.\n\\end{slogan}\nLet $\\mathcal{A}$ be an abelian category.\nLet $f : U \\to V$ be a morphism of\nsimplicial objects of $\\mathcal{A}$.\nIf the induced morphisms $N(f)_i : N(U)_i \\to N(V)_i$\nare injective for all $i$, then $f_i$ is\ninjective for all $i$. Same holds with ``injective'' replaced\nwith ``surjective'', or ``isomorphism''.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Splitting simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017W","source_file":"simplicial.tex","source_line":2056,"source_end_line":2069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2056-L2069","statement_sha256":"de9a3c11772a24ad1db304e2752b77d54e22879af31f640e29af834a8e8cb72f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2734,"rank":2734,"depth":3,"x":2450.298,"y":131.463,"cluster":"homological-algebra"},{"id":"stacks:017X","tag":"017X","title":"Splitting simplicial objects · Lemma 017X","summary":"Let A be an abelian category. Let U be a simplicial object in A. Let N(U_m) as in Lemma [Tag 017V] above. Then d^m_m(N(U_m)) ⊂ N(U_m - 1).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $U$ be a simplicial object in $\\mathcal{A}$.\nLet $N(U_m)$ as in Lemma \\ref{lemma-splitting-abelian-category} above.\nThen $d^m_m(N(U_m)) \\subset N(U_{m - 1})$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Splitting simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017X","source_file":"simplicial.tex","source_line":2077,"source_end_line":2083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2077-L2083","statement_sha256":"3d2138bd3e8cc580545ff29449ed1ada9fac307e444e600890cc1b5f034a9520","origin":"The Stacks Project","memory_eligible":false,"source_rank":2735,"rank":2735,"depth":3,"x":2573.303,"y":268.538,"cluster":"homological-algebra"},{"id":"stacks:017Y","tag":"017Y","title":"Splitting simplicial objects · Lemma 017Y","summary":"Let A be an abelian category. Let U be a simplicial object of A. Let n ≥ 0 be an integer. The rule U'_m = ∑_φ : [m] → [i], i≤ n Im(U(φ)) defines a sub simplicial object U' ⊂ U with U'_i = U_i for i ≤ n. Moreover, N(U'_m) = 0 for all m > n.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $U$ be a simplicial object of $\\mathcal{A}$.\nLet $n \\geq 0$ be an integer.\nThe rule\n$$\nU'_m = \\sum\\nolimits_{\\varphi : [m] \\to [i], \\ i\\leq n} \\Im(U(\\varphi))\n$$\ndefines a sub simplicial object $U' \\subset U$ with $U'_i = U_i$\nfor $i \\leq n$.\nMoreover, $N(U'_m) = 0$ for all $m > n$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Splitting simplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/017Y","source_file":"simplicial.tex","source_line":2092,"source_end_line":2104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2092-L2104","statement_sha256":"60364485086e9e7c2db7338f7697278a4350e2803ac7b3fd196f7863c4fd4eb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2736,"rank":2736,"depth":3,"x":2371.933,"y":237.188,"cluster":"homological-algebra"},{"id":"stacks:0183","tag":"0183","title":"Coskeleton functors · Lemma 0183","summary":"If the category C has finite limits, then cosk_m functors exist for all m. Moreover, for any m-truncated simplicial object U the simplicial object cosk_mU is described by the formula (cosk_mU)_n = lim_(Δ/[n])_≤ m^opp U(n) and for φ : [n] → [n'] the map cosk_mU(φ) comes from the identification U(n') ∘ overlineφ = U(n) above via Categories, Lemma [Tag 002L].","statement_latex":"If the category $\\mathcal{C}$ has finite limits, then\n$\\text{cosk}_m$ functors exist for all $m$. Moreover,\nfor any $m$-truncated simplicial object $U$ the\nsimplicial object $\\text{cosk}_mU$ is described\nby the formula\n$$\n(\\text{cosk}_mU)_n = \\lim_{(\\Delta/[n])_{\\leq m}^{opp}} U(n)\n$$\nand for $\\varphi : [n] \\to [n']$ the map\n$\\text{cosk}_mU(\\varphi)$ comes from the\nidentification $U(n') \\circ \\overline{\\varphi} = U(n)$ above\nvia Categories, Lemma \\ref{categories-lemma-functorial-limit}.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0183","source_file":"simplicial.tex","source_line":2233,"source_end_line":2247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2233-L2247","statement_sha256":"868edc316d09b2bb59b4867279b64fae16c772fe21a6dbe4c22d23a4d83dc87e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2737,"rank":2737,"depth":1,"x":2546.006,"y":145.846,"cluster":"homological-algebra"},{"id":"stacks:0184","tag":"0184","title":"Coskeleton functors · Lemma 0184","summary":"Let C be a category. Let U be an m-truncated simplicial object of C. For n ≤ m the limit lim_(Δ/[n])_≤ m^opp U(n) exists and is canonically isomorphic to U_n.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$.\nFor $n \\leq m$ the limit $\\lim_{(\\Delta/[n])_{\\leq m}^{opp}} U(n)$\nexists and is canonically isomorphic to $U_n$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0184","source_file":"simplicial.tex","source_line":2291,"source_end_line":2297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2291-L2297","statement_sha256":"f5182768237fa3dab40ef07820d3c0650a4344e9a8b79040d260a7d6782f23b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2738,"rank":2738,"depth":0,"x":2490.984,"y":312.331,"cluster":"homological-algebra"},{"id":"stacks:0185","tag":"0185","title":"Coskeleton functors · Lemma 0185","summary":"Let C be a category with finite limits. Let U be an n-truncated simplicial object of C. The morphism sk_n cosk_n U → U is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category with finite limits.\nLet $U$ be an $n$-truncated simplicial object of $\\mathcal{C}$.\nThe morphism $\\text{sk}_n \\text{cosk}_n U \\to U$\nis an isomorphism.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0185","source_file":"simplicial.tex","source_line":2305,"source_end_line":2311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2305-L2311","statement_sha256":"2aafb2de21ff06aff0994404cb2386ee6cc2b056278e5196e4906ccc802d302b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2739,"rank":2739,"depth":2,"x":2397.475,"y":158.017,"cluster":"homological-algebra"},{"id":"stacks:0186","tag":"0186","title":"Coskeleton functors · Lemma 0186","summary":"Let n be an integer ≥ 1. Let U be a n-truncated simplicial object of C. Consider the contravariant functor from C to Sets which associates to an object T the set ( (f_0, …, f_n + 1) ∈ Mor_C(T, U_n) mid d^n_j - 1 ∘ f_i = d^n_i ∘ f_j ∀ 0≤ i < j≤ n + 1) If this functor is representable by some object U_n + 1 of C, then U_n + 1 = lim_(Δ/[n + 1])_≤ n^opp U(n)","statement_latex":"Let $n$ be an integer $\\geq 1$.\nLet $U$ be a $n$-truncated simplicial object of $\\mathcal{C}$.\nConsider the contravariant functor from $\\mathcal{C}$ to\n$\\textit{Sets}$ which associates to an object $T$ the set\n$$\n\\{ (f_0, \\ldots, f_{n + 1}) \\in \\Mor_\\mathcal{C}(T, U_n)\n\\mid\nd^n_{j - 1} \\circ f_i = d^n_i \\circ f_j\n\\ \\forall\\ 0\\leq i < j\\leq n + 1\\}\n$$\nIf this functor is representable by some object $U_{n + 1}$\nof $\\mathcal{C}$, then\n$$\nU_{n + 1} = \\lim_{(\\Delta/[n + 1])_{\\leq n}^{opp}} U(n)\n$$","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0186","source_file":"simplicial.tex","source_line":2336,"source_end_line":2353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2336-L2353","statement_sha256":"cad93399547d0d37cc7a928f8c7d6162a99bcc8034842c6699d657da1460e323","origin":"The Stacks Project","memory_eligible":false,"source_rank":2740,"rank":2740,"depth":0,"x":2590.933,"y":218.876,"cluster":"homological-algebra"},{"id":"stacks:0187","tag":"0187","title":"Coskeleton functors · Lemma 0187","summary":"Let n be an integer ≥ 1. Let U be a n-truncated simplicial object of C. Consider the contravariant functor from C to Sets which associates to an object T the set ( (f_0, …, f_n + 1) ∈ Mor_C(T, U_n) mid d^n_j - 1 ∘ f_i = d^n_i ∘ f_j ∀ 0≤ i < j≤ n + 1) If this functor is representable by some object U_n + 1 of C, then there exists an (n + 1)-truncated simplicial object tilde U, with sk_n tilde U = U and tilde U_n + 1 = U_n + 1 such that the following adjointness holds…","statement_latex":"Let $n$ be an integer $\\geq 1$. Let $U$ be a $n$-truncated\nsimplicial object of $\\mathcal{C}$. Consider the\ncontravariant functor from $\\mathcal{C}$ to $\\textit{Sets}$\nwhich associates to an object $T$ the set\n$$\n\\{ (f_0, \\ldots, f_{n + 1}) \\in \\Mor_\\mathcal{C}(T, U_n)\n\\mid\nd^n_{j - 1} \\circ f_i = d^n_i \\circ f_j\n\\ \\forall\\ 0\\leq i < j\\leq n + 1\\}\n$$\nIf this functor is representable by some object $U_{n + 1}$\nof $\\mathcal{C}$, then there exists an $(n + 1)$-truncated\nsimplicial object $\\tilde U$, with $\\text{sk}_n \\tilde U = U$\nand $\\tilde U_{n + 1} = U_{n + 1}$ such that the following\nadjointness holds\n$$\n\\Mor_{\\text{Simp}_{n + 1}(\\mathcal{C})}(V, \\tilde U)\n=\n\\Mor_{\\text{Simp}_n(\\mathcal{C})}(\\text{sk}_nV, U)\n$$","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0187","source_file":"simplicial.tex","source_line":2427,"source_end_line":2449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2427-L2449","statement_sha256":"9556754e18c0840106229021a83647b7a8a0411528f7a6a6fa9466efe0c60df2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2741,"rank":2741,"depth":2,"x":2398.932,"y":283.91,"cluster":"homological-algebra"},{"id":"stacks:018B","tag":"018B","title":"Coskeleton functors · Lemma 018B","summary":"Let C be a category which has finite limits. • For every n the functor sk_n : Simp(C) → Simp_n(C) has a right adjoint cosk_n. • For every n' ≥ n the functor sk_n : Simp_n'(C) → Simp_n(C) has a right adjoint, namely sk_n'cosk_n. • For every m ≥ n ≥ 0 and every n-truncated simplicial object U of C we have cosk_m sk_m cosk_n U = cosk_n U. • If U is a simplicial object of C such that the canonical map U → cosk_n sk_nU is an isomorphism for some n ≥ 0, then the canonical map U…","statement_latex":"Let $\\mathcal{C}$ be a category which has finite limits.\n\\begin{enumerate}\n\\item For every $n$ the functor $\\text{sk}_n : \\text{Simp}(\\mathcal{C})\n\\to \\text{Simp}_n(\\mathcal{C})$ has a right adjoint $\\text{cosk}_n$.\n\\item For every $n' \\geq n$ the functor\n$\\text{sk}_n : \\text{Simp}_{n'}(\\mathcal{C}) \\to \\text{Simp}_n(\\mathcal{C})$\nhas a right adjoint, namely $\\text{sk}_{n'}\\text{cosk}_n$.\n\\item For every $m \\geq n \\geq 0$ and every $n$-truncated simplicial\nobject $U$ of $\\mathcal{C}$ we have\n$\\text{cosk}_m \\text{sk}_m \\text{cosk}_n U = \\text{cosk}_n U$.\n\\item If $U$ is a simplicial object of $\\mathcal{C}$ such that\nthe canonical map\n$U \\to \\text{cosk}_n \\text{sk}_nU$\nis an isomorphism for some $n \\geq 0$, then the canonical map\n$U \\to \\text{cosk}_m \\text{sk}_mU$\nis an isomorphism for all $m \\geq n$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018B","source_file":"simplicial.tex","source_line":2601,"source_end_line":2620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2601-L2620","statement_sha256":"c94ac7ca0e27eee0c51369f51c748778d4dcc0c139216ff971af94ebba0b9fdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":2742,"rank":2742,"depth":2,"x":2488.401,"y":126.677,"cluster":"homological-algebra"},{"id":"stacks:018C","tag":"018C","title":"Coskeleton functors · Lemma 018C","summary":"Let U, V be n-truncated simplicial objects of a category C. Then cosk_n (U × V) = cosk_nU × cosk_nV whenever the left and right hand sides exist.","statement_latex":"Let $U$, $V$ be $n$-truncated simplicial objects of a\ncategory $\\mathcal{C}$. Then\n$$\n\\text{cosk}_n (U \\times V) = \\text{cosk}_nU \\times \\text{cosk}_nV\n$$\nwhenever the left and right hand sides exist.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018C","source_file":"simplicial.tex","source_line":2650,"source_end_line":2658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2650-L2658","statement_sha256":"08dd100cf82bb804b17627e934e8c3962ae3e53910245b693da954b4b6650bc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2743,"rank":2743,"depth":0,"x":2548.997,"y":293.736,"cluster":"homological-algebra"},{"id":"stacks:018D","tag":"018D","title":"Coskeleton functors · Lemma 018D","summary":"Assume C has fibre products. Let U → V and W → V be morphisms of n-truncated simplicial objects of the category C. Then cosk_n (U ×_V W) = cosk_nU ×_cosk_n V cosk_nW whenever the left and right hand side exist.","statement_latex":"Assume $\\mathcal{C}$ has fibre products.\nLet $U \\to V$ and $W \\to V$ be morphisms\nof $n$-truncated simplicial objects of the\ncategory $\\mathcal{C}$. Then\n$$\n\\text{cosk}_n (U \\times_V W)\n=\n\\text{cosk}_nU \\times_{\\text{cosk}_n V} \\text{cosk}_nW\n$$\nwhenever the left and right hand side exist.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018D","source_file":"simplicial.tex","source_line":2680,"source_end_line":2692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2680-L2692","statement_sha256":"081bf57d7b1a114374a7e7da69e462686f6629322989fc72f2a0a5864ca73f9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2744,"rank":2744,"depth":1,"x":2369.596,"y":204.747,"cluster":"homological-algebra"},{"id":"stacks:08NJ","tag":"08NJ","title":"Coskeleton functors · Lemma 08NJ","summary":"Let C be a category with finite limits. Let X ∈ Ob(C). The functor C/X → C commutes with the coskeleton functors cosk_k for k ≥ 1.","statement_latex":"Let $\\mathcal{C}$ be a category with finite limits.\nLet $X \\in \\Ob(\\mathcal{C})$.\nThe functor $\\mathcal{C}/X \\to \\mathcal{C}$ commutes with\nthe coskeleton functors $\\text{cosk}_k$ for $k \\geq 1$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NJ","source_file":"simplicial.tex","source_line":2699,"source_end_line":2705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2699-L2705","statement_sha256":"a6263838dc3a3df1967cc4ca7de82920cb0c61847fa7b1829bbaa5182dc1c6b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2745,"rank":2745,"depth":3,"x":2573.872,"y":168.495,"cluster":"homological-algebra"},{"id":"stacks:018E","tag":"018E","title":"Coskeleton functors · Lemma 018E","summary":"The canonical map Δ[n] → cosk_1 sk_1 Δ[n] is an isomorphism.","statement_latex":"The canonical map\n$\\Delta[n] \\to \\text{cosk}_1 \\text{sk}_1 \\Delta[n]$\nis an isomorphism.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Coskeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018E","source_file":"simplicial.tex","source_line":2721,"source_end_line":2726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2721-L2726","statement_sha256":"4b16c2b898b6994fed3f876ca52401f77161c1098af3cb93d489dd8c72ff14bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":2746,"rank":2746,"depth":0,"x":2452.142,"y":311.433,"cluster":"homological-algebra"},{"id":"stacks:018G","tag":"018G","title":"Augmentations · Definition 018G","summary":"Let C be a category. Let U be a simplicial object of C. An augmentation ε : U → X of U towards an object X of C is a morphism from U into the constant simplicial object X.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U$ be a simplicial object of $\\mathcal{C}$.\nAn {\\it augmentation $\\epsilon : U \\to X$ of\n$U$ towards an object $X$ of $\\mathcal{C}$}\nis a morphism from $U$ into the constant simplicial\nobject $X$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Augmentations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018G","source_file":"simplicial.tex","source_line":2776,"source_end_line":2784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2776-L2784","statement_sha256":"acc54c75a98e3872cfc7c9b1dfa5af1a25b9c3cf6837b8d7c0df98334d1bda7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2747,"rank":2747,"depth":0,"x":2426.904,"y":136.599,"cluster":"homological-algebra"},{"id":"stacks:018H","tag":"018H","title":"Augmentations · Lemma 018H","summary":"Let C be a category. Let X ∈ Ob(C). Let U be a simplicial object of C. To give an augmentation of U towards X is the same as giving a morphism ε_0 : U_0 → X such that ε_0 ∘ d^1_0 = ε_0 ∘ d^1_1.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $X \\in \\Ob(\\mathcal{C})$.\nLet $U$ be a simplicial object of $\\mathcal{C}$.\nTo give an augmentation of $U$ towards $X$ is\nthe same as giving a morphism $\\epsilon_0 : U_0 \\to X$\nsuch that $\\epsilon_0 \\circ d^1_0 = \\epsilon_0 \\circ d^1_1$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Augmentations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018H","source_file":"simplicial.tex","source_line":2786,"source_end_line":2794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2786-L2794","statement_sha256":"29448b0ecb106b9c2269614db4cdee36b790283c20aed31c8d6c3f1f733dac41","origin":"The Stacks Project","memory_eligible":false,"source_rank":2748,"rank":2748,"depth":0,"x":2586.439,"y":251.435,"cluster":"homological-algebra"},{"id":"stacks:018I","tag":"018I","title":"Augmentations · Lemma 018I","summary":"Let C be a category with fibred products. Let f : Y→ X be a morphism of C. Let U be the simplicial object of C whose nth term is the (n + 1)fold fibred product Y ×_X Y ×_X … ×_X Y. See Example [Tag 016E]. For any simplicial object V of C we have Mor_Simp(C)(V, U) & = Mor_Simp_1(C)(sk_1 V, sk_1 U) & = (g_0 : V_0 → Y mid f ∘ g_0 ∘ d^1_0 = f ∘ g_0 ∘ d^1_1) In particular we have U = cosk_1 sk_1 U.","statement_latex":"Let $\\mathcal{C}$ be a category with fibred products.\nLet $f : Y\\to X$ be a morphism of $\\mathcal{C}$. Let $U$ be the\nsimplicial object of $\\mathcal{C}$ whose $n$th term\nis the $(n + 1)$fold fibred product\n$Y \\times_X Y \\times_X \\ldots \\times_X Y$.\nSee Example \\ref{example-fibre-products-simplicial-object}.\nFor any simplicial object $V$ of $\\mathcal{C}$ we have\n\\begin{align*}\n\\Mor_{\\text{Simp}(\\mathcal{C})}(V, U)\n& =\n\\Mor_{\\text{Simp}_1(\\mathcal{C})}(\\text{sk}_1 V, \\text{sk}_1 U) \\\\\n& =\n\\{g_0 : V_0 \\to Y \\mid f \\circ g_0 \\circ d^1_0 = f \\circ g_0 \\circ d^1_1\\}\n\\end{align*}\nIn particular we have $U = \\text{cosk}_1 \\text{sk}_1 U$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Augmentations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018I","source_file":"simplicial.tex","source_line":2813,"source_end_line":2830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2813-L2830","statement_sha256":"982848ed833ca09bcaa0d5b8d6fd279e135774e57a51819ac7057632c9d0e728","origin":"The Stacks Project","memory_eligible":false,"source_rank":2749,"rank":2749,"depth":0,"x":2376.022,"y":257.292,"cluster":"homological-algebra"},{"id":"stacks:018L","tag":"018L","title":"Left adjoints to the skeleton functors · Lemma 018L","summary":"Let C be a category which has finite colimits. The functors i_m! exist for all m. Let U be an m-truncated simplicial object of C. The simplicial object i_m!U is described by the formula (i_m!U)_n = colim_([n]/Δ)_≤ m^opp U(n) and for φ : [n] → [n'] the map i_m!U(φ) comes from the identification U(n) ∘ underlineφ = U(n') above via Categories, Lemma [Tag 002K].","statement_latex":"Let $\\mathcal{C}$ be a category which has finite colimits.\nThe functors $i_{m!}$ exist for all $m$.\nLet $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$.\nThe simplicial object $i_{m!}U$\nis described by the formula\n$$\n(i_{m!}U)_n = \\colim_{([n]/\\Delta)_{\\leq m}^{opp}} U(n)\n$$\nand for $\\varphi : [n] \\to [n']$ the map\n$i_{m!}U(\\varphi)$ comes from the\nidentification $U(n) \\circ \\underline{\\varphi} = U(n')$ above\nvia Categories, Lemma \\ref{categories-lemma-functorial-colimit}.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018L","source_file":"simplicial.tex","source_line":2934,"source_end_line":2948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2934-L2948","statement_sha256":"5e69307e0744d5fbd8269a14c00962b736670bbac788264cd18f7c2c5846d260","origin":"The Stacks Project","memory_eligible":false,"source_rank":2750,"rank":2750,"depth":1,"x":2526.777,"y":133.325,"cluster":"homological-algebra"},{"id":"stacks:018M","tag":"018M","title":"Left adjoints to the skeleton functors · Lemma 018M","summary":"Let C be a category. Let U be an m-truncated simplicial object of C. For any n ≤ m the colimit colim_([n]/Δ)_≤ m^opp U(n) exists and is equal to U_n.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$.\nFor any $n \\leq m$ the colimit\n$$\n\\colim_{([n]/\\Delta)_{\\leq m}^{opp}} U(n)\n$$\nexists and is equal to $U_n$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018M","source_file":"simplicial.tex","source_line":2988,"source_end_line":2997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L2988-L2997","statement_sha256":"7b39abb954325fbadd11e8166d480405d3bfa431387bcb95cf6bab643b2709d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2751,"rank":2751,"depth":0,"x":2515.28,"y":310.639,"cluster":"homological-algebra"},{"id":"stacks:018N","tag":"018N","title":"Left adjoints to the skeleton functors · Lemma 018N","summary":"Let C be a category which has finite colimits. Let U be an m-truncated simplicial object of C. The map U → sk_m i_m!U is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category which has finite colimits.\nLet $U$ be an $m$-truncated simplicial object of $\\mathcal{C}$.\nThe map $U \\to \\text{sk}_m i_{m!}U$ is an isomorphism.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018N","source_file":"simplicial.tex","source_line":3004,"source_end_line":3009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3004-L3009","statement_sha256":"4ee726670810ef1634f830039511021c8eb8014c4b66b67988390039e3534b7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2752,"rank":2752,"depth":2,"x":2380.896,"y":173.088,"cluster":"homological-algebra"},{"id":"stacks:018O","tag":"018O","title":"Left adjoints to the skeleton functors · Lemma 018O","summary":"If U is an m-truncated simplicial set and n > m then all n-simplices of i_m!U are degenerate.","statement_latex":"If $U$ is an $m$-truncated simplicial set and $n > m$\nthen all $n$-simplices of $i_{m!}U$ are degenerate.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018O","source_file":"simplicial.tex","source_line":3015,"source_end_line":3019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3015-L3019","statement_sha256":"53804b00a0f7079c111aa9148765cea859b948db97843df1c2fb40c1cebeb805","origin":"The Stacks Project","memory_eligible":false,"source_rank":2753,"rank":2753,"depth":2,"x":2591.026,"y":198.314,"cluster":"homological-algebra"},{"id":"stacks:018P","tag":"018P","title":"Left adjoints to the skeleton functors · Lemma 018P","summary":"Let U be a simplicial set. Let n ≥ 0 be an integer. The morphism i_n! sk_n U → U identifies i_n! sk_n U with the simplicial set U' ⊂ U defined in Lemma [Tag 017T].","statement_latex":"Let $U$ be a simplicial set.\nLet $n \\geq 0$ be an integer.\nThe morphism $i_{n!} \\text{sk}_n U \\to U$ identifies\n$i_{n!} \\text{sk}_n U$ with the simplicial set\n$U' \\subset U$ defined in Lemma \\ref{lemma-simplicial-set-n-skel-sub}.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018P","source_file":"simplicial.tex","source_line":3034,"source_end_line":3041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3034-L3041","statement_sha256":"447f3a40f10a1bbedb313ebd85ff1915c7649683e8182fa2fd94cfa4ed7be9a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2754,"rank":2754,"depth":3,"x":2415.441,"y":299.148,"cluster":"homological-algebra"},{"id":"stacks:018R","tag":"018R","title":"Left adjoints to the skeleton functors · Lemma 018R","summary":"Let U ⊂ V be simplicial sets. Suppose n ≥ 0 and x ∈ V_n, x not ∈ U_n are such that • V_i = U_i for i < n, • V_n = U_n ∪ (x), • any z ∈ V_j, z not ∈ U_j for j > n is degenerate. Let Δ[n] → V be the unique morphism mapping the nondegenerate n-simplex of Δ[n] to x. In this case the diagram xymatrix Δ[n] ar[r] & V i_(n - 1)! sk_n - 1 Δ[n] ar[r] ar[u] & U ar[u] is a pushout diagram.","statement_latex":"Let $U \\subset V$ be simplicial sets.\nSuppose $n \\geq 0$ and $x \\in V_n$, $x \\not \\in U_n$ are such that\n\\begin{enumerate}\n\\item $V_i = U_i$ for $i < n$,\n\\item $V_n = U_n \\cup \\{x\\}$,\n\\item any $z \\in V_j$, $z \\not \\in U_j$ for $j > n$\nis degenerate.\n\\end{enumerate}\nLet $\\Delta[n] \\to V$ be the unique morphism mapping the\nnondegenerate $n$-simplex of $\\Delta[n]$ to $x$.\nIn this case the diagram\n$$\n\\xymatrix{\n\\Delta[n] \\ar[r] & V \\\\\ni_{(n - 1)!} \\text{sk}_{n - 1} \\Delta[n] \\ar[r] \\ar[u] & U \\ar[u]\n}\n$$\nis a pushout diagram.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018R","source_file":"simplicial.tex","source_line":3083,"source_end_line":3103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3083-L3103","statement_sha256":"aca4783eec3f1f3b038e85a0343604cca08b02e3f862a92105ee29420cad84bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2755,"rank":2755,"depth":2,"x":2463.924,"y":124.816,"cluster":"homological-algebra"},{"id":"stacks:018S","tag":"018S","title":"Left adjoints to the skeleton functors · Lemma 018S","summary":"Let U ⊂ V be simplicial sets, with U_n, V_n finite nonempty for all n. Assume that U and V have finitely many nondegenerate simplices. Then there exists a sequence of sub simplicial sets U = W^0 ⊂ W^1 ⊂ W^2 ⊂ … W^r = V such that Lemma [Tag 018R] applies to each of the inclusions W^i ⊂ W^i + 1.","statement_latex":"Let $U \\subset V$ be simplicial sets, with $U_n, V_n$\nfinite nonempty for all $n$.\nAssume that $U$ and $V$ have finitely many nondegenerate simplices.\nThen there exists a sequence of sub simplicial sets\n$$\nU = W^0 \\subset W^1 \\subset W^2 \\subset \\ldots W^r = V\n$$\nsuch that Lemma \\ref{lemma-glue-simplex} applies to each of\nthe inclusions $W^i \\subset W^{i + 1}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018S","source_file":"simplicial.tex","source_line":3128,"source_end_line":3139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3128-L3139","statement_sha256":"dc6d708c8c9f5020e3800a8e3a3f6bad90a4efbecddfa19b399c81ffd2a2b1b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2756,"rank":2756,"depth":3,"x":2568.574,"y":281.187,"cluster":"homological-algebra"},{"id":"stacks:018T","tag":"018T","title":"Left adjoints to the skeleton functors · Lemma 018T","summary":"Let A be an abelian category Let U be an m-truncated simplicial object of A. For n > m we have N(i_m!U)_n = 0.","statement_latex":"Let $\\mathcal{A}$ be an abelian category\nLet $U$ be an $m$-truncated simplicial object of\n$\\mathcal{A}$. For $n > m$ we have $N(i_{m!}U)_n = 0$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018T","source_file":"simplicial.tex","source_line":3157,"source_end_line":3162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3157-L3162","statement_sha256":"908ff9d4bf31a5085996ee21b29c12ddbd08ce37bf1f54441f8c06bf22473284","origin":"The Stacks Project","memory_eligible":false,"source_rank":2757,"rank":2757,"depth":4,"x":2365.255,"y":225.151,"cluster":"homological-algebra"},{"id":"stacks:018U","tag":"018U","title":"Left adjoints to the skeleton functors · Lemma 018U","summary":"Let A be an abelian category. Let U be a simplicial object of A. Let n ≥ 0 be an integer. The morphism i_n! sk_n U → U identifies i_n! sk_n U with the simplicial subobject U' ⊂ U defined in Lemma [Tag 017Y].","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $U$ be a simplicial object of $\\mathcal{A}$.\nLet $n \\geq 0$ be an integer.\nThe morphism $i_{n!} \\text{sk}_n U \\to U$ identifies\n$i_{n!} \\text{sk}_n U$ with the simplicial subobject\n$U' \\subset U$ defined in Lemma \\ref{lemma-simplicial-abelian-n-skel-sub}.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018U","source_file":"simplicial.tex","source_line":3174,"source_end_line":3182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3174-L3182","statement_sha256":"d5fe03732763ca3f26156cb53639a79d1f5b1e179e1dcfddcf144e73573f43c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2758,"rank":2758,"depth":5,"x":2560.628,"y":150.957,"cluster":"homological-algebra"},{"id":"stacks:018V","tag":"018V","title":"Left adjoints to the skeleton functors · Lemma 018V","summary":"Let C be a category with finite coproducts and finite limits. Let V be a simplicial object of C. In this case (cosk_n sk_n V)_n + 1 = Hom(i_n !sk_n Δ[n + 1], V)_0.","statement_latex":"Let $\\mathcal{C}$ be a category with finite coproducts\nand finite limits. Let $V$ be a simplicial object of $\\mathcal{C}$.\nIn this case\n$$\n(\\text{cosk}_n \\text{sk}_n V)_{n + 1}\n=\n\\Hom(i_{n !}\\text{sk}_n \\Delta[n + 1], V)_0.\n$$","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Left adjoints to the skeleton functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018V","source_file":"simplicial.tex","source_line":3201,"source_end_line":3211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3201-L3211","statement_sha256":"a739c137a00efe6ac22110cf6d5a0bbb7845c38746dc3367bc3e7cb07fa14636","origin":"The Stacks Project","memory_eligible":false,"source_rank":2759,"rank":2759,"depth":1,"x":2476.061,"y":316.852,"cluster":"homological-algebra"},{"id":"stacks:018Z","tag":"018Z","title":"Simplicial objects in abelian categories · Lemma 018Z","summary":"Let A be an abelian category. • The categories Simp(A) and CoSimp(A) are abelian. • A morphism of (co)simplicial objects f : A → B is injective if and only if each f_n : A_n → B_n is injective. • A morphism of (co)simplicial objects f : A → B is surjective if and only if each f_n : A_n → B_n is surjective. • A sequence of (co)simplicial objects A xrightarrowf B xrightarrowg C is exact at B if and only if each sequence A_i xrightarrowf_i B_i xrightarrowg_i C_i is exact at B_i.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item The categories $\\text{Simp}(\\mathcal{A})$ and\n$\\text{CoSimp}(\\mathcal{A})$ are abelian.\n\\item A morphism of (co)simplicial objects\n$f : A \\to B$ is injective\nif and only if each $f_n : A_n \\to B_n$ is injective.\n\\item A morphism of (co)simplicial objects\n$f : A \\to B$ is surjective\nif and only if each $f_n : A_n \\to B_n$ is surjective.\n\\item A sequence of (co)simplicial objects\n$$\nA \\xrightarrow{f} B \\xrightarrow{g} C\n$$\nis exact at $B$ if and only if each sequence\n$$\nA_i \\xrightarrow{f_i} B_i \\xrightarrow{g_i} C_i\n$$\nis exact at $B_i$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects in abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/018Z","source_file":"simplicial.tex","source_line":3270,"source_end_line":3292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3270-L3292","statement_sha256":"43771d0601582d899f182192b7a3f865495538a55e6f8c15c81df442802475d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2760,"rank":2760,"depth":1,"x":2404.872,"y":146.204,"cluster":"homological-algebra"},{"id":"stacks:0190","tag":"0190","title":"Simplicial objects in abelian categories · Lemma 0190","summary":"With A, k and U as above, so U_i = 0, i < k and U_k = A. • Given a k-truncated simplicial object V we have Mor(U, V) = ( f : A → V_k mid d^k_i ∘ f = 0, i = 0, …, k ) and Mor(V, U) = ( f : V_k → A mid f ∘ s^k - 1_i = 0, i = 0, …, k - 1 ). • The object i_k! U has nth term equal to bigoplus_α A where α runs over all surjective morphisms α : [n] → [k]. • For any φ : [m] → [n] the map i_k! U(φ) is described as the mapping bigoplus_α A → bigoplus_α' A which maps to component…","statement_latex":"With $A$, $k$ and $U$ as above, so $U_i = 0$, $i < k$ and $U_k = A$.\n\\begin{enumerate}\n\\item Given a $k$-truncated simplicial object $V$\nwe have\n$$\n\\Mor(U, V)\n=\n\\{ f : A \\to V_k \\mid d^k_i \\circ f = 0, \\ i = 0, \\ldots, k \\}\n$$\nand\n$$\n\\Mor(V, U)\n=\n\\{ f : V_k \\to A \\mid f \\circ s^{k - 1}_i = 0, \\ i = 0, \\ldots, k - 1 \\}.\n$$\n\\item The object $i_{k!} U$ has $n$th term equal to\n$\\bigoplus_\\alpha A$ where $\\alpha$ runs over all\nsurjective morphisms $\\alpha : [n] \\to [k]$.\n\\item For any $\\varphi : [m] \\to [n]$ the map\n$i_{k!} U(\\varphi)$ is described as the mapping\n$\\bigoplus_\\alpha A \\to \\bigoplus_{\\alpha'} A$\nwhich maps to component corresponding to $\\alpha : [n] \\to [k]$\nto zero if $\\alpha \\circ \\varphi$ is not surjective and\nby the identity to the component corresponding to\n$\\alpha \\circ \\varphi$ if it is surjective.\n\\item The object $\\text{cosk}_k U$ has $n$th term equal to\n$\\bigoplus_\\beta A$, where $\\beta$ runs over all\ninjective morphisms $\\beta : [k] \\to [n]$.\n\\item For any $\\varphi : [m] \\to [n]$ the map\n$\\text{cosk}_k U(\\varphi)$ is described as the mapping\n$\\bigoplus_\\beta A \\to \\bigoplus_{\\beta'} A$\nwhich maps to component corresponding to $\\beta : [k] \\to [n]$\nto zero if $\\beta$ does not factor through $\\varphi$ and\nby the identity to each of the components corresponding to\n$\\beta'$ such that $\\beta = \\varphi \\circ \\beta'$\nif it does.\n\\item The canonical map\n$\nc : i_{k !} U \\to \\text{cosk}_k U\n$\nin degree $n$ has $(\\alpha, \\beta)$ coefficient $A \\to A$\nequal to zero if $\\alpha \\circ \\beta$ is not the identity\nand equal to $\\text{id}_A$ if it is.\n\\item The canonical map\n$\nc : i_{k !} U \\to \\text{cosk}_k U\n$\nis injective.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects in abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0190","source_file":"simplicial.tex","source_line":3319,"source_end_line":3370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3319-L3370","statement_sha256":"804bc8adbc457a3e9ccb35ea5567c9bf6eefe969ff2a54c5f60246d9353c849e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2761,"rank":2761,"depth":5,"x":2594.966,"y":231.81,"cluster":"homological-algebra"},{"id":"stacks:0191","tag":"0191","title":"Simplicial objects in abelian categories · Definition 0191","summary":"Let A be an abelian category. Let A be an object of A and let k be an integer ≥ 0. The Eilenberg-Maclane object K(A, k) is given by the object K(A, k) = i_k!U which is described in Lemma [Tag 0190] above.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A$ be an object of $\\mathcal{A}$ and\nlet $k$ be an integer $\\geq 0$.\nThe {\\it Eilenberg-Maclane object $K(A, k)$}\nis given by the object $K(A, k) = i_{k!}U$\nwhich is described in\nLemma \\ref{lemma-eilenberg-maclane-object} above.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects in abelian categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0191","source_file":"simplicial.tex","source_line":3543,"source_end_line":3552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3543-L3552","statement_sha256":"82eaca994c239bab924cc56a39bc09a98ceacb808e623e612bcc6950f3b34d9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2762,"rank":2762,"depth":6,"x":2385.546,"y":276.636,"cluster":"homological-algebra"},{"id":"stacks:0192","tag":"0192","title":"Simplicial objects in abelian categories · Lemma 0192","summary":"Let A be an abelian category. Let A be an object of A and let k be an integer ≥ 0. Consider the simplicial object E defined by the following rules • E_n = bigoplus_α A, where the sum is over α : [n] → [k + 1] whose image is either [k] or [k + 1]. • Given φ : [m] → [n] the map E_n → E_m maps the summand corresponding to α via id_A to the summand corresponding to α ∘ φ, provided Im(α ∘ φ) is equal to [k] or [k + 1]. Then there exists a short exact sequence 0 → K(A, k) → E →…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A$ be an object of $\\mathcal{A}$ and\nlet $k$ be an integer $\\geq 0$. Consider the\nsimplicial object $E$ defined by the following rules\n\\begin{enumerate}\n\\item $E_n = \\bigoplus_\\alpha A$, where the\nsum is over $\\alpha : [n] \\to [k + 1]$ whose\nimage is either $[k]$ or $[k + 1]$.\n\\item Given $\\varphi : [m] \\to [n]$ the map\n$E_n \\to E_m$ maps the summand corresponding\nto $\\alpha$ via $\\text{id}_A$ to the summand\ncorresponding to $\\alpha \\circ \\varphi$,\nprovided $\\Im(\\alpha \\circ \\varphi)$\nis equal to $[k]$ or $[k + 1]$.\n\\end{enumerate}\nThen there exists a short exact sequence\n$$\n0 \\to K(A, k) \\to E \\to K(A, k + 1) \\to 0\n$$\nwhich is term by term split exact.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects in abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0192","source_file":"simplicial.tex","source_line":3555,"source_end_line":3577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3555-L3577","statement_sha256":"6e843cea878dd914c0b08d3f5e9b3b81250fc75ad5baec20222aac4b900e19a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2763,"rank":2763,"depth":0,"x":2504.151,"y":124.457,"cluster":"homological-algebra"},{"id":"stacks:0193","tag":"0193","title":"Simplicial objects in abelian categories · Lemma 0193","summary":"Let A be an abelian category. For any simplicial object V of A we have V = colim_n i_n!sk_n V where all the transition maps are injections.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nFor any simplicial object $V$ of $\\mathcal{A}$ we have\n$$\nV = \\colim_n i_{n!}\\text{sk}_n V\n$$\nwhere all the transition maps are injections.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects in abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0193","source_file":"simplicial.tex","source_line":3587,"source_end_line":3595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3587-L3595","statement_sha256":"96bd997e7e608c668e384e3ffba53c8bae087eda95f7a72df69fcd83665ca126","origin":"The Stacks Project","memory_eligible":false,"source_rank":2764,"rank":2764,"depth":6,"x":2539.138,"y":304.317,"cluster":"homological-algebra"},{"id":"stacks:0195","tag":"0195","title":"Simplicial objects and chain complexes · Lemma 0195","summary":"The functor s is exact.","statement_latex":"The functor $s$ is exact.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects and chain complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0195","source_file":"simplicial.tex","source_line":3647,"source_end_line":3650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3647-L3650","statement_sha256":"3575e23c96592ac9fab4b0b46c29b2d1b249fdf67f55be007e694a27aab2a728","origin":"The Stacks Project","memory_eligible":false,"source_rank":2765,"rank":2765,"depth":2,"x":2368.373,"y":191.327,"cluster":"homological-algebra"},{"id":"stacks:0196","tag":"0196","title":"Simplicial objects and chain complexes · Lemma 0196","summary":"Let A be an abelian category. Let A be an object of A and let k be an integer. Let E be the object described in Lemma [Tag 0192]. Then the complex s(E) is acyclic.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A$ be an object of $\\mathcal{A}$ and\nlet $k$ be an integer. Let $E$ be the object\ndescribed in Lemma \\ref{lemma-extension}.\nThen the complex $s(E)$ is acyclic.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects and chain complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0196","source_file":"simplicial.tex","source_line":3656,"source_end_line":3663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3656-L3663","statement_sha256":"b07192397add8a4552b22b1ace6f771ee1feae367f5aba9ce354ce5cf41b7147","origin":"The Stacks Project","memory_eligible":false,"source_rank":2766,"rank":2766,"depth":1,"x":2585.571,"y":177.723,"cluster":"homological-algebra"},{"id":"stacks:0197","tag":"0197","title":"Simplicial objects and chain complexes · Lemma 0197","summary":"Let A be an abelian category. Let A be an object of A and let k be an integer. We have H_i(s(K(A, k))) = A if i = k and 0 else.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A$ be an object of $\\mathcal{A}$ and\nlet $k$ be an integer. We have\n$H_i(s(K(A, k))) = A$ if $i = k$ and\n$0$ else.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects and chain complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0197","source_file":"simplicial.tex","source_line":3716,"source_end_line":3723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3716-L3723","statement_sha256":"5d183f1392e887b42ab486c6acf72cdf4cbe4fbc149d1b354afd8723e9a8b47e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2767,"rank":2767,"depth":7,"x":2436.068,"y":311.252,"cluster":"homological-algebra"},{"id":"stacks:0198","tag":"0198","title":"Simplicial objects and chain complexes · Lemma 0198","summary":"Let A be an abelian category. Let U be a simplicial object of A. The canonical map N(U_n) → U_n gives rise to a morphism of complexes N(U) → s(U).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $U$ be a simplicial object of $\\mathcal{A}$.\nThe canonical map $N(U_n) \\to U_n$ gives rise to\na morphism of complexes $N(U) \\to s(U)$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects and chain complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0198","source_file":"simplicial.tex","source_line":3779,"source_end_line":3785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3779-L3785","statement_sha256":"5d77c65eff7ce2c301d0ff0c9ba207e122de46691c28e02546dd1b01b564070b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2768,"rank":2768,"depth":0,"x":2438.935,"y":127.609,"cluster":"homological-algebra"},{"id":"stacks:0199","tag":"0199","title":"Simplicial objects and chain complexes · Lemma 0199","summary":"Let A be an abelian category. Let A be an object of A and let k be an integer. We have N(K(A, k))_i = A if i = k and 0 else.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $A$ be an object of $\\mathcal{A}$ and\nlet $k$ be an integer. We have\n$N(K(A, k))_i = A$ if $i = k$ and\n$0$ else.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects and chain complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0199","source_file":"simplicial.tex","source_line":3795,"source_end_line":3802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3795-L3802","statement_sha256":"e98a933be2a35f333f630f96efdba7e26acde16a3bd25ac5a7a22afa284dd747","origin":"The Stacks Project","memory_eligible":false,"source_rank":2769,"rank":2769,"depth":7,"x":2584.776,"y":264.911,"cluster":"homological-algebra"},{"id":"stacks:019A","tag":"019A","title":"Simplicial objects and chain complexes · Lemma 019A","summary":"Let A be an abelian category. Let U be a simplicial object of A. The canonical morphism of chain complexes N(U) → s(U) is split. In fact, s(U) = N(U) ⊕ D(U) for some complex D(U). The construction U ↦ D(U) is functorial.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $U$ be a simplicial object of $\\mathcal{A}$.\nThe canonical morphism of chain complexes\n$N(U) \\to s(U)$ is split. In fact,\n$$\ns(U) = N(U) \\oplus D(U)\n$$\nfor some complex $D(U)$. The construction $U \\mapsto D(U)$\nis functorial.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects and chain complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019A","source_file":"simplicial.tex","source_line":3815,"source_end_line":3826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3815-L3826","statement_sha256":"233d0f7bef654dfa4141aa116ce07b628d08a221944c38b39c78999b236e7af7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2770,"rank":2770,"depth":3,"x":2366.41,"y":246.386,"cluster":"homological-algebra"},{"id":"stacks:019B","tag":"019B","title":"Simplicial objects and chain complexes · Lemma 019B","summary":"The functor N is exact.","statement_latex":"The functor $N$ is exact.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects and chain complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019B","source_file":"simplicial.tex","source_line":3917,"source_end_line":3920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3917-L3920","statement_sha256":"9b965f75176fad15811767a7286407370eda7cb8acc343a752ab8f030bc23835","origin":"The Stacks Project","memory_eligible":false,"source_rank":2771,"rank":2771,"depth":4,"x":2542.658,"y":135.933,"cluster":"homological-algebra"},{"id":"stacks:019C","tag":"019C","title":"Simplicial objects and chain complexes · Lemma 019C","summary":"Let A be an abelian category. Let V be a simplicial object of A. The canonical morphism of chain complexes N(V) → s(V) is a quasi-isomorphism. In other words, the complex D(V) of Lemma [Tag 019A] is acyclic.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $V$ be a simplicial object of $\\mathcal{A}$.\nThe canonical morphism of chain complexes\n$N(V) \\to s(V)$ is a quasi-isomorphism.\nIn other words, the complex $D(V)$ of Lemma\n\\ref{lemma-decompose-associated-complexes} is acyclic.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Simplicial objects and chain complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019C","source_file":"simplicial.tex","source_line":3927,"source_end_line":3935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3927-L3935","statement_sha256":"e8ea8fc364913f67cb402353b1f6bef7af914ddf3112cfa8c053fd4fe9dc073e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2772,"rank":2772,"depth":8,"x":2501.442,"y":317.726,"cluster":"homological-algebra"},{"id":"stacks:019E","tag":"019E","title":"Dold-Kan · Lemma 019E","summary":"Let A be an abelian category. The functor N is faithful, and reflects isomorphisms, injections and surjections.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nThe functor $N$ is faithful, and reflects\nisomorphisms, injections and surjections.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Dold-Kan","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019E","source_file":"simplicial.tex","source_line":3999,"source_end_line":4004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L3999-L4004","statement_sha256":"660981e7b42fb1f4a2be03358c03fa7006d2d3159afa84c033033aab4a906312","origin":"The Stacks Project","memory_eligible":false,"source_rank":2773,"rank":2773,"depth":4,"x":2385.425,"y":159.994,"cluster":"homological-algebra"},{"id":"stacks:019F","tag":"019F","title":"Dold-Kan · Lemma 019F","summary":"Let A and B be abelian categories. Let N : A → B, and S : B → A be functors. Suppose that • the functors S and N are exact, • there is an isomorphism g : N ∘ S → id_B to the identity functor of B, • N is faithful, and • S is essentially surjective. Then S and N are quasi-inverse equivalences of categories.","statement_latex":"Let $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $N : \\mathcal{A} \\to \\mathcal{B}$, and\n$S : \\mathcal{B} \\to \\mathcal{A}$ be functors.\nSuppose that\n\\begin{enumerate}\n\\item the functors $S$ and $N$ are exact,\n\\item there is an isomorphism $g : N \\circ S \\to \\text{id}_\\mathcal{B}$\nto the identity functor of $\\mathcal{B}$,\n\\item $N$ is faithful, and\n\\item $S$ is essentially surjective.\n\\end{enumerate}\nThen $S$ and $N$ are quasi-inverse equivalences of categories.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Dold-Kan","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019F","source_file":"simplicial.tex","source_line":4014,"source_end_line":4028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4014-L4028","statement_sha256":"2229c4119130176f78ab5e1196391f0c049df9a3a09b8cd3cfeb35c06612fa45","origin":"The Stacks Project","memory_eligible":false,"source_rank":2774,"rank":2774,"depth":0,"x":2598.212,"y":210.566,"cluster":"homological-algebra"},{"id":"stacks:019G","tag":"019G","title":"Dold-Kan · Theorem 019G","summary":"Let A be an abelian category. The functor N induces an equivalence of categories N : Simp(A) → Ch_≥ 0(A)","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nThe functor $N$ induces an equivalence of\ncategories\n$$\nN :\n\\text{Simp}(\\mathcal{A})\n\\longrightarrow\n\\text{Ch}_{\\geq 0}(\\mathcal{A})\n$$","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Dold-Kan","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019G","source_file":"simplicial.tex","source_line":4044,"source_end_line":4055,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4044-L4055","statement_sha256":"7aedaf5887c143c51c945fb6d2eaa212d35950c9f9cdc09a9ea701c30fda2882","origin":"The Stacks Project","memory_eligible":false,"source_rank":2775,"rank":2775,"depth":4,"x":2400.272,"y":294.169,"cluster":"homological-algebra"},{"id":"stacks:019I","tag":"019I","title":"Dold-Kan for cosimplicial objects · Lemma 019I","summary":"Let A be an abelian category. • The functor s : CoSimp(A) → CoCh_≥ 0(A) is exact. • The maps s(U)^n → Q(U)^n define a morphism of cochain complexes. • There exists a functorial direct sum decomposition s(U) = D(U) ⊕ Q(U) in CoCh_≥ 0(A). • The functor Q is exact. • The morphism of complexes s(U) → Q(U) is a quasi-isomorphism. • The functor U ↦ Q(U)^bullet defines an equivalence of categories CoSimp(A) → CoCh_≥ 0(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item The functor\n$s : \\text{CoSimp}(\\mathcal{A}) \\to \\text{CoCh}_{\\geq 0}(\\mathcal{A})$\nis exact.\n\\item The maps $s(U)^n \\to Q(U)^n$ define a morphism\nof cochain complexes.\n\\item There exists a functorial direct sum decomposition\n$s(U) = D(U) \\oplus Q(U)$ in $\\text{CoCh}_{\\geq 0}(\\mathcal{A})$.\n\\item The functor $Q$ is exact.\n\\item The morphism of complexes $s(U) \\to Q(U)$ is a quasi-isomorphism.\n\\item The functor $U \\mapsto Q(U)^\\bullet$ defines\nan equivalence of categories\n$\\text{CoSimp}(\\mathcal{A}) \\to \\text{CoCh}_{\\geq 0}(\\mathcal{A})$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Dold-Kan for cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019I","source_file":"simplicial.tex","source_line":4369,"source_end_line":4386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4369-L4386","statement_sha256":"447faeba31e6c50f78ebb30f2e40c5db20c6fde908fab72537b80774db6e5eed","origin":"The Stacks Project","memory_eligible":false,"source_rank":2776,"rank":2776,"depth":9,"x":2479.144,"y":119.885,"cluster":"homological-algebra"},{"id":"stacks:019K","tag":"019K","title":"Homotopies · Definition 019K","summary":"Let C be a category having finite coproducts. Suppose that U and V are two simplicial objects of C. Let a, b : U → V be two morphisms. • We say a morphism h : U × Δ[1] → V is a homotopy from a to b if a = h ∘ e_0 and b = h ∘ e_1. • We say the morphisms a and b are homotopic or are in the same homotopy class if there exists a sequence of morphisms a = a_0, a_1, …, a_n = b from U to V such that for each i = 1, …, n there either exists a homotopy from a_i - 1 to a_i or there…","statement_latex":"Let $\\mathcal{C}$ be a category having finite coproducts.\nSuppose that $U$ and $V$ are two simplicial objects of $\\mathcal{C}$.\nLet $a, b : U \\to V$ be two morphisms.\n\\begin{enumerate}\n\\item We say a morphism\n$$\nh : U \\times \\Delta[1] \\longrightarrow V\n$$\nis a {\\it homotopy from $a$ to $b$} if $a = h \\circ e_0$ and\n$b = h \\circ e_1$.\n\\item We say the morphisms $a$ and $b$ are {\\it homotopic} or are\n{\\it in the same homotopy class}\nif there exists a sequence of morphisms $a = a_0, a_1, \\ldots, a_n = b$\nfrom $U$ to $V$ such that for each $i = 1, \\ldots, n$ there either exists\na homotopy from $a_{i - 1}$ to $a_i$ or there exists a homotopy\nfrom $a_i$ to $a_{i - 1}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019K","source_file":"simplicial.tex","source_line":4422,"source_end_line":4441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4422-L4441","statement_sha256":"32b5992eb732c19beefb7a36277568d0c866e24955f868059262a9562a737a3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2777,"rank":2777,"depth":0,"x":2561.289,"y":293.472,"cluster":"homological-algebra"},{"id":"stacks:019L","tag":"019L","title":"Homotopies · Lemma 019L","summary":"In the situation above, we have the following relations: • We have h_n, 0 = b_n and h_n, n + 1 = a_n. • We have d^n_j ∘ h_n, i = h_n - 1, i - 1 ∘ d^n_j for i > j. • We have d^n_j ∘ h_n, i = h_n - 1, i ∘ d^n_j for i ≤ j. • We have s^n_j ∘ h_n, i = h_n + 1, i + 1 ∘ s^n_j for i > j. • We have s^n_j ∘ h_n, i = h_n + 1, i ∘ s^n_j for i ≤ j. Conversely, given a system of maps h_n, i satisfying the properties listed above, then these define a morphism h which is a homotopy from…","statement_latex":"In the situation above, we have the following relations:\n\\begin{enumerate}\n\\item We have $h_{n, 0} = b_n$ and $h_{n, n + 1} = a_n$.\n\\item We have $d^n_j \\circ h_{n, i} = h_{n - 1, i - 1} \\circ d^n_j$\nfor $i > j$.\n\\item We have $d^n_j \\circ h_{n, i} = h_{n - 1, i} \\circ d^n_j$\nfor $i \\leq j$.\n\\item We have $s^n_j \\circ h_{n, i} = h_{n + 1, i + 1} \\circ s^n_j$\nfor $i > j$.\n\\item We have $s^n_j \\circ h_{n, i} = h_{n + 1, i} \\circ s^n_j$\nfor $i \\leq j$.\n\\end{enumerate}\nConversely, given a system of maps $h_{n, i}$ satisfying the\nproperties listed above, then these define a morphism\n$h$ which is a homotopy from $a$ to $b$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019L","source_file":"simplicial.tex","source_line":4486,"source_end_line":4503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4486-L4503","statement_sha256":"768f547f28c7ac639ee157c5a27e12e8fd81d1a50757adcd582165303cef1261","origin":"The Stacks Project","memory_eligible":false,"source_rank":2778,"rank":2778,"depth":2,"x":2360.757,"y":211.934,"cluster":"homological-algebra"},{"id":"stacks:019N","tag":"019N","title":"Homotopies · Definition 019N","summary":"Let U and V be two simplicial objects of a category C. We say a morphism a : U → V is a homotopy equivalence if there exists a morphism b : V → U such that a ∘ b is homotopic to id_V and b ∘ a is homotopic to id_U. We say U and V are homotopy equivalent if there exists a homotopy equivalence a : U → V.","statement_latex":"Let $U$ and $V$ be two simplicial objects of a category $\\mathcal{C}$.\nWe say a morphism $a : U \\to V$ is a {\\it homotopy equivalence}\nif there exists a morphism $b : V \\to U$ such that $a \\circ b$ is\nhomotopic to $\\text{id}_V$ and $b \\circ a$ is homotopic to $\\text{id}_U$.\nWe say $U$ and $V$ are {\\it homotopy equivalent} if there\nexists a homotopy equivalence $a : U \\to V$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019N","source_file":"simplicial.tex","source_line":4570,"source_end_line":4578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4570-L4578","statement_sha256":"1fadfb24ca39a5008743d16140dee8c2390743d82edd5036abecea9b3216b157","origin":"The Stacks Project","memory_eligible":false,"source_rank":2779,"rank":2779,"depth":0,"x":2574.586,"y":158.175,"cluster":"homological-algebra"},{"id":"stacks:019O","tag":"019O","title":"Homotopies · Lemma 019O","summary":"Let C be a category with finite coproducts. Let U be a simplicial object of C. Consider the maps e_1, e_0 : U → U × Δ[1], and π : U × Δ[1] → U, see Lemma [Tag 017E]. • We have π ∘ e_1 = π ∘ e_0 = id_U, and • The morphisms id_U × Δ[1], and e_0 ∘ π are homotopic. • The morphisms id_U × Δ[1], and e_1 ∘ π are homotopic.","statement_latex":"Let $\\mathcal{C}$ be a category with finite coproducts.\nLet $U$ be a simplicial object of $\\mathcal{C}$.\nConsider the maps $e_1, e_0 : U \\to U \\times \\Delta[1]$,\nand $\\pi : U \\times \\Delta[1] \\to U$, see\nLemma \\ref{lemma-back-to-U}.\n\\begin{enumerate}\n\\item We have $\\pi \\circ e_1 = \\pi \\circ e_0 = \\text{id}_U$, and\n\\item The morphisms $\\text{id}_{U \\times \\Delta[1]}$,\nand $e_0 \\circ \\pi$ are homotopic.\n\\item The morphisms $\\text{id}_{U \\times \\Delta[1]}$,\nand $e_1 \\circ \\pi$ are homotopic.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019O","source_file":"simplicial.tex","source_line":4613,"source_end_line":4627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4613-L4627","statement_sha256":"28e53d98ed6a44d2ce7aded792cf7d2d077934779f79304aa74be227e1e53512","origin":"The Stacks Project","memory_eligible":false,"source_rank":2780,"rank":2780,"depth":1,"x":2459.936,"y":319.439,"cluster":"homological-algebra"},{"id":"stacks:019P","tag":"019P","title":"Homotopies · Lemma 019P","summary":"Let f : Y → X be a morphism of a category C with fibre products. Assume f has a section s. Consider the simplicial object U constructed in Example [Tag 016E] starting with f. The morphism U → U which in each degree is the self map (s ∘ f)^n + 1 of Y ×_X … ×_X Y given by s ∘ f on each factor is homotopic to the identity on U. In particular, U is homotopy equivalent to the constant simplicial object X.","statement_latex":"Let $f : Y \\to X$ be a morphism of a category $\\mathcal{C}$ with\nfibre products. Assume $f$ has a section $s$. Consider the\nsimplicial object $U$ constructed in\nExample \\ref{example-fibre-products-simplicial-object}\nstarting with $f$. The morphism $U \\to U$ which in each degree\nis the self map $(s \\circ f)^{n + 1}$ of $Y \\times_X \\ldots \\times_X Y$\ngiven by $s \\circ f$ on each factor is homotopic to the identity on $U$.\nIn particular, $U$ is homotopy equivalent to the constant\nsimplicial object $X$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019P","source_file":"simplicial.tex","source_line":4656,"source_end_line":4667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4656-L4667","statement_sha256":"f8b1c09683fc5ed6575ceb8d765fb72046c1eb22433bb3786f2ac1c9de520f80","origin":"The Stacks Project","memory_eligible":false,"source_rank":2781,"rank":2781,"depth":3,"x":2414.711,"y":135.137,"cluster":"homological-algebra"},{"id":"stacks:08Q4","tag":"08Q4","title":"Homotopies · Lemma 08Q4","summary":"Let C be a category. Let T be a set. For t ∈ T let X_t, Y_t be simplicial objects of C. Assume X = ∏_t ∈ T X_t and Y = ∏_t ∈ T Y_t exist. • If X_t and Y_t are homotopy equivalent for all t ∈ T and T is finite, then X and Y are homotopy equivalent. For t ∈ T let a_t, b_t : X_t → Y_t be morphisms. Set a = ∏ a_t : X → Y and b = ∏ b_t : X → Y. • [(2)] If there exists a homotopy from a_t to b_t for all t ∈ T, then there exists a homotopy from a to b. • [(3)] If T is finite and…","statement_latex":"Let $\\mathcal{C}$ be a category. Let $T$ be a set. For $t \\in T$\nlet $X_t$, $Y_t$ be simplicial objects of $\\mathcal{C}$. Assume\n$X = \\prod_{t \\in T} X_t$ and $Y = \\prod_{t \\in T} Y_t$ exist.\n\\begin{enumerate}\n\\item If $X_t$ and $Y_t$ are homotopy equivalent for all $t \\in T$\nand $T$ is finite, then $X$ and $Y$ are homotopy equivalent.\n\\end{enumerate}\nFor $t \\in T$ let $a_t, b_t : X_t \\to Y_t$ be morphisms.\nSet $a = \\prod a_t : X \\to Y$ and $b = \\prod b_t : X \\to Y$.\n\\begin{enumerate}\n\\item[(2)] If there exists a homotopy from $a_t$ to $b_t$ for\nall $t \\in T$, then there exists a homotopy from $a$ to $b$.\n\\item[(3)] If $T$ is finite and $a_t, b_t : X_t \\to Y_t$ for $t \\in T$\nare homotopic, then $a$ and $b$ are homotopic.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Q4","source_file":"simplicial.tex","source_line":4691,"source_end_line":4708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4691-L4708","statement_sha256":"a8b6e78dec2113b96af0f716905bb1b004c158067e2078facd67df498ad2898a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2782,"rank":2782,"depth":0,"x":2596.597,"y":245.576,"cluster":"homological-algebra"},{"id":"stacks:019S","tag":"019S","title":"Homotopies in abelian categories · Lemma 019S","summary":"Let A be an additive category. Let a, b : U → V be morphisms of simplicial objects of A. If a, b are homotopic, then s(a), s(b) : s(U) → s(V) are homotopic maps of chain complexes. If A is abelian, then also N(a), N(b) : N(U) → N(V) are homotopic maps of chain complexes.","statement_latex":"Let $\\mathcal{A}$ be an additive category. Let $a, b : U \\to V$ be morphisms\nof simplicial objects of $\\mathcal{A}$. If $a$, $b$ are homotopic,\nthen $s(a), s(b) : s(U) \\to s(V)$ are homotopic maps of chain complexes.\nIf $\\mathcal{A}$ is abelian, then also $N(a), N(b) : N(U) \\to N(V)$ are\nhomotopic maps of chain complexes.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies in abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019S","source_file":"simplicial.tex","source_line":4816,"source_end_line":4823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4816-L4823","statement_sha256":"2e2e8613b520ec295155aa59241fd1c2d784c1a6229a7f566d9d43198d677dfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2783,"rank":2783,"depth":4,"x":2373.265,"y":267.385,"cluster":"homological-algebra"},{"id":"stacks:019T","tag":"019T","title":"Homotopies in abelian categories · Lemma 019T","summary":"Let A be an additive category. Let a : U → V be a morphism of simplicial objects of A. If a is a homotopy equivalence, then s(a) : s(U) → s(V) is a homotopy equivalence of chain complexes. If in addition A is abelian, then also N(a) : N(U) → N(V) is a homotopy equivalence of chain complexes.","statement_latex":"Let $\\mathcal{A}$ be an additive category. Let $a : U \\to V$ be a morphism\nof simplicial objects of $\\mathcal{A}$. If $a$ is a homotopy equivalence,\nthen $s(a) : s(U) \\to s(V)$ is a homotopy equivalence of chain complexes.\nIf in addition $\\mathcal{A}$ is abelian, then also\n$N(a) : N(U) \\to N(V)$ is a homotopy equivalence of chain complexes.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies in abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019T","source_file":"simplicial.tex","source_line":4851,"source_end_line":4858,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4851-L4858","statement_sha256":"68fba485ebcfe03ce9eb9a3d7bf0a5fe9a79af6f5520a8a8eed45c98f5d0a850","origin":"The Stacks Project","memory_eligible":false,"source_rank":2784,"rank":2784,"depth":5,"x":2520.671,"y":124.324,"cluster":"homological-algebra"},{"id":"stacks:019W","tag":"019W","title":"Homotopies and cosimplicial objects · Definition 019W","summary":"Let C be a category having finite products. Let U and V be two cosimplicial objects of C. Let a, b : U → V be two morphisms of cosimplicial objects of C. • We say a morphism h : U → Hom(Δ[1], V) such that a = e_0 ∘ h and b = e_1 ∘ h is a homotopy from a to b. • We say a and b are homotopic or are in the same homotopy class if there exists a sequence a = a_0, a_1, …, a_n = b of morphisms from U to V such that for each i = 1, …, n there either exists a homotopy from a_i to…","statement_latex":"Let $\\mathcal{C}$ be a category having finite products.\nLet $U$ and $V$ be two cosimplicial objects of $\\mathcal{C}$.\nLet $a, b : U \\to V$ be two morphisms of cosimplicial objects\nof $\\mathcal{C}$.\n\\begin{enumerate}\n\\item We say a morphism\n$$\nh : U \\longrightarrow \\Hom(\\Delta[1], V)\n$$\nsuch that $a = e_0 \\circ h$ and $b = e_1 \\circ h$ is a\n{\\it homotopy from $a$ to $b$}.\n\\item We say $a$ and $b$ are {\\it homotopic} or are\n{\\it in the same homotopy class} if there exists a sequence\n$a = a_0, a_1, \\ldots, a_n = b$ of morphisms from $U$ to $V$\nsuch that for each $i = 1, \\ldots, n$ there either exists a\nhomotopy from $a_i$ to $a_{i - 1}$ or there exists a homotopy\nfrom $a_{i - 1}$ to $a_i$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies and cosimplicial objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019W","source_file":"simplicial.tex","source_line":4877,"source_end_line":4897,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4877-L4897","statement_sha256":"926f11daae42513c33b597a0107bde85cad2e95efb8b653eb211cffba83b2d43","origin":"The Stacks Project","memory_eligible":false,"source_rank":2785,"rank":2785,"depth":0,"x":2527.033,"y":313.794,"cluster":"homological-algebra"},{"id":"stacks:019X","tag":"019X","title":"Homotopies and cosimplicial objects · Lemma 019X","summary":"Let C be a category. Suppose that U and V are two cosimplicial objects of C. Let a, b : U → V be morphisms of cosimplicial objects. Recall that U, V correspond to simplicial objects U', V' of C^opp. Moreover a, b correspond to morphisms a', b' : V' → U'. The following are equivalent • There exists a homotopy h = (h_n, α) from a to b as in Remark [Tag 0FKJ]. • There exists a homotopy h = (h_n, i) from a' to b' as in Remark [Tag 019M]. Thus a is homotopic to b as in Remark…","statement_latex":"Let $\\mathcal{C}$ be a category. Suppose that $U$ and $V$ are two\ncosimplicial objects of $\\mathcal{C}$. Let $a, b : U \\to V$ be morphisms\nof cosimplicial objects. Recall that $U$, $V$ correspond\nto simplicial objects $U'$, $V'$ of $\\mathcal{C}^{opp}$.\nMoreover $a, b$ correspond to morphisms $a', b' : V' \\to U'$.\nThe following are equivalent\n\\begin{enumerate}\n\\item There exists a homotopy $h = \\{h_{n, \\alpha}\\}$ from\n$a$ to $b$ as in Remark \\ref{remark-homotopy-cosimplicial-better}.\n\\item There exists a homotopy $h = \\{h_{n, i}\\}$ from $a'$ to $b'$\nas in Remark \\ref{remark-homotopy-better}.\n\\end{enumerate}\nThus $a$ is homotopic to $b$ as in\nRemark \\ref{remark-homotopy-cosimplicial-better}\nif and only if $a'$ is homotopic to $b'$ as in\nRemark \\ref{remark-homotopy-better}.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies and cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019X","source_file":"simplicial.tex","source_line":4958,"source_end_line":4976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L4958-L4976","statement_sha256":"76bacb57d7fee7af2eb8ab088a66b9815458b58d55d26ca2930f153bd5166ba3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2786,"rank":2786,"depth":3,"x":2369.697,"y":177.45,"cluster":"homological-algebra"},{"id":"stacks:019Y","tag":"019Y","title":"Homotopies and cosimplicial objects · Lemma 019Y","summary":"Functors preserve homotopic morphisms of (co)simplicial objects. Let C, C', D, D' be categories. With terminology as in Remarks [Tag 0FKJ] and [Tag 019M]. • Let a, b : U → V be morphisms of simplicial objects of D. Let F : D → D' be a covariant functor. If a and b are homotopic, then F(a), F(b) are homotopic morphisms F(U) → F(V) of simplicial objects. • Let a, b : U → V be morphisms of cosimplicial objects of C. Let F : C → C' be a covariant functor. If a and b are…","statement_latex":"\\begin{slogan}\nFunctors preserve homotopic morphisms of (co)simplicial objects.\n\\end{slogan}\nLet $\\mathcal{C}, \\mathcal{C}', \\mathcal{D}, \\mathcal{D}'$ be categories.\nWith terminology as in Remarks \\ref{remark-homotopy-cosimplicial-better} and\n\\ref{remark-homotopy-better}.\n\\begin{enumerate}\n\\item Let $a, b : U \\to V$ be morphisms of simplicial objects\nof $\\mathcal{D}$. Let $F : \\mathcal{D} \\to \\mathcal{D}'$ be a covariant\nfunctor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$\nare homotopic morphisms $F(U) \\to F(V)$ of simplicial objects.\n\\item Let $a, b : U \\to V$ be morphisms of cosimplicial objects\nof $\\mathcal{C}$. Let $F : \\mathcal{C} \\to \\mathcal{C}'$ be a covariant\nfunctor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$\nare homotopic morphisms $F(U) \\to F(V)$ of cosimplicial objects.\n\\item Let $a, b : U \\to V$ be morphisms of simplicial objects of $\\mathcal{D}$.\nLet $F : \\mathcal{D} \\to \\mathcal{C}$ be a contravariant\nfunctor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$\nare homotopic morphisms $F(V) \\to F(U)$ of cosimplicial objects.\n\\item Let $a, b : U \\to V$ be morphisms of cosimplicial objects of\n$\\mathcal{C}$.\nLet $F : \\mathcal{C} \\to \\mathcal{D}$ be a contravariant\nfunctor. If $a$ and $b$ are homotopic, then $F(a)$, $F(b)$\nare homotopic morphisms $F(V) \\to F(U)$ of simplicial objects.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies and cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019Y","source_file":"simplicial.tex","source_line":5005,"source_end_line":5032,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5005-L5032","statement_sha256":"a5f6ea5ad5e03f1a4abb7cba83535011b4706962652bb3815cb5ba1089a45de9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2787,"rank":2787,"depth":4,"x":2595.757,"y":188.732,"cluster":"homological-algebra"},{"id":"stacks:019Z","tag":"019Z","title":"Homotopies and cosimplicial objects · Lemma 019Z","summary":"Let f : X → Y be a morphism of a category C with pushouts. Assume there is a morphism s : Y → X with s ∘ f = id_X. Consider the cosimplicial object U constructed in Example [Tag 016N] starting with f. The morphism U → U which in each degree is the self map of Y amalg_X … amalg_X Y given by f ∘ s on each factor is homotopic to the identity on U. In particular, U is homotopy equivalent to the constant cosimplicial object X.","statement_latex":"Let $f : X \\to Y$ be a morphism of a category $\\mathcal{C}$ with\npushouts. Assume there is a morphism $s : Y \\to X$ with\n$s \\circ f = \\text{id}_X$. Consider the cosimplicial object $U$ constructed in\nExample \\ref{example-push-outs-simplicial-object}\nstarting with $f$. The morphism $U \\to U$ which in each degree\nis the self map of $Y \\amalg_X \\ldots \\amalg_X Y$\ngiven by $f \\circ s$ on each factor is homotopic to the identity on $U$.\nIn particular, $U$ is homotopy equivalent to the constant\ncosimplicial object $X$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies and cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/019Z","source_file":"simplicial.tex","source_line":5041,"source_end_line":5052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5041-L5052","statement_sha256":"5e4098af82b132356279f1a33e332e8a8397bd7179cbb5eeb89d7a1abef4a445","origin":"The Stacks Project","memory_eligible":false,"source_rank":2788,"rank":2788,"depth":4,"x":2419.681,"y":308.896,"cluster":"homological-algebra"},{"id":"stacks:01A0","tag":"01A0","title":"Homotopies and cosimplicial objects · Lemma 01A0","summary":"The (cosimplicial) Dold-Kan functor carries homotopic maps to homotopic maps. Let A be an additive category. Let a, b : U → V be morphisms of cosimplicial objects of A. If a, b are homotopic, then s(a), s(b) : s(U) → s(V) are homotopic maps of cochain complexes. If in addition A is abelian, then Q(a), Q(b) : Q(U) → Q(V) are homotopic maps of cochain complexes.","statement_latex":"\\begin{slogan}\nThe (cosimplicial) Dold-Kan functor carries homotopic maps to homotopic maps.\n\\end{slogan}\nLet $\\mathcal{A}$ be an additive category. Let $a, b : U \\to V$ be morphisms\nof cosimplicial objects of $\\mathcal{A}$. If $a$, $b$ are homotopic,\nthen $s(a), s(b) : s(U) \\to s(V)$ are homotopic maps of cochain complexes.\nIf in addition $\\mathcal{A}$ is abelian, then $Q(a), Q(b) : Q(U) \\to Q(V)$\nare homotopic maps of cochain complexes.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies and cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01A0","source_file":"simplicial.tex","source_line":5061,"source_end_line":5071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5061-L5071","statement_sha256":"47edfac5bcb16824762c0de9c025af665ef6e1adbd58f9ea32c40a1a47c57273","origin":"The Stacks Project","memory_eligible":false,"source_rank":2789,"rank":2789,"depth":5,"x":2452.943,"y":120.048,"cluster":"homological-algebra"},{"id":"stacks:0FKK","tag":"0FKK","title":"Homotopies and cosimplicial objects · Lemma 0FKK","summary":"Let A be an additive category. Let a : U → V be a morphism of cosimplicial objects of A. If a is a homotopy equivalence, then s(a) : s(U) → s(V) is a homotopy equivalence of chain complexes. If in addition A is abelian, then also Q(a) : Q(U) → Q(V) is a homotopy equivalence of chain complexes.","statement_latex":"Let $\\mathcal{A}$ be an additive category. Let $a : U \\to V$ be a morphism\nof cosimplicial objects of $\\mathcal{A}$. If $a$ is a homotopy equivalence,\nthen $s(a) : s(U) \\to s(V)$ is a homotopy equivalence of chain complexes.\nIf in addition $\\mathcal{A}$ is abelian, then also\n$Q(a) : Q(U) \\to Q(V)$ is a homotopy equivalence of chain complexes.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Homotopies and cosimplicial objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKK","source_file":"simplicial.tex","source_line":5088,"source_end_line":5095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5088-L5095","statement_sha256":"e6e407ac0371828608fdfcd078c326d64468286083b2b86a6f5b7e4137608df1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2790,"rank":2790,"depth":6,"x":2580.505,"y":278.452,"cluster":"homological-algebra"},{"id":"stacks:01A2","tag":"01A2","title":"More homotopies in abelian categories · Lemma 01A2","summary":"Let A be an abelian category. Let A be a chain complex. Consider the covariant functor B ↦ ( (a, b, h) mid a, b : A → B and h a homotopy between a, b ) There exists a chain complex diamond A such that Mor_Ch(A)(diamond A, -) is isomorphic to the displayed functor. The construction A ↦ diamond A is functorial.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $A$ be a chain complex.\nConsider the covariant functor\n$$\nB \\longmapsto\n\\{\n(a, b, h)\n\\mid\na, b : A \\to B\\text{ and }h\\text{ a homotopy between }a, b\n\\}\n$$\nThere exists a chain complex $\\diamond A$\nsuch that $\\Mor_{\\text{Ch}(\\mathcal{A})}(\\diamond A, -)$\nis isomorphic to the displayed functor.\nThe construction $A \\mapsto \\diamond A$ is functorial.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"More homotopies in abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01A2","source_file":"simplicial.tex","source_line":5117,"source_end_line":5133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5117-L5133","statement_sha256":"660c067ea5a25d61687d7ecb7609e2ba5ede3a206b16a917477bcd7187756d40","origin":"The Stacks Project","memory_eligible":false,"source_rank":2791,"rank":2791,"depth":0,"x":2358.672,"y":233.952,"cluster":"homological-algebra"},{"id":"stacks:01A3","tag":"01A3","title":"More homotopies in abelian categories · Lemma 01A3","summary":"Let A be an abelian category. Let 0 → A ⊕ A → B → C → 0 be a short exact sequence of chain complexes of A. Suppose given in addition morphisms s_n : C_n → B_n splitting the associated short exact sequence in degree n. Let δ(s) : C → (A ⊕ A)[-1] = A[-1] ⊕ A[-1] be the associated morphism of complexes, see Homology, Lemma [Tag 011D]. If δ(s) factors through the morphism (1, -1) : A[-1] → A[-1] ⊕ A[-1], then there is a unique morphism B → diamond A fitting into a commutative…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet\n$$\n0 \\to A \\oplus A \\to B \\to C \\to 0\n$$\nbe a short exact sequence of chain complexes of $\\mathcal{A}$.\nSuppose given in addition morphisms $s_n : C_n \\to B_n$\nsplitting the associated short exact sequence in degree $n$.\nLet $\\delta(s) : C \\to (A \\oplus A)[-1] = A[-1] \\oplus A[-1]$\nbe the associated morphism of complexes, see\nHomology, Lemma \\ref{homology-lemma-ses-termwise-split}.\nIf $\\delta(s)$ factors through the morphism\n$(1, -1) : A[-1] \\to A[-1] \\oplus A[-1]$, then\nthere is a unique morphism $B \\to \\diamond A$\nfitting into a commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\nA \\oplus A \\ar[d] \\ar[r] &\nB \\ar[r] \\ar[d] &\nC \\ar[d] \\ar[r] &\n0 \\\\\n0 \\ar[r] &\nA \\oplus A \\ar[r] &\n\\diamond A \\ar[r] &\nA[-1] \\ar[r] &\n0\n}\n$$\nwhere the vertical maps are compatible with the splittings\n$s_n$ and the splittings of $\\diamond A_n \\to A[-1]_n$\nas well.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"More homotopies in abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01A3","source_file":"simplicial.tex","source_line":5216,"source_end_line":5250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5216-L5250","statement_sha256":"8ff7e6346d127b80c7487b95ce3e935ad75fe011d9f447e21edf5eb53797218f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2792,"rank":2792,"depth":6,"x":2558.381,"y":140.731,"cluster":"homological-algebra"},{"id":"stacks:01A4","tag":"01A4","title":"More homotopies in abelian categories · Lemma 01A4","summary":"Let A be an abelian category. Let U, V be simplicial objects of A. Let a, b : U → V be a pair of morphisms. Assume the corresponding maps of chain complexes N(a), N(b) : N(U) → N(V) are homotopic by a homotopy (N_n : N(U)_n → N(V)_n + 1). Then there exists a homotopy from a to b as in Definition [Tag 019K]. Moreover, one can choose the homotopy h : U × Δ[1] → V such that N_n = N(h)_n where N(h) is the homotopy coming from h as in Section [Tag 019Q].","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $U$, $V$ be simplicial objects of $\\mathcal{A}$.\nLet $a, b : U \\to V$ be a pair of morphisms.\nAssume the corresponding maps of chain complexes\n$N(a), N(b) : N(U) \\to N(V)$ are homotopic by\na homotopy $\\{N_n : N(U)_n \\to N(V)_{n + 1}\\}$.\nThen there exists a homotopy from $a$ to $b$ as in\nDefinition \\ref{definition-homotopy}. Moreover, one can choose the\nhomotopy $h : U \\times \\Delta[1] \\to V$ such that\n$N_n = N(h)_n$ where $N(h)$ is the homotopy coming\nfrom $h$ as in Section \\ref{section-homotopy-abelian}.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"More homotopies in abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01A4","source_file":"simplicial.tex","source_line":5289,"source_end_line":5302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5289-L5302","statement_sha256":"f2820bf78723c6a24f8b2f31e2aa423ad30b28c4477c864d9ed1d19f7f466b58","origin":"The Stacks Project","memory_eligible":false,"source_rank":2793,"rank":2793,"depth":7,"x":2485.959,"y":323.105,"cluster":"homological-algebra"},{"id":"stacks:08NL","tag":"08NL","title":"Trivial Kan fibrations · Definition 08NL","summary":"A map X → Y of simplicial sets is called a trivial Kan fibration if X_0 → Y_0 is surjective and for all n ≥ 1 and any commutative solid diagram xymatrix ∂ Δ[n] ar[r] ar[d] & X ar[d] Δ[n] ar[r] ar@-->[ru] & Y a dotted arrow exists making the diagram commute.","statement_latex":"A map $X \\to Y$ of simplicial sets is called a {\\it trivial Kan fibration}\nif $X_0 \\to Y_0$ is surjective and for all $n \\geq 1$ and any commutative\nsolid diagram\n$$\n\\xymatrix{\n\\partial \\Delta[n] \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Delta[n] \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\na dotted arrow exists making the diagram commute.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Trivial Kan fibrations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NL","source_file":"simplicial.tex","source_line":5533,"source_end_line":5545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5533-L5545","statement_sha256":"20e9e0bd85e6aa3fac6edce861a5d8527fb8a2f3f64023902d25da081c1eab1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2794,"rank":2794,"depth":0,"x":2392.541,"y":147.228,"cluster":"homological-algebra"},{"id":"stacks:08NM","tag":"08NM","title":"Trivial Kan fibrations · Lemma 08NM","summary":"Let f : X → Y be a trivial Kan fibration of simplicial sets. For any solid commutative diagram xymatrix Z ar[r]_b ar[d] & X ar[d] W ar[r]^a ar@-->[ru] & Y of simplicial sets with Z → W (termwise) injective a dotted arrow exists making the diagram commute.","statement_latex":"Let $f : X \\to Y$ be a trivial Kan fibration of simplicial sets.\nFor any solid commutative diagram\n$$\n\\xymatrix{\nZ \\ar[r]_b \\ar[d] & X \\ar[d] \\\\\nW \\ar[r]^a \\ar@{-->}[ru] & Y\n}\n$$\nof simplicial sets with $Z \\to W$ (termwise) injective\na dotted arrow exists making the diagram commute.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Trivial Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NM","source_file":"simplicial.tex","source_line":5550,"source_end_line":5562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5550-L5562","statement_sha256":"a7e5f912ae5229cf57ceeb3e9736edee3242bef96a5a2376caf24558cad67a13","origin":"The Stacks Project","memory_eligible":false,"source_rank":2795,"rank":2795,"depth":3,"x":2603.223,"y":224.042,"cluster":"homological-algebra"},{"id":"stacks:08NN","tag":"08NN","title":"Trivial Kan fibrations · Lemma 08NN","summary":"Let f : X → Y be a trivial Kan fibration of simplicial sets. Let Y' → Y be a morphism of simplicial sets. Then X ×_Y Y' → Y' is a trivial Kan fibration.","statement_latex":"Let $f : X \\to Y$ be a trivial Kan fibration of simplicial sets.\nLet $Y' \\to Y$ be a morphism of simplicial sets.\nThen $X \\times_Y Y' \\to Y'$ is a trivial Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Trivial Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NN","source_file":"simplicial.tex","source_line":5585,"source_end_line":5590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5585-L5590","statement_sha256":"4259c4f81625ff5bb7e9bbca919072dd23737b140caf649c3f9c22ea5e8b6942","origin":"The Stacks Project","memory_eligible":false,"source_rank":2796,"rank":2796,"depth":1,"x":2385.726,"y":287.054,"cluster":"homological-algebra"},{"id":"stacks:08NP","tag":"08NP","title":"Trivial Kan fibrations · Lemma 08NP","summary":"The composition of two trivial Kan fibrations is a trivial Kan fibration.","statement_latex":"The composition of two trivial Kan fibrations is a trivial Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Trivial Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NP","source_file":"simplicial.tex","source_line":5597,"source_end_line":5600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5597-L5600","statement_sha256":"5997e911816afe84795dccffc2838ca9793e668ebca637593e1724b0679797b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2797,"rank":2797,"depth":0,"x":2495.62,"y":116.887,"cluster":"homological-algebra"},{"id":"stacks:08NQ","tag":"08NQ","title":"Trivial Kan fibrations · Lemma 08NQ","summary":"Let … → U^2 → U^1 → U^0 be a sequence of trivial Kan fibrations. Let U = lim U^t defined by taking U_n = lim U_n^t. Then U → U^0 is a trivial Kan fibration.","statement_latex":"Let $\\ldots \\to U^2 \\to U^1 \\to U^0$ be a sequence of trivial Kan\nfibrations. Let $U = \\lim U^t$ defined by taking $U_n = \\lim U_n^t$.\nThen $U \\to U^0$ is a trivial Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Trivial Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NQ","source_file":"simplicial.tex","source_line":5606,"source_end_line":5611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5606-L5611","statement_sha256":"de28f2e097182272ddbde6edc8f03d5d7c90103f9d428bafccc5717a7735e2cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2798,"rank":2798,"depth":0,"x":2551.524,"y":305.041,"cluster":"homological-algebra"},{"id":"stacks:08NR","tag":"08NR","title":"Trivial Kan fibrations · Lemma 08NR","summary":"Products of trivial Kan fibrations are trivial Kan fibrations. Let X_i → Y_i be a set of trivial Kan fibrations. Then ∏ X_i → ∏ Y_i is a trivial Kan fibration.","statement_latex":"\\begin{slogan}\nProducts of trivial Kan fibrations are trivial Kan fibrations.\n\\end{slogan}\nLet $X_i \\to Y_i$ be a set of trivial Kan fibrations. Then\n$\\prod X_i \\to \\prod Y_i$ is a trivial Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Trivial Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NR","source_file":"simplicial.tex","source_line":5618,"source_end_line":5625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5618-L5625","statement_sha256":"a8d05de2ca2b109a70447aef8462e8a7e0d35b26360940157b2c3871716902ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":2799,"rank":2799,"depth":0,"x":2358.666,"y":197.839,"cluster":"homological-algebra"},{"id":"stacks:08Q5","tag":"08Q5","title":"Trivial Kan fibrations · Lemma 08Q5","summary":"A filtered colimit of trivial Kan fibrations is a trivial Kan fibration.","statement_latex":"A filtered colimit of trivial Kan fibrations is a trivial Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Trivial Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Q5","source_file":"simplicial.tex","source_line":5631,"source_end_line":5634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5631-L5634","statement_sha256":"9f4297fd08b416bec029ab6630f967e2d6abf3103e1f4b7fa69a5a6644bceca3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2800,"rank":2800,"depth":0,"x":2587.472,"y":167.405,"cluster":"homological-algebra"},{"id":"stacks:08NS","tag":"08NS","title":"Trivial Kan fibrations · Lemma 08NS","summary":"Let f : X → Y be a trivial Kan fibration of simplicial sets. Then f is a homotopy equivalence.","statement_latex":"Let $f : X \\to Y$ be a trivial Kan fibration of simplicial sets.\nThen $f$ is a homotopy equivalence.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Trivial Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NS","source_file":"simplicial.tex","source_line":5641,"source_end_line":5645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5641-L5645","statement_sha256":"ba46b7cf438f44141b37bbd22279078c0401c18ca5e5675e67e49ef2af0b5436","origin":"The Stacks Project","memory_eligible":false,"source_rank":2801,"rank":2801,"depth":4,"x":2442.985,"y":319.933,"cluster":"homological-algebra"},{"id":"stacks:08NU","tag":"08NU","title":"Kan fibrations · Definition 08NU","summary":"A map X → Y of simplicial sets is called a Kan fibration if for all k, n with 1 ≤ n, 0 ≤ k ≤ n and any commutative solid diagram xymatrix Lambda_k[n] ar[r] ar[d] & X ar[d] Δ[n] ar[r] ar@-->[ru] & Y a dotted arrow exists making the diagram commute. A Kan complex is a simplicial set X such that X → * is a Kan fibration, where * is the constant simplicial set on a singleton.","statement_latex":"A map $X \\to Y$ of simplicial sets is called a {\\it Kan fibration}\nif for all $k, n$ with $1 \\leq n$, $0 \\leq k \\leq n$ and any commutative\nsolid diagram\n$$\n\\xymatrix{\n\\Lambda_k[n] \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Delta[n] \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\na dotted arrow exists making the diagram commute. A {\\it Kan complex}\nis a simplicial set $X$ such that $X \\to *$ is a Kan fibration, where\n$*$ is the constant simplicial set on a singleton.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Kan fibrations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NU","source_file":"simplicial.tex","source_line":5687,"source_end_line":5701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5687-L5701","statement_sha256":"0fe4b3b2de8fd953465a0a2e231f95e45d3e0ba877b98023ea257243d3f51992","origin":"The Stacks Project","memory_eligible":false,"source_rank":2802,"rank":2802,"depth":0,"x":2426.842,"y":125.15,"cluster":"homological-algebra"},{"id":"stacks:08NV","tag":"08NV","title":"Kan fibrations · Lemma 08NV","summary":"Let f : X → Y be a Kan fibration of simplicial sets. Let Y' → Y be a morphism of simplicial sets. Then X ×_Y Y' → Y' is a Kan fibration.","statement_latex":"Let $f : X \\to Y$ be a Kan fibration of simplicial sets.\nLet $Y' \\to Y$ be a morphism of simplicial sets.\nThen $X \\times_Y Y' \\to Y'$ is a Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NV","source_file":"simplicial.tex","source_line":5709,"source_end_line":5714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5709-L5714","statement_sha256":"b2f72103adc7c4800fda5954d24ad426d29876bd510eb0e6ff24e01663f4bfc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2803,"rank":2803,"depth":1,"x":2595.667,"y":259.845,"cluster":"homological-algebra"},{"id":"stacks:08NW","tag":"08NW","title":"Kan fibrations · Lemma 08NW","summary":"The composition of two Kan fibrations is a Kan fibration.","statement_latex":"The composition of two Kan fibrations is a Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NW","source_file":"simplicial.tex","source_line":5721,"source_end_line":5724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5721-L5724","statement_sha256":"53844fa17f8d1d46f6361f10ed311b32fcdcbc697b8ebed46877f54986439462","origin":"The Stacks Project","memory_eligible":false,"source_rank":2804,"rank":2804,"depth":0,"x":2362.472,"y":256.312,"cluster":"homological-algebra"},{"id":"stacks:08NX","tag":"08NX","title":"Kan fibrations · Lemma 08NX","summary":"Let … → U^2 → U^1 → U^0 be a sequence of Kan fibrations. Let U = lim U^t defined by taking U_n = lim U_n^t. Then U → U^0 is a Kan fibration.","statement_latex":"Let $\\ldots \\to U^2 \\to U^1 \\to U^0$ be a sequence of Kan\nfibrations. Let $U = \\lim U^t$ defined by taking $U_n = \\lim U_n^t$.\nThen $U \\to U^0$ is a Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NX","source_file":"simplicial.tex","source_line":5730,"source_end_line":5735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5730-L5735","statement_sha256":"1c95082f549d5449d51e75cae088daeb3a4b421283b251f9095e62dd00000ffe","origin":"The Stacks Project","memory_eligible":false,"source_rank":2805,"rank":2805,"depth":0,"x":2537.558,"y":126.381,"cluster":"homological-algebra"},{"id":"stacks:08NY","tag":"08NY","title":"Kan fibrations · Lemma 08NY","summary":"Let X_i → Y_i be a set of Kan fibrations. Then ∏ X_i → ∏ Y_i is a Kan fibration.","statement_latex":"Let $X_i \\to Y_i$ be a set of Kan fibrations. Then\n$\\prod X_i \\to \\prod Y_i$ is a Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NY","source_file":"simplicial.tex","source_line":5742,"source_end_line":5746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5742-L5746","statement_sha256":"a4b9935e4d50d4c3fe216789b9513827f581fbbd722c58cff7907a749fe09260","origin":"The Stacks Project","memory_eligible":false,"source_rank":2806,"rank":2806,"depth":0,"x":2512.898,"y":321.861,"cluster":"homological-algebra"},{"id":"stacks:08NZ","tag":"08NZ","title":"Kan fibrations · Lemma 08NZ","summary":"Let X be a simplicial group. Then X is a Kan complex.","statement_latex":"Let $X$ be a simplicial group. Then $X$ is a Kan complex.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NZ","source_file":"simplicial.tex","source_line":5755,"source_end_line":5758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5755-L5758","statement_sha256":"2894bc13dfdf9e51227bfbc79c1c137d46dde08a94b7604073ff47d5a0df5a55","origin":"The Stacks Project","memory_eligible":false,"source_rank":2807,"rank":2807,"depth":0,"x":2373.653,"y":163.463,"cluster":"homological-algebra"},{"id":"stacks:08P0","tag":"08P0","title":"Kan fibrations · Lemma 08P0","summary":"Let f : X → Y be a homomorphism of simplicial abelian groups which is termwise surjective. Then f is a Kan fibration of simplicial sets.","statement_latex":"Let $f : X \\to Y$ be a homomorphism of simplicial abelian groups\nwhich is termwise surjective. Then $f$ is a Kan fibration of\nsimplicial sets.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08P0","source_file":"simplicial.tex","source_line":5805,"source_end_line":5810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5805-L5810","statement_sha256":"a7c8d1d0c78cff24a641038727d0adc12d7bf2838555e2139913692ec92497ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":2808,"rank":2808,"depth":1,"x":2604.087,"y":201.315,"cluster":"homological-algebra"},{"id":"stacks:08P1","tag":"08P1","title":"Kan fibrations · Lemma 08P1","summary":"Let f : X → Y be a homomorphism of simplicial abelian groups which is termwise surjective and induces a quasi-isomorphism on associated chain complexes. Then f is a trivial Kan fibration of simplicial sets.","statement_latex":"Let $f : X \\to Y$ be a homomorphism of simplicial abelian groups\nwhich is termwise surjective and induces a quasi-isomorphism on\nassociated chain complexes. Then $f$ is a trivial Kan fibration of\nsimplicial sets.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08P1","source_file":"simplicial.tex","source_line":5836,"source_end_line":5842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5836-L5842","statement_sha256":"e6fb89474814d2edc1cec3d9887b7416381e51a669f14c41f29fa2c960de2b3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2809,"rank":2809,"depth":9,"x":2403.401,"y":304.326,"cluster":"homological-algebra"},{"id":"stacks:08P2","tag":"08P2","title":"Kan fibrations · Lemma 08P2","summary":"Let f : X → Y be a map of simplicial abelian groups. If f is a homotopy equivalence of simplicial sets, then f induces a quasi-isomorphism of associated chain complexes.","statement_latex":"Let $f : X \\to Y$ be a map of simplicial abelian groups. If $f$ is a\nhomotopy equivalence of simplicial sets, then $f$ induces a\nquasi-isomorphism of associated chain complexes.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Kan fibrations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08P2","source_file":"simplicial.tex","source_line":5884,"source_end_line":5889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5884-L5889","statement_sha256":"bc608a60bc88585fac763fde9e7e29a2d74eb3dea25dfffdb97308f0f5784fb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2810,"rank":2810,"depth":6,"x":2468.652,"y":114.183,"cluster":"homological-algebra"},{"id":"stacks:01A6","tag":"01A6","title":"A homotopy equivalence · Lemma 01A6","summary":"Let f : V → U be a morphism of simplicial sets. Let n ≥ 0 be an integer. Assume • The map f_i : V_i → U_i is a bijection for i < n. • The map f_n : V_n → U_n is a surjection. • The canonical morphism U → cosk_n sk_n U is an isomorphism. • The canonical morphism V → cosk_n sk_n V is an isomorphism. Then f is a trivial Kan fibration.","statement_latex":"Let $f : V \\to U$ be a morphism of simplicial sets. Let $n \\geq 0$ be an\ninteger. Assume\n\\begin{enumerate}\n\\item The map $f_i : V_i \\to U_i$ is a bijection for $i < n$.\n\\item The map $f_n : V_n \\to U_n$ is a surjection.\n\\item The canonical morphism $U \\to \\text{cosk}_n \\text{sk}_n U$\nis an isomorphism.\n\\item The canonical morphism $V \\to \\text{cosk}_n \\text{sk}_n V$\nis an isomorphism.\n\\end{enumerate}\nThen $f$ is a trivial Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"A homotopy equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01A6","source_file":"simplicial.tex","source_line":5981,"source_end_line":5994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L5981-L5994","statement_sha256":"beece269c72d0172234bea31a104526b93252338f685927bc4a52513d1560b39","origin":"The Stacks Project","memory_eligible":false,"source_rank":2811,"rank":2811,"depth":1,"x":2573.614,"y":291.705,"cluster":"homological-algebra"},{"id":"stacks:01A9","tag":"01A9","title":"A homotopy equivalence · Lemma 01A9","summary":"Let f^0, f^1 : V → U be maps of simplicial sets. Let n ≥ 0 be an integer. Assume • The maps f^j_i : V_i → U_i, j = 0, 1 are equal for i < n. • The canonical morphism U → cosk_n sk_n U is an isomorphism. • The canonical morphism V → cosk_n sk_n V is an isomorphism. Then f^0 is homotopic to f^1.","statement_latex":"Let $f^0, f^1 : V \\to U$ be maps of simplicial sets.\nLet $n \\geq 0$ be an integer.\nAssume\n\\begin{enumerate}\n\\item The maps $f^j_i : V_i \\to U_i$, $j = 0, 1$ are equal for $i < n$.\n\\item The canonical morphism $U \\to \\text{cosk}_n \\text{sk}_n U$\nis an isomorphism.\n\\item The canonical morphism $V \\to \\text{cosk}_n \\text{sk}_n V$\nis an isomorphism.\n\\end{enumerate}\nThen $f^0$ is homotopic to $f^1$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"A homotopy equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01A9","source_file":"simplicial.tex","source_line":6043,"source_end_line":6056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L6043-L6056","statement_sha256":"cd09b9452ec4ef3b1d23e94ff5004d737e42b6641552bb072971fefec7d1f69e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2812,"rank":2812,"depth":4,"x":2353.102,"y":220.246,"cluster":"homological-algebra"},{"id":"stacks:01AB","tag":"01AB","title":"A homotopy equivalence · Lemma 01AB","summary":"Let A, B be sets, and that f : A → B is a map. Consider the simplicial set U with n-simplices A ×_B A ×_B … ×_B A (n + 1 factors). see Example [Tag 016E]. If f is surjective, the morphism U → B where B indicates the constant simplicial set with value B is a trivial Kan fibration.","statement_latex":"Let $A$, $B$ be sets, and that $f : A \\to B$ is a map. Consider the simplicial\nset $U$ with $n$-simplices\n$$\nA \\times_B A \\times_B \\ldots \\times_B A\\ (n + 1 \\text{ factors)}.\n$$\nsee Example \\ref{example-fibre-products-simplicial-object}.\nIf $f$ is surjective, the morphism $U \\to B$\nwhere $B$ indicates the constant simplicial set with value $B$\nis a trivial Kan fibration.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"A homotopy equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AB","source_file":"simplicial.tex","source_line":6150,"source_end_line":6161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L6150-L6161","statement_sha256":"e1a81bd7c6359148552505aa2299efc8cf465e63d683006a3c3ccfa25e54d78d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2813,"rank":2813,"depth":2,"x":2573.526,"y":147.698,"cluster":"homological-algebra"},{"id":"stacks:0G5N","tag":"0G5N","title":"Preparation for standard resolutions · Lemma 0G5N","summary":"In Example [Tag 0G5M] if 1_Y = (d star 1_Y) ∘ s = (1_Y star d) ∘ s and (s star 1) ∘ s = (1 star s) ∘ s then X = (X_n, d^n_j, s^n_j) is a simplicial object in the category of endofunctors of C and d : X_0 = Y → id_C defines an augmentation.","statement_latex":"In Example \\ref{example-godement} if\n$$\n1_Y = (d \\star 1_Y) \\circ s = (1_Y \\star d) \\circ s\n\\quad\\text{and}\\quad\n(s \\star 1) \\circ s = (1 \\star s) \\circ s\n$$\nthen $X = (X_n, d^n_j, s^n_j)$ is a simplicial object in the category\nof endofunctors of $\\mathcal{C}$ and $d : X_0 = Y \\to \\text{id}_\\mathcal{C}$\ndefines an augmentation.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Preparation for standard resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5N","source_file":"simplicial.tex","source_line":6225,"source_end_line":6236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L6225-L6236","statement_sha256":"eacc922a4fa7793cf6131864afa0e9b6b9ea62c62768ae440e4d1c3567f65ecc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2814,"rank":2814,"depth":3,"x":2469.162,"y":326.555,"cluster":"homological-algebra"},{"id":"stacks:0G5Q","tag":"0G5Q","title":"Preparation for standard resolutions · Lemma 0G5Q","summary":"Let A, B, C, Y, d, s, F, G be as in Example [Tag 0G5P]. Given a transformation of functors h_0 : G ∘ F → G ∘ Y ∘ F such that 1_G ∘ F = (1_G star d star 1_F) ∘ h_0 Then there is a morphism h : G ∘ F → G ∘ X ∘ F of simplicial objects such that ε ∘ h = id where ε : G ∘ X ∘ F → G ∘ F is the augmentation.","statement_latex":"Let $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$, $Y$, $d$, $s$, $F$, $G$\nbe as in Example \\ref{example-godement-functorial}. Given a transformation\nof functors $h_0 : G \\circ F \\to G \\circ Y \\circ F$ such that\n$$\n1_{G \\circ F} = (1_G \\star d \\star 1_F) \\circ h_0\n$$\nThen there is a morphism $h : G \\circ F \\to G \\circ X \\circ F$\nof simplicial objects such that $\\epsilon \\circ h = \\text{id}$ where\n$\\epsilon : G \\circ X \\circ F \\to G \\circ F$ is the augmentation.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Preparation for standard resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5Q","source_file":"simplicial.tex","source_line":6356,"source_end_line":6367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L6356-L6367","statement_sha256":"56efc22ec38f92a6bfbfdbd1e09a1c31ec7ff85292241b392ae5eea1b6f511e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2815,"rank":2815,"depth":0,"x":2402.179,"y":135.143,"cluster":"homological-algebra"},{"id":"stacks:0G5R","tag":"0G5R","title":"Preparation for standard resolutions · Lemma 0G5R","summary":"Let A, B, C, Y, d, s, F, G be as in Example [Tag 0G5P]. Let F' : A → C and G' : C → B be two functors. Let (a_n) : G ∘ X → G' ∘ X be a morphism of simplicial objects compatible via augmentations with a : G → G'. Let (b_n) : X ∘ F → X ∘ F' be a morphism of simplicial objects compatible via augmentations with b : F → F'. Then the two maps a star (b_n), (a_n) star b : G ∘ X ∘ F → G' ∘ X ∘ F' are homotopic.","statement_latex":"Let $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$, $Y$, $d$, $s$, $F$, $G$\nbe as in Example \\ref{example-godement-functorial}. Let\n$F' : \\mathcal{A} \\to \\mathcal{C}$ and\n$G' : \\mathcal{C} \\to \\mathcal{B}$ be two functors.\nLet $(a_n) : G \\circ X \\to G' \\circ X$ be a morphism of simplicial objects\ncompatible via augmentations with $a : G \\to G'$.\nLet $(b_n) : X \\circ F \\to X \\circ F'$ be a morphism of simplicial objects\ncompatible via augmentations with $b : F \\to F'$. Then the two maps\n$$\na \\star (b_n), (a_n) \\star b : G \\circ X \\circ F \\to G' \\circ X \\circ F'\n$$\nare homotopic.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Preparation for standard resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5R","source_file":"simplicial.tex","source_line":6394,"source_end_line":6408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L6394-L6408","statement_sha256":"b9b6d57066c38a905586a946197aa3ce610b9704c90977fab08eefa839bab5c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2816,"rank":2816,"depth":4,"x":2605.825,"y":238.445,"cluster":"homological-algebra"},{"id":"stacks:0G5S","tag":"0G5S","title":"Preparation for standard resolutions · Lemma 0G5S","summary":"Let C, Y, d, s be as in Example [Tag 0G5M] satisfying the equations of Lemma [Tag 0G5N]. Let f : id_C → id_C be an endomorphism of the identity functor. Then f star 1_X, 1_X star f : X → X are maps of simplicial objects compatible with f via the augmentation ε : X → id_C. Moreover, f star 1_X and 1_X star f are homotopic.","statement_latex":"Let $\\mathcal{C}$, $Y$, $d$, $s$ be as in Example \\ref{example-godement}\nsatisfying the equations of Lemma \\ref{lemma-godement}. Let\n$f : \\text{id}_\\mathcal{C} \\to \\text{id}_\\mathcal{C}$ be an endomorphism\nof the identity functor. Then $f \\star 1_X, 1_X \\star f : X \\to X$\nare maps of simplicial objects compatible with $f$ via the augmentation\n$\\epsilon : X \\to \\text{id}_\\mathcal{C}$. Moreover, $f \\star 1_X$\nand $1_X \\star f$ are homotopic.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Preparation for standard resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5S","source_file":"simplicial.tex","source_line":6533,"source_end_line":6542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L6533-L6542","statement_sha256":"cf08b3c25edca2c72cd96e28b925d48150ebffe3fae0ab656d7585b1352be04a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2817,"rank":2817,"depth":5,"x":2372.215,"y":277.887,"cluster":"homological-algebra"},{"id":"stacks:08NC","tag":"08NC","title":"Standard resolutions · Lemma 08NC","summary":"In Situation [Tag 08N9] the system X = (X_n, d^n_j, s^n_j) is a simplicial object of Fun(A, A) and ε_0 defines an augmentation ε from X to the constant simplicial object with value X_-1 = id_A.","statement_latex":"In Situation \\ref{situation-adjoint-functors}\nthe system $X = (X_n, d^n_j, s^n_j)$\nis a simplicial object of $\\text{Fun}(\\mathcal{A}, \\mathcal{A})$\nand $\\epsilon_0$ defines an augmentation $\\epsilon$ from $X$\nto the constant simplicial object with value $X_{-1} = \\text{id}_\\mathcal{A}$.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Standard resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NC","source_file":"simplicial.tex","source_line":6674,"source_end_line":6681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L6674-L6681","statement_sha256":"d25969803625e7a673e9e2b451c9f94467bebe81cf295c0aebdb91a9eec45bb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2818,"rank":2818,"depth":4,"x":2512.98,"y":115.99,"cluster":"homological-algebra"},{"id":"stacks:08ND","tag":"08ND","title":"Standard resolutions · Lemma 08ND","summary":"In Situation [Tag 08N9] the maps 1_V star ε : V ∘ X → V, and ε star 1_U : X ∘ U → U are homotopy equivalences.","statement_latex":"In Situation \\ref{situation-adjoint-functors} the maps\n$$\n1_V \\star \\epsilon : V \\circ X \\to V,\n\\quad\\text{and}\\quad\n\\epsilon \\star 1_U : X \\circ U \\to U\n$$\nare homotopy equivalences.","area":"Homological Algebra","chapter":"Simplicial Methods","chapter_id":"simplicial","section":"Standard resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ND","source_file":"simplicial.tex","source_line":6728,"source_end_line":6737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/simplicial.tex#L6728-L6737","statement_sha256":"1a890ac0f06ad5e90d038e76184fc9c545b98df0e086eb55f9e85b0fa24372bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":2819,"rank":2819,"depth":5,"x":2539.417,"y":315.56,"cluster":"homological-algebra"},{"id":"stacks:0BC3","tag":"0BC3","title":"Stably free modules · Definition 0BC3","summary":"Let R be a ring. • Two modules M, N over R are said to be stably isomorphic if there exist n, m ≥ 0 such that M ⊕ R^⊕ m ≅ N ⊕ R^⊕ n as R-modules. • A module M is stably free if it is stably isomorphic to a free module.","statement_latex":"Let $R$ be a ring. \n\\begin{enumerate}\n\\item Two modules $M$, $N$ over $R$ are said to be\n{\\it stably isomorphic} if there exist $n, m \\geq 0$ such\nthat $M \\oplus R^{\\oplus m} \\cong N \\oplus R^{\\oplus n}$\nas $R$-modules.\n\\item A module $M$ is {\\it stably free} if it is stably isomorphic\nto a free module.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Stably free modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BC3","source_file":"more-algebra.tex","source_line":53,"source_end_line":64,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L53-L64","statement_sha256":"6c5f5f794d5ce88ee4072fd528c5b97bc1b59579e75976ac7de5e9e09756aaef","origin":"The Stacks Project","memory_eligible":false,"source_rank":2820,"rank":2820,"depth":0,"x":683.996,"y":680.0,"cluster":"advanced-algebra"},{"id":"stacks:0BC4","tag":"0BC4","title":"Stably free modules · Lemma 0BC4","summary":"Let R be a ring. Let 0 → P' → P → P\" → 0 be a short exact sequence of finite projective R-modules. If 2 out of 3 of these modules are stably free, then so is the third.","statement_latex":"Let $R$ be a ring. Let $0 \\to P' \\to P \\to P'' \\to 0$ be a short\nexact sequence of finite projective $R$-modules. If $2$ out of $3$\nof these modules are stably free, then so is the third.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Stably free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BC4","source_file":"more-algebra.tex","source_line":69,"source_end_line":74,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L69-L74","statement_sha256":"0910b642e38ed70aa6a089226f54b53499bcc0bd027c2e1c4372e68e0aaeee54","origin":"The Stacks Project","memory_eligible":false,"source_rank":2821,"rank":2821,"depth":0,"x":674.897,"y":683.927,"cluster":"advanced-algebra"},{"id":"stacks:0BC5","tag":"0BC5","title":"Stably free modules · Lemma 0BC5","summary":"Let R be a ring. Let I ⊂ R be an ideal. Assume that every element of 1 + I is a unit (in other words I is contained in the Jacobson radical of R). For every finite stably free R/I-module E there exists a finite stably free R-module M such that M/IM ≅ E.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Assume that\nevery element of $1 + I$ is a unit (in other words $I$ is contained\nin the Jacobson radical of $R$). For every finite stably free $R/I$-module $E$\nthere exists a finite stably free $R$-module $M$ such that $M/IM \\cong E$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Stably free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BC5","source_file":"more-algebra.tex","source_line":92,"source_end_line":98,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L92-L98","statement_sha256":"a2b63cad15a63ee2413875fabde3b0b4a2ddb67d33afa67a0f711b5b58e25765","origin":"The Stacks Project","memory_eligible":false,"source_rank":2822,"rank":2822,"depth":0,"x":680.781,"y":672.523,"cluster":"advanced-algebra"},{"id":"stacks:0D48","tag":"0D48","title":"Stably free modules · Lemma 0D48","summary":"Let R be a ring. Let I ⊂ R be an ideal. Assume that every element of 1 + I is a unit (in other words I is contained in the Jacobson radical of R). Let M be a finite flat R-module such that M/IM is a projective R/I-module. Then M is a finite projective R-module.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nAssume that every element of $1 + I$ is a unit\n(in other words $I$ is contained in the Jacobson radical of $R$).\nLet $M$ be a finite flat $R$-module such that\n$M/IM$ is a projective $R/I$-module.\nThen $M$ is a finite projective $R$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Stably free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D48","source_file":"more-algebra.tex","source_line":113,"source_end_line":121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L113-L121","statement_sha256":"416a2dffa4b5bfffb111b1504beb4ff44043c6e2210dc3abc2a0c9aa7f53c926","origin":"The Stacks Project","memory_eligible":false,"source_rank":2823,"rank":2823,"depth":5,"x":686.432,"y":687.048,"cluster":"advanced-algebra"},{"id":"stacks:0BC6","tag":"0BC6","title":"Stably free modules · Lemma 0BC6","summary":"Let R be a ring. Let I ⊂ R be an ideal. Assume that every element of 1 + I is a unit (in other words I is contained in the Jacobson radical of R). If P and P' are finite projective R-modules, then • if φ : P → P' is an R-module map inducing an isomorphism overlineφ : P/IP → P'/IP', then φ is an isomorphism, • if P/IP ≅ P'/IP', then P ≅ P'.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Assume that\nevery element of $1 + I$ is a unit (in other words $I$ is contained\nin the Jacobson radical of $R$). If $P$ and $P'$ are finite\nprojective $R$-modules, then\n\\begin{enumerate}\n\\item if $\\varphi : P \\to P'$ is an $R$-module map inducing an\nisomorphism $\\overline{\\varphi} : P/IP \\to P'/IP'$, then $\\varphi$\nis an isomorphism,\n\\item if $P/IP \\cong P'/IP'$, then $P \\cong P'$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Stably free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BC6","source_file":"more-algebra.tex","source_line":144,"source_end_line":156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L144-L156","statement_sha256":"686b8f64edcee347bd178c99459e0cd0de0aacde0c401b84d04083757015115a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2824,"rank":2824,"depth":3,"x":668.196,"y":678.246,"cluster":"advanced-algebra"},{"id":"stacks:07VE","tag":"07VE","title":"A comment on the Artin-Rees property · Lemma 07VE","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal contained in the Jacobson radical of A. Let S : L xrightarrowf M xrightarrowg N and S' : L xrightarrowf' M xrightarrowg' N be two complexes of finite A-modules as shown. Assume that • c works in the Artin-Rees lemma for f and g, • the complex S is exact, and • f' = f bmod I^c + 1M and g' = g bmod I^c + 1N. Then c works in the Artin-Rees lemma for g' and the complex S' is exact.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal contained in\nthe Jacobson radical of $A$. Let\n$$\nS : L \\xrightarrow{f} M \\xrightarrow{g} N\n\\quad\\text{and}\\quad\nS' : L \\xrightarrow{f'} M \\xrightarrow{g'} N\n$$\nbe two complexes of finite $A$-modules as shown. Assume that\n\\begin{enumerate}\n\\item $c$ works in the Artin-Rees lemma for $f$ and $g$,\n\\item the complex $S$ is exact, and\n\\item $f' = f \\bmod I^{c + 1}M$ and $g' = g \\bmod I^{c + 1}N$.\n\\end{enumerate}\nThen $c$ works in the Artin-Rees lemma for $g'$ and the\ncomplex $S'$ is exact.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"A comment on the Artin-Rees property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VE","source_file":"more-algebra.tex","source_line":194,"source_end_line":211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L194-L211","statement_sha256":"8e07d60c7b823d5fccdaea5b1906e3c29be13ddac242b7bc55387ddb758eee78","origin":"The Stacks Project","memory_eligible":false,"source_rank":2825,"rank":2825,"depth":5,"x":691.182,"y":674.025,"cluster":"advanced-algebra"},{"id":"stacks:07VF","tag":"07VF","title":"A comment on the Artin-Rees property · Lemma 07VF","summary":"Assumptions as in Lemma [Tag 07VE]. Let Q = Coker(g) and Q' = Coker(g'). Then Gr_I(Q) ≅ Gr_I(Q') as graded Gr_I(A)-modules.","statement_latex":"Assumptions as in Lemma \\ref{lemma-approximate-complex}.\nLet $Q = \\Coker(g)$ and $Q' = \\Coker(g')$. Then\n$\\text{Gr}_I(Q) \\cong \\text{Gr}_I(Q')$\nas graded $\\text{Gr}_I(A)$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"A comment on the Artin-Rees property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VF","source_file":"more-algebra.tex","source_line":255,"source_end_line":261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L255-L261","statement_sha256":"abd91807afdc51249d97023adde72a9621d5b49c912327cb4d1be620d402b8e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2826,"rank":2826,"depth":6,"x":676.26,"y":691.687,"cluster":"advanced-algebra"},{"id":"stacks:07VG","tag":"07VG","title":"A comment on the Artin-Rees property · Lemma 07VG","summary":"Let A → B be a flat map of Noetherian rings. Let I ⊂ A be an ideal. Let f : M → N be a homomorphism of finite A-modules. Assume that c works for f in the Artin-Rees lemma. Then c works for f ⊗ 1 : M ⊗_A B → N ⊗_A B in the Artin-Rees lemma for the ideal IB.","statement_latex":"Let $A \\to B$ be a flat map of Noetherian rings. Let $I \\subset A$ be\nan ideal. Let $f : M \\to N$ be a homomorphism of finite $A$-modules.\nAssume that $c$ works for $f$ in the Artin-Rees lemma. Then $c$ works for\n$f \\otimes 1 : M \\otimes_A B \\to N \\otimes_A B$ in the Artin-Rees lemma\nfor the ideal $IB$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"A comment on the Artin-Rees property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VG","source_file":"more-algebra.tex","source_line":283,"source_end_line":290,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L283-L290","statement_sha256":"60f240d46457f63807c1614964f4452e84173790ace46d5ab448008c3e123faf","origin":"The Stacks Project","memory_eligible":false,"source_rank":2827,"rank":2827,"depth":0,"x":672.867,"y":668.463,"cluster":"advanced-algebra"},{"id":"stacks:00IT","tag":"00IT","title":"Fibre products of rings, I · Lemma 00IT","summary":"Let R be a ring. Let A → B and C → B be R-algebra maps. Assume • R is Noetherian, • A, B, C are of finite type over R, • A → B is surjective, and • B is finite over C. Then A ×_B C is of finite type over R.","statement_latex":"Let $R$ be a ring. Let $A \\to B$ and $C \\to B$ be $R$-algebra maps.\nAssume\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $A$, $B$, $C$ are of finite type over $R$,\n\\item $A \\to B$ is surjective, and\n\\item $B$ is finite over $C$.\n\\end{enumerate}\nThen $A \\times_B C$ is of finite type over $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00IT","source_file":"more-algebra.tex","source_line":326,"source_end_line":337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L326-L337","statement_sha256":"a52ddbaa927f5f28307cffd25142f6641cd78ed17be3fe270c7ca74a7a99b1d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2828,"rank":2828,"depth":1,"x":695.476,"y":684.747,"cluster":"advanced-algebra"},{"id":"stacks:08NI","tag":"08NI","title":"Fibre products of rings, I · Lemma 08NI","summary":"Let R be a Noetherian ring. Let I be a finite set. Suppose given a cartesian diagram xymatrix ∏ B_i & ∏ A_i ar[l]^∏ φ_i Q ar[u]^∏ ψ_i & P ar[u] ar[l] with ψ_i and φ_i surjective, and Q, A_i, B_i of finite type over R. Then P is of finite type over R.","statement_latex":"Let $R$ be a Noetherian ring. Let $I$ be a finite set. Suppose given a\ncartesian diagram\n$$\n\\xymatrix{\n\\prod B_i &\n\\prod A_i \\ar[l]^{\\prod \\varphi_i} \\\\\nQ \\ar[u]^{\\prod \\psi_i} &\nP \\ar[u] \\ar[l]\n}\n$$\nwith $\\psi_i$ and $\\varphi_i$ surjective, and $Q$, $A_i$, $B_i$ of\nfinite type over $R$. Then $P$ is of finite type over $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08NI","source_file":"more-algebra.tex","source_line":364,"source_end_line":378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L364-L378","statement_sha256":"15febb75785d4da424ef0bfaae94589df1385b48e25327f49986d74ad0a241fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":2829,"rank":2829,"depth":2,"x":663.9,"y":685.582,"cluster":"advanced-algebra"},{"id":"stacks:01Z8","tag":"01Z8","title":"Fibre products of rings, I · Lemma 01Z8","summary":"Suppose given a cartesian diagram of rings xymatrix R & R' ar[l]^t B ar[u]_s & B'ar[u] ar[l] i.e., B' = B ×_R R'. If h ∈ B' corresponds to g ∈ B and f ∈ R' such that s(g) = t(f), then the diagram xymatrix R_s(g) = R_t(f) & (R')_f ar[l]^-t B_g ar[u]_s & (B')_h ar[u] ar[l] is cartesian too.","statement_latex":"Suppose given a cartesian diagram of rings\n$$\n\\xymatrix{\nR &\nR' \\ar[l]^t \\\\\nB \\ar[u]_s &\nB'\\ar[u] \\ar[l]\n}\n$$\ni.e., $B' = B \\times_R R'$. If $h \\in B'$ corresponds to $g \\in B$\nand $f \\in R'$ such that $s(g) = t(f)$, then the diagram\n$$\n\\xymatrix{\nR_{s(g)} = R_{t(f)} &\n(R')_f \\ar[l]^-t \\\\\nB_g \\ar[u]_s &\n(B')_h \\ar[u] \\ar[l]\n}\n$$\nis cartesian too.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Z8","source_file":"more-algebra.tex","source_line":397,"source_end_line":419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L397-L419","statement_sha256":"c2b0db4de630bf047e46d215582036ce42ada01f089d5718fdef0d6aa702077d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2830,"rank":2830,"depth":1,"x":687.761,"y":666.068,"cluster":"advanced-algebra"},{"id":"stacks:0D2F","tag":"0D2F","title":"Fibre products of rings, I · Lemma 0D2F","summary":"Given a commutative diagram of rings xymatrix R & R' ar[l] B ar[u] & B' ar[u] ar[l] the functor ([Tag 0D2E]) has a right adjoint, namely the functor F : (N, M', φ) ↦ N ×_φ M' (see proof for elucidation).","statement_latex":"Given a commutative diagram of rings\n$$\n\\xymatrix{\nR &\nR' \\ar[l] \\\\\nB \\ar[u] &\nB' \\ar[u] \\ar[l]\n}\n$$\nthe functor (\\ref{equation-modules}) has a right adjoint,\nnamely the functor\n$$\nF : (N, M', \\varphi) \\longmapsto N \\times_\\varphi M'\n$$\n(see proof for elucidation).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2F","source_file":"more-algebra.tex","source_line":462,"source_end_line":479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L462-L479","statement_sha256":"2371c124de7ee62e4527a7af16074d90f9f07c21d419fefaa891b9f549a428d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2831,"rank":2831,"depth":1,"x":685.735,"y":695.36,"cluster":"advanced-algebra"},{"id":"stacks:0B7J","tag":"0B7J","title":"Fibre products of rings, II · Lemma 0B7J","summary":"In Situation [Tag 08KH] we have Spec(B') = Spec(B) amalg_Spec(A) Spec(A') as topological spaces.","statement_latex":"In Situation \\ref{situation-module-over-fibre-product}\nwe have\n$$\n\\Spec(B') = \\Spec(B) \\amalg_{\\Spec(A)} \\Spec(A')\n$$\nas topological spaces.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7J","source_file":"more-algebra.tex","source_line":543,"source_end_line":551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L543-L551","statement_sha256":"e3502b6f06e7f0d51b9d9391c8c153741bdbc04c44d5ee21219fd48723787448","origin":"The Stacks Project","memory_eligible":false,"source_rank":2832,"rank":2832,"depth":2,"x":662.714,"y":671.585,"cluster":"advanced-algebra"},{"id":"stacks:0E1S","tag":"0E1S","title":"Fibre products of rings, II · Lemma 0E1S","summary":"In Situation [Tag 08KH] if B → A is integral, then B' → A' is integral.","statement_latex":"In Situation \\ref{situation-module-over-fibre-product}\nif $B \\to A$ is integral, then $B' \\to A'$ is integral.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1S","source_file":"more-algebra.tex","source_line":606,"source_end_line":610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L606-L610","statement_sha256":"8fe2ac356f71a95298008a64f87f52c2b3f314063297f88fce545716ebd3aa3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2833,"rank":2833,"depth":0,"x":700.279,"y":676.255,"cluster":"advanced-algebra"},{"id":"stacks:07RU","tag":"07RU","title":"Fibre products of rings, II · Lemma 07RU","summary":"In Situation [Tag 08KH] the functor ([Tag 08KI]) has a right adjoint, namely the functor F : (N, M', φ) ↦ N ×_φ, M M' where M = M'/IM'. Moreover, the composition of F with ([Tag 08KI]) is the identity functor on Mod_B ×_Mod_A Mod_A'. In other words, setting N' = N ×_φ, M M' we have N' ⊗_B' B = N and N' ⊗_B' A' = M'.","statement_latex":"In Situation \\ref{situation-module-over-fibre-product}\nthe functor (\\ref{equation-functor}) has a right adjoint, namely\nthe functor\n$$\nF : (N, M', \\varphi) \\longmapsto N \\times_{\\varphi, M} M'\n$$\nwhere $M = M'/IM'$. Moreover, the composition of $F$ with\n(\\ref{equation-functor}) is the identity functor on\n$\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$. In other words,\nsetting $N' = N \\times_{\\varphi, M} M'$ we have\n$N' \\otimes_{B'} B = N$ and $N' \\otimes_{B'} A' = M'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RU","source_file":"more-algebra.tex","source_line":643,"source_end_line":656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L643-L656","statement_sha256":"7f45bd043b98f69a6e099f559f8ad268cf5895371ea8640c6e5966e999afe6f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2834,"rank":2834,"depth":2,"x":667.624,"y":694.787,"cluster":"advanced-algebra"},{"id":"stacks:08IG","tag":"08IG","title":"Fibre products of rings, II · Lemma 08IG","summary":"In the situation of Lemma [Tag 07RU] for a B'-module L' the adjunction map L' → (L' ⊗_B' B) ×_(L' ⊗_B' A) (L' ⊗_B' A') is surjective but in general not injective.","statement_latex":"In the situation of Lemma \\ref{lemma-module-over-fibre-product}\nfor a $B'$-module $L'$ the adjunction map\n$$\nL' \\longrightarrow \n(L' \\otimes_{B'} B) \\times_{(L' \\otimes_{B'} A)} (L' \\otimes_{B'} A')\n$$\nis surjective but in general not injective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IG","source_file":"more-algebra.tex","source_line":717,"source_end_line":726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L717-L726","statement_sha256":"16bbe26d49b51851e0905088b13676494018c7893565bf66e4664911dfbadef4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2835,"rank":2835,"depth":3,"x":677.141,"y":661.466,"cluster":"advanced-algebra"},{"id":"stacks:08KJ","tag":"08KJ","title":"Fibre products of rings, II · Lemma 08KJ","summary":"In Situation [Tag 08KH] let (N_1, M'_1, φ_1) → (N_2, M'_2, φ_2) be a morphism of Mod_B ×_Mod_A Mod_A' with N_1 → N_2 and M'_1 → M'_2 surjective. Then N_1 ×_φ_1, M_1 M'_1 → N_2 ×_φ_2, M_2 M'_2 where M_1 = M'_1/IM'_1 and M_2 = M'_2/IM'_2 is surjective.","statement_latex":"In Situation \\ref{situation-module-over-fibre-product}\nlet $(N_1, M'_1, \\varphi_1) \\to (N_2, M'_2, \\varphi_2)$ be a morphism\nof $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$\nwith $N_1 \\to N_2$ and $M'_1 \\to M'_2$ surjective. Then\n$$\nN_1 \\times_{\\varphi_1, M_1} M'_1 \\to N_2 \\times_{\\varphi_2, M_2} M'_2\n$$\nwhere $M_1 = M'_1/IM'_1$ and $M_2 = M'_2/IM'_2$ is surjective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KJ","source_file":"more-algebra.tex","source_line":750,"source_end_line":760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L750-L760","statement_sha256":"e01725056fc1a1503496621d72c6106f186e535f003f8d77cccc0e6344f09666","origin":"The Stacks Project","memory_eligible":false,"source_rank":2836,"rank":2836,"depth":0,"x":697.552,"y":692.426,"cluster":"advanced-algebra"},{"id":"stacks:0D2H","tag":"0D2H","title":"Fibre products of rings, II · Lemma 0D2H","summary":"Let A, A', B, B', I, M, M', N, φ be as in Lemma [Tag 07RU]. If N finite over B and M' finite over A', then N' = N ×_φ, M M' is finite over B'.","statement_latex":"Let $A, A', B, B', I, M, M', N, \\varphi$ be as in\nLemma \\ref{lemma-module-over-fibre-product}.\nIf $N$ finite over $B$ and $M'$ finite over $A'$, then\n$N' = N \\times_{\\varphi, M} M'$ is finite over $B'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2H","source_file":"more-algebra.tex","source_line":774,"source_end_line":780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L774-L780","statement_sha256":"ff884ae080d0367ebaadc279a898de32a8027d5b89be7aabf9203a856ab8ff5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2837,"rank":2837,"depth":3,"x":656.38,"y":680.82,"cluster":"advanced-algebra"},{"id":"stacks:0D2I","tag":"0D2I","title":"Fibre products of rings, II · Lemma 0D2I","summary":"With A, A', B, B', I as in Situation [Tag 08KH]. • Let (N, M', φ) be an object of Mod_B ×_Mod_A Mod_A'. If M' is flat over A' and N is flat over B, then N' = N ×_φ, M M' is flat over B'. • If L' is a flat B'-module, then L' = (L ⊗_B' B) ×_(L ⊗_B' A) (L ⊗_B' A'). • The category of flat B'-modules is equivalent to the full subcategory of Mod_B ×_Mod_A Mod_A' consisting of triples (N, M', φ) with N flat over B and M' flat over A'.","statement_latex":"With $A, A', B, B', I$ as in\nSituation \\ref{situation-module-over-fibre-product}.\n\\begin{enumerate}\n\\item Let $(N, M', \\varphi)$ be an object of\n$\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$.\nIf $M'$ is flat over $A'$ and $N$ is flat over $B$, then\n$N' = N \\times_{\\varphi, M} M'$ is flat over $B'$.\n\\item If $L'$ is a flat $B'$-module, then\n$L' = (L \\otimes_{B'} B) \\times_{(L \\otimes_{B'} A)} (L \\otimes_{B'} A')$.\n\\item The category of flat $B'$-modules is equivalent to the\nfull subcategory of $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$\nconsisting of triples\n$(N, M', \\varphi)$ with $N$ flat over $B$ and $M'$ flat over $A'$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2I","source_file":"more-algebra.tex","source_line":806,"source_end_line":822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L806-L822","statement_sha256":"8746c6ffaa395823c039750810f4c6c5b86c8694bc88f5229e62420ca7cc5dad","origin":"The Stacks Project","memory_eligible":false,"source_rank":2838,"rank":2838,"depth":3,"x":697.229,"y":665.598,"cluster":"advanced-algebra"},{"id":"stacks:0D2J","tag":"0D2J","title":"Fibre products of rings, II · Lemma 0D2J","summary":"Let A, A', B, B', I be as in Situation [Tag 08KH]. The category of finite projective B'-modules is equivalent to the full subcategory of Mod_B ×_Mod_A Mod_A' consisting of triples (N, M', φ) with N finite projective over B and M' finite projective over A'.","statement_latex":"Let $A, A', B, B', I$ be as in\nSituation \\ref{situation-module-over-fibre-product}.\nThe category of finite projective $B'$-modules\nis equivalent to the full subcategory\nof $\\text{Mod}_B \\times_{\\text{Mod}_A} \\text{Mod}_{A'}$\nconsisting of triples $(N, M', \\varphi)$\nwith $N$ finite projective over $B$ and $M'$ finite projective over $A'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2J","source_file":"more-algebra.tex","source_line":891,"source_end_line":900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L891-L900","statement_sha256":"8a0cb5ebcc1e230215f1cc546485243a8d52b3d1309accfc9caf6e27b525f966","origin":"The Stacks Project","memory_eligible":false,"source_rank":2839,"rank":2839,"depth":5,"x":678.847,"y":700.939,"cluster":"advanced-algebra"},{"id":"stacks:08KM","tag":"08KM","title":"Fibre products of rings, III · Lemma 08KM","summary":"In Situation [Tag 08KK] the functor ([Tag 08KL]) has a right adjoint, namely the functor F : (N, M', φ) ↦ N ×_φ, M M' where M = M'/IM'. Moreover, the composition of F with ([Tag 08KL]) is the identity functor on Mod_D ×_Mod_C Mod_C'. In other words, setting N' = N ×_φ, M M' we have N' ⊗_D' D = N and N' ⊗_D' C' = M'.","statement_latex":"In Situation \\ref{situation-relative-module-over-fibre-product}\nthe functor (\\ref{equation-relative-functor}) has a right adjoint, namely\nthe functor\n$$\nF : (N, M', \\varphi) \\longmapsto N \\times_{\\varphi, M} M'\n$$\nwhere $M = M'/IM'$. Moreover, the composition of $F$ with\n(\\ref{equation-relative-functor}) is the identity functor on\n$\\text{Mod}_D \\times_{\\text{Mod}_C} \\text{Mod}_{C'}$. In other words,\nsetting $N' = N \\times_{\\varphi, M} M'$ we have\n$N' \\otimes_{D'} D = N$ and $N' \\otimes_{D'} C' = M'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KM","source_file":"more-algebra.tex","source_line":1001,"source_end_line":1014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1001-L1014","statement_sha256":"04636d577bf165cc1e499a254d8b3c7247d210e7b105cf7cd1071aaf755cf6e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2840,"rank":2840,"depth":3,"x":663.607,"y":663.498,"cluster":"advanced-algebra"},{"id":"stacks:08KN","tag":"08KN","title":"Fibre products of rings, III · Lemma 08KN","summary":"In Situation [Tag 08KK] the map JD' → IC' is surjective where J = Ker(B' → B).","statement_latex":"In Situation \\ref{situation-relative-module-over-fibre-product}\nthe map $JD' \\to IC'$ is surjective where $J = \\Ker(B' \\to B)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KN","source_file":"more-algebra.tex","source_line":1027,"source_end_line":1031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1027-L1031","statement_sha256":"29908a09659f5bed7cf7c20144e3a2125831d86972044cbb2f393cdb7c2f7120","origin":"The Stacks Project","memory_eligible":false,"source_rank":2841,"rank":2841,"depth":0,"x":705.969,"y":682.935,"cluster":"advanced-algebra"},{"id":"stacks:08IH","tag":"08IH","title":"Fibre products of rings, III · Lemma 08IH","summary":"Let A, A', B, B', C, C', D, D', I, M', M, N, φ be as in Lemma [Tag 08KM]. If N finite over D and M' finite over C', then N' = N ×_φ, M M' is finite over D'.","statement_latex":"Let $A, A', B, B', C, C', D, D', I, M', M, N, \\varphi$ be as in\nLemma \\ref{lemma-relative-module-over-fibre-product}.\nIf $N$ finite over $D$ and $M'$ finite over $C'$, then\n$N' = N \\times_{\\varphi, M} M'$ is finite over $D'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IH","source_file":"more-algebra.tex","source_line":1039,"source_end_line":1045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1039-L1045","statement_sha256":"911443784cad3b7e916e3b01ea5cde437a578bc464f752a38085fb1540725152","origin":"The Stacks Project","memory_eligible":false,"source_rank":2842,"rank":2842,"depth":4,"x":657.997,"y":692.86,"cluster":"advanced-algebra"},{"id":"stacks:07RW","tag":"07RW","title":"Fibre products of rings, III · Lemma 07RW","summary":"With A, A', B, B', C, C', D, D', I as in Situation [Tag 08KK]. • Let (N, M', φ) be an object of Mod_D ×_Mod_C Mod_C'. If M' is flat over A' and N is flat over B, then N' = N ×_φ, M M' is flat over B'. • If L' is a D'-module flat over B', then L' = (L ⊗_D' D) ×_(L ⊗_D' C) (L ⊗_D' C'). • The category of D'-modules flat over B' is equivalent to the categories of objects (N, M', φ) of Mod_D ×_Mod_C Mod_C' with N flat over B and M' flat over A'.","statement_latex":"With $A, A', B, B', C, C', D, D', I$ as in\nSituation \\ref{situation-relative-module-over-fibre-product}.\n\\begin{enumerate}\n\\item Let $(N, M', \\varphi)$ be an object of\n$\\text{Mod}_D \\times_{\\text{Mod}_C} \\text{Mod}_{C'}$.\nIf $M'$ is flat over $A'$ and $N$ is flat over $B$, then\n$N' = N \\times_{\\varphi, M} M'$ is flat over $B'$.\n\\item If $L'$ is a $D'$-module flat over $B'$, then\n$L' = (L \\otimes_{D'} D) \\times_{(L \\otimes_{D'} C)} (L \\otimes_{D'} C')$.\n\\item The category of $D'$-modules flat over $B'$\nis equivalent to the categories of objects $(N, M', \\varphi)$\nof $\\text{Mod}_D \\times_{\\text{Mod}_C} \\text{Mod}_{C'}$\nwith $N$ flat over $B$ and $M'$ flat over $A'$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RW","source_file":"more-algebra.tex","source_line":1058,"source_end_line":1074,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1058-L1074","statement_sha256":"4ee51c89666d5b67715ab431a757c46f000b514ce302944bb321ac00268e4387","origin":"The Stacks Project","memory_eligible":false,"source_rank":2843,"rank":2843,"depth":4,"x":686.013,"y":657.55,"cluster":"advanced-algebra"},{"id":"stacks:08KP","tag":"08KP","title":"Fibre products of rings, III · Lemma 08KP","summary":"Let A, A', B, B', C, C', D, D', I, M', M, N, φ be as in Lemma [Tag 08KM]. If • N is finitely presented over D and flat over B, • M' finitely presented over C' and flat over A', and • the ring map B' → D' factors as B' → D\" → D' with B' → D\" flat and D\" → D' of finite presentation, then N' = N ×_M M' is finitely presented over D'.","statement_latex":"Let $A, A', B, B', C, C', D, D', I, M', M, N, \\varphi$ be as in\nLemma \\ref{lemma-relative-module-over-fibre-product}. If\n\\begin{enumerate}\n\\item $N$ is finitely presented over $D$ and flat over $B$,\n\\item $M'$ finitely presented over $C'$ and flat over $A'$, and\n\\item the ring map $B' \\to D'$ factors as $B' \\to D'' \\to D'$\nwith $B' \\to D''$ flat and $D'' \\to D'$ of finite presentation,\n\\end{enumerate}\nthen $N' = N \\times_M M'$ is finitely presented over $D'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KP","source_file":"more-algebra.tex","source_line":1098,"source_end_line":1109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1098-L1109","statement_sha256":"80bc010eda7af2cfba7fdb45cf6c0108a25dff5c0c8b96008761ed00d006fd6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2844,"rank":2844,"depth":5,"x":693.907,"y":700.386,"cluster":"advanced-algebra"},{"id":"stacks:08KQ","tag":"08KQ","title":"Fibre products of rings, III · Lemma 08KQ","summary":"Let A, A', B, B', I be as in Situation [Tag 08KH]. Let (D, C', φ) be a system consisting of an B-algebra D, a A'-algebra C' and an isomorphism D ⊗_B A → C'/IC' = C. Set D' = D ×_C C' (as in Lemma [Tag 07RU]). Then • B' → D' is finite type if and only if B → D and A' → C' are finite type, • B' → D' is flat if and only if B → D and A' → C' are flat, • B' → D' is flat and of finite presentation if and only if B → D and A' → C' are flat and of finite presentation, • B' → D'…","statement_latex":"Let $A, A', B, B', I$ be as in\nSituation \\ref{situation-module-over-fibre-product}.\nLet $(D, C', \\varphi)$ be a system consisting of an $B$-algebra $D$,\na $A'$-algebra $C'$ and an isomorphism $D \\otimes_B A \\to C'/IC' = C$.\nSet $D' = D \\times_C C'$ (as in\nLemma \\ref{lemma-module-over-fibre-product}). Then\n\\begin{enumerate}\n\\item $B' \\to D'$ is finite type if and only if $B \\to D$ and\n$A' \\to C'$ are finite type,\n\\item $B' \\to D'$ is flat if and only if $B \\to D$ and $A' \\to C'$ are flat,\n\\item $B' \\to D'$ is flat and of finite presentation if and only if\n$B \\to D$ and $A' \\to C'$ are flat and of finite presentation,\n\\item $B' \\to D'$ is smooth if and only if $B \\to D$ and $A' \\to C'$ are smooth,\n\\item $B' \\to D'$ is \\'etale if and only if $B \\to D$ and $A' \\to C'$\nare \\'etale.\n\\end{enumerate}\nMoreover, if $D'$ is a flat $B'$-algebra, then\n$D' \\to (D' \\otimes_{B'} B) \\times_{(D' \\otimes_{B'} A)} (D' \\otimes_{B'} A')$\nis an isomorphism. In this way the category of flat $B'$-algebras\nis equivalent to the categories of systems $(D, C', \\varphi)$ as above\nwith $D$ flat over $B$ and $C'$ flat over $A'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fibre products of rings, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KQ","source_file":"more-algebra.tex","source_line":1158,"source_end_line":1181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1158-L1181","statement_sha256":"03398882c2249487cd12c8e70285f227a05e99526b6c2c2d92df5a02c47922d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2845,"rank":2845,"depth":38,"x":652.814,"y":672.715,"cluster":"advanced-algebra"},{"id":"stacks:07Z7","tag":"07Z7","title":"Fitting ideals · Lemma 07Z7","summary":"Let R be a ring. Let A be an n × m matrix with coefficients in R. Let I_r(A) be the ideal generated by the r × r-minors of A with the convention that I_0(A) = R and I_r(A) = 0 if r > min(n, m). Then • I_0(A) ⊃ I_1(A) ⊃ I_2(A) ⊃ …, • if B is an (n + n') × m matrix, and A is the first n rows of B, then I_r + n'(B) ⊂ I_r(A), • if C is an n × n matrix then I_r(CA) ⊂ I_r(A). • If A is a block matrix ( A_1 & 0 0 & A_2 ) then I_r(A) = ∑_r_1 + r_2 = r I_r_1(A_1) I_r_2(A_2). • Add…","statement_latex":"Let $R$ be a ring. Let $A$ be an $n \\times m$ matrix with coefficients\nin $R$. Let $I_r(A)$ be the ideal generated by the $r \\times r$-minors\nof $A$ with the convention that $I_0(A) = R$ and $I_r(A) = 0$ if\n$r > \\min(n, m)$. Then\n\\begin{enumerate}\n\\item $I_0(A) \\supset I_1(A) \\supset I_2(A) \\supset \\ldots$,\n\\item if $B$ is an $(n + n') \\times m$ matrix, and $A$ is the first\n$n$ rows of $B$, then $I_{r + n'}(B) \\subset I_r(A)$,\n\\item if $C$ is an $n \\times n$ matrix then $I_r(CA) \\subset I_r(A)$.\n\\item If $A$ is a block matrix\n$$\n\\left(\n\\begin{matrix}\nA_1 & 0 \\\\\n0 & A_2 \n\\end{matrix}\n\\right)\n$$\nthen $I_r(A) = \\sum_{r_1 + r_2 = r} I_{r_1}(A_1) I_{r_2}(A_2)$.\n\\item Add more here.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Z7","source_file":"more-algebra.tex","source_line":1325,"source_end_line":1348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1325-L1348","statement_sha256":"b7b3595d7ebf49e278e78b563aea3bd48d2f0802288d40783522c9780d4cd6c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2846,"rank":2846,"depth":0,"x":706.408,"y":669.751,"cluster":"advanced-algebra"},{"id":"stacks:07Z8","tag":"07Z8","title":"Fitting ideals · Lemma 07Z8","summary":"Let R be a ring. Let M be a finite R-module. Choose a presentation bigoplus_j ∈ J R → R^⊕ n → M → 0. of M. Let A = (a_ij)_i = 1, …, n, j ∈ J be the matrix of the map bigoplus_j ∈ J R → R^⊕ n. The ideal Fit_k(M) generated by the (n - k) × (n - k) minors of A is independent of the choice of the presentation.","statement_latex":"Let $R$ be a ring. Let $M$ be a finite $R$-module. Choose a presentation\n$$\n\\bigoplus\\nolimits_{j \\in J} R \\longrightarrow R^{\\oplus n}\n\\longrightarrow M \\longrightarrow 0.\n$$\nof $M$. Let $A = (a_{ij})_{i = 1, \\ldots, n, j \\in J}$ be the matrix\nof the map $\\bigoplus_{j \\in J} R \\to R^{\\oplus n}$.\nThe ideal $\\text{Fit}_k(M)$ generated by the\n$(n - k) \\times (n - k)$ minors of\n$A$ is independent of the choice of the presentation.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Z8","source_file":"more-algebra.tex","source_line":1355,"source_end_line":1367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1355-L1367","statement_sha256":"9e308f0167b0d3906fae05adde00c37c77cec8b1442539f10a2f16d20f5e85aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":2847,"rank":2847,"depth":1,"x":668.559,"y":702.963,"cluster":"advanced-algebra"},{"id":"stacks:07Z9","tag":"07Z9","title":"Fitting ideals · Definition 07Z9","summary":"Let R be a ring. Let M be a finite R-module. Let k ≥ 0. The kth Fitting ideal of M is the ideal Fit_k(M) constructed in Lemma [Tag 07Z8]. Set Fit_-1(M) = 0.","statement_latex":"Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$.\nThe {\\it $k$th Fitting ideal} of $M$ is the ideal $\\text{Fit}_k(M)$\nconstructed in Lemma \\ref{lemma-fitting-ideal}. Set $\\text{Fit}_{-1}(M) = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fitting ideals","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Z9","source_file":"more-algebra.tex","source_line":1413,"source_end_line":1418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1413-L1418","statement_sha256":"0ea077a99d501404432af749ff7d24f630674b81fae61b8b24293941481013c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2848,"rank":2848,"depth":2,"x":669.789,"y":656.154,"cluster":"advanced-algebra"},{"id":"stacks:07ZA","tag":"07ZA","title":"Fitting ideals · Lemma 07ZA","summary":"Let R be a ring. Let M be a finite R-module. • If M can be generated by n elements, then Fit_n(M) = R. • Given a second finite R-module M' we have Fit_0(M ⊕ M') = Fit_0(M)Fit_0(M') and more generally Fit_l(M ⊕ M') = ∑_k + k' = l Fit_k(M)Fit_k'(M') • If R → R' is a ring map, then Fit_k(M ⊗_R R') is the ideal of R' generated by the image of Fit_k(M). • If M is of finite presentation, then Fit_k(M) is a finitely generated ideal. • If M → M' is a surjection, then Fit_k(M) ⊂…","statement_latex":"Let $R$ be a ring. Let $M$ be a finite $R$-module.\n\\begin{enumerate}\n\\item If $M$ can be generated by $n$ elements, then\n$\\text{Fit}_n(M) = R$.\n\\item Given a second finite $R$-module $M'$ we have\n$\\text{Fit}_0(M \\oplus M') = \\text{Fit}_0(M)\\text{Fit}_0(M')$ and\nmore generally\n$$\n\\text{Fit}_l(M \\oplus M') =\n\\sum\\nolimits_{k + k' = l} \\text{Fit}_k(M)\\text{Fit}_{k'}(M')\n$$\n\\item If $R \\to R'$ is a ring map, then $\\text{Fit}_k(M \\otimes_R R')$\nis the ideal of $R'$ generated by the image of $\\text{Fit}_k(M)$.\n\\item If $M$ is of finite presentation, then $\\text{Fit}_k(M)$\nis a finitely generated ideal.\n\\item If $M \\to M'$ is a surjection, then\n$\\text{Fit}_k(M) \\subset \\text{Fit}_k(M')$.\n\\item We have $\\text{Fit}_0(M) \\subset \\text{Ann}_R(M)$.\n\\item We have $V(\\text{Fit}_0(M)) = \\text{Supp}(M)$.\n\\item If $I$ is an ideal of $R$, then $\\text{Fit}_0(R/I) = I$.\n\\item Add more here.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZA","source_file":"more-algebra.tex","source_line":1431,"source_end_line":1455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1431-L1455","statement_sha256":"eaa1a4e9b605dcba1c08a9fbe472bd33d383c5aaf1feceecce32895778ae0dd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2849,"rank":2849,"depth":3,"x":707.169,"y":691.995,"cluster":"advanced-algebra"},{"id":"stacks:07ZC","tag":"07ZC","title":"Fitting ideals · Lemma 07ZC","summary":"Let R be a ring. Let M be a finite R-module. Let k ≥ 0. Let p ⊂ R be a prime ideal. The following are equivalent • Fit_k(M) not ⊂ p, • dim_kappa( p) M ⊗_R kappa( p) ≤ k, • M_ p can be generated by k elements over R_ p, and • M_f can be generated by k elements over R_f for some f ∈ R, f not ∈ p.","statement_latex":"Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$.\nLet $\\mathfrak p \\subset R$ be a prime ideal. The following\nare equivalent\n\\begin{enumerate}\n\\item $\\text{Fit}_k(M) \\not \\subset \\mathfrak p$,\n\\item $\\dim_{\\kappa(\\mathfrak p)} M \\otimes_R \\kappa(\\mathfrak p) \\leq k$,\n\\item $M_\\mathfrak p$ can be generated by $k$ elements over $R_\\mathfrak p$, and\n\\item $M_f$ can be generated by $k$ elements over $R_f$\nfor some $f \\in R$, $f \\not \\in \\mathfrak p$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZC","source_file":"more-algebra.tex","source_line":1523,"source_end_line":1535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1523-L1535","statement_sha256":"dd272e63d1b19d6ca112a9b052073ed64574e0a3f4634632f54a65c9652c4066","origin":"The Stacks Project","memory_eligible":false,"source_rank":2850,"rank":2850,"depth":4,"x":649.822,"y":686.682,"cluster":"advanced-algebra"},{"id":"stacks:07ZD","tag":"07ZD","title":"Fitting ideals · Lemma 07ZD","summary":"Let R be a ring. Let M be a finite R-module. Let r ≥ 0. The following are equivalent • M is finite locally free of rank r (Algebra, Definition [Tag 00NW]), • Fit_r - 1(M) = 0 and Fit_r(M) = R, and • Fit_k(M) = 0 for k < r and Fit_k(M) = R for k ≥ r.","statement_latex":"Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $r \\geq 0$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ is finite locally free of rank $r$\n(Algebra, Definition \\ref{algebra-definition-locally-free}),\n\\item $\\text{Fit}_{r - 1}(M) = 0$ and $\\text{Fit}_r(M) = R$, and\n\\item $\\text{Fit}_k(M) = 0$ for $k < r$ and $\\text{Fit}_k(M) = R$\nfor $k \\geq r$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZD","source_file":"more-algebra.tex","source_line":1549,"source_end_line":1560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1549-L1560","statement_sha256":"08aafc28409fe1ec22ac10924ca43c28c94c99bce576d63e5da7798af928355e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2851,"rank":2851,"depth":5,"x":697.153,"y":657.591,"cluster":"advanced-algebra"},{"id":"stacks:080Z","tag":"080Z","title":"Fitting ideals · Lemma 080Z","summary":"Let R be a local ring. Let M be a finite R-module. Let k ≥ 0. Assume that Fit_k(M) = (f) for some f ∈ R. Let M' be the quotient of M by (x ∈ M mid fx = 0). Then M' can be generated by k elements.","statement_latex":"Let $R$ be a local ring. Let $M$ be a finite $R$-module. Let $k \\geq 0$.\nAssume that $\\text{Fit}_k(M) = (f)$ for some $f \\in R$.\nLet $M'$ be the quotient of $M$ by $\\{x \\in M \\mid fx = 0\\}$. Then\n$M'$ can be generated by $k$ elements.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080Z","source_file":"more-algebra.tex","source_line":1579,"source_end_line":1585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1579-L1585","statement_sha256":"5bec556f5c57f43d356a4dca6827546ab0c31245882081a26f3348e6984b0750","origin":"The Stacks Project","memory_eligible":false,"source_rank":2852,"rank":2852,"depth":3,"x":685.457,"y":706.67,"cluster":"advanced-algebra"},{"id":"stacks:0F7M","tag":"0F7M","title":"Fitting ideals · Lemma 0F7M","summary":"Let R be a ring. Let M be a finitely presented R-module. Let k ≥ 0. Assume that Fit_k(M) = (f) for some nonzerodivisor f ∈ R and Fit_k - 1(M) = 0. Then • M has projective dimension ≤ 1, • M' = Ker(f : M → M) is the f-power torsion submodule of M, • M' has projective dimension ≤ 1, • M/M' is finite locally free of rank k, and • M ≅ M/M' ⊕ M'.","statement_latex":"Let $R$ be a ring. Let $M$ be a finitely presented $R$-module. Let $k \\geq 0$.\nAssume that $\\text{Fit}_k(M) = (f)$ for some nonzerodivisor $f \\in R$\nand $\\text{Fit}_{k - 1}(M) = 0$. Then\n\\begin{enumerate}\n\\item $M$ has projective dimension $\\leq 1$,\n\\item $M' = \\Ker(f : M \\to M)$ is the $f$-power torsion submodule of $M$,\n\\item $M'$ has projective dimension $\\leq 1$,\n\\item $M/M'$ is finite locally free of rank $k$, and\n\\item $M \\cong M/M' \\oplus M'$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7M","source_file":"more-algebra.tex","source_line":1615,"source_end_line":1627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1615-L1627","statement_sha256":"49387de517669a7035f7a2c2d826e6bd3930a8b883129d894eb6d62691e7b723","origin":"The Stacks Project","memory_eligible":false,"source_rank":2853,"rank":2853,"depth":5,"x":654.14,"y":663.177,"cluster":"advanced-algebra"},{"id":"stacks:07LX","tag":"07LX","title":"Lifting · Lemma 07LX","summary":"Let A be a ring, let I ⊂ A be an ideal, let overlineu ∈ A/I be an invertible element. There exists an étale ring map A → A' which induces an isomorphism A/I → A'/IA' and an invertible element u' ∈ A' lifting overlineu.","statement_latex":"Let $A$ be a ring, let $I \\subset A$ be an ideal, let $\\overline{u} \\in A/I$\nbe an invertible element. There exists an \\'etale ring map $A \\to A'$ which\ninduces an isomorphism $A/I \\to A'/IA'$ and an invertible element $u' \\in A'$\nlifting $\\overline{u}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LX","source_file":"more-algebra.tex","source_line":1771,"source_end_line":1777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1771-L1777","statement_sha256":"092d70b429084e41f585fddc36eae21076c3dc3aeb38aa249d29a9b3b08d11ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":2854,"rank":2854,"depth":0,"x":713.079,"y":677.698,"cluster":"advanced-algebra"},{"id":"stacks:07LY","tag":"07LY","title":"Lifting · Lemma 07LY","summary":"Let A be a ring, let I ⊂ A be an ideal, let overlinee ∈ A/I be an idempotent. There exists an étale ring map A → A' which induces an isomorphism A/I → A'/IA' and an idempotent e' ∈ A' lifting overlinee.","statement_latex":"Let $A$ be a ring, let $I \\subset A$ be an ideal, let $\\overline{e} \\in A/I$\nbe an idempotent. There exists an \\'etale ring map $A \\to A'$ which\ninduces an isomorphism $A/I \\to A'/IA'$ and an idempotent $e' \\in A'$\nlifting $\\overline{e}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LY","source_file":"more-algebra.tex","source_line":1784,"source_end_line":1790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1784-L1790","statement_sha256":"fd7aac03acebee162002b91bf09658525715124920c2e328c4ec8b41fc89bbc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2855,"rank":2855,"depth":0,"x":657.135,"y":700.761,"cluster":"advanced-algebra"},{"id":"stacks:07LZ","tag":"07LZ","title":"Lifting · Lemma 07LZ","summary":"Let A be a ring, let I ⊂ A be an ideal. Let Spec(A/I) = coprod_j ∈ J overlineU_j be a finite disjoint open covering. Then there exists an étale ring map A → A' which induces an isomorphism A/I → A'/IA' and a finite disjoint open covering Spec(A') = coprod_j ∈ J U'_j lifting the given covering.","statement_latex":"Let $A$ be a ring, let $I \\subset A$ be an ideal. Let\n$\\Spec(A/I) = \\coprod_{j \\in J} \\overline{U}_j$ be a finite disjoint open\ncovering. Then there exists an \\'etale ring map $A \\to A'$ which\ninduces an isomorphism $A/I \\to A'/IA'$ and a finite disjoint open covering\n$\\Spec(A') = \\coprod_{j \\in J} U'_j$ lifting the given covering.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LZ","source_file":"more-algebra.tex","source_line":1805,"source_end_line":1812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1805-L1812","statement_sha256":"30f7e0aa6e4df9fdfb72644be0780fe7d8a2d66b56ec650d47b148cb877975c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":2856,"rank":2856,"depth":3,"x":680.167,"y":651.322,"cluster":"advanced-algebra"},{"id":"stacks:07M0","tag":"07M0","title":"Lifting · Lemma 07M0","summary":"Let A → B be a ring map and J ⊂ B an ideal. If A → B is étale at every prime of V(J), then there exists a g ∈ B mapping to an invertible element of B/J such that A' = B_g is étale over A.","statement_latex":"Let $A \\to B$ be a ring map and $J \\subset B$ an ideal. If\n$A \\to B$ is \\'etale at every prime of $V(J)$, then there exists\na $g \\in B$ mapping to an invertible element\nof $B/J$ such that $A' = B_g$ is \\'etale over $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07M0","source_file":"more-algebra.tex","source_line":1820,"source_end_line":1826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1820-L1826","statement_sha256":"c851ec514be00efd137916064f2bad5618e06b61abf9bc80ce44e80865f822eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2857,"rank":2857,"depth":1,"x":703.251,"y":701.53,"cluster":"advanced-algebra"},{"id":"stacks:0ALH","tag":"0ALH","title":"Lifting · Lemma 0ALH","summary":"Let A be a ring, let I ⊂ A be an ideal. Let f ∈ A[x] be a monic polynomial. Let overlinef = overlineg overlineh be a factorization of f in A/I[x] such that overlineg and overlineh are monic and generate the unit ideal in A/I[x]. Then there exists an étale ring map A → A' which induces an isomorphism A/I → A'/IA' and a factorization f = g' h' in A'[x] with g', h' monic lifting the given factorization over A/I.","statement_latex":"Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $f \\in A[x]$ be a\nmonic polynomial. Let $\\overline{f} = \\overline{g} \\overline{h}$ be a\nfactorization of $f$ in $A/I[x]$ such that $\\overline{g}$ and $\\overline{h}$\nare monic and generate the unit ideal in $A/I[x]$. Then there exists an\n\\'etale ring map $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$\nand a factorization $f = g' h'$ in $A'[x]$ with $g'$, $h'$ monic\nlifting the given factorization over $A/I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALH","source_file":"more-algebra.tex","source_line":1843,"source_end_line":1852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1843-L1852","statement_sha256":"a2f074735a27e002896ed0ec452cb3a5822f86cdb231b617dded82a1c0b3a277","origin":"The Stacks Project","memory_eligible":false,"source_rank":2858,"rank":2858,"depth":42,"x":645.086,"y":677.282,"cluster":"advanced-algebra"},{"id":"stacks:07M1","tag":"07M1","title":"Lifting · Lemma 07M1","summary":"Let A be a ring, let I ⊂ A be an ideal. Let f ∈ A[x] be a monic polynomial. Let overlinef = overlineg overlineh be a factorization of f in A/I[x] and assume • the leading coefficient of overlineg is an invertible element of A/I, and • overlineg, overlineh generate the unit ideal in A/I[x]. Then there exists an étale ring map A → A' which induces an isomorphism A/I → A'/IA' and a factorization f = g' h' in A'[x] lifting the given factorization over A/I.","statement_latex":"Let $A$ be a ring, let $I \\subset A$ be an ideal. Let $f \\in A[x]$ be a\nmonic polynomial. Let $\\overline{f} = \\overline{g} \\overline{h}$ be a\nfactorization of $f$ in $A/I[x]$ and assume\n\\begin{enumerate}\n\\item the leading coefficient of $\\overline{g}$ is an invertible element\nof $A/I$, and\n\\item $\\overline{g}$, $\\overline{h}$ generate the unit ideal in $A/I[x]$.\n\\end{enumerate}\nThen there exists an \\'etale ring map $A \\to A'$ which induces an\nisomorphism $A/I \\to A'/IA'$ and a factorization $f = g' h'$ in $A'[x]$\nlifting the given factorization over $A/I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07M1","source_file":"more-algebra.tex","source_line":1899,"source_end_line":1912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1899-L1912","statement_sha256":"3fba69a94b97e1f2cfacb2ec20cb7915278e63c7ec90c99b87eada4ca11fce65","origin":"The Stacks Project","memory_eligible":false,"source_rank":2859,"rank":2859,"depth":43,"x":708.291,"y":661.964,"cluster":"advanced-algebra"},{"id":"stacks:07M3","tag":"07M3","title":"Lifting · Lemma 07M3","summary":"Let R → S be a ring map. Let I ⊂ R be an ideal of R and let J ⊂ S be an ideal of S. If the closure of the image of V(J) in Spec(R) is disjoint from V(I), then there exists an element f ∈ R which maps to 1 in R/I and to an element of J in S.","statement_latex":"Let $R \\to S$ be a ring map. Let $I \\subset R$ be an ideal of $R$\nand let $J \\subset S$ be an ideal of $S$. If the closure of the image\nof $V(J)$ in $\\Spec(R)$ is disjoint from $V(I)$, then there exists\nan element $f \\in R$ which maps to $1$ in $R/I$ and to an element\nof $J$ in $S$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07M3","source_file":"more-algebra.tex","source_line":1938,"source_end_line":1945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1938-L1945","statement_sha256":"63a9f92321b9ce2fbf93c8340dcb103a27c594f3f12021800b038c99648ccf99","origin":"The Stacks Project","memory_eligible":false,"source_rank":2860,"rank":2860,"depth":1,"x":673.563,"y":709.721,"cluster":"advanced-algebra"},{"id":"stacks:09XG","tag":"09XG","title":"Lifting · Lemma 09XG","summary":"Let I be an ideal of a ring A. Let A → B be an integral ring map. Let b ∈ B map to an idempotent in B/IB. Then there exists a monic f ∈ A[x] with f(b) = 0 and f bmod I = x^d(x - 1)^d for some d ≥ 1.","statement_latex":"Let $I$ be an ideal of a ring $A$. Let $A \\to B$ be an integral ring map.\nLet $b \\in B$ map to an idempotent in $B/IB$. Then there exists a\nmonic $f \\in A[x]$ with $f(b) = 0$ and $f \\bmod I = x^d(x - 1)^d$\nfor some $d \\geq 1$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XG","source_file":"more-algebra.tex","source_line":1959,"source_end_line":1965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1959-L1965","statement_sha256":"3916bc111d76e8b3696df31c97cb01bafe51b241c612a5f6057221a542fea5ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":2861,"rank":2861,"depth":8,"x":660.611,"y":654.119,"cluster":"advanced-algebra"},{"id":"stacks:07M4","tag":"07M4","title":"Lifting · Lemma 07M4","summary":"Let A be a ring, let I ⊂ A be an ideal. Let A → B be an integral ring map. Let overlinee ∈ B/IB be an idempotent. Then there exists an étale ring map A → A' which induces an isomorphism A/I → A'/IA' and an idempotent e' ∈ B ⊗_A A' lifting overlinee.","statement_latex":"Let $A$ be a ring, let $I \\subset A$ be an ideal.\nLet $A \\to B$ be an integral ring map.\nLet $\\overline{e} \\in B/IB$ be an idempotent.\nThen there exists an \\'etale ring map $A \\to A'$\nwhich induces an isomorphism $A/I \\to A'/IA'$ and an idempotent\n$e' \\in B \\otimes_A A'$ lifting $\\overline{e}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07M4","source_file":"more-algebra.tex","source_line":1977,"source_end_line":1985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L1977-L1985","statement_sha256":"7b427cc699f8bc797da8a4e35ae4359ff0099e9100601fcf8a2d465173f48b0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2862,"rank":2862,"depth":44,"x":715.53,"y":688.179,"cluster":"advanced-algebra"},{"id":"stacks:07M5","tag":"07M5","title":"Lifting · Lemma 07M5","summary":"Let A be a ring, let I ⊂ A be an ideal. Let overlineP be a finite projective A/I-module. Then there exists an étale ring map A → A' which induces an isomorphism A/I → A'/IA' and a finite projective A'-module P' lifting overlineP.","statement_latex":"Let $A$ be a ring, let $I \\subset A$ be an ideal.\nLet $\\overline{P}$ be a finite projective $A/I$-module.\nThen there exists an \\'etale ring map $A \\to A'$ which induces\nan isomorphism $A/I \\to A'/IA'$ and a finite projective\n$A'$-module $P'$ lifting $\\overline{P}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07M5","source_file":"more-algebra.tex","source_line":2019,"source_end_line":2026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2019-L2026","statement_sha256":"1668b48899f81bb3993dafc37057bcc95e79a2c05ad9befede0667e186fa1213","origin":"The Stacks Project","memory_eligible":false,"source_rank":2863,"rank":2863,"depth":45,"x":646.84,"y":694.294,"cluster":"advanced-algebra"},{"id":"stacks:07EV","tag":"07EV","title":"Lifting · Lemma 07EV","summary":"Let A be a ring. Let 0 → K → A^⊕ m → M → 0 be a sequence of A-modules. Consider the A-algebra C = Sym^*_A(M) with its presentation α : A[y_1, …, y_m] → C coming from the surjection A^⊕ m → M. Then NL(α) = (K ⊗_A C → bigoplus_j = 1, …, m C dy_j) (see Algebra, Section [Tag 00S0]) in particular Ω_C/A = M ⊗_A C.","statement_latex":"Let $A$ be a ring. Let $0 \\to K \\to A^{\\oplus m} \\to M \\to 0$\nbe a sequence of $A$-modules. Consider the $A$-algebra\n$C = \\text{Sym}^*_A(M)$ with its presentation\n$\\alpha : A[y_1, \\ldots, y_m] \\to C$\ncoming from the surjection $A^{\\oplus m} \\to M$. Then\n$$\n\\NL(\\alpha) =\n(K \\otimes_A C \\to \\bigoplus\\nolimits_{j = 1, \\ldots, m} C \\text{d}y_j)\n$$\n(see Algebra, Section \\ref{algebra-section-netherlander})\nin particular $\\Omega_{C/A} = M \\otimes_A C$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EV","source_file":"more-algebra.tex","source_line":2066,"source_end_line":2079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2066-L2079","statement_sha256":"aca50dc89a2259885b8be0684c35541c8f06ca62ecefe6167d195e8e28a53467","origin":"The Stacks Project","memory_eligible":false,"source_rank":2864,"rank":2864,"depth":1,"x":693.104,"y":650.309,"cluster":"advanced-algebra"},{"id":"stacks:07M6","tag":"07M6","title":"Lifting · Lemma 07M6","summary":"Let A be a ring. Let M be an A-module. Then C = Sym_A^*(M) is smooth over A if and only if M is a finite projective A-module.","statement_latex":"Let $A$ be a ring. Let $M$ be an $A$-module. Then $C = \\text{Sym}_A^*(M)$\nis smooth over $A$ if and only if $M$ is a finite projective $A$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07M6","source_file":"more-algebra.tex","source_line":2111,"source_end_line":2115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2111-L2115","statement_sha256":"0c7e15158f5b4d567c98282b5abc15f9dd72fba3f12d8ea586371573fcafe8a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2865,"rank":2865,"depth":7,"x":694.372,"y":709.656,"cluster":"advanced-algebra"},{"id":"stacks:07M7","tag":"07M7","title":"Lifting · Lemma 07M7","summary":"Let A be a ring, let I ⊂ A be an ideal. Consider a commutative diagram xymatrix B ar[rd] A ar[u] ar[r] & A/I where B is a smooth A-algebra. Then there exists an étale ring map A → A' which induces an isomorphism A/I → A'/IA' and an A-algebra map B → A' lifting the ring map B → A/I.","statement_latex":"Let $A$ be a ring, let $I \\subset A$ be an ideal. Consider a commutative\ndiagram\n$$\n\\xymatrix{\nB \\ar[rd] \\\\\nA \\ar[u] \\ar[r] & A/I\n}\n$$\nwhere $B$ is a smooth $A$-algebra. Then there exists an \\'etale ring\nmap $A \\to A'$ which induces an isomorphism $A/I \\to A'/IA'$ and an\n$A$-algebra map $B \\to A'$ lifting the ring map $B \\to A/I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07M7","source_file":"more-algebra.tex","source_line":2138,"source_end_line":2151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2138-L2151","statement_sha256":"f29a9ef0e1563e9d04c5f7e1d13aa1bf882450da708ae73e1737de4a77e08348","origin":"The Stacks Project","memory_eligible":false,"source_rank":2866,"rank":2866,"depth":46,"x":645.178,"y":666.138,"cluster":"advanced-algebra"},{"id":"stacks:0ELY","tag":"0ELY","title":"Zariski pairs · Definition 0ELY","summary":"A Zariski pair is a pair (A, I) such that I is contained in the Jacobson radical of A.","statement_latex":"A {\\it Zariski pair} is a pair $(A, I)$ such that\n$I$ is contained in the Jacobson radical of $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Zariski pairs","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ELY","source_file":"more-algebra.tex","source_line":2245,"source_end_line":2249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2245-L2249","statement_sha256":"da984a842544275b0b3929963c2ae489393003ce0f5293b93c6c1e72811a2cdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2867,"rank":2867,"depth":0,"x":717.218,"y":670.361,"cluster":"advanced-algebra"},{"id":"stacks:09XF","tag":"09XF","title":"Zariski pairs · Lemma 09XF","summary":"Let (A, I) be a Zariski pair. Then the map from idempotents of A to idempotents of A/I is injective.","statement_latex":"Let $(A, I)$ be a Zariski pair. Then the map from\nidempotents of $A$ to idempotents of $A/I$ is injective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Zariski pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XF","source_file":"more-algebra.tex","source_line":2251,"source_end_line":2255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2251-L2255","statement_sha256":"2cbe407e21c3680853d44385874a6c00fb03c7f23bfe621739dc4e9dd5d28fd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2868,"rank":2868,"depth":2,"x":660.101,"y":708.521,"cluster":"advanced-algebra"},{"id":"stacks:0ELZ","tag":"0ELZ","title":"Zariski pairs · Lemma 0ELZ","summary":"Let (A, I) be a Zariski pair. Let A → B be a flat, integral, finitely presented ring map such that A/I → B/IB is an isomorphism. Then A → B is an isomorphism.","statement_latex":"Let $(A, I)$ be a Zariski pair. Let $A \\to B$ be a flat,\nintegral, finitely presented ring map such that $A/I \\to B/IB$\nis an isomorphism. Then $A \\to B$ is an isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Zariski pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ELZ","source_file":"more-algebra.tex","source_line":2264,"source_end_line":2269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2264-L2269","statement_sha256":"f6bc1e57c6662739f64e12c0f68f267bafc294f520c0cf7a653a7374f4028959","origin":"The Stacks Project","memory_eligible":false,"source_rank":2869,"rank":2869,"depth":6,"x":671.653,"y":647.347,"cluster":"advanced-algebra"},{"id":"stacks:0EM0","tag":"0EM0","title":"Zariski pairs · Lemma 0EM0","summary":"Let (A, I) be a Zariski pair. Let A → B be a finite ring map. Assume • B/IB = B_1 × B_2 is a product of A/I-algebras • A/I → B_1/IB_1 is surjective, • b ∈ B maps to (1, 0) in the product. Then there exists a monic f ∈ A[x] with f(b) = 0 and f bmod I = (x - 1)x^d for some d ≥ 1.","statement_latex":"Let $(A, I)$ be a Zariski pair. Let $A \\to B$ be a finite ring map.\nAssume\n\\begin{enumerate}\n\\item $B/IB = B_1 \\times B_2$ is a product of $A/I$-algebras\n\\item $A/I \\to B_1/IB_1$ is surjective,\n\\item $b \\in B$ maps to $(1, 0)$ in the product.\n\\end{enumerate}\nThen there exists a monic $f \\in A[x]$ with $f(b) = 0$ and\n$f \\bmod I = (x - 1)x^d$ for some $d \\geq 1$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Zariski pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EM0","source_file":"more-algebra.tex","source_line":2289,"source_end_line":2300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2289-L2300","statement_sha256":"e8f227fb6c00baba965812cf851a5ee204bc1c3f47ef993b1df05e36870dc29e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2870,"rank":2870,"depth":45,"x":712.739,"y":699.535,"cluster":"advanced-algebra"},{"id":"stacks:0GED","tag":"0GED","title":"Zariski pairs · Lemma 0GED","summary":"Let (A, I) be a Zariski pair with A Noetherian. Let f ∈ I. Then A_f is a Jacobson ring.","statement_latex":"Let $(A, I)$ be a Zariski pair with $A$ Noetherian. Let $f \\in I$.\nThen $A_f$ is a Jacobson ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Zariski pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GED","source_file":"more-algebra.tex","source_line":2346,"source_end_line":2350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2346-L2350","statement_sha256":"11a891899afe219d6e772b30e21d77b011514450be2ac9e99ba029e3750427cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2871,"rank":2871,"depth":14,"x":639.757,"y":684.213,"cluster":"advanced-algebra"},{"id":"stacks:09XE","tag":"09XE","title":"Henselian pairs · Definition 09XE","summary":"A henselian pair is a pair (A, I) satisfying • I is contained in the Jacobson radical of A, and • for any monic polynomial f ∈ A[T] and factorization overlinef = g_0h_0 with g_0, h_0 ∈ A/I[T] monic generating the unit ideal in A/I[T], there exists a factorization f = gh in A[T] with g, h monic and g_0 = overlineg and h_0 = overlineh.","statement_latex":"A {\\it henselian pair} is a pair $(A, I)$ satisfying\n\\begin{enumerate}\n\\item $I$ is contained in the Jacobson radical of $A$, and\n\\item for any monic polynomial $f \\in A[T]$ and factorization\n$\\overline{f} = g_0h_0$ with $g_0, h_0 \\in A/I[T]$ monic\ngenerating the unit ideal in $A/I[T]$, there\nexists a factorization $f = gh$ in $A[T]$ with $g, h$ monic\nand $g_0 = \\overline{g}$ and $h_0 = \\overline{h}$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XE","source_file":"more-algebra.tex","source_line":2393,"source_end_line":2404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2393-L2404","statement_sha256":"7a426dc8364e0d7790994ff1e04fda744bd21341b7685cadbd87ede8d63763e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2872,"rank":2872,"depth":0,"x":706.54,"y":653.809,"cluster":"advanced-algebra"},{"id":"stacks:0ALI","tag":"0ALI","title":"Henselian pairs · Lemma 0ALI","summary":"Let (A, I) be a pair with I locally nilpotent. Then the functor B ↦ B/IB induces an equivalence between the category of étale algebras over A and the category of étale algebras over A/I. Moreover, the pair is henselian.","statement_latex":"Let $(A, I)$ be a pair with $I$ locally nilpotent. Then the functor\n$B \\mapsto B/IB$ induces an equivalence between the category of\n\\'etale algebras over $A$ and the category of \\'etale algebras over $A/I$.\nMoreover, the pair is henselian.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALI","source_file":"more-algebra.tex","source_line":2415,"source_end_line":2421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2415-L2421","statement_sha256":"8ef69645b1fed2b2fdad7436bd36a299a61f1bc03cf4a4e33bf4f8e343d9bde4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2873,"rank":2873,"depth":43,"x":681.506,"y":714.697,"cluster":"advanced-algebra"},{"id":"stacks:0CT7","tag":"0CT7","title":"Henselian pairs · Lemma 0CT7","summary":"Let A = lim A_n where (A_n) is an inverse system of rings whose transition maps are surjective and have locally nilpotent kernels. Then (A, I_n) is a henselian pair, where I_n = Ker(A → A_n).","statement_latex":"Let $A = \\lim A_n$ where $(A_n)$ is an inverse system of rings\nwhose transition maps are surjective and have locally nilpotent kernels.\nThen $(A, I_n)$ is a henselian pair, where $I_n = \\Ker(A \\to A_n)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CT7","source_file":"more-algebra.tex","source_line":2452,"source_end_line":2457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2452-L2457","statement_sha256":"8c0b673113deef971eacbce57f280b7a0b00298553e621e8ec67097729424c5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2874,"rank":2874,"depth":44,"x":650.717,"y":655.04,"cluster":"advanced-algebra"},{"id":"stacks:0ALJ","tag":"0ALJ","title":"Henselian pairs · Lemma 0ALJ","summary":"Let (A, I) be a pair. If A is I-adically complete, then the pair is henselian.","statement_latex":"Let $(A, I)$ be a pair. If $A$ is $I$-adically complete, then\nthe pair is henselian.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALJ","source_file":"more-algebra.tex","source_line":2481,"source_end_line":2485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2481-L2485","statement_sha256":"5bd1f507e4657a1b8811bda5188f6566b0a534593da83ac86f2ac94476140d1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2875,"rank":2875,"depth":44,"x":722.044,"y":681.806,"cluster":"advanced-algebra"},{"id":"stacks:09XH","tag":"09XH","title":"Henselian pairs · Lemma 09XH","summary":"Let (A, I) be a pair. Let A → B be a finite type ring map such that B/IB = C_1 × C_2 with A/I → C_1 finite. Let B' be the integral closure of A in B. Then we can write B'/IB' = C_1 × C'_2 such that the map B'/IB' → B/IB preserves product decompositions and there exists a g ∈ B' mapping to (1, 0) in C_1 × C'_2 with B'_g → B_g an isomorphism.","statement_latex":"Let $(A, I)$ be a pair. Let $A \\to B$ be a finite type ring map\nsuch that $B/IB = C_1 \\times C_2$ with $A/I \\to C_1$ finite.\nLet $B'$ be the integral closure of $A$ in $B$.\nThen we can write $B'/IB' = C_1 \\times C'_2$ such that\nthe map $B'/IB' \\to B/IB$ preserves product decompositions\nand there exists a $g \\in B'$ mapping to $(1, 0)$ in\n$C_1 \\times C'_2$ with $B'_g \\to B_g$ an isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XH","source_file":"more-algebra.tex","source_line":2501,"source_end_line":2510,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2501-L2510","statement_sha256":"c1ad8439ae34f11f100dd52a2c6c93c0990259397db84db2099e8def3237836c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2876,"rank":2876,"depth":28,"x":647.255,"y":702.727,"cluster":"advanced-algebra"},{"id":"stacks:09XI","tag":"09XI","title":"Henselian pairs · Lemma 09XI","summary":"[Henselian] and [Gabber-henselian] Let (A, I) be a pair. The following are equivalent • (A, I) is a henselian pair, • given an étale ring map A → A' and an A-algebra map σ : A' → A/I, there exists an A-algebra map A' → A lifting σ, • for any finite A-algebra B the map B → B/IB induces a bijection on idempotents, • for any integral A-algebra B the map B → B/IB induces a bijection on idempotents, and • (Gabber) I is contained in the Jacobson radical of A and every monic…","statement_latex":"\\begin{reference}\n\\cite[Chapter XI]{Henselian} and \\cite[Proposition 1]{Gabber-henselian}\n\\end{reference}\nLet $(A, I)$ be a pair. The following are equivalent\n\\begin{enumerate}\n\\item $(A, I)$ is a henselian pair,\n\\item given an \\'etale ring map $A \\to A'$ and an $A$-algebra map\n$\\sigma : A' \\to A/I$, there exists an $A$-algebra map $A' \\to A$\nlifting $\\sigma$,\n\\item for any finite $A$-algebra $B$ the map $B \\to B/IB$ induces\na bijection on idempotents,\n\\item for any integral $A$-algebra $B$ the map $B \\to B/IB$ induces\na bijection on idempotents, and\n\\item (Gabber) $I$ is contained in the Jacobson radical of $A$ and\nevery monic polynomial $f(T) \\in A[T]$ of the form\n$$\nf(T) = T^n(T - 1) + a_n T^n + \\ldots + a_1 T + a_0\n$$\nwith $a_n, \\ldots, a_0 \\in I$ and $n \\ge 1$ has a root $\\alpha \\in 1 + I$.\n\\end{enumerate}\nMoreover, in part (5) the root is unique.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XI","source_file":"more-algebra.tex","source_line":2552,"source_end_line":2575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2552-L2575","statement_sha256":"6167f09769719c176d0760d263217b0d2f7388444004f9afe7396b7779cf2a16","origin":"The Stacks Project","memory_eligible":false,"source_rank":2877,"rank":2877,"depth":46,"x":685.921,"y":644.35,"cluster":"advanced-algebra"},{"id":"stacks:09XJ","tag":"09XJ","title":"Henselian pairs · Lemma 09XJ","summary":"Let A be a ring. Let I, J ⊂ A be ideals with V(I) = V(J). Then (A, I) is henselian if and only if (A, J) is henselian.","statement_latex":"Let $A$ be a ring. Let $I, J \\subset A$ be ideals with $V(I) = V(J)$.\nThen $(A, I)$ is henselian if and only if $(A, J)$ is henselian.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XJ","source_file":"more-algebra.tex","source_line":2736,"source_end_line":2740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2736-L2740","statement_sha256":"4a0fbdf6917b1a42b60d3486a03be4fbd05a618105c90b267f14005c01891e41","origin":"The Stacks Project","memory_eligible":false,"source_rank":2878,"rank":2878,"depth":47,"x":704.512,"y":709.903,"cluster":"advanced-algebra"},{"id":"stacks:09XK","tag":"09XK","title":"Henselian pairs · Lemma 09XK","summary":"Let (A, I) be a henselian pair and let A → B be an integral ring map. Then (B, IB) is a henselian pair.","statement_latex":"Let $(A, I)$ be a henselian pair and let $A \\to B$ be an integral ring\nmap. Then $(B, IB)$ is a henselian pair.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XK","source_file":"more-algebra.tex","source_line":2752,"source_end_line":2756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2752-L2756","statement_sha256":"d237ed1d100e488a398473d995b387287166909adcc4dca280ab0789220cdac0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2879,"rank":2879,"depth":47,"x":637.52,"y":671.789,"cluster":"advanced-algebra"},{"id":"stacks:0DYD","tag":"0DYD","title":"Henselian pairs · Lemma 0DYD","summary":"Let I ⊂ J ⊂ A be ideals of a ring A. The following are equivalent • (A, I) and (A/I, J/I) are henselian pairs, and • (A, J) is an henselian pair.","statement_latex":"Let $I \\subset J \\subset A$ be ideals of a ring $A$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $(A, I)$ and $(A/I, J/I)$ are henselian pairs, and\n\\item $(A, J)$ is an henselian pair.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYD","source_file":"more-algebra.tex","source_line":2764,"source_end_line":2772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2764-L2772","statement_sha256":"b67cba8d2383f1e63c91fc212bc1765b46d6e214d3d25ba93a6bb8983e196f90","origin":"The Stacks Project","memory_eligible":false,"source_rank":2880,"rank":2880,"depth":48,"x":718.243,"y":661.8,"cluster":"advanced-algebra"},{"id":"stacks:0G1R","tag":"0G1R","title":"Henselian pairs · Lemma 0G1R","summary":"Let A be a ring and let (A, I) and (A, I') be henselian pairs. Then (A, I + I') is an henselian pair.","statement_latex":"Let $A$ be a ring and let $(A, I)$ and $(A, I')$ be henselian pairs.\nThen $(A, I + I')$ is an henselian pair.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1R","source_file":"more-algebra.tex","source_line":2797,"source_end_line":2801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2797-L2801","statement_sha256":"b286ec3c2891ecb1e0c8d737798f03ae525d1f49d5c8658416724f77ece03bb2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2881,"rank":2881,"depth":49,"x":666.325,"y":715.409,"cluster":"advanced-algebra"},{"id":"stacks:0ATD","tag":"0ATD","title":"Henselian pairs · Lemma 0ATD","summary":"Let J be a set and let ( (A_j, I_j))_j ∈ J be a collection of pairs. Then (∏_j ∈ J A_j, ∏_j∈ J I_j) is Henselian if and only if so is each (A_j, I_j).","statement_latex":"Let $J$ be a set and let $\\{ (A_j, I_j)\\}_{j \\in J}$ be a collection\nof pairs. Then $(\\prod_{j \\in J} A_j, \\prod_{j\\in J} I_j)$ is Henselian\nif and only if so is each $(A_j, I_j)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATD","source_file":"more-algebra.tex","source_line":2809,"source_end_line":2814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2809-L2814","statement_sha256":"20ecc47a93646d0b7b959873a73767cf384c87191cd1b950076fe19b4ea4c08e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2882,"rank":2882,"depth":48,"x":661.46,"y":645.857,"cluster":"advanced-algebra"},{"id":"stacks:0EM6","tag":"0EM6","title":"Henselian pairs · Lemma 0EM6","summary":"The property of being Henselian is preserved under limits of pairs. More precisely, let J be a preordered set and let (A_j, I_j) be an inverse system of henselian pairs over J. Then A = lim A_j equipped with the ideal I = lim I_j is a henselian pair (A, I).","statement_latex":"The property of being Henselian is preserved under limits of pairs.\nMore precisely, let $J$ be a preordered set and let $(A_j, I_j)$\nbe an inverse system of henselian pairs over $J$.\nThen $A = \\lim A_j$ equipped with the ideal $I = \\lim I_j$\nis a henselian pair $(A, I)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EM6","source_file":"more-algebra.tex","source_line":2842,"source_end_line":2849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2842-L2849","statement_sha256":"e2430cf2325e3a323d7e8bd687553045d255ed7862ead9fc6814b538bc46af12","origin":"The Stacks Project","memory_eligible":false,"source_rank":2883,"rank":2883,"depth":49,"x":721.455,"y":694.773,"cluster":"advanced-algebra"},{"id":"stacks:0FWT","tag":"0FWT","title":"Henselian pairs · Lemma 0FWT","summary":"The property of being Henselian is preserved under filtered colimits of pairs. More precisely, let J be a directed set and let (A_j, I_j) be a system of henselian pairs over J. Then A = colim A_j equipped with the ideal I = colim I_j is a henselian pair (A, I).","statement_latex":"The property of being Henselian is preserved under filtered colimits of pairs.\nMore precisely, let $J$ be a directed set and let $(A_j, I_j)$\nbe a system of henselian pairs over $J$.\nThen $A = \\colim A_j$ equipped with the ideal $I = \\colim I_j$\nis a henselian pair $(A, I)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWT","source_file":"more-algebra.tex","source_line":2884,"source_end_line":2891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2884-L2891","statement_sha256":"4ae06ac2e1573006910dbc35c489e6c22451ef001a0db95316b9fcd3315f4858","origin":"The Stacks Project","memory_eligible":false,"source_rank":2884,"rank":2884,"depth":2,"x":637.219,"y":692.728,"cluster":"advanced-algebra"},{"id":"stacks:0G1S","tag":"0G1S","title":"Henselian pairs · Lemma 0G1S","summary":"Let A be a ring. There exists a largest ideal I ⊂ A such that (A, I) is a henselian pair.","statement_latex":"Let $A$ be a ring. There exists a largest ideal $I \\subset A$ such that\n$(A, I)$ is a henselian pair.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1S","source_file":"more-algebra.tex","source_line":2936,"source_end_line":2940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2936-L2940","statement_sha256":"14f0cc6a8f39d5a75194acf6ae133aa0ed949de252cff42b586f6927c9e08cb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":2885,"rank":2885,"depth":50,"x":701.475,"y":646.081,"cluster":"advanced-algebra"},{"id":"stacks:09Y6","tag":"09Y6","title":"Henselian pairs · Lemma 09Y6","summary":"Let (A, I) be a henselian pair. Let p ⊂ A be a prime ideal. Then V( p + I) is connected.","statement_latex":"Let $(A, I)$ be a henselian pair. Let $\\mathfrak p \\subset A$\nbe a prime ideal. Then $V(\\mathfrak p + I)$ is connected.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Y6","source_file":"more-algebra.tex","source_line":2947,"source_end_line":2951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2947-L2951","statement_sha256":"ad08f7bfd050e78f2e0d5f11022e42b24e4697138c750685f8b465dc41853415","origin":"The Stacks Project","memory_eligible":false,"source_rank":2886,"rank":2886,"depth":48,"x":691.529,"y":717.479,"cluster":"advanced-algebra"},{"id":"stacks:0A02","tag":"0A02","title":"Henselization of pairs · Lemma 0A02","summary":"The inclusion functor category of henselian pairs → category of pairs has a left adjoint (A, I) ↦ (A^h, I^h).","statement_latex":"The inclusion functor\n$$\n\\text{category of henselian pairs}\n\\longrightarrow\n\\text{category of pairs}\n$$\nhas a left adjoint $(A, I) \\mapsto (A^h, I^h)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselization of pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A02","source_file":"more-algebra.tex","source_line":2979,"source_end_line":2988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L2979-L2988","statement_sha256":"27733e4ce474803d4540769e986efe9c3ca3eeb412318d8386d191573c9d7251","origin":"The Stacks Project","memory_eligible":false,"source_rank":2887,"rank":2887,"depth":47,"x":641.071,"y":658.748,"cluster":"advanced-algebra"},{"id":"stacks:0AGU","tag":"0AGU","title":"Henselization of pairs · Lemma 0AGU","summary":"Let (A, I) be a pair. Let (A^h, I^h) be as in Lemma [Tag 0A02]. Then A → A^h is flat, I^h = IA^h and A/I^n → A^h/I^nA^h is an isomorphism for all n.","statement_latex":"Let $(A, I)$ be a pair. Let $(A^h, I^h)$ be as in \nLemma \\ref{lemma-henselization}. Then $A \\to A^h$ is flat,\n$I^h = IA^h$ and $A/I^n \\to A^h/I^nA^h$ is an isomorphism\nfor all $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselization of pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGU","source_file":"more-algebra.tex","source_line":3086,"source_end_line":3092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3086-L3092","statement_sha256":"64fc723280a158aa30f51123cd4bf31cd8db48962fc44ef9546e4a9ea1fe047a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2888,"rank":2888,"depth":48,"x":726.133,"y":673.535,"cluster":"advanced-algebra"},{"id":"stacks:0A03","tag":"0A03","title":"Henselization of pairs · Lemma 0A03","summary":"Compatibility henselization of pairs and of local rings. The functor of Lemma [Tag 0A02] associates to a local ring (A, m) its henselization.","statement_latex":"\\begin{slogan}\nCompatibility henselization of pairs and of local rings.\n\\end{slogan}\nThe functor of Lemma \\ref{lemma-henselization} associates to a local ring\n$(A, \\mathfrak m)$ its henselization.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselization of pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A03","source_file":"more-algebra.tex","source_line":3108,"source_end_line":3115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3108-L3115","statement_sha256":"867b123f6e681ac82595f30e97bffa5ec7cdb15e724926bac463e76a4d43f399","origin":"The Stacks Project","memory_eligible":false,"source_rank":2889,"rank":2889,"depth":49,"x":650.973,"y":711.169,"cluster":"advanced-algebra"},{"id":"stacks:0AGV","tag":"0AGV","title":"Henselization of pairs · Lemma 0AGV","summary":"The henselization of a Noetherian pair is a Noetherian pair with the same completion Let (A, I) be a pair with A Noetherian. Let (A^h, I^h) be as in Lemma [Tag 0A02]. Then the map of I-adic completions A^wedge → (A^h)^wedge is an isomorphism. Moreover, A^h is Noetherian, the maps A → A^h → A^wedge are flat, and A^h → A^wedge is faithfully flat.","statement_latex":"\\begin{slogan}\nThe henselization of a Noetherian pair is a Noetherian pair\nwith the same completion\n\\end{slogan}\nLet $(A, I)$ be a pair with $A$ Noetherian.  Let $(A^h, I^h)$ be as in \nLemma \\ref{lemma-henselization}. Then the map of $I$-adic completions\n$$\nA^\\wedge \\to (A^h)^\\wedge\n$$\nis an isomorphism. Moreover, $A^h$ is Noetherian, the maps\n$A \\to A^h \\to A^\\wedge$ are flat, and $A^h \\to A^\\wedge$ is\nfaithfully flat.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselization of pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGV","source_file":"more-algebra.tex","source_line":3141,"source_end_line":3155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3141-L3155","statement_sha256":"eeaba9461d685d590b13f8f080b800fc0d64babbd16a526471aed6205d2996db","origin":"The Stacks Project","memory_eligible":false,"source_rank":2890,"rank":2890,"depth":49,"x":676.313,"y":640.264,"cluster":"advanced-algebra"},{"id":"stacks:0A04","tag":"0A04","title":"Henselization of pairs · Lemma 0A04","summary":"Let (A, I) = colim (A_i, I_i) be a filtered colimit of pairs. The functor of Lemma [Tag 0A02] gives A^h = colim A_i^h and I^h = colim I_i^h.","statement_latex":"Let $(A, I) = \\colim (A_i, I_i)$ be a filtered colimit of pairs. The functor of\nLemma \\ref{lemma-henselization} gives\n$A^h = \\colim A_i^h$ and $I^h = \\colim I_i^h$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselization of pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A04","source_file":"more-algebra.tex","source_line":3180,"source_end_line":3185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3180-L3185","statement_sha256":"b957cd5a6a39cbbcfc077fe275a1227f1687a87509bf61e9e6546f49c388171b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2891,"rank":2891,"depth":48,"x":714.918,"y":707.401,"cluster":"advanced-algebra"},{"id":"stacks:0F0L","tag":"0F0L","title":"Henselization of pairs · Lemma 0F0L","summary":"The henselization of a pair only depends on the radical of the ideal Let A be a ring with ideals I and J. If V(I) = V(J) then the functor of Lemma [Tag 0A02] produces the same ring for the pair (A, I) as for the pair (A, J).","statement_latex":"\\begin{slogan}\nThe henselization of a pair only depends on the radical of the ideal\n\\end{slogan}\nLet $A$ be a ring with ideals $I$ and $J$. If $V(I) = V(J)$ then the functor\nof Lemma \\ref{lemma-henselization} produces the same ring for the pair\n$(A, I)$ as for the pair $(A, J)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselization of pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0L","source_file":"more-algebra.tex","source_line":3209,"source_end_line":3217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3209-L3217","statement_sha256":"02f06e63fcc8a44ab56b795bfb48cd831bc05c9a4d4a8693131d1b5c03533f40","origin":"The Stacks Project","memory_eligible":false,"source_rank":2892,"rank":2892,"depth":49,"x":631.885,"y":679.606,"cluster":"advanced-algebra"},{"id":"stacks:0DYE","tag":"0DYE","title":"Henselization of pairs · Lemma 0DYE","summary":"Henselization commutes with integral base change Let (A, I) → (B, J) be a map of pairs such that V(J) = V(IB). Let (A^h , I^h) → (B^h, J^h) be the induced map on henselizations (Lemma [Tag 0A02]). If A → B is integral, then the induced map A^h ⊗_A B → B^h is an isomorphism.","statement_latex":"\\begin{slogan}\nHenselization commutes with integral base change\n\\end{slogan}\nLet $(A, I) \\to (B, J)$ be a map of pairs such that $V(J) = V(IB)$.\nLet $(A^h , I^h) \\to (B^h, J^h)$ be the induced map\non henselizations (Lemma \\ref{lemma-henselization}).\nIf $A \\to B$ is integral, then the induced map\n$A^h \\otimes_A B \\to B^h$ is an isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselization of pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYE","source_file":"more-algebra.tex","source_line":3231,"source_end_line":3241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3231-L3241","statement_sha256":"1b64b0a6c3d4bade7a5585bc68f1788885b9228e7c2a75ffa2d8113b931e2bff","origin":"The Stacks Project","memory_eligible":false,"source_rank":2893,"rank":2893,"depth":50,"x":716.041,"y":652.804,"cluster":"advanced-algebra"},{"id":"stacks:0H7Q","tag":"0H7Q","title":"Henselization of pairs · Lemma 0H7Q","summary":"Let I_1, I_2, dots, I_n ⊂ A be ideals in a ring A. Suppose I_i + I_j = A for i not = j, 1 ≤ i, j ≤ n. Denote A^h the henselization of the pair (A, I_1 ∩ … ∩ I_n) and A_i^h the henselization of the pair (A, I_i). Then the natural morphism A^h → ∏_i = 1, …, n A^h_i is an isomorphism.","statement_latex":"Let $I_1, I_2, \\dots, I_n \\subset A$ be ideals in a ring $A$.\nSuppose $I_i + I_j = A$ for $i \\not = j$, $1 \\leq i, j \\leq n$.\nDenote $A^h$ the henselization of the pair $(A, I_1 \\cap \\ldots \\cap I_n)$\nand $A_i^h$ the henselization of the pair $(A, I_i)$.\nThen the natural morphism\n$$\nA^h \\to \\prod\\nolimits_{i = 1, \\ldots, n} A^h_i\n$$\nis an isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Henselization of pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7Q","source_file":"more-algebra.tex","source_line":3253,"source_end_line":3264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3253-L3264","statement_sha256":"91f6deec942e451cce433b6587a9b30c893c28db79004521f6569f3f028df6df","origin":"The Stacks Project","memory_eligible":false,"source_rank":2894,"rank":2894,"depth":48,"x":675.262,"y":720.778,"cluster":"advanced-algebra"},{"id":"stacks:0D4A","tag":"0D4A","title":"Lifting and henselian pairs · Lemma 0D4A","summary":"Let (R, I) be a henselian pair. The map P → P/IP induces a bijection between the sets of isomorphism classes of finite projective R-modules and finite projective R/I-modules. In particular, any finite projective R/I-module is isomorphic to P/IP for some finite projective R-module P.","statement_latex":"Let $(R, I)$ be a henselian pair. The map\n$$\nP \\longrightarrow P/IP\n$$\ninduces a bijection between the sets of isomorphism classes of finite\nprojective $R$-modules and finite projective $R/I$-modules. In particular,\nany finite projective $R/I$-module is isomorphic to $P/IP$\nfor some finite projective $R$-module $P$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting and henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4A","source_file":"more-algebra.tex","source_line":3347,"source_end_line":3357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3347-L3357","statement_sha256":"a4580402aa45986532889f335f447141f05235a5dbd0e2d91f94b0f1028c910f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2895,"rank":2895,"depth":47,"x":650.506,"y":647.024,"cluster":"advanced-algebra"},{"id":"stacks:09ZL","tag":"09ZL","title":"Lifting and henselian pairs · Lemma 09ZL","summary":"Let (A, I) be a henselian pair. The functor B → B/IB determines an equivalence between finite étale A-algebras and finite étale A/I-algebras.","statement_latex":"Let $(A, I)$ be a henselian pair. The functor $B \\to B/IB$ determines\nan equivalence between finite \\'etale $A$-algebras and finite \\'etale\n$A/I$-algebras.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting and henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZL","source_file":"more-algebra.tex","source_line":3382,"source_end_line":3387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3382-L3387","statement_sha256":"318e8f24537d1bdc79ab2f1741ba4e9665e76d22aa45799d71a919102ef34791","origin":"The Stacks Project","memory_eligible":false,"source_rank":2896,"rank":2896,"depth":47,"x":728.585,"y":687.63,"cluster":"advanced-algebra"},{"id":"stacks:0H74","tag":"0H74","title":"Lifting and henselian pairs · Lemma 0H74","summary":"Let R be a ring and S a smooth R-algebra. Assume that A is an R-algebra and (A,I) is a henselian pair. Then any R-algebra map S → A/I can be lifted to an R-algebra map S → A.","statement_latex":"Let $R$ be a ring and $S$ a smooth $R$-algebra. Assume that $A$\nis an $R$-algebra and $(A,I)$ is a henselian pair. Then any $R$-algebra\nmap $S \\to A/I$ can be lifted to an $R$-algebra map $S \\to A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting and henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H74","source_file":"more-algebra.tex","source_line":3439,"source_end_line":3444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3439-L3444","statement_sha256":"a4d9622667232e7ce73748fb77f3219407434a4688af60e707720595c56f9098","origin":"The Stacks Project","memory_eligible":false,"source_rank":2897,"rank":2897,"depth":47,"x":637.766,"y":702.084,"cluster":"advanced-algebra"},{"id":"stacks:0D4B","tag":"0D4B","title":"Lifting and henselian pairs · Lemma 0D4B","summary":"Let A = lim A_n be a limit of an inverse system (A_n) of rings. Suppose given A_n-modules M_n and A_n + 1-module maps M_n + 1 → M_n. Assume • the transition maps A_n + 1 → A_n are surjective with locally nilpotent kernels, • M_1 is a finite projective A_1-module, • M_n is a finite flat A_n-module, and • the maps induce isomorphisms M_n + 1 ⊗_A_n + 1 A_n → M_n. Then M = lim M_n is a finite projective A-module and M ⊗_A A_n → M_n is an isomorphism for all n.","statement_latex":"Let $A = \\lim A_n$ be a limit of an inverse system $(A_n)$ of rings.\nSuppose given $A_n$-modules $M_n$ and $A_{n + 1}$-module maps\n$M_{n + 1} \\to M_n$. Assume\n\\begin{enumerate}\n\\item the transition maps $A_{n + 1} \\to A_n$ are surjective\nwith locally nilpotent kernels,\n\\item $M_1$ is a finite projective $A_1$-module,\n\\item $M_n$ is a finite flat $A_n$-module, and\n\\item the maps induce isomorphisms\n$M_{n + 1} \\otimes_{A_{n + 1}} A_n \\to M_n$.\n\\end{enumerate}\nThen $M = \\lim M_n$ is a finite projective $A$-module\nand $M \\otimes_A A_n \\to M_n$ is an isomorphism for all $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting and henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4B","source_file":"more-algebra.tex","source_line":3460,"source_end_line":3475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3460-L3475","statement_sha256":"b22362e80f175a3c2069315be10d8912780e9eafe915626cf15bfd994c053b25","origin":"The Stacks Project","memory_eligible":false,"source_rank":2898,"rank":2898,"depth":48,"x":693.469,"y":639.493,"cluster":"advanced-algebra"},{"id":"stacks:0DCL","tag":"0DCL","title":"Absolute integral closure · Definition 0DCL","summary":"A ring A is absolutely integrally closed if every monic f ∈ A[T] is a product of linear factors.","statement_latex":"A ring $A$ is {\\it absolutely integrally closed} if every\nmonic $f \\in A[T]$ is a product of linear factors.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Absolute integral closure","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCL","source_file":"more-algebra.tex","source_line":3509,"source_end_line":3513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3509-L3513","statement_sha256":"fcdd874c1a6ee12b880380c5f73171b1b86700758f4090538c905cce1282b846","origin":"The Stacks Project","memory_eligible":false,"source_rank":2899,"rank":2899,"depth":0,"x":702.786,"y":717.749,"cluster":"advanced-algebra"},{"id":"stacks:0DCM","tag":"0DCM","title":"Absolute integral closure · Lemma 0DCM","summary":"Let A be a ring. The following are equivalent • A is absolutely integrally closed, and • any monic f ∈ A[T] has a root in A.","statement_latex":"Let $A$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $A$ is absolutely integrally closed, and\n\\item any monic $f \\in A[T]$ has a root in $A$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Absolute integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCM","source_file":"more-algebra.tex","source_line":3519,"source_end_line":3526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3519-L3526","statement_sha256":"3b3bc839a2f251bd749e9deaf1d0437c5a40a9b6057cd25e66c4ca2d9295d619","origin":"The Stacks Project","memory_eligible":false,"source_rank":2900,"rank":2900,"depth":0,"x":632.546,"y":665.0,"cluster":"advanced-algebra"},{"id":"stacks:0DCN","tag":"0DCN","title":"Absolute integral closure · Lemma 0DCN","summary":"Let A be absolutely integrally closed. • Any quotient ring A/I of A is absolutely integrally closed. • Any localization S^-1A is absolutely integrally closed.","statement_latex":"Let $A$ be absolutely integrally closed.\n\\begin{enumerate}\n\\item Any quotient ring $A/I$ of $A$ is absolutely integrally closed.\n\\item Any localization $S^{-1}A$ is absolutely integrally closed.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Absolute integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCN","source_file":"more-algebra.tex","source_line":3532,"source_end_line":3539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3532-L3539","statement_sha256":"7caebb843b2048681bda56412c5b53fddb5e8a2b0d0c51d282b4a37b56e39d7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2901,"rank":2901,"depth":0,"x":727.344,"y":664.036,"cluster":"advanced-algebra"},{"id":"stacks:0DCP","tag":"0DCP","title":"Absolute integral closure · Lemma 0DCP","summary":"Let A be a ring. Let S ⊂ A be a multiplicative subset consisting of nonzerodivisors. If S^-1A is absolutely integrally closed and A ⊂ S^-1A is integrally closed in S^-1A, then A is absolutely integrally closed.","statement_latex":"Let $A$ be a ring. Let $S \\subset A$ be a multiplicative subset\nconsisting of nonzerodivisors.\nIf $S^{-1}A$ is absolutely integrally closed and $A \\subset S^{-1}A$\nis integrally closed in $S^{-1}A$, then $A$ is absolutely integrally closed.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Absolute integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCP","source_file":"more-algebra.tex","source_line":3545,"source_end_line":3551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3545-L3551","statement_sha256":"12374f3c622262a7ece3cee402b5a4caa51cf604cc637742283ecb0c257e8f37","origin":"The Stacks Project","memory_eligible":false,"source_rank":2902,"rank":2902,"depth":0,"x":657.792,"y":718.871,"cluster":"advanced-algebra"},{"id":"stacks:0DCQ","tag":"0DCQ","title":"Absolute integral closure · Lemma 0DCQ","summary":"Let A be a normal domain. Then A is absolutely integrally closed if and only if its fraction field is algebraically closed.","statement_latex":"Let $A$ be a normal domain. Then $A$ is absolutely integrally closed\nif and only if its fraction field is algebraically closed.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Absolute integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCQ","source_file":"more-algebra.tex","source_line":3557,"source_end_line":3561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3557-L3561","statement_sha256":"876eb5784ce1a64be74649a9059dfffe386f3adf5db506f19e64bb87dff4196d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2903,"rank":2903,"depth":1,"x":665.027,"y":638.487,"cluster":"advanced-algebra"},{"id":"stacks:0DCR","tag":"0DCR","title":"Absolute integral closure · Lemma 0DCR","summary":"For any ring A there exists an extension A ⊂ B such that • B is a filtered colimit of finite free A-algebras, • B is free as an A-module, and • B is absolutely integrally closed.","statement_latex":"For any ring $A$ there exists an extension $A \\subset B$ such that\n\\begin{enumerate}\n\\item $B$ is a filtered colimit of finite free $A$-algebras,\n\\item $B$ is free as an $A$-module, and\n\\item $B$ is absolutely integrally closed.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Absolute integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCR","source_file":"more-algebra.tex","source_line":3570,"source_end_line":3578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3570-L3578","statement_sha256":"6024653866d18cde62449ff01d18b8d5c6ea2de02f482d8c1c078a3506ade4df","origin":"The Stacks Project","memory_eligible":false,"source_rank":2904,"rank":2904,"depth":1,"x":724.689,"y":702.246,"cluster":"advanced-algebra"},{"id":"stacks:0DCS","tag":"0DCS","title":"Absolute integral closure · Lemma 0DCS","summary":"Let A be absolutely integrally closed. Let p ⊂ A be a prime. Then the local ring A_ p is strictly henselian.","statement_latex":"Let $A$ be absolutely integrally closed. Let $\\mathfrak p \\subset A$\nbe a prime. Then the local ring $A_\\mathfrak p$ is strictly henselian.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Absolute integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCS","source_file":"more-algebra.tex","source_line":3604,"source_end_line":3608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3604-L3608","statement_sha256":"b32e494cc924af9e2cbb00733da0d31b347e4ac87a172d0d6d2d5bbd0d32878a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2905,"rank":2905,"depth":1,"x":628.859,"y":689.006,"cluster":"advanced-algebra"},{"id":"stacks:0DCT","tag":"0DCT","title":"Absolute integral closure · Lemma 0DCT","summary":"Let A be absolutely integrally closed. Let I ⊂ A be an ideal. Then (A, I) is a henselian pair if (and only if) the following conditions hold • I is contained in the Jacobson radical of A, • A → A/I induces a bijection on idempotents.","statement_latex":"Let $A$ be absolutely integrally closed. Let $I \\subset A$ be an ideal.\nThen $(A, I)$ is a henselian pair if (and only if) the following\nconditions hold\n\\begin{enumerate}\n\\item $I$ is contained in the Jacobson radical of $A$,\n\\item $A \\to A/I$ induces a bijection on idempotents.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Absolute integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCT","source_file":"more-algebra.tex","source_line":3619,"source_end_line":3628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3619-L3628","statement_sha256":"0444da73c7fa494ab18df8e03909f442fa3130f4dfa1b3351a219185280c2dbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":2906,"rank":2906,"depth":0,"x":710.646,"y":644.133,"cluster":"advanced-algebra"},{"id":"stacks:05GM","tag":"05GM","title":"Auto-associated rings · Definition 05GM","summary":"A ring R is said to be auto-associated if R is local and its maximal ideal m is weakly associated to R.","statement_latex":"A ring $R$ is said to be {\\it auto-associated} if $R$ is local and its\nmaximal ideal $\\mathfrak m$ is weakly associated to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Auto-associated rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GM","source_file":"more-algebra.tex","source_line":3686,"source_end_line":3690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3686-L3690","statement_sha256":"53313904dc46bd5caeae78edf6dbdda17112a27439dbdacfd96562f8de11c741","origin":"The Stacks Project","memory_eligible":false,"source_rank":2907,"rank":2907,"depth":0,"x":686.281,"y":724.088,"cluster":"advanced-algebra"},{"id":"stacks:05GN","tag":"05GN","title":"Auto-associated rings · Lemma 05GN","summary":"An auto-associated ring R has the following property: (P) Every proper finitely generated ideal I ⊂ R has a nonzero annihilator.","statement_latex":"An auto-associated ring $R$ has the following property: (P)\nEvery proper finitely generated ideal $I \\subset R$ has a nonzero\nannihilator.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Auto-associated rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GN","source_file":"more-algebra.tex","source_line":3692,"source_end_line":3697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3692-L3697","statement_sha256":"4c7d936a2c2446333d66f421fd0cdf6c9e8d92bdd87721d5c8de34715b0543aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":2908,"rank":2908,"depth":1,"x":639.686,"y":650.89,"cluster":"advanced-algebra"},{"id":"stacks:05GP","tag":"05GP","title":"Auto-associated rings · Lemma 05GP","summary":"Let R be a ring having property (P) of Lemma [Tag 05GN]. Let u : N → M be a homomorphism of projective R-modules. Then u is universally injective if and only if u is injective.","statement_latex":"Let $R$ be a ring having property (P) of\nLemma \\ref{lemma-auto-ass-implies-P}.\nLet $u : N \\to M$ be a homomorphism of projective $R$-modules.\nThen $u$ is universally injective if and only if $u$ is injective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Auto-associated rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GP","source_file":"more-algebra.tex","source_line":3708,"source_end_line":3714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3708-L3714","statement_sha256":"2a448bcfa2b082641e81306f2818c4663600942316f736db9216740c0ec39481","origin":"The Stacks Project","memory_eligible":false,"source_rank":2909,"rank":2909,"depth":2,"x":733.435,"y":678.582,"cluster":"advanced-algebra"},{"id":"stacks:05GQ","tag":"05GQ","title":"Auto-associated rings · Lemma 05GQ","summary":"Let R be a ring. The following are equivalent • R has property (P) of Lemma [Tag 05GN], • any injective map of projective R-modules is universally injective, • if u : N → M is injective and N, M are finite projective R-modules then Coker(u) is a finite projective R-module, • if N ⊂ M and N, M are finite projective as R-modules, then N is a direct summand of M, and • any injective map R → R^⊕ n is a split injection.","statement_latex":"Let $R$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $R$ has property (P) of\nLemma \\ref{lemma-auto-ass-implies-P},\n\\item any injective map of projective $R$-modules is\nuniversally injective,\n\\item if $u : N \\to M$ is injective and $N$, $M$ are finite projective\n$R$-modules then $\\Coker(u)$ is a finite projective $R$-module,\n\\item if $N \\subset M$ and $N$, $M$ are finite projective as $R$-modules, then\n$N$ is a direct summand of $M$, and\n\\item any injective map $R \\to R^{\\oplus n}$ is a split injection.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Auto-associated rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GQ","source_file":"more-algebra.tex","source_line":3756,"source_end_line":3770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3756-L3770","statement_sha256":"27ed1ab31773c740ca01392325126049a087c6d973225fbdb9c9b478b8ee9ba4","origin":"The Stacks Project","memory_eligible":false,"source_rank":2910,"rank":2910,"depth":3,"x":641.526,"y":711.54,"cluster":"advanced-algebra"},{"id":"stacks:00MX","tag":"00MX","title":"Auto-associated rings · Lemma 00MX","summary":"Let (R, m) be a local ring. Suppose that φ : R^m → R^n is a map of finite free modules. The following are equivalent • φ is injective, • the rank of φ is m and the annihilator of I(φ) in R is zero. If R is Noetherian these are also equivalent to • [(3)] the rank of φ is m and either I(φ) = R or it contains a nonzerodivisor. Here the rank of φ and I(φ) are defined as in Algebra, Definition [Tag 00MV].","statement_latex":"Let $(R, \\mathfrak m)$ be a local ring.\nSuppose that $\\varphi : R^m \\to R^n$ is a map\nof finite free modules. The following are equivalent\n\\begin{enumerate}\n\\item $\\varphi$ is injective,\n\\item the rank of $\\varphi$ is $m$ and the annihilator of\n$I(\\varphi)$ in $R$ is zero.\n\\end{enumerate}\nIf $R$ is Noetherian these are also equivalent to\n\\begin{enumerate}\n\\item[(3)] the rank of $\\varphi$ is $m$ and\neither $I(\\varphi) = R$ or it contains a nonzerodivisor.\n\\end{enumerate}\nHere the rank of $\\varphi$ and $I(\\varphi)$ are defined\nas in Algebra, Definition \\ref{algebra-definition-rank}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Auto-associated rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/00MX","source_file":"more-algebra.tex","source_line":3824,"source_end_line":3841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3824-L3841","statement_sha256":"e39a6dc35faa8380c2fddb8983eda1e40720600ae936dd8edab1ab5e6ecfbe7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2911,"rank":2911,"depth":15,"x":683.024,"y":634.665,"cluster":"advanced-algebra"},{"id":"stacks:0EWY","tag":"0EWY","title":"Auto-associated rings · Lemma 0EWY","summary":"Let R be a ring. Suppose that φ : R^n → R^n be an injective map of finite free modules of the same rank. Then Hom_R(Coker(φ), R) = 0.","statement_latex":"Let $R$ be a ring. Suppose that $\\varphi : R^n \\to R^n$ be an injective\nmap of finite free modules of the same rank. Then\n$\\Hom_R(\\Coker(\\varphi), R) = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Auto-associated rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWY","source_file":"more-algebra.tex","source_line":3898,"source_end_line":3903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3898-L3903","statement_sha256":"51d6b0d1007cd1ebe5a841cf4e656ed9eeefca19220e478c9aba4df0a22ae412","origin":"The Stacks Project","memory_eligible":false,"source_rank":2912,"rank":2912,"depth":16,"x":714.414,"y":715.336,"cluster":"advanced-algebra"},{"id":"stacks:0522","tag":"0522","title":"Flattening stratification · Lemma 0522","summary":"Let R be a ring. Let M be an R-module. Let I_1, I_2 be ideals of R. If M/I_1M is flat over R/I_1 and M/I_2M is flat over R/I_2, then M/(I_1 ∩ I_2)M is flat over R/(I_1 ∩ I_2).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. Let $I_1$, $I_2$ be ideals of $R$.\nIf $M/I_1M$ is flat over $R/I_1$ and $M/I_2M$ is flat over $R/I_2$,\nthen $M/(I_1 \\cap I_2)M$ is flat over $R/(I_1 \\cap I_2)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening stratification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0522","source_file":"more-algebra.tex","source_line":3952,"source_end_line":3957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L3952-L3957","statement_sha256":"8cd0c31d93536d38ae504e8202180affd25b29b07aec229a2a23b8fef6860917","origin":"The Stacks Project","memory_eligible":false,"source_rank":2913,"rank":2913,"depth":3,"x":625.919,"y":673.436,"cluster":"advanced-algebra"},{"id":"stacks:0524","tag":"0524","title":"Flattening over an Artinian ring · Lemma 0524","summary":"Let R be an Artinian ring. Let M be an R-module. Then there exists a smallest ideal I ⊂ R such that M/IM is flat over R/I.","statement_latex":"Let $R$ be an Artinian ring.\nLet $M$ be an $R$-module.\nThen there exists a smallest ideal $I \\subset R$ such that\n$M/IM$ is flat over $R/I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over an Artinian ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0524","source_file":"more-algebra.tex","source_line":4005,"source_end_line":4011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4005-L4011","statement_sha256":"92f0469818e8bd500f8878bcbf0497edcf76e289e9e5902f87642996dade1d2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2914,"rank":2914,"depth":4,"x":725.397,"y":654.016,"cluster":"advanced-algebra"},{"id":"stacks:0525","tag":"0525","title":"Flattening over an Artinian ring · Lemma 0525","summary":"Let R be an Artinian ring. Let M be an R-module. Let I ⊂ R be the smallest ideal I ⊂ R such that M/IM is flat over R/I. Then I has the following universal property: For every ring map φ : R → R' we have R' ⊗_R M is flat over R' ⇔ we have φ(I) = 0.","statement_latex":"Let $R$ be an Artinian ring. Let $M$ be an $R$-module.\nLet $I \\subset R$ be the smallest ideal $I \\subset R$ such that\n$M/IM$ is flat over $R/I$.\nThen $I$ has the following universal property:\nFor every ring map $\\varphi : R \\to R'$ we have\n$$\nR' \\otimes_R M\\text{ is flat over }R'\n\\Leftrightarrow\n\\text{we have }\\varphi(I) = 0.\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over an Artinian ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0525","source_file":"more-algebra.tex","source_line":4022,"source_end_line":4034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4022-L4034","statement_sha256":"d47f474c0b42ea392dfaa9f72cf01d88d37abe9874b6f2ba360ac24212928914","origin":"The Stacks Project","memory_eligible":false,"source_rank":2915,"rank":2915,"depth":8,"x":667.354,"y":725.156,"cluster":"advanced-algebra"},{"id":"stacks:052X","tag":"052X","title":"Flattening over a closed subset of the base · Lemma 052X","summary":"Let R → S be a ring map. Let I ⊂ R be an ideal. Let M be an S-module. Let R → R' be a ring map and IR' ⊂ I' ⊂ R' an ideal. If ([Tag 052W]) holds for (R → S, I, M), then ([Tag 052W]) holds for (R' → S ⊗_R R', I', M ⊗_R R').","statement_latex":"Let $R \\to S$ be a ring map.\nLet $I \\subset R$ be an ideal.\nLet $M$ be an $S$-module.\nLet $R \\to R'$ be a ring map and $IR' \\subset I' \\subset R'$ an ideal.\nIf (\\ref{equation-flat-at-primes-over}) holds for\n$(R \\to S, I, M)$, then (\\ref{equation-flat-at-primes-over})\nholds for $(R' \\to S \\otimes_R R', I', M \\otimes_R R')$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a closed subset of the base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052X","source_file":"more-algebra.tex","source_line":4080,"source_end_line":4089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4080-L4089","statement_sha256":"1f647820cc47692c9ab9713af5fc8aeebbceba1c94a27b3a688923f0daf3225c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2916,"rank":2916,"depth":3,"x":652.871,"y":639.317,"cluster":"advanced-algebra"},{"id":"stacks:05LL","tag":"05LL","title":"Flattening over a closed subset of the base · Lemma 05LL","summary":"Let R → S be a ring map. Let I ⊂ R be an ideal. Let M be an S-module. Let R → R' be a ring map and IR' ⊂ I' ⊂ R' an ideal such that • the map V(I') → V(I) induced by Spec(R') → Spec(R) is surjective, and • R'_ p' is flat over R for all primes p' ∈ V(I'). If ([Tag 052W]) holds for (R' → S ⊗_R R', I', M ⊗_R R'), then ([Tag 052W]) holds for (R → S, I, M).","statement_latex":"Let $R \\to S$ be a ring map.\nLet $I \\subset R$ be an ideal.\nLet $M$ be an $S$-module.\nLet $R \\to R'$ be a ring map and $IR' \\subset I' \\subset R'$ an ideal\nsuch that\n\\begin{enumerate}\n\\item the map $V(I') \\to V(I)$ induced by\n$\\Spec(R') \\to \\Spec(R)$ is surjective, and\n\\item $R'_{\\mathfrak p'}$ is flat over $R$ for all primes\n$\\mathfrak p' \\in V(I')$.\n\\end{enumerate}\nIf (\\ref{equation-flat-at-primes-over}) holds for\n$(R' \\to S \\otimes_R R', I', M \\otimes_R R')$, then\n(\\ref{equation-flat-at-primes-over}) holds for $(R \\to S, I, M)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a closed subset of the base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LL","source_file":"more-algebra.tex","source_line":4104,"source_end_line":4120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4104-L4120","statement_sha256":"0d824ecead959ac30cb478df598ab3f071781c8c10c2079e3f60a6af2d0d059e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2917,"rank":2917,"depth":5,"x":732.992,"y":694.681,"cluster":"advanced-algebra"},{"id":"stacks:05LM","tag":"05LM","title":"Flattening over a closed subset of the base · Lemma 05LM","summary":"Let R → S be a ring map of finite presentation. Let M be an S-module of finite presentation. Let R' = colim_λ ∈ Lambda R_λ be a directed colimit of R-algebras. Let I_λ ⊂ R_λ be ideals such that I_λ R_μ ⊂ I_μ for all μ ≥ λ and set I' = colim_λ I_λ. If ([Tag 052W]) holds for (R' → S ⊗_R R', I', M ⊗_R R'), then there exists a λ ∈ Lambda such that ([Tag 052W]) holds for (R_λ → S ⊗_R R_λ, I_λ, M ⊗_R R_λ).","statement_latex":"Let $R \\to S$ be a ring map of finite presentation.\nLet $M$ be an $S$-module of finite presentation.\nLet $R' = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$\nbe a directed colimit of $R$-algebras. Let $I_\\lambda \\subset R_\\lambda$\nbe ideals such that $I_\\lambda R_\\mu \\subset I_\\mu$ for all $\\mu \\geq \\lambda$\nand set $I' = \\colim_\\lambda I_\\lambda$.\nIf (\\ref{equation-flat-at-primes-over}) holds for\n$(R' \\to S \\otimes_R R', I', M \\otimes_R R')$, then there exists\na $\\lambda \\in \\Lambda$ such that\n(\\ref{equation-flat-at-primes-over}) holds for\n$(R_\\lambda \\to S \\otimes_R R_\\lambda, I_\\lambda, M \\otimes_R R_\\lambda)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a closed subset of the base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LM","source_file":"more-algebra.tex","source_line":4142,"source_end_line":4155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4142-L4155","statement_sha256":"908eb810fa5f6bd05ad1829dc9f0f1ead96dd1c7e8ff77f3dbc73fdbb0b226ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":2918,"rank":2918,"depth":35,"x":628.859,"y":699.341,"cluster":"advanced-algebra"},{"id":"stacks:05LR","tag":"05LR","title":"Flattening over a closed subsets of source and base · Lemma 05LR","summary":"In Situation [Tag 05LP] let R' → R\" be an R-algebra map. Let I' ⊂ R' and I'R\" ⊂ I\" ⊂ R\" be ideals. If ([Tag 05LQ]) holds for (R', I'), then ([Tag 05LQ]) holds for (R\", I\").","statement_latex":"In Situation \\ref{situation-flattening-general}\nlet $R' \\to R''$ be an $R$-algebra map.\nLet $I' \\subset R'$ and $I'R'' \\subset I'' \\subset R''$ be ideals.\nIf (\\ref{equation-flat-at-primes}) holds for\n$(R', I')$, then (\\ref{equation-flat-at-primes})\nholds for $(R'', I'')$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a closed subsets of source and base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LR","source_file":"more-algebra.tex","source_line":4232,"source_end_line":4240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4232-L4240","statement_sha256":"ee2af58ef25df9c59dadf25f9cb24c280506bd4f7e066a758cb3994d5d1f7100","origin":"The Stacks Project","memory_eligible":false,"source_rank":2919,"rank":2919,"depth":3,"x":702.268,"y":636.501,"cluster":"advanced-algebra"},{"id":"stacks:05LS","tag":"05LS","title":"Flattening over a closed subsets of source and base · Lemma 05LS","summary":"In Situation [Tag 05LP] let R' → R\" be an R-algebra map. Let I' ⊂ R' and I'R\" ⊂ I\" ⊂ R\" be ideals. Assume • the map V(I\") → V(I') induced by Spec(R\") → Spec(R') is surjective, and • R\"_ p\" is flat over R' for all primes p\" ∈ V(I\"). If ([Tag 05LQ]) holds for (R\", I\"), then ([Tag 05LQ]) holds for (R', I').","statement_latex":"In Situation \\ref{situation-flattening-general}\nlet $R' \\to R''$ be an $R$-algebra map.\nLet $I' \\subset R'$ and $I'R'' \\subset I'' \\subset R''$ be ideals.\nAssume\n\\begin{enumerate}\n\\item the map $V(I'') \\to V(I')$ induced by\n$\\Spec(R'') \\to \\Spec(R')$ is surjective, and\n\\item $R''_{\\mathfrak p''}$ is flat over $R'$ for all primes\n$\\mathfrak p'' \\in V(I'')$.\n\\end{enumerate}\nIf (\\ref{equation-flat-at-primes}) holds for\n$(R'', I'')$, then (\\ref{equation-flat-at-primes}) holds for $(R', I')$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a closed subsets of source and base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LS","source_file":"more-algebra.tex","source_line":4256,"source_end_line":4270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4256-L4270","statement_sha256":"c04933ca05ad5bf0f9dc379500cf9654b672f880ee15bdc77dfe726b47444304","origin":"The Stacks Project","memory_eligible":false,"source_rank":2920,"rank":2920,"depth":5,"x":698.653,"y":724.934,"cluster":"advanced-algebra"},{"id":"stacks:05LT","tag":"05LT","title":"Flattening over a closed subsets of source and base · Lemma 05LT","summary":"In Situation [Tag 05LP] assume R → S is essentially of finite presentation and M is an S-module of finite presentation. Let R' = colim_λ ∈ Lambda R_λ be a directed colimit of R-algebras. Let I_λ ⊂ R_λ be ideals such that I_λ R_μ ⊂ I_μ for all μ ≥ λ and set I' = colim_λ I_λ. If ([Tag 05LQ]) holds for (R', I'), then there exists a λ ∈ Lambda such that ([Tag 05LQ]) holds for (R_λ, I_λ).","statement_latex":"In Situation \\ref{situation-flattening-general}\nassume $R \\to S$ is essentially of finite presentation\nand $M$ is an $S$-module of finite presentation. Let\n$R' = \\colim_{\\lambda \\in \\Lambda} R_\\lambda$\nbe a directed colimit of $R$-algebras. Let $I_\\lambda \\subset R_\\lambda$\nbe ideals such that $I_\\lambda R_\\mu \\subset I_\\mu$ for all\n$\\mu \\geq \\lambda$ and set $I' = \\colim_\\lambda I_\\lambda$.\nIf (\\ref{equation-flat-at-primes}) holds for\n$(R', I')$, then there exists a $\\lambda \\in \\Lambda$ such that\n(\\ref{equation-flat-at-primes}) holds for $(R_\\lambda, I_\\lambda)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a closed subsets of source and base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LT","source_file":"more-algebra.tex","source_line":4294,"source_end_line":4306,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4294-L4306","statement_sha256":"93b534fc463fa2fc6a8e950b6998be52ac492a5a0c33f49b253b526b47ba892d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2921,"rank":2921,"depth":35,"x":629.865,"y":657.339,"cluster":"advanced-algebra"},{"id":"stacks:05LU","tag":"05LU","title":"Flattening over a closed subsets of source and base · Lemma 05LU","summary":"In Situation [Tag 05LP]. Let I ⊂ R be an ideal. Assume • R is a Noetherian ring, • S is a Noetherian ring, • M is a finite S-module, and • for each n ≥ 1 and any prime q ∈ V(J + IS) the module (M/I^n M)_ q is flat over R/I^n. Then ([Tag 05LQ]) holds for (R, I), i.e., for every prime q ∈ V(J + IS) the localization M_ q is flat over R.","statement_latex":"In Situation \\ref{situation-flattening-general}.\nLet $I \\subset R$ be an ideal. Assume\n\\begin{enumerate}\n\\item $R$ is a Noetherian ring,\n\\item $S$ is a Noetherian ring,\n\\item $M$ is a finite $S$-module, and\n\\item for each $n \\geq 1$ and any prime\n$\\mathfrak q \\in V(J + IS)$ the module $(M/I^n M)_{\\mathfrak q}$\nis flat over $R/I^n$.\n\\end{enumerate}\nThen (\\ref{equation-flat-at-primes}) holds for $(R, I)$, i.e.,\nfor every prime $\\mathfrak q \\in V(J + IS)$\nthe localization $M_{\\mathfrak q}$ is flat over $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a closed subsets of source and base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LU","source_file":"more-algebra.tex","source_line":4369,"source_end_line":4384,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4369-L4384","statement_sha256":"d1127be1d03252ed31d75737814e1ed09c634a7395796a8d0710df779e739fd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":2922,"rank":2922,"depth":10,"x":735.462,"y":668.204,"cluster":"advanced-algebra"},{"id":"stacks:0526","tag":"0526","title":"Flattening over a Noetherian complete local ring · Lemma 0526","summary":"Let R → S be a ring map. Let M be an S-module. Assume • (R, m) is a complete local Noetherian ring, • S is a Noetherian ring, and • M is finite over S. Then there exists an ideal I ⊂ m such that • (M/IM)_ q is flat over R/I for all primes q of S/IS lying over m, and • if J ⊂ R is an ideal such that (M/JM)_ q is flat over R/J for all primes q lying over m, then I ⊂ J. In other words, I is the smallest ideal of R such that ([Tag 052W]) holds for (overlineR → overlineS,…","statement_latex":"Let $R \\to S$ be a ring map.\nLet $M$ be an $S$-module.\nAssume\n\\begin{enumerate}\n\\item $(R, \\mathfrak m)$ is a complete local Noetherian ring,\n\\item $S$ is a Noetherian ring, and\n\\item $M$ is finite over $S$.\n\\end{enumerate}\nThen there exists an ideal $I \\subset \\mathfrak m$ such that\n\\begin{enumerate}\n\\item $(M/IM)_{\\mathfrak q}$ is flat over $R/I$ for all\nprimes $\\mathfrak q$ of $S/IS$ lying over $\\mathfrak m$, and\n\\item if $J \\subset R$ is an ideal such that $(M/JM)_{\\mathfrak q}$\nis flat over $R/J$ for all primes $\\mathfrak q$ lying over\n$\\mathfrak m$, then $I \\subset J$.\n\\end{enumerate}\nIn other words, $I$ is the smallest ideal of $R$ such that\n(\\ref{equation-flat-at-primes-over}) holds for\n$(\\overline{R} \\to \\overline{S}, \\overline{\\mathfrak m}, \\overline{M})$\nwhere $\\overline{R} = R/I$, $\\overline{S} = S/IS$,\n$\\overline{\\mathfrak m} = \\mathfrak m/I$ and $\\overline{M} = M/IM$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a Noetherian complete local ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0526","source_file":"more-algebra.tex","source_line":4411,"source_end_line":4434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4411-L4434","statement_sha256":"f015d1542ccc0e832a8d20d93b4c23e766d4707df3aea26d38dd4c168536fb16","origin":"The Stacks Project","memory_eligible":false,"source_rank":2923,"rank":2923,"depth":10,"x":648.436,"y":720.363,"cluster":"advanced-algebra"},{"id":"stacks:0527","tag":"0527","title":"Flattening over a Noetherian complete local ring · Lemma 0527","summary":"With notation R → S, M, and I and assumptions as in Lemma [Tag 0526]. Consider a local homomorphism of local rings φ : (R, m) → (R', m') such that R' is Noetherian. Then the following are equivalent • condition ([Tag 052W]) holds for (R' → S ⊗_R R', m', M ⊗_R R'), and • φ(I) = 0.","statement_latex":"With notation $R \\to S$, $M$, and $I$ and assumptions as in\nLemma \\ref{lemma-flattening-complete-local-noetherian}.\nConsider a local homomorphism of local rings\n$\\varphi : (R, \\mathfrak m) \\to (R', \\mathfrak m')$\nsuch that $R'$ is Noetherian. Then the following are equivalent\n\\begin{enumerate}\n\\item condition (\\ref{equation-flat-at-primes-over}) holds\nfor $(R' \\to S \\otimes_R R', \\mathfrak m', M \\otimes_R R')$, and\n\\item $\\varphi(I) = 0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a Noetherian complete local ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0527","source_file":"more-algebra.tex","source_line":4461,"source_end_line":4473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4461-L4473","statement_sha256":"121b3f9a768166b0e07262c8fcfcba861e32c1c38586f8831dbe6e7bcaddd57e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2924,"rank":2924,"depth":11,"x":670.771,"y":632.098,"cluster":"advanced-algebra"},{"id":"stacks:0528","tag":"0528","title":"Flattening over a Noetherian complete local ring · Lemma 0528","summary":"With notation R → S, M, and I and assumptions as in Lemma [Tag 0526]. In addition assume that R → S is of finite type. Then for any local homomorphism of local rings φ : (R, m) → (R', m') the following are equivalent • condition ([Tag 052W]) holds for (R' → S ⊗_R R', m', M ⊗_R R'), and • φ(I) = 0.","statement_latex":"With notation $R \\to S$, $M$, and $I$ and assumptions as in\nLemma \\ref{lemma-flattening-complete-local-noetherian}.\nIn addition assume that $R \\to S$ is of finite type.\nThen for any local homomorphism of local rings\n$\\varphi : (R, \\mathfrak m) \\to (R', \\mathfrak m')$\nthe following are equivalent\n\\begin{enumerate}\n\\item condition (\\ref{equation-flat-at-primes-over}) holds\nfor $(R' \\to S \\otimes_R R', \\mathfrak m', M \\otimes_R R')$, and\n\\item $\\varphi(I) = 0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flattening over a Noetherian complete local ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0528","source_file":"more-algebra.tex","source_line":4508,"source_end_line":4521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4508-L4521","statement_sha256":"af82b49747d3b148fb637cc72dbc5dda8aeb327be71a8e5c51370122de43313f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2925,"rank":2925,"depth":36,"x":725.542,"y":710.229,"cluster":"advanced-algebra"},{"id":"stacks:052Z","tag":"052Z","title":"Descent of flatness along integral maps · Lemma 052Z","summary":"Let R be a ring. Let P(T) be a monic polynomial with coefficients in R. Let α ∈ R be such that P(α) = 0. Then P(T) = (T - α)Q(T) for some monic polynomial Q(T) ∈ R[T].","statement_latex":"Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients\nin $R$. Let $\\alpha \\in R$ be such that $P(\\alpha) = 0$. Then\n$P(T) = (T - \\alpha)Q(T)$ for some monic polynomial $Q(T) \\in R[T]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Descent of flatness along integral maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052Z","source_file":"more-algebra.tex","source_line":4551,"source_end_line":4556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4551-L4556","statement_sha256":"44cf8d650903aed6281bfaa060cbe4d79964c32bf3fd67765ca3052b374261c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2926,"rank":2926,"depth":0,"x":621.837,"y":683.568,"cluster":"advanced-algebra"},{"id":"stacks:0530","tag":"0530","title":"Descent of flatness along integral maps · Lemma 0530","summary":"Let R be a ring. Let P(T) be a monic polynomial with coefficients in R. There exists a finite free ring map R → R' such that P(T) = (T - α)Q(T) for some α ∈ R' and some monic polynomial Q(T) ∈ R'[T].","statement_latex":"Let $R$ be a ring. Let $P(T)$ be a monic polynomial with coefficients\nin $R$. There exists a finite free ring map $R \\to R'$ such that\n$P(T) = (T - \\alpha)Q(T)$ for some $\\alpha \\in R'$ and some\nmonic polynomial $Q(T) \\in R'[T]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Descent of flatness along integral maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0530","source_file":"more-algebra.tex","source_line":4567,"source_end_line":4573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4567-L4573","statement_sha256":"d86ff3dc0e76b2a7c4ddd3d9b41f0664ef7e332d6dc5bc8e260929ad2ca2a8fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2927,"rank":2927,"depth":1,"x":720.206,"y":644.2,"cluster":"advanced-algebra"},{"id":"stacks:0531","tag":"0531","title":"Descent of flatness along integral maps · Lemma 0531","summary":"Let R → S be a finite ring map. There exists a finite free ring extension R ⊂ R' such that S ⊗_R R' is a quotient of a ring of the form R'[T_1, …, T_n]/(P_1(T_1), …, P_n(T_n)) with P_i(T) = ∏_j = 1, …, d_i (T - α_ij) for some α_ij ∈ R'.","statement_latex":"Let $R \\to S$ be a finite ring map.\nThere exists a finite free ring extension $R \\subset R'$ such\nthat $S \\otimes_R R'$ is a quotient of a ring of the form\n$$\nR'[T_1, \\ldots, T_n]/(P_1(T_1), \\ldots, P_n(T_n))\n$$\nwith $P_i(T) = \\prod_{j = 1, \\ldots, d_i} (T - \\alpha_{ij})$ for some\n$\\alpha_{ij} \\in R'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Descent of flatness along integral maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0531","source_file":"more-algebra.tex","source_line":4583,"source_end_line":4593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4583-L4593","statement_sha256":"048ce44e3f42af25eaef9c8c9e9eb5508fb77152303d5af3e3e9ee9e9c63970b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2928,"rank":2928,"depth":4,"x":679.139,"y":729.44,"cluster":"advanced-algebra"},{"id":"stacks:0532","tag":"0532","title":"Descent of flatness along integral maps · Lemma 0532","summary":"Let R be a ring. Let S = R[T_1, …, T_n]/J. Assume J contains elements of the form P_i(T_i) with P_i(T) = ∏_j = 1, …, d_i (T - α_ij) for some α_ij ∈ R. For underlinek = (k_1, …, k_n) with 1 ≤ k_i ≤ d_i consider the ring map Φ_underlinek : R[T_1, …, T_n] → R, T_i ↦ α_ik_i Set J_underlinek = Φ_underlinek(J). Then the image of Spec(S) → Spec(R) is equal to V(⋂ J_underlinek).","statement_latex":"Let $R$ be a ring.\nLet $S = R[T_1, \\ldots, T_n]/J$.\nAssume $J$ contains elements of the form $P_i(T_i)$\nwith $P_i(T) = \\prod_{j = 1, \\ldots, d_i} (T - \\alpha_{ij})$ for some\n$\\alpha_{ij} \\in R$. For $\\underline{k} = (k_1, \\ldots, k_n)$\nwith $1 \\leq k_i \\leq d_i$ consider the ring map\n$$\n\\Phi_{\\underline{k}} : R[T_1, \\ldots, T_n] \\to R,\n\\quad\nT_i \\longmapsto \\alpha_{ik_i}\n$$\nSet $J_{\\underline{k}} = \\Phi_{\\underline{k}}(J)$.\nThen the image of $\\Spec(S) \\to \\Spec(R)$ is equal to\n$V(\\bigcap J_{\\underline{k}})$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Descent of flatness along integral maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0532","source_file":"more-algebra.tex","source_line":4613,"source_end_line":4629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4613-L4629","statement_sha256":"be8aca22e80a2e93667d4b5d246136b7ad6f5e51201f1ea2d41492438c769436","origin":"The Stacks Project","memory_eligible":false,"source_rank":2929,"rank":2929,"depth":0,"x":640.698,"y":642.886,"cluster":"advanced-algebra"},{"id":"stacks:0533","tag":"0533","title":"Descent of flatness along integral maps · Lemma 0533","summary":"Let R → S be a finite injective homomorphism of Noetherian rings. Let M be an R-module. If M ⊗_R S is a flat S-module, then M is a flat R-module.","statement_latex":"Let $R \\to S$ be a finite injective homomorphism of Noetherian rings.\nLet $M$ be an $R$-module. If $M \\otimes_R S$ is a flat $S$-module,\nthen $M$ is a flat $R$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Descent of flatness along integral maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0533","source_file":"more-algebra.tex","source_line":4639,"source_end_line":4644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4639-L4644","statement_sha256":"eaeb96fdf8d1d11c63224e7cc8609c5784f7feda7414d70a26f7e0ce17ed5b8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2930,"rank":2930,"depth":6,"x":739.093,"y":685.089,"cluster":"advanced-algebra"},{"id":"stacks:0534","tag":"0534","title":"Descent of flatness along integral maps · Lemma 0534","summary":"Let R → S be an injective integral ring map. Let M be a finitely presented module over R[x_1, …, x_n]. If M ⊗_R S is flat over S, then M is flat over R.","statement_latex":"Let $R \\to S$ be an injective integral ring map.\nLet $M$ be a finitely presented module over $R[x_1, \\ldots, x_n]$.\nIf $M \\otimes_R S$ is flat over $S$, then $M$ is flat over $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Descent of flatness along integral maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0534","source_file":"more-algebra.tex","source_line":4683,"source_end_line":4688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4683-L4688","statement_sha256":"9c1e5c38274224ebc346fd6a4d9f63b584f506ecb75059f44f0fe226338b9e8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2931,"rank":2931,"depth":35,"x":632.118,"y":709.912,"cluster":"advanced-algebra"},{"id":"stacks:0H7R","tag":"0H7R","title":"Descent of flatness along integral maps · Lemma 0H7R","summary":"Let R → S be a finite injective homomorphism of Noetherian rings. Let P be an R-module. If P ⊗_R S is a projective S-module, then P is a projective R-module.","statement_latex":"Let $R \\to S$ be a finite injective homomorphism of Noetherian rings.\nLet $P$ be an $R$-module. If $P \\otimes_R S$ is a projective $S$-module,\nthen $P$ is a projective $R$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Descent of flatness along integral maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7R","source_file":"more-algebra.tex","source_line":4721,"source_end_line":4726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4721-L4726","statement_sha256":"b4997e49dc0e5e5c2a2cb1b3a86d47ee0e449f9a099e20e985e189679567c180","origin":"The Stacks Project","memory_eligible":false,"source_rank":2932,"rank":2932,"depth":10,"x":691.303,"y":630.555,"cluster":"advanced-algebra"},{"id":"stacks:0536","tag":"0536","title":"Torsion free modules · Definition 0536","summary":"Let R be a domain. Let M be an R-module. • We say an element x ∈ M is torsion if there exists a nonzero f ∈ R such that fx = 0. • We say M is torsion free if the only torsion element of M is 0. • We say M is a torsion module if every element of M is torsion.","statement_latex":"Let $R$ be a domain. Let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item We say an element $x \\in M$ is {\\it torsion} if there exists\na nonzero $f \\in R$ such that $fx = 0$.\n\\item We say $M$ is {\\it torsion free} if the only torsion element of $M$\nis $0$.\n\\item We say $M$ is a {\\it torsion module} if every element of $M$\nis torsion.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0536","source_file":"more-algebra.tex","source_line":4781,"source_end_line":4792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4781-L4792","statement_sha256":"4103c1362d4dd6d5480e1575feee3236d3e144d5b8538e75d21307c059c711e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2933,"rank":2933,"depth":0,"x":711.566,"y":723.063,"cluster":"advanced-algebra"},{"id":"stacks:0537","tag":"0537","title":"Torsion free modules · Lemma 0537","summary":"Let R be a domain. Let M be an R-module. The set of torsion elements M_tors of M is the kernel of the map M → M ⊗_R K. Thus M_tors is an R-submodule of M. The quotient module M/M_tors is torsion free.","statement_latex":"Let $R$ be a domain. Let $M$ be an $R$-module. The set of torsion elements\n$M_{tors}$ of $M$ is the kernel of the map $M \\to M \\otimes_R K$.\nThus $M_{tors}$ is an $R$-submodule of $M$.\nThe quotient module $M/M_{tors}$ is torsion free.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0537","source_file":"more-algebra.tex","source_line":4801,"source_end_line":4807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4801-L4807","statement_sha256":"a06f21588f2008e34f18e0419ec2cbcb6393c35c0ffc610173e897917d89ec13","origin":"The Stacks Project","memory_eligible":false,"source_rank":2934,"rank":2934,"depth":0,"x":621.84,"y":666.097,"cluster":"advanced-algebra"},{"id":"stacks:0AUR","tag":"0AUR","title":"Torsion free modules · Lemma 0AUR","summary":"Let R be a domain. Let M be a torsion free R-module. For any multiplicative set S ⊂ R the module S^-1M is a torsion free S^-1R-module.","statement_latex":"Let $R$ be a domain. Let $M$ be a torsion free $R$-module.\nFor any multiplicative set $S \\subset R$ the module\n$S^{-1}M$ is a torsion free $S^{-1}R$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUR","source_file":"more-algebra.tex","source_line":4813,"source_end_line":4818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4813-L4818","statement_sha256":"b9fb2c1d9a00e6cd4914f51a2e1c46f2d72779654a32bccb1123cd81614169c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2935,"rank":2935,"depth":0,"x":734.301,"y":657.152,"cluster":"advanced-algebra"},{"id":"stacks:0AXM","tag":"0AXM","title":"Torsion free modules · Lemma 0AXM","summary":"Let R → R' be a flat homomorphism of domains. If M is a torsion free R-module, then M ⊗_R R' is a torsion free R'-module.","statement_latex":"Let $R \\to R'$ be a flat homomorphism of domains. If $M$ is a torsion\nfree $R$-module, then $M \\otimes_R R'$ is a torsion free $R'$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXM","source_file":"more-algebra.tex","source_line":4824,"source_end_line":4828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4824-L4828","statement_sha256":"16157753842ce71e9513b24f8a2840921bc2ef8208230291d8109ad15c548683","origin":"The Stacks Project","memory_eligible":false,"source_rank":2936,"rank":2936,"depth":0,"x":658.24,"y":727.864,"cluster":"advanced-algebra"},{"id":"stacks:0AUS","tag":"0AUS","title":"Torsion free modules · Lemma 0AUS","summary":"Let R be a domain. Let 0 → M → M' → M\" → 0 be a short exact sequence of R-modules. If M and M\" are torsion free, then M' is torsion free.","statement_latex":"Let $R$ be a domain. Let $0 \\to M \\to M' \\to M'' \\to 0$\nbe a short exact sequence of $R$-modules. If $M$ and $M''$\nare torsion free, then $M'$ is torsion free.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUS","source_file":"more-algebra.tex","source_line":4839,"source_end_line":4844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4839-L4844","statement_sha256":"5a303e0c8894f1ec2853022f11486f2b94b9a735f14260936980ae32e83612f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":2937,"rank":2937,"depth":0,"x":657.459,"y":632.156,"cluster":"advanced-algebra"},{"id":"stacks:0AUT","tag":"0AUT","title":"Torsion free modules · Lemma 0AUT","summary":"Let R be a domain. Let M be an R-module. Then M is torsion free if and only if M_ m is a torsion free R_ m-module for all maximal ideals m of R.","statement_latex":"Let $R$ be a domain. Let $M$ be an $R$-module.\nThen $M$ is torsion free if and only if $M_\\mathfrak m$ is a\ntorsion free $R_\\mathfrak m$-module for all maximal ideals\n$\\mathfrak m$ of $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUT","source_file":"more-algebra.tex","source_line":4850,"source_end_line":4856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4850-L4856","statement_sha256":"d3c476691900b345e950a5c1c7b8ab260c5cbe42780ae9377a4924d510c77017","origin":"The Stacks Project","memory_eligible":false,"source_rank":2938,"rank":2938,"depth":2,"x":735.33,"y":702.584,"cluster":"advanced-algebra"},{"id":"stacks:0AUU","tag":"0AUU","title":"Torsion free modules · Lemma 0AUU","summary":"Let R be a domain. Let M be a finite R-module. Then M is torsion free if and only if M is a submodule of a finite free module.","statement_latex":"Let $R$ be a domain. Let $M$ be a finite $R$-module.\nThen $M$ is torsion free if and only if $M$ is a\nsubmodule of a finite free module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUU","source_file":"more-algebra.tex","source_line":4863,"source_end_line":4868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4863-L4868","statement_sha256":"b1e9fc5d50d00d4b7b7436b4b20b0f9e4b4b783b3eb9a930a3b95df0518a653c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2939,"rank":2939,"depth":0,"x":620.792,"y":694.804,"cluster":"advanced-algebra"},{"id":"stacks:0AUV","tag":"0AUV","title":"Torsion free modules · Lemma 0AUV","summary":"Let R be a Noetherian domain. Let M be a nonzero finite R-module. The following are equivalent • M is torsion free, • M is a submodule of a finite free module, • (0) is the only associated prime of M, • (0) is in the support of M and M has property (S_1), and • (0) is in the support of M and M has no embedded associated prime.","statement_latex":"Let $R$ be a Noetherian domain. Let $M$ be a nonzero finite $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ is torsion free,\n\\item $M$ is a submodule of a finite free module,\n\\item $(0)$ is the only associated prime of $M$,\n\\item $(0)$ is in the support of $M$ and $M$ has property $(S_1)$, and\n\\item $(0)$ is in the support of $M$ and $M$ has no embedded associated prime.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUV","source_file":"more-algebra.tex","source_line":4883,"source_end_line":4894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4883-L4894","statement_sha256":"6fb56d65ab9684801ed6303ca68bb90b078637c67c5d490bfea6b86a00660800","origin":"The Stacks Project","memory_eligible":false,"source_rank":2940,"rank":2940,"depth":12,"x":711.886,"y":635.303,"cluster":"advanced-algebra"},{"id":"stacks:0538","tag":"0538","title":"Torsion free modules · Lemma 0538","summary":"Let R be a domain. Any flat R-module is torsion free.","statement_latex":"Let $R$ be a domain. Any flat $R$-module is torsion free.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0538","source_file":"more-algebra.tex","source_line":4913,"source_end_line":4916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4913-L4916","statement_sha256":"cbe696a9b22157cc68eaa314379f9dada5b5251038cabcee6e2c81dc1c3856ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":2941,"rank":2941,"depth":0,"x":692.483,"y":731.262,"cluster":"advanced-algebra"},{"id":"stacks:0539","tag":"0539","title":"Torsion free modules · Lemma 0539","summary":"Let A be a valuation ring. An A-module M is flat over A if and only if M is torsion free.","statement_latex":"Let $A$ be a valuation ring.\nAn $A$-module $M$ is flat over $A$ if and only if $M$ is torsion free.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0539","source_file":"more-algebra.tex","source_line":4924,"source_end_line":4928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4924-L4928","statement_sha256":"430612a55df82dbeee9c9ea69f869f15dd7bad05613e7412bb2e60d212f63554","origin":"The Stacks Project","memory_eligible":false,"source_rank":2942,"rank":2942,"depth":1,"x":629.366,"y":649.158,"cluster":"advanced-algebra"},{"id":"stacks:0AUW","tag":"0AUW","title":"Torsion free modules · Lemma 0AUW","summary":"Let A be a Dedekind domain (for example a discrete valuation ring or more generally a PID). • An A-module is flat if and only if it is torsion free. • A finite torsion free A-module is finite locally free. • A finite torsion free A-module is finite free if A is a PID.","statement_latex":"Let $A$ be a Dedekind domain (for example a discrete valuation ring\nor more generally a PID).\n\\begin{enumerate}\n\\item An $A$-module is flat if and only if it is torsion free.\n\\item A finite torsion free $A$-module is finite locally free.\n\\item A finite torsion free $A$-module is finite free if\n$A$ is a PID.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUW","source_file":"more-algebra.tex","source_line":4953,"source_end_line":4963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4953-L4963","statement_sha256":"c0aef7f2679e9311ec2d22a5d235a12be490fd351af60fd70a096ab13230ac2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2943,"rank":2943,"depth":17,"x":742.391,"y":673.987,"cluster":"advanced-algebra"},{"id":"stacks:0AUX","tag":"0AUX","title":"Torsion free modules · Lemma 0AUX","summary":"Let R be a domain. Let M, N be R-modules. If N is torsion free, so is Hom_R(M, N).","statement_latex":"Let $R$ be a domain. Let $M$, $N$ be $R$-modules.\nIf $N$ is torsion free, so is $\\Hom_R(M, N)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUX","source_file":"more-algebra.tex","source_line":4995,"source_end_line":4999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L4995-L4999","statement_sha256":"540e3bad2405084202fcf50d1f2fc0ab375dcd8361e91288c60bd6c8e217f217","origin":"The Stacks Project","memory_eligible":false,"source_rank":2944,"rank":2944,"depth":0,"x":638.664,"y":719.996,"cluster":"advanced-algebra"},{"id":"stacks:0H9U","tag":"0H9U","title":"Ranks of modules · Definition 0H9U","summary":"Let R be a domain with fraction field K. The rank of an R-module M is dim_K(M ⊗_R K).","statement_latex":"Let $R$ be a domain with fraction field $K$.\nThe {\\it rank} of an $R$-module $M$ is $\\dim_K(M \\otimes_R K)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ranks of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9U","source_file":"more-algebra.tex","source_line":5015,"source_end_line":5019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5015-L5019","statement_sha256":"8e84a3ea4b3c8f63a38d16debf4183633120ba5aa4aa49f944aeead099f80680","origin":"The Stacks Project","memory_eligible":false,"source_rank":2945,"rank":2945,"depth":0,"x":678.31,"y":626.841,"cluster":"advanced-algebra"},{"id":"stacks:0H9V","tag":"0H9V","title":"Ranks of modules · Lemma 0H9V","summary":"Let R be a domain. If M → M' is a map of R-modules whose kernel and cokernel are torsion, then the rank of M equals the rank of M'.","statement_latex":"Let $R$ be a domain. If $M \\to M'$ is a map of $R$-modules\nwhose kernel and cokernel are torsion, then the rank of $M$\nequals the rank of $M'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9V","source_file":"more-algebra.tex","source_line":5026,"source_end_line":5031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5026-L5031","statement_sha256":"4aead533dc30609da9acced1c87057923cb75d0eacdf50abe92df838e5728366","origin":"The Stacks Project","memory_eligible":false,"source_rank":2946,"rank":2946,"depth":0,"x":724.169,"y":718.391,"cluster":"advanced-algebra"},{"id":"stacks:0H9W","tag":"0H9W","title":"Ranks of modules · Lemma 0H9W","summary":"Let R be a domain. Let 0 → M → M' → M\" → 0 be a short exact sequence of R-modules. Then the rank of M' is the sum of the ranks of M and M\".","statement_latex":"Let $R$ be a domain. Let $0 \\to M \\to M' \\to M'' \\to 0$ be a short \nexact sequence of $R$-modules. Then the rank of $M'$ is the\nsum of the ranks of $M$ and $M''$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9W","source_file":"more-algebra.tex","source_line":5038,"source_end_line":5043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5038-L5043","statement_sha256":"da96a40d489c14703d3b28d6a10bce9c1920bb3ba25fb62fb52c4e9197ab8273","origin":"The Stacks Project","memory_eligible":false,"source_rank":2947,"rank":2947,"depth":0,"x":616.309,"y":676.741,"cluster":"advanced-algebra"},{"id":"stacks:0H9X","tag":"0H9X","title":"Ranks of modules · Lemma 0H9X","summary":"Let R be a domain. Let M and N be R-modules. • The rank of M ⊗_R N is the product of the ranks of M and N. • If M is a finitely presented R-module, then the rank of Hom_R(M, N) is the product of the ranks of M and N.","statement_latex":"Let $R$ be a domain. Let $M$ and $N$ be $R$-modules.\n\\begin{enumerate}\n\\item The rank of $M \\otimes_R N$ is the product of the\nranks of $M$ and $N$.\n\\item If $M$ is a finitely presented $R$-module, then\nthe rank of $\\Hom_R(M, N)$ is the product of the\nranks of $M$ and $N$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9X","source_file":"more-algebra.tex","source_line":5049,"source_end_line":5059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5049-L5059","statement_sha256":"f5598e099e27ae617ec9d8b83b266feb8fe9603c17bf92a299dfdb3d62f0fab6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2948,"rank":2948,"depth":2,"x":729.779,"y":646.132,"cluster":"advanced-algebra"},{"id":"stacks:0H9Y","tag":"0H9Y","title":"Ranks of modules · Lemma 0H9Y","summary":"Let R ⊂ R' be an extension of domains. If M is an R-module, then the rank of M over R is equal to the rank of M ⊗_R R' over R'.","statement_latex":"Let $R \\subset R'$ be an extension of domains. If $M$ is an $R$-module,\nthen the rank of $M$ over $R$ is equal to the rank of $M \\otimes_R R'$\nover $R'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9Y","source_file":"more-algebra.tex","source_line":5071,"source_end_line":5076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5071-L5076","statement_sha256":"38ff0fda50ffc864e2f12960d17c70dd035a2e742614fabe3b8d507390479d3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2949,"rank":2949,"depth":0,"x":670.493,"y":733.425,"cluster":"advanced-algebra"},{"id":"stacks:0H9Z","tag":"0H9Z","title":"Ranks of modules · Lemma 0H9Z","summary":"Let R be a domain. Let M be a finite R-module. Then • M has finite rank r ≥ 0, • there is a map M → R^⊕ r whose kernel and cokernel are torsion modules, • there is an f ∈ R, f not = 0 such that M_f is free of rank r, and • there is an injective map R^⊕ r → M whose cokernel is a torsion module.","statement_latex":"Let $R$ be a domain. Let $M$ be a finite $R$-module. Then\n\\begin{enumerate}\n\\item $M$ has finite rank $r \\geq 0$,\n\\item there is a map $M \\to R^{\\oplus r}$ whose kernel and cokernel\nare torsion modules,\n\\item there is an $f \\in R$, $f \\not = 0$ such that $M_f$\nis free of rank $r$, and\n\\item there is an injective map $R^{\\oplus r} \\to M$\nwhose cokernel is a torsion module.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9Z","source_file":"more-algebra.tex","source_line":5083,"source_end_line":5095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5083-L5095","statement_sha256":"721e6559f17087a77dc20db5f6a94438c3c3f1bd8789804b0986c33af4cda6d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2950,"rank":2950,"depth":0,"x":643.91,"y":635.039,"cluster":"advanced-algebra"},{"id":"stacks:0AUZ","tag":"0AUZ","title":"Reflexive modules · Definition 0AUZ","summary":"Let R be a domain. We say an R-module M is reflexive if the natural map j : M → Hom_R(Hom_R(M, R), R) which sends m ∈ M to the map sending φ ∈ Hom_R(M, R) to φ(m) ∈ R is an isomorphism.","statement_latex":"Let $R$ be a domain. We say an $R$-module $M$ is {\\it reflexive} if\nthe natural map\n$$\nj : M \\longrightarrow \\Hom_R(\\Hom_R(M, R), R)\n$$\nwhich sends $m \\in M$ to the map sending $\\varphi \\in \\Hom_R(M, R)$\nto $\\varphi(m) \\in R$ is an isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUZ","source_file":"more-algebra.tex","source_line":5148,"source_end_line":5157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5148-L5157","statement_sha256":"5b761c17dd3636e574c1d0eeb864d77904e912336dfcf3fdbbc749fb64db9475","origin":"The Stacks Project","memory_eligible":false,"source_rank":2951,"rank":2951,"depth":0,"x":743.007,"y":692.723,"cluster":"advanced-algebra"},{"id":"stacks:0AV0","tag":"0AV0","title":"Reflexive modules · Lemma 0AV0","summary":"Let R be a domain and let M be an R-module. • If M is reflexive, then M is torsion free. • If M is finite, then the kernel and cokernel of j : M → Hom_R(Hom_R(M, R), R) are torsion modules. • If M is finite, then j is injective if and only if M is torsion free.","statement_latex":"Let $R$ be a domain and let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item If $M$ is reflexive, then $M$ is torsion free.\n\\item If $M$ is finite, then the kernel and cokernel of\n$j : M \\to \\Hom_R(\\Hom_R(M, R), R)$ are torsion modules.\n\\item If $M$ is finite, then $j$ is injective\nif and only if $M$ is torsion free.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV0","source_file":"more-algebra.tex","source_line":5165,"source_end_line":5175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5165-L5175","statement_sha256":"f6ab4500c77a803817f1bd540deb2c211de693832b45dc85cd42e27146066212","origin":"The Stacks Project","memory_eligible":false,"source_rank":2952,"rank":2952,"depth":1,"x":623.094,"y":706.47,"cluster":"advanced-algebra"},{"id":"stacks:0B36","tag":"0B36","title":"Reflexive modules · Lemma 0B36","summary":"Let R be a discrete valuation ring and let M be a finite R-module. Then the map j : M → Hom_R(Hom_R(M, R), R) is surjective.","statement_latex":"Let $R$ be a discrete valuation ring and let $M$ be a finite $R$-module.\nThen the map $j : M \\to \\Hom_R(\\Hom_R(M, R), R)$ is surjective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B36","source_file":"more-algebra.tex","source_line":5220,"source_end_line":5224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5220-L5224","statement_sha256":"914d1064041071d543f39bc7085e441ee5dcb3fe3c0a5f4213f40611851a94b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2953,"rank":2953,"depth":18,"x":700.753,"y":627.998,"cluster":"advanced-algebra"},{"id":"stacks:0AV1","tag":"0AV1","title":"Reflexive modules · Lemma 0AV1","summary":"Let R be a Noetherian domain. Let M be a finite R-module. The following are equivalent: • M is reflexive, • M_ p is a reflexive R_ p-module for all primes p ⊂ R, and • M_ m is a reflexive R_ m-module for all maximal ideals m of R.","statement_latex":"Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $M$ is reflexive,\n\\item $M_\\mathfrak p$ is a reflexive $R_\\mathfrak p$-module\nfor all primes $\\mathfrak p \\subset R$, and\n\\item $M_\\mathfrak m$ is a reflexive $R_\\mathfrak m$-module\nfor all maximal ideals $\\mathfrak m$ of $R$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV1","source_file":"more-algebra.tex","source_line":5233,"source_end_line":5244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5233-L5244","statement_sha256":"243183d53682d210d4142d645d25bb8d148802badbfd5cba69edc8cd9d03b08a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2954,"rank":2954,"depth":2,"x":706.614,"y":730.308,"cluster":"advanced-algebra"},{"id":"stacks:0EB8","tag":"0EB8","title":"Reflexive modules · Lemma 0EB8","summary":"Let R be a Noetherian domain. Let 0 → M → M' → M\" an exact sequence of finite R-modules. If M' is reflexive and M\" is torsion free, then M is reflexive.","statement_latex":"Let $R$ be a Noetherian domain. Let $0 \\to M \\to M' \\to M''$\nan exact sequence of finite $R$-modules. If $M'$ is reflexive\nand $M''$ is torsion free, then $M$ is reflexive.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EB8","source_file":"more-algebra.tex","source_line":5255,"source_end_line":5260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5255-L5260","statement_sha256":"9b67b189f47e8d08502eb51bb143cf94de6a1f0c939350c3d1786b913f5dabd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2955,"rank":2955,"depth":2,"x":619.697,"y":657.924,"cluster":"advanced-algebra"},{"id":"stacks:0AV2","tag":"0AV2","title":"Reflexive modules · Lemma 0AV2","summary":"Let R be a Noetherian domain. Let M be a finite R-module. The following are equivalent • M is reflexive, • there exists a short exact sequence 0 → M → F → N → 0 with F finite free and N torsion free.","statement_latex":"Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ is reflexive,\n\\item there exists a short exact sequence $0 \\to M \\to F \\to N \\to 0$\nwith $F$ finite free and $N$ torsion free.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV2","source_file":"more-algebra.tex","source_line":5308,"source_end_line":5317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5308-L5317","statement_sha256":"1f246f56ba8af18e93663dfb44d5b2bae098677a6b014e824b0e1b71413ef465","origin":"The Stacks Project","memory_eligible":false,"source_rank":2956,"rank":2956,"depth":3,"x":742.447,"y":661.995,"cluster":"advanced-algebra"},{"id":"stacks:0EB9","tag":"0EB9","title":"Reflexive modules · Lemma 0EB9","summary":"Let R → R' be a flat homomorphism of Noetherian domains. If M is a finite reflexive R-module, then M ⊗_R R' is a finite reflexive R'-module.","statement_latex":"Let $R \\to R'$ be a flat homomorphism of Noetherian domains.\nIf $M$ is a finite reflexive $R$-module, then $M \\otimes_R R'$\nis a finite reflexive $R'$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EB9","source_file":"more-algebra.tex","source_line":5332,"source_end_line":5337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5332-L5337","statement_sha256":"3ad78f4ab0dee5afd12923d5be9cd12928863f9cd1a27b48a240122f6e3e678d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2957,"rank":2957,"depth":4,"x":648.317,"y":728.887,"cluster":"advanced-algebra"},{"id":"stacks:0AV3","tag":"0AV3","title":"Reflexive modules · Lemma 0AV3","summary":"Let R be a Noetherian domain. Let M be a finite R-module. Let N be a finite reflexive R-module. Then Hom_R(M, N) is reflexive.","statement_latex":"Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.\nLet $N$ be a finite reflexive $R$-module. Then $\\Hom_R(M, N)$ is reflexive.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV3","source_file":"more-algebra.tex","source_line":5350,"source_end_line":5354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5350-L5354","statement_sha256":"844137e815d100a92956ee13282266c2162a79afb4cf73f4eceb0a635607e990","origin":"The Stacks Project","memory_eligible":false,"source_rank":2958,"rank":2958,"depth":3,"x":663.991,"y":625.778,"cluster":"advanced-algebra"},{"id":"stacks:0AV4","tag":"0AV4","title":"Reflexive modules · Definition 0AV4","summary":"Let R be a Noetherian domain. Let M be a finite R-module. The module M^** = Hom_R(Hom_R(M, R), R) is called the reflexive hull of M.","statement_latex":"Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.\nThe module $M^{**} = \\Hom_R(\\Hom_R(M, R), R)$ is called the\n{\\it reflexive hull} of $M$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV4","source_file":"more-algebra.tex","source_line":5366,"source_end_line":5371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5366-L5371","statement_sha256":"aadef762e81f9679b129c386408481f5bdf5590695f444a772ab7f3e7d8aa036","origin":"The Stacks Project","memory_eligible":false,"source_rank":2959,"rank":2959,"depth":0,"x":735.606,"y":711.009,"cluster":"advanced-algebra"},{"id":"stacks:0AV5","tag":"0AV5","title":"Reflexive modules · Lemma 0AV5","summary":"Let R be a Noetherian local ring. Let M, N be finite R-modules. • If N has depth ≥ 1, then Hom_R(M, N) has depth ≥ 1. • If N has depth ≥ 2, then Hom_R(M, N) has depth ≥ 2.","statement_latex":"Let $R$ be a Noetherian local ring. Let $M$, $N$ be finite $R$-modules.\n\\begin{enumerate}\n\\item If $N$ has depth $\\geq 1$, then $\\Hom_R(M, N)$ has depth $\\geq 1$.\n\\item If $N$ has depth $\\geq 2$, then $\\Hom_R(M, N)$ has depth $\\geq 2$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV5","source_file":"more-algebra.tex","source_line":5386,"source_end_line":5393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5386-L5393","statement_sha256":"791797dacbcc540912ae816fc414a188a212219081d9204cd5e7968a8f334245","origin":"The Stacks Project","memory_eligible":false,"source_rank":2960,"rank":2960,"depth":13,"x":613.825,"y":688.718,"cluster":"advanced-algebra"},{"id":"stacks:0AV6","tag":"0AV6","title":"Reflexive modules · Lemma 0AV6","summary":"Let R be a Noetherian ring. Let M, N be finite R-modules. • If N has property (S_1), then Hom_R(M, N) has property (S_1). • If N has property (S_2), then Hom_R(M, N) has property (S_2). • If R is a domain, N is torsion free and (S_2), then Hom_R(M, N) is torsion free and has property (S_2).","statement_latex":"Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules.\n\\begin{enumerate}\n\\item If $N$ has property $(S_1)$, then $\\Hom_R(M, N)$ has property $(S_1)$.\n\\item If $N$ has property $(S_2)$, then $\\Hom_R(M, N)$ has property $(S_2)$.\n\\item If $R$ is a domain, $N$ is torsion free and $(S_2)$, then\n$\\Hom_R(M, N)$ is torsion free and has property $(S_2)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV6","source_file":"more-algebra.tex","source_line":5410,"source_end_line":5419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5410-L5419","statement_sha256":"b6e1c7e6e16f409f0572255069077eac464105b1aba8177cc8551614d5e3a7ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":2961,"rank":2961,"depth":14,"x":721.933,"y":635.868,"cluster":"advanced-algebra"},{"id":"stacks:0AV7","tag":"0AV7","title":"Reflexive modules · Lemma 0AV7","summary":"Let R be a Noetherian ring. Let φ : M → N be a map of R-modules. Assume that for every prime p of R at least one of the following happens • M_ p → N_ p is injective, or • p not ∈ Ass(M). Then φ is injective.","statement_latex":"Let $R$ be a Noetherian ring. Let $\\varphi : M \\to N$ be a map of\n$R$-modules. Assume that for every prime $\\mathfrak p$\nof $R$ at least one of the following happens\n\\begin{enumerate}\n\\item $M_\\mathfrak p \\to N_\\mathfrak p$ is injective, or\n\\item $\\mathfrak p \\not \\in \\text{Ass}(M)$.\n\\end{enumerate}\nThen $\\varphi$ is injective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV7","source_file":"more-algebra.tex","source_line":5429,"source_end_line":5439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5429-L5439","statement_sha256":"a5d7df574ce5c2c85e1ea8df7675db4831a81491bf43b006829151aefe49650e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2962,"rank":2962,"depth":9,"x":684.585,"y":736.534,"cluster":"advanced-algebra"},{"id":"stacks:0AV8","tag":"0AV8","title":"Reflexive modules · Lemma 0AV8","summary":"Let R be a Noetherian ring. Let φ : M → N be a map of R-modules. Assume M is finite and that for every prime p of R one of the following happens • M_ p → N_ p is an isomorphism, or • depth(M_ p) ≥ 2 and p not ∈ Ass(N). Then φ is an isomorphism.","statement_latex":"Let $R$ be a Noetherian ring. Let $\\varphi : M \\to N$ be a map of\n$R$-modules. Assume $M$ is finite and that for every prime $\\mathfrak p$\nof $R$ one of the following happens\n\\begin{enumerate}\n\\item $M_\\mathfrak p \\to N_\\mathfrak p$ is an isomorphism, or\n\\item $\\text{depth}(M_\\mathfrak p) \\geq 2$ and\n$\\mathfrak p \\not \\in \\text{Ass}(N)$.\n\\end{enumerate}\nThen $\\varphi$ is an isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV8","source_file":"more-algebra.tex","source_line":5452,"source_end_line":5463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5452-L5463","statement_sha256":"ce54c03ea5a4ea3f45143fa9aa1571e1ac4ba9604c9a5ac375fc69a54145191f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2963,"rank":2963,"depth":13,"x":630.986,"y":640.778,"cluster":"advanced-algebra"},{"id":"stacks:0AV9","tag":"0AV9","title":"Reflexive modules · Lemma 0AV9","summary":"Let R be a Noetherian domain. Let φ : M → N be a map of R-modules. Assume M is finite, N is torsion free, and that for every prime p of R one of the following happens • M_ p → N_ p is an isomorphism, or • depth(M_ p) ≥ 2. Then φ is an isomorphism.","statement_latex":"Let $R$ be a Noetherian domain. Let $\\varphi : M \\to N$ be a map of\n$R$-modules. Assume $M$ is finite, $N$ is torsion free, and\nthat for every prime $\\mathfrak p$ of $R$ one of the following happens\n\\begin{enumerate}\n\\item $M_\\mathfrak p \\to N_\\mathfrak p$ is an isomorphism, or\n\\item $\\text{depth}(M_\\mathfrak p) \\geq 2$.\n\\end{enumerate}\nThen $\\varphi$ is an isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AV9","source_file":"more-algebra.tex","source_line":5489,"source_end_line":5499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5489-L5499","statement_sha256":"c2976a6a1e4fdd9924fe1883636e5f674d5a97830e9c28bb090cc663223aeee3","origin":"The Stacks Project","memory_eligible":false,"source_rank":2964,"rank":2964,"depth":14,"x":747.917,"y":681.113,"cluster":"advanced-algebra"},{"id":"stacks:0AVA","tag":"0AVA","title":"Reflexive modules · Lemma 0AVA","summary":"Let R be a Noetherian domain. Let M be a finite R-module. The following are equivalent • M is reflexive, • for every prime p of R one of the following happens • M_ p is a reflexive R_ p-module, or • depth(M_ p) ≥ 2.","statement_latex":"Let $R$ be a Noetherian domain. Let $M$ be a finite $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M$ is reflexive,\n\\item for every prime $\\mathfrak p$ of $R$ one of the following happens\n\\begin{enumerate}\n\\item $M_\\mathfrak p$ is a reflexive $R_\\mathfrak p$-module, or\n\\item $\\text{depth}(M_\\mathfrak p) \\geq 2$.\n\\end{enumerate}\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVA","source_file":"more-algebra.tex","source_line":5506,"source_end_line":5518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5506-L5518","statement_sha256":"41d8519621ec2bc9d9b6b24c5e3665d5e34dd6972323c115c4a4d69b0de9ed5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":2965,"rank":2965,"depth":15,"x":628.849,"y":717.846,"cluster":"advanced-algebra"},{"id":"stacks:0EBA","tag":"0EBA","title":"Reflexive modules · Lemma 0EBA","summary":"Let R be a Noetherian domain. Let M be a finite reflexive R-module. Let p ⊂ R be a prime ideal. • If depth(R_ p) ≥ 2, then depth(M_ p) ≥ 2. • If R is (S_2), then M is (S_2).","statement_latex":"Let $R$ be a Noetherian domain. Let $M$ be a finite reflexive $R$-module.\nLet $\\mathfrak p \\subset R$ be a prime ideal.\n\\begin{enumerate}\n\\item If $\\text{depth}(R_\\mathfrak p) \\geq 2$, then\n$\\text{depth}(M_\\mathfrak p) \\geq 2$.\n\\item If $R$ is $(S_2)$, then $M$ is $(S_2)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBA","source_file":"more-algebra.tex","source_line":5539,"source_end_line":5548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5539-L5548","statement_sha256":"6d28ba3278c2518d02456601e5ba8cebbd06f69b0e07c6488d1ef9378b6d152c","origin":"The Stacks Project","memory_eligible":false,"source_rank":2966,"rank":2966,"depth":15,"x":687.308,"y":622.874,"cluster":"advanced-algebra"},{"id":"stacks:0AVB","tag":"0AVB","title":"Reflexive modules · Lemma 0AVB","summary":"Let R be a Noetherian normal domain with fraction field K. Let M be a finite R-module. The following are equivalent • M is reflexive, • M is torsion free and has property (S_2), • M is torsion free and M = ⋂_height( p) = 1 M_ p where the intersection happens in M_K = M ⊗_R K.","statement_latex":"Let $R$ be a Noetherian normal domain with fraction field $K$.\nLet $M$ be a finite $R$-module. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is reflexive,\n\\item $M$ is torsion free and has property $(S_2)$,\n\\item $M$ is torsion free and\n$M = \\bigcap_{\\text{height}(\\mathfrak p) = 1} M_{\\mathfrak p}$\nwhere the intersection happens in $M_K = M \\otimes_R K$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVB","source_file":"more-algebra.tex","source_line":5577,"source_end_line":5588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5577-L5588","statement_sha256":"4dc20da4db17fa535e9ec6aa8f1dc91bbd895c72f3c5f182ab1fe0f1a0b82269","origin":"The Stacks Project","memory_eligible":false,"source_rank":2967,"rank":2967,"depth":18,"x":720.688,"y":726.427,"cluster":"advanced-algebra"},{"id":"stacks:0AVC","tag":"0AVC","title":"Reflexive modules · Lemma 0AVC","summary":"Let R be a Noetherian normal domain. Let M be a finite R-module. Then the reflexive hull of M is the intersection M^** = ⋂_height( p) = 1 M_ p/(M_ p)_tors = ⋂_height( p) = 1 (M/M_tors)_ p taken in M ⊗_R K.","statement_latex":"Let $R$ be a Noetherian normal domain. Let $M$ be a finite $R$-module.\nThen the reflexive hull of $M$ is the intersection\n$$\nM^{**} =\n\\bigcap\\nolimits_{\\text{height}(\\mathfrak p) = 1}\nM_{\\mathfrak p}/(M_\\mathfrak p)_{tors} =\n\\bigcap\\nolimits_{\\text{height}(\\mathfrak p) = 1}\n(M/M_{tors})_\\mathfrak p\n$$\ntaken in $M \\otimes_R K$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVC","source_file":"more-algebra.tex","source_line":5635,"source_end_line":5647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5635-L5647","statement_sha256":"4b248ad976719d761b99cb214e028a06e1298af7486316fc10c139e4d20d6fde","origin":"The Stacks Project","memory_eligible":false,"source_rank":2968,"rank":2968,"depth":19,"x":612.436,"y":668.815,"cluster":"advanced-algebra"},{"id":"stacks:0BM4","tag":"0BM4","title":"Reflexive modules · Lemma 0BM4","summary":"Let A be a Noetherian normal domain with fraction field K. Let L be a finite extension of K. If the integral closure B of A in L is finite over A, then B is reflexive as an A-module.","statement_latex":"Let $A$ be a Noetherian normal domain with fraction field $K$.\nLet $L$ be a finite extension of $K$. If the integral closure\n$B$ of $A$ in $L$ is finite over $A$, then $B$ is reflexive as an $A$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BM4","source_file":"more-algebra.tex","source_line":5664,"source_end_line":5669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5664-L5669","statement_sha256":"1446405ef48ca2f6ccedea62d3c5113b94acefa3bf33bdaeb131176f22489c40","origin":"The Stacks Project","memory_eligible":false,"source_rank":2969,"rank":2969,"depth":19,"x":739.012,"y":649.81,"cluster":"advanced-algebra"},{"id":"stacks:0ASA","tag":"0ASA","title":"Content ideals · Definition 0ASA","summary":"Let A be a ring. Let M be a flat A-module. Let x ∈ M. If the set of ideals I in A such that x ∈ IM has a smallest element, we call it the content ideal of x.","statement_latex":"Let $A$ be a ring. Let $M$ be a flat $A$-module. Let $x \\in M$.\nIf the set of ideals $I$ in $A$ such that $x \\in IM$ has a\nsmallest element, we call it the {\\it content ideal of $x$}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Content ideals","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASA","source_file":"more-algebra.tex","source_line":5699,"source_end_line":5704,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5699-L5704","statement_sha256":"70564541d2dd81c79709faf0448fe7fd21525236ac1f60759a1fd8ab3dd90df6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2970,"rank":2970,"depth":0,"x":660.7,"y":735.932,"cluster":"advanced-algebra"},{"id":"stacks:0ASB","tag":"0ASB","title":"Content ideals · Lemma 0ASB","summary":"Let A be a ring. Let M be a flat A-module. Let x ∈ M. The content ideal of x, if it exists, is finitely generated.","statement_latex":"Let $A$ be a ring. Let $M$ be a flat $A$-module. Let $x \\in M$.\nThe content ideal of $x$, if it exists, is finitely generated.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Content ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASB","source_file":"more-algebra.tex","source_line":5712,"source_end_line":5716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5712-L5716","statement_sha256":"51a75177f8736dfa212de94d6c070e18d7ff73d3c7d0ad7fafad588371a9c57f","origin":"The Stacks Project","memory_eligible":false,"source_rank":2971,"rank":2971,"depth":0,"x":649.152,"y":627.633,"cluster":"advanced-algebra"},{"id":"stacks:0ASC","tag":"0ASC","title":"Content ideals · Lemma 0ASC","summary":"Let (A, m) be a local ring. Let u : M → N be a map of flat A-modules such that overlineu : M/ m M → N/ m N is injective. If x ∈ M has content ideal I, then u(x) has content ideal I as well.","statement_latex":"Let $(A, \\mathfrak m)$ be a local ring. Let $u : M \\to N$ be a map of flat\n$A$-modules such that $\\overline{u} : M/\\mathfrak m M \\to N/\\mathfrak m N$\nis injective. If $x \\in M$ has content ideal $I$, then $u(x)$ has content\nideal $I$ as well.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Content ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASC","source_file":"more-algebra.tex","source_line":5724,"source_end_line":5730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5724-L5730","statement_sha256":"e9335e2506132faaaeb0a93e07195a6c08896ad39ec985681fbc08917a0d4b0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2972,"rank":2972,"depth":3,"x":745.071,"y":701.179,"cluster":"advanced-algebra"},{"id":"stacks:0ASD","tag":"0ASD","title":"Content ideals · Lemma 0ASD","summary":"Let A be a ring. Let M be a flat Mittag-Leffler module. Then every element of M has a content ideal.","statement_latex":"Let $A$ be a ring. Let $M$ be a flat Mittag-Leffler module.\nThen every element of $M$ has a content ideal.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Content ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASD","source_file":"more-algebra.tex","source_line":5757,"source_end_line":5761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5757-L5761","statement_sha256":"2caf8d7648be3980700a25f681ed498c29e5da418c979c94c67c640cb164bdc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":2973,"rank":2973,"depth":8,"x":614.774,"y":701.375,"cluster":"advanced-algebra"},{"id":"stacks:053A","tag":"053A","title":"Flatness and finiteness conditions · Lemma 053A","summary":"Let R be a ring. Let S = R[x_1, …, x_n] be a polynomial ring over R. Let M be an S-module. Assume • there exist finitely many primes p_1, …, p_m of R such that the map R → ∏ R_ p_j is injective, • M is a finite S-module, • M flat over R, and • for every prime p of R the module M_ p is of finite presentation over S_ p. Then M is of finite presentation over S.","statement_latex":"Let $R$ be a ring. Let $S = R[x_1, \\ldots, x_n]$ be a polynomial\nring over $R$. Let $M$ be an $S$-module.\nAssume\n\\begin{enumerate}\n\\item there exist finitely many primes $\\mathfrak p_1, \\ldots, \\mathfrak p_m$\nof $R$ such that the map $R \\to \\prod R_{\\mathfrak p_j}$ is injective,\n\\item $M$ is a finite $S$-module,\n\\item $M$ flat over $R$, and\n\\item for every prime $\\mathfrak p$ of $R$ the module $M_{\\mathfrak p}$\nis of finite presentation over $S_{\\mathfrak p}$.\n\\end{enumerate}\nThen $M$ is of finite presentation over $S$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flatness and finiteness conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053A","source_file":"more-algebra.tex","source_line":5785,"source_end_line":5799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5785-L5799","statement_sha256":"320be41796d7fa791264bc26abc44c5e12128ede7eaf476e7cfba61dac1b7955","origin":"The Stacks Project","memory_eligible":false,"source_rank":2974,"rank":2974,"depth":34,"x":711.007,"y":627.057,"cluster":"advanced-algebra"},{"id":"stacks:053B","tag":"053B","title":"Flatness and finiteness conditions · Lemma 053B","summary":"Let R → S be a ring homomorphism. Assume • there exist finitely many primes p_1, …, p_m of R such that the map R → ∏ R_ p_j is injective, • R → S is of finite type, • S flat over R, and • for every prime p of R the ring S_ p is of finite presentation over R_ p. Then S is of finite presentation over R.","statement_latex":"Let $R \\to S$ be a ring homomorphism.\nAssume\n\\begin{enumerate}\n\\item there exist finitely many primes\n$\\mathfrak p_1, \\ldots, \\mathfrak p_m$ of $R$ such that\nthe map $R \\to \\prod R_{\\mathfrak p_j}$ is injective,\n\\item $R \\to S$ is of finite type,\n\\item $S$ flat over $R$, and\n\\item for every prime $\\mathfrak p$ of $R$ the ring $S_{\\mathfrak p}$\nis of finite presentation over $R_{\\mathfrak p}$.\n\\end{enumerate}\nThen $S$ is of finite presentation over $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flatness and finiteness conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053B","source_file":"more-algebra.tex","source_line":5833,"source_end_line":5847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5833-L5847","statement_sha256":"744edfb23460b24be43c892fd88406b521751942f911e8f0b02de548604798d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2975,"rank":2975,"depth":35,"x":699.774,"y":736.815,"cluster":"advanced-algebra"},{"id":"stacks:053C","tag":"053C","title":"Flatness and finiteness conditions · Lemma 053C","summary":"Let R be a ring. Let S = R[x_1, …, x_n] be a graded polynomial algebra over R, i.e., deg(x_i) > 0 but not necessarily equal to 1. Let M be a graded S-module. Assume • R is a local ring, • M is a finite S-module, and • M is flat over R. Then M is finitely presented as an S-module.","statement_latex":"Let $R$ be a ring.\nLet $S = R[x_1, \\ldots, x_n]$ be a graded polynomial algebra over $R$,\ni.e., $\\deg(x_i) > 0$ but not necessarily equal to $1$.\nLet $M$ be a graded $S$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is a local ring,\n\\item $M$ is a finite $S$-module, and\n\\item $M$ is flat over $R$.\n\\end{enumerate}\nThen $M$ is finitely presented as an $S$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flatness and finiteness conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053C","source_file":"more-algebra.tex","source_line":5855,"source_end_line":5868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5855-L5868","statement_sha256":"b687f5ff496bc768813eb34617b6a63baa5d81c48462b042644745e09ea6f1c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":2976,"rank":2976,"depth":4,"x":619.538,"y":649.228,"cluster":"advanced-algebra"},{"id":"stacks:053D","tag":"053D","title":"Flatness and finiteness conditions · Lemma 053D","summary":"Let R be a ring. Let S = bigoplus_n ≥ 0 S_n be a graded R-algebra. Let M = bigoplus_d ∈ Z M_d be a graded S-module. Assume S is finitely generated as an R-algebra, assume S_0 is a finite R-algebra, and assume there exist finitely many primes p_j, i = 1, …, m such that R → ∏ R_ p_j is injective. • If S is flat over R, then S is a finitely presented R-algebra. • If M is flat as an R-module and finite as an S-module, then M is finitely presented as an S-module.","statement_latex":"Let $R$ be a ring. Let $S = \\bigoplus_{n \\geq 0} S_n$ be a graded $R$-algebra.\nLet $M = \\bigoplus_{d \\in \\mathbf{Z}} M_d$ be a graded $S$-module.\nAssume $S$ is finitely generated as an $R$-algebra, assume $S_0$ is a finite\n$R$-algebra, and assume there exist finitely many primes\n$\\mathfrak p_j$, $i = 1, \\ldots, m$ such that\n$R \\to \\prod R_{\\mathfrak p_j}$ is injective.\n\\begin{enumerate}\n\\item If $S$ is flat over $R$, then $S$ is a finitely presented $R$-algebra.\n\\item If $M$ is flat as an $R$-module and finite as an $S$-module,\nthen $M$ is finitely presented as an $S$-module.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flatness and finiteness conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053D","source_file":"more-algebra.tex","source_line":5912,"source_end_line":5925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5912-L5925","statement_sha256":"34e4ad31c629bcb41a7b90d29242fd83e0fc65264bd11e883f55c82aa3f5b451","origin":"The Stacks Project","memory_eligible":false,"source_rank":2977,"rank":2977,"depth":35,"x":749.55,"y":668.346,"cluster":"advanced-algebra"},{"id":"stacks:053E","tag":"053E","title":"Flatness and finiteness conditions · Lemma 053E","summary":"[Nagata-Finitely] Let A be a valuation ring. Let A → B be a ring map of finite type. Let M be a finite B-module. • If B is flat over A, then B is a finitely presented A-algebra. • If M is flat as an A-module, then M is finitely presented as a B-module.","statement_latex":"\\begin{reference}\n\\cite[Theorem 3]{Nagata-Finitely}\n\\end{reference}\nLet $A$ be a valuation ring. Let $A \\to B$ be a ring map of finite type.\nLet $M$ be a finite $B$-module.\n\\begin{enumerate}\n\\item If $B$ is flat over $A$, then $B$ is a finitely presented $A$-algebra.\n\\item If $M$ is flat as an $A$-module, then $M$ is finitely presented\nas a $B$-module.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flatness and finiteness conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053E","source_file":"more-algebra.tex","source_line":5966,"source_end_line":5978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L5966-L5978","statement_sha256":"37b205258c9008bae456bf5fb5aea3a90d15b29e86911ebd70a4d97f0cd276a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":2978,"rank":2978,"depth":36,"x":637.955,"y":728.209,"cluster":"advanced-algebra"},{"id":"stacks:0GSE","tag":"0GSE","title":"Flatness and finiteness conditions · Lemma 0GSE","summary":"Let A be a valuation ring. Let A → B be a local homomorphism which is essentially of finite type. Let M be a finite B-module. • If B is flat over A, then B is essentially of finite presentation over A. • If M is flat as an A-module, then M is finitely presented as a B-module.","statement_latex":"Let $A$ be a valuation ring. Let $A \\to B$ be a local homomorphism\nwhich is essentially of finite type. Let $M$ be a finite $B$-module.\n\\begin{enumerate}\n\\item If $B$ is flat over $A$, then $B$ is essentially of finite\npresentation over $A$.\n\\item If $M$ is flat as an $A$-module, then $M$ is finitely presented\nas a $B$-module.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Flatness and finiteness conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSE","source_file":"more-algebra.tex","source_line":6004,"source_end_line":6014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6004-L6014","statement_sha256":"48384ddbd177628625d7c09de13522e2d6bfd6fb97a3390616fc611dc4635b7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2979,"rank":2979,"depth":37,"x":672.211,"y":620.408,"cluster":"advanced-algebra"},{"id":"stacks:053H","tag":"053H","title":"Blowing up and flatness · Definition 053H","summary":"Let R be a ring. Let I ⊂ R be an ideal and a ∈ I. Let R[fracIa] be the affine blowup algebra, see Algebra, Definition [Tag 052Q]. Let M be an R-module. The strict transform of M along R → R[fracIa] is the R[fracIa]-module M' = (M ⊗_R R[textstylefracIa])/a-power torsion","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal and $a \\in I$.\nLet $R[\\frac{I}{a}]$ be the affine blowup algebra, see\nAlgebra, Definition \\ref{algebra-definition-blow-up}.\nLet $M$ be an $R$-module.\nThe {\\it strict transform of $M$ along $R \\to R[\\frac{I}{a}]$} is\nthe $R[\\frac{I}{a}]$-module\n$$\nM' = \\left(M \\otimes_R R[\\textstyle{\\frac{I}{a}}]\\right)/a\\text{-power torsion}\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Blowing up and flatness","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053H","source_file":"more-algebra.tex","source_line":6060,"source_end_line":6071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6060-L6071","statement_sha256":"c74ff814c88f0fa20ec698bd63c744c1fbdd3a8ca43293e95e686629966bbf8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2980,"rank":2980,"depth":1,"x":733.832,"y":719.645,"cluster":"advanced-algebra"},{"id":"stacks:053I","tag":"053I","title":"Blowing up and flatness · Lemma 053I","summary":"Let (R, m) be a local domain with fraction field K. Let S be a finite type R-algebra. Let M be a finite S-module. For every valuation ring A ⊂ K dominating R there exists an ideal I ⊂ m and a nonzero element a ∈ I such that • I is finitely generated, • A has center on R[fracIa], • the fibre ring of R → R[fracIa] at m is not zero, and • the strict transform S_I, a of S along R → R[fracIa] is flat and of finite presentation over R, and the strict transform M_I, a of M along…","statement_latex":"Let $(R, \\mathfrak m)$ be a local domain with fraction field $K$.\nLet $S$ be a finite type $R$-algebra.\nLet $M$ be a finite $S$-module.\nFor every valuation ring $A \\subset K$ dominating $R$\nthere exists an ideal $I \\subset \\mathfrak m$ and a nonzero\nelement $a \\in I$ such that\n\\begin{enumerate}\n\\item $I$ is finitely generated,\n\\item $A$ has center on $R[\\frac{I}{a}]$,\n\\item the fibre ring of $R \\to R[\\frac{I}{a}]$ at $\\mathfrak m$\nis not zero, and\n\\item the strict transform $S_{I, a}$ of $S$ along $R \\to R[\\frac{I}{a}]$\nis flat and of finite presentation over $R$, and the strict transform\n$M_{I, a}$ of $M$ along $R \\to R[\\frac{I}{a}]$ is flat over $R$ and\nfinitely presented over $S_{I, a}$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053I","source_file":"more-algebra.tex","source_line":6077,"source_end_line":6095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6077-L6095","statement_sha256":"fb363cb39635b3674c10e4e740c6e72ff830fa8db8798b18660ebdf2bb0facdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":2981,"rank":2981,"depth":37,"x":608.202,"y":681.316,"cluster":"advanced-algebra"},{"id":"stacks:0CZM","tag":"0CZM","title":"Blowing up and flatness · Lemma 0CZM","summary":"Let R be a ring. Let M be a finite R-module. Let k ≥ 0 and I = Fit_k(M). For every a ∈ I with R' = R[fracIa] the strict transform M' = (M ⊗_R R')/a-power torsion has Fit_k(M') = R'.","statement_latex":"Let $R$ be a ring. Let $M$ be a finite $R$-module.\nLet $k \\geq 0$ and $I = \\text{Fit}_k(M)$. For every $a \\in I$\nwith $R' = R[\\frac{I}{a}]$ the strict transform\n$$\nM' = (M \\otimes_R R')/a\\text{-power torsion}\n$$\nhas $\\text{Fit}_k(M') = R'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZM","source_file":"more-algebra.tex","source_line":6173,"source_end_line":6182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6173-L6182","statement_sha256":"4748032c01833fa12c75afea0b6b71bb3a68575f65cfd6535760faeb7abed18a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2982,"rank":2982,"depth":5,"x":732.044,"y":638.162,"cluster":"advanced-algebra"},{"id":"stacks:0CZN","tag":"0CZN","title":"Blowing up and flatness · Lemma 0CZN","summary":"Let R be a ring. Let M be a finite R-module. Let k ≥ 0 and I = Fit_k(M). Assume that M_ p is free of rank k for every p not ∈ V(I). Then for every a ∈ I with R' = R[fracIa] the strict transform M' = (M ⊗_R R')/a-power torsion is locally free of rank k.","statement_latex":"Let $R$ be a ring. Let $M$ be a finite $R$-module.\nLet $k \\geq 0$ and $I = \\text{Fit}_k(M)$. Assume that\n$M_\\mathfrak p$ is free of rank $k$ for every\n$\\mathfrak p \\not \\in V(I)$. Then for every $a \\in I$\nwith $R' = R[\\frac{I}{a}]$ the strict transform\n$$\nM' = (M \\otimes_R R')/a\\text{-power torsion}\n$$\nis locally free of rank $k$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZN","source_file":"more-algebra.tex","source_line":6197,"source_end_line":6208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6197-L6208","statement_sha256":"8e390f2b4b1222f17bb6de256f08e60e43b5ce41858520d233b8ca79245cdf1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":2983,"rank":2983,"depth":6,"x":675.255,"y":740.566,"cluster":"advanced-algebra"},{"id":"stacks:0BBJ","tag":"0BBJ","title":"Blowing up and flatness · Lemma 0BBJ","summary":"Let R be a ring. Let M be a finite R-module. Let f ∈ R be an element such that M_f is finite locally free of rank r. Then there exists a finitely generated ideal I ⊂ R with V(f) = V(I) such that for all a ∈ I with R' = R[fracIa] the strict transform M' = (M ⊗_R R')/a-power torsion is locally free of rank r.","statement_latex":"Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $f \\in R$\nbe an element such that $M_f$ is finite locally free of rank $r$.\nThen there exists a finitely generated ideal $I \\subset R$ with\n$V(f) = V(I)$ such that for all $a \\in I$ with $R' = R[\\frac{I}{a}]$\nthe strict transform\n$$\nM' = (M \\otimes_R R')/a\\text{-power torsion}\n$$\nis locally free of rank $r$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBJ","source_file":"more-algebra.tex","source_line":6228,"source_end_line":6239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6228-L6239","statement_sha256":"89613009639785a06e7a0ec81609051d806729f6076bfbd6e5bb1f70bacbfd5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2984,"rank":2984,"depth":6,"x":634.657,"y":632.503,"cluster":"advanced-algebra"},{"id":"stacks:05BC","tag":"05BC","title":"Completion and flatness · Lemma 05BC","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let A be a set. Assume R is Noetherian and complete with respect to I. There is a canonical map (bigoplus_α ∈ A R)^wedge → ∏_α ∈ A R from the I-adic completion of the direct sum into the product which is universally injective.","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\nLet $A$ be a set.\nAssume $R$ is Noetherian and complete with respect to $I$. There is a\ncanonical map\n$$\n\\left(\\bigoplus\\nolimits_{\\alpha \\in A} R\\right)^\\wedge\n\\longrightarrow\n\\prod\\nolimits_{\\alpha \\in A} R\n$$\nfrom the $I$-adic completion of the direct sum into the product\nwhich is universally injective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Completion and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BC","source_file":"more-algebra.tex","source_line":6317,"source_end_line":6331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6317-L6331","statement_sha256":"c81ab67c4f0e95bd0c73079cec55d2311ef128ac062abeac0f957293acc699fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":2985,"rank":2985,"depth":2,"x":751.847,"y":689.322,"cluster":"advanced-algebra"},{"id":"stacks:06LE","tag":"06LE","title":"Completion and flatness · Lemma 06LE","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let A be a set. Assume R is Noetherian. The completion (bigoplus_α ∈ A R)^wedge is a flat R-module.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $A$ be a set.\nAssume $R$ is Noetherian. The completion\n$(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge$\nis a flat $R$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Completion and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LE","source_file":"more-algebra.tex","source_line":6411,"source_end_line":6417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6411-L6417","statement_sha256":"835e6a1c41905b8fb8745a906c3650e8f6699bc8b5632519c6fed8d9a01c7392","origin":"The Stacks Project","memory_eligible":false,"source_rank":2986,"rank":2986,"depth":6,"x":619.343,"y":713.995,"cluster":"advanced-algebra"},{"id":"stacks:0911","tag":"0911","title":"Completion and flatness · Lemma 0911","summary":"This is [quillenhomology]; note that the author forgot the word \"strict\" in the statement although it was clearly intended. Let A be a Noetherian ring. Let I be an ideal of A. Let M be a finite A-module. For every p > 0 there exists a c > 0 such that Tor_p^A(M, A/I^n) → Tor_p^A(M, A/I^n - c) is zero for all n ≥ c.","statement_latex":"\\begin{reference}\nThis is \\cite[Lemma 9.9]{quillenhomology}; note that\nthe author forgot the word ``strict'' in the statement\nalthough it was clearly intended.\n\\end{reference}\nLet $A$ be a Noetherian ring. Let $I$ be an ideal of $A$.\nLet $M$ be a finite $A$-module. For every $p > 0$ there exists a $c > 0$\nsuch that $\\text{Tor}_p^A(M, A/I^n) \\to \\text{Tor}_p^A(M, A/I^{n - c})$\nis zero for all $n \\geq c$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Completion and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0911","source_file":"more-algebra.tex","source_line":6449,"source_end_line":6460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6449-L6460","statement_sha256":"7f59d210a9470b52ccbde7034f248c3116fe5f92b57c28bfb10ed7150ed0fd60","origin":"The Stacks Project","memory_eligible":false,"source_rank":2987,"rank":2987,"depth":2,"x":697.441,"y":620.338,"cluster":"advanced-algebra"},{"id":"stacks:0912","tag":"0912","title":"Completion and flatness · Lemma 0912","summary":"Let A be a Noetherian ring. Let I be an ideal of A. Let (M_n) be an inverse system of A-modules such that • M_n is a flat A/I^n-module, • M_n + 1 → M_n is surjective. Then M = lim M_n is a flat A-module and Q ⊗_A M = lim Q ⊗_A M_n for every finite A-module Q.","statement_latex":"Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let\n$(M_n)$ be an inverse system of $A$-modules such that\n\\begin{enumerate}\n\\item $M_n$ is a flat $A/I^n$-module,\n\\item $M_{n + 1} \\to M_n$ is surjective.\n\\end{enumerate}\nThen $M = \\lim M_n$ is a flat $A$-module and\n$Q \\otimes_A M = \\lim Q \\otimes_A M_n$ for every finite $A$-module $Q$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Completion and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0912","source_file":"more-algebra.tex","source_line":6474,"source_end_line":6484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6474-L6484","statement_sha256":"4bf2294864e9c4a49694c76907d9dbe1ae0e7a45f60d5e664953a31e79176005","origin":"The Stacks Project","memory_eligible":false,"source_rank":2988,"rank":2988,"depth":5,"x":715.222,"y":734.05,"cluster":"advanced-algebra"},{"id":"stacks:0AGW","tag":"0AGW","title":"Completion and flatness · Lemma 0AGW","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. Assume • I is finitely generated, • R/I is Noetherian, • M/IM is flat over R/I, • Tor_1^R(M, R/I) = 0. Then the I-adic completion R^wedge is a Noetherian ring and M^wedge is flat over R^wedge.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be\nan $R$-module. Assume\n\\begin{enumerate}\n\\item $I$ is finitely generated,\n\\item $R/I$ is Noetherian,\n\\item $M/IM$ is flat over $R/I$,\n\\item $\\text{Tor}_1^R(M, R/I) = 0$.\n\\end{enumerate}\nThen the $I$-adic completion $R^\\wedge$\nis a Noetherian ring and $M^\\wedge$ is flat over $R^\\wedge$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Completion and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGW","source_file":"more-algebra.tex","source_line":6526,"source_end_line":6538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6526-L6538","statement_sha256":"2fc3c58c3719c2663217c594574ffeac3a9447766dfdb5a8429be2d82590a1ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":2989,"rank":2989,"depth":6,"x":610.358,"y":660.071,"cluster":"advanced-algebra"},{"id":"stacks:0622","tag":"0622","title":"The Koszul complex · Definition 0622","summary":"Let R be a ring. Let φ : E → R be an R-module map. The Koszul complex K_bullet(φ) associated to φ is the commutative differential graded algebra defined as follows: • the underlying graded algebra is the exterior algebra K_bullet(φ) = wedge(E), • the differential d : K_bullet(φ) → K_bullet(φ) is the unique derivation such that d(e) = φ(e) for all e ∈ E = K_1(φ).","statement_latex":"Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map. The\n{\\it Koszul complex} $K_\\bullet(\\varphi)$ associated to $\\varphi$\nis the commutative differential graded algebra defined as follows:\n\\begin{enumerate}\n\\item the underlying graded algebra is the exterior algebra\n$K_\\bullet(\\varphi) = \\wedge(E)$,\n\\item the differential $d : K_\\bullet(\\varphi) \\to K_\\bullet(\\varphi)$\nis the unique derivation such that $d(e) = \\varphi(e)$ for all\n$e \\in E = K_1(\\varphi)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0622","source_file":"more-algebra.tex","source_line":6565,"source_end_line":6577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6565-L6577","statement_sha256":"a816053544050a8bf6ec78cf3ecbb6f1991c7e973bbb1938c47d2ee8c6ade1a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":2990,"rank":2990,"depth":0,"x":747.576,"y":655.106,"cluster":"advanced-algebra"},{"id":"stacks:0623","tag":"0623","title":"The Koszul complex · Definition 0623","summary":"Let R be a ring and let f_1, …, f_r ∈ R. The Koszul complex on f_1, …, f_r is the Koszul complex associated to the map (f_1, …, f_r) : R^⊕ r → R. Notation K_bullet(f_bullet), K_bullet(f_1, …, f_r), K_bullet(R, f_1, …, f_r), or K_bullet(R, f_bullet).","statement_latex":"Let $R$ be a ring and let $f_1, \\ldots, f_r \\in R$. The\n{\\it Koszul complex on $f_1, \\ldots, f_r$} is the Koszul complex\nassociated to the map $(f_1, \\ldots, f_r) : R^{\\oplus r} \\to R$.\nNotation $K_\\bullet(f_\\bullet)$, $K_\\bullet(f_1, \\ldots, f_r)$,\n$K_\\bullet(R, f_1, \\ldots, f_r)$, or $K_\\bullet(R, f_\\bullet)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0623","source_file":"more-algebra.tex","source_line":6596,"source_end_line":6603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6596-L6603","statement_sha256":"c280016b601f44671f1b85422ec009f144f36561542d75c00ad62da3ad4e0a89","origin":"The Stacks Project","memory_eligible":false,"source_rank":2991,"rank":2991,"depth":0,"x":650.103,"y":736.866,"cluster":"advanced-algebra"},{"id":"stacks:0624","tag":"0624","title":"The Koszul complex · Lemma 0624","summary":"Let φ : E → R and φ' : E' → R be R-module maps. Let ψ : E → E' be an R-module map such that φ' ∘ ψ = φ. Then ψ induces a homomorphism of differential graded algebras K_bullet(φ) → K_bullet(φ').","statement_latex":"Let $\\varphi : E \\to R$ and $\\varphi' : E' \\to R$ be $R$-module maps.\nLet $\\psi : E \\to E'$ be an $R$-module map such that\n$\\varphi' \\circ \\psi = \\varphi$. Then $\\psi$ induces a\nhomomorphism of differential graded algebras\n$K_\\bullet(\\varphi) \\to K_\\bullet(\\varphi')$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0624","source_file":"more-algebra.tex","source_line":6611,"source_end_line":6618,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6611-L6618","statement_sha256":"3ee9c9b929cbabd27e2a0618fe7400e98f0b5f3922bad8d97e30722fc32466ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":2992,"rank":2992,"depth":0,"x":656.247,"y":620.934,"cluster":"advanced-algebra"},{"id":"stacks:0625","tag":"0625","title":"The Koszul complex · Lemma 0625","summary":"Let f_1, …, f_r ∈ R be a sequence. Let (x_ij) be an invertible r × r-matrix with coefficients in R. Then the complexes K_bullet(f_bullet) and K_bullet(∑ x_1jf_j, ∑ x_2jf_j, …, ∑ x_rjf_j) are isomorphic.","statement_latex":"Let $f_1, \\ldots, f_r \\in R$ be a sequence.\nLet $(x_{ij})$ be an invertible $r \\times r$-matrix with\ncoefficients in $R$. Then the complexes\n$K_\\bullet(f_\\bullet)$ and\n$$\nK_\\bullet(\\sum x_{1j}f_j, \\sum x_{2j}f_j, \\ldots, \\sum x_{rj}f_j)\n$$\nare isomorphic.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0625","source_file":"more-algebra.tex","source_line":6624,"source_end_line":6634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6624-L6634","statement_sha256":"e335a02c93284efc6294e2382b212662cdcdbdd17103574dfc41cecb97c564cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":2993,"rank":2993,"depth":1,"x":745.201,"y":710.163,"cluster":"advanced-algebra"},{"id":"stacks:0626","tag":"0626","title":"The Koszul complex · Lemma 0626","summary":"Let R be a ring. Let φ : E → R be an R-module map. Let e ∈ E with image f = φ(e) in R. Then f = de + ed as endomorphisms of K_bullet(φ).","statement_latex":"Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map.\nLet $e \\in E$ with image $f = \\varphi(e)$ in $R$. Then\n$$\nf = de + ed\n$$\nas endomorphisms of $K_\\bullet(\\varphi)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0626","source_file":"more-algebra.tex","source_line":6645,"source_end_line":6653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6645-L6653","statement_sha256":"fd82a0f70951df2c95883329d5d873add60d959249003b0d0a21c86b7e2baa7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2994,"rank":2994,"depth":0,"x":607.459,"y":694.797,"cluster":"advanced-algebra"},{"id":"stacks:0663","tag":"0663","title":"The Koszul complex · Lemma 0663","summary":"Let R be a ring. Let f_1, …, f_r ∈ R be a sequence. Multiplication by f_i on K_bullet(f_bullet) is homotopic to zero, and in particular the cohomology modules H_i(K_bullet(f_bullet)) are annihilated by the ideal (f_1, …, f_r).","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be a sequence.\nMultiplication by $f_i$ on $K_\\bullet(f_\\bullet)$ is homotopic to\nzero, and in particular the cohomology modules $H_i(K_\\bullet(f_\\bullet))$\nare annihilated by the ideal $(f_1, \\ldots, f_r)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0663","source_file":"more-algebra.tex","source_line":6659,"source_end_line":6665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6659-L6665","statement_sha256":"56bf7af71eddc39c08c6e9de8dede8dcac3cf122b55ef7e842592d6f24bdfa2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":2995,"rank":2995,"depth":1,"x":721.71,"y":627.779,"cluster":"advanced-algebra"},{"id":"stacks:0628","tag":"0628","title":"The Koszul complex · Lemma 0628","summary":"Let R be a ring. Let φ : E → R be an R-module map. Let f ∈ R. Set E' = E ⊕ R and define φ' : E' → R by φ on E and multiplication by f on R. The complex K_bullet(φ') is isomorphic to the cone of the map of complexes f : K_bullet(φ) → K_bullet(φ).","statement_latex":"Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map.\nLet $f \\in R$. Set $E' = E \\oplus R$ and define $\\varphi' : E' \\to R$\nby $\\varphi$ on $E$ and multiplication by $f$ on $R$.\nThe complex $K_\\bullet(\\varphi')$ is isomorphic to the\ncone of the map of complexes\n$$\nf :\nK_\\bullet(\\varphi)\n\\longrightarrow\nK_\\bullet(\\varphi).\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0628","source_file":"more-algebra.tex","source_line":6694,"source_end_line":6707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6694-L6707","statement_sha256":"76b1740b0e95ef1cbad569ce5419790f93fd19c6d57a9bf50d6be09d43dbb940","origin":"The Stacks Project","memory_eligible":false,"source_rank":2996,"rank":2996,"depth":0,"x":691.27,"y":742.349,"cluster":"advanced-algebra"},{"id":"stacks:0629","tag":"0629","title":"The Koszul complex · Lemma 0629","summary":"Let R be a ring. Let f_1, …, f_r be a sequence of elements of R. The complex K_bullet(f_1, …, f_r) is isomorphic to the cone of the map of complexes f_r : K_bullet(f_1, …, f_r - 1) → K_bullet(f_1, …, f_r - 1).","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r$ be a sequence of elements\nof $R$. The complex $K_\\bullet(f_1, \\ldots, f_r)$ is isomorphic to the\ncone of the map of complexes\n$$\nf_r :\nK_\\bullet(f_1, \\ldots, f_{r - 1})\n\\longrightarrow\nK_\\bullet(f_1, \\ldots, f_{r - 1}).\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0629","source_file":"more-algebra.tex","source_line":6750,"source_end_line":6761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6750-L6761","statement_sha256":"67244a2a0753559470a3a1aec8382df7d48a4daa1dfc2d208531642fd382f29d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2997,"rank":2997,"depth":1,"x":621.386,"y":640.308,"cluster":"advanced-algebra"},{"id":"stacks:062A","tag":"062A","title":"The Koszul complex · Lemma 062A","summary":"Let R be a ring. Let A_bullet be a complex of R-modules. Let f, g ∈ R. Let C(f)_bullet be the cone of f : A_bullet → A_bullet. Define similarly C(g)_bullet and C(fg)_bullet. Then C(fg)_bullet is homotopy equivalent to the cone of a map C(f)_bullet[1] → C(g)_bullet","statement_latex":"Let $R$ be a ring. Let $A_\\bullet$ be a complex of $R$-modules.\nLet $f, g \\in R$. Let $C(f)_\\bullet$ be the cone of\n$f : A_\\bullet \\to A_\\bullet$. Define similarly $C(g)_\\bullet$ and\n$C(fg)_\\bullet$. Then $C(fg)_\\bullet$ is homotopy equivalent to the\ncone of a map\n$$\nC(f)_\\bullet[1] \\longrightarrow C(g)_\\bullet\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062A","source_file":"more-algebra.tex","source_line":6768,"source_end_line":6778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6768-L6778","statement_sha256":"4f5ae5153d213d344171682e9d83b5fd82b66bf833cdfb477bcafcc89c3990ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":2998,"rank":2998,"depth":0,"x":755.35,"y":675.998,"cluster":"advanced-algebra"},{"id":"stacks:062B","tag":"062B","title":"The Koszul complex · Lemma 062B","summary":"Let R be a ring. Let φ : E → R be an R-module map. Let f, g ∈ R. Set E' = E ⊕ R and define φ'_f, φ'_g, φ'_fg : E' → R by φ on E and multiplication by f, g, fg on R. The complex K_bullet(φ'_fg) is homotopy equivalent to the cone of a map of complexes K_bullet(φ'_f)[1] → K_bullet(φ'_g).","statement_latex":"Let $R$ be a ring. Let $\\varphi : E \\to R$ be an $R$-module map.\nLet $f, g \\in R$. Set $E' = E \\oplus R$ and define\n$\\varphi'_f, \\varphi'_g, \\varphi'_{fg} : E' \\to R$\nby $\\varphi$ on $E$ and multiplication by $f, g, fg$ on $R$.\nThe complex $K_\\bullet(\\varphi'_{fg})$ is homotopy equivalent to the\ncone of a map of complexes\n$$\nK_\\bullet(\\varphi'_f)[1]\n\\longrightarrow\nK_\\bullet(\\varphi'_g).\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062B","source_file":"more-algebra.tex","source_line":6843,"source_end_line":6856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6843-L6856","statement_sha256":"8d252a9e8d210b7736d6d11dd1a0f0cd091c2e278d089bcba390bcc88b3e8c7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":2999,"rank":2999,"depth":1,"x":627.511,"y":725.833,"cluster":"advanced-algebra"},{"id":"stacks:062C","tag":"062C","title":"The Koszul complex · Lemma 062C","summary":"Let R be a ring. Let f_1, …, f_r - 1 be a sequence of elements of R. Let f, g ∈ R. The complex K_bullet(f_1, …, f_r - 1, fg) is homotopy equivalent to the cone of a map of complexes K_bullet(f_1, …, f_r - 1, f)[1] → K_bullet(f_1, …, f_r - 1, g)","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_{r - 1}$ be a sequence of elements\nof $R$. Let $f, g \\in R$. The complex\n$K_\\bullet(f_1, \\ldots, f_{r - 1}, fg)$\nis homotopy equivalent to the cone of a map of complexes\n$$\nK_\\bullet(f_1, \\ldots, f_{r - 1}, f)[1]\n\\longrightarrow\nK_\\bullet(f_1, \\ldots, f_{r - 1}, g)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062C","source_file":"more-algebra.tex","source_line":6867,"source_end_line":6878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6867-L6878","statement_sha256":"e21db10adce0bd81b5174a692a76bc4536736bccd109a93e4b1b663bb22ba4c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3000,"rank":3000,"depth":2,"x":681.852,"y":616.245,"cluster":"advanced-algebra"},{"id":"stacks:0664","tag":"0664","title":"The Koszul complex · Lemma 0664","summary":"Let R be a ring. Let f_1, …, f_r, g_1, …, g_s be elements of R. Then there is an isomorphism of Koszul complexes K_bullet(R, f_1, …, f_r, g_1, …, g_s) = Tot(K_bullet(R, f_1, …, f_r) ⊗_R K_bullet(R, g_1, …, g_s)).","statement_latex":"Let $R$ be a ring.\nLet $f_1, \\ldots, f_r$, $g_1, \\ldots, g_s$ be elements of $R$.\nThen there is an isomorphism of Koszul complexes\n$$\nK_\\bullet(R, f_1, \\ldots, f_r, g_1, \\ldots, g_s) =\n\\text{Tot}(K_\\bullet(R, f_1, \\ldots, f_r) \\otimes_R\nK_\\bullet(R, g_1, \\ldots, g_s)).\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Koszul complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0664","source_file":"more-algebra.tex","source_line":6885,"source_end_line":6895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6885-L6895","statement_sha256":"67a0d802d0a0cba0927afd8b0919b651fb6af08ae34adaaed65d604094c4dd5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3001,"rank":3001,"depth":0,"x":730.042,"y":728.195,"cluster":"advanced-algebra"},{"id":"stacks:0G6G","tag":"0G6G","title":"The extended alternating v Cech complex · Lemma 0G6G","summary":"The extended alternating v Cech complexes defined above are complexes of R-modules.","statement_latex":"The extended alternating {\\v C}ech complexes defined above\nare complexes of $R$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The extended alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6G","source_file":"more-algebra.tex","source_line":6945,"source_end_line":6949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6945-L6949","statement_sha256":"3026684180c53b815491d397cd1e2cdad672b9a6c3556fbdcac911e2f0b21e47","origin":"The Stacks Project","memory_eligible":false,"source_rank":3002,"rank":3002,"depth":0,"x":604.136,"y":672.837,"cluster":"advanced-algebra"},{"id":"stacks:0G6H","tag":"0G6H","title":"The extended alternating v Cech complex · Lemma 0G6H","summary":"Let R be a ring. Let f_1, …, f_r ∈ R. Let M be an R-module. The extended alternating v Cech complex of M is the tensor product over R of M with the extended alternating v Cech complex of R.","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module.\nThe extended alternating {\\v C}ech complex of $M$ is the tensor product\nover $R$ of $M$ with the extended alternating {\\v C}ech complex of $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The extended alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6H","source_file":"more-algebra.tex","source_line":6955,"source_end_line":6960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6955-L6960","statement_sha256":"71579c0672b29ae8a67d1c1ca6d8ee403cf7b5504691051581cc17752a49d1bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3003,"rank":3003,"depth":0,"x":741.868,"y":642.132,"cluster":"advanced-algebra"},{"id":"stacks:0G6I","tag":"0G6I","title":"The extended alternating v Cech complex · Lemma 0G6I","summary":"Let R be a ring. Let f_1, …, f_r ∈ R. Let M be an R-module. Let R → S be a ring map, denote g_1, …, g_r ∈ S the images of f_1, …, f_r, and set N = M ⊗_R S. The extended alternating v Cech complex constructed using S, g_1, …, g_r, and N is the tensor product of the extended alternating v Cech complex of M with S over R.","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module.\nLet $R \\to S$ be a ring map, denote $g_1, \\ldots, g_r \\in S$ the images\nof $f_1, \\ldots, f_r$, and set $N = M \\otimes_R S$.\nThe extended alternating {\\v C}ech complex constructed using\n$S$, $g_1, \\ldots, g_r$, and $N$ is the tensor product of the\nextended alternating {\\v C}ech complex of $M$ with $S$ over $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The extended alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6I","source_file":"more-algebra.tex","source_line":6966,"source_end_line":6974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6966-L6974","statement_sha256":"28bfbd054cb8765bca10961dff6af43b2adc0cbe28b27b750b474d5880c6c202","origin":"The Stacks Project","memory_eligible":false,"source_rank":3004,"rank":3004,"depth":0,"x":664.791,"y":743.199,"cluster":"advanced-algebra"},{"id":"stacks:0G6J","tag":"0G6J","title":"The extended alternating v Cech complex · Lemma 0G6J","summary":"Let R be a ring. Let f_1, …, f_r ∈ R. Let M be an R-module. If there exists an i ∈ (1, …, r) such that f_i is a unit, then the extended alternating v Cech complex of M is homotopy equivalent to 0.","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module.\nIf there exists an $i \\in \\{1, \\ldots, r\\}$ such that $f_i$ is a unit, then\nthe extended alternating {\\v C}ech\ncomplex of $M$ is homotopy equivalent to $0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The extended alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6J","source_file":"more-algebra.tex","source_line":6980,"source_end_line":6986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L6980-L6986","statement_sha256":"f15f45d292741ef73e0661171a628789e0b3d237ff8e27109b46572487a2c0ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":3005,"rank":3005,"depth":0,"x":640.286,"y":624.62,"cluster":"advanced-algebra"},{"id":"stacks:0G6K","tag":"0G6K","title":"The extended alternating v Cech complex · Lemma 0G6K","summary":"Let R be a ring. Let f_1, …, f_r ∈ R. Let M be an R-module. Let H^q be the qth cohomology module of the extended alternation v Cech complex of M. Then • H^q = 0 if q not ∈ [0, r], • for x ∈ H^i there exists an n ≥ 1 such that f_i^n x = 0 for i = 1, …, r, • the support of H^q is contained in V(f_1, …, f_r), • if there is an f ∈ (f_1, …, f_r) which acts invertibly on M, then H^q = 0.","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. Let $M$ be an $R$-module.\nLet $H^q$ be the $q$th cohomology module of the extended alternation\n{\\v C}ech complex of $M$. Then\n\\begin{enumerate}\n\\item $H^q = 0$ if $q \\not \\in [0, r]$,\n\\item for $x \\in H^i$ there exists an $n \\geq 1$ such that $f_i^n x = 0$\nfor $i = 1, \\ldots, r$,\n\\item the support of $H^q$ is contained in $V(f_1, \\ldots, f_r)$,\n\\item if there is an $f \\in (f_1, \\ldots, f_r)$ which acts invertibly\non $M$, then $H^q = 0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The extended alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6K","source_file":"more-algebra.tex","source_line":7065,"source_end_line":7078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7065-L7078","statement_sha256":"dc9b29bbd06cf8a0acc318e5efd7ab41c43daadabf6c0284fafc61f149a90e54","origin":"The Stacks Project","memory_eligible":false,"source_rank":3006,"rank":3006,"depth":1,"x":754.017,"y":698.351,"cluster":"advanced-algebra"},{"id":"stacks:0913","tag":"0913","title":"The extended alternating v Cech complex · Lemma 0913","summary":"Let R be a ring. Let f_1, …, f_r ∈ R. The extended alternating v Cech complex R → bigoplus_i_0 R_f_i_0 → bigoplus_i_0 < i_1 R_f_i_0f_i_1 → … → R_f_1… f_r is a colimit of the Koszul complexes K(R, f_1^n, …, f_r^n); see proof for a precise statement.","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$. The extended alternating\n{\\v C}ech complex\n$$\nR \\to \\bigoplus\\nolimits_{i_0} R_{f_{i_0}} \\to\n\\bigoplus\\nolimits_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \\to\n\\ldots \\to R_{f_1\\ldots f_r}\n$$\nis a colimit of the Koszul complexes $K(R, f_1^n, \\ldots, f_r^n)$; see\nproof for a precise statement.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The extended alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0913","source_file":"more-algebra.tex","source_line":7099,"source_end_line":7110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7099-L7110","statement_sha256":"fdd6ec4efa2626d961b706b758403a0f1f732d75a477ac1e0fa89d6ca5b81fc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3007,"rank":3007,"depth":1,"x":610.48,"y":708.543,"cluster":"advanced-algebra"},{"id":"stacks:062E","tag":"062E","title":"Koszul regular sequences · Definition 062E","summary":"Let R be a ring. Let r ≥ 0 and let f_1, …, f_r ∈ R be a sequence of elements. Let M be an R-module. The sequence f_1, …, f_r is called • M-Koszul-regular if H_i(K_bullet(f_1, …, f_r) ⊗_R M) = 0 for all i not = 0, • M-H_1-regular if H_1(K_bullet(f_1, …, f_r) ⊗_R M) = 0, • Koszul-regular if H_i(K_bullet(f_1, …, f_r)) = 0 for all i not = 0, and • H_1-regular if H_1(K_bullet(f_1, …, f_r)) = 0.","statement_latex":"Let $R$ be a ring. Let $r \\geq 0$ and let $f_1, \\ldots, f_r \\in R$\nbe a sequence of elements. Let $M$ be an $R$-module.\nThe sequence $f_1, \\ldots, f_r$ is called\n\\begin{enumerate}\n\\item {\\it $M$-Koszul-regular} if\n$H_i(K_\\bullet(f_1, \\ldots, f_r) \\otimes_R M) = 0$ for\nall $i \\not = 0$,\n\\item {\\it $M$-$H_1$-regular} if\n$H_1(K_\\bullet(f_1, \\ldots, f_r) \\otimes_R M) = 0$,\n\\item {\\it Koszul-regular} if $H_i(K_\\bullet(f_1, \\ldots, f_r)) = 0$ for\nall $i \\not = 0$, and\n\\item {\\it $H_1$-regular} if $H_1(K_\\bullet(f_1, \\ldots, f_r)) = 0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062E","source_file":"more-algebra.tex","source_line":7216,"source_end_line":7231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7216-L7231","statement_sha256":"d27161868af40144657dfd7f80d570beecb122777dbfdcbf94a3bb5cd1392a74","origin":"The Stacks Project","memory_eligible":false,"source_rank":3008,"rank":3008,"depth":0,"x":708.385,"y":619.346,"cluster":"advanced-algebra"},{"id":"stacks:062F","tag":"062F","title":"Koszul regular sequences · Lemma 062F","summary":"Let R be a ring, M an R-module, and f_1, …, f_r ∈ R such that for i = 1, …, r multiplication by f_i is injective on M/(f_1, …, f_i - 1)M. Then f_1, …, f_r is M-Koszul regular. In particular, an M-regular sequence is M-Koszul-regular and any regular sequence is Koszul-regular.","statement_latex":"Let $R$ be a ring, $M$ an $R$-module, and $f_1, \\ldots, f_r \\in R$\nsuch that for $i = 1, \\ldots, r$ multiplication by $f_i$\nis injective on $M/(f_1, \\ldots, f_{i - 1})M$. Then $f_1, \\ldots, f_r$\nis $M$-Koszul regular. In particular, an $M$-regular sequence\nis $M$-Koszul-regular and any regular sequence is Koszul-regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062F","source_file":"more-algebra.tex","source_line":7264,"source_end_line":7271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7264-L7271","statement_sha256":"b73845a0adf6e31ea63d34c25c4454b355276e4c9a206e702b2c9e8c3cfa6006","origin":"The Stacks Project","memory_eligible":false,"source_rank":3009,"rank":3009,"depth":2,"x":707.919,"y":740.992,"cluster":"advanced-algebra"},{"id":"stacks:0CEM","tag":"0CEM","title":"Koszul regular sequences · Lemma 0CEM","summary":"A M-Koszul-regular sequence is M-H_1-regular. A Koszul-regular sequence is H_1-regular.","statement_latex":"A $M$-Koszul-regular sequence is $M$-$H_1$-regular.\nA Koszul-regular sequence is $H_1$-regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEM","source_file":"more-algebra.tex","source_line":7300,"source_end_line":7304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7300-L7304","statement_sha256":"72f6c48c71b15fd456bb798311f1337b01141d734182121676f99162d978734e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3010,"rank":3010,"depth":0,"x":610.183,"y":650.792,"cluster":"advanced-algebra"},{"id":"stacks:062G","tag":"062G","title":"Koszul regular sequences · Lemma 062G","summary":"Let f_1, …, f_r - 1 ∈ R be a sequence and f, g ∈ R. Let M be an R-module. • If f_1, …, f_r - 1, f and f_1, …, f_r - 1, g are M-H_1-regular then f_1, …, f_r - 1, fg is M-H_1-regular too. • If f_1, …, f_r - 1, f and f_1, …, f_r - 1, g are M-Koszul-regular then f_1, …, f_r - 1, fg is M-Koszul-regular too.","statement_latex":"Let $f_1, \\ldots, f_{r - 1} \\in R$ be a sequence and $f, g \\in R$.\nLet $M$ be an $R$-module.\n\\begin{enumerate}\n\\item If $f_1, \\ldots, f_{r - 1}, f$ and $f_1, \\ldots, f_{r - 1}, g$\nare $M$-$H_1$-regular then $f_1, \\ldots, f_{r - 1}, fg$ is\n$M$-$H_1$-regular too.\n\\item If $f_1, \\ldots, f_{r - 1}, f$ and $f_1, \\ldots, f_{r - 1}, g$\nare $M$-Koszul-regular then $f_1, \\ldots, f_{r - 1}, fg$ is\n$M$-Koszul-regular too.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062G","source_file":"more-algebra.tex","source_line":7310,"source_end_line":7322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7310-L7322","statement_sha256":"06909968efd60685214a97bc9a781253cfb00a5bf6394fbec80c8d3620f4c8f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3011,"rank":3011,"depth":3,"x":755.165,"y":661.875,"cluster":"advanced-algebra"},{"id":"stacks:062H","tag":"062H","title":"Koszul regular sequences · Lemma 062H","summary":"Let φ : R → S be a flat ring map. Let f_1, …, f_r ∈ R. Let M be an R-module and set N = M ⊗_R S. • If f_1, …, f_r in R is an M-H_1-regular sequence, then φ(f_1), …, φ(f_r) is an N-H_1-regular sequence in S. • If f_1, …, f_r is an M-Koszul-regular sequence in R, then φ(f_1), …, φ(f_r) is an N-Koszul-regular sequence in S.","statement_latex":"Let $\\varphi : R \\to S$ be a flat ring map. Let $f_1, \\ldots, f_r \\in R$.\nLet $M$ be an $R$-module and set $N = M \\otimes_R S$.\n\\begin{enumerate}\n\\item If $f_1, \\ldots, f_r$ in $R$ is an $M$-$H_1$-regular sequence, then\n$\\varphi(f_1), \\ldots, \\varphi(f_r)$ is an $N$-$H_1$-regular\nsequence in $S$.\n\\item If $f_1, \\ldots, f_r$ is an $M$-Koszul-regular sequence in $R$, then\n$\\varphi(f_1), \\ldots, \\varphi(f_r)$ is an $N$-Koszul-regular\nsequence in $S$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062H","source_file":"more-algebra.tex","source_line":7336,"source_end_line":7348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7336-L7348","statement_sha256":"95a7ca929a1b3d8ed59db86f0e5895d6b86e136eb8cc517ef06e5745c976a093","origin":"The Stacks Project","memory_eligible":false,"source_rank":3012,"rank":3012,"depth":0,"x":639.044,"y":736.161,"cluster":"advanced-algebra"},{"id":"stacks:062I","tag":"062I","title":"Koszul regular sequences · Lemma 062I","summary":"An M-H_1-regular sequence is M-quasi-regular.","statement_latex":"An $M$-$H_1$-regular sequence is $M$-quasi-regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062I","source_file":"more-algebra.tex","source_line":7359,"source_end_line":7362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7359-L7362","statement_sha256":"56ec6bc584a7e38f4fbf375d7255b8ddd967ea56266808a0bce33b9e1f5f4b3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3013,"rank":3013,"depth":4,"x":664.999,"y":615.182,"cluster":"advanced-algebra"},{"id":"stacks:09CC","tag":"09CC","title":"Koszul regular sequences · Lemma 09CC","summary":"Let (R, m) be a Noetherian local ring. Let M be a nonzero finite R-module. Let f_1, …, f_r ∈ m. The following are equivalent • f_1, …, f_r is an M-regular sequence, • f_1, …, f_r is a M-Koszul-regular sequence, • f_1, …, f_r is an M-H_1-regular sequence, • f_1, …, f_r is an M-quasi-regular sequence. In particular the sequence f_1, …, f_r is a regular sequence in R if and only if it is a Koszul regular sequence, if and only if it is a H_1-regular sequence, if and only if…","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let $M$ be a nonzero\nfinite $R$-module. Let $f_1, \\ldots, f_r \\in \\mathfrak m$. The following\nare equivalent\n\\begin{enumerate}\n\\item $f_1, \\ldots, f_r$ is an $M$-regular sequence,\n\\item $f_1, \\ldots, f_r$ is a $M$-Koszul-regular sequence,\n\\item $f_1, \\ldots, f_r$ is an $M$-$H_1$-regular sequence,\n\\item $f_1, \\ldots, f_r$ is an $M$-quasi-regular sequence.\n\\end{enumerate}\nIn particular the sequence $f_1, \\ldots, f_r$ is a regular sequence\nin $R$ if and only if it is a Koszul regular sequence, if and only if\nit is a $H_1$-regular sequence, if and only if it is a quasi-regular sequence.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CC","source_file":"more-algebra.tex","source_line":7444,"source_end_line":7458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7444-L7458","statement_sha256":"ba8de15d29f3001ea3385ef59639ae1bc153c1ef48f1b933a7ae033bf0800660","origin":"The Stacks Project","memory_eligible":false,"source_rank":3014,"rank":3014,"depth":5,"x":743.348,"y":719.384,"cluster":"advanced-algebra"},{"id":"stacks:0665","tag":"0665","title":"Koszul regular sequences · Lemma 0665","summary":"Let A be a ring. Let I ⊂ A be an ideal. Let g_1, …, g_m be a sequence in A whose image in A/I is H_1-regular. Then I ∩ (g_1, …, g_m) = I(g_1, …, g_m).","statement_latex":"Let $A$ be a ring. Let $I \\subset A$ be an ideal.\nLet $g_1, \\ldots, g_m$ be a sequence in $A$ whose image in\n$A/I$ is $H_1$-regular. Then $I \\cap (g_1, \\ldots, g_m) =\nI(g_1, \\ldots, g_m)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0665","source_file":"more-algebra.tex","source_line":7471,"source_end_line":7477,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7471-L7477","statement_sha256":"672413961de93d8017f5a6e63815bc36d12c4e2b238bc2e6d3071154614885a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3015,"rank":3015,"depth":0,"x":601.417,"y":686.922,"cluster":"advanced-algebra"},{"id":"stacks:0666","tag":"0666","title":"Koszul regular sequences · Lemma 0666","summary":"Let A be a ring. Let I ⊂ J ⊂ A be ideals. Assume that J/I ⊂ A/I is generated by an H_1-regular sequence. Then I ∩ J^2 = IJ.","statement_latex":"Let $A$ be a ring. Let $I \\subset J \\subset A$ be ideals.\nAssume that $J/I \\subset A/I$ is generated by an $H_1$-regular sequence.\nThen $I \\cap J^2 = IJ$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0666","source_file":"more-algebra.tex","source_line":7496,"source_end_line":7501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7496-L7501","statement_sha256":"06c3e1c75cac1dc0e3aa021ea74294cc787a79bf42d9d8433ea083e14eebdf55","origin":"The Stacks Project","memory_eligible":false,"source_rank":3016,"rank":3016,"depth":1,"x":732.512,"y":630.18,"cluster":"advanced-algebra"},{"id":"stacks:0667","tag":"0667","title":"Koszul regular sequences · Lemma 0667","summary":"Let A be a ring. Let I be an ideal generated by a quasi-regular sequence f_1, …, f_n in A. Let g_1, …, g_m ∈ A be elements whose images overlineg_1, …, overlineg_m form an H_1-regular sequence in A/I. Then f_1, …, f_n, g_1, …, g_m is a quasi-regular sequence in A.","statement_latex":"Let $A$ be a ring. Let $I$ be an ideal generated by a quasi-regular\nsequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be\nelements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form an\n$H_1$-regular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$\nis a quasi-regular sequence in $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0667","source_file":"more-algebra.tex","source_line":7522,"source_end_line":7529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7522-L7529","statement_sha256":"73c9988bbc2d33e114d36d78380a9b18bd3e1b933bc92327b9bbae1372fde98e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3017,"rank":3017,"depth":5,"x":681.345,"y":746.701,"cluster":"advanced-algebra"},{"id":"stacks:0668","tag":"0668","title":"Koszul regular sequences · Lemma 0668","summary":"Let A be a ring. Let I be an ideal generated by an H_1-regular sequence f_1, …, f_n in A. Let g_1, …, g_m ∈ A be elements whose images overlineg_1, …, overlineg_m form an H_1-regular sequence in A/I. Then f_1, …, f_n, g_1, …, g_m is an H_1-regular sequence in A.","statement_latex":"Let $A$ be a ring. Let $I$ be an ideal generated by an\n$H_1$-regular sequence $f_1, \\ldots, f_n$ in $A$.\nLet $g_1, \\ldots, g_m \\in A$ be elements whose images\n$\\overline{g}_1, \\ldots, \\overline{g}_m$ form an $H_1$-regular sequence\nin $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$ is an $H_1$-regular\nsequence in $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0668","source_file":"more-algebra.tex","source_line":7575,"source_end_line":7583,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7575-L7583","statement_sha256":"6c8fc0fcf77f344d61b5d00eac972406f59872a5b4d83bb10f262723621652ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":3018,"rank":3018,"depth":0,"x":625.232,"y":631.458,"cluster":"advanced-algebra"},{"id":"stacks:068L","tag":"068L","title":"Koszul regular sequences · Lemma 068L","summary":"Let A be a ring. Let f_1, …, f_n, g_1, …, g_m ∈ A be an H_1-regular sequence. Then the images overlineg_1, …, overlineg_m in A/(f_1, …, f_n) form an H_1-regular sequence.","statement_latex":"Let $A$ be a ring. Let $f_1, \\ldots, f_n, g_1, \\ldots, g_m \\in A$\nbe an $H_1$-regular sequence. Then the images\n$\\overline{g}_1, \\ldots, \\overline{g}_m$ in $A/(f_1, \\ldots, f_n)$\nform an $H_1$-regular sequence.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068L","source_file":"more-algebra.tex","source_line":7618,"source_end_line":7624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7618-L7624","statement_sha256":"b78ccc019c3208e8e2681139ab2d2b97910e8b1b643670b110c6d531829f27c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3019,"rank":3019,"depth":0,"x":759.619,"y":684.73,"cluster":"advanced-algebra"},{"id":"stacks:0669","tag":"0669","title":"Koszul regular sequences · Lemma 0669","summary":"Let A be a ring. Let I be an ideal generated by a Koszul-regular sequence f_1, …, f_n in A. Let g_1, …, g_m ∈ A be elements whose images overlineg_1, …, overlineg_m form a Koszul-regular sequence in A/I. Then f_1, …, f_n, g_1, …, g_m is a Koszul-regular sequence in A.","statement_latex":"Let $A$ be a ring. Let $I$ be an ideal generated by a Koszul-regular\nsequence $f_1, \\ldots, f_n$ in $A$. Let $g_1, \\ldots, g_m \\in A$ be\nelements whose images $\\overline{g}_1, \\ldots, \\overline{g}_m$ form a\nKoszul-regular sequence in $A/I$. Then $f_1, \\ldots, f_n, g_1, \\ldots, g_m$\nis a Koszul-regular sequence in $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0669","source_file":"more-algebra.tex","source_line":7639,"source_end_line":7646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7639-L7646","statement_sha256":"26f87be611164250a303b7736204cc4474feaeca66b424221f95dd441797392f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3020,"rank":3020,"depth":6,"x":617.332,"y":721.794,"cluster":"advanced-algebra"},{"id":"stacks:068M","tag":"068M","title":"Koszul regular sequences · Lemma 068M","summary":"Let A be a ring. Let f_1, …, f_n, g_1, …, g_m ∈ A. If both f_1, …, f_n and f_1, …, f_n, g_1, …, g_m are Koszul-regular sequences in A, then overlineg_1, …, overlineg_m in A/(f_1, …, f_n) form a Koszul-regular sequence.","statement_latex":"Let $A$ be a ring. Let $f_1, \\ldots, f_n, g_1, \\ldots, g_m \\in A$.\nIf both $f_1, \\ldots, f_n$ and $f_1, \\ldots, f_n, g_1, \\ldots, g_m$\nare Koszul-regular sequences in $A$, then\n$\\overline{g}_1, \\ldots, \\overline{g}_m$ in $A/(f_1, \\ldots, f_n)$\nform a Koszul-regular sequence.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068M","source_file":"more-algebra.tex","source_line":7679,"source_end_line":7686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7679-L7686","statement_sha256":"b0bdb2f3737ddafb5f4fdccb2742b61750e97064a6ff866b2ba56eb66ee50627","origin":"The Stacks Project","memory_eligible":false,"source_rank":3021,"rank":3021,"depth":6,"x":692.632,"y":613.459,"cluster":"advanced-algebra"},{"id":"stacks:066A","tag":"066A","title":"Koszul regular sequences · Lemma 066A","summary":"Let R be a ring. Let I be an ideal generated by f_1, …, f_r ∈ R. • If I can be generated by a quasi-regular sequence of length r, then f_1, …, f_r is a quasi-regular sequence. • If I can be generated by an H_1-regular sequence of length r, then f_1, …, f_r is an H_1-regular sequence. • If I can be generated by a Koszul-regular sequence of length r, then f_1, …, f_r is a Koszul-regular sequence.","statement_latex":"Let $R$ be a ring. Let $I$ be an ideal generated by $f_1, \\ldots, f_r \\in R$.\n\\begin{enumerate}\n\\item If $I$ can be generated by a quasi-regular sequence of length $r$,\nthen $f_1, \\ldots, f_r$ is a quasi-regular sequence.\n\\item If $I$ can be generated by an $H_1$-regular sequence of length $r$,\nthen $f_1, \\ldots, f_r$ is an $H_1$-regular sequence.\n\\item If $I$ can be generated by a Koszul-regular sequence of length $r$,\nthen $f_1, \\ldots, f_r$ is a Koszul-regular sequence.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066A","source_file":"more-algebra.tex","source_line":7711,"source_end_line":7722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7711-L7722","statement_sha256":"d2e1e140a2fd09f37a88d6a9ea33628a954e09724a208e5e136538a0a600181a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3022,"rank":3022,"depth":5,"x":724.304,"y":736.373,"cluster":"advanced-algebra"},{"id":"stacks:068P","tag":"068P","title":"Koszul regular sequences · Lemma 068P","summary":"This is a particular case of [McCoy] Let R be a ring. Let a_1, …, a_n ∈ R be elements such that R → R^⊕ n, x ↦ (xa_1, …, xa_n) is injective. Then the element ∑ a_i t_i of the polynomial ring R[t_1, …, t_n] is a nonzerodivisor.","statement_latex":"\\begin{reference}\nThis is a particular case of \\cite[Corollary]{McCoy}\n\\end{reference}\nLet $R$ be a ring. Let $a_1, \\ldots, a_n \\in R$ be elements such\nthat $R \\to R^{\\oplus n}$, $x \\mapsto (xa_1, \\ldots, xa_n)$ is injective.\nThen the element $\\sum a_i t_i$ of the polynomial ring $R[t_1, \\ldots, t_n]$\nis a nonzerodivisor.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068P","source_file":"more-algebra.tex","source_line":7757,"source_end_line":7766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7757-L7766","statement_sha256":"64937fdaac46bc63a0163329f0a58c915d2d83904370c8f3968c36dd5555a814","origin":"The Stacks Project","memory_eligible":false,"source_rank":3023,"rank":3023,"depth":10,"x":601.806,"y":663.531,"cluster":"advanced-algebra"},{"id":"stacks:068Q","tag":"068Q","title":"Koszul regular sequences · Lemma 068Q","summary":"Let R be a ring. Let f_1, …, f_n be a Koszul-regular sequence in R such that (f_1, …, f_n) not = R. Consider the faithfully flat, smooth ring map R → S = R[(t_ij)_i ≤ j, t_11^-1, t_22^-1, …, t_nn^-1] For 1 ≤ i ≤ n set g_i = ∑_i ≤ j t_ij f_j ∈ S. Then g_1, …, g_n is a regular sequence in S and (f_1, …, f_n)S = (g_1, …, g_n).","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_n$ be a Koszul-regular sequence\nin $R$ such that $(f_1, \\ldots, f_n) \\not = R$.\nConsider the faithfully flat, smooth ring map\n$$\nR \\longrightarrow\nS = R[\\{t_{ij}\\}_{i \\leq j}, t_{11}^{-1}, t_{22}^{-1}, \\ldots, t_{nn}^{-1}]\n$$\nFor $1 \\leq i \\leq n$ set\n$$\ng_i = \\sum\\nolimits_{i \\leq j} t_{ij} f_j \\in S.\n$$\nThen $g_1, \\ldots, g_n$ is a regular sequence in $S$ and\n$(f_1, \\ldots, f_n)S = (g_1, \\ldots, g_n)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068Q","source_file":"more-algebra.tex","source_line":7798,"source_end_line":7813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7798-L7813","statement_sha256":"63f3fea7fc687a63630f4d8a5b5c9736481c5d6951f9055420c7aa2b613d9bd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3024,"rank":3024,"depth":11,"x":751.075,"y":647.697,"cluster":"advanced-algebra"},{"id":"stacks:0G6L","tag":"0G6L","title":"More on Koszul regular sequences · Lemma 0G6L","summary":"Let R be a ring. Let f_1, …, f_r ∈ R be an Koszul-regular sequence. Then the extended alternating v Cech complex R → bigoplus_i_0 R_f_i_0 → bigoplus_i_0 < i_1 R_f_i_0f_i_1 → … → R_f_1… f_r from Section [Tag 0G6F] only has cohomology in degree r.","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be an\nKoszul-regular sequence. Then the extended alternating\n{\\v C}ech complex $R \\to \\bigoplus\\nolimits_{i_0} R_{f_{i_0}} \\to\n\\bigoplus\\nolimits_{i_0 < i_1} R_{f_{i_0}f_{i_1}} \\to\n\\ldots \\to R_{f_1\\ldots f_r}$ from Section \\ref{section-alternating-cech}\nonly has cohomology in degree $r$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"More on Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6L","source_file":"more-algebra.tex","source_line":7862,"source_end_line":7870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7862-L7870","statement_sha256":"4eb4c30e29b54d688702ba9c236314a8c746e078e958b25db6791b24c49caf92","origin":"The Stacks Project","memory_eligible":false,"source_rank":3025,"rank":3025,"depth":4,"x":653.504,"y":744.305,"cluster":"advanced-algebra"},{"id":"stacks:0BIQ","tag":"0BIQ","title":"More on Koszul regular sequences · Lemma 0BIQ","summary":"Let a, a_2, …, a_r be an H_1-regular sequence in a ring R (for example a Koszul regular sequence or a regular sequence, see Lemmas [Tag 062F] and [Tag 0CEM]). With I = (a, a_2, …, a_r) the blowup algebra R' = R[fracIa] is isomorphic to R\" = R[y_2, …, y_r]/(a y_i - a_i).","statement_latex":"Let $a, a_2, \\ldots, a_r$ be an $H_1$-regular sequence in a ring $R$\n(for example a Koszul regular sequence or a regular sequence, see\nLemmas \\ref{lemma-regular-koszul-regular} and\n\\ref{lemma-koszul-regular-H1-regular}).\nWith $I = (a, a_2, \\ldots, a_r)$ the blowup algebra $R' = R[\\frac{I}{a}]$\nis isomorphic to $R'' = R[y_2, \\ldots, y_r]/(a y_i - a_i)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"More on Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIQ","source_file":"more-algebra.tex","source_line":7883,"source_end_line":7891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7883-L7891","statement_sha256":"eac82c0009dbf71097d1a7f4979fa20b7721c7b42640f4809a72f674dc44fa80","origin":"The Stacks Project","memory_eligible":false,"source_rank":3026,"rank":3026,"depth":3,"x":647.748,"y":617.397,"cluster":"advanced-algebra"},{"id":"stacks:063Q","tag":"063Q","title":"More on Koszul regular sequences · Lemma 063Q","summary":"Let A → B be a ring map. Let f_1, …, f_r be a sequence in B such that B/(f_1, …, f_r) is A-flat. Let A → A' be a ring map. Then the canonical map H_1(K_bullet(B, f_1, …, f_r)) ⊗_A A' → H_1(K_bullet(B', f'_1, …, f'_r)) is surjective. Here B' = B ⊗_A A' and f_i' ∈ B' is the image of f_i.","statement_latex":"Let $A \\to B$ be a ring map.\nLet $f_1, \\ldots, f_r$ be a sequence in $B$ such that $B/(f_1, \\ldots, f_r)$\nis $A$-flat. Let $A \\to A'$ be a ring map. Then the canonical map\n$$\nH_1(K_\\bullet(B, f_1, \\ldots, f_r)) \\otimes_A A'\n\\longrightarrow\nH_1(K_\\bullet(B', f'_1, \\ldots, f'_r))\n$$\nis surjective. Here $B' = B \\otimes_A A'$ and $f_i' \\in B'$ is the image\nof $f_i$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"More on Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063Q","source_file":"more-algebra.tex","source_line":7934,"source_end_line":7946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7934-L7946","statement_sha256":"91f15bcb185b9ec9ea007cce6d0472ff2ba1323effb6d04353b3623836c12080","origin":"The Stacks Project","memory_eligible":false,"source_rank":3027,"rank":3027,"depth":1,"x":754.303,"y":707.929,"cluster":"advanced-algebra"},{"id":"stacks:0CEP","tag":"0CEP","title":"More on Koszul regular sequences · Lemma 0CEP","summary":"Let A → B and A → A' be ring maps. Set B' = B ⊗_A A'. Let f_1, …, f_r ∈ B. Assume B/(f_1, …, f_r)B is flat over A • If f_1, …, f_r is a quasi-regular sequence, then the image in B' is a quasi-regular sequence. • If f_1, …, f_r is a H_1-regular sequence, then the image in B' is a H_1-regular sequence.","statement_latex":"Let $A \\to B$ and $A \\to A'$ be ring maps. Set $B' = B \\otimes_A A'$.\nLet $f_1, \\ldots, f_r \\in B$. Assume $B/(f_1, \\ldots, f_r)B$ is flat over $A$\n\\begin{enumerate}\n\\item If $f_1, \\ldots, f_r$ is a quasi-regular sequence, then\nthe image in $B'$ is a quasi-regular sequence.\n\\item If $f_1, \\ldots, f_r$ is a $H_1$-regular sequence, then\nthe image in $B'$ is a $H_1$-regular sequence.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"More on Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEP","source_file":"more-algebra.tex","source_line":7991,"source_end_line":8001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L7991-L8001","statement_sha256":"9138e2893a7538f70d57510c01c58f38e64e6d7cb0e8231b0054242bcf6b5fa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3028,"rank":3028,"depth":4,"x":602.567,"y":701.618,"cluster":"advanced-algebra"},{"id":"stacks:0CEQ","tag":"0CEQ","title":"More on Koszul regular sequences · Lemma 0CEQ","summary":"Let A' → B' be a ring map. Let I ⊂ A' be an ideal. Set A = A'/I and B = B'/IB'. Let f'_1, …, f'_r ∈ B'. Assume • A' → B' is flat and of finite presentation, • I is locally nilpotent, • the images f_1, …, f_r ∈ B form a quasi-regular sequence, • B/(f_1, …, f_r) is flat over A. Then B'/(f'_1, …, f'_r) is flat over A'.","statement_latex":"Let $A' \\to B'$ be a ring map. Let $I \\subset A'$ be an ideal.\nSet $A = A'/I$ and $B = B'/IB'$. Let $f'_1, \\ldots, f'_r \\in B'$. Assume\n\\begin{enumerate}\n\\item $A' \\to B'$ is flat and of finite presentation,\n\\item $I$ is locally nilpotent,\n\\item the images $f_1, \\ldots, f_r \\in B$ form a quasi-regular sequence,\n\\item $B/(f_1, \\ldots, f_r)$ is flat over $A$.\n\\end{enumerate}\nThen $B'/(f'_1, \\ldots, f'_r)$ is flat over $A'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"More on Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEQ","source_file":"more-algebra.tex","source_line":8023,"source_end_line":8034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8023-L8034","statement_sha256":"762b028de391a420d6b937a68d17f1993fc5390b566c290523ce54f5dc4367e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3029,"rank":3029,"depth":14,"x":719.808,"y":619.979,"cluster":"advanced-algebra"},{"id":"stacks:0CER","tag":"0CER","title":"More on Koszul regular sequences · Lemma 0CER","summary":"Let A' → B' be a ring map. Let I ⊂ A' be an ideal. Set A = A'/I and B = B'/IB'. Let f'_1, …, f'_r ∈ B'. Assume • A' → B' is flat and of finite presentation (for example smooth), • I is locally nilpotent, • the images f_1, …, f_r ∈ B form a quasi-regular sequence, • B/(f_1, …, f_r) is smooth over A. Then B'/(f'_1, …, f'_r) is smooth over A'.","statement_latex":"Let $A' \\to B'$ be a ring map. Let $I \\subset A'$ be an ideal.\nSet $A = A'/I$ and $B = B'/IB'$. Let $f'_1, \\ldots, f'_r \\in B'$. Assume\n\\begin{enumerate}\n\\item $A' \\to B'$ is flat and of finite presentation (for example smooth),\n\\item $I$ is locally nilpotent,\n\\item the images $f_1, \\ldots, f_r \\in B$ form a quasi-regular sequence,\n\\item $B/(f_1, \\ldots, f_r)$ is smooth over $A$.\n\\end{enumerate}\nThen $B'/(f'_1, \\ldots, f'_r)$ is smooth over $A'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"More on Koszul regular sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CER","source_file":"more-algebra.tex","source_line":8061,"source_end_line":8072,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8061-L8072","statement_sha256":"6f705e673c0862de6aa20099e81f1dd705ee766505a9e151896564266997f12c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3030,"rank":3030,"depth":36,"x":698.958,"y":747.004,"cluster":"advanced-algebra"},{"id":"stacks:07CV","tag":"07CV","title":"Regular ideals · Definition 07CV","summary":"Let R be a ring and let I ⊂ R be an ideal. • We say I is a regular ideal if for every p ∈ V(I) there exists a g ∈ R, g not ∈ p and a regular sequence f_1, …, f_r ∈ R_g such that I_g is generated by f_1, …, f_r. • We say I is a Koszul-regular ideal if for every p ∈ V(I) there exists a g ∈ R, g not ∈ p and a Koszul-regular sequence f_1, …, f_r ∈ R_g such that I_g is generated by f_1, …, f_r. • We say I is a H_1-regular ideal if for every p ∈ V(I) there exists a g ∈ R, g not…","statement_latex":"Let $R$ be a ring and let $I \\subset R$ be an ideal.\n\\begin{enumerate}\n\\item We say $I$ is a {\\it regular ideal} if for every\n$\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$\nand a regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$\nis generated by $f_1, \\ldots, f_r$.\n\\item We say $I$ is a {\\it Koszul-regular ideal} if for every\n$\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$\nand a Koszul-regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$\nis generated by $f_1, \\ldots, f_r$.\n\\item We say $I$ is a {\\it $H_1$-regular ideal} if for every\n$\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$\nand an $H_1$-regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$\nis generated by $f_1, \\ldots, f_r$.\n\\item We say $I$ is a {\\it quasi-regular ideal} if for every\n$\\mathfrak p \\in V(I)$ there exists a $g \\in R$, $g \\not \\in \\mathfrak p$\nand a quasi-regular sequence $f_1, \\ldots, f_r \\in R_g$ such that $I_g$\nis generated by $f_1, \\ldots, f_r$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ideals","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CV","source_file":"more-algebra.tex","source_line":8100,"source_end_line":8121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8100-L8121","statement_sha256":"1bf853ab1d120ffb4b10ace91779b1c20d5b16479baf61d6205a955115122f77","origin":"The Stacks Project","memory_eligible":false,"source_rank":3031,"rank":3031,"depth":0,"x":611.978,"y":641.259,"cluster":"advanced-algebra"},{"id":"stacks:07CW","tag":"07CW","title":"Regular ideals · Lemma 07CW","summary":"A quasi-regular ideal is finitely generated.","statement_latex":"A quasi-regular ideal is finitely generated.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CW","source_file":"more-algebra.tex","source_line":8139,"source_end_line":8142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8139-L8142","statement_sha256":"c2424b0746305a2359f40b42c19e79c9a4c0bfdcc253199b59bb32d967df1cac","origin":"The Stacks Project","memory_eligible":false,"source_rank":3032,"rank":3032,"depth":3,"x":761.503,"y":669.946,"cluster":"advanced-algebra"},{"id":"stacks:08RK","tag":"08RK","title":"Regular ideals · Lemma 08RK","summary":"Let I ⊂ R be a quasi-regular ideal of a ring. Then I/I^2 is a finite projective R/I-module.","statement_latex":"Let $I \\subset R$ be a quasi-regular ideal of a ring.\nThen $I/I^2$ is a finite projective $R/I$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RK","source_file":"more-algebra.tex","source_line":8159,"source_end_line":8163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8159-L8163","statement_sha256":"e002d98ddb46a07831f9404c31c93ac5d2f3331664e650b196062eca89a932e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3033,"rank":3033,"depth":5,"x":627.864,"y":733.785,"cluster":"advanced-algebra"},{"id":"stacks:068N","tag":"068N","title":"Regular ideals · Lemma 068N","summary":"Let A → B be a faithfully flat ring map. Let I ⊂ A be an ideal. If IB is a Koszul-regular (resp. H_1-regular, resp. quasi-regular) ideal in B, then I is a Koszul-regular (resp. H_1-regular, resp. quasi-regular) ideal in A.","statement_latex":"Let $A \\to B$ be a faithfully flat ring map. Let $I \\subset A$ be an ideal.\nIf $IB$ is a Koszul-regular\n(resp.\\ $H_1$-regular, resp.\\ quasi-regular) ideal in $B$, then\n$I$ is a Koszul-regular (resp.\\ $H_1$-regular, resp.\\ quasi-regular)\nideal in $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068N","source_file":"more-algebra.tex","source_line":8174,"source_end_line":8181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8174-L8181","statement_sha256":"764fba45dfaeca49404af0e4ec76ad4e32b039eb95ff7a009bf2430f66fc69fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3034,"rank":3034,"depth":7,"x":675.18,"y":610.596,"cluster":"advanced-algebra"},{"id":"stacks:07CX","tag":"07CX","title":"Regular ideals · Lemma 07CX","summary":"Let A be a ring. Let I ⊂ J ⊂ A be ideals. Assume that J/I ⊂ A/I is a H_1-regular ideal. Then I ∩ J^2 = IJ.","statement_latex":"Let $A$ be a ring. Let $I \\subset J \\subset A$ be ideals.\nAssume that $J/I \\subset A/I$ is a $H_1$-regular ideal.\nThen $I \\cap J^2 = IJ$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CX","source_file":"more-algebra.tex","source_line":8234,"source_end_line":8239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8234-L8239","statement_sha256":"c4f2b860f99973b1974ff439d295d98e526f96bd429021add1c94262b020f78d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3035,"rank":3035,"depth":2,"x":739.504,"y":728.554,"cluster":"advanced-algebra"},{"id":"stacks:07CZ","tag":"07CZ","title":"Local complete intersection maps · Lemma 07CZ","summary":"Let A → B be a finite type ring map. If for some presentation α : A[x_1, …, x_n] → B the kernel I is a Koszul-regular ideal then for any presentation β : A[y_1, …, y_m] → B the kernel J is a Koszul-regular ideal.","statement_latex":"Let $A \\to B$ be a finite type ring map. If for some presentation\n$\\alpha : A[x_1, \\ldots, x_n] \\to B$ the kernel $I$ is a Koszul-regular ideal\nthen for any presentation $\\beta : A[y_1, \\ldots, y_m] \\to B$ the kernel\n$J$ is a Koszul-regular ideal.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CZ","source_file":"more-algebra.tex","source_line":8259,"source_end_line":8265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8259-L8265","statement_sha256":"3674a99a9ac868ae045c0f5d080380b6bc10c5d429d496854f34a4e85adac48a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3036,"rank":3036,"depth":7,"x":596.886,"y":677.956,"cluster":"advanced-algebra"},{"id":"stacks:07D0","tag":"07D0","title":"Local complete intersection maps · Definition 07D0","summary":"A ring map A → B is called a local complete intersection if it is of finite type and for some (equivalently any) presentation B = A[x_1, …, x_n]/I the ideal I is Koszul-regular.","statement_latex":"A ring map $A \\to B$ is called a {\\it local complete intersection}\nif it is of finite type and for some (equivalently any) presentation\n$B = A[x_1, \\ldots, x_n]/I$ the ideal $I$ is Koszul-regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local complete intersection maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07D0","source_file":"more-algebra.tex","source_line":8337,"source_end_line":8342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8337-L8342","statement_sha256":"a77d2e44cd9cabcf1d673131730887a83b4f48b531ca977c4d71328504d91a43","origin":"The Stacks Project","memory_eligible":false,"source_rank":3037,"rank":3037,"depth":0,"x":743.074,"y":634.242,"cluster":"advanced-algebra"},{"id":"stacks:07D1","tag":"07D1","title":"Local complete intersection maps · Lemma 07D1","summary":"Let R → S be a ring map. Let g_1, …, g_m ∈ S generate the unit ideal. If each R → S_g_j is a local complete intersection so is R → S.","statement_latex":"Let $R \\to S$ be a ring map. Let $g_1, \\ldots, g_m \\in S$\ngenerate the unit ideal. If each $R \\to S_{g_j}$ is a local\ncomplete intersection so is $R \\to S$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07D1","source_file":"more-algebra.tex","source_line":8347,"source_end_line":8352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8347-L8352","statement_sha256":"a379ac98d4e6e928f1faf97e1eb5bb6d565494f19df5557036f34e453e24e0fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":3038,"rank":3038,"depth":7,"x":670.265,"y":749.689,"cluster":"advanced-algebra"},{"id":"stacks:07D2","tag":"07D2","title":"Local complete intersection maps · Lemma 07D2","summary":"Let R be a ring. If R[x_1, …, x_n]/(f_1, …, f_c) is a relative global complete intersection, then f_1, …, f_c is a Koszul regular sequence.","statement_latex":"Let $R$ be a ring. If $R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$\nis a relative global complete intersection, then $f_1, \\ldots, f_c$\nis a Koszul regular sequence.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07D2","source_file":"more-algebra.tex","source_line":8385,"source_end_line":8390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8385-L8390","statement_sha256":"656274033ad167028a9ce1a14924d71262114836184498df9b5f8dd728486d15","origin":"The Stacks Project","memory_eligible":false,"source_rank":3039,"rank":3039,"depth":33,"x":631.026,"y":622.96,"cluster":"advanced-algebra"},{"id":"stacks:07D3","tag":"07D3","title":"Local complete intersection maps · Lemma 07D3","summary":"Let R → S be a ring map. The following are equivalent • R → S is syntomic (Algebra, Definition [Tag 00SL]), and • R → S is flat and a local complete intersection.","statement_latex":"Let $R \\to S$ be a ring map. The following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is syntomic\n(Algebra, Definition \\ref{algebra-definition-lci}), and\n\\item $R \\to S$ is flat and a local complete intersection.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07D3","source_file":"more-algebra.tex","source_line":8404,"source_end_line":8412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8404-L8412","statement_sha256":"1a0961f7806d1228d1282e27e0298e2e519776be50a381bf531cf67c6680e77f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3040,"rank":3040,"depth":35,"x":762.168,"y":694.304,"cluster":"advanced-algebra"},{"id":"stacks:07D4","tag":"07D4","title":"Local complete intersection maps · Lemma 07D4","summary":"Let A → B → C be ring maps. Assume B → C is a local complete intersection homomorphism. Choose a presentation α : A[x_s, s ∈ S] → B with kernel I. Choose a presentation β : B[y_1, …, y_m] → C with kernel J. Let γ : A[x_s, y_t] → C be the induced presentation of C with kernel K. Then we get a canonical commutative diagram xymatrix 0 ar[r] & Ω_A[x_s]/A ⊗ C ar[r] & Ω_A[x_s, y_t]/A ⊗ C ar[r] & Ω_B[y_t]/B ⊗ C ar[r] & 0 0 ar[r] & I/I^2 ⊗ C ar[r] ar[u] & K/K^2 ar[r] ar[u] &…","statement_latex":"Let $A \\to B \\to C$ be ring maps. Assume $B \\to C$ is a local complete\nintersection homomorphism. Choose a presentation\n$\\alpha : A[x_s, s \\in S] \\to B$ with kernel $I$. Choose a presentation\n$\\beta : B[y_1, \\ldots, y_m] \\to C$ with kernel $J$. Let\n$\\gamma : A[x_s, y_t] \\to C$ be the induced presentation of $C$ with kernel\n$K$. Then we get a canonical commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\n\\Omega_{A[x_s]/A} \\otimes C \\ar[r] &\n\\Omega_{A[x_s, y_t]/A} \\otimes C \\ar[r] &\n\\Omega_{B[y_t]/B} \\otimes C \\ar[r] &\n0 \\\\\n0 \\ar[r] &\nI/I^2 \\otimes C \\ar[r] \\ar[u] &\nK/K^2 \\ar[r] \\ar[u] &\nJ/J^2 \\ar[r] \\ar[u] &\n0\n}\n$$\nwith exact rows. In particular, the six term exact sequence of\nAlgebra, Lemma \\ref{algebra-lemma-exact-sequence-NL}\ncan be completed with a zero on the left, i.e., the sequence\n$$\n0 \\to H_1(\\NL_{B/A} \\otimes_B C) \\to\nH_1(L_{C/A}) \\to\nH_1(L_{C/B}) \\to\n\\Omega_{B/A} \\otimes_B C \\to\n\\Omega_{C/A} \\to\n\\Omega_{C/B} \\to 0\n$$\nis exact.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07D4","source_file":"more-algebra.tex","source_line":8458,"source_end_line":8492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8458-L8492","statement_sha256":"0268b9851f3ce6b19f88b88b99a3ac0d50432cde2be979b6a1adbe31c09a7673","origin":"The Stacks Project","memory_eligible":false,"source_rank":3041,"rank":3041,"depth":3,"x":607.746,"y":716.158,"cluster":"advanced-algebra"},{"id":"stacks:07D5","tag":"07D5","title":"Local complete intersection maps · Lemma 07D5","summary":"Let A → B → C be ring maps. If B → C is a filtered colimit of local complete intersection homomorphisms then the conclusion of Lemma [Tag 07D4] remains valid.","statement_latex":"Let $A \\to B \\to C$ be ring maps.\nIf $B \\to C$ is a filtered colimit of local complete intersection\nhomomorphisms then the conclusion of\nLemma \\ref{lemma-transitive-lci-at-end}\nremains valid.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07D5","source_file":"more-algebra.tex","source_line":8507,"source_end_line":8514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8507-L8514","statement_sha256":"aeaeda62835adbf13278a3a7084fe5d6cd98a3184c0c3bc3d904ad49f7d9af2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3042,"rank":3042,"depth":4,"x":704.256,"y":612.188,"cluster":"advanced-algebra"},{"id":"stacks:0D08","tag":"0D08","title":"Local complete intersection maps · Lemma 0D08","summary":"Let A → B be a local homomorphism of local rings. Let A^h → B^h, resp. A^sh → B^sh be the induced map on henselizations, resp. strict henselizations (Algebra, Lemma [Tag 04GS], resp. Lemma [Tag 04GU]). Then NL_B/A ⊗_B B^h → NL_B^h/A^h and NL_B/A ⊗_B B^sh → NL_B^sh/A^sh induce isomorphisms on cohomology groups.","statement_latex":"Let $A \\to B$ be a local homomorphism of local rings.\nLet $A^h \\to B^h$, resp.\\ $A^{sh} \\to B^{sh}$ be the induced\nmap on henselizations, resp.\\ strict henselizations\n(Algebra, Lemma \\ref{algebra-lemma-henselian-functorial},\nresp.\\ Lemma \\ref{algebra-lemma-strictly-henselian-functorial}).\nThen $\\NL_{B/A} \\otimes_B B^h \\to \\NL_{B^h/A^h}$ and\n$\\NL_{B/A} \\otimes_B B^{sh} \\to \\NL_{B^{sh}/A^{sh}}$\ninduce isomorphisms on cohomology groups.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D08","source_file":"more-algebra.tex","source_line":8523,"source_end_line":8533,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8523-L8533","statement_sha256":"8e4cc2b83b67c84168f6f4ac9ef64485f3c6e4d97af0d4393397f00f0ca61cd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3043,"rank":3043,"depth":47,"x":716.728,"y":743.908,"cluster":"advanced-algebra"},{"id":"stacks:07E1","tag":"07E1","title":"Cartier equality · Lemma 07E1","summary":"Let K/k be a finitely generated field extension. Then Ω_K/k and H_1(L_K/k) are finite dimensional and trdeg_k(K) = dim_K Ω_K/k - dim_K H_1(L_K/k).","statement_latex":"Let $K/k$ be a finitely generated field extension.\nThen $\\Omega_{K/k}$ and $H_1(L_{K/k})$ are finite dimensional and\n$\\text{trdeg}_k(K) = \\dim_K \\Omega_{K/k} - \\dim_K H_1(L_{K/k})$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Cartier's equality and geometric regularity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07E1","source_file":"more-algebra.tex","source_line":8572,"source_end_line":8577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8572-L8577","statement_sha256":"021bb2b0c0c945f58c68668f15ba383c7e71d8372812d522360338f69f8223ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":3044,"rank":3044,"depth":41,"x":601.35,"y":653.657,"cluster":"advanced-algebra"},{"id":"stacks:07E2","tag":"07E2","title":"Cartier's equality and geometric regularity · Lemma 07E2","summary":"Let M/L/K be field extensions. Then the Jacobi-Zariski sequence 0 → H_1(L_L/K) ⊗_L M → H_1(L_M/K) → H_1(L_M/L) → Ω_L/K ⊗_L M → Ω_M/K → Ω_M/L → 0 is exact.","statement_latex":"Let $M/L/K$ be field extensions. Then the Jacobi-Zariski\nsequence\n$$\n0 \\to H_1(L_{L/K}) \\otimes_L M \\to\nH_1(L_{M/K}) \\to\nH_1(L_{M/L}) \\to\n\\Omega_{L/K} \\otimes_L M \\to\n\\Omega_{M/K} \\to\n\\Omega_{M/L} \\to 0\n$$\nis exact.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Cartier's equality and geometric regularity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07E2","source_file":"more-algebra.tex","source_line":8598,"source_end_line":8611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8598-L8611","statement_sha256":"db1ffd1046871dac78a394e9522a0f950c5b1cbe921c7d39b7abdf1199fd1252","origin":"The Stacks Project","memory_eligible":false,"source_rank":3045,"rank":3045,"depth":41,"x":759.354,"y":654.742,"cluster":"advanced-algebra"},{"id":"stacks:07E3","tag":"07E3","title":"Cartier's equality and geometric regularity · Lemma 07E3","summary":"Given a commutative diagram of fields xymatrix K ar[r] & K' k ar[u] ar[r] & k' ar[u] with k'/k and K'/K finitely generated field extensions the kernel and cokernel of the maps α : Ω_K/k ⊗_K K' → Ω_K'/k' and β : H_1(L_K/k) ⊗_K K' → H_1(L_K'/k') are finite dimensional and dim Ker(α) - dim Coker(α) -dim Ker(β) + dim Coker(β) = trdeg_k(k') - trdeg_K(K')","statement_latex":"Given a commutative diagram of fields\n$$\n\\xymatrix{\nK \\ar[r] & K' \\\\\nk \\ar[u] \\ar[r] & k' \\ar[u]\n}\n$$\nwith $k'/k$ and $K'/K$ finitely generated field extensions\nthe kernel and cokernel of the maps\n$$\n\\alpha : \\Omega_{K/k} \\otimes_K K' \\to \\Omega_{K'/k'}\n\\quad\\text{and}\\quad\n\\beta : H_1(L_{K/k}) \\otimes_K K' \\to H_1(L_{K'/k'})\n$$\nare finite dimensional and\n$$\n\\dim \\Ker(\\alpha) - \\dim \\Coker(\\alpha)\n-\\dim \\Ker(\\beta) + \\dim \\Coker(\\beta)\n=\n\\text{trdeg}_k(k') - \\text{trdeg}_K(K')\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Cartier's equality and geometric regularity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07E3","source_file":"more-algebra.tex","source_line":8620,"source_end_line":8643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8620-L8643","statement_sha256":"2311bbc5c2bb1f1dc0ffffa5d33caa368e1628da74a2aa8ccd117b461620862d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3046,"rank":3046,"depth":42,"x":641.714,"y":743.792,"cluster":"advanced-algebra"},{"id":"stacks:07E5","tag":"07E5","title":"Geometric regularity · Proposition 07E5","summary":"Let k be a field of characteristic p > 0. Let (A, m, K) be a Noetherian local k-algebra. The following are equivalent • A is geometrically regular over k, • for all k ⊂ k' ⊂ k^1/p finite over k the ring A ⊗_k k' is regular, • A is regular and the canonical map H_1(L_K/k) → m/ m^2 is injective, and • A is regular and the map Ω_k/F_p ⊗_k K → Ω_A/F_p ⊗_A K is injective.","statement_latex":"Let $k$ be a field of characteristic $p > 0$.\nLet $(A, \\mathfrak m, K)$ be a Noetherian local\n$k$-algebra. The following are equivalent\n\\begin{enumerate}\n\\item $A$ is geometrically regular over $k$,\n\\item for all $k \\subset k' \\subset k^{1/p}$\nfinite over $k$ the ring $A \\otimes_k k'$ is regular,\n\\item $A$ is regular and the canonical map\n$H_1(L_{K/k}) \\to \\mathfrak m/\\mathfrak m^2$ is injective, and\n\\item $A$ is regular and the map\n$\\Omega_{k/\\mathbf{F}_p} \\otimes_k K \\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A K$\nis injective.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Geometric regularity","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07E5","source_file":"more-algebra.tex","source_line":8696,"source_end_line":8711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8696-L8711","statement_sha256":"c91de0dbed71d251daff72162f14939bbd2e54723a5a6c00d3cdb55ea3d61f54","origin":"The Stacks Project","memory_eligible":false,"source_rank":3047,"rank":3047,"depth":43,"x":656.882,"y":611.086,"cluster":"advanced-algebra"},{"id":"stacks:07E6","tag":"07E6","title":"Geometric regularity · Lemma 07E6","summary":"Let k be a field of characteristic p > 0. Let (A, m, K) be a Noetherian local k-algebra. Assume A is geometrically regular over k. Let K/F/k be a finitely generated subextension. Let φ : k[y_1, …, y_m] → A be a k-algebra map such that y_i maps to an element of F in K and such that dy_1, …, dy_m map to a basis of Ω_F/k. Set p = φ^-1( m). Then k[y_1, …, y_m]_ p → A is flat and A/ pA is regular.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $(A, \\mathfrak m, K)$\nbe a Noetherian local $k$-algebra. Assume $A$ is geometrically regular\nover $k$. Let $K/F/k$ be a finitely generated subextension.\nLet $\\varphi : k[y_1, \\ldots, y_m] \\to A$ be a $k$-algebra map\nsuch that $y_i$ maps to an element of $F$ in $K$ and such that\n$\\text{d}y_1, \\ldots, \\text{d}y_m$ map to a basis of $\\Omega_{F/k}$.\nSet $\\mathfrak p = \\varphi^{-1}(\\mathfrak m)$. Then\n$$\nk[y_1, \\ldots, y_m]_\\mathfrak p \\to A\n$$\nis flat and $A/\\mathfrak pA$ is regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Geometric regularity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07E6","source_file":"more-algebra.tex","source_line":8850,"source_end_line":8863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8850-L8863","statement_sha256":"15bb1ac211f5b1e0e95c14b702c82bdc5ef94f2de4d5df7886f77c364747ae71","origin":"The Stacks Project","memory_eligible":false,"source_rank":3048,"rank":3048,"depth":44,"x":752.623,"y":717.78,"cluster":"advanced-algebra"},{"id":"stacks:07E8","tag":"07E8","title":"Topological rings · Definition 07E8","summary":"[EGA1] Let R be a ring and let M be an R-module. • We say R is a topological ring if R is endowed with a topology such that both addition and multiplication are continuous as maps R × R → R where R × R has the product topology. In this case we say M is a topological module if M is endowed with a topology such that addition M × M → M and scalar multiplication R × M → M are continuous. • A homomorphism of topological modules is just a continuous R-module map. A homomorphism…","statement_latex":"\\begin{reference}\n\\cite[Chapter 0, Sections 7.1 and 7.2]{EGA1}\n\\end{reference}\nLet $R$ be a ring and let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item We say $R$ is a {\\it topological ring} if $R$ is endowed with a topology\nsuch that both addition and multiplication are continuous as maps\n$R \\times R \\to R$ where $R \\times R$ has the product topology.\nIn this case we say $M$ is a {\\it topological module} if $M$ is endowed\nwith a topology such that addition $M \\times M \\to M$ and\nscalar multiplication $R \\times M \\to M$ are continuous.\n\\item A {\\it homomorphism of topological modules} is just a continuous\n$R$-module map. A {\\it homomorphism of topological rings} is a\nring homomorphism which is continuous for the given topologies.\n\\item We say $M$ is {\\it linearly topologized} if $0$ has a fundamental\nsystem of neighbourhoods consisting of submodules. We say $R$ is\n{\\it linearly topologized} if $0$ has a fundamental system of neighbourhoods\nconsisting of ideals.\n\\item If $R$ is linearly topologized, we say that $I \\subset R$ is an\n{\\it ideal of definition} if $I$ is open and if every neighbourhood\nof $0$ contains $I^n$ for some $n$.\n\\item If $R$ is linearly topologized, we say that $R$ is {\\it pre-admissible}\nif $R$ has an ideal of definition.\n\\item If $R$ is linearly topologized, we say that $R$ is {\\it admissible} if\nit is pre-admissible and\ncomplete\\footnote{By our conventions this includes separated.}.\n\\item If $R$ is linearly topologized, we say that $R$ is {\\it pre-adic} if\nthere exists an ideal of definition $I$ such that $\\{I^n\\}_{n \\geq 0}$\nforms a fundamental system of neighbourhoods of $0$.\n\\item If $R$ is linearly topologized, we say that $R$ is {\\it adic} if\n$R$ is pre-adic and complete.\n\\end{enumerate}\nNote that a (pre)adic topological ring is the same thing as a (pre)admissible\ntopological ring which has an ideal of definition $I$ such that $I^n$ is\nopen for all $n \\geq 1$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Topological rings and modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07E8","source_file":"more-algebra.tex","source_line":8958,"source_end_line":8995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L8958-L8995","statement_sha256":"248ec06457b8ae85a13da2a9ca72dfd537b73c6eb36fc9d67eeba3dd57ceb8ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":3049,"rank":3049,"depth":0,"x":595.885,"y":693.379,"cluster":"advanced-algebra"},{"id":"stacks:07E9","tag":"07E9","title":"Topological rings and modules · Lemma 07E9","summary":"Let φ : R → S be a ring map. Let I ⊂ R and J ⊂ S be ideals and endow R with the I-adic topology and S with the J-adic topology. Then φ is a homomorphism of topological rings if and only if φ(I^n) ⊂ J for some n ≥ 1.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\nLet $I \\subset R$ and $J \\subset S$ be ideals\nand endow $R$ with the $I$-adic topology and $S$ with the $J$-adic\ntopology. Then $\\varphi$ is a homomorphism of topological rings\nif and only if $\\varphi(I^n) \\subset J$ for some $n \\geq 1$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07E9","source_file":"more-algebra.tex","source_line":9024,"source_end_line":9031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9024-L9031","statement_sha256":"078c8cbb504f94581281cf814740a8ce000f9684f234e1bfc2c44bb763c293af","origin":"The Stacks Project","memory_eligible":false,"source_rank":3050,"rank":3050,"depth":0,"x":731.376,"y":622.282,"cluster":"advanced-algebra"},{"id":"stacks:0CQU","tag":"0CQU","title":"Baire category theorem · Lemma 0CQU","summary":"Let M be a topological abelian group. Assume M is linearly topologized, complete, and has a countable fundamental system of neighbourhoods of 0. If U_n ⊂ M, n ≥ 1 are open dense subsets, then ⋂_n ≥ 1 U_n is dense.","statement_latex":"Let $M$ be a topological abelian group. Assume $M$ is linearly\ntopologized, complete, and has a countable fundamental system of\nneighbourhoods of $0$. If $U_n \\subset M$, $n \\geq 1$\nare open dense subsets, then $\\bigcap_{n \\geq 1} U_n$ is dense.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQU","source_file":"more-algebra.tex","source_line":9037,"source_end_line":9043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9037-L9043","statement_sha256":"f7740368747420bbb11a5da4bc371ba2699a3802e98be52971127345b75831a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3051,"rank":3051,"depth":0,"x":688.549,"y":751.866,"cluster":"advanced-algebra"},{"id":"stacks:0CQV","tag":"0CQV","title":"Topological rings and modules · Lemma 0CQV","summary":"With same assumptions as Lemma [Tag 0CQU] if M = ⋃_n ≥ 1 N_n for some closed subgroups N_n, then N_n is open for some n.","statement_latex":"With same assumptions as Lemma \\ref{lemma-baire-category-complete-module}\nif $M = \\bigcup_{n \\geq 1} N_n$ for some closed subgroups $N_n$,\nthen $N_n$ is open for some $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQV","source_file":"more-algebra.tex","source_line":9077,"source_end_line":9082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9077-L9082","statement_sha256":"2d1f8b81fccd2c5a9164f18156acda9c7dec3da578b7f8230ca1020d43906a44","origin":"The Stacks Project","memory_eligible":false,"source_rank":3052,"rank":3052,"depth":1,"x":615.766,"y":631.755,"cluster":"advanced-algebra"},{"id":"stacks:0CQW","tag":"0CQW","title":"Open mapping lemma · Lemma 0CQW","summary":"Let u : N → M be a continuous map of topologized abelian groups. Assume that M separated and that N is complete, linearly topologized, and has a countable fundamental system of neighbourhoods of 0. Then exactly one of the following holds • u is open, or • for some open subgroup N' ⊂ N the image u(N') is nowhere dense in M.","statement_latex":"Let $u : N \\to M$ be a continuous map of topologized\nabelian groups. Assume that $M$ separated and that $N$ is complete, linearly\ntopologized, and has a countable fundamental system of neighbourhoods of $0$.\nThen exactly one of the following holds\n\\begin{enumerate}\n\\item $u$ is open, or\n\\item for some open subgroup $N' \\subset N$ the image\n$u(N')$ is nowhere dense in $M$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQW","source_file":"more-algebra.tex","source_line":9089,"source_end_line":9100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9089-L9100","statement_sha256":"1ae08da578dc0f44aecb674affad16aaf449591a23ff4fa084bb333df74ab9ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":3053,"rank":3053,"depth":0,"x":766.346,"y":679.125,"cluster":"advanced-algebra"},{"id":"stacks:07EB","tag":"07EB","title":"Formally smooth maps of topological rings · Definition 07EB","summary":"Let R → S be a homomorphism of topological rings with R and S linearly topologized. We say S is formally smooth over R if for every commutative solid diagram xymatrix S ar[r] ar@-->[rd] & A/J R ar[r] ar[u] & A ar[u] of homomorphisms of topological rings where A is a discrete ring and J ⊂ A is an ideal of square zero, a dotted arrow exists which makes the diagram commute.","statement_latex":"Let $R \\to S$ be a homomorphism of topological rings with $R$ and $S$\nlinearly topologized. We say $S$ is {\\it formally smooth over $R$} if\nfor every commutative solid diagram\n$$\n\\xymatrix{\nS \\ar[r] \\ar@{-->}[rd] & A/J \\\\\nR \\ar[r] \\ar[u] & A \\ar[u]\n}\n$$\nof homomorphisms of topological rings where $A$ is a discrete ring and\n$J \\subset A$ is an ideal of square zero, a dotted arrow exists which\nmakes the diagram commute.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of topological rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EB","source_file":"more-algebra.tex","source_line":9139,"source_end_line":9153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9139-L9153","statement_sha256":"2dc09d85c89fa67535b321cdb10ceb94b288caa095b2d56a1548002954c662fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3054,"rank":3054,"depth":0,"x":616.9,"y":729.745,"cluster":"advanced-algebra"},{"id":"stacks:07EC","tag":"07EC","title":"Formally smooth maps of topological rings · Lemma 07EC","summary":"Let φ : R → S be a ring map. • If R → S is formally smooth in the sense of Algebra, Definition [Tag 00TI], then R → S is formally smooth for any linear topology on R and any pre-adic topology on S such that R → S is continuous. • Let n ⊂ S and m ⊂ R ideals such that φ is continuous for the m-adic topology on R and the n-adic topology on S. Then the following are equivalent • φ is formally smooth for the m-adic topology on R and the n-adic topology on S, and • φ is…","statement_latex":"Let $\\varphi : R \\to S$ be a ring map.\n\\begin{enumerate}\n\\item If $R \\to S$ is formally smooth in\nthe sense of Algebra, Definition \\ref{algebra-definition-formally-smooth},\nthen $R \\to S$ is formally smooth for any linear topology on $R$ and\nany pre-adic topology on $S$ such that $R \\to S$ is continuous.\n\\item Let $\\mathfrak n \\subset S$ and $\\mathfrak m \\subset R$\nideals such that $\\varphi$ is continuous for the $\\mathfrak m$-adic\ntopology on $R$ and the $\\mathfrak n$-adic topology\non $S$. Then the following are equivalent\n\\begin{enumerate}\n\\item $\\varphi$ is formally smooth for the $\\mathfrak m$-adic topology on\n$R$ and the $\\mathfrak n$-adic topology on $S$, and\n\\item $\\varphi$ is formally smooth for the discrete topology\non $R$ and the $\\mathfrak n$-adic topology on $S$.\n\\end{enumerate}\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of topological rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EC","source_file":"more-algebra.tex","source_line":9164,"source_end_line":9183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9164-L9183","statement_sha256":"99434713f730dc859e38534176a3d8d23034f10f7b901a937e9c27b7336008e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3055,"rank":3055,"depth":1,"x":686.539,"y":607.362,"cluster":"advanced-algebra"},{"id":"stacks:07NI","tag":"07NI","title":"Formally smooth maps of topological rings · Definition 07NI","summary":"Let R → S be a ring map. Let n ⊂ S be an ideal. If the equivalent conditions (2)(a) and (2)(b) of Lemma [Tag 07EC] hold, then we say R → S is formally smooth for the n-adic topology.","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak n \\subset S$ be an\nideal. If the equivalent conditions (2)(a) and (2)(b) of\nLemma \\ref{lemma-formally-smooth} hold, then we say\n$R \\to S$ is {\\it formally smooth for the $\\mathfrak n$-adic topology}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of topological rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NI","source_file":"more-algebra.tex","source_line":9212,"source_end_line":9218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9212-L9218","statement_sha256":"ff281a493a9667d8b09670e9b479223cdb49aa935c080afd65ecf01021c7e972","origin":"The Stacks Project","memory_eligible":false,"source_rank":3056,"rank":3056,"depth":2,"x":733.704,"y":737.393,"cluster":"advanced-algebra"},{"id":"stacks:07ED","tag":"07ED","title":"Formally smooth maps of topological rings · Lemma 07ED","summary":"Let (R, m) and (S, n) be rings endowed with finitely generated ideals. Endow R and S with the m-adic and n-adic topologies. Let R → S be a homomorphism of topological rings. The following are equivalent • R → S is formally smooth for the n-adic topology, • R → S^wedge is formally smooth for the n^wedge-adic topology, • R^wedge → S^wedge is formally smooth for the n^wedge-adic topology. Here R^wedge and S^wedge are the m-adic and n-adic completions of R and S.","statement_latex":"Let $(R, \\mathfrak m)$ and $(S, \\mathfrak n)$ be rings endowed\nwith finitely generated ideals. Endow $R$ and $S$ with the\n$\\mathfrak m$-adic and $\\mathfrak n$-adic topologies.\nLet $R \\to S$ be a homomorphism of topological rings.\nThe following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is formally smooth for the $\\mathfrak n$-adic topology,\n\\item $R \\to S^\\wedge$ is formally smooth for the $\\mathfrak n^\\wedge$-adic\ntopology,\n\\item $R^\\wedge \\to S^\\wedge$ is formally smooth for the\n$\\mathfrak n^\\wedge$-adic topology.\n\\end{enumerate}\nHere $R^\\wedge$ and $S^\\wedge$ are the $\\mathfrak m$-adic and\n$\\mathfrak n$-adic completions of $R$ and $S$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of topological rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ED","source_file":"more-algebra.tex","source_line":9223,"source_end_line":9239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9223-L9239","statement_sha256":"d4255e638ea92bd17fd90fde0c1d9d6fe586522c57d5b3a103163797f2c88a61","origin":"The Stacks Project","memory_eligible":false,"source_rank":3057,"rank":3057,"depth":5,"x":594.066,"y":668.127,"cluster":"advanced-algebra"},{"id":"stacks:07NJ","tag":"07NJ","title":"Formally smooth maps of topological rings · Lemma 07NJ","summary":"Let R → S be a ring map. Let n be an ideal of S. Assume that R → S is formally smooth in the n-adic topology. Consider a solid commutative diagram xymatrix S ar[r]_ψ ar@-->[rd] & A/J R ar[r] ar[u] & A ar[u] of homomorphisms of topological rings where A is adic and A/J is the quotient (as topological ring) of A by a closed ideal J ⊂ A such that J^t is contained in an ideal of definition of A for some t ≥ 1. Then there exists a dotted arrow in the category of topological…","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak n$ be an ideal of $S$.\nAssume that $R \\to S$ is formally smooth in the $\\mathfrak n$-adic\ntopology. Consider a solid commutative diagram\n$$\n\\xymatrix{\nS \\ar[r]_\\psi \\ar@{-->}[rd] & A/J \\\\\nR \\ar[r] \\ar[u] & A \\ar[u]\n}\n$$\nof homomorphisms of topological rings where $A$ is adic\nand $A/J$ is the quotient (as topological ring) of $A$ by a closed ideal\n$J \\subset A$ such that $J^t$ is contained in an ideal of definition\nof $A$ for some $t \\geq 1$. Then there exists a dotted arrow in the category of\ntopological rings which makes the diagram commute.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of topological rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NJ","source_file":"more-algebra.tex","source_line":9256,"source_end_line":9272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9256-L9272","statement_sha256":"562c656a6834f695c791a4c28b2bc4f912a70988c28e3e1cace190c9e5cd5398","origin":"The Stacks Project","memory_eligible":false,"source_rank":3058,"rank":3058,"depth":1,"x":753.066,"y":639.91,"cluster":"advanced-algebra"},{"id":"stacks:07EE","tag":"07EE","title":"Formally smooth maps of topological rings · Lemma 07EE","summary":"Let R → S be a ring map. Let n ⊂ n' ⊂ S be ideals. If R → S is formally smooth for the n-adic topology, then R → S is formally smooth for the n'-adic topology.","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak n \\subset \\mathfrak n' \\subset S$\nbe ideals. If $R \\to S$ is formally smooth for the $\\mathfrak n$-adic\ntopology, then $R \\to S$ is formally smooth for the $\\mathfrak n'$-adic\ntopology.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of topological rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EE","source_file":"more-algebra.tex","source_line":9322,"source_end_line":9328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9322-L9328","statement_sha256":"08a9e7f2459f30a9a1c3871bb4e3a7cc0a0a339e88bb786cc41b235a5158371c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3059,"rank":3059,"depth":0,"x":658.317,"y":751.168,"cluster":"advanced-algebra"},{"id":"stacks:07EF","tag":"07EF","title":"Formally smooth maps of topological rings · Lemma 07EF","summary":"A composition of formally smooth continuous homomorphisms of linearly topologized rings is formally smooth.","statement_latex":"A composition of formally smooth continuous homomorphisms of linearly\ntopologized rings is formally smooth.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of topological rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EF","source_file":"more-algebra.tex","source_line":9334,"source_end_line":9338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9334-L9338","statement_sha256":"501dd0e5ae095ee6458b5e2422d028128ddd58054e4928dfa2c0040710254729","origin":"The Stacks Project","memory_eligible":false,"source_rank":3060,"rank":3060,"depth":0,"x":638.673,"y":615.085,"cluster":"advanced-algebra"},{"id":"stacks:07EG","tag":"07EG","title":"Formally smooth maps of topological rings · Lemma 07EG","summary":"Let R, S be rings. Let n ⊂ S be an ideal. Let R → S be formally smooth for the n-adic topology. Let R → R' be any ring map. Then R' → S' = S ⊗_R R' is formally smooth in the n' = nS'-adic topology.","statement_latex":"Let $R$, $S$ be rings. Let $\\mathfrak n \\subset S$ be an ideal.\nLet $R \\to S$ be formally smooth for the $\\mathfrak n$-adic topology.\nLet $R \\to R'$ be any ring map. Then $R' \\to S' = S \\otimes_R R'$\nis formally smooth in the $\\mathfrak n' = \\mathfrak nS'$-adic\ntopology.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of topological rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EG","source_file":"more-algebra.tex","source_line":9345,"source_end_line":9352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9345-L9352","statement_sha256":"ab1a4d12d00c06be3a0684a557b04e0be5fe45884ffedd10ba0ebedf8a55c8ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":3061,"rank":3061,"depth":1,"x":762.847,"y":704.468,"cluster":"advanced-algebra"},{"id":"stacks:07EH","tag":"07EH","title":"Formally smooth maps of topological rings · Lemma 07EH","summary":"Let R, S be rings. Let n ⊂ S be an ideal. Let R → R' be a ring map. Set S' = S ⊗_R R' and n' = nS. If • the map R → R' embeds R as a direct summand of R' as an R-module, and • R' → S' is formally smooth for the n'-adic topology, then R → S is formally smooth in the n-adic topology.","statement_latex":"Let $R$, $S$ be rings. Let $\\mathfrak n \\subset S$ be an ideal.\nLet $R \\to R'$ be a ring map. Set $S' = S \\otimes_R R'$ and\n$\\mathfrak n' = \\mathfrak nS$. If\n\\begin{enumerate}\n\\item the map $R \\to R'$ embeds $R$ as a direct summand of $R'$\nas an $R$-module, and\n\\item $R' \\to S'$ is formally smooth for the $\\mathfrak n'$-adic topology,\n\\end{enumerate}\nthen $R \\to S$ is formally smooth in the $\\mathfrak n$-adic topology.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of topological rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EH","source_file":"more-algebra.tex","source_line":9375,"source_end_line":9386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9375-L9386","statement_sha256":"ecd18db9ea384d503e12da7a18eb3ce822a3a929c3c2e309b414b3307a3f2202","origin":"The Stacks Project","memory_eligible":false,"source_rank":3062,"rank":3062,"depth":1,"x":599.068,"y":709.027,"cluster":"advanced-algebra"},{"id":"stacks:0DYG","tag":"0DYG","title":"Formally smooth maps of local rings · Lemma 0DYG","summary":"Let (R, m) → (S, n) be a local homomorphism of local rings. The following are equivalent • R → S is formally smooth in the n-adic topology, • for every solid commutative diagram xymatrix S ar[r] ar@-->[rd] & A/J R ar[r] ar[u] & A ar[u] of local homomorphisms of local rings where J ⊂ A is an ideal of square zero, m_A^n = 0 for some n > 0, and S → A/J induces an isomorphism on residue fields, a dotted arrow exists which makes the diagram commute. If S is Noetherian these…","statement_latex":"Let $(R, \\mathfrak m) \\to (S, \\mathfrak n)$ be a local homomorphism\nof local rings. The following are equivalent\n\\begin{enumerate}\n\\item $R \\to S$ is formally smooth in the $\\mathfrak n$-adic topology,\n\\item for every solid commutative diagram\n$$\n\\xymatrix{\nS \\ar[r] \\ar@{-->}[rd] & A/J \\\\\nR \\ar[r] \\ar[u] & A \\ar[u]\n}\n$$\nof local homomorphisms of local rings where $J \\subset A$ is\nan ideal of square zero, $\\mathfrak m_A^n = 0$ for some $n > 0$, and\n$S \\to A/J$ induces an isomorphism on residue fields, a dotted\narrow exists which makes the diagram commute.\n\\end{enumerate}\nIf $S$ is Noetherian these conditions are also equivalent to\n\\begin{enumerate}\n\\item[(3)] same as in (2) but only for diagrams where in addition\n$A \\to A/J$ is a small extension\n(Algebra, Definition \\ref{algebra-definition-small-extension}).\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYG","source_file":"more-algebra.tex","source_line":9435,"source_end_line":9459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9435-L9459","statement_sha256":"9ce2df05db4cfaf6bde233a647c79a73e53ac430f1497c6a7d8effaa7dc76295","origin":"The Stacks Project","memory_eligible":false,"source_rank":3063,"rank":3063,"depth":1,"x":716.41,"y":612.536,"cluster":"advanced-algebra"},{"id":"stacks:07EI","tag":"07EI","title":"Formally smooth maps of local rings · Lemma 07EI","summary":"Let k be a field and let (A, m, K) be a Noetherian local k-algebra. If k → A is formally smooth for the m-adic topology, then A is a regular local ring.","statement_latex":"Let $k$ be a field and let $(A, \\mathfrak m, K)$ be a Noetherian\nlocal $k$-algebra. If $k \\to A$ is formally smooth for the\n$\\mathfrak m$-adic topology, then $A$ is a regular local ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EI","source_file":"more-algebra.tex","source_line":9534,"source_end_line":9539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9534-L9539","statement_sha256":"e3e949200bf6f14f999eea2134c57621f1979fec860aa1a5c547df83be9371ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":3064,"rank":3064,"depth":18,"x":707.46,"y":750.549,"cluster":"advanced-algebra"},{"id":"stacks:0C34","tag":"0C34","title":"Formally smooth maps of local rings · Lemma 0C34","summary":"Let k be a field. Let (A, m, kappa) be a complete local k-algebra. If kappa/k is separable, then there exists a k-algebra map kappa → A such that kappa → A → kappa is id_kappa.","statement_latex":"Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a complete\nlocal $k$-algebra. If $\\kappa/k$ is separable, then there exists\na $k$-algebra map $\\kappa \\to A$ such that $\\kappa \\to A \\to \\kappa$\nis $\\text{id}_\\kappa$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C34","source_file":"more-algebra.tex","source_line":9589,"source_end_line":9595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9589-L9595","statement_sha256":"d9d172dcdca8bb2ec7f00c38b84e8efb3db403bab18c7bac46b7a6310d0d951b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3065,"rank":3065,"depth":12,"x":602.862,"y":643.486,"cluster":"advanced-algebra"},{"id":"stacks:0C35","tag":"0C35","title":"Formally smooth maps of local rings · Lemma 0C35","summary":"Let k be a field. Let (A, m, kappa) be a complete local k-algebra. If kappa/k is separable and A regular, then there exists an isomorphism of A ≅ kappa[[t_1, …, t_d]] as k-algebras.","statement_latex":"Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a complete\nlocal $k$-algebra. If $\\kappa/k$ is separable and $A$ regular, then\nthere exists an isomorphism of $A \\cong \\kappa[[t_1, \\ldots, t_d]]$\nas $k$-algebras.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C35","source_file":"more-algebra.tex","source_line":9607,"source_end_line":9613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9607-L9613","statement_sha256":"e3097b07ae9fb91b65233292750375dfeb18e634039a3f754f07df9cc3bc1ba9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3066,"rank":3066,"depth":19,"x":766.418,"y":663.121,"cluster":"advanced-algebra"},{"id":"stacks:07EJ","tag":"07EJ","title":"Formally smooth maps of local rings · Lemma 07EJ","summary":"Let k be a field. Let (A, m, K) be a regular local k-algebra such that K/k is separable. Then k → A is formally smooth in the m-adic topology.","statement_latex":"Let $k$ be a field. Let $(A, \\mathfrak m, K)$ be a regular local\n$k$-algebra such that $K/k$ is separable. Then $k \\to A$\nis formally smooth in the $\\mathfrak m$-adic topology.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EJ","source_file":"more-algebra.tex","source_line":9625,"source_end_line":9630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9625-L9630","statement_sha256":"3fb1258b7e1183baa778561d9ba7046fc80d8a0993cacf5ec84a6819c9927382","origin":"The Stacks Project","memory_eligible":false,"source_rank":3067,"rank":3067,"depth":20,"x":629.75,"y":741.605,"cluster":"advanced-algebra"},{"id":"stacks:07VH","tag":"07VH","title":"Formally smooth maps of local rings · Lemma 07VH","summary":"Let A → B be a finite type ring map with A Noetherian. Let q ⊂ B be a prime ideal lying over p ⊂ A. The following are equivalent • A → B is smooth at q, and • A_ p → B_ q is formally smooth in the q-adic topology.","statement_latex":"Let $A \\to B$ be a finite type ring map with $A$ Noetherian.\nLet $\\mathfrak q \\subset B$ be a prime ideal lying over\n$\\mathfrak p \\subset A$. The following are equivalent\n\\begin{enumerate}\n\\item $A \\to B$ is smooth at $\\mathfrak q$, and\n\\item $A_\\mathfrak p \\to B_\\mathfrak q$ is formally smooth in\nthe $\\mathfrak q$-adic topology.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally smooth maps of local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VH","source_file":"more-algebra.tex","source_line":9654,"source_end_line":9664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9654-L9664","statement_sha256":"967bfd0fb5909f73cfea43163c9d1ad46fc9b56e3285bf756294c148f478f724","origin":"The Stacks Project","memory_eligible":false,"source_rank":3068,"rank":3068,"depth":7,"x":667.487,"y":605.913,"cluster":"advanced-algebra"},{"id":"stacks:07NL","tag":"07NL","title":"Some results on power series rings · Lemma 07NL","summary":"Let K be a field of characteristic 0 and A = K[[x_1, …, x_n]]. Let L be a field of characteristic p > 0 and B = L[[x_1, …, x_n]]. Let Lambda be a Cohen ring. Let C = Lambda[[x_1, …, x_n]]. • Q → A is formally smooth in the m_A-adic topology. • F_p → B is formally smooth in the m_B-adic topology. • Z → C is formally smooth in the m_C-adic topology.","statement_latex":"Let $K$ be a field of characteristic $0$ and $A = K[[x_1, \\ldots, x_n]]$.\nLet $L$ be a field of characteristic $p > 0$ and $B = L[[x_1, \\ldots, x_n]]$.\nLet $\\Lambda$ be a Cohen ring. Let $C = \\Lambda[[x_1, \\ldots, x_n]]$.\n\\begin{enumerate}\n\\item $\\mathbf{Q} \\to A$ is formally smooth in the \n$\\mathfrak m_A$-adic topology.\n\\item $\\mathbf{F}_p \\to B$ is formally smooth in the \n$\\mathfrak m_B$-adic topology.\n\\item $\\mathbf{Z} \\to C$ is formally smooth in the\n$\\mathfrak m_C$-adic topology.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Some results on power series rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NL","source_file":"more-algebra.tex","source_line":9703,"source_end_line":9716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9703-L9716","statement_sha256":"8308260bd5eeadbf5e817ae5c525f4db1d0b2b58c48d3d31edb2f901899463af","origin":"The Stacks Project","memory_eligible":false,"source_rank":3069,"rank":3069,"depth":13,"x":748.942,"y":727.625,"cluster":"advanced-algebra"},{"id":"stacks:07NM","tag":"07NM","title":"Some results on power series rings · Lemma 07NM","summary":"Let K be a field and A = K[[x_1, …, x_n]]. Let Lambda be a Cohen ring and let B = Lambda[[x_1, …, x_n]]. • If y_1, …, y_n ∈ A is a regular system of parameters then K[[y_1, …, y_n]] → A is an isomorphism. • If z_1, …, z_r ∈ A form part of a regular system of parameters for A, then r ≤ n and A/(z_1, …, z_r) ≅ K[[y_1, …, y_n - r]]. • If p, y_1, …, y_n ∈ B is a regular system of parameters then Lambda[[y_1, …, y_n]] → B is an isomorphism. • If p, z_1, …, z_r ∈ B form part of…","statement_latex":"Let $K$ be a field and $A = K[[x_1, \\ldots, x_n]]$.\nLet $\\Lambda$ be a Cohen ring and let $B = \\Lambda[[x_1, \\ldots, x_n]]$.\n\\begin{enumerate}\n\\item If $y_1, \\ldots, y_n \\in A$ is a regular system of parameters\nthen $K[[y_1, \\ldots, y_n]] \\to A$ is an isomorphism.\n\\item If $z_1, \\ldots, z_r \\in A$ form part of a regular system of\nparameters for $A$, then $r \\leq n$ and\n$A/(z_1, \\ldots, z_r) \\cong K[[y_1, \\ldots, y_{n - r}]]$.\n\\item If $p, y_1, \\ldots, y_n \\in B$ is a regular system of parameters\nthen $\\Lambda[[y_1, \\ldots, y_n]] \\to B$ is an isomorphism.\n\\item If $p, z_1, \\ldots, z_r \\in B$ form part of a regular system of\nparameters for $B$, then $r \\leq n$ and\n$B/(z_1, \\ldots, z_r) \\cong \\Lambda[[y_1, \\ldots, y_{n - r}]]$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Some results on power series rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NM","source_file":"more-algebra.tex","source_line":9736,"source_end_line":9752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9736-L9752","statement_sha256":"8898a319ccac748f1c063d341cb9749c04aa424f45bfaf3acf7b360c64d370a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3070,"rank":3070,"depth":0,"x":590.687,"y":684.009,"cluster":"advanced-algebra"},{"id":"stacks:07NN","tag":"07NN","title":"Some results on power series rings · Lemma 07NN","summary":"Let A → B be a local homomorphism of Noetherian complete local rings. Then there exists a commutative diagram xymatrix S ar[r] & B R ar[u] ar[r] & A ar[u] with the following properties: • the horizontal arrows are surjective, • if the characteristic of A/ m_A is zero, then S and R are power series rings over fields, • if the characteristic of A/ m_A is p > 0, then S and R are power series rings over Cohen rings, and • R → S maps a regular system of parameters of R to part…","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian complete local rings.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\nS \\ar[r] & B \\\\\nR \\ar[u] \\ar[r] & A \\ar[u]\n}\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item the horizontal arrows are surjective,\n\\item if the characteristic of $A/\\mathfrak m_A$ is zero, then $S$ and $R$\nare power series rings over fields,\n\\item if the characteristic of $A/\\mathfrak m_A$ is $p > 0$, then $S$ and $R$\nare power series rings over Cohen rings, and\n\\item $R \\to S$ maps a regular system of parameters of $R$ to part of a\nregular system of parameters of $S$.\n\\end{enumerate}\nIn particular $R \\to S$ is flat (see Algebra,\nLemma \\ref{algebra-lemma-flat-over-regular}) with regular fibre\n$S/\\mathfrak m_R S$ (see Algebra, Lemma \\ref{algebra-lemma-regular-ring-CM}).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Some results on power series rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NN","source_file":"more-algebra.tex","source_line":9768,"source_end_line":9791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9768-L9791","statement_sha256":"d246c80146bac401bd69dc14967aba32e9a61df61e9daa56d1ce611705c74ff9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3071,"rank":3071,"depth":18,"x":742.757,"y":626.26,"cluster":"advanced-algebra"},{"id":"stacks:09Q8","tag":"09Q8","title":"Some results on power series rings · Lemma 09Q8","summary":"Let S → R and S' → R be surjective maps of complete Noetherian local rings. Then S ×_R S' is a complete Noetherian local ring.","statement_latex":"Let $S \\to R$ and $S' \\to R$ be surjective maps of complete Noetherian local\nrings. Then $S \\times_R S'$ is a complete Noetherian local ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Some results on power series rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Q8","source_file":"more-algebra.tex","source_line":9821,"source_end_line":9825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9821-L9825","statement_sha256":"841cdd1258998d07a991af2a06d9eecd5ffa63c65f7cbcf680bd5dd16f5f3047","origin":"The Stacks Project","memory_eligible":false,"source_rank":3072,"rank":3072,"depth":18,"x":676.934,"y":755.385,"cluster":"advanced-algebra"},{"id":"stacks:07EL","tag":"07EL","title":"Geometric regularity and formal smoothness · Theorem 07EL","summary":"Let k be a field. Let (A, m, K) be a Noetherian local k-algebra. If the characteristic of k is zero then the following are equivalent • A is a regular local ring, and • k → A is formally smooth in the m-adic topology. If the characteristic of k is p > 0 then the following are equivalent • A is geometrically regular over k, • k → A is formally smooth in the m-adic topology. • for all k ⊂ k' ⊂ k^1/p finite over k the ring A ⊗_k k' is regular, • A is regular and the…","statement_latex":"Let $k$ be a field. Let $(A, \\mathfrak m, K)$ be a Noetherian local\n$k$-algebra. If the characteristic of $k$ is zero then the following\nare equivalent\n\\begin{enumerate}\n\\item $A$ is a regular local ring, and\n\\item $k \\to A$ is formally smooth in the $\\mathfrak m$-adic topology.\n\\end{enumerate}\nIf the characteristic of $k$ is $p > 0$ then the following are equivalent\n\\begin{enumerate}\n\\item $A$ is geometrically regular over $k$,\n\\item $k \\to A$ is formally smooth in the $\\mathfrak m$-adic topology.\n\\item for all $k \\subset k' \\subset k^{1/p}$\nfinite over $k$ the ring $A \\otimes_k k'$ is regular,\n\\item $A$ is regular and the canonical map\n$H_1(L_{K/k}) \\to \\mathfrak m/\\mathfrak m^2$ is injective, and\n\\item $A$ is regular and the map\n$\\Omega_{k/\\mathbf{F}_p} \\otimes_k K \\to \\Omega_{A/\\mathbf{F}_p} \\otimes_A K$\nis injective.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Geometric regularity and formal smoothness","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EL","source_file":"more-algebra.tex","source_line":9871,"source_end_line":9892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9871-L9892","statement_sha256":"ae7fb88640f1a5a1ddc2684be9ebd64acd73a9f40edd0b7b4e8433f9e9d2064d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3073,"rank":3073,"depth":44,"x":621.524,"y":622.56,"cluster":"advanced-algebra"},{"id":"stacks:07NP","tag":"07NP","title":"Geometric regularity and formal smoothness · Lemma 07NP","summary":"Let A → B be a local homomorphism of Noetherian local rings. Assume A → B is formally smooth in the m_B-adic topology. Then A → B is flat.","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian local rings.\nAssume $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic\ntopology. Then $A \\to B$ is flat.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Geometric regularity and formal smoothness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NP","source_file":"more-algebra.tex","source_line":9980,"source_end_line":9985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L9980-L9985","statement_sha256":"0f035c1215f19867e6771f6314d110d5f0fa4d3d056b286a7b5b6462adc7c35d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3074,"rank":3074,"depth":19,"x":769.485,"y":689.192,"cluster":"advanced-algebra"},{"id":"stacks:0DYH","tag":"0DYH","title":"Geometric regularity and formal smoothness · Lemma 0DYH","summary":"Let A → B be a local homomorphism of Noetherian local rings. Assume A → B is formally smooth in the m_B-adic topology. Let K be the residue field of B. Then the Jacobi-Zariski sequence for A → B → K gives an exact sequence 0 → H_1(NL_K/A) → m_B/ m_B^2 → Ω_B/A ⊗_B K → Ω_K/A → 0","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian local rings.\nAssume $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic\ntopology. Let $K$ be the residue field of $B$. Then\nthe Jacobi-Zariski sequence for $A \\to B \\to K$ gives an exact sequence\n$$\n0 \\to H_1(\\NL_{K/A}) \\to \\mathfrak m_B/\\mathfrak m_B^2\n\\to \\Omega_{B/A} \\otimes_B K \\to \\Omega_{K/A} \\to 0\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Geometric regularity and formal smoothness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYH","source_file":"more-algebra.tex","source_line":10016,"source_end_line":10026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10016-L10026","statement_sha256":"2f2acf48d2b8c87e47b9d9f19bc051a33b64f3d912f27414a53979babaaa35b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3075,"rank":3075,"depth":45,"x":606.481,"y":724.083,"cluster":"advanced-algebra"},{"id":"stacks:07NQ","tag":"07NQ","title":"Geometric regularity and formal smoothness · Proposition 07NQ","summary":"Let A → B be a local homomorphism of Noetherian local rings. Let k be the residue field of A and overlineB = B ⊗_A k the special fibre. The following are equivalent • A → B is flat and overlineB is geometrically regular over k, • A → B is flat and k → overlineB is formally smooth in the m_overlineB-adic topology, and • A → B is formally smooth in the m_B-adic topology.","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian local rings.\nLet $k$ be the residue field of $A$ and $\\overline{B} = B \\otimes_A k$\nthe special fibre. The following are equivalent\n\\begin{enumerate}\n\\item $A \\to B$ is flat and $\\overline{B}$ is geometrically regular\nover $k$,\n\\item $A \\to B$ is flat and $k \\to \\overline{B}$ is formally smooth\nin the $\\mathfrak m_{\\overline{B}}$-adic topology, and\n\\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic\ntopology.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Geometric regularity and formal smoothness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NQ","source_file":"more-algebra.tex","source_line":10080,"source_end_line":10093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10080-L10093","statement_sha256":"9aa1a59dcf76c61c072973aeb5aa1b1dc4b9db20f0598bb6a75bf0c27fc5e513","origin":"The Stacks Project","memory_eligible":false,"source_rank":3076,"rank":3076,"depth":45,"x":698.798,"y":605.634,"cluster":"advanced-algebra"},{"id":"stacks:07NR","tag":"07NR","title":"Geometric regularity and formal smoothness · Lemma 07NR","summary":"Let A be a Noetherian complete local ring with residue field k. Let B be a Noetherian complete local k-algebra. Assume k → B is formally smooth in the m_B-adic topology. Then there exists a Noetherian complete local ring C and a local homomorphism A → C which is formally smooth in the m_C-adic topology such that C ⊗_A k ≅ B.","statement_latex":"Let $A$ be a Noetherian complete local ring with residue field $k$.\nLet $B$ be a Noetherian complete local $k$-algebra. Assume $k \\to B$\nis formally smooth in the $\\mathfrak m_B$-adic topology.\nThen there exists a Noetherian complete local ring $C$\nand a local homomorphism $A \\to C$ which is formally smooth\nin the $\\mathfrak m_C$-adic topology such that $C \\otimes_A k \\cong B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Geometric regularity and formal smoothness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NR","source_file":"more-algebra.tex","source_line":10255,"source_end_line":10263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10255-L10263","statement_sha256":"4730bd5fc37ccb9e77b246001f4289738e1f3c0bc02f147b2758d3f59cdb9b3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3077,"rank":3077,"depth":46,"x":726.031,"y":745.629,"cluster":"advanced-algebra"},{"id":"stacks:07BZ","tag":"07BZ","title":"Regular ring maps · Definition 07BZ","summary":"A ring map R → Lambda is regular if it is flat and for every prime p ⊂ R the fibre ring Lambda ⊗_R kappa( p) = Lambda_ p/ pLambda_ p is Noetherian and geometrically regular over kappa( p).","statement_latex":"A ring map $R \\to \\Lambda$ is {\\it regular} if it is flat and\nfor every prime $\\mathfrak p \\subset R$ the fibre ring\n$$\n\\Lambda \\otimes_R \\kappa(\\mathfrak p) =\n\\Lambda_\\mathfrak p/\\mathfrak p\\Lambda_\\mathfrak p\n$$\nis Noetherian and geometrically regular over $\\kappa(\\mathfrak p)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BZ","source_file":"more-algebra.tex","source_line":10374,"source_end_line":10383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10374-L10383","statement_sha256":"f40ae56972fe2a767d84b236d79a3ca916deeae15592a95afff0e36fdb2c7124","origin":"The Stacks Project","memory_eligible":false,"source_rank":3078,"rank":3078,"depth":0,"x":593.114,"y":657.682,"cluster":"advanced-algebra"},{"id":"stacks:07C0","tag":"07C0","title":"Regular is a local property · Lemma 07C0","summary":"Let R → Lambda be a ring map with Lambda Noetherian. The following are equivalent • R → Lambda is regular, • R_ p → Lambda_ q is regular for all q ⊂ Lambda lying over p ⊂ R, and • R_ m → Lambda_ m' is regular for all maximal ideals m' ⊂ Lambda lying over m in R.","statement_latex":"Let $R \\to \\Lambda$ be a ring map with $\\Lambda$ Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $R \\to \\Lambda$ is regular,\n\\item $R_\\mathfrak p \\to \\Lambda_\\mathfrak q$ is regular for all\n$\\mathfrak q \\subset \\Lambda$ lying over $\\mathfrak p \\subset R$, and\n\\item $R_\\mathfrak m \\to \\Lambda_{\\mathfrak m'}$ is regular for\nall maximal ideals $\\mathfrak m' \\subset \\Lambda$\nlying over $\\mathfrak m$ in $R$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07C0","source_file":"more-algebra.tex","source_line":10389,"source_end_line":10401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10389-L10401","statement_sha256":"bd326c2e54edbef5be0b59858fe041f13031406e56307c3b363193a5db269851","origin":"The Stacks Project","memory_eligible":false,"source_rank":3079,"rank":3079,"depth":3,"x":762.172,"y":647.093,"cluster":"advanced-algebra"},{"id":"stacks:07C1","tag":"07C1","title":"Regular maps and base change · Lemma 07C1","summary":"Let R → Lambda be a regular ring map. For any finite type ring map R → R' the base change R' → Lambda ⊗_R R' is regular too.","statement_latex":"Let $R \\to \\Lambda$ be a regular ring map.\nFor any finite type ring map $R \\to R'$ the base change\n$R' \\to \\Lambda \\otimes_R R'$ is regular too.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07C1","source_file":"more-algebra.tex","source_line":10412,"source_end_line":10417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10412-L10417","statement_sha256":"7babc66d797a34f77d8705dd13c6f00b156b5bdf6301b0894c35d89b6df57711","origin":"The Stacks Project","memory_eligible":false,"source_rank":3080,"rank":3080,"depth":41,"x":645.805,"y":751.027,"cluster":"advanced-algebra"},{"id":"stacks:07QI","tag":"07QI","title":"Composition of regular maps · Lemma 07QI","summary":"Let A → B and B → C be regular ring maps. If the fibre rings of A → C are Noetherian, then A → C is regular.","statement_latex":"Let $A \\to B$ and $B \\to C$ be regular ring maps.\nIf the fibre rings of $A \\to C$ are Noetherian, then\n$A \\to C$ is regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QI","source_file":"more-algebra.tex","source_line":10439,"source_end_line":10444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10439-L10444","statement_sha256":"2ce96c4c6484901fd885d271a93cada5f6106857e3b5087c93050f767266762c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3081,"rank":3081,"depth":42,"x":648.037,"y":608.087,"cluster":"advanced-algebra"},{"id":"stacks:07EP","tag":"07EP","title":"Regular ring maps · Lemma 07EP","summary":"Let R be a ring. Let (A_i, φ_ii') be a directed system of smooth R-algebras. Set Lambda = colim A_i. If the fibre rings Lambda ⊗_R kappa( p) are Noetherian for all p ⊂ R, then R → Lambda is regular.","statement_latex":"Let $R$ be a ring. Let $(A_i, \\varphi_{ii'})$ be a directed system\nof smooth $R$-algebras. Set $\\Lambda = \\colim A_i$. If the fibre\nrings $\\Lambda \\otimes_R \\kappa(\\mathfrak p)$ are Noetherian for all\n$\\mathfrak p \\subset R$, then $R \\to \\Lambda$ is regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EP","source_file":"more-algebra.tex","source_line":10457,"source_end_line":10463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10457-L10463","statement_sha256":"2a4055c6482caa40a694f6c8a31869ebb33d193b383cb2ed049ec2e0acad6bd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3082,"rank":3082,"depth":37,"x":761.553,"y":714.956,"cluster":"advanced-algebra"},{"id":"stacks:07EQ","tag":"07EQ","title":"Regular ring maps · Lemma 07EQ","summary":"Let K/k be a field extension. Then k → K is a regular ring map if and only if K is a separable field extension of k.","statement_latex":"Let $K/k$ be a field extension. Then $k \\to K$ is a regular\nring map if and only if $K$ is a separable field extension of $k$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EQ","source_file":"more-algebra.tex","source_line":10488,"source_end_line":10492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10488-L10492","statement_sha256":"c86b46988e4215a9f78fd5495b3085df76363be26e031f098eb3c12a1294aed1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3083,"rank":3083,"depth":41,"x":591.587,"y":700.538,"cluster":"advanced-algebra"},{"id":"stacks:07NT","tag":"07NT","title":"Regular ring maps · Lemma 07NT","summary":"Let A → B → C be ring maps. If A → C is regular and B → C is flat and surjective on spectra, then A → B is regular.","statement_latex":"Let $A \\to B \\to C$ be ring maps. If $A \\to C$ is regular and $B \\to C$\nis flat and surjective on spectra, then $A \\to B$ is regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NT","source_file":"more-algebra.tex","source_line":10506,"source_end_line":10510,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10506-L10510","statement_sha256":"16abca23755529b3ca968ab602ad7fad5d66b65100797134b87de32cb1985bb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3084,"rank":3084,"depth":20,"x":728.77,"y":614.566,"cluster":"advanced-algebra"},{"id":"stacks:07QK","tag":"07QK","title":"Ascending properties along regular ring maps · Lemma 07QK","summary":"Let φ : R → S be a ring map. Assume • φ is regular, • S is Noetherian, and • R is Noetherian and reduced. Then S is reduced.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is regular,\n\\item $S$ is Noetherian, and\n\\item $R$ is Noetherian and reduced.\n\\end{enumerate}\nThen $S$ is reduced.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ascending properties along regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QK","source_file":"more-algebra.tex","source_line":10532,"source_end_line":10541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10532-L10541","statement_sha256":"36e21ca195e7d91fe35b1309576aff07ac528f46444876294a63ef814f149dda","origin":"The Stacks Project","memory_eligible":false,"source_rank":3085,"rank":3085,"depth":17,"x":696.689,"y":756.065,"cluster":"advanced-algebra"},{"id":"stacks:0BFK","tag":"0BFK","title":"Ascending properties along regular ring maps · Lemma 0BFK","summary":"Let φ : R → S be a ring map. Assume • φ is regular, • S is Noetherian, and • R is Noetherian and normal. Then S is normal.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is regular,\n\\item $S$ is Noetherian, and\n\\item $R$ is Noetherian and normal.\n\\end{enumerate}\nThen $S$ is normal.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ascending properties along regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFK","source_file":"more-algebra.tex","source_line":10552,"source_end_line":10561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10552-L10561","statement_sha256":"d4c4be7710667fc4bfbb9a597a48e5ff9f7fa88f777aee0d8e6e2030ae9a37e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3086,"rank":3086,"depth":17,"x":606.387,"y":633.294,"cluster":"advanced-algebra"},{"id":"stacks:0H7S","tag":"0H7S","title":"Ascending properties along regular ring maps · Lemma 0H7S","summary":"Let φ : R → S be a ring map. Assume • φ is regular, • S is Noetherian, and • R is Noetherian and regular. Then S is regular.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is regular,\n\\item $S$ is Noetherian, and\n\\item $R$ is Noetherian and regular.\n\\end{enumerate}\nThen $S$ is regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ascending properties along regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7S","source_file":"more-algebra.tex","source_line":10572,"source_end_line":10581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10572-L10581","statement_sha256":"206a81a72bbe0eadb37cc2daa871b216233f8afce05d96475b1005772acbeaa2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3087,"rank":3087,"depth":14,"x":772.011,"y":672.657,"cluster":"advanced-algebra"},{"id":"stacks:0H7T","tag":"0H7T","title":"Ascending properties along regular ring maps · Lemma 0H7T","summary":"Let φ : R → S be a ring map. Assume • φ is regular, • S is Noetherian, and • R is Noetherian and Cohen-Macaulay. Then S is Cohen-Macaulay.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is regular,\n\\item $S$ is Noetherian, and\n\\item $R$ is Noetherian and Cohen-Macaulay.\n\\end{enumerate}\nThen $S$ is Cohen-Macaulay.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Ascending properties along regular ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7T","source_file":"more-algebra.tex","source_line":10589,"source_end_line":10598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10589-L10598","statement_sha256":"ea2aff913fb5574ba088437719ddd26fe3e501a48acfffc7cba77fcd0bdd6b1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3088,"rank":3088,"depth":17,"x":617.943,"y":737.73,"cluster":"advanced-algebra"},{"id":"stacks:07NV","tag":"07NV","title":"Permanence of properties under completion · Lemma 07NV","summary":"Let A be a Noetherian local ring. Then dim(A) = dim(A^wedge).","statement_latex":"Let $A$ be a Noetherian local ring.\nThen $\\dim(A) = \\dim(A^\\wedge)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NV","source_file":"more-algebra.tex","source_line":10623,"source_end_line":10627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10623-L10627","statement_sha256":"16e3c108614e5124c17ce060406b5c88be99171f8c24649c524d0fff886b6177","origin":"The Stacks Project","memory_eligible":false,"source_rank":3089,"rank":3089,"depth":12,"x":679.334,"y":602.075,"cluster":"advanced-algebra"},{"id":"stacks:07NW","tag":"07NW","title":"Permanence of properties under completion · Lemma 07NW","summary":"Let A be a Noetherian local ring. Then depth(A) = depth(A^wedge).","statement_latex":"Let $A$ be a Noetherian local ring. Then\n$\\text{depth}(A) = \\text{depth}(A^\\wedge)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NW","source_file":"more-algebra.tex","source_line":10644,"source_end_line":10648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10644-L10648","statement_sha256":"410edec6e66ec1d156c987be9b76e6696b67445142950d7fb7bc785a2b678cde","origin":"The Stacks Project","memory_eligible":false,"source_rank":3090,"rank":3090,"depth":16,"x":743.272,"y":737.187,"cluster":"advanced-algebra"},{"id":"stacks:07NX","tag":"07NX","title":"Permanence of properties under completion · Lemma 07NX","summary":"Let A be a Noetherian local ring. Then A is Cohen-Macaulay if and only if A^wedge is so.","statement_latex":"Let $A$ be a Noetherian local ring.\nThen $A$ is Cohen-Macaulay if and only if $A^\\wedge$ is so.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NX","source_file":"more-algebra.tex","source_line":10654,"source_end_line":10658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10654-L10658","statement_sha256":"de26599ec89383b69179f54efc0958cae790c38c05b1e95db3cd197b6116903a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3091,"rank":3091,"depth":17,"x":587.187,"y":673.722,"cluster":"advanced-algebra"},{"id":"stacks:07NY","tag":"07NY","title":"Permanence of properties under completion · Lemma 07NY","summary":"Let A be a Noetherian local ring. Then A is regular if and only if A^wedge is so.","statement_latex":"Let $A$ be a Noetherian local ring.\nThen $A$ is regular if and only if $A^\\wedge$ is so.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NY","source_file":"more-algebra.tex","source_line":10667,"source_end_line":10671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10667-L10671","statement_sha256":"8ff2f9fd4f0890e129604952793c681e5d5a57862227d2649af4a230e75aabf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3092,"rank":3092,"depth":18,"x":753.622,"y":631.878,"cluster":"advanced-algebra"},{"id":"stacks:0AP1","tag":"0AP1","title":"Permanence of properties under completion · Lemma 0AP1","summary":"Let A be a Noetherian local ring. Then A is a discrete valuation ring if and only if A^wedge is so.","statement_latex":"Let $A$ be a Noetherian local ring.\nThen $A$ is a discrete valuation ring if and only if $A^\\wedge$ is so.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AP1","source_file":"more-algebra.tex","source_line":10686,"source_end_line":10690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10686-L10690","statement_sha256":"8143ca8f817db7b644bef518c56be59d3cc68344b31420d2856e965aad42f65e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3093,"rank":3093,"depth":19,"x":664.383,"y":757.399,"cluster":"advanced-algebra"},{"id":"stacks:07NZ","tag":"07NZ","title":"Permanence of properties under completion · Lemma 07NZ","summary":"Let A be a Noetherian local ring. • If A^wedge is reduced, then so is A. • In general A reduced does not imply A^wedge is reduced. • If A is Nagata, then A is reduced if and only if A^wedge is reduced.","statement_latex":"Let $A$ be a Noetherian local ring.\n\\begin{enumerate}\n\\item If $A^\\wedge$ is reduced, then so is $A$.\n\\item In general $A$ reduced does not imply $A^\\wedge$ is reduced.\n\\item If $A$ is Nagata, then $A$ is reduced if and only if $A^\\wedge$\nis reduced.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07NZ","source_file":"more-algebra.tex","source_line":10698,"source_end_line":10707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10698-L10707","statement_sha256":"4ced9000de339e0a589126a1cf3e3aaf6fc86b2b75845016408f2700ebb2e2d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3094,"rank":3094,"depth":32,"x":629.182,"y":613.946,"cluster":"advanced-algebra"},{"id":"stacks:0FIZ","tag":"0FIZ","title":"Permanence of properties under completion · Lemma 0FIZ","summary":"Let A be a Noetherian local ring. If A^wedge is normal, then so is A.","statement_latex":"Let $A$ be a Noetherian local ring. If $A^\\wedge$ is normal, then so is $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIZ","source_file":"more-algebra.tex","source_line":10720,"source_end_line":10723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10720-L10723","statement_sha256":"10eb296afd4fb9ba14c2a3a31c03be827f4797aa2d24f2aa66274a4e5d179323","origin":"The Stacks Project","memory_eligible":false,"source_rank":3095,"rank":3095,"depth":5,"x":770.754,"y":699.907,"cluster":"advanced-algebra"},{"id":"stacks:0C4G","tag":"0C4G","title":"Permanence of properties under completion · Lemma 0C4G","summary":"Let A → B be a local homomorphism of Noetherian local rings. Then the induced map of completions A^wedge → B^wedge is flat if and only if A → B is flat.","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian local rings.\nThen the induced map of completions $A^\\wedge \\to B^\\wedge$\nis flat if and only if $A \\to B$ is flat.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4G","source_file":"more-algebra.tex","source_line":10730,"source_end_line":10735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10730-L10735","statement_sha256":"f4adb49b0de301bad8faa432aa0f84e4cf9c30e0f4da626bad67567a16f7a139","origin":"The Stacks Project","memory_eligible":false,"source_rank":3096,"rank":3096,"depth":10,"x":596.922,"y":716.883,"cluster":"advanced-algebra"},{"id":"stacks:0AGX","tag":"0AGX","title":"Permanence of properties under completion · Lemma 0AGX","summary":"Let A → B be a flat local homomorphism of Noetherian local rings such that m_A B = m_B and kappa( m_A) = kappa( m_B). Then A → B induces an isomorphism A^wedge → B^wedge of completions.","statement_latex":"Let $A \\to B$ be a flat local homomorphism of Noetherian local rings\nsuch that $\\mathfrak m_A B = \\mathfrak m_B$ and\n$\\kappa(\\mathfrak m_A) = \\kappa(\\mathfrak m_B)$.\nThen $A \\to B$ induces an isomorphism $A^\\wedge \\to B^\\wedge$\nof completions.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGX","source_file":"more-algebra.tex","source_line":10763,"source_end_line":10770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10763-L10770","statement_sha256":"7c30ecc6c04432f122e78f037b0d2f7be1bee9690c6677db820a9b763d7b4925","origin":"The Stacks Project","memory_eligible":false,"source_rank":3097,"rank":3097,"depth":9,"x":711.657,"y":605.53,"cluster":"advanced-algebra"},{"id":"stacks:0AGZ","tag":"0AGZ","title":"Permanence of properties under étale maps · Lemma 0AGZ","summary":"If A → B is an étale ring map and q is a prime of B lying over p ⊂ A, then A_ p is Noetherian if and only if B_ q is Noetherian.","statement_latex":"If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of\n$B$ lying over $\\mathfrak p \\subset A$, then\n$A_{\\mathfrak p}$ is Noetherian if and only if $B_{\\mathfrak q}$ is\nNoetherian.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGZ","source_file":"more-algebra.tex","source_line":10812,"source_end_line":10818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10812-L10818","statement_sha256":"c74b3d5108e17f76f16813a918e6813408cc9c2f37ca1789c706511ae1337042","origin":"The Stacks Project","memory_eligible":false,"source_rank":3098,"rank":3098,"depth":2,"x":716.608,"y":753.005,"cluster":"advanced-algebra"},{"id":"stacks:07QP","tag":"07QP","title":"Permanence of properties under étale maps · Lemma 07QP","summary":"If A → B is an étale ring map and q is a prime of B lying over p ⊂ A, then dim(A_ p) = dim(B_ q).","statement_latex":"If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of\n$B$ lying over $\\mathfrak p \\subset A$, then\n$\\dim(A_{\\mathfrak p}) = \\dim(B_{\\mathfrak q})$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QP","source_file":"more-algebra.tex","source_line":10830,"source_end_line":10835,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10830-L10835","statement_sha256":"ff4210ac984322303148a3308cca83fbefdbe7faac3abf0ae8816090ce503dbf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3099,"rank":3099,"depth":13,"x":594.145,"y":646.881,"cluster":"advanced-algebra"},{"id":"stacks:0AH0","tag":"0AH0","title":"Permanence of properties under étale maps · Lemma 0AH0","summary":"If A → B is an étale ring map and q is a prime of B lying over p ⊂ A, then A_ p is regular if and only if B_ q is regular.","statement_latex":"If $A \\to B$ is an \\'etale ring map and $\\mathfrak q$ is a prime of\n$B$ lying over $\\mathfrak p \\subset A$, then\n$A_{\\mathfrak p}$ is regular if and only if $B_{\\mathfrak q}$ is regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AH0","source_file":"more-algebra.tex","source_line":10857,"source_end_line":10862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10857-L10862","statement_sha256":"8c58f7bf2b9b568446b45af1b54ba94ffc710de7bb3dd9363d5091fbdb89e5a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3100,"rank":3100,"depth":41,"x":770.101,"y":655.662,"cluster":"advanced-algebra"},{"id":"stacks:0AP2","tag":"0AP2","title":"Permanence of properties under étale maps · Lemma 0AP2","summary":"If A → B is an étale ring map and A is a Dedekind domain, then B is a finite product of Dedekind domains. In particular, the localizations B_ q for q ⊂ B maximal are discrete valuation rings.","statement_latex":"If $A \\to B$ is an \\'etale ring map and $A$ is a Dedekind domain, then\n$B$ is a finite product of Dedekind domains. In particular, the\nlocalizations $B_\\mathfrak q$ for $\\mathfrak q \\subset B$ maximal\nare discrete valuation rings.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under étale maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AP2","source_file":"more-algebra.tex","source_line":10880,"source_end_line":10886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10880-L10886","statement_sha256":"94129ec2ab12dc48945ac27f3f0ed4bfc8e5850f0cd56bb0b77fefcec4320e75","origin":"The Stacks Project","memory_eligible":false,"source_rank":3101,"rank":3101,"depth":42,"x":633.05,"y":749.193,"cluster":"advanced-algebra"},{"id":"stacks:07QM","tag":"07QM","title":"Permanence of properties under henselization · Lemma 07QM","summary":"Let (R, m, kappa) be a local ring. Then we have the following • R → R^h → R^sh are faithfully flat ring maps, • m R^h = m^h and m R^sh = m^h R^sh = m^sh, • R/ m^n = R^h/ m^nR^h for all n, • there exist elements x_i ∈ R^sh such that R^sh/ m^nR^sh is a free R/ m^n-module on x_i bmod m^nR^sh.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Then we have the following\n\\begin{enumerate}\n\\item $R \\to R^h \\to R^{sh}$ are faithfully flat ring maps,\n\\item $\\mathfrak m R^h = \\mathfrak m^h$ and\n$\\mathfrak m R^{sh} = \\mathfrak m^h R^{sh} = \\mathfrak m^{sh}$,\n\\item $R/\\mathfrak m^n = R^h/\\mathfrak m^nR^h$ for all $n$,\n\\item there exist elements $x_i \\in R^{sh}$ such that\n$R^{sh}/\\mathfrak m^nR^{sh}$ is a free $R/\\mathfrak m^n$-module\non $x_i \\bmod \\mathfrak m^nR^{sh}$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QM","source_file":"more-algebra.tex","source_line":10912,"source_end_line":10924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10912-L10924","statement_sha256":"d0420debe79f0ef63a6b9a0eb5da22f3b7664bee6a4faa912edb092f65fea3e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3102,"rank":3102,"depth":44,"x":658.94,"y":602.202,"cluster":"advanced-algebra"},{"id":"stacks:07QN","tag":"07QN","title":"Permanence of properties under henselization · Lemma 07QN","summary":"Let (R, m, kappa) be a local ring. Then • R → R^h, R^h → R^sh, and R → R^sh are formally étale, • R → R^h, R^h → R^sh, resp. R → R^sh are formally smooth in the m^h, m^sh, resp. m^sh-topology.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Then\n\\begin{enumerate}\n\\item $R \\to R^h$, $R^h \\to R^{sh}$, and $R \\to R^{sh}$ are formally \\'etale,\n\\item $R \\to R^h$, $R^h \\to R^{sh}$, resp.\\ $R \\to R^{sh}$ are formally\nsmooth in the $\\mathfrak m^h$, $\\mathfrak m^{sh}$,\nresp.\\ $\\mathfrak m^{sh}$-topology.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QN","source_file":"more-algebra.tex","source_line":10956,"source_end_line":10965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10956-L10965","statement_sha256":"e2939af0f9764cb78ca58c6d81d4ba91138e8e6dbfe8f0ecb07a258110ce4ef2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3103,"rank":3103,"depth":7,"x":758.229,"y":725.497,"cluster":"advanced-algebra"},{"id":"stacks:06LJ","tag":"06LJ","title":"Permanence of properties under henselization · Lemma 06LJ","summary":"[EGA] Let R be a local ring. The following are equivalent • R is Noetherian, • R^h is Noetherian, and • R^sh is Noetherian. In this case we have • [(a)] (R^h)^wedge and (R^sh)^wedge are Noetherian complete local rings, • [(b)] R^wedge → (R^h)^wedge is an isomorphism, • [(c)] R^h → (R^h)^wedge and R^sh → (R^sh)^wedge are flat, • [(d)] R^wedge → (R^sh)^wedge is formally smooth in the m_(R^sh)^wedge-adic topology, • [(e)] (R^wedge)^sh = R^wedge ⊗_R^h R^sh, and • [(f)]…","statement_latex":"\\begin{reference}\n\\cite[IV, Theorem 18.6.6 and Proposition 18.8.8]{EGA}\n\\end{reference}\nLet $R$ be a local ring. The following are equivalent\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $R^h$ is Noetherian, and\n\\item $R^{sh}$ is Noetherian.\n\\end{enumerate}\nIn this case we have\n\\begin{enumerate}\n\\item[(a)] $(R^h)^\\wedge$ and $(R^{sh})^\\wedge$ are Noetherian complete\nlocal rings,\n\\item[(b)] $R^\\wedge \\to (R^h)^\\wedge$ is an isomorphism,\n\\item[(c)] $R^h \\to (R^h)^\\wedge$ and $R^{sh} \\to (R^{sh})^\\wedge$ are flat,\n\\item[(d)] $R^\\wedge \\to (R^{sh})^\\wedge$ is formally smooth in\nthe $\\mathfrak m_{(R^{sh})^\\wedge}$-adic topology,\n\\item[(e)] $(R^\\wedge)^{sh} = R^\\wedge \\otimes_{R^h} R^{sh}$, and\n\\item[(f)] $((R^\\wedge)^{sh})^\\wedge = (R^{sh})^\\wedge$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LJ","source_file":"more-algebra.tex","source_line":10978,"source_end_line":11000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L10978-L11000","statement_sha256":"27feb9631fc6c75e2d9529568c6a5dd9d7d25d3728c4ee79f7a5cc3d5f7ec9ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":3104,"rank":3104,"depth":47,"x":585.564,"y":690.859,"cluster":"advanced-algebra"},{"id":"stacks:06DH","tag":"06DH","title":"Permanence of properties under henselization · Lemma 06DH","summary":"Reducedness passes to the (strict) henselization. Let R be a local ring. The following are equivalent: R is reduced, the henselization R^h of R is reduced, and the strict henselization R^sh of R is reduced.","statement_latex":"\\begin{slogan}\nReducedness passes to the (strict) henselization.\n\\end{slogan}\nLet $R$ be a local ring.\nThe following are equivalent: $R$ is reduced,\nthe henselization $R^h$ of $R$ is reduced, and\nthe strict henselization $R^{sh}$ of $R$ is reduced.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DH","source_file":"more-algebra.tex","source_line":11066,"source_end_line":11075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11066-L11075","statement_sha256":"0dbc6525bf401d4f54e5f6fb4de8c0f0110b3ff343e0dd5f5ca6bf33af2d133a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3105,"rank":3105,"depth":19,"x":741.009,"y":618.3,"cluster":"advanced-algebra"},{"id":"stacks:0ASE","tag":"0ASE","title":"Permanence of properties under henselization · Lemma 0ASE","summary":"Let R be a local ring. Let nil(R) denote the ideal of nilpotent elements of R. Then nil(R)R^h = nil(R^h) and nil(R)R^sh = nil(R^sh).","statement_latex":"Let $R$ be a local ring. Let $nil(R)$ denote the ideal of\nnilpotent elements of $R$. Then $nil(R)R^h = nil(R^h)$ and\n$nil(R)R^{sh} = nil(R^{sh})$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASE","source_file":"more-algebra.tex","source_line":11087,"source_end_line":11092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11087-L11092","statement_sha256":"faea77a4dcaa869f41363191460a830122405ef4f1836db5b8e9df40904b0b25","origin":"The Stacks Project","memory_eligible":false,"source_rank":3106,"rank":3106,"depth":51,"x":684.638,"y":760.253,"cluster":"advanced-algebra"},{"id":"stacks:06DI","tag":"06DI","title":"Permanence of properties under henselization · Lemma 06DI","summary":"Let R be a local ring. The following are equivalent: R is a normal domain, the henselization R^h of R is a normal domain, and the strict henselization R^sh of R is a normal domain.","statement_latex":"Let $R$ be a local ring.\nThe following are equivalent: $R$ is a normal domain,\nthe henselization $R^h$ of $R$ is a normal domain, and\nthe strict henselization $R^{sh}$ of $R$ is a normal domain.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DI","source_file":"more-algebra.tex","source_line":11109,"source_end_line":11115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11109-L11115","statement_sha256":"dc694adc1f1fb727977724c77f0b9ece464aae182543093091d6b9ec575380a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3107,"rank":3107,"depth":33,"x":611.926,"y":623.357,"cluster":"advanced-algebra"},{"id":"stacks:06LK","tag":"06LK","title":"Permanence of properties under henselization · Lemma 06LK","summary":"Given any local ring R we have dim(R) = dim(R^h) = dim(R^sh).","statement_latex":"Given any local ring $R$ we have $\\dim(R) = \\dim(R^h) = \\dim(R^{sh})$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LK","source_file":"more-algebra.tex","source_line":11131,"source_end_line":11134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11131-L11134","statement_sha256":"0d06436c62159482d2c4fa90e3859ae77a9b6f15879f6569cb9273f785c80072","origin":"The Stacks Project","memory_eligible":false,"source_rank":3108,"rank":3108,"depth":45,"x":775.912,"y":683.146,"cluster":"advanced-algebra"},{"id":"stacks:06LL","tag":"06LL","title":"Permanence of properties under henselization · Lemma 06LL","summary":"Given a Noetherian local ring R we have depth(R) = depth(R^h) = depth(R^sh).","statement_latex":"Given a Noetherian local ring $R$ we have\n$\\text{depth}(R) = \\text{depth}(R^h) = \\text{depth}(R^{sh})$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LL","source_file":"more-algebra.tex","source_line":11159,"source_end_line":11163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11159-L11163","statement_sha256":"6d11b9cfd867fd9d4821a0b48d67d657f67970c71fefb20109be291609c50d2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3109,"rank":3109,"depth":48,"x":606.621,"y":732.192,"cluster":"advanced-algebra"},{"id":"stacks:06LM","tag":"06LM","title":"Permanence of properties under henselization · Lemma 06LM","summary":"Let R be a Noetherian local ring. The following are equivalent: R is Cohen-Macaulay, the henselization R^h of R is Cohen-Macaulay, and the strict henselization R^sh of R is Cohen-Macaulay.","statement_latex":"Let $R$ be a Noetherian local ring. The following are equivalent:\n$R$ is Cohen-Macaulay, the henselization $R^h$ of $R$ is Cohen-Macaulay,\nand the strict henselization $R^{sh}$ of $R$ is Cohen-Macaulay.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LM","source_file":"more-algebra.tex","source_line":11173,"source_end_line":11178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11173-L11178","statement_sha256":"4423fa2d4afb04e6b55d887d07850a94679f41da18f68ae648c15e5cd2a39238","origin":"The Stacks Project","memory_eligible":false,"source_rank":3110,"rank":3110,"depth":49,"x":692.158,"y":599.741,"cluster":"advanced-algebra"},{"id":"stacks:06LN","tag":"06LN","title":"Permanence of properties under henselization · Lemma 06LN","summary":"Let R be a Noetherian local ring. The following are equivalent: R is a regular local ring, the henselization R^h of R is a regular local ring, and the strict henselization R^sh of R is a regular local ring.","statement_latex":"Let $R$ be a Noetherian local ring. The following are equivalent:\n$R$ is a regular local ring, the henselization $R^h$ of $R$ is a regular\nlocal ring, and the strict henselization $R^{sh}$ of $R$ is a regular\nlocal ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LN","source_file":"more-algebra.tex","source_line":11194,"source_end_line":11200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11194-L11200","statement_sha256":"e744d15eaeadf82261c3743cd14a8694c46f6c7c010a680f42e6db624936fc5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3111,"rank":3111,"depth":48,"x":735.672,"y":746.193,"cluster":"advanced-algebra"},{"id":"stacks:0AP3","tag":"0AP3","title":"Permanence of properties under henselization · Lemma 0AP3","summary":"Let R be a Noetherian local ring. Then R is a discrete valuation ring if and only if R^h is a discrete valuation ring if and only if R^sh is a discrete valuation ring.","statement_latex":"Let $R$ be a Noetherian local ring. Then $R$ is a discrete valuation ring\nif and only if $R^h$ is a discrete valuation ring if and only if\n$R^{sh}$ is a discrete valuation ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AP3","source_file":"more-algebra.tex","source_line":11222,"source_end_line":11227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11222-L11227","statement_sha256":"a32b5bd51f9303e6d5405df1c54d65efc224e5e325dcbbd51bfe3758ce413864","origin":"The Stacks Project","memory_eligible":false,"source_rank":3112,"rank":3112,"depth":49,"x":585.559,"y":662.751,"cluster":"advanced-algebra"},{"id":"stacks:0AH1","tag":"0AH1","title":"Permanence of properties under henselization · Lemma 0AH1","summary":"Let A be a ring. Let B be a filtered colimit of étale A-algebras. Let p be a prime of A. If B is Noetherian, then there are finitely many primes q_1, …, q_r lying over p, we have B ⊗_A kappa( p) = ∏ kappa( q_i), and each of the field extensions kappa( q_i)/kappa( p) is separable algebraic.","statement_latex":"Let $A$ be a ring. Let $B$ be a filtered colimit of \\'etale $A$-algebras.\nLet $\\mathfrak p$ be a prime of $A$. If $B$ is Noetherian, then\nthere are finitely many primes $\\mathfrak q_1, \\ldots, \\mathfrak q_r$\nlying over $\\mathfrak p$, we have\n$B \\otimes_A \\kappa(\\mathfrak p) = \\prod \\kappa(\\mathfrak q_i)$, and\neach of the field extensions\n$\\kappa(\\mathfrak q_i)/\\kappa(\\mathfrak p)$ is separable algebraic.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AH1","source_file":"more-algebra.tex","source_line":11235,"source_end_line":11244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11235-L11244","statement_sha256":"00871c0de0b0b98457ae6cf7b281c23367a8cd3eedf5cc063f2f272df10680e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3113,"rank":3113,"depth":40,"x":763.651,"y":639.062,"cluster":"advanced-algebra"},{"id":"stacks:07QQ","tag":"07QQ","title":"Permanence of properties under henselization · Lemma 07QQ","summary":"Let R be a Noetherian local ring. Let p ⊂ R be a prime. Then R^h ⊗_R kappa( p) = ∏_i = 1, …, t kappa( q_i) resp. R^sh ⊗_R kappa( p) = ∏_i = 1, …, s kappa( r_i) where q_1, …, q_t, resp. r_1, …, r_s are the prime of R^h, resp. R^sh lying over p. Moreover, the field extensions kappa( q_i)/kappa( p) resp. kappa( r_i)/kappa( p) are separable algebraic.","statement_latex":"Let $R$ be a Noetherian local ring. Let $\\mathfrak p \\subset R$ be a prime.\nThen\n$$\nR^h \\otimes_R \\kappa(\\mathfrak p) =\n\\prod\\nolimits_{i = 1, \\ldots, t} \\kappa(\\mathfrak q_i)\n\\quad\\text{resp.}\\quad\nR^{sh} \\otimes_R \\kappa(\\mathfrak p) =\n\\prod\\nolimits_{i = 1, \\ldots, s} \\kappa(\\mathfrak r_i)\n$$\nwhere $\\mathfrak q_1, \\ldots, \\mathfrak q_t$,\nresp.\\ $\\mathfrak r_1, \\ldots, \\mathfrak r_s$\nare the prime of $R^h$, resp.\\ $R^{sh}$ lying over $\\mathfrak p$.\nMoreover, the field extensions\n$\\kappa(\\mathfrak q_i)/\\kappa(\\mathfrak p)$\nresp.\\ $\\kappa(\\mathfrak r_i)/\\kappa(\\mathfrak p)$\nare separable algebraic.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Permanence of properties under henselization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QQ","source_file":"more-algebra.tex","source_line":11269,"source_end_line":11287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11269-L11287","statement_sha256":"1474f3ab61478603f736379eceb847a8824d08f58a59e72ef05902425a96641e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3114,"rank":3114,"depth":51,"x":651.189,"y":757.783,"cluster":"advanced-algebra"},{"id":"stacks:07P1","tag":"07P1","title":"Field extensions, revisited · Definition 07P1","summary":"Let p be a prime number. Let k → K be an extension of fields of characteristic p. Denote kK^p the compositum of k and K^p in K. • A subset (x_i) ⊂ K is called p-independent over k if the elements x^E = ∏ x_i^e_i where 0 ≤ e_i < p are linearly independent over kK^p. • A subset (x_i) of K is called a p-basis of K over k if the elements x^E form a basis of K over kK^p.","statement_latex":"Let $p$ be a prime number. Let $k \\to K$ be an extension of fields\nof characteristic $p$. Denote $kK^p$ the compositum of $k$ and $K^p$\nin $K$.\n\\begin{enumerate}\n\\item A subset $\\{x_i\\} \\subset K$ is called {\\it p-independent\nover $k$} if the elements $x^E = \\prod x_i^{e_i}$ where\n$0 \\leq e_i < p$ are linearly independent over $kK^p$.\n\\item A subset $\\{x_i\\}$ of $K$ is called a\n{\\it p-basis of $K$ over $k$} if the elements\n$x^E$ form a basis of $K$ over $kK^p$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Field extensions, revisited","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07P1","source_file":"more-algebra.tex","source_line":11343,"source_end_line":11356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11343-L11356","statement_sha256":"5d41b60e4a2305e9f06c4bbfc73d94850e56d5fb963e882344640577338e5e50","origin":"The Stacks Project","memory_eligible":false,"source_rank":3115,"rank":3115,"depth":0,"x":638.624,"y":606.173,"cluster":"advanced-algebra"},{"id":"stacks:07P2","tag":"07P2","title":"Field extensions, revisited · Lemma 07P2","summary":"Let K/k be a field extension. Assume k has characteristic p > 0. Let (x_i) be a subset of K. The following are equivalent • the elements (x_i) are p-independent over k, and • the elements dx_i are K-linearly independent in Ω_K/k. Any p-independent collection can be extended to a p-basis of K over k. In particular, the field K has a p-basis over k. Moreover, the following are equivalent: • [(a)] (x_i) is a p-basis of K over k, and • [(b)] dx_i is a basis of the K-vector…","statement_latex":"Let $K/k$ be a field extension. Assume $k$ has characteristic\n$p > 0$. Let $\\{x_i\\}$ be a subset of $K$. The following are equivalent\n\\begin{enumerate}\n\\item the elements $\\{x_i\\}$ are $p$-independent over $k$, and\n\\item the elements $\\text{d}x_i$ are $K$-linearly independent\nin $\\Omega_{K/k}$.\n\\end{enumerate}\nAny $p$-independent collection can be extended to a $p$-basis of $K$ over $k$.\nIn particular, the field $K$ has a $p$-basis over $k$.\nMoreover, the following are equivalent:\n\\begin{enumerate}\n\\item[(a)] $\\{x_i\\}$ is a $p$-basis of $K$ over $k$, and\n\\item[(b)] $\\text{d}x_i$ is a basis of the $K$-vector space $\\Omega_{K/k}$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Field extensions, revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07P2","source_file":"more-algebra.tex","source_line":11362,"source_end_line":11378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11362-L11378","statement_sha256":"2f53f0f84285da731886fc4022e21abad738b4f0fca6a3353bab21e70ca22ead","origin":"The Stacks Project","memory_eligible":false,"source_rank":3116,"rank":3116,"depth":0,"x":770.03,"y":711.013,"cluster":"advanced-algebra"},{"id":"stacks:07P3","tag":"07P3","title":"Field extensions, revisited · Lemma 07P3","summary":"Let K/k be a field extension. Let (K_α)_α ∈ A be a collection of subfields of K with the following properties • k ⊂ K_α for all α ∈ A, • k = ⋂_α ∈ A K_α, • for α, α' ∈ A there exists an α\" ∈ A such that K_α\" ⊂ K_α ∩ K_α'. Then for n ≥ 1 and V ⊂ K^⊕ n a K-vector space we have V ∩ k^⊕ n not = 0 if and only if V ∩ K_α^⊕ n not = 0 for all α ∈ A.","statement_latex":"Let $K/k$ be a field extension. Let $\\{K_\\alpha\\}_{\\alpha \\in A}$\nbe a collection of subfields of $K$ with the following properties\n\\begin{enumerate}\n\\item $k \\subset K_\\alpha$ for all $\\alpha \\in A$,\n\\item $k = \\bigcap_{\\alpha \\in A} K_\\alpha$,\n\\item for $\\alpha, \\alpha' \\in A$ there exists an $\\alpha'' \\in A$\nsuch that $K_{\\alpha''} \\subset K_\\alpha \\cap K_{\\alpha'}$.\n\\end{enumerate}\nThen for $n \\geq 1$ and $V \\subset K^{\\oplus n}$ a $K$-vector space\nwe have $V \\cap k^{\\oplus n} \\not = 0$ if and only if\n$V \\cap K_\\alpha^{\\oplus n} \\not = 0$ for all $\\alpha \\in A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Field extensions, revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07P3","source_file":"more-algebra.tex","source_line":11417,"source_end_line":11430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11417-L11430","statement_sha256":"880f098581cd0018fe1f6c942568ed5df6ed1e0b1b20a6a17c53c36bc8d82c4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3117,"rank":3117,"depth":0,"x":588.521,"y":708.263,"cluster":"advanced-algebra"},{"id":"stacks:07P4","tag":"07P4","title":"Field extensions, revisited · Lemma 07P4","summary":"Let K be a field of characteristic p. Let (K_α)_α ∈ A be a collection of subfields of K with the following properties • K^p ⊂ K_α for all α ∈ A, • K^p = ⋂_α ∈ A K_α, • for α, α' ∈ A there exists an α\" ∈ A such that K_α\" ⊂ K_α ∩ K_α'. Then • the intersection of the kernels of the maps Ω_K/F_p → Ω_K/K_α is zero, • for any finite extension L/K we have L^p = ⋂_α ∈ A L^pK_α.","statement_latex":"Let $K$ be a field of characteristic $p$. Let $\\{K_\\alpha\\}_{\\alpha \\in A}$\nbe a collection of subfields of $K$ with the following properties\n\\begin{enumerate}\n\\item $K^p \\subset K_\\alpha$ for all $\\alpha \\in A$,\n\\item $K^p = \\bigcap_{\\alpha \\in A} K_\\alpha$,\n\\item for $\\alpha, \\alpha' \\in A$ there exists an $\\alpha'' \\in A$\nsuch that $K_{\\alpha''} \\subset K_\\alpha \\cap K_{\\alpha'}$.\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item the intersection of the kernels of the maps\n$\\Omega_{K/\\mathbf{F}_p} \\to \\Omega_{K/K_\\alpha}$ is zero,\n\\item for any finite extension $L/K$ we have\n$L^p = \\bigcap_{\\alpha \\in A} L^pK_\\alpha$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Field extensions, revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07P4","source_file":"more-algebra.tex","source_line":11457,"source_end_line":11474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11457-L11474","statement_sha256":"890fc6852e793f2ca0630ead77ac72bc467b8db293610bf399af35783f12de9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3118,"rank":3118,"depth":1,"x":724.8,"y":607.131,"cluster":"advanced-algebra"},{"id":"stacks:07P5","tag":"07P5","title":"Field extensions, revisited · Lemma 07P5","summary":"Let k be a field of characteristic p > 0. Let (x_i)_i ∈ I be a p-basis for k. Let n, m ≥ 0. Let K be the fraction field of A = k[[x_1, …, x_n]][y_1, …, y_m]. Let J be a finite subset of I. Consider the subfield k/k_J/k^p generated by k^p and x_i with i ∈ I setminus J. The fraction fields K_J of A_J = k_J[[x_1^p, …, x_n^p]][y_1^p, …, y_m^p] form a family of subfields of K as in Lemma [Tag 07P4]. Moreover, each of the ring extensions A_J ⊂ A is finite.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $\\{x_i\\}_{i \\in I}$\nbe a $p$-basis for $k$. Let $n, m \\geq 0$.\nLet $K$ be the fraction field of $A = k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_m]$.\nLet $J$ be a finite subset of $I$. Consider the subfield\n$k/k_J/k^p$ generated by $k^p$ and $x_i$ with $i \\in I \\setminus J$.\nThe fraction fields $K_J$ of\n$$\nA_J = k_J[[x_1^p, \\ldots, x_n^p]][y_1^p, \\ldots, y_m^p]\n$$\nform a family of subfields of $K$ as in\nLemma \\ref{lemma-intersection-subfields}. Moreover, each of the ring extensions\n$A_J \\subset A$ is finite.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Field extensions, revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07P5","source_file":"more-algebra.tex","source_line":11520,"source_end_line":11534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11520-L11534","statement_sha256":"c0c7c01ea534c8085282e823c5153955ff7bae9be74a0cd19c96a879b40eecfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":3119,"rank":3119,"depth":9,"x":705.606,"y":759.284,"cluster":"advanced-algebra"},{"id":"stacks:07P7","tag":"07P7","title":"The singular locus · Definition 07P7","summary":"[MatCA] Let R be a Noetherian ring. Let X = Spec(R). • We say R is J-0 if Reg(X) contains a nonempty open. • We say R is J-1 if Reg(X) is open. • We say R is J-2 if any finite type R-algebra is J-1.","statement_latex":"\\begin{reference}\n\\cite[(32.B)]{MatCA}\n\\end{reference}\nLet $R$ be a Noetherian ring. Let $X = \\Spec(R)$.\n\\begin{enumerate}\n\\item We say $R$ is {\\it J-0} if $\\text{Reg}(X)$ contains a nonempty open.\n\\item We say $R$ is {\\it J-1} if $\\text{Reg}(X)$ is open.\n\\item We say $R$ is {\\it J-2} if any finite type $R$-algebra is J-1.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The singular locus","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07P7","source_file":"more-algebra.tex","source_line":11582,"source_end_line":11593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11582-L11593","statement_sha256":"efb5510c7b7ae2b770d6543370928d6376b8432d17f4e078fa91b78d7ab3780e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3120,"rank":3120,"depth":0,"x":597.224,"y":635.994,"cluster":"advanced-algebra"},{"id":"stacks:07P8","tag":"07P8","title":"The singular locus · Lemma 07P8","summary":"Let R be a Noetherian ring. Let X = Spec(R). The ring R is J-1 if and only if V( p) ∩ Reg(X) contains a nonempty open subset of V( p) for all p ∈ Reg(X).","statement_latex":"Let $R$ be a Noetherian ring. Let $X = \\Spec(R)$.\nThe ring $R$ is J-1 if and only if $V(\\mathfrak p) \\cap \\text{Reg}(X)$\ncontains a nonempty open subset of $V(\\mathfrak p)$ for all\n$\\mathfrak p \\in \\text{Reg}(X)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The singular locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07P8","source_file":"more-algebra.tex","source_line":11601,"source_end_line":11607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11601-L11607","statement_sha256":"683b54c8efdc4e7751cddb41fed592f53b70dbb58e76c3eab3e263778984dd88","origin":"The Stacks Project","memory_eligible":false,"source_rank":3121,"rank":3121,"depth":18,"x":776.584,"y":665.456,"cluster":"advanced-algebra"},{"id":"stacks:07P9","tag":"07P9","title":"The singular locus · Lemma 07P9","summary":"Let R be a Noetherian ring. Let X = Spec(R). Assume that for all primes p ⊂ R the ring R/ p is J-0. Then R is J-1.","statement_latex":"Let $R$ be a Noetherian ring. Let $X = \\Spec(R)$. Assume that for all primes\n$\\mathfrak p \\subset R$ the ring $R/\\mathfrak p$ is J-0.\nThen $R$ is J-1.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The singular locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07P9","source_file":"more-algebra.tex","source_line":11616,"source_end_line":11621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11616-L11621","statement_sha256":"a60545f3f14d8fdb27cff914c776542a84213a2812907207adc01928d3454852","origin":"The Stacks Project","memory_eligible":false,"source_rank":3122,"rank":3122,"depth":19,"x":620.379,"y":745.635,"cluster":"advanced-algebra"},{"id":"stacks:07PA","tag":"07PA","title":"The singular locus · Lemma 07PA","summary":"Let R → S be a ring map. Assume that • R is a Noetherian domain, • R → S is injective and of finite type, and • S is a domain and J-0. Then R is J-0.","statement_latex":"Let $R \\to S$ be a ring map. Assume that\n\\begin{enumerate}\n\\item $R$ is a Noetherian domain,\n\\item $R \\to S$ is injective and of finite type, and\n\\item $S$ is a domain and J-0.\n\\end{enumerate}\nThen $R$ is J-0.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The singular locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PA","source_file":"more-algebra.tex","source_line":11646,"source_end_line":11655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11646-L11655","statement_sha256":"837ca402074688c021f095ae4062cbbf6b3adf36e3e8d11beef6af515b3d1ebb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3123,"rank":3123,"depth":19,"x":671.167,"y":597.637,"cluster":"advanced-algebra"},{"id":"stacks:07PB","tag":"07PB","title":"The singular locus · Lemma 07PB","summary":"Let R → S be a ring map. Assume that • R is a Noetherian domain and J-0, • R → S is injective and of finite type, and • S is a domain, and • the induced extension of fraction fields is separable. Then S is J-0.","statement_latex":"Let $R \\to S$ be a ring map. Assume that\n\\begin{enumerate}\n\\item $R$ is a Noetherian domain and J-0,\n\\item $R \\to S$ is injective and of finite type, and\n\\item $S$ is a domain, and\n\\item the induced extension of fraction fields is separable.\n\\end{enumerate}\nThen $S$ is J-0.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The singular locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PB","source_file":"more-algebra.tex","source_line":11666,"source_end_line":11676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11666-L11676","statement_sha256":"fd63cfdfe2e583f0a898ceafe3d475ca7ed15b2387e60b39ddcadcaa79a28c54","origin":"The Stacks Project","memory_eligible":false,"source_rank":3124,"rank":3124,"depth":39,"x":752.865,"y":735.812,"cluster":"advanced-algebra"},{"id":"stacks:07PC","tag":"07PC","title":"The singular locus · Lemma 07PC","summary":"Let R be a Noetherian ring. The following are equivalent • R is J-2, • every finite type R-algebra which is a domain is J-0, • every finite R-algebra is J-1, • for every prime p and every finite purely inseparable extension L/kappa( p) there exists a finite R-algebra R' which is a domain, which is J-0, and whose field of fractions is L.","statement_latex":"Let $R$ be a Noetherian ring. The following are equivalent\n\\begin{enumerate}\n\\item $R$ is J-2,\n\\item every finite type $R$-algebra which is a domain is J-0,\n\\item every finite $R$-algebra is J-1,\n\\item for every prime $\\mathfrak p$ and every finite purely inseparable\nextension $L/\\kappa(\\mathfrak p)$ there exists a finite\n$R$-algebra $R'$ which is a domain, which is J-0, and whose field\nof fractions is $L$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The singular locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PC","source_file":"more-algebra.tex","source_line":11688,"source_end_line":11700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11688-L11700","statement_sha256":"ae8bf9d8cdbd94299c7b8c82472fa0b23a08a22ebff14970d148527a7b1989ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":3125,"rank":3125,"depth":40,"x":581.228,"y":680.191,"cluster":"advanced-algebra"},{"id":"stacks:07PE","tag":"07PE","title":"Regularity and derivations · Lemma 07PE","summary":"Let R be a ring. Let D : R → R be a derivation. • For any ideal I ⊂ R the derivation D extends canonically to a derivation D^wedge : R^wedge → R^wedge on the I-adic completion. • For any multiplicative subset S ⊂ R the derivation D extends uniquely to the localization S^-1R of R. If R ⊂ R' is a finite type extension of rings such that R_g ≅ R'_g for some g ∈ R which is a nonzerodivisor in R', then g^ND extends to R' for some N ≥ 0.","statement_latex":"Let $R$ be a ring. Let $D : R \\to R$ be a derivation.\n\\begin{enumerate}\n\\item For any ideal $I \\subset R$ the derivation $D$ extends\ncanonically to a derivation $D^\\wedge : R^\\wedge \\to R^\\wedge$\non the $I$-adic completion.\n\\item For any multiplicative subset $S \\subset R$ the derivation\n$D$ extends uniquely to the localization $S^{-1}R$ of $R$.\n\\end{enumerate}\nIf $R \\subset R'$ is a finite type extension of rings such that\n$R_g \\cong R'_g$ for some $g \\in R$ which is a nonzerodivisor in $R'$,\nthen $g^ND$ extends to $R'$ for some $N \\geq 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regularity and derivations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PE","source_file":"more-algebra.tex","source_line":11753,"source_end_line":11766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11753-L11766","statement_sha256":"636e8738fec0db6b9b6a31ffd26165dbd1e337cf3f8a0b32bc3754d5351a8d00","origin":"The Stacks Project","memory_eligible":false,"source_rank":3126,"rank":3126,"depth":0,"x":752.796,"y":623.723,"cluster":"advanced-algebra"},{"id":"stacks:07PF","tag":"07PF","title":"Regularity and derivations · Lemma 07PF","summary":"The Jacobian criterion for hypersurfaces, done right. Let R be a regular ring. Let f ∈ R. Assume there exists a derivation D : R → R such that D(f) is a unit of R/(f). Then R/(f) is regular.","statement_latex":"\\begin{slogan}\nThe Jacobian criterion for hypersurfaces, done right.\n\\end{slogan}\nLet $R$ be a regular ring. Let $f \\in R$. Assume there exists a\nderivation $D : R \\to R$ such that $D(f)$ is a unit of $R/(f)$.\nThen $R/(f)$ is regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regularity and derivations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PF","source_file":"more-algebra.tex","source_line":11797,"source_end_line":11805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11797-L11805","statement_sha256":"0d4392893ed4d6ba8e6466117d8cb3e42ccbc737a2c3932ede1d9799f4338eaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3127,"rank":3127,"depth":13,"x":671.564,"y":762.938,"cluster":"advanced-algebra"},{"id":"stacks:0GEE","tag":"0GEE","title":"Regularity and derivations · Lemma 0GEE","summary":"Let (R, m, kappa) be a regular local ring. Let m ≥ 1. Let f_1, …, f_m ∈ m. Assume there exist derivations D_1, …, D_m : R → R such that det_1 ≤ i, j ≤ m(D_i(f_j)) is a unit of R. Then R/(f_1, …, f_m) is regular and f_1, …, f_m is a regular sequence.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a regular local ring. Let $m \\geq 1$. Let\n$f_1, \\ldots, f_m \\in \\mathfrak m$. Assume there exist derivations\n$D_1, \\ldots, D_m : R \\to R$ such that\n$\\det_{1 \\leq i, j \\leq m}(D_i(f_j))$ is a unit of $R$.\nThen $R/(f_1, \\ldots, f_m)$ is regular and $f_1, \\ldots, f_m$\nis a regular sequence.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regularity and derivations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEE","source_file":"more-algebra.tex","source_line":11816,"source_end_line":11824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11816-L11824","statement_sha256":"698103282e64476f57a800116b73a2497f80d215a78a128368ef1eb4ec89038c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3128,"rank":3128,"depth":13,"x":619.428,"y":613.951,"cluster":"advanced-algebra"},{"id":"stacks:07PG","tag":"07PG","title":"Regularity and derivations · Lemma 07PG","summary":"Let R be a regular ring. Let f ∈ R. Assume there exists a derivation D : R → R such that D(f) is a unit of R. Then R[z]/(z^n - f) is regular for any integer n ≥ 1. More generally, R[z]/(p(z) - f) is regular for any p ∈ Z[z].","statement_latex":"Let $R$ be a regular ring. Let $f \\in R$.\nAssume there exists a derivation $D : R \\to R$ such that $D(f)$ is a unit\nof $R$. Then $R[z]/(z^n - f)$ is regular for any integer $n \\geq 1$.\nMore generally, $R[z]/(p(z) - f)$ is regular for any $p \\in \\mathbf{Z}[z]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regularity and derivations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PG","source_file":"more-algebra.tex","source_line":11843,"source_end_line":11849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11843-L11849","statement_sha256":"17d778bd87b5d9614cfb6ea99470b5e692e8dcba3fe4f569650c9a6d38318af6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3129,"rank":3129,"depth":38,"x":777.937,"y":694.356,"cluster":"advanced-algebra"},{"id":"stacks:07PH","tag":"07PH","title":"Regularity and derivations · Lemma 07PH","summary":"Let p be a prime number. Let B be a domain with p = 0 in B. Let f ∈ B be an element which is not a pth power in the fraction field of B. If B is of finite type over a Noetherian complete local ring, then there exists a derivation D : B → B such that D(f) is not zero.","statement_latex":"Let $p$ be a prime number. Let $B$ be a domain with $p = 0$ in $B$.\nLet $f \\in B$ be an element which is not a $p$th power in the fraction\nfield of $B$. If $B$ is of finite type over a Noetherian complete\nlocal ring, then there exists a derivation $D : B \\to B$ such that $D(f)$\nis not zero.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regularity and derivations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PH","source_file":"more-algebra.tex","source_line":11859,"source_end_line":11866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11859-L11866","statement_sha256":"6e076db0e7954b7d8749959e9a73ecbdda9c820eb8dc3fdec82389611234abbf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3130,"rank":3130,"depth":12,"x":596.105,"y":725.057,"cluster":"advanced-algebra"},{"id":"stacks:07PI","tag":"07PI","title":"Regularity and derivations · Lemma 07PI","summary":"Let A be a Noetherian complete local domain. Then A is J-0.","statement_latex":"Let $A$ be a Noetherian complete local domain. Then $A$ is J-0.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regularity and derivations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PI","source_file":"more-algebra.tex","source_line":11912,"source_end_line":11915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11912-L11915","statement_sha256":"6986b8d6506f69a75ab48102936f2579d273f8f03b368b6a3d2a9e15349b8485","origin":"The Stacks Project","memory_eligible":false,"source_rank":3131,"rank":3131,"depth":40,"x":705.67,"y":599.043,"cluster":"advanced-algebra"},{"id":"stacks:07PJ","tag":"07PJ","title":"Regularity and derivations · Proposition 07PJ","summary":"The following types of rings are J-2: • fields, • Noetherian complete local rings, • Z, • Noetherian local rings of dimension 1, • Nagata rings of dimension 1, • Dedekind domains with fraction field of characteristic zero, • finite type ring extensions of any of the above.","statement_latex":"The following types of rings are J-2:\n\\begin{enumerate}\n\\item fields,\n\\item Noetherian complete local rings,\n\\item $\\mathbf{Z}$,\n\\item Noetherian local rings of dimension $1$,\n\\item Nagata rings of dimension $1$,\n\\item Dedekind domains with fraction field of characteristic zero,\n\\item finite type ring extensions of any of the above.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Regularity and derivations","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PJ","source_file":"more-algebra.tex","source_line":11944,"source_end_line":11956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L11944-L11956","statement_sha256":"ea16bedb6cd93fab3069f0b03e3b134b6549787df869121f63ee61a033ca1170","origin":"The Stacks Project","memory_eligible":false,"source_rank":3132,"rank":3132,"depth":41,"x":726.247,"y":754.38,"cluster":"advanced-algebra"},{"id":"stacks:07PL","tag":"07PL","title":"Formal smoothness and regularity · Lemma 07PL","summary":"Let A → B be a local homomorphism of Noetherian local rings. Let D : A → A be a derivation. Assume that B is complete and A → B is formally smooth in the m_B-adic topology. Then there exists an extension D' : B → B of D.","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian local rings.\nLet $D : A \\to A$ be a derivation. Assume that $B$ is complete\nand $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology.\nThen there exists an extension $D' : B \\to B$ of $D$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal smoothness and regularity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PL","source_file":"more-algebra.tex","source_line":12002,"source_end_line":12008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12002-L12008","statement_sha256":"1a415a0fc93202542e8377914fdea9fcd9a359f58cd769843f0c30b108fae7ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":3133,"rank":3133,"depth":2,"x":585.936,"y":651.35,"cluster":"advanced-algebra"},{"id":"stacks:07PM","tag":"07PM","title":"Formal smoothness and regularity · Proposition 07PM","summary":"Let A → B be a local homomorphism of Noetherian complete local rings. Let k be the residue field of A and overlineB = B ⊗_A k the special fibre. The following are equivalent • A → B is regular, • A → B is flat and overlineB is geometrically regular over k, • A → B is flat and k → overlineB is formally smooth in the m_overlineB-adic topology, and • A → B is formally smooth in the m_B-adic topology.","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian complete local rings.\nLet $k$ be the residue field of $A$ and $\\overline{B} = B \\otimes_A k$\nthe special fibre.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A \\to B$ is regular,\n\\item $A \\to B$ is flat and $\\overline{B}$ is geometrically regular\nover $k$,\n\\item $A \\to B$ is flat and $k \\to \\overline{B}$ is formally smooth\nin the $\\mathfrak m_{\\overline{B}}$-adic topology, and\n\\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic\ntopology.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal smoothness and regularity","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PM","source_file":"more-algebra.tex","source_line":12027,"source_end_line":12042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12027-L12042","statement_sha256":"24a5b0ecf97d9b96da5c53d554cf37812709a34ec9561ef0b63d78bc1a68316b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3134,"rank":3134,"depth":46,"x":772.546,"y":647.701,"cluster":"advanced-algebra"},{"id":"stacks:0H7U","tag":"0H7U","title":"André · Theorem 0H7U","summary":"The implication (4) ⇒ (1) is the main result of [Andre-smooth]. Let A → B be a local homomorphism of Noetherian local rings. Let k be the residue field of A and overlineB = B ⊗_A k the special fibre. Assume A → A^wedge is regular. The following are equivalent • A → B is regular, • A → B is flat and overlineB is geometrically regular over k, • A → B is flat and k → overlineB is formally smooth in the m_overlineB-adic topology, and • A → B is formally smooth in the m_B-adic…","statement_latex":"\\begin{reference}\nThe implication (4) $\\Rightarrow$ (1) is the main result of \\cite{Andre-smooth}.\n\\end{reference}\nLet $A \\to B$ be a local homomorphism of Noetherian local rings.\nLet $k$ be the residue field of $A$ and $\\overline{B} = B \\otimes_A k$\nthe special fibre. Assume $A \\to A^\\wedge$ is regular\\footnote{For example,\nif $A$ is a G-ring or (quasi-)excellent, see Sections \\ref{section-G-ring} and\n\\ref{section-excellent}.}.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A \\to B$ is regular,\n\\item $A \\to B$ is flat and $\\overline{B}$ is geometrically regular\nover $k$,\n\\item $A \\to B$ is flat and $k \\to \\overline{B}$ is formally smooth\nin the $\\mathfrak m_{\\overline{B}}$-adic topology, and\n\\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic\ntopology.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal smoothness and regularity","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7U","source_file":"more-algebra.tex","source_line":12118,"source_end_line":12138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12118-L12138","statement_sha256":"9ae9e46614c9fe6162d4742b3cb2a50ff781338e76bb1a8c8f71e130d811e0f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3135,"rank":3135,"depth":47,"x":637.666,"y":756.45,"cluster":"advanced-algebra"},{"id":"stacks:07GH","tag":"07GH","title":"G-rings · Definition 07GH","summary":"A ring R is called a G-ring if R is Noetherian and for every prime p of R the ring map R_ p → (R_ p)^wedge is regular.","statement_latex":"A ring $R$ is called a {\\it G-ring} if $R$ is Noetherian and for every\nprime $\\mathfrak p$ of $R$ the ring map\n$R_\\mathfrak p \\to (R_\\mathfrak p)^\\wedge$ is regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GH","source_file":"more-algebra.tex","source_line":12178,"source_end_line":12183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12178-L12183","statement_sha256":"5e2b0ce32b0de8f16f7ccea33fd2591154ab92d7d118f93f533240bd0c526f34","origin":"The Stacks Project","memory_eligible":false,"source_rank":3136,"rank":3136,"depth":0,"x":649.691,"y":599.48,"cluster":"advanced-algebra"},{"id":"stacks:07PN","tag":"07PN","title":"G-rings · Lemma 07PN","summary":"Let R be a Noetherian ring. Then R is a G-ring if and only if for every pair of primes q ⊂ p ⊂ R the algebra (R/ q)_ p^wedge ⊗_R/ q kappa( q) is geometrically regular over kappa( q).","statement_latex":"Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if\nfor every pair of primes $\\mathfrak q \\subset \\mathfrak p \\subset R$\nthe algebra\n$$\n(R/\\mathfrak q)_\\mathfrak p^\\wedge \\otimes_{R/\\mathfrak q} \\kappa(\\mathfrak q)\n$$\nis geometrically regular over $\\kappa(\\mathfrak q)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PN","source_file":"more-algebra.tex","source_line":12193,"source_end_line":12202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12193-L12202","statement_sha256":"4c7c70a95bb484b5fe5278c925f257dd1ce5a61cde00658db207257ce6c288ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":3137,"rank":3137,"depth":0,"x":767.237,"y":722.242,"cluster":"advanced-algebra"},{"id":"stacks:07PP","tag":"07PP","title":"G-rings · Lemma 07PP","summary":"Let R → R' be a finite type map of Noetherian rings and let xymatrix q' ar[r] & p' ar[r] & R' q ar[r] ar@-[u] & p ar[r] ar@-[u] & R ar[u] be primes. Assume R → R' is quasi-finite at p'. • If the formal fibre R_ p^wedge ⊗_R kappa( q) is geometrically regular over kappa( q), then the formal fibre (R'_ p')^wedge ⊗_R' kappa( q') is geometrically regular over kappa( q'). • If the formal fibres of R_ p are geometrically regular, then the formal fibres of R'_ p' are…","statement_latex":"Let $R \\to R'$ be a finite type map of Noetherian rings and let\n$$\n\\xymatrix{\n\\mathfrak q' \\ar[r] & \\mathfrak p' \\ar[r] & R' \\\\\n\\mathfrak q \\ar[r] \\ar@{-}[u] &\n\\mathfrak p \\ar[r] \\ar@{-}[u] & R \\ar[u]\n}\n$$\nbe primes. Assume $R \\to R'$ is quasi-finite at $\\mathfrak p'$.\n\\begin{enumerate}\n\\item If the formal fibre $R_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)$\nis geometrically regular over $\\kappa(\\mathfrak q)$, then the formal fibre\n$(R'_{\\mathfrak p'})^\\wedge \\otimes_{R'} \\kappa(\\mathfrak q')$\nis geometrically regular over $\\kappa(\\mathfrak q')$.\n\\item If the formal fibres of $R_\\mathfrak p$ are geometrically regular,\nthen the formal fibres of $R'_{\\mathfrak p'}$ are geometrically regular.\n\\item If $R \\to R'$ is quasi-finite and $R$ is a G-ring, then $R'$ is\na G-ring.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PP","source_file":"more-algebra.tex","source_line":12213,"source_end_line":12234,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12213-L12234","statement_sha256":"8a417261a6713587a58c294f06103b028e2236e477a13d6d993bee73d2ff8820","origin":"The Stacks Project","memory_eligible":false,"source_rank":3138,"rank":3138,"depth":41,"x":581.551,"y":698.381,"cluster":"advanced-algebra"},{"id":"stacks:07PQ","tag":"07PQ","title":"G-rings · Lemma 07PQ","summary":"Let R be a Noetherian ring. Then R is a G-ring if and only if for every finite free ring map R → S the formal fibres of S are regular rings.","statement_latex":"Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if\nfor every finite free ring map $R \\to S$ the formal fibres of $S$\nare regular rings.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PQ","source_file":"more-algebra.tex","source_line":12262,"source_end_line":12267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12262-L12267","statement_sha256":"56a7dd65a858225f27e6321138c9d226d429a5609b74b6339ac1164fad7e8ab1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3139,"rank":3139,"depth":42,"x":737.903,"y":610.477,"cluster":"advanced-algebra"},{"id":"stacks:07PR","tag":"07PR","title":"G-rings · Lemma 07PR","summary":"Let k be a field of characteristic p. Let A = k[[x_1, …, x_n]][y_1, …, y_m] and denote K the fraction field of A. Let p ⊂ A be a prime. Then A_ p^wedge ⊗_A K is geometrically regular over K.","statement_latex":"Let $k$ be a field of characteristic $p$.\nLet $A = k[[x_1, \\ldots, x_n]][y_1, \\ldots, y_m]$ and denote $K$\nthe fraction field of $A$.\nLet $\\mathfrak p \\subset A$ be a prime. Then\n$A_\\mathfrak p^\\wedge \\otimes_A K$ is geometrically regular over $K$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PR","source_file":"more-algebra.tex","source_line":12305,"source_end_line":12312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12305-L12312","statement_sha256":"364e0f1398e13f367db8ee42d952b90a12aac0979f47bbc29b0622edb38481d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3140,"rank":3140,"depth":39,"x":693.232,"y":764.251,"cluster":"advanced-algebra"},{"id":"stacks:07PS","tag":"07PS","title":"G-rings · Proposition 07PS","summary":"A Noetherian complete local ring is a G-ring.","statement_latex":"A Noetherian complete local ring is a G-ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PS","source_file":"more-algebra.tex","source_line":12355,"source_end_line":12358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12355-L12358","statement_sha256":"8c9e85963aad3b81a8bd6b3075dd7dd58a5f37b8a073b41b8ca6d81ca02f0327","origin":"The Stacks Project","memory_eligible":false,"source_rank":3141,"rank":3141,"depth":42,"x":602.371,"y":625.299,"cluster":"advanced-algebra"},{"id":"stacks:07PT","tag":"07PT","title":"G-rings · Lemma 07PT","summary":"Let R be a Noetherian ring. Then R is a G-ring if and only if R_ m has geometrically regular formal fibres for every maximal ideal m of R.","statement_latex":"Let $R$ be a Noetherian ring. Then $R$ is a G-ring if and only if\n$R_\\mathfrak m$ has geometrically regular formal fibres for every\nmaximal ideal $\\mathfrak m$ of $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PT","source_file":"more-algebra.tex","source_line":12391,"source_end_line":12396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12391-L12396","statement_sha256":"cddec82a50fabbe69410af3db19964efc218a76e2c5292fe8a715bf56e425194","origin":"The Stacks Project","memory_eligible":false,"source_rank":3142,"rank":3142,"depth":43,"x":781.386,"y":676.282,"cluster":"advanced-algebra"},{"id":"stacks:07QR","tag":"07QR","title":"G-rings · Lemma 07QR","summary":"Let R be a Noetherian local ring which is a G-ring. Then the henselization R^h and the strict henselization R^sh are G-rings.","statement_latex":"Let $R$ be a Noetherian local ring which is a G-ring.\nThen the henselization $R^h$ and the strict henselization $R^{sh}$\nare G-rings.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QR","source_file":"more-algebra.tex","source_line":12429,"source_end_line":12434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12429-L12434","statement_sha256":"dd0c84b13ad7e2e10f0f5e61ab43a8a7e23404c14353dad1d76301ce99314750","origin":"The Stacks Project","memory_eligible":false,"source_rank":3143,"rank":3143,"depth":52,"x":608.12,"y":740.363,"cluster":"advanced-algebra"},{"id":"stacks:07PU","tag":"07PU","title":"G-rings · Lemma 07PU","summary":"Let p be a prime number. Let A be a Noetherian complete local domain with fraction field K of characteristic p. Let q ⊂ A[x] be a maximal ideal lying over the maximal ideal of A and let (0) not = r ⊂ q be a prime lying over (0) ⊂ A. Then A[x]_ q^wedge ⊗_A[x] kappa( r) is geometrically regular over kappa( r).","statement_latex":"Let $p$ be a prime number. Let $A$ be a Noetherian complete local domain\nwith fraction field $K$ of characteristic $p$. Let $\\mathfrak q \\subset A[x]$\nbe a maximal ideal lying over the maximal ideal of $A$ and let\n$(0) \\not = \\mathfrak r \\subset \\mathfrak q$ be a prime lying over\n$(0) \\subset A$. Then\n$A[x]_\\mathfrak q^\\wedge \\otimes_{A[x]} \\kappa(\\mathfrak r)$ is geometrically\nregular over $\\kappa(\\mathfrak r)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PU","source_file":"more-algebra.tex","source_line":12493,"source_end_line":12502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12493-L12502","statement_sha256":"fcc258f5bb795e80e1e944ae4d8818004dbe53b8c4b8cb2cc42faec15a7a468f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3144,"rank":3144,"depth":47,"x":684.468,"y":594.573,"cluster":"advanced-algebra"},{"id":"stacks:07PV","tag":"07PV","title":"G-rings · Proposition 07PV","summary":"Let R be a G-ring. If R → S is essentially of finite type then S is a G-ring.","statement_latex":"Let $R$ be a G-ring. If $R \\to S$ is essentially of finite type\nthen $S$ is a G-ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PV","source_file":"more-algebra.tex","source_line":12551,"source_end_line":12555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12551-L12555","statement_sha256":"ec5effc51633723cca32ae3c3efa60181aca0f2280a6649e8abf91255d48bf5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3145,"rank":3145,"depth":48,"x":745.503,"y":745.627,"cluster":"advanced-algebra"},{"id":"stacks:07PX","tag":"07PX","title":"G-rings · Proposition 07PX","summary":"The following types of rings are G-rings: • fields, • Noetherian complete local rings, • Z, • Dedekind domains with fraction field of characteristic zero, • finite type ring extensions of any of the above.","statement_latex":"The following types of rings are G-rings:\n\\begin{enumerate}\n\\item fields,\n\\item Noetherian complete local rings,\n\\item $\\mathbf{Z}$,\n\\item Dedekind domains with fraction field of characteristic zero,\n\\item finite type ring extensions of any of the above.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07PX","source_file":"more-algebra.tex","source_line":12651,"source_end_line":12661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12651-L12661","statement_sha256":"4993b5c87911f6aab93d2cbb56baaa82d0c3bbc05daa125aece1639a60081256","origin":"The Stacks Project","memory_eligible":false,"source_rank":3146,"rank":3146,"depth":49,"x":578.77,"y":668.758,"cluster":"advanced-algebra"},{"id":"stacks:0A41","tag":"0A41","title":"G-rings · Lemma 0A41","summary":"Let (A, m) be a henselian local ring. Then A is a filtered colimit of a system of henselian local G-rings with local transition maps.","statement_latex":"Let $(A, \\mathfrak m)$ be a henselian local ring.\nThen $A$ is a filtered colimit of a system\nof henselian local G-rings with local\ntransition maps.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A41","source_file":"more-algebra.tex","source_line":12673,"source_end_line":12679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12673-L12679","statement_sha256":"2f3aea25a619fa633a706128fc76b7c02e1517427d665e2b3fa7e6c456f4683d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3147,"rank":3147,"depth":53,"x":763.812,"y":630.775,"cluster":"advanced-algebra"},{"id":"stacks:0AH2","tag":"0AH2","title":"G-rings · Lemma 0AH2","summary":"[MatCA] Let A be a G-ring. Let I ⊂ A be an ideal and let A^wedge be the completion of A with respect to I. Then A → A^wedge is regular.","statement_latex":"\\begin{reference}\n\\cite[Theorem 79]{MatCA}\n\\end{reference}\nLet $A$ be a G-ring. Let $I \\subset A$ be an ideal\nand let $A^\\wedge$ be the completion of $A$ with respect to $I$.\nThen $A \\to A^\\wedge$ is regular.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AH2","source_file":"more-algebra.tex","source_line":12693,"source_end_line":12701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12693-L12701","statement_sha256":"2e2b5c64e7b306ebc021d8861c368b158092944c32196977e30d91ab03042709","origin":"The Stacks Project","memory_eligible":false,"source_rank":3148,"rank":3148,"depth":21,"x":657.75,"y":763.981,"cluster":"advanced-algebra"},{"id":"stacks:0AH3","tag":"0AH3","title":"G-rings · Lemma 0AH3","summary":"Being a G-ring is stable under Henselizations along ideals [Greco] Let A be a G-ring. Let I ⊂ A be an ideal. Let (A^h, I^h) be the henselization of the pair (A, I), see Lemma [Tag 0A02]. Then A^h is a G-ring.","statement_latex":"\\begin{slogan}\nBeing a G-ring is stable under Henselizations along ideals\n\\end{slogan}\n\\begin{reference}\n\\cite[Theorem 5.3 i)]{Greco}\n\\end{reference}\nLet $A$ be a G-ring. Let $I \\subset A$ be an ideal.\nLet $(A^h, I^h)$ be the henselization of the pair $(A, I)$, see\nLemma \\ref{lemma-henselization}.\nThen $A^h$ is a G-ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AH3","source_file":"more-algebra.tex","source_line":12741,"source_end_line":12753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12741-L12753","statement_sha256":"255068cf956896e3f392cb17c57b0befc6252c5b20e8b36f0addb736aef387e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3149,"rank":3149,"depth":50,"x":628.795,"y":605.337,"cluster":"advanced-algebra"},{"id":"stacks:0BIS","tag":"0BIS","title":"Properties of formal fibres · Lemma 0BIS","summary":"Let R be a Noetherian ring. Let P be a property as above. Then R is a P-ring if and only if for every pair of primes q ⊂ p ⊂ R the kappa( q)-algebra (R/ q)_ p^wedge ⊗_R/ q kappa( q) has property P.","statement_latex":"Let $R$ be a Noetherian ring. Let $P$ be a property as above.\nThen $R$ is a $P$-ring if and only if\nfor every pair of primes $\\mathfrak q \\subset \\mathfrak p \\subset R$\nthe $\\kappa(\\mathfrak q)$-algebra\n$$\n(R/\\mathfrak q)_\\mathfrak p^\\wedge \\otimes_{R/\\mathfrak q} \\kappa(\\mathfrak q)\n$$\nhas property $P$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIS","source_file":"more-algebra.tex","source_line":12849,"source_end_line":12859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12849-L12859","statement_sha256":"67c62d7094383eac1a228f71556c788fe52d3d13ccea755b4441994bc9e1437d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3150,"rank":3150,"depth":0,"x":777.946,"y":706.039,"cluster":"advanced-algebra"},{"id":"stacks:0BK8","tag":"0BK8","title":"Properties of formal fibres · Lemma 0BK8","summary":"Let R → Lambda be a homomorphism of Noetherian rings. Assume P has property (B). The following are equivalent • the fibres of R → Lambda have P, • the fibres of R_ p → Lambda_ q have P for all q ⊂ Lambda lying over p ⊂ R, and • the fibres of R_ m → Lambda_ m' have P for all maximal ideals m' ⊂ Lambda lying over m in R.","statement_latex":"Let $R \\to \\Lambda$ be a homomorphism of Noetherian rings.\nAssume $P$ has property (B). The following are equivalent\n\\begin{enumerate}\n\\item the fibres of $R \\to \\Lambda$ have $P$,\n\\item the fibres of $R_\\mathfrak p \\to \\Lambda_\\mathfrak q$ have $P$\nfor all $\\mathfrak q \\subset \\Lambda$ lying over $\\mathfrak p \\subset R$, and\n\\item the fibres of $R_\\mathfrak m \\to \\Lambda_{\\mathfrak m'}$ have $P$\nfor all maximal ideals $\\mathfrak m' \\subset \\Lambda$\nlying over $\\mathfrak m$ in $R$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BK8","source_file":"more-algebra.tex","source_line":12870,"source_end_line":12882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12870-L12882","statement_sha256":"fcc85f624b958c4f1eeb04fdda6c09a44e61380f4f2b1ba32fa841781f5adef7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3151,"rank":3151,"depth":0,"x":586.698,"y":716.43,"cluster":"advanced-algebra"},{"id":"stacks:0BIT","tag":"0BIT","title":"Properties of formal fibres · Lemma 0BIT","summary":"Let R → R' be a finite type map of Noetherian rings and let xymatrix q' ar[r] & p' ar[r] & R' q ar[r] ar@-[u] & p ar[r] ar@-[u] & R ar[u] be primes. Assume R → R' is quasi-finite at p'. Assume P satisfies (A) and (B). • If kappa( q) → R_ p^wedge ⊗_R kappa( q) has P, then kappa( q') → (R'_ p')^wedge ⊗_R' kappa( q') has P. • If the formal fibres of R_ p have P, then the formal fibres of R'_ p' have P. • If R → R' is quasi-finite and R is a P-ring, then R' is a P-ring.","statement_latex":"Let $R \\to R'$ be a finite type map of Noetherian rings and let\n$$\n\\xymatrix{\n\\mathfrak q' \\ar[r] & \\mathfrak p' \\ar[r] & R' \\\\\n\\mathfrak q \\ar[r] \\ar@{-}[u] &\n\\mathfrak p \\ar[r] \\ar@{-}[u] & R \\ar[u]\n}\n$$\nbe primes. Assume $R \\to R'$ is quasi-finite at $\\mathfrak p'$.\nAssume $P$ satisfies (A) and (B).\n\\begin{enumerate}\n\\item If $\\kappa(\\mathfrak q) \\to\nR_\\mathfrak p^\\wedge \\otimes_R \\kappa(\\mathfrak q)$\nhas $P$, then\n$\\kappa(\\mathfrak q') \\to\n(R'_{\\mathfrak p'})^\\wedge \\otimes_{R'} \\kappa(\\mathfrak q')$\nhas $P$.\n\\item If the formal fibres of $R_\\mathfrak p$ have $P$,\nthen the formal fibres of $R'_{\\mathfrak p'}$ have $P$.\n\\item If $R \\to R'$ is quasi-finite and $R$ is a $P$-ring, then $R'$ is\na $P$-ring.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIT","source_file":"more-algebra.tex","source_line":12903,"source_end_line":12927,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12903-L12927","statement_sha256":"c044a0d86d5fcfa90d4a5bb883ff2379b24948a1fdf78e4c089a0436be2963c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3152,"rank":3152,"depth":30,"x":719.562,"y":600.076,"cluster":"advanced-algebra"},{"id":"stacks:0BIU","tag":"0BIU","title":"Properties of formal fibres · Lemma 0BIU","summary":"Let R be a Noetherian ring. Assume P satisfies (C) and (D). Then R is a P-ring if and only if the formal fibres of R_ m have P for every maximal ideal m of R.","statement_latex":"Let $R$ be a Noetherian ring. Assume $P$ satisfies (C) and (D).\nThen $R$ is a $P$-ring if and only if the formal fibres of\n$R_\\mathfrak m$ have $P$ for every\nmaximal ideal $\\mathfrak m$ of $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIU","source_file":"more-algebra.tex","source_line":12953,"source_end_line":12959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12953-L12959","statement_sha256":"d7ae80361744c3bda9aa27c3ad3fb0af259ef9c1b79073a472a9c153a56b5854","origin":"The Stacks Project","memory_eligible":false,"source_rank":3153,"rank":3153,"depth":43,"x":715.152,"y":761.504,"cluster":"advanced-algebra"},{"id":"stacks:0BIV","tag":"0BIV","title":"Properties of formal fibres · Proposition 0BIV","summary":"Let R be a P-ring where P satisfies (A), (B), (C), and (D). If R → S is essentially of finite type then S is a P-ring.","statement_latex":"Let $R$ be a $P$-ring where $P$ satisfies (A), (B), (C), and (D).\nIf $R \\to S$ is essentially of finite type then $S$ is a $P$-ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIV","source_file":"more-algebra.tex","source_line":12998,"source_end_line":13002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L12998-L13002","statement_sha256":"569e4243e8f9ff5ab6065c9b66db43d035eedc817587b5086fcf9603a6622ca8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3154,"rank":3154,"depth":49,"x":588.402,"y":639.787,"cluster":"advanced-algebra"},{"id":"stacks:0BK9","tag":"0BK9","title":"Properties of formal fibres · Lemma 0BK9","summary":"Let A be a P-ring where P satisfies (B) and (D). Let I ⊂ A be an ideal and let A^wedge be the completion of A with respect to I. Then the fibres of A → A^wedge have P.","statement_latex":"Let $A$ be a $P$-ring where $P$ satisfies (B) and (D).\nLet $I \\subset A$ be an ideal and let $A^\\wedge$ be the completion of $A$\nwith respect to $I$. Then the fibres of $A \\to A^\\wedge$ have $P$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BK9","source_file":"more-algebra.tex","source_line":13047,"source_end_line":13052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13047-L13052","statement_sha256":"869a02bd3df2b98a3e8d29c048066dce0a6d8dedc85e83625f405ac7a234981c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3155,"rank":3155,"depth":10,"x":780.028,"y":657.644,"cluster":"advanced-algebra"},{"id":"stacks:0BKA","tag":"0BKA","title":"Properties of formal fibres · Lemma 0BKA","summary":"Henselization of a ring inherits good properties of formal fibers Let A be a P-ring where P satisfies (B), (C), (D), and (E). Let I ⊂ A be an ideal. Let (A^h, I^h) be the henselization of the pair (A, I), see Lemma [Tag 0A02]. Then A^h is a P-ring.","statement_latex":"\\begin{slogan}\nHenselization of a ring inherits good properties of formal fibers\n\\end{slogan}\nLet $A$ be a $P$-ring where $P$ satisfies (B), (C), (D), and (E).\nLet $I \\subset A$ be an ideal. Let $(A^h, I^h)$ be the henselization\nof the pair $(A, I)$, see Lemma \\ref{lemma-henselization}.\nThen $A^h$ is a $P$-ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKA","source_file":"more-algebra.tex","source_line":13093,"source_end_line":13102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13093-L13102","statement_sha256":"eab7f61bf3d080c3d5e5d1f51004b6f7192445a3661b1055276fde46af810f24","origin":"The Stacks Project","memory_eligible":false,"source_rank":3156,"rank":3156,"depth":50,"x":624.136,"y":753.351,"cluster":"advanced-algebra"},{"id":"stacks:0C36","tag":"0C36","title":"Properties of formal fibres · Lemma 0C36","summary":"Let R be a Noetherian local ring which is a P-ring where P satisfies (B), (C), (D), and (E). Then the henselization R^h and the strict henselization R^sh are P-rings.","statement_latex":"Let $R$ be a Noetherian local ring which is a $P$-ring where $P$\nsatisfies (B), (C), (D), and (E). Then the henselization $R^h$\nand the strict henselization $R^{sh}$ are $P$-rings.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C36","source_file":"more-algebra.tex","source_line":13154,"source_end_line":13159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13154-L13159","statement_sha256":"4c294285bb8ebfeb053679513fe0b09d77b2339dbff249112042c1ee50a066fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3157,"rank":3157,"depth":52,"x":662.18,"y":594.089,"cluster":"advanced-algebra"},{"id":"stacks:0BIW","tag":"0BIW","title":"Properties of formal fibres · Lemma 0BIW","summary":"Properties (A), (B), (C), (D), and (E) hold for P(k → R) =\"R is geometrically reduced over k\".","statement_latex":"Properties (A), (B), (C), (D), and (E) hold for\n$P(k \\to R) =$``$R$ is geometrically reduced over $k$''.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIW","source_file":"more-algebra.tex","source_line":13197,"source_end_line":13201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13197-L13201","statement_sha256":"655e8fe10d8f1d6bce7c2548e11fcad9bf1fb6fdeeb0f5fa1a21ee1ecd4e5e31","origin":"The Stacks Project","memory_eligible":false,"source_rank":3158,"rank":3158,"depth":18,"x":762.348,"y":733.316,"cluster":"advanced-algebra"},{"id":"stacks:0BIX","tag":"0BIX","title":"Properties of formal fibres · Lemma 0BIX","summary":"Properties (A), (B), (C), (D), and (E) hold for P(k → R) =\"R is geometrically normal over k\".","statement_latex":"Properties (A), (B), (C), (D), and (E) hold for\n$P(k \\to R) =$``$R$ is geometrically normal over $k$''.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIX","source_file":"more-algebra.tex","source_line":13214,"source_end_line":13218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13214-L13218","statement_sha256":"e3f036910ec5fda15c3945718bcc09ab7f27c717af4376bd59af8b3cca9743d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3159,"rank":3159,"depth":42,"x":576.252,"y":687.422,"cluster":"advanced-algebra"},{"id":"stacks:0BIY","tag":"0BIY","title":"Properties of formal fibres · Lemma 0BIY","summary":"Fix n ≥ 1. Properties (A), (B), (C), (D), and (E) hold for P(k → R) =\"R has (S_n)\".","statement_latex":"Fix $n \\geq 1$. Properties (A), (B), (C), (D), and (E) hold for\n$P(k \\to R) =$``$R$ has $(S_n)$''.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIY","source_file":"more-algebra.tex","source_line":13231,"source_end_line":13235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13231-L13235","statement_sha256":"f3b46035d818f5fe2f357b10e92c4f2030cb2a73fd3fb30cea05af5b5c1a269f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3160,"rank":3160,"depth":17,"x":750.636,"y":615.565,"cluster":"advanced-algebra"},{"id":"stacks:0BJ9","tag":"0BJ9","title":"Properties of formal fibres · Lemma 0BJ9","summary":"Properties (A), (B), (C), (D), and (E) hold for P(k → R) =\"R is Cohen-Macaulay\".","statement_latex":"Properties (A), (B), (C), (D), and (E) hold for\n$P(k \\to R) =$``$R$ is Cohen-Macaulay''.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJ9","source_file":"more-algebra.tex","source_line":13257,"source_end_line":13261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13257-L13261","statement_sha256":"97ee2b7d3fc768bc4589f57ad990d8da278b417b771776f65f60825b7ca0e26d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3161,"rank":3161,"depth":18,"x":679.731,"y":767.721,"cluster":"advanced-algebra"},{"id":"stacks:0BIZ","tag":"0BIZ","title":"Properties of formal fibres · Lemma 0BIZ","summary":"Fix n ≥ 0. Properties (A), (B), (C), (D), and (E) hold for P(k → R) =\"R ⊗_k k' has (R_n) for all finite extensions k'/k\".","statement_latex":"Fix $n \\geq 0$. Properties (A), (B), (C), (D), and (E) hold for\n$P(k \\to R) =$``$R \\otimes_k k'$ has $(R_n)$ for all finite\nextensions $k'/k$''.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Properties of formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIZ","source_file":"more-algebra.tex","source_line":13269,"source_end_line":13274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13269-L13274","statement_sha256":"b060944baba94b5ba5fb42518abb3fd6110bf794c06a1174909ca0fdbb263ee1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3162,"rank":3162,"depth":40,"x":609.554,"y":615.07,"cluster":"advanced-algebra"},{"id":"stacks:07QT","tag":"07QT","title":"Excellent rings · Definition 07QT","summary":"Let R be a ring. • We say R is quasi-excellent if R is Noetherian, a G-ring, and J-2. • We say R is excellent if R is quasi-excellent and universally catenary.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item We say $R$ is {\\it quasi-excellent} if $R$ is Noetherian,\na G-ring, and J-2.\n\\item We say $R$ is {\\it excellent} if $R$ is quasi-excellent\nand universally catenary.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Excellent rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QT","source_file":"more-algebra.tex","source_line":13351,"source_end_line":13360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13351-L13360","statement_sha256":"cda22e980a3ea68f1f77506641460329b45597a9f91c58906fdfebe2b5996915","origin":"The Stacks Project","memory_eligible":false,"source_rank":3163,"rank":3163,"depth":0,"x":784.31,"y":687.917,"cluster":"advanced-algebra"},{"id":"stacks:07QU","tag":"07QU","title":"Excellent rings · Lemma 07QU","summary":"Any localization of a finite type ring over a (quasi-)excellent ring is (quasi-)excellent.","statement_latex":"Any localization of a finite type ring over a (quasi-)excellent ring\nis (quasi-)excellent.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Excellent rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QU","source_file":"more-algebra.tex","source_line":13372,"source_end_line":13376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13372-L13376","statement_sha256":"5c8ba70027e61733a1dea510cc552c7eed2ba00faf8e19c117a1f3da005c7e2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3164,"rank":3164,"depth":49,"x":596.597,"y":733.427,"cluster":"advanced-algebra"},{"id":"stacks:07QW","tag":"07QW","title":"Excellent rings · Proposition 07QW","summary":"The following types of rings are excellent: • fields, • Noetherian complete local rings, • Z, • Dedekind domains with fraction field of characteristic zero, • finite type ring extensions of any of the above.","statement_latex":"The following types of rings are excellent:\n\\begin{enumerate}\n\\item fields,\n\\item Noetherian complete local rings,\n\\item $\\mathbf{Z}$,\n\\item Dedekind domains with fraction field of characteristic zero,\n\\item finite type ring extensions of any of the above.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Excellent rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QW","source_file":"more-algebra.tex","source_line":13385,"source_end_line":13395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13385-L13395","statement_sha256":"720b7ded1a5aa7c348ff59dc37e5f43ec809e56d8acc736ecddc1634e1f6206f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3165,"rank":3165,"depth":50,"x":698.562,"y":593.155,"cluster":"advanced-algebra"},{"id":"stacks:0BJ0","tag":"0BJ0","title":"Excellent rings · Lemma 0BJ0","summary":"Let (A, m) be a Noetherian local ring. The following are equivalent • A is Nagata, and • the formal fibres of A are geometrically reduced.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A$ is Nagata, and\n\\item the formal fibres of $A$ are geometrically reduced.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Excellent rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJ0","source_file":"more-algebra.tex","source_line":13412,"source_end_line":13420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13412-L13420","statement_sha256":"3b51859af30082d3bed6af9426125529d8de97fcf20e11443ab822dc5364cabc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3166,"rank":3166,"depth":50,"x":736.23,"y":754.677,"cluster":"advanced-algebra"},{"id":"stacks:07QV","tag":"07QV","title":"Excellent rings · Lemma 07QV","summary":"A quasi-excellent ring is Nagata.","statement_latex":"A quasi-excellent ring is Nagata.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Excellent rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QV","source_file":"more-algebra.tex","source_line":13463,"source_end_line":13466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13463-L13466","statement_sha256":"1021325384cee455b5ac8b21f4ebbf0c4f194f9255dabb2b879de0c7b1a7379b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3167,"rank":3167,"depth":50,"x":578.339,"y":656.808,"cluster":"advanced-algebra"},{"id":"stacks:0C23","tag":"0C23","title":"Excellent rings · Lemma 0C23","summary":"Let (A, m) be a Noetherian local ring. If A is normal and the formal fibres of A are normal (for example if A is excellent or quasi-excellent), then A^wedge is normal.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is normal\nand the formal fibres of $A$ are normal (for example if $A$\nis excellent or quasi-excellent), then $A^\\wedge$ is normal.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Excellent rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C23","source_file":"more-algebra.tex","source_line":13484,"source_end_line":13489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13484-L13489","statement_sha256":"9a39b7527d32a5be12417a7d992ac0628f3efece002681e82b023b0a7c11efcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3168,"rank":3168,"depth":18,"x":773.746,"y":639.359,"cluster":"advanced-algebra"},{"id":"stacks:01D7","tag":"01D7","title":"Injective abelian groups · Lemma 01D7","summary":"An abelian group J is an injective object in the category of abelian groups if and only if J is divisible.","statement_latex":"An abelian group $J$ is an injective object in\nthe category of abelian groups if and only if $J$\nis divisible.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01D7","source_file":"more-algebra.tex","source_line":13552,"source_end_line":13557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13552-L13557","statement_sha256":"f4df01054f966d3931f946ad4176ba38918f7fb9359f37e9618b2628717f128e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3169,"rank":3169,"depth":0,"x":643.504,"y":763.279,"cluster":"advanced-algebra"},{"id":"stacks:0AVD","tag":"0AVD","title":"Injective modules · Definition 0AVD","summary":"Let R be a ring. An R-module J is injective if and only if the functor Hom_R(-, J) : Mod_R → Mod_R is an exact functor.","statement_latex":"Let $R$ be a ring. An $R$-module $J$ is {\\it injective} if and only if\nthe functor $\\Hom_R(-, J) : \\text{Mod}_R \\to \\text{Mod}_R$ is\nan exact functor.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVD","source_file":"more-algebra.tex","source_line":13622,"source_end_line":13627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13622-L13627","statement_sha256":"02888c2b6e287c698deb38ad0be3e5bb0651cebb55008d711a2d338b68bbf724","origin":"The Stacks Project","memory_eligible":false,"source_rank":3170,"rank":3170,"depth":0,"x":639.884,"y":597.767,"cluster":"advanced-algebra"},{"id":"stacks:0AUL","tag":"0AUL","title":"Injective modules · Lemma 0AUL","summary":"Let R be a ring. Let A be the abelian category of R-modules. There is a canonical isomorphism Ext_A(M, N) = Ext^1_R(M, N) compatible with the long exact sequences of Algebra, Lemmas [Tag 00LU] and [Tag 065P] and the 6-term exact sequences of Homology, Lemma [Tag 05E2].","statement_latex":"Let $R$ be a ring. Let $\\mathcal{A}$ be the abelian category of\n$R$-modules. There is a canonical isomorphism\n$\\Ext_\\mathcal{A}(M, N) = \\Ext^1_R(M, N)$\ncompatible with the long exact sequences of\nAlgebra, Lemmas \\ref{algebra-lemma-long-exact-seq-ext} and\n\\ref{algebra-lemma-reverse-long-exact-seq-ext}\nand the $6$-term exact sequences of\nHomology, Lemma \\ref{homology-lemma-six-term-sequence-ext}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUL","source_file":"more-algebra.tex","source_line":13641,"source_end_line":13651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13641-L13651","statement_sha256":"68c5c95e94e38bb63f1b0a44202446db8098bdefef1487d65f9f8cf93da3ebb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3171,"rank":3171,"depth":1,"x":775.844,"y":717.928,"cluster":"advanced-algebra"},{"id":"stacks:0AVE","tag":"0AVE","title":"Injective modules · Lemma 0AVE","summary":"Let R be a ring. Let J be an R-module. The following are equivalent • J is injective, • Ext^1_R(M, J) = 0 for every R-module M.","statement_latex":"Let $R$ be a ring. Let $J$ be an $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $J$ is injective,\n\\item $\\Ext^1_R(M, J) = 0$ for every $R$-module $M$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVE","source_file":"more-algebra.tex","source_line":13657,"source_end_line":13665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13657-L13665","statement_sha256":"a8a3cd17c8ce7251537113f8a2af9c61a2cf22146d430789e0c0f1a64e2edaa4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3172,"rank":3172,"depth":2,"x":578.683,"y":706.454,"cluster":"advanced-algebra"},{"id":"stacks:0AVF","tag":"0AVF","title":"Injective modules · Lemma 0AVF","summary":"Let R be a ring. Let J be an R-module. The following are equivalent • J is injective, • Ext^1_R(R/I, J) = 0 for every ideal I ⊂ R, and • for an ideal I ⊂ R and module map I → J there exists an extension R → J.","statement_latex":"Let $R$ be a ring. Let $J$ be an $R$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $J$ is injective,\n\\item $\\Ext^1_R(R/I, J) = 0$ for every ideal $I \\subset R$, and\n\\item for an ideal $I \\subset R$ and module map $I \\to J$\nthere exists an extension $R \\to J$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVF","source_file":"more-algebra.tex","source_line":13691,"source_end_line":13701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13691-L13701","statement_sha256":"459f8234795dbae70036e885a2269efd8893583e49d4cf3d9a9283ffc460bd8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3173,"rank":3173,"depth":1,"x":733.51,"y":602.896,"cluster":"advanced-algebra"},{"id":"stacks:01D9","tag":"01D9","title":"Injective modules · Definition 01D9","summary":"Let R be a ring. • For any R-module M over R we denote M^vee = Hom(M, Q/Z) with its natural R-module structure. We think of M ↦ M^vee as a contravariant functor from the category of R-modules to itself. • For any R-module M we denote F(M) = bigoplus_m ∈ M R[m] the free module with basis given by the elements [m] with m ∈ M. We let F(M)→ M, ∑ f_i [m_i] ↦ ∑ f_i m_i be the natural surjection of R-modules. We think of M ↦ (F(M) → M) as a functor from the category of R-modules…","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item For any $R$-module $M$ over $R$ we denote\n$M^\\vee = \\Hom(M, \\mathbf{Q}/\\mathbf{Z})$\nwith its natural $R$-module structure. We think\nof {\\it $M \\mapsto M^\\vee$} as a contravariant functor\nfrom the category of $R$-modules to itself.\n\\item For any $R$-module $M$ we denote\n$$\nF(M) = \\bigoplus\\nolimits_{m \\in M} R[m]\n$$\nthe {\\it free module} with basis given by the elements $[m]$ with\n$m \\in M$. We let $F(M)\\to M$, $\\sum f_i [m_i] \\mapsto \\sum f_i m_i$\nbe the natural surjection of $R$-modules.\nWe think of $M \\mapsto (F(M) \\to M)$ as a functor from\nthe category of $R$-modules to the category of\narrows in $R$-modules.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01D9","source_file":"more-algebra.tex","source_line":13764,"source_end_line":13784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13764-L13784","statement_sha256":"a2506a26e22a0fb7470bfa4c1fbb4b2d72d30da6f42b2cb2709d85e7deb7b7ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":3174,"rank":3174,"depth":0,"x":702.578,"y":767.34,"cluster":"advanced-algebra"},{"id":"stacks:01DA","tag":"01DA","title":"Injective modules · Lemma 01DA","summary":"Let R be a ring. The functor M ↦ M^vee is exact.","statement_latex":"Let $R$ be a ring.\nThe functor $M \\mapsto M^\\vee$ is exact.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DA","source_file":"more-algebra.tex","source_line":13786,"source_end_line":13790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13786-L13790","statement_sha256":"cef384e2d8ff50012681a62670aaf54a38dcd5e7729f37ef2446e69e7ad6d0ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":3175,"rank":3175,"depth":1,"x":592.995,"y":628.337,"cluster":"advanced-algebra"},{"id":"stacks:01DB","tag":"01DB","title":"Injective modules · Lemma 01DB","summary":"For any R-module M the evaluation map ev : M → (M^vee)^vee is injective.","statement_latex":"For any $R$-module $M$ the evaluation map\n$ev : M \\to (M^\\vee)^\\vee$ is injective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DB","source_file":"more-algebra.tex","source_line":13803,"source_end_line":13807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13803-L13807","statement_sha256":"100ea40afa463855fe5b1df510dca11d60248c9dc1bd3c75d140b81a9b562802","origin":"The Stacks Project","memory_eligible":false,"source_rank":3176,"rank":3176,"depth":0,"x":785.849,"y":668.711,"cluster":"advanced-algebra"},{"id":"stacks:01DC","tag":"01DC","title":"Injective modules · Lemma 01DC","summary":"Let R be a ring. For every R-module M the R-module J(M) is injective.","statement_latex":"Let $R$ be a ring. For every $R$-module $M$ the\n$R$-module $J(M)$ is injective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DC","source_file":"more-algebra.tex","source_line":13828,"source_end_line":13832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13828-L13832","statement_sha256":"68fa1e1d8ba458cc2b2e5e7b391d2ec5f2ece3198922fc73d41fb809a99415ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":3177,"rank":3177,"depth":2,"x":610.932,"y":748.48,"cluster":"advanced-algebra"},{"id":"stacks:01DD","tag":"01DD","title":"Injective modules · Lemma 01DD","summary":"Let R be a ring. The construction above defines a covariant functor M ↦ (M → J(M)) from the category of R-modules to the category of arrows of R-modules such that for every module M the output M → J(M) is an injective map of M into an injective R-module J(M).","statement_latex":"Let $R$ be a ring.\nThe construction above defines a covariant functor\n$M \\mapsto (M \\to J(M))$ from the category of\n$R$-modules to the category of arrows of $R$-modules\nsuch that for every module $M$ the output\n$M \\to J(M)$ is an injective map of $M$ into\nan injective $R$-module $J(M)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DD","source_file":"more-algebra.tex","source_line":13848,"source_end_line":13857,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13848-L13857","statement_sha256":"a0efe8c213c40febfa1e6395096b2925ea553bc04fa194167679fc50191f015a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3178,"rank":3178,"depth":0,"x":675.854,"y":590.19,"cluster":"advanced-algebra"},{"id":"stacks:0915","tag":"0915","title":"Derived categories of modules · Lemma 0915","summary":"Let R → S be a flat ring map. If I^bullet is a K-injective complex of S-modules, then I^bullet is K-injective as a complex of R-modules.","statement_latex":"Let $R \\to S$ be a flat ring map. If $I^\\bullet$ is a K-injective\ncomplex of $S$-modules, then $I^\\bullet$ is K-injective as a\ncomplex of $R$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0915","source_file":"more-algebra.tex","source_line":13925,"source_end_line":13930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13925-L13930","statement_sha256":"8501367aa0b04298535180534867ec48203730fc901e9c364b56fb1aa5865323","origin":"The Stacks Project","memory_eligible":false,"source_rank":3179,"rank":3179,"depth":1,"x":755.383,"y":743.96,"cluster":"advanced-algebra"},{"id":"stacks:0916","tag":"0916","title":"Derived categories of modules · Lemma 0916","summary":"Let R → S be an epimorphism of rings. Let I^bullet be a complex of S-modules. If I^bullet is K-injective as a complex of R-modules, then I^bullet is a K-injective complex of S-modules.","statement_latex":"Let $R \\to S$ be an epimorphism of rings. Let $I^\\bullet$ be a complex\nof $S$-modules. If $I^\\bullet$ is K-injective as a complex of\n$R$-modules, then $I^\\bullet$ is a K-injective complex of $S$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0916","source_file":"more-algebra.tex","source_line":13940,"source_end_line":13945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13940-L13945","statement_sha256":"b740027523148e081f359dc88ada2257e0954935ba20f2ee01728953f6c2f5be","origin":"The Stacks Project","memory_eligible":false,"source_rank":3180,"rank":3180,"depth":2,"x":572.832,"y":675.605,"cluster":"advanced-algebra"},{"id":"stacks:0917","tag":"0917","title":"Derived categories of modules · Lemma 0917","summary":"Let A → B be a ring map. If I^bullet is a K-injective complex of A-modules, then Hom_A(B, I^bullet) is a K-injective complex of B-modules.","statement_latex":"Let $A \\to B$ be a ring map. If $I^\\bullet$ is a K-injective complex of\n$A$-modules, then $\\Hom_A(B, I^\\bullet)$ is a K-injective complex of\n$B$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0917","source_file":"more-algebra.tex","source_line":13955,"source_end_line":13960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L13955-L13960","statement_sha256":"2911afb0491a9d78491f61700f4d86999870e2c6fd47805331f7595aa49b5d73","origin":"The Stacks Project","memory_eligible":false,"source_rank":3181,"rank":3181,"depth":1,"x":762.671,"y":622.353,"cluster":"advanced-algebra"},{"id":"stacks:0FNI","tag":"0FNI","title":"Tensor products of complexes · Lemma 0FNI","summary":"Let R be a ring. The category Comp(R) of complexes of R-modules endowed with the functor (L^bullet, M^bullet) ↦ Tot(L^bullet ⊗_R M^bullet) and associativity and commutativity constraints as above is a symmetric monoidal category.","statement_latex":"Let $R$ be a ring. The category $\\text{Comp}(R)$ of complexes of $R$-modules\nendowed with the functor\n$(L^\\bullet, M^\\bullet) \\mapsto \\text{Tot}(L^\\bullet \\otimes_R M^\\bullet)$\nand associativity and commutativity constraints as above is a symmetric\nmonoidal category.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tensor products of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNI","source_file":"more-algebra.tex","source_line":14082,"source_end_line":14089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14082-L14089","statement_sha256":"ff16882ee10771b62cd902ab611daae4b5761c3814c964bc7d520b043645e58e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3182,"rank":3182,"depth":1,"x":665.378,"y":769.539,"cluster":"advanced-algebra"},{"id":"stacks:064I","tag":"064I","title":"Tensor products of complexes · Lemma 064I","summary":"Let R be a ring. Let P^bullet be a complex of R-modules. Let α, β : L^bullet → M^bullet be homotopic maps of complexes. Then α and β induce homotopic maps Tot(α ⊗ id_P), Tot(β ⊗ id_P) : Tot(L^bullet ⊗_R P^bullet) → Tot(M^bullet ⊗_R P^bullet). In particular the construction L^bullet ↦ Tot(L^bullet ⊗_R P^bullet) defines an endo-functor of the homotopy category of complexes.","statement_latex":"Let $R$ be a ring. Let $P^\\bullet$ be a complex of $R$-modules.\nLet $\\alpha, \\beta : L^\\bullet \\to M^\\bullet$ be homotopic\nmaps of complexes. Then $\\alpha$ and $\\beta$ induce homotopic maps\n$$\n\\text{Tot}(\\alpha \\otimes \\text{id}_P),\n\\text{Tot}(\\beta \\otimes \\text{id}_P) :\n\\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)\n\\longrightarrow\n\\text{Tot}(M^\\bullet \\otimes_R P^\\bullet).\n$$\nIn particular the construction\n$L^\\bullet \\mapsto \\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)$\ndefines an endo-functor of the homotopy category of complexes.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tensor products of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064I","source_file":"more-algebra.tex","source_line":14103,"source_end_line":14118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14103-L14118","statement_sha256":"00afe09c28006284b7ddbdfcae11153577766f290f585363ef13a19d5f78ef95","origin":"The Stacks Project","memory_eligible":false,"source_rank":3183,"rank":3183,"depth":0,"x":618.694,"y":605.577,"cluster":"advanced-algebra"},{"id":"stacks:0GWP","tag":"0GWP","title":"Tensor products of complexes · Lemma 0GWP","summary":"Let R be a ring. The homotopy category K(R) of complexes of R-modules endowed with the functor (L^bullet, M^bullet) ↦ Tot(L^bullet ⊗_R M^bullet) and associativity and commutativity constraints as above is a symmetric monoidal category.","statement_latex":"Let $R$ be a ring. The homotopy category $K(R)$ of complexes of $R$-modules\nendowed with the functor\n$(L^\\bullet, M^\\bullet) \\mapsto \\text{Tot}(L^\\bullet \\otimes_R M^\\bullet)$\nand associativity and commutativity constraints as above is a symmetric\nmonoidal category.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tensor products of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWP","source_file":"more-algebra.tex","source_line":14138,"source_end_line":14145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14138-L14145","statement_sha256":"ed3eb2122d16e09e347ef5ce4891e34405b3fbec8431220c90f9799c47281429","origin":"The Stacks Project","memory_eligible":false,"source_rank":3184,"rank":3184,"depth":2,"x":785.197,"y":700.119,"cluster":"advanced-algebra"},{"id":"stacks:064J","tag":"064J","title":"Tensor products of complexes · Lemma 064J","summary":"Let R be a ring. Let P^bullet be a complex of R-modules. The functors K(R) → K(R), L^bullet ↦ Tot(P^bullet ⊗_R L^bullet) and K(R) → K(R), L^bullet ↦ Tot(L^bullet ⊗_R P^bullet) are exact functors of triangulated categories.","statement_latex":"Let $R$ be a ring. Let $P^\\bullet$ be a complex of $R$-modules.\nThe functors\n$$\nK(R) \\longrightarrow K(R), \\quad\nL^\\bullet \\longmapsto \\text{Tot}(P^\\bullet \\otimes_R L^\\bullet)\n$$\nand\n$$\nK(R) \\longrightarrow K(R), \\quad\nL^\\bullet \\longmapsto \\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)\n$$\nare exact functors of triangulated categories.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tensor products of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064J","source_file":"more-algebra.tex","source_line":14152,"source_end_line":14166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14152-L14166","statement_sha256":"71467659d7a49762181ab9b4b2e45ef7f8547bd7fdede189f145991341476097","origin":"The Stacks Project","memory_eligible":false,"source_rank":3185,"rank":3185,"depth":0,"x":586.124,"y":724.917,"cluster":"advanced-algebra"},{"id":"stacks:06XZ","tag":"06XZ","title":"Derived tensor product · Definition 06XZ","summary":"Let R be a ring. A complex K^bullet is called K-flat if for every acyclic complex M^bullet the total complex Tot(M^bullet ⊗_R K^bullet) is acyclic.","statement_latex":"Let $R$ be a ring. A complex $K^\\bullet$ is called {\\it K-flat}\nif for every acyclic complex $M^\\bullet$ the total complex\n$\\text{Tot}(M^\\bullet \\otimes_R K^\\bullet)$ is acyclic.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XZ","source_file":"more-algebra.tex","source_line":14184,"source_end_line":14189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14184-L14189","statement_sha256":"8ae256003e328625a6a86eb7affe3a1c55dfb1827ad0e6a49ae0a41f4cea6d15","origin":"The Stacks Project","memory_eligible":false,"source_rank":3186,"rank":3186,"depth":0,"x":713.146,"y":593.495,"cluster":"advanced-algebra"},{"id":"stacks:06Y0","tag":"06Y0","title":"Derived tensor product · Lemma 06Y0","summary":"Let R be a ring. Let K^bullet be a K-flat complex. Then the functor K(R) → K(R), L^bullet ↦ Tot(L^bullet ⊗_R K^bullet) transforms quasi-isomorphisms into quasi-isomorphisms.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a K-flat complex.\nThen the functor\n$$\nK(R) \\longrightarrow K(R), \\quad\nL^\\bullet \\longmapsto \\text{Tot}(L^\\bullet \\otimes_R K^\\bullet)\n$$\ntransforms quasi-isomorphisms into quasi-isomorphisms.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Y0","source_file":"more-algebra.tex","source_line":14191,"source_end_line":14200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14191-L14200","statement_sha256":"80fcd7d8f3b178b8df3d132c8cadd0eaf44c499be4853872c5d8f19ecce0b10d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3187,"rank":3187,"depth":1,"x":725.184,"y":762.706,"cluster":"advanced-algebra"},{"id":"stacks:06Y1","tag":"06Y1","title":"Derived tensor product · Lemma 06Y1","summary":"Let R → R' be a ring map. If K^bullet is a K-flat complex of R-modules, then K^bullet ⊗_R R' is a K-flat complex of R'-modules.","statement_latex":"Let $R \\to R'$ be a ring map. If $K^\\bullet$ is a K-flat complex\nof $R$-modules, then $K^\\bullet \\otimes_R R'$ is a K-flat complex\nof $R'$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Y1","source_file":"more-algebra.tex","source_line":14209,"source_end_line":14214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14209-L14214","statement_sha256":"f58e701e2f9f02618be66e165aaf4fd3169b335b6a395103e579ac738eaa65aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":3188,"rank":3188,"depth":0,"x":580.039,"y":644.605,"cluster":"advanced-algebra"},{"id":"stacks:0795","tag":"0795","title":"Derived tensor product · Lemma 0795","summary":"Let R be a ring. If K^bullet, L^bullet are K-flat complexes of R-modules, then Tot(K^bullet ⊗_R L^bullet) is a K-flat complex of R-modules.","statement_latex":"Let $R$ be a ring. If $K^\\bullet$, $L^\\bullet$ are K-flat complexes\nof $R$-modules, then $\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet)$ is a\nK-flat complex of $R$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0795","source_file":"more-algebra.tex","source_line":14223,"source_end_line":14228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14223-L14228","statement_sha256":"c9ec5772334cb89dcabc9f666276d20810763f51e20d27eee204b619ddfb0686","origin":"The Stacks Project","memory_eligible":false,"source_rank":3189,"rank":3189,"depth":0,"x":782.31,"y":649.339,"cluster":"advanced-algebra"},{"id":"stacks:06Y2","tag":"06Y2","title":"Derived tensor product · Lemma 06Y2","summary":"Let R be a ring. Let (K_1^bullet, K_2^bullet, K_3^bullet) be a distinguished triangle in K(R). If two out of three of K_i^bullet are K-flat, so is the third.","statement_latex":"Let $R$ be a ring. Let $(K_1^\\bullet, K_2^\\bullet, K_3^\\bullet)$ be\na distinguished triangle in $K(R)$. If two out of three\nof $K_i^\\bullet$ are K-flat, so is the third.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Y2","source_file":"more-algebra.tex","source_line":14240,"source_end_line":14245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14240-L14245","statement_sha256":"2ce6150ba0c345a488f4044e47182a745bf1f08e3529887a3578da8987a63711","origin":"The Stacks Project","memory_eligible":false,"source_rank":3190,"rank":3190,"depth":1,"x":629.147,"y":760.77,"cluster":"advanced-algebra"},{"id":"stacks:0BYH","tag":"0BYH","title":"Derived tensor product · Lemma 0BYH","summary":"Let R be a ring. Let 0 → K_1^bullet → K_2^bullet → K_3^bullet → 0 be a short exact sequence of complexes. If K_3^n is flat for all n ∈ Z and two out of three of K_i^bullet are K-flat, so is the third.","statement_latex":"Let $R$ be a ring. Let\n$0 \\to K_1^\\bullet \\to K_2^\\bullet \\to K_3^\\bullet \\to 0$ be\na short exact sequence of complexes. If $K_3^n$ is flat for\nall $n \\in \\mathbf{Z}$ and two out of three\nof $K_i^\\bullet$ are K-flat, so is the third.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYH","source_file":"more-algebra.tex","source_line":14254,"source_end_line":14261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14254-L14261","statement_sha256":"613da19758d0fe49bfb5acd427e2cbe4963a1e1cc44e6b3188bcc5a09435ccd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3191,"rank":3191,"depth":1,"x":652.509,"y":591.469,"cluster":"advanced-algebra"},{"id":"stacks:064K","tag":"064K","title":"Derived tensor product · Lemma 064K","summary":"Let R be a ring. Let P^bullet be a bounded above complex of flat R-modules. Then P^bullet is K-flat.","statement_latex":"Let $R$ be a ring. Let $P^\\bullet$ be a bounded above complex of\nflat $R$-modules. Then $P^\\bullet$ is K-flat.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064K","source_file":"more-algebra.tex","source_line":14283,"source_end_line":14287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14283-L14287","statement_sha256":"ccc6b03085ccdf4074e9245f11a2af1fb38af942bdc0ee50c7b2632c2929865c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3192,"rank":3192,"depth":5,"x":771.587,"y":729.748,"cluster":"advanced-algebra"},{"id":"stacks:06Y3","tag":"06Y3","title":"Derived tensor product · Lemma 06Y3","summary":"Let R be a ring. Let K_1^bullet → K_2^bullet → … be a system of K-flat complexes. Then colim_i K_i^bullet is K-flat. More generally any filtered colimit of K-flat complexes is K-flat.","statement_latex":"Let $R$ be a ring.\nLet $K_1^\\bullet \\to K_2^\\bullet \\to \\ldots$\nbe a system of K-flat complexes.\nThen $\\colim_i K_i^\\bullet$ is K-flat.\nMore generally any filtered colimit of K-flat complexes\nis K-flat.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Y3","source_file":"more-algebra.tex","source_line":14313,"source_end_line":14321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14313-L14321","statement_sha256":"cd18b455fadfae4514ffedf38f409dbab6254e9600b8ad3c02ac51b25348f6b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3193,"rank":3193,"depth":4,"x":572.318,"y":695.306,"cluster":"advanced-algebra"},{"id":"stacks:0E8F","tag":"0E8F","title":"Derived tensor product · Lemma 0E8F","summary":"Let R be a ring. Let K^bullet be a complex of R-modules. If K^bullet ⊗_R M is acyclic for all finitely presented R-modules M, then K^bullet is K-flat.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.\nIf $K^\\bullet \\otimes_R M$ is acyclic for all finitely presented\n$R$-modules $M$, then $K^\\bullet$ is K-flat.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8F","source_file":"more-algebra.tex","source_line":14334,"source_end_line":14339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14334-L14339","statement_sha256":"454a472e19c9c6c82bd70ca47a1bf6e891c2cb7d0b1bb7424d96901991b97c03","origin":"The Stacks Project","memory_eligible":false,"source_rank":3194,"rank":3194,"depth":4,"x":747.183,"y":607.517,"cluster":"advanced-algebra"},{"id":"stacks:06Y4","tag":"06Y4","title":"Derived tensor product · Lemma 06Y4","summary":"Let R be a ring. For any complex M^bullet there exists a K-flat complex K^bullet whose terms are flat R-modules and a quasi-isomorphism K^bullet → M^bullet which is termwise surjective.","statement_latex":"Let $R$ be a ring. For any complex $M^\\bullet$ there exists a\nK-flat complex $K^\\bullet$ whose terms are flat $R$-modules\nand a quasi-isomorphism $K^\\bullet \\to M^\\bullet$ which is termwise\nsurjective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Y4","source_file":"more-algebra.tex","source_line":14379,"source_end_line":14385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14379-L14385","statement_sha256":"2f0ea4ec3942b1e7ba58dd0a96d2497db9b125697773fe5857a6168c086463fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":3195,"rank":3195,"depth":14,"x":688.76,"y":771.689,"cluster":"advanced-algebra"},{"id":"stacks:064L","tag":"064L","title":"Derived tensor product · Lemma 064L","summary":"Let R be a ring. Let α : P^bullet → Q^bullet be a quasi-isomorphism of K-flat complexes of R-modules. For every complex L^bullet of R-modules the induced map Tot(id_L ⊗ α) : Tot(L^bullet ⊗_R P^bullet) → Tot(L^bullet ⊗_R Q^bullet) is a quasi-isomorphism.","statement_latex":"Let $R$ be a ring. Let\n$\\alpha : P^\\bullet \\to Q^\\bullet$ be a quasi-isomorphism of\nK-flat complexes of $R$-modules. For every complex $L^\\bullet$\nof $R$-modules the induced map\n$$\n\\text{Tot}(\\text{id}_L \\otimes \\alpha) :\n\\text{Tot}(L^\\bullet \\otimes_R P^\\bullet)\n\\longrightarrow\n\\text{Tot}(L^\\bullet \\otimes_R Q^\\bullet)\n$$\nis a quasi-isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064L","source_file":"more-algebra.tex","source_line":14456,"source_end_line":14469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14456-L14469","statement_sha256":"ea8bbf33ae50eb9f6159a3af194c3d9dcbb7797d5350fccd4dc4ecaceafa0e52","origin":"The Stacks Project","memory_eligible":false,"source_rank":3196,"rank":3196,"depth":15,"x":599.701,"y":617.278,"cluster":"advanced-algebra"},{"id":"stacks:064M","tag":"064M","title":"Derived tensor product · Definition 064M","summary":"Let R be a ring. Let M^bullet be an object of D(R). The derived tensor product - ⊗_R^L M^bullet : D(R) → D(R) is the exact functor of triangulated categories described above.","statement_latex":"Let $R$ be a ring. Let $M^\\bullet$ be an object of $D(R)$.\nThe {\\it derived tensor product}\n$$\n- \\otimes_R^{\\mathbf{L}} M^\\bullet : D(R) \\longrightarrow D(R)\n$$\nis the exact functor of triangulated categories described above.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064M","source_file":"more-algebra.tex","source_line":14511,"source_end_line":14519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14511-L14519","statement_sha256":"2d0a5ca643f66bafd3c398ee1b402664d23f0c08d53cf709c377a0b411ffcedf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3197,"rank":3197,"depth":0,"x":789.793,"y":680.687,"cluster":"advanced-algebra"},{"id":"stacks:0BYI","tag":"0BYI","title":"Derived tensor product · Lemma 0BYI","summary":"Let R be a ring. Let K^bullet, L^bullet be complexes of R-modules. There is a canonical isomorphism K^bullet ⊗_R^L L^bullet → L^bullet ⊗_R^L K^bullet functorial in both complexes which uses a sign of (-1)^pq for the map K^p ⊗_R L^q → L^q ⊗_R K^p (see proof for explanation).","statement_latex":"Let $R$ be a ring. Let $K^\\bullet, L^\\bullet$ be complexes of $R$-modules.\nThere is a canonical isomorphism\n$$\nK^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet \\longrightarrow\nL^\\bullet \\otimes_R^\\mathbf{L} K^\\bullet\n$$\nfunctorial in both complexes which uses a sign of $(-1)^{pq}$\nfor the map $K^p \\otimes_R L^q \\to L^q \\otimes_R K^p$ (see proof\nfor explanation).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYI","source_file":"more-algebra.tex","source_line":14535,"source_end_line":14546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14535-L14546","statement_sha256":"3b4eaa20ee1f6376eb450125e18e5e69b65375aaa1709702e21641f906f15bc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3198,"rank":3198,"depth":0,"x":598.381,"y":741.873,"cluster":"advanced-algebra"},{"id":"stacks:0BYJ","tag":"0BYJ","title":"Derived tensor product · Lemma 0BYJ","summary":"Let R be a ring. Let K^bullet, L^bullet, M^bullet be complexes of R-modules. There is a canonical isomorphism (K^bullet ⊗_R^L L^bullet) ⊗_R^L M^bullet = K^bullet ⊗_R^L (L^bullet ⊗_R^L M^bullet) functorial in all three complexes.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet, L^\\bullet, M^\\bullet$\nbe complexes of $R$-modules. There is a canonical isomorphism\n$$\n(K^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet) \\otimes_R^\\mathbf{L} M^\\bullet\n=\nK^\\bullet \\otimes_R^\\mathbf{L} (L^\\bullet \\otimes_R^\\mathbf{L} M^\\bullet)\n$$\nfunctorial in all three complexes.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYJ","source_file":"more-algebra.tex","source_line":14554,"source_end_line":14564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14554-L14564","statement_sha256":"706099a92f821e9eb281baec834c8b47e319e5394b8e3110bd749040d669edf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3199,"rank":3199,"depth":0,"x":690.442,"y":587.944,"cluster":"advanced-algebra"},{"id":"stacks:0G6M","tag":"0G6M","title":"Derived tensor product · Lemma 0G6M","summary":"Let R be a ring. Let a : K^bullet → L^bullet be a map of complexes of R-modules. If K^bullet is K-flat, then there exist a complex N^bullet and maps of complexes b : K^bullet → N^bullet and c : N^bullet → L^bullet such that • N^bullet is K-flat, • c is a quasi-isomorphism, • a is homotopic to c ∘ b. If the terms of K^bullet are flat, then we may choose N^bullet, b, and c such that the same is true for N^bullet.","statement_latex":"Let $R$ be a ring. Let $a : K^\\bullet \\to L^\\bullet$ be a map of complexes\nof $R$-modules. If $K^\\bullet$ is K-flat, then there exist a complex\n$N^\\bullet$ and maps of complexes $b : K^\\bullet \\to N^\\bullet$\nand $c : N^\\bullet \\to L^\\bullet$ such that\n\\begin{enumerate}\n\\item $N^\\bullet$ is K-flat,\n\\item $c$ is a quasi-isomorphism,\n\\item $a$ is homotopic to $c \\circ b$.\n\\end{enumerate}\nIf the terms of $K^\\bullet$ are flat, then we may choose\n$N^\\bullet$, $b$, and $c$\nsuch that the same is true for $N^\\bullet$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6M","source_file":"more-algebra.tex","source_line":14571,"source_end_line":14585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14571-L14585","statement_sha256":"89fc5829bd877c09baf29096588d81e13195cc351b82f661493829d9def829bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3200,"rank":3200,"depth":15,"x":746.415,"y":753.901,"cluster":"advanced-algebra"},{"id":"stacks:06Y6","tag":"06Y6","title":"Derived change of rings · Lemma 06Y6","summary":"The construction above is independent of choices and defines an exact functor of triangulated categories - ⊗_R^L N^bullet : D(R) → D(A). There is a functorial isomorphism E^bullet ⊗_R^L N^bullet = (E^bullet ⊗_R^L A) ⊗_A^L N^bullet for E^bullet in D(R).","statement_latex":"The construction above is independent of choices and defines an exact\nfunctor of triangulated categories\n$- \\otimes_R^\\mathbf{L} N^\\bullet : D(R) \\to D(A)$.\nThere is a functorial isomorphism\n$$\nE^\\bullet \\otimes_R^\\mathbf{L} N^\\bullet =\n(E^\\bullet \\otimes_R^\\mathbf{L} A) \\otimes_A^\\mathbf{L} N^\\bullet\n$$\nfor $E^\\bullet$ in $D(R)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived change of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Y6","source_file":"more-algebra.tex","source_line":14665,"source_end_line":14676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14665-L14676","statement_sha256":"bb31f98dd2a59a689822970b9128db0410b30d2d9859a149d379962721229962","origin":"The Stacks Project","memory_eligible":false,"source_rank":3201,"rank":3201,"depth":16,"x":571.457,"y":663.17,"cluster":"advanced-algebra"},{"id":"stacks:0BYK","tag":"0BYK","title":"Derived change of rings · Lemma 0BYK","summary":"Let R → A be a ring map. Let f : L^bullet → N^bullet be a map of complexes of A-modules. Then f induces a transformation of functors 1 ⊗ f : - ⊗_A^L L^bullet → - ⊗_A^L N^bullet If f is a quasi-isomorphism, then 1 ⊗ f is an isomorphism of functors.","statement_latex":"Let $R \\to A$ be a ring map. Let $f : L^\\bullet \\to N^\\bullet$ be a\nmap of complexes of $A$-modules. Then $f$ induces a transformation\nof functors\n$$\n1 \\otimes f :\n- \\otimes_A^\\mathbf{L} L^\\bullet\n\\longrightarrow\n- \\otimes_A^\\mathbf{L} N^\\bullet\n$$\nIf $f$ is a quasi-isomorphism, then $1 \\otimes f$ is an isomorphism\nof functors.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived change of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYK","source_file":"more-algebra.tex","source_line":14727,"source_end_line":14740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14727-L14740","statement_sha256":"c1963aac8a89d55923e264a5236067e8eb53d6c62eae9e0b7e98f45d0e30bfba","origin":"The Stacks Project","memory_eligible":false,"source_rank":3202,"rank":3202,"depth":2,"x":773.692,"y":630.757,"cluster":"advanced-algebra"},{"id":"stacks:0GMT","tag":"0GMT","title":"Derived change of rings · Lemma 0GMT","summary":"Let R → A be a ring map. The functor D(R) → D(A), E ↦ E ⊗_R^L A of Lemma [Tag 06Y6] is left adjoint to the restriction functor D(A) → D(R).","statement_latex":"Let $R \\to A$ be a ring map. The functor\n$D(R) \\to D(A)$, $E \\mapsto E \\otimes_R^\\mathbf{L} A$\nof Lemma \\ref{lemma-derived-base-change}\nis left adjoint to the restriction functor\n$D(A) \\to D(R)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived change of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMT","source_file":"more-algebra.tex","source_line":14753,"source_end_line":14760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14753-L14760","statement_sha256":"694e02f5fb9c0b06313dd58c54aa08edfcfc5b6516446f21289dbfd9b2b1b402","origin":"The Stacks Project","memory_eligible":false,"source_rank":3203,"rank":3203,"depth":17,"x":650.475,"y":769.589,"cluster":"advanced-algebra"},{"id":"stacks:08YU","tag":"08YU","title":"Derived change of rings · Lemma 08YU","summary":"Let A → B → C be ring maps. Let N^bullet be a complex of B-modules and K^bullet a complex of C-modules. The compositions of the functors D(A) xrightarrow- ⊗_A^L N^bullet D(B) xrightarrow- ⊗_B^L K^bullet D(C) is the functor - ⊗_A^L (N^bullet ⊗_B^L K^bullet) : D(A) → D(C). If M, N, K are modules over A, B, C, then we have (M ⊗_A^L N) ⊗_B^L K = M ⊗_A^L (N ⊗_B^L K) = (M ⊗_A^L C) ⊗_C^L (N ⊗_B^L K) in D(C). We also have a canonical isomorphism (M ⊗_A^L N) ⊗_B^L K → (M ⊗_A^L K)…","statement_latex":"Let $A \\to B \\to C$ be ring maps. Let $N^\\bullet$ be a complex of\n$B$-modules and $K^\\bullet$ a complex of $C$-modules.\nThe compositions of the functors\n$$\nD(A) \\xrightarrow{- \\otimes_A^\\mathbf{L} N^\\bullet}\nD(B) \\xrightarrow{- \\otimes_B^\\mathbf{L} K^\\bullet} D(C)\n$$\nis the functor\n$- \\otimes_A^\\mathbf{L} (N^\\bullet \\otimes_B^\\mathbf{L} K^\\bullet) :\nD(A) \\to D(C)$. If $M$, $N$, $K$ are modules over $A$, $B$, $C$, then\nwe have\n$$\n(M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} K =\nM \\otimes_A^\\mathbf{L} (N \\otimes_B^\\mathbf{L} K) =\n(M \\otimes_A^\\mathbf{L} C) \\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} K)\n$$\nin $D(C)$. We also have a canonical isomorphism\n$$\n(M \\otimes_A^\\mathbf{L} N) \\otimes_B^\\mathbf{L} K \\longrightarrow\n(M \\otimes_A^\\mathbf{L} K) \\otimes_C^\\mathbf{L} (N \\otimes_B^\\mathbf{L} C)\n$$\nusing signs. Similar results holds for complexes.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived change of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YU","source_file":"more-algebra.tex","source_line":14795,"source_end_line":14819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14795-L14819","statement_sha256":"3bcf38dda8a2a2687dfa4f22d34c1703fab055349fd26564cb1e77f49b03a89f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3204,"rank":3204,"depth":17,"x":629.661,"y":597.079,"cluster":"advanced-algebra"},{"id":"stacks:0660","tag":"0660","title":"Tor independence · Definition 0660","summary":"Let R be a ring. Let A, B be R-algebras. We say A and B are Tor independent over R if Tor_p^R(A, B) = 0 for all p > 0.","statement_latex":"Let $R$ be a ring. Let $A$, $B$ be $R$-algebras. We say\n$A$ and $B$ are {\\it Tor independent over $R$} if\n$\\text{Tor}_p^R(A, B) = 0$ for all $p > 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor independence","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0660","source_file":"more-algebra.tex","source_line":14947,"source_end_line":14952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14947-L14952","statement_sha256":"826bc5520a9c2a9be070d2da530755b21495071793cea3e902cdc8fce9c14354","origin":"The Stacks Project","memory_eligible":false,"source_rank":3205,"rank":3205,"depth":0,"x":783.934,"y":712.623,"cluster":"advanced-algebra"},{"id":"stacks:0661","tag":"0661","title":"Tor independence · Lemma 0661","summary":"The comparison map ([Tag 065Z]) is an isomorphism if A' = A ⊗_R R' and A and R' are Tor independent over R.","statement_latex":"The comparison map (\\ref{equation-comparison-map}) is an isomorphism\nif $A' = A \\otimes_R R'$ and $A$ and $R'$ are Tor independent over $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor independence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0661","source_file":"more-algebra.tex","source_line":14954,"source_end_line":14958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14954-L14958","statement_sha256":"30e7f25f817141c2c9638dfc7256b526df48ded9ee2794891deb73c27840c69c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3206,"rank":3206,"depth":0,"x":576.995,"y":714.964,"cluster":"advanced-algebra"},{"id":"stacks:08HW","tag":"08HW","title":"Tor independence · Lemma 08HW","summary":"Consider a commutative diagram of rings xymatrix A' & R' ar[r] ar[l] & B' A ar[u] & R ar[l] ar[u] ar[r] & B ar[u] Assume that R' is flat over R and A' is flat over A ⊗_R R' and B' is flat over R' ⊗_R B. Then Tor_i^R(A, B) ⊗_(A ⊗_R B) (A' ⊗_R' B') = Tor_i^R'(A', B')","statement_latex":"Consider a commutative diagram of rings\n$$\n\\xymatrix{\nA' & R' \\ar[r] \\ar[l] & B' \\\\\nA \\ar[u] & R \\ar[l] \\ar[u] \\ar[r] & B \\ar[u]\n}\n$$\nAssume that $R'$ is flat over $R$ and $A'$ is flat over $A \\otimes_R R'$\nand $B'$ is flat over $R' \\otimes_R B$. Then\n$$\n\\text{Tor}_i^R(A, B) \\otimes_{(A \\otimes_R B)} (A' \\otimes_{R'} B') =\n\\text{Tor}_i^{R'}(A', B')\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor independence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HW","source_file":"more-algebra.tex","source_line":14976,"source_end_line":14991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L14976-L14991","statement_sha256":"e84bb0de5082b6a324755d02613ef754ca147df9f01d4b1c62474375c44e5d36","origin":"The Stacks Project","memory_eligible":false,"source_rank":3207,"rank":3207,"depth":4,"x":727.899,"y":595.664,"cluster":"advanced-algebra"},{"id":"stacks:0FXF","tag":"0FXF","title":"Tor independence · Lemma 0FXF","summary":"Let R → A and R → B be ring maps. Let R → R' be a ring map and set A' = A ⊗_R R' and B' = B ⊗_R R'. If A and B are tor independent over R and R → R' is flat, then A' and B' are tor independent over R'.","statement_latex":"Let $R \\to A$ and $R \\to B$ be ring maps. Let $R \\to R'$ be a\nring map and set $A' = A \\otimes_R R'$ and $B' = B \\otimes_R R'$.\nIf $A$ and $B$ are tor independent over $R$ and $R \\to R'$ is flat,\nthen $A'$ and $B'$ are tor independent over $R'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor independence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXF","source_file":"more-algebra.tex","source_line":15025,"source_end_line":15031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15025-L15031","statement_sha256":"6120f460f647482e91c00af324c954c653bc212322f674561eca8fba431eddf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3208,"rank":3208,"depth":5,"x":712.542,"y":769.481,"cluster":"advanced-algebra"},{"id":"stacks:0DJD","tag":"0DJD","title":"Tor independence · Lemma 0DJD","summary":"Assumptions as in Lemma [Tag 08HW]. For M ∈ D(A) there are canonical isomorphisms H^i((M ⊗_A^L A') ⊗_R'^L B') = H^i(M ⊗_R^L B) ⊗_(A ⊗_R B) (A' ⊗_R' B') of A' ⊗_R' B'-modules.","statement_latex":"Assumptions as in Lemma \\ref{lemma-tor-independent-flat}.\nFor $M \\in D(A)$ there are canonical isomorphisms\n$$\nH^i((M \\otimes_A^\\mathbf{L} A') \\otimes_{R'}^\\mathbf{L} B') =\nH^i(M \\otimes_R^\\mathbf{L} B) \\otimes_{(A \\otimes_R B)} (A' \\otimes_{R'} B')\n$$\nof $A' \\otimes_{R'} B'$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor independence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJD","source_file":"more-algebra.tex","source_line":15038,"source_end_line":15047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15038-L15047","statement_sha256":"71b726344e799ee68726bb9d1bb3f8829df7f66f72b31e3f1c1fc5ba392bec9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3209,"rank":3209,"depth":5,"x":583.925,"y":632.424,"cluster":"advanced-algebra"},{"id":"stacks:08HX","tag":"08HX","title":"Tor independence · Lemma 08HX","summary":"Let R be a ring. Let A, B be R-algebras. The following are equivalent • A and B are Tor independent over R, • for every pair of primes p ⊂ A and q ⊂ B lying over the same prime r ⊂ R the rings A_ p and B_ q are Tor independent over R_ r, and • For every prime s of A ⊗_R B the module Tor_i^R(A, B)_ s = Tor_i^R_ r(A_ p, B_ q)_ s (where p = A ∩ s, q = B ∩ s and r = R ∩ s) is zero.","statement_latex":"Let $R$ be a ring. Let $A$, $B$ be $R$-algebras. The following are equivalent\n\\begin{enumerate}\n\\item $A$ and $B$ are Tor independent over $R$,\n\\item for every pair of primes $\\mathfrak p \\subset A$ and\n$\\mathfrak q \\subset B$ lying over the same prime $\\mathfrak r \\subset R$\nthe rings $A_\\mathfrak p$ and $B_\\mathfrak q$ are Tor independent over\n$R_\\mathfrak r$, and\n\\item For every prime $\\mathfrak s$ of $A \\otimes_R B$ the module\n$$\n\\text{Tor}_i^R(A, B)_\\mathfrak s =\n\\text{Tor}_i^{R_\\mathfrak r}(A_\\mathfrak p, B_\\mathfrak q)_\\mathfrak s\n$$\n(where $\\mathfrak p = A \\cap \\mathfrak s$, $\\mathfrak q = B \\cap \\mathfrak s$\nand $\\mathfrak r = R \\cap \\mathfrak s$) is zero.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor independence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HX","source_file":"more-algebra.tex","source_line":15101,"source_end_line":15118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15101-L15118","statement_sha256":"012d105215873c1036cf4594690feb536191b7fb380d22f861f9067aec29fab5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3210,"rank":3210,"depth":5,"x":789.242,"y":660.542,"cluster":"advanced-algebra"},{"id":"stacks:068K","tag":"068K","title":"Products and Tor · Lemma 068K","summary":"Let R be a ring. Let A, B, C be R-algebras and let B → C be an R-algebra map. Then the induced map Tor^R_star(B, A) → Tor^R_star(C, A) is an A-algebra homomorphism.","statement_latex":"Let $R$ be a ring. Let $A, B, C$ be $R$-algebras and let $B \\to C$ be an\n$R$-algebra map. Then the induced map\n$$\n\\text{Tor}^R_{\\star}(B, A)\n\\longrightarrow\n\\text{Tor}^R_{\\star}(C, A)\n$$\nis an $A$-algebra homomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Products and Tor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068K","source_file":"more-algebra.tex","source_line":15402,"source_end_line":15412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15402-L15412","statement_sha256":"f24fa4711e00d72e7d3d096e04cb3c11c620ffc533ed9d5985616b344ca9f828","origin":"The Stacks Project","memory_eligible":false,"source_rank":3211,"rank":3211,"depth":0,"x":615.013,"y":756.43,"cluster":"advanced-algebra"},{"id":"stacks:0H7W","tag":"0H7W","title":"K\\\"unneth spectral sequence · Lemma 0H7W","summary":"Under the assumptions above, if in addition • the filtration on K^bullet is finite and the filtration on L^bullet is finite, or • more generally the following are true • F^iK^bullet is acyclic for i gg 0, • F^iK^bullet → K^bullet is a quasi-isomorphism for i ll 0, • F^jL^bullet is acyclic for j gg 0, and • F^jL^bullet → L^bullet is a quasi-isomorphism for j ll 0. Then the spectral sequence is bounded, the associated filtration on each H^n(T^bullet) = H^n(K^bullet ⊗_R^L…","statement_latex":"Under the assumptions above, if in addition\n\\begin{enumerate}\n\\item the filtration on $K^\\bullet$ is finite and the filtration on\n$L^\\bullet$ is finite, or\n\\item more generally the following are true\n\\begin{enumerate}\n\\item $F^iK^\\bullet$ is acyclic for $i \\gg 0$,\n\\item $F^iK^\\bullet \\to K^\\bullet$ is a quasi-isomorphism for $i \\ll 0$,\n\\item $F^jL^\\bullet$ is acyclic for $j \\gg 0$, and\n\\item $F^jL^\\bullet \\to L^\\bullet$ is a quasi-isomorphism for $j \\ll 0$.\n\\end{enumerate}\n\\end{enumerate}\nThen the spectral sequence is bounded, the associated filtration on each\n$H^n(T^\\bullet) = H^n(K^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet)$ is finite\nand we have convergence\n$$\nE_1^{p, q} =\n\\bigoplus\\nolimits_{i + j = p}\nH^{p + q}(\\text{gr}^iK^\\bullet \\otimes_R^\\mathbf{L} \\text{gr}^jL^\\bullet)\n\\Rightarrow\nH^n(K^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"K\\\"unneth spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7W","source_file":"more-algebra.tex","source_line":15517,"source_end_line":15541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15517-L15541","statement_sha256":"2ed26161ae3b659eb6f03f68cf69a693375b1a94ec8ab817bb1dfc9e3083b74b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3212,"rank":3212,"depth":4,"x":666.44,"y":586.649,"cluster":"advanced-algebra"},{"id":"stacks:0H7X","tag":"0H7X","title":"K\\\"unneth spectral sequence · Lemma 0H7X","summary":"Let R be a ring. Let K^bullet be a filtered complex. There exists a map f : P^bullet → K^bullet of filtered complexes such that • each P^n, F^iP^n, gr^iP^n is a free R-module, • the complexes of R-modules P^bullet, F^iP^bullet, and gr^iP^bullet are K-flat, • f induces quasi-isomorphisms P^bullet → K^bullet, F^iP^bullet → F^iK^bullet, and gr^iP^bullet → gr^iK^bullet.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a filtered complex. There exists a\nmap $f : P^\\bullet \\to K^\\bullet$ of filtered complexes such that\n\\begin{enumerate}\n\\item each $P^n$, $F^iP^n$, $\\text{gr}^iP^n$ is a free $R$-module,\n\\item the complexes of $R$-modules $P^\\bullet$, $F^iP^\\bullet$, and\n$\\text{gr}^iP^\\bullet$ are K-flat,\n\\item $f$ induces quasi-isomorphisms\n$P^\\bullet \\to K^\\bullet$,\n$F^iP^\\bullet \\to F^iK^\\bullet$, and\n$\\text{gr}^iP^\\bullet \\to \\text{gr}^iK^\\bullet$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"K\\\"unneth spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7X","source_file":"more-algebra.tex","source_line":15645,"source_end_line":15658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15645-L15658","statement_sha256":"d28c42ce11ae93865e9ea4b1262d4bd390595cd48262577b731d039822e472cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3213,"rank":3213,"depth":5,"x":765.175,"y":741.218,"cluster":"advanced-algebra"},{"id":"stacks:0H7Y","tag":"0H7Y","title":"K\\\"unneth spectral sequence · Proposition 0H7Y","summary":"Let R be a ring. Let K^bullet and L^bullet be filtered complexes of R-modules. Then there exists a filtered complex T^bullet representing K^bullet ⊗_R^L L^bullet in D(R) such that the associated spectral sequence has E_1-page E_1^p, q = bigoplus_i + j = p H^p + q(gr^iK^bullet ⊗_R^L gr^jL^bullet) If • F^iK^bullet is acyclic for i gg 0, • F^iK^bullet → K^bullet is a quasi-isomorphism for i ll 0, • F^jL^bullet is acyclic for j gg 0, and • F^jL^bullet → L^bullet is a…","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ and $L^\\bullet$ be filtered complexes\nof $R$-modules. Then there exists a filtered complex $T^\\bullet$\nrepresenting $K^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet$ in $D(R)$\nsuch that the associated spectral sequence has $E_1$-page\n$$\nE_1^{p, q} =\n\\bigoplus\\nolimits_{i + j = p}\nH^{p + q}(\\text{gr}^iK^\\bullet \\otimes_R^\\mathbf{L} \\text{gr}^jL^\\bullet)\n$$\nIf\n\\begin{enumerate}\n\\item $F^iK^\\bullet$ is acyclic for $i \\gg 0$,\n\\item $F^iK^\\bullet \\to K^\\bullet$ is a quasi-isomorphism for $i \\ll 0$,\n\\item $F^jL^\\bullet$ is acyclic for $j \\gg 0$, and\n\\item $F^jL^\\bullet \\to L^\\bullet$ is a quasi-isomorphism for $j \\ll 0$.\n\\end{enumerate}\nthen the spectral sequence is bounded, the associated filtration on each\n$H^n(K^\\bullet \\otimes_R^\\mathbf{L} L^\\bullet)$ is finite\nand the spectral sequence convergences.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"K\\\"unneth spectral sequence","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7Y","source_file":"more-algebra.tex","source_line":15766,"source_end_line":15787,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15766-L15787","statement_sha256":"6d32dd88208a0c5a3bf1057fe7aa79c792f033cff112150f8eddce9c0ef85114","origin":"The Stacks Project","memory_eligible":false,"source_rank":3214,"rank":3214,"depth":6,"x":567.823,"y":683.193,"cluster":"advanced-algebra"},{"id":"stacks:0H7Z","tag":"0H7Z","title":"K\\\"unneth Spectral Sequence · Lemma 0H7Z","summary":"Let R be a ring. Let K and L be objects of D^b(R). There exists a bigraded bounded spectral sequence (E_r)_r ≥ 2 with E_2^p, q = bigoplus_i + j = q Tor^R_-p(H^i(K), H^j(L)) and d_r of bidegree (r, -r + 1) converging to H^p + q(K ⊗_R^L L).","statement_latex":"Let $R$ be a ring. Let $K$ and $L$ be objects of $D^b(R)$.\nThere exists a bigraded bounded spectral sequence $\\{E_r\\}_{r \\geq 2}$\nwith\n$$\nE_2^{p, q} = \\bigoplus\\nolimits_{i + j = q} \\text{Tor}^R_{-p}(H^i(K), H^j(L))\n$$\nand $d_r$ of bidegree $(r, -r + 1)$\nconverging to $H^{p + q}(K \\otimes_R^\\mathbf{L} L)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"K\\\"unneth spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7Z","source_file":"more-algebra.tex","source_line":15800,"source_end_line":15810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15800-L15810","statement_sha256":"4c39e45c6b6ff68c0dadf5de61a870c31104be57f9e25da30c5f5644cef63fe4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3215,"rank":3215,"depth":7,"x":760.249,"y":613.911,"cluster":"advanced-algebra"},{"id":"stacks:064Q","tag":"064Q","title":"Pseudo-coherent modules, I · Definition 064Q","summary":"Let R be a ring. Denote D(R) its derived category. Let m ∈ Z. • An object K^bullet of D(R) is m-pseudo-coherent if there exists a bounded complex E^bullet of finite free R-modules and a morphism α : E^bullet → K^bullet such that H^i(α) is an isomorphism for i > m and H^m(α) is surjective. • An object K^bullet of D(R) is pseudo-coherent if it is quasi-isomorphic to a bounded above complex of finite free R-modules. • An R-module M is called m-pseudo-coherent if M[0] is an…","statement_latex":"Let $R$ be a ring. Denote $D(R)$ its derived category.\nLet $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item An object $K^\\bullet$ of $D(R)$ is {\\it $m$-pseudo-coherent}\nif there exists a bounded complex $E^\\bullet$ of finite free $R$-modules\nand a morphism $\\alpha : E^\\bullet \\to K^\\bullet$ such that\n$H^i(\\alpha)$ is an isomorphism for $i > m$ and $H^m(\\alpha)$\nis surjective.\n\\item An object $K^\\bullet$ of $D(R)$ is {\\it pseudo-coherent}\nif it is quasi-isomorphic to a bounded above complex of finite\nfree $R$-modules.\n\\item An $R$-module $M$ is called {\\it $m$-pseudo-coherent}\nif $M[0]$ is an $m$-pseudo-coherent object of $D(R)$.\n\\item An $R$-module $M$ is called\n{\\it pseudo-coherent}\\footnote{This clashes with what is meant by\na pseudo-coherent module in \\cite{Bourbaki-CA}.}\nif $M[0]$ is a pseudo-coherent object of $D(R)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064Q","source_file":"more-algebra.tex","source_line":15873,"source_end_line":15893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15873-L15893","statement_sha256":"04a29b874f9c0ff9ed2a8ec5bac4a71a6a9c50c36a018e3248f5fde1d1cb583c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3216,"rank":3216,"depth":0,"x":673.965,"y":774.385,"cluster":"advanced-algebra"},{"id":"stacks:064R","tag":"064R","title":"Pseudo-coherent modules, I · Lemma 064R","summary":"Let R be a ring and m ∈ Z. Let (K^bullet, L^bullet, M^bullet, f, g, h) be a distinguished triangle in D(R). • If K^bullet is (m + 1)-pseudo-coherent and L^bullet is m-pseudo-coherent then M^bullet is m-pseudo-coherent. • If K^bullet, M^bullet are m-pseudo-coherent, then L^bullet is m-pseudo-coherent. • If L^bullet is (m + 1)-pseudo-coherent and M^bullet is m-pseudo-coherent, then K^bullet is (m + 1)-pseudo-coherent.","statement_latex":"Let $R$ be a ring and $m \\in \\mathbf{Z}$.\nLet $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$ be a distinguished\ntriangle in $D(R)$.\n\\begin{enumerate}\n\\item If $K^\\bullet$ is $(m + 1)$-pseudo-coherent and\n$L^\\bullet$ is $m$-pseudo-coherent then $M^\\bullet$ is\n$m$-pseudo-coherent.\n\\item If $K^\\bullet, M^\\bullet$ are $m$-pseudo-coherent, then\n$L^\\bullet$ is $m$-pseudo-coherent.\n\\item If $L^\\bullet$ is $(m + 1)$-pseudo-coherent and $M^\\bullet$\nis $m$-pseudo-coherent, then $K^\\bullet$ is $(m + 1)$-pseudo-coherent.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064R","source_file":"more-algebra.tex","source_line":15909,"source_end_line":15923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15909-L15923","statement_sha256":"1098bd664bf72b9fdc1968cc14d67849671e3faf6d659191a3533b5a8d253f21","origin":"The Stacks Project","memory_eligible":false,"source_rank":3217,"rank":3217,"depth":8,"x":608.46,"y":606.887,"cluster":"advanced-algebra"},{"id":"stacks:064S","tag":"064S","title":"Pseudo-coherent modules, I · Lemma 064S","summary":"Let R be a ring. Let K^bullet be a complex of R-modules. Let m ∈ Z. • If K^bullet is m-pseudo-coherent and H^i(K^bullet) = 0 for i > m, then H^m(K^bullet) is a finite type R-module. • If K^bullet is m-pseudo-coherent and H^i(K^bullet) = 0 for i > m + 1, then H^m + 1(K^bullet) is a finitely presented R-module.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.\nLet $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $K^\\bullet$ is $m$-pseudo-coherent and $H^i(K^\\bullet) = 0$\nfor $i > m$, then $H^m(K^\\bullet)$ is a finite type $R$-module.\n\\item If $K^\\bullet$ is $m$-pseudo-coherent and $H^i(K^\\bullet) = 0$\nfor $i > m + 1$, then $H^{m + 1}(K^\\bullet)$ is a finitely presented\n$R$-module.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064S","source_file":"more-algebra.tex","source_line":15964,"source_end_line":15975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L15964-L15975","statement_sha256":"9c8953c3e2c3108b0ca2f4b2b6eb74370dd5c6c428bf6ef24a22e0287c597568","origin":"The Stacks Project","memory_eligible":false,"source_rank":3218,"rank":3218,"depth":0,"x":791.686,"y":693.335,"cluster":"advanced-algebra"},{"id":"stacks:064T","tag":"064T","title":"Pseudo-coherent modules, I · Lemma 064T","summary":"Let R be a ring. Let M be an R-module. Then • M is 0-pseudo-coherent if and only if M is a finite R-module, • M is (-1)-pseudo-coherent if and only if M is a finitely presented R-module, • M is (-d)-pseudo-coherent if and only if there exists a resolution R^⊕ a_d → R^⊕ a_d - 1 → … → R^⊕ a_0 → M → 0 of length d, and • M is pseudo-coherent if and only if there exists an infinite resolution … → R^⊕ a_1 → R^⊕ a_0 → M → 0 by finite free R-modules.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nThen\n\\begin{enumerate}\n\\item $M$ is $0$-pseudo-coherent if and only if $M$ is a finite\n$R$-module,\n\\item $M$ is $(-1)$-pseudo-coherent if and only if $M$ is a finitely\npresented $R$-module,\n\\item $M$ is $(-d)$-pseudo-coherent if and only if there exists a\nresolution\n$$\nR^{\\oplus a_d} \\to R^{\\oplus a_{d - 1}} \\to \\ldots \\to R^{\\oplus a_0} \\to\nM \\to 0\n$$\nof length $d$, and\n\\item $M$ is pseudo-coherent if and only if there exists an\ninfinite resolution\n$$\n\\ldots \\to R^{\\oplus a_1} \\to R^{\\oplus a_0} \\to M \\to 0\n$$\nby finite free $R$-modules.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064T","source_file":"more-algebra.tex","source_line":16002,"source_end_line":16025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16002-L16025","statement_sha256":"3de7cd2eea0edd24b9c6014506be1ab72be314fa650823fe5ae27421ed4287a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3219,"rank":3219,"depth":9,"x":586.806,"y":733.606,"cluster":"advanced-algebra"},{"id":"stacks:064U","tag":"064U","title":"Pseudo-coherent modules, I · Lemma 064U","summary":"Let R be a ring. Let K^bullet be a complex of R-modules. The following are equivalent • K^bullet is pseudo-coherent, • K^bullet is m-pseudo-coherent for every m ∈ Z, and • K^bullet is quasi-isomorphic to a bounded above complex of finite projective R-modules. If (1), (2), and (3) hold and H^i(K^bullet) = 0 for i > b, then we can find a quasi-isomorphism F^bullet → K^bullet with F^i finite free R-modules and F^i = 0 for i > b.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K^\\bullet$ is pseudo-coherent,\n\\item $K^\\bullet$ is $m$-pseudo-coherent for every $m \\in \\mathbf{Z}$, and\n\\item $K^\\bullet$ is quasi-isomorphic to a bounded above complex of finite\nprojective $R$-modules.\n\\end{enumerate}\nIf (1), (2), and (3) hold and $H^i(K^\\bullet) = 0$ for $i > b$, then\nwe can find a quasi-isomorphism $F^\\bullet \\to K^\\bullet$ with\n$F^i$ finite free $R$-modules and $F^i = 0$ for $i > b$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064U","source_file":"more-algebra.tex","source_line":16047,"source_end_line":16060,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16047-L16060","statement_sha256":"6b414b064d889c3ba00ba68526d5e638a4b04fe7d2c631cfa384f6073b852598","origin":"The Stacks Project","memory_eligible":false,"source_rank":3220,"rank":3220,"depth":9,"x":705.643,"y":587.477,"cluster":"advanced-algebra"},{"id":"stacks:064V","tag":"064V","title":"Pseudo-coherent modules, I · Lemma 064V","summary":"Let R be a ring. Let (K^bullet, L^bullet, M^bullet, f, g, h) be a distinguished triangle in D(R). If two out of three of K^bullet, L^bullet, M^bullet are pseudo-coherent then the third is also pseudo-coherent.","statement_latex":"Let $R$ be a ring. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$\nbe a distinguished triangle in $D(R)$. If two out of three of\n$K^\\bullet, L^\\bullet, M^\\bullet$ are\npseudo-coherent then the third is also pseudo-coherent.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064V","source_file":"more-algebra.tex","source_line":16144,"source_end_line":16150,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16144-L16150","statement_sha256":"952a686eff8fc7643545d5b120415be50dffb8a0c70ebc79630fbe5ced56768d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3221,"rank":3221,"depth":10,"x":735.563,"y":762.877,"cluster":"advanced-algebra"},{"id":"stacks:064W","tag":"064W","title":"Pseudo-coherent modules, I · Lemma 064W","summary":"Let R be a ring. Let K^bullet be a complex of R-modules. Let m ∈ Z. • If H^i(K^bullet) = 0 for all i ≥ m, then K^bullet is m-pseudo-coherent. • If H^i(K^bullet) = 0 for i > m and H^m(K^bullet) is a finite R-module, then K^bullet is m-pseudo-coherent. • If H^i(K^bullet) = 0 for i > m + 1, the module H^m + 1(K^bullet) is of finite presentation, and H^m(K^bullet) is of finite type, then K^bullet is m-pseudo-coherent.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.\nLet $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $H^i(K^\\bullet) = 0$ for all $i \\geq m$, then\n$K^\\bullet$ is $m$-pseudo-coherent.\n\\item If $H^i(K^\\bullet) = 0$ for $i > m$ and $H^m(K^\\bullet)$ is\na finite $R$-module, then $K^\\bullet$ is $m$-pseudo-coherent.\n\\item If $H^i(K^\\bullet) = 0$ for $i > m + 1$, the module\n$H^{m + 1}(K^\\bullet)$ is of finite presentation, and\n$H^m(K^\\bullet)$ is of finite type, then $K^\\bullet$ is\n$m$-pseudo-coherent.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064W","source_file":"more-algebra.tex","source_line":16157,"source_end_line":16171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16157-L16171","statement_sha256":"c9e5d8c56046941a32dfc85ca2fe1b43568eb5aa255926b45bc5e889865c02f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3222,"rank":3222,"depth":10,"x":572.25,"y":650.38,"cluster":"advanced-algebra"},{"id":"stacks:064X","tag":"064X","title":"Pseudo-coherent modules, I · Lemma 064X","summary":"Let R be a ring. Let m ∈ Z. If K^bullet ⊕ L^bullet is m-pseudo-coherent (resp. pseudo-coherent) so are K^bullet and L^bullet.","statement_latex":"Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. If $K^\\bullet \\oplus L^\\bullet$\nis $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nso are $K^\\bullet$ and $L^\\bullet$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064X","source_file":"more-algebra.tex","source_line":16195,"source_end_line":16200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16195-L16200","statement_sha256":"0f2210baac96ff293695330adbb247cfe9268afa9ac84065303c656886a21504","origin":"The Stacks Project","memory_eligible":false,"source_rank":3223,"rank":3223,"depth":11,"x":783.399,"y":640.653,"cluster":"advanced-algebra"},{"id":"stacks:064Y","tag":"064Y","title":"Pseudo-coherent modules, I · Lemma 064Y","summary":"Let R be a ring. Let m ∈ Z. Let K^bullet be a bounded above complex of R-modules such that K^i is (m - i)-pseudo-coherent for all i. Then K^bullet is m-pseudo-coherent. In particular, if K^bullet is a bounded above complex of pseudo-coherent R-modules, then K^bullet is pseudo-coherent.","statement_latex":"Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a bounded\nabove complex of $R$-modules such that $K^i$ is $(m - i)$-pseudo-coherent\nfor all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent.\nIn particular, if $K^\\bullet$ is a bounded above complex of\npseudo-coherent $R$-modules, then $K^\\bullet$ is pseudo-coherent.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064Y","source_file":"more-algebra.tex","source_line":16230,"source_end_line":16237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16230-L16237","statement_sha256":"d9a6a5b37e441bf010b8efbe3cb4c1346a2b620964aa0407a6841f6802421f85","origin":"The Stacks Project","memory_eligible":false,"source_rank":3224,"rank":3224,"depth":10,"x":635.342,"y":767.791,"cluster":"advanced-algebra"},{"id":"stacks:066B","tag":"066B","title":"Pseudo-coherent modules, I · Lemma 066B","summary":"Let R be a ring. Let m ∈ Z. Let K^bullet ∈ D^-(R) such that H^i(K^bullet) is (m - i)-pseudo-coherent (resp. pseudo-coherent) for all i. Then K^bullet is m-pseudo-coherent (resp. pseudo-coherent).","statement_latex":"Let $R$ be a ring. Let $m \\in \\mathbf{Z}$.\nLet $K^\\bullet \\in D^{-}(R)$ such that $H^i(K^\\bullet)$ is\n$(m - i)$-pseudo-coherent (resp.\\ pseudo-coherent) for all $i$.\nThen $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066B","source_file":"more-algebra.tex","source_line":16252,"source_end_line":16258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16252-L16258","statement_sha256":"4752120d8177bea2f8da91f4f4d025489fb1e6908a10c2aacf843c019d6f4da4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3225,"rank":3225,"depth":11,"x":642.285,"y":589.815,"cluster":"advanced-algebra"},{"id":"stacks:064Z","tag":"064Z","title":"Pseudo-coherent modules, I · Lemma 064Z","summary":"Let A → B be a ring map. Assume that B is pseudo-coherent as an A-module. Let K^bullet be a complex of B-modules. The following are equivalent • K^bullet is m-pseudo-coherent as a complex of B-modules, and • K^bullet is m-pseudo-coherent as a complex of A-modules. The same equivalence holds for pseudo-coherence.","statement_latex":"Let $A \\to B$ be a ring map. Assume that $B$ is pseudo-coherent as an\n$A$-module. Let $K^\\bullet$ be a complex of $B$-modules.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K^\\bullet$ is $m$-pseudo-coherent\nas a complex of $B$-modules, and\n\\item $K^\\bullet$ is $m$-pseudo-coherent\nas a complex of $A$-modules.\n\\end{enumerate}\nThe same equivalence holds for pseudo-coherence.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/064Z","source_file":"more-algebra.tex","source_line":16282,"source_end_line":16294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16282-L16294","statement_sha256":"58eb376832599d9fbe6cb321badcbb35a791c1f6cc623120e7d14a0b96fc3d2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3226,"rank":3226,"depth":11,"x":780.457,"y":725.155,"cluster":"advanced-algebra"},{"id":"stacks:0650","tag":"0650","title":"Pseudo-coherent modules, I · Lemma 0650","summary":"Let A → B be a ring map. Let K^bullet be an m-pseudo-coherent (resp. pseudo-coherent) complex of A-modules. Then K^bullet ⊗_A^L B is an m-pseudo-coherent (resp. pseudo-coherent) complex of B-modules.","statement_latex":"Let $A \\to B$ be a ring map.\nLet $K^\\bullet$ be an $m$-pseudo-coherent (resp.\\ pseudo-coherent)\ncomplex of $A$-modules. Then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$\nis an $m$-pseudo-coherent (resp.\\ pseudo-coherent) complex of $B$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0650","source_file":"more-algebra.tex","source_line":16344,"source_end_line":16350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16344-L16350","statement_sha256":"8151597c09acd3d6bc025c1ee24469ef38779c5013f49288eede3b3eb0ed8ddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3227,"rank":3227,"depth":10,"x":569.479,"y":703.733,"cluster":"advanced-algebra"},{"id":"stacks:066C","tag":"066C","title":"Pseudo-coherent modules, I · Lemma 066C","summary":"Let A → B be a flat ring map. Let M be an m-pseudo-coherent (resp. pseudo-coherent) A-module. Then M ⊗_A B is an m-pseudo-coherent (resp. pseudo-coherent) B-module.","statement_latex":"Let $A \\to B$ be a flat ring map.\nLet $M$ be an $m$-pseudo-coherent (resp.\\ pseudo-coherent)\n$A$-module. Then $M \\otimes_A B$\nis an $m$-pseudo-coherent (resp.\\ pseudo-coherent) $B$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066C","source_file":"more-algebra.tex","source_line":16384,"source_end_line":16390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16384-L16390","statement_sha256":"69c7d8d608e2928d7d371f13f39e448260c06da3c80a635fc5464cbdaf461a11","origin":"The Stacks Project","memory_eligible":false,"source_rank":3228,"rank":3228,"depth":11,"x":742.486,"y":599.69,"cluster":"advanced-algebra"},{"id":"stacks:066D","tag":"066D","title":"Pseudo-coherent modules, I · Lemma 066D","summary":"Let R be a ring. Let f_1, …, f_r ∈ R be elements which generate the unit ideal. Let m ∈ Z. Let K^bullet be a complex of R-modules. If for each i the complex K^bullet ⊗_R R_f_i is m-pseudo-coherent (resp. pseudo-coherent), then K^bullet is m-pseudo-coherent (resp. pseudo-coherent).","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be elements which\ngenerate the unit ideal. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$\nbe a complex of $R$-modules. If for each $i$ the complex\n$K^\\bullet \\otimes_R R_{f_i}$ is $m$-pseudo-coherent\n(resp.\\ pseudo-coherent), then $K^\\bullet$ is $m$-pseudo-coherent\n(resp.\\ pseudo-coherent).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066D","source_file":"more-algebra.tex","source_line":16403,"source_end_line":16411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16403-L16411","statement_sha256":"8a2ca12b5dafc422046f292b2c1c926feb4a9d3d4bd4c9526162dc2c58e03fcf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3229,"rank":3229,"depth":11,"x":698.528,"y":774.789,"cluster":"advanced-algebra"},{"id":"stacks:068R","tag":"068R","title":"Pseudo-coherent modules, I · Lemma 068R","summary":"Let R be a ring. Let m ∈ Z. Let K^bullet be a complex of R-modules. Let R → R' be a faithfully flat ring map. If the complex K^bullet ⊗_R R' is m-pseudo-coherent (resp. pseudo-coherent), then K^bullet is m-pseudo-coherent (resp. pseudo-coherent).","statement_latex":"Let $R$ be a ring. Let $m \\in \\mathbf{Z}$. Let $K^\\bullet$\nbe a complex of $R$-modules. Let $R \\to R'$ be a faithfully flat\nring map. If the complex $K^\\bullet \\otimes_R R'$ is $m$-pseudo-coherent\n(resp.\\ pseudo-coherent), then $K^\\bullet$ is $m$-pseudo-coherent\n(resp.\\ pseudo-coherent).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068R","source_file":"more-algebra.tex","source_line":16447,"source_end_line":16454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16447-L16454","statement_sha256":"9ab4c1c14d10390dcc0821338d99fe078fed2f4270ec5ac6934f10cedeb4e885","origin":"The Stacks Project","memory_eligible":false,"source_rank":3230,"rank":3230,"depth":11,"x":590.003,"y":620.546,"cluster":"advanced-algebra"},{"id":"stacks:0DJE","tag":"0DJE","title":"Pseudo-coherent modules, I · Lemma 0DJE","summary":"Let R be a ring. Let K, L be objects of D(R). • If K is n-pseudo-coherent and H^i(K) = 0 for i > a and L is m-pseudo-coherent and H^j(L) = 0 for j > b, then K ⊗_R^L L is t-pseudo-coherent with t = max(m + a, n + b). • If K and L are pseudo-coherent, then K ⊗_R^L L is pseudo-coherent.","statement_latex":"Let $R$ be a ring. Let $K, L$ be objects of $D(R)$.\n\\begin{enumerate}\n\\item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$\nand $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then\n$K \\otimes_R^\\mathbf{L} L$ is $t$-pseudo-coherent with $t = \\max(m + a, n + b)$.\n\\item If $K$ and $L$ are pseudo-coherent, then\n$K \\otimes_R^\\mathbf{L} L$ is pseudo-coherent.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJE","source_file":"more-algebra.tex","source_line":16486,"source_end_line":16496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16486-L16496","statement_sha256":"aac9199efbbea9fe309afa31b66d096216b639dd4469e03bd1f65d8bc0223bfc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3231,"rank":3231,"depth":10,"x":794.31,"y":672.765,"cluster":"advanced-algebra"},{"id":"stacks:066E","tag":"066E","title":"Pseudo-coherent modules, I · Lemma 066E","summary":"Let R be a Noetherian ring. Then • A complex of R-modules K^bullet is m-pseudo-coherent if and only if K^bullet ∈ D^-(R) and H^i(K^bullet) is a finite R-module for i ≥ m. • A complex of R-modules K^bullet is pseudo-coherent if and only if K^bullet ∈ D^-(R) and H^i(K^bullet) is a finite R-module for all i. • An R-module is pseudo-coherent if and only if it is finite.","statement_latex":"Let $R$ be a Noetherian ring. Then\n\\begin{enumerate}\n\\item A complex of $R$-modules $K^\\bullet$ is $m$-pseudo-coherent\nif and only if $K^\\bullet \\in D^{-}(R)$ and\n$H^i(K^\\bullet)$ is a finite $R$-module for $i \\geq m$.\n\\item A complex of $R$-modules $K^\\bullet$ is pseudo-coherent\nif and only if $K^\\bullet \\in D^{-}(R)$ and\n$H^i(K^\\bullet)$ is a finite $R$-module for all $i$.\n\\item An $R$-module is pseudo-coherent if and only if it is finite.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066E","source_file":"more-algebra.tex","source_line":16516,"source_end_line":16528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16516-L16528","statement_sha256":"547ce6754d1cddf100415c2a861ba43376338b95901a44497d04ad68ce939b4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3232,"rank":3232,"depth":12,"x":601.434,"y":750.281,"cluster":"advanced-algebra"},{"id":"stacks:0EWZ","tag":"0EWZ","title":"Pseudo-coherent modules, I · Lemma 0EWZ","summary":"Let R be a coherent ring (Algebra, Definition [Tag 05CV]). Let K ∈ D^-(R). The following are equivalent • K is m-pseudo-coherent, • H^m(K) is a finite R-module and H^i(K) is coherent for i > m, and • H^m(K) is a finite R-module and H^i(K) is finitely presented for i > m. Thus K is pseudo-coherent if and only if H^i(K) is a coherent module for all i.","statement_latex":"Let $R$ be a coherent ring\n(Algebra, Definition \\ref{algebra-definition-coherent}).\nLet $K \\in D^-(R)$. The following are equivalent\n\\begin{enumerate}\n\\item $K$ is $m$-pseudo-coherent,\n\\item $H^m(K)$ is a finite $R$-module and $H^i(K)$ is coherent for $i > m$, and\n\\item $H^m(K)$ is a finite $R$-module and\n$H^i(K)$ is finitely presented for $i > m$.\n\\end{enumerate}\nThus $K$ is pseudo-coherent if and only if $H^i(K)$\nis a coherent module for all $i$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWZ","source_file":"more-algebra.tex","source_line":16548,"source_end_line":16561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16548-L16561","statement_sha256":"9710347ca2a44b8ef8aca56f84e678eb927cd91334e06f126999422900beb080","origin":"The Stacks Project","memory_eligible":false,"source_rank":3233,"rank":3233,"depth":12,"x":681.417,"y":583.481,"cluster":"advanced-algebra"},{"id":"stacks:0G8W","tag":"0G8W","title":"Pseudo-coherent modules, II · Lemma 0G8W","summary":"Let R be a ring. Let M = colim M_i be a filtered colimit of R-modules. Let K ∈ D(R) be m-pseudo-coherent. Then colim Ext^n_R(K, M_i) = Ext^n_R(K, M) for n < -m and colim Ext^-m_R(K, M_i) → Ext^-m_R(K, M) is injective.","statement_latex":"Let $R$ be a ring. Let $M = \\colim M_i$ be a filtered colimit of $R$-modules.\nLet $K \\in D(R)$ be $m$-pseudo-coherent. Then\n$\\colim \\Ext^n_R(K, M_i) = \\Ext^n_R(K, M)$ for $n < -m$ and\n$\\colim \\Ext^{-m}_R(K, M_i) \\to \\Ext^{-m}_R(K, M)$ is injective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8W","source_file":"more-algebra.tex","source_line":16595,"source_end_line":16601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16595-L16601","statement_sha256":"664d887f52e7f93bac499962e8fbaba0b1ce6ad78d746992f6b9ef396ef0678b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3234,"rank":3234,"depth":5,"x":756.664,"y":752.061,"cluster":"advanced-algebra"},{"id":"stacks:0G8X","tag":"0G8X","title":"Pseudo-coherent modules, II · Lemma 0G8X","summary":"Let R be a ring. Let K ∈ D^-(R). Let m ∈ Z. Then K is m-pseudo-coherent if and only if for any filtered colimit M = colim M_i of R-modules we have colim Ext^n_R(K, M_i) = Ext^n_R(K, M) for n < -m and colim Ext^-m_R(K, M_i) → Ext^-m_R(K, M) is injective.","statement_latex":"Let $R$ be a ring. Let $K \\in D^-(R)$. Let $m \\in \\mathbf{Z}$.\nThen $K$ is $m$-pseudo-coherent if and only if\nfor any filtered colimit $M = \\colim M_i$ of $R$-modules we have\n$\\colim \\Ext^n_R(K, M_i) = \\Ext^n_R(K, M)$ for $n < -m$ and\n$\\colim \\Ext^{-m}_R(K, M_i) \\to \\Ext^{-m}_R(K, M)$ is injective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8X","source_file":"more-algebra.tex","source_line":16627,"source_end_line":16634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16627-L16634","statement_sha256":"94c3c1f737bff0712425602728d123610e9d23b9a8001257171fc420c896e2dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3235,"rank":3235,"depth":11,"x":565.384,"y":670.353,"cluster":"advanced-algebra"},{"id":"stacks:087Q","tag":"087Q","title":"Pseudo-coherent modules, II · Lemma 087Q","summary":"Let R be a ring. Let L, M, N be R-modules. • If M is finitely presented and L is flat, then the canonical map Hom_R(M, N) ⊗_R L → Hom_R(M, N ⊗_R L) is an isomorphism. • If M is (-m)-pseudo-coherent and L is flat, then the canonical map Ext^i_R(M, N) ⊗_R L → Ext^i_R(M, N ⊗_R L) is an isomorphism for i < m.","statement_latex":"Let $R$ be a ring. Let $L$, $M$, $N$ be $R$-modules.\n\\begin{enumerate}\n\\item If $M$ is finitely presented and $L$ is flat, then the canonical map\n$\\Hom_R(M, N) \\otimes_R L \\to \\Hom_R(M, N \\otimes_R L)$\nis an isomorphism.\n\\item If $M$ is $(-m)$-pseudo-coherent and $L$ is flat, then the canonical map\n$\\Ext^i_R(M, N) \\otimes_R L \\to \\Ext^i_R(M, N \\otimes_R L)$\nis an isomorphism for $i < m$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087Q","source_file":"more-algebra.tex","source_line":16666,"source_end_line":16677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16666-L16677","statement_sha256":"c31cd777a5aafb81bd840bc90f16398c29310ebf08a8ac030318732003298ab6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3236,"rank":3236,"depth":10,"x":772.383,"y":622.009,"cluster":"advanced-algebra"},{"id":"stacks:087R","tag":"087R","title":"Pseudo-coherent modules, II · Lemma 087R","summary":"Let R → R' be a flat ring map. Let M, N be R-modules. • If M is a finitely presented R-module, then Hom_R(M, N) ⊗_R R' = Hom_R'(M ⊗_R R', N ⊗_R R'). • If M is (-m)-pseudo-coherent, then Ext^i_R(M, N) ⊗_R R' = Ext^i_R'(M ⊗_R R', N ⊗_R R') for i < m. In particular if R is Noetherian and M is a finite module this holds for all i.","statement_latex":"Let $R \\to R'$ be a flat ring map. Let $M$, $N$ be $R$-modules.\n\\begin{enumerate}\n\\item If $M$ is a finitely presented $R$-module, then\n$\\Hom_R(M, N) \\otimes_R R' = \\Hom_{R'}(M \\otimes_R R', N \\otimes_R R')$.\n\\item If $M$ is $(-m)$-pseudo-coherent, then\n$\\Ext^i_R(M, N) \\otimes_R R' = \\Ext^i_{R'}(M \\otimes_R R', N \\otimes_R R')$\nfor $i < m$.\n\\end{enumerate}\nIn particular if $R$ is Noetherian and $M$ is a finite module this\nholds for all $i$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087R","source_file":"more-algebra.tex","source_line":16708,"source_end_line":16720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16708-L16720","statement_sha256":"2f5e1e1ad65abb3e240474c4bfe599cef061b899424d7136fb2aafe5ad0cd26d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3237,"rank":3237,"depth":13,"x":658.487,"y":775.294,"cluster":"advanced-algebra"},{"id":"stacks:0CYB","tag":"0CYB","title":"Pseudo-coherent modules, II · Lemma 0CYB","summary":"Let R be a ring. Let K ∈ D^-(R). The following are equivalent: • K is pseudo-coherent, • for every family (Q_α)_α ∈ A of R-modules, the canonical map α : K ⊗_R^L ( ∏_α Q_α ) → ∏_α (K ⊗_R^L Q_α) is an isomorphism in D(R), • for every R-module Q and every set A, the canonical map β : K ⊗_R^L Q^A → (K ⊗_R^L Q)^A is an isomorphism in D(R), and • for every set A, the canonical map γ : K ⊗_R^L R^A → K^A is an isomorphism in D(R). Given m ∈ Z the following are equivalent • [(a)]…","statement_latex":"Let $R$ be a ring. Let $K \\in D^-(R)$. The following are equivalent:\n\\begin{enumerate}\n\\item $K$ is pseudo-coherent,\n\\item for every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules, the\ncanonical map\n$$\n\\alpha :\nK \\otimes_R^\\mathbf{L} \\left( \\prod\\nolimits_\\alpha Q_{\\alpha} \\right)\n\\longrightarrow\n\\prod\\nolimits_\\alpha (K \\otimes_R^\\mathbf{L} Q_{\\alpha})\n$$\nis an isomorphism in $D(R)$,\n\\item for every $R$-module $Q$ and every set $A$, the canonical map\n$$\n\\beta : K \\otimes_R^\\mathbf{L} Q^A \\longrightarrow (K \\otimes_R^\\mathbf{L} Q)^A\n$$\nis an isomorphism in $D(R)$, and\n\\item for every set $A$, the canonical map\n$$\n\\gamma : K \\otimes_R^\\mathbf{L} R^A \\longrightarrow K^A\n$$\nis an isomorphism in $D(R)$.\n\\end{enumerate}\nGiven $m \\in \\mathbf{Z}$ the following are equivalent\n\\begin{enumerate}\n\\item[(a)] $K$ is $m$-pseudo-coherent,\n\\item[(b)] for every family $(Q_{\\alpha})_{\\alpha \\in A}$ of $R$-modules,\nwith $\\alpha$ as above\n$H^i(\\alpha)$ is an isomorphism for $i > m$ and surjective for $i = m$,\n\\item[(c)] for every $R$-module $Q$ and every set $A$, with $\\beta$ as above\n$H^i(\\beta)$ is an isomorphism for $i > m$ and surjective for $i = m$,\n\\item[(d)] for every set $A$, with $\\gamma$ as above\n$H^i(\\gamma)$ is an isomorphism for $i > m$ and surjective for $i = m$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYB","source_file":"more-algebra.tex","source_line":16731,"source_end_line":16767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16731-L16767","statement_sha256":"fcaac5f5be297818383106cd1b674b8800dfd76670e9beeabf92c435e757d42e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3238,"rank":3238,"depth":10,"x":619.159,"y":597.428,"cluster":"advanced-algebra"},{"id":"stacks:0G8Y","tag":"0G8Y","title":"Pseudo-coherent modules, II · Lemma 0G8Y","summary":"Let R be a ring. Let K ∈ D(R) be pseudo-coherent. Let i ∈ Z. There exists a finitely presented R-module M and a map K → M[-i] in D(R) which induces an injection H^i(K) → M.","statement_latex":"Let $R$ be a ring. Let $K \\in D(R)$ be pseudo-coherent.\nLet $i \\in \\mathbf{Z}$. There exists a finitely presented\n$R$-module $M$ and a map $K \\to M[-i]$ in $D(R)$ which induces\nan injection $H^i(K) \\to M$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8Y","source_file":"more-algebra.tex","source_line":16810,"source_end_line":16816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16810-L16816","statement_sha256":"ea90a0c52b4e8291a14b861d4bb346c17e0e9342b5c9a0c545de37c834de104b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3239,"rank":3239,"depth":1,"x":791.396,"y":706.396,"cluster":"advanced-algebra"},{"id":"stacks:0A7D","tag":"0A7D","title":"Pseudo-coherent modules, II · Lemma 0A7D","summary":"Let A be a Noetherian ring. Let K ∈ D(A) be pseudo-coherent, i.e., K ∈ D^-(A) with finite cohomology modules. Let m be a maximal ideal of A. If H^i(K)/ m H^i(K) not = 0, then there exists a finite A-module E annihilated by a power of m and a map K → E[-i] which is nonzero on H^i(K).","statement_latex":"Let $A$ be a Noetherian ring. Let $K \\in D(A)$ be pseudo-coherent,\ni.e., $K \\in D^-(A)$ with finite cohomology modules.\nLet $\\mathfrak m$ be a maximal ideal of $A$.\nIf $H^i(K)/\\mathfrak m H^i(K) \\not = 0$, then there exists a finite\n$A$-module $E$ annihilated by a power of $\\mathfrak m$\nand a map $K \\to E[-i]$ which is nonzero on $H^i(K)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7D","source_file":"more-algebra.tex","source_line":16824,"source_end_line":16832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16824-L16832","statement_sha256":"4286680c345b5be41c46fd589eae4ede19676890b12f44d6ac8d1d15d7e3d769","origin":"The Stacks Project","memory_eligible":false,"source_rank":3240,"rank":3240,"depth":13,"x":576.511,"y":723.796,"cluster":"advanced-algebra"},{"id":"stacks:0652","tag":"0652","title":"Tor dimension · Definition 0652","summary":"Let R be a ring. Denote D(R) its derived category. Let a, b ∈ Z. • An object K^bullet of D(R) has tor-amplitude in [a, b] if H^i(K^bullet ⊗_R^L M) = 0 for all R-modules M and all i not ∈ [a, b]. • An object K^bullet of D(R) has finite tor dimension if it has tor-amplitude in [a, b] for some a, b. • An R-module M has tor dimension ≤ d if M[0] as an object of D(R) has tor-amplitude in [-d, 0]. • An R-module M has finite tor dimension if M[0] as an object of D(R) has finite…","statement_latex":"Let $R$ be a ring. Denote $D(R)$ its derived category.\nLet $a, b \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item An object $K^\\bullet$ of $D(R)$ has\n{\\it tor-amplitude in $[a, b]$}\nif $H^i(K^\\bullet \\otimes_R^\\mathbf{L} M) = 0$ for all $R$-modules\n$M$ and all $i \\not \\in [a, b]$.\n\\item An object $K^\\bullet$ of $D(R)$ has {\\it finite tor dimension}\nif it has tor-amplitude in $[a, b]$ for some $a, b$.\n\\item An $R$-module $M$ has {\\it tor dimension $\\leq d$}\nif $M[0]$ as an object of $D(R)$ has tor-amplitude in $[-d, 0]$.\n\\item An $R$-module $M$ has {\\it finite tor dimension}\nif $M[0]$ as an object of $D(R)$ has finite tor dimension.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0652","source_file":"more-algebra.tex","source_line":16860,"source_end_line":16876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16860-L16876","statement_sha256":"d1de35fcbfb99fccd3f7cf03792f08289623ac18646bcf04d5a748646a845cf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3241,"rank":3241,"depth":0,"x":721.14,"y":588.877,"cluster":"advanced-algebra"},{"id":"stacks:0653","tag":"0653","title":"Tor dimension · Lemma 0653","summary":"Let R be a ring. Let K^bullet be a bounded above complex of flat R-modules with tor-amplitude in [a, b]. Then Coker(d_K^a - 1) is a flat R-module.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a bounded above complex of\nflat $R$-modules with tor-amplitude in $[a, b]$.\nThen $\\Coker(d_K^{a - 1})$ is a flat $R$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0653","source_file":"more-algebra.tex","source_line":16882,"source_end_line":16887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16882-L16887","statement_sha256":"ef8cc723cb97ef9bbc695b190319a8ce80c666a388327fc6945569dabbf2fb5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3242,"rank":3242,"depth":4,"x":722.993,"y":770.642,"cluster":"advanced-algebra"},{"id":"stacks:0654","tag":"0654","title":"Tor dimension · Lemma 0654","summary":"Let R be a ring. Let K^bullet be an object of D(R). Let a, b ∈ Z. The following are equivalent • K^bullet has tor-amplitude in [a, b]. • K^bullet is quasi-isomorphic to a complex E^bullet of flat R-modules with E^i = 0 for i not ∈ [a, b].","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be an object of $D(R)$.\nLet $a, b \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $K^\\bullet$ has tor-amplitude in $[a, b]$.\n\\item $K^\\bullet$ is quasi-isomorphic to a complex\n$E^\\bullet$ of flat $R$-modules with $E^i = 0$ for $i \\not \\in [a, b]$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0654","source_file":"more-algebra.tex","source_line":16905,"source_end_line":16914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16905-L16914","statement_sha256":"059c21b87e278d0329382fc458f670eee5adde10baec6e8a1e86f2a9dd5a0cae","origin":"The Stacks Project","memory_eligible":false,"source_rank":3243,"rank":3243,"depth":5,"x":575.285,"y":637.508,"cluster":"advanced-algebra"},{"id":"stacks:0BYL","tag":"0BYL","title":"Tor dimension · Lemma 0BYL","summary":"Let R be a ring. Let a ∈ Z and let K be an object of D(R). The following are equivalent • K has tor-amplitude in [a, ∞], and • K is quasi-isomorphic to a K-flat complex E^bullet whose terms are flat R-modules with E^i = 0 for i not ∈ [a, ∞].","statement_latex":"Let $R$ be a ring. Let $a \\in \\mathbf{Z}$ and let $K$ be an object of $D(R)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ has tor-amplitude in $[a, \\infty]$, and\n\\item $K$ is quasi-isomorphic to a K-flat complex\n$E^\\bullet$ whose terms are flat $R$-modules with\n$E^i = 0$ for $i \\not \\in [a, \\infty]$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYL","source_file":"more-algebra.tex","source_line":16929,"source_end_line":16939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16929-L16939","statement_sha256":"b933fca64af8b9b83635767fde0bd00ed3ddbed74890ca6a4cb95fb9b8953c80","origin":"The Stacks Project","memory_eligible":false,"source_rank":3244,"rank":3244,"depth":15,"x":791.515,"y":651.882,"cluster":"advanced-algebra"},{"id":"stacks:0655","tag":"0655","title":"Tor dimension · Lemma 0655","summary":"Let R be a ring. Let (K^bullet, L^bullet, M^bullet, f, g, h) be a distinguished triangle in D(R). Let a, b ∈ Z. • If K^bullet has tor-amplitude in [a + 1, b + 1] and L^bullet has tor-amplitude in [a, b] then M^bullet has tor-amplitude in [a, b]. • If K^bullet, M^bullet have tor-amplitude in [a, b], then L^bullet has tor-amplitude in [a, b]. • If L^bullet has tor-amplitude in [a + 1, b + 1] and M^bullet has tor-amplitude in [a, b], then K^bullet has tor-amplitude in [a +…","statement_latex":"Let $R$ be a ring.\nLet $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$ be a distinguished\ntriangle in $D(R)$. Let $a, b \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $K^\\bullet$ has tor-amplitude in $[a + 1, b + 1]$ and\n$L^\\bullet$ has tor-amplitude in $[a, b]$ then $M^\\bullet$ has\ntor-amplitude in $[a, b]$.\n\\item If $K^\\bullet, M^\\bullet$ have tor-amplitude in $[a, b]$, then\n$L^\\bullet$ has tor-amplitude in $[a, b]$.\n\\item If $L^\\bullet$ has tor-amplitude in $[a + 1, b + 1]$\nand $M^\\bullet$ has tor-amplitude in $[a, b]$, then\n$K^\\bullet$ has tor-amplitude in $[a + 1, b + 1]$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0655","source_file":"more-algebra.tex","source_line":16971,"source_end_line":16986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16971-L16986","statement_sha256":"d79cb60d05a7c3d5a09a38579b1fee3355556f3c1a27d275f90ecd619088cf3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3245,"rank":3245,"depth":0,"x":620.313,"y":764.107,"cluster":"advanced-algebra"},{"id":"stacks:066F","tag":"066F","title":"Tor dimension · Lemma 066F","summary":"Let R be a ring. Let M be an R-module. Let d ≥ 0. The following are equivalent • M has tor dimension ≤ d, and • there exists a resolution 0 → F_d → … → F_1 → F_0 → M → 0 with F_i a flat R-module. In particular an R-module has tor dimension 0 if and only if it is a flat R-module.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $d \\geq 0$. The following are equivalent\n\\begin{enumerate}\n\\item $M$ has tor dimension $\\leq d$, and\n\\item there exists a resolution\n$$\n0 \\to F_d \\to \\ldots \\to F_1 \\to F_0 \\to M \\to 0\n$$\nwith $F_i$ a flat $R$-module.\n\\end{enumerate}\nIn particular an $R$-module has tor dimension $0$ if and only if\nit is a flat $R$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066F","source_file":"more-algebra.tex","source_line":16996,"source_end_line":17010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L16996-L17010","statement_sha256":"b9738d0a9b1eabb667afbc78b740e106c08bf12e86093604f8ec72080083516c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3246,"rank":3246,"depth":6,"x":656.349,"y":584.002,"cluster":"advanced-algebra"},{"id":"stacks:066G","tag":"066G","title":"Tor dimension · Lemma 066G","summary":"Let R be a ring. Let a, b ∈ Z. If K^bullet ⊕ L^bullet has tor amplitude in [a, b] so do K^bullet and L^bullet.","statement_latex":"Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$.\nIf $K^\\bullet \\oplus L^\\bullet$ has tor amplitude in $[a, b]$\nso do $K^\\bullet$ and $L^\\bullet$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066G","source_file":"more-algebra.tex","source_line":17024,"source_end_line":17029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17024-L17029","statement_sha256":"cd969df6507daae4f86ed823e4bc1a06a9dcd221e71c2c1dcf6533933bde5ead","origin":"The Stacks Project","memory_eligible":false,"source_rank":3247,"rank":3247,"depth":0,"x":774.748,"y":737.433,"cluster":"advanced-algebra"},{"id":"stacks:066H","tag":"066H","title":"Tor dimension · Lemma 066H","summary":"Let R be a ring. Let K^bullet be a bounded complex of R-modules such that K^i has tor amplitude in [a - i, b - i] for all i. Then K^bullet has tor amplitude in [a, b]. In particular if K^bullet is a finite complex of R-modules of finite tor dimension, then K^bullet has finite tor dimension.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a bounded complex of $R$-modules\nsuch that $K^i$ has tor amplitude in $[a - i, b - i]$ for all $i$.\nThen $K^\\bullet$ has tor amplitude in $[a, b]$. In particular\nif $K^\\bullet$ is a finite complex of $R$-modules of finite tor dimension,\nthen $K^\\bullet$ has finite tor dimension.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066H","source_file":"more-algebra.tex","source_line":17035,"source_end_line":17042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17035-L17042","statement_sha256":"755fa97ffccb9d806e1ca61e5094f774c6973bc1186a7e07140c5738ad113cb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3248,"rank":3248,"depth":1,"x":563.815,"y":691.425,"cluster":"advanced-algebra"},{"id":"stacks:066I","tag":"066I","title":"Tor dimension · Lemma 066I","summary":"Let R be a ring. Let a, b ∈ Z. Let K^bullet ∈ D^b(R) such that H^i(K^bullet) has tor amplitude in [a - i, b - i] for all i. Then K^bullet has tor amplitude in [a, b]. In particular if K^bullet ∈ D^b(R) and all its cohomology groups have finite tor dimension then K^bullet has finite tor dimension.","statement_latex":"Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet \\in D^b(R)$\nsuch that $H^i(K^\\bullet)$ has tor amplitude in $[a - i, b - i]$\nfor all $i$. Then $K^\\bullet$ has tor amplitude in $[a, b]$. In particular\nif $K^\\bullet \\in D^b(R)$ and all its cohomology groups have finite tor\ndimension then $K^\\bullet$ has finite tor dimension.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066I","source_file":"more-algebra.tex","source_line":17050,"source_end_line":17057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17050-L17057","statement_sha256":"e72454cb1e75c9a3983bac7d83a45170d6472729068a2fc79cb5efda80c5ec01","origin":"The Stacks Project","memory_eligible":false,"source_rank":3249,"rank":3249,"depth":1,"x":756.573,"y":605.564,"cluster":"advanced-algebra"},{"id":"stacks:0B66","tag":"0B66","title":"Tor dimension · Lemma 0B66","summary":"Let A → B be a ring map. Let K^bullet and L^bullet be complexes of B-modules. Let a, b, c, d ∈ Z. If • K^bullet as a complex of B-modules has tor amplitude in [a, b], • L^bullet as a complex of A-modules has tor amplitude in [c, d], then K^bullet ⊗^L_B L^bullet as a complex of A-modules has tor amplitude in [a + c, b + d].","statement_latex":"Let $A \\to B$ be a ring map. Let $K^\\bullet$ and $L^\\bullet$ be complexes\nof $B$-modules. Let $a, b, c, d \\in \\mathbf{Z}$. If\n\\begin{enumerate}\n\\item $K^\\bullet$ as a complex of $B$-modules has tor amplitude in $[a, b]$,\n\\item $L^\\bullet$ as a complex of $A$-modules has tor amplitude in $[c, d]$,\n\\end{enumerate}\nthen $K^\\bullet \\otimes^\\mathbf{L}_B L^\\bullet$ as a complex of $A$-modules\nhas tor amplitude in $[a + c, b + d]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B66","source_file":"more-algebra.tex","source_line":17065,"source_end_line":17075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17065-L17075","statement_sha256":"1d8771a45742a65c7cb3478606c7f1ab4ab1ca398d74c78469f82360802c6739","origin":"The Stacks Project","memory_eligible":false,"source_rank":3250,"rank":3250,"depth":6,"x":683.4,"y":778.449,"cluster":"advanced-algebra"},{"id":"stacks:066J","tag":"066J","title":"Tor dimension · Lemma 066J","summary":"Let A → B be a ring map. Assume that B is flat as an A-module. Let K^bullet be a complex of B-modules. Let a, b ∈ Z. If K^bullet as a complex of B-modules has tor amplitude in [a, b], then K^bullet as a complex of A-modules has tor amplitude in [a, b].","statement_latex":"Let $A \\to B$ be a ring map. Assume that $B$ is flat as an\n$A$-module. Let $K^\\bullet$ be a complex of $B$-modules.\nLet $a, b \\in \\mathbf{Z}$. If $K^\\bullet$ as a complex of $B$-modules\nhas tor amplitude in $[a, b]$, then $K^\\bullet$ as a complex of\n$A$-modules has tor amplitude in $[a, b]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066J","source_file":"more-algebra.tex","source_line":17099,"source_end_line":17106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17099-L17106","statement_sha256":"f24e23fc3e361f479106620aa738fc7aee4bf56f58b0ce55120718a9f9d0d530","origin":"The Stacks Project","memory_eligible":false,"source_rank":3251,"rank":3251,"depth":7,"x":598.23,"y":609.254,"cluster":"advanced-algebra"},{"id":"stacks:066K","tag":"066K","title":"Tor dimension · Lemma 066K","summary":"Let A → B be a ring map. Assume that B has tor dimension ≤ d as an A-module. Let K^bullet be a complex of B-modules. Let a, b ∈ Z. If K^bullet as a complex of B-modules has tor amplitude in [a, b], then K^bullet as a complex of A-modules has tor amplitude in [a - d, b].","statement_latex":"Let $A \\to B$ be a ring map. Assume that $B$ has tor dimension $\\leq d$\nas an $A$-module. Let $K^\\bullet$ be a complex of $B$-modules.\nLet $a, b \\in \\mathbf{Z}$. If $K^\\bullet$ as a complex of $B$-modules\nhas tor amplitude in $[a, b]$, then $K^\\bullet$ as a complex of\n$A$-modules has tor amplitude in $[a - d, b]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066K","source_file":"more-algebra.tex","source_line":17118,"source_end_line":17125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17118-L17125","statement_sha256":"f33fea0579f86adb58a1b42741e07a4a750fe18bf824792a1b4304f395fdbfc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3252,"rank":3252,"depth":7,"x":797.321,"y":685.776,"cluster":"advanced-algebra"},{"id":"stacks:066L","tag":"066L","title":"Tor dimension · Lemma 066L","summary":"Let A → B be a ring map. Let a, b ∈ Z. Let K^bullet be a complex of A-modules with tor amplitude in [a, b]. Then K^bullet ⊗_A^L B as a complex of B-modules has tor amplitude in [a, b].","statement_latex":"Let $A \\to B$ be a ring map.\nLet $a, b \\in \\mathbf{Z}$.\nLet $K^\\bullet$ be a complex of $A$-modules with tor amplitude in $[a, b]$.\nThen $K^\\bullet \\otimes_A^{\\mathbf{L}} B$ as a complex of $B$-modules\nhas tor amplitude in $[a, b]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066L","source_file":"more-algebra.tex","source_line":17147,"source_end_line":17154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17147-L17154","statement_sha256":"c82f6eb46e35e550d6fd3f54bdc68485f0e54abb71f2afb1d4df07e41391a7d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3253,"rank":3253,"depth":6,"x":588.741,"y":742.383,"cluster":"advanced-algebra"},{"id":"stacks:066M","tag":"066M","title":"Tor dimension · Lemma 066M","summary":"Let A → B be a flat ring map. Let d ≥ 0. Let M be an A-module of tor dimension ≤ d. Then M ⊗_A B is a B-module of tor dimension ≤ d.","statement_latex":"Let $A \\to B$ be a flat ring map. Let $d \\geq 0$.\nLet $M$ be an $A$-module of tor dimension $\\leq d$.\nThen $M \\otimes_A B$ is a $B$-module of tor dimension $\\leq d$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066M","source_file":"more-algebra.tex","source_line":17169,"source_end_line":17174,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17169-L17174","statement_sha256":"1ed048ee425baaa4ce32cc3a4e192ca61824bd3a81cb0f017dadad257b1383db","origin":"The Stacks Project","memory_eligible":false,"source_rank":3254,"rank":3254,"depth":7,"x":697.146,"y":582.107,"cluster":"advanced-algebra"},{"id":"stacks:0B67","tag":"0B67","title":"Tor dimension · Lemma 0B67","summary":"Let A → B be a ring map. Let K^bullet be a complex of B-modules. Let a, b ∈ Z. The following are equivalent • K^bullet has tor amplitude in [a, b] as a complex of A-modules, • K^bullet_ q has tor amplitude in [a, b] as a complex of A_ p-modules for every prime q ⊂ B with p = A ∩ q, • K^bullet_ m has tor amplitude in [a, b] as a complex of A_ p-modules for every maximal ideal m ⊂ B with p = A ∩ m.","statement_latex":"Let $A  \\to B$ be a ring map. Let $K^\\bullet$ be a complex of $B$-modules.\nLet $a, b \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $K^\\bullet$ has tor amplitude in $[a, b]$ as a complex of $A$-modules,\n\\item $K^\\bullet_\\mathfrak q$ has tor amplitude in $[a, b]$ as a complex\nof $A_\\mathfrak p$-modules for every prime $\\mathfrak q \\subset B$\nwith $\\mathfrak p = A \\cap \\mathfrak q$,\n\\item $K^\\bullet_\\mathfrak m$ has tor amplitude in $[a, b]$ as a complex\nof $A_\\mathfrak p$-modules for every maximal ideal $\\mathfrak m \\subset B$\nwith $\\mathfrak p = A \\cap \\mathfrak m$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B67","source_file":"more-algebra.tex","source_line":17183,"source_end_line":17196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17183-L17196","statement_sha256":"8d6f0a019bce34d84a0232b92ded0f821233d385989f9e9b209276ca7c2224d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3255,"rank":3255,"depth":2,"x":746.155,"y":762.006,"cluster":"advanced-algebra"},{"id":"stacks:066N","tag":"066N","title":"Tor dimension · Lemma 066N","summary":"Let R be a ring. Let f_1, …, f_r ∈ R be elements which generate the unit ideal. Let a, b ∈ Z. Let K^bullet be a complex of R-modules. If for each i the complex K^bullet ⊗_R R_f_i has tor amplitude in [a, b], then K^bullet has tor amplitude in [a, b].","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be elements which\ngenerate the unit ideal. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$\nbe a complex of $R$-modules. If for each $i$ the complex\n$K^\\bullet \\otimes_R R_{f_i}$ has tor amplitude in $[a, b]$,\nthen $K^\\bullet$ has tor amplitude in $[a, b]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066N","source_file":"more-algebra.tex","source_line":17209,"source_end_line":17216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17209-L17216","statement_sha256":"557c832d8e4f9017779f83d4073c3234e117b1cace78264495502b909da2afc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3256,"rank":3256,"depth":3,"x":565.142,"y":657.042,"cluster":"advanced-algebra"},{"id":"stacks:068S","tag":"068S","title":"Tor dimension · Lemma 068S","summary":"Let R be a ring. Let a, b ∈ Z. Let K^bullet be a complex of R-modules. Let R → R' be a faithfully flat ring map. If the complex K^bullet ⊗_R R' has tor amplitude in [a, b], then K^bullet has tor amplitude in [a, b].","statement_latex":"Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$. Let $K^\\bullet$\nbe a complex of $R$-modules. Let $R \\to R'$ be a faithfully flat\nring map. If the complex $K^\\bullet \\otimes_R R'$ has tor amplitude\nin $[a, b]$, then $K^\\bullet$ has tor amplitude in $[a, b]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068S","source_file":"more-algebra.tex","source_line":17237,"source_end_line":17243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17237-L17243","statement_sha256":"a57ae8a5c84f2dff39c7675cc7a02408f4b6efca6335073125bc44eb6143081a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3257,"rank":3257,"depth":0,"x":783.273,"y":631.702,"cluster":"advanced-algebra"},{"id":"stacks:0DJF","tag":"0DJF","title":"Tor dimension · Lemma 0DJF","summary":"Given ring maps R → A → B with A → B faithfully flat and K ∈ D(A) the tor amplitude of K over R is the same as the tor amplitude of K ⊗_A^L B over R.","statement_latex":"Given ring maps $R \\to A \\to B$ with $A \\to B$ faithfully flat\nand $K \\in D(A)$ the tor amplitude of $K$ over $R$\nis the same as the tor amplitude of $K \\otimes_A^\\mathbf{L} B$\nover $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJF","source_file":"more-algebra.tex","source_line":17261,"source_end_line":17267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17261-L17267","statement_sha256":"d5bab36b773b114a67ea97afb95c440dfcbe0224670ebeca0368382f0987c495","origin":"The Stacks Project","memory_eligible":false,"source_rank":3258,"rank":3258,"depth":0,"x":642.646,"y":774.319,"cluster":"advanced-algebra"},{"id":"stacks:066P","tag":"066P","title":"Tor dimension · Lemma 066P","summary":"Let R be a ring of finite global dimension d. Then • every module has tor dimension ≤ d, • a complex of R-modules K^bullet with H^i(K^bullet) not = 0 only if i ∈ [a, b] has tor amplitude in [a - d, b], and • a complex of R-modules K^bullet has finite tor dimension if and only if K^bullet ∈ D^b(R).","statement_latex":"Let $R$ be a ring of finite global dimension $d$. Then\n\\begin{enumerate}\n\\item every module has tor dimension $\\leq d$,\n\\item a complex of $R$-modules $K^\\bullet$ with $H^i(K^\\bullet) \\not = 0$\nonly if $i \\in [a, b]$ has tor amplitude in $[a - d, b]$, and\n\\item a complex of $R$-modules $K^\\bullet$ has finite tor dimension if and only\nif $K^\\bullet \\in D^b(R)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066P","source_file":"more-algebra.tex","source_line":17286,"source_end_line":17296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17286-L17296","statement_sha256":"b71bb8b14ba1ad8513710a11712f2f4469711e8ed97aa625a4c2152168961dc9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3259,"rank":3259,"depth":2,"x":631.641,"y":589.154,"cluster":"advanced-algebra"},{"id":"stacks:0H75","tag":"0H75","title":"Tor dimension · Lemma 0H75","summary":"Let R' → R be a surjective ring map whose kernel is a nilpotent ideal. Let K' ∈ D(R') and set K = K' ⊗_R'^L R. Let a, b ∈ Z. Then K has tor amplitude in [a, b] if and only if K' does.","statement_latex":"Let $R' \\to R$ be a surjective ring map whose kernel is a nilpotent ideal.\nLet $K' \\in D(R')$ and set $K = K' \\otimes_{R'}^\\mathbf{L} R$.\nLet $a, b \\in \\mathbf{Z}$.\nThen $K$ has tor amplitude in $[a, b]$ if and only if $K'$ does.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H75","source_file":"more-algebra.tex","source_line":17306,"source_end_line":17312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17306-L17312","statement_sha256":"8298e7f3922b69dd32518471de72f51d8a5b0bf5e0bbf478cd16f0b70cd5c72d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3260,"rank":3260,"depth":7,"x":788.837,"y":719.593,"cluster":"advanced-algebra"},{"id":"stacks:0A5N","tag":"0A5N","title":"Projective dimension · Definition 0A5N","summary":"Let R be a ring. Let K be an object of D(R). We say K has finite projective dimension if K can be represented by a bounded complex of projective modules. We say K has projective-amplitude in [a, b] if K is quasi-isomorphic to a complex … → 0 → P^a → P^a + 1 → … → P^b - 1 → P^b → 0 → … where P^i is a projective R-module for all i ∈ Z.","statement_latex":"Let $R$ be a ring. Let $K$ be an object of $D(R)$. We say $K$ has\n{\\it finite projective dimension} if $K$ can be represented by a\nbounded complex of projective modules. We say $K$ has\n{\\it projective-amplitude in $[a, b]$} if  $K$ is quasi-isomorphic\nto a complex\n$$\n\\ldots \\to 0 \\to P^a \\to P^{a + 1} \\to \\ldots \\to\nP^{b - 1} \\to P^b \\to 0 \\to \\ldots\n$$\nwhere $P^i$ is a projective $R$-module for all $i \\in \\mathbf{Z}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Projective dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5N","source_file":"more-algebra.tex","source_line":17393,"source_end_line":17405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17393-L17405","statement_sha256":"f7bab06c45704205901fa157869a747658cb3b19bb71b928dd3eef3a70d46635","origin":"The Stacks Project","memory_eligible":false,"source_rank":3261,"rank":3261,"depth":0,"x":567.781,"y":712.598,"cluster":"advanced-algebra"},{"id":"stacks:0A5P","tag":"0A5P","title":"Projective dimension · Lemma 0A5P","summary":"Let R be a ring. Let K be an object of D(R). Let a, b ∈ Z. The following are equivalent • K has projective-amplitude in [a, b], • Ext^i_R(K, N) = 0 for all R-modules N and all i not ∈ [-b, -a], • H^n(K) = 0 for n > b and Ext^i_R(K, N) = 0 for all R-modules N and all i > -a, and • H^n(K) = 0 for n not ∈ [a - 1, b] and Ext^-a + 1_R(K, N) = 0 for all R-modules N.","statement_latex":"Let $R$ be a ring. Let $K$ be an object of $D(R)$. Let $a, b \\in \\mathbf{Z}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ has projective-amplitude in $[a, b]$,\n\\item $\\Ext^i_R(K, N) = 0$ for all $R$-modules $N$ and all\n$i \\not \\in [-b, -a]$,\n\\item $H^n(K) = 0$ for $n > b$ and\n$\\Ext^i_R(K, N) = 0$ for all $R$-modules $N$ and all $i > -a$, and\n\\item $H^n(K) = 0$ for $n \\not \\in [a - 1, b]$ and\n$\\Ext^{-a + 1}_R(K, N) = 0$ for all $R$-modules $N$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Projective dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5P","source_file":"more-algebra.tex","source_line":17413,"source_end_line":17426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17413-L17426","statement_sha256":"6ef268427914759daabd40dcd0f1f2c281f8f01535120ba27bc9a93a919c6f3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3262,"rank":3262,"depth":1,"x":736.597,"y":592.19,"cluster":"advanced-algebra"},{"id":"stacks:0A5S","tag":"0A5S","title":"Injective dimension · Definition 0A5S","summary":"Let R be a ring. Let K be an object of D(R). We say K has finite injective dimension if K can be represented by a finite complex of injective R-modules. We say K has injective-amplitude in [a, b] if K is isomorphic to a complex … → 0 → I^a → I^a + 1 → … → I^b - 1 → I^b → 0 → … with I^i an injective R-module for all i ∈ Z.","statement_latex":"Let $R$ be a ring. Let $K$ be an object of $D(R)$.\nWe say $K$ has {\\it finite injective dimension} if $K$ can be\nrepresented by a finite complex of injective $R$-modules.\nWe say $K$ has {\\it injective-amplitude in $[a, b]$}\nif $K$ is isomorphic to a complex\n$$\n\\ldots \\to 0 \\to I^a \\to I^{a + 1} \\to \\ldots \\to\nI^{b - 1} \\to I^b \\to 0 \\to \\ldots\n$$\nwith $I^i$ an injective $R$-module for all $i \\in \\mathbf{Z}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5S","source_file":"more-algebra.tex","source_line":17509,"source_end_line":17521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17509-L17521","statement_sha256":"0b168031c1bd5801364a7ae803599d0b73ad1111062a29a3caf31d82d683763e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3263,"rank":3263,"depth":0,"x":708.913,"y":776.972,"cluster":"advanced-algebra"},{"id":"stacks:0A5T","tag":"0A5T","title":"Injective dimension · Lemma 0A5T","summary":"Let R be a ring. Let K be an object of D(R). Let a, b ∈ Z. The following are equivalent • K has injective-amplitude in [a, b], • Ext^i_R(N, K) = 0 for all R-modules N and all i not ∈ [a, b], • Ext^i(R/I, K) = 0 for all ideals I ⊂ R and all i not ∈ [a, b].","statement_latex":"Let $R$ be a ring. Let $K$ be an object of $D(R)$. Let $a, b \\in \\mathbf{Z}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ has injective-amplitude in $[a, b]$,\n\\item $\\Ext^i_R(N, K) = 0$ for all $R$-modules $N$ and all\n$i \\not \\in [a, b]$,\n\\item $\\Ext^i(R/I, K) = 0$ for all ideals $I \\subset R$ and\nall $i \\not \\in [a, b]$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5T","source_file":"more-algebra.tex","source_line":17529,"source_end_line":17540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17529-L17540","statement_sha256":"586d69b53afd239f61e01dea1fb5671661905ff53063780770795f10138265fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3264,"rank":3264,"depth":2,"x":580.589,"y":624.839,"cluster":"advanced-algebra"},{"id":"stacks:0A5V","tag":"0A5V","title":"Injective dimension · Lemma 0A5V","summary":"Let R be a ring. Let K ∈ D(R). • If K is in D^b(R) and H^i(K) has finite injective dimension for all i, then K has finite injective dimension. • If K^bullet represents K, is a bounded complex of R-modules, and K^i has finite injective dimension for all i, then K has finite injective dimension.","statement_latex":"Let $R$ be a ring. Let $K \\in D(R)$.\n\\begin{enumerate}\n\\item If $K$ is in $D^b(R)$ and $H^i(K)$ has finite injective dimension\nfor all $i$, then $K$ has finite injective dimension.\n\\item If $K^\\bullet$ represents $K$, is a bounded complex of $R$-modules,\nand $K^i$ has finite injective dimension for all $i$, then $K$ has finite\ninjective dimension.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5V","source_file":"more-algebra.tex","source_line":17616,"source_end_line":17626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17616-L17626","statement_sha256":"e60d6457108714911aa7e09c435f5d3da414542d0d6e8bfe08a562b15dc40893","origin":"The Stacks Project","memory_eligible":false,"source_rank":3265,"rank":3265,"depth":7,"x":797.793,"y":664.249,"cluster":"advanced-algebra"},{"id":"stacks:0DW2","tag":"0DW2","title":"Injective dimension · Lemma 0DW2","summary":"Let R be a Noetherian ring. Let I ⊂ R be an ideal contained in the Jacobson radical of R. Let K ∈ D^+(R) have finite cohomology modules. Then the following are equivalent • K has finite injective dimension, and • there exists a b such that Ext^i_R(R/J, K) = 0 for i > b and any ideal J ⊃ I.","statement_latex":"Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal contained\nin the Jacobson radical of $R$. Let $K \\in D^+(R)$ have\nfinite cohomology modules. Then the following are equivalent\n\\begin{enumerate}\n\\item $K$ has finite injective dimension, and\n\\item there exists a $b$ such that $\\Ext^i_R(R/J, K) = 0$ for $i > b$\nand any ideal $J \\supset I$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DW2","source_file":"more-algebra.tex","source_line":17636,"source_end_line":17646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17636-L17646","statement_sha256":"1b22060a535f89e39130a873abf3efd4eeec54bd4e7615bb4ebc1d9623ea50ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":3266,"rank":3266,"depth":3,"x":605.726,"y":758.539,"cluster":"advanced-algebra"},{"id":"stacks:0AVJ","tag":"0AVJ","title":"Injective dimension · Lemma 0AVJ","summary":"Let (R, m, kappa) be a local Noetherian ring. Let K ∈ D^+(R) have finite cohomology modules. Then the following are equivalent • K has finite injective dimension, and • Ext^i_R(kappa, K) = 0 for i gg 0.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local Noetherian ring.\nLet $K \\in D^+(R)$ have finite cohomology modules.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $K$ has finite injective dimension, and\n\\item $\\Ext^i_R(\\kappa, K) = 0$ for $i \\gg 0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Injective dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVJ","source_file":"more-algebra.tex","source_line":17682,"source_end_line":17691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17682-L17691","statement_sha256":"3b31fefa5ba7f58580e62640cf8f9bab59b752599b958c8000bdba048c7e436d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3267,"rank":3267,"depth":4,"x":671.6,"y":579.832,"cluster":"advanced-algebra"},{"id":"stacks:0G90","tag":"0G90","title":"Modules which are close to being projective · Lemma 0G90","summary":"Let R be a ring. Let M, N be R-modules. • Given an R-module map φ : M → N the following are equivalent: (a) φ factors through a projective R-module, and (b) φ factors through a free R-module. • The set of φ : M → N satisfying the equivalent conditions of (1) is an R-submodule of Hom_R(M, N). • Given maps ψ : M' → M and xi : N → N', if φ : M → N satisfies the equivalent conditions of (1), then xi ∘ φ ∘ ψ : M' → N' does too.","statement_latex":"Let $R$ be a ring. Let $M$, $N$ be $R$-modules.\n\\begin{enumerate}\n\\item Given an $R$-module map $\\varphi : M \\to N$ the following are\nequivalent: (a) $\\varphi$ factors through a projective $R$-module, and\n(b) $\\varphi$ factors through a free $R$-module.\n\\item The set of $\\varphi : M \\to N$ satisfying the equivalent\nconditions of (1) is an $R$-submodule of $\\Hom_R(M, N)$.\n\\item Given maps $\\psi : M' \\to M$ and $\\xi : N \\to N'$, if\n$\\varphi : M \\to N$ satisfies the equivalent conditions\nof (1), then $\\xi \\circ \\varphi \\circ \\psi : M' \\to N'$ does too.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Modules which are close to being projective","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G90","source_file":"more-algebra.tex","source_line":17711,"source_end_line":17724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17711-L17724","statement_sha256":"bfbc67e77c1a4a7f608bf227401068974100326ab63b9c33669a2aedeade4e9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3268,"rank":3268,"depth":1,"x":766.842,"y":749.172,"cluster":"advanced-algebra"},{"id":"stacks:0G91","tag":"0G91","title":"Modules which are close to being projective · Lemma 0G91","summary":"Let R be a ring. Let φ : M → N be an R-module map. If φ factors through a projective module and M is a finite R-module, then φ factors through a finite projective module.","statement_latex":"Let $R$ be a ring. Let $\\varphi : M \\to N$ be an $R$-module map.\nIf $\\varphi$ factors through a projective module and $M$ is a\nfinite $R$-module, then $\\varphi$ factors through a finite projective\nmodule.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Modules which are close to being projective","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G91","source_file":"more-algebra.tex","source_line":17737,"source_end_line":17743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17737-L17743","statement_sha256":"49cf7cf508dd71b8d175ebfc61d8768d02b2e0ab090846866efb8327f133477f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3269,"rank":3269,"depth":2,"x":560.208,"y":678.268,"cluster":"advanced-algebra"},{"id":"stacks:0G92","tag":"0G92","title":"Modules which are close to being projective · Lemma 0G92","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. The following conditions are equivalent • for every a ∈ I the map a : M → M factors through a projective R-module, • for every a ∈ I the map a : M → M factors through a free R-module, and • Ext^1_R(M, N) is annihilated by I for every R-module N.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.\nThe following conditions are equivalent\n\\begin{enumerate}\n\\item for every $a \\in I$ the map $a : M \\to M$ factors through a projective\n$R$-module,\n\\item for every $a \\in I$ the map $a : M \\to M$ factors through a free\n$R$-module, and\n\\item $\\Ext^1_R(M, N)$ is annihilated by $I$ for every $R$-module $N$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Modules which are close to being projective","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G92","source_file":"more-algebra.tex","source_line":17759,"source_end_line":17770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17759-L17770","statement_sha256":"3d5551b8a22b2359dfa55e8277c7d70fec876976a8d1bbfffa0f4d2ec4286f53","origin":"The Stacks Project","memory_eligible":false,"source_rank":3270,"rank":3270,"depth":2,"x":769.824,"y":613.231,"cluster":"advanced-algebra"},{"id":"stacks:0G93","tag":"0G93","title":"Modules which are close to being projective · Definition 0G93","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. We say M is I-projective if the equivalent conditions of Lemma [Tag 0G92] hold.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.\nWe say $M$ is {\\it $I$-projective}\\footnote{This is nonstandard notation.}\nif the equivalent conditions of Lemma \\ref{lemma-near-projective} hold.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Modules which are close to being projective","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G93","source_file":"more-algebra.tex","source_line":17789,"source_end_line":17794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17789-L17794","statement_sha256":"45c01692cb1b6d001e376e46bdd7e0efccc5134c0b0cf899fcc710504cc34be5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3271,"rank":3271,"depth":3,"x":667.445,"y":780.311,"cluster":"advanced-algebra"},{"id":"stacks:0G94","tag":"0G94","title":"Modules which are close to being projective · Lemma 0G94","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. If M is annihilated by I, then M is I-projective.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $M$ be an $R$-module.\nIf $M$ is annihilated by $I$, then $M$ is $I$-projective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Modules which are close to being projective","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G94","source_file":"more-algebra.tex","source_line":17799,"source_end_line":17803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17799-L17803","statement_sha256":"8798aff3a600c1f2de0adfc6ecb08b1ed6bc028d7e3f50066523e0bf9cf3475b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3272,"rank":3272,"depth":0,"x":608.513,"y":598.82,"cluster":"advanced-algebra"},{"id":"stacks:0G95","tag":"0G95","title":"Modules which are close to being projective · Lemma 0G95","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let 0 → K → P → M → 0 be a short exact sequence of R-modules. If M is I-projective and P is projective, then K is I-projective.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let\n$$\n0 \\to K \\to P \\to M \\to 0\n$$\nbe a short exact sequence of $R$-modules.\nIf $M$ is $I$-projective and $P$ is projective, then $K$ is $I$-projective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Modules which are close to being projective","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G95","source_file":"more-algebra.tex","source_line":17809,"source_end_line":17817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17809-L17817","statement_sha256":"e29ba6fdce8a7936374695010e5ac3f628549c40107ad0e95075cbfb28e1c4c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3273,"rank":3273,"depth":0,"x":798.124,"y":699.318,"cluster":"advanced-algebra"},{"id":"stacks:0G96","tag":"0G96","title":"Modules which are close to being projective · Lemma 0G96","summary":"Let R be a ring. Let I ⊂ R be an ideal. If M is a finite, I-projective R-module, then M^vee = Hom_R(M, R) is I-projective.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nIf $M$ is a finite, $I$-projective $R$-module, then\n$M^\\vee = \\Hom_R(M, R)$ is $I$-projective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Modules which are close to being projective","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G96","source_file":"more-algebra.tex","source_line":17826,"source_end_line":17831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17826-L17831","statement_sha256":"c1ac1eea8101520094f6f61e5a9edbceb234363c2b5e3b6b4254a51da3f0bfae","origin":"The Stacks Project","memory_eligible":false,"source_rank":3274,"rank":3274,"depth":0,"x":577.251,"y":732.839,"cluster":"advanced-algebra"},{"id":"stacks:0A5Y","tag":"0A5Y","title":"Hom complexes · Lemma 0A5Y","summary":"Let R be a ring. Given complexes K^bullet, L^bullet, M^bullet of R-modules there is a canonical isomorphism Hom^bullet(K^bullet, Hom^bullet(L^bullet, M^bullet)) = Hom^bullet(Tot(K^bullet ⊗_R L^bullet), M^bullet) of complexes of R-modules.","statement_latex":"Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$\nof $R$-modules there is a canonical isomorphism\n$$\n\\Hom^\\bullet(K^\\bullet, \\Hom^\\bullet(L^\\bullet, M^\\bullet))\n=\n\\Hom^\\bullet(\\text{Tot}(K^\\bullet \\otimes_R L^\\bullet), M^\\bullet)\n$$\nof complexes of $R$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5Y","source_file":"more-algebra.tex","source_line":17880,"source_end_line":17890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17880-L17890","statement_sha256":"8cd30598ca8ff2c12815e49bbe397d982fb55bb654d205fd5f1130864320d4fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3275,"rank":3275,"depth":1,"x":713.31,"y":582.631,"cluster":"advanced-algebra"},{"id":"stacks:0A8I","tag":"0A8I","title":"Hom complexes · Lemma 0A8I","summary":"Let R be a ring. Given complexes K^bullet, L^bullet, M^bullet of R-modules there is a canonical morphism Tot( Hom^bullet(L^bullet, M^bullet) ⊗_R Hom^bullet(K^bullet, L^bullet) ) → Hom^bullet(K^bullet, M^bullet) of complexes of R-modules.","statement_latex":"Let $R$ be a ring. Given complexes\n$K^\\bullet, L^\\bullet, M^\\bullet$\nof $R$-modules there is a canonical morphism\n$$\n\\text{Tot}\\left(\n\\Hom^\\bullet(L^\\bullet, M^\\bullet) \\otimes_R \\Hom^\\bullet(K^\\bullet, L^\\bullet)\n\\right)\n\\longrightarrow\n\\Hom^\\bullet(K^\\bullet, M^\\bullet)\n$$\nof complexes of $R$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8I","source_file":"more-algebra.tex","source_line":17965,"source_end_line":17978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L17965-L17978","statement_sha256":"9e166d16ffb6d1e95b41e1ef368d040cb439b80a83584c33a1cac3e98a86a460","origin":"The Stacks Project","memory_eligible":false,"source_rank":3276,"rank":3276,"depth":0,"x":733.797,"y":770.797,"cluster":"advanced-algebra"},{"id":"stacks:0BYM","tag":"0BYM","title":"Hom complexes · Lemma 0BYM","summary":"Let R be a ring. Given complexes K^bullet, L^bullet, M^bullet of R-modules there is a canonical morphism Tot(K^bullet ⊗_R Hom^bullet(M^bullet, L^bullet)) → Hom^bullet(M^bullet, Tot(K^bullet ⊗_R L^bullet)) of complexes of R-modules functorial in all three complexes.","statement_latex":"Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$\nof $R$-modules there is a canonical morphism\n$$\n\\text{Tot}(K^\\bullet \\otimes_R \\Hom^\\bullet(M^\\bullet, L^\\bullet))\n\\longrightarrow\n\\Hom^\\bullet(M^\\bullet, \\text{Tot}(K^\\bullet \\otimes_R L^\\bullet))\n$$\nof complexes of $R$-modules functorial in all three complexes.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYM","source_file":"more-algebra.tex","source_line":18035,"source_end_line":18045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18035-L18045","statement_sha256":"6316ae6b2d2b721d840982539c867a7492477f096648e90c9cb743fa5554daa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3277,"rank":3277,"depth":0,"x":567.193,"y":643.534,"cluster":"advanced-algebra"},{"id":"stacks:0A62","tag":"0A62","title":"Hom complexes · Lemma 0A62","summary":"Let R be a ring. Given complexes K^bullet, L^bullet of R-modules there is a canonical morphism K^bullet → Hom^bullet(L^bullet, Tot(K^bullet ⊗_R L^bullet)) of complexes of R-modules functorial in both complexes.","statement_latex":"Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet$\nof $R$-modules there is a canonical morphism\n$$\nK^\\bullet\n\\longrightarrow\n\\Hom^\\bullet(L^\\bullet, \\text{Tot}(K^\\bullet \\otimes_R L^\\bullet))\n$$\nof complexes of $R$-modules functorial in both complexes.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A62","source_file":"more-algebra.tex","source_line":18113,"source_end_line":18123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18113-L18123","statement_sha256":"be2e7fa8f7a5cf11a5e9a0ef6a74b579749ec181657e3e64095b9caf18c73d53","origin":"The Stacks Project","memory_eligible":false,"source_rank":3278,"rank":3278,"depth":0,"x":792.627,"y":642.84,"cluster":"advanced-algebra"},{"id":"stacks:0A60","tag":"0A60","title":"Hom complexes · Lemma 0A60","summary":"Let R be a ring. Given complexes K^bullet, L^bullet, M^bullet of R-modules there is a canonical morphism Tot(Hom^bullet(L^bullet, M^bullet) ⊗_R K^bullet) → Hom^bullet(Hom^bullet(K^bullet, L^bullet), M^bullet) of complexes of R-modules functorial in all three complexes.","statement_latex":"Let $R$ be a ring. Given complexes $K^\\bullet, L^\\bullet, M^\\bullet$\nof $R$-modules there is a canonical morphism\n$$\n\\text{Tot}(\\Hom^\\bullet(L^\\bullet, M^\\bullet) \\otimes_R K^\\bullet)\n\\longrightarrow\n\\Hom^\\bullet(\\Hom^\\bullet(K^\\bullet, L^\\bullet), M^\\bullet)\n$$\nof complexes of $R$-modules functorial in all three complexes.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A60","source_file":"more-algebra.tex","source_line":18163,"source_end_line":18173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18163-L18173","statement_sha256":"e4f7a2830697cd70b5499df731473435de6d8530c62b6c66076b9210d8dfc2e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3279,"rank":3279,"depth":0,"x":626.777,"y":771.406,"cluster":"advanced-algebra"},{"id":"stacks:0FNJ","tag":"0FNJ","title":"Sign rules · Lemma 0FNJ","summary":"Let R be a ring. Let M be an R-module. Let N, eta, ε be a left dual of M in the monoidal category of R-modules, see Categories, Definition [Tag 0FFP]. Then • M and N are finite projective R-modules, • the map e : Hom_R(M, R) → N, λ ↦ (λ ⊗ 1)(eta) is an isomorphism, • we have ε(n, m) = e^-1(n)(m) for n ∈ N and m ∈ M.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. Let  $N, \\eta, \\epsilon$\nbe a left dual of $M$ in the monoidal category of $R$-modules, see\nCategories, Definition \\ref{categories-definition-dual}. Then\n\\begin{enumerate}\n\\item $M$ and $N$ are finite projective $R$-modules,\n\\item the map\n$e : \\Hom_R(M, R) \\to N$, $\\lambda \\mapsto (\\lambda \\otimes 1)(\\eta)$\nis an isomorphism,\n\\item we have $\\epsilon(n, m) = e^{-1}(n)(m)$ for $n \\in N$ and $m \\in M$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Sign rules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNJ","source_file":"more-algebra.tex","source_line":18418,"source_end_line":18430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18418-L18430","statement_sha256":"39e3cad33247c29fc84a2207417638f61a8de446f609edbc6e6ae7e833b300cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3280,"rank":3280,"depth":1,"x":645.703,"y":582.294,"cluster":"advanced-algebra"},{"id":"stacks:0FNK","tag":"0FNK","title":"Sign rules · Lemma 0FNK","summary":"Let R be a ring. Let M^bullet be a complex of R-modules. Let N^bullet, eta, ε be a left dual of M^bullet in the monoidal category of complexes of R-modules. Then • M^bullet and N^bullet are bounded, • M^n and N^n are finite projective R-modules, • writing ε = ∑ ε_n with ε_n : N^-n ⊗_R M^n → R and eta = ∑ eta_n with eta_n : R → M^n ⊗_R N^-n then (N^-n, eta_n, ε_n) is the left dual of M^n as in Lemma [Tag 0FNJ], • the differential d_N^n : N^n → N^n + 1 is equal to -(-1)^n…","statement_latex":"Let $R$ be a ring. Let $M^\\bullet$ be a complex of $R$-modules.\nLet $N^\\bullet, \\eta, \\epsilon$ be a left dual of $M^\\bullet$\nin the monoidal category of complexes of $R$-modules.\nThen\n\\begin{enumerate}\n\\item $M^\\bullet$ and $N^\\bullet$ are bounded,\n\\item $M^n$ and $N^n$ are finite projective $R$-modules,\n\\item writing $\\epsilon = \\sum \\epsilon_n$\nwith $\\epsilon_n : N^{-n} \\otimes_R M^n \\to R$ and $\\eta = \\sum \\eta_n$\nwith $\\eta_n : R \\to  M^n \\otimes_R N^{-n}$ then $(N^{-n}, \\eta_n, \\epsilon_n)$\nis the left dual of $M^n$ as in Lemma \\ref{lemma-left-dual-module},\n\\item the differential $d_N^n : N^n \\to N^{n + 1}$ is equal\nto $-(-1)^n$ times the map\n$$\nN^n = \\Hom_R(M^{-n}, R)\n\\xrightarrow{d_M^{-n - 1}}\n\\Hom_R(M^{-n - 1}, R) = N^{n + 1}\n$$\nwhere the equality signs are the identifications from\nLemma \\ref{lemma-left-dual-module} part (2).\n\\end{enumerate}\nConversely, given a bounded complex $M^\\bullet$ of finite projective\n$R$-modules, setting $N^n = \\Hom_R(M^{-n}, R)$ with differentials as above,\nsetting $\\epsilon = \\sum \\epsilon_n$ with\n$\\epsilon_n : N^{-n} \\otimes_R M^n \\to R$ given by evaluation, and\nsetting $\\eta = \\sum \\eta_n$ with $\\eta_n : R \\to M^n \\otimes_R N^{-n}$\nmapping $1$ to $\\text{id}_{M_n}$ we obtain a left dual of $M^\\bullet$\nin the monoidal category of complexes of $R$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Sign rules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNK","source_file":"more-algebra.tex","source_line":18461,"source_end_line":18491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18461-L18491","statement_sha256":"23c4953a4119b262a9a6033aaa9be6b44c58a0549d004d4d69982f9f1ad20b09","origin":"The Stacks Project","memory_eligible":false,"source_rank":3281,"rank":3281,"depth":2,"x":783.973,"y":732.642,"cluster":"advanced-algebra"},{"id":"stacks:0A65","tag":"0A65","title":"Derived hom · Lemma 0A65","summary":"Let R be a ring. Let K, L, M be objects of D(R). There is a canonical isomorphism RHom_R(K, RHom_R(L, M)) = RHom_R(K ⊗_R^L L, M) in D(R) functorial in K, L, M which recovers ([Tag 0A63]) by taking H^0.","statement_latex":"Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical\nisomorphism\n$$\nR\\Hom_R(K, R\\Hom_R(L, M)) = R\\Hom_R(K \\otimes_R^\\mathbf{L} L, M)\n$$\nin $D(R)$ functorial in $K, L, M$ which recovers\n(\\ref{equation-internal-hom}) by taking $H^0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A65","source_file":"more-algebra.tex","source_line":18681,"source_end_line":18690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18681-L18690","statement_sha256":"7e0ad366ec5c1b25506a6addfe5264b00fcb13bd4e2a7a07cafa4db5dc046131","origin":"The Stacks Project","memory_eligible":false,"source_rank":3282,"rank":3282,"depth":2,"x":560.873,"y":700.201,"cluster":"advanced-algebra"},{"id":"stacks:0A66","tag":"0A66","title":"Derived hom · Lemma 0A66","summary":"Let R be a ring. Let P^bullet be a bounded above complex of projective R-modules. Let L^bullet be a complex of R-modules. Then RHom_R(P^bullet, L^bullet) is represented by the complex Hom^bullet(P^bullet, L^bullet).","statement_latex":"Let $R$ be a ring. Let $P^\\bullet$ be a bounded above complex\nof projective $R$-modules. Let $L^\\bullet$ be a complex of $R$-modules.\nThen $R\\Hom_R(P^\\bullet, L^\\bullet)$ is represented by the complex\n$\\Hom^\\bullet(P^\\bullet, L^\\bullet)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A66","source_file":"more-algebra.tex","source_line":18707,"source_end_line":18713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18707-L18713","statement_sha256":"c8435ae6322ff571fe7f6004557f6cd9c0a604ad3f7722f2d23e89df4e638271","origin":"The Stacks Project","memory_eligible":false,"source_rank":3283,"rank":3283,"depth":5,"x":751.674,"y":597.421,"cluster":"advanced-algebra"},{"id":"stacks:0A67","tag":"0A67","title":"Derived hom · Lemma 0A67","summary":"Let R be a ring. Let K, L, M be objects of D(R). There is a canonical morphism RHom_R(L, M) ⊗_R^L K → RHom_R(RHom_R(K, L), M) in D(R) functorial in K, L, M.","statement_latex":"Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$.\nThere is a canonical morphism\n$$\nR\\Hom_R(L, M) \\otimes_R^\\mathbf{L} K\n\\longrightarrow\nR\\Hom_R(R\\Hom_R(K, L), M)\n$$\nin $D(R)$ functorial in $K, L, M$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A67","source_file":"more-algebra.tex","source_line":18731,"source_end_line":18741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18731-L18741","statement_sha256":"649d7762d4d474a232436b27c50a119e0e378c1947bcdd45b92f65da455b2aae","origin":"The Stacks Project","memory_eligible":false,"source_rank":3284,"rank":3284,"depth":1,"x":693.571,"y":781.669,"cluster":"advanced-algebra"},{"id":"stacks:0A8J","tag":"0A8J","title":"Derived hom · Lemma 0A8J","summary":"Let R be a ring. Given K, L, M in D(R) there is a canonical morphism RHom_R(L, M) ⊗_R^L RHom_R(K, L) → RHom_R(K, M) in D(R) functorial in K, L, M.","statement_latex":"Let $R$ be a ring. Given $K, L, M$ in $D(R)$ there is a canonical morphism\n$$\nR\\Hom_R(L, M) \\otimes_R^\\mathbf{L} R\\Hom_R(K, L) \\longrightarrow R\\Hom_R(K, M)\n$$\nin $D(R)$ functorial in $K, L, M$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8J","source_file":"more-algebra.tex","source_line":18758,"source_end_line":18765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18758-L18765","statement_sha256":"6baba08fd63ad334992e8033e9c60927f68f429ff929724372730b153584207e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3285,"rank":3285,"depth":1,"x":588.137,"y":612.66,"cluster":"advanced-algebra"},{"id":"stacks:0BYN","tag":"0BYN","title":"Derived hom · Lemma 0BYN","summary":"Let R be a ring. Given complexes K, L, M in D(R) there is a canonical morphism K ⊗_R^L RHom_R(M, L) → RHom_R(M, K ⊗_R^L L) in D(R) functorial in K, L, M.","statement_latex":"Let $R$ be a ring. Given complexes $K, L, M$ in $D(R)$\nthere is a canonical morphism\n$$\nK \\otimes_R^\\mathbf{L} R\\Hom_R(M, L)\n\\longrightarrow\nR\\Hom_R(M, K \\otimes_R^\\mathbf{L} L)\n$$\nin $D(R)$ functorial in $K$, $L$, $M$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYN","source_file":"more-algebra.tex","source_line":18798,"source_end_line":18808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18798-L18808","statement_sha256":"d737376a707f1ea099ffef86fd41673c4b8f51686a2e6661e1c100e5312e6edb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3286,"rank":3286,"depth":1,"x":802.019,"y":677.527,"cluster":"advanced-algebra"},{"id":"stacks:0A6B","tag":"0A6B","title":"Derived hom · Lemma 0A6B","summary":"Let R be a ring. Given complexes K, L in D(R) there is a canonical morphism K → RHom_R(L, K ⊗_R^L L) in D(R) functorial in both K and L.","statement_latex":"Let $R$ be a ring. Given complexes $K, L$ in $D(R)$\nthere is a canonical morphism\n$$\nK \\longrightarrow R\\Hom_R(L, K \\otimes_R^\\mathbf{L} L)\n$$\nin $D(R)$ functorial in both $K$ and $L$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6B","source_file":"more-algebra.tex","source_line":18829,"source_end_line":18837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18829-L18837","statement_sha256":"3a5fdcc8cad9f1759b4678bd4306e07b4e450a433a54f2360b8be7fcd74b6fe9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3287,"rank":3287,"depth":2,"x":591.921,"y":751.135,"cluster":"advanced-algebra"},{"id":"stacks:0657","tag":"0657","title":"Perfect complexes · Definition 0657","summary":"Let R be a ring. Denote D(R) the derived category of the abelian category of R-modules. • An object K of D(R) is perfect if it is quasi-isomorphic to a bounded complex of finite projective R-modules. • An R-module M is perfect if M[0] is a perfect object in D(R).","statement_latex":"Let $R$ be a ring. Denote $D(R)$ the derived category of the abelian\ncategory of $R$-modules.\n\\begin{enumerate}\n\\item An object $K$ of $D(R)$ is {\\it perfect} if it is quasi-isomorphic\nto a bounded complex of finite projective $R$-modules.\n\\item An $R$-module $M$ is {\\it perfect} if $M[0]$ is a perfect object\nin $D(R)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0657","source_file":"more-algebra.tex","source_line":18870,"source_end_line":18880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18870-L18880","statement_sha256":"caa84dea1e6487fbe7eaf96e33de87bf12b088d9daf894a8566fe6331e67da1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3288,"rank":3288,"depth":0,"x":687.752,"y":577.461,"cluster":"advanced-algebra"},{"id":"stacks:0658","tag":"0658","title":"Perfect complexes · Lemma 0658","summary":"Let K^bullet be an object of D(R). The following are equivalent • K^bullet is perfect, and • K^bullet is pseudo-coherent and has finite tor dimension. If (1) and (2) hold and K^bullet has tor-amplitude in [a, b], then K^bullet is quasi-isomorphic to a complex E^bullet of finite projective R-modules with E^i = 0 for i not ∈ [a, b].","statement_latex":"Let $K^\\bullet$ be an object of $D(R)$. The following are equivalent\n\\begin{enumerate}\n\\item $K^\\bullet$ is perfect, and\n\\item $K^\\bullet$ is pseudo-coherent and has finite tor dimension.\n\\end{enumerate}\nIf (1) and (2) hold and $K^\\bullet$ has tor-amplitude\nin $[a, b]$, then $K^\\bullet$ is quasi-isomorphic to a complex\n$E^\\bullet$ of finite projective $R$-modules with $E^i = 0$\nfor $i \\not \\in [a, b]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0658","source_file":"more-algebra.tex","source_line":18888,"source_end_line":18899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18888-L18899","statement_sha256":"eb1bf4cc49ce8c507f9991bc93726d52d3fc0dcec1e5291b143cbbd8b2d62d42","origin":"The Stacks Project","memory_eligible":false,"source_rank":3289,"rank":3289,"depth":10,"x":756.823,"y":760.093,"cluster":"advanced-algebra"},{"id":"stacks:066Q","tag":"066Q","title":"Perfect complexes · Lemma 066Q","summary":"Let M be a module over a ring R. The following are equivalent • M is a perfect module, and • there exists a resolution 0 → F_d → … → F_1 → F_0 → M → 0 with each F_i a finite projective R-module.","statement_latex":"Let $M$ be a module over a ring $R$. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is a perfect module, and\n\\item there exists a resolution\n$$\n0 \\to F_d \\to \\ldots \\to F_1 \\to F_0 \\to M \\to 0\n$$\nwith each $F_i$ a finite projective $R$-module.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066Q","source_file":"more-algebra.tex","source_line":18916,"source_end_line":18927,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18916-L18927","statement_sha256":"b9611d3b16ade0e97a043160de9fd71c55929d3c891fa7cdf9d6354a9dd046ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":3290,"rank":3290,"depth":11,"x":558.817,"y":664.516,"cluster":"advanced-algebra"},{"id":"stacks:066R","tag":"066R","title":"Perfect complexes · Lemma 066R","summary":"Let R be a ring. Let (K^bullet, L^bullet, M^bullet, f, g, h) be a distinguished triangle in D(R). If two out of three of K^bullet, L^bullet, M^bullet are perfect then the third is also perfect.","statement_latex":"Let $R$ be a ring. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$\nbe a distinguished triangle in $D(R)$. If two out of three of\n$K^\\bullet, L^\\bullet, M^\\bullet$ are\nperfect then the third is also perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066R","source_file":"more-algebra.tex","source_line":18948,"source_end_line":18954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18948-L18954","statement_sha256":"580ee68e2d0d8c30180400016a5c874b9dcb0f1a1f208175b7c704161fcdff29","origin":"The Stacks Project","memory_eligible":false,"source_rank":3291,"rank":3291,"depth":11,"x":781.916,"y":622.596,"cluster":"advanced-algebra"},{"id":"stacks:066S","tag":"066S","title":"Perfect complexes · Lemma 066S","summary":"Let R be a ring. If K^bullet ⊕ L^bullet is perfect, then so are K^bullet and L^bullet.","statement_latex":"Let $R$ be a ring. If $K^\\bullet \\oplus L^\\bullet$ is perfect, then\nso are $K^\\bullet$ and $L^\\bullet$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066S","source_file":"more-algebra.tex","source_line":18962,"source_end_line":18966,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18962-L18966","statement_sha256":"43032d242ab379620c2ed6e74ca0ca60be2831b4c1d3058c5ba6b5d0f92a74b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3292,"rank":3292,"depth":12,"x":650.981,"y":780.263,"cluster":"advanced-algebra"},{"id":"stacks:066T","tag":"066T","title":"Perfect complexes · Lemma 066T","summary":"Let R be a ring. Let K^bullet be a bounded complex of perfect R-modules. Then K^bullet is a perfect complex.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a bounded complex of perfect\n$R$-modules. Then $K^\\bullet$ is a perfect complex.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066T","source_file":"more-algebra.tex","source_line":18974,"source_end_line":18978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18974-L18978","statement_sha256":"82d88d984926108bb7cd8c73f40eb830800684686076f45f78a38360bfc79826","origin":"The Stacks Project","memory_eligible":false,"source_rank":3293,"rank":3293,"depth":12,"x":620.708,"y":589.508,"cluster":"advanced-algebra"},{"id":"stacks:066U","tag":"066U","title":"Perfect complexes · Lemma 066U","summary":"Let R be a ring. If K^bullet ∈ D^b(R) and all its cohomology modules are perfect, then K^bullet is perfect.","statement_latex":"Let $R$ be a ring. If $K^\\bullet \\in D^b(R)$ and all its cohomology\nmodules are perfect, then $K^\\bullet$ is perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066U","source_file":"more-algebra.tex","source_line":18986,"source_end_line":18990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18986-L18990","statement_sha256":"3146ec0bfc6cd476ad054a156bedf6f3220e4ba3482ae1a9ea53c4349f330157","origin":"The Stacks Project","memory_eligible":false,"source_rank":3294,"rank":3294,"depth":12,"x":796.612,"y":713.118,"cluster":"advanced-algebra"},{"id":"stacks:066V","tag":"066V","title":"Perfect complexes · Lemma 066V","summary":"Let A → B be a ring map. Assume that B is perfect as an A-module. Let K^bullet be a perfect complex of B-modules. Then K^bullet is perfect as a complex of A-modules.","statement_latex":"Let $A \\to B$ be a ring map. Assume that $B$ is perfect as\nan $A$-module. Let $K^\\bullet$ be a perfect complex of $B$-modules.\nThen $K^\\bullet$ is perfect as a complex of $A$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066V","source_file":"more-algebra.tex","source_line":18998,"source_end_line":19003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L18998-L19003","statement_sha256":"f1513306a997a4e7e372222dbef72c1863b0d8680ef521c4d2f77a475e10db95","origin":"The Stacks Project","memory_eligible":false,"source_rank":3295,"rank":3295,"depth":12,"x":567.263,"y":721.791,"cluster":"advanced-algebra"},{"id":"stacks:066W","tag":"066W","title":"Perfect complexes · Lemma 066W","summary":"Let A → B be a ring map. Let K^bullet be a perfect complex of A-modules. Then K^bullet ⊗_A^L B is a perfect complex of B-modules.","statement_latex":"Let $A \\to B$ be a ring map.\nLet $K^\\bullet$ be a perfect\ncomplex of $A$-modules. Then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$\nis a perfect complex of $B$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066W","source_file":"more-algebra.tex","source_line":19016,"source_end_line":19022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19016-L19022","statement_sha256":"ae78bac8542c558084d24eb72b5032cb0dc2d1a187060295c72f84e162906fa6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3296,"rank":3296,"depth":11,"x":729.574,"y":585.117,"cluster":"advanced-algebra"},{"id":"stacks:066X","tag":"066X","title":"Perfect complexes · Lemma 066X","summary":"Let A → B be a flat ring map. Let M be a perfect A-module. Then M ⊗_A B is a perfect B-module.","statement_latex":"Let $A \\to B$ be a flat ring map. Let $M$ be a perfect $A$-module.\nThen $M \\otimes_A B$ is a perfect $B$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066X","source_file":"more-algebra.tex","source_line":19035,"source_end_line":19039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19035-L19039","statement_sha256":"b29bcf90160504542b2bf3878e651116b49337481524105cd8969d90cfcccafb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3297,"rank":3297,"depth":12,"x":719.788,"y":778.196,"cluster":"advanced-algebra"},{"id":"stacks:0GM0","tag":"0GM0","title":"Perfect complexes · Lemma 0GM0","summary":"Let R be a ring. If K and L are perfect objects of D(R), then K ⊗_R^L L is a perfect object too.","statement_latex":"Let $R$ be a ring. If $K$ and $L$ are perfect objects of $D(R)$, then\n$K \\otimes_R^\\mathbf{L} L$ is a perfect object too.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GM0","source_file":"more-algebra.tex","source_line":19050,"source_end_line":19054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19050-L19054","statement_sha256":"5fb51ceaaf3dab70050d422792fd7ae43ec5339c6b4c92c9f2c62a14a960b4f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3298,"rank":3298,"depth":11,"x":571.584,"y":630.118,"cluster":"advanced-algebra"},{"id":"stacks:066Y","tag":"066Y","title":"Perfect complexes · Lemma 066Y","summary":"Let R be a ring. Let f_1, …, f_r ∈ R be elements which generate the unit ideal. Let K^bullet be a complex of R-modules. If for each i the complex K^bullet ⊗_R R_f_i is perfect, then K^bullet is perfect.","statement_latex":"Let $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$ be elements which\ngenerate the unit ideal. Let $K^\\bullet$\nbe a complex of $R$-modules. If for each $i$ the complex\n$K^\\bullet \\otimes_R R_{f_i}$ is perfect,\nthen $K^\\bullet$ is perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066Y","source_file":"more-algebra.tex","source_line":19076,"source_end_line":19083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19076-L19083","statement_sha256":"f4507fe57ae81dc2eb487ce1c25c51bc64183661be760fecc8a6a8a2d212bf73","origin":"The Stacks Project","memory_eligible":false,"source_rank":3299,"rank":3299,"depth":12,"x":800.182,"y":655.239,"cluster":"advanced-algebra"},{"id":"stacks:068T","tag":"068T","title":"Perfect complexes · Lemma 068T","summary":"Let R be a ring. Let K^bullet be a complex of R-modules. Let R → R' be a faithfully flat ring map. If the complex K^bullet ⊗_R R' is perfect, then K^bullet is perfect.","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.\nLet $R \\to R'$ be a faithfully flat ring map. If the complex\n$K^\\bullet \\otimes_R R'$ is perfect, then $K^\\bullet$ is perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068T","source_file":"more-algebra.tex","source_line":19096,"source_end_line":19101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19096-L19101","statement_sha256":"9802d96be6b775921cf98827b6196c22471ff05e07f01233432255d41f71b2cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3300,"rank":3300,"depth":12,"x":611.222,"y":766.54,"cluster":"advanced-algebra"},{"id":"stacks:066Z","tag":"066Z","title":"Perfect complexes · Lemma 066Z","summary":"Let R be a regular ring. Then • an R-module is perfect if and only if it is a finite R-module, and • a complex of R-modules K^bullet is perfect if and only if K^bullet ∈ D^b(R) and each H^i(K^bullet) is a finite R-module.","statement_latex":"Let $R$ be a regular ring. Then\n\\begin{enumerate}\n\\item an $R$-module is perfect if and only if it is a finite $R$-module, and\n\\item a complex of $R$-modules $K^\\bullet$ is perfect if and only\nif $K^\\bullet \\in D^b(R)$ and each $H^i(K^\\bullet)$ is a finite $R$-module.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/066Z","source_file":"more-algebra.tex","source_line":19114,"source_end_line":19122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19114-L19122","statement_sha256":"b8ed5722e8c4c4c92a616c9bfbb0fb4436dcc2f36081474a1779c4d152a5aa5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3301,"rank":3301,"depth":16,"x":661.103,"y":577.055,"cluster":"advanced-algebra"},{"id":"stacks:07VI","tag":"07VI","title":"Perfect complexes · Lemma 07VI","summary":"Let A be a ring. Let K ∈ D(A) be perfect. Then K^vee = RHom_A(K, A) is a perfect complex and K ≅ (K^vee)^vee. There are functorial isomorphisms L ⊗_A^L K^vee = RHom_A(K, L) and H^0(L ⊗_A^L K^vee) = Ext_A^0(K, L) for L ∈ D(A).","statement_latex":"Let $A$ be a ring. Let $K \\in D(A)$ be perfect. Then $K^\\vee = R\\Hom_A(K, A)$\nis a perfect complex and $K \\cong (K^\\vee)^\\vee$. There are functorial\nisomorphisms\n$$\nL \\otimes_A^\\mathbf{L} K^\\vee = R\\Hom_A(K, L)\n\\quad\\text{and}\\quad\nH^0(L \\otimes_A^\\mathbf{L} K^\\vee) = \\Ext_A^0(K, L)\n$$\nfor $L \\in D(A)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VI","source_file":"more-algebra.tex","source_line":19175,"source_end_line":19186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19175-L19186","statement_sha256":"54705ff9ee34eedcccc765f65e1ac7cce8d5e27e123559932bf525906f6b9628","origin":"The Stacks Project","memory_eligible":false,"source_rank":3302,"rank":3302,"depth":6,"x":776.818,"y":745.254,"cluster":"advanced-algebra"},{"id":"stacks:0BKB","tag":"0BKB","title":"Perfect complexes · Lemma 0BKB","summary":"Trivial duality for systems of perfect objects. Let A be a ring. Let (K_n)_n ∈ N be a system of perfect objects of D(A). Let K = hocolim K_n be the derived colimit (Derived Categories, Definition [Tag 090Z]). Then for any object E of D(A) we have RHom_A(K, E) = Rlim E ⊗^L_A K_n^vee where (K_n^vee) is the inverse system of dual perfect complexes.","statement_latex":"\\begin{slogan}\nTrivial duality for systems of perfect objects.\n\\end{slogan}\nLet $A$ be a ring. Let $(K_n)_{n \\in \\mathbf{N}}$ be a system of\nperfect objects of $D(A)$. Let $K = \\text{hocolim} K_n$ be the derived colimit\n(Derived Categories, Definition \\ref{derived-definition-derived-colimit}).\nThen for any object $E$ of $D(A)$ we have\n$$\nR\\Hom_A(K, E) = R\\lim E \\otimes^\\mathbf{L}_A K_n^\\vee\n$$\nwhere $(K_n^\\vee)$ is the inverse system of dual perfect complexes.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKB","source_file":"more-algebra.tex","source_line":19235,"source_end_line":19248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19235-L19248","statement_sha256":"52d43dccd666c9e63cc81db37e74fa645d1c3d1d25ac127e19c67833e51627eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3303,"rank":3303,"depth":7,"x":556.007,"y":686.827,"cluster":"advanced-algebra"},{"id":"stacks:0BC7","tag":"0BC7","title":"Perfect complexes · Lemma 0BC7","summary":"Let R = colim_i ∈ I R_i be a filtered colimit of rings. • Given a perfect K in D(R) there exists an i ∈ I and a perfect K_i in D(R_i) such that K ≅ K_i ⊗_R_i^L R in D(R). • Given 0 ∈ I and K_0, L_0 ∈ D(R_0) with K_0 perfect, we have Hom_D(R)(K_0 ⊗_R_0^L R, L_0 ⊗_R_0^L R) = colim_i ≥ 0 Hom_D(R_i)(K_0 ⊗_R_0^L R_i, L_0 ⊗_R_0^L R_i) In other words, the triangulated category of perfect complexes over R is the colimit of the triangulated categories of perfect complexes over R_i.","statement_latex":"Let $R = \\colim_{i \\in I} R_i$ be a filtered colimit of rings.\n\\begin{enumerate}\n\\item Given a perfect $K$ in $D(R)$ there exists an $i \\in I$\nand a perfect $K_i$ in $D(R_i)$ such that\n$K \\cong K_i \\otimes_{R_i}^\\mathbf{L} R$ in $D(R)$.\n\\item Given $0 \\in I$ and $K_0, L_0 \\in D(R_0)$ with $K_0$ perfect,\nwe have\n$$\n\\Hom_{D(R)}(K_0 \\otimes_{R_0}^\\mathbf{L} R, L_0 \\otimes_{R_0}^\\mathbf{L} R) =\n\\colim_{i \\geq 0}\n\\Hom_{D(R_i)}(K_0 \\otimes_{R_0}^\\mathbf{L} R_i,\nL_0 \\otimes_{R_0}^\\mathbf{L} R_i)\n$$\n\\end{enumerate}\nIn other words, the triangulated category of perfect complexes over $R$\nis the colimit of the triangulated categories of perfect complexes over $R_i$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BC7","source_file":"more-algebra.tex","source_line":19267,"source_end_line":19285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19267-L19285","statement_sha256":"8152402afc79bc3d698e2be412bfe0394106c91cb0bd00aec0335cbd71bb4353","origin":"The Stacks Project","memory_eligible":false,"source_rank":3304,"rank":3304,"depth":15,"x":766.028,"y":604.533,"cluster":"advanced-algebra"},{"id":"stacks:0BC9","tag":"0BC9","title":"Lifting complexes · Lemma 0BC9","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let P be a class of R-modules. Assume • each P ∈ P is a projective R-module, • if P_1 ∈ P and P_1 ⊕ P_2 ∈ P, then P_2 ∈ P, and • if f : P_1 → P_2, P_1, P_2 ∈ P is surjective modulo I, then f is surjective. Then given any bounded above acyclic complex E^bullet whose terms are of the form P/IP for P ∈ P there exists a bounded above acyclic complex P^bullet whose terms are in P lifting E^bullet.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $\\mathcal{P}$\nbe a class of $R$-modules. Assume\n\\begin{enumerate}\n\\item each $P \\in \\mathcal{P}$ is a projective $R$-module,\n\\item if $P_1 \\in \\mathcal{P}$ and $P_1 \\oplus P_2 \\in \\mathcal{P}$, then\n$P_2 \\in \\mathcal{P}$, and\n\\item if $f : P_1 \\to P_2$, $P_1, P_2 \\in \\mathcal{P}$ is surjective\nmodulo $I$, then $f$ is surjective.\n\\end{enumerate}\nThen given any bounded above acyclic complex $E^\\bullet$\nwhose terms are of the form $P/IP$ for $P \\in \\mathcal{P}$ there\nexists a bounded above acyclic complex $P^\\bullet$ whose terms\nare in $\\mathcal{P}$ lifting $E^\\bullet$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BC9","source_file":"more-algebra.tex","source_line":19342,"source_end_line":19357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19342-L19357","statement_sha256":"857852fe268ceb985260eb04d753875fc35dac802dbf053ff51d6dca847d83be","origin":"The Stacks Project","memory_eligible":false,"source_rank":3305,"rank":3305,"depth":0,"x":677.25,"y":784.567,"cluster":"advanced-algebra"},{"id":"stacks:0BCA","tag":"0BCA","title":"Lifting complexes · Lemma 0BCA","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let P be a class of R-modules. Let K ∈ D(R) and let E^bullet be a complex of R/I-modules representing K ⊗_R^L R/I. Assume • each P ∈ P is a projective R-module, • P_1 ∈ P and P_1 ⊕ P_2 ∈ P if and only if P_1, P_2 ∈ P, • if f : P_1 → P_2, P_1, P_2 ∈ P is surjective modulo I, then f is surjective, • E^bullet is bounded above and E^i is of the form P/IP for P ∈ P, and • K can be represented by a bounded above complex whose terms are in…","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $\\mathcal{P}$\nbe a class of $R$-modules.\nLet $K \\in D(R)$ and let $E^\\bullet$ be a complex of $R/I$-modules\nrepresenting $K \\otimes_R^\\mathbf{L} R/I$. Assume\n\\begin{enumerate}\n\\item each $P \\in \\mathcal{P}$ is a projective $R$-module,\n\\item $P_1 \\in \\mathcal{P}$ and $P_1 \\oplus P_2 \\in \\mathcal{P}$\nif and only if $P_1, P_2 \\in \\mathcal{P}$,\n\\item if $f : P_1 \\to P_2$, $P_1, P_2 \\in \\mathcal{P}$ is surjective\nmodulo $I$, then $f$ is surjective,\n\\item $E^\\bullet$ is bounded above and $E^i$ is of the form $P/IP$\nfor $P \\in \\mathcal{P}$, and\n\\item $K$ can be represented by a bounded above complex\nwhose terms are in $\\mathcal{P}$.\n\\end{enumerate}\nThen there exists a bounded above complex $P^\\bullet$ whose terms\nare in $\\mathcal{P}$ with $P^\\bullet/IP^\\bullet$ isomorphic to\n$E^\\bullet$ and representing $K$ in $D(R)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCA","source_file":"more-algebra.tex","source_line":19389,"source_end_line":19409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19389-L19409","statement_sha256":"83d95f0ad7f07d8a361e68bdfcd36d753606d8f4e9dc9092aec2f4bf6e8807d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3306,"rank":3306,"depth":5,"x":597.855,"y":601.254,"cluster":"advanced-algebra"},{"id":"stacks:09AR","tag":"09AR","title":"Lifting complexes · Lemma 09AR","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let E^bullet be a complex of R/I-modules. Let K be an object of D(R). Assume that • E^bullet is a bounded above complex of projective R/I-modules, • K ⊗_R^L R/I is represented by E^bullet in D(R/I), and • I is a nilpotent ideal. Then there exists a bounded above complex P^bullet of projective R-modules representing K in D(R) such that P^bullet ⊗_R R/I is isomorphic to E^bullet.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $E^\\bullet$\nbe a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that\n\\begin{enumerate}\n\\item $E^\\bullet$ is a bounded above complex of projective $R/I$-modules,\n\\item $K \\otimes_R^\\mathbf{L} R/I$ is represented by $E^\\bullet$ in\n$D(R/I)$, and\n\\item $I$ is a nilpotent ideal.\n\\end{enumerate}\nThen there exists a bounded above complex $P^\\bullet$ of projective\n$R$-modules representing $K$ in $D(R)$ such that $P^\\bullet \\otimes_R R/I$\nis isomorphic to $E^\\bullet$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AR","source_file":"more-algebra.tex","source_line":19468,"source_end_line":19481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19468-L19481","statement_sha256":"b061029a16a4e09b2a3af468ffab94fdcc9dd902f31e5cab1ca6eb8ffa64581d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3307,"rank":3307,"depth":8,"x":804.023,"y":691.467,"cluster":"advanced-algebra"},{"id":"stacks:0H76","tag":"0H76","title":"Lifting complexes · Lemma 0H76","summary":"Let R' → R be a surjective ring map whose kernel is a nilpotent ideal. Let K' ∈ D(R') and set K = K' ⊗_R'^L R. Then K is pseudo-coherent if and only if K' is pseudo-coherent.","statement_latex":"Let $R' \\to R$ be a surjective ring map whose kernel is a nilpotent ideal.\nLet $K' \\in D(R')$ and set $K = K' \\otimes_{R'}^\\mathbf{L} R$.\nThen $K$ is pseudo-coherent if and only if $K'$ is pseudo-coherent.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H76","source_file":"more-algebra.tex","source_line":19497,"source_end_line":19502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19497-L19502","statement_sha256":"9ed5c66be97a8d9d5f229367c8d7f78e3f4044b51bc65a150dcb1ad6957616d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3308,"rank":3308,"depth":11,"x":579.225,"y":741.98,"cluster":"advanced-algebra"},{"id":"stacks:0BCB","tag":"0BCB","title":"Lifting complexes · Lemma 0BCB","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let E^bullet be a complex of R/I-modules. Let K be an object of D(R). Assume that • E^bullet is a bounded above complex of finite stably free R/I-modules, • K ⊗_R^L R/I is represented by E^bullet in D(R/I), • K^bullet is pseudo-coherent, and • every element of 1 + I is invertible. Then there exists a bounded above complex P^bullet of finite stably free R-modules representing K in D(R) such that P^bullet ⊗_R R/I is isomorphic to…","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $E^\\bullet$\nbe a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that\n\\begin{enumerate}\n\\item $E^\\bullet$ is a bounded above complex of\nfinite stably free $R/I$-modules,\n\\item $K \\otimes_R^\\mathbf{L} R/I$ is represented by $E^\\bullet$ in $D(R/I)$,\n\\item $K^\\bullet$ is pseudo-coherent, and\n\\item every element of $1 + I$ is invertible.\n\\end{enumerate}\nThen there exists a bounded above complex $P^\\bullet$ of finite stably free\n$R$-modules representing $K$ in $D(R)$ such that $P^\\bullet \\otimes_R R/I$\nis isomorphic to $E^\\bullet$. Moreover, if $E^i$ is free, then $P^i$ is free.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCB","source_file":"more-algebra.tex","source_line":19516,"source_end_line":19530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19516-L19530","statement_sha256":"107b8cc2fcdc4a4122e7291e7f4364e04305a0d5eb7251a699ac31a34378f7df","origin":"The Stacks Project","memory_eligible":false,"source_rank":3309,"rank":3309,"depth":6,"x":704.491,"y":577.012,"cluster":"advanced-algebra"},{"id":"stacks:0BCC","tag":"0BCC","title":"Lifting complexes · Lemma 0BCC","summary":"Let (R, m, kappa) be a local ring. Let K ∈ D(R) be pseudo-coherent. Set d_i = dim_kappa H^i(K ⊗_R^L kappa). Then d_i < ∞ and for some b ∈ Z we have d_i = 0 for i > b. Then there exists a complex … → R^⊕ d_b - 2 → R^⊕ d_b - 1 → R^⊕ d_b → 0 → … representing K in D(R). Moreover, this complex is unique up to isomorphism(!).","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Let $K \\in D(R)$\nbe pseudo-coherent. Set\n$d_i = \\dim_\\kappa H^i(K \\otimes_R^\\mathbf{L} \\kappa)$.\nThen $d_i < \\infty$ and for some $b \\in \\mathbf{Z}$ we have\n$d_i = 0$ for $i > b$.\nThen there exists a complex\n$$\n\\ldots \\to\nR^{\\oplus d_{b - 2}} \\to\nR^{\\oplus d_{b - 1}} \\to\nR^{\\oplus d_b} \\to 0 \\to \\ldots\n$$\nrepresenting $K$ in $D(R)$. Moreover, this complex is unique up to\nisomorphism(!).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCC","source_file":"more-algebra.tex","source_line":19547,"source_end_line":19563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19547-L19563","statement_sha256":"71cfb460d687d340c170b2945eadb5d87ef3d7c598557766deeaee382a8b5ad2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3310,"rank":3310,"depth":11,"x":744.825,"y":769.929,"cluster":"advanced-algebra"},{"id":"stacks:0BCD","tag":"0BCD","title":"Lifting complexes · Lemma 0BCD","summary":"Let R be a ring. Let p ⊂ R be a prime. Let K ∈ D(R) be perfect. Set d_i = dim_kappa( p) H^i(K ⊗_R^L kappa( p)). Then d_i < ∞ and only a finite number are nonzero. Then there exists an f ∈ R, f not ∈ p and a complex … → 0 → R_f^⊕ d_a → R_f^⊕ d_a + 1 → … → R_f^⊕ d_b - 1 → R_f^⊕ d_b → 0 → … representing K ⊗_R^L R_f in D(R_f).","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime. Let $K \\in D(R)$\nbe perfect. Set\n$d_i =\n\\dim_{\\kappa(\\mathfrak p)} H^i(K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p))$.\nThen $d_i < \\infty$ and only a finite number are nonzero.\nThen there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ and a\ncomplex\n$$\n\\ldots \\to 0 \\to R_f^{\\oplus d_a} \\to R_f^{\\oplus d_{a + 1}} \\to\n\\ldots \\to\nR_f^{\\oplus d_{b - 1}} \\to\nR_f^{\\oplus d_b} \\to 0\n\\to \\ldots\n$$\nrepresenting $K \\otimes_R^\\mathbf{L} R_f$ in $D(R_f)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCD","source_file":"more-algebra.tex","source_line":19593,"source_end_line":19610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19593-L19610","statement_sha256":"16f51a7cbc78529513a115260d2589c9a499797dc7006c03f0d32e3ab2e4b39e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3311,"rank":3311,"depth":16,"x":559.761,"y":650.442,"cluster":"advanced-algebra"},{"id":"stacks:0F9V","tag":"0F9V","title":"Lifting complexes · Lemma 0F9V","summary":"Let R be a ring. Let p ⊂ R be a prime. Let M^bullet and N^bullet be bounded complexes of finite projective R-modules representing the same object of D(R). Then there exists an f ∈ R, f not ∈ p such that there is an isomorphism (!) of complexes M^bullet_f ⊕ P^bullet ≅ N^bullet_f ⊕ Q^bullet where P^bullet and Q^bullet are finite direct sums of trivial complexes, i.e., complexes of the form the form … → 0 → R_f xrightarrow1 R_f → 0 → … (placed in arbitrary degrees).","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime. Let $M^\\bullet$\nand $N^\\bullet$ be bounded complexes of finite projective $R$-modules\nrepresenting the same object of $D(R)$. Then there exists an $f \\in R$,\n$f \\not \\in \\mathfrak p$ such that there is an isomorphism (!)\nof complexes\n$$\nM^\\bullet_f \\oplus P^\\bullet \\cong N^\\bullet_f \\oplus Q^\\bullet\n$$\nwhere $P^\\bullet$ and $Q^\\bullet$ are finite direct sums of\ntrivial complexes, i.e., complexes of the form\nthe form $\\ldots \\to 0 \\to R_f \\xrightarrow{1} R_f \\to 0 \\to \\ldots$\n(placed in arbitrary degrees).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9V","source_file":"more-algebra.tex","source_line":19627,"source_end_line":19641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19627-L19641","statement_sha256":"7a12550b8ea2f7aae3c7e8684b5608b1da5b96703f501fe08046c974400af7a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3312,"rank":3312,"depth":12,"x":792.544,"y":633.523,"cluster":"advanced-algebra"},{"id":"stacks:0BCE","tag":"0BCE","title":"Lifting complexes · Lemma 0BCE","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let E^bullet be a complex of R/I-modules. Let K be an object of D(R). Assume that • E^bullet is a bounded above complex of finite projective R/I-modules, • K ⊗_R^L R/I is represented by E^bullet in D(R/I), • K is pseudo-coherent, and • (R, I) is a henselian pair. Then there exists a bounded above complex P^bullet of finite projective R-modules representing K in D(R) such that P^bullet ⊗_R R/I is isomorphic to E^bullet. Moreover, if…","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $E^\\bullet$\nbe a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that\n\\begin{enumerate}\n\\item $E^\\bullet$ is a bounded above complex of finite projective $R/I$-modules,\n\\item $K \\otimes_R^\\mathbf{L} R/I$ is represented by $E^\\bullet$ in $D(R/I)$,\n\\item $K$ is pseudo-coherent, and\n\\item $(R, I)$ is a henselian pair.\n\\end{enumerate}\nThen there exists a bounded above complex $P^\\bullet$ of finite projective\n$R$-modules representing $K$ in $D(R)$ such that $P^\\bullet \\otimes_R R/I$\nis isomorphic to $E^\\bullet$. Moreover, if $E^i$ is free, then $P^i$ is free.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCE","source_file":"more-algebra.tex","source_line":19656,"source_end_line":19669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19656-L19669","statement_sha256":"db51113c5bab0c7ee313965f4a81d2e3fe92ad07e873f4cb9c7e7274709ecea5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3313,"rank":3313,"depth":48,"x":634.342,"y":778.229,"cluster":"advanced-algebra"},{"id":"stacks:0BCG","tag":"0BCG","title":"Splitting complexes · Lemma 0BCG","summary":"Let R be a ring. Let K and L be objects of D(R). Assume L has projective-amplitude in [a, b], for example if L is perfect of tor-amplitude in [a, b]. • If H^i(K) = 0 for i ≥ a, then Hom_D(R)(L, K) = 0. • If H^i(K) = 0 for i ≥ a + 1, then given any distinguished triangle K → M → L → K[1] there is an isomorphism M ≅ K ⊕ L in D(R) compatible with the maps in the distinguished triangle. • If H^i(K) = 0 for i ≥ a, then the isomorphism in (2) exists and is unique.","statement_latex":"Let $R$ be a ring. Let $K$ and $L$ be objects of $D(R)$.\nAssume $L$ has projective-amplitude in $[a, b]$, for example if $L$\nis perfect of tor-amplitude in $[a, b]$.\n\\begin{enumerate}\n\\item If $H^i(K) = 0$ for $i \\geq a$, then\n$\\Hom_{D(R)}(L, K) = 0$.\n\\item If $H^i(K) = 0$ for $i \\geq a + 1$, then given any distinguished\ntriangle $K \\to M \\to L \\to K[1]$\nthere is an isomorphism $M \\cong K \\oplus L$\nin $D(R)$ compatible with the maps in the distinguished triangle.\n\\item If $H^i(K) = 0$ for $i \\geq a$, then the isomorphism\nin (2) exists and is unique.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCG","source_file":"more-algebra.tex","source_line":19703,"source_end_line":19718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19703-L19718","statement_sha256":"9fa63af944fe85accc1e44003157ab37341014db670a3e3969fd94edf03d3fdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3314,"rank":3314,"depth":11,"x":634.629,"y":581.562,"cluster":"advanced-algebra"},{"id":"stacks:0A1U","tag":"0A1U","title":"Splitting complexes · Lemma 0A1U","summary":"Let R be a ring. Let p ⊂ R be a prime ideal. Let K^bullet be a pseudo-coherent complex of R-modules. Assume that for some i ∈ Z the map H^i(K^bullet) ⊗_R kappa( p) → H^i(K^bullet ⊗_R^L kappa( p)) is surjective. Then there exists an f ∈ R, f not ∈ p such that τ_≥ i + 1(K^bullet ⊗_R R_f) is a perfect object of D(R_f) with tor amplitude in [i + 1, ∞] and a canonical isomorphism K^bullet ⊗_R R_f ≅ τ_≤ i(K^bullet ⊗_R R_f) ⊕ τ_≥ i + 1(K^bullet ⊗_R R_f) in D(R_f).","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal.\nLet $K^\\bullet$ be a pseudo-coherent complex of $R$-modules.\nAssume that for some $i \\in \\mathbf{Z}$ the map\n$$\nH^i(K^\\bullet) \\otimes_R \\kappa(\\mathfrak p)\n\\longrightarrow\nH^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p))\n$$\nis surjective. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$\nsuch that $\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)$ is a perfect\nobject of $D(R_f)$ with tor amplitude in $[i + 1, \\infty]$ and\na canonical isomorphism\n$$\nK^\\bullet \\otimes_R R_f \\cong\n\\tau_{\\leq i}(K^\\bullet \\otimes_R R_f) \\oplus\n\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)\n$$\nin $D(R_f)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1U","source_file":"more-algebra.tex","source_line":19743,"source_end_line":19763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19743-L19763","statement_sha256":"9b0fb98eee1f2d172e0040adfbd6a914a55b7cbf9daa031320bc22b398056786","origin":"The Stacks Project","memory_eligible":false,"source_rank":3315,"rank":3315,"depth":12,"x":792.728,"y":726.888,"cluster":"advanced-algebra"},{"id":"stacks:0A1V","tag":"0A1V","title":"Splitting complexes · Lemma 0A1V","summary":"Let R be a ring. Let p ⊂ R be a prime ideal. Let K^bullet be a pseudo-coherent complex of R-modules. Assume that for some i ∈ Z the maps H^i(K^bullet) ⊗_R kappa( p) → H^i(K^bullet ⊗_R^L kappa( p)) and H^i - 1(K^bullet) ⊗_R kappa( p) → H^i - 1(K^bullet ⊗_R^L kappa( p)) are surjective. Then there exists an f ∈ R, f not ∈ p such that • τ_≥ i + 1(K^bullet ⊗_R R_f) is a perfect object of D(R_f) with tor amplitude in [i + 1, ∞], • H^i(K^bullet)_f is a finite free R_f-module,…","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal.\nLet $K^\\bullet$ be a pseudo-coherent complex of $R$-modules.\nAssume that for some $i \\in \\mathbf{Z}$ the maps\n$$\nH^i(K^\\bullet) \\otimes_R \\kappa(\\mathfrak p)\n\\longrightarrow\nH^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p))\n\\quad\\text{and}\\quad\nH^{i - 1}(K^\\bullet) \\otimes_R \\kappa(\\mathfrak p)\n\\longrightarrow\nH^{i - 1}(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p))\n$$\nare surjective. Then there exists an $f \\in R$, $f \\not \\in \\mathfrak p$\nsuch that\n\\begin{enumerate}\n\\item $\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)$ is a perfect\nobject of $D(R_f)$ with tor amplitude in $[i + 1, \\infty]$,\n\\item $H^i(K^\\bullet)_f$ is a finite free $R_f$-module, and\n\\item there is a canonical direct sum decomposition\n$$\nK^\\bullet \\otimes_R R_f \\cong\n\\tau_{\\leq i - 1}(K^\\bullet \\otimes_R R_f) \\oplus\nH^i(K^\\bullet)_f[-i] \\oplus\n\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)\n$$\nin $D(R_f)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1V","source_file":"more-algebra.tex","source_line":19821,"source_end_line":19850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19821-L19850","statement_sha256":"b589b2fa2eca1a9d38c4f96e0576f992ae64c2192c893c54d1cb6cdfe24578bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3316,"rank":3316,"depth":13,"x":559.05,"y":709.419,"cluster":"advanced-algebra"},{"id":"stacks:068U","tag":"068U","title":"Splitting complexes · Lemma 068U","summary":"Let R be a ring. Let p ⊂ R be a prime ideal. Let i ∈ Z. Let K^bullet be a pseudo-coherent complex of R-modules such that H^i(K^bullet ⊗_R^L kappa( p)) = 0. Then there exists an f ∈ R, f not ∈ p and a canonical direct sum decomposition K^bullet ⊗_R R_f = τ_≥ i + 1(K^bullet ⊗_R R_f) ⊕ τ_≤ i - 1(K^bullet ⊗_R R_f) in D(R_f) with τ_≥ i + 1(K^bullet ⊗_R R_f) a perfect complex with tor-amplitude in [i + 1, ∞].","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p \\subset R$ be a prime ideal.\nLet $i \\in \\mathbf{Z}$. Let $K^\\bullet$ be a pseudo-coherent complex\nof $R$-modules such that\n$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) = 0$.\nThen there exists an $f \\in R$, $f \\not \\in \\mathfrak p$\nand a canonical direct sum decomposition\n$$\nK^\\bullet \\otimes_R R_f =\n\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f) \\oplus\n\\tau_{\\leq i - 1}(K^\\bullet \\otimes_R R_f)\n$$\nin $D(R_f)$ with $\\tau_{\\geq i + 1}(K^\\bullet \\otimes_R R_f)$ a perfect\ncomplex with tor-amplitude in $[i + 1, \\infty]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068U","source_file":"more-algebra.tex","source_line":19866,"source_end_line":19881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19866-L19881","statement_sha256":"6ebd98f40fddd6235a88c46bdf4b5d14c584527c4b1c64e11abd88a65b2aeda8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3317,"rank":3317,"depth":13,"x":745.593,"y":589.588,"cluster":"advanced-algebra"},{"id":"stacks:0G97","tag":"0G97","title":"Splitting complexes · Lemma 0G97","summary":"Let R be a ring. Let K ∈ D^-(R). Let a ∈ Z. Assume that for any injective R-module map M → M' the map Ext^-a_R(K, M) → Ext^-a_R(K, M') is injective. Then there is a unique direct sum decomposition K ≅ τ_≤ aK ⊕ τ_≥ a + 1K and τ_≥ a + 1K has projective-amplitude in [a + 1, b] for some b.","statement_latex":"Let $R$ be a ring. Let $K \\in D^-(R)$. Let $a \\in \\mathbf{Z}$.\nAssume that for any injective $R$-module map $M \\to M'$ the map\n$\\Ext^{-a}_R(K, M) \\to \\Ext^{-a}_R(K, M')$ is injective.\nThen there is a unique direct sum decomposition\n$K \\cong \\tau_{\\leq a}K \\oplus \\tau_{\\geq a + 1}K$\nand $\\tau_{\\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$\nfor some $b$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G97","source_file":"more-algebra.tex","source_line":19917,"source_end_line":19926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19917-L19926","statement_sha256":"79fee789e5e7db626eacbcb32a540a89713e31242888de747362c502540707fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3318,"rank":3318,"depth":12,"x":704.363,"y":783.989,"cluster":"advanced-algebra"},{"id":"stacks:0G98","tag":"0G98","title":"Splitting complexes · Lemma 0G98","summary":"Let R be a ring. Let K ∈ D^-(R). Let a ∈ Z. Assume Ext^-a_R(K, M) = 0 for any R-module M. Then there is a unique direct sum decomposition K ≅ τ_≤ a - 1K ⊕ τ_≥ a + 1K and τ_≥ a + 1K has projective-amplitude in [a + 1, b] for some b.","statement_latex":"Let $R$ be a ring. Let $K \\in D^-(R)$. Let $a \\in \\mathbf{Z}$.\nAssume $\\Ext^{-a}_R(K, M) = 0$ for any $R$-module $M$.\nThen there is a unique direct sum decomposition\n$K \\cong \\tau_{\\leq a - 1}K \\oplus \\tau_{\\geq a + 1}K$\nand $\\tau_{\\geq a + 1}K$ has projective-amplitude in $[a + 1, b]$\nfor some $b$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G98","source_file":"more-algebra.tex","source_line":19960,"source_end_line":19968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19960-L19968","statement_sha256":"52e7a68d67d854af5b8bb5000ba152121183068de037d9667357cd5074863f86","origin":"The Stacks Project","memory_eligible":false,"source_rank":3319,"rank":3319,"depth":13,"x":578.31,"y":617.082,"cluster":"advanced-algebra"},{"id":"stacks:0BYP","tag":"0BYP","title":"Recognizing perfect complexes · Lemma 0BYP","summary":"Let R be a ring and let p ⊂ R be a prime. Let K be pseudo-coherent and bounded below. Set d_i = dim_kappa( p) H^i(K ⊗_R^L kappa( p)). If there exists an a ∈ Z such that d_i = 0 for i < a, then there exists an f ∈ R, f not ∈ p and a complex … → 0 → R_f^⊕ d_a → R_f^⊕ d_a + 1 → … → R_f^⊕ d_b - 1 → R_f^⊕ d_b → 0 → … representing K ⊗_R^L R_f in D(R_f). In particular K ⊗_R^L R_f is perfect.","statement_latex":"Let $R$ be a ring and let $\\mathfrak p \\subset R$ be a prime.\nLet $K$ be pseudo-coherent and bounded below. Set\n$d_i = \\dim_{\\kappa(\\mathfrak p)}\nH^i(K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p))$.\nIf there exists an $a \\in \\mathbf{Z}$ such that $d_i = 0$ for $i < a$,\nthen there exists an $f \\in R$, $f \\not \\in \\mathfrak p$ and a\ncomplex\n$$\n\\ldots \\to 0 \\to R_f^{\\oplus d_a} \\to R_f^{\\oplus d_{a + 1}} \\to\n\\ldots \\to\nR_f^{\\oplus d_{b - 1}} \\to\nR_f^{\\oplus d_b} \\to 0\n\\to \\ldots\n$$\nrepresenting $K \\otimes_R^\\mathbf{L} R_f$ in $D(R_f)$.\nIn particular $K \\otimes_R^\\mathbf{L} R_f$ is perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Recognizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYP","source_file":"more-algebra.tex","source_line":19999,"source_end_line":20017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L19999-L20017","statement_sha256":"8e867dc587f9eb75ac993423a89dd588c5cbb28af18edd6e5ca0b9cdba8f635b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3320,"rank":3320,"depth":17,"x":805.704,"y":668.682,"cluster":"advanced-algebra"},{"id":"stacks:068V","tag":"068V","title":"Recognizing perfect complexes · Lemma 068V","summary":"Let R be a ring. Let a, b ∈ Z. Let K^bullet be a pseudo-coherent complex of R-modules. The following are equivalent • K^bullet is perfect with tor amplitude in [a, b], • for every prime p we have H^i(K^bullet ⊗_R^L kappa( p)) = 0 for all i not ∈ [a, b], and • for every maximal ideal m we have H^i(K^bullet ⊗_R^L kappa( m)) = 0 for all i not ∈ [a, b].","statement_latex":"Let $R$ be a ring. Let $a, b \\in \\mathbf{Z}$.\nLet $K^\\bullet$ be a pseudo-coherent complex of $R$-modules.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K^\\bullet$ is perfect with tor amplitude in $[a, b]$,\n\\item for every prime $\\mathfrak p$ we have\n$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) = 0$ for all\n$i \\not \\in [a, b]$, and\n\\item for every maximal ideal $\\mathfrak m$ we have\n$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak m)) = 0$ for all\n$i \\not \\in [a, b]$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Recognizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068V","source_file":"more-algebra.tex","source_line":20031,"source_end_line":20045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20031-L20045","statement_sha256":"e3162827622869ed04f0acfb789d70a0ae7ddfa9b0e881069a472a1219fac1bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3321,"rank":3321,"depth":14,"x":596.328,"y":759.751,"cluster":"advanced-algebra"},{"id":"stacks:068W","tag":"068W","title":"Recognizing perfect complexes · Lemma 068W","summary":"Let R be a ring. Let K^bullet be a pseudo-coherent complex of R-modules. Consider the following conditions • K^bullet is perfect, • for every prime ideal p the complex K^bullet ⊗_R R_ p is perfect, • for every maximal ideal m the complex K^bullet ⊗_R R_ m is perfect, • for every prime p we have H^i(K^bullet ⊗_R^L kappa( p)) = 0 for all i ll 0, • for every maximal ideal m we have H^i(K^bullet ⊗_R^L kappa( m)) = 0 for all i ll 0. We always have the implications (1) ⇒ (2) ⇔…","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a pseudo-coherent\ncomplex of $R$-modules. Consider the following conditions\n\\begin{enumerate}\n\\item $K^\\bullet$ is perfect,\n\\item for every prime ideal $\\mathfrak p$ the complex\n$K^\\bullet \\otimes_R R_{\\mathfrak p}$ is perfect,\n\\item for every maximal ideal $\\mathfrak m$ the complex\n$K^\\bullet \\otimes_R R_{\\mathfrak m}$ is perfect,\n\\item for every prime $\\mathfrak p$ we have\n$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak p)) = 0$ for all\n$i \\ll 0$,\n\\item for every maximal ideal $\\mathfrak m$ we have\n$H^i(K^\\bullet \\otimes_R^{\\mathbf{L}} \\kappa(\\mathfrak m)) = 0$ for all\n$i \\ll 0$.\n\\end{enumerate}\nWe always have the implications\n$$\n(1) \\Rightarrow (2) \\Leftrightarrow (3) \\Leftrightarrow (3)\n\\Leftrightarrow (4) \\Leftrightarrow (5)\n$$\nIf $K^\\bullet$ is bounded below, then all conditions are equivalent.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Recognizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068W","source_file":"more-algebra.tex","source_line":20063,"source_end_line":20086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20063-L20086","statement_sha256":"01afa0913898c545b090d19bc20c0eb6569ddfeba1c15c8afab33d7ba397aa00","origin":"The Stacks Project","memory_eligible":false,"source_rank":3322,"rank":3322,"depth":18,"x":677.563,"y":573.611,"cluster":"advanced-algebra"},{"id":"stacks:0G9A","tag":"0G9A","title":"Recognizing perfect complexes · Lemma 0G9A","summary":"Let R be a ring. Let K be a pseudo-coherent object of D(R). Let a, b ∈ Z. The following are equivalent • K has projective-amplitude in [a, b], • K is perfect of tor-amplitude in [a, b], • Ext^i_R(K, N) = 0 for all finitely presented R-modules N and all i not ∈ [-b, -a], • H^n(K) = 0 for n > b and Ext^i_R(K, N) = 0 for all finitely presented R-modules N and all i > -a, and • H^n(K) = 0 for n not ∈ [a - 1, b] and Ext^-a + 1_R(K, N) = 0 for all finitely presented R-modules N.","statement_latex":"Let $R$ be a ring. Let $K$ be a pseudo-coherent object of $D(R)$.\nLet $a, b \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $K$ has projective-amplitude in $[a, b]$,\n\\item $K$ is perfect of tor-amplitude in $[a, b]$,\n\\item $\\Ext^i_R(K, N) = 0$ for all finitely presented $R$-modules $N$ and all\n$i \\not \\in [-b, -a]$,\n\\item $H^n(K) = 0$ for $n > b$ and\n$\\Ext^i_R(K, N) = 0$ for all finitely presented $R$-modules $N$ and\nall $i > -a$, and\n\\item $H^n(K) = 0$ for $n \\not \\in [a - 1, b]$ and\n$\\Ext^{-a + 1}_R(K, N) = 0$ for all finitely presented $R$-modules $N$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Recognizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9A","source_file":"more-algebra.tex","source_line":20135,"source_end_line":20150,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20135-L20150","statement_sha256":"a81e5ffc520cbefa28f59d9812e540b671f5431219dba8b532d7aed563a2d72b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3323,"rank":3323,"depth":11,"x":767.437,"y":757.141,"cluster":"advanced-algebra"},{"id":"stacks:068X","tag":"068X","title":"Recognizing perfect complexes · Lemma 068X","summary":"Let A → B be a ring map. Let a, b ∈ Z. Let d ≥ 0. Let K^bullet be a complex of B-modules. Assume • the ring map A → B is flat, • for every prime p ⊂ A the ring B ⊗_A kappa( p) has finite global dimension ≤ d, • K^bullet is pseudo-coherent as a complex of B-modules, and • K^bullet has tor amplitude in [a, b] as a complex of A-modules. Then K^bullet is perfect as a complex of B-modules with tor amplitude in [a - d, b].","statement_latex":"Let $A \\to B$ be a ring map. Let $a, b \\in \\mathbf{Z}$. Let $d \\geq 0$.\nLet $K^\\bullet$ be a complex of $B$-modules. Assume\n\\begin{enumerate}\n\\item the ring map $A \\to B$ is flat,\n\\item for every prime $\\mathfrak p \\subset A$ the ring\n$B \\otimes_A \\kappa(\\mathfrak p)$ has finite global dimension $\\leq d$,\n\\item $K^\\bullet$ is pseudo-coherent as a complex of $B$-modules, and\n\\item $K^\\bullet$ has tor amplitude in $[a, b]$ as a complex\nof $A$-modules.\n\\end{enumerate}\nThen $K^\\bullet$ is perfect as a complex of $B$-modules\nwith tor amplitude in $[a - d, b]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Recognizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068X","source_file":"more-algebra.tex","source_line":20174,"source_end_line":20188,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20174-L20188","statement_sha256":"e5cc285dc8f2398d55615403df13ea0db7e4d7a181009dc08f91272ffe52569d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3324,"rank":3324,"depth":15,"x":553.367,"y":672.724,"cluster":"advanced-algebra"},{"id":"stacks:09PC","tag":"09PC","title":"Recognizing perfect complexes · Lemma 09PC","summary":"Let A → B be a local ring homomorphism. Let a, b ∈ Z. Let d ≥ 0. Let K^bullet be a complex of B-modules. Assume • the ring map A → B is flat, • the ring B/ m_AB is regular of dimension d, • K^bullet is pseudo-coherent as a complex of B-modules, and • K^bullet has tor amplitude in [a, b] as a complex of A-modules, in fact it suffices if H^i(K^bullet ⊗_A^L kappa( m_A)) is nonzero only for i ∈ [a, b]. Then K^bullet is perfect as a complex of B-modules with tor amplitude in…","statement_latex":"Let $A \\to B$ be a local ring homomorphism.\nLet $a, b \\in \\mathbf{Z}$. Let $d \\geq 0$.\nLet $K^\\bullet$ be a complex of $B$-modules. Assume\n\\begin{enumerate}\n\\item the ring map $A \\to B$ is flat,\n\\item the ring $B/\\mathfrak m_AB$ is regular of dimension $d$,\n\\item $K^\\bullet$ is pseudo-coherent as a complex of $B$-modules, and\n\\item $K^\\bullet$ has tor amplitude in $[a, b]$ as a complex\nof $A$-modules, in fact it suffices if\n$H^i(K^\\bullet \\otimes_A^\\mathbf{L} \\kappa(\\mathfrak m_A))$\nis nonzero only for $i \\in [a, b]$.\n\\end{enumerate}\nThen $K^\\bullet$ is perfect as a complex of $B$-modules\nwith tor amplitude in $[a - d, b]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Recognizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PC","source_file":"more-algebra.tex","source_line":20228,"source_end_line":20244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20228-L20244","statement_sha256":"4732caaaef889e839d027ca42c077c9c0fe67a55b71164b0641c77672b3814cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3325,"rank":3325,"depth":16,"x":779.324,"y":613.446,"cluster":"advanced-algebra"},{"id":"stacks:0ATI","tag":"0ATI","title":"Characterizing perfect complexes · Lemma 0ATI","summary":"Let R be a ring. The full subcategory D_perf(R) ⊂ D(R) of perfect objects is the smallest strictly full, saturated, triangulated subcategory containing R = R[0]. In other words D_perf(R) = langle R rangle. In particular, R is a classical generator for D_perf(R).","statement_latex":"Let $R$ be a ring. The full subcategory $D_{perf}(R) \\subset D(R)$ of perfect\nobjects is the smallest strictly full, saturated, triangulated subcategory\ncontaining $R = R[0]$. In other words $D_{perf}(R) = \\langle R \\rangle$.\nIn particular, $R$ is a classical generator for $D_{perf}(R)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Characterizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATI","source_file":"more-algebra.tex","source_line":20287,"source_end_line":20293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20287-L20293","statement_sha256":"d324a5a74ee932ce30d691bd0881f7bbe6d75c56df52ed6fdbccabecf549bc4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3326,"rank":3326,"depth":13,"x":660.261,"y":785.537,"cluster":"advanced-algebra"},{"id":"stacks:07LR","tag":"07LR","title":"Characterizing perfect complexes · Lemma 07LR","summary":"Let R be a ring. Let K ∈ D(R) be an object such that for every countable set of objects E_n ∈ D(R) the canonical map bigoplus Hom_D(R)(K, E_n) → Hom_D(R)(K, bigoplus E_n) is a bijection. Then, given any system L_n^bullet of complexes over N we have that colim Hom_D(R)(K, L^bullet_n) → Hom_D(R)(K, L^bullet) is a bijection, where L^bullet is the termwise colimit, i.e., L^m = colim L_n^m for all m ∈ Z.","statement_latex":"Let $R$ be a ring. Let $K \\in D(R)$ be an object such that for every\ncountable set of objects $E_n \\in D(R)$ the canonical map\n$$\n\\bigoplus \\Hom_{D(R)}(K, E_n) \\longrightarrow \\Hom_{D(R)}(K, \\bigoplus E_n)\n$$\nis a bijection. Then, given any system $L_n^\\bullet$ of complexes over\n$\\mathbf{N}$ we have that\n$$\n\\colim \\Hom_{D(R)}(K, L^\\bullet_n) \\longrightarrow \\Hom_{D(R)}(K, L^\\bullet)\n$$\nis a bijection, where $L^\\bullet$ is the termwise colimit, i.e.,\n$L^m = \\colim L_n^m$ for all $m \\in \\mathbf{Z}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Characterizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LR","source_file":"more-algebra.tex","source_line":20331,"source_end_line":20345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20331-L20345","statement_sha256":"fea2b37adb46ebc41eddd10675f917e91ebde882f00d9e62a0bec26bbb4be0e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3327,"rank":3327,"depth":2,"x":609.618,"y":590.893,"cluster":"advanced-algebra"},{"id":"stacks:07LT","tag":"07LT","title":"Characterizing perfect complexes · Proposition 07LT","summary":"Let R be a ring. For an object K of D(R) the following are equivalent • K is perfect, and • K is a compact object of D(R).","statement_latex":"Let $R$ be a ring. For an object $K$ of $D(R)$ the following are equivalent\n\\begin{enumerate}\n\\item $K$ is perfect, and\n\\item $K$ is a compact object of $D(R)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Characterizing perfect complexes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LT","source_file":"more-algebra.tex","source_line":20388,"source_end_line":20395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20388-L20395","statement_sha256":"533b26f3be822a7746aebcda703f4a44003cdb3ac4ad3c627f8da4ba92121ac0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3328,"rank":3328,"depth":13,"x":803.676,"y":705.794,"cluster":"advanced-algebra"},{"id":"stacks:07LU","tag":"07LU","title":"Characterizing perfect complexes · Lemma 07LU","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let K be an object of D(R). Assume that • K ⊗_R^L R/I is perfect in D(R/I), and • I is a nilpotent ideal. Then K is perfect in D(R).","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nLet $K$ be an object of $D(R)$. Assume that\n\\begin{enumerate}\n\\item $K \\otimes_R^\\mathbf{L} R/I$ is perfect in $D(R/I)$, and\n\\item $I$ is a nilpotent ideal.\n\\end{enumerate}\nThen $K$ is perfect in $D(R)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Characterizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LU","source_file":"more-algebra.tex","source_line":20508,"source_end_line":20517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20508-L20517","statement_sha256":"ed7420dc9968a34cf06715f9678d319409c99658a62e789676454414108e7201","origin":"The Stacks Project","memory_eligible":false,"source_rank":3329,"rank":3329,"depth":9,"x":567.953,"y":731.205,"cluster":"advanced-algebra"},{"id":"stacks:09AS","tag":"09AS","title":"Characterizing perfect complexes · Lemma 09AS","summary":"Let R be a ring. Let I, J ⊂ R be ideals. Let K be an object of D(R). Assume that • K ⊗_R^L R/I is perfect in D(R/I), and • K ⊗_R^L R/J is perfect in D(R/J). Then K ⊗_R^L R/IJ is perfect in D(R/IJ).","statement_latex":"Let $R$ be a ring. Let $I, J \\subset R$ be ideals.\nLet $K$ be an object of $D(R)$. Assume that\n\\begin{enumerate}\n\\item $K \\otimes_R^\\mathbf{L} R/I$ is perfect in $D(R/I)$, and\n\\item $K \\otimes_R^\\mathbf{L} R/J$ is perfect in $D(R/J)$.\n\\end{enumerate}\nThen $K \\otimes_R^\\mathbf{L} R/IJ$ is perfect in $D(R/IJ)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Characterizing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AS","source_file":"more-algebra.tex","source_line":20531,"source_end_line":20540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20531-L20540","statement_sha256":"3064763e9d4fa18b85a38725758d5101bd74dcb3d62e9af90a72d85135e42d0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3330,"rank":3330,"depth":15,"x":721.484,"y":578.567,"cluster":"advanced-algebra"},{"id":"stacks:0FXH","tag":"0FXH","title":"Strong generators and regular rings · Lemma 0FXH","summary":"[Kelly] Let R be a ring. Let n ≥ 1. Let K ∈ langle R rangle_n with notation as in Derived Categories, Section [Tag 09SI]. Consider maps K xrightarrowf_1 K_1 xrightarrowf_2 K_2 xrightarrowf_3 … xrightarrowf_n K_n in D(R). If H^i(f_j) = 0 for all i, j, then f_n ∘ … ∘ f_1 = 0.","statement_latex":"\\begin{reference}\n\\cite{Kelly}\n\\end{reference}\nLet $R$ be a ring. Let $n \\geq 1$. Let $K \\in \\langle R \\rangle_n$ with\nnotation as in Derived Categories, Section \\ref{derived-section-generators}.\nConsider maps\n$$\nK \\xrightarrow{f_1} K_1 \\xrightarrow{f_2} K_2\n\\xrightarrow{f_3} \\ldots \\xrightarrow{f_n} K_n\n$$\nin $D(R)$. If $H^i(f_j) = 0$ for all $i, j$, then\n$f_n \\circ \\ldots \\circ f_1 = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Strong generators and regular rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXH","source_file":"more-algebra.tex","source_line":20663,"source_end_line":20677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20663-L20677","statement_sha256":"5db370bcb947ee53215bd46723a478272a1e172dfa1b4861c22e3b8171f20b34","origin":"The Stacks Project","memory_eligible":false,"source_rank":3331,"rank":3331,"depth":2,"x":731.029,"y":778.429,"cluster":"advanced-algebra"},{"id":"stacks:0FXI","tag":"0FXI","title":"Strong generators and regular rings · Lemma 0FXI","summary":"Let R be a Noetherian ring. If R is a strong generator for D_perf(R), then R is regular of finite dimension.","statement_latex":"Let $R$ be a Noetherian ring. If $R$ is\na strong generator for $D_{perf}(R)$, then $R$ is regular\nof finite dimension.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Strong generators and regular rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXI","source_file":"more-algebra.tex","source_line":20710,"source_end_line":20715,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20710-L20715","statement_sha256":"82f967399cbe942780e011fb3959df1da61e6aa0c4447868dcedfef78d7de438","origin":"The Stacks Project","memory_eligible":false,"source_rank":3332,"rank":3332,"depth":18,"x":563.107,"y":636.334,"cluster":"advanced-algebra"},{"id":"stacks:0FXJ","tag":"0FXJ","title":"Strong generators and regular rings · Lemma 0FXJ","summary":"Let R be a Noetherian regular ring of dimension d < ∞. Let K, L ∈ D^-(R). Assume there exists an k such that H^i(K) = 0 for i ≤ k and H^i(L) = 0 for i ≥ k - d + 1. Then Hom_D(R)(K, L) = 0.","statement_latex":"Let $R$ be a Noetherian regular ring of dimension $d < \\infty$.\nLet $K, L \\in D^-(R)$. Assume there exists an $k$ such that\n$H^i(K) = 0$ for $i \\leq k$ and $H^i(L) = 0$ for $i \\geq k - d + 1$.\nThen $\\Hom_{D(R)}(K, L) = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Strong generators and regular rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXJ","source_file":"more-algebra.tex","source_line":20766,"source_end_line":20772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20766-L20772","statement_sha256":"05706d0bf6201fde59c4f5416e4972151a2c29c30f34e728d9475e0e8d6a04dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3333,"rank":3333,"depth":18,"x":801.426,"y":645.838,"cluster":"advanced-algebra"},{"id":"stacks:0FXK","tag":"0FXK","title":"Strong generators and regular rings · Lemma 0FXK","summary":"Let R be a Noetherian regular ring of dimension 1 ≤ d < ∞. Let K ∈ D(R) be perfect and let k ∈ Z such that H^i(K) = 0 for i = k - d + 2, …, k (empty condition if d = 1). Then K = τ_≤ k - d + 1K ⊕ τ_≥ k + 1K.","statement_latex":"Let $R$ be a Noetherian regular ring of dimension $1 \\leq d < \\infty$.\nLet $K \\in D(R)$ be perfect and let $k \\in \\mathbf{Z}$ such that\n$H^i(K) = 0$ for $i = k - d + 2, \\ldots, k$ (empty condition if $d = 1$).\nThen $K = \\tau_{\\leq k - d + 1}K \\oplus \\tau_{\\geq k + 1}K$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Strong generators and regular rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXK","source_file":"more-algebra.tex","source_line":20799,"source_end_line":20805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20799-L20805","statement_sha256":"a4a70bdeae36f7724e95ee56fbbc1affee938108a5630af74aa5b6685412e7bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3334,"rank":3334,"depth":19,"x":617.875,"y":774.18,"cluster":"advanced-algebra"},{"id":"stacks:0FXL","tag":"0FXL","title":"Strong generators and regular rings · Lemma 0FXL","summary":"Let R be a Noetherian regular ring of finite dimension. Then R is a strong generator for the full subcategory D_perf(R) ⊂ D(R) of perfect objects.","statement_latex":"Let $R$ be a Noetherian regular ring of finite dimension.\nThen $R$ is a strong generator for the full subcategory\n$D_{perf}(R) \\subset D(R)$ of perfect objects.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Strong generators and regular rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXL","source_file":"more-algebra.tex","source_line":20819,"source_end_line":20824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20819-L20824","statement_sha256":"847ac51eda84e96a52a3eeaaa0ea09b0b985bfdfe0fa56e9c62242364d0bffd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3335,"rank":3335,"depth":18,"x":650.045,"y":575.202,"cluster":"advanced-algebra"},{"id":"stacks:0FXM","tag":"0FXM","title":"Strong generators and regular rings · Proposition 0FXM","summary":"Let R be a Noetherian ring. The following are equivalent • R is regular of finite dimension, • D_perf(R) has a strong generator, and • R is a strong generator for D_perf(R).","statement_latex":"Let $R$ be a Noetherian ring. The following are equivalent\n\\begin{enumerate}\n\\item $R$ is regular of finite dimension,\n\\item $D_{perf}(R)$ has a strong generator, and\n\\item $R$ is a strong generator for $D_{perf}(R)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Strong generators and regular rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXM","source_file":"more-algebra.tex","source_line":20882,"source_end_line":20890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20882-L20890","statement_sha256":"4406df3b4aa13d600dc11475ae1444934f58c3728807c0ce41c26fa3f04a1362","origin":"The Stacks Project","memory_eligible":false,"source_rank":3336,"rank":3336,"depth":19,"x":786.465,"y":740.336,"cluster":"advanced-algebra"},{"id":"stacks:05GY","tag":"05GY","title":"Relatively finitely presented modules · Lemma 05GY","summary":"Let R → A be a ring map of finite type. Let M be an A-module. The following are equivalent • for some presentation α : R[x_1, …, x_n] → A the module M is a finitely presented R[x_1, …, x_n]-module, • for all presentations α : R[x_1, …, x_n] → A the module M is a finitely presented R[x_1, …, x_n]-module, and • for any surjection A' → A where A' is a finitely presented R-algebra, the module M is finitely presented as A'-module. In this case M is a finitely presented A-module.","statement_latex":"Let $R \\to A$ be a ring map of finite type.\nLet $M$ be an $A$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some presentation $\\alpha : R[x_1, \\ldots, x_n] \\to A$\nthe module $M$ is a finitely presented $R[x_1, \\ldots, x_n]$-module,\n\\item for all presentations $\\alpha : R[x_1, \\ldots, x_n] \\to A$\nthe module $M$ is a finitely presented $R[x_1, \\ldots, x_n]$-module, and\n\\item for any surjection $A' \\to A$ where $A'$ is a finitely presented\n$R$-algebra, the module $M$ is finitely presented as $A'$-module.\n\\end{enumerate}\nIn this case $M$ is a finitely presented $A$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GY","source_file":"more-algebra.tex","source_line":20922,"source_end_line":20936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20922-L20936","statement_sha256":"c8da311fe03f63d4b3af322eed1988112a53600e189baa341c9b7cbed39b9de5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3337,"rank":3337,"depth":4,"x":552.854,"y":695.935,"cluster":"advanced-algebra"},{"id":"stacks:05GZ","tag":"05GZ","title":"Relatively finitely presented modules · Definition 05GZ","summary":"Let R → A be a finite type ring map. Let M be an A-module. We say M is an A-module finitely presented relative to R if the equivalent conditions of Lemma [Tag 05GY] hold.","statement_latex":"Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module.\nWe say $M$ is an $A$-module {\\it finitely presented relative to $R$}\nif the equivalent conditions of\nLemma \\ref{lemma-relatively-finitely-presented}\nhold.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GZ","source_file":"more-algebra.tex","source_line":20962,"source_end_line":20969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20962-L20969","statement_sha256":"f9ef87383b3a6c29d30cb3946497b2fdabdd90c3c9db3bef25b3d6047f29971a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3338,"rank":3338,"depth":5,"x":761.017,"y":596.025,"cluster":"advanced-algebra"},{"id":"stacks:05H0","tag":"05H0","title":"Relatively finitely presented modules · Lemma 05H0","summary":"Let R be a ring. Let A → B be a finite map of finite type R-algebras. Let M be a B-module. Then M is an A-module finitely presented relative to R if and only if M is a B-module finitely presented relative to R.","statement_latex":"Let $R$ be a ring. Let $A \\to B$ be a finite map of finite type $R$-algebras.\nLet $M$ be a $B$-module. Then\n$M$ is an $A$-module finitely presented relative to $R$\nif and only if\n$M$ is a $B$-module finitely presented relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05H0","source_file":"more-algebra.tex","source_line":20979,"source_end_line":20986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L20979-L20986","statement_sha256":"ff23749c90d38ded0eadea55ad42cb41d3e5dd36346d4ae97d54994e99caaba7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3339,"rank":3339,"depth":4,"x":687.797,"y":787.995,"cluster":"advanced-algebra"},{"id":"stacks:065A","tag":"065A","title":"Relatively finitely presented modules · Lemma 065A","summary":"Let R be a ring, f ∈ R an element, R_f → A is a finite type ring map, g ∈ A, and M an A-module. If M of finite presentation relative to R_f, then M_g is an A_g-module of finite presentation relative to R.","statement_latex":"Let $R$ be a ring, $f \\in R$ an element, $R_f \\to A$ is a finite type ring map,\n$g \\in A$, and $M$ an $A$-module. If $M$ of finite presentation relative\nto $R_f$, then $M_g$ is an $A_g$-module of finite presentation relative\nto $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065A","source_file":"more-algebra.tex","source_line":21014,"source_end_line":21020,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21014-L21020","statement_sha256":"480b1974dfcc79ce4e8eaabf435b4d1d233691597efea33289fbbdf19e0e2415","origin":"The Stacks Project","memory_eligible":false,"source_rank":3340,"rank":3340,"depth":0,"x":587.317,"y":604.72,"cluster":"advanced-algebra"},{"id":"stacks:065B","tag":"065B","title":"Relatively finitely presented modules · Lemma 065B","summary":"Let R → A be a finite type ring map. Let M be an A-module finitely presented relative to R. For any ring map R → R' the A ⊗_R R'-module M ⊗_A A' = M ⊗_R R' is finitely presented relative to R'.","statement_latex":"Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module finitely\npresented relative to $R$. For any ring map $R \\to R'$ the\n$A \\otimes_R R'$-module\n$$\nM \\otimes_A A' = M \\otimes_R R'\n$$\nis finitely presented relative to $R'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065B","source_file":"more-algebra.tex","source_line":21047,"source_end_line":21056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21047-L21056","statement_sha256":"13aea312ee3e9feea487eda82bbc183fac0f128a650d08eebf6df36b2fe3d80c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3341,"rank":3341,"depth":0,"x":809.002,"y":682.923,"cluster":"advanced-algebra"},{"id":"stacks:0670","tag":"0670","title":"Relatively finitely presented modules · Lemma 0670","summary":"Let R → A be a finite type ring map. Let M be an A-module finitely presented relative to R. Let A → A' be a ring map of finite presentation. The A'-module M ⊗_A A' is finitely presented relative to R.","statement_latex":"Let $R \\to A$ be a finite type ring map.\nLet $M$ be an $A$-module finitely presented relative to $R$.\nLet $A \\to A'$ be a ring map of finite presentation.\nThe $A'$-module $M \\otimes_A A'$ is finitely presented relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0670","source_file":"more-algebra.tex","source_line":21072,"source_end_line":21078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21072-L21078","statement_sha256":"0f35d5abf29a506248e7edcb66586dffd7e269596b981cf6a2e32a9d41241105","origin":"The Stacks Project","memory_eligible":false,"source_rank":3342,"rank":3342,"depth":0,"x":582.434,"y":751.11,"cluster":"advanced-algebra"},{"id":"stacks:065C","tag":"065C","title":"Relatively finitely presented modules · Lemma 065C","summary":"Let R → A → B be finite type ring maps. Let M be a B-module. If M is finitely presented relative to A and A is of finite presentation over R, then M is finitely presented relative to R.","statement_latex":"Let $R \\to A \\to B$ be finite type ring maps. Let $M$ be a $B$-module.\nIf $M$ is finitely presented relative to $A$ and $A$ is of finite presentation\nover $R$, then $M$ is finitely presented relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065C","source_file":"more-algebra.tex","source_line":21101,"source_end_line":21106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21101-L21106","statement_sha256":"30f9f84b96557ca83bfc422e50aa898836d627eb969c96f39402ecc41f9afe59","origin":"The Stacks Project","memory_eligible":false,"source_rank":3343,"rank":3343,"depth":0,"x":694.773,"y":572.102,"cluster":"advanced-algebra"},{"id":"stacks:065D","tag":"065D","title":"Relatively finitely presented modules · Lemma 065D","summary":"Let R → A be a finite type ring map. Let M be an A-module. Let f_1, …, f_r ∈ A generate the unit ideal. The following are equivalent • each M_f_i is finitely presented relative to R, and • M is finitely presented relative to R.","statement_latex":"Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module.\nLet $f_1, \\ldots, f_r \\in A$ generate the unit ideal.\nThe following are equivalent\n\\begin{enumerate}\n\\item each $M_{f_i}$ is finitely presented relative to $R$, and\n\\item $M$ is finitely presented relative to $R$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065D","source_file":"more-algebra.tex","source_line":21130,"source_end_line":21139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21130-L21139","statement_sha256":"26911a6c623f1719dd50af1f7fc6bf873ea9335f29e0220ec83d9daeca433d77","origin":"The Stacks Project","memory_eligible":false,"source_rank":3344,"rank":3344,"depth":5,"x":755.946,"y":768.027,"cluster":"advanced-algebra"},{"id":"stacks:0671","tag":"0671","title":"Relatively finitely presented modules · Lemma 0671","summary":"Let R → A be a finite type ring map. Let 0 → M' → M → M\" → 0 be a short exact sequence of A-modules. • If M', M\" are finitely presented relative to R, then so is M. • If M' is a finite type A-module and M is finitely presented relative to R, then M\" is finitely presented relative to R.","statement_latex":"Let $R \\to A$ be a finite type ring map. Let $0 \\to M' \\to M \\to M'' \\to 0$\nbe a short exact sequence of $A$-modules.\n\\begin{enumerate}\n\\item If $M', M''$ are finitely presented relative to $R$, then so is $M$.\n\\item If $M'$ is a finite type $A$-module and $M$ is finitely presented\nrelative to $R$, then $M''$ is finitely presented relative to $R$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0671","source_file":"more-algebra.tex","source_line":21177,"source_end_line":21186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21177-L21186","statement_sha256":"e8a19c7747b62ec00d7ed7a28268012dff94de71a280be39e9dcfbc462d22362","origin":"The Stacks Project","memory_eligible":false,"source_rank":3345,"rank":3345,"depth":2,"x":553.092,"y":658.164,"cluster":"advanced-algebra"},{"id":"stacks:0672","tag":"0672","title":"Relatively finitely presented modules · Lemma 0672","summary":"Let R → A be a finite type ring map. Let M, M' be A-modules. If M ⊕ M' is finitely presented relative to R, then so are M and M'.","statement_latex":"Let $R \\to A$ be a finite type ring map.\nLet $M, M'$ be $A$-modules. If $M \\oplus M'$ is\nfinitely presented relative to $R$, then so are $M$ and $M'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0672","source_file":"more-algebra.tex","source_line":21193,"source_end_line":21198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21193-L21198","statement_sha256":"fb5d6ffd2ccb7be7f5f8013ec7a8750909440138d36f2a446c77f2b96e2921fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3346,"rank":3346,"depth":0,"x":791.244,"y":624.039,"cluster":"advanced-algebra"},{"id":"stacks:065F","tag":"065F","title":"Relatively pseudo-coherent modules · Lemma 065F","summary":"Let R be a ring. Let K^bullet be a complex of R-modules. Consider the R-algebra map R[x] → R which maps x to zero. Then K^bullet ⊗_R[x]^L R ≅ K^bullet ⊕ K^bullet[1] in D(R).","statement_latex":"Let $R$ be a ring. Let $K^\\bullet$ be a complex of $R$-modules.\nConsider the $R$-algebra map $R[x] \\to R$ which maps $x$ to zero.\nThen\n$$\nK^\\bullet \\otimes_{R[x]}^{\\mathbf{L}} R \\cong K^\\bullet \\oplus K^\\bullet[1]\n$$\nin $D(R)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065F","source_file":"more-algebra.tex","source_line":21217,"source_end_line":21226,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21217-L21226","statement_sha256":"3bdffb6c35c78b432e8f6f0e8125f870879a0050762bff2e6d0f92285c1354ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":3347,"rank":3347,"depth":15,"x":642.939,"y":784.484,"cluster":"advanced-algebra"},{"id":"stacks:065G","tag":"065G","title":"Relatively pseudo-coherent modules · Lemma 065G","summary":"Let R be a ring and K^bullet a complex of R-modules. Let m ∈ Z. Consider the R-algebra map R[x] → R which maps x to zero. Then K^bullet is m-pseudo-coherent as a complex of R-modules if and only if K^bullet is m-pseudo-coherent as a complex of R[x]-modules.","statement_latex":"Let $R$ be a ring and $K^\\bullet$ a complex of $R$-modules.\nLet $m \\in \\mathbf{Z}$. Consider the $R$-algebra map $R[x] \\to R$\nwhich maps $x$ to zero. Then $K^\\bullet$ is $m$-pseudo-coherent as\na complex of $R$-modules if and only if $K^\\bullet$ is $m$-pseudo-coherent\nas a complex of $R[x]$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065G","source_file":"more-algebra.tex","source_line":21258,"source_end_line":21265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21258-L21265","statement_sha256":"33629a3792157621397367849241764858a4843ed9556937ff0ef184bef2b871","origin":"The Stacks Project","memory_eligible":false,"source_rank":3348,"rank":3348,"depth":16,"x":623.253,"y":581.835,"cluster":"advanced-algebra"},{"id":"stacks:065H","tag":"065H","title":"Relatively pseudo-coherent modules · Lemma 065H","summary":"Let R → A be a ring map of finite type. Let K^bullet be a complex of A-modules. Let m ∈ Z. The following are equivalent • for some presentation α : R[x_1, …, x_n] → A the complex K^bullet is an m-pseudo-coherent complex of R[x_1, …, x_n]-modules, • for all presentations α : R[x_1, …, x_n] → A the complex K^bullet is an m-pseudo-coherent complex of R[x_1, …, x_n]-modules. In particular the same equivalence holds for pseudo-coherence.","statement_latex":"Let $R \\to A$ be a ring map of finite type.\nLet $K^\\bullet$ be a complex of $A$-modules.\nLet $m \\in \\mathbf{Z}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some presentation $\\alpha : R[x_1, \\ldots, x_n] \\to A$\nthe complex $K^\\bullet$ is an $m$-pseudo-coherent complex of\n$R[x_1, \\ldots, x_n]$-modules,\n\\item for all presentations $\\alpha : R[x_1, \\ldots, x_n] \\to A$\nthe complex $K^\\bullet$ is an $m$-pseudo-coherent complex of\n$R[x_1, \\ldots, x_n]$-modules.\n\\end{enumerate}\nIn particular the same equivalence holds for pseudo-coherence.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065H","source_file":"more-algebra.tex","source_line":21290,"source_end_line":21305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21290-L21305","statement_sha256":"10015055b89fbfd40fbe87a8ad412f44b8809d7cd1469359feb71ab8bd97f37e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3349,"rank":3349,"depth":17,"x":800.898,"y":720.223,"cluster":"advanced-algebra"},{"id":"stacks:065I","tag":"065I","title":"Relatively pseudo-coherent modules · Definition 065I","summary":"Let R → A be a finite type ring map. Let K^bullet be a complex of A-modules. Let M be an A-module. Let m ∈ Z. • We say K^bullet is m-pseudo-coherent relative to R if the equivalent conditions of Lemma [Tag 065H] hold. • We say K^bullet is pseudo-coherent relative to R if K^bullet is m-pseudo-coherent relative to R for all m ∈ Z. • We say M is m-pseudo-coherent relative to R if M[0] is m-pseudo-coherent relative to R. • We say M is pseudo-coherent relative to R if M[0] is…","statement_latex":"Let $R \\to A$ be a finite type ring map.\nLet $K^\\bullet$ be a complex of $A$-modules.\nLet $M$ be an $A$-module.\nLet $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item We say $K^\\bullet$ is {\\it $m$-pseudo-coherent relative to $R$}\nif the equivalent conditions of\nLemma \\ref{lemma-relatively-pseudo-coherent}\nhold.\n\\item We say $K^\\bullet$ is {\\it pseudo-coherent relative to $R$}\nif $K^\\bullet$ is $m$-pseudo-coherent relative to $R$ for all\n$m \\in \\mathbf{Z}$.\n\\item We say $M$ is {\\it $m$-pseudo-coherent relative to $R$}\nif $M[0]$ is $m$-pseudo-coherent relative to $R$.\n\\item We say $M$ is {\\it pseudo-coherent relative to $R$}\nif $M[0]$ is pseudo-coherent relative to $R$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065I","source_file":"more-algebra.tex","source_line":21331,"source_end_line":21350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21331-L21350","statement_sha256":"fd57f44b2ec34434609d209861f378b7a1907881c09363cd21a004147faf72f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3350,"rank":3350,"depth":18,"x":558.394,"y":718.977,"cluster":"advanced-algebra"},{"id":"stacks:0673","tag":"0673","title":"Relatively pseudo-coherent modules · Lemma 0673","summary":"Let R be a ring. Let A → B be a finite map of finite type R-algebras. Let m ∈ Z. Let K^bullet be a complex of B-modules. Then K^bullet is m-pseudo-coherent (resp. pseudo-coherent) relative to R if and only if K^bullet seen as a complex of A-modules is m-pseudo-coherent (pseudo-coherent) relative to R.","statement_latex":"Let $R$ be a ring. Let $A \\to B$ be a finite map of finite type $R$-algebras.\nLet $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $B$-modules.\nThen $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nrelative to $R$ if and only if $K^\\bullet$ seen as a complex of $A$-modules\nis $m$-pseudo-coherent (pseudo-coherent) relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0673","source_file":"more-algebra.tex","source_line":21359,"source_end_line":21366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21359-L21366","statement_sha256":"378a131d560ef7cf4114f7ca0831c96d3696be62f12c6ba9ce797dcfd702a810","origin":"The Stacks Project","memory_eligible":false,"source_rank":3351,"rank":3351,"depth":12,"x":738.381,"y":582.167,"cluster":"advanced-algebra"},{"id":"stacks:0674","tag":"0674","title":"Relatively pseudo-coherent modules · Lemma 0674","summary":"Let R be a ring. Let R → A be a finite type ring map. Let m ∈ Z. Let (K^bullet, L^bullet, M^bullet, f, g, h) be a distinguished triangle in D(A). • If K^bullet is (m + 1)-pseudo-coherent relative to R and L^bullet is m-pseudo-coherent relative to R then M^bullet is m-pseudo-coherent relative to R. • If K^bullet, M^bullet are m-pseudo-coherent relative to R, then L^bullet is m-pseudo-coherent relative to R. • If L^bullet is (m + 1)-pseudo-coherent relative to R and…","statement_latex":"Let $R$ be a ring. Let $R \\to A$ be a finite type ring map.\nLet $m \\in \\mathbf{Z}$. Let $(K^\\bullet, L^\\bullet, M^\\bullet, f, g, h)$\nbe a distinguished triangle in $D(A)$.\n\\begin{enumerate}\n\\item If $K^\\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$ and\n$L^\\bullet$ is $m$-pseudo-coherent relative to $R$ then $M^\\bullet$ is\n$m$-pseudo-coherent relative to $R$.\n\\item If $K^\\bullet, M^\\bullet$ are $m$-pseudo-coherent relative to $R$,\nthen $L^\\bullet$ is $m$-pseudo-coherent relative to $R$.\n\\item If $L^\\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$\nand $M^\\bullet$ is $m$-pseudo-coherent relative to $R$, then\n$K^\\bullet$ is $(m + 1)$-pseudo-coherent relative to $R$.\n\\end{enumerate}\nMoreover, if two out of three of $K^\\bullet, L^\\bullet, M^\\bullet$\nare pseudo-coherent relative to $R$, the so is the third.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0674","source_file":"more-algebra.tex","source_line":21392,"source_end_line":21409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21392-L21409","statement_sha256":"11a971d2663ce9ab3cc31dff00ccf82429c4265733605be78987baf621c472e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3352,"rank":3352,"depth":9,"x":715.658,"y":785.364,"cluster":"advanced-algebra"},{"id":"stacks:0675","tag":"0675","title":"Relatively pseudo-coherent modules · Lemma 0675","summary":"Let R → A be a finite type ring map. Let M be an A-module. Then • M is 0-pseudo-coherent relative to R if and only if M is a finite type A-module, • M is (-1)-pseudo-coherent relative to R if and only if M is a finitely presented relative to R, • M is (-d)-pseudo-coherent relative to R if and only if for every surjection R[x_1, …, x_n] → A there exists a resolution R[x_1, …, x_n]^⊕ a_d → R[x_1, …, x_n]^⊕ a_d - 1 → … → R[x_1, …, x_n]^⊕ a_0 → M → 0 of length d, and • M is…","statement_latex":"Let $R \\to A$ be a finite type ring map. Let $M$ be an $A$-module.\nThen\n\\begin{enumerate}\n\\item $M$ is $0$-pseudo-coherent relative to $R$ if and only if\n$M$ is a finite type $A$-module,\n\\item $M$ is $(-1)$-pseudo-coherent relative to $R$ if and only if\n$M$ is a finitely presented relative to $R$,\n\\item $M$ is $(-d)$-pseudo-coherent relative to $R$ if and only if\nfor every surjection $R[x_1, \\ldots, x_n] \\to A$ there exists a\nresolution\n$$\nR[x_1, \\ldots, x_n]^{\\oplus a_d} \\to R[x_1, \\ldots, x_n]^{\\oplus a_{d - 1}}\n\\to \\ldots \\to R[x_1, \\ldots, x_n]^{\\oplus a_0} \\to M \\to 0\n$$\nof length $d$, and\n\\item $M$ is pseudo-coherent relative to $R$ if and only if\nfor every presentation $R[x_1, \\ldots, x_n] \\to A$ there exists an\ninfinite resolution\n$$\n\\ldots \\to R[x_1, \\ldots, x_n]^{\\oplus a_1} \\to\nR[x_1, \\ldots, x_n]^{\\oplus a_0} \\to M \\to 0\n$$\nby finite free $R[x_1, \\ldots, x_n]$-modules.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0675","source_file":"more-algebra.tex","source_line":21417,"source_end_line":21443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21417-L21443","statement_sha256":"fc40a87e7fee137d2405f71d22d12392d851c818d7dd57a5e938823f1cdb7bc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3353,"rank":3353,"depth":10,"x":568.873,"y":622.487,"cluster":"advanced-algebra"},{"id":"stacks:0676","tag":"0676","title":"Relatively pseudo-coherent modules · Lemma 0676","summary":"Let R → A be a finite type ring map. Let m ∈ Z. Let K^bullet, L^bullet ∈ D(A). If K^bullet ⊕ L^bullet is m-pseudo-coherent (resp. pseudo-coherent) relative to R so are K^bullet and L^bullet.","statement_latex":"Let $R \\to A$ be a finite type ring map.\nLet $m \\in \\mathbf{Z}$. Let $K^\\bullet, L^\\bullet \\in D(A)$.\nIf $K^\\bullet \\oplus L^\\bullet$\nis $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$\nso are $K^\\bullet$ and $L^\\bullet$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0676","source_file":"more-algebra.tex","source_line":21451,"source_end_line":21458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21451-L21458","statement_sha256":"18736d41ab1544192f9c633f14ce807346d7ce1eac6353ecc3e2bdbafc868c70","origin":"The Stacks Project","memory_eligible":false,"source_rank":3354,"rank":3354,"depth":12,"x":808.311,"y":659.334,"cluster":"advanced-algebra"},{"id":"stacks:0677","tag":"0677","title":"Relatively pseudo-coherent modules · Lemma 0677","summary":"Let R → A be a finite type ring map. Let m ∈ Z. Let K^bullet be a bounded above complex of A-modules such that K^i is (m - i)-pseudo-coherent relative to R for all i. Then K^bullet is m-pseudo-coherent relative to R. In particular, if K^bullet is a bounded above complex of A-modules pseudo-coherent relative to R, then K^bullet is pseudo-coherent relative to R.","statement_latex":"Let $R \\to A$ be a finite type ring map.\nLet $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a bounded\nabove complex of $A$-modules such that $K^i$ is $(m - i)$-pseudo-coherent\nrelative to $R$ for all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent\nrelative to $R$. In particular, if $K^\\bullet$ is a bounded above complex of\n$A$-modules pseudo-coherent relative to $R$, then $K^\\bullet$ is\npseudo-coherent relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0677","source_file":"more-algebra.tex","source_line":21466,"source_end_line":21475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21466-L21475","statement_sha256":"51f420ff3d953b76ab6320055a95619e99812287fef49a721536d48fcd58e2b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3355,"rank":3355,"depth":11,"x":601.933,"y":768.126,"cluster":"advanced-algebra"},{"id":"stacks:0678","tag":"0678","title":"Relatively pseudo-coherent modules · Lemma 0678","summary":"Let R → A be a finite type ring map. Let m ∈ Z. Let K^bullet ∈ D^-(A) such that H^i(K^bullet) is (m - i)-pseudo-coherent (resp. pseudo-coherent) relative to R for all i. Then K^bullet is m-pseudo-coherent (resp. pseudo-coherent) relative to R.","statement_latex":"Let $R \\to A$ be a finite type ring map. Let $m \\in \\mathbf{Z}$.\nLet $K^\\bullet \\in D^{-}(A)$ such that $H^i(K^\\bullet)$ is\n$(m - i)$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$\nfor all $i$. Then $K^\\bullet$ is $m$-pseudo-coherent\n(resp.\\ pseudo-coherent) relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0678","source_file":"more-algebra.tex","source_line":21483,"source_end_line":21490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21483-L21490","statement_sha256":"7114358a04581559cedcb9ad3bdda6b5b3f166be267e9d13c954cc60b64a9ee2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3356,"rank":3356,"depth":12,"x":666.685,"y":570.621,"cluster":"advanced-algebra"},{"id":"stacks:0679","tag":"0679","title":"Relatively pseudo-coherent modules · Lemma 0679","summary":"Let R be a ring, f ∈ R an element, R_f → A is a finite type ring map, g ∈ A, and K^bullet a complex of A-modules. If K^bullet is m-pseudo-coherent (resp. pseudo-coherent) relative to R_f, then K^bullet ⊗_A A_g is m-pseudo-coherent (resp. pseudo-coherent) relative to R.","statement_latex":"Let $R$ be a ring, $f \\in R$ an element, $R_f \\to A$ is a finite type ring map,\n$g \\in A$, and $K^\\bullet$ a complex of $A$-modules.\nIf $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nrelative to $R_f$, then $K^\\bullet \\otimes_A A_g$ is\n$m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0679","source_file":"more-algebra.tex","source_line":21498,"source_end_line":21505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21498-L21505","statement_sha256":"58f8633813d56311702d25095c60fadab3566af64bb40abda488b8da671a6e51","origin":"The Stacks Project","memory_eligible":false,"source_rank":3357,"rank":3357,"depth":12,"x":777.867,"y":753.166,"cluster":"advanced-algebra"},{"id":"stacks:067A","tag":"067A","title":"Relatively pseudo-coherent modules · Lemma 067A","summary":"Let R → A be a finite type ring map. Let m ∈ Z. Let K^bullet be a complex of A-modules which is m-pseudo-coherent (resp. pseudo-coherent) relative to R. Let R → R' be a ring map such that A and R' are Tor independent over R. Set A' = A ⊗_R R'. Then K^bullet ⊗_A^L A' is m-pseudo-coherent (resp. pseudo-coherent) relative to R'.","statement_latex":"Let $R \\to A$ be a finite type ring map. Let $m \\in \\mathbf{Z}$.\nLet $K^\\bullet$ be a complex of $A$-modules which is $m$-pseudo-coherent\n(resp.\\ pseudo-coherent) relative to $R$. Let $R \\to R'$ be a ring\nmap such that $A$ and $R'$ are Tor independent over $R$. Set\n$A' = A \\otimes_R R'$. Then\n$K^\\bullet \\otimes_A^{\\mathbf{L}} A'$\nis $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067A","source_file":"more-algebra.tex","source_line":21542,"source_end_line":21551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21542-L21551","statement_sha256":"6b57406d97882fc76e855afd035ab976c9cef574229cbd8b6fde270b76b16be3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3358,"rank":3358,"depth":11,"x":548.878,"y":681.582,"cluster":"advanced-algebra"},{"id":"stacks:067B","tag":"067B","title":"Relatively pseudo-coherent modules · Lemma 067B","summary":"Let R → A → B be finite type ring maps. Let m ∈ Z. Let K^bullet be a complex of A-modules. Assume B as a B-module is pseudo-coherent relative to A. If K^bullet is m-pseudo-coherent (resp. pseudo-coherent) relative to R, then K^bullet ⊗_A^L B is m-pseudo-coherent (resp. pseudo-coherent) relative to R.","statement_latex":"Let $R \\to A \\to B$ be finite type ring maps.\nLet $m \\in \\mathbf{Z}$.\nLet $K^\\bullet$ be a complex of $A$-modules.\nAssume $B$ as a $B$-module is pseudo-coherent relative to $A$.\nIf $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nrelative to $R$, then $K^\\bullet \\otimes_A^{\\mathbf{L}} B$ is\n$m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067B","source_file":"more-algebra.tex","source_line":21567,"source_end_line":21576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21567-L21576","statement_sha256":"63b8de777cbf004ce1fbf3f53b98970d8484e773944f6a894fefa589dbeb9682","origin":"The Stacks Project","memory_eligible":false,"source_rank":3359,"rank":3359,"depth":12,"x":775.502,"y":604.363,"cluster":"advanced-algebra"},{"id":"stacks:067C","tag":"067C","title":"Relatively pseudo-coherent modules · Lemma 067C","summary":"Let R → A → B be finite type ring maps. Let m ∈ Z. Let M be an A-module. Assume B is flat over A and B as a B-module is pseudo-coherent relative to A. If M is m-pseudo-coherent (resp. pseudo-coherent) relative to R, then M ⊗_A B is m-pseudo-coherent (resp. pseudo-coherent) relative to R.","statement_latex":"Let $R \\to A \\to B$ be finite type ring maps.\nLet $m \\in \\mathbf{Z}$. Let $M$ be an $A$-module.\nAssume $B$ is flat over $A$ and $B$ as a $B$-module is\npseudo-coherent relative to $A$.\nIf $M$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nrelative to $R$, then $M \\otimes_A B$ is\n$m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067C","source_file":"more-algebra.tex","source_line":21621,"source_end_line":21630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21621-L21630","statement_sha256":"700614d0aee26ccd5338500291268c1ab0f1a00f0eeef43261dfd987e9f87f10","origin":"The Stacks Project","memory_eligible":false,"source_rank":3360,"rank":3360,"depth":13,"x":670.395,"y":790.063,"cluster":"advanced-algebra"},{"id":"stacks:067D","tag":"067D","title":"Relatively pseudo-coherent modules · Lemma 067D","summary":"Let R be a ring. Let A → B be a map of finite type R-algebras. Let m ∈ Z. Let K^bullet be a complex of B-modules. Assume A is pseudo-coherent relative to R. Then the following are equivalent • K^bullet is m-pseudo-coherent (resp. pseudo-coherent) relative to A, and • K^bullet is m-pseudo-coherent (resp. pseudo-coherent) relative to R.","statement_latex":"Let $R$ be a ring. Let $A \\to B$ be a map of finite type $R$-algebras.\nLet $m \\in \\mathbf{Z}$. Let $K^\\bullet$ be a complex of $B$-modules.\nAssume $A$ is pseudo-coherent relative to $R$. Then the following are\nequivalent\n\\begin{enumerate}\n\\item $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nrelative to $A$, and\n\\item $K^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nrelative to $R$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067D","source_file":"more-algebra.tex","source_line":21637,"source_end_line":21649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21637-L21649","statement_sha256":"10f43bcf049ab514e7cdf4023274165659f5e2778b6f24d36d35b18a9020e3ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":3361,"rank":3361,"depth":12,"x":598.499,"y":593.313,"cluster":"advanced-algebra"},{"id":"stacks:067E","tag":"067E","title":"Relatively pseudo-coherent modules · Lemma 067E","summary":"Let R → A be a finite type ring map. Let K^bullet be a complex of A-modules. Let m ∈ Z. Let f_1, …, f_r ∈ A generate the unit ideal. The following are equivalent • each K^bullet ⊗_A A_f_i is m-pseudo-coherent relative to R, and • K^bullet is m-pseudo-coherent relative to R. The same equivalence holds for pseudo-coherence relative to R.","statement_latex":"Let $R \\to A$ be a finite type ring map.\nLet $K^\\bullet$ be a complex of $A$-modules.\nLet $m \\in \\mathbf{Z}$.\nLet $f_1, \\ldots, f_r \\in A$ generate the unit ideal.\nThe following are equivalent\n\\begin{enumerate}\n\\item each $K^\\bullet \\otimes_A A_{f_i}$ is\n$m$-pseudo-coherent relative to $R$, and\n\\item $K^\\bullet$ is $m$-pseudo-coherent relative to $R$.\n\\end{enumerate}\nThe same equivalence holds for pseudo-coherence relative to $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067E","source_file":"more-algebra.tex","source_line":21668,"source_end_line":21681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21668-L21681","statement_sha256":"b303dc5bd3fdbfa0fe748ff9bcc74ac7bb763271735aac6409d75b689e065bfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3362,"rank":3362,"depth":13,"x":809.926,"y":697.692,"cluster":"advanced-algebra"},{"id":"stacks:067F","tag":"067F","title":"Relatively pseudo-coherent modules · Lemma 067F","summary":"Let R be a Noetherian ring. Let R → A be a finite type ring map. Then • A complex of A-modules K^bullet is m-pseudo-coherent relative to R if and only if K^bullet ∈ D^-(A) and H^i(K^bullet) is a finite A-module for i ≥ m. • A complex of A-modules K^bullet is pseudo-coherent relative to R if and only if K^bullet ∈ D^-(A) and H^i(K^bullet) is a finite A-module for all i. • An A-module is pseudo-coherent relative to R if and only if it is finite.","statement_latex":"Let $R$ be a Noetherian ring. Let $R \\to A$ be a finite type ring map. Then\n\\begin{enumerate}\n\\item A complex of $A$-modules $K^\\bullet$ is $m$-pseudo-coherent\nrelative to $R$ if and only if $K^\\bullet \\in D^{-}(A)$ and\n$H^i(K^\\bullet)$ is a finite $A$-module for $i \\geq m$.\n\\item A complex of $A$-modules $K^\\bullet$ is pseudo-coherent relative to $R$\nif and only if $K^\\bullet \\in D^{-}(A)$ and\n$H^i(K^\\bullet)$ is a finite $A$-module for all $i$.\n\\item An $A$-module is pseudo-coherent relative to $R$\nif and only if it is finite.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067F","source_file":"more-algebra.tex","source_line":21714,"source_end_line":21727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21714-L21727","statement_sha256":"9e07dad20186bcf1512e65358276f47842a5f8d2d73b15136f2f5af37ef95a31","origin":"The Stacks Project","memory_eligible":false,"source_rank":3363,"rank":3363,"depth":13,"x":569.868,"y":740.732,"cluster":"advanced-algebra"},{"id":"stacks:067H","tag":"067H","title":"Pseudo-coherent and perfect ring maps · Definition 067H","summary":"Let A → B be a ring map. • We say A → B is a pseudo-coherent ring map if it is of finite type and B, as a B-module, is pseudo-coherent relative to A. • We say A → B is a perfect ring map if it is a pseudo-coherent ring map such that B as an A-module has finite tor dimension.","statement_latex":"Let $A \\to B$ be a ring map.\n\\begin{enumerate}\n\\item We say $A \\to B$ is a {\\it pseudo-coherent ring map} if it is of finite\ntype and $B$, as a $B$-module, is pseudo-coherent relative to $A$.\n\\item We say $A \\to B$ is a {\\it perfect ring map} if it is a\npseudo-coherent ring map such that $B$ as an $A$-module has finite\ntor dimension.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent and perfect ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067H","source_file":"more-algebra.tex","source_line":21750,"source_end_line":21760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21750-L21760","statement_sha256":"8c5a1a34a5c3de5d00a8a768be5eedc93f41e27b9733cc742035363e6e589e2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3364,"rank":3364,"depth":0,"x":712.4,"y":572.63,"cluster":"advanced-algebra"},{"id":"stacks:068Y","tag":"068Y","title":"Pseudo-coherent and perfect ring maps · Lemma 068Y","summary":"A ring map A → B is perfect if and only if B = A[x_1, …, x_n]/I and B as an A[x_1, …, x_n]-module has a finite resolution by finite projective A[x_1, …, x_n]-modules.","statement_latex":"A ring map $A \\to B$ is perfect if and only if $B = A[x_1, \\ldots, x_n]/I$\nand $B$ as an $A[x_1, \\ldots, x_n]$-module has a finite resolution by\nfinite projective $A[x_1, \\ldots, x_n]$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent and perfect ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068Y","source_file":"more-algebra.tex","source_line":21773,"source_end_line":21778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21773-L21778","statement_sha256":"3f3c299f4ae6dcc25ad4f418ea9aa92badfb978084aa90f8c8c4643b603fc1ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":3365,"rank":3365,"depth":16,"x":742.509,"y":777.645,"cluster":"advanced-algebra"},{"id":"stacks:067I","tag":"067I","title":"Pseudo-coherent and perfect ring maps · Lemma 067I","summary":"A finite type ring map of Noetherian rings is pseudo-coherent.","statement_latex":"A finite type ring map of Noetherian rings is pseudo-coherent.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent and perfect ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067I","source_file":"more-algebra.tex","source_line":21804,"source_end_line":21807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21804-L21807","statement_sha256":"364af1c8c9120bfbc3d3079e43ea8aa64499b0dc88a50cfe10d653d35519f13e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3366,"rank":3366,"depth":14,"x":555.272,"y":643.434,"cluster":"advanced-algebra"},{"id":"stacks:067J","tag":"067J","title":"Pseudo-coherent and perfect ring maps · Lemma 067J","summary":"A ring map which is flat and of finite presentation is perfect.","statement_latex":"A ring map which is flat and of finite presentation is perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent and perfect ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067J","source_file":"more-algebra.tex","source_line":21814,"source_end_line":21817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21814-L21817","statement_sha256":"f1afdd9e6fa8573f166ee5c7e704b9aa251a49bac1c30baa657fee02bf73a1b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3367,"rank":3367,"depth":35,"x":801.486,"y":636.15,"cluster":"advanced-algebra"},{"id":"stacks:067K","tag":"067K","title":"Pseudo-coherent and perfect ring maps · Lemma 067K","summary":"Let A → B be a finite type ring map with A a regular ring of finite dimension. Then A → B is perfect.","statement_latex":"Let $A \\to B$ be a finite type ring map with $A$ a regular ring\nof finite dimension. Then $A \\to B$ is perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent and perfect ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067K","source_file":"more-algebra.tex","source_line":21834,"source_end_line":21838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21834-L21838","statement_sha256":"29ee037b28b7bf47c83b8dc6235c26ad35f9f4ed5f17e10550754d598b133345","origin":"The Stacks Project","memory_eligible":false,"source_rank":3368,"rank":3368,"depth":18,"x":625.632,"y":781.359,"cluster":"advanced-algebra"},{"id":"stacks:07EN","tag":"07EN","title":"Pseudo-coherent and perfect ring maps · Lemma 07EN","summary":"A local complete intersection homomorphism is perfect.","statement_latex":"A local complete intersection homomorphism is perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent and perfect ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EN","source_file":"more-algebra.tex","source_line":21851,"source_end_line":21854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21851-L21854","statement_sha256":"e3f54e3677d7000f0a258dfc359e0f116f1d6138c1581435811a9a81655d4244","origin":"The Stacks Project","memory_eligible":false,"source_rank":3369,"rank":3369,"depth":17,"x":638.543,"y":574.316,"cluster":"advanced-algebra"},{"id":"stacks:0DHQ","tag":"0DHQ","title":"Pseudo-coherent and perfect ring maps · Lemma 0DHQ","summary":"Let R → A be a pseudo-coherent ring map. Let K ∈ D(A). The following are equivalent • K is m-pseudo-coherent (resp. pseudo-coherent) relative to R, and • K is m-pseudo-coherent (resp. pseudo-coherent) in D(A).","statement_latex":"Let $R \\to A$ be a pseudo-coherent ring map. Let $K \\in D(A)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) relative to $R$, and\n\\item $K$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent) in $D(A)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent and perfect ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHQ","source_file":"more-algebra.tex","source_line":21869,"source_end_line":21877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21869-L21877","statement_sha256":"a05e797c362660d2b1f8a93c89ba88471dcdd3285d2b12983bcf360afbebfe7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3370,"rank":3370,"depth":13,"x":795.66,"y":734.454,"cluster":"advanced-algebra"},{"id":"stacks:0E1T","tag":"0E1T","title":"Pseudo-coherent and perfect ring maps · Lemma 0E1T","summary":"Let R → B → A be ring maps with φ : B → A surjective and R → B and R → A flat and of finite presentation. For K ∈ D(A) denote φ_*K ∈ D(B) the restriction. The following are equivalent • K is pseudo-coherent, • K is pseudo-coherent relative to R, • K is pseudo-coherent relative to A, • φ_*K is pseudo-coherent, • φ_*K is pseudo-coherent relative to R. Similar holds for m-pseudo-coherence.","statement_latex":"Let $R \\to B \\to A$ be ring maps with $\\varphi : B \\to A$ surjective and\n$R \\to B$ and $R \\to A$ flat and of finite presentation. For $K \\in D(A)$\ndenote $\\varphi_*K \\in D(B)$ the restriction.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ is pseudo-coherent,\n\\item $K$ is pseudo-coherent relative to $R$,\n\\item $K$ is pseudo-coherent relative to $A$,\n\\item $\\varphi_*K$ is pseudo-coherent,\n\\item $\\varphi_*K$ is pseudo-coherent relative to $R$.\n\\end{enumerate}\nSimilar holds for $m$-pseudo-coherence.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Pseudo-coherent and perfect ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1T","source_file":"more-algebra.tex","source_line":21884,"source_end_line":21898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21884-L21898","statement_sha256":"d903c14e6ead6eca3218fe6d589264e3bbc59775abd74346e66dfd0f7a748c5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3371,"rank":3371,"depth":36,"x":550.809,"y":705.497,"cluster":"advanced-algebra"},{"id":"stacks:0DHS","tag":"0DHS","title":"Relatively perfect modules · Definition 0DHS","summary":"Let R → A be a flat ring map of finite presentation. An object K of D(A) is R-perfect or perfect relative to R if K is pseudo-coherent (Definition [Tag 064Q]) and has finite tor dimension over R (Definition [Tag 0652]).","statement_latex":"Let $R \\to A$ be a flat ring map of finite presentation.\nAn object $K$ of $D(A)$ is {\\it $R$-perfect} or {\\it perfect relative to $R$}\nif $K$ is pseudo-coherent\n(Definition \\ref{definition-pseudo-coherent})\nand has finite tor dimension over $R$\n(Definition \\ref{definition-tor-amplitude}).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHS","source_file":"more-algebra.tex","source_line":21932,"source_end_line":21940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21932-L21940","statement_sha256":"1b7b5ddd89485daee72c7148c2123849ebd2b679b6592cfecbcd724fe0e6d6f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3372,"rank":3372,"depth":1,"x":754.826,"y":587.811,"cluster":"advanced-algebra"},{"id":"stacks:0DHT","tag":"0DHT","title":"Relatively perfect modules · Lemma 0DHT","summary":"Let R → A be a flat ring map of finite presentation. The R-perfect objects of D(A) form a saturated triangulated strictly full subcategory.","statement_latex":"Let $R \\to A$ be a flat ring map of finite presentation.\nThe $R$-perfect objects of $D(A)$ form a\nsaturated\\footnote{Derived Categories, Definition\n\\ref{derived-definition-saturated}.} triangulated\nstrictly full subcategory.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHT","source_file":"more-algebra.tex","source_line":21948,"source_end_line":21955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21948-L21955","statement_sha256":"e157f33c306810a9d65dc4c456da5a1f7bb66284970852da9506731be089043a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3373,"rank":3373,"depth":12,"x":698.977,"y":790.534,"cluster":"advanced-algebra"},{"id":"stacks:0DHU","tag":"0DHU","title":"Relatively perfect modules · Lemma 0DHU","summary":"Let R → A be a flat ring map of finite presentation. A perfect object of D(A) is R-perfect. If K, M ∈ D(A) then K ⊗_A^L M is R-perfect if K is perfect and M is R-perfect.","statement_latex":"Let $R \\to A$ be a flat ring map of finite presentation.\nA perfect object of $D(A)$ is $R$-perfect. If $K, M \\in D(A)$\nthen $K \\otimes_A^\\mathbf{L} M$ is $R$-perfect if $K$ is perfect\nand $M$ is $R$-perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHU","source_file":"more-algebra.tex","source_line":21965,"source_end_line":21971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21965-L21971","statement_sha256":"9fc1086738b58da61f8004bc180f4da3e5be5fa70923175c39e9d16df7ec26d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3374,"rank":3374,"depth":11,"x":577.027,"y":609.2,"cluster":"advanced-algebra"},{"id":"stacks:0DHV","tag":"0DHV","title":"Relatively perfect modules · Lemma 0DHV","summary":"Let R → A be a flat ring map of finite presentation. Let K ∈ D(A). The following are equivalent • K is R-perfect, and • K is isomorphic to a finite complex of R-flat, finitely presented A-modules.","statement_latex":"Let $R \\to A$ be a flat ring map of finite presentation.\nLet $K \\in D(A)$. The following are equivalent\n\\begin{enumerate}\n\\item $K$ is $R$-perfect, and\n\\item $K$ is isomorphic to a finite complex of $R$-flat,\nfinitely presented $A$-modules.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHV","source_file":"more-algebra.tex","source_line":21979,"source_end_line":21988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L21979-L21988","statement_sha256":"6b337b7fade3deaf1dbc2959e063264efda44d58317ceac581ab8d6ebe0bb143","origin":"The Stacks Project","memory_eligible":false,"source_rank":3375,"rank":3375,"depth":35,"x":812.983,"y":673.772,"cluster":"advanced-algebra"},{"id":"stacks:0DHW","tag":"0DHW","title":"Relatively perfect modules · Lemma 0DHW","summary":"Let R → A be a flat ring map of finite presentation. Let R → R' be a ring map and set A' = A ⊗_R R'. If K ∈ D(A) is R-perfect, then K ⊗_A^L A' is R'-perfect.","statement_latex":"Let $R \\to A$ be a flat ring map of finite presentation.\nLet $R \\to R'$ be a ring map and set $A' = A \\otimes_R R'$.\nIf $K \\in D(A)$ is $R$-perfect, then $K \\otimes_A^\\mathbf{L} A'$ is\n$R'$-perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHW","source_file":"more-algebra.tex","source_line":22021,"source_end_line":22027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22021-L22027","statement_sha256":"03c66bb465dd1bfba396e658fccef6647ff460237b1a47ee6897013dd9bfa7ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":3376,"rank":3376,"depth":11,"x":586.867,"y":760.12,"cluster":"advanced-algebra"},{"id":"stacks:0E1U","tag":"0E1U","title":"Relatively perfect modules · Lemma 0E1U","summary":"Let R → A be a flat ring map. Let K, L ∈ D(A) with K pseudo-coherent and L finite tor dimension over R. We may choose • a bounded above complex P^bullet of finite free A-modules representing K, and • a bounded complex of R-flat A-modules F^bullet representing L. Given these choices we have • [(a)] E^bullet = Hom^bullet(P^bullet, F^bullet) is a bounded below complex of R-flat A-modules representing RHom_A(K, L), • [(b)] for any ring map R → R' with A' = A ⊗_R R' the…","statement_latex":"Let $R \\to A$ be a flat ring map. Let $K, L \\in D(A)$ with $K$\npseudo-coherent and $L$ finite tor dimension over $R$. We may choose\n\\begin{enumerate}\n\\item a bounded above complex $P^\\bullet$\nof finite free $A$-modules representing $K$, and\n\\item a bounded complex of $R$-flat $A$-modules\n$F^\\bullet$ representing $L$.\n\\end{enumerate}\nGiven these choices we have\n\\begin{enumerate}\n\\item[(a)] $E^\\bullet = \\Hom^\\bullet(P^\\bullet, F^\\bullet)$\nis a bounded below complex\nof $R$-flat $A$-modules representing $R\\Hom_A(K, L)$,\n\\item[(b)] for any ring map $R \\to R'$ with $A' = A \\otimes_R R'$\nthe complex $E^\\bullet \\otimes_R R'$ represents\n$R\\Hom_{A'}(K \\otimes_A^\\mathbf{L} A', L \\otimes_A^\\mathbf{L} A')$.\n\\end{enumerate}\nIf in addition $R \\to A$ is of finite presentation and $L$ is\n$R$-perfect, then we may choose $F^p$ to be finitely presented\n$A$-modules and consequently $E^n$ will be finitely presented $A$-modules\nas well.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1U","source_file":"more-algebra.tex","source_line":22039,"source_end_line":22062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22039-L22062","statement_sha256":"f5e23b7141a3569d0ad7a5db4f78e8e303d92c00cf6e84ae114df7ca21a429c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3377,"rank":3377,"depth":6,"x":684.248,"y":567.976,"cluster":"advanced-algebra"},{"id":"stacks:0DHX","tag":"0DHX","title":"Relatively perfect modules · Lemma 0DHX","summary":"Let R = colim_i ∈ I R_i be a filtered colimit of rings. Let 0 ∈ I and R_0 → A_0 be a flat ring map of finite presentation. For i ≥ 0 set A_i = R_i ⊗_R_0 A_0 and set A = R ⊗_R_0 A_0. • Given an R-perfect K in D(A) there exists an i ∈ I and an R_i-perfect K_i in D(A_i) such that K ≅ K_i ⊗_A_i^L A in D(A). • Given K_0, L_0 ∈ D(A_0) with K_0 pseudo-coherent and L_0 finite tor dimension over R_0, then we have Hom_D(A)(K_0 ⊗_A_0^L A, L_0 ⊗_A_0^L A) = colim_i ≥ 0 Hom_D(A_i)(K_0…","statement_latex":"Let $R = \\colim_{i \\in I} R_i$ be a filtered colimit of rings.\nLet $0 \\in I$ and $R_0 \\to A_0$ be a flat ring map of\nfinite presentation. For $i \\geq 0$ set $A_i = R_i \\otimes_{R_0} A_0$\nand set $A = R \\otimes_{R_0} A_0$.\n\\begin{enumerate}\n\\item Given an $R$-perfect $K$ in $D(A)$ there exists an $i \\in I$\nand an $R_i$-perfect $K_i$ in $D(A_i)$ such that\n$K \\cong K_i \\otimes_{A_i}^\\mathbf{L} A$ in $D(A)$.\n\\item Given $K_0, L_0 \\in D(A_0)$ with $K_0$ pseudo-coherent\nand $L_0$ finite tor dimension over $R_0$, then\nwe have\n$$\n\\Hom_{D(A)}(K_0 \\otimes_{A_0}^\\mathbf{L} A, L_0 \\otimes_{A_0}^\\mathbf{L} A) =\n\\colim_{i \\geq 0}\n\\Hom_{D(A_i)}(K_0 \\otimes_{A_0}^\\mathbf{L} A_i,\nL_0 \\otimes_{A_0}^\\mathbf{L} A_i)\n$$\n\\end{enumerate}\nIn particular, the triangulated category of $R$-perfect complexes over $A$\nis the colimit of the triangulated categories of\n$R_i$-perfect complexes over $A_i$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHX","source_file":"more-algebra.tex","source_line":22104,"source_end_line":22127,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22104-L22127","statement_sha256":"46b1c975174487c268e072f7092c9b4a4380e1f7d5f95314bc450feb9ecbd6f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3378,"rank":3378,"depth":36,"x":767.03,"y":765.09,"cluster":"advanced-algebra"},{"id":"stacks:0DJG","tag":"0DJG","title":"Relatively perfect modules · Lemma 0DJG","summary":"Let R' → A' be a flat ring map of finite presentation. Let R' → R be a surjective ring map whose kernel is a nilpotent ideal. Set A = A' ⊗_R' R. Let K' ∈ D(A') and set K = K' ⊗_A'^L A in D(A). If K is R-perfect, then K' is R'-perfect.","statement_latex":"Let $R' \\to A'$ be a flat ring map of finite presentation.\nLet $R' \\to R$ be a surjective ring map whose kernel is a nilpotent ideal.\nSet $A = A' \\otimes_{R'} R$. Let $K' \\in D(A')$ and set\n$K = K' \\otimes_{A'}^\\mathbf{L} A$ in $D(A)$.\nIf $K$ is $R$-perfect, then $K'$ is $R'$-perfect.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJG","source_file":"more-algebra.tex","source_line":22163,"source_end_line":22170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22163-L22170","statement_sha256":"4b76a63c752d0be6db0615cc8dae387e057999ec50bfadc18066c349b6d6d1f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3379,"rank":3379,"depth":12,"x":547.283,"y":666.627,"cluster":"advanced-algebra"},{"id":"stacks:0DJH","tag":"0DJH","title":"Relatively perfect modules · Lemma 0DJH","summary":"Let R be a ring. Let A = R[x_1, …, x_d]/I be flat and of finite presentation over R. Let q ⊂ A be a prime ideal lying over p ⊂ R. Let K ∈ D(A) be pseudo-coherent. Let a, b ∈ Z. If H^i(K_ q ⊗_R_ p^L kappa( p)) is nonzero only for i ∈ [a, b], then K_ q has tor amplitude in [a - d, b] over R.","statement_latex":"Let $R$ be a ring. Let $A = R[x_1, \\ldots, x_d]/I$\nbe flat and of finite presentation over $R$.\nLet $\\mathfrak q \\subset A$ be a prime ideal lying over\n$\\mathfrak p \\subset R$. Let $K \\in D(A)$ be pseudo-coherent.\nLet $a, b \\in \\mathbf{Z}$. If\n$H^i(K_\\mathfrak q \\otimes_{R_\\mathfrak p}^\\mathbf{L} \\kappa(\\mathfrak p))$\nis nonzero only for $i \\in [a, b]$, then\n$K_\\mathfrak q$ has tor amplitude in $[a - d, b]$ over $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJH","source_file":"more-algebra.tex","source_line":22180,"source_end_line":22190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22180-L22190","statement_sha256":"da321246c49917c17871aff4061d2364f674e4685bc075afbcdcec8cb02f2d75","origin":"The Stacks Project","memory_eligible":false,"source_rank":3380,"rank":3380,"depth":37,"x":788.712,"y":614.497,"cluster":"advanced-algebra"},{"id":"stacks:0GHJ","tag":"0GHJ","title":"Relatively perfect modules · Lemma 0GHJ","summary":"Let R → A be a ring map which is flat and of finite presentation. Let K ∈ D(A) be pseudo-coherent. The following are equivalent • K is R-perfect, and • K is bounded below and for every prime ideal p ⊂ R the object K ⊗_R^L kappa( p) is bounded below.","statement_latex":"Let $R \\to A$ be a ring map which is flat and of finite presentation.\nLet $K \\in D(A)$ be pseudo-coherent. The following are equivalent\n\\begin{enumerate}\n\\item $K$ is $R$-perfect, and\n\\item $K$ is bounded below and for every prime ideal $\\mathfrak p \\subset R$\nthe object $K \\otimes_R^\\mathbf{L} \\kappa(\\mathfrak p)$ is bounded below.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Relatively perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHJ","source_file":"more-algebra.tex","source_line":22204,"source_end_line":22213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22204-L22213","statement_sha256":"5d15e5bb806231bd89aabd7f380aa4cf9f904e508a29d134bf90fb2dc3f25268","origin":"The Stacks Project","memory_eligible":false,"source_rank":3381,"rank":3381,"depth":37,"x":652.489,"y":790.083,"cluster":"advanced-algebra"},{"id":"stacks:0G9C","tag":"0G9C","title":"Two term complexes · Lemma 0G9C","summary":"Let R be a ring. Let K ∈ D(R) with H^i(K) = 0 for i not ∈ (-1, 0). The following are equivalent • H^-1(K) = 0 and H^0(K) is a projective module and • Ext^1_R(K, M) = 0 for every R-module M. If R is Noetherian and H^i(K) is a finite R-module for i = -1, 0, then these are also equivalent to • [(3)] Ext^1_R(K, M) = 0 for every finite R-module M.","statement_latex":"Let $R$ be a ring. Let $K \\in D(R)$ with $H^i(K) = 0$ for\n$i \\not \\in \\{-1, 0\\}$. The following are equivalent\n\\begin{enumerate}\n\\item $H^{-1}(K) = 0$ and $H^0(K)$ is a projective module and\n\\item $\\Ext^1_R(K, M) = 0$ for every $R$-module $M$.\n\\end{enumerate}\nIf $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for\n$i = -1, 0$, then these are also equivalent to\n\\begin{enumerate}\n\\item[(3)] $\\Ext^1_R(K, M) = 0$ for every finite $R$-module $M$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9C","source_file":"more-algebra.tex","source_line":22264,"source_end_line":22277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22264-L22277","statement_sha256":"306a591b29e1046c67deb97e9c6bec3a8bec0026f23355306986d25c65843eb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3382,"rank":3382,"depth":13,"x":611.701,"y":583.132,"cluster":"advanced-algebra"},{"id":"stacks:0G9E","tag":"0G9E","title":"Two term complexes · Lemma 0G9E","summary":"Let R be a ring. Let K be an object of D(R) with H^i(K) = 0 for i not ∈ (-1, 0). Then • K can be represented by a two term complex K^-1 → K^0 with K^0 a free module, and • if R is Noetherian and H^i(K) is a finite R-module for i = -1, 0, then K can be represented by a two term complex K^-1 → K^0 with K^0 a finite free module and K^-1 finite.","statement_latex":"Let $R$ be a ring. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$\nfor $i \\not \\in \\{-1, 0\\}$. Then\n\\begin{enumerate}\n\\item $K$ can be represented by a two term complex\n$K^{-1} \\to K^0$ with $K^0$ a free module, and\n\\item if $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for\n$i = -1, 0$, then $K$ can be represented by a two term complex\n$K^{-1} \\to K^0$ with $K^0$ a finite free module and $K^{-1}$ finite.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9E","source_file":"more-algebra.tex","source_line":22303,"source_end_line":22314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22303-L22314","statement_sha256":"4a3e84d639a1c143342dcaf1f418956c246a187b0084d0ab877057c39ca94171","origin":"The Stacks Project","memory_eligible":false,"source_rank":3383,"rank":3383,"depth":10,"x":808.372,"y":712.703,"cluster":"advanced-algebra"},{"id":"stacks:0ALN","tag":"0ALN","title":"Two term complexes · Lemma 0ALN","summary":"Let R be a ring. Let M^bullet be a complex of modules over R with M^i = 0 for i > 0 and M^0 a projective R-module. Let K^bullet be a second complex. • Assume K^i = 0 for i ≤ -2. Then Hom_D(R)(M^bullet, K^bullet) = Hom_K(R)(M^bullet, K^bullet). • Assume K^i = 0 for i not ∈ [-1, 0] and K^0 a projective R-module. Then for a map of complexes a^bullet : M^bullet → K^bullet, the following are equivalent • a^bullet induces the zero map Ext^1_R(K^bullet, N) → Ext^1_R(M^bullet, N)…","statement_latex":"Let $R$ be a ring. Let $M^\\bullet$ be a complex of modules over $R$\nwith $M^i = 0$ for $i > 0$ and $M^0$ a projective $R$-module.\nLet $K^\\bullet$ be a second complex.\n\\begin{enumerate}\n\\item Assume $K^i = 0$ for $i \\leq -2$. Then\n$\\Hom_{D(R)}(M^\\bullet, K^\\bullet) = \\Hom_{K(R)}(M^\\bullet, K^\\bullet)$.\n\\item Assume $K^i = 0$ for $i \\not \\in [-1, 0]$ and\n$K^0$ a projective $R$-module. Then for a map of complexes\n$a^\\bullet : M^\\bullet \\to K^\\bullet$, the following are equivalent\n\\begin{enumerate}\n\\item $a^\\bullet$ induces the zero map $\\Ext^1_R(K^\\bullet, N) \\to\n\\Ext^1_R(M^\\bullet, N)$ for all $R$-modules $N$, and\n\\item there is a map $h^0 : M^0 \\to K^{-1}$ such that\n$a^{-1} + h^0 \\circ d^{-1}_K = 0$.\n\\end{enumerate}\n\\item Assume $K^i = 0$ for $i \\leq -3$. Let\n$\\alpha \\in \\Hom_{D(R)}(M^\\bullet, K^\\bullet)$. If the\ncomposition of $\\alpha$ with\n$K^\\bullet \\to K^{-2}[2]$ comes from an $R$-module map\n$a : M^{-2} \\to K^{-2}$ with $a \\circ d_M^{-3} = 0$, then\n$\\alpha$ can be represented by a map of complexes\n$a^\\bullet : M^\\bullet \\to K^\\bullet$ with $a^{-2} = a$.\n\\item In (2) for any second map of complexes\n$(a')^\\bullet : M^\\bullet \\to K^\\bullet$\nrepresenting $\\alpha$ with $a = (a')^{-2}$\nthere exist $h^i : M^i \\to K^{i - 1}$ for $i = 0, -1$ such that\n$$\nh^{-1} \\circ d_M^{-2} = 0, \\quad\n(a')^{-1} = a^{-1} + d_K^{-2} \\circ h^{-1} + h^0 \\circ d_M^{-1},\\quad\n(a')^0 = a^0 + d_K^{-1} \\circ h^0\n$$\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALN","source_file":"more-algebra.tex","source_line":22337,"source_end_line":22371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22337-L22371","statement_sha256":"83388dd9bcb9717a73af09c47d7989096bbd28a0c42869b2240ab1acd89897da","origin":"The Stacks Project","memory_eligible":false,"source_rank":3384,"rank":3384,"depth":5,"x":558.937,"y":728.769,"cluster":"advanced-algebra"},{"id":"stacks:0G9F","tag":"0G9F","title":"Two term complexes · Lemma 0G9F","summary":"Let R be a ring and let I ⊂ R be an ideal. Let K ∈ D(R). Assume H^i(K) = 0 for i not ∈ (-1, 0). The following are equivalent • Ext^1_R(K, N) is annihilated by I for all R-modules N, • K can be represented by a complex K^-1 → K^0 with K^0 free such that for any a ∈ I the map a : K^-1 → K^-1 factors through d_K^-1 : K^-1 → K^0, • whenever K is represented by a two term complex K^-1 → K^0 with K^0 projective, then for any a ∈ I the map a : K^-1 → K^-1 factors through d_K^-1…","statement_latex":"Let $R$ be a ring and let $I \\subset R$ be an ideal.\nLet $K \\in D(R)$. Assume $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\Ext^1_R(K, N)$ is annihilated by $I$ for all $R$-modules $N$,\n\\item $K$ can be represented by a complex $K^{-1} \\to K^0$\nwith $K^0$ free such that for any $a \\in I$ the map\n$a : K^{-1} \\to K^{-1}$ factors through $d_K^{-1} : K^{-1} \\to K^0$,\n\\item whenever $K$ is represented by a two term complex\n$K^{-1} \\to K^0$ with $K^0$ projective, then for any $a \\in I$ the map\n$a : K^{-1} \\to K^{-1}$ factors through $d_K^{-1} : K^{-1} \\to K^0$.\n\\end{enumerate}\nIf $R$ is Noetherian and $H^i(K)$ is a finite $R$-module for $i = -1, 0$,\nthen these are also equivalent to\n\\begin{enumerate}\n\\item[(4)] $\\Ext^1_R(K, N)$ is annihilated by $I$ for every finite\n$R$-module $N$,\n\\item[(5)] $K$ can be represented by a complex $K^{-1} \\to K^0$\nwith $K^0$ finite free and $K^{-1}$ finite such that for any $a \\in I$ the map\n$a : K^{-1} \\to K^{-1}$ factors through $d_K^{-1} : K^{-1} \\to K^0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9F","source_file":"more-algebra.tex","source_line":22451,"source_end_line":22474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22451-L22474","statement_sha256":"cc2854b75f8761a4e889941bb26144b81733898c939aa82d7dde01003acc0381","origin":"The Stacks Project","memory_eligible":false,"source_rank":3385,"rank":3385,"depth":11,"x":730.094,"y":575.254,"cluster":"advanced-algebra"},{"id":"stacks:0G9G","tag":"0G9G","title":"Two term complexes · Lemma 0G9G","summary":"Let R be a ring. Let K be an object of D(R) with H^i(K) = 0 for i not ∈ (-1, 0). Let K^-1 → K^0 be a two term complex of R-modules representing K such that K^0 is a flat R-module (for example projective or free). Let R → R' be a ring map. Then the complex K^bullet ⊗_R R' represents τ_≥ -1(K ⊗_R^L R').","statement_latex":"Let $R$ be a ring. Let $K$ be an object of $D(R)$ with $H^i(K) = 0$\nfor $i \\not \\in \\{-1, 0\\}$. Let $K^{-1} \\to K^0$ be a two term complex\nof $R$-modules representing $K$ such that $K^0$ is a flat $R$-module\n(for example projective or free). Let $R \\to R'$ be a ring map.\nThen the complex $K^\\bullet \\otimes_R R'$ represents\n$\\tau_{\\geq -1}(K \\otimes_R^\\mathbf{L} R')$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9G","source_file":"more-algebra.tex","source_line":22519,"source_end_line":22527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22519-L22527","statement_sha256":"c57fd3e13f0a8be46c90051a30c8825e31c5ed4f67782de4161172f3f9a02b73","origin":"The Stacks Project","memory_eligible":false,"source_rank":3386,"rank":3386,"depth":0,"x":727.336,"y":785.754,"cluster":"advanced-algebra"},{"id":"stacks:0G9H","tag":"0G9H","title":"Two term complexes · Lemma 0G9H","summary":"Let I be an ideal of a ring R. Let K be an object of D(R) with H^i(K) = 0 for i not ∈ (-1, 0). Let R → R' be a ring map. If K satisfies the equivalent conditions (1), (2), and (3) of Lemma [Tag 0G9F] with respect to (R, I), then τ_≥ -1(K ⊗_R^L R') satisfies the equivalent conditions (1), (2), and (3) of Lemma [Tag 0G9F] with respect to (R', IR')","statement_latex":"Let $I$ be an ideal of a ring $R$. Let $K$ be an object of $D(R)$ with\n$H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$. Let $R \\to R'$ be a ring map.\nIf $K$ satisfies the equivalent conditions (1), (2), and (3)\nof Lemma \\ref{lemma-ext-1-annihilated-definite} with respect to $(R, I)$,\nthen $\\tau_{\\geq -1}(K \\otimes_R^\\mathbf{L} R')$\nsatisfies the equivalent conditions (1), (2), and (3)\nof Lemma \\ref{lemma-ext-1-annihilated-definite} with respect to $(R', IR')$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9H","source_file":"more-algebra.tex","source_line":22552,"source_end_line":22561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22552-L22561","statement_sha256":"c6078debb72db2eb6aa18825234ce202bc738b3a93a2d24f2baa76bee03fdbdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3387,"rank":3387,"depth":12,"x":559.948,"y":628.834,"cluster":"advanced-algebra"},{"id":"stacks:0G9I","tag":"0G9I","title":"Two term complexes · Lemma 0G9I","summary":"Let R be a ring. Let α : K → K' be a morphism of D(R). Assume • H^i(K) = H^i(K') = 0 for i not ∈ (-1, 0) • H^0(α) is an isomorphism and H^-1(α) is surjective. For any f ∈ R if f : K → K is 0, then f : K' → K' is 0.","statement_latex":"Let $R$ be a ring. Let $\\alpha : K \\to K'$ be a morphism of $D(R)$. Assume\n\\begin{enumerate}\n\\item $H^i(K) = H^i(K') = 0$ for $i \\not \\in \\{-1, 0\\}$\n\\item $H^0(\\alpha)$ is an isomorphism and $H^{-1}(\\alpha)$ is surjective.\n\\end{enumerate}\nFor any $f \\in R$ if $f : K \\to K$ is $0$, then $f : K' \\to K'$ is $0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9I","source_file":"more-algebra.tex","source_line":22577,"source_end_line":22585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22577-L22585","statement_sha256":"cf5a61c305b49c1db0fcffd57990e75904561a29463ead3fc836252f850db4f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3388,"rank":3388,"depth":1,"x":809.782,"y":649.582,"cluster":"advanced-algebra"},{"id":"stacks:0G9J","tag":"0G9J","title":"Two term complexes · Lemma 0G9J","summary":"Let I be an ideal of a ring R. Let α : K → K' be a morphism of D(R). Assume • H^i(K) = H^i(K') = 0 for i not ∈ (-1, 0) • H^0(α) is an isomorphism and H^-1(α) is surjective. If K satisfies the equivalent conditions (1), (2), and (3) of Lemma [Tag 0G9F], then K' does too.","statement_latex":"Let $I$ be an ideal of a ring $R$. Let $\\alpha : K \\to K'$\nbe a morphism of $D(R)$. Assume\n\\begin{enumerate}\n\\item $H^i(K) = H^i(K') = 0$ for $i \\not \\in \\{-1, 0\\}$\n\\item $H^0(\\alpha)$ is an isomorphism and $H^{-1}(\\alpha)$ is surjective.\n\\end{enumerate}\nIf $K$ satisfies the equivalent conditions (1), (2), and (3)\nof Lemma \\ref{lemma-ext-1-annihilated-definite},\nthen $K'$ does too.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9J","source_file":"more-algebra.tex","source_line":22599,"source_end_line":22610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22599-L22610","statement_sha256":"8614290c3cc564bed2f3321d17613b41601a9c023ff96723d145e210a20aab2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3389,"rank":3389,"depth":12,"x":608.701,"y":776.153,"cluster":"advanced-algebra"},{"id":"stacks:0G9K","tag":"0G9K","title":"Two term complexes · Lemma 0G9K","summary":"Let R be ring and let I ⊂ R be an ideal. Let K ∈ D(R) with H^i(K) = 0 for i not ∈ (-1, 0). The following are equivalent • there exists a c ≥ 0 such that the equivalent conditions (1), (2), (3) of Lemma [Tag 0G9F] hold for K and the ideal I^c, • there exists a c ≥ 0 such that (a) I^c annihilates H^-1(K) and (b) H^0(K) is an I^c-projective module (see Section [Tag 0G8Z]). If R is Noetherian and H^i(K) is a finite R-module for i = -1, 0, then these are also equivalent to •…","statement_latex":"Let $R$ be ring and let $I \\subset R$ be an ideal.\nLet $K \\in D(R)$ with $H^i(K) = 0$ for $i \\not \\in \\{-1, 0\\}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists a $c \\geq 0$ such that the equivalent\nconditions (1), (2), (3) of Lemma \\ref{lemma-ext-1-annihilated-definite}\nhold for $K$ and the ideal $I^c$,\n\\item there exists a $c \\geq 0$ such that (a) $I^c$ annihilates\n$H^{-1}(K)$ and (b) $H^0(K)$ is an $I^c$-projective module (see\nSection \\ref{section-near-projective}).\n\\end{enumerate}\nIf $R$ is Noetherian and $H^i(K)$ is a finite $R$-module\nfor $i = -1, 0$, then these are also equivalent to\n\\begin{enumerate}\n\\item[(3)] there exists a $c \\geq 0$ such that the equivalent\nconditions (4), (5) of Lemma \\ref{lemma-ext-1-annihilated-definite}\nhold for $K$ and the ideal $I^c$,\n\\item[(4)] $H^{-1}(K)$ is $I$-power torsion and there exist\n$f_1, \\ldots, f_s \\in R$ with $V(f_1, \\ldots, f_s) \\subset V(I)$\nsuch that the localizations $H^0(K)_{f_i}$ are projective\n$R_{f_i}$-modules,\n\\item[(5)] $H^{-1}(K)$ is $I$-power torsion and there exist\n$f_1, \\ldots, f_s \\in I$ with $V(f_1, \\ldots, f_s) = V(I)$\nsuch that the localizations $H^0(K)_{f_i}$ are projective\n$R_{f_i}$-modules, and\n\\item[(6)] $H^{-1}(K)$ is $I$-power torsion and for any\n$f_1, \\ldots, f_s \\in I$ with $V(f_1, \\ldots, f_s) = V(I)$\nthe localizations $H^0(K)_{f_i}$ are projective $R_{f_i}$-modules.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9K","source_file":"more-algebra.tex","source_line":22629,"source_end_line":22660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22629-L22660","statement_sha256":"3bb66f08a957c702e345dba88d501d32cbf41215911007a345fd33c64482a131","origin":"The Stacks Project","memory_eligible":false,"source_rank":3390,"rank":3390,"depth":14,"x":655.23,"y":568.546,"cluster":"advanced-algebra"},{"id":"stacks:0AJT","tag":"0AJT","title":"Two term complexes · Lemma 0AJT","summary":"Let R be a ring. Let K_j ∈ D(R), j = 1, 2, 3 with H^i(K_j) = 0 for i not ∈ (-1, 0). Let φ : K_1 → K_2 and ψ : K_2 → K_3 be maps in D(R). If H^0(φ) = 0 and H^-1(ψ) = 0, then φ ∘ ψ = 0.","statement_latex":"Let $R$ be a ring. Let $K_j \\in D(R)$, $j = 1, 2, 3$ with $H^i(K_j) = 0$\nfor $i \\not \\in \\{-1, 0\\}$. Let $\\varphi : K_1 \\to K_2$ and\n$\\psi : K_2 \\to K_3$ be maps in $D(R)$.\nIf $H^0(\\varphi) = 0$ and $H^{-1}(\\psi) = 0$, then\n$\\varphi \\circ \\psi = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJT","source_file":"more-algebra.tex","source_line":22711,"source_end_line":22718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22711-L22718","statement_sha256":"380e92fca92ef5802b634297f7cfb800de4b1a644834f1940e5ba6202a9d5498","origin":"The Stacks Project","memory_eligible":false,"source_rank":3391,"rank":3391,"depth":3,"x":787.986,"y":748.188,"cluster":"advanced-algebra"},{"id":"stacks:0G9L","tag":"0G9L","title":"Two term complexes · Lemma 0G9L","summary":"Let R be a ring. Let K ∈ D(R) be given by a two term complex of the form R^⊕ n → R^⊕ n. Denote A ∈ Mat(n × n, R) the matrix of the differential. Then det(a) : K → K is zero in D(R).","statement_latex":"Let $R$ be a ring. Let $K \\in D(R)$ be given by a two term complex\nof the form $R^{\\oplus n} \\to R^{\\oplus n}$. Denote\n$A \\in \\text{Mat}(n \\times n, R)$ the matrix of the differential.\nThen $\\det(a) : K \\to K$ is zero in $D(R)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Two term complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9L","source_file":"more-algebra.tex","source_line":22725,"source_end_line":22731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22725-L22731","statement_sha256":"05656c7fcd603b49eb28526dfc7a1daad466d1e92ce6b224b8ec7c6e7f084a18","origin":"The Stacks Project","memory_eligible":false,"source_rank":3392,"rank":3392,"depth":0,"x":545.424,"y":691.002,"cluster":"advanced-algebra"},{"id":"stacks:0FUY","tag":"0FUY","title":"The naive cotangent complex · Lemma 0FUY","summary":"Let R → S and S → S' be ring maps. The canonical map NL_S/R ⊗_S^L S' → NL_S/R ⊗_S S' induces an isomorphism τ_≥ -1(NL_S/R ⊗_S^L S') → NL_S/R ⊗_S S' in D(S'). Similarly, given a presentation α of S over R the canonical map NL(α) ⊗_S^L S' → NL(α) ⊗_S S' induces an isomorphism τ_≥ -1(NL(α) ⊗_S^L S') → NL(α) ⊗_S S' in D(S').","statement_latex":"Let $R \\to S$ and $S \\to S'$ be ring maps. The canonical map\n$\\NL_{S/R} \\otimes_S^\\mathbf{L} S' \\to \\NL_{S/R} \\otimes_S S'$\ninduces an isomorphism\n$\\tau_{\\geq -1}(\\NL_{S/R} \\otimes_S^\\mathbf{L} S') \\to \\NL_{S/R} \\otimes_S S'$\nin $D(S')$. Similarly, given a presentation $\\alpha$ of $S$ over $R$\nthe canonical map\n$\\NL(\\alpha) \\otimes_S^\\mathbf{L} S' \\to \\NL(\\alpha) \\otimes_S S'$\ninduces an isomorphism $\\tau_{\\geq -1}(\\NL(\\alpha) \\otimes_S^\\mathbf{L} S') \\to\n\\NL(\\alpha) \\otimes_S S'$ in $D(S')$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUY","source_file":"more-algebra.tex","source_line":22753,"source_end_line":22764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22753-L22764","statement_sha256":"66dfee38e44d0d28f2601b891c1f2ad2510f011d7473d09e9fa2cecb9d193b9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3393,"rank":3393,"depth":1,"x":770.464,"y":595.453,"cluster":"advanced-algebra"},{"id":"stacks:0FUZ","tag":"0FUZ","title":"The naive cotangent complex · Lemma 0FUZ","summary":"Let R → S and R → R' be ring maps. Let α : P → S be a presentation of S over R. Then α' : P ⊗_R R' → S ⊗_R R' is a presentation of S' = S ⊗_R R' over R'. The canonical map NL(α) ⊗_S S' → NL(α') is an isomorphism on H^0 and surjective on H^-1. In particular, the canonical map NL_S/R ⊗_S S' → NL_S'/R' is an isomorphism on H^0 and surjective on H^-1.","statement_latex":"Let $R \\to S$ and $R \\to R'$ be ring maps.\nLet $\\alpha : P \\to S$ be a presentation of $S$ over $R$.\nThen $\\alpha' : P \\otimes_R R' \\to S \\otimes_R R'$ is a\npresentation of $S' = S \\otimes_R R'$ over $R'$.\nThe canonical map\n$$\nNL(\\alpha) \\otimes_S S' \\to \\NL(\\alpha')\n$$\nis an isomorphism on $H^0$ and surjective on $H^{-1}$. In particular,\nthe canonical map\n$$\n\\NL_{S/R} \\otimes_S S' \\to \\NL_{S'/R'}\n$$\nis an isomorphism on $H^0$ and surjective on $H^{-1}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUZ","source_file":"more-algebra.tex","source_line":22770,"source_end_line":22786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22770-L22786","statement_sha256":"661d5bf7089b9a0ae1454c3ea705e4dbd8c67ec6c654a6031634c18c4610cbf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3394,"rank":3394,"depth":0,"x":681.285,"y":793.772,"cluster":"advanced-algebra"},{"id":"stacks:0FJU","tag":"0FJU","title":"The naive cotangent complex · Lemma 0FJU","summary":"Consider a cocartesian diagram of rings xymatrix B ar[r] & B' A ar[r] ar[u] & A' ar[u] If B is flat over A, then the canonical map NL_B/A ⊗_B B' → NL_B'/A' is a quasi-isomorphism. If in addition NL_B/A has tor-amplitude in [-1, 0] then NL_B/A ⊗_B^L B' → NL_B'/A' is a quasi-isomorphism too.","statement_latex":"Consider a cocartesian diagram of rings\n$$\n\\xymatrix{\nB \\ar[r] & B' \\\\\nA \\ar[r] \\ar[u] & A' \\ar[u]\n}\n$$\nIf $B$ is flat over $A$, then the canonical map\n$\\NL_{B/A} \\otimes_B B' \\to \\NL_{B'/A'}$ is a quasi-isomorphism.\nIf in addition $\\NL_{B/A}$ has tor-amplitude in $[-1, 0]$\nthen $\\NL_{B/A} \\otimes_B^\\mathbf{L} B' \\to \\NL_{B'/A'}$\nis a quasi-isomorphism too.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJU","source_file":"more-algebra.tex","source_line":22809,"source_end_line":22823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22809-L22823","statement_sha256":"94781aa1be74f7c50c979de26e6980322d4ec7671db73c21f95cb2df707fc67d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3395,"rank":3395,"depth":5,"x":587.482,"y":596.765,"cluster":"advanced-algebra"},{"id":"stacks:0FV0","tag":"0FV0","title":"The naive cotangent complex · Lemma 0FV0","summary":"Let A → B be a local complete intersection as in Definition [Tag 07D0]. Then NL_B/A is a perfect object of D(B) with tor amplitude in [-1, 0].","statement_latex":"Let $A \\to B$ be a local complete intersection as in\nDefinition \\ref{definition-local-complete-intersection}.\nThen $\\NL_{B/A}$ is a perfect object of\n$D(B)$ with tor amplitude in $[-1, 0]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV0","source_file":"more-algebra.tex","source_line":22855,"source_end_line":22861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22855-L22861","statement_sha256":"c6f93014b6c033eb779090921faaa486defda9d95d0cade2da96b1928f2b5084","origin":"The Stacks Project","memory_eligible":false,"source_rank":3396,"rank":3396,"depth":6,"x":815.271,"y":688.887,"cluster":"advanced-algebra"},{"id":"stacks:0FV1","tag":"0FV1","title":"The naive cotangent complex · Lemma 0FV1","summary":"Consider a cocartesian diagram of rings xymatrix B ar[r] & B' A ar[r] ar[u] & A' ar[u] If A → B and A' → B' are local complete intersections as in Definition [Tag 07D0], then the kernel of H^-1(NL_B/A ⊗_B B') → H^-1(NL_B'/A') is a finite projective B'-module.","statement_latex":"Consider a cocartesian diagram of rings\n$$\n\\xymatrix{\nB \\ar[r] & B' \\\\\nA \\ar[r] \\ar[u] & A' \\ar[u]\n}\n$$\nIf $A \\to B$ and $A' \\to B'$ are local complete intersections as in\nDefinition \\ref{definition-local-complete-intersection}, then\nthe kernel of $H^{-1}(\\NL_{B/A} \\otimes_B B') \\to H^{-1}(\\NL_{B'/A'})$\nis a finite projective $B'$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV1","source_file":"more-algebra.tex","source_line":22880,"source_end_line":22893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22880-L22893","statement_sha256":"01f46b14a2baf8000761419ab5e95789d3ba869901a4390a093336821e181346","origin":"The Stacks Project","memory_eligible":false,"source_rank":3397,"rank":3397,"depth":12,"x":573.016,"y":750.262,"cluster":"advanced-algebra"},{"id":"stacks:07KW","tag":"07KW","title":"Rlim of abelian groups · Lemma 07KW","summary":"The functor lim : Ab(N) → Ab has a right derived functor Rlim : D(Ab(N)) → D(Ab) As usual we set R^plim(K) = H^p(Rlim(K)). Moreover, we have • for any (A_n) in Ab(N) we have R^plim A_n = 0 for p > 1, • the object Rlim A_n of D(Ab) is represented by the complex ∏ A_n → ∏ A_n, (x_n) ↦ (x_n - f_n + 1(x_n + 1)) sitting in degrees 0 and 1, • if (A_n) is ML, then R^1lim A_n = 0, i.e., (A_n) is right acyclic for lim, • every K^bullet ∈ D(Ab(N)) is quasi-isomorphic to a complex…","statement_latex":"The functor $\\lim : \\textit{Ab}(\\mathbf{N}) \\to \\textit{Ab}$\nhas a right derived functor\n\\begin{equation}\n\nR\\lim : D(\\textit{Ab}(\\mathbf{N})) \\longrightarrow D(\\textit{Ab})\n\\end{equation}\nAs usual we set $R^p\\lim(K) = H^p(R\\lim(K))$. Moreover, we have\n\\begin{enumerate}\n\\item for any $(A_n)$ in $\\textit{Ab}(\\mathbf{N})$ we have\n$R^p\\lim A_n = 0$ for $p > 1$,\n\\item the object $R\\lim A_n$ of $D(\\textit{Ab})$ is represented\nby the complex\n$$\n\\prod A_n \\to \\prod A_n,\\quad (x_n) \\mapsto (x_n - f_{n + 1}(x_{n + 1}))\n$$\nsitting in degrees $0$ and $1$,\n\\item if $(A_n)$ is ML, then $R^1\\lim A_n = 0$, i.e., $(A_n)$\nis right acyclic for $\\lim$,\n\\item every $K^\\bullet \\in D(\\textit{Ab}(\\mathbf{N}))$ is quasi-isomorphic\nto a complex whose terms are right acyclic for $\\lim$, and\n\\item if each $K^p = (K^p_n)$ is right acyclic for $\\lim$, i.e.,\nof $R^1\\lim_n K^p_n = 0$, then $R\\lim K$ is represented by the\ncomplex whose term in degree $p$ is $\\lim_n K_n^p$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KW","source_file":"more-algebra.tex","source_line":22942,"source_end_line":22968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L22942-L22968","statement_sha256":"de7e8639b92c92d0a3ee9e960fe0073c311087559f0a3c6f3f5eb7be9b7ec19d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3398,"rank":3398,"depth":16,"x":702.404,"y":567.39,"cluster":"advanced-algebra"},{"id":"stacks:0H31","tag":"0H31","title":"Rlim of abelian groups · Lemma 0H31","summary":"Let 0 → (A_i) → (B_i) → (C_i) → 0 be a short exact sequence of inverse systems of abelian groups. Then there is an associated 6 term exact sequence 0 → lim A_i → lim B_i → lim C_i → R^1lim A_i → R^1lim B_i → R^1lim C_i → 0.","statement_latex":"Let\n$$\n0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0\n$$\nbe a short exact sequence of inverse systems of abelian groups.\nThen there is an associated $6$ term exact sequence\n$0 \\to \\lim A_i \\to \\lim B_i \\to \\lim C_i \\to\nR^1\\lim A_i \\to R^1\\lim B_i \\to R^1\\lim C_i \\to 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H31","source_file":"more-algebra.tex","source_line":23002,"source_end_line":23012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23002-L23012","statement_sha256":"a5dff626db633ee8522fdbbb8279cb17aa85078aba8f06cc997042cb31c37011","origin":"The Stacks Project","memory_eligible":false,"source_rank":3399,"rank":3399,"depth":17,"x":754.1,"y":775.83,"cluster":"advanced-algebra"},{"id":"stacks:0918","tag":"0918","title":"Rlim of abelian groups · Lemma 0918","summary":"Let (A^-2_n → A^-1_n → A^0_n → A^1_n) be an inverse system of complexes of abelian groups and denote A^-2 → A^-1 → A^0 → A^1 its limit. Denote (H_n^-1), (H_n^0) the inverse systems of cohomologies, and denote H^-1, H^0 the cohomologies of A^-2 → A^-1 → A^0 → A^1. If • (A^-2_n) and (A^-1_n) have vanishing R^1lim, • (H^-1_n) has vanishing R^1lim, then H^0 = lim H_n^0.","statement_latex":"Let\n$$\n(A^{-2}_n \\to A^{-1}_n \\to A^0_n \\to A^1_n)\n$$\nbe an inverse system of complexes of abelian groups and denote\n$A^{-2} \\to A^{-1} \\to A^0 \\to A^1$ its limit. Denote\n$(H_n^{-1})$, $(H_n^0)$ the inverse systems of cohomologies, and\ndenote $H^{-1}$, $H^0$ the cohomologies of $A^{-2} \\to A^{-1} \\to A^0 \\to A^1$.\nIf\n\\begin{enumerate}\n\\item $(A^{-2}_n)$ and $(A^{-1}_n)$ have vanishing $R^1\\lim$,\n\\item $(H^{-1}_n)$ has vanishing $R^1\\lim$,\n\\end{enumerate}\nthen $H^0 = \\lim H_n^0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0918","source_file":"more-algebra.tex","source_line":23022,"source_end_line":23038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23022-L23038","statement_sha256":"d63be9df5143b75a6e7835b1a786996b8cc05d8eed57251c5c4f4d7b1b2d729a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3400,"rank":3400,"depth":7,"x":548.185,"y":651.358,"cluster":"advanced-algebra"},{"id":"stacks:0H9K","tag":"0H9K","title":"Rlim of abelian groups · Lemma 0H9K","summary":"Let (A_n) and (B_n) be inverse systems of abelian groups. A morphism of pro-systems φ : (A_n) → (B_n) determines maps lim A_n → lim B_n and R^1lim A_n → R^1lim B_n. These maps are isomorphisms if φ is a pro-isomorphism.","statement_latex":"Let $(A_n)$ and $(B_n)$ be inverse systems of abelian groups.\nA morphism of pro-systems $\\varphi : (A_n) \\to (B_n)$ determines\nmaps $\\lim A_n \\to \\lim B_n$ and $R^1\\lim A_n \\to R^1\\lim B_n$.\nThese maps are isomorphisms if $\\varphi$ is a pro-isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9K","source_file":"more-algebra.tex","source_line":23072,"source_end_line":23078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23072-L23078","statement_sha256":"04b96b77e3609b3d2deeaa17660ec53e4dcfe4fb52381b3dc93aeac0d7ff7245","origin":"The Stacks Project","memory_eligible":false,"source_rank":3401,"rank":3401,"depth":17,"x":800.332,"y":626.28,"cluster":"advanced-algebra"},{"id":"stacks:0919","tag":"0919","title":"Rlim of abelian groups · Lemma 0919","summary":"Let D be a triangulated category. Let (K_n) be an inverse system of objects of D. Let K be a derived limit of the system (K_n). Then for every L in D we have a short exact sequence 0 → R^1lim Hom_D(L, K_n[-1]) → Hom_D(L, K) → lim Hom_D(L, K_n) → 0","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$(K_n)$ be an inverse system of objects of $\\mathcal{D}$.\nLet $K$ be a derived limit of the system $(K_n)$.\nThen for every $L$ in $\\mathcal{D}$ we have a short exact sequence\n$$\n0 \\to R^1\\lim \\Hom_\\mathcal{D}(L, K_n[-1]) \\to\n\\Hom_\\mathcal{D}(L, K) \\to\n\\lim \\Hom_\\mathcal{D}(L, K_n) \\to 0\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0919","source_file":"more-algebra.tex","source_line":23148,"source_end_line":23159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23148-L23159","statement_sha256":"7d08b9ec0a2f93829dffec78b1a312a8016e5dfafb77ab1a84776eddf82cff96","origin":"The Stacks Project","memory_eligible":false,"source_rank":3402,"rank":3402,"depth":17,"x":634.431,"y":787.982,"cluster":"advanced-algebra"},{"id":"stacks:0H9L","tag":"0H9L","title":"Rlim of abelian groups · Lemma 0H9L","summary":"Let D be a triangulated category. Let (K_n) and (M_n) be inverse systems of objects of D with derived limits K and M. Let a : (K_n) → (M_n) be a pro-isomorphism of pro-objects. Then a can be used to produce a (non-canonical) isomorphism K → M.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$(K_n)$ and $(M_n)$ be inverse systems of objects of $\\mathcal{D}$\nwith derived limits $K$ and $M$. Let $a : (K_n) \\to (M_n)$ be a\npro-isomorphism of pro-objects. Then $a$ can be used\nto produce a (non-canonical) isomorphism $K \\to M$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9L","source_file":"more-algebra.tex","source_line":23169,"source_end_line":23176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23169-L23176","statement_sha256":"749118fd8db6e96919e9b5fd3ab55cc5ebab450d9ec313d3cef747001f2ee9ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":3403,"rank":3403,"depth":18,"x":626.721,"y":574.43,"cluster":"advanced-algebra"},{"id":"stacks:0CQX","tag":"0CQX","title":"Rlim of abelian groups · Lemma 0CQX","summary":"Let D be a triangulated category. Let (K_n) be a system of objects of D. Let K be a derived colimit of the system (K_n). Then for every L in D we have a short exact sequence 0 → R^1lim Hom_D(K_n, L[-1]) → Hom_D(K, L) → lim Hom_D(K_n, L) → 0","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$(K_n)$ be a system of objects of $\\mathcal{D}$.\nLet $K$ be a derived colimit of the system $(K_n)$.\nThen for every $L$ in $\\mathcal{D}$ we have a short exact sequence\n$$\n0 \\to R^1\\lim \\Hom_\\mathcal{D}(K_n, L[-1]) \\to\n\\Hom_\\mathcal{D}(K, L) \\to\n\\lim \\Hom_\\mathcal{D}(K_n, L) \\to 0\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQX","source_file":"more-algebra.tex","source_line":23219,"source_end_line":23230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23219-L23230","statement_sha256":"bf729a4562e79f776e193318ed15f19248e6d72c0c07ba2ccf5b927b06ccd8ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":3404,"rank":3404,"depth":17,"x":804.287,"y":727.654,"cluster":"advanced-algebra"},{"id":"stacks:07KX","tag":"07KX","title":"Rlim of abelian groups · Lemma 07KX","summary":"Let K = (K_n^bullet) be an object of D(Ab(N)). There exists a canonical distinguished triangle Rlim K → ∏_n K_n^bullet → ∏_n K_n^bullet → Rlim K[1] in D(Ab). In other words, Rlim K is a derived limit of the inverse system (K_n^bullet) of D(Ab), see Derived Categories, Definition [Tag 08TC].","statement_latex":"Let $K = (K_n^\\bullet)$ be an object of $D(\\textit{Ab}(\\mathbf{N}))$.\nThere exists a canonical distinguished triangle\n$$\nR\\lim K \\to \\prod\\nolimits_n K_n^\\bullet \\to \\prod\\nolimits_n K_n^\\bullet\n\\to R\\lim K[1]\n$$\nin $D(\\textit{Ab})$. In other words, $R\\lim K$ is a derived limit\nof the inverse system $(K_n^\\bullet)$ of $D(\\textit{Ab})$, see\nDerived Categories, Definition \\ref{derived-definition-derived-limit}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KX","source_file":"more-algebra.tex","source_line":23269,"source_end_line":23280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23269-L23280","statement_sha256":"bb098824dc73c0f9133c6775831b93267de169360aaa9b7b5409050aafcc2349","origin":"The Stacks Project","memory_eligible":false,"source_rank":3405,"rank":3405,"depth":17,"x":549.923,"y":715.414,"cluster":"advanced-algebra"},{"id":"stacks:07KY","tag":"07KY","title":"Rlim of abelian groups · Lemma 07KY","summary":"With notation as in Lemma [Tag 07KX] the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences 0 → R^1lim_n H^p - 1(K_n^bullet) → H^p(Rlim K) → lim_n H^p(K_n^bullet) → 0","statement_latex":"With notation as in Lemma \\ref{lemma-distinguished-triangle-Rlim}\nthe long exact cohomology sequence associated to the distinguished\ntriangle breaks up into short exact sequences\n$$\n0 \\to R^1\\lim_n H^{p - 1}(K_n^\\bullet) \\to\nH^p(R\\lim K) \\to\n\\lim_n H^p(K_n^\\bullet) \\to 0\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KY","source_file":"more-algebra.tex","source_line":23302,"source_end_line":23312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23302-L23312","statement_sha256":"ef7ecc243057b88c7589c97fc9efd9b69035883a2b6ec19ca4abee0ade2707b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3406,"rank":3406,"depth":18,"x":747.494,"y":579.994,"cluster":"advanced-algebra"},{"id":"stacks:0CQ9","tag":"0CQ9","title":"Rlim of abelian groups · Lemma 0CQ9","summary":"Let (K_n) be an inverse system of objects of D(Ab). Then there exists an object M = (M_n^bullet) of D(Ab(N)) and isomorphisms M_n^bullet → K_n in D(Ab) such that the diagrams xymatrix M_n + 1^bullet ar[d] ar[r] & M_n^bullet ar[d] K_n + 1 ar[r] & K_n commute in D(Ab).","statement_latex":"Let $(K_n)$ be an inverse system of objects of $D(\\textit{Ab})$.\nThen there exists an object $M = (M_n^\\bullet)$\nof $D(\\textit{Ab}(\\mathbf{N}))$ and isomorphisms\n$M_n^\\bullet \\to K_n$ in $D(\\textit{Ab})$ such that the diagrams\n$$\n\\xymatrix{\nM_{n + 1}^\\bullet \\ar[d] \\ar[r] &\nM_n^\\bullet \\ar[d] \\\\\nK_{n + 1} \\ar[r] & K_n\n}\n$$\ncommute in $D(\\textit{Ab})$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQ9","source_file":"more-algebra.tex","source_line":23334,"source_end_line":23348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23334-L23348","statement_sha256":"fb98b6270cd4a904ab4a315231b03db29658db3cc3a3fa5c96e8c59b22e37a56","origin":"The Stacks Project","memory_eligible":false,"source_rank":3407,"rank":3407,"depth":0,"x":710.678,"y":792.134,"cluster":"advanced-algebra"},{"id":"stacks:091B","tag":"091B","title":"Rlim of abelian groups · Lemma 091B","summary":"Let E → D be a morphism of D(Ab(N)). Let (E_n), resp. (D_n) be the system of objects of D(Ab) associated to E, resp. D. If (E_n) → (D_n) is an isomorphism of pro-objects, then Rlim E → Rlim D is an isomorphism in D(Ab).","statement_latex":"Let $E \\to D$ be a morphism of $D(\\textit{Ab}(\\mathbf{N}))$.\nLet $(E_n)$, resp.\\ $(D_n)$ be the system of objects of\n$D(\\textit{Ab})$ associated to $E$, resp.\\ $D$.\nIf $(E_n) \\to (D_n)$ is an isomorphism of pro-objects, then\n$R\\lim E \\to R\\lim D$ is an isomorphism in $D(\\textit{Ab})$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091B","source_file":"more-algebra.tex","source_line":23404,"source_end_line":23411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23404-L23411","statement_sha256":"c2600fecbe7400bb2cfd64826d7b17b1933b6dd08c2e99ff2789dd3ab6983fc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3408,"rank":3408,"depth":19,"x":567.11,"y":614.668,"cluster":"advanced-algebra"},{"id":"stacks:0CQA","tag":"0CQA","title":"Emmanouil · Lemma 0CQA","summary":"Taken from [Emmanouil]. Let (A_n) be an inverse system of abelian groups. The following are equivalent • (A_n) is Mittag-Leffler, • R^1lim A_n = 0 and the same holds for bigoplus_i ∈ N (A_n).","statement_latex":"\\begin{reference}\nTaken from \\cite{Emmanouil}.\n\\end{reference}\nLet $(A_n)$ be an inverse system of abelian groups.\nThe following are equivalent\n\\begin{enumerate}\n\\item $(A_n)$ is Mittag-Leffler,\n\\item $R^1\\lim A_n = 0$ and\nthe same holds for $\\bigoplus_{i \\in \\mathbf{N}} (A_n)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQA","source_file":"more-algebra.tex","source_line":23421,"source_end_line":23433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23421-L23433","statement_sha256":"f3dd7a212bdc6a1985342a286014ba66e245e8963f12e71d03aad6eec284e43c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3409,"rank":3409,"depth":17,"x":815.894,"y":664.105,"cluster":"advanced-algebra"},{"id":"stacks:0CQB","tag":"0CQB","title":"Rlim of abelian groups · Lemma 0CQB","summary":"Let 0 → (A_i) → (B_i) → (C_i) → 0 be a short exact sequence of inverse systems of abelian groups. If (A_i) and (C_i) are ML, then so is (B_i).","statement_latex":"Let\n$$\n0 \\to (A_i) \\to (B_i) \\to (C_i) \\to 0\n$$\nbe a short exact sequence of inverse systems of abelian groups.\nIf $(A_i)$ and $(C_i)$ are ML, then so is $(B_i)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQB","source_file":"more-algebra.tex","source_line":23510,"source_end_line":23518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23510-L23518","statement_sha256":"43b7c647f902dc978d31aa097ceadc8cab0410619de74e1b89ef602d325b2d65","origin":"The Stacks Project","memory_eligible":false,"source_rank":3410,"rank":3410,"depth":18,"x":592.503,"y":768.903,"cluster":"advanced-algebra"},{"id":"stacks:091C","tag":"091C","title":"Rlim of abelian groups · Lemma 091C","summary":"Let (A_n) be an inverse system of abelian groups. The following are equivalent • (A_n) is zero as a pro-object, • lim A_n = 0 and R^1lim A_n = 0 and the same holds for bigoplus_i ∈ N (A_n).","statement_latex":"Let $(A_n)$ be an inverse system of abelian groups.\nThe following are equivalent\n\\begin{enumerate}\n\\item $(A_n)$ is zero as a pro-object,\n\\item $\\lim A_n = 0$ and $R^1\\lim A_n = 0$ and\nthe same holds for $\\bigoplus_{i \\in \\mathbf{N}} (A_n)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of abelian groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091C","source_file":"more-algebra.tex","source_line":23526,"source_end_line":23535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23526-L23535","statement_sha256":"168d57f2889f995d06ddc452692664f37680f1f0a8745577a1187c13d00cae73","origin":"The Stacks Project","memory_eligible":false,"source_rank":3411,"rank":3411,"depth":18,"x":673.019,"y":564.701,"cluster":"advanced-algebra"},{"id":"stacks:091D","tag":"091D","title":"Rlim of modules · Lemma 091D","summary":"In the situation above. The functor lim : Mod(N, (A_n)) → Mod_A has a right derived functor Rlim : D(Mod(N, (A_n))) → D(A) As usual we set R^plim(K) = H^p(Rlim(K)). Moreover, we have • for any (M_n) in Mod(N, (A_n)) we have R^plim M_n = 0 for p > 1, • the object Rlim M_n of D(Mod_A) is represented by the complex ∏ M_n → ∏ M_n, (x_n) ↦ (x_n - f_n + 1(x_n + 1)) sitting in degrees 0 and 1, • if (M_n) is ML, then R^1lim M_n = 0, i.e., (M_n) is right acyclic for lim, • every…","statement_latex":"In the situation above. The functor\n$\\lim : \\textit{Mod}(\\mathbf{N}, (A_n)) \\to \\text{Mod}_A$\nhas a right derived functor\n$$\nR\\lim :\nD(\\textit{Mod}(\\mathbf{N}, (A_n)))\n\\longrightarrow\nD(A)\n$$\nAs usual we set $R^p\\lim(K) = H^p(R\\lim(K))$. Moreover, we have\n\\begin{enumerate}\n\\item for any $(M_n)$ in $\\textit{Mod}(\\mathbf{N}, (A_n))$ we have\n$R^p\\lim M_n = 0$ for $p > 1$,\n\\item the object $R\\lim M_n$ of $D(\\text{Mod}_A)$ is represented\nby the complex\n$$\n\\prod M_n \\to \\prod M_n,\\quad\n(x_n) \\mapsto (x_n - f_{n + 1}(x_{n + 1}))\n$$\nsitting in degrees $0$ and $1$,\n\\item if $(M_n)$ is ML, then $R^1\\lim M_n = 0$, i.e., $(M_n)$\nis right acyclic for $\\lim$,\n\\item every $K^\\bullet \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$\nis quasi-isomorphic to a complex whose terms are right acyclic for $\\lim$, and\n\\item if each $K^p = (K^p_n)$ is right acyclic for $\\lim$, i.e.,\nof $R^1\\lim_n K^p_n = 0$, then $R\\lim K$ is represented by the\ncomplex whose term in degree $p$ is $\\lim_n K_n^p$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091D","source_file":"more-algebra.tex","source_line":23573,"source_end_line":23603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23573-L23603","statement_sha256":"373d699c48f984b984d134a16460d27fd48cc0c3e26cd0fdbb11b498c94f6703","origin":"The Stacks Project","memory_eligible":false,"source_rank":3412,"rank":3412,"depth":17,"x":777.948,"y":761.125,"cluster":"advanced-algebra"},{"id":"stacks:0CQD","tag":"0CQD","title":"Rlim of modules · Lemma 0CQD","summary":"Let K = (K_n^bullet) be an object of D(Mod(N, (A_n))). There exists a canonical distinguished triangle Rlim K → ∏_n K_n^bullet → ∏_n K_n^bullet → Rlim K[1] in D(A). In other words, Rlim K is a derived limit of the inverse system (K_n^bullet) of D(A), see Derived Categories, Definition [Tag 08TC].","statement_latex":"Let $K = (K_n^\\bullet)$ be an object of\n$D(\\textit{Mod}(\\mathbf{N}, (A_n)))$.\nThere exists a canonical distinguished triangle\n$$\nR\\lim K \\to \\prod\\nolimits_n K_n^\\bullet \\to \\prod\\nolimits_n K_n^\\bullet\n\\to R\\lim K[1]\n$$\nin $D(A)$. In other words, $R\\lim K$ is a derived limit\nof the inverse system $(K_n^\\bullet)$ of $D(A)$, see\nDerived Categories, Definition \\ref{derived-definition-derived-limit}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQD","source_file":"more-algebra.tex","source_line":23631,"source_end_line":23643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23631-L23643","statement_sha256":"d3e86934475629b187e6a672394432af507b62d28c5f74eab8c918fb78cea1a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3413,"rank":3413,"depth":18,"x":542.423,"y":675.754,"cluster":"advanced-algebra"},{"id":"stacks:0CQE","tag":"0CQE","title":"Rlim of modules · Lemma 0CQE","summary":"With notation as in Lemma [Tag 0CQD] the long exact cohomology sequence associated to the distinguished triangle breaks up into short exact sequences 0 → R^1lim_n H^p - 1(K_n^bullet) → H^p(Rlim K) → lim_n H^p(K_n^bullet) → 0 of A-modules.","statement_latex":"With notation as in Lemma \\ref{lemma-distinguished-triangle-Rlim-modules}\nthe long exact cohomology sequence associated to the distinguished\ntriangle breaks up into short exact sequences\n$$\n0 \\to R^1\\lim_n H^{p - 1}(K_n^\\bullet) \\to\nH^p(R\\lim K) \\to\n\\lim_n H^p(K_n^\\bullet) \\to 0\n$$\nof $A$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQE","source_file":"more-algebra.tex","source_line":23652,"source_end_line":23663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23652-L23663","statement_sha256":"972d555236e0ae19a510b33c3cb19fae27b916309c7e4232fd0a21162c776863","origin":"The Stacks Project","memory_eligible":false,"source_rank":3414,"rank":3414,"depth":19,"x":784.948,"y":605.005,"cluster":"advanced-algebra"},{"id":"stacks:091I","tag":"091I","title":"Rlim of modules · Lemma 091I","summary":"Let (A_n) be an inverse system of rings. Suppose that we are given • for every n an object K_n of D(A_n), and • for every n a map φ_n : K_n + 1 → K_n of D(A_n + 1) where we think of K_n as an object of D(A_n + 1) by restriction via A_n + 1 → A_n. There exists an object M = (M_n^bullet) ∈ D(Mod(N, (A_n))) and isomorphisms ψ_n : M_n^bullet → K_n in D(A_n) such that the diagrams xymatrix M_n + 1^bullet ar[d]_ψ_n + 1 ar[r] & M_n^bullet ar[d]^ψ_n K_n + 1 ar[r]^φ_n & K_n…","statement_latex":"Let $(A_n)$ be an inverse system of rings. Suppose that we are given\n\\begin{enumerate}\n\\item for every $n$ an object $K_n$ of $D(A_n)$, and\n\\item for every $n$ a map $\\varphi_n : K_{n + 1} \\to K_n$ of\n$D(A_{n + 1})$ where we think of $K_n$ as an object of $D(A_{n + 1})$\nby restriction via $A_{n + 1} \\to A_n$.\n\\end{enumerate}\nThere exists an object\n$M = (M_n^\\bullet) \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$\nand isomorphisms $\\psi_n : M_n^\\bullet \\to K_n$ in $D(A_n)$\nsuch that the diagrams\n$$\n\\xymatrix{\nM_{n + 1}^\\bullet \\ar[d]_{\\psi_{n + 1}} \\ar[r] & M_n^\\bullet \\ar[d]^{\\psi_n} \\\\\nK_{n + 1} \\ar[r]^{\\varphi_n} & K_n\n}\n$$\ncommute in $D(A_{n + 1})$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091I","source_file":"more-algebra.tex","source_line":23681,"source_end_line":23701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23681-L23701","statement_sha256":"54c4114808667512a0a370eeb621e6ff689f2834dffd77eeed68865635ce32ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":3415,"rank":3415,"depth":0,"x":662.908,"y":794.944,"cluster":"advanced-algebra"},{"id":"stacks:091F","tag":"091F","title":"Rlim of modules · Lemma 091F","summary":"Let (A_n) be an inverse system of rings. Every K ∈ D(Mod(N, (A_n))) can be represented by a system of complexes (M_n^bullet) such that all the transition maps M_n + 1^bullet → M_n^bullet are surjective.","statement_latex":"Let $(A_n)$ be an inverse system of rings. Every\n$K \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$\ncan be represented by a system of complexes $(M_n^\\bullet)$\nsuch that all the transition maps $M_{n + 1}^\\bullet \\to M_n^\\bullet$\nare surjective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091F","source_file":"more-algebra.tex","source_line":23779,"source_end_line":23786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23779-L23786","statement_sha256":"fd7985b576be4c5541df40fe079cab28855b3991f9aa55411dd0554c632b478c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3416,"rank":3416,"depth":3,"x":600.103,"y":585.466,"cluster":"advanced-algebra"},{"id":"stacks:091G","tag":"091G","title":"Rlim of modules · Lemma 091G","summary":"Let (A_n) be an inverse system of rings. Every K ∈ D(Mod(N, (A_n))) can be represented by a system of complexes (K_n^bullet) such that each K_n^bullet is K-flat.","statement_latex":"Let $(A_n)$ be an inverse system of rings. Every\n$K \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$\ncan be represented by a system of complexes $(K_n^\\bullet)$\nsuch that each $K_n^\\bullet$ is K-flat.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091G","source_file":"more-algebra.tex","source_line":23815,"source_end_line":23821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23815-L23821","statement_sha256":"72f6098319a903049f9a46dd6bf6529353bfc7421be5938994eb157eda7b67e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3417,"rank":3417,"depth":15,"x":815.046,"y":704.392,"cluster":"advanced-algebra"},{"id":"stacks:091H","tag":"091H","title":"Rlim of modules · Lemma 091H","summary":"Let (A_n) be an inverse system of rings. Given K, L ∈ D(Mod(N, (A_n))) there is a canonical derived tensor product K ⊗^L L in D(N, (A_n)) compatible with the maps to D(A_n). The construction is symmetric in K and L and an exact functor of triangulated categories in each variable.","statement_latex":"Let $(A_n)$ be an inverse system of rings. Given\n$K, L \\in D(\\textit{Mod}(\\mathbf{N}, (A_n)))$ there is a canonical derived\ntensor product $K \\otimes^\\mathbf{L} L$ in $D(\\mathbf{N}, (A_n))$\ncompatible with the maps to $D(A_n)$. The construction is symmetric\nin $K$ and $L$ and an exact functor of triangulated categories in\neach variable.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091H","source_file":"more-algebra.tex","source_line":23858,"source_end_line":23866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23858-L23866","statement_sha256":"b3a3c82b3262deb33698a12906d1725e496a46817935d19aabe672abda7f8819","origin":"The Stacks Project","memory_eligible":false,"source_rank":3418,"rank":3418,"depth":16,"x":560.706,"y":738.69,"cluster":"advanced-algebra"},{"id":"stacks:091K","tag":"091K","title":"Rlim of modules · Lemma 091K","summary":"Let A be a ring. Let E → D → F → E[1] be a distinguished triangle of D(N, A). Let (E_n), resp. (D_n), resp. (F_n) be the system of objects of D(A) associated to E, resp. D, resp. F. Then for every K ∈ D(A) there is a canonical distinguished triangle Rlim (K ⊗^L_A E_n) → Rlim (K ⊗^L_A D_n) → Rlim (K ⊗^L_A F_n) → Rlim (K ⊗^L_A E_n)[1] in D(A) with notation as in Remark [Tag 091J].","statement_latex":"Let $A$ be a ring. Let $E \\to D \\to F \\to E[1]$ be a distinguished\ntriangle of $D(\\mathbf{N}, A)$. Let $(E_n)$, resp.\\ $(D_n)$, resp.\\ $(F_n)$\nbe the system of objects of $D(A)$ associated to $E$, resp.\\ $D$, resp.\\ $F$.\nThen for every $K \\in D(A)$ there is a canonical distinguished triangle\n$$\nR\\lim (K \\otimes^\\mathbf{L}_A E_n) \\to\nR\\lim (K \\otimes^\\mathbf{L}_A D_n) \\to\nR\\lim (K \\otimes^\\mathbf{L}_A F_n) \\to\nR\\lim (K \\otimes^\\mathbf{L}_A E_n)[1]\n$$\nin $D(A)$ with notation as in\nRemark \\ref{remark-constructing-tensor-with-limits-functorially}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091K","source_file":"more-algebra.tex","source_line":23914,"source_end_line":23928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23914-L23928","statement_sha256":"ee2ecd4896a547145b08a28d5334cfba022e4e473e7313bd382adfb950503b5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3419,"rank":3419,"depth":0,"x":720.802,"y":568.942,"cluster":"advanced-algebra"},{"id":"stacks:091L","tag":"091L","title":"Rlim of modules · Lemma 091L","summary":"Let A be a ring. Let E → D be a morphism of D(N, A). Let (E_n), resp. (D_n) be the system of objects of D(A) associated to E, resp. D. If (E_n) → (D_n) is an isomorphism of pro-objects, then for every K ∈ D(A) the corresponding map Rlim (K ⊗^L_A E_n) → Rlim (K ⊗^L_A D_n) in D(A) is an isomorphism (notation as in Remark [Tag 091J]).","statement_latex":"Let $A$ be a ring. Let $E \\to D$ be a morphism of\n$D(\\mathbf{N}, A)$. Let $(E_n)$, resp.\\ $(D_n)$\nbe the system of objects of $D(A)$ associated to $E$, resp.\\ $D$.\nIf $(E_n) \\to (D_n)$ is an isomorphism of pro-objects, then for every\n$K \\in D(A)$ the corresponding map\n$$\nR\\lim (K \\otimes^\\mathbf{L}_A E_n)\n\\longrightarrow\nR\\lim (K \\otimes^\\mathbf{L}_A D_n)\n$$\nin $D(A)$ is an isomorphism\n(notation as in\nRemark \\ref{remark-constructing-tensor-with-limits-functorially}).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Rlim of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091L","source_file":"more-algebra.tex","source_line":23938,"source_end_line":23953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23938-L23953","statement_sha256":"c1a0f0d45ba00a21117ccb9652d5c611986a1d658b1e7ec9392d9d0008d87164","origin":"The Stacks Project","memory_eligible":false,"source_rank":3420,"rank":3420,"depth":20,"x":739.271,"y":785.129,"cluster":"advanced-algebra"},{"id":"stacks:05E6","tag":"05E6","title":"Torsion modules · Definition 05E6","summary":"Let R be a ring. Let M be an R-module. • Let I ⊂ R be an ideal. We say M is an I-power torsion module if for every m ∈ M there exists an n > 0 such that I^n m = 0. • Let f ∈ R. We say M is an f-power torsion module if for each m ∈ M, there exists an n > 0 such that f^n m = 0.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\n\\begin{enumerate}\n\\item Let $I \\subset R$ be an ideal. We say $M$ is an\n{\\it $I$-power torsion module} if for every $m \\in M$ there exists an $n > 0$\nsuch that $I^n m = 0$.\n\\item Let $f \\in R$. We say $M$ is\n{\\it an $f$-power torsion module} if for each\n$m \\in M$, there exists an $n > 0$ such that $f^n m = 0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05E6","source_file":"more-algebra.tex","source_line":23975,"source_end_line":23986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L23975-L23986","statement_sha256":"4b9a5127ab450c0fb3b26c9911b22fd0410dff0bd7e985c259c37b76f1ed4158","origin":"The Stacks Project","memory_eligible":false,"source_rank":3421,"rank":3421,"depth":0,"x":551.648,"y":636.076,"cluster":"advanced-algebra"},{"id":"stacks:05E8","tag":"05E8","title":"Torsion modules · Lemma 05E8","summary":"Let R be a ring. Let I be an ideal of R. Let M be an I-power torsion module. Then M admits a resolution … → K_2 → K_1 → K_0 → M → 0 with each K_i a direct sum of copies of R/I^n for n variable.","statement_latex":"Let $R$ be a ring.\nLet $I$ be an ideal of $R$.\nLet $M$ be an $I$-power torsion module.\nThen $M$ admits a resolution\n$$\n\\ldots \\to K_2 \\to K_1 \\to K_0 \\to M \\to 0\n$$\nwith each $K_i$ a direct sum of copies of $R/I^n$ for $n$ variable.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05E8","source_file":"more-algebra.tex","source_line":24001,"source_end_line":24011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24001-L24011","statement_sha256":"2a72060d1f28a48559380f8276c728258f107223c556c57d6304778f3ccc3529","origin":"The Stacks Project","memory_eligible":false,"source_rank":3422,"rank":3422,"depth":0,"x":810.073,"y":639.527,"cluster":"advanced-algebra"},{"id":"stacks:05EA","tag":"05EA","title":"Torsion modules · Lemma 05EA","summary":"Let R be a ring. Let I be an ideal of R. For any R-module M set M[I^n] = (m ∈ M mid I^nm = 0). If I is finitely generated then the following are equivalent • M[I] = 0, • M[I^n] = 0 for all n ≥ 1, and • if I = (f_1, …, f_t), then the map M → bigoplus M_f_i is injective.","statement_latex":"Let $R$ be a ring. Let $I$ be an ideal of $R$.\nFor any $R$-module $M$ set $M[I^n] = \\{m \\in M \\mid I^nm = 0\\}$.\nIf $I$ is finitely generated then the following are equivalent\n\\begin{enumerate}\n\\item $M[I] = 0$,\n\\item $M[I^n] = 0$ for all $n \\geq 1$, and\n\\item if $I = (f_1, \\ldots, f_t)$, then the map\n$M \\to \\bigoplus M_{f_i}$ is injective.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EA","source_file":"more-algebra.tex","source_line":24023,"source_end_line":24034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24023-L24034","statement_sha256":"3ff6bf888337d84d3a089d7a08e5490fc8b5ba2747da3afe851320d84815b0ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":3423,"rank":3423,"depth":1,"x":616.583,"y":783.734,"cluster":"advanced-algebra"},{"id":"stacks:05EB","tag":"05EB","title":"Torsion modules · Lemma 05EB","summary":"Let R be a ring. Let I be a finitely generated ideal of R. • For any R-module M we have (M/M[I^∞])[I] = 0. • An extension of I-power torsion modules is I-power torsion.","statement_latex":"Let $R$ be a ring. Let $I$ be a finitely generated ideal of $R$.\n\\begin{enumerate}\n\\item For any $R$-module $M$ we have $(M/M[I^\\infty])[I] = 0$.\n\\item An extension of $I$-power torsion modules is $I$-power torsion.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EB","source_file":"more-algebra.tex","source_line":24041,"source_end_line":24048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24041-L24048","statement_sha256":"5841b7ae0530dd5c7e0f5277f4cc3c4302c10565636d1d956e3e1c1acd67a48f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3424,"rank":3424,"depth":2,"x":643.313,"y":567.433,"cluster":"advanced-algebra"},{"id":"stacks:0A6K","tag":"0A6K","title":"Torsion modules · Lemma 0A6K","summary":"Let I be a finitely generated ideal of a ring R. The I-power torsion modules form a Serre subcategory of the abelian category Mod_R, see Homology, Definition [Tag 02MO].","statement_latex":"Let $I$ be a finitely generated ideal of a ring $R$.\nThe $I$-power torsion modules form a Serre subcategory of\nthe abelian category $\\text{Mod}_R$, see\nHomology, Definition \\ref{homology-definition-serre-subcategory}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6K","source_file":"more-algebra.tex","source_line":24068,"source_end_line":24074,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24068-L24074","statement_sha256":"cd0f1be8b56c2701a9ec4f211056e1abd74d2a4335389de9be8f9174d51e61f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3425,"rank":3425,"depth":3,"x":797.671,"y":742.238,"cluster":"advanced-algebra"},{"id":"stacks:0953","tag":"0953","title":"Torsion modules · Lemma 0953","summary":"Let R be a ring and let I ⊂ R be a finitely generated ideal. The subcategory I^∞-torsion ⊂ Mod_R depends only on the closed subset Z = V(I) ⊂ Spec(R). In fact, an R-module M is I-power torsion if and only if its support is contained in Z.","statement_latex":"Let $R$ be a ring and let $I \\subset R$ be a finitely generated ideal.\nThe subcategory $I^\\infty\\text{-torsion} \\subset \\text{Mod}_R$\ndepends only on the closed subset $Z = V(I) \\subset \\Spec(R)$.\nIn fact, an $R$-module $M$ is $I$-power torsion if and only if its\nsupport is contained in $Z$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0953","source_file":"more-algebra.tex","source_line":24085,"source_end_line":24092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24085-L24092","statement_sha256":"d75f3d861c34e3d7791fbd3bda2f59bd7868602aabd473ee398c8540b941c9a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3426,"rank":3426,"depth":0,"x":543.071,"y":700.892,"cluster":"advanced-algebra"},{"id":"stacks:0H82","tag":"0H82","title":"Torsion modules · Lemma 0H82","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let K be an object of D(R) such that H^i(K ⊗_R^L R/I) = 0 for i > 0. Then • H^i(K ⊗_R^L R/I^n) = 0 for all n ≥ 1 and i > 0, • H^i(K ⊗_R^L N) = 0 for any I-power torsion R-module N and i > 0, and • for any M ∈ D^b(R) whose cohomology modules H^i(M) are I-power torsion and 0 for i > 0 we have H^i(K ⊗_R^L M) = 0 for i > 0.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $K$ be an object of\n$D(R)$ such that $H^i(K \\otimes_R^\\mathbf{L} R/I) = 0$ for $i > 0$. Then\n\\begin{enumerate}\n\\item $H^i(K \\otimes_R^\\mathbf{L} R/I^n) = 0$ for all $n \\geq 1$\nand $i > 0$,\n\\item $H^i(K \\otimes_R^\\mathbf{L} N) = 0$ for any $I$-power torsion\n$R$-module $N$ and $i > 0$, and\n\\item for any $M \\in D^b(R)$ whose cohomology modules $H^i(M)$\nare $I$-power torsion and $0$ for $i > 0$ we have\n$H^i(K \\otimes_R^\\mathbf{L} M) = 0$ for $i > 0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H82","source_file":"more-algebra.tex","source_line":24109,"source_end_line":24122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24109-L24122","statement_sha256":"5d42d6ca078369257cf81d16db5d1d2ae411cf217d4ce921198b903111420b71","origin":"The Stacks Project","memory_eligible":false,"source_rank":3427,"rank":3427,"depth":8,"x":764.236,"y":586.823,"cluster":"advanced-algebra"},{"id":"stacks:0G1T","tag":"0G1T","title":"Torsion modules · Lemma 0G1T","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let K be an object of D(R) such that K ⊗_R^L R/I = 0 in D(R). Then • K ⊗_R^L R/I^n = 0 for all n ≥ 1, • K ⊗_R^L N = 0 for any I-power torsion R-module N, • K ⊗_R^L M = 0 for any M ∈ D^b(R) whose cohomology modules are I-power torsion.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $K$ be an object of\n$D(R)$ such that $K \\otimes_R^\\mathbf{L} R/I = 0$ in $D(R)$. Then\n\\begin{enumerate}\n\\item $K \\otimes_R^\\mathbf{L} R/I^n = 0$ for all $n \\geq 1$,\n\\item $K \\otimes_R^\\mathbf{L} N = 0$ for any $I$-power torsion\n$R$-module $N$,\n\\item $K \\otimes_R^\\mathbf{L} M = 0$ for any $M \\in D^b(R)$ whose\ncohomology modules are $I$-power torsion.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1T","source_file":"more-algebra.tex","source_line":24154,"source_end_line":24165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24154-L24165","statement_sha256":"d97bbc7e7a980a37941228498fffc1be5f2afde59636277458c7752dbd0a0887","origin":"The Stacks Project","memory_eligible":false,"source_rank":3428,"rank":3428,"depth":9,"x":692.827,"y":796.598,"cluster":"advanced-algebra"},{"id":"stacks:0BNK","tag":"0BNK","title":"Torsion modules · Lemma 0BNK","summary":"Slight generalization of [Beauville-Laszlo]. Let R → R' be a ring map. Let I ⊂ R be an ideal such that R/I^n → R'/I^nR' is an isomorphism for n > 0. For any I-power torsion R-module M the map M → M ⊗_R R' is an isomorphism. For example, if I is finitely generated and R^wedge is the completion of R with respect to I, then we have M ≅ M ⊗_R R^wedge.","statement_latex":"\\begin{reference}\nSlight generalization of \\cite[Lemme~1]{Beauville-Laszlo}.\n\\end{reference}\nLet $R \\to R'$ be a ring map. Let $I \\subset R$ be an ideal such that\n$R/I^n \\to R'/I^nR'$ is an isomorphism for $n > 0$.\nFor any $I$-power torsion $R$-module $M$ the map $M \\to M \\otimes_R R'$\nis an isomorphism.\nFor example, if $I$ is finitely generated and $R^\\wedge$ is the completion\nof $R$ with respect to $I$, then we have $M \\cong M \\otimes_R R^\\wedge$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Torsion modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNK","source_file":"more-algebra.tex","source_line":24172,"source_end_line":24183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24172-L24183","statement_sha256":"aea2e927712d32f93c5a908211129e64dc0257345be24e4de7b1bfcc35e09852","origin":"The Stacks Project","memory_eligible":false,"source_rank":3429,"rank":3429,"depth":5,"x":576.694,"y":601.237,"cluster":"advanced-algebra"},{"id":"stacks:05E7","tag":"05E7","title":"Formal glueing of module categories · Lemma 05E7","summary":"Let φ : R → S be a ring map. Let I ⊂ R be an ideal. The following are equivalent • φ is flat and R/I → S/IS is faithfully flat, • φ is flat, and the map Spec(S/IS) → Spec(R/I) is surjective. • φ is flat, and the base change functor M ↦ M ⊗_R S is faithful on modules annihilated by I, and • φ is flat, and the base change functor M ↦ M ⊗_R S is faithful on I-power torsion modules.","statement_latex":"Let $\\varphi : R \\to S$ be a ring map. Let $I \\subset R$ be an ideal.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\varphi$ is flat and $R/I \\to S/IS$ is faithfully flat,\n\\item $\\varphi$ is flat, and the map\n$\\Spec(S/IS) \\to \\Spec(R/I)$ is surjective.\n\\item $\\varphi$ is flat, and the base change functor\n$M \\mapsto M \\otimes_R S$ is faithful on modules annihilated by $I$, and\n\\item $\\varphi$ is flat, and the base change functor\n$M \\mapsto M \\otimes_R S$ is faithful on $I$-power torsion modules.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05E7","source_file":"more-algebra.tex","source_line":24226,"source_end_line":24239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24226-L24239","statement_sha256":"0c121f20f26b1b7c2eea1db79fd02dd5b722559c2b7d1a99ae73e64c127f6e0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3430,"rank":3430,"depth":3,"x":819.626,"y":679.46,"cluster":"advanced-algebra"},{"id":"stacks:05E9","tag":"05E9","title":"Formal glueing of module categories · Lemma 05E9","summary":"Assume (φ : R → S, I) satisfies the equivalent conditions of Lemma [Tag 05E7]. The following are equivalent • for any I-power torsion module M, the natural map M → M ⊗_R S is an isomorphism, and • R/I → S/IS is an isomorphism.","statement_latex":"Assume $(\\varphi : R \\to S, I)$ satisfies the equivalent conditions of\nLemma \\ref{lemma-characterize-flatness-on-torsion}.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any $I$-power torsion module $M$, the natural map\n$M \\to M \\otimes_R S$ is an isomorphism, and\n\\item $R/I \\to S/IS$ is an isomorphism.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05E9","source_file":"more-algebra.tex","source_line":24263,"source_end_line":24273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24263-L24273","statement_sha256":"a937bfa3bb6475b2bcf89475585aa8feefc1edbb437120ec1955a0e81b9fb45e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3431,"rank":3431,"depth":4,"x":577.395,"y":759.689,"cluster":"advanced-algebra"},{"id":"stacks:05EC","tag":"05EC","title":"Formal glueing of module categories · Lemma 05EC","summary":"Assume φ : R → S is a flat ring map and I ⊂ R is a finitely generated ideal such that R/I → S/IS is an isomorphism. Then • for any R-module M the map M → M ⊗_R S induces an isomorphism M[I^∞] → (M ⊗_R S)[(IS)^∞] of I-power torsion submodules, • the natural map Hom_R(M, N) → Hom_S(M ⊗_R S, N ⊗_R S) is an isomorphism if either M or N is I-power torsion, and • the base change functor M ↦ M ⊗_R S defines an equivalence of categories between I-power torsion modules and…","statement_latex":"Assume $\\varphi : R \\to S$ is a flat ring map and $I \\subset R$ is a\nfinitely generated ideal such that $R/I \\to S/IS$ is an isomorphism. Then\n\\begin{enumerate}\n\\item for any $R$-module $M$ the map $M \\to M \\otimes_R S$ induces\nan isomorphism\n$M[I^\\infty] \\to (M \\otimes_R S)[(IS)^\\infty]$ of $I$-power\ntorsion submodules,\n\\item the natural map\n$$\n\\Hom_R(M, N) \\longrightarrow \\Hom_S(M \\otimes_R S, N \\otimes_R S)\n$$\nis an isomorphism if either $M$ or $N$ is $I$-power torsion, and\n\\item the base change functor $M \\mapsto M \\otimes_R S$ defines an\nequivalence of categories between $I$-power torsion modules\nand $IS$-power torsion modules.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EC","source_file":"more-algebra.tex","source_line":24300,"source_end_line":24318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24300-L24318","statement_sha256":"8b9c106fc18783d0b87ec4dd7faee4a009ef09b87b6332caab1bb698dbf7311e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3432,"rank":3432,"depth":5,"x":691.585,"y":562.925,"cluster":"advanced-algebra"},{"id":"stacks:091M","tag":"091M","title":"Formal glueing of module categories · Lemma 091M","summary":"Assume φ : R → S is a flat ring map and I ⊂ R is a finitely generated ideal such that R/I → S/IS is an isomorphism. For any f_1, …, f_r ∈ R such that V(f_1, …, f_r) = V(I) • the map of Koszul complexes K(R, f_1, …, f_r) → K(S, f_1, …, f_r) is a quasi-isomorphism, and • The map of extended alternating v Cech complexes xymatrix R → ∏_i_0 R_f_i_0 → ∏_i_0 < i_1 R_f_i_0f_i_1 → … → R_f_1… f_r ar[d] S → ∏_i_0 S_f_i_0 → ∏_i_0 < i_1 S_f_i_0f_i_1 → … → S_f_1… f_r is a…","statement_latex":"Assume $\\varphi : R \\to S$ is a flat ring map and $I \\subset R$ is a\nfinitely generated ideal such that $R/I \\to S/IS$ is an isomorphism.\nFor any $f_1, \\ldots, f_r \\in R$ such that $V(f_1, \\ldots, f_r) = V(I)$\n\\begin{enumerate}\n\\item the map of Koszul complexes\n$K(R, f_1, \\ldots, f_r) \\to K(S, f_1, \\ldots, f_r)$ is a quasi-isomorphism, and\n\\item The map of extended alternating {\\v C}ech complexes\n$$\n\\xymatrix{\nR \\to \\prod_{i_0} R_{f_{i_0}} \\to \\prod_{i_0 < i_1} R_{f_{i_0}f_{i_1}}\n\\to \\ldots \\to R_{f_1\\ldots f_r} \\ar[d] \\\\\nS \\to \\prod_{i_0} S_{f_{i_0}} \\to \\prod_{i_0 < i_1} S_{f_{i_0}f_{i_1}}\n\\to \\ldots \\to S_{f_1\\ldots f_r}\n}\n$$\nis a quasi-isomorphism.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091M","source_file":"more-algebra.tex","source_line":24362,"source_end_line":24381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24362-L24381","statement_sha256":"7970009956045907fadd83e17fdd232b369bd6d9af62af4d8fd49dcf05e4600d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3433,"rank":3433,"depth":5,"x":765.674,"y":772.977,"cluster":"advanced-algebra"},{"id":"stacks:05ED","tag":"05ED","title":"Formal glueing of module categories · Lemma 05ED","summary":"Let R be a ring. Let I = (f_1, …, f_n) be a finitely generated ideal of R. Let M be the R-module generated by elements e_1, …, e_n subject to the relations f_i e_j - f_j e_i = 0. There exists a short exact sequence 0 → K → M → I → 0 such that K is annihilated by I.","statement_latex":"Let $R$ be a ring. Let $I = (f_1, \\ldots, f_n)$ be a finitely generated ideal\nof $R$. Let $M$ be the $R$-module generated by elements\n$e_1, \\ldots, e_n$ subject to the relations $f_i e_j - f_j e_i = 0$.\nThere exists a short exact sequence\n$$\n0 \\to K \\to M \\to I \\to 0\n$$\nsuch that $K$ is annihilated by $I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ED","source_file":"more-algebra.tex","source_line":24393,"source_end_line":24403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24393-L24403","statement_sha256":"0e7cd0c015f3055d3de92181f6bf4dc516f8d4fdcb4977684bc2f223852878e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3434,"rank":3434,"depth":0,"x":541.946,"y":660.038,"cluster":"advanced-algebra"},{"id":"stacks:05EE","tag":"05EE","title":"Formal glueing of module categories · Lemma 05EE","summary":"Let R be a ring. Let I = (f_1, …, f_n) be a finitely generated ideal of R. For any R-module N set H_1(N, f_bullet) = frac((x_1, …, x_n) ∈ N^⊕ n mid f_i x_j = f_j x_i ) (f_1x, …, f_nx) mid x ∈ N) For any R-module N there exists a canonical short exact sequence 0 → Ext_R(R/I, N) → H_1(N, f_bullet) → Hom_R(K, N) where K is as in Lemma [Tag 05ED].","statement_latex":"Let $R$ be a ring. Let $I = (f_1, \\ldots, f_n)$ be a finitely generated ideal\nof $R$. For any $R$-module $N$ set\n$$\nH_1(N, f_\\bullet) =\n\\frac{\\{(x_1, \\ldots, x_n) \\in N^{\\oplus n} \\mid f_i x_j = f_j x_i \\}}\n{\\{f_1x, \\ldots, f_nx) \\mid x \\in N\\}}\n$$\nFor any $R$-module $N$ there exists a canonical short exact sequence\n$$\n0 \\to \\Ext_R(R/I, N) \\to H_1(N, f_\\bullet) \\to \\Hom_R(K, N)\n$$\nwhere $K$ is as in\nLemma \\ref{lemma-naive-Koszul-complex}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EE","source_file":"more-algebra.tex","source_line":24412,"source_end_line":24427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24412-L24427","statement_sha256":"bd2197265e08e709a055632fe9edb0c4538b58692189f168d5f1c1780b2aa7e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3435,"rank":3435,"depth":1,"x":797.945,"y":616.334,"cluster":"advanced-algebra"},{"id":"stacks:05EF","tag":"05EF","title":"Formal glueing of module categories · Lemma 05EF","summary":"Let R be a ring. Let I = (f_1, …, f_n) be a finitely generated ideal of R. For any R-module N the Koszul homology group H_1(N, f_bullet) defined in Lemma [Tag 05EE] is annihilated by I.","statement_latex":"Let $R$ be a ring. Let $I = (f_1, \\ldots, f_n)$ be a finitely generated ideal\nof $R$. For any $R$-module $N$ the Koszul homology group\n$H_1(N, f_\\bullet)$ defined in\nLemma \\ref{lemma-explicit-ext}\nis annihilated by $I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EF","source_file":"more-algebra.tex","source_line":24453,"source_end_line":24460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24453-L24460","statement_sha256":"564d8576fc613671ac7c5c68d2c82f9c3a96f4489392c73541d8f53ad1e99b45","origin":"The Stacks Project","memory_eligible":false,"source_rank":3436,"rank":3436,"depth":2,"x":644.199,"y":793.961,"cluster":"advanced-algebra"},{"id":"stacks:05EG","tag":"05EG","title":"Formal glueing of module categories · Lemma 05EG","summary":"Assume φ : R → S is a flat ring map and I ⊂ R is a finitely generated ideal such that R/I → S/IS is an isomorphism. Let M, N be R-modules. Assume M is I-power torsion. Given an short exact sequence 0 → N ⊗_R S → tilde E → M ⊗_R S → 0 there exists a commutative diagram xymatrix 0 ar[r] & N ar[r] ar[d] & E ar[r] ar[d] & M ar[r] ar[d] & 0 0 ar[r] & N ⊗_R S ar[r] & tilde E ar[r] & M ⊗_R S ar[r] & 0 with exact rows.","statement_latex":"Assume $\\varphi : R \\to S$ is a flat ring map and $I \\subset R$ is a\nfinitely generated ideal such that $R/I \\to S/IS$ is an isomorphism.\nLet $M$, $N$ be $R$-modules. Assume $M$ is $I$-power torsion.\nGiven an short exact sequence\n$$\n0 \\to N \\otimes_R S \\to \\tilde E \\to M \\otimes_R S \\to 0\n$$\nthere exists a commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\nN \\ar[r] \\ar[d] &\nE \\ar[r] \\ar[d] &\nM \\ar[r] \\ar[d] &\n0 \\\\\n0 \\ar[r] &\nN \\otimes_R S \\ar[r] &\n\\tilde E \\ar[r] &\nM \\otimes_R S \\ar[r] &\n0\n}\n$$\nwith exact rows.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EG","source_file":"more-algebra.tex","source_line":24476,"source_end_line":24501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24476-L24501","statement_sha256":"9d6a638da1f4c83a70775ce48ca9cd1c15db5feec0ec2569f91e09c8cca2ce80","origin":"The Stacks Project","memory_eligible":false,"source_rank":3437,"rank":3437,"depth":6,"x":614.703,"y":575.57,"cluster":"advanced-algebra"},{"id":"stacks:05EK","tag":"05EK","title":"Formal glueing of module categories · Lemma 05EK","summary":"Assume φ : R → S is a flat ring map and I = (f_1, …, f_t) ⊂ R is an ideal such that R/I → S/IS is an isomorphism. Let M be an R-module. Then the complex ([Tag 05EJ]) is exact.","statement_latex":"Assume $\\varphi : R \\to S$ is a flat ring map and\n$I = (f_1, \\ldots, f_t) \\subset R$ is an ideal such that\n$R/I \\to S/IS$ is an isomorphism.\nLet $M$ be an $R$-module. Then the\ncomplex (\\ref{equation-glueing-complex})\nis exact.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EK","source_file":"more-algebra.tex","source_line":24578,"source_end_line":24586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24578-L24586","statement_sha256":"267c5e259ff58a54dc6ec6166ced7870a3d1e7ff1df70bf0628c3c17e0d4b726","origin":"The Stacks Project","memory_eligible":false,"source_rank":3438,"rank":3438,"depth":16,"x":812.232,"y":719.986,"cluster":"advanced-algebra"},{"id":"stacks:0H77","tag":"0H77","title":"Formal glueing of module categories · Lemma 0H77","summary":"In Remark [Tag 05EL] the functor H^0 : Glue(R → S, f_1, …, f_t) → Mod_R is a right adjoint to the functor Can : Mod_R → Glue(R → S, f_1, …, f_t).","statement_latex":"In Remark \\ref{remark-glueing-data} the functor\n$H^0 : \\text{Glue}(R \\to S, f_1, \\ldots, f_t) \\to \\text{Mod}_R$\nis a right adjoint to the functor\n$\\text{Can} : \\text{Mod}_R \\to \\text{Glue}(R \\to S, f_1, \\ldots, f_t)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H77","source_file":"more-algebra.tex","source_line":24701,"source_end_line":24707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24701-L24707","statement_sha256":"15134ce528eb5b690a7ab9f835a0e3b95042d0b17c5b191044742cdb9e1163ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":3439,"rank":3439,"depth":0,"x":550.236,"y":725.583,"cluster":"advanced-algebra"},{"id":"stacks:05EM","tag":"05EM","title":"Formal glueing of module categories · Lemma 05EM","summary":"Assume φ : R → S is a flat ring map and I = (f_1, …, f_t) ⊂ R is an ideal such that R/I → S/IS is an isomorphism. Then the functor H^0 is a left quasi-inverse to the functor Can of Remark [Tag 05EL].","statement_latex":"Assume $\\varphi : R \\to S$ is a flat ring map and\n$I = (f_1, \\ldots, f_t) \\subset R$ is an ideal such that\n$R/I \\to S/IS$ is an isomorphism. Then the functor $H^0$\nis a left quasi-inverse to the functor $\\text{Can}$ of\nRemark \\ref{remark-glueing-data}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EM","source_file":"more-algebra.tex","source_line":24728,"source_end_line":24735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24728-L24735","statement_sha256":"b6293147a51643dd59d4020bc759293de8d7f7671977efe4cda06c348d19e798","origin":"The Stacks Project","memory_eligible":false,"source_rank":3440,"rank":3440,"depth":17,"x":739.075,"y":572.673,"cluster":"advanced-algebra"},{"id":"stacks:05EN","tag":"05EN","title":"Formal glueing of module categories · Lemma 05EN","summary":"Assume φ : R → S is a flat ring map and let I = (f_1, …, f_t) ⊂ R be an ideal. Then Glue(R → S, f_1, …, f_t) is an abelian category, and the functor Can is exact and commutes with arbitrary colimits.","statement_latex":"Assume $\\varphi : R \\to S$ is a flat ring map and let\n$I = (f_1, \\ldots, f_t) \\subset R$ be an ideal.\nThen $\\text{Glue}(R \\to S, f_1, \\ldots, f_t)$ is an abelian category, and\nthe functor $\\text{Can}$ is exact and commutes with arbitrary colimits.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EN","source_file":"more-algebra.tex","source_line":24742,"source_end_line":24748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24742-L24748","statement_sha256":"b2cd6e44415b68c4ecd3e1631c0709c1bf62f66e3d7bad1a1fa87c810f335609","origin":"The Stacks Project","memory_eligible":false,"source_rank":3441,"rank":3441,"depth":0,"x":722.782,"y":792.751,"cluster":"advanced-algebra"},{"id":"stacks:05EP","tag":"05EP","title":"Formal glueing of module categories · Lemma 05EP","summary":"Let φ : R → S be a flat ring map and (f_1, …, f_t) = R. Then Can and H^0 are quasi-inverse equivalences of categories Mod_R = Glue(R → S, f_1, …, f_t)","statement_latex":"Let $\\varphi : R \\to S$ be a flat ring map and $(f_1, \\ldots, f_t) = R$.\nThen $\\text{Can}$ and $H^0$ are quasi-inverse equivalences of categories\n$$\n\\text{Mod}_R = \\text{Glue}(R \\to S, f_1, \\ldots, f_t)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EP","source_file":"more-algebra.tex","source_line":24767,"source_end_line":24774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24767-L24774","statement_sha256":"dff366708fbc4e0077bf41e03e67752be358730c5e4362286cf9a3a25bd8433a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3442,"rank":3442,"depth":18,"x":557.687,"y":621.089,"cluster":"advanced-algebra"},{"id":"stacks:05EQ","tag":"05EQ","title":"Formal glueing of module categories · Lemma 05EQ","summary":"Let φ : R → S be a flat ring map and I = (f_1, …, f_t) and ideal. Let R → R' be a flat ring map, and set S' = S ⊗_R R'. Then we obtain a commutative diagram of categories and functors xymatrix Mod_R ar[r]_-Can ar[d]_-⊗_R R' & Glue(R → S, f_1, …, f_t) ar[r]_-H^0 ar[d]^-⊗_R R' & Mod_R ar[d]^-⊗_R R' Mod_R' ar[r]^-Can & Glue(R' → S', f_1, …, f_t) ar[r]^-H^0 & Mod_R'","statement_latex":"Let $\\varphi : R \\to S$ be a flat ring map and $I = (f_1, \\ldots, f_t)$\nand ideal. Let $R \\to R'$ be a flat ring map, and set $S' = S \\otimes_R R'$.\nThen we obtain a commutative diagram of categories and functors\n$$\n\\xymatrix{\n\\text{Mod}_R \\ar[r]_-{\\text{Can}} \\ar[d]_{-\\otimes_R R'} &\n\\text{Glue}(R \\to S, f_1, \\ldots, f_t) \\ar[r]_-{H^0} \\ar[d]^{-\\otimes_R R'} &\n\\text{Mod}_R \\ar[d]^{-\\otimes_R R'} \\\\\n\\text{Mod}_{R'} \\ar[r]^-{\\text{Can}} &\n\\text{Glue}(R' \\to S', f_1, \\ldots, f_t) \\ar[r]^-{H^0} &\n\\text{Mod}_{R'}\n}\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EQ","source_file":"more-algebra.tex","source_line":24791,"source_end_line":24806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24791-L24806","statement_sha256":"cfa2b492f2c53bd36273ea9bca68deee91b18ca203761f5532f5c7bcc4996b37","origin":"The Stacks Project","memory_eligible":false,"source_rank":3443,"rank":3443,"depth":0,"x":817.674,"y":654.016,"cluster":"advanced-algebra"},{"id":"stacks:05ER","tag":"05ER","title":"Formal glueing of module categories · Proposition 05ER","summary":"Assume φ : R → S is a flat ring map and I = (f_1, …, f_t) ⊂ R is an ideal such that R/I → S/IS is an isomorphism. Then Can and H^0 are quasi-inverse equivalences of categories Mod_R = Glue(R → S, f_1, …, f_t)","statement_latex":"Assume $\\varphi : R \\to S$ is a flat ring map and\n$I = (f_1, \\ldots, f_t) \\subset R$ is an ideal such that\n$R/I \\to S/IS$ is an isomorphism. Then $\\text{Can}$ and\n$H^0$ are quasi-inverse equivalences of categories\n$$\n\\text{Mod}_R = \\text{Glue}(R \\to S, f_1, \\ldots, f_t)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ER","source_file":"more-algebra.tex","source_line":24812,"source_end_line":24821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24812-L24821","statement_sha256":"18f366791279a100db7bdf316bfe8eb5b27f8a42ad1d8ccb926a8ef3b5e19991","origin":"The Stacks Project","memory_eligible":false,"source_rank":3444,"rank":3444,"depth":19,"x":599.313,"y":777.355,"cluster":"advanced-algebra"},{"id":"stacks:0ALK","tag":"0ALK","title":"Formal glueing of module categories · Lemma 0ALK","summary":"Let φ : R → S be a flat ring map and let I ⊂ R be a finitely generated ideal such that R/I → S/IS is an isomorphism. • Given an R-module N, an S-module M' and an S-module map φ : M' → N ⊗_R S whose kernel and cokernel are I-power torsion, there exists an R-module map ψ : M → N and an isomorphism M ⊗_R S = M' compatible with φ and ψ. • Given an R-module M, an S-module N' and an S-module map φ : M ⊗_R S → N' whose kernel and cokernel are I-power torsion, there exists an…","statement_latex":"Let $\\varphi : R \\to S$ be a flat ring map and let $I \\subset R$ be a\nfinitely generated ideal such that $R/I \\to S/IS$ is an isomorphism. \n\\begin{enumerate}\n\\item Given an $R$-module $N$, an $S$-module $M'$ and an $S$-module\nmap $\\varphi : M' \\to N \\otimes_R S$ whose kernel and cokernel are\n$I$-power torsion, there exists an $R$-module map\n$\\psi : M \\to N$ and an isomorphism $M \\otimes_R S = M'$\ncompatible with $\\varphi$ and $\\psi$.\n\\item Given an $R$-module $M$, an $S$-module $N'$ and an $S$-module\nmap $\\varphi : M \\otimes_R S \\to N'$ whose kernel and cokernel are\n$I$-power torsion, there exists an $R$-module map\n$\\psi : M \\to N$ and an isomorphism $N \\otimes_R S = N'$\ncompatible with $\\varphi$ and $\\psi$.\n\\end{enumerate}\nIn both cases we have $\\Ker(\\varphi) \\cong \\Ker(\\psi)$ and\n$\\Coker(\\varphi) \\cong \\Coker(\\psi)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALK","source_file":"more-algebra.tex","source_line":24881,"source_end_line":24899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24881-L24899","statement_sha256":"1ee37b7ba55fa7a063957d2a5010890213aa538b8642429dfdcecc586b66e615","origin":"The Stacks Project","memory_eligible":false,"source_rank":3445,"rank":3445,"depth":20,"x":661.192,"y":562.337,"cluster":"advanced-algebra"},{"id":"stacks:05ES","tag":"05ES","title":"Formal glueing of module categories · Theorem 05ES","summary":"Let R be a ring, and let f ∈ R. Let φ : R → S be a flat ring map inducing an isomorphism R/fR → S/fS. Then the functor Mod_R → Mod_S ×_Mod_S_f Mod_R_f, M ↦ (M ⊗_R S, M_f, can) is an equivalence.","statement_latex":"Let $R$ be a ring, and let $f \\in R$.\nLet $\\varphi : R \\to S$ be a flat ring map inducing an isomorphism\n$R/fR \\to S/fS$. Then the functor\n$$\n\\text{Mod}_R\n\\longrightarrow\n\\text{Mod}_S \\times_{\\text{Mod}_{S_f}} \\text{Mod}_{R_f},\n\\quad\nM\n\\longmapsto\n(M \\otimes_R S, M_f, \\text{can})\n$$\nis an equivalence.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ES","source_file":"more-algebra.tex","source_line":24961,"source_end_line":24976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24961-L24976","statement_sha256":"f267eb430ff036c5c76cf8f2c06a3e69e916bfe93f14bef652ebced4bddbed4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3446,"rank":3446,"depth":20,"x":788.575,"y":756.15,"cluster":"advanced-algebra"},{"id":"stacks:05ET","tag":"05ET","title":"Formal glueing of module categories · Proposition 05ET","summary":"Let R be a Noetherian ring. Let f ∈ R be an element. Let R^wedge be the f-adic completion of R. Then the functor M ↦ (M^wedge, M_f, can) defines an equivalence Mod^fg_R → Mod^fg_R^wedge ×_Mod^fg_(R^wedge)_f Mod^fg_R_f","statement_latex":"Let $R$ be a Noetherian ring.\nLet $f \\in R$ be an element.\nLet $R^\\wedge$ be the $f$-adic completion of $R$.\nThen the functor $M \\mapsto (M^\\wedge, M_f, \\text{can})$\ndefines an equivalence\n$$\n\\text{Mod}^{fg}_R\n\\longrightarrow\n\\text{Mod}^{fg}_{R^\\wedge}\n\\times_{\\text{Mod}^{fg}_{(R^\\wedge)_f}}\n\\text{Mod}^{fg}_{R_f}\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formal glueing of module categories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ET","source_file":"more-algebra.tex","source_line":24998,"source_end_line":25012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L24998-L25012","statement_sha256":"624648aa444a11c23cb6f6cbdf0eaa3863a0a0e7132faf27d68d63ac0dba7a48","origin":"The Stacks Project","memory_eligible":false,"source_rank":3447,"rank":3447,"depth":21,"x":538.591,"y":685.46,"cluster":"advanced-algebra"},{"id":"stacks:0BNJ","tag":"0BNJ","title":"The Beauville-Laszlo theorem · Lemma 0BNJ","summary":"Let R be a ring and let f ∈ R. For every positive integer n the map R/f^nR → R^wedge/f^n R^wedge is an isomorphism.","statement_latex":"Let $R$ be a ring and let $f \\in R$. For every positive integer $n$ the map\n$R/f^nR \\to R^\\wedge/f^n R^\\wedge$ is an isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNJ","source_file":"more-algebra.tex","source_line":25128,"source_end_line":25132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25128-L25132","statement_sha256":"7aaf9a9d4ca22a1c6a860bb8559577d184f3b9046f7addabb619ecab101994b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3448,"rank":3448,"depth":5,"x":779.959,"y":595.669,"cluster":"advanced-algebra"},{"id":"stacks:0BNL","tag":"0BNL","title":"The Beauville-Laszlo theorem · Lemma 0BNL","summary":"Let R be a ring, let f ∈ R, and let R → R' be a ring map which induces isomorphisms R/f^nR → R'/f^nR' for n > 0. The R-module R' ⊕ R_f is faithful: for every nonzero R-module M, the module M ⊗_R (R' ⊕ R_f) is also nonzero. For example, if M is nonzero, then M ⊗_R (R^wedge ⊕ R_f) is nonzero.","statement_latex":"Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map\nwhich induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$.\nThe $R$-module $R' \\oplus R_f$ is faithful: for every nonzero\n$R$-module $M$, the module $M \\otimes_R (R' \\oplus R_f)$\nis also nonzero. For example, if $M$ is nonzero, then\n$M \\otimes_R (R^\\wedge \\oplus R_f)$ is nonzero.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNL","source_file":"more-algebra.tex","source_line":25148,"source_end_line":25156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25148-L25156","statement_sha256":"bff532d8aaaa52fc7483274b0fcd8c6370ea46cd68ed11c1e5f52241bf119161","origin":"The Stacks Project","memory_eligible":false,"source_rank":3449,"rank":3449,"depth":0,"x":674.103,"y":798.995,"cluster":"advanced-algebra"},{"id":"stacks:0BNM","tag":"0BNM","title":"The Beauville-Laszlo theorem · Lemma 0BNM","summary":"Let R be a ring, let f ∈ R, and let R → R' be a ring map which induces an isomorphism R/fR → R'/fR'. The map Spec(R') amalg Spec(R_f) → Spec(R) is surjective. For example, the map Spec(R^wedge) amalg Spec(R_f) → Spec(R) is surjective.","statement_latex":"Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map\nwhich induces an isomorphism $R/fR \\to R'/fR'$.\nThe map $\\Spec(R') \\amalg \\Spec(R_f) \\to \\Spec(R)$ is surjective.\nFor example, the map\n$\\Spec(R^\\wedge) \\amalg \\Spec(R_f) \\to \\Spec(R)$ is surjective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNM","source_file":"more-algebra.tex","source_line":25169,"source_end_line":25176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25169-L25176","statement_sha256":"f1e3e6423ba72be1eb55fdec9087df6ca01e05e922f2d8a9d01129bf70cecf7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3450,"rank":3450,"depth":6,"x":588.585,"y":588.838,"cluster":"advanced-algebra"},{"id":"stacks:0BNN","tag":"0BNN","title":"The Beauville-Laszlo theorem · Lemma 0BNN","summary":"Slight generalization of [Beauville-Laszlo]. Let R be a ring, let f ∈ R, and let R → R' be a ring map which induces isomorphisms R/f^nR → R'/f^nR' for n > 0. An R-module M is finitely generated if and only if the (R' ⊕ R_f)-module M ⊗_R (R' ⊕ R_f) is finitely generated. For example, if M ⊗_R (R^wedge ⊕ R_f) is finitely generated as a module over R^wedge ⊕ R_f, then M is a finitely generated R-module.","statement_latex":"\\begin{reference}\nSlight generalization of \\cite[Lemme~2(a)]{Beauville-Laszlo}.\n\\end{reference}\nLet $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map\nwhich induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$.\nAn $R$-module $M$ is finitely generated if and only if the\n($R' \\oplus R_f$)-module $M \\otimes_R (R' \\oplus R_f)$ is finitely generated.\nFor example, if $M \\otimes_R (R^\\wedge \\oplus R_f)$ is finitely generated\nas a module over $R^\\wedge \\oplus R_f$, then $M$ is a finitely generated\n$R$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNN","source_file":"more-algebra.tex","source_line":25189,"source_end_line":25201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25189-L25201","statement_sha256":"f8dbef9806b47a7f03a2c535a902ff4209783bcb32c4220dd74ccdf31083e3ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":3451,"rank":3451,"depth":6,"x":820.826,"y":695.362,"cluster":"advanced-algebra"},{"id":"stacks:0BNR","tag":"0BNR","title":"The Beauville-Laszlo theorem · Lemma 0BNR","summary":"Let R be a ring, let f ∈ R, and let R → R' be a ring map which induces isomorphisms R/f^nR → R'/f^nR' for n > 0. The sequence ([Tag 0F1Q]) is • exact on the right, • exact on the left if and only if R[f^∞] → R'[f^∞] is injective, and • exact in the middle if and only if R[f^∞] → R'[f^∞] is surjective. In particular, (R → R', f) is a glueing pair if and only if R[f^∞] → R'[f^∞] is bijective. For example, (R, f) is a glueing pair if and only if R[f^∞] → R^wedge[f^∞] is…","statement_latex":"Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map\nwhich induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$.\nThe sequence (\\ref{equation-BL-cech-re}) is\n\\begin{enumerate}\n\\item exact on the right,\n\\item exact on the left if and only if $R[f^\\infty] \\to R'[f^\\infty]$\nis injective, and\n\\item exact in the middle if and only if $R[f^\\infty] \\to R'[f^\\infty]$\nis surjective.\n\\end{enumerate}\nIn particular, $(R \\to R', f)$ is a glueing pair if and only if\n$R[f^\\infty] \\to R'[f^\\infty]$ is bijective. For example, $(R, f)$\nis a glueing pair if and only if\n$R[f^\\infty] \\to R^\\wedge[f^\\infty]$ is bijective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNR","source_file":"more-algebra.tex","source_line":25254,"source_end_line":25270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25254-L25270","statement_sha256":"e47e9411a53388a741ffa16ee075af46d0964045c943968585bb3e7815b509e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3452,"rank":3452,"depth":6,"x":563.714,"y":748.633,"cluster":"advanced-algebra"},{"id":"stacks:0BNW","tag":"0BNW","title":"The Beauville-Laszlo theorem · Lemma 0BNW","summary":"Let R be a ring, let f ∈ R, and let R → R' be a ring map which induces isomorphisms R/f^nR → R'/f^nR' for n > 0. The sequence ([Tag 0F1R]) is • exact on the right, • exact on the left if and only if M[f^∞] → (M ⊗_R R')[f^∞] is injective, and • exact in the middle if and only if M[f^∞] → (M ⊗_R R')[f^∞] is surjective. Thus M is glueable for (R → R', f) if and only if M[f^∞] → (M ⊗_R R')[f^∞] is bijective. If (R → R', f) is a glueing pair, then M is glueable for (R → R', f)…","statement_latex":"Let $R$ be a ring, let $f \\in R$, and let $R \\to R'$ be a ring map\nwhich induces isomorphisms $R/f^nR \\to R'/f^nR'$ for $n > 0$.\nThe sequence (\\ref{equation-BL-cech-mod-re}) is\n\\begin{enumerate}\n\\item exact on the right,\n\\item exact on the left if and only if\n$M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$\nis injective, and\n\\item exact in the middle if and only if\n$M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$\nis surjective.\n\\end{enumerate}\nThus $M$ is glueable for $(R \\to R', f)$ if and only if\n$M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$ is bijective.\nIf $(R \\to R', f)$ is a glueing pair, then $M$ is glueable for $(R \\to R', f)$\nif and only if $M[f^\\infty] \\to (M \\otimes_R R')[f^\\infty]$ is injective.\nFor example, if $(R, f)$ is a glueing pair, then $M$ is glueable\nif and only if $M[f^\\infty] \\to (M \\otimes_R R^\\wedge)[f^\\infty]$ is\ninjective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNW","source_file":"more-algebra.tex","source_line":25364,"source_end_line":25385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25364-L25385","statement_sha256":"52edc23cdb564d514fcbd3a245bd4b4ca63fdb5fd61341d98cbc1d548c4d6c68","origin":"The Stacks Project","memory_eligible":false,"source_rank":3453,"rank":3453,"depth":7,"x":710.578,"y":563.317,"cluster":"advanced-algebra"},{"id":"stacks:0BNZ","tag":"0BNZ","title":"The Beauville-Laszlo theorem · Lemma 0BNZ","summary":"Let (R → R', f) be a glueing pair. Then Tor^R_1(R', f^n R) = 0 for each n > 0.","statement_latex":"Let $(R \\to R', f)$ be a glueing pair. Then\n$\\text{Tor}^R_1(R', f^n R) = 0$ for each $n > 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNZ","source_file":"more-algebra.tex","source_line":25471,"source_end_line":25475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25471-L25475","statement_sha256":"6fc745f09ace5bd19bd03d6115623fa73b93763499d6343e648c3a2f6e556973","origin":"The Stacks Project","memory_eligible":false,"source_rank":3454,"rank":3454,"depth":7,"x":751.34,"y":783.47,"cluster":"advanced-algebra"},{"id":"stacks:0BP0","tag":"0BP0","title":"The Beauville-Laszlo theorem · Lemma 0BP0","summary":"Let (R → R',f) be a glueing pair. Then Tor^R_1(R', R/R[f^∞]) = 0.","statement_latex":"Let $(R \\to R',f)$ be a glueing pair.\nThen $\\text{Tor}^R_1(R', R/R[f^\\infty]) = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BP0","source_file":"more-algebra.tex","source_line":25485,"source_end_line":25489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25485-L25489","statement_sha256":"1f1537f9fe3ed21a6e1b154b8234a8fdee54e5d6c0f3d43b0804190f08b11ee2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3455,"rank":3455,"depth":8,"x":544.083,"y":644.155,"cluster":"advanced-algebra"},{"id":"stacks:0BP1","tag":"0BP1","title":"The Beauville-Laszlo theorem · Lemma 0BP1","summary":"Slight generalization of [Beauville-Laszlo] Let (R → R', f) be a glueing pair. For every R-module M, we have Tor^R_1(R', Coker(M → M_f)) = 0.","statement_latex":"\\begin{reference}\nSlight generalization of \\cite[Lemme 3(a)]{Beauville-Laszlo}\n\\end{reference}\nLet $(R \\to R', f)$ be a glueing pair. For every $R$-module $M$, we have\n$\\text{Tor}^R_1(R', \\Coker(M \\to M_f)) = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BP1","source_file":"more-algebra.tex","source_line":25498,"source_end_line":25505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25498-L25505","statement_sha256":"a22f95ae1a1e48b604ba458b56222045adf48048bad243a852b9f880baef1d7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3456,"rank":3456,"depth":9,"x":809.147,"y":629.27,"cluster":"advanced-algebra"},{"id":"stacks:0BP2","tag":"0BP2","title":"The Beauville-Laszlo theorem · Theorem 0BP2","summary":"Slight generalization of the main theorem of [Beauville-Laszlo]. Let (R → R',f) be a glueing pair. The functor Can : Mod_R → Glue(R → R', f) determines an equivalence of the category of R-modules glueable for (R → R', f) and the category Glue(R → R', f) of glueing data.","statement_latex":"\\begin{reference}\nSlight generalization of the main theorem of \\cite{Beauville-Laszlo}.\n\\end{reference}\nLet $(R \\to R',f)$ be a glueing pair. The functor\n$\\text{Can} : \\text{Mod}_R \\longrightarrow \\text{Glue}(R \\to R', f)$\ndetermines an equivalence of the category of $R$-modules glueable\nfor $(R \\to R', f)$ and the category $\\text{Glue}(R \\to R', f)$\nof glueing data.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BP2","source_file":"more-algebra.tex","source_line":25578,"source_end_line":25588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25578-L25588","statement_sha256":"97e9d5f388e2ddb6896a883eaddfe0c035213369e624be4e7e97d8db1e9777c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3457,"rank":3457,"depth":10,"x":625.523,"y":790.773,"cluster":"advanced-algebra"},{"id":"stacks:0BP7","tag":"0BP7","title":"The Beauville-Laszlo theorem · Lemma 0BP7","summary":"Let (R → R', f) be a glueing pair. Let M be an R-module which is not necessarily glueable for (R → R', f). Then M is flat over R if and only if M ⊗_R R' is flat over R' and M_f is flat over R_f.","statement_latex":"Let $(R \\to R', f)$ be a glueing pair. Let $M$ be an $R$-module\nwhich is not necessarily glueable for $(R \\to R', f)$. Then $M$\nis flat over $R$ if and only if $M \\otimes_R R'$ is flat over $R'$\nand $M_f$ is flat over $R_f$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BP7","source_file":"more-algebra.tex","source_line":25707,"source_end_line":25713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25707-L25713","statement_sha256":"1072016896c4aad282fc4c013df0838c06966ab866e38aa2f16a8cbc7fd6ed91","origin":"The Stacks Project","memory_eligible":false,"source_rank":3458,"rank":3458,"depth":4,"x":631.053,"y":567.32,"cluster":"advanced-algebra"},{"id":"stacks:0BP6","tag":"0BP6","title":"The Beauville-Laszlo theorem · Lemma 0BP6","summary":"Let (R → R', f) be a glueing pair. Let M be an R-module which is not necessarily glueable for (R → R', f). Then M is a finite projective R-module if and only if M ⊗_R R' is finite projective over R' and M_f is finite projective over R_f.","statement_latex":"Let $(R \\to R', f)$ be a glueing pair. Let $M$ be an $R$-module\nwhich is not necessarily glueable for $(R \\to R', f)$. Then\n$M$ is a finite projective $R$-module if and only if\n$M \\otimes_R R'$ is finite projective over $R'$ and\n$M_f$ is finite projective over $R_f$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BP6","source_file":"more-algebra.tex","source_line":25777,"source_end_line":25784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25777-L25784","statement_sha256":"dfdaa98dc141d8d25918cb555abacb539af89caaccc0eb8f36456c2fcd4014a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3459,"rank":3459,"depth":7,"x":806.803,"y":735.357,"cluster":"advanced-algebra"},{"id":"stacks:091P","tag":"091P","title":"Derived Completion · Lemma 091P","summary":"Let A be a ring. Let f ∈ A. Let K ∈ D(A). The following are equivalent • Ext^n_A(A_f, K) = 0 for all n, • Hom_D(A)(E, K) = 0 for all E in D(A_f), • T(K, f) = 0, • for every p ∈ Z we have T(H^p(K), f) = 0, • for every p ∈ Z we have Hom_A(A_f, H^p(K)) = 0 and Ext^1_A(A_f, H^p(K)) = 0, • RHom_A(A_f, K) = 0, • the map ∏_n ≥ 0 K → ∏_n ≥ 0 K, (x_0, x_1, …) ↦ (x_0 - fx_1, x_1 - fx_2, …) is an isomorphism in D(A), and • add more here.","statement_latex":"Let $A$ be a ring. Let $f \\in A$. Let $K \\in D(A)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\Ext^n_A(A_f, K) = 0$ for all $n$,\n\\item $\\Hom_{D(A)}(E, K) = 0$ for all $E$ in $D(A_f)$,\n\\item $T(K, f) = 0$,\n\\item for every $p \\in \\mathbf{Z}$ we have $T(H^p(K), f) = 0$,\n\\item for every $p \\in \\mathbf{Z}$ we have\n$\\Hom_A(A_f, H^p(K)) = 0$ and $\\Ext^1_A(A_f, H^p(K)) = 0$,\n\\item $R\\Hom_A(A_f, K) = 0$,\n\\item the map $\\prod_{n \\geq 0} K \\to \\prod_{n \\geq 0} K$,\n$(x_0, x_1, \\ldots) \\mapsto (x_0 - fx_1, x_1 - fx_2, \\ldots)$\nis an isomorphism in $D(A)$, and\n\\item add more here.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091P","source_file":"more-algebra.tex","source_line":25849,"source_end_line":25866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25849-L25866","statement_sha256":"d071f86356a9aa87ca5584cd46bf6d53ba7c27dd2a5613607d8e05b7c159a076","origin":"The Stacks Project","memory_eligible":false,"source_rank":3460,"rank":3460,"depth":2,"x":541.876,"y":711.156,"cluster":"advanced-algebra"},{"id":"stacks:091Q","tag":"091Q","title":"Derived Completion · Lemma 091Q","summary":"Let A be a ring. Let K ∈ D(A). The set I of f ∈ A such that T(K, f) = 0 is a radical ideal of A.","statement_latex":"Let $A$ be a ring. Let $K \\in D(A)$. The set $I$ of $f \\in A$ such that\n$T(K, f) = 0$ is a radical ideal of $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091Q","source_file":"more-algebra.tex","source_line":25935,"source_end_line":25939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25935-L25939","statement_sha256":"dc9162375bfdb8e93594503b1489c134862f53064e90848629efd3f84cd39094","origin":"The Stacks Project","memory_eligible":false,"source_rank":3461,"rank":3461,"depth":4,"x":756.854,"y":578.574,"cluster":"advanced-algebra"},{"id":"stacks:091R","tag":"091R","title":"Derived Completion · Lemma 091R","summary":"Let A be a ring. Let I ⊂ A be an ideal. Let M be an A-module. • If M is I-adically complete, then T(M, f) = 0 for all f ∈ I. • Conversely, if T(M, f) = 0 for all f ∈ I and I is finitely generated, then M → lim M/I^nM is surjective.","statement_latex":"Let $A$ be a ring. Let $I \\subset A$ be an ideal. Let $M$ be an $A$-module.\n\\begin{enumerate}\n\\item If $M$ is $I$-adically complete, then $T(M, f) = 0$ for all $f \\in I$.\n\\item Conversely, if $T(M, f) = 0$ for all $f \\in I$ and $I$ is finitely\ngenerated, then $M \\to \\lim M/I^nM$ is surjective.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091R","source_file":"more-algebra.tex","source_line":25962,"source_end_line":25970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L25962-L25970","statement_sha256":"25dbf4be1b56e5b272a568b3ac05ff37bffaa813ac5edf07b4d40b08dd970247","origin":"The Stacks Project","memory_eligible":false,"source_rank":3462,"rank":3462,"depth":3,"x":704.912,"y":798.488,"cluster":"advanced-algebra"},{"id":"stacks:091S","tag":"091S","title":"Derived Completion · Definition 091S","summary":"Let A be a ring. Let K ∈ D(A). Let I ⊂ A be an ideal. We say K is derived complete with respect to I if for every f ∈ I we have T(K, f) = 0. If M is an A-module, then we say M is derived complete with respect to I if M[0] ∈ D(A) is derived complete with respect to I.","statement_latex":"Let $A$ be a ring. Let $K \\in D(A)$. Let $I \\subset A$ be an ideal.\nWe say $K$ is {\\it derived complete with respect to $I$}\nif for every $f \\in I$ we have $T(K, f) = 0$.\nIf $M$ is an $A$-module, then we say $M$ is\n{\\it derived complete with respect to $I$}\nif $M[0] \\in D(A)$ is derived complete with respect to $I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091S","source_file":"more-algebra.tex","source_line":26020,"source_end_line":26028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26020-L26028","statement_sha256":"2bf58cdabdaab126253eebbe9a4344bbc8d92ce9701d2a33e046f0c0d328308b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3463,"rank":3463,"depth":0,"x":566.26,"y":606.709,"cluster":"advanced-algebra"},{"id":"stacks:091T","tag":"091T","title":"Derived Completion · Proposition 091T","summary":"Let I ⊂ A be a finitely generated ideal of a ring A. Let M be an A-module. The following are equivalent • M is I-adically complete, and • M is derived complete with respect to I and ⋂ I^nM = 0.","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring $A$.\nLet $M$ be an $A$-module. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is $I$-adically complete, and\n\\item $M$ is derived complete with respect to $I$ and $\\bigcap I^nM = 0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091T","source_file":"more-algebra.tex","source_line":26046,"source_end_line":26054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26046-L26054","statement_sha256":"994c656c76aeb655f5547a30f0ec1d197a53383f3f1ed58c6ab93124c5c586c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3464,"rank":3464,"depth":4,"x":822.917,"y":669.497,"cluster":"advanced-algebra"},{"id":"stacks:091U","tag":"091U","title":"Derived Completion · Lemma 091U","summary":"Let I be an ideal of a ring A. • The derived complete A-modules form a weak Serre subcategory C of Mod_A. • D_C(A) ⊂ D(A) is the full subcategory of derived complete objects.","statement_latex":"Let $I$ be an ideal of a ring $A$.\n\\begin{enumerate}\n\\item The derived complete $A$-modules form a weak Serre\nsubcategory $\\mathcal{C}$ of $\\text{Mod}_A$.\n\\item $D_\\mathcal{C}(A) \\subset D(A)$ is the full subcategory\nof derived complete objects.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091U","source_file":"more-algebra.tex","source_line":26067,"source_end_line":26076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26067-L26076","statement_sha256":"14791a3a457cd7e2ea6b757aa5a985a6c079bf16fad7ef8c00bb92195e5adce1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3465,"rank":3465,"depth":3,"x":582.988,"y":768.906,"cluster":"advanced-algebra"},{"id":"stacks:09B9","tag":"09B9","title":"Derived Completion · Lemma 09B9","summary":"Let I be a finitely generated ideal of a ring A. Let M be a derived complete A-module. If M/IM = 0, then M = 0.","statement_latex":"Let $I$ be a finitely generated ideal of a ring $A$.\nLet $M$ be a derived complete $A$-module.\nIf $M/IM = 0$, then $M = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09B9","source_file":"more-algebra.tex","source_line":26099,"source_end_line":26104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26099-L26104","statement_sha256":"0de197d94689b5d8c31001db701466f3b211c2d59090ab005590ca2614c55e9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3466,"rank":3466,"depth":4,"x":680.039,"y":559.304,"cluster":"advanced-algebra"},{"id":"stacks:0A05","tag":"0A05","title":"Derived Completion · Lemma 0A05","summary":"Let A be a ring and I ⊂ A an ideal. If A is derived complete (eg. I-adically complete) then any pseudo-coherent object of D(A) is derived complete.","statement_latex":"Let $A$ be a ring and $I \\subset A$ an ideal. If $A$ is derived complete\n(eg.\\ $I$-adically complete)\nthen any pseudo-coherent object of $D(A)$ is derived complete.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A05","source_file":"more-algebra.tex","source_line":26125,"source_end_line":26130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26125-L26130","statement_sha256":"836fd9e0874f8065544ed1e14c8bf60c7c46474fb0ab1d55483898beaf50d487","origin":"The Stacks Project","memory_eligible":false,"source_rank":3467,"rank":3467,"depth":4,"x":777.104,"y":769.088,"cluster":"advanced-algebra"},{"id":"stacks:0A6C","tag":"0A6C","title":"Derived Completion · Lemma 0A6C","summary":"Let A be a ring. Let f, g ∈ A. Then for K ∈ D(A) we have RHom_A(A_f, RHom_A(A_g, K)) = RHom_A(A_fg, K).","statement_latex":"Let $A$ be a ring. Let $f, g \\in A$. Then for $K \\in D(A)$ we have\n$R\\Hom_A(A_f, R\\Hom_A(A_g, K)) = R\\Hom_A(A_{fg}, K)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6C","source_file":"more-algebra.tex","source_line":26143,"source_end_line":26147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26143-L26147","statement_sha256":"582e2afc774e614d4573dae7861d7b0ed618a843666c0deceada30f51baadfc9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3468,"rank":3468,"depth":3,"x":536.647,"y":669.399,"cluster":"advanced-algebra"},{"id":"stacks:091V","tag":"091V","title":"Derived Completion · Lemma 091V","summary":"Derived completions along finitely generated ideals exist, and can be computed by a v Cech procedure. Let I be a finitely generated ideal of a ring A. The inclusion functor D_comp(A, I) → D(A) has a left adjoint, i.e., given any object K of D(A) there exists a map K → K^wedge of K into a derived complete object of D(A) such that the map Hom_D(A)(K^wedge, E) → Hom_D(A)(K, E) is bijective whenever E is a derived complete object of D(A). In fact, if I is generated by f_1, …,…","statement_latex":"\\begin{slogan}\nDerived completions along finitely generated ideals exist, and can\nbe computed by a {\\v C}ech procedure.\n\\end{slogan}\nLet $I$ be a finitely generated ideal of a ring $A$.\nThe inclusion functor $D_{comp}(A, I) \\to D(A)$ has a\nleft adjoint, i.e., given any object $K$ of $D(A)$ there\nexists a map $K \\to K^\\wedge$ of $K$ into a derived complete\nobject of $D(A)$ such that the map\n$$\n\\Hom_{D(A)}(K^\\wedge, E) \\longrightarrow \\Hom_{D(A)}(K, E)\n$$\nis bijective whenever $E$ is a derived complete object of $D(A)$.\nIn fact, if $I$ is generated by $f_1, \\ldots, f_r \\in A$, then we have\n$$\nK^\\wedge = R\\Hom\\left((A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to\n\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}}\n\\to \\ldots \\to A_{f_1\\ldots f_r}), K\\right)\n$$\nfunctorially in $K$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091V","source_file":"more-algebra.tex","source_line":26153,"source_end_line":26175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26153-L26175","statement_sha256":"9f3e7cf8e2e17f0da02729c86d466c4adb340efb70a3f9c5ef9bedcf25230432","origin":"The Stacks Project","memory_eligible":false,"source_rank":3469,"rank":3469,"depth":4,"x":794.317,"y":606.42,"cluster":"advanced-algebra"},{"id":"stacks:0A6D","tag":"0A6D","title":"Derived Completion · Lemma 0A6D","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. Let K^bullet be a complex of A-modules such that f : K^bullet → K^bullet is an isomorphism for some f ∈ I, i.e., K^bullet is a complex of A_f-modules. Then the derived completion of K^bullet is zero.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nLet $K^\\bullet$ be a complex of $A$-modules such that\n$f : K^\\bullet \\to K^\\bullet$ is an isomorphism for some\n$f \\in I$, i.e., $K^\\bullet$ is a complex of $A_f$-modules. Then\nthe derived completion of $K^\\bullet$ is zero.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6D","source_file":"more-algebra.tex","source_line":26243,"source_end_line":26250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26243-L26250","statement_sha256":"4f763e5674958b21cbdf0531354ca1ddf48ef85f065c6b2c4b8a8f316f606c29","origin":"The Stacks Project","memory_eligible":false,"source_rank":3470,"rank":3470,"depth":5,"x":654.857,"y":799.212,"cluster":"advanced-algebra"},{"id":"stacks:0A6E","tag":"0A6E","title":"Derived Completion · Lemma 0A6E","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. Let K, L ∈ D(A). Then RHom_A(K, L)^wedge = RHom_A(K, L^wedge) = RHom_A(K^wedge, L^wedge)","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nLet $K, L \\in D(A)$. Then\n$$\nR\\Hom_A(K, L)^\\wedge = R\\Hom_A(K, L^\\wedge) = R\\Hom_A(K^\\wedge, L^\\wedge)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6E","source_file":"more-algebra.tex","source_line":26259,"source_end_line":26266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26259-L26266","statement_sha256":"a17af1288f52980736c4687a69bcb8eb3ef1f53dc03aaa52e425a6d1ee8ae181","origin":"The Stacks Project","memory_eligible":false,"source_rank":3471,"rank":3471,"depth":5,"x":602.615,"y":577.752,"cluster":"advanced-algebra"},{"id":"stacks:091W","tag":"091W","title":"Derived Completion · Lemma 091W","summary":"Let A be a ring and let I ⊂ A be an ideal. Let (K_n) be an inverse system of objects of D(A) such that for all f ∈ I and n there exists an e = e(n, f) such that f^e is zero on K_n. Then for K ∈ D(A) the object K' = Rlim (K ⊗_A^L K_n) is derived complete with respect to I.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be an ideal. Let $(K_n)$ be an inverse\nsystem of objects of $D(A)$ such that for all $f \\in I$ and $n$\nthere exists an $e = e(n, f)$ such that $f^e$ is zero on $K_n$.\nThen for $K \\in D(A)$ the object $K' = R\\lim (K \\otimes_A^\\mathbf{L} K_n)$\nis derived complete with respect to $I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091W","source_file":"more-algebra.tex","source_line":26287,"source_end_line":26294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26287-L26294","statement_sha256":"0e164a13fffb4762bc096a179c4aa9c5520955b72424359ef33881a7d444395e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3472,"rank":3472,"depth":0,"x":819.391,"y":711.51,"cluster":"advanced-algebra"},{"id":"stacks:091Y","tag":"091Y","title":"Derived Completion · Lemma 091Y","summary":"In Situation [Tag 0BKC]. For K ∈ D(A) the object K' = Rlim (K ⊗_A^L K_n^bullet) is derived complete with respect to I.","statement_latex":"In Situation \\ref{situation-koszul}. For\n$K \\in D(A)$ the object $K' = R\\lim (K \\otimes_A^\\mathbf{L} K_n^\\bullet)$\nis derived complete with respect to $I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091Y","source_file":"more-algebra.tex","source_line":26328,"source_end_line":26333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26328-L26333","statement_sha256":"5f6c1df321aea9dd0584a4ca8c2c6b7c3d204fbee7df0ed8d2ff878024dde519","origin":"The Stacks Project","memory_eligible":false,"source_rank":3473,"rank":3473,"depth":2,"x":551.78,"y":735.901,"cluster":"advanced-algebra"},{"id":"stacks:091Z","tag":"091Z","title":"Derived Completion · Lemma 091Z","summary":"In Situation [Tag 0BKC]. Let K ∈ D(A). The following are equivalent • K is derived complete with respect to I, and • the canonical map K → Rlim (K ⊗_A^L K_n^bullet) is an isomorphism of D(A).","statement_latex":"In Situation \\ref{situation-koszul}. Let $K \\in D(A)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ is derived complete with respect to $I$, and\n\\item the canonical map $K \\to R\\lim (K \\otimes_A^\\mathbf{L} K_n^\\bullet)$\nis an isomorphism of $D(A)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091Z","source_file":"more-algebra.tex","source_line":26341,"source_end_line":26350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26341-L26350","statement_sha256":"53e001f0d983d2d6d05ef61d8f2accd64fca6b358020d3ebf9d6c632325186d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3474,"rank":3474,"depth":3,"x":729.631,"y":565.94,"cluster":"advanced-algebra"},{"id":"stacks:0920","tag":"0920","title":"Derived Completion · Lemma 0920","summary":"In Situation [Tag 0BKC]. The functor which sends K ∈ D(A) to the derived limit K' = Rlim( K ⊗_A^L K_n^bullet ) is the left adjoint to the inclusion functor D_comp(A) → D(A) constructed in Lemma [Tag 091V].","statement_latex":"In Situation \\ref{situation-koszul}.\nThe functor which sends $K \\in D(A)$ to the derived limit\n$K' = R\\lim( K \\otimes_A^\\mathbf{L} K_n^\\bullet )$ is the left\nadjoint to the inclusion functor $D_{comp}(A) \\to D(A)$\nconstructed in Lemma \\ref{lemma-derived-completion}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0920","source_file":"more-algebra.tex","source_line":26387,"source_end_line":26394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26387-L26394","statement_sha256":"9e43ccf17a7ff7560c6706ac8d32b345753cd5067e2188a9393c0af8b3a4697f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3475,"rank":3475,"depth":5,"x":735.168,"y":792.351,"cluster":"advanced-algebra"},{"id":"stacks:0H83","tag":"0H83","title":"Derived Completion · Lemma 0H83","summary":"Let I = (f_1, …, f_r) be a finitely generated ideal of a ring A. Let K be a derived complete object of D(A). The following are equivalent • H^i(K) = 0 for i > 0, • H^i(K ⊗_A^L A/I) = 0 for i > 0, • H^i(K ⊗_A^L K_1^bullet) = 0 for i > 0 where K_1^bullet is as in Situation [Tag 0BKC].","statement_latex":"Let $I = (f_1, \\ldots, f_r)$ be a finitely generated ideal of a ring $A$.\nLet $K$ be a derived complete object of $D(A)$. The following are equivalent\n\\begin{enumerate}\n\\item $H^i(K) = 0$ for $i > 0$,\n\\item $H^i(K \\otimes_A^\\mathbf{L} A/I) = 0$ for $i > 0$,\n\\item $H^i(K \\otimes_A^\\mathbf{L} K_1^\\bullet) = 0$ for $i > 0$\nwhere $K_1^\\bullet$ is as in Situation \\ref{situation-koszul}.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H83","source_file":"more-algebra.tex","source_line":26436,"source_end_line":26446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26436-L26446","statement_sha256":"0cad74b56759f21baa4854f67f0c29d2e406f7cf5dad20249167d4608d34c25c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3476,"rank":3476,"depth":9,"x":548.873,"y":628.419,"cluster":"advanced-algebra"},{"id":"stacks:0G1U","tag":"0G1U","title":"Derived Completion · Lemma 0G1U","summary":"Derived Nakayama A related result is [Dwyer-Greenlees]. The derived Nakayama lemma can for example be found in Bhatt's 3rd lecture on Prismatic cohomology at Columbia University in Fall 2018 as Section 2 property (2). Leonid Positselski proposed a proof in urlhttps://mathoverflow.net/a/331501. However, we follow the proof suggested by Anonymous in the comments. Let I be a finitely generated ideal of a ring A. Let K be a derived complete object of D(A). If K ⊗_A^L A/I = 0,…","statement_latex":"\\begin{slogan}\nDerived Nakayama\n\\end{slogan}\n\\begin{reference}\nA related result is \\cite[Proposition 6.5]{Dwyer-Greenlees}.\nThe derived Nakayama lemma can for example be found in Bhatt's 3rd lecture\non Prismatic cohomology at Columbia University in Fall 2018\nas Section 2 property (2). Leonid Positselski proposed a proof in\n\\url{https://mathoverflow.net/a/331501}. However, we follow the proof\nsuggested by Anonymous in the comments.\n\\end{reference}\nLet $I$ be a finitely generated ideal of a ring $A$.\nLet $K$ be a derived complete object of $D(A)$.\nIf $K \\otimes_A^\\mathbf{L} A/I = 0$, then $K = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1U","source_file":"more-algebra.tex","source_line":26471,"source_end_line":26487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26471-L26487","statement_sha256":"6012bb5202fe0a834d8adefc4ea6af0f274f238650b34898a5ff18f9f426d441","origin":"The Stacks Project","memory_eligible":false,"source_rank":3477,"rank":3477,"depth":10,"x":818.273,"y":643.603,"cluster":"advanced-algebra"},{"id":"stacks:0AAJ","tag":"0AAJ","title":"Derived Completion · Lemma 0AAJ","summary":"Let A be a ring and let I ⊂ A be an ideal which can be generated by r elements. Then derived completion has finite cohomological dimension: • Let K → L be a morphism in D(A) such that H^i(K) → H^i(L) is an isomorphism for i ≥ 1 and surjective for i = 0. Then H^i(K^wedge) → H^i(L^wedge) is an isomorphism for i ≥ 1 and surjective for i = 0. • Let K → L be a morphism of D(A) such that H^i(K) → H^i(L) is an isomorphism for i ≤ -1 and injective for i = 0. Then H^i(K^wedge) →…","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be an ideal which can be\ngenerated by $r$ elements. Then derived completion has finite\ncohomological dimension:\n\\begin{enumerate}\n\\item Let $K \\to L$ be a morphism in $D(A)$ such that $H^i(K) \\to H^i(L)$\nis an isomorphism for $i \\geq 1$ and surjective for $i = 0$.\nThen $H^i(K^\\wedge) \\to H^i(L^\\wedge)$ is an isomorphism for $i \\geq 1$\nand surjective for $i = 0$.\n\\item Let $K \\to L$ be a morphism of $D(A)$ such that $H^i(K) \\to H^i(L)$\nis an isomorphism for $i \\leq -1$ and injective for $i = 0$.\nThen $H^i(K^\\wedge) \\to H^i(L^\\wedge)$ is an isomorphism for $i \\leq -r - 1$\nand injective for $i = -r$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAJ","source_file":"more-algebra.tex","source_line":26507,"source_end_line":26522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26507-L26522","statement_sha256":"ec7352d9cddd2ec2bb5696255ca40f89b3458ebffaa10d0ea6be72c4df745b42","origin":"The Stacks Project","memory_eligible":false,"source_rank":3478,"rank":3478,"depth":20,"x":607.255,"y":785.375,"cluster":"advanced-algebra"},{"id":"stacks:0BKD","tag":"0BKD","title":"Derived Completion · Lemma 0BKD","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. Let K^bullet be a filtered complex of A-modules. There exists a canonical spectral sequence (E_r, d_r)_r ≥ 1 of bigraded derived complete A-modules with d_r of bidegree (r, -r + 1) and with E_1^p, q = H^p + q((gr^pK^bullet)^wedge) If the filtration on each K^n is finite, then the spectral sequence is bounded and converges to H^*((K^bullet)^wedge).","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nLet $K^\\bullet$ be a filtered complex of $A$-modules. There exists a\ncanonical spectral sequence $(E_r, \\text{d}_r)_{r \\geq 1}$\nof bigraded derived complete $A$-modules with $d_r$ of bidegree\n$(r, -r + 1)$ and with\n$$\nE_1^{p, q} = H^{p + q}((\\text{gr}^pK^\\bullet)^\\wedge)\n$$\nIf the filtration on each $K^n$ is finite, then the spectral sequence is\nbounded and converges to $H^*((K^\\bullet)^\\wedge)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKD","source_file":"more-algebra.tex","source_line":26555,"source_end_line":26567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26555-L26567","statement_sha256":"05030c91aa914c1eab5453866479f0fbee63f1b89b7496bb27886dec6457bbd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3479,"rank":3479,"depth":21,"x":648.878,"y":560.933,"cluster":"advanced-algebra"},{"id":"stacks:0924","tag":"0924","title":"Derived Completion · Lemma 0924","summary":"Let A → B be a ring map. Let I ⊂ A be an ideal. The inverse image of D_comp(A, I) under the restriction functor D(B) → D(A) is D_comp(B, IB).","statement_latex":"Let $A \\to B$ be a ring map. Let $I \\subset A$ be an ideal. The inverse\nimage of $D_{comp}(A, I)$ under the restriction functor $D(B) \\to D(A)$ is\n$D_{comp}(B, IB)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0924","source_file":"more-algebra.tex","source_line":26615,"source_end_line":26620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26615-L26620","statement_sha256":"5dfcc16b9b7bd57b2fd5000fbd83276bc2a004d72ed7a978b5897e0a67568aac","origin":"The Stacks Project","memory_eligible":false,"source_rank":3480,"rank":3480,"depth":5,"x":798.787,"y":750.19,"cluster":"advanced-algebra"},{"id":"stacks:0925","tag":"0925","title":"Derived Completion · Lemma 0925","summary":"Let A → B be a ring map. Let I ⊂ A be a finitely generated ideal. If A → B is flat and A/I ≅ B/IB, then the restriction functor D(B) → D(A) induces an equivalence D_comp(B, IB) → D_comp(A, I).","statement_latex":"Let $A \\to B$ be a ring map. Let $I \\subset A$ be a finitely generated ideal.\nIf $A \\to B$ is flat and $A/I \\cong B/IB$, then the restriction functor\n$D(B) \\to D(A)$ induces an equivalence\n$D_{comp}(B, IB) \\to D_{comp}(A, I)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived Completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0925","source_file":"more-algebra.tex","source_line":26634,"source_end_line":26640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26634-L26640","statement_sha256":"81ac9b14430df881db2fc23822d172d267c9565f6850a381223f98e23988c8f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3481,"rank":3481,"depth":6,"x":535.857,"y":695.657,"cluster":"advanced-algebra"},{"id":"stacks:0GLP","tag":"0GLP","title":"The category of derived complete modules · Lemma 0GLP","summary":"Let A be a ring and let I ⊂ A be an ideal. The category C of derived complete modules is abelian, has arbitrary limits, and the inclusion functor F : C → Mod_A is exact and commutes with limits. If I is finitely generated, then C has arbitrary colimits and F has a left adjoint","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be an ideal. The category $\\mathcal{C}$\nof derived complete modules is abelian, has arbitrary limits, and the\ninclusion functor $F : \\mathcal{C} \\to \\text{Mod}_A$ is exact and commutes\nwith limits. If $I$ is finitely generated, then $\\mathcal{C}$ has arbitrary\ncolimits and $F$ has a left adjoint","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"The category of derived complete modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLP","source_file":"more-algebra.tex","source_line":26754,"source_end_line":26761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26754-L26761","statement_sha256":"6fc3595eb060c9d1edad3db61b4246a2d5a7e05c4e04be3aecf4b6ab1e9f4342","origin":"The Stacks Project","memory_eligible":false,"source_rank":3482,"rank":3482,"depth":0,"x":773.767,"y":586.596,"cluster":"advanced-algebra"},{"id":"stacks:091X","tag":"091X","title":"Derived completion for a principal ideal · Lemma 091X","summary":"Let A be a ring. Let f ∈ A. If there exists an integer c ≥ 1 such that A[f^c] = A[f^c + 1] = A[f^c + 2] = … (for example if A is Noetherian), then for all n ≥ 1 there exist maps (A xrightarrowf^n A) → A/(f^n), and A/(f^n + c) → (A xrightarrowf^n A) in D(A) inducing an isomorphism of the pro-objects (A/(f^n)) and ((f^n : A → A)) in D(A).","statement_latex":"Let $A$ be a ring. Let $f \\in A$. If there exists an integer $c \\geq 1$\nsuch that $A[f^c] = A[f^{c + 1}] = A[f^{c + 2}] = \\ldots$ (for example\nif $A$ is Noetherian), then for all $n \\geq 1$ there exist maps\n$$\n(A \\xrightarrow{f^n} A) \\longrightarrow A/(f^n),\n\\quad\\text{and}\\quad\nA/(f^{n + c}) \\longrightarrow (A \\xrightarrow{f^n} A)\n$$\nin $D(A)$ inducing an isomorphism of the pro-objects $\\{A/(f^n)\\}$ and\n$\\{(f^n : A \\to A)\\}$ in $D(A)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for a principal ideal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/091X","source_file":"more-algebra.tex","source_line":26781,"source_end_line":26793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26781-L26793","statement_sha256":"da711db6d73e3834cc3071de7834c0831cb2892af760fea24cb8072114017ffd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3483,"rank":3483,"depth":0,"x":685.975,"y":802.169,"cluster":"advanced-algebra"},{"id":"stacks:0923","tag":"0923","title":"Derived completion for a principal ideal · Lemma 0923","summary":"Let A be a ring and f ∈ A. Set I = (f). In this situation we have the naive derived completion K ↦ K' = Rlim (K ⊗_A^L A/f^nA) and the derived completion K ↦ K^wedge = Rlim (K ⊗_A^L (A xrightarrowf^n A)) of Lemma [Tag 0920]. The natural transformation of functors K^wedge → K' is an isomorphism if and only if the f-power torsion of A is bounded.","statement_latex":"Let $A$ be a ring and $f \\in A$. Set $I = (f)$. In this situation\nwe have the naive derived completion\n$K \\mapsto K' = R\\lim (K \\otimes_A^\\mathbf{L} A/f^nA)$ and the\nderived completion\n$$\nK \\mapsto K^\\wedge = R\\lim (K \\otimes_A^\\mathbf{L} (A \\xrightarrow{f^n} A))\n$$\nof Lemma \\ref{lemma-derived-completion-koszul}.\nThe natural transformation of functors $K^\\wedge \\to K'$\nis an isomorphism if and only if the $f$-power torsion of $A$ is bounded.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for a principal ideal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0923","source_file":"more-algebra.tex","source_line":26810,"source_end_line":26822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26810-L26822","statement_sha256":"cfa14cb6d581bab3528ed71175e6a3b30f7236b27a8a6dfcef37ec96e91be06e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3484,"rank":3484,"depth":20,"x":577.274,"y":593.241,"cluster":"advanced-algebra"},{"id":"stacks:0H32","tag":"0H32","title":"Derived completion for a principal ideal · Lemma 0H32","summary":"Let A be a ring and let f ∈ A. Let K be an object of D(A). Denote K_n = K ⊗_A^L (A xrightarrowf^n A). For all p ∈ Z there is a commutative diagram xymatrix & 0 & 0 0 ar[r] & widehatH^p(K) ar[r] ar[u] & lim H^p(K_n) ar[r] ar[u] & T_f(H^p + 1(K)) ar[r] & 0 0 ar[r] & H^0(H^p(K)^wedge) ar[r] ar[u] & H^p(K^wedge) ar[r] ar[u] & T_f(H^p + 1(K)) ar[r] ar@=[u] & 0 & R^1lim H^p(K)[f^n] ar[u] ar[r]^≅ & R^1lim H^p - 1(K_n) ar[u] & 0 ar[u] & 0 ar[u] with exact rows and columns where…","statement_latex":"Let $A$ be a ring and let $f \\in A$. Let $K$ be an object of $D(A)$.\nDenote $K_n = K \\otimes_A^\\mathbf{L} (A \\xrightarrow{f^n} A)$.\nFor all $p \\in \\mathbf{Z}$ there is a commutative diagram\n$$\n\\xymatrix{\n& 0 & 0 \\\\\n0 \\ar[r] &\n\\widehat{H^p(K)} \\ar[r] \\ar[u] &\n\\lim H^p(K_n) \\ar[r] \\ar[u] &\nT_f(H^{p + 1}(K)) \\ar[r] &\n0 \\\\\n0 \\ar[r] &\nH^0(H^p(K)^\\wedge) \\ar[r] \\ar[u] &\nH^p(K^\\wedge) \\ar[r] \\ar[u] &\nT_f(H^{p + 1}(K)) \\ar[r] \\ar@{=}[u] &\n0 \\\\\n&\nR^1\\lim H^p(K)[f^n] \\ar[u] \\ar[r]^\\cong &\nR^1\\lim H^{p - 1}(K_n) \\ar[u] \\\\\n& 0 \\ar[u] & 0 \\ar[u]\n}\n$$\nwith exact rows and columns where $\\widehat{H^p(K)} = \\lim H^p(K)/f^nH^p(K)$\nis the usual $f$-adic completion. The left vertical short exact sequence\nand the middle horizontal short exact sequence are taken from\nExample \\ref{example-spectral-sequence-principal}\nThe middle vertical short exact sequence is the one from\nLemma \\ref{lemma-break-long-exact-sequence-modules}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for a principal ideal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H32","source_file":"more-algebra.tex","source_line":26927,"source_end_line":26957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L26927-L26957","statement_sha256":"dbfad79589a18fcafc32a5a2ca3bb7da584132a52ef32169069c5b900b48f233","origin":"The Stacks Project","memory_eligible":false,"source_rank":3485,"rank":3485,"depth":20,"x":825.624,"y":685.69,"cluster":"advanced-algebra"},{"id":"stacks:0CQY","tag":"0CQY","title":"Bhatt · Lemma 0CQY","summary":"Let I be a finitely generated ideal in a ring A. Let M be a derived complete A-module. If M is an I-power torsion module, then I^nM = 0 for some n.","statement_latex":"Let $I$ be a finitely generated ideal in a ring $A$.\nLet $M$ be a derived complete $A$-module. If $M$ is\nan $I$-power torsion module, then $I^nM = 0$ for some $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for a principal ideal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQY","source_file":"more-algebra.tex","source_line":27014,"source_end_line":27019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27014-L27019","statement_sha256":"4fcaa155a8931aa448a6ec3aafbcddaa0f00ade4723244f4d55e228f45c34371","origin":"The Stacks Project","memory_eligible":false,"source_rank":3486,"rank":3486,"depth":6,"x":567.962,"y":758.492,"cluster":"advanced-algebra"},{"id":"stacks:0G3G","tag":"0G3G","title":"Derived completion for a principal ideal · Lemma 0G3G","summary":"Let f ∈ A be an element of a ring. Set J = ⋂ f^nA. Let M be an A-module derived complete with respect to f. Then JM' = 0 where M' = Ker(M → lim M/f^nM). In particular, if A is derived complete then J is an ideal of square zero.","statement_latex":"Let $f \\in A$ be an element of a ring. Set $J = \\bigcap f^nA$.\nLet $M$ be an $A$-module derived complete with respect to $f$.\nThen $JM' = 0$ where $M' = \\Ker(M \\to \\lim M/f^nM)$. In particular,\nif $A$ is derived complete then $J$ is an ideal of square zero.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for a principal ideal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3G","source_file":"more-algebra.tex","source_line":27066,"source_end_line":27072,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27066-L27072","statement_sha256":"44a94009a372a07657e3b0e0a83255cefe53e0e89c9e9bd4742db2d80d78873a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3487,"rank":3487,"depth":3,"x":699.508,"y":558.459,"cluster":"advanced-algebra"},{"id":"stacks:0G3H","tag":"0G3H","title":"Derived completion for a principal ideal · Lemma 0G3H","summary":"Let A be a ring derived complete with respect to an ideal I. Then (A, I) is a henselian pair.","statement_latex":"Let $A$ be a ring derived complete with respect to an ideal $I$.\nThen $(A, I)$ is a henselian pair.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for a principal ideal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3H","source_file":"more-algebra.tex","source_line":27086,"source_end_line":27090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27086-L27090","statement_sha256":"a3a6871887e4018346758bfb3e4f4f4c99b329ff3e452cdfe675a43378cabba7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3488,"rank":3488,"depth":51,"x":763.415,"y":780.765,"cluster":"advanced-algebra"},{"id":"stacks:0G3I","tag":"0G3I","title":"Derived completion for a principal ideal · Lemma 0G3I","summary":"Let A be a ring derived complete with respect to an ideal I. Set J = ⋂ I^n. If I can be generated by r elements then J^N = 0 where N = 2^r.","statement_latex":"Let $A$ be a ring derived complete with respect to an ideal $I$.\nSet $J = \\bigcap I^n$. If $I$ can be generated by $r$ elements\nthen $J^N = 0$ where $N = 2^r$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for a principal ideal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3I","source_file":"more-algebra.tex","source_line":27106,"source_end_line":27111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27106-L27111","statement_sha256":"d36b7f774e143a973bdb33943da9dfbdd40727294178a1abea94626d3574d4b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3489,"rank":3489,"depth":4,"x":537.355,"y":653.01,"cluster":"advanced-algebra"},{"id":"stacks:0G5V","tag":"0G5V","title":"Derived completion for a principal ideal · Lemma 0G5V","summary":"Let A be a reduced ring derived complete with respect to a finitely generated ideal I. Then A is I-adically complete.","statement_latex":"Let $A$ be a reduced ring derived complete with respect to\na finitely generated ideal $I$. Then $A$ is $I$-adically complete.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for a principal ideal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5V","source_file":"more-algebra.tex","source_line":27125,"source_end_line":27129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27125-L27129","statement_sha256":"6336ef6a2079bc7ec74f4f3f101f905f123c17986314fb536293f020cc0012a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3490,"rank":3490,"depth":5,"x":806.981,"y":618.918,"cluster":"advanced-algebra"},{"id":"stacks:0921","tag":"0921","title":"Derived completion for Noetherian rings · Lemma 0921","summary":"In Situation [Tag 0BKC]. If A is Noetherian, then the pro-objects (K_n^bullet) and (A/(f_1^n, …, f_r^n)) of D(A) are isomorphic.","statement_latex":"In Situation \\ref{situation-koszul}. If $A$ is Noetherian, then the pro-objects\n$\\{K_n^\\bullet\\}$ and $\\{A/(f_1^n, \\ldots, f_r^n)\\}$ of $D(A)$ are\nisomorphic\\footnote{In particular, for every $n$ there exists an $m \\geq n$\nsuch that $K_m^\\bullet \\to K_n^\\bullet$ factors through the map\n$K_m^\\bullet \\to A/(f_1^m, \\ldots, f_r^m)$.}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0921","source_file":"more-algebra.tex","source_line":27171,"source_end_line":27178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27171-L27178","statement_sha256":"f9306c16e2e9b9b16a3e2fa4306c123141800a683cbb504c473f932a8da32a11","origin":"The Stacks Project","memory_eligible":false,"source_rank":3491,"rank":3491,"depth":7,"x":635.455,"y":797.178,"cluster":"advanced-algebra"},{"id":"stacks:0922","tag":"0922","title":"Derived completion for Noetherian rings · Proposition 0922","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. The functor which sends K ∈ D(A) to the derived limit K' = Rlim( K ⊗_A^L A/I^n ) is the left adjoint to the inclusion functor D_comp(A) → D(A) constructed in Lemma [Tag 091V].","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nThe functor which sends $K \\in D(A)$ to the derived limit\n$K' = R\\lim( K \\otimes_A^\\mathbf{L} A/I^n )$ is the left\nadjoint to the inclusion functor $D_{comp}(A) \\to D(A)$\nconstructed in Lemma \\ref{lemma-derived-completion}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for Noetherian rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0922","source_file":"more-algebra.tex","source_line":27217,"source_end_line":27224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27217-L27224","statement_sha256":"39435fa2c139db3414649678e91196de44bc61e3950e6987671fa7655b62a6df","origin":"The Stacks Project","memory_eligible":false,"source_rank":3492,"rank":3492,"depth":21,"x":618.571,"y":568.238,"cluster":"advanced-algebra"},{"id":"stacks:0EET","tag":"0EET","title":"Derived completion for Noetherian rings · Lemma 0EET","summary":"Let I be an ideal of a Noetherian ring A. Let M be an A-module with derived completion M^wedge. Then there are short exact sequences 0 → R^1lim Tor_i + 1^A(M, A/I^n) → H^-i(M^wedge) → lim Tor_i^A(M, A/I^n) → 0 A similar result holds for M ∈ D^-(A).","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let $M$ be an $A$-module\nwith derived completion $M^\\wedge$. Then there are short exact sequences\n$$\n0 \\to R^1\\lim \\text{Tor}_{i + 1}^A(M, A/I^n) \\to\nH^{-i}(M^\\wedge) \\to \\lim \\text{Tor}_i^A(M, A/I^n) \\to 0\n$$\nA similar result holds for $M \\in D^-(A)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EET","source_file":"more-algebra.tex","source_line":27244,"source_end_line":27253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27244-L27253","statement_sha256":"83f661fa35d7ba175a8236599a6cd9df565395b15641301d11c06dba7e2935dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3493,"rank":3493,"depth":22,"x":815.27,"y":727.59,"cluster":"advanced-algebra"},{"id":"stacks:0A06","tag":"0A06","title":"Derived completion for Noetherian rings · Lemma 0A06","summary":"Let A be a Noetherian ring and I ⊂ A an ideal. Let K be an object of D(A) such that H^n(K) a finite A-module for all n ∈ Z. Then the cohomology modules H^n(K^wedge) of the derived completion are the I-adic completions of the cohomology modules H^n(K).","statement_latex":"Let $A$ be a Noetherian ring and $I \\subset A$ an ideal. Let $K$ be an\nobject of $D(A)$ such that $H^n(K)$ a finite $A$-module for all\n$n \\in \\mathbf{Z}$. Then the cohomology modules $H^n(K^\\wedge)$ of\nthe derived completion are the $I$-adic\ncompletions of the cohomology modules $H^n(K)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A06","source_file":"more-algebra.tex","source_line":27265,"source_end_line":27272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27265-L27272","statement_sha256":"e0b2bff821d3f1fbf09c077890605f60d7b1eb4915df667e10f529d949c09199","origin":"The Stacks Project","memory_eligible":false,"source_rank":3494,"rank":3494,"depth":22,"x":541.884,"y":721.693,"cluster":"advanced-algebra"},{"id":"stacks:09BA","tag":"09BA","title":"Derived completion for Noetherian rings · Lemma 09BA","summary":"Let I be an ideal of a Noetherian ring A. Let M be a derived complete A-module. If M/IM is a finite A/I-module, then M = lim M/I^nM and M is a finite A^wedge-module.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$.\nLet $M$ be a derived complete $A$-module.\nIf $M/IM$ is a finite $A/I$-module, then\n$M = \\lim M/I^nM$ and $M$ is a finite $A^\\wedge$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BA","source_file":"more-algebra.tex","source_line":27291,"source_end_line":27297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27291-L27297","statement_sha256":"841391e5e1e76b08d12c9cb5d46c57b54e91fcc2175400bb9d4ce6ca1cf54b52","origin":"The Stacks Project","memory_eligible":false,"source_rank":3495,"rank":3495,"depth":22,"x":748.365,"y":570.807,"cluster":"advanced-algebra"},{"id":"stacks:0EEU","tag":"0EEU","title":"Derived completion for Noetherian rings · Lemma 0EEU","summary":"Let I be an ideal in a Noetherian ring A. • If M is a finite A-module and N is a flat A-module, then the derived I-adic completion of M ⊗_A N is the usual I-adic completion of M ⊗_A N. • If M is a finite A-module and f ∈ A, then the derived I-adic completion of M_f is the usual I-adic completion of M_f.","statement_latex":"Let $I$ be an ideal in a Noetherian ring $A$.\n\\begin{enumerate}\n\\item If $M$ is a finite $A$-module and $N$ is a flat $A$-module, then the\nderived $I$-adic completion of $M \\otimes_A N$ is the usual\n$I$-adic completion of $M \\otimes_A N$.\n\\item If $M$ is a finite $A$-module and $f \\in A$, then the derived\n$I$-adic completion of $M_f$ is the usual $I$-adic completion\nof $M_f$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEU","source_file":"more-algebra.tex","source_line":27317,"source_end_line":27328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27317-L27328","statement_sha256":"c2e3aa1c300da44d01f98c50d968577902f1056fbcfbf940773f6c00c760a3ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":3496,"rank":3496,"depth":23,"x":717.426,"y":799.395,"cluster":"advanced-algebra"},{"id":"stacks:0EEV","tag":"0EEV","title":"Derived completion for Noetherian rings · Lemma 0EEV","summary":"Let I be an ideal in a Noetherian ring A. Let ^wedge denote derived completion with respect to I. Let K ∈ D^-(A). • If M is a finite A-module, then (K ⊗_A^L M)^wedge = K^wedge ⊗_A^L M. • If L ∈ D(A) is pseudo-coherent, then (K ⊗_A^L L)^wedge = K^wedge ⊗_A^L L.","statement_latex":"Let $I$ be an ideal in a Noetherian ring $A$.\nLet ${}^\\wedge$ denote derived completion with respect to $I$.\nLet $K \\in D^-(A)$.\n\\begin{enumerate}\n\\item If $M$ is a finite $A$-module, then\n$(K \\otimes_A^\\mathbf{L} M)^\\wedge = K^\\wedge \\otimes_A^\\mathbf{L} M$.\n\\item If $L \\in D(A)$ is pseudo-coherent, then\n$(K \\otimes_A^\\mathbf{L} L)^\\wedge = K^\\wedge \\otimes_A^\\mathbf{L} L$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Derived completion for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEV","source_file":"more-algebra.tex","source_line":27346,"source_end_line":27357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27346-L27357","statement_sha256":"d20d999e62a520844d16ae72df5874f1b0f7aa09eeeab812bac76501c4059171","origin":"The Stacks Project","memory_eligible":false,"source_rank":3497,"rank":3497,"depth":24,"x":556.3,"y":613.148,"cluster":"advanced-algebra"},{"id":"stacks:0F7P","tag":"0F7P","title":"An operator introduced by Berthelot and Ogus · Lemma 0F7P","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. Let M^bullet be a complex of f-torsion free A-modules. There is a canonical isomorphism f^i : H^i(M^bullet)/H^i(M^bullet)[f] → H^i(eta_fM^bullet) given by multiplication by f^i.","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be\na complex of $f$-torsion free $A$-modules. There is a canonical isomorphism\n$$\nf^i : H^i(M^\\bullet)/H^i(M^\\bullet)[f] \\longrightarrow H^i(\\eta_fM^\\bullet)\n$$\ngiven by multiplication by $f^i$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7P","source_file":"more-algebra.tex","source_line":27493,"source_end_line":27501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27493-L27501","statement_sha256":"50fbc8c4b834f808f66f8fe5069ea0ce70a1cf59b2865d88ae8d4d15c387c1ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":3498,"rank":3498,"depth":0,"x":825.079,"y":659.09,"cluster":"advanced-algebra"},{"id":"stacks:0F7Q","tag":"0F7Q","title":"An operator introduced by Berthelot and Ogus · Lemma 0F7Q","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. If M^bullet → N^bullet is a quasi-isomorphism of complexes of f-torsion free A-modules, then the induced map eta_fM^bullet → eta_fN^bullet is a quasi-isomorphism too.","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor.\nIf $M^\\bullet \\to N^\\bullet$ is a quasi-isomorphism of complexes of\n$f$-torsion free $A$-modules,\nthen the induced map $\\eta_fM^\\bullet \\to \\eta_fN^\\bullet$\nis a quasi-isomorphism too.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7Q","source_file":"more-algebra.tex","source_line":27517,"source_end_line":27524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27517-L27524","statement_sha256":"ce28ffc3c3bba713df4d6110ed49ed35af08f3444b1d788a2a80fb8a5f8d7f98","origin":"The Stacks Project","memory_eligible":false,"source_rank":3499,"rank":3499,"depth":1,"x":589.772,"y":777.809,"cluster":"advanced-algebra"},{"id":"stacks:0F7R","tag":"0F7R","title":"An operator introduced by Berthelot and Ogus · Lemma 0F7R","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. There is an additive functor Leta_f : D(A) → D(A) such that if M ∈ D(A) is represented by a complex M^bullet of f-torsion free A-modules, then Leta_fM = eta_fM^bullet and similarly for morphisms.","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. There is an additive\nfunctor\\footnote{Beware that this functor isn't exact, i.e.,\ndoes not transform distinguished triangles into distinguished triangles.\nSee Example \\ref{example-eta-not-distinguished}.}\n$L\\eta_f : D(A) \\to D(A)$ such that if $M \\in D(A)$ is\nrepresented by a complex $M^\\bullet$ of $f$-torsion free $A$-modules,\nthen $L\\eta_fM = \\eta_fM^\\bullet$ and similarly for morphisms.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7R","source_file":"more-algebra.tex","source_line":27531,"source_end_line":27540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27531-L27540","statement_sha256":"53d1dcede9a84e92dcad50962230d8fa2e327ab6c91d92736f6e76e4fa30360d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3500,"rank":3500,"depth":15,"x":667.869,"y":556.591,"cluster":"advanced-algebra"},{"id":"stacks:0F7T","tag":"0F7T","title":"An operator introduced by Berthelot and Ogus · Lemma 0F7T","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. Let M^bullet be a complex of f-torsion free A-modules. There is a canonical map of complexes eta_fM^bullet ⊗_A A/fA → H^bullet(M^bullet/f) which is a quasi-isomorphism where the right hand side is the complex ([Tag 0GSQ]).","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be\na complex of $f$-torsion free $A$-modules.\nThere is a canonical map of complexes\n$$\n\\eta_fM^\\bullet \\otimes_A A/fA\n\\longrightarrow\nH^\\bullet(M^\\bullet/f)\n$$\nwhich is a quasi-isomorphism where the right hand side is the complex\n(\\ref{equation-complex-bocksteins}).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7T","source_file":"more-algebra.tex","source_line":27738,"source_end_line":27750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27738-L27750","statement_sha256":"e55ae00b5970e4422e1a407dbe0e6e3d48c241a4eac43de2d4ee3f9bebf1e5bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3501,"rank":3501,"depth":0,"x":788.265,"y":764.176,"cluster":"advanced-algebra"},{"id":"stacks:0F7Y","tag":"0F7Y","title":"An operator introduced by Berthelot and Ogus · Lemma 0F7Y","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. Let M^bullet be a complex of f-torsion free A-modules. For i ∈ Z the following are equivalent • Ker(d^i bmod f^2) surjects onto Ker(d^i bmod f), • β : H^i(M^bullet ⊗_A f^iA/f^i + 1A) → H^i + 1(M^bullet ⊗_A f^i + 1A/f^i + 2A) is zero. These equivalent conditions are implied by the condition H^i + 1(M^bullet)[f] = 0.","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let\n$M^\\bullet$ be a complex of $f$-torsion free $A$-modules.\nFor $i \\in \\mathbf{Z}$ the following are equivalent\n\\begin{enumerate}\n\\item $\\Ker(d^i \\bmod f^2)$ surjects onto $\\Ker(d^i \\bmod f)$,\n\\item $\\beta : H^i(M^\\bullet \\otimes_A f^iA/f^{i + 1}A) \\to\nH^{i + 1}(M^\\bullet \\otimes_A f^{i + 1}A/f^{i + 2}A)$ is zero.\n\\end{enumerate}\nThese equivalent conditions are implied by the condition\n$H^{i + 1}(M^\\bullet)[f] = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7Y","source_file":"more-algebra.tex","source_line":27782,"source_end_line":27794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27782-L27794","statement_sha256":"57b8b0d7b4488c22c72fa6b3ff3fb630bb31911006019c94b4c2cd2813e745eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3502,"rank":3502,"depth":0,"x":532.37,"y":679.361,"cluster":"advanced-algebra"},{"id":"stacks:0F7Z","tag":"0F7Z","title":"An operator introduced by Berthelot and Ogus · Lemma 0F7Z","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. Let M^bullet be a complex of f-torsion free A-modules. If Ker(d^i bmod f^2) surjects onto Ker(d^i bmod f), then the canonical map (1, d^i) : (eta_fM)^i / f(eta_fM)^i → f^iM^i/f^i + 1M^i ⊕ f^i + 1M^i + 1/f^i + 2M^i + 1 identifies the left hand side with a direct sum of submodules of the right hand side.","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let\n$M^\\bullet$ be a complex of $f$-torsion free $A$-modules.\nIf $\\Ker(d^i \\bmod f^2)$ surjects onto $\\Ker(d^i \\bmod f)$,\nthen the canonical map\n$$\n(1, d^i) :\n(\\eta_fM)^i / f(\\eta_fM)^i \\longrightarrow\nf^iM^i/f^{i + 1}M^i \\oplus f^{i + 1}M^{i + 1}/f^{i + 2}M^{i + 1}\n$$\nidentifies the left hand side with a direct sum of submodules of\nthe right hand side.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7Z","source_file":"more-algebra.tex","source_line":27806,"source_end_line":27819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27806-L27819","statement_sha256":"5db3eb969bfb82822ae851568b1cc64b5444167b72c60eab7b5c9964a5ce91c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3503,"rank":3503,"depth":0,"x":789.451,"y":596.643,"cluster":"advanced-algebra"},{"id":"stacks:0F7U","tag":"0F7U","title":"An operator introduced by Berthelot and Ogus · Lemma 0F7U","summary":"Let A be a ring and let f, g ∈ A be nonzerodivisors. Let M^bullet be a complex of A-modules such that fg is a nonzerodivisor on all M^i. Then eta_feta_gM^bullet = eta_fgM^bullet.","statement_latex":"Let $A$ be a ring and let $f, g \\in A$ be nonzerodivisors. Let $M^\\bullet$ be\na complex of $A$-modules such that $fg$ is a nonzerodivisor on all $M^i$.\nThen $\\eta_f\\eta_gM^\\bullet = \\eta_{fg}M^\\bullet$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7U","source_file":"more-algebra.tex","source_line":27839,"source_end_line":27844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27839-L27844","statement_sha256":"155c18a624dc3632cf36e18e51b054a95c98dc2d2102a7c0c949cfce5cc5e981","origin":"The Stacks Project","memory_eligible":false,"source_rank":3504,"rank":3504,"depth":0,"x":666.316,"y":803.659,"cluster":"advanced-algebra"},{"id":"stacks:0GSR","tag":"0GSR","title":"An operator introduced by Berthelot and Ogus · Lemma 0GSR","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. Let A → B be a flat ring map and let g ∈ B the image of f. Let M^bullet be a complex of f-torsion free A-modules. Then g is a nonzerodivisor, M^bullet ⊗_A B is a complex of g-torsion free modules, and eta_fM^bullet ⊗_A B = eta_g(M^bullet ⊗_A B).","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let\n$A \\to B$ be a flat ring map and let $g \\in B$ the image of $f$.\nLet $M^\\bullet$ be a complex of $f$-torsion free $A$-modules.\nThen $g$ is a nonzerodivisor, $M^\\bullet \\otimes_A B$\nis a complex of $g$-torsion free modules, and\n$\\eta_fM^\\bullet \\otimes_A B = \\eta_g(M^\\bullet \\otimes_A B)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSR","source_file":"more-algebra.tex","source_line":27851,"source_end_line":27859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27851-L27859","statement_sha256":"06fbc3b3b7c5437a13a84f249a6555f4fb3f3dbe6a6a9f0fcbbacc1d99927114","origin":"The Stacks Project","memory_eligible":false,"source_rank":3505,"rank":3505,"depth":0,"x":590.584,"y":580.981,"cluster":"advanced-algebra"},{"id":"stacks:0GSS","tag":"0GSS","title":"Perfect complexes and the eta operator · Lemma 0GSS","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. Let M^bullet and N^bullet be two bounded complexes of finite free A-modules representing the same object of D(A). Then f^m I_i(M^bullet, f) = f^n I_i(N^bullet, f) as ideals of A for integers n, m ≥ 0 such that m + ∑_j ≥ i (-1)^j - irk(M^j) = n + ∑_j ≥ i (-1)^j - irk(N^j)","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor.\nLet $M^\\bullet$ and $N^\\bullet$ be two bounded complexes\nof finite free $A$-modules representing the same object of $D(A)$.\nThen\n$$\nf^m I_i(M^\\bullet, f) = f^n I_i(N^\\bullet, f)\n$$\nas ideals of $A$ for integers $n, m \\geq 0$ such that\n$$\nm + \\sum\\nolimits_{j \\geq i} (-1)^{j - i}rk(M^j) =\nn + \\sum\\nolimits_{j \\geq i} (-1)^{j - i}rk(N^j)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and the eta operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSS","source_file":"more-algebra.tex","source_line":27892,"source_end_line":27906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L27892-L27906","statement_sha256":"cf6dfd3ee1f3384834a0642fa0035cc737dac09d4338958926341f0f329f0e01","origin":"The Stacks Project","memory_eligible":false,"source_rank":3506,"rank":3506,"depth":13,"x":825.666,"y":702.294,"cluster":"advanced-algebra"},{"id":"stacks:0GST","tag":"0GST","title":"Perfect complexes and the eta operator · Lemma 0GST","summary":"Let f ∈ A be a nonzerodivisor of a ring A. Let u ∈ A be a unit. Let M^bullet be a bounded complex of finite free A-modules. Then I_i(M^bullet, f) = I_i(M^bullet, uf).","statement_latex":"Let $f \\in A$ be a nonzerodivisor of a ring $A$. Let $u \\in A$ be a unit. Let\n$M^\\bullet$ be a bounded complex of finite free $A$-modules. Then\n$I_i(M^\\bullet, f) = I_i(M^\\bullet, uf)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and the eta operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GST","source_file":"more-algebra.tex","source_line":28001,"source_end_line":28006,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28001-L28006","statement_sha256":"a5c5b5fad0461ffe3be31835d864d3758ef87998e68e87327bc57cc2e958ac7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3507,"rank":3507,"depth":0,"x":554.571,"y":746.262,"cluster":"advanced-algebra"},{"id":"stacks:0GSU","tag":"0GSU","title":"Perfect complexes and the eta operator · Lemma 0GSU","summary":"Let A → B be a ring map. Let f ∈ A be a nonzerodivisor. Let M^bullet be a bounded complex of finite free A-modules. Assume f maps to a nonzerodivisor g in B. Then I_i(M^bullet, f)B = I_i(M^bullet ⊗_A B, g).","statement_latex":"Let $A \\to B$ be a ring map. Let $f \\in A$ be a nonzerodivisor. Let\n$M^\\bullet$ be a bounded complex of finite free $A$-modules. Assume\n$f$ maps to a nonzerodivisor $g$ in $B$. Then\n$I_i(M^\\bullet, f)B = I_i(M^\\bullet \\otimes_A B, g)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and the eta operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSU","source_file":"more-algebra.tex","source_line":28012,"source_end_line":28018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28012-L28018","statement_sha256":"c135f1a669bf3b338904489a2dfa3ed55c3be6926fc6568e07ff02fd40af9e07","origin":"The Stacks Project","memory_eligible":false,"source_rank":3508,"rank":3508,"depth":0,"x":719.231,"y":559.884,"cluster":"advanced-algebra"},{"id":"stacks:0GSV","tag":"0GSV","title":"Perfect complexes and the eta operator · Lemma 0GSV","summary":"Let A be a ring, let p ⊂ A be a prime ideal, and let f ∈ A be a nonzerodivisor. Let M^bullet be a bounded complex of finite free A-modules. If H^i(M^bullet)_ p is free for all i, then I_i(M^bullet, f)_ p is a principal ideal and in fact generated by a power of f for all i.","statement_latex":"Let $A$ be a ring, let $\\mathfrak p \\subset A$ be a prime ideal, and\nlet $f \\in A$ be a nonzerodivisor. Let $M^\\bullet$ be a bounded complex\nof finite free $A$-modules. If $H^i(M^\\bullet)_\\mathfrak p$\nis free for all $i$, then $I_i(M^\\bullet, f)_\\mathfrak p$ is a\nprincipal ideal and in fact generated by a power of $f$ for all $i$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and the eta operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSV","source_file":"more-algebra.tex","source_line":28026,"source_end_line":28033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28026-L28033","statement_sha256":"5a25f363376fa957d3d0a4a98a5da7468f46f0bb43c4798c0a1117ef9fdd5d46","origin":"The Stacks Project","memory_eligible":false,"source_rank":3509,"rank":3509,"depth":14,"x":747.714,"y":790.91,"cluster":"advanced-algebra"},{"id":"stacks:0F7W","tag":"0F7W","title":"Perfect complexes and the eta operator · Lemma 0F7W","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. Let M^bullet be a bounded complex of finite free A-modules. Assume I_i(M^bullet, f) is a principal ideal. Then (eta_fM)^i is locally free of rank r_i and the map (1, d^i) : (eta_fM)^i → f^iM^i ⊕ f^i + 1M^i + 1 is the inclusion of a direct summand.","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let\n$M^\\bullet$ be a bounded complex of finite free $A$-modules. Assume\n$I_i(M^\\bullet, f)$ is a principal ideal.\nThen $(\\eta_fM)^i$ is locally free of rank $r_i$ and the map\n$(1, d^i) : (\\eta_fM)^i \\to f^iM^i \\oplus f^{i + 1}M^{i + 1}$\nis the inclusion of a direct summand.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and the eta operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7W","source_file":"more-algebra.tex","source_line":28049,"source_end_line":28057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28049-L28057","statement_sha256":"c1f8669267038e8fb6a271071cfe8c02fbfc2eda8bc3b8f9bfac7d1cd14301ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":3510,"rank":3510,"depth":6,"x":540.78,"y":636.608,"cluster":"advanced-algebra"},{"id":"stacks:0F7X","tag":"0F7X","title":"Perfect complexes and the eta operator · Lemma 0F7X","summary":"Let A → B be a ring map. Let f ∈ A be a nonzerodivisor. Let M^bullet be a bounded complex of finite free A-modules. Assume f maps to a nonzerodivisor g in B and I_i(M^bullet, f) is a principal ideal for all i ∈ Z. Then there is a canonical isomorphism eta_fM^bullet ⊗_A B = eta_g(M^bullet ⊗_A B).","statement_latex":"Let $A \\to B$ be a ring map. Let $f \\in A$ be a nonzerodivisor. Let\n$M^\\bullet$ be a bounded complex of finite free $A$-modules. Assume\n$f$ maps to a nonzerodivisor $g$ in $B$ and $I_i(M^\\bullet, f)$\nis a principal ideal for all $i \\in \\mathbf{Z}$.\nThen there is a canonical isomorphism\n$\\eta_fM^\\bullet \\otimes_A B = \\eta_g(M^\\bullet \\otimes_A B)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and the eta operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7X","source_file":"more-algebra.tex","source_line":28104,"source_end_line":28112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28104-L28112","statement_sha256":"be94e9ef38ef649ca2759190a6a5bd47d1dfaba4d06f7e03260caa06c9ff6f00","origin":"The Stacks Project","memory_eligible":false,"source_rank":3511,"rank":3511,"depth":7,"x":817.65,"y":632.967,"cluster":"advanced-algebra"},{"id":"stacks:0F80","tag":"0F80","title":"Perfect complexes and the eta operator · Lemma 0F80","summary":"Let A be a ring. Let M, N_1, N_2 be finite projective A-modules. Let s : M → N_1 ⊕ N_2 be a split injection. There exists a finitely generated ideal J ⊂ A with the following property: a ring map A → B factors through A/J if and only if s ⊗ id_B identifies M ⊗_A B with a direct sum of submodules of N_1 ⊗_A B ⊕ N_2 ⊗_A B.","statement_latex":"Let $A$ be a ring. Let $M$, $N_1$, $N_2$ be finite projective $A$-modules.\nLet $s : M \\to N_1 \\oplus N_2$ be a split injection. There exists a\nfinitely generated ideal $J \\subset A$ with the following property:\na ring map $A \\to B$ factors through $A/J$ if and only if\n$s \\otimes \\text{id}_B$\nidentifies $M \\otimes_A B$ with a direct sum of submodules of\n$N_1 \\otimes_A B \\oplus N_2 \\otimes_A B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and the eta operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F80","source_file":"more-algebra.tex","source_line":28126,"source_end_line":28135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28126-L28135","statement_sha256":"628273275dd2553bb24aa0f2de83281d7e4e106372b50a7eb17510d77be2e1ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":3512,"rank":3512,"depth":0,"x":616.277,"y":792.866,"cluster":"advanced-algebra"},{"id":"stacks:0GSW","tag":"0GSW","title":"Perfect complexes and the eta operator · Lemma 0GSW","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. Let M^bullet and N^bullet be two bounded complexes of finite free A-modules representing the same object in D(A). Assume I_i(M^bullet, f) is a principal ideal for all i ∈ Z. Then J_i(M^bullet, f) = J_i(N^bullet, f) as ideals in A/fA.","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let\n$M^\\bullet$ and $N^\\bullet$ be two bounded complexes of finite\nfree $A$-modules representing the same object in $D(A)$.\nAssume $I_i(M^\\bullet, f)$ is a principal ideal\nfor all $i \\in \\mathbf{Z}$. Then $J_i(M^\\bullet, f) = J_i(N^\\bullet, f)$\nas ideals in $A/fA$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and the eta operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSW","source_file":"more-algebra.tex","source_line":28163,"source_end_line":28171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28163-L28171","statement_sha256":"2b77f6f2d6d2d5ac4fafc860f94621542b5d18ff15fb8a6b7d1740178fc5adcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3513,"rank":3513,"depth":14,"x":636.193,"y":560.533,"cluster":"advanced-algebra"},{"id":"stacks:0F81","tag":"0F81","title":"Perfect complexes and the eta operator · Lemma 0F81","summary":"Let A be a ring and let f ∈ A be a nonzerodivisor. Let M^bullet be a bounded complex of finite free A-modules. Assume I_i(M^bullet, f) is a principal ideal for all i ∈ Z. Consider the ideal J(M^bullet, f) = ∑_i J_i(M^bullet, f) of A/fA. Consider the set of prime ideals E & = (f ∈ p ⊂ A mid Ker(d^i bmod f^2)_ p surjects onto Ker(d^i bmod f)_ p for all i ∈ Z) & = (f ∈ p ⊂ A mid the localizations β_ p of the Bockstein operators are zero) Then we have • J(M^bullet, f) is…","statement_latex":"Let $A$ be a ring and let $f \\in A$ be a nonzerodivisor. Let\n$M^\\bullet$ be a bounded complex of finite free $A$-modules.\nAssume $I_i(M^\\bullet, f)$ is a principal ideal\nfor all $i \\in \\mathbf{Z}$. Consider the ideal\n$J(M^\\bullet, f) = \\sum_i J_i(M^\\bullet, f)$ of $A/fA$.\nConsider the set of prime ideals\n\\begin{align*}\nE\n& =\n\\{f \\in \\mathfrak p \\subset A \\mid\n\\Ker(d^i \\bmod f^2)_\\mathfrak p \\text{ surjects onto }\n\\Ker(d^i \\bmod f)_\\mathfrak p \\text{ for all }i \\in \\mathbf{Z}\\} \\\\\n& =\n\\{f \\in \\mathfrak p \\subset A \\mid\n\\text{the localizations }\\beta_\\mathfrak p\n\\text{ of the Bockstein operators are zero}\\}\n\\end{align*}\nThen we have\n\\begin{enumerate}\n\\item $J(M^\\bullet, f)$ is finitely generated,\n\\item $A/fA \\to C = (A/fA)/J(M^\\bullet, f)$\nis surjective of finite presentation,\n\\item $J(M^\\bullet, f)_\\mathfrak p = 0$ for $\\mathfrak p \\in E$,\n\\item if $f \\in \\mathfrak p$ and\n$H^i(M^\\bullet)_\\mathfrak p$ is free for all $i \\in \\mathbf{Z}$,\nthen $\\mathfrak p \\in E$, and\n\\item the cohomology modules of\n$\\eta_f M^\\bullet \\otimes_A C$ are finite locally free $C$-modules.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and the eta operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F81","source_file":"more-algebra.tex","source_line":28197,"source_end_line":28228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28197-L28228","statement_sha256":"ecb56665844b80ed96b55b785e62e88b587e6679a95012513d46d12277d980c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3514,"rank":3514,"depth":7,"x":808.464,"y":743.281,"cluster":"advanced-algebra"},{"id":"stacks:0CQF","tag":"0CQF","title":"Taking limits of complexes · Lemma 0CQF","summary":"Let A = lim A_n be a limit of an inverse system (A_n) of rings. Suppose given K_n ∈ D(A_n) and maps K_n + 1 → K_n in D(A_n + 1). Assume • the transition maps A_n + 1 → A_n are surjective with locally nilpotent kernels, • either all K_n are pseudo-coherent or there exists an integer n_0 such that K_n_0 is pseudo-coherent and the kernels of A_n + 1 → A_n are nilpotent ideals for n ≥ n_0, • the maps induce isomorphisms K_n + 1 ⊗_A_n + 1^L A_n → K_n. Then K = Rlim K_n is a…","statement_latex":"Let $A = \\lim A_n$ be a limit of an inverse system $(A_n)$ of rings.\nSuppose given $K_n \\in D(A_n)$ and maps $K_{n + 1} \\to K_n$\nin $D(A_{n + 1})$. Assume\n\\begin{enumerate}\n\\item the transition maps $A_{n + 1} \\to A_n$ are surjective\nwith locally nilpotent kernels,\n\\item either all $K_n$ are pseudo-coherent or there exists an integer\n$n_0$ such that $K_{n_0}$ is pseudo-coherent and the kernels of\n$A_{n + 1} \\to A_n$ are nilpotent ideals for $n \\geq n_0$,\n\\item the maps induce isomorphisms\n$K_{n + 1} \\otimes_{A_{n + 1}}^\\mathbf{L} A_n \\to K_n$.\n\\end{enumerate}\nThen $K = R\\lim K_n$ is a pseudo-coherent object of $D(A)$\nand $K \\otimes_A^\\mathbf{L} A_n \\to K_n$ is an isomorphism for all $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Taking limits of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQF","source_file":"more-algebra.tex","source_line":28335,"source_end_line":28351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28335-L28351","statement_sha256":"7f36013d147cd2414ef915700e8d871e76652fe9e43b4d17a84d0e3010145ee7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3515,"rank":3515,"depth":18,"x":534.282,"y":706.25,"cluster":"advanced-algebra"},{"id":"stacks:09AV","tag":"09AV","title":"Taking limits of complexes · Lemma 09AV","summary":"Let A be a ring and I ⊂ A an ideal. Suppose given K_n ∈ D(A/I^n) and maps K_n + 1 → K_n in D(A/I^n + 1). Assume • A is I-adically complete, • K_1 is pseudo-coherent, and • the maps induce isomorphisms K_n + 1 ⊗_A/I^n + 1^L A/I^n → K_n. Then K = Rlim K_n is a pseudo-coherent, derived complete object of D(A) and K ⊗_A^L A/I^n → K_n is an isomorphism for all n.","statement_latex":"Let $A$ be a ring and $I \\subset A$ an ideal.\nSuppose given $K_n \\in D(A/I^n)$ and maps $K_{n + 1} \\to K_n$\nin $D(A/I^{n + 1})$. Assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete,\n\\item $K_1$ is pseudo-coherent, and\n\\item the maps induce isomorphisms\n$K_{n + 1} \\otimes_{A/I^{n + 1}}^\\mathbf{L} A/I^n \\to K_n$.\n\\end{enumerate}\nThen $K = R\\lim K_n$ is a pseudo-coherent, derived complete object of $D(A)$\nand $K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Taking limits of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AV","source_file":"more-algebra.tex","source_line":28377,"source_end_line":28390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28377-L28390","statement_sha256":"abb68bd104847de54121261e6a24e7f6de9bb7823935a4d21bba0ecbb77ddc36","origin":"The Stacks Project","memory_eligible":false,"source_rank":3516,"rank":3516,"depth":19,"x":766.401,"y":577.889,"cluster":"advanced-algebra"},{"id":"stacks:0CQG","tag":"0CQG","title":"Taking limits of complexes · Lemma 0CQG","summary":"[Bhatt-Algebraize] Let A = lim A_n be a limit of an inverse system (A_n) of rings. Suppose given K_n ∈ D(A_n) and maps K_n + 1 → K_n in D(A_n + 1). Assume • the transition maps A_n + 1 → A_n are surjective with locally nilpotent kernels, • either all K_n are perfect or there exists an integer n_0 such that K_n_0 is perfect and the kernels A_n + 1 → A_n are nilpotent for n ≥ n_0, and • the maps induce isomorphisms K_n + 1 ⊗_A_n + 1^L A_n → K_n. Then K = Rlim K_n is a…","statement_latex":"\\begin{reference}\n\\cite[Lemma 4.2]{Bhatt-Algebraize}\n\\end{reference}\nLet $A = \\lim A_n$ be a limit of an inverse system $(A_n)$ of rings.\nSuppose given $K_n \\in D(A_n)$ and maps $K_{n + 1} \\to K_n$\nin $D(A_{n + 1})$. Assume\n\\begin{enumerate}\n\\item the transition maps $A_{n + 1} \\to A_n$ are surjective with\nlocally nilpotent kernels,\n\\item either all $K_n$ are perfect or there exists an integer $n_0$\nsuch that $K_{n_0}$ is perfect and the kernels $A_{n + 1} \\to A_n$\nare nilpotent for $n \\geq n_0$, and\n\\item the maps induce isomorphisms\n$K_{n + 1} \\otimes_{A_{n + 1}}^\\mathbf{L} A_n \\to K_n$.\n\\end{enumerate}\nThen $K = R\\lim K_n$ is a perfect object of $D(A)$\nand $K \\otimes_A^\\mathbf{L} A_n \\to K_n$ is an isomorphism for all $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Taking limits of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQG","source_file":"more-algebra.tex","source_line":28400,"source_end_line":28419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28400-L28419","statement_sha256":"cc3372b3c38ba00b4f4eaf73969af7084faa19b4d41d6499a63ed698dfab38fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":3517,"rank":3517,"depth":19,"x":698.417,"y":804.408,"cluster":"advanced-algebra"},{"id":"stacks:09AW","tag":"09AW","title":"Taking limits of complexes · Lemma 09AW","summary":"Let A be a ring and I ⊂ A an ideal. Suppose given K_n ∈ D(A/I^n) and maps K_n + 1 → K_n in D(A/I^n + 1). Assume • A is I-adically complete, • K_1 is a perfect object, and • the maps induce isomorphisms K_n + 1 ⊗_A/I^n + 1^L A/I^n → K_n. Then K = Rlim K_n is a perfect, derived complete object of D(A) and K ⊗_A^L A/I^n → K_n is an isomorphism for all n.","statement_latex":"Let $A$ be a ring and $I \\subset A$ an ideal.\nSuppose given $K_n \\in D(A/I^n)$ and maps $K_{n + 1} \\to K_n$\nin $D(A/I^{n + 1})$. Assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete,\n\\item $K_1$ is a perfect object, and\n\\item the maps induce isomorphisms\n$K_{n + 1} \\otimes_{A/I^{n + 1}}^\\mathbf{L} A/I^n \\to K_n$.\n\\end{enumerate}\nThen $K = R\\lim K_n$ is a perfect, derived complete object of $D(A)$\nand $K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Taking limits of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AW","source_file":"more-algebra.tex","source_line":28446,"source_end_line":28459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28446-L28459","statement_sha256":"3e878a8b741ee4c88555f15b2a361bf49e379aa018433174398ddb785049e8be","origin":"The Stacks Project","memory_eligible":false,"source_rank":3518,"rank":3518,"depth":20,"x":566.295,"y":598.657,"cluster":"advanced-algebra"},{"id":"stacks:09AU","tag":"09AU","title":"Taking limits of complexes · Lemma 09AU","summary":"Let A be a ring and I ⊂ A an ideal. Suppose given K_n ∈ D(A/I^n) and maps K_n + 1 → K_n in D(A/I^n + 1). If • A is Noetherian, • K_1 is bounded above, and • the maps induce isomorphisms K_n + 1 ⊗_A/I^n + 1^L A/I^n → K_n, then K = Rlim K_n is a derived complete object of D^-(A) and K ⊗_A^L A/I^n → K_n is an isomorphism for all n.","statement_latex":"Let $A$ be a ring and $I \\subset A$ an ideal. Suppose\ngiven $K_n \\in D(A/I^n)$ and maps $K_{n + 1} \\to K_n$ in\n$D(A/I^{n + 1})$. If\n\\begin{enumerate}\n\\item $A$ is Noetherian,\n\\item $K_1$ is bounded above, and\n\\item the maps induce isomorphisms\n$K_{n + 1} \\otimes_{A/I^{n + 1}}^\\mathbf{L} A/I^n \\to K_n$,\n\\end{enumerate}\nthen $K = R\\lim K_n$ is a derived complete object of $D^-(A)$ and\n$K \\otimes_A^\\mathbf{L} A/I^n \\to K_n$ is an isomorphism for all $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Taking limits of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AU","source_file":"more-algebra.tex","source_line":28470,"source_end_line":28483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28470-L28483","statement_sha256":"ece328a2de11fe9ca2af86855c90721c2916ad7eb8876c5c0f49aee289912a3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3519,"rank":3519,"depth":18,"x":829.361,"y":675.459,"cluster":"advanced-algebra"},{"id":"stacks:0EGS","tag":"0EGS","title":"Koll\\'ar-Kov\\'acs · Lemma 0EGS","summary":"Email from Kovacs of 23/02/2018. Let I be an ideal of a Noetherian ring A. Let K ∈ D(A). Set K_n = K ⊗_A^L A/I^n. Assume for all i ∈ Z we have • H^i(K) is a finite A-module, and • the system H^i(K_n) satisfies Mittag-Leffler. Then lim H^i(K)/I^nH^i(K) is equal to lim H^i(K_n) for all i ∈ Z.","statement_latex":"\\begin{reference}\nEmail from Kovacs of 23/02/2018.\n\\end{reference}\nLet $I$ be an ideal of a Noetherian ring $A$. Let $K \\in D(A)$.\nSet $K_n = K \\otimes_A^\\mathbf{L} A/I^n$. Assume for all\n$i \\in \\mathbf{Z}$ we have\n\\begin{enumerate}\n\\item $H^i(K)$ is a finite $A$-module, and\n\\item the system $H^i(K_n)$ satisfies Mittag-Leffler.\n\\end{enumerate}\nThen $\\lim H^i(K)/I^nH^i(K)$ is equal to $\\lim H^i(K_n)$ for all\n$i \\in \\mathbf{Z}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Taking limits of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EGS","source_file":"more-algebra.tex","source_line":28522,"source_end_line":28536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28522-L28536","statement_sha256":"8e603f5fd432b8c874dacb9e9596854105eaa82dbc253b5e602fc50cb4525e0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3520,"rank":3520,"depth":23,"x":573.441,"y":768.161,"cluster":"advanced-algebra"},{"id":"stacks:0A68","tag":"0A68","title":"Some evaluation maps · Lemma 0A68","summary":"Let R be a ring. Let K, L, M be objects of D(R). the map RHom_R(L, M) ⊗_R^L K → RHom_R(RHom_R(K, L), M) of Lemma [Tag 0A67] is an isomorphism in the following two cases • K perfect, or • K is pseudo-coherent, L ∈ D^+(R), and M finite injective dimension.","statement_latex":"Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$.\nthe map\n$$\nR\\Hom_R(L, M) \\otimes_R^\\mathbf{L} K \\longrightarrow R\\Hom_R(R\\Hom_R(K, L), M)\n$$\nof Lemma \\ref{lemma-internal-hom-evaluate} is an isomorphism\nin the following two cases\n\\begin{enumerate}\n\\item $K$ perfect, or\n\\item $K$ is pseudo-coherent, $L \\in D^+(R)$, and $M$ finite injective\ndimension.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Some evaluation maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A68","source_file":"more-algebra.tex","source_line":28561,"source_end_line":28575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28561-L28575","statement_sha256":"4122599c880ba3cc2fe2c08c53e8bd91babb52bd2f066cd864ee35454ad20c6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3521,"rank":3521,"depth":2,"x":687.684,"y":554.441,"cluster":"advanced-algebra"},{"id":"stacks:0A69","tag":"0A69","title":"Some evaluation maps · Lemma 0A69","summary":"Let R be a ring. Let K, L, M be objects of D(R). the map RHom_R(L, M) ⊗_R^L K → RHom_R(RHom_R(K, L), M) of Lemma [Tag 0A67] is an isomorphism if the following three conditions are satisfied • L, M have finite injective dimension, • RHom_R(L, M) has finite tor dimension, • for every n ∈ Z the truncation τ_≤ nK is pseudo-coherent","statement_latex":"Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$.\nthe map\n$$\nR\\Hom_R(L, M) \\otimes_R^\\mathbf{L} K \\longrightarrow R\\Hom_R(R\\Hom_R(K, L), M)\n$$\nof Lemma \\ref{lemma-internal-hom-evaluate} is an isomorphism\nif the following three conditions are satisfied\n\\begin{enumerate}\n\\item $L, M$ have finite injective dimension,\n\\item $R\\Hom_R(L, M)$ has finite tor dimension,\n\\item for every $n \\in \\mathbf{Z}$ the truncation $\\tau_{\\leq n}K$\nis pseudo-coherent\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Some evaluation maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A69","source_file":"more-algebra.tex","source_line":28616,"source_end_line":28631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28616-L28631","statement_sha256":"16491382b5424ca3de9019a5eb66360c282a52eee8fe74959c31ed8d6226f576","origin":"The Stacks Project","memory_eligible":false,"source_rank":3522,"rank":3522,"depth":3,"x":775.371,"y":777.012,"cluster":"advanced-algebra"},{"id":"stacks:0ATK","tag":"0ATK","title":"Some evaluation maps · Lemma 0ATK","summary":"Let R be a ring. Let K, L, M be objects of D(R). The map K ⊗_R^L RHom_R(M, L) → RHom_R(M, K ⊗_R^L L) of Lemma [Tag 0BYN] is an isomorphism in the following cases • M perfect, or • K is perfect, or • M is pseudo-coherent, L ∈ D^+(R), and K has tor amplitude in [a, ∞].","statement_latex":"Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. The map\n$$\nK \\otimes_R^\\mathbf{L} R\\Hom_R(M, L) \\longrightarrow\nR\\Hom_R(M, K \\otimes_R^\\mathbf{L} L)\n$$\nof Lemma \\ref{lemma-internal-hom-diagonal-better}\nis an isomorphism in the following cases\n\\begin{enumerate}\n\\item $M$ perfect, or\n\\item $K$ is perfect, or\n\\item $M$ is pseudo-coherent, $L \\in D^+(R)$, and $K$ has\ntor amplitude in $[a, \\infty]$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Some evaluation maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATK","source_file":"more-algebra.tex","source_line":28652,"source_end_line":28667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28652-L28667","statement_sha256":"7b3441fc1682323b8f4b71eea08afdc395373cd2d659b46647a48ffb5b99e470","origin":"The Stacks Project","memory_eligible":false,"source_rank":3523,"rank":3523,"depth":16,"x":531.558,"y":662.569,"cluster":"advanced-algebra"},{"id":"stacks:0BYQ","tag":"0BYQ","title":"Some evaluation maps · Lemma 0BYQ","summary":"Let R be a ring. Let P^bullet be a bounded above complex of projective R-modules. Let K^bullet be a K-flat complex of R-modules. If P^bullet is a perfect object of D(R), then Hom^bullet(P^bullet, K^bullet) is K-flat and represents RHom_R(P^bullet, K^bullet).","statement_latex":"Let $R$ be a ring. Let $P^\\bullet$ be a bounded above complex\nof projective $R$-modules. Let $K^\\bullet$ be a K-flat complex\nof $R$-modules. If $P^\\bullet$ is a perfect object of $D(R)$,\nthen $\\Hom^\\bullet(P^\\bullet, K^\\bullet)$ is K-flat and\nrepresents $R\\Hom_R(P^\\bullet, K^\\bullet)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Some evaluation maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYQ","source_file":"more-algebra.tex","source_line":28710,"source_end_line":28717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28710-L28717","statement_sha256":"6fa1341e9ad0d35edcd33ad0a10073d7e70cf3ea9371c7b3e14b16bb6e225490","origin":"The Stacks Project","memory_eligible":false,"source_rank":3524,"rank":3524,"depth":7,"x":803.562,"y":608.575,"cluster":"advanced-algebra"},{"id":"stacks:0E1W","tag":"0E1W","title":"Base change for derived hom · Lemma 0E1W","summary":"Let R → R' be a ring map. For K ∈ D(R) and M ∈ D(R') there is a canonical isomorphism RHom_R(K, M) = RHom_R'(K ⊗_R^L R', M)","statement_latex":"Let $R \\to R'$ be a ring map. For $K \\in D(R)$ and\n$M \\in D(R')$ there is a canonical isomorphism\n$$\nR\\Hom_R(K, M) = R\\Hom_{R'}(K \\otimes_R^\\mathbf{L} R', M)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Base change for derived hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1W","source_file":"more-algebra.tex","source_line":28762,"source_end_line":28769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28762-L28769","statement_sha256":"2e5ada155be9a51afe42a4857a7b617142a07c20ae7746ab3cd20204671697bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3525,"rank":3525,"depth":18,"x":646.302,"y":802.865,"cluster":"advanced-algebra"},{"id":"stacks:0A6A","tag":"0A6A","title":"Base change for derived hom · Lemma 0A6A","summary":"Let R → R' be a ring map. Let K, M ∈ D(R). The map ([Tag 0E1X]) RHom_R(K, M) ⊗_R^L R' → RHom_R'(K ⊗_R^L R', M ⊗_R^L R') is an isomorphism in D(R') in the following cases • K is perfect, • R' is perfect as an R-module, • R → R' is flat, K is pseudo-coherent, and M ∈ D^+(R), or • R' has finite tor dimension as an R-module, K is pseudo-coherent, and M ∈ D^+(R)","statement_latex":"Let $R \\to R'$ be a ring map. Let $K, M \\in D(R)$. The map\n(\\ref{equation-base-change-RHom})\n$$\nR\\Hom_R(K, M) \\otimes_R^\\mathbf{L} R'\n\\longrightarrow\nR\\Hom_{R'}(K \\otimes_R^\\mathbf{L} R', M \\otimes_R^\\mathbf{L} R')\n$$\nis an isomorphism in $D(R')$ in the following cases\n\\begin{enumerate}\n\\item $K$ is perfect,\n\\item $R'$ is perfect as an $R$-module,\n\\item $R \\to R'$ is flat, $K$ is pseudo-coherent, and $M \\in D^{+}(R)$, or\n\\item $R'$ has finite tor dimension as an $R$-module,\n$K$ is pseudo-coherent, and $M \\in D^{+}(R)$\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Base change for derived hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6A","source_file":"more-algebra.tex","source_line":28819,"source_end_line":28836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28819-L28836","statement_sha256":"34715ddc3897eea790a42827f7d72799ea507a936873b4ecbd52996d6718e9eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3526,"rank":3526,"depth":19,"x":605.993,"y":570.205,"cluster":"advanced-algebra"},{"id":"stacks:0EGU","tag":"0EGU","title":"Systems of modules · Lemma 0EGU","summary":"Let I be an ideal of a Noetherian ring A. Let K xrightarrowα L xrightarrowβ M be a complex of finite A-modules. Set H = Ker(β)/Im(α). For n ≥ 0 let K/I^nK xrightarrowα_n L/I^nL xrightarrowβ_n M/I^nM be the induced complex. Set H_n = Ker(β_n)/Im(α_n). Then there are canonical A-module maps giving a commutative diagram xymatrix & & & H ar[lld] ar[ld] ar[d] … ar[r] & H_3 ar[r] & H_2 ar[r] & H_1 Moreover, there exists a c > 0 and canonical A-module maps H_n → H/I^n - cH for n…","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let\n$\nK \\xrightarrow{\\alpha} L \\xrightarrow{\\beta} M\n$\nbe a complex of finite $A$-modules. Set $H = \\Ker(\\beta)/\\Im(\\alpha)$.\nFor $n \\geq 0$ let\n$$\nK/I^nK \\xrightarrow{\\alpha_n} L/I^nL \\xrightarrow{\\beta_n} M/I^nM\n$$\nbe the induced complex. Set $H_n = \\Ker(\\beta_n)/\\Im(\\alpha_n)$.\nThen there are canonical $A$-module maps giving a commutative diagram\n$$\n\\xymatrix{\n& & & H \\ar[lld] \\ar[ld] \\ar[d] \\\\\n\\ldots \\ar[r] & H_3 \\ar[r] & H_2 \\ar[r] & H_1\n}\n$$\nMoreover, there exists a $c > 0$ and canonical $A$-module maps\n$H_n \\to H/I^{n - c}H$ for $n \\geq c$ such that the compositions\n$$\nH/I^n H \\to H_n \\to  H/I^{n - c}H\n\\quad\\text{and}\\quad\nH_n \\to H/I^{n - c}H \\to H_{n - c}\n$$\nare the canonical ones. Moreover, we have\n\\begin{enumerate}\n\\item $(H_n)$ and $(H/I^nH)$ are isomorphic as pro-objects of $\\text{Mod}_A$,\n\\item $\\lim H_n = \\lim H/I^n H$,\n\\item the inverse system $(H_n)$ is Mittag-Leffler,\n\\item the image of $H_{n + c} \\to H_n$ is equal to the image of $H \\to H_n$,\n\\item the composition $I^cH_n \\to H_n \\to H/I^{n - c}H \\to H_n/I^{n - c}H_n$\nis the inclusion $I^cH_n \\to H_n$ followed by the quotient map\n$H_n \\to H_n/I^{n - c}H_n$, and\n\\item the kernel and cokernel of $H/I^nH \\to H_n$ is annihilated by $I^c$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EGU","source_file":"more-algebra.tex","source_line":28866,"source_end_line":28903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28866-L28903","statement_sha256":"b84d6cca2551edf9a10abdc0e6323a1d97d54684fce93b2c4b9e593402fefdc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3527,"rank":3527,"depth":3,"x":822.963,"y":718.994,"cluster":"advanced-algebra"},{"id":"stacks:0EGV","tag":"0EGV","title":"Systems of modules · Lemma 0EGV","summary":"Email from Kovacs of 23/02/2018. Let I be an ideal of a Noetherian ring A. Let K ∈ D(A) be pseudo-coherent. Set K_n = K ⊗_A^L A/I^n. Then for all i ∈ Z the system H^i(K_n) satisfies Mittag-Leffler and lim H^i(K)/I^nH^i(K) is equal to lim H^i(K_n).","statement_latex":"\\begin{reference}\nEmail from Kovacs of 23/02/2018.\n\\end{reference}\nLet $I$ be an ideal of a Noetherian ring $A$. Let $K \\in D(A)$\nbe pseudo-coherent. Set $K_n = K \\otimes_A^\\mathbf{L} A/I^n$.\nThen for all $i \\in \\mathbf{Z}$ the system $H^i(K_n)$\nsatisfies Mittag-Leffler and $\\lim H^i(K)/I^nH^i(K)$ is equal to\n$\\lim H^i(K_n)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EGV","source_file":"more-algebra.tex","source_line":28993,"source_end_line":29003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L28993-L29003","statement_sha256":"674812a0ab54107c08bedd327817ac9e2f2e6f1a12cb26008b374f255e001935","origin":"The Stacks Project","memory_eligible":false,"source_rank":3528,"rank":3528,"depth":24,"x":543.129,"y":732.403,"cluster":"advanced-algebra"},{"id":"stacks:0G9M","tag":"0G9M","title":"Systems of modules · Lemma 0G9M","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let M^bullet be a bounded complex of finite A-modules. The inverse system of maps M^bullet ⊗_A^L A/I^n → M^bullet/I^nM^bullet defines an isomorphism of pro-objects of D(A).","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let\n$M^\\bullet$ be a bounded complex of finite $A$-modules. The\ninverse system of maps\n$$\nM^\\bullet \\otimes_A^\\mathbf{L} A/I^n \\longrightarrow M^\\bullet/I^nM^\\bullet\n$$\ndefines an isomorphism of pro-objects of $D(A)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9M","source_file":"more-algebra.tex","source_line":29014,"source_end_line":29023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29014-L29023","statement_sha256":"7e80209386867d2696782c4da6df610deb3e172b234b07df32cd38f12c224e3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3529,"rank":3529,"depth":8,"x":738.826,"y":563.615,"cluster":"advanced-algebra"},{"id":"stacks:09BB","tag":"09BB","title":"Systems of modules · Lemma 09BB","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let M, N be finite A-modules. Set M_n = M/I^nM and N_n = N/I^nN. Then • the systems (Hom_A(M_n, N_n)) and (Isom_A(M_n, N_n)) are Mittag-Leffler, • there exists a c ≥ 0 such that the kernels and cokernels of Hom_A(M, N)/I^nHom_A(M, N) → Hom_A(M_n, N_n) are killed by I^c for all n, • we have lim Hom_A(M_n, N_n) =Hom_A(M, N)^wedge = Hom_A^wedge(M^wedge, N^wedge) • lim Isom_A(M_n, N_n) = Isom_A^wedge(M^wedge, N^wedge). Here…","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$ be\nfinite $A$-modules. Set $M_n = M/I^nM$ and $N_n = N/I^nN$. Then\n\\begin{enumerate}\n\\item the systems $(\\Hom_A(M_n, N_n))$ and $(\\text{Isom}_A(M_n, N_n))$\nare Mittag-Leffler,\n\\item there exists a $c \\geq 0$ such that the kernels and cokernels of\n$$\n\\Hom_A(M, N)/I^n\\Hom_A(M, N) \\to \\Hom_A(M_n, N_n)\n$$\nare killed by $I^c$ for all $n$,\n\\item we have\n$\\lim \\Hom_A(M_n, N_n) =\\Hom_A(M, N)^\\wedge =\n\\Hom_{A^\\wedge}(M^\\wedge, N^\\wedge)$\n\\item $\\lim \\text{Isom}_A(M_n, N_n) =\n\\text{Isom}_{A^\\wedge}(M^\\wedge, N^\\wedge)$.\n\\end{enumerate}\nHere ${}^\\wedge$ denotes usual $I$-adic completion.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BB","source_file":"more-algebra.tex","source_line":29059,"source_end_line":29078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29059-L29078","statement_sha256":"13196ef97378a236a8d4477929f9a33165a3094498f061e4ae9b410a318d2885","origin":"The Stacks Project","memory_eligible":false,"source_rank":3530,"rank":3530,"depth":6,"x":730.25,"y":799.281,"cluster":"advanced-algebra"},{"id":"stacks:09BC","tag":"09BC","title":"Systems of modules · Lemma 09BC","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let M, N be finite A-modules. Set M_n = M/I^nM and N_n = N/I^nN. If M_n ≅ N_n for all n, then M^wedge ≅ N^wedge as A^wedge-modules.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$ be\nfinite $A$-modules. Set $M_n = M/I^nM$ and $N_n = N/I^nN$. If\n$M_n \\cong N_n$ for all $n$, then $M^\\wedge \\cong N^\\wedge$\nas $A^\\wedge$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BC","source_file":"more-algebra.tex","source_line":29123,"source_end_line":29129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29123-L29129","statement_sha256":"2a39e72262c7f9f6c975f2203a37007125e3b3c8d4f47dc702083b726cbb3b84","origin":"The Stacks Project","memory_eligible":false,"source_rank":3531,"rank":3531,"depth":7,"x":546.933,"y":620.517,"cluster":"advanced-algebra"},{"id":"stacks:0EGX","tag":"0EGX","title":"Systems of modules · Lemma 0EGX","summary":"A morphism (c, φ_n) of the category of Remark [Tag 0EGW] is an isomorphism if and only if there exists a c' ≥ 0 such that Ker(φ_n) and Coker(φ_n) are I^c'-torsion for all n gg 0.","statement_latex":"A morphism $(c, \\varphi_n)$ of the category of\nRemark \\ref{remark-weird-systems} is an\nisomorphism if and only if there exists a $c' \\geq 0$ such that\n$\\Ker(\\varphi_n)$ and $\\Coker(\\varphi_n)$ are $I^{c'}$-torsion for\nall $n \\gg 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EGX","source_file":"more-algebra.tex","source_line":29167,"source_end_line":29174,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29167-L29174","statement_sha256":"fe4b2d61feb6a8baec840c489972c4c5870fb52019180e28962f0b8702b55c7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3532,"rank":3532,"depth":0,"x":826.055,"y":648.335,"cluster":"advanced-algebra"},{"id":"stacks:0EGY","tag":"0EGY","title":"Systems of modules · Lemma 0EGY","summary":"Email correspondence between Janos Kollar, Sandor Kovacs, and Johan de Jong of 23/02/2018. Let I be an ideal of the Noetherian ring A. Let M and N be finite A-modules. Write A_n = A/I^n, M_n = M/I^nM, and N_n = N/I^nN. For every i ≥ 0 the objects (Ext^i_A(M, N)/I^nExt^i_A(M, N))_n ≥ 1 and (Ext^i_A_n(M_n, N_n))_n ≥ 1 are isomorphic in the category C of Remark [Tag 0EGW].","statement_latex":"\\begin{reference}\nEmail correspondence between Janos Kollar, Sandor Kovacs, and\nJohan de Jong of 23/02/2018.\n\\end{reference}\nLet $I$ be an ideal of the Noetherian ring $A$. Let $M$ and $N$\nbe finite $A$-modules. Write $A_n = A/I^n$, $M_n = M/I^nM$, and\n$N_n = N/I^nN$.\nFor every $i \\geq 0$ the objects\n$$\n\\{\\Ext^i_A(M, N)/I^n\\Ext^i_A(M, N)\\}_{n \\geq 1}\n\\quad\\text{and}\\quad\n\\{\\Ext^i_{A_n}(M_n, N_n)\\}_{n \\geq 1}\n$$\nare isomorphic in the category $\\mathcal{C}$ of\nRemark \\ref{remark-weird-systems}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EGY","source_file":"more-algebra.tex","source_line":29193,"source_end_line":29210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29193-L29210","statement_sha256":"2649388be75e199eeab8bfaa9564bf0e08080c35948ca64951c4e6932e86e6d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3533,"rank":3533,"depth":7,"x":597.709,"y":786.297,"cluster":"advanced-algebra"},{"id":"stacks:0EH0","tag":"0EH0","title":"Systems of modules · Lemma 0EH0","summary":"Email correspondence between Janos Kollar, Sandor Kovacs, and Johan de Jong of 23/02/2018. Let A → B be a flat homomorphism of Noetherian rings. Let I ⊂ A be an ideal. Let M, N be A-modules. Set B_n = B/I^nB, M_n = M/I^nM, N_n = N/I^nN. If M is flat over A, then we have lim Ext^i_B(M, N)/I^n Ext^i_B(M, N) = lim Ext^i_B_n(M_n, N_n) for all i ∈ Z.","statement_latex":"\\begin{reference}\nEmail correspondence between Janos Kollar, Sandor Kovacs, and\nJohan de Jong of 23/02/2018.\n\\end{reference}\nLet $A \\to B$ be a flat homomorphism of Noetherian rings.\nLet $I \\subset A$ be an ideal. Let $M, N$ be $A$-modules.\nSet $B_n = B/I^nB$, $M_n = M/I^nM$, $N_n = N/I^nN$.\nIf $M$ is flat over $A$, then we have\n$$\n\\lim \\Ext^i_B(M, N)/I^n \\Ext^i_B(M, N) =\n\\lim \\Ext^i_{B_n}(M_n, N_n)\n$$\nfor all $i \\in \\mathbf{Z}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EH0","source_file":"more-algebra.tex","source_line":29370,"source_end_line":29385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29370-L29385","statement_sha256":"68a39333084f9ff4e3fcb5e9ef7a5bee9c8b033de227dc19e32daa22d87930de","origin":"The Stacks Project","memory_eligible":false,"source_rank":3534,"rank":3534,"depth":4,"x":655.182,"y":554.84,"cluster":"advanced-algebra"},{"id":"stacks:0G3K","tag":"0G3K","title":"Systems of modules, bis · Lemma 0G3K","summary":"Let I be an ideal of a Noetherian ring A. Let K xrightarrowα L xrightarrowβ M be a complex of finite A-modules. Set H = Ker(β)/Im(α). For n ≥ 0 let I^nK xrightarrowα_n I^nL xrightarrowβ_n I^nM be the induced complex. Set H_n = Ker(β_n)/Im(α_n). Then there are canonical A-module maps … → H_3 → H_2 → H_1 → H There exists a c > 0 such that for n ≥ c the image of H_n → H is contained in I^n - cH and there is a canonical A-module map I^nH → H_n - c such that the compositions…","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let\n$\nK \\xrightarrow{\\alpha} L \\xrightarrow{\\beta} M\n$\nbe a complex of finite $A$-modules. Set $H = \\Ker(\\beta)/\\Im(\\alpha)$.\nFor $n \\geq 0$ let\n$$\nI^nK \\xrightarrow{\\alpha_n} I^nL \\xrightarrow{\\beta_n} I^nM\n$$\nbe the induced complex. Set $H_n = \\Ker(\\beta_n)/\\Im(\\alpha_n)$.\nThen there are canonical $A$-module maps\n$$\n\\ldots \\to H_3 \\to H_2 \\to H_1 \\to H\n$$\nThere exists a $c > 0$ such that for $n \\geq c$ the image of $H_n \\to H$ is\ncontained in $I^{n - c}H$ and there is a canonical $A$-module map\n$I^nH \\to H_{n - c}$ such that the compositions\n$$\nI^n H \\to H_{n - c} \\to  I^{n - 2c}H\n\\quad\\text{and}\\quad\nH_n \\to I^{n - c}H \\to H_{n - 2c}\n$$\nare the canonical ones. In particular, the inverse systems\n$(H_n)$ and $(I^nH)$ are isomorphic as pro-objects of $\\text{Mod}_A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3K","source_file":"more-algebra.tex","source_line":29431,"source_end_line":29457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29431-L29457","statement_sha256":"2f7ad028a8ff2ec729ec28aad7f30e1f939d63f9c924d92ea263bc57197d4c50","origin":"The Stacks Project","memory_eligible":false,"source_rank":3535,"rank":3535,"depth":3,"x":799.031,"y":758.262,"cluster":"advanced-algebra"},{"id":"stacks:0G3L","tag":"0G3L","title":"Systems of modules, bis · Lemma 0G3L","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let M, N be A-modules with M finite. For each p > 0 there exists a c ≥ 0 such that for n ≥ c the map Ext_A^p(M, N) → Ext_A^p(I^nM, N) factors through Ext^p_A(I^nM, I^n - cN) → Ext_A^p(I^nM, N).","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nLet $M$, $N$ be $A$-modules with $M$ finite. For each $p > 0$ there exists a\n$c \\geq 0$ such that for $n \\geq c$ the map\n$\\Ext_A^p(M, N) \\to \\Ext_A^p(I^nM, N)$\nfactors through $\\Ext^p_A(I^nM, I^{n - c}N) \\to \\Ext_A^p(I^nM, N)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3L","source_file":"more-algebra.tex","source_line":29492,"source_end_line":29499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29492-L29499","statement_sha256":"9f354b01e2620f7865b547bc61308a1c8f45bf7d3184a8757e173a389c301f62","origin":"The Stacks Project","memory_eligible":false,"source_rank":3536,"rank":3536,"depth":2,"x":529.19,"y":689.839,"cluster":"advanced-algebra"},{"id":"stacks:0927","tag":"0927","title":"Systems of modules, bis · Lemma 0927","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let M, N be A-modules with M finite and N annihilated by a power of I. For each p > 0 there exists an n such that the map Ext_A^p(M, N) → Ext_A^p(I^nM, N) is zero.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$, $N$\nbe $A$-modules with $M$ finite and $N$ annihilated by a power of $I$.\nFor each $p > 0$ there exists an $n$ such that the map\n$\\Ext_A^p(M, N) \\to \\Ext_A^p(I^nM, N)$ is zero.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0927","source_file":"more-algebra.tex","source_line":29570,"source_end_line":29576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29570-L29576","statement_sha256":"0512bceeee845a8e670cef0dbc8baba3785b978f6dd5a8c45026a5b06a1769b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3537,"rank":3537,"depth":3,"x":783.363,"y":587.109,"cluster":"advanced-algebra"},{"id":"stacks:0DYI","tag":"0DYI","title":"Systems of modules, bis · Lemma 0DYI","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let K ∈ D(A) be pseudo-coherent and let M be a finite A-module. For each p ∈ Z there exists an c such that the image of Ext_A^p(K, I^nM) → Ext_A^p(K, M) is contained in I^n - cExt_A^p(K, M) for n ≥ c.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nLet $K \\in D(A)$ be pseudo-coherent and let $M$ be a finite\n$A$-module. For each $p \\in \\mathbf{Z}$ there exists an $c$\nsuch that the image of $\\Ext_A^p(K, I^nM) \\to \\Ext_A^p(K, M)$\nis contained in $I^{n - c}\\Ext_A^p(K, M)$ for $n \\geq c$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYI","source_file":"more-algebra.tex","source_line":29583,"source_end_line":29590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29583-L29590","statement_sha256":"f24e7bcf2c4e6d2b9c66623da106d753b1f61e3861183cbc2378a95827a0b18a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3538,"rank":3538,"depth":4,"x":678.481,"y":807.233,"cluster":"advanced-algebra"},{"id":"stacks:0G3M","tag":"0G3M","title":"Systems of modules, bis · Lemma 0G3M","summary":"In Situation [Tag 0BKC] assume A is Noetherian. With notation as above, the inverse system (I^n) is pro-isomorphic in D(A) to the inverse system (I_n^bullet).","statement_latex":"In Situation \\ref{situation-koszul} assume $A$ is Noetherian. With\nnotation as above, the inverse system $(I^n)$ is pro-isomorphic\nin $D(A)$ to the inverse system $(I_n^\\bullet)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3M","source_file":"more-algebra.tex","source_line":29647,"source_end_line":29652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29647-L29652","statement_sha256":"a690f8cc4cfce216bff4cde9504dc5c94da36c09b84778fe69a0dfb504df3b14","origin":"The Stacks Project","memory_eligible":false,"source_rank":3539,"rank":3539,"depth":8,"x":578.735,"y":585.255,"cluster":"advanced-algebra"},{"id":"stacks:0G3N","tag":"0G3N","title":"Systems of modules, bis · Lemma 0G3N","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let M^bullet be a bounded complex of finite A-modules. The inverse system of maps I^n ⊗_A^L M^bullet → I^nM^bullet defines an isomorphism of pro-objects of D(A).","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let\n$M^\\bullet$ be a bounded complex of finite $A$-modules. The\ninverse system of maps\n$$\nI^n \\otimes_A^\\mathbf{L} M^\\bullet \\longrightarrow I^nM^\\bullet\n$$\ndefines an isomorphism of pro-objects of $D(A)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3N","source_file":"more-algebra.tex","source_line":29676,"source_end_line":29685,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29676-L29685","statement_sha256":"330216e09104909b641c5c974981dc9debc750c55e788e6a90f22d3d1939e707","origin":"The Stacks Project","memory_eligible":false,"source_rank":3540,"rank":3540,"depth":9,"x":830.965,"y":692.412,"cluster":"advanced-algebra"},{"id":"stacks:0928","tag":"0928","title":"Systems of modules, bis · Lemma 0928","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let M be a finite A-module. There exists an integer n > 0 such that I^nM → M factors through the map I ⊗_A^L M → M in D(A).","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$\nbe a finite $A$-module. There exists an integer $n > 0$ such that\n$I^nM \\to M$ factors through the map $I \\otimes_A^\\mathbf{L} M \\to M$\nin $D(A)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Systems of modules, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0928","source_file":"more-algebra.tex","source_line":29736,"source_end_line":29742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29736-L29742","statement_sha256":"3fab4f55443643b7068621fc7dc206d67e2fc59b145883c0adc71c10bd7f2f9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3541,"rank":3541,"depth":10,"x":558.618,"y":756.56,"cluster":"advanced-algebra"},{"id":"stacks:0DYJ","tag":"0DYJ","title":"Miscellany · Lemma 0DYJ","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let K ∈ D(A) be pseudo-coherent. Let a ∈ Z. Assume that for every finite A-module M the modules Ext^i_A(K, M) are I-power torsion for i ≥ a. Then for i ≥ a and M finite the system Ext^i_A(K, M/I^nM) is essentially constant with value Ext^i_A(K, M) = lim Ext^i_A(K, M/I^nM)","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nLet $K \\in D(A)$ be pseudo-coherent. Let $a \\in \\mathbf{Z}$.\nAssume that for every finite $A$-module $M$ the modules\n$\\Ext^i_A(K, M)$ are $I$-power torsion for $i \\geq a$.\nThen for $i \\geq a$ and $M$ finite\nthe system $\\Ext^i_A(K, M/I^nM)$\nis essentially constant with value\n$$\n\\Ext^i_A(K, M) = \\lim \\Ext^i_A(K, M/I^nM)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYJ","source_file":"more-algebra.tex","source_line":29781,"source_end_line":29793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29781-L29793","statement_sha256":"0ca9cbf0a2ec4b9e7867f686a7e51557b296505368c685bd70709a25aca23b3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3542,"rank":3542,"depth":5,"x":707.956,"y":554.587,"cluster":"advanced-algebra"},{"id":"stacks:0FXN","tag":"0FXN","title":"Miscellany · Lemma 0FXN","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let M be a finite A-module. Let N be an A-module annihilated by I. There exists an integer n > 0 such that Tor^A_p(I^nM, N) → Tor^A_p(M, N) is zero for all p ≥ 0.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Let $M$\nbe a finite $A$-module. Let $N$ be an $A$-module annihilated by $I$.\nThere exists an integer $n > 0$ such that\n$\\text{Tor}^A_p(I^nM, N) \\to \\text{Tor}^A_p(M, N)$ is zero\nfor all $p \\geq 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXN","source_file":"more-algebra.tex","source_line":29815,"source_end_line":29822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29815-L29822","statement_sha256":"9459db2e821dddb5a26ff8c774e887f9f1ef5cc8045857527ca37c6abdcfd91d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3543,"rank":3543,"depth":11,"x":760.293,"y":788.414,"cluster":"advanced-algebra"},{"id":"stacks:0D2L","tag":"0D2L","title":"Miscellany · Lemma 0D2L","summary":"Let R be a ring. Let K ∈ D(R) be pseudo-coherent. Let (M_n) be an inverse system of R-modules. Then Rlim K ⊗_R^L M_n = K ⊗_R^L Rlim M_n.","statement_latex":"Let $R$ be a ring. Let $K \\in D(R)$ be pseudo-coherent.\nLet $(M_n)$ be an inverse system of $R$-modules.\nThen $R\\lim K \\otimes_R^\\mathbf{L} M_n = K \\otimes_R^\\mathbf{L} R\\lim M_n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2L","source_file":"more-algebra.tex","source_line":29845,"source_end_line":29850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29845-L29850","statement_sha256":"e04ab1b1be0f2da0037bab5c8592102448ca5cd2a828f21f2cfd2e5084fbdea6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3544,"rank":3544,"depth":11,"x":533.512,"y":645.595,"cluster":"advanced-algebra"},{"id":"stacks:0929","tag":"0929","title":"Miscellany · Lemma 0929","summary":"Let R be a Noetherian local ring. Let I ⊂ R be an ideal and let E be a nonzero finite module over R/I. If R/I has finite projective dimension and E has finite projective dimension over R/I, then E has finite projective dimension over R and pd_R(E) = pd_R(R/I) + pd_R/I(E)","statement_latex":"Let $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal and\nlet $E$ be a nonzero finite module over $R/I$. If $R/I$ has finite projective\ndimension and $E$ has finite projective dimension over $R/I$, then\n$E$ has finite projective dimension over $R$ and\n$$\n\\text{pd}_R(E) = \\text{pd}_R(R/I) + \\text{pd}_{R/I}(E)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0929","source_file":"more-algebra.tex","source_line":29860,"source_end_line":29869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29860-L29869","statement_sha256":"0c4e6aefa61a4a76d7f232ec7e95c35cfec7b13948ff4409e4845dc6bfe351ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":3545,"rank":3545,"depth":16,"x":815.777,"y":622.21,"cluster":"advanced-algebra"},{"id":"stacks:0GYI","tag":"0GYI","title":"Miscellany · Lemma 0GYI","summary":"Let A → B be a ring map. There exists a cardinal kappa = kappa(A → B) with the following property: Let M^bullet, resp. N^bullet be a complex of A-modules, resp. B-modules. Let a : M^bullet → N^bullet be a map of complexes of A-modules which induces an isomorphism M^bullet ⊗_A^L B → N^bullet in D(B). Let M_1^bullet ⊂ M^bullet, resp. N_1^bullet ⊂ N^bullet be a subcomplex of A-modules, resp. B-modules such that a(M_1^bullet) ⊂ N_1^bullet. Then there exist subcomplexes…","statement_latex":"Let $A \\to B$ be a ring map. There exists a cardinal $\\kappa = \\kappa(A \\to B)$\nwith the following property: Let $M^\\bullet$, resp.\\ $N^\\bullet$\nbe a complex of $A$-modules, resp.\\ $B$-modules. Let\n$a : M^\\bullet \\to N^\\bullet$ be a map of complexes of $A$-modules\nwhich induces an isomorphism\n$M^\\bullet \\otimes_A^\\mathbf{L} B \\to N^\\bullet$ in $D(B)$.\nLet $M_1^\\bullet \\subset M^\\bullet$, resp.\\ $N_1^\\bullet \\subset N^\\bullet$\nbe a subcomplex of $A$-modules, resp.\\ $B$-modules such that\n$a(M_1^\\bullet) \\subset N_1^\\bullet$. Then there exist\nsubcomplexes\n$$\nM_1^\\bullet \\subset M_2^\\bullet \\subset M^\\bullet\n\\quad\\text{and}\\quad\nN_1^\\bullet \\subset N_2^\\bullet \\subset N^\\bullet\n$$\nsuch that $a(M_2^\\bullet) \\subset N_2^\\bullet$\nwith the following properties:\n\\begin{enumerate}\n\\item $\\Ker(H^i(M_1^\\bullet \\otimes_A^\\mathbf{L} B) \\to H^i(N_1^\\bullet))$\nmaps to zero in $H^i(M_2^\\bullet \\otimes_A^\\mathbf{L} B)$,\n\\item $\\Im(H^i(N_1^\\bullet) \\to H^i(N_2^\\bullet))$ is contained in\n$\\Im(H^i(M_2^\\bullet \\otimes_A^\\mathbf{L} B) \\to H^2(N_2^\\bullet))$,\n\\item $|\\bigcup M_2^i \\cup \\bigcup N_2^i| \\leq\n\\max(\\kappa, |\\bigcup M_1^i \\cup \\bigcup N_1^i|)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYI","source_file":"more-algebra.tex","source_line":29894,"source_end_line":29921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L29894-L29921","statement_sha256":"9377ccb0035394dfbd24b2a0e1c4383652208c4b29584c09f72e5a9620df9b48","origin":"The Stacks Project","memory_eligible":false,"source_rank":3546,"rank":3546,"depth":0,"x":626.317,"y":799.736,"cluster":"advanced-algebra"},{"id":"stacks:0H84","tag":"0H84","title":"Miscellany · Lemma 0H84","summary":"Let R be a ring and f ∈ R. Let M ∈ D(R) and let C be the cone of f : M → M. If H^i(M)_f = 0 for i < 0 and H^i(C) = 0 for i < -1, then H^i(M) = 0 for i < 0.","statement_latex":"Let $R$ be a ring and $f \\in R$. Let $M \\in D(R)$ and let\n$C$ be the cone of $f : M \\to M$.\nIf $H^i(M)_f = 0$ for $i < 0$ and $H^i(C) = 0$ for $i < -1$,\nthen $H^i(M) = 0$ for $i < 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H84","source_file":"more-algebra.tex","source_line":30011,"source_end_line":30017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30011-L30017","statement_sha256":"1bbe7a4e28783344eb14e9c4ede8b856bbdd40c733d78a59ee4b4c470bbcc807","origin":"The Stacks Project","memory_eligible":false,"source_rank":3547,"rank":3547,"depth":1,"x":623.259,"y":561.169,"cluster":"advanced-algebra"},{"id":"stacks:0H85","tag":"0H85","title":"Miscellany · Lemma 0H85","summary":"Let R be a ring and f ∈ R. Let M ∈ D(R). Assume • H^i(M) = 0 for i > 0, • M ⊗_R^L R_f is isomorphic to a flat R_f-module placed in degree 0, • M ⊗_R^L R/fR is isomorphic to a flat R/fR-module placed in degree 0. Then M is isomorphic to a flat R-module placed in degree 0.","statement_latex":"Let $R$ be a ring and $f \\in R$. Let $M \\in D(R)$. Assume\n\\begin{enumerate}\n\\item $H^i(M) = 0$ for $i > 0$,\n\\item $M \\otimes_R^\\mathbf{L} R_f$ is isomorphic to a flat $R_f$-module\nplaced in degree $0$,\n\\item $M \\otimes_R^\\mathbf{L} R/fR$ is isomorphic to a flat $R/fR$-module\nplaced in degree $0$.\n\\end{enumerate}\nThen $M$ is isomorphic to a flat $R$-module placed in degree $0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H85","source_file":"more-algebra.tex","source_line":30032,"source_end_line":30043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30032-L30043","statement_sha256":"af12457d166f9e603fa0fc344f13e75e7742a43f0245e7b04ddd417c49618c7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3548,"rank":3548,"depth":2,"x":817.493,"y":735.465,"cluster":"advanced-algebra"},{"id":"stacks:0H0Q","tag":"0H0Q","title":"Tricks with double complexes · Lemma 0H0Q","summary":"Let A_0^bullet → A_1^bullet → A_2^bullet → … be a complex of complexes of abelian groups. Assume H^-p(A_p^bullet) = 0 for all p ≥ 0. Set A^p, q = A_p^q and view A^bullet, bullet as a double complex. Then H^0(Tot_π(A^bullet, bullet)) = 0.","statement_latex":"Let $A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots$\nbe a complex of complexes of abelian groups. Assume\n$H^{-p}(A_p^\\bullet) = 0$ for all $p \\geq 0$.\nSet $A^{p, q} = A_p^q$ and view $A^{\\bullet, \\bullet}$ as a double complex.\nThen $H^0(\\text{Tot}_\\pi(A^{\\bullet, \\bullet})) = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tricks with double complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0Q","source_file":"more-algebra.tex","source_line":30095,"source_end_line":30102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30095-L30102","statement_sha256":"bcb8b1936c3e1883059ce3cc60b82157f8ef84b5d24f68ad02dcf00c49ce774d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3549,"rank":3549,"depth":0,"x":533.915,"y":717.142,"cluster":"advanced-algebra"},{"id":"stacks:0EYX","tag":"0EYX","title":"Tricks with double complexes · Lemma 0EYX","summary":"Let (A_0^bullet → A_1^bullet → A_2^bullet → …) → (B_0^bullet → B_1^bullet → B_2^bullet → …) be a map between two complexes of complexes of abelian groups. Set A^p, q = A_p^q, B^p, q = B_p^q to obtain double complexes. Let Tot_π(A^bullet, bullet) and Tot_π(B^bullet, bullet) be the product total complexes associated to the double complexes. If each A_p^bullet → B_p^bullet is a quasi-isomorphism, then Tot_π(A^bullet, bullet) → Tot_π(B^bullet, bullet) is a quasi-isomorphism.","statement_latex":"Let\n$$\n(A_0^\\bullet \\to A_1^\\bullet \\to A_2^\\bullet \\to \\ldots)\n\\longrightarrow\n(B_0^\\bullet \\to B_1^\\bullet \\to B_2^\\bullet \\to \\ldots)\n$$\nbe a map between two complexes of complexes of abelian groups.\nSet $A^{p, q} = A_p^q$, $B^{p, q} = B_p^q$ to obtain double complexes.\nLet $\\text{Tot}_\\pi(A^{\\bullet, \\bullet})$\nand $\\text{Tot}_\\pi(B^{\\bullet, \\bullet})$ be the\nproduct total complexes associated to the double complexes.\nIf each $A_p^\\bullet \\to B_p^\\bullet$ is a\nquasi-isomorphism, then\n$\\text{Tot}_\\pi(A^{\\bullet, \\bullet}) \\to \\text{Tot}_\\pi(B^{\\bullet, \\bullet})$\nis a quasi-isomorphism.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Tricks with double complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYX","source_file":"more-algebra.tex","source_line":30130,"source_end_line":30147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30130-L30147","statement_sha256":"38fd455f1d7f1fa8b54a8e5d406203cda85d99439819964ffb1926b2e9eba150","origin":"The Stacks Project","memory_eligible":false,"source_rank":3550,"rank":3550,"depth":1,"x":757.904,"y":569.646,"cluster":"advanced-algebra"},{"id":"stacks:092B","tag":"092B","title":"Weakly étale ring maps · Definition 092B","summary":"A ring A is called absolutely flat if every A-module is flat over A. A ring map A → B is weakly étale or absolutely flat if both A → B and B ⊗_A B → B are flat.","statement_latex":"A ring $A$ is called {\\it absolutely flat} if every $A$-module is flat over\n$A$. A ring map $A \\to B$ is {\\it weakly \\'etale} or {\\it absolutely flat}\nif both $A \\to B$ and $B \\otimes_A B \\to B$ are flat.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092B","source_file":"more-algebra.tex","source_line":30188,"source_end_line":30193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30188-L30193","statement_sha256":"2090d45a0cfe5588cc9a579944925ed9d248cfc71c94fb6c299df4ddc582343a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3551,"rank":3551,"depth":0,"x":711.319,"y":805.661,"cluster":"advanced-algebra"},{"id":"stacks:092C","tag":"092C","title":"Weakly étale ring maps · Lemma 092C","summary":"Let A → B be a ring map such that B ⊗_A B → B is flat. Let N be a B-module. If N is flat as an A-module, then N is flat as a B-module.","statement_latex":"Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat.\nLet $N$ be a $B$-module. If $N$ is flat as an $A$-module, then\n$N$ is flat as a $B$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092C","source_file":"more-algebra.tex","source_line":30201,"source_end_line":30206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30201-L30206","statement_sha256":"29fa961ae6c248da432e7866c6c504dbade55eeea9748e99688374c5ad7e7d2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3552,"rank":3552,"depth":0,"x":555.771,"y":605.061,"cluster":"advanced-algebra"},{"id":"stacks:092D","tag":"092D","title":"Weakly étale ring maps · Definition 092D","summary":"Let A be a ring. Let d ≥ 0 be an integer. We say that A has weak dimension ≤ d if every A-module has tor dimension ≤ d.","statement_latex":"Let $A$ be a ring. Let $d \\geq 0$ be an integer.\nWe say that $A$ has {\\it weak dimension $\\leq d$}\nif every $A$-module has tor dimension $\\leq d$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092D","source_file":"more-algebra.tex","source_line":30223,"source_end_line":30228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30223-L30228","statement_sha256":"ef35e7b75b8d047fa773a0a4675379abe7825c7a99a632d145177b4c9cb4093a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3553,"rank":3553,"depth":0,"x":831.969,"y":664.758,"cluster":"advanced-algebra"},{"id":"stacks:092E","tag":"092E","title":"Weakly étale ring maps · Lemma 092E","summary":"Let A → B be a weakly étale ring map. If A has weak dimension at most d, then so does B.","statement_latex":"Let $A \\to B$ be a weakly \\'etale ring map.\nIf $A$ has weak dimension at most $d$, then so does $B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092E","source_file":"more-algebra.tex","source_line":30230,"source_end_line":30234,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30230-L30234","statement_sha256":"9a94dd8de4104d2599f5aca28db6617cb58fbfaed3cdcdbb8e224070db31b07d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3554,"rank":3554,"depth":5,"x":580.132,"y":777.534,"cluster":"advanced-algebra"},{"id":"stacks:092F","tag":"092F","title":"Weakly étale ring maps · Lemma 092F","summary":"Let A be a ring. The following are equivalent • A has weak dimension ≤ 0, • A is absolutely flat, • A is reduced and every prime is maximal, and • every local ring of A is a field.","statement_latex":"Let $A$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $A$ has weak dimension $\\leq 0$,\n\\item $A$ is absolutely flat,\n\\item $A$ is reduced and every prime is maximal, and\n\\item every local ring of $A$ is a field.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092F","source_file":"more-algebra.tex","source_line":30249,"source_end_line":30258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30249-L30258","statement_sha256":"9243b07660aed02d36b9fde391f6f9ae8e0acfff7c8bc4863d5583e29bd8460c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3555,"rank":3555,"depth":6,"x":675.204,"y":551.327,"cluster":"advanced-algebra"},{"id":"stacks:092G","tag":"092G","title":"Weakly étale ring maps · Lemma 092G","summary":"A product of fields is an absolutely flat ring.","statement_latex":"A product of fields is an absolutely flat ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092G","source_file":"more-algebra.tex","source_line":30288,"source_end_line":30291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30288-L30291","statement_sha256":"421d16441a2ea156fb8da39f9e7c087c85d5062de31f3af37ececeff12051ac6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3556,"rank":3556,"depth":7,"x":787.082,"y":772.22,"cluster":"advanced-algebra"},{"id":"stacks:092H","tag":"092H","title":"Weakly étale ring maps · Lemma 092H","summary":"Let A → B and A → A' be ring maps. Let B' = B ⊗_A A' be the base change of B. • If B ⊗_A B → B is flat, then B' ⊗_A' B' → B' is flat. • If A → B is weakly étale, then A' → B' is weakly étale.","statement_latex":"Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$\nbe the base change of $B$.\n\\begin{enumerate}\n\\item If $B \\otimes_A B \\to B$ is flat, then $B' \\otimes_{A'} B' \\to B'$\nis flat.\n\\item If $A \\to B$ is weakly \\'etale, then $A' \\to B'$ is weakly \\'etale.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092H","source_file":"more-algebra.tex","source_line":30302,"source_end_line":30311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30302-L30311","statement_sha256":"ae57f43f675c790af31c2c52e8e940478a3c8a161f855e43c067fb63e5b6a238","origin":"The Stacks Project","memory_eligible":false,"source_rank":3557,"rank":3557,"depth":1,"x":526.777,"y":672.754,"cluster":"advanced-algebra"},{"id":"stacks:092I","tag":"092I","title":"Weakly étale ring maps · Lemma 092I","summary":"Let A → B be a ring map such that B ⊗_A B → B is flat. • If A is an absolutely flat ring, then so is B. • If A is reduced and A → B is weakly étale, then B is reduced.","statement_latex":"Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat.\n\\begin{enumerate}\n\\item If $A$ is an absolutely flat ring, then so is $B$.\n\\item If $A$ is reduced and $A \\to B$ is weakly \\'etale, then $B$ is reduced.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092I","source_file":"more-algebra.tex","source_line":30322,"source_end_line":30329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30322-L30329","statement_sha256":"31d62681a0f625452153ac1473843dd81c6dd4e167032b5c6d9d2d387794c43e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3558,"rank":3558,"depth":8,"x":798.889,"y":598.347,"cluster":"advanced-algebra"},{"id":"stacks:092J","tag":"092J","title":"Weakly étale ring maps · Lemma 092J","summary":"Let A → B and B → C be ring maps. • If B ⊗_A B → B and C ⊗_B C → C are flat, then C ⊗_A C → C is flat. • If A → B and B → C are weakly étale, then A → C is weakly étale.","statement_latex":"Let $A \\to B$ and $B \\to C$ be ring maps.\n\\begin{enumerate}\n\\item If $B \\otimes_A B \\to B$ and $C \\otimes_B C \\to C$\nare flat, then $C \\otimes_A C \\to C$ is flat.\n\\item If $A \\to B$ and $B \\to C$ are weakly \\'etale, then $A \\to C$\nis weakly \\'etale.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092J","source_file":"more-algebra.tex","source_line":30346,"source_end_line":30355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30346-L30355","statement_sha256":"313efe7bede34e56a171c94eb4842947af546a3a665a26ba8362f96cda25c923","origin":"The Stacks Project","memory_eligible":false,"source_rank":3559,"rank":3559,"depth":3,"x":657.982,"y":807.754,"cluster":"advanced-algebra"},{"id":"stacks:092K","tag":"092K","title":"Weakly étale ring maps · Lemma 092K","summary":"Let A → B → C be ring maps. • If B → C is faithfully flat and C ⊗_A C → C is flat, then B ⊗_A B → B is flat. • If B → C is faithfully flat and A → C is weakly étale, then A → B is weakly étale.","statement_latex":"Let $A \\to B \\to C$ be ring maps.\n\\begin{enumerate}\n\\item If $B \\to C$ is faithfully flat and $C \\otimes_A C \\to C$ is flat,\nthen $B \\otimes_A B \\to B$ is flat.\n\\item If $B \\to C$ is faithfully flat and $A \\to C$ is weakly \\'etale,\nthen $A \\to B$ is weakly \\'etale.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092K","source_file":"more-algebra.tex","source_line":30374,"source_end_line":30383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30374-L30383","statement_sha256":"6067272ed06d1820b37d791a47bf177f0ed7cae658f2118fe230967ab2dc741f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3560,"rank":3560,"depth":4,"x":593.443,"y":573.233,"cluster":"advanced-algebra"},{"id":"stacks:092L","tag":"092L","title":"Weakly étale ring maps · Lemma 092L","summary":"Let A be a ring. Let B → C be an A-algebra map of weakly étale A-algebras. Then B → C is weakly étale.","statement_latex":"Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra map of weakly \\'etale\n$A$-algebras. Then $B \\to C$ is weakly \\'etale.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092L","source_file":"more-algebra.tex","source_line":30404,"source_end_line":30408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30404-L30408","statement_sha256":"c8fe690bdcf2f1a8706f55e9b595f682f91589ff2d4d2f7e1a196d26e056c85a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3561,"rank":3561,"depth":1,"x":829.782,"y":709.633,"cluster":"advanced-algebra"},{"id":"stacks:092M","tag":"092M","title":"Weakly étale ring maps · Lemma 092M","summary":"Let A → B be a ring map such that B ⊗_A B → B is flat. Then Ω_B/A = 0, i.e., B is formally unramified over A.","statement_latex":"Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat.\nThen $\\Omega_{B/A} = 0$, i.e., $B$ is formally unramified over $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092M","source_file":"more-algebra.tex","source_line":30417,"source_end_line":30421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30417-L30421","statement_sha256":"e9f51d3b7f3adf7af6320e91e2356ccd13571816fe2b7218d3dc46da7d489ec4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3562,"rank":3562,"depth":5,"x":545.635,"y":743.18,"cluster":"advanced-algebra"},{"id":"stacks:0CKP","tag":"0CKP","title":"Weakly étale ring maps · Lemma 0CKP","summary":"Let A → B be a ring map such that B ⊗_A B → B is flat. • If A → B is of finite type, then A → B is unramified. • If A → B is of finite presentation and flat, then A → B is étale. In particular a weakly étale ring map of finite presentation is étale.","statement_latex":"Let $A \\to B$ be a ring map such that $B \\otimes_A B \\to B$ is flat.\n\\begin{enumerate}\n\\item If $A \\to B$ is of finite type, then $A \\to B$ is unramified.\n\\item If $A \\to B$ is of finite presentation and flat, then\n$A \\to B$ is \\'etale.\n\\end{enumerate}\nIn particular a weakly \\'etale ring map of finite presentation is \\'etale.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKP","source_file":"more-algebra.tex","source_line":30435,"source_end_line":30444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30435-L30444","statement_sha256":"96b6b2a458029d605e1c788d3f6daab0e118b1abee4b4a9c1ecaed68ce94c02d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3563,"rank":3563,"depth":42,"x":728.303,"y":557.09,"cluster":"advanced-algebra"},{"id":"stacks:092N","tag":"092N","title":"Weakly étale ring maps · Lemma 092N","summary":"Let A → B be a ring map. Then A → B is weakly étale in each of the following cases • B = S^-1A is a localization of A, • A → B is étale, • B is a filtered colimit of weakly étale A-algebras.","statement_latex":"Let $A \\to B$ be a ring map. Then $A \\to B$ is weakly \\'etale in each\nof the following cases\n\\begin{enumerate}\n\\item $B = S^{-1}A$ is a localization of $A$,\n\\item $A \\to B$ is \\'etale,\n\\item $B$ is a filtered colimit of weakly \\'etale $A$-algebras.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092N","source_file":"more-algebra.tex","source_line":30453,"source_end_line":30462,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30453-L30462","statement_sha256":"7dba838eb5f62d167a02f75d66c5ab85da1c8fc9228bed84da28da42db940af8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3564,"rank":3564,"depth":42,"x":743.265,"y":798.117,"cluster":"advanced-algebra"},{"id":"stacks:092P","tag":"092P","title":"Weakly étale ring maps · Lemma 092P","summary":"Let L/K be an extension of fields. If L ⊗_K L → L is flat, then L is an algebraic separable extension of K.","statement_latex":"Let $L/K$ be an extension of fields. If $L \\otimes_K L \\to L$\nis flat, then $L$ is an algebraic separable extension of $K$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092P","source_file":"more-algebra.tex","source_line":30487,"source_end_line":30491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30487-L30491","statement_sha256":"10fa2d40483a6c6407459d47b73c2a10b395d408bc396d75a5c13fc53f522d60","origin":"The Stacks Project","memory_eligible":false,"source_rank":3565,"rank":3565,"depth":42,"x":538.271,"y":628.767,"cluster":"advanced-algebra"},{"id":"stacks:092Q","tag":"092Q","title":"Weakly étale ring maps · Lemma 092Q","summary":"Let B be an algebra over a field K. The following are equivalent • B ⊗_K B → B is flat, • K → B is weakly étale, and • B is a filtered colimit of étale K-algebras. Moreover, every finitely generated K-subalgebra of B is étale over K.","statement_latex":"Let $B$ be an algebra over a field $K$. The following are\nequivalent\n\\begin{enumerate}\n\\item $B \\otimes_K B \\to B$ is flat,\n\\item $K \\to B$ is weakly \\'etale, and\n\\item $B$ is a filtered colimit of \\'etale $K$-algebras.\n\\end{enumerate}\nMoreover, every finitely generated $K$-subalgebra of $B$\nis \\'etale over $K$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092Q","source_file":"more-algebra.tex","source_line":30503,"source_end_line":30514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30503-L30514","statement_sha256":"ca45b177372696ffb6ff4cbb4cb8afe98e258f8de8f185dd2332b4ad42d451dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3566,"rank":3566,"depth":43,"x":825.803,"y":637.331,"cluster":"advanced-algebra"},{"id":"stacks:092R","tag":"092R","title":"Weakly étale ring maps · Lemma 092R","summary":"Let A → B be a ring map. If A → B is weakly étale, then A → B induces separable algebraic residue field extensions.","statement_latex":"Let $A \\to B$ be a ring map. If $A \\to B$ is weakly \\'etale, then\n$A \\to B$ induces separable algebraic residue field extensions.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092R","source_file":"more-algebra.tex","source_line":30555,"source_end_line":30559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30555-L30559","statement_sha256":"ac9e0aa3313562a3d1b79e9d5c096f31a026865e492620944709ab21190e7140","origin":"The Stacks Project","memory_eligible":false,"source_rank":3567,"rank":3567,"depth":44,"x":606.753,"y":794.27,"cluster":"advanced-algebra"},{"id":"stacks:092S","tag":"092S","title":"Weakly étale ring maps · Lemma 092S","summary":"Let A be a ring. The following are equivalent • A has weak dimension ≤ 1, • every ideal of A is flat, • every finitely generated ideal of A is flat, • every submodule of a flat A-module is flat, and • every local ring of A is a valuation ring.","statement_latex":"Let $A$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $A$ has weak dimension $\\leq 1$,\n\\item every ideal of $A$ is flat,\n\\item every finitely generated ideal of $A$ is flat,\n\\item every submodule of a flat $A$-module is flat, and\n\\item every local ring of $A$ is a valuation ring.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092S","source_file":"more-algebra.tex","source_line":30574,"source_end_line":30584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30574-L30584","statement_sha256":"54b48bd3f27683c71dc61b183b6b7ef03c3b301a7ee4ab9dcedbce1b5a36b70b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3568,"rank":3568,"depth":43,"x":642.094,"y":554.095,"cluster":"advanced-algebra"},{"id":"stacks:092T","tag":"092T","title":"Weakly étale ring maps · Lemma 092T","summary":"Let J be a set. For each j ∈ J let A_j be a valuation ring with fraction field K_j. Set A = ∏ A_j and K = ∏ K_j. Then A has weak dimension at most 1 and A → K is a localization.","statement_latex":"Let $J$ be a set. For each $j \\in J$ let\n$A_j$ be a valuation ring with fraction field $K_j$.\nSet $A = \\prod A_j$ and $K = \\prod K_j$.\nThen $A$ has weak dimension at most $1$ and $A \\to K$ is\na localization.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092T","source_file":"more-algebra.tex","source_line":30647,"source_end_line":30654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30647-L30654","statement_sha256":"38d1980a58e386e7607ca7475f2be2d987eff6e39566604660e86e0ae12abf0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3569,"rank":3569,"depth":44,"x":809.284,"y":751.377,"cluster":"advanced-algebra"},{"id":"stacks:092U","tag":"092U","title":"Weakly étale ring maps · Lemma 092U","summary":"Let A be a normal domain with fraction field K. There exists a cartesian diagram xymatrix A ar[d] ar[r] & K ar[d] V ar[r] & L of rings where V has weak dimension at most 1 and V → L is a flat, injective, epimorphism of rings.","statement_latex":"Let $A$ be a normal domain with fraction field $K$.\nThere exists a cartesian diagram\n$$\n\\xymatrix{\nA \\ar[d] \\ar[r] & K \\ar[d] \\\\\nV \\ar[r] & L\n}\n$$\nof rings where $V$ has weak dimension at most $1$\nand $V \\to L$ is a flat, injective, epimorphism of rings.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092U","source_file":"more-algebra.tex","source_line":30678,"source_end_line":30690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30678-L30690","statement_sha256":"b38cf77121321cf320f89863dace0eb3d9b2800dcd13e029a7fe7c457a94961d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3570,"rank":3570,"depth":45,"x":527.17,"y":700.74,"cluster":"advanced-algebra"},{"id":"stacks:092V","tag":"092V","title":"Weakly étale ring maps · Lemma 092V","summary":"Let A be a ring of weak dimension at most 1. If A → B is a flat, injective, epimorphism of rings, then A is integrally closed in B.","statement_latex":"Let $A$ be a ring of weak dimension at most $1$.\nIf $A \\to B$ is a flat, injective, epimorphism of rings, then\n$A$ is integrally closed in $B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092V","source_file":"more-algebra.tex","source_line":30705,"source_end_line":30710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30705-L30710","statement_sha256":"cfc1f413b647d1d54f0ca47d81e89019ac4246b70ccd01f2aa715e5854695b0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3571,"rank":3571,"depth":44,"x":776.078,"y":577.922,"cluster":"advanced-algebra"},{"id":"stacks:092W","tag":"092W","title":"Weakly étale ring maps · Lemma 092W","summary":"Let A be a normal domain with fraction field K. Let A → B be weakly étale. Then B is integrally closed in B ⊗_A K.","statement_latex":"Let $A$ be a normal domain with fraction field $K$.\nLet $A \\to B$ be weakly \\'etale. Then\n$B$ is integrally closed in $B \\otimes_A K$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092W","source_file":"more-algebra.tex","source_line":30727,"source_end_line":30732,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30727-L30732","statement_sha256":"fb2f9f0c6822eeb9eff8585b838d2d44a09b3a93a0c346cd891a5960245c1943","origin":"The Stacks Project","memory_eligible":false,"source_rank":3572,"rank":3572,"depth":46,"x":691.249,"y":809.872,"cluster":"advanced-algebra"},{"id":"stacks:092X","tag":"092X","title":"Weakly étale ring maps · Lemma 092X","summary":"Let A → B be a ring homomorphism. Assume • A is a henselian local ring, • A → B is integral, • B is a domain. Then B is a henselian local ring and A → B is a local homomorphism. If A is strictly henselian, then B is a strictly henselian local ring and the extension kappa( m_B)/kappa( m_A) of residue fields is purely inseparable.","statement_latex":"Let $A \\to B$ be a ring homomorphism.\nAssume\n\\begin{enumerate}\n\\item $A$ is a henselian local ring,\n\\item $A \\to B$ is integral,\n\\item $B$ is a domain.\n\\end{enumerate}\nThen $B$ is a henselian local ring and $A \\to B$ is a local homomorphism.\nIf $A$ is strictly henselian, then $B$ is a strictly henselian local ring\nand the extension $\\kappa(\\mathfrak m_B)/\\kappa(\\mathfrak m_A)$\nof residue fields is purely inseparable.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092X","source_file":"more-algebra.tex","source_line":30756,"source_end_line":30769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30756-L30769","statement_sha256":"0f9ed94e58d88720e3b27ada6785ecf1f4e82b28c0d7044371d3953f50af9f6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3573,"rank":3573,"depth":45,"x":567.193,"y":590.56,"cluster":"advanced-algebra"},{"id":"stacks:092Z","tag":"092Z","title":"Olivier · Theorem 092Z","summary":"Let A → B be a local homomorphism of local rings. If A is strictly henselian and A → B is weakly étale, then A = B.","statement_latex":"Let $A \\to B$ be a local homomorphism of local rings.\nIf $A$ is strictly henselian and $A \\to B$ is weakly \\'etale, then\n$A = B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale ring maps","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/092Z","source_file":"more-algebra.tex","source_line":30787,"source_end_line":30792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30787-L30792","statement_sha256":"a61b33f8a7bc70faecf95c0e857e753935cadb4ab06b9d8f28b95f32f85f0867","origin":"The Stacks Project","memory_eligible":false,"source_rank":3574,"rank":3574,"depth":47,"x":835.207,"y":681.944,"cluster":"advanced-algebra"},{"id":"stacks:0CKR","tag":"0CKR","title":"Weakly étale algebras over fields · Lemma 0CKR","summary":"Let K be a field. If B is weakly étale over K, then • B is reduced, • B is integral over K, • any finitely generated K-subalgebra of B is a finite product of finite separable extensions of K, • B is a field if and only if B does not have nontrivial idempotents and in this case it is a separable algebraic extension of K, • any sub or quotient K-algebra of B is weakly étale over K, • if B' is weakly étale over K, then B ⊗_K B' is weakly étale over K.","statement_latex":"Let $K$ be a field. If $B$ is weakly \\'etale over $K$, then\n\\begin{enumerate}\n\\item $B$ is reduced,\n\\item $B$ is integral over $K$,\n\\item any finitely generated $K$-subalgebra of $B$ is a finite product\nof finite separable extensions of $K$,\n\\item $B$ is a field if and only if $B$ does not have nontrivial idempotents\nand in this case it is a separable algebraic extension of $K$,\n\\item any sub or quotient $K$-algebra of $B$ is weakly \\'etale over $K$,\n\\item if $B'$ is weakly \\'etale over $K$, then $B \\otimes_K B'$ is\nweakly \\'etale over $K$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKR","source_file":"more-algebra.tex","source_line":30845,"source_end_line":30859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30845-L30859","statement_sha256":"6df86b5de7f508d3b13ea16b067199ce8076df81e922ba9a319c581c047bd8fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3575,"rank":3575,"depth":44,"x":563.916,"y":766.69,"cluster":"advanced-algebra"},{"id":"stacks:0CKS","tag":"0CKS","title":"Weakly étale algebras over fields · Lemma 0CKS","summary":"Let K be a field. Let A be a K-algebra. There exists a maximal weakly étale K-subalgebra B_max ⊂ A.","statement_latex":"Let $K$ be a field. Let $A$ be a $K$-algebra. There exists\na maximal weakly \\'etale $K$-subalgebra $B_{max} \\subset A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKS","source_file":"more-algebra.tex","source_line":30891,"source_end_line":30895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30891-L30895","statement_sha256":"6600ec265508bd3b285890215264ef523cd00633fef04132cfcd11158470ca06","origin":"The Stacks Project","memory_eligible":false,"source_rank":3576,"rank":3576,"depth":45,"x":695.895,"y":550.124,"cluster":"advanced-algebra"},{"id":"stacks:0CKT","tag":"0CKT","title":"Weakly étale algebras over fields · Lemma 0CKT","summary":"Let K be a field. For a K-algebra A denote B_max(A) the maximal weakly étale K-subalgebra of A as in Lemma [Tag 0CKS]. Then • any K-algebra map A' → A induces a K-algebra map B_max(A') → B_max(A), • if A' ⊂ A, then B_max(A') = B_max(A) ∩ A', • if A = colim A_i is a filtered colimit, then B_max(A) = colim B_max(A_i), • the map B_max(A) → B_max(A_red) is an isomorphism, • B_max(A_1 × … × A_n) = B_max(A_1) × … × B_max(A_n), • if A has no nontrivial idempotents, then B_max(A)…","statement_latex":"Let $K$ be a field. For a $K$-algebra $A$ denote $B_{max}(A)$\nthe maximal weakly \\'etale $K$-subalgebra of $A$ as in\nLemma \\ref{lemma-max-weakly-etale-subalgebra}. Then\n\\begin{enumerate}\n\\item any $K$-algebra map $A' \\to A$ induces a $K$-algebra map\n$B_{max}(A') \\to B_{max}(A)$,\n\\item if $A' \\subset A$, then $B_{max}(A') = B_{max}(A) \\cap A'$,\n\\item if $A = \\colim A_i$ is a filtered colimit, then\n$B_{max}(A) = \\colim B_{max}(A_i)$,\n\\item the map $B_{max}(A) \\to B_{max}(A_{red})$ is an isomorphism,\n\\item $B_{max}(A_1 \\times \\ldots \\times A_n) =\nB_{max}(A_1) \\times \\ldots \\times B_{max}(A_n)$,\n\\item if $A$ has no nontrivial idempotents, then $B_{max}(A)$ is a\nfield and a separable algebraic extension of $K$,\n\\item add more here.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKT","source_file":"more-algebra.tex","source_line":30912,"source_end_line":30930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30912-L30930","statement_sha256":"cb56196d826a70a3c7dfb7e97c10ead38df017d28c692f91fd8837d659cec1e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3577,"rank":3577,"depth":46,"x":772.781,"y":784.855,"cluster":"advanced-algebra"},{"id":"stacks:0CKU","tag":"0CKU","title":"Weakly étale algebras over fields · Lemma 0CKU","summary":"Let L/K be an extension of fields. Let A be a K-algebra. Let B ⊂ A be the maximal weakly étale K-subalgebra of A as in Lemma [Tag 0CKS]. Then B ⊗_K L is the maximal weakly étale L-subalgebra of A ⊗_K L.","statement_latex":"Let $L/K$ be an extension of fields. Let $A$ be a $K$-algebra.\nLet $B \\subset A$ be the maximal weakly \\'etale $K$-subalgebra of\n$A$ as in Lemma \\ref{lemma-max-weakly-etale-subalgebra}.\nThen $B \\otimes_K L$ is the maximal weakly \\'etale $L$-subalgebra\nof $A \\otimes_K L$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Weakly étale algebras over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKU","source_file":"more-algebra.tex","source_line":30979,"source_end_line":30986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L30979-L30986","statement_sha256":"27df0d47a72e5044dd3d1ca6f4e3fb4fdc6348f626abffdc90b5c6cb719020c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3578,"rank":3578,"depth":47,"x":527.165,"y":655.312,"cluster":"advanced-algebra"},{"id":"stacks:0BPZ","tag":"0BPZ","title":"Local irreducibility · Definition 0BPZ","summary":"[EGA4] Let A be a local ring. We say A is unibranch if the reduction A_red is a domain and if the integral closure A' of A_red in its field of fractions is local. We say A is geometrically unibranch if A is unibranch and moreover the residue field of A' is purely inseparable over the residue field of A.","statement_latex":"\\begin{reference}\n\\cite[Chapter 0 (23.2.1)]{EGA4}\n\\end{reference}\nLet $A$ be a local ring. We say $A$ is {\\it unibranch}\nif the reduction $A_{red}$ is a domain and if the integral closure\n$A'$ of $A_{red}$ in its field of fractions is local.\nWe say $A$ is {\\it geometrically unibranch} if $A$ is unibranch\nand moreover the residue field of $A'$ is purely inseparable over\nthe residue field of $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local irreducibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPZ","source_file":"more-algebra.tex","source_line":31123,"source_end_line":31134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31123-L31134","statement_sha256":"6e93275fbbc94d74ec6580d5b833e6f1409f2227cc23fa1b53350c6dd5d9f303","origin":"The Stacks Project","memory_eligible":false,"source_rank":3579,"rank":3579,"depth":0,"x":812.636,"y":611.439,"cluster":"advanced-algebra"},{"id":"stacks:0C24","tag":"0C24","title":"Local irreducibility · Lemma 0C24","summary":"Let A be a local ring. Assume A has finitely many minimal prime ideals. Let A' be the integral closure of A in the total ring of fractions of A_red. Let A^h be the henselization of A. Consider the maps Spec(A') ← Spec((A')^h) → Spec(A^h) where (A')^h = A' ⊗_A A^h. Then • the left arrow is bijective on maximal ideals, • the right arrow is bijective on minimal primes, • every minimal prime of (A')^h is contained in a unique maximal ideal and every maximal ideal contains…","statement_latex":"Let $A$ be a local ring. Assume $A$ has finitely many minimal prime ideals.\nLet $A'$ be the integral closure of $A$ in the total ring of fractions\nof $A_{red}$.\nLet $A^h$ be the henselization of $A$.\nConsider the maps\n$$\n\\Spec(A') \\leftarrow \\Spec((A')^h) \\rightarrow \\Spec(A^h)\n$$\nwhere $(A')^h = A' \\otimes_A A^h$. Then\n\\begin{enumerate}\n\\item the left arrow is bijective on maximal ideals,\n\\item the right arrow is bijective on minimal primes,\n\\item every minimal prime of $(A')^h$ is contained in a unique\nmaximal ideal and every maximal ideal contains exactly one minimal prime.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C24","source_file":"more-algebra.tex","source_line":31154,"source_end_line":31171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31154-L31171","statement_sha256":"adee4d169d5d095df8fb8dc0c63c0c64fab02bf6a918c378a81cbec5c9712967","origin":"The Stacks Project","memory_eligible":false,"source_rank":3580,"rank":3580,"depth":50,"x":637.304,"y":805.897,"cluster":"advanced-algebra"},{"id":"stacks:0BQ0","tag":"0BQ0","title":"Local irreducibility · Lemma 0BQ0","summary":"[EGA4] Let A be a local ring. Let A^h be the henselization of A. The following are equivalent • A is unibranch, and • A^h has a unique minimal prime.","statement_latex":"\\begin{reference}\n\\cite[Chapter IV Proposition 18.6.12]{EGA4}\n\\end{reference}\nLet $A$ be a local ring. Let $A^h$ be the henselization of $A$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A$ is unibranch, and\n\\item $A^h$ has a unique minimal prime.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQ0","source_file":"more-algebra.tex","source_line":31248,"source_end_line":31259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31248-L31259","statement_sha256":"5390dbae34480c79cae3ee135ff215a844c91914dfaea190dc0f76512960867e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3581,"rank":3581,"depth":51,"x":610.196,"y":562.865,"cluster":"advanced-algebra"},{"id":"stacks:0C25","tag":"0C25","title":"Local irreducibility · Lemma 0C25","summary":"Let (A, m, kappa) be a local ring. Assume A has finitely many minimal prime ideals. Let A' be the integral closure of A in the total ring of fractions of A_red. Choose an algebraic closure overlinekappa of kappa and denote kappa^sep ⊂ overlinekappa the separable algebraic closure of kappa. Let A^sh be the strict henselization of A with respect to kappa^sep. Consider the maps Spec(A') xleftarrowc Spec((A')^sh) xrightarrowe Spec(A^sh) where (A')^sh = A' ⊗_A A^sh. Then • for…","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local ring.\nAssume $A$ has finitely many minimal prime ideals.\nLet $A'$ be the integral closure of $A$ in the total ring of fractions\nof $A_{red}$. Choose an algebraic closure $\\overline{\\kappa}$ of\n$\\kappa$ and denote $\\kappa^{sep} \\subset \\overline{\\kappa}$\nthe separable algebraic closure of $\\kappa$.\nLet $A^{sh}$ be the strict henselization of $A$\nwith respect to $\\kappa^{sep}$.\nConsider the maps\n$$\n\\Spec(A') \\xleftarrow{c} \\Spec((A')^{sh}) \\xrightarrow{e} \\Spec(A^{sh})\n$$\nwhere $(A')^{sh} = A' \\otimes_A A^{sh}$. Then\n\\begin{enumerate}\n\\item for $\\mathfrak m' \\subset A'$ maximal the residue field\n$\\kappa'$ is algebraic over $\\kappa$ and the fibre of $c$\nover $\\mathfrak m'$ can be canonically identified\nwith $\\Hom_\\kappa(\\kappa', \\overline{\\kappa})$,\n\\item the right arrow is bijective on minimal primes,\n\\item every minimal prime of $(A')^{sh}$ is contained in a unique\nmaximal ideal and every maximal ideal contains a unique minimal prime.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C25","source_file":"more-algebra.tex","source_line":31338,"source_end_line":31362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31338-L31362","statement_sha256":"07857d82cff71b8e8c2c8e71d7bd19de914876e54bf0fadbfed599243a1f5838","origin":"The Stacks Project","memory_eligible":false,"source_rank":3582,"rank":3582,"depth":51,"x":825.762,"y":726.795,"cluster":"advanced-algebra"},{"id":"stacks:06DM","tag":"06DM","title":"Local irreducibility · Lemma 06DM","summary":"[Etale-coverings] and [EGA4] Let A be a local ring. Let A^sh be a strict henselization of A. The following are equivalent • A is geometrically unibranch, and • A^sh has a unique minimal prime.","statement_latex":"\\begin{reference}\n\\cite[Lemma 2.2]{Etale-coverings} and\n\\cite[Chapter IV Proposition 18.8.15]{EGA4}\n\\end{reference}\nLet $A$ be a local ring. Let $A^{sh}$ be a strict henselization of $A$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A$ is geometrically unibranch, and\n\\item $A^{sh}$ has a unique minimal prime.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DM","source_file":"more-algebra.tex","source_line":31452,"source_end_line":31464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31452-L31464","statement_sha256":"40782ffd5ffbd0eec4ab4ed811956c6b267765a2377bc2330aef39989578ec7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3583,"rank":3583,"depth":52,"x":534.794,"y":728.233,"cluster":"advanced-algebra"},{"id":"stacks:0C26","tag":"0C26","title":"Local irreducibility · Definition 0C26","summary":"Let A be a local ring with henselization A^h and strict henselization A^sh. The number of branches of A is the number of minimal primes of A^h if finite and ∞ otherwise. The number of geometric branches of A is the number of minimal primes of A^sh if finite and ∞ otherwise.","statement_latex":"Let $A$ be a local ring with henselization $A^h$ and\nstrict henselization $A^{sh}$. The {\\it number of branches of $A$}\nis the number of minimal primes of $A^h$ if finite and $\\infty$\notherwise. The {\\it number of geometric branches of $A$}\nis the number of minimal primes of $A^{sh}$ if finite and $\\infty$\notherwise.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local irreducibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C26","source_file":"more-algebra.tex","source_line":31552,"source_end_line":31560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31552-L31560","statement_sha256":"cc52f11a2cb357f16edf129028088b4e5197815928c0c2aeb7313da446edc99b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3584,"rank":3584,"depth":0,"x":748.328,"y":561.966,"cluster":"advanced-algebra"},{"id":"stacks:0C37","tag":"0C37","title":"Local irreducibility · Lemma 0C37","summary":"Let (A, m, kappa) be a local ring. • If A has infinitely many minimal prime ideals, then the number of (geometric) branches of A is ∞. • The number of branches of A is 1 if and only if A is unibranch. • The number of geometric branches of A is 1 if and only if A is geometrically unibranch. Assume A has finitely many minimal primes and let A' be the integral closure of A in the total ring of fractions of A_red. Then • [(4)] the number of branches of A is the number of…","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local ring.\n\\begin{enumerate}\n\\item If $A$ has infinitely many minimal prime ideals, then\nthe number of (geometric) branches of $A$ is $\\infty$.\n\\item The number of branches of $A$ is $1$ if and only if\n$A$ is unibranch.\n\\item The number of geometric branches of $A$ is $1$ if and only if\n$A$ is geometrically unibranch.\n\\end{enumerate}\nAssume $A$ has finitely many minimal primes and let $A'$ be the\nintegral closure of $A$ in the total ring of fractions of $A_{red}$.\nThen\n\\begin{enumerate}\n\\item[(4)] the number of branches of $A$ is the number of maximal ideals\n$\\mathfrak m'$ of $A'$,\n\\item[(5)] to get the number of geometric branches of $A$ we have to count\neach maximal ideal $\\mathfrak m'$ of $A'$ with multiplicity given by the\nseparable degree of $\\kappa(\\mathfrak m')/\\kappa$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C37","source_file":"more-algebra.tex","source_line":31565,"source_end_line":31586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31565-L31586","statement_sha256":"c1bcbacacc04e58734d40c0f8bd79f12f1f7849bcaf84beac2d2dfce2aad3dfc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3585,"rank":3585,"depth":52,"x":724.564,"y":805.887,"cluster":"advanced-algebra"},{"id":"stacks:0DQ1","tag":"0DQ1","title":"Local irreducibility · Lemma 0DQ1","summary":"Let A → B be a local homomorphism of local rings which is the localization of a smooth ring map. • The number of geometric branches of A is equal to the number of geometric branches of B. • If A → B induces a purely inseparable extension of residue fields, then the number of branches of A is the number of branches of B.","statement_latex":"Let $A \\to B$ be a local homomorphism of local rings which\nis the localization of a smooth ring map.\n\\begin{enumerate}\n\\item The number of geometric branches of $A$ is equal to the number\nof geometric branches of $B$.\n\\item If $A \\to B$ induces a purely inseparable extension of residue fields,\nthen the number of branches of $A$ is the number of branches of $B$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQ1","source_file":"more-algebra.tex","source_line":31595,"source_end_line":31605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31595-L31605","statement_sha256":"970dddf38a9d0e7fc0c194c32c9a257c392d8243b973c2af2b941e465b70be88","origin":"The Stacks Project","memory_eligible":false,"source_rank":3586,"rank":3586,"depth":53,"x":545.819,"y":612.417,"cluster":"advanced-algebra"},{"id":"stacks:0GS5","tag":"0GS5","title":"Miscellaneous on branches · Lemma 0GS5","summary":"Let A and B be domains and let A → B be a ring map. Assume A → B has additionally at least one of the following properties • it is the localization of an étale ring map, • it is flat and the localization of an unramified ring map, • it is flat and the localization of a quasi-finite ring map, • it is flat and the localization of an integral ring map, • it is flat and there are no nontrivial specializations between points of fibres of Spec(B) → Spec(A), • Spec(B) → Spec(A)…","statement_latex":"Let $A$ and $B$ be domains and let $A \\to B$ be a ring map.\nAssume $A \\to B$ has additionally at least one of the following properties\n\\begin{enumerate}\n\\item it is the localization of an \\'etale ring map,\n\\item it is flat and the localization of an unramified ring map,\n\\item it is flat and the localization of a quasi-finite ring map,\n\\item it is flat and the localization of an integral ring map,\n\\item it is flat and there are no nontrivial specializations between points\nof fibres of $\\Spec(B) \\to \\Spec(A)$,\n\\item $\\Spec(B) \\to \\Spec(A)$ maps the generic point to the generic\npoint and there are no nontrivial specializations between points of fibres, or\n\\item exactly one point of $\\Spec(B)$ is mapped to the generic\npoint of $\\Spec(A)$.\n\\end{enumerate}\nThen $A \\cap J$ is nonzero for every nonzero ideal $J$ of $B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellaneous on branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GS5","source_file":"more-algebra.tex","source_line":31701,"source_end_line":31718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31701-L31718","statement_sha256":"520c037c73ac06bdde741ebaabccab63293394ed55718bee4bd744483562cd0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3587,"rank":3587,"depth":42,"x":833.388,"y":653.681,"cluster":"advanced-algebra"},{"id":"stacks:0GSC","tag":"0GSC","title":"Miscellaneous on branches · Lemma 0GSC","summary":"Let A → B be a ring map. Let q ⊂ B be a prime ideal lying over the prime p ⊂ A. Assume • A is a domain, • A_ p is geometrically unibranch, • A → B is unramified at q, and • A_ p → B_ q is injective. Then there exists a g ∈ B, g not ∈ q such that B_g is étale over A.","statement_latex":"Let $A \\to B$ be a ring map. Let $\\mathfrak q \\subset B$ be a prime\nideal lying over the prime $\\mathfrak p \\subset A$. Assume\n\\begin{enumerate}\n\\item $A$ is a domain,\n\\item $A_\\mathfrak p$ is geometrically unibranch,\n\\item $A \\to B$ is unramified at $\\mathfrak q$, and\n\\item $A_\\mathfrak p \\to B_\\mathfrak q$ is injective.\n\\end{enumerate}\nThen there exists a $g \\in B$, $g \\not \\in \\mathfrak q$ such that\n$B_g$ is \\'etale over $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellaneous on branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSC","source_file":"more-algebra.tex","source_line":31762,"source_end_line":31774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31762-L31774","statement_sha256":"0736d6ee8f1e777069716f1b394286eba6e943545ea5fced617ed5227f0a89ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":3588,"rank":3588,"depth":54,"x":588.001,"y":786.51,"cluster":"advanced-algebra"},{"id":"stacks:0GS6","tag":"0GS6","title":"Miscellaneous on branches · Lemma 0GS6","summary":"Generalization of [SGA1] Let (A, m) be a geometrically unibranch local domain. Let A → B be an injective local homomorphism of local rings, which is essentially of finite type. If m B is the maximal ideal of B and the induced extension of residue fields is separable, then A → B is the localization of an étale ring map.","statement_latex":"\\begin{reference}\nGeneralization of \\cite[Expose I, Theorem 9.5 part (ii)]{SGA1}\n\\end{reference}\nLet $(A, \\mathfrak m)$ be a geometrically unibranch local domain. Let\n$A \\to B$ be an injective local homomorphism of local rings,\nwhich is essentially of finite type. If $\\mathfrak m B$\nis the maximal ideal of $B$ and the induced extension of\nresidue fields is separable, then $A \\to B$ is the localization\nof an \\'etale ring map.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellaneous on branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GS6","source_file":"more-algebra.tex","source_line":31817,"source_end_line":31828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31817-L31828","statement_sha256":"8ef06af7c0a09ac38d2c8a002b2f89695f64eb57f43cb800dd01602fef309284","origin":"The Stacks Project","memory_eligible":false,"source_rank":3589,"rank":3589,"depth":54,"x":662.174,"y":549.176,"cluster":"advanced-algebra"},{"id":"stacks:06DU","tag":"06DU","title":"Miscellaneous on branches · Lemma 06DU","summary":"Let k be an algebraically closed field. Let A, B be strictly henselian local k-algebras with residue field equal to k. Let C be the strict henselization of A ⊗_k B at the maximal ideal m_A ⊗_k B + A ⊗_k m_B. Then the minimal primes of C correspond 1-to-1 to pairs of minimal primes of A and B.","statement_latex":"Let $k$ be an algebraically closed field. Let $A$, $B$ be strictly\nhenselian local $k$-algebras with residue field equal to $k$.\nLet $C$ be the strict henselization of $A \\otimes_k B$ at the maximal\nideal $\\mathfrak m_A \\otimes_k B + A \\otimes_k \\mathfrak m_B$.\nThen the minimal primes of $C$ correspond $1$-to-$1$ to pairs of\nminimal primes of $A$ and $B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Miscellaneous on branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DU","source_file":"more-algebra.tex","source_line":31855,"source_end_line":31863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31855-L31863","statement_sha256":"92f10040752694590bcec09718864d80a7b6f12cebc1c20d58e71082b7addc3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3590,"rank":3590,"depth":53,"x":798.423,"y":766.407,"cluster":"advanced-algebra"},{"id":"stacks:0C28","tag":"0C28","title":"Branches of the completion · Lemma 0C28","summary":"Let (A, m) be a Noetherian local ring. • The map A^h → A^wedge defines a surjective map from minimal primes of A^wedge to minimal primes of A^h. • The number of branches of A is at most the number of branches of A^wedge. • The number of geometric branches of A is at most the number of geometric branches of A^wedge.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\n\\begin{enumerate}\n\\item The map $A^h \\to A^\\wedge$ defines a surjective map from minimal\nprimes of $A^\\wedge$ to minimal primes of $A^h$.\n\\item The number of branches of $A$ is at most the number of branches\nof $A^\\wedge$.\n\\item The number of geometric branches of $A$ is at most the number\nof geometric branches of $A^\\wedge$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Branches of the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C28","source_file":"more-algebra.tex","source_line":31935,"source_end_line":31946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31935-L31946","statement_sha256":"ddf43f3e18f61bddb3fbef10f3e2f790918b5598645598bbb444a232c1d38687","origin":"The Stacks Project","memory_eligible":false,"source_rank":3591,"rank":3591,"depth":48,"x":523.092,"y":683.483,"cluster":"advanced-algebra"},{"id":"stacks:0C29","tag":"0C29","title":"Branches of the completion · Lemma 0C29","summary":"Let (A, m) be a Noetherian local ring. The number of branches of A is the same as the number of branches of A^wedge if and only if sqrt qA^wedge is prime for every minimal prime q ⊂ A^h of the henselization.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. The number of\nbranches of $A$ is the same as the number of branches of $A^\\wedge$\nif and only if $\\sqrt{\\mathfrak qA^\\wedge}$ is prime for every\nminimal prime $\\mathfrak q \\subset A^h$ of the henselization.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Branches of the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C29","source_file":"more-algebra.tex","source_line":31963,"source_end_line":31969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31963-L31969","statement_sha256":"ed1a520637d42d51981711a8f6702894b06bad3aa5166479fcc52e6be213cda1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3592,"rank":3592,"depth":49,"x":792.971,"y":588.341,"cluster":"advanced-algebra"},{"id":"stacks:0C2A","tag":"0C2A","title":"Branches of the completion · Lemma 0C2A","summary":"Let A be a ring and let I be a finitely generated ideal. Let A → C be a ring map such that for all f ∈ I the ring map A_f → C_f is localization at an idempotent. Then there exists a surjection A → C' such that A_f → (C × C')_f is an isomorphism for all f ∈ I.","statement_latex":"Let $A$ be a ring and let $I$ be a finitely generated ideal.\nLet $A \\to C$ be a ring map such that for all $f \\in I$\nthe ring map $A_f \\to C_f$ is localization at an idempotent.\nThen there exists a surjection $A \\to C'$ such that\n$A_f \\to (C \\times C')_f$ is an isomorphism for all $f \\in I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Branches of the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2A","source_file":"more-algebra.tex","source_line":31983,"source_end_line":31990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L31983-L31990","statement_sha256":"8f2f48b98b810bdf7a566f171ae884cb7a3c7b4a90e809691220a089a909f8f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3593,"rank":3593,"depth":3,"x":670.401,"y":811.773,"cluster":"advanced-algebra"},{"id":"stacks:0ALR","tag":"0ALR","title":"Branches of the completion · Lemma 0ALR","summary":"Let A be a Noetherian ring and I an ideal. Let B be a finite type A-algebra. Let B^wedge → C be a surjective ring map with kernel J where B^wedge is the I-adic completion. If J/J^2 is annihilated by I^c for some c ≥ 0, then C is isomorphic to the completion of a finite type A-algebra.","statement_latex":"Let $A$ be a Noetherian ring and $I$ an ideal. Let $B$\nbe a finite type $A$-algebra. Let $B^\\wedge \\to C$ be a surjective\nring map with kernel $J$ where $B^\\wedge$ is the $I$-adic completion.\nIf $J/J^2$ is annihilated by $I^c$ for some $c \\geq 0$, then $C$ is\nisomorphic to the completion of a finite type $A$-algebra.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Branches of the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALR","source_file":"more-algebra.tex","source_line":32036,"source_end_line":32043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32036-L32043","statement_sha256":"7190bc1d0277357e14634f9b69643f3eaf494836c9763465cf6d9a265a3ccca3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3594,"rank":3594,"depth":47,"x":581.048,"y":577.322,"cluster":"advanced-algebra"},{"id":"stacks:0C2B","tag":"0C2B","title":"Branches of the completion · Lemma 0C2B","summary":"Let (A, m) be a Noetherian local ring with henselization A^h. Let q ⊂ A^wedge be a minimal prime with dim(A^wedge/ q) = 1. Then there exists a minimal prime q^h of A^h such that q = sqrt q^hA^wedge.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring with henselization $A^h$.\nLet $\\mathfrak q \\subset A^\\wedge$ be a minimal prime with\n$\\dim(A^\\wedge/\\mathfrak q) = 1$. Then there exists a minimal\nprime $\\mathfrak q^h$ of $A^h$ such that\n$\\mathfrak q = \\sqrt{\\mathfrak q^hA^\\wedge}$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Branches of the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2B","source_file":"more-algebra.tex","source_line":32099,"source_end_line":32106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32099-L32106","statement_sha256":"ae58d6aca2efb85ae9975794925e36c1b90b03b93e01b5dbe054bde0503b823a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3595,"rank":3595,"depth":48,"x":835.634,"y":699.578,"cluster":"advanced-algebra"},{"id":"stacks:0C2C","tag":"0C2C","title":"Branches of the completion · Lemma 0C2C","summary":"Let (A, m) be a Noetherian local ring. The punctured spectrum of A^wedge is disconnected if and only if the punctured spectrum of A^h is disconnected.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. The punctured spectrum\nof $A^\\wedge$ is disconnected if and only if the punctured spectrum of $A^h$\nis disconnected.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Branches of the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2C","source_file":"more-algebra.tex","source_line":32144,"source_end_line":32149,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32144-L32149","statement_sha256":"090a756d47c4828a453783d1330a1c4854af01d72d7f42b6840b38eb61aefdfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3596,"rank":3596,"depth":49,"x":549.413,"y":753.92,"cluster":"advanced-algebra"},{"id":"stacks:0C2D","tag":"0C2D","title":"Branches of the completion · Lemma 0C2D","summary":"Let (A, m) be a Noetherian local ring of dimension 1. Then the number of (geometric) branches of A and A^wedge is the same.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.\nThen the number of (geometric) branches of $A$ and $A^\\wedge$ is the same.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Branches of the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2D","source_file":"more-algebra.tex","source_line":32209,"source_end_line":32213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32209-L32213","statement_sha256":"84dd30c048c026872134b4b20e859b3d56518a461b3dc7b80fa4d0fea14f8b2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3597,"rank":3597,"depth":50,"x":716.872,"y":551.314,"cluster":"advanced-algebra"},{"id":"stacks:0C2E","tag":"0C2E","title":"Branches of the completion · Lemma 0C2E","summary":"[Beddani] Let (A, m) be a Noetherian local ring. If the formal fibres of A are geometrically normal (for example if A is excellent or quasi-excellent), then A is Nagata and the number of (geometric) branches of A and A^wedge is the same.","statement_latex":"\\begin{reference}\n\\cite[Theorem 2.3]{Beddani}\n\\end{reference}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring. If the formal\nfibres of $A$ are geometrically normal (for example if $A$ is\nexcellent or quasi-excellent), then $A$ is Nagata\nand the number of (geometric) branches of $A$ and $A^\\wedge$ is the same.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Branches of the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2E","source_file":"more-algebra.tex","source_line":32230,"source_end_line":32239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32230-L32239","statement_sha256":"cd6aeb5908e756fbc9806aea1d357485bc1fb9220b3c16f02920c811abe7e370","origin":"The Stacks Project","memory_eligible":false,"source_rank":3598,"rank":3598,"depth":53,"x":756.344,"y":795.885,"cluster":"advanced-algebra"},{"id":"stacks:0AW2","tag":"0AW2","title":"Formally catenary rings · Definition 0AW2","summary":"A Noetherian local ring A is formally catenary if for every minimal prime p ⊂ A the spectrum of A^wedge/ p A^wedge is equidimensional.","statement_latex":"A Noetherian local ring $A$ is {\\it formally catenary}\nif for every minimal prime $\\mathfrak p \\subset A$ the spectrum of\n$A^\\wedge/\\mathfrak p A^\\wedge$ is equidimensional.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally catenary rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AW2","source_file":"more-algebra.tex","source_line":32309,"source_end_line":32314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32309-L32314","statement_sha256":"a7c7bef9780070a142c3e24e941f4d358b45bdcd58fb55acb0888f350d686521","origin":"The Stacks Project","memory_eligible":false,"source_rank":3599,"rank":3599,"depth":0,"x":530.421,"y":637.842,"cluster":"advanced-algebra"},{"id":"stacks:0AW3","tag":"0AW3","title":"Formally catenary rings · Lemma 0AW3","summary":"Let (A, m) be a Noetherian local ring which is not formally catenary. Then A is not universally catenary.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring which is not\nformally catenary. Then $A$ is not universally catenary.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AW3","source_file":"more-algebra.tex","source_line":32324,"source_end_line":32328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32324-L32328","statement_sha256":"431fe7b32634f46cd9a6ff9373455d3e8c0959ed04f4a684f4c64775c3fd6c85","origin":"The Stacks Project","memory_eligible":false,"source_rank":3600,"rank":3600,"depth":48,"x":824.289,"y":626.179,"cluster":"advanced-algebra"},{"id":"stacks:0AW4","tag":"0AW4","title":"Formally catenary rings · Lemma 0AW4","summary":"Let A → B be a flat local ring map of local Noetherian rings. Assume B is catenary and is Spec(B) equidimensional. Then • Spec(B/ p B) is equidimensional for all p ⊂ A and • A is catenary and Spec(A) is equidimensional.","statement_latex":"Let $A \\to B$ be a flat local ring map of local Noetherian rings.\nAssume $B$ is catenary and is $\\Spec(B)$ equidimensional. Then\n\\begin{enumerate}\n\\item $\\Spec(B/\\mathfrak p B)$ is equidimensional for all\n$\\mathfrak p \\subset A$ and\n\\item $A$ is catenary and $\\Spec(A)$ is equidimensional.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AW4","source_file":"more-algebra.tex","source_line":32388,"source_end_line":32397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32388-L32397","statement_sha256":"003e5d23d6e1ccdc34dd4e48dd2bb5f0fcef2e070444e88806e387aba03233b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3601,"rank":3601,"depth":12,"x":616.846,"y":801.635,"cluster":"advanced-algebra"},{"id":"stacks:0AW5","tag":"0AW5","title":"Formally catenary rings · Lemma 0AW5","summary":"Let A be a formally catenary Noetherian local ring. Then A is universally catenary.","statement_latex":"Let $A$ be a formally catenary Noetherian local ring.\nThen $A$ is universally catenary.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AW5","source_file":"more-algebra.tex","source_line":32437,"source_end_line":32441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32437-L32441","statement_sha256":"16a13563c907bba04d05b0f9bf3a864e088f15e1672ebbed28260a47add571fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3602,"rank":3602,"depth":13,"x":628.721,"y":554.395,"cluster":"advanced-algebra"},{"id":"stacks:0AW6","tag":"0AW6","title":"Ratliff · Proposition 0AW6","summary":"[Ratliff] A Noetherian local ring is universally catenary if and only if it is formally catenary.","statement_latex":"\\begin{reference}\n\\cite{Ratliff}\n\\end{reference}\nA Noetherian local ring is universally catenary if and only if\nit is formally catenary.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally catenary rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AW6","source_file":"more-algebra.tex","source_line":32483,"source_end_line":32490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32483-L32490","statement_sha256":"73b22cd37f1d1ac976ca2609a6008171ffca7de3628ce3d2335caf2b2177d88b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3603,"rank":3603,"depth":49,"x":818.906,"y":743.562,"cluster":"advanced-algebra"},{"id":"stacks:0C2F","tag":"0C2F","title":"Formally catenary rings · Lemma 0C2F","summary":"[Heinzer-Rotthaus-Wiegand] Let (A, m) be a Noetherian local ring with geometrically normal formal fibres. Then • A^h is universally catenary, and • if A is unibranch (for example normal), then A is universally catenary.","statement_latex":"\\begin{reference}\n\\cite[Corollary 2.3]{Heinzer-Rotthaus-Wiegand}\n\\end{reference}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring with\ngeometrically normal formal fibres. Then\n\\begin{enumerate}\n\\item $A^h$ is universally catenary, and\n\\item if $A$ is unibranch (for example normal), then\n$A$ is universally catenary.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Formally catenary rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2F","source_file":"more-algebra.tex","source_line":32497,"source_end_line":32509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32497-L32509","statement_sha256":"b0530881ad323ff4ef436c5c643329961bf2460f64bcb0013bc651437f477ff2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3604,"rank":3604,"depth":54,"x":526.364,"y":711.969,"cluster":"advanced-algebra"},{"id":"stacks:0BRF","tag":"0BRF","title":"Group actions and integral closure · Lemma 0BRF","summary":"Let φ : A → B be a surjection of rings. Let G be a finite group of order n acting on φ : A → B. If b ∈ B^G, then there exists a monic polynomial P ∈ A^G[T] which maps to (T - b)^n in B^G[T].","statement_latex":"Let $\\varphi : A \\to B$ be a surjection of rings. Let $G$ be a finite group\nof order $n$ acting on $\\varphi : A \\to B$. If $b \\in B^G$, then\nthere exists a monic polynomial $P \\in A^G[T]$ which maps to\n$(T - b)^n$ in $B^G[T]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRF","source_file":"more-algebra.tex","source_line":32539,"source_end_line":32545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32539-L32545","statement_sha256":"bbc786f7374eb4338a0b4c075b65ca657d1411a6a0d5e37406f3ca13774faa88","origin":"The Stacks Project","memory_eligible":false,"source_rank":3605,"rank":3605,"depth":0,"x":767.634,"y":569.182,"cluster":"advanced-algebra"},{"id":"stacks:09EG","tag":"09EG","title":"Group actions and integral closure · Lemma 09EG","summary":"Let R be a ring. Let G be a finite group acting on R. Let I ⊂ R be an ideal such that σ(I) ⊂ I for all σ ∈ G. Then R^G/I^G ⊂ (R/I)^G is an integral extension of rings which induces a homeomorphism on spectra and purely inseparable extensions of residue fields.","statement_latex":"Let $R$ be a ring. Let $G$ be a finite group acting on $R$. Let $I \\subset R$\nbe an ideal such that $\\sigma(I) \\subset I$ for all $\\sigma \\in G$.\nThen $R^G/I^G \\subset (R/I)^G$ is an integral extension of rings which\ninduces a homeomorphism on spectra and purely inseparable extensions of\nresidue fields.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EG","source_file":"more-algebra.tex","source_line":32552,"source_end_line":32559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32552-L32559","statement_sha256":"05757c353ab8463f34d255299d58052dca258ea5c137232798fe74b81ffb03d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3606,"rank":3606,"depth":9,"x":704.512,"y":811.522,"cluster":"advanced-algebra"},{"id":"stacks:0H34","tag":"0H34","title":"Group actions and integral closure · Lemma 0H34","summary":"Let G be a finite group of order n acting on a ring R. Let J ⊂ R^G be an ideal. For x ∈ JR we have ∏_σ ∈ G (T - σ(x)) = T^n + a_1 T^n - 1 + … + a_n with a_i ∈ J.","statement_latex":"Let $G$ be a finite group of order $n$ acting on a ring $R$.\nLet $J \\subset R^G$ be an ideal. For $x \\in JR$\nwe have $\\prod_{\\sigma \\in G} (T - \\sigma(x)) =\nT^n + a_1 T^{n - 1} + \\ldots + a_n$ with $a_i \\in J$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H34","source_file":"more-algebra.tex","source_line":32568,"source_end_line":32574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32568-L32574","statement_sha256":"4d9e651f2de4235b4cde2a44c037aaef31788fc406369cfc18461ac5f7107fe8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3607,"rank":3607,"depth":0,"x":556.083,"y":596.875,"cluster":"advanced-algebra"},{"id":"stacks:0H35","tag":"0H35","title":"Group actions and integral closure · Lemma 0H35","summary":"Let R be a ring. Let G be a finite group of order n acting on R. Let J ⊂ R^G be an ideal. Then R^G/J → (R/JR)^G is ring map such that • for b ∈ (R/JR)^G there is a monic polynomial P ∈ R^G/J[T] whose image in (R/JR)^G[T] is (T - b)^n, • for a ∈ Ker(R^G/J → (R/JR)^G) we have (T - a)^n = T^n in R^G/J[T]. In particular, R^G/J → (R/JR)^G is an integral ring map which induces homeomorphisms on spectra and purely inseparable extensions of residue fields.","statement_latex":"Let $R$ be a ring. Let $G$ be a finite group of order $n$ acting on $R$.\nLet $J \\subset R^G$ be an ideal. Then $R^G/J \\to (R/JR)^G$ is ring map\nsuch that\n\\begin{enumerate}\n\\item for $b \\in (R/JR)^G$ there is a monic polynomial\n$P \\in R^G/J[T]$ whose image in $(R/JR)^G[T]$ is $(T - b)^n$,\n\\item for $a \\in \\Ker(R^G/J \\to (R/JR)^G)$\nwe have $(T - a)^n = T^n$ in $R^G/J[T]$.\n\\end{enumerate}\nIn particular, $R^G/J \\to (R/JR)^G$ is an integral ring map which\ninduces homeomorphisms on spectra and purely inseparable extensions of\nresidue fields.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H35","source_file":"more-algebra.tex","source_line":32602,"source_end_line":32616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32602-L32616","statement_sha256":"1c5aa9ba1fcee73661c5fdaeb7cb14f496f0cf5385c5a2cbfc337984e9c3ba84","origin":"The Stacks Project","memory_eligible":false,"source_rank":3608,"rank":3608,"depth":9,"x":838.319,"y":670.976,"cluster":"advanced-algebra"},{"id":"stacks:0BRG","tag":"0BRG","title":"Group actions and integral closure · Lemma 0BRG","summary":"Let R be a ring. Let G be a finite group of order n acting on R. Let A be an R^G-algebra. • for b ∈ (A ⊗_R^G R)^G there exists a monic polynomial P ∈ A[T] whose image in (A ⊗_R^G R)^G[T] is (T - b)^n, • for a ∈ Ker(A → (A ⊗_R^G R)^G) we have (T - a)^n = T^n in A[T].","statement_latex":"Let $R$ be a ring. Let $G$ be a finite group of order $n$ acting on $R$.\nLet $A$ be an $R^G$-algebra.\n\\begin{enumerate}\n\\item for $b \\in (A \\otimes_{R^G} R)^G$ there exists a monic polynomial\n$P \\in A[T]$ whose image in $(A \\otimes_{R^G} R)^G[T]$ is $(T - b)^n$,\n\\item for $a \\in \\Ker(A \\to (A \\otimes_{R^G} R)^G)$ we have\n$(T - a)^n = T^n$ in $A[T]$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRG","source_file":"more-algebra.tex","source_line":32634,"source_end_line":32644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32634-L32644","statement_sha256":"cee0e79ca343ff5db228a04c617be7a7d067390ca3a248bdb21e13f2e66da2fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3609,"rank":3609,"depth":10,"x":570.448,"y":776.547,"cluster":"advanced-algebra"},{"id":"stacks:0BRH","tag":"0BRH","title":"Group actions and integral closure · Lemma 0BRH","summary":"Let R be a ring. Let G be a finite group acting on R. Let R^G → A be a ring map. The map A → (A ⊗_R^G R)^G is an isomorphism if R^G → A is flat. In general the map is integral, induces a homeomorphism on spectra, and induces purely inseparable residue field extensions.","statement_latex":"Let $R$ be a ring. Let $G$ be a finite group acting on $R$.\nLet $R^G \\to A$ be a ring map. The map\n$$\nA \\to (A \\otimes_{R^G} R)^G\n$$\nis an isomorphism if $R^G \\to A$ is flat. In general the map\nis integral, induces a homeomorphism on spectra, and\ninduces purely inseparable residue field extensions.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRH","source_file":"more-algebra.tex","source_line":32655,"source_end_line":32665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32655-L32665","statement_sha256":"0dc4822d12e52136730b0e3c53f67bfbbf4ffff6798ff0c49943fe13a1ce2391","origin":"The Stacks Project","memory_eligible":false,"source_rank":3610,"rank":3610,"depth":11,"x":683.143,"y":546.564,"cluster":"advanced-algebra"},{"id":"stacks:0BRI","tag":"0BRI","title":"Group actions and integral closure · Lemma 0BRI","summary":"Let G be a finite group acting on a ring R. For any two primes q, q' ⊂ R lying over the same prime in R^G there exists a σ ∈ G with σ( q) = q'.","statement_latex":"Let $G$ be a finite group acting on a ring $R$. For any two primes\n$\\mathfrak q, \\mathfrak q' \\subset R$ lying over the same prime in $R^G$\nthere exists a $\\sigma \\in G$ with $\\sigma(\\mathfrak q) = \\mathfrak q'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRI","source_file":"more-algebra.tex","source_line":32679,"source_end_line":32684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32679-L32684","statement_sha256":"5b459b7c0d1bf918e453abcb8d0838eb41f971f13f395441d932cb6918bb01f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3611,"rank":3611,"depth":8,"x":785.052,"y":780.239,"cluster":"advanced-algebra"},{"id":"stacks:0BRJ","tag":"0BRJ","title":"Group actions and integral closure · Lemma 0BRJ","summary":"Let G be a finite group acting on a ring R. Let q ⊂ R be a prime lying over p ⊂ R^G. Then kappa( q)/kappa( p) is an algebraic normal extension and the map D = (σ ∈ G mid σ( q) = q) → Aut(kappa( q)/kappa( p)) is surjective only in the case of Galois extensions..","statement_latex":"Let $G$ be a finite group acting on a ring $R$. Let $\\mathfrak q \\subset R$\nbe a prime lying over $\\mathfrak p \\subset R^G$. Then\n$\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$ is an algebraic normal\nextension and the map\n$$\nD = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\\}\n\\longrightarrow\n\\text{Aut}(\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p))\n$$\nis surjective\\footnote{Recall that we use the notation $\\text{Gal}$\nonly in the case of Galois extensions.}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRJ","source_file":"more-algebra.tex","source_line":32703,"source_end_line":32716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32703-L32716","statement_sha256":"4730e03e2e543de15fa06ef4d7bed71c872d0f006b91e425ae0e0e498f4e8dcc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3612,"rank":3612,"depth":12,"x":521.831,"y":665.686,"cluster":"advanced-algebra"},{"id":"stacks:0BRK","tag":"0BRK","title":"Group actions and integral closure · Lemma 0BRK","summary":"Let A be a normal domain with fraction field K. Let L/K be a (possibly infinite) Galois extension. Let G = Gal(L/K) and let B be the integral closure of A in L. • For any two primes q, q' ⊂ B lying over the same prime in A there exists a σ ∈ G with σ( q) = q'. • Let q ⊂ B be a prime lying over p ⊂ A. Then kappa( q)/kappa( p) is an algebraic normal extension and the map D = (σ ∈ G mid σ( q) = q) → Aut(kappa( q)/kappa( p)) is surjective.","statement_latex":"Let $A$ be a normal domain with fraction field $K$.\nLet $L/K$ be a (possibly infinite) Galois extension.\nLet $G = \\text{Gal}(L/K)$ and let\n$B$ be the integral closure of $A$ in $L$.\n\\begin{enumerate}\n\\item For any two primes\n$\\mathfrak q, \\mathfrak q' \\subset B$ lying over the same prime in $A$\nthere exists a $\\sigma \\in G$ with $\\sigma(\\mathfrak q) = \\mathfrak q'$.\n\\item Let $\\mathfrak q \\subset B$ be a prime lying over\n$\\mathfrak p \\subset A$. Then $\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p)$\nis an algebraic normal extension and the map\n$$\nD = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\\}\n\\longrightarrow\n\\text{Aut}(\\kappa(\\mathfrak q)/\\kappa(\\mathfrak p))\n$$\nis surjective.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRK","source_file":"more-algebra.tex","source_line":32776,"source_end_line":32796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32776-L32796","statement_sha256":"c2b9c011a0dd1d9469106a2ac698c365cc50fce350d63e76180dac66714d3837","origin":"The Stacks Project","memory_eligible":false,"source_rank":3613,"rank":3613,"depth":13,"x":808.221,"y":600.758,"cluster":"advanced-algebra"},{"id":"stacks:0BSX","tag":"0BSX","title":"Group actions and integral closure · Lemma 0BSX","summary":"Let A be a normal domain with fraction field K. Let M/L/K be a tower of (possibly infinite) Galois extensions of K. Let H = Gal(M/K) and G = Gal(L/K) and let C and B be the integral closure of A in M and L. Let r ⊂ C and q = B ∩ r. Set D_ r = (τ ∈ H mid τ( r) = r) and I_ r = (τ ∈ D_ r mid τ bmod r = id_kappa( r)) and similarly for D_ q and I_ q. Under the map H → G the induced maps D_ r → D_ q and I_ r → I_ q are surjective.","statement_latex":"Let $A$ be a normal domain with fraction field $K$.\nLet $M/L/K$ be a tower of (possibly infinite) Galois extensions of $K$.\nLet $H = \\text{Gal}(M/K)$ and $G = \\text{Gal}(L/K)$ and let\n$C$ and $B$ be the integral closure of $A$ in $M$ and $L$.\nLet $\\mathfrak r \\subset C$ and $\\mathfrak q = B \\cap \\mathfrak r$.\nSet\n$D_\\mathfrak r = \\{\\tau \\in H \\mid \\tau(\\mathfrak r) = \\mathfrak r\\}$\nand\n$I_\\mathfrak r = \\{\\tau \\in D_\\mathfrak r \\mid\n\\tau \\bmod \\mathfrak r = \\text{id}_{\\kappa(\\mathfrak r)}\\}$\nand similarly for $D_\\mathfrak q$ and $I_\\mathfrak q$.\nUnder the map $H \\to G$ the induced maps\n$D_\\mathfrak r \\to D_\\mathfrak q$ and\n$I_\\mathfrak r \\to I_\\mathfrak q$ are surjective.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSX","source_file":"more-algebra.tex","source_line":32868,"source_end_line":32884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32868-L32884","statement_sha256":"a1e77dd7c2b0e6944eafbd96359bcebe923759180e751f58c9074e1106a1a9ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":3614,"rank":3614,"depth":14,"x":649.158,"y":811.267,"cluster":"advanced-algebra"},{"id":"stacks:09E4","tag":"09E4","title":"Extensions of discrete valuation rings · Definition 09E4","summary":"We say that A → B or A ⊂ B is an extension of discrete valuation rings if A and B are discrete valuation rings and A → B is injective and local. In particular, if π_A and π_B are uniformizers of A and B, then π_A = u π_B^e for some e ≥ 1 and unit u of B. The integer e does not depend on the choice of the uniformizers as it is also the unique integer ≥ 1 such that m_A B = m_B^e The integer e is called the ramification index of B over A. We say that B is weakly unramified…","statement_latex":"We say that $A \\to B$ or $A \\subset B$ is an\n{\\it extension of discrete valuation rings} if $A$ and $B$ are\ndiscrete valuation rings and $A \\to B$ is injective and local.\nIn particular, if $\\pi_A$ and $\\pi_B$ are uniformizers of\n$A$ and $B$, then $\\pi_A = u \\pi_B^e$ for some $e \\geq 1$ and unit\n$u$ of $B$. The integer $e$ does not depend on the choice of\nthe uniformizers as it is also the unique integer $\\geq 1$ such that\n$$\n\\mathfrak m_A B = \\mathfrak m_B^e\n$$\nThe integer $e$ is called the {\\it ramification index} of $B$ over $A$.\nWe say that $B$ is {\\it weakly unramified} over $A$ if $e = 1$.\nIf the extension of residue fields\n$\\kappa_A = A/\\mathfrak m_A \\subset \\kappa_B = B/\\mathfrak m_B$\nis finite, then we set $f = [\\kappa_B : \\kappa_A]$ and we\ncall it the {\\it residual degree} or {\\it residue degree}\nof the extension $A \\subset B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of discrete valuation rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09E4","source_file":"more-algebra.tex","source_line":32911,"source_end_line":32930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32911-L32930","statement_sha256":"1fccae02e514c93c92d0a744be86bba0448272b5ac6d020ed605934c0540caf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3615,"rank":3615,"depth":0,"x":597.131,"y":565.635,"cluster":"advanced-algebra"},{"id":"stacks:09E5","tag":"09E5","title":"Extensions of discrete valuation rings · Lemma 09E5","summary":"Let A ⊂ B be an extension of discrete valuation rings with fraction fields K ⊂ L. If the extension L/K is finite, then the residue field extension is finite and we have ef ≤ [L : K].","statement_latex":"Let $A \\subset B$ be an extension of discrete valuation rings with\nfraction fields $K \\subset L$. If the extension $L/K$\nis finite, then the residue field extension is finite and we have\n$ef \\leq [L : K]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of discrete valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09E5","source_file":"more-algebra.tex","source_line":32935,"source_end_line":32941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32935-L32941","statement_sha256":"24dc8de626908fa9649f7ce95734454835ff3b03f4583f06fea10c3f67b8b6f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3616,"rank":3616,"depth":8,"x":833.168,"y":717.331,"cluster":"advanced-algebra"},{"id":"stacks:0BRL","tag":"0BRL","title":"Extensions of discrete valuation rings · Lemma 0BRL","summary":"Let A ⊂ B ⊂ C be extensions of discrete valuation rings. Then the ramification indices of B/A and C/B multiply to give the ramification index of C/A. In a formula e_C/A = e_B/A e_C/B. Similarly for the residual degrees in case they are finite.","statement_latex":"Let $A \\subset B \\subset C$ be extensions of discrete valuation rings.\nThen the ramification indices of $B/A$ and $C/B$ multiply to give\nthe ramification index of $C/A$. In a formula $e_{C/A} = e_{B/A} e_{C/B}$.\nSimilarly for the residual degrees in case they are finite.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of discrete valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRL","source_file":"more-algebra.tex","source_line":32951,"source_end_line":32957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32951-L32957","statement_sha256":"d56308c5b047f467f434aeb87cd0af3642b9abfed7da4a7582693a271eef9a21","origin":"The Stacks Project","memory_eligible":false,"source_rank":3617,"rank":3617,"depth":1,"x":536.948,"y":739.42,"cluster":"advanced-algebra"},{"id":"stacks:09E6","tag":"09E6","title":"Extensions of discrete valuation rings · Lemma 09E6","summary":"Let A ⊂ B be an extension of discrete valuation rings inducing the field extension K ⊂ L. If the characteristic of K is p > 0 and L is purely inseparable over K, then the ramification index e is a power of p.","statement_latex":"Let $A \\subset B$ be an extension of discrete valuation rings\ninducing the field extension $K \\subset L$. If the characteristic\nof $K$ is $p > 0$ and $L$ is purely inseparable over $K$, then\nthe ramification index $e$ is a power of $p$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of discrete valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09E6","source_file":"more-algebra.tex","source_line":32964,"source_end_line":32970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32964-L32970","statement_sha256":"241c10dac92c077699aa41d9ecb1cef039d749c0e00e75ade3abaa84aec51da2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3618,"rank":3618,"depth":0,"x":737.735,"y":554.938,"cluster":"advanced-algebra"},{"id":"stacks:09E7","tag":"09E7","title":"Extensions of discrete valuation rings · Lemma 09E7","summary":"Let A ⊂ B be an extension of discrete valuation rings. The following are equivalent • A → B is formally smooth in the m_B-adic topology, and • A → B is weakly unramified and kappa_B/kappa_A is a separable field extension.","statement_latex":"Let $A \\subset B$ be an extension of discrete valuation rings.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A \\to B$ is formally smooth in the $\\mathfrak m_B$-adic topology, and\n\\item $A \\to B$ is weakly unramified and $\\kappa_B/\\kappa_A$\nis a separable field extension.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of discrete valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09E7","source_file":"more-algebra.tex","source_line":32988,"source_end_line":32997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L32988-L32997","statement_sha256":"adf0fb0a3396555ee3ea289a5dca0de3e8a952c7e265992ea2f87a3015458e6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3619,"rank":3619,"depth":46,"x":738.033,"y":805.055,"cluster":"advanced-algebra"},{"id":"stacks:09E9","tag":"09E9","title":"Extensions of discrete valuation rings · Definition 09E9","summary":"Let A be a discrete valuation ring with fraction field K. Let L/K be a finite separable extension. With B and m_i, i = 1, …, n as in Remark [Tag 09E8] we say the extension L/K is • unramified with respect to A if e_i = 1 and the extension kappa( m_i)/kappa_A is separable for all i, • tamely ramified with respect to A if either the characteristic of kappa_A is 0 or the characteristic of kappa_A is p > 0, the field extensions kappa( m_i)/kappa_A are separable, and the…","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$\nbe a finite separable extension. With $B$ and\n$\\mathfrak m_i$, $i = 1, \\ldots, n$\nas in Remark \\ref{remark-finite-separable-extension} we say the extension\n$L/K$ is\n\\begin{enumerate}\n\\item {\\it unramified with respect to $A$} if $e_i = 1$ and the extension\n$\\kappa(\\mathfrak m_i)/\\kappa_A$ is separable for all $i$,\n\\item {\\it tamely ramified with respect to $A$}\nif either the characteristic of $\\kappa_A$\nis $0$ or the characteristic of $\\kappa_A$ is $p > 0$, the field extensions\n$\\kappa(\\mathfrak m_i)/\\kappa_A$ are separable,\nand the ramification indices $e_i$ are prime to $p$, and\n\\item {\\it totally ramified with respect to $A$}\nif $n = 1$ and the residue field extension\n$\\kappa(\\mathfrak m_1)/\\kappa_A$ is trivial.\n\\end{enumerate}\nIf the discrete valuation ring $A$ is clear from context, then we sometimes\nsay $L/K$ is unramified, totally ramified, or tamely ramified for short.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of discrete valuation rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09E9","source_file":"more-algebra.tex","source_line":33040,"source_end_line":33061,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33040-L33061","statement_sha256":"5727f868d716764f12690f6213b52e4e551d84ec9a75065d297bfe059789cfa4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3620,"rank":3620,"depth":0,"x":536.555,"y":620.68,"cluster":"advanced-algebra"},{"id":"stacks:0EXR","tag":"0EXR","title":"Extensions of discrete valuation rings · Lemma 0EXR","summary":"Let A be a discrete valuation ring with fraction field K. • If M/L/K are finite separable extensions and M is unramified with respect to A, then L is unramified with respect to A. • If L/K is a finite separable extension which is unramified with respect to A, then there exists a Galois extension M/K containing L which is unramified with respect to A. • If L_1/K, L_2/K are finite separable extensions which are unramified with respect to A, then there exists a a finite…","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\n\\begin{enumerate}\n\\item If $M/L/K$ are finite separable extensions and\n$M$ is unramified with respect to $A$, then $L$ is unramified\nwith respect to $A$.\n\\item If $L/K$ is a finite separable extension which is\nunramified with respect to $A$, then there exists a Galois\nextension $M/K$ containing $L$ which is unramified with respect to $A$.\n\\item If $L_1/K$, $L_2/K$ are finite separable extensions which are\nunramified with respect to $A$, then there exists a a finite\nseparable extension $L/K$ which is unramified with respect\nto $A$ containing $L_1$ and $L_2$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of discrete valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXR","source_file":"more-algebra.tex","source_line":33066,"source_end_line":33081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33066-L33081","statement_sha256":"3138efe3a19737c15912b5e984f870ba1e8b8bc523a2a8a16162bcc73be66f16","origin":"The Stacks Project","memory_eligible":false,"source_rank":3621,"rank":3621,"depth":43,"x":833.57,"y":642.325,"cluster":"advanced-algebra"},{"id":"stacks:0EXS","tag":"0EXS","title":"Extensions of discrete valuation rings · Lemma 0EXS","summary":"Let A be a discrete valuation ring with fraction field K. Let M/L/K be finite separable extensions. Let B be the integral closure of A in L. If L/K is unramified with respect to A and M/L is unramified with respect to B_ m for every maximal ideal m of B, then M/K is unramified with respect to A.","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $M/L/K$ be finite separable extensions.\nLet $B$ be the integral closure of $A$ in $L$.\nIf $L/K$ is unramified with respect to $A$\nand $M/L$ is unramified with respect to $B_\\mathfrak m$\nfor every maximal ideal $\\mathfrak m$ of $B$, then\n$M/K$ is unramified with respect to $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of discrete valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXS","source_file":"more-algebra.tex","source_line":33135,"source_end_line":33144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33135-L33144","statement_sha256":"1a9e00f8d56b7ed1b4d08c2f7e427697abffa029c080d300df9c6bd63401343a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3622,"rank":3622,"depth":2,"x":597.008,"y":794.99,"cluster":"advanced-algebra"},{"id":"stacks:09EA","tag":"09EA","title":"Galois extensions and ramification · Lemma 09EA","summary":"Let A be a discrete valuation ring with fraction field K. Let L/K be a finite Galois extension with Galois group G. Then G acts on the ring B of Remark [Tag 09E8] and acts transitively on the set of maximal ideals of B.","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $L/K$ be a finite Galois extension with Galois group $G$.\nThen $G$ acts on the ring $B$ of Remark \\ref{remark-finite-separable-extension}\nand acts transitively on the set of maximal ideals of $B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Galois extensions and ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EA","source_file":"more-algebra.tex","source_line":33178,"source_end_line":33184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33178-L33184","statement_sha256":"92457f2c24c60f1cc12be359ed32b92799c6c887bc7566773784a3413c88398d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3623,"rank":3623,"depth":9,"x":648.707,"y":548.037,"cluster":"advanced-algebra"},{"id":"stacks:09EB","tag":"09EB","title":"Galois extensions and ramification · Lemma 09EB","summary":"Let A be a discrete valuation ring with fraction field K. Let L/K be a finite Galois extension. Then there are e ≥ 1 and f ≥ 1 such that e_i = e and f_i = f for all i (notation as in Remark [Tag 09E8]). In particular [L : K] = n e f.","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $L/K$ be a finite Galois extension. Then there are $e \\geq 1$ and\n$f \\geq 1$ such that $e_i = e$ and $f_i = f$ for all $i$ (notation\nas in Remark \\ref{remark-finite-separable-extension}). In particular\n$[L : K] = n e f$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Galois extensions and ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EB","source_file":"more-algebra.tex","source_line":33191,"source_end_line":33198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33191-L33198","statement_sha256":"61db0207821c49cc8e8e4d30cddde89b1c08bac52027a56926753a7d592a99fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3624,"rank":3624,"depth":10,"x":809.273,"y":759.599,"cluster":"advanced-algebra"},{"id":"stacks:09EC","tag":"09EC","title":"Galois extensions and ramification · Definition 09EC","summary":"Let A be a discrete valuation ring with fraction field K. Let L/K be a finite Galois extension with Galois group G. Let B be the integral closure of A in L. Let m ⊂ B be a maximal ideal. • The decomposition group of m is the subgroup D = (σ ∈ G mid σ( m) = m). • The inertia group of m is the kernel I of the map D → Aut(kappa( m)/kappa_A).","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $L/K$ be a finite Galois extension with Galois group $G$.\nLet $B$ be the integral closure of $A$ in $L$.\nLet $\\mathfrak m \\subset B$ be a maximal ideal.\n\\begin{enumerate}\n\\item The {\\it decomposition group of $\\mathfrak m$}\nis the subgroup $D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m) = \\mathfrak m\\}$.\n\\item The {\\it inertia group of $\\mathfrak m$} is the kernel $I$ of the map\n$D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Galois extensions and ramification","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EC","source_file":"more-algebra.tex","source_line":33204,"source_end_line":33216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33204-L33216","statement_sha256":"67d6fbfc67912f4cc6118c7d12044e16bde7e84645b89d09a68d6b8cbe335529","origin":"The Stacks Project","memory_eligible":false,"source_rank":3625,"rank":3625,"depth":0,"x":520.569,"y":694.667,"cluster":"advanced-algebra"},{"id":"stacks:09ED","tag":"09ED","title":"Galois extensions and ramification · Lemma 09ED","summary":"Let A be a discrete valuation ring with fraction field K and residue field kappa. Let L/K be a finite Galois extension with Galois group G. Let B be the integral closure of A in L. Let m be a maximal ideal of B. Then • the field extension kappa( m)/kappa is normal, and • D → Aut(kappa( m)/kappa) is surjective. If for some (equivalently all) maximal ideal(s) m ⊂ B the field extension kappa( m)/kappa is separable, then • [(3)] kappa( m)/kappa is Galois, and • [(4)] D →…","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$ and residue field\n$\\kappa$. Let $L/K$ be a finite Galois extension with Galois group $G$.\nLet $B$ be the integral closure of $A$ in $L$. Let $\\mathfrak m$ be a maximal\nideal of $B$. Then\n\\begin{enumerate}\n\\item the field extension $\\kappa(\\mathfrak m)/\\kappa$ is normal, and\n\\item $D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa)$ is surjective.\n\\end{enumerate}\nIf for some (equivalently all) maximal ideal(s) $\\mathfrak m \\subset B$\nthe field extension $\\kappa(\\mathfrak m)/\\kappa$ is separable, then\n\\begin{enumerate}\n\\item[(3)] $\\kappa(\\mathfrak m)/\\kappa$ is Galois, and\n\\item[(4)] $D \\to \\text{Gal}(\\kappa(\\mathfrak m)/\\kappa)$ is surjective.\n\\end{enumerate}\nHere $D \\subset G$ is the decomposition group of $\\mathfrak m$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Galois extensions and ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ED","source_file":"more-algebra.tex","source_line":33223,"source_end_line":33240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33223-L33240","statement_sha256":"35864c5cfce68ffb287ee7f810a8b05ae00aa634a0ea8c6ffb227dc073443383","origin":"The Stacks Project","memory_eligible":false,"source_rank":3626,"rank":3626,"depth":13,"x":785.831,"y":578.659,"cluster":"advanced-algebra"},{"id":"stacks:09EE","tag":"09EE","title":"Galois extensions and ramification · Lemma 09EE","summary":"Let A be a discrete valuation ring with fraction field K. Let L/K be a finite Galois extension with Galois group G. Let B be the integral closure of A in L. Let m ⊂ B be a maximal ideal. The inertia group I of m sits in a canonical exact sequence 1 → P → I → I_t → 1 such that • if D is the decomposition group we have P = (σ ∈ I mid σ|_ m/ m^2 = id_ m/ m^2) = (σ ∈ D mid σ acts trivially on Gr_ m(B)) • P is a normal subgroup of D, • P is a p-group if the characteristic of…","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $L/K$ be a finite Galois extension with Galois group $G$.\nLet $B$ be the integral closure of $A$ in $L$. Let $\\mathfrak m \\subset B$\nbe a maximal ideal. The inertia group $I$ of $\\mathfrak m$\nsits in a canonical exact sequence\n$$\n1 \\to P \\to I \\to I_t \\to 1\n$$\nsuch that\n\\begin{enumerate}\n\\item if $D$ is the decomposition group we have\n$$\nP = \\{\\sigma \\in I \\mid \\sigma|_{\\mathfrak m/\\mathfrak m^2} =\n\\text{id}_{\\mathfrak m/\\mathfrak m^2}\\}\n=\n\\{\\sigma \\in D \\mid \\sigma \\text{ acts trivially on }\\text{Gr}_\\mathfrak m(B)\\}\n$$\n\\item $P$ is a normal subgroup of $D$,\n\\item $P$ is a $p$-group if the characteristic of $\\kappa_A$ is\n$p > 0$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$ is zero,\n\\item $I_t$ is cyclic of order the prime to $p$ part of the integer $e$,\n\\item there is a canonical isomorphism\n$\\theta : I_t \\to \\mu_e(\\kappa(\\mathfrak m))$, and\n\\item\n$P =\n\\{\\sigma \\in D \\mid \\sigma|_{B/\\mathfrak m^2} = \\text{id}_{B/\\mathfrak m^2}\\}$\nif $\\kappa(m)$ is separable over the residue field of $A$.\n\\end{enumerate}\nHere $e$ is the integer of Lemma \\ref{lemma-galois-conclusion}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Galois extensions and ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EE","source_file":"more-algebra.tex","source_line":33250,"source_end_line":33281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33250-L33281","statement_sha256":"153f9744645aa8cb06968ffa0c6625abb61d851e20c698675c6fcb4bc810e0b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3627,"rank":3627,"depth":42,"x":683.459,"y":814.859,"cluster":"advanced-algebra"},{"id":"stacks:0BU4","tag":"0BU4","title":"Galois extensions and ramification · Definition 0BU4","summary":"With assumptions and notation as in Lemma [Tag 09EE]. • The wild inertia group of m is the subgroup P. • The tame inertia group of m is the quotient I → I_t. We denote theta : I → μ_e(kappa( m)) the surjective map ([Tag 0BU3]) whose kernel is P and which induces the isomorphism I_t → μ_e(kappa( m)).","statement_latex":"With assumptions and notation as in Lemma \\ref{lemma-galois-inertia}.\n\\begin{enumerate}\n\\item The {\\it wild inertia group of $\\mathfrak m$} is the subgroup $P$.\n\\item The {\\it tame inertia group of $\\mathfrak m$} is the\nquotient $I \\to I_t$.\n\\end{enumerate}\nWe denote $\\theta : I \\to \\mu_e(\\kappa(\\mathfrak m))$ the surjective map\n(\\ref{equation-inertia-character}) whose kernel is $P$ and which\ninduces the isomorphism $I_t \\to \\mu_e(\\kappa(\\mathfrak m))$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Galois extensions and ramification","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BU4","source_file":"more-algebra.tex","source_line":33417,"source_end_line":33428,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33417-L33428","statement_sha256":"e3e0d043c295ddc1bf594f4fbb8ae074a08b21232e6bf5429ca3001bcd8c8571","origin":"The Stacks Project","memory_eligible":false,"source_rank":3628,"rank":3628,"depth":43,"x":568.933,"y":582.462,"cluster":"advanced-algebra"},{"id":"stacks:0BU5","tag":"0BU5","title":"Galois extensions and ramification · Lemma 0BU5","summary":"With assumptions and notation as in Lemma [Tag 09EE]. The inertia character theta : I → μ_e(kappa( m)) satisfies the following property theta(τ σ τ^-1) = τ(theta(σ)) for τ ∈ D and σ ∈ I.","statement_latex":"With assumptions and notation as in Lemma \\ref{lemma-galois-inertia}.\nThe inertia character $\\theta : I \\to \\mu_e(\\kappa(\\mathfrak m))$\nsatisfies the following property\n$$\n\\theta(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta(\\sigma))\n$$\nfor $\\tau \\in D$ and $\\sigma \\in I$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Galois extensions and ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BU5","source_file":"more-algebra.tex","source_line":33430,"source_end_line":33439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33430-L33439","statement_sha256":"d6915e5fae7c10f6f69cb78c8e4f033058ef1ddf7396e6281df4ff26cfe5ff78","origin":"The Stacks Project","memory_eligible":false,"source_rank":3629,"rank":3629,"depth":43,"x":840.432,"y":688.906,"cluster":"advanced-algebra"},{"id":"stacks:09EH","tag":"09EH","title":"Galois extensions and ramification · Lemma 09EH","summary":"Let A be a discrete valuation ring with fraction field K. Let L/K be a finite Galois extension. Let m ⊂ B be a maximal ideal of the integral closure of A in L. Let I ⊂ G be the inertia group of m. Then B^I is the integral closure of A in L^I and A → (B^I)_B^I ∩ m is étale.","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $L/K$ be a finite Galois extension. Let $\\mathfrak m \\subset B$\nbe a maximal ideal of the integral closure of $A$ in $L$.\nLet $I \\subset G$ be the inertia group of $\\mathfrak m$.\nThen $B^I$ is the integral closure of $A$ in $L^I$ and\n$A \\to (B^I)_{B^I \\cap \\mathfrak m}$ is \\'etale.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Galois extensions and ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EH","source_file":"more-algebra.tex","source_line":33474,"source_end_line":33482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33474-L33482","statement_sha256":"b010a6fb867bb316eedb743b64e11b0642eb8640f4f5a8bb3310307a325aaabb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3630,"rank":3630,"depth":37,"x":554.463,"y":764.516,"cluster":"advanced-algebra"},{"id":"stacks:0BU7","tag":"0BU7","title":"Galois extensions and ramification · Lemma 0BU7","summary":"Let A be a discrete valuation ring with fraction field K. Let M/L/K be a tower with M/K and L/K finite Galois. Let C, B be the integral closure of A in M, L. Let m' ⊂ C be a maximal ideal and set m = m' ∩ B. Let P ⊂ I ⊂ D ⊂ Gal(L/K) and P' ⊂ I' ⊂ D' ⊂ Gal(M/K) be the wild inertia, inertia, decomposition group of m and m'. Then the canonical surjection Gal(M/K) → Gal(L/K) induces surjections P' → P, I' → I, and D' → D. Moreover these fit into commutative diagrams vcenter…","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $M/L/K$ be a tower with $M/K$ and $L/K$ finite Galois.\nLet $C$, $B$ be the integral closure of $A$ in $M$, $L$.\nLet $\\mathfrak m' \\subset C$ be a maximal ideal and set\n$\\mathfrak m = \\mathfrak m' \\cap B$. Let\n$$\nP \\subset I \\subset D \\subset \\text{Gal}(L/K)\n\\quad\\text{and}\\quad\nP' \\subset I' \\subset D' \\subset \\text{Gal}(M/K)\n$$\nbe the wild inertia, inertia, decomposition group of\n$\\mathfrak m$ and $\\mathfrak m'$.\nThen the canonical surjection $\\text{Gal}(M/K) \\to \\text{Gal}(L/K)$\ninduces surjections $P' \\to P$, $I' \\to I$, and $D' \\to D$. Moreover\nthese fit into commutative diagrams\n$$\n\\vcenter{\n\\xymatrix{\nD' \\ar[r] \\ar[d] &\n\\text{Aut}(\\kappa(\\mathfrak m')/\\kappa_A) \\ar[d] \\\\\nD \\ar[r] &\n\\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A)\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nI' \\ar[r]_-{\\theta'} \\ar[d] &\n\\mu_{e'}(\\kappa(\\mathfrak m')) \\ar[d]^{(-)^{e'/e}} \\\\\nI \\ar[r]^-\\theta &\n\\mu_e(\\kappa(\\mathfrak m))\n}\n}\n$$\nwhere $e'$ and $e$ are the ramification indices of\n$A \\to C_{\\mathfrak m'}$ and $A \\to B_\\mathfrak m$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Galois extensions and ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BU7","source_file":"more-algebra.tex","source_line":33619,"source_end_line":33657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33619-L33657","statement_sha256":"92f8b2f2b5739aa3e3ee3c115e233a04ac0ea20a5034325ad1d77a7e40b6fdd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3631,"rank":3631,"depth":15,"x":704.618,"y":546.367,"cluster":"advanced-algebra"},{"id":"stacks:09EI","tag":"09EI","title":"Krasner's lemma · Lemma 09EI","summary":"Let A be a complete local domain of dimension 1. Let P(t) ∈ A[t] be a polynomial with coefficients in A. Let α ∈ A be a root of P but not a root of the derivative P' = dP/dt. For every c ≥ 0 there exists an integer n such that for any Q ∈ A[t] whose coefficients are in m_A^n the polynomial P + Q has a root β ∈ A with β - α ∈ m_A^c.","statement_latex":"Let $A$ be a complete local domain of dimension $1$. Let $P(t) \\in A[t]$\nbe a polynomial with coefficients in $A$. Let $\\alpha \\in A$ be a root\nof $P$ but not a root of the derivative $P' = \\text{d}P/\\text{d}t$.\nFor every $c \\geq 0$ there exists an integer $n$ such that for any\n$Q \\in A[t]$ whose coefficients are in $\\mathfrak m_A^n$ the polynomial\n$P + Q$ has a root $\\beta \\in A$ with $\\beta - \\alpha \\in \\mathfrak m_A^c$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Krasner's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EI","source_file":"more-algebra.tex","source_line":33747,"source_end_line":33755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33747-L33755","statement_sha256":"abfc58f75e935b8e234a7ec2c689890a6632ef61676e5daaefa4741da9bf0e11","origin":"The Stacks Project","memory_eligible":false,"source_rank":3632,"rank":3632,"depth":0,"x":769.364,"y":792.575,"cluster":"advanced-algebra"},{"id":"stacks:09EJ","tag":"09EJ","title":"Krasner's lemma · Lemma 09EJ","summary":"Let A be a discrete valuation ring with field of fractions K. Let A^wedge be the completion of A with fraction field K^wedge. If M/K^wedge is a finite separable extension, then there exists a finite separable extension L/K such that M = K^wedge ⊗_K L.","statement_latex":"Let $A$ be a discrete valuation ring with field of fractions $K$.\nLet $A^\\wedge$ be the completion of $A$ with fraction field $K^\\wedge$.\nIf $M/K^\\wedge$ is a finite separable extension, then\nthere exists a finite separable extension $L/K$\nsuch that $M = K^\\wedge \\otimes_K L$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Krasner's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EJ","source_file":"more-algebra.tex","source_line":33813,"source_end_line":33820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33813-L33820","statement_sha256":"92dc587bf9af4149922da07ef4825b1eb50b0fd02321e572a5ad2d90c5d12760","origin":"The Stacks Project","memory_eligible":false,"source_rank":3633,"rank":3633,"depth":19,"x":523.482,"y":647.677,"cluster":"advanced-algebra"},{"id":"stacks:09EK","tag":"09EK","title":"Krasner's lemma · Definition 09EK","summary":"Let A be a discrete valuation ring. We say A has mixed characteristic if the characteristic of the residue field of A is p > 0 and the characteristic of the fraction field of A is 0. In this case we obtain an extension of discrete valuation rings Z_(p) ⊂ A and the absolute ramification index of A is the ramification index of this extension.","statement_latex":"Let $A$ be a discrete valuation ring. We say $A$ has {\\it mixed characteristic}\nif the characteristic of the residue field of $A$ is $p > 0$ and the\ncharacteristic of the fraction field of $A$ is $0$.\nIn this case we obtain an extension of discrete valuation rings\n$\\mathbf{Z}_{(p)} \\subset A$ and the {\\it absolute ramification index}\nof $A$ is the ramification index of this extension.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Krasner's lemma","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EK","source_file":"more-algebra.tex","source_line":33854,"source_end_line":33862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33854-L33862","statement_sha256":"4e486229021a4a1aa70724699d9bc441378f645136dc28ca90fc7518db4ce863","origin":"The Stacks Project","memory_eligible":false,"source_rank":3634,"rank":3634,"depth":0,"x":821.491,"y":614.984,"cluster":"advanced-algebra"},{"id":"stacks:09EV","tag":"09EV","title":"Abhyankar's lemma and tame ramification · Lemma 09EV","summary":"Let A be a discrete valuation ring with uniformizer π. Let n ≥ 2. Let K_1 = K[π^1/n]. Then • [K_1 : K] = n, • the integral closure A_1 of A in K_1 is the ring A[π^1/n], • A_1 is a discrete valuation ring, • the ramification index of A_1 over A is n, • K_1 is totally ramified with respect to A, and • if n is prime to the residue characteristic of A, then K_1/K is tamely ramified and any subextension of K_1/K is generated by π^1/d for some divisor d of n.","statement_latex":"Let $A$ be a discrete valuation ring with uniformizer $\\pi$. Let $n \\geq 2$.\nLet $K_1 = K[\\pi^{1/n}]$. Then\n\\begin{enumerate}\n\\item $[K_1 : K] = n$,\n\\item the integral closure $A_1$ of $A$ in $K_1$ is the ring $A[\\pi^{1/n}]$,\n\\item $A_1$ is a discrete valuation ring,\n\\item the ramification index of $A_1$ over $A$ is $n$,\n\\item $K_1$ is totally ramified with respect to $A$, and\n\\item if $n$ is prime to the residue characteristic of $A$, then\n$K_1/K$ is tamely ramified and any subextension of $K_1/K$ is\ngenerated by $\\pi^{1/d}$ for some divisor $d$ of $n$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Abhyankar's lemma and tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EV","source_file":"more-algebra.tex","source_line":33953,"source_end_line":33967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L33953-L33967","statement_sha256":"b9228a3d5a7e555a5e21401e16fa913771978ab3742a3de8f1e110fdc6850880","origin":"The Stacks Project","memory_eligible":false,"source_rank":3635,"rank":3635,"depth":1,"x":627.92,"y":808.303,"cluster":"advanced-algebra"},{"id":"stacks:09EQ","tag":"09EQ","title":"Abhyankar's lemma and tame ramification · Lemma 09EQ","summary":"Let A → B be an extension of discrete valuation rings with fraction fields K ⊂ L. Assume that A → B is formally smooth in the m_B-adic topology. Then for any finite extension K_1/K we have L_1 = L ⊗_K K_1, B_1 = B ⊗_A A_1, and each extension (A_1)_ m_i ⊂ (B_1)_ m_ij (see Remark [Tag 09EM]) is formally smooth in the m_ij-adic topology.","statement_latex":"Let $A \\to B$ be an extension of discrete valuation rings with fraction fields\n$K \\subset L$. Assume that $A \\to B$ is formally smooth in the\n$\\mathfrak m_B$-adic topology. Then for any finite extension $K_1/K$\nwe have $L_1 = L \\otimes_K K_1$, $B_1 = B \\otimes_A A_1$, and each extension\n$(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ (see\nRemark \\ref{remark-construction}) is formally smooth in the\n$\\mathfrak m_{ij}$-adic topology.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Abhyankar's lemma and tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EQ","source_file":"more-algebra.tex","source_line":34005,"source_end_line":34014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34005-L34014","statement_sha256":"2ac236f056f77ea492ef86a90d4b0e8e6391d919f3402b98b069fa33f63b2bf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3636,"rank":3636,"depth":47,"x":615.187,"y":555.766,"cluster":"advanced-algebra"},{"id":"stacks:0BRM","tag":"0BRM","title":"Abhyankar's lemma · Lemma 0BRM","summary":"Let A ⊂ B be an extension of discrete valuation rings. Assume that either the residue characteristic of A is 0 or it is p, the ramification index e is prime to p, and kappa_B/kappa_A is a separable field extension. Let K_1/K be a finite extension. Using the notation of Remark [Tag 09EM] assume e divides the ramification index of A ⊂ (A_1)_ m_i for some i. Then (A_1)_ m_i ⊂ (B_1)_ m_ij is formally smooth in the m_ij-adic topology for all j = 1, …, m_i.","statement_latex":"Let $A \\subset B$ be an extension of discrete valuation rings.\nAssume that either the residue characteristic of $A$ is $0$\nor it is $p$, the ramification index $e$ is prime to $p$, and\n$\\kappa_B/\\kappa_A$ is a separable field extension.\nLet $K_1/K$ be a finite extension. Using the notation of\nRemark \\ref{remark-construction}\nassume $e$ divides the ramification index of $A \\subset (A_1)_{\\mathfrak m_i}$\nfor some $i$. Then $(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$\nis formally smooth in the $\\mathfrak m_{ij}$-adic topology\nfor all $j = 1, \\ldots, m_i$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Abhyankar's lemma and tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRM","source_file":"more-algebra.tex","source_line":34036,"source_end_line":34048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34036-L34048","statement_sha256":"5389be981b1649acaf7b04a59bccb42265061da9d530fd80b78e8fb571241907","origin":"The Stacks Project","memory_eligible":false,"source_rank":3637,"rank":3637,"depth":48,"x":827.785,"y":734.864,"cluster":"advanced-algebra"},{"id":"stacks:0EXU","tag":"0EXU","title":"Abhyankar's lemma and tame ramification · Lemma 0EXU","summary":"Let A be a discrete valuation ring with fraction field K. Let M/L/K be finite separable extensions. Let B be the integral closure of A in L. If L/K is tamely ramified with respect to A and M/L is tamely ramified with respect to B_ m for every maximal ideal m of B, then M/K is tamely ramified with respect to A.","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $M/L/K$ be finite separable extensions.\nLet $B$ be the integral closure of $A$ in $L$.\nIf $L/K$ is tamely ramified with respect to $A$\nand $M/L$ is tamely ramified with respect to $B_\\mathfrak m$\nfor every maximal ideal $\\mathfrak m$ of $B$, then\n$M/K$ is tamely ramified with respect to $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Abhyankar's lemma and tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXU","source_file":"more-algebra.tex","source_line":34137,"source_end_line":34146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34137-L34146","statement_sha256":"cc0e3c3b8ab8b6d93a558e299beed6c6890e48096ce3373d926624cb7f2d7f11","origin":"The Stacks Project","memory_eligible":false,"source_rank":3638,"rank":3638,"depth":2,"x":526.815,"y":723.427,"cluster":"advanced-algebra"},{"id":"stacks:0EXV","tag":"0EXV","title":"Abhyankar's lemma and tame ramification · Lemma 0EXV","summary":"Let A be a discrete valuation ring with fraction field K. If M/L/K are finite separable extensions and M is tamely ramified with respect to A, then L is tamely ramified with respect to A.","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nIf $M/L/K$ are finite separable extensions and\n$M$ is tamely ramified with respect to $A$, then\n$L$ is tamely ramified with respect to $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Abhyankar's lemma and tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXV","source_file":"more-algebra.tex","source_line":34159,"source_end_line":34165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34159-L34165","statement_sha256":"214591586fafccbdd12f6cd13850787af6af6da26dd04b2e3e31f5d4e3a6b9d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3639,"rank":3639,"depth":2,"x":758.08,"y":560.987,"cluster":"advanced-algebra"},{"id":"stacks:0EXW","tag":"0EXW","title":"Abhyankar's lemma and tame ramification · Lemma 0EXW","summary":"Let A be a discrete valuation ring with fraction field K. Let π ∈ A be a uniformizer. Let L/K be a finite separable extension. The following are equivalent • L is tamely ramified with respect to A, • there exists an e ≥ 1 invertible in kappa_A and an extension L'/K' = K[π^1/e] unramified with respect to A' = A[π^1/e] such that L is contained in L', and • there exists an e_0 ≥ 1 invertible in kappa_A such that for every d ≥ 1 invertible in kappa_A (2) holds with e = de_0.","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $\\pi \\in A$ be a uniformizer.\nLet $L/K$ be a finite separable extension.\nThe following are equivalent\n\\begin{enumerate}\n\\item $L$ is tamely ramified with respect to $A$,\n\\item there exists an $e \\geq 1$ invertible in $\\kappa_A$\nand an extension $L'/K' = K[\\pi^{1/e}]$ unramified with respect to\n$A' = A[\\pi^{1/e}]$ such that $L$ is contained in $L'$, and\n\\item there exists an $e_0 \\geq 1$ invertible in $\\kappa_A$\nsuch that for every $d \\geq 1$ invertible in $\\kappa_A$\n(2) holds with $e = de_0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Abhyankar's lemma and tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXW","source_file":"more-algebra.tex","source_line":34184,"source_end_line":34199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34184-L34199","statement_sha256":"61168c0396334fda1236089025e6a1093825da8ee04c29c72357207d3f19cc9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3640,"rank":3640,"depth":49,"x":718.155,"y":812.14,"cluster":"advanced-algebra"},{"id":"stacks:0EXX","tag":"0EXX","title":"Abhyankar's lemma and tame ramification · Lemma 0EXX","summary":"Let A be a discrete valuation ring with fraction field K. • If L/K is a finite separable extension which is tamely ramified with respect to A, then there exists a Galois extension M/K containing L which is tamely ramified with respect to A. • If L_1/K, L_2/K are finite separable extensions which are tamely ramified with respect to A, then there exists a finite separable extension L/K which is tamely ramified with respect to A containing L_1 and L_2.","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\n\\begin{enumerate}\n\\item If $L/K$ is a finite separable extension which is tamely\nramified with respect to $A$, then there exists a Galois\nextension $M/K$ containing $L$ which is tamely ramified\nwith respect to $A$.\n\\item If $L_1/K$, $L_2/K$ are finite separable extensions which are tamely\nramified with respect to $A$, then there exists a finite\nseparable extension $L/K$ which is tamely ramified with respect\nto $A$ containing $L_1$ and $L_2$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Abhyankar's lemma and tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXX","source_file":"more-algebra.tex","source_line":34242,"source_end_line":34255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34242-L34255","statement_sha256":"b9a4988a28b24aa3887b8934b57f28943726b6698c19088d19ff7cd94e8b7b16","origin":"The Stacks Project","memory_eligible":false,"source_rank":3641,"rank":3641,"depth":50,"x":545.523,"y":604.167,"cluster":"advanced-algebra"},{"id":"stacks:0EXY","tag":"0EXY","title":"Abhyankar's lemma and tame ramification · Lemma 0EXY","summary":"Let A ⊂ B be an extension of discrete valuation rings. Denote L/K the corresponding extension of fraction fields. Let K'/K be a finite separable extension. Then K' ⊗_K L = ∏ L'_i is a finite product of fields and the following is true • If K' is unramified with respect to A, then each L'_i is unramified with respect to B. • If K' is tamely ramified with respect to A, then each L'_i is tamely ramified with respect to B.","statement_latex":"Let $A \\subset B$ be an extension of discrete valuation rings.\nDenote $L/K$ the corresponding extension of fraction fields.\nLet $K'/K$ be a finite separable extension.\nThen\n$$\nK' \\otimes_K L = \\prod L'_i\n$$\nis a finite product of fields and the following is true\n\\begin{enumerate}\n\\item If $K'$ is unramified with respect to $A$, then\neach $L'_i$ is unramified with respect to $B$.\n\\item If $K'$ is tamely ramified with respect to $A$, then\neach $L'_i$ is tamely ramified with respect to $B$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Abhyankar's lemma and tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXY","source_file":"more-algebra.tex","source_line":34329,"source_end_line":34345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34329-L34345","statement_sha256":"804d679765fee143b434863f26f2e4d529281ed6212e841ad45fb30ca8c7fdaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":3642,"rank":3642,"depth":50,"x":840.238,"y":659.601,"cluster":"advanced-algebra"},{"id":"stacks:09EN","tag":"09EN","title":"Eliminating ramification · Definition 09EN","summary":"Let A → B be an extension of discrete valuation rings with fraction fields K ⊂ L. • We say a finite field extension K_1/K is a weak solution for A ⊂ B if all the extensions (A_1)_ m_i ⊂ (B_1)_ m_ij of Remark [Tag 09EM] are weakly unramified. • We say a finite field extension K_1/K is a solution for A ⊂ B if each extension (A_1)_ m_i ⊂ (B_1)_ m_ij of Remark [Tag 09EM] is formally smooth in the m_ij-adic topology. We say a solution K_1/K is a separable solution if K_1/K is…","statement_latex":"Let $A \\to B$ be an extension of discrete valuation rings with fraction\nfields $K \\subset L$.\n\\begin{enumerate}\n\\item We say a finite field extension $K_1/K$ is a\n{\\it weak solution for $A \\subset B$} if all the extensions\n$(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ of\nRemark \\ref{remark-construction} are weakly unramified.\n\\item We say a finite field extension $K_1/K$ is a\n{\\it solution for $A \\subset B$} if each extension\n$(A_1)_{\\mathfrak m_i} \\subset (B_1)_{\\mathfrak m_{ij}}$ of\nRemark \\ref{remark-construction} is formally smooth in\nthe $\\mathfrak m_{ij}$-adic topology.\n\\end{enumerate}\nWe say a solution $K_1/K$ is a {\\it separable solution}\nif $K_1/K$ is separable.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EN","source_file":"more-algebra.tex","source_line":34444,"source_end_line":34461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34444-L34461","statement_sha256":"835fa0ab8430b9ae24fcc791567c11e849bcac2eec50b6e89b521d88d643eb22","origin":"The Stacks Project","memory_eligible":false,"source_rank":3643,"rank":3643,"depth":0,"x":578.188,"y":786.027,"cluster":"advanced-algebra"},{"id":"stacks:09ER","tag":"09ER","title":"Eliminating ramification · Lemma 09ER","summary":"Let A → B be an extension of discrete valuation rings with fraction fields K ⊂ L. Assume that A → B is weakly unramified. Then for any finite separable extension K_1/K totally ramified with respect to A we have that L_1 = L ⊗_K K_1 is a field, A_1 and B_1 = B ⊗_A A_1 are discrete valuation rings, and the extension A_1 ⊂ B_1 (see Remark [Tag 09EM]) is weakly unramified.","statement_latex":"Let $A \\to B$ be an extension of discrete valuation rings with fraction\nfields $K \\subset L$. Assume that $A \\to B$ is weakly unramified. Then for\nany finite separable extension $K_1/K$ totally ramified with respect to $A$\nwe have that $L_1 = L \\otimes_K K_1$ is a field, $A_1$ and\n$B_1 = B \\otimes_A A_1$ are discrete valuation rings, and the extension\n$A_1 \\subset B_1$ (see\nRemark \\ref{remark-construction}) is weakly unramified.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ER","source_file":"more-algebra.tex","source_line":34489,"source_end_line":34498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34489-L34498","statement_sha256":"46b39dcd563e25a61b854f2f7572b04950b52ede3fd4b361364c692170450e89","origin":"The Stacks Project","memory_eligible":false,"source_rank":3644,"rank":3644,"depth":0,"x":669.804,"y":543.967,"cluster":"advanced-algebra"},{"id":"stacks:09ES","tag":"09ES","title":"Eliminating ramification · Lemma 09ES","summary":"Let A → B → C be extensions of discrete valuation rings with fraction fields K ⊂ L ⊂ M. Let K_1/K be a finite extension. • If K_1 is a (weak) solution for A → C, then K_1 is a (weak) solution for A → B. • If K_1 is a (weak) solution for A → B and L_1 = (L ⊗_K K_1)_red is a product of fields which are (weak) solutions for B → C, then K_1 is a (weak) solution for A → C.","statement_latex":"Let $A \\to B \\to C$ be extensions of discrete valuation rings with fraction\nfields $K \\subset L \\subset M$. Let $K_1/K$ be a finite extension.\n\\begin{enumerate}\n\\item If $K_1$ is a (weak) solution for $A \\to C$, then $K_1$ is a (weak)\nsolution for $A \\to B$.\n\\item If $K_1$ is a (weak) solution for $A \\to B$ and\n$L_1 = (L \\otimes_K K_1)_{red}$ is a product of fields which are\n(weak) solutions for $B \\to C$, then $K_1$ is a (weak) solution for $A \\to C$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ES","source_file":"more-algebra.tex","source_line":34516,"source_end_line":34527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34516-L34527","statement_sha256":"4a1900ed1c37a2529b63168752cf0b4cb44206658f191fd34b47a3424a1b74f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3645,"rank":3645,"depth":47,"x":796.98,"y":774.579,"cluster":"advanced-algebra"},{"id":"stacks:09ET","tag":"09ET","title":"Eliminating ramification · Lemma 09ET","summary":"Let A → B be an extension of discrete valuation rings. There exists a commutative diagram xymatrix B ar[r] & B' A ar[r] ar[u] & A' ar[u] of extensions of discrete valuation rings such that • the extensions K'/K and L'/L of fraction fields are separable algebraic, • the residue fields of A' and B' are separable algebraic closures of the residue fields of A and B, and • if a solution, weak solution, or separable solution exists for A' → B', then a solution, weak solution,…","statement_latex":"Let $A \\to B$ be an extension of discrete valuation rings. There exists\na commutative diagram\n$$\n\\xymatrix{\nB \\ar[r] & B' \\\\\nA \\ar[r] \\ar[u] & A' \\ar[u]\n}\n$$\nof extensions of discrete valuation rings such that\n\\begin{enumerate}\n\\item the extensions $K'/K$ and $L'/L$ of fraction fields\nare separable algebraic,\n\\item the residue fields of $A'$ and $B'$ are separable algebraic\nclosures of the residue fields of $A$ and $B$, and\n\\item if a solution, weak solution, or separable solution exists for\n$A' \\to B'$, then a solution, weak solution, or separable solution exists\nfor $A \\to B$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ET","source_file":"more-algebra.tex","source_line":34554,"source_end_line":34574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34554-L34574","statement_sha256":"829460e845fda59cfd34cac6e812f0d01c16b8dc70ea6137a28488bac20ac4ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":3646,"rank":3646,"depth":48,"x":517.588,"y":676.635,"cluster":"advanced-algebra"},{"id":"stacks:09EU","tag":"09EU","title":"Eliminating ramification · Lemma 09EU","summary":"Let A → B be an extension of discrete valuation rings with fraction fields K ⊂ L. Let K_1/K be a normal extension. Say G = Aut(K_1/K). Then G acts on the rings K_1, L_1, A_1 and B_1 of Remark [Tag 09EM] and acts transitively on the set of maximal ideals of B_1.","statement_latex":"Let $A \\to B$ be an extension of discrete valuation rings with fraction fields\n$K \\subset L$. Let $K_1/K$ be a normal extension. Say\n$G = \\text{Aut}(K_1/K)$. Then $G$ acts on the rings $K_1$, $L_1$,\n$A_1$ and $B_1$ of Remark \\ref{remark-construction}\nand acts transitively on the set of maximal ideals of $B_1$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EU","source_file":"more-algebra.tex","source_line":34608,"source_end_line":34615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34608-L34615","statement_sha256":"5ba7f63c1922475789198146714093dda6a352389d55d54b7b157af8cd8e626a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3647,"rank":3647,"depth":0,"x":802.538,"y":590.273,"cluster":"advanced-algebra"},{"id":"stacks:09EW","tag":"09EW","title":"Eliminating ramification · Lemma 09EW","summary":"Let A be a discrete valuation ring with uniformizer π. If the residue characteristic of A is p > 0, then for every n > 1 and p-power q there exists a degree q separable extension L/K totally ramified with respect to A such that the integral closure B of A in L has ramification index q and a uniformizer π_B such that π_B^q = π + π^n b and π_B^q = π + (π_B)^nqb' for some b, b' ∈ B.","statement_latex":"Let $A$ be a discrete valuation ring with uniformizer $\\pi$. If the residue\ncharacteristic of $A$ is $p > 0$, then for every $n > 1$ and $p$-power $q$\nthere exists a degree $q$ separable extension $L/K$\ntotally ramified with respect to $A$\nsuch that the integral closure $B$ of $A$ in $L$ has ramification index\n$q$ and a uniformizer $\\pi_B$ such that\n$\\pi_B^q = \\pi + \\pi^n b$ and $\\pi_B^q = \\pi + (\\pi_B)^{nq}b'$\nfor some $b, b' \\in B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EW","source_file":"more-algebra.tex","source_line":34640,"source_end_line":34650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34640-L34650","statement_sha256":"63bead8c1e7d858831410ae180a67ec98bf32e3ea97249b331866fca8e8db48e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3648,"rank":3648,"depth":2,"x":661.788,"y":815.773,"cluster":"advanced-algebra"},{"id":"stacks:09EX","tag":"09EX","title":"Eliminating ramification · Lemma 09EX","summary":"Let A be a discrete valuation ring. Assume the reside field kappa_A has characteristic p > 0 and that a ∈ A is an element whose residue class in kappa_A is not a pth power. Then a is not a pth power in K and the integral closure of A in K[a^1/p] is the ring A[a^1/p] which is a discrete valuation ring weakly unramified over A.","statement_latex":"Let $A$ be a discrete valuation ring. Assume the reside field $\\kappa_A$ has\ncharacteristic $p > 0$ and that $a \\in A$ is an element whose residue\nclass in $\\kappa_A$ is not a $p$th power. Then $a$ is not a $p$th power in $K$\nand the integral closure of $A$ in $K[a^{1/p}]$ is the ring $A[a^{1/p}]$\nwhich is a discrete valuation ring weakly unramified over $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EX","source_file":"more-algebra.tex","source_line":34662,"source_end_line":34669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34662-L34669","statement_sha256":"e9b20b1867b21a6b43b051762d38fdf72f4e89341df8164b5a8773f6c31074b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3649,"rank":3649,"depth":0,"x":584.188,"y":569.484,"cluster":"advanced-algebra"},{"id":"stacks:09EY","tag":"09EY","title":"Eliminating ramification · Lemma 09EY","summary":"Let A ⊂ B ⊂ C be extensions of discrete valuation rings with fractions fields K ⊂ L ⊂ M. Let π ∈ A be a uniformizer. Assume • B is a Nagata ring, • A ⊂ B is weakly unramified, • M is a degree p purely inseparable extension of L. Then either • A → C is weakly unramified, or • C = B[π^1/p], or • there exists a degree p separable extension K_1/K totally ramified with respect to A such that L_1 = L ⊗_K K_1 and M_1 = M ⊗_K K_1 are fields and the maps of integral closures A_1 →…","statement_latex":"Let $A \\subset B \\subset C$ be extensions of discrete valuation rings\nwith fractions fields $K \\subset L \\subset M$. Let $\\pi \\in A$ be a\nuniformizer. Assume\n\\begin{enumerate}\n\\item $B$ is a Nagata ring,\n\\item $A \\subset B$ is weakly unramified,\n\\item $M$ is a degree $p$ purely inseparable extension of $L$.\n\\end{enumerate}\nThen either\n\\begin{enumerate}\n\\item $A \\to C$ is weakly unramified, or\n\\item $C = B[\\pi^{1/p}]$, or\n\\item there exists a degree $p$ separable extension $K_1/K$\ntotally ramified with respect to $A$\nsuch that $L_1 = L \\otimes_K K_1$ and $M_1 = M \\otimes_K K_1$\nare fields and the maps of integral closures $A_1 \\to B_1 \\to C_1$\nare weakly unramified extensions of discrete valuation rings.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EY","source_file":"more-algebra.tex","source_line":34675,"source_end_line":34695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34675-L34695","statement_sha256":"e8eabc6efd06e434df77e87b8c74c1cc7b003082a57a59e35c841193a93c22b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3650,"rank":3650,"depth":34,"x":839.616,"y":707.142,"cluster":"advanced-algebra"},{"id":"stacks:09EZ","tag":"09EZ","title":"Eliminating ramification · Lemma 09EZ","summary":"Let A be a local ring annihilated by a prime p whose maximal ideal is nilpotent. There exists a ring map σ : kappa_A → A which is a section to the residue map A → kappa_A. If A → A' is a local homomorphism of local rings, then we can choose a similar ring map σ' : kappa_A' → A' compatible with σ provided that the extension kappa_A'/kappa_A is separable.","statement_latex":"Let $A$ be a local ring annihilated by a prime $p$ whose maximal ideal is\nnilpotent. There exists a ring map $\\sigma : \\kappa_A \\to A$\nwhich is a section to the residue map $A \\to \\kappa_A$. If $A \\to A'$ is\na local homomorphism of local rings, then we can choose a similar\nring map $\\sigma' : \\kappa_{A'} \\to A'$ compatible with $\\sigma$ provided\nthat the extension $\\kappa_{A'}/\\kappa_A$ is separable.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09EZ","source_file":"more-algebra.tex","source_line":34746,"source_end_line":34754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34746-L34754","statement_sha256":"e94d4e084e622409210a3acc30ff772156c514be57d7b1a23de729a1abfbb573","origin":"The Stacks Project","memory_eligible":false,"source_rank":3651,"rank":3651,"depth":12,"x":540.393,"y":750.597,"cluster":"advanced-algebra"},{"id":"stacks:09F0","tag":"09F0","title":"Eliminating ramification · Lemma 09F0","summary":"Let A be a discrete valuation ring with fraction field K of characteristic p > 0. Let xi ∈ K. Let L be an extension of K obtained by adjoining a root of z^p - z = xi. Then L/K is Galois and one of the following happens • L = K, • L/K is unramified with respect to A of degree p, • L/K is totally ramified with respect to A with ramification index p, and • the integral closure B of A in L is a discrete valuation ring, A ⊂ B is weakly unramified, and A → B induces a purely…","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$ of characteristic\n$p > 0$. Let $\\xi \\in K$. Let $L$ be an extension of $K$ obtained by\nadjoining a root of $z^p - z = \\xi$. Then $L/K$ is Galois and one of the\nfollowing happens\n\\begin{enumerate}\n\\item $L = K$,\n\\item $L/K$ is unramified with respect to $A$ of degree $p$,\n\\item $L/K$ is totally ramified with respect to $A$\nwith ramification index $p$, and\n\\item the integral closure $B$ of $A$ in $L$ is a discrete valuation ring,\n$A \\subset B$ is weakly unramified, and $A \\to B$ induces a purely inseparable\nresidue field extension of degree $p$.\n\\end{enumerate}\nLet $\\pi$ be a uniformizer of $A$. We have the following implications:\n\\begin{enumerate}\n\\item[(A)] If $\\xi \\in A$, then we are in case (1) or (2).\n\\item[(B)] If $\\xi = \\pi^{-n}a$ where $n > 0$ is not divisible by\n$p$ and $a$ is a unit in $A$, then we are in case (3)\n\\item[(C)] If $\\xi = \\pi^{-n} a$ where $n > 0$ is divisible by $p$ and\nthe image of $a$ in $\\kappa_A$ is not a $p$th power, then we are in case (4).\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09F0","source_file":"more-algebra.tex","source_line":34764,"source_end_line":34787,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34764-L34787","statement_sha256":"da370d87dc85d66149ab8f6592ce58bc9c5ddf9377b1ef63eb8d7e888ff2fc69","origin":"The Stacks Project","memory_eligible":false,"source_rank":3652,"rank":3652,"depth":0,"x":726.199,"y":548.651,"cluster":"advanced-algebra"},{"id":"stacks:09F1","tag":"09F1","title":"Eliminating ramification · Lemma 09F1","summary":"Let A ⊂ B ⊂ C be extensions of discrete valuation rings with fractions fields K ⊂ L ⊂ M. Assume • A ⊂ B weakly unramified, • the characteristic of K is p, • M is a degree p Galois extension of L, and • kappa_A = ⋂_n ≥ 1 kappa_B^p^n. Then there exists a finite Galois extension K_1/K totally ramified with respect to A which is a weak solution for A → C.","statement_latex":"Let $A \\subset B \\subset C$ be extensions of discrete valuation rings\nwith fractions fields $K \\subset L \\subset M$. Assume\n\\begin{enumerate}\n\\item $A \\subset B$ weakly unramified,\n\\item the characteristic of $K$ is $p$,\n\\item $M$ is a degree $p$ Galois extension of $L$, and\n\\item $\\kappa_A = \\bigcap_{n \\geq 1} \\kappa_B^{p^n}$.\n\\end{enumerate}\nThen there exists a finite Galois extension $K_1/K$\ntotally ramified with respect to $A$\nwhich is a weak solution for $A \\to C$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09F1","source_file":"more-algebra.tex","source_line":34817,"source_end_line":34830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34817-L34830","statement_sha256":"bc2d81f9f9b3ed0147ae10913e1715f7a467774f96fce37e89f2f04d98b8b5fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":3653,"rank":3653,"depth":13,"x":751.603,"y":803.14,"cluster":"advanced-algebra"},{"id":"stacks:09F2","tag":"09F2","title":"Eliminating ramification · Lemma 09F2","summary":"Let A be a ring which contains a primitive pth root of unity zeta. Set w = 1 - zeta. Then P(z) = frac(1 + wz)^p - 1w^p = z^p - z + ∑_0 < i < p a_i z^i is an element of A[z] and in fact a_i ∈ (w). Moreover, we have P(z_1 + z_2 + w z_1 z_2) = P(z_1) + P(z_2) + w^p P(z_1) P(z_2) in the polynomial ring A[z_1, z_2].","statement_latex":"Let $A$ be a ring which contains a primitive $p$th root of unity $\\zeta$.\nSet $w = 1 - \\zeta$. Then\n$$\nP(z) = \\frac{(1 + wz)^p - 1}{w^p} =\nz^p - z + \\sum\\nolimits_{0 < i < p} a_i z^i\n$$\nis an element of $A[z]$ and in fact $a_i \\in (w)$. Moreover, we have\n$$\nP(z_1 + z_2 + w z_1 z_2) = P(z_1) + P(z_2) + w^p P(z_1) P(z_2)\n$$\nin the polynomial ring $A[z_1, z_2]$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09F2","source_file":"more-algebra.tex","source_line":34976,"source_end_line":34989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L34976-L34989","statement_sha256":"0891d51f256fa9d6c157a88939b962a502458ad0cfcaae6b0cdfc02e083d1d42","origin":"The Stacks Project","memory_eligible":false,"source_rank":3654,"rank":3654,"depth":0,"x":528.087,"y":629.798,"cluster":"advanced-algebra"},{"id":"stacks:09F3","tag":"09F3","title":"Eliminating ramification · Lemma 09F3","summary":"Let A be a discrete valuation ring of mixed characteristic (0, p) which contains a primitive pth root of 1. Let P(t) ∈ A[t] be the polynomial of Lemma [Tag 09F2]. Let xi ∈ K. Let L be an extension of K obtained by adjoining a root of P(z) = xi. Then L/K is Galois and one of the following happens • L = K, • L/K is unramified with respect to A of degree p, • L/K is totally ramified with respect to A with ramification index p, and • the integral closure B of A in L is a…","statement_latex":"Let $A$ be a discrete valuation ring of mixed characteristic $(0, p)$\nwhich contains a primitive $p$th root of $1$.\nLet $P(t) \\in A[t]$ be the polynomial of Lemma \\ref{lemma-prepare}.\nLet $\\xi \\in K$.\nLet $L$ be an extension of $K$ obtained by\nadjoining a root of $P(z) = \\xi$. Then $L/K$ is Galois and one of the\nfollowing happens\n\\begin{enumerate}\n\\item $L = K$,\n\\item $L/K$ is unramified with respect to $A$ of degree $p$,\n\\item $L/K$ is totally ramified with respect to $A$\nwith ramification index $p$, and\n\\item the integral closure $B$ of $A$ in $L$ is a discrete valuation ring,\n$A \\subset B$ is weakly unramified, and $A \\to B$ induces a purely inseparable\nresidue field extension of degree $p$.\n\\end{enumerate}\nLet $\\pi$ be a uniformizer of $A$. We have the following implications:\n\\begin{enumerate}\n\\item[(A)] If $\\xi \\in A$, then we are in case (1) or (2).\n\\item[(B)] If $\\xi = \\pi^{-n}a$ where $n > 0$ is not divisible by\n$p$ and $a$ is a unit in $A$, then we are in case (3)\n\\item[(C)] If $\\xi = \\pi^{-n} a$ where $n > 0$ is divisible by $p$ and\nthe image of $a$ in $\\kappa_A$ is not a $p$th power, then we are in case (4).\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09F3","source_file":"more-algebra.tex","source_line":35016,"source_end_line":35042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35016-L35042","statement_sha256":"cedeb4b19fbaa7ad8d94c5222b86ec5ece0cb45364a85d8708f956f189aa3d04","origin":"The Stacks Project","memory_eligible":false,"source_rank":3655,"rank":3655,"depth":1,"x":832.477,"y":630.791,"cluster":"advanced-algebra"},{"id":"stacks:09F4","tag":"09F4","title":"Eliminating ramification · Lemma 09F4","summary":"Let A ⊂ B ⊂ C be extensions of discrete valuation rings with fractions fields K ⊂ L ⊂ M. Assume that • A has mixed characteristic (0, p), • A ⊂ B is weakly unramified, • B contains a primitive pth root of 1, and • M/L is Galois of degree p. Then there exists a finite Galois extension K_1/K totally ramified with respect to A which is either a weak solution for A → C or is such that M_1/L_1 is a degree p extension of finite level.","statement_latex":"Let $A \\subset B \\subset C$ be extensions of discrete valuation rings\nwith fractions fields $K \\subset L \\subset M$. Assume that\n\\begin{enumerate}\n\\item $A$ has mixed characteristic $(0, p)$,\n\\item $A \\subset B$ is weakly unramified,\n\\item $B$ contains a primitive $p$th root of $1$, and\n\\item $M/L$ is Galois of degree $p$.\n\\end{enumerate}\nThen there exists a finite Galois extension $K_1/K$ totally ramified\nwith respect to $A$ which is either a weak solution for $A \\to C$\nor is such that $M_1/L_1$ is a degree $p$ extension of finite level.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09F4","source_file":"more-algebra.tex","source_line":35087,"source_end_line":35100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35087-L35100","statement_sha256":"005844fc778b577f8ea082131c70497c8b17db867217329c87ce044a6c6017a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3656,"rank":3656,"depth":3,"x":607.096,"y":802.876,"cluster":"advanced-algebra"},{"id":"stacks:09F5","tag":"09F5","title":"Eliminating ramification · Lemma 09F5","summary":"Let A ⊂ B ⊂ C be extensions of discrete valuation rings with fractions fields K ⊂ L ⊂ M. Assume • A has mixed characteristic (0, p), • A ⊂ B weakly unramified, • B contains a primitive pth root of 1, • M/L is a degree p extension of finite level l > 0, • kappa_A = ⋂_n ≥ 1 kappa_B^p^n. Then there exists a finite separable extension K_1 of K totally ramified with respect to A such that either K_1 is a weak solution for A → C, or the extension M_1/L_1 is a degree p extension…","statement_latex":"Let $A \\subset B \\subset C$ be extensions of discrete valuation rings\nwith fractions fields $K \\subset L \\subset M$. Assume\n\\begin{enumerate}\n\\item $A$ has mixed characteristic $(0, p)$,\n\\item $A \\subset B$ weakly unramified,\n\\item $B$ contains a primitive $p$th root of $1$,\n\\item $M/L$ is a degree $p$ extension of finite level $l > 0$,\n\\item $\\kappa_A = \\bigcap_{n \\geq 1} \\kappa_B^{p^n}$.\n\\end{enumerate}\nThen there exists a finite separable extension $K_1$ of $K$\ntotally ramified with respect to $A$\nsuch that either $K_1$ is a weak solution for $A \\to C$, or the extension\n$M_1/L_1$ is a degree $p$ extension of finite level\n$\\leq \\max(0, l - 1, 2l - p)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09F5","source_file":"more-algebra.tex","source_line":35164,"source_end_line":35180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35164-L35180","statement_sha256":"36fa46bf21492f59d5a5dff81c751bb175cf740be83efefdfaec45391fe2a883","origin":"The Stacks Project","memory_eligible":false,"source_rank":3657,"rank":3657,"depth":14,"x":634.918,"y":547.95,"cluster":"advanced-algebra"},{"id":"stacks:09F8","tag":"09F8","title":"Eliminating ramification · Lemma 09F8","summary":"Let A ⊂ B ⊂ C be extensions of discrete valuation rings with fraction fields K ⊂ L ⊂ M. Assume • the residue field k of A is algebraically closed of characteristic p > 0, • A and B are complete, • A → B is weakly unramified, • M is a finite extension of L, • k = ⋂_n ≥ 1 kappa_B^p^n Then there exists a finite extension K_1/K which is a weak solution for A → C.","statement_latex":"Let $A \\subset B \\subset C$ be extensions of discrete valuation rings\nwith fraction fields $K \\subset L \\subset M$. Assume\n\\begin{enumerate}\n\\item the residue field $k$ of $A$ is algebraically closed of\ncharacteristic $p > 0$,\n\\item $A$ and $B$ are complete,\n\\item $A \\to B$ is weakly unramified,\n\\item $M$ is a finite extension of $L$,\n\\item $k = \\bigcap\\nolimits_{n \\geq 1} \\kappa_B^{p^n}$\n\\end{enumerate}\nThen there exists a finite extension $K_1/K$ which\nis a weak solution for $A \\to C$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09F8","source_file":"more-algebra.tex","source_line":35369,"source_end_line":35383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35369-L35383","statement_sha256":"eaeb76221a142551afd7498f6f8fce72f871299436eb135dbb8d16cc3f802b32","origin":"The Stacks Project","memory_eligible":false,"source_rank":3658,"rank":3658,"depth":49,"x":819.514,"y":751.833,"cluster":"advanced-algebra"},{"id":"stacks:09F9","tag":"09F9","title":"Epp · Theorem 09F9","summary":"Let A ⊂ B be an extension of discrete valuation rings with fraction fields K ⊂ L. If the characteristic of kappa_A is p > 0, assume that every element of ⋂_n ≥ 1 kappa_B^p^n is separable algebraic over kappa_A. Then there exists a finite extension K_1/K which is a weak solution for A → B as defined in Definition [Tag 09EN].","statement_latex":"Let $A \\subset B$ be an extension of discrete valuation rings with fraction\nfields $K \\subset L$. If the characteristic of $\\kappa_A$ is $p > 0$,\nassume that every element of\n$$\n\\bigcap\\nolimits_{n \\geq 1} \\kappa_B^{p^n}\n$$\nis separable algebraic over $\\kappa_A$. Then there exists a finite extension\n$K_1/K$ which is a weak solution for $A \\to B$ as defined in\nDefinition \\ref{definition-solution}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09F9","source_file":"more-algebra.tex","source_line":35517,"source_end_line":35528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35517-L35528","statement_sha256":"49822ee4f057e6288a17c3d361407a01d9d3149344cc35ac2cc00feb9672596e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3659,"rank":3659,"depth":50,"x":519.267,"y":706.21,"cluster":"advanced-algebra"},{"id":"stacks:0GLR","tag":"0GLR","title":"Eliminating ramification, II · Lemma 0GLR","summary":"Let A → B be an extension of discrete valuation rings with fraction fields K ⊂ L. If K_1/K is a solution for A ⊂ B, then for any finite extension K_2/K_1 the extension K_2/K is a solution for A ⊂ B.","statement_latex":"Let $A \\to B$ be an extension of discrete valuation rings with fraction\nfields $K \\subset L$. If $K_1/K$ is a solution for $A \\subset B$,\nthen for any finite extension $K_2/K_1$ the extension $K_2/K$ is a\nsolution for $A \\subset B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLR","source_file":"more-algebra.tex","source_line":35654,"source_end_line":35660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35654-L35660","statement_sha256":"26c2948055566e49422a491445ed84f6b29b61afc9d6f16d1d66539d459c9411","origin":"The Stacks Project","memory_eligible":false,"source_rank":3660,"rank":3660,"depth":48,"x":777.501,"y":569.406,"cluster":"advanced-algebra"},{"id":"stacks:0GLS","tag":"0GLS","title":"Eliminating ramification, II · Lemma 0GLS","summary":"Let A ⊂ B be an extension of discrete valuation rings. If B is Nagata and the extension L/K of fraction fields is separable, then A is Nagata.","statement_latex":"Let $A \\subset B$ be an extension of discrete valuation rings.\nIf $B$ is Nagata and the extension $L/K$ of fraction fields is\nseparable, then $A$ is Nagata.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLS","source_file":"more-algebra.tex","source_line":35666,"source_end_line":35671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35666-L35671","statement_sha256":"2ad18c0216b6e4ae1da58fad8194a6caab62f36d1fc5f0924f46430cc8a8eb6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3661,"rank":3661,"depth":11,"x":697.051,"y":816.953,"cluster":"advanced-algebra"},{"id":"stacks:0GLT","tag":"0GLT","title":"Eliminating ramification, II · Lemma 0GLT","summary":"Let A' ⊂ A be an extension of rings. Let f ∈ A'. Assume that (a) A is finite over A', (b) f is a nonzerodivisor on A, and (c) A'_f = A_f. Then there exists an integer n_0 > 0 such that for all n ≥ n_0 the following is true: given a ring B', a nonzerodivisor g ∈ B', and an isomorphism φ' : A'/f^n A' → B'/g^n B' with φ'(f) equiv g, there is a finite extension B' ⊂ B and an isomorphism φ : A/fA → B/gB compatible with φ'.","statement_latex":"Let $A' \\subset A$ be an extension of rings. Let $f \\in A'$.\nAssume that (a) $A$ is finite over $A'$, (b) $f$ is a nonzerodivisor\non $A$, and (c) $A'_f = A_f$. Then there exists an integer $n_0 > 0$\nsuch that for all $n \\geq n_0$ the following is true: given a ring\n$B'$, a nonzerodivisor $g \\in B'$, and an isomorphism\n$\\varphi' : A'/f^n A' \\to B'/g^n B'$ with $\\varphi'(f) \\equiv g$, there is a\nfinite extension $B' \\subset B$ and an isomorphism\n$\\varphi : A/fA \\to B/gB$ compatible with $\\varphi'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLT","source_file":"more-algebra.tex","source_line":35690,"source_end_line":35700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35690-L35700","statement_sha256":"bd7104ebd3ca72a939523440d0bcf41e175cad65f9456354500bc92678fb51de","origin":"The Stacks Project","memory_eligible":false,"source_rank":3662,"rank":3662,"depth":0,"x":557.222,"y":588.636,"cluster":"advanced-algebra"},{"id":"stacks:0GLV","tag":"0GLV","title":"Eliminating ramification, II · Lemma 0GLV","summary":"Let p be a prime number. Let A ⊂ B be an extension of discrete valuation rings with fraction field extension L/K. Let K_2/K_1/K be a tower of finite field extensions. Assume • K has characteristic p, • L/K is separable, • B is Nagata, • K_2 is a solution for A ⊂ B, • K_2/K_1 is purely inseparable of degree p. Then there exists a separable extension K_3/K_1 which is a solution for A ⊂ B.","statement_latex":"Let $p$ be a prime number. Let $A \\subset B$ be an extension of discrete\nvaluation rings with fraction field extension $L/K$. Let $K_2/K_1/K$ be\na tower of finite field extensions. Assume\n\\begin{enumerate}\n\\item $K$ has characteristic $p$,\n\\item $L/K$ is separable,\n\\item $B$ is Nagata,\n\\item $K_2$ is a solution for $A \\subset B$,\n\\item $K_2/K_1$ is purely inseparable of degree $p$.\n\\end{enumerate}\nThen there exists a separable extension $K_3/K_1$\nwhich is a solution for $A \\subset B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLV","source_file":"more-algebra.tex","source_line":35774,"source_end_line":35788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35774-L35788","statement_sha256":"5087e17b333fca03048e5bc6ba9eb07ab8e688c8358c7e6435d7d0cb928e9b3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3663,"rank":3663,"depth":47,"x":844.101,"y":677.702,"cluster":"advanced-algebra"},{"id":"stacks:0BRN","tag":"0BRN","title":"Eliminating ramification, II · Lemma 0BRN","summary":"Let A ⊂ B be an extension of discrete valuation rings. Assume • the extension L/K of fraction fields is separable, • B is Nagata, and • there exists a solution for A ⊂ B. Then there exists a separable solution for A ⊂ B.","statement_latex":"Let $A \\subset B$ be an extension of discrete valuation rings.\nAssume\n\\begin{enumerate}\n\\item the extension $L/K$ of fraction fields is separable,\n\\item $B$ is Nagata, and\n\\item there exists a solution for $A \\subset B$.\n\\end{enumerate}\nThen there exists a separable solution for $A \\subset B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRN","source_file":"more-algebra.tex","source_line":35892,"source_end_line":35902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35892-L35902","statement_sha256":"207fe839283d61272a809b607cdf0738af6a5da9c13217cd85373778aecf88d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3664,"rank":3664,"depth":48,"x":560.774,"y":774.863,"cluster":"advanced-algebra"},{"id":"stacks:09IH","tag":"09IH","title":"Eliminating ramification, II · Lemma 09IH","summary":"Let A → B be an extension of discrete valuation rings with fraction fields K ⊂ L. Assume B is essentially of finite type over A. Let K'/K be an algebraic extension of fields such that the integral closure A' of A in K' is Noetherian. Then the integral closure B' of B in L' = (L ⊗_K K')_red is Noetherian as well. Moreover, the map Spec(B') → Spec(A') is surjective and the corresponding residue field extensions are finitely generated field extensions.","statement_latex":"Let $A \\to B$ be an extension of discrete valuation rings with fraction\nfields $K \\subset L$. Assume $B$ is essentially of finite type over $A$.\nLet $K'/K$ be an algebraic extension of fields such that\nthe integral closure $A'$ of $A$ in $K'$ is Noetherian. Then the integral\nclosure $B'$ of $B$ in $L' = (L \\otimes_K K')_{red}$ is Noetherian\nas well. Moreover, the map $\\Spec(B') \\to \\Spec(A')$\nis surjective and the corresponding residue field extensions are finitely\ngenerated field extensions.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IH","source_file":"more-algebra.tex","source_line":35925,"source_end_line":35935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35925-L35935","statement_sha256":"0638495b5461dbba26071aec30e542ce77337e90cde6138e37ffc96627d59d6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3665,"rank":3665,"depth":11,"x":691.636,"y":542.319,"cluster":"advanced-algebra"},{"id":"stacks:09II","tag":"09II","title":"Eliminating ramification, II · Proposition 09II","summary":"See [alterations] for a special case. Let A → B be an extension of discrete valuation rings with fraction fields K ⊂ L. If B is essentially of finite type over A, then there exists a finite extension K_1/K which is a solution for A → B as defined in Definition [Tag 09EN].","statement_latex":"\\begin{reference}\nSee \\cite[Lemma 2.13]{alterations} for a special case.\n\\end{reference}\nLet $A \\to B$ be an extension of discrete valuation rings with fraction\nfields $K \\subset L$. If $B$ is essentially of finite type over $A$, then\nthere exists a finite extension $K_1/K$ which is a solution for\n$A \\to B$ as defined in\nDefinition \\ref{definition-solution}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09II","source_file":"more-algebra.tex","source_line":35960,"source_end_line":35970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L35960-L35970","statement_sha256":"01e73fb17920fe1bebf8316ea6f7601e72afc97cfdfe61abfebe81994aa86445","origin":"The Stacks Project","memory_eligible":false,"source_rank":3666,"rank":3666,"depth":51,"x":782.197,"y":788.188,"cluster":"advanced-algebra"},{"id":"stacks:0BRP","tag":"0BRP","title":"Eliminating ramification, II · Lemma 0BRP","summary":"Let A → B be an extension of discrete valuation rings with fraction fields K ⊂ L. Assume • B is essentially of finite type over A, • either A or B is a Nagata ring, and • L/K is separable. Then there exists a separable solution for A → B (Definition [Tag 09EN]).","statement_latex":"Let $A \\to B$ be an extension of discrete valuation rings with fraction\nfields $K \\subset L$. Assume\n\\begin{enumerate}\n\\item $B$ is essentially of finite type over $A$,\n\\item either $A$ or $B$ is a Nagata ring, and\n\\item $L/K$ is separable.\n\\end{enumerate}\nThen there exists a separable solution for $A \\to B$\n(Definition \\ref{definition-solution}).","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Eliminating ramification, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRP","source_file":"more-algebra.tex","source_line":36071,"source_end_line":36082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36071-L36082","statement_sha256":"500688785bf487e524f048ae1e02feb2c9b07732e81ece90d14ab00886ce19f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3667,"rank":3667,"depth":52,"x":517.547,"y":658.201,"cluster":"advanced-algebra"},{"id":"stacks:0B8H","tag":"0B8H","title":"Picard groups of rings · Definition 0B8H","summary":"Let R be a ring. An R-module M is invertible if the functor Mod_R → Mod_R, N ↦ M ⊗_R N is an equivalence of categories. An invertible R-module is said to be trivial if it is isomorphic to R as an R-module.","statement_latex":"Let $R$ be a ring. An $R$-module $M$ is {\\it invertible} if the functor\n$$\n\\text{Mod}_R \\longrightarrow \\text{Mod}_R,\\quad\nN \\longmapsto M \\otimes_R N\n$$\nis an equivalence of categories. An invertible $R$-module is said to be\n{\\it trivial} if it is isomorphic to $R$ as an $R$-module.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Picard groups of rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8H","source_file":"more-algebra.tex","source_line":36106,"source_end_line":36115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36106-L36115","statement_sha256":"2ffdd72bf05f46f83923687c4378dd6ece1f3d0aef5d9f18f731fb98c138acc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3668,"rank":3668,"depth":0,"x":817.398,"y":603.852,"cluster":"advanced-algebra"},{"id":"stacks:0B8I","tag":"0B8I","title":"Picard groups of rings · Lemma 0B8I","summary":"Let R be a ring. Let M be an R-module. Equivalent are • M is finite locally free module of rank 1, • M is invertible, and • there exists an R-module N such that M ⊗_R N ≅ R. Moreover, in this case the module N in (3) is isomorphic to Hom_R(M, R).","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. Equivalent are\n\\begin{enumerate}\n\\item $M$ is finite locally free module of rank $1$,\n\\item $M$ is invertible, and\n\\item there exists an $R$-module $N$ such that $M \\otimes_R N \\cong R$.\n\\end{enumerate}\nMoreover, in this case the module $N$ in (3) is isomorphic\nto $\\Hom_R(M, R)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Picard groups of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8I","source_file":"more-algebra.tex","source_line":36117,"source_end_line":36127,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36117-L36127","statement_sha256":"b002352061725352776a132bc0df4e019161ebc412b9ac16b3019533af8efd28","origin":"The Stacks Project","memory_eligible":false,"source_rank":3669,"rank":3669,"depth":5,"x":639.898,"y":814.19,"cluster":"advanced-algebra"},{"id":"stacks:0BCH","tag":"0BCH","title":"Picard groups of rings · Lemma 0BCH","summary":"Let R be a UFD. Then Pic(R) is trivial.","statement_latex":"Let $R$ be a UFD. Then $\\Pic(R)$ is trivial.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Picard groups of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCH","source_file":"more-algebra.tex","source_line":36183,"source_end_line":36186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36183-L36186","statement_sha256":"f48070cad74329e85234c4a7b0b7f14421a01ab7ec59bc3bb3c2c45228318f25","origin":"The Stacks Project","memory_eligible":false,"source_rank":3670,"rank":3670,"depth":6,"x":601.614,"y":558.227,"cluster":"advanced-algebra"},{"id":"stacks:0FJB","tag":"0FJB","title":"Determinants · Lemma 0FJB","summary":"Let R be a ring. Let 0 → M' → M → M\" → 0 be a short exact sequence of finite projective R-modules. Then there is a canonical isomorphism γ : det(M') ⊗ det(M\") → det(M)","statement_latex":"Let $R$ be a ring. Let\n$$\n0 \\to M' \\to M \\to M'' \\to 0\n$$\nbe a short exact sequence of finite projective $R$-modules. Then there\nis a canonical isomorphism\n$$\n\\gamma : \\det(M') \\otimes \\det(M'') \\longrightarrow \\det(M)\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJB","source_file":"more-algebra.tex","source_line":36282,"source_end_line":36293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36282-L36293","statement_sha256":"02c7c0f9260989426e5b4d7ae6ff1c00f2988c49f0994093e178d58f87cd37d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3671,"rank":3671,"depth":0,"x":835.815,"y":725.341,"cluster":"advanced-algebra"},{"id":"stacks:0FJC","tag":"0FJC","title":"Determinants · Lemma 0FJC","summary":"Let R be a ring. Let xymatrix 0 ar[r] & M' ar[r] ar[d]^u & M ar[r] ar[d]^v & M\" ar[r] ar[d]^w & 0 0 ar[r] & K' ar[r] & K ar[r] & K\" ar[r] & 0 be a commutative diagram of finite projective R-modules whose vertical arrows are isomorphisms. Then we get a commutative diagram of isomorphisms xymatrix det(M') ⊗ det(M\") ar[r]_-γ ar[d]_det(u) ⊗ det(w) & det(M) ar[d]^det(v) det(K') ⊗ det(K\") ar[r]^-γ & det(K) where the horizontal arrows are the ones constructed in Lemma [Tag 0FJB].","statement_latex":"Let $R$ be a ring. Let\n$$\n\\xymatrix{\n0 \\ar[r] &\nM' \\ar[r] \\ar[d]^u &\nM \\ar[r] \\ar[d]^v &\nM'' \\ar[r] \\ar[d]^w &\n0 \\\\\n0 \\ar[r] &\nK' \\ar[r] &\nK \\ar[r] &\nK'' \\ar[r] &\n0\n}\n$$\nbe a commutative diagram of finite projective $R$-modules\nwhose vertical arrows are isomorphisms. Then we get a commutative\ndiagram of isomorphisms\n$$\n\\xymatrix{\n\\det(M') \\otimes \\det(M'') \\ar[r]_-\\gamma \\ar[d]_{\\det(u) \\otimes \\det(w)} &\n\\det(M) \\ar[d]^{\\det(v)} \\\\\n\\det(K') \\otimes \\det(K'') \\ar[r]^-\\gamma & \\det(K)\n}\n$$\nwhere the horizontal arrows are the ones constructed\nin Lemma \\ref{lemma-det-ses}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJC","source_file":"more-algebra.tex","source_line":36336,"source_end_line":36365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36336-L36365","statement_sha256":"a763847b4cb85379ad90ecebdcae665b8482f733bca89e86ac05a3adfa88c236","origin":"The Stacks Project","memory_eligible":false,"source_rank":3672,"rank":3672,"depth":1,"x":528.556,"y":735.01,"cluster":"advanced-algebra"},{"id":"stacks:0FJD","tag":"0FJD","title":"Determinants · Lemma 0FJD","summary":"Let R be a ring. Let K ⊂ L ⊂ M be R-modules such that K, L/K, and M/L are finite projective R-modules. Then the diagram xymatrix det(K) ⊗ det(L/K) ⊗ det(M/L) ar[r] ar[d] & det(L) ⊗ det(M/L) ar[d] det(K) ⊗ det(M/K) ar[r] & det(M) commutes where the maps are those of Lemma [Tag 0FJB].","statement_latex":"Let $R$ be a ring. Let\n$$\nK \\subset L \\subset M\n$$\nbe $R$-modules such that $K$, $L/K$, and $M/L$ are finite projective\n$R$-modules. Then the diagram\n$$\n\\xymatrix{\n\\det(K) \\otimes \\det(L/K) \\otimes \\det(M/L) \\ar[r] \\ar[d] &\n\\det(L) \\otimes \\det(M/L) \\ar[d] \\\\\n\\det(K) \\otimes \\det(M/K) \\ar[r] &\n\\det(M)\n}\n$$\ncommutes where the maps are those of Lemma \\ref{lemma-det-ses}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJD","source_file":"more-algebra.tex","source_line":36372,"source_end_line":36389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36372-L36389","statement_sha256":"50fab61dfc2b56a4415f8d83c55c273677bdfdc010f2d51cba9cdfe75d5958df","origin":"The Stacks Project","memory_eligible":false,"source_rank":3673,"rank":3673,"depth":1,"x":747.473,"y":553.432,"cluster":"advanced-algebra"},{"id":"stacks:0FJE","tag":"0FJE","title":"Determinants · Lemma 0FJE","summary":"Let R be a ring. Let M' and M\" be two finite projective R-modules. Then the diagram xymatrix det(M') ⊗ det(M\") ar[r] ar[d]_ε · (switch tensors) & det(M' ⊕ M\") ar[d]^det(switch summands) det(M\") ⊗ det(M') ar[r] & det(M\" ⊕ M') commutes where ε = det( -id_M' ⊗ M\") ∈ R^* and the horizontal arrows are those of Lemma [Tag 0FJB].","statement_latex":"Let $R$ be a ring. Let $M'$ and $M''$ be two finite projective\n$R$-modules. Then the diagram\n$$\n\\xymatrix{\n\\det(M') \\otimes \\det(M'') \\ar[r]\n\\ar[d]_{\\epsilon \\cdot (\\text{switch tensors})} &\n\\det(M' \\oplus M'') \\ar[d]^{\\det(\\text{switch summands})} \\\\\n\\det(M'') \\otimes \\det(M') \\ar[r] &\n\\det(M'' \\oplus M')\n}\n$$\ncommutes where $\\epsilon = \\det( -\\text{id}_{M' \\otimes M''}) \\in R^*$\nand the horizontal arrows are those of Lemma \\ref{lemma-det-ses}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJE","source_file":"more-algebra.tex","source_line":36398,"source_end_line":36413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36398-L36413","statement_sha256":"07c219f52bd43777f6aa28016e738d6b43807496b2d001a90912c06909b6bcd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3674,"rank":3674,"depth":1,"x":732.059,"y":811.689,"cluster":"advanced-algebra"},{"id":"stacks:0FJF","tag":"0FJF","title":"Determinants · Lemma 0FJF","summary":"Let R be a ring. Let M, N be finite projective R-modules. Let a : M → N and b : N → M be R-linear maps. Then det(id + a ∘ b) = det(id + b ∘ a) as elements of R.","statement_latex":"Let $R$ be a ring. Let $M$, $N$ be finite projective $R$-modules.\nLet $a : M \\to N$ and $b : N \\to M$ be $R$-linear maps.\nThen\n$$\n\\det(\\text{id} + a \\circ b) = \\det(\\text{id} + b \\circ a)\n$$\nas elements of $R$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJF","source_file":"more-algebra.tex","source_line":36419,"source_end_line":36428,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36419-L36428","statement_sha256":"eb8e353e77a874deba0aad0ff4959adfae9fe4b856def05756cfd707ebe11538","origin":"The Stacks Project","memory_eligible":false,"source_rank":3675,"rank":3675,"depth":0,"x":535.631,"y":612.396,"cluster":"advanced-algebra"},{"id":"stacks:0AFX","tag":"0AFX","title":"Determinants · Lemma 0AFX","summary":"Let R be a ring. There is a map det : K_0(R) → Pic(R) which maps [M] to the class of the invertible module wedge^n(M) if M is a finite locally free module of rank n.","statement_latex":"Let $R$ be a ring. There is a map\n$$\n\\det : K_0(R) \\longrightarrow \\Pic(R)\n$$\nwhich maps $[M]$ to the class of the invertible module\n$\\wedge^n(M)$ if $M$ is a finite locally free module of rank $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFX","source_file":"more-algebra.tex","source_line":36463,"source_end_line":36471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36463-L36471","statement_sha256":"949d46a748513bf85a63c12994b4b08b9e10335fa51e9010c528c874f52d1b7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3676,"rank":3676,"depth":1,"x":840.912,"y":647.914,"cluster":"advanced-algebra"},{"id":"stacks:0AFY","tag":"0AFY","title":"Perfect complexes and K-groups · Lemma 0AFY","summary":"Let R be a ring. There is a map c : perfect complexes over R → K_0(R) with the following properties • c(K[n]) = (-1)^nc(K) for a perfect complex K, • if K → L → M → K[1] is a distinguished triangle of perfect complexes, then c(L) = c(K) + c(M), • if K is represented by a finite complex M^bullet consisting of finite projective modules, then c(K) = ∑ (-1)^i[M_i].","statement_latex":"Let $R$ be a ring. There is a map\n$$\nc : \\text{perfect complexes over }R \\longrightarrow K_0(R)\n$$\nwith the following properties\n\\begin{enumerate}\n\\item $c(K[n]) = (-1)^nc(K)$ for a perfect complex $K$,\n\\item if $K \\to L \\to M \\to K[1]$ is a distinguished triangle of\nperfect complexes, then $c(L) = c(K) + c(M)$,\n\\item if $K$ is represented by a finite complex $M^\\bullet$\nconsisting of finite projective modules, then\n$c(K) = \\sum (-1)^i[M_i]$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFY","source_file":"more-algebra.tex","source_line":36490,"source_end_line":36505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36490-L36505","statement_sha256":"32fd2afd7945e15e1f8721d784785fb8dd82801660e876cd43fe5a283922eea3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3677,"rank":3677,"depth":5,"x":587.097,"y":795.03,"cluster":"advanced-algebra"},{"id":"stacks:0FCU","tag":"0FCU","title":"Perfect complexes and K-groups · Lemma 0FCU","summary":"Let R be a ring. Let D_perf(R) be the derived category of perfect objects, see Lemma [Tag 0ATI]. The map c of Lemma [Tag 0AFY] gives an isomorphism K_0(D_perf(R)) = K_0(R).","statement_latex":"Let $R$ be a ring. Let $D_{perf}(R)$ be the derived category of\nperfect objects, see Lemma \\ref{lemma-perfect-ring-classical-generator}.\nThe map $c$ of Lemma \\ref{lemma-perfect-to-K-group} gives an isomorphism\n$K_0(D_{perf}(R)) = K_0(R)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Perfect complexes and K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCU","source_file":"more-algebra.tex","source_line":36579,"source_end_line":36585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36579-L36585","statement_sha256":"f439de8ad26ccbe946071d147a702ff1591267c7e943b6a40c102ee66b404196","origin":"The Stacks Project","memory_eligible":false,"source_rank":3678,"rank":3678,"depth":14,"x":655.988,"y":542.386,"cluster":"advanced-algebra"},{"id":"stacks:0GSY","tag":"0GSY","title":"Determinants of endomorphisms of finite length modules · Lemma 0GSY","summary":"Let (R, m, kappa) be a local ring. Let 0 → (M, φ) → (M', φ') → (M\", φ\") → 0 be a short exact sequence in the category discussed above. Then det_kappa(φ') = det_kappa(φ)det_kappa(φ\"), Trace_kappa(φ') = Trace_kappa(φ) + Trace_kappa(φ\") Also, the characteristic polynomial of φ' over kappa is the product of the characteristic polynomials of φ and φ\".","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring.\nLet $0 \\to (M, \\varphi) \\to (M', \\varphi') \\to (M'', \\varphi'') \\to 0$\nbe a short exact sequence in the category discussed above.\nThen\n$$\n\\det\\nolimits_\\kappa(\\varphi') =\n\\det\\nolimits_\\kappa(\\varphi)\\det\\nolimits_\\kappa(\\varphi''),\\quad\n\\text{Trace}_\\kappa(\\varphi') = \\text{Trace}_\\kappa(\\varphi) + \n\\text{Trace}_\\kappa(\\varphi'')\n$$\nAlso, the characteristic polynomial\nof $\\varphi'$ over $\\kappa$ is the product of the characteristic polynomials\nof $\\varphi$ and $\\varphi''$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants of endomorphisms of finite length modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSY","source_file":"more-algebra.tex","source_line":36677,"source_end_line":36692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36677-L36692","statement_sha256":"3ad27b70c8b10a18f5c284a882b8d175814529aba65127373c3d7a55fed69744","origin":"The Stacks Project","memory_eligible":false,"source_rank":3679,"rank":3679,"depth":0,"x":808.443,"y":767.899,"cluster":"advanced-algebra"},{"id":"stacks:0GSZ","tag":"0GSZ","title":"Determinants of endomorphisms of finite length modules · Lemma 0GSZ","summary":"Let (R, m, kappa) → (R', m', kappa') be a local homomorphism of local rings. Assume that kappa'/kappa is a finite extension. Let u ∈ R'. Then for any finite length R'-module M' we have det_kappa(u : M' → M') = Norm_kappa'/kappa(u bmod m')^m where m = length_R'(M').","statement_latex":"Let $(R, \\mathfrak m, \\kappa) \\to (R', \\mathfrak m', \\kappa')$\nbe a local homomorphism of local rings. Assume that $\\kappa'/\\kappa$\nis a finite extension. Let $u \\in R'$. Then for\nany finite length $R'$-module $M'$ we have\n$$\n\\det\\nolimits_\\kappa(u : M' \\to M') =\n\\text{Norm}_{\\kappa'/\\kappa}(u \\bmod \\mathfrak m')^m\n$$\nwhere $m = \\text{length}_{R'}(M')$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants of endomorphisms of finite length modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSZ","source_file":"more-algebra.tex","source_line":36698,"source_end_line":36709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36698-L36709","statement_sha256":"65abb8b28a3a3b7fea664894d5b6f039556a2b00146ddcaef3b580144e492cae","origin":"The Stacks Project","memory_eligible":false,"source_rank":3680,"rank":3680,"depth":1,"x":514.509,"y":688.071,"cluster":"advanced-algebra"},{"id":"stacks:0GT0","tag":"0GT0","title":"Determinants of endomorphisms of finite length modules · Lemma 0GT0","summary":"Let (R, m, kappa) → (R', m', kappa') be a flat local homomorphism of local rings such that m = length_R'(R'/ mR') < ∞. For any (M, φ) as above, the element det_kappa(φ)^m maps to det_kappa'(φ ⊗ 1 : M ⊗_R R' → M ⊗_R R') in kappa'.","statement_latex":"Let $(R, \\mathfrak m, \\kappa) \\to (R', \\mathfrak m', \\kappa')$\nbe a flat local homomorphism of local rings such that\n$m = \\text{length}_{R'}(R'/\\mathfrak mR') < \\infty$.\nFor any $(M, \\varphi)$ as above, the element\n$\\det_\\kappa(\\varphi)^m$ maps to\n$\\det_{\\kappa'}(\\varphi \\otimes 1 : M \\otimes_R R' \\to M \\otimes_R R')$\nin $\\kappa'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants of endomorphisms of finite length modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GT0","source_file":"more-algebra.tex","source_line":36721,"source_end_line":36730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36721-L36730","statement_sha256":"d5751d21319e267fc1ffe7d969407f716b774fa5f6e3be77f6e0daab535b24fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":3681,"rank":3681,"depth":1,"x":795.604,"y":580.089,"cluster":"advanced-algebra"},{"id":"stacks:0AFZ","tag":"0AFZ","title":"A regular local ring is a UFD · Lemma 0AFZ","summary":"Let R be a regular local ring. Let f ∈ R. Then Pic(R_f) = 0.","statement_latex":"Let $R$ be a regular local ring. Let $f \\in R$.\nThen $\\Pic(R_f) = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"A regular local ring is a UFD","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFZ","source_file":"more-algebra.tex","source_line":36779,"source_end_line":36783,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36779-L36783","statement_sha256":"fc9d6d3c8de5a920efa866557c57d9b27183446b304a79d8ba3bf7bb700b1815","origin":"The Stacks Project","memory_eligible":false,"source_rank":3682,"rank":3682,"depth":16,"x":675.098,"y":819.347,"cluster":"advanced-algebra"},{"id":"stacks:0AG0","tag":"0AG0","title":"A regular local ring is a UFD · Lemma 0AG0","summary":"A regular local ring is a UFD.","statement_latex":"A regular local ring is a UFD.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"A regular local ring is a UFD","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AG0","source_file":"more-algebra.tex","source_line":36800,"source_end_line":36803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36800-L36803","statement_sha256":"ac28a3adcd89a2a565650a4321e839fb8a8c35e94e032a8feb8579b83a1a3237","origin":"The Stacks Project","memory_eligible":false,"source_rank":3683,"rank":3683,"depth":18,"x":571.495,"y":574.407,"cluster":"advanced-algebra"},{"id":"stacks:0DLQ","tag":"0DLQ","title":"A regular local ring is a UFD · Lemma 0DLQ","summary":"Let R be a valuation ring with fraction field K and residue field kappa. Let R → A be a homomorphism of rings such that • A is local and R → A is local, • A is flat and essentially of finite type over R, • A ⊗_R kappa regular. Then Pic(A ⊗_R K) = 0.","statement_latex":"Let $R$ be a valuation ring with fraction field $K$\nand residue field $\\kappa$. Let $R \\to A$ be a\nhomomorphism of rings such that\n\\begin{enumerate}\n\\item $A$ is local and $R \\to A$ is local,\n\\item $A$ is flat and essentially of finite type over $R$,\n\\item $A \\otimes_R \\kappa$ regular.\n\\end{enumerate}\nThen $\\Pic(A \\otimes_R K) = 0$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"A regular local ring is a UFD","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLQ","source_file":"more-algebra.tex","source_line":36845,"source_end_line":36856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36845-L36856","statement_sha256":"a1c60d4233d9098c94520808438d95606396a945088f8890fd696ddad5372aa7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3684,"rank":3684,"depth":37,"x":845.017,"y":696.303,"cluster":"advanced-algebra"},{"id":"stacks:0FJJ","tag":"0FJJ","title":"Determinants of complexes · Lemma 0FJJ","summary":"Let R be a ring. Let a^bullet : K^bullet → L^bullet be a map of complexes of R-modules satisfying (1), (2), (3) above. If L^bullet has rank 0, then det(a^bullet) maps the canonical element δ(K^bullet) to δ(L^bullet).","statement_latex":"Let $R$ be a ring. Let $a^\\bullet : K^\\bullet \\to L^\\bullet$ be a map of\ncomplexes of $R$-modules satisfying (1), (2), (3) above. If $L^\\bullet$\nhas rank $0$, then $\\det(a^\\bullet)$ maps the\ncanonical element $\\delta(K^\\bullet)$ to $\\delta(L^\\bullet)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJJ","source_file":"more-algebra.tex","source_line":36984,"source_end_line":36990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L36984-L36990","statement_sha256":"f18faf1d0f9198f78a6aab0c5586c06f0d3b0163b22507aab2aafa6fa7acac24","origin":"The Stacks Project","memory_eligible":false,"source_rank":3685,"rank":3685,"depth":2,"x":545.133,"y":761.658,"cluster":"advanced-algebra"},{"id":"stacks:0FJK","tag":"0FJK","title":"Determinants of complexes · Lemma 0FJK","summary":"Let R be a ring. Let a^bullet : K^bullet → L^bullet be a map of complexes of R-modules satisfying (1), (2), (3) above. Let h : K^0 → L^-1 be a map such that b^0 = a^0 + d ∘ h and b^-1 = a^-1 + h ∘ d are surjective. Then det(a^bullet) = det(b^bullet) as maps det(K^bullet) → det(L^bullet).","statement_latex":"Let $R$ be a ring. Let $a^\\bullet : K^\\bullet \\to L^\\bullet$ be a map of\ncomplexes of $R$-modules satisfying (1), (2), (3) above.\nLet $h : K^0 \\to L^{-1}$ be a map such that\n$b^0 = a^0 + d \\circ h$ and $b^{-1} = a^{-1} + h \\circ d$ are surjective.\nThen $\\det(a^\\bullet) = \\det(b^\\bullet)$ as maps\n$\\det(K^\\bullet) \\to \\det(L^\\bullet)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJK","source_file":"more-algebra.tex","source_line":37015,"source_end_line":37023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37015-L37023","statement_sha256":"d87478a056c4eb449a97859fc176c98d1827831c4c0aac9ebcfac822b82d629b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3686,"rank":3686,"depth":3,"x":713.8,"y":543.184,"cluster":"advanced-algebra"},{"id":"stacks:0FJL","tag":"0FJL","title":"Determinants of complexes · Lemma 0FJL","summary":"Let R be a ring. Let a^bullet : K^bullet → L^bullet and b^bullet : L^bullet → M^bullet be maps of complexes of R-modules satisfying (1), (2), (3) above. Then we have det(b^bullet) ∘ det(a^bullet) = det(b^bullet ∘ a^bullet) as maps det(M^bullet) → det(K^bullet).","statement_latex":"Let $R$ be a ring. Let $a^\\bullet : K^\\bullet \\to L^\\bullet$\nand $b^\\bullet : L^\\bullet \\to M^\\bullet$ be maps of\ncomplexes of $R$-modules satisfying (1), (2), (3) above.\nThen we have $\\det(b^\\bullet) \\circ \\det(a^\\bullet) = \n\\det(b^\\bullet \\circ a^\\bullet)$ as maps\n$\\det(M^\\bullet) \\to \\det(K^\\bullet)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJL","source_file":"more-algebra.tex","source_line":37107,"source_end_line":37115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37107-L37115","statement_sha256":"8da0d11774e599a891ee88b98b07328c5ad37755233948e4176737a10a4a4849","origin":"The Stacks Project","memory_eligible":false,"source_rank":3687,"rank":3687,"depth":2,"x":765.147,"y":800.132,"cluster":"advanced-algebra"},{"id":"stacks:0FJM","tag":"0FJM","title":"Determinants of complexes · Lemma 0FJM","summary":"Let R be a ring. The constructions above determine a functor det : ( category of perfect complexes with tor amplitude in [-1, 0] morphisms are isomorphisms ) → ( category of invertible modules morphisms are isomorphisms ) Moreover, given a rank 0 perfect object L of D(R) with tor-amplitude in [-1, 0] there is a canonical element δ(L) ∈ det(L) such that for any isomorphism a : L → K in D(R) we have det(a)(δ(L)) = δ(K).","statement_latex":"Let $R$ be a ring. The constructions above determine a functor\n$$\n\\det :\n\\left\\{\n\\begin{matrix}\n\\text{category of perfect complexes} \\\\\n\\text{with tor amplitude in }[-1, 0] \\\\\n\\text{morphisms are isomorphisms}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{category of invertible modules} \\\\\n\\text{morphisms are isomorphisms}\n\\end{matrix}\n\\right\\}\n$$\nMoreover, given a rank $0$ perfect object $L$ of $D(R)$ with\ntor-amplitude in $[-1, 0]$ there is a canonical element\n$\\delta(L) \\in \\det(L)$ such that for any isomorphism\n$a : L \\to K$ in $D(R)$ we have $\\det(a)(\\delta(L)) = \\delta(K)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Determinants of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJM","source_file":"more-algebra.tex","source_line":37123,"source_end_line":37147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37123-L37147","statement_sha256":"d22f1b68ed7633e5867d594cc290ee27e5de0e2c3b1839ab06e6a3efa7ca6c17","origin":"The Stacks Project","memory_eligible":false,"source_rank":3688,"rank":3688,"depth":11,"x":520.519,"y":639.709,"cluster":"advanced-algebra"},{"id":"stacks:0ASG","tag":"0ASG","title":"Extensions of valuation rings · Definition 0ASG","summary":"We say that A → B or A ⊂ B is an extension of valuation rings if A and B are valuation rings and A → B is injective and local. Such an extension induces a commutative diagram xymatrix A setminus (0) ar[r] ar[d]_v & B setminus (0) ar[d]^v Γ_A ar[r] & Γ_B where Γ_A and Γ_B are the value groups. We say that B is weakly unramified over A if the lower horizontal arrow is a bijection. If the extension of residue fields kappa_A = A/ m_A ⊂ kappa_B = B/ m_B is finite, then we set…","statement_latex":"We say that $A \\to B$ or $A \\subset B$ is an\n{\\it extension of valuation rings} if $A$ and $B$ are\nvaluation rings and $A \\to B$ is injective and local.\nSuch an extension induces a commutative diagram\n$$\n\\xymatrix{\nA \\setminus \\{0\\} \\ar[r] \\ar[d]_v & B \\setminus \\{0\\} \\ar[d]^v \\\\\n\\Gamma_A \\ar[r] & \\Gamma_B\n}\n$$\nwhere $\\Gamma_A$ and $\\Gamma_B$ are the value groups.\nWe say that $B$ is {\\it weakly unramified} over $A$ if\nthe lower horizontal arrow is a bijection.\nIf the extension of residue fields\n$\\kappa_A = A/\\mathfrak m_A \\subset \\kappa_B = B/\\mathfrak m_B$\nis finite, then we set $f = [\\kappa_B : \\kappa_A]$ and we\ncall it the {\\it residual degree} or {\\it residue degree}\nof the extension $A \\subset B$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of valuation rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASG","source_file":"more-algebra.tex","source_line":37305,"source_end_line":37325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37305-L37325","statement_sha256":"17b40f18e9200e737b355c5d8f0e650e2799e4954bb60d2ef145a790f3073328","origin":"The Stacks Project","memory_eligible":false,"source_rank":3689,"rank":3689,"depth":0,"x":830.083,"y":619.183,"cluster":"advanced-algebra"},{"id":"stacks:0ASH","tag":"0ASH","title":"Extensions of valuation rings · Lemma 0ASH","summary":"Let A ⊂ B be an extension of valuation rings with fraction fields K ⊂ L. If the extension L/K is finite, then the residue field extension is finite, the index of Γ_A in Γ_B is finite, and [Γ_B : Γ_A] [kappa_B : kappa_A] ≤ [L : K].","statement_latex":"Let $A \\subset B$ be an extension of valuation rings with\nfraction fields $K \\subset L$. If the extension $L/K$\nis finite, then the residue field extension is finite,\nthe index of $\\Gamma_A$ in $\\Gamma_B$ is finite, and\n$$\n[\\Gamma_B : \\Gamma_A] [\\kappa_B : \\kappa_A] \\leq [L : K].\n$$","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASH","source_file":"more-algebra.tex","source_line":37333,"source_end_line":37342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37333-L37342","statement_sha256":"e97fa2f3bc76a75f3c6a5c1a282472e688500a9b6977541956e86ca5c94a0ccf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3690,"rank":3690,"depth":0,"x":618.204,"y":810.078,"cluster":"advanced-algebra"},{"id":"stacks:0H37","tag":"0H37","title":"Extensions of valuation rings · Lemma 0H37","summary":"Let A be a valuation ring with fraction field K of characteristic p > 0. Let L/K be a purely inseparable extension. Then the integral closure B of A in L is a valuation ring with fraction field L and A ⊂ B is an extension of valuation rings.","statement_latex":"Let $A$ be a valuation ring with fraction field $K$ of characteristic $p > 0$.\nLet $L/K$ be a purely inseparable extension.\nThen the integral closure $B$ of $A$ in $L$ is a valuation ring\nwith fraction field $L$\nand $A \\subset B$ is an extension of valuation rings.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H37","source_file":"more-algebra.tex","source_line":37375,"source_end_line":37382,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37375-L37382","statement_sha256":"0f0c440ba4f82b65cd8417fcc7865d286cda9766a36529fd657beefee6351596","origin":"The Stacks Project","memory_eligible":false,"source_rank":3691,"rank":3691,"depth":6,"x":620.929,"y":548.945,"cluster":"advanced-algebra"},{"id":"stacks:0ASI","tag":"0ASI","title":"Extensions of valuation rings · Lemma 0ASI","summary":"Let A → B be a flat local homomorphism of Noetherian local normal domains. Let f ∈ A and h ∈ B such that f = w h^n for some n > 1 and some unit w of B. Assume that for every height 1 prime p ⊂ A there is a height 1 prime q ⊂ B lying over p such that the extension A_ p ⊂ B_ q is weakly unramified. Then f = u g^n for some g ∈ A and unit u of A.","statement_latex":"Let $A \\to B$ be a flat local homomorphism of Noetherian local normal domains.\nLet $f \\in A$ and $h \\in B$ such that $f = w h^n$ for some $n > 1$ and some\nunit $w$ of $B$. Assume that for every height $1$ prime\n$\\mathfrak p \\subset A$ there is a height $1$ prime\n$\\mathfrak q \\subset B$ lying over $\\mathfrak p$\nsuch that the extension $A_\\mathfrak p \\subset B_\\mathfrak q$ is\nweakly unramified. Then $f = u g^n$ for some $g \\in A$ and unit $u$ of $A$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASI","source_file":"more-algebra.tex","source_line":37390,"source_end_line":37399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37390-L37399","statement_sha256":"3891c6823ae0b8e808761ab5d09181a533875b287c4b1e222e1a0aa9cf4796d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3692,"rank":3692,"depth":18,"x":829.031,"y":743.154,"cluster":"advanced-algebra"},{"id":"stacks:0ASJ","tag":"0ASJ","title":"Extensions of valuation rings · Lemma 0ASJ","summary":"Let A be a valuation ring. Let A → B be an étale ring map and let m ⊂ B be a prime lying over the maximal ideal of A. Then A ⊂ B_ m is an extension of valuation rings which is weakly unramified.","statement_latex":"Let $A$ be a valuation ring. Let $A \\to B$ be an \\'etale ring map\nand let $\\mathfrak m \\subset B$ be a prime lying over the maximal\nideal of $A$. Then $A \\subset B_\\mathfrak m$ is an extension of\nvaluation rings which is weakly unramified.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASJ","source_file":"more-algebra.tex","source_line":37479,"source_end_line":37485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37479-L37485","statement_sha256":"93406d463898084396b28af8c93a7c1d90542b504a353f21d299a3118ec35115","origin":"The Stacks Project","memory_eligible":false,"source_rank":3693,"rank":3693,"depth":44,"x":519.232,"y":718.015,"cluster":"advanced-algebra"},{"id":"stacks:0ASK","tag":"0ASK","title":"Extensions of valuation rings · Lemma 0ASK","summary":"Let A be a valuation ring. Let A^h, resp. A^sh be its henselization, resp. strict henselization. Then A ⊂ A^h ⊂ A^sh are extensions of valuation rings which induce bijections on value groups, i.e., which are weakly unramified.","statement_latex":"Let $A$ be a valuation ring. Let $A^h$, resp.\\ $A^{sh}$ be its\nhenselization, resp.\\ strict henselization. Then\n$$\nA \\subset A^h \\subset A^{sh}\n$$\nare extensions of valuation rings which induce bijections on\nvalue groups, i.e., which are weakly unramified.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Extensions of valuation rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASK","source_file":"more-algebra.tex","source_line":37534,"source_end_line":37543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37534-L37543","statement_sha256":"1c65d2145847d65000e9d6d927008a9b9b90d4f1610a5cce07cdf7617dd5db52","origin":"The Stacks Project","memory_eligible":false,"source_rank":3694,"rank":3694,"depth":46,"x":768.026,"y":560.678,"cluster":"advanced-algebra"},{"id":"stacks:0ASM","tag":"0ASM","title":"Structure of modules over a PID · Lemma 0ASM","summary":"[Warfield-Purity] Let P be a module over a ring R. The following are equivalent • P is a direct summand of a direct sum of modules of the form R/fR, for f ∈ R varying. • for every short exact sequence 0 → A → B → C → 0 of R-modules such that fA = A ∩ fB for all f ∈ R the map Hom_R(P, B) → Hom_R(P, C) is surjective.","statement_latex":"\\begin{reference}\n\\cite[Corollary 1]{Warfield-Purity}\n\\end{reference}\nLet $P$ be a module over a ring $R$. The following are equivalent\n\\begin{enumerate}\n\\item $P$ is a direct summand of a direct sum of modules of the\nform $R/fR$, for $f \\in R$ varying.\n\\item for every short exact sequence $0 \\to A \\to B \\to C \\to 0$\nof $R$-modules such that $fA = A \\cap fB$ for all $f \\in R$\nthe map $\\Hom_R(P, B) \\to \\Hom_R(P, C)$ is surjective.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASM","source_file":"more-algebra.tex","source_line":37585,"source_end_line":37598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37585-L37598","statement_sha256":"aa3d6b9a5524715e906160727da7f3dd1bbd04f65aa8cc1f25108b6100ed9051","origin":"The Stacks Project","memory_eligible":false,"source_rank":3695,"rank":3695,"depth":0,"x":711.063,"y":818.01,"cluster":"advanced-algebra"},{"id":"stacks:0ASN","tag":"0ASN","title":"Generalized valuation rings · Lemma 0ASN","summary":"[Warfield-Decomposition] Let R be a nonzero ring. The following are equivalent • For a, b ∈ R either a divides b or b divides a. • Every finitely generated ideal is principal and R is local. • The set of ideals of R is linearly ordered by inclusion. This holds in particular if R is a valuation ring.","statement_latex":"\\begin{reference}\n\\cite{Warfield-Decomposition}\n\\end{reference}\nLet $R$ be a nonzero ring. The following are equivalent\n\\begin{enumerate}\n\\item For $a, b \\in R$ either $a$ divides $b$ or $b$ divides $a$.\n\\item Every finitely generated ideal is principal and $R$ is local.\n\\item The set of ideals of $R$ is linearly ordered by inclusion.\n\\end{enumerate}\nThis holds in particular if $R$ is a valuation ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASN","source_file":"more-algebra.tex","source_line":37637,"source_end_line":37649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37637-L37649","statement_sha256":"cdac6fc0132d6cba67d73069df085d64b958e7db6434d326b99b66098bca9a8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3696,"rank":3696,"depth":8,"x":546.037,"y":595.814,"cluster":"advanced-algebra"},{"id":"stacks:0ASP","tag":"0ASP","title":"Structure of modules over a PID · Lemma 0ASP","summary":"[Warfield-Decomposition] Let R be a ring satisfying the equivalent conditions of Lemma [Tag 0ASN]. Then every finitely presented R-module is isomorphic to a finite direct sum of modules of the form R/fR.","statement_latex":"\\begin{reference}\n\\cite[Theorem 1]{Warfield-Decomposition}\n\\end{reference}\nLet $R$ be a ring satisfying the equivalent conditions of\nLemma \\ref{lemma-generalized-valuation-ring}.\nThen every finitely presented $R$-module\nis isomorphic to a finite direct sum of modules of the form $R/fR$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASP","source_file":"more-algebra.tex","source_line":37674,"source_end_line":37683,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37674-L37683","statement_sha256":"629788cf2b659bfb65bfe662e532999eaf63affffec263de5b9aa59066673bf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3697,"rank":3697,"depth":9,"x":846.574,"y":666.056,"cluster":"advanced-algebra"},{"id":"stacks:0ASQ","tag":"0ASQ","title":"Structure of modules over a PID · Lemma 0ASQ","summary":"[Warfield-Decomposition] Let R be a ring such that every local ring of R at a maximal ideal satisfies the equivalent conditions of Lemma [Tag 0ASN]. Then every finitely presented R-module is a summand of a finite direct sum of modules of the form R/fR for f in R varying.","statement_latex":"\\begin{reference}\n\\cite[Theorem 3]{Warfield-Decomposition}\n\\end{reference}\nLet $R$ be a ring such that every local ring of $R$ at a maximal\nideal satisfies the equivalent conditions of\nLemma \\ref{lemma-generalized-valuation-ring}.\nThen every finitely presented $R$-module is a summand of a \nfinite direct sum of modules of the form $R/fR$ for $f$ in $R$ varying.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASQ","source_file":"more-algebra.tex","source_line":37736,"source_end_line":37746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37736-L37746","statement_sha256":"719e693ffe709e5fd7c85e61ad5e83a6596869c5dec7ee6f4437dcfe3c112704","origin":"The Stacks Project","memory_eligible":false,"source_rank":3698,"rank":3698,"depth":10,"x":568.323,"y":784.857,"cluster":"advanced-algebra"},{"id":"stacks:0ASR","tag":"0ASR","title":"Structure of modules over a PID · Definition 0ASR","summary":"Let R be a domain. • We say R is a Bézout domain if every finitely generated ideal of R is principal. • We say R is an elementary divisor domain if for all n , m ≥ 1 and every n × m matrix A, there exist invertible matrices U, V of size n × n, m × m such that U A V = ( f_1 & 0 & 0 & … 0 & f_2 & 0 & … 0 & 0 & f_3 & … … & … & … & … ) with f_1, …, f_min(n, m) ∈ R and f_1 | f_2 | ….","statement_latex":"Let $R$ be a domain.\n\\begin{enumerate}\n\\item We say $R$ is a {\\it B\\'ezout domain} if every finitely generated\nideal of $R$ is principal.\n\\item We say $R$ is an {\\it elementary divisor domain} if for\nall $n , m \\geq 1$ and every $n \\times m$ matrix $A$, there\nexist invertible matrices $U, V$ of size $n \\times n, m \\times m$\nsuch that\n$$\nU A V =\n\\left(\n\\begin{matrix}\nf_1 & 0 & 0 & \\ldots \\\\\n0 & f_2 & 0 & \\ldots \\\\\n0 & 0 & f_3 & \\ldots \\\\\n\\ldots & \\ldots & \\ldots & \\ldots\n\\end{matrix}\n\\right)\n$$\nwith $f_1, \\ldots, f_{\\min(n, m)} \\in R$ and $f_1 | f_2 | \\ldots$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASR","source_file":"more-algebra.tex","source_line":37788,"source_end_line":37811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37788-L37811","statement_sha256":"b9bb1ab8d4b2e7c3fd96022d67b88ee3f80f7acaa05470a435b275033657a88c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3699,"rank":3699,"depth":0,"x":678.025,"y":539.235,"cluster":"advanced-algebra"},{"id":"stacks:0ASS","tag":"0ASS","title":"Structure of modules over a PID · Lemma 0ASS","summary":"An elementary divisor domain is Bézout.","statement_latex":"An elementary divisor domain is B\\'ezout.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASS","source_file":"more-algebra.tex","source_line":37819,"source_end_line":37822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37819-L37822","statement_sha256":"ede572385aabfbfcbaabefeb4f1bf400e232533f17e746689c32ccb7c38b56da","origin":"The Stacks Project","memory_eligible":false,"source_rank":3700,"rank":3700,"depth":0,"x":794.719,"y":782.733,"cluster":"advanced-algebra"},{"id":"stacks:0AST","tag":"0AST","title":"Structure of modules over a PID · Lemma 0AST","summary":"The localization of a Bézout domain is Bézout. Every local ring of a Bézout domain is a valuation ring. A local domain is Bézout if and only if it is a valuation ring.","statement_latex":"The localization of a B\\'ezout domain is B\\'ezout.\nEvery local ring of a B\\'ezout domain is a valuation ring.\nA local domain is B\\'ezout if and only if it is a valuation ring.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AST","source_file":"more-algebra.tex","source_line":37833,"source_end_line":37838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37833-L37838","statement_sha256":"cb906b166cda2b8e9ec98a4a5af46e4f40d0bab7da8b5fe3b214af18eafe46d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3701,"rank":3701,"depth":2,"x":512.701,"y":669.334,"cluster":"advanced-algebra"},{"id":"stacks:0ASU","tag":"0ASU","title":"Structure of modules over a PID · Lemma 0ASU","summary":"Let R be a Bézout domain. • Every finite submodule of a free module is finite free. • Every finitely presented R-module M is a direct sum of a finite free module and a torsion module M_tors which is a summand of a module of the form bigoplus_i = 1, …, n R/f_iR with f_1, …, f_n ∈ R nonzero.","statement_latex":"Let $R$ be a B\\'ezout domain.\n\\begin{enumerate}\n\\item Every finite submodule of a free module is finite free.\n\\item Every finitely presented $R$-module $M$ is a direct sum of a\nfinite free module and a torsion module $M_{tors}$ which is a\nsummand of a module of the form $\\bigoplus_{i = 1, \\ldots, n} R/f_iR$\nwith $f_1, \\ldots, f_n \\in R$ nonzero.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASU","source_file":"more-algebra.tex","source_line":37846,"source_end_line":37856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37846-L37856","statement_sha256":"07a9c6bfcfee2eb3873452db3bc01562398863445a6ee2a8f1988c0a88a497bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3702,"rank":3702,"depth":11,"x":812.012,"y":592.888,"cluster":"advanced-algebra"},{"id":"stacks:0ASV","tag":"0ASV","title":"Structure of modules over a PID · Lemma 0ASV","summary":"Let R be a PID. Every finite R-module M is of isomorphic to a module of the form R^⊕ r ⊕ bigoplus_i = 1, …, n R/f_iR for some r, n ≥ 0 and f_1, …, f_n ∈ R nonzero.","statement_latex":"Let $R$ be a PID. Every finite $R$-module $M$ is of isomorphic\nto a module of the form\n$$\nR^{\\oplus r} \\oplus \\bigoplus\\nolimits_{i = 1, \\ldots, n} R/f_iR\n$$\nfor some $r, n \\geq 0$ and $f_1, \\ldots, f_n \\in R$ nonzero.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASV","source_file":"more-algebra.tex","source_line":37886,"source_end_line":37894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37886-L37894","statement_sha256":"b1a4182e0b27a00e18f87370b8ca6d98f5e0cdfc7ef63990ff14244f5ba75575","origin":"The Stacks Project","memory_eligible":false,"source_rank":3703,"rank":3703,"depth":12,"x":652.694,"y":819.218,"cluster":"advanced-algebra"},{"id":"stacks:0ASW","tag":"0ASW","title":"Structure of modules over a PID · Lemma 0ASW","summary":"Let R be a Bézout domain. Let n ≥ 1 and f_1, …, f_n ∈ R generate the unit ideal. There exists an invertible n × n matrix in R whose first row is f_1 … f_n.","statement_latex":"Let $R$ be a B\\'ezout domain. Let $n \\geq 1$ and $f_1, \\ldots, f_n \\in R$\ngenerate the unit ideal. There exists an invertible $n \\times n$ matrix in\n$R$ whose first row is $f_1 \\ldots f_n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Structure of modules over a PID","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASW","source_file":"more-algebra.tex","source_line":37925,"source_end_line":37930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37925-L37930","statement_sha256":"f179d1a4df7c7a451e85567500d7dc774a2ce7f5a75c5fe2312d4b1890dd10a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3704,"rank":3704,"depth":12,"x":588.129,"y":561.785,"cluster":"advanced-algebra"},{"id":"stacks:0BWS","tag":"0BWS","title":"Principal radical ideals · Lemma 0BWS","summary":"Let (R, m) be a Noetherian local ring of dimension one, and let x∈ m be an element not contained in any minimal prime of R. Then • the function P : n ↦ length_R(R/x^n R) satisfies P(n) ≤ n P(1) for n ≥ 0, • if x is a nonzerodivisor, then P(n) = nP(1) for n ≥ 0.","statement_latex":"Let $(R,\\mathfrak m)$ be a Noetherian local ring of dimension one, and\nlet $x\\in\\mathfrak m$ be an element not contained in any minimal prime\nof $R$. Then\n\\begin{enumerate}\n\\item the function $P : n \\mapsto \\text{length}_R(R/x^n R)$\nsatisfies $P(n) \\leq n P(1)$ for $n \\geq 0$,\n\\item if $x$ is a nonzerodivisor, then $P(n) = nP(1)$ for $n \\geq 0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWS","source_file":"more-algebra.tex","source_line":37978,"source_end_line":37988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L37978-L37988","statement_sha256":"417a53954e889ab3c7bcff020f648231c0c6f98d9b9ba367f28f6ea9461e2517","origin":"The Stacks Project","memory_eligible":false,"source_rank":3705,"rank":3705,"depth":6,"x":842.898,"y":715.06,"cluster":"advanced-algebra"},{"id":"stacks:0BWT","tag":"0BWT","title":"Principal radical ideals · Lemma 0BWT","summary":"Let (R, m) be a Noetherian local ring of dimension 1. Let x ∈ m be an element not contained in any minimal prime of R. Let t be the number of minimal prime ideals of R. Then t ≤ length_R(R/xR).","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.\nLet $x \\in \\mathfrak m$ be an element not contained in any minimal\nprime of $R$. Let $t$ be the number of minimal prime ideals of $R$.\nThen $t \\leq \\text{length}_R(R/xR)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWT","source_file":"more-algebra.tex","source_line":38015,"source_end_line":38021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38015-L38021","statement_sha256":"ec30ff78f67a20cb5d744455d0aa405792a634d79b7500e0d74d97013e1a0d1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3706,"rank":3706,"depth":7,"x":531.607,"y":746.616,"cluster":"advanced-algebra"},{"id":"stacks:0BWU","tag":"0BWU","title":"Principal radical ideals · Lemma 0BWU","summary":"Let (R, m) be a Noetherian local ring of dimension d > 1, let f ∈ m be an element not contained in any minimal prime ideal of R, and let k∈N. Then there exist elements g_1, …, g_d - 1 ∈ m^k such that f, g_1, …, g_d - 1 is a system of parameters.","statement_latex":"Let $(R,\\mathfrak m)$ be a Noetherian local ring of dimension $d > 1$,\nlet $f \\in \\mathfrak m$ be an element not contained in any minimal prime\nideal of $R$, and let $k\\in\\mathbf{N}$. Then there exist elements\n$g_1, \\ldots, g_{d - 1} \\in \\mathfrak m^k$ such that\n$f, g_1, \\ldots, g_{d - 1}$ is a system of parameters.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWU","source_file":"more-algebra.tex","source_line":38085,"source_end_line":38092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38085-L38092","statement_sha256":"7e66c0b37017a0224744e7f91ff42ba7a2b4b8fb7991c63b610e54704024a632","origin":"The Stacks Project","memory_eligible":false,"source_rank":3707,"rank":3707,"depth":11,"x":735.882,"y":546.604,"cluster":"advanced-algebra"},{"id":"stacks:0BWV","tag":"0BWV","title":"Principal radical ideals · Lemma 0BWV","summary":"Let (R, m) be a Noetherian local ring of dimension two, and let f ∈ m be an element not contained in any minimal prime ideal of R. Then there exist g ∈ m and N ∈ N such that • [(a)] f,g form a system of parameters for R. • [(b)] If h ∈ m^N, then f + h, g is a system of parameters and length_R (R/(f, g)) = length_R(R/(f + h, g)).","statement_latex":"Let $(R,\\mathfrak m)$ be a Noetherian local ring of dimension\ntwo, and let $f \\in \\mathfrak m$ be an element not contained in\nany minimal prime ideal of $R$. Then there exist\n$g \\in \\mathfrak m$ and $N \\in \\mathbf{N}$ such that\n\\begin{enumerate}\n\\item[(a)] $f,g$ form a system of parameters for $R$.\n\\item[(b)] If $h \\in \\mathfrak m^N$, then $f + h, g$ is a\nsystem of parameters and\n$\\text{length}_R (R/(f, g)) = \\text{length}_R(R/(f + h, g))$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWV","source_file":"more-algebra.tex","source_line":38107,"source_end_line":38119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38107-L38119","statement_sha256":"6bc57f93ea3639b21154b42a3f9520a1efdd5d64d8d4936fdf52d3939043b990","origin":"The Stacks Project","memory_eligible":false,"source_rank":3708,"rank":3708,"depth":12,"x":746.102,"y":810.143,"cluster":"advanced-algebra"},{"id":"stacks:0AXH","tag":"0AXH","title":"Principal radical ideals · Lemma 0AXH","summary":"Let R be a Noetherian local normal domain of dimension 2. Let p_1, …, p_r be pairwise distinct primes of height 1. There exists a nonzero element f ∈ p_1 ∩ … ∩ p_r such that R/fR is reduced.","statement_latex":"Let $R$ be a Noetherian local normal domain of dimension $2$.\nLet $\\mathfrak p_1, \\ldots, \\mathfrak p_r$ be pairwise distinct\nprimes of height $1$. There exists a nonzero element\n$f \\in \\mathfrak p_1 \\cap \\ldots \\cap \\mathfrak p_r$ such\nthat $R/fR$ is reduced.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXH","source_file":"more-algebra.tex","source_line":38137,"source_end_line":38144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38137-L38144","statement_sha256":"90e7923055fb3a2d48bdcf848ef8058a5e73d4605caebadd2028f594c14493a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3709,"rank":3709,"depth":18,"x":526.517,"y":621.51,"cluster":"advanced-algebra"},{"id":"stacks:0AXI","tag":"0AXI","title":"Principal radical ideals · Lemma 0AXI","summary":"Let (A, m, kappa) be a Noetherian normal local domain of dimension 2. If a ∈ m is nonzero, then there exists an element c ∈ A such that A/cA is reduced and such that a divides c^n for some n.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian normal local domain\nof dimension $2$. If $a \\in \\mathfrak m$ is nonzero, then there exists an\nelement $c \\in A$ such that $A/cA$ is reduced and such that $a$ divides\n$c^n$ for some $n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXI","source_file":"more-algebra.tex","source_line":38238,"source_end_line":38244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38238-L38244","statement_sha256":"a22360bc8586b815850cd4ac9f81aca26b39d99e256d6911f9b19a09b11b192c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3710,"rank":3710,"depth":19,"x":840.299,"y":636.015,"cluster":"advanced-algebra"},{"id":"stacks:0BWW","tag":"0BWW","title":"Principal radical ideals · Lemma 0BWW","summary":"Let (R, m) be a Noetherian local ring of dimension d, let g_1, …, g_d be a system of parameters, and let I = (g_1, …, g_d). If e_I/d! is the leading coefficient of the numerical polynomial n ↦ length_R(R/I^n+1), then e_I ≤ length_R(R/I).","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$, let\n$g_1, \\ldots, g_d$ be a system of parameters, and let\n$I = (g_1, \\ldots, g_d)$. If $e_I/d!$ is the leading coefficient of the\nnumerical polynomial\n$n \\mapsto \\text{length}_R(R/I^{n+1})$, then $e_I \\leq \\text{length}_R(R/I)$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWW","source_file":"more-algebra.tex","source_line":38260,"source_end_line":38267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38260-L38267","statement_sha256":"b5112753c0400575687fe28c3dbc8976c8a50728aecc572e29f7b34d9ae76135","origin":"The Stacks Project","memory_eligible":false,"source_rank":3711,"rank":3711,"depth":10,"x":597.125,"y":803.458,"cluster":"advanced-algebra"},{"id":"stacks:0BWX","tag":"0BWX","title":"Principal radical ideals · Lemma 0BWX","summary":"Let (R, m) be a Noetherian local ring of dimension d, let t be the number of minimal prime ideals of R of dimension d, and let (g_1,…,g_d) be a system of parameters. Then t ≤ length_R(R/(g_1,…,g_n)).","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$, let $t$\nbe the number of minimal prime ideals of $R$ of dimension $d$, and let\n$(g_1,\\ldots,g_d)$ be a system of parameters. Then\n$t \\leq \\text{length}_R(R/(g_1,\\ldots,g_n))$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWX","source_file":"more-algebra.tex","source_line":38296,"source_end_line":38302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38296-L38302","statement_sha256":"29a1c20bbf72ea8c0235f9fa63f46daf394c1d28bf7e3eb2be859afd7cd49d37","origin":"The Stacks Project","memory_eligible":false,"source_rank":3712,"rank":3712,"depth":11,"x":641.809,"y":541.864,"cluster":"advanced-algebra"},{"id":"stacks:0BWY","tag":"0BWY","title":"Principal radical ideals · Lemma 0BWY","summary":"Let (R, m) be a Noetherian local ring of dimension d, and let f ∈ m be an element not contained in any minimal prime ideal of R. Then there exist elements g_1, …, g_d - 1 ∈ m and N ∈ N such that • f, g_1, …, g_d - 1 form a system of parameters for R • If h ∈ m^N, then f + h, g_1, …, g_d - 1 is a system of parameters and we have length_R R/(f, g_1, …, g_d-1) = length_R R/(f + h, g_1, …, g_d-1).","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $d$, and let\n$f \\in \\mathfrak m$ be an element not contained in any minimal\nprime ideal of $R$. Then there exist elements\n$g_1, \\ldots, g_{d - 1} \\in \\mathfrak m$ and $N \\in \\mathbf{N}$ such that\n\\begin{enumerate}\n\\item $f, g_1, \\ldots, g_{d - 1}$ form a system of parameters for $R$\n\\item If $h \\in \\mathfrak m^N$, then $f + h, g_1, \\ldots, g_{d - 1}$ is a\nsystem of parameters and we have\n$\\text{length}_R R/(f, g_1, \\ldots, g_{d-1}) =\n\\text{length}_R R/(f + h, g_1, \\ldots, g_{d-1})$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWY","source_file":"more-algebra.tex","source_line":38347,"source_end_line":38360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38347-L38360","statement_sha256":"94c774079e967b5bb0ce88c395540af433b0eee42b9e1500f0109ed6a300d6b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3713,"rank":3713,"depth":12,"x":819.321,"y":760.232,"cluster":"advanced-algebra"},{"id":"stacks:0BWZ","tag":"0BWZ","title":"Principal radical ideals · Proposition 0BWZ","summary":"[Artin-Lipman] has this result without the assumption that the ring is catenary Let R be a catenary Noetherian local normal domain. Let J ⊂ R be a radical ideal. Then there exists a nonzero element f ∈ J such that R/fR is reduced.","statement_latex":"\\begin{reference}\n\\cite[Lemma 3.14]{Artin-Lipman} has this result without the\nassumption that the ring is catenary\n\\end{reference}\nLet $R$ be a catenary Noetherian local normal domain.\nLet $J \\subset R$ be a radical ideal.\nThen there exists a nonzero element $f \\in J$\nsuch that $R/fR$ is reduced.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Principal radical ideals","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWZ","source_file":"more-algebra.tex","source_line":38386,"source_end_line":38396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38386-L38396","statement_sha256":"42b9da9a27d47a20e89213d7206c5a1c6f21ce3d6d8d2961da5f9f36446ddc0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3714,"rank":3714,"depth":19,"x":512.657,"y":699.903,"cluster":"advanced-algebra"},{"id":"stacks:0FNQ","tag":"0FNQ","title":"Invertible objects in the derived category · Lemma 0FNQ","summary":"Let R be a ring. The derived category D(R) of R is a symmetric monoidal category with tensor product given by derived tensor product and associativity and commutativity constraints as in Section [Tag 0FNG].","statement_latex":"Let $R$ be a ring. The derived category $D(R)$ of $R$\nis a symmetric monoidal category with tensor product\ngiven by derived tensor product and associativity and\ncommutativity constraints as in Section \\ref{section-sign-rules}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Invertible objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNQ","source_file":"more-algebra.tex","source_line":38529,"source_end_line":38535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38529-L38535","statement_sha256":"2f760eeddb8e26970ae25a4b2025ff09e8b454cdcd36217fc8afa1c55a37ad6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3715,"rank":3715,"depth":1,"x":787.449,"y":570.31,"cluster":"advanced-algebra"},{"id":"stacks:0FNR","tag":"0FNR","title":"Invertible objects in the derived category · Lemma 0FNR","summary":"Let R be a ring. Let F^bullet be a bounded above complex of free R-modules. Given pairs (n_i, f_i), i = 1, …, N with n_i ∈ Z and f_i ∈ F^n_i there exists a subcomplex G^bullet ⊂ F^bullet containing all f_i which is bounded and consists of finite free R-modules.","statement_latex":"Let $R$ be a ring. Let $F^\\bullet$ be a bounded above complex of\nfree $R$-modules. Given pairs $(n_i, f_i)$, $i = 1, \\ldots, N$\nwith $n_i \\in \\mathbf{Z}$ and $f_i \\in F^{n_i}$ there exists\na subcomplex $G^\\bullet \\subset F^\\bullet$ containing\nall $f_i$ which is bounded and consists of finite free $R$-modules.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Invertible objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNR","source_file":"more-algebra.tex","source_line":38552,"source_end_line":38559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38552-L38559","statement_sha256":"ead7cca9ede88914983e262febdba65c8c7ad945680ac2a4d97caafdc5e95403","origin":"The Stacks Project","memory_eligible":false,"source_rank":3716,"rank":3716,"depth":0,"x":688.983,"y":821.929,"cluster":"advanced-algebra"},{"id":"stacks:0FNS","tag":"0FNS","title":"Invertible objects in the derived category · Lemma 0FNS","summary":"Let R be a ring. Let M be an object of D(R). The following are equivalent • M has a left dual in D(R) as in Categories, Definition [Tag 0FFP], • M is a perfect object of D(R). Moreover, in this case the left dual of M is the object M^vee of Lemma [Tag 07VI].","statement_latex":"Let $R$ be a ring. Let $M$ be an object of $D(R)$. The following\nare equivalent\n\\begin{enumerate}\n\\item $M$ has a left dual in $D(R)$ as in\nCategories, Definition \\ref{categories-definition-dual},\n\\item $M$ is a perfect object of $D(R)$.\n\\end{enumerate}\nMoreover, in this case the left dual of $M$ is the object\n$M^\\vee$ of Lemma \\ref{lemma-dual-perfect-complex}.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Invertible objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNS","source_file":"more-algebra.tex","source_line":38580,"source_end_line":38591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38580-L38591","statement_sha256":"892c3f2a4b5ea57cbaaa9b6bd9042cf6f5ee4fae124616c654310aebd72e523f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3717,"rank":3717,"depth":15,"x":559.176,"y":580.388,"cluster":"advanced-algebra"},{"id":"stacks:0FNT","tag":"0FNT","title":"Invertible objects in the derived category · Lemma 0FNT","summary":"Let R be a ring. Let M be an object of D(R). The following are equivalent • M is invertible in D(R), see Categories, Definition [Tag 0FFN], and • for every prime ideal p ⊂ R there exists an f ∈ R, f not ∈ p such that M_f ≅ R_f[-n] for some n ∈ Z. Moreover, in this case • [(a)] M is a perfect object of D(R), • [(b)] M = bigoplus H^n(M)[-n] in D(R), • [(c)] each H^n(M) is a finite projective R-module, • [(d)] we can write R = ∏_a ≤ n ≤ b R_n such that H^n(M) corresponds to…","statement_latex":"Let $R$ be a ring. Let $M$ be an object of $D(R)$. The following\nare equivalent\n\\begin{enumerate}\n\\item $M$ is invertible in $D(R)$, see\nCategories, Definition \\ref{categories-definition-invertible}, and\n\\item for every prime ideal $\\mathfrak p \\subset R$ there\nexists an $f \\in R$, $f \\not \\in \\mathfrak p$ such that\n$M_f \\cong R_f[-n]$ for some $n \\in \\mathbf{Z}$.\n\\end{enumerate}\nMoreover, in this case\n\\begin{enumerate}\n\\item[(a)] $M$ is a perfect object of $D(R)$,\n\\item[(b)] $M = \\bigoplus H^n(M)[-n]$ in $D(R)$,\n\\item[(c)] each $H^n(M)$ is a finite projective $R$-module,\n\\item[(d)] we can write $R = \\prod_{a \\leq n \\leq b} R_n$\nsuch that $H^n(M)$ corresponds to an invertible $R_n$-module.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Invertible objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNT","source_file":"more-algebra.tex","source_line":38656,"source_end_line":38675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38656-L38675","statement_sha256":"f051481ade18a098004dad3606bdbb2882df6b736fb8dab88c837ac46d1791f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3718,"rank":3718,"depth":17,"x":849.29,"y":684.897,"cluster":"advanced-algebra"},{"id":"stacks:0GV9","tag":"0GV9","title":"Splitting off a free module · Lemma 0GV9","summary":"In Situation [Tag 0GV8] let x ∈ Ω. There exists a canonical short exact sequence 0 → B(x) → M(x) → V(x) → 0 of kappa(x)-vector spaces which the following property: for s_1, …, s_r ∈ M the following are equivalent • there exists an f ∈ R, f not ∈ x such that the map s_1, …, s_r : R^⊕ r → M becomes the inclusion of a direct summand after inverting f, and • s_1(x), …, s_r(x) map to linearly independent elements of V(x).","statement_latex":"In Situation \\ref{situation-splitting} let $x \\in \\Omega$. There exists a\ncanonical short exact sequence\n$$\n0 \\to B(x) \\to M(x) \\to V(x) \\to 0\n$$\nof $\\kappa(x)$-vector spaces which the following\nproperty: for $s_1, \\ldots, s_r \\in M$ the following are equivalent\n\\begin{enumerate}\n\\item there exists an $f \\in R$, $f \\not \\in x$ such that\nthe map $s_1, \\ldots, s_r : R^{\\oplus r} \\to M$ becomes the\ninclusion of a direct summand after inverting $f$, and\n\\item $s_1(x), \\ldots, s_r(x)$ map to linearly independent\nelements of $V(x)$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting off a free module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GV9","source_file":"more-algebra.tex","source_line":38730,"source_end_line":38746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38730-L38746","statement_sha256":"b42c696832dac963949af2239023fb9b64173c2eb61f75d249f7144152f05db8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3719,"rank":3719,"depth":2,"x":551.162,"y":772.498,"cluster":"advanced-algebra"},{"id":"stacks:0GVA","tag":"0GVA","title":"Splitting off a free module · Lemma 0GVA","summary":"In Situation [Tag 0GV8] let x_1, …, x_n ∈ Ω be pairwise distinct. Let v_i ∈ V(x_i). Then there exists an s ∈ M such that s(x_i) maps to v_i for i = 1, …, n.","statement_latex":"In Situation \\ref{situation-splitting} let $x_1, \\ldots, x_n \\in \\Omega$\nbe pairwise distinct. Let $v_i \\in V(x_i)$. Then there exists\nan $s \\in M$ such that $s(x_i)$ maps to $v_i$ for $i = 1, \\ldots, n$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting off a free module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVA","source_file":"more-algebra.tex","source_line":38794,"source_end_line":38799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38794-L38799","statement_sha256":"616197dd2dfced6227c4cc7efd0980b47b9e6f392ca6aff5a9028e75fe1300d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3720,"rank":3720,"depth":1,"x":700.63,"y":538.612,"cluster":"advanced-algebra"},{"id":"stacks:0GVB","tag":"0GVB","title":"Splitting off a free module · Proposition 0GVB","summary":"[Serre-projective] In Situation [Tag 0GV8] assume Ω is a Noetherian topological space. Let s_1, …, s_h ∈ M. Let Z(s_1, …, s_h) ⊂ F ⊂ Ω be closed. Let x_1, …, x_n ∈ F be pairwise distinct. Let v_i ∈ V(x_i). Let k ≥ 0 be an integer such that (*) h + k ≤ dim_kappa(x) V(x) for all x ∈ Ω Then there exist s ∈ M and F' ⊂ Ω closed such that • [(a)] s(x_i) maps to v_i, • [(b)] Z(s_1, …, s_h, s) ⊂ F ∪ F', and • [(c)] every irreducible component of F' has codimension ≥ k in Ω.","statement_latex":"\\begin{reference}\n\\cite[Theorem 2]{Serre-projective}\n\\end{reference}\nIn Situation \\ref{situation-splitting} assume $\\Omega$ is\na Noetherian topological space. Let $s_1, \\ldots, s_h \\in M$.\nLet $Z(s_1, \\ldots, s_h) \\subset F \\subset \\Omega$ be closed.\nLet $x_1, \\ldots, x_n \\in F$ be pairwise distinct.\nLet $v_i  \\in V(x_i)$.\nLet $k \\geq 0$ be an integer such that\n$$\n(*)\\quad h + k \\leq \\dim_{\\kappa(x)} V(x)\\text{ for all }x \\in \\Omega\n$$\nThen there exist $s \\in M$ and $F' \\subset \\Omega$ closed such that\n\\begin{enumerate}\n\\item[(a)] $s(x_i)$ maps to $v_i$,\n\\item[(b)] $Z(s_1, \\ldots, s_h, s) \\subset F \\cup F'$, and\n\\item[(c)] every irreducible component of $F'$ has\ncodimension $\\geq k$ in $\\Omega$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting off a free module","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVB","source_file":"more-algebra.tex","source_line":38808,"source_end_line":38829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38808-L38829","statement_sha256":"05314f277c9dfa0256de5669221c683c73ff71d8be3cdc42250a54ff1acf971a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3721,"rank":3721,"depth":2,"x":778.54,"y":796.025,"cluster":"advanced-algebra"},{"id":"stacks:0GVC","tag":"0GVC","title":"Splitting off a free module · Theorem 0GVC","summary":"[Serre-projective] Let R be a ring whose max spectrum Ω ⊂ Spec(R) is a Noetherian topological space of dimension d < ∞. Let M be a finitely presented R-module such that for all m ∈ Ω the R_ m-module M_ m has a free direct summand of rank > d. Then M ≅ R ⊕ M'.","statement_latex":"\\begin{reference}\n\\cite[Theorem 1]{Serre-projective}\n\\end{reference}\nLet $R$ be a ring whose max spectrum $\\Omega \\subset \\Spec(R)$\nis a Noetherian topological space of dimension $d < \\infty$.\nLet $M$ be a finitely presented $R$-module such that\nfor all $\\mathfrak m \\in \\Omega$ the $R_\\mathfrak m$-module\n$M_\\mathfrak m$ has a free direct summand of rank $> d$.\nThen $M \\cong R \\oplus M'$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Splitting off a free module","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVC","source_file":"more-algebra.tex","source_line":38919,"source_end_line":38930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38919-L38930","statement_sha256":"623644d58b8c52dbf28f0bd0d6e3888dd455fe1b53acec639beca2a37ce9f34c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3722,"rank":3722,"depth":3,"x":513.945,"y":650.343,"cluster":"advanced-algebra"},{"id":"stacks:0GVF","tag":"0GVF","title":"Eilenberg's lemma · Lemma 0GVF","summary":"Eilenberg swindle In [Bass] we find: \"...is an elegant little swindle, observed several years ago by Eilenberg, and which might well have sprung from the brow of Barry Mazur.\" [Bass] If P ⊕ Q ≅ F with F a nonfinitely generated free module, then P ⊕ F ≅ F.","statement_latex":"\\begin{slogan}\nEilenberg swindle\n\\end{slogan}\n\\begin{history}\nIn \\cite{Bass} we find: ``...is an elegant little swindle,\nobserved several years ago by Eilenberg, and which\nmight well have sprung from the brow of Barry Mazur.''\n\\end{history}\n\\begin{reference}\n\\cite[Eilenberg's lemma]{Bass}\n\\end{reference}\nIf $P \\oplus Q \\cong F$ with $F$ a nonfinitely generated free module,\nthen $P \\oplus F \\cong F$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Big projective modules are free","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVF","source_file":"more-algebra.tex","source_line":38970,"source_end_line":38985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38970-L38985","statement_sha256":"cd1896abf893408deffcf3d70eb28e9a0ce848740426c2dca3fe56061f4a8292","origin":"The Stacks Project","memory_eligible":false,"source_rank":3723,"rank":3723,"depth":0,"x":826.373,"y":607.606,"cluster":"advanced-algebra"},{"id":"stacks:0GVG","tag":"0GVG","title":"Big projective modules are free · Lemma 0GVG","summary":"Let R be a ring. Let P be a projective module. There exists a free module F such that P ⊕ F is free.","statement_latex":"Let $R$ be a ring. Let $P$ be a projective module.\nThere exists a free module $F$ such that $P \\oplus F$ is free.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Big projective modules are free","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVG","source_file":"more-algebra.tex","source_line":38995,"source_end_line":38999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L38995-L38999","statement_sha256":"42521c2430f17ac0eccdd6435204cdabe45c650dd1cfcfab4c1b25a7671b8705","origin":"The Stacks Project","memory_eligible":false,"source_rank":3724,"rank":3724,"depth":1,"x":630.257,"y":816.51,"cluster":"advanced-algebra"},{"id":"stacks:0GVH","tag":"0GVH","title":"Big projective modules are free · Lemma 0GVH","summary":"Let R be a ring. Let P be a projective module. Let s ∈ P. There exists a finite free module F and a finite free direct summand K ⊂ F ⊕ P with (0, s) ∈ K.","statement_latex":"Let $R$ be a ring. Let $P$ be a projective module.\nLet $s \\in P$. There exists a finite free module $F$\nand a finite free direct summand $K \\subset F \\oplus P$\nwith $(0, s) \\in K$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Big projective modules are free","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVH","source_file":"more-algebra.tex","source_line":39008,"source_end_line":39014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L39008-L39014","statement_sha256":"f2c5c0efd3feb6e9d385c36e224d3ddb21b66cbd30445ceaddbe6a0598f4e8f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3725,"rank":3725,"depth":2,"x":606.863,"y":551.046,"cluster":"advanced-algebra"},{"id":"stacks:0GVI","tag":"0GVI","title":"Big projective modules are free · Lemma 0GVI","summary":"Let R be a ring with Jacobson radical J such that R/J is Noetherian. Let P be a projective R-module such that P_ m has infinite rank for all maximal ideals m of R. Let s ∈ P and M ⊂ P such that Rs + M = P. Then we can find m ∈ M such that R(s + m) is a free direct summand of P.","statement_latex":"Let $R$ be a ring with Jacobson radical $J$ such that $R/J$\nis Noetherian. Let $P$ be a projective $R$-module\nsuch that $P_\\mathfrak m$ has infinite rank for all maximal ideals\n$\\mathfrak m$ of $R$. Let $s \\in P$ and $M \\subset P$ such that\n$Rs + M = P$. Then we can find $m \\in M$ such that $R(s + m)$\nis a free direct summand of $P$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Big projective modules are free","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVI","source_file":"more-algebra.tex","source_line":39025,"source_end_line":39033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L39025-L39033","statement_sha256":"1cec1a86f57a2efe0d98db5a605b443fcea7f390ab6e58fb8bdd65f6d7eaebae","origin":"The Stacks Project","memory_eligible":false,"source_rank":3726,"rank":3726,"depth":4,"x":837.715,"y":733.618,"cluster":"advanced-algebra"},{"id":"stacks:0GVJ","tag":"0GVJ","title":"Big projective modules are free · Lemma 0GVJ","summary":"Let R be a ring with Jacobson radical J such that R/J is Noetherian. Let P be a projective R-module such that P_ m has infinite rank for all maximal ideals m of R. Let s ∈ P. Then we can find a finite stably free direct summand M ⊂ P such that s ∈ M.","statement_latex":"Let $R$ be a ring with Jacobson radical $J$ such that $R/J$\nis Noetherian. Let $P$ be a projective $R$-module\nsuch that $P_\\mathfrak m$ has infinite rank for all maximal ideals\n$\\mathfrak m$ of $R$. Let $s \\in P$. Then we can find\na finite stably free direct summand $M \\subset P$ such that $s \\in M$.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Big projective modules are free","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVJ","source_file":"more-algebra.tex","source_line":39107,"source_end_line":39114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L39107-L39114","statement_sha256":"e5973acfb38b068c8008d44fa01a951887a7946311b19115a388605d925da0ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":3727,"rank":3727,"depth":5,"x":520.501,"y":729.981,"cluster":"advanced-algebra"},{"id":"stacks:0GVK","tag":"0GVK","title":"Big projective modules are free · Theorem 0GVK","summary":"Commutative case of [Bass] Let R be a ring with Jacobson radical J such that R/J is Noetherian. Let P be a countably generated projective R-module such that P_ m has infinite rank for all maximal ideals m of R. Then P is free.","statement_latex":"\\begin{reference}\nCommutative case of \\cite[Theorem 3.1]{Bass}\n\\end{reference}\nLet $R$ be a ring with Jacobson radical $J$ such that $R/J$ is Noetherian.\nLet $P$ be a countably generated projective $R$-module such that\n$P_\\mathfrak m$ has infinite rank for all maximal ideals\n$\\mathfrak m$ of $R$. Then $P$ is free.","area":"Advanced Algebra","chapter":"More on Algebra","chapter_id":"more-algebra","section":"Big projective modules are free","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVK","source_file":"more-algebra.tex","source_line":39152,"source_end_line":39161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-algebra.tex#L39152-L39161","statement_sha256":"cb4669d21e3d20e7eff269e9c65c35da66c529ae8a68ca83bc33c6bcce2a2179","origin":"The Stacks Project","memory_eligible":false,"source_rank":3728,"rank":3728,"depth":6,"x":757.46,"y":552.574,"cluster":"advanced-algebra"},{"id":"stacks:07C5","tag":"07C5","title":"Singular ideals · Definition 07C5","summary":"Let R → A be a ring map. The singular ideal of A over R, denoted H_A/R is the unique radical ideal H_A/R ⊂ A with V(H_A/R) = ( q ∈ Spec(A) mid R → A not smooth at q)","statement_latex":"Let $R \\to A$ be a ring map. The {\\it singular ideal of $A$ over $R$},\ndenoted $H_{A/R}$ is the unique radical ideal $H_{A/R} \\subset A$ with\n$$\nV(H_{A/R}) = \\{\\mathfrak q \\in \\Spec(A) \\mid R \\to A\n\\text{ not smooth at }\\mathfrak q\\}\n$$","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Singular ideals","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07C5","source_file":"smoothing.tex","source_line":124,"source_end_line":132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L124-L132","statement_sha256":"8187a3db56b2ba23400205fa9f25456d984dc7e38ae2f391c1704f2a22e26dbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3729,"rank":3729,"depth":0,"x":725.379,"y":817.988,"cluster":"advanced-algebra"},{"id":"stacks:07C6","tag":"07C6","title":"Singular ideals · Lemma 07C6","summary":"Let R be a ring. Let A = R[x_1, …, x_n]/(f_1, …, f_m). Let q ⊂ A be a prime ideal. Assume R → A is smooth at q. Then there exists an a ∈ A, a not ∈ q, an integer c, 0 ≤ c ≤ min(n, m), subsets U ⊂ (1, …, n), V ⊂ (1, …, m) of cardinality c such that a = a' det(∂ f_j/∂ x_i)_j ∈ V, i ∈ U for some a' ∈ A and a f_ℓ ∈ (f_j, j ∈ V) + (f_1, …, f_m)^2 for all ℓ ∈ (1, …, m).","statement_latex":"Let $R$ be a ring. Let $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.\nLet $\\mathfrak q \\subset A$ be a prime ideal. Assume $R \\to A$ is smooth\nat $\\mathfrak q$. Then there exists an $a \\in A$, $a \\not \\in \\mathfrak q$,\nan integer $c$, $0 \\leq c \\leq \\min(n, m)$, subsets\n$U \\subset \\{1, \\ldots, n\\}$, $V \\subset \\{1, \\ldots, m\\}$\nof cardinality $c$ such that\n$$\na = a' \\det(\\partial f_j/\\partial x_i)_{j \\in V, i \\in U}\n$$\nfor some $a' \\in A$ and\n$$\na f_\\ell \\in (f_j, j \\in V) + (f_1, \\ldots, f_m)^2\n$$\nfor all $\\ell \\in \\{1, \\ldots, m\\}$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Singular ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07C6","source_file":"smoothing.tex","source_line":141,"source_end_line":157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L141-L157","statement_sha256":"e8cd6d366e69b246665caa9c3968802e10bc17a112a05cca2ef66002e52dcf39","origin":"The Stacks Project","memory_eligible":false,"source_rank":3730,"rank":3730,"depth":5,"x":535.496,"y":603.959,"cluster":"advanced-algebra"},{"id":"stacks:07C7","tag":"07C7","title":"Singular ideals · Definition 07C7","summary":"Let R → A be a ring map of finite presentation. We say an element a ∈ A is elementary standard in A over R if there exists a presentation A = R[x_1, …, x_n]/(f_1, …, f_m) and 0 ≤ c ≤ min(n, m) such that a = a' det(∂ f_j/∂ x_i)_i, j = 1, …, c for some a' ∈ A and a f_c + j ∈ (f_1, …, f_c) + (f_1, …, f_m)^2 for j = 1, …, m - c. We say a ∈ A is strictly standard in A over R if there exists a presentation A = R[x_1, …, x_n]/(f_1, …, f_m) and 0 ≤ c ≤ min(n, m) such that a = ∑_I…","statement_latex":"Let $R \\to A$ be a ring map of finite presentation.\nWe say an element $a \\in A$ is {\\it elementary standard in $A$ over $R$}\nif there exists a presentation\n$A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$\nand $0 \\leq c \\leq \\min(n, m)$ such that\n\\begin{equation}\n\na = a' \\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}\n\\end{equation}\nfor some $a' \\in A$ and\n\\begin{equation}\n\na f_{c + j} \\in (f_1, \\ldots, f_c) + (f_1, \\ldots, f_m)^2\n\\end{equation}\nfor $j = 1, \\ldots, m - c$. We say $a \\in A$ is\n{\\it strictly standard in $A$ over $R$} if there exists a presentation\n$A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$\nand $0 \\leq c \\leq \\min(n, m)$ such that\n\\begin{equation}\n\na = \\sum\\nolimits_{I \\subset \\{1, \\ldots, n\\},\\ |I| = c}\na_I \\det(\\partial f_j/\\partial x_i)_{j = 1, \\ldots, c,\\ i \\in I}\n\\end{equation}\nfor some $a_I \\in A$ and\n\\begin{equation}\n\na f_{c + j} \\in (f_1, \\ldots, f_c) + (f_1, \\ldots, f_m)^2\n\\end{equation}\nfor $j = 1, \\ldots, m - c$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Singular ideals","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07C7","source_file":"smoothing.tex","source_line":191,"source_end_line":222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L191-L222","statement_sha256":"75de4702e996175c3ca779efcdab6afa4b4db9ac40815bc52de918b55fd01b60","origin":"The Stacks Project","memory_eligible":false,"source_rank":3731,"rank":3731,"depth":0,"x":847.794,"y":654.063,"cluster":"advanced-algebra"},{"id":"stacks:07ET","tag":"07ET","title":"Singular ideals · Lemma 07ET","summary":"Let R be a ring. Let A = R[x_1, …, x_n]/(f_1, …, f_m) and write I = (f_1, …, f_m). Let a ∈ A. Then ([Tag 07ER]) implies there exists an A-linear map ψ : bigoplus_i = 1, …, n A dx_i → A^⊕ c such that the composition A^⊕ c xrightarrow(f_1, …, f_c) I/I^2 xrightarrowf ↦ df bigoplus_i = 1, …, n A dx_i xrightarrowψ A^⊕ c is multiplication by a. Conversely, if such a ψ exists, then a^c satisfies ([Tag 07ER]).","statement_latex":"Let $R$ be a ring. Let $A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$\nand write $I = (f_1, \\ldots, f_m)$. Let $a \\in A$. Then\n(\\ref{equation-strictly-standard-one}) implies\nthere exists an $A$-linear map\n$\\psi : \\bigoplus\\nolimits_{i = 1, \\ldots, n} A \\text{d}x_i \\to A^{\\oplus c}$\nsuch that the composition\n$$\nA^{\\oplus c} \\xrightarrow{(f_1, \\ldots, f_c)}\nI/I^2 \\xrightarrow{f \\mapsto \\text{d}f}\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} A \\text{d}x_i\n\\xrightarrow{\\psi}\nA^{\\oplus c}\n$$\nis multiplication by $a$. Conversely, if such a $\\psi$ exists, then\n$a^c$ satisfies (\\ref{equation-strictly-standard-one}).","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Singular ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ET","source_file":"smoothing.tex","source_line":228,"source_end_line":245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L228-L245","statement_sha256":"a652c18c542feefdeb1fa4a566e14455ffc5d31afc3d5a631348a5a893b45c69","origin":"The Stacks Project","memory_eligible":false,"source_rank":3732,"rank":3732,"depth":1,"x":577.075,"y":794.396,"cluster":"advanced-algebra"},{"id":"stacks:07CA","tag":"07CA","title":"Elkik · Lemma 07CA","summary":"Let R → A be a ring map of finite presentation. The singular ideal H_A/R is the radical of the ideal generated by strictly standard elements in A over R and also the radical of the ideal generated by elementary standard elements in A over R.","statement_latex":"Let $R \\to A$ be a ring map of finite presentation.\nThe singular ideal $H_{A/R}$ is the radical of the ideal\ngenerated by strictly standard elements in $A$ over $R$\nand also the radical of the ideal generated by elementary\nstandard elements in $A$ over $R$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Singular ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CA","source_file":"smoothing.tex","source_line":252,"source_end_line":259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L252-L259","statement_sha256":"8fd8461b93d3d42bf379a7e0dd06c8f0072828fd6d18b6928bea6cb3ea263549","origin":"The Stacks Project","memory_eligible":false,"source_rank":3733,"rank":3733,"depth":6,"x":663.893,"y":537.169,"cluster":"advanced-algebra"},{"id":"stacks:07CC","tag":"07CC","title":"Singular ideals · Lemma 07CC","summary":"Let R → A be a ring map of finite presentation. Let R → R' be a ring map. If a ∈ A is elementary, resp. strictly standard in A over R, then a ⊗ 1 is elementary, resp. strictly standard in A ⊗_R R' over R'.","statement_latex":"Let $R \\to A$ be a ring map of finite presentation.\nLet $R \\to R'$ be a ring map. If $a \\in A$ is elementary,\nresp.\\ strictly standard in $A$ over $R$, then $a \\otimes 1$\nis elementary, resp.\\ strictly standard in $A \\otimes_R R'$ over $R'$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Singular ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CC","source_file":"smoothing.tex","source_line":294,"source_end_line":300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L294-L300","statement_sha256":"fc9d41a23b7505a74b000e53dffdac5f83506babade14219d988bd91ba324a32","origin":"The Stacks Project","memory_eligible":false,"source_rank":3734,"rank":3734,"depth":0,"x":806.805,"y":776.232,"cluster":"advanced-algebra"},{"id":"stacks:07EU","tag":"07EU","title":"Singular ideals · Lemma 07EU","summary":"Let R → A → Lambda be ring maps with A of finite presentation over R. Assume that H_A/R Lambda = Lambda. Then there exists a factorization A → B → Lambda with B smooth over R.","statement_latex":"Let $R \\to A \\to \\Lambda$ be ring maps with $A$ of finite presentation\nover $R$. Assume that $H_{A/R} \\Lambda = \\Lambda$. Then there exists\na factorization $A \\to B \\to \\Lambda$ with $B$ smooth over $R$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Singular ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EU","source_file":"smoothing.tex","source_line":311,"source_end_line":316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L311-L316","statement_sha256":"4039bb9e69ceb2f97131ec0d2ef5d645dbd2933ce5103862a01ce0f26e7eb070","origin":"The Stacks Project","memory_eligible":false,"source_rank":3735,"rank":3735,"depth":0,"x":509.019,"y":680.992,"cluster":"advanced-algebra"},{"id":"stacks:07CE","tag":"07CE","title":"Presentations of algebras · Lemma 07CE","summary":"Let R be a ring and let A be a finitely presented R-algebra. There exists finite type R-algebra map A → C which has a retraction with the following two properties • for each a ∈ A such that R → A_a is a local complete intersection (More on Algebra, Definition [Tag 07D0]) the ring C_a is smooth over A_a and has a presentation C_a = R[y_1, …, y_m]/J such that J/J^2 is free over C_a, and • for each a ∈ A such that A_a is smooth over R the module Ω_C_a/R is free over C_a.","statement_latex":"Let $R$ be a ring and let $A$ be a finitely presented $R$-algebra.\nThere exists finite type $R$-algebra map $A \\to C$ which has a\nretraction with the following two properties\n\\begin{enumerate}\n\\item for each $a \\in A$ such that $R \\to A_a$ is a local complete\nintersection (More on Algebra, Definition\n\\ref{more-algebra-definition-local-complete-intersection})\nthe ring $C_a$ is smooth over $A_a$ and has a presentation\n$C_a = R[y_1, \\ldots, y_m]/J$ such that $J/J^2$ is free over $C_a$, and\n\\item for each $a \\in A$ such that $A_a$ is smooth over $R$ the\nmodule $\\Omega_{C_a/R}$ is free over $C_a$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Presentations of algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CE","source_file":"smoothing.tex","source_line":341,"source_end_line":355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L341-L355","statement_sha256":"24140fbbb3283574eeb79097198e14fc4937391b9c7c96abb5cc6037e06fe281","origin":"The Stacks Project","memory_eligible":false,"source_rank":3736,"rank":3736,"depth":6,"x":805.346,"y":582.198,"cluster":"advanced-algebra"},{"id":"stacks:07M8","tag":"07M8","title":"Presentations of algebras · Proposition 07M8","summary":"Smooth and syntomic algebras lift along surjections Let R → R_0 be a surjective ring map with kernel I. • If R_0 → A_0 is a syntomic ring map, then there exists a syntomic ring map R → A such that A/IA ≅ A_0. • If R_0 → A_0 is a smooth ring map, then there exists a smooth ring map R → A such that A/IA ≅ A_0.","statement_latex":"\\begin{slogan}\nSmooth and syntomic algebras lift along surjections\n\\end{slogan}\nLet $R \\to R_0$ be a surjective ring map with kernel $I$.\n\\begin{enumerate}\n\\item If $R_0 \\to A_0$ is a syntomic ring map, then there exists a syntomic\nring map $R \\to A$ such that $A/IA \\cong A_0$.\n\\item If $R_0 \\to A_0$ is a smooth ring map, then there exists a smooth\nring map $R \\to A$ such that $A/IA \\cong A_0$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Presentations of algebras","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07M8","source_file":"smoothing.tex","source_line":447,"source_end_line":459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L447-L459","statement_sha256":"8998ed7776221d64ca50874e678eaebe61c4234ee02478c6846f62f059b90b64","origin":"The Stacks Project","memory_eligible":false,"source_rank":3737,"rank":3737,"depth":46,"x":666.214,"y":823.317,"cluster":"advanced-algebra"},{"id":"stacks:07CG","tag":"07CG","title":"Presentations of algebras · Lemma 07CG","summary":"Let R → A be a syntomic ring map. Then there exists a smooth R-algebra map A → C with a retraction such that C is a global relative complete intersection over R, i.e., C ≅ R[x_1, …, x_n]/(f_1, …, f_c) flat over R and all fibres of dimension n - c.","statement_latex":"Let $R \\to A$ be a syntomic ring map. Then there exists a smooth $R$-algebra\nmap $A \\to C$ with a retraction such that $C$ is a global relative complete\nintersection over $R$, i.e.,\n$$\nC \\cong R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)\n$$\nflat over $R$ and all fibres of dimension $n - c$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Presentations of algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CG","source_file":"smoothing.tex","source_line":566,"source_end_line":575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L566-L575","statement_sha256":"080baf94ed7bc6bf4a69193e0013838eec7cb5a86f75980d8afe495c8fa89a6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3738,"rank":3738,"depth":25,"x":574.859,"y":566.438,"cluster":"advanced-algebra"},{"id":"stacks:07CH","tag":"07CH","title":"Presentations of algebras · Lemma 07CH","summary":"Let R → A be a smooth ring map. Then there exists a smooth R-algebra map A → B with a retraction such that B is standard smooth over R, i.e., B ≅ R[x_1, …, x_n]/(f_1, …, f_c) and det(∂ f_j/∂ x_i)_i, j = 1, …, c is invertible in B.","statement_latex":"Let $R \\to A$ be a smooth ring map. Then there exists a smooth $R$-algebra\nmap $A \\to B$ with a retraction such that $B$ is standard smooth over\n$R$, i.e.,\n$$\nB \\cong R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)\n$$\nand $\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$\nis invertible in $B$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Presentations of algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CH","source_file":"smoothing.tex","source_line":589,"source_end_line":599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L589-L599","statement_sha256":"0dad80839bc6db2c6b9adbe23398c17509a1294a4857cf08284ed1d176225282","origin":"The Stacks Project","memory_eligible":false,"source_rank":3739,"rank":3739,"depth":37,"x":848.941,"y":704.092,"cluster":"advanced-algebra"},{"id":"stacks:07CI","tag":"07CI","title":"Presentations of algebras · Lemma 07CI","summary":"Let R → Lambda be a ring map. If Lambda is a filtered colimit of smooth R-algebras, then Lambda is a filtered colimit of standard smooth R-algebras.","statement_latex":"Let $R \\to \\Lambda$ be a ring map. If $\\Lambda$ is a filtered colimit of\nsmooth $R$-algebras, then $\\Lambda$ is a filtered colimit of standard\nsmooth $R$-algebras.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Presentations of algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CI","source_file":"smoothing.tex","source_line":674,"source_end_line":679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L674-L679","statement_sha256":"6806009b7db6b09534cf49642596f0350ed7735343eb9ae97919f26fb9ef9b7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3740,"rank":3740,"depth":38,"x":535.976,"y":758.137,"cluster":"advanced-algebra"},{"id":"stacks:07EY","tag":"07EY","title":"Presentations of algebras · Lemma 07EY","summary":"Let R → A be a standard smooth ring map. Let E ⊂ A be a finite subset of order |E| = n. Then there exists a presentation A = R[x_1, …, x_n + m]/(f_1, …, f_c) with c ≥ n, with det(∂ f_j/∂ x_i)_i, j = 1, …, c invertible in A, and such that E is the set of congruence classes of x_1, …, x_n.","statement_latex":"Let $R \\to A$ be a standard smooth ring map.\nLet $E \\subset A$ be a finite subset of order $|E| = n$.\nThen there exists a presentation\n$A = R[x_1, \\ldots, x_{n + m}]/(f_1, \\ldots, f_c)$ with $c \\geq n$,\nwith $\\det(\\partial f_j/\\partial x_i)_{i, j = 1, \\ldots, c}$\ninvertible in $A$, and such that $E$ is the set of congruence classes of\n$x_1, \\ldots, x_n$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Presentations of algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EY","source_file":"smoothing.tex","source_line":693,"source_end_line":702,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L693-L702","statement_sha256":"0a9e8edc9e3d6138d17fde0819318b223c00dac91f2be3428a8fed5565fb2945","origin":"The Stacks Project","memory_eligible":false,"source_rank":3741,"rank":3741,"depth":0,"x":723.388,"y":540.588,"cluster":"advanced-algebra"},{"id":"stacks:07EZ","tag":"07EZ","title":"Presentations of algebras · Lemma 07EZ","summary":"Let R → A be a ring map of finite presentation. Let a ∈ A. Consider the following conditions on a: • A_a is smooth over R, • A_a is smooth over R and Ω_A_a/R is stably free, • A_a is smooth over R and Ω_A_a/R is free, • A_a is standard smooth over R, • a is strictly standard in A over R, • a is elementary standard in A over R. Then we have • [(a)] (4) ⇒ (3) ⇒ (2) ⇒ (1), • [(b)] (6) ⇒ (5), • [(c)] (6) ⇒ (4), • [(d)] (5) ⇒ (2), • [(e)] (2) ⇒ the elements a^e, e ≥ e_0 are…","statement_latex":"Let $R \\to A$ be a ring map of finite presentation.\nLet $a \\in A$. Consider the following conditions on $a$:\n\\begin{enumerate}\n\\item $A_a$ is smooth over $R$,\n\\item $A_a$ is smooth over $R$ and $\\Omega_{A_a/R}$ is stably free,\n\\item $A_a$ is smooth over $R$ and $\\Omega_{A_a/R}$ is free,\n\\item $A_a$ is standard smooth over $R$,\n\\item $a$ is strictly standard in $A$ over $R$,\n\\item $a$ is elementary standard in $A$ over $R$.\n\\end{enumerate}\nThen we have\n\\begin{enumerate}\n\\item[(a)] (4) $\\Rightarrow$ (3) $\\Rightarrow$ (2) $\\Rightarrow$ (1),\n\\item[(b)] (6) $\\Rightarrow$ (5),\n\\item[(c)] (6) $\\Rightarrow$ (4),\n\\item[(d)] (5) $\\Rightarrow$ (2),\n\\item[(e)] (2) $\\Rightarrow$ the elements $a^e$, $e \\geq e_0$ are\nstrictly standard in $A$ over $R$,\n\\item[(f)] (4) $\\Rightarrow$ the elements $a^e$, $e \\geq e_0$ are\nelementary standard in $A$ over $R$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Presentations of algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07EZ","source_file":"smoothing.tex","source_line":718,"source_end_line":741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L718-L741","statement_sha256":"f0b3d10db2b135f5b2f89e5b8f01ec73b0bf3a6991b1bec99626813802793df2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3742,"rank":3742,"depth":31,"x":760.159,"y":807.486,"cluster":"advanced-algebra"},{"id":"stacks:0BJ3","tag":"0BJ3","title":"Intermezzo: Néron desingularization · Lemma 0BJ3","summary":"In Situation [Tag 0BJ2] Néron's blowup is functorial in the following sense • if a ∈ A, a not ∈ p, then Néron's blowup of A_a is A'_a, and • if B → A is a surjection of flat finite type R-algebras with kernel I, then A' is the quotient of B'/IB' by its π-power torsion.","statement_latex":"In Situation \\ref{situation-neron} N\\'eron's blowup is functorial\nin the following sense\n\\begin{enumerate}\n\\item if $a \\in A$, $a \\not \\in \\mathfrak p$, then N\\'eron's blowup\nof $A_a$ is $A'_a$, and\n\\item if $B \\to A$ is a surjection of flat finite type $R$-algebras\nwith kernel $I$, then $A'$ is the quotient of $B'/IB'$ by its\n$\\pi$-power torsion.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Intermezzo: Néron desingularization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJ3","source_file":"smoothing.tex","source_line":906,"source_end_line":917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L906-L917","statement_sha256":"33abdba52172d5062dd8d30fbfc62a97652f71a15644f412023091c6c8a54743","origin":"The Stacks Project","memory_eligible":false,"source_rank":3743,"rank":3743,"depth":1,"x":518.287,"y":631.453,"cluster":"advanced-algebra"},{"id":"stacks:0BJ4","tag":"0BJ4","title":"Intermezzo: Néron desingularization · Lemma 0BJ4","summary":"In Situation [Tag 0BJ2] assume that R → A is smooth at p and that R/π R ⊂ Lambda/π Lambda is a separable field extension. Then R → A' is smooth at p' and there is a short exact sequence 0 → Ω_A/R ⊗_A A'_ p' → Ω_A'/R, p' → (A'/π A')_ p'^⊕ c → 0 where c = dim((A/π A)_ p).","statement_latex":"In Situation \\ref{situation-neron} assume that $R \\to A$ is smooth\nat $\\mathfrak p$ and that $R/\\pi R \\subset \\Lambda/\\pi \\Lambda$\nis a separable field extension. Then $R \\to A'$ is smooth at\n$\\mathfrak p'$ and there is a short exact sequence\n$$\n0 \\to\n\\Omega_{A/R} \\otimes_A A'_{\\mathfrak p'} \\to\n\\Omega_{A'/R, \\mathfrak p'} \\to\n(A'/\\pi A')_{\\mathfrak p'}^{\\oplus c} \\to 0\n$$\nwhere $c = \\dim((A/\\pi A)_\\mathfrak p)$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Intermezzo: Néron desingularization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJ4","source_file":"smoothing.tex","source_line":927,"source_end_line":940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L927-L940","statement_sha256":"e48c108d2996d96fdc13b7dff0aa97b0f78e7aec6a2854193d53640d4281481e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3744,"rank":3744,"depth":37,"x":838.367,"y":624.009,"cluster":"advanced-algebra"},{"id":"stacks:0BJ5","tag":"0BJ5","title":"Intermezzo: Néron desingularization · Lemma 0BJ5","summary":"In Situation [Tag 0BJ2] assume that R → A is smooth at q and that we have a surjection of R-algebras B → A with kernel I. Assume R → B smooth at p_B = (B → A)^-1 p. If the cokernel of I/I^2 ⊗_A Lambda → Ω_B/R ⊗_B Lambda is a free Lambda-module, then R → A is smooth at p.","statement_latex":"In Situation \\ref{situation-neron} assume that $R \\to A$ is smooth\nat $\\mathfrak q$ and that we have a surjection of $R$-algebras\n$B \\to A$ with kernel $I$. Assume $R \\to B$ smooth at\n$\\mathfrak p_B = (B \\to A)^{-1}\\mathfrak p$. If the cokernel of\n$$\nI/I^2 \\otimes_A \\Lambda \\to \\Omega_{B/R} \\otimes_B \\Lambda\n$$\nis a free $\\Lambda$-module, then $R \\to A$ is smooth at $\\mathfrak p$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Intermezzo: Néron desingularization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJ5","source_file":"smoothing.tex","source_line":1007,"source_end_line":1017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1007-L1017","statement_sha256":"8dd14240d2b296e535ad27b2cefc65e91963b20bbcbf3bebb74dc071088aaf9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3745,"rank":3745,"depth":37,"x":608.212,"y":811.217,"cluster":"advanced-algebra"},{"id":"stacks:0BJ6","tag":"0BJ6","title":"Intermezzo: Néron desingularization · Lemma 0BJ6","summary":"In Situation [Tag 0BJ2] assume that R → A is smooth at q and that R/π R ⊂ Lambda/π Lambda is a separable extension of fields. Then after a finite number of affine Néron blowups the algebra A becomes smooth over R at p.","statement_latex":"In Situation \\ref{situation-neron}\nassume that $R \\to A$ is smooth at $\\mathfrak q$\nand that $R/\\pi R \\subset \\Lambda/\\pi \\Lambda$ is a separable\nextension of fields. Then after a finite number of affine N\\'eron\nblowups the algebra $A$ becomes smooth over $R$ at $\\mathfrak p$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Intermezzo: Néron desingularization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJ6","source_file":"smoothing.tex","source_line":1040,"source_end_line":1047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1040-L1047","statement_sha256":"a6e51e7f69ac2c21a6dd378061233442647ea0c2b3b7ed2838f55dcd3da601e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3746,"rank":3746,"depth":38,"x":627.388,"y":542.437,"cluster":"advanced-algebra"},{"id":"stacks:0BJ7","tag":"0BJ7","title":"Intermezzo: Néron desingularization · Lemma 0BJ7","summary":"Unramified extensions of DVRs are ind-smooth AKA Néron desingularization Let R ⊂ Lambda be an extension of discrete valuation rings which has ramification index 1 and induces a separable extension of residue fields and of fraction fields. Then Lambda is a filtered colimit of smooth R-algebras.","statement_latex":"\\begin{slogan}\nUnramified extensions of DVRs are ind-smooth AKA N\\'eron desingularization\n\\end{slogan}\nLet $R \\subset \\Lambda$ be an extension of discrete valuation\nrings which has ramification index $1$ and induces a separable\nextension of residue fields and of fraction fields.\nThen $\\Lambda$ is a filtered colimit of smooth $R$-algebras.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Intermezzo: Néron desingularization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJ7","source_file":"smoothing.tex","source_line":1188,"source_end_line":1197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1188-L1197","statement_sha256":"c4d05ecfff8812bb6762d072d74d47442a23bde5ac84a2dec90cd2ce2b5659e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3747,"rank":3747,"depth":39,"x":829.497,"y":751.62,"cluster":"advanced-algebra"},{"id":"stacks:07CK","tag":"07CK","title":"The lifting problem · Lemma 07CK","summary":"Let R → Lambda be a ring map. Let I ⊂ R be an ideal. Assume that • I^2 = 0, and • Lambda/ILambda is a filtered colimit of smooth R/I-algebras. Let φ : A → Lambda be an R-algebra map with A of finite presentation over R. Then there exists a factorization A → B/J → Lambda where B is a smooth R-algebra and J ⊂ IB is a finitely generated ideal.","statement_latex":"Let $R \\to \\Lambda$ be a ring map. Let $I \\subset R$ be an ideal.\nAssume that\n\\begin{enumerate}\n\\item $I^2 = 0$, and\n\\item $\\Lambda/I\\Lambda$ is a filtered colimit of smooth $R/I$-algebras.\n\\end{enumerate}\nLet $\\varphi : A \\to \\Lambda$ be an $R$-algebra map with $A$ of finite\npresentation over $R$. Then there exists a factorization\n$$\nA \\to B/J \\to \\Lambda\n$$\nwhere $B$ is a smooth $R$-algebra and $J \\subset IB$ is a finitely generated\nideal.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The lifting problem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CK","source_file":"smoothing.tex","source_line":1237,"source_end_line":1252,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1237-L1252","statement_sha256":"e13d1489642d5a6e0ba9b04d54717c629865fec74215389f674c9e40b0d77db6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3748,"rank":3748,"depth":39,"x":512.081,"y":712.033,"cluster":"advanced-algebra"},{"id":"stacks:07CL","tag":"07CL","title":"The lifting problem · Lemma 07CL","summary":"Let R → Lambda be a ring map. Let I ⊂ R be an ideal. Assume that • I^2 = 0, • Lambda/ILambda is a filtered colimit of smooth R/I-algebras, and • R → Lambda is flat. Let φ : B → Lambda be an R-algebra map with B smooth over R. Let J ⊂ IB be a finitely generated ideal such that φ(J) = 0. Then there exists R-algebra maps B xrightarrowα B' xrightarrowβ Lambda such that B' is smooth over R, such that α(J) = 0 and such that β ∘ α = φ.","statement_latex":"Let $R \\to \\Lambda$ be a ring map. Let $I \\subset R$ be an ideal.\nAssume that\n\\begin{enumerate}\n\\item $I^2 = 0$,\n\\item $\\Lambda/I\\Lambda$ is a filtered colimit of smooth $R/I$-algebras, and\n\\item $R \\to \\Lambda$ is flat.\n\\end{enumerate}\nLet $\\varphi : B \\to \\Lambda$ be an $R$-algebra map with $B$\nsmooth over $R$. Let $J \\subset IB$ be a finitely generated ideal\nsuch that $\\varphi(J) = 0$.\nThen there exists $R$-algebra maps\n$$\nB \\xrightarrow{\\alpha} B' \\xrightarrow{\\beta} \\Lambda\n$$\nsuch that $B'$ is smooth over $R$, such that $\\alpha(J) = 0$ and\nsuch that $\\beta \\circ \\alpha = \\varphi$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The lifting problem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CL","source_file":"smoothing.tex","source_line":1307,"source_end_line":1325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1307-L1325","statement_sha256":"2746ad97cb63dc46ca251006cd03625aa00b491353c1c2262480ec659067505f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3749,"rank":3749,"depth":40,"x":778.111,"y":561.037,"cluster":"advanced-algebra"},{"id":"stacks:07CM","tag":"07CM","title":"The lifting problem · Proposition 07CM","summary":"Ind-smoothness of an algebra is stable under infinitesimal deformations Let R → Lambda be a ring map. Let I ⊂ R be an ideal. Assume that • I is nilpotent, • Lambda/ILambda is a filtered colimit of smooth R/I-algebras, and • R → Lambda is flat. Then Lambda is a filtered colimit of smooth R-algebras.","statement_latex":"\\begin{slogan}\nInd-smoothness of an algebra is stable under infinitesimal deformations\n\\end{slogan}\nLet $R \\to \\Lambda$ be a ring map. Let $I \\subset R$ be an ideal.\nAssume that\n\\begin{enumerate}\n\\item $I$ is nilpotent,\n\\item $\\Lambda/I\\Lambda$ is a filtered colimit of smooth $R/I$-algebras, and\n\\item $R \\to \\Lambda$ is flat.\n\\end{enumerate}\nThen $\\Lambda$ is a filtered colimit of smooth $R$-algebras.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The lifting problem","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CM","source_file":"smoothing.tex","source_line":1373,"source_end_line":1386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1373-L1386","statement_sha256":"94edf5ca088dd394e09783bf8756b7a023ad7f6ae9d95e60216d5e1722545168","origin":"The Stacks Project","memory_eligible":false,"source_rank":3750,"rank":3750,"depth":41,"x":703.333,"y":823.467,"cluster":"advanced-algebra"},{"id":"stacks:07CP","tag":"07CP","title":"The lifting lemma · Lemma 07CP","summary":"Let R be a Noetherian ring. Let Lambda be an R-algebra. Let π ∈ R and assume that Ann_R(π) = Ann_R(π^2) and Ann_Lambda(π) = Ann_Lambda(π^2). Suppose we have R-algebra maps R/π^2R → bar C → Lambda/π^2Lambda with bar C of finite presentation. Then there exists an R-algebra homomorphism D → Lambda and a commutative diagram xymatrix R/π^2R ar[r] ar[d] & bar C ar[r] ar[d] & Lambda/π^2Lambda ar[d] R/π R ar[r] & D/π D ar[r] & Lambda/π Lambda with the following properties • [(a)]…","statement_latex":"Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra.\nLet $\\pi \\in R$ and assume that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$ and\n$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$.\nSuppose we have $R$-algebra maps\n$R/\\pi^2R \\to \\bar C \\to \\Lambda/\\pi^2\\Lambda$\nwith $\\bar C$ of finite presentation.\nThen there exists an $R$-algebra homomorphism\n$D \\to \\Lambda$ and a commutative diagram\n$$\n\\xymatrix{\nR/\\pi^2R \\ar[r] \\ar[d] &\n\\bar C \\ar[r] \\ar[d] &\n\\Lambda/\\pi^2\\Lambda \\ar[d] \\\\\nR/\\pi R \\ar[r] &\nD/\\pi D \\ar[r] &\n\\Lambda/\\pi \\Lambda\n}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item[(a)] $D$ is of finite presentation,\n\\item[(b)] $R \\to D$ is smooth at any prime $\\mathfrak q$ with\n$\\pi \\not \\in \\mathfrak q$,\n\\item[(c)] $R \\to D$ is smooth at any prime $\\mathfrak q$ with\n$\\pi \\in \\mathfrak q$ lying over a prime of $\\bar C$ where\n$R/\\pi^2 R \\to \\bar C$ is smooth, and\n\\item[(d)] $\\bar C/\\pi \\bar C \\to D/\\pi D$ is smooth at any prime\nlying over a prime of $\\bar C$ where $R/\\pi^2R \\to \\bar C$ is smooth.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The lifting lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CP","source_file":"smoothing.tex","source_line":1420,"source_end_line":1451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1420-L1451","statement_sha256":"41ae8d54053d05877673ad23d71905df4277dbad91c1fe0d198e186ffbcda292","origin":"The Stacks Project","memory_eligible":false,"source_rank":3751,"rank":3751,"depth":36,"x":547.354,"y":587.402,"cluster":"advanced-algebra"},{"id":"stacks:07CR","tag":"07CR","title":"The desingularization lemma · Lemma 07CR","summary":"Let R be a Noetherian ring. Let Lambda be an R-algebra. Let π ∈ R and assume that Ann_Lambda(π) = Ann_Lambda(π^2). Let A → Lambda be an R-algebra map with A of finite presentation. Assume • the image of π is strictly standard in A over R, and • there exists a section ρ : A/π^4 A → R/π^4 R which is compatible with the map to Lambda/π^4 Lambda. Then we can find R-algebra maps A → B → Lambda with B of finite presentation such that a B ⊂ H_B/R where a =…","statement_latex":"Let $R$ be a Noetherian ring.\nLet $\\Lambda$ be an $R$-algebra. Let $\\pi \\in R$ and\nassume that $\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$. Let\n$A \\to \\Lambda$ be an $R$-algebra map with $A$ of finite\npresentation. Assume\n\\begin{enumerate}\n\\item the image of $\\pi$ is strictly standard in $A$ over $R$, and\n\\item there exists a section $\\rho : A/\\pi^4 A \\to R/\\pi^4 R$\nwhich is compatible with the map to $\\Lambda/\\pi^4 \\Lambda$.\n\\end{enumerate}\nThen we can find $R$-algebra maps $A \\to B \\to \\Lambda$ with $B$\nof finite presentation such that $\\mathfrak a B \\subset H_{B/R}$ where\n$\\mathfrak a = \\text{Ann}_R(\\text{Ann}_R(\\pi^2)/\\text{Ann}_R(\\pi))$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The desingularization lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CR","source_file":"smoothing.tex","source_line":1664,"source_end_line":1679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1664-L1679","statement_sha256":"d8001b190b0ae7b35565ce559c86fba22074401393ba6c55fd4ddab3f4a63633","origin":"The Stacks Project","memory_eligible":false,"source_rank":3752,"rank":3752,"depth":36,"x":852.364,"y":673.01,"cluster":"advanced-algebra"},{"id":"stacks:07CT","tag":"07CT","title":"The desingularization lemma · Lemma 07CT","summary":"Let R be a Noetherian ring. Let Lambda be an R-algebra. Let π ∈ R and assume that Ann_R(π) = Ann_R(π^2) and Ann_Lambda(π) = Ann_Lambda(π^2). Let A → Lambda and D → Lambda be R-algebra maps with A and D of finite presentation. Assume • π is strictly standard in A over R, and • there exists an R-algebra map A/π^4 A → D/π^4 D compatible with the maps to Lambda/π^4 Lambda. Then we can find an R-algebra map B → Lambda with B of finite presentation and R-algebra maps A → B and…","statement_latex":"Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra.\nLet $\\pi \\in R$ and assume that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$ and\n$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$.\nLet $A \\to \\Lambda$ and $D \\to \\Lambda$ be $R$-algebra maps with\n$A$ and $D$ of finite presentation. Assume\n\\begin{enumerate}\n\\item $\\pi$ is strictly standard in $A$ over $R$, and\n\\item there exists an $R$-algebra map $A/\\pi^4 A \\to D/\\pi^4 D$ compatible\nwith the maps to $\\Lambda/\\pi^4 \\Lambda$.\n\\end{enumerate}\nThen we can find an $R$-algebra map $B \\to \\Lambda$ with $B$ of finite\npresentation and $R$-algebra maps $A \\to B$ and $D \\to B$\ncompatible with the maps to $\\Lambda$ such that $H_{D/R}B \\subset H_{B/D}$\nand $H_{D/R}B \\subset H_{B/R}$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The desingularization lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07CT","source_file":"smoothing.tex","source_line":1811,"source_end_line":1827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1811-L1827","statement_sha256":"3d88f49016eb5b88122a01441586f6a8c9752ec9fd6f5809cc11f8e9a0752cf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3753,"rank":3753,"depth":37,"x":558.46,"y":783.011,"cluster":"advanced-algebra"},{"id":"stacks:07F0","tag":"07F0","title":"The desingularization lemma · Lemma 07F0","summary":"Let R be a Noetherian ring. Let Lambda be an R-algebra. Let π ∈ R and assume that Ann_R(π) = Ann_R(π^2) and Ann_Lambda(π) = Ann_Lambda(π^2). Let A → Lambda be an R-algebra map with A of finite presentation and assume π is strictly standard in A over R. Let A/π^8A → bar C → Lambda/π^8Lambda be a factorization with bar C of finite presentation. Then we can find a factorization A → B → Lambda with B of finite presentation such that R_π → B_π is smooth and such that H_bar…","statement_latex":"Let $R$ be a Noetherian ring. Let $\\Lambda$ be an $R$-algebra.\nLet $\\pi \\in R$ and assume that $\\text{Ann}_R(\\pi) = \\text{Ann}_R(\\pi^2)$ and\n$\\text{Ann}_\\Lambda(\\pi) = \\text{Ann}_\\Lambda(\\pi^2)$.\nLet $A \\to \\Lambda$ be an $R$-algebra map with\n$A$ of finite presentation and assume $\\pi$ is strictly standard\nin $A$ over $R$. Let\n$$\nA/\\pi^8A \\to \\bar C \\to \\Lambda/\\pi^8\\Lambda\n$$\nbe a factorization with $\\bar C$ of finite presentation.\nThen we can find a factorization $A \\to B \\to \\Lambda$ with $B$ of finite\npresentation such that $R_\\pi \\to B_\\pi$ is smooth and such that\n$$\nH_{\\bar C/(R/\\pi^8 R)} \\cdot \\Lambda/\\pi^8\\Lambda\n\\subset\n\\sqrt{H_{B/R} \\Lambda} \\bmod \\pi^8\\Lambda.\n$$","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The desingularization lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07F0","source_file":"smoothing.tex","source_line":1861,"source_end_line":1880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1861-L1880","statement_sha256":"9968790c03d452f2cc30d77e14f5ad56392ac68d5ca27cb1d8ac9d4849ceb481","origin":"The Stacks Project","memory_eligible":false,"source_rank":3754,"rank":3754,"depth":38,"x":686.787,"y":535.002,"cluster":"advanced-algebra"},{"id":"stacks:07F3","tag":"07F3","title":"Warmup: reduction to a base field · Lemma 07F3","summary":"Let R_i → Lambda_i, i = 1, 2 be as in Situation [Tag 07F2]. If PT holds for R_i → Lambda_i, i = 1, 2, then PT holds for R_1 × R_2 → Lambda_1 × Lambda_2.","statement_latex":"Let $R_i \\to \\Lambda_i$, $i = 1, 2$ be as in Situation \\ref{situation-global}.\nIf PT holds for $R_i \\to \\Lambda_i$, $i = 1, 2$, then PT holds for\n$R_1 \\times R_2 \\to \\Lambda_1 \\times \\Lambda_2$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Warmup: reduction to a base field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07F3","source_file":"smoothing.tex","source_line":1925,"source_end_line":1930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1925-L1930","statement_sha256":"e74b841be40f90ff7e4479c9738357fc5590e2bf6f896638ce1b9e892cbf00ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":3755,"rank":3755,"depth":0,"x":791.656,"y":790.827,"cluster":"advanced-algebra"},{"id":"stacks:07F4","tag":"07F4","title":"Warmup: reduction to a base field · Lemma 07F4","summary":"Let R → A → Lambda be ring maps with A of finite presentation over R. Let S ⊂ R be a multiplicative set. Let S^-1A → B' → S^-1Lambda be a factorization with B' smooth over S^-1R. Then we can find a factorization A → B → Lambda such that some s ∈ S maps to an elementary standard element (Definition [Tag 07C7]) in B over R.","statement_latex":"Let $R \\to A \\to \\Lambda$ be ring maps with $A$ of finite presentation\nover $R$. Let $S \\subset R$ be a multiplicative\nset. Let $S^{-1}A \\to B' \\to S^{-1}\\Lambda$ be a factorization with\n$B'$ smooth over $S^{-1}R$. Then we can find a factorization\n$A \\to B \\to \\Lambda$ such that some $s \\in S$ maps to an\nelementary standard element (Definition \\ref{definition-strictly-standard})\nin $B$ over $R$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Warmup: reduction to a base field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07F4","source_file":"smoothing.tex","source_line":1936,"source_end_line":1945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L1936-L1945","statement_sha256":"b13d675611761ecf26335e4de1f0d116294ad106f56608fc07de1a4be7450cc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3756,"rank":3756,"depth":38,"x":508.454,"y":661.625,"cluster":"advanced-algebra"},{"id":"stacks:07F5","tag":"07F5","title":"Warmup: reduction to a base field · Lemma 07F5","summary":"Proving Popescu approximation reduces to algebras over a field If for every Situation [Tag 07F2] where R is a field PT holds, then PT holds in general.","statement_latex":"\\begin{slogan}\nProving Popescu approximation reduces to algebras over a field\n\\end{slogan}\nIf for every Situation \\ref{situation-global} where $R$\nis a field PT holds, then PT holds in general.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Warmup: reduction to a base field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07F5","source_file":"smoothing.tex","source_line":2002,"source_end_line":2009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2002-L2009","statement_sha256":"2bd0027de00b4551ad55959e8a33bf371876173113c521cc437b861910def0fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":3757,"rank":3757,"depth":42,"x":821.345,"y":596.168,"cluster":"advanced-algebra"},{"id":"stacks:07F8","tag":"07F8","title":"Local tricks · Lemma 07F8","summary":"Let R → A → Lambda ⊃ q be as in Situation [Tag 07F7]. Let r ≥ 1 and π_1, …, π_r ∈ R map to elements of q. Assume • for i = 1, …, r we have Ann_R/(π_1^8, …, π_i - 1^8)R(π_i) = Ann_R/(π_1^8, …, π_i - 1^8)R(π_i^2) and Ann_Lambda/(π_1^8, …, π_i - 1^8)Lambda(π_i) = Ann_Lambda/(π_1^8, …, π_i - 1^8)Lambda(π_i^2) • for i = 1, …, r the element π_i maps to a strictly standard element in A over R. Then, if R/(π_1^8, …, π_r^8)R → A/(π_1^8, …, π_r^8)A → Lambda/(π_1^8, …, π_r^8)Lambda…","statement_latex":"Let $R \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in\nSituation \\ref{situation-local}. Let $r \\geq 1$ and\n$\\pi_1, \\ldots, \\pi_r \\in R$ map to elements of $\\mathfrak q$. Assume\n\\begin{enumerate}\n\\item for $i = 1, \\ldots, r$ we have\n$$\n\\text{Ann}_{R/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)R}(\\pi_i)\n=\n\\text{Ann}_{R/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)R}(\\pi_i^2)\n$$\nand\n$$\n\\text{Ann}_{\\Lambda/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)\\Lambda}(\\pi_i)\n=\n\\text{Ann}_{\\Lambda/(\\pi_1^8, \\ldots, \\pi_{i - 1}^8)\\Lambda}(\\pi_i^2)\n$$\n\\item for $i = 1, \\ldots, r$ the element $\\pi_i$ maps to a strictly\nstandard element in $A$ over $R$.\n\\end{enumerate}\nThen, if\n$$\nR/(\\pi_1^8, \\ldots, \\pi_r^8)R \\to A/(\\pi_1^8, \\ldots, \\pi_r^8)A\n\\to \\Lambda/(\\pi_1^8, \\ldots, \\pi_r^8)\\Lambda \\supset\n\\mathfrak q/(\\pi_1^8, \\ldots, \\pi_r^8)\\Lambda\n$$\ncan be resolved, so can $R \\to A \\to \\Lambda \\supset \\mathfrak q$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Local tricks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07F8","source_file":"smoothing.tex","source_line":2096,"source_end_line":2124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2096-L2124","statement_sha256":"8b27018bec943334db7430e9f86d9e5a2041918d3f918c642909c67efacc6b20","origin":"The Stacks Project","memory_eligible":false,"source_rank":3758,"rank":3758,"depth":39,"x":643.171,"y":822.092,"cluster":"advanced-algebra"},{"id":"stacks:07F9","tag":"07F9","title":"Local tricks · Lemma 07F9","summary":"[swan] or [popescu-GND] Let R → A → Lambda ⊃ q be as in Situation [Tag 07F7]. Let p = R ∩ q. Assume that q is minimal over h_A and that R_ p → A_ p → Lambda_ q ⊃ qLambda_ q can be resolved. Then there exists a factorization A → C → Lambda with C of finite presentation such that H_C/R Lambda not ⊂ q.","statement_latex":"\\begin{reference}\n\\cite[Lemma 12.2]{swan} or\n\\cite[Lemma 2]{popescu-GND}\n\\end{reference}\nLet $R \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in\nSituation \\ref{situation-local}. Let $\\mathfrak p = R \\cap \\mathfrak q$.\nAssume that $\\mathfrak q$ is minimal over $\\mathfrak h_A$ and that\n$R_\\mathfrak p \\to A_\\mathfrak p \\to \\Lambda_\\mathfrak q\n\\supset \\mathfrak q\\Lambda_\\mathfrak q$ can be resolved.\nThen there exists a factorization $A \\to C \\to \\Lambda$ with $C$ of\nfinite presentation such that $H_{C/R} \\Lambda \\not \\subset \\mathfrak q$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Local tricks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07F9","source_file":"smoothing.tex","source_line":2147,"source_end_line":2160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2147-L2160","statement_sha256":"60fd0e85932511a7e0b5daf331cf4f1f0b8e94704f92f37b830f2c8857699422","origin":"The Stacks Project","memory_eligible":false,"source_rank":3759,"rank":3759,"depth":38,"x":592.846,"y":554.262,"cluster":"advanced-algebra"},{"id":"stacks:07FA","tag":"07FA","title":"Local tricks · Lemma 07FA","summary":"Let R → A → Lambda ⊃ q be as in Situation [Tag 07F7]. Let p = R ∩ q. Assume • q is minimal over h_A, • R_ p → A_ p → Lambda_ q ⊃ qLambda_ q can be resolved, and • dim(Lambda_ q) = 0. Then R → A → Lambda ⊃ q can be resolved.","statement_latex":"Let $R \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in\nSituation \\ref{situation-local}. Let $\\mathfrak p = R \\cap \\mathfrak q$.\nAssume\n\\begin{enumerate}\n\\item $\\mathfrak q$ is minimal over $\\mathfrak h_A$,\n\\item $R_\\mathfrak p \\to A_\\mathfrak p \\to \\Lambda_\\mathfrak q\n\\supset \\mathfrak q\\Lambda_\\mathfrak q$ can be resolved, and\n\\item $\\dim(\\Lambda_\\mathfrak q) = 0$.\n\\end{enumerate}\nThen $R \\to A \\to \\Lambda \\supset \\mathfrak q$ can be resolved.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Local tricks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07FA","source_file":"smoothing.tex","source_line":2229,"source_end_line":2241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2229-L2241","statement_sha256":"c136511e2153b3660462389a1c3db872781c871ef1e46d5191cfdc2e3623c1f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3760,"rank":3760,"depth":39,"x":845.465,"y":723.286,"cluster":"advanced-algebra"},{"id":"stacks:07FC","tag":"07FC","title":"Ogoma · Lemma 07FC","summary":"Let A be a Noetherian ring and let M be a finite A-module. Let S ⊂ A be a multiplicative set. If π ∈ A and Ker(π : S^-1M → S^-1M) = Ker(π^2 : S^-1M → S^-1M) then there exists an s ∈ S such that for any n > 0 we have Ker(s^nπ : M → M) = Ker((s^nπ)^2 : M → M).","statement_latex":"Let $A$ be a Noetherian ring and let $M$ be a finite $A$-module.\nLet $S \\subset A$ be a multiplicative set. If $\\pi \\in A$ and\n$\\Ker(\\pi : S^{-1}M \\to S^{-1}M) =\n\\Ker(\\pi^2 : S^{-1}M \\to S^{-1}M)$\nthen there exists an $s \\in S$ such that for any $n > 0$ we have\n$\\Ker(s^n\\pi : M \\to M) = \\Ker((s^n\\pi)^2 : M \\to M)$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Separable residue fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07FC","source_file":"smoothing.tex","source_line":2286,"source_end_line":2294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2286-L2294","statement_sha256":"f05e3926f14b70e0ff197d2bd80e7e0f94128454d2da2ecdad5ba85023158fe5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3761,"rank":3761,"depth":0,"x":523.099,"y":742.002,"cluster":"advanced-algebra"},{"id":"stacks:07FD","tag":"07FD","title":"Separable residue fields · Lemma 07FD","summary":"Let Lambda be a Noetherian ring. Let I ⊂ Lambda be an ideal. Let I ⊂ q be a prime. Let n, e be positive integers Assume that q^nLambda_ q ⊂ ILambda_ q and that Lambda_ q is a regular local ring of dimension d. Then there exists an n > 0 and π_1, …, π_d ∈ Lambda such that • (π_1, …, π_d)Lambda_ q = qLambda_ q, • π_1^n, …, π_d^n ∈ I, and • for i = 1, …, d we have Ann_Lambda/(π_1^e, …, π_i - 1^e)Lambda(π_i) = Ann_Lambda/(π_1^e, …, π_i - 1^e)Lambda(π_i^2).","statement_latex":"Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal.\nLet $I \\subset \\mathfrak q$ be a prime. Let $n, e$ be positive integers\nAssume that $\\mathfrak q^n\\Lambda_\\mathfrak q \\subset I\\Lambda_\\mathfrak q$\nand that $\\Lambda_\\mathfrak q$ is a regular local ring of dimension $d$.\nThen there exists an $n > 0$ and\n$\\pi_1, \\ldots, \\pi_d \\in \\Lambda$ such that\n\\begin{enumerate}\n\\item $(\\pi_1, \\ldots, \\pi_d)\\Lambda_\\mathfrak q =\n\\mathfrak q\\Lambda_\\mathfrak q$,\n\\item $\\pi_1^n, \\ldots, \\pi_d^n \\in I$, and\n\\item for $i = 1, \\ldots, d$ we have\n$$\n\\text{Ann}_{\\Lambda/(\\pi_1^e, \\ldots, \\pi_{i - 1}^e)\\Lambda}(\\pi_i) =\n\\text{Ann}_{\\Lambda/(\\pi_1^e, \\ldots, \\pi_{i - 1}^e)\\Lambda}(\\pi_i^2).\n$$\n\\end{enumerate}","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Separable residue fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07FD","source_file":"smoothing.tex","source_line":2305,"source_end_line":2323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2305-L2323","statement_sha256":"4374f11aa8c512b18ba937ad751007fd9b50e2cf0ba43e477737d322e1389988","origin":"The Stacks Project","memory_eligible":false,"source_rank":3762,"rank":3762,"depth":13,"x":745.87,"y":545.183,"cluster":"advanced-algebra"},{"id":"stacks:07FE","tag":"07FE","title":"Separable residue fields · Lemma 07FE","summary":"Let k → A → Lambda ⊃ q be as in Situation [Tag 07F7] where • k is a field, • Lambda is Noetherian, • q is minimal over h_A, • Lambda_ q is a regular local ring, and • the field extension kappa( q)/k is separable. Then k → A → Lambda ⊃ q can be resolved.","statement_latex":"Let $k \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in\nSituation \\ref{situation-local} where\n\\begin{enumerate}\n\\item $k$ is a field,\n\\item $\\Lambda$ is Noetherian,\n\\item $\\mathfrak q$ is minimal over $\\mathfrak h_A$,\n\\item $\\Lambda_\\mathfrak q$ is a regular local ring, and\n\\item the field extension $\\kappa(\\mathfrak q)/k$ is separable.\n\\end{enumerate}\nThen $k \\to A \\to \\Lambda \\supset \\mathfrak q$ can be resolved.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Separable residue fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07FE","source_file":"smoothing.tex","source_line":2351,"source_end_line":2363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2351-L2363","statement_sha256":"ec7b0c638fbfcd692999f4452b62d111656a4d116aa9aab9b8c8c831c9ba3526","origin":"The Stacks Project","memory_eligible":false,"source_rank":3763,"rank":3763,"depth":42,"x":739.876,"y":816.858,"cluster":"advanced-algebra"},{"id":"stacks:07FG","tag":"07FG","title":"Inseparable residue fields · Lemma 07FG","summary":"Let k be a field of characteristic p > 0. Let (Lambda, m, K) be an Artinian local k-algebra. Assume that dim H_1(L_K/k) < ∞. Then Lambda is a filtered colimit of Artinian local k-algebras A with each map A → Lambda flat, with m_A Lambda = m, and with A essentially of finite type over k.","statement_latex":"Let $k$ be a field of characteristic $p > 0$.\nLet $(\\Lambda, \\mathfrak m, K)$ be an Artinian local $k$-algebra.\nAssume that $\\dim H_1(L_{K/k}) < \\infty$.\nThen $\\Lambda$ is a filtered colimit of Artinian\nlocal $k$-algebras $A$ with each map $A \\to \\Lambda$ flat, with\n$\\mathfrak m_A \\Lambda = \\mathfrak m$, and with\n$A$ essentially of finite type over $k$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Inseparable residue fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07FG","source_file":"smoothing.tex","source_line":2444,"source_end_line":2453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2444-L2453","statement_sha256":"996cc45bd7c356e2786a9b823a58c740c82f5421237d27821acba381100975ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":3764,"rank":3764,"depth":8,"x":525.713,"y":613.024,"cluster":"advanced-algebra"},{"id":"stacks:07FH","tag":"07FH","title":"Inseparable residue fields · Lemma 07FH","summary":"Let k be a field of characteristic p > 0. Let Lambda be a Noetherian geometrically regular k-algebra. Let q ⊂ Lambda be a prime ideal. Let n ≥ 1 be an integer and let E ⊂ Lambda_ q/ q^nLambda_ q be a finite subset. Then we can find m ≥ 0 and φ : k[y_1, …, y_m] → Lambda with the following properties • setting p = φ^-1( q) we have qLambda_ q = p Lambda_ q and k[y_1, …, y_m]_ p → Lambda_ q is flat, • there is a factorization by homomorphisms of local Artinian rings k[y_1, …,…","statement_latex":"Let $k$ be a field of characteristic $p > 0$.\nLet $\\Lambda$ be a Noetherian geometrically regular $k$-algebra.\nLet $\\mathfrak q \\subset \\Lambda$ be a prime ideal.\nLet $n \\geq 1$ be an integer and let\n$E \\subset \\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q$\nbe a finite subset.\nThen we can find $m \\geq 0$ and\n$\\varphi : k[y_1, \\ldots, y_m] \\to \\Lambda$ with the following properties\n\\begin{enumerate}\n\\item setting $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$ we have\n$\\mathfrak q\\Lambda_\\mathfrak q = \\mathfrak p \\Lambda_\\mathfrak q$\nand $k[y_1, \\ldots, y_m]_\\mathfrak p \\to \\Lambda_\\mathfrak q$ is flat,\n\\item there is a factorization by homomorphisms of local Artinian rings\n$$\nk[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n k[y_1, \\ldots, y_m]_\\mathfrak p\n\\to D \\to\n\\Lambda_\\mathfrak q/\\mathfrak q^n\\Lambda_\\mathfrak q\n$$\nwhere the first arrow is essentially smooth and the second is flat,\n\\item $E$ is contained in $D$ modulo $\\mathfrak q^n\\Lambda_\\mathfrak q$.\n\\end{enumerate}","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Inseparable residue fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07FH","source_file":"smoothing.tex","source_line":2551,"source_end_line":2574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2551-L2574","statement_sha256":"dcf7979c6a1ae4a2f8c0dfe698dbe8b7d38b8c0e477c4a1b2945232c3ede0100","origin":"The Stacks Project","memory_eligible":false,"source_rank":3765,"rank":3765,"depth":45,"x":847.714,"y":641.821,"cluster":"advanced-algebra"},{"id":"stacks:07FI","tag":"07FI","title":"Inseparable residue fields · Lemma 07FI","summary":"Let φ : k[y_1, …, y_m] → Lambda, n, q, p and k[y_1, …, y_m]_ p/ p^n → D → Lambda_ q/ q^n Lambda_ q be as in Lemma [Tag 07FH]. Then for any λ ∈ Lambda setminus q there exists an integer q > 0 and a factorization k[y_1, …, y_m]_ p/ p^n → D → D' → Lambda_ q/ q^n Lambda_ q such that D → D' is an essentially smooth map of local Artinian rings, the last arrow is flat, and λ^q is in D'.","statement_latex":"Let $\\varphi : k[y_1, \\ldots, y_m] \\to \\Lambda$, $n$, $\\mathfrak q$,\n$\\mathfrak p$ and\n$$\nk[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n \\to\nD \\to \\Lambda_\\mathfrak q/\\mathfrak q^n \\Lambda_\\mathfrak q\n$$\nbe as in Lemma \\ref{lemma-solution-modulo}. Then for any\n$\\lambda \\in \\Lambda \\setminus \\mathfrak q$\nthere exists an integer $q > 0$ and a factorization\n$$\nk[y_1, \\ldots, y_m]_\\mathfrak p/\\mathfrak p^n \\to\nD \\to D' \\to \\Lambda_\\mathfrak q/\\mathfrak q^n \\Lambda_\\mathfrak q\n$$\nsuch that $D \\to D'$ is an essentially smooth map of local Artinian rings,\nthe last arrow is flat, and $\\lambda^q$ is in $D'$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Inseparable residue fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07FI","source_file":"smoothing.tex","source_line":2623,"source_end_line":2640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2623-L2640","statement_sha256":"fb4a4a5f4b8b96a6eaaf711a3f3ee593ab07c44abf7533cecce704052b2024fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":3766,"rank":3766,"depth":46,"x":586.985,"y":803.38,"cluster":"advanced-algebra"},{"id":"stacks:07FJ","tag":"07FJ","title":"Inseparable residue fields · Lemma 07FJ","summary":"Let k → A → Lambda ⊃ q be as in Situation [Tag 07F7] where • k is a field of characteristic p > 0, • Lambda is Noetherian and geometrically regular over k, • q is minimal over h_A. Then k → A → Lambda ⊃ q can be resolved.","statement_latex":"Let $k \\to A \\to \\Lambda \\supset \\mathfrak q$ be as in\nSituation \\ref{situation-local} where\n\\begin{enumerate}\n\\item $k$ is a field of characteristic $p > 0$,\n\\item $\\Lambda$ is Noetherian and geometrically regular over $k$,\n\\item $\\mathfrak q$ is minimal over $\\mathfrak h_A$.\n\\end{enumerate}\nThen $k \\to A \\to \\Lambda \\supset \\mathfrak q$ can be resolved.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Inseparable residue fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07FJ","source_file":"smoothing.tex","source_line":2671,"source_end_line":2681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L2671-L2681","statement_sha256":"5c54dbde480c077308bedcbd9064cad7abcf88a0a62d4620a61c1b2d36ffc268","origin":"The Stacks Project","memory_eligible":false,"source_rank":3767,"rank":3767,"depth":47,"x":649.354,"y":536.17,"cluster":"advanced-algebra"},{"id":"stacks:07GC","tag":"07GC","title":"Popescu · Theorem 07GC","summary":"Any regular homomorphism of Noetherian rings is a filtered colimit of smooth ring maps.","statement_latex":"Any regular homomorphism of Noetherian rings is a filtered colimit\nof smooth ring maps.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The main theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GC","source_file":"smoothing.tex","source_line":3112,"source_end_line":3116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L3112-L3116","statement_sha256":"cc07d8f6252d2753a1a7770397555eced871a05ac789982ea9d5c0b588991b0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3768,"rank":3768,"depth":48,"x":818.333,"y":768.714,"cluster":"advanced-algebra"},{"id":"stacks:07QY","tag":"07QY","title":"The approximation property for G-rings · Theorem 07QY","summary":"Let R be a Noetherian local ring. Let f_1, …, f_m ∈ R[x_1, …, x_n]. Suppose that (a_1, …, a_n) ∈ (R^wedge)^n is a solution in R^wedge. If R is a henselian G-ring, then for every integer N there exists a solution (b_1, …, b_n) ∈ R^n in R such that a_i - b_i ∈ m^NR^wedge.","statement_latex":"Let $R$ be a Noetherian local ring. Let\n$f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$.\nSuppose that $(a_1, \\ldots, a_n) \\in (R^\\wedge)^n$ is a solution\nin $R^\\wedge$. If $R$ is a henselian G-ring, then for every integer\n$N$ there exists a solution $(b_1, \\ldots, b_n) \\in R^n$ in $R$ such that\n$a_i - b_i \\in \\mathfrak m^NR^\\wedge$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The approximation property for G-rings","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QY","source_file":"smoothing.tex","source_line":3171,"source_end_line":3179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L3171-L3179","statement_sha256":"fa417b1c009365efbf245ad6e0c01173be66fd3b000c9a2d0e9cdba03e18b71d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3769,"rank":3769,"depth":49,"x":506.567,"y":693.084,"cluster":"advanced-algebra"},{"id":"stacks:07QZ","tag":"07QZ","title":"The approximation property for G-rings · Theorem 07QZ","summary":"Let R be a Noetherian local ring. Let f_1, …, f_m ∈ R[x_1, …, x_n]. Suppose that (a_1, …, a_n) ∈ (R^wedge)^n is a solution. If R is a G-ring, then for every integer N there exist • an étale ring map R → R', • a maximal ideal m' ⊂ R' lying over m • a solution (b_1, …, b_n) ∈ (R')^n in R' such that kappa( m) = kappa( m') and a_i - b_i ∈ ( m')^NR^wedge.","statement_latex":"Let $R$ be a Noetherian local ring. Let\n$f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$.\nSuppose that $(a_1, \\ldots, a_n) \\in (R^\\wedge)^n$ is a solution.\nIf $R$ is a G-ring, then for every integer $N$ there exist\n\\begin{enumerate}\n\\item an \\'etale ring map $R \\to R'$,\n\\item a maximal ideal $\\mathfrak m' \\subset R'$ lying over $\\mathfrak m$\n\\item a solution $(b_1, \\ldots, b_n) \\in (R')^n$ in $R'$\n\\end{enumerate}\nsuch that $\\kappa(\\mathfrak m) = \\kappa(\\mathfrak m')$ and\n$a_i - b_i \\in (\\mathfrak m')^NR^\\wedge$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The approximation property for G-rings","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QZ","source_file":"smoothing.tex","source_line":3237,"source_end_line":3250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L3237-L3250","statement_sha256":"158c2472aef915f34f61f46724e2f9f032e26d5b37ef11f495663731258e25b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3770,"rank":3770,"depth":53,"x":797.425,"y":571.888,"cluster":"advanced-algebra"},{"id":"stacks:0CAR","tag":"0CAR","title":"The approximation property for G-rings · Lemma 0CAR","summary":"Let R be a Noetherian ring. Let p ⊂ R be a prime ideal. Let f_1, …, f_m ∈ R[x_1, …, x_n]. Suppose that (a_1, …, a_n) ∈ ((R_ p)^wedge)^n is a solution. If R_ p is a G-ring, then for every integer N there exist • an étale ring map R → R', • a prime ideal p' ⊂ R' lying over p • a solution (b_1, …, b_n) ∈ (R')^n in R' such that kappa( p) = kappa( p') and a_i - b_i ∈ ( p')^N(R'_ p')^wedge.","statement_latex":"Let $R$ be a Noetherian ring. Let $\\mathfrak p \\subset R$ be a prime ideal. Let\n$f_1, \\ldots, f_m \\in R[x_1, \\ldots, x_n]$.\nSuppose that $(a_1, \\ldots, a_n) \\in ((R_\\mathfrak p)^\\wedge)^n$ is a solution.\nIf $R_\\mathfrak p$ is a G-ring, then for every integer $N$ there exist\n\\begin{enumerate}\n\\item an \\'etale ring map $R \\to R'$,\n\\item a prime ideal $\\mathfrak p' \\subset R'$ lying over $\\mathfrak p$\n\\item a solution $(b_1, \\ldots, b_n) \\in (R')^n$ in $R'$\n\\end{enumerate}\nsuch that $\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$ and\n$a_i - b_i \\in (\\mathfrak p')^N(R'_{\\mathfrak p'})^\\wedge$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"The approximation property for G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAR","source_file":"smoothing.tex","source_line":3346,"source_end_line":3359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L3346-L3359","statement_sha256":"a5b830849016ea2f72a68b813420bced602c961abb2fe2e6b8c398bc04a7cabd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3771,"rank":3771,"depth":54,"x":680.354,"y":826.424,"cluster":"advanced-algebra"},{"id":"stacks:0AH5","tag":"0AH5","title":"Approximation for henselian pairs · Lemma 0AH5","summary":"Approximation for henselian pairs. Let (A, I) be a henselian pair with A Noetherian. Let A^wedge be the I-adic completion of A. Assume at least one of the following conditions holds • A → A^wedge is a regular ring map, • A is a Noetherian G-ring, or • (A, I) is the henselization (More on Algebra, Lemma [Tag 0A02]) of a pair (B, J) where B is a Noetherian G-ring. Given f_1, …, f_m ∈ A[x_1, …, x_n] and hata_1, …, hata_n ∈ A^wedge such that f_j(hata_1, …, hata_n) = 0 for j =…","statement_latex":"\\begin{slogan}\nApproximation for henselian pairs.\n\\end{slogan}\nLet $(A, I)$ be a henselian pair with $A$ Noetherian.\nLet $A^\\wedge$ be the $I$-adic completion\nof $A$. Assume at least one of the following\nconditions holds\n\\begin{enumerate}\n\\item $A \\to A^\\wedge$ is a regular ring map,\n\\item $A$ is a Noetherian G-ring, or\n\\item $(A, I)$ is the henselization\n(More on Algebra, Lemma \\ref{more-algebra-lemma-henselization})\nof a pair $(B, J)$ where $B$ is a Noetherian G-ring.\n\\end{enumerate}\nGiven $f_1, \\ldots, f_m \\in A[x_1, \\ldots, x_n]$\nand $\\hat{a}_1, \\ldots, \\hat{a}_n \\in A^\\wedge$ such that\n$f_j(\\hat{a}_1, \\ldots, \\hat{a}_n) = 0$\nfor $j = 1, \\ldots, m$, for every $N \\geq 1$ there exist\n$a_1, \\ldots, a_n \\in A$ such that\n$\\hat{a}_i - a_i \\in I^N$ and such that $f_j(a_1, \\ldots, a_n) = 0$\nfor $j = 1, \\ldots, m$.","area":"Advanced Algebra","chapter":"Smoothing Ring Maps","chapter_id":"smoothing","section":"Approximation for henselian pairs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AH5","source_file":"smoothing.tex","source_line":3388,"source_end_line":3411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/smoothing.tex#L3388-L3411","statement_sha256":"99ba6341c0ec9fc810983d98230d0fecbed2c9d8216599f7250e6634a1b38371","origin":"The Stacks Project","memory_eligible":false,"source_rank":3772,"rank":3772,"depth":51,"x":561.929,"y":572.176,"cluster":"advanced-algebra"},{"id":"stacks:01AG","tag":"01AG","title":"The abelian category of sheaves of modules · Lemma 01AG","summary":"Let (X, O_X) be a ringed space. The category Mod(O_X) is an abelian category. Moreover a complex F → G → H is exact at G if and only if for all x ∈ X the complex F_x → G_x → H_x is exact at G_x.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. The category\n$\\textit{Mod}(\\mathcal{O}_X)$ is an abelian category. Moreover\na complex\n$$\n\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}\n$$\nis exact at $\\mathcal{G}$ if and only if for all $x \\in X$ the\ncomplex\n$$\n\\mathcal{F}_x \\to \\mathcal{G}_x \\to \\mathcal{H}_x\n$$\nis exact at $\\mathcal{G}_x$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The abelian category of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AG","source_file":"modules.tex","source_line":148,"source_end_line":162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L148-L162","statement_sha256":"796551528b3f3d0659cff10715a2b5dcd43c8fb1605e3b845087e7415dc6739e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3773,"rank":3773,"depth":1,"x":1253.92,"y":234.796,"cluster":"sheaves-sites"},{"id":"stacks:01AH","tag":"01AH","title":"The abelian category of sheaves of modules · Lemma 01AH","summary":"Let (X, O_X) be a ringed space. • All limits exist in Mod(O_X). Limits are the same as the corresponding limits of presheaves of O_X-modules (i.e., commute with taking sections over opens). • All colimits exist in Mod(O_X). Colimits are the sheafification of the corresponding colimit in the category of presheaves. Taking colimits commutes with taking stalks. • Filtered colimits are exact. • Finite direct sums are the same as the corresponding finite direct sums of…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\n\\begin{enumerate}\n\\item All limits exist in $\\textit{Mod}(\\mathcal{O}_X)$.\nLimits are the same as the corresponding limits of presheaves of\n$\\mathcal{O}_X$-modules (i.e., commute with taking\nsections over opens).\n\\item All colimits exist in $\\textit{Mod}(\\mathcal{O}_X)$.\nColimits are the sheafification of the corresponding colimit in\nthe category of presheaves. Taking colimits commutes with taking\nstalks.\n\\item Filtered colimits are exact.\n\\item Finite direct sums are the same as the corresponding\nfinite direct sums of presheaves of $\\mathcal{O}_X$-modules.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The abelian category of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AH","source_file":"modules.tex","source_line":203,"source_end_line":219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L203-L219","statement_sha256":"56ab22b4a767368faa5afed55069a32e17af54e2d43c588ebf246136263dd05f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3774,"rank":3774,"depth":3,"x":1026.598,"y":279.478,"cluster":"sheaves-sites"},{"id":"stacks:01AJ","tag":"01AJ","title":"The abelian category of sheaves of modules · Lemma 01AJ","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. • The functor f_* : Mod(O_X) → Mod(O_Y) is left exact. In fact it commutes with all limits. • The functor f^* : Mod(O_Y) → Mod(O_X) is right exact. In fact it commutes with all colimits. • Pullback f^-1 : Ab(Y) → Ab(X) on abelian sheaves is exact.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\n\\begin{enumerate}\n\\item The functor\n$f_* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_Y)$\nis left exact. In fact it commutes with all limits.\n\\item The functor\n$f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$\nis right exact. In fact it commutes with all colimits.\n\\item Pullback $f^{-1} : \\textit{Ab}(Y) \\to \\textit{Ab}(X)$\non abelian sheaves is exact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The abelian category of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AJ","source_file":"modules.tex","source_line":252,"source_end_line":266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L252-L266","statement_sha256":"a51a31699852cbc2e844d7dad2c35627a05fbfb240ea278fa199e0372af28102","origin":"The Stacks Project","memory_eligible":false,"source_rank":3775,"rank":3775,"depth":5,"x":1158.452,"y":117.343,"cluster":"sheaves-sites"},{"id":"stacks:01AK","tag":"01AK","title":"The abelian category of sheaves of modules · Lemma 01AK","summary":"Let j : U → X be an open immersion of topological spaces. The functor j_! : Ab(U) → Ab(X) is exact.","statement_latex":"Let $j : U \\to X$ be an open immersion of topological spaces.\nThe functor $j_! : \\textit{Ab}(U) \\to \\textit{Ab}(X)$\nis exact.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The abelian category of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AK","source_file":"modules.tex","source_line":280,"source_end_line":285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L280-L285","statement_sha256":"268314c3ccb2b9f41e73c0183df85954db1a6d018bcfe46cdcf0aa026cd6aeb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3776,"rank":3776,"depth":2,"x":1191.649,"y":311.955,"cluster":"sheaves-sites"},{"id":"stacks:01AI","tag":"01AI","title":"The abelian category of sheaves of modules · Lemma 01AI","summary":"Let (X, O_X) be a ringed space. Let I be a set. For i ∈ I, let F_i be a sheaf of O_X-modules. For U ⊂ X quasi-compact open the map bigoplus_i ∈ I F_i(U) → (bigoplus_i ∈ I F_i)(U) is bijective.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $I$ be a set. For $i \\in I$, let $\\mathcal{F}_i$\nbe a sheaf of $\\mathcal{O}_X$-modules.\nFor $U \\subset X$ quasi-compact open the map\n$$\n\\bigoplus\\nolimits_{i \\in I} \\mathcal{F}_i(U)\n\\longrightarrow\n\\left(\\bigoplus\\nolimits_{i \\in I} \\mathcal{F}_i\\right)(U)\n$$\nis bijective.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The abelian category of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AI","source_file":"modules.tex","source_line":292,"source_end_line":304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L292-L304","statement_sha256":"dccc02e0154de7ea440c38259c980ee1416ba59ba78a93edb15878f071553515","origin":"The Stacks Project","memory_eligible":false,"source_rank":3777,"rank":3777,"depth":4,"x":1010.447,"y":187.135,"cluster":"sheaves-sites"},{"id":"stacks:01AM","tag":"01AM","title":"Sections of sheaves of modules · Definition 01AM","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. We say that F is generated by global sections if there exist a set I, and global sections s_i ∈ Γ(X, F), i ∈ I such that the map bigoplus_i ∈ I O_X → F which is the map associated to s_i on the summand corresponding to i, is surjective. In this case we say that the sections s_i generate F.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nWe say that $\\mathcal{F}$ is {\\it generated by global\nsections} if there exist a set $I$, and\nglobal sections $s_i \\in \\Gamma(X, \\mathcal{F})$, $i \\in I$\nsuch that the map\n$$\n\\bigoplus\\nolimits_{i \\in I}\n\\mathcal{O}_X \\longrightarrow \\mathcal{F}\n$$\nwhich is the map associated to $s_i$ on the summand corresponding to $i$,\nis surjective. In this case we say that the sections $s_i$\n{\\it generate} $\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Sections of sheaves of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AM","source_file":"modules.tex","source_line":340,"source_end_line":355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L340-L355","statement_sha256":"cd4ef3a4f4c816f2c37c471604d28ac4e3d30377e9f9aa686573a1478892c0c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3778,"rank":3778,"depth":0,"x":1244.725,"y":176.344,"cluster":"sheaves-sites"},{"id":"stacks:01AN","tag":"01AN","title":"Sections of sheaves of modules · Lemma 01AN","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. Let I be a set. Let s_i ∈ Γ(X, F), i ∈ I be global sections. The sections s_i generate F if and only if for all x∈ X the elements s_i, x ∈ F_x generate the O_X, x-module F_x.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nLet $I$ be a set. Let\n$s_i \\in \\Gamma(X, \\mathcal{F})$, $i \\in I$\nbe global sections. The sections $s_i$ generate\n$\\mathcal{F}$ if and only if for all $x\\in X$ the\nelements $s_{i, x} \\in \\mathcal{F}_x$ generate\nthe $\\mathcal{O}_{X, x}$-module $\\mathcal{F}_x$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Sections of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AN","source_file":"modules.tex","source_line":364,"source_end_line":374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L364-L374","statement_sha256":"d27a221655afffc4a06725035c69608e5e10336ccba088b95841cd2252d590d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3779,"rank":3779,"depth":0,"x":1080.451,"y":317.408,"cluster":"sheaves-sites"},{"id":"stacks:01AO","tag":"01AO","title":"Sections of sheaves of modules · Lemma 01AO","summary":"The tensor product of globally generated sheaves of modules is globally generated. Let (X, O_X) be a ringed space. Let F, G be sheaves of O_X-modules. If F and G are generated by global sections then so is F ⊗_O_X G.","statement_latex":"\\begin{slogan}\nThe tensor product of globally generated sheaves of modules is\nglobally generated.\n\\end{slogan}\nLet $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}$ and $\\mathcal{G}$ are generated by global sections\nthen so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Sections of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AO","source_file":"modules.tex","source_line":380,"source_end_line":390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L380-L390","statement_sha256":"96afdc67716b2ddbf47cea86d93bfe471ae9935f6340bd92d8cafe827f2c8b8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3780,"rank":3780,"depth":0,"x":1088.154,"y":119.936,"cluster":"sheaves-sites"},{"id":"stacks:01AP","tag":"01AP","title":"Sections of sheaves of modules · Lemma 01AP","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. Let I be a set. Let s_i, i ∈ I be a collection of local sections of F, i.e., s_i ∈ F(U_i) for some opens U_i ⊂ X. There exists a unique smallest subsheaf of O_X-modules G such that each s_i corresponds to a local section of G.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nLet $I$ be a set. Let $s_i$, $i \\in I$ be a collection\nof local sections of $\\mathcal{F}$, i.e., $s_i \\in \\mathcal{F}(U_i)$\nfor some opens $U_i \\subset X$. There exists a unique smallest\nsubsheaf of $\\mathcal{O}_X$-modules $\\mathcal{G}$ such\nthat each $s_i$ corresponds to a local section of\n$\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Sections of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AP","source_file":"modules.tex","source_line":396,"source_end_line":406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L396-L406","statement_sha256":"1558a0716ee302c8cd5d8aa38102e1af314ad4c5c231d0d347c97101d7073ea1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3781,"rank":3781,"depth":1,"x":1241.46,"y":270.102,"cluster":"sheaves-sites"},{"id":"stacks:01AQ","tag":"01AQ","title":"Sections of sheaves of modules · Definition 01AQ","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. Given a set I, and local sections s_i, i ∈ I of F we say that the subsheaf G of Lemma [Tag 01AP] above is the subsheaf generated by the s_i.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nGiven a set $I$, and\nlocal sections $s_i$, $i \\in I$ of $\\mathcal{F}$\nwe say that the subsheaf $\\mathcal{G}$ of\nLemma \\ref{lemma-generated-by-local-sections}\nabove is the {\\it subsheaf generated by the $s_i$}.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Sections of sheaves of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AQ","source_file":"modules.tex","source_line":428,"source_end_line":437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L428-L437","statement_sha256":"813e0dcbdc745d67dffd41459167823e5daa7ea9720b2dbe19fb851f53f99db1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3782,"rank":3782,"depth":2,"x":1007.373,"y":246.333,"cluster":"sheaves-sites"},{"id":"stacks:01AR","tag":"01AR","title":"Sections of sheaves of modules · Lemma 01AR","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. Given a set I, and local sections s_i, i ∈ I of F. Let G be the subsheaf generated by the s_i and let x∈ X. Then G_x is the O_X, x-submodule of F_x generated by the elements s_i, x for those i such that s_i is defined at x.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nGiven a set $I$, and\nlocal sections $s_i$, $i \\in I$ of $\\mathcal{F}$.\nLet $\\mathcal{G}$ be the subsheaf generated by the\n$s_i$ and let $x\\in X$.\nThen $\\mathcal{G}_x$ is the $\\mathcal{O}_{X, x}$-submodule of\n$\\mathcal{F}_x$ generated by the elements $s_{i, x}$\nfor those $i$ such that $s_i$ is defined at $x$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Sections of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AR","source_file":"modules.tex","source_line":439,"source_end_line":450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L439-L450","statement_sha256":"49aa2cf9be53a9ccb2b18aecc93a59991615e60cb40d098de4ab10d195be2483","origin":"The Stacks Project","memory_eligible":false,"source_rank":3783,"rank":3783,"depth":2,"x":1199.331,"y":130.894,"cluster":"sheaves-sites"},{"id":"stacks:01AT","tag":"01AT","title":"Supports of modules and sections · Definition 01AT","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. • The support of F is the set of points x ∈ X such that F_x not = 0. • We denote Supp(F) the support of F. • Let s ∈ Γ(X, F) be a global section. The support of s is the set of points x ∈ X such that the image s_x ∈ F_x of s is not zero.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item The {\\it support of $\\mathcal{F}$} is the set of\npoints $x \\in X$ such that $\\mathcal{F}_x \\not = 0$.\n\\item We denote $\\text{Supp}(\\mathcal{F})$ the support of $\\mathcal{F}$.\n\\item Let $s \\in \\Gamma(X, \\mathcal{F})$ be a global section.\nThe {\\it support of $s$} is the set of points $x \\in X$\nsuch that the image $s_x \\in \\mathcal{F}_x$ of $s$ is\nnot zero.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Supports of modules and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AT","source_file":"modules.tex","source_line":469,"source_end_line":482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L469-L482","statement_sha256":"1d88caa493e8e398d5ed1c08a8540d490401c30a42dc3337081a80c0fcd40cb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3784,"rank":3784,"depth":0,"x":1150.558,"y":325.172,"cluster":"sheaves-sites"},{"id":"stacks:01AU","tag":"01AU","title":"Supports of modules and sections · Lemma 01AU","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. Let U ⊂ X open. • The support of s ∈ F(U) is closed in U. • The support of fs is contained in the intersections of the supports of f ∈ O_X(U) and s ∈ F(U). • The support of s + s' is contained in the union of the supports of s, s' ∈ F(U). • The support of F is the union of the supports of all local sections of F. • If φ : F → G is a morphism of O_X-modules, then the support of φ(s) is contained in the…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nLet $U \\subset X$ open.\n\\begin{enumerate}\n\\item The support of $s \\in \\mathcal{F}(U)$ is closed in $U$.\n\\item The support of $fs$ is contained in the intersections\nof the supports of $f \\in \\mathcal{O}_X(U)$ and $s \\in \\mathcal{F}(U)$.\n\\item The support of $s + s'$ is contained in the union of\nthe supports of $s, s' \\in \\mathcal{F}(U)$.\n\\item The support of $\\mathcal{F}$ is the union of the supports\nof all local sections of $\\mathcal{F}$.\n\\item If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of\n$\\mathcal{O}_X$-modules, then the support of $\\varphi(s)$ is\ncontained in the support of $s \\in \\mathcal{F}(U)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Supports of modules and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AU","source_file":"modules.tex","source_line":489,"source_end_line":506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L489-L506","statement_sha256":"71c5f476e0301dd31a471952372f92adba2115213111328cff1691c339333438","origin":"The Stacks Project","memory_eligible":false,"source_rank":3785,"rank":3785,"depth":0,"x":1030.145,"y":154.034,"cluster":"sheaves-sites"},{"id":"stacks:01AV","tag":"01AV","title":"Supports of modules and sections · Lemma 01AV","summary":"Let X be a topological space. The support of a sheaf of rings is closed.","statement_latex":"Let $X$ be a topological space.\nThe support of a sheaf of rings is closed.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Supports of modules and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AV","source_file":"modules.tex","source_line":530,"source_end_line":534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L530-L534","statement_sha256":"96b6f05297361829b1f0d2fac85d21a4c5dcc7775191cc41246a6c0c51beaf09","origin":"The Stacks Project","memory_eligible":false,"source_rank":3786,"rank":3786,"depth":0,"x":1256.831,"y":211.973,"cluster":"sheaves-sites"},{"id":"stacks:01AX","tag":"01AX","title":"Closed immersions and abelian sheaves · Lemma 01AX","summary":"Let X be a topological space. Let Z ⊂ X be a closed subset. Denote i : Z → X the inclusion map. The functor i_* : Ab(Z) → Ab(X) is exact, fully faithful, with essential image exactly those abelian sheaves whose support is contained in Z. The functor i^-1 is a left inverse to i_*.","statement_latex":"Let $X$ be a topological space. Let $Z \\subset X$ be a closed subset.\nDenote $i : Z \\to X$ the inclusion map. The functor\n$$\ni_* : \\textit{Ab}(Z) \\longrightarrow \\textit{Ab}(X)\n$$\nis exact, fully faithful, with essential image exactly those\nabelian sheaves whose support is contained in $Z$. The functor $i^{-1}$\nis a left inverse to $i_*$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Closed immersions and abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AX","source_file":"modules.tex","source_line":556,"source_end_line":566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L556-L566","statement_sha256":"acc0c520afdbc6c000590e2d3e8b9964004d321d133e9bb41861a816ff91833d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3787,"rank":3787,"depth":2,"x":1042.828,"y":297.979,"cluster":"sheaves-sites"},{"id":"stacks:01AZ","tag":"01AZ","title":"Closed immersions and abelian sheaves · Lemma 01AZ","summary":"Let i : Z → X be the inclusion of a closed subset into the topological space X. The functor Ab(X) → Ab(Z), F ↦ H_Z(F) of Remark [Tag 01AY] is a right adjoint to i_* : Ab(Z) → Ab(X). In particular i_* commutes with arbitrary colimits.","statement_latex":"Let $i : Z \\to X$ be the inclusion of a closed subset into the\ntopological space $X$. The functor $\\textit{Ab}(X) \\to \\textit{Ab}(Z)$,\n$\\mathcal{F} \\mapsto \\mathcal{H}_Z(\\mathcal{F})$ of\nRemark \\ref{remark-sections-support-in-closed}\nis a right adjoint to $i_* : \\textit{Ab}(Z) \\to \\textit{Ab}(X)$.\nIn particular $i_*$ commutes with arbitrary colimits.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Closed immersions and abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01AZ","source_file":"modules.tex","source_line":608,"source_end_line":616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L608-L616","statement_sha256":"114e7b4183a3513ac3727e99519145fcb4dd5de268bbe3750d3f65a7386c25ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":3788,"rank":3788,"depth":0,"x":1131.572,"y":112.909,"cluster":"sheaves-sites"},{"id":"stacks:02UT","tag":"02UT","title":"A canonical exact sequence · Lemma 02UT","summary":"Let X be a topological space. Let U ⊂ X be an open subset with complement Z ⊂ X. Denote j : U → X the open immersion and i : Z → X the closed immersion. For any sheaf of abelian groups F on X the adjunction mappings j_!j^-1F → F and F → i_*i^-1F give a short exact sequence 0 → j_!j^-1F → F → i_*i^-1F → 0 of sheaves of abelian groups. For any morphism φ : F → G of abelian sheaves on X we obtain a morphism of short exact sequences xymatrix 0 ar[r] & j_!j^-1F ar[r] ar[d] & F…","statement_latex":"Let $X$ be a topological space.\nLet $U \\subset X$ be an open subset with complement $Z \\subset X$.\nDenote $j : U \\to X$ the open immersion and\n$i : Z \\to X$ the closed immersion.\nFor any sheaf of abelian groups $\\mathcal{F}$ on $X$\nthe adjunction mappings $j_{!}j^{-1}\\mathcal{F} \\to \\mathcal{F}$ and\n$\\mathcal{F} \\to i_*i^{-1}\\mathcal{F}$ give a short exact\nsequence\n$$\n0 \\to j_{!}j^{-1}\\mathcal{F} \\to \\mathcal{F} \\to i_*i^{-1}\\mathcal{F} \\to 0\n$$\nof sheaves of abelian groups. For any morphism\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of abelian sheaves on $X$\nwe obtain a morphism of short exact sequences\n$$\n\\xymatrix{\n0 \\ar[r] &\nj_{!}j^{-1}\\mathcal{F} \\ar[r] \\ar[d] &\n\\mathcal{F} \\ar[r] \\ar[d] &\ni_*i^{-1}\\mathcal{F} \\ar[r] \\ar[d] &\n0 \\\\\n0 \\ar[r] &\nj_{!}j^{-1}\\mathcal{G} \\ar[r] &\n\\mathcal{G} \\ar[r] &\ni_*i^{-1}\\mathcal{G} \\ar[r] &\n0\n}\n$$","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"A canonical exact sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UT","source_file":"modules.tex","source_line":656,"source_end_line":686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L656-L686","statement_sha256":"419c17c8e218c996fdc58a0e4fe770d36b10d6fddb10a37d491f4ccb3ccee9fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3789,"rank":3789,"depth":2,"x":1215.061,"y":299.954,"cluster":"sheaves-sites"},{"id":"stacks:01B2","tag":"01B2","title":"Modules locally generated by sections · Definition 01B2","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. We say that F is locally generated by sections if for every x ∈ X there exists an open neighbourhood U of x such that F|_U is globally generated as a sheaf of O_U-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nWe say that $\\mathcal{F}$ is {\\it locally generated by sections}\nif for every $x \\in X$ there exists an open\nneighbourhood $U$ of $x$ such that $\\mathcal{F}|_U$\nis globally generated as a sheaf of $\\mathcal{O}_U$-modules.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules locally generated by sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01B2","source_file":"modules.tex","source_line":727,"source_end_line":735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L727-L735","statement_sha256":"9f7054fe01cdec442a0590251d4823a2774b9b9fb529b76b77f6761cf975afe5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3790,"rank":3790,"depth":0,"x":1002.83,"y":209.296,"cluster":"sheaves-sites"},{"id":"stacks:01B3","tag":"01B3","title":"Modules locally generated by sections · Lemma 01B3","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The pullback f^*G is locally generated by sections if G is locally generated by sections.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\nThe pullback $f^*\\mathcal{G}$ is locally generated by sections\nif $\\mathcal{G}$ is locally generated by sections.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules locally generated by sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01B3","source_file":"modules.tex","source_line":748,"source_end_line":754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L748-L754","statement_sha256":"2b677b2c4d9a78e1178a9baeed782c364ac3524ac82ab2449ce53e90a132b9d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3791,"rank":3791,"depth":6,"x":1232.502,"y":155.658,"cluster":"sheaves-sites"},{"id":"stacks:01B5","tag":"01B5","title":"Modules of finite type · Definition 01B5","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. We say that F is of finite type if for every x ∈ X there exists an open neighbourhood U such that F|_U is generated by finitely many sections.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nWe say that $\\mathcal{F}$ is of {\\it finite type}\nif for every $x \\in X$ there exists an open\nneighbourhood $U$ such that $\\mathcal{F}|_U$\nis generated by finitely many sections.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01B5","source_file":"modules.tex","source_line":809,"source_end_line":817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L809-L817","statement_sha256":"7a64c1c7deb98255f3dcc707ad733933fe308b9bb760a62b629d8fde82faf367","origin":"The Stacks Project","memory_eligible":false,"source_rank":3792,"rank":3792,"depth":0,"x":1106.131,"y":325.732,"cluster":"sheaves-sites"},{"id":"stacks:01B6","tag":"01B6","title":"Modules of finite type · Lemma 01B6","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The pullback f^*G of a finite type O_Y-module is a finite type O_X-module.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\nThe pullback $f^*\\mathcal{G}$ of a finite type\n$\\mathcal{O}_Y$-module is a finite type $\\mathcal{O}_X$-module.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01B6","source_file":"modules.tex","source_line":819,"source_end_line":825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L819-L825","statement_sha256":"0be5fda20d6cf226b198b3d4ce121fd5582197e9ab2abd0a522135733315f4af","origin":"The Stacks Project","memory_eligible":false,"source_rank":3793,"rank":3793,"depth":7,"x":1062.495,"y":128.384,"cluster":"sheaves-sites"},{"id":"stacks:01B7","tag":"01B7","title":"Modules of finite type · Lemma 01B7","summary":"Let X be a ringed space. The image of a morphism of O_X-modules of finite type is of finite type. Let 0 → F_1 → F_2 → F_3 → 0 be a short exact sequence of O_X-modules. If F_1 and F_3 are of finite type, so is F_2.","statement_latex":"Let $X$ be a ringed space.\nThe image of a morphism of $\\mathcal{O}_X$-modules of finite\ntype is of finite type.\nLet\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nbe a short exact sequence of $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}_1$ and $\\mathcal{F}_3$ are of finite type,\nso is $\\mathcal{F}_2$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01B7","source_file":"modules.tex","source_line":854,"source_end_line":864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L854-L864","statement_sha256":"bb191325b046dce9721a21e24de33d9bc622586af736f4720fbd181f5e8babd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3794,"rank":3794,"depth":0,"x":1253.597,"y":249.287,"cluster":"sheaves-sites"},{"id":"stacks:01B8","tag":"01B8","title":"Modules of finite type · Lemma 01B8","summary":"Let X be a ringed space. Let φ : G → F be a homomorphism of O_X-modules. Let x ∈ X. Assume F of finite type and the map on stalks φ_x : G_x → F_x surjective. Then there exists an open neighbourhood x ∈ U ⊂ X such that φ|_U is surjective.","statement_latex":"Let $X$ be a ringed space.\nLet $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism\nof $\\mathcal{O}_X$-modules.\nLet $x \\in X$. Assume $\\mathcal{F}$ of finite type and\nthe map on stalks\n$\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ surjective.\nThen there exists an open neighbourhood\n$x \\in U \\subset X$ such that $\\varphi|_U$ is surjective.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01B8","source_file":"modules.tex","source_line":875,"source_end_line":885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L875-L885","statement_sha256":"1547093c994f5a8cb824ffb0d425e5e03954209c1021187d65901b7a82707b2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3795,"rank":3795,"depth":0,"x":1015.175,"y":268.593,"cluster":"sheaves-sites"},{"id":"stacks:01B9","tag":"01B9","title":"Modules of finite type · Lemma 01B9","summary":"Let X be a ringed space. Let F be an O_X-module. Let x ∈ X. Assume F of finite type and F_x = 0. Then there exists an open neighbourhood x ∈ U ⊂ X such that F|_U is zero.","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nLet $x \\in X$.\nAssume $\\mathcal{F}$ of finite type and $\\mathcal{F}_x = 0$.\nThen there exists an open neighbourhood\n$x \\in U \\subset X$ such that $\\mathcal{F}|_U$ is zero.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01B9","source_file":"modules.tex","source_line":896,"source_end_line":904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L896-L904","statement_sha256":"b3f5ff7116e038879fd132c186eeeb729c1af706c45611d47ccd3c063c7207a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3796,"rank":3796,"depth":1,"x":1175.646,"y":118.896,"cluster":"sheaves-sites"},{"id":"stacks:01BA","tag":"01BA","title":"Modules of finite type · Lemma 01BA","summary":"Over any ringed space, sheaves of modules of finite type have closed support. Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. If F is of finite type then support of F is closed.","statement_latex":"\\begin{slogan}\nOver any ringed space, sheaves of modules of finite type have closed support.\n\\end{slogan}\nLet $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}$ is of finite type then support of $\\mathcal{F}$ is closed.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BA","source_file":"modules.tex","source_line":912,"source_end_line":920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L912-L920","statement_sha256":"281621c71e22727fe19b37b46e20fe643465a33714766f0654b39263a59c88b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3797,"rank":3797,"depth":2,"x":1177.702,"y":320.57,"cluster":"sheaves-sites"},{"id":"stacks:01BB","tag":"01BB","title":"Modules of finite type · Lemma 01BB","summary":"Let X be a ringed space. Let I be a preordered set and let (F_i, f_ii') be a system over I consisting of sheaves of O_X-modules (see Categories, Section [Tag 002Z]). Let F = colim F_i be the colimit. Assume (a) I is directed, (b) F is a finite type O_X-module, and (c) X is quasi-compact. Then there exists an i such that F_i → F is surjective. If the transition maps f_ii' are injective then we conclude that F = F_i for some i ∈ I.","statement_latex":"Let $X$ be a ringed space. Let $I$ be a preordered set and\nlet $(\\mathcal{F}_i, f_{ii'})$ be a system over $I$ consisting of sheaves\nof $\\mathcal{O}_X$-modules (see\nCategories, Section \\ref{categories-section-posets-limits}).\nLet $\\mathcal{F} = \\colim \\mathcal{F}_i$ be the colimit. Assume\n(a) $I$ is directed,\n(b) $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, and\n(c) $X$ is quasi-compact. Then there exists an $i$ such that\n$\\mathcal{F}_i \\to \\mathcal{F}$ is surjective.\nIf the transition maps $f_{ii'}$ are injective\nthen we conclude that $\\mathcal{F} = \\mathcal{F}_i$ for some $i \\in I$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BB","source_file":"modules.tex","source_line":926,"source_end_line":939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L926-L939","statement_sha256":"89606292a39bb3ba911a19e42b6226c63e4c1554765fb46c7fdcb497556e2546","origin":"The Stacks Project","memory_eligible":false,"source_rank":3798,"rank":3798,"depth":0,"x":1013.815,"y":172.854,"cluster":"sheaves-sites"},{"id":"stacks:01BC","tag":"01BC","title":"Modules of finite type · Lemma 01BC","summary":"Let X be a ringed space. There exists a set of O_X-modules (F_i)_i ∈ I of finite type such that each finite type O_X-module on X is isomorphic to exactly one of the F_i.","statement_latex":"Let $X$ be a ringed space.\nThere exists a set of $\\mathcal{O}_X$-modules\n$\\{\\mathcal{F}_i\\}_{i \\in I}$ of finite type\nsuch that each finite type $\\mathcal{O}_X$-module\non $X$ is isomorphic to exactly one of the $\\mathcal{F}_i$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BC","source_file":"modules.tex","source_line":957,"source_end_line":964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L957-L964","statement_sha256":"460b18d073bf5e57343f34580cc2a16a3e83f8793d5452ddb1142217debcbaf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3799,"rank":3799,"depth":0,"x":1253.73,"y":188.803,"cluster":"sheaves-sites"},{"id":"stacks:01BE","tag":"01BE","title":"Quasi-coherent modules · Definition 01BE","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. We say that F is a quasi-coherent sheaf of O_X-modules if for every point x ∈ X there exists an open neighbourhood x∈ U ⊂ X such that F|_U is isomorphic to the cokernel of a map bigoplus_j ∈ J O_U → bigoplus_i ∈ I O_U The category of quasi-coherent O_X-modules is denoted QCoh(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nWe say that $\\mathcal{F}$ is a {\\it quasi-coherent\nsheaf of $\\mathcal{O}_X$-modules} if for every\npoint $x \\in X$ there exists an open neighbourhood\n$x\\in U \\subset X$ such that $\\mathcal{F}|_U$\nis isomorphic to the cokernel of a map\n$$\n\\bigoplus\\nolimits_{j \\in J}\n\\mathcal{O}_U\n\\longrightarrow\n\\bigoplus\\nolimits_{i \\in I}\n\\mathcal{O}_U\n$$\nThe category of quasi-coherent $\\mathcal{O}_X$-modules\nis denoted $\\QCoh(\\mathcal{O}_X)$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Quasi-coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BE","source_file":"modules.tex","source_line":1020,"source_end_line":1038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1020-L1038","statement_sha256":"5f3f71f2310d20e7b5879b322081a925151995ecdbe8fa3dac5270950bbd4a99","origin":"The Stacks Project","memory_eligible":false,"source_rank":3800,"rank":3800,"depth":0,"x":1063.775,"y":313.32,"cluster":"sheaves-sites"},{"id":"stacks:01BF","tag":"01BF","title":"Quasi-coherent modules · Lemma 01BF","summary":"Let (X, O_X) be a ringed space. The direct sum of two quasi-coherent O_X-modules is a quasi-coherent O_X-module.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nThe direct sum of two quasi-coherent $\\mathcal{O}_X$-modules is\na quasi-coherent $\\mathcal{O}_X$-module.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BF","source_file":"modules.tex","source_line":1066,"source_end_line":1071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1066-L1071","statement_sha256":"3d339595830d5fe0919b744fe3c4a7a9bb1769d8cc5bc8c7a076181eed34570f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3801,"rank":3801,"depth":0,"x":1103.758,"y":113.487,"cluster":"sheaves-sites"},{"id":"stacks:01BG","tag":"01BG","title":"Quasi-coherent modules · Lemma 01BG","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The pullback f^*G of a quasi-coherent O_Y-module is quasi-coherent.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\nThe pullback $f^*\\mathcal{G}$ of a quasi-coherent\n$\\mathcal{O}_Y$-module is quasi-coherent.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BG","source_file":"modules.tex","source_line":1085,"source_end_line":1091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1085-L1091","statement_sha256":"eea4edc759c835b2b640add34c16bd01f03f5dcbcbe4856e61b07e757e299339","origin":"The Stacks Project","memory_eligible":false,"source_rank":3802,"rank":3802,"depth":7,"x":1235.126,"y":283.724,"cluster":"sheaves-sites"},{"id":"stacks:01BH","tag":"01BH","title":"Quasi-coherent modules · Lemma 01BH","summary":"Let (X, O_X) be ringed space. Let α : R → Γ(X, O_X) be a ring homomorphism from a ring R into the ring of global sections on X. Let M be an R-module. The following three constructions give canonically isomorphic sheaves of O_X-modules: • Let π : (X, O_X) → ((*), R) be the morphism of ringed spaces with π : X → (*) the unique map and with π-map π^sharp the given map α : R → Γ(X, O_X). Set F_1 = π^*M. • Choose a presentation bigoplus_j ∈ J R → bigoplus_i ∈ I R → M → 0. Set…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be ringed space.\nLet $\\alpha : R \\to \\Gamma(X, \\mathcal{O}_X)$ be a ring homomorphism from\na ring $R$ into the ring of global sections on $X$.\nLet $M$ be an $R$-module.\nThe following three constructions give canonically isomorphic\nsheaves of $\\mathcal{O}_X$-modules:\n\\begin{enumerate}\n\\item Let $\\pi : (X, \\mathcal{O}_X) \\longrightarrow (\\{*\\}, R)$\nbe the morphism of ringed spaces with $\\pi : X \\to \\{*\\}$\nthe unique map and with $\\pi$-map $\\pi^\\sharp$ the given map\n$\\alpha : R \\to \\Gamma(X, \\mathcal{O}_X)$. Set $\\mathcal{F}_1 = \\pi^*M$.\n\\item Choose a presentation\n$\\bigoplus_{j \\in J} R \\to \\bigoplus_{i \\in I} R \\to M \\to 0$.\nSet\n$$\n\\mathcal{F}_2 = \\Coker\\left(\n\\bigoplus\\nolimits_{j \\in J} \\mathcal{O}_X\n\\to\n\\bigoplus\\nolimits_{i \\in I} \\mathcal{O}_X\n\\right).\n$$\nHere the map on the component $\\mathcal{O}_X$ corresponding to $j \\in J$\ngiven by the section $\\sum_i \\alpha(r_{ij})$ where the $r_{ij}$\nare the matrix coefficients of the map in the presentation of $M$.\n\\item Set $\\mathcal{F}_3$ equal to the sheaf associated to the presheaf\n$U \\mapsto \\mathcal{O}_X(U) \\otimes_R M$, where the map\n$R \\to \\mathcal{O}_X(U)$ is the composition of $\\alpha$ and\nthe restriction map $\\mathcal{O}_X(X) \\to \\mathcal{O}_X(U)$.\n\\end{enumerate}\nThis construction has the following properties:\n\\begin{enumerate}\n\\item The resulting sheaf of $\\mathcal{O}_X$-modules\n$\\mathcal{F}_M = \\mathcal{F}_1 = \\mathcal{F}_2 = \\mathcal{F}_3$\nis quasi-coherent.\n\\item The construction gives a functor from\nthe category of $R$-modules to the category of quasi-coherent\nsheaves on $X$ which commutes with arbitrary colimits.\n\\item For any $x \\in X$ we have\n$\\mathcal{F}_{M, x} = \\mathcal{O}_{X, x} \\otimes_R M$\nfunctorial in $M$.\n\\item Given any $\\mathcal{O}_X$-module\n$\\mathcal{G}$ we have\n$$\n\\Mor_{\\mathcal{O}_X}(\\mathcal{F}_M, \\mathcal{G})\n=\n\\Hom_R(M, \\Gamma(X, \\mathcal{G}))\n$$\nwhere the $R$-module structure on $\\Gamma(X, \\mathcal{G})$\ncomes from the $\\Gamma(X, \\mathcal{O}_X)$-module structure via\n$\\alpha$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BH","source_file":"modules.tex","source_line":1129,"source_end_line":1182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1129-L1182","statement_sha256":"df938852cc97436e1aced0ad264b8300ed71faffe478934c15e4c846b94bcbb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3803,"rank":3803,"depth":6,"x":1001.089,"y":232.676,"cluster":"sheaves-sites"},{"id":"stacks:01BI","tag":"01BI","title":"Quasi-coherent modules · Definition 01BI","summary":"In the situation of Lemma [Tag 01BH] we say F_M is the sheaf associated to the module M and the ring map α. If R = Γ(X, O_X) and α = id_R we simply say F_M is the sheaf associated to the module M.","statement_latex":"In the situation of Lemma \\ref{lemma-construct-quasi-coherent-sheaves}\nwe say $\\mathcal{F}_M$ is the {\\it sheaf associated to the module $M$\nand the ring map $\\alpha$}. If $R = \\Gamma(X, \\mathcal{O}_X)$\nand $\\alpha = \\text{id}_R$ we simply say $\\mathcal{F}_M$ is the\n{\\it sheaf associated to the module $M$}.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Quasi-coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BI","source_file":"modules.tex","source_line":1206,"source_end_line":1213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1206-L1213","statement_sha256":"a333308733c481a74734b4f6e56c216849d25f6be907f5b65d880909ec94629c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3804,"rank":3804,"depth":7,"x":1214.96,"y":137.411,"cluster":"sheaves-sites"},{"id":"stacks:01BJ","tag":"01BJ","title":"Quasi-coherent modules · Lemma 01BJ","summary":"Let (X, O_X) be a ringed space. Set R = Γ(X, O_X). Let M be an R-module. Let F_M be the quasi-coherent sheaf of O_X-modules associated to M. If g : (Y, O_Y) → (X, O_X) is a morphism of ringed spaces, then g^*F_M is the sheaf associated to the Γ(Y, O_Y)-module Γ(Y, O_Y) ⊗_R M.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nSet $R = \\Gamma(X, \\mathcal{O}_X)$.\nLet $M$ be an $R$-module.\nLet $\\mathcal{F}_M$ be the quasi-coherent sheaf of\n$\\mathcal{O}_X$-modules associated to $M$.\nIf $g : (Y, \\mathcal{O}_Y) \\to (X, \\mathcal{O}_X)$\nis a morphism of ringed spaces, then\n$g^*\\mathcal{F}_M$ is the sheaf associated\nto the $\\Gamma(Y, \\mathcal{O}_Y)$-module\n$\\Gamma(Y, \\mathcal{O}_Y) \\otimes_R M$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BJ","source_file":"modules.tex","source_line":1216,"source_end_line":1228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1216-L1228","statement_sha256":"fb36540fe343d4c80c291ec43a08a41a9ed1618820b052569b899ac968022924","origin":"The Stacks Project","memory_eligible":false,"source_rank":3805,"rank":3805,"depth":7,"x":1133.772,"y":329.233,"cluster":"sheaves-sites"},{"id":"stacks:01BK","tag":"01BK","title":"Quasi-coherent modules · Lemma 01BK","summary":"Let (X, O_X) be a ringed space. Let x ∈ X be a point. Assume that x has a fundamental system of quasi-compact neighbourhoods. Consider any quasi-coherent O_X-module F. Then there exists an open neighbourhood U of x such that F|_U is isomorphic to the sheaf of modules F_M on (U, O_U) associated to some Γ(U, O_U)-module M.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $x \\in X$ be a point.\nAssume that $x$ has a fundamental system of quasi-compact neighbourhoods.\nConsider any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$.\nThen there exists an open neighbourhood $U$ of $x$\nsuch that $\\mathcal{F}|_U$ is isomorphic to the\nsheaf of modules $\\mathcal{F}_M$ on $(U, \\mathcal{O}_U)$\nassociated to some $\\Gamma(U, \\mathcal{O}_U)$-module $M$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BK","source_file":"modules.tex","source_line":1246,"source_end_line":1256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1246-L1256","statement_sha256":"9153e4a3cfda48b35fb4e9674f1dc0e3739a52c1b1f55800f2833d1fa14e5965","origin":"The Stacks Project","memory_eligible":false,"source_rank":3806,"rank":3806,"depth":7,"x":1039.274,"y":141.504,"cluster":"sheaves-sites"},{"id":"stacks:01BN","tag":"01BN","title":"Modules of finite presentation · Definition 01BN","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. We say that F is of finite presentation if for every point x ∈ X there exists an open neighbourhood x∈ U ⊂ X, and n, m ∈ N such that F|_U is isomorphic to the cokernel of a map bigoplus_j = 1, …, m O_U → bigoplus_i = 1, …, n O_U","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nWe say that $\\mathcal{F}$ is of {\\it finite presentation}\nif for every point $x \\in X$ there exists an open neighbourhood\n$x\\in U \\subset X$, and  $n, m \\in \\mathbf{N}$ such that $\\mathcal{F}|_U$\nis isomorphic to the cokernel of a map\n$$\n\\bigoplus\\nolimits_{j = 1, \\ldots, m}\n\\mathcal{O}_U\n\\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n}\n\\mathcal{O}_U\n$$","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BN","source_file":"modules.tex","source_line":1377,"source_end_line":1392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1377-L1392","statement_sha256":"0a2d970cbc61822875287bf7974466be54f82dd272f45cd917026dca57ac2024","origin":"The Stacks Project","memory_eligible":false,"source_rank":3807,"rank":3807,"depth":0,"x":1260.172,"y":226.408,"cluster":"sheaves-sites"},{"id":"stacks:01BO","tag":"01BO","title":"Modules of finite presentation · Lemma 01BO","summary":"Let (X, O_X) be a ringed space. Any O_X-module of finite presentation is quasi-coherent.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nAny $\\mathcal{O}_X$-module of finite presentation\nis quasi-coherent.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BO","source_file":"modules.tex","source_line":1420,"source_end_line":1425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1420-L1425","statement_sha256":"3d7b6859512b35c13d13eb05f998621032378eb98451aed6c8626195f1ef88f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3808,"rank":3808,"depth":0,"x":1028.745,"y":289.216,"cluster":"sheaves-sites"},{"id":"stacks:0H96","tag":"0H96","title":"Modules of finite presentation · Lemma 0H96","summary":"Let (X,O_X) be a ringed space. Let F be a finitely presented O_X-module. Let φ : G → F be a morphism of O_X-modules. If G is finite type, then Coker(φ) is finitely presented.","statement_latex":"Let $(X,\\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a\nfinitely presented $\\mathcal{O}_X$-module. Let\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a morphism of\n$\\mathcal{O}_X$-modules. If $\\mathcal{G}$ is finite type, then\n$\\Coker(\\varphi)$ is finitely presented.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H96","source_file":"modules.tex","source_line":1431,"source_end_line":1438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1431-L1438","statement_sha256":"ce06c80154a599166a9fded35bf772633463f672c5aff2916609b2ea9c4ac329","origin":"The Stacks Project","memory_eligible":false,"source_rank":3809,"rank":3809,"depth":1,"x":1149.024,"y":111.384,"cluster":"sheaves-sites"},{"id":"stacks:01BP","tag":"01BP","title":"Modules of finite presentation · Lemma 01BP","summary":"Let (X, O_X) be a ringed space. Let F be an O_X-module of finite presentation. • If ψ : O_X^⊕ r → F is a surjection, then Ker(ψ) is of finite type. • If theta : G → F is surjective with G of finite type, then Ker(theta) is of finite type.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation.\n\\begin{enumerate}\n\\item If $\\psi : \\mathcal{O}_X^{\\oplus r} \\to \\mathcal{F}$ is a surjection,\nthen $\\Ker(\\psi)$ is of finite type.\n\\item If $\\theta : \\mathcal{G} \\to \\mathcal{F}$ is surjective with\n$\\mathcal{G}$ of finite type, then $\\Ker(\\theta)$ is of finite type.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BP","source_file":"modules.tex","source_line":1456,"source_end_line":1466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1456-L1466","statement_sha256":"63836d11306dfef5d5e2985367d69ec0e7d3e3da3b05419d6e65cd7ad2df38c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3810,"rank":3810,"depth":0,"x":1203.401,"y":310.989,"cluster":"sheaves-sites"},{"id":"stacks:01BQ","tag":"01BQ","title":"Modules of finite presentation · Lemma 01BQ","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The pullback f^*G of a module of finite presentation is of finite presentation.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces.\nThe pullback $f^*\\mathcal{G}$ of a module of finite presentation\nis of finite presentation.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BQ","source_file":"modules.tex","source_line":1506,"source_end_line":1512,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1506-L1512","statement_sha256":"fef239eb8a8b59d3ad93e50638a53210c3cb28aa0e8fe1b54a70832af40b6d56","origin":"The Stacks Project","memory_eligible":false,"source_rank":3811,"rank":3811,"depth":8,"x":1002.561,"y":194.527,"cluster":"sheaves-sites"},{"id":"stacks:01BR","tag":"01BR","title":"Modules of finite presentation · Lemma 01BR","summary":"Let (X, O_X) be a ringed space. Set R = Γ(X, O_X). Let M be an R-module. The O_X-module F_M associated to M is a directed colimit of finitely presented O_X-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nSet $R = \\Gamma(X, \\mathcal{O}_X)$.\nLet $M$ be an $R$-module.\nThe $\\mathcal{O}_X$-module $\\mathcal{F}_M$ associated to $M$\nis a directed colimit of finitely presented $\\mathcal{O}_X$-modules.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BR","source_file":"modules.tex","source_line":1519,"source_end_line":1526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1519-L1526","statement_sha256":"a52b912b817725bc70065d20d291179a61edfaf53d4a2f4427d02735b01c79a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3812,"rank":3812,"depth":7,"x":1244.585,"y":166.412,"cluster":"sheaves-sites"},{"id":"stacks:0B8J","tag":"0B8J","title":"Modules of finite presentation · Lemma 0B8J","summary":"Let (X, O_X) be a ringed space. Let F be a finitely presented O_X-module. Let x ∈ X such that F_x ≅ O_X, x^⊕ r. Then there exists an open neighbourhood U of x such that F|_U ≅ O_U^⊕ r.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be\na finitely presented $\\mathcal{O}_X$-module. Let $x \\in X$ such that\n$\\mathcal{F}_x \\cong \\mathcal{O}_{X, x}^{\\oplus r}$. Then there exists\nan open neighbourhood $U$ of $x$ such that\n$\\mathcal{F}|_U \\cong \\mathcal{O}_U^{\\oplus r}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8J","source_file":"modules.tex","source_line":1536,"source_end_line":1543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1536-L1543","statement_sha256":"6f38c6d6077f6000a80b611587a2aada902582a4d81135df74cdd55fce12dcd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3813,"rank":3813,"depth":2,"x":1088.555,"y":324.65,"cluster":"sheaves-sites"},{"id":"stacks:01BV","tag":"01BV","title":"Coherent modules · Definition 01BV","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. We say that F is a coherent O_X-module if the following two conditions hold: • F is of finite type, and • for every open U ⊂ X and every finite collection s_i ∈ F(U), i = 1, …, n the kernel of the associated map bigoplus_i = 1, …, n O_U → F|_U is of finite type. The category of coherent O_X-modules is denoted Coh(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nWe say that $\\mathcal{F}$ is a {\\it coherent $\\mathcal{O}_X$-module}\nif the following two conditions hold:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is of finite type, and\n\\item for every open $U \\subset X$ and every finite\ncollection $s_i \\in \\mathcal{F}(U)$, $i = 1, \\ldots, n$\nthe kernel of the associated map\n$\\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_U \\to \\mathcal{F}|_U$\nis of finite type.\n\\end{enumerate}\nThe category of coherent $\\mathcal{O}_X$-modules is denoted\n$\\textit{Coh}(\\mathcal{O}_X)$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BV","source_file":"modules.tex","source_line":1577,"source_end_line":1593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1577-L1593","statement_sha256":"404ea8deb819b9b4cb57c8279664df44b6eb66fbb6a212310b9101d56a30ff62","origin":"The Stacks Project","memory_eligible":false,"source_rank":3814,"rank":3814,"depth":0,"x":1076.344,"y":119.203,"cluster":"sheaves-sites"},{"id":"stacks:01BW","tag":"01BW","title":"Coherent modules · Lemma 01BW","summary":"Let (X, O_X) be a ringed space. Any coherent O_X-module is of finite presentation and hence quasi-coherent.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nAny coherent $\\mathcal{O}_X$-module is of finite presentation\nand hence quasi-coherent.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BW","source_file":"modules.tex","source_line":1595,"source_end_line":1600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1595-L1600","statement_sha256":"b28f206db5aaa9baaf285c959ee33ef3edac95c7a46be9d4ca112c3361207f6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3815,"rank":3815,"depth":0,"x":1250.758,"y":263.93,"cluster":"sheaves-sites"},{"id":"stacks:01BY","tag":"01BY","title":"Coherent modules · Lemma 01BY","summary":"Let (X, O_X) be a ringed space. • Any finite type subsheaf of a coherent sheaf is coherent. • Let φ : F → G be a morphism from a finite type sheaf F to a coherent sheaf G. Then Ker(φ) is of finite type. • Let φ : F → G be a morphism of coherent O_X-modules. Then Ker(φ) and Coker(φ) are coherent. • Given a short exact sequence of O_X-modules 0 → F_1 → F_2 → F_3 → 0 if two out of three are coherent so is the third. • The category Coh(O_X) is a weak Serre subcategory of…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\n\\begin{enumerate}\n\\item Any finite type subsheaf of a coherent sheaf is coherent.\n\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nbe a morphism from a finite type sheaf $\\mathcal{F}$\nto a coherent sheaf $\\mathcal{G}$. Then $\\Ker(\\varphi)$ is of finite type.\n\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism\nof coherent $\\mathcal{O}_X$-modules. Then\n$\\Ker(\\varphi)$ and\n$\\Coker(\\varphi)$ are coherent.\n\\item Given a short exact sequence of $\\mathcal{O}_X$-modules\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nif two out of three are coherent so is the third.\n\\item The category $\\textit{Coh}(\\mathcal{O}_X)$ is a weak Serre subcategory\nof $\\textit{Mod}(\\mathcal{O}_X)$. In particular, the category of\ncoherent modules is abelian and the inclusion functor\n$\\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$\nis exact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BY","source_file":"modules.tex","source_line":1640,"source_end_line":1661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1640-L1661","statement_sha256":"828e99e76e84308ec59e813af2bf60b25229d943870b92d92775630fe5ecd16a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3816,"rank":3816,"depth":1,"x":1005.49,"y":256.168,"cluster":"sheaves-sites"},{"id":"stacks:01BZ","tag":"01BZ","title":"Coherent modules · Lemma 01BZ","summary":"Let (X, O_X) be a ringed space. Let F be an O_X-module. Assume O_X is a coherent O_X-module. Then F is coherent if and only if it is of finite presentation.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nAssume $\\mathcal{O}_X$ is a coherent $\\mathcal{O}_X$-module.\nThen $\\mathcal{F}$ is coherent if and only if it is\nof finite presentation.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BZ","source_file":"modules.tex","source_line":1767,"source_end_line":1774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1767-L1774","statement_sha256":"4af249c5136434c08357018708a0f103ba6baf195a4dfe922827a86bfea10e9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3817,"rank":3817,"depth":0,"x":1192.796,"y":122.572,"cluster":"sheaves-sites"},{"id":"stacks:01C0","tag":"01C0","title":"Coherent modules · Lemma 01C0","summary":"Let X be a ringed space. Let φ : G → F be a homomorphism of O_X-modules. Let x ∈ X. Assume G of finite type, F coherent and the map on stalks φ_x : G_x → F_x injective. Then there exists an open neighbourhood x ∈ U ⊂ X such that φ|_U is injective.","statement_latex":"Let $X$ be a ringed space.\nLet $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism\nof $\\mathcal{O}_X$-modules.\nLet $x \\in X$. Assume $\\mathcal{G}$ of finite type,\n$\\mathcal{F}$ coherent and the map on stalks\n$\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ injective.\nThen there exists an open neighbourhood\n$x \\in U \\subset X$ such that $\\varphi|_U$ is injective.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01C0","source_file":"modules.tex","source_line":1780,"source_end_line":1790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1780-L1790","statement_sha256":"218516d51411d661287fce4974d8554f18a893e1f4d31506a716ae6614133f4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3818,"rank":3818,"depth":2,"x":1162.079,"y":327.593,"cluster":"sheaves-sites"},{"id":"stacks:01C2","tag":"01C2","title":"Closed immersions of ringed spaces · Definition 01C2","summary":"A closed immersion of ringed spaces is a morphism i : (Z, O_Z) → (X, O_X) with the following properties: • The map i is a closed immersion of topological spaces. • The associated map O_X → i_*O_Z is surjective. Denote the kernel by I. • The O_X-module I is locally generated by sections.","statement_latex":"A {\\it closed immersion of ringed spaces}\\footnote{This is\nnonstandard notation; see discussion above.} is a morphism\n$i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$\nwith the following properties:\n\\begin{enumerate}\n\\item The map $i$ is a closed immersion of topological spaces.\n\\item The associated map $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$\nis surjective. Denote the kernel by $\\mathcal{I}$.\n\\item The $\\mathcal{O}_X$-module $\\mathcal{I}$ is locally\ngenerated by sections.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Closed immersions of ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01C2","source_file":"modules.tex","source_line":1855,"source_end_line":1868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1855-L1868","statement_sha256":"18321f87631b1d4979b1c881548159962974a35ee04b392e915e3f0d2d903462","origin":"The Stacks Project","memory_eligible":false,"source_rank":3819,"rank":3819,"depth":0,"x":1019.7,"y":158.798,"cluster":"sheaves-sites"},{"id":"stacks:01C3","tag":"01C3","title":"Closed immersions of ringed spaces · Lemma 01C3","summary":"Let i : (Z, O_Z) → (X, O_X) be a closed immersion of ringed spaces. Let F be a quasi-coherent O_Z-module. Then i_*F is locally on X the cokernel of a map of quasi-coherent O_X-modules.","statement_latex":"Let $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$\nbe a closed immersion of ringed spaces.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_Z$-module.\nThen $i_*\\mathcal{F}$ is locally on $X$ the cokernel of\na map of quasi-coherent $\\mathcal{O}_X$-modules.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Closed immersions of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01C3","source_file":"modules.tex","source_line":1879,"source_end_line":1886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1879-L1886","statement_sha256":"43f39d884558769db2db1b78dd8e1702e0bcfb426856b595a88c5df4e5c244c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3820,"rank":3820,"depth":1,"x":1260.695,"y":202.524,"cluster":"sheaves-sites"},{"id":"stacks:01C4","tag":"01C4","title":"Closed immersions of ringed spaces · Lemma 01C4","summary":"Let i : (Z, O_Z) → (X, O_X) be a morphism of ringed spaces. Assume i is a homeomorphism onto a closed subset of X and that O_X → i_*O_Z is surjective. Let F be an O_Z-module. Then i_*F is of finite type if and only if F is of finite type.","statement_latex":"Let $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$ be a morphism of ringed\nspaces. Assume $i$ is a homeomorphism onto a closed subset of $X$ and\nthat $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_Z$-module.\nThen $i_*\\mathcal{F}$ is of finite type if and only if\n$\\mathcal{F}$ is of finite type.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Closed immersions of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01C4","source_file":"modules.tex","source_line":1898,"source_end_line":1906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1898-L1906","statement_sha256":"b9436ac241df1bed9865243968e6535c9f9ca79eea98b0f62acda111319b2a41","origin":"The Stacks Project","memory_eligible":false,"source_rank":3821,"rank":3821,"depth":0,"x":1047.591,"y":307.142,"cluster":"sheaves-sites"},{"id":"stacks:08KS","tag":"08KS","title":"Closed immersions of ringed spaces · Lemma 08KS","summary":"Let i : (Z, O_Z) → (X, O_X) be a morphism of ringed spaces. Assume i is a homeomorphism onto a closed subset of X and i^sharp : O_X → i_*O_Z is surjective. Denote I ⊂ O_X the kernel of i^sharp. The functor i_* : Mod(O_Z) → Mod(O_X) is exact, fully faithful, with essential image those O_X-modules G such that IG = 0.","statement_latex":"Let $i : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$ be a morphism\nof ringed spaces. Assume $i$ is a homeomorphism onto a closed subset of $X$\nand $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective.\nDenote $\\mathcal{I} \\subset \\mathcal{O}_X$ the kernel of $i^\\sharp$.\nThe functor\n$$\ni_* :\n\\textit{Mod}(\\mathcal{O}_Z)\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_X)\n$$\nis exact, fully faithful, with essential image those\n$\\mathcal{O}_X$-modules $\\mathcal{G}$ such that $\\mathcal{I}\\mathcal{G} = 0$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Closed immersions of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KS","source_file":"modules.tex","source_line":1945,"source_end_line":1960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L1945-L1960","statement_sha256":"42fdcf7a8176f50dd014b8381ef56f68587c06fadf3d6001b82166a10ce1c3b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3822,"rank":3822,"depth":3,"x":1120.68,"y":108.86,"cluster":"sheaves-sites"},{"id":"stacks:0G6P","tag":"0G6P","title":"Closed immersions of ringed spaces · Lemma 0G6P","summary":"Let (X, O_X) be a ringed space. Let i : Z → X be the inclusion of a closed subset. The functor H_Z : Mod(O_X) → Mod(O_X|_Z) of Remark [Tag 0G6N] is right adjoint to i_* : Mod(O_X|_Z) → Mod(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the\ninclusion of a closed subset. The functor\n$\\mathcal{H}_Z : \\textit{Mod}(\\mathcal{O}_X) \\to\n\\textit{Mod}(\\mathcal{O}_X|_Z)$ of\nRemark \\ref{remark-sections-support-in-closed-modules}\nis right adjoint to\n$i_* : \\textit{Mod}(\\mathcal{O}_X|_Z) \\to \\textit{Mod}(\\mathcal{O}_X)$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Closed immersions of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6P","source_file":"modules.tex","source_line":2038,"source_end_line":2047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2038-L2047","statement_sha256":"898f502d6b141e15236a0bc04f1324d1be8420312f1ea93f311fe18b3b21bcc6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3823,"rank":3823,"depth":0,"x":1226.354,"y":296.748,"cluster":"sheaves-sites"},{"id":"stacks:01C6","tag":"01C6","title":"Locally free sheaves · Definition 01C6","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. • We say F is locally free if for every point x ∈ X there exist a set I and an open neighbourhood x ∈ U ⊂ X such that F|_U is isomorphic to bigoplus_i ∈ I O_X|_U as an O_X|_U-module. • We say F is finite locally free if we may choose the index sets I to be finite. • We say F is finite locally free of rank r if we may choose the index sets I to have cardinality r.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ is {\\it locally free} if for every\npoint $x \\in X$ there exist a set $I$ and an open\nneighbourhood $x \\in U \\subset X$\nsuch that $\\mathcal{F}|_U$ is isomorphic to\n$\\bigoplus_{i \\in I} \\mathcal{O}_X|_U$ as an $\\mathcal{O}_X|_U$-module.\n\\item We say $\\mathcal{F}$ is {\\it finite locally free} if we may\nchoose the index sets $I$ to be finite.\n\\item We say $\\mathcal{F}$ is {\\it finite locally free of rank $r$}\nif we may choose the index sets $I$ to have cardinality $r$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Locally free sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01C6","source_file":"modules.tex","source_line":2079,"source_end_line":2094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2079-L2094","statement_sha256":"a26a3a40f79c45079abcda4b1df96cf2bd7503c7c1de914ecf3edf3fb2a7d3c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3824,"rank":3824,"depth":0,"x":997.086,"y":218.078,"cluster":"sheaves-sites"},{"id":"stacks:01C7","tag":"01C7","title":"Locally free sheaves · Lemma 01C7","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. If F is locally free then it is quasi-coherent.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}$ is locally free then it is quasi-coherent.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Locally free sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01C7","source_file":"modules.tex","source_line":2101,"source_end_line":2106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2101-L2106","statement_sha256":"9f08e11ea14c723de14890e2bac897251f93ad9f75dc300490913c428f55a611","origin":"The Stacks Project","memory_eligible":false,"source_rank":3825,"rank":3825,"depth":0,"x":1229.662,"y":145.918,"cluster":"sheaves-sites"},{"id":"stacks:01C8","tag":"01C8","title":"Locally free sheaves · Lemma 01C8","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. If G is a locally free O_Y-module, then f^*G is a locally free O_X-module.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces. If $\\mathcal{G}$ is\na locally free $\\mathcal{O}_Y$-module, then\n$f^*\\mathcal{G}$ is a locally free $\\mathcal{O}_X$-module.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Locally free sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01C8","source_file":"modules.tex","source_line":2112,"source_end_line":2118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2112-L2118","statement_sha256":"5af55309b161f912e618f6883201ea94970c12661b062f3a4f33fde5490425bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":3826,"rank":3826,"depth":0,"x":1116.07,"y":331.299,"cluster":"sheaves-sites"},{"id":"stacks:01C9","tag":"01C9","title":"Locally free sheaves · Lemma 01C9","summary":"Let (X, O_X) be a ringed space. Suppose that the support of O_X is X, i.e., all stalks of O_X are nonzero rings. Let F be a locally free sheaf of O_X-modules. There exists a locally constant function rank_F : X → (0, 1, 2, …)∪(∞) such that for any point x ∈ X the cardinality of any set I such that F is isomorphic to bigoplus_i∈ I O_X in a neighbourhood of x is rank_F(x).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nSuppose that the support of $\\mathcal{O}_X$ is $X$,\ni.e., all stalks of $\\mathcal{O}_X$ are nonzero rings.\nLet $\\mathcal{F}$ be a locally free sheaf of $\\mathcal{O}_X$-modules.\nThere exists a locally constant function\n$$\n\\text{rank}_\\mathcal{F} :\nX \\longrightarrow \\{0, 1, 2, \\ldots\\}\\cup\\{\\infty\\}\n$$\nsuch that for any point $x \\in X$ the cardinality of any\nset $I$ such that $\\mathcal{F}$ is isomorphic to\n$\\bigoplus_{i\\in I} \\mathcal{O}_X$ in a neighbourhood\nof $x$ is $\\text{rank}_\\mathcal{F}(x)$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Locally free sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01C9","source_file":"modules.tex","source_line":2124,"source_end_line":2139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2124-L2139","statement_sha256":"8152519593767a6b1e818d8244e3866e26c15e9b2d9acc0b18a00515e7c09092","origin":"The Stacks Project","memory_eligible":false,"source_rank":3827,"rank":3827,"depth":0,"x":1050.682,"y":129.928,"cluster":"sheaves-sites"},{"id":"stacks:089Q","tag":"089Q","title":"Locally free sheaves · Lemma 089Q","summary":"Let (X, O_X) be a ringed space. Let r ≥ 0. Let φ : F → G be a map of finite locally free O_X-modules of rank r. Then φ is an isomorphism if and only if φ is surjective.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $r \\geq 0$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of finite\nlocally free $\\mathcal{O}_X$-modules of rank $r$.\nThen $\\varphi$ is an isomorphism if and only if $\\varphi$\nis surjective.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Locally free sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089Q","source_file":"modules.tex","source_line":2147,"source_end_line":2154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2147-L2154","statement_sha256":"03ec0478e4c4f3f8760519b89f7d04e66f958acac48bc441a65ea831f383e489","origin":"The Stacks Project","memory_eligible":false,"source_rank":3828,"rank":3828,"depth":3,"x":1261.063,"y":241.434,"cluster":"sheaves-sites"},{"id":"stacks:0BCI","tag":"0BCI","title":"Locally free sheaves · Lemma 0BCI","summary":"Let (X, O_X) be a ringed space. If all stalks O_X, x are local rings, then any direct summand of a finite locally free O_X-module is finite locally free.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. If all stalks $\\mathcal{O}_{X, x}$\nare local rings, then any direct summand of a finite locally free\n$\\mathcal{O}_X$-module is finite locally free.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Locally free sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCI","source_file":"modules.tex","source_line":2175,"source_end_line":2180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2175-L2180","statement_sha256":"7bee3d725f677d4a71d08f77cb93cefba386d4d246e45ecb9c6a1fddde0576fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3829,"rank":3829,"depth":5,"x":1015.997,"y":278.626,"cluster":"sheaves-sites"},{"id":"stacks:01CB","tag":"01CB","title":"Tensor product · Lemma 01CB","summary":"Let (X, O_X) be a ringed space. Let F, G be O_X-modules. Let x ∈ X. There is a canonical isomorphism of O_X, x-modules (F ⊗_O_X G)_x = F_x ⊗_O_X, x G_x functorial in F and G.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.\nLet $x \\in X$. There is a canonical isomorphism\nof $\\mathcal{O}_{X, x}$-modules\n$$\n(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G})_x\n=\n\\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\mathcal{G}_x\n$$\nfunctorial in $\\mathcal{F}$ and $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CB","source_file":"modules.tex","source_line":2332,"source_end_line":2344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2332-L2344","statement_sha256":"e69e938e44810ade61e681efeaebc018566f19b5d9a66a5fc32dbe28b4a5956a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3830,"rank":3830,"depth":0,"x":1166.958,"y":111.965,"cluster":"sheaves-sites"},{"id":"stacks:05NA","tag":"05NA","title":"Tensor product · Lemma 05NA","summary":"Let (X, O_X) be a ringed space. Let F', G' be presheaves of O_X-modules with sheafifications F, G. Then F ⊗_O_X G = (F' ⊗_p, O_X G')^\\#.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}'$, $\\mathcal{G}'$ be presheaves of $\\mathcal{O}_X$-modules\nwith sheafifications $\\mathcal{F}$, $\\mathcal{G}$. Then\n$\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G} =\n(\\mathcal{F}' \\otimes_{p, \\mathcal{O}_X} \\mathcal{G}')^\\#$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NA","source_file":"modules.tex","source_line":2350,"source_end_line":2357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2350-L2357","statement_sha256":"29dda82570958a46775ad690beb6ba5585bbf819dbc1de6264a1ad8bc19f474e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3831,"rank":3831,"depth":0,"x":1189.69,"y":320.743,"cluster":"sheaves-sites"},{"id":"stacks:01CC","tag":"01CC","title":"Tensor product · Lemma 01CC","summary":"Let (X, O_X) be a ringed space. Let G be an O_X-module. If F_1 → F_2 → F_3 → 0 is an exact sequence of O_X-modules then the induced sequence F_1 ⊗_O_X G → F_2 ⊗_O_X G → F_3 ⊗_O_X G → 0 is exact.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{G}$ be an $\\mathcal{O}_X$-module.\nIf\n$\\mathcal{F}_1\n\\to \\mathcal{F}_2\n\\to \\mathcal{F}_3\n\\to 0$\nis an exact sequence of $\\mathcal{O}_X$-modules then\nthe induced sequence\n$$\n\\mathcal{F}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to\n\\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to\n\\mathcal{F}_3 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to\n0\n$$\nis exact.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CC","source_file":"modules.tex","source_line":2364,"source_end_line":2382,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2364-L2382","statement_sha256":"5954179fb8f1494dd39c37666ba87a323fceedae73774b34f42106aa0a666977","origin":"The Stacks Project","memory_eligible":false,"source_rank":3832,"rank":3832,"depth":2,"x":1004.837,"y":179.54,"cluster":"sheaves-sites"},{"id":"stacks:01CD","tag":"01CD","title":"Tensor product · Lemma 01CD","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let F, G be O_Y-modules. Then f^*(F ⊗_O_Y G) = f^*F ⊗_O_X f^*G functorially in F, G.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be\na morphism of ringed spaces. Let $\\mathcal{F}$, $\\mathcal{G}$\nbe $\\mathcal{O}_Y$-modules. Then\n$f^*(\\mathcal{F} \\otimes_{\\mathcal{O}_Y} \\mathcal{G})\n= f^*\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}$\nfunctorially in $\\mathcal{F}$, $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CD","source_file":"modules.tex","source_line":2392,"source_end_line":2400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2392-L2400","statement_sha256":"55e455bd85f96b2974519b3ce9287cc5d66a7ce43b32a6586c85010d4e2fc099","origin":"The Stacks Project","memory_eligible":false,"source_rank":3833,"rank":3833,"depth":0,"x":1254.964,"y":178.77,"cluster":"sheaves-sites"},{"id":"stacks:05NB","tag":"05NB","title":"Tensor product · Lemma 05NB","summary":"Let (X, O_X) be a ringed space. For any O_X-module F the functor Mod(O_X) → Mod(O_X) , G ↦ F ⊗_O_X G commutes with arbitrary colimits.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nFor any $\\mathcal{O}_X$-module $\\mathcal{F}$ the functor\n$$\n\\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X)\n, \\quad\n\\mathcal{G} \\longmapsto \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}\n$$\ncommutes with arbitrary colimits.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NB","source_file":"modules.tex","source_line":2406,"source_end_line":2416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2406-L2416","statement_sha256":"87d98c8e9d26e8fb664adccbef4ede120da357b492fd2633ee257abb12e398de","origin":"The Stacks Project","memory_eligible":false,"source_rank":3834,"rank":3834,"depth":4,"x":1070.947,"y":321.418,"cluster":"sheaves-sites"},{"id":"stacks:01CE","tag":"01CE","title":"Tensor product · Lemma 01CE","summary":"Let (X, O_X) be a ringed space. Let F, G be O_X-modules. • If F, G are locally generated by sections, so is F ⊗_O_X G. • If F, G are of finite type, so is F ⊗_O_X G. • If F, G are quasi-coherent, so is F ⊗_O_X G. • If F, G are of finite presentation, so is F ⊗_O_X G. • If F is of finite presentation and G is coherent, then F ⊗_O_X G is coherent. • If F, G are coherent, so is F ⊗_O_X G. • If F, G are locally free, so is F ⊗_O_X G.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally generated\nby sections, so is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite type,\nso is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are quasi-coherent,\nso is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite presentation,\nso is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.\n\\item If $\\mathcal{F}$ is of finite presentation and $\\mathcal{G}$ is coherent,\nthen $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$ is coherent.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are coherent,\nso is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally free,\nso is $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CE","source_file":"modules.tex","source_line":2439,"source_end_line":2459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2439-L2459","statement_sha256":"979e62e47d98b1f7238f7ff56d8e71ea1b0077d20b5544ea170dec9be1d7660d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3835,"rank":3835,"depth":5,"x":1091.947,"y":111.592,"cluster":"sheaves-sites"},{"id":"stacks:05ND","tag":"05ND","title":"Flat modules · Definition 05ND","summary":"Let (X, O_X) be a ringed space. An O_X-module F is flat if the functor Mod(O_X) → Mod(O_X), G ↦ G ⊗_O_X F is exact.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nAn $\\mathcal{O}_X$-module $\\mathcal{F}$ is {\\it flat} if the functor\n$$\n\\textit{Mod}(\\mathcal{O}_X)\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_X), \\quad\n\\mathcal{G} \\mapsto \\mathcal{G} \\otimes_{\\mathcal{O}_X} \\mathcal{F}\n$$\nis exact.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ND","source_file":"modules.tex","source_line":2533,"source_end_line":2544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2533-L2544","statement_sha256":"b76a0bfa7ac5696ee5c491a82f719eae7673defeccf1c606db952348c33e5356","origin":"The Stacks Project","memory_eligible":false,"source_rank":3836,"rank":3836,"depth":0,"x":1245.361,"y":278.408,"cluster":"sheaves-sites"},{"id":"stacks:05NE","tag":"05NE","title":"Flat modules · Lemma 05NE","summary":"Let (X, O_X) be a ringed space. An O_X-module F is flat if and only if the stalk F_x is a flat O_X, x-module for all x ∈ X.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nAn $\\mathcal{O}_X$-module $\\mathcal{F}$ is flat\nif and only if the stalk $\\mathcal{F}_x$ is a flat\n$\\mathcal{O}_{X, x}$-module for all $x \\in X$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NE","source_file":"modules.tex","source_line":2549,"source_end_line":2555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2549-L2555","statement_sha256":"48c6eaed907e7e31a29cb62a195022ec613783bcee33e680fd00b3ba44e5198b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3837,"rank":3837,"depth":1,"x":997.824,"y":242.413,"cluster":"sheaves-sites"},{"id":"stacks:05NF","tag":"05NF","title":"Flat modules · Definition 05NF","summary":"Let (X, O_X) be a ringed space. Let x ∈ X. An O_X-module F is flat at x if F_x is a flat O_X, x-module.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $x \\in X$.\nAn $\\mathcal{O}_X$-module $\\mathcal{F}$ is\n{\\it flat at $x$} if $\\mathcal{F}_x$ is a flat\n$\\mathcal{O}_{X, x}$-module.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NF","source_file":"modules.tex","source_line":2584,"source_end_line":2590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2584-L2590","statement_sha256":"5cc8ceb7466587ef3d24329b3ca056d26d3fd340f05b84927998ea4decf87e07","origin":"The Stacks Project","memory_eligible":false,"source_rank":3838,"rank":3838,"depth":0,"x":1209.524,"y":128.376,"cluster":"sheaves-sites"},{"id":"stacks:0H97","tag":"0H97","title":"Flat modules · Lemma 0H97","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. If G is a flat O_Y-module, then f^*G is a flat O_X-module.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces. If $\\mathcal{G}$ is a flat\n$\\mathcal{O}_Y$-module, then $f^*\\mathcal{G}$ is a flat\n$\\mathcal{O}_X$-module.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H97","source_file":"modules.tex","source_line":2596,"source_end_line":2602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2596-L2602","statement_sha256":"5d45985651f89cd3dfa647776bb3b03c0d25480791dc303b63b557fa572ffc53","origin":"The Stacks Project","memory_eligible":false,"source_rank":3839,"rank":3839,"depth":3,"x":1145.057,"y":332.805,"cluster":"sheaves-sites"},{"id":"stacks:05NG","tag":"05NG","title":"Flat modules · Lemma 05NG","summary":"Let (X, O_X) be a ringed space. A filtered colimit of flat O_X-modules is flat. A direct sum of flat O_X-modules is flat.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nA filtered colimit of flat $\\mathcal{O}_X$-modules is flat.\nA direct sum of flat $\\mathcal{O}_X$-modules is flat.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NG","source_file":"modules.tex","source_line":2610,"source_end_line":2615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2610-L2615","statement_sha256":"7a785ff679321bd63e726d3a92d335a7402b6fac62307e56af7ab9de71c75d44","origin":"The Stacks Project","memory_eligible":false,"source_rank":3840,"rank":3840,"depth":5,"x":1028.075,"y":145.284,"cluster":"sheaves-sites"},{"id":"stacks:05NH","tag":"05NH","title":"Flat modules · Lemma 05NH","summary":"Let (X, O_X) be a ringed space. Let U ⊂ X be open. The sheaf j_U!O_U is a flat sheaf of O_X-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $U \\subset X$ be open. The sheaf $j_{U!}\\mathcal{O}_U$\nis a flat sheaf of $\\mathcal{O}_X$-modules.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NH","source_file":"modules.tex","source_line":2625,"source_end_line":2630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2625-L2630","statement_sha256":"11edc2f922f3749341de430c56a60877f9ed5485653c183c06eb197d7fbff6d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3841,"rank":3841,"depth":2,"x":1265.384,"y":217.257,"cluster":"sheaves-sites"},{"id":"stacks:05NI","tag":"05NI","title":"Flat modules · Lemma 05NI","summary":"Let (X, O_X) be a ringed space. • Any sheaf of O_X-modules is a quotient of a direct sum bigoplus j_U_i!O_U_i. • Any O_X-module is a quotient of a flat O_X-module.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\n\\begin{enumerate}\n\\item Any sheaf of $\\mathcal{O}_X$-modules is a quotient of\na direct sum $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$.\n\\item Any $\\mathcal{O}_X$-module is a quotient of\na flat $\\mathcal{O}_X$-module.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NI","source_file":"modules.tex","source_line":2638,"source_end_line":2647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2638-L2647","statement_sha256":"9e6559ec8ab48074e463938653cb8b4447ec93acf407bc95360aa8359d917961","origin":"The Stacks Project","memory_eligible":false,"source_rank":3842,"rank":3842,"depth":6,"x":1032.273,"y":298.926,"cluster":"sheaves-sites"},{"id":"stacks:05NJ","tag":"05NJ","title":"Flat modules · Lemma 05NJ","summary":"Let (X, O_X) be a ringed space. Let 0 → F\" → F' → F → 0 be a short exact sequence of O_X-modules. Assume F is flat. Then for any O_X-module G the sequence 0 → F\" ⊗_O G → F' ⊗_O G → F ⊗_O G → 0 is exact.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet\n$$\n0 \\to \\mathcal{F}'' \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0\n$$\nbe a short exact sequence of $\\mathcal{O}_X$-modules.\nAssume $\\mathcal{F}$ is flat. Then for any $\\mathcal{O}_X$-module\n$\\mathcal{G}$ the sequence\n$$\n0 \\to\n\\mathcal{F}'' \\otimes_\\mathcal{O} \\mathcal{G} \\to\n\\mathcal{F}' \\otimes_\\mathcal{O} \\mathcal{G} \\to\n\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G} \\to 0\n$$\nis exact.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NJ","source_file":"modules.tex","source_line":2665,"source_end_line":2682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2665-L2682","statement_sha256":"878fb52b37b464f33531b7890e5a08eeaf8339d7201df836d721408d429122c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3843,"rank":3843,"depth":1,"x":1138.601,"y":106.23,"cluster":"sheaves-sites"},{"id":"stacks:05NK","tag":"05NK","title":"Flat modules · Lemma 05NK","summary":"Kernels of epimorphisms and extensions of flat sheaves of modules over a ringed space are again flat. Let (X, O_X) be a ringed space. Let 0 → F_2 → F_1 → F_0 → 0 be a short exact sequence of O_X-modules. • If F_2 and F_0 are flat so is F_1. • If F_1 and F_0 are flat so is F_2.","statement_latex":"\\begin{slogan}\nKernels of epimorphisms and extensions of flat sheaves of modules over\na ringed space are again flat.\n\\end{slogan}\nLet $(X, \\mathcal{O}_X)$ be a ringed space.\nLet\n$$\n0 \\to\n\\mathcal{F}_2 \\to\n\\mathcal{F}_1 \\to\n\\mathcal{F}_0 \\to 0\n$$\nbe a short exact sequence of $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}_2$ and $\\mathcal{F}_0$ are flat so is\n$\\mathcal{F}_1$.\n\\item If $\\mathcal{F}_1$ and $\\mathcal{F}_0$ are flat so is\n$\\mathcal{F}_2$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NK","source_file":"modules.tex","source_line":2691,"source_end_line":2712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2691-L2712","statement_sha256":"42cd410764d4462ebc74d3fc09de182bc16c621d97baad6bf329c023b67e68ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":3844,"rank":3844,"depth":4,"x":1215.238,"y":308.866,"cluster":"sheaves-sites"},{"id":"stacks:05NL","tag":"05NL","title":"Flat modules · Lemma 05NL","summary":"Let (X, O_X) be a ringed space. Let … → F_2 → F_1 → F_0 → Q → 0 be an exact complex of O_X-modules. If Q and all F_i are flat O_X-modules, then for any O_X-module G the complex … → F_2 ⊗_O_X G → F_1 ⊗_O_X G → F_0 ⊗_O_X G → Q ⊗_O_X G → 0 is exact also.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet\n$$\n\\ldots \\to\n\\mathcal{F}_2 \\to\n\\mathcal{F}_1 \\to\n\\mathcal{F}_0 \\to\n\\mathcal{Q} \\to 0\n$$\nbe an exact complex of $\\mathcal{O}_X$-modules.\nIf $\\mathcal{Q}$ and all $\\mathcal{F}_i$ are flat $\\mathcal{O}_X$-modules,\nthen for any $\\mathcal{O}_X$-module $\\mathcal{G}$ the complex\n$$\n\\ldots \\to\n\\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to\n\\mathcal{F}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to\n\\mathcal{F}_0 \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to\n\\mathcal{Q} \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to 0\n$$\nis exact also.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NL","source_file":"modules.tex","source_line":2720,"source_end_line":2742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2720-L2742","statement_sha256":"2c61ac615ebd50272b37dc47d6c0ec6ef3ec2af4afb4856b3e03e096ff3627bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3845,"rank":3845,"depth":5,"x":995.543,"y":202.82,"cluster":"sheaves-sites"},{"id":"stacks:08BK","tag":"08BK","title":"Flat modules · Lemma 08BK","summary":"Let (X, O_X) be a ringed space. Let F be a flat O_X-module. Let U ⊂ X be open and let O_U xrightarrow(f_1, …, f_n) O_U^⊕ n xrightarrow(s_1, …, s_n) F|_U be a complex of O_U-modules. For every x ∈ U there exists an open neighbourhood V ⊂ U of x and a factorization O_V^⊕ n xrightarrowA O_V^⊕ m xrightarrow(t_1, …, t_m) F|_V of (s_1, …, s_n)|_V such that A ∘ (f_1, …, f_n)|_V = 0.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a flat\n$\\mathcal{O}_X$-module. Let $U \\subset X$ be open and let\n$$\n\\mathcal{O}_U \\xrightarrow{(f_1, \\ldots, f_n)}\n\\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)}\n\\mathcal{F}|_U\n$$\nbe a complex of $\\mathcal{O}_U$-modules. For every $x \\in U$ there\nexists an open neighbourhood $V \\subset U$ of $x$ and a factorization\n$$\n\\mathcal{O}_V^{\\oplus n}\n\\xrightarrow{A}\n\\mathcal{O}_V^{\\oplus m} \\xrightarrow{(t_1, \\ldots, t_m)}\n\\mathcal{F}|_V\n$$\nof $(s_1, \\ldots, s_n)|_V$ such that $A \\circ (f_1, \\ldots, f_n)|_V = 0$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BK","source_file":"modules.tex","source_line":2755,"source_end_line":2773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2755-L2773","statement_sha256":"b0deeec123a985b288ba582661a6ba84460ff7de6a0848a15e68c6909de0b403","origin":"The Stacks Project","memory_eligible":false,"source_rank":3846,"rank":3846,"depth":1,"x":1243.08,"y":156.308,"cluster":"sheaves-sites"},{"id":"stacks:0FNW","tag":"0FNW","title":"Duals · Lemma 0FNW","summary":"Let (X, O_X) be a ringed space. Let F be an O_X-module. Let G, eta, ε be a left dual of F in the monoidal category of O_X-modules, see Categories, Definition [Tag 0FFP]. Then • F is locally a direct summand of a finite free O_X-module, • the map e : SheafHom_O_X(F, O_X) → G sending a local section λ to (λ ⊗ 1)(eta) is an isomorphism, • we have ε(f, g) = e^-1(g)(f) for local sections f and g of F and G.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be an\n$\\mathcal{O}_X$-module. Let $\\mathcal{G}, \\eta, \\epsilon$\nbe a left dual of $\\mathcal{F}$ in the monoidal category of\n$\\mathcal{O}_X$-modules, see\nCategories, Definition \\ref{categories-definition-dual}. Then\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is locally a direct summand of a finite free\n$\\mathcal{O}_X$-module,\n\\item the map\n$e : \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X) \\to \\mathcal{G}$\nsending a local section $\\lambda$ to $(\\lambda \\otimes 1)(\\eta)$\nis an isomorphism,\n\\item we have $\\epsilon(f, g) = e^{-1}(g)(f)$ for local sections\n$f$ and $g$ of $\\mathcal{F}$ and $\\mathcal{G}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNW","source_file":"modules.tex","source_line":2868,"source_end_line":2885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2868-L2885","statement_sha256":"75133b002130ddd327a124e01c1ef18e12de738eae7c4c4635ec2836b8c4cc57","origin":"The Stacks Project","memory_eligible":false,"source_rank":3847,"rank":3847,"depth":1,"x":1097.803,"y":331.245,"cluster":"sheaves-sites"},{"id":"stacks:08BL","tag":"08BL","title":"Duals · Lemma 08BL","summary":"Let (X, O_X) be a ringed space. Let F be a flat O_X-module of finite presentation. Then F is locally a direct summand of a finite free O_X-module.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a\nflat $\\mathcal{O}_X$-module of finite presentation. Then $\\mathcal{F}$ is\nlocally a direct summand of a finite free $\\mathcal{O}_X$-module.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BL","source_file":"modules.tex","source_line":2936,"source_end_line":2941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2936-L2941","statement_sha256":"fcc84214f2fe2c32c4cb52e6004957aa5cee2b67d443622b886b3bdc2de4ca45","origin":"The Stacks Project","memory_eligible":false,"source_rank":3848,"rank":3848,"depth":2,"x":1064.214,"y":119.596,"cluster":"sheaves-sites"},{"id":"stacks:0CAH","tag":"0CAH","title":"Constructible sheaves of sets · Lemma 0CAH","summary":"Let X be a topological space. Let B be a basis for the topology on X. Let F be a sheaf of sets on X. There exists a set I and for each i ∈ I an element U_i ∈ B and a finite set S_i such that there exists a surjection coprod_i ∈ I j_U_i!underlineS_i → F.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the\ntopology on $X$. Let $\\mathcal{F}$ be a sheaf of sets on $X$.\nThere exists a set $I$ and for each $i \\in I$ an element\n$U_i \\in \\mathcal{B}$ and a finite set $S_i$ such that there exists\na surjection $\\coprod_{i \\in I} j_{U_i!}\\underline{S_i} \\to \\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Constructible sheaves of sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAH","source_file":"modules.tex","source_line":2991,"source_end_line":2998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L2991-L2998","statement_sha256":"4a47c49bb0f56ccda204a6d1ade45edf3eb16d0740fc503adb8c572db06bf97f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3849,"rank":3849,"depth":2,"x":1259.385,"y":256.745,"cluster":"sheaves-sites"},{"id":"stacks:0CAI","tag":"0CAI","title":"Constructible sheaves of sets · Lemma 0CAI","summary":"Let X be a topological space. Let B be a basis for the topology of X and assume that each U ∈ B is quasi-compact. Then every sheaf of sets on X is a filtered colimit of sheaves of the form Coequalizer( xymatrix coprod_b = 1, …, m j_V_b!underlineS_b ar@<1ex>[r] ar@<-1ex>[r] & coprod_a = 1, …, n j_U_a!underlineS_a ) with U_a and V_b in B and S_a and S_b finite sets.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{B}$ be a basis for the\ntopology of $X$ and assume that each $U \\in \\mathcal{B}$ is quasi-compact.\nThen every sheaf of sets on $X$ is a filtered colimit of sheaves of the form\n\\begin{equation}\n\n\\text{Coequalizer}\\left(\n\\xymatrix{\n\\coprod\\nolimits_{b = 1, \\ldots, m} j_{V_b!}\\underline{S_b}\n\\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\coprod\\nolimits_{a = 1, \\ldots, n} j_{U_a!}\\underline{S_a}\n}\n\\right)\n\\end{equation}\nwith $U_a$ and $V_b$ in $\\mathcal{B}$ and $S_a$ and $S_b$ finite sets.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Constructible sheaves of sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAI","source_file":"modules.tex","source_line":3015,"source_end_line":3031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3015-L3031","statement_sha256":"1012b3810bcd6813cf3a285c716e4583cf5225e42df9a998ec560857a9bc5ffb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3850,"rank":3850,"depth":3,"x":1004.915,"y":266.369,"cluster":"sheaves-sites"},{"id":"stacks:0CAK","tag":"0CAK","title":"Constructible sheaves of sets · Lemma 0CAK","summary":"Let X be a spectral topological space. Let B be the set of quasi-compact open subsets of X. Let F be a sheaf of sets as in Equation ([Tag 0CAJ]). Then there exists a continuous spectral map f : X → Y to a finite sober topological space Y and a sheaf of sets G on Y with finite stalks such that f^-1G ≅ F.","statement_latex":"Let $X$ be a spectral topological space. Let $\\mathcal{B}$ be\nthe set of quasi-compact open subsets of $X$.\nLet $\\mathcal{F}$ be a sheaf of sets as in\nEquation (\\ref{equation-towards-constructible-sets}).\nThen there exists a continuous spectral map $f : X \\to Y$\nto a finite sober topological space $Y$ and a sheaf\nof sets $\\mathcal{G}$ on $Y$ with finite stalks\nsuch that $f^{-1}\\mathcal{G} \\cong \\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Constructible sheaves of sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAK","source_file":"modules.tex","source_line":3059,"source_end_line":3069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3059-L3069","statement_sha256":"5d3fc8f36d28122c8985a97c5357881cb91b5557928eb18431e86da782c967a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3851,"rank":3851,"depth":8,"x":1185.004,"y":114.724,"cluster":"sheaves-sites"},{"id":"stacks:0CAL","tag":"0CAL","title":"Constructible sheaves of sets · Lemma 0CAL","summary":"Let X be a spectral topological space. Let B be the set of quasi-compact open subsets of X. Let F be a sheaf of sets as in Equation ([Tag 0CAJ]). Then there exist finitely many constructible closed subsets Z_1, …, Z_n ⊂ X and finite sets S_i such that F is isomorphic to a subsheaf of ∏ (Z_i → X)_*underlineS_i.","statement_latex":"Let $X$ be a spectral topological space. Let $\\mathcal{B}$ be\nthe set of quasi-compact open subsets of $X$.\nLet $\\mathcal{F}$ be a sheaf of sets as in\nEquation (\\ref{equation-towards-constructible-sets}).\nThen there exist finitely many constructible closed subsets\n$Z_1, \\ldots, Z_n \\subset X$ and finite sets $S_i$\nsuch that $\\mathcal{F}$ is isomorphic to a subsheaf of\n$\\prod (Z_i \\to X)_*\\underline{S_i}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Constructible sheaves of sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAL","source_file":"modules.tex","source_line":3132,"source_end_line":3142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3132-L3142","statement_sha256":"5f1a0fe0b1ee6e32c6cccee6bb33b4fd3a98b86154abcae0708cee49c25f6317","origin":"The Stacks Project","memory_eligible":false,"source_rank":3852,"rank":3852,"depth":9,"x":1174.146,"y":328.952,"cluster":"sheaves-sites"},{"id":"stacks:02N3","tag":"02N3","title":"Flat morphisms of ringed spaces · Definition 02N3","summary":"Let f : X → Y be a morphism of ringed spaces. Let x ∈ X. We say f is flat at x if the map of rings O_Y, f(x) → O_X, x is flat. We say f is flat if f is flat at every x ∈ X.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $x \\in X$. We say $f$ is {\\it flat at $x$}\nif the map of rings $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is flat.\nWe say $f$ is {\\it flat} if $f$ is flat at every $x \\in X$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat morphisms of ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02N3","source_file":"modules.tex","source_line":3172,"source_end_line":3178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3172-L3178","statement_sha256":"ad2b7c38e0d88b06c6131383e9155768887763a5dce93474a189a19a1d9ad6f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3853,"rank":3853,"depth":0,"x":1009.708,"y":164.654,"cluster":"sheaves-sites"},{"id":"stacks:02N4","tag":"02N4","title":"Flat morphisms of ringed spaces · Lemma 02N4","summary":"Let f : X → Y be a flat morphism of ringed spaces. Then the pullback functor f^* : Mod(O_Y) → Mod(O_X) is exact.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of ringed spaces.\nThen the pullback functor\n$f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$\nis exact.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat morphisms of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02N4","source_file":"modules.tex","source_line":3190,"source_end_line":3196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3190-L3196","statement_sha256":"1ffc2d3baec97d894049a9ab8fa88d8f73ced418088fb9f1a7695d1c507564d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3854,"rank":3854,"depth":1,"x":1263.345,"y":192.527,"cluster":"sheaves-sites"},{"id":"stacks:08KT","tag":"08KT","title":"Flat morphisms of ringed spaces · Definition 08KT","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let F be a sheaf of O_X-modules. • We say that F is flat over Y at a point x ∈ X if the stalk F_x is a flat O_Y, f(x)-module. • We say that F is flat over Y if F is flat over Y at every point x of X.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of\nringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item We say that $\\mathcal{F}$ is {\\it flat over $Y$ at a point $x \\in X$}\nif the stalk $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{Y, f(x)}$-module.\n\\item We say that $\\mathcal{F}$ is {\\it flat over $Y$} if\n$\\mathcal{F}$ is flat over $Y$ at every point $x$ of $X$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat morphisms of ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KT","source_file":"modules.tex","source_line":3211,"source_end_line":3221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3211-L3221","statement_sha256":"16638721efc158ca01c2ffc01709bb051dd7597a80d31f223f45f8d381b7a339","origin":"The Stacks Project","memory_eligible":false,"source_rank":3855,"rank":3855,"depth":0,"x":1053.688,"y":316.019,"cluster":"sheaves-sites"},{"id":"stacks:0GMU","tag":"0GMU","title":"Flat morphisms of ringed spaces · Lemma 0GMU","summary":"Let f : X → Y be a morphism of ringed spaces. Let F be an O_X-module flat over Y. Then the functor Mod(O_Y) → Mod(O_X), G ↦ f^*G ⊗_O_X F is exact.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces. Let $\\mathcal{F}$\nbe an $\\mathcal{O}_X$-module flat over $Y$. Then the functor\n$$\n\\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X),\\quad\n\\mathcal{G} \\longmapsto f^*\\mathcal{G} \\otimes_{\\mathcal{O}_X} \\mathcal{F}\n$$\nis exact.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Flat morphisms of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMU","source_file":"modules.tex","source_line":3230,"source_end_line":3239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3230-L3239","statement_sha256":"2ce27a4fd244cea8806ad9acae9f1c7e16c74b33114408951b0af45e4988bbd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3856,"rank":3856,"depth":0,"x":1109.034,"y":105.779,"cluster":"sheaves-sites"},{"id":"stacks:01CG","tag":"01CG","title":"Symmetric and exterior powers · Lemma 01CG","summary":"In the situation described above. The sheaf wedge^nF is the sheafification of the presheaf U ↦ wedge^n_O_X(U)(F(U)). See Algebra, Section [Tag 00DM]. Similarly, the sheaf Sym^nF is the sheafification of the presheaf U ↦ Sym^n_O_X(U)(F(U)).","statement_latex":"In the situation described above.\nThe sheaf $\\wedge^n\\mathcal{F}$ is the sheafification of the\npresheaf\n$$\nU \\longmapsto \\wedge^n_{\\mathcal{O}_X(U)}(\\mathcal{F}(U)).\n$$\nSee Algebra, Section \\ref{algebra-section-tensor-algebra}.\nSimilarly, the sheaf $\\text{Sym}^n\\mathcal{F}$ is the sheafification\nof the presheaf\n$$\nU \\longmapsto \\text{Sym}^n_{\\mathcal{O}_X(U)}(\\mathcal{F}(U)).\n$$","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Symmetric and exterior powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CG","source_file":"modules.tex","source_line":3302,"source_end_line":3316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3302-L3316","statement_sha256":"0134b0f3a72dccf9d5a8bab90fa868af8d415ded7a7482845ca049b4b4cc7758","origin":"The Stacks Project","memory_eligible":false,"source_rank":3857,"rank":3857,"depth":0,"x":1237.423,"y":292.401,"cluster":"sheaves-sites"},{"id":"stacks:01CH","tag":"01CH","title":"Symmetric and exterior powers · Lemma 01CH","summary":"In the situation described above. Let x ∈ X. There are canonical isomorphisms of O_X, x-modules T(F)_x = T(F_x), Sym(F)_x = Sym(F_x), and wedge(F)_x = wedge(F_x).","statement_latex":"In the situation described above. Let $x \\in X$.\nThere are canonical isomorphisms of $\\mathcal{O}_{X, x}$-modules\n$\\text{T}(\\mathcal{F})_x = \\text{T}(\\mathcal{F}_x)$,\n$\\text{Sym}(\\mathcal{F})_x = \\text{Sym}(\\mathcal{F}_x)$, and\n$\\wedge(\\mathcal{F})_x = \\wedge(\\mathcal{F}_x)$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Symmetric and exterior powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CH","source_file":"modules.tex","source_line":3324,"source_end_line":3331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3324-L3331","statement_sha256":"70b31714dee3c424fac6ed5185d9dc4fec27b61239309e78711e542057fd9135","origin":"The Stacks Project","memory_eligible":false,"source_rank":3858,"rank":3858,"depth":5,"x":992.426,"y":227.574,"cluster":"sheaves-sites"},{"id":"stacks:01CI","tag":"01CI","title":"Symmetric and exterior powers · Lemma 01CI","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let F be a sheaf of O_Y-modules. Then f^*T(F) = T(f^*F), and similarly for the exterior and symmetric algebras associated to F.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of\nringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules.\nThen $f^*\\text{T}(\\mathcal{F}) = \\text{T}(f^*\\mathcal{F})$,\nand similarly for the exterior and symmetric algebras associated\nto $\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Symmetric and exterior powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CI","source_file":"modules.tex","source_line":3338,"source_end_line":3345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3338-L3345","statement_sha256":"7820a5a7e1aab4ea63ab1bd0a04b839b40f063d108941c84eb93c1509516cb36","origin":"The Stacks Project","memory_eligible":false,"source_rank":3859,"rank":3859,"depth":0,"x":1225.451,"y":136.267,"cluster":"sheaves-sites"},{"id":"stacks:01CJ","tag":"01CJ","title":"Symmetric and exterior powers · Lemma 01CJ","summary":"Let (X, O_X) be a ringed space. Let F_2 → F_1 → F → 0 be an exact sequence of sheaves of O_X-modules. For each n ≥ 1 there is an exact sequence F_2 ⊗_O_X Sym^n - 1(F_1) → Sym^n(F_1) → Sym^n(F) → 0 and similarly an exact sequence F_2 ⊗_O_X wedge^n - 1(F_1) → wedge^n(F_1) → wedge^n(F) → 0","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to 0$\nbe an exact sequence of sheaves of $\\mathcal{O}_X$-modules.\nFor each $n \\geq 1$ there is an exact sequence\n$$\n\\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\text{Sym}^{n - 1}(\\mathcal{F}_1)\n\\to\n\\text{Sym}^n(\\mathcal{F}_1)\n\\to\n\\text{Sym}^n(\\mathcal{F})\n\\to\n0\n$$\nand similarly an exact sequence\n$$\n\\mathcal{F}_2 \\otimes_{\\mathcal{O}_X} \\wedge^{n - 1}(\\mathcal{F}_1)\n\\to\n\\wedge^n(\\mathcal{F}_1)\n\\to\n\\wedge^n(\\mathcal{F})\n\\to\n0\n$$","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Symmetric and exterior powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CJ","source_file":"modules.tex","source_line":3351,"source_end_line":3376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3351-L3376","statement_sha256":"1159cd9adf249c310ad933e05046f22289e092ab7bb1c62dce318b682dcc5900","origin":"The Stacks Project","memory_eligible":false,"source_rank":3860,"rank":3860,"depth":1,"x":1126.949,"y":336.021,"cluster":"sheaves-sites"},{"id":"stacks:01CK","tag":"01CK","title":"Symmetric and exterior powers · Lemma 01CK","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. • If F is locally generated by sections, then so is each T^n(F), wedge^n(F), and Sym^n(F). • If F is of finite type, then so is each T^n(F), wedge^n(F), and Sym^n(F). • If F is of finite presentation, then so is each T^n(F), wedge^n(F), and Sym^n(F). • If F is coherent, then for n > 0 each T^n(F), wedge^n(F), and Sym^n(F) is coherent. • If F is quasi-coherent, then so is each T^n(F), wedge^n(F), and…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is locally generated by sections,\nthen so is each $\\text{T}^n(\\mathcal{F})$,\n$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.\n\\item If $\\mathcal{F}$ is of finite type,\nthen so is each $\\text{T}^n(\\mathcal{F})$,\n$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.\n\\item If $\\mathcal{F}$ is of finite presentation,\nthen so is each $\\text{T}^n(\\mathcal{F})$,\n$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.\n\\item If $\\mathcal{F}$ is coherent,\nthen for $n > 0$ each $\\text{T}^n(\\mathcal{F})$,\n$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$\nis coherent.\n\\item If $\\mathcal{F}$ is quasi-coherent,\nthen so is each $\\text{T}^n(\\mathcal{F})$,\n$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.\n\\item If $\\mathcal{F}$ is locally free,\nthen so is each $\\text{T}^n(\\mathcal{F})$,\n$\\wedge^n(\\mathcal{F})$, and $\\text{Sym}^n(\\mathcal{F})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Symmetric and exterior powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CK","source_file":"modules.tex","source_line":3382,"source_end_line":3407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3382-L3407","statement_sha256":"2e9a2fbc03849880b656e0378e2565eac633901a51ea71d94718e2e3d519ebc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3861,"rank":3861,"depth":6,"x":1038.857,"y":132.628,"cluster":"sheaves-sites"},{"id":"stacks:01CL","tag":"01CL","title":"Symmetric and exterior powers · Lemma 01CL","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. • If F is quasi-coherent, then so is each T(F), wedge(F), and Sym(F). • If F is locally free, then so is each T(F), wedge(F), and Sym(F).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is quasi-coherent,\nthen so is each $\\text{T}(\\mathcal{F})$,\n$\\wedge(\\mathcal{F})$, and $\\text{Sym}(\\mathcal{F})$.\n\\item If $\\mathcal{F}$ is locally free,\nthen so is each $\\text{T}(\\mathcal{F})$,\n$\\wedge(\\mathcal{F})$, and $\\text{Sym}(\\mathcal{F})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Symmetric and exterior powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CL","source_file":"modules.tex","source_line":3460,"source_end_line":3472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3460-L3472","statement_sha256":"f53082e30eb47f25b68ceaa5b923a716d3231cf4b077b2e3d9d31d1678e165df","origin":"The Stacks Project","memory_eligible":false,"source_rank":3862,"rank":3862,"depth":7,"x":1267.607,"y":232.723,"cluster":"sheaves-sites"},{"id":"stacks:01CN","tag":"01CN","title":"Internal Hom · Lemma 01CN","summary":"Let (X, O_X) be a ringed space. Let F, G, H be O_X-modules. There is a canonical isomorphism SheafHom_O_X (F ⊗_O_X G, H) → SheafHom_O_X (F, SheafHom_O_X(G, H)) which is functorial in all three entries (sheaf Hom in all three spots). In particular, to give a morphism F ⊗_O_X G → H is the same as giving a morphism F → SheafHom_O_X(G, H).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$, $\\mathcal{G}$, $\\mathcal{H}$ be $\\mathcal{O}_X$-modules.\nThere is a canonical isomorphism\n$$\n\\SheafHom_{\\mathcal{O}_X}\n(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}, \\mathcal{H})\n\\longrightarrow\n\\SheafHom_{\\mathcal{O}_X}\n(\\mathcal{F}, \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{H}))\n$$\nwhich is functorial in all three entries (sheaf Hom in\nall three spots). In particular, to give a\nmorphism $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G} \\to \\mathcal{H}$\nis the same as giving a morphism\n$\\mathcal{F} \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{H})$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CN","source_file":"modules.tex","source_line":3533,"source_end_line":3550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3533-L3550","statement_sha256":"6c832de059f5fdcfa8d7393465a071cf58f1cdc29f347c5b53430832bc0e3792","origin":"The Stacks Project","memory_eligible":false,"source_rank":3863,"rank":3863,"depth":1,"x":1018.187,"y":288.769,"cluster":"sheaves-sites"},{"id":"stacks:01CO","tag":"01CO","title":"Internal Hom · Lemma 01CO","summary":"Let (X, O_X) be a ringed space. Let F, G be O_X-modules. • If F_2 → F_1 → F → 0 is an exact sequence of O_X-modules, then 0 → SheafHom_O_X(F, G) → SheafHom_O_X(F_1, G) → SheafHom_O_X(F_2, G) is exact. • If 0 → G → G_1 → G_2 is an exact sequence of O_X-modules, then 0 → SheafHom_O_X(F, G) → SheafHom_O_X(F, G_1) → SheafHom_O_X(F, G_2) is exact.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to 0$\nis an exact sequence of $\\mathcal{O}_X$-modules, then\n$$\n0 \\to\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) \\to\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}_1, \\mathcal{G}) \\to\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}_2, \\mathcal{G})\n$$\nis exact.\n\\item If $0 \\to \\mathcal{G} \\to \\mathcal{G}_1 \\to \\mathcal{G}_2$\nis an exact sequence of $\\mathcal{O}_X$-modules, then\n$$\n0 \\to\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) \\to\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}_1) \\to\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}_2)\n$$\nis exact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CO","source_file":"modules.tex","source_line":3558,"source_end_line":3582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3558-L3582","statement_sha256":"6a7d65633051e7886982d6f36621741483560c9470239e554bdbeaffaefd6f6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3864,"rank":3864,"depth":2,"x":1157.174,"y":105.731,"cluster":"sheaves-sites"},{"id":"stacks:0A6F","tag":"0A6F","title":"Internal Hom · Lemma 0A6F","summary":"Let X be a topological space. Let O_1 → O_2 be a homomorphism of sheaves of rings. Then we have Hom_O_1(F_O_1, G) = Hom_O_2(F, SheafHom_O_1(O_2, G)) bifunctorially in F ∈ Mod(O_2) and G ∈ Mod(O_1).","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. Then we have\n$$\n\\Hom_{\\mathcal{O}_1}(\\mathcal{F}_{\\mathcal{O}_1}, \\mathcal{G}) =\n\\Hom_{\\mathcal{O}_2}(\\mathcal{F},\n\\SheafHom_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{G}))\n$$\nbifunctorially in $\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O}_2)$\nand $\\mathcal{G} \\in \\textit{Mod}(\\mathcal{O}_1)$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6F","source_file":"modules.tex","source_line":3599,"source_end_line":3610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3599-L3610","statement_sha256":"132d8490448df8436220d3de992c0ec154de22141684c128476e8bffd53c2603","origin":"The Stacks Project","memory_eligible":false,"source_rank":3865,"rank":3865,"depth":1,"x":1201.926,"y":319.779,"cluster":"sheaves-sites"},{"id":"stacks:01CP","tag":"01CP","title":"Internal Hom · Lemma 01CP","summary":"Let (X, O_X) be a ringed space. Let F, G be O_X-modules. If F is of finite type then the canonical map SheafHom_O_X(F, G)_x → Hom_O_X, x(F_x, G_x) is injective. If F is finitely presented, this canonical morphism is an isomorphism.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}$ is of finite type then the canonical map\n$$\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x\n\\to\n\\Hom_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, \\mathcal{G}_x)\n$$\nis injective.\nIf $\\mathcal{F}$ is finitely presented,\nthis canonical morphism is an isomorphism.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CP","source_file":"modules.tex","source_line":3618,"source_end_line":3631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3618-L3631","statement_sha256":"fdc44a2304f2cd81078beab8500e8cf5d69ffaea097360b24acf416b7023c86a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3866,"rank":3866,"depth":3,"x":996.59,"y":187.204,"cluster":"sheaves-sites"},{"id":"stacks:0C6I","tag":"0C6I","title":"Internal Hom · Lemma 0C6I","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let F, G be O_Y-modules. If F is finitely presented and f is flat, then the canonical map f^*SheafHom_O_Y(F, G) → SheafHom_O_X(f^*F, f^*G) is an isomorphism.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism\nof ringed spaces. Let $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_Y$-modules.\nIf $\\mathcal{F}$ is finitely presented and $f$ is flat,\nthen the canonical map\n$$\nf^*\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{F}, \\mathcal{G})\n\\longrightarrow\n\\SheafHom_{\\mathcal{O}_X}(f^*\\mathcal{F}, f^*\\mathcal{G})\n$$\nis an isomorphism.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6I","source_file":"modules.tex","source_line":3710,"source_end_line":3722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3710-L3722","statement_sha256":"37cd34bff12001dfdb8fcac60f5ba04b6fe655f27b043b27dbf1a16bc33d2319","origin":"The Stacks Project","memory_eligible":false,"source_rank":3867,"rank":3867,"depth":14,"x":1254.872,"y":168.432,"cluster":"sheaves-sites"},{"id":"stacks:01CQ","tag":"01CQ","title":"Internal Hom · Lemma 01CQ","summary":"Let (X, O_X) be a ringed space. Let F, G be O_X-modules. If F is finitely presented then the sheaf SheafHom_O_X(F, G) is locally a kernel of a map between finite direct sums of copies of G. In particular, if G is coherent then SheafHom_O_X(F, G) is coherent too.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}$ is finitely presented then the sheaf\n$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is\nlocally a kernel of a map between finite direct sums\nof copies of $\\mathcal{G}$.\nIn particular, if $\\mathcal{G}$ is coherent then\n$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$\nis coherent too.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CQ","source_file":"modules.tex","source_line":3738,"source_end_line":3749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3738-L3749","statement_sha256":"9e0b464df0c78cde0f57f70004715362161e8ab4b469a0f3224d96e11407bcf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3868,"rank":3868,"depth":4,"x":1079.341,"y":328.989,"cluster":"sheaves-sites"},{"id":"stacks:0GMV","tag":"0GMV","title":"Internal Hom · Lemma 0GMV","summary":"Let X be a ringed space. Let F be an O_X-module of finite presentation. Let G = colim_λ ∈ Lambda G_λ be a filtered colimit of O_X-modules. Then the canonical map colim_λ SheafHom_O_X(F, G_λ) → SheafHom_O_X(F, G) is an isomorphism.","statement_latex":"Let $X$ be a ringed space. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module\nof finite presentation. Let\n$\\mathcal{G} = \\colim_{\\lambda \\in \\Lambda} \\mathcal{G}_\\lambda$\nbe a filtered colimit of $\\mathcal{O}_X$-modules. Then the canonical map\n$$\n\\colim_\\lambda \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}_\\lambda)\n\\longrightarrow\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})\n$$\nis an isomorphism.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMV","source_file":"modules.tex","source_line":3758,"source_end_line":3770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3758-L3770","statement_sha256":"4b76b5e5c62bb56f8b5541a2b50d45283f1815da16d94f95d1b4be4a7db1cba3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3869,"rank":3869,"depth":3,"x":1079.66,"y":110.78,"cluster":"sheaves-sites"},{"id":"stacks:01BS","tag":"01BS","title":"Internal Hom · Lemma 01BS","summary":"Let X be a ringed space. Let I be a preordered set and let (F_i, φ_ii') be a system over I consisting of sheaves of O_X-modules (see Categories, Section [Tag 002Z]). Assume • I is directed, • G is an O_X-module of finite presentation, and • X has a cofinal system of open coverings U : X = ⋃_j∈ J U_j with J finite and U_j ∩ U_j' quasi-compact for all j, j' ∈ J. Then we have colim_i Hom_X(G, F_i) = Hom_X(G, colim_i F_i).","statement_latex":"Let $X$ be a ringed space.\nLet $I$ be a preordered set and\nlet $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system over $I$\nconsisting of sheaves of $\\mathcal{O}_X$-modules\n(see Categories, Section \\ref{categories-section-posets-limits}).\nAssume\n\\begin{enumerate}\n\\item $I$ is directed,\n\\item $\\mathcal{G}$ is an $\\mathcal{O}_X$-module of finite presentation, and\n\\item $X$ has a cofinal system of open coverings\n$\\mathcal{U} : X = \\bigcup_{j\\in J} U_j$ with\n$J$ finite and $U_j \\cap U_{j'}$ quasi-compact\nfor all $j, j' \\in J$.\n\\end{enumerate}\nThen we have\n$$\n\\colim_i \\Hom_X(\\mathcal{G}, \\mathcal{F}_i)\n=\n\\Hom_X(\\mathcal{G}, \\colim_i \\mathcal{F}_i).\n$$","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01BS","source_file":"modules.tex","source_line":3817,"source_end_line":3839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3817-L3839","statement_sha256":"a042c8565d693ad2ce6f03b379b0f5d8bd933a2e891c9db72f1676cfa0c6cb41","origin":"The Stacks Project","memory_eligible":false,"source_rank":3870,"rank":3870,"depth":4,"x":1255.076,"y":272.024,"cluster":"sheaves-sites"},{"id":"stacks:0H2H","tag":"0H2H","title":"The annihilator of a sheaf of modules · Definition 0H2H","summary":"Let (X, O_X) be a ringed space and let F be an O_X-module. The annihilator of F, denoted Ann_O_X(F) is the kernel of the map O_X → SheafHom_O_X(F, F) discussed above.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space and let $\\mathcal{F}$\nbe an $\\mathcal{O}_X$-module. The {\\it annihilator} of $\\mathcal{F}$,\ndenoted $\\text{Ann}_{\\mathcal{O}_X}(\\mathcal{F})$\nis the kernel of the map\n$\\mathcal{O}_X \\to \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{F})$\ndiscussed above.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The annihilator of a sheaf of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2H","source_file":"modules.tex","source_line":3881,"source_end_line":3889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3881-L3889","statement_sha256":"c5fe5ab457526a060b0b43c4c8809affa2e1b834a5572d1d89aec5645410fe05","origin":"The Stacks Project","memory_eligible":false,"source_rank":3871,"rank":3871,"depth":0,"x":995.802,"y":252.641,"cluster":"sheaves-sites"},{"id":"stacks:0H2J","tag":"0H2J","title":"The annihilator of a sheaf of modules · Lemma 0H2J","summary":"Let (X, O_X) be a ringed space and let F be a sheaf of O_X-modules. If F is of finite type, then (Ann_O_X(F))_x = Ann_O_X, x(F_x).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space and let\n$\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}$ is of finite type, then\n$(\\text{Ann}_{\\mathcal{O}_X}(\\mathcal{F}))_x =\n\\text{Ann}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x)$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The annihilator of a sheaf of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2J","source_file":"modules.tex","source_line":3904,"source_end_line":3911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3904-L3911","statement_sha256":"00a62c5587fc00a4e4832837116d13a8a42ce27cef0684e6d08519c9c57d08df","origin":"The Stacks Project","memory_eligible":false,"source_rank":3872,"rank":3872,"depth":4,"x":1202.778,"y":119.685,"cluster":"sheaves-sites"},{"id":"stacks:0H2K","tag":"0H2K","title":"The annihilator of a sheaf of modules · Lemma 0H2K","summary":"Let (X, O_X) be a ringed space, let F be an O_X-module and let I ⊂ O_X be an ideal sheaf. If I ⊂ Ann_O_X(F), then F has a natural O_X/I-module structure which agrees with the usual commutative algebra construction on stalks.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space, let $\\mathcal{F}$ be an\n$\\mathcal{O}_X$-module and let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe an ideal sheaf. If\n$\\mathcal{I} \\subset \\text{Ann}_{\\mathcal{O}_X}(\\mathcal{F})$,\nthen $\\mathcal{F}$ has a natural $\\mathcal{O}_X/\\mathcal{I}$-module structure\nwhich agrees with the usual commutative algebra construction on stalks.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The annihilator of a sheaf of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2K","source_file":"modules.tex","source_line":3926,"source_end_line":3934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3926-L3934","statement_sha256":"ffd94b89a5703e4f0846d00f858afa89ee3e06bbedce5f482fb9ad2451314a4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3873,"rank":3873,"depth":2,"x":1157.032,"y":335.38,"cluster":"sheaves-sites"},{"id":"stacks:0H2L","tag":"0H2L","title":"The annihilator of a sheaf of modules · Lemma 0H2L","summary":"Let (X,O_X) be a ringed space. If O_X and F are coherent, then so is Ann_O_X(F).","statement_latex":"Let $(X,\\mathcal{O}_X)$ be a ringed space.\nIf $\\mathcal{O}_X$ and $\\mathcal{F}$ are coherent,\nthen so is $\\text{Ann}_{\\mathcal{O}_X}(\\mathcal{F})$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The annihilator of a sheaf of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2L","source_file":"modules.tex","source_line":3960,"source_end_line":3965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3960-L3965","statement_sha256":"520443cefc270b0ec699e439cbd31a8e734b2ea751204dc24d8a21be3d0ce18d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3874,"rank":3874,"depth":5,"x":1017.171,"y":150.191,"cluster":"sheaves-sites"},{"id":"stacks:062K","tag":"062K","title":"Koszul complexes · Definition 062K","summary":"Let X be a ringed space. Let φ : E → O_X be an O_X-module map. The Koszul complex K_bullet(φ) associated to φ is the sheaf of commutative differential graded algebras defined as follows: • the underlying graded algebra is the exterior algebra K_bullet(φ) = wedge(E), • the differential d : K_bullet(φ) → K_bullet(φ) is the unique derivation such that d(e) = φ(e) for all local sections e of E = K_1(φ).","statement_latex":"Let $X$ be a ringed space. Let $\\varphi : \\mathcal{E} \\to \\mathcal{O}_X$\nbe an $\\mathcal{O}_X$-module map. The\n{\\it Koszul complex} $K_\\bullet(\\varphi)$ associated to $\\varphi$\nis the sheaf of commutative differential graded algebras defined as follows:\n\\begin{enumerate}\n\\item the underlying graded algebra is the exterior algebra\n$K_\\bullet(\\varphi) = \\wedge(\\mathcal{E})$,\n\\item the differential $d : K_\\bullet(\\varphi) \\to K_\\bullet(\\varphi)$\nis the unique derivation such that $d(e) = \\varphi(e)$ for all\nlocal sections $e$ of $\\mathcal{E} = K_1(\\varphi)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Koszul complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062K","source_file":"modules.tex","source_line":3991,"source_end_line":4004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L3991-L4004","statement_sha256":"30db4c3dca4de1d1cc6304d98ca4dc6a6e14d5201a3975a5c22ae8a21a27572d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3875,"rank":3875,"depth":0,"x":1269.473,"y":207.442,"cluster":"sheaves-sites"},{"id":"stacks:062L","tag":"062L","title":"Koszul complexes · Definition 062L","summary":"Let X be a ringed space and let f_1, …, f_n ∈ Γ(X, O_X). The Koszul complex on f_1, …, f_n is the Koszul complex associated to the map (f_1, …, f_n) : O_X^⊕ n → O_X. Notation K_bullet(O_X, f_1, …, f_n), or K_bullet(O_X, f_bullet).","statement_latex":"Let $X$ be a ringed space and let\n$f_1, \\ldots, f_n \\in \\Gamma(X, \\mathcal{O}_X)$. The\n{\\it Koszul complex on $f_1, \\ldots, f_n$} is the Koszul complex\nassociated to the map\n$(f_1, \\ldots, f_n) : \\mathcal{O}_X^{\\oplus n} \\to \\mathcal{O}_X$.\nNotation $K_\\bullet(\\mathcal{O}_X, f_1, \\ldots, f_n)$,\nor $K_\\bullet(\\mathcal{O}_X, f_\\bullet)$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Koszul complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062L","source_file":"modules.tex","source_line":4018,"source_end_line":4027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4018-L4027","statement_sha256":"c16f10f724c52f68da89a9c25aa8f7651d09b08bc3aa9d7764763ea59a686b2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3876,"rank":3876,"depth":0,"x":1037.163,"y":308.487,"cluster":"sheaves-sites"},{"id":"stacks:01CS","tag":"01CS","title":"Invertible modules · Definition 01CS","summary":"Let (X, O_X) be a ringed space. An invertible O_X-module is a sheaf of O_X-modules L such that the functor Mod(O_X) → Mod(O_X), F ↦ L ⊗_O_X F is an equivalence of categories. We say that L is trivial if it is isomorphic as an O_X-module to O_X.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. An\n{\\it invertible $\\mathcal{O}_X$-module} is a sheaf\nof $\\mathcal{O}_X$-modules $\\mathcal{L}$ such that\nthe functor\n$$\n\\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\textit{Mod}(\\mathcal{O}_X),\\quad\n\\mathcal{F} \\longmapsto \\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{F}\n$$\nis an equivalence of categories. We say that $\\mathcal{L}$ is\n{\\it trivial} if it is isomorphic as an $\\mathcal{O}_X$-module\nto $\\mathcal{O}_X$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CS","source_file":"modules.tex","source_line":4046,"source_end_line":4059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4046-L4059","statement_sha256":"10b97ad85fa7acb6295867f803ca52521da6e0d0f709122b33b0943b736a9dbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3877,"rank":3877,"depth":0,"x":1127.296,"y":101.958,"cluster":"sheaves-sites"},{"id":"stacks:0B8K","tag":"0B8K","title":"Invertible modules · Lemma 0B8K","summary":"Let (X, O_X) be a ringed space. Let L be an O_X-module. Equivalent are • L is invertible, and • there exists an O_X-module N such that L ⊗_O_X N ≅ O_X. In this case L is locally a direct summand of a finite free O_X-module and the module N in (2) is isomorphic to SheafHom_O_X(L, O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{L}$\nbe an $\\mathcal{O}_X$-module. Equivalent are\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is invertible, and\n\\item there exists an $\\mathcal{O}_X$-module $\\mathcal{N}$\nsuch that\n$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N} \\cong \\mathcal{O}_X$.\n\\end{enumerate}\nIn this case $\\mathcal{L}$ is locally a direct summand of a finite free\n$\\mathcal{O}_X$-module and the module $\\mathcal{N}$ in (2) is isomorphic to\n$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X)$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8K","source_file":"modules.tex","source_line":4066,"source_end_line":4079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4066-L4079","statement_sha256":"402748b73f2d158b008a1e773f60c18fa8f609663d35da9b72bb93bab6839236","origin":"The Stacks Project","memory_eligible":false,"source_rank":3878,"rank":3878,"depth":6,"x":1227.014,"y":305.591,"cluster":"sheaves-sites"},{"id":"stacks:0B8L","tag":"0B8L","title":"Invertible modules · Lemma 0B8L","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The pullback f^*L of an invertible O_Y-module is invertible.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a\nmorphism of ringed spaces. The pullback $f^*\\mathcal{L}$ of an\ninvertible $\\mathcal{O}_Y$-module is invertible.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8L","source_file":"modules.tex","source_line":4142,"source_end_line":4147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4142-L4147","statement_sha256":"08f3af209effc78f65c8b4538a5bc4f91964415fae5739914fa4398b485ba000","origin":"The Stacks Project","memory_eligible":false,"source_rank":3879,"rank":3879,"depth":7,"x":989.497,"y":211.927,"cluster":"sheaves-sites"},{"id":"stacks:0B8M","tag":"0B8M","title":"Invertible modules · Lemma 0B8M","summary":"Let (X, O_X) be a ringed space. Any locally free O_X-module of rank 1 is invertible. If all stalks O_X, x are local rings, then the converse holds as well (but in general this is not the case).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Any locally free\n$\\mathcal{O}_X$-module of rank $1$ is invertible.\nIf all stalks $\\mathcal{O}_{X, x}$ are local rings, then\nthe converse holds as well (but in general this is not the case).","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8M","source_file":"modules.tex","source_line":4159,"source_end_line":4165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4159-L4165","statement_sha256":"b3467d012eb885ac24752617c4e578ae0734c6963b1e101a809bcd2a36bc696c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3880,"rank":3880,"depth":7,"x":1240.204,"y":146.157,"cluster":"sheaves-sites"},{"id":"stacks:01CT","tag":"01CT","title":"Invertible modules · Lemma 01CT","summary":"Let (X, O_X) be a ringed space. • If L, N are invertible O_X-modules, then so is L ⊗_O_X N. • If L is an invertible O_X-module, then so is SheafHom_O_X(L, O_X) and the evaluation map L ⊗_O_X SheafHom_O_X(L, O_X) → O_X is an isomorphism.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\n\\begin{enumerate}\n\\item If $\\mathcal{L}$, $\\mathcal{N}$ are invertible\n$\\mathcal{O}_X$-modules, then so is\n$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$.\n\\item If $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module, then so is\n$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X)$ and the evaluation map\n$\\mathcal{L} \\otimes_{\\mathcal{O}_X}\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{L}, \\mathcal{O}_X) \\to \\mathcal{O}_X$\nis an isomorphism.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CT","source_file":"modules.tex","source_line":4200,"source_end_line":4213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4200-L4213","statement_sha256":"a13fed87bb46623e7d9e3d42cd4266774545134849f68430c496d7c81d461037","origin":"The Stacks Project","memory_eligible":false,"source_rank":3881,"rank":3881,"depth":7,"x":1108.099,"y":337.096,"cluster":"sheaves-sites"},{"id":"stacks:01CU","tag":"01CU","title":"Invertible modules · Definition 01CU","summary":"Let (X, O_X) be a ringed space. Given an invertible sheaf L on X and n ∈ Z we define the nth tensor power L^⊗ n of L as the image of O_X under applying the equivalence F ↦F ⊗_O_X L exactly n times.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Given an invertible sheaf\n$\\mathcal{L}$ on $X$ and $n \\in \\mathbf{Z}$ we define the\n$n$th {\\it tensor power} $\\mathcal{L}^{\\otimes n}$ of $\\mathcal{L}$\nas the image of $\\mathcal{O}_X$ under applying the equivalence\n$\\mathcal{F} \\mapsto\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}$\nexactly $n$ times.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CU","source_file":"modules.tex","source_line":4220,"source_end_line":4228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4220-L4228","statement_sha256":"a6b6e29a036f7125bfd47cc9270185878175aeabc73d667660d7cad77c9145e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3882,"rank":3882,"depth":0,"x":1051.908,"y":121.132,"cluster":"sheaves-sites"},{"id":"stacks:01CV","tag":"01CV","title":"Invertible modules · Definition 01CV","summary":"Let (X, O_X) be a ringed space. Given an invertible sheaf L on X we define the associated graded ring to be Γ_*(X, L) = bigoplus_n ≥ 0 Γ(X, L^⊗ n) Given a sheaf of O_X-modules F we set Γ_*(X, L, F) = bigoplus_n ∈ Z Γ(X, F ⊗_O_X L^⊗ n) which we think of as a graded Γ_*(X, L)-module.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nGiven an invertible sheaf $\\mathcal{L}$ on $X$ we define\nthe {\\it associated graded ring} to be\n$$\n\\Gamma_*(X, \\mathcal{L})\n=\n\\bigoplus\\nolimits_{n \\geq 0} \\Gamma(X, \\mathcal{L}^{\\otimes n})\n$$\nGiven a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we set\n$$\n\\Gamma_*(X, \\mathcal{L}, \\mathcal{F})\n=\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\Gamma(X,\n\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})\n$$\nwhich we think of as a graded $\\Gamma_*(X, \\mathcal{L})$-module.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CV","source_file":"modules.tex","source_line":4269,"source_end_line":4287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4269-L4287","statement_sha256":"45575b9e5b2d918f4f5dc464b9f7b05b332059ffaed9215f9bb16efa89b10821","origin":"The Stacks Project","memory_eligible":false,"source_rank":3883,"rank":3883,"depth":0,"x":1267.222,"y":248.62,"cluster":"sheaves-sites"},{"id":"stacks:01CW","tag":"01CW","title":"Invertible modules · Lemma 01CW","summary":"Let (X, O_X) be a ringed space. There exists a set of invertible modules (L_i)_i ∈ I such that each invertible module on X is isomorphic to exactly one of the L_i.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nThere exists a set of invertible modules $\\{\\mathcal{L}_i\\}_{i \\in I}$\nsuch that each invertible module on $X$ is isomorphic to exactly\none of the $\\mathcal{L}_i$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CW","source_file":"modules.tex","source_line":4314,"source_end_line":4320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4314-L4320","statement_sha256":"75bd52e2b5959f58bc4e9d470839c5a7d3b2ef9426562a5951f1536c49c06828","origin":"The Stacks Project","memory_eligible":false,"source_rank":3884,"rank":3884,"depth":7,"x":1005.68,"y":276.814,"cluster":"sheaves-sites"},{"id":"stacks:01CX","tag":"01CX","title":"Invertible modules · Definition 01CX","summary":"Let (X, O_X) be a ringed space. The Picard group Pic(X) of X is the abelian group whose elements are isomorphism classes of invertible O_X-modules, with addition corresponding to tensor product.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nThe {\\it Picard group} $\\Pic(X)$ of $X$ is the\nabelian group whose elements are isomorphism classes of\ninvertible $\\mathcal{O}_X$-modules, with addition\ncorresponding to tensor product.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CX","source_file":"modules.tex","source_line":4350,"source_end_line":4357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4350-L4357","statement_sha256":"62f84fbb64dd38172ba11af833dfc5d388eb8f206637cbe9cb282e742a26eb7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3885,"rank":3885,"depth":0,"x":1176.028,"y":107.456,"cluster":"sheaves-sites"},{"id":"stacks:01CY","tag":"01CY","title":"Invertible modules · Lemma 01CY","summary":"A (local) trivialisation of a linebundle is the same as a (local) nonvanishing section. Let X be a ringed space. Assume that each stalk O_X, x is a local ring with maximal ideal m_x. Let L be an invertible O_X-module. For any section s ∈ Γ(X, L) the set X_s = (x ∈ X mid image s not∈ m_xL_x) is open in X. The map s : O_X_s → L|_X_s is an isomorphism, and there exists a section s' of L^⊗ -1 over X_s such that s' (s|_X_s) = 1.","statement_latex":"\\begin{slogan}\nA (local) trivialisation of a linebundle\nis the same as a (local) nonvanishing section.\n\\end{slogan}\nLet $X$ be a ringed space. Assume that each stalk $\\mathcal{O}_{X, x}$\nis a local ring with maximal ideal $\\mathfrak m_x$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nFor any section $s \\in \\Gamma(X, \\mathcal{L})$ the set\n$$\nX_s = \\{x \\in X \\mid \\text{image }s \\not\\in \\mathfrak m_x\\mathcal{L}_x\\}\n$$\nis open in $X$. The map $s : \\mathcal{O}_{X_s} \\to \\mathcal{L}|_{X_s}$\nis an isomorphism, and there exists a section $s'$\nof $\\mathcal{L}^{\\otimes -1}$ over $X_s$ such that $s' (s|_{X_s}) = 1$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01CY","source_file":"modules.tex","source_line":4359,"source_end_line":4375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4359-L4375","statement_sha256":"f99082ae3c2f08a097cbb7a95c578e066f1ba9346ff8c28a14737bf82d2c5f62","origin":"The Stacks Project","memory_eligible":false,"source_rank":3886,"rank":3886,"depth":8,"x":1186.619,"y":329.21,"cluster":"sheaves-sites"},{"id":"stacks:0B38","tag":"0B38","title":"Rank and determinant · Lemma 0B38","summary":"Let X be a ringed space. Let 0 → E' → E → E\" → 0 be a short exact sequence of finite locally free O_X-modules. Then there is a canonical isomorphism det(E') ⊗_O_Xdet(E\") → det(E) of O_X-modules.","statement_latex":"Let $X$ be a ringed space. Let\n$0 \\to \\mathcal{E}' \\to \\mathcal{E} \\to \\mathcal{E}'' \\to 0$\nbe a short exact sequence of finite locally free $\\mathcal{O}_X$-modules.\nThen there is a canonical isomorphism\n$$\n\\det(\\mathcal{E}') \\otimes_{\\mathcal{O}_X}\\det(\\mathcal{E}'')\n\\longrightarrow\n\\det(\\mathcal{E})\n$$\nof $\\mathcal{O}_X$-modules.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Rank and determinant","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B38","source_file":"modules.tex","source_line":4477,"source_end_line":4489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4477-L4489","statement_sha256":"90936366f7963235ad0e6a8a02101fa1d7365a574e07c6d2908817744eedd1d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3887,"rank":3887,"depth":0,"x":1000.303,"y":171.551,"cluster":"sheaves-sites"},{"id":"stacks:0FJN","tag":"0FJN","title":"Rank and determinant · Lemma 0FJN","summary":"Let (X, O_X) be a ringed space. Let F be a flat and finitely presented O_X-module. Denote det(F) ⊂ wedge^*_O_X(F) the annihilator of F ⊂ wedge^*_O_X(F). Then det(F) is an invertible O_X-module.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$\nbe a flat and finitely presented $\\mathcal{O}_X$-module.\nDenote\n$$\n\\det(\\mathcal{F}) \\subset\n\\wedge^*_{\\mathcal{O}_X}(\\mathcal{F})\n$$\nthe annihilator of $\\mathcal{F} \\subset \\wedge^*_{\\mathcal{O}_X}(\\mathcal{F})$.\nThen $\\det(\\mathcal{F})$ is an invertible $\\mathcal{O}_X$-module.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Rank and determinant","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJN","source_file":"modules.tex","source_line":4538,"source_end_line":4549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4538-L4549","statement_sha256":"37fc62369d32618b04bdf23603720e18fe1fab0809daba7c8b8905e46434af57","origin":"The Stacks Project","memory_eligible":false,"source_rank":3888,"rank":3888,"depth":7,"x":1264.726,"y":182.096,"cluster":"sheaves-sites"},{"id":"stacks:01D0","tag":"01D0","title":"Localizing sheaves of rings · Lemma 01D0","summary":"Let X be a topological space and let O_X be a presheaf of rings. Let S ⊂ O_X be a pre-sheaf of sets contained in O_X. Suppose that for every open U ⊂ X the set S(U) ⊂ O_X(U) is a multiplicative subset. • There is a map of presheaves of rings O_X → S^-1O_X such that every local section of S maps to an invertible section of O_X. • For any homomorphism of presheaves of rings O_X → A such that each local section of S maps to an invertible section of A there exists a unique…","statement_latex":"Let $X$ be a topological space and let $\\mathcal{O}_X$ be a\npresheaf of rings. Let $\\mathcal{S} \\subset \\mathcal{O}_X$\nbe a pre-sheaf of sets contained in $\\mathcal{O}_X$.\nSuppose that for every open $U \\subset X$ the set\n$\\mathcal{S}(U) \\subset \\mathcal{O}_X(U)$ is a multiplicative subset.\n\\begin{enumerate}\n\\item There is a map of presheaves of rings\n$\\mathcal{O}_X \\to \\mathcal{S}^{-1}\\mathcal{O}_X$\nsuch that every local section of $\\mathcal{S}$ maps to an invertible\nsection of $\\mathcal{O}_X$.\n\\item For any homomorphism of presheaves of rings\n$\\mathcal{O}_X \\to \\mathcal{A}$ such that each local section\nof $\\mathcal{S}$ maps to an invertible section of $\\mathcal{A}$\nthere exists a unique factorization\n$\\mathcal{S}^{-1}\\mathcal{O}_X \\to \\mathcal{A}$.\n\\item For any $x \\in X$ we have\n$$\n(\\mathcal{S}^{-1}\\mathcal{O}_X)_x = \\mathcal{S}_x^{-1} \\mathcal{O}_{X, x}.\n$$\n\\item The sheafification $(\\mathcal{S}^{-1}\\mathcal{O}_X)^\\#$ is a sheaf\nof rings with a map of sheaves of rings\n$(\\mathcal{O}_X)^\\# \\to (\\mathcal{S}^{-1}\\mathcal{O}_X)^\\#$\nwhich is universal for maps of $(\\mathcal{O}_X)^\\#$ into sheaves\nof rings such that each local section of $\\mathcal{S}$ maps\nto an invertible section.\n\\item For any $x \\in X$ we have\n$$\n(\\mathcal{S}^{-1}\\mathcal{O}_X)^\\#_x = \\mathcal{S}_x^{-1} \\mathcal{O}_{X, x}.\n$$\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Localizing sheaves of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01D0","source_file":"modules.tex","source_line":4618,"source_end_line":4650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4618-L4650","statement_sha256":"c645fb312082ea067d50f07cc00ff2a68048639a6705dbaa7ce7b50de4e923b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3889,"rank":3889,"depth":0,"x":1061.07,"y":324.496,"cluster":"sheaves-sites"},{"id":"stacks:01D1","tag":"01D1","title":"Localizing sheaves of rings · Lemma 01D1","summary":"Let X be a topological space. Let O_X be a presheaf of rings. Let S ⊂ O_X be a pre-sheaf of sets contained in O_X. Suppose that for every open U ⊂ X the set S(U) ⊂ O_X(U) is a multiplicative subset. For any presheaf of O_X-modules F we have S^-1F = S^-1O_X ⊗_p, O_X F (see Sheaves, Section [Tag 006P] for notation) and if F and O_X are sheaves then (S^-1F)^\\# = (S^-1O_X)^\\# ⊗_O_X F (see Sheaves, Section [Tag 0088] for notation).","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{O}_X$ be a presheaf of rings.\nLet $\\mathcal{S} \\subset \\mathcal{O}_X$ be a pre-sheaf of sets contained\nin $\\mathcal{O}_X$. Suppose that for every open $U \\subset X$ the set\n$\\mathcal{S}(U) \\subset \\mathcal{O}_X(U)$ is a multiplicative subset.\nFor any presheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we\nhave\n$$\n\\mathcal{S}^{-1}\\mathcal{F}\n=\n\\mathcal{S}^{-1}\\mathcal{O}_X \\otimes_{p, \\mathcal{O}_X} \\mathcal{F}\n$$\n(see Sheaves, Section \\ref{sheaves-section-presheaves-modules} for notation)\nand if $\\mathcal{F}$ and $\\mathcal{O}_X$ are sheaves then\n$$\n(\\mathcal{S}^{-1}\\mathcal{F})^\\#\n=\n(\\mathcal{S}^{-1}\\mathcal{O}_X)^\\# \\otimes_{\\mathcal{O}_X} \\mathcal{F}\n$$\n(see Sheaves, Section \\ref{sheaves-section-sheafification-presheaves-modules}\nfor notation).","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Localizing sheaves of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01D1","source_file":"modules.tex","source_line":4673,"source_end_line":4696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4673-L4696","statement_sha256":"b92ac1ec6336d46cbb8140c23373ea27f719e3663685274ff554dd370cbd2d0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3890,"rank":3890,"depth":0,"x":1096.763,"y":103.724,"cluster":"sheaves-sites"},{"id":"stacks:01UN","tag":"01UN","title":"Modules of differentials · Definition 01UN","summary":"Let X be a topological space. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings. Let F be an O_2-module. An O_1-derivation or more precisely a φ-derivation into F is a map D : O_2 → F which is additive, annihilates the image of O_1 → O_2, and satisfies the Leibniz rule D(ab) = aD(b) + D(a)b for all a, b local sections of O_2 (wherever they are both defined). We denote Der_O_1(O_2, F) the set of φ-derivations into F.","statement_latex":"Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. Let $\\mathcal{F}$ be an\n$\\mathcal{O}_2$-module. An {\\it $\\mathcal{O}_1$-derivation} or more precisely\na {\\it $\\varphi$-derivation} into $\\mathcal{F}$ is a map\n$D : \\mathcal{O}_2 \\to \\mathcal{F}$ which is additive, annihilates the image\nof $\\mathcal{O}_1 \\to \\mathcal{O}_2$, and satisfies the\n{\\it Leibniz rule}\n$$\nD(ab) = aD(b) + D(a)b\n$$\nfor all $a, b$ local sections of $\\mathcal{O}_2$ (wherever they are both\ndefined). We denote $\\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$\nthe set of $\\varphi$-derivations into $\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UN","source_file":"modules.tex","source_line":4718,"source_end_line":4733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4718-L4733","statement_sha256":"d3d418d98a667c77e921780c0512c395cfcda940f4141e3aa78edbe07cd246a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3891,"rank":3891,"depth":0,"x":1248.126,"y":286.945,"cluster":"sheaves-sites"},{"id":"stacks:08RM","tag":"08RM","title":"Modules of differentials · Lemma 08RM","summary":"Let X be a topological space. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings. The functor Mod(O_2) → Ab, F ↦ Der_O_1(O_2, F) is representable.","statement_latex":"Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. The functor\n$$\n\\textit{Mod}(\\mathcal{O}_2) \\longrightarrow \\textit{Ab}, \\quad\n\\mathcal{F} \\longmapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})\n$$\nis representable.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RM","source_file":"modules.tex","source_line":4754,"source_end_line":4763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4754-L4763","statement_sha256":"2406579f06317b41c309523285067adc364b943a35abf4d30cca9d1e34333f34","origin":"The Stacks Project","memory_eligible":false,"source_rank":3892,"rank":3892,"depth":0,"x":988.927,"y":237.68,"cluster":"sheaves-sites"},{"id":"stacks:08RP","tag":"08RP","title":"Modules of differentials · Definition 08RP","summary":"Let X be a topological space. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings on X. The module of differentials of φ is the object representing the functor F ↦ Der_O_1(O_2, F) which exists by Lemma [Tag 08RM]. It is denoted Ω_O_2/O_1, and the universal φ-derivation is denoted d : O_2 → Ω_O_2/O_1.","statement_latex":"Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings on $X$. The {\\it module of differentials}\nof $\\varphi$ is the object representing the functor\n$\\mathcal{F} \\mapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$\nwhich exists by Lemma \\ref{lemma-universal-module}.\nIt is denoted $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$, and the {\\it universal\n$\\varphi$-derivation} is denoted\n$\\text{d} : \\mathcal{O}_2 \\to \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RP","source_file":"modules.tex","source_line":4817,"source_end_line":4827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4817-L4827","statement_sha256":"a3274e72b97a14fcffac3738c3d554801bf708abf64671d86d5006555f4754ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":3893,"rank":3893,"depth":1,"x":1219.892,"y":126.827,"cluster":"sheaves-sites"},{"id":"stacks:08TD","tag":"08TD","title":"Modules of differentials · Lemma 08TD","summary":"Let X be a topological space. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings on X. Then Ω_O_2/O_1 is the sheaf associated to the presheaf U ↦ Ω_O_2(U)/O_1(U).","statement_latex":"Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings on $X$. Then\n$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$ is the sheaf associated to the\npresheaf $U \\mapsto \\Omega_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TD","source_file":"modules.tex","source_line":4835,"source_end_line":4841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4835-L4841","statement_sha256":"5fcdea2c580bff9e9b38b47189c4f18c2e6ea1aa9e9d20016c87fc46bb024aff","origin":"The Stacks Project","memory_eligible":false,"source_rank":3894,"rank":3894,"depth":1,"x":1138.651,"y":339.824,"cluster":"sheaves-sites"},{"id":"stacks:08RQ","tag":"08RQ","title":"Modules of differentials · Lemma 08RQ","summary":"Let X be a topological space. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings. For U ⊂ X open there is a canonical isomorphism Ω_O_2/O_1|_U = Ω_(O_2|_U)/(O_1|_U) compatible with universal derivations.","statement_latex":"Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. For $U \\subset X$ open\nthere is a canonical isomorphism\n$$\n\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}|_U =\n\\Omega_{(\\mathcal{O}_2|_U)/(\\mathcal{O}_1|_U)}\n$$\ncompatible with universal derivations.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RQ","source_file":"modules.tex","source_line":4861,"source_end_line":4871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4861-L4871","statement_sha256":"a2d40e98c85653e6b5ccd462571e1568471ecd888f03116d378c35c6fcbba6d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3895,"rank":3895,"depth":1,"x":1027.165,"y":136.474,"cluster":"sheaves-sites"},{"id":"stacks:08RR","tag":"08RR","title":"Modules of differentials · Lemma 08RR","summary":"Let f : Y → X be a continuous map of topological spaces. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings on X. Then there is a canonical identification f^-1Ω_O_2/O_1 = Ω_f^-1O_2/f^-1O_1 compatible with universal derivations.","statement_latex":"Let $f : Y \\to X$ be a continuous map of topological spaces.\nLet $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings on $X$.\nThen there is a canonical identification\n$f^{-1}\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} =\n\\Omega_{f^{-1}\\mathcal{O}_2/f^{-1}\\mathcal{O}_1}$\ncompatible with universal derivations.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RR","source_file":"modules.tex","source_line":4878,"source_end_line":4887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4878-L4887","statement_sha256":"74aae29e90756d648db41604355535272404366ce2e75ba9791409f7a14e5a0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3896,"rank":3896,"depth":1,"x":1273.133,"y":223.243,"cluster":"sheaves-sites"},{"id":"stacks:08TE","tag":"08TE","title":"Modules of differentials · Lemma 08TE","summary":"Let X be a topological space. Let O_1 → O_2 be a homomorphism of sheaves of rings on X. Let x ∈ X. Then we have Ω_O_2/O_1, x = Ω_O_2, x/O_1, x.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings on $X$. Let $x \\in X$. Then we have\n$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1, x} =\n\\Omega_{\\mathcal{O}_{2, x}/\\mathcal{O}_{1, x}}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TE","source_file":"modules.tex","source_line":4903,"source_end_line":4909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4903-L4909","statement_sha256":"17dd8a767aa1bc12cfd0778c191d174fea49afef91f48a70e9cc9dc86b37aa17","origin":"The Stacks Project","memory_eligible":false,"source_rank":3897,"rank":3897,"depth":2,"x":1021.747,"y":298.899,"cluster":"sheaves-sites"},{"id":"stacks:08RS","tag":"08RS","title":"Modules of differentials · Lemma 08RS","summary":"Let X be a topological space. Let xymatrix O_2 ar[r]_φ & O_2' O_1 ar[r] ar[u] & O'_1 ar[u] be a commutative diagram of sheaves of rings on X. The map O_2 → O'_2 composed with the map d : O'_2 → Ω_O'_2/O'_1 is a O_1-derivation. Hence we obtain a canonical map of O_2-modules Ω_O_2/O_1 → Ω_O'_2/O'_1. It is uniquely characterized by the property that d(f) ↦ d(φ(f)) for any local section f of O_2. In this way Ω_-/- becomes a functor on the category of arrows of sheaves of rings.","statement_latex":"Let $X$ be a topological space. Let\n$$\n\\xymatrix{\n\\mathcal{O}_2 \\ar[r]_\\varphi & \\mathcal{O}_2' \\\\\n\\mathcal{O}_1 \\ar[r] \\ar[u] & \\mathcal{O}'_1 \\ar[u]\n}\n$$\nbe a commutative diagram of sheaves of rings on $X$. The map\n$\\mathcal{O}_2 \\to \\mathcal{O}'_2$ composed with the map\n$\\text{d} : \\mathcal{O}'_2 \\to \\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$\nis a $\\mathcal{O}_1$-derivation. Hence we obtain a canonical map of\n$\\mathcal{O}_2$-modules\n$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\to\n\\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$.\nIt is uniquely characterized by the property that\n$\\text{d}(f) \\mapsto \\text{d}(\\varphi(f))$\nfor any local section $f$ of $\\mathcal{O}_2$.\nIn this way $\\Omega_{-/-}$ becomes a functor on the category\nof arrows of sheaves of rings.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RS","source_file":"modules.tex","source_line":4919,"source_end_line":4940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4919-L4940","statement_sha256":"53af802516597996001d6e75cd5748eb8c073ac23f14fd9a7caeda2acb4eeafc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3898,"rank":3898,"depth":0,"x":1146.391,"y":100.284,"cluster":"sheaves-sites"},{"id":"stacks:08TF","tag":"08TF","title":"Modules of differentials · Lemma 08TF","summary":"In Lemma [Tag 08RS] suppose that O_2 → O'_2 is surjective with kernel I ⊂ O_2 and assume that O_1 = O'_1. Then there is a canonical exact sequence of O'_2-modules I/I^2 → Ω_O_2/O_1 ⊗_O_2 O'_2 → Ω_O'_2/O_1 → 0 The leftmost map is characterized by the rule that a local section f of I maps to df ⊗ 1.","statement_latex":"In Lemma \\ref{lemma-functoriality-differentials} suppose that\n$\\mathcal{O}_2 \\to \\mathcal{O}'_2$ is surjective with kernel\n$\\mathcal{I} \\subset \\mathcal{O}_2$ and assume that\n$\\mathcal{O}_1 = \\mathcal{O}'_1$. Then there is a canonical exact\nsequence of $\\mathcal{O}'_2$-modules\n$$\n\\mathcal{I}/\\mathcal{I}^2\n\\longrightarrow\n\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{O}'_2\n\\longrightarrow\n\\Omega_{\\mathcal{O}'_2/\\mathcal{O}_1}\n\\longrightarrow\n0\n$$\nThe leftmost map is characterized by the rule that a local section\n$f$ of $\\mathcal{I}$ maps to $\\text{d}f \\otimes 1$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TF","source_file":"modules.tex","source_line":4946,"source_end_line":4964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4946-L4964","statement_sha256":"1377438519af38aa8d4de71e42dbc5250b55a4200ce52aee72c98f42aa4dbf19","origin":"The Stacks Project","memory_eligible":false,"source_rank":3899,"rank":3899,"depth":3,"x":1214.265,"y":317.669,"cluster":"sheaves-sites"},{"id":"stacks:08RT","tag":"08RT","title":"Modules of differentials · Definition 08RT","summary":"Let (f, f^sharp) : (X, O_X) → (S, O_S) be a morphism of ringed spaces. • Let F be an O_X-module. An S-derivation into F is a f^-1O_S-derivation, or more precisely a f^sharp-derivation in the sense of Definition [Tag 01UN]. We denote Der_S(O_X, F) the set of S-derivations into F. • The sheaf of differentials Ω_X/S of X over S is the module of differentials Ω_O_X/f^-1O_S endowed with its universal S-derivation d_X/S : O_X → Ω_X/S.","statement_latex":"Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$\nbe a morphism of ringed spaces.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. An {\\it $S$-derivation}\ninto $\\mathcal{F}$ is a $f^{-1}\\mathcal{O}_S$-derivation, or more\nprecisely a $f^\\sharp$-derivation in the sense of\nDefinition \\ref{definition-derivation}.\nWe denote $\\text{Der}_S(\\mathcal{O}_X, \\mathcal{F})$\nthe set of $S$-derivations into $\\mathcal{F}$.\n\\item The {\\it sheaf of differentials $\\Omega_{X/S}$ of $X$ over $S$}\nis the module of differentials $\\Omega_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_S}$\nendowed with its universal\n$S$-derivation $\\text{d}_{X/S} : \\mathcal{O}_X \\to \\Omega_{X/S}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RT","source_file":"modules.tex","source_line":4985,"source_end_line":5001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L4985-L5001","statement_sha256":"1aeaae47823d4c37fe072b3f974928772291311bd6a17bb53c5d85ab3f4c6f6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3900,"rank":3900,"depth":1,"x":989.191,"y":195.772,"cluster":"sheaves-sites"},{"id":"stacks:01UP","tag":"01UP","title":"Modules of differentials · Lemma 01UP","summary":"Let (f, f^sharp) : (X, O_X) → (S, O_S) be a morphism of ringed spaces. Consider a short exact sequence 0 → I → A → O_X → 0 Here A is a sheaf of f^-1O_S-algebras, π : A → O_X is a surjection of sheaves of f^-1O_S-algebras, and I = Ker(π) is its kernel. Assume I an ideal sheaf with square zero in A. So I has a natural structure of an O_X-module. A section s : O_X → A of π is a f^-1O_S-algebra map such that π ∘ s = id. Given any section s : O_X → A of π and any S-derivation…","statement_latex":"Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$\nbe a morphism of ringed spaces. Consider a short exact sequence\n$$\n0 \\to \\mathcal{I} \\to \\mathcal{A} \\to \\mathcal{O}_X \\to 0\n$$\nHere $\\mathcal{A}$ is a sheaf of $f^{-1}\\mathcal{O}_S$-algebras,\n$\\pi : \\mathcal{A} \\to \\mathcal{O}_X$ is a surjection\nof sheaves of $f^{-1}\\mathcal{O}_S$-algebras, and\n$\\mathcal{I} = \\Ker(\\pi)$ is its kernel. Assume $\\mathcal{I}$ an ideal\nsheaf with square zero in $\\mathcal{A}$. So $\\mathcal{I}$\nhas a natural structure of an $\\mathcal{O}_X$-module.\nA section $s : \\mathcal{O}_X \\to \\mathcal{A}$ of $\\pi$\nis a $f^{-1}\\mathcal{O}_S$-algebra map such that $\\pi \\circ s = \\text{id}$.\nGiven any section $s : \\mathcal{O}_X \\to \\mathcal{A}$\nof $\\pi$ and any $S$-derivation $D : \\mathcal{O}_X \\to \\mathcal{I}$\nthe map\n$$\ns + D : \\mathcal{O}_X \\to \\mathcal{A}\n$$\nis a section of $\\pi$ and every section $s'$ is of the form $s + D$\nfor a unique $S$-derivation $D$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UP","source_file":"modules.tex","source_line":5007,"source_end_line":5030,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5007-L5030","statement_sha256":"19b56d2ca9fd6cb349c2e8886fde7f9c193accbb19df55a0004fab223b97c68e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3901,"rank":3901,"depth":0,"x":1253.429,"y":157.909,"cluster":"sheaves-sites"},{"id":"stacks:08RU","tag":"08RU","title":"Modules of differentials · Lemma 08RU","summary":"Let xymatrix X' ar[d]_h' ar[r]_f & X ar[d]^h S' ar[r]^g & S be a commutative diagram of ringed spaces. • The canonical map O_X → f_*O_X' composed with f_*d_X'/S' : f_*O_X' → f_*Ω_X'/S' is a S-derivation and we obtain a canonical map of O_X-modules Ω_X/S → f_*Ω_X'/S'. • The commutative diagram xymatrix f^-1O_X ar[r] & O_X' f^-1h^-1O_S ar[u] ar[r] & (h')^-1O_S' ar[u] induces by Lemmas [Tag 08RR] and [Tag 08RS] a canonical map f^-1Ω_X/S → Ω_X'/S'. These two maps correspond…","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[d]_{h'} \\ar[r]_f & X \\ar[d]^h \\\\\nS' \\ar[r]^g & S\n}\n$$\nbe a commutative diagram of ringed spaces.\n\\begin{enumerate}\n\\item The canonical map $\\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with\n$f_*\\text{d}_{X'/S'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/S'}$ is a\n$S$-derivation and we obtain a canonical map of $\\mathcal{O}_X$-modules\n$\\Omega_{X/S} \\to f_*\\Omega_{X'/S'}$.\n\\item The commutative diagram\n$$\n\\xymatrix{\nf^{-1}\\mathcal{O}_X \\ar[r] & \\mathcal{O}_{X'} \\\\\nf^{-1}h^{-1}\\mathcal{O}_S \\ar[u] \\ar[r] & (h')^{-1}\\mathcal{O}_{S'} \\ar[u]\n}\n$$\ninduces by Lemmas \\ref{lemma-pullback-differentials} and\n\\ref{lemma-functoriality-differentials}\na canonical map $f^{-1}\\Omega_{X/S} \\to \\Omega_{X'/S'}$.\n\\end{enumerate}\nThese two maps correspond (via adjointness of $f_*$ and $f^*$ and\nvia $f^*\\Omega_{X/S} =\nf^{-1}\\Omega_{X/S} \\otimes_{f^{-1}\\mathcal{O}_X} \\mathcal{O}_{X'}$ and\nSheaves, Lemma \\ref{sheaves-lemma-adjointness-tensor-restrict})\nto the same $\\mathcal{O}_{X'}$-module homomorphism\n$$\nc_f : f^*\\Omega_{X/S} \\longrightarrow \\Omega_{X'/S'}\n$$\nwhich is uniquely characterized by the property that\n$f^*\\text{d}_{X/S}(a)$ maps to $\\text{d}_{X'/S'}(f^*a)$\nfor any local section $a$ of $\\mathcal{O}_X$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RU","source_file":"modules.tex","source_line":5053,"source_end_line":5090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5053-L5090","statement_sha256":"2fb240118f68ae28a14ef5b1583957b49c0a562ebf3b0211b06583bd9e1745b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3902,"rank":3902,"depth":2,"x":1088.879,"y":335.929,"cluster":"sheaves-sites"},{"id":"stacks:01UW","tag":"01UW","title":"Modules of differentials · Lemma 01UW","summary":"Let xymatrix X\" ar[d] ar[r]_g & X' ar[d] ar[r]_f & X ar[d] S\" ar[r] & S' ar[r] & S be a commutative diagram of ringed spaces. With notation as in Lemma [Tag 08RU] we have c_f ∘ g = c_g ∘ g^* c_f as maps (f ∘ g)^*Ω_X/S → Ω_X\"/S\".","statement_latex":"Let\n$$\n\\xymatrix{\nX'' \\ar[d] \\ar[r]_g & X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\\nS'' \\ar[r] & S' \\ar[r] & S\n}\n$$\nbe a commutative diagram of ringed spaces. With notation as in\nLemma \\ref{lemma-functoriality-differentials-ringed-spaces} we have\n$$\nc_{f \\circ g} = c_g \\circ g^* c_f\n$$\nas maps $(f \\circ g)^*\\Omega_{X/S} \\to \\Omega_{X''/S''}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UW","source_file":"modules.tex","source_line":5096,"source_end_line":5111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5096-L5111","statement_sha256":"b560f3d8707129d19b835efaea96e20d2afd6620151d6f29e6524ce24d950143","origin":"The Stacks Project","memory_eligible":false,"source_rank":3903,"rank":3903,"depth":3,"x":1067.037,"y":111.082,"cluster":"sheaves-sites"},{"id":"stacks:0H79","tag":"0H79","title":"Modules of differentials · Lemma 0H79","summary":"Let f : X → Y, g : Y → S be morphisms of ringed spaces Then there is a canonical exact sequence f^*Ω_Y/S → Ω_X/S → Ω_X/Y → 0 where the maps come from applications of Lemma [Tag 08RU].","statement_latex":"Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of ringed spaces\nThen there is a canonical exact sequence\n$$\nf^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0\n$$\nwhere the maps come from applications of\nLemma \\ref{lemma-functoriality-differentials-ringed-spaces}.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H79","source_file":"modules.tex","source_line":5117,"source_end_line":5126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5117-L5126","statement_sha256":"232bebc8c54514654e2930592365e99b719fa6f81a47b5b13fb0b6f7381dedde","origin":"The Stacks Project","memory_eligible":false,"source_rank":3904,"rank":3904,"depth":3,"x":1264.14,"y":264.629,"cluster":"sheaves-sites"},{"id":"stacks:0G3Q","tag":"0G3Q","title":"Finite order differential operators · Definition 0G3Q","summary":"Let X be a topological space. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings on X. Let k ≥ 0 be an integer. Let F, G be sheaves of O_2-modules. A differential operator D : F → G of order k is an is an O_1-linear map such that for all local sections g of O_2 the map s ↦ D(gs) - gD(s) is a differential operator of order k - 1. For the base case k = 0 we define a differential operator of order 0 to be an O_2-linear map.","statement_latex":"Let $X$ be a topological space. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings on $X$. Let $k \\geq 0$ be an integer.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}_2$-modules.\nA {\\it differential operator $D : \\mathcal{F} \\to \\mathcal{G}$ of order $k$}\nis an is an $\\mathcal{O}_1$-linear map such that for all local sections\n$g$ of $\\mathcal{O}_2$ the map $s \\mapsto D(gs) - gD(s)$ is a\ndifferential operator of order $k - 1$. For the base case $k = 0$\nwe define a differential operator of order $0$ to be an\n$\\mathcal{O}_2$-linear map.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Finite order differential operators","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3Q","source_file":"modules.tex","source_line":5173,"source_end_line":5184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5173-L5184","statement_sha256":"d9520d542c0d944de4b9ee1fffa49e829fe6f6d902c000cf4064f92fb9a20bfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":3905,"rank":3905,"depth":0,"x":995.074,"y":263.246,"cluster":"sheaves-sites"},{"id":"stacks:0G3R","tag":"0G3R","title":"Finite order differential operators · Lemma 0G3R","summary":"Let X be a topological space. Let O_1 → O_2 be a map of sheaves of rings on X. Let E, F, G be sheaves of O_2-modules. If D : E → F and D' : F → G are differential operators of order k and k', then D' ∘ D is a differential operator of order k + k'.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on $X$.\nLet $\\mathcal{E}, \\mathcal{F}, \\mathcal{G}$ be sheaves of\n$\\mathcal{O}_2$-modules.\nIf $D : \\mathcal{E} \\to \\mathcal{F}$ and $D' : \\mathcal{F} \\to \\mathcal{G}$\nare differential operators of order $k$ and $k'$, then $D' \\circ D$ is a\ndifferential operator of order $k + k'$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3R","source_file":"modules.tex","source_line":5207,"source_end_line":5216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5207-L5216","statement_sha256":"df8e43a1e7a0cfa7d1386ce6333cf3880daf392a28b4a232b7cd19fffa458c37","origin":"The Stacks Project","memory_eligible":false,"source_rank":3906,"rank":3906,"depth":0,"x":1194.775,"y":111.451,"cluster":"sheaves-sites"},{"id":"stacks:0G3S","tag":"0G3S","title":"Finite order differential operators · Lemma 0G3S","summary":"Let X be a topological space. Let O_1 → O_2 be a map of sheaves of rings on X. Let F be a sheaf of O_2-modules. Let k ≥ 0. There exists a sheaf of O_2-modules P^k_O_2/O_1(F) and a canonical isomorphism Diff^k_O_2/O_1(F, G) = Hom_O_2( P^k_O_2/O_1(F), G) functorial in the O_2-module G.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on $X$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules.\nLet $k \\geq 0$. There exists a sheaf of $\\mathcal{O}_2$-modules\n$\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$\nand a canonical isomorphism\n$$\n\\text{Diff}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}, \\mathcal{G}) =\n\\Hom_{\\mathcal{O}_2}(\n\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}), \\mathcal{G})\n$$\nfunctorial in the $\\mathcal{O}_2$-module $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3S","source_file":"modules.tex","source_line":5228,"source_end_line":5242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5228-L5242","statement_sha256":"86bfee37fabd974cfde572e5360eb023e679ba47af7d64473c64ed8b87f24a5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3907,"rank":3907,"depth":1,"x":1169.564,"y":336.905,"cluster":"sheaves-sites"},{"id":"stacks:0G3T","tag":"0G3T","title":"Finite order differential operators · Definition 0G3T","summary":"Let X be a topological space. Let O_1 → O_2 be a map of sheaves of rings on X. Let F be a sheaf of O_2-modules. The module P^k_O_2/O_1(F) constructed in Lemma [Tag 0G3S] is called the module of principal parts of order k of F.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings on $X$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules.\nThe module $\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$\nconstructed in Lemma \\ref{lemma-module-principal-parts}\nis called the {\\it module of principal parts of order $k$} of $\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Finite order differential operators","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3T","source_file":"modules.tex","source_line":5285,"source_end_line":5293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5285-L5293","statement_sha256":"66f1d5d4fb5e5d1453a6022d4dc6ca3cf8dfac52cfb153f8ab46ca883a5b7639","origin":"The Stacks Project","memory_eligible":false,"source_rank":3908,"rank":3908,"depth":2,"x":1006.701,"y":156.187,"cluster":"sheaves-sites"},{"id":"stacks:0G3U","tag":"0G3U","title":"Finite order differential operators · Lemma 0G3U","summary":"Let X be a topological space. Let O_1 → O_2 be a homomorphism of presheaves of rings on X. Let F be a presheaf of O_2-modules. Then P^k_O_2^\\#/O_1^\\#(F^\\#) is the sheaf associated to the presheaf U ↦ P^k_O_2(U)/O_1(U)(F(U)).","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of presheaves of rings on $X$. Let $\\mathcal{F}$ be a\npresheaf of $\\mathcal{O}_2$-modules. Then\n$\\mathcal{P}^k_{\\mathcal{O}_2^\\#/\\mathcal{O}_1^\\#}(\\mathcal{F}^\\#)$\nis the sheaf associated to the presheaf\n$U \\mapsto P^k_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}(\\mathcal{F}(U))$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3U","source_file":"modules.tex","source_line":5310,"source_end_line":5318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5310-L5318","statement_sha256":"2da507b34849ea1ba875326addf50914073ad8973179d27c305acdcad2b105dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":3909,"rank":3909,"depth":2,"x":1272.365,"y":197.071,"cluster":"sheaves-sites"},{"id":"stacks:0G3V","tag":"0G3V","title":"Finite order differential operators · Lemma 0G3V","summary":"Let X be a topological space. Let O_1 → O_2 be a homomorphism of sheaves of rings on X. Let F be a sheaf of O_2-modules. There is a canonical short exact sequence 0 → Ω_O_2/O_1 ⊗_O_2 F → P^1_O_2/O_1(F) → F → 0 functorial in F called the sequence of principal parts.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings on $X$. Let $\\mathcal{F}$ be a\nsheaf of $\\mathcal{O}_2$-modules. There is a\ncanonical short exact sequence\n$$\n0 \\to\n\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{F} \\to\n\\mathcal{P}^1_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}) \\to\n\\mathcal{F} \\to 0\n$$\nfunctorial in $\\mathcal{F}$ called the {\\it sequence of principal parts}.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3V","source_file":"modules.tex","source_line":5328,"source_end_line":5341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5328-L5341","statement_sha256":"0d67629dba16ffa18862a0cebaf7472c1d9cf5754eb831b03f0e8ce6964fdbf3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3910,"rank":3910,"depth":3,"x":1043.382,"y":317.778,"cluster":"sheaves-sites"},{"id":"stacks:0G3X","tag":"0G3X","title":"Finite order differential operators · Definition 0G3X","summary":"Let (f, f^sharp) : (X, O_X) → (S, O_S) be a morphism of ringed spaces. Let F and G be O_X-modules. Let k ≥ 0 be an integer. A differential operator of order k on X/S is a differential operator D : F → G with respect to f^sharp : f^-1O_S → O_X We denote Diff^k_X/S(F, G) the set of these differential operators.","statement_latex":"Let $(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$\nbe a morphism of ringed spaces.\nLet $\\mathcal{F}$ and $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.\nLet $k \\geq 0$ be an integer.\nA {\\it differential operator of order $k$ on $X/S$}\nis a differential operator $D : \\mathcal{F} \\to \\mathcal{G}$\nwith respect to $f^\\sharp : f^{-1}\\mathcal{O}_S \\to \\mathcal{O}_X$\nWe denote $\\text{Diff}^k_{X/S}(\\mathcal{F}, \\mathcal{G})$\nthe set of these differential operators.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"Finite order differential operators","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3X","source_file":"modules.tex","source_line":5389,"source_end_line":5400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5389-L5400","statement_sha256":"7cd5632492e88d7eee21466bbfcf193f484aa34a2bff980159160307fae4c358","origin":"The Stacks Project","memory_eligible":false,"source_rank":3911,"rank":3911,"depth":0,"x":1115.227,"y":98.64,"cluster":"sheaves-sites"},{"id":"stacks:0FKM","tag":"0FKM","title":"The de Rham complex · Definition 0FKM","summary":"In the situation above, the de Rham complex of B over A is the unique complex Ω_B/A^0 → Ω_B/A^1 → Ω_B/A^2 → … of sheaves of A-modules whose differential in degree 0 is given by d : B → Ω_B/A and whose differentials in higher degrees have the following property d(b_0db_1 wedge … wedge db_p) = db_0 wedge db_1 wedge … wedge db_p where b_0, …, b_p ∈ B(U) are sections over a common open U ⊂ X.","statement_latex":"In the situation above, the\n{\\it de Rham complex of $\\mathcal{B}$ over $\\mathcal{A}$}\nis the unique complex\n$$\n\\Omega_{\\mathcal{B}/\\mathcal{A}}^0 \\to\n\\Omega_{\\mathcal{B}/\\mathcal{A}}^1 \\to\n\\Omega_{\\mathcal{B}/\\mathcal{A}}^2 \\to \\ldots\n$$\nof sheaves of $\\mathcal{A}$-modules whose differential in degree\n$0$ is given by $\\text{d} : \\mathcal{B} \\to \\Omega_{\\mathcal{B}/\\mathcal{A}}$\nand whose differentials in higher degrees have the following property\n\\begin{equation}\n\n\\text{d}\\left(b_0\\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_p\\right) =\n\\text{d}b_0 \\wedge \\text{d}b_1 \\wedge \\ldots \\wedge \\text{d}b_p\n\\end{equation}\nwhere $b_0, \\ldots, b_p \\in \\mathcal{B}(U)$ are sections over a common\nopen $U \\subset X$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The de Rham complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKM","source_file":"modules.tex","source_line":5435,"source_end_line":5455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5435-L5455","statement_sha256":"37d67f56f27aabe8b3f3cbf0e0a0b75437965d75d307072323725e2562029412","origin":"The Stacks Project","memory_eligible":false,"source_rank":3912,"rank":3912,"depth":0,"x":1238.587,"y":301.18,"cluster":"sheaves-sites"},{"id":"stacks:0FKP","tag":"0FKP","title":"The de Rham complex · Lemma 0FKP","summary":"Let f : Y → X be a continuous map of topological spaces. Let A → B be a homomorphism of sheaves of rings on X. Then there is a canonical identification f^-1Ω^bullet_B/A = Ω^bullet_f^-1B/f^-1A of de Rham complexes.","statement_latex":"Let $f : Y \\to X$ be a continuous map of topological spaces.\nLet $\\mathcal{A} \\to \\mathcal{B}$\nbe a homomorphism of sheaves of rings on $X$.\nThen there is a canonical identification\n$f^{-1}\\Omega^\\bullet_{\\mathcal{B}/\\mathcal{A}} =\n\\Omega^\\bullet_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$\nof de Rham complexes.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The de Rham complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKP","source_file":"modules.tex","source_line":5475,"source_end_line":5484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5475-L5484","statement_sha256":"20d7dfc548c493306b135ce9f5e8f5d9d2ca4f46f1799925377c6d844b8a3613","origin":"The Stacks Project","memory_eligible":false,"source_rank":3913,"rank":3913,"depth":2,"x":984.515,"y":221.755,"cluster":"sheaves-sites"},{"id":"stacks:0G3Y","tag":"0G3Y","title":"The de Rham complex · Lemma 0G3Y","summary":"Let X be a topological space. Let A → B be a homomorphism of sheaves of rings on X. The differentials d : Ω^i_B/A → Ω^i + 1_B/A are differential operators of order 1.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$\nbe a homomorphism of sheaves of rings on $X$. The differentials\n$\\text{d} : \\Omega^i_{\\mathcal{B}/\\mathcal{A}}  \\to\n\\Omega^{i + 1}_{\\mathcal{B}/\\mathcal{A}}$\nare differential operators of order $1$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The de Rham complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3Y","source_file":"modules.tex","source_line":5490,"source_end_line":5497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5490-L5497","statement_sha256":"b299b2ad33ccdc1269295e19b92a90a18706abeb9109f6c54d5185776dc6d651","origin":"The Stacks Project","memory_eligible":false,"source_rank":3914,"rank":3914,"depth":1,"x":1235.963,"y":136.078,"cluster":"sheaves-sites"},{"id":"stacks:0FKQ","tag":"0FKQ","title":"The de Rham complex · Definition 0FKQ","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The de Rham complex of f or of X over Y is the complex Ω^bullet_X/Y = Ω^bullet_O_X/f^-1O_Y","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism\nof ringed spaces. The {\\it de Rham complex} of $f$ or of $X$ over $Y$\nis the complex\n$$\n\\Omega^\\bullet_{X/Y} = \\Omega^\\bullet_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_Y}\n$$","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The de Rham complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKQ","source_file":"modules.tex","source_line":5532,"source_end_line":5540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5532-L5540","statement_sha256":"04c192d76f4bbd876901a728bd8b9091cfb00bbd96bc0c6e57ce8922ef432db9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3915,"rank":3915,"depth":0,"x":1119.343,"y":342.119,"cluster":"sheaves-sites"},{"id":"stacks:0G3Z","tag":"0G3Z","title":"The de Rham complex · Lemma 0G3Z","summary":"Let f : X → Y be a morphism of ringed spaces. The differentials d : Ω^i_X/Y → Ω^i + 1_X/Y are differential operators of order 1 on X/Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces. The differentials\n$\\text{d} : \\Omega^i_{X/Y}  \\to \\Omega^{i + 1}_{X/Y}$\nare differential operators of order $1$ on $X/Y$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The de Rham complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G3Z","source_file":"modules.tex","source_line":5573,"source_end_line":5578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5573-L5578","statement_sha256":"c4d7cc73a63d7b60cb584f9f612e9e5f01e9962ad070c5ac7a27fcc16e652063","origin":"The Stacks Project","memory_eligible":false,"source_rank":3916,"rank":3916,"depth":2,"x":1039.572,"y":123.818,"cluster":"sheaves-sites"},{"id":"stacks:08TJ","tag":"08TJ","title":"The naive cotangent complex · Definition 08TJ","summary":"Let X be a topological space. Let A → B be a homomorphism of sheaves of rings. The naive cotangent complex NL_B/A is the chain complex ([Tag 08TI]) NL_B/A = (I/I^2 → Ω_A[B]/A ⊗_A[B] B) with I/I^2 placed in degree -1 and Ω_A[B]/A ⊗_A[B] B placed in degree 0.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$ be a\nhomomorphism of sheaves of rings. The {\\it naive cotangent complex}\n$\\NL_{\\mathcal{B}/\\mathcal{A}}$ is the chain complex\n(\\ref{equation-naive-cotangent-complex})\n$$\n\\NL_{\\mathcal{B}/\\mathcal{A}} =\n\\left(\\mathcal{I}/\\mathcal{I}^2\n\\longrightarrow\n\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}}\n\\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}\\right)\n$$\nwith $\\mathcal{I}/\\mathcal{I}^2$ placed in degree $-1$ and\n$\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}}\n\\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}$\nplaced in degree $0$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The naive cotangent complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TJ","source_file":"modules.tex","source_line":5641,"source_end_line":5658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5641-L5658","statement_sha256":"ce1751b5e34c2477b7439442e804671055ebb0a4b7e21b708802787de81d8083","origin":"The Stacks Project","memory_eligible":false,"source_rank":3917,"rank":3917,"depth":0,"x":1274.157,"y":239.63,"cluster":"sheaves-sites"},{"id":"stacks:08TL","tag":"08TL","title":"The naive cotangent complex · Lemma 08TL","summary":"In the situation above there is a canonical isomorphism NL(α) = NL_B/A in D(B).","statement_latex":"In the situation above there is a canonical isomorphism\n$\\NL(\\alpha) = \\NL_{\\mathcal{B}/\\mathcal{A}}$ in $D(\\mathcal{B})$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TL","source_file":"modules.tex","source_line":5725,"source_end_line":5729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5725-L5729","statement_sha256":"33eeb8aff661ef20d7c66c09eb510884502851356201b911f5f8463150fd7f2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3918,"rank":3918,"depth":3,"x":1007.805,"y":287.384,"cluster":"sheaves-sites"},{"id":"stacks:08TM","tag":"08TM","title":"The naive cotangent complex · Lemma 08TM","summary":"Let f : X → Y be a continuous map of topological spaces. Let A → B be a homomorphism of sheaves of rings on Y. Then f^-1NL_B/A = NL_f^-1B/f^-1A.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings\non $Y$. Then $f^{-1}\\NL_{\\mathcal{B}/\\mathcal{A}} =\n\\NL_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TM","source_file":"modules.tex","source_line":5760,"source_end_line":5766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5760-L5766","statement_sha256":"49358c226cbce8f6182545b6e275097b02c765d49ca7317ea204e78e19d524d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3919,"rank":3919,"depth":2,"x":1165.949,"y":100.869,"cluster":"sheaves-sites"},{"id":"stacks:0D09","tag":"0D09","title":"The naive cotangent complex · Lemma 0D09","summary":"Let X be a topological space. Let A → B be a homomorphism of sheaves of rings on X. Let x ∈ X. Then we have NL_B/A, x = NL_B_x/A_x.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$\nbe a homomorphism of sheaves of rings on $X$. Let $x \\in X$.\nThen we have $\\NL_{\\mathcal{B}/\\mathcal{A}, x} =\n\\NL_{\\mathcal{B}_x/\\mathcal{A}_x}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D09","source_file":"modules.tex","source_line":5772,"source_end_line":5778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5772-L5778","statement_sha256":"08d97e7cdc446f5da387c9bfd7a3c5725013518d1a3cafc6dc8eacfa32769b3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3920,"rank":3920,"depth":3,"x":1199.356,"y":328.341,"cluster":"sheaves-sites"},{"id":"stacks:0E1Y","tag":"0E1Y","title":"The naive cotangent complex · Lemma 0E1Y","summary":"Let X be a topological space. Let A → B → C be maps of sheaves of rings. Let C be the cone (Derived Categories, Definition [Tag 014E]) of the map of complexes NL_C/A → NL_C/B. There is a canonical map c : NL_B/A ⊗_B C → C[-1] of complexes of C-modules which produces a canonical six term exact sequence xymatrix H^0(NL_B/A ⊗_B C) ar[r] & H^0(NL_C/A) ar[r] & H^0(NL_C/B) ar[r] & 0 H^-1(NL_B/A ⊗_B C) ar[r] & H^-1(NL_C/A) ar[r] & H^-1(NL_C/B) ar[llu] of cohomology sheaves.","statement_latex":"Let $X$ be a topological space. Let\n$\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$\nbe maps of sheaves of rings. Let $C$ be the cone\n(Derived Categories, Definition \\ref{derived-definition-cone})\nof the map of complexes\n$\\NL_{\\mathcal{C}/\\mathcal{A}} \\to \\NL_{\\mathcal{C}/\\mathcal{B}}$.\nThere is a canonical map\n$$\nc :\n\\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C}\n\\longrightarrow\nC[-1]\n$$\nof complexes of $\\mathcal{C}$-modules\nwhich produces a canonical six term exact sequence\n$$\n\\xymatrix{\nH^0(\\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C}) \\ar[r] &\nH^0(\\NL_{\\mathcal{C}/\\mathcal{A}}) \\ar[r] &\nH^0(\\NL_{\\mathcal{C}/\\mathcal{B}}) \\ar[r] &\n0 \\\\\nH^{-1}(\\NL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B} \\mathcal{C}) \\ar[r] &\nH^{-1}(\\NL_{\\mathcal{C}/\\mathcal{A}}) \\ar[r] &\nH^{-1}(\\NL_{\\mathcal{C}/\\mathcal{B}}) \\ar[llu]\n}\n$$\nof cohomology sheaves.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1Y","source_file":"modules.tex","source_line":5785,"source_end_line":5814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5785-L5814","statement_sha256":"ae13132889eccd37ec84cbaac42ac581cc0519999b378278bd7c3b3d5972d56e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3921,"rank":3921,"depth":4,"x":991.61,"y":179.429,"cluster":"sheaves-sites"},{"id":"stacks:08TN","tag":"08TN","title":"The naive cotangent complex · Definition 08TN","summary":"The naive cotangent complex NL_f = NL_X/Y of a morphism of ringed spaces f : (X, O_X) → (Y, O_Y) is NL_O_X/f^-1O_Y.","statement_latex":"The {\\it naive cotangent complex} $\\NL_f = \\NL_{X/Y}$ of a morphism of ringed\nspaces $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ is\n$\\NL_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_Y}$.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The naive cotangent complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TN","source_file":"modules.tex","source_line":5850,"source_end_line":5855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5850-L5855","statement_sha256":"2377975bba3b2848cee4f77e5032b409eaf18c8aed0f23674b76f09dab3c646f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3922,"rank":3922,"depth":0,"x":1264.793,"y":171.347,"cluster":"sheaves-sites"},{"id":"stacks:0E1Z","tag":"0E1Z","title":"The naive cotangent complex · Lemma 0E1Z","summary":"Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Let C be the cone of the map NL_X/Z → NL_X/Y of complexes of O_X-modules. There is a canonical map f^*NL_Y/Z → C[-1] which produces a canonical six term exact sequence xymatrix H^0(f^*NL_Y/Z) ar[r] & H^0(NL_X/Z) ar[r] & H^0(NL_X/Y) ar[r] & 0 H^-1(f^*NL_Y/Z) ar[r] & H^-1(NL_X/Z) ar[r] & H^-1(NL_X/Y) ar[llu] of cohomology sheaves.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.\nLet $C$ be the cone of the map $\\NL_{X/Z} \\to \\NL_{X/Y}$ of complexes\nof $\\mathcal{O}_X$-modules. There is a canonical map\n$$\nf^*\\NL_{Y/Z} \\to C[-1]\n$$\nwhich produces a canonical six term exact sequence\n$$\n\\xymatrix{\nH^0(f^*\\NL_{Y/Z}) \\ar[r] &\nH^0(\\NL_{X/Z}) \\ar[r] &\nH^0(\\NL_{X/Y}) \\ar[r] &\n0 \\\\\nH^{-1}(f^*\\NL_{Y/Z}) \\ar[r] &\nH^{-1}(\\NL_{X/Z}) \\ar[r] &\nH^{-1}(\\NL_{X/Y}) \\ar[llu]\n}\n$$\nof cohomology sheaves.","area":"Sheaves & Sites","chapter":"Sheaves of Modules","chapter_id":"modules","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1Z","source_file":"modules.tex","source_line":5897,"source_end_line":5918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/modules.tex#L5897-L5918","statement_sha256":"a255d924e0c3d9c56072b73db2e45086df4fed88e61562a522643d772862d6f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3923,"rank":3923,"depth":5,"x":1069.677,"y":332.461,"cluster":"sheaves-sites"},{"id":"stacks:03CL","tag":"03CL","title":"Abelian presheaves · Lemma 03CL","summary":"Let C be a category. • All limits and colimits exist in PAb(C). • All limits and colimits commute with taking sections over objects of C.","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item All limits and colimits exist in $\\textit{PAb}(\\mathcal{C})$.\n\\item All limits and colimits commute with taking sections over objects of\n$\\mathcal{C}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Abelian presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CL","source_file":"sites-modules.tex","source_line":72,"source_end_line":80,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L72-L80","statement_sha256":"b259513f4df1a749c2885da3c6248621767d19ba3360366523b6964297186167","origin":"The Stacks Project","memory_eligible":false,"source_rank":3924,"rank":3924,"depth":0,"x":1084.002,"y":102.74,"cluster":"sheaves-sites"},{"id":"stacks:03CN","tag":"03CN","title":"Abelian sheaves · Lemma 03CN","summary":"Let C be a site. Let φ : F → G be a morphism of abelian sheaves on C. • The category Ab(C) is an abelian category. • The kernel Ker(φ) of φ is the same as the kernel of φ as a morphism of presheaves. • The morphism φ is injective (Homology, Definition [Tag 010B]) if and only if φ is injective as a map of presheaves (Sites, Definition [Tag 00V6]), if and only if φ is injective as a map of sheaves (Sites, Definition [Tag 00WM]). • The cokernel Coker(φ) of φ is the…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nbe a morphism of abelian sheaves on $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The category $\\textit{Ab}(\\mathcal{C})$ is an abelian category.\n\\item The kernel $\\Ker(\\varphi)$ of $\\varphi$ is the same as the\nkernel of $\\varphi$ as a morphism of presheaves.\n\\item The morphism $\\varphi$ is injective\n(Homology, Definition \\ref{homology-definition-injective-surjective})\nif and only if $\\varphi$ is injective as a map of presheaves\n(Sites, Definition \\ref{sites-definition-presheaves-injective-surjective}),\nif and only if $\\varphi$ is injective as a map of sheaves\n(Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}).\n\\item The cokernel $\\Coker(\\varphi)$ of $\\varphi$ is the sheafification\nof the cokernel of $\\varphi$ as a morphism of presheaves.\n\\item The morphism $\\varphi$ is surjective\n(Homology, Definition \\ref{homology-definition-injective-surjective})\nif and only if $\\varphi$ is surjective as a map of sheaves\n(Sites, Definition \\ref{sites-definition-sheaves-injective-surjective}).\n\\item A complex of abelian sheaves\n$$\n\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}\n$$\nis exact at $\\mathcal{G}$ if and only if for all\n$U \\in \\Ob(\\mathcal{C})$ and all $s \\in \\mathcal{G}(U)$\nmapping to zero in $\\mathcal{H}(U)$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ in $\\mathcal{C}$ such that each\n$s|_{U_i}$ is in the image of $\\mathcal{F}(U_i) \\to \\mathcal{G}(U_i)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CN","source_file":"sites-modules.tex","source_line":124,"source_end_line":154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L124-L154","statement_sha256":"a1acc7e0489ae34017519a5671ff2d760e28b80feedae27ba72b51b9ce1a03d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3925,"rank":3925,"depth":4,"x":1258.329,"y":280.419,"cluster":"sheaves-sites"},{"id":"stacks:03CO","tag":"03CO","title":"Abelian sheaves · Lemma 03CO","summary":"Let C be a site. • All limits and colimits exist in Ab(C). • Limits are the same as the corresponding limits of abelian presheaves over C (i.e., commute with taking sections over objects of C). • Finite direct sums are the same as the corresponding finite direct sums in the category of abelian pre-sheaves over C. • A colimit is the sheafification of the corresponding colimit in the category of abelian presheaves. • Filtered colimits are exact.","statement_latex":"Let $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item All limits and colimits exist in $\\textit{Ab}(\\mathcal{C})$.\n\\item Limits are the same as the corresponding limits of abelian presheaves\nover $\\mathcal{C}$ (i.e., commute with taking sections over objects of\n$\\mathcal{C}$).\n\\item Finite direct sums are the same as the corresponding finite direct sums\nin the category of abelian pre-sheaves over $\\mathcal{C}$.\n\\item A colimit is the sheafification of the corresponding colimit in\nthe category of abelian presheaves.\n\\item Filtered colimits are exact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CO","source_file":"sites-modules.tex","source_line":205,"source_end_line":219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L205-L219","statement_sha256":"73745ef0df31f099ff7ada7b93e32fa87dc2661302826d4193be7e37171fc28b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3926,"rank":3926,"depth":2,"x":986.658,"y":248.29,"cluster":"sheaves-sites"},{"id":"stacks:03A7","tag":"03A7","title":"Free abelian presheaves · Definition 03A7","summary":"Let C be a category. Let G be a presheaf of sets. The free abelian presheaf Z_G on G is the abelian presheaf defined by the rule U ↦ Z[G(U)]. In the special case G = h_X of a representable presheaf associated to an object X of C we use the notation Z_X = Z_h_X. In other words Z_X(U) = Z[Mor_C(U, X)].","statement_latex":"Let $\\mathcal{C}$ be a category. Let $\\mathcal{G}$ be a presheaf of sets.\nThe {\\it free abelian presheaf} $\\mathbf{Z}_\\mathcal{G}$ on $\\mathcal{G}$\nis the abelian presheaf defined by the rule\n$$\nU \\longmapsto \\mathbf{Z}[\\mathcal{G}(U)].\n$$\nIn the special case $\\mathcal{G} = h_X$ of a representable presheaf\nassociated to an object $X$ of $\\mathcal{C}$\nwe use the notation $\\mathbf{Z}_X = \\mathbf{Z}_{h_X}$. In other words\n$$\n\\mathbf{Z}_X(U) = \\mathbf{Z}[\\Mor_\\mathcal{C}(U, X)].\n$$","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Free abelian presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03A7","source_file":"sites-modules.tex","source_line":283,"source_end_line":297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L283-L297","statement_sha256":"05c63454b56d2f1a378bbb97e627859f22d7cddea48f618499dd4c47fc2537b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3927,"rank":3927,"depth":0,"x":1213.022,"y":117.713,"cluster":"sheaves-sites"},{"id":"stacks:03A8","tag":"03A8","title":"Free abelian presheaves · Lemma 03A8","summary":"Let C be a category. Let G, F be a presheaves of sets. Let A be an abelian presheaf. Let U be an object of C. Then we have Mor_PSh(C)(h_U, F) & = F(U), Mor_PAb(C)(Z_G, A) & = Mor_PSh(C)(G, A), Mor_PAb(C)(Z_U, A) & = A(U). All of these equalities are functorial.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{G}$, $\\mathcal{F}$ be a presheaves of sets.\nLet $\\mathcal{A}$ be an abelian presheaf.\nLet $U$ be an object of $\\mathcal{C}$. Then\nwe have\n\\begin{align*}\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F})\n& =\n\\mathcal{F}(U), \\\\\n\\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_\\mathcal{G}, \\mathcal{A})\n& =\n\\Mor_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{G}, \\mathcal{A}), \\\\\n\\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_U, \\mathcal{A})\n& =\n\\mathcal{A}(U).\n\\end{align*}\nAll of these equalities are functorial.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Free abelian presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03A8","source_file":"sites-modules.tex","source_line":305,"source_end_line":324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L305-L324","statement_sha256":"0ea85517f82267a1a7e1420e69a934af540c1da3d791f3f2a183261e06efe0c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3928,"rank":3928,"depth":0,"x":1151.056,"y":342.641,"cluster":"sheaves-sites"},{"id":"stacks:03A9","tag":"03A9","title":"Free abelian presheaves · Lemma 03A9","summary":"Let C be a category. Let I be a set. For each i ∈ I let G_i be a presheaf of sets. Then Z_coprod_i G_i = bigoplus_i ∈ I Z_G_i in PAb(C).","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $I$ be a set. For each $i \\in I$ let\n$\\mathcal{G}_i$ be a presheaf of sets.\nThen\n$$\n\\mathbf{Z}_{\\coprod_i \\mathcal{G}_i}\n=\n\\bigoplus\\nolimits_{i \\in I} \\mathbf{Z}_{\\mathcal{G}_i}\n$$\nin $\\textit{PAb}(\\mathcal{C})$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Free abelian presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03A9","source_file":"sites-modules.tex","source_line":330,"source_end_line":342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L330-L342","statement_sha256":"5766afea25f6b2eaea80357c5498d1c8b881072c7c8a5071427a7eaacf08be0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3929,"rank":3929,"depth":0,"x":1015.748,"y":141.445,"cluster":"sheaves-sites"},{"id":"stacks:03AA","tag":"03AA","title":"Free abelian sheaves · Definition 03AA","summary":"Let C be a site. Let G be a presheaf of sets. The free abelian sheaf Z_G^\\# on G is the abelian sheaf Z_G^\\# which is the sheafification of the free abelian presheaf on G. In the special case G = h_X of a representable presheaf associated to an object X of C we use the notation Z_X^\\#.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be a presheaf of sets.\nThe {\\it free abelian sheaf} $\\mathbf{Z}_\\mathcal{G}^\\#$\non $\\mathcal{G}$ is the abelian sheaf $\\mathbf{Z}_\\mathcal{G}^\\#$\nwhich is the sheafification of the free abelian presheaf on $\\mathcal{G}$.\nIn the special case $\\mathcal{G} = h_X$ of a representable presheaf\nassociated to an object $X$ of $\\mathcal{C}$\nwe use the notation $\\mathbf{Z}_X^\\#$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Free abelian sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AA","source_file":"sites-modules.tex","source_line":356,"source_end_line":365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L356-L365","statement_sha256":"de649eade570f00e224dd35a2f6bd826e6784e4ca778e24737cef7105681b482","origin":"The Stacks Project","memory_eligible":false,"source_rank":3930,"rank":3930,"depth":0,"x":1277.549,"y":213.09,"cluster":"sheaves-sites"},{"id":"stacks:03AB","tag":"03AB","title":"Free abelian sheaves · Lemma 03AB","summary":"Let C be a site. Let G, F be a sheaves of sets. Let A be an abelian sheaf. Let U be an object of C. Then we have Mor_Sh(C)(h_U^\\#, F) & = F(U), Mor_Ab(C)(Z_G^\\#, A) & = Mor_Sh(C)(G, A), Mor_Ab(C)(Z_U^\\#, A) & = A(U). All of these equalities are functorial.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{G}$, $\\mathcal{F}$ be a sheaves of sets.\nLet $\\mathcal{A}$ be an abelian sheaf.\nLet $U$ be an object of $\\mathcal{C}$. Then\nwe have\n\\begin{align*}\n\\Mor_{\\Sh(\\mathcal{C})}(h_U^\\#, \\mathcal{F})\n& =\n\\mathcal{F}(U), \\\\\n\\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathbf{Z}_\\mathcal{G}^\\#,\n\\mathcal{A})\n& =\n\\Mor_{\\Sh(\\mathcal{C})}(\\mathcal{G}, \\mathcal{A}), \\\\\n\\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathbf{Z}_U^\\#, \\mathcal{A})\n& =\n\\mathcal{A}(U).\n\\end{align*}\nAll of these equalities are functorial.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Free abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AB","source_file":"sites-modules.tex","source_line":373,"source_end_line":393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L373-L393","statement_sha256":"0e524af013ff54eb87e825dad329e47d8a8376cc92c6ea5aeae1f135f6a972f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":3931,"rank":3931,"depth":0,"x":1026.665,"y":308.896,"cluster":"sheaves-sites"},{"id":"stacks:03AC","tag":"03AC","title":"Free abelian sheaves · Lemma 03AC","summary":"Let C be a site. Let G be a presheaf of sets. Then Z_G^\\# = (Z_G^\\#)^\\#.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{G}$ be a presheaf of sets.\nThen $\\mathbf{Z}_\\mathcal{G}^\\# = (\\mathbf{Z}_{\\mathcal{G}^\\#})^\\#$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Free abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AC","source_file":"sites-modules.tex","source_line":399,"source_end_line":404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L399-L404","statement_sha256":"35798d7d82a6a104b506d52a55b048c9dea4e679a660554634226d423192f1e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":3932,"rank":3932,"depth":0,"x":1134.714,"y":95.706,"cluster":"sheaves-sites"},{"id":"stacks:03AD","tag":"03AD","title":"Ringed sites · Definition 03AD","summary":"Ringed sites. • A ringed site is a pair (C, O) where C is a site and O is a sheaf of rings on C. The sheaf O is called the structure sheaf of the ringed site. • Let (C, O), (C', O') be ringed sites. A morphism of ringed sites (f, f^sharp) : (C, O) → (C', O') is given by a morphism of sites f : C → C' (see Sites, Definition [Tag 00X1]) together with a map of sheaves of rings f^sharp : f^-1O' → O, which by adjunction is the same thing as a map of sheaves of rings f^sharp :…","statement_latex":"Ringed sites.\n\\begin{enumerate}\n\\item A {\\it ringed site} is a pair $(\\mathcal{C}, \\mathcal{O})$\nwhere $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a sheaf of rings\non $\\mathcal{C}$. The sheaf $\\mathcal{O}$ is called the\n{\\it structure sheaf} of the ringed site.\n\\item Let $(\\mathcal{C}, \\mathcal{O})$, $(\\mathcal{C}', \\mathcal{O}')$ be ringed\nsites. A {\\it morphism of ringed sites}\n$$\n(f, f^\\sharp) :\n(\\mathcal{C}, \\mathcal{O})\n\\longrightarrow\n(\\mathcal{C}', \\mathcal{O}')\n$$\nis given by a morphism of sites $f : \\mathcal{C} \\to \\mathcal{C}'$\n(see Sites, Definition \\ref{sites-definition-morphism-sites})\ntogether with a map of sheaves of rings\n$f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$, which by adjunction\nis the same thing as a map of sheaves of rings\n$f^\\sharp : \\mathcal{O}' \\to f_*\\mathcal{O}$.\n\\item Let\n$(f, f^\\sharp) :\n(\\mathcal{C}_1, \\mathcal{O}_1) \\to (\\mathcal{C}_2, \\mathcal{O}_2)$ and\n$(g, g^\\sharp) :\n(\\mathcal{C}_2, \\mathcal{O}_2) \\to (\\mathcal{C}_3, \\mathcal{O}_3)$\nbe morphisms of ringed sites. Then we define\nthe {\\it composition of morphisms of ringed sites}\nby the rule\n$$\n(g, g^\\sharp) \\circ (f, f^\\sharp) = (g \\circ f, f^\\sharp \\circ g^\\sharp).\n$$\nHere we use composition of morphisms of sites defined in\nSites, Definition \\ref{sites-definition-composition-morphisms-sites}\nand $f^\\sharp \\circ g^\\sharp$ indicates the morphism of sheaves of\nrings\n$$\n\\mathcal{O}_3 \\xrightarrow{g^\\sharp} g_*\\mathcal{O}_2\n\\xrightarrow{g_*f^\\sharp} g_*f_*\\mathcal{O}_1 = (g \\circ f)_*\\mathcal{O}_1\n$$\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Ringed sites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AD","source_file":"sites-modules.tex","source_line":424,"source_end_line":466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L424-L466","statement_sha256":"c0a879365d9932380978f202506247bf07a3611647cd34a91ea17169af990166","origin":"The Stacks Project","memory_eligible":false,"source_rank":3933,"rank":3933,"depth":1,"x":1226.563,"y":314.41,"cluster":"sheaves-sites"},{"id":"stacks:01D3","tag":"01D3","title":"Ringed topoi · Definition 01D3","summary":"Ringed topoi. • A ringed topos is a pair (Sh(C), O) where C is a site and O is a sheaf of rings on C. The sheaf O is called the structure sheaf of the ringed topos. • Let (Sh(C), O), (Sh(C'), O') be ringed topoi. A morphism of ringed topoi (f, f^sharp) : (Sh(C), O) → (Sh(C'), O') is given by a morphism of topoi f : Sh(C) → Sh(C') (see Sites, Definition [Tag 00XA]) together with a map of sheaves of rings f^sharp : f^-1O' → O, which by adjunction is the same thing as a map…","statement_latex":"Ringed topoi.\n\\begin{enumerate}\n\\item A {\\it ringed topos} is a pair\n$(\\Sh(\\mathcal{C}), \\mathcal{O})$\nwhere $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a sheaf of rings\non $\\mathcal{C}$. The sheaf $\\mathcal{O}$ is called the\n{\\it structure sheaf} of the ringed topos.\n\\item Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$,\n$(\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be ringed topoi.\nA {\\it morphism of ringed topoi}\n$$\n(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\longrightarrow\n(\\Sh(\\mathcal{C}'), \\mathcal{O}')\n$$\nis given by a morphism of topoi $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$\n(see Sites, Definition \\ref{sites-definition-topos})\ntogether with a map of sheaves of rings\n$f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$, which by adjunction\nis the same thing as a map of sheaves of rings\n$f^\\sharp : \\mathcal{O}' \\to f_*\\mathcal{O}$.\n\\item Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}_1), \\mathcal{O}_1)\n\\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and\n$(g, g^\\sharp) :\n(\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to\n(\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$\nbe morphisms of ringed topoi. Then we define\nthe {\\it composition of morphisms of ringed topoi}\nby the rule\n$$\n(g, g^\\sharp) \\circ (f, f^\\sharp) = (g \\circ f, f^\\sharp \\circ g^\\sharp).\n$$\nHere we use composition of morphisms of topoi defined in\nSites, Definition \\ref{sites-definition-topos}\nand $f^\\sharp \\circ g^\\sharp$ indicates the morphism of sheaves of\nrings\n$$\n\\mathcal{O}_3 \\xrightarrow{g^\\sharp} g_*\\mathcal{O}_2\n\\xrightarrow{g_*f^\\sharp} g_*f_*\\mathcal{O}_1 = (g \\circ f)_*\\mathcal{O}_1\n$$\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Ringed topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01D3","source_file":"sites-modules.tex","source_line":481,"source_end_line":527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L481-L527","statement_sha256":"f3941d3d3c1c6a9d9a2c1afaaaa2091536e4e65f20694c6ddf7726270b56c87f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3934,"rank":3934,"depth":1,"x":982.746,"y":205.163,"cluster":"sheaves-sites"},{"id":"stacks:03CR","tag":"03CR","title":"Ringed topoi · Lemma 03CR","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. There exists a factorization xymatrix (Sh(C), O_C) ar[rr]_(f, f^sharp) ar[d]_(g, g^sharp) & & (Sh(D), O_D) ar[d]^(e, e^sharp) (Sh(C'), O_C') ar[rr]^(h, h^sharp) & & (Sh(D'), O_D') where • g : Sh(C) → Sh(C') is an equivalence of topoi induced by a special cocontinuous functor C → C' (see Sites, Definition [Tag 03CG]), • e : Sh(D) → Sh(D') is an equivalence of topoi induced by a special…","statement_latex":"Let $(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi. There exists a factorization\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\ar[rr]_{(f, f^\\sharp)}\n\\ar[d]_{(g, g^\\sharp)}\n& &\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\ar[d]^{(e, e^\\sharp)}\n\\\\\n(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})\n\\ar[rr]^{(h, h^\\sharp)} & &\n(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})\n}\n$$\nwhere\n\\begin{enumerate}\n\\item $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$\nis an equivalence of topoi induced by a special cocontinuous functor\n$\\mathcal{C} \\to \\mathcal{C}'$ (see\nSites, Definition \\ref{sites-definition-special-cocontinuous-functor}),\n\\item $e : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{D}')$\nis an equivalence of topoi induced by a special cocontinuous functor\n$\\mathcal{D} \\to \\mathcal{D}'$ (see\nSites, Definition \\ref{sites-definition-special-cocontinuous-functor}),\n\\item $\\mathcal{O}_{\\mathcal{C}'} = g_*\\mathcal{O}_\\mathcal{C}$\nand $g^\\sharp$ is the obvious map,\n\\item $\\mathcal{O}_{\\mathcal{D}'} = e_*\\mathcal{O}_\\mathcal{D}$\nand $e^\\sharp$ is the obvious map,\n\\item the sites $\\mathcal{C}'$ and $\\mathcal{D}'$ have final objects\nand fibre products (i.e., all finite limits),\n\\item $h$ is a morphism of sites induced by a continuous functor\n$u : \\mathcal{D}' \\to \\mathcal{C}'$ which commutes with all finite limits\n(i.e., it satisfies the assumptions of\nSites, Proposition \\ref{sites-proposition-get-morphism}), and\n\\item given any set of sheaves $\\mathcal{F}_i$ (resp.\\ $\\mathcal{G}_j$)\non $\\mathcal{C}$ (resp.\\ $\\mathcal{D}$) we may assume each of these is\na representable sheaf on $\\mathcal{C}'$ (resp.\\ $\\mathcal{D}'$).\n\\end{enumerate}\nMoreover, if $(f, f^\\sharp)$ is an equivalence of ringed topoi,\nthen we can choose the diagram such that\n$\\mathcal{C}' = \\mathcal{D}'$,\n$\\mathcal{O}_{\\mathcal{C}'} = \\mathcal{O}_{\\mathcal{D}'}$\nand $(h, h^\\sharp)$ is the identity.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CR","source_file":"sites-modules.tex","source_line":534,"source_end_line":582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L534-L582","statement_sha256":"62491859a470eac8617fc04998de33caeb8831159b01a4a9d19630a97f713c8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3935,"rank":3935,"depth":9,"x":1250.62,"y":147.322,"cluster":"sheaves-sites"},{"id":"stacks:04IC","tag":"04IC","title":"2-morphisms of ringed topoi · Definition 04IC","summary":"Let f, g : (Sh(C), O_C) → (Sh(D), O_D) be two morphisms of ringed topoi. A 2-morphism from f to g is given by a transformation of functors t : f_* → g_* such that xymatrix & O_D ar[ld]_f^sharp ar[rd]^g^sharp f_*O_C ar[rr]^t & & g_*O_C is commutative.","statement_latex":"Let\n$f, g :\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe two morphisms of ringed topoi. A {\\it 2-morphism from $f$ to $g$}\nis given by a transformation of functors $t : f_* \\to g_*$ such that\n$$\n\\xymatrix{\n& \\mathcal{O}_\\mathcal{D}\n\\ar[ld]_{f^\\sharp}\n\\ar[rd]^{g^\\sharp} \\\\\nf_*\\mathcal{O}_\\mathcal{C} \\ar[rr]^t & &\ng_*\\mathcal{O}_\\mathcal{C}\n}\n$$\nis commutative.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"2-morphisms of ringed topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IC","source_file":"sites-modules.tex","source_line":607,"source_end_line":626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L607-L626","statement_sha256":"377695ce2efa8e71991736b1794db5da36ce14795c9491fecd966b5097cf79e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":3936,"rank":3936,"depth":0,"x":1099.476,"y":342.14,"cluster":"sheaves-sites"},{"id":"stacks:03CT","tag":"03CT","title":"Presheaves of modules · Definition 03CT","summary":"Let C be a category, and let O be a presheaf of rings on C. • A presheaf of O-modules is given by an abelian presheaf F together with a map of presheaves of sets O × F → F such that for every object U of C the map O(U) × F(U) → F(U) defines the structure of an O(U)-module structure on the abelian group F(U). • A morphism φ : F → G of presheaves of O-modules is a morphism of abelian presheaves φ : F → G such that the diagram xymatrix O × F ar[r] ar[d]_id × φ & F ar[d]^φ O…","statement_latex":"Let $\\mathcal{C}$ be a category, and\nlet $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$.\n\\begin{enumerate}\n\\item A {\\it presheaf of $\\mathcal{O}$-modules}\nis given by an abelian presheaf $\\mathcal{F}$ together with a\nmap of presheaves of sets\n$$\n\\mathcal{O} \\times \\mathcal{F} \\longrightarrow \\mathcal{F}\n$$\nsuch that for every object $U$ of $\\mathcal{C}$ the map\n$\\mathcal{O}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$\ndefines the structure of an $\\mathcal{O}(U)$-module\nstructure on the abelian group $\\mathcal{F}(U)$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves of $\\mathcal{O}$-modules} is a morphism of abelian presheaves\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ such that\nthe diagram\n$$\n\\xymatrix{\n\\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d]_{\\text{id} \\times \\varphi} &\n\\mathcal{F} \\ar[d]^{\\varphi} \\\\\n\\mathcal{O} \\times \\mathcal{G} \\ar[r] &\n\\mathcal{G}\n}\n$$\ncommutes.\n\\item The set of $\\mathcal{O}$-module morphisms as above is\ndenoted $\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$.\n\\item The category of presheaves of $\\mathcal{O}$-modules is denoted\n$\\textit{PMod}(\\mathcal{O})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Presheaves of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CT","source_file":"sites-modules.tex","source_line":675,"source_end_line":708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L675-L708","statement_sha256":"357876d2377036577a71709d8ab8590759fd5995ef8b61c206a99dbbc0b190db","origin":"The Stacks Project","memory_eligible":false,"source_rank":3937,"rank":3937,"depth":0,"x":1054.22,"y":112.522,"cluster":"sheaves-sites"},{"id":"stacks:03CU","tag":"03CU","title":"Presheaves of modules · Lemma 03CU","summary":"With C, O_1 → O_2, F and G as above there exists a canonical bijection Hom_O_1(G, F_O_1) = Hom_O_2( O_2 ⊗_p, O_1 G, F ) In other words, the restriction and change of rings functors defined above are adjoint to each other.","statement_latex":"With $\\mathcal{C}$, $\\mathcal{O}_1 \\to \\mathcal{O}_2$, $\\mathcal{F}$ and\n$\\mathcal{G}$ as above there exists a canonical bijection\n$$\n\\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1})\n=\n\\Hom_{\\mathcal{O}_2}(\n\\mathcal{O}_2 \\otimes_{p, \\mathcal{O}_1} \\mathcal{G},\n\\mathcal{F}\n)\n$$\nIn other words, the restriction and change of rings functors defined\nabove are adjoint to each other.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CU","source_file":"sites-modules.tex","source_line":755,"source_end_line":769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L755-L769","statement_sha256":"4499d0cd46d45719d69593e4a14c1263bcadba0a6aec5aa59f94e77d8072d713","origin":"The Stacks Project","memory_eligible":false,"source_rank":3938,"rank":3938,"depth":0,"x":1272.433,"y":256.285,"cluster":"sheaves-sites"},{"id":"stacks:03CW","tag":"03CW","title":"Sheaves of modules · Definition 03CW","summary":"Let C be a site. Let O be a sheaf of rings on C. • A sheaf of O-modules is a presheaf of O-modules F, see Definition [Tag 03CT], such that the underlying presheaf of abelian groups F is a sheaf. • A morphism of sheaves of O-modules is a morphism of presheaves of O-modules. • Given sheaves of O-modules F and G we denote Hom_O(F, G) the set of morphism of sheaves of O-modules. • The category of sheaves of O-modules is denoted Mod(O).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.\n\\begin{enumerate}\n\\item A {\\it sheaf of $\\mathcal{O}$-modules} is a presheaf\nof $\\mathcal{O}$-modules $\\mathcal{F}$,\nsee Definition \\ref{definition-presheaf-modules},\nsuch that the underlying presheaf of abelian groups $\\mathcal{F}$\nis a sheaf.\n\\item A {\\it morphism of sheaves of $\\mathcal{O}$-modules}\nis a morphism of presheaves of $\\mathcal{O}$-modules.\n\\item Given sheaves of $\\mathcal{O}$-modules\n$\\mathcal{F}$ and $\\mathcal{G}$ we denote\n$\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$\nthe set of morphism of sheaves of $\\mathcal{O}$-modules.\n\\item The category of sheaves of $\\mathcal{O}$-modules\nis denoted $\\textit{Mod}(\\mathcal{O})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Sheaves of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CW","source_file":"sites-modules.tex","source_line":781,"source_end_line":800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L781-L800","statement_sha256":"a8a7170f9ef45fcbca0652d47dfe961254bdc6628a1ca8ee547c775f76acabdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3939,"rank":3939,"depth":1,"x":995.677,"y":274.111,"cluster":"sheaves-sites"},{"id":"stacks:03CY","tag":"03CY","title":"Sheafification of presheaves of modules · Lemma 03CY","summary":"Let C be a site. Let O be a presheaf of rings on C. Let F be a presheaf of O-modules. Let O^\\# be the sheafification of O as a presheaf of rings, see Sites, Section [Tag 00YR]. Let F^\\# be the sheafification of F as a presheaf of abelian groups. There exists a unique map of sheaves of sets O^\\# × F^\\# → F^\\# which makes the diagram xymatrix O × F ar[r] ar[d] & F ar[d] O^\\# × F^\\# ar[r] & F^\\# commute and which makes F^\\# into a sheaf of O^\\#-modules. In addition, if G is…","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.\nLet $\\mathcal{O}^\\#$ be the sheafification of $\\mathcal{O}$ as a presheaf\nof rings, see Sites, Section \\ref{sites-section-sheaves-algebraic-structures}.\nLet $\\mathcal{F}^\\#$ be the sheafification of $\\mathcal{F}$\nas a presheaf of abelian groups. There exists a unique map of\nsheaves of sets\n$$\n\\mathcal{O}^\\# \\times \\mathcal{F}^\\#\n\\longrightarrow\n\\mathcal{F}^\\#\n$$\nwhich makes the diagram\n$$\n\\xymatrix{\n\\mathcal{O} \\times \\mathcal{F} \\ar[r] \\ar[d] &\n\\mathcal{F} \\ar[d] \\\\\n\\mathcal{O}^\\# \\times \\mathcal{F}^\\# \\ar[r] &\n\\mathcal{F}^\\#\n}\n$$\ncommute and which makes $\\mathcal{F}^\\#$ into a sheaf\nof $\\mathcal{O}^\\#$-modules. In addition, if $\\mathcal{G}$\nis a sheaf of $\\mathcal{O}^\\#$-modules, then any morphism\nof presheaves of $\\mathcal{O}$-modules $\\mathcal{F} \\to \\mathcal{G}$\n(into the restriction of $\\mathcal{G}$ to a $\\mathcal{O}$-module)\nfactors uniquely as $\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$\nwhere $\\mathcal{F}^\\# \\to \\mathcal{G}$ is a morphism of\n$\\mathcal{O}^\\#$-modules.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Sheafification of presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CY","source_file":"sites-modules.tex","source_line":814,"source_end_line":846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L814-L846","statement_sha256":"e12bae80a381c7c30a9027f540b72375715774637a14e1ad7f44f10f6d8a4186","origin":"The Stacks Project","memory_eligible":false,"source_rank":3940,"rank":3940,"depth":0,"x":1185.581,"y":103.781,"cluster":"sheaves-sites"},{"id":"stacks:03EI","tag":"03EI","title":"Sheafification of presheaves of modules · Lemma 03EI","summary":"Let C be a site. Let O be a presheaf of rings on C The sheafification functor PMod(O) → Mod(O^\\#), F ↦ F^\\# is exact.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$\nThe sheafification functor\n$$\n\\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}^\\#), \\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\#\n$$\nis exact.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Sheafification of presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EI","source_file":"sites-modules.tex","source_line":873,"source_end_line":883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L873-L883","statement_sha256":"6e9989ce94ce5bcbdc0224df95491e773d6427eee1482979c5d0d0754faf5da2","origin":"The Stacks Project","memory_eligible":false,"source_rank":3941,"rank":3941,"depth":0,"x":1182.52,"y":337.337,"cluster":"sheaves-sites"},{"id":"stacks:03CZ","tag":"03CZ","title":"Sheafification of presheaves of modules · Lemma 03CZ","summary":"With X, O_1, O_2, F and G as above there exists a canonical bijection Hom_O_1(G, F_O_1) = Hom_O_2( O_2 ⊗_O_1 G, F ) In other words, the restriction and change of rings functors are adjoint to each other.","statement_latex":"With $X$, $\\mathcal{O}_1$, $\\mathcal{O}_2$, $\\mathcal{F}$ and\n$\\mathcal{G}$ as above there exists a canonical bijection\n$$\n\\Hom_{\\mathcal{O}_1}(\\mathcal{G}, \\mathcal{F}_{\\mathcal{O}_1})\n=\n\\Hom_{\\mathcal{O}_2}(\n\\mathcal{O}_2 \\otimes_{\\mathcal{O}_1} \\mathcal{G},\n\\mathcal{F}\n)\n$$\nIn other words, the restriction and change of rings functors\nare adjoint to each other.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Sheafification of presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03CZ","source_file":"sites-modules.tex","source_line":932,"source_end_line":946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L932-L946","statement_sha256":"44282dddee264d459336630e31e244969bdd97d0c8a18a7b6b1e0b70efd95f92","origin":"The Stacks Project","memory_eligible":false,"source_rank":3942,"rank":3942,"depth":1,"x":996.798,"y":163.23,"cluster":"sheaves-sites"},{"id":"stacks:0930","tag":"0930","title":"Sheafification of presheaves of modules · Lemma 0930","summary":"Let C be a site. Let O → O' be an epimorphism of sheaves of rings. Let G_1, G_2 be O'-modules. Then Hom_O'(G_1, G_2) = Hom_O(G_1, G_2). In other words, the restriction functor Mod(O') → Mod(O) is fully faithful.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O} \\to \\mathcal{O}'$ be an epimorphism of sheaves of rings.\nLet $\\mathcal{G}_1, \\mathcal{G}_2$ be $\\mathcal{O}'$-modules.\nThen\n$$\n\\Hom_{\\mathcal{O}'}(\\mathcal{G}_1, \\mathcal{G}_2) =\n\\Hom_\\mathcal{O}(\\mathcal{G}_1, \\mathcal{G}_2).\n$$\nIn other words, the restriction functor\n$\\textit{Mod}(\\mathcal{O}') \\to \\textit{Mod}(\\mathcal{O})$ is fully faithful.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Sheafification of presheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0930","source_file":"sites-modules.tex","source_line":964,"source_end_line":976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L964-L976","statement_sha256":"5f041eb62dbe6331c6036a2e506308a58114ab71af3578f07ef9df5b5ce20147","origin":"The Stacks Project","memory_eligible":false,"source_rank":3943,"rank":3943,"depth":2,"x":1273.998,"y":186.251,"cluster":"sheaves-sites"},{"id":"stacks:03D1","tag":"03D1","title":"Morphisms of topoi and sheaves of modules · Lemma 03D1","summary":"Let C, D be sites. Let f : Sh(C) → Sh(D) be a morphism of topoi. Let O be a sheaf of rings on C. Let F be a sheaf of O-modules. There is a natural map of sheaves of sets f_*O × f_*F → f_*F which turns f_*F into a sheaf of f_*O-modules. This construction is functorial in F.","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe a morphism of topoi.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nThere is a natural map of sheaves of sets\n$$\nf_*\\mathcal{O} \\times f_*\\mathcal{F}\n\\longrightarrow\nf_*\\mathcal{F}\n$$\nwhich turns $f_*\\mathcal{F}$ into a sheaf of $f_*\\mathcal{O}$-modules.\nThis construction is functorial in $\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Morphisms of topoi and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03D1","source_file":"sites-modules.tex","source_line":995,"source_end_line":1010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L995-L1010","statement_sha256":"b81584c6db1eb2e79aeb440f3eadd080eeb10a3707f971224a160355bbf0c364","origin":"The Stacks Project","memory_eligible":false,"source_rank":3944,"rank":3944,"depth":0,"x":1050.891,"y":326.685,"cluster":"sheaves-sites"},{"id":"stacks:03D2","tag":"03D2","title":"Morphisms of topoi and sheaves of modules · Lemma 03D2","summary":"Let C, D be sites. Let f : Sh(C) → Sh(D) be a morphism of topoi. Let O be a sheaf of rings on D. Let G be a sheaf of O-modules. There is a natural map of sheaves of sets f^-1O × f^-1G → f^-1G which turns f^-1G into a sheaf of f^-1O-modules. This construction is functorial in G.","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe a morphism of topoi.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{D}$.\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules.\nThere is a natural map of sheaves of sets\n$$\nf^{-1}\\mathcal{O} \\times f^{-1}\\mathcal{G}\n\\longrightarrow\nf^{-1}\\mathcal{G}\n$$\nwhich turns $f^{-1}\\mathcal{G}$ into a sheaf of $f^{-1}\\mathcal{O}$-modules.\nThis construction is functorial in $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Morphisms of topoi and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03D2","source_file":"sites-modules.tex","source_line":1019,"source_end_line":1034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1019-L1034","statement_sha256":"8e8e9930ef4dc6bd173518c52a89078bda7aa69d52074fb92e6cec47b35c47ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":3945,"rank":3945,"depth":0,"x":1102.517,"y":96.338,"cluster":"sheaves-sites"},{"id":"stacks:03D3","tag":"03D3","title":"Morphisms of topoi and sheaves of modules · Lemma 03D3","summary":"Let C, D be sites. Let f : Sh(C) → Sh(D) be a morphism of topoi. Let O be a sheaf of rings on D. Let G be a sheaf of O-modules. Let F be a sheaf of f^-1O-modules. Then Mor_Mod(f^-1O)(f^-1G, F) = Mor_Mod(O)(G, f_*F). Here we use Lemmas [Tag 03D2] and [Tag 03D1], and we think of f_*F as an O-module by restriction via O → f_*f^-1O.","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe a morphism of topoi.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{D}$.\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules.\nLet $\\mathcal{F}$ be a sheaf of $f^{-1}\\mathcal{O}$-modules.\nThen\n$$\n\\Mor_{\\textit{Mod}(f^{-1}\\mathcal{O})}(f^{-1}\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).\n$$\nHere we use\nLemmas \\ref{lemma-pullback-module}\nand \\ref{lemma-pushforward-module}, and we think of\n$f_*\\mathcal{F}$ as an $\\mathcal{O}$-module by restriction via\n$\\mathcal{O} \\to f_*f^{-1}\\mathcal{O}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Morphisms of topoi and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03D3","source_file":"sites-modules.tex","source_line":1043,"source_end_line":1062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1043-L1062","statement_sha256":"0011e2d3956f439806eb4f72ae2ba33c1ef39779d97a1827479ef7a42502e295","origin":"The Stacks Project","memory_eligible":false,"source_rank":3946,"rank":3946,"depth":2,"x":1249.814,"y":295.657,"cluster":"sheaves-sites"},{"id":"stacks:03D4","tag":"03D4","title":"Morphisms of topoi and sheaves of modules · Lemma 03D4","summary":"Let C, D be sites. Let f : Sh(C) → Sh(D) be a morphism of topoi. Let O be a sheaf of rings on C. Let F be a sheaf of O-modules. Let G be a sheaf of f_*O-modules. Then Mor_Mod(O)( O ⊗_f^-1f_*O f^-1G, F) = Mor_Mod(f_*O)(G, f_*F). Here we use Lemmas [Tag 03D2] and [Tag 03D1], and we use the canonical map f^-1f_*O → O in the definition of the tensor product.","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites.\nLet $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe a morphism of topoi.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nLet $\\mathcal{G}$ be a sheaf of $f_*\\mathcal{O}$-modules.\nThen\n$$\n\\Mor_{\\textit{Mod}(\\mathcal{O})}(\n\\mathcal{O} \\otimes_{f^{-1}f_*\\mathcal{O}} f^{-1}\\mathcal{G}, \\mathcal{F})\n=\n\\Mor_{\\textit{Mod}(f_*\\mathcal{O})}(\\mathcal{G}, f_*\\mathcal{F}).\n$$\nHere we use\nLemmas \\ref{lemma-pullback-module}\nand \\ref{lemma-pushforward-module}, and we use\nthe canonical map $f^{-1}f_*\\mathcal{O} \\to \\mathcal{O}$\nin the definition of the tensor product.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Morphisms of topoi and sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03D4","source_file":"sites-modules.tex","source_line":1097,"source_end_line":1117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1097-L1117","statement_sha256":"cc419a429b44089c1e61855965cc0bb2a66bf0c62885770c3d0ce3685b1cfeaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":3947,"rank":3947,"depth":3,"x":980.683,"y":232.207,"cluster":"sheaves-sites"},{"id":"stacks:03D6","tag":"03D6","title":"Morphisms of ringed topoi and modules · Definition 03D6","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi or ringed sites. • Let F be a sheaf of O_C-modules. We define the pushforward of F as the sheaf of O_D-modules which as a sheaf of abelian groups equals f_*F and with module structure given by the restriction via f^sharp : O_D → f_*O_C of the module structure f_*O_C × f_*F → f_*F from Lemma [Tag 03D1]. • Let G be a sheaf of O_D-modules. We define the pullback f^*G to be the sheaf of O_C-modules…","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi or ringed sites.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{C}$-modules.\nWe define the {\\it pushforward} of $\\mathcal{F}$ as the\nsheaf of $\\mathcal{O}_\\mathcal{D}$-modules which as a sheaf\nof abelian groups equals $f_*\\mathcal{F}$ and with\nmodule structure given by the restriction\nvia $f^\\sharp : \\mathcal{O}_\\mathcal{D} \\to f_*\\mathcal{O}_\\mathcal{C}$\nof the module structure\n$$\nf_*\\mathcal{O}_\\mathcal{C} \\times f_*\\mathcal{F}\n\\longrightarrow\nf_*\\mathcal{F}\n$$\nfrom Lemma \\ref{lemma-pushforward-module}.\n\\item Let $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_\\mathcal{D}$-modules.\nWe define the {\\it pullback} $f^*\\mathcal{G}$ to be the\nsheaf of $\\mathcal{O}_\\mathcal{C}$-modules defined by the formula\n$$\nf^*\\mathcal{G}\n=\n\\mathcal{O}_\\mathcal{C} \\otimes_{f^{-1}\\mathcal{O}_\\mathcal{D}}\nf^{-1}\\mathcal{G}\n$$\nwhere the ring map\n$f^{-1}\\mathcal{O}_\\mathcal{D} \\to \\mathcal{O}_\\mathcal{C}$\nis $f^\\sharp$, and where the  module\nstructure is given by Lemma \\ref{lemma-pullback-module}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Morphisms of ringed topoi and modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03D6","source_file":"sites-modules.tex","source_line":1145,"source_end_line":1181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1145-L1181","statement_sha256":"ed4c5a5dbf048ae6fe81cb086e12cc690b7e7037f52bee582ccdc4b9eb40b468","origin":"The Stacks Project","memory_eligible":false,"source_rank":3948,"rank":3948,"depth":1,"x":1230.373,"y":126.192,"cluster":"sheaves-sites"},{"id":"stacks:03D7","tag":"03D7","title":"Morphisms of ringed topoi and modules · Lemma 03D7","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi or ringed sites. Let F be a sheaf of O_C-modules. Let G be a sheaf of O_D-modules. There is a canonical bijection Hom_O_C(f^*G, F) = Hom_O_D(G, f_*F). In other words: the functor f^* is the left adjoint to f_*.","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi or ringed sites.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{C}$-modules.\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}_\\mathcal{D}$-modules.\nThere is a canonical bijection\n$$\n\\Hom_{\\mathcal{O}_\\mathcal{C}}(f^*\\mathcal{G}, \\mathcal{F})\n=\n\\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, f_*\\mathcal{F}).\n$$\nIn other words: the functor $f^*$ is the left adjoint to\n$f_*$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Morphisms of ringed topoi and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03D7","source_file":"sites-modules.tex","source_line":1196,"source_end_line":1214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1196-L1214","statement_sha256":"c81ebf3409f0b5e0fde0945d940122eddbd235a8c7a8e9232f3dc3960dfdc39a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3949,"rank":3949,"depth":3,"x":1131.425,"y":346.233,"cluster":"sheaves-sites"},{"id":"stacks:03D8","tag":"03D8","title":"Morphisms of ringed topoi and modules · Lemma 03D8","summary":"(f, f^sharp) : (Sh(C_1), O_1) → (Sh(C_2), O_2) and (g, g^sharp) : (Sh(C_2), O_2) → (Sh(C_3), O_3) be morphisms of ringed topoi. There are canonical isomorphisms of functors (g ∘ f)_* ≅ g_* ∘ f_* and (g ∘ f)^* ≅ f^* ∘ g^*.","statement_latex":"$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}_1), \\mathcal{O}_1)\n\\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and\n$(g, g^\\sharp) :\n(\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to\n(\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$\nbe morphisms of ringed topoi.\nThere are canonical isomorphisms of functors\n$(g \\circ f)_* \\cong g_* \\circ f_*$ and\n$(g \\circ f)^* \\cong f^* \\circ g^*$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Morphisms of ringed topoi and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03D8","source_file":"sites-modules.tex","source_line":1235,"source_end_line":1247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1235-L1247","statement_sha256":"1662d7f04a9e1437953563c92fcb35fe6163011e41b198cc47b0edd01236ed2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3950,"rank":3950,"depth":0,"x":1027.349,"y":127.648,"cluster":"sheaves-sites"},{"id":"stacks:03DA","tag":"03DA","title":"The abelian category of sheaves of modules · Lemma 03DA","summary":"Let (Sh(C), O) be a ringed topos. The category Mod(O) is an abelian category. The forgetful functor Mod(O) → Ab(C) is exact, hence kernels, cokernels and exactness of O-modules, correspond to the corresponding notions for abelian sheaves.","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.\nThe category $\\textit{Mod}(\\mathcal{O})$ is an abelian category.\nThe forgetful functor\n$\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}(\\mathcal{C})$\nis exact, hence kernels, cokernels and exactness of\n$\\mathcal{O}$-modules, correspond to the corresponding notions\nfor abelian sheaves.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"The abelian category of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DA","source_file":"sites-modules.tex","source_line":1336,"source_end_line":1345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1336-L1345","statement_sha256":"1aca0c64b11e6300e2501c6144723f308858be91654951c9f8d03a72e5824855","origin":"The Stacks Project","memory_eligible":false,"source_rank":3951,"rank":3951,"depth":1,"x":1280.088,"y":229.86,"cluster":"sheaves-sites"},{"id":"stacks:03DB","tag":"03DB","title":"The abelian category of sheaves of modules · Lemma 03DB","summary":"Let (Sh(C), O) be a ringed topos. All limits and colimits exist in Mod(O) and the forgetful functor Mod(O) → Ab(C) commutes with them. Moreover, filtered colimits are exact.","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.\nAll limits and colimits exist in $\\textit{Mod}(\\mathcal{O})$\nand the forgetful functor\n$\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}(\\mathcal{C})$\ncommutes with them. Moreover, filtered colimits are exact.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"The abelian category of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DB","source_file":"sites-modules.tex","source_line":1357,"source_end_line":1364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1357-L1364","statement_sha256":"8704bc55f5bf0babe10b383066cb683d890c21907b49e42de1f0508ba4a68b27","origin":"The Stacks Project","memory_eligible":false,"source_rank":3952,"rank":3952,"depth":3,"x":1011.298,"y":297.958,"cluster":"sheaves-sites"},{"id":"stacks:03DC","tag":"03DC","title":"The abelian category of sheaves of modules · Lemma 03DC","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. • The functor f_* is left exact. In fact it commutes with all limits. • The functor f^* is right exact. In fact it commutes with all colimits.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi.\n\\begin{enumerate}\n\\item The functor $f_*$ is left exact. In fact it commutes with\nall limits.\n\\item The functor $f^*$ is right exact. In fact it commutes\nwith all colimits.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"The abelian category of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DC","source_file":"sites-modules.tex","source_line":1411,"source_end_line":1422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1411-L1422","statement_sha256":"1ae6c6f23fd72aed6916e3af241dbf76020ce816e997c9b49e5f0d4835d9e780","origin":"The Stacks Project","memory_eligible":false,"source_rank":3953,"rank":3953,"depth":4,"x":1154.858,"y":95.055,"cluster":"sheaves-sites"},{"id":"stacks:05V3","tag":"05V3","title":"The abelian category of sheaves of modules · Lemma 05V3","summary":"Let C be a site. If (p_i)_i ∈ I is a conservative family of points, then we may check exactness of a sequence of abelian sheaves on the stalks at the points p_i, i ∈ I. If C has enough points, then exactness of a sequence of abelian sheaves may be checked on stalks.","statement_latex":"Let $\\mathcal{C}$ be a site. If $\\{p_i\\}_{i \\in I}$ is a conservative\nfamily of points, then we may check exactness of a sequence of abelian\nsheaves on the stalks at the points $p_i$, $i \\in I$.\nIf $\\mathcal{C}$ has enough points, then\nexactness of a sequence of abelian sheaves may\nbe checked on stalks.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"The abelian category of sheaves of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05V3","source_file":"sites-modules.tex","source_line":1431,"source_end_line":1439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1431-L1439","statement_sha256":"669e82b5ee5fecf981889e43e4ecff124c433a68a74e9b55ecc1b4ffd21aa6b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":3954,"rank":3954,"depth":3,"x":1212.216,"y":326.327,"cluster":"sheaves-sites"},{"id":"stacks:04DA","tag":"04DA","title":"Exactness of pushforward · Lemma 04DA","summary":"Let f : Sh(C) → Sh(D) be a morphism of topoi. The following are equivalent: • f^-1f_*F → F is surjective for all F in Ab(C), and • f_* : Ab(C) → Ab(D) reflects surjections. In this case the functor f_* : Ab(C) → Ab(D) is faithful.","statement_latex":"Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be\na morphism of topoi. The following are equivalent:\n\\begin{enumerate}\n\\item $f^{-1}f_*\\mathcal{F} \\to \\mathcal{F}$ is surjective for\nall $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$, and\n\\item $f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$\nreflects surjections.\n\\end{enumerate}\nIn this case the functor\n$f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$\nis faithful.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Exactness of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DA","source_file":"sites-modules.tex","source_line":1456,"source_end_line":1469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1456-L1469","statement_sha256":"eb4ba8fdb78159d2f46a23c81b10b8059f0ed302c6739b79eb5ec90707ddf114","origin":"The Stacks Project","memory_eligible":false,"source_rank":3955,"rank":3955,"depth":0,"x":983.747,"y":188.221,"cluster":"sheaves-sites"},{"id":"stacks:04DB","tag":"04DB","title":"Exactness of pushforward · Lemma 04DB","summary":"Let f : Sh(C) → Sh(D) be a morphism of topoi. Assume at least one of the following properties holds • f_* transforms surjections of sheaves of sets into surjections, • f_* transforms surjections of abelian sheaves into surjections, • f_* commutes with coequalizers on sheaves of sets, • f_* commutes with pushouts on sheaves of sets, Then f_* : Ab(C) → Ab(D) is exact.","statement_latex":"Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be\na morphism of topoi. Assume at least one of the following properties\nholds\n\\begin{enumerate}\n\\item $f_*$ transforms surjections of sheaves of sets into surjections,\n\\item $f_*$ transforms surjections of abelian sheaves into surjections,\n\\item $f_*$ commutes with coequalizers on sheaves of sets,\n\\item $f_*$ commutes with pushouts on sheaves of sets,\n\\end{enumerate}\nThen $f_* : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$\nis exact.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Exactness of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DB","source_file":"sites-modules.tex","source_line":1491,"source_end_line":1504,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1491-L1504","statement_sha256":"f0d9d79af1f776a2bcda5d107be3df3845d9116fd55d5ff72d77f66a61b960d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":3956,"rank":3956,"depth":4,"x":1263.513,"y":160.396,"cluster":"sheaves-sites"},{"id":"stacks:04BD","tag":"04BD","title":"Exactness of pushforward · Lemma 04BD","summary":"Let f : D → C be a morphism of sites associated to the continuous functor u : C → D. Assume u is almost cocontinuous. Then • f_* : Ab(D) → Ab(C) is exact. • if f^sharp : f^-1O_C → O_D is given so that f becomes a morphism of ringed sites, then f_* : Mod(O_D) → Mod(O_C) is exact.","statement_latex":"Let $f : \\mathcal{D} \\to \\mathcal{C}$ be a morphism of sites\nassociated to the continuous functor $u : \\mathcal{C} \\to \\mathcal{D}$.\nAssume $u$ is almost cocontinuous. Then\n\\begin{enumerate}\n\\item $f_* : \\textit{Ab}(\\mathcal{D}) \\to \\textit{Ab}(\\mathcal{C})$ is exact.\n\\item if $f^\\sharp : f^{-1}\\mathcal{O}_\\mathcal{C} \\to \\mathcal{O}_\\mathcal{D}$\nis given so that $f$ becomes a morphism of ringed sites, then\n$f_* : \\textit{Mod}(\\mathcal{O}_\\mathcal{D}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{C})$ is exact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Exactness of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BD","source_file":"sites-modules.tex","source_line":1526,"source_end_line":1538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1526-L1538","statement_sha256":"49efb7cba5780543a1378f48e4a5b09fb19ce041ffbde282a955b20732ddc15d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3957,"rank":3957,"depth":5,"x":1079.439,"y":339.81,"cluster":"sheaves-sites"},{"id":"stacks:04BF","tag":"04BF","title":"Exactness of lower shriek · Definition 04BF","summary":"With u : C → D satisfying (a), (b) above. For F ∈ PAb(C) we define g_p!F as the presheaf V ↦ colim_V → u(U) F(U) with colimits over (I_V^u)^opp taken in Ab. For F ∈ PAb(C) we set g_!F = (g_p!F)^\\#.","statement_latex":"With $u : \\mathcal{C} \\to \\mathcal{D}$ satisfying (a), (b) above.\nFor $\\mathcal{F} \\in \\textit{PAb}(\\mathcal{C})$ we define\n{\\it $g_{p!}\\mathcal{F}$} as the presheaf\n$$\nV \\longmapsto \\colim_{V \\to u(U)} \\mathcal{F}(U)\n$$\nwith colimits over $(\\mathcal{I}_V^u)^{opp}$ taken in $\\textit{Ab}$. For\n$\\mathcal{F} \\in \\textit{PAb}(\\mathcal{C})$ we set\n{\\it $g_!\\mathcal{F} = (g_{p!}\\mathcal{F})^\\#$}.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Exactness of lower shriek","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BF","source_file":"sites-modules.tex","source_line":1582,"source_end_line":1593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1582-L1593","statement_sha256":"5f27efa3f67f8a5a18eaf88c50f2b09cfa997c513c15a302c4f9ef4baa26a09b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3958,"rank":3958,"depth":0,"x":1070.885,"y":102.867,"cluster":"sheaves-sites"},{"id":"stacks:04BG","tag":"04BG","title":"Exactness of lower shriek · Lemma 04BG","summary":"The functor g_p! is a left adjoint to the functor u^p. The functor g_! is a left adjoint to the functor g^-1. In other words the formulas Mor_PAb(C)(F, u^pG) & = Mor_PAb(D)(g_p!F, G), Mor_Ab(C)(F, g^-1G) & = Mor_Ab(D)(g_!F, G) hold bifunctorially in F and G.","statement_latex":"The functor $g_{p!}$ is a left adjoint to the functor $u^p$.\nThe functor $g_!$ is a left adjoint to the functor $g^{-1}$.\nIn other words the formulas\n\\begin{align*}\n\\Mor_{\\textit{PAb}(\\mathcal{C})}(\\mathcal{F}, u^p\\mathcal{G})\n& =\n\\Mor_{\\textit{PAb}(\\mathcal{D})}(g_{p!}\\mathcal{F}, \\mathcal{G}), \\\\\n\\Mor_{\\textit{Ab}(\\mathcal{C})}(\\mathcal{F}, g^{-1}\\mathcal{G})\n& =\n\\Mor_{\\textit{Ab}(\\mathcal{D})}(g_!\\mathcal{F}, \\mathcal{G})\n\\end{align*}\nhold bifunctorially in $\\mathcal{F}$ and $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Exactness of lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BG","source_file":"sites-modules.tex","source_line":1600,"source_end_line":1614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1600-L1614","statement_sha256":"9f713f11c28a026f154c8e5f7007d61eeec7ee81bc5abb4872528c9baa0c5f0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3959,"rank":3959,"depth":1,"x":1267.901,"y":272.873,"cluster":"sheaves-sites"},{"id":"stacks:04BH","tag":"04BH","title":"Exactness of lower shriek · Lemma 04BH","summary":"Let C and D be sites. Let u : C → D be a functor. Assume that • [(a)] u is cocontinuous, • [(b)] u is continuous, and • [(c)] fibre products and equalizers exist in C and u commutes with them. In this case the functor g_! : Ab(C) → Ab(D) is exact.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item[(a)] $u$ is cocontinuous,\n\\item[(b)] $u$ is continuous, and\n\\item[(c)] fibre products and equalizers exist in $\\mathcal{C}$ and\n$u$ commutes with them.\n\\end{enumerate}\nIn this case the functor\n$g_! : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}(\\mathcal{D})$\nis exact.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Exactness of lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BH","source_file":"sites-modules.tex","source_line":1657,"source_end_line":1671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1657-L1671","statement_sha256":"5d8d0a165fef7ca4a35bccb35c0f5b9ed889fb9615fb2f2fdcfbc8ee04107702","origin":"The Stacks Project","memory_eligible":false,"source_rank":3960,"rank":3960,"depth":5,"x":985.675,"y":259.293,"cluster":"sheaves-sites"},{"id":"stacks:077I","tag":"077I","title":"Exactness of lower shriek · Lemma 077I","summary":"Let C and D be sites. Let u : C → D be a functor. Assume that • [(a)] u is cocontinuous, • [(b)] u is continuous, and • [(c)] u is fully faithful. For g_!, g^-1, g_* as above the canonical maps F → g^-1g_!F and g^-1g_*F → F are isomorphisms for all abelian sheaves F on C.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item[(a)] $u$ is cocontinuous,\n\\item[(b)] $u$ is continuous, and\n\\item[(c)] $u$ is fully faithful.\n\\end{enumerate}\nFor $g_!, g^{-1}, g_*$ as above the canonical maps\n$\\mathcal{F} \\to g^{-1}g_!\\mathcal{F}$ and\n$g^{-1}g_*\\mathcal{F} \\to \\mathcal{F}$\nare isomorphisms\nfor all abelian sheaves $\\mathcal{F}$ on $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Exactness of lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077I","source_file":"sites-modules.tex","source_line":1697,"source_end_line":1712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1697-L1712","statement_sha256":"8eed542ae6929ef0c230f09de6938dd3691d691eb70f6bd55d868e6a5008eb62","origin":"The Stacks Project","memory_eligible":false,"source_rank":3961,"rank":3961,"depth":7,"x":1204.887,"y":109.04,"cluster":"sheaves-sites"},{"id":"stacks:08P3","tag":"08P3","title":"Exactness of lower shriek · Lemma 08P3","summary":"Let C and D be sites. Let g : Sh(C) → Sh(D) be the morphism of topoi associated to a continuous and cocontinuous functor u : C → D. • If u has a left adjoint w, then g_! agrees with g_!^Sh on underlying sheaves of sets and g_! is exact. • If in addition w is cocontinuous, then g_! = h^-1 and g^-1 = h_* where h : Sh(D) → Sh(C) is the morphism of topoi associated to w.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites. Let\n$g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the morphism of topoi\nassociated to a continuous and cocontinuous functor\n$u : \\mathcal{C} \\to \\mathcal{D}$.\n\\begin{enumerate}\n\\item If $u$ has a left adjoint $w$, then $g_!$ agrees with $g_!^{\\Sh}$\non underlying sheaves of sets and $g_!$ is exact.\n\\item If in addition $w$ is cocontinuous, then $g_! = h^{-1}$ and\n$g^{-1} = h_*$ where\n$h : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ is the morphism of topoi\nassociated to $w$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Exactness of lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08P3","source_file":"sites-modules.tex","source_line":1761,"source_end_line":1775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1761-L1775","statement_sha256":"cbe70cea18497f3dce72d6d1725fdf0f03a826b18b2bb54c638a0deb149a4c34","origin":"The Stacks Project","memory_eligible":false,"source_rank":3962,"rank":3962,"depth":8,"x":1164.038,"y":344.416,"cluster":"sheaves-sites"},{"id":"stacks:03DE","tag":"03DE","title":"Global types of modules · Definition 03DE","summary":"Let (Sh(C), O) be a ringed topos. Let F be a sheaf of O-modules. • We say F is a free O-module if F is isomorphic as an O-module to a sheaf of the form bigoplus_i ∈ I O. • We say F is finite free if F is isomorphic as an O-module to a sheaf of the form bigoplus_i ∈ I O with a finite index set I. • We say F is generated by global sections if there exists a surjection bigoplus_i ∈ I O → F from a free O-module onto F. • Given r ≥ 0 we say F is generated by r global sections…","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ is a {\\it free $\\mathcal{O}$-module}\nif $\\mathcal{F}$ is isomorphic as an $\\mathcal{O}$-module\nto a sheaf of the form $\\bigoplus_{i \\in I} \\mathcal{O}$.\n\\item We say $\\mathcal{F}$ is {\\it finite free} if\n$\\mathcal{F}$ is isomorphic as an $\\mathcal{O}$-module\nto a sheaf of the form $\\bigoplus_{i \\in I} \\mathcal{O}$\nwith a finite index set $I$.\n\\item We say $\\mathcal{F}$ is {\\it generated by global sections}\nif there exists a surjection\n$$\n\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow \\mathcal{F}\n$$\nfrom a free $\\mathcal{O}$-module onto $\\mathcal{F}$.\n\\item Given $r \\geq 0$ we say $\\mathcal{F}$ is\n{\\it generated by $r$ global sections} if there exists a surjection\n$\\mathcal{O}^{\\oplus r} \\to \\mathcal{F}$.\n\\item We say $\\mathcal{F}$ is {\\it generated by finitely many global sections}\nif it is generated by $r$ global sections for some $r \\geq 0$.\n\\item We say $\\mathcal{F}$ has a {\\it global presentation}\nif there exists an exact sequence\n$$\n\\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\longrightarrow\n\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow\n\\mathcal{F} \\longrightarrow 0\n$$\nof $\\mathcal{O}$-modules.\n\\item We say $\\mathcal{F}$ has a {\\it global finite presentation}\nif there exists an exact sequence\n$$\n\\bigoplus\\nolimits_{j \\in J} \\mathcal{O} \\longrightarrow\n\\bigoplus\\nolimits_{i \\in I} \\mathcal{O} \\longrightarrow\n\\mathcal{F} \\longrightarrow 0\n$$\nof $\\mathcal{O}$-modules with $I$ and $J$ finite sets.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Global types of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DE","source_file":"sites-modules.tex","source_line":1797,"source_end_line":1837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1797-L1837","statement_sha256":"86313a3ad27fdc70270250091e4defd1256c124487d58cd737bb4b3f4b63e8de","origin":"The Stacks Project","memory_eligible":false,"source_rank":3963,"rank":3963,"depth":0,"x":1004.745,"y":147.512,"cluster":"sheaves-sites"},{"id":"stacks:03DF","tag":"03DF","title":"Global types of modules · Lemma 03DF","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let F be an O_D-module. • If F is free then f^*F is free. • If F is finite free then f^*F is finite free. • If F is generated by global sections then f^*F is generated by global sections. • Given r ≥ 0 if F is generated by r global sections, then f^*F is generated by r global sections. • If F is generated by finitely many global sections then f^*F is generated by finitely many global sections. •…","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{D}$-module.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is free then $f^*\\mathcal{F}$ is free.\n\\item If $\\mathcal{F}$ is finite free then $f^*\\mathcal{F}$ is finite free.\n\\item If $\\mathcal{F}$ is generated by global sections\nthen $f^*\\mathcal{F}$ is generated by global sections.\n\\item Given $r \\geq 0$ if $\\mathcal{F}$ is generated by $r$ global\nsections, then $f^*\\mathcal{F}$ is generated by $r$ global sections.\n\\item If $\\mathcal{F}$ is generated by finitely many global sections\nthen $f^*\\mathcal{F}$ is generated by finitely many global sections.\n\\item If $\\mathcal{F}$ has a global presentation then\n$f^*\\mathcal{F}$ has a global presentation.\n\\item If $\\mathcal{F}$ has a finite global presentation\nthen $f^*\\mathcal{F}$ has a finite global presentation.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Global types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DF","source_file":"sites-modules.tex","source_line":1847,"source_end_line":1870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L1847-L1870","statement_sha256":"496d5f1fb4c9ba90445003028367be4277bf2e433640a6cd229908b42ab08da0","origin":"The Stacks Project","memory_eligible":false,"source_rank":3964,"rank":3964,"depth":5,"x":1280.779,"y":202.364,"cluster":"sheaves-sites"},{"id":"stacks:04IX","tag":"04IX","title":"Localization of ringed sites · Definition 04IX","summary":"Let (C, O) be a ringed site. Let U ∈ Ob(C). • The ringed site (C/U, O_U) is called the localization of the ringed site (C, O) at the object U. • The morphism of ringed topoi (j_U, j_U^sharp) : (Sh(C/U), O_U) → (Sh(C), O) is called the localization morphism. • The functor j_U* : Mod(O_U) → Mod(O) is called the direct image functor. • For a sheaf of O-modules F on C the sheaf j_U^*F is called the restriction of F to C/U. We will sometimes denote it by F|_C/U or even F|_U.…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $U \\in \\Ob(\\mathcal{C})$.\n\\begin{enumerate}\n\\item The ringed site $(\\mathcal{C}/U, \\mathcal{O}_U)$ is called the\n{\\it localization of the ringed site $(\\mathcal{C}, \\mathcal{O})$\nat the object $U$}.\n\\item The morphism of ringed topoi\n$(j_U, j_U^\\sharp) :\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n\\to\n(\\Sh(\\mathcal{C}), \\mathcal{O})$\nis called the {\\it localization morphism}.\n\\item The functor\n$j_{U*} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$\nis called the {\\it direct image functor}.\n\\item For a sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on $\\mathcal{C}$\nthe sheaf $j_U^*\\mathcal{F}$ is called the\n{\\it restriction of $\\mathcal{F}$ to $\\mathcal{C}/U$}.\nWe will sometimes denote it by\n$\\mathcal{F}|_{\\mathcal{C}/U}$ or even $\\mathcal{F}|_U$.\nIt is described by the simple rule $j_U^*(\\mathcal{F})(X/U) = \\mathcal{F}(X)$.\n\\item The left adjoint\n$j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$\nof restriction is called {\\it extension by zero}. It exists and is\nexact by\nLemmas \\ref{lemma-extension-by-zero} and\n\\ref{lemma-extension-by-zero-exact}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed sites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IX","source_file":"sites-modules.tex","source_line":2018,"source_end_line":2048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2018-L2048","statement_sha256":"2348d9dc158f01b7a900153f6c3013edbd8a0185b3b1a0e470324cf4b62b5cbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":3965,"rank":3965,"depth":0,"x":1032.92,"y":318.641,"cluster":"sheaves-sites"},{"id":"stacks:03DI","tag":"03DI","title":"Localization of ringed sites · Lemma 03DI","summary":"Let (C, O) be a ringed site. Let U ∈ Ob(C). The restriction functor j_U^* : Mod(O) → Mod(O_U) has a left adjoint j_U! : Mod(O_U) → Mod(O). So Mor_Mod(O_U)(G, j_U^*F) = Mor_Mod(O)(j_U!G, F) for F ∈ Ob(Mod(O)) and G ∈ Ob(Mod(O_U)). Moreover, the extension by zero j_U!G of G is the sheaf associated to the presheaf V ↦ bigoplus_φ ∈ Mor_C(V, U) G(V xrightarrowφ U) with obvious restriction mappings and an obvious O-module structure.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $U \\in \\Ob(\\mathcal{C})$.\nThe restriction functor\n$j_U^* : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_U)$\nhas a left adjoint\n$j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$.\nSo\n$$\n\\Mor_{\\textit{Mod}(\\mathcal{O}_U)}(\\mathcal{G}, j_U^*\\mathcal{F})\n=\n\\Mor_{\\textit{Mod}(\\mathcal{O})}(j_{U!}\\mathcal{G}, \\mathcal{F})\n$$\nfor $\\mathcal{F} \\in \\Ob(\\textit{Mod}(\\mathcal{O}))$\nand $\\mathcal{G} \\in \\Ob(\\textit{Mod}(\\mathcal{O}_U))$.\nMoreover, the extension by zero $j_{U!}\\mathcal{G}$ of $\\mathcal{G}$\nis the sheaf associated to the presheaf\n$$\nV\n\\longmapsto\n\\bigoplus\\nolimits_{\\varphi \\in \\Mor_\\mathcal{C}(V, U)}\n\\mathcal{G}(V \\xrightarrow{\\varphi} U)\n$$\nwith obvious restriction mappings and an obvious $\\mathcal{O}$-module\nstructure.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DI","source_file":"sites-modules.tex","source_line":2057,"source_end_line":2083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2057-L2083","statement_sha256":"fbf7a76b8ec81c91a436d619626e64c2e0802c9eca69260c7cdd32285782b763","origin":"The Stacks Project","memory_eligible":false,"source_rank":3966,"rank":3966,"depth":1,"x":1122.255,"y":92.072,"cluster":"sheaves-sites"},{"id":"stacks:03DJ","tag":"03DJ","title":"Localization of ringed sites · Lemma 03DJ","summary":"Let (C, O) be a ringed site. Let U ∈ Ob(C). The functor j_U! : Mod(O_U) → Mod(O) is exact.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $U \\in \\Ob(\\mathcal{C})$.\nThe functor\n$j_{U!} : \\textit{Mod}(\\mathcal{O}_U) \\to \\textit{Mod}(\\mathcal{O})$\nis exact.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DJ","source_file":"sites-modules.tex","source_line":2159,"source_end_line":2166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2159-L2166","statement_sha256":"d75c82726e32e9fff38f44eeaefbf57f3efdc18f2988d8c4897295965ba6b4e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3967,"rank":3967,"depth":3,"x":1238.677,"y":310.011,"cluster":"sheaves-sites"},{"id":"stacks:0E8G","tag":"0E8G","title":"Localization of ringed sites · Lemma 0E8G","summary":"Let (C, O) be a ringed site. Let U ∈ Ob(C). A complex of O_U-modules G_1 → G_2 → G_3 is exact if and only if j_U!G_1 → j_U!G_2 → j_U!G_3 is exact as a sequence of O-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $U \\in \\Ob(\\mathcal{C})$. A complex of $\\mathcal{O}_U$-modules\n$\\mathcal{G}_1 \\to \\mathcal{G}_2 \\to \\mathcal{G}_3$ is exact\nif and only if\n$j_{U!}\\mathcal{G}_1 \\to j_{U!}\\mathcal{G}_2 \\to j_{U!}\\mathcal{G}_3$\nis exact as a sequence of $\\mathcal{O}$-modules.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8G","source_file":"sites-modules.tex","source_line":2192,"source_end_line":2200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2192-L2200","statement_sha256":"8758c65926f76c1ba57d047fa91e072284dfdef0c5cd9e9fad521b3f8c7f05f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":3968,"rank":3968,"depth":4,"x":977.354,"y":215.289,"cluster":"sheaves-sites"},{"id":"stacks:04IY","tag":"04IY","title":"Localization of ringed sites · Lemma 04IY","summary":"Let (C, O) be a ringed site. Let f : V → U be a morphism of C. Then there exists a commutative diagram xymatrix (Sh(C/V), O_V) ar[rd]_(j_V, j_V^sharp) ar[rr]_(j, j^sharp) & & (Sh(C/U), O_U) ar[ld]^(j_U, j_U^sharp) & (Sh(C), O) & of ringed topoi. Here (j, j^sharp) is the localization morphism associated to the object V/U of the ringed site (C/V, O_V).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $f : V \\to U$ be a morphism of $\\mathcal{C}$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}/V), \\mathcal{O}_V)\n\\ar[rd]_{(j_V, j_V^\\sharp)} \\ar[rr]_{(j, j^\\sharp)} & &\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n\\ar[ld]^{(j_U, j_U^\\sharp)} \\\\\n& (\\Sh(\\mathcal{C}), \\mathcal{O}) &\n}\n$$\nof ringed topoi. Here $(j, j^\\sharp)$ is the localization morphism\nassociated to the object $V/U$ of the ringed site\n$(\\mathcal{C}/V, \\mathcal{O}_V)$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IY","source_file":"sites-modules.tex","source_line":2243,"source_end_line":2260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2243-L2260","statement_sha256":"e8732bda7fe4167275314944c3a7163189782bc9058dd27a9a8838e9e972e282","origin":"The Stacks Project","memory_eligible":false,"source_rank":3969,"rank":3969,"depth":6,"x":1246.443,"y":136.79,"cluster":"sheaves-sites"},{"id":"stacks:0F6Z","tag":"0F6Z","title":"Localization of ringed sites · Lemma 0F6Z","summary":"Let C be a site. Let U ∈ Ob(C). Assume that every X in C has at most one morphism to U. Let F be an abelian sheaf on C/U. The canonical maps F → j_U^-1j_U!F and j_U^-1j_U*F → F are isomorphisms.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U \\in \\Ob(\\mathcal{C})$.\nAssume that every $X$ in $\\mathcal{C}$ has at most\none morphism to $U$. Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}/U$.\nThe canonical maps $\\mathcal{F} \\to j_U^{-1}j_{U!}\\mathcal{F}$\nand $j_U^{-1}j_{U*}\\mathcal{F} \\to \\mathcal{F}$ are\nisomorphisms.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6Z","source_file":"sites-modules.tex","source_line":2337,"source_end_line":2345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2337-L2345","statement_sha256":"3ce12a19208fc8c4062a94e21ca52cd6705cce299b15bcf9d46d39e08881cc2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3970,"rank":3970,"depth":8,"x":1111.036,"y":347.535,"cluster":"sheaves-sites"},{"id":"stacks:04J0","tag":"04J0","title":"Localization of morphisms of ringed sites · Lemma 04J0","summary":"Let (f, f^sharp) : (C, O) → (D, O') be a morphism of ringed sites where f is given by the continuous functor u : D → C. Let V be an object of D and set U = u(V). Then there is a canonical map of sheaves of rings (f')^sharp such that the diagram of Sites, Lemma [Tag 03CF] is turned into a commutative diagram of ringed topoi xymatrix (Sh(C/U), O_U) ar[rr]_(j_U, j_U^sharp) ar[d]_(f', (f')^sharp) & & (Sh(C), O) ar[d]^(f, f^sharp) (Sh(D/V), O'_V) ar[rr]^(j_V, j_V^sharp) & &…","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\mathcal{C}, \\mathcal{O})\n\\longrightarrow\n(\\mathcal{D}, \\mathcal{O}')$\nbe a morphism of ringed sites where $f$ is given by the continuous\nfunctor $u : \\mathcal{D} \\to \\mathcal{C}$.\nLet $V$ be an object of $\\mathcal{D}$ and set $U = u(V)$.\nThen there is a canonical map of sheaves of rings $(f')^\\sharp$\nsuch that the diagram of\nSites, Lemma \\ref{sites-lemma-localize-morphism}\nis turned into a commutative diagram of ringed topoi\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n\\ar[rr]_{(j_U, j_U^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} & &\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\ar[d]^{(f, f^\\sharp)} \\\\\n(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)\n\\ar[rr]^{(j_V, j_V^\\sharp)} & &\n(\\Sh(\\mathcal{D}), \\mathcal{O}').\n}\n$$\nMoreover, in this situation we have $f'_*j_U^{-1} = j_V^{-1}f_*$\nand $f'_*j_U^* = j_V^*f_*$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of morphisms of ringed sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04J0","source_file":"sites-modules.tex","source_line":2368,"source_end_line":2395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2368-L2395","statement_sha256":"4e814d7134b8602ff30592d49b795c831406887a7c0acfd5b16ec73bd9ba494e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3971,"rank":3971,"depth":9,"x":1041.351,"y":115.112,"cluster":"sheaves-sites"},{"id":"stacks:04J1","tag":"04J1","title":"Localization of morphisms of ringed sites · Lemma 04J1","summary":"Let (f, f^sharp) : (C, O) → (D, O') be a morphism of ringed sites where f is given by the continuous functor u : D → C. Let V ∈ Ob(D), U ∈ Ob(C) and c : U → u(V) a morphism of C. There exists a commutative diagram of ringed topoi xymatrix (Sh(C/U), O_U) ar[rr]_(j_U, j_U^sharp) ar[d]_(f_c, f_c^sharp) & & (Sh(C), O) ar[d]^(f, f^sharp) (Sh(D/V), O'_V) ar[rr]^(j_V, j_V^sharp) & & (Sh(D), O'). The morphism (f_c, f_c^sharp) is equal to the composition of the morphism (f',…","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\mathcal{C}, \\mathcal{O})\n\\longrightarrow\n(\\mathcal{D}, \\mathcal{O}')$\nbe a morphism of ringed sites where $f$ is given by the continuous\nfunctor $u : \\mathcal{D} \\to \\mathcal{C}$.\nLet $V \\in \\Ob(\\mathcal{D})$, $U \\in \\Ob(\\mathcal{C})$\nand $c : U \\to u(V)$ a morphism of $\\mathcal{C}$.\nThere exists a commutative diagram of ringed topoi\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n\\ar[rr]_{(j_U, j_U^\\sharp)} \\ar[d]_{(f_c, f_c^\\sharp)} & &\n(\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d]^{(f, f^\\sharp)} \\\\\n(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)\n\\ar[rr]^{(j_V, j_V^\\sharp)} & &\n(\\Sh(\\mathcal{D}), \\mathcal{O}').\n}\n$$\nThe morphism $(f_c, f_c^\\sharp)$\nis equal to the composition of the morphism\n$$\n(f', (f')^\\sharp) :\n(\\Sh(\\mathcal{C}/u(V)), \\mathcal{O}_{u(V)})\n\\longrightarrow\n(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)\n$$\nof\nLemma \\ref{lemma-localize-morphism-ringed-sites}\nand the morphism\n$$\n(j, j^\\sharp) :\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n\\to\n(\\Sh(\\mathcal{C}/u(V)), \\mathcal{O}_{u(V)})\n$$\nof\nLemma \\ref{lemma-relocalize}.\nGiven any morphisms $b : V' \\to V$, $a : U' \\to U$ and\n$c' : U' \\to u(V')$ such that\n$$\n\\xymatrix{\nU' \\ar[r]_-{c'} \\ar[d]_a & u(V') \\ar[d]^{u(b)} \\\\\nU \\ar[r]^-c & u(V)\n}\n$$\ncommutes, then the following diagram of ringed topoi\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}/U'), \\mathcal{O}_{U'})\n\\ar[rr]_{(j_{U'/U}, j_{U'/U}^\\sharp)} \\ar[d]_{(f_{c'}, f_{c'}^\\sharp)} & &\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n\\ar[d]^{(f_c, f_c^\\sharp)} \\\\\n(\\Sh(\\mathcal{D}/V'), \\mathcal{O}'_{V'})\n\\ar[rr]^{(j_{V'/V}, j_{V'/V}^\\sharp)} & &\n(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_{V'})\n}\n$$\ncommutes.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of morphisms of ringed sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04J1","source_file":"sites-modules.tex","source_line":2411,"source_end_line":2473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2411-L2473","statement_sha256":"d02ad5d357f01cb7bdd98483c3fca0c6f0fe7fed221e1cd329d95b6798802465","origin":"The Stacks Project","memory_eligible":false,"source_rank":3972,"rank":3972,"depth":10,"x":1279.839,"y":247.064,"cluster":"sheaves-sites"},{"id":"stacks:04IE","tag":"04IE","title":"Localization of ringed topoi · Lemma 04IE","summary":"Let (Sh(C), O) be a ringed topos. Let F ∈ Sh(C) be a sheaf. For a sheaf H on C denote H_F the sheaf H × F seen as an object of the category Sh(C)/F. The pair (Sh(C)/F, O_F) is a ringed topos and there is a canonical morphism of ringed topoi (j_F, j_F^sharp) : (Sh(C)/F, O_F) → (Sh(C), O) which is a localization as in Section [Tag 03DH] such that • the functor j_F^-1 is the functor H ↦ H_F, • the functor j_F^* is the functor H ↦ H_F, • the functor j_F! on sheaves of sets is…","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.\nLet $\\mathcal{F} \\in \\Sh(\\mathcal{C})$ be a sheaf.\nFor a sheaf $\\mathcal{H}$ on $\\mathcal{C}$ denote\n$\\mathcal{H}_\\mathcal{F}$ the sheaf $\\mathcal{H} \\times \\mathcal{F}$\nseen as an object of the category $\\Sh(\\mathcal{C})/\\mathcal{F}$.\nThe pair\n$(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$\nis a ringed topos and there is a canonical morphism of ringed topoi\n$$\n(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) :\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\longrightarrow\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n$$\nwhich is a localization as in\nSection \\ref{section-localize}\nsuch that\n\\begin{enumerate}\n\\item the functor $j_\\mathcal{F}^{-1}$ is the functor\n$\\mathcal{H} \\mapsto \\mathcal{H}_\\mathcal{F}$,\n\\item the functor $j_\\mathcal{F}^*$ is the functor\n$\\mathcal{H} \\mapsto \\mathcal{H}_\\mathcal{F}$,\n\\item the functor $j_{\\mathcal{F}!}$ on sheaves of sets is the forgetful\nfunctor $\\mathcal{G}/\\mathcal{F} \\mapsto \\mathcal{G}$,\n\\item the functor $j_{\\mathcal{F}!}$ on sheaves of modules associates\nto the $\\mathcal{O}_\\mathcal{F}$-module\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ the $\\mathcal{O}$-module\nwhich is the sheafification of the presheaf\n$$\nV \\longmapsto\n\\bigoplus\\nolimits_{s \\in \\mathcal{F}(V)}\n\\{\\sigma \\in \\mathcal{G}(V) \\mid \\varphi(\\sigma) = s \\}\n$$\nfor $V \\in \\Ob(\\mathcal{C})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IE","source_file":"sites-modules.tex","source_line":2508,"source_end_line":2545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2508-L2545","statement_sha256":"77af6e51f901d0ec14d093067d1cb1c1db9eb9a66eaacdc28e2169becd169da6","origin":"The Stacks Project","memory_eligible":false,"source_rank":3973,"rank":3973,"depth":10,"x":997.64,"y":285.119,"cluster":"sheaves-sites"},{"id":"stacks:04J2","tag":"04J2","title":"Localization of ringed topoi · Definition 04J2","summary":"Let (Sh(C), O) be a ringed topos. Let F ∈ Sh(C). • The ringed topos (Sh(C)/F, O_F) is called the localization of the ringed topos (Sh(C), O) at F. • The morphism of ringed topoi (j_F, j_F^sharp) : (Sh(C)/F, O_F) → (Sh(C), O) of Lemma [Tag 04IE] is called the localization morphism.","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.\nLet $\\mathcal{F} \\in \\Sh(\\mathcal{C})$.\n\\begin{enumerate}\n\\item The ringed topos\n$(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$\nis called the\n{\\it localization of the ringed topos\n$(\\Sh(\\mathcal{C}), \\mathcal{O})$ at $\\mathcal{F}$}.\n\\item The morphism of ringed topoi\n$(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) :\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\to\n(\\Sh(\\mathcal{C}), \\mathcal{O})$ of\nLemma \\ref{lemma-localize-ringed-topos}\nis called the {\\it localization morphism}.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04J2","source_file":"sites-modules.tex","source_line":2567,"source_end_line":2585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2567-L2585","statement_sha256":"25558e85a8e62e8bc6a5e0cfd02bbe43b5ec85ab91acb631f3d9c318e2568d0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3974,"rank":3974,"depth":11,"x":1175.271,"y":96.778,"cluster":"sheaves-sites"},{"id":"stacks:04J3","tag":"04J3","title":"Localization of ringed topoi · Lemma 04J3","summary":"With (Sh(C), O) and F ∈ Sh(C) as in Lemma [Tag 04IE]. If F = h_U^\\# for some object U of C then via the identification Sh(C/U) = Sh(C)/h_U^\\# of Sites, Lemma [Tag 00Y1] we have • canonically O_U = O_F, and • with these identifications we have (j_F, j_F^sharp) = (j_U, j_U^sharp).","statement_latex":"With\n$(\\Sh(\\mathcal{C}), \\mathcal{O})$ and\n$\\mathcal{F} \\in \\Sh(\\mathcal{C})$ as in\nLemma \\ref{lemma-localize-ringed-topos}.\nIf $\\mathcal{F} = h_U^\\#$ for some object $U$ of $\\mathcal{C}$\nthen via the identification\n$\\Sh(\\mathcal{C}/U) = \\Sh(\\mathcal{C})/h_U^\\#$ of\nSites, Lemma \\ref{sites-lemma-essential-image-j-shriek}\nwe have\n\\begin{enumerate}\n\\item canonically $\\mathcal{O}_U = \\mathcal{O}_\\mathcal{F}$, and\n\\item with these identifications\nwe have $(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) = (j_U, j_U^\\sharp)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04J3","source_file":"sites-modules.tex","source_line":2592,"source_end_line":2608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2592-L2608","statement_sha256":"c4427448f0684763e6ee99d9f6c82f2a6a8ce703ef18cb23d42daab23c46c0e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":3975,"rank":3975,"depth":11,"x":1195.763,"y":336.644,"cluster":"sheaves-sites"},{"id":"stacks:04J4","tag":"04J4","title":"Localization of ringed topoi · Lemma 04J4","summary":"Let (Sh(C), O) be a ringed topos. If s : G → F is a morphism of sheaves on C then there exists a natural commutative diagram of morphisms of ringed topoi xymatrix (Sh(C)/G, O_G) ar[rd]_(j_G, j_G^sharp) ar[rr]_(j, j^sharp) & & (Sh(C)/F, O_F) ar[ld]^(j_F, j_F^sharp) & (Sh(C), O) & where (j, j^sharp) is the localization morphism of the ringed topos (Sh(C)/F, O_F) at the object G/F.","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.\nIf $s : \\mathcal{G} \\to \\mathcal{F}$ is a morphism of sheaves\non $\\mathcal{C}$ then there exists a natural commutative diagram of\nmorphisms of ringed topoi\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C})/\\mathcal{G}, \\mathcal{O}_\\mathcal{G})\n\\ar[rd]_{(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp)} \\ar[rr]_{(j, j^\\sharp)} & &\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\ar[ld]^{(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp)} \\\\\n& (\\Sh(\\mathcal{C}), \\mathcal{O}) &\n}\n$$\nwhere $(j, j^\\sharp)$ is the localization morphism of the ringed topos\n$(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$\nat the object $\\mathcal{G}/\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04J4","source_file":"sites-modules.tex","source_line":2632,"source_end_line":2650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2632-L2650","statement_sha256":"03f1f59ff4e2d3cd73bf674b0599c090d450de28b42f0d4ffbb97937d12628ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":3976,"rank":3976,"depth":11,"x":987.59,"y":171.264,"cluster":"sheaves-sites"},{"id":"stacks:04J5","tag":"04J5","title":"Localization of ringed topoi · Lemma 04J5","summary":"With (Sh(C), O), s : G → F as in Lemma [Tag 04J4]. If there exist a morphism f : V → U of C such that G = h_V^\\# and F = h_U^\\# and s is induced by f, then the diagrams of Lemma [Tag 04IY] and Lemma [Tag 04J4] agree via the identifications (j_F, j_F^sharp) = (j_U, j_U^sharp) and (j_G, j_G^sharp) = (j_V, j_V^sharp) of Lemma [Tag 04J3].","statement_latex":"With $(\\Sh(\\mathcal{C}), \\mathcal{O})$,\n$s : \\mathcal{G} \\to \\mathcal{F}$ as in\nLemma \\ref{lemma-relocalize-ringed-topos}.\nIf there exist a morphism $f : V \\to U$ of $\\mathcal{C}$\nsuch that $\\mathcal{G} = h_V^\\#$ and $\\mathcal{F} = h_U^\\#$\nand $s$ is induced by $f$, then the\ndiagrams of\nLemma \\ref{lemma-relocalize}\nand\nLemma \\ref{lemma-relocalize-ringed-topos}\nagree via the identifications\n$(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) = (j_U, j_U^\\sharp)$\nand\n$(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp) = (j_V, j_V^\\sharp)$\nof\nLemma \\ref{lemma-localize-compare}.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04J5","source_file":"sites-modules.tex","source_line":2660,"source_end_line":2678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2660-L2678","statement_sha256":"6cdb17d6e6aac44efa829509930cbebba49947fbfa2cce6b616d61b1de104a0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3977,"rank":3977,"depth":12,"x":1274.32,"y":175.093,"cluster":"sheaves-sites"},{"id":"stacks:04IF","tag":"04IF","title":"Localization of morphisms of ringed topoi · Lemma 04IF","summary":"Let f : (Sh(C), O) → (Sh(D), O') be a morphism of ringed topoi. Let G be a sheaf on D. Set F = f^-1G. Then there exists a commutative diagram of ringed topoi xymatrix (Sh(C)/F, O_F) ar[rr]_(j_F, j_F^sharp) ar[d]_(f', (f')^sharp) & & (Sh(C), O) ar[d]^(f, f^sharp) (Sh(D)/G, O'_G) ar[rr]^(j_G, j_G^sharp) & & (Sh(D), O') We have f'_*j_F^-1 = j_G^-1f_* and f'_*j_F^* = j_G^*f_*. Moreover, the morphism f' is characterized by the rule (f')^-1(H xrightarrowφ G) = (f^-1H…","statement_latex":"Let\n$$\nf :\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\longrightarrow\n(\\Sh(\\mathcal{D}), \\mathcal{O}')\n$$\nbe a morphism of ringed topoi. Let $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$.\nSet $\\mathcal{F} = f^{-1}\\mathcal{G}$.\nThen there exists a commutative diagram of ringed topoi\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\ar[rr]_{(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp)}\n\\ar[d]_{(f', (f')^\\sharp)} & &\n(\\Sh(\\mathcal{C}), \\mathcal{O}) \\ar[d]^{(f, f^\\sharp)} \\\\\n(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G})\n\\ar[rr]^{(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp)} & &\n(\\Sh(\\mathcal{D}), \\mathcal{O}')\n}\n$$\nWe have $f'_*j_\\mathcal{F}^{-1} = j_\\mathcal{G}^{-1}f_*$\nand $f'_*j_\\mathcal{F}^* = j_\\mathcal{G}^*f_*$. Moreover, the\nmorphism $f'$ is characterized by the rule\n$$\n(f')^{-1}(\\mathcal{H} \\xrightarrow{\\varphi} \\mathcal{G})\n=\n(f^{-1}\\mathcal{H} \\xrightarrow{f^{-1}\\varphi} \\mathcal{F}).\n$$","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of morphisms of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IF","source_file":"sites-modules.tex","source_line":2704,"source_end_line":2735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2704-L2735","statement_sha256":"cc9082c074ad4458208cc62c4ae66be13298d23071d2075b2d360f0c4554ef34","origin":"The Stacks Project","memory_eligible":false,"source_rank":3978,"rank":3978,"depth":12,"x":1059.636,"y":335.097,"cluster":"sheaves-sites"},{"id":"stacks:04J7","tag":"04J7","title":"Localization of morphisms of ringed topoi · Lemma 04J7","summary":"Let f : (Sh(C), O) → (Sh(D), O') be a morphism of ringed topoi. Let G be a sheaf on D. Set F = f^-1G. If f is given by a continuous functor u : D → C and G = h_V^\\#, then the commutative diagrams of Lemma [Tag 04J0] and Lemma [Tag 04IF] agree via the identifications of Lemma [Tag 04J3].","statement_latex":"Let\n$$\nf :\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\longrightarrow\n(\\Sh(\\mathcal{D}), \\mathcal{O}')\n$$\nbe a morphism of ringed topoi.\nLet $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$.\nSet $\\mathcal{F} = f^{-1}\\mathcal{G}$.\nIf $f$ is given by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$\nand $\\mathcal{G} = h_V^\\#$, then the commutative diagrams of\nLemma \\ref{lemma-localize-morphism-ringed-sites}\nand\nLemma \\ref{lemma-localize-morphism-ringed-topoi}\nagree via the identifications of\nLemma \\ref{lemma-localize-compare}.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of morphisms of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04J7","source_file":"sites-modules.tex","source_line":2775,"source_end_line":2794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2775-L2794","statement_sha256":"280748fc23b8c478c2c376fe55a6b76d54046687879920a5d64d92de76593e71","origin":"The Stacks Project","memory_eligible":false,"source_rank":3979,"rank":3979,"depth":13,"x":1089.294,"y":95.103,"cluster":"sheaves-sites"},{"id":"stacks:04J8","tag":"04J8","title":"Localization of morphisms of ringed topoi · Lemma 04J8","summary":"Let (f, f^sharp) : (Sh(C), O) → (Sh(D), O') be a morphism of ringed topoi. Let G be a sheaf on D, let F be a sheaf on C, and let s : F → f^-1G a morphism of sheaves. There exists a commutative diagram of ringed topoi xymatrix (Sh(C)/F, O_F) ar[rr]_(j_F, j_F^sharp) ar[d]_(f_c, f_c^sharp) & & (Sh(C), O) ar[d]^(f, f^sharp) (Sh(D)/G, O'_G) ar[rr]^(j_G, j_G^sharp) & & (Sh(D), O'). The morphism (f_s, f_s^sharp) is equal to the composition of the morphism (f', (f')^sharp) :…","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a morphism of ringed topoi.\nLet $\\mathcal{G}$ be a sheaf on $\\mathcal{D}$,\nlet $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$,\nand let $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ a morphism of sheaves.\nThere exists a commutative diagram of ringed topoi\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\ar[rr]_{(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp)}\n\\ar[d]_{(f_c, f_c^\\sharp)} & &\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\ar[d]^{(f, f^\\sharp)} \\\\\n(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G})\n\\ar[rr]^{(j_\\mathcal{G}, j_\\mathcal{G}^\\sharp)} & &\n(\\Sh(\\mathcal{D}), \\mathcal{O}').\n}\n$$\nThe morphism $(f_s, f_s^\\sharp)$\nis equal to the composition of the morphism\n$$\n(f', (f')^\\sharp) :\n(\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}, \\mathcal{O}_{f^{-1}\\mathcal{G}})\n\\longrightarrow\n(\\Sh(\\mathcal{D})/{\\mathcal{G}}, \\mathcal{O}'_\\mathcal{G})\n$$\nof\nLemma \\ref{lemma-localize-morphism-ringed-topoi}\nand the morphism\n$$\n(j, j^\\sharp) :\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\to\n(\\Sh(\\mathcal{C})/f^{-1}\\mathcal{G}, \\mathcal{O}_{f^{-1}\\mathcal{G}})\n$$\nof\nLemma \\ref{lemma-relocalize-ringed-topos}.\nGiven any morphisms $b : \\mathcal{G}' \\to \\mathcal{G}$,\n$a : \\mathcal{F}' \\to \\mathcal{F}$, and\n$s' : \\mathcal{F}' \\to f^{-1}\\mathcal{G}'$ such that\n$$\n\\xymatrix{\n\\mathcal{F}' \\ar[r]_-{s'} \\ar[d]_a &\nf^{-1}\\mathcal{G}' \\ar[d]^{f^{-1}b} \\\\\n\\mathcal{F} \\ar[r]^-s &\nf^{-1}\\mathcal{G}\n}\n$$\ncommutes, then the following diagram of ringed topoi\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C})/\\mathcal{F}', \\mathcal{O}_{\\mathcal{F}'})\n\\ar[rr]_{(j_{\\mathcal{F}'/\\mathcal{F}}, j_{\\mathcal{F}'/\\mathcal{F}}^\\sharp)}\n\\ar[d]_{(f_{s'}, f_{s'}^\\sharp)} & &\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\ar[d]^{(f_s, f_s^\\sharp)} \\\\\n(\\Sh(\\mathcal{D})/\\mathcal{G}', \\mathcal{O}'_{\\mathcal{G}'})\n\\ar[rr]^{(j_{\\mathcal{G}'/\\mathcal{G}}, j_{\\mathcal{G}'/\\mathcal{G}}^\\sharp)}\n& &\n(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_{\\mathcal{G}'})\n}\n$$\ncommutes.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of morphisms of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04J8","source_file":"sites-modules.tex","source_line":2807,"source_end_line":2876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2807-L2876","statement_sha256":"7023118f502895f6621ca0b822ad4a2ff01d02a6665860819acc53e12e6a2e24","origin":"The Stacks Project","memory_eligible":false,"source_rank":3980,"rank":3980,"depth":13,"x":1260.56,"y":289.056,"cluster":"sheaves-sites"},{"id":"stacks:04J9","tag":"04J9","title":"Localization of morphisms of ringed topoi · Lemma 04J9","summary":"Let (f, f^sharp) : (Sh(C), O) → (Sh(D), O'), s : F → f^-1G be as in Lemma [Tag 04J8]. If f is given by a continuous functor u : D → C and G = h_V^\\#, F = h_U^\\# and s comes from a morphism c : U → u(V), then the commutative diagrams of Lemma [Tag 04J1] and Lemma [Tag 04J8] agree via the identifications of Lemma [Tag 04J3].","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}')$,\n$s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ be as in\nLemma \\ref{lemma-relocalize-morphism-ringed-topoi}.\nIf $f$ is given by a continuous functor\n$u : \\mathcal{D} \\to \\mathcal{C}$\nand $\\mathcal{G} = h_V^\\#$,\n$\\mathcal{F} = h_U^\\#$ and $s$ comes from a morphism\n$c : U \\to u(V)$, then\nthe commutative diagrams of\nLemma \\ref{lemma-relocalize-morphism-ringed-sites}\nand\nLemma \\ref{lemma-relocalize-morphism-ringed-topoi}\nagree via the identifications of\nLemma \\ref{lemma-localize-compare}.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization of morphisms of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04J9","source_file":"sites-modules.tex","source_line":2887,"source_end_line":2907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2887-L2907","statement_sha256":"77ec11741b3be6109ac9e84abbc53f9070d3d1f79af1caa3d1704b3454e37594","origin":"The Stacks Project","memory_eligible":false,"source_rank":3981,"rank":3981,"depth":14,"x":978.073,"y":243.181,"cluster":"sheaves-sites"},{"id":"stacks:03DL","tag":"03DL","title":"Local types of modules · Definition 03DL","summary":"Let (C, O) be a ringed site. Let F be a sheaf of O-modules. We will freely use the notions defined in Definition [Tag 03DE]. • We say F is locally free if for every object U of C there exists a covering (U_i → U)_i ∈ I of C such that each restriction F|_C/U_i is a free O_U_i-module. • We say F is finite locally free if for every object U of C there exists a covering (U_i → U)_i ∈ I of C such that each restriction F|_C/U_i is a finite free O_U_i-module. • We say F is…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nWe will freely use the notions defined in\nDefinition \\ref{definition-global}.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ is {\\it locally free}\nif for every object $U$ of $\\mathcal{C}$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction\n$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is a free\n$\\mathcal{O}_{U_i}$-module.\n\\item We say $\\mathcal{F}$ is {\\it finite locally free}\nif for every object $U$ of $\\mathcal{C}$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction\n$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is a finite free\n$\\mathcal{O}_{U_i}$-module.\n\\item We say $\\mathcal{F}$ is {\\it locally generated by sections}\nif for every object $U$ of $\\mathcal{C}$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction\n$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an\n$\\mathcal{O}_{U_i}$-module generated by global sections.\n\\item Given $r \\geq 0$ we sat $\\mathcal{F}$ is {\\it locally generated\nby $r$ sections} if for every object $U$ of $\\mathcal{C}$ there exists\na covering $\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each\nrestriction $\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an\n$\\mathcal{O}_{U_i}$-module generated by $r$ global sections.\n\\item We say $\\mathcal{F}$ is {\\it of finite type}\nif for every object $U$ of $\\mathcal{C}$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction\n$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an\n$\\mathcal{O}_{U_i}$-module generated by finitely many global sections.\n\\item We say $\\mathcal{F}$ is {\\it quasi-coherent}\nif for every object $U$ of $\\mathcal{C}$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction\n$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an\n$\\mathcal{O}_{U_i}$-module which has a global presentation.\n\\item We say $\\mathcal{F}$ is {\\it of finite presentation}\nif for every object $U$ of $\\mathcal{C}$ there exists a covering\n$\\{U_i \\to U\\}_{i \\in I}$ of $\\mathcal{C}$ such that each restriction\n$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is an\n$\\mathcal{O}_{U_i}$-module which has a finite global presentation.\n\\item We say $\\mathcal{F}$ is {\\it coherent} if and only if\n$\\mathcal{F}$ is of finite type, and for every object\n$U$ of $\\mathcal{C}$ and any $s_1, \\ldots, s_n \\in \\mathcal{F}(U)$\nthe kernel of the map\n$\\bigoplus_{i = 1, \\ldots, n} \\mathcal{O}_U \\to \\mathcal{F}|_U$\nis of finite type on $(\\mathcal{C}/U, \\mathcal{O}_U)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Local types of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DL","source_file":"sites-modules.tex","source_line":2940,"source_end_line":2989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2940-L2989","statement_sha256":"f73eaa83c11d83977baa30d979f15a6dfb48d873d88a589f076682516d7fbc19","origin":"The Stacks Project","memory_eligible":false,"source_rank":3982,"rank":3982,"depth":1,"x":1223.463,"y":116.617,"cluster":"sheaves-sites"},{"id":"stacks:03DM","tag":"03DM","title":"Local types of modules · Lemma 03DM","summary":"Any of the properties (1) -- (8) of Definition [Tag 03DL] is intrinsic (see discussion in Section [Tag 03DG]).","statement_latex":"Any of the properties (1) -- (8) of Definition \\ref{definition-site-local}\nis intrinsic (see discussion in Section \\ref{section-intrinsic}).","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Local types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DM","source_file":"sites-modules.tex","source_line":2991,"source_end_line":2995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L2991-L2995","statement_sha256":"e56bdecc131921c5d78e7ee3dde1de135474ed6c34b21d19cf7dbfc1aa6aeabe","origin":"The Stacks Project","memory_eligible":false,"source_rank":3983,"rank":3983,"depth":8,"x":1144.232,"y":349.37,"cluster":"sheaves-sites"},{"id":"stacks:03DN","tag":"03DN","title":"Local types of modules · Lemma 03DN","summary":"Let (Sh(C), O) be a ringed topos. Let F be an O-module. Assume that the site C has a final object X. Then • The following are equivalent • F is locally free, • there exists a covering (X_i → X) in C such that each restriction F|_C/X_i is a locally free O_X_i-module, and • there exists a covering (X_i → X) in C such that each restriction F|_C/X_i is a free O_X_i-module. • The following are equivalent • F is finite locally free, • there exists a covering (X_i → X) in C such…","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$\nbe a ringed topos. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module.\nAssume that the site $\\mathcal{C}$ has a final object $X$.\nThen\n\\begin{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is locally free,\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that\neach restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a locally free\n$\\mathcal{O}_{X_i}$-module, and\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$ such that\neach restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$ is a free\n$\\mathcal{O}_{X_i}$-module.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is finite locally free,\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis a finite locally free $\\mathcal{O}_{X_i}$-module, and\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis a finite free $\\mathcal{O}_{X_i}$-module.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is locally generated by sections,\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis an $\\mathcal{O}_{X_i}$-module locally generated by sections, and\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis an $\\mathcal{O}_{X_i}$-module globally generated by sections.\n\\end{enumerate}\n\\item Given $r \\geq 0$, the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is locally generated by $r$ sections,\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis an $\\mathcal{O}_{X_i}$-module locally generated by $r$ sections, and\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis an $\\mathcal{O}_{X_i}$-module globally generated by $r$ sections.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is of finite type,\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis an $\\mathcal{O}_{X_i}$-module of finite type, and\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis an $\\mathcal{O}_{X_i}$-module globally generated by finitely many sections.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is quasi-coherent,\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis a quasi-coherent $\\mathcal{O}_{X_i}$-module, and\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis an $\\mathcal{O}_{X_i}$-module which has a global presentation.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is of finite presentation,\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis an $\\mathcal{O}_{X_i}$-module of finite presentation, and\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis an $\\mathcal{O}_{X_i}$-module has a finite global presentation.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is coherent, and\n\\item there exists a covering $\\{X_i \\to X\\}$ in $\\mathcal{C}$\nsuch that each restriction $\\mathcal{F}|_{\\mathcal{C}/X_i}$\nis a coherent $\\mathcal{O}_{X_i}$-module.\n\\end{enumerate}\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Local types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DN","source_file":"sites-modules.tex","source_line":3121,"source_end_line":3206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3121-L3206","statement_sha256":"facdcff3de5a09c3a596c1d774597d189d0fed5090d78c612cb516aad9167e4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":3984,"rank":3984,"depth":6,"x":1015.379,"y":132.61,"cluster":"sheaves-sites"},{"id":"stacks:03DO","tag":"03DO","title":"Local types of modules · Lemma 03DO","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let F be an O_D-module. • If F is locally free then f^*F is locally free. • If F is finite locally free then f^*F is finite locally free. • If F is locally generated by sections then f^*F is locally generated by sections. • If F is locally generated by r sections then f^*F is locally generated by r sections. • If F is of finite type then f^*F is of finite type. • If F is quasi-coherent then f^*F…","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{D}$-module.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is locally free then $f^*\\mathcal{F}$ is locally free.\n\\item If $\\mathcal{F}$ is finite locally free then $f^*\\mathcal{F}$ is\nfinite locally free.\n\\item If $\\mathcal{F}$ is locally generated by sections\nthen $f^*\\mathcal{F}$ is locally generated by sections.\n\\item If $\\mathcal{F}$ is locally generated by $r$ sections\nthen $f^*\\mathcal{F}$ is locally generated by $r$ sections.\n\\item If $\\mathcal{F}$ is of finite type\nthen $f^*\\mathcal{F}$ is of finite type.\n\\item If $\\mathcal{F}$ is quasi-coherent then\n$f^*\\mathcal{F}$ is quasi-coherent.\n\\item If $\\mathcal{F}$ is of finite presentation\nthen $f^*\\mathcal{F}$ is of finite presentation.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Local types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DO","source_file":"sites-modules.tex","source_line":3223,"source_end_line":3247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3223-L3247","statement_sha256":"85737886d4e2e7c54a342b48fe0217dd2276b82ed4e3440cdea2e12b5cb2d938","origin":"The Stacks Project","memory_eligible":false,"source_rank":3985,"rank":3985,"depth":10,"x":1284.92,"y":219.401,"cluster":"sheaves-sites"},{"id":"stacks:0H98","tag":"0H98","title":"Basic results on local types of modules · Lemma 0H98","summary":"Let (C,O) be a ringed site. Let F be a finitely presented O-module. Let φ : G → F be a morphism of O-modules. If G is finite type, then Coker(φ) is finitely presented.","statement_latex":"Let $(\\mathcal{C},\\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be a\nfinitely presented $\\mathcal{O}$-module. Let\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a morphism of\n$\\mathcal{O}$-modules. If $\\mathcal{G}$ is finite type, then\n$\\Coker(\\varphi)$ is finitely presented.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Basic results on local types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H98","source_file":"sites-modules.tex","source_line":3295,"source_end_line":3302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3295-L3302","statement_sha256":"abf860251bfa0f8a3343bebdc3d0b6b403fd4ba667644ebaec612fbdf764c96e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3986,"rank":3986,"depth":2,"x":1016.156,"y":308.418,"cluster":"sheaves-sites"},{"id":"stacks:082T","tag":"082T","title":"Basic results on local types of modules · Lemma 082T","summary":"Let (C, O) be a ringed site. Let theta : G → F be a surjective O-module map with F of finite presentation and G of finite type. Then Ker(theta) is of finite type.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\theta : \\mathcal{G} \\to \\mathcal{F}$ be a surjective\n$\\mathcal{O}$-module map with $\\mathcal{F}$ of finite presentation\nand $\\mathcal{G}$ of finite type. Then $\\Ker(\\theta)$ is of finite type.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Basic results on local types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082T","source_file":"sites-modules.tex","source_line":3309,"source_end_line":3315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3309-L3315","statement_sha256":"9764259d633bf1dcf8ab040305cb88e81716185f895f8cbe377722bbc4f8767b","origin":"The Stacks Project","memory_eligible":false,"source_rank":3987,"rank":3987,"depth":1,"x":1142.854,"y":90.101,"cluster":"sheaves-sites"},{"id":"stacks:0GZN","tag":"0GZN","title":"Basic results on local types of modules · Lemma 0GZN","summary":"Let C be a category viewed as a site with the chaotic topology, see Sites, Example [Tag 07GE]. Let O be a sheaf of rings on C and let F be a sheaf of O-modules. Then F is quasi-coherent if and only if for all U → V in C the canonical map F(V) ⊗_O(V) O(U) → F(U) is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category viewed as a site with the chaotic\ntopology, see Sites, Example \\ref{sites-example-indiscrete}. Let $\\mathcal{O}$\nbe a sheaf of rings on $\\mathcal{C}$ and let $\\mathcal{F}$ be a sheaf\nof $\\mathcal{O}$-modules. Then $\\mathcal{F}$ is quasi-coherent if\nand only if for all $U \\to V$ in $\\mathcal{C}$ the canonical map\n$$\n\\mathcal{F}(V) \\otimes_{\\mathcal{O}(V)} \\mathcal{O}(U)\n\\longrightarrow\n\\mathcal{F}(U)\n$$\nis an isomorphism.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Basic results on local types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZN","source_file":"sites-modules.tex","source_line":3322,"source_end_line":3335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3322-L3335","statement_sha256":"7b9ce6102bd1ba7f49ae8ac46e9c6f8f8e3afe093f03313df5d6b10a697a421c","origin":"The Stacks Project","memory_eligible":false,"source_rank":3988,"rank":3988,"depth":2,"x":1225.058,"y":323.161,"cluster":"sheaves-sites"},{"id":"stacks:0GZP","tag":"0GZP","title":"Basic results on local types of modules · Lemma 0GZP","summary":"Let C be a category viewed as a site with the chaotic topology, see Sites, Example [Tag 07GE]. Let O be a sheaf of rings on C. Assume for all U → V in C the restriction map O(V) → O(U) is a flat ring map. Then the category of quasi-coherent O-modules is a weak Serre subcategory of Mod(O).","statement_latex":"Let $\\mathcal{C}$ be a category viewed as a site with the chaotic\ntopology, see Sites, Example \\ref{sites-example-indiscrete}.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.\nAssume for all $U \\to V$ in $\\mathcal{C}$ the restriction map\n$\\mathcal{O}(V) \\to \\mathcal{O}(U)$ is a flat ring map.\nThen the category of quasi-coherent $\\mathcal{O}$-modules is\na weak Serre subcategory of $\\textit{Mod}(\\mathcal{O})$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Basic results on local types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZP","source_file":"sites-modules.tex","source_line":3378,"source_end_line":3387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3378-L3387","statement_sha256":"28947db4201f65a633032ad898502c8ddfb610ab2eb35988064def40260bfd9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":3989,"rank":3989,"depth":8,"x":976.825,"y":197.851,"cluster":"sheaves-sites"},{"id":"stacks:08M3","tag":"08M3","title":"Closed immersions of ringed topoi · Lemma 08M3","summary":"Let i : (Sh(C), O) → (Sh(D), O') be a morphism of ringed topoi. Assume i is a closed immersion of topoi and i^sharp : O' → i_*O is surjective. Denote I ⊂ O' the kernel of i^sharp. The functor i_* : Mod(O) → Mod(O') is exact, fully faithful, with essential image those O'-modules G such that IG = 0.","statement_latex":"Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a morphism of ringed topoi. Assume $i$ is a closed immersion of topoi\nand $i^\\sharp : \\mathcal{O}' \\to i_*\\mathcal{O}$ is surjective.\nDenote $\\mathcal{I} \\subset \\mathcal{O}'$ the kernel of $i^\\sharp$.\nThe functor\n$$\ni_* :\n\\textit{Mod}(\\mathcal{O})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}')\n$$\nis exact, fully faithful, with essential image those\n$\\mathcal{O}'$-modules $\\mathcal{G}$ such that $\\mathcal{I}\\mathcal{G} = 0$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Closed immersions of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08M3","source_file":"sites-modules.tex","source_line":3450,"source_end_line":3465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3450-L3465","statement_sha256":"e06be32ddc357d37aaa91f8891338fdb15963ee9baf81badf68fda6289fcd949","origin":"The Stacks Project","memory_eligible":false,"source_rank":3990,"rank":3990,"depth":11,"x":1260.864,"y":149.361,"cluster":"sheaves-sites"},{"id":"stacks:0GMW","tag":"0GMW","title":"Tensor product · Lemma 0GMW","summary":"Let C be a site. Let O be a presheaf of rings. Let F, G be presheaves of O-modules. Then F^\\# ⊗_O^\\# G^\\# is equal to (F ⊗_p, O G)^\\#.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a presheaf of rings.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be presheaves of $\\mathcal{O}$-modules.\nThen\n$\\mathcal{F}^\\# \\otimes_{\\mathcal{O}^\\#} \\mathcal{G}^\\#$\nis equal to\n$(\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G})^\\#$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMW","source_file":"sites-modules.tex","source_line":3605,"source_end_line":3613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3605-L3613","statement_sha256":"9024adc50c24781cdb460d9542f1656bd7f3aa12a22f8ac1f3eb87e0ed3b8c21","origin":"The Stacks Project","memory_eligible":false,"source_rank":3991,"rank":3991,"depth":0,"x":1090.277,"y":346.444,"cluster":"sheaves-sites"},{"id":"stacks:03EL","tag":"03EL","title":"Tensor product · Lemma 03EL","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let F, G be O_D-modules. Then f^*(F ⊗_O_D G) = f^*F ⊗_O_C f^*G functorially in F, G.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be\na morphism of ringed topoi. Let $\\mathcal{F}$, $\\mathcal{G}$\nbe $\\mathcal{O}_\\mathcal{D}$-modules. Then\n$f^*(\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{D}} \\mathcal{G})\n= f^*\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{C}} f^*\\mathcal{G}$\nfunctorially in $\\mathcal{F}$, $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EL","source_file":"sites-modules.tex","source_line":3648,"source_end_line":3657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3648-L3657","statement_sha256":"f07da19dba69889277b05de622065e99a104db2988ba63a6a01777e89da0ad57","origin":"The Stacks Project","memory_eligible":false,"source_rank":3992,"rank":3992,"depth":0,"x":1057.551,"y":104.132,"cluster":"sheaves-sites"},{"id":"stacks:03L6","tag":"03L6","title":"Tensor product · Lemma 03L6","summary":"Let (C, O) be a ringed site. Let F, G be sheaves of O-modules. • If F, G are locally free, so is F ⊗_O G. • If F, G are finite locally free, so is F ⊗_O G. • If F, G are locally generated by sections, so is F ⊗_O G. • If F, G are of finite type, so is F ⊗_O G. • If F, G are quasi-coherent, so is F ⊗_O G. • If F, G are of finite presentation, so is F ⊗_O G. • If F is of finite presentation and G is coherent, then F ⊗_O G is coherent. • If F, G are coherent, so is F ⊗_O G.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally free,\nso is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are finite locally free,\nso is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are locally generated\nby sections, so is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite type,\nso is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are quasi-coherent,\nso is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are of finite presentation,\nso is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.\n\\item If $\\mathcal{F}$ is of finite presentation and $\\mathcal{G}$ is coherent,\nthen $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$ is coherent.\n\\item If $\\mathcal{F}$, $\\mathcal{G}$ are coherent,\nso is $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03L6","source_file":"sites-modules.tex","source_line":3689,"source_end_line":3711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3689-L3711","statement_sha256":"51324b97712d6f34aef5f11d28393d7847b986572a574483cb1a2fa442e17bed","origin":"The Stacks Project","memory_eligible":false,"source_rank":3993,"rank":3993,"depth":6,"x":1276.716,"y":264.365,"cluster":"sheaves-sites"},{"id":"stacks:03EM","tag":"03EM","title":"Internal Hom · Lemma 03EM","summary":"If C is a site, O is a sheaf of rings, F is a presheaf of O-modules, and G is a sheaf of O-modules, then SheafHom_O(F, G) is a sheaf of O-modules.","statement_latex":"If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,\n$\\mathcal{F}$ is a presheaf of $\\mathcal{O}$-modules, and\n$\\mathcal{G}$ is a sheaf of $\\mathcal{O}$-modules, then\n$\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$\nis a sheaf of $\\mathcal{O}$-modules.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EM","source_file":"sites-modules.tex","source_line":3756,"source_end_line":3763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3756-L3763","statement_sha256":"fe1e4ea3a4d8cb7035c571ebcdf08d02f752c100839b881012064fe27933b383","origin":"The Stacks Project","memory_eligible":false,"source_rank":3994,"rank":3994,"depth":7,"x":986.023,"y":270.576,"cluster":"sheaves-sites"},{"id":"stacks:0E8H","tag":"0E8H","title":"Internal Hom · Lemma 0E8H","summary":"Let (C, O) be a ringed site. Let F, G be sheaves of O-modules. Then formation of SheafHom_O(F, G) commutes with restriction to U for U ∈ Ob(C).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}, \\mathcal{G}$ be sheaves of $\\mathcal{O}$-modules.\nThen formation of $\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$\ncommutes with restriction to $U$ for $U \\in \\Ob(\\mathcal{C})$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8H","source_file":"sites-modules.tex","source_line":3774,"source_end_line":3780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3774-L3780","statement_sha256":"70bdf79aebe5477b4706adf590ad09653bacfddfecb4f8ff693b587fc939ea68","origin":"The Stacks Project","memory_eligible":false,"source_rank":3995,"rank":3995,"depth":0,"x":1195.546,"y":100.917,"cluster":"sheaves-sites"},{"id":"stacks:03EN","tag":"03EN","title":"Internal Hom · Lemma 03EN","summary":"Internal hom and (co)limits. Let C be a category and let O be a presheaf of rings. • For any presheaf of O-modules F the functor PMod(O) → PMod(O) , G ↦ SheafHom_O(F, G) commutes with arbitrary limits. • For any presheaf of O-modules G the functor PMod(O) → PMod(O)^opp , F ↦ SheafHom_O(F, G) commutes with arbitrary colimits, in a formula SheafHom_O(colim_i F_i, G) = lim_i SheafHom_O(F_i, G). Suppose that C is a site, and O is a sheaf of rings. • [(3)] For any sheaf of…","statement_latex":"Internal hom and (co)limits.\nLet $\\mathcal{C}$ be a category and let $\\mathcal{O}$ be a presheaf of rings.\n\\begin{enumerate}\n\\item For any presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor\n$$\n\\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O})\n, \\quad\n\\mathcal{G} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})\n$$\ncommutes with arbitrary limits.\n\\item For any presheaf of $\\mathcal{O}$-modules $\\mathcal{G}$ the functor\n$$\n\\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O})^{opp}\n, \\quad\n\\mathcal{F} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})\n$$\ncommutes with arbitrary colimits, in a formula\n$$\n\\SheafHom_\\mathcal{O}(\\colim_i \\mathcal{F}_i, \\mathcal{G})\n=\n\\lim_i \\SheafHom_\\mathcal{O}(\\mathcal{F}_i, \\mathcal{G}).\n$$\n\\end{enumerate}\nSuppose that $\\mathcal{C}$ is a site, and $\\mathcal{O}$ is a sheaf of rings.\n\\begin{enumerate}\n\\item[(3)] For any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor\n$$\n\\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O})\n, \\quad\n\\mathcal{G} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})\n$$\ncommutes with arbitrary limits.\n\\item[(4)] For any sheaf of $\\mathcal{O}$-modules $\\mathcal{G}$ the functor\n$$\n\\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O})^{opp}\n, \\quad\n\\mathcal{F} \\longmapsto \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})\n$$\ncommutes with arbitrary colimits, in a formula\n$$\n\\SheafHom_\\mathcal{O}(\\colim_i \\mathcal{F}_i, \\mathcal{G})\n=\n\\lim_i \\SheafHom_\\mathcal{O}(\\mathcal{F}_i, \\mathcal{G}).\n$$\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EN","source_file":"sites-modules.tex","source_line":3837,"source_end_line":3884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3837-L3884","statement_sha256":"7b60d512577edf9437b797335da6e18c1a1204de78febda20c95bc153eb2b07a","origin":"The Stacks Project","memory_eligible":false,"source_rank":3996,"rank":3996,"depth":0,"x":1177.468,"y":345.101,"cluster":"sheaves-sites"},{"id":"stacks:0GMY","tag":"0GMY","title":"Internal Hom · Lemma 0GMY","summary":"Let (C, O) be a ringed site. Let F, G be O-modules. • If F_2 → F_1 → F → 0 is an exact sequence of O-modules, then 0 → SheafHom_O(F, G) → SheafHom_O(F_1, G) → SheafHom_O(F_2, G) is exact. • If 0 → G → G_1 → G_2 is an exact sequence of O-modules, then 0 → SheafHom_O(F, G) → SheafHom_O(F, G_1) → SheafHom_O(F, G_2) is exact.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}_2 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to 0$\nis an exact sequence of $\\mathcal{O}$-modules, then\n$$\n0 \\to\n\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) \\to\n\\SheafHom_\\mathcal{O}(\\mathcal{F}_1, \\mathcal{G}) \\to\n\\SheafHom_\\mathcal{O}(\\mathcal{F}_2, \\mathcal{G})\n$$\nis exact.\n\\item If $0 \\to \\mathcal{G} \\to \\mathcal{G}_1 \\to \\mathcal{G}_2$\nis an exact sequence of $\\mathcal{O}$-modules, then\n$$\n0 \\to\n\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) \\to\n\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}_1) \\to\n\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}_2)\n$$\nis exact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMY","source_file":"sites-modules.tex","source_line":3917,"source_end_line":3941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3917-L3941","statement_sha256":"de1a6182ddc9a6b07389a77994ccd458cbd74cd52f36b8e374a3a771c5fa0d44","origin":"The Stacks Project","memory_eligible":false,"source_rank":3997,"rank":3997,"depth":7,"x":994.289,"y":154.636,"cluster":"sheaves-sites"},{"id":"stacks:03EO","tag":"03EO","title":"Internal Hom · Lemma 03EO","summary":"Let C be a category. Let O be a presheaf of rings. • Let F, G, H be presheaves of O-modules. There is a canonical isomorphism SheafHom_O (F ⊗_p, O G, H) → SheafHom_O (F, SheafHom_O(G, H)) which is functorial in all three entries (sheaf Hom in all three spots). In particular, Mor_PMod(O)( F ⊗_p, O G, H) = Mor_PMod(O)( F, SheafHom_O(G, H)) • Suppose that C is a site, O is a sheaf of rings, and F, G, H are sheaves of O-modules. There is a canonical isomorphism SheafHom_O (F…","statement_latex":"Let $\\mathcal{C}$ be a category. Let $\\mathcal{O}$ be a presheaf of\nrings.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$, $\\mathcal{G}$, $\\mathcal{H}$ be\npresheaves of $\\mathcal{O}$-modules. There is a canonical isomorphism\n$$\n\\SheafHom_\\mathcal{O}\n(\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}, \\mathcal{H})\n\\longrightarrow\n\\SheafHom_\\mathcal{O}\n(\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H}))\n$$\nwhich is functorial in all three entries (sheaf Hom in\nall three spots). In particular,\n$$\n\\Mor_{\\textit{PMod}(\\mathcal{O})}(\n\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}, \\mathcal{H})\n=\n\\Mor_{\\textit{PMod}(\\mathcal{O})}(\n\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H}))\n$$\n\\item\nSuppose that $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,\nand $\\mathcal{F}$, $\\mathcal{G}$, $\\mathcal{H}$ are sheaves of\n$\\mathcal{O}$-modules. There is a canonical isomorphism\n$$\n\\SheafHom_\\mathcal{O}\n(\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}, \\mathcal{H})\n\\longrightarrow\n\\SheafHom_\\mathcal{O}\n(\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H}))\n$$\nwhich is functorial in all three entries (sheaf Hom in\nall three spots). In particular,\n$$\n\\Mor_{\\textit{Mod}(\\mathcal{O})}(\n\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}, \\mathcal{H})\n=\n\\Mor_{\\textit{Mod}(\\mathcal{O})}(\n\\mathcal{F}, \\SheafHom_\\mathcal{O}(\\mathcal{G}, \\mathcal{H}))\n$$\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EO","source_file":"sites-modules.tex","source_line":3948,"source_end_line":3992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L3948-L3992","statement_sha256":"996855076636598dfc560e28a95cce655ad05cd9b55f820a3d1c49b8294c112e","origin":"The Stacks Project","memory_eligible":false,"source_rank":3998,"rank":3998,"depth":1,"x":1282.754,"y":191.17,"cluster":"sheaves-sites"},{"id":"stacks:03EP","tag":"03EP","title":"Internal Hom · Lemma 03EP","summary":"Tensor product and colimits. Let C be a category and let O be a presheaf of rings. • For any presheaf of O-modules F the functor PMod(O) → PMod(O) , G ↦ F ⊗_p, O G commutes with arbitrary colimits. • Suppose that C is a site, and O is a sheaf of rings. For any sheaf of O-modules F the functor Mod(O) → Mod(O) , G ↦ F ⊗_O G commutes with arbitrary colimits.","statement_latex":"Tensor product and colimits.\nLet $\\mathcal{C}$ be a category and let $\\mathcal{O}$ be a presheaf of rings.\n\\begin{enumerate}\n\\item For any presheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor\n$$\n\\textit{PMod}(\\mathcal{O}) \\longrightarrow \\textit{PMod}(\\mathcal{O})\n, \\quad\n\\mathcal{G} \\longmapsto \\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G}\n$$\ncommutes with arbitrary colimits.\n\\item\nSuppose that $\\mathcal{C}$ is a site, and $\\mathcal{O}$ is a sheaf of rings.\nFor any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ the functor\n$$\n\\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O})\n, \\quad\n\\mathcal{G} \\longmapsto \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}\n$$\ncommutes with arbitrary colimits.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EP","source_file":"sites-modules.tex","source_line":4000,"source_end_line":4022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4000-L4022","statement_sha256":"35329d308aa995fc62cb9fff922b982806daf6f54b367cdd94c35a13c10b30ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":3999,"rank":3999,"depth":2,"x":1040.476,"y":328.019,"cluster":"sheaves-sites"},{"id":"stacks:0932","tag":"0932","title":"Internal Hom · Lemma 0932","summary":"Let C be a category, resp. a site Let O → O' be a map of presheaves, resp. sheaves of rings. Then Hom_O(G, F) = Hom_O'(G, SheafHom_O(O', F)) for any O'-module G and O-module F.","statement_latex":"Let $\\mathcal{C}$ be a category, resp.\\ a site\nLet $\\mathcal{O} \\to \\mathcal{O}'$ be a map of presheaves, resp.\\ sheaves\nof rings. Then\n$$\n\\Hom_\\mathcal{O}(\\mathcal{G}, \\mathcal{F}) =\n\\Hom_{\\mathcal{O}'}(\\mathcal{G},\n\\SheafHom_\\mathcal{O}(\\mathcal{O}', \\mathcal{F}))\n$$\nfor any $\\mathcal{O}'$-module $\\mathcal{G}$ and $\\mathcal{O}$-module\n$\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0932","source_file":"sites-modules.tex","source_line":4030,"source_end_line":4042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4030-L4042","statement_sha256":"a84951bee2f65f542cb53cfae4c517a33e41daf65c487208f9172f8764fa2371","origin":"The Stacks Project","memory_eligible":false,"source_rank":4000,"rank":4000,"depth":1,"x":1109.132,"y":89.448,"cluster":"sheaves-sites"},{"id":"stacks:0E8I","tag":"0E8I","title":"Internal Hom · Lemma 0E8I","summary":"Let (C, O) be a ringed site. Let U ∈ Ob(C). For G in Mod(O_U) and F in Mod(O) we have j_U!G ⊗_O F = j_U!(G ⊗_O_U F|_U).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $U \\in \\Ob(\\mathcal{C})$.\nFor $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_U)$\nand $\\mathcal{F}$ in $\\textit{Mod}(\\mathcal{O})$\nwe have $j_{U!}\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F} =\nj_{U!}(\\mathcal{G} \\otimes_{\\mathcal{O}_U} \\mathcal{F}|_U)$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8I","source_file":"sites-modules.tex","source_line":4050,"source_end_line":4058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4050-L4058","statement_sha256":"792bb93dd1f3673376f7876027f3088dc6eb1e36a918066f56daa90e4d376ae4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4001,"rank":4001,"depth":2,"x":1250.468,"y":304.491,"cluster":"sheaves-sites"},{"id":"stacks:0GMZ","tag":"0GMZ","title":"Internal Hom · Lemma 0GMZ","summary":"Let (C, O) be a ringed site. Let F be an O-module of finite presentation. Let G = colim_λ ∈ Lambda G_λ be a filtered colimit of O-modules. Then the canonical map colim_λ SheafHom_O(F, G_λ) → SheafHom_O(F, G) is an isomorphism.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$ be an $\\mathcal{O}$-module of finite presentation. Let\n$\\mathcal{G} = \\colim_{\\lambda \\in \\Lambda} \\mathcal{G}_\\lambda$\nbe a filtered colimit of $\\mathcal{O}$-modules. Then the canonical map\n$$\n\\colim_\\lambda \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}_\\lambda)\n\\longrightarrow\n\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})\n$$\nis an isomorphism.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMZ","source_file":"sites-modules.tex","source_line":4128,"source_end_line":4140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4128-L4140","statement_sha256":"a1621bbceca7068516544882770c0263a825a2df5e92e568987346cfbb0804d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4002,"rank":4002,"depth":8,"x":973.102,"y":226.058,"cluster":"sheaves-sites"},{"id":"stacks:0GN0","tag":"0GN0","title":"Internal Hom · Lemma 0GN0","summary":"Let (C, O) be a ringed site. Let G = colim_λ ∈ Lambda G_λ be a filtered colimit of O-modules. Let F be an O-module of finite presentation. Then we have colim_λ Hom_O(F, G_λ) = Hom_O(F, G). if the hypotheses of Sites, Lemma [Tag 0GMR] part (4) are satisfied for the site C; please see Sites, Remark [Tag 0GMS].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{G} = \\colim_{\\lambda \\in \\Lambda} \\mathcal{G}_\\lambda$\nbe a filtered colimit of $\\mathcal{O}$-modules.\nLet $\\mathcal{F}$ be an $\\mathcal{O}$-module of finite presentation.\nThen we have\n$$\n\\colim_\\lambda \\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}_\\lambda)\n=\n\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G}).\n$$\nif the hypotheses of\nSites, Lemma \\ref{sites-lemma-directed-colimits-global-sections} part (4)\nare satisfied for the site $\\mathcal{C}$; please see\nSites, Remark \\ref{sites-remark-stronger-conditions}.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Internal Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GN0","source_file":"sites-modules.tex","source_line":4193,"source_end_line":4209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4193-L4209","statement_sha256":"c23640e881b85cc681f5629a5a4454bd21dfad62fa62d6cedf3271740f47ebb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4003,"rank":4003,"depth":9,"x":1240.908,"y":126.432,"cluster":"sheaves-sites"},{"id":"stacks:03ER","tag":"03ER","title":"Flat modules · Definition 03ER","summary":"Let C be a category. Let O be a presheaf of rings. • A presheaf F of O-modules is called flat if the functor PMod(O) → PMod(O), G ↦ G ⊗_p, O F is exact. • A map O → O' of presheaves of rings is called flat if O' is flat as a presheaf of O-modules. • If C is a site, O is a sheaf of rings and F is a sheaf of O-modules, then we say F is flat if the functor Mod(O) → Mod(O), G ↦ G ⊗_O F is exact. • A map O → O' of sheaves of rings on a site is called flat if O' is flat as a…","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{O}$ be a presheaf of rings.\n\\begin{enumerate}\n\\item A presheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules is called\n{\\it flat} if the functor\n$$\n\\textit{PMod}(\\mathcal{O})\n\\longrightarrow\n\\textit{PMod}(\\mathcal{O}), \\quad\n\\mathcal{G} \\mapsto \\mathcal{G} \\otimes_{p, \\mathcal{O}} \\mathcal{F}\n$$\nis exact.\n\\item A map $\\mathcal{O} \\to \\mathcal{O}'$ of presheaves of rings\nis called {\\it flat} if $\\mathcal{O}'$ is flat as a presheaf of\n$\\mathcal{O}$-modules.\n\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings\nand $\\mathcal{F}$ is a sheaf of $\\mathcal{O}$-modules, then we\nsay $\\mathcal{F}$ is {\\it flat} if the functor\n$$\n\\textit{Mod}(\\mathcal{O})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}), \\quad\n\\mathcal{G} \\mapsto \\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}\n$$\nis exact.\n\\item A map $\\mathcal{O} \\to \\mathcal{O}'$ of sheaves of rings on a site\nis called {\\it flat} if $\\mathcal{O}'$ is flat as a sheaf of\n$\\mathcal{O}$-modules.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ER","source_file":"sites-modules.tex","source_line":4242,"source_end_line":4273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4242-L4273","statement_sha256":"fe2014bf6c917247038483097b3a9d37d9f2750bbd51a2d3064ab12a900726ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":4004,"rank":4004,"depth":0,"x":1123.457,"y":352.029,"cluster":"sheaves-sites"},{"id":"stacks:03ES","tag":"03ES","title":"Flat modules · Lemma 03ES","summary":"Let C be a category. Let O be a presheaf of rings. Let F be a presheaf of O-modules. If each F(U) is a flat O(U)-module, then F is flat.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{O}$ be a presheaf of rings.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.\nIf each $\\mathcal{F}(U)$ is a flat $\\mathcal{O}(U)$-module,\nthen $\\mathcal{F}$ is flat.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ES","source_file":"sites-modules.tex","source_line":4279,"source_end_line":4286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4279-L4286","statement_sha256":"9dbdefb1f4d50214f08bec36c72db18945d9c6aedec88d65efa5ceec628a79c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4005,"rank":4005,"depth":0,"x":1028.572,"y":118.853,"cluster":"sheaves-sites"},{"id":"stacks:03ET","tag":"03ET","title":"Flat modules · Lemma 03ET","summary":"Let C be a site. Let O be a presheaf of rings. Let F be a presheaf of O-modules. If F is a flat O-module, then F^\\# is a flat O^\\#-module.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O}$ be a presheaf of rings.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules.\nIf $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, then\n$\\mathcal{F}^\\#$ is a flat $\\mathcal{O}^\\#$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ET","source_file":"sites-modules.tex","source_line":4292,"source_end_line":4299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4292-L4299","statement_sha256":"b155f202785d363243434cb7717493779d323a6ac4961f29cc876a5eb72fb9d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4006,"rank":4006,"depth":0,"x":1286.252,"y":237.045,"cluster":"sheaves-sites"},{"id":"stacks:0GN1","tag":"0GN1","title":"Flat modules · Lemma 0GN1","summary":"Let C be a site. Let O be a presheaf of rings. Let F be a presheaf of O-modules. Assume that every object U of C has a covering (U_i → U)_i ∈ I such that F(U_i) is a flat O(U_i)-module. Then F^\\# is a flat O^\\#-module.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a presheaf of rings.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules. Assume\nthat every object $U$ of $\\mathcal{C}$ has a covering $\\{U_i \\to U\\}_{i \\in I}$\nsuch that $\\mathcal{F}(U_i)$ is a flat $\\mathcal{O}(U_i)$-module.\nThen $\\mathcal{F}^\\#$ is a flat $\\mathcal{O}^\\#$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GN1","source_file":"sites-modules.tex","source_line":4305,"source_end_line":4312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4305-L4312","statement_sha256":"d22bdefe01f4f2ff68412baf8ed29bcf2d1a9e67300795c4c9785fc01387173d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4007,"rank":4007,"depth":1,"x":1000.976,"y":296.151,"cluster":"sheaves-sites"},{"id":"stacks:03EU","tag":"03EU","title":"Flat modules · Lemma 03EU","summary":"Colimits and tensor product. • A filtered colimit of flat presheaves of modules is flat. A direct sum of flat presheaves of modules is flat. • A filtered colimit of flat sheaves of modules is flat. A direct sum of flat sheaves of modules is flat.","statement_latex":"Colimits and tensor product.\n\\begin{enumerate}\n\\item A filtered colimit of flat presheaves of modules\nis flat. A direct sum of flat presheaves of modules is flat.\n\\item A filtered colimit of flat sheaves of modules is flat.\nA direct sum of flat sheaves of modules is flat.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EU","source_file":"sites-modules.tex","source_line":4341,"source_end_line":4350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4341-L4350","statement_sha256":"6ae2313f523bab7f9a4e5e9f84a297a937801002c94869e29a83a27d4e5cabdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4008,"rank":4008,"depth":3,"x":1163.927,"y":90.537,"cluster":"sheaves-sites"},{"id":"stacks:0E8J","tag":"0E8J","title":"Flat modules · Lemma 0E8J","summary":"Let (C, O) be a ringed site. Let U be an object of C. If F is a flat O-module, then F|_U is a flat O_U-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $U$ be an object of $\\mathcal{C}$.\nIf $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, then\n$\\mathcal{F}|_U$ is a flat $\\mathcal{O}_U$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8J","source_file":"sites-modules.tex","source_line":4362,"source_end_line":4368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4362-L4368","statement_sha256":"26907c72992f00de31a9535c43da0553fb02d8d6b4b3d41e8b2f84b4048b3283","origin":"The Stacks Project","memory_eligible":false,"source_rank":4009,"rank":4009,"depth":5,"x":1209.154,"y":334.803,"cluster":"sheaves-sites"},{"id":"stacks:03EV","tag":"03EV","title":"Flat modules · Lemma 03EV","summary":"Let C be a category. Let O be a presheaf of rings. Let U be an object of C. Consider the functor j_U : C/U → C. • The presheaf of O-modules j_U!O_U (see Remark [Tag 03EJ]) is flat. • If C is a site, O is a sheaf of rings, j_U!O_U is a flat sheaf of O-modules.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{O}$ be a presheaf of rings.\nLet $U$ be an object of $\\mathcal{C}$.\nConsider the functor $j_U : \\mathcal{C}/U \\to \\mathcal{C}$.\n\\begin{enumerate}\n\\item The presheaf of $\\mathcal{O}$-modules\n$j_{U!}\\mathcal{O}_U$ (see\nRemark \\ref{remark-localize-presheaves})\nis flat.\n\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,\n$j_{U!}\\mathcal{O}_U$ is a flat sheaf of $\\mathcal{O}$-modules.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EV","source_file":"sites-modules.tex","source_line":4388,"source_end_line":4402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4388-L4402","statement_sha256":"f1edfba9024c16b1689ef2107234135c794766bf847107930b40450108d24f61","origin":"The Stacks Project","memory_eligible":false,"source_rank":4010,"rank":4010,"depth":1,"x":979.196,"y":180.229,"cluster":"sheaves-sites"},{"id":"stacks:03EW","tag":"03EW","title":"Flat modules · Lemma 03EW","summary":"Let C be a category. Let O be a presheaf of rings. • Any presheaf of O-modules is a quotient of a direct sum bigoplus j_U_i!O_U_i. • Any presheaf of O-modules is a quotient of a flat presheaf of O-modules. • If C is a site, O is a sheaf of rings, then any sheaf of O-modules is a quotient of a direct sum bigoplus j_U_i!O_U_i. • If C is a site, O is a sheaf of rings, then any sheaf of O-modules is a quotient of a flat sheaf of O-modules.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{O}$ be a presheaf of rings.\n\\begin{enumerate}\n\\item Any presheaf of $\\mathcal{O}$-modules is a quotient of\na direct sum $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$.\n\\item Any presheaf of $\\mathcal{O}$-modules is a quotient of\na flat presheaf of $\\mathcal{O}$-modules.\n\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,\nthen any sheaf of $\\mathcal{O}$-modules is a quotient of\na direct sum $\\bigoplus j_{U_i!}\\mathcal{O}_{U_i}$.\n\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$ is a sheaf of rings,\nthen any sheaf of $\\mathcal{O}$-modules is a quotient of\na flat sheaf of $\\mathcal{O}$-modules.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EW","source_file":"sites-modules.tex","source_line":4421,"source_end_line":4437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4421-L4437","statement_sha256":"4d695ee9169a859db972b654fb244471603e7f78cddf53ad9f5c9c7827baa0da","origin":"The Stacks Project","memory_eligible":false,"source_rank":4011,"rank":4011,"depth":4,"x":1273.293,"y":163.714,"cluster":"sheaves-sites"},{"id":"stacks:03EX","tag":"03EX","title":"Flat modules · Lemma 03EX","summary":"Let C be a category. Let O be a presheaf of rings. Let 0 → F\" → F' → F → 0 be a short exact sequence of presheaves of O-modules. Let G be a presheaf of O-modules. • If F is a flat presheaf of modules, then the sequence 0 → F\" ⊗_p, O G → F' ⊗_p, O G → F ⊗_p, O G → 0 is exact. • If C is a site, O, F, F', F\", and G are sheaves, and F is flat as a sheaf of modules, then the sequence 0 → F\" ⊗_O G → F' ⊗_O G → F ⊗_O G → 0 is exact.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{O}$ be a presheaf of rings.\nLet\n$$\n0 \\to \\mathcal{F}'' \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0\n$$\nbe a short exact sequence of presheaves of $\\mathcal{O}$-modules.\nLet $\\mathcal{G}$ be a presheaf of $\\mathcal{O}$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is a flat presheaf of modules, then\nthe sequence\n$$\n0 \\to\n\\mathcal{F}'' \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to\n\\mathcal{F}' \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to\n\\mathcal{F} \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to 0\n$$\nis exact.\n\\item If $\\mathcal{C}$ is a site, $\\mathcal{O}$,\n$\\mathcal{F}$, $\\mathcal{F}'$, $\\mathcal{F}''$, and\n$\\mathcal{G}$ are sheaves, and $\\mathcal{F}$ is flat\nas a sheaf of modules, then the sequence\n$$\n0 \\to\n\\mathcal{F}'' \\otimes_\\mathcal{O} \\mathcal{G} \\to\n\\mathcal{F}' \\otimes_\\mathcal{O} \\mathcal{G} \\to\n\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G} \\to 0\n$$\nis exact.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EX","source_file":"sites-modules.tex","source_line":4456,"source_end_line":4488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4456-L4488","statement_sha256":"e58940d54f1a07e0bcdce88255c6585fd212046da9764c3e95a32dc11dd90b05","origin":"The Stacks Project","memory_eligible":false,"source_rank":4012,"rank":4012,"depth":5,"x":1069.556,"y":342.906,"cluster":"sheaves-sites"},{"id":"stacks:03EY","tag":"03EY","title":"Flat modules · Lemma 03EY","summary":"Let C be a category. Let O be a presheaf of rings. Let 0 → F_2 → F_1 → F_0 → 0 be a short exact sequence of presheaves of O-modules. • If F_2 and F_0 are flat so is F_1. • If F_1 and F_0 are flat so is F_2. If C is a site and O is a sheaf of rings then the same result holds in Mod(O).","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{O}$ be a presheaf of rings.\nLet\n$$\n0 \\to\n\\mathcal{F}_2 \\to\n\\mathcal{F}_1 \\to\n\\mathcal{F}_0 \\to 0\n$$\nbe a short exact sequence of presheaves of $\\mathcal{O}$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}_2$ and $\\mathcal{F}_0$ are flat so is\n$\\mathcal{F}_1$.\n\\item If $\\mathcal{F}_1$ and $\\mathcal{F}_0$ are flat so is\n$\\mathcal{F}_2$.\n\\end{enumerate}\nIf $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a\nsheaf of rings then the same result holds in $\\textit{Mod}(\\mathcal{O})$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EY","source_file":"sites-modules.tex","source_line":4521,"source_end_line":4541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4521-L4541","statement_sha256":"9e7bf336765cca3e714f019f06933453ce966d8f23e2de967423ec31f8381334","origin":"The Stacks Project","memory_eligible":false,"source_rank":4013,"rank":4013,"depth":6,"x":1075.691,"y":94.978,"cluster":"sheaves-sites"},{"id":"stacks:03EZ","tag":"03EZ","title":"Flat modules · Lemma 03EZ","summary":"Let C be a category. Let O be a presheaf of rings. Let … → F_2 → F_1 → F_0 → Q → 0 be an exact complex of presheaves of O-modules. If Q and all F_i are flat O-modules, then for any presheaf G of O-modules the complex … → F_2 ⊗_p, O G → F_1 ⊗_p, O G → F_0 ⊗_p, O G → Q ⊗_p, O G → 0 is exact also. If C is a site and O is a sheaf of rings then the same result holds Mod(O).","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{O}$ be a presheaf of rings.\nLet\n$$\n\\ldots \\to\n\\mathcal{F}_2 \\to\n\\mathcal{F}_1 \\to\n\\mathcal{F}_0 \\to\n\\mathcal{Q} \\to 0\n$$\nbe an exact complex of presheaves of $\\mathcal{O}$-modules.\nIf $\\mathcal{Q}$ and all $\\mathcal{F}_i$ are flat $\\mathcal{O}$-modules,\nthen for any presheaf $\\mathcal{G}$ of $\\mathcal{O}$-modules the\ncomplex\n$$\n\\ldots \\to\n\\mathcal{F}_2 \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to\n\\mathcal{F}_1 \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to\n\\mathcal{F}_0 \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to\n\\mathcal{Q} \\otimes_{p, \\mathcal{O}} \\mathcal{G} \\to 0\n$$\nis exact also. If $\\mathcal{C}$ is a site and $\\mathcal{O}$ is a\nsheaf of rings then the same result holds $\\textit{Mod}(\\mathcal{O})$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EZ","source_file":"sites-modules.tex","source_line":4558,"source_end_line":4583,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4558-L4583","statement_sha256":"80b8cc331fa57d86b20db9de49c166ac10d579e81ca4fac18d35fa2e620dc40b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4014,"rank":4014,"depth":7,"x":1270.693,"y":281.42,"cluster":"sheaves-sites"},{"id":"stacks:0G6Q","tag":"0G6Q","title":"Flat modules · Lemma 0G6Q","summary":"Let (C, O) be a ringed site. If G and F are flat O-modules, then G ⊗_O F is a flat O-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. If $\\mathcal{G}$ and\n$\\mathcal{F}$ are flat $\\mathcal{O}$-modules, then\n$\\mathcal{G} \\otimes_\\mathcal{O} \\mathcal{F}$ is a flat $\\mathcal{O}$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6Q","source_file":"sites-modules.tex","source_line":4592,"source_end_line":4597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4592-L4597","statement_sha256":"dbfe4e9573dcb42868acce08ef469a6f5d0813dad59c2ad43d37ecbb1e850f9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4015,"rank":4015,"depth":0,"x":976.747,"y":254.569,"cluster":"sheaves-sites"},{"id":"stacks:05V4","tag":"05V4","title":"Flat modules · Lemma 05V4","summary":"Let O_1 → O_2 be a map of sheaves of rings on a site C. If G is a flat O_1-module, then G ⊗_O_1 O_2 is a flat O_2-module.","statement_latex":"Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves\nof rings on a site $\\mathcal{C}$. If $\\mathcal{G}$ is a\nflat $\\mathcal{O}_1$-module, then\n$\\mathcal{G} \\otimes_{\\mathcal{O}_1} \\mathcal{O}_2$\nis a flat $\\mathcal{O}_2$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05V4","source_file":"sites-modules.tex","source_line":4609,"source_end_line":4616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4609-L4616","statement_sha256":"ce88c2b7e5476c36036016fc65340b7a030a85b97525b94ad2a675d0142a30e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4016,"rank":4016,"depth":0,"x":1215.272,"y":107.465,"cluster":"sheaves-sites"},{"id":"stacks:08FC","tag":"08FC","title":"Flat modules · Lemma 08FC","summary":"Let (C, O) be a ringed site. Let F be an O-module. The following are equivalent • F is a flat O-module. • Let U be an object of C and let O_U xrightarrow(f_1, …, f_n) O_U^⊕ n xrightarrow(s_1, …, s_n) F|_U be a complex of O_U-modules. Then there exists a covering (U_i → U) and for each i a factorization O_U_i^⊕ n xrightarrowB_i O_U_i^⊕ l_i xrightarrow(t_i1, …, t_il_i) F|_U_i of (s_1, …, s_n)|_U_i such that B_i ∘ (f_1, …, f_n)|_U_i = 0. • Let U be an object of C and let…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be an\n$\\mathcal{O}$-module. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a flat $\\mathcal{O}$-module.\n\\item Let $U$ be an object of $\\mathcal{C}$ and let\n$$\n\\mathcal{O}_U \\xrightarrow{(f_1, \\ldots, f_n)}\n\\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)}\n\\mathcal{F}|_U\n$$\nbe a complex of $\\mathcal{O}_U$-modules. Then there exists a covering\n$\\{U_i \\to U\\}$ and for each $i$ a factorization\n$$\n\\mathcal{O}_{U_i}^{\\oplus n}\n\\xrightarrow{B_i}\n\\mathcal{O}_{U_i}^{\\oplus l_i} \\xrightarrow{(t_{i1}, \\ldots, t_{il_i})}\n\\mathcal{F}|_{U_i}\n$$\nof $(s_1, \\ldots, s_n)|_{U_i}$ such that\n$B_i \\circ (f_1, \\ldots, f_n)|_{U_i} = 0$.\n\\item Let $U$ be an object of $\\mathcal{C}$ and let\n$$\n\\mathcal{O}_U^{\\oplus m} \\xrightarrow{A}\n\\mathcal{O}_U^{\\oplus n} \\xrightarrow{(s_1, \\ldots, s_n)}\n\\mathcal{F}|_U\n$$\nbe a complex of $\\mathcal{O}_U$-modules. Then there exists a covering\n$\\{U_i \\to U\\}$ and for each $i$ a factorization\n$$\n\\mathcal{O}_{U_i}^{\\oplus n}\n\\xrightarrow{B_i}\n\\mathcal{O}_{U_i}^{\\oplus l_i} \\xrightarrow{(t_{i1}, \\ldots, t_{il_i})}\n\\mathcal{F}|_{U_i}\n$$\nof $(s_1, \\ldots, s_n)|_{U_i}$ such that\n$B_i \\circ A|_{U_i} = 0$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FC","source_file":"sites-modules.tex","source_line":4633,"source_end_line":4672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4633-L4672","statement_sha256":"0ac5327964657222734558967b3eb12348175a52003cebd9353089887f089317","origin":"The Stacks Project","memory_eligible":false,"source_rank":4017,"rank":4017,"depth":6,"x":1157.64,"y":351.467,"cluster":"sheaves-sites"},{"id":"stacks:08M4","tag":"08M4","title":"Flat modules · Lemma 08M4","summary":"Let C be a site. Let O' → O be a surjection of sheaves of rings whose kernel I is an ideal of square zero. Let F' be an O'-module and set F = F'/IF'. The following are equivalent • F' is a flat O'-module, and • F is a flat O-module and I ⊗_O F → F' is injective.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}' \\to \\mathcal{O}$\nbe a surjection of sheaves of rings whose kernel $\\mathcal{I}$ is\nan ideal of square zero. Let $\\mathcal{F}'$ be an $\\mathcal{O}'$-module\nand set $\\mathcal{F} = \\mathcal{F}'/\\mathcal{I}\\mathcal{F}'$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}'$ is a flat $\\mathcal{O}'$-module, and\n\\item $\\mathcal{F}$ is a flat $\\mathcal{O}$-module and\n$\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{F}'$\nis injective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08M4","source_file":"sites-modules.tex","source_line":4786,"source_end_line":4799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4786-L4799","statement_sha256":"7015c362289b326c9c46a1208e96100b6ca8b4f2cc8e4655c71de05ffef371ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":4018,"rank":4018,"depth":6,"x":1003.801,"y":138.681,"cluster":"sheaves-sites"},{"id":"stacks:0GLY","tag":"0GLY","title":"Flat modules · Lemma 0GLY","summary":"Let C be a site. Let O → O' be a flat homomorphism of sheaves of rings. Let I ⊂ O be a sheaf of ideals such that the induced map O/I → O'/IO' is an isomorphism. For any O-module F annihilated by I^n for some n ≥ 0 the map id ⊗ 1 : F → F ⊗_O O' is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$\nbe a flat homomorphism of sheaves of rings. Let\n$\\mathcal{I} \\subset \\mathcal{O}$\nbe a sheaf of ideals such that the induced map\n$\\mathcal{O}/\\mathcal{I} \\to \\mathcal{O}'/\\mathcal{I}\\mathcal{O}'$\nis an isomorphism. For any $\\mathcal{O}$-module $\\mathcal{F}$\nannihilated by $\\mathcal{I}^n$ for some $n \\geq 0$ the map\n$\\text{id} \\otimes 1 :\n\\mathcal{F} \\to \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{O}'$\nis an isomorphism.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLY","source_file":"sites-modules.tex","source_line":4840,"source_end_line":4852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4840-L4852","statement_sha256":"82766b8e68057e924f26272edfe5864f78a3e9d5760eec9d1841b89a9dbe9b47","origin":"The Stacks Project","memory_eligible":false,"source_rank":4019,"rank":4019,"depth":5,"x":1288.571,"y":208.347,"cluster":"sheaves-sites"},{"id":"stacks:0FNZ","tag":"0FNZ","title":"Duals · Lemma 0FNZ","summary":"Let (C, O) be a ringed site. Let F be a O-module. Let G, eta, ε be a left dual of F in the monoidal category of O-modules, see Categories, Definition [Tag 0FFP]. Then • for every object U of C there exists a covering (U_i → U) such that F|_U_i is a direct summand of a finite free O|_U_i-module, • the map e : SheafHom_O(F, O) → G sending a local section λ to (λ ⊗ 1)(eta) is an isomorphism, • we have ε(f, g) = e^-1(g)(f) for local sections f and g of F and G.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$ be a\n$\\mathcal{O}$-module. Let $\\mathcal{G}, \\eta, \\epsilon$\nbe a left dual of $\\mathcal{F}$ in the monoidal category of\n$\\mathcal{O}$-modules, see\nCategories, Definition \\ref{categories-definition-dual}. Then\n\\begin{enumerate}\n\\item for every object $U$ of $\\mathcal{C}$\nthere exists a covering $\\{U_i \\to U\\}$ such that $\\mathcal{F}|_{U_i}$\nis a direct summand of a finite free $\\mathcal{O}|_{U_i}$-module,\n\\item the map\n$e : \\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{O}) \\to \\mathcal{G}$\nsending a local section $\\lambda$ to $(\\lambda \\otimes 1)(\\eta)$\nis an isomorphism,\n\\item we have $\\epsilon(f, g) = e^{-1}(g)(f)$ for local sections\n$f$ and $g$ of $\\mathcal{F}$ and $\\mathcal{G}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FNZ","source_file":"sites-modules.tex","source_line":4929,"source_end_line":4947,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L4929-L4947","statement_sha256":"959a44635189fdf41517ebd50585e8e5b4d7292b04000f8b66b8d6ad8ca72ccf","origin":"The Stacks Project","memory_eligible":false,"source_rank":4020,"rank":4020,"depth":1,"x":1022.362,"y":318.645,"cluster":"sheaves-sites"},{"id":"stacks:08FD","tag":"08FD","title":"Duals · Lemma 08FD","summary":"Let (C, O) be a ringed site. Let F be locally of finite presentation and flat. Then given an object U of C there exists a covering (U_i → U) such that F|_U_i is a direct summand of a finite free O_U_i-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}$\nbe locally of finite presentation and flat. Then given an object\n$U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that\n$\\mathcal{F}|_{U_i}$ is a direct summand of a finite free\n$\\mathcal{O}_{U_i}$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FD","source_file":"sites-modules.tex","source_line":5000,"source_end_line":5007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5000-L5007","statement_sha256":"78f388944b83af2dcdc20fc60621fb12b2456104da09571b1c06f19d8bc8dad4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4021,"rank":4021,"depth":7,"x":1130.043,"y":86.084,"cluster":"sheaves-sites"},{"id":"stacks:0934","tag":"0934","title":"Towards constructible modules · Lemma 0934","summary":"Let (C, O) be a ringed site. Let (U_i → U) be a covering of C. Then the sequence bigoplus j_U_i ×_U U_j!O_U_i ×_U U_j → bigoplus j_U_i!O_U_i → j_!O_U → 0 is exact.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\{U_i \\to U\\}$\nbe a covering of $\\mathcal{C}$. Then the sequence\n$$\n\\bigoplus j_{U_i \\times_U U_j!}\\mathcal{O}_{U_i \\times_U U_j} \\to\n\\bigoplus j_{U_i!}\\mathcal{O}_{U_i} \\to j_!\\mathcal{O}_U \\to 0\n$$\nis exact.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0934","source_file":"sites-modules.tex","source_line":5052,"source_end_line":5061,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5052-L5061","statement_sha256":"3835db7de1d2d0eb2015677fe64744da3ce9ca32895eadf6e8edec8f9306673a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4022,"rank":4022,"depth":2,"x":1237.741,"y":318.847,"cluster":"sheaves-sites"},{"id":"stacks:0G1W","tag":"0G1W","title":"Towards constructible modules · Lemma 0G1W","summary":"Let (C, O) be a ringed site. Let U = (U_i → U)_i ∈ I be covering of C. If U is quasi-compact, then there exist a finite subset I' ⊂ I such that the sequence bigoplus_i, i' ∈ I' j_U_i ×_U U_i'!O_U_i ×_U U_i' → bigoplus_i ∈ I' j_U_i!O_U_i → j_!O_U → 0 is exact.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be covering of $\\mathcal{C}$.\nIf $U$ is quasi-compact, then there exist a finite subset\n$I' \\subset I$ such that the sequence\n$$\n\\bigoplus\\nolimits_{i, i' \\in I'}\nj_{U_i \\times_U U_{i'}!}\\mathcal{O}_{U_i \\times_U U_{i'}} \\to\n\\bigoplus\\nolimits_{i \\in I'}\nj_{U_i!}\\mathcal{O}_{U_i} \\to\nj_!\\mathcal{O}_U \\to 0\n$$\nis exact.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1W","source_file":"sites-modules.tex","source_line":5073,"source_end_line":5087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5073-L5087","statement_sha256":"c1f9812edfc2e46c9dcbc4a5105c7ebe1df1f0453712eb8bee698ff879837cdb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4023,"rank":4023,"depth":3,"x":970.945,"y":208.238,"cluster":"sheaves-sites"},{"id":"stacks:0935","tag":"0935","title":"Towards constructible modules · Lemma 0935","summary":"Let C be a site. Let W be a quasi-compact object of C. • The functor Sh(C) → Sets, F ↦ F(W) commutes with coproducts. • Let O be a sheaf of rings on C. The functor Mod(O) → Ab, F ↦ F(W) commutes with direct sums.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $W$ be a quasi-compact\nobject of $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The functor $\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,\n$\\mathcal{F} \\mapsto \\mathcal{F}(W)$ commutes with coproducts.\n\\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. The functor\n$\\textit{Mod}(\\mathcal{O}) \\to \\textit{Ab}$,\n$\\mathcal{F} \\mapsto \\mathcal{F}(W)$\ncommutes with direct sums.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0935","source_file":"sites-modules.tex","source_line":5119,"source_end_line":5131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5119-L5131","statement_sha256":"f24dbcff613cb20cc2cf336b3346f1c4c90be161488205089fb451dd0d7d9fc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4024,"rank":4024,"depth":4,"x":1256.839,"y":138.36,"cluster":"sheaves-sites"},{"id":"stacks:0936","tag":"0936","title":"Towards constructible modules · Lemma 0936","summary":"Let (C, O) be a ringed site. Let U be a quasi-compact object of C. Then the functor Hom_O(j_!O_U, -) commutes with direct sums.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be a quasi-compact\nobject of $\\mathcal{C}$. Then the functor\n$\\Hom_\\mathcal{O}(j_!\\mathcal{O}_U, -)$ commutes with direct sums.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0936","source_file":"sites-modules.tex","source_line":5162,"source_end_line":5167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5162-L5167","statement_sha256":"06c20791caee4be72d9046efe3d9867bcf95d0b1ac7eb1af760279d07c7e3fbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4025,"rank":4025,"depth":5,"x":1102.103,"y":352.27,"cluster":"sheaves-sites"},{"id":"stacks:093B","tag":"093B","title":"Towards constructible modules · Lemma 093B","summary":"In Situation [Tag 0937] assume ([Tag 0938]) holds. • Every sheaf of sets is the target of a surjective map whose source is a coproduct coprod h_U_i^\\# with U_i in B. • If O is a sheaf of rings, then every O-module is a quotient of a direct sum bigoplus j_U_i!O_U_i with U_i in B.","statement_latex":"In Situation \\ref{situation-quasi-compact-objects} assume\n(\\ref{item-enough}) holds.\n\\begin{enumerate}\n\\item Every sheaf of sets is the target of a surjective map\nwhose source is a coproduct $\\coprod h_{U_i}^\\#$ with $U_i$ in $\\mathcal{B}$.\n\\item If $\\mathcal{O}$ is a sheaf of rings, then every $\\mathcal{O}$-module\nis a quotient of a direct sum $\\bigoplus\\nolimits j_{U_i!}\\mathcal{O}_{U_i}$\nwith $U_i$ in $\\mathcal{B}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093B","source_file":"sites-modules.tex","source_line":5202,"source_end_line":5213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5202-L5213","statement_sha256":"0d10e2655b22f2eb590bc487d314c1c8ee13569b45c26db8f9e059f0208dcdde","origin":"The Stacks Project","memory_eligible":false,"source_rank":4026,"rank":4026,"depth":5,"x":1044.139,"y":106.552,"cluster":"sheaves-sites"},{"id":"stacks:093C","tag":"093C","title":"Towards constructible modules · Lemma 093C","summary":"In Situation [Tag 0937] assume ([Tag 0938]) and ([Tag 0939]) hold. • Every sheaf of sets is a filtered colimit of sheaves of the form Coequalizer( xymatrix coprod_j = 1, …, m h_V_j^\\# ar@<1ex>[r] ar@<-1ex>[r] & coprod_i = 1, …, n h_U_i^\\# ) with U_i and V_j in B. • If O is a sheaf of rings, then every O-module is a filtered colimit of sheaves of the form Coker( bigoplus_j = 1, …, m j_V_j!O_V_j → bigoplus_i = 1, …, n j_U_i!O_U_i ) with U_i and V_j in B.","statement_latex":"In Situation \\ref{situation-quasi-compact-objects} assume\n(\\ref{item-enough}) and (\\ref{item-enough-qc}) hold.\n\\begin{enumerate}\n\\item Every sheaf of sets is a filtered colimit of sheaves of the form\n\\begin{equation}\n\n\\text{Coequalizer}\\left(\n\\xymatrix{\n\\coprod\\nolimits_{j = 1, \\ldots, m} h_{V_j}^\\#\n\\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\coprod\\nolimits_{i = 1, \\ldots, n} h_{U_i}^\\#\n}\n\\right)\n\\end{equation}\nwith $U_i$ and $V_j$ in $\\mathcal{B}$.\n\\item If $\\mathcal{O}$ is a sheaf of rings, then every $\\mathcal{O}$-module\nis a filtered colimit of sheaves of the form\n\\begin{equation}\n\n\\Coker\\left(\n\\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j}\n\\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}\n\\right)\n\\end{equation}\nwith $U_i$ and $V_j$ in $\\mathcal{B}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093C","source_file":"sites-modules.tex","source_line":5223,"source_end_line":5252,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5223-L5252","statement_sha256":"22f740c4543b8a8f5468cca7ac19895bde4566d68b3c58f676411004e56af3b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4027,"rank":4027,"depth":6,"x":1284.66,"y":254.961,"cluster":"sheaves-sites"},{"id":"stacks:093E","tag":"093E","title":"Towards constructible modules · Lemma 093E","summary":"In Situation [Tag 0937] assume ([Tag 0938]) and ([Tag 0939]) hold. Let O be a sheaf of rings. Then a cokernel of a map between modules as in ([Tag 093D]) is another module as in ([Tag 093D]).","statement_latex":"In Situation \\ref{situation-quasi-compact-objects} assume\n(\\ref{item-enough}) and (\\ref{item-enough-qc}) hold.\nLet $\\mathcal{O}$ be a sheaf of rings.\nThen a cokernel of a map between modules as in\n(\\ref{equation-towards-constructible}) is another module as\nin (\\ref{equation-towards-constructible}).","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093E","source_file":"sites-modules.tex","source_line":5300,"source_end_line":5308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5300-L5308","statement_sha256":"84d444174d010a81a8693a16f656f40947c0ccb8df16e935ecc65163c1a1272f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4028,"rank":4028,"depth":7,"x":987.736,"y":282.024,"cluster":"sheaves-sites"},{"id":"stacks:093F","tag":"093F","title":"Towards constructible modules · Lemma 093F","summary":"In Situation [Tag 0937] assume ([Tag 0938]), ([Tag 0939]), and ([Tag 093A]) hold. Let O be a sheaf of rings. Assume given a map bigoplus_j = 1, …, m j_V_j!O_V_j → bigoplus_i = 1, …, n j_U_i!O_U_i with U_i and V_j in B, and coverings (U_ik → U_i)_k ∈ K_i with U_ik ∈ B. Then there exist finite subsets K'_i ⊂ K_i and a finite set L of W_l ∈ B and a commutative diagram xymatrix bigoplus_l ∈ L j_W_l!O_W_l ar[d] ar[r] & bigoplus_i = 1, …, n bigoplus_k ∈ K'_i j_U_ik!O_U_ik ar[d]…","statement_latex":"In Situation \\ref{situation-quasi-compact-objects} assume\n(\\ref{item-enough}), (\\ref{item-enough-qc}), and (\\ref{item-enough-qc-qs})\nhold. Let $\\mathcal{O}$ be a sheaf of rings. Assume given a map\n$$\n\\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j}\n\\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}\n$$\nwith $U_i$ and $V_j$ in $\\mathcal{B}$, and coverings\n$\\{U_{ik} \\to U_i\\}_{k \\in K_i}$ with $U_{ik} \\in \\mathcal{B}$.\nThen there exist finite subsets $K'_i \\subset K_i$ and\na finite set $L$ of $W_l \\in \\mathcal{B}$ and a commutative diagram\n$$\n\\xymatrix{\n\\bigoplus_{l \\in L} j_{W_l!}\\mathcal{O}_{W_l} \\ar[d] \\ar[r] &\n\\bigoplus_{i = 1, \\ldots, n} \\bigoplus_{k \\in K'_i}\nj_{U_{ik}!}\\mathcal{O}_{U_{ik}} \\ar[d] \\\\\n\\bigoplus_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j} \\ar[r] &\n\\bigoplus_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}\n}\n$$\ninducing an isomorphism on cokernels of the horizontal maps.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093F","source_file":"sites-modules.tex","source_line":5340,"source_end_line":5364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5340-L5364","statement_sha256":"18127dba48f03e06128e531a9066805eba758b43b38a4614cc907e87653425a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4029,"rank":4029,"depth":4,"x":1185.067,"y":93.446,"cluster":"sheaves-sites"},{"id":"stacks:093G","tag":"093G","title":"Towards constructible modules · Lemma 093G","summary":"In Situation [Tag 0937] assume ([Tag 0938]), ([Tag 0939]), and ([Tag 093A]) hold. Let O be a sheaf of rings. Then an extension of modules as in ([Tag 093D]) is another module as in ([Tag 093D]).","statement_latex":"In Situation \\ref{situation-quasi-compact-objects} assume\n(\\ref{item-enough}), (\\ref{item-enough-qc}), and (\\ref{item-enough-qc-qs})\nhold. Let $\\mathcal{O}$ be a sheaf of rings.\nThen an extension of modules as in (\\ref{equation-towards-constructible})\nis another module as in (\\ref{equation-towards-constructible}).","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093G","source_file":"sites-modules.tex","source_line":5409,"source_end_line":5416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5409-L5416","statement_sha256":"7ebfb8555de9bf841b785b6599035b261e3db4b53e71474edb5146008f3828ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":4030,"rank":4030,"depth":7,"x":1191.211,"y":344.657,"cluster":"sheaves-sites"},{"id":"stacks:093H","tag":"093H","title":"Towards constructible modules · Lemma 093H","summary":"In Situation [Tag 0937] assume ([Tag 0938]), ([Tag 0939]), and ([Tag 093A]) hold. Let O be a sheaf of rings. Let A ⊂ Mod(O) be the full subcategory of modules isomorphic to a cokernel as in ([Tag 093D]). If the kernel of every map of O-modules of the form bigoplus_j = 1, …, m j_V_j!O_V_j → bigoplus_i = 1, …, n j_U_i!O_U_i with U_i and V_j in B, is in A, then A is weak Serre subcategory of Mod(O).","statement_latex":"In Situation \\ref{situation-quasi-compact-objects} assume\n(\\ref{item-enough}), (\\ref{item-enough-qc}), and (\\ref{item-enough-qc-qs})\nhold. Let $\\mathcal{O}$ be a sheaf of rings.\nLet $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$ be the full\nsubcategory of modules isomorphic to a cokernel as in\n(\\ref{equation-towards-constructible}).\nIf the kernel of every map of $\\mathcal{O}$-modules of the form\n$$\n\\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\mathcal{O}_{V_j}\n\\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\mathcal{O}_{U_i}\n$$\nwith $U_i$ and $V_j$ in $\\mathcal{B}$, is in $\\mathcal{A}$, then\n$\\mathcal{A}$ is weak Serre subcategory of $\\textit{Mod}(\\mathcal{O})$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Towards constructible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093H","source_file":"sites-modules.tex","source_line":5455,"source_end_line":5471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5455-L5471","statement_sha256":"364570218b0333b0d4c42c79ffeaa357ee1303f77b027df1dac4c301b9212288","origin":"The Stacks Project","memory_eligible":false,"source_rank":4031,"rank":4031,"depth":8,"x":984.51,"y":162.77,"cluster":"sheaves-sites"},{"id":"stacks:04JB","tag":"04JB","title":"Flat morphisms · Definition 04JB","summary":"Let (f, f^sharp) : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. We say (f, f^sharp) is flat if the ring map f^sharp : f^-1O' → O is flat. We say a morphism of ringed sites is flat if the associated morphism of ringed topoi is flat.","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\longrightarrow\n(\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi. We say $(f, f^\\sharp)$ is\n{\\it flat} if the ring map $f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$\nis flat. We say a morphism of ringed sites is {\\it flat}\nif the associated morphism of ringed topoi is flat.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JB","source_file":"sites-modules.tex","source_line":5526,"source_end_line":5537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5526-L5537","statement_sha256":"d4cda0046e8af8b244e70ee828acebdd5a5577c9a5db2ac9f7a378a12f9aec3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4032,"rank":4032,"depth":0,"x":1283.418,"y":179.617,"cluster":"sheaves-sites"},{"id":"stacks:04JC","tag":"04JC","title":"Flat morphisms · Lemma 04JC","summary":"Let f : Sh(C) → Sh(C') be a morphism of ringed topoi. Then f^-1 : Ab(C') → Ab(C), F ↦ f^-1F is exact. If (f, f^sharp) : (Sh(C), O) → (Sh(C'), O') is a flat morphism of ringed topoi then f^* : Mod(O') → Mod(O), F ↦ f^*F is exact.","statement_latex":"Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}')$\nbe a morphism of ringed topoi. Then\n$$\nf^{-1} : \\textit{Ab}(\\mathcal{C}') \\longrightarrow \\textit{Ab}(\\mathcal{C}),\n\\quad\n\\mathcal{F} \\longmapsto f^{-1}\\mathcal{F}\n$$\nis exact. If\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\to\n(\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nis a flat morphism of ringed topoi then\n$$\nf^* : \\textit{Mod}(\\mathcal{O}') \\longrightarrow \\textit{Mod}(\\mathcal{O}),\n\\quad\n\\mathcal{F} \\longmapsto f^*\\mathcal{F}\n$$\nis exact.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JC","source_file":"sites-modules.tex","source_line":5539,"source_end_line":5560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5539-L5560","statement_sha256":"d2391c2ce5478be05a209c9094d635113b2cb793a1d03031a6368ddd17a64abc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4033,"rank":4033,"depth":1,"x":1049.287,"y":336.918,"cluster":"sheaves-sites"},{"id":"stacks:08M5","tag":"08M5","title":"Flat morphisms · Definition 08M5","summary":"Let f : (Sh(C), O) → (Sh(D), O') be a morphism of ringed topoi. Let F be a sheaf of O-modules. We say that F is flat over (Sh(D), O') if F is flat as an f^-1O'-module.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a morphism of ringed topoi. Let $\\mathcal{F}$ be a sheaf of\n$\\mathcal{O}$-modules. We say that $\\mathcal{F}$ is\n{\\it flat over $(\\Sh(\\mathcal{D}), \\mathcal{O}')$} if\n$\\mathcal{F}$ is flat as an $f^{-1}\\mathcal{O}'$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08M5","source_file":"sites-modules.tex","source_line":5579,"source_end_line":5586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5579-L5586","statement_sha256":"d208c653840c61f82c5af7b3cf78f587f45bf96e845eed356a3889d58c762a9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4034,"rank":4034,"depth":0,"x":1095.469,"y":87.891,"cluster":"sheaves-sites"},{"id":"stacks:0GN2","tag":"0GN2","title":"Flat morphisms · Lemma 0GN2","summary":"Let f : (C, O_C) → (D, O_D) be a morphism of ringed sites. Let F, G be O_D-modules. If F is finitely presented and f is flat, then the canonical map f^*SheafHom_O_D(F, G) → SheafHom_O_C(f^*F, f^*G) of Remark [Tag 0GMX] is an isomorphism.","statement_latex":"Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism\nof ringed sites. Let $\\mathcal{F}$, $\\mathcal{G}$ be\n$\\mathcal{O}_\\mathcal{D}$-modules.\nIf $\\mathcal{F}$ is finitely presented and $f$ is flat,\nthen the canonical map\n$$\nf^*\\SheafHom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{F}, \\mathcal{G})\n\\longrightarrow\n\\SheafHom_{\\mathcal{O}_\\mathcal{C}}(f^*\\mathcal{F}, f^*\\mathcal{G})\n$$\nof Remark \\ref{remark-pullback-internal-hom} is an isomorphism.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GN2","source_file":"sites-modules.tex","source_line":5593,"source_end_line":5607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5593-L5607","statement_sha256":"44aeb577d1d242caca33e861bf8a2126e2ed4f089656c5e6675bd8220e54adf3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4035,"rank":4035,"depth":8,"x":1261.798,"y":297.879,"cluster":"sheaves-sites"},{"id":"stacks:0409","tag":"0409","title":"Invertible modules · Definition 0409","summary":"Let (C, O) be a ringed site. • A finite locally free O-module F is said to have rank r if for every object U of C there exists a covering (U_i → U) of U such that F|_U_i is isomorphic to O_U_i^⊕ r as an O_U_i-module. • An O-module L is invertible if the functor Mod(O) → Mod(O), F ↦ F ⊗_O L is an equivalence. • The sheaf O^* is the subsheaf of O defined by the rule U ↦ O^*(U) = (f ∈ O(U) mid ∃ g ∈ O(U) such that fg = 1) It is a sheaf of abelian groups with multiplication…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\n\\begin{enumerate}\n\\item A finite locally free $\\mathcal{O}$-module $\\mathcal{F}$ is said\nto have {\\it rank $r$} if for every object $U$ of $\\mathcal{C}$ there\nexists a covering $\\{U_i \\to U\\}$ of $U$ such that $\\mathcal{F}|_{U_i}$\nis isomorphic to $\\mathcal{O}_{U_i}^{\\oplus r}$ as an\n$\\mathcal{O}_{U_i}$-module.\n\\item An $\\mathcal{O}$-module $\\mathcal{L}$ is {\\it invertible}\nif the functor\n$$\n\\textit{Mod}(\\mathcal{O}) \\longrightarrow \\textit{Mod}(\\mathcal{O}),\\quad\n\\mathcal{F}  \\longmapsto \\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{L}\n$$\nis an equivalence.\n\\item The sheaf {\\it $\\mathcal{O}^*$} is the subsheaf of\n$\\mathcal{O}$ defined by the rule\n$$\nU \\longmapsto \\mathcal{O}^*(U) = \\{f \\in \\mathcal{O}(U) \\mid\n\\exists g \\in \\mathcal{O}(U)\\text{ such that }fg = 1\\}\n$$\nIt is a sheaf of abelian groups with multiplication as the group law.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Invertible modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0409","source_file":"sites-modules.tex","source_line":5674,"source_end_line":5698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5674-L5698","statement_sha256":"f48c51555a4060444929e6914440e2dc6e9555ee772a255e4dec1b0bee565b2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4036,"rank":4036,"depth":0,"x":970.069,"y":237.372,"cluster":"sheaves-sites"},{"id":"stacks:0B8N","tag":"0B8N","title":"Invertible modules · Lemma 0B8N","summary":"Let (C, O) be a ringed site. Let L be an O-module. The following are equivalent: • L is invertible, and • there exists an O-module N such that L ⊗_O N ≅ O. In this case we have • [(a)] L is a flat O-module of finite presentation, • [(b)] for every object U of C there exists a covering (U_i → U) such that L|_U_i is a direct summand of a finite free module, and • [(c)] the module N in (2) is isomorphic to SheafHom_O(L, O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{L}$\nbe an $\\mathcal{O}$-module. The following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is invertible, and\n\\item there exists an $\\mathcal{O}$-module $\\mathcal{N}$\nsuch that\n$\\mathcal{L} \\otimes_\\mathcal{O} \\mathcal{N} \\cong \\mathcal{O}$.\n\\end{enumerate}\nIn this case we have\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{L}$ is a flat $\\mathcal{O}$-module of finite presentation,\n\\item[(b)] for every object $U$ of $\\mathcal{C}$ there exists a\ncovering $\\{U_i \\to U\\}$ such that $\\mathcal{L}|_{U_i}$\nis a direct summand of a finite free module, and\n\\item[(c)] the module $\\mathcal{N}$ in (2) is isomorphic to\n$\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8N","source_file":"sites-modules.tex","source_line":5705,"source_end_line":5724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5705-L5724","statement_sha256":"ebcce6bd7318236d770535e13d5d2d0049f98555abb1e0fa48879354152bf0e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4037,"rank":4037,"depth":2,"x":1234.037,"y":116.366,"cluster":"sheaves-sites"},{"id":"stacks:0B8P","tag":"0B8P","title":"Invertible modules · Lemma 0B8P","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. The pullback f^*L of an invertible O_D-module is invertible.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a\nmorphism of ringed topoi. The pullback $f^*\\mathcal{L}$ of an\ninvertible $\\mathcal{O}_\\mathcal{D}$-module is invertible.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8P","source_file":"sites-modules.tex","source_line":5793,"source_end_line":5799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5793-L5799","statement_sha256":"20b116bf66ea674b66283729231faf610c1b6fe8e6d1b589b34b70c27df96182","origin":"The Stacks Project","memory_eligible":false,"source_rank":4038,"rank":4038,"depth":3,"x":1136.629,"y":355.551,"cluster":"sheaves-sites"},{"id":"stacks:040A","tag":"040A","title":"Invertible modules · Lemma 040A","summary":"Let (C, O) be a ringed site. • If L, N are invertible O-modules, then so is L ⊗_O N. • If L is an invertible O-module, then so is SheafHom_O(L, O) and the evaluation map L ⊗_O SheafHom_O(L, O) → O is an isomorphism.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\n\\begin{enumerate}\n\\item If $\\mathcal{L}$, $\\mathcal{N}$ are invertible\n$\\mathcal{O}$-modules, then so is\n$\\mathcal{L} \\otimes_\\mathcal{O} \\mathcal{N}$.\n\\item If $\\mathcal{L}$ is an invertible\n$\\mathcal{O}$-module, then so is\n$\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O})$ and the evaluation map\n$\\mathcal{L} \\otimes_\\mathcal{O}\n\\SheafHom_\\mathcal{O}(\\mathcal{L}, \\mathcal{O}) \\to \\mathcal{O}$\nis an isomorphism.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040A","source_file":"sites-modules.tex","source_line":5812,"source_end_line":5826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5812-L5826","statement_sha256":"c989ef146caba03515dc6b83ed5cad45d341c18d7c6cf087f238520926ebb5fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":4039,"rank":4039,"depth":3,"x":1016.022,"y":123.738,"cluster":"sheaves-sites"},{"id":"stacks:040B","tag":"040B","title":"Invertible modules · Lemma 040B","summary":"Let (C, O) be a ringed site. There exists a set of invertible modules (L_i)_i ∈ I such that each invertible module on (C, O) is isomorphic to exactly one of the L_i.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nThere exists a set of invertible modules $\\{\\mathcal{L}_i\\}_{i \\in I}$\nsuch that each invertible module on $(\\mathcal{C}, \\mathcal{O})$\nis isomorphic to exactly one of the $\\mathcal{L}_i$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040B","source_file":"sites-modules.tex","source_line":5833,"source_end_line":5839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5833-L5839","statement_sha256":"3c667e1a89f20ca2592d049aa2083e430c3492183dfbc5d05e93edf234da7002","origin":"The Stacks Project","memory_eligible":false,"source_rank":4040,"rank":4040,"depth":8,"x":1291.575,"y":226.313,"cluster":"sheaves-sites"},{"id":"stacks:040C","tag":"040C","title":"Invertible modules · Definition 040C","summary":"Let (C, O) be a ringed site. The Picard group Pic(O) of the ringed site is the abelian group whose elements are isomorphism classes of invertible O-modules, with addition corresponding to tensor product.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nThe {\\it Picard group} $\\Pic(\\mathcal{O})$ of\nthe ringed site is the\nabelian group whose elements are isomorphism classes of\ninvertible $\\mathcal{O}$-modules, with addition\ncorresponding to tensor product.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Invertible modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040C","source_file":"sites-modules.tex","source_line":5885,"source_end_line":5893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5885-L5893","statement_sha256":"8603ce5ddd52eb74a41e2709eca3d3a3c613d962ea4315662afbabb31514a3be","origin":"The Stacks Project","memory_eligible":false,"source_rank":4041,"rank":4041,"depth":0,"x":1005.69,"y":307.09,"cluster":"sheaves-sites"},{"id":"stacks:04BK","tag":"04BK","title":"Modules of differentials · Definition 04BK","summary":"Let C be a site. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings. Let F be an O_2-module. A O_1-derivation or more precisely a φ-derivation into F is a map D : O_2 → F which is additive, annihilates the image of O_1 → O_2, and satisfies the Leibniz rule D(ab) = aD(b) + D(a)b for all a, b local sections of O_2 (wherever they are both defined). We denote Der_O_1(O_2, F) the set of φ-derivations into F.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. Let $\\mathcal{F}$\nbe an $\\mathcal{O}_2$-module. A {\\it $\\mathcal{O}_1$-derivation}\nor more precisely a {\\it $\\varphi$-derivation} into $\\mathcal{F}$\nis a map $D : \\mathcal{O}_2 \\to \\mathcal{F}$ which is additive, annihilates\nthe image of $\\mathcal{O}_1 \\to \\mathcal{O}_2$, and satisfies the\n{\\it Leibniz rule}\n$$\nD(ab) = aD(b) + D(a)b\n$$\nfor all $a, b$ local sections of $\\mathcal{O}_2$\n(wherever they are both defined). We denote\n$\\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$\nthe set of $\\varphi$-derivations into $\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BK","source_file":"sites-modules.tex","source_line":5913,"source_end_line":5929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5913-L5929","statement_sha256":"60283f032c592ffdb76cc6310859d81aecf57aa1528f4dd9b97771ad8288c3da","origin":"The Stacks Project","memory_eligible":false,"source_rank":4042,"rank":4042,"depth":0,"x":1151.645,"y":85.146,"cluster":"sheaves-sites"},{"id":"stacks:04BL","tag":"04BL","title":"Modules of differentials · Lemma 04BL","summary":"Let C be a site. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings. The functor Mod(O_2) → Ab, F ↦ Der_O_1(O_2, F) is representable.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. The functor\n$$\n\\textit{Mod}(\\mathcal{O}_2) \\longrightarrow \\textit{Ab}, \\quad\n\\mathcal{F} \\longmapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})\n$$\nis representable.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BL","source_file":"sites-modules.tex","source_line":5950,"source_end_line":5959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L5950-L5959","statement_sha256":"0ea46cdf0dcdd09827b2410b77349f3bfe00a7fee533df5ae7d635e35557c0b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4043,"rank":4043,"depth":0,"x":1222.552,"y":331.802,"cluster":"sheaves-sites"},{"id":"stacks:04BN","tag":"04BN","title":"Modules of differentials · Definition 04BN","summary":"Let C be a site. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings. The module of differentials of the ring map φ is the object representing the functor F ↦ Der_O_1(O_2, F) which exists by Lemma [Tag 04BL]. It is denoted Ω_O_2/O_1, and the universal φ-derivation is denoted d : O_2 → Ω_O_2/O_1.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. The {\\it module of differentials}\nof the ring map $\\varphi$ is the object representing the functor\n$\\mathcal{F} \\mapsto \\text{Der}_{\\mathcal{O}_1}(\\mathcal{O}_2, \\mathcal{F})$\nwhich exists by Lemma \\ref{lemma-universal-module}.\nIt is denoted $\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$, and the {\\it universal\n$\\varphi$-derivation} is denoted\n$\\text{d} : \\mathcal{O}_2 \\to \\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BN","source_file":"sites-modules.tex","source_line":6013,"source_end_line":6023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6013-L6023","statement_sha256":"318a33e054c906ecb8cabc2f3a12713cdfd4dd33ced766aa97a2b5cd77d36f9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4044,"rank":4044,"depth":1,"x":971.73,"y":190.054,"cluster":"sheaves-sites"},{"id":"stacks:08TP","tag":"08TP","title":"Modules of differentials · Lemma 08TP","summary":"Let C be a site. Let φ : O_1 → O_2 be a homomorphism of presheaves of rings. Then Ω_O_2^\\#/O_1^\\# is the sheaf associated to the presheaf U ↦ Ω_O_2(U)/O_1(U).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of presheaves of rings. Then\n$\\Omega_{\\mathcal{O}_2^\\#/\\mathcal{O}_1^\\#}$ is the sheaf associated to the\npresheaf $U \\mapsto \\Omega_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TP","source_file":"sites-modules.tex","source_line":6035,"source_end_line":6041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6035-L6041","statement_sha256":"49089a53ada0a9dcbcfb5c94395cce856cc5e04ba3d6f8abbbf7bd7f97f0f555","origin":"The Stacks Project","memory_eligible":false,"source_rank":4045,"rank":4045,"depth":1,"x":1270.89,"y":152.227,"cluster":"sheaves-sites"},{"id":"stacks:08TQ","tag":"08TQ","title":"Modules of differentials · Lemma 08TQ","summary":"Let f : Sh(D) → Sh(C) be a morphism of topoi. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings on C. Then there is a canonical identification f^-1Ω_O_2/O_1 = Ω_f^-1O_2/f^-1O_1 compatible with universal derivations.","statement_latex":"Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.\nLet $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings on $\\mathcal{C}$.\nThen there is a canonical identification\n$f^{-1}\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} =\n\\Omega_{f^{-1}\\mathcal{O}_2/f^{-1}\\mathcal{O}_1}$\ncompatible with universal derivations.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TQ","source_file":"sites-modules.tex","source_line":6061,"source_end_line":6070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6061-L6070","statement_sha256":"aa6aa4130a1793179a6ac1b9cce4f4dc8c3a807bd56e7a981572b244805b8cc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4046,"rank":4046,"depth":1,"x":1080.575,"y":350.013,"cluster":"sheaves-sites"},{"id":"stacks:04BO","tag":"04BO","title":"Modules of differentials · Lemma 04BO","summary":"Let C be a site. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings. For any object U of C there is a canonical isomorphism Ω_O_2/O_1|_U = Ω_(O_2|_U)/(O_1|_U) compatible with universal derivations.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. For any object $U$ of $\\mathcal{C}$\nthere is a canonical isomorphism\n$$\n\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1}|_U =\n\\Omega_{(\\mathcal{O}_2|_U)/(\\mathcal{O}_1|_U)}\n$$\ncompatible with universal derivations.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BO","source_file":"sites-modules.tex","source_line":6086,"source_end_line":6096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6086-L6096","statement_sha256":"01d8fa9dba4515275ab294edddfaca9d9976ca946b873535aeb825fc7fa8f796","origin":"The Stacks Project","memory_eligible":false,"source_rank":4047,"rank":4047,"depth":2,"x":1061.843,"y":95.996,"cluster":"sheaves-sites"},{"id":"stacks:08TR","tag":"08TR","title":"Modules of differentials · Lemma 08TR","summary":"Let C be a site. Let xymatrix O_2 ar[r]_φ & O_2' O_1 ar[r] ar[u] & O'_1 ar[u] be a commutative diagram of sheaves of rings on C. The map O_2 → O'_2 composed with the map d : O'_2 → Ω_O'_2/O'_1 is a O_1-derivation. Hence we obtain a canonical map of O_2-modules Ω_O_2/O_1 → Ω_O'_2/O'_1. It is uniquely characterized by the property that d(f) maps to d(φ(f)) for any local section f of O_2. In this way Ω_-/- becomes a functor on the category of arrows of sheaves of rings.","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$$\n\\xymatrix{\n\\mathcal{O}_2 \\ar[r]_\\varphi & \\mathcal{O}_2' \\\\\n\\mathcal{O}_1 \\ar[r] \\ar[u] & \\mathcal{O}'_1 \\ar[u]\n}\n$$\nbe a commutative diagram of sheaves of rings on $\\mathcal{C}$. The map\n$\\mathcal{O}_2 \\to \\mathcal{O}'_2$ composed with the map\n$\\text{d} : \\mathcal{O}'_2 \\to \\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$\nis a $\\mathcal{O}_1$-derivation. Hence we obtain a canonical map of\n$\\mathcal{O}_2$-modules\n$\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\to\n\\Omega_{\\mathcal{O}'_2/\\mathcal{O}'_1}$.\nIt is uniquely characterized by the property that\n$\\text{d}(f)$ maps to $\\text{d}(\\varphi(f))$\nfor any local section $f$ of $\\mathcal{O}_2$.\nIn this way $\\Omega_{-/-}$ becomes a functor on the category\nof arrows of sheaves of rings.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TR","source_file":"sites-modules.tex","source_line":6102,"source_end_line":6123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6102-L6123","statement_sha256":"35b981e6506beb80b323e8e37982bc7a739953f6b21490da35a8e89575b1a457","origin":"The Stacks Project","memory_eligible":false,"source_rank":4048,"rank":4048,"depth":0,"x":1280.087,"y":272.803,"cluster":"sheaves-sites"},{"id":"stacks:08TS","tag":"08TS","title":"Modules of differentials · Lemma 08TS","summary":"In Lemma [Tag 08TR] suppose that O_2 → O'_2 is surjective with kernel I ⊂ O_2 and assume that O_1 = O'_1. Then there is a canonical exact sequence of O'_2-modules I/I^2 → Ω_O_2/O_1 ⊗_O_2 O'_2 → Ω_O'_2/O_1 → 0 The leftmost map is characterized by the rule that a local section f of I maps to df ⊗ 1.","statement_latex":"In Lemma \\ref{lemma-functoriality-differentials} suppose that\n$\\mathcal{O}_2 \\to \\mathcal{O}'_2$ is surjective with kernel\n$\\mathcal{I} \\subset \\mathcal{O}_2$ and assume that\n$\\mathcal{O}_1 = \\mathcal{O}'_1$. Then there is a canonical exact\nsequence of $\\mathcal{O}'_2$-modules\n$$\n\\mathcal{I}/\\mathcal{I}^2\n\\longrightarrow\n\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{O}'_2\n\\longrightarrow\n\\Omega_{\\mathcal{O}'_2/\\mathcal{O}_1}\n\\longrightarrow\n0\n$$\nThe leftmost map is characterized by the rule that a local section\n$f$ of $\\mathcal{I}$ maps to $\\text{d}f \\otimes 1$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TS","source_file":"sites-modules.tex","source_line":6129,"source_end_line":6147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6129-L6147","statement_sha256":"1377438519af38aa8d4de71e42dbc5250b55a4200ce52aee72c98f42aa4dbf19","origin":"The Stacks Project","memory_eligible":false,"source_rank":4049,"rank":4049,"depth":2,"x":976.756,"y":266.26,"cluster":"sheaves-sites"},{"id":"stacks:04BP","tag":"04BP","title":"Modules of differentials · Lemma 04BP","summary":"Let C be a site. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings. Consider a short exact sequence 0 → F → A → O_2 → 0 Here A is a sheaf of O_1-algebras, π : A → O_2 is a surjection of sheaves of O_1-algebras, and F = Ker(π) is its kernel. Assume F an ideal sheaf with square zero in A. So F has a natural structure of an O_2-module. A section s : O_2 → A of π is a O_1-algebra map such that π ∘ s = id. Given any section s : O_2 → F of π and any φ-derivation D : O_1 →…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings.\nConsider a short exact sequence\n$$\n0 \\to \\mathcal{F} \\to \\mathcal{A} \\to \\mathcal{O}_2 \\to 0\n$$\nHere $\\mathcal{A}$ is a sheaf of $\\mathcal{O}_1$-algebras,\n$\\pi : \\mathcal{A} \\to \\mathcal{O}_2$ is a surjection\nof sheaves of $\\mathcal{O}_1$-algebras, and\n$\\mathcal{F} = \\Ker(\\pi)$ is its kernel. Assume $\\mathcal{F}$ an ideal\nsheaf with square zero in $\\mathcal{A}$. So $\\mathcal{F}$\nhas a natural structure of an $\\mathcal{O}_2$-module.\nA section $s : \\mathcal{O}_2 \\to \\mathcal{A}$ of $\\pi$\nis a $\\mathcal{O}_1$-algebra map such that $\\pi \\circ s = \\text{id}$.\nGiven any section $s : \\mathcal{O}_2 \\to \\mathcal{F}$\nof $\\pi$ and any $\\varphi$-derivation $D : \\mathcal{O}_1 \\to \\mathcal{F}$\nthe map\n$$\ns + D : \\mathcal{O}_1 \\to \\mathcal{A}\n$$\nis a section of $\\pi$ and every section $s'$ is of the form $s + D$\nfor a unique $\\varphi$-derivation $D$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BP","source_file":"sites-modules.tex","source_line":6191,"source_end_line":6215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6191-L6215","statement_sha256":"b463c3d1a8575e0f6c39df7755bbfeed5a9ef4d509b9577aac00398dcc89faa8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4050,"rank":4050,"depth":0,"x":1205.853,"y":98.847,"cluster":"sheaves-sites"},{"id":"stacks:04BQ","tag":"04BQ","title":"Modules of differentials · Definition 04BQ","summary":"Let X = (Sh(C), O) and Y = (Sh(C'), O') be ringed topoi. Let (f, f^sharp) : X → Y be a morphism of ringed topoi. In this situation • for a sheaf F of O-modules a Y-derivation D : O → F is just a f^sharp-derivation, and • the sheaf of differentials Ω_X/Y of X over Y is the module of differentials of f^sharp : f^-1O' → O, see Definition [Tag 04BN]. Thus Ω_X/Y comes equipped with a universal Y-derivation d_X/Y : O → Ω_X/Y. We sometimes write Ω_X/Y = Ω_f.","statement_latex":"Let $X = (\\Sh(\\mathcal{C}), \\mathcal{O})$ and\n$Y = (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be ringed topoi.\nLet $(f, f^\\sharp) : X \\to Y$ be a morphism of ringed topoi.\nIn this situation\n\\begin{enumerate}\n\\item for a sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules a\n{\\it $Y$-derivation} $D : \\mathcal{O} \\to \\mathcal{F}$ is just a\n$f^\\sharp$-derivation, and\n\\item the {\\it sheaf of differentials $\\Omega_{X/Y}$ of $X$ over $Y$}\nis the module of differentials of\n$f^\\sharp : f^{-1}\\mathcal{O}' \\to \\mathcal{O}$,\nsee Definition \\ref{definition-module-differentials}.\n\\end{enumerate}\nThus $\\Omega_{X/Y}$ comes equipped with a {\\it universal $Y$-derivation}\n$\\text{d}_{X/Y} : \\mathcal{O} \\longrightarrow \\Omega_{X/Y}$. We sometimes\nwrite $\\Omega_{X/Y} = \\Omega_f$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BQ","source_file":"sites-modules.tex","source_line":6238,"source_end_line":6256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6238-L6256","statement_sha256":"5f0f73d36765c5df60bdd47aa2107869271313bda74a7a8e6358b89da9230556","origin":"The Stacks Project","memory_eligible":false,"source_rank":4051,"rank":4051,"depth":2,"x":1171.525,"y":352.473,"cluster":"sheaves-sites"},{"id":"stacks:04BR","tag":"04BR","title":"Modules of differentials · Lemma 04BR","summary":"Let X = (Sh(C_X), O_X), Y = (Sh(C_Y), O_Y), X' = (Sh(C_X'), O_X'), and Y' = (Sh(C_Y'), O_Y') be ringed topoi. Let xymatrix X' ar[d] ar[r]_f & X ar[d] Y' ar[r] & Y be a commutative diagram of morphisms of ringed topoi. The map f^sharp : O_X → f_*O_X' composed with the map f_*d_X'/Y' : f_*O_X' → f_*Ω_X'/Y' is a Y-derivation. Hence we obtain a canonical map of O_X-modules Ω_X/Y → f_*Ω_X'/Y', and by adjointness of f_* and f^* a canonical O_X'-module homomorphism c_f :…","statement_latex":"Let\n$X = (\\Sh(\\mathcal{C}_X), \\mathcal{O}_X)$,\n$Y = (\\Sh(\\mathcal{C}_Y), \\mathcal{O}_Y)$,\n$X' = (\\Sh(\\mathcal{C}_{X'}), \\mathcal{O}_{X'})$, and\n$Y' = (\\Sh(\\mathcal{C}_{Y'}), \\mathcal{O}_{Y'})$ be ringed topoi.\nLet\n$$\n\\xymatrix{\nX' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\\nY' \\ar[r] & Y\n}\n$$\nbe a commutative diagram of morphisms of ringed topoi. The map\n$f^\\sharp : \\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with the map\n$f_*\\text{d}_{X'/Y'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/Y'}$ is a\n$Y$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_X$-modules\n$\\Omega_{X/Y} \\to f_*\\Omega_{X'/Y'}$, and by\nadjointness of $f_*$ and $f^*$ a\ncanonical $\\mathcal{O}_{X'}$-module homomorphism\n$$\nc_f : f^*\\Omega_{X/Y} \\longrightarrow \\Omega_{X'/Y'}.\n$$\nIt is uniquely characterized by the property that\n$f^*\\text{d}_{X/Y}(t)$ maps to $\\text{d}_{X'/Y'}(f^* t)$\nfor any local section $t$ of $\\mathcal{O}_X$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Modules of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BR","source_file":"sites-modules.tex","source_line":6262,"source_end_line":6289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6262-L6289","statement_sha256":"433ad5feb90265140f2c0f329a3ef4affe68ebef77c7755856e2a2fac52a04a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4052,"rank":4052,"depth":0,"x":992.751,"y":145.826,"cluster":"sheaves-sites"},{"id":"stacks:09CR","tag":"09CR","title":"Finite order differential operators · Definition 09CR","summary":"Let C be a site. Let φ : O_1 → O_2 be a homomorphism of sheaves of rings. Let k ≥ 0 be an integer. Let F, G be sheaves of O_2-modules. A differential operator D : F → G of order k is an is an O_1-linear map such that for all local sections g of O_2 the map s ↦ D(gs) - gD(s) is a differential operator of order k - 1. For the base case k = 0 we define a differential operator of order 0 to be an O_2-linear map.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\varphi : \\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. Let $k \\geq 0$ be an integer.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be sheaves of $\\mathcal{O}_2$-modules.\nA {\\it differential operator $D : \\mathcal{F} \\to \\mathcal{G}$ of order $k$}\nis an is an $\\mathcal{O}_1$-linear map such that for all local sections\n$g$ of $\\mathcal{O}_2$ the map $s \\mapsto D(gs) - gD(s)$ is a\ndifferential operator of order $k - 1$. For the base case $k = 0$\nwe define a differential operator of order $0$ to be an\n$\\mathcal{O}_2$-linear map.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Finite order differential operators","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CR","source_file":"sites-modules.tex","source_line":6348,"source_end_line":6359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6348-L6359","statement_sha256":"af885b562d783fb031ed8d8f65b94dc1b62bafeb6d57e34d5c0c98ffa2ccf7e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4053,"rank":4053,"depth":0,"x":1290.97,"y":196.8,"cluster":"sheaves-sites"},{"id":"stacks:09CS","tag":"09CS","title":"Finite order differential operators · Lemma 09CS","summary":"Let C be a site. Let O_1 → O_2 be a map of sheaves of rings. Let E, F, G be sheaves of O_2-modules. If D : E → F and D' : F → G are differential operators of order k and k', then D' ∘ D is a differential operator of order k + k'.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings.\nLet $\\mathcal{E}, \\mathcal{F}, \\mathcal{G}$ be sheaves of\n$\\mathcal{O}_2$-modules.\nIf $D : \\mathcal{E} \\to \\mathcal{F}$ and $D' : \\mathcal{F} \\to \\mathcal{G}$\nare differential operators of order $k$ and $k'$, then $D' \\circ D$ is a\ndifferential operator of order $k + k'$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CS","source_file":"sites-modules.tex","source_line":6382,"source_end_line":6391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6382-L6391","statement_sha256":"0448b200e95ccf9dde6761a145be756545758d14e088163490b4757692141c91","origin":"The Stacks Project","memory_eligible":false,"source_rank":4054,"rank":4054,"depth":0,"x":1029.889,"y":328.523,"cluster":"sheaves-sites"},{"id":"stacks:09CT","tag":"09CT","title":"Finite order differential operators · Lemma 09CT","summary":"Let C be a site. Let O_1 → O_2 be a map of sheaves of rings. Let F be a sheaf of O_2-modules. Let k ≥ 0. There exists a sheaf of O_2-modules P^k_O_2/O_1(F) and a canonical isomorphism Diff^k_O_2/O_1(F, G) = Hom_O_2( P^k_O_2/O_1(F), G) functorial in the O_2-module G.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules.\nLet $k \\geq 0$. There exists a sheaf of $\\mathcal{O}_2$-modules\n$\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$\nand a canonical isomorphism\n$$\n\\text{Diff}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}, \\mathcal{G}) =\n\\Hom_{\\mathcal{O}_2}(\n\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}), \\mathcal{G})\n$$\nfunctorial in the $\\mathcal{O}_2$-module $\\mathcal{G}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CT","source_file":"sites-modules.tex","source_line":6403,"source_end_line":6417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6403-L6417","statement_sha256":"c9202363ca9039bf7bdcc9c0e5d1bd40dbf8c010294eb335be4bb5af2d6ff477","origin":"The Stacks Project","memory_eligible":false,"source_rank":4055,"rank":4055,"depth":1,"x":1116.54,"y":83.074,"cluster":"sheaves-sites"},{"id":"stacks:09CU","tag":"09CU","title":"Finite order differential operators · Definition 09CU","summary":"Let C be a site. Let O_1 → O_2 be a map of sheaves of rings. Let F be a sheaf of O_2-modules. The module P^k_O_2/O_1(F) constructed in Lemma [Tag 09CT] is called the module of principal parts of order k of F.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O}_1 \\to \\mathcal{O}_2$ be a map of sheaves of rings.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_2$-modules.\nThe module $\\mathcal{P}^k_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F})$\nconstructed in Lemma \\ref{lemma-module-principal-parts}\nis called the {\\it module of principal parts of order $k$} of $\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Finite order differential operators","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CU","source_file":"sites-modules.tex","source_line":6460,"source_end_line":6468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6460-L6468","statement_sha256":"19335f477f06e5172790e563d2a8cd280f4b99c64b18f99763d0b8fb68434b37","origin":"The Stacks Project","memory_eligible":false,"source_rank":4056,"rank":4056,"depth":2,"x":1250.123,"y":313.396,"cluster":"sheaves-sites"},{"id":"stacks:09CV","tag":"09CV","title":"Finite order differential operators · Lemma 09CV","summary":"Let C be a site. Let O_1 → O_2 be a homomorphism of presheaves of rings. Let F be a presheaf of O_2-modules. Then P^k_O_2^\\#/O_1^\\#(F^\\#) is the sheaf associated to the presheaf U ↦ P^k_O_2(U)/O_1(U)(F(U)).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of presheaves of rings. Let $\\mathcal{F}$ be a presheaf\nof $\\mathcal{O}_2$-modules. Then\n$\\mathcal{P}^k_{\\mathcal{O}_2^\\#/\\mathcal{O}_1^\\#}(\\mathcal{F}^\\#)$\nis the sheaf associated to the presheaf\n$U \\mapsto P^k_{\\mathcal{O}_2(U)/\\mathcal{O}_1(U)}(\\mathcal{F}(U))$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CV","source_file":"sites-modules.tex","source_line":6485,"source_end_line":6493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6485-L6493","statement_sha256":"419efd1219f22a3444f582049f0e456995af4831d70ee15391de6a1a35b85e34","origin":"The Stacks Project","memory_eligible":false,"source_rank":4057,"rank":4057,"depth":2,"x":966.2,"y":219.291,"cluster":"sheaves-sites"},{"id":"stacks:09CW","tag":"09CW","title":"Finite order differential operators · Lemma 09CW","summary":"Let C be a site. Let O_1 → O_2 be a homomorphism of sheaves of rings. Let F be a sheaf of O_2-modules. There is a canonical short exact sequence 0 → Ω_O_2/O_1 ⊗_O_2 F → P^1_O_2/O_1(F) → F → 0 functorial in F called the sequence of principal parts.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}_1 \\to \\mathcal{O}_2$\nbe a homomorphism of sheaves of rings. Let $\\mathcal{F}$ be a sheaf\nof $\\mathcal{O}_2$-modules. There is a\ncanonical short exact sequence\n$$\n0 \\to\n\\Omega_{\\mathcal{O}_2/\\mathcal{O}_1} \\otimes_{\\mathcal{O}_2} \\mathcal{F} \\to\n\\mathcal{P}^1_{\\mathcal{O}_2/\\mathcal{O}_1}(\\mathcal{F}) \\to\n\\mathcal{F} \\to 0\n$$\nfunctorial in $\\mathcal{F}$ called the {\\it sequence of principal parts}.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CW","source_file":"sites-modules.tex","source_line":6503,"source_end_line":6516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6503-L6516","statement_sha256":"3eefc13a0c2ac4f7e6c3f31b32ea5963e4f2453d7203770a3093c6f302320f4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4058,"rank":4058,"depth":3,"x":1251.44,"y":127.513,"cluster":"sheaves-sites"},{"id":"stacks:08TW","tag":"08TW","title":"The naive cotangent complex · Definition 08TW","summary":"Let C be a site. Let A → B be a homomorphism of sheaves of rings on C. The naive cotangent complex NL_B/A is the chain complex ([Tag 08TV]) NL_B/A = (I/I^2 → Ω_A[B]/A ⊗_A[B] B) with I/I^2 placed in degree -1 and Ω_A[B]/A ⊗_A[B] B placed in degree 0.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a\nhomomorphism of sheaves of rings on $\\mathcal{C}$.\nThe {\\it naive cotangent complex} $\\NL_{\\mathcal{B}/\\mathcal{A}}$\nis the chain complex (\\ref{equation-naive-cotangent-complex})\n$$\n\\NL_{\\mathcal{B}/\\mathcal{A}} =\n\\left(\\mathcal{I}/\\mathcal{I}^2\n\\longrightarrow\n\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}}\n\\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}\\right)\n$$\nwith $\\mathcal{I}/\\mathcal{I}^2$ placed in degree $-1$ and\n$\\Omega_{\\mathcal{A}[\\mathcal{B}]/\\mathcal{A}}\n\\otimes_{\\mathcal{A}[\\mathcal{B}]} \\mathcal{B}$\nplaced in degree $0$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"The naive cotangent complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TW","source_file":"sites-modules.tex","source_line":6606,"source_end_line":6623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6606-L6623","statement_sha256":"6bf37662c60672be4c242245c451a87cd3bfa04dba317d02152997359d1c3241","origin":"The Stacks Project","memory_eligible":false,"source_rank":4059,"rank":4059,"depth":0,"x":1114.817,"y":357.204,"cluster":"sheaves-sites"},{"id":"stacks:08TY","tag":"08TY","title":"The naive cotangent complex · Lemma 08TY","summary":"In the situation above there is a canonical isomorphism NL(α) = NL_B/A in D(B).","statement_latex":"In the situation above there is a canonical isomorphism\n$\\NL(\\alpha) = \\NL_{\\mathcal{B}/\\mathcal{A}}$ in $D(\\mathcal{B})$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TY","source_file":"sites-modules.tex","source_line":6669,"source_end_line":6673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6669-L6673","statement_sha256":"33eeb8aff661ef20d7c66c09eb510884502851356201b911f5f8463150fd7f2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4060,"rank":4060,"depth":3,"x":1030.79,"y":110.135,"cluster":"sheaves-sites"},{"id":"stacks:08TZ","tag":"08TZ","title":"The naive cotangent complex · Lemma 08TZ","summary":"Let f : Sh(C) → Sh(D) be morphism of topoi. Let A → B be a homomorphism of sheaves of rings on D. Then f^-1NL_B/A = NL_f^-1B/f^-1A.","statement_latex":"Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be morphism of topoi.\nLet $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings\non $\\mathcal{D}$. Then $f^{-1}\\NL_{\\mathcal{B}/\\mathcal{A}} =\n\\NL_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08TZ","source_file":"sites-modules.tex","source_line":6753,"source_end_line":6759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6753-L6759","statement_sha256":"523daae3c7aa8da61afadcea5b3b4d0f0f21194ef71fac40572d19f26d0c60a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4061,"rank":4061,"depth":2,"x":1291.621,"y":244.736,"cluster":"sheaves-sites"},{"id":"stacks:08U0","tag":"08U0","title":"The naive cotangent complex · Definition 08U0","summary":"Let X = (Sh(C), O) and Y = (Sh(C'), O') be ringed topoi. Let (f, f^sharp) : X → Y be a morphism of ringed topoi. The naive cotangent complex NL_f = NL_X/Y of the given morphism of ringed topoi is NL_O/f^-1O'. We sometimes write NL_X/Y = NL_O/O'.","statement_latex":"Let $X = (\\Sh(\\mathcal{C}), \\mathcal{O})$ and\n$Y = (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ be ringed topoi.\nLet $(f, f^\\sharp) : X \\to Y$ be a morphism of ringed topoi.\nThe {\\it naive cotangent complex} $\\NL_f = \\NL_{X/Y}$\nof the given morphism of ringed topoi is\n$\\NL_{\\mathcal{O}/f^{-1}\\mathcal{O}'}$.\nWe sometimes write $\\NL_{X/Y} = \\NL_{\\mathcal{O}/\\mathcal{O}'}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"The naive cotangent complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08U0","source_file":"sites-modules.tex","source_line":6769,"source_end_line":6778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6769-L6778","statement_sha256":"8b603c0df78cbb8976832489f4f119dd82f0194d7e40533c3767faa3eb929bcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4062,"rank":4062,"depth":0,"x":990.833,"y":293.52,"cluster":"sheaves-sites"},{"id":"stacks:04EN","tag":"04EN","title":"Stalks of modules · Lemma 04EN","summary":"Let C be a site. Let p be a point of C. • We have (F^\\#)_p = F_p for any presheaf of sets on C. • The stalk functor Sh(C) → Sets, F ↦ F_p is exact (see Categories, Definition [Tag 0034]) and commutes with arbitrary colimits. • The stalk functor PSh(C) → Sets, F ↦ F_p is exact (see Categories, Definition [Tag 0034]) and commutes with arbitrary colimits.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p$ be a point of $\\mathcal{C}$.\n\\begin{enumerate}\n\\item We have $(\\mathcal{F}^\\#)_p = \\mathcal{F}_p$\nfor any presheaf of sets on $\\mathcal{C}$.\n\\item The stalk functor\n$\\Sh(\\mathcal{C}) \\to \\textit{Sets}$,\n$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact (see\nCategories, Definition \\ref{categories-definition-exact})\nand commutes with arbitrary colimits.\n\\item The stalk functor\n$\\textit{PSh}(\\mathcal{C}) \\to \\textit{Sets}$,\n$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact (see\nCategories, Definition \\ref{categories-definition-exact})\nand commutes with arbitrary colimits.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Stalks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EN","source_file":"sites-modules.tex","source_line":6801,"source_end_line":6819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6801-L6819","statement_sha256":"bd31f733f7f801e334d6ccf09e7d74df39549324ab6e0a4ae6fb7be0dcde8d30","origin":"The Stacks Project","memory_eligible":false,"source_rank":4063,"rank":4063,"depth":4,"x":1173.528,"y":86.727,"cluster":"sheaves-sites"},{"id":"stacks:04EP","tag":"04EP","title":"Stalks of modules · Lemma 04EP","summary":"Let C be a site. Let p be a point of C. • The functor Ab(C) → Ab, F ↦ F_p is exact. • The stalk functor PAb(C) → Ab, F ↦ F_p is exact. • For F ∈ Ob(PAb(C)) we have F_p = F^\\#_p.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $p$ be a point of $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The functor\n$\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$,\n$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact.\n\\item The stalk functor\n$\\textit{PAb}(\\mathcal{C}) \\to \\textit{Ab}$,\n$\\mathcal{F}  \\mapsto  \\mathcal{F}_p$\nis exact.\n\\item For $\\mathcal{F} \\in \\Ob(\\textit{PAb}(\\mathcal{C}))$\nwe have $\\mathcal{F}_p = \\mathcal{F}^\\#_p$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Stalks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EP","source_file":"sites-modules.tex","source_line":6884,"source_end_line":6899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6884-L6899","statement_sha256":"26eba35de6faec1027d2dec4dbdd34a2e6c89e15b7beb452f48b762dfcece55e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4064,"rank":4064,"depth":5,"x":1205.13,"y":343.059,"cluster":"sheaves-sites"},{"id":"stacks:04EQ","tag":"04EQ","title":"Stalks of modules · Lemma 04EQ","summary":"Let (C, O) be a ringed site. Let p be a point of C. • The functor Mod(O) → Mod(O_p), F ↦ F_p is exact. • The stalk functor PMod(O) → Mod(O_p), F ↦ F_p is exact. • For F ∈ Ob(PMod(O)) we have F_p = F^\\#_p.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $p$ be a point of $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The functor\n$\\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_p)$,\n$\\mathcal{F} \\mapsto \\mathcal{F}_p$ is exact.\n\\item The stalk functor\n$\\textit{PMod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_p)$,\n$\\mathcal{F} \\mapsto \\mathcal{F}_p$\nis exact.\n\\item For $\\mathcal{F} \\in \\Ob(\\textit{PMod}(\\mathcal{O}))$\nwe have $\\mathcal{F}_p = \\mathcal{F}^\\#_p$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Stalks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EQ","source_file":"sites-modules.tex","source_line":6944,"source_end_line":6959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6944-L6959","statement_sha256":"26231b4edb8971daa38e90cc3ebb001a4c02fb9675453a641e676a4b6f77ee3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4065,"rank":4065,"depth":6,"x":975.531,"y":171.855,"cluster":"sheaves-sites"},{"id":"stacks:05V5","tag":"05V5","title":"Stalks of modules · Lemma 05V5","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi or ringed sites. Let p be a point of C or Sh(C) and set q = f ∘ p. Then (f^*F)_p = F_q ⊗_O_D, q O_C, p for any O_D-module F.","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi or ringed sites.\nLet $p$ be a point of $\\mathcal{C}$ or $\\Sh(\\mathcal{C})$\nand set $q = f \\circ p$. Then\n$$\n(f^*\\mathcal{F})_p =\n\\mathcal{F}_q \\otimes_{\\mathcal{O}_{\\mathcal{D}, q}}\n\\mathcal{O}_{\\mathcal{C}, p}\n$$\nfor any $\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{F}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Stalks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05V5","source_file":"sites-modules.tex","source_line":6968,"source_end_line":6984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L6968-L6984","statement_sha256":"da034ff7dd481274bdbf501081811414ef66b9c675b727c7bfcf9ca6eb45a2b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4066,"rank":4066,"depth":4,"x":1282.727,"y":167.815,"cluster":"sheaves-sites"},{"id":"stacks:05V7","tag":"05V7","title":"Skyscraper sheaves · Lemma 05V7","summary":"Let C be a site. Let p be a point of C or of its associated topos. • The functor p_* : Ab → Ab(C), A ↦ p_*A is exact. • There is a functorial direct sum decomposition p^-1p_*A = A ⊕ I(A) for A ∈ Ob(Ab).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p$ be a point of\n$\\mathcal{C}$ or of its associated topos.\n\\begin{enumerate}\n\\item The functor $p_* : \\textit{Ab} \\to \\textit{Ab}(\\mathcal{C})$,\n$A \\mapsto p_*A$ is exact.\n\\item There is a functorial direct sum decomposition\n$$\np^{-1}p_*A = A \\oplus I(A)\n$$\nfor $A \\in \\Ob(\\textit{Ab})$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Skyscraper sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05V7","source_file":"sites-modules.tex","source_line":7020,"source_end_line":7033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7020-L7033","statement_sha256":"f205f5e5ae49d76aeafba4444261d6370a7f672d6f3ceb7df5cc005f94127db5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4067,"rank":4067,"depth":6,"x":1059.297,"y":345.229,"cluster":"sheaves-sites"},{"id":"stacks:05V8","tag":"05V8","title":"Skyscraper sheaves · Lemma 05V8","summary":"Let (Sh(C), O) be a ringed topos. Let p be a point of the topos Sh(C). • The functor p_* : Mod(O_p) → Mod(O), M ↦ p_*M is exact. • The canonical surjection p^-1p_*M → M is O_p-linear. • The functorial direct sum decomposition p^-1p_*M = M ⊕ I(M) of Lemma [Tag 05V7] is not O_p-linear in general.","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos.\nLet $p$ be a point of the topos $\\Sh(\\mathcal{C})$.\n\\begin{enumerate}\n\\item The functor\n$p_* : \\textit{Mod}(\\mathcal{O}_p) \\to \\textit{Mod}(\\mathcal{O})$,\n$M \\mapsto p_*M$ is exact.\n\\item The canonical surjection $p^{-1}p_*M \\to M$ is $\\mathcal{O}_p$-linear.\n\\item The functorial direct sum decomposition\n$p^{-1}p_*M = M \\oplus I(M)$ of Lemma \\ref{lemma-skyscraper-exact}\nis {\\bf not} $\\mathcal{O}_p$-linear in general.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Skyscraper sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05V8","source_file":"sites-modules.tex","source_line":7066,"source_end_line":7079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7066-L7079","statement_sha256":"ab2429a9b131b070f2bf294211b7f53a833684e4a0745be6a3e197a8137fbf78","origin":"The Stacks Project","memory_eligible":false,"source_rank":4068,"rank":4068,"depth":7,"x":1081.395,"y":87.447,"cluster":"sheaves-sites"},{"id":"stacks:0710","tag":"0710","title":"Localization and points · Lemma 0710","summary":"Let (C, O) be a ringed site. Let p be a point of C. Let U be an object of C. For G in Mod(O_U) we have (j_U!G)_p = bigoplus_q G_q where the coproduct is over the points q of C/U lying over p, see Sites, Lemma [Tag 04H3].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $p$ be a point of $\\mathcal{C}$. Let $U$ be an object of $\\mathcal{C}$.\nFor $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_U)$ we have\n$$\n(j_{U!}\\mathcal{G})_p =\n\\bigoplus\\nolimits_q \\mathcal{G}_q\n$$\nwhere the coproduct is over the points $q$ of $\\mathcal{C}/U$\nlying over $p$, see\nSites, Lemma \\ref{sites-lemma-points-above-point}.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localization and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0710","source_file":"sites-modules.tex","source_line":7141,"source_end_line":7153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7141-L7153","statement_sha256":"0c01df13901610f0e73a61112eba31df8c1ad3673863fee5c1e0a44a7c4b8e07","origin":"The Stacks Project","memory_eligible":false,"source_rank":4069,"rank":4069,"depth":11,"x":1272.536,"y":290.212,"cluster":"sheaves-sites"},{"id":"stacks:05VD","tag":"05VD","title":"Pullbacks of flat modules · Lemma 05VD","summary":"[SGA4] Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi or ringed sites. Then f^*F is a flat O_C-module whenever F is a flat O_D-module.","statement_latex":"\\begin{reference}\n\\cite[Expos\\'e V, Corollary 1.7.1]{SGA4}\n\\end{reference}\nLet\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi or ringed sites.\nThen $f^*\\mathcal{F}$ is a flat $\\mathcal{O}_\\mathcal{C}$-module\nwhenever $\\mathcal{F}$ is a flat $\\mathcal{O}_\\mathcal{D}$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Pullbacks of flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VD","source_file":"sites-modules.tex","source_line":7211,"source_end_line":7224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7211-L7224","statement_sha256":"260de91c20ef153070a6d2bd3b541d6fc4787fbdd088298e4efdf8903b73d789","origin":"The Stacks Project","memory_eligible":false,"source_rank":4070,"rank":4070,"depth":10,"x":968.321,"y":249.125,"cluster":"sheaves-sites"},{"id":"stacks:05VB","tag":"05VB","title":"Pullbacks of flat modules · Lemma 05VB","summary":"Let (C, O) be a ringed site. Let p be a point of C. If F is a flat O-module, then F_p is a flat O_p-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $p$ be a point of $\\mathcal{C}$.\nIf $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, then\n$\\mathcal{F}_p$ is a flat $\\mathcal{O}_p$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Pullbacks of flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VB","source_file":"sites-modules.tex","source_line":7326,"source_end_line":7332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7326-L7332","statement_sha256":"786f3b2cc76bd2f2761a940625c18d115d8596a0ade16e648806631ab3b0ff62","origin":"The Stacks Project","memory_eligible":false,"source_rank":4071,"rank":4071,"depth":11,"x":1225.865,"y":106.704,"cluster":"sheaves-sites"},{"id":"stacks:05VC","tag":"05VC","title":"Pullbacks of flat modules · Lemma 05VC","summary":"Let (C, O) be a ringed site. Let F be a sheaf of O-modules. Let (p_i)_i ∈ I be a conservative family of points of C. Then F is flat if and only if F_p_i is a flat O_p_i-module for all i ∈ I.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nLet $\\{p_i\\}_{i \\in I}$ be a conservative family of points of $\\mathcal{C}$.\nThen $\\mathcal{F}$ is flat if and only if $\\mathcal{F}_{p_i}$ is\na flat $\\mathcal{O}_{p_i}$-module for all $i \\in I$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Pullbacks of flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VC","source_file":"sites-modules.tex","source_line":7349,"source_end_line":7356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7349-L7356","statement_sha256":"913b4cfc5199feac58546d977d2f2c81782a7757580e4a9a5c915348326c7356","origin":"The Stacks Project","memory_eligible":false,"source_rank":4072,"rank":4072,"depth":12,"x":1150.434,"y":358.035,"cluster":"sheaves-sites"},{"id":"stacks:0G6R","tag":"0G6R","title":"Pullbacks of flat modules · Lemma 0G6R","summary":"Let f : (Sh(C'), O') → (Sh(C), O) be a morphism of ringed topoi. Let 0 → F → G → H → 0 be a short exact sequence of O-modules with H a flat O-module. Then the sequence 0 → f^*F → f^*G → f^*H → 0 is exact as well.","statement_latex":"Let\n$f : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nbe a morphism of ringed topoi. Let\n$0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$\nbe a short exact sequence of $\\mathcal{O}$-modules with $\\mathcal{H}$\na flat $\\mathcal{O}$-module. Then the sequence\n$0 \\to f^*\\mathcal{F} \\to f^*\\mathcal{G} \\to f^*\\mathcal{H} \\to 0$\nis exact as well.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Pullbacks of flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6R","source_file":"sites-modules.tex","source_line":7371,"source_end_line":7381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7371-L7381","statement_sha256":"3a436f3968f6382513c05083ed01a276fd4605423ddd22ec61a8e264814f1629","origin":"The Stacks Project","memory_eligible":false,"source_rank":4073,"rank":4073,"depth":11,"x":1003.841,"y":129.748,"cluster":"sheaves-sites"},{"id":"stacks:04ES","tag":"04ES","title":"Locally ringed topoi · Lemma 04ES","summary":"Let (C, O) be a ringed site. The following are equivalent • For every object U of C and f ∈ O(U) there exists a covering (U_j → U) such that for each j either f|_U_j is invertible or (1 - f)|_U_j is invertible. • For U ∈ Ob(C), n ≥ 1, and f_1, …, f_n ∈ O(U) which generate the unit ideal in O(U) there exists a covering (U_j → U) such that for each j there exists an i such that f_i|_U_j is invertible. • The map of sheaves of sets (O × O) amalg (O × O) → O × O which maps (f,…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The following\nare equivalent\n\\begin{enumerate}\n\\item For every object $U$ of $\\mathcal{C}$ and $f \\in \\mathcal{O}(U)$\nthere exists a covering $\\{U_j \\to U\\}$ such that for each $j$\neither $f|_{U_j}$ is invertible or $(1 - f)|_{U_j}$ is invertible.\n\\item For $U \\in \\Ob(\\mathcal{C})$, $n \\geq 1$, and\n$f_1, \\ldots, f_n \\in \\mathcal{O}(U)$ which generate the unit ideal\nin $\\mathcal{O}(U)$ there exists a covering $\\{U_j \\to U\\}$\nsuch that for each $j$ there exists an $i$ such that $f_i|_{U_j}$\nis invertible.\n\\item The map of sheaves of sets\n$$\n(\\mathcal{O} \\times \\mathcal{O})\n\\amalg\n(\\mathcal{O} \\times \\mathcal{O})\n\\longrightarrow\n\\mathcal{O} \\times \\mathcal{O}\n$$\nwhich maps $(f, a)$ in the first component to $(f, af)$ and\n$(f, b)$ in the second component to $(f, b(1 - f))$ is surjective.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ES","source_file":"sites-modules.tex","source_line":7408,"source_end_line":7432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7408-L7432","statement_sha256":"d2f4f48bbc0aee5b961d353e9656f1431135dddaec347c2b83368858d863d2ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":4074,"rank":4074,"depth":0,"x":1295.722,"y":214.961,"cluster":"sheaves-sites"},{"id":"stacks:04ET","tag":"04ET","title":"Locally ringed topoi · Lemma 04ET","summary":"Let (C, O) be a ringed site. Consider the following conditions • For every object U of C and f ∈ O(U) there exists a covering (U_j → U) such that for each j either f|_U_j is invertible or (1 - f)|_U_j is invertible. • For every point p of C the stalk O_p is either the zero ring or a local ring. We always have (1) ⇒ (2). If C has enough points then (1) and (2) are equivalent.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nConsider the following conditions\n\\begin{enumerate}\n\\item For every object $U$ of $\\mathcal{C}$ and $f \\in \\mathcal{O}(U)$\nthere exists a covering $\\{U_j \\to U\\}$ such that for each $j$\neither $f|_{U_j}$ is invertible or $(1 - f)|_{U_j}$ is invertible.\n\\item For every point $p$ of $\\mathcal{C}$ the stalk $\\mathcal{O}_p$\nis either the zero ring or a local ring.\n\\end{enumerate}\nWe always have (1) $\\Rightarrow$ (2). If $\\mathcal{C}$ has enough points\nthen (1) and (2) are equivalent.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ET","source_file":"sites-modules.tex","source_line":7471,"source_end_line":7484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7471-L7484","statement_sha256":"0b2443a8414eb6e63e262575f77d5a3d9297b3c065fe927a8e05f1186dccf316","origin":"The Stacks Project","memory_eligible":false,"source_rank":4075,"rank":4075,"depth":2,"x":1011.77,"y":317.817,"cluster":"sheaves-sites"},{"id":"stacks:05D8","tag":"05D8","title":"Locally ringed topoi · Lemma 05D8","summary":"Let (C, O) be a ringed site. Consider the statements • ([Tag 05D7]) is an isomorphism, and • for every point p of C the stalk O_p is not the zero ring. We always have (1) ⇒ (2) and if C has enough points then (1) ⇔ (2).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Consider the statements\n\\begin{enumerate}\n\\item (\\ref{equation-one-is-never-zero}) is an isomorphism, and\n\\item for every point $p$ of $\\mathcal{C}$ the stalk $\\mathcal{O}_p$\nis not the zero ring.\n\\end{enumerate}\nWe always have (1) $\\Rightarrow$ (2) and if $\\mathcal{C}$ has enough points\nthen (1) $\\Leftrightarrow$ (2).","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05D8","source_file":"sites-modules.tex","source_line":7548,"source_end_line":7558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7548-L7558","statement_sha256":"5db51eacd13c97302fbbc2007f0ed361f9a2986a74f55a18b88c07226ed42416","origin":"The Stacks Project","memory_eligible":false,"source_rank":4076,"rank":4076,"depth":0,"x":1138.525,"y":80.688,"cluster":"sheaves-sites"},{"id":"stacks:04EU","tag":"04EU","title":"Locally ringed topoi · Definition 04EU","summary":"A ringed site (C, O) is said to be locally ringed site if ([Tag 05D7]) is an isomorphism, and the equivalent properties of Lemma [Tag 04ES] are satisfied.","statement_latex":"A ringed site $(\\mathcal{C}, \\mathcal{O})$ is said to be\n{\\it locally ringed site} if (\\ref{equation-one-is-never-zero})\nis an isomorphism, and the equivalent properties of\nLemma \\ref{lemma-locally-ringed}\nare satisfied.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EU","source_file":"sites-modules.tex","source_line":7570,"source_end_line":7577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7570-L7577","statement_sha256":"ac0f382a22d127b851c565578913e019ac6cf49cc1aa6d9969c025b830e1c18f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4077,"rank":4077,"depth":1,"x":1235.818,"y":327.638,"cluster":"sheaves-sites"},{"id":"stacks:04H7","tag":"04H7","title":"Locally ringed topoi · Lemma 04H7","summary":"Being a locally ringed site is an intrinsic property. More precisely, • if f : Sh(C') → Sh(C) is a morphism of topoi and (C, O) is a locally ringed site, then (C', f^-1O) is a locally ringed site, and • if (f, f^sharp) : (Sh(C'), O') → (Sh(C), O) is an equivalence of ringed topoi, then (C, O) is locally ringed if and only if (C', O') is locally ringed.","statement_latex":"Being a locally ringed site is an intrinsic property.\nMore precisely,\n\\begin{enumerate}\n\\item if $f : \\Sh(\\mathcal{C}') \\to \\Sh(\\mathcal{C})$\nis a morphism of topoi and $(\\mathcal{C}, \\mathcal{O})$ is\na locally ringed site, then $(\\mathcal{C}', f^{-1}\\mathcal{O})$\nis a locally ringed site, and\n\\item if\n$(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')\n\\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nis an equivalence of ringed topoi, then\n$(\\mathcal{C}, \\mathcal{O})$ is locally ringed if and only if\n$(\\mathcal{C}', \\mathcal{O}')$\nis locally ringed.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04H7","source_file":"sites-modules.tex","source_line":7586,"source_end_line":7603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7586-L7603","statement_sha256":"2829415ff04f4f90840373ba38187f919a85bf66f6df185a41e04371961275ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":4078,"rank":4078,"depth":1,"x":965.298,"y":200.659,"cluster":"sheaves-sites"},{"id":"stacks:04H8","tag":"04H8","title":"Locally ringed topoi · Definition 04H8","summary":"A ringed topos (Sh(C), O) is said to be locally ringed if the underlying ringed site (C, O) is locally ringed.","statement_latex":"A ringed topos $(\\Sh(\\mathcal{C}), \\mathcal{O})$ is said to be\n{\\it locally ringed} if the underlying ringed site\n$(\\mathcal{C}, \\mathcal{O})$ is locally ringed.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04H8","source_file":"sites-modules.tex","source_line":7624,"source_end_line":7629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7624-L7629","statement_sha256":"9959b1c7d30d83bad85a82e945b43407d658eda10e433ee295c648f3e2421889","origin":"The Stacks Project","memory_eligible":false,"source_rank":4079,"rank":4079,"depth":0,"x":1267.096,"y":140.751,"cluster":"sheaves-sites"},{"id":"stacks:0B8Q","tag":"0B8Q","title":"Locally ringed topoi · Lemma 0B8Q","summary":"Let (Sh(C), O) be a ringed topos. Any locally free O-module of rank 1 is invertible. If (C, O) is locally ringed, then the converse holds as well (but in general this is not the case).","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a ringed topos. Any locally free\n$\\mathcal{O}$-module of rank $1$ is invertible.\nIf $(\\mathcal{C}, \\mathcal{O})$ is locally ringed, then\nthe converse holds as well (but in general this is not the case).","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8Q","source_file":"sites-modules.tex","source_line":7634,"source_end_line":7640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7634-L7640","statement_sha256":"19ae4b4d4794a7214111d830674314eaf37a8eaca88436136d679fc960e00cff","origin":"The Stacks Project","memory_eligible":false,"source_rank":4080,"rank":4080,"depth":5,"x":1092.611,"y":356.322,"cluster":"sheaves-sites"},{"id":"stacks:04H9","tag":"04H9","title":"Locally ringed topoi · Lemma 04H9","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Consider the following conditions • The diagram of sheaves xymatrix f^-1(O^*_D) ar[r]_-f^sharp ar[d] & O^*_C ar[d] f^-1(O_D) ar[r]^-f^sharp & O_C is cartesian. • For any point p of C, setting q = f ∘ p, the diagram xymatrix O^*_D, q ar[r] ar[d] & O^*_C, p ar[d] O_D, q ar[r] & O_C, p of sets is cartesian. We always have (1) ⇒ (2). If C has enough points then (1) and (2) are equivalent. If (Sh(C),…","statement_latex":"Let\n$(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi.\nConsider the following conditions\n\\begin{enumerate}\n\\item The diagram of sheaves\n$$\n\\xymatrix{\nf^{-1}(\\mathcal{O}^*_\\mathcal{D}) \\ar[r]_-{f^\\sharp} \\ar[d] &\n\\mathcal{O}^*_\\mathcal{C} \\ar[d] \\\\\nf^{-1}(\\mathcal{O}_\\mathcal{D}) \\ar[r]^-{f^\\sharp} &\n\\mathcal{O}_\\mathcal{C}\n}\n$$\nis cartesian.\n\\item For any point $p$ of $\\mathcal{C}$, setting $q = f \\circ p$,\nthe diagram\n$$\n\\xymatrix{\n\\mathcal{O}^*_{\\mathcal{D}, q} \\ar[r] \\ar[d] &\n\\mathcal{O}^*_{\\mathcal{C}, p} \\ar[d] \\\\\n\\mathcal{O}_{\\mathcal{D}, q} \\ar[r] &\n\\mathcal{O}_{\\mathcal{C}, p}\n}\n$$\nof sets is cartesian.\n\\end{enumerate}\nWe always have (1) $\\Rightarrow$ (2). If $\\mathcal{C}$ has enough points\nthen (1) and (2) are equivalent. If\n$(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$\nand\n$(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nare locally ringed topoi then (2) is equivalent to\n\\begin{enumerate}\n\\item[(3)] For any point $p$ of $\\mathcal{C}$, setting $q = f \\circ p$,\nthe ring map $\\mathcal{O}_{\\mathcal{D}, q} \\to \\mathcal{O}_{\\mathcal{C}, p}$\nis a local ring map.\n\\end{enumerate}\nIn fact, properties (2), or (3) for a conservative\nfamily of points implies (1).","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04H9","source_file":"sites-modules.tex","source_line":7687,"source_end_line":7730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7687-L7730","statement_sha256":"fff440c4b0f206742ca651170db921a0e114de8ad7542ca392cc176e3a860867","origin":"The Stacks Project","memory_eligible":false,"source_rank":4081,"rank":4081,"depth":0,"x":1047.887,"y":98.179,"cluster":"sheaves-sites"},{"id":"stacks:04HA","tag":"04HA","title":"Locally ringed topoi · Definition 04HA","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Assume (Sh(C), O_C) and (Sh(D), O_D) are locally ringed topoi. We say that (f, f^sharp) is a morphism of locally ringed topoi if and only if the diagram of sheaves xymatrix f^-1(O^*_D) ar[r]_-f^sharp ar[d] & O^*_C ar[d] f^-1(O_D) ar[r]^-f^sharp & O_C (see Lemma [Tag 04H9]) is cartesian. If (f, f^sharp) is a morphism of ringed sites, then we say that it is a morphism of locally ringed sites if…","statement_latex":"Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi. Assume\n$(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$\nand\n$(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nare locally ringed topoi. We say that $(f, f^\\sharp)$ is a\n{\\it morphism of locally ringed topoi} if and only if the\ndiagram of sheaves\n$$\n\\xymatrix{\nf^{-1}(\\mathcal{O}^*_\\mathcal{D}) \\ar[r]_-{f^\\sharp} \\ar[d] &\n\\mathcal{O}^*_\\mathcal{C} \\ar[d] \\\\\nf^{-1}(\\mathcal{O}_\\mathcal{D}) \\ar[r]^-{f^\\sharp} &\n\\mathcal{O}_\\mathcal{C}\n}\n$$\n(see\nLemma \\ref{lemma-locally-ringed-morphism})\nis cartesian. If $(f, f^\\sharp)$ is a morphism of ringed sites, then\nwe say that it is a {\\it morphism of locally ringed sites} if\nthe associated morphism of ringed topoi is a morphism of locally ringed\ntopoi.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HA","source_file":"sites-modules.tex","source_line":7737,"source_end_line":7762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7737-L7762","statement_sha256":"784264e4cd35c1563b66493ef08e367472fcdefb7d01b01d6cd75d494ea7f68e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4082,"rank":4082,"depth":1,"x":1288.623,"y":263.266,"cluster":"sheaves-sites"},{"id":"stacks:04IG","tag":"04IG","title":"Locally ringed topoi · Lemma 04IG","summary":"Let (f, f^sharp) : (Sh(C_1), O_1) → (Sh(C_2), O_2) and (g, g^sharp) : (Sh(C_2), O_2) → (Sh(C_3), O_3) be morphisms of locally ringed topoi. Then the composition (g, g^sharp) ∘ (f, f^sharp) (see Definition [Tag 01D3]) is also a morphism of locally ringed topoi.","statement_latex":"Let\n$(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}_1), \\mathcal{O}_1)\n\\to (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and\n$(g, g^\\sharp) :\n(\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to\n(\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$\nbe morphisms of locally ringed topoi. Then the composition\n$(g, g^\\sharp) \\circ (f, f^\\sharp)$ (see\nDefinition \\ref{definition-ringed-topos})\nis also a morphism of locally ringed topoi.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IG","source_file":"sites-modules.tex","source_line":7768,"source_end_line":7781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7768-L7781","statement_sha256":"559f4d47506125b917f5413882b24bfe80286a83f34dc147e4b5f86daa4988ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":4083,"rank":4083,"depth":2,"x":978.137,"y":278.143,"cluster":"sheaves-sites"},{"id":"stacks:04KR","tag":"04KR","title":"Locally ringed topoi · Lemma 04KR","summary":"Let f : Sh(C') → Sh(C) be a morphism of topoi. If O is a sheaf of rings on C, then f^-1(O^*) = (f^-1O)^*. In particular, if O turns C into a locally ringed site, then setting f^sharp = id the morphism of ringed topoi (f, f^sharp) : (Sh(C'), f^-1O) → (Sh(C, O) is a morphism of locally ringed topoi.","statement_latex":"Let $f : \\Sh(\\mathcal{C}') \\to \\Sh(\\mathcal{C})$\nbe a morphism of topoi. If $\\mathcal{O}$ is a sheaf of rings\non $\\mathcal{C}$, then\n$$\nf^{-1}(\\mathcal{O}^*) = (f^{-1}\\mathcal{O})^*.\n$$\nIn particular, if $\\mathcal{O}$ turns $\\mathcal{C}$ into a locally\nringed site, then setting $f^\\sharp = \\text{id}$\nthe morphism of ringed topoi\n$$\n(f, f^\\sharp) :\n(\\Sh(\\mathcal{C}'), f^{-1}\\mathcal{O})\n\\to\n(\\Sh(\\mathcal{C}, \\mathcal{O})\n$$\nis a morphism of locally ringed topoi.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KR","source_file":"sites-modules.tex","source_line":7787,"source_end_line":7805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7787-L7805","statement_sha256":"947ba3bf9bb28be6032abe963e06dfd79cf44e15faf8659c59f77df460d945b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4084,"rank":4084,"depth":2,"x":1195.269,"y":90.867,"cluster":"sheaves-sites"},{"id":"stacks:04IH","tag":"04IH","title":"Locally ringed topoi · Lemma 04IH","summary":"Localization of locally ringed sites and topoi. • Let (C, O) be a locally ringed site. Let U be an object of C. Then the localization (C/U, O_U) is a locally ringed site, and the localization morphism (j_U, j_U^sharp) : (Sh(C/U), O_U) → (Sh(C), O) is a morphism of locally ringed topoi. • Let (C, O) be a locally ringed site. Let f : V → U be a morphism of C. Then the morphism (j, j^sharp) : (Sh(C/V), O_V) → (Sh(C/U), O_U) of Lemma [Tag 04IY] is a morphism of locally ringed…","statement_latex":"Localization of locally ringed sites and topoi.\n\\begin{enumerate}\n\\item Let $(\\mathcal{C}, \\mathcal{O})$ be a locally ringed site.\nLet $U$ be an object of $\\mathcal{C}$. Then the localization\n$(\\mathcal{C}/U, \\mathcal{O}_U)$ is a locally ringed site, and\nthe localization morphism\n$$\n(j_U, j_U^\\sharp) :\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n\\to\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n$$\nis a morphism of locally ringed topoi.\n\\item Let $(\\mathcal{C}, \\mathcal{O})$ be a locally ringed site.\nLet $f : V \\to U$ be a morphism of $\\mathcal{C}$.\nThen the morphism\n$$\n(j, j^\\sharp) :\n(\\Sh(\\mathcal{C}/V), \\mathcal{O}_V)\n\\to\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n$$\nof\nLemma \\ref{lemma-relocalize}\nis a morphism of locally ringed topoi.\n\\item Let\n$(f, f^\\sharp) :\n(\\mathcal{C}, \\mathcal{O})\n\\longrightarrow\n(\\mathcal{D}, \\mathcal{O}')$\nbe a morphism of locally ringed sites where $f$ is given by the continuous\nfunctor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V$ be an object of\n$\\mathcal{D}$ and let $U = u(V)$. Then the morphism\n$$\n(f', (f')^\\sharp) :\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n\\to\n(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)\n$$\nof\nLemma \\ref{lemma-localize-morphism-ringed-sites}\nis a morphism of locally ringed sites.\n\\item Let\n$(f, f^\\sharp) :\n(\\mathcal{C}, \\mathcal{O})\n\\longrightarrow\n(\\mathcal{D}, \\mathcal{O}')$\nbe a morphism of locally ringed sites where $f$ is given by the continuous\nfunctor $u : \\mathcal{D} \\to \\mathcal{C}$. Let $V \\in \\Ob(\\mathcal{D})$,\n$U \\in \\Ob(\\mathcal{C})$, and $c : U \\to u(V)$. Then the morphism\n$$\n(f_c, (f_c)^\\sharp) :\n(\\Sh(\\mathcal{C}/U), \\mathcal{O}_U)\n\\to\n(\\Sh(\\mathcal{D}/V), \\mathcal{O}'_V)\n$$\nof\nLemma \\ref{lemma-relocalize-morphism-ringed-sites}\nis a morphism of locally ringed topoi.\n\\item Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a locally\nringed topos. Let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$.\nThen the localization\n$(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})$\nis a locally ringed topos and the localization morphism\n$$\n(j_\\mathcal{F}, j_\\mathcal{F}^\\sharp) :\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\to\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n$$\nis a morphism of locally ringed topoi.\n\\item  Let $(\\Sh(\\mathcal{C}), \\mathcal{O})$ be a locally\nringed topos. Let $s : \\mathcal{G} \\to \\mathcal{F}$ be a map of sheaves\non $\\mathcal{C}$. Then the morphism\n$$\n(j, j^\\sharp) :\n(\\Sh(\\mathcal{C})/\\mathcal{G}, \\mathcal{O}_\\mathcal{G})\n\\longrightarrow\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n$$\nof\nLemma \\ref{lemma-relocalize-ringed-topos}\nis a morphism of locally ringed topoi.\n\\item Let\n$f :\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\longrightarrow\n(\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a morphism of locally ringed topoi. Let $\\mathcal{G}$ be a sheaf\non $\\mathcal{D}$. Set $\\mathcal{F} = f^{-1}\\mathcal{G}$.\nThen the morphism\n$$\n(f', (f')^\\sharp) :\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\longrightarrow\n(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G})\n$$\nof\nLemma \\ref{lemma-localize-morphism-ringed-topoi}\nis a morphism of locally ringed topoi.\n\\item  Let\n$f :\n(\\Sh(\\mathcal{C}), \\mathcal{O})\n\\longrightarrow\n(\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a morphism of locally ringed topoi. Let $\\mathcal{G}$ be a sheaf\non $\\mathcal{D}$, let $\\mathcal{F}$ be a sheaf on $\\mathcal{C}$, and\nlet $s : \\mathcal{F} \\to f^{-1}\\mathcal{G}$ be a morphism of sheaves.\nThen the morphism\n$$\n(f_s, (f_s)^\\sharp) :\n(\\Sh(\\mathcal{C})/\\mathcal{F}, \\mathcal{O}_\\mathcal{F})\n\\longrightarrow\n(\\Sh(\\mathcal{D})/\\mathcal{G}, \\mathcal{O}'_\\mathcal{G})\n$$\nof\nLemma \\ref{lemma-relocalize-morphism-ringed-topoi}\nis a morphism of locally ringed topoi.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IH","source_file":"sites-modules.tex","source_line":7835,"source_end_line":7956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L7835-L7956","statement_sha256":"bf580cffddf850583835f203deb8cce864f69456023b7b1a1e20bfaa78235d78","origin":"The Stacks Project","memory_eligible":false,"source_rank":4085,"rank":4085,"depth":15,"x":1185.754,"y":352.346,"cluster":"sheaves-sites"},{"id":"stacks:0797","tag":"0797","title":"Lower shriek for modules · Lemma 0797","summary":"Let u : C → D be a continuous and cocontinuous functor between sites. Denote g : Sh(C) → Sh(D) the associated morphism of topoi. Let O_D be a sheaf of rings on D. Set O_C = g^-1O_D. Hence g becomes a morphism of ringed topoi with g^* = g^-1. In this case there exists a functor g_! : Mod(O_C) → Mod(O_D) which is left adjoint to g^*.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous\nfunctor between sites. Denote\n$g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ the associated\nmorphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$ be a sheaf of rings\non $\\mathcal{D}$. Set\n$\\mathcal{O}_\\mathcal{C} = g^{-1}\\mathcal{O}_\\mathcal{D}$.\nHence $g$ becomes a morphism of ringed topoi with $g^* = g^{-1}$.\nIn this case there exists a functor\n$$\ng_! :\n\\textit{Mod}(\\mathcal{O}_\\mathcal{C})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_\\mathcal{D})\n$$\nwhich is left adjoint to $g^*$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Lower shriek for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0797","source_file":"sites-modules.tex","source_line":8000,"source_end_line":8017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8000-L8017","statement_sha256":"5e9afc9e42ddbb5e0df5a2261b2d7f3881554f031f502b9c0ef612048cde59de","origin":"The Stacks Project","memory_eligible":false,"source_rank":4086,"rank":4086,"depth":6,"x":982.358,"y":154.002,"cluster":"sheaves-sites"},{"id":"stacks:0FN3","tag":"0FN3","title":"Lower shriek for modules · Lemma 0FN3","summary":"Assume given a commutative diagram xymatrix (Sh(C'), O_C') ar[r]_(g', (g')^sharp) ar[d]_(f', (f')^sharp) & (Sh(C), O_C) ar[d]^(f, f^sharp) (Sh(D'), O_D') ar[r]^(g, g^sharp) & (Sh(D), O_D) of ringed topoi. Assume • f, f', g, and g' correspond to cocontinuous functors u, u', v, and v' as in Sites, Lemma [Tag 00XO], • v ∘ u' = u ∘ v', • v and v' are continuous as well as cocontinuous, • for any object V' of D' the functor ^u'_V'I → ^ u_v(V')I given by v is cofinal, and •…","statement_latex":"Assume given a commutative diagram\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})\n\\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} &\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\\n(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} &\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})\n}\n$$\nof ringed topoi. Assume\n\\begin{enumerate}\n\\item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors\n$u$, $u'$, $v$, and $v'$ as in\nSites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi},\n\\item $v \\circ u' = u \\circ v'$,\n\\item $v$ and $v'$ are continuous as well as cocontinuous,\n\\item for any object $V'$ of $\\mathcal{D}'$ the functor\n${}^{u'}_{V'}\\mathcal{I} \\to {}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$\ngiven by $v$ is cofinal, and\n\\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$\nand $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$.\n\\end{enumerate}\nThen we have $f'_* \\circ (g')^* = g^* \\circ f_*$ and\n$g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$ on modules.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Lower shriek for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FN3","source_file":"sites-modules.tex","source_line":8093,"source_end_line":8120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8093-L8120","statement_sha256":"eeffced0cce4e5661b948358dcf7d2405fb4cddc836a9d8fa5b1bc5b350e41b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4087,"rank":4087,"depth":7,"x":1292.053,"y":184.867,"cluster":"sheaves-sites"},{"id":"stacks:0FN4","tag":"0FN4","title":"Lower shriek for modules · Lemma 0FN4","summary":"Consider a commutative diagram xymatrix (Sh(C'), O_C') ar[r]_(g', (g')^sharp) ar[d]_(f', (f')^sharp) & (Sh(C), O_C) ar[d]^(f, f^sharp) (Sh(D'), O_D') ar[r]^(g, g^sharp) & (Sh(D), O_D) of ringed topoi and suppose we have functors xymatrix C' ar[r]_v' & C D' ar[r]^v ar[u]^u' & D ar[u]_u such that (with notation as in Sites, Sections [Tag 00X0] and [Tag 00XN]) we have • u and u' are continuous and give rise to the morphisms f and f', • v and v' are cocontinuous giving rise…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})\n\\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} &\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\\n(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} &\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})\n}\n$$\nof ringed topoi and suppose we have functors\n$$\n\\xymatrix{\n\\mathcal{C}' \\ar[r]_{v'} &\n\\mathcal{C} \\\\\n\\mathcal{D}' \\ar[r]^v \\ar[u]^{u'} &\n\\mathcal{D} \\ar[u]_u\n}\n$$\nsuch that (with notation as in\nSites, Sections \\ref{sites-section-morphism-sites} and\n\\ref{sites-section-cocontinuous-morphism-topoi}) we have\n\\begin{enumerate}\n\\item $u$ and $u'$ are continuous and give rise to the morphisms\n$f$ and $f'$,\n\\item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$,\n\\item $u \\circ v = v' \\circ u'$,\n\\item $v$ and $v'$ are continuous as well as cocontinuous, and\n\\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$\nand $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$.\n\\end{enumerate}\nThen $f'_* \\circ (g')^* = g^* \\circ f_*$ and\n$g'_! \\circ (f')^{-1} = f^{-1} \\circ g_!$ on modules.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Lower shriek for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FN4","source_file":"sites-modules.tex","source_line":8133,"source_end_line":8168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8133-L8168","statement_sha256":"2bcbbe3911fb35d076a6787d05818fc8e09a6f8ecff097867177051c284370a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4088,"rank":4088,"depth":7,"x":1038.695,"y":337.94,"cluster":"sheaves-sites"},{"id":"stacks:093J","tag":"093J","title":"Constant sheaves · Lemma 093J","summary":"Let C be a site. If 0 → A → B → C → 0 is a short exact sequence of abelian groups, then 0 → underlineA → underlineB → underlineC → 0 is an exact sequence of abelian sheaves and in fact it is even exact as a sequence of abelian presheaves.","statement_latex":"Let $\\mathcal{C}$ be a site. If $0 \\to A \\to B \\to C \\to 0$\nis a short exact sequence of abelian groups, then\n$0 \\to \\underline{A} \\to \\underline{B} \\to \\underline{C} \\to 0$\nis an exact sequence of abelian sheaves and in fact it is even\nexact as a sequence of abelian presheaves.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093J","source_file":"sites-modules.tex","source_line":8197,"source_end_line":8204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8197-L8204","statement_sha256":"dcf1e0b053ff2376bbc80e14f6d839470d2d0d90dc2aea0accc19a8bee224286","origin":"The Stacks Project","memory_eligible":false,"source_rank":4089,"rank":4089,"depth":0,"x":1102.464,"y":81.13,"cluster":"sheaves-sites"},{"id":"stacks:093K","tag":"093K","title":"Constant sheaves · Lemma 093K","summary":"Let C be a site. Let Lambda be a ring and let M and Q be Lambda-modules. If Q is a finitely presented Lambda-module, then we have underlineM ⊗_Lambda Q(U) = underlineM(U) ⊗_Lambda Q for all U ∈ Ob(C).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring and let $M$\nand $Q$ be $\\Lambda$-modules. If $Q$ is a finitely presented\n$\\Lambda$-module, then we have\n$\\underline{M \\otimes_\\Lambda Q}(U) = \\underline{M}(U) \\otimes_\\Lambda Q$\nfor all $U \\in \\Ob(\\mathcal{C})$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093K","source_file":"sites-modules.tex","source_line":8219,"source_end_line":8226,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8219-L8226","statement_sha256":"1ff234a3b83c725daf6a693ca9a1ffb1c3ace0746c48472417e9538b3167ee57","origin":"The Stacks Project","memory_eligible":false,"source_rank":4090,"rank":4090,"depth":1,"x":1262.069,"y":306.835,"cluster":"sheaves-sites"},{"id":"stacks:093L","tag":"093L","title":"Constant sheaves · Lemma 093L","summary":"Let C be a site. Let Lambda be a coherent ring. Let M be a flat Lambda-module. For U ∈ Ob(C) the module underlineM(U) is a flat Lambda-module.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a coherent ring.\nLet $M$ be a flat $\\Lambda$-module. For $U \\in \\Ob(\\mathcal{C})$ the\nmodule $\\underline{M}(U)$ is a flat $\\Lambda$-module.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093L","source_file":"sites-modules.tex","source_line":8242,"source_end_line":8247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8242-L8247","statement_sha256":"0cb1fd2085f909fb2a3106cefe8adbe6fe91a50390a24f0b3d0582981d2574e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4091,"rank":4091,"depth":3,"x":962.671,"y":230.916,"cluster":"sheaves-sites"},{"id":"stacks:093M","tag":"093M","title":"Constant sheaves · Lemma 093M","summary":"Let C be a site. Let Lambda be a Noetherian ring. Let I ⊂ Lambda be an ideal. The sheaf underlineLambda^wedge = lim underlineLambda/I^n is a flat underlineLambda-algebra. Moreover we have canonical identifications underlineLambda/IunderlineLambda = underlineLambda/underlineI = underlineLambda^wedge/IunderlineLambda^wedge = underlineLambda^wedge/underlineI · underlineLambda^wedge = underlineLambda^wedge/underlineI^wedge = underlineLambda/I where underlineI^wedge = lim…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring.\nLet $I \\subset \\Lambda$ be an ideal. The sheaf\n$\\underline{\\Lambda}^\\wedge = \\lim \\underline{\\Lambda/I^n}$\nis a flat $\\underline{\\Lambda}$-algebra.\nMoreover we have canonical identifications\n$$\n\\underline{\\Lambda}/I\\underline{\\Lambda} =\n\\underline{\\Lambda}/\\underline{I} =\n\\underline{\\Lambda}^\\wedge/I\\underline{\\Lambda}^\\wedge =\n\\underline{\\Lambda}^\\wedge/\\underline{I} \\cdot \\underline{\\Lambda}^\\wedge =\n\\underline{\\Lambda}^\\wedge/\\underline{I}^\\wedge =\n\\underline{\\Lambda/I}\n$$\nwhere $\\underline{I}^\\wedge = \\lim \\underline{I/I^n}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093M","source_file":"sites-modules.tex","source_line":8260,"source_end_line":8276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8260-L8276","statement_sha256":"9bc80d85ad40c934e3dd65c1ec2b33127102d86a3d0baf2b247c4f64394e55d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4092,"rank":4092,"depth":6,"x":1244.685,"y":116.934,"cluster":"sheaves-sites"},{"id":"stacks:093N","tag":"093N","title":"Constant sheaves · Lemma 093N","summary":"Let C be a site. Let Lambda be a ring and let M be a Lambda-module. Assume Sh(C) is not the empty topos. Then • underlineM is a finite type sheaf of underlineLambda-modules if and only if M is a finite Lambda-module, and • underlineM is a finitely presented sheaf of underlineLambda-modules if and only if M is a finitely presented Lambda-module.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring and let $M$ be a\n$\\Lambda$-module. Assume $\\Sh(\\mathcal{C})$ is not the empty topos. Then\n\\begin{enumerate}\n\\item $\\underline{M}$ is a finite type sheaf of $\\underline{\\Lambda}$-modules\nif and only if $M$ is a finite $\\Lambda$-module, and\n\\item $\\underline{M}$ is a finitely presented sheaf of\n$\\underline{\\Lambda}$-modules if and only if $M$ is a \nfinitely presented $\\Lambda$-module.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093N","source_file":"sites-modules.tex","source_line":8346,"source_end_line":8357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8346-L8357","statement_sha256":"31993f558cc95f554a4ae611f1fdb681d2523284bb1a6af80320715203a616d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4093,"rank":4093,"depth":2,"x":1128.315,"y":361.169,"cluster":"sheaves-sites"},{"id":"stacks:093Q","tag":"093Q","title":"Locally constant sheaves · Definition 093Q","summary":"Let C be a site. Let F be a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring Lambda, etc. • We say F is a constant sheaf of sets, groups, abelian groups, rings, modules over a fixed ring Lambda, etc if it is isomorphic as a sheaf of sets, groups, abelian groups, rings, modules over a fixed ring Lambda, etc to a constant sheaf underlineE as in Section [Tag 093I]. • We say F is locally constant if for every object U of C there exists a covering (U_i →…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be a sheaf of sets, groups,\nabelian groups, rings, modules over a fixed ring $\\Lambda$, etc.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ is a\n{\\it constant sheaf} of\nsets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc\nif it is isomorphic as a sheaf of\nsets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc\nto a constant sheaf $\\underline{E}$ as in Section \\ref{section-constant}.\n\\item We say $\\mathcal{F}$ is {\\it locally constant} if for every object\n$U$ of $\\mathcal{C}$ there exists a\ncovering $\\{U_i \\to U\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf.\n\\item If $\\mathcal{F}$ is a sheaf of sets or groups, then we say $\\mathcal{F}$\nis {\\it finite locally constant} if the constant values are finite sets or\nfinite groups.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally constant sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093Q","source_file":"sites-modules.tex","source_line":8414,"source_end_line":8432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8414-L8432","statement_sha256":"f177185becf4572514444969c75f53ab6421b4b4977122915b7fbfa585fa006d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4094,"rank":4094,"depth":0,"x":1017.643,"y":114.877,"cluster":"sheaves-sites"},{"id":"stacks:093R","tag":"093R","title":"Locally constant sheaves · Lemma 093R","summary":"Let f : Sh(C) → Sh(D) be a morphism of topoi. If G is a locally constant sheaf of sets, groups, abelian groups, rings, modules over a fixed ring Lambda, etc on D, the same is true for f^-1G on C.","statement_latex":"Let $f : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be a morphism of topoi.\nIf $\\mathcal{G}$ is a locally constant sheaf of\nsets, groups, abelian groups, rings, modules over a fixed ring $\\Lambda$, etc\non $\\mathcal{D}$, the same is true for $f^{-1}\\mathcal{G}$\non $\\mathcal{C}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093R","source_file":"sites-modules.tex","source_line":8434,"source_end_line":8441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8434-L8441","statement_sha256":"fd12aea6923400c3fb63250a5f5d1b78e9450772dc37654b1c67aecae959ca68","origin":"The Stacks Project","memory_eligible":false,"source_rank":4095,"rank":4095,"depth":0,"x":1297.5,"y":233.771,"cluster":"sheaves-sites"},{"id":"stacks:093S","tag":"093S","title":"Locally constant sheaves · Lemma 093S","summary":"Let C be a site with a final object X. • Let φ : F → G be a map of locally constant sheaves of sets on C. If F is finite locally constant, there exists a covering (U_i → X) such that φ|_U_i is the map of constant sheaves associated to a map of sets. • Let φ : F → G be a map of locally constant sheaves of abelian groups on C. If F is finite locally constant, there exists a covering (U_i → X) such that φ|_U_i is the map of constant abelian sheaves associated to a map of…","statement_latex":"Let $\\mathcal{C}$ be a site with a final object $X$.\n\\begin{enumerate}\n\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map\nof locally constant sheaves of sets on $\\mathcal{C}$.\nIf $\\mathcal{F}$ is finite locally constant, there exists a\ncovering $\\{U_i \\to X\\}$ such that\n$\\varphi|_{U_i}$ is the map of constant sheaves associated to\na map of sets.\n\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map\nof locally constant sheaves of abelian groups on $\\mathcal{C}$.\nIf $\\mathcal{F}$ is finite locally constant, there exists a\ncovering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of\nconstant abelian sheaves associated to a map of abelian groups.\n\\item Let $\\Lambda$ be a ring.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map\nof locally constant sheaves of $\\Lambda$-modules on $\\mathcal{C}$.\nIf $\\mathcal{F}$ is of finite type, then there exists a covering\n$\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant\nsheaves of $\\Lambda$-modules associated to a map of $\\Lambda$-modules.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093S","source_file":"sites-modules.tex","source_line":8447,"source_end_line":8469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8447-L8469","statement_sha256":"51e292afa306c7af16feae6ff7e8721c1b54904ed9a592bf4951b69322f64f93","origin":"The Stacks Project","memory_eligible":false,"source_rank":4096,"rank":4096,"depth":0,"x":995.323,"y":304.946,"cluster":"sheaves-sites"},{"id":"stacks:093T","tag":"093T","title":"Locally constant sheaves · Lemma 093T","summary":"Let C be a site. Let Lambda be a ring. Let M, N be Lambda-modules. Let F, G be a locally constant sheaves of Lambda-modules. • If M is of finite presentation, then underlineHom_Lambda(M, N) = SheafHom_underlineLambda(underlineM, underlineN) • If M and N are both of finite presentation, then underlineIsom_Lambda(M, N) = mathitIsom_underlineLambda(underlineM, underlineN) • If F is of finite presentation, then SheafHom_underlineLambda(F, G) is a locally constant sheaf of…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring.\nLet $M$, $N$ be $\\Lambda$-modules.\nLet $\\mathcal{F}, \\mathcal{G}$ be a locally constant sheaves of\n$\\Lambda$-modules.\n\\begin{enumerate}\n\\item If $M$ is of finite presentation, then\n$$\n\\underline{\\Hom_\\Lambda(M, N)} =\n\\SheafHom_{\\underline{\\Lambda}}(\\underline{M}, \\underline{N})\n$$\n\\item If $M$ and $N$ are both of finite presentation, then\n$$\n\\underline{\\text{Isom}_\\Lambda(M, N)} =\n\\mathit{Isom}_{\\underline{\\Lambda}}(\\underline{M}, \\underline{N})\n$$\n\\item If $\\mathcal{F}$ is of finite presentation, then\n$\\SheafHom_{\\underline{\\Lambda}}(\\mathcal{F}, \\mathcal{G})$\nis a locally constant sheaf of $\\Lambda$-modules.\n\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are both of finite presentation, then\n$\\mathit{Isom}_{\\underline{\\Lambda}}(\\mathcal{F}, \\mathcal{G})$\nis a locally constant sheaf of sets.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093T","source_file":"sites-modules.tex","source_line":8475,"source_end_line":8499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8475-L8499","statement_sha256":"22984a5da70e8dbfa5c5d2192f5744c5235b1779b8e78b1176d0fe52d41cb442","origin":"The Stacks Project","memory_eligible":false,"source_rank":4097,"rank":4097,"depth":3,"x":1161.018,"y":80.85,"cluster":"sheaves-sites"},{"id":"stacks:093U","tag":"093U","title":"Locally constant sheaves · Lemma 093U","summary":"Let C be a site. • The category of finite locally constant sheaves of sets is closed under finite limits and colimits inside Sh(C). • The category of finite locally constant abelian sheaves is a weak Serre subcategory of Ab(C). • Let Lambda be a Noetherian ring. The category of finite type, locally constant sheaves of Lambda-modules on C is a weak Serre subcategory of Mod(C, Lambda).","statement_latex":"Let $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item The category of finite locally constant sheaves of sets\nis closed under finite limits and colimits inside $\\Sh(\\mathcal{C})$.\n\\item The category of finite locally constant abelian sheaves is a\nweak Serre subcategory of $\\textit{Ab}(\\mathcal{C})$.\n\\item Let $\\Lambda$ be a Noetherian ring. The category of\nfinite type, locally constant sheaves of $\\Lambda$-modules on\n$\\mathcal{C}$ is a weak Serre subcategory of\n$\\textit{Mod}(\\mathcal{C}, \\Lambda)$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093U","source_file":"sites-modules.tex","source_line":8541,"source_end_line":8554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8541-L8554","statement_sha256":"c2dea6730f29e61f0be1e9b3768d37d12e20259b860f87c2b35b25326823e2fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4098,"rank":4098,"depth":1,"x":1219.088,"y":340.289,"cluster":"sheaves-sites"},{"id":"stacks:093V","tag":"093V","title":"Locally constant sheaves · Lemma 093V","summary":"Let C be a site. Let Lambda be a ring. The tensor product of two locally constant sheaves of Lambda-modules on C is a locally constant sheaf of Lambda-modules.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a ring.\nThe tensor product of two locally constant sheaves of $\\Lambda$-modules\non $\\mathcal{C}$ is a locally constant sheaf of $\\Lambda$-modules.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093V","source_file":"sites-modules.tex","source_line":8609,"source_end_line":8614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8609-L8614","statement_sha256":"ec117c1900c5d8032ab6923e58e4eef23c957d717a6cd5a48343ff76d3d0e657","origin":"The Stacks Project","memory_eligible":false,"source_rank":4099,"rank":4099,"depth":0,"x":967.466,"y":181.826,"cluster":"sheaves-sites"},{"id":"stacks:0EMC","tag":"0EMC","title":"Localizing sheaves of rings · Lemma 0EMC","summary":"In the situation above the map to the sheafification O → (S^-1O)^\\# is a homomorphism of sheaves of rings with the following universal property: for any homomorphism of sheaves of rings O → A such that each local section of S maps to an invertible section of A there exists a unique factorization (S^-1O)^\\# → A.","statement_latex":"In the situation above the map to the sheafification\n$$\n\\mathcal{O} \\longrightarrow (\\mathcal{S}^{-1}\\mathcal{O})^\\#\n$$\nis a homomorphism of sheaves of rings with the following\nuniversal property: for any homomorphism of sheaves of rings\n$\\mathcal{O} \\to \\mathcal{A}$ such that each local section\nof $\\mathcal{S}$ maps to an invertible section of $\\mathcal{A}$\nthere exists a unique factorization\n$(\\mathcal{S}^{-1}\\mathcal{O})^\\# \\to \\mathcal{A}$.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localizing sheaves of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMC","source_file":"sites-modules.tex","source_line":8648,"source_end_line":8660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8648-L8660","statement_sha256":"7a468e90a20f9b52afac5a74f05aae2302ca78d77e0914dc1760891e8c76003e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4100,"rank":4100,"depth":0,"x":1280.649,"y":155.88,"cluster":"sheaves-sites"},{"id":"stacks:0EMD","tag":"0EMD","title":"Localizing sheaves of rings · Lemma 0EMD","summary":"In the situation above the map to the sheafification F → (S^-1F)^\\# has the following universal property: for any homomorphism of O-modules F → G such that each local section of S acts invertibly on G there exists a unique factorization (S^-1F)^\\# → G. Moreover we have (S^-1F)^\\# = (S^-1O)^\\# ⊗_O F as sheaves of (S^-1O)^\\#-modules.","statement_latex":"In the situation above the map to the sheafification\n$$\n\\mathcal{F} \\longrightarrow (\\mathcal{S}^{-1}\\mathcal{F})^\\#\n$$\nhas the following universal property: for any homomorphism\nof $\\mathcal{O}$-modules $\\mathcal{F} \\to \\mathcal{G}$ such\nthat each local section of $\\mathcal{S}$ acts invertibly on $\\mathcal{G}$\nthere exists a unique factorization\n$(\\mathcal{S}^{-1}\\mathcal{F})^\\# \\to \\mathcal{G}$.\nMoreover we have\n$$\n(\\mathcal{S}^{-1}\\mathcal{F})^\\#\n=\n(\\mathcal{S}^{-1}\\mathcal{O})^\\# \\otimes_\\mathcal{O} \\mathcal{F}\n$$\nas sheaves of $(\\mathcal{S}^{-1}\\mathcal{O})^\\#$-modules.","area":"Sheaves & Sites","chapter":"Modules on Sites","chapter_id":"sites-modules","section":"Localizing sheaves of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMD","source_file":"sites-modules.tex","source_line":8683,"source_end_line":8701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-modules.tex#L8683-L8701","statement_sha256":"fef67b3645f55a3f70ef345686805249ef67e3f52e108b21ffaf2245445634f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4101,"rank":4101,"depth":0,"x":1070.437,"y":352.851,"cluster":"sheaves-sites"},{"id":"stacks:05NR","tag":"05NR","title":"Baer's argument for modules · Lemma 05NR","summary":"Suppose that, in ([Tag 05NN]), C is the category of sets and A is a finite set, then the map is a bijection.","statement_latex":"Suppose that, in (\\ref{equation-compare}), $\\mathcal{C}$ is the category\nof sets and $A$ is a {\\it finite set}, then the map is a bijection.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Baer's argument for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NR","source_file":"injectives.tex","source_line":108,"source_end_line":112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L108-L112","statement_sha256":"65911f4194789c767c1bf9d2169609ba0a05438caa4fe3b213313d1d8a8e62d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4102,"rank":4102,"depth":0,"x":1067.043,"y":88.153,"cluster":"sheaves-sites"},{"id":"stacks:05NS","tag":"05NS","title":"Baer's argument for modules · Definition 05NS","summary":"Let C be a category, let I ⊂ Arrows(C), and let α be an ordinal. An object A of C is said to be α-small with respect to I if whenever (B_β) is a system over α with transition maps in I, then the map ([Tag 05NN]) is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a category, let $I \\subset \\text{Arrows}(\\mathcal{C})$,\nand let $\\alpha$ be an ordinal. An object $A$ of $\\mathcal{C}$ is said to\nbe {\\it $\\alpha$-small with respect to $I$} if whenever $\\{B_\\beta\\}$ is\na system over $\\alpha$ with transition maps in $I$, then\nthe map (\\ref{equation-compare}) is an isomorphism.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Baer's argument for modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NS","source_file":"injectives.tex","source_line":143,"source_end_line":150,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L143-L150","statement_sha256":"cda4ac54c1d49e4357743cdfff8d0d40b88951bcbf1b35642d185c26c9423874","origin":"The Stacks Project","memory_eligible":false,"source_rank":4103,"rank":4103,"depth":0,"x":1282.553,"y":281.54,"cluster":"sheaves-sites"},{"id":"stacks:05NT","tag":"05NT","title":"Baer's argument for modules · Proposition 05NT","summary":"Let R be a ring. Let M be an R-module. Let kappa the cardinality of the set of submodules of M. If α is an ordinal whose cofinality is bigger than kappa, then M is α-small with respect to injections.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $\\kappa$ the cardinality of the set of submodules of $M$.\nIf $\\alpha$ is an ordinal whose cofinality is bigger than $\\kappa$,\nthen $M$ is $\\alpha$-small with respect to injections.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Baer's argument for modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NT","source_file":"injectives.tex","source_line":169,"source_end_line":175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L169-L175","statement_sha256":"322df63d8028eb0295bf57230f7f19f9d5f21735c7c71313afd1c7e03698f9a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4104,"rank":4104,"depth":1,"x":967.913,"y":261.211,"cluster":"sheaves-sites"},{"id":"stacks:05NU","tag":"05NU","title":"Baer's criterion · Lemma 05NU","summary":"[Baer] Let R be a ring. An R-module Q is injective if and only if in every commutative diagram xymatrix a ar[d] ar[r] & Q R ar@-->[ru] for a ⊂ R an ideal, the dotted arrow exists.","statement_latex":"\\begin{reference}\n\\cite[Theorem 1]{Baer}\n\\end{reference}\nLet $R$ be a ring. An $R$-module $Q$ is injective if and only if in every\ncommutative diagram\n$$\n\\xymatrix{\n\\mathfrak{a} \\ar[d] \\ar[r] &  Q \\\\\nR \\ar@{-->}[ru]\n}\n$$\nfor $\\mathfrak{a} \\subset R$ an ideal, the dotted arrow exists.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Baer's argument for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NU","source_file":"injectives.tex","source_line":219,"source_end_line":233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L219-L233","statement_sha256":"cb755f4227a783607c5d3791ebb0ac54b5637dc245b8cb32be7677edaf438166","origin":"The Stacks Project","memory_eligible":false,"source_rank":4105,"rank":4105,"depth":2,"x":1216.437,"y":97.559,"cluster":"sheaves-sites"},{"id":"stacks:05NW","tag":"05NW","title":"Baer's argument for modules · Lemma 05NW","summary":"Let R be a ring. • The construction M ↦ (M → M(M)) is functorial in M. • The map M → M(M) is injective. • For any ideal a and any R-module map φ : a → M there is an R-module map φ' : R → M(M) such that xymatrix a ar[d] ar[r]_φ & M ar[d] R ar[r]^φ' & M(M) commutes.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item The construction $M \\mapsto (M \\to \\mathbf{M}(M))$\nis functorial in $M$.\n\\item The map $M \\to \\mathbf{M}(M)$ is injective.\n\\item For any ideal $\\mathfrak{a}$ and any $R$-module map\n$\\varphi : \\mathfrak a \\to M$ there is an $R$-module map\n$\\varphi' : R \\to \\mathbf{M}(M)$ such that\n$$\n\\xymatrix{\n\\mathfrak{a} \\ar[d] \\ar[r]_\\varphi &  M \\ar[d] \\\\\nR \\ar[r]^{\\varphi'} & \\mathbf{M}(M)\n}\n$$\ncommutes.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Baer's argument for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NW","source_file":"injectives.tex","source_line":282,"source_end_line":300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L282-L300","statement_sha256":"f5f7fc0772bf1a55d8343f3451fe5deb9b684accff3029e71c37724ac4e13aa6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4106,"rank":4106,"depth":0,"x":1164.75,"y":359.425,"cluster":"sheaves-sites"},{"id":"stacks:05NX","tag":"05NX","title":"Baer's argument for modules · Theorem 05NX","summary":"Let kappa be the cardinality of the set of ideals in R, and let α be an ordinal whose cofinality is greater than kappa. Then M_α(N) is an injective R-module, and N → M_α(N) is a functorial injective embedding.","statement_latex":"Let $\\kappa$ be the cardinality of the set of ideals in $R$, and\nlet $\\alpha$ be an ordinal whose cofinality is greater than\n$\\kappa$. Then $\\mathbf{M}_\\alpha(N)$ is an injective $R$-module,\nand $N \\to \\mathbf{M}_\\alpha(N)$ is a functorial injective embedding.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Baer's argument for modules","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05NX","source_file":"injectives.tex","source_line":349,"source_end_line":355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L349-L355","statement_sha256":"c5af5dd91a41c386786bcf03d2f378cd8552549b0f6c121776058a3a727b4d7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4107,"rank":4107,"depth":3,"x":992.163,"y":136.853,"cluster":"sheaves-sites"},{"id":"stacks:04JF","tag":"04JF","title":"G-modules · Lemma 04JF","summary":"Let G be a topological group. Let R be a ring. The category Mod_R, G of R-G-modules, see Étale Cohomology, Definition [Tag 04JP], has functorial injective embeddings. In particular this holds for the category of discrete G-modules.","statement_latex":"Let $G$ be a topological group. Let $R$ be a ring.\nThe category $\\text{Mod}_{R, G}$ of $R\\text{-}G$-modules, see\n\\'Etale Cohomology, Definition\n\\ref{etale-cohomology-definition-G-module-continuous},\nhas functorial injective embeddings. In particular this holds\nfor the category of discrete $G$-modules.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"G-modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JF","source_file":"injectives.tex","source_line":408,"source_end_line":416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L408-L416","statement_sha256":"04ac9820244d59af957f63be15291418c86c058f0a4ff59864ced9585e91fdf9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4108,"rank":4108,"depth":10,"x":1298.615,"y":203.088,"cluster":"sheaves-sites"},{"id":"stacks:01DG","tag":"01DG","title":"Abelian sheaves on a space · Lemma 01DG","summary":"Let X be a topological space. The category of abelian sheaves on X has enough injectives. In fact it has functorial injective embeddings.","statement_latex":"Let $X$ be a topological space.\nThe category of abelian sheaves on $X$ has enough injectives.\nIn fact it has functorial injective embeddings.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Abelian sheaves on a space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DG","source_file":"injectives.tex","source_line":438,"source_end_line":443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L438-L443","statement_sha256":"3e912e55aacc7eafda104273a9a1ee8e47175bdfb87e1642421a708653c9c443","origin":"The Stacks Project","memory_eligible":false,"source_rank":4109,"rank":4109,"depth":1,"x":1019.193,"y":328.218,"cluster":"sheaves-sites"},{"id":"stacks:01DI","tag":"01DI","title":"Sheaves of modules on a ringed space · Lemma 01DI","summary":"Let (X, O_X) be a ringed space, see Sheaves, Section [Tag 0090]. The category of sheaves of O_X-modules on X has enough injectives. In fact it has functorial injective embeddings.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space, see\nSheaves, Section \\ref{sheaves-section-ringed-spaces}.\nThe category of sheaves of $\\mathcal{O}_X$-modules on $X$\nhas enough injectives. In fact it has functorial injective embeddings.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Sheaves of modules on a ringed space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DI","source_file":"injectives.tex","source_line":497,"source_end_line":503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L497-L503","statement_sha256":"f8e74392f13479ba617f7f14be6d61da92b3ca7e4ab4509d90aeae07299222d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4110,"rank":4110,"depth":8,"x":1124.678,"y":77.233,"cluster":"sheaves-sites"},{"id":"stacks:01DK","tag":"01DK","title":"Abelian presheaves on a category · Proposition 01DK","summary":"For abelian presheaves on a category there is a functorial injective embedding.","statement_latex":"For abelian presheaves on a category there is a functorial injective\nembedding.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Abelian presheaves on a category","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DK","source_file":"injectives.tex","source_line":646,"source_end_line":650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L646-L650","statement_sha256":"c5e124c18605de1a19257d36287906ffc294c2d56a14226b7b228b69cb87c591","origin":"The Stacks Project","memory_eligible":false,"source_rank":4111,"rank":4111,"depth":0,"x":1248.811,"y":322.322,"cluster":"sheaves-sites"},{"id":"stacks:01DM","tag":"01DM","title":"Abelian Sheaves on a site · Lemma 01DM","summary":"With notation as above. Suppose that G_1 → G_2 is an injective map of abelian sheaves on C. Let α be an ordinal and let G_1 → J_α(F) be a morphism of sheaves. There exists a morphism G_2 → J_α + 1(F) such that the following diagram commutes xymatrix G_1 ar[d] ar[r] & G_2 ar[d] J_α(F) ar[r] & J_α + 1(F)","statement_latex":"With notation as above.\nSuppose that $\\mathcal{G}_1 \\to \\mathcal{G}_2$ is an injective\nmap of abelian sheaves on $\\mathcal{C}$. Let $\\alpha$ be an ordinal\nand let $\\mathcal{G}_1 \\to J_\\alpha(\\mathcal{F})$ be a morphism\nof sheaves. There exists a morphism $\\mathcal{G}_2 \\to\nJ_{\\alpha + 1}(\\mathcal{F})$ such that the following diagram commutes\n$$\n\\xymatrix{\n\\mathcal{G}_1 \\ar[d] \\ar[r] & \\mathcal{G}_2 \\ar[d] \\\\\nJ_{\\alpha}(\\mathcal{F}) \\ar[r] & J_{\\alpha + 1}(\\mathcal{F}) }\n$$","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Abelian Sheaves on a site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DM","source_file":"injectives.tex","source_line":709,"source_end_line":722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L709-L722","statement_sha256":"cdbcbb99950ae6f9cbb863980f3ab04d5f9bc05d34a79e2105cca8726bea1beb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4112,"rank":4112,"depth":0,"x":959.994,"y":211.961,"cluster":"sheaves-sites"},{"id":"stacks:01DN","tag":"01DN","title":"Abelian Sheaves on a site · Lemma 01DN","summary":"Suppose that G_i, i∈ I is set of abelian sheaves on C. There exists an ordinal β such that for any sheaf F, any i∈ I, and any map φ : G_i → J_β(F) there exists an α < β such that φ factors through J_α(F).","statement_latex":"Suppose that $\\mathcal{G}_i$, $i\\in I$ is set of abelian sheaves\non $\\mathcal{C}$. There exists an ordinal $\\beta$ such that\nfor any sheaf $\\mathcal{F}$, any $i\\in I$, and any map\n$\\varphi : \\mathcal{G}_i \\to J_\\beta(\\mathcal{F})$ there exists an\n$\\alpha < \\beta$ such that $ \\varphi $ factors through\n$J_\\alpha(\\mathcal{F})$.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Abelian Sheaves on a site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DN","source_file":"injectives.tex","source_line":739,"source_end_line":747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L739-L747","statement_sha256":"67533217bc39be65724d84ee6c48bfb814996110ec08335c029e947582a3c323","origin":"The Stacks Project","memory_eligible":false,"source_rank":4113,"rank":4113,"depth":4,"x":1261.911,"y":129.403,"cluster":"sheaves-sites"},{"id":"stacks:01DO","tag":"01DO","title":"Abelian Sheaves on a site · Lemma 01DO","summary":"Suppose J is a sheaf of abelian groups with the following property: For all X∈ Ob(C), for any abelian subsheaf S ⊂ Z_X^\\# and any morphism φ : S → J, there exists a morphism Z_X^\\# → J extending φ. Then J is an injective sheaf of abelian groups.","statement_latex":"Suppose $\\mathcal{J}$ is a sheaf of abelian groups with the following\nproperty: For all $X\\in \\Ob(\\mathcal{C})$, for any abelian subsheaf\n$\\mathcal{S} \\subset \\mathbf{Z}_X^\\#$ and any morphism\n$\\varphi : \\mathcal{S} \\to \\mathcal{J}$, there exists a morphism\n$\\mathbf{Z}_X^\\# \\to \\mathcal{J}$ extending $\\varphi$.\nThen $\\mathcal{J}$ is an injective sheaf of abelian groups.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Abelian Sheaves on a site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DO","source_file":"injectives.tex","source_line":800,"source_end_line":808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L800-L808","statement_sha256":"fda8c19a10d84dca0ebf26555e88dc3ee9847aab194e6a2956562e0347a7970b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4114,"rank":4114,"depth":1,"x":1105.57,"y":361.747,"cluster":"sheaves-sites"},{"id":"stacks:01DP","tag":"01DP","title":"Abelian Sheaves on a site · Theorem 01DP","summary":"The category of sheaves of abelian groups on a site has enough injectives. In fact there exists a functorial injective embedding, see Homology, Definition [Tag 0139].","statement_latex":"The category of sheaves of abelian groups on a\nsite has enough injectives. In fact there exists\na functorial injective embedding, see\nHomology, Definition \\ref{homology-definition-functorial-injective-embedding}.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Abelian Sheaves on a site","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DP","source_file":"injectives.tex","source_line":841,"source_end_line":847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L841-L847","statement_sha256":"2fd706694cfa4ea6ead571c2f8ae8d95577d20cc45b33b61b734014af21885e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4115,"rank":4115,"depth":5,"x":1033.962,"y":101.538,"cluster":"sheaves-sites"},{"id":"stacks:01DR","tag":"01DR","title":"Modules on a ringed site · Lemma 01DR","summary":"The functor F ↦ F^vee is exact.","statement_latex":"The functor $\\mathcal{F} \\mapsto \\mathcal{F}^\\vee$ is exact.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Modules on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DR","source_file":"injectives.tex","source_line":930,"source_end_line":933,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L930-L933","statement_sha256":"97271d3972d956970fe34185c85100146c35da9d9cda26eb79c17fb7306d025e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4116,"rank":4116,"depth":0,"x":1296.189,"y":252.879,"cluster":"sheaves-sites"},{"id":"stacks:01DS","tag":"01DS","title":"Modules on a ringed site · Lemma 01DS","summary":"For any O-module F the evaluation map ev : F → (F^vee)^vee is injective.","statement_latex":"For any $\\mathcal{O}$-module $\\mathcal{F}$ the evaluation map\n$ev : \\mathcal{F} \\to (\\mathcal{F}^\\vee)^\\vee$ is injective.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Modules on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DS","source_file":"injectives.tex","source_line":946,"source_end_line":950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L946-L950","statement_sha256":"38b6d0731516c754ecdc6d7c66ae8e2fbaa584d5020620177e6091991773b830","origin":"The Stacks Project","memory_eligible":false,"source_rank":4117,"rank":4117,"depth":0,"x":980.917,"y":290.101,"cluster":"sheaves-sites"},{"id":"stacks:01DT","tag":"01DT","title":"Modules on a ringed site · Lemma 01DT","summary":"Let O be a sheaf of rings. For every O-module F the O-module J(F) is injective.","statement_latex":"Let $\\mathcal{O}$ be a sheaf of rings.\nFor every $\\mathcal{O}$-module $\\mathcal{F}$ the\n$\\mathcal{O}$-module $J(\\mathcal{F})$ is injective.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Modules on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DT","source_file":"injectives.tex","source_line":977,"source_end_line":982,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L977-L982","statement_sha256":"aea9b6a7b77d72c14e6dedb078ca485a9f00348c76c4bef20399adbc3bed022f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4118,"rank":4118,"depth":0,"x":1183.593,"y":83.627,"cluster":"sheaves-sites"},{"id":"stacks:01DU","tag":"01DU","title":"Modules on a ringed site · Theorem 01DU","summary":"Let C be a site. Let O be a sheaf of rings on C. The category of sheaves of O-modules on a site has enough injectives. In fact there exists a functorial injective embedding, see Homology, Definition [Tag 0139].","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.\nThe category of sheaves of $\\mathcal{O}$-modules on a\nsite has enough injectives. In fact there exists\na functorial injective embedding, see\nHomology, Definition \\ref{homology-definition-functorial-injective-embedding}.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Modules on a ringed site","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DU","source_file":"injectives.tex","source_line":1002,"source_end_line":1010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1002-L1010","statement_sha256":"364cd305336706be8e39e0cf36b0271ecb8a605b401806401c6ea3e636e39cc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4119,"rank":4119,"depth":1,"x":1200.194,"y":351.055,"cluster":"sheaves-sites"},{"id":"stacks:01DV","tag":"01DV","title":"Modules on a ringed site · Proposition 01DV","summary":"Let C be a category. Let O be a presheaf of rings on C. The category PMod(O) of presheaves of O-modules has functorial injective embeddings.","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$.\nThe category $\\textit{PMod}(\\mathcal{O})$ of presheaves of\n$\\mathcal{O}$-modules has functorial injective embeddings.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Modules on a ringed site","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01DV","source_file":"injectives.tex","source_line":1016,"source_end_line":1022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1016-L1022","statement_sha256":"058602f3885d3bb0bd83d9866076880be0f0c93479b5424135ce9b243aae9619","origin":"The Stacks Project","memory_eligible":false,"source_rank":4120,"rank":4120,"depth":0,"x":972.747,"y":163.155,"cluster":"sheaves-sites"},{"id":"stacks:05PM","tag":"05PM","title":"Embedding abelian categories · Lemma 05PM","summary":"Let A be an abelian category. Let Cov = ((f : V → U) mid f is surjective). Then (A, Cov) is a site, see Sites, Definition [Tag 00VH].","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet\n$$\n\\text{Cov} = \\{\\{f : V \\to U\\} \\mid f\\text{ is surjective}\\}.\n$$\nThen $(\\mathcal{A}, \\text{Cov})$ is a site, see\nSites, Definition \\ref{sites-definition-site}.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Embedding abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PM","source_file":"injectives.tex","source_line":1049,"source_end_line":1058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1049-L1058","statement_sha256":"e1758dbb4825dda274b6d517421b3617ed18df2ba16834b1b073b599b04f9a0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4121,"rank":4121,"depth":4,"x":1291.774,"y":172.657,"cluster":"sheaves-sites"},{"id":"stacks:05PN","tag":"05PN","title":"Embedding abelian categories · Lemma 05PN","summary":"Let A be an abelian category. Let C = (A, Cov) be the site defined in Lemma [Tag 05PM]. Then X ↦ h_X defines a fully faithful, exact functor A → Ab(C). Moreover, the site C has enough points.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\mathcal{C} = (\\mathcal{A}, \\text{Cov})$ be the\nsite defined in\nLemma \\ref{lemma-site-abelian-category}.\nThen $X \\mapsto h_X$ defines a fully faithful, exact functor\n$$\n\\mathcal{A} \\longrightarrow \\textit{Ab}(\\mathcal{C}).\n$$\nMoreover, the site $\\mathcal{C}$ has enough points.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Embedding abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PN","source_file":"injectives.tex","source_line":1074,"source_end_line":1085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1074-L1085","statement_sha256":"14e4befabd4e3fdb0f5e10962a121620d82bce7f662660befe95ef727ce6d23d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4122,"rank":4122,"depth":6,"x":1048.73,"y":346.787,"cluster":"sheaves-sites"},{"id":"stacks:079B","tag":"079B","title":"Grothendieck's AB conditions · Definition 079B","summary":"Let A be an abelian category. We name some conditions • [AB3] A has direct sums, • [AB4] A has AB3 and direct sums are exact, • [AB5] A has AB3 and filtered colimits are exact. Here are the dual notions • [AB3*] A has products, • [AB4*] A has AB3* and products are exact, • [AB5*] A has AB3* and cofiltered limits are exact. We say an object U of A is a generator if for every N ⊂ M, N not = M in A there exists a morphism U → M which does not factor through N. We say A is a…","statement_latex":"Let $\\mathcal{A}$ be an abelian category. We name some conditions\n\\begin{enumerate}\n\\item[AB3] $\\mathcal{A}$ has direct sums,\n\\item[AB4] $\\mathcal{A}$ has AB3 and direct sums are exact,\n\\item[AB5] $\\mathcal{A}$ has AB3 and filtered colimits are exact.\n\\end{enumerate}\nHere are the dual notions\n\\begin{enumerate}\n\\item[AB3*] $\\mathcal{A}$ has products,\n\\item[AB4*] $\\mathcal{A}$ has AB3* and products are exact,\n\\item[AB5*] $\\mathcal{A}$ has AB3* and cofiltered limits are exact.\n\\end{enumerate}\nWe say an object $U$ of $\\mathcal{A}$ is a {\\it generator} if\nfor every $N \\subset M$, $N \\not = M$ in $\\mathcal{A}$ there exists a morphism\n$U \\to M$ which does not factor through $N$.\nWe say $\\mathcal{A}$ is a {\\it Grothendieck abelian category} if\nit has AB5 and a generator.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Grothendieck's AB conditions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079B","source_file":"injectives.tex","source_line":1272,"source_end_line":1291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1272-L1291","statement_sha256":"f9c636de392ec231a23d5734bd6fe71c7b9ac679dbc281bec5d646dd5ed77913","origin":"The Stacks Project","memory_eligible":false,"source_rank":4123,"rank":4123,"depth":0,"x":1087.942,"y":80.305,"cluster":"sheaves-sites"},{"id":"stacks:0E8N","tag":"0E8N","title":"Injectives in Grothendieck categories · Lemma 0E8N","summary":"Let A be an abelian category with a generator U and X an object of A. If kappa is the cardinality of Mor(U, X) then • There does not exist a strictly increasing (or strictly decreasing) chain of subobjects of X indexed by a cardinal bigger than kappa. • If α is an ordinal of cofinality > kappa then any increasing (or decreasing) sequence of subobjects of X indexed by α is eventually constant. • The cardinality of the set of subobjects of X is ≤ 2^kappa.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with a generator $U$ and\n$X$ an object of $\\mathcal{A}$. If $\\kappa$ is the cardinality of\n$\\Mor(U, X)$ then\n\\begin{enumerate}\n\\item There does not exist a strictly increasing\n(or strictly decreasing) chain of subobjects\nof $X$ indexed by a cardinal bigger than $\\kappa$.\n\\item If $\\alpha$ is an ordinal of cofinality $> \\kappa$\nthen any increasing (or decreasing) sequence of subobjects\nof $X$ indexed by $\\alpha$ is eventually constant.\n\\item The cardinality of the set of subobjects of $X$\nis $\\leq 2^\\kappa$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Injectives in Grothendieck categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8N","source_file":"injectives.tex","source_line":1341,"source_end_line":1356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1341-L1356","statement_sha256":"3ab1eebe02b35ea2b1373174b974d4844f1514bf7f9679107abedafd0ba8c350","origin":"The Stacks Project","memory_eligible":false,"source_rank":4124,"rank":4124,"depth":0,"x":1273.445,"y":299.196,"cluster":"sheaves-sites"},{"id":"stacks:079C","tag":"079C","title":"Injectives in Grothendieck categories · Definition 079C","summary":"Let A be a Grothendieck abelian category. Let M be an object of A. The size |M| of M is the cardinality of the set of subobjects of M.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nLet $M$ be an object of $\\mathcal{A}$.\nThe {\\it size} $|M|$ of $M$ is the cardinality of the set of subobjects\nof $M$.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Injectives in Grothendieck categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079C","source_file":"injectives.tex","source_line":1379,"source_end_line":1385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1379-L1385","statement_sha256":"67d510b3176a3cd5daf93bdadbb66bc9fb7c73fdd4a1dd769273cc500bba7d96","origin":"The Stacks Project","memory_eligible":false,"source_rank":4125,"rank":4125,"depth":0,"x":960.43,"y":243.011,"cluster":"sheaves-sites"},{"id":"stacks:079D","tag":"079D","title":"Injectives in Grothendieck categories · Lemma 079D","summary":"Let A be a Grothendieck abelian category. If 0 → M' → M → M\" → 0 is a short exact sequence of A, then |M'|, |M\"| ≤ |M|.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nIf $0 \\to M' \\to M \\to M'' \\to 0$ is a short exact sequence of\n$\\mathcal{A}$, then $|M'|, |M''| \\leq |M|$.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Injectives in Grothendieck categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079D","source_file":"injectives.tex","source_line":1387,"source_end_line":1392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1387-L1392","statement_sha256":"83f9b6d73557e2ca2e016ad7b7759fe33fb483e17fda90c2f1a3cad55520023e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4126,"rank":4126,"depth":0,"x":1236.602,"y":106.741,"cluster":"sheaves-sites"},{"id":"stacks:079E","tag":"079E","title":"Injectives in Grothendieck categories · Lemma 079E","summary":"Let A be a Grothendieck abelian category with generator U. • If |M| ≤ kappa, then M is the quotient of a direct sum of at most kappa copies of U. • For every cardinal kappa the isomorphism classes of objects M with |M| ≤ kappa form a set.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category with generator $U$.\n\\begin{enumerate}\n\\item If $|M| \\leq \\kappa$, then $M$ is the quotient of a direct\nsum of at most $\\kappa$ copies of $U$.\n\\item For every cardinal $\\kappa$ the isomorphism classes\nof objects $M$ with $|M| \\leq \\kappa$ form a set.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Injectives in Grothendieck categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079E","source_file":"injectives.tex","source_line":1398,"source_end_line":1407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1398-L1407","statement_sha256":"c3bd300e11dc9cf18dd0b936d4f1725296aa590241395f7287a63d7396546721","origin":"The Stacks Project","memory_eligible":false,"source_rank":4127,"rank":4127,"depth":0,"x":1142.481,"y":364.097,"cluster":"sheaves-sites"},{"id":"stacks:079F","tag":"079F","title":"Injectives in Grothendieck categories · Proposition 079F","summary":"Let A be a Grothendieck abelian category. Let M be an object of A. Let kappa = |M|. If α is an ordinal whose cofinality is bigger than kappa, then M is α-small with respect to injections.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $M$ be an\nobject of $\\mathcal{A}$. Let $\\kappa = |M|$.\nIf $\\alpha$ is an ordinal whose cofinality is bigger than $\\kappa$,\nthen $M$ is $\\alpha$-small with respect to injections.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Injectives in Grothendieck categories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079F","source_file":"injectives.tex","source_line":1416,"source_end_line":1422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1416-L1422","statement_sha256":"8dc71fe653c018eea6b733c6f5e14519236ce33ff34b88e6061aef558dd617b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4128,"rank":4128,"depth":2,"x":1004.837,"y":120.763,"cluster":"sheaves-sites"},{"id":"stacks:079G","tag":"079G","title":"Injectives in Grothendieck categories · Lemma 079G","summary":"To check that an object is injective, one only needs to check that lifting holds for subobjects of a generator. Let A be a Grothendieck abelian category with generator U. An object I of A is injective if and only if in every commutative diagram xymatrix M ar[d] ar[r] & I U ar@-->[ru] for M ⊂ U a subobject, the dotted arrow exists.","statement_latex":"\\begin{slogan}\nTo check that an object is injective, one only needs to check that lifting\nholds for subobjects of a generator.\n\\end{slogan}\nLet $\\mathcal{A}$ be a Grothendieck abelian category with generator $U$.\nAn object $I$ of $\\mathcal{A}$ is injective if and only if in every\ncommutative diagram\n$$\n\\xymatrix{\nM \\ar[d] \\ar[r] &  I \\\\\nU \\ar@{-->}[ru]\n}\n$$\nfor $M \\subset U$ a subobject, the dotted arrow exists.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Injectives in Grothendieck categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079G","source_file":"injectives.tex","source_line":1453,"source_end_line":1469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1453-L1469","statement_sha256":"ec6095e4b42e33d747ca689820bb2ae653995f1148c9504924b203d8ba9558db","origin":"The Stacks Project","memory_eligible":false,"source_rank":4129,"rank":4129,"depth":3,"x":1302.207,"y":222.156,"cluster":"sheaves-sites"},{"id":"stacks:079H","tag":"079H","title":"Injectives in Grothendieck categories · Theorem 079H","summary":"Let A be a Grothendieck abelian category. Then A has functorial injective embeddings.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nThen $\\mathcal{A}$ has functorial injective embeddings.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Injectives in Grothendieck categories","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079H","source_file":"injectives.tex","source_line":1498,"source_end_line":1502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1498-L1502","statement_sha256":"e496630eea8a5e21f9c98cdd1537797c0013a1747a11aaf746c1a6efcf833c72","origin":"The Stacks Project","memory_eligible":false,"source_rank":4130,"rank":4130,"depth":4,"x":1001.201,"y":316.186,"cluster":"sheaves-sites"},{"id":"stacks:079J","tag":"079J","title":"K-injectives in Grothendieck categories · Lemma 079J","summary":"Let A be a Grothendieck abelian category with generator U. Let c be the function on cardinals defined by c(kappa) = |bigoplus_α ∈ kappa U|. If π : M → N is a surjection then there exists a subobject M' ⊂ M which surjects onto N with |M'| ≤ c(|N|).","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category with generator $U$.\nLet $c$ be the function on cardinals defined by\n$c(\\kappa) = |\\bigoplus_{\\alpha \\in \\kappa} U|$. If $\\pi : M \\to N$ is a\nsurjection then there exists a subobject $M' \\subset M$ which surjects\nonto $N$ with $|M'| \\leq c(|N|)$.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"K-injectives in Grothendieck categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079J","source_file":"injectives.tex","source_line":1569,"source_end_line":1576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1569-L1576","statement_sha256":"631b3432c3dcaedd7151abeae7aefd068a009dc09f73cd03127f72f0a0a0f53a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4131,"rank":4131,"depth":0,"x":1147.636,"y":75.898,"cluster":"sheaves-sites"},{"id":"stacks:079K","tag":"079K","title":"K-injectives in Grothendieck categories · Lemma 079K","summary":"Let A be a Grothendieck abelian category. There exists a cardinal kappa such that given any acyclic complex M^bullet we have • if M^bullet is nonzero, there is a nonzero subcomplex N^bullet which is bounded above, acyclic, and |N^n| ≤ kappa, • there exists a surjection of complexes bigoplus_i ∈ I M_i^bullet → M^bullet where M_i^bullet is bounded above, acyclic, and |M_i^n| ≤ kappa.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category. There exists a cardinal\n$\\kappa$ such that given any acyclic complex $M^\\bullet$ we have\n\\begin{enumerate}\n\\item if $M^\\bullet$ is nonzero, there is a nonzero subcomplex\n$N^\\bullet$ which is bounded above, acyclic, and $|N^n| \\leq \\kappa$,\n\\item there exists a surjection of complexes\n$$\n\\bigoplus\\nolimits_{i \\in I} M_i^\\bullet \\longrightarrow M^\\bullet\n$$\nwhere $M_i^\\bullet$ is bounded above, acyclic, and $|M_i^n| \\leq \\kappa$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"K-injectives in Grothendieck categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079K","source_file":"injectives.tex","source_line":1590,"source_end_line":1603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1590-L1603","statement_sha256":"b0bd71f5f01950895796db103d6f9c9bf4afa45db2a741b33583c5a207bc102d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4132,"rank":4132,"depth":1,"x":1232.944,"y":336.34,"cluster":"sheaves-sites"},{"id":"stacks:079L","tag":"079L","title":"K-injectives in Grothendieck categories · Lemma 079L","summary":"Let A be a Grothendieck abelian category. Let kappa be a cardinal as in Lemma [Tag 079K]. Suppose that I^bullet is a complex such that • each I^j is injective, and • for every bounded above acyclic complex M^bullet such that |M^n| ≤ kappa we have Hom_K(A)(M^bullet, I^bullet) = 0. Then I^bullet is an K-injective complex.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nLet $\\kappa$ be a cardinal as in\nLemma \\ref{lemma-acyclic-quotient-complexes-bounded-size}.\nSuppose that $I^\\bullet$ is a complex such that\n\\begin{enumerate}\n\\item each $I^j$ is injective, and\n\\item for every bounded above acyclic complex $M^\\bullet$\nsuch that $|M^n| \\leq \\kappa$\nwe have $\\Hom_{K(\\mathcal{A})}(M^\\bullet, I^\\bullet) = 0$.\n\\end{enumerate}\nThen $I^\\bullet$ is an $K$-injective complex.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"K-injectives in Grothendieck categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079L","source_file":"injectives.tex","source_line":1632,"source_end_line":1645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1632-L1645","statement_sha256":"fad2ef862f3976de94451353ccac9a9b756794df8d219b213e0e4891163b5b0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4133,"rank":4133,"depth":2,"x":960.425,"y":192.608,"cluster":"sheaves-sites"},{"id":"stacks:079M","tag":"079M","title":"K-injectives in Grothendieck categories · Lemma 079M","summary":"Let A be a Grothendieck abelian category. Let (K_i^bullet)_i ∈ I be a set of acyclic complexes. There exists a functor M^bullet ↦ M^bullet(M^bullet) and a natural transformation j_M^bullet : M^bullet → M^bullet(M^bullet) such • j_M^bullet is a (termwise) injective quasi-isomorphism, and • for every i ∈ I and w : K_i^bullet → M^bullet the morphism j_M^bullet ∘ w is homotopic to zero.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nLet $(K_i^\\bullet)_{i \\in I}$ be a set of acyclic complexes.\nThere exists a functor $M^\\bullet \\mapsto \\mathbf{M}^\\bullet(M^\\bullet)$\nand a natural transformation\n$j_{M^\\bullet} : M^\\bullet \\to \\mathbf{M}^\\bullet(M^\\bullet)$\nsuch\n\\begin{enumerate}\n\\item $j_{M^\\bullet}$ is a (termwise) injective quasi-isomorphism, and\n\\item for every $i \\in I$ and $w : K_i^\\bullet \\to M^\\bullet$\nthe morphism $j_{M^\\bullet} \\circ w$ is homotopic to zero.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"K-injectives in Grothendieck categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079M","source_file":"injectives.tex","source_line":1704,"source_end_line":1717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1704-L1717","statement_sha256":"1e5116a987cdafad1ec0335fd43073861818d6ec19632bf96608abc3544d6109","origin":"The Stacks Project","memory_eligible":false,"source_rank":4134,"rank":4134,"depth":0,"x":1277.164,"y":143.929,"cluster":"sheaves-sites"},{"id":"stacks:079N","tag":"079N","title":"K-injectives in Grothendieck categories · Lemma 079N","summary":"Let A be a Grothendieck abelian category. There exists a functor M^bullet ↦ N^bullet(M^bullet) and a natural transformation j_M^bullet : M^bullet → N^bullet(M^bullet) such • j_M^bullet is a (termwise) injective quasi-isomorphism, and • for every n ∈ Z the map M^n → N^n(M^bullet) factors through a subobject I^n ⊂ N^n(M^bullet) where I^n is an injective object of A.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nThere exists a functor $M^\\bullet \\mapsto \\mathbf{N}^\\bullet(M^\\bullet)$\nand a natural transformation\n$j_{M^\\bullet} : M^\\bullet \\to \\mathbf{N}^\\bullet(M^\\bullet)$\nsuch\n\\begin{enumerate}\n\\item $j_{M^\\bullet}$ is a (termwise) injective quasi-isomorphism, and\n\\item for every $n \\in \\mathbf{Z}$ the map $M^n \\to \\mathbf{N}^n(M^\\bullet)$\nfactors through a subobject $I^n \\subset \\mathbf{N}^n(M^\\bullet)$ where $I^n$\nis an injective object of $\\mathcal{A}$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"K-injectives in Grothendieck categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079N","source_file":"injectives.tex","source_line":1742,"source_end_line":1755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1742-L1755","statement_sha256":"8557b428c125edb437e5140ae4ba2a99c425a98eb1182acbcbd1ae7516cfacc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4135,"rank":4135,"depth":5,"x":1082.628,"y":359.687,"cluster":"sheaves-sites"},{"id":"stacks:079P","tag":"079P","title":"K-injectives in Grothendieck categories · Theorem 079P","summary":"Existence of K-injective complexes for Grothendieck abelian categories. Let A be a Grothendieck abelian category. For every complex M^bullet there exists a quasi-isomorphism M^bullet → I^bullet such that M^n → I^n is injective and I^n is an injective object of A for all n and I^bullet is a K-injective complex. Moreover, the construction is functorial in M^bullet.","statement_latex":"\\begin{slogan}\nExistence of K-injective complexes for Grothendieck abelian categories.\n\\end{slogan}\nLet $\\mathcal{A}$ be a Grothendieck abelian category.\nFor every complex $M^\\bullet$ there exists a quasi-isomorphism\n$M^\\bullet \\to I^\\bullet$ such that $M^n \\to I^n$ is injective and $I^n$\nis an injective object of $\\mathcal{A}$ for all $n$ and $I^\\bullet$\nis a K-injective complex. Moreover, the construction is functorial in\n$M^\\bullet$.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"K-injectives in Grothendieck categories","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079P","source_file":"injectives.tex","source_line":1806,"source_end_line":1817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1806-L1817","statement_sha256":"eec811c7f01e30e408b266a2f04d3237958fa4947a73edea9359ef4e564c144c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4136,"rank":4136,"depth":6,"x":1052.55,"y":90.035,"cluster":"sheaves-sites"},{"id":"stacks:07D7","tag":"07D7","title":"Additional remarks on Grothendieck abelian categories · Lemma 07D7","summary":"Let A be a Grothendieck abelian category. Let F : A^opp → Sets be a functor. Then F is representable if and only if F commutes with colimits, i.e., F(colim_i N_i) = lim F(N_i) for any diagram I → A, i ∈ I.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nLet $F : \\mathcal{A}^{opp} \\to \\textit{Sets}$ be a functor.\nThen $F$ is representable if and only if $F$ commutes with colimits, i.e.,\n$$\nF(\\colim_i N_i) = \\lim F(N_i)\n$$\nfor any diagram $\\mathcal{I} \\to \\mathcal{A}$, $i \\in \\mathcal{I}$.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Additional remarks on Grothendieck abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07D7","source_file":"injectives.tex","source_line":1902,"source_end_line":1911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1902-L1911","statement_sha256":"45b4b07738e111a44ea11028b0fec2b4b3d4e12a51c5100ae2d44a768951c4e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4137,"rank":4137,"depth":0,"x":1291.728,"y":271.921,"cluster":"sheaves-sites"},{"id":"stacks:07D8","tag":"07D8","title":"Additional remarks on Grothendieck abelian categories · Lemma 07D8","summary":"A Grothendieck abelian category has Ab3*.","statement_latex":"A Grothendieck abelian category has Ab3*.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Additional remarks on Grothendieck abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07D8","source_file":"injectives.tex","source_line":1971,"source_end_line":1974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L1971-L1974","statement_sha256":"9b83c76448ac12ae1376a278a22fb29dd839043d54174b461ffd9cc08003cbe8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4138,"rank":4138,"depth":1,"x":968.889,"y":273.517,"cluster":"sheaves-sites"},{"id":"stacks:07D9","tag":"07D9","title":"Additional remarks on Grothendieck abelian categories · Lemma 07D9","summary":"Let A be a Grothendieck abelian category. Then • D(A) has both direct sums and products, • direct sums are obtained by taking termwise direct sums of any complexes, • products are obtained by taking termwise products of K-injective complexes.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nThen\n\\begin{enumerate}\n\\item $D(\\mathcal{A})$ has both direct sums and products,\n\\item direct sums are obtained by taking termwise direct sums of\nany complexes,\n\\item products are obtained by taking termwise products of\nK-injective complexes.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Additional remarks on Grothendieck abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07D9","source_file":"injectives.tex","source_line":2008,"source_end_line":2019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2008-L2019","statement_sha256":"595d3ec3a3099daf0218eef33c19fbd2ab5d405e34dc435ab4070810429cb49d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4139,"rank":4139,"depth":7,"x":1205.812,"y":89.037,"cluster":"sheaves-sites"},{"id":"stacks:08U1","tag":"08U1","title":"Additional remarks on Grothendieck abelian categories · Lemma 08U1","summary":"Let F : A → B be an additive functor of abelian categories. Assume • A is a Grothendieck abelian category, • B has exact countable products, and • F commutes with countable products. Then RF : D(A) → D(B) commutes with derived limits.","statement_latex":"Let $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor of\nabelian categories. Assume\n\\begin{enumerate}\n\\item $\\mathcal{A}$ is a Grothendieck abelian category,\n\\item $\\mathcal{B}$ has exact countable products, and\n\\item $F$ commutes with countable products.\n\\end{enumerate}\nThen\n$RF : D(\\mathcal{A}) \\to D(\\mathcal{B})$ commutes with derived limits.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Additional remarks on Grothendieck abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08U1","source_file":"injectives.tex","source_line":2065,"source_end_line":2076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2065-L2076","statement_sha256":"909c47d5dee48dbb118bf3664fc1c3209df3a686bb2ba62fa06175924256318d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4140,"rank":4140,"depth":8,"x":1179.446,"y":359.676,"cluster":"sheaves-sites"},{"id":"stacks:0BKI","tag":"0BKI","title":"Additional remarks on Grothendieck abelian categories · Lemma 0BKI","summary":"Let A be a Grothendieck abelian category. Let K^bullet be a filtered complex of A, see Homology, Definition [Tag 012L]. Then there exists a morphism j : K^bullet → J^bullet of filtered complexes of A such that • J^n, F^pJ^n, J^n/F^pJ^n and F^pJ^n/F^p'J^n are injective objects of A, • J^bullet, F^pJ^bullet, J^bullet/F^pJ^bullet, and F^pJ^bullet/F^p'J^bullet are K-injective complexes, • j induces quasi-isomorphisms K^bullet → J^bullet, F^pK^bullet → F^pJ^bullet,…","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nLet $K^\\bullet$ be a filtered complex of $\\mathcal{A}$, see\nHomology, Definition \\ref{homology-definition-filtered-complex}.\nThen there exists a morphism $j : K^\\bullet \\to J^\\bullet$\nof filtered complexes of $\\mathcal{A}$ such that\n\\begin{enumerate}\n\\item $J^n$, $F^pJ^n$, $J^n/F^pJ^n$ and $F^pJ^n/F^{p'}J^n$ are injective\nobjects of $\\mathcal{A}$,\n\\item $J^\\bullet$, $F^pJ^\\bullet$, $J^\\bullet/F^pJ^\\bullet$, and\n$F^pJ^\\bullet/F^{p'}J^\\bullet$ are K-injective complexes,\n\\item $j$ induces quasi-isomorphisms\n$K^\\bullet \\to J^\\bullet$,\n$F^pK^\\bullet \\to F^pJ^\\bullet$,\n$K^\\bullet/F^pK^\\bullet \\to J^\\bullet/F^pJ^\\bullet$, and\n$F^pK^\\bullet/F^{p'}K^\\bullet \\to F^pJ^\\bullet/F^{p'}J^\\bullet$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Additional remarks on Grothendieck abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKI","source_file":"injectives.tex","source_line":2107,"source_end_line":2125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2107-L2125","statement_sha256":"a813646b756a0229b6c99baf5f2bce141c2fa1ea6ac8b9aa64250ac7ab873cb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4141,"rank":4141,"depth":13,"x":981.122,"y":145.014,"cluster":"sheaves-sites"},{"id":"stacks:0ESJ","tag":"0ESJ","title":"Additional remarks on Grothendieck abelian categories · Lemma 0ESJ","summary":"Let A be a Grothendieck abelian category. Suppose given an object E ∈ D(A) and an inverse system (E^i)_i ∈ Z of objects of D(A) over Z together with a compatible system of maps E^i → E. Picture: … → E^i + 1 → E^i → E^i - 1 → … → E Then there exists a filtered complex K^bullet of A (Homology, Definition [Tag 012L]) such that K^bullet represents E and F^iK^bullet represents E^i compatibly with the given maps.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category. Suppose given an object\n$E \\in D(\\mathcal{A})$ and an inverse system $\\{E^i\\}_{i \\in \\mathbf{Z}}$\nof objects of $D(\\mathcal{A})$ over $\\mathbf{Z}$ together with\na compatible system of maps $E^i \\to E$. Picture:\n$$\n\\ldots \\to E^{i + 1} \\to E^i \\to E^{i - 1} \\to \\ldots \\to E\n$$\nThen there exists a filtered complex $K^\\bullet$ of $\\mathcal{A}$\n(Homology, Definition \\ref{homology-definition-filtered-complex})\nsuch that $K^\\bullet$ represents $E$\nand $F^iK^\\bullet$ represents $E^i$ compatibly with the given maps.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Additional remarks on Grothendieck abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESJ","source_file":"injectives.tex","source_line":2384,"source_end_line":2397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2384-L2397","statement_sha256":"500730ede0642cfb9392c8dff8ac2ce2c7cd16d2b181023daa69560e0dbb0e87","origin":"The Stacks Project","memory_eligible":false,"source_rank":4142,"rank":4142,"depth":8,"x":1300.189,"y":190.798,"cluster":"sheaves-sites"},{"id":"stacks:0ESK","tag":"0ESK","title":"Additional remarks on Grothendieck abelian categories · Lemma 0ESK","summary":"In the situation of Lemma [Tag 0ESJ] assume we have a second inverse system ((E')^i)_i ∈ Z and a compatible system of maps (E')^i → E. Then there exists a bi-filtered complex K^bullet of A such that K^bullet represents E, F^iK^bullet represents E^i, and (F')^iK^bullet represents (E')^i compatibly with the given maps.","statement_latex":"In the situation of Lemma \\ref{lemma-represent-by-filtered-complex}\nassume we have a second inverse system $\\{(E')^i\\}_{i \\in \\mathbf{Z}}$\nand a compatible system of maps $(E')^i \\to E$.\nThen there exists a bi-filtered complex $K^\\bullet$ of $\\mathcal{A}$\nsuch that $K^\\bullet$ represents $E$, $F^iK^\\bullet$ represents $E^i$,\nand $(F')^iK^\\bullet$ represents $(E')^i$ compatibly with the given maps.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Additional remarks on Grothendieck abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESK","source_file":"injectives.tex","source_line":2488,"source_end_line":2496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2488-L2496","statement_sha256":"f85608551b43d0890297a60086b5ae53aaaad6fcd4a17cd11b388d89e4f15141","origin":"The Stacks Project","memory_eligible":false,"source_rank":4143,"rank":4143,"depth":14,"x":1027.925,"y":338.177,"cluster":"sheaves-sites"},{"id":"stacks:0F5S","tag":"0F5S","title":"The Gabriel-Popescu theorem · Lemma 0F5S","summary":"The functor G above has a left adjoint F : Mod_R → A.","statement_latex":"The functor $G$ above has a left adjoint\n$F : \\text{Mod}_R \\to \\mathcal{A}$.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"The Gabriel-Popescu theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5S","source_file":"injectives.tex","source_line":2545,"source_end_line":2549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2545-L2549","statement_sha256":"ff1832fd7967d2c7f93d0b2abf6d5f6b3a6489414b3a263ab0b7d9a7d1604c2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4144,"rank":4144,"depth":2,"x":1110.222,"y":74.847,"cluster":"sheaves-sites"},{"id":"stacks:0F5T","tag":"0F5T","title":"The Gabriel-Popescu theorem · Lemma 0F5T","summary":"Let f : M → G(A) be an injective map in Mod_R. Then the adjoint map f' : F(M) → A is injective too.","statement_latex":"Let $f : M \\to G(A)$ be an injective map in $\\text{Mod}_R$.\nThen the adjoint map $f' : F(M) \\to A$ is injective too.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"The Gabriel-Popescu theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5T","source_file":"injectives.tex","source_line":2613,"source_end_line":2617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2613-L2617","statement_sha256":"d5fd3ad26972869a16b372e62966be5497f9c23ae138cbf84a5932b6b990b69f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4145,"rank":4145,"depth":3,"x":1261.395,"y":315.871,"cluster":"sheaves-sites"},{"id":"stacks:0F5U","tag":"0F5U","title":"The Gabriel-Popescu theorem · Theorem 0F5U","summary":"Let A be a Grothendieck abelian category. Then there exists a (noncommutative) ring R and functors G : A → Mod_R and F : Mod_R → A such that • F is the left adjoint to G, • G is fully faithful, and • F is exact. Moreover, the functors are the ones constructed above.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category. Then there exists\na (noncommutative) ring $R$ and functors $G : \\mathcal{A} \\to \\text{Mod}_R$\nand $F : \\text{Mod}_R \\to \\mathcal{A}$ such that\n\\begin{enumerate}\n\\item $F$ is the left adjoint to $G$,\n\\item $G$ is fully faithful, and\n\\item $F$ is exact.\n\\end{enumerate}\nMoreover, the functors are the ones constructed above.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"The Gabriel-Popescu theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5U","source_file":"injectives.tex","source_line":2660,"source_end_line":2671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2660-L2671","statement_sha256":"f2d8357a52837c581d086c65f4abd171e05304998bc5a794a53df05f49a0bea5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4146,"rank":4146,"depth":4,"x":955.905,"y":223.865,"cluster":"sheaves-sites"},{"id":"stacks:0F5V","tag":"0F5V","title":"The Gabriel-Popescu theorem · Lemma 0F5V","summary":"[serpe] Let A be a Grothendieck abelian category. Let R, F, G be as in the Gabriel-Popescu theorem (Theorem [Tag 0F5U]). Then we obtain derived functors RG : D(A) → D(Mod_R) and F : D(Mod_R) → D(A) such that F is left adjoint to RG, RG is fully faithful, and F ∘ RG = id.","statement_latex":"\\begin{reference}\n\\cite[Corollary 4.1]{serpe}\n\\end{reference}\nLet $\\mathcal{A}$ be a Grothendieck abelian category. Let\n$R$, $F$, $G$ be as in the Gabriel-Popescu theorem\n(Theorem \\ref{theorem-gabriel-popescu}). Then we obtain\nderived functors\n$$\nRG : D(\\mathcal{A}) \\to D(\\text{Mod}_R)\n\\quad\\text{and}\\quad\nF : D(\\text{Mod}_R) \\to D(\\mathcal{A})\n$$\nsuch that $F$ is left adjoint to $RG$, $RG$ is fully faithful,\nand $F \\circ RG = \\text{id}$.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"The Gabriel-Popescu theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5V","source_file":"injectives.tex","source_line":2704,"source_end_line":2720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2704-L2720","statement_sha256":"e579cec19bc7ed24e472f6c207acb3a59d0f5f93bd716ffc62ae8cfc2d0b1649","origin":"The Stacks Project","memory_eligible":false,"source_rank":4147,"rank":4147,"depth":7,"x":1255.345,"y":118.301,"cluster":"sheaves-sites"},{"id":"stacks:0F5X","tag":"0F5X","title":"Brown representability and Grothendieck abelian categories · Lemma 0F5X","summary":"Let A be a Grothendieck abelian category. Let H : D(A) → Ab be a contravariant cohomological functor which transforms direct sums into products. Then H is representable.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category.\nLet $H : D(\\mathcal{A}) \\to \\textit{Ab}$ be a contravariant\ncohomological functor which transforms direct sums into products.\nThen $H$ is representable.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Brown representability and Grothendieck abelian categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5X","source_file":"injectives.tex","source_line":2757,"source_end_line":2763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2757-L2763","statement_sha256":"ad1b9abd6c0372e45555476fcbc097f341ba193fb1e3d2fd5924a0abfa036077","origin":"The Stacks Project","memory_eligible":false,"source_rank":4148,"rank":4148,"depth":8,"x":1119.349,"y":366.207,"cluster":"sheaves-sites"},{"id":"stacks:0F5Y","tag":"0F5Y","title":"Brown representability and Grothendieck abelian categories · Proposition 0F5Y","summary":"Let A be a Grothendieck abelian category. Let D be a triangulated category. Let F : D(A) → D be an exact functor of triangulated categories which transforms direct sums into direct sums. Then F has an exact right adjoint.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category. Let $\\mathcal{D}$\nbe a triangulated category. Let $F : D(\\mathcal{A}) \\to \\mathcal{D}$ be an\nexact functor of triangulated categories which transforms direct sums\ninto direct sums. Then $F$ has an exact right adjoint.","area":"Sheaves & Sites","chapter":"Injectives","chapter_id":"injectives","section":"Brown representability and Grothendieck abelian categories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5Y","source_file":"injectives.tex","source_line":2803,"source_end_line":2809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/injectives.tex#L2803-L2809","statement_sha256":"b340985c2afd3bbf52ed1b943be8f8ab13d7aa4c1f09a197b9315d3d62885043","origin":"The Stacks Project","memory_eligible":false,"source_rank":4149,"rank":4149,"depth":9,"x":1020.21,"y":106.075,"cluster":"sheaves-sites"},{"id":"stacks:02FO","tag":"02FO","title":"First cohomology and torsors · Definition 02FO","summary":"Let X be a topological space. Let G be a sheaf of (possibly non-commutative) groups on X. A pseudo torsor, or more precisely a pseudo G-torsor, is a sheaf of sets F on X endowed with an action G × F → F such that • whenever F(U) is nonempty the action G(U) × F(U) → F(U) is simply transitive A morphism of pseudo G-torsors F → F' is a morphism of sheaves of sets compatible with the G-actions. A torsor, or more precisely a G-torsor, is a pseudo G-torsor such that in addition…","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$.\nA {\\it pseudo torsor}, or more precisely a {\\it pseudo $\\mathcal{G}$-torsor},\nis a sheaf of sets $\\mathcal{F}$ on $X$ endowed with an action\n$\\mathcal{G} \\times \\mathcal{F} \\to \\mathcal{F}$ such that\n\\begin{enumerate}\n\\item whenever $\\mathcal{F}(U)$ is nonempty the action\n$\\mathcal{G}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$\nis simply transitive\n\\end{enumerate}\nA {\\it morphism of pseudo $\\mathcal{G}$-torsors} $\\mathcal{F} \\to \\mathcal{F}'$\nis a morphism of sheaves of sets compatible with the\n$\\mathcal{G}$-actions. A {\\it torsor}, or more precisely a\n{\\it $\\mathcal{G}$-torsor}, is a pseudo $\\mathcal{G}$-torsor such\nthat in addition\n\\begin{enumerate}\n\\item[(2)] for every $x \\in X$ the stalk $\\mathcal{F}_x$ is nonempty.\n\\end{enumerate}\nA {\\it morphism of $\\mathcal{G}$-torsors} is a morphism of\npseudo $\\mathcal{G}$-torsors. The {\\it trivial $\\mathcal{G}$-torsor}\nis the sheaf $\\mathcal{G}$ endowed with the obvious left\n$\\mathcal{G}$-action.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"First cohomology and torsors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FO","source_file":"cohomology.tex","source_line":226,"source_end_line":250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L226-L250","statement_sha256":"4701f1ac0ca363e64820db13053a901cd811eb7e7c75d5239b25a4c185254dd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4150,"rank":4150,"depth":0,"x":1135.215,"y":680.0,"cluster":"sheaf-cohomology"},{"id":"stacks:02FP","tag":"02FP","title":"First cohomology and torsors · Lemma 02FP","summary":"Let X be a topological space. Let G be a sheaf of (possibly non-commutative) groups on X. A G-torsor F is trivial if and only if F(X) not = ∅.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$.\nA $\\mathcal{G}$-torsor $\\mathcal{F}$ is trivial if and only if\n$\\mathcal{F}(X) \\not = \\emptyset$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"First cohomology and torsors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FP","source_file":"cohomology.tex","source_line":255,"source_end_line":261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L255-L261","statement_sha256":"e01efdf08dea2695256575c53f8dfebbee1a1dc895430c894cfdc2d2825677ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":4151,"rank":4151,"depth":0,"x":1123.339,"y":685.125,"cluster":"sheaf-cohomology"},{"id":"stacks:02FQ","tag":"02FQ","title":"First cohomology and torsors · Lemma 02FQ","summary":"Let X be a topological space. Let H be an abelian sheaf on X. There is a canonical bijection between the set of isomorphism classes of H-torsors and H^1(X, H).","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{H}$ be an abelian sheaf on $X$.\nThere is a canonical bijection between the set of isomorphism\nclasses of $\\mathcal{H}$-torsors and $H^1(X, \\mathcal{H})$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"First cohomology and torsors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FQ","source_file":"cohomology.tex","source_line":267,"source_end_line":273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L267-L273","statement_sha256":"da29dff57c0c442de14e91814a8de3995e3954e838581816f952027deaa21330","origin":"The Stacks Project","memory_eligible":false,"source_rank":4152,"rank":4152,"depth":19,"x":1131.02,"y":670.242,"cluster":"sheaf-cohomology"},{"id":"stacks:0B3A","tag":"0B3A","title":"First cohomology and extensions · Lemma 0B3A","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. There is a canonical bijection Ext^1_Mod(O_X)(O_X, F) → H^1(X, F) which associates to the extension 0 → F → E → O_X → 0 the image of 1 ∈ Γ(X, O_X) in H^1(X, F).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf of\n$\\mathcal{O}_X$-modules. There is a canonical bijection\n$$\n\\Ext^1_{\\textit{Mod}(\\mathcal{O}_X)}(\\mathcal{O}_X, \\mathcal{F})\n\\longrightarrow\nH^1(X, \\mathcal{F})\n$$\nwhich associates to the extension\n$$\n0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{O}_X \\to 0\n$$\nthe image of $1 \\in \\Gamma(X, \\mathcal{O}_X)$ in $H^1(X, \\mathcal{F})$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"First cohomology and extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3A","source_file":"cohomology.tex","source_line":347,"source_end_line":361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L347-L361","statement_sha256":"468f34f7d7d15dff325f0d2810ae84f0d46808dfa89b406a341d7e0748feb4ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":4153,"rank":4153,"depth":0,"x":1138.395,"y":689.198,"cluster":"sheaf-cohomology"},{"id":"stacks:09NU","tag":"09NU","title":"First cohomology and invertible sheaves · Lemma 09NU","summary":"Let (X, O_X) be a ringed space. If all stalks O_X, x are local rings, then there is a canonical isomorphism H^1(X, O_X^*) = Pic(X). of abelian groups.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. If all stalks\n$\\mathcal{O}_{X, x}$ are local rings, then there is a canonical isomorphism\n$$\nH^1(X, \\mathcal{O}_X^*) = \\Pic(X).\n$$\nof abelian groups.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"First cohomology and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NU","source_file":"cohomology.tex","source_line":405,"source_end_line":413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L405-L413","statement_sha256":"ba596af1006d83b03c314a6ff5e46b64e898c917a5efca89376c8e04698fa078","origin":"The Stacks Project","memory_eligible":false,"source_rank":4154,"rank":4154,"depth":20,"x":1114.594,"y":677.711,"cluster":"sheaf-cohomology"},{"id":"stacks:01E1","tag":"01E1","title":"Locality of cohomology · Lemma 01E1","summary":"Let X be a ringed space. Let U ⊂ X be an open subspace. • If I is an injective O_X-module then I|_U is an injective O_U-module. • For any sheaf of O_X-modules F we have H^p(U, F) = H^p(U, F|_U).","statement_latex":"Let $X$ be a ringed space.\nLet $U \\subset X$ be an open subspace.\n\\begin{enumerate}\n\\item If $\\mathcal{I}$ is an injective $\\mathcal{O}_X$-module\nthen $\\mathcal{I}|_U$ is an injective $\\mathcal{O}_U$-module.\n\\item For any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we have\n$H^p(U, \\mathcal{F}) = H^p(U, \\mathcal{F}|_U)$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Locality of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01E1","source_file":"cohomology.tex","source_line":495,"source_end_line":505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L495-L505","statement_sha256":"f6eb22adc86f4ffa7bb9fc62583c5d27835101194bbee2aeccf0adf171b0a09f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4155,"rank":4155,"depth":8,"x":1144.594,"y":672.202,"cluster":"sheaf-cohomology"},{"id":"stacks:01E3","tag":"01E3","title":"Locality of cohomology · Lemma 01E3","summary":"Let X be a ringed space. Let F be a sheaf of O_X-modules. Let U ⊂ X be an open subspace. Let n > 0 and let xi ∈ H^n(U, F). Then there exists an open covering U = ⋃_i∈ I U_i such that xi|_U_i = 0 for all i ∈ I.","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nLet $U \\subset X$ be an open subspace.\nLet $n > 0$ and let $\\xi \\in H^n(U, \\mathcal{F})$.\nThen there exists an open covering\n$U = \\bigcup_{i\\in I} U_i$ such that $\\xi|_{U_i} = 0$ for\nall $i \\in I$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Locality of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01E3","source_file":"cohomology.tex","source_line":558,"source_end_line":567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L558-L567","statement_sha256":"23d2fb1987f330da268eca774216ceb480f50892673a33b8d07a78a19e84515e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4156,"rank":4156,"depth":0,"x":1125.119,"y":695.253,"cluster":"sheaf-cohomology"},{"id":"stacks:01E4","tag":"01E4","title":"Locality of cohomology · Lemma 01E4","summary":"Let f : X → Y be a morphism of ringed spaces. Let F be a O_X-module. The sheaves R^if_*F are the sheaves associated to the presheaves V ↦ H^i(f^-1(V), F) with restriction mappings as in Equation ([Tag 01E2]). There is a similar statement for R^if_* applied to a bounded below complex F^bullet.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $\\mathcal{F}$ be a $\\mathcal{O}_X$-module.\nThe sheaves $R^if_*\\mathcal{F}$ are the sheaves\nassociated to the presheaves\n$$\nV \\longmapsto H^i(f^{-1}(V), \\mathcal{F})\n$$\nwith restriction mappings as in Equation (\\ref{equation-restriction-mapping}).\nThere is a similar statement for $R^if_*$ applied to a\nbounded below complex $\\mathcal{F}^\\bullet$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Locality of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01E4","source_file":"cohomology.tex","source_line":591,"source_end_line":603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L591-L603","statement_sha256":"dbd4b62c1aa8a041c8ef7b7fde14df2d2fdfb5e754adc08f1f1131bcd62b63f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4157,"rank":4157,"depth":0,"x":1120.69,"y":664.943,"cluster":"sheaf-cohomology"},{"id":"stacks:01E5","tag":"01E5","title":"Locality of cohomology · Lemma 01E5","summary":"Let f : X → Y be a morphism of ringed spaces. Let F be an O_X-module. Let V ⊂ Y be an open subspace. Denote g : f^-1(V) → V the restriction of f. Then we have R^pg_*(F|_f^-1(V)) = (R^pf_*F)|_V There is a similar statement for the derived image Rf_*F^bullet where F^bullet is a bounded below complex of O_X-modules.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nLet $V \\subset Y$ be an open subspace.\nDenote $g : f^{-1}(V) \\to V$ the restriction of $f$.\nThen we have\n$$\nR^pg_*(\\mathcal{F}|_{f^{-1}(V)}) = (R^pf_*\\mathcal{F})|_V\n$$\nThere is a similar statement for the\nderived image $Rf_*\\mathcal{F}^\\bullet$ where $\\mathcal{F}^\\bullet$\nis a bounded below complex of $\\mathcal{O}_X$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Locality of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01E5","source_file":"cohomology.tex","source_line":629,"source_end_line":642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L629-L642","statement_sha256":"50137e2d8145157225281b08f4720c252f0d10621a75be9a196b0d6f283d9d31","origin":"The Stacks Project","memory_eligible":false,"source_rank":4158,"rank":4158,"depth":9,"x":1150.198,"y":686.196,"cluster":"sheaf-cohomology"},{"id":"stacks:01EA","tag":"01EA","title":"Mayer-Vietoris · Lemma 01EA","summary":"Injectives are flasque. Let X be a ringed space. Let U' ⊂ U ⊂ X be open subspaces. For any injective O_X-module I the restriction mapping I(U) → I(U') is surjective.","statement_latex":"\\begin{slogan}\nInjectives are flasque.\n\\end{slogan}\nLet $X$ be a ringed space.\nLet $U' \\subset U \\subset X$ be open subspaces.\nFor any injective $\\mathcal{O}_X$-module $\\mathcal{I}$ the\nrestriction mapping\n$\\mathcal{I}(U) \\to \\mathcal{I}(U')$ is surjective.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EA","source_file":"cohomology.tex","source_line":714,"source_end_line":724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L714-L724","statement_sha256":"2cd0fa5b9ad5e0b467c6fe6ebfd055f85ecb94a3fb2cb3d23ee5d4a4e5981e17","origin":"The Stacks Project","memory_eligible":false,"source_rank":4159,"rank":4159,"depth":1,"x":1108.987,"y":687.286,"cluster":"sheaf-cohomology"},{"id":"stacks:01EB","tag":"01EB","title":"Mayer-Vietoris · Lemma 01EB","summary":"Let X be a ringed space. Suppose that X = U ∪ V is a union of two open subsets. For every O_X-module F there exists a long exact cohomology sequence 0 → H^0(X, F) → H^0(U, F) ⊕ H^0(V, F) → H^0(U ∩ V, F) → H^1(X, F) → … This long exact sequence is functorial in F.","statement_latex":"Let $X$ be a ringed space. Suppose that $X = U \\cup V$ is a\nunion of two open subsets. For every $\\mathcal{O}_X$-module $\\mathcal{F}$\nthere exists a long exact cohomology sequence\n$$\n0 \\to\nH^0(X, \\mathcal{F}) \\to\nH^0(U, \\mathcal{F}) \\oplus H^0(V, \\mathcal{F}) \\to\nH^0(U \\cap V, \\mathcal{F}) \\to\nH^1(X, \\mathcal{F}) \\to \\ldots\n$$\nThis long exact sequence is functorial in $\\mathcal{F}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EB","source_file":"cohomology.tex","source_line":766,"source_end_line":779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L766-L779","statement_sha256":"39d4db4751713a45b308cc0973828943959bbbb4ce372b84b2c49a94a2fdc150","origin":"The Stacks Project","memory_eligible":false,"source_rank":4160,"rank":4160,"depth":7,"x":1140.129,"y":661.817,"cluster":"sheaf-cohomology"},{"id":"stacks:01EC","tag":"01EC","title":"Relative Mayer-Vietoris · Lemma 01EC","summary":"Let f : X → Y be a morphism of ringed spaces. Suppose that X = U ∪ V is a union of two open subsets. Denote a = f|_U : U → Y, b = f|_V : V → Y, and c = f|_U ∩ V : U ∩ V → Y. For every O_X-module F there exists a long exact sequence 0 → f_*F → a_*(F|_U) ⊕ b_*(F|_V) → c_*(F|_U ∩ V) → R^1f_*F → … This long exact sequence is functorial in F.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nSuppose that $X = U \\cup V$ is a union of two open subsets.\nDenote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and\n$c = f|_{U \\cap V} : U \\cap V \\to Y$.\nFor every $\\mathcal{O}_X$-module $\\mathcal{F}$\nthere exists a long exact sequence\n$$\n0 \\to\nf_*\\mathcal{F} \\to\na_*(\\mathcal{F}|_U) \\oplus b_*(\\mathcal{F}|_V) \\to\nc_*(\\mathcal{F}|_{U \\cap V}) \\to\nR^1f_*\\mathcal{F} \\to \\ldots\n$$\nThis long exact sequence is functorial in $\\mathcal{F}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EC","source_file":"cohomology.tex","source_line":803,"source_end_line":819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L803-L819","statement_sha256":"b3fe2e419be31e69aa121866893e2009b1a3e82cf3d275ce18486d2af63223d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4161,"rank":4161,"depth":9,"x":1137.485,"y":700.046,"cluster":"sheaf-cohomology"},{"id":"stacks:01EF","tag":"01EF","title":"The v Cech complex and v Cech cohomology · Definition 01EF","summary":"Let X be a topological space. Let U : U = ⋃_i ∈ I U_i be an open covering. Let F be an abelian presheaf on X. The complex checkC^bullet(U, F) is the v Cech complex associated to F and the open covering U. Its cohomology groups H^i(checkC^bullet(U, F)) are called the v Cech cohomology groups associated to F and the covering U. They are denoted check H^i(U, F).","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it {\\v C}ech complex} associated to $\\mathcal{F}$ and the\nopen covering $\\mathcal{U}$. Its cohomology groups\n$H^i(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}))$ are\ncalled the {\\it {\\v C}ech cohomology groups} associated to\n$\\mathcal{F}$ and the covering $\\mathcal{U}$.\nThey are denoted $\\check H^i(\\mathcal{U}, \\mathcal{F})$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The v Cech complex and v Cech cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EF","source_file":"cohomology.tex","source_line":912,"source_end_line":924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L912-L924","statement_sha256":"19417bbf54ca5162cf44e39b2165778f941c15d9c8d787d89334de94de7f59b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4162,"rank":4162,"depth":0,"x":1107.439,"y":669.017,"cluster":"sheaf-cohomology"},{"id":"stacks:01EG","tag":"01EG","title":"The v Cech complex and v Cech cohomology · Lemma 01EG","summary":"Let X be a topological space. Let F be an abelian presheaf on X. The following are equivalent • F is an abelian sheaf and • for every open covering U : U = ⋃_i ∈ I U_i the natural map F(U) → checkH^0(U, F) is bijective.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf and\n\\item for every open covering $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$\nthe natural map\n$$\n\\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F})\n$$\nis bijective.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The v Cech complex and v Cech cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EG","source_file":"cohomology.tex","source_line":926,"source_end_line":940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L926-L940","statement_sha256":"49afc7a7533836dc98dbc12b1f87023a93a14a2a9a898ad149b7a9ec2e57a9b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4163,"rank":4163,"depth":0,"x":1156.467,"y":675.112,"cluster":"sheaf-cohomology"},{"id":"stacks:0G6S","tag":"0G6S","title":"The v Cech complex and v Cech cohomology · Lemma 0G6S","summary":"Let X be a topological space. Let F be an abelian presheaf on X. Let U : U = ⋃_i ∈ I U_i be an open covering. If U_i = U for some i ∈ I, then the extended v Cech complex F(U) → checkC^bullet(U, F) obtained by putting F(U) in degree -1 with differential given by the canonical map of F(U) into checkC^0(U, F) is homotopy equivalent to 0.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian presheaf on $X$.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. If\n$U_i = U$ for some $i \\in I$, then the extended {\\v C}ech complex\n$$\n\\mathcal{F}(U) \\to \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nobtained by putting $\\mathcal{F}(U)$ in degree $-1$ with differential given by\nthe canonical map of $\\mathcal{F}(U)$ into\n$\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{F})$\nis homotopy equivalent to $0$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The v Cech complex and v Cech cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6S","source_file":"cohomology.tex","source_line":948,"source_end_line":960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L948-L960","statement_sha256":"427e0ced0293c129cd29b98e41c0e7a0620c6204b693988d907332abe8ebcc64","origin":"The Stacks Project","memory_eligible":false,"source_rank":4164,"rank":4164,"depth":0,"x":1113.848,"y":699.299,"cluster":"sheaf-cohomology"},{"id":"stacks:01EJ","tag":"01EJ","title":"v Cech cohomology as a functor on presheaves · Lemma 01EJ","summary":"The functor given by Equation ([Tag 01EI]) is an exact functor (see Homology, Lemma [Tag 010N]).","statement_latex":"The functor given by Equation (\\ref{equation-cech-functor})\nis an exact functor (see Homology, Lemma \\ref{homology-lemma-exact-functor}).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EJ","source_file":"cohomology.tex","source_line":1048,"source_end_line":1052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1048-L1052","statement_sha256":"318532492bd1c8b8dab5ce6f3150f3ed0b810d6cf2dfea28ea0756b05b13f63e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4165,"rank":4165,"depth":7,"x":1126.268,"y":655.811,"cluster":"sheaf-cohomology"},{"id":"stacks:01EK","tag":"01EK","title":"v Cech cohomology as a functor on presheaves · Lemma 01EK","summary":"Let X be a ringed space. Let U : U = ⋃_i ∈ I U_i be an open covering. The functors F ↦ checkH^n(U, F) form a δ-functor from the abelian category of presheaves of O_X-modules to the category of O_X(U)-modules (see Homology, Definition [Tag 010Q]).","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nThe functors $\\mathcal{F} \\mapsto \\check{H}^n(\\mathcal{U}, \\mathcal{F})$\nform a $\\delta$-functor from the abelian category of\npresheaves of $\\mathcal{O}_X$-modules to the category\nof $\\mathcal{O}_X(U)$-modules (see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EK","source_file":"cohomology.tex","source_line":1065,"source_end_line":1074,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1065-L1074","statement_sha256":"3bf22879a12fd3003b9c4df5887887667fe6ba516cfc8555d993ae71c4d3732c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4166,"rank":4166,"depth":8,"x":1152.908,"y":696.218,"cluster":"sheaf-cohomology"},{"id":"stacks:01EL","tag":"01EL","title":"v Cech cohomology as a functor on presheaves · Lemma 01EL","summary":"Let X be a ringed space. Let U : U = ⋃_i ∈ I U_i be an open covering. Denote j_i_0… i_p : U_i_0 … i_p → X the open immersion. Consider the chain complex K(U)_bullet of presheaves of O_X-modules … → bigoplus_i_0i_1i_2 (j_i_0i_1i_2)_p!O_U_i_0i_1i_2 → bigoplus_i_0i_1 (j_i_0i_1)_p!O_U_i_0i_1 → bigoplus_i_0 (j_i_0)_p!O_U_i_0 → 0 → … where the last nonzero term is placed in degree 0 and where the map (j_i_0… i_p + 1)_p!O_U_i_0… i_p + 1 → (j_i_0… hat i_j … i_p + 1)_p! O_U_i_0……","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nDenote $j_{i_0\\ldots i_p} : U_{i_0 \\ldots i_p} \\to X$ the open immersion.\nConsider the chain complex $K(\\mathcal{U})_\\bullet$\nof presheaves of $\\mathcal{O}_X$-modules\n$$\n\\ldots\n\\to\n\\bigoplus_{i_0i_1i_2} (j_{i_0i_1i_2})_{p!}\\mathcal{O}_{U_{i_0i_1i_2}}\n\\to\n\\bigoplus_{i_0i_1} (j_{i_0i_1})_{p!}\\mathcal{O}_{U_{i_0i_1}}\n\\to\n\\bigoplus_{i_0} (j_{i_0})_{p!}\\mathcal{O}_{U_{i_0}}\n\\to 0 \\to \\ldots\n$$\nwhere the last nonzero term is placed in degree $0$\nand where the map\n$$\n(j_{i_0\\ldots i_{p + 1}})_{p!}\\mathcal{O}_{U_{i_0\\ldots i_{p + 1}}}\n\\longrightarrow\n(j_{i_0\\ldots \\hat i_j \\ldots i_{p + 1}})_{p!}\n\\mathcal{O}_{U_{i_0\\ldots \\hat i_j \\ldots i_{p + 1}}}\n$$\nis given by $(-1)^j$ times the canonical map.\nThen there is an isomorphism\n$$\n\\Hom_{\\mathcal{O}_X}(K(\\mathcal{U})_\\bullet, \\mathcal{F})\n=\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nfunctorial in $\\mathcal{F} \\in \\Ob(\\textit{PMod}(\\mathcal{O}_X))$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EL","source_file":"cohomology.tex","source_line":1137,"source_end_line":1170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1137-L1170","statement_sha256":"8ecc2cbad9c415a54488bee3f48338eae92ca36d3dddce04e2937dfb6332ab9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4167,"rank":4167,"depth":1,"x":1099.173,"y":681.071,"cluster":"sheaf-cohomology"},{"id":"stacks:01EM","tag":"01EM","title":"v Cech cohomology as a functor on presheaves · Lemma 01EM","summary":"Let X be a ringed space. Let U : U = ⋃_i ∈ I U_i be an open covering. Let O_U ⊂ O_X be the image presheaf of the map bigoplus j_p!O_U_i → O_X. The chain complex K(U)_bullet of presheaves of Lemma [Tag 01EL] above has homology presheaves H_i(K(U)_bullet) = ( 0 & if & i not = 0 O_U & if & i = 0 .","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nLet $\\mathcal{O}_\\mathcal{U} \\subset \\mathcal{O}_X$\nbe the image presheaf of the map\n$\\bigoplus j_{p!}\\mathcal{O}_{U_i} \\to \\mathcal{O}_X$.\nThe chain complex $K(\\mathcal{U})_\\bullet$ of presheaves\nof Lemma \\ref{lemma-cech-map-into} above has homology presheaves\n$$\nH_i(K(\\mathcal{U})_\\bullet) =\n\\left\\{\n\\begin{matrix}\n0 & \\text{if} & i \\not = 0 \\\\\n\\mathcal{O}_\\mathcal{U} & \\text{if} & i = 0\n\\end{matrix}\n\\right.\n$$","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EM","source_file":"cohomology.tex","source_line":1198,"source_end_line":1216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1198-L1216","statement_sha256":"bff61796d691d69badc4c67273288251176b7d50e72382ac58bf7929f3b297b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4168,"rank":4168,"depth":2,"x":1152.486,"y":661.204,"cluster":"sheaf-cohomology"},{"id":"stacks:01EN","tag":"01EN","title":"v Cech cohomology as a functor on presheaves · Lemma 01EN","summary":"Let X be a ringed space. Let U : U = ⋃_i ∈ I U_i be an open covering of U ⊂ X. The v Cech cohomology functors checkH^p(U, -) are canonically isomorphic as a δ-functor to the right derived functors of the functor checkH^0(U, -) : PMod(O_X) → Mod_O_X(U). Moreover, there is a functorial quasi-isomorphism checkC^bullet(U, F) → RcheckH^0(U, F) where the right hand side indicates the right derived functor RcheckH^0(U, -) : D^+(PMod(O_X)) → D^+(O_X(U)) of the left exact functor…","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$\nbe an open covering of $U \\subset X$.\nThe {\\v C}ech cohomology functors $\\check{H}^p(\\mathcal{U}, -)$\nare canonically isomorphic as a $\\delta$-functor to\nthe right derived functors of the functor\n$$\n\\check{H}^0(\\mathcal{U}, -) :\n\\textit{PMod}(\\mathcal{O}_X)\n\\longrightarrow\n\\text{Mod}_{\\mathcal{O}_X(U)}.\n$$\nMoreover, there is a functorial quasi-isomorphism\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})\n\\longrightarrow\nR\\check{H}^0(\\mathcal{U}, \\mathcal{F})\n$$\nwhere the right hand side indicates the right derived functor\n$$\nR\\check{H}^0(\\mathcal{U}, -) :\nD^{+}(\\textit{PMod}(\\mathcal{O}_X))\n\\longrightarrow\nD^{+}(\\mathcal{O}_X(U))\n$$\nof the left exact functor $\\check{H}^0(\\mathcal{U}, -)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EN","source_file":"cohomology.tex","source_line":1286,"source_end_line":1314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1286-L1314","statement_sha256":"53ede4f9d99dc6eda568a38b966304a9ad3d66f2d50cc88746a3d2d652232b91","origin":"The Stacks Project","memory_eligible":false,"source_rank":4169,"rank":4169,"depth":19,"x":1128.496,"y":707.329,"cluster":"sheaf-cohomology"},{"id":"stacks:01EP","tag":"01EP","title":"v Cech cohomology and cohomology · Lemma 01EP","summary":"Let X be a ringed space. Let U : U = ⋃_i ∈ I U_i be an open covering. Let I be an injective O_X-module. Then checkH^p(U, I) = ( I(U) & if & p = 0 0 & if & p > 0 .","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nLet $\\mathcal{I}$ be an injective $\\mathcal{O}_X$-module.\nThen\n$$\n\\check{H}^p(\\mathcal{U}, \\mathcal{I}) =\n\\left\\{\n\\begin{matrix}\n\\mathcal{I}(U) & \\text{if} & p = 0 \\\\\n0 & \\text{if} & p > 0\n\\end{matrix}\n\\right.\n$$","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EP","source_file":"cohomology.tex","source_line":1407,"source_end_line":1422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1407-L1422","statement_sha256":"00db6804d5af0b2bc9b57342c4a33da02207a5879d85c150d5c8baac34bd84f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4170,"rank":4170,"depth":20,"x":1108.605,"y":658.463,"cluster":"sheaf-cohomology"},{"id":"stacks:01EQ","tag":"01EQ","title":"v Cech cohomology and cohomology · Lemma 01EQ","summary":"Let X be a ringed space. Let U : U = ⋃_i ∈ I U_i be an open covering. There is a transformation checkC^bullet(U, -) → RΓ(U, -) of functors Mod(O_X) → D^+(O_X(U)). In particular this provides canonical maps checkH^p(U, F) → H^p(U, F) for F ranging over Mod(O_X).","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nThere is a transformation\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, -)\n\\longrightarrow\nR\\Gamma(U, -)\n$$\nof functors\n$\\textit{Mod}(\\mathcal{O}_X) \\to D^{+}(\\mathcal{O}_X(U))$.\nIn particular this provides canonical maps\n$\\check{H}^p(\\mathcal{U}, \\mathcal{F}) \\to H^p(U, \\mathcal{F})$ for\n$\\mathcal{F}$ ranging over $\\textit{Mod}(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EQ","source_file":"cohomology.tex","source_line":1433,"source_end_line":1448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1433-L1448","statement_sha256":"ebde3dbee251d1700394f15c3298b05885c3ad7104ceaa344d8a9c76232db4fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4171,"rank":4171,"depth":21,"x":1163.893,"y":683.831,"cluster":"sheaf-cohomology"},{"id":"stacks:0B8R","tag":"0B8R","title":"v Cech cohomology and cohomology · Lemma 0B8R","summary":"Let X be a topological space. Let H be an abelian sheaf on X. Let U : X = ⋃_i ∈ I U_i be an open covering. The map checkH^1(U, H) → H^1(X, H) is injective and identifies checkH^1(U, H) via the bijection of Lemma [Tag 02FQ] with the set of isomorphism classes of H-torsors which restrict to trivial torsors over each U_i.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{H}$ be an abelian sheaf\non $X$. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering.\nThe map\n$$\n\\check{H}^1(\\mathcal{U}, \\mathcal{H}) \\longrightarrow H^1(X, \\mathcal{H})\n$$\nis injective and identifies $\\check{H}^1(\\mathcal{U}, \\mathcal{H})$ via\nthe bijection of Lemma \\ref{lemma-torsors-h1}\nwith the set of isomorphism classes of $\\mathcal{H}$-torsors\nwhich restrict to trivial torsors over each $U_i$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8R","source_file":"cohomology.tex","source_line":1485,"source_end_line":1497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1485-L1497","statement_sha256":"20072112601e0a4feee49c2edb41886044d0c8bdbc8d3b6ed3d09e88afcd7f09","origin":"The Stacks Project","memory_eligible":false,"source_rank":4172,"rank":4172,"depth":20,"x":1101.283,"y":696.784,"cluster":"sheaf-cohomology"},{"id":"stacks:01ER","tag":"01ER","title":"v Cech cohomology and cohomology · Lemma 01ER","summary":"Let X be a ringed space. Consider the functor i : Mod(O_X) → PMod(O_X). It is a left exact functor with right derived functors given by R^pi(F) = underlineH^p(F) : U ↦ H^p(U, F) see discussion in Section [Tag 01E0].","statement_latex":"Let $X$ be a ringed space.\nConsider the functor\n$i : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{PMod}(\\mathcal{O}_X)$.\nIt is a left exact functor with right derived functors given by\n$$\nR^pi(\\mathcal{F}) = \\underline{H}^p(\\mathcal{F}) :\nU \\longmapsto H^p(U, \\mathcal{F})\n$$\nsee discussion in Section \\ref{section-locality}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ER","source_file":"cohomology.tex","source_line":1514,"source_end_line":1525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1514-L1525","statement_sha256":"49a7d1b02f1ab6ab4371aa968e911252a07177906323532f79292e3cd5af763d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4173,"rank":4173,"depth":0,"x":1137.847,"y":650.699,"cluster":"sheaf-cohomology"},{"id":"stacks:01ES","tag":"01ES","title":"v Cech cohomology and cohomology · Lemma 01ES","summary":"Let X be a ringed space. Let U : U = ⋃_i ∈ I U_i be an open covering. For any sheaf of O_X-modules F there is a spectral sequence (E_r, d_r)_r ≥ 0 with E_2^p, q = checkH^p(U, underlineH^q(F)) converging to H^p + q(U, F). This spectral sequence is functorial in F.","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nFor any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ there\nis a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))\n$$\nconverging to $H^{p + q}(U, \\mathcal{F})$.\nThis spectral sequence is functorial in $\\mathcal{F}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ES","source_file":"cohomology.tex","source_line":1540,"source_end_line":1551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1540-L1551","statement_sha256":"f2f727c08f6516df4e54cabcee8e1e1b103ba6208195b9d8cacea5aa6f3c1878","origin":"The Stacks Project","memory_eligible":false,"source_rank":4174,"rank":4174,"depth":21,"x":1148.15,"y":706.607,"cluster":"sheaf-cohomology"},{"id":"stacks:01ET","tag":"01ET","title":"v Cech cohomology and cohomology · Lemma 01ET","summary":"Let X be a ringed space. Let U : U = ⋃_i ∈ I U_i be an open covering. Let F be an O_X-module. Assume that H^i(U_i_0 … i_p, F) = 0 for all i > 0, all p ≥ 0 and all i_0, …, i_p ∈ I. Then checkH^p(U, F) = H^p(U, F) as O_X(U)-modules.","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nAssume that $H^i(U_{i_0 \\ldots i_p}, \\mathcal{F}) = 0$\nfor all $i > 0$, all $p \\geq 0$ and all $i_0, \\ldots, i_p \\in I$.\nThen $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(U, \\mathcal{F})$\nas $\\mathcal{O}_X(U)$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ET","source_file":"cohomology.tex","source_line":1573,"source_end_line":1582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1573-L1582","statement_sha256":"a86ad35ec770fdefcbe03140ae9248c8b756556704e7cabeacb2bbfb93899ec8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4175,"rank":4175,"depth":22,"x":1094.518,"y":670.491,"cluster":"sheaf-cohomology"},{"id":"stacks:01EU","tag":"01EU","title":"v Cech cohomology and cohomology · Lemma 01EU","summary":"Let X be a ringed space. Let 0 → F → G → H → 0 be a short exact sequence of O_X-modules. Let U ⊂ X be an open subset. If there exists a cofinal system of open coverings U of U such that checkH^1(U, F) = 0, then the map G(U) → H(U) is surjective.","statement_latex":"Let $X$ be a ringed space.\nLet\n$$\n0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0\n$$\nbe a short exact sequence of $\\mathcal{O}_X$-modules.\nLet $U \\subset X$ be an open subset.\nIf there exists a cofinal system of open coverings $\\mathcal{U}$\nof $U$ such that $\\check{H}^1(\\mathcal{U}, \\mathcal{F}) = 0$,\nthen the map $\\mathcal{G}(U) \\to \\mathcal{H}(U)$ is\nsurjective.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EU","source_file":"cohomology.tex","source_line":1592,"source_end_line":1605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1592-L1605","statement_sha256":"5f7f08405a1dd81b852cc23dcc43f58c79a9a1b3ee5fa806cc13e582b5b25f52","origin":"The Stacks Project","memory_eligible":false,"source_rank":4176,"rank":4176,"depth":0,"x":1164.466,"y":666.624,"cluster":"sheaf-cohomology"},{"id":"stacks:01EV","tag":"01EV","title":"v Cech cohomology and cohomology · Lemma 01EV","summary":"If higher v Cech cohomology of an abelian sheaf vanishes for all open covers, then higher cohomology vanishes. Let X be a ringed space. Let F be an O_X-module such that checkH^p(U, F) = 0 for all p > 0 and any open covering U : U = ⋃_i ∈ I U_i of an open of X. Then H^p(U, F) = 0 for all p > 0 and any open U ⊂ X.","statement_latex":"\\begin{slogan}\nIf higher {\\v C}ech cohomology of an abelian sheaf vanishes for all open covers,\nthen higher cohomology vanishes.\n\\end{slogan}\nLet $X$ be a ringed space.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module such that\n$$\n\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0\n$$\nfor all $p > 0$ and any open covering $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$\nof an open of $X$. Then $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$\nand any open $U \\subset X$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EV","source_file":"cohomology.tex","source_line":1630,"source_end_line":1644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1630-L1644","statement_sha256":"ece565f5b9e2a29f3d6cbd1fe438661414be8bfb7b15bf4236b7fa6c30eeae38","origin":"The Stacks Project","memory_eligible":false,"source_rank":4177,"rank":4177,"depth":21,"x":1115.068,"y":709.97,"cluster":"sheaf-cohomology"},{"id":"stacks:01EW","tag":"01EW","title":"v Cech cohomology and cohomology · Lemma 01EW","summary":"(Variant of Lemma [Tag 01EV].) Let X be a ringed space. Let B be a basis for the topology on X. Let F be an O_X-module. Assume there exists a set of open coverings Cov with the following properties: • For every U ∈ Cov with U : U = ⋃_i ∈ I U_i we have U, U_i ∈ B and every U_i_0 … i_p ∈ B. • For every U ∈ B the open coverings of U occurring in Cov is a cofinal system of open coverings of U. • For every U ∈ Cov we have checkH^p(U, F) = 0 for all p > 0. Then H^p(U, F) = 0…","statement_latex":"(Variant of Lemma \\ref{lemma-cech-vanish}.)\nLet $X$ be a ringed space.\nLet $\\mathcal{B}$ be a basis for the topology on $X$.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nAssume there exists a set of open coverings $\\text{Cov}$\nwith the following properties:\n\\begin{enumerate}\n\\item For every $\\mathcal{U} \\in \\text{Cov}$\nwith $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ we have\n$U, U_i \\in \\mathcal{B}$ and every $U_{i_0 \\ldots i_p} \\in \\mathcal{B}$.\n\\item For every $U \\in \\mathcal{B}$ the open coverings of $U$\noccurring in $\\text{Cov}$ is a cofinal system of open coverings\nof $U$.\n\\item For every $\\mathcal{U} \\in \\text{Cov}$ we have\n$\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for all $p > 0$.\n\\end{enumerate}\nThen $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$ and any $U \\in \\mathcal{B}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EW","source_file":"cohomology.tex","source_line":1695,"source_end_line":1714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1695-L1714","statement_sha256":"56f846203d0528f7fdf3e5c8f3243365b56eba2cbb15fec3a6482751bbdc864e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4178,"rank":4178,"depth":22,"x":1116.674,"y":648.878,"cluster":"sheaf-cohomology"},{"id":"stacks:01EX","tag":"01EX","title":"v Cech cohomology and cohomology · Lemma 01EX","summary":"Let f : X → Y be a morphism of ringed spaces. Let I be an injective O_X-module. Then • checkH^p(V, f_*I) = 0 for all p > 0 and any open covering V : V = ⋃_j ∈ J V_j of Y. • H^p(V, f_*I) = 0 for all p > 0 and every open V ⊂ Y. In other words, f_*I is right acyclic for Γ(V, -) (see Derived Categories, Definition [Tag 0157]) for any V ⊂ Y open.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $\\mathcal{I}$ be an injective $\\mathcal{O}_X$-module.\nThen\n\\begin{enumerate}\n\\item $\\check{H}^p(\\mathcal{V}, f_*\\mathcal{I}) = 0$\nfor all $p > 0$ and any open covering\n$\\mathcal{V} : V = \\bigcup_{j \\in J} V_j$ of $Y$.\n\\item $H^p(V, f_*\\mathcal{I}) = 0$ for all $p > 0$ and\nevery open $V \\subset Y$.\n\\end{enumerate}\nIn other words, $f_*\\mathcal{I}$ is right acyclic for $\\Gamma(V, -)$\n(see\nDerived Categories, Definition \\ref{derived-definition-derived-functor})\nfor any $V \\subset Y$ open.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EX","source_file":"cohomology.tex","source_line":1778,"source_end_line":1794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1778-L1794","statement_sha256":"e8c8cf1b1bba893679c461ec7a8aaff73cb38bb13aab152bc047442dafbba5fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4179,"rank":4179,"depth":22,"x":1165.459,"y":695.655,"cluster":"sheaf-cohomology"},{"id":"stacks:02N5","tag":"02N5","title":"v Cech cohomology and cohomology · Lemma 02N5","summary":"Let f : X → Y be a morphism of ringed spaces. Assume f is flat. Then f_*I is an injective O_Y-module for any injective O_X-module I.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nAssume $f$ is flat.\nThen $f_*\\mathcal{I}$ is an injective $\\mathcal{O}_Y$-module\nfor any injective $\\mathcal{O}_X$-module $\\mathcal{I}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02N5","source_file":"cohomology.tex","source_line":1820,"source_end_line":1826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1820-L1826","statement_sha256":"4c689a17b6d7031f5b75cf259e173303e2ee0312b92183f4c9480ba372fcfab8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4180,"rank":4180,"depth":8,"x":1090.614,"y":688.721,"cluster":"sheaf-cohomology"},{"id":"stacks:0D0A","tag":"0D0A","title":"v Cech cohomology and cohomology · Lemma 0D0A","summary":"Let (X, O_X) be a ringed space. Let I be a set. For i ∈ I let F_i be an O_X-module. Let U ⊂ X be open. The canonical map H^p(U, ∏_i ∈ I F_i) → ∏_i ∈ I H^p(U, F_i) is an isomorphism for p = 0 and injective for p = 1.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $I$ be a set.\nFor $i \\in I$ let  $\\mathcal{F}_i$ be an $\\mathcal{O}_X$-module.\nLet $U \\subset X$ be open. The canonical map\n$$\nH^p(U, \\prod\\nolimits_{i \\in I} \\mathcal{F}_i)\n\\longrightarrow\n\\prod\\nolimits_{i \\in I} H^p(U, \\mathcal{F}_i)\n$$\nis an isomorphism for $p = 0$ and injective for $p = 1$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0A","source_file":"cohomology.tex","source_line":1835,"source_end_line":1846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1835-L1846","statement_sha256":"b584620501b195b26123622766f2d35dfba7f0c0e01e696ecd06255318e7433e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4181,"rank":4181,"depth":21,"x":1152.388,"y":650.753,"cluster":"sheaf-cohomology"},{"id":"stacks:09SW","tag":"09SW","title":"Flasque sheaves · Definition 09SW","summary":"Let X be a topological space. We say a presheaf of sets F is flasque or flabby if for every U ⊂ V open in X the restriction map F(V) → F(U) is surjective.","statement_latex":"Let $X$ be a topological space. We say a presheaf of sets\n$\\mathcal{F}$ is {\\it flasque} or {\\it flabby} if for every\n$U \\subset V$ open in $X$ the restriction map\n$\\mathcal{F}(V) \\to \\mathcal{F}(U)$ is surjective.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flasque sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SW","source_file":"cohomology.tex","source_line":1887,"source_end_line":1893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1887-L1893","statement_sha256":"350531deab953fd9a95076feb0756a115bcaf112956f6c07084b3c2676c39d29","origin":"The Stacks Project","memory_eligible":false,"source_rank":4182,"rank":4182,"depth":0,"x":1137.122,"y":714.808,"cluster":"sheaf-cohomology"},{"id":"stacks:09SX","tag":"09SX","title":"Flasque sheaves · Lemma 09SX","summary":"Let (X, O_X) be a ringed space. Then any injective O_X-module is flasque.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nThen any injective $\\mathcal{O}_X$-module is flasque.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flasque sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SX","source_file":"cohomology.tex","source_line":1902,"source_end_line":1906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1902-L1906","statement_sha256":"c6db3688fa87afb607f898361d8cf87e287d59b6fc98ce95d729c3f579e62d32","origin":"The Stacks Project","memory_eligible":false,"source_rank":4183,"rank":4183,"depth":2,"x":1096.25,"y":658.044,"cluster":"sheaf-cohomology"},{"id":"stacks:09SY","tag":"09SY","title":"Flasque sheaves · Lemma 09SY","summary":"Let (X, O_X) be a ringed space. Any flasque O_X-module is acyclic for RΓ(X, -) as well as RΓ(U, -) for any open U of X.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Any flasque $\\mathcal{O}_X$-module\nis acyclic for $R\\Gamma(X, -)$ as well as $R\\Gamma(U, -)$ for any\nopen $U$ of $X$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flasque sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SY","source_file":"cohomology.tex","source_line":1912,"source_end_line":1917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1912-L1917","statement_sha256":"f69b3f32287876275d18071ef79c16a6a8df0004e9afbe8acdda44db6417dae6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4184,"rank":4184,"depth":3,"x":1173.173,"y":676.996,"cluster":"sheaf-cohomology"},{"id":"stacks:09SZ","tag":"09SZ","title":"Flasque sheaves · Lemma 09SZ","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. Let U : U = ⋃ U_i be an open covering. If F is flasque, then checkH^p(U, F) = 0 for p > 0.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nLet $\\mathcal{U} : U = \\bigcup U_i$ be an open covering.\nIf $\\mathcal{F}$ is flasque, then\n$\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for $p > 0$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flasque sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SZ","source_file":"cohomology.tex","source_line":1963,"source_end_line":1970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1963-L1970","statement_sha256":"874f5f88160cd3992a7f9516298cefe306b15e26b598608a8ee29a2f2a574a6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4185,"rank":4185,"depth":22,"x":1100.159,"y":707.096,"cluster":"sheaf-cohomology"},{"id":"stacks:09T0","tag":"09T0","title":"Flasque sheaves · Lemma 09T0","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let F be a sheaf of O_X-modules. If F is flasque, then R^pf_*F = 0 for p > 0.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism\nof ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}$ is flasque, then $R^pf_*\\mathcal{F} = 0$ for $p > 0$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flasque sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09T0","source_file":"cohomology.tex","source_line":1980,"source_end_line":1985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1980-L1985","statement_sha256":"d04b9a5e02f1485d44d88226bc4cfab9bbc682ead4f1d2ff8f3264a1d00b78e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4186,"rank":4186,"depth":4,"x":1130.217,"y":642.571,"cluster":"sheaf-cohomology"},{"id":"stacks:0A36","tag":"0A36","title":"Flasque sheaves · Lemma 0A36","summary":"Let X be a topological space. Let F be an abelian sheaf on X. Let U : U = ⋃_i ∈ I U_i be an open covering. Assume the restriction mappings F(U) → F(U') are surjective for U' an arbitrary union of opens of the form U_i_0 … i_p. Then checkH^p(U, F) vanishes for p > 0.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian sheaf\non $X$. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an\nopen covering. Assume the restriction mappings\n$\\mathcal{F}(U) \\to \\mathcal{F}(U')$ are surjective\nfor $U'$ an arbitrary union of opens of the form $U_{i_0 \\ldots i_p}$.\nThen $\\check{H}^p(\\mathcal{U}, \\mathcal{F})$\nvanishes for $p > 0$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flasque sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A36","source_file":"cohomology.tex","source_line":1998,"source_end_line":2007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L1998-L2007","statement_sha256":"e9329aff02f4578d99cd7155a2a140269ffe3cfbc46e7658d0844dbad2b300a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4187,"rank":4187,"depth":23,"x":1160.346,"y":708.099,"cluster":"sheaf-cohomology"},{"id":"stacks:01EZ","tag":"01EZ","title":"The Leray spectral sequence · Lemma 01EZ","summary":"Let f : X → Y be a morphism of ringed spaces. There is a commutative diagram xymatrix D^+(X) ar[rr]_-RΓ(X, -) ar[d]_Rf_* & & D^+(O_X(X)) ar[d]^restriction D^+(Y) ar[rr]^-RΓ(Y, -) & & D^+(O_Y(Y)) More generally for any V ⊂ Y open and U = f^-1(V) there is a commutative diagram xymatrix D^+(X) ar[rr]_-RΓ(U, -) ar[d]_Rf_* & & D^+(O_X(U)) ar[d]^restriction D^+(Y) ar[rr]^-RΓ(V, -) & & D^+(O_Y(V)) See also Remark [Tag 01F0] for more explanation.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nThere is a commutative diagram\n$$\n\\xymatrix{\nD^{+}(X) \\ar[rr]_-{R\\Gamma(X, -)} \\ar[d]_{Rf_*} & &\nD^{+}(\\mathcal{O}_X(X)) \\ar[d]^{\\text{restriction}} \\\\\nD^{+}(Y) \\ar[rr]^-{R\\Gamma(Y, -)} & &\nD^{+}(\\mathcal{O}_Y(Y))\n}\n$$\nMore generally for any $V \\subset Y$ open and $U = f^{-1}(V)$ there\nis a commutative diagram\n$$\n\\xymatrix{\nD^{+}(X) \\ar[rr]_-{R\\Gamma(U, -)} \\ar[d]_{Rf_*} & &\nD^{+}(\\mathcal{O}_X(U)) \\ar[d]^{\\text{restriction}} \\\\\nD^{+}(Y) \\ar[rr]^-{R\\Gamma(V, -)} & &\nD^{+}(\\mathcal{O}_Y(V))\n}\n$$\nSee also Remark \\ref{remark-elucidate-lemma} for more explanation.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01EZ","source_file":"cohomology.tex","source_line":2091,"source_end_line":2114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2091-L2114","statement_sha256":"99042fe3ee5084d941636fe2db35541127d2aef21481c1979d9fc2ae4c747273","origin":"The Stacks Project","memory_eligible":false,"source_rank":4188,"rank":4188,"depth":23,"x":1084.432,"y":676.453,"cluster":"sheaf-cohomology"},{"id":"stacks:01F1","tag":"01F1","title":"The Leray spectral sequence · Lemma 01F1","summary":"Let X be a ringed space. Let F be an O_X-module. • The cohomology groups H^i(U, F) for U ⊂ X open of F computed as an O_X-module, or computed as an abelian sheaf are identical. • Let f : X → Y be a morphism of ringed spaces. The higher direct images R^if_*F of F computed as an O_X-module, or computed as an abelian sheaf are identical. There are similar statements in the case of bounded below complexes of O_X-modules.","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item The cohomology groups $H^i(U, \\mathcal{F})$ for $U \\subset X$ open\nof $\\mathcal{F}$ computed as an $\\mathcal{O}_X$-module, or computed as an\nabelian sheaf are identical.\n\\item Let $f : X \\to Y$ be a morphism of ringed spaces.\nThe higher direct images $R^if_*\\mathcal{F}$ of $\\mathcal{F}$\ncomputed as an $\\mathcal{O}_X$-module, or computed as an abelian sheaf\nare identical.\n\\end{enumerate}\nThere are similar statements in the case of bounded below\ncomplexes of $\\mathcal{O}_X$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01F1","source_file":"cohomology.tex","source_line":2174,"source_end_line":2189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2174-L2189","statement_sha256":"0e5377567b124799a98a96f310ffd49680091c3c8ac5859fdeb976ca18f5e0ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":4189,"rank":4189,"depth":1,"x":1166.923,"y":656.46,"cluster":"sheaf-cohomology"},{"id":"stacks:01F2","tag":"01F2","title":"Leray spectral sequence · Lemma 01F2","summary":"Let f : X → Y be a morphism of ringed spaces. Let F^bullet be a bounded below complex of O_X-modules. There is a spectral sequence E_2^p, q = H^p(Y, R^qf_*(F^bullet)) converging to H^p + q(X, F^bullet).","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $\\mathcal{F}^\\bullet$ be\na bounded below complex of $\\mathcal{O}_X$-modules.\nThere is a spectral sequence\n$$\nE_2^{p, q} = H^p(Y, R^qf_*(\\mathcal{F}^\\bullet))\n$$\nconverging to $H^{p + q}(X, \\mathcal{F}^\\bullet)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01F2","source_file":"cohomology.tex","source_line":2212,"source_end_line":2222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2212-L2222","statement_sha256":"a2351186a5eb123da171a9fca9b64840c27bc92323877e332d59d7514dde36df","origin":"The Stacks Project","memory_eligible":false,"source_rank":4190,"rank":4190,"depth":24,"x":1121.599,"y":718.79,"cluster":"sheaf-cohomology"},{"id":"stacks:01F4","tag":"01F4","title":"The Leray spectral sequence · Lemma 01F4","summary":"Let f : X → Y be a morphism of ringed spaces. Let F be an O_X-module. • If R^qf_*F = 0 for q > 0, then H^p(X, F) = H^p(Y, f_*F) for all p. • If H^p(Y, R^qf_*F) = 0 for all q and p > 0, then H^q(X, F) = H^0(Y, R^qf_*F) for all q.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If $R^qf_*\\mathcal{F} = 0$ for $q > 0$, then\n$H^p(X, \\mathcal{F}) = H^p(Y, f_*\\mathcal{F})$ for all $p$.\n\\item If $H^p(Y, R^qf_*\\mathcal{F}) = 0$ for all $q$ and $p > 0$, then\n$H^q(X, \\mathcal{F}) = H^0(Y, R^qf_*\\mathcal{F})$ for all $q$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01F4","source_file":"cohomology.tex","source_line":2254,"source_end_line":2264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2254-L2264","statement_sha256":"802e81bb44d70ebf91a471f458b936d7bc18c0225b8698a80bad5d3298311d30","origin":"The Stacks Project","memory_eligible":false,"source_rank":4191,"rank":4191,"depth":0,"x":1104.695,"y":646.221,"cluster":"sheaf-cohomology"},{"id":"stacks:01F5","tag":"01F5","title":"The Leray spectral sequence · Lemma 01F5","summary":"The total derived functor of a composition is the composition of the total derived functors. Let f : X → Y and g : Y → Z be morphisms of ringed spaces. In this case Rg_* ∘ Rf_* = R(g ∘ f)_* as functors from D^+(X) → D^+(Z).","statement_latex":"\\begin{slogan}\nThe total derived functor of a composition is the\ncomposition of the total derived functors.\n\\end{slogan}\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.\nIn this case $Rg_* \\circ Rf_* = R(g \\circ f)_*$ as functors\nfrom $D^{+}(X) \\to D^{+}(Z)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01F5","source_file":"cohomology.tex","source_line":2272,"source_end_line":2281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2272-L2281","statement_sha256":"594e79ef6f7c0d0d5001c0d581d4a7ee9e0b4511038d6b41dbd1f366307d5faf","origin":"The Stacks Project","memory_eligible":false,"source_rank":4192,"rank":4192,"depth":23,"x":1176.372,"y":690.675,"cluster":"sheaf-cohomology"},{"id":"stacks:01F6","tag":"01F6","title":"Relative Leray spectral sequence · Lemma 01F6","summary":"Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Let F be an O_X-module. There is a spectral sequence with E_2^p, q = R^pg_*(R^qf_*F) converging to R^p + q(g ∘ f)_*F. This spectral sequence is functorial in F, and there is a version for bounded below complexes of O_X-modules.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nThere is a spectral sequence with\n$$\nE_2^{p, q} = R^pg_*(R^qf_*\\mathcal{F})\n$$\nconverging to $R^{p + q}(g \\circ f)_*\\mathcal{F}$.\nThis spectral sequence is functorial in $\\mathcal{F}$, and there\nis a version for bounded below complexes of $\\mathcal{O}_X$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01F6","source_file":"cohomology.tex","source_line":2295,"source_end_line":2306,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2295-L2306","statement_sha256":"e0543fb01b76ac5e6615fbe3ae155231b1ff5f58e89dfecfcca32614d176dbe9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4193,"rank":4193,"depth":24,"x":1086.722,"y":698.656,"cluster":"sheaf-cohomology"},{"id":"stacks:01F8","tag":"01F8","title":"Functoriality of cohomology · Lemma 01F8","summary":"Let f : X → Y be a morphism of ringed spaces. Let G^bullet, resp. F^bullet be a bounded below complex of O_Y-modules, resp. O_X-modules. Let φ : G^bullet → f_*F^bullet be a morphism of complexes. There is a canonical morphism G^bullet → Rf_*(F^bullet) in D^+(Y). Moreover this construction is functorial in the triple (G^bullet, F^bullet, φ).","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $\\mathcal{G}^\\bullet$, resp.\\ $\\mathcal{F}^\\bullet$ be\na bounded below complex of $\\mathcal{O}_Y$-modules,\nresp.\\ $\\mathcal{O}_X$-modules. Let\n$\\varphi : \\mathcal{G}^\\bullet \\to f_*\\mathcal{F}^\\bullet$\nbe a morphism of complexes. There is a canonical morphism\n$$\n\\mathcal{G}^\\bullet\n\\longrightarrow\nRf_*(\\mathcal{F}^\\bullet)\n$$\nin $D^{+}(Y)$. Moreover this construction is functorial in the triple\n$(\\mathcal{G}^\\bullet, \\mathcal{F}^\\bullet, \\varphi)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Functoriality of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01F8","source_file":"cohomology.tex","source_line":2330,"source_end_line":2345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2330-L2345","statement_sha256":"24154603f0e75e907cc3e273075322ccaadfe352872ff6b5252e2c0d46e8fe4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4194,"rank":4194,"depth":0,"x":1147.103,"y":641.249,"cluster":"sheaf-cohomology"},{"id":"stacks:01FD","tag":"01FD","title":"Refinements and v Cech cohomology · Lemma 01FD","summary":"Let f : X → Y be a morphism of ringed spaces. Let φ : f^*G → F be an f-map from an O_Y-module G to an O_X-module F. Let U : X = ⋃_i ∈ I U_i and V : Y = ⋃_j ∈ J V_j be open coverings. Assume that U is a refinement of f^-1V : X = ⋃_j ∈ J f^-1(V_j). In this case there exists a commutative diagram xymatrix checkC^bullet(U, F) ar[r] & RΓ(X, F) checkC^bullet(V, G) ar[r] ar[u]^γ & RΓ(Y, G) ar[u] in D^+(O_X(X)) with horizontal arrows given by Lemma [Tag 01EQ] and right vertical…","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $\\varphi : f^*\\mathcal{G} \\to \\mathcal{F}$ be an $f$-map\nfrom an $\\mathcal{O}_Y$-module $\\mathcal{G}$ to an\n$\\mathcal{O}_X$-module $\\mathcal{F}$.\nLet $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ and\n$\\mathcal{V} : Y = \\bigcup_{j \\in J} V_j$ be open coverings.\nAssume that $\\mathcal{U}$ is a refinement of\n$f^{-1}\\mathcal{V} : X = \\bigcup_{j \\in J} f^{-1}(V_j)$.\nIn this case there exists a commutative diagram\n$$\n\\xymatrix{\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\ar[r] &\nR\\Gamma(X, \\mathcal{F}) \\\\\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{G}) \\ar[r]\n\\ar[u]^\\gamma &\nR\\Gamma(Y, \\mathcal{G}) \\ar[u]\n}\n$$\nin $D^{+}(\\mathcal{O}_X(X))$ with horizontal arrows given by\nLemma \\ref{lemma-cech-cohomology} and right vertical arrow by\n(\\ref{equation-functorial-derived}).\nIn particular we get commutative diagrams of cohomology groups\n$$\n\\xymatrix{\n\\check{H}^p(\\mathcal{U}, \\mathcal{F}) \\ar[r] &\nH^p(X, \\mathcal{F}) \\\\\n\\check{H}^p(\\mathcal{V}, \\mathcal{G}) \\ar[r]\n\\ar[u]^\\gamma &\nH^p(Y, \\mathcal{G}) \\ar[u]\n}\n$$\nwhere the right vertical arrow is (\\ref{equation-functorial})","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Refinements and v Cech cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01FD","source_file":"cohomology.tex","source_line":2519,"source_end_line":2553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2519-L2553","statement_sha256":"71ff013923fa91eb626a3227ff6fe5ce949b225dbf9c3df77463db166cbfdbe1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4195,"rank":4195,"depth":22,"x":1148.758,"y":718.705,"cluster":"sheaf-cohomology"},{"id":"stacks:09V1","tag":"09V1","title":"Cohomology on Hausdorff quasi-compact spaces · Lemma 09V1","summary":"Let X be a topological space. Let F be an abelian sheaf. Then the map checkH^1(X, F) → H^1(X, F) defined in ([Tag 09UZ]) is an isomorphism.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian sheaf. Then\nthe map $\\check{H}^1(X, \\mathcal{F}) \\to H^1(X, \\mathcal{F})$ defined\nin (\\ref{equation-cech-to-cohomology}) is an isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology on Hausdorff quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09V1","source_file":"cohomology.tex","source_line":2642,"source_end_line":2647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2642-L2647","statement_sha256":"019adb2aa4c89f976aafd1b125d4ff4375ccd2d11f7ba7cc6a2c3397434f80b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4196,"rank":4196,"depth":22,"x":1084.552,"y":661.908,"cluster":"sheaf-cohomology"},{"id":"stacks:09V2","tag":"09V2","title":"Cohomology on Hausdorff quasi-compact spaces · Lemma 09V2","summary":"Let X be a Hausdorff and quasi-compact topological space. Let F be an abelian sheaf on X. Then the map checkH^n(X, F) → H^n(X, F) defined in ([Tag 09UZ]) is an isomorphism for all n.","statement_latex":"Let $X$ be a Hausdorff and quasi-compact topological space. Let\n$\\mathcal{F}$ be an abelian sheaf on $X$. Then\nthe map $\\check{H}^n(X, \\mathcal{F}) \\to H^n(X, \\mathcal{F})$ defined\nin (\\ref{equation-cech-to-cohomology}) is an isomorphism for\nall $n$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology on Hausdorff quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09V2","source_file":"cohomology.tex","source_line":2666,"source_end_line":2673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2666-L2673","statement_sha256":"b1b23a91ffd90ecfeb25c69471e4b8690e651b0dea0460f4ac1fcdfd2067954e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4197,"rank":4197,"depth":23,"x":1178.575,"y":667.42,"cluster":"sheaf-cohomology"},{"id":"stacks:09V3","tag":"09V3","title":"Cohomology on Hausdorff quasi-compact spaces · Lemma 09V3","summary":"[SGA4] Let X be a topological space. Let Z ⊂ X be a quasi-compact subset such that any two points of Z have disjoint open neighbourhoods in X. For every abelian sheaf F on X the canonical map colim H^p(U, F) → H^p(Z, F|_Z) where the colimit is over open neighbourhoods U of Z in X is an isomorphism.","statement_latex":"\\begin{reference}\n\\cite[Expose V bis, 4.1.3]{SGA4}\n\\end{reference}\nLet $X$ be a topological space. Let $Z \\subset X$ be a quasi-compact subset\nsuch that any two points of $Z$ have disjoint open neighbourhoods in $X$.\nFor every abelian sheaf $\\mathcal{F}$ on $X$ the canonical\nmap\n$$\n\\colim H^p(U, \\mathcal{F})\n\\longrightarrow\nH^p(Z, \\mathcal{F}|_Z)\n$$\nwhere the colimit is over open neighbourhoods $U$ of $Z$ in $X$\nis an isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology on Hausdorff quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09V3","source_file":"cohomology.tex","source_line":2847,"source_end_line":2863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2847-L2863","statement_sha256":"add2ed5755241d67e3c12e1201048ae5f48d1558835ba1eb4b04f831843c48d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4198,"rank":4198,"depth":24,"x":1104.029,"y":717.224,"cluster":"sheaf-cohomology"},{"id":"stacks:02N7","tag":"02N7","title":"The base change map · Lemma 02N7","summary":"Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a commutative diagram of ringed spaces. Let F^bullet be a bounded below complex of O_X-modules. Assume both g and g' are flat. Then there exists a canonical base change map g^*Rf_*F^bullet → R(f')_*(g')^*F^bullet in D^+(S').","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nS' \\ar[r]^g &\nS\n}\n$$\nbe a commutative diagram of ringed spaces.\nLet $\\mathcal{F}^\\bullet$ be a bounded below complex of\n$\\mathcal{O}_X$-modules.\nAssume both $g$ and $g'$ are flat.\nThen there exists a canonical base change map\n$$\ng^*Rf_*\\mathcal{F}^\\bullet\n\\longrightarrow\nR(f')_*(g')^*\\mathcal{F}^\\bullet\n$$\nin $D^{+}(S')$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The base change map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02N7","source_file":"cohomology.tex","source_line":2976,"source_end_line":2998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L2976-L2998","statement_sha256":"2b60c673cd2d97107f31e06fea4c142aa00376690ec70ac6f8ac923fc21b56f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4199,"rank":4199,"depth":9,"x":1119.106,"y":637.384,"cluster":"sheaf-cohomology"},{"id":"stacks:09V5","tag":"09V5","title":"Proper base change in topology · Lemma 09V5","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let y ∈ Y. Assume that • f is closed, • f is separated, and • f^-1(y) is quasi-compact. Then for E in D^+(O_X) we have (Rf_*E)_y = RΓ(f^-1(y), E|_f^-1(y)) in D^+(O_Y, y).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of\nringed spaces. Let $y \\in Y$. Assume that\n\\begin{enumerate}\n\\item $f$ is closed,\n\\item $f$ is separated, and\n\\item $f^{-1}(y)$ is quasi-compact.\n\\end{enumerate}\nThen for $E$ in $D^+(\\mathcal{O}_X)$\nwe have $(Rf_*E)_y = R\\Gamma(f^{-1}(y), E|_{f^{-1}(y)})$ in\n$D^+(\\mathcal{O}_{Y, y})$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Proper base change in topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09V5","source_file":"cohomology.tex","source_line":3058,"source_end_line":3070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3058-L3070","statement_sha256":"2ef6d8c9dacb517c3eb641f84d6ca2792d0fa711d14d9c3517d8d561f5d822e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4200,"rank":4200,"depth":25,"x":1172.729,"y":705.496,"cluster":"sheaf-cohomology"},{"id":"stacks:09V6","tag":"09V6","title":"Proper base change · Theorem 09V6","summary":"[SGA4] Consider a cartesian square of topological spaces xymatrix X' = Y' ×_Y X ar[d]_f' ar[r]_-g' & X ar[d]^f Y' ar[r]^g & Y Assume that f is proper. Let E be an object of D^+(X). Then the base change map g^-1Rf_*E → Rf'_*(g')^-1E of Lemma [Tag 02N7] is an isomorphism in D^+(Y').","statement_latex":"\\begin{reference}\n\\cite[Expose V bis, 4.1.1]{SGA4}\n\\end{reference}\nConsider a cartesian square of topological spaces\n$$\n\\xymatrix{\nX' = Y' \\times_Y X \\ar[d]_{f'} \\ar[r]_-{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nAssume that $f$ is proper.\nLet $E$ be an object of $D^+(X)$. Then the base change map\n$$\ng^{-1}Rf_*E \\longrightarrow Rf'_*(g')^{-1}E\n$$\nof Lemma \\ref{lemma-base-change-map-flat-case} is an isomorphism\nin $D^+(Y')$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Proper base change in topology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09V6","source_file":"cohomology.tex","source_line":3100,"source_end_line":3119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3100-L3119","statement_sha256":"e405ec47098d267a9109967ca126429829683a73c8c06710e4f719291a46c873","origin":"The Stacks Project","memory_eligible":false,"source_rank":4201,"rank":4201,"depth":26,"x":1077.478,"y":685.498,"cluster":"sheaf-cohomology"},{"id":"stacks:0D90","tag":"0D90","title":"Proper base change for sheaves of sets · Lemma 0D90","summary":"Consider a cartesian square of topological spaces xymatrix X' ar[d]_f' ar[r]_-g' & X ar[d]^f Y' ar[r]^g & Y Assume that f is proper. Then g^-1f_*F = f'_*(g')^-1F for any sheaf of sets F on X.","statement_latex":"Consider a cartesian square of topological spaces\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_-{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nAssume that $f$ is proper. Then\n$g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$\nfor any sheaf of sets $\\mathcal{F}$ on $X$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Proper base change in topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D90","source_file":"cohomology.tex","source_line":3142,"source_end_line":3154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3142-L3154","statement_sha256":"ed8cad2e91996c9d4b93979af6f494da3a56c97c545ce0e872c91453bc2f4559","origin":"The Stacks Project","memory_eligible":false,"source_rank":4202,"rank":4202,"depth":27,"x":1164.638,"y":645.817,"cluster":"sheaf-cohomology"},{"id":"stacks:01FF","tag":"01FF","title":"Cohomology and colimits · Lemma 01FF","summary":"Let X be a ringed space. Assume that the underlying topological space of X has the following properties: • there exists a basis of quasi-compact open subsets, and • the intersection of any two quasi-compact opens is quasi-compact. Then for any directed system (F_i, φ_ii') of sheaves of O_X-modules and for any quasi-compact open U ⊂ X the canonical map colim_i H^q(U, F_i) → H^q(U, colim_i F_i) is an isomorphism for every q ≥ 0.","statement_latex":"Let $X$ be a ringed space. Assume that the underlying topological space\nof $X$ has the following properties:\n\\begin{enumerate}\n\\item there exists a basis of quasi-compact open subsets, and\n\\item the intersection of any two quasi-compact opens is quasi-compact.\n\\end{enumerate}\nThen for any directed system $(\\mathcal{F}_i, \\varphi_{ii'})$\nof sheaves of $\\mathcal{O}_X$-modules and for any quasi-compact open\n$U \\subset X$ the canonical map\n$$\n\\colim_i H^q(U, \\mathcal{F}_i)\n\\longrightarrow\nH^q(U, \\colim_i \\mathcal{F}_i)\n$$\nis an isomorphism for every $q \\geq 0$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01FF","source_file":"cohomology.tex","source_line":3193,"source_end_line":3210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3193-L3210","statement_sha256":"e6be8449dda65f6b36f1b492de2ff61799fc115f18f4039bb9f6bbf28eb9b532","origin":"The Stacks Project","memory_eligible":false,"source_rank":4203,"rank":4203,"depth":23,"x":1131.966,"y":725.285,"cluster":"sheaf-cohomology"},{"id":"stacks:0H7A","tag":"0H7A","title":"Cohomology and colimits · Lemma 0H7A","summary":"Let f : X → Y be a continuous map of topological spaces. Let (F_i, φ_ii') be a system of abelian sheaves on X. Set F = colim F_i. Let p ≥ 0 be an integer. Assume the set of opens V ⊂ Y such that H^p(f^-1(V), F) = colim H^p(f^-1(V), F_i) is a basis for the topology on Y. Then R^pf_*F = colim R^pf_*F_i.","statement_latex":"Let $f : X \\to Y$ be a continuous map of topological spaces.\nLet $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system of\nabelian sheaves on $X$. Set $\\mathcal{F} = \\colim \\mathcal{F}_i$.\nLet $p \\geq 0$ be an integer. Assume the set of opens $V \\subset Y$ such that\n$H^p(f^{-1}(V), \\mathcal{F}) = \\colim H^p(f^{-1}(V), \\mathcal{F}_i)$\nis a basis for the topology on $Y$. Then\n$R^pf_*\\mathcal{F} = \\colim R^pf_*\\mathcal{F}_i$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7A","source_file":"cohomology.tex","source_line":3305,"source_end_line":3314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3305-L3314","statement_sha256":"c01ec31779d387f9b8d7ad7011a35457f8553045f82e3334659eaca0e0df1530","origin":"The Stacks Project","memory_eligible":false,"source_rank":4204,"rank":4204,"depth":2,"x":1091.782,"y":647.424,"cluster":"sheaf-cohomology"},{"id":"stacks:0A37","tag":"0A37","title":"Cohomology and colimits · Lemma 0A37","summary":"In the situation discussed above. Let i ∈ Ob(I) and let U_i ⊂ X_i be quasi-compact open. Then colim_a : j → i H^p(f_a^-1(U_i), F_j) = H^p(p_i^-1(U_i), F) for all p ≥ 0. In particular we have H^p(X, F) = colim H^p(X_i, F_i).","statement_latex":"In the situation discussed above.\nLet $i \\in \\Ob(\\mathcal{I})$ and let $U_i \\subset X_i$ be quasi-compact open.\nThen\n$$\n\\colim_{a : j \\to i} H^p(f_a^{-1}(U_i), \\mathcal{F}_j) =\nH^p(p_i^{-1}(U_i), \\mathcal{F})\n$$\nfor all $p \\geq 0$. In particular we have\n$H^p(X, \\mathcal{F}) = \\colim H^p(X_i, \\mathcal{F}_i)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A37","source_file":"cohomology.tex","source_line":3359,"source_end_line":3370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3359-L3370","statement_sha256":"1a734a52930142ef65beb136701177cf05fe85f376b9f666d9bae8cb5f51a2a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4205,"rank":4205,"depth":24,"x":1184.874,"y":682.357,"cluster":"sheaf-cohomology"},{"id":"stacks:02UV","tag":"02UV","title":"Vanishing on Noetherian topological spaces · Lemma 02UV","summary":"Let i : Z → X be a closed immersion of topological spaces. For any abelian sheaf F on Z we have H^p(Z, F) = H^p(X, i_*F).","statement_latex":"Let $i : Z \\to X$ be a closed immersion of topological spaces.\nFor any abelian sheaf $\\mathcal{F}$ on $Z$ we have\n$H^p(Z, \\mathcal{F}) = H^p(X, i_*\\mathcal{F})$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Vanishing on Noetherian topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UV","source_file":"cohomology.tex","source_line":3446,"source_end_line":3451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3446-L3451","statement_sha256":"b95b6ceecdbd8a6e5f3dd74a658e79b42fa582a8bcd6f9de984bb3ac52cae6d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4206,"rank":4206,"depth":3,"x":1087.263,"y":709.662,"cluster":"sheaf-cohomology"},{"id":"stacks:02UW","tag":"02UW","title":"Vanishing on Noetherian topological spaces · Lemma 02UW","summary":"Let X be an irreducible topological space. Then H^p(X, underlineA) = 0 for all p > 0 and any abelian group A.","statement_latex":"Let $X$ be an irreducible topological space.\nThen $H^p(X, \\underline{A}) = 0$ for all $p > 0$\nand any abelian group $A$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Vanishing on Noetherian topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UW","source_file":"cohomology.tex","source_line":3461,"source_end_line":3466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3461-L3466","statement_sha256":"aced3e54cbe407e3d12e89b298dac7549b0b27c1afa7cfc53c6bf9eb0be5e421","origin":"The Stacks Project","memory_eligible":false,"source_rank":4207,"rank":4207,"depth":4,"x":1137.728,"y":633.472,"cluster":"sheaf-cohomology"},{"id":"stacks:0A38","tag":"0A38","title":"Vanishing on Noetherian topological spaces · Lemma 0A38","summary":"[Tohoku]. Let X be a topological space such that the intersection of any two quasi-compact opens is quasi-compact. Let F ⊂ underlineZ be a subsheaf generated by finitely many sections over quasi-compact opens. Then there exists a finite filtration (0) = F_0 ⊂ F_1 ⊂ … ⊂ F_n = F by abelian subsheaves such that for each 0 < i ≤ n there exists a short exact sequence 0 → j'_!underlineZ_V → j_!underlineZ_U → F_i/F_i - 1 → 0 with j : U → X and j' : V → X the inclusion of…","statement_latex":"\\begin{reference}\n\\cite[Page 168]{Tohoku}.\n\\end{reference}\nLet $X$ be a topological space such that the intersection of any\ntwo quasi-compact opens is quasi-compact. Let\n$\\mathcal{F} \\subset \\underline{\\mathbf{Z}}$\nbe a subsheaf generated by finitely many sections over quasi-compact opens.\nThen there exists a finite filtration\n$$\n(0) = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset \\ldots \\subset\n\\mathcal{F}_n = \\mathcal{F}\n$$\nby abelian subsheaves such that for each $0 < i \\leq n$\nthere exists a short exact sequence\n$$\n0 \\to j'_!\\underline{\\mathbf{Z}}_V \\to j_!\\underline{\\mathbf{Z}}_U \\to\n\\mathcal{F}_i/\\mathcal{F}_{i - 1} \\to 0\n$$\nwith $j : U \\to X$ and $j' : V \\to X$ the inclusion of quasi-compact opens\ninto $X$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Vanishing on Noetherian topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A38","source_file":"cohomology.tex","source_line":3479,"source_end_line":3501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3479-L3501","statement_sha256":"89f68ca1222d01a35f105e2b0a04e57da8735b50afd159e39b9d0976a899b660","origin":"The Stacks Project","memory_eligible":false,"source_rank":4208,"rank":4208,"depth":0,"x":1161.992,"y":719.028,"cluster":"sheaf-cohomology"},{"id":"stacks:02UX","tag":"02UX","title":"Vanishing on Noetherian topological spaces · Lemma 02UX","summary":"This is a special case of [Tohoku]. Let X be a topological space. Let d ≥ 0 be an integer. Assume • X is quasi-compact, • the quasi-compact opens form a basis for X, and • the intersection of two quasi-compact opens is quasi-compact. • H^p(X, j_!underlineZ_U) = 0 for all p > d and any quasi-compact open j : U → X. Then H^p(X, F) = 0 for all p > d and any abelian sheaf F on X.","statement_latex":"\\begin{reference}\nThis is a special case of \\cite[Proposition 3.6.1]{Tohoku}.\n\\end{reference}\nLet $X$ be a topological space. Let $d \\geq 0$ be an integer. Assume\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item the quasi-compact opens form a basis for $X$, and\n\\item the intersection of two quasi-compact opens is quasi-compact.\n\\item $H^p(X, j_!\\underline{\\mathbf{Z}}_U) = 0$ for all $p > d$\nand any quasi-compact open $j : U \\to X$.\n\\end{enumerate}\nThen $H^p(X, \\mathcal{F}) = 0$ for all $p > d$\nand any abelian sheaf $\\mathcal{F}$ on $X$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Vanishing on Noetherian topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UX","source_file":"cohomology.tex","source_line":3530,"source_end_line":3545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3530-L3545","statement_sha256":"9ad3203130bcdf565e8544901daf1405676d8cdaf4d199ba99d2ff4821040346","origin":"The Stacks Project","memory_eligible":false,"source_rank":4209,"rank":4209,"depth":24,"x":1074.558,"y":669.284,"cluster":"sheaf-cohomology"},{"id":"stacks:02UY","tag":"02UY","title":"Vanishing on Noetherian topological spaces · Lemma 02UY","summary":"Let X be an irreducible topological space. Let H ⊂ underlineZ be an abelian subsheaf of the constant sheaf. Then there exists a nonempty open U ⊂ X such that H|_U = underlinedZ_U for some d ∈ Z.","statement_latex":"Let $X$ be an irreducible topological space.\nLet $\\mathcal{H} \\subset \\underline{\\mathbf{Z}}$ be\nan abelian subsheaf of the constant sheaf.\nThen there exists a nonempty open $U \\subset X$ such\nthat $\\mathcal{H}|_U = \\underline{d\\mathbf{Z}}_U$\nfor some $d \\in \\mathbf{Z}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Vanishing on Noetherian topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UY","source_file":"cohomology.tex","source_line":3634,"source_end_line":3642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3634-L3642","statement_sha256":"ad77e1dd1c27934c4197bbd4ef58862ce2a71f81df6c75cd67bf89103afc5dee","origin":"The Stacks Project","memory_eligible":false,"source_rank":4210,"rank":4210,"depth":5,"x":1179.913,"y":656.246,"cluster":"sheaf-cohomology"},{"id":"stacks:02UZ","tag":"02UZ","title":"Grothendieck · Proposition 02UZ","summary":"[Tohoku]. Let X be a Noetherian topological space. If dim(X) ≤ d, then H^p(X, F) = 0 for all p > d and any abelian sheaf F on X.","statement_latex":"\\begin{reference}\n\\cite[Theorem 3.6.5]{Tohoku}.\n\\end{reference}\nLet $X$ be a Noetherian topological space.\nIf $\\dim(X) \\leq d$, then $H^p(X, \\mathcal{F}) = 0$\nfor all $p > d$ and any abelian sheaf $\\mathcal{F}$\non $X$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Vanishing on Noetherian topological spaces","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UZ","source_file":"cohomology.tex","source_line":3665,"source_end_line":3674,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3665-L3674","statement_sha256":"0a6bc84a86f870b4369042c617dc8d59dbf19b700bfe4b4eca9e84afea0ab2da","origin":"The Stacks Project","memory_eligible":false,"source_rank":4211,"rank":4211,"depth":25,"x":1112.152,"y":726.214,"cluster":"sheaf-cohomology"},{"id":"stacks:0A3A","tag":"0A3A","title":"Cohomology with support in a closed subset · Lemma 0A3A","summary":"Let i : Z → X be the inclusion of a closed subset. Let I be an injective abelian sheaf on X. Then H_Z(I) is an injective abelian sheaf on Z.","statement_latex":"Let $i : Z \\to X$ be the inclusion of a closed subset.\nLet $\\mathcal{I}$ be an injective abelian sheaf on $X$.\nThen $\\mathcal{H}_Z(\\mathcal{I})$ is an injective abelian sheaf on $Z$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3A","source_file":"cohomology.tex","source_line":3839,"source_end_line":3844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3839-L3844","statement_sha256":"2fde827333dffc0d375c06d3c1bb84cdca2515b1bc55b225cced9787812a678f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4212,"rank":4212,"depth":8,"x":1105.803,"y":635.438,"cluster":"sheaf-cohomology"},{"id":"stacks:0A3D","tag":"0A3D","title":"Cohomology on spectral spaces · Lemma 0A3D","summary":"Let X be a spectral space. Let F be an abelian sheaf on X. Let E ⊂ X be a quasi-compact subset. Let W ⊂ X be the set of points of X which specialize to a point of E. • H^p(W, F|_W) = colim H^p(U, F) where the colimit is over quasi-compact open neighbourhoods of E, • H^p(W setminus E, F|_W setminus E) = colim H^p(U setminus E, F|_U setminus E) if E is a constructible subset.","statement_latex":"Let $X$ be a spectral space. Let $\\mathcal{F}$ be an abelian sheaf on $X$.\nLet $E \\subset X$ be a quasi-compact subset. Let $W \\subset X$ be the set of\npoints of $X$ which specialize to a point of $E$.\n\\begin{enumerate}\n\\item $H^p(W, \\mathcal{F}|_W) = \\colim H^p(U, \\mathcal{F})$\nwhere the colimit is over quasi-compact open neighbourhoods of $E$,\n\\item $H^p(W \\setminus E, \\mathcal{F}|_{W \\setminus E}) =\n\\colim H^p(U \\setminus E, \\mathcal{F}|_{U \\setminus E})$\nif $E$ is a constructible subset.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology on spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3D","source_file":"cohomology.tex","source_line":3869,"source_end_line":3881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3869-L3881","statement_sha256":"e3401b6240685cfa7fe53691324ffb20149927e5cce7704d53029d063e076ba3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4213,"rank":4213,"depth":25,"x":1184.104,"y":699.281,"cluster":"sheaf-cohomology"},{"id":"stacks:0A3E","tag":"0A3E","title":"Cohomology on spectral spaces · Lemma 0A3E","summary":"Let f : X → Y be a spectral map of spectral spaces. Let y ∈ Y. Let E ⊂ Y be the set of points specializing to y. Let F be an abelian sheaf on X. Then (R^pf_*F)_y = H^p(f^-1(E), F|_f^-1(E)).","statement_latex":"Let $f : X \\to Y$ be a spectral map of spectral spaces. Let $y \\in Y$.\nLet $E \\subset Y$ be the set of points specializing to $y$.\nLet $\\mathcal{F}$ be an abelian sheaf on $X$.\nThen $(R^pf_*\\mathcal{F})_y = H^p(f^{-1}(E), \\mathcal{F}|_{f^{-1}(E)})$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology on spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3E","source_file":"cohomology.tex","source_line":3893,"source_end_line":3899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3893-L3899","statement_sha256":"4eb3e7cdba870d509fe7728bbb69c74c697e3875ea1b099e2571055ea79fa50b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4214,"rank":4214,"depth":25,"x":1074.166,"y":696.611,"cluster":"sheaf-cohomology"},{"id":"stacks:0A3F","tag":"0A3F","title":"Cohomology on spectral spaces · Lemma 0A3F","summary":"Let X be a profinite topological space. Then H^q(X, F) = 0 for all q > 0 and all abelian sheaves F.","statement_latex":"Let $X$ be a profinite topological space. Then $H^q(X, \\mathcal{F}) = 0$\nfor all $q > 0$ and all abelian sheaves $\\mathcal{F}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology on spectral spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3F","source_file":"cohomology.tex","source_line":3916,"source_end_line":3920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3916-L3920","statement_sha256":"103f81be214e13834df3cc76dc24b8ec1aab4ce4db404827bc963bad478d2f9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4215,"rank":4215,"depth":5,"x":1158.027,"y":635.731,"cluster":"sheaf-cohomology"},{"id":"stacks:0A3G","tag":"0A3G","title":"Cohomology on spectral spaces · Proposition 0A3G","summary":"Part (1) is the main theorem of [Scheiderer]. Let X be a spectral space of Krull dimension d. Let F be an abelian sheaf on X. • H^q(X, F) = 0 for q > d, • H^d(X, F) → H^d(U, F) is surjective for every quasi-compact open U ⊂ X, • H^q_Z(X, F) = 0 for q > d and any constructible closed subset Z ⊂ X.","statement_latex":"\\begin{reference}\nPart (1) is the main theorem of \\cite{Scheiderer}.\n\\end{reference}\nLet $X$ be a spectral space of Krull dimension $d$.\nLet $\\mathcal{F}$ be an abelian sheaf on $X$.\n\\begin{enumerate}\n\\item $H^q(X, \\mathcal{F}) = 0$ for $q > d$,\n\\item $H^d(X, \\mathcal{F}) \\to H^d(U, \\mathcal{F})$ is surjective\nfor every quasi-compact open $U \\subset X$,\n\\item $H^q_Z(X, \\mathcal{F}) = 0$ for $q > d$ and any constructible\nclosed subset $Z \\subset X$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology on spectral spaces","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3G","source_file":"cohomology.tex","source_line":3939,"source_end_line":3953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L3939-L3953","statement_sha256":"a9bc786c36c715b330231397684317bdc7240f12f90ec4383b5f2e09866ddc30","origin":"The Stacks Project","memory_eligible":false,"source_rank":4216,"rank":4216,"depth":26,"x":1145.046,"y":728.915,"cluster":"sheaf-cohomology"},{"id":"stacks:01FH","tag":"01FH","title":"The alternating v Cech complex · Definition 01FH","summary":"Let X be a topological space. Let U : U = ⋃_i ∈ I U_i be an open covering. Let F be an abelian presheaf on X. The complex checkC_alt^bullet(U, F) is the alternating v Cech complex associated to F and the open covering U.","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$\nbe an open covering. Let $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe complex $\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it alternating {\\v C}ech complex} associated to $\\mathcal{F}$ and the\nopen covering $\\mathcal{U}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The alternating v Cech complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01FH","source_file":"cohomology.tex","source_line":4102,"source_end_line":4109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L4102-L4109","statement_sha256":"f09f5ee99b905993979ec6a4d97fc10e1e6cd6c4747545c87f36efbe099abce1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4217,"rank":4217,"depth":0,"x":1079.193,"y":652.263,"cluster":"sheaf-cohomology"},{"id":"stacks:01FI","tag":"01FI","title":"The alternating v Cech complex · Definition 01FI","summary":"Let X be a topological space. Let U : U = ⋃_i ∈ I U_i be an open covering. Assume given a total ordering on I. Let F be an abelian presheaf on X. The complex checkC_ord^bullet(U, F) is the ordered v Cech complex associated to F, the open covering U and the given total ordering on I.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume given a total ordering on $I$.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThe complex $\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it ordered {\\v C}ech complex} associated to $\\mathcal{F}$, the\nopen covering $\\mathcal{U}$ and the given total ordering on $I$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The alternating v Cech complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01FI","source_file":"cohomology.tex","source_line":4152,"source_end_line":4161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L4152-L4161","statement_sha256":"50ebf1204ceeb35aecb826f9f05f6be8745cda8196b5875e11dafea20e3b5481","origin":"The Stacks Project","memory_eligible":false,"source_rank":4218,"rank":4218,"depth":0,"x":1190.21,"y":671.562,"cluster":"sheaf-cohomology"},{"id":"stacks:01FJ","tag":"01FJ","title":"The alternating v Cech complex · Lemma 01FJ","summary":"Let X be a topological space. Let U : U = ⋃_i ∈ I U_i be an open covering. Assume I comes equipped with a total ordering. The map c is a morphism of complexes. In fact it induces an isomorphism c : checkC_ord^bullet(U, F) → checkC_alt^bullet(U, F) of complexes.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume $I$ comes equipped with a total ordering.\nThe map $c$ is a morphism of complexes. In fact it induces\nan isomorphism\n$$\nc : \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})\n\\to \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nof complexes.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01FJ","source_file":"cohomology.tex","source_line":4193,"source_end_line":4205,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L4193-L4205","statement_sha256":"520c2bf048c6e4b693e1e8513f626362a8ffaa157892fea6ea53001a4eba4928","origin":"The Stacks Project","memory_eligible":false,"source_rank":4219,"rank":4219,"depth":0,"x":1092.115,"y":720.68,"cluster":"sheaf-cohomology"},{"id":"stacks:01FK","tag":"01FK","title":"The alternating v Cech complex · Lemma 01FK","summary":"Let X be a topological space. Let U : U = ⋃_i ∈ I U_i be an open covering. Assume I comes equipped with a total ordering. The map π : checkC^bullet(U, F) → checkC_ord^bullet(U, F) is a morphism of complexes. It induces an isomorphism π : checkC_alt^bullet(U, F) → checkC_ord^bullet(U, F) of complexes which is a left inverse to the morphism c.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume $I$ comes equipped with a total ordering.\nThe map $\\pi : \\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})\n\\to \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis a morphism of complexes. It induces an isomorphism\n$$\n\\pi : \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})\n\\to \\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nof complexes which is a left inverse to the morphism $c$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01FK","source_file":"cohomology.tex","source_line":4225,"source_end_line":4238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L4225-L4238","statement_sha256":"bb22add5b640607003297f0d70fdde544990169c26de2507a716879c6b83224d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4220,"rank":4220,"depth":0,"x":1125.188,"y":628.139,"cluster":"sheaf-cohomology"},{"id":"stacks:01FM","tag":"01FM","title":"The alternating v Cech complex · Lemma 01FM","summary":"Let X be a topological space. Let U : U = ⋃_i ∈ I U_i be an open covering. Assume I comes equipped with a total ordering. The map c ∘ π is homotopic to the identity on checkC^bullet(U, F). In particular the inclusion map checkC_alt^bullet(U, F) → checkC^bullet(U, F) is a homotopy equivalence.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering.\nAssume $I$ comes equipped with a total ordering.\nThe map $c \\circ \\pi$ is homotopic to the identity on\n$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$.\nIn particular the inclusion map\n$\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}) \\to\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis a homotopy equivalence.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01FM","source_file":"cohomology.tex","source_line":4262,"source_end_line":4273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L4262-L4273","statement_sha256":"0f0107f2e047eb46649ff2073352b1e9167244b9637ea0c40b36007745ff18a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4221,"rank":4221,"depth":1,"x":1175.572,"y":715.761,"cluster":"sheaf-cohomology"},{"id":"stacks:0G6T","tag":"0G6T","title":"The alternating v Cech complex · Lemma 0G6T","summary":"Let X be a topological space. Let F be an abelian presheaf on X. Let U : U = ⋃_i ∈ I U_i be an open covering. If U_i = U for some i ∈ I, then the extended alternating v Cech complex F(U) → checkC_alt^bullet(U, F) obtained by putting F(U) in degree -1 with differential given by the canonical map of F(U) into checkC^0(U, F) is homotopy equivalent to 0. Similarly, for any total ordering on I the extended ordered v Cech complex F(U) → checkC_ord^bullet(U, F) is homotopy…","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{F}$ be an abelian presheaf on $X$.\nLet $\\mathcal{U} : U = \\bigcup_{i \\in I} U_i$ be an open covering. If\n$U_i = U$ for some $i \\in I$, then the extended alternating {\\v C}ech complex\n$$\n\\mathcal{F}(U) \\to \\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nobtained by putting $\\mathcal{F}(U)$ in degree $-1$ with differential given by\nthe canonical map of $\\mathcal{F}(U)$ into\n$\\check{\\mathcal{C}}^0(\\mathcal{U}, \\mathcal{F})$\nis homotopy equivalent to $0$. Similarly, for any total ordering on $I$\nthe extended ordered {\\v C}ech complex\n$$\n\\mathcal{F}(U) \\to\n\\check{\\mathcal{C}}_{ord}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nis homotopy equivalent to $0$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"The alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6T","source_file":"cohomology.tex","source_line":4455,"source_end_line":4473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L4455-L4473","statement_sha256":"88fc0761c727469d5228b40dc53f73988e6f0f373e60ab661559dca1b509205c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4222,"rank":4222,"depth":2,"x":1067.204,"y":679.485,"cluster":"sheaf-cohomology"},{"id":"stacks:02FU","tag":"02FU","title":"Alternative view of the v Cech complex · Lemma 02FU","summary":"Let X be a ringed space. Let U : X = ⋃_i ∈ I U_i be an open covering of X. Let F be an O_X-module. Denote F_i_0 … i_p the restriction of F to U_i_0 … i_p. There exists a complex C^bullet(U, F) of O_X-modules with C^p(U, F) = ∏_i_0 … i_p (j_i_0 … i_p)_* F_i_0 … i_p and differential d : C^p(U, F) → C^p + 1(U, F) as in Equation ([Tag 01EE]). Moreover, there exists a canonical map F → C^bullet(U, F) which is a quasi-isomorphism, i.e., C^bullet(U, F) is a resolution of F.","statement_latex":"Let $X$ be a ringed space. Let $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$\nbe an open covering of $X$. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nDenote $\\mathcal{F}_{i_0 \\ldots i_p}$ the restriction of\n$\\mathcal{F}$ to $U_{i_0 \\ldots i_p}$. There exists a complex\n${\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nof $\\mathcal{O}_X$-modules with\n$$\n{\\mathfrak C}^p(\\mathcal{U}, \\mathcal{F}) =\n\\prod\\nolimits_{i_0 \\ldots i_p}\n(j_{i_0 \\ldots i_p})_* \\mathcal{F}_{i_0 \\ldots i_p}\n$$\nand differential\n$d : {\\mathfrak C}^p(\\mathcal{U}, \\mathcal{F})\n\\to {\\mathfrak C}^{p + 1}(\\mathcal{U}, \\mathcal{F})$\nas in Equation (\\ref{equation-d-cech}). Moreover, there exists a canonical\nmap\n$$\n\\mathcal{F} \\to {\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nwhich is a quasi-isomorphism, i.e.,\n${\\mathfrak C}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis a resolution of $\\mathcal{F}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Alternative view of the v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FU","source_file":"cohomology.tex","source_line":4571,"source_end_line":4595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L4571-L4595","statement_sha256":"be46b20dfdeaee113a797f621ed372cac57580630d9f1b84ebe4c8672b009dcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4223,"rank":4223,"depth":0,"x":1177.039,"y":644.506,"cluster":"sheaf-cohomology"},{"id":"stacks:02FS","tag":"02FS","title":"Alternative view of the v Cech complex · Definition 02FS","summary":"Let X be a topological space. An open covering X = ⋃_i ∈ I U_i is said to be locally finite if for every x ∈ X there exists an open neighbourhood W of x such that (i ∈ I mid W ∩ U_i not = ∅) is finite.","statement_latex":"Let $X$ be a topological space.\nAn open covering $X = \\bigcup_{i \\in I} U_i$ is said to be\n{\\it locally finite} if for every $x \\in X$ there exists an open neighbourhood\n$W$ of $x$ such that $\\{i \\in I \\mid W \\cap U_i \\not = \\emptyset\\}$ is finite.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Alternative view of the v Cech complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FS","source_file":"cohomology.tex","source_line":4696,"source_end_line":4702,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L4696-L4702","statement_sha256":"fe45c96c0a641bdaa0d86fb3ce7de0c8400ddc01ef41a2903471f4702e61f79e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4224,"rank":4224,"depth":0,"x":1123.816,"y":733.221,"cluster":"sheaf-cohomology"},{"id":"stacks:08BN","tag":"08BN","title":"v Cech cohomology of complexes · Lemma 08BN","summary":"Let (X, O_X) be a ringed space. Let U : X = ⋃_i ∈ I U_i be an open covering. For a bounded below complex F^bullet of O_X-modules there is a canonical map Tot(checkC^bullet(U, F^bullet)) → RΓ(X, F^bullet) functorial in F^bullet and compatible with ([Tag 07M9]) and ([Tag 08BM]). There is a spectral sequence (E_r, d_r)_r ≥ 0 with E_2^p, q = H^p(Tot(checkC^bullet(U, underlineH^q(F^bullet))) converging to H^p + q(X, F^bullet).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be\nan open covering. For a bounded below complex $\\mathcal{F}^\\bullet$\nof $\\mathcal{O}_X$-modules there is a canonical map\n$$\n\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))\n\\longrightarrow\nR\\Gamma(X, \\mathcal{F}^\\bullet)\n$$\nfunctorial in $\\mathcal{F}^\\bullet$ and compatible with\n(\\ref{equation-global-sections-to-cech}) and (\\ref{equation-transformation}).\nThere is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_2^{p, q} =\nH^p(\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U},\n\\underline{H}^q(\\mathcal{F}^\\bullet)))\n$$\nconverging to $H^{p + q}(X, \\mathcal{F}^\\bullet)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BN","source_file":"cohomology.tex","source_line":4938,"source_end_line":4958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L4938-L4958","statement_sha256":"00e8558f9b41f4f0f93b09a807595c462ecdb651d51bce2879e2eb162a905b47","origin":"The Stacks Project","memory_eligible":false,"source_rank":4225,"rank":4225,"depth":21,"x":1091.506,"y":636.962,"cluster":"sheaf-cohomology"},{"id":"stacks:0FLH","tag":"0FLH","title":"v Cech cohomology of complexes · Lemma 0FLH","summary":"Let (X, O_X) be a ringed space. Let U : X = ⋃_i ∈ I U_i be an open covering. Let F^bullet be a bounded below complex of O_X-modules. If H^i(U_i_0 … i_p, F^q) = 0 for all i > 0 and all p, i_0, …, i_p, q, then the map Tot(checkC^bullet(U, F^bullet)) → RΓ(X, F^bullet) of Lemma [Tag 08BN] is an isomorphism.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be\nan open covering. Let $\\mathcal{F}^\\bullet$ be a bounded below complex\nof $\\mathcal{O}_X$-modules. If $H^i(U_{i_0 \\ldots i_p}, \\mathcal{F}^q) = 0$\nfor all $i > 0$ and all $p, i_0, \\ldots, i_p, q$, then the map\n$\n\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))\n\\to\nR\\Gamma(X, \\mathcal{F}^\\bullet)\n$\nof Lemma \\ref{lemma-cech-complex-complex} is an isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLH","source_file":"cohomology.tex","source_line":5021,"source_end_line":5034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5021-L5034","statement_sha256":"aa717264e34c22b9da9e513de5fc03a5ba1db184aeeab6bf2a2965dbe27124cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4226,"rank":4226,"depth":22,"x":1193.409,"y":689.959,"cluster":"sheaf-cohomology"},{"id":"stacks:07MC","tag":"07MC","title":"v Cech cohomology of complexes · Lemma 07MC","summary":"In the situation above, assume v Cech cohomology agrees with cohomology for the sheaves F_i^p and G_j^q. Let a_3 ∈ H^n(X, F_3^bullet) and b_1 ∈ H^m(X, G_1^bullet). Then we have γ_1( ∂ a_3 ∪ b_1) = (-1)^n + 1 γ_3( a_3 ∪ ∂ b_1) in H^n + m + 1(X, H^bullet) where ∂ indicates the boundary map on cohomology associated to the short exact sequences of complexes above.","statement_latex":"In the situation above, assume {\\v C}ech cohomology agrees with cohomology\nfor the sheaves $\\mathcal{F}_i^p$ and $\\mathcal{G}_j^q$.\nLet $a_3 \\in H^n(X, \\mathcal{F}_3^\\bullet)$ and\n$b_1 \\in H^m(X, \\mathcal{G}_1^\\bullet)$. Then we have\n$$\n\\gamma_1( \\partial a_3 \\cup b_1) =\n(-1)^{n + 1} \\gamma_3( a_3 \\cup \\partial b_1)\n$$\nin $H^{n + m + 1}(X, \\mathcal{H}^\\bullet)$ where $\\partial$ indicates the\nboundary map on cohomology associated to the short exact sequences of\ncomplexes above.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07MC","source_file":"cohomology.tex","source_line":5656,"source_end_line":5669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5656-L5669","statement_sha256":"4170ff42099dac11a2c21480adc9c5f1a1efe8492a1df9f2c5d1f6e1120e60ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":4227,"rank":4227,"depth":0,"x":1074.879,"y":708.823,"cluster":"sheaf-cohomology"},{"id":"stacks:0B8S","tag":"0B8S","title":"v Cech cohomology of complexes · Lemma 0B8S","summary":"Let X be a topological space. Let O' → O be a surjection of sheaves of rings whose kernel I ⊂ O' has square zero. Then M = H^1(X, I) is a R = H^0(X, O)-module and the boundary map ∂ : R → M associated to the short exact sequence 0 → I → O' → O → 0 is a derivation (Algebra, Definition [Tag 00RN]).","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{O}' \\to \\mathcal{O}$ be a\nsurjection of sheaves of rings whose kernel $\\mathcal{I} \\subset \\mathcal{O}'$\nhas square zero. Then $M = H^1(X, \\mathcal{I})$ is a\n$R = H^0(X, \\mathcal{O})$-module and the boundary map\n$\\partial : R \\to M$ associated to the short exact sequence\n$$\n0 \\to \\mathcal{I} \\to \\mathcal{O}' \\to \\mathcal{O} \\to 0\n$$\nis a derivation (Algebra, Definition \\ref{algebra-definition-derivation}).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8S","source_file":"cohomology.tex","source_line":5741,"source_end_line":5752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5741-L5752","statement_sha256":"fec473b0f3529d65619f7cf97da826bb03b0b649292ca4032ace532bcbab9856","origin":"The Stacks Project","memory_eligible":false,"source_rank":4228,"rank":4228,"depth":21,"x":1147.579,"y":627.133,"cluster":"sheaf-cohomology"},{"id":"stacks:06Y8","tag":"06Y8","title":"Flat resolutions · Lemma 06Y8","summary":"Let (X, O_X) be a ringed space. Let G^bullet be a complex of O_X-modules. The functors K(Mod(O_X)) → K(Mod(O_X)), F^bullet ↦ Tot(G^bullet ⊗_O_X F^bullet) and K(Mod(O_X)) → K(Mod(O_X)), F^bullet ↦ Tot(F^bullet ⊗_O_X G^bullet) are exact functors of triangulated categories.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{G}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.\nThe functors\n$$\nK(\\textit{Mod}(\\mathcal{O}_X))\n\\longrightarrow\nK(\\textit{Mod}(\\mathcal{O}_X)),\n\\quad\n\\mathcal{F}^\\bullet \\longmapsto\n\\text{Tot}(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{F}^\\bullet)\n$$\nand\n$$\nK(\\textit{Mod}(\\mathcal{O}_X))\n\\longrightarrow\nK(\\textit{Mod}(\\mathcal{O}_X)),\n\\quad\n\\mathcal{F}^\\bullet \\longmapsto\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)\n$$\nare exact functors of triangulated categories.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Y8","source_file":"cohomology.tex","source_line":5807,"source_end_line":5830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5807-L5830","statement_sha256":"b859a1e6802652372e73c6f833139f33798ad2b7495505f2423beea1c661692f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4229,"rank":4229,"depth":0,"x":1159.739,"y":729.268,"cluster":"sheaf-cohomology"},{"id":"stacks:06Y9","tag":"06Y9","title":"Flat resolutions · Definition 06Y9","summary":"Let (X, O_X) be a ringed space. A complex K^bullet of O_X-modules is called K-flat if for every acyclic complex F^bullet of O_X-modules the complex Tot(F^bullet ⊗_O_X K^bullet) is acyclic.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nA complex $\\mathcal{K}^\\bullet$ of $\\mathcal{O}_X$-modules is\ncalled {\\it K-flat} if for every acyclic complex $\\mathcal{F}^\\bullet$\nof $\\mathcal{O}_X$-modules the complex\n$$\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet)\n$$\nis acyclic.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Y9","source_file":"cohomology.tex","source_line":5837,"source_end_line":5847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5837-L5847","statement_sha256":"be184ef65241f15abac13bec4a988274d03a04a031e0ed9741b4b0624cfb9029","origin":"The Stacks Project","memory_eligible":false,"source_rank":4230,"rank":4230,"depth":0,"x":1068.067,"y":660.424,"cluster":"sheaf-cohomology"},{"id":"stacks:06YA","tag":"06YA","title":"Flat resolutions · Lemma 06YA","summary":"Let (X, O_X) be a ringed space. Let K^bullet be a K-flat complex. Then the functor K(Mod(O_X)) → K(Mod(O_X)), F^bullet ↦ Tot(F^bullet ⊗_O_X K^bullet) transforms quasi-isomorphisms into quasi-isomorphisms.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{K}^\\bullet$ be a K-flat complex.\nThen the functor\n$$\nK(\\textit{Mod}(\\mathcal{O}_X))\n\\longrightarrow\nK(\\textit{Mod}(\\mathcal{O}_X)), \\quad\n\\mathcal{F}^\\bullet\n\\longmapsto\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet)\n$$\ntransforms quasi-isomorphisms into quasi-isomorphisms.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YA","source_file":"cohomology.tex","source_line":5849,"source_end_line":5863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5849-L5863","statement_sha256":"5add6bae21100c53a1f1847b333848d854f9baf8e4025bf53a658dde94482ffc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4231,"rank":4231,"depth":1,"x":1191.791,"y":659.165,"cluster":"sheaf-cohomology"},{"id":"stacks:06YB","tag":"06YB","title":"Flat resolutions · Lemma 06YB","summary":"Let (X, O_X) be a ringed space. Let K^bullet be a complex of O_X-modules. Then K^bullet is K-flat if and only if for all x ∈ X the complex K_x^bullet of O_X, x-modules is K-flat (More on Algebra, Definition [Tag 06XZ]).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{K}^\\bullet$\nbe a complex of $\\mathcal{O}_X$-modules. Then $\\mathcal{K}^\\bullet$\nis K-flat if and only if for all $x \\in X$ the complex\n$\\mathcal{K}_x^\\bullet$ of $\\mathcal{O}_{X, x}$-modules is K-flat\n(More on Algebra, Definition \\ref{more-algebra-definition-K-flat}).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YB","source_file":"cohomology.tex","source_line":5872,"source_end_line":5879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5872-L5879","statement_sha256":"1df04d249f46fd10296b41fcd8526fb241097236f96472e6b0f2a725e182f140","origin":"The Stacks Project","memory_eligible":false,"source_rank":4232,"rank":4232,"depth":2,"x":1101.016,"y":730.732,"cluster":"sheaf-cohomology"},{"id":"stacks:079R","tag":"079R","title":"Flat resolutions · Lemma 079R","summary":"Let (X, O_X) be a ringed space. If K^bullet, L^bullet are K-flat complexes of O_X-modules, then Tot(K^bullet ⊗_O_X L^bullet) is a K-flat complex of O_X-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nIf $\\mathcal{K}^\\bullet$, $\\mathcal{L}^\\bullet$ are K-flat complexes\nof $\\mathcal{O}_X$-modules, then\n$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet)$\nis a K-flat complex of $\\mathcal{O}_X$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079R","source_file":"cohomology.tex","source_line":5897,"source_end_line":5904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5897-L5904","statement_sha256":"080d29cba6ea6feb9bd9742ea141a5f114c9ef746b09f35124f6d8c5e3a8bd54","origin":"The Stacks Project","memory_eligible":false,"source_rank":4233,"rank":4233,"depth":0,"x":1110.458,"y":625.82,"cluster":"sheaf-cohomology"},{"id":"stacks:079S","tag":"079S","title":"Flat resolutions · Lemma 079S","summary":"Let (X, O_X) be a ringed space. Let (K_1^bullet, K_2^bullet, K_3^bullet) be a distinguished triangle in K(Mod(O_X)). If two out of three of K_i^bullet are K-flat, so is the third.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $(\\mathcal{K}_1^\\bullet, \\mathcal{K}_2^\\bullet, \\mathcal{K}_3^\\bullet)$\nbe a distinguished triangle in $K(\\textit{Mod}(\\mathcal{O}_X))$.\nIf two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079S","source_file":"cohomology.tex","source_line":5918,"source_end_line":5924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5918-L5924","statement_sha256":"b5b4311ad3c1498e6f2ddd1a2bd8188d4e87017589082f141aafca830e4ac0ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":4234,"rank":4234,"depth":1,"x":1188.325,"y":709.034,"cluster":"sheaf-cohomology"},{"id":"stacks:0G6U","tag":"0G6U","title":"Flat resolutions · Lemma 0G6U","summary":"Let (X, O_X) be a ringed space. Let 0 → K_1^bullet → K_2^bullet → K_3^bullet → 0 be a short exact sequence of complexes such that the terms of K_3^bullet are flat O_X-modules. If two out of three of K_i^bullet are K-flat, so is the third.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let\n$0 \\to \\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to\n\\mathcal{K}_3^\\bullet \\to 0$ be a short exact sequence of complexes\nsuch that the terms of $\\mathcal{K}_3^\\bullet$ are flat $\\mathcal{O}_X$-modules.\nIf two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6U","source_file":"cohomology.tex","source_line":5934,"source_end_line":5941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5934-L5941","statement_sha256":"264aea86c6696e41e56e753539ab971f2d0a7af9ad182cb2ad23a9bdabe95be6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4235,"rank":4235,"depth":2,"x":1063.254,"y":691.754,"cluster":"sheaf-cohomology"},{"id":"stacks:06YC","tag":"06YC","title":"Flat resolutions · Lemma 06YC","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The pullback of a K-flat complex of O_Y-modules is a K-flat complex of O_X-modules.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of\nringed spaces. The pullback of a K-flat complex of $\\mathcal{O}_Y$-modules\nis a K-flat complex of $\\mathcal{O}_X$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YC","source_file":"cohomology.tex","source_line":5960,"source_end_line":5965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5960-L5965","statement_sha256":"3b0112d172a2813f606bdf4d3e4e410483dc1ae3982f99395e24b3f3f3962096","origin":"The Stacks Project","memory_eligible":false,"source_rank":4236,"rank":4236,"depth":3,"x":1169.997,"y":633.189,"cluster":"sheaf-cohomology"},{"id":"stacks:06YD","tag":"06YD","title":"Flat resolutions · Lemma 06YD","summary":"Let (X, O_X) be a ringed space. A bounded above complex of flat O_X-modules is K-flat.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. A bounded above complex\nof flat $\\mathcal{O}_X$-modules is K-flat.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YD","source_file":"cohomology.tex","source_line":5976,"source_end_line":5980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5976-L5980","statement_sha256":"bf0c325f4a5cbe7140297a1921b863eb1ac38e9bf0c5f0753a31c62682c8660f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4237,"rank":4237,"depth":6,"x":1138.198,"y":737.541,"cluster":"sheaf-cohomology"},{"id":"stacks:06YE","tag":"06YE","title":"Flat resolutions · Lemma 06YE","summary":"Let (X, O_X) be a ringed space. Let K_1^bullet → K_2^bullet → … be a system of K-flat complexes. Then colim_i K_i^bullet is K-flat.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\ldots$\nbe a system of K-flat complexes.\nThen $\\colim_i \\mathcal{K}_i^\\bullet$ is K-flat.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YE","source_file":"cohomology.tex","source_line":5995,"source_end_line":6001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L5995-L6001","statement_sha256":"c74395c7d2ba16bdd766ba3d9b971bf7afe9476543f069297576d55d905b363c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4238,"rank":4238,"depth":0,"x":1077.385,"y":642.007,"cluster":"sheaf-cohomology"},{"id":"stacks:079T","tag":"079T","title":"Flat resolutions · Lemma 079T","summary":"Let (X, O_X) be a ringed space. For any complex G^bullet of O_X-modules there exists a commutative diagram of complexes of O_X-modules xymatrix K_1^bullet ar[d] ar[r] & K_2^bullet ar[d] ar[r] & … τ_≤ 1G^bullet ar[r] & τ_≤ 2G^bullet ar[r] & … with the following properties: (1) the vertical arrows are quasi-isomorphisms and termwise surjective, (2) each K_n^bullet is a bounded above complex whose terms are direct sums of O_X-modules of the form j_U!O_U, and (3) the maps…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nFor any complex $\\mathcal{G}^\\bullet$ of $\\mathcal{O}_X$-modules\nthere exists a commutative diagram of complexes of $\\mathcal{O}_X$-modules\n$$\n\\xymatrix{\n\\mathcal{K}_1^\\bullet \\ar[d] \\ar[r] &\n\\mathcal{K}_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\\n\\tau_{\\leq 1}\\mathcal{G}^\\bullet \\ar[r] &\n\\tau_{\\leq 2}\\mathcal{G}^\\bullet \\ar[r] & \\ldots\n}\n$$\nwith the following properties: (1) the vertical arrows are quasi-isomorphisms\nand termwise surjective,\n(2) each $\\mathcal{K}_n^\\bullet$ is a bounded above complex whose terms\nare direct sums of $\\mathcal{O}_X$-modules of the form\n$j_{U!}\\mathcal{O}_U$, and\n(3) the maps $\\mathcal{K}_n^\\bullet \\to \\mathcal{K}_{n + 1}^\\bullet$ are\ntermwise split injections whose cokernels are direct sums of\n$\\mathcal{O}_X$-modules of the form $j_{U!}\\mathcal{O}_U$. Moreover, the map\n$\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$ is a quasi-isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079T","source_file":"cohomology.tex","source_line":6016,"source_end_line":6038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6016-L6038","statement_sha256":"76f0bcd3ccf912509bd48e84fc42aed6e8f8a794cf7cc4d68212a692a719094f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4239,"rank":4239,"depth":14,"x":1199.74,"y":678.15,"cluster":"sheaf-cohomology"},{"id":"stacks:06YF","tag":"06YF","title":"Flat resolutions · Lemma 06YF","summary":"Let (X, O_X) be a ringed space. For any complex G^bullet there exists a K-flat complex K^bullet whose terms are flat O_X-modules and a quasi-isomorphism K^bullet → G^bullet which is termwise surjective.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nFor any complex $\\mathcal{G}^\\bullet$ there exists a $K$-flat complex\n$\\mathcal{K}^\\bullet$ whose terms are flat $\\mathcal{O}_X$-modules\nand a quasi-isomorphism $\\mathcal{K}^\\bullet \\to \\mathcal{G}^\\bullet$\nwhich is termwise surjective.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YF","source_file":"cohomology.tex","source_line":6051,"source_end_line":6058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6051-L6058","statement_sha256":"13995b72cb2c8356728676fa3b80f24087ec2d11bf0375b86dc9111d1dd1b6fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4240,"rank":4240,"depth":15,"x":1079.786,"y":721.163,"cluster":"sheaf-cohomology"},{"id":"stacks:06YG","tag":"06YG","title":"Flat resolutions · Lemma 06YG","summary":"Let (X, O_X) be a ringed space. Let α : P^bullet → Q^bullet be a quasi-isomorphism of K-flat complexes of O_X-modules. For every complex F^bullet of O_X-modules the induced map Tot(id_F^bullet ⊗ α) : Tot(F^bullet ⊗_O_X P^bullet) → Tot(F^bullet ⊗_O_X Q^bullet) is a quasi-isomorphism.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let\n$\\alpha : \\mathcal{P}^\\bullet \\to \\mathcal{Q}^\\bullet$ be a\nquasi-isomorphism of K-flat complexes of $\\mathcal{O}_X$-modules.\nFor every complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}_X$-modules\nthe induced map\n$$\n\\text{Tot}(\\text{id}_{\\mathcal{F}^\\bullet} \\otimes \\alpha) :\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{P}^\\bullet)\n\\longrightarrow\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{Q}^\\bullet)\n$$\nis a quasi-isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YG","source_file":"cohomology.tex","source_line":6078,"source_end_line":6092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6078-L6092","statement_sha256":"1c503edcaac5d5b869ac6188a35a8e609ac76f3699777ecdf160786a8c9f02fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4241,"rank":4241,"depth":16,"x":1133.946,"y":620.831,"cluster":"sheaf-cohomology"},{"id":"stacks:06YH","tag":"06YH","title":"Flat resolutions · Definition 06YH","summary":"Let (X, O_X) be a ringed space. Let F^bullet be an object of D(O_X). The derived tensor product - ⊗_O_X^L F^bullet : D(O_X) → D(O_X) is the exact functor of triangulated categories described above.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}^\\bullet$ be an object of $D(\\mathcal{O}_X)$.\nThe {\\it derived tensor product}\n$$\n- \\otimes_{\\mathcal{O}_X}^{\\mathbf{L}} \\mathcal{F}^\\bullet :\nD(\\mathcal{O}_X)\n\\longrightarrow\nD(\\mathcal{O}_X)\n$$\nis the exact functor of triangulated categories described above.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YH","source_file":"cohomology.tex","source_line":6144,"source_end_line":6156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6144-L6156","statement_sha256":"f80e7ea0bfe86ce44e7a619ff81fa77249de3525e3c5d56ea04aca96ab92f318","origin":"The Stacks Project","memory_eligible":false,"source_rank":4242,"rank":4242,"depth":0,"x":1174.915,"y":726.118,"cluster":"sheaf-cohomology"},{"id":"stacks:08BP","tag":"08BP","title":"Flat resolutions · Definition 08BP","summary":"Let (X, O_X) be a ringed space. Let F, G be O_X-modules. The Tor's of F and G are define by the formula Tor_p^O_X(F, G) = H^-p(F ⊗_O_X^L G) with derived tensor product as defined above.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}_X$-modules.\nThe {\\it Tor}'s of $\\mathcal{F}$ and $\\mathcal{G}$ are define by\nthe formula\n$$\n\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) =\nH^{-p}(\\mathcal{F} \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G})\n$$\nwith derived tensor product as defined above.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BP","source_file":"cohomology.tex","source_line":6173,"source_end_line":6184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6173-L6184","statement_sha256":"8fe396fc86979db47685ef9cc30de68aa2e3c2b05eebcfaf96ba31299490de5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4243,"rank":4243,"depth":0,"x":1059.417,"y":671.433,"cluster":"sheaf-cohomology"},{"id":"stacks:08BQ","tag":"08BQ","title":"Flat resolutions · Lemma 08BQ","summary":"Tor measures the deviation of flatness. Let (X, O_X) be a ringed space. Let F be an O_X-module. The following are equivalent • F is a flat O_X-module, and • Tor_1^O_X(F, G) = 0 for every O_X-module G.","statement_latex":"\\begin{slogan}\nTor measures the deviation of flatness.\n\\end{slogan}\nLet $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module, and\n\\item $\\text{Tor}_1^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G}) = 0$\nfor every $\\mathcal{O}_X$-module $\\mathcal{G}$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BQ","source_file":"cohomology.tex","source_line":6204,"source_end_line":6217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6204-L6217","statement_sha256":"6e82c933a46126f67c582c083454746619ead28c0060b0625af7207c2de64049","origin":"The Stacks Project","memory_eligible":false,"source_rank":4244,"rank":4244,"depth":0,"x":1189.25,"y":646.087,"cluster":"sheaf-cohomology"},{"id":"stacks:0G6V","tag":"0G6V","title":"Flat resolutions · Lemma 0G6V","summary":"Let (X, O_X) be a ringed space. Let a : K^bullet → L^bullet be a map of complexes of O_X-modules. If K^bullet is K-flat, then there exist a complex N^bullet and maps of complexes b : K^bullet → N^bullet and c : N^bullet → L^bullet such that • N^bullet is K-flat, • c is a quasi-isomorphism, • a is homotopic to c ∘ b. If the terms of K^bullet are flat, then we may choose N^bullet, b, and c such that the same is true for N^bullet.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $a : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$ be a map of complexes\nof $\\mathcal{O}_X$-modules. If $\\mathcal{K}^\\bullet$ is K-flat, then\nthere exist a complex $\\mathcal{N}^\\bullet$ and maps of complexes\n$b : \\mathcal{K}^\\bullet \\to \\mathcal{N}^\\bullet$\nand $c : \\mathcal{N}^\\bullet \\to \\mathcal{L}^\\bullet$ such that\n\\begin{enumerate}\n\\item $\\mathcal{N}^\\bullet$ is K-flat,\n\\item $c$ is a quasi-isomorphism,\n\\item $a$ is homotopic to $c \\circ b$.\n\\end{enumerate}\nIf the terms of $\\mathcal{K}^\\bullet$ are flat, then we may choose\n$\\mathcal{N}^\\bullet$, $b$, and $c$\nsuch that the same is true for $\\mathcal{N}^\\bullet$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6V","source_file":"cohomology.tex","source_line":6232,"source_end_line":6248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6232-L6248","statement_sha256":"7330499facb77cc19b822191953255085ba53a62ec8c959178e16e1dc4e6729a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4245,"rank":4245,"depth":16,"x":1113.496,"y":738.934,"cluster":"sheaf-cohomology"},{"id":"stacks:06YJ","tag":"06YJ","title":"Derived pullback · Lemma 06YJ","summary":"The construction above is independent of choices and defines an exact functor of triangulated categories Lf^* : D(O_Y) → D(O_X).","statement_latex":"The construction above is independent of choices and defines an exact\nfunctor of triangulated categories\n$Lf^* : D(\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YJ","source_file":"cohomology.tex","source_line":6329,"source_end_line":6334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6329-L6334","statement_sha256":"fb6fa9fe12e26466ee2ff9e5ea50aceead9c5299f67525af8c8629efb151ad19","origin":"The Stacks Project","memory_eligible":false,"source_rank":4246,"rank":4246,"depth":17,"x":1094.594,"y":626.903,"cluster":"sheaf-cohomology"},{"id":"stacks:0D5S","tag":"0D5S","title":"Derived pullback · Lemma 0D5S","summary":"Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Then Lf^* ∘ Lg^* = L(g ∘ f)^* as functors D(O_Z) → D(O_X).","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.\nThen $Lf^* \\circ Lg^* = L(g \\circ f)^*$ as functors\n$D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5S","source_file":"cohomology.tex","source_line":6392,"source_end_line":6397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6392-L6397","statement_sha256":"e9b7a0a82db0948549563b8863f93cbdcb15245056d0debefd3dd40fa0fa7b23","origin":"The Stacks Project","memory_eligible":false,"source_rank":4247,"rank":4247,"depth":4,"x":1199.162,"y":699.161,"cluster":"sheaf-cohomology"},{"id":"stacks:079U","tag":"079U","title":"Derived pullback · Lemma 079U","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. There is a canonical bifunctorial isomorphism Lf^*( F^bullet ⊗_O_Y^L G^bullet ) = Lf^*F^bullet ⊗_O_X^L Lf^*G^bullet for F^bullet, G^bullet ∈ Ob(D(O_Y)).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces. There is a canonical bifunctorial\nisomorphism\n$$\nLf^*(\n\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_Y}^{\\mathbf{L}} \\mathcal{G}^\\bullet\n) =\nLf^*\\mathcal{F}^\\bullet \n\\otimes_{\\mathcal{O}_X}^{\\mathbf{L}}\nLf^*\\mathcal{G}^\\bullet \n$$\nfor $\\mathcal{F}^\\bullet, \\mathcal{G}^\\bullet \\in \\Ob(D(\\mathcal{O}_Y))$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079U","source_file":"cohomology.tex","source_line":6410,"source_end_line":6424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6410-L6424","statement_sha256":"5d20360d5082566e13b4dfe7671d2266d644f3ce4635426b308ad12d20804155","origin":"The Stacks Project","memory_eligible":false,"source_rank":4248,"rank":4248,"depth":1,"x":1063.255,"y":705.243,"cluster":"sheaf-cohomology"},{"id":"stacks:08DE","tag":"08DE","title":"Derived pullback · Lemma 08DE","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. There is a canonical bifunctorial isomorphism F^bullet ⊗_O_X^L Lf^*G^bullet = F^bullet ⊗_f^-1O_Y^L f^-1G^bullet for F^bullet in D(O_X) and G^bullet in D(O_Y).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces. There is a canonical bifunctorial\nisomorphism\n$$\n\\mathcal{F}^\\bullet\n\\otimes_{\\mathcal{O}_X}^{\\mathbf{L}}\nLf^*\\mathcal{G}^\\bullet\n=\n\\mathcal{F}^\\bullet \n\\otimes_{f^{-1}\\mathcal{O}_Y}^{\\mathbf{L}}\nf^{-1}\\mathcal{G}^\\bullet \n$$\nfor $\\mathcal{F}^\\bullet$ in $D(\\mathcal{O}_X)$ and\n$\\mathcal{G}^\\bullet$ in $D(\\mathcal{O}_Y)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DE","source_file":"cohomology.tex","source_line":6449,"source_end_line":6465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6449-L6465","statement_sha256":"0d53f719006ee19bc04f0dcb0d6abf2e8b08861d87914196da91a88195f8a841","origin":"The Stacks Project","memory_eligible":false,"source_rank":4249,"rank":4249,"depth":0,"x":1159.063,"y":623.228,"cluster":"sheaf-cohomology"},{"id":"stacks:0FP0","tag":"0FP0","title":"Derived pullback · Lemma 0FP0","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let K^bullet and M^bullet be complexes of O_Y-modules. The diagram xymatrix Lf^*(K^bullet ⊗_O_Y^L M^bullet) ar[r] ar[d] & Lf^*Tot(K^bullet ⊗_O_Y M^bullet) ar[d] Lf^*K^bullet ⊗_O_X^L Lf^*M^bullet ar[d] & f^*Tot(K^bullet ⊗_O_Y M^bullet) ar[d] f^*K^bullet ⊗_O_X^L f^*M^bullet ar[r] & Tot(f^*K^bullet ⊗_O_X f^*M^bullet) commutes.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism\nof ringed spaces. Let $\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$\nbe complexes of $\\mathcal{O}_Y$-modules. The diagram\n$$\n\\xymatrix{\nLf^*(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}\n\\mathcal{M}^\\bullet) \\ar[r] \\ar[d] &\nLf^*\\text{Tot}(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_Y}\n\\mathcal{M}^\\bullet) \\ar[d] \\\\\nLf^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L}\nLf^*\\mathcal{M}^\\bullet \\ar[d] &\nf^*\\text{Tot}(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_Y}\n\\mathcal{M}^\\bullet) \\ar[d] \\\\\nf^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L}\nf^*\\mathcal{M}^\\bullet \\ar[r] &\n\\text{Tot}(f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}\nf^*\\mathcal{M}^\\bullet)\n}\n$$\ncommutes.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FP0","source_file":"cohomology.tex","source_line":6478,"source_end_line":6503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6478-L6503","statement_sha256":"f6a64faf1e3115e8c8ecc91122217dbd3f41490e998e972d48f859e91e597339","origin":"The Stacks Project","memory_eligible":false,"source_rank":4250,"rank":4250,"depth":4,"x":1154.344,"y":738.645,"cluster":"sheaf-cohomology"},{"id":"stacks:079W","tag":"079W","title":"Cohomology of unbounded complexes · Lemma 079W","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The functor Rf_* defined above and the functor Lf^* defined in Lemma [Tag 06YJ] are adjoint: Hom_D(O_X)(Lf^*G^bullet, F^bullet) = Hom_D(O_Y)(G^bullet, Rf_*F^bullet) bifunctorially in F^bullet ∈ Ob(D(O_X)) and G^bullet ∈ Ob(D(O_Y)).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of\nringed spaces. The functor $Rf_*$ defined above and the functor $Lf^*$\ndefined in Lemma \\ref{lemma-derived-base-change} are adjoint:\n$$\n\\Hom_{D(\\mathcal{O}_X)}(Lf^*\\mathcal{G}^\\bullet, \\mathcal{F}^\\bullet)\n=\n\\Hom_{D(\\mathcal{O}_Y)}(\\mathcal{G}^\\bullet, Rf_*\\mathcal{F}^\\bullet)\n$$\nbifunctorially in $\\mathcal{F}^\\bullet \\in \\Ob(D(\\mathcal{O}_X))$ and\n$\\mathcal{G}^\\bullet \\in \\Ob(D(\\mathcal{O}_Y))$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology of unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079W","source_file":"cohomology.tex","source_line":6610,"source_end_line":6622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6610-L6622","statement_sha256":"59e9d2f0548ebc73ca2dd653b9c62d3523bb5d7c6b4314e053fd02a882b7a1a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4251,"rank":4251,"depth":18,"x":1064.567,"y":650.424,"cluster":"sheaf-cohomology"},{"id":"stacks:0D5T","tag":"0D5T","title":"Cohomology of unbounded complexes · Lemma 0D5T","summary":"Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Then Rg_* ∘ Rf_* = R(g ∘ f)_* as functors D(O_X) → D(O_Z).","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.\nThen $Rg_* \\circ Rf_* = R(g \\circ f)_*$ as functors\n$D(\\mathcal{O}_X) \\to D(\\mathcal{O}_Z)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology of unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5T","source_file":"cohomology.tex","source_line":6629,"source_end_line":6634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6629-L6634","statement_sha256":"dffba5d90b024c8f5e76a5f59b5d110ede58d47f88e6e876b01f5c8106f824d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4252,"rank":4252,"depth":19,"x":1202.386,"y":664.605,"cluster":"sheaf-cohomology"},{"id":"stacks:0FP1","tag":"0FP1","title":"Cohomology of unbounded complexes · Lemma 0FP1","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let K^bullet be a complex of O_X-modules. The diagram xymatrix Lf^*f_*K^bullet ar[r] ar[d] & f^*f_*K^bullet ar[d] Lf^*Rf_*K^bullet ar[r] & K^bullet coming from Lf^* → f^* on complexes, f_* → Rf_* on complexes, and adjunction Lf^* ∘ Rf_* → id commutes in D(O_X).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces. Let $\\mathcal{K}^\\bullet$\nbe a complex of $\\mathcal{O}_X$-modules.\nThe diagram\n$$\n\\xymatrix{\nLf^*f_*\\mathcal{K}^\\bullet \\ar[r] \\ar[d] &\nf^*f_*\\mathcal{K}^\\bullet \\ar[d] \\\\\nLf^*Rf_*\\mathcal{K}^\\bullet \\ar[r] &\n\\mathcal{K}^\\bullet\n}\n$$\ncoming from $Lf^* \\to f^*$ on complexes, $f_* \\to Rf_*$ on complexes,\nand adjunction $Lf^* \\circ Rf_* \\to \\text{id}$\ncommutes in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology of unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FP1","source_file":"cohomology.tex","source_line":6723,"source_end_line":6740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6723-L6740","statement_sha256":"946fc08a7411a52121de976966a21cf6b4aea09becad800f9ee73e1441fd94bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4253,"rank":4253,"depth":4,"x":1088.805,"y":732.679,"cluster":"sheaf-cohomology"},{"id":"stacks:0BKK","tag":"0BKK","title":"Cohomology of filtered complexes · Lemma 0BKK","summary":"Let (X, O_X) be a ringed space. Let F^bullet be a filtered complex of O_X-modules. There exists a canonical spectral sequence (E_r, d_r)_r ≥ 1 of bigraded Γ(X, O_X)-modules with d_r of bidegree (r, -r + 1) and E_1^p, q = H^p + q(X, gr^pF^bullet) If for every n we have H^n(X, F^pF^bullet) = 0 for p gg 0 and H^n(X, F^pF^bullet) = H^n(X, F^bullet) for p ll 0 then the spectral sequence is bounded and converges to H^*(X, F^bullet).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$ be a\nfiltered complex of $\\mathcal{O}_X$-modules. There exists a canonical\nspectral sequence $(E_r, \\text{d}_r)_{r \\geq 1}$ of bigraded\n$\\Gamma(X, \\mathcal{O}_X)$-modules with $d_r$ of bidegree $(r, -r + 1)$ and\n$$\nE_1^{p, q} = H^{p + q}(X, \\text{gr}^p\\mathcal{F}^\\bullet)\n$$\nIf for every $n$ we have\n$$\nH^n(X, F^p\\mathcal{F}^\\bullet) = 0\\text{ for }p \\gg 0\n\\quad\\text{and}\\quad\nH^n(X, F^p\\mathcal{F}^\\bullet) = H^n(X, \\mathcal{F}^\\bullet)\\text{ for }p \\ll 0\n$$\nthen the spectral sequence is bounded and converges to\n$H^*(X, \\mathcal{F}^\\bullet)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology of filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKK","source_file":"cohomology.tex","source_line":6821,"source_end_line":6838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6821-L6838","statement_sha256":"f6ae18d8f041ac1c504e6d3b114ddcb4031f76d02b4bd37c9e936cd0d633a008","origin":"The Stacks Project","memory_eligible":false,"source_rank":4254,"rank":4254,"depth":14,"x":1117.955,"y":617.482,"cluster":"sheaf-cohomology"},{"id":"stacks:0FLL","tag":"0FLL","title":"Cohomology of filtered complexes · Lemma 0FLL","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let F^bullet be a filtered complex of O_X-modules. There exists a canonical spectral sequence (E_r, d_r)_r ≥ 1 of bigraded O_Y-modules with d_r of bidegree (r, -r + 1) and E_1^p, q = R^p + qf_*gr^pF^bullet If for every n we have R^nf_*F^pF^bullet = 0 for p gg 0 and R^nf_*F^pF^bullet = R^nf_*F^bullet for p ll 0 then the spectral sequence is bounded and converges to Rf_*F^bullet.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of\nringed spaces. Let $\\mathcal{F}^\\bullet$ be a filtered complex of\n$\\mathcal{O}_X$-modules. There exists a canonical spectral sequence\n$(E_r, \\text{d}_r)_{r \\geq 1}$ of bigraded\n$\\mathcal{O}_Y$-modules with $d_r$ of bidegree $(r, -r + 1)$ and\n$$\nE_1^{p, q} = R^{p + q}f_*\\text{gr}^p\\mathcal{F}^\\bullet\n$$\nIf for every $n$ we have\n$$\nR^nf_*F^p\\mathcal{F}^\\bullet = 0 \\text{ for }p \\gg 0\n\\quad\\text{and}\\quad\nR^nf_*F^p\\mathcal{F}^\\bullet = R^nf_*\\mathcal{F}^\\bullet \\text{ for }p \\ll 0\n$$\nthen the spectral sequence is bounded and converges to\n$Rf_*\\mathcal{F}^\\bullet$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology of filtered complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLL","source_file":"cohomology.tex","source_line":6947,"source_end_line":6965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L6947-L6965","statement_sha256":"308c7576b87f5c4bf43194bc6212004e9f03e26c17f09e90917c30c9437cfcb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4255,"rank":4255,"depth":15,"x":1189.438,"y":719.452,"cluster":"sheaf-cohomology"},{"id":"stacks:0FKS","tag":"0FKS","title":"Godement resolution · Lemma 0FKS","summary":"Let (X, O_X) be a ringed space. For every sheaf of O_X-modules F there is a resolution 0 → F → f_*f^*F → f_*f^*f_*f^*F → f_*f^*f_*f^*f_*f^*F → … functorial in F such that each term f_*f^* … f_*f^*F is a flasque O_X-module and such that for all x ∈ X the map F_x[0] → Big( (f_*f^*F)_x → (f_*f^*f_*f^*F)_x → (f_*f^*f_*f^*f_*f^*F)_x → … Big) is a homotopy equivalence in the category of complexes of O_X, x-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. For every sheaf of\n$\\mathcal{O}_X$-modules $\\mathcal{F}$ there is a resolution\n$$\n0 \\to\n\\mathcal{F} \\to\nf_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots\n$$\nfunctorial in $\\mathcal{F}$ such that each term\n$f_*f^* \\ldots f_*f^*\\mathcal{F}$ is a flasque\n$\\mathcal{O}_X$-module and such that for all $x \\in X$ the\nmap\n$$\n\\mathcal{F}_x[0] \\to \\Big(\n(f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*f_*f^*\\mathcal{F})_x \\to \\ldots\n\\Big)\n$$\nis a homotopy equivalence in the category of complexes\nof $\\mathcal{O}_{X, x}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Godement resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKS","source_file":"cohomology.tex","source_line":7014,"source_end_line":7038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7014-L7038","statement_sha256":"6188aa9449bfc7e6158cc6fb1b84a1081bef2ad4bc0a5939ccf4db8231f4e89a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4256,"rank":4256,"depth":7,"x":1054.089,"y":684.657,"cluster":"sheaf-cohomology"},{"id":"stacks:0FKT","tag":"0FKT","title":"Godement resolution · Lemma 0FKT","summary":"Let (X, O_X) be a ringed space. Let F^bullet be a bounded below complex of O_X-modules. There exists a quasi-isomorphism F^bullet → G^bullet where G^bullet be a bounded below complex of flasque O_X-modules and for all x ∈ X the map F^bullet_x → G^bullet_x is a homotopy equivalence in the category of complexes of O_X, x-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let\n$\\mathcal{F}^\\bullet$ be a bounded below complex of\n$\\mathcal{O}_X$-modules. There exists a quasi-isomorphism\n$\\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$\nwhere $\\mathcal{G}^\\bullet$ be a bounded below complex of flasque\n$\\mathcal{O}_X$-modules and for all $x \\in X$ the\nmap $\\mathcal{F}^\\bullet_x \\to \\mathcal{G}^\\bullet_x$\nis a homotopy equivalence in the category of complexes\nof $\\mathcal{O}_{X, x}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Godement resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKT","source_file":"cohomology.tex","source_line":7080,"source_end_line":7091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7080-L7091","statement_sha256":"285caaada0079c7f2b8809169f4a3a372d616fdee93c0a57d8f8a23adec773bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4257,"rank":4257,"depth":8,"x":1182.474,"y":633.276,"cluster":"sheaf-cohomology"},{"id":"stacks:0FP2","tag":"0FP2","title":"Cup product · Lemma 0FP2","summary":"This construction gives the cup product.","statement_latex":"This construction gives the cup product.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FP2","source_file":"cohomology.tex","source_line":7214,"source_end_line":7217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7214-L7217","statement_sha256":"574de39040baa3fd52118fec507e590d616b427500f48a2d08b751549640fe8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4258,"rank":4258,"depth":0,"x":1128.876,"y":744.526,"cluster":"sheaf-cohomology"},{"id":"stacks:0FP3","tag":"0FP3","title":"Cup product · Lemma 0FP3","summary":"In the situation above the following diagram commutes xymatrix f_*K^bullet ⊗_O_Y^L f_*M^bullet ar[r] ar[d] & Rf_*K^bullet ⊗_O_Y^L Rf_*M^bullet ar[d]^Remark [Tag 0B68] Tot( f_*K^bullet ⊗_O_Y f_*M^bullet) ar[d]_naive cup product & Rf_*(K^bullet ⊗_O_X^L M^bullet) ar[d] f_*Tot(K^bullet ⊗_O_X M^bullet) ar[r] & Rf_*Tot(K^bullet ⊗_O_X M^bullet)","statement_latex":"In the situation above the following diagram commutes\n$$\n\\xymatrix{\nf_*\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}\nf_*\\mathcal{M}^\\bullet \\ar[r] \\ar[d]\n&\nRf_*\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_Y}^\\mathbf{L}\nRf_*\\mathcal{M}^\\bullet \\ar[d]^{\\text{Remark \\ref{remark-cup-product}}} \\\\\n\\text{Tot}(\nf_*\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_Y}\nf_*\\mathcal{M}^\\bullet) \\ar[d]_{\\text{naive cup product}} &\nRf_*(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_X}^\\mathbf{L}\n\\mathcal{M}^\\bullet) \\ar[d] \\\\\nf_*\\text{Tot}(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_X}\n\\mathcal{M}^\\bullet) \\ar[r] &\nRf_*\\text{Tot}(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_X}\n\\mathcal{M}^\\bullet)\n}\n$$","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FP3","source_file":"cohomology.tex","source_line":7311,"source_end_line":7338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7311-L7338","statement_sha256":"432aefd4e78e4a900950c00ff9005a4d5db5a2ef90c9cb68630049b8266c097f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4259,"rank":4259,"depth":5,"x":1078.706,"y":631.561,"cluster":"sheaf-cohomology"},{"id":"stacks:0FKV","tag":"0FKV","title":"Cup product · Lemma 0FKV","summary":"Let (X, O_X) be a ringed space. Let K^bullet and M^bullet be bounded below complexes of O_X-modules. Let U : X = ⋃_i ∈ I U_i be an open covering Then xymatrix Tot(checkC^bullet(U, K^bullet)) ⊗_A^L Tot(checkC^bullet(U, M^bullet)) ar[d] ar[r] & RΓ(X, K^bullet) ⊗_A^L RΓ(X, M^bullet) ar[d]^μ Tot( Tot(checkC^bullet(U, K^bullet)) ⊗_A Tot(checkC^bullet(U, M^bullet))) ar[d]^([Tag 07MB]) & RΓ(X, K^bullet ⊗_O_X^L M^bullet) ar[d] Tot( checkC^bullet( U, Tot(K^bullet ⊗_O_X M^bullet)…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let\n$\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$\nbe bounded below complexes of $\\mathcal{O}_X$-modules.\nLet $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering\nThen\n$$\n\\xymatrix{\n\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{K}^\\bullet))\n\\otimes_A^\\mathbf{L}\n\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{M}^\\bullet))\n\\ar[d] \\ar[r] &\nR\\Gamma(X, \\mathcal{K}^\\bullet)\n\\otimes_A^\\mathbf{L}\nR\\Gamma(X, \\mathcal{M}^\\bullet) \\ar[d]^\\mu \\\\\n\\text{Tot}(\n\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{K}^\\bullet))\n\\otimes_A\n\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{M}^\\bullet)))\n\\ar[d]^{(\\ref{equation-needs-signs})} &\nR\\Gamma(X,\n\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{M}^\\bullet)\n\\ar[d] \\\\\n\\text{Tot}(\n\\check{\\mathcal{C}}^\\bullet({\\mathcal U},\n\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet)\n)) \\ar[r] &\nR\\Gamma(X,\n\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{M}^\\bullet))\n}\n$$\nwhere the horizontal arrows are the ones in\nLemma \\ref{lemma-cech-complex-complex}\ncommutes in $D(A)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKV","source_file":"cohomology.tex","source_line":7543,"source_end_line":7578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7543-L7578","statement_sha256":"d70f13cb40bca3b7415cbd6d4fcc8160782f1da7b8618253c972ffce9e082ec2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4260,"rank":4260,"depth":23,"x":1207.125,"y":686.642,"cluster":"sheaf-cohomology"},{"id":"stacks:0FP4","tag":"0FP4","title":"Cup product · Lemma 0FP4","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The relative cup product of Remark [Tag 0B68] is associative in the sense that the diagram xymatrix Rf_*K ⊗_O_Y^L Rf_*L ⊗_O_Y^L Rf_*M ar[r] ar[d] & Rf_*(K ⊗_O_X^L L) ⊗_O_Y^L Rf_*M ar[d] Rf_*K ⊗_O_Y^L Rf_*(L ⊗_O_X^L M) ar[r] & Rf_*(K ⊗_O_X^L L ⊗_O_X^L M) is commutative in D(O_Y) for all K, L, M in D(O_X).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces. The relative cup product of\nRemark \\ref{remark-cup-product} is associative in the sense that\nthe diagram\n$$\n\\xymatrix{\nRf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}\nRf_*L \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}\nRf_*M \\ar[r] \\ar[d] &\nRf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)\n\\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*M \\ar[d] \\\\\nRf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}\nRf_*(L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[r] &\nRf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \nL \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)\n}\n$$\nis commutative in $D(\\mathcal{O}_Y)$ for all $K, L, M$ in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FP4","source_file":"cohomology.tex","source_line":7659,"source_end_line":7679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7659-L7679","statement_sha256":"002efe3c143aff2250ea4fe79d0aaa2ab2dae8a4f577ea54b07cd6810e977126","origin":"The Stacks Project","memory_eligible":false,"source_rank":4261,"rank":4261,"depth":0,"x":1067.508,"y":719.039,"cluster":"sheaf-cohomology"},{"id":"stacks:0FP5","tag":"0FP5","title":"Cup product · Lemma 0FP5","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. The relative cup product of Remark [Tag 0B68] is commutative in the sense that the diagram xymatrix Rf_*K ⊗_O_Y^L Rf_*L ar[r] ar[d]_ψ & Rf_*(K ⊗_O_X^L L) ar[d]^Rf_*ψ Rf_*L ⊗_O_Y^L Rf_*K ar[r] & Rf_*(L ⊗_O_X^L K) is commutative in D(O_Y) for all K, L in D(O_X). Here ψ is the commutativity constraint on the derived category (Lemma [Tag 0FPB]).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces. The relative cup product of\nRemark \\ref{remark-cup-product} is commutative in the sense that\nthe diagram\n$$\n\\xymatrix{\nRf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L \\ar[r] \\ar[d]_\\psi &\nRf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\ar[d]^{Rf_*\\psi} \\\\\nRf_*L \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*K \\ar[r] &\nRf_*(L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K)\n}\n$$\nis commutative in $D(\\mathcal{O}_Y)$ for all $K, L$ in $D(\\mathcal{O}_X)$.\nHere $\\psi$ is the commutativity constraint on the derived category\n(Lemma \\ref{lemma-symmetric-monoidal-derived}).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FP5","source_file":"cohomology.tex","source_line":7697,"source_end_line":7714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7697-L7714","statement_sha256":"6565559afdae0397a30f4e6d18f79964831cb2341e4fb93514cf8541354ca0d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4262,"rank":4262,"depth":0,"x":1144.752,"y":615.468,"cluster":"sheaf-cohomology"},{"id":"stacks:0FP6","tag":"0FP6","title":"Cup product · Lemma 0FP6","summary":"Let f : (X, O_X) → (Y, O_Y) and g : (Y, O_Y) → (Z, O_Z) be morphisms of ringed spaces. The relative cup product of Remark [Tag 0B68] is compatible with compositions in the sense that the diagram xymatrix R(g ∘ f)_*K ⊗_O_Z^L R(g ∘ f)_*L ar@=[rr] ar[d] & & Rg_*Rf_*K ⊗_O_Z^L Rg_*Rf_*L ar[d] R(g ∘ f)_*(K ⊗_O_X^L L) ar@=[r] & Rg_*Rf_*(K ⊗_O_X^L L) & Rg_*(Rf_*K ⊗_O_Y^L Rf_*L) ar[l] is commutative in D(O_Z) for all K, L in D(O_X).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ and\n$g : (Y, \\mathcal{O}_Y) \\to (Z, \\mathcal{O}_Z)$\nbe morphisms of ringed spaces. The relative cup product of\nRemark \\ref{remark-cup-product} is compatible with compositions\nin the sense that the diagram\n$$\n\\xymatrix{\nR(g \\circ f)_*K \\otimes_{\\mathcal{O}_Z}^\\mathbf{L} R(g \\circ f)_*L\n\\ar@{=}[rr] \\ar[d] & &\nRg_*Rf_*K \\otimes_{\\mathcal{O}_Z}^\\mathbf{L} Rg_*Rf_*L \\ar[d] \\\\\nR(g \\circ f)_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\ar@{=}[r] &\nRg_*Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) &\nRg_*(Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}  Rf_*L) \\ar[l]\n}\n$$\nis commutative in $D(\\mathcal{O}_Z)$ for all $K, L$ in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FP6","source_file":"cohomology.tex","source_line":7720,"source_end_line":7738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7720-L7738","statement_sha256":"05d8c2dd9ff02bb1ccdad16125d03f748a9eb9bcb31c232f78e47d5dd54c3100","origin":"The Stacks Project","memory_eligible":false,"source_rank":4263,"rank":4263,"depth":2,"x":1171.198,"y":736.203,"cluster":"sheaf-cohomology"},{"id":"stacks:0H99","tag":"0H99","title":"Cup product · Lemma 0H99","summary":"Consider a commutative square xymatrix (X', O_X') ar[r]_g' ar[d]_f' & (X, O_X) ar[d]^f (Y', O_Y') ar[r]^g & (Y, O_Y) of ringed spaces. Let K, L in D(O_X). The relative cup product is compatible with the square in the sense that the diagram xymatrix Lg^*(Rf_*K ⊗_O_Y^L Rf_*L) ar[r] ar@=[d] & Lg^*(Rf_*(K ⊗_O_X^L L)) ar[d] Lg^*Rf_*K ⊗_O_Y'^L Lg^*Rf_*L ar[d] & R(f')_*L(g')^*(K ⊗_O_X^L L) ar@=[d] R(f')_*(L(g')^*K ⊗_O_Y' R(f')_*(L(g')^*L ar[r] & R(f')_*(L(g')^*K ⊗_O_X'^L…","statement_latex":"Consider a commutative square\n$$\n\\xymatrix{\n(X', \\mathcal{O}_{X'}) \\ar[r]_{g'} \\ar[d]_{f'} &\n(X, \\mathcal{O}_X) \\ar[d]^f \\\\\n(Y', \\mathcal{O}_{Y'}) \\ar[r]^g & (Y, \\mathcal{O}_Y) \n}\n$$\nof ringed spaces. Let $K, L$ in $D(\\mathcal{O}_X)$.\nThe relative cup product\nis compatible with the square in the sense that the diagram\n$$\n\\xymatrix{\nLg^*(Rf_*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*L)\n\\ar[r] \\ar@{=}[d] &\nLg^*(Rf_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)) \\ar[d] \\\\\nLg^*Rf_*K \\otimes_{\\mathcal{O}_{Y'}}^\\mathbf{L} Lg^*Rf_*L \\ar[d] &\nR(f')_*L(g')^*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L) \\ar@{=}[d] \\\\\nR(f')_*(L(g')^*K \\otimes_{\\mathcal{O}_{Y'}} R(f')_*(L(g')^*L \\ar[r] &\nR(f')_*(L(g')^*K \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} L(g')^*L)\n}\n$$\nis commutative in $D(\\mathcal{O}_{Y'})$. The horizontal arrows are given\nby the relative cup product (Remark \\ref{remark-cup-product})\nand the vertical arrows are given\nby the base change map (Remark \\ref{remark-base-change})\nand Lemma \\ref{lemma-pullback-tensor-product}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H99","source_file":"cohomology.tex","source_line":7765,"source_end_line":7794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7765-L7794","statement_sha256":"13aff1e8876f87a18678c0e730d285eac60bc1dd4738b3a514cce55067844a25","origin":"The Stacks Project","memory_eligible":false,"source_rank":4264,"rank":4264,"depth":2,"x":1054.094,"y":661.855,"cluster":"sheaf-cohomology"},{"id":"stacks:08BS","tag":"08BS","title":"Some properties of K-injective complexes · Lemma 08BS","summary":"Let X be a ringed space. Let U ⊂ X be an open subspace. The restriction of a K-injective complex of O_X-modules to U is a K-injective complex of O_U-modules.","statement_latex":"Let $X$ be a ringed space. Let $U \\subset X$ be an open subspace.\nThe restriction of a K-injective complex of $\\mathcal{O}_X$-modules\nto $U$ is a K-injective complex of $\\mathcal{O}_U$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BS","source_file":"cohomology.tex","source_line":7839,"source_end_line":7844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7839-L7844","statement_sha256":"768b9caebf2ea2b089f9f4be84a18673d08bffd6458652dbc98cfc8cd3297b62","origin":"The Stacks Project","memory_eligible":false,"source_rank":4265,"rank":4265,"depth":3,"x":1200.87,"y":650.18,"cluster":"sheaf-cohomology"},{"id":"stacks:0D5V","tag":"0D5V","title":"Some properties of K-injective complexes · Lemma 0D5V","summary":"Let X be a ringed space. Let U ⊂ X be an open subspace. For K in D(O_X) we have H^p(U, K) = H^p(U, K|_U).","statement_latex":"Let $X$ be a ringed space. Let $U \\subset X$ be an open subspace.\nFor $K$ in $D(\\mathcal{O}_X)$ we have\n$H^p(U, K) = H^p(U, K|_U)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5V","source_file":"cohomology.tex","source_line":7856,"source_end_line":7861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7856-L7861","statement_sha256":"5f6728b0b358eb472c7ffc5f18e75aa0c3ebafda32476e6600543be480db397b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4266,"rank":4266,"depth":4,"x":1101.601,"y":742.469,"cluster":"sheaf-cohomology"},{"id":"stacks:0BKJ","tag":"0BKJ","title":"Some properties of K-injective complexes · Lemma 0BKJ","summary":"Let (X, O_X) be a ringed space. Let K be an object of D(O_X). The sheafification of U ↦ H^q(U, K) = H^q(U, K|_U) is the qth cohomology sheaf H^q(K) of K.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ be an object of\n$D(\\mathcal{O}_X)$. The sheafification of\n$$\nU \\mapsto H^q(U, K) = H^q(U, K|_U)\n$$\nis the $q$th cohomology sheaf $H^q(K)$ of $K$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKJ","source_file":"cohomology.tex","source_line":7875,"source_end_line":7883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7875-L7883","statement_sha256":"1fc4e6a97db2e9885c88cfec102de0a63bce6d445e62b993f3831994ab57f965","origin":"The Stacks Project","memory_eligible":false,"source_rank":4267,"rank":4267,"depth":5,"x":1100.58,"y":617.557,"cluster":"sheaf-cohomology"},{"id":"stacks:08FE","tag":"08FE","title":"Some properties of K-injective complexes · Lemma 08FE","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Given an open subspace V ⊂ Y, set U = f^-1(V) and denote g : U → V the induced morphism. Then (Rf_*E)|_V = Rg_*(E|_U) for E in D(O_X).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed\nspaces. Given an open subspace $V \\subset Y$, set $U = f^{-1}(V)$ and denote\n$g : U \\to V$ the induced morphism. Then\n$(Rf_*E)|_V = Rg_*(E|_U)$ for $E$ in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FE","source_file":"cohomology.tex","source_line":7900,"source_end_line":7906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7900-L7906","statement_sha256":"c556acfc57304572c5fbe1eef7ac0e3c9ee9159c266583a0b4f4e3a620bb9fb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4268,"rank":4268,"depth":4,"x":1202.213,"y":709.475,"cluster":"sheaf-cohomology"},{"id":"stacks:0D5W","tag":"0D5W","title":"Some properties of K-injective complexes · Lemma 0D5W","summary":"Let f : X → Y be a morphism of ringed spaces. Then RΓ(Y, -) ∘ Rf_* = RΓ(X, -) as functors D(O_X) → D(Γ(Y, O_Y)). More generally for V ⊂ Y open and U = f^-1(V) we have RΓ(U, -) = RΓ(V, -) ∘ Rf_*.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nThen $R\\Gamma(Y, -) \\circ Rf_* = R\\Gamma(X, -)$ as functors\n$D(\\mathcal{O}_X) \\to D(\\Gamma(Y, \\mathcal{O}_Y))$.\nMore generally for $V \\subset Y$ open and $U = f^{-1}(V)$\nwe have $R\\Gamma(U, -) = R\\Gamma(V, -) \\circ Rf_*$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5W","source_file":"cohomology.tex","source_line":7917,"source_end_line":7924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7917-L7924","statement_sha256":"1bf715af063b4b6aec84d1af31af65eb387fb16ee64149221973d3330ada5511","origin":"The Stacks Project","memory_eligible":false,"source_rank":4269,"rank":4269,"depth":20,"x":1052.726,"y":699.321,"cluster":"sheaf-cohomology"},{"id":"stacks:0D5X","tag":"0D5X","title":"Some properties of K-injective complexes · Lemma 0D5X","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let K be in D(O_X). Then H^i(Rf_*K) is the sheaf associated to the presheaf V ↦ H^i(f^-1(V), K) = H^i(V, Rf_*K)","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed\nspaces. Let $K$ be in $D(\\mathcal{O}_X)$. Then $H^i(Rf_*K)$ is the sheaf\nassociated to the presheaf\n$$\nV \\mapsto H^i(f^{-1}(V), K) = H^i(V, Rf_*K)\n$$","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5X","source_file":"cohomology.tex","source_line":7941,"source_end_line":7949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7941-L7949","statement_sha256":"cba838099a548cdfa28198d042d11f102bdeedc07ebbd1368166f2ccece8a491","origin":"The Stacks Project","memory_eligible":false,"source_rank":4270,"rank":4270,"depth":21,"x":1171.615,"y":621.664,"cluster":"sheaf-cohomology"},{"id":"stacks:0D5Y","tag":"0D5Y","title":"Some properties of K-injective complexes · Lemma 0D5Y","summary":"Let X be a ringed space. Let K be an object of D(O_X) and denote K_ab its image in D(underlineZ_X). • For any open U ⊂ X there is a canonical map RΓ(U, K) → RΓ(U, K_ab) which is an isomorphism in D(Ab). • Let f : X → Y be a morphism of ringed spaces. There is a canonical map Rf_*K → Rf_*(K_ab) which is an isomorphism in D(underlineZ_Y).","statement_latex":"Let $X$ be a ringed space. Let $K$ be an object of $D(\\mathcal{O}_X)$\nand denote $K_{ab}$ its image in $D(\\underline{\\mathbf{Z}}_X)$.\n\\begin{enumerate}\n\\item For any open $U \\subset X$ there is a canonical map\n$R\\Gamma(U, K) \\to R\\Gamma(U, K_{ab})$\nwhich is an isomorphism in $D(\\textit{Ab})$.\n\\item Let $f : X \\to Y$ be a morphism of ringed spaces.\nThere is a canonical map $Rf_*K \\to Rf_*(K_{ab})$ which\nis an isomorphism in $D(\\underline{\\mathbf{Z}}_Y)$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5Y","source_file":"cohomology.tex","source_line":7958,"source_end_line":7970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L7958-L7970","statement_sha256":"e31b85d6a03ccbfa22103df7b261420239832b096ea7ba5c501f3657aeb12f7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4271,"rank":4271,"depth":22,"x":1146.292,"y":746.904,"cluster":"sheaf-cohomology"},{"id":"stacks:08BT","tag":"08BT","title":"Some properties of K-injective complexes · Lemma 08BT","summary":"Let (X, O_X) be a ringed space. Let U ⊂ X be an open subset. Denote j : (U, O_U) → (X, O_X) the corresponding open immersion. The restriction functor D(O_X) → D(O_U) is a right adjoint to extension by zero j_! : D(O_U) → D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $U \\subset X$ be an\nopen subset. Denote $j : (U, \\mathcal{O}_U) \\to (X, \\mathcal{O}_X)$\nthe corresponding open immersion. The restriction functor\n$D(\\mathcal{O}_X) \\to D(\\mathcal{O}_U)$ is a right adjoint to\nextension by zero $j_! : D(\\mathcal{O}_U) \\to D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BT","source_file":"cohomology.tex","source_line":8037,"source_end_line":8044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8037-L8044","statement_sha256":"d673a2e6faa974c0aa13d514e766eee79f75bb586556526d1d65fd19aa465932","origin":"The Stacks Project","memory_eligible":false,"source_rank":4272,"rank":4272,"depth":5,"x":1063.915,"y":639.747,"cluster":"sheaf-cohomology"},{"id":"stacks:0D5Z","tag":"0D5Z","title":"Some properties of K-injective complexes · Lemma 0D5Z","summary":"Let f : X → Y be a flat morphism of ringed spaces. If I^bullet is a K-injective complex of O_X-modules, then f_*I^bullet is K-injective as a complex of O_Y-modules.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of ringed spaces.\nIf $\\mathcal{I}^\\bullet$ is a K-injective complex of $\\mathcal{O}_X$-modules,\nthen $f_*\\mathcal{I}^\\bullet$ is K-injective as a complex of\n$\\mathcal{O}_Y$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5Z","source_file":"cohomology.tex","source_line":8052,"source_end_line":8058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8052-L8058","statement_sha256":"e084c44a293074017440346f6cebb1fa1c4809c4d2512cbc6c9c425c6522e675","origin":"The Stacks Project","memory_eligible":false,"source_rank":4273,"rank":4273,"depth":5,"x":1211.429,"y":672.152,"cluster":"sheaf-cohomology"},{"id":"stacks:08BU","tag":"08BU","title":"Unbounded Mayer-Vietoris · Lemma 08BU","summary":"Let (X, O_X) be a ringed space. Let X = U ∪ V be the union of two open subspaces. For any object E of D(O_X) we have a distinguished triangle j_U ∩ V!E|_U ∩ V → j_U!E|_U ⊕ j_V!E|_V → E → j_U ∩ V!E|_U ∩ V[1] in D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $X = U \\cup V$ be the union of two open subspaces.\nFor any object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished\ntriangle\n$$\nj_{U \\cap V!}E|_{U \\cap V} \\to\nj_{U!}E|_U \\oplus j_{V!}E|_V \\to E \\to \nj_{U \\cap V!}E|_{U \\cap V}[1]\n$$\nin $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Unbounded Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BU","source_file":"cohomology.tex","source_line":8083,"source_end_line":8095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8083-L8095","statement_sha256":"e150de7664e587702d9fbaff32edebcd4e1dcf044901df896687ba0fd9ec47a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4274,"rank":4274,"depth":2,"x":1076.051,"y":732.2,"cluster":"sheaf-cohomology"},{"id":"stacks:08BV","tag":"08BV","title":"Unbounded Mayer-Vietoris · Lemma 08BV","summary":"Let (X, O_X) be a ringed space. Let X = U ∪ V be the union of two open subspaces. For any object E of D(O_X) we have a distinguished triangle E → Rj_U, *E|_U ⊕ Rj_V, *E|_V → Rj_U ∩ V, *E|_U ∩ V → E[1] in D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $X = U \\cup V$ be the union of two open subspaces.\nFor any object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished\ntriangle\n$$\nE \\to \nRj_{U, *}E|_U \\oplus Rj_{V, *}E|_V \\to\nRj_{U \\cap V, *}E|_{U \\cap V} \\to\nE[1]\n$$\nin $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Unbounded Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BV","source_file":"cohomology.tex","source_line":8118,"source_end_line":8131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8118-L8131","statement_sha256":"e051323821ff58d7942f37872deb7ec277cf5eece1ea25b03b6b0274ffb71cb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4275,"rank":4275,"depth":8,"x":1127.795,"y":610.621,"cluster":"sheaf-cohomology"},{"id":"stacks:08BW","tag":"08BW","title":"Unbounded Mayer-Vietoris · Lemma 08BW","summary":"Let (X, O_X) be a ringed space. Let X = U ∪ V be the union of two open subspaces of X. For objects E, F of D(O_X) we have a Mayer-Vietoris sequence xymatrix & … ar[r] & Ext^-1(E_U ∩ V, F_U ∩ V) ar[lld] Hom(E, F) ar[r] & Hom(E_U, F_U) ⊕ Hom(E_V, F_V) ar[r] & Hom(E_U ∩ V, F_U ∩ V) where the subscripts denote restrictions to the relevant opens and the Hom's and Ext's are taken in the relevant derived categories.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $X = U \\cup V$ be\nthe union of two open subspaces of $X$.\nFor objects $E$, $F$ of $D(\\mathcal{O}_X)$ we have a\nMayer-Vietoris sequence\n$$\n\\xymatrix{\n& \\ldots \\ar[r] & \\Ext^{-1}(E_{U \\cap V}, F_{U \\cap V}) \\ar[lld] \\\\\n\\Hom(E, F) \\ar[r] &\n\\Hom(E_U, F_U) \\oplus\n\\Hom(E_V, F_V) \\ar[r] &\n\\Hom(E_{U \\cap V}, F_{U \\cap V})\n}\n$$\nwhere the subscripts denote restrictions to the relevant opens\nand the $\\Hom$'s and $\\Ext$'s are taken in the relevant\nderived categories.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Unbounded Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BW","source_file":"cohomology.tex","source_line":8166,"source_end_line":8184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8166-L8184","statement_sha256":"1ad66388ec254f9b64d6eb5e167609d9e7664072a2810f55f752092b168997a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4276,"rank":4276,"depth":6,"x":1187.646,"y":730.105,"cluster":"sheaf-cohomology"},{"id":"stacks:08BX","tag":"08BX","title":"Unbounded Mayer-Vietoris · Lemma 08BX","summary":"Let (X, O_X) be a ringed space. Suppose that X = U ∪ V is a union of two open subsets. For an object E of D(O_X) we have a distinguished triangle RΓ(X, E) → RΓ(U, E) ⊕ RΓ(V, E) → RΓ(U ∩ V, E) → RΓ(X, E)[1] and in particular a long exact cohomology sequence … → H^n(X, E) → H^n(U, E) ⊕ H^0(V, E) → H^n(U ∩ V, E) → H^n + 1(X, E) → … The construction of the distinguished triangle and the long exact sequence is functorial in E.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Suppose that\n$X = U \\cup V$ is a union of two open subsets. For an object $E$\nof $D(\\mathcal{O}_X)$ we have a distinguished triangle\n$$\nR\\Gamma(X, E) \\to R\\Gamma(U, E) \\oplus R\\Gamma(V, E) \\to\nR\\Gamma(U \\cap V, E) \\to R\\Gamma(X, E)[1]\n$$\nand in particular a long exact cohomology sequence\n$$\n\\ldots \\to\nH^n(X, E) \\to\nH^n(U, E) \\oplus H^0(V, E) \\to\nH^n(U \\cap V, E) \\to\nH^{n + 1}(X, E) \\to \\ldots\n$$\nThe construction of the distinguished triangle and the\nlong exact sequence is functorial in $E$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Unbounded Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BX","source_file":"cohomology.tex","source_line":8199,"source_end_line":8218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8199-L8218","statement_sha256":"30a4e32b5f782acac8a003fb85dbdb487b3d05b5cf0e2fd994fc7846b9f5c6bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4277,"rank":4277,"depth":8,"x":1046.874,"y":675.746,"cluster":"sheaf-cohomology"},{"id":"stacks:08HZ","tag":"08HZ","title":"Unbounded Mayer-Vietoris · Lemma 08HZ","summary":"Let f : X → Y be a morphism of ringed spaces. Suppose that X = U ∪ V is a union of two open subsets. Denote a = f|_U : U → Y, b = f|_V : V → Y, and c = f|_U ∩ V : U ∩ V → Y. For every object E of D(O_X) there exists a distinguished triangle Rf_*E → Ra_*(E|_U) ⊕ Rb_*(E|_V) → Rc_*(E|_U ∩ V) → Rf_*E[1] This triangle is functorial in E.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nSuppose that $X = U \\cup V$ is a union of two open subsets.\nDenote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and\n$c = f|_{U \\cap V} : U \\cap V \\to Y$.\nFor every object $E$ of $D(\\mathcal{O}_X)$ there exists a\ndistinguished triangle\n$$\nRf_*E \\to\nRa_*(E|_U) \\oplus Rb_*(E|_V) \\to\nRc_*(E|_{U \\cap V}) \\to\nRf_*E[1]\n$$\nThis triangle is functorial in $E$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Unbounded Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HZ","source_file":"cohomology.tex","source_line":8249,"source_end_line":8264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8249-L8264","statement_sha256":"4e00ec963acbebb5f4e995e5c7e179e0321874668fd84dd142c5de6e93b42918","origin":"The Stacks Project","memory_eligible":false,"source_rank":4278,"rank":4278,"depth":10,"x":1194.968,"y":635.798,"cluster":"sheaf-cohomology"},{"id":"stacks:08DF","tag":"08DF","title":"Unbounded Mayer-Vietoris · Lemma 08DF","summary":"Let (X, O_X) be a ringed space. Let j : U → X be an open subspace. Let T ⊂ X be a closed subset contained in U. • If E is an object of D(O_X) whose cohomology sheaves are supported on T, then E → Rj_*(E|_U) is an isomorphism. • If F is an object of D(O_U) whose cohomology sheaves are supported on T, then j_!F → Rj_*F is an isomorphism.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $j : U \\to X$ be an\nopen subspace. Let $T \\subset X$ be a closed subset contained in $U$.\n\\begin{enumerate}\n\\item If $E$ is an object of $D(\\mathcal{O}_X)$ whose cohomology sheaves\nare supported on $T$, then $E \\to Rj_*(E|_U)$ is an isomorphism.\n\\item If $F$ is an object of $D(\\mathcal{O}_U)$ whose cohomology sheaves\nare supported on $T$, then $j_!F \\to Rj_*F$ is an isomorphism.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Unbounded Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DF","source_file":"cohomology.tex","source_line":8297,"source_end_line":8307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8297-L8307","statement_sha256":"2c6639d647be967240912e34ec1d6a176b1a7d93b8435e10dd6be83ccc3a1aaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4279,"rank":4279,"depth":5,"x":1117.592,"y":749.727,"cluster":"sheaf-cohomology"},{"id":"stacks:0G6X","tag":"0G6X","title":"Unbounded Mayer-Vietoris · Lemma 0G6X","summary":"Let (X, O_X) be a ringed space. Set A = Γ(X, O_X). Suppose that X = U ∪ V is a union of two open subsets. For objects K and M of D(O_X) we have a map of distinguished triangles xymatrix RΓ(X, K) ⊗_A^L RΓ(X, M) ar[r] ar[d] & RΓ(X, K ⊗_O_X^L M) ar[d] RΓ(X, K) ⊗_A^L (RΓ(U, M) ⊕ RΓ(V, M)) ar[r] ar[d] & RΓ(U, K ⊗_O_X^L M) ⊕ RΓ(V, K ⊗_O_X^L M)) ar[d] RΓ(X, K) ⊗_A^L RΓ(U ∩ V, M) ar[r] ar[d] & RΓ(U ∩ V, K ⊗_O_X^L M) ar[d] RΓ(X, K) ⊗_A^L RΓ(X, M)[1] ar[r] & RΓ(X, K ⊗_O_X^L M)[1]…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Set $A = \\Gamma(X, \\mathcal{O}_X)$.\nSuppose that $X = U \\cup V$ is a union of two open subsets. For objects\n$K$ and $M$ of $D(\\mathcal{O}_X)$ we have a map of distinguished triangles\n$$\n\\xymatrix{\nR\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(X, M) \\ar[r] \\ar[d] &\nR\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[d] \\\\\nR\\Gamma(X, K) \\otimes_A^\\mathbf{L}\n(R\\Gamma(U, M) \\oplus R\\Gamma(V, M)) \\ar[r] \\ar[d] &\nR\\Gamma(U, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)\n\\oplus R\\Gamma(V, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)) \\ar[d] \\\\\nR\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(U \\cap V, M) \\ar[r] \\ar[d] &\nR\\Gamma(U \\cap V, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[d] \\\\\nR\\Gamma(X, K) \\otimes_A^\\mathbf{L} R\\Gamma(X, M)[1] \\ar[r] &\nR\\Gamma(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)[1]\n}\n$$\nwhere\n\\begin{enumerate}\n\\item the horizontal arrows are given by cup product,\n\\item on the right hand side we have the distinguished triangle\nof Lemma \\ref{lemma-unbounded-mayer-vietoris} for\n$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$, and\n\\item on the left hand side we have the exact functor\n$R\\Gamma(X, K) \\otimes_A^\\mathbf{L} - $ applied to the\ndistinguished triangle of Lemma \\ref{lemma-unbounded-mayer-vietoris} for $M$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Unbounded Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6X","source_file":"cohomology.tex","source_line":8326,"source_end_line":8355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8326-L8355","statement_sha256":"9b2f40c216e0eb6c8a1774be167c879be24019240fa5c2256e54c3642cdfe368","origin":"The Stacks Project","memory_eligible":false,"source_rank":4280,"rank":4280,"depth":17,"x":1082.897,"y":621.32,"cluster":"sheaf-cohomology"},{"id":"stacks:0A3B","tag":"0A3B","title":"Cohomology with support in a closed subset, II · Lemma 0A3B","summary":"Let (X, O_X) be a ringed space. Let i : Z → X be the inclusion of a closed subset. • RH_Z : D(O_X) → D(O_X|_Z) is right adjoint to i_* : D(O_X|_Z) → D(O_X). • For K in D(O_X|_Z) we have RH_Z(i_*K) = K. • Let G be a sheaf of O_X|_Z-modules on Z. Then H^p_Z(i_*G) = 0 for p > 0.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the\ninclusion of a closed subset.\n\\begin{enumerate}\n\\item $R\\mathcal{H}_Z : D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X|_Z)$\nis right adjoint to $i_* : D(\\mathcal{O}_X|_Z) \\to D(\\mathcal{O}_X)$.\n\\item For $K$ in $D(\\mathcal{O}_X|_Z)$ we have $R\\mathcal{H}_Z(i_*K) = K$.\n\\item Let $\\mathcal{G}$ be a sheaf of\n$\\mathcal{O}_X|_Z$-modules on $Z$. Then\n$\\mathcal{H}^p_Z(i_*\\mathcal{G}) = 0$ for $p > 0$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3B","source_file":"cohomology.tex","source_line":8498,"source_end_line":8510,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8498-L8510","statement_sha256":"806187c38e65a625afba964aeca9c8a4d326968949dc625d963de1be233a9961","origin":"The Stacks Project","memory_eligible":false,"source_rank":4281,"rank":4281,"depth":6,"x":1212.233,"y":696.605,"cluster":"sheaf-cohomology"},{"id":"stacks:0AEF","tag":"0AEF","title":"Cohomology with support in a closed subset, II · Lemma 0AEF","summary":"Let (X, O_X) be a ringed space. Let i : Z → X be the inclusion of a closed subset. • For K in D(O_X|_Z) we have i_*K in D_Z(O_X). • The functor i_* : D(O_X|_Z) → D_Z(O_X) is an equivalence with quasi-inverse i^-1|_D_Z(O_X) = RH_Z|_D_Z(O_X). • The functor i_* ∘ RH_Z : D(O_X) → D_Z(O_X) is right adjoint to the inclusion functor D_Z(O_X) → D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the\ninclusion of a closed subset.\n\\begin{enumerate}\n\\item For $K$ in $D(\\mathcal{O}_X|_Z)$ we have $i_*K$ in $D_Z(\\mathcal{O}_X)$.\n\\item The functor $i_* : D(\\mathcal{O}_X|_Z) \\to D_Z(\\mathcal{O}_X)$\nis an equivalence with quasi-inverse\n$i^{-1}|_{D_Z(\\mathcal{O}_X)} = R\\mathcal{H}_Z|_{D_Z(\\mathcal{O}_X)}$.\n\\item The functor\n$i_* \\circ R\\mathcal{H}_Z : D(\\mathcal{O}_X) \\to D_Z(\\mathcal{O}_X)$\nis right adjoint to the inclusion functor\n$D_Z(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEF","source_file":"cohomology.tex","source_line":8542,"source_end_line":8556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8542-L8556","statement_sha256":"53aee6182ed2e721728319f64126d0177683931a32149369816f2bc872b4a9c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4282,"rank":4282,"depth":7,"x":1055.73,"y":714.546,"cluster":"sheaf-cohomology"},{"id":"stacks:0G6Z","tag":"0G6Z","title":"Cohomology with support in a closed subset, II · Lemma 0G6Z","summary":"Let (X, O_X) be a ringed space. Let i : Z → X be the inclusion of a closed subset. If I^bullet is a K-injective complex of O_X-modules, then H_Z(I^bullet) is K-injective complex of O_X|_Z-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$\nbe the inclusion of a closed subset. If $\\mathcal{I}^\\bullet$ is a K-injective\ncomplex of $\\mathcal{O}_X$-modules, then\n$\\mathcal{H}_Z(\\mathcal{I}^\\bullet)$ is K-injective complex of\n$\\mathcal{O}_X|_Z$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G6Z","source_file":"cohomology.tex","source_line":8594,"source_end_line":8601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8594-L8601","statement_sha256":"42c936df43081d01e8b960de8a47d6e030e63a9ba2e25c70504944da33628b0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4283,"rank":4283,"depth":1,"x":1157.086,"y":612.13,"cluster":"sheaf-cohomology"},{"id":"stacks:0G70","tag":"0G70","title":"Cohomology with support in a closed subset, II · Lemma 0G70","summary":"Let (X, O_X) be a ringed space. Let i : Z → X be the inclusion of a closed subset. Then RΓ(Z, - ) ∘ RH_Z = RΓ_Z(X, - ) as functors D(O_X) → D(O_X(X)).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the\ninclusion of a closed subset. Then\n$R\\Gamma(Z, - ) \\circ R\\mathcal{H}_Z = R\\Gamma_Z(X, - )$\nas functors $D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X(X))$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G70","source_file":"cohomology.tex","source_line":8611,"source_end_line":8617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8611-L8617","statement_sha256":"8d9c0872ba5a3b32124c5397107e90e119d0b0837eb6f04595fbce93c3e45ff3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4284,"rank":4284,"depth":2,"x":1164.735,"y":745.658,"cluster":"sheaf-cohomology"},{"id":"stacks:0G71","tag":"0G71","title":"Cohomology with support in a closed subset, II · Lemma 0G71","summary":"Let (X, O_X) be a ringed space. Let i : Z → X be the inclusion of a closed subset. Let U = X setminus Z. There is a distinguished triangle RΓ_Z(X, K) → RΓ(X, K) → RΓ(U, K) → RΓ_Z(X, K)[1] in D(O_X(X)) functorial for K in D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the\ninclusion of a closed subset. Let $U = X \\setminus Z$.\nThere is a distinguished triangle\n$$\nR\\Gamma_Z(X, K) \\to R\\Gamma(X, K) \\to R\\Gamma(U, K) \\to\nR\\Gamma_Z(X, K)[1]\n$$\nin $D(\\mathcal{O}_X(X))$ functorial for $K$ in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G71","source_file":"cohomology.tex","source_line":8625,"source_end_line":8635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8625-L8635","statement_sha256":"4504ba682e3331f667bbec0bbf9560fcada95c37ccfa4f1cd046cb02359ee087","origin":"The Stacks Project","memory_eligible":false,"source_rank":4285,"rank":4285,"depth":4,"x":1051.297,"y":651.188,"cluster":"sheaf-cohomology"},{"id":"stacks:0G72","tag":"0G72","title":"Cohomology with support in a closed subset, II · Lemma 0G72","summary":"Let (X, O_X) be a ringed space. Let i : Z → X be the inclusion of a closed subset. Denote j : U = X setminus Z → X the inclusion of the complement. There is a distinguished triangle i_*RH_Z(K) → K → Rj_*(K|_U) → i_*RH_Z(K)[1] in D(O_X) functorial for K in D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $i : Z \\to X$ be the\ninclusion of a closed subset. Denote $j : U = X \\setminus Z \\to X$\nthe inclusion of the complement. There is a distinguished triangle\n$$\ni_*R\\mathcal{H}_Z(K) \\to K \\to Rj_*(K|_U) \\to\ni_*R\\mathcal{H}_Z(K)[1]\n$$\nin $D(\\mathcal{O}_X)$ functorial for $K$ in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G72","source_file":"cohomology.tex","source_line":8656,"source_end_line":8666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8656-L8666","statement_sha256":"aee64e58604d53aae9b4d9443086f0a27b881aed78e46caa855d195ec6dce245","origin":"The Stacks Project","memory_eligible":false,"source_rank":4286,"rank":4286,"depth":4,"x":1211.502,"y":656.502,"cluster":"sheaf-cohomology"},{"id":"stacks:0G73","tag":"0G73","title":"Cohomology with support in a closed subset, II · Lemma 0G73","summary":"Let (X, O_X) be a ringed space. Let Z ⊂ X be a closed subset. Let j : U → X be the inclusion of an open subset with U ∩ Z = ∅. Then RH_Z(Rj_*K) = 0 for all K in D(O_U).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $Z \\subset X$\nbe a closed subset. Let $j : U \\to X$ be the inclusion of\nan open subset with $U \\cap Z = \\emptyset$. Then\n$R\\mathcal{H}_Z(Rj_*K) = 0$ for all $K$ in $D(\\mathcal{O}_U)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G73","source_file":"cohomology.tex","source_line":8687,"source_end_line":8693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8687-L8693","statement_sha256":"7194e16ace36b1e1f0b28f51551a02b6cbc6a5afcbfbb9556613cbb37f177be6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4287,"rank":4287,"depth":6,"x":1088.649,"y":743.804,"cluster":"sheaf-cohomology"},{"id":"stacks:0G74","tag":"0G74","title":"Cohomology with support in a closed subset, II · Lemma 0G74","summary":"Let (X, O_X) be a ringed space. Let Z ⊂ X be a closed subset. Let K be an object of D(O_X) and denote K_ab its image in D(underlineZ_X). • There is a canonical map RΓ_Z(X, K) → RΓ_Z(X, K_ab) which is an isomorphism in D(Ab). • There is a canonical map RH_Z(K) → RH_Z(K_ab) which is an isomorphism in D(underlineZ_Z).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $Z \\subset X$\nbe a closed subset. Let $K$ be an object of $D(\\mathcal{O}_X)$\nand denote $K_{ab}$ its image in $D(\\underline{\\mathbf{Z}}_X)$.\n\\begin{enumerate}\n\\item There is a canonical map $R\\Gamma_Z(X, K) \\to R\\Gamma_Z(X, K_{ab})$\nwhich is an isomorphism in $D(\\textit{Ab})$.\n\\item There is a canonical map\n$R\\mathcal{H}_Z(K) \\to R\\mathcal{H}_Z(K_{ab})$\nwhich is an isomorphism in $D(\\underline{\\mathbf{Z}}_Z)$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G74","source_file":"cohomology.tex","source_line":8710,"source_end_line":8722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8710-L8722","statement_sha256":"5c736c37fd068a9c817827100c7ed2059d415e75775b05f4c868c9e23663436c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4288,"rank":4288,"depth":23,"x":1109.107,"y":609.234,"cluster":"sheaf-cohomology"},{"id":"stacks:0G77","tag":"0G77","title":"Cohomology with support in a closed subset, II · Lemma 0G77","summary":"With notation as in Remark [Tag 0G75] the diagram xymatrix H^i(X, K) × H^j_Z(X, M) ar[r] ar[d] & H^i + j_Z(X, K ⊗_O_X^L M) ar[d] H^i(X, K) × H^j(X, M) ar[r] & H^i + j(X, K ⊗_O_X^L M) commutes where the top horizontal arrow is the cup product of Remark [Tag 0G76].","statement_latex":"With notation as in Remark \\ref{remark-support-cup-product} the diagram\n$$\n\\xymatrix{\nH^i(X, K) \\times H^j_Z(X, M) \\ar[r] \\ar[d] &\nH^{i + j}_Z(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M) \\ar[d] \\\\\nH^i(X, K) \\times H^j(X, M) \\ar[r] &\nH^{i + j}(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)\n}\n$$\ncommutes where the top horizontal arrow is the cup product of\nRemark \\ref{remark-support-cup-product-global}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G77","source_file":"cohomology.tex","source_line":8831,"source_end_line":8844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8831-L8844","statement_sha256":"48f81360d564d6a413894db5f49a4ec266d4b09a24cc414af4371a7df7deb170","origin":"The Stacks Project","memory_eligible":false,"source_rank":4289,"rank":4289,"depth":0,"x":1202.574,"y":720.471,"cluster":"sheaf-cohomology"},{"id":"stacks:0G79","tag":"0G79","title":"Cohomology with support in a closed subset, II · Lemma 0G79","summary":"With notation and assumptions as in Remark [Tag 0G78] the diagram xymatrix H^p_Z(X, K) ar[r] ar[d] & H^p_Z'(X', Lf^*K) ar[d] H^p(X, K) ar[r] & H^p(X', Lf^*K) commutes. Here the top horizontal arrow comes from the identifications H^p_Z(X, K) = H^p(Z, RH_Z(K)) and H^p_Z'(X', Lf^*K) = H^p(Z', RH_Z'(K')), the pullback map H^p(Z, RH_Z(K)) → H^p(Z', L(f|_Z')^*RH_Z(K)), and the map constructed in Remark [Tag 0G78].","statement_latex":"With notation and assumptions as in Remark \\ref{remark-support-functorial}\nthe diagram\n$$\n\\xymatrix{\nH^p_Z(X, K) \\ar[r] \\ar[d] & H^p_{Z'}(X', Lf^*K) \\ar[d] \\\\\nH^p(X, K) \\ar[r] & H^p(X', Lf^*K)\n}\n$$\ncommutes. Here the top horizontal arrow comes from the identifications\n$H^p_Z(X, K) = H^p(Z, R\\mathcal{H}_Z(K))$ and\n$H^p_{Z'}(X', Lf^*K) = H^p(Z', R\\mathcal{H}_{Z'}(K'))$,\nthe pullback map\n$H^p(Z, R\\mathcal{H}_Z(K)) \\to H^p(Z', L(f|_{Z'})^*R\\mathcal{H}_Z(K))$,\nand the map constructed in Remark \\ref{remark-support-functorial}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Cohomology with support in a closed subset, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G79","source_file":"cohomology.tex","source_line":8892,"source_end_line":8908,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8892-L8908","statement_sha256":"907176922b8bbb66e44783b6d8645f8e9b67f30576434d3f5c7913cb029c9ef5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4290,"rank":4290,"depth":0,"x":1043.633,"y":691.378,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYK","tag":"0GYK","title":"Inverse systems and cohomology, I · Lemma 0GYK","summary":"Let I be an ideal of a ring A. Let X be a topological space. Let … → F_3 → F_2 → F_1 be an inverse system of sheaves of A-modules on X such that F_n = F_n + 1/I^nF_n + 1. Let p ≥ 0. Assume bigoplus_n ≥ 0 H^p + 1(X, I^nF_n + 1) satisfies the ascending chain condition as a graded bigoplus_n ≥ 0 I^n/I^n + 1-module. Then the inverse system M_n = H^p(X, F_n) satisfies the Mittag-Leffler condition) is the stable image for all n ≥ c..","statement_latex":"Let $I$ be an ideal of a ring $A$. Let $X$ be a topological space.\nLet\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of sheaves of $A$-modules on $X$\nsuch that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nLet $p \\geq 0$. Assume\n$$\n\\bigoplus\\nolimits_{n \\geq 0} H^{p + 1}(X, I^n\\mathcal{F}_{n + 1})\n$$\nsatisfies the ascending chain condition as a graded\n$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module.\nThen the inverse system $M_n = H^p(X, \\mathcal{F}_n)$ satisfies the\nMittag-Leffler condition\\footnote{In fact, there exists\na $c \\geq 0$ such that $\\Im(M_n \\to M_{n - c})$ is the stable image\nfor all $n \\geq c$.}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYK","source_file":"cohomology.tex","source_line":8930,"source_end_line":8949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8930-L8949","statement_sha256":"11b319f9f077da4f2ba1a29a8d9a639742eeb2d08acd5c352e8a94e96c501d30","origin":"The Stacks Project","memory_eligible":false,"source_rank":4291,"rank":4291,"depth":0,"x":1184.728,"y":622.401,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYL","tag":"0GYL","title":"Inverse systems and cohomology, I · Lemma 0GYL","summary":"Let I be an ideal of a ring A. Let X be a topological space. Let … → F_3 → F_2 → F_1 be an inverse system of A-modules on X such that F_n = F_n + 1/I^nF_n + 1. Let p ≥ 0. Given n define N_n = ⋂_m ≥ n Im( H^p + 1(X, I^nF_m + 1) → H^p + 1(X, I^nF_n + 1) ) If bigoplus N_n satisfies the ascending chain condition as a graded bigoplus_n ≥ 0 I^n/I^n + 1-module, then the inverse system M_n = H^p(X, F_n) satisfies the Mittag-Leffler condition) is the stable image for all n ≥ c..","statement_latex":"Let $I$ be an ideal of a ring $A$. Let $X$ be a topological space. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of $A$-modules on $X$\nsuch that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nLet $p \\geq 0$. Given $n$ define\n$$\nN_n =\n\\bigcap\\nolimits_{m \\geq n}\n\\Im\\left(\nH^{p + 1}(X, I^n\\mathcal{F}_{m + 1}) \\to H^{p + 1}(X, I^n\\mathcal{F}_{n + 1})\n\\right)\n$$\nIf $\\bigoplus N_n$ satisfies the ascending chain condition as a graded\n$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module, then the inverse system\n$M_n = H^p(X, \\mathcal{F}_n)$ satisfies the Mittag-Leffler\ncondition\\footnote{In fact, there exists\na $c \\geq 0$ such that $\\Im(M_n \\to M_{n - c})$ is the stable image\nfor all $n \\geq c$.}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYL","source_file":"cohomology.tex","source_line":8988,"source_end_line":9010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L8988-L9010","statement_sha256":"a7e7c1365c5b5e7af127e884da9a4ad7344b836c388c399f3f4617fdbb8d63c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4292,"rank":4292,"depth":1,"x":1135.984,"y":753.784,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYM","tag":"0GYM","title":"Inverse systems and cohomology, I · Lemma 0GYM","summary":"Let I be an ideal of a ring A. Let X be a topological space. Let … → F_3 → F_2 → F_1 be an inverse system of sheaves of A-modules on X such that F_n = F_n + 1/I^nF_n + 1. Let p ≥ 0. Assume bigoplus_n ≥ 0 H^p(X, I^nF_n + 1) satisfies the ascending chain condition as a graded bigoplus_n ≥ 0 I^n/I^n + 1-module. Then the limit topology on M = lim H^p(X, F_n) is the I-adic topology.","statement_latex":"Let $I$ be an ideal of a ring $A$. Let $X$ be a topological space. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of sheaves of $A$-modules on $X$ such that\n$\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nLet $p \\geq 0$. Assume\n$$\n\\bigoplus\\nolimits_{n \\geq 0} H^p(X, I^n\\mathcal{F}_{n + 1})\n$$\nsatisfies the ascending chain condition as a graded\n$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module.\nThen the limit topology on $M = \\lim H^p(X, \\mathcal{F}_n)$\nis the $I$-adic topology.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYM","source_file":"cohomology.tex","source_line":9063,"source_end_line":9079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9063-L9079","statement_sha256":"14f0bb3da2108d051d9fc746abee8b2ac8611eb6fd86f0d3cd10f93de51da2d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4293,"rank":4293,"depth":1,"x":1066.03,"y":628.811,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYN","tag":"0GYN","title":"Inverse systems and cohomology, I · Lemma 0GYN","summary":"Let I be an ideal of a ring A. Let X be a topological space. Let … → F_3 → F_2 → F_1 be an inverse system of sheaves of A-modules on X such that F_n = F_n + 1/I^nF_n + 1. Let p ≥ 0. Given n define N_n = ⋂_m ≥ n Im( H^p(X, I^nF_m + 1) → H^p(X, I^nF_n + 1) ) If bigoplus N_n satisfies the ascending chain condition as a graded bigoplus_n ≥ 0 I^n/I^n + 1-module, then the limit topology on M = lim H^p(X, F_n) is the I-adic topology.","statement_latex":"Let $I$ be an ideal of a ring $A$. Let $X$ be a topological space. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of sheaves of $A$-modules on $X$ such that\n$\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nLet $p \\geq 0$. Given $n$ define\n$$\nN_n =\n\\bigcap\\nolimits_{m \\geq n}\n\\Im\\left(\nH^p(X, I^n\\mathcal{F}_{m + 1}) \\to H^p(X, I^n\\mathcal{F}_{n + 1})\n\\right)\n$$\nIf $\\bigoplus N_n$ satisfies the ascending chain condition as a graded\n$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module, then\nthe limit topology on $M = \\lim H^p(X, \\mathcal{F}_n)$\nis the $I$-adic topology.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYN","source_file":"cohomology.tex","source_line":9137,"source_end_line":9157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9137-L9157","statement_sha256":"adca901a9432dc0dbc101c450de5cacbed5ec6996157837d351768d1213a31ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":4294,"rank":4294,"depth":2,"x":1218.641,"y":681.453,"cluster":"sheaf-cohomology"},{"id":"stacks:0H39","tag":"0H39","title":"Inverse systems and cohomology, II · Lemma 0H39","summary":"Let (X, O_X) be a ringed space. Let f ∈ Γ(X, O_X). Let … → F_3 → F_2 → F_1 be inverse system of O_X-modules. Consider the conditions • for all n ≥ 1 the map f : F_n + 1 → F_n + 1 factors through F_n + 1 → F_n to give a short exact sequence 0 → F_n → F_n + 1 → F_1 → 0, • for all n ≥ 1 the map f^n : F_n + 1 → F_n + 1 factors through F_n + 1 → F_1 to give a short exact sequence 0 → F_1 → F_n + 1 → F_n → 0 • there exists an O_X-module G which is f-divisible such that F_n =…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $f \\in \\Gamma(X, \\mathcal{O}_X)$. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe inverse system of $\\mathcal{O}_X$-modules. Consider the conditions\n\\begin{enumerate}\n\\item for all $n \\geq 1$ the map\n$f : \\mathcal{F}_{n + 1} \\to \\mathcal{F}_{n + 1}$ factors\nthrough $\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n$ to give a\nshort exact sequence\n$0 \\to \\mathcal{F}_n \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_1 \\to 0$,\n\\item for all $n \\geq 1$ the map\n$f^n : \\mathcal{F}_{n + 1} \\to \\mathcal{F}_{n + 1}$\nfactors through $\\mathcal{F}_{n + 1} \\to \\mathcal{F}_1$\nto give a short exact sequence\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_n \\to 0$\n\\item there exists an $\\mathcal{O}_X$-module $\\mathcal{G}$\nwhich is $f$-divisible such that $\\mathcal{F}_n = \\mathcal{G}[f^n]$, and\n\\item there exists an $\\mathcal{O}_X$-module $\\mathcal{F}$\nwhich is $f$-torsion free such that\n$\\mathcal{F}_n = \\mathcal{F}/f^n\\mathcal{F}$.\n\\end{enumerate}\nThen (4) $\\Rightarrow$ (3) $\\Leftrightarrow$ (2) $\\Leftrightarrow$ (1).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H39","source_file":"cohomology.tex","source_line":9207,"source_end_line":9233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9207-L9233","statement_sha256":"cbfc3ec0530419c494985db059d3a3a415b8dfa856a6edb16b80420fe8a250c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4295,"rank":4295,"depth":0,"x":1063.24,"y":729.395,"cluster":"sheaf-cohomology"},{"id":"stacks:0EHA","tag":"0EHA","title":"Inverse systems and cohomology, II · Lemma 0EHA","summary":"Suppose X, f, (F_n) satisfy condition (1) of Lemma [Tag 0H39]. Let p ≥ 0 and set H^p = lim H^p(X, F_n). Then f^cH^p is the kernel of H^p → H^p(X, F_c) for all c ≥ 1. Thus the limit topology on H^p is the f-adic topology.","statement_latex":"Suppose $X$, $f$, $(\\mathcal{F}_n)$ satisfy condition (1) of\nLemma \\ref{lemma-equivalent-f-good}. Let $p \\geq 0$ and set\n$H^p = \\lim H^p(X, \\mathcal{F}_n)$.\nThen $f^cH^p$ is the kernel of $H^p \\to H^p(X, \\mathcal{F}_c)$ for all\n$c \\geq 1$. Thus the limit topology on $H^p$ is the $f$-adic topology.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHA","source_file":"cohomology.tex","source_line":9261,"source_end_line":9268,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9261-L9268","statement_sha256":"9f97f28fe5ff40b3d79bf114b6e32f2257f3a57dab38555746a2d2aac2b8f7d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4296,"rank":4296,"depth":1,"x":1139.538,"y":605.443,"cluster":"sheaf-cohomology"},{"id":"stacks:0BLB","tag":"0BLB","title":"Inverse systems and cohomology, II · Lemma 0BLB","summary":"Let A be a Noetherian ring complete with respect to a principal ideal (f). Let X be a topological space. Let … → F_3 → F_2 → F_1 be an inverse system of sheaves of A-modules. Assume • Γ(X, F_1) is a finite A-module, • X, f, (F_n) satisfy condition (1) of Lemma [Tag 0H39]. Then M = lim Γ(X, F_n) is a finite A-module, f is a nonzerodivisor on M, and M/fM is the image of M in Γ(X, F_1).","statement_latex":"Let $A$ be a Noetherian ring complete with respect to a principal ideal $(f)$.\nLet $X$ be a topological space. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of sheaves of $A$-modules. Assume\n\\begin{enumerate}\n\\item $\\Gamma(X, \\mathcal{F}_1)$ is a finite $A$-module,\n\\item $X$, $f$, $(\\mathcal{F}_n)$ satisfy condition (1) of\nLemma \\ref{lemma-equivalent-f-good}.\n\\end{enumerate}\nThen\n$$\nM = \\lim \\Gamma(X, \\mathcal{F}_n)\n$$\nis a finite $A$-module, $f$ is a nonzerodivisor on $M$, and\n$M/fM$ is the image of $M$ in $\\Gamma(X, \\mathcal{F}_1)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLB","source_file":"cohomology.tex","source_line":9294,"source_end_line":9313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9294-L9313","statement_sha256":"3d8cdfb6afd8d6fed59f1c5b49e7f8549b0d01a5fa5bc88d158b38c2870d4e61","origin":"The Stacks Project","memory_eligible":false,"source_rank":4297,"rank":4297,"depth":6,"x":1183.103,"y":740.594,"cluster":"sheaf-cohomology"},{"id":"stacks:0BLC","tag":"0BLC","title":"Inverse systems and cohomology, II · Lemma 0BLC","summary":"Let A be a ring. Let f ∈ A. Let X be a topological space. Let … → F_3 → F_2 → F_1 be an inverse system of sheaves of A-modules. Let p ≥ 0. Assume • either H^p + 1(X, F_1) is an A-module of finite length or A is Noetherian and H^p + 1(X, F_1) is a finite A-module, • X, f, (F_n) satisfy condition (1) of Lemma [Tag 0H39]. Then the inverse system M_n = H^p(X, F_n) satisfies the Mittag-Leffler condition.","statement_latex":"Let $A$ be a ring. Let $f \\in A$. Let $X$ be a topological space. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of sheaves of $A$-modules. Let $p \\geq 0$. Assume\n\\begin{enumerate}\n\\item either $H^{p + 1}(X, \\mathcal{F}_1)$ is an $A$-module of finite length\nor $A$ is Noetherian and $H^{p + 1}(X, \\mathcal{F}_1)$ is a finite $A$-module,\n\\item $X$, $f$, $(\\mathcal{F}_n)$ satisfy condition (1) of\nLemma \\ref{lemma-equivalent-f-good}.\n\\end{enumerate}\nThen the inverse system $M_n = H^p(X, \\mathcal{F}_n)$ satisfies the\nMittag-Leffler condition.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLC","source_file":"cohomology.tex","source_line":9331,"source_end_line":9346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9331-L9346","statement_sha256":"d4c20dcbaff10205d89424cb9b999d5944fd059f24be595bc1041d263b37980d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4298,"rank":4298,"depth":1,"x":1041.82,"y":665.402,"cluster":"sheaf-cohomology"},{"id":"stacks:0DXG","tag":"0DXG","title":"Inverse systems and cohomology, II · Lemma 0DXG","summary":"Let A be a ring. Let f ∈ A. Let X be a topological space. Let … → F_3 → F_2 → F_1 be an inverse system of sheaves of A-modules. Let p ≥ 0. Assume • either there is an m ≥ 1 such that the image of H^p + 1(X, F_m) → H^p + 1(X, F_1) is an A-module of finite length or A is Noetherian and the intersection of the images of H^p + 1(X, F_m) → H^p + 1(X, F_1) is a finite A-module, • X, f, (F_n) satisfy condition (1) of Lemma [Tag 0H39]. Then the inverse system M_n = H^p(X, F_n)…","statement_latex":"Let $A$ be a ring. Let $f \\in A$. Let $X$ be a topological space. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of sheaves of $A$-modules. Let $p \\geq 0$. Assume\n\\begin{enumerate}\n\\item either there is an $m \\geq 1$ such that the image of\n$H^{p + 1}(X, \\mathcal{F}_m) \\to H^{p + 1}(X, \\mathcal{F}_1)$\nis an $A$-module of finite length or $A$ is Noetherian\nand the intersection of the images of\n$H^{p + 1}(X, \\mathcal{F}_m) \\to H^{p + 1}(X, \\mathcal{F}_1)$\nis a finite $A$-module,\n\\item $X$, $f$, $(\\mathcal{F}_n)$ satisfy condition (1) of\nLemma \\ref{lemma-equivalent-f-good}.\n\\end{enumerate}\nThen the inverse system $M_n = H^p(X, \\mathcal{F}_n)$ satisfies the\nMittag-Leffler condition.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXG","source_file":"cohomology.tex","source_line":9364,"source_end_line":9383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9364-L9383","statement_sha256":"8ed4fd5a9e36b2855efcb284862e1e7d9ae69c333f0d41acc36484a56fed24b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4299,"rank":4299,"depth":2,"x":1207.019,"y":640.598,"cluster":"sheaf-cohomology"},{"id":"stacks:0D60","tag":"0D60","title":"Derived limits · Lemma 0D60","summary":"Let (X, O_X) be a ringed space. For U ⊂ X open the functor RΓ(U, -) commutes with Rlim. Moreover, there are short exact sequences 0 → R^1lim H^m - 1(U, K_n) → H^m(U, Rlim K_n) → lim H^m(U, K_n) → 0 for any inverse system (K_n) in D(O_X) and any m ∈ Z.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. For $U \\subset X$ open the\nfunctor $R\\Gamma(U, -)$ commutes with $R\\lim$. Moreover, there are\nshort exact sequences\n$$\n0 \\to\nR^1\\lim H^{m - 1}(U, K_n) \\to H^m(U, R\\lim K_n) \\to\n\\lim H^m(U, K_n) \\to 0\n$$\nfor any inverse system $(K_n)$ in $D(\\mathcal{O}_X)$ and any $m \\in \\mathbf{Z}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D60","source_file":"cohomology.tex","source_line":9456,"source_end_line":9467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9456-L9467","statement_sha256":"eb2d0647f872fd3957b3caa6eb92f8f679cde6ac3f101a4f5615b5fb7a0ad598","origin":"The Stacks Project","memory_eligible":false,"source_rank":4300,"rank":4300,"depth":9,"x":1104.811,"y":752.998,"cluster":"sheaf-cohomology"},{"id":"stacks:0BKP","tag":"0BKP","title":"Derived limits · Lemma 0BKP","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Then Rf_* commutes with Rlim, i.e., Rf_* commutes with derived limits.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed\nspaces. Then $Rf_*$ commutes with $R\\lim$, i.e., $Rf_*$ commutes with\nderived limits.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKP","source_file":"cohomology.tex","source_line":9478,"source_end_line":9483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9478-L9483","statement_sha256":"394241a72184e7371c697dacc5113fab470ed609d99fb6fc354d0950ce2c3ae7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4301,"rank":4301,"depth":19,"x":1089.739,"y":611.655,"cluster":"sheaf-cohomology"},{"id":"stacks:0BKS","tag":"0BKS","title":"Derived limits · Lemma 0BKS","summary":"Let (X, O_X) be a ringed space. Let (F_n) be an inverse system of O_X-modules. Let B be a set of opens of X. Assume • every open of X has a covering whose members are elements of B, • H^p(U, F_n) = 0 for p > 0 and U ∈ B, • the inverse system F_n(U) has vanishing R^1lim for U ∈ B. Then Rlim F_n = lim F_n and we have H^p(U, lim F_n) = 0 for p > 0 and U ∈ B.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(\\mathcal{F}_n)$ be an\ninverse system of $\\mathcal{O}_X$-modules. Let $\\mathcal{B}$ be a set\nof opens of $X$. Assume\n\\begin{enumerate}\n\\item every open of $X$ has a covering whose members are elements of\n$\\mathcal{B}$,\n\\item $H^p(U, \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$,\n\\item the inverse system $\\mathcal{F}_n(U)$ has vanishing $R^1\\lim$\nfor $U \\in \\mathcal{B}$.\n\\end{enumerate}\nThen $R\\lim \\mathcal{F}_n = \\lim \\mathcal{F}_n$ and we have\n$H^p(U, \\lim \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKS","source_file":"cohomology.tex","source_line":9553,"source_end_line":9567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9553-L9567","statement_sha256":"c209797a0889d901a2c124edff428a05e51a6dfefd8c16d5584bc007e6c05b69","origin":"The Stacks Project","memory_eligible":false,"source_rank":4302,"rank":4302,"depth":0,"x":1214.927,"y":707.642,"cluster":"sheaf-cohomology"},{"id":"stacks:0D61","tag":"0D61","title":"Derived limits · Lemma 0D61","summary":"Let (X, O_X) be a ringed space. Let (K_n) be an inverse system in D(O_X). Let x ∈ X and m ∈ Z. Assume there exist an integer n(x) and a fundamental system U_x of open neighbourhoods of x such that for U ∈ U_x • R^1lim H^m - 1(U, K_n) = 0, and • H^m(U, K_n) → H^m(U, K_n(x)) is injective for n ≥ n(x). Then the map on stalks H^m(Rlim K_n)_x → H^m(K_n(x))_x is injective.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K_n)$ be an\ninverse system in $D(\\mathcal{O}_X)$. Let $x \\in X$ and $m \\in \\mathbf{Z}$.\nAssume there exist an integer $n(x)$ and a fundamental system $\\mathfrak{U}_x$\nof open neighbourhoods of $x$ such that for $U \\in \\mathfrak{U}_x$\n\\begin{enumerate}\n\\item $R^1\\lim H^{m - 1}(U, K_n) = 0$, and\n\\item $H^m(U, K_n) \\to H^m(U, K_{n(x)})$ is injective\nfor $n \\geq n(x)$.\n\\end{enumerate}\nThen the map on stalks $H^m(R\\lim K_n)_x \\to H^m(K_{n(x)})_x$ is injective.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D61","source_file":"cohomology.tex","source_line":9584,"source_end_line":9596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9584-L9596","statement_sha256":"f77d2395f4a51501849c0e88d8156f12366e61a5a2377e38b397868aeb484282","origin":"The Stacks Project","memory_eligible":false,"source_rank":4303,"rank":4303,"depth":10,"x":1044.872,"y":707.897,"cluster":"sheaf-cohomology"},{"id":"stacks:0D62","tag":"0D62","title":"Derived limits · Lemma 0D62","summary":"Let (X, O_X) be a ringed space. Let E ∈ D(O_X). Assume that for every x ∈ X there exist a function p(x, -) : Z → Z and a fundamental system U_x of open neighbourhoods of x such that H^p(U, H^m - p(E)) = 0 for U ∈ U_x and p > p(x, m) Then the map E → Rlim τ_≥ -n E of Derived Categories, Remark [Tag 0H72] is an isomorphism in D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$.\nAssume that for every $x \\in X$ there exist\na function $p(x, -) : \\mathbf{Z} \\to \\mathbf{Z}$ and\na fundamental system $\\mathfrak{U}_x$ of open neighbourhoods of $x$\nsuch that\n$$\nH^p(U, H^{m - p}(E)) = 0 \\text{ for }\nU \\in \\mathfrak{U}_x \\text{ and } p > p(x, m)\n$$\nThen the map $E \\to R\\lim \\tau_{\\geq -n} E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D62","source_file":"cohomology.tex","source_line":9623,"source_end_line":9638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9623-L9638","statement_sha256":"eda6da872802b2c30437bae267e338af0b92bff537380a5ed17d6e690c804be9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4304,"rank":4304,"depth":11,"x":1170.469,"y":610.903,"cluster":"sheaf-cohomology"},{"id":"stacks:0D63","tag":"0D63","title":"Derived limits · Lemma 0D63","summary":"[Spaltenstein] Let (X, O_X) be a ringed space. Let E ∈ D(O_X). Assume that for every x ∈ X there exist an integer d_x ≥ 0 and a fundamental system U_x of open neighbourhoods of x such that H^p(U, H^q(E)) = 0 for U ∈ U_x, p > d_x, and q < 0 Then the map E → Rlim τ_≥ -n E of Derived Categories, Remark [Tag 0H72] is an isomorphism in D(O_X).","statement_latex":"\\begin{reference}\n\\cite[Proposition 3.13]{Spaltenstein}\n\\end{reference}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$.\nAssume that for every $x \\in X$ there exist an integer $d_x \\geq 0$ and\na fundamental system $\\mathfrak{U}_x$ of open neighbourhoods of $x$\nsuch that\n$$\nH^p(U, H^q(E)) = 0 \\text{ for }\nU \\in \\mathfrak{U}_x,\\ p > d_x, \\text{ and }q < 0\n$$\nThen the map $E \\to R\\lim \\tau_{\\geq -n} E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D63","source_file":"cohomology.tex","source_line":9689,"source_end_line":9706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9689-L9706","statement_sha256":"cafe020547f25022cf0175e324f859ca5e9dd503d59527922eb251dd568255f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4305,"rank":4305,"depth":12,"x":1155.808,"y":754.151,"cluster":"sheaf-cohomology"},{"id":"stacks:08U2","tag":"08U2","title":"Derived limits · Lemma 08U2","summary":"Let (X, O_X) be a ringed space. Let E ∈ D(O_X). Assume there exist a function p(-) : Z → Z and a set B of opens of X such that • every open in X has a covering whose members are elements of B, and • H^p(U, H^m - p(E)) = 0 for p > p(m) and U ∈ B. Then the map E → Rlim τ_≥ -n E of Derived Categories, Remark [Tag 0H72] is an isomorphism in D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$.\nAssume there exist a function $p(-) : \\mathbf{Z} \\to \\mathbf{Z}$\nand a set $\\mathcal{B}$ of opens of $X$ such that\n\\begin{enumerate}\n\\item every open in $X$ has a covering whose members are\nelements of $\\mathcal{B}$, and\n\\item $H^p(U, H^{m - p}(E)) = 0$ for $p > p(m)$ and $U \\in \\mathcal{B}$.\n\\end{enumerate}\nThen the map $E \\to R\\lim \\tau_{\\geq -n} E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08U2","source_file":"cohomology.tex","source_line":9713,"source_end_line":9727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9713-L9727","statement_sha256":"a4d0d788945aa0288911f7ee4839df2628b9a1be5955404dd9b349915cb02286","origin":"The Stacks Project","memory_eligible":false,"source_rank":4306,"rank":4306,"depth":12,"x":1051.088,"y":639.838,"cluster":"sheaf-cohomology"},{"id":"stacks:0D64","tag":"0D64","title":"Derived limits · Lemma 0D64","summary":"Let (X, O_X) be a ringed space. Let E ∈ D(O_X). Assume there exist an integer d ≥ 0 and a basis B for the topology of X such that H^p(U, H^q(E)) = 0 for U ∈ B, p > d, and q < 0 Then the map E → Rlim τ_≥ -n E of Derived Categories, Remark [Tag 0H72] is an isomorphism in D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E \\in D(\\mathcal{O}_X)$.\nAssume there exist an integer $d \\geq 0$ and a basis $\\mathcal{B}$ for the\ntopology of $X$ such that\n$$\nH^p(U, H^q(E)) = 0 \\text{ for }\nU \\in \\mathcal{B},\\ p > d, \\text{ and }q < 0\n$$\nThen the map $E \\to R\\lim \\tau_{\\geq -n} E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D64","source_file":"cohomology.tex","source_line":9735,"source_end_line":9748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9735-L9748","statement_sha256":"5a12a592b7ca054aa2ecfe7ca2829210787de1dc5a0a966ad5ce28fe9bd4ac27","origin":"The Stacks Project","memory_eligible":false,"source_rank":4307,"rank":4307,"depth":13,"x":1220.772,"y":664.79,"cluster":"sheaf-cohomology"},{"id":"stacks:0BKT","tag":"0BKT","title":"Derived limits · Lemma 0BKT","summary":"Let (X, O_X) be a ringed space. Let K be an object of D(O_X). Let B be a set of opens of X. Assume • every open of X has a covering whose members are elements of B, • H^p(U, H^q(K)) = 0 for all p > 0, q ∈ Z, and U ∈ B. Then H^q(U, K) = H^0(U, H^q(K)) for q ∈ Z and U ∈ B.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$\nbe an object of $D(\\mathcal{O}_X)$.\nLet $\\mathcal{B}$ be a set of opens of $X$. Assume\n\\begin{enumerate}\n\\item every open of $X$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item $H^p(U, H^q(K)) = 0$ for all $p > 0$, $q \\in \\mathbf{Z}$, and\n$U \\in \\mathcal{B}$.\n\\end{enumerate}\nThen $H^q(U, K) = H^0(U, H^q(K))$ for $q \\in \\mathbf{Z}$\nand $U \\in \\mathcal{B}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKT","source_file":"cohomology.tex","source_line":9759,"source_end_line":9772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9759-L9772","statement_sha256":"0a0a6865aa109f44e44bdc442a908c96bbf1d25a9996f2f53d80061613b6fc55","origin":"The Stacks Project","memory_eligible":false,"source_rank":4308,"rank":4308,"depth":14,"x":1075.125,"y":742.919,"cluster":"sheaf-cohomology"},{"id":"stacks:0BKU","tag":"0BKU","title":"Derived limits · Lemma 0BKU","summary":"Let (X, O_X) be a ringed space. Let (K_n) be an inverse system of objects of D(O_X). Let B be a set of opens of X. Assume • every open of X has a covering whose members are elements of B, • for all U ∈ B and all q ∈ Z we have • H^p(U, H^q(K_n)) = 0 for p > 0, • the inverse system H^0(U, H^q(K_n)) has vanishing R^1lim. Then H^q(Rlim K_n) = lim H^q(K_n) for q ∈ Z.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K_n)$\nbe an inverse system of objects of $D(\\mathcal{O}_X)$.\nLet $\\mathcal{B}$ be a set of opens of $X$. Assume\n\\begin{enumerate}\n\\item every open of $X$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item for all $U \\in \\mathcal{B}$ and all $q \\in \\mathbf{Z}$ we have\n\\begin{enumerate}\n\\item $H^p(U, H^q(K_n)) = 0$ for $p > 0$,\n\\item the inverse system $H^0(U, H^q(K_n))$ has vanishing $R^1\\lim$.\n\\end{enumerate}\n\\end{enumerate}\nThen $H^q(R\\lim K_n) = \\lim H^q(K_n)$ for $q \\in \\mathbf{Z}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Derived limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKU","source_file":"cohomology.tex","source_line":9797,"source_end_line":9812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9797-L9812","statement_sha256":"9bf2f2dbb7da24b19dafa6bc6b2642ae3d514eb7345447603361444ea0929093","origin":"The Stacks Project","memory_eligible":false,"source_rank":4309,"rank":4309,"depth":15,"x":1119.834,"y":602.225,"cluster":"sheaf-cohomology"},{"id":"stacks:071B","tag":"071B","title":"Producing K-injective resolutions · Lemma 071B","summary":"In the situation described above. Denote H^m = H^m(F^bullet) the mth cohomology sheaf. Let B be a set of open subsets of X. Let d ∈ N. Assume • every open in X has a covering whose members are elements of B, • for every U ∈ B we have H^p(U, H^q) = 0 for p > d and q < 0^m - p) = 0 for p > p(m), see Lemma [Tag 08U2].. Then ([Tag 071A]) is a quasi-isomorphism.","statement_latex":"In the situation described above.\nDenote $\\mathcal{H}^m = H^m(\\mathcal{F}^\\bullet)$ the $m$th cohomology sheaf.\nLet $\\mathcal{B}$ be a set of open subsets of $X$.\nLet $d \\in \\mathbf{N}$.\nAssume\n\\begin{enumerate}\n\\item every open in $X$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item for every $U \\in \\mathcal{B}$ we have $H^p(U, \\mathcal{H}^q) = 0$\nfor $p > d$ and $q < 0$\\footnote{It suffices if\n$\\forall m$, $\\exists p(m)$, $H^p(U. \\mathcal{H}^{m - p}) = 0$ for\n$p > p(m)$, see Lemma \\ref{lemma-is-limit}.}.\n\\end{enumerate}\nThen (\\ref{equation-into-candidate-K-injective}) is a quasi-isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Producing K-injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/071B","source_file":"cohomology.tex","source_line":9873,"source_end_line":9889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9873-L9889","statement_sha256":"284deddd0524013c2d50d2c4db04b44d43ead3ac93c7b0797989b15566b65e4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4310,"rank":4310,"depth":16,"x":1200.258,"y":731.742,"cluster":"sheaf-cohomology"},{"id":"stacks:08BY","tag":"08BY","title":"Producing K-injective resolutions · Lemma 08BY","summary":"Let (X, O_X) be a ringed space. Let (F_n^bullet) be an inverse system of complexes of O_X-modules. Let m ∈ Z. Assume there exist a set B of open subsets of X and an integer n_0 such that • every open in X has a covering whose members are elements of B, • for every U ∈ B • the systems of abelian groups F_n^m - 2(U) and F_n^m - 1(U) have vanishing R^1lim (for example these have the Mittag-Leffler condition), • the system of abelian groups H^m - 1(F_n^bullet(U)) has…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(\\mathcal{F}_n^\\bullet)$\nbe an inverse system of complexes of $\\mathcal{O}_X$-modules.\nLet $m \\in \\mathbf{Z}$. Assume there exist a set $\\mathcal{B}$\nof open subsets of $X$ and an integer $n_0$ such that\n\\begin{enumerate}\n\\item every open in $X$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item for every $U \\in \\mathcal{B}$\n\\begin{enumerate}\n\\item the systems of abelian groups\n$\\mathcal{F}_n^{m - 2}(U)$ and $\\mathcal{F}_n^{m - 1}(U)$\nhave vanishing $R^1\\lim$ (for example these have the Mittag-Leffler\ncondition),\n\\item the system of abelian groups $H^{m - 1}(\\mathcal{F}_n^\\bullet(U))$\nhas vanishing $R^1\\lim$ (for example it has the Mittag-Leffler condition), and\n\\item we have\n$H^m(\\mathcal{F}_n^\\bullet(U)) = H^m(\\mathcal{F}_{n_0}^\\bullet(U))$\nfor all $n \\geq n_0$.\n\\end{enumerate}\n\\end{enumerate}\nThen the maps\n$H^m(\\mathcal{F}^\\bullet) \\to \\lim H^m(\\mathcal{F}_n^\\bullet) \\to\nH^m(\\mathcal{F}_{n_0}^\\bullet)$\nare isomorphisms of sheaves where\n$\\mathcal{F}^\\bullet = \\lim \\mathcal{F}_n^\\bullet$ is the termwise\ninverse limit.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Producing K-injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BY","source_file":"cohomology.tex","source_line":9904,"source_end_line":9932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9904-L9932","statement_sha256":"35c798fd52d501451945d3954cddeedd6ddd262bacde369c47ee9fecc17fe7a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4311,"rank":4311,"depth":8,"x":1036.294,"y":681.718,"cluster":"sheaf-cohomology"},{"id":"stacks:0H3C","tag":"0H3C","title":"Inverse systems and cohomology, III · Lemma 0H3C","summary":"Let (X, O_X) be a ringed space. Let A → Γ(X, O_X) be a ring map and let f ∈ A. Let E be an object of D(O_X). Denote E_n = E ⊗_O_X (O_X xrightarrowf^n O_X) and set E^wedge = Rlim E_n. For p ∈ Z is a canonical commutative diagram xymatrix & 0 & 0 0 ar[r] & widehatH^p(X, E) ar[r] ar[u] & lim H^p(X, E_n) ar[r] ar[u] & T_f(H^p + 1(X, E)) ar[r] & 0 0 ar[r] & H^0(H^p(X, E)^wedge) ar[r] ar[u] & H^p(X, E^wedge) ar[r] ar[u] & T_f(H^p + 1(X, E)) ar[r] ar@=[u] & 0 & R^1lim H^p(X,…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $A \\to \\Gamma(X, \\mathcal{O}_X)$\nbe a ring map and let $f \\in A$. Let $E$ be an object of $D(\\mathcal{O}_X)$.\nDenote\n$$\nE_n = E \\otimes_{\\mathcal{O}_X} (\\mathcal{O}_X \\xrightarrow{f^n} \\mathcal{O}_X)\n$$\nand set $E^\\wedge = R\\lim E_n$. For $p \\in \\mathbf{Z}$\nis a canonical commutative diagram\n$$\n\\xymatrix{\n& 0 & 0 \\\\\n0 \\ar[r] &\n\\widehat{H^p(X, E)} \\ar[r] \\ar[u] &\n\\lim H^p(X, E_n) \\ar[r] \\ar[u] &\nT_f(H^{p + 1}(X, E)) \\ar[r] &\n0 \\\\\n0 \\ar[r] &\nH^0(H^p(X, E)^\\wedge) \\ar[r] \\ar[u] &\nH^p(X, E^\\wedge) \\ar[r] \\ar[u] &\nT_f(H^{p + 1}(X, E)) \\ar[r] \\ar@{=}[u] &\n0 \\\\\n&\nR^1\\lim H^p(X, E)[f^n] \\ar[u] \\ar[r]^\\cong &\nR^1\\lim H^{p - 1}(X, E_n) \\ar[u] \\\\\n& 0 \\ar[u] & 0 \\ar[u]\n}\n$$\nwith exact rows and columns\nwhere $\\widehat{H^p(X, E)} = \\lim H^p(X, E)/f^n H^p(X, E)$ is the usual\n$f$-adic completion, $H^p(X, E)^\\wedge$ is the derived $f$-adic completion,\nand $T_f(H^{p + 1}(X, E))$ is the $f$-adic Tate module, see\nMore on Algebra, Example\n\\ref{more-algebra-example-spectral-sequence-principal}.\nFinally, we have $H^p(X, E^\\wedge) = H^p(R\\Gamma(X, E)^\\wedge)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3C","source_file":"cohomology.tex","source_line":9981,"source_end_line":10017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L9981-L10017","statement_sha256":"011935891b658bd5bad3c4ea49093d74e78e2cf68a0eb035ea917dffd0fbbb90","origin":"The Stacks Project","memory_eligible":false,"source_rank":4312,"rank":4312,"depth":21,"x":1197.924,"y":625.395,"cluster":"sheaf-cohomology"},{"id":"stacks:0H3D","tag":"0H3D","title":"Inverse systems and cohomology, III · Lemma 0H3D","summary":"Let A be an abelian category. Let f : M → M be a morphism of A. If M[f^n] = Ker(f^n : M → M) stabilizes, then the inverse systems (M xrightarrowf^n M) and Coker(f^n : M → M) are pro-isomorphic in D(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $f : M \\to M$ be a morphism\nof $\\mathcal{A}$. If $M[f^n] = \\Ker(f^n : M \\to M)$ stabilizes, then the\ninverse systems\n$$\n(M \\xrightarrow{f^n} M)\n\\quad\\text{and}\\quad\n\\Coker(f^n : M \\to M)\n$$\nare pro-isomorphic in $D(\\mathcal{A})$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Inverse systems and cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3D","source_file":"cohomology.tex","source_line":10033,"source_end_line":10044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10033-L10044","statement_sha256":"3f7550d606cb1bc4dbc62e618654bc0ffaa0a4864f2678f9dafc85e8658e0378","origin":"The Stacks Project","memory_eligible":false,"source_rank":4313,"rank":4313,"depth":0,"x":1123.807,"y":759.047,"cluster":"sheaf-cohomology"},{"id":"stacks:08C1","tag":"08C1","title":"v Cech cohomology of unbounded complexes · Lemma 08C1","summary":"Let (X, O_X) be a ringed space. Let U : X = ⋃_i ∈ I U_i be a finite open covering. For a complex F^bullet of O_X-modules there is a canonical map Tot(checkC^bullet_alt(U, F^bullet)) → RΓ(X, F^bullet) functorial in F^bullet and compatible with ([Tag 08C0]).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be\na finite open covering. For a complex $\\mathcal{F}^\\bullet$\nof $\\mathcal{O}_X$-modules there is a canonical map\n$$\n\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U}, \\mathcal{F}^\\bullet))\n\\longrightarrow\nR\\Gamma(X, \\mathcal{F}^\\bullet)\n$$\nfunctorial in $\\mathcal{F}^\\bullet$ and compatible with\n(\\ref{equation-global-sections-to-alternating-cech}).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology of unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08C1","source_file":"cohomology.tex","source_line":10156,"source_end_line":10169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10156-L10169","statement_sha256":"59402dfd806ab3df7a15097e9f0816f074b21ecb9cb52f0696353c19443a6fd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4314,"rank":4314,"depth":21,"x":1070.821,"y":618.011,"cluster":"sheaf-cohomology"},{"id":"stacks:08C2","tag":"08C2","title":"v Cech cohomology of unbounded complexes · Lemma 08C2","summary":"Let (X, O_X) be a ringed space. Let U : X = ⋃_i ∈ I U_i be a finite open covering. Let F^bullet be a complex of O_X-modules. Let B be a set of open subsets of X. Assume • every open in X has a covering whose members are elements of B, • we have U_i_0… i_p ∈ B for all i_0, …, i_p ∈ I, • for every U ∈ B and p > 0 we have • H^p(U, F^q) = 0, • H^p(U, Coker(F^q - 1 → F^q)) = 0, and • H^p(U, H^q(F)) = 0. Then the map Tot(checkC^bullet_alt(U, F^bullet)) → RΓ(X, F^bullet) of…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let\n$\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be a finite open covering. Let\n$\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.\nLet $\\mathcal{B}$ be a set of open subsets of $X$. Assume\n\\begin{enumerate}\n\\item every open in $X$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item we have $U_{i_0\\ldots i_p} \\in \\mathcal{B}$ for all\n$i_0, \\ldots, i_p \\in I$,\n\\item for every $U \\in \\mathcal{B}$ and $p > 0$ we have\n\\begin{enumerate}\n\\item $H^p(U, \\mathcal{F}^q) = 0$,\n\\item $H^p(U, \\Coker(\\mathcal{F}^{q - 1} \\to \\mathcal{F}^q)) = 0$, and\n\\item $H^p(U, H^q(\\mathcal{F})) = 0$.\n\\end{enumerate}\n\\end{enumerate}\nThen the map\n$$\n\\text{Tot}(\\check{\\mathcal{C}}^\\bullet_{alt}(\\mathcal{U}, \\mathcal{F}^\\bullet))\n\\longrightarrow\nR\\Gamma(X, \\mathcal{F}^\\bullet)\n$$\nof Lemma \\ref{lemma-alternating-cech-complex-complex}\nis an isomorphism in $D(\\textit{Ab})$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"v Cech cohomology of unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08C2","source_file":"cohomology.tex","source_line":10203,"source_end_line":10229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10203-L10229","statement_sha256":"b239412aa09a9e7d7d70d1f935e65ec3eb28c5b9ae4a8d11ff3ae79216510b6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4315,"rank":4315,"depth":23,"x":1223.77,"y":692.167,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8M","tag":"0A8M","title":"Hom complexes · Lemma 0A8M","summary":"Let (X, O_X) be a ringed space. Given complexes K^bullet, L^bullet, M^bullet of O_X-modules there is an isomorphism SheafHom^bullet(K^bullet, SheafHom^bullet(L^bullet, M^bullet)) = SheafHom^bullet(Tot(K^bullet ⊗_O_X L^bullet), M^bullet) of complexes of O_X-modules functorial in K^bullet, L^bullet, M^bullet.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nGiven complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$\nof $\\mathcal{O}_X$-modules there is an isomorphism\n$$\n\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,\n\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet))\n=\n\\SheafHom^\\bullet(\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}\n\\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet)\n$$\nof complexes of $\\mathcal{O}_X$-modules functorial in\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8M","source_file":"cohomology.tex","source_line":10328,"source_end_line":10342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10328-L10342","statement_sha256":"b0c5deec31e63eeee54d93208ca77d586c708b1284cdad9a1fd86cfa0d912814","origin":"The Stacks Project","memory_eligible":false,"source_rank":4316,"rank":4316,"depth":2,"x":1050.835,"y":724.368,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8N","tag":"0A8N","title":"Hom complexes · Lemma 0A8N","summary":"Let (X, O_X) be a ringed space. Given complexes K^bullet, L^bullet, M^bullet of O_X-modules there is a canonical morphism Tot( SheafHom^bullet(L^bullet, M^bullet) ⊗_O_X SheafHom^bullet(K^bullet, L^bullet) ) → SheafHom^bullet(K^bullet, M^bullet) of complexes of O_X-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$\nof $\\mathcal{O}_X$-modules there is a canonical morphism\n$$\n\\text{Tot}\\left(\n\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet)\n\\otimes_{\\mathcal{O}_X}\n\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet)\n\\right)\n\\longrightarrow\n\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{M}^\\bullet)\n$$\nof complexes of $\\mathcal{O}_X$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8N","source_file":"cohomology.tex","source_line":10349,"source_end_line":10364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10349-L10364","statement_sha256":"08b875ab7422844a481d939a2e46a36495086deb157f6632d1d4be2487b012cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4317,"rank":4317,"depth":1,"x":1152.763,"y":602.133,"cluster":"sheaf-cohomology"},{"id":"stacks:0BYR","tag":"0BYR","title":"Hom complexes · Lemma 0BYR","summary":"Let (X, O_X) be a ringed space. Given complexes K^bullet, L^bullet, M^bullet of O_X-modules there is a canonical morphism Tot( K^bullet ⊗_O_X SheafHom^bullet(M^bullet, L^bullet) ) → SheafHom^bullet(M^bullet, Tot(K^bullet ⊗_O_X L^bullet)) of complexes of O_X-modules functorial in all three complexes.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$\nof $\\mathcal{O}_X$-modules there is a canonical morphism\n$$\n\\text{Tot}\\left(\n\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X}\n\\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{L}^\\bullet)\n\\right)\n\\longrightarrow\n\\SheafHom^\\bullet(\\mathcal{M}^\\bullet,\n\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet))\n$$\nof complexes of $\\mathcal{O}_X$-modules functorial in all three complexes.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYR","source_file":"cohomology.tex","source_line":10371,"source_end_line":10386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10371-L10386","statement_sha256":"92835e7f31066d7b1d2c6a633882b2388ad8a4fe6fd048250630d8fe7ec3e3c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4318,"rank":4318,"depth":1,"x":1175.969,"y":750.543,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8Q","tag":"0A8Q","title":"Hom complexes · Lemma 0A8Q","summary":"Let (X, O_X) be a ringed space. Given complexes K^bullet, L^bullet of O_X-modules there is a canonical morphism K^bullet → SheafHom^bullet(L^bullet, Tot(K^bullet ⊗_O_X L^bullet)) of complexes of O_X-modules functorial in both complexes.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet$\nof $\\mathcal{O}_X$-modules there is a canonical morphism\n$$\n\\mathcal{K}^\\bullet\n\\longrightarrow\n\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,\n\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{L}^\\bullet))\n$$\nof complexes of $\\mathcal{O}_X$-modules functorial in both complexes.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8Q","source_file":"cohomology.tex","source_line":10393,"source_end_line":10405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10393-L10405","statement_sha256":"fb5a45b7098f5ea13886899a46101564f4b2eecbf1899e7a3db54a0322e9da94","origin":"The Stacks Project","memory_eligible":false,"source_rank":4319,"rank":4319,"depth":1,"x":1039.108,"y":653.991,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8P","tag":"0A8P","title":"Hom complexes · Lemma 0A8P","summary":"Let (X, O_X) be a ringed space. Given complexes K^bullet, L^bullet, M^bullet of O_X-modules there is a canonical morphism Tot(SheafHom^bullet(L^bullet, M^bullet) ⊗_O_X K^bullet) → SheafHom^bullet(SheafHom^bullet(K^bullet, L^bullet), M^bullet) of complexes of O_X-modules functorial in all three complexes.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Given complexes\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$\nof $\\mathcal{O}_X$-modules there is a canonical morphism\n$$\n\\text{Tot}(\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,\n\\mathcal{M}^\\bullet) \\otimes_{\\mathcal{O}_X} \\mathcal{K}^\\bullet)\n\\longrightarrow\n\\SheafHom^\\bullet(\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,\n\\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet)\n$$\nof complexes of $\\mathcal{O}_X$-modules functorial in all three complexes.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8P","source_file":"cohomology.tex","source_line":10412,"source_end_line":10425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10412-L10425","statement_sha256":"c8c9e7618b0b231f392f6a2664c7964c15d90c890f81f4f2351b7d80797c661d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4320,"rank":4320,"depth":1,"x":1218.196,"y":647.51,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8R","tag":"0A8R","title":"Hom complexes · Lemma 0A8R","summary":"Let (X, O_X) be a ringed space. Let L and M be objects of D(O_X). Let I^bullet be a K-injective complex of O_X-modules representing M. Let L^bullet be a complex of O_X-modules representing L. Then H^0(Γ(U, SheafHom^bullet(L^bullet, I^bullet))) = Hom_D(O_U)(L|_U, M|_U) for all U ⊂ X open.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $L$ and $M$\nbe objects of $D(\\mathcal{O}_X)$. Let $\\mathcal{I}^\\bullet$\nbe a K-injective complex of $\\mathcal{O}_X$-modules representing $M$. Let\n$\\mathcal{L}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules\nrepresenting $L$. Then\n$$\nH^0(\\Gamma(U, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))) =\n\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)\n$$\nfor all $U \\subset X$ open.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8R","source_file":"cohomology.tex","source_line":10432,"source_end_line":10444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10432-L10444","statement_sha256":"db69459e83150518aa05164ae250bef1e150a25ecfc6416b4dc6feea0dfc465d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4321,"rank":4321,"depth":4,"x":1090.98,"y":754.217,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8S","tag":"0A8S","title":"Hom complexes · Lemma 0A8S","summary":"Let (X, O_X) be a ringed space. Let (I')^bullet → I^bullet be a quasi-isomorphism of K-injective complexes of O_X-modules. Let (L')^bullet → L^bullet be a quasi-isomorphism of complexes of O_X-modules. Then SheafHom^bullet(L^bullet, (I')^bullet) → SheafHom^bullet((L')^bullet, I^bullet) is a quasi-isomorphism.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let\n$(\\mathcal{I}')^\\bullet \\to \\mathcal{I}^\\bullet$\nbe a quasi-isomorphism of K-injective complexes of $\\mathcal{O}_X$-modules.\nLet $(\\mathcal{L}')^\\bullet \\to \\mathcal{L}^\\bullet$\nbe a quasi-isomorphism of complexes of $\\mathcal{O}_X$-modules.\nThen\n$$\n\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, (\\mathcal{I}')^\\bullet)\n\\longrightarrow\n\\SheafHom^\\bullet((\\mathcal{L}')^\\bullet, \\mathcal{I}^\\bullet)\n$$\nis a quasi-isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8S","source_file":"cohomology.tex","source_line":10460,"source_end_line":10474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10460-L10474","statement_sha256":"0fc244a08a2965eae5bb6e9aeabcd87fd1c55b0df21fc7b0dedf5396765b6a89","origin":"The Stacks Project","memory_eligible":false,"source_rank":4322,"rank":4322,"depth":5,"x":1099.0,"y":602.911,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8T","tag":"0A8T","title":"Hom complexes · Lemma 0A8T","summary":"Let (X, O_X) be a ringed space. Let I^bullet be a K-injective complex of O_X-modules. Let L^bullet be a K-flat complex of O_X-modules. Then SheafHom^bullet(L^bullet, I^bullet) is a K-injective complex of O_X-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{I}^\\bullet$\nbe a K-injective complex of $\\mathcal{O}_X$-modules. Let\n$\\mathcal{L}^\\bullet$ be a K-flat complex of $\\mathcal{O}_X$-modules.\nThen $\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)$\nis a K-injective complex of $\\mathcal{O}_X$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8T","source_file":"cohomology.tex","source_line":10495,"source_end_line":10502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10495-L10502","statement_sha256":"03f161060b62b82b3d3ac423bfbab2c2118827038508a24328a703a5c71726ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":4323,"rank":4323,"depth":3,"x":1215.096,"y":719.367,"cluster":"sheaf-cohomology"},{"id":"stacks:08DK","tag":"08DK","title":"Internal hom in the derived category · Lemma 08DK","summary":"Let (X, O_X) be a ringed space. Let L, M be objects of D(O_X). For every open U we have H^0(U, RSheafHom(L, M)) = Hom_D(O_U)(L|_U, M|_U) and in particular H^0(X, RSheafHom(L, M)) = Hom_D(O_X)(L, M).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $L, M$ be objects\nof $D(\\mathcal{O}_X)$. For every open $U$ we have\n$$\nH^0(U, R\\SheafHom(L, M)) =\n\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)\n$$\nand in particular $H^0(X, R\\SheafHom(L, M)) = \\Hom_{D(\\mathcal{O}_X)}(L, M)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DK","source_file":"cohomology.tex","source_line":10580,"source_end_line":10589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10580-L10589","statement_sha256":"622e9ba172a6d1e926b463ee66bd7063b3f2c728120e215a2b06c0a46e65ee24","origin":"The Stacks Project","memory_eligible":false,"source_rank":4324,"rank":4324,"depth":5,"x":1035.324,"y":699.312,"cluster":"sheaf-cohomology"},{"id":"stacks:08DJ","tag":"08DJ","title":"Internal hom in the derived category · Lemma 08DJ","summary":"Let (X, O_X) be a ringed space. Let K, L, M be objects of D(O_X). With the construction as described above there is a canonical isomorphism RSheafHom(K, RSheafHom(L, M)) = RSheafHom(K ⊗_O_X^L L, M) in D(O_X) functorial in K, L, M which recovers ([Tag 08DI]) by taking H^0(X, -).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L, M$ be objects\nof $D(\\mathcal{O}_X)$. With the construction as described above\nthere is a canonical isomorphism\n$$\nR\\SheafHom(K, R\\SheafHom(L, M)) =\nR\\SheafHom(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L, M)\n$$\nin $D(\\mathcal{O}_X)$ functorial in $K, L, M$\nwhich recovers (\\ref{equation-internal-hom}) by taking $H^0(X, -)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DJ","source_file":"cohomology.tex","source_line":10601,"source_end_line":10612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10601-L10612","statement_sha256":"a17f709244deb060a04224e6baa84ac7ec6284043e07ed5b66ed1b6897c33b36","origin":"The Stacks Project","memory_eligible":false,"source_rank":4325,"rank":4325,"depth":6,"x":1184.437,"y":611.845,"cluster":"sheaf-cohomology"},{"id":"stacks:08DL","tag":"08DL","title":"Internal hom in the derived category · Lemma 08DL","summary":"Let (X, O_X) be a ringed space. Let K, L be objects of D(O_X). The construction of RSheafHom(K, L) commutes with restrictions to opens, i.e., for every open U we have RSheafHom(K|_U, L|_U) = RSheafHom(K, L)|_U.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects\nof $D(\\mathcal{O}_X)$. The construction of $R\\SheafHom(K, L)$\ncommutes with restrictions to opens, i.e.,\nfor every open $U$ we have\n$R\\SheafHom(K|_U, L|_U) = R\\SheafHom(K, L)|_U$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DL","source_file":"cohomology.tex","source_line":10639,"source_end_line":10646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10639-L10646","statement_sha256":"415d608851e974a3c0fa6e7e0f13848a29546933f16a557e032d92fef58a1708","origin":"The Stacks Project","memory_eligible":false,"source_rank":4326,"rank":4326,"depth":4,"x":1144.709,"y":761.374,"cluster":"sheaf-cohomology"},{"id":"stacks:08I0","tag":"08I0","title":"Internal hom in the derived category · Lemma 08I0","summary":"Let (X, O_X) be a ringed space. The bifunctor RSheafHom(- , -) transforms distinguished triangles into distinguished triangles in both variables.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. The bifunctor $R\\SheafHom(- , -)$\ntransforms distinguished triangles into distinguished triangles in both\nvariables.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08I0","source_file":"cohomology.tex","source_line":10653,"source_end_line":10658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10653-L10658","statement_sha256":"41b8c8a223175440265373b050a5a5a07dc0dee707217dd4c8524a1da2214a8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4327,"rank":4327,"depth":0,"x":1053.501,"y":628.197,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8V","tag":"0A8V","title":"Internal hom in the derived category · Lemma 0A8V","summary":"Let (X, O_X) be a ringed space. Given K, L, M in D(O_X) there is a canonical morphism RSheafHom(L, M) ⊗_O_X^L RSheafHom(K, L) → RSheafHom(K, M) in D(O_X) functorial in K, L, M.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Given $K, L, M$ in\n$D(\\mathcal{O}_X)$ there is a canonical morphism\n$$\nR\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(K, L)\n\\longrightarrow R\\SheafHom(K, M)\n$$\nin $D(\\mathcal{O}_X)$ functorial in $K, L, M$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8V","source_file":"cohomology.tex","source_line":10670,"source_end_line":10679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10670-L10679","statement_sha256":"bc09cce6cd3f5c5b0bdc28cafa0c46c4aa3a442765a0c07f34276344e115e6e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4328,"rank":4328,"depth":2,"x":1228.341,"y":674.777,"cluster":"sheaf-cohomology"},{"id":"stacks:0BYS","tag":"0BYS","title":"Internal hom in the derived category · Lemma 0BYS","summary":"Let (X, O_X) be a ringed space. Given K, L, M in D(O_X) there is a canonical morphism K ⊗_O_X^L RSheafHom(M, L) → RSheafHom(M, K ⊗_O_X^L L) in D(O_X) functorial in K, L, M.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Given $K, L, M$\nin $D(\\mathcal{O}_X)$ there is a canonical morphism\n$$\nK \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, L)\n\\longrightarrow\nR\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)\n$$\nin $D(\\mathcal{O}_X)$ functorial in $K, L, M$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYS","source_file":"cohomology.tex","source_line":10722,"source_end_line":10732,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10722-L10732","statement_sha256":"f7a6e2088515f847433190b8fee7083ba8b001747d3e9ce8af263a7036cc4e86","origin":"The Stacks Project","memory_eligible":false,"source_rank":4329,"rank":4329,"depth":2,"x":1061.495,"y":739.818,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8W","tag":"0A8W","title":"Internal hom in the derived category · Lemma 0A8W","summary":"Let (X, O_X) be a ringed space. Given K, L in D(O_X) there is a canonical morphism K → RSheafHom(L, K ⊗_O_X^L L) in D(O_X) functorial in both K and L.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Given $K, L$ in $D(\\mathcal{O}_X)$\nthere is a canonical morphism\n$$\nK \\longrightarrow R\\SheafHom(L, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)\n$$\nin $D(\\mathcal{O}_X)$ functorial in both $K$ and $L$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8W","source_file":"cohomology.tex","source_line":10756,"source_end_line":10764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10756-L10764","statement_sha256":"11da2a4abbad52deec33b5ba89dca01451f3abbf39db2adf40b7fbae6a515289","origin":"The Stacks Project","memory_eligible":false,"source_rank":4330,"rank":4330,"depth":2,"x":1132.417,"y":596.791,"cluster":"sheaf-cohomology"},{"id":"stacks:08I1","tag":"08I1","title":"Internal hom in the derived category · Lemma 08I1","summary":"Let (X, O_X) be a ringed space. Let L be an object of D(O_X). Set L^vee = RSheafHom(L, O_X). For M in D(O_X) there is a canonical map M ⊗^L_O_X L^vee → RSheafHom(L, M) which induces a canonical map H^0(X, M ⊗^L_O_X L^vee) → Hom_D(O_X)(L, M) functorial in M in D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $L$ be an\nobject of $D(\\mathcal{O}_X)$. Set $L^\\vee = R\\SheafHom(L, \\mathcal{O}_X)$.\nFor $M$ in $D(\\mathcal{O}_X)$ there is a canonical map\n\\begin{equation}\n\nM \\otimes^\\mathbf{L}_{\\mathcal{O}_X} L^\\vee\n\\longrightarrow\nR\\SheafHom(L, M)\n\\end{equation}\nwhich induces a canonical map\n$$\nH^0(X, M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} L^\\vee)\n\\longrightarrow\n\\Hom_{D(\\mathcal{O}_X)}(L, M)\n$$\nfunctorial in $M$ in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08I1","source_file":"cohomology.tex","source_line":10783,"source_end_line":10801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10783-L10801","statement_sha256":"9c65e8cf9108064c067f489721f95e67a202609b3ad4e7ce158058a0adcbe338","origin":"The Stacks Project","memory_eligible":false,"source_rank":4331,"rank":4331,"depth":3,"x":1195.311,"y":742.901,"cluster":"sheaf-cohomology"},{"id":"stacks:0A8U","tag":"0A8U","title":"Internal hom in the derived category · Lemma 0A8U","summary":"Let (X, O_X) be a ringed space. Let K, L, M be objects of D(O_X). There is a canonical morphism RSheafHom(L, M) ⊗_O_X^L K → RSheafHom(RSheafHom(K, L), M) in D(O_X) functorial in K, L, M.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L, M$ be objects of\n$D(\\mathcal{O}_X)$. There is a canonical morphism\n$$\nR\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K\n\\longrightarrow\nR\\SheafHom(R\\SheafHom(K, L), M)\n$$\nin $D(\\mathcal{O}_X)$ functorial in $K, L, M$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8U","source_file":"cohomology.tex","source_line":10809,"source_end_line":10819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L10809-L10819","statement_sha256":"a0a9b3a79da7479c94b37903d20b61404f39aee4c4c409dc4cfa490fb0d6ecd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4332,"rank":4332,"depth":2,"x":1030.988,"y":670.652,"cluster":"sheaf-cohomology"},{"id":"stacks:08DG","tag":"08DG","title":"Glueing complexes · Lemma 08DG","summary":"Let (X, O_X) be a ringed space. Let X = U ∪ V be the union of two open subspaces of X. Suppose given • an object A of D(O_U), • an object B of D(O_V), and • an isomorphism c : A|_U ∩ V → B|_U ∩ V. Then there exists an object F of D(O_X) and isomorphisms f : F|_U → A, g : F|_V → B such that c = g|_U ∩ V ∘ f^-1|_U ∩ V. Moreover, given • an object E of D(O_X), • a morphism a : A → E|_U of D(O_U), • a morphism b : B → E|_V of D(O_V), such that a|_U ∩ V = b|_U ∩ V ∘ c. Then…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $X = U \\cup V$ be\nthe union of two open subspaces of $X$. Suppose given\n\\begin{enumerate}\n\\item an object $A$ of $D(\\mathcal{O}_U)$,\n\\item an object $B$ of $D(\\mathcal{O}_V)$, and\n\\item an isomorphism $c : A|_{U \\cap V} \\to B|_{U \\cap V}$.\n\\end{enumerate}\nThen there exists an object $F$ of $D(\\mathcal{O}_X)$\nand isomorphisms $f : F|_U \\to A$, $g : F|_V \\to B$ such\nthat $c = g|_{U \\cap V} \\circ f^{-1}|_{U \\cap V}$.\nMoreover, given\n\\begin{enumerate}\n\\item an object $E$ of $D(\\mathcal{O}_X)$,\n\\item a morphism $a : A \\to E|_U$ of $D(\\mathcal{O}_U)$,\n\\item a morphism $b : B \\to E|_V$ of $D(\\mathcal{O}_V)$, \n\\end{enumerate}\nsuch that\n$$\na|_{U \\cap V}  = b|_{U \\cap V} \\circ c.\n$$\nThen there exists a morphism $F \\to E$ in $D(\\mathcal{O}_X)$\nwhose restriction to $U$ is $a \\circ f$\nand whose restriction to $V$ is $b \\circ g$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Glueing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DG","source_file":"cohomology.tex","source_line":11070,"source_end_line":11095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11070-L11095","statement_sha256":"cf72fcef885c0f4ab5f1574a7acb21339456c8e539bd3325bbfcee9e71dbec61","origin":"The Stacks Project","memory_eligible":false,"source_rank":4333,"rank":4333,"depth":7,"x":1210.746,"y":630.577,"cluster":"sheaf-cohomology"},{"id":"stacks:0D66","tag":"0D66","title":"Glueing complexes · Lemma 0D66","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let B be a basis for the topology on Y. • Assume K is in D(O_X) such that for V ∈ B we have H^i(f^-1(V), K) = 0 for i < 0. Then Rf_*K has vanishing cohomology sheaves in negative degrees, H^i(f^-1(V), K) = 0 for i < 0 for all opens V ⊂ Y, and the rule V ↦ H^0(f^-1V, K) is a sheaf on Y. • Assume K, L are in D(O_X) such that for V ∈ B we have Ext^i(K|_f^-1V, L|_f^-1V) = 0 for i < 0. Then Ext^i(K|_f^-1V, L|_f^-1V) =…","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism\nof ringed spaces. Let $\\mathcal{B}$ be a basis for the topology on $Y$.\n\\begin{enumerate}\n\\item Assume $K$ is in $D(\\mathcal{O}_X)$ such that\nfor $V \\in \\mathcal{B}$ we have $H^i(f^{-1}(V), K) = 0$ for $i < 0$.\nThen $Rf_*K$ has vanishing cohomology sheaves in negative degrees,\n$H^i(f^{-1}(V), K) = 0$ for $i < 0$ for all opens $V \\subset Y$, and\nthe rule $V \\mapsto H^0(f^{-1}V, K)$ is a sheaf on $Y$.\n\\item Assume $K, L$ are in $D(\\mathcal{O}_X)$ such that\nfor $V \\in \\mathcal{B}$ we have\n$\\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$.\nThen $\\Ext^i(K|_{f^{-1}V}, L|_{f^{-1}V}) = 0$ for $i < 0$\nfor all opens $V \\subset Y$ and\nthe rule $V \\mapsto \\Hom(K|_{f^{-1}V}, L|_{f^{-1}V})$ is a sheaf on $Y$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Glueing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D66","source_file":"cohomology.tex","source_line":11126,"source_end_line":11143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11126-L11143","statement_sha256":"ed3a33f185a81204fa682fc2373c5d4e772c6eabe8e6447fbf3ef4f9e9d60e31","origin":"The Stacks Project","memory_eligible":false,"source_rank":4334,"rank":4334,"depth":22,"x":1110.15,"y":762.483,"cluster":"sheaf-cohomology"},{"id":"stacks:0D68","tag":"0D68","title":"Glueing complexes · Lemma 0D68","summary":"In Situation [Tag 0D67] assume • X = ⋃_U ∈ B U and for U, V ∈ B we have U ∩ V = ⋃_W ∈ B, W ⊂ U ∩ V W, • for any U ∈ B we have Ext^i(K_U, K_U) = 0 for i < 0. If a solution (K, ρ_U) exists, then it is unique up to unique isomorphism and moreover Ext^i(K, K) = 0 for i < 0.","statement_latex":"In Situation \\ref{situation-locally-given} assume\n\\begin{enumerate}\n\\item $X = \\bigcup_{U \\in \\mathcal{B}} U$ and\nfor $U, V \\in \\mathcal{B}$ we have\n$U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$,\n\\item for any $U \\in \\mathcal{B}$ we have $\\Ext^i(K_U, K_U) = 0$\nfor $i < 0$.\n\\end{enumerate}\nIf a solution $(K, \\rho_U)$ exists, then it is unique up to unique isomorphism\nand moreover $\\Ext^i(K, K) = 0$ for $i < 0$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Glueing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D68","source_file":"cohomology.tex","source_line":11187,"source_end_line":11199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11187-L11199","statement_sha256":"a24e4faa990f3dc230f228bc0acceb7dfb60feac2b64b186884a2a2d3368c4db","origin":"The Stacks Project","memory_eligible":false,"source_rank":4335,"rank":4335,"depth":23,"x":1078.168,"y":607.721,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6A","tag":"0D6A","title":"Glueing complexes · Lemma 0D6A","summary":"In Situation [Tag 0D67] assume • X = U_1 ∪ … ∪ U_n with U_i ∈ B, • for U, V ∈ B we have U ∩ V = ⋃_W ∈ B, W ⊂ U ∩ V W, • for any U ∈ B we have Ext^i(K_U, K_U) = 0 for i < 0. Then a solution exists and is unique up to unique isomorphism.","statement_latex":"In Situation \\ref{situation-locally-given} assume\n\\begin{enumerate}\n\\item $X = U_1 \\cup \\ldots \\cup U_n$ with $U_i \\in \\mathcal{B}$,\n\\item for $U, V \\in \\mathcal{B}$ we have\n$U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$,\n\\item for any $U \\in \\mathcal{B}$ we have $\\Ext^i(K_U, K_U) = 0$\nfor $i < 0$.\n\\end{enumerate}\nThen a solution exists and is unique up to unique isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Glueing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6A","source_file":"cohomology.tex","source_line":11270,"source_end_line":11281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11270-L11281","statement_sha256":"a2daadae166722939db07c820b0e6d36ba31ca63bc00139c76b77e90e56d6eb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4336,"rank":4336,"depth":24,"x":1226.602,"y":703.95,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6B","tag":"0D6B","title":"Glueing complexes · Lemma 0D6B","summary":"Let X be a ringed space. Let E be a well ordered set and let X = ⋃_α ∈ E W_α be an open covering with W_α ⊂ W_α + 1 and W_α = ⋃_β < α W_β if α is not a successor. Let K_α be an object of D(O_W_α) with Ext^i(K_α, K_α) = 0 for i < 0. Assume given isomorphisms ρ_β^α : K_α|_W_β → K_β in D(O_W_β) for all β < α with ρ_γ^α = ρ_γ^β ∘ ρ^α_β|_W_γ for γ < β < α. Then there exists an object K in D(O_X) and isomorphisms K|_W_α → K_α for α ∈ E compatible with the isomorphisms ρ_β^α.","statement_latex":"Let $X$ be a ringed space. Let $E$ be a well ordered set and let\n$$\nX = \\bigcup\\nolimits_{\\alpha \\in E} W_\\alpha\n$$\nbe an open covering with $W_\\alpha \\subset W_{\\alpha + 1}$\nand $W_\\alpha = \\bigcup_{\\beta < \\alpha} W_\\beta$ if $\\alpha$ is not\na successor. Let $K_\\alpha$ be an object of $D(\\mathcal{O}_{W_\\alpha})$\nwith $\\Ext^i(K_\\alpha, K_\\alpha) = 0$ for $i < 0$.\nAssume given isomorphisms\n$\\rho_\\beta^\\alpha :  K_\\alpha|_{W_\\beta} \\to K_\\beta$ in\n$D(\\mathcal{O}_{W_\\beta})$ for all $\\beta < \\alpha$ with\n$\\rho_\\gamma^\\alpha = \\rho_\\gamma^\\beta \\circ \\rho^\\alpha_\\beta|_{W_\\gamma}$\nfor $\\gamma < \\beta < \\alpha$.\nThen there exists an object\n$K$ in $D(\\mathcal{O}_X)$ and isomorphisms\n$K|_{W_\\alpha} \\to K_\\alpha$ for $\\alpha \\in E$\ncompatible with the isomorphisms $\\rho_\\beta^\\alpha$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Glueing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6B","source_file":"cohomology.tex","source_line":11342,"source_end_line":11361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11342-L11361","statement_sha256":"bf14cbc63f5d0d8ff5a3fdcd93a56cf58092a8dc396bfe757cde8173d7261d34","origin":"The Stacks Project","memory_eligible":false,"source_rank":4337,"rank":4337,"depth":25,"x":1039.267,"y":717.252,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6C","tag":"0D6C","title":"BBD gluing lemma · Theorem 0D6C","summary":"Special case of [BBD] without boundedness assumption. In Situation [Tag 0D67] assume • X = ⋃_U ∈ B U, • for U, V ∈ B we have U ∩ V = ⋃_W ∈ B, W ⊂ U ∩ V W, • for any U ∈ B we have Ext^i(K_U, K_U) = 0 for i < 0. Then there exists an object K of D(O_X) and isomorphisms ρ_U : K|_U → K_U in D(O_U) for U ∈ B such that ρ^U_V ∘ ρ_U|_V = ρ_V for all V ⊂ U with U, V ∈ B. The pair (K, ρ_U) is unique up to unique isomorphism.","statement_latex":"\\begin{reference}\nSpecial case of \\cite[Theorem 3.2.4]{BBD}\nwithout boundedness assumption.\n\\end{reference}\nIn Situation \\ref{situation-locally-given} assume\n\\begin{enumerate}\n\\item $X = \\bigcup_{U \\in \\mathcal{B}} U$,\n\\item for $U, V \\in \\mathcal{B}$ we have\n$U \\cap V = \\bigcup_{W \\in \\mathcal{B}, W \\subset U \\cap V} W$,\n\\item for any $U \\in \\mathcal{B}$ we have $\\Ext^i(K_U, K_U) = 0$\nfor $i < 0$.\n\\end{enumerate}\nThen there exists an object $K$ of $D(\\mathcal{O}_X)$\nand isomorphisms $\\rho_U : K|_U \\to K_U$ in $D(\\mathcal{O}_U)$ for\n$U \\in \\mathcal{B}$ such that $\\rho^U_V \\circ \\rho_U|_V = \\rho_V$\nfor all $V \\subset U$ with $U, V \\in \\mathcal{B}$.\nThe pair $(K, \\rho_U)$ is unique up to unique isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Glueing complexes","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6C","source_file":"cohomology.tex","source_line":11457,"source_end_line":11476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11457-L11476","statement_sha256":"5c09b99b130821d2d7da620837876a8d0c7afb13eb3cdbb4ff8a58b893e6686d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4338,"rank":4338,"depth":26,"x":1167.046,"y":600.838,"cluster":"sheaf-cohomology"},{"id":"stacks:08C4","tag":"08C4","title":"Strictly perfect complexes · Definition 08C4","summary":"Let (X, O_X) be a ringed space. Let E^bullet be a complex of O_X-modules. We say E^bullet is strictly perfect if E^i is zero for all but finitely many i and E^i is a direct summand of a finite free O_X-module for all i.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.\nWe say $\\mathcal{E}^\\bullet$ is {\\it strictly perfect}\nif $\\mathcal{E}^i$ is zero for all but finitely many $i$ and\n$\\mathcal{E}^i$ is a direct summand of a finite free\n$\\mathcal{O}_X$-module for all $i$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08C4","source_file":"cohomology.tex","source_line":11601,"source_end_line":11609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11601-L11609","statement_sha256":"2d058a37ae44973a5789c860f8b65c7f2ce1ebf35b4e5d96ed6d29cb96ab653e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4339,"rank":4339,"depth":0,"x":1166.439,"y":759.602,"cluster":"sheaf-cohomology"},{"id":"stacks:08C5","tag":"08C5","title":"Strictly perfect complexes · Lemma 08C5","summary":"The cone on a morphism of strictly perfect complexes is strictly perfect.","statement_latex":"The cone on a morphism of strictly perfect complexes is\nstrictly perfect.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08C5","source_file":"cohomology.tex","source_line":11616,"source_end_line":11620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11616-L11620","statement_sha256":"0dcd61eb62b03fe8d9a38ae2f16845251ed27b8104e57237dfa123a73dd9b054","origin":"The Stacks Project","memory_eligible":false,"source_rank":4340,"rank":4340,"depth":0,"x":1038.879,"y":641.879,"cluster":"sheaf-cohomology"},{"id":"stacks:09J2","tag":"09J2","title":"Strictly perfect complexes · Lemma 09J2","summary":"The total complex associated to the tensor product of two strictly perfect complexes is strictly perfect.","statement_latex":"The total complex associated to the tensor product of two\nstrictly perfect complexes is strictly perfect.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09J2","source_file":"cohomology.tex","source_line":11626,"source_end_line":11630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11626-L11630","statement_sha256":"66baba13494d7020988b6cb0bb3d7d34eec68fbb34bbf9658f99f516a676013e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4341,"rank":4341,"depth":0,"x":1228.101,"y":656.344,"cluster":"sheaf-cohomology"},{"id":"stacks:09U6","tag":"09U6","title":"Strictly perfect complexes · Lemma 09U6","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. If F^bullet is a strictly perfect complex of O_Y-modules, then f^*F^bullet is a strictly perfect complex of O_X-modules.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces. If $\\mathcal{F}^\\bullet$ is a strictly\nperfect complex of $\\mathcal{O}_Y$-modules, then\n$f^*\\mathcal{F}^\\bullet$ is a strictly perfect complex of\n$\\mathcal{O}_X$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09U6","source_file":"cohomology.tex","source_line":11636,"source_end_line":11643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11636-L11643","statement_sha256":"fb38e5fc88eb756c6f7e9d1594777ac662ecf9eb14e9db9c97dd4501bd18801b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4342,"rank":4342,"depth":0,"x":1076.547,"y":753.297,"cluster":"sheaf-cohomology"},{"id":"stacks:08C6","tag":"08C6","title":"Strictly perfect complexes · Lemma 08C6","summary":"Let (X, O_X) be a ringed space. Given a solid diagram of O_X-modules xymatrix E ar@..>[dr] ar[r] & F & G ar[u]_p with E a direct summand of a finite free O_X-module and p surjective, then a dotted arrow making the diagram commute exists locally on X.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nGiven a solid diagram of $\\mathcal{O}_X$-modules\n$$\n\\xymatrix{\n\\mathcal{E} \\ar@{..>}[dr] \\ar[r] & \\mathcal{F} \\\\\n& \\mathcal{G} \\ar[u]_p\n}\n$$\nwith $\\mathcal{E}$ a direct summand of a finite free\n$\\mathcal{O}_X$-module and $p$ surjective, then a dotted arrow\nmaking the diagram commute exists locally on $X$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08C6","source_file":"cohomology.tex","source_line":11650,"source_end_line":11663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11650-L11663","statement_sha256":"cd7d2fde3008a6194b969a55822da828dee4045c0541b2b263e3993c9532f05c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4343,"rank":4343,"depth":0,"x":1110.421,"y":595.404,"cluster":"sheaf-cohomology"},{"id":"stacks:08C7","tag":"08C7","title":"Strictly perfect complexes · Lemma 08C7","summary":"Let (X, O_X) be a ringed space. • Let α : E^bullet → F^bullet be a morphism of complexes of O_X-modules with E^bullet strictly perfect and F^bullet acyclic. Then α is locally on X homotopic to zero. • Let α : E^bullet → F^bullet be a morphism of complexes of O_X-modules with E^bullet strictly perfect, E^i = 0 for i < a, and H^i(F^bullet) = 0 for i ≥ a. Then α is locally on X homotopic to zero.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\n\\begin{enumerate}\n\\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$\nbe a morphism of complexes of $\\mathcal{O}_X$-modules\nwith $\\mathcal{E}^\\bullet$ strictly perfect and $\\mathcal{F}^\\bullet$\nacyclic. Then $\\alpha$ is locally on $X$ homotopic to zero.\n\\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$\nbe a morphism of complexes of $\\mathcal{O}_X$-modules\nwith $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^i = 0$\nfor $i < a$, and $H^i(\\mathcal{F}^\\bullet) = 0$ for $i \\geq a$.\nThen $\\alpha$ is locally on $X$ homotopic to zero.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08C7","source_file":"cohomology.tex","source_line":11673,"source_end_line":11687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11673-L11687","statement_sha256":"3e28b3cacae4b126ab9d4197f2ae46b7b534357bdb3a850f624e28af6d189513","origin":"The Stacks Project","memory_eligible":false,"source_rank":4344,"rank":4344,"depth":1,"x":1212.678,"y":731.401,"cluster":"sheaf-cohomology"},{"id":"stacks:08C8","tag":"08C8","title":"Strictly perfect complexes · Lemma 08C8","summary":"Let (X, O_X) be a ringed space. Given a solid diagram of complexes of O_X-modules xymatrix E^bullet ar@..>[dr] ar[r]_α & F^bullet & G^bullet ar[u]_f with E^bullet strictly perfect, E^j = 0 for j < a and H^j(f) an isomorphism for j > a and surjective for j = a, then a dotted arrow making the diagram commute up to homotopy exists locally on X.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nGiven a solid diagram of complexes of $\\mathcal{O}_X$-modules\n$$\n\\xymatrix{\n\\mathcal{E}^\\bullet \\ar@{..>}[dr] \\ar[r]_\\alpha & \\mathcal{F}^\\bullet \\\\\n& \\mathcal{G}^\\bullet \\ar[u]_f\n}\n$$\nwith $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^j = 0$ for\n$j < a$ and $H^j(f)$ an isomorphism for $j > a$ and surjective for $j = a$,\nthen a dotted arrow making the diagram commute up to homotopy\nexists locally on $X$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08C8","source_file":"cohomology.tex","source_line":11721,"source_end_line":11735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11721-L11735","statement_sha256":"9a7608babf4e36fd0e8fb560b8a9e4880032592f277bef7e75ba3923543d926b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4345,"rank":4345,"depth":2,"x":1027.439,"y":689.034,"cluster":"sheaf-cohomology"},{"id":"stacks:08C9","tag":"08C9","title":"Strictly perfect complexes · Lemma 08C9","summary":"Let (X, O_X) be a ringed space. Let E^bullet, F^bullet be complexes of O_X-modules with E^bullet strictly perfect. • For any element α ∈ Hom_D(O_X)(E^bullet, F^bullet) there exists an open covering X = ⋃ U_i such that α|_U_i is given by a morphism of complexes α_i : E^bullet|_U_i → F^bullet|_U_i. • Given a morphism of complexes α : E^bullet → F^bullet whose image in the group Hom_D(O_X)(E^bullet, F^bullet) is zero, there exists an open covering X = ⋃ U_i such that α|_U_i…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes\nof $\\mathcal{O}_X$-modules with $\\mathcal{E}^\\bullet$ strictly perfect.\n\\begin{enumerate}\n\\item For any element\n$\\alpha \\in \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$\nthere exists an open covering $X = \\bigcup U_i$ such that\n$\\alpha|_{U_i}$ is given by a morphism of complexes\n$\\alpha_i : \\mathcal{E}^\\bullet|_{U_i} \\to \\mathcal{F}^\\bullet|_{U_i}$.\n\\item Given a morphism of complexes\n$\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$\nwhose image in the group\n$\\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$\nis zero, there exists an open covering $X = \\bigcup U_i$ such that\n$\\alpha|_{U_i}$ is homotopic to zero.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08C9","source_file":"cohomology.tex","source_line":11754,"source_end_line":11772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11754-L11772","statement_sha256":"873de7505cc74a3d3ad04180303a6f55295c50b838252e52d06e5af99427caec","origin":"The Stacks Project","memory_eligible":false,"source_rank":4346,"rank":4346,"depth":3,"x":1198.536,"y":614.979,"cluster":"sheaf-cohomology"},{"id":"stacks:08DM","tag":"08DM","title":"Strictly perfect complexes · Lemma 08DM","summary":"Let (X, O_X) be a ringed space. Let E^bullet, F^bullet be complexes of O_X-modules with E^bullet strictly perfect. Then the internal hom RSheafHom(E^bullet, F^bullet) is represented by the complex H^bullet with terms H^n = bigoplus_n = p + q SheafHom_O_X(E^-q, F^p) and differential as described in Section [Tag 0A8K].","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes\nof $\\mathcal{O}_X$-modules with $\\mathcal{E}^\\bullet$ strictly perfect.\nThen the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$\nis represented by the complex $\\mathcal{H}^\\bullet$ with terms\n$$\n\\mathcal{H}^n =\n\\bigoplus\\nolimits_{n = p + q}\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{F}^p)\n$$\nand differential as described in Section \\ref{section-hom-complexes}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DM","source_file":"cohomology.tex","source_line":11784,"source_end_line":11797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11784-L11797","statement_sha256":"adf4fc6cdcf5c99186a40c4872c8c152c8b97baeb41f28cfc3b4fb000abafbef","origin":"The Stacks Project","memory_eligible":false,"source_rank":4347,"rank":4347,"depth":4,"x":1131.756,"y":767.053,"cluster":"sheaf-cohomology"},{"id":"stacks:0GM5","tag":"0GM5","title":"Strictly perfect complexes · Lemma 0GM5","summary":"In the situation of Lemma [Tag 08DM] if F^bullet is K-flat, then H^bullet is K-flat.","statement_latex":"In the situation of Lemma \\ref{lemma-Rhom-strictly-perfect}\nif $\\mathcal{F}^\\bullet$ is K-flat, then $\\mathcal{H}^\\bullet$ is K-flat.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GM5","source_file":"cohomology.tex","source_line":11831,"source_end_line":11835,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11831-L11835","statement_sha256":"5f45a60450df441efd066625047d031d7a477f9442a1a9d8d02af42348cc8cc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4348,"rank":4348,"depth":5,"x":1058.521,"y":616.646,"cluster":"sheaf-cohomology"},{"id":"stacks:08I5","tag":"08I5","title":"Strictly perfect complexes · Lemma 08I5","summary":"Let (X, O_X) be a ringed space. Let E^bullet, F^bullet be complexes of O_X-modules with • F^n = 0 for n ll 0, • E^n = 0 for n gg 0, and • E^n isomorphic to a direct summand of a finite free O_X-module. Then the internal hom RSheafHom(E^bullet, F^bullet) is represented by the complex H^bullet with terms H^n = bigoplus_n = p + q SheafHom_O_X(E^-q, F^p) and differential as described in Section [Tag 08DH].","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes\nof $\\mathcal{O}_X$-modules with\n\\begin{enumerate}\n\\item $\\mathcal{F}^n = 0$ for $n \\ll 0$,\n\\item $\\mathcal{E}^n = 0$ for $n \\gg 0$, and\n\\item $\\mathcal{E}^n$ isomorphic to a direct summand of a finite\nfree $\\mathcal{O}_X$-module.\n\\end{enumerate}\nThen the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$\nis represented by the complex $\\mathcal{H}^\\bullet$ with terms\n$$\n\\mathcal{H}^n =\n\\bigoplus\\nolimits_{n = p + q}\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{E}^{-q}, \\mathcal{F}^p)\n$$\nand differential as described in Section \\ref{section-internal-hom}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08I5","source_file":"cohomology.tex","source_line":11881,"source_end_line":11900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11881-L11900","statement_sha256":"1f330bdb861fedc4d21350fc2649dddebbc75a049fa1a1d9567bd250830267ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":4349,"rank":4349,"depth":5,"x":1233.914,"y":686.173,"cluster":"sheaf-cohomology"},{"id":"stacks:08CB","tag":"08CB","title":"Pseudo-coherent modules · Definition 08CB","summary":"Let (X, O_X) be a ringed space. Let E^bullet be a complex of O_X-modules. Let m ∈ Z. • We say E^bullet is m-pseudo-coherent if there exists an open covering X = ⋃ U_i and for each i a morphism of complexes α_i : E_i^bullet → E^bullet|_U_i where E_i^bullet is strictly perfect on U_i and H^j(α_i) is an isomorphism for j > m and H^m(α_i) is surjective. • We say E^bullet is pseudo-coherent if it is m-pseudo-coherent for all m. • We say an object E of D(O_X) is…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{E}^\\bullet$\nbe a complex of $\\mathcal{O}_X$-modules. Let $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item We say $\\mathcal{E}^\\bullet$ is {\\it $m$-pseudo-coherent}\nif there exists an open covering $X = \\bigcup U_i$ and for each $i$\na morphism of complexes\n$\\alpha_i : \\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$\nwhere $\\mathcal{E}_i^\\bullet$ is strictly perfect on $U_i$ and\n$H^j(\\alpha_i)$ is an isomorphism for $j > m$ and $H^m(\\alpha_i)$\nis surjective.\n\\item We say $\\mathcal{E}^\\bullet$ is {\\it pseudo-coherent}\nif it is $m$-pseudo-coherent for all $m$.\n\\item We say an object $E$ of $D(\\mathcal{O}_X)$ is\n{\\it $m$-pseudo-coherent} (resp.\\ {\\it pseudo-coherent})\nif and only if it can be represented by a $m$-pseudo-coherent\n(resp.\\ pseudo-coherent) complex of $\\mathcal{O}_X$-modules.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CB","source_file":"cohomology.tex","source_line":11957,"source_end_line":11976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11957-L11976","statement_sha256":"aa7fe8a69e4e96be6f4b955029926ace7af42c84abd9a8c35da85f50c4f70b3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4350,"rank":4350,"depth":0,"x":1048.209,"y":734.546,"cluster":"sheaf-cohomology"},{"id":"stacks:08CC","tag":"08CC","title":"Pseudo-coherent modules · Lemma 08CC","summary":"Let (X, O_X) be a ringed space. Let E be an object of D(O_X). • If there exists an open covering X = ⋃ U_i, strictly perfect complexes E_i^bullet on U_i, and maps α_i : E_i^bullet → E|_U_i in D(O_U_i) with H^j(α_i) an isomorphism for j > m and H^m(α_i) surjective, then E is m-pseudo-coherent. • If E is m-pseudo-coherent, then any complex representing E is m-pseudo-coherent.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object\nof $D(\\mathcal{O}_X)$.\n\\begin{enumerate}\n\\item If there exists an open covering $X = \\bigcup U_i$,\nstrictly perfect complexes $\\mathcal{E}_i^\\bullet$ on $U_i$, and\nmaps $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ in\n$D(\\mathcal{O}_{U_i})$ with $H^j(\\alpha_i)$ an isomorphism for $j > m$\nand $H^m(\\alpha_i)$ surjective, then $E$ is $m$-pseudo-coherent.\n\\item If $E$ is $m$-pseudo-coherent, then any complex representing\n$E$ is $m$-pseudo-coherent.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CC","source_file":"cohomology.tex","source_line":11983,"source_end_line":11996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L11983-L11996","statement_sha256":"995a170766e56d2c76da3329367dd0656ca2d3bbb3583115f07ceb2424f7ad76","origin":"The Stacks Project","memory_eligible":false,"source_rank":4351,"rank":4351,"depth":4,"x":1146.487,"y":593.154,"cluster":"sheaf-cohomology"},{"id":"stacks:09U7","tag":"09U7","title":"Pseudo-coherent modules · Lemma 09U7","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let E be an object of D(O_Y). If E is m-pseudo-coherent, then Lf^*E is m-pseudo-coherent.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nbe a morphism of ringed spaces. Let $E$ be an object of\n$D(\\mathcal{O}_Y)$. If $E$ is $m$-pseudo-coherent,\nthen $Lf^*E$ is $m$-pseudo-coherent.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09U7","source_file":"cohomology.tex","source_line":12014,"source_end_line":12020,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12014-L12020","statement_sha256":"cc0d8a7f08b670e164237ea205835bacc62df2564b2d42e87d4d82f777040e94","origin":"The Stacks Project","memory_eligible":false,"source_rank":4352,"rank":4352,"depth":10,"x":1187.823,"y":753.574,"cluster":"sheaf-cohomology"},{"id":"stacks:08CD","tag":"08CD","title":"Pseudo-coherent modules · Lemma 08CD","summary":"Let (X, O_X) be a ringed space and m ∈ Z. Let (K, L, M, f, g, h) be a distinguished triangle in D(O_X). • If K is (m + 1)-pseudo-coherent and L is m-pseudo-coherent then M is m-pseudo-coherent. • If K and M are m-pseudo-coherent, then L is m-pseudo-coherent. • If L is (m + 1)-pseudo-coherent and M is m-pseudo-coherent, then K is (m + 1)-pseudo-coherent.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space and $m \\in \\mathbf{Z}$.\nLet $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O}_X)$.\n\\begin{enumerate}\n\\item If $K$ is $(m + 1)$-pseudo-coherent and $L$ is $m$-pseudo-coherent\nthen $M$ is $m$-pseudo-coherent.\n\\item If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent.\n\\item If $L$ is $(m + 1)$-pseudo-coherent and $M$\nis $m$-pseudo-coherent, then $K$ is $(m + 1)$-pseudo-coherent.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CD","source_file":"cohomology.tex","source_line":12059,"source_end_line":12070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12059-L12070","statement_sha256":"f6c2bcbd73e173b383fa24e44221e6794526f2252c015c87dd79ed8c1e4f3fda","origin":"The Stacks Project","memory_eligible":false,"source_rank":4353,"rank":4353,"depth":8,"x":1027.947,"y":658.505,"cluster":"sheaf-cohomology"},{"id":"stacks:09J3","tag":"09J3","title":"Pseudo-coherent modules · Lemma 09J3","summary":"Let (X, O_X) be a ringed space. Let K, L be objects of D(O_X). • If K is n-pseudo-coherent and H^i(K) = 0 for i > a and L is m-pseudo-coherent and H^j(L) = 0 for j > b, then K ⊗_O_X^L L is t-pseudo-coherent with t = max(m + a, n + b). • If K and L are pseudo-coherent, then K ⊗_O_X^L L is pseudo-coherent.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects\nof $D(\\mathcal{O}_X)$.\n\\begin{enumerate}\n\\item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$\nand $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then\n$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ is $t$-pseudo-coherent\nwith $t = \\max(m + a, n + b)$.\n\\item If $K$ and $L$ are pseudo-coherent, then\n$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ is pseudo-coherent.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09J3","source_file":"cohomology.tex","source_line":12120,"source_end_line":12132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12120-L12132","statement_sha256":"2a17f6ad208f8281da3b72b802f78beea45bc9bf862ac6bd97ae6c6ab69e45c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4354,"rank":4354,"depth":0,"x":1222.763,"y":637.84,"cluster":"sheaf-cohomology"},{"id":"stacks:08CE","tag":"08CE","title":"Pseudo-coherent modules · Lemma 08CE","summary":"Let (X, O_X) be a ringed space. Let m ∈ Z. If K ⊕ L is m-pseudo-coherent (resp. pseudo-coherent) in D(O_X) so are K and L.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $m \\in \\mathbf{Z}$.\nIf $K \\oplus L$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nin $D(\\mathcal{O}_X)$ so are $K$ and $L$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CE","source_file":"cohomology.tex","source_line":12158,"source_end_line":12163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12158-L12163","statement_sha256":"3d78631f4bc9173056fd689db0eb88f41f17374354ed1dc3daf6267f0f3c3140","origin":"The Stacks Project","memory_eligible":false,"source_rank":4355,"rank":4355,"depth":9,"x":1095.419,"y":763.927,"cluster":"sheaf-cohomology"},{"id":"stacks:09V7","tag":"09V7","title":"Pseudo-coherent modules · Lemma 09V7","summary":"Let (X, O_X) be a ringed space. Let m ∈ Z. Let F^bullet be a (locally) bounded above complex of O_X-modules such that F^i is (m - i)-pseudo-coherent for all i. Then F^bullet is m-pseudo-coherent.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $m \\in \\mathbf{Z}$. Let $\\mathcal{F}^\\bullet$ be a (locally) bounded\nabove complex of $\\mathcal{O}_X$-modules such that\n$\\mathcal{F}^i$ is $(m - i)$-pseudo-coherent for all $i$.\nThen $\\mathcal{F}^\\bullet$ is $m$-pseudo-coherent.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09V7","source_file":"cohomology.tex","source_line":12191,"source_end_line":12198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12191-L12198","statement_sha256":"0c0e8a8a581edddeb12988e0869c2572ab64f6cf56954b79c7c25723d902caf3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4356,"rank":4356,"depth":11,"x":1087.907,"y":598.295,"cluster":"sheaf-cohomology"},{"id":"stacks:09V8","tag":"09V8","title":"Pseudo-coherent modules · Lemma 09V8","summary":"Let (X, O_X) be a ringed space. Let m ∈ Z. Let E be an object of D(O_X). If E is (locally) bounded above and H^i(E) is (m - i)-pseudo-coherent for all i, then E is m-pseudo-coherent.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $m \\in \\mathbf{Z}$. Let\n$E$ be an object of $D(\\mathcal{O}_X)$. If $E$ is (locally) bounded above\nand $H^i(E)$ is $(m - i)$-pseudo-coherent for all $i$, then\n$E$ is $m$-pseudo-coherent.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09V8","source_file":"cohomology.tex","source_line":12206,"source_end_line":12212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12206-L12212","statement_sha256":"3d0cb26b95caf2c53ce7d1ac6add7df0bd1a152ee4b7957c7ce7c66f359674be","origin":"The Stacks Project","memory_eligible":false,"source_rank":4357,"rank":4357,"depth":12,"x":1226.975,"y":716.451,"cluster":"sheaf-cohomology"},{"id":"stacks:08DN","tag":"08DN","title":"Pseudo-coherent modules · Lemma 08DN","summary":"Let (X, O_X) be a ringed space. Let K be an object of D(O_X). Let m ∈ Z. • If K is m-pseudo-coherent and H^i(K) = 0 for i > m, then H^m(K) is a finite type O_X-module. • If K is m-pseudo-coherent and H^i(K) = 0 for i > m + 1, then H^m + 1(K) is a finitely presented O_X-module.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $K$ be an object of $D(\\mathcal{O}_X)$.\nLet $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$\nfor $i > m$, then $H^m(K)$ is a finite type $\\mathcal{O}_X$-module.\n\\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$\nfor $i > m + 1$, then $H^{m + 1}(K)$ is a finitely presented\n$\\mathcal{O}_X$-module.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DN","source_file":"cohomology.tex","source_line":12220,"source_end_line":12232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12220-L12232","statement_sha256":"50034229ba29a8bf1b55cd1128461f81e66b69c3cee0c2c0b981e0fc8d098067","origin":"The Stacks Project","memory_eligible":false,"source_rank":4358,"rank":4358,"depth":1,"x":1028.939,"y":708.215,"cluster":"sheaf-cohomology"},{"id":"stacks:09V9","tag":"09V9","title":"Pseudo-coherent modules · Lemma 09V9","summary":"Let (X, O_X) be a ringed space. Let F be a sheaf of O_X-modules. • F viewed as an object of D(O_X) is 0-pseudo-coherent if and only if F is a finite type O_X-module, and • F viewed as an object of D(O_X) is (-1)-pseudo-coherent if and only if F is an O_X-module of finite presentation.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be a sheaf\nof $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item $\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is\n$0$-pseudo-coherent if and only if $\\mathcal{F}$ is a finite type\n$\\mathcal{O}_X$-module, and\n\\item $\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is\n$(-1)$-pseudo-coherent if and only if $\\mathcal{F}$ is an\n$\\mathcal{O}_X$-module of finite presentation.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09V9","source_file":"cohomology.tex","source_line":12264,"source_end_line":12276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12264-L12276","statement_sha256":"ba4aaa11d3f0e3af1552e11307ade53a3cdc6c89d38f0dd33ebe70236416d82f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4359,"rank":4359,"depth":13,"x":1181.954,"y":601.665,"cluster":"sheaf-cohomology"},{"id":"stacks:08CG","tag":"08CG","title":"Tor dimension · Definition 08CG","summary":"Let (X, O_X) be a ringed space. Let E be an object of D(O_X). Let a, b ∈ Z with a ≤ b. • We say E has tor-amplitude in [a, b] if H^i(E ⊗_O_X^L F) = 0 for all O_X-modules F and all i not ∈ [a, b]. • We say E has finite tor dimension if it has tor-amplitude in [a, b] for some a, b. • We say E locally has finite tor dimension if there exists an open covering X = ⋃ U_i such that E|_U_i has finite tor dimension for all i. An O_X-module F has tor dimension ≤ d if F[0] viewed as…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\nLet $a, b \\in \\mathbf{Z}$ with $a \\leq b$.\n\\begin{enumerate}\n\\item We say $E$ has {\\it tor-amplitude in $[a, b]$}\nif $H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}) = 0$\nfor all $\\mathcal{O}_X$-modules $\\mathcal{F}$ and all $i \\not \\in [a, b]$.\n\\item We say $E$ has {\\it finite tor dimension}\nif it has tor-amplitude in $[a, b]$ for some $a, b$.\n\\item We say $E$ {\\it locally has finite tor dimension}\nif there exists an open covering $X = \\bigcup U_i$ such that\n$E|_{U_i}$ has finite tor dimension for all $i$.\n\\end{enumerate}\nAn $\\mathcal{O}_X$-module $\\mathcal{F}$ has {\\it tor dimension $\\leq d$}\nif $\\mathcal{F}[0]$ viewed as an object of $D(\\mathcal{O}_X)$ has\ntor-amplitude in $[-d, 0]$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Tor dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CG","source_file":"cohomology.tex","source_line":12294,"source_end_line":12312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12294-L12312","statement_sha256":"6355248f44e680d5fcd3e150521ddd3b2e1dd82669a8869732f3b002e47081d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4360,"rank":4360,"depth":0,"x":1154.743,"y":767.449,"cluster":"sheaf-cohomology"},{"id":"stacks:08CH","tag":"08CH","title":"Tor dimension · Lemma 08CH","summary":"Let (X, O_X) be a ringed space. Let E^bullet be a bounded above complex of flat O_X-modules with tor-amplitude in [a, b]. Then Coker(d_E^bullet^a - 1) is a flat O_X-module.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{E}^\\bullet$ be a bounded above complex of flat\n$\\mathcal{O}_X$-modules with tor-amplitude in $[a, b]$.\nThen $\\Coker(d_{\\mathcal{E}^\\bullet}^{a - 1})$ is a flat\n$\\mathcal{O}_X$-module.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CH","source_file":"cohomology.tex","source_line":12320,"source_end_line":12327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12320-L12327","statement_sha256":"b45f74897a07a8dc1496e577a10dd5d6d397be5135f08fa0752426a3908f6152","origin":"The Stacks Project","memory_eligible":false,"source_rank":4361,"rank":4361,"depth":1,"x":1041.222,"y":629.437,"cluster":"sheaf-cohomology"},{"id":"stacks:08CI","tag":"08CI","title":"Tor dimension · Lemma 08CI","summary":"Let (X, O_X) be a ringed space. Let E be an object of D(O_X). Let a, b ∈ Z with a ≤ b. The following are equivalent • E has tor-amplitude in [a, b]. • E is represented by a complex E^bullet of flat O_X-modules with E^i = 0 for i not ∈ [a, b].","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\nLet $a, b \\in \\mathbf{Z}$ with $a \\leq b$. The following are equivalent\n\\begin{enumerate}\n\\item $E$ has tor-amplitude in $[a, b]$.\n\\item $E$ is represented by a complex\n$\\mathcal{E}^\\bullet$ of flat $\\mathcal{O}_X$-modules with\n$\\mathcal{E}^i = 0$ for $i \\not \\in [a, b]$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CI","source_file":"cohomology.tex","source_line":12349,"source_end_line":12360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12349-L12360","statement_sha256":"cf19eac3fcd2bb023eb432db88e4d7258e8c0082e21aae8f619a7c1a0a8ef6b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4362,"rank":4362,"depth":5,"x":1236.373,"y":666.879,"cluster":"sheaf-cohomology"},{"id":"stacks:09U8","tag":"09U8","title":"Tor dimension · Lemma 09U8","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let E be an object of D(O_Y). If E has tor amplitude in [a, b], then Lf^*E has tor amplitude in [a, b].","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed\nspaces. Let $E$ be an object of $D(\\mathcal{O}_Y)$.\nIf $E$ has tor amplitude in $[a, b]$, then $Lf^*E$ has tor amplitude in\n$[a, b]$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09U8","source_file":"cohomology.tex","source_line":12389,"source_end_line":12395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12389-L12395","statement_sha256":"bc39cbbe9ad53fec92ad78db36b72f6556dd76cc6ba72a4c728d353fb71b60c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4363,"rank":4363,"depth":6,"x":1061.956,"y":750.197,"cluster":"sheaf-cohomology"},{"id":"stacks:09U9","tag":"09U9","title":"Tor dimension · Lemma 09U9","summary":"Let (X, O_X) be a ringed space. Let E be an object of D(O_X). Let a, b ∈ Z with a ≤ b. The following are equivalent • E has tor-amplitude in [a, b]. • for every x ∈ X the object E_x of D(O_X, x) has tor-amplitude in [a, b].","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\nLet $a, b \\in \\mathbf{Z}$ with $a \\leq b$. The following are equivalent\n\\begin{enumerate}\n\\item $E$ has tor-amplitude in $[a, b]$.\n\\item for every $x \\in X$ the object $E_x$ of $D(\\mathcal{O}_{X, x})$\nhas tor-amplitude in $[a, b]$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09U9","source_file":"cohomology.tex","source_line":12410,"source_end_line":12420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12410-L12420","statement_sha256":"ece3b489f9636c1ebc499f1fa70da11b856822e6773be56188d2c870c5a526aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4364,"rank":4364,"depth":7,"x":1123.71,"y":589.419,"cluster":"sheaf-cohomology"},{"id":"stacks:08CJ","tag":"08CJ","title":"Tor dimension · Lemma 08CJ","summary":"Let (X, O_X) be a ringed space. Let (K, L, M, f, g, h) be a distinguished triangle in D(O_X). Let a, b ∈ Z. • If K has tor-amplitude in [a + 1, b + 1] and L has tor-amplitude in [a, b] then M has tor-amplitude in [a, b]. • If K and M have tor-amplitude in [a, b], then L has tor-amplitude in [a, b]. • If L has tor-amplitude in [a + 1, b + 1] and M has tor-amplitude in [a, b], then K has tor-amplitude in [a + 1, b + 1].","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $(K, L, M, f, g, h)$ be a distinguished\ntriangle in $D(\\mathcal{O}_X)$. Let $a, b \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $K$ has tor-amplitude in $[a + 1, b + 1]$ and\n$L$ has tor-amplitude in $[a, b]$ then $M$ has\ntor-amplitude in $[a, b]$.\n\\item If $K$ and $M$ have tor-amplitude in $[a, b]$, then\n$L$ has tor-amplitude in $[a, b]$.\n\\item If $L$ has tor-amplitude in $[a + 1, b + 1]$\nand $M$ has tor-amplitude in $[a, b]$, then\n$K$ has tor-amplitude in $[a + 1, b + 1]$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CJ","source_file":"cohomology.tex","source_line":12446,"source_end_line":12461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12446-L12461","statement_sha256":"add913d54966e89ed96a8d6669140143e5f43d8a980e73ce0944e6859a961255","origin":"The Stacks Project","memory_eligible":false,"source_rank":4365,"rank":4365,"depth":0,"x":1207.66,"y":743.37,"cluster":"sheaf-cohomology"},{"id":"stacks:09J4","tag":"09J4","title":"Tor dimension · Lemma 09J4","summary":"Let (X, O_X) be a ringed space. Let K, L be objects of D(O_X). If K has tor-amplitude in [a, b] and L has tor-amplitude in [c, d] then K ⊗_O_X^L L has tor amplitude in [a + c, b + d].","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L$ be objects of\n$D(\\mathcal{O}_X)$. If $K$ has tor-amplitude in $[a, b]$ and\n$L$ has tor-amplitude in $[c, d]$ then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$\nhas tor amplitude in $[a + c, b + d]$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09J4","source_file":"cohomology.tex","source_line":12472,"source_end_line":12478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12472-L12478","statement_sha256":"895a5abf926e09f2d9734691100a69e5dccbbba6460c2845a46be2b5d1db6753","origin":"The Stacks Project","memory_eligible":false,"source_rank":4366,"rank":4366,"depth":0,"x":1021.526,"y":677.332,"cluster":"sheaf-cohomology"},{"id":"stacks:08CK","tag":"08CK","title":"Tor dimension · Lemma 08CK","summary":"Let (X, O_X) be a ringed space. Let a, b ∈ Z. For K, L objects of D(O_X) if K ⊕ L has tor amplitude in [a, b] so do K and L.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $a, b \\in \\mathbf{Z}$.\nFor $K$, $L$ objects of $D(\\mathcal{O}_X)$ if $K \\oplus L$ has tor\namplitude in $[a, b]$ so do $K$ and $L$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CK","source_file":"cohomology.tex","source_line":12484,"source_end_line":12489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12484-L12489","statement_sha256":"f01a223e71be002699905503fd8c5768e50bb56b8dd1a627fb10c7de9e9686ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":4367,"rank":4367,"depth":0,"x":1212.32,"y":620.28,"cluster":"sheaf-cohomology"},{"id":"stacks:08CM","tag":"08CM","title":"Perfect complexes · Definition 08CM","summary":"Let (X, O_X) be a ringed space. Let E^bullet be a complex of O_X-modules. We say E^bullet is perfect if there exists an open covering X = ⋃ U_i such that for each i there exists a morphism of complexes E_i^bullet → E^bullet|_U_i which is a quasi-isomorphism with E_i^bullet a strictly perfect complex of O_U_i-modules. An object E of D(O_X) is perfect if it can be represented by a perfect complex of O_X-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}_X$-modules.\nWe say $\\mathcal{E}^\\bullet$ is {\\it perfect} if there exists\nan open covering $X = \\bigcup U_i$ such that for each $i$\nthere exists a morphism of complexes\n$\\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$\nwhich is a quasi-isomorphism with $\\mathcal{E}_i^\\bullet$\na strictly perfect complex of $\\mathcal{O}_{U_i}$-modules.\nAn object $E$ of $D(\\mathcal{O}_X)$ is {\\it perfect}\nif it can be represented by a perfect complex of $\\mathcal{O}_X$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CM","source_file":"cohomology.tex","source_line":12507,"source_end_line":12519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12507-L12519","statement_sha256":"f76208d4f038151813c0c3825ececdb522fd3424e4db875c3296ef719bd612ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":4368,"rank":4368,"depth":0,"x":1117.295,"y":770.953,"cluster":"sheaf-cohomology"},{"id":"stacks:08CN","tag":"08CN","title":"Perfect complexes · Lemma 08CN","summary":"Let (X, O_X) be a ringed space. Let E be an object of D(O_X). • If there exists an open covering X = ⋃ U_i and strictly perfect complexes E_i^bullet on U_i such that E_i^bullet represents E|_U_i in D(O_U_i), then E is perfect. • If E is perfect, then any complex representing E is perfect.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\n\\begin{enumerate}\n\\item If there exists an open covering $X = \\bigcup U_i$ and\nstrictly perfect complexes $\\mathcal{E}_i^\\bullet$ on $U_i$\nsuch that $\\mathcal{E}_i^\\bullet$ represents $E|_{U_i}$ in\n$D(\\mathcal{O}_{U_i})$, then $E$ is perfect.\n\\item If $E$ is perfect, then any complex representing $E$ is perfect.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CN","source_file":"cohomology.tex","source_line":12526,"source_end_line":12537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12526-L12537","statement_sha256":"b07986403ae3fccc10f60835b1ddd7de1f4a3eced58be74e27e2e9076f974c26","origin":"The Stacks Project","memory_eligible":false,"source_rank":4369,"rank":4369,"depth":5,"x":1066.082,"y":605.555,"cluster":"sheaf-cohomology"},{"id":"stacks:0BCJ","tag":"0BCJ","title":"Perfect complexes · Lemma 0BCJ","summary":"Let (X, O_X) be a ringed space. Let E be an object of D(O_X). Assume that all stalks O_X, x are local rings. Then the following are equivalent • E is perfect, • there exists an open covering X = ⋃ U_i such that E|_U_i can be represented by a finite complex of finite locally free O_U_i-modules, and • there exists an open covering X = ⋃ U_i such that E|_U_i can be represented by a finite complex of finite free O_U_i-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ be an object of\n$D(\\mathcal{O}_X)$. Assume that all stalks $\\mathcal{O}_{X, x}$\nare local rings. Then the following are equivalent\n\\begin{enumerate}\n\\item $E$ is perfect,\n\\item there exists an open covering $X = \\bigcup U_i$ such that\n$E|_{U_i}$ can be represented by a finite complex of finite locally\nfree $\\mathcal{O}_{U_i}$-modules, and\n\\item there exists an open covering $X = \\bigcup U_i$ such that\n$E|_{U_i}$ can be represented by a finite complex of finite\nfree $\\mathcal{O}_{U_i}$-modules.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCJ","source_file":"cohomology.tex","source_line":12544,"source_end_line":12558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12544-L12558","statement_sha256":"5f99c99731e605a43b504b4bc994e6dcb3074ac4c841c8f70d8ab0ad58e7d07c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4370,"rank":4370,"depth":6,"x":1237.24,"y":698.668,"cluster":"sheaf-cohomology"},{"id":"stacks:08CP","tag":"08CP","title":"Perfect complexes · Lemma 08CP","summary":"Let (X, O_X) be a ringed space. Let E be an object of D(O_X). Let a ≤ b be integers. If E has tor amplitude in [a, b] and is (a - 1)-pseudo-coherent, then E is perfect.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\nLet $a \\leq b$ be integers. If $E$ has tor amplitude in $[a, b]$\nand is $(a - 1)$-pseudo-coherent, then $E$ is perfect.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CP","source_file":"cohomology.tex","source_line":12567,"source_end_line":12573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12567-L12573","statement_sha256":"7dc60ad1f1792c33025e6652580ef66df8c66fd8abf7f9b028b575e0419abc0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4371,"rank":4371,"depth":3,"x":1035.699,"y":727.191,"cluster":"sheaf-cohomology"},{"id":"stacks:08CQ","tag":"08CQ","title":"Perfect complexes · Lemma 08CQ","summary":"Let (X, O_X) be a ringed space. Let E be an object of D(O_X). The following are equivalent • E is perfect, and • E is pseudo-coherent and locally has finite tor dimension.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $E$ is perfect, and\n\\item $E$ is pseudo-coherent and locally has finite tor dimension.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CQ","source_file":"cohomology.tex","source_line":12606,"source_end_line":12615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12606-L12615","statement_sha256":"4263f79b195fc6c6ae8aa0eb3d5ff61824354ed5ce1efac950c57ad0bc569dff","origin":"The Stacks Project","memory_eligible":false,"source_rank":4372,"rank":4372,"depth":6,"x":1161.657,"y":591.497,"cluster":"sheaf-cohomology"},{"id":"stacks:09UA","tag":"09UA","title":"Perfect complexes · Lemma 09UA","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. Let E be an object of D(O_Y). If E is perfect in D(O_Y), then Lf^*E is perfect in D(O_X).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed\nspaces. Let $E$ be an object of $D(\\mathcal{O}_Y)$. If $E$ is perfect in\n$D(\\mathcal{O}_Y)$, then $Lf^*E$ is perfect in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UA","source_file":"cohomology.tex","source_line":12634,"source_end_line":12639,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12634-L12639","statement_sha256":"5dce82e19b565e9b5d872f9cb9df2ccbc12cd8aa97f491327c32b989e9e86d5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4373,"rank":4373,"depth":11,"x":1177.935,"y":763.409,"cluster":"sheaf-cohomology"},{"id":"stacks:08CR","tag":"08CR","title":"Perfect complexes · Lemma 08CR","summary":"Let (X, O_X) be a ringed space. Let (K, L, M, f, g, h) be a distinguished triangle in D(O_X). If two out of three of K, L, M are perfect then the third is also perfect.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $(K, L, M, f, g, h)$\nbe a distinguished triangle in $D(\\mathcal{O}_X)$. If two out of three of\n$K, L, M$ are perfect then the third is also perfect.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CR","source_file":"cohomology.tex","source_line":12649,"source_end_line":12654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12649-L12654","statement_sha256":"96b6229999d6c605b45bc2ebf624601a4707e6eb716804f4aa815aff6ee33cd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4374,"rank":4374,"depth":9,"x":1027.352,"y":645.619,"cluster":"sheaf-cohomology"},{"id":"stacks:09J5","tag":"09J5","title":"Perfect complexes · Lemma 09J5","summary":"Let (X, O_X) be a ringed space. If K, L are perfect objects of D(O_X), then so is K ⊗_O_X^L L.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nIf $K, L$ are perfect objects of $D(\\mathcal{O}_X)$, then\nso is $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09J5","source_file":"cohomology.tex","source_line":12671,"source_end_line":12676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12671-L12676","statement_sha256":"79c63aa2c26861c908d46180ff4bdd52e934d80f4ae395d7cdac55e5f99d9de3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4375,"rank":4375,"depth":7,"x":1233.567,"y":647.035,"cluster":"sheaf-cohomology"},{"id":"stacks:08CS","tag":"08CS","title":"Perfect complexes · Lemma 08CS","summary":"Let (X, O_X) be a ringed space. If K ⊕ L is a perfect object of D(O_X), then so are K and L.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nIf $K \\oplus L$ is a perfect object of $D(\\mathcal{O}_X)$, then\nso are $K$ and $L$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CS","source_file":"cohomology.tex","source_line":12684,"source_end_line":12689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12684-L12689","statement_sha256":"307705e8cbcd8172288c7e5c883bec05f34a6d69cba33180a3f3f814e4c2d7d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4376,"rank":4376,"depth":10,"x":1080.032,"y":763.257,"cluster":"sheaf-cohomology"},{"id":"stacks:08DP","tag":"08DP","title":"Perfect complexes · Lemma 08DP","summary":"Let (X, O_X) be a ringed space. Let j : U → X be an open subspace. Let E be a perfect object of D(O_U) whose cohomology sheaves are supported on a closed subset T ⊂ U with j(T) closed in X. Then Rj_*E is a perfect object of D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $j : U \\to X$ be an\nopen subspace. Let $E$ be a perfect object of $D(\\mathcal{O}_U)$\nwhose cohomology\nsheaves are supported on a closed subset $T \\subset U$ with $j(T)$\nclosed in $X$. Then $Rj_*E$ is a perfect object of $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DP","source_file":"cohomology.tex","source_line":12697,"source_end_line":12704,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12697-L12704","statement_sha256":"cb1a31cd308e590dd6bf08a692f390ef16d5e641d37e715a112fbc3697483df6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4377,"rank":4377,"depth":0,"x":1099.827,"y":590.058,"cluster":"sheaf-cohomology"},{"id":"stacks:0GT1","tag":"0GT1","title":"Perfect complexes · Lemma 0GT1","summary":"Let (X, O_X) be a ringed space. Let E in D(O_X) be perfect. Assume that all stalks O_X, x are local rings. Then the set U = (x ∈ X mid H^i(E)_x is a finite free O_X, x-module for all i∈ Z) is open in X and is the maximal open set U ⊂ X such that H^i(E)|_U is finite locally free for all i ∈ Z.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $E$ in $D(\\mathcal{O}_X)$\nbe perfect. Assume that all stalks $\\mathcal{O}_{X, x}$ are local rings.\nThen the set\n$$\nU =\n\\{x \\in X \\mid\nH^i(E)_x\\text{ is a finite free }\n\\mathcal{O}_{X, x}\\text{-module for all }i\\in \\mathbf{Z}\\}\n$$\nis open in $X$ and is the maximal open set $U \\subset X$ such that\n$H^i(E)|_U$ is finite locally free for all $i \\in \\mathbf{Z}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GT1","source_file":"cohomology.tex","source_line":12713,"source_end_line":12726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12713-L12726","statement_sha256":"38e7c6f38f43a2c5dbb8465976e9eb5f97170b6526bc83597e05fd07874a5fb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4378,"rank":4378,"depth":3,"x":1224.783,"y":729.308,"cluster":"sheaf-cohomology"},{"id":"stacks:0FP8","tag":"0FP8","title":"Duals · Lemma 0FP8","summary":"Let (X, O_X) be a ringed space. The category of complexes of O_X-modules with tensor product defined by F^bullet ⊗ G^bullet = Tot(F^bullet ⊗_O_X G^bullet) is a symmetric monoidal category (for sign rules, see More on Algebra, Section [Tag 0FNG]).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. The category of complexes\nof $\\mathcal{O}_X$-modules with tensor product defined by\n$\\mathcal{F}^\\bullet \\otimes \\mathcal{G}^\\bullet =\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)$\nis a symmetric monoidal category (for sign rules, see\nMore on Algebra, Section \\ref{more-algebra-section-sign-rules}).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FP8","source_file":"cohomology.tex","source_line":12773,"source_end_line":12781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12773-L12781","statement_sha256":"7efdd73d3bf8e28140d56ced2d6dba67c515545f5f6129394c242b6800221581","origin":"The Stacks Project","memory_eligible":false,"source_rank":4379,"rank":4379,"depth":1,"x":1020.219,"y":697.461,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPA","tag":"0FPA","title":"Duals · Lemma 0FPA","summary":"Let (X, O_X) be a ringed space. Let F^bullet be a complex of O_X-modules. If F^bullet has a left dual in the monoidal category of complexes of O_X-modules (Categories, Definition [Tag 0FFP]) then F^bullet is a locally bounded complex whose terms are locally direct summands of finite free O_X-modules and the left dual is as constructed in Example [Tag 0FP9].","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}^\\bullet$\nbe a complex of $\\mathcal{O}_X$-modules. If $\\mathcal{F}^\\bullet$\nhas a left dual in the monoidal category of complexes of\n$\\mathcal{O}_X$-modules\n(Categories, Definition \\ref{categories-definition-dual})\nthen $\\mathcal{F}^\\bullet$ is a locally bounded complex whose terms are\nlocally direct summands of finite free $\\mathcal{O}_X$-modules\nand the left dual is as constructed in Example \\ref{example-dual}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPA","source_file":"cohomology.tex","source_line":12838,"source_end_line":12848,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12838-L12848","statement_sha256":"9713e0cc7536c90a8d209d03d3381e448e659f98a1eb95ef8e97df89b40e7fa1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4380,"rank":4380,"depth":2,"x":1197.053,"y":604.67,"cluster":"sheaf-cohomology"},{"id":"stacks:0G40","tag":"0G40","title":"Duals · Lemma 0G40","summary":"Let (X, O_X) be a ringed space. Let K, L, M ∈ D(O_X). If K is perfect, then the map RSheafHom(L, M) ⊗_O_X^L K → RSheafHom(RSheafHom(K, L), M) of Lemma [Tag 0A8U] is an isomorphism.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K, L, M \\in D(\\mathcal{O}_X)$.\nIf $K$ is perfect, then the map\n$$\nR\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K\n\\longrightarrow\nR\\SheafHom(R\\SheafHom(K, L), M)\n$$\nof Lemma \\ref{lemma-internal-hom-evaluate} is an isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G40","source_file":"cohomology.tex","source_line":12877,"source_end_line":12887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12877-L12887","statement_sha256":"3dcaeb992e8e037686a1970bae1d9f81703968c84d2e754bd5525275bb917250","origin":"The Stacks Project","memory_eligible":false,"source_rank":4381,"rank":4381,"depth":9,"x":1141.158,"y":773.795,"cluster":"sheaf-cohomology"},{"id":"stacks:08DQ","tag":"08DQ","title":"Duals · Lemma 08DQ","summary":"Let (X, O_X) be a ringed space. Let K be a perfect object of D(O_X). Then K^vee = RSheafHom(K, O_X) is a perfect object too and (K^vee)^vee ≅ K. There are functorial isomorphisms M ⊗^L_O_X K^vee = RSheafHom(K, M) and H^0(X, M ⊗^L_O_X K^vee) = Hom_D(O_X)(K, M) for M in D(O_X).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ be a perfect object of\n$D(\\mathcal{O}_X)$. Then $K^\\vee = R\\SheafHom(K, \\mathcal{O}_X)$ is a\nperfect object too and $(K^\\vee)^\\vee \\cong K$. There are\nfunctorial isomorphisms\n$$\nM \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K^\\vee = R\\SheafHom(K, M)\n$$\nand\n$$\nH^0(X, M \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K^\\vee) =\n\\Hom_{D(\\mathcal{O}_X)}(K, M)\n$$\nfor $M$ in $D(\\mathcal{O}_X)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DQ","source_file":"cohomology.tex","source_line":12932,"source_end_line":12947,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L12932-L12947","statement_sha256":"ca64f27df86e449d010c47e5ea2a646a5537e2159cce0edaaf1b5c4a7f5e9c86","origin":"The Stacks Project","memory_eligible":false,"source_rank":4382,"rank":4382,"depth":10,"x":1046.166,"y":617.034,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPB","tag":"0FPB","title":"Duals · Lemma 0FPB","summary":"Let (X, O_X) be a ringed space. The derived category D(O_X) is a symmetric monoidal category with tensor product given by derived tensor product with usual associativity and commutativity constraints (for sign rules, see More on Algebra, Section [Tag 0FNG]).","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. The derived category\n$D(\\mathcal{O}_X)$ is a symmetric monoidal category with tensor product\ngiven by derived tensor product with usual associativity and\ncommutativity constraints (for sign rules, see\nMore on Algebra, Section \\ref{more-algebra-section-sign-rules}).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPB","source_file":"cohomology.tex","source_line":13014,"source_end_line":13021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13014-L13021","statement_sha256":"925152c51862df01655a2f0e38b7172ccd28755b094bbb7421b4c1cd41948baa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4383,"rank":4383,"depth":2,"x":1242.693,"y":678.858,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPD","tag":"0FPD","title":"Duals · Lemma 0FPD","summary":"Let (X, O_X) be a ringed space. Let M be an object of D(O_X). If M has a left dual in the monoidal category D(O_X) (Categories, Definition [Tag 0FFP]) then M is perfect and the left dual is as constructed in Example [Tag 0FPC].","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $M$ be an object\nof $D(\\mathcal{O}_X)$. If $M$ has a left dual in the monoidal category\n$D(\\mathcal{O}_X)$ (Categories, Definition \\ref{categories-definition-dual})\nthen $M$ is perfect and the left dual is as constructed in\nExample \\ref{example-dual-derived}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPD","source_file":"cohomology.tex","source_line":13064,"source_end_line":13071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13064-L13071","statement_sha256":"9f1099f1dfdbb32a0f32b31cc2932fb287790bd4253be31343ac044f961a8fb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4384,"rank":4384,"depth":16,"x":1047.646,"y":744.924,"cluster":"sheaf-cohomology"},{"id":"stacks:0DJI","tag":"0DJI","title":"Miscellany · Lemma 0DJI","summary":"Let (X, O_X) be a ringed space. Let (K_n)_n ∈ N be a system of perfect objects of D(O_X). Let K = hocolim K_n be the derived colimit (Derived Categories, Definition [Tag 090Z]). Then for any object E of D(O_X) we have RSheafHom(K, E) = Rlim E ⊗^L_O_X K_n^vee where (K_n^vee) is the inverse system of dual perfect complexes.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let\n$(K_n)_{n \\in \\mathbf{N}}$ be a system of perfect objects of $D(\\mathcal{O}_X)$.\nLet $K = \\text{hocolim} K_n$ be the derived colimit\n(Derived Categories, Definition \\ref{derived-definition-derived-colimit}).\nThen for any object $E$ of $D(\\mathcal{O}_X)$ we have\n$$\nR\\SheafHom(K, E) = R\\lim E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} K_n^\\vee\n$$\nwhere $(K_n^\\vee)$ is the inverse system of dual perfect complexes.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJI","source_file":"cohomology.tex","source_line":13293,"source_end_line":13304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13293-L13304","statement_sha256":"53f9a800a81028ff333bf2c1d155a2cd8462b3ca95ba2c85345d8f33315f663d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4385,"rank":4385,"depth":11,"x":1138.535,"y":585.197,"cluster":"sheaf-cohomology"},{"id":"stacks:0FVB","tag":"0FVB","title":"Miscellany · Lemma 0FVB","summary":"Let (X, O_X) be a ringed space. Let K and E be objects of D(O_X) with E perfect. The diagram xymatrix H^0(X, K ⊗_O_X^L E^vee) × H^0(X, E) ar[r] ar[d] & H^0(X, K ⊗_O_X^L E^vee ⊗_O_X^L E) ar[d] Hom_X(E, K) × H^0(X, E) ar[r] & H^0(X, K) commutes where the top horizontal arrow is the cup product, the right vertical arrow uses ε : E^vee ⊗_O_X^L E → O_X (Example [Tag 0FPC]), the left vertical arrow uses Lemma [Tag 08DQ], and the bottom horizontal arrow is the obvious one.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $K$ and $E$ be objects\nof $D(\\mathcal{O}_X)$ with $E$ perfect. The diagram\n$$\n\\xymatrix{\nH^0(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E^\\vee) \\times H^0(X, E)\n\\ar[r] \\ar[d] &\nH^0(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E^\\vee\n\\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) \\ar[d] \\\\\n\\Hom_X(E, K) \\times H^0(X, E) \\ar[r] &\nH^0(X, K)\n}\n$$\ncommutes where the top horizontal arrow is the cup product, the\nright vertical arrow uses\n$\\epsilon : E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E \\to \\mathcal{O}_X$\n(Example \\ref{example-dual-derived}), the left vertical arrow uses\nLemma \\ref{lemma-dual-perfect-complex}, and the bottom horizontal\narrow is the obvious one.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVB","source_file":"cohomology.tex","source_line":13323,"source_end_line":13343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13323-L13343","statement_sha256":"c4ee7ae11bbe2505c048d2421ba6b07b5ca914d1d6f4dae0bc31419405f4423c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4386,"rank":4386,"depth":11,"x":1200.091,"y":754.906,"cluster":"sheaf-cohomology"},{"id":"stacks:0GM7","tag":"0GM7","title":"Miscellany · Lemma 0GM7","summary":"Let h : X → Y be a morphism of ringed spaces. Let K, M be objects of D(O_Y). The canonical map Lh^*RSheafHom(K, M) → RSheafHom(Lh^*K, Lh^*M) of Remark [Tag 08I3] is an isomorphism in the following cases • K is perfect, • h is flat, K is pseudo-coherent, and M is (locally) bounded below, • O_X has finite tor dimension over h^-1O_Y, K is pseudo-coherent, and M is (locally) bounded below,","statement_latex":"Let $h : X \\to Y$ be a morphism of ringed spaces.\nLet $K, M$ be objects of $D(\\mathcal{O}_Y)$. The\ncanonical map\n$$\nLh^*R\\SheafHom(K, M) \\longrightarrow R\\SheafHom(Lh^*K, Lh^*M)\n$$\nof Remark \\ref{remark-prepare-fancy-base-change}\nis an isomorphism in the following cases\n\\begin{enumerate}\n\\item $K$ is perfect,\n\\item $h$ is flat, $K$ is pseudo-coherent, and $M$ is (locally) bounded below,\n\\item $\\mathcal{O}_X$ has finite tor dimension over $h^{-1}\\mathcal{O}_Y$,\n$K$ is pseudo-coherent, and $M$ is (locally) bounded below,\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GM7","source_file":"cohomology.tex","source_line":13407,"source_end_line":13423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13407-L13423","statement_sha256":"81cbfde324a1df15b95ac0bfa6cc4ab93c5a2c35188090577a3163d39586920f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4387,"rank":4387,"depth":6,"x":1017.845,"y":664.505,"cluster":"sheaf-cohomology"},{"id":"stacks:0GM8","tag":"0GM8","title":"Miscellany · Lemma 0GM8","summary":"Let X be a ringed space. Let K, M be objects of D(O_X). Let x ∈ X. The canonical map RSheafHom(K, M)_x → RHom_O_X, x(K_x, M_x) is an isomorphism in the following cases • K is perfect, • K is pseudo-coherent and M is (locally) bounded below.","statement_latex":"Let $X$ be a ringed space. Let $K, M$ be objects of $D(\\mathcal{O}_X)$.\nLet $x \\in X$. The canonical map\n$$\nR\\SheafHom(K, M)_x \\longrightarrow\nR\\Hom_{\\mathcal{O}_{X, x}}(K_x, M_x)\n$$\nis an isomorphism in the following cases\n\\begin{enumerate}\n\\item $K$ is perfect,\n\\item $K$ is pseudo-coherent and $M$ is (locally) bounded below.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GM8","source_file":"cohomology.tex","source_line":13482,"source_end_line":13495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13482-L13495","statement_sha256":"4350e0427dd2e40248d5b15770af7a3400183fade48d9bc6ea44f0ef73f86926","origin":"The Stacks Project","memory_eligible":false,"source_rank":4388,"rank":4388,"depth":7,"x":1225.36,"y":627.678,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPF","tag":"0FPF","title":"Invertible objects in the derived category · Lemma 0FPF","summary":"Let (X, O_X) be a ringed space. Set R = Γ(X, O_X). The category of O_X-modules which are summands of finite free O_X-modules is equivalent to the category of finite projective R-modules.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\nSet $R = \\Gamma(X, \\mathcal{O}_X)$. The category of\n$\\mathcal{O}_X$-modules which are summands of finite free\n$\\mathcal{O}_X$-modules is equivalent to the category of\nfinite projective $R$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Invertible objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPF","source_file":"cohomology.tex","source_line":13518,"source_end_line":13525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13518-L13525","statement_sha256":"ee2821aef0e87c0f23bce97c9895c2c500240f96501dc44bba4328352534d511","origin":"The Stacks Project","memory_eligible":false,"source_rank":4389,"rank":4389,"depth":7,"x":1101.7,"y":772.884,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPG","tag":"0FPG","title":"Invertible objects in the derived category · Lemma 0FPG","summary":"Let (X, O_X) be a ringed space. Let M be an object of D(O_X). The following are equivalent • M is invertible in D(O_X), see Categories, Definition [Tag 0FFN], and • there is a locally finite direct product decomposition O_X = ∏_n ∈ Z O_n and for each n there is an invertible O_n-module H^n (Modules, Definition [Tag 01CS]) and M = bigoplus H^n[-n] in D(O_X). If (1) and (2) hold, then M is a perfect object of D(O_X). If O_X, x is a local ring for all x ∈ X these condition…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $M$ be an object\nof $D(\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is invertible in $D(\\mathcal{O}_X)$, see\nCategories, Definition \\ref{categories-definition-invertible}, and\n\\item there is a locally finite direct product decomposition\n$$\n\\mathcal{O}_X = \\prod\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_n\n$$\nand for each $n$ there is an invertible $\\mathcal{O}_n$-module\n$\\mathcal{H}^n$ (Modules, Definition \\ref{modules-definition-invertible})\nand $M = \\bigoplus \\mathcal{H}^n[-n]$ in $D(\\mathcal{O}_X)$.\n\\end{enumerate}\nIf (1) and (2) hold, then $M$ is a perfect object of $D(\\mathcal{O}_X)$. If\n$\\mathcal{O}_{X, x}$ is a local ring for all $x \\in X$ these condition\nare also equivalent to\n\\begin{enumerate}\n\\item[(3)] there exists an open covering $X = \\bigcup U_i$\nand for each $i$ an integer $n_i$ such that $M|_{U_i}$\nis represented by an invertible $\\mathcal{O}_{U_i}$-module\nplaced in degree $n_i$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Invertible objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPG","source_file":"cohomology.tex","source_line":13535,"source_end_line":13559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13535-L13559","statement_sha256":"9f1511a0a369939971f75574d7873f028ccd67f3a7ab7d593aa7c52d92007c02","origin":"The Stacks Project","memory_eligible":false,"source_rank":4390,"rank":4390,"depth":18,"x":1076.062,"y":595.277,"cluster":"sheaf-cohomology"},{"id":"stacks:0F5Z","tag":"0F5Z","title":"Compact objects · Lemma 0F5Z","summary":"Let X be a ringed space. Let j : U → X be the inclusion of an open. The O_X-module j_!O_U is a compact object of D(O_X) if there exists an integer d such that • H^p(U, F) = 0 for all p > d, and • the functors F ↦ H^p(U, F) commute with direct sums.","statement_latex":"Let $X$ be a ringed space. Let $j : U \\to X$ be the\ninclusion of an open. The $\\mathcal{O}_X$-module $j_!\\mathcal{O}_U$ is a\ncompact object of $D(\\mathcal{O}_X)$ if there exists an integer $d$ such that\n\\begin{enumerate}\n\\item $H^p(U, \\mathcal{F}) = 0$ for all $p > d$, and\n\\item the functors $\\mathcal{F} \\mapsto H^p(U, \\mathcal{F})$\ncommute with direct sums.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5Z","source_file":"cohomology.tex","source_line":13689,"source_end_line":13699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13689-L13699","statement_sha256":"b16d22a4bfcfae1cb5815280c05b7e3a4ba9681c756789a8f87f0b332ebb9e8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4391,"rank":4391,"depth":16,"x":1238.127,"y":711.934,"cluster":"sheaf-cohomology"},{"id":"stacks:09J7","tag":"09J7","title":"Compact objects · Lemma 09J7","summary":"Let X be a ringed space. Assume that the underlying topological space of X has the following properties: • X is quasi-compact, • there exists a basis of quasi-compact open subsets, and • the intersection of any two quasi-compact opens is quasi-compact. Let K be a perfect object of D(O_X). Then • [(a)] K is a compact object of D^+(O_X) in the following sense: if M = bigoplus_i ∈ I M_i is bounded below, then Hom(K, M) = bigoplus_i ∈ I Hom(K, M_i). • [(b)] If X has finite…","statement_latex":"Let $X$ be a ringed space. Assume that the underlying topological space\nof $X$ has the following properties:\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item there exists a basis of quasi-compact open subsets, and\n\\item the intersection of any two quasi-compact opens is quasi-compact.\n\\end{enumerate}\nLet $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then\n\\begin{enumerate}\n\\item[(a)] $K$ is a compact object of $D^+(\\mathcal{O}_X)$\nin the following sense: if $M = \\bigoplus_{i \\in I} M_i$ is\nbounded below, then $\\Hom(K, M) = \\bigoplus_{i \\in I} \\Hom(K, M_i)$.\n\\item[(b)] If $X$ has finite cohomological dimension, i.e., if there exists\na $d$ such that $H^i(X, \\mathcal{F}) = 0$ for $i > d$, then\n$K$ is a compact object of $D(\\mathcal{O}_X)$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09J7","source_file":"cohomology.tex","source_line":13723,"source_end_line":13741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13723-L13741","statement_sha256":"74d04313227767ce726a2cc09f2b8df0bbd12849613c19327b9fd0766daa8542","origin":"The Stacks Project","memory_eligible":false,"source_rank":4392,"rank":4392,"depth":24,"x":1024.373,"y":717.884,"cluster":"sheaf-cohomology"},{"id":"stacks:01E7","tag":"01E7","title":"Projection formula · Lemma 01E7","summary":"Let X be a ringed space. Let I be an injective O_X-module. Let E be an O_X-module. Assume E is finite locally free on X, see Modules, Definition [Tag 01C6]. Then E ⊗_O_X I is an injective O_X-module.","statement_latex":"Let $X$ be a ringed space.\nLet $\\mathcal{I}$ be an injective $\\mathcal{O}_X$-module.\nLet $\\mathcal{E}$ be an $\\mathcal{O}_X$-module.\nAssume $\\mathcal{E}$ is finite locally free on $X$, see\nModules, Definition \\ref{modules-definition-locally-free}.\nThen $\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{I}$ is\nan injective $\\mathcal{O}_X$-module.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Projection formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01E7","source_file":"cohomology.tex","source_line":13815,"source_end_line":13824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13815-L13824","statement_sha256":"2a2f189f63fca3c1fd85a3b10d611f32152a4ac12ff094fdd25c233269cf6353","origin":"The Stacks Project","memory_eligible":false,"source_rank":4393,"rank":4393,"depth":1,"x":1177.519,"y":591.951,"cluster":"sheaf-cohomology"},{"id":"stacks:01E8","tag":"01E8","title":"Projection formula · Lemma 01E8","summary":"Let f : X → Y be a morphism of ringed spaces. Let F be an O_X-module. Let E be an O_Y-module. Assume E is finite locally free on Y, see Modules, Definition [Tag 01C6]. Then there exist isomorphisms E ⊗_O_Y R^qf_*F → R^qf_*(f^*E ⊗_O_X F) for all q ≥ 0. In fact there exists an isomorphism E ⊗_O_Y Rf_*F → Rf_*(f^*E ⊗_O_X F) in D^+(Y) functorial in F.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nLet $\\mathcal{E}$ be an $\\mathcal{O}_Y$-module.\nAssume $\\mathcal{E}$ is finite locally free on $Y$, see\nModules, Definition \\ref{modules-definition-locally-free}.\nThen there exist isomorphisms\n$$\n\\mathcal{E} \\otimes_{\\mathcal{O}_Y} R^qf_*\\mathcal{F}\n\\longrightarrow\nR^qf_*(f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F})\n$$\nfor all $q \\geq 0$. In fact there exists an isomorphism\n$$\n\\mathcal{E} \\otimes_{\\mathcal{O}_Y} Rf_*\\mathcal{F}\n\\longrightarrow\nRf_*(f^*\\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{F})\n$$\nin $D^{+}(Y)$ functorial in $\\mathcal{F}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Projection formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01E8","source_file":"cohomology.tex","source_line":13842,"source_end_line":13862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13842-L13862","statement_sha256":"c492ad3b616cd396bd2784cbfad47201806a433e0bee0e0921438919fa322ccc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4394,"rank":4394,"depth":2,"x":1165.839,"y":772.076,"cluster":"sheaf-cohomology"},{"id":"stacks:0B54","tag":"0B54","title":"Projection formula · Lemma 0B54","summary":"Let f : X → Y be a morphism of ringed spaces. Let E ∈ D(O_X) and K ∈ D(O_Y). If K is perfect, then Rf_*E ⊗^L_O_Y K = Rf_*(E ⊗^L_O_X Lf^*K) in D(O_Y).","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces.\nLet $E \\in D(\\mathcal{O}_X)$ and $K \\in D(\\mathcal{O}_Y)$.\nIf $K$ is perfect, then\n$$\nRf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} K =\nRf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} Lf^*K)\n$$\nin $D(\\mathcal{O}_Y)$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Projection formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B54","source_file":"cohomology.tex","source_line":13910,"source_end_line":13920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13910-L13920","statement_sha256":"f43bb6ae01e3b2779c599bbca24702a2ce71f3bd3b798a9b57c5134fe997125e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4395,"rank":4395,"depth":0,"x":1029.324,"y":632.344,"cluster":"sheaf-cohomology"},{"id":"stacks:0B55","tag":"0B55","title":"Projection formula · Lemma 0B55","summary":"Let f : X → Y be a morphism of ringed spaces such that f is a homeomorphism onto a closed subset. Then ([Tag 0B53]) is an isomorphism always.","statement_latex":"Let $f : X \\to Y$ be a morphism of ringed spaces such that $f$ is a\nhomeomorphism onto a closed subset. Then\n(\\ref{equation-projection-formula-map}) is an isomorphism always.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"Projection formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B55","source_file":"cohomology.tex","source_line":13945,"source_end_line":13950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L13945-L13950","statement_sha256":"e2352f7299fe92c5390fe50e7bce23dd1655d620a7d03d0fa2d13efd2d890d8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4396,"rank":4396,"depth":4,"x":1242.787,"y":657.97,"cluster":"sheaf-cohomology"},{"id":"stacks:0GT3","tag":"0GT3","title":"An operator introduced by Berthelot and Ogus · Lemma 0GT3","summary":"Let (X, O_X) be a ringed space. Let I ⊂ O_X be a sheaf of ideals. Consider the following two conditions • for every x ∈ X there exists an open neighbourhood U ⊂ X of x and f ∈ I(U) such that I|_U = O_U · f and f : O_U → O_U is injective, and • I is invertible as an O_X-module. Then (1) implies (2) and the converse is true if all stalks O_X, x of the structure sheaf are local rings.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let\n$\\mathcal{I} \\subset \\mathcal{O}_X$ be a sheaf of ideals.\nConsider the following two conditions\n\\begin{enumerate}\n\\item for every $x \\in X$ there exists an open neighbourhood\n$U \\subset X$ of $x$ and $f \\in \\mathcal{I}(U)$ such that\n$\\mathcal{I}|_U = \\mathcal{O}_U \\cdot f$ and\n$f : \\mathcal{O}_U \\to \\mathcal{O}_U$ is injective, and\n\\item $\\mathcal{I}$ is invertible as an $\\mathcal{O}_X$-module.\n\\end{enumerate}\nThen (1) implies (2) and the converse is true if all stalks\n$\\mathcal{O}_{X, x}$ of the structure sheaf are local rings.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GT3","source_file":"cohomology.tex","source_line":14049,"source_end_line":14063,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L14049-L14063","statement_sha256":"23bb78777440d3dd315c21ae7e8f8e16f62c79b7a9a48d15bcdc994e88140858","origin":"The Stacks Project","memory_eligible":false,"source_rank":4397,"rank":4397,"depth":8,"x":1064.417,"y":760.403,"cluster":"sheaf-cohomology"},{"id":"stacks:0GT5","tag":"0GT5","title":"An operator introduced by Berthelot and Ogus · Lemma 0GT5","summary":"In Situation [Tag 0GT4] let F be an O_X-module. The following are equivalent • the subsheaf F[I] ⊂ F of sections annihilated by I is zero, • the subsheaf F[I^n] is zero for all n ≥ 1, • the multiplication map I ⊗_O_X F → F is injective, • for every open U ⊂ X such that I|_U = O_U · f for some f ∈ I(U) the map f : F|_U → F|_U is injective, • for every x ∈ X and generator f of the ideal I_x ⊂ O_X, x the element f is a nonzerodivisor on the stalk F_x.","statement_latex":"In Situation \\ref{situation-eta}\nlet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item the subsheaf $\\mathcal{F}[\\mathcal{I}] \\subset \\mathcal{F}$\nof sections annihilated by $\\mathcal{I}$ is zero,\n\\item the subsheaf $\\mathcal{F}[\\mathcal{I}^n]$ is zero for all $n \\geq 1$,\n\\item the multiplication map\n$\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{F}$\nis injective,\n\\item for every open $U \\subset X$ such that\n$\\mathcal{I}|_U = \\mathcal{O}_U \\cdot f$\nfor some $f \\in \\mathcal{I}(U)$\nthe map $f : \\mathcal{F}|_U \\to \\mathcal{F}|_U$ is injective,\n\\item for every $x \\in X$ and generator $f$ of the ideal\n$\\mathcal{I}_x \\subset \\mathcal{O}_{X, x}$ the element $f$\nis a nonzerodivisor on the stalk $\\mathcal{F}_x$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GT5","source_file":"cohomology.tex","source_line":14081,"source_end_line":14101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L14081-L14101","statement_sha256":"c4ec75016e3513a5c2f637fa1ee0c0e403f044d772b6b60ef8bd4c775990a361","origin":"The Stacks Project","memory_eligible":false,"source_rank":4398,"rank":4398,"depth":0,"x":1113.669,"y":583.306,"cluster":"sheaf-cohomology"},{"id":"stacks:0GT6","tag":"0GT6","title":"An operator introduced by Berthelot and Ogus · Lemma 0GT6","summary":"In Situation [Tag 0GT4] let F^bullet be a complex of I-torsion free O_X-modules. For x ∈ X choose a generator f ∈ I_x. Then the stalk (eta_IF^bullet)_x is canonically isomorphic to the complex eta_fF^bullet_x constructed in More on Algebra, Section [Tag 0F7N].","statement_latex":"In Situation \\ref{situation-eta}\nlet $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{I}$-torsion free\n$\\mathcal{O}_X$-modules.\nFor $x \\in X$ choose a generator $f \\in \\mathcal{I}_x$. Then\nthe stalk $(\\eta_\\mathcal{I}\\mathcal{F}^\\bullet)_x$ is canonically\nisomorphic to the complex $\\eta_f\\mathcal{F}^\\bullet_x$ constructed\nin More on Algebra, Section \\ref{more-algebra-section-eta}.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GT6","source_file":"cohomology.tex","source_line":14204,"source_end_line":14213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L14204-L14213","statement_sha256":"b6266e21840f623535495e8d2e0b8cf8a46c56e1c7863cf118be8fe5d1cf81cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4399,"rank":4399,"depth":0,"x":1219.979,"y":742.158,"cluster":"sheaf-cohomology"},{"id":"stacks:0F8N","tag":"0F8N","title":"An operator introduced by Berthelot and Ogus · Lemma 0F8N","summary":"In Situation [Tag 0GT4] let F^bullet be a complex of I-torsion free O_X-modules. There is a canonical isomorphism I^⊗ i ⊗_O_X ( H^i(F^bullet)/H^i(F^bullet)[I] ) → H^i(eta_IF^bullet) of cohomology sheaves.","statement_latex":"In Situation \\ref{situation-eta}\nlet $\\mathcal{F}^\\bullet$ be a complex of $\\mathcal{I}$-torsion free\n$\\mathcal{O}_X$-modules. There is a canonical isomorphism\n$$\n\\mathcal{I}^{\\otimes i} \\otimes_{\\mathcal{O}_X}\n\\left(\nH^i(\\mathcal{F}^\\bullet)/H^i(\\mathcal{F}^\\bullet)[\\mathcal{I}]\n\\right)\n\\longrightarrow H^i(\\eta_\\mathcal{I}\\mathcal{F}^\\bullet)\n$$\nof cohomology sheaves.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8N","source_file":"cohomology.tex","source_line":14219,"source_end_line":14232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L14219-L14232","statement_sha256":"df42e38f4d5a7792642dc14ac1a246a5b68fbec88cd9b953c801595e769e9741","origin":"The Stacks Project","memory_eligible":false,"source_rank":4400,"rank":4400,"depth":1,"x":1013.435,"y":685.233,"cluster":"sheaf-cohomology"},{"id":"stacks:0F8P","tag":"0F8P","title":"An operator introduced by Berthelot and Ogus · Lemma 0F8P","summary":"In Situation [Tag 0GT4] let F^bullet → G^bullet be a map of complexes of I-torsion free O_X-modules. Then the induced map eta_IF^bullet → eta_IG^bullet is a quasi-isomorphism too.","statement_latex":"In Situation \\ref{situation-eta}\nlet $\\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ be a map of\ncomplexes of $\\mathcal{I}$-torsion free $\\mathcal{O}_X$-modules.\nThen the induced map\n$\\eta_\\mathcal{I}\\mathcal{F}^\\bullet \\to \\eta_\\mathcal{I}\\mathcal{G}^\\bullet$\nis a quasi-isomorphism too.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8P","source_file":"cohomology.tex","source_line":14257,"source_end_line":14265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L14257-L14265","statement_sha256":"72a8e30ace305d16be5de9c847bfdd08cc155558dcc3608bcd21278448440a69","origin":"The Stacks Project","memory_eligible":false,"source_rank":4401,"rank":4401,"depth":2,"x":1211.907,"y":609.862,"cluster":"sheaf-cohomology"},{"id":"stacks:0F8Q","tag":"0F8Q","title":"An operator introduced by Berthelot and Ogus · Lemma 0F8Q","summary":"In Situation [Tag 0GT4] there is an additive functor Leta_I : D(O_X) → D(O_X) such that if M in D(O_X) is represented by a complex F^bullet of I-torsion free O_X-modules, then Leta_IM = eta_IF^bullet. Similarly for morphisms.","statement_latex":"In Situation \\ref{situation-eta} there is an additive\nfunctor\\footnote{Beware that this functor isn't exact, i.e.,\ndoes not transform distinguished triangles into distinguished triangles.}\n$L\\eta_\\mathcal{I} : D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$\nsuch that if $M$ in $D(\\mathcal{O}_X)$ is represented by a complex\n$\\mathcal{F}^\\bullet$ of $\\mathcal{I}$-torsion free $\\mathcal{O}_X$-modules,\nthen $L\\eta_\\mathcal{I}M = \\eta_\\mathcal{I}\\mathcal{F}^\\bullet$.\nSimilarly for morphisms.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8Q","source_file":"cohomology.tex","source_line":14272,"source_end_line":14282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L14272-L14282","statement_sha256":"7ecb59b34bdedee6468a19f488e805f87abf4e57b5e95fab7fb0136fb39fa8f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4402,"rank":4402,"depth":16,"x":1125.999,"y":778.388,"cluster":"sheaf-cohomology"},{"id":"stacks:0GT9","tag":"0GT9","title":"An operator introduced by Berthelot and Ogus · Lemma 0GT9","summary":"In Situation [Tag 0GT4] let M be an object of D(O_X). There is a canonical isomorphism Leta_IM ⊗^L O_X/I → H^bullet(M/I) in D(O_X) where the right hand side is the complex ([Tag 0GT8]).","statement_latex":"In Situation \\ref{situation-eta} let $M$ be an object of\n$D(\\mathcal{O}_X)$. There is a canonical isomorphism\n$$\nL\\eta_\\mathcal{I}M \\otimes^\\mathbf{L} \\mathcal{O}_X/\\mathcal{I}\n\\longrightarrow\nH^\\bullet(M/\\mathcal{I})\n$$\nin $D(\\mathcal{O}_X)$ where the right hand side is the complex\n(\\ref{equation-complex-bocksteins}).","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GT9","source_file":"cohomology.tex","source_line":14428,"source_end_line":14439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L14428-L14439","statement_sha256":"1f8f611294f85d00bb3766c2a219419318ec655a87f9c2a5785e4c38a3e24733","origin":"The Stacks Project","memory_eligible":false,"source_rank":4403,"rank":4403,"depth":3,"x":1053.681,"y":605.034,"cluster":"sheaf-cohomology"},{"id":"stacks:0F9W","tag":"0F9W","title":"An operator introduced by Berthelot and Ogus · Lemma 0F9W","summary":"In Situation [Tag 0GT4] let F^bullet be a complex of I-torsion free O_X-modules. Let L be an invertible O_X-module. Then eta_I(F^bullet ⊗ L) = (eta_IF^bullet) ⊗ L.","statement_latex":"In Situation \\ref{situation-eta}\nlet $\\mathcal{F}^\\bullet$ be a complex of\n$\\mathcal{I}$-torsion free $\\mathcal{O}_X$-modules.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen $\\eta_\\mathcal{I}(\\mathcal{F}^\\bullet \\otimes \\mathcal{L}) =\n(\\eta_\\mathcal{I}\\mathcal{F}^\\bullet) \\otimes \\mathcal{L}$.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9W","source_file":"cohomology.tex","source_line":14482,"source_end_line":14490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L14482-L14490","statement_sha256":"b0a5c8b571c20a9a9034ac4e702e4e15b7fef690a5960b48be39618fa34119e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4404,"rank":4404,"depth":0,"x":1246.79,"y":691.997,"cluster":"sheaf-cohomology"},{"id":"stacks:0GTA","tag":"0GTA","title":"An operator introduced by Berthelot and Ogus · Lemma 0GTA","summary":"In Situation [Tag 0GT4] let M be an object of D(O_X). Let x ∈ X with O_X, x nonzero. If H^i(M)_x is finite free over O_X, x, then H^i(Leta_IM)_x is finite free over O_X, x of the same rank.","statement_latex":"In Situation \\ref{situation-eta} let $M$ be an object of $D(\\mathcal{O}_X)$.\nLet $x \\in X$ with $\\mathcal{O}_{X, x}$ nonzero. If $H^i(M)_x$\nis finite free over $\\mathcal{O}_{X, x}$, then $H^i(L\\eta_\\mathcal{I}M)_x$\nis finite free over $\\mathcal{O}_{X, x}$ of the same rank.","area":"Sheaf Cohomology","chapter":"Cohomology of Sheaves","chapter_id":"cohomology","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTA","source_file":"cohomology.tex","source_line":14496,"source_end_line":14502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cohomology.tex#L14496-L14502","statement_sha256":"1589a3050576136e96ca5510da5e74c584694989ad2ea06947d7a0a3636a5735","origin":"The Stacks Project","memory_eligible":false,"source_rank":4405,"rank":4405,"depth":2,"x":1034.048,"y":737.534,"cluster":"sheaf-cohomology"},{"id":"stacks:03AH","tag":"03AH","title":"First cohomology and torsors · Definition 03AH","summary":"Let C be a site. Let G be a sheaf of (possibly non-commutative) groups on C. A pseudo torsor, or more precisely a pseudo G-torsor, is a sheaf of sets F on C endowed with an action G × F → F such that • whenever F(U) is nonempty the action G(U) × F(U) → F(U) is simply transitive. A morphism of pseudo G-torsors F → F' is a morphism of sheaves of sets compatible with the G-actions. A torsor, or more precisely a G-torsor, is a pseudo G-torsor such that in addition • [(2)] for…","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{G}$ be a sheaf of (possibly non-commutative)\ngroups on $\\mathcal{C}$.\nA {\\it pseudo torsor}, or more precisely a\n{\\it pseudo $\\mathcal{G}$-torsor}, is a sheaf\nof sets $\\mathcal{F}$ on $\\mathcal{C}$ endowed with an action\n$\\mathcal{G} \\times \\mathcal{F} \\to \\mathcal{F}$ such that\n\\begin{enumerate}\n\\item whenever $\\mathcal{F}(U)$ is nonempty the action\n$\\mathcal{G}(U) \\times \\mathcal{F}(U) \\to \\mathcal{F}(U)$\nis simply transitive.\n\\end{enumerate}\nA {\\it morphism of pseudo $\\mathcal{G}$-torsors}\n$\\mathcal{F} \\to \\mathcal{F}'$\nis a morphism of sheaves of sets compatible with the\n$\\mathcal{G}$-actions.\nA {\\it torsor}, or more precisely a\n{\\it $\\mathcal{G}$-torsor}, is a pseudo $\\mathcal{G}$-torsor such that\nin addition\n\\begin{enumerate}\n\\item[(2)] for every $U \\in \\Ob(\\mathcal{C})$\nthere exists a covering $\\{U_i \\to U\\}_{i \\in I}$ of $U$\nsuch that $\\mathcal{F}(U_i)$ is nonempty for all $i \\in I$.\n\\end{enumerate}\nA {\\it morphism of $\\mathcal{G}$-torsors} is a morphism of\npseudo $\\mathcal{G}$-torsors.\nThe {\\it trivial $\\mathcal{G}$-torsor}\nis the sheaf $\\mathcal{G}$ endowed with the obvious left\n$\\mathcal{G}$-action.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"First cohomology and torsors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AH","source_file":"sites-cohomology.tex","source_line":241,"source_end_line":272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L241-L272","statement_sha256":"b5340dba9729d9434e59ef6f6b8f81d24686f725d51d4da8d1e6ca7c27b9d37c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4406,"rank":4406,"depth":0,"x":1154.534,"y":582.942,"cluster":"sheaf-cohomology"},{"id":"stacks:03AI","tag":"03AI","title":"First cohomology and torsors · Lemma 03AI","summary":"Let C be a site. Let G be a sheaf of (possibly non-commutative) groups on C. A G-torsor F is trivial if and only if Γ(C, F) not = ∅.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{G}$ be a sheaf of (possibly non-commutative)\ngroups on $\\mathcal{C}$.\nA $\\mathcal{G}$-torsor $\\mathcal{F}$ is trivial if and only if\n$\\Gamma(\\mathcal{C}, \\mathcal{F}) \\not = \\emptyset$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"First cohomology and torsors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AI","source_file":"sites-cohomology.tex","source_line":277,"source_end_line":284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L277-L284","statement_sha256":"8d5042767e27f389f5f7a2d0fb8cfb8e4ca75375da8479963e0715bc116edfd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4407,"rank":4407,"depth":0,"x":1190.076,"y":765.655,"cluster":"sheaf-cohomology"},{"id":"stacks:03AJ","tag":"03AJ","title":"First cohomology and torsors · Lemma 03AJ","summary":"Let C be a site. Let H be an abelian sheaf on C. There is a canonical bijection between the set of isomorphism classes of H-torsors and H^1(C, H).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{H}$ be an abelian sheaf on $\\mathcal{C}$.\nThere is a canonical bijection between the set of isomorphism\nclasses of $\\mathcal{H}$-torsors and $H^1(\\mathcal{C}, \\mathcal{H})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"First cohomology and torsors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AJ","source_file":"sites-cohomology.tex","source_line":290,"source_end_line":296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L290-L296","statement_sha256":"0b626dd9f26364e77c216f554bfcd698c5c6d57a54ae0536dbc2ac27a5d1e5ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":4408,"rank":4408,"depth":19,"x":1016.603,"y":650.872,"cluster":"sheaf-cohomology"},{"id":"stacks:03F1","tag":"03F1","title":"First cohomology and extensions · Lemma 03F1","summary":"Let (C, O) be a ringed site. Let F be a sheaf of O-modules on C. There is a canonical bijection Ext^1_Mod(O)(O, F) → H^1(C, F) which associates to the extension 0 → F → E → O → 0 the image of 1 ∈ Γ(C, O) in H^1(C, F).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $\\mathcal{C}$.\nThere is a canonical bijection\n$$\n\\Ext^1_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{O}, \\mathcal{F})\n\\longrightarrow\nH^1(\\mathcal{C}, \\mathcal{F})\n$$\nwhich associates to the extension\n$$\n0 \\to \\mathcal{F} \\to \\mathcal{E} \\to \\mathcal{O} \\to 0\n$$\nthe image of $1 \\in \\Gamma(\\mathcal{C}, \\mathcal{O})$ in\n$H^1(\\mathcal{C}, \\mathcal{F})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"First cohomology and extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03F1","source_file":"sites-cohomology.tex","source_line":371,"source_end_line":387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L371-L387","statement_sha256":"1a21ef88c2c8d88d927d6bc31a154e6556aced426dc67b98388fb4c81addf75c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4409,"rank":4409,"depth":0,"x":1237.246,"y":637.052,"cluster":"sheaf-cohomology"},{"id":"stacks:03F2","tag":"03F2","title":"First cohomology and extensions · Lemma 03F2","summary":"Let (C, O) be a ringed site. Let F be a sheaf of O-modules on C. Let F_ab denote the underlying sheaf of abelian groups. Then there is a functorial isomorphism H^1(C, F_ab) = H^1(C, F) where the left hand side is cohomology computed in Ab(C) and the right hand side is cohomology computed in Mod(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules on $\\mathcal{C}$.\nLet $\\mathcal{F}_{ab}$ denote the underlying sheaf of abelian\ngroups. Then there is a functorial isomorphism\n$$\nH^1(\\mathcal{C}, \\mathcal{F}_{ab})\n=\nH^1(\\mathcal{C}, \\mathcal{F})\n$$\nwhere the left hand side is cohomology computed in\n$\\textit{Ab}(\\mathcal{C})$ and the right hand side\nis cohomology computed in $\\textit{Mod}(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"First cohomology and extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03F2","source_file":"sites-cohomology.tex","source_line":421,"source_end_line":435,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L421-L435","statement_sha256":"ac9e56fde9d284b4630597dbe8ffacb0b04a0ec107791cb4d2ed297e33ea57b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4410,"rank":4410,"depth":6,"x":1085.371,"y":772.699,"cluster":"sheaf-cohomology"},{"id":"stacks:040E","tag":"040E","title":"First cohomology and invertible sheaves · Lemma 040E","summary":"Let (C, O) be a locally ringed site. There is a canonical isomorphism H^1(C, O^*) = Pic(O). of abelian groups.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a locally ringed site.\nThere is a canonical isomorphism\n$$\nH^1(\\mathcal{C}, \\mathcal{O}^*) = \\Pic(\\mathcal{O}).\n$$\nof abelian groups.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"First cohomology and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040E","source_file":"sites-cohomology.tex","source_line":520,"source_end_line":528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L520-L528","statement_sha256":"90d6604169829e615d1ac54572c4b093388b56d3928e669d211d5c7d1f439ded","origin":"The Stacks Project","memory_eligible":false,"source_rank":4411,"rank":4411,"depth":20,"x":1088.283,"y":586.144,"cluster":"sheaf-cohomology"},{"id":"stacks:03F3","tag":"03F3","title":"Locality of cohomology · Lemma 03F3","summary":"Let (C, O) be a ringed site. Let U be an object of C. • If I is an injective O-module then I|_U is an injective O_U-module. • For any sheaf of O-modules F we have H^p(U, F) = H^p(C/U, F|_U).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $U$ be an object of $\\mathcal{C}$.\n\\begin{enumerate}\n\\item If $\\mathcal{I}$ is an injective $\\mathcal{O}$-module\nthen $\\mathcal{I}|_U$ is an injective $\\mathcal{O}_U$-module.\n\\item For any sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ we have\n$H^p(U, \\mathcal{F}) = H^p(\\mathcal{C}/U, \\mathcal{F}|_U)$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Locality of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03F3","source_file":"sites-cohomology.tex","source_line":607,"source_end_line":617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L607-L617","statement_sha256":"db4a2c8eb27e3aa78fb9354b35ddabeb136a8290882fccdbc68222ca7106a21f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4412,"rank":4412,"depth":8,"x":1236.438,"y":725.623,"cluster":"sheaf-cohomology"},{"id":"stacks:03YU","tag":"03YU","title":"Locality of cohomology · Lemma 03YU","summary":"Let C and D be sites. Let u : C → D be a functor. Assume u satisfies the hypotheses of Sites, Lemma [Tag 00XU]. Let g : Sh(C) → Sh(D) be the associated morphism of topoi. For any abelian sheaf F on D we have isomorphisms RΓ(C, g^-1F) = RΓ(D, F), in particular H^p(C, g^-1F) = H^p(D, F) and for any U ∈ Ob(C) we have isomorphisms RΓ(U, g^-1F) = RΓ(u(U), F), in particular H^p(U, g^-1F) = H^p(u(U), F). All of these isomorphisms are functorial in F.","statement_latex":"Let $\\mathcal{C}$ and $\\mathcal{D}$ be sites.\nLet $u : \\mathcal{C} \\to \\mathcal{D}$ be a functor.\nAssume $u$ satisfies the hypotheses of\nSites, Lemma \\ref{sites-lemma-bigger-site}.\nLet $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe the associated morphism of topoi.\nFor any abelian sheaf $\\mathcal{F}$ on $\\mathcal{D}$ we have\nisomorphisms\n$$\nR\\Gamma(\\mathcal{C}, g^{-1}\\mathcal{F}) = R\\Gamma(\\mathcal{D}, \\mathcal{F}),\n$$\nin particular\n$H^p(\\mathcal{C}, g^{-1}\\mathcal{F}) = H^p(\\mathcal{D}, \\mathcal{F})$\nand for any $U \\in \\Ob(\\mathcal{C})$ we have isomorphisms\n$$\nR\\Gamma(U, g^{-1}\\mathcal{F}) = R\\Gamma(u(U), \\mathcal{F}),\n$$\nin particular\n$H^p(U, g^{-1}\\mathcal{F}) = H^p(u(U), \\mathcal{F})$. All of these\nisomorphisms are functorial in $\\mathcal{F}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Locality of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YU","source_file":"sites-cohomology.tex","source_line":645,"source_end_line":667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L645-L667","statement_sha256":"53adb1d47204834c4436efa83c2c80474f3399f2b3ef8988c60b85cc6778d223","origin":"The Stacks Project","memory_eligible":false,"source_rank":4413,"rank":4413,"depth":8,"x":1014.609,"y":706.804,"cluster":"sheaf-cohomology"},{"id":"stacks:01FW","tag":"01FW","title":"Locality of cohomology · Lemma 01FW","summary":"Let (C, O) be a ringed site. Let F be a sheaf of O-modules. Let U be an object of C. Let n > 0 and let xi ∈ H^n(U, F). Then there exists a covering (U_i → U) of C such that xi|_U_i = 0 for all i ∈ I.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nLet $U$ be an object of $\\mathcal{C}$.\nLet $n > 0$ and let $\\xi \\in H^n(U, \\mathcal{F})$.\nThen there exists a covering $\\{U_i \\to U\\}$ of $\\mathcal{C}$\nsuch that $\\xi|_{U_i} = 0$ for all $i \\in I$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Locality of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01FW","source_file":"sites-cohomology.tex","source_line":714,"source_end_line":722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L714-L722","statement_sha256":"d10f02275bb34a66f57e7ff7e53034697601b5a68055179800d77cd4d93682ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":4414,"rank":4414,"depth":0,"x":1193.652,"y":594.599,"cluster":"sheaf-cohomology"},{"id":"stacks:072W","tag":"072W","title":"Locality of cohomology · Lemma 072W","summary":"Let f : (C, O_C) → (D, O_D) be a morphism of ringed sites corresponding to the continuous functor u : D → C. For any F ∈ Ob(Mod(O_C)) the sheaf R^if_*F is the sheaf associated to the presheaf V ↦ H^i(u(V), F)","statement_latex":"Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites\ncorresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nFor any $\\mathcal{F} \\in \\Ob(\\textit{Mod}(\\mathcal{O}_\\mathcal{C}))$\nthe sheaf $R^if_*\\mathcal{F}$ is the sheaf associated to the\npresheaf\n$$\nV \\longmapsto H^i(u(V), \\mathcal{F})\n$$","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Locality of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072W","source_file":"sites-cohomology.tex","source_line":746,"source_end_line":757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L746-L757","statement_sha256":"b665a6b82916711fc2d21a22113c76c068d67dd7604227b00ccd251f182b8162","origin":"The Stacks Project","memory_eligible":false,"source_rank":4415,"rank":4415,"depth":0,"x":1151.782,"y":779.275,"cluster":"sheaf-cohomology"},{"id":"stacks:03AM","tag":"03AM","title":"The v Cech complex and v Cech cohomology · Definition 03AM","summary":"Let C be a category. Let U = (U_i → U)_i ∈ I be a family of morphisms with fixed target such that all fibre products U_i_0 ×_U … ×_U U_i_p exist in C. Let F be an abelian presheaf on C. The complex checkC^bullet(U, F) is the v Cech complex associated to F and the family U. Its cohomology groups H^i(checkC^bullet(U, F)) are called the v Cech cohomology groups of F with respect to U. They are denoted check H^i(U, F).","statement_latex":"Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$\nbe a family of morphisms with fixed target such that all fibre products\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.\nLet $\\mathcal{F}$ be an abelian presheaf on $\\mathcal{C}$.\nThe complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\nis the {\\it {\\v C}ech complex} associated to $\\mathcal{F}$ and the\nfamily $\\mathcal{U}$. Its cohomology groups\n$H^i(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}))$ are\ncalled the {\\it {\\v C}ech cohomology groups} of $\\mathcal{F}$ with respect\nto $\\mathcal{U}$. They are denoted $\\check H^i(\\mathcal{U}, \\mathcal{F})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"The v Cech complex and v Cech cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AM","source_file":"sites-cohomology.tex","source_line":832,"source_end_line":844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L832-L844","statement_sha256":"13001c08a857456e80ef7bf0f9fd36c506a1aacc94148e80a8e0563937f54296","origin":"The Stacks Project","memory_eligible":false,"source_rank":4416,"rank":4416,"depth":0,"x":1033.926,"y":619.042,"cluster":"sheaf-cohomology"},{"id":"stacks:03AN","tag":"03AN","title":"The v Cech complex and v Cech cohomology · Lemma 03AN","summary":"Let C be a site. Let F be an abelian presheaf on C. The following are equivalent • F is an abelian sheaf on C and • for every covering U = (U_i → U)_i ∈ I of the site C the natural map F(U) → checkH^0(U, F) (see Sites, Section [Tag 00W1]) is bijective.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{F}$ be an abelian presheaf on $\\mathcal{C}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{C}$ and\n\\item for every covering $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$\nof the site $\\mathcal{C}$ the natural map\n$$\n\\mathcal{F}(U) \\to \\check{H}^0(\\mathcal{U}, \\mathcal{F})\n$$\n(see Sites, Section \\ref{sites-section-sheafification}) is bijective.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"The v Cech complex and v Cech cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AN","source_file":"sites-cohomology.tex","source_line":850,"source_end_line":864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L850-L864","statement_sha256":"042a3c6bcd3046a510a94ea7c1202ec2498d789eb62a4dc8b0cf784b1fba480f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4417,"rank":4417,"depth":0,"x":1250.087,"y":670.417,"cluster":"sheaf-cohomology"},{"id":"stacks:03AQ","tag":"03AQ","title":"v Cech cohomology as a functor on presheaves · Lemma 03AQ","summary":"The functor given by Equation ([Tag 03AP]) is an exact functor (see Homology, Lemma [Tag 010N]).","statement_latex":"The functor given by Equation (\\ref{equation-cech-functor})\nis an exact functor (see Homology, Lemma \\ref{homology-lemma-exact-functor}).","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AQ","source_file":"sites-cohomology.tex","source_line":925,"source_end_line":929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L925-L929","statement_sha256":"318532492bd1c8b8dab5ce6f3150f3ed0b810d6cf2dfea28ea0756b05b13f63e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4418,"rank":4418,"depth":7,"x":1049.007,"y":755.345,"cluster":"sheaf-cohomology"},{"id":"stacks:03AR","tag":"03AR","title":"v Cech cohomology as a functor on presheaves · Lemma 03AR","summary":"Let C be a category. Let U = (U_i → U)_i ∈ I be a family of morphisms with fixed target such that all fibre products U_i_0 ×_U … ×_U U_i_p exist in C. The functors F ↦ checkH^n(U, F) form a δ-functor from the abelian category PAb(C) to the category of Z-modules (see Homology, Definition [Tag 010Q]).","statement_latex":"Let $\\mathcal{C}$ be a category.\nLet $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a family of morphisms\nwith fixed target such that all fibre products\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.\nThe functors $\\mathcal{F} \\mapsto \\check{H}^n(\\mathcal{U}, \\mathcal{F})$\nform a $\\delta$-functor from the abelian category $\\textit{PAb}(\\mathcal{C})$\nto the category of $\\mathbf{Z}$-modules (see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}).","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AR","source_file":"sites-cohomology.tex","source_line":941,"source_end_line":951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L941-L951","statement_sha256":"64e97aae294d01a119b85e0fb3b128ad78c70e6b13684fa5589d84ed21552fa2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4419,"rank":4419,"depth":8,"x":1129.13,"y":578.298,"cluster":"sheaf-cohomology"},{"id":"stacks:03AS","tag":"03AS","title":"v Cech cohomology as a functor on presheaves · Lemma 03AS","summary":"Let C be a category. Let U = (U_i → U)_i ∈ I be a family of morphisms with fixed target such that all fibre products U_i_0 ×_U … ×_U U_i_p exist in C. Consider the chain complex Z_U, bullet of abelian presheaves … → bigoplus_i_0i_1i_2 Z_U_i_0 ×_U U_i_1 ×_U U_i_2 → bigoplus_i_0i_1 Z_U_i_0 ×_U U_i_1 → bigoplus_i_0 Z_U_i_0 → 0 → … where the last nonzero term is placed in degree 0 and where the map Z_U_i_0 ×_U … ×_u U_i_p + 1 → Z_U_i_0 ×_U … widehatU_i_j … ×_U U_i_p + 1 is…","statement_latex":"Let $\\mathcal{C}$ be a category. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$\nbe a family of morphisms with fixed target such that all fibre products\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.\nConsider the chain complex $\\mathbf{Z}_{\\mathcal{U}, \\bullet}$\nof abelian presheaves\n$$\n\\ldots\n\\to\n\\bigoplus_{i_0i_1i_2} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2}}\n\\to\n\\bigoplus_{i_0i_1} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}}\n\\to\n\\bigoplus_{i_0} \\mathbf{Z}_{U_{i_0}}\n\\to 0 \\to \\ldots\n$$\nwhere the last nonzero term is placed in degree $0$\nand where the map\n$$\n\\mathbf{Z}_{U_{i_0} \\times_U \\ldots \\times_u U_{i_{p + 1}}}\n\\longrightarrow\n\\mathbf{Z}_{U_{i_0} \\times_U\n\\ldots \\widehat{U_{i_j}} \\ldots \\times_U U_{i_{p + 1}}}\n$$\nis given by $(-1)^j$ times the canonical map.\nThen there is an isomorphism\n$$\n\\Hom_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_{\\mathcal{U}, \\bullet}, \\mathcal{F})\n=\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nfunctorial in $\\mathcal{F} \\in \\Ob(\\textit{PAb}(\\mathcal{C}))$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AS","source_file":"sites-cohomology.tex","source_line":970,"source_end_line":1003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L970-L1003","statement_sha256":"3943395b71c76ab0a42506fb485ad092a158630b658930463fce029521a194ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":4420,"rank":4420,"depth":1,"x":1212.578,"y":754.637,"cluster":"sheaf-cohomology"},{"id":"stacks:03AT","tag":"03AT","title":"v Cech cohomology as a functor on presheaves · Lemma 03AT","summary":"Let C be a category. Let U = (f_i : U_i → U)_i ∈ I be a family of morphisms with fixed target such that all fibre products U_i_0 ×_U … ×_U U_i_p exist in C. The chain complex Z_U, bullet of presheaves of Lemma [Tag 03AS] above is exact in positive degrees, i.e., the homology presheaves H_i(Z_U, bullet) are zero for i > 0.","statement_latex":"Let $\\mathcal{C}$ be a category. Let\n$\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms\nwith fixed target such that all fibre products\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.\nThe chain complex $\\mathbf{Z}_{\\mathcal{U}, \\bullet}$ of presheaves\nof Lemma \\ref{lemma-cech-map-into} above is exact in positive\ndegrees, i.e., the homology presheaves\n$H_i(\\mathbf{Z}_{\\mathcal{U}, \\bullet})$ are zero for $i > 0$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AT","source_file":"sites-cohomology.tex","source_line":1024,"source_end_line":1034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1024-L1034","statement_sha256":"5ee370b7e5798197f5d15ed835efa864327d4b54e49e96fdd16beca94e73ded0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4421,"rank":4421,"depth":2,"x":1008.866,"y":671.806,"cluster":"sheaf-cohomology"},{"id":"stacks:03F5","tag":"03F5","title":"v Cech cohomology as a functor on presheaves · Lemma 03F5","summary":"The integral presheaf v Cech complex is a flat resolution of the constant presheaf of integers. Let C be a category. Let U = (f_i : U_i → U)_i ∈ I be a family of morphisms with fixed target such that all fibre products U_i_0 ×_U … ×_U U_i_p exist in C. Let O be a presheaf of rings on C. The chain complex Z_U, bullet ⊗_p, Z O is exact in positive degrees. Here Z_U, bullet is the chain complex of Lemma [Tag 03AS], and the tensor product is over the constant presheaf of…","statement_latex":"\\begin{slogan}\nThe integral presheaf {\\v C}ech complex is a flat resolution of the\nconstant presheaf of integers.\n\\end{slogan}\nLet $\\mathcal{C}$ be a category. Let\n$\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms\nwith fixed target such that all fibre products\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.\nLet $\\mathcal{O}$ be a presheaf of rings on $\\mathcal{C}$.\nThe chain complex\n$$\n\\mathbf{Z}_{\\mathcal{U}, \\bullet}\n\\otimes_{p, \\mathbf{Z}}\n\\mathcal{O}\n$$\nis exact in positive degrees. Here $\\mathbf{Z}_{\\mathcal{U}, \\bullet}$\nis the chain complex of Lemma \\ref{lemma-cech-map-into}, and\nthe tensor product is over the constant presheaf of rings\nwith value $\\mathbf{Z}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03F5","source_file":"sites-cohomology.tex","source_line":1149,"source_end_line":1170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1149-L1170","statement_sha256":"9e613e0102bbb61c8f05903e808079f57869861483c5dc0c614de4715d327808","origin":"The Stacks Project","memory_eligible":false,"source_rank":4422,"rank":4422,"depth":8,"x":1226.086,"y":617.194,"cluster":"sheaf-cohomology"},{"id":"stacks:03AU","tag":"03AU","title":"v Cech cohomology as a functor on presheaves · Lemma 03AU","summary":"Let C be a category. Let U = (f_i : U_i → U)_i ∈ I be a family of morphisms with fixed target such that all fibre products U_i_0 ×_U … ×_U U_i_p exist in C. The v Cech cohomology functors checkH^p(U, -) are canonically isomorphic as a δ-functor to the right derived functors of the functor checkH^0(U, -) : PAb(C) → Ab. Moreover, there is a functorial quasi-isomorphism checkC^bullet(U, F) → RcheckH^0(U, F) where the right hand side indicates the derived functor RcheckH^0(U,…","statement_latex":"Let $\\mathcal{C}$ be a category. Let\n$\\mathcal{U} = \\{f_i : U_i \\to U\\}_{i \\in I}$ be a family of morphisms\nwith fixed target such that all fibre products\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exist in $\\mathcal{C}$.\nThe {\\v C}ech cohomology functors $\\check{H}^p(\\mathcal{U}, -)$\nare canonically isomorphic as a $\\delta$-functor to\nthe right derived functors of the functor\n$$\n\\check{H}^0(\\mathcal{U}, -) :\n\\textit{PAb}(\\mathcal{C})\n\\longrightarrow\n\\textit{Ab}.\n$$\nMoreover, there is a functorial quasi-isomorphism\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})\n\\longrightarrow\nR\\check{H}^0(\\mathcal{U}, \\mathcal{F})\n$$\nwhere the right hand side indicates the derived functor\n$$\nR\\check{H}^0(\\mathcal{U}, -) :\nD^{+}(\\textit{PAb}(\\mathcal{C}))\n\\longrightarrow\nD^{+}(\\mathbf{Z})\n$$\nof the left exact functor $\\check{H}^0(\\mathcal{U}, -)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology as a functor on presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AU","source_file":"sites-cohomology.tex","source_line":1188,"source_end_line":1217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1188-L1217","statement_sha256":"fa4917209f51bcf406608825f8544b520acb30c41d29edbcd267d015d7088062","origin":"The Stacks Project","memory_eligible":false,"source_rank":4423,"rank":4423,"depth":19,"x":1109.618,"y":781.016,"cluster":"sheaf-cohomology"},{"id":"stacks:03F6","tag":"03F6","title":"v Cech cohomology and cohomology · Lemma 03F6","summary":"Let C be a site. An injective abelian sheaf is also injective as an object in the category PAb(C).","statement_latex":"Let $\\mathcal{C}$ be a site. An injective abelian sheaf is also injective as an\nobject in the category $\\textit{PAb}(\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03F6","source_file":"sites-cohomology.tex","source_line":1318,"source_end_line":1322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1318-L1322","statement_sha256":"aa374f5c56b7c70a8c9e91daf7aa0b1dfba209e81bd73222c2046ee61aa0e52e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4424,"rank":4424,"depth":8,"x":1063.676,"y":593.791,"cluster":"sheaf-cohomology"},{"id":"stacks:03AW","tag":"03AW","title":"v Cech cohomology and cohomology · Lemma 03AW","summary":"Let C be a site. Let U = (U_i → U)_i ∈ I be a covering of C. Let I be an injective abelian sheaf, i.e., an injective object of Ab(C). Then checkH^p(U, I) = ( I(U) & if & p = 0 0 & if & p > 0 .","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.\nLet $\\mathcal{I}$ be an injective abelian sheaf, i.e., an injective\nobject of $\\textit{Ab}(\\mathcal{C})$.\nThen\n$$\n\\check{H}^p(\\mathcal{U}, \\mathcal{I}) =\n\\left\\{\n\\begin{matrix}\n\\mathcal{I}(U) & \\text{if} & p = 0 \\\\\n0 & \\text{if} & p > 0\n\\end{matrix}\n\\right.\n$$","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AW","source_file":"sites-cohomology.tex","source_line":1333,"source_end_line":1349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1333-L1349","statement_sha256":"18732a1dac24976269462bb5108a08087ad23bf349a4cba8178c2698e3caeef7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4425,"rank":4425,"depth":20,"x":1248.446,"y":705.982,"cluster":"sheaf-cohomology"},{"id":"stacks:03AX","tag":"03AX","title":"v Cech cohomology and cohomology · Lemma 03AX","summary":"Let C be a site. Let U = (U_i → U)_i ∈ I be a covering of C. There is a transformation checkC^bullet(U, -) → RΓ(U, -) of functors Ab(C) → D^+(Z). In particular this gives a transformation of functors checkH^p(U, F) → H^p(U, F) for F ranging over Ab(C).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.\nThere is a transformation\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, -)\n\\longrightarrow\nR\\Gamma(U, -)\n$$\nof functors\n$\\textit{Ab}(\\mathcal{C}) \\to D^{+}(\\mathbf{Z})$.\nIn particular this gives a transformation of functors\n$\\check{H}^p(\\mathcal{U}, \\mathcal{F}) \\to H^p(U, \\mathcal{F})$ for\n$\\mathcal{F}$ ranging over $\\textit{Ab}(\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AX","source_file":"sites-cohomology.tex","source_line":1360,"source_end_line":1375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1360-L1375","statement_sha256":"228c34ba85eb5db36826d3df7314fa8821ed4ff152a4342f8ce69b328a35ff94","origin":"The Stacks Project","memory_eligible":false,"source_rank":4426,"rank":4426,"depth":21,"x":1021.572,"y":728.137,"cluster":"sheaf-cohomology"},{"id":"stacks:0A6G","tag":"0A6G","title":"v Cech cohomology and cohomology · Lemma 0A6G","summary":"Let C be a site. Let G be an abelian sheaf on C. Let U = (U_i → U)_i ∈ I be a covering of C. The map checkH^1(U, G) → H^1(U, G) is injective and identifies checkH^1(U, G) via the bijection of Lemma [Tag 03AJ] with the set of isomorphism classes of G|_U-torsors which restrict to trivial torsors over each U_i.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{G}$ be an abelian sheaf\non $\\mathcal{C}$. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a\ncovering of $\\mathcal{C}$. The map\n$$\n\\check{H}^1(\\mathcal{U}, \\mathcal{G})\n\\longrightarrow\nH^1(U, \\mathcal{G})\n$$\nis injective and identifies $\\check{H}^1(\\mathcal{U}, \\mathcal{G})$ via\nthe bijection of Lemma \\ref{lemma-torsors-h1}\nwith the set of isomorphism classes of $\\mathcal{G}|_U$-torsors\nwhich restrict to trivial torsors over each $U_i$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6G","source_file":"sites-cohomology.tex","source_line":1415,"source_end_line":1429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1415-L1429","statement_sha256":"4d8e7ba4753daa5ab2fc906f4a08463327072004210b7cdfa7ed1a223ebcd0c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4427,"rank":4427,"depth":20,"x":1171.316,"y":582.807,"cluster":"sheaf-cohomology"},{"id":"stacks:03AY","tag":"03AY","title":"v Cech cohomology and cohomology · Lemma 03AY","summary":"Let C be a site. Consider the functor i : Ab(C) → PAb(C). It is a left exact functor with right derived functors given by R^pi(F) = underlineH^p(F) : U ↦ H^p(U, F) see discussion in Section [Tag 01FU].","statement_latex":"Let $\\mathcal{C}$ be a site.\nConsider the functor\n$i : \\textit{Ab}(\\mathcal{C}) \\to \\textit{PAb}(\\mathcal{C})$.\nIt is a left exact functor with right derived functors given by\n$$\nR^pi(\\mathcal{F}) = \\underline{H}^p(\\mathcal{F}) :\nU \\longmapsto H^p(U, \\mathcal{F})\n$$\nsee discussion in Section \\ref{section-locality}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AY","source_file":"sites-cohomology.tex","source_line":1447,"source_end_line":1458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1447-L1458","statement_sha256":"294cb2ee502e2d89356df6b46cc01dffba84ca751d6de3d97d19146b1867fa42","origin":"The Stacks Project","memory_eligible":false,"source_rank":4428,"rank":4428,"depth":0,"x":1177.779,"y":775.282,"cluster":"sheaf-cohomology"},{"id":"stacks:03AZ","tag":"03AZ","title":"v Cech cohomology and cohomology · Lemma 03AZ","summary":"Let C be a site. Let U = (U_i → U)_i ∈ I be a covering of C. For any abelian sheaf F there is a spectral sequence (E_r, d_r)_r ≥ 0 with E_2^p, q = checkH^p(U, underlineH^q(F)) converging to H^p + q(U, F). This spectral sequence is functorial in F.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$\nbe a covering of $\\mathcal{C}$. For any abelian sheaf $\\mathcal{F}$ there\nis a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_2^{p, q} = \\check{H}^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))\n$$\nconverging to $H^{p + q}(U, \\mathcal{F})$.\nThis spectral sequence is functorial in $\\mathcal{F}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03AZ","source_file":"sites-cohomology.tex","source_line":1474,"source_end_line":1484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1474-L1484","statement_sha256":"203220aebc1d1e2be707f77a2760d185204d4ec7b43ad9639dc03952da94177f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4429,"rank":4429,"depth":21,"x":1017.948,"y":636.775,"cluster":"sheaf-cohomology"},{"id":"stacks:03F7","tag":"03F7","title":"v Cech cohomology and cohomology · Lemma 03F7","summary":"Let C be a site. Let U = (U_i → U)_i ∈ I be a covering. Let F ∈ Ob(Ab(C)). Assume that H^i(U_i_0 ×_U … ×_U U_i_p, F) = 0 for all i > 0, all p ≥ 0 and all i_0, …, i_p ∈ I. Then checkH^p(U, F) = H^p(U, F).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering.\nLet $\\mathcal{F} \\in \\Ob(\\textit{Ab}(\\mathcal{C}))$.\nAssume that $H^i(U_{i_0} \\times_U \\ldots \\times_U U_{i_p}, \\mathcal{F}) = 0$\nfor all $i > 0$, all $p \\geq 0$ and all $i_0, \\ldots, i_p \\in I$.\nThen $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(U, \\mathcal{F})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03F7","source_file":"sites-cohomology.tex","source_line":1505,"source_end_line":1513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1505-L1513","statement_sha256":"909ea16d4566d1cad70310b93bd11c333725c550190a145a75b90518813e3629","origin":"The Stacks Project","memory_eligible":false,"source_rank":4430,"rank":4430,"depth":22,"x":1247.593,"y":648.236,"cluster":"sheaf-cohomology"},{"id":"stacks:03F8","tag":"03F8","title":"v Cech cohomology and cohomology · Lemma 03F8","summary":"Let C be a site. Let 0 → F → G → H → 0 be a short exact sequence of abelian sheaves on C. Let U be an object of C. If there exists a cofinal system of coverings U of U such that checkH^1(U, F) = 0, then the map G(U) → H(U) is surjective.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet\n$$\n0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0\n$$\nbe a short exact sequence of abelian sheaves on $\\mathcal{C}$.\nLet $U$ be an object of $\\mathcal{C}$. If there exists a cofinal system\nof coverings $\\mathcal{U}$ of $U$ such that\n$\\check{H}^1(\\mathcal{U}, \\mathcal{F}) = 0$,\nthen the map $\\mathcal{G}(U) \\to \\mathcal{H}(U)$ is\nsurjective.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03F8","source_file":"sites-cohomology.tex","source_line":1523,"source_end_line":1536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1523-L1536","statement_sha256":"9f5ea4de4ed5985aa33c4032d00692d29be7cea0d10e2aaceaa6036782bd613c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4431,"rank":4431,"depth":0,"x":1068.724,"y":770.306,"cluster":"sheaf-cohomology"},{"id":"stacks:03F9","tag":"03F9","title":"v Cech cohomology and cohomology · Lemma 03F9","summary":"(Variant of Cohomology, Lemma [Tag 01EV].) Let C be a site. Let Cov_C be the set of coverings of C (see Sites, Definition [Tag 00VH]). Let B ⊂ Ob(C), and Cov ⊂ Cov_C be subsets. Let F be an abelian sheaf on C. Assume that • For every U ∈ Cov, U = (U_i → U)_i ∈ I we have U, U_i ∈ B and every U_i_0 ×_U … ×_U U_i_p ∈ B. • For every U ∈ B the coverings of U occurring in Cov is a cofinal system of coverings of U. • For every U ∈ Cov we have checkH^p(U, F) = 0 for all p > 0.…","statement_latex":"(Variant of Cohomology, Lemma \\ref{cohomology-lemma-cech-vanish}.)\nLet $\\mathcal{C}$ be a site. Let $\\text{Cov}_\\mathcal{C}$ be the set\nof coverings of $\\mathcal{C}$ (see\nSites, Definition \\ref{sites-definition-site}). Let\n$\\mathcal{B} \\subset \\Ob(\\mathcal{C})$, and\n$\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$\nbe subsets. Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$.\nAssume that\n\\begin{enumerate}\n\\item For every $\\mathcal{U} \\in \\text{Cov}$,\n$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ we have\n$U, U_i \\in \\mathcal{B}$ and every\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p} \\in \\mathcal{B}$.\n\\item For every $U \\in \\mathcal{B}$ the coverings of $U$\noccurring in $\\text{Cov}$ is a cofinal system of coverings of $U$.\n\\item For every $\\mathcal{U} \\in \\text{Cov}$ we have\n$\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for all $p > 0$.\n\\end{enumerate}\nThen $H^p(U, \\mathcal{F}) = 0$ for all $p > 0$ and any $U \\in \\mathcal{B}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"v Cech cohomology and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03F9","source_file":"sites-cohomology.tex","source_line":1564,"source_end_line":1585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1564-L1585","statement_sha256":"1d118745bc2505ba79605e94bbcce5b82c6bb1c0067059cfd16836ab3851a93b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4432,"rank":4432,"depth":22,"x":1102.514,"y":578.463,"cluster":"sheaf-cohomology"},{"id":"stacks:0CK0","tag":"0CK0","title":"Second cohomology and gerbes · Lemma 0CK0","summary":"Let C be a site. Let p : S → C be a gerbe over a site whose automorphism sheaves are abelian. Let G be the sheaf of abelian groups constructed in Stacks, Lemma [Tag 0CJY]. Let U be an object of C such that • there exists a cofinal system of coverings (U_i → U) of U in C such that H^1(U_i, G) = 0 and H^1(U_i ×_U U_j, G) = 0 for all i, j, and • H^2(U, G) = 0. Then there exists an object of S lying over U.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $p : \\mathcal{S} \\to \\mathcal{C}$\nbe a gerbe over a site whose automorphism sheaves are abelian.\nLet $\\mathcal{G}$ be the sheaf of abelian groups constructed\nin Stacks, Lemma \\ref{stacks-lemma-gerbe-abelian-auts}.\nLet $U$ be an object of $\\mathcal{C}$ such that\n\\begin{enumerate}\n\\item there exists a cofinal system of coverings $\\{U_i \\to U\\}$\nof $U$ in $\\mathcal{C}$ such that $H^1(U_i, \\mathcal{G}) = 0$ and\n$H^1(U_i \\times_U U_j, \\mathcal{G}) = 0$\nfor all $i, j$, and\n\\item $H^2(U, \\mathcal{G}) = 0$.\n\\end{enumerate}\nThen there exists an object of $\\mathcal{S}$ lying over $U$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Second cohomology and gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CK0","source_file":"sites-cohomology.tex","source_line":1676,"source_end_line":1691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1676-L1691","statement_sha256":"86172bc8a037e750dde6d271370343124334bdece24bceae7bdaf3aa8226cb18","origin":"The Stacks Project","memory_eligible":false,"source_rank":4433,"rank":4433,"depth":22,"x":1232.099,"y":739.379,"cluster":"sheaf-cohomology"},{"id":"stacks:03FB","tag":"03FB","title":"Cohomology of modules · Lemma 03FB","summary":"Let (C, O) be a ringed site. An injective sheaf of modules is also injective as an object in the category PMod(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nAn injective sheaf of modules is also injective as an\nobject in the category $\\textit{PMod}(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FB","source_file":"sites-cohomology.tex","source_line":1769,"source_end_line":1774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1769-L1774","statement_sha256":"89bab63e6397b276355748b53648101228c37ecb135f69914dc9453aeca9a39f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4434,"rank":4434,"depth":8,"x":1006.748,"y":694.173,"cluster":"sheaf-cohomology"},{"id":"stacks:06YK","tag":"06YK","title":"Cohomology of modules · Lemma 06YK","summary":"Let (C, O) be a ringed site. Consider the functor i : Mod(C) → PMod(C). It is a left exact functor with right derived functors given by R^pi(F) = underlineH^p(F) : U ↦ H^p(U, F) see discussion in Section [Tag 01FU].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nConsider the functor\n$i : \\textit{Mod}(\\mathcal{C}) \\to \\textit{PMod}(\\mathcal{C})$.\nIt is a left exact functor with right derived functors given by\n$$\nR^pi(\\mathcal{F}) = \\underline{H}^p(\\mathcal{F}) :\nU \\longmapsto H^p(U, \\mathcal{F})\n$$\nsee discussion in\nSection \\ref{section-locality}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YK","source_file":"sites-cohomology.tex","source_line":1786,"source_end_line":1798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1786-L1798","statement_sha256":"b30601fc9f55fc66601cb7d5d0dea9c595bc2ad83d95c1eadf231d7aa3c6a691","origin":"The Stacks Project","memory_eligible":false,"source_rank":4435,"rank":4435,"depth":0,"x":1209.624,"y":599.474,"cluster":"sheaf-cohomology"},{"id":"stacks:03FC","tag":"03FC","title":"Cohomology of modules · Lemma 03FC","summary":"Let (C, O) be a ringed site. Let U = (U_i → U)_i ∈ I be a covering of C. Let I be an injective O-module, i.e., an injective object of Mod(O). Then checkH^p(U, I) = ( I(U) & if & p = 0 0 & if & p > 0 .","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a covering of $\\mathcal{C}$.\nLet $\\mathcal{I}$ be an injective $\\mathcal{O}$-module, i.e., an injective\nobject of $\\textit{Mod}(\\mathcal{O})$. Then\n$$\n\\check{H}^p(\\mathcal{U}, \\mathcal{I}) =\n\\left\\{\n\\begin{matrix}\n\\mathcal{I}(U) & \\text{if} & p = 0 \\\\\n0 & \\text{if} & p > 0\n\\end{matrix}\n\\right.\n$$","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FC","source_file":"sites-cohomology.tex","source_line":1815,"source_end_line":1830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1815-L1830","statement_sha256":"380629c0081e6eaba7b235948bce372ce59fec8595c630714fe8cd36406d2891","origin":"The Stacks Project","memory_eligible":false,"source_rank":4436,"rank":4436,"depth":9,"x":1136.054,"y":784.74,"cluster":"sheaf-cohomology"},{"id":"stacks:03FD","tag":"03FD","title":"Cohomology of modules · Lemma 03FD","summary":"Let C be a site. Let O be a sheaf of rings on C. Let F be an O-module, and denote F_ab the underlying sheaf of abelian groups. Then we have H^i(C, F_ab) = H^i(C, F) and for any object U of C we also have H^i(U, F_ab) = H^i(U, F). Here the left hand side is cohomology computed in Ab(C) and the right hand side is cohomology computed in Mod(O).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.\nLet $\\mathcal{F}$ be an $\\mathcal{O}$-module, and denote\n$\\mathcal{F}_{ab}$ the underlying sheaf of abelian groups.\nThen we have\n$$\nH^i(\\mathcal{C}, \\mathcal{F}_{ab})\n=\nH^i(\\mathcal{C}, \\mathcal{F})\n$$\nand for any object $U$ of $\\mathcal{C}$ we also have\n$$\nH^i(U, \\mathcal{F}_{ab})\n=\nH^i(U, \\mathcal{F}).\n$$\nHere the left hand side is cohomology computed in\n$\\textit{Ab}(\\mathcal{C})$ and the right hand side\nis cohomology computed in $\\textit{Mod}(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FD","source_file":"sites-cohomology.tex","source_line":1861,"source_end_line":1882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1861-L1882","statement_sha256":"09ea03000bd474b5b2cb0eca90cf15a4e80efeccc4393dbf859f565c9d4ff10f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4437,"rank":4437,"depth":23,"x":1041.154,"y":606.074,"cluster":"sheaf-cohomology"},{"id":"stacks:060L","tag":"060L","title":"Cohomology of modules · Lemma 060L","summary":"Let C be a site. Let I be a set. For i ∈ I let F_i be an abelian sheaf on C. Let U ∈ Ob(C). The canonical map H^p(U, ∏_i ∈ I F_i) → ∏_i ∈ I H^p(U, F_i) is an isomorphism for p = 0 and injective for p = 1.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $I$ be a set. For $i \\in I$ let \n$\\mathcal{F}_i$ be an abelian sheaf on $\\mathcal{C}$. Let\n$U \\in \\Ob(\\mathcal{C})$. The canonical map\n$$\nH^p(U, \\prod\\nolimits_{i \\in I} \\mathcal{F}_i)\n\\longrightarrow\n\\prod\\nolimits_{i \\in I} H^p(U, \\mathcal{F}_i)\n$$\nis an isomorphism for $p = 0$ and injective for $p = 1$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060L","source_file":"sites-cohomology.tex","source_line":1921,"source_end_line":1932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1921-L1932","statement_sha256":"848efa80049ba557bcfda8c0b85f67bbb97a8ed1e081690cc6d8e7d5c101de74","origin":"The Stacks Project","memory_eligible":false,"source_rank":4438,"rank":4438,"depth":21,"x":1255.177,"y":684.106,"cluster":"sheaf-cohomology"},{"id":"stacks:093X","tag":"093X","title":"Cohomology of modules · Lemma 093X","summary":"Let (C, O) be a ringed site. Let a : U' → U be a monomorphism in C. Then for any injective O-module I the restriction mapping I(U) → I(U') is surjective.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $a : U' \\to U$ be a\nmonomorphism in $\\mathcal{C}$. Then for any injective $\\mathcal{O}$-module\n$\\mathcal{I}$ the restriction mapping $\\mathcal{I}(U) \\to \\mathcal{I}(U')$\nis surjective.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093X","source_file":"sites-cohomology.tex","source_line":1961,"source_end_line":1967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L1961-L1967","statement_sha256":"4bda264c3a2dab4798c715dc71ff0825919afe9457b60fb0ae40dd50f8f52f88","origin":"The Stacks Project","memory_eligible":false,"source_rank":4439,"rank":4439,"depth":2,"x":1034.231,"y":748.117,"cluster":"sheaf-cohomology"},{"id":"stacks:079Y","tag":"079Y","title":"Totally acyclic sheaves · Lemma 079Y","summary":"Let (C, O) be a ringed site. Let K be a presheaf of sets on C. Let F be an O-module and denote F_ab the underlying sheaf of abelian groups. Then H^p(K, F) = H^p(K, F_ab).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $K$ be a presheaf of sets on $\\mathcal{C}$.\nLet $\\mathcal{F}$ be an $\\mathcal{O}$-module and denote\n$\\mathcal{F}_{ab}$ the underlying sheaf of abelian groups.\nThen $H^p(K, \\mathcal{F}) = H^p(K, \\mathcal{F}_{ab})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Totally acyclic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079Y","source_file":"sites-cohomology.tex","source_line":2052,"source_end_line":2059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2052-L2059","statement_sha256":"9c5974e80674cccd1eb0ae6c05c502fa72461af576ba8a879df6a6f723367c72","origin":"The Stacks Project","memory_eligible":false,"source_rank":4440,"rank":4440,"depth":24,"x":1145.867,"y":575.251,"cluster":"sheaf-cohomology"},{"id":"stacks:079Z","tag":"079Z","title":"Totally acyclic sheaves · Lemma 079Z","summary":"Let C be a site. Let K' → K be a map of presheaves of sets on C whose sheafification is surjective. Set K'_p = K' ×_K … ×_K K' (p + 1-factors). For every abelian sheaf F there is a spectral sequence with E_1^p, q = H^q(K'_p, F) converging to H^p + q(K, F).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $K' \\to K$ be a map of presheaves\nof sets on $\\mathcal{C}$ whose sheafification is surjective. Set\n$K'_p = K' \\times_K \\ldots \\times_K K'$ ($p + 1$-factors).\nFor every abelian sheaf $\\mathcal{F}$ there is a spectral sequence\nwith $E_1^{p, q} = H^q(K'_p, \\mathcal{F})$ converging to\n$H^{p + q}(K, \\mathcal{F})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Totally acyclic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/079Z","source_file":"sites-cohomology.tex","source_line":2072,"source_end_line":2080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2072-L2080","statement_sha256":"c9bc263640ffb627da2bda9484583731f92591373519bb7e117a4c0ea94a8157","origin":"The Stacks Project","memory_eligible":false,"source_rank":4441,"rank":4441,"depth":22,"x":1202.659,"y":766.39,"cluster":"sheaf-cohomology"},{"id":"stacks:07A0","tag":"07A0","title":"Totally acyclic sheaves · Lemma 07A0","summary":"Let C be a site. Let K be a sheaf of sets on C. Consider the morphism of topoi j : Sh(C/K) → Sh(C), see Sites, Lemma [Tag 0791]. Then j^-1 preserves injectives and H^p(K, F) = H^p(C/K, j^-1F) for any abelian sheaf F on C.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $K$ be a sheaf of sets on $\\mathcal{C}$.\nConsider the morphism of topoi\n$j : \\Sh(\\mathcal{C}/K) \\to \\Sh(\\mathcal{C})$, see\nSites, Lemma \\ref{sites-lemma-localize-topos-site}.\nThen $j^{-1}$ preserves injectives and\n$H^p(K, \\mathcal{F}) = H^p(\\mathcal{C}/K, j^{-1}\\mathcal{F})$\nfor any abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Totally acyclic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07A0","source_file":"sites-cohomology.tex","source_line":2096,"source_end_line":2105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2096-L2105","statement_sha256":"782e678c1ea069976cab9c18bbfce33fb72a62d8786bddeb211b6f4b25831868","origin":"The Stacks Project","memory_eligible":false,"source_rank":4442,"rank":4442,"depth":10,"x":1006.742,"y":657.488,"cluster":"sheaf-cohomology"},{"id":"stacks:072Y","tag":"072Y","title":"Totally acyclic sheaves · Definition 072Y","summary":"Let C be a site. We say an abelian sheaf F is totally acyclicstacks.project@gmail.com if you have a better suggestion. if for every sheaf of sets K we have H^p(K, F) = 0 for all p ≥ 1.","statement_latex":"Let $\\mathcal{C}$ be a site.\nWe say an abelian sheaf $\\mathcal{F}$ is\n{\\it totally acyclic}\\footnote{Although this terminology is is used in\n\\cite[Vbis, Proposition 1.3.10]{SGA4} this is probably nonstandard notation.\nIn \\cite[V, Definition 4.1]{SGA4} this property is dubbed ``flasque'', but\nwe cannot use this because it would clash with our definition\nof flasque sheaves on topological spaces. Please email\n\\href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}\nif you have a better suggestion.}\nif for every sheaf of sets $K$ we have $H^p(K, \\mathcal{F}) = 0$\nfor all $p \\geq 1$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Totally acyclic sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072Y","source_file":"sites-cohomology.tex","source_line":2121,"source_end_line":2134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2121-L2134","statement_sha256":"9d57e9d794bd0c940b1bc0eb0dd87e230c33af93308eb1c8921bb16597e76b2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4443,"rank":4443,"depth":0,"x":1239.176,"y":626.571,"cluster":"sheaf-cohomology"},{"id":"stacks:07A1","tag":"07A1","title":"Totally acyclic sheaves · Lemma 07A1","summary":"Let C be a site. Let F be an abelian sheaf. If • H^p(U, F) = 0 for p > 0 and U ∈ Ob(C), and • for every surjection K' → K of sheaves of sets the extended v Cech complex 0 → H^0(K, F) → H^0(K', F) → H^0(K' ×_K K', F) → … is exact, then F is totally acyclic (and the converse holds too).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{F}$ be an abelian sheaf. If\n\\begin{enumerate}\n\\item $H^p(U, \\mathcal{F}) = 0$ for $p > 0$ and $U \\in \\Ob(\\mathcal{C})$, and\n\\item for every surjection $K' \\to K$ of sheaves of sets the\nextended {\\v C}ech complex\n$$\n0 \\to H^0(K, \\mathcal{F}) \\to H^0(K', \\mathcal{F}) \\to\nH^0(K' \\times_K K', \\mathcal{F}) \\to \\ldots\n$$\nis exact,\n\\end{enumerate}\nthen $\\mathcal{F}$ is totally acyclic (and the converse holds too).","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Totally acyclic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07A1","source_file":"sites-cohomology.tex","source_line":2144,"source_end_line":2158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2144-L2158","statement_sha256":"7b416576372976991dc933d27d84d2d201a8092b680594a3a4b02c25d519c3b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4444,"rank":4444,"depth":23,"x":1092.398,"y":781.518,"cluster":"sheaf-cohomology"},{"id":"stacks:072Z","tag":"072Z","title":"The Leray spectral sequence · Lemma 072Z","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Then for any injective object I in Mod(O_C) the pushforward f_*I is totally acyclic.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nThen for any injective object $\\mathcal{I}$ in\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{C})$\nthe pushforward $f_*\\mathcal{I}$ is totally acyclic.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072Z","source_file":"sites-cohomology.tex","source_line":2211,"source_end_line":2218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2211-L2218","statement_sha256":"61eab786e142eeefd4019b4e803260071cd584ac5e9dce0161835c920509d766","origin":"The Stacks Project","memory_eligible":false,"source_rank":4445,"rank":4445,"depth":23,"x":1075.999,"y":583.645,"cluster":"sheaf-cohomology"},{"id":"stacks:0730","tag":"0730","title":"The Leray spectral sequence · Lemma 0730","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. If f is flat, then f_*I is an injective O_D-module for any injective O_C-module I.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nIf $f$ is flat, then $f_*\\mathcal{I}$ is an injective\n$\\mathcal{O}_\\mathcal{D}$-module\nfor any injective $\\mathcal{O}_\\mathcal{C}$-module $\\mathcal{I}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0730","source_file":"sites-cohomology.tex","source_line":2256,"source_end_line":2263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2256-L2263","statement_sha256":"91b4dfcf3320f9818ba7670701264c6da0573894380ed209a68dd5d53655e9ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":4446,"rank":4446,"depth":8,"x":1247.501,"y":720.477,"cluster":"sheaf-cohomology"},{"id":"stacks:0731","tag":"0731","title":"The Leray spectral sequence · Lemma 0731","summary":"Let (Sh(C), O_C) be a ringed topos. A totally acyclic sheaf is right acyclic for the following functors: • the functor H^0(U, -) for any object U of C, • the functor F ↦ F(K) for any presheaf of sets K, • the functor Γ(C, -) of global sections, • the functor f_* for any morphism f : (Sh(C), O_C) → (Sh(D), O_D) of ringed topoi.","statement_latex":"Let $(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ be a ringed topos.\nA totally acyclic sheaf is right acyclic for the following functors:\n\\begin{enumerate}\n\\item the functor $H^0(U, -)$ for any object $U$ of $\\mathcal{C}$,\n\\item the functor $\\mathcal{F} \\mapsto \\mathcal{F}(K)$ for any\npresheaf of sets $K$,\n\\item the functor $\\Gamma(\\mathcal{C}, -)$ of global sections,\n\\item the functor $f_*$ for any morphism\n$f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ of ringed topoi.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0731","source_file":"sites-cohomology.tex","source_line":2272,"source_end_line":2285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2272-L2285","statement_sha256":"5684673e6fc803f463b2ec14446de0b4302040b600a4be023b971c92bcfbb6fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4447,"rank":4447,"depth":10,"x":1010.608,"y":716.887,"cluster":"sheaf-cohomology"},{"id":"stacks:0732","tag":"0732","title":"Leray spectral sequence · Lemma 0732","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let F^bullet be a bounded below complex of O_C-modules. There is a spectral sequence E_2^p, q = H^p(D, R^qf_*(F^bullet)) converging to H^p + q(C, F^bullet).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nLet $\\mathcal{F}^\\bullet$ be a bounded below complex of\n$\\mathcal{O}_\\mathcal{C}$-modules. There is a spectral sequence\n$$\nE_2^{p, q} = H^p(\\mathcal{D}, R^qf_*(\\mathcal{F}^\\bullet))\n$$\nconverging to $H^{p + q}(\\mathcal{C}, \\mathcal{F}^\\bullet)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0732","source_file":"sites-cohomology.tex","source_line":2322,"source_end_line":2332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2322-L2332","statement_sha256":"ec7f3a4694e6a6f8f7ddc8e2e0c9f17c8ab788d4e5a154f4cb869490fa0d7de0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4448,"rank":4448,"depth":24,"x":1188.471,"y":584.897,"cluster":"sheaf-cohomology"},{"id":"stacks:0733","tag":"0733","title":"The Leray spectral sequence · Lemma 0733","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let F be an O_C-module. • If R^qf_*F = 0 for q > 0, then H^p(C, F) = H^p(D, f_*F) for all p. • If H^p(D, R^qf_*F) = 0 for all q and p > 0, then H^q(C, F) = H^0(D, R^qf_*F) for all q.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{C}$-module.\n\\begin{enumerate}\n\\item If $R^qf_*\\mathcal{F} = 0$ for $q > 0$, then\n$H^p(\\mathcal{C}, \\mathcal{F}) = H^p(\\mathcal{D}, f_*\\mathcal{F})$ for all $p$.\n\\item If $H^p(\\mathcal{D}, R^qf_*\\mathcal{F}) = 0$ for all $q$ and $p > 0$,\nthen $H^q(\\mathcal{C}, \\mathcal{F}) = H^0(\\mathcal{D}, R^qf_*\\mathcal{F})$\nfor all $q$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0733","source_file":"sites-cohomology.tex","source_line":2346,"source_end_line":2358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2346-L2358","statement_sha256":"d421d8120637e348ee82398f37366fc0499ba8812b31b6a4a9a3ef9ddd6aa908","origin":"The Stacks Project","memory_eligible":false,"source_rank":4449,"rank":4449,"depth":0,"x":1163.419,"y":783.476,"cluster":"sheaf-cohomology"},{"id":"stacks:0734","tag":"0734","title":"Relative Leray spectral sequence · Lemma 0734","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) and g : (Sh(D), O_D) → (Sh(E), O_E) be morphisms of ringed topoi. Let F be an O_C-module. There is a spectral sequence with E_2^p, q = R^pg_*(R^qf_*F) converging to R^p + q(g ∘ f)_*F. This spectral sequence is functorial in F, and there is a version for bounded below complexes of O_C-modules.","statement_latex":"Let\n$f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nand\n$g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to\n(\\Sh(\\mathcal{E}), \\mathcal{O}_\\mathcal{E})$\nbe morphisms of ringed topoi.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{C}$-module.\nThere is a spectral sequence with\n$$\nE_2^{p, q} = R^pg_*(R^qf_*\\mathcal{F})\n$$\nconverging to $R^{p + q}(g \\circ f)_*\\mathcal{F}$.\nThis spectral sequence is functorial in $\\mathcal{F}$, and there\nis a version for bounded below complexes of $\\mathcal{O}_\\mathcal{C}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0734","source_file":"sites-cohomology.tex","source_line":2366,"source_end_line":2383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2366-L2383","statement_sha256":"d57eb4db9b78ad272691ea237dc8b00c8303612486abab7efb68ed8513427393","origin":"The Stacks Project","memory_eligible":false,"source_rank":4450,"rank":4450,"depth":24,"x":1021.967,"y":622.567,"cluster":"sheaf-cohomology"},{"id":"stacks:0736","tag":"0736","title":"The base change map · Lemma 0736","summary":"Let xymatrix (Sh(C'), O_C') ar[r]_g' ar[d]_f' & (Sh(C), O_C) ar[d]^f (Sh(D'), O_D') ar[r]^g & (Sh(D), O_D) be a commutative diagram of ringed topoi. Let F^bullet be a bounded below complex of O_C-modules. Assume both g and g' are flat. Then there exists a canonical base change map g^*Rf_*F^bullet → R(f')_*(g')^*F^bullet in D^+(O_D').","statement_latex":"Let\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})\n\\ar[r]_{g'} \\ar[d]_{f'} &\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^f \\\\\n(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'})\n\\ar[r]^g &\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})\n}\n$$\nbe a commutative diagram of ringed topoi.\nLet $\\mathcal{F}^\\bullet$ be a bounded below complex of\n$\\mathcal{O}_\\mathcal{C}$-modules.\nAssume both $g$ and $g'$ are flat.\nThen there exists a canonical base change map\n$$\ng^*Rf_*\\mathcal{F}^\\bullet\n\\longrightarrow\nR(f')_*(g')^*\\mathcal{F}^\\bullet\n$$\nin $D^{+}(\\mathcal{O}_{\\mathcal{D}'})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"The base change map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0736","source_file":"sites-cohomology.tex","source_line":2426,"source_end_line":2450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2426-L2450","statement_sha256":"dde5abccded6eb4600682bafc6213d62f8016e1d7f28a4addac568a0a78c244c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4451,"rank":4451,"depth":9,"x":1256.055,"y":661.019,"cluster":"sheaf-cohomology"},{"id":"stacks:0739","tag":"0739","title":"Cohomology and colimits · Lemma 0739","summary":"Let C be a site. Let Cov_C be the set of coverings of C (see Sites, Definition [Tag 00VH]). Let B ⊂ Ob(C), and Cov ⊂ Cov_C be subsets. Assume that • For every U ∈ Cov we have U = (U_i → U)_i ∈ I with I finite, U, U_i ∈ B and every U_i_0 ×_U … ×_U U_i_p ∈ B. • For every U ∈ B the coverings of U occurring in Cov is a cofinal system of coverings of U. Then the map colim_i H^p(U, F_i) → H^p(U, colim_i F_i) is an isomorphism for every p ≥ 0, every U ∈ B, and every filtered…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\text{Cov}_\\mathcal{C}$ be the set\nof coverings of $\\mathcal{C}$ (see\nSites, Definition \\ref{sites-definition-site}). Let\n$\\mathcal{B} \\subset \\Ob(\\mathcal{C})$, and\n$\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$\nbe subsets. Assume that\n\\begin{enumerate}\n\\item For every $\\mathcal{U} \\in \\text{Cov}$ we have\n$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ with $I$ finite,\n$U, U_i \\in \\mathcal{B}$ and every\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p} \\in \\mathcal{B}$.\n\\item For every $U \\in \\mathcal{B}$ the coverings of $U$\noccurring in $\\text{Cov}$ is a cofinal system of coverings of $U$.\n\\end{enumerate}\nThen the map\n$$\n\\colim_i H^p(U, \\mathcal{F}_i)\n\\longrightarrow\nH^p(U, \\colim_i \\mathcal{F}_i)\n$$\nis an isomorphism for every $p \\geq 0$, every $U \\in \\mathcal{B}$, and\nevery filtered diagram $\\mathcal{I} \\to \\textit{Ab}(\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0739","source_file":"sites-cohomology.tex","source_line":2517,"source_end_line":2541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2517-L2541","statement_sha256":"9bf980f4d4bd568b1e6932b127590e4e49b461fdc9523e540e4b600251fe7a9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4452,"rank":4452,"depth":23,"x":1052.186,"y":765.663,"cluster":"sheaf-cohomology"},{"id":"stacks:0GN3","tag":"0GN3","title":"Cohomology and colimits · Lemma 0GN3","summary":"Let C be a site. Let S ⊂ Ob(Sh(C)) be a subset. Denote * the final object of Sh(C). Assume • for some K ∈ S the map K → * is surjective, • given a surjective map of sheaves F → K with K ∈ S there exists a K' ∈ S and a map K' → F such that the composition K' → K is surjective, • given K, K' ∈ S there is a surjection K\" → K × K' with K\" ∈ S, • given a, b : K → K' with K, K' ∈ S there exists a surjection K\" → Equalizer(a, b) with K\" ∈ S, and • every K ∈ S is quasi-compact…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $S \\subset \\Ob(\\Sh(\\mathcal{C}))$\nbe a subset. Denote $*$ the final object of $\\Sh(\\mathcal{C})$. Assume\n\\begin{enumerate}\n\\item for some $K \\in S$ the map $K \\to *$ is surjective,\n\\item given a surjective map of sheaves $\\mathcal{F} \\to K$ with $K \\in S$\nthere exists a $K' \\in S$ and a map $K' \\to \\mathcal{F}$ such\nthat the composition $K' \\to K$ is surjective,\n\\item given $K, K' \\in S$ there is a surjection $K'' \\to K \\times K'$\nwith $K'' \\in S$,\n\\item given $a, b : K \\to K'$ with $K, K' \\in S$ there exists a\nsurjection $K'' \\to \\text{Equalizer}(a, b)$ with $K'' \\in S$, and\n\\item every $K \\in S$ is quasi-compact\n(Sites, Definition \\ref{sites-definition-quasi-compact-topos}).\n\\end{enumerate}\nThen for all $p \\geq 0$ the map\n$$\n\\colim_\\lambda H^p(\\mathcal{C}, \\mathcal{F}_\\lambda)\n\\longrightarrow\nH^p(\\mathcal{C}, \\colim_\\lambda \\mathcal{F}_\\lambda)\n$$\nis an isomorphism for every filtered diagram\n$\\Lambda \\to \\textit{Ab}(\\mathcal{C})$, $\\lambda \\mapsto \\mathcal{F}_\\lambda$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GN3","source_file":"sites-cohomology.tex","source_line":2629,"source_end_line":2653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2629-L2653","statement_sha256":"7473284a71b5e74057bb07bc771d8b6477b8999e38c4ddb6211fb9ed206367f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4453,"rank":4453,"depth":24,"x":1118.472,"y":572.505,"cluster":"sheaf-cohomology"},{"id":"stacks:0H7B","tag":"0H7B","title":"Cohomology and colimits · Lemma 0H7B","summary":"Let f : C → D be a morphism of sites corresponding to the continuous functor u : D → C. Let (F_i, φ_ii') be a system of abelian sheaves on C. Set F = colim F_i. Let p ≥ 0 be an integer. Denote B the set of V ∈ Ob(D) such that H^p(u(V), F) = colim H^p(u(V), F_i). If every object of D has a covering by elements of B, then R^pf_*F = colim R^pf_*F_i.","statement_latex":"Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites\ncorresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nLet $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system of\nabelian sheaves on $\\mathcal{C}$. Set $\\mathcal{F} = \\colim \\mathcal{F}_i$.\nLet $p \\geq 0$ be an integer. Denote $\\mathcal{B}$ the set of\n$V \\in \\Ob(\\mathcal{D})$ such that\n$H^p(u(V), \\mathcal{F}) = \\colim H^p(u(V), \\mathcal{F}_i)$.\nIf every object of $\\mathcal{D}$ has a covering by elements of $\\mathcal{B}$,\nthen $R^pf_*\\mathcal{F} = \\colim R^pf_*\\mathcal{F}_i$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7B","source_file":"sites-cohomology.tex","source_line":2821,"source_end_line":2832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2821-L2832","statement_sha256":"2d3e68969d4ac494d837b5103bc8fcb15dcc74959f7a20dafc8b3607073f5fcc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4454,"rank":4454,"depth":2,"x":1225.099,"y":752.842,"cluster":"sheaf-cohomology"},{"id":"stacks:0EXZ","tag":"0EXZ","title":"Cohomology and colimits · Lemma 0EXZ","summary":"Let I be a cofiltered index category and let (C_i, f_a) be an inverse system of sites over I as in Sites, Situation [Tag 0A34]. Set C = colim C_i as in Sites, Lemmas [Tag 09YL] and [Tag 0A35]. Moreover, assume given • an abelian sheaf F_i on C_i for all i ∈ Ob(I), • for a : j → i a map φ_a : f_a^-1F_i → F_j of abelian sheaves on C_j such that φ_c = φ_b ∘ f_b^-1φ_a whenever c = a ∘ b. Then there exists a map of systems (F_i, φ_a) → (G_i, ψ_a) such that F_i → G_i is…","statement_latex":"Let $\\mathcal{I}$ be a cofiltered index category and let\n$(\\mathcal{C}_i, f_a)$ be an inverse system of sites over $\\mathcal{I}$\nas in Sites, Situation \\ref{sites-situation-inverse-limit-sites}.\nSet $\\mathcal{C} = \\colim \\mathcal{C}_i$ as in Sites,\nLemmas \\ref{sites-lemma-colimit-sites} and\n\\ref{sites-lemma-compute-pullback-to-limit}.\nMoreover, assume given\n\\begin{enumerate}\n\\item an abelian sheaf $\\mathcal{F}_i$ on $\\mathcal{C}_i$ for all\n$i \\in \\Ob(\\mathcal{I})$,\n\\item for $a : j \\to i$ a map\n$\\varphi_a : f_a^{-1}\\mathcal{F}_i \\to \\mathcal{F}_j$\nof abelian sheaves on $\\mathcal{C}_j$\n\\end{enumerate}\nsuch that $\\varphi_c = \\varphi_b \\circ f_b^{-1}\\varphi_a$\nwhenever $c = a \\circ b$. Then there exists a map of systems\n$(\\mathcal{F}_i, \\varphi_a) \\to (\\mathcal{G}_i, \\psi_a)$\nsuch that $\\mathcal{F}_i \\to \\mathcal{G}_i$ is injective and\n$\\mathcal{G}_i$ is an injective abelian sheaf.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXZ","source_file":"sites-cohomology.tex","source_line":2854,"source_end_line":2875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2854-L2875","statement_sha256":"9058c251c1db5b80fb6216472ff0e974962d17c3eb7f5aba0627bd3036c2c7e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4455,"rank":4455,"depth":9,"x":1001.089,"y":680.249,"cluster":"sheaf-cohomology"},{"id":"stacks:09YP","tag":"09YP","title":"Cohomology and colimits · Lemma 09YP","summary":"In the situation of Lemma [Tag 0EXZ] set F = colim f_i^-1F_i. Let i ∈ Ob(I), X_i ∈ Ob(C_i). Then colim_a : j → i H^p(u_a(X_i), F_j) = H^p(u_i(X_i), F) for all p ≥ 0.","statement_latex":"In the situation of Lemma \\ref{lemma-colim-sites-injective} set\n$\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$.\nLet $i \\in \\Ob(\\mathcal{I})$, $X_i \\in \\text{Ob}(\\mathcal{C}_i)$. Then\n$$\n\\colim_{a : j \\to i} H^p(u_a(X_i), \\mathcal{F}_j) =\nH^p(u_i(X_i), \\mathcal{F})\n$$\nfor all $p \\geq 0$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology and colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YP","source_file":"sites-cohomology.tex","source_line":2904,"source_end_line":2914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2904-L2914","statement_sha256":"7141f311ede9086c196e614db652b6a9cd1bf86d3df0d1724f7d095eb596a93c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4456,"rank":4456,"depth":24,"x":1225.009,"y":606.551,"cluster":"sheaf-cohomology"},{"id":"stacks:06YM","tag":"06YM","title":"Flat resolutions · Lemma 06YM","summary":"Let (C, O) be a ringed site. Let G^bullet be a complex of O-modules. The functors K(Mod(O)) → K(Mod(O)), F^bullet ↦ Tot(G^bullet ⊗_O F^bullet) and K(Mod(O)) → K(Mod(O)), F^bullet ↦ Tot(F^bullet ⊗_O G^bullet) are exact functors of triangulated categories.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{G}^\\bullet$ be a complex of $\\mathcal{O}$-modules.\nThe functors\n$$\nK(\\textit{Mod}(\\mathcal{O}))\n\\longrightarrow\nK(\\textit{Mod}(\\mathcal{O})),\n\\quad\n\\mathcal{F}^\\bullet \\longmapsto\n\\text{Tot}(\\mathcal{G}^\\bullet \\otimes_\\mathcal{O} \\mathcal{F}^\\bullet)\n$$\nand\n$$\nK(\\textit{Mod}(\\mathcal{O}))\n\\longrightarrow\nK(\\textit{Mod}(\\mathcal{O})),\n\\quad\n\\mathcal{F}^\\bullet \\longmapsto\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{G}^\\bullet)\n$$\nare exact functors of triangulated categories.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YM","source_file":"sites-cohomology.tex","source_line":2968,"source_end_line":2991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2968-L2991","statement_sha256":"c02d82fec1347b315ae9cae660cfccf490edd8e862d6322c70711d6914965b73","origin":"The Stacks Project","memory_eligible":false,"source_rank":4457,"rank":4457,"depth":0,"x":1118.99,"y":788.245,"cluster":"sheaf-cohomology"},{"id":"stacks:06YN","tag":"06YN","title":"Flat resolutions · Definition 06YN","summary":"Let (C, O) be a ringed site. A complex K^bullet of O-modules is called K-flat if for every acyclic complex F^bullet of O-modules the complex Tot(F^bullet ⊗_O K^bullet) is acyclic.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nA complex $\\mathcal{K}^\\bullet$ of $\\mathcal{O}$-modules is\ncalled {\\it K-flat} if for every acyclic complex $\\mathcal{F}^\\bullet$\nof $\\mathcal{O}$-modules the complex\n$$\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet)\n$$\nis acyclic.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YN","source_file":"sites-cohomology.tex","source_line":2998,"source_end_line":3008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L2998-L3008","statement_sha256":"81269234be4a14ecc39cfc9521e457b43ff463cc6359a2d16afc0121a27b00bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4458,"rank":4458,"depth":0,"x":1050.945,"y":593.797,"cluster":"sheaf-cohomology"},{"id":"stacks:06YP","tag":"06YP","title":"Flat resolutions · Lemma 06YP","summary":"Let (C, O) be a ringed site. Let K^bullet be a K-flat complex. Then the functor K(Mod(O)) → K(Mod(O)), F^bullet ↦ Tot(F^bullet ⊗_O K^bullet) transforms quasi-isomorphisms into quasi-isomorphisms.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{K}^\\bullet$ be a K-flat complex.\nThen the functor\n$$\nK(\\textit{Mod}(\\mathcal{O}))\n\\longrightarrow\nK(\\textit{Mod}(\\mathcal{O})), \\quad\n\\mathcal{F}^\\bullet\n\\longmapsto\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet)\n$$\ntransforms quasi-isomorphisms into quasi-isomorphisms.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YP","source_file":"sites-cohomology.tex","source_line":3010,"source_end_line":3024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3010-L3024","statement_sha256":"c197a7ffd5880f67afae627fe17e3b6769193ff15171bbfc3a7241a86db2ba49","origin":"The Stacks Project","memory_eligible":false,"source_rank":4459,"rank":4459,"depth":1,"x":1257.82,"y":698.737,"cluster":"sheaf-cohomology"},{"id":"stacks:0E8K","tag":"0E8K","title":"Flat resolutions · Lemma 0E8K","summary":"Let (C, O) be a ringed site. Let U be an object of C. If K^bullet is a K-flat complex of O-modules, then K^bullet|_U is a K-flat complex of O_U-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $U$ be an object of $\\mathcal{C}$.\nIf $\\mathcal{K}^\\bullet$ is a K-flat complex of $\\mathcal{O}$-modules, then\n$\\mathcal{K}^\\bullet|_U$ is a K-flat complex of $\\mathcal{O}_U$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8K","source_file":"sites-cohomology.tex","source_line":3033,"source_end_line":3039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3033-L3039","statement_sha256":"544e36954a95142631ee15a045a1f9fc4afab5c1b8957ca5b1530ce4e43d57e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4460,"rank":4460,"depth":5,"x":1020.505,"y":738.805,"cluster":"sheaf-cohomology"},{"id":"stacks:07A2","tag":"07A2","title":"Flat resolutions · Lemma 07A2","summary":"Let (C, O) be a ringed site. If K^bullet, L^bullet are K-flat complexes of O-modules, then Tot(K^bullet ⊗_O L^bullet) is a K-flat complex of O-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nIf $\\mathcal{K}^\\bullet$, $\\mathcal{L}^\\bullet$ are K-flat complexes\nof $\\mathcal{O}$-modules, then\n$\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet)$\nis a K-flat complex of $\\mathcal{O}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07A2","source_file":"sites-cohomology.tex","source_line":3059,"source_end_line":3066,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3059-L3066","statement_sha256":"bb9205dd6fdc66177291195b1c236460698200eb515a6ea94b7fde29eca3468e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4461,"rank":4461,"depth":0,"x":1163.503,"y":574.34,"cluster":"sheaf-cohomology"},{"id":"stacks:07A3","tag":"07A3","title":"Flat resolutions · Lemma 07A3","summary":"Let (C, O) be a ringed site. Let (K_1^bullet, K_2^bullet, K_3^bullet) be a distinguished triangle in K(Mod(O)). If two out of three of K_i^bullet are K-flat, so is the third.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{K}_1^\\bullet, \\mathcal{K}_2^\\bullet, \\mathcal{K}_3^\\bullet)$\nbe a distinguished triangle in $K(\\textit{Mod}(\\mathcal{O}))$.\nIf two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07A3","source_file":"sites-cohomology.tex","source_line":3080,"source_end_line":3086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3080-L3086","statement_sha256":"ff455ce5ff420ee9d392faafb569ea08f23c6065ad2895377aa18dbe7d468308","origin":"The Stacks Project","memory_eligible":false,"source_rank":4462,"rank":4462,"depth":1,"x":1190.359,"y":777.076,"cluster":"sheaf-cohomology"},{"id":"stacks:0G7B","tag":"0G7B","title":"Flat resolutions · Lemma 0G7B","summary":"Let (C, O) be a ringed site. Let 0 → K_1^bullet → K_2^bullet → K_3^bullet → 0 be a short exact sequence of complexes such that the terms of K_3^bullet are flat O-modules. If two out of three of K_i^bullet are K-flat, so is the third.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$0 \\to \\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to\n\\mathcal{K}_3^\\bullet \\to 0$ be a short exact sequence of complexes\nsuch that the terms of $\\mathcal{K}_3^\\bullet$ are flat $\\mathcal{O}$-modules.\nIf two out of three of $\\mathcal{K}_i^\\bullet$ are K-flat, so is the third.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7B","source_file":"sites-cohomology.tex","source_line":3096,"source_end_line":3103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3096-L3103","statement_sha256":"edc2e5998ffce0ab24e55800d2d2932fe524d942470a07cd94bfc5c3c189d67f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4463,"rank":4463,"depth":6,"x":1007.233,"y":642.608,"cluster":"sheaf-cohomology"},{"id":"stacks:06YQ","tag":"06YQ","title":"Flat resolutions · Lemma 06YQ","summary":"Let (C, O) be a ringed site. A bounded above complex of flat O-modules is K-flat.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A bounded above complex\nof flat $\\mathcal{O}$-modules is K-flat.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YQ","source_file":"sites-cohomology.tex","source_line":3119,"source_end_line":3123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3119-L3123","statement_sha256":"9f077663802d11f79f1f87ab65fd392b5b8c6fca8681def2bf64e076c9c72cb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4464,"rank":4464,"depth":5,"x":1250.785,"y":637.845,"cluster":"sheaf-cohomology"},{"id":"stacks:06YR","tag":"06YR","title":"Flat resolutions · Lemma 06YR","summary":"Let (C, O) be a ringed site. Let K_1^bullet → K_2^bullet → … be a system of K-flat complexes. Then colim_i K_i^bullet is K-flat.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{K}_1^\\bullet \\to \\mathcal{K}_2^\\bullet \\to \\ldots$\nbe a system of K-flat complexes.\nThen $\\colim_i \\mathcal{K}_i^\\bullet$ is K-flat.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YR","source_file":"sites-cohomology.tex","source_line":3151,"source_end_line":3157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3151-L3157","statement_sha256":"f00f55fb2c65c8ddb71e4c571b443b40ec6d80fd70d411499b95a5c027c58658","origin":"The Stacks Project","memory_eligible":false,"source_rank":4465,"rank":4465,"depth":0,"x":1074.748,"y":779.777,"cluster":"sheaf-cohomology"},{"id":"stacks:077J","tag":"077J","title":"Flat resolutions · Lemma 077J","summary":"Let (C, O) be a ringed site. For any complex G^bullet of O-modules there exists a commutative diagram of complexes of O-modules xymatrix K_1^bullet ar[d] ar[r] & K_2^bullet ar[d] ar[r] & … τ_≤ 1G^bullet ar[r] & τ_≤ 2G^bullet ar[r] & … with the following properties: (1) the vertical arrows are quasi-isomorphisms and termwise surjective, (2) each K_n^bullet is a bounded above complex whose terms are direct sums of O-modules of the form j_U!O_U, and (3) the maps K_n^bullet →…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nFor any complex $\\mathcal{G}^\\bullet$ of $\\mathcal{O}$-modules\nthere exists a commutative diagram of complexes of $\\mathcal{O}$-modules\n$$\n\\xymatrix{\n\\mathcal{K}_1^\\bullet \\ar[d] \\ar[r] &\n\\mathcal{K}_2^\\bullet \\ar[d] \\ar[r] & \\ldots \\\\\n\\tau_{\\leq 1}\\mathcal{G}^\\bullet \\ar[r] &\n\\tau_{\\leq 2}\\mathcal{G}^\\bullet \\ar[r] & \\ldots\n}\n$$\nwith the following properties: (1) the vertical arrows are quasi-isomorphisms\nand termwise surjective,\n(2) each $\\mathcal{K}_n^\\bullet$ is a bounded above complex whose terms\nare direct sums of $\\mathcal{O}$-modules of the form $j_{U!}\\mathcal{O}_U$, and\n(3) the maps $\\mathcal{K}_n^\\bullet \\to \\mathcal{K}_{n + 1}^\\bullet$ are\ntermwise split injections whose cokernels are direct sums of\n$\\mathcal{O}$-modules of the form $j_{U!}\\mathcal{O}_U$. Moreover, the map\n$\\colim \\mathcal{K}_n^\\bullet \\to \\mathcal{G}^\\bullet$ is a quasi-isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077J","source_file":"sites-cohomology.tex","source_line":3172,"source_end_line":3193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3172-L3193","statement_sha256":"7edadaee4b16238ab3c442df5ce3b87cc1b8b37d2f93259fd67444a3b7eadd17","origin":"The Stacks Project","memory_eligible":false,"source_rank":4466,"rank":4466,"depth":14,"x":1090.442,"y":574.911,"cluster":"sheaf-cohomology"},{"id":"stacks:06YS","tag":"06YS","title":"Flat resolutions · Lemma 06YS","summary":"Let (C, O) be a ringed site. For any complex G^bullet there exists a K-flat complex K^bullet whose terms are flat O-modules and a quasi-isomorphism K^bullet → G^bullet which is termwise surjective.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. For any complex\n$\\mathcal{G}^\\bullet$ there exists a $K$-flat complex $\\mathcal{K}^\\bullet$\nwhose terms are flat $\\mathcal{O}$-modules and a quasi-isomorphism\n$\\mathcal{K}^\\bullet \\to \\mathcal{G}^\\bullet$ which is termwise surjective.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YS","source_file":"sites-cohomology.tex","source_line":3206,"source_end_line":3212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3206-L3212","statement_sha256":"c19da130ec3ebadb91e05167bfa155fe7a70b300d740c6180309856d0c0e7e89","origin":"The Stacks Project","memory_eligible":false,"source_rank":4467,"rank":4467,"depth":15,"x":1243.856,"y":735.131,"cluster":"sheaf-cohomology"},{"id":"stacks:06YT","tag":"06YT","title":"Flat resolutions · Lemma 06YT","summary":"Let (C, O) be a ringed site. Let α : P^bullet → Q^bullet be a quasi-isomorphism of K-flat complexes of O-modules. For every complex F^bullet of O-modules the induced map Tot(id_F^bullet ⊗ α) : Tot(F^bullet ⊗_O P^bullet) → Tot(F^bullet ⊗_O Q^bullet) is a quasi-isomorphism.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$\\alpha : \\mathcal{P}^\\bullet \\to \\mathcal{Q}^\\bullet$ be a\nquasi-isomorphism of K-flat complexes of $\\mathcal{O}$-modules.\nFor every complex $\\mathcal{F}^\\bullet$ of $\\mathcal{O}$-modules\nthe induced map\n$$\n\\text{Tot}(\\text{id}_{\\mathcal{F}^\\bullet} \\otimes \\alpha) :\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{P}^\\bullet)\n\\longrightarrow\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{Q}^\\bullet)\n$$\nis a quasi-isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YT","source_file":"sites-cohomology.tex","source_line":3232,"source_end_line":3246,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3232-L3246","statement_sha256":"871c7e9172de925cd152d5d55897819ec51e091e4576a295802f5b48cf004190","origin":"The Stacks Project","memory_eligible":false,"source_rank":4468,"rank":4468,"depth":16,"x":1001.51,"y":703.989,"cluster":"sheaf-cohomology"},{"id":"stacks:06YU","tag":"06YU","title":"Flat resolutions · Definition 06YU","summary":"Let (C, O) be a ringed site. Let F^bullet be an object of D(O). The derived tensor product - ⊗_O^L F^bullet : D(O) → D(O) is the exact functor of triangulated categories described above.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}^\\bullet$ be an object of $D(\\mathcal{O})$.\nThe {\\it derived tensor product}\n$$\n- \\otimes_\\mathcal{O}^{\\mathbf{L}} \\mathcal{F}^\\bullet :\nD(\\mathcal{O})\n\\longrightarrow\nD(\\mathcal{O})\n$$\nis the exact functor of triangulated categories described above.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YU","source_file":"sites-cohomology.tex","source_line":3299,"source_end_line":3311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3299-L3311","statement_sha256":"bbdfb844057681bdbd09508aa7cf2fa21f755fb5e52d41455a5868019d0b11ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":4469,"rank":4469,"depth":0,"x":1205.572,"y":589.263,"cluster":"sheaf-cohomology"},{"id":"stacks:08FF","tag":"08FF","title":"Flat resolutions · Definition 08FF","summary":"Let (C, O) be a ringed site. Let F, G be O-modules. The Tor's of F and G are defined by the formula Tor_p^O(F, G) = H^-p(F ⊗_O^L G) with derived tensor product as defined above.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be $\\mathcal{O}$-modules.\nThe {\\it Tor}'s of $\\mathcal{F}$ and $\\mathcal{G}$ are defined by\nthe formula\n$$\n\\text{Tor}_p^\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) =\nH^{-p}(\\mathcal{F} \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{G})\n$$\nwith derived tensor product as defined above.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FF","source_file":"sites-cohomology.tex","source_line":3328,"source_end_line":3339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3328-L3339","statement_sha256":"5b85e81c8aa72f308be11260231786e83ecc459de26763bdfc846c8d77c5f5a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4470,"rank":4470,"depth":0,"x":1147.27,"y":789.959,"cluster":"sheaf-cohomology"},{"id":"stacks:08FG","tag":"08FG","title":"Flat resolutions · Lemma 08FG","summary":"Let (C, O) be a ringed site. Let F be an O-module. The following are equivalent • F is a flat O-module, and • Tor_1^O(F, G) = 0 for every O-module G.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{F}$ be an $\\mathcal{O}$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a flat $\\mathcal{O}$-module, and\n\\item $\\text{Tor}_1^\\mathcal{O}(\\mathcal{F}, \\mathcal{G}) = 0$\nfor every $\\mathcal{O}$-module $\\mathcal{G}$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FG","source_file":"sites-cohomology.tex","source_line":3359,"source_end_line":3369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3359-L3369","statement_sha256":"055461713280070b818216932b5d291de3c922db438f281ac9fa98eb77376782","origin":"The Stacks Project","memory_eligible":false,"source_rank":4471,"rank":4471,"depth":0,"x":1028.684,"y":608.608,"cluster":"sheaf-cohomology"},{"id":"stacks:0G7C","tag":"0G7C","title":"Flat resolutions · Lemma 0G7C","summary":"Let (C, O) be a ringed site. Let K^bullet be a K-flat, acyclic complex with flat terms. Then F = Ker(K^n → K^n + 1) is a flat O-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{K}^\\bullet$\nbe a K-flat, acyclic complex with flat terms. Then\n$\\mathcal{F} = \\Ker(\\mathcal{K}^n \\to \\mathcal{K}^{n + 1})$\nis a flat $\\mathcal{O}$-module.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7C","source_file":"sites-cohomology.tex","source_line":3384,"source_end_line":3390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3384-L3390","statement_sha256":"296dbebfb606d5317d8566ed74c655270c1e881fab0b0c319dfe2a0398b94a93","origin":"The Stacks Project","memory_eligible":false,"source_rank":4472,"rank":4472,"depth":6,"x":1262.323,"y":675.147,"cluster":"sheaf-cohomology"},{"id":"stacks:0G7D","tag":"0G7D","title":"Flat resolutions · Lemma 0G7D","summary":"Let (C, O) be a ringed site. Let a : K^bullet → L^bullet be a map of complexes of O-modules. If K^bullet is K-flat, then there exist a complex N^bullet and maps of complexes b : K^bullet → N^bullet and c : N^bullet → L^bullet such that • N^bullet is K-flat, • c is a quasi-isomorphism, • a is homotopic to c ∘ b. If the terms of K^bullet are flat, then we may choose N^bullet, b, and c such that the same is true for N^bullet.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $a : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$ be a map of complexes\nof $\\mathcal{O}$-modules. If $\\mathcal{K}^\\bullet$ is K-flat, then\nthere exist a complex $\\mathcal{N}^\\bullet$ and maps of complexes\n$b : \\mathcal{K}^\\bullet \\to \\mathcal{N}^\\bullet$\nand $c : \\mathcal{N}^\\bullet \\to \\mathcal{L}^\\bullet$ such that\n\\begin{enumerate}\n\\item $\\mathcal{N}^\\bullet$ is K-flat,\n\\item $c$ is a quasi-isomorphism,\n\\item $a$ is homotopic to $c \\circ b$.\n\\end{enumerate}\nIf the terms of $\\mathcal{K}^\\bullet$ are flat, then we may choose\n$\\mathcal{N}^\\bullet$, $b$, and $c$\nsuch that the same is true for $\\mathcal{N}^\\bullet$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7D","source_file":"sites-cohomology.tex","source_line":3417,"source_end_line":3433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3417-L3433","statement_sha256":"68dd90e2c73ed97d31fea745d3a9036aa56f2aa2a292fc7556433a219ca1abfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4473,"rank":4473,"depth":16,"x":1036.187,"y":758.782,"cluster":"sheaf-cohomology"},{"id":"stacks:0G7E","tag":"0G7E","title":"Derived pullback · Lemma 0G7E","summary":"Let f : (Sh(C'), O') → (Sh(C), O) be a morphism of ringed topoi. Let K^bullet be a K-flat complex of O-modules whose terms are flat O-modules. Then f^*K^bullet is a K-flat complex of O'-modules whose terms are flat O'-modules.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nbe a morphism of ringed topoi. Let $\\mathcal{K}^\\bullet$ be a K-flat complex\nof $\\mathcal{O}$-modules whose terms are flat $\\mathcal{O}$-modules. Then\n$f^*\\mathcal{K}^\\bullet$ is a K-flat complex of $\\mathcal{O}'$-modules whose\nterms are flat $\\mathcal{O}'$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7E","source_file":"sites-cohomology.tex","source_line":3507,"source_end_line":3514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3507-L3514","statement_sha256":"269024e561f0af0bd00cc1481b7d55afe556223b326e9cec468ce5a43caa6353","origin":"The Stacks Project","memory_eligible":false,"source_rank":4474,"rank":4474,"depth":15,"x":1135.831,"y":568.506,"cluster":"sheaf-cohomology"},{"id":"stacks:06YY","tag":"06YY","title":"Derived pullback · Lemma 06YY","summary":"Let f : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. There exists an exact functor Lf^* : D(O') → D(O) of triangulated categories so that Lf^*K^bullet = f^*K^bullet for any K-flat complex K^bullet with flat terms and in particular for any bounded above complex of flat O'-modules.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi. There exists an exact functor\n$$\nLf^* : D(\\mathcal{O}') \\longrightarrow D(\\mathcal{O})\n$$\nof triangulated categories so that\n$Lf^*\\mathcal{K}^\\bullet = f^*\\mathcal{K}^\\bullet$ for any\nK-flat complex $\\mathcal{K}^\\bullet$ with flat terms and\nin particular for any bounded above complex of flat $\\mathcal{O}'$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06YY","source_file":"sites-cohomology.tex","source_line":3584,"source_end_line":3595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3584-L3595","statement_sha256":"ed5096950dc17a2c7faef3173e9369d16bdc4a2cb7213e70e3aa695b59e6d8ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":4475,"rank":4475,"depth":17,"x":1215.49,"y":765.653,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6D","tag":"0D6D","title":"Derived pullback · Lemma 0D6D","summary":"Consider morphisms of ringed topoi f : (Sh(C), O_C) → (Sh(D), O_D) and g : (Sh(D), O_D) → (Sh(E), O_E). Then Lf^* ∘ Lg^* = L(g ∘ f)^* as functors D(O_E) → D(O_C).","statement_latex":"Consider morphisms of ringed topoi\n$f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nand\n$g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to\n(\\Sh(\\mathcal{E}), \\mathcal{O}_\\mathcal{E})$.\nThen $Lf^* \\circ Lg^* = L(g \\circ f)^*$ as functors\n$D(\\mathcal{O}_\\mathcal{E}) \\to D(\\mathcal{O}_\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6D","source_file":"sites-cohomology.tex","source_line":3657,"source_end_line":3667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3657-L3667","statement_sha256":"3be1832f7617d1b7de1436a212a0d19678f1ba77c9a3091cd3406a1de0787ddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4476,"rank":4476,"depth":18,"x":997.882,"y":665.328,"cluster":"sheaf-cohomology"},{"id":"stacks:07A4","tag":"07A4","title":"Derived pullback · Lemma 07A4","summary":"Let f : (Sh(C), O) → (Sh(D), O') be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism Lf^*( F^bullet ⊗_O'^L G^bullet ) = Lf^*F^bullet ⊗_O^L Lf^*G^bullet for F^bullet, G^bullet ∈ Ob(D(O')).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a morphism of ringed topoi.\nThere is a canonical bifunctorial isomorphism\n$$\nLf^*(\n\\mathcal{F}^\\bullet \\otimes_{\\mathcal{O}'}^{\\mathbf{L}} \\mathcal{G}^\\bullet\n) =\nLf^*\\mathcal{F}^\\bullet \n\\otimes_{\\mathcal{O}}^{\\mathbf{L}}\nLf^*\\mathcal{G}^\\bullet \n$$\nfor $\\mathcal{F}^\\bullet, \\mathcal{G}^\\bullet \\in \\Ob(D(\\mathcal{O}'))$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07A4","source_file":"sites-cohomology.tex","source_line":3683,"source_end_line":3697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3683-L3697","statement_sha256":"4bd9b76d4b1f9e9796237c4c86fe0184124e1fa6330216963a881e205c439bea","origin":"The Stacks Project","memory_eligible":false,"source_rank":4477,"rank":4477,"depth":18,"x":1239.386,"y":615.755,"cluster":"sheaf-cohomology"},{"id":"stacks:08I6","tag":"08I6","title":"Derived pullback · Lemma 08I6","summary":"Let f : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism F^bullet ⊗_O^L Lf^*G^bullet = F^bullet ⊗_f^-1O'^L f^-1G^bullet for F^bullet in D(O) and G^bullet in D(O').","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi. There is a canonical bifunctorial\nisomorphism\n$$\n\\mathcal{F}^\\bullet\n\\otimes_\\mathcal{O}^{\\mathbf{L}}\nLf^*\\mathcal{G}^\\bullet\n=\n\\mathcal{F}^\\bullet \n\\otimes_{f^{-1}\\mathcal{O}'}^{\\mathbf{L}}\nf^{-1}\\mathcal{G}^\\bullet \n$$\nfor $\\mathcal{F}^\\bullet$ in $D(\\mathcal{O})$ and\n$\\mathcal{G}^\\bullet$ in $D(\\mathcal{O}')$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08I6","source_file":"sites-cohomology.tex","source_line":3727,"source_end_line":3743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3727-L3743","statement_sha256":"24ea0310682fd0c950977dc48969e9259cc9dd414a70e909ed7492456a412474","origin":"The Stacks Project","memory_eligible":false,"source_rank":4478,"rank":4478,"depth":0,"x":1100.961,"y":789.606,"cluster":"sheaf-cohomology"},{"id":"stacks:0DEN","tag":"0DEN","title":"Derived pullback · Lemma 0DEN","summary":"Let (C, O) be a ringed site. Let K^bullet be a complex of O-modules. • If K^bullet is K-flat, then for every point p of the site C the complex of O_p-modules K_p^bullet is K-flat in the sense of More on Algebra, Definition [Tag 06XZ] • If C has enough points, then the converse is true.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{K}^\\bullet$ be a complex of $\\mathcal{O}$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{K}^\\bullet$ is K-flat, then for every point $p$\nof the site $\\mathcal{C}$ the complex of $\\mathcal{O}_p$-modules\n$\\mathcal{K}_p^\\bullet$ is K-flat in the sense of\nMore on Algebra, Definition \\ref{more-algebra-definition-K-flat}\n\\item If $\\mathcal{C}$ has enough points, then the converse is true.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEN","source_file":"sites-cohomology.tex","source_line":3766,"source_end_line":3777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3766-L3777","statement_sha256":"c8713f9ed8c195855aa29e20863edc3781aa7bca6760351d0d99069689f01375","origin":"The Stacks Project","memory_eligible":false,"source_rank":4479,"rank":4479,"depth":16,"x":1063.171,"y":582.555,"cluster":"sheaf-cohomology"},{"id":"stacks:0DEP","tag":"0DEP","title":"Derived pullback · Lemma 0DEP","summary":"Let f : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. If C has enough points, then the pullback of a K-flat complex of O'-modules is a K-flat complex of O-modules.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi. If $\\mathcal{C}$ has enough points, then\nthe pullback of a K-flat complex of\n$\\mathcal{O}'$-modules is a K-flat complex of $\\mathcal{O}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEP","source_file":"sites-cohomology.tex","source_line":3833,"source_end_line":3839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3833-L3839","statement_sha256":"1c0ebee16fd51eaa4d272b9fe2ab10d667a44c9346b07e9c9533a4cb5c58a844","origin":"The Stacks Project","memory_eligible":false,"source_rank":4480,"rank":4480,"depth":17,"x":1257.832,"y":713.985,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPH","tag":"0FPH","title":"Derived pullback · Lemma 0FPH","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let K^bullet and M^bullet be complexes of O_D-modules. The diagram xymatrix Lf^*(K^bullet ⊗_O_D^L M^bullet) ar[r] ar[d] & Lf^*Tot(K^bullet ⊗_O_D M^bullet) ar[d] Lf^*K^bullet ⊗_O_C^L Lf^*M^bullet ar[d] & f^*Tot(K^bullet ⊗_O_D M^bullet) ar[d] f^*K^bullet ⊗_O_C^L f^*M^bullet ar[r] & Tot(f^*K^bullet ⊗_O_C f^*M^bullet) commutes.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nLet $\\mathcal{K}^\\bullet$ and $\\mathcal{M}^\\bullet$\nbe complexes of $\\mathcal{O}_\\mathcal{D}$-modules.\nThe diagram\n$$\n\\xymatrix{\nLf^*(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}\n\\mathcal{M}^\\bullet) \\ar[r] \\ar[d] &\nLf^*\\text{Tot}(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_\\mathcal{D}}\n\\mathcal{M}^\\bullet) \\ar[d] \\\\\nLf^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L}\nLf^*\\mathcal{M}^\\bullet \\ar[d] &\nf^*\\text{Tot}(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_\\mathcal{D}}\n\\mathcal{M}^\\bullet) \\ar[d] \\\\\nf^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L}\nf^*\\mathcal{M}^\\bullet \\ar[r] &\n\\text{Tot}(f^*\\mathcal{K}^\\bullet \\otimes_{\\mathcal{O}_\\mathcal{C}}\nf^*\\mathcal{M}^\\bullet)\n}\n$$\ncommutes.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPH","source_file":"sites-cohomology.tex","source_line":3848,"source_end_line":3875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3848-L3875","statement_sha256":"891f40b27a7195ec18a51d3e02f37464796f046ca9d992377b2ae64cfb339ffd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4481,"rank":4481,"depth":18,"x":1008.228,"y":727.546,"cluster":"sheaf-cohomology"},{"id":"stacks:07A6","tag":"07A6","title":"Cohomology of unbounded complexes · Lemma 07A6","summary":"Let f : (Sh(C), O) → (Sh(D), O') be a morphism of ringed topoi. The functor Rf_* defined above and the functor Lf^* defined in Lemma [Tag 06YY] are adjoint: Hom_D(O)(Lf^*G^bullet, F^bullet) = Hom_D(O')(G^bullet, Rf_*F^bullet) bifunctorially in F^bullet ∈ Ob(D(O)) and G^bullet ∈ Ob(D(O')).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a morphism of ringed topoi.\nThe functor $Rf_*$ defined above and\nthe functor $Lf^*$ defined in\nLemma \\ref{lemma-derived-base-change} are adjoint:\n$$\n\\Hom_{D(\\mathcal{O})}(Lf^*\\mathcal{G}^\\bullet, \\mathcal{F}^\\bullet)\n=\n\\Hom_{D(\\mathcal{O}')}(\\mathcal{G}^\\bullet, Rf_*\\mathcal{F}^\\bullet)\n$$\nbifunctorially in $\\mathcal{F}^\\bullet \\in \\Ob(D(\\mathcal{O}))$ and\n$\\mathcal{G}^\\bullet \\in \\Ob(D(\\mathcal{O}'))$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07A6","source_file":"sites-cohomology.tex","source_line":3989,"source_end_line":4003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L3989-L4003","statement_sha256":"9103d340b11487448e0589c4377d8cb9cd0c51658b70feef80c7d4b1dc0b9232","origin":"The Stacks Project","memory_eligible":false,"source_rank":4482,"rank":4482,"depth":18,"x":1181.634,"y":575.689,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6E","tag":"0D6E","title":"Cohomology of unbounded complexes · Lemma 0D6E","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) and g : (Sh(D), O_D) → (Sh(E), O_E) be morphisms of ringed topoi. Then Rg_* ∘ Rf_* = R(g ∘ f)_* as functors D(O_C) → D(O_E).","statement_latex":"Let\n$f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nand\n$g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to\n(\\Sh(\\mathcal{E}), \\mathcal{O}_\\mathcal{E})$\nbe morphisms of ringed topoi.\nThen $Rg_* \\circ Rf_* = R(g \\circ f)_*$ as functors\n$D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_\\mathcal{E})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6E","source_file":"sites-cohomology.tex","source_line":4010,"source_end_line":4021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4010-L4021","statement_sha256":"238ff050d7fcd2a07601becbb3228786e17f32e3b25b60be344e0106cc195a08","origin":"The Stacks Project","memory_eligible":false,"source_rank":4483,"rank":4483,"depth":19,"x":1175.878,"y":786.373,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPI","tag":"0FPI","title":"Cohomology of unbounded complexes · Lemma 0FPI","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let K^bullet be a complex of O_C-modules. The diagram xymatrix Lf^*f_*K^bullet ar[r] ar[d] & f^*f_*K^bullet ar[d] Lf^*Rf_*K^bullet ar[r] & K^bullet coming from Lf^* → f^* on complexes, f_* → Rf_* on complexes, and adjunction Lf^* ∘ Rf_* → id commutes in D(O_C).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nLet $\\mathcal{K}^\\bullet$\nbe a complex of $\\mathcal{O}_\\mathcal{C}$-modules.\nThe diagram\n$$\n\\xymatrix{\nLf^*f_*\\mathcal{K}^\\bullet \\ar[r] \\ar[d] &\nf^*f_*\\mathcal{K}^\\bullet \\ar[d] \\\\\nLf^*Rf_*\\mathcal{K}^\\bullet \\ar[r] &\n\\mathcal{K}^\\bullet\n}\n$$\ncoming from $Lf^* \\to f^*$ on complexes, $f_* \\to Rf_*$ on complexes,\nand adjunction $Lf^* \\circ Rf_* \\to \\text{id}$\ncommutes in $D(\\mathcal{O}_\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPI","source_file":"sites-cohomology.tex","source_line":4128,"source_end_line":4146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4128-L4146","statement_sha256":"340e8bcdbe64f5e6f4b04d97702b552c5f017b5ab3d4a99dce89a24664296945","origin":"The Stacks Project","memory_eligible":false,"source_rank":4484,"rank":4484,"depth":18,"x":1010.452,"y":627.517,"cluster":"sheaf-cohomology"},{"id":"stacks:0DD7","tag":"0DD7","title":"Cohomology of unbounded complexes · Lemma 0DD7","summary":"Let C be a site. Let A ⊂ Ab(C) denote the Serre subcategory consisting of torsion abelian sheaves. Then the functor D(A) → D_A(C) is an equivalence.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\subset \\textit{Ab}(\\mathcal{C})$\ndenote the Serre subcategory consisting of torsion abelian sheaves.\nThen the functor $D(\\mathcal{A}) \\to D_\\mathcal{A}(\\mathcal{C})$\nis an equivalence.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology of unbounded complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DD7","source_file":"sites-cohomology.tex","source_line":4217,"source_end_line":4223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4217-L4223","statement_sha256":"ca6aabcaa1c9aba367fcf75054246075cb9c617144d394a137ebd95f66b82ad4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4485,"rank":4485,"depth":7,"x":1260.55,"y":650.823,"cluster":"sheaf-cohomology"},{"id":"stacks:08FI","tag":"08FI","title":"Some properties of K-injective complexes · Lemma 08FI","summary":"Let (C, O) be a ringed site. Let U be an object of C. The restriction of a K-injective complex of O-modules to C/U is a K-injective complex of O_U-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of\n$\\mathcal{C}$. The restriction of a K-injective complex of\n$\\mathcal{O}$-modules to $\\mathcal{C}/U$ is a K-injective complex of\n$\\mathcal{O}_U$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FI","source_file":"sites-cohomology.tex","source_line":4313,"source_end_line":4319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4313-L4319","statement_sha256":"e9000a70fbb134b1e7a18b3d4533a9f1104dafac50389ce43947329280a070e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4486,"rank":4486,"depth":1,"x":1057.091,"y":775.732,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6F","tag":"0D6F","title":"Some properties of K-injective complexes · Lemma 0D6F","summary":"Let (C, O) be a ringed site. Let U ∈ Ob(C). For K in D(O) we have H^p(U, K) = H^p(C/U, K|_C/U).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U \\in \\Ob(\\mathcal{C})$.\nFor $K$ in $D(\\mathcal{O})$ we have\n$H^p(U, K) = H^p(\\mathcal{C}/U, K|_{\\mathcal{C}/U})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6F","source_file":"sites-cohomology.tex","source_line":4328,"source_end_line":4333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4328-L4333","statement_sha256":"241fdc5f97d449e2aa9b3331e899192cea93abfebb34230cfcd7831dbbdafef4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4487,"rank":4487,"depth":2,"x":1106.743,"y":567.874,"cluster":"sheaf-cohomology"},{"id":"stacks:0BKV","tag":"0BKV","title":"Some properties of K-injective complexes · Lemma 0BKV","summary":"Let (C, O) be a ringed site. Let K be an object of D(O). The sheafification of U ↦ H^q(U, K) = H^q(C/U, K|_C/U) is the qth cohomology sheaf H^q(K) of K.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$ be an object of\n$D(\\mathcal{O})$. The sheafification of\n$$\nU \\mapsto H^q(U, K) = H^q(\\mathcal{C}/U, K|_{\\mathcal{C}/U})\n$$\nis the $q$th cohomology sheaf $H^q(K)$ of $K$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKV","source_file":"sites-cohomology.tex","source_line":4348,"source_end_line":4356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4348-L4356","statement_sha256":"3226df9b124abb627089945711b7e8ce76dc7fe84525554654f112b6ac3d240c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4488,"rank":4488,"depth":3,"x":1237.475,"y":749.585,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6G","tag":"0D6G","title":"Some properties of K-injective complexes · Lemma 0D6G","summary":"Let f : (C, O_C) → (D, O_D) be a morphism of ringed sites corresponding to the continuous functor u : D → C. Given V ∈ D, set U = u(V) and denote g : (C/U, O_U) → (D/V, O_V) the induced morphism of ringed sites (Modules on Sites, Lemma [Tag 04J0]). Then (Rf_*E)|_D/V = Rg_*(E|_C/U) for E in D(O_C).","statement_latex":"Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites\ncorresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nGiven $V \\in \\mathcal{D}$, set $U = u(V)$ and denote\n$g : (\\mathcal{C}/U, \\mathcal{O}_U) \\to (\\mathcal{D}/V, \\mathcal{O}_V)$\nthe induced morphism of ringed sites\n(Modules on Sites, Lemma\n\\ref{sites-modules-lemma-localize-morphism-ringed-sites}).\nThen $(Rf_*E)|_{\\mathcal{D}/V} = Rg_*(E|_{\\mathcal{C}/U})$\nfor $E$ in $D(\\mathcal{O}_\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6G","source_file":"sites-cohomology.tex","source_line":4373,"source_end_line":4385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4373-L4385","statement_sha256":"483ca7063eff94454e051351c2c5b6f36762fa6e3652c68b023341f2bd3b6a89","origin":"The Stacks Project","memory_eligible":false,"source_rank":4489,"rank":4489,"depth":10,"x":994.595,"y":689.687,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6H","tag":"0D6H","title":"Some properties of K-injective complexes · Lemma 0D6H","summary":"Let f : (C, O_C) → (D, O_D) be a morphism of ringed sites corresponding to the continuous functor u : D → C. Then RΓ(D, -) ∘ Rf_* = RΓ(C, -) as functors D(O_C) → D(Γ(O_D)). More generally, for V ∈ D with U = u(V) we have RΓ(U, -) = RΓ(V, -) ∘ Rf_*.","statement_latex":"Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites\ncorresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nThen $R\\Gamma(\\mathcal{D}, -) \\circ Rf_* = R\\Gamma(\\mathcal{C}, -)$ as\nfunctors $D(\\mathcal{O}_\\mathcal{C}) \\to D(\\Gamma(\\mathcal{O}_\\mathcal{D}))$.\nMore generally, for $V \\in \\mathcal{D}$ with $U = u(V)$\nwe have $R\\Gamma(U, -) = R\\Gamma(V, -) \\circ Rf_*$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6H","source_file":"sites-cohomology.tex","source_line":4402,"source_end_line":4411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4402-L4411","statement_sha256":"edfe5757caab0e765e3584e4ed52bf8bc2562a854c247b02d81a55f25b73b159","origin":"The Stacks Project","memory_eligible":false,"source_rank":4490,"rank":4490,"depth":20,"x":1222.189,"y":595.903,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6I","tag":"0D6I","title":"Some properties of K-injective complexes · Lemma 0D6I","summary":"Let f : (C, O_C) → (D, O_D) be a morphism of ringed sites corresponding to the continuous functor u : D → C. Let K be in D(O_C). Then H^i(Rf_*K) is the sheaf associated to the presheaf V ↦ H^i(u(V), K) = H^i(V, Rf_*K)","statement_latex":"Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites\ncorresponding to the continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nLet $K$ be in $D(\\mathcal{O}_\\mathcal{C})$. Then $H^i(Rf_*K)$ is the sheaf\nassociated to the presheaf\n$$\nV \\mapsto H^i(u(V), K) = H^i(V, Rf_*K)\n$$","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6I","source_file":"sites-cohomology.tex","source_line":4433,"source_end_line":4443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4433-L4443","statement_sha256":"0f96708ce72491d0ee2b1b2ba0948581d6112d2be2fdb353f925f8311d694acf","origin":"The Stacks Project","memory_eligible":false,"source_rank":4491,"rank":4491,"depth":21,"x":1129.649,"y":794.487,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6J","tag":"0D6J","title":"Some properties of K-injective complexes · Lemma 0D6J","summary":"Let (C, O_C) be a ringed site. Let K be an object of D(O_C) and denote K_ab its image in D(underlineZ_C). • There is a canonical map RΓ(C, K) → RΓ(C, K_ab) which is an isomorphism in D(Ab). • For any U ∈ C there is a canonical map RΓ(U, K) → RΓ(U, K_ab) which is an isomorphism in D(Ab). • Let f : (C, O_C) → (D, O_D) be a morphism of ringed sites. There is a canonical map Rf_*K → Rf_*(K_ab) which is an isomorphism in D(underlineZ_D).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O}_\\mathcal{C})$ be a ringed site.\nLet $K$ be an object of $D(\\mathcal{O}_\\mathcal{C})$\nand denote $K_{ab}$ its image in $D(\\underline{\\mathbf{Z}}_\\mathcal{C})$.\n\\begin{enumerate}\n\\item There is a canonical map\n$R\\Gamma(\\mathcal{C}, K) \\to R\\Gamma(\\mathcal{C}, K_{ab})$\nwhich is an isomorphism in $D(\\textit{Ab})$.\n\\item For any $U \\in \\mathcal{C}$ there is a canonical map\n$R\\Gamma(U, K) \\to R\\Gamma(U, K_{ab})$\nwhich is an isomorphism in $D(\\textit{Ab})$.\n\\item Let $f : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed sites.\nThere is a canonical map $Rf_*K \\to Rf_*(K_{ab})$ which\nis an isomorphism in $D(\\underline{\\mathbf{Z}}_\\mathcal{D})$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6J","source_file":"sites-cohomology.tex","source_line":4452,"source_end_line":4469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4452-L4469","statement_sha256":"75d74545f9c43d79268b87f57ba5e1e5e76b08c1ae1e91b4ba36ef6928e457b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4492,"rank":4492,"depth":22,"x":1038.059,"y":595.258,"cluster":"sheaf-cohomology"},{"id":"stacks:08FJ","tag":"08FJ","title":"Some properties of K-injective complexes · Lemma 08FJ","summary":"Let (C, O) be a ringed site. Let U be an object of C. Denote j : (Sh(C/U), O_U) → (Sh(C), O) the corresponding localization morphism. The restriction functor D(O) → D(O_U) is a right adjoint to extension by zero j_! : D(O_U) → D(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an\nobject of $\\mathcal{C}$. Denote\n$j : (\\Sh(\\mathcal{C}/U), \\mathcal{O}_U) \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nthe corresponding localization morphism. The restriction functor\n$D(\\mathcal{O}) \\to D(\\mathcal{O}_U)$ is a right adjoint to\nextension by zero $j_! : D(\\mathcal{O}_U) \\to D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FJ","source_file":"sites-cohomology.tex","source_line":4546,"source_end_line":4554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4546-L4554","statement_sha256":"1f305ed55439b877f162f582cd254fcfa820d1965b84a8a895e1f32c1be61bbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4493,"rank":4493,"depth":2,"x":1266.139,"y":690.333,"cluster":"sheaf-cohomology"},{"id":"stacks:0GL1","tag":"0GL1","title":"Some properties of K-injective complexes · Lemma 0GL1","summary":"Let (C, O) be a ringed site. Let U ∈ Ob(C). For L in D(O_U) and K in D(O) we have j_!L ⊗_O^L K = j_!(L ⊗_O_U^L K|_U).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U \\in \\Ob(\\mathcal{C})$.\nFor $L$ in $D(\\mathcal{O}_U)$ and $K$ in $D(\\mathcal{O})$ we have\n$j_!L \\otimes_\\mathcal{O}^\\mathbf{L} K =\nj_!(L \\otimes_{\\mathcal{O}_U}^\\mathbf{L} K|_U)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GL1","source_file":"sites-cohomology.tex","source_line":4577,"source_end_line":4583,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4577-L4583","statement_sha256":"03d8ec62314067535e751c22bca38d3f035ad5d9b5e2f8119ad41d6b7ca31987","origin":"The Stacks Project","memory_eligible":false,"source_rank":4494,"rank":4494,"depth":3,"x":1021.148,"y":749.729,"cluster":"sheaf-cohomology"},{"id":"stacks:093Y","tag":"093Y","title":"Some properties of K-injective complexes · Lemma 093Y","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a flat morphism of ringed topoi. If I^bullet is a K-injective complex of O_C-modules, then f_*I^bullet is K-injective as a complex of O_D-modules.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a flat morphism\nof ringed topoi. If $\\mathcal{I}^\\bullet$ is a K-injective\ncomplex of $\\mathcal{O}_\\mathcal{C}$-modules, then\n$f_*\\mathcal{I}^\\bullet$ is K-injective\nas a complex of $\\mathcal{O}_\\mathcal{D}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093Y","source_file":"sites-cohomology.tex","source_line":4592,"source_end_line":4600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4592-L4600","statement_sha256":"5a60b7b7956a978dd6ebb55372bbe8acf6bf5ce7663f5ddee07fbe22c1f348e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4495,"rank":4495,"depth":4,"x":1154.226,"y":566.656,"cluster":"sheaf-cohomology"},{"id":"stacks:093Z","tag":"093Z","title":"Some properties of K-injective complexes · Lemma 093Z","summary":"Let C be a site. Let O → O' be a map of sheaves of rings. If I^bullet is a K-injective complex of O-modules, then SheafHom_O(O', I^bullet) is a K-injective complex of O'-modules.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$ be a map\nof sheaves of rings. If $\\mathcal{I}^\\bullet$ is a K-injective complex of\n$\\mathcal{O}$-modules, then\n$\\SheafHom_\\mathcal{O}(\\mathcal{O}', \\mathcal{I}^\\bullet)$\nis a K-injective complex of $\\mathcal{O}'$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Some properties of K-injective complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/093Z","source_file":"sites-cohomology.tex","source_line":4615,"source_end_line":4622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4615-L4622","statement_sha256":"b96a062f10774a9169a87349fb814866a750d23ff233b6478a193e152afeefe0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4496,"rank":4496,"depth":2,"x":1203.388,"y":777.463,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZ0","tag":"0EZ0","title":"Localization and cohomology · Lemma 0EZ0","summary":"Let C be a site. Let xymatrix X' ar[d] ar[r] & X ar[d] Y' ar[r] & Y be a cartesian diagram of C. Then we have j_Y'/Y^-1 ∘ Rj_X/Y, * = Rj_X'/Y', * ∘ j_X'/X^-1 as functors D(C/X) → D(C/Y').","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$$\n\\xymatrix{\nX' \\ar[d] \\ar[r] & X \\ar[d] \\\\\nY' \\ar[r] & Y\n}\n$$\nbe a cartesian diagram of $\\mathcal{C}$. Then we have\n$j_{Y'/Y}^{-1} \\circ Rj_{X/Y, *} = Rj_{X'/Y', *} \\circ j_{X'/X}^{-1}$\nas functors $D(\\mathcal{C}/X) \\to D(\\mathcal{C}/Y')$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Localization and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZ0","source_file":"sites-cohomology.tex","source_line":4651,"source_end_line":4663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4651-L4663","statement_sha256":"c261f60406c1d96b48004e2be2365703e548000ab5ecba6a82210c55abfc6023","origin":"The Stacks Project","memory_eligible":false,"source_rank":4497,"rank":4497,"depth":10,"x":997.319,"y":649.731,"cluster":"sheaf-cohomology"},{"id":"stacks:0FN5","tag":"0FN5","title":"Localization and cohomology · Lemma 0FN5","summary":"Let (C, O) be a ringed site. Let xymatrix X' ar[d] ar[r] & X ar[d] Y' ar[r] & Y be a cartesian diagram of C. Then we have j_Y'/Y^* ∘ Rj_X/Y, * = Rj_X'/Y', * ∘ j_X'/X^* as functors D(O_X) → D(O_Y').","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$$\n\\xymatrix{\nX' \\ar[d] \\ar[r] & X \\ar[d] \\\\\nY' \\ar[r] & Y\n}\n$$\nbe a cartesian diagram of $\\mathcal{C}$. Then we have\n$j_{Y'/Y}^* \\circ Rj_{X/Y, *} = Rj_{X'/Y', *} \\circ j_{X'/X}^*$\nas functors\n$D(\\mathcal{O}_X) \\to D(\\mathcal{O}_{Y'})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Localization and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FN5","source_file":"sites-cohomology.tex","source_line":4688,"source_end_line":4701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4688-L4701","statement_sha256":"4e9faf0de990fe4fc244e97f53e2b0213b337f3720929233a1b470b8c977048b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4498,"rank":4498,"depth":23,"x":1252.351,"y":626.958,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYQ","tag":"0GYQ","title":"Inverse systems and cohomology · Lemma 0GYQ","summary":"Let I be an ideal of a ring A. Let C be a site. Let … → F_3 → F_2 → F_1 be an inverse system of sheaves of A-modules on C such that F_n = F_n + 1/I^nF_n + 1. Let p ≥ 0. Assume bigoplus_n ≥ 0 H^p + 1(C, I^nF_n + 1) satisfies the ascending chain condition as a graded bigoplus_n ≥ 0 I^n/I^n + 1-module. Then the inverse system M_n = H^p(C, F_n) satisfies the Mittag-Leffler condition) is the stable image for all n ≥ c..","statement_latex":"Let $I$ be an ideal of a ring $A$. Let $\\mathcal{C}$ be a site.\nLet\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of sheaves of $A$-modules on $\\mathcal{C}$\nsuch that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nLet $p \\geq 0$. Assume\n$$\n\\bigoplus\\nolimits_{n \\geq 0} H^{p + 1}(\\mathcal{C}, I^n\\mathcal{F}_{n + 1})\n$$\nsatisfies the ascending chain condition as a graded\n$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module.\nThen the inverse system $M_n = H^p(\\mathcal{C}, \\mathcal{F}_n)$ satisfies the\nMittag-Leffler condition\\footnote{In fact, there exists\na $c \\geq 0$ such that $\\Im(M_n \\to M_{n - c})$ is the stable image\nfor all $n \\geq c$.}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Inverse systems and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYQ","source_file":"sites-cohomology.tex","source_line":4730,"source_end_line":4749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4730-L4749","statement_sha256":"51884a187c84a0c46f00dcc6e48d5dc8ec0556aaf651374b14b198e6d2e14c37","origin":"The Stacks Project","memory_eligible":false,"source_rank":4499,"rank":4499,"depth":0,"x":1082.368,"y":788.69,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYR","tag":"0GYR","title":"Inverse systems and cohomology · Lemma 0GYR","summary":"Let I be an ideal of a ring A. Let C be a site. Let … → F_3 → F_2 → F_1 be an inverse system of A-modules on C such that F_n = F_n + 1/I^nF_n + 1. Let p ≥ 0. Given n define N_n = ⋂_m ≥ n Im( H^p + 1(C, I^nF_m + 1) → H^p + 1(C, I^nF_n + 1) ) If bigoplus N_n satisfies the ascending chain condition as a graded bigoplus_n ≥ 0 I^n/I^n + 1-module, then the inverse system M_n = H^p(C, F_n) satisfies the Mittag-Leffler condition) is the stable image for all n ≥ c..","statement_latex":"Let $I$ be an ideal of a ring $A$. Let $\\mathcal{C}$ be a site. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of $A$-modules on $\\mathcal{C}$\nsuch that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nLet $p \\geq 0$. Given $n$ define\n$$\nN_n =\n\\bigcap\\nolimits_{m \\geq n}\n\\Im\\left(\nH^{p + 1}(\\mathcal{C}, I^n\\mathcal{F}_{m + 1}) \\to\nH^{p + 1}(\\mathcal{C}, I^n\\mathcal{F}_{n + 1})\n\\right)\n$$\nIf $\\bigoplus N_n$ satisfies the ascending chain condition as a graded\n$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module, then the inverse system\n$M_n = H^p(\\mathcal{C}, \\mathcal{F}_n)$ satisfies the Mittag-Leffler\ncondition\\footnote{In fact, there exists\na $c \\geq 0$ such that $\\Im(M_n \\to M_{n - c})$ is the stable image\nfor all $n \\geq c$.}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Inverse systems and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYR","source_file":"sites-cohomology.tex","source_line":4788,"source_end_line":4811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4788-L4811","statement_sha256":"3390da094bb8954b5b43d457b03874c6fb413dbfd434057524f9c054fd6ca5e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4500,"rank":4500,"depth":1,"x":1077.644,"y":572.675,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYS","tag":"0GYS","title":"Inverse systems and cohomology · Lemma 0GYS","summary":"Let I be an ideal of a ring A. Let C be a site. Let … → F_3 → F_2 → F_1 be an inverse system of sheaves of A-modules on C such that F_n = F_n + 1/I^nF_n + 1. Let p ≥ 0. Assume bigoplus_n ≥ 0 H^p(C, I^nF_n + 1) satisfies the ascending chain condition as a graded bigoplus_n ≥ 0 I^n/I^n + 1-module. Then the limit topology on M = lim H^p(C, F_n) is the I-adic topology.","statement_latex":"Let $I$ be an ideal of a ring $A$. Let $\\mathcal{C}$ be a site. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of sheaves of $A$-modules on $\\mathcal{C}$ such that\n$\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nLet $p \\geq 0$. Assume\n$$\n\\bigoplus\\nolimits_{n \\geq 0} H^p(\\mathcal{C}, I^n\\mathcal{F}_{n + 1})\n$$\nsatisfies the ascending chain condition as a graded\n$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module.\nThen the limit topology on $M = \\lim H^p(\\mathcal{C}, \\mathcal{F}_n)$\nis the $I$-adic topology.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Inverse systems and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYS","source_file":"sites-cohomology.tex","source_line":4864,"source_end_line":4880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4864-L4880","statement_sha256":"0e95c72d53a5bb31511ef2e2d6cd5cfae8ac89d4f90600909b92bb9b90626f91","origin":"The Stacks Project","memory_eligible":false,"source_rank":4501,"rank":4501,"depth":1,"x":1255.09,"y":729.501,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYT","tag":"0GYT","title":"Inverse systems and cohomology · Lemma 0GYT","summary":"Let I be an ideal of a ring A. Let C be a site. Let … → F_3 → F_2 → F_1 be an inverse system of sheaves of A-modules on C such that F_n = F_n + 1/I^nF_n + 1. Let p ≥ 0. Given n define N_n = ⋂_m ≥ n Im( H^p(C, I^nF_m + 1) → H^p(C, I^nF_n + 1) ) If bigoplus N_n satisfies the ascending chain condition as a graded bigoplus_n ≥ 0 I^n/I^n + 1-module, then the limit topology on M = lim H^p(C, F_n) is the I-adic topology.","statement_latex":"Let $I$ be an ideal of a ring $A$. Let $\\mathcal{C}$ be a site. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of sheaves of $A$-modules on $\\mathcal{C}$ such that\n$\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nLet $p \\geq 0$. Given $n$ define\n$$\nN_n =\n\\bigcap\\nolimits_{m \\geq n}\n\\Im\\left(\nH^p(\\mathcal{C}, I^n\\mathcal{F}_{m + 1}) \\to\nH^p(\\mathcal{C}, I^n\\mathcal{F}_{n + 1})\n\\right)\n$$\nIf $\\bigoplus N_n$ satisfies the ascending chain condition as a graded\n$\\bigoplus_{n \\geq 0} I^n/I^{n + 1}$-module, then\nthe limit topology on $M = \\lim H^p(\\mathcal{C}, \\mathcal{F}_n)$\nis the $I$-adic topology.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Inverse systems and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYT","source_file":"sites-cohomology.tex","source_line":4940,"source_end_line":4961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L4940-L4961","statement_sha256":"b7c8c6723d4a6b9e28a535d477b8fbd90aa4c5b88edb52e54ed7a64d32dd410a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4502,"rank":4502,"depth":2,"x":997.768,"y":714.526,"cluster":"sheaf-cohomology"},{"id":"stacks:0941","tag":"0941","title":"Derived and homotopy limits · Lemma 0941","summary":"Let C be a site. Let K be an object of D(C × N). Set K_n = i_n^-1K as above. Then Rlim K ≅ Rlim K_n in D(C).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $K$ be an object of\n$D(\\mathcal{C} \\times \\mathbf{N})$. Set $K_n = i_n^{-1}K$ as above.\nThen\n$$\nR\\lim K \\cong R\\lim K_n\n$$\nin $D(\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0941","source_file":"sites-cohomology.tex","source_line":5066,"source_end_line":5075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5066-L5075","statement_sha256":"c9df28fa0b2aad4850297c299b0d641f0145640cef884e5c4a9f2894060398af","origin":"The Stacks Project","memory_eligible":false,"source_rank":4503,"rank":4503,"depth":8,"x":1199.838,"y":579.369,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6K","tag":"0D6K","title":"Derived and homotopy limits · Lemma 0D6K","summary":"Let (C, O) be a ringed site. The functors RΓ(C, -) and RΓ(U, -) for U ∈ Ob(C) commute with Rlim. Moreover, there are short exact sequences 0 → R^1lim H^m - 1(U, K_n) → H^m(U, Rlim K_n) → lim H^m(U, K_n) → 0 for any inverse system (K_n) in D(O) and m ∈ Z. Similar for H^m(C, Rlim K_n).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The functors\n$R\\Gamma(\\mathcal{C}, -)$ and $R\\Gamma(U, -)$ for $U \\in \\Ob(\\mathcal{C})$\ncommute with $R\\lim$. Moreover, there are\nshort exact sequences\n$$\n0 \\to\nR^1\\lim H^{m - 1}(U, K_n) \\to H^m(U, R\\lim K_n) \\to\n\\lim H^m(U, K_n) \\to 0\n$$\nfor any inverse system $(K_n)$ in $D(\\mathcal{O})$ and $m \\in \\mathbf{Z}$.\nSimilar for $H^m(\\mathcal{C}, R\\lim K_n)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6K","source_file":"sites-cohomology.tex","source_line":5125,"source_end_line":5138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5125-L5138","statement_sha256":"3ebf8dc30edb33abc6d15410d869601871c9f118a1949b80611b78c3c91c0535","origin":"The Stacks Project","memory_eligible":false,"source_rank":4504,"rank":4504,"depth":9,"x":1159.468,"y":793.99,"cluster":"sheaf-cohomology"},{"id":"stacks:0A07","tag":"0A07","title":"Derived and homotopy limits · Lemma 0A07","summary":"Let f : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. Then Rf_* commutes with Rlim, i.e., Rf_* commutes with derived limits.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi. Then $Rf_*$ commutes with $R\\lim$, i.e.,\n$Rf_*$ commutes with derived limits.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A07","source_file":"sites-cohomology.tex","source_line":5149,"source_end_line":5154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5149-L5154","statement_sha256":"e48fad6af8d95c3435cad6d17d2f55665e7ab412156119f8464f58bef1251cfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4505,"rank":4505,"depth":19,"x":1016.446,"y":612.573,"cluster":"sheaf-cohomology"},{"id":"stacks:0BKY","tag":"0BKY","title":"Derived and homotopy limits · Lemma 0BKY","summary":"Let (C, O) be a ringed site. Let (F_n) be an inverse system of O-modules. Let B ⊂ Ob(C) be a subset. Assume • every object of C has a covering whose members are elements of B, • H^p(U, F_n) = 0 for p > 0 and U ∈ B, • the inverse system F_n(U) has vanishing R^1lim for U ∈ B. Then Rlim F_n = lim F_n and we have H^p(U, lim F_n) = 0 for p > 0 and U ∈ B.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(\\mathcal{F}_n)$ be an\ninverse system of $\\mathcal{O}$-modules. Let\n$\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume\n\\begin{enumerate}\n\\item every object of $\\mathcal{C}$ has a covering whose members are elements\nof $\\mathcal{B}$,\n\\item $H^p(U, \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$,\n\\item the inverse system $\\mathcal{F}_n(U)$ has vanishing $R^1\\lim$\nfor $U \\in \\mathcal{B}$.\n\\end{enumerate}\nThen $R\\lim \\mathcal{F}_n = \\lim \\mathcal{F}_n$ and we have\n$H^p(U, \\lim \\mathcal{F}_n) = 0$ for $p > 0$ and $U \\in \\mathcal{B}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKY","source_file":"sites-cohomology.tex","source_line":5246,"source_end_line":5260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5246-L5260","statement_sha256":"3a5836a6dcb7b52e64918a91c63022ab2828b6bc91f1f30b965590b8eb2f2f28","origin":"The Stacks Project","memory_eligible":false,"source_rank":4506,"rank":4506,"depth":0,"x":1268.147,"y":665.266,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6L","tag":"0D6L","title":"Derived and homotopy limits · Lemma 0D6L","summary":"Let (C, O) be a ringed site. Let (K_n) be an inverse system in D(O). Let V ∈ Ob(C) and m ∈ Z. Assume there exist an integer n(V) and a cofinal system Cov_V of coverings of V such that for (V_i → V) ∈ Cov_V • R^1lim H^m - 1(V_i, K_n) = 0, and • H^m(V_i, K_n) → H^m(V_i, K_n(V)) is injective for n ≥ n(V). Then the map on sections H^m(Rlim K_n)(V) → H^m(K_n(V))(V) is injective.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)$ be an\ninverse system in $D(\\mathcal{O})$. Let $V \\in \\Ob(\\mathcal{C})$\nand $m \\in \\mathbf{Z}$. Assume there exist an integer $n(V)$\nand a cofinal system $\\text{Cov}_V$ of coverings of $V$ such that\nfor $\\{V_i \\to V\\} \\in \\text{Cov}_V$\n\\begin{enumerate}\n\\item $R^1\\lim H^{m - 1}(V_i, K_n) = 0$, and\n\\item $H^m(V_i, K_n) \\to H^m(V_i, K_{n(V)})$ is injective\nfor $n \\geq n(V)$.\n\\end{enumerate}\nThen the map on sections $H^m(R\\lim K_n)(V) \\to H^m(K_{n(V)})(V)$ is injective.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6L","source_file":"sites-cohomology.tex","source_line":5278,"source_end_line":5291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5278-L5291","statement_sha256":"a17afb75a890ac0e181e75db9b687bd5423d50ccf8452b8e4e6fb7b6165f52ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":4507,"rank":4507,"depth":10,"x":1039.857,"y":769.375,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6M","tag":"0D6M","title":"Derived and homotopy limits · Lemma 0D6M","summary":"Let (C, O) be a ringed site. Let E ∈ D(O). Let B ⊂ Ob(C) be a subset. Assume • every object of C has a covering whose members are elements of B, and • for every V ∈ B there exist a function p(V, -) : Z → Z and a cofinal system Cov_V of coverings of V such that H^p(V_i, H^m - p(E)) = 0 for all (V_i → V) ∈ Cov_V and all integers p, m satisfying p > p(V, m). Then the map E → Rlim τ_≥ -n E of Derived Categories, Remark [Tag 0H72] is an isomorphism in D(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$.\nLet $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume\n\\begin{enumerate}\n\\item every object of $\\mathcal{C}$ has a covering whose members\nare elements of $\\mathcal{B}$, and\n\\item for every $V \\in \\mathcal{B}$ there exist a function\n$p(V, -) : \\mathbf{Z} \\to \\mathbf{Z}$ and a cofinal system $\\text{Cov}_V$\nof coverings of $V$ such that\n$$\nH^p(V_i, H^{m - p}(E)) = 0\n$$\nfor all $\\{V_i \\to V\\} \\in \\text{Cov}_V$ and all integers $p, m$\nsatisfying $p > p(V, m)$.\n\\end{enumerate}\nThen the map $E \\to R\\lim \\tau_{\\geq -n} E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6M","source_file":"sites-cohomology.tex","source_line":5319,"source_end_line":5339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5319-L5339","statement_sha256":"e4dcdcf0d7e33a5cb869d9dc783f8202f715f94ac0d64978b1a9d1082db8d0dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4508,"rank":4508,"depth":11,"x":1124.589,"y":562.785,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6N","tag":"0D6N","title":"Derived and homotopy limits · Lemma 0D6N","summary":"Let (C, O) be a ringed site. Let E ∈ D(O). Let B ⊂ Ob(C) be a subset. Assume • every object of C has a covering whose members are elements of B, and • for every V ∈ B there exist an integer d_V ≥ 0 and a cofinal system Cov_V of coverings of V such that H^p(V_i, H^q(E)) = 0 for (V_i → V) ∈ Cov_V, p > d_V, and q < 0 Then the map E → Rlim τ_≥ -n E of Derived Categories, Remark [Tag 0H72] is an isomorphism in D(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$.\nLet $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume\n\\begin{enumerate}\n\\item every object of $\\mathcal{C}$ has a covering whose members are\nelements of $\\mathcal{B}$, and\n\\item for every $V \\in \\mathcal{B}$ there exist an integer $d_V \\geq 0$ and\na cofinal system $\\text{Cov}_V$ of coverings of $V$ such that\n$$\nH^p(V_i, H^q(E)) = 0 \\text{ for }\n\\{V_i \\to V\\} \\in \\text{Cov}_V,\\ p > d_V, \\text{ and }q < 0\n$$\n\\end{enumerate}\nThen the map $E \\to R\\lim \\tau_{\\geq -n} E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6N","source_file":"sites-cohomology.tex","source_line":5391,"source_end_line":5409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5391-L5409","statement_sha256":"cf0951fb4d1f3c8bb572b719bc4269605922fa3849cc7fdd96581f99c89c4fc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4509,"rank":4509,"depth":12,"x":1228.385,"y":763.477,"cluster":"sheaf-cohomology"},{"id":"stacks:08U3","tag":"08U3","title":"Derived and homotopy limits · Lemma 08U3","summary":"Let (C, O) be a ringed site. Let E ∈ D(O). Assume there exists a function p(-) : Z → Z and a subset B ⊂ Ob(C) such that • every object of C has a covering whose members are elements of B, • H^p(V, H^m - p(E)) = 0 for p > p(m) and V ∈ B. Then the map E → Rlim τ_≥ -n E of Derived Categories, Remark [Tag 0H72] is an isomorphism in D(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$.\nAssume there exists a function $p(-) : \\mathbf{Z} \\to \\mathbf{Z}$\nand a subset $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ such that\n\\begin{enumerate}\n\\item every object of $\\mathcal{C}$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item $H^p(V, H^{m - p}(E)) = 0$ for $p > p(m)$ and $V \\in \\mathcal{B}$.\n\\end{enumerate}\nThen the map $E \\to R\\lim \\tau_{\\geq -n} E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08U3","source_file":"sites-cohomology.tex","source_line":5416,"source_end_line":5430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5416-L5430","statement_sha256":"93190a704f8ea8326f77df10f02f60115a9bbb5a4adade5e4c2676e932c55baa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4510,"rank":4510,"depth":12,"x":990.132,"y":674.264,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6P","tag":"0D6P","title":"Derived and homotopy limits · Lemma 0D6P","summary":"Let (C, O) be a ringed site. Let E ∈ D(O). Assume there exists an integer d ≥ 0 and a subset B ⊂ Ob(C) such that • every object of C has a covering whose members are elements of B, • H^p(V, H^q(E)) = 0 for p > d, q < 0, and V ∈ B. Then the map E → Rlim τ_≥ -n E of Derived Categories, Remark [Tag 0H72] is an isomorphism in D(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E \\in D(\\mathcal{O})$.\nAssume there exists an integer $d \\geq 0$\nand a subset $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ such that\n\\begin{enumerate}\n\\item every object of $\\mathcal{C}$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item $H^p(V, H^q(E)) = 0$ for $p > d$, $q < 0$, and $V \\in \\mathcal{B}$.\n\\end{enumerate}\nThen the map $E \\to R\\lim \\tau_{\\geq -n} E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6P","source_file":"sites-cohomology.tex","source_line":5439,"source_end_line":5453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5439-L5453","statement_sha256":"37b82b4bd31aae676f6dbb5462682d575d2d14288d8f9d4d0de2dc19dad68470","origin":"The Stacks Project","memory_eligible":false,"source_rank":4511,"rank":4511,"depth":13,"x":1237.897,"y":604.762,"cluster":"sheaf-cohomology"},{"id":"stacks:0BKZ","tag":"0BKZ","title":"Derived and homotopy limits · Lemma 0BKZ","summary":"Let (C, O) be a ringed site. Let K be an object of D(O). Let B ⊂ Ob(C) be a subset. Assume • every object of C has a covering whose members are elements of B, • H^p(U, H^q(K)) = 0 for all p > 0, q ∈ Z, and U ∈ B. Then H^q(U, K) = H^0(U, H^q(K)) for q ∈ Z and U ∈ B.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$\nbe an object of $D(\\mathcal{O})$.\nLet $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume\n\\begin{enumerate}\n\\item every object of $\\mathcal{C}$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item $H^p(U, H^q(K)) = 0$ for all $p > 0$, $q \\in \\mathbf{Z}$, and\n$U \\in \\mathcal{B}$.\n\\end{enumerate}\nThen $H^q(U, K) = H^0(U, H^q(K))$ for $q \\in \\mathbf{Z}$\nand $U \\in \\mathcal{B}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BKZ","source_file":"sites-cohomology.tex","source_line":5466,"source_end_line":5479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5466-L5479","statement_sha256":"c015028adfbd39b41e50226610c1d6ad2b259fe90f328febede305d99f663868","origin":"The Stacks Project","memory_eligible":false,"source_rank":4512,"rank":4512,"depth":14,"x":1110.917,"y":796.861,"cluster":"sheaf-cohomology"},{"id":"stacks:0A09","tag":"0A09","title":"Derived and homotopy limits · Lemma 0A09","summary":"Let (C, O) be a ringed site. Let (K_n) be an inverse system of objects of D(O). Let B ⊂ Ob(C) be a subset. Assume • every object of C has a covering whose members are elements of B, • for all U ∈ B and all q ∈ Z we have • H^p(U, H^q(K_n)) = 0 for p > 0, • the inverse system H^0(U, H^q(K_n)) has vanishing R^1lim. Then H^q(Rlim K_n) = lim H^q(K_n) for q ∈ Z.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K_n)$\nbe an inverse system of objects of $D(\\mathcal{O})$.\nLet $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume\n\\begin{enumerate}\n\\item every object of $\\mathcal{C}$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item for all $U \\in \\mathcal{B}$ and all $q \\in \\mathbf{Z}$ we have\n\\begin{enumerate}\n\\item $H^p(U, H^q(K_n)) = 0$ for $p > 0$,\n\\item the inverse system $H^0(U, H^q(K_n))$ has vanishing $R^1\\lim$.\n\\end{enumerate}\n\\end{enumerate}\nThen $H^q(R\\lim K_n) = \\lim H^q(K_n)$ for $q \\in \\mathbf{Z}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived and homotopy limits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A09","source_file":"sites-cohomology.tex","source_line":5504,"source_end_line":5519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5504-L5519","statement_sha256":"b004c72b162a51a7aa8b38f7d2a369f321adf5e3efd690392b58103cdf68d8ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":4513,"rank":4513,"depth":15,"x":1049.987,"y":582.869,"cluster":"sheaf-cohomology"},{"id":"stacks:070Q","tag":"070Q","title":"Producing K-injective resolutions · Lemma 070Q","summary":"In the situation described above. Denote H^m = H^m(F^bullet) the mth cohomology sheaf. Let B ⊂ Ob(C) be a subset. Let d ∈ N. Assume • every object of C has a covering whose members are elements of B, • for every U ∈ B we have H^p(U, H^q) = 0 for p > d and q < 0^m - p) = 0 for p > p(m), see Lemma [Tag 08U3].. Then ([Tag 070P]) is a quasi-isomorphism.","statement_latex":"In the situation described above. Denote\n$\\mathcal{H}^m = H^m(\\mathcal{F}^\\bullet)$ the $m$th cohomology sheaf.\nLet $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset.\nLet $d \\in \\mathbf{N}$.\nAssume\n\\begin{enumerate}\n\\item every object of $\\mathcal{C}$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item for every $U \\in \\mathcal{B}$ we have $H^p(U, \\mathcal{H}^q) = 0$\nfor $p > d$ and $q < 0$\\footnote{It suffices if\n$\\forall m$, $\\exists p(m)$, $H^p(U. \\mathcal{H}^{m - p}) = 0$ for\n$p > p(m)$, see Lemma \\ref{lemma-is-limit}.}.\n\\end{enumerate}\nThen (\\ref{equation-into-candidate-K-injective}) is a quasi-isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Producing K-injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070Q","source_file":"sites-cohomology.tex","source_line":5585,"source_end_line":5601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5585-L5601","statement_sha256":"640bef60025569909365a85d3a60fe44e197a285a635b6e8abacd328a292d4cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4514,"rank":4514,"depth":16,"x":1267.296,"y":706.257,"cluster":"sheaf-cohomology"},{"id":"stacks:08CT","tag":"08CT","title":"Producing K-injective resolutions · Lemma 08CT","summary":"Let (C, O) be a ringed site. Let (F_n^bullet) be an inverse system of complexes of O-modules. Let m ∈ Z. Suppose given B ⊂ Ob(C) and an integer n_0 such that • every object of C has a covering whose members are elements of B, • for every U ∈ B • the systems of abelian groups F_n^m - 2(U) and F_n^m - 1(U) have vanishing R^1lim (for example these have the Mittag-Leffler property), • the system of abelian groups H^m - 1(F_n^bullet(U)) has vanishing R^1lim (for example it has…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{F}_n^\\bullet)$ be an inverse system of complexes of\n$\\mathcal{O}$-modules. Let $m \\in \\mathbf{Z}$. Suppose given\n$\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ and an integer\n$n_0$ such that\n\\begin{enumerate}\n\\item every object of $\\mathcal{C}$ has a covering whose members are\nelements of $\\mathcal{B}$,\n\\item for every $U \\in \\mathcal{B}$\n\\begin{enumerate}\n\\item the systems of abelian groups\n$\\mathcal{F}_n^{m - 2}(U)$ and $\\mathcal{F}_n^{m - 1}(U)$\nhave vanishing $R^1\\lim$ (for example these have the Mittag-Leffler property),\n\\item the system of abelian groups $H^{m - 1}(\\mathcal{F}_n^\\bullet(U))$\nhas vanishing $R^1\\lim$ (for example it has the Mittag-Leffler property), and\n\\item we have\n$H^m(\\mathcal{F}_n^\\bullet(U)) = H^m(\\mathcal{F}_{n_0}^\\bullet(U))$\nfor all $n \\geq n_0$.\n\\end{enumerate}\n\\end{enumerate}\nThen the maps $H^m(\\mathcal{F}^\\bullet) \\to \\lim H^m(\\mathcal{F}_n^\\bullet)\n\\to H^m(\\mathcal{F}_{n_0}^\\bullet)$ are isomorphisms of sheaves where\n$\\mathcal{F}^\\bullet = \\lim \\mathcal{F}_n^\\bullet$ is the termwise\ninverse limit.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Producing K-injective resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08CT","source_file":"sites-cohomology.tex","source_line":5616,"source_end_line":5642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5616-L5642","statement_sha256":"f4f98dfc87bd5d1f5aab27cd418f89aa856fccd116ab07714ca3860710ea1651","origin":"The Stacks Project","memory_eligible":false,"source_rank":4515,"rank":4515,"depth":8,"x":1007.479,"y":738.622,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6S","tag":"0D6S","title":"Bounded cohomological dimension · Lemma 0D6S","summary":"This is [six-I] with slightly changed hypotheses; it is the analogue of [Spaltenstein] for sites. In Situation [Tag 0D6R] for any E ∈ D_A(O) the map E → Rlim τ_≥ -n E of Derived Categories, Remark [Tag 0H72] is an isomorphism in D(O).","statement_latex":"\\begin{reference}\nThis is \\cite[Proposition 2.1.4]{six-I} with slightly changed\nhypotheses; it is the analogue of \\cite[Proposition 3.13]{Spaltenstein}\nfor sites.\n\\end{reference}\nIn Situation \\ref{situation-olsson-laszlo} for any\n$E \\in D_\\mathcal{A}(\\mathcal{O})$ the map\n$E \\to R\\lim \\tau_{\\geq -n} E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism in $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Bounded cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6S","source_file":"sites-cohomology.tex","source_line":5706,"source_end_line":5719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5706-L5719","statement_sha256":"789f036f3ff42c7c0703efe4304fe29c9b0dd178c8a58d994efa931a182fb0f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4516,"rank":4516,"depth":13,"x":1173.26,"y":567.1,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6T","tag":"0D6T","title":"Bounded cohomological dimension · Lemma 0D6T","summary":"In Situation [Tag 0D6R] let (K_n) be an inverse system in D_A^+(O). Assume that for every j the inverse system (H^j(K_n)) in A is eventually constant with value H^j. Then H^j(Rlim K_n) = H^j for all j.","statement_latex":"In Situation \\ref{situation-olsson-laszlo} let\n$(K_n)$ be an inverse system in $D_\\mathcal{A}^+(\\mathcal{O})$.\nAssume that for every $j$ the inverse system $(H^j(K_n))$\nin $\\mathcal{A}$ is eventually constant with value $\\mathcal{H}^j$. Then\n$H^j(R\\lim K_n) = \\mathcal{H}^j$ for all $j$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Bounded cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6T","source_file":"sites-cohomology.tex","source_line":5725,"source_end_line":5732,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5725-L5732","statement_sha256":"97945603b5b391a0f76f2b4f0045db46b98877cdd19b8b26137bf74f64582554","origin":"The Stacks Project","memory_eligible":false,"source_rank":4517,"rank":4517,"depth":11,"x":1188.971,"y":787.943,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6U","tag":"0D6U","title":"Bounded cohomological dimension · Lemma 0D6U","summary":"This is a version of [six-I] with slightly changed hypotheses. Let f : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. Let A ⊂ Mod(O) and A' ⊂ Mod(O') be weak Serre subcategories. Assume there is an integer N such that • C, O, A satisfy the assumption of Situation [Tag 0D6R], • C', O', A' satisfy the assumption of Situation [Tag 0D6R], • R^pf_*F ∈ Ob(A') for p ≥ 0 and F ∈ Ob(A), • R^pf_*F = 0 for p > N and F ∈ Ob(A), Then for K in D_A(O) we have • [(a)] Rf_*K is…","statement_latex":"\\begin{reference}\nThis is a version of \\cite[Lemma 2.1.10]{six-I} with slightly changed\nhypotheses.\n\\end{reference}\nLet $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi.\nLet $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$\nand $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$\nbe weak Serre subcategories. Assume there is an integer $N$ such that\n\\begin{enumerate}\n\\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the\nassumption of Situation \\ref{situation-olsson-laszlo},\n\\item $\\mathcal{C}', \\mathcal{O}', \\mathcal{A}'$ satisfy the\nassumption of Situation \\ref{situation-olsson-laszlo},\n\\item $R^pf_*\\mathcal{F} \\in \\Ob(\\mathcal{A}')$ for\n$p \\geq 0$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$,\n\\item $R^pf_*\\mathcal{F} = 0$ for\n$p > N$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$,\n\\end{enumerate}\nThen for $K$ in $D_\\mathcal{A}(\\mathcal{O})$ we have\n\\begin{enumerate}\n\\item[(a)] $Rf_*K$ is in $D_{\\mathcal{A}'}(\\mathcal{O}')$,\n\\item[(b)] the map\n$H^j(Rf_*K) \\to H^j(Rf_*(\\tau_{\\geq -n}K))$ is an isomorphism\nfor $j \\geq N - n$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Bounded cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6U","source_file":"sites-cohomology.tex","source_line":5795,"source_end_line":5823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5795-L5823","statement_sha256":"3443fedd43f086b2d21c2394d97f43a7289b4460903e71d18a5ce2d0a4438798","origin":"The Stacks Project","memory_eligible":false,"source_rank":4518,"rank":4518,"depth":20,"x":999.537,"y":633.804,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6W","tag":"0D6W","title":"Bounded cohomological dimension · Lemma 0D6W","summary":"This is a version of [six-I] with slightly changed hypotheses. Let f : (C, O) → (C', O') be a morphism of ringed sites. assume moreover there is an integer N such that • C, O, A satisfy the assumption of Situation [Tag 0D6R], • f : (C, O) → (C', O') and A satisfy the assumption of Situation [Tag 0D6V], • R^pf_*F = 0 for p > N and F ∈ Ob(A), Then for K in D_A(O) the map H^j(Rf_*K) → H^j(Rf_*(τ_≥ -nK)) is an isomorphism for j ≥ N - n.","statement_latex":"\\begin{reference}\nThis is a version of \\cite[Lemma 2.1.10]{six-I} with slightly changed\nhypotheses.\n\\end{reference}\nLet $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$\nbe a morphism of ringed sites.\nassume moreover there is an integer $N$ such that\n\\begin{enumerate}\n\\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the\nassumption of Situation \\ref{situation-olsson-laszlo},\n\\item $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$\nand $\\mathcal{A}$ satisfy the assumption of\nSituation \\ref{situation-olsson-laszlo-prime},\n\\item $R^pf_*\\mathcal{F} = 0$ for\n$p > N$ and $\\mathcal{F} \\in \\Ob(\\mathcal{A})$,\n\\end{enumerate}\nThen for $K$ in $D_\\mathcal{A}(\\mathcal{O})$ the map\n$H^j(Rf_*K) \\to H^j(Rf_*(\\tau_{\\geq -n}K))$ is an isomorphism\nfor $j \\geq N - n$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Bounded cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6W","source_file":"sites-cohomology.tex","source_line":5875,"source_end_line":5896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L5875-L5896","statement_sha256":"52415039fc70e1ca8beda86343424f04a2492570e74e04abb5352def3526d799","origin":"The Stacks Project","memory_eligible":false,"source_rank":4519,"rank":4519,"depth":20,"x":1263.528,"y":639.983,"cluster":"sheaf-cohomology"},{"id":"stacks:0F16","tag":"0F16","title":"Mayer-Vietoris · Lemma 0F16","summary":"In the situation above, choose a K-injective complex I^bullet of O-modules representing K. Using -1 times the canonical map for one of the four arrows we get maps of complexes I^bullet(X) xrightarrowα I^bullet(Z) ⊕ I^bullet(Y) xrightarrowβ I^bullet(E) with β ∘ α = 0. Thus a canonical map c^K_X, Z, Y, E : I^bullet(X) → C(β)^bullet[-1] This map is canonical in the sense that a different choice of K-injective complex representing K determines an isomorphic arrow in the…","statement_latex":"In the situation above, choose a K-injective complex $\\mathcal{I}^\\bullet$\nof $\\mathcal{O}$-modules representing $K$. Using $-1$ times the canonical map\nfor one of the four arrows we get maps of complexes\n$$\n\\mathcal{I}^\\bullet(X) \\xrightarrow{\\alpha}\n\\mathcal{I}^\\bullet(Z) \\oplus\n\\mathcal{I}^\\bullet(Y) \\xrightarrow{\\beta}\n\\mathcal{I}^\\bullet(E)\n$$\nwith $\\beta \\circ \\alpha = 0$. Thus a canonical map\n$$\nc^K_{X, Z, Y, E} :\n\\mathcal{I}^\\bullet(X)\n\\longrightarrow\nC(\\beta)^\\bullet[-1]\n$$\nThis map is canonical in the sense that a different choice\nof K-injective complex representing $K$ determines an isomorphic\narrow in the derived category of abelian groups. If $c^K_{X, Z, Y, E}$\nis an isomorphism, then using its inverse we obtain a canonical\ndistinguished triangle\n$$\nR\\Gamma(X, K) \\to\nR\\Gamma(Z, K) \\oplus\nR\\Gamma(Y, K) \\to\nR\\Gamma(E, K) \\to\nR\\Gamma(X, K)[1]\n$$\nAll of these constructions are functorial in $K$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F16","source_file":"sites-cohomology.tex","source_line":6000,"source_end_line":6031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6000-L6031","statement_sha256":"5c6dbb8f0b78c3079be83f23ead1584c16e36565225315818cb233efa3f39786","origin":"The Stacks Project","memory_eligible":false,"source_rank":4520,"rank":4520,"depth":0,"x":1063.631,"y":785.415,"cluster":"sheaf-cohomology"},{"id":"stacks:0EWP","tag":"0EWP","title":"Mayer-Vietoris · Lemma 0EWP","summary":"In the situation above, let K_1 → K_2 → K_3 → K_1[1] be a distinguished triangle in D(O). If c^K_i_X, Z, Y, E is a quasi-isomorphism for two i out of (1, 2, 3), then it is a quasi-isomorphism for the third i.","statement_latex":"In the situation above, let $K_1 \\to K_2 \\to K_3 \\to K_1[1]$ be a distinguished\ntriangle in $D(\\mathcal{O})$.\nIf $c^{K_i}_{X, Z, Y, E}$ is a quasi-isomorphism for\ntwo $i$ out of $\\{1, 2, 3\\}$, then it is a quasi-isomorphism\nfor the third $i$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWP","source_file":"sites-cohomology.tex","source_line":6045,"source_end_line":6052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6045-L6052","statement_sha256":"2d75c54079294d9272d52779072b37953634f9f463dd21e0889f79c36e1c4702","origin":"The Stacks Project","memory_eligible":false,"source_rank":4521,"rank":4521,"depth":13,"x":1094.12,"y":564.456,"cluster":"sheaf-cohomology"},{"id":"stacks:0EVY","tag":"0EVY","title":"Mayer-Vietoris · Lemma 0EVY","summary":"In the situation above assume • h_X^\\# = h_Y^\\# amalg_h_E^\\# h_Z^\\#, and • h_E^\\# → h_Y^\\# is injective. Then the construction of Lemma [Tag 0F16] produces a distinguished triangle RΓ(X, K) → RΓ(Z, K) ⊕ RΓ(Y, K) → RΓ(E, K) → RΓ(X, K)[1] functorial for K in D(C).","statement_latex":"In the situation above assume\n\\begin{enumerate}\n\\item $h_X^\\# = h_Y^\\# \\amalg_{h_E^\\#} h_Z^\\#$, and\n\\item $h_E^\\# \\to h_Y^\\#$ is injective.\n\\end{enumerate}\nThen the construction of Lemma \\ref{lemma-c-square}\nproduces a distinguished triangle\n$$\nR\\Gamma(X, K) \\to\nR\\Gamma(Z, K) \\oplus\nR\\Gamma(Y, K) \\to\nR\\Gamma(E, K) \\to R\\Gamma(X, K)[1]\n$$\nfunctorial for $K$ in $D(\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVY","source_file":"sites-cohomology.tex","source_line":6073,"source_end_line":6089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6073-L6089","statement_sha256":"ebaa2bddcb0ebb3af13c949317a94a7d5e28171e041ec8165958d7d66a46fee3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4522,"rank":4522,"depth":1,"x":1249.533,"y":744.927,"cluster":"sheaf-cohomology"},{"id":"stacks:0EVZ","tag":"0EVZ","title":"Mayer-Vietoris · Lemma 0EVZ","summary":"Let C be a site. Consider a commutative diagram xymatrix D ar[r] ar[d] & F ar[d] E ar[r] & G of presheaves of sets on C and assume that • G^\\# = E^\\# amalg_D^\\# F^\\#, and • D^\\# → F^\\# is injective. Then there is a canonical distinguished triangle RΓ(G, K) → RΓ(E, K) ⊕ RΓ(F, K) → RΓ(D, K) → RΓ(G, K)[1] functorial in K ∈ D(C) where RΓ(G, -) is the cohomology discussed in Section [Tag 079X].","statement_latex":"Let $\\mathcal{C}$ be a site. Consider a commutative diagram\n$$\n\\xymatrix{\n\\mathcal{D} \\ar[r] \\ar[d] & \\mathcal{F} \\ar[d] \\\\\n\\mathcal{E} \\ar[r] & \\mathcal{G}\n}\n$$\nof presheaves of sets on $\\mathcal{C}$ and assume that\n\\begin{enumerate}\n\\item $\\mathcal{G}^\\# =\n\\mathcal{E}^\\# \\amalg_{\\mathcal{D}^\\#} \\mathcal{F}^\\#$, and\n\\item $\\mathcal{D}^\\# \\to \\mathcal{F}^\\#$ is injective.\n\\end{enumerate}\nThen there is a canonical distinguished triangle\n$$\nR\\Gamma(\\mathcal{G}, K) \\to\nR\\Gamma(\\mathcal{E}, K) \\oplus\nR\\Gamma(\\mathcal{F}, K) \\to\nR\\Gamma(\\mathcal{D}, K) \\to\nR\\Gamma(\\mathcal{G}, K)[1]\n$$\nfunctorial in $K \\in D(\\mathcal{C})$ where $R\\Gamma(\\mathcal{G}, -)$\nis the cohomology discussed in Section \\ref{section-limp}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVZ","source_file":"sites-cohomology.tex","source_line":6125,"source_end_line":6150,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6125-L6150","statement_sha256":"3410f9e153d7bc97bbf404467a59e5b9add4b7d0685af4478af27dfb6d0b7d5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4523,"rank":4523,"depth":9,"x":989.46,"y":699.976,"cluster":"sheaf-cohomology"},{"id":"stacks:0D7Q","tag":"0D7Q","title":"Formalities on cohomological descent · Lemma 0D7Q","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Consider the full subcategory D' ⊂ D(O_D) consisting of objects K such that K → Rf_*Lf^*K is an isomorphism. Then D' is a saturated triangulated strictly full subcategory of D(O_D) and the functor Lf^* : D' → D(O_C) is fully faithful.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nConsider the full subcategory $D' \\subset D(\\mathcal{O}_\\mathcal{D})$\nconsisting of objects $K$ such that\n$$\nK \\longrightarrow Rf_*Lf^*K\n$$\nis an isomorphism. Then $D'$ is a saturated triangulated strictly full\nsubcategory of $D(\\mathcal{O}_\\mathcal{D})$ and the functor\n$Lf^* : D' \\to D(\\mathcal{O}_\\mathcal{C})$ is fully faithful.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Formalities on cohomological descent","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7Q","source_file":"sites-cohomology.tex","source_line":6285,"source_end_line":6297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6285-L6297","statement_sha256":"1948f35114e88c2b9053dd98b8b8ffb1ba48a7aa2e9f7168cd1ec241f08d694d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4524,"rank":4524,"depth":19,"x":1217.683,"y":585.4,"cluster":"sheaf-cohomology"},{"id":"stacks:0D7R","tag":"0D7R","title":"Formalities on cohomological descent · Lemma 0D7R","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Consider the full subcategory D' ⊂ D(O_C) consisting of objects K such that Lf^*Rf_*K → K is an isomorphism. Then D' is a saturated triangulated strictly full subcategory of D(O_C) and the functor Rf_* : D' → D(O_D) is fully faithful.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nConsider the full subcategory $D' \\subset D(\\mathcal{O}_\\mathcal{C})$\nconsisting of objects $K$ such that\n$$\nLf^*Rf_*K \\longrightarrow K\n$$\nis an isomorphism. Then $D'$ is a saturated triangulated strictly full\nsubcategory of $D(\\mathcal{O}_\\mathcal{C})$ and the functor\n$Rf_* : D' \\to D(\\mathcal{O}_\\mathcal{D})$ is fully faithful.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Formalities on cohomological descent","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7R","source_file":"sites-cohomology.tex","source_line":6314,"source_end_line":6326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6314-L6326","statement_sha256":"185e19e7082c50b8e7c1929ca6f588b2aaf2224c77127457846f2eea6a01b4b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4525,"rank":4525,"depth":19,"x":1141.433,"y":799.667,"cluster":"sheaf-cohomology"},{"id":"stacks:0D7S","tag":"0D7S","title":"Formalities on cohomological descent · Lemma 0D7S","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let K be an object of D(O_C). Assume • f is flat, • K is bounded below, • f^*Rf_*H^q(K) → H^q(K) is an isomorphism. Then f^*Rf_*K → K is an isomorphism.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nLet $K$ be an object of $D(\\mathcal{O}_\\mathcal{C})$. Assume\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $K$ is bounded below,\n\\item $f^*Rf_*H^q(K) \\to H^q(K)$ is an isomorphism.\n\\end{enumerate}\nThen $f^*Rf_*K \\to K$ is an isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Formalities on cohomological descent","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7S","source_file":"sites-cohomology.tex","source_line":6343,"source_end_line":6354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6343-L6354","statement_sha256":"d5bbb38c50368cab1cf142bbda0091b8626c12e9f2a066810ed2f4df90ef17b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4526,"rank":4526,"depth":20,"x":1025.199,"y":598.14,"cluster":"sheaf-cohomology"},{"id":"stacks:0D7T","tag":"0D7T","title":"Formalities on cohomological descent · Lemma 0D7T","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let K be an object of D(O_D). Assume • f is flat, • K is bounded below, • H^q(K) → Rf_*f^*H^q(K) is an isomorphism. Then K → Rf_*f^*K is an isomorphism.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nLet $K$ be an object of $D(\\mathcal{O}_\\mathcal{D})$. Assume\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $K$ is bounded below,\n\\item $H^q(K) \\to Rf_*f^*H^q(K)$ is an isomorphism.\n\\end{enumerate}\nThen $K \\to Rf_*f^*K$ is an isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Formalities on cohomological descent","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7T","source_file":"sites-cohomology.tex","source_line":6367,"source_end_line":6378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6367-L6378","statement_sha256":"ae627cdd278fc69028fdf5ad1ef64ebab45e8c2e54b0348edc476f55a86a9765","origin":"The Stacks Project","memory_eligible":false,"source_rank":4527,"rank":4527,"depth":20,"x":1273.295,"y":680.897,"cluster":"sheaf-cohomology"},{"id":"stacks:0D7U","tag":"0D7U","title":"Formalities on cohomological descent · Lemma 0D7U","summary":"Let f : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. Let A ⊂ Mod(O) and A' ⊂ Mod(O') be weak Serre subcategories. Assume • f is flat, • f^* induces an equivalence of categories A' → A, • F' → Rf_*f^*F' is an isomorphism for F' ∈ Ob(A'). Then f^* : D_A'^+(O') → D_A^+(O) is an equivalence of categories with quasi-inverse given by Rf_* : D_A^+(O) → D_A'^+(O').","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi.\nLet $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$\nand $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$\nbe weak Serre subcategories. Assume\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $f^*$ induces an equivalence of categories\n$\\mathcal{A}' \\to \\mathcal{A}$,\n\\item $\\mathcal{F}' \\to Rf_*f^*\\mathcal{F}'$ is an isomorphism\nfor $\\mathcal{F}' \\in \\Ob(\\mathcal{A}')$.\n\\end{enumerate}\nThen\n$f^* : D_{\\mathcal{A}'}^+(\\mathcal{O}') \\to D_\\mathcal{A}^+(\\mathcal{O})$\nis an equivalence of categories with quasi-inverse given by\n$Rf_* : D_\\mathcal{A}^+(\\mathcal{O}) \\to D_{\\mathcal{A}'}^+(\\mathcal{O}')$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Formalities on cohomological descent","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7U","source_file":"sites-cohomology.tex","source_line":6390,"source_end_line":6408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6390-L6408","statement_sha256":"5a2396d33372dcb9f9ade5a7ae4db578de1f1db79b20ff1007a90df99f7d5f07","origin":"The Stacks Project","memory_eligible":false,"source_rank":4528,"rank":4528,"depth":21,"x":1023.477,"y":760.752,"cluster":"sheaf-cohomology"},{"id":"stacks:0D7V","tag":"0D7V","title":"Formalities on cohomological descent · Lemma 0D7V","summary":"This is analogous to [six-I]. Let f : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. Let A ⊂ Mod(O) and A' ⊂ Mod(O') be weak Serre subcategories. Assume • f is flat, • f^* induces an equivalence of categories A' → A, • F' → Rf_*f^*F' is an isomorphism for F' ∈ Ob(A'), • C, O, A satisfy the assumption of Situation [Tag 0D6R], • C', O', A' satisfy the assumption of Situation [Tag 0D6R]. Then f^* : D_A'(O') → D_A(O) is an equivalence of categories with…","statement_latex":"\\begin{reference}\nThis is analogous to \\cite[Theorem 2.2.3]{six-I}.\n\\end{reference}\nLet $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi.\nLet $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$\nand $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$\nbe weak Serre subcategories. Assume\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $f^*$ induces an equivalence of categories\n$\\mathcal{A}' \\to \\mathcal{A}$,\n\\item $\\mathcal{F}' \\to Rf_*f^*\\mathcal{F}'$ is an isomorphism\nfor $\\mathcal{F}' \\in \\Ob(\\mathcal{A}')$,\n\\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the\nassumption of Situation \\ref{situation-olsson-laszlo},\n\\item $\\mathcal{C}', \\mathcal{O}', \\mathcal{A}'$ satisfy the\nassumption of Situation \\ref{situation-olsson-laszlo}.\n\\end{enumerate}\nThen $f^* : D_{\\mathcal{A}'}(\\mathcal{O}') \\to D_\\mathcal{A}(\\mathcal{O})$\nis an equivalence of categories with quasi-inverse given by\n$Rf_* : D_\\mathcal{A}(\\mathcal{O}) \\to D_{\\mathcal{A}'}(\\mathcal{O}')$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Formalities on cohomological descent","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7V","source_file":"sites-cohomology.tex","source_line":6433,"source_end_line":6457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6433-L6457","statement_sha256":"dd9b8c5b708593b2a0dfa554bcc4b4b3b413b096432818ca5ce040d9fbee7ad5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4529,"rank":4529,"depth":22,"x":1143.628,"y":559.855,"cluster":"sheaf-cohomology"},{"id":"stacks:0D7W","tag":"0D7W","title":"Formalities on cohomological descent · Lemma 0D7W","summary":"This is analogous to [six-I]. Let f : (C, O) → (C', O') be a morphism of ringed sites. Let A ⊂ Mod(O) and A' ⊂ Mod(O') be weak Serre subcategories. Assume • f is flat, • f^* induces an equivalence of categories A' → A, • F' → Rf_*f^*F' is an isomorphism for F' ∈ Ob(A'), • C, O, A satisfy the assumption of Situation [Tag 0D6R], • f : (C, O) → (C', O') and A satisfy the assumption of Situation [Tag 0D6V]. Then f^* : D_A'(O') → D_A(O) is an equivalence of categories with…","statement_latex":"\\begin{reference}\nThis is analogous to \\cite[Theorem 2.2.3]{six-I}.\n\\end{reference}\nLet $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$\nbe a morphism of ringed sites.\nLet $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$\nand $\\mathcal{A}' \\subset \\textit{Mod}(\\mathcal{O}')$\nbe weak Serre subcategories. Assume\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $f^*$ induces an equivalence of categories\n$\\mathcal{A}' \\to \\mathcal{A}$,\n\\item $\\mathcal{F}' \\to Rf_*f^*\\mathcal{F}'$ is an isomorphism\nfor $\\mathcal{F}' \\in \\Ob(\\mathcal{A}')$,\n\\item $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ satisfy the\nassumption of Situation \\ref{situation-olsson-laszlo},\n\\item $f : (\\mathcal{C}, \\mathcal{O}) \\to (\\mathcal{C}', \\mathcal{O}')$\nand $\\mathcal{A}$ satisfy the assumption of\nSituation \\ref{situation-olsson-laszlo-prime}.\n\\end{enumerate}\nThen $f^* : D_{\\mathcal{A}'}(\\mathcal{O}') \\to D_\\mathcal{A}(\\mathcal{O})$\nis an equivalence of categories with quasi-inverse given by\n$Rf_* : D_\\mathcal{A}(\\mathcal{O}) \\to D_{\\mathcal{A}'}(\\mathcal{O}')$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Formalities on cohomological descent","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7W","source_file":"sites-cohomology.tex","source_line":6481,"source_end_line":6506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6481-L6506","statement_sha256":"c665bcb61a278c2a421cd70fe35b1b06f371cdadcaf398ed5da6a5739b1e866d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4530,"rank":4530,"depth":23,"x":1216.681,"y":776.451,"cluster":"sheaf-cohomology"},{"id":"stacks:07A8","tag":"07A8","title":"Comparing two topologies, II · Lemma 07A8","summary":"With ε : (C_τ, O_τ) → (C_τ', O_τ') as above. Let B ⊂ Ob(C) be a subset. Let A ⊂ PMod(O) be a full subcategory. Assume • every object of A is a sheaf for the τ-topology, • A is a weak Serre subcategory of Mod(O_τ), • every object of C has a τ'-covering whose members are elements of B, and • for every U ∈ B we have H^p_τ(U, F) = 0, p > 0 for all F ∈ A. Then A is a weak Serre subcategory of Mod(O_τ') and there is an equivalence of triangulated categories D_A(O_τ) = D_A(O_τ')…","statement_latex":"With $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to\n(\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ as above.\nLet $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset.\nLet $\\mathcal{A} \\subset \\textit{PMod}(\\mathcal{O})$\nbe a full subcategory. Assume\n\\begin{enumerate}\n\\item every object of $\\mathcal{A}$ is a sheaf for the $\\tau$-topology,\n\\item $\\mathcal{A}$ is a weak Serre subcategory of\n$\\textit{Mod}(\\mathcal{O}_\\tau)$,\n\\item every object of $\\mathcal{C}$ has a $\\tau'$-covering whose\nmembers are elements of $\\mathcal{B}$, and\n\\item for every $U \\in \\mathcal{B}$ we have $H^p_\\tau(U, \\mathcal{F}) = 0$,\n$p > 0$ for all $\\mathcal{F} \\in \\mathcal{A}$.\n\\end{enumerate}\nThen $\\mathcal{A}$ is a weak Serre subcategory of\n$\\textit{Mod}(\\mathcal{O}_{\\tau'})$ and there is an equivalence\nof triangulated categories\n$D_\\mathcal{A}(\\mathcal{O}_\\tau) = D_\\mathcal{A}(\\mathcal{O}_{\\tau'})$\ngiven by $\\epsilon^*$ and $R\\epsilon_*$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing two topologies, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07A8","source_file":"sites-cohomology.tex","source_line":6550,"source_end_line":6571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6550-L6571","statement_sha256":"d0111d48e8a778ff92e497a3e1252b931fc8a2238b37eab99da98eb882ba814a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4531,"rank":4531,"depth":24,"x":988.337,"y":658.035,"cluster":"sheaf-cohomology"},{"id":"stacks:0F18","tag":"0F18","title":"Comparing two topologies, II · Lemma 0F18","summary":"With ε : (C_τ, O_τ) → (C_τ', O_τ') as above. Let A be a set and for α ∈ A let xymatrix E_α ar[d] ar[r] & Y_α ar[d] Z_α ar[r] & X_α be a commutative diagram in the category C. Assume that • a τ'-sheaf F' is a τ-sheaf if F'(X_α) = F'(Z_α) ×_F'(E_α) F'(Y_α) for all α, • for K' in D(O_τ') in the essential image of Rε_* the maps c^K'_X_α, Z_α, Y_α, E_α of Lemma [Tag 0F16] are isomorphisms for all α. Then K' ∈ D^+(O_τ') is in the essential image of Rε_* if and only if the maps…","statement_latex":"With $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to\n(\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ as above.\nLet $A$ be a set and for $\\alpha \\in A$ let\n$$\n\\xymatrix{\nE_\\alpha \\ar[d] \\ar[r] & Y_\\alpha \\ar[d] \\\\\nZ_\\alpha \\ar[r] & X_\\alpha\n}\n$$\nbe a commutative diagram in the category $\\mathcal{C}$. Assume that\n\\begin{enumerate}\n\\item a $\\tau'$-sheaf $\\mathcal{F}'$ is a $\\tau$-sheaf if\n$\\mathcal{F}'(X_\\alpha) =\n\\mathcal{F}'(Z_\\alpha) \\times_{\\mathcal{F}'(E_\\alpha)} \n\\mathcal{F}'(Y_\\alpha)$ for all $\\alpha$,\n\\item for $K'$ in $D(\\mathcal{O}_{\\tau'})$ in the essential image\nof $R\\epsilon_*$ the maps $c^{K'}_{X_\\alpha, Z_\\alpha, Y_\\alpha, E_\\alpha}$\nof Lemma \\ref{lemma-c-square}\nare isomorphisms for all $\\alpha$.\n\\end{enumerate}\nThen $K' \\in D^+(\\mathcal{O}_{\\tau'})$ is in\nthe essential image of $R\\epsilon_*$ if and only if\nthe maps $c^{K'}_{X_\\alpha, Z_\\alpha, Y_\\alpha, E_\\alpha}$\nare isomorphisms for all $\\alpha$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing two topologies, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F18","source_file":"sites-cohomology.tex","source_line":6669,"source_end_line":6695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6669-L6695","statement_sha256":"3b00a2af708eb240e862cfb71676b6fd251a6141d9897b789db50c8184494b41","origin":"The Stacks Project","memory_eligible":false,"source_rank":4532,"rank":4532,"depth":14,"x":1252.281,"y":615.731,"cluster":"sheaf-cohomology"},{"id":"stacks:0F19","tag":"0F19","title":"Comparing two topologies, II · Lemma 0F19","summary":"With ε : (C_τ, O_τ) → (C_τ', O_τ') as above. Let xymatrix E ar[d] ar[r] & Y ar[d] Z ar[r] & X be a commutative diagram in the category C such that • h_X^\\# = h_Y^\\# amalg_h_E^\\# h_Z^\\#, and • h_E^\\# → h_Y^\\# is injective where ^\\# denotes τ-sheafification. Then for K' ∈ D(O_τ') in the essential image of Rε_* the map c^K'_X, Z, Y, E of Lemma [Tag 0F16] (using the τ'-topology) is an isomorphism.","statement_latex":"With $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to\n(\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ as above. Let\n$$\n\\xymatrix{\nE \\ar[d] \\ar[r] & Y \\ar[d] \\\\\nZ \\ar[r] & X\n}\n$$\nbe a commutative diagram in the category $\\mathcal{C}$ such that\n\\begin{enumerate}\n\\item $h_X^\\# = h_Y^\\# \\amalg_{h_E^\\#} h_Z^\\#$, and\n\\item $h_E^\\# \\to h_Y^\\#$ is injective\n\\end{enumerate}\nwhere ${}^\\#$ denotes $\\tau$-sheafification. Then for\n$K' \\in D(\\mathcal{O}_{\\tau'})$ in the essential image of\n$R\\epsilon_*$ the map $c^{K'}_{X, Z, Y, E}$ of Lemma \\ref{lemma-c-square}\n(using the $\\tau'$-topology) is an isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing two topologies, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F19","source_file":"sites-cohomology.tex","source_line":6757,"source_end_line":6776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6757-L6776","statement_sha256":"c0a45a59cb639351b6f7e79f01ab974580bf7c6a8704c81837c424c178dfbc86","origin":"The Stacks Project","memory_eligible":false,"source_rank":4533,"rank":4533,"depth":5,"x":1091.466,"y":796.926,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZ9","tag":"0EZ9","title":"Comparing cohomology · Lemma 0EZ9","summary":"In Situation [Tag 0EZ3] for X in C denote A_X the objects of Ab(C_τ/X) of the form ε_X^-1F' with F' in A'_X. Then • for F in Ab(C_τ/X) we have F ∈ A_X ⇔ ε_X, *F ∈ A'_X, and • f_τ^-1 sends A_Y into A_X for any morphism f : X → Y of C.","statement_latex":"In Situation \\ref{situation-compare} for $X$ in $\\mathcal{C}$\ndenote $\\mathcal{A}_X$\nthe objects of $\\textit{Ab}(\\mathcal{C}_\\tau/X)$ of the form\n$\\epsilon_X^{-1}\\mathcal{F}'$ with $\\mathcal{F}'$ in $\\mathcal{A}'_X$.\nThen\n\\begin{enumerate}\n\\item for $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C}_\\tau/X)$\nwe have $\\mathcal{F} \\in \\mathcal{A}_X \\Leftrightarrow\n\\epsilon_{X, *}\\mathcal{F} \\in \\mathcal{A}'_X$, and\n\\item $f_\\tau^{-1}$ sends $\\mathcal{A}_Y$ into $\\mathcal{A}_X$\nfor any morphism $f : X \\to Y$ of $\\mathcal{C}$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZ9","source_file":"sites-cohomology.tex","source_line":6881,"source_end_line":6895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6881-L6895","statement_sha256":"45dc2abd8987dc2b15ee2f8d0c045780d5587bbb72b255e6ff49b2d6170d5c9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4534,"rank":4534,"depth":0,"x":1064.302,"y":571.777,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZA","tag":"0EZA","title":"Comparing cohomology · Lemma 0EZA","summary":"In Situation [Tag 0EZ3] assume (V_n) holds. For f : X → Y in P and F in A_X we have R^if_τ', *ε_X, *F = ε_Y, *R^if_τ, *F for i ≤ n.","statement_latex":"In Situation \\ref{situation-compare} assume $(V_n)$ holds.\nFor $f : X \\to Y$ in $\\mathcal{P}$ and $\\mathcal{F}$ in $\\mathcal{A}_X$\nwe have $R^if_{\\tau', *}\\epsilon_{X, *}\\mathcal{F} =\n\\epsilon_{Y, *}R^if_{\\tau, *}\\mathcal{F}$ for $i \\leq n$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZA","source_file":"sites-cohomology.tex","source_line":6913,"source_end_line":6919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L6913-L6919","statement_sha256":"fdea9e8d7e33fe08a3ecd6b36fafee54808becb9cf63698ff772fd44c7238421","origin":"The Stacks Project","memory_eligible":false,"source_rank":4535,"rank":4535,"depth":25,"x":1265.648,"y":722.578,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZB","tag":"0EZB","title":"Comparing cohomology · Lemma 0EZB","summary":"In Situation [Tag 0EZ3] if (V_n) holds, then for X in C and L ∈ D(C_τ'/X) with H^i(L) = 0 for i < 0 and H^i(L) in A'_X for 0 ≤ i ≤ n we have H^n_τ'(X, L) = H^n_τ(X, ε_X^-1L).","statement_latex":"In Situation \\ref{situation-compare} if $(V_n)$ holds, then\nfor $X$ in $\\mathcal{C}$ and $L \\in D(\\mathcal{C}_{\\tau'}/X)$\nwith $H^i(L) = 0$ for $i < 0$ and $H^i(L)$ in $\\mathcal{A}'_X$\nfor $0 \\leq i \\leq n$ we have\n$H^n_{\\tau'}(X, L) = H^n_\\tau(X, \\epsilon_X^{-1}L)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZB","source_file":"sites-cohomology.tex","source_line":7028,"source_end_line":7035,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7028-L7035","statement_sha256":"ec4aacdcdfe43396c15ad0ae9d82fffd71beb102a757e974565b4fa348903c29","origin":"The Stacks Project","memory_eligible":false,"source_rank":4536,"rank":4536,"depth":21,"x":995.564,"y":725.632,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZC","tag":"0EZC","title":"Comparing cohomology · Lemma 0EZC","summary":"In Situation [Tag 0EZ3] if (V_n) holds, then for X in C and F in A_X the map H^n + 1_τ'(X, ε_X, *F) → H^n + 1_τ(X, F) is injective with image those classes which become trivial on a τ'-covering of X.","statement_latex":"In Situation \\ref{situation-compare} if $(V_n)$ holds, then for\n$X$ in $\\mathcal{C}$ and $\\mathcal{F}$ in $\\mathcal{A}_X$ the map\n$H^{n + 1}_{\\tau'}(X, \\epsilon_{X, *}\\mathcal{F}) \\to\nH^{n + 1}_\\tau(X, \\mathcal{F})$\nis injective with image those classes which become trivial on\na $\\tau'$-covering of $X$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZC","source_file":"sites-cohomology.tex","source_line":7060,"source_end_line":7068,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7060-L7068","statement_sha256":"598e614ca5313404c7c83e7bd7bf7d133dbe9b0135ff16abae8df7dd33742b09","origin":"The Stacks Project","memory_eligible":false,"source_rank":4537,"rank":4537,"depth":25,"x":1192.514,"y":569.93,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZD","tag":"0EZD","title":"Comparing cohomology · Lemma 0EZD","summary":"In Situation [Tag 0EZ3] let f : X → Y be in P such that (X → Y) is a τ-covering. Let F' be in A'_Y. If n ≥ 0 and theta ∈ Equalizer( xymatrix H^n + 1_τ'(X, F') ar@<1ex>[r] ar@<-1ex>[r] & H^n + 1_τ'(X ×_Y X, F') ) then there exists a τ'-covering (Y_i → Y) such that theta restricts to zero in H^n + 1_τ'(Y_i ×_Y X, F').","statement_latex":"In Situation \\ref{situation-compare} let $f : X \\to Y$\nbe in $\\mathcal{P}$ such that $\\{X \\to Y\\}$ is a $\\tau$-covering.\nLet $\\mathcal{F}'$ be in $\\mathcal{A}'_Y$. If $n \\geq 0$ and\n$$\n\\theta \\in\n\\text{Equalizer}\\left(\n\\xymatrix{\nH^{n + 1}_{\\tau'}(X, \\mathcal{F}')\n\\ar@<1ex>[r] \\ar@<-1ex>[r] &\nH^{n + 1}_{\\tau'}(X \\times_Y X, \\mathcal{F}')\n}\n\\right)\n$$\nthen there exists a $\\tau'$-covering $\\{Y_i \\to Y\\}$\nsuch that $\\theta$ restricts to zero in\n$H^{n + 1}_{\\tau'}(Y_i \\times_Y X, \\mathcal{F}')$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZD","source_file":"sites-cohomology.tex","source_line":7089,"source_end_line":7107,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7089-L7107","statement_sha256":"e22df5f839495fb9b9b091a70e72afbac0951ff34aaf8078fa5777704e502efb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4538,"rank":4538,"depth":11,"x":1172.472,"y":796.784,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZE","tag":"0EZE","title":"Comparing cohomology · Lemma 0EZE","summary":"In Situation [Tag 0EZ3] we have (V_n) ⇒ (V_n + 1).","statement_latex":"In Situation \\ref{situation-compare} we have $(V_n) \\Rightarrow (V_{n + 1})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZE","source_file":"sites-cohomology.tex","source_line":7147,"source_end_line":7150,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7147-L7150","statement_sha256":"a771a7b087ee1e034289032bc1046bd284f440c7e295d435fe04a198256b4919","origin":"The Stacks Project","memory_eligible":false,"source_rank":4539,"rank":4539,"depth":26,"x":1004.609,"y":617.907,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZF","tag":"0EZF","title":"Comparing cohomology · Lemma 0EZF","summary":"In Situation [Tag 0EZ3] we have that (V_n) is true for all n. Moreover: • For X in C and K' ∈ D^+_A'_X(C_τ'/X) the map K' → Rε_X, *(ε_X^-1K') is an isomorphism. • For f : X → Y in P and K' ∈ D^+_A'_X(C_τ'/X) we have Rf_τ', *K' ∈ D^+_A'_X(C_τ'/Y) and ε_Y^-1(Rf_τ', *K') = Rf_τ, *(ε_X^-1K').","statement_latex":"In Situation \\ref{situation-compare} we have that\n$(V_n)$ is true for all $n$. Moreover:\n\\begin{enumerate}\n\\item For $X$ in $\\mathcal{C}$ and\n$K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$ the map\n$K' \\to R\\epsilon_{X, *}(\\epsilon_X^{-1}K')$ is an isomorphism.\n\\item For $f : X \\to Y$ in $\\mathcal{P}$ and\n$K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$ we have\n$Rf_{\\tau', *}K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/Y)$ and\n$\\epsilon_Y^{-1}(Rf_{\\tau', *}K') = Rf_{\\tau, *}(\\epsilon_X^{-1}K')$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZF","source_file":"sites-cohomology.tex","source_line":7291,"source_end_line":7304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7291-L7304","statement_sha256":"458f294ecdb293978fc64a07ca4b33b5e7db07c4fba0fe06f1324777441cf8fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4540,"rank":4540,"depth":27,"x":1272.575,"y":654.605,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZG","tag":"0EZG","title":"Comparing cohomology · Lemma 0EZG","summary":"In Situation [Tag 0EZ3]. For any X in C the category A_X ⊂ Ab(C_τ/X) is a weak Serre subcategory and the functor Rε_X, * : D^+_A_X(C_τ/X) → D^+_A'_X(C_τ'/X) is an equivalence with quasi-inverse given by ε_X^-1.","statement_latex":"In Situation \\ref{situation-compare}. For any $X$ in\n$\\mathcal{C}$ the category\n$\\mathcal{A}_X \\subset \\textit{Ab}(\\mathcal{C}_\\tau/X)$\nis a weak Serre subcategory and the functor\n$$\nR\\epsilon_{X, *} :\nD^+_{\\mathcal{A}_X}(\\mathcal{C}_\\tau/X)\n\\longrightarrow\nD^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)\n$$\nis an equivalence with quasi-inverse given by $\\epsilon_X^{-1}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZG","source_file":"sites-cohomology.tex","source_line":7360,"source_end_line":7373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7360-L7373","statement_sha256":"a2d97916fd2579164dbb09bfc9700c94c5a5ecf3cbbb84b4c8d94729106cb04c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4541,"rank":4541,"depth":28,"x":1045.183,"y":779.752,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZH","tag":"0EZH","title":"Comparing cohomology · Lemma 0EZH","summary":"In Situation [Tag 0EZ3]. Let X be in C. • for F' in A'_X we have H^n_τ'(X, F') = H^n_τ(X, ε_X^-1F'), • for K' ∈ D^+_A'_X(C_τ'/X) we have H^n_τ'(X, K') = H^n_τ(X, ε_X^-1K').","statement_latex":"In Situation \\ref{situation-compare}. Let $X$ be in $\\mathcal{C}$.\n\\begin{enumerate}\n\\item for $\\mathcal{F}'$ in $\\mathcal{A}'_X$ we have\n$H^n_{\\tau'}(X, \\mathcal{F}') = H^n_\\tau(X, \\epsilon_X^{-1}\\mathcal{F}')$,\n\\item for $K' \\in D^+_{\\mathcal{A}'_X}(\\mathcal{C}_{\\tau'}/X)$\nwe have $H^n_{\\tau'}(X, K') = H^n_\\tau(X, \\epsilon_X^{-1}K')$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Comparing cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZH","source_file":"sites-cohomology.tex","source_line":7424,"source_end_line":7433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7424-L7433","statement_sha256":"26cbce90cff83dde01ff23d7f8e1338bb191f1e401ecb3c2a5e4f4cd977e335c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4542,"rank":4542,"depth":28,"x":1112.303,"y":558.164,"cluster":"sheaf-cohomology"},{"id":"stacks:09WZ","tag":"09WZ","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 09WZ","summary":"The category LC has fibre products and a final object and hence has arbitrary finite limits. Given morphisms X → Z and Y → Z in LC with X and Y quasi-compact, then X ×_Z Y is quasi-compact.","statement_latex":"The category $\\textit{LC}$ has fibre products and a final object and hence\nhas arbitrary finite limits. Given morphisms $X \\to Z$ and $Y \\to Z$\nin $\\textit{LC}$ with\n$X$ and $Y$ quasi-compact, then $X \\times_Z Y$ is quasi-compact.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WZ","source_file":"sites-cohomology.tex","source_line":7473,"source_end_line":7479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7473-L7479","statement_sha256":"2e820a8b9766e178a1333a8e44beb83740ce33685610f170eaca98699145eb58","origin":"The Stacks Project","memory_eligible":false,"source_rank":4543,"rank":4543,"depth":2,"x":1241.165,"y":759.898,"cluster":"sheaf-cohomology"},{"id":"stacks:09X0","tag":"09X0","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Definition 09X0","summary":"Let (f_i : X_i → X) be a family of morphisms with fixed target in the category LC. We say this family is a qc covering if for every x ∈ X there exist i_1, …, i_n ∈ I and quasi-compact subsets E_j ⊂ X_i_j such that ⋃ f_i_j(E_j) is a neighbourhood of x.","statement_latex":"Let $\\{f_i : X_i \\to X\\}$ be a family of morphisms with fixed target\nin the category $\\textit{LC}$. We say this family is a\n{\\it qc covering}\\footnote{This is nonstandard notation.\nWe chose it to remind the reader of fpqc coverings of schemes.}\nif for every $x \\in X$ there exist $i_1, \\ldots, i_n \\in I$ and\nquasi-compact subsets $E_j \\subset X_{i_j}$ such that\n$\\bigcup f_{i_j}(E_j)$ is a neighbourhood of $x$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09X0","source_file":"sites-cohomology.tex","source_line":7508,"source_end_line":7517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7508-L7517","statement_sha256":"e58039450cdb401c9ff34b209efee893a8de2bef27e3f0809924b4442a7cf730","origin":"The Stacks Project","memory_eligible":false,"source_rank":4544,"rank":4544,"depth":0,"x":983.594,"y":684.168,"cluster":"sheaf-cohomology"},{"id":"stacks:09X1","tag":"09X1","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 09X1","summary":"Let X be a Hausdorff and locally quasi-compact space, in other words, an object of LC. • If X' → X is an isomorphism in LC then (X' → X) is a qc covering. • If (f_i : X_i → X)_i∈ I is a qc covering and for each i we have a qc covering (g_ij : X_ij → X_i)_j∈ J_i, then (X_ij → X)_i ∈ I, j∈ J_i is a qc covering. • If (X_i → X)_i∈ I is a qc covering and X' → X is a morphism of LC then (X' ×_X X_i → X')_i∈ I is a qc covering.","statement_latex":"Let $X$ be a Hausdorff and locally quasi-compact space, in other words,\nan object of $\\textit{LC}$.\n\\begin{enumerate}\n\\item If $X' \\to X$ is an isomorphism in $\\textit{LC}$ then\n$\\{X' \\to X\\}$ is a qc covering.\n\\item If $\\{f_i : X_i \\to X\\}_{i\\in I}$ is a qc covering and for each\n$i$ we have a qc covering $\\{g_{ij} : X_{ij} \\to X_i\\}_{j\\in J_i}$, then\n$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a qc covering.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a qc covering\nand $X' \\to X$ is a morphism of $\\textit{LC}$ then\n$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a qc covering.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09X1","source_file":"sites-cohomology.tex","source_line":7524,"source_end_line":7538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7524-L7538","statement_sha256":"59d2fd3545a3a597f188bcd113132b25b1b674e21e0f8f4f8aaf120a90b120b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4545,"rank":4545,"depth":0,"x":1234.736,"y":593.745,"cluster":"sheaf-cohomology"},{"id":"stacks:09X5","tag":"09X5","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 09X5","summary":"Let f : X → Y be a morphism of LC. If f is proper and surjective, then (f : X → Y) is a qc covering.","statement_latex":"Let $f : X \\to Y$ be a morphism of $\\textit{LC}$.\nIf $f$ is proper and surjective, then $\\{f : X \\to Y\\}$\nis a qc covering.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09X5","source_file":"sites-cohomology.tex","source_line":7584,"source_end_line":7589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7584-L7589","statement_sha256":"d145fa5d108133c6440976e23a5c73ea9bdad53a35e3a229fe1394f6ff693673","origin":"The Stacks Project","memory_eligible":false,"source_rank":4546,"rank":4546,"depth":3,"x":1122.123,"y":803.185,"cluster":"sheaf-cohomology"},{"id":"stacks:09X3","tag":"09X3","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 09X3","summary":"Let X be an object of LC_qc. Let F be a sheaf on X. The rule LC_qc/X → Sets, (f : Y → X) ↦ Γ(Y, f^-1F) is a sheaf and a fortiori also a sheaf on LC_Zar/X. This sheaf is equal to π_X^-1F on LC_Zar/X and ε_X^-1π_X^-1F on LC_qc/X.","statement_latex":"Let $X$ be an object of $\\textit{LC}_{qc}$. Let $\\mathcal{F}$ be a\nsheaf on $X$. The rule\n$$\n\\textit{LC}_{qc}/X \\longrightarrow \\textit{Sets},\\quad\n(f : Y \\to X) \\longmapsto \\Gamma(Y, f^{-1}\\mathcal{F})\n$$\nis a sheaf and a fortiori also a sheaf on $\\textit{LC}_{Zar}/X$.\nThis sheaf is equal to\n$\\pi_X^{-1}\\mathcal{F}$ on $\\textit{LC}_{Zar}/X$ and\n$\\epsilon_X^{-1}\\pi_X^{-1}\\mathcal{F}$ on $\\textit{LC}_{qc}/X$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09X3","source_file":"sites-cohomology.tex","source_line":7674,"source_end_line":7686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7674-L7686","statement_sha256":"f3e3b95b72d0eaf1f448c0b475c3571a5b6e5771729d14b043c797d411dbde8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4547,"rank":4547,"depth":3,"x":1036.631,"y":584.577,"cluster":"sheaf-cohomology"},{"id":"stacks:0DCU","tag":"0DCU","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 0DCU","summary":"Let X be an object of LC_Zar. Then • for F ∈ Ab(X) we have H^n_Zar(X, π_X^-1F) = H^n(X, F), • π_X, * : Ab(LC_Zar/X) → Ab(X) is exact, • the unit id → π_X, * ∘ π_X^-1 of the adjunction is an isomorphism, and • for K ∈ D(X) the canonical map K → Rπ_X, * π_X^-1K is an isomorphism. Let f : X → Y be a morphism of LC_Zar. Then • [(5)] there is a commutative diagram xymatrix Sh(LC_Zar/X) ar[r]_f_Zar ar[d]_π_X & Sh(LC_Zar/Y) ar[d]^π_Y Sh(X_Zar) ar[r]^f & Sh(Y_Zar) of topoi, •…","statement_latex":"Let $X$ be an object of $\\textit{LC}_{Zar}$. Then\n\\begin{enumerate}\n\\item for $\\mathcal{F} \\in \\textit{Ab}(X)$ we have\n$H^n_{Zar}(X, \\pi_X^{-1}\\mathcal{F}) = H^n(X, \\mathcal{F})$,\n\\item $\\pi_{X, *} : \\textit{Ab}(\\textit{LC}_{Zar}/X) \\to \\textit{Ab}(X)$\nis exact,\n\\item the unit $\\text{id} \\to \\pi_{X, *} \\circ \\pi_X^{-1}$\nof the adjunction is an isomorphism, and\n\\item for $K \\in D(X)$ the canonical map\n$K \\to R\\pi_{X, *} \\pi_X^{-1}K$ is an isomorphism.\n\\end{enumerate}\nLet $f : X \\to Y$ be a morphism of $\\textit{LC}_{Zar}$. Then\n\\begin{enumerate}\n\\item[(5)] there is a commutative diagram\n$$\n\\xymatrix{\n\\Sh(\\textit{LC}_{Zar}/X) \\ar[r]_{f_{Zar}} \\ar[d]_{\\pi_X} &\n\\Sh(\\textit{LC}_{Zar}/Y) \\ar[d]^{\\pi_Y} \\\\\n\\Sh(X_{Zar}) \\ar[r]^f &\n\\Sh(Y_{Zar})\n}\n$$\nof topoi,\n\\item[(6)] for $L \\in D^+(Y)$ we have\n$H^n_{Zar}(X, \\pi_Y^{-1}L) = H^n(X, f^{-1}L)$,\n\\item[(7)] if $f$ is proper, then we have\n\\begin{enumerate}\n\\item $\\pi_Y^{-1} \\circ f_* = f_{Zar, *} \\circ \\pi_X^{-1}$ as functors\n$\\Sh(X) \\to \\Sh(\\textit{LC}_{Zar}/Y)$,\n\\item $\\pi_Y^{-1} \\circ Rf_* = Rf_{Zar, *} \\circ \\pi_X^{-1}$ as\nfunctors $D^+(X) \\to D^+(\\textit{LC}_{Zar}/Y)$.\n\\end{enumerate}\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCU","source_file":"sites-cohomology.tex","source_line":7769,"source_end_line":7804,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7769-L7804","statement_sha256":"d60899f27ee3a52b9cf5ad9a04992c610b9c966c7d7bf9ed3aa5a6501be04417","origin":"The Stacks Project","memory_eligible":false,"source_rank":4548,"rank":4548,"depth":28,"x":1275.765,"y":697.405,"cluster":"sheaf-cohomology"},{"id":"stacks:0D92","tag":"0D92","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 0D92","summary":"Let f : X → Y be a morphism of LC_qc. Then there are commutative diagrams of topoi vcenter xymatrix Sh(LC_qc/X) ar[r]_f_qc ar[d]_ε_X & Sh(LC_qc/Y) ar[d]^ε_Y Sh(LC_Zar/X) ar[r]^f_Zar & Sh(LC_Zar/Y) and vcenter xymatrix Sh(LC_qc/X) ar[r]_f_qc ar[d]_a_X & Sh(LC_qc/Y) ar[d]^a_Y Sh(X) ar[r]^f & Sh(Y) with a_X = π_X ∘ ε_X, a_Y = π_X ∘ ε_X. If f is proper, then a_Y^-1 ∘ f_* = f_qc, * ∘ a_X^-1.","statement_latex":"Let $f : X \\to Y$ be a morphism of $\\textit{LC}_{qc}$.\nThen there are commutative diagrams of topoi\n$$\n\\vcenter{\n\\xymatrix{\n\\Sh(\\textit{LC}_{qc}/X) \\ar[r]_{f_{qc}} \\ar[d]_{\\epsilon_X} &\n\\Sh(\\textit{LC}_{qc}/Y) \\ar[d]^{\\epsilon_Y} \\\\\n\\Sh(\\textit{LC}_{Zar}/X) \\ar[r]^{f_{Zar}} &\n\\Sh(\\textit{LC}_{Zar}/Y)\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\n\\Sh(\\textit{LC}_{qc}/X) \\ar[r]_{f_{qc}} \\ar[d]_{a_X} &\n\\Sh(\\textit{LC}_{qc}/Y) \\ar[d]^{a_Y} \\\\\n\\Sh(X) \\ar[r]^f &\n\\Sh(Y)\n}\n}\n$$\nwith $a_X = \\pi_X \\circ \\epsilon_X$, $a_Y = \\pi_X \\circ \\epsilon_X$.\nIf $f$ is proper, then $a_Y^{-1} \\circ f_* = f_{qc, *} \\circ a_X^{-1}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D92","source_file":"sites-cohomology.tex","source_line":7925,"source_end_line":7950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7925-L7950","statement_sha256":"8e725a68f6e514988ad1d06e8480b29cf461273b9362e3643a0a5cb50d71e18a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4549,"rank":4549,"depth":29,"x":1008.369,"y":749.963,"cluster":"sheaf-cohomology"},{"id":"stacks:0EZI","tag":"0EZI","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 0EZI","summary":"Consider the comparison morphism ε : LC_qc → LC_Zar. Let P denote the class of proper maps of topological spaces. For X in LC_Zar denote A'_X ⊂ Ab(LC_Zar/X) the full subcategory consisting of sheaves of the form π_X^-1F with F in Ab(X). Then ([Tag 0EZ4]), ([Tag 0EZ5]), ([Tag 0EZ6]), ([Tag 0EZ7]), and ([Tag 0EZ8]) of Situation [Tag 0EZ3] hold.","statement_latex":"Consider the comparison morphism\n$\\epsilon : \\textit{LC}_{qc} \\to \\textit{LC}_{Zar}$.\nLet $\\mathcal{P}$ denote the class of proper maps of topological spaces.\nFor $X$ in $\\textit{LC}_{Zar}$ denote\n$\\mathcal{A}'_X \\subset \\textit{Ab}(\\textit{LC}_{Zar}/X)$\nthe full subcategory consisting of sheaves of the form\n$\\pi_X^{-1}\\mathcal{F}$ with $\\mathcal{F}$ in $\\textit{Ab}(X)$.\nThen\n(\\ref{item-base-change-P}),\n(\\ref{item-restriction-A}),\n(\\ref{item-A-sheaf}),\n(\\ref{item-A-and-P}), and\n(\\ref{item-refine-tau-by-P})\nof Situation \\ref{situation-compare} hold.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZI","source_file":"sites-cohomology.tex","source_line":7985,"source_end_line":8001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L7985-L8001","statement_sha256":"219f009c04706f7e65e03e0cd76541f0e2c0159637525f5c2e76edfabbbe999c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4550,"rank":4550,"depth":29,"x":1163.468,"y":559.246,"cluster":"sheaf-cohomology"},{"id":"stacks:0DCY","tag":"0DCY","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 0DCY","summary":"With notation as above. • For X ∈ Ob(LC_qc) and an abelian sheaf F on X we have ε_X, *a_X^-1F = π_X^-1F and R^iε_X, *(a_X^-1F) = 0 for i > 0. • For a proper morphism f : X → Y in LC_qc and abelian sheaf F on X we have a_Y^-1(R^if_*F) = R^if_qc, *(a_X^-1F) for all i. • For X ∈ Ob(LC_qc) and K in D^+(X) the map π_X^-1K → Rε_X, *(a_X^-1K) is an isomorphism. • For a proper morphism f : X → Y in LC_qc and K in D^+(X) we have a_Y^-1(Rf_*K) = Rf_qc, *(a_X^-1K).","statement_latex":"With notation as above.\n\\begin{enumerate}\n\\item For $X \\in \\Ob(\\textit{LC}_{qc})$ and an abelian sheaf $\\mathcal{F}$\non $X$ we have $\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$\nand $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.\n\\item For a proper morphism $f : X \\to Y$ in $\\textit{LC}_{qc}$\nand abelian sheaf $\\mathcal{F}$ on $X$ we have\n$a_Y^{-1}(R^if_*\\mathcal{F}) = R^if_{qc, *}(a_X^{-1}\\mathcal{F})$\nfor all $i$.\n\\item For $X \\in \\Ob(\\textit{LC}_{qc})$ and $K$ in $D^+(X)$ the map\n$\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism.\n\\item For a proper morphism $f : X \\to Y$ in $\\textit{LC}_{qc}$\nand $K$ in $D^+(X)$ we have $a_Y^{-1}(Rf_*K) = Rf_{qc, *}(a_X^{-1}K)$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCY","source_file":"sites-cohomology.tex","source_line":8048,"source_end_line":8064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8048-L8064","statement_sha256":"6fdb08fbb12c191579877bdd0f711d96ddcf619011e6f413e0295a865866b567","origin":"The Stacks Project","memory_eligible":false,"source_rank":4551,"rank":4551,"depth":30,"x":1202.518,"y":788.165,"cluster":"sheaf-cohomology"},{"id":"stacks:0D91","tag":"0D91","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 0D91","summary":"Let X be an object of LC_qc. For K ∈ D^+(X) the map K → Ra_X, *a_X^-1K is an isomorphism with a_X : Sh(LC_qc/X) → Sh(X) as above.","statement_latex":"Let $X$ be an object of $\\textit{LC}_{qc}$. For $K \\in D^+(X)$ the map\n$$\nK \\longrightarrow Ra_{X, *}a_X^{-1}K\n$$\nis an isomorphism with $a_X : \\Sh(\\textit{LC}_{qc}/X) \\to \\Sh(X)$ as above.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D91","source_file":"sites-cohomology.tex","source_line":8092,"source_end_line":8099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8092-L8099","statement_sha256":"798308723215f660673e63137db569a24f321c295c3b75ceb4cb13b6ecf7bfba","origin":"The Stacks Project","memory_eligible":false,"source_rank":4552,"rank":4552,"depth":31,"x":989.371,"y":641.342,"cluster":"sheaf-cohomology"},{"id":"stacks:09X4","tag":"09X4","title":"Cohomology on Hausdorff and locally quasi-compact spaces · Lemma 09X4","summary":"With X ∈ Ob(LC_qc) and a_X : Sh(LC_qc/X) → Sh(X) as above: • for an abelian sheaf F on X we have H^n(X, F) = H^n_qc(X, a_X^-1F), • for K ∈ D^+(X) we have H^n(X, K) = H^n_qc(X, a_X^-1K). For example, if A is an abelian group, then we have H^n(X, underlineA) = H^n_qc(X, underlineA).","statement_latex":"With $X \\in \\Ob(\\textit{LC}_{qc})$ and\n$a_X : \\Sh(\\textit{LC}_{qc}/X) \\to \\Sh(X)$ as above:\n\\begin{enumerate}\n\\item for an abelian sheaf $\\mathcal{F}$ on $X$ we have\n$H^n(X, \\mathcal{F}) = H^n_{qc}(X, a_X^{-1}\\mathcal{F})$,\n\\item for $K \\in D^+(X)$ we have $H^n(X, K) = H^n_{qc}(X, a_X^{-1}K)$.\n\\end{enumerate}\nFor example, if $A$ is an abelian group, then we have\n$H^n(X, \\underline{A}) = H^n_{qc}(X, \\underline{A})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cohomology on Hausdorff and locally quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09X4","source_file":"sites-cohomology.tex","source_line":8127,"source_end_line":8138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8127-L8138","statement_sha256":"29aad6661e4b1ed6cd0e65e383332aa25d30099e6c44d50ee48c398b7fe682a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4553,"rank":4553,"depth":32,"x":1264.95,"y":628.648,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPK","tag":"0FPK","title":"Cup product · Lemma 0FPK","summary":"In the situation above the following diagram commutes xymatrix f_*K^bullet ⊗_O_D^L f_*M^bullet ar[r] ar[d] & Rf_*K^bullet ⊗_O_D^L Rf_*M^bullet ar[d]^Remark [Tag 0B6C] Tot( f_*K^bullet ⊗_O_D f_*M^bullet) ar[d]_naive cup product & Rf_*(K^bullet ⊗_O_C^L M^bullet) ar[d] f_*Tot(K^bullet ⊗_O_C M^bullet) ar[r] & Rf_*Tot(K^bullet ⊗_O_C M^bullet)","statement_latex":"In the situation above the following diagram commutes\n$$\n\\xymatrix{\nf_*\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}\nf_*\\mathcal{M}^\\bullet \\ar[r] \\ar[d]\n&\nRf_*\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_\\mathcal{D}}^\\mathbf{L}\nRf_*\\mathcal{M}^\\bullet \\ar[d]^{\\text{Remark \\ref{remark-cup-product}}} \\\\\n\\text{Tot}(\nf_*\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_\\mathcal{D}}\nf_*\\mathcal{M}^\\bullet) \\ar[d]_{\\text{naive cup product}} &\nRf_*(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_\\mathcal{C}}^\\mathbf{L}\n\\mathcal{M}^\\bullet) \\ar[d] \\\\\nf_*\\text{Tot}(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_\\mathcal{C}}\n\\mathcal{M}^\\bullet) \\ar[r] &\nRf_*\\text{Tot}(\\mathcal{K}^\\bullet\n\\otimes_{\\mathcal{O}_\\mathcal{C}}\n\\mathcal{M}^\\bullet)\n}\n$$","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPK","source_file":"sites-cohomology.tex","source_line":8243,"source_end_line":8270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8243-L8270","statement_sha256":"b8a931e5c9718c6c19509e0573992a55d2bcc2abaedc8d7f74b60fea9980c1fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":4554,"rank":4554,"depth":19,"x":1071.715,"y":794.58,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPL","tag":"0FPL","title":"Cup product · Lemma 0FPL","summary":"Let f : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. The relative cup product of Remark [Tag 0B6C] is associative in the sense that the diagram xymatrix Rf_*K ⊗_O'^L Rf_*L ⊗_O'^L Rf_*M ar[r] ar[d] & Rf_*(K ⊗_O^L L) ⊗_O'^L Rf_*M ar[d] Rf_*K ⊗_O'^L Rf_*(L ⊗_O^L M) ar[r] & Rf_*(K ⊗_O^L L ⊗_O^L M) is commutative in D(O') for all K, L, M in D(O).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi. The relative cup product of\nRemark \\ref{remark-cup-product} is associative in the sense that\nthe diagram\n$$\n\\xymatrix{\nRf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L}\nRf_*L \\otimes_{\\mathcal{O}'}^\\mathbf{L}\nRf_*M \\ar[r] \\ar[d] &\nRf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L)\n\\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*M \\ar[d] \\\\\nRf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L}\nRf_*(L \\otimes_\\mathcal{O}^\\mathbf{L} M) \\ar[r] &\nRf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} \nL \\otimes_\\mathcal{O}^\\mathbf{L} M)\n}\n$$\nis commutative in $D(\\mathcal{O}')$ for all $K, L, M$ in $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPL","source_file":"sites-cohomology.tex","source_line":8421,"source_end_line":8441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8421-L8441","statement_sha256":"629c8b2d746f4c05e2e53f5b0f71c105e147499bd12bf20c06502ce19a3be44a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4555,"rank":4555,"depth":0,"x":1080.777,"y":562.296,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPM","tag":"0FPM","title":"Cup product · Lemma 0FPM","summary":"Let f : (Sh(C), O) → (Sh(C'), O') be a morphism of ringed topoi. The relative cup product of Remark [Tag 0B6C] is commutative in the sense that the diagram xymatrix Rf_*K ⊗_O'^L Rf_*L ar[r] ar[d]_ψ & Rf_*(K ⊗_O^L L) ar[d]^Rf_*ψ Rf_*L ⊗_O'^L Rf_*K ar[r] & Rf_*(L ⊗_O^L K) is commutative in D(O') for all K, L in D(O). Here ψ is the commutativity constraint on the derived category (Lemma [Tag 0FPT]).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nbe a morphism of ringed topoi. The relative cup product of\nRemark \\ref{remark-cup-product} is commutative in the sense that\nthe diagram\n$$\n\\xymatrix{\nRf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*L \\ar[r] \\ar[d]_\\psi &\nRf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) \\ar[d]^{Rf_*\\psi} \\\\\nRf_*L \\otimes_{\\mathcal{O}'}^\\mathbf{L} Rf_*K \\ar[r] &\nRf_*(L \\otimes_\\mathcal{O}^\\mathbf{L} K)\n}\n$$\nis commutative in $D(\\mathcal{O}')$ for all $K, L$ in $D(\\mathcal{O})$.\nHere $\\psi$ is the commutativity constraint on the derived category\n(Lemma \\ref{lemma-symmetric-monoidal-derived}).","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPM","source_file":"sites-cohomology.tex","source_line":8459,"source_end_line":8476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8459-L8476","statement_sha256":"6de9aad2d53923ee025807cdf9758f10559a7a8b784d051292f1848f5d3f0db3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4556,"rank":4556,"depth":0,"x":1261.109,"y":738.934,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPN","tag":"0FPN","title":"Cup product · Lemma 0FPN","summary":"Let f : (Sh(C), O) → (Sh(C'), O') and f' : (Sh(C'), O') → (Sh(C\"), O\") be morphisms of ringed topoi. The relative cup product of Remark [Tag 0B6C] is compatible with compositions in the sense that the diagram xymatrix R(f' ∘ f)_*K ⊗_O\"^L R(f' ∘ f)_*L ar@=[rr] ar[d] & & Rf'_*Rf_*K ⊗_O\"^L Rf'_*Rf_*L ar[d] R(f' ∘ f)_*(K ⊗_O^L L) ar@=[r] & Rf'_*Rf_*(K ⊗_O^L L) & Rf'_*(Rf_*K ⊗_O'^L Rf_*L) ar[l] is commutative in D(O\") for all K, L in D(O).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$\nand $f' : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to\n(\\Sh(\\mathcal{C}''), \\mathcal{O}'')$\nbe morphisms of ringed topoi. The relative cup product of\nRemark \\ref{remark-cup-product} is compatible with compositions\nin the sense that the diagram\n$$\n\\xymatrix{\nR(f' \\circ f)_*K \\otimes_{\\mathcal{O}''}^\\mathbf{L} R(f' \\circ f)_*L\n\\ar@{=}[rr] \\ar[d] & &\nRf'_*Rf_*K \\otimes_{\\mathcal{O}''}^\\mathbf{L} Rf'_*Rf_*L \\ar[d] \\\\\nR(f' \\circ f)_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) \\ar@{=}[r] &\nRf'_*Rf_*(K \\otimes_\\mathcal{O}^\\mathbf{L} L) &\nRf'_*(Rf_*K \\otimes_{\\mathcal{O}'}^\\mathbf{L}  Rf_*L) \\ar[l]\n}\n$$\nis commutative in $D(\\mathcal{O}'')$ for all $K, L$ in $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPN","source_file":"sites-cohomology.tex","source_line":8482,"source_end_line":8501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8482-L8501","statement_sha256":"3242545cac357a47a0cbbc1411111f01c7544ac64d64c406a5fa47a16ce72877","origin":"The Stacks Project","memory_eligible":false,"source_rank":4557,"rank":4557,"depth":2,"x":985.755,"y":710.975,"cluster":"sheaf-cohomology"},{"id":"stacks:0H9A","tag":"0H9A","title":"Cup product · Lemma 0H9A","summary":"Consider a commutative square xymatrix (Sh(C'), O_C') ar[r]_g' ar[d]_f' & (Sh(C, O_C) ar[d]^f (Sh(D'), O_D') ar[r]^g & (Sh(D), O_D) of ringed topoi. Let K, L in D(O_C). The relative cup product is compatible with the square in the sense that the diagram xymatrix Lg^*(Rf_*K ⊗_O_D^L Rf_*L) ar[r] ar@=[d] & Lg^*(Rf_*(K ⊗_O_C^L L)) ar[d] Lg^*Rf_*K ⊗_O_D'^L Lg^*Rf_*L ar[d] & R(f')_*L(g')^*(K ⊗_O_C^L L) ar@=[d] R(f')_*(L(g')^*K ⊗_O_D' R(f')_*(L(g')^*L ar[r] & R(f')_*(L(g')^*K…","statement_latex":"Consider a commutative square\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'}) \\ar[r]_{g'} \\ar[d]_{f'} &\n(\\Sh(\\mathcal{C}, \\mathcal{O}_{\\mathcal{C}}) \\ar[d]^f \\\\\n(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^g &\n(\\Sh(\\mathcal{D}), \\mathcal{O}_{\\mathcal{D}}) \n}\n$$\nof ringed topoi. Let $K, L$ in $D(\\mathcal{O}_{\\mathcal{C}})$.\nThe relative cup product is compatible with the square\nin the sense that the diagram\n$$\n\\xymatrix{\nLg^*(Rf_*K \\otimes_{\\mathcal{O}_{\\mathcal{D}}}^\\mathbf{L} Rf_*L)\n\\ar[r] \\ar@{=}[d] &\nLg^*(Rf_*(K \\otimes_{\\mathcal{O}_{\\mathcal{C}}}^\\mathbf{L} L)) \\ar[d] \\\\\nLg^*Rf_*K \\otimes_{\\mathcal{O}_{\\mathcal{D}'}}^\\mathbf{L} Lg^*Rf_*L \\ar[d] &\nR(f')_*L(g')^*(K \\otimes_{\\mathcal{O}_{\\mathcal{C}}}^\\mathbf{L} L) \\ar@{=}[d] \\\\\nR(f')_*(L(g')^*K \\otimes_{\\mathcal{O}_{\\mathcal{D}'}} R(f')_*(L(g')^*L \\ar[r] &\nR(f')_*(L(g')^*K \\otimes_{\\mathcal{O}_{\\mathcal{C}'}}^\\mathbf{L} L(g')^*L)\n}\n$$\nis commutative in $D(\\mathcal{O}_{\\mathcal{D}'})$.\nThe horizontal arrows are given\nby the relative cup product (Remark \\ref{remark-cup-product})\nand the vertical arrows are given\nby the base change map (Remark \\ref{remark-base-change})\nand Lemma \\ref{lemma-pullback-tensor-product}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9A","source_file":"sites-cohomology.tex","source_line":8528,"source_end_line":8559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8528-L8559","statement_sha256":"88eea095aa9822d78d088e6a135bff69b3fb623c9917bb93e1b0afeab3f4370a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4558,"rank":4558,"depth":19,"x":1211.553,"y":575.185,"cluster":"sheaf-cohomology"},{"id":"stacks:0A90","tag":"0A90","title":"Hom complexes · Lemma 0A90","summary":"Let (C, O) be a ringed site. Given complexes K^bullet, L^bullet, M^bullet of O-modules there is an isomorphism SheafHom^bullet(K^bullet, SheafHom^bullet(L^bullet, M^bullet)) = SheafHom^bullet(Tot(K^bullet ⊗_O L^bullet), M^bullet) of complexes of O-modules functorial in K^bullet, L^bullet, M^bullet.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nGiven complexes $\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$\nof $\\mathcal{O}$-modules there is an isomorphism\n$$\n\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,\n\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet))\n=\n\\SheafHom^\\bullet(\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O}\n\\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet)\n$$\nof complexes of $\\mathcal{O}$-modules functorial in\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A90","source_file":"sites-cohomology.tex","source_line":8615,"source_end_line":8629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8615-L8629","statement_sha256":"64514282a43cffb6fc145341eadadcd136721add8bfa17ded2020356bb283636","origin":"The Stacks Project","memory_eligible":false,"source_rank":4559,"rank":4559,"depth":2,"x":1154.182,"y":803.712,"cluster":"sheaf-cohomology"},{"id":"stacks:0A91","tag":"0A91","title":"Hom complexes · Lemma 0A91","summary":"Let (C, O) be a ringed site. Given complexes K^bullet, L^bullet, M^bullet of O-modules there is a canonical morphism Tot( SheafHom^bullet(L^bullet, M^bullet) ⊗_O SheafHom^bullet(K^bullet, L^bullet) ) → SheafHom^bullet(K^bullet, M^bullet) of complexes of O-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$\nof $\\mathcal{O}$-modules there is a canonical morphism\n$$\n\\text{Tot}\\left(\n\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet)\n\\otimes_\\mathcal{O}\n\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet)\n\\right)\n\\longrightarrow\n\\SheafHom^\\bullet(\\mathcal{K}^\\bullet, \\mathcal{M}^\\bullet)\n$$\nof complexes of $\\mathcal{O}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A91","source_file":"sites-cohomology.tex","source_line":8636,"source_end_line":8651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8636-L8651","statement_sha256":"cbeaab31e1069a8aa82787cd3f2736fbf67120de57617ebe647968f7637430ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":4560,"rank":4560,"depth":1,"x":1012.542,"y":602.405,"cluster":"sheaf-cohomology"},{"id":"stacks:0BYT","tag":"0BYT","title":"Hom complexes · Lemma 0BYT","summary":"Let (C, O) be a ringed site. Given complexes K^bullet, L^bullet, M^bullet of O-modules there is a canonical morphism Tot( K^bullet ⊗_O SheafHom^bullet(M^bullet, L^bullet) ) → SheafHom^bullet(M^bullet, Tot(K^bullet ⊗_O L^bullet)) of complexes of O-modules functorial in all three complexes.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$\nof $\\mathcal{O}$-modules there is a canonical morphism\n$$\n\\text{Tot}\\left(\n\\mathcal{K}^\\bullet \\otimes_\\mathcal{O}\n\\SheafHom^\\bullet(\\mathcal{M}^\\bullet, \\mathcal{L}^\\bullet)\n\\right)\n\\longrightarrow\n\\SheafHom^\\bullet(\\mathcal{M}^\\bullet,\n\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet))\n$$\nof complexes of $\\mathcal{O}$-modules functorial in all three complexes.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYT","source_file":"sites-cohomology.tex","source_line":8658,"source_end_line":8673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8658-L8673","statement_sha256":"040cbb9afeca43cf8932b71f49dc2d5d08ac45f12de0e2a694d807a8d863c7f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4561,"rank":4561,"depth":1,"x":1279.19,"y":670.557,"cluster":"sheaf-cohomology"},{"id":"stacks:0A93","tag":"0A93","title":"Hom complexes · Lemma 0A93","summary":"Let (C, O) be a ringed site. Given complexes K^bullet, L^bullet, M^bullet of O-modules there is a canonical morphism K^bullet → SheafHom^bullet(L^bullet, Tot(K^bullet ⊗_O L^bullet)) of complexes of O-modules functorial in both complexes.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$\nof $\\mathcal{O}$-modules there is a canonical morphism\n$$\n\\mathcal{K}^\\bullet\n\\longrightarrow\n\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,\n\\text{Tot}(\\mathcal{K}^\\bullet \\otimes_\\mathcal{O} \\mathcal{L}^\\bullet))\n$$\nof complexes of $\\mathcal{O}$-modules functorial in both complexes.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A93","source_file":"sites-cohomology.tex","source_line":8680,"source_end_line":8692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8680-L8692","statement_sha256":"e1c7daa5425cfe8e8e25f3226200f88c8c50635b22d8e28fafce182b48b1b7f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4562,"rank":4562,"depth":1,"x":1027.461,"y":771.726,"cluster":"sheaf-cohomology"},{"id":"stacks:0A92","tag":"0A92","title":"Hom complexes · Lemma 0A92","summary":"Let (C, O) be a ringed site. Given complexes K^bullet, L^bullet, M^bullet of O-modules there is a canonical morphism Tot(SheafHom^bullet(L^bullet, M^bullet) ⊗_O K^bullet) → SheafHom^bullet(SheafHom^bullet(K^bullet, L^bullet), M^bullet) of complexes of O-modules functorial in all three complexes.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given complexes\n$\\mathcal{K}^\\bullet, \\mathcal{L}^\\bullet, \\mathcal{M}^\\bullet$\nof $\\mathcal{O}$-modules there is a canonical morphism\n$$\n\\text{Tot}(\\SheafHom^\\bullet(\\mathcal{L}^\\bullet,\n\\mathcal{M}^\\bullet) \\otimes_\\mathcal{O} \\mathcal{K}^\\bullet)\n\\longrightarrow\n\\SheafHom^\\bullet(\\SheafHom^\\bullet(\\mathcal{K}^\\bullet,\n\\mathcal{L}^\\bullet), \\mathcal{M}^\\bullet)\n$$\nof complexes of $\\mathcal{O}$-modules functorial in all three complexes.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A92","source_file":"sites-cohomology.tex","source_line":8699,"source_end_line":8712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8699-L8712","statement_sha256":"9fa50f6722c1b096d5fc6b68bae7bcdd16dc0dddcf3ac0cf8eb464f5bfa70eaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4563,"rank":4563,"depth":1,"x":1131.849,"y":554.03,"cluster":"sheaf-cohomology"},{"id":"stacks:0A94","tag":"0A94","title":"Hom complexes · Lemma 0A94","summary":"Let (C, O) be a ringed site. Let L and M be objects of D(O). Let I^bullet be a K-injective complex of O-modules representing M. Let L^bullet be a complex of O-modules representing L. Then H^0(Γ(U, SheafHom^bullet(L^bullet, I^bullet))) = Hom_D(O_U)(L|_U, M|_U) for all U ∈ Ob(C). Similarly, H^0(Γ(C, SheafHom^bullet(L^bullet, I^bullet))) = Hom_D(O)(L, M).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $L$ and $M$\nbe objects of $D(\\mathcal{O})$. Let $\\mathcal{I}^\\bullet$\nbe a K-injective complex of $\\mathcal{O}$-modules representing $M$. Let\n$\\mathcal{L}^\\bullet$ be a complex of $\\mathcal{O}$-modules\nrepresenting $L$.\nThen\n$$\nH^0(\\Gamma(U, \\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))) =\n\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)\n$$\nfor all $U \\in \\Ob(\\mathcal{C})$. Similarly,\n$H^0(\\Gamma(\\mathcal{C},\n\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet))) =\n\\Hom_{D(\\mathcal{O})}(L, M)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A94","source_file":"sites-cohomology.tex","source_line":8719,"source_end_line":8735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8719-L8735","statement_sha256":"37429b45b6c6dc34e5d202c67005a502df19960c5f4d768fabecb60f208c75c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4564,"rank":4564,"depth":2,"x":1230.057,"y":774.049,"cluster":"sheaf-cohomology"},{"id":"stacks:0A95","tag":"0A95","title":"Hom complexes · Lemma 0A95","summary":"Let (C, O) be a ringed site. Let (I')^bullet → I^bullet be a quasi-isomorphism of K-injective complexes of O-modules. Let (L')^bullet → L^bullet be a quasi-isomorphism of complexes of O-modules. Then SheafHom^bullet(L^bullet, (I')^bullet) → SheafHom^bullet((L')^bullet, I^bullet) is a quasi-isomorphism.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{I}')^\\bullet \\to \\mathcal{I}^\\bullet$\nbe a quasi-isomorphism of K-injective complexes of $\\mathcal{O}$-modules.\nLet $(\\mathcal{L}')^\\bullet \\to \\mathcal{L}^\\bullet$\nbe a quasi-isomorphism of complexes of $\\mathcal{O}$-modules.\nThen\n$$\n\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, (\\mathcal{I}')^\\bullet)\n\\longrightarrow\n\\SheafHom^\\bullet((\\mathcal{L}')^\\bullet, \\mathcal{I}^\\bullet)\n$$\nis a quasi-isomorphism.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A95","source_file":"sites-cohomology.tex","source_line":8753,"source_end_line":8767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8753-L8767","statement_sha256":"0722168660b2c3377340c824e75b00fa4ebcbf75efe9d870de6442b315404947","origin":"The Stacks Project","memory_eligible":false,"source_rank":4565,"rank":4565,"depth":3,"x":980.411,"y":667.409,"cluster":"sheaf-cohomology"},{"id":"stacks:0A96","tag":"0A96","title":"Hom complexes · Lemma 0A96","summary":"Let (C, O) be a ringed site. Let I^bullet be a K-injective complex of O-modules. Let L^bullet be a K-flat complex of O-modules. Then SheafHom^bullet(L^bullet, I^bullet) is a K-injective complex of O-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{I}^\\bullet$\nbe a K-injective complex of $\\mathcal{O}$-modules. Let\n$\\mathcal{L}^\\bullet$ be a K-flat complex of $\\mathcal{O}$-modules.\nThen $\\SheafHom^\\bullet(\\mathcal{L}^\\bullet, \\mathcal{I}^\\bullet)$\nis a K-injective complex of $\\mathcal{O}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Hom complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A96","source_file":"sites-cohomology.tex","source_line":8788,"source_end_line":8795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8788-L8795","statement_sha256":"cbb9f67fccb500b66b31e1ff3e1d5a7b1eff62b7e65f12d8c05453d1785f9ee1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4566,"rank":4566,"depth":3,"x":1250.572,"y":604.315,"cluster":"sheaf-cohomology"},{"id":"stacks:08JA","tag":"08JA","title":"Internal hom in the derived category · Lemma 08JA","summary":"Let (C, O) be a ringed site. Let L, M be objects of D(O). For every object U of C we have H^0(U, RSheafHom(L, M)) = Hom_D(O_U)(L|_U, M|_U) and we have H^0(C, RSheafHom(L, M)) = Hom_D(O)(L, M).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $L, M$ be objects\nof $D(\\mathcal{O})$. For every object $U$ of $\\mathcal{C}$ we have\n$$\nH^0(U, R\\SheafHom(L, M)) =\n\\Hom_{D(\\mathcal{O}_U)}(L|_U, M|_U)\n$$\nand we have $H^0(\\mathcal{C}, R\\SheafHom(L, M)) =\n\\Hom_{D(\\mathcal{O})}(L, M)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JA","source_file":"sites-cohomology.tex","source_line":8873,"source_end_line":8883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8873-L8883","statement_sha256":"bf9a01bbeec5213e1e185f873f60204b2aa52a603795f9ea9ba15196b95aa2f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4567,"rank":4567,"depth":4,"x":1101.923,"y":804.371,"cluster":"sheaf-cohomology"},{"id":"stacks:08J9","tag":"08J9","title":"Internal hom in the derived category · Lemma 08J9","summary":"Let (C, O) be a ringed site. Let K, L, M be objects of D(O). With the construction as described above there is a canonical isomorphism RSheafHom(K, RSheafHom(L, M)) = RSheafHom(K ⊗_O^L L, M) in D(O) functorial in K, L, M which recovers ([Tag 08J8]) on taking H^0(C, -).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L, M$ be objects\nof $D(\\mathcal{O})$. With the construction as described above\nthere is a canonical isomorphism\n$$\nR\\SheafHom(K, R\\SheafHom(L, M)) =\nR\\SheafHom(K \\otimes_\\mathcal{O}^\\mathbf{L} L, M)\n$$\nin $D(\\mathcal{O})$ functorial in $K, L, M$\nwhich recovers (\\ref{equation-internal-hom}) on taking $H^0(\\mathcal{C}, -)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08J9","source_file":"sites-cohomology.tex","source_line":8895,"source_end_line":8906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8895-L8906","statement_sha256":"4d0c594e4139022362804061f4351d3b5c7ade7fdb5b21f031b4470133eb9b2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4568,"rank":4568,"depth":5,"x":1050.594,"y":572.232,"cluster":"sheaf-cohomology"},{"id":"stacks:08JB","tag":"08JB","title":"Internal hom in the derived category · Lemma 08JB","summary":"Let (C, O) be a ringed site. Let K, L be objects of D(O). The construction of RSheafHom(K, L) commutes with restrictions, i.e., for every object U of C we have RSheafHom(K|_U, L|_U) = RSheafHom(K, L)|_U.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects\nof $D(\\mathcal{O})$. The construction of $R\\SheafHom(K, L)$\ncommutes with restrictions, i.e.,\nfor every object $U$ of $\\mathcal{C}$ we have\n$R\\SheafHom(K|_U, L|_U) = R\\SheafHom(K, L)|_U$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JB","source_file":"sites-cohomology.tex","source_line":8933,"source_end_line":8940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8933-L8940","statement_sha256":"6297c955c8ca26356a5c08ca50565e5882f0f12c2cbab1e364b595bb1b28e9ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":4569,"rank":4569,"depth":2,"x":1275.386,"y":714.45,"cluster":"sheaf-cohomology"},{"id":"stacks:08JC","tag":"08JC","title":"Internal hom in the derived category · Lemma 08JC","summary":"Let (C, O) be a ringed site. The bifunctor RSheafHom(- , -) transforms distinguished triangles into distinguished triangles in both variables.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The bifunctor\n$R\\SheafHom(- , -)$ transforms distinguished triangles into\ndistinguished triangles in both variables.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JC","source_file":"sites-cohomology.tex","source_line":8947,"source_end_line":8952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8947-L8952","statement_sha256":"515e8e257ea24c959b42a5cdc76b0b13a3b2b07784a4135b8b40934517f28b0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4570,"rank":4570,"depth":0,"x":994.933,"y":737.16,"cluster":"sheaf-cohomology"},{"id":"stacks:0A97","tag":"0A97","title":"Internal hom in the derived category · Lemma 0A97","summary":"Let (C, O) be a ringed site. Let K, L, M be objects of D(O). There is a canonical morphism RSheafHom(L, M) ⊗_O^L K → RSheafHom(RSheafHom(K, L), M) in D(O) functorial in K, L, M.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L, M$ be objects of\n$D(\\mathcal{O})$. There is a canonical morphism\n$$\nR\\SheafHom(L, M) \\otimes_\\mathcal{O}^\\mathbf{L} K\n\\longrightarrow\nR\\SheafHom(R\\SheafHom(K, L), M)\n$$\nin $D(\\mathcal{O})$ functorial in $K, L, M$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A97","source_file":"sites-cohomology.tex","source_line":8964,"source_end_line":8974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8964-L8974","statement_sha256":"db94604a1dab60caf6bd620eb376153a9963dd809470d4922be06d167ac86f12","origin":"The Stacks Project","memory_eligible":false,"source_rank":4571,"rank":4571,"depth":2,"x":1183.693,"y":561.072,"cluster":"sheaf-cohomology"},{"id":"stacks:0A98","tag":"0A98","title":"Internal hom in the derived category · Lemma 0A98","summary":"Composition on RSheafHom. Let (C, O) be a ringed site. Given K, L, M in D(O) there is a canonical morphism RSheafHom(L, M) ⊗_O^L RSheafHom(K, L) → RSheafHom(K, M) in D(O).","statement_latex":"\\begin{slogan}\nComposition on RSheafHom.\n\\end{slogan}\nLet $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given $K, L, M$ in\n$D(\\mathcal{O})$ there is a canonical morphism\n$$\nR\\SheafHom(L, M) \\otimes_\\mathcal{O}^\\mathbf{L} R\\SheafHom(K, L)\n\\longrightarrow R\\SheafHom(K, M)\n$$\nin $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A98","source_file":"sites-cohomology.tex","source_line":8997,"source_end_line":9009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L8997-L9009","statement_sha256":"92b5067fd6ea47f050d277361a70a1720bc992bf0e97c67d3760f5f53021cd03","origin":"The Stacks Project","memory_eligible":false,"source_rank":4572,"rank":4572,"depth":2,"x":1186.111,"y":798.3,"cluster":"sheaf-cohomology"},{"id":"stacks:0BYU","tag":"0BYU","title":"Internal hom in the derived category · Lemma 0BYU","summary":"Let (C, O) be a ringed site. Given K, L, M in D(O) there is a canonical morphism K ⊗_O^L RSheafHom(M, L) → RSheafHom(M, K ⊗_O^L L) in D(O) functorial in K, L, M.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Given $K, L, M$\nin $D(\\mathcal{O})$ there is a canonical morphism\n$$\nK \\otimes_\\mathcal{O}^\\mathbf{L} R\\SheafHom(M, L)\n\\longrightarrow\nR\\SheafHom(M, K \\otimes_\\mathcal{O}^\\mathbf{L} L)\n$$\nin $D(\\mathcal{O})$ functorial in $K, L, M$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYU","source_file":"sites-cohomology.tex","source_line":9052,"source_end_line":9062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9052-L9062","statement_sha256":"72a53e53df0b3fe3fc7b2f2ffe6b42ba9297d0b1e98a331ce84dced61e7a35f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4573,"rank":4573,"depth":2,"x":993.333,"y":624.542,"cluster":"sheaf-cohomology"},{"id":"stacks:0A99","tag":"0A99","title":"Internal hom in the derived category · Lemma 0A99","summary":"Let (C, O) be a ringed site. Given K, L in D(O) there is a canonical morphism K → RSheafHom(L, K ⊗_O^L L) in D(O) functorial in both K and L.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nGiven $K, L$ in $D(\\mathcal{O})$ there is a canonical morphism\n$$\nK \\longrightarrow R\\SheafHom(L, K \\otimes_\\mathcal{O}^\\mathbf{L} L)\n$$\nin $D(\\mathcal{O})$ functorial in both $K$ and $L$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A99","source_file":"sites-cohomology.tex","source_line":9086,"source_end_line":9094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9086-L9094","statement_sha256":"e9e42c93a3613300ce161bc5e50301d344cf2cc1074e8b8c5c2dcba1145b8cff","origin":"The Stacks Project","memory_eligible":false,"source_rank":4574,"rank":4574,"depth":2,"x":1275.542,"y":643.303,"cluster":"sheaf-cohomology"},{"id":"stacks:08JD","tag":"08JD","title":"Internal hom in the derived category · Lemma 08JD","summary":"Let (C, O) be a ringed site. Let L be an object of D(O). Set L^vee = RSheafHom(L, O). For M in D(O) there is a canonical map M ⊗^L_O L^vee → RSheafHom(L, M) which induces a canonical map H^0(C, M ⊗_O^L L^vee) → Hom_D(O)(L, M) functorial in M in D(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $L$ be an\nobject of $D(\\mathcal{O})$. Set $L^\\vee = R\\SheafHom(L, \\mathcal{O})$.\nFor $M$ in $D(\\mathcal{O})$ there is a canonical map\n\\begin{equation}\n\nM \\otimes^\\mathbf{L}_\\mathcal{O} L^\\vee \\longrightarrow R\\SheafHom(L, M)\n\\end{equation}\nwhich induces a canonical map\n$$\nH^0(\\mathcal{C}, M \\otimes_\\mathcal{O}^\\mathbf{L} L^\\vee)\n\\longrightarrow\n\\Hom_{D(\\mathcal{O})}(L, M)\n$$\nfunctorial in $M$ in $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Internal hom in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JD","source_file":"sites-cohomology.tex","source_line":9113,"source_end_line":9129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9113-L9129","statement_sha256":"6092fd45976fb5763e3629d2589ad2b12af3dce4b8feb6d87a859257342558d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4575,"rank":4575,"depth":3,"x":1052.1,"y":789.771,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6X","tag":"0D6X","title":"Derived lower shriek · Lemma 0D6X","summary":"Let u : C → D be a continuous and cocontinuous functor of sites. Let g : Sh(C) → Sh(D) be the corresponding morphism of topoi. Let O_D be a sheaf of rings and let I be an injective O_D-module. Then H^p(U, g^-1I) = 0 for all p > 0 and U ∈ Ob(C).","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous\nfunctor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe the corresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$\nbe a sheaf of rings and let $\\mathcal{I}$ be an injective\n$\\mathcal{O}_\\mathcal{D}$-module. Then\n$H^p(U, g^{-1}\\mathcal{I}) = 0$ for all $p > 0$ and $U \\in \\Ob(\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6X","source_file":"sites-cohomology.tex","source_line":9277,"source_end_line":9285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9277-L9285","statement_sha256":"7e21272d52ab67b701a5cc2988a3750ddb21abc799df976c311adcc9721186a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4576,"rank":4576,"depth":23,"x":1099.132,"y":554.71,"cluster":"sheaf-cohomology"},{"id":"stacks:07AC","tag":"07AC","title":"Derived lower shriek · Lemma 07AC","summary":"Let u : C → D be a continuous and cocontinuous functor of sites. Let g : Sh(C) → Sh(D) be the corresponding morphism of topoi. Let O_D be a sheaf of rings and set O_C = g^-1O_D. The functor g_! : Mod(O_C) → Mod(O_D) (see Modules on Sites, Lemma [Tag 0797]) has a left derived functor Lg_! : D(O_C) → D(O_D) which is left adjoint to g^*. Moreover, for U ∈ Ob(C) we have Lg_!(j_U!O_U) = g_!j_U!O_U = j_u(U)! O_u(U). where j_U! and j_u(U)! are extension by zero associated to the…","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous\nfunctor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$ be the\ncorresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$\nbe a sheaf of rings and set\n$\\mathcal{O}_\\mathcal{C} = g^{-1}\\mathcal{O}_\\mathcal{D}$.\nThe functor $g_! : \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{D})$\n(see\nModules on Sites, Lemma \\ref{sites-modules-lemma-lower-shriek-modules})\nhas a left derived functor\n$$\nLg_! : D(\\mathcal{O}_\\mathcal{C}) \\longrightarrow D(\\mathcal{O}_\\mathcal{D})\n$$\nwhich is left adjoint to $g^*$. Moreover, for $U \\in \\Ob(\\mathcal{C})$ we\nhave\n$$\nLg_!(j_{U!}\\mathcal{O}_U) =\ng_!j_{U!}\\mathcal{O}_U =\nj_{u(U)!} \\mathcal{O}_{u(U)}.\n$$\nwhere $j_{U!}$ and $j_{u(U)!}$ are extension by zero associated to the\nlocalization morphism\n$j_U : \\mathcal{C}/U \\to \\mathcal{C}$ and\n$j_{u(U)} : \\mathcal{D}/u(U) \\to \\mathcal{D}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AC","source_file":"sites-cohomology.tex","source_line":9309,"source_end_line":9335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9309-L9335","statement_sha256":"19749f348742b7aed3ee067948ce5cddf812404c5a83910c786fae127141bffd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4577,"rank":4577,"depth":24,"x":1253.658,"y":754.958,"cluster":"sheaf-cohomology"},{"id":"stacks:0D6Y","tag":"0D6Y","title":"Derived lower shriek · Lemma 0D6Y","summary":"Let u : C → D be a continuous and cocontinuous functor of sites. Let g : Sh(C) → Sh(D) be the corresponding morphism of topoi. Let O_D be a sheaf of rings and let I be an injective O_D-module. If g_!^Sh : Sh(C) → Sh(D) commutes with fibre products has finite connected limits and u commutes with them, see Sites, Lemma [Tag 00XS]., then g^-1I is totally acyclic.","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous\nfunctor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe the corresponding morphism of topoi. Let $\\mathcal{O}_\\mathcal{D}$\nbe a sheaf of rings and let $\\mathcal{I}$ be an injective\n$\\mathcal{O}_\\mathcal{D}$-module. If\n$g_!^{Sh} : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\ncommutes with fibre products\\footnote{Holds if $\\mathcal{C}$\nhas finite connected limits and $u$ commutes with them, see\nSites, Lemma \\ref{sites-lemma-preserve-equalizers}.}, then\n$g^{-1}\\mathcal{I}$ is totally acyclic.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D6Y","source_file":"sites-cohomology.tex","source_line":9470,"source_end_line":9482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9470-L9482","statement_sha256":"9c68e4047e82b1aac57962f0f3f49df867f83169e99a4a500c1ecabfb747774d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4578,"rank":4578,"depth":24,"x":978.364,"y":694.911,"cluster":"sheaf-cohomology"},{"id":"stacks:0DD8","tag":"0DD8","title":"Derived lower shriek · Lemma 0DD8","summary":"Let u : C → D be a continuous and cocontinuous functor of sites. Let g : Sh(C) → Sh(D) be the corresponding morphism of topoi. Let U ∈ Ob(C). • For M in D(D) we have RΓ(U, g^-1M) = RΓ(u(U), M). • If O_D is a sheaf of rings and O_C = g^-1O_D, then for M in D(O_D) we have RΓ(U, g^*M) = RΓ(u(U), M).","statement_latex":"Let $u : \\mathcal{C} \\to \\mathcal{D}$ be a continuous and cocontinuous\nfunctor of sites. Let $g : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{D})$\nbe the corresponding morphism of topoi. Let $U \\in \\Ob(\\mathcal{C})$.\n\\begin{enumerate}\n\\item For $M$ in $D(\\mathcal{D})$ we have\n$R\\Gamma(U, g^{-1}M) = R\\Gamma(u(U), M)$.\n\\item If $\\mathcal{O}_\\mathcal{D}$ is a sheaf of rings and\n$\\mathcal{O}_\\mathcal{C} = g^{-1}\\mathcal{O}_\\mathcal{D}$, then\nfor $M$ in $D(\\mathcal{O}_\\mathcal{D})$ we have\n$R\\Gamma(U, g^*M) = R\\Gamma(u(U), M)$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DD8","source_file":"sites-cohomology.tex","source_line":9510,"source_end_line":9523,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9510-L9523","statement_sha256":"37ba819c6891e5ccab477d663333baa523599fca167bd89840ea1f8b8024081e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4579,"rank":4579,"depth":25,"x":1229.937,"y":582.851,"cluster":"sheaf-cohomology"},{"id":"stacks:0FN6","tag":"0FN6","title":"Derived lower shriek · Lemma 0FN6","summary":"Assume given a commutative diagram xymatrix (Sh(C'), O_C') ar[r]_(g', (g')^sharp) ar[d]_(f', (f')^sharp) & (Sh(C), O_C) ar[d]^(f, f^sharp) (Sh(D'), O_D') ar[r]^(g, g^sharp) & (Sh(D), O_D) of ringed topoi. Assume • f, f', g, and g' correspond to cocontinuous functors u, u', v, and v' as in Sites, Lemma [Tag 00XO], • v ∘ u' = u ∘ v', • v and v' are continuous as well as cocontinuous, • for any object V' of D' the functor ^u'_V'I → ^ u_v(V')I given by v is cofinal, • g^-1O_D…","statement_latex":"Assume given a commutative diagram\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})\n\\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} &\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\\n(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} &\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})\n}\n$$\nof ringed topoi. Assume\n\\begin{enumerate}\n\\item $f$, $f'$, $g$, and $g'$ correspond to cocontinuous functors\n$u$, $u'$, $v$, and $v'$ as in\nSites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi},\n\\item $v \\circ u' = u \\circ v'$,\n\\item $v$ and $v'$ are continuous as well as cocontinuous,\n\\item for any object $V'$ of $\\mathcal{D}'$ the functor\n${}^{u'}_{V'}\\mathcal{I} \\to {}^{\\ \\ \\ u}_{v(V')}\\mathcal{I}$\ngiven by $v$ is cofinal,\n\\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$\nand $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$, and\n\\item $g'_! : \\textit{Ab}(\\mathcal{C}') \\to \\textit{Ab}(\\mathcal{C})$\nis exact\\footnote{Holds if fibre products and equalizers exist in\n$\\mathcal{C}'$ and $v'$ commutes with them, see\nModules on Sites, Lemma \\ref{sites-modules-lemma-exactness-lower-shriek}.}.\n\\end{enumerate}\nThen we have $Rf'_* \\circ (g')^* = g^* \\circ Rf_*$ as functors\n$D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_{\\mathcal{D}'})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FN6","source_file":"sites-cohomology.tex","source_line":9542,"source_end_line":9573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9542-L9573","statement_sha256":"6d7ad3fed498dd7430b7be51facda29856af006902eddc48117b5281fd0cdf0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4580,"rank":4580,"depth":23,"x":1134.437,"y":808.489,"cluster":"sheaf-cohomology"},{"id":"stacks:0FN7","tag":"0FN7","title":"Derived lower shriek · Lemma 0FN7","summary":"Consider a commutative diagram xymatrix (Sh(C'), O_C') ar[r]_(g', (g')^sharp) ar[d]_(f', (f')^sharp) & (Sh(C), O_C) ar[d]^(f, f^sharp) (Sh(D'), O_D') ar[r]^(g, g^sharp) & (Sh(D), O_D) of ringed topoi and suppose we have functors xymatrix C' ar[r]_v' & C D' ar[r]^v ar[u]^u' & D ar[u]_u such that (with notation as in Sites, Sections [Tag 00X0] and [Tag 00XN]) we have • u and u' are continuous and give rise to the morphisms f and f', • v and v' are cocontinuous giving rise…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}'), \\mathcal{O}_{\\mathcal{C}'})\n\\ar[r]_{(g', (g')^\\sharp)} \\ar[d]_{(f', (f')^\\sharp)} &\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\ar[d]^{(f, f^\\sharp)} \\\\\n(\\Sh(\\mathcal{D}'), \\mathcal{O}_{\\mathcal{D}'}) \\ar[r]^{(g, g^\\sharp)} &\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})\n}\n$$\nof ringed topoi and suppose we have functors\n$$\n\\xymatrix{\n\\mathcal{C}' \\ar[r]_{v'} &\n\\mathcal{C} \\\\\n\\mathcal{D}' \\ar[r]^v \\ar[u]^{u'} &\n\\mathcal{D} \\ar[u]_u\n}\n$$\nsuch that (with notation as in\nSites, Sections \\ref{sites-section-morphism-sites} and\n\\ref{sites-section-cocontinuous-morphism-topoi}) we have\n\\begin{enumerate}\n\\item $u$ and $u'$ are continuous and give rise to the morphisms\n$f$ and $f'$,\n\\item $v$ and $v'$ are cocontinuous giving rise to the morphisms $g$ and $g'$,\n\\item $u \\circ v = v' \\circ u'$,\n\\item $v$ and $v'$ are continuous as well as cocontinuous, and\n\\item $g^{-1}\\mathcal{O}_{\\mathcal{D}} = \\mathcal{O}_{\\mathcal{D}'}$\nand $(g')^{-1}\\mathcal{O}_{\\mathcal{C}} = \\mathcal{O}_{\\mathcal{C}'}$.\n\\end{enumerate}\nThen $Rf'_* \\circ (g')^* = g^* \\circ Rf_*$ as functors\n$D^+(\\mathcal{O}_\\mathcal{C}) \\to D^+(\\mathcal{O}_{\\mathcal{D}'})$.\nIf in addition\n\\begin{enumerate}\n\\item[(6)] $g'_! : \\textit{Ab}(\\mathcal{C}') \\to \\textit{Ab}(\\mathcal{C})$\nis exact\\footnote{Holds if fibre products and equalizers exist in\n$\\mathcal{C}'$ and $v'$ commutes with them, see\nModules on Sites, Lemma \\ref{sites-modules-lemma-exactness-lower-shriek}.},\n\\end{enumerate}\nthen $Rf'_* \\circ (g')^* = g^* \\circ Rf_*$ as functors\n$D(\\mathcal{O}_\\mathcal{C}) \\to D(\\mathcal{O}_{\\mathcal{D}'})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FN7","source_file":"sites-cohomology.tex","source_line":9594,"source_end_line":9638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9594-L9638","statement_sha256":"00aa51e4d92f6d9a6f7fa921e19239c713dafc44d42b10f9e5f2d2f8072e94c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4581,"rank":4581,"depth":24,"x":1023.279,"y":587.666,"cluster":"sheaf-cohomology"},{"id":"stacks:08P9","tag":"08P9","title":"Derived lower shriek for fibred categories · Lemma 08P9","summary":"Assumptions and notation as in Situation [Tag 08P8]. For U ∈ Ob(C) consider the induced morphism of topoi π_U : Sh(C/U) → Sh(D/p(U)) Then there exists a morphism of topoi σ : Sh(D/p(U)) → Sh(C/U) such that π_U ∘ σ = id and σ^-1 = π_U, *.","statement_latex":"Assumptions and notation as in Situation \\ref{situation-fibred-category}.\nFor $U \\in \\Ob(\\mathcal{C})$ consider the induced morphism\nof topoi\n$$\n\\pi_U : \\Sh(\\mathcal{C}/U) \\longrightarrow \\Sh(\\mathcal{D}/p(U))\n$$\nThen there exists a morphism of topoi\n$$\n\\sigma : \\Sh(\\mathcal{D}/p(U)) \\to \\Sh(\\mathcal{C}/U)\n$$\nsuch that $\\pi_U \\circ \\sigma = \\text{id}$ and $\\sigma^{-1} = \\pi_{U, *}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek for fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08P9","source_file":"sites-cohomology.tex","source_line":9719,"source_end_line":9732,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9719-L9732","statement_sha256":"35a160b05326e0f3200db9cd467cbf55d8d9b03f815121952e8d0d237307af1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4582,"rank":4582,"depth":8,"x":1283.12,"y":687.538,"cluster":"sheaf-cohomology"},{"id":"stacks:08PB","tag":"08PB","title":"Derived lower shriek for fibred categories · Lemma 08PB","summary":"Assumptions and notation as in Situation [Tag 08PA]. For U' ∈ Ob(C') set U = u(U') and V = p'(U') and consider the induced morphisms of ringed topoi xymatrix (Sh(C'/U'), O_U') ar[rd]_π'_U' ar[rr]_g' & & (Sh(C), O_U) ar[ld]^π_U & (Sh(D/V), O_V) Then there exists a morphism of topoi σ' : Sh(D/V) → Sh(C'/U'), such that setting σ = g' ∘ σ' we have π'_U' ∘ σ' = id, π_U ∘ σ = id, (σ')^-1 = π'_U', *, and σ^-1 = π_U, *.","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-morphism-fibred-categories}.\nFor $U' \\in \\Ob(\\mathcal{C}')$ set $U = u(U')$ and $V = p'(U')$ and\nconsider the induced morphisms of ringed topoi\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}'/U'), \\mathcal{O}_{U'}) \\ar[rd]_{\\pi'_{U'}} \\ar[rr]_{g'} & &\n(\\Sh(\\mathcal{C}), \\mathcal{O}_U) \\ar[ld]^{\\pi_U} \\\\\n& (\\Sh(\\mathcal{D}/V), \\mathcal{O}_V)\n}\n$$\nThen there exists a morphism of topoi\n$$\n\\sigma' : \\Sh(\\mathcal{D}/V) \\to \\Sh(\\mathcal{C}'/U'),\n$$\nsuch that setting $\\sigma = g' \\circ \\sigma'$ we have\n$\\pi'_{U'} \\circ \\sigma' = \\text{id}$, $\\pi_U \\circ \\sigma = \\text{id}$,\n$(\\sigma')^{-1} = \\pi'_{U', *}$, and $\\sigma^{-1} = \\pi_{U, *}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek for fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PB","source_file":"sites-cohomology.tex","source_line":9794,"source_end_line":9814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9794-L9814","statement_sha256":"1abf89232d5506f879306a884f197ab29c61db9c446eeeb66a5f2efe0754d823","origin":"The Stacks Project","memory_eligible":false,"source_rank":4583,"rank":4583,"depth":9,"x":1010.895,"y":761.418,"cluster":"sheaf-cohomology"},{"id":"stacks:08PC","tag":"08PC","title":"Derived lower shriek for fibred categories · Lemma 08PC","summary":"Assumption and notation as in Situation [Tag 08PA]. • There are left adjoints g_! : Mod(O_C') → Mod(O_C) and g_!^Ab : Ab(C') → Ab(C) to g^* = g^-1 on modules and on abelian sheaves. • The diagram xymatrix Mod(O_C') ar[d] ar[r]_g_! & Mod(O_C) ar[d] Ab(C') ar[r]^g_!^Ab & Ab(C) commutes. • There are left adjoints Lg_! : D(O_C') → D(O_C) and Lg_!^Ab : D(C') → D(C) to g^* = g^-1 on derived categories of modules and abelian sheaves. • The diagram xymatrix D(O_C') ar[d]…","statement_latex":"Assumption and notation as in\nSituation \\ref{situation-morphism-fibred-categories}.\n\\begin{enumerate}\n\\item There are left adjoints\n$g_! : \\textit{Mod}(\\mathcal{O}_{\\mathcal{C}'}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{C})$ and\n$g_!^{\\textit{Ab}} : \\textit{Ab}(\\mathcal{C}') \\to \\textit{Ab}(\\mathcal{C})$\nto $g^* = g^{-1}$ on modules and on abelian sheaves.\n\\item The diagram\n$$\n\\xymatrix{\n\\textit{Mod}(\\mathcal{O}_{\\mathcal{C}'}) \\ar[d] \\ar[r]_{g_!} &\n\\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\ar[d] \\\\\n\\textit{Ab}(\\mathcal{C}') \\ar[r]^{g_!^{\\textit{Ab}}} &\n\\textit{Ab}(\\mathcal{C})\n}\n$$\ncommutes.\n\\item There are left adjoints\n$Lg_! : D(\\mathcal{O}_{\\mathcal{C}'}) \\to D(\\mathcal{O}_\\mathcal{C})$\nand\n$Lg_!^{\\textit{Ab}} : D(\\mathcal{C}') \\to D(\\mathcal{C})$\nto $g^* = g^{-1}$ on derived categories of modules and abelian sheaves.\n\\item The diagram\n$$\n\\xymatrix{\nD(\\mathcal{O}_{\\mathcal{C}'}) \\ar[d] \\ar[r]_{Lg_!} &\nD(\\mathcal{O}_\\mathcal{C}) \\ar[d] \\\\\nD(\\mathcal{C}') \\ar[r]^{Lg_!^{\\textit{Ab}}} &\nD(\\mathcal{C})\n}\n$$\ncommutes.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek for fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PC","source_file":"sites-cohomology.tex","source_line":9828,"source_end_line":9864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9828-L9864","statement_sha256":"376325d1f47836215246ace5581b6f7df83a6d4cf8a3ad7cc91a87f1d0bf9bde","origin":"The Stacks Project","memory_eligible":false,"source_rank":4584,"rank":4584,"depth":10,"x":1152.378,"y":552.236,"cluster":"sheaf-cohomology"},{"id":"stacks:08PE","tag":"08PE","title":"Derived lower shriek for fibred categories · Lemma 08PE","summary":"Assumptions and notation as in Situation [Tag 08P8]. For F in Ab(C) the sheaf π_!F is the sheaf associated to the presheaf V ↦ colim_C_V^opp F|_C_V with restriction maps as indicated in the proof.","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-fibred-category}.\nFor $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$\nthe sheaf $\\pi_!\\mathcal{F}$ is the\nsheaf associated to the presheaf\n$$\nV \\longmapsto \\colim_{\\mathcal{C}_V^{opp}} \\mathcal{F}|_{\\mathcal{C}_V}\n$$\nwith restriction maps as indicated in the proof.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Derived lower shriek for fibred categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PE","source_file":"sites-cohomology.tex","source_line":9959,"source_end_line":9970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L9959-L9970","statement_sha256":"dbd126c6e4f4319396805f757684ea8419c593f31acfa47c4406d120c721e466","origin":"The Stacks Project","memory_eligible":false,"source_rank":4585,"rank":4585,"depth":1,"x":1216.341,"y":787.029,"cluster":"sheaf-cohomology"},{"id":"stacks:08Q7","tag":"08Q7","title":"Homology on a category · Lemma 08Q7","summary":"Notation and assumptions as in Example [Tag 08PF]. If C has either an initial or a final object, then Lπ_! ∘ π^-1 = id on D(Ab), resp. D(B).","statement_latex":"Notation and assumptions as in Example \\ref{example-category-to-point}.\nIf $\\mathcal{C}$ has either an initial or a final object, then\n$L\\pi_! \\circ \\pi^{-1} = \\text{id}$ on $D(\\textit{Ab})$, resp.\\ $D(B)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Homology on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Q7","source_file":"sites-cohomology.tex","source_line":10188,"source_end_line":10193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10188-L10193","statement_sha256":"ee86bb5be009f0ab859f1f81d43c7751ad99f96ce31a61f31998af7da3284f80","origin":"The Stacks Project","memory_eligible":false,"source_rank":4586,"rank":4586,"depth":3,"x":980.095,"y":650.036,"cluster":"sheaf-cohomology"},{"id":"stacks:08Q8","tag":"08Q8","title":"Homology on a category · Lemma 08Q8","summary":"Notation and assumptions as in Example [Tag 08PF]. Let B → B' be a ring map. Consider the commutative diagram of ringed topoi xymatrix (Sh(C), underlineB) ar[d]_π & (Sh(C), underlineB') ar[d]^π' ar[l]^h (*, B) & (*, B') ar[l]_f Then Lπ_! ∘ Lh^* = Lf^* ∘ Lπ'_!.","statement_latex":"Notation and assumptions as in Example \\ref{example-category-to-point}.\nLet $B \\to B'$ be a ring map. Consider the commutative diagram\nof ringed topoi\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}), \\underline{B}) \\ar[d]_\\pi &\n(\\Sh(\\mathcal{C}), \\underline{B'}) \\ar[d]^{\\pi'} \\ar[l]^h \\\\\n(*, B) & (*, B') \\ar[l]_f\n}\n$$\nThen $L\\pi_! \\circ Lh^* = Lf^* \\circ L\\pi'_!$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Homology on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Q8","source_file":"sites-cohomology.tex","source_line":10208,"source_end_line":10221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10208-L10221","statement_sha256":"6e657761eef7bea0238f9d799d33448aa988420ea5067ad5a358bc0baae0a8e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4587,"rank":4587,"depth":0,"x":1264.785,"y":616.965,"cluster":"sheaf-cohomology"},{"id":"stacks:08Q9","tag":"08Q9","title":"Homology on a category · Lemma 08Q9","summary":"Notation and assumptions as in Example [Tag 08PF]. Let U_bullet be a cosimplicial object in C such that for every U ∈ Ob(C) the simplicial set Mor_C(U_bullet, U) is homotopy equivalent to the constant simplicial set on a singleton. Then Lπ_!(F) = F(U_bullet) in D(Ab), resp. D(B) functorially in F in Ab(C), resp. Mod(underlineB).","statement_latex":"Notation and assumptions as in Example \\ref{example-category-to-point}.\nLet $U_\\bullet$ be a cosimplicial object in $\\mathcal{C}$ such that\nfor every $U \\in \\Ob(\\mathcal{C})$ the simplicial set\n$\\Mor_\\mathcal{C}(U_\\bullet, U)$\nis homotopy equivalent to the constant simplicial set on a singleton. Then\n$$\nL\\pi_!(\\mathcal{F}) = \\mathcal{F}(U_\\bullet)\n$$\nin $D(\\textit{Ab})$, resp.\\ $D(B)$ functorially in $\\mathcal{F}$ in\n$\\textit{Ab}(\\mathcal{C})$, resp.\\ $\\textit{Mod}(\\underline{B})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Homology on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Q9","source_file":"sites-cohomology.tex","source_line":10228,"source_end_line":10240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10228-L10240","statement_sha256":"18bec4fd5d07014aa5d0d8afd679519403f5c4d8eec5bd73db88b76358271456","origin":"The Stacks Project","memory_eligible":false,"source_rank":4588,"rank":4588,"depth":11,"x":1081.248,"y":803.099,"cluster":"sheaf-cohomology"},{"id":"stacks:08QA","tag":"08QA","title":"Homology on a category · Lemma 08QA","summary":"Notation and assumptions as in Example [Tag 08PH]. If there exists a cosimplicial object U'_bullet of C' such that Lemma [Tag 08Q9] applies to both U'_bullet in C' and u(U'_bullet) in C, then we have Lπ'_! ∘ g^-1 = Lπ_! as functors D(C) → D(Ab), resp. D(C, underlineB) → D(B).","statement_latex":"Notation and assumptions as in Example \\ref{example-morphism-categories}.\nIf there exists a cosimplicial object $U'_\\bullet$ of $\\mathcal{C}'$\nsuch that Lemma \\ref{lemma-compute-by-cosimplicial-resolution}\napplies to both $U'_\\bullet$ in $\\mathcal{C}'$\nand $u(U'_\\bullet)$ in $\\mathcal{C}$, then we have\n$L\\pi'_! \\circ g^{-1} = L\\pi_!$ as functors\n$D(\\mathcal{C}) \\to D(\\textit{Ab})$,\nresp.\\ $D(\\mathcal{C}, \\underline{B}) \\to D(B)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Homology on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QA","source_file":"sites-cohomology.tex","source_line":10297,"source_end_line":10307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10297-L10307","statement_sha256":"82bf01fd772a3caf3ddb45d69d4a3d3303235fe700bf83cc788d7f54b763ca7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4589,"rank":4589,"depth":12,"x":1066.885,"y":561.433,"cluster":"sheaf-cohomology"},{"id":"stacks:08QB","tag":"08QB","title":"Homology on a category · Lemma 08QB","summary":"Let C_i, i = 1, 2 be categories. Let u_i : C_1 × C_2 → C_i be the projection functors. Let B be a ring. Let g_i : (Sh(C_1 × C_2), underlineB) → (Sh(C_i), underlineB) be the corresponding morphisms of ringed topoi, see Example [Tag 08PH]. For K_i ∈ D(C_i, B) we have L(π_1 × π_2)_!( g_1^-1K_1 ⊗_underlineB^L g_2^-1K_2) = Lπ_1, !(K_1) ⊗_B^L Lπ_2, !(K_2) in D(B) with obvious notation.","statement_latex":"Let $\\mathcal{C}_i$, $i = 1, 2$ be categories. Let\n$u_i : \\mathcal{C}_1 \\times \\mathcal{C}_2 \\to \\mathcal{C}_i$ be the\nprojection functors. Let $B$ be a ring. Let\n$g_i : (\\Sh(\\mathcal{C}_1 \\times \\mathcal{C}_2), \\underline{B}) \\to\n(\\Sh(\\mathcal{C}_i), \\underline{B})$ be the corresponding morphisms\nof ringed topoi, see Example \\ref{example-morphism-categories}. For\n$K_i \\in D(\\mathcal{C}_i, B)$ we have\n$$\nL(\\pi_1 \\times \\pi_2)_!(\ng_1^{-1}K_1 \\otimes_{\\underline{B}}^\\mathbf{L} g_2^{-1}K_2)\n=\nL\\pi_{1, !}(K_1) \\otimes_B^\\mathbf{L} L\\pi_{2, !}(K_2)\n$$\nin $D(B)$ with obvious notation.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Homology on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QB","source_file":"sites-cohomology.tex","source_line":10315,"source_end_line":10331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10315-L10331","statement_sha256":"364928c82e5683ae3320c762d06a823dd1f3248c4731698cebce1d42ddbb0caa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4590,"rank":4590,"depth":25,"x":1272.046,"y":731.674,"cluster":"sheaf-cohomology"},{"id":"stacks:08QC","tag":"08QC","title":"Homology on a category · Lemma 08QC","summary":"Notation and assumptions as in Example [Tag 08PF]. If there exists a cosimplicial object U_bullet of C such that Lemma [Tag 08Q9] applies, then Lπ_!(K_1 ⊗^L_underlineB K_2) = Lπ_!(K_1) ⊗^L_B Lπ_!(K_2) for all K_i ∈ D(underlineB).","statement_latex":"Notation and assumptions as in Example \\ref{example-category-to-point}.\nIf there exists a cosimplicial object $U_\\bullet$ of $\\mathcal{C}$\nsuch that Lemma \\ref{lemma-compute-by-cosimplicial-resolution}\napplies, then\n$$\nL\\pi_!(K_1 \\otimes^\\mathbf{L}_{\\underline{B}} K_2) =\nL\\pi_!(K_1) \\otimes^\\mathbf{L}_B L\\pi_!(K_2)\n$$\nfor all $K_i \\in D(\\underline{B})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Homology on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QC","source_file":"sites-cohomology.tex","source_line":10350,"source_end_line":10361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10350-L10361","statement_sha256":"a04310fb158a5c74096cbec55d8659dee03194606320cdd2c9c2c87ebc215ddb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4591,"rank":4591,"depth":26,"x":983.54,"y":722.544,"cluster":"sheaf-cohomology"},{"id":"stacks:08RX","tag":"08RX","title":"Homology on a category · Lemma 08RX","summary":"Let C be a category (endowed with chaotic topology). Let O → O' be a map of sheaves of rings on C. Assume • there exists a cosimplicial object U_bullet in C as in Lemma [Tag 08Q9], and • Lπ_!O → Lπ_!O' is an isomorphism. For K in D(O) we have Lπ_!(K) = Lπ_!(K ⊗_O^L O') in D(Ab).","statement_latex":"Let $\\mathcal{C}$ be a category (endowed with chaotic topology).\nLet $\\mathcal{O} \\to \\mathcal{O}'$ be a map of sheaves of rings on\n$\\mathcal{C}$. Assume\n\\begin{enumerate}\n\\item there exists a cosimplicial object $U_\\bullet$ in $\\mathcal{C}$\nas in Lemma \\ref{lemma-compute-by-cosimplicial-resolution}, and\n\\item $L\\pi_!\\mathcal{O} \\to L\\pi_!\\mathcal{O}'$ is an isomorphism.\n\\end{enumerate}\nFor $K$ in $D(\\mathcal{O})$ we have\n$$\nL\\pi_!(K) = L\\pi_!(K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}')\n$$\nin $D(\\textit{Ab})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Homology on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RX","source_file":"sites-cohomology.tex","source_line":10421,"source_end_line":10436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10421-L10436","statement_sha256":"53eeff220ed7c89656dfb9685c09f07e5bb796516775eb32e9d838922fc3b936","origin":"The Stacks Project","memory_eligible":false,"source_rank":4592,"rank":4592,"depth":17,"x":1203.867,"y":565.396,"cluster":"sheaf-cohomology"},{"id":"stacks:08PI","tag":"08PI","title":"Calculating derived lower shriek · Lemma 08PI","summary":"Assumptions and notation as in Situation [Tag 08P8]. For F in PAb(C) and n ≥ 0 consider the abelian sheaf L_n(F) on D which is the sheaf associated to the presheaf V ↦ H_n(C_V, F|_C_V) with restriction maps as indicated in the proof. Then L_n(F) = L_n(F^\\#).","statement_latex":"Assumptions and notation as in Situation \\ref{situation-fibred-category}.\nFor $\\mathcal{F}$ in $\\textit{PAb}(\\mathcal{C})$ and $n \\geq 0$\nconsider the abelian sheaf $L_n(\\mathcal{F})$ on $\\mathcal{D}$\nwhich is the sheaf associated to the presheaf\n$$\nV \\longmapsto H_n(\\mathcal{C}_V, \\mathcal{F}|_{\\mathcal{C}_V})\n$$\nwith restriction maps as indicated in the proof. Then\n$L_n(\\mathcal{F}) = L_n(\\mathcal{F}^\\#)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Calculating derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PI","source_file":"sites-cohomology.tex","source_line":10525,"source_end_line":10536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10525-L10536","statement_sha256":"8a6b1991689b833d87bd11bd408973b70a099f7070dba6ea6c7c085351c1827c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4593,"rank":4593,"depth":3,"x":1167.735,"y":806.561,"cluster":"sheaf-cohomology"},{"id":"stacks:08PJ","tag":"08PJ","title":"Calculating derived lower shriek · Lemma 08PJ","summary":"Assumptions and notation as in Situation [Tag 08P8]. For F in Ab(C) and n ≥ 0 the sheaf L_nπ_!(F) is equal to the sheaf L_n(F) constructed in Lemma [Tag 08PI].","statement_latex":"Assumptions and notation as in Situation \\ref{situation-fibred-category}.\nFor $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{C})$ and $n \\geq 0$\nthe sheaf $L_n\\pi_!(\\mathcal{F})$ is equal to the sheaf\n$L_n(\\mathcal{F})$ constructed in\nLemma \\ref{lemma-compute-left-derived-pi-shriek-pre}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Calculating derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PJ","source_file":"sites-cohomology.tex","source_line":10614,"source_end_line":10621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10614-L10621","statement_sha256":"7ea6750757160e400e1520420c8b32fce4e2be04789a9e275f39aebdac1d843e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4594,"rank":4594,"depth":18,"x":1000.255,"y":608.008,"cluster":"sheaf-cohomology"},{"id":"stacks:08PK","tag":"08PK","title":"Calculating derived lower shriek · Lemma 08PK","summary":"Assumptions and notation as in Situation [Tag 08PA]. For an abelian sheaf F' on C' the sheaf L_ng_!(F') is the sheaf associated to the presheaf U ↦ H_n(I_U, F'_U) For notation and restriction maps see proof.","statement_latex":"Assumptions and notation as in\nSituation \\ref{situation-morphism-fibred-categories}.\nFor an abelian sheaf $\\mathcal{F}'$ on $\\mathcal{C}'$ the sheaf\n$L_ng_!(\\mathcal{F}')$ is the sheaf associated to the presheaf\n$$\nU \\longmapsto H_n(\\mathcal{I}_U, \\mathcal{F}'_U)\n$$\nFor notation and restriction maps see proof.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Calculating derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PK","source_file":"sites-cohomology.tex","source_line":10677,"source_end_line":10687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10677-L10687","statement_sha256":"7d47db2e8ddafe9a6f28a05aa86a87ece1e2fcc62e46970a46bd3e4d1bb23edb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4595,"rank":4595,"depth":19,"x":1283.735,"y":659.443,"cluster":"sheaf-cohomology"},{"id":"stacks:09D1","tag":"09D1","title":"Simplicial modules · Definition 09D1","summary":"Let C be a site. Let A_bullet be a simplicial sheaf of rings on C. A simplicial A_bullet-module F_bullet (sometimes called a simplicial sheaf of A_bullet-modules) is a sheaf of modules over the sheaf of rings on Δ × C associated to A_bullet.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A}_\\bullet$ be a simplicial\nsheaf of rings on $\\mathcal{C}$. A\n{\\it simplicial $\\mathcal{A}_\\bullet$-module} $\\mathcal{F}_\\bullet$\n(sometimes called a\n{\\it simplicial sheaf of $\\mathcal{A}_\\bullet$-modules})\nis a sheaf of modules over the sheaf of rings on $\\Delta \\times \\mathcal{C}$\nassociated to $\\mathcal{A}_\\bullet$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Simplicial modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09D1","source_file":"sites-cohomology.tex","source_line":10787,"source_end_line":10796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10787-L10796","statement_sha256":"b9b20e5225afb833f41a6698e158a1e715e34737098efeb7bf7aee21d2c456cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4596,"rank":4596,"depth":0,"x":1033.063,"y":782.504,"cluster":"sheaf-cohomology"},{"id":"stacks:09D2","tag":"09D2","title":"Simplicial modules · Lemma 09D2","summary":"Let C be a site. Let A_bullet → B_bullet be a homomorphism of simplicial sheaves of rings on C. If Lπ_!A_bullet → Lπ_!B_bullet is an isomorphism in D(C), then we have Lπ_!(K) = Lπ_!(K ⊗^L_A_bullet B_bullet) for all K in D(A_bullet).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A}_\\bullet \\to \\mathcal{B}_\\bullet$\nbe a homomorphism of simplicial sheaves of rings on $\\mathcal{C}$.\nIf $L\\pi_!\\mathcal{A}_\\bullet \\to L\\pi_!\\mathcal{B}_\\bullet$ is an\nisomorphism in $D(\\mathcal{C})$, then we have\n$$\nL\\pi_!(K) =\nL\\pi_!(K \\otimes^\\mathbf{L}_{\\mathcal{A}_\\bullet} \\mathcal{B}_\\bullet)\n$$\nfor all $K$ in $D(\\mathcal{A}_\\bullet)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Simplicial modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09D2","source_file":"sites-cohomology.tex","source_line":10822,"source_end_line":10833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L10822-L10833","statement_sha256":"3e7ab0c48a6b7b194a823a34775a1d18bf7bcacaa811824da1b3e30405052610","origin":"The Stacks Project","memory_eligible":false,"source_rank":4597,"rank":4597,"depth":19,"x":1119.037,"y":549.267,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYV","tag":"0GYV","title":"Modules on a category · Definition 0GYV","summary":"In the situation above, we denote mathitQC(C, O) or simply mathitQC(O) the full subcategory of D(O) = D(C, O) consisting of objects K such that for all U → V in C the canonical map RΓ(V, K) ⊗_O(V)^L O(U) → RΓ(U, K) is an isomorphism in D(O(U)).","statement_latex":"In the situation above, we denote $\\mathit{QC}(\\mathcal{C}, \\mathcal{O})$\nor simply {\\it $\\mathit{QC}(\\mathcal{O})$}\nthe full subcategory of $D(\\mathcal{O}) = D(\\mathcal{C}, \\mathcal{O})$\nconsisting of objects $K$ such that\nfor all $U \\to V$ in $\\mathcal{C}$ the canonical map\n$$\nR\\Gamma(V, K) \\otimes_{\\mathcal{O}(V)}^\\mathbf{L} \\mathcal{O}(U)\n\\longrightarrow\nR\\Gamma(U, K)\n$$\nis an isomorphism in $D(\\mathcal{O}(U))$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYV","source_file":"sites-cohomology.tex","source_line":11097,"source_end_line":11110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11097-L11110","statement_sha256":"4608c9cd35fc5c363b1730a02b2893d04790de74c8cd5946234b836f5cfd92c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4598,"rank":4598,"depth":0,"x":1243.34,"y":770.278,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYW","tag":"0GYW","title":"Modules on a category · Lemma 0GYW","summary":"In the situation above, the subcategory mathitQC(O) is a strictly full, saturated, triangulated subcategory of D(O) preserved by arbitrary direct sums.","statement_latex":"In the situation above, the subcategory $\\mathit{QC}(\\mathcal{O})$ is a\nstrictly full, saturated, triangulated subcategory of $D(\\mathcal{O})$\npreserved by arbitrary direct sums.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYW","source_file":"sites-cohomology.tex","source_line":11112,"source_end_line":11117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11112-L11117","statement_sha256":"097ac7dc9bdfd5f2a33639e94597dce6b3b8e73e28f3cb10b5ddeab7f171f9e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4599,"rank":4599,"depth":8,"x":973.655,"y":677.74,"cluster":"sheaf-cohomology"},{"id":"stacks:0GZQ","tag":"0GZQ","title":"Modules on a category · Lemma 0GZQ","summary":"In the situation above, suppose that M is an object of mathitQC(O) and b ∈ Z such that H^i(M) = 0 for all i > b. Then H^b(M) is a quasi-coherent module on (C, O) in the sense of Modules on Sites, Definition [Tag 03DL].","statement_latex":"In the situation above, suppose that $M$ is an object of\n$\\mathit{QC}(\\mathcal{O})$ and $b \\in \\mathbf{Z}$ such that $H^i(M) = 0$\nfor all $i > b$. Then $H^b(M)$ is a quasi-coherent module on\n$(\\mathcal{C}, \\mathcal{O})$ in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZQ","source_file":"sites-cohomology.tex","source_line":11135,"source_end_line":11142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11135-L11142","statement_sha256":"e1fdc8c1f6b0489ad97f55738c888441fcc1109917f083a08487820ac45aace2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4600,"rank":4600,"depth":3,"x":1247.232,"y":592.858,"cluster":"sheaf-cohomology"},{"id":"stacks:0H0R","tag":"0H0R","title":"Modules on a category · Lemma 0H0R","summary":"In the situation above, suppose that C has a final object X. Set R = O(X) and denote f : (C, O) → (pt, R) the obvious morphism of sites. Then mathitQC(O) = D(R) given by Lf^* and Rf_*.","statement_latex":"In the situation above, suppose that $\\mathcal{C}$ has a final object $X$.\nSet $R = \\mathcal{O}(X)$ and denote\n$f : (\\mathcal{C}, \\mathcal{O}) \\to (pt, R)$ the obvious morphism of sites.\nThen $\\mathit{QC}(\\mathcal{O}) = D(R)$ given by $Lf^*$ and $Rf_*$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0R","source_file":"sites-cohomology.tex","source_line":11160,"source_end_line":11166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11160-L11166","statement_sha256":"3ab0011fd0ab11fb18e48438c42d25bfa42e921868c523031e403e82612ab2d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4601,"rank":4601,"depth":0,"x":1113.614,"y":810.92,"cluster":"sheaf-cohomology"},{"id":"stacks:0H0S","tag":"0H0S","title":"Modules on a category · Lemma 0H0S","summary":"In the situation above, suppose that K is an object of mathitQC(O) and M arbitrary in D(O). For every object U of C we have Hom_D(O_U)(K|_U, M|_U) = RHom_O(U)(RΓ(U, K), RΓ(U, M))","statement_latex":"In the situation above, suppose that $K$ is an object of\n$\\mathit{QC}(\\mathcal{O})$ and $M$ arbitrary in $D(\\mathcal{O})$.\nFor every object $U$ of $\\mathcal{C}$ we have\n$$\n\\Hom_{D(\\mathcal{O}_U)}(K|_U, M|_U) =\nR\\Hom_{\\mathcal{O}(U)}(R\\Gamma(U, K), R\\Gamma(U, M))\n$$","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0S","source_file":"sites-cohomology.tex","source_line":11172,"source_end_line":11181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11172-L11181","statement_sha256":"8faf5fa1f24ca6affc41a2a6f9016a553087263b6c0053f09ff2a66952180b8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4602,"rank":4602,"depth":1,"x":1036.699,"y":574.049,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYX","tag":"0GYX","title":"Modules on a category · Lemma 0GYX","summary":"In the situation above, there exists a cardinal kappa with the following property: given a complex F^bullet of O-modules and subsets Ω^i_U ⊂ F^i(U) there exists a subcomplex H^bullet ⊂ F^bullet with Ω^i_U ⊂ H^i(U) and |H^bullet| ≤ max(kappa, |⋃ Ω^i_U|).","statement_latex":"In the situation above, there exists a cardinal $\\kappa$ with the\nfollowing property: given a complex $\\mathcal{F}^\\bullet$ of\n$\\mathcal{O}$-modules and subsets $\\Omega^i_U \\subset \\mathcal{F}^i(U)$\nthere exists a subcomplex $\\mathcal{H}^\\bullet \\subset \\mathcal{F}^\\bullet$\nwith $\\Omega^i_U \\subset \\mathcal{H}^i(U)$ and\n$|\\mathcal{H}^\\bullet| \\leq \\max(\\kappa, |\\bigcup \\Omega^i_U|)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYX","source_file":"sites-cohomology.tex","source_line":11216,"source_end_line":11224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11216-L11224","statement_sha256":"85f1ebdc1a04fbceb3b2394a34578e9cca548c5116119d120ee601317968cb27","origin":"The Stacks Project","memory_eligible":false,"source_rank":4603,"rank":4603,"depth":0,"x":1284.168,"y":705.213,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYY","tag":"0GYY","title":"Modules on a category · Lemma 0GYY","summary":"In the situation above, there exists a cardinal kappa with the following property: given a complex F^bullet of O-modules representing an object K of D(O) there exists a subcomplex H^bullet ⊂ F^bullet such that H^bullet represents K and such that |H^bullet| ≤ max(kappa, |K|).","statement_latex":"In the situation above, there exists a cardinal $\\kappa$ with the\nfollowing property: given a complex $\\mathcal{F}^\\bullet$ of\n$\\mathcal{O}$-modules representing an object $K$ of $D(\\mathcal{O})$\nthere exists a subcomplex $\\mathcal{H}^\\bullet \\subset \\mathcal{F}^\\bullet$\nsuch that $\\mathcal{H}^\\bullet$ represents $K$ and such that\n$|\\mathcal{H}^\\bullet| \\leq \\max(\\kappa, |K|)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYY","source_file":"sites-cohomology.tex","source_line":11237,"source_end_line":11245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11237-L11245","statement_sha256":"803868eee766dad5a34958cbb7a7966cc1483f0f24b58d8b3f851e8da5362ce0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4604,"rank":4604,"depth":1,"x":995.899,"y":748.962,"cluster":"sheaf-cohomology"},{"id":"stacks:0GYZ","tag":"0GYZ","title":"Modules on a category · Lemma 0GYZ","summary":"In the situation above, there exists a cardinal kappa with the following properties: • for every nonzero object K of mathitQC(O) there exists a nonzero morphism E → K of mathitQC(O) such that |E| ≤ kappa, • for every morphism α : E → bigoplus_n K_n of mathitQC(O) such that |E| ≤ kappa, there exist morphisms E_n → K_n in mathitQC(O) with |E_n| ≤ kappa such that α factors through bigoplus E_n → bigoplus K_n.","statement_latex":"In the situation above, there exists a cardinal $\\kappa$ with the following\nproperties:\n\\begin{enumerate}\n\\item for every nonzero object $K$ of $\\mathit{QC}(\\mathcal{O})$ there exists a\nnonzero morphism $E \\to K$ of $\\mathit{QC}(\\mathcal{O})$ such that\n$|E| \\leq \\kappa$,\n\\item for every morphism $\\alpha : E \\to \\bigoplus_n K_n$ of\n$\\mathit{QC}(\\mathcal{O})$ such that $|E| \\leq \\kappa$, there\nexist morphisms $E_n \\to K_n$\nin $\\mathit{QC}(\\mathcal{O})$ with $|E_n| \\leq \\kappa$\nsuch that $\\alpha$ factors through $\\bigoplus E_n \\to \\bigoplus K_n$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GYZ","source_file":"sites-cohomology.tex","source_line":11295,"source_end_line":11309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11295-L11309","statement_sha256":"52dbe82650051defbe5eef2518dd43296196c164405e2803dddd1d966e72764d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4605,"rank":4605,"depth":2,"x":1173.474,"y":552.92,"cluster":"sheaf-cohomology"},{"id":"stacks:0GZ0","tag":"0GZ0","title":"Modules on a category · Proposition 0GZ0","summary":"Let C be a category viewed as a site with the chaotic topology. Let O be a sheaf of rings on C. With mathitQC(O) as in Definition [Tag 0GYV] we have • mathitQC(O) is a strictly full, saturated, triangulated subcategory of D(O) preserved by arbitrary direct sums, • any contravariant cohomological functor H : mathitQC(O) → Ab which transforms direct sums into products is representable, • any exact functor F : mathitQC(O) → D of triangulated categories which transforms…","statement_latex":"Let $\\mathcal{C}$ be a category viewed as a site with\nthe chaotic topology. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.\nWith $\\mathit{QC}(\\mathcal{O})$ as in\nDefinition \\ref{definition-cartesian} we have\n\\begin{enumerate}\n\\item $\\mathit{QC}(\\mathcal{O})$ is a strictly full, saturated, triangulated\nsubcategory of $D(\\mathcal{O})$ preserved by arbitrary direct sums,\n\\item any contravariant cohomological functor\n$H : \\mathit{QC}(\\mathcal{O}) \\to \\textit{Ab}$\nwhich transforms direct sums into products is representable,\n\\item any exact functor $F : \\mathit{QC}(\\mathcal{O}) \\to \\mathcal{D}$\nof triangulated categories which transforms direct sums into direct sums\nhas an exact right adjoint, and\n\\item the inclusion functor $\\mathit{QC}(\\mathcal{O}) \\to D(\\mathcal{O})$ has\nan exact right adjoint.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZ0","source_file":"sites-cohomology.tex","source_line":11457,"source_end_line":11475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11457-L11475","statement_sha256":"c2477dfe9660ebb5b69396b2011642b028b490326ad41c7039165de7e3f7ac5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4606,"rank":4606,"depth":9,"x":1200.213,"y":798.502,"cluster":"sheaf-cohomology"},{"id":"stacks:0GZ1","tag":"0GZ1","title":"Modules on a category · Lemma 0GZ1","summary":"Let g : (Sh(C'), O') → (Sh(C), O) be as above. Then the functor Lg^* : D(O) → D(O') maps mathitQC(O) into mathitQC(O').","statement_latex":"Let $g : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nbe as above. Then the functor $Lg^* : D(\\mathcal{O}) \\to D(\\mathcal{O}')$\nmaps $\\mathit{QC}(\\mathcal{O})$ into $\\mathit{QC}(\\mathcal{O}')$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZ1","source_file":"sites-cohomology.tex","source_line":11500,"source_end_line":11505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11500-L11505","statement_sha256":"135ce3492e539d1b03e79c96808e4a10b3549272e2bc7ff1a44d1588727e7711","origin":"The Stacks Project","memory_eligible":false,"source_rank":4607,"rank":4607,"depth":19,"x":982.772,"y":632.408,"cluster":"sheaf-cohomology"},{"id":"stacks:0GZR","tag":"0GZR","title":"Modules on a category · Lemma 0GZR","summary":"Let C be a category viewed as a site with the chaotic topology. Let O be a sheaf of rings on C. Assume for all U → V in C the restriction map O(V) → O(U) is a flat ring map. Then mathitQC(O) agrees with the subcategory D_QCoh(O) ⊂ D(O) of complexes whose cohomology sheaves are quasi-coherent.","statement_latex":"Let $\\mathcal{C}$ be a category viewed as a site with the chaotic\ntopology. Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$.\nAssume for all $U \\to V$ in $\\mathcal{C}$ the restriction map\n$\\mathcal{O}(V) \\to \\mathcal{O}(U)$ is a flat ring map.\nThen $\\mathit{QC}(\\mathcal{O})$ agrees with the subcategory\n$D_\\QCoh(\\mathcal{O}) \\subset D(\\mathcal{O})$ of complexes\nwhose cohomology sheaves are quasi-coherent.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZR","source_file":"sites-cohomology.tex","source_line":11552,"source_end_line":11561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11552-L11561","statement_sha256":"8ee32f10e252eea884b47f76b3c464c2d25bd196870397fc0a607550cbc54522","origin":"The Stacks Project","memory_eligible":false,"source_rank":4608,"rank":4608,"depth":9,"x":1276.994,"y":631.501,"cluster":"sheaf-cohomology"},{"id":"stacks:0GZS","tag":"0GZS","title":"Modules on a category · Lemma 0GZS","summary":"Let ε : (C_τ, O_τ) → (C_τ', O_τ') be as in Section [Tag 0EWK]. Assume • τ' is the chaotic topology on the category C, • for all U ∈ Ob(C) and all K-flat complexes of O(U)-modules M^bullet the map M^bullet → RΓ((C/U)_τ, (M^bullet ⊗_O(U) O_U)^\\#) is a quasi-isomorphism (see proof for an explanation). Then ε^* and Rε_* define mutually quasi-inverse equivalences between mathitQC(O) and the full subcategory of D(C_τ, O_τ) consisting of objects K such that Rε_*K is in…","statement_latex":"Let $\\epsilon : (\\mathcal{C}_\\tau, \\mathcal{O}_\\tau) \\to\n(\\mathcal{C}_{\\tau'}, \\mathcal{O}_{\\tau'})$ be as in\nSection \\ref{section-compare}. Assume\n\\begin{enumerate}\n\\item $\\tau'$ is the chaotic topology on the category $\\mathcal{C}$,\n\\item for all $U \\in \\Ob(\\mathcal{C})$ and all K-flat complexes of\n$\\mathcal{O}(U)$-modules $M^\\bullet$ the map\n$$\nM^\\bullet \\longrightarrow\nR\\Gamma((\\mathcal{C}/U)_\\tau,\n(M^\\bullet \\otimes_{\\mathcal{O}(U)} \\mathcal{O}_U)^\\#)\n$$\nis a quasi-isomorphism (see proof for an explanation).\n\\end{enumerate}\nThen $\\epsilon^*$ and $R\\epsilon_*$ define mutually quasi-inverse\nequivalences between $\\mathit{QC}(\\mathcal{O})$ and\nthe full subcategory of $D(\\mathcal{C}_\\tau, \\mathcal{O}_\\tau)$\nconsisting of objects $K$ such that $R\\epsilon_*K$ is in\n$\\mathit{QC}(\\mathcal{O})$\\footnote{This means that\n$R\\Gamma(V, K) \\otimes_{\\mathcal{O}(V)}^\\mathbf{L} \\mathcal{O}(U)\n\\to R\\Gamma(U, K)$ is an isomorphism for all $U \\to V$ in $\\mathcal{C}$.}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZS","source_file":"sites-cohomology.tex","source_line":11607,"source_end_line":11630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11607-L11630","statement_sha256":"a3a27850cb556f34b03d24eb6ed9e3c52495cf8e26425bb79405c1c55acc2728","origin":"The Stacks Project","memory_eligible":false,"source_rank":4609,"rank":4609,"depth":16,"x":1060.536,"y":799.297,"cluster":"sheaf-cohomology"},{"id":"stacks:0H0T","tag":"0H0T","title":"Modules on a category · Lemma 0H0T","summary":"Notation and assumptions as in Lemma [Tag 0GZS]. Suppose that K is an object of mathitQC(O) and M arbitrary in D(O_τ). For every object U of C we have Hom_D((O_U)_τ)(ε^*K|_U, M|_U) = RHom_O(U)(RΓ(U, K), RΓ(U, M))","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-cartesion-plus-topology}.\nSuppose that $K$ is an object of\n$\\mathit{QC}(\\mathcal{O})$ and $M$ arbitrary in $D(\\mathcal{O}_\\tau)$.\nFor every object $U$ of $\\mathcal{C}$ we have\n$$\n\\Hom_{D((\\mathcal{O}_U)_\\tau)}(\\epsilon^*K|_U, M|_U) =\nR\\Hom_{\\mathcal{O}(U)}(R\\Gamma(U, K), R\\Gamma(U, M))\n$$","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0T","source_file":"sites-cohomology.tex","source_line":11677,"source_end_line":11687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11677-L11687","statement_sha256":"dc9853f239876adbb75f590d0e28525fb42d4cb96445f40e6db7647e8183e008","origin":"The Stacks Project","memory_eligible":false,"source_rank":4610,"rank":4610,"depth":17,"x":1085.238,"y":552.481,"cluster":"sheaf-cohomology"},{"id":"stacks:08FL","tag":"08FL","title":"Strictly perfect complexes · Definition 08FL","summary":"Let (C, O) be a ringed site. Let E^bullet be a complex of O-modules. We say E^bullet is strictly perfect if E^i is zero for all but finitely many i and E^i is a direct summand of a finite free O-module for all i.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}$-modules.\nWe say $\\mathcal{E}^\\bullet$ is {\\it strictly perfect}\nif $\\mathcal{E}^i$ is zero for all but finitely many $i$ and\n$\\mathcal{E}^i$ is a direct summand of a finite free\n$\\mathcal{O}$-module for all $i$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FL","source_file":"sites-cohomology.tex","source_line":11716,"source_end_line":11724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11716-L11724","statement_sha256":"8e12b42b7ae8727e81ecc4ca49400195f13ab6ec3f52a9086e91faca6a434e12","origin":"The Stacks Project","memory_eligible":false,"source_rank":4611,"rank":4611,"depth":0,"x":1265.698,"y":748.705,"cluster":"sheaf-cohomology"},{"id":"stacks:08FM","tag":"08FM","title":"Strictly perfect complexes · Lemma 08FM","summary":"The cone on a morphism of strictly perfect complexes is strictly perfect.","statement_latex":"The cone on a morphism of strictly perfect complexes is\nstrictly perfect.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FM","source_file":"sites-cohomology.tex","source_line":11734,"source_end_line":11738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11734-L11738","statement_sha256":"0dcd61eb62b03fe8d9a38ae2f16845251ed27b8104e57237dfa123a73dd9b054","origin":"The Stacks Project","memory_eligible":false,"source_rank":4612,"rank":4612,"depth":0,"x":974.523,"y":706.365,"cluster":"sheaf-cohomology"},{"id":"stacks:09J8","tag":"09J8","title":"Strictly perfect complexes · Lemma 09J8","summary":"The total complex associated to the tensor product of two strictly perfect complexes is strictly perfect.","statement_latex":"The total complex associated to the tensor product of two\nstrictly perfect complexes is strictly perfect.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09J8","source_file":"sites-cohomology.tex","source_line":11744,"source_end_line":11748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11744-L11748","statement_sha256":"66baba13494d7020988b6cb0bb3d7d34eec68fbb34bbf9658f99f516a676013e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4613,"rank":4613,"depth":0,"x":1223.544,"y":572.224,"cluster":"sheaf-cohomology"},{"id":"stacks:08H3","tag":"08H3","title":"Strictly perfect complexes · Lemma 08H3","summary":"Let (f, f^sharp) : (C, O_C) → (D, O_D) be a morphism of ringed topoi. If F^bullet is a strictly perfect complex of O_D-modules, then f^*F^bullet is a strictly perfect complex of O_C-modules.","statement_latex":"Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi. If $\\mathcal{F}^\\bullet$ is a strictly\nperfect complex of $\\mathcal{O}_\\mathcal{D}$-modules, then\n$f^*\\mathcal{F}^\\bullet$ is a strictly perfect complex of\n$\\mathcal{O}_\\mathcal{C}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08H3","source_file":"sites-cohomology.tex","source_line":11754,"source_end_line":11762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11754-L11762","statement_sha256":"554ced1308c416a3e9f69777042a0b1ef8c630795e84c9d06f0392191c24afb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4614,"rank":4614,"depth":6,"x":1147.712,"y":812.692,"cluster":"sheaf-cohomology"},{"id":"stacks:08FN","tag":"08FN","title":"Strictly perfect complexes · Lemma 08FN","summary":"Let (C, O) be a ringed site. Let U be an object of C. Given a solid diagram of O_U-modules xymatrix E ar@..>[dr] ar[r] & F & G ar[u]_p with E a direct summand of a finite free O_U-module and p surjective, then there exists a covering (U_i → U) such that a dotted arrow making the diagram commute exists over each U_i.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of\n$\\mathcal{C}$. Given a solid diagram of $\\mathcal{O}_U$-modules\n$$\n\\xymatrix{\n\\mathcal{E} \\ar@{..>}[dr] \\ar[r] & \\mathcal{F} \\\\\n& \\mathcal{G} \\ar[u]_p\n}\n$$\nwith $\\mathcal{E}$ a direct summand of a finite free\n$\\mathcal{O}_U$-module and $p$ surjective, then there exists a\ncovering $\\{U_i \\to U\\}$ such that a dotted arrow\nmaking the diagram commute exists over each $U_i$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FN","source_file":"sites-cohomology.tex","source_line":11771,"source_end_line":11785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11771-L11785","statement_sha256":"ad42b22a2fad826e694a438f132a9650745e141b99da848d602bacd2b9c7be23","origin":"The Stacks Project","memory_eligible":false,"source_rank":4615,"rank":4615,"depth":0,"x":1010.107,"y":592.113,"cluster":"sheaf-cohomology"},{"id":"stacks:08FP","tag":"08FP","title":"Strictly perfect complexes · Lemma 08FP","summary":"Let (C, O) be a ringed site. Let U be an object of C. • Let α : E^bullet → F^bullet be a morphism of complexes of O_U-modules with E^bullet strictly perfect and F^bullet acyclic. Then there exists a covering (U_i → U) such that each α|_U_i is homotopic to zero. • Let α : E^bullet → F^bullet be a morphism of complexes of O_U-modules with E^bullet strictly perfect, E^i = 0 for i < a, and H^i(F^bullet) = 0 for i ≥ a. Then there exists a covering (U_i → U) such that each…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object\nof $\\mathcal{C}$.\n\\begin{enumerate}\n\\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$\nbe a morphism of complexes of $\\mathcal{O}_U$-modules\nwith $\\mathcal{E}^\\bullet$ strictly perfect and $\\mathcal{F}^\\bullet$\nacyclic. Then there exists a covering $\\{U_i \\to U\\}$ such that each\n$\\alpha|_{U_i}$ is homotopic to zero.\n\\item Let $\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$\nbe a morphism of complexes of $\\mathcal{O}_U$-modules\nwith $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^i = 0$\nfor $i < a$, and $H^i(\\mathcal{F}^\\bullet) = 0$ for $i \\geq a$.\nThen there exists a covering $\\{U_i \\to U\\}$ such that each\n$\\alpha|_{U_i}$ is homotopic to zero.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FP","source_file":"sites-cohomology.tex","source_line":11795,"source_end_line":11812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11795-L11812","statement_sha256":"2de9a0c2f0bc06831639f87ba2fc426dd49a4c84098fdc7961b3c987c4fe2bec","origin":"The Stacks Project","memory_eligible":false,"source_rank":4616,"rank":4616,"depth":1,"x":1289.251,"y":676.773,"cluster":"sheaf-cohomology"},{"id":"stacks:08FQ","tag":"08FQ","title":"Strictly perfect complexes · Lemma 08FQ","summary":"Let (C, O) be a ringed site. Let U be an object of C. Given a solid diagram of complexes of O_U-modules xymatrix E^bullet ar@..>[dr] ar[r]_α & F^bullet & G^bullet ar[u]_f with E^bullet strictly perfect, E^j = 0 for j < a and H^j(f) an isomorphism for j > a and surjective for j = a, then there exists a covering (U_i → U) and for each i a dotted arrow over U_i making the diagram commute up to homotopy.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of\n$\\mathcal{C}$. Given a solid diagram of complexes of $\\mathcal{O}_U$-modules\n$$\n\\xymatrix{\n\\mathcal{E}^\\bullet \\ar@{..>}[dr] \\ar[r]_\\alpha & \\mathcal{F}^\\bullet \\\\\n& \\mathcal{G}^\\bullet \\ar[u]_f\n}\n$$\nwith $\\mathcal{E}^\\bullet$ strictly perfect, $\\mathcal{E}^j = 0$ for\n$j < a$ and $H^j(f)$ an isomorphism for $j > a$ and surjective for\n$j = a$, then there exists a covering $\\{U_i \\to U\\}$ and for each $i$\na dotted arrow over $U_i$ making the diagram commute up to homotopy.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FQ","source_file":"sites-cohomology.tex","source_line":11847,"source_end_line":11861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11847-L11861","statement_sha256":"fed698aa7c8fcf656f72c7082cce71c357cf2b12a93cbc48832bcd684707c792","origin":"The Stacks Project","memory_eligible":false,"source_rank":4617,"rank":4617,"depth":2,"x":1015.045,"y":772.84,"cluster":"sheaf-cohomology"},{"id":"stacks:08FR","tag":"08FR","title":"Strictly perfect complexes · Lemma 08FR","summary":"Let (C, O) be a ringed site. Let U be an object of C. Let E^bullet, F^bullet be complexes of O_U-modules with E^bullet strictly perfect. • For any element α ∈ Hom_D(O_U)(E^bullet, F^bullet) there exists a covering (U_i → U) such that α|_U_i is given by a morphism of complexes α_i : E^bullet|_U_i → F^bullet|_U_i. • Given a morphism of complexes α : E^bullet → F^bullet whose image in the group Hom_D(O_U)(E^bullet, F^bullet) is zero, there exists a covering (U_i → U) such…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object\nof $\\mathcal{C}$. Let $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be\ncomplexes of $\\mathcal{O}_U$-modules with $\\mathcal{E}^\\bullet$ strictly\nperfect.\n\\begin{enumerate}\n\\item For any element\n$\\alpha \\in \\Hom_{D(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$\nthere exists a covering $\\{U_i \\to U\\}$ such that\n$\\alpha|_{U_i}$ is given by a morphism of complexes\n$\\alpha_i : \\mathcal{E}^\\bullet|_{U_i} \\to \\mathcal{F}^\\bullet|_{U_i}$.\n\\item Given a morphism of complexes\n$\\alpha : \\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$\nwhose image in the group\n$\\Hom_{D(\\mathcal{O}_U)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$\nis zero, there exists a covering $\\{U_i \\to U\\}$ such that\n$\\alpha|_{U_i}$ is homotopic to zero.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FR","source_file":"sites-cohomology.tex","source_line":11880,"source_end_line":11899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11880-L11899","statement_sha256":"d948b021bb0988de04b3a8e36aeb24fc87a5203b7f628732f632d43491a5964c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4618,"rank":4618,"depth":3,"x":1140.117,"y":546.173,"cluster":"sheaf-cohomology"},{"id":"stacks:08JH","tag":"08JH","title":"Strictly perfect complexes · Lemma 08JH","summary":"Let (C, O) be a ringed site. Let E^bullet, F^bullet be complexes of O-modules with E^bullet strictly perfect. Then the internal hom RSheafHom(E^bullet, F^bullet) is represented by the complex H^bullet with terms H^n = bigoplus_n = p + q SheafHom_O(E^-q, F^p) and differential as described in Section [Tag 08J7].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes\nof $\\mathcal{O}$-modules with $\\mathcal{E}^\\bullet$ strictly perfect.\nThen the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$\nis represented by the complex $\\mathcal{H}^\\bullet$ with terms\n$$\n\\mathcal{H}^n =\n\\bigoplus\\nolimits_{n = p + q}\n\\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{F}^p)\n$$\nand differential as described in Section \\ref{section-internal-hom}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JH","source_file":"sites-cohomology.tex","source_line":11911,"source_end_line":11924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11911-L11924","statement_sha256":"83aa9c88f8ffeb72850dcb1b06cce020093b0b0fa71d7754c85d2b7151498a3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4619,"rank":4619,"depth":4,"x":1230.264,"y":784.532,"cluster":"sheaf-cohomology"},{"id":"stacks:08JI","tag":"08JI","title":"Strictly perfect complexes · Lemma 08JI","summary":"Let (C, O) be a ringed site. Let E^bullet, F^bullet be complexes of O-modules with • F^n = 0 for n ll 0, • E^n = 0 for n gg 0, and • E^n isomorphic to a direct summand of a finite free O-module. Then the internal hom RSheafHom(E^bullet, F^bullet) is represented by the complex H^bullet with terms H^n = bigoplus_n = p + q SheafHom_O(E^-q, F^p) and differential as described in Section [Tag 08J7].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{E}^\\bullet$, $\\mathcal{F}^\\bullet$ be complexes\nof $\\mathcal{O}$-modules with\n\\begin{enumerate}\n\\item $\\mathcal{F}^n = 0$ for $n \\ll 0$,\n\\item $\\mathcal{E}^n = 0$ for $n \\gg 0$, and\n\\item $\\mathcal{E}^n$ isomorphic to a direct summand of a finite\nfree $\\mathcal{O}$-module.\n\\end{enumerate}\nThen the internal hom $R\\SheafHom(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$\nis represented by the complex $\\mathcal{H}^\\bullet$ with terms\n$$\n\\mathcal{H}^n =\n\\bigoplus\\nolimits_{n = p + q}\n\\SheafHom_\\mathcal{O}(\\mathcal{E}^{-q}, \\mathcal{F}^p)\n$$\nand differential as described in Section \\ref{section-internal-hom}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Strictly perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JI","source_file":"sites-cohomology.tex","source_line":11958,"source_end_line":11977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L11958-L11977","statement_sha256":"60a7eb9e40194e02b98ff65bac68c25e17ea035b150e7a5920217987f75c0194","origin":"The Stacks Project","memory_eligible":false,"source_rank":4620,"rank":4620,"depth":5,"x":971.84,"y":659.791,"cluster":"sheaf-cohomology"},{"id":"stacks:08FT","tag":"08FT","title":"Pseudo-coherent modules · Definition 08FT","summary":"Let (C, O) be a ringed site. Let E^bullet be a complex of O-modules. Let m ∈ Z. • We say E^bullet is m-pseudo-coherent if for every object U of C there exists a covering (U_i → U) and for each i a morphism of complexes α_i : E_i^bullet → E^bullet|_U_i where E_i^bullet is a strictly perfect complex of O_U_i-modules and H^j(α_i) is an isomorphism for j > m and H^m(α_i) is surjective. • We say E^bullet is pseudo-coherent if it is m-pseudo-coherent for all m. • We say an…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{E}^\\bullet$\nbe a complex of $\\mathcal{O}$-modules. Let $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item We say $\\mathcal{E}^\\bullet$ is {\\it $m$-pseudo-coherent}\nif for every object $U$ of $\\mathcal{C}$ there exists a covering\n$\\{U_i \\to U\\}$ and for each $i$ a morphism of complexes\n$\\alpha_i : \\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$\nwhere $\\mathcal{E}_i^\\bullet$ is a strictly perfect complex of\n$\\mathcal{O}_{U_i}$-modules and $H^j(\\alpha_i)$ is an isomorphism\nfor $j > m$ and $H^m(\\alpha_i)$ is surjective.\n\\item We say $\\mathcal{E}^\\bullet$ is {\\it pseudo-coherent}\nif it is $m$-pseudo-coherent for all $m$.\n\\item We say an object $E$ of $D(\\mathcal{O})$ is\n{\\it $m$-pseudo-coherent} (resp.\\ {\\it pseudo-coherent})\nif and only if it can be represented by a $m$-pseudo-coherent\n(resp.\\ pseudo-coherent) complex of $\\mathcal{O}$-modules.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Pseudo-coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FT","source_file":"sites-cohomology.tex","source_line":12038,"source_end_line":12057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12038-L12057","statement_sha256":"5b7eafde35d5c28b053d4242b7312c34951b074b7c943180db5f48295f1193b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4621,"rank":4621,"depth":0,"x":1263.014,"y":605.08,"cluster":"sheaf-cohomology"},{"id":"stacks:08FU","tag":"08FU","title":"Pseudo-coherent modules · Lemma 08FU","summary":"Let (C, O) be a ringed site. Let E be an object of D(O). • If C has a final object X and if there exist a covering (U_i → X), strictly perfect complexes E_i^bullet of O_U_i-modules, and maps α_i : E_i^bullet → E|_U_i in D(O_U_i) with H^j(α_i) an isomorphism for j > m and H^m(α_i) surjective, then E is m-pseudo-coherent. • If E is m-pseudo-coherent, then any complex of O-modules representing E is m-pseudo-coherent. • If for every object U of C there exists a covering (U_i…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $E$ be an object of $D(\\mathcal{O})$.\n\\begin{enumerate}\n\\item If $\\mathcal{C}$ has a final object $X$ and if there exist a covering\n$\\{U_i \\to X\\}$, strictly perfect complexes $\\mathcal{E}_i^\\bullet$ of\n$\\mathcal{O}_{U_i}$-modules, and\nmaps $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ in\n$D(\\mathcal{O}_{U_i})$ with $H^j(\\alpha_i)$ an isomorphism for $j > m$\nand $H^m(\\alpha_i)$ surjective, then $E$ is $m$-pseudo-coherent.\n\\item If $E$ is $m$-pseudo-coherent, then any complex of $\\mathcal{O}$-modules\nrepresenting $E$ is $m$-pseudo-coherent.\n\\item If for every object $U$ of $\\mathcal{C}$ there exists a covering\n$\\{U_i \\to U\\}$ such that $E|_{U_i}$ is $m$-pseudo-coherent, then\n$E$ is $m$-pseudo-coherent.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FU","source_file":"sites-cohomology.tex","source_line":12065,"source_end_line":12082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12065-L12082","statement_sha256":"1c0ea24d61db22f9694a39bd720aaeb95d43082aa38fe8ca1830cb2b180b0069","origin":"The Stacks Project","memory_eligible":false,"source_rank":4622,"rank":4622,"depth":4,"x":1092.127,"y":810.856,"cluster":"sheaf-cohomology"},{"id":"stacks:08H4","tag":"08H4","title":"Pseudo-coherent modules · Lemma 08H4","summary":"Let (f, f^sharp) : (C, O_C) → (D, O_D) be a morphism of ringed sites. Let E be an object of D(O_C). If E is m-pseudo-coherent, then Lf^*E is m-pseudo-coherent.","statement_latex":"Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed sites. Let $E$ be an object of\n$D(\\mathcal{O}_\\mathcal{C})$. If $E$ is $m$-pseudo-coherent,\nthen $Lf^*E$ is $m$-pseudo-coherent.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08H4","source_file":"sites-cohomology.tex","source_line":12109,"source_end_line":12116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12109-L12116","statement_sha256":"13f22af37df68a7e9829a746fe44b0527522c47f175b52d40f51c90bd9750f42","origin":"The Stacks Project","memory_eligible":false,"source_rank":4623,"rank":4623,"depth":10,"x":1052.616,"y":561.896,"cluster":"sheaf-cohomology"},{"id":"stacks:08FV","tag":"08FV","title":"Pseudo-coherent modules · Lemma 08FV","summary":"Let (C, O) be a ringed site and m ∈ Z. Let (K, L, M, f, g, h) be a distinguished triangle in D(O). • If K is (m + 1)-pseudo-coherent and L is m-pseudo-coherent then M is m-pseudo-coherent. • If K and M are m-pseudo-coherent, then L is m-pseudo-coherent. • If L is (m + 1)-pseudo-coherent and M is m-pseudo-coherent, then K is (m + 1)-pseudo-coherent.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site and $m \\in \\mathbf{Z}$.\nLet $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\\mathcal{O})$.\n\\begin{enumerate}\n\\item If $K$ is $(m + 1)$-pseudo-coherent and $L$ is $m$-pseudo-coherent\nthen $M$ is $m$-pseudo-coherent.\n\\item If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent.\n\\item If $L$ is $(m + 1)$-pseudo-coherent and $M$\nis $m$-pseudo-coherent, then $K$ is $(m + 1)$-pseudo-coherent.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FV","source_file":"sites-cohomology.tex","source_line":12176,"source_end_line":12187,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12176-L12187","statement_sha256":"36f7e2c6a643b8b020145986d64b0111805d42990759ebe66310b600c661dd4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4624,"rank":4624,"depth":8,"x":1282.195,"y":723.223,"cluster":"sheaf-cohomology"},{"id":"stacks:09J9","tag":"09J9","title":"Pseudo-coherent modules · Lemma 09J9","summary":"Let (C, O) be a ringed site. Let K, L be objects of D(O). • If K is n-pseudo-coherent and H^i(K) = 0 for i > a and L is m-pseudo-coherent and H^j(L) = 0 for j > b, then K ⊗_O^L L is t-pseudo-coherent with t = max(m + a, n + b). • If K and L are pseudo-coherent, then K ⊗_O^L L is pseudo-coherent.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects\nof $D(\\mathcal{O})$.\n\\begin{enumerate}\n\\item If $K$ is $n$-pseudo-coherent and $H^i(K) = 0$ for $i > a$\nand $L$ is $m$-pseudo-coherent and $H^j(L) = 0$ for $j > b$, then\n$K \\otimes_\\mathcal{O}^\\mathbf{L} L$ is $t$-pseudo-coherent\nwith $t = \\max(m + a, n + b)$.\n\\item If $K$ and $L$ are pseudo-coherent, then\n$K \\otimes_\\mathcal{O}^\\mathbf{L} L$ is pseudo-coherent.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09J9","source_file":"sites-cohomology.tex","source_line":12237,"source_end_line":12249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12237-L12249","statement_sha256":"a3e44e20f8b0e83757ae83d8424adb0548b5ec9668495536c6ea2342f2626c49","origin":"The Stacks Project","memory_eligible":false,"source_rank":4625,"rank":4625,"depth":0,"x":982.863,"y":734.543,"cluster":"sheaf-cohomology"},{"id":"stacks:08FW","tag":"08FW","title":"Pseudo-coherent modules · Lemma 08FW","summary":"Let (C, O) be a ringed site. Let m ∈ Z. If K ⊕ L is m-pseudo-coherent (resp. pseudo-coherent) in D(O) so are K and L.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $m \\in \\mathbf{Z}$.\nIf $K \\oplus L$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nin $D(\\mathcal{O})$ so are $K$ and $L$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FW","source_file":"sites-cohomology.tex","source_line":12279,"source_end_line":12284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12279-L12284","statement_sha256":"8e5430ca10d1839f812912e525c8d394f9bf60a6ef49c6aca106afa19924327d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4626,"rank":4626,"depth":9,"x":1194.701,"y":556.165,"cluster":"sheaf-cohomology"},{"id":"stacks:08FX","tag":"08FX","title":"Pseudo-coherent modules · Lemma 08FX","summary":"Let (C, O) be a ringed site. Let K be an object of D(O). Let m ∈ Z. • If K is m-pseudo-coherent and H^i(K) = 0 for i > m, then H^m(K) is a finite type O-module. • If K is m-pseudo-coherent and H^i(K) = 0 for i > m + 1, then H^m + 1(K) is a finitely presented O-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K$ be an object of\n$D(\\mathcal{O})$. Let $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$\nfor $i > m$, then $H^m(K)$ is a finite type $\\mathcal{O}$-module.\n\\item If $K$ is $m$-pseudo-coherent and $H^i(K) = 0$\nfor $i > m + 1$, then $H^{m + 1}(K)$ is a finitely presented\n$\\mathcal{O}$-module.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Pseudo-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FX","source_file":"sites-cohomology.tex","source_line":12311,"source_end_line":12322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12311-L12322","statement_sha256":"5527a96fa183d693c00ae0d9128861fe2ece357a41ae5cd077ec1c9bd310943e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4627,"rank":4627,"depth":2,"x":1181.929,"y":808.159,"cluster":"sheaf-cohomology"},{"id":"stacks:08FZ","tag":"08FZ","title":"Tor dimension · Definition 08FZ","summary":"Let (C, O) be a ringed site. Let E be an object of D(O). Let a, b ∈ Z with a ≤ b. • We say E has tor-amplitude in [a, b] if H^i(E ⊗_O^L F) = 0 for all O-modules F and all i not ∈ [a, b]. • We say E has finite tor dimension if it has tor-amplitude in [a, b] for some a, b. • We say E locally has finite tor dimension if for any object U of C there exists a covering (U_i → U) such that E|_U_i has finite tor dimension for all i. An O-module F has tor dimension ≤ d if F[0]…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $E$ be an object of $D(\\mathcal{O})$.\nLet $a, b \\in \\mathbf{Z}$ with $a \\leq b$.\n\\begin{enumerate}\n\\item We say $E$ has {\\it tor-amplitude in $[a, b]$}\nif $H^i(E \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{F}) = 0$\nfor all $\\mathcal{O}$-modules $\\mathcal{F}$ and all $i \\not \\in [a, b]$.\n\\item We say $E$ has {\\it finite tor dimension}\nif it has tor-amplitude in $[a, b]$ for some $a, b$.\n\\item We say $E$ {\\it locally has finite tor dimension} if for any\nobject $U$ of $\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$\nsuch that $E|_{U_i}$ has finite tor dimension for all $i$.\n\\end{enumerate}\nAn $\\mathcal{O}$-module $\\mathcal{F}$ has {\\it tor dimension $\\leq d$}\nif $\\mathcal{F}[0]$ viewed as an object of $D(\\mathcal{O})$ has\ntor-amplitude in $[-d, 0]$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FZ","source_file":"sites-cohomology.tex","source_line":12376,"source_end_line":12394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12376-L12394","statement_sha256":"eef21da5f2106d48ad6f593e9b839118a2857f24dc645feca7af678e24be50e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4628,"rank":4628,"depth":0,"x":988.502,"y":614.897,"cluster":"sheaf-cohomology"},{"id":"stacks:08G0","tag":"08G0","title":"Tor dimension · Lemma 08G0","summary":"Let (C, O) be a ringed site. Let E^bullet be a bounded above complex of flat O-modules with tor-amplitude in [a, b]. Then Coker(d_E^bullet^a - 1) is a flat O-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{E}^\\bullet$ be a bounded above complex of flat\n$\\mathcal{O}$-modules with tor-amplitude in $[a, b]$.\nThen $\\Coker(d_{\\mathcal{E}^\\bullet}^{a - 1})$ is a flat\n$\\mathcal{O}$-module.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08G0","source_file":"sites-cohomology.tex","source_line":12402,"source_end_line":12409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12402-L12409","statement_sha256":"69b7b78d5107b2d42976b70d2715fdf31a078e94bebf4baf83c8a85c523783d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4629,"rank":4629,"depth":1,"x":1286.853,"y":647.684,"cluster":"sheaf-cohomology"},{"id":"stacks:08G1","tag":"08G1","title":"Tor dimension · Lemma 08G1","summary":"Let (C, O) be a ringed site. Let E be an object of D(O). Let a, b ∈ Z with a ≤ b. The following are equivalent • E has tor-amplitude in [a, b]. • E is represented by a complex E^bullet of flat O-modules with E^i = 0 for i not ∈ [a, b].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of\n$D(\\mathcal{O})$. Let $a, b \\in \\mathbf{Z}$ with $a \\leq b$. The following\nare equivalent\n\\begin{enumerate}\n\\item $E$ has tor-amplitude in $[a, b]$.\n\\item $E$ is represented by a complex\n$\\mathcal{E}^\\bullet$ of flat $\\mathcal{O}$-modules with\n$\\mathcal{E}^i = 0$ for $i \\not \\in [a, b]$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08G1","source_file":"sites-cohomology.tex","source_line":12431,"source_end_line":12442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12431-L12442","statement_sha256":"661e0048da924ab7c67b0ae04c79ebc3fb7883269f6ad30f267061343be090f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4630,"rank":4630,"depth":7,"x":1040.235,"y":792.947,"cluster":"sheaf-cohomology"},{"id":"stacks:0F1M","tag":"0F1M","title":"Tor dimension · Lemma 0F1M","summary":"Let (C, O) be a ringed site. Let E be an object of D(O). Let a ∈ Z. The following are equivalent • E has tor-amplitude in [a, ∞]. • E can be represented by a K-flat complex E^bullet of flat O-modules with E^i = 0 for i not ∈ [a, ∞]. Moreover, we can choose E^bullet such that any pullback by a morphism of ringed sites is a K-flat complex with flat terms.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $E$ be an object of\n$D(\\mathcal{O})$. Let $a \\in \\mathbf{Z}$. The following\nare equivalent\n\\begin{enumerate}\n\\item $E$ has tor-amplitude in $[a, \\infty]$.\n\\item $E$ can be represented by a K-flat complex $\\mathcal{E}^\\bullet$\nof flat $\\mathcal{O}$-modules with $\\mathcal{E}^i = 0$ for\n$i \\not \\in [a, \\infty]$.\n\\end{enumerate}\nMoreover, we can choose $\\mathcal{E}^\\bullet$ such that any pullback\nby a morphism of ringed sites is a K-flat complex with flat terms.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1M","source_file":"sites-cohomology.tex","source_line":12471,"source_end_line":12484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12471-L12484","statement_sha256":"5cb7ba5734f16183f2c30d6ca96373e49cd6b110269abb8f464344771ddc965f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4631,"rank":4631,"depth":16,"x":1105.337,"y":545.643,"cluster":"sheaf-cohomology"},{"id":"stacks:08H5","tag":"08H5","title":"Tor dimension · Lemma 08H5","summary":"Let (f, f^sharp) : (C, O_C) → (D, O_D) be a morphism of ringed sites. Let E be an object of D(O_D). If E has tor amplitude in [a, b], then Lf^*E has tor amplitude in [a, b].","statement_latex":"Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed sites.\nLet $E$ be an object of $D(\\mathcal{O}_\\mathcal{D})$.\nIf $E$ has tor amplitude in $[a, b]$,\nthen $Lf^*E$ has tor amplitude in $[a, b]$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08H5","source_file":"sites-cohomology.tex","source_line":12542,"source_end_line":12550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12542-L12550","statement_sha256":"6dcced4f338b9eb60413151a12284f06129e0ef627e5bb925f44c0bce7decff8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4632,"rank":4632,"depth":11,"x":1256.36,"y":765.165,"cluster":"sheaf-cohomology"},{"id":"stacks:08G2","tag":"08G2","title":"Tor dimension · Lemma 08G2","summary":"Let (C, O) be a ringed site. Let (K, L, M, f, g, h) be a distinguished triangle in D(O). Let a, b ∈ Z. • If K has tor-amplitude in [a + 1, b + 1] and L has tor-amplitude in [a, b] then M has tor-amplitude in [a, b]. • If K and M have tor-amplitude in [a, b], then L has tor-amplitude in [a, b]. • If L has tor-amplitude in [a + 1, b + 1] and M has tor-amplitude in [a, b], then K has tor-amplitude in [a + 1, b + 1].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(K, L, M, f, g, h)$ be a distinguished\ntriangle in $D(\\mathcal{O})$. Let $a, b \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $K$ has tor-amplitude in $[a + 1, b + 1]$ and\n$L$ has tor-amplitude in $[a, b]$ then $M$ has\ntor-amplitude in $[a, b]$.\n\\item If $K$ and $M$ have tor-amplitude in $[a, b]$, then\n$L$ has tor-amplitude in $[a, b]$.\n\\item If $L$ has tor-amplitude in $[a + 1, b + 1]$\nand $M$ has tor-amplitude in $[a, b]$, then\n$K$ has tor-amplitude in $[a + 1, b + 1]$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08G2","source_file":"sites-cohomology.tex","source_line":12565,"source_end_line":12580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12565-L12580","statement_sha256":"b3cfa08091d45d6e7555ba61bcb13920be623428f923c516fdd4c045a85cb5cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":4633,"rank":4633,"depth":0,"x":968.173,"y":688.91,"cluster":"sheaf-cohomology"},{"id":"stacks:09JA","tag":"09JA","title":"Tor dimension · Lemma 09JA","summary":"Let (C, O) be a ringed site. Let K, L be objects of D(O). If K has tor-amplitude in [a, b] and L has tor-amplitude in [c, d] then K ⊗_O^L L has tor amplitude in [a + c, b + d].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K, L$ be objects of\n$D(\\mathcal{O})$. If $K$ has tor-amplitude in $[a, b]$ and\n$L$ has tor-amplitude in $[c, d]$ then $K \\otimes_\\mathcal{O}^\\mathbf{L} L$\nhas tor amplitude in $[a + c, b + d]$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JA","source_file":"sites-cohomology.tex","source_line":12591,"source_end_line":12597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12591-L12597","statement_sha256":"3fdc2fafe085273661d868afdca943afb2fca37a2e0b10bc6655bd970ade8d5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4634,"rank":4634,"depth":0,"x":1242.278,"y":581.506,"cluster":"sheaf-cohomology"},{"id":"stacks:08G3","tag":"08G3","title":"Tor dimension · Lemma 08G3","summary":"Let (C, O) be a ringed site. Let a, b ∈ Z. For K, L objects of D(O) if K ⊕ L has tor amplitude in [a, b] so do K and L.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $a, b \\in \\mathbf{Z}$.\nFor $K$, $L$ objects of $D(\\mathcal{O})$ if $K \\oplus L$ has tor\namplitude in $[a, b]$ so do $K$ and $L$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08G3","source_file":"sites-cohomology.tex","source_line":12603,"source_end_line":12608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12603-L12608","statement_sha256":"21b6ced5da6fc2cb714c6d1a6dc12bf754ae6e46c8eaace61881c515040909c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4635,"rank":4635,"depth":0,"x":1126.411,"y":816.475,"cluster":"sheaf-cohomology"},{"id":"stacks:0942","tag":"0942","title":"Tor dimension · Lemma 0942","summary":"Let (C, O) be a ringed site. Let I ⊂ O be a sheaf of ideals. Let K be an object of D(O). • If K ⊗_O^L O/I is bounded above, then K ⊗_O^L O/I^n is uniformly bounded above for all n. • If K ⊗_O^L O/I as an object of D(O/I) has tor amplitude in [a, b], then K ⊗_O^L O/I^n as an object of D(O/I^n) has tor amplitude in [a, b] for all n.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals.\nLet $K$ be an object of $D(\\mathcal{O})$.\n\\begin{enumerate}\n\\item If $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}$\nis bounded above, then\n$K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$\nis uniformly bounded above for all $n$.\n\\item If $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}$\nas an object of $D(\\mathcal{O}/\\mathcal{I})$ has tor amplitude in $[a, b]$,\nthen $K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}/\\mathcal{I}^n$\nas an object of $D(\\mathcal{O}/\\mathcal{I}^n)$\nhas tor amplitude in $[a, b]$ for all $n$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0942","source_file":"sites-cohomology.tex","source_line":12614,"source_end_line":12630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12614-L12630","statement_sha256":"e7e630090da52a539f38a944e006e220dc157175df7fc72cd005f05d457a651f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4636,"rank":4636,"depth":0,"x":1022.79,"y":577.226,"cluster":"sheaf-cohomology"},{"id":"stacks:0DJJ","tag":"0DJJ","title":"Tor dimension · Lemma 0DJJ","summary":"Let (C, O) be a ringed site. Let E be an object of D(O). Let a, b ∈ Z. • If E has tor amplitude in [a, b], then for every point p of the site C the object E_p of D(O_p) has tor amplitude in [a, b]. • If C has enough points, then the converse is true.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $E$ be an object of $D(\\mathcal{O})$.\nLet $a, b \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $E$ has tor amplitude in $[a, b]$, then for every point $p$\nof the site $\\mathcal{C}$ the object $E_p$ of $D(\\mathcal{O}_p)$\nhas tor amplitude in $[a, b]$.\n\\item If $\\mathcal{C}$ has enough points, then the converse is true.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Tor dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJJ","source_file":"sites-cohomology.tex","source_line":12689,"source_end_line":12700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12689-L12700","statement_sha256":"a1870dfb99bf9f5f2bd66dab853033bdf0683786a3d4d098267e2c38a8795974","origin":"The Stacks Project","memory_eligible":false,"source_rank":4637,"rank":4637,"depth":12,"x":1291.866,"y":694.966,"cluster":"sheaf-cohomology"},{"id":"stacks:08G5","tag":"08G5","title":"Perfect complexes · Definition 08G5","summary":"Let (C, O) be a ringed site. Let E^bullet be a complex of O-modules. We say E^bullet is perfect if for every object U of C there exists a covering (U_i → U) such that for each i there exists a morphism of complexes E_i^bullet → E^bullet|_U_i which is a quasi-isomorphism with E_i^bullet strictly perfect. An object E of D(O) is perfect if it can be represented by a perfect complex of O-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{E}^\\bullet$ be a complex of $\\mathcal{O}$-modules.\nWe say $\\mathcal{E}^\\bullet$ is {\\it perfect} if for every object $U$ of\n$\\mathcal{C}$ there exists a covering $\\{U_i \\to U\\}$ such that for each $i$\nthere exists a morphism of complexes\n$\\mathcal{E}_i^\\bullet \\to \\mathcal{E}^\\bullet|_{U_i}$\nwhich is a quasi-isomorphism with $\\mathcal{E}_i^\\bullet$\nstrictly perfect.\nAn object $E$ of $D(\\mathcal{O})$ is {\\it perfect}\nif it can be represented by a perfect complex of $\\mathcal{O}$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Perfect complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08G5","source_file":"sites-cohomology.tex","source_line":12734,"source_end_line":12746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12734-L12746","statement_sha256":"d36e5c04588f94d9ada672f2e7509853e7821069cded9c7509f662980e60702a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4638,"rank":4638,"depth":0,"x":998.476,"y":760.893,"cluster":"sheaf-cohomology"},{"id":"stacks:08G6","tag":"08G6","title":"Perfect complexes · Lemma 08G6","summary":"Let (C, O) be a ringed site. Let E be an object of D(O). • If C has a final object X and there exist a covering (U_i → X), strictly perfect complexes E_i^bullet of O_U_i-modules, and isomorphisms α_i : E_i^bullet → E|_U_i in D(O_U_i), then E is perfect. • If E is perfect, then any complex representing E is perfect.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $E$ be an object of $D(\\mathcal{O})$.\n\\begin{enumerate}\n\\item If $\\mathcal{C}$ has a final object $X$ and there exist a\ncovering $\\{U_i \\to X\\}$, strictly perfect complexes $\\mathcal{E}_i^\\bullet$\nof $\\mathcal{O}_{U_i}$-modules, and isomorphisms\n $\\alpha_i : \\mathcal{E}_i^\\bullet \\to E|_{U_i}$ in\n$D(\\mathcal{O}_{U_i})$, then $E$ is perfect.\n\\item If $E$ is perfect, then any complex representing $E$ is perfect.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08G6","source_file":"sites-cohomology.tex","source_line":12754,"source_end_line":12766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12754-L12766","statement_sha256":"3036fba1be9c4ede0eedef3c17fcfc6a197ac5a946d1de6554d8767743a9e15f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4639,"rank":4639,"depth":5,"x":1161.965,"y":545.586,"cluster":"sheaf-cohomology"},{"id":"stacks:08G7","tag":"08G7","title":"Perfect complexes · Lemma 08G7","summary":"Let (C, O) be a ringed site. Let E be an object of D(O). Let a ≤ b be integers. If E has tor amplitude in [a, b] and is (a - 1)-pseudo-coherent, then E is perfect.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $E$ be an object of $D(\\mathcal{O})$.\nLet $a \\leq b$ be integers. If $E$ has tor amplitude in $[a, b]$\nand is $(a - 1)$-pseudo-coherent, then $E$ is perfect.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08G7","source_file":"sites-cohomology.tex","source_line":12773,"source_end_line":12779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12773-L12779","statement_sha256":"9966ffb2b27a6b23bec1e65041fcf42aa19b616e93bb91bfb8cc02e3440880f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4640,"rank":4640,"depth":8,"x":1214.606,"y":797.369,"cluster":"sheaf-cohomology"},{"id":"stacks:08G8","tag":"08G8","title":"Perfect complexes · Lemma 08G8","summary":"Let (C, O) be a ringed site. Let E be an object of D(O). The following are equivalent • E is perfect, and • E is pseudo-coherent and locally has finite tor dimension.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $E$ be an object of $D(\\mathcal{O})$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $E$ is perfect, and\n\\item $E$ is pseudo-coherent and locally has finite tor dimension.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08G8","source_file":"sites-cohomology.tex","source_line":12817,"source_end_line":12826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12817-L12826","statement_sha256":"769ece8d47e1d42658c209efd45b2c41b059876f735c2ca744385810c658a7bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4641,"rank":4641,"depth":9,"x":973.072,"y":641.423,"cluster":"sheaf-cohomology"},{"id":"stacks:08H6","tag":"08H6","title":"Perfect complexes · Lemma 08H6","summary":"Let (f, f^sharp) : (C, O_C) → (D, O_D) be a morphism of ringed sites. Let E be an object of D(O_D). If E is perfect in D(O_D), then Lf^*E is perfect in D(O_C).","statement_latex":"Let $(f, f^\\sharp) : (\\mathcal{C}, \\mathcal{O}_\\mathcal{C}) \\to\n(\\mathcal{D}, \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed sites.\nLet $E$ be an object of $D(\\mathcal{O}_\\mathcal{D})$.\nIf $E$ is perfect in $D(\\mathcal{O}_\\mathcal{D})$,\nthen $Lf^*E$ is perfect in $D(\\mathcal{O}_\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08H6","source_file":"sites-cohomology.tex","source_line":12846,"source_end_line":12854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12846-L12854","statement_sha256":"6d19c637f51c265386c56b9c3c5f973b73c19186eae81d0553d9d2f34fa5f6ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":4642,"rank":4642,"depth":12,"x":1276.885,"y":619.341,"cluster":"sheaf-cohomology"},{"id":"stacks:08G9","tag":"08G9","title":"Perfect complexes · Lemma 08G9","summary":"Let (C, O) be a ringed site. Let (K, L, M, f, g, h) be a distinguished triangle in D(O). If two out of three of K, L, M are perfect then the third is also perfect.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$\nbe a distinguished triangle in $D(\\mathcal{O})$. If two out of three of\n$K, L, M$ are perfect then the third is also perfect.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08G9","source_file":"sites-cohomology.tex","source_line":12862,"source_end_line":12867,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12862-L12867","statement_sha256":"37990bddf755effb7aff62cf7c699dc7445d5d834e25b7a148ed2597d297ce1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4643,"rank":4643,"depth":10,"x":1070.41,"y":808.203,"cluster":"sheaf-cohomology"},{"id":"stacks:09JB","tag":"09JB","title":"Perfect complexes · Lemma 09JB","summary":"Let (C, O) be a ringed site. If K, L are perfect objects of D(O), then so is K ⊗_O^L L.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nIf $K, L$ are perfect objects of $D(\\mathcal{O})$, then\nso is $K \\otimes_\\mathcal{O}^\\mathbf{L} L$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JB","source_file":"sites-cohomology.tex","source_line":12885,"source_end_line":12890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12885-L12890","statement_sha256":"f843633ee227aa00703e8995ff9b9729063b6bac51ca776af379040425ec08d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4644,"rank":4644,"depth":10,"x":1070.785,"y":551.525,"cluster":"sheaf-cohomology"},{"id":"stacks:08GA","tag":"08GA","title":"Perfect complexes · Lemma 08GA","summary":"Let (C, O) be a ringed site. If K ⊕ L is a perfect object of D(O), then so are K and L.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nIf $K \\oplus L$ is a perfect object of $D(\\mathcal{O})$, then\nso are $K$ and $L$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GA","source_file":"sites-cohomology.tex","source_line":12898,"source_end_line":12903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12898-L12903","statement_sha256":"97ae8a7e58f33dd3a89aaa52f3fcfd36df1eaca3ae14ff88d92cf3555104f606","origin":"The Stacks Project","memory_eligible":false,"source_rank":4645,"rank":4645,"depth":10,"x":1277.125,"y":741.196,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPQ","tag":"0FPQ","title":"Duals · Lemma 0FPQ","summary":"Let (C, O) be a ringed site. The category of complexes of O-modules with tensor product defined by F^bullet ⊗ G^bullet = Tot(F^bullet ⊗_O G^bullet) is a symmetric monoidal category.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The category of complexes\nof $\\mathcal{O}$-modules with tensor product defined by\n$\\mathcal{F}^\\bullet \\otimes \\mathcal{G}^\\bullet =\n\\text{Tot}(\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{G}^\\bullet)$\nis a symmetric monoidal category.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPQ","source_file":"sites-cohomology.tex","source_line":12935,"source_end_line":12942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12935-L12942","statement_sha256":"336d680a89553362fa2c18eb5400085e314dd2e1b3faa2daabae198123a5e420","origin":"The Stacks Project","memory_eligible":false,"source_rank":4646,"rank":4646,"depth":1,"x":972.145,"y":718.396,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPS","tag":"0FPS","title":"Duals · Lemma 0FPS","summary":"Let (C, O) be a ringed site. Let F^bullet be a complex of O-modules. If F^bullet has a left dual in the monoidal category of complexes of O-modules (Categories, Definition [Tag 0FFP]) then for every object U of C there exists a covering (U_i → U) such that F^bullet|_U_i is strictly perfect and the left dual is as constructed in Example [Tag 0FPR].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{F}^\\bullet$\nbe a complex of $\\mathcal{O}$-modules. If $\\mathcal{F}^\\bullet$\nhas a left dual in the monoidal category of complexes of\n$\\mathcal{O}$-modules\n(Categories, Definition \\ref{categories-definition-dual})\nthen for every object $U$ of $\\mathcal{C}$ there exists a\ncovering $\\{U_i \\to U\\}$ such that $\\mathcal{F}^\\bullet|_{U_i}$\nis strictly perfect and the left dual is as constructed in\nExample \\ref{example-dual}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPS","source_file":"sites-cohomology.tex","source_line":12999,"source_end_line":13010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L12999-L13010","statement_sha256":"01b4054d1e96522230210c2eada8994ba58bead63076f7adeb704f979d2e84c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4647,"rank":4647,"depth":2,"x":1215.608,"y":562.001,"cluster":"sheaf-cohomology"},{"id":"stacks:08JJ","tag":"08JJ","title":"Duals · Lemma 08JJ","summary":"Let (C, O) be a ringed site. Let K be a perfect object of D(O). Then K^vee = RSheafHom(K, O) is a perfect object too and (K^vee)^vee ≅ K. There are functorial isomorphisms M ⊗^L_O K^vee = RSheafHom_O(K, M) and H^0(C, M ⊗^L_O K^vee) = Hom_D(O)(K, M) for M in D(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $K$ be a perfect object of $D(\\mathcal{O})$.\nThen $K^\\vee = R\\SheafHom(K, \\mathcal{O})$ is a\nperfect object too and $(K^\\vee)^\\vee \\cong K$. There are\nfunctorial isomorphisms\n$$\nM \\otimes^\\mathbf{L}_\\mathcal{O} K^\\vee = R\\SheafHom_\\mathcal{O}(K, M)\n$$\nand\n$$\nH^0(\\mathcal{C}, M \\otimes^\\mathbf{L}_\\mathcal{O} K^\\vee) =\n\\Hom_{D(\\mathcal{O})}(K, M)\n$$\nfor $M$ in $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JJ","source_file":"sites-cohomology.tex","source_line":13047,"source_end_line":13063,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13047-L13063","statement_sha256":"1d4e6d7cc9d0e517cc0441cb15971a2c00fadb9f842260f68ead713b90001c3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4648,"rank":4648,"depth":5,"x":1161.797,"y":815.72,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPT","tag":"0FPT","title":"Duals · Lemma 0FPT","summary":"Let (C, O) be a ringed site. The derived category D(O) is a symmetric monoidal category with tensor product given by derived tensor product with usual associativity and commutativity constraints (for sign rules, see More on Algebra, Section [Tag 0FNG]).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. The derived category\n$D(\\mathcal{O})$ is a symmetric monoidal category with tensor product\ngiven by derived tensor product with usual associativity and\ncommutativity constraints (for sign rules, see\nMore on Algebra, Section \\ref{more-algebra-section-sign-rules}).","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPT","source_file":"sites-cohomology.tex","source_line":13135,"source_end_line":13142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13135-L13142","statement_sha256":"5cca4dc18fd7610c637a68e04c5b06206dd6cfced4bced0115a6a63ebde54837","origin":"The Stacks Project","memory_eligible":false,"source_rank":4649,"rank":4649,"depth":2,"x":997.281,"y":597.884,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPV","tag":"0FPV","title":"Duals · Lemma 0FPV","summary":"Let (C, O) be a ringed site. Let M be an object of D(O). If M has a left dual in the monoidal category D(O) (Categories, Definition [Tag 0FFP]) then M is perfect and the left dual is as constructed in Example [Tag 0FPU].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $M$ be an object\nof $D(\\mathcal{O})$. If $M$ has a left dual in the monoidal category\n$D(\\mathcal{O})$ (Categories, Definition \\ref{categories-definition-dual})\nthen $M$ is perfect and the left dual is as constructed in\nExample \\ref{example-dual-derived}.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPV","source_file":"sites-cohomology.tex","source_line":13185,"source_end_line":13192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13185-L13192","statement_sha256":"a0ac6878ad1e8d70a848bc677634008c37201a9692fa48125795a2f2122cf3b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4650,"rank":4650,"depth":16,"x":1294.061,"y":665.228,"cluster":"sheaf-cohomology"},{"id":"stacks:0A0A","tag":"0A0A","title":"Duals · Lemma 0A0A","summary":"Trivial duality for systems of perfect objects. Let (C, O) be a ringed site. Let (K_n)_n ∈ N be a system of perfect objects of D(O). Let K = hocolim K_n be the derived colimit (Derived Categories, Definition [Tag 090Z]). Then for any object E of D(O) we have RSheafHom(K, E) = Rlim E ⊗^L_O K_n^vee where (K_n^vee) is the inverse system of dual perfect complexes.","statement_latex":"\\begin{slogan}\nTrivial duality for systems of perfect objects.\n\\end{slogan}\nLet $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(K_n)_{n \\in \\mathbf{N}}$ be a system of perfect objects of $D(\\mathcal{O})$.\nLet $K = \\text{hocolim} K_n$ be the derived colimit\n(Derived Categories, Definition \\ref{derived-definition-derived-colimit}).\nThen for any object $E$ of $D(\\mathcal{O})$ we have\n$$\nR\\SheafHom(K, E) = R\\lim E \\otimes^\\mathbf{L}_\\mathcal{O} K_n^\\vee\n$$\nwhere $(K_n^\\vee)$ is the inverse system of dual perfect complexes.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Duals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0A","source_file":"sites-cohomology.tex","source_line":13415,"source_end_line":13429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13415-L13429","statement_sha256":"4ff253823d9ccf9cd2b185a549013653a7d5c1f8584b8e565d5378902d05248d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4651,"rank":4651,"depth":6,"x":1020.796,"y":784.086,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPX","tag":"0FPX","title":"Invertible objects in the derived category · Lemma 0FPX","summary":"Let (C, O) be a ringed site. Set R = Γ(C, O). The category of O-modules which are summands of finite free O-modules is equivalent to the category of finite projective R-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nSet $R = \\Gamma(\\mathcal{C}, \\mathcal{O})$. The category of\n$\\mathcal{O}$-modules which are summands of finite free\n$\\mathcal{O}$-modules is equivalent to the category of\nfinite projective $R$-modules.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Invertible objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPX","source_file":"sites-cohomology.tex","source_line":13461,"source_end_line":13468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13461-L13468","statement_sha256":"e8df268d667deb30cef4810960211bb214d36b34197e6138de53198a9a6e14f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4652,"rank":4652,"depth":0,"x":1126.819,"y":541.148,"cluster":"sheaf-cohomology"},{"id":"stacks:0FPY","tag":"0FPY","title":"Invertible objects in the derived category · Lemma 0FPY","summary":"Let (C, O) be a ringed site. Let M be an object of D(O). The following are equivalent • M is invertible in D(O), see Categories, Definition [Tag 0FFN], and • there is a locally finite there is a covering (U_i → U) such that for every i the sheaf O_n|_U_i is nonzero for only a finite number of n. direct product decomposition O = ∏_n ∈ Z O_n and for each n there is an invertible O_n-module H^n (Modules on Sites, Definition [Tag 0409]) and M = bigoplus H^n[-n] in D(O). If…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $M$ be an object\nof $D(\\mathcal{O})$. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is invertible in $D(\\mathcal{O})$, see\nCategories, Definition \\ref{categories-definition-invertible}, and\n\\item there is a locally finite\\footnote{This means that for every\nobject $U$ of $\\mathcal{C}$ there is a covering $\\{U_i \\to U\\}$\nsuch that for every $i$ the sheaf $\\mathcal{O}_n|_{U_i}$ is nonzero\nfor only a finite number of $n$.} direct product decomposition\n$$\n\\mathcal{O} = \\prod\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_n\n$$\nand for each $n$ there is an invertible $\\mathcal{O}_n$-module\n$\\mathcal{H}^n$\n(Modules on Sites, Definition \\ref{sites-modules-definition-invertible-sheaf})\nand $M = \\bigoplus \\mathcal{H}^n[-n]$ in $D(\\mathcal{O})$.\n\\end{enumerate}\nIf (1) and (2) hold, then $M$ is a perfect object of $D(\\mathcal{O})$. If\n$(\\mathcal{C}, \\mathcal{O})$ is a locally ringed site these condition\nare also equivalent to\n\\begin{enumerate}\n\\item[(3)] for every object $U$ of $\\mathcal{C}$ there exists a\ncovering $\\{U_i \\to U\\}$ and for each $i$ an integer $n_i$ such that\n$M|_{U_i}$ is represented by an invertible $\\mathcal{O}_{U_i}$-module\nplaced in degree $n_i$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Invertible objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FPY","source_file":"sites-cohomology.tex","source_line":13486,"source_end_line":13514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13486-L13514","statement_sha256":"be131c74d3b2d6be7d0c7242f9a07e0f07d4adf6da735cfe179ef4566a6dd4ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":4653,"rank":4653,"depth":18,"x":1244.117,"y":780.68,"cluster":"sheaf-cohomology"},{"id":"stacks:0944","tag":"0944","title":"Projection formula · Lemma 0944","summary":"Let f : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let E ∈ D(O_C) and K ∈ D(O_D). If K is perfect, then Rf_*E ⊗^L_O_D K = Rf_*(E ⊗^L_O_C Lf^*K) in D(O_D).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nLet $E \\in D(\\mathcal{O}_\\mathcal{C})$ and $K \\in D(\\mathcal{O}_\\mathcal{D})$.\nIf $K$ is perfect, then\n$$\nRf_*E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{D}} K =\nRf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_\\mathcal{C}} Lf^*K)\n$$\nin $D(\\mathcal{O}_\\mathcal{D})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Projection formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0944","source_file":"sites-cohomology.tex","source_line":13682,"source_end_line":13693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13682-L13693","statement_sha256":"82dfd39f663dcdd481d81238690a44db1b73e8d8db99114229544d98d9b1fc38","origin":"The Stacks Project","memory_eligible":false,"source_rank":4654,"rank":4654,"depth":0,"x":964.727,"y":670.504,"cluster":"sheaf-cohomology"},{"id":"stacks:0946","tag":"0946","title":"Weakly contractible objects · Lemma 0946","summary":"Let C be a site. Let U be a weakly contractible object of C. Then • the functor F ↦ F(U) is an exact functor Ab(C) → Ab, • H^p(U, F) = 0 for every abelian sheaf F and all p ≥ 1, and • for any sheaf of groups G any G-torsor has a section over U.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $U$ be a weakly contractible\nobject of $\\mathcal{C}$. Then\n\\begin{enumerate}\n\\item the functor $\\mathcal{F} \\mapsto \\mathcal{F}(U)$ is an exact\nfunctor $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$,\n\\item $H^p(U, \\mathcal{F}) = 0$\nfor every abelian sheaf $\\mathcal{F}$ and all $p \\geq 1$, and\n\\item for any sheaf of groups $\\mathcal{G}$ any $\\mathcal{G}$-torsor\nhas a section over $U$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Weakly contractible objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0946","source_file":"sites-cohomology.tex","source_line":13772,"source_end_line":13784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13772-L13784","statement_sha256":"d2e6ae0a0f3f7c3ff5ce5311f0e5055c13167acd61e6ac748581b726bc3c51f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4655,"rank":4655,"depth":1,"x":1259.632,"y":593.139,"cluster":"sheaf-cohomology"},{"id":"stacks:0947","tag":"0947","title":"Weakly contractible objects · Proposition 0947","summary":"Let C be a site. Let B ⊂ Ob(C) such that every U ∈ B is weakly contractible and every object of C has a covering by elements of B. Let O be a sheaf of rings on C. Then • A complex F_1 → F_2 → F_3 of O-modules is exact, if and only if F_1(U) → F_2(U) → F_3(U) is exact for all U ∈ B. • Every object K of D(O) is a derived limit of its canonical truncations: K = Rlim τ_≥ -n K. • Given an inverse system … → F_3 → F_2 → F_1 with surjective transition maps, the projection lim…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$\nsuch that every $U \\in \\mathcal{B}$ is weakly contractible and\nevery object of $\\mathcal{C}$ has a covering by elements of $\\mathcal{B}$.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}$. Then\n\\begin{enumerate}\n\\item A complex $\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3$\nof $\\mathcal{O}$-modules is exact, if and only if\n$\\mathcal{F}_1(U) \\to \\mathcal{F}_2(U) \\to \\mathcal{F}_3(U)$\nis exact for all $U \\in \\mathcal{B}$.\n\\item Every object $K$ of $D(\\mathcal{O})$ is a derived limit\nof its canonical truncations: $K = R\\lim \\tau_{\\geq -n} K$.\n\\item Given an inverse system\n$\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1$\nwith surjective transition maps, the projection\n$\\lim \\mathcal{F}_n \\to \\mathcal{F}_1$ is surjective.\n\\item Products are exact on $\\textit{Mod}(\\mathcal{O})$.\n\\item Products on $D(\\mathcal{O})$ can be computed by taking\nproducts of any representative complexes.\n\\item If $(\\mathcal{F}_n)$ is an inverse system of $\\mathcal{O}$-modules,\nthen $R^p\\lim \\mathcal{F}_n = 0$ for all $p > 1$ and\n$$\nR^1\\lim \\mathcal{F}_n  =\n\\Coker(\\prod \\mathcal{F}_n \\to \\prod \\mathcal{F}_n)\n$$\nwhere the map is $(x_n) \\mapsto (x_n - f(x_{n + 1}))$.\n\\item If $(K_n)$ is an inverse system of objects of $D(\\mathcal{O})$,\nthen there are short exact sequences\n$$\n0 \\to R^1\\lim H^{p - 1}(K_n) \\to H^p(R\\lim K_n) \\to\n\\lim H^p(K_n) \\to 0\n$$\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Weakly contractible objects","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0947","source_file":"sites-cohomology.tex","source_line":13800,"source_end_line":13834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13800-L13834","statement_sha256":"5c4f236f6bc2cb167dcc6c7a73540741a1fedb818ec34ef9d9de58cff019f524","origin":"The Stacks Project","memory_eligible":false,"source_rank":4656,"rank":4656,"depth":14,"x":1104.238,"y":817.739,"cluster":"sheaf-cohomology"},{"id":"stacks:094B","tag":"094B","title":"Compact objects · Lemma 094B","summary":"Let A be a Grothendieck abelian category. Let S ⊂ Ob(A) be a set of objects such that • any object of A is a quotient of a direct sum of elements of S, and • for any E ∈ S the functor Hom_A(E, -) commutes with direct sums. Then every compact object of D(A) is a direct summand in D(A) of a finite complex of finite direct sums of elements of S.","statement_latex":"Let $\\mathcal{A}$ be a Grothendieck abelian category. Let\n$S \\subset \\Ob(\\mathcal{A})$ be a set of objects such that\n\\begin{enumerate}\n\\item any object of $\\mathcal{A}$ is a quotient of a direct sum\nof elements of $S$, and\n\\item for any $E \\in S$ the functor $\\Hom_\\mathcal{A}(E, -)$\ncommutes with direct sums.\n\\end{enumerate}\nThen every compact object of $D(\\mathcal{A})$ is a direct summand\nin $D(\\mathcal{A})$ of a finite complex of finite direct sums of\nelements of $S$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094B","source_file":"sites-cohomology.tex","source_line":13933,"source_end_line":13946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L13933-L13946","statement_sha256":"826d7d79007a2c9d297fa6ce55a370157b84539d66d44eb5b4387de06e0547ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":4657,"rank":4657,"depth":6,"x":1038.142,"y":563.703,"cluster":"sheaf-cohomology"},{"id":"stacks:094C","tag":"094C","title":"Compact objects · Lemma 094C","summary":"Let (C, O) be a ringed site. Assume every object of C has a covering by quasi-compact objects. Then every compact object of D(O) is a direct summand in D(O) of a finite complex whose terms are finite direct sums of O-modules of the form j_!O_U where U is a quasi-compact object of C.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Assume every object of\n$\\mathcal{C}$ has a covering by quasi-compact objects. Then every\ncompact object of $D(\\mathcal{O})$ is a direct summand in $D(\\mathcal{O})$\nof a finite complex whose terms are finite direct sums of\n$\\mathcal{O}$-modules of the form $j_!\\mathcal{O}_U$\nwhere $U$ is a quasi-compact object of $\\mathcal{C}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094C","source_file":"sites-cohomology.tex","source_line":14034,"source_end_line":14042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14034-L14042","statement_sha256":"2b5038969f071e4f9ffc79679d5f4041979a51b448ea5548c80cf69b4e7613c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4658,"rank":4658,"depth":7,"x":1291.413,"y":713.665,"cluster":"sheaf-cohomology"},{"id":"stacks:0G21","tag":"0G21","title":"Compact objects · Lemma 0G21","summary":"Let (C, O) be a ringed site. Let U be an object of C. Assume the functors F ↦ H^p(U, F) commute with direct sums. Then O-module j_!O_U is a compact object of D^+(O) in the following sense: if M = bigoplus_i ∈ I M_i in D(O) is bounded below, then Hom(j_U!O_U, M) = bigoplus_i ∈ I Hom(j_U!O_U, M_i).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of\n$\\mathcal{C}$. Assume the functors $\\mathcal{F} \\mapsto H^p(U, \\mathcal{F})$\ncommute with direct sums. Then $\\mathcal{O}$-module $j_!\\mathcal{O}_U$ is a\ncompact object of $D^+(\\mathcal{O})$ in the following sense:\nif $M = \\bigoplus_{i \\in I} M_i$ in $D(\\mathcal{O})$ is\nbounded below, then $\\Hom(j_{U!}\\mathcal{O}_U, M) =\n\\bigoplus_{i \\in I} \\Hom(j_{U!}\\mathcal{O}_U, M_i)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G21","source_file":"sites-cohomology.tex","source_line":14066,"source_end_line":14075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14066-L14075","statement_sha256":"d431d07aec4e9a4f03b9b6feee55a8d7f14956679dd8f696a14812aa5c227410","origin":"The Stacks Project","memory_eligible":false,"source_rank":4659,"rank":4659,"depth":11,"x":983.763,"y":746.83,"cluster":"sheaf-cohomology"},{"id":"stacks:0G22","tag":"0G22","title":"Compact objects · Lemma 0G22","summary":"Let (C, O) be a ringed site with set of coverings Cov_C. Let B ⊂ Ob(C), and Cov ⊂ Cov_C be subsets. Assume that • For every U ∈ Cov we have U = (U_i → U)_i ∈ I with I finite, U, U_i ∈ B and every U_i_0 ×_U … ×_U U_i_p ∈ B. • For every U ∈ B the coverings of U occurring in Cov is a cofinal system of coverings of U. Then for U ∈ B the object j_U!O_U is a compact object of D^+(O) in the following sense: if M = bigoplus_i ∈ I M_i in D(O) is bounded below, then Hom(j_U!O_U, M)…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site\nwith set of coverings $\\text{Cov}_\\mathcal{C}$.\nLet $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$, and\n$\\text{Cov} \\subset \\text{Cov}_\\mathcal{C}$\nbe subsets. Assume that\n\\begin{enumerate}\n\\item For every $\\mathcal{U} \\in \\text{Cov}$ we have\n$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ with $I$ finite,\n$U, U_i \\in \\mathcal{B}$ and every\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p} \\in \\mathcal{B}$.\n\\item For every $U \\in \\mathcal{B}$ the coverings of $U$\noccurring in $\\text{Cov}$ is a cofinal system of coverings of $U$.\n\\end{enumerate}\nThen for $U \\in \\mathcal{B}$ the object $j_{U!}\\mathcal{O}_U$ is\na compact object of $D^+(\\mathcal{O})$ in the following sense:\nif $M = \\bigoplus_{i \\in I} M_i$ in $D(\\mathcal{O})$ is\nbounded below, then $\\Hom(j_{U!}\\mathcal{O}_U, M) =\n\\bigoplus_{i \\in I} \\Hom(j_{U!}\\mathcal{O}_U, M_i)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G22","source_file":"sites-cohomology.tex","source_line":14108,"source_end_line":14128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14108-L14128","statement_sha256":"28b8d62dd382f0fce57dd03054b251bf0d0a46993047dd74b542fb29fdebc5ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":4660,"rank":4660,"depth":24,"x":1184.142,"y":547.616,"cluster":"sheaf-cohomology"},{"id":"stacks:094D","tag":"094D","title":"Compact objects · Lemma 094D","summary":"Let (C, O) be a ringed site. Let U be an object of C. The O-module j_!O_U is a compact object of D(O) if there exists an integer d such that • H^p(U, F) = 0 for all p > d, and • the functors F ↦ H^p(U, F) commute with direct sums.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$ be an object of\n$\\mathcal{C}$. The $\\mathcal{O}$-module $j_!\\mathcal{O}_U$ is a\ncompact object of $D(\\mathcal{O})$ if there exists an integer $d$ such that\n\\begin{enumerate}\n\\item $H^p(U, \\mathcal{F}) = 0$ for all $p > d$, and\n\\item the functors $\\mathcal{F} \\mapsto H^p(U, \\mathcal{F})$\ncommute with direct sums.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094D","source_file":"sites-cohomology.tex","source_line":14135,"source_end_line":14145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14135-L14145","statement_sha256":"65318f3681ddf33b0ba1706faaefcac2dd47738c82bc27b108decf5f8bd34e19","origin":"The Stacks Project","memory_eligible":false,"source_rank":4661,"rank":4661,"depth":16,"x":1196.6,"y":808.462,"cluster":"sheaf-cohomology"},{"id":"stacks:094E","tag":"094E","title":"Compact objects · Lemma 094E","summary":"Let (C, O) be a ringed site. Let U be an object of C which is quasi-compact and weakly contractible. Then j_!O_U is a compact object of D(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $U$\nbe an object of $\\mathcal{C}$ which is quasi-compact and\nweakly contractible. Then\n$j_!\\mathcal{O}_U$ is a compact object of $D(\\mathcal{O})$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094E","source_file":"sites-cohomology.tex","source_line":14166,"source_end_line":14172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14166-L14172","statement_sha256":"5a4fc08acfb93e971ecbbc49af61b31d368e956ec906e1211213befbb2cc12db","origin":"The Stacks Project","memory_eligible":false,"source_rank":4662,"rank":4662,"depth":17,"x":977.439,"y":623.01,"cluster":"sheaf-cohomology"},{"id":"stacks:09JC","tag":"09JC","title":"Compact objects · Lemma 09JC","summary":"Let (C, O) be a ringed site. Assume C has the following properties • C has a quasi-compact final object X, • every quasi-compact object of C has a cofinal system of coverings which are finite and consist of quasi-compact objects, • for a finite covering (U_i → U)_i ∈ I with U, U_i quasi-compact the fibre products U_i ×_U U_j are quasi-compact. Let K be a perfect object of D(O). Then • [(a)] K is a compact object of D^+(O) in the following sense: if M = bigoplus_i ∈ I M_i…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Assume\n$\\mathcal{C}$ has the following properties\n\\begin{enumerate}\n\\item $\\mathcal{C}$ has a quasi-compact final object $X$,\n\\item every quasi-compact object of $\\mathcal{C}$\nhas a cofinal system of coverings which are finite\nand consist of quasi-compact objects,\n\\item for a finite covering $\\{U_i \\to U\\}_{i \\in I}$\nwith $U$, $U_i$ quasi-compact the fibre products $U_i \\times_U U_j$ are\nquasi-compact.\n\\end{enumerate}\nLet $K$ be a perfect object of $D(\\mathcal{O})$. Then\n\\begin{enumerate}\n\\item[(a)] $K$ is a compact object of $D^+(\\mathcal{O})$\nin the following sense: if $M = \\bigoplus_{i \\in I} M_i$ is\nbounded below, then $\\Hom(K, M) = \\bigoplus_{i \\in I} \\Hom(K, M_i)$.\n\\item[(b)] If $(\\mathcal{C}, \\mathcal{O})$\nhas finite cohomological dimension, i.e., if there exists\na $d$ such that $H^i(X, \\mathcal{F}) = 0$ for $i > d$ for\nany $\\mathcal{O}$-module $\\mathcal{F}$, then\n$K$ is a compact object of $D(\\mathcal{O})$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JC","source_file":"sites-cohomology.tex","source_line":14180,"source_end_line":14204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14180-L14204","statement_sha256":"999dd87984dd526e01c05fe8b485cc556a76a57732c1ba36f5fa701a76c5c828","origin":"The Stacks Project","memory_eligible":false,"source_rank":4663,"rank":4663,"depth":25,"x":1288.477,"y":635.414,"cluster":"sheaf-cohomology"},{"id":"stacks:094G","tag":"094G","title":"Complexes with locally constant cohomology sheaves · Lemma 094G","summary":"Let C be a site with final object X. Let Lambda be a Noetherian ring. Let K ∈ D^b(C, Lambda) with H^i(K) locally constant sheaves of Lambda-modules of finite type. Then there exists a covering (U_i → X) such that each K|_U_i is represented by a complex of locally constant sheaves of Lambda-modules of finite type.","statement_latex":"Let $\\mathcal{C}$ be a site with final object $X$.\nLet $\\Lambda$ be a Noetherian ring.\nLet $K \\in D^b(\\mathcal{C}, \\Lambda)$\nwith $H^i(K)$ locally constant sheaves of $\\Lambda$-modules\nof finite type. Then there exists a covering $\\{U_i \\to X\\}$\nsuch that each $K|_{U_i}$ is represented by\na complex of locally constant sheaves of $\\Lambda$-modules\nof finite type.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Complexes with locally constant cohomology sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094G","source_file":"sites-cohomology.tex","source_line":14246,"source_end_line":14256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14246-L14256","statement_sha256":"75a0fb944514097bd541dddf7f27c788bc5dbf23cbf9a2ae683341e820b058e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4664,"rank":4664,"depth":3,"x":1048.919,"y":802.918,"cluster":"sheaf-cohomology"},{"id":"stacks:09BD","tag":"09BD","title":"Complexes with locally constant cohomology sheaves · Lemma 09BD","summary":"Let C be a site with final object X. Let Lambda be a ring. Let • K a perfect object of D(Lambda), • a finite complex K^bullet of finite projective Lambda-modules representing K, • L^bullet a complex of sheaves of Lambda-modules, and • φ : underlineK → L^bullet a map in D(C, Lambda). Then there exists a covering (U_i → X) and maps of complexes α_i : underlineK^bullet|_U_i → L^bullet|_U_i representing φ|_U_i.","statement_latex":"Let $\\mathcal{C}$ be a site with final object $X$. Let $\\Lambda$ be a ring. Let\n\\begin{enumerate}\n\\item $K$ a perfect object of $D(\\Lambda)$,\n\\item a finite complex $K^\\bullet$ of finite projective $\\Lambda$-modules\nrepresenting $K$,\n\\item $\\mathcal{L}^\\bullet$ a complex of sheaves of $\\Lambda$-modules, and\n\\item $\\varphi : \\underline{K} \\to \\mathcal{L}^\\bullet$ a map in\n$D(\\mathcal{C}, \\Lambda)$.\n\\end{enumerate}\nThen there exists a covering $\\{U_i \\to X\\}$ and maps of complexes\n$\\alpha_i : \\underline{K}^\\bullet|_{U_i} \\to \\mathcal{L}^\\bullet|_{U_i}$\nrepresenting $\\varphi|_{U_i}$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Complexes with locally constant cohomology sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BD","source_file":"sites-cohomology.tex","source_line":14320,"source_end_line":14334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14320-L14334","statement_sha256":"7cab74c01cc7440735484c87ab148632f89537dc00fc8b89396eb67abe5edec6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4665,"rank":4665,"depth":4,"x":1090.904,"y":543.225,"cluster":"sheaf-cohomology"},{"id":"stacks:09BE","tag":"09BE","title":"Complexes with locally constant cohomology sheaves · Lemma 09BE","summary":"Let C be a site with final object X. Let Lambda be a ring. Let K, L be objects of D(Lambda) with K perfect. Let φ : underlineK → underlineL be map in D(C, Lambda). There exists a covering (U_i → X) such that φ|_U_i is equal to underlineα_i for some map α_i : K → L in D(Lambda).","statement_latex":"Let $\\mathcal{C}$ be a site with final object $X$.\nLet $\\Lambda$ be a ring. Let $K, L$ be objects of\n$D(\\Lambda)$ with $K$ perfect. Let $\\varphi : \\underline{K} \\to \\underline{L}$\nbe map in $D(\\mathcal{C}, \\Lambda)$. There exists a covering $\\{U_i \\to X\\}$\nsuch that $\\varphi|_{U_i}$ is equal to $\\underline{\\alpha_i}$\nfor some map $\\alpha_i : K \\to L$ in $D(\\Lambda)$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Complexes with locally constant cohomology sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BE","source_file":"sites-cohomology.tex","source_line":14340,"source_end_line":14348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14340-L14348","statement_sha256":"71efd02533d3465ff21651060dc8c8bb3dc66c43dabacb8645f5a978966bf405","origin":"The Stacks Project","memory_eligible":false,"source_rank":4666,"rank":4666,"depth":5,"x":1268.951,"y":758.746,"cluster":"sheaf-cohomology"},{"id":"stacks:094H","tag":"094H","title":"Complexes with locally constant cohomology sheaves · Lemma 094H","summary":"Let C be a site. Let Lambda be a Noetherian ring. Let K, L ∈ D^-(C, Lambda). If the cohomology sheaves of K and L are locally constant sheaves of Lambda-modules of finite type, then the cohomology sheaves of K ⊗_Lambda^L L are locally constant sheaves of Lambda-modules of finite type.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring.\nLet $K, L \\in D^-(\\mathcal{C}, \\Lambda)$. If the cohomology sheaves of\n$K$ and $L$ are locally constant sheaves of $\\Lambda$-modules of\nfinite type, then the cohomology sheaves of\n$K \\otimes_\\Lambda^\\mathbf{L} L$\nare locally constant sheaves of $\\Lambda$-modules of finite type.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Complexes with locally constant cohomology sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094H","source_file":"sites-cohomology.tex","source_line":14355,"source_end_line":14363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14355-L14363","statement_sha256":"5f91912973e844c262e12e1f401486109b7e83b9824b809c7ec9c243477b233b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4667,"rank":4667,"depth":4,"x":964.057,"y":700.797,"cluster":"sheaf-cohomology"},{"id":"stacks:094I","tag":"094I","title":"Complexes with locally constant cohomology sheaves · Lemma 094I","summary":"Let C be a site. Let Lambda be a Noetherian ring. Let I ⊂ Lambda be an ideal. Let K ∈ D^-(C, Lambda). If the cohomology sheaves of K ⊗_Lambda^L underlineLambda/I are locally constant sheaves of Lambda/I-modules of finite type, then the cohomology sheaves of K ⊗_Lambda^L underlineLambda/I^n are locally constant sheaves of Lambda/I^n-modules of finite type for all n ≥ 1.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring.\nLet $I \\subset \\Lambda$ be an ideal.\nLet $K \\in D^-(\\mathcal{C}, \\Lambda)$. If the cohomology sheaves of\n$K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ are locally constant\nsheaves of $\\Lambda/I$-modules of finite type, then the cohomology sheaves of\n$K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$\nare locally constant sheaves of $\\Lambda/I^n$-modules of finite type for all\n$n \\geq 1$.","area":"Sheaf Cohomology","chapter":"Cohomology on Sites","chapter_id":"sites-cohomology","section":"Complexes with locally constant cohomology sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094I","source_file":"sites-cohomology.tex","source_line":14378,"source_end_line":14388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sites-cohomology.tex#L14378-L14388","statement_sha256":"ac723b46d68bda7c698dd05e223f9086e8b5c9604a1ffbafc0955ab48ff9e444","origin":"The Stacks Project","memory_eligible":false,"source_rank":4668,"rank":4668,"depth":5,"x":1235.739,"y":570.401,"cluster":"sheaf-cohomology"},{"id":"stacks:061V","tag":"061V","title":"Differential graded algebras · Definition 061V","summary":"Let R be a commutative ring. A differential graded algebra over R is either • a chain complex A_bullet of R-modules endowed with R-bilinear maps A_n × A_m → A_n + m, (a, b) ↦ ab such that d_n + m(ab) = d_n(a)b + (-1)^n ad_m(b) and such that bigoplus A_n becomes an associative and unital R-algebra, or • a cochain complex A^bullet of R-modules endowed with R-bilinear maps A^n × A^m → A^n + m, (a, b) ↦ ab such that d^n + m(ab) = d^n(a)b + (-1)^n ad^m(b) and such that…","statement_latex":"Let $R$ be a commutative ring. A {\\it differential graded algebra over $R$}\nis either\n\\begin{enumerate}\n\\item a chain complex $A_\\bullet$ of $R$-modules endowed with\n$R$-bilinear maps $A_n \\times A_m \\to A_{n + m}$,\n$(a, b) \\mapsto ab$ such that\n$$\n\\text{d}_{n + m}(ab) = \\text{d}_n(a)b + (-1)^n a\\text{d}_m(b)\n$$\nand such that $\\bigoplus A_n$ becomes an associative and unital\n$R$-algebra, or\n\\item a cochain complex $A^\\bullet$ of $R$-modules endowed with\n$R$-bilinear maps $A^n \\times A^m \\to A^{n + m}$, $(a, b) \\mapsto ab$\nsuch that\n$$\n\\text{d}^{n + m}(ab) = \\text{d}^n(a)b + (-1)^n a\\text{d}^m(b)\n$$\nand such that $\\bigoplus A^n$ becomes an associative and unital $R$-algebra.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061V","source_file":"dga.tex","source_line":64,"source_end_line":85,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L64-L85","statement_sha256":"1ecd49a879f807a7d38027f3db17d16dc525eec2d65f6a5f3cede024b06fe22f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4669,"rank":4669,"depth":0,"x":2359.15,"y":183.192,"cluster":"homological-algebra"},{"id":"stacks:061X","tag":"061X","title":"Differential graded algebras · Definition 061X","summary":"A homomorphism of differential graded algebras f : (A, d) → (B, d) is an algebra map f : A → B compatible with the gradings and d.","statement_latex":"A {\\it homomorphism of differential graded algebras}\n$f : (A, \\text{d}) \\to (B, \\text{d})$ is an algebra map $f : A \\to B$\ncompatible with the gradings and $\\text{d}$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061X","source_file":"dga.tex","source_line":103,"source_end_line":108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L103-L108","statement_sha256":"9af041429305bd16b0aaed9c43c4198a299fe7a63716304d186295a3510badc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4670,"rank":4670,"depth":0,"x":2598.9,"y":178.503,"cluster":"homological-algebra"},{"id":"stacks:061W","tag":"061W","title":"Differential graded algebras · Definition 061W","summary":"A differential graded algebra (A, d) is commutative if ab = (-1)^nmba for a in degree n and b in degree m. We say A is strictly commutative if in addition a^2 = 0 for deg(a) odd.","statement_latex":"A differential graded algebra $(A, \\text{d})$ is {\\it commutative} if\n$ab = (-1)^{nm}ba$ for $a$ in degree $n$ and $b$ in degree $m$.\nWe say $A$ is {\\it strictly commutative} if in addition $a^2 = 0$\nfor $\\deg(a)$ odd.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061W","source_file":"dga.tex","source_line":110,"source_end_line":116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L110-L116","statement_sha256":"4833f48570408e6890180631f333b60859d8e4f7cec49096760374c0cae08387","origin":"The Stacks Project","memory_eligible":false,"source_rank":4671,"rank":4671,"depth":0,"x":2425.611,"y":318.219,"cluster":"homological-algebra"},{"id":"stacks:065W","tag":"065W","title":"Differential graded algebras · Definition 065W","summary":"Let R be a ring. Let (A, d), (B, d) be differential graded algebras over R. The tensor product differential graded algebra of A and B is the algebra A ⊗_R B with multiplication defined by (a ⊗ b)(a' ⊗ b') = (-1)^deg(a')deg(b) aa' ⊗ bb' endowed with differential d defined by the rule d(a ⊗ b) = d(a) ⊗ b + (-1)^m a ⊗ d(b) where m = deg(a).","statement_latex":"Let $R$ be a ring.\nLet $(A, \\text{d})$, $(B, \\text{d})$ be differential graded algebras over $R$.\nThe {\\it tensor product differential graded algebra} of $A$ and $B$\nis the algebra $A \\otimes_R B$ with multiplication defined by\n$$\n(a \\otimes b)(a' \\otimes b') = (-1)^{\\deg(a')\\deg(b)} aa' \\otimes bb'\n$$\nendowed with differential $\\text{d}$ defined by the rule\n$\\text{d}(a \\otimes b) = \\text{d}(a) \\otimes b + (-1)^m a \\otimes \\text{d}(b)$\nwhere $m = \\deg(a)$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065W","source_file":"dga.tex","source_line":123,"source_end_line":135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L123-L135","statement_sha256":"6f42263e877b513d8816e5ff3bf0eecf1764a70676d0a2392f37f4d06f30a4a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4672,"rank":4672,"depth":0,"x":2441.06,"y":116.552,"cluster":"homological-algebra"},{"id":"stacks:065X","tag":"065X","title":"Differential graded algebras · Lemma 065X","summary":"Let R be a ring. Let (A, d), (B, d) be differential graded algebras over R. Denote A^bullet, B^bullet the underlying cochain complexes. As cochain complexes of R-modules we have (A ⊗_R B)^bullet = Tot(A^bullet ⊗_R B^bullet).","statement_latex":"Let $R$ be a ring.\nLet $(A, \\text{d})$, $(B, \\text{d})$ be differential graded algebras over $R$.\nDenote $A^\\bullet$, $B^\\bullet$ the underlying cochain complexes.\nAs cochain complexes of $R$-modules we have\n$$\n(A \\otimes_R B)^\\bullet = \\text{Tot}(A^\\bullet \\otimes_R B^\\bullet).\n$$","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/065X","source_file":"dga.tex","source_line":137,"source_end_line":146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L137-L146","statement_sha256":"6ba09c0ce86dabb4e01b264efe2804d68a5d17e97abb83495204d575c006bc90","origin":"The Stacks Project","memory_eligible":false,"source_rank":4673,"rank":4673,"depth":1,"x":2592.078,"y":274.27,"cluster":"homological-algebra"},{"id":"stacks:09JI","tag":"09JI","title":"Differential graded modules · Definition 09JI","summary":"Let R be a ring. Let (A, d) be a differential graded algebra over R. A (right) differential graded module M over A is a right A-module M which has a grading M = bigoplus M^n and a differential d such that M^n A^m ⊂ M^n + m, such that d(M^n) ⊂ M^n + 1, and such that d(ma) = d(m)a + (-1)^n md(a) for a ∈ A and m ∈ M^n. A homomorphism of differential graded modules f : M → N is an A-module map compatible with gradings and differentials. The category of (right) differential…","statement_latex":"Let $R$ be a ring.\nLet $(A, \\text{d})$ be a differential graded algebra over $R$.\nA (right) {\\it differential graded module} $M$ over $A$ is a right $A$-module\n$M$ which has a grading $M = \\bigoplus M^n$ and a differential $\\text{d}$\nsuch that $M^n A^m \\subset M^{n + m}$, such that\n$\\text{d}(M^n) \\subset M^{n + 1}$, and such that\n$$\n\\text{d}(ma) = \\text{d}(m)a + (-1)^n m\\text{d}(a)\n$$\nfor $a \\in A$ and $m \\in M^n$. A\n{\\it homomorphism of differential graded modules} $f : M \\to N$\nis an $A$-module map compatible with gradings and differentials.\nThe category of (right) differential graded $A$-modules is denoted\n$\\text{Mod}_{(A, \\text{d})}$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JI","source_file":"dga.tex","source_line":168,"source_end_line":184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L168-L184","statement_sha256":"9b12583cb7d46a923498ef388ec6a8a5c8ea8e4851644c24eb5298433d8cad96","origin":"The Stacks Project","memory_eligible":false,"source_rank":4674,"rank":4674,"depth":0,"x":2353.516,"y":243.615,"cluster":"homological-algebra"},{"id":"stacks:09JJ","tag":"09JJ","title":"Differential graded modules · Lemma 09JJ","summary":"Let (A, d) be a differential graded algebra. The category Mod_(A, d) is abelian and has arbitrary limits and colimits.","statement_latex":"Let $(A, d)$ be a differential graded algebra. The category\n$\\text{Mod}_{(A, \\text{d})}$ is abelian and has arbitrary limits and colimits.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JJ","source_file":"dga.tex","source_line":202,"source_end_line":206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L202-L206","statement_sha256":"5cf72cf74d6ba3e85273953b7f183b7e08bdbcf73b57cd96b7b4e240a8a15018","origin":"The Stacks Project","memory_eligible":false,"source_rank":4675,"rank":4675,"depth":1,"x":2554.392,"y":130.679,"cluster":"homological-algebra"},{"id":"stacks:09JL","tag":"09JL","title":"Differential graded modules · Definition 09JL","summary":"Let (A, d) be a differential graded algebra. Let M be a differential graded module whose underlying complex of R-modules is M^bullet. For any k ∈ Z we define the k-shifted module M[k] as follows • the underlying complex of R-modules of M[k] is M^bullet[k], i.e., we have M[k]^n = M^n + k and d_M[k] = (-1)^kd_M and • as A-module the multiplication (M[k])^n × A^m → (M[k])^n + m is equal to the given multiplication M^n + k × A^m → M^n + k + m. For a morphism f : M → N of…","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $M$ be a differential graded module whose underlying complex\nof $R$-modules is $M^\\bullet$. For any $k \\in \\mathbf{Z}$\nwe define the {\\it $k$-shifted module} $M[k]$ as follows\n\\begin{enumerate}\n\\item the underlying complex of $R$-modules of $M[k]$ is $M^\\bullet[k]$,\ni.e., we have $M[k]^n = M^{n + k}$ and\n$\\text{d}_{M[k]} = (-1)^k\\text{d}_M$ and\n\\item as $A$-module the multiplication\n$$\n(M[k])^n \\times A^m \\longrightarrow (M[k])^{n + m}\n$$\nis equal to the given multiplication $M^{n + k} \\times A^m \\to M^{n + k + m}$.\n\\end{enumerate}\nFor a morphism $f : M \\to N$ of differential graded $A$-modules\nwe let $f[k] : M[k] \\to N[k]$ be the map equal to $f$ on underlying\n$A$-modules. This defines a functor\n$[k] : \\text{Mod}_{(A, \\text{d})} \\to \\text{Mod}_{(A, \\text{d})}$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JL","source_file":"dga.tex","source_line":249,"source_end_line":269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L249-L269","statement_sha256":"3402d3a4ccd6d88cb79b9436a13521a8d6e9ad093b2f263d537578b1ca33a980","origin":"The Stacks Project","memory_eligible":false,"source_rank":4676,"rank":4676,"depth":0,"x":2497.0,"y":328.242,"cluster":"homological-algebra"},{"id":"stacks:09JN","tag":"09JN","title":"The homotopy category · Definition 09JN","summary":"Let (A, d) be a differential graded algebra. Let f, g : M → N be homomorphisms of differential graded A-modules. A homotopy between f and g is an A-module map h : M → N such that • h(M^n) ⊂ N^n - 1 for all n, and • f(x) - g(x) = d_N(h(x)) + h(d_M(x)) for all x ∈ M. If a homotopy exists, then we say f and g are homotopic.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Let\n$f, g : M \\to N$ be homomorphisms of differential graded $A$-modules.\nA {\\it homotopy between $f$ and $g$} is an $A$-module map $h : M \\to N$\nsuch that\n\\begin{enumerate}\n\\item $h(M^n) \\subset N^{n - 1}$ for all $n$, and\n\\item $f(x) - g(x) = \\text{d}_N(h(x)) + h(\\text{d}_M(x))$ for\nall $x \\in M$.\n\\end{enumerate}\nIf a homotopy exists, then we say $f$ and $g$ are {\\it homotopic}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The homotopy category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JN","source_file":"dga.tex","source_line":324,"source_end_line":336,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L324-L336","statement_sha256":"88cc3e00bf6d7925b7d26516606309f49050c4c5859c9afdfb9a931556ea509d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4677,"rank":4677,"depth":0,"x":2380.266,"y":149.722,"cluster":"homological-algebra"},{"id":"stacks:09JP","tag":"09JP","title":"The homotopy category · Lemma 09JP","summary":"Let (A, d) be a differential graded algebra. Let f, g : L → M be homomorphisms of differential graded A-modules. Suppose given further homomorphisms a : K → L, and c : M → N. If h : L → M is an A-module map which defines a homotopy between f and g, then c ∘ h ∘ a defines a homotopy between c ∘ f ∘ a and c ∘ g ∘ a.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $f, g : L \\to M$ be homomorphisms of differential graded $A$-modules.\nSuppose given further homomorphisms $a : K \\to L$, and $c : M \\to N$.\nIf $h : L \\to M$ is an $A$-module map which defines a homotopy between\n$f$ and $g$, then $c \\circ h \\circ a$ defines a homotopy between\n$c \\circ f \\circ a$ and $c \\circ g \\circ a$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The homotopy category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JP","source_file":"dga.tex","source_line":344,"source_end_line":352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L344-L352","statement_sha256":"c61a4871a6e2194dd0fc570366de5bedc9b19fc2541821e3cc92edc536adff24","origin":"The Stacks Project","memory_eligible":false,"source_rank":4678,"rank":4678,"depth":1,"x":2610.257,"y":215.223,"cluster":"homological-algebra"},{"id":"stacks:09JQ","tag":"09JQ","title":"The homotopy category · Definition 09JQ","summary":"Let (A, d) be a differential graded algebra. The homotopy category, denoted K(Mod_(A, d)), is the category whose objects are the objects of Mod_(A, d) and whose morphisms are homotopy classes of homomorphisms of differential graded A-modules.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nThe {\\it homotopy category}, denoted $K(\\text{Mod}_{(A, \\text{d})})$, is\nthe category whose objects are the objects of\n$\\text{Mod}_{(A, \\text{d})}$ and whose morphisms are homotopy classes\nof homomorphisms of differential graded $A$-modules.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The homotopy category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JQ","source_file":"dga.tex","source_line":361,"source_end_line":368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L361-L368","statement_sha256":"e4b1abff99e021bc152e7989fc9c4b63544a4240af054733bafa942cc92b65d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4679,"rank":4679,"depth":0,"x":2387.651,"y":297.552,"cluster":"homological-algebra"},{"id":"stacks:09JR","tag":"09JR","title":"The homotopy category · Lemma 09JR","summary":"Let (A, d) be a differential graded algebra. The homotopy category K(Mod_(A, d)) has direct sums and products.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nThe homotopy category $K(\\text{Mod}_{(A, \\text{d})})$\nhas direct sums and products.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The homotopy category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JR","source_file":"dga.tex","source_line":374,"source_end_line":379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L374-L379","statement_sha256":"af3abadbe1d1466e9c4b59b714da3cd349c31e3c07c88576aaecf640ddbf316d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4680,"rank":4680,"depth":2,"x":2485.74,"y":110.247,"cluster":"homological-algebra"},{"id":"stacks:09KA","tag":"09KA","title":"Cones · Definition 09KA","summary":"Let (A, d) be a differential graded algebra. Let f : K → L be a homomorphism of differential graded A-modules. The cone of f is the differential graded A-module C(f) given by C(f) = L ⊕ K with grading C(f)^n = L^n ⊕ K^n + 1 and differential d_C(f) = ( d_L & f 0 & -d_K ) It comes equipped with canonical morphisms of complexes i : L → C(f) and p : C(f) → K[1] induced by the obvious maps L → C(f) and C(f) → K.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $f : K \\to L$ be a homomorphism of differential graded $A$-modules.\nThe {\\it cone} of $f$ is the differential graded $A$-module\n$C(f)$ given by $C(f) = L \\oplus K$ with grading\n$C(f)^n = L^n \\oplus K^{n + 1}$ and\ndifferential\n$$\nd_{C(f)} =\n\\left(\n\\begin{matrix}\n\\text{d}_L & f \\\\\n0 & -\\text{d}_K\n\\end{matrix}\n\\right)\n$$\nIt comes equipped with canonical morphisms of complexes $i : L \\to C(f)$\nand $p : C(f) \\to K[1]$ induced by the obvious maps $L \\to C(f)$\nand $C(f) \\to K$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Cones","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KA","source_file":"dga.tex","source_line":401,"source_end_line":421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L401-L421","statement_sha256":"567aa0b94d4b6407199a97757dc8a6a58cd187992b9b72cb6feb8e98fa41c961","origin":"The Stacks Project","memory_eligible":false,"source_rank":4681,"rank":4681,"depth":0,"x":2564.155,"y":304.315,"cluster":"homological-algebra"},{"id":"stacks:09KD","tag":"09KD","title":"Cones · Lemma 09KD","summary":"Let (A, d) be a differential graded algebra. Suppose that xymatrix K_1 ar[r]_f_1 ar[d]_a & L_1 ar[d]^b K_2 ar[r]^f_2 & L_2 is a diagram of homomorphisms of differential graded A-modules which is commutative up to homotopy. Then there exists a morphism c : C(f_1) → C(f_2) which gives rise to a morphism of triangles (a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) → (K_1, L_1, C(f_1), f_2, i_2, p_2) in K(Mod_(A, d)).","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nSuppose that\n$$\n\\xymatrix{\nK_1 \\ar[r]_{f_1} \\ar[d]_a & L_1 \\ar[d]^b \\\\\nK_2 \\ar[r]^{f_2} & L_2\n}\n$$\nis a diagram of homomorphisms of differential graded $A$-modules which is\ncommutative up to homotopy.\nThen there exists a morphism $c : C(f_1) \\to C(f_2)$ which gives rise to\na morphism of triangles\n$$\n(a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) \\to\n(K_1, L_1, C(f_1), f_2, i_2, p_2)\n$$\nin $K(\\text{Mod}_{(A, \\text{d})})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Cones","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KD","source_file":"dga.tex","source_line":426,"source_end_line":445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L426-L445","statement_sha256":"c0bbfb7619b7a127ab6f74ac12b43cbade2f6262355e4c0494d9262379c06a83","origin":"The Stacks Project","memory_eligible":false,"source_rank":4682,"rank":4682,"depth":0,"x":2349.944,"y":205.557,"cluster":"homological-algebra"},{"id":"stacks:09JT","tag":"09JT","title":"Admissible short exact sequences · Definition 09JT","summary":"Let (A, d) be a differential graded algebra. • A homomorphism K → L of differential graded A-modules is an admissible monomorphism if there exists a graded A-module map L → K which is left inverse to K → L. • A homomorphism L → M of differential graded A-modules is an admissible epimorphism if there exists a graded A-module map M → L which is right inverse to L → M. • A short exact sequence 0 → K → L → M → 0 of differential graded A-modules is an admissible short exact…","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\n\\begin{enumerate}\n\\item A homomorphism $K \\to L$ of differential graded $A$-modules\nis an {\\it admissible monomorphism} if there exists a graded $A$-module\nmap $L \\to K$ which is left inverse to $K \\to L$.\n\\item A homomorphism $L \\to M$ of differential graded $A$-modules\nis an {\\it admissible epimorphism} if there exists a graded $A$-module\nmap $M \\to L$ which is right inverse to $L \\to M$.\n\\item A short exact sequence $0 \\to K \\to L \\to M \\to 0$ of differential\ngraded $A$-modules is an {\\it admissible short exact sequence}\nif it is split as a sequence of graded $A$-modules.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Admissible short exact sequences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JT","source_file":"dga.tex","source_line":482,"source_end_line":496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L482-L496","statement_sha256":"5e428bf382326ec0aee6ed3889da159a95d72dd49ed69908913c93eb47bb8cce","origin":"The Stacks Project","memory_eligible":false,"source_rank":4683,"rank":4683,"depth":0,"x":2587.678,"y":156.758,"cluster":"homological-algebra"},{"id":"stacks:09JU","tag":"09JU","title":"Admissible short exact sequences · Lemma 09JU","summary":"Let (A, d) be a differential graded algebra. Let 0 → K → L → M → 0 be an admissible short exact sequence of differential graded A-modules. Let s : M → L and π : L → K be splittings such that Ker(π) = Im(s). Then we obtain a morphism δ = π ∘ d_L ∘ s : M → K[1] of Mod_(A, d) which induces the boundary maps in the long exact sequence of cohomology ([Tag 09JK]).","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $0 \\to K \\to L \\to M \\to 0$ be an admissible short exact sequence\nof differential graded $A$-modules. Let $s : M \\to L$ and $\\pi : L \\to K$\nbe splittings such that $\\Ker(\\pi) = \\Im(s)$.\nThen we obtain a morphism\n$$\n\\delta = \\pi \\circ \\text{d}_L \\circ s : M \\to K[1]\n$$\nof $\\text{Mod}_{(A, \\text{d})}$ which induces the boundary maps\nin the long exact sequence of cohomology (\\ref{equation-les}).","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Admissible short exact sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JU","source_file":"dga.tex","source_line":504,"source_end_line":516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L504-L516","statement_sha256":"687d69f1c1fc55655cf8f08e383badb398b9066aa4666a923dfbf05e50cbfdd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4684,"rank":4684,"depth":8,"x":2451.415,"y":327.895,"cluster":"homological-algebra"},{"id":"stacks:09JV","tag":"09JV","title":"Admissible short exact sequences · Lemma 09JV","summary":"Let (A, d) be a differential graded algebra. Let xymatrix K ar[r]_f ar[d]_a & L ar[d]^b M ar[r]^g & N be a diagram of homomorphisms of differential graded A-modules commuting up to homotopy. • If f is an admissible monomorphism, then b is homotopic to a homomorphism which makes the diagram commute. • If g is an admissible epimorphism, then a is homotopic to a morphism which makes the diagram commute.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Let\n$$\n\\xymatrix{\nK \\ar[r]_f \\ar[d]_a & L \\ar[d]^b \\\\\nM \\ar[r]^g & N\n}\n$$\nbe a diagram of homomorphisms of differential graded $A$-modules\ncommuting up to homotopy.\n\\begin{enumerate}\n\\item If $f$ is an admissible monomorphism, then $b$ is homotopic to a\nhomomorphism which makes the diagram commute.\n\\item If $g$ is an admissible epimorphism, then $a$ is homotopic to a\nmorphism which makes the diagram commute.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Admissible short exact sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JV","source_file":"dga.tex","source_line":530,"source_end_line":547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L530-L547","statement_sha256":"645465e44cb05fb7f8d8d401f7d39d0905eb8b5ed07a0f9c70b1fea244374cea","origin":"The Stacks Project","memory_eligible":false,"source_rank":4685,"rank":4685,"depth":0,"x":2414.214,"y":124.076,"cluster":"homological-algebra"},{"id":"stacks:09JW","tag":"09JW","title":"Admissible short exact sequences · Lemma 09JW","summary":"Let (A, d) be a differential graded algebra. Let α : K → L be a homomorphism of differential graded A-modules. There exists a factorization xymatrix K ar[r]^tilde α ar@/_1pc/[rr]_α & tilde L ar[r]^π & L in Mod_(A, d) such that • tilde α is an admissible monomorphism (see Definition [Tag 09JT]), • there is a morphism s : L → tilde L such that π ∘ s = id_L and such that s ∘ π is homotopic to id_tilde L.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $\\alpha : K \\to L$ be a homomorphism of differential graded\n$A$-modules. There exists a factorization\n$$\n\\xymatrix{\nK \\ar[r]^{\\tilde \\alpha} \\ar@/_1pc/[rr]_\\alpha &\n\\tilde L \\ar[r]^\\pi & L\n}\n$$\nin $\\text{Mod}_{(A, \\text{d})}$ such that\n\\begin{enumerate}\n\\item $\\tilde \\alpha$ is an admissible monomorphism (see\nDefinition \\ref{definition-admissible-ses}),\n\\item there is a morphism $s : L \\to \\tilde L$\nsuch that $\\pi \\circ s = \\text{id}_L$ and such that\n$s \\circ \\pi$ is homotopic to $\\text{id}_{\\tilde L}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Admissible short exact sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JW","source_file":"dga.tex","source_line":559,"source_end_line":578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L559-L578","statement_sha256":"d903531c600be95c8bf4ae87de36dae68e4e3793099ccbbfbcf013742724abc9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4686,"rank":4686,"depth":2,"x":2605.836,"y":253.454,"cluster":"homological-algebra"},{"id":"stacks:09JX","tag":"09JX","title":"Admissible short exact sequences · Lemma 09JX","summary":"Let (A, d) be a differential graded algebra. Let L_1 → L_2 → … → L_n be a sequence of composable homomorphisms of differential graded A-modules. There exists a commutative diagram xymatrix L_1 ar[r] & L_2 ar[r] & … ar[r] & L_n M_1 ar[r] ar[u] & M_2 ar[r] ar[u] & … ar[r] & M_n ar[u] in Mod_(A, d) such that each M_i → M_i + 1 is an admissible monomorphism and each M_i → L_i is a homotopy equivalence.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $L_1 \\to L_2 \\to \\ldots \\to L_n$\nbe a sequence of composable homomorphisms of\ndifferential graded $A$-modules.\nThere exists a commutative diagram\n$$\n\\xymatrix{\nL_1 \\ar[r] &\nL_2 \\ar[r] &\n\\ldots \\ar[r] &\nL_n \\\\\nM_1 \\ar[r] \\ar[u] &\nM_2 \\ar[r] \\ar[u] &\n\\ldots \\ar[r] &\nM_n \\ar[u]\n}\n$$\nin $\\text{Mod}_{(A, \\text{d})}$ such that each $M_i \\to M_{i + 1}$\nis an admissible monomorphism and each $M_i \\to L_i$\nis a homotopy equivalence.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Admissible short exact sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JX","source_file":"dga.tex","source_line":587,"source_end_line":609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L587-L609","statement_sha256":"d6a0fbb773761de228b30a3c3e3512e824e17c2866762f21a690bf2af0812312","origin":"The Stacks Project","memory_eligible":false,"source_rank":4687,"rank":4687,"depth":3,"x":2360.129,"y":266.804,"cluster":"homological-algebra"},{"id":"stacks:09JY","tag":"09JY","title":"Admissible short exact sequences · Lemma 09JY","summary":"Let (A, d) be a differential graded algebra. Let 0 → K_i → L_i → M_i → 0, i = 1, 2, 3 be admissible short exact sequence of differential graded A-modules. Let b : L_1 → L_2 and b' : L_2 → L_3 be homomorphisms of differential graded modules such that vcenter xymatrix K_1 ar[d]_0 ar[r] & L_1 ar[r] ar[d]_b & M_1 ar[d]_0 K_2 ar[r] & L_2 ar[r] & M_2 and vcenter xymatrix K_2 ar[d]^0 ar[r] & L_2 ar[r] ar[d]^b' & M_2 ar[d]^0 K_3 ar[r] & L_3 ar[r] & M_3 commute up to homotopy.…","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $0 \\to K_i \\to L_i \\to M_i \\to 0$, $i = 1, 2, 3$\nbe admissible short exact sequence of differential graded $A$-modules.\nLet $b : L_1 \\to L_2$ and $b' : L_2 \\to L_3$\nbe homomorphisms of differential graded modules such that\n$$\n\\vcenter{\n\\xymatrix{\nK_1 \\ar[d]_0 \\ar[r] &\nL_1 \\ar[r] \\ar[d]_b &\nM_1 \\ar[d]_0 \\\\\nK_2 \\ar[r] & L_2 \\ar[r] & M_2\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nK_2 \\ar[d]^0 \\ar[r] &\nL_2 \\ar[r] \\ar[d]^{b'} &\nM_2 \\ar[d]^0 \\\\\nK_3 \\ar[r] & L_3 \\ar[r] & M_3\n}\n}\n$$\ncommute up to homotopy. Then $b' \\circ b$ is homotopic to $0$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Admissible short exact sequences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JY","source_file":"dga.tex","source_line":623,"source_end_line":650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L623-L650","statement_sha256":"c248aa16b3d1085ed1ee17e28554f0e848d492295e717febb88b19fffd541e8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4688,"rank":4688,"depth":1,"x":2530.828,"y":117.318,"cluster":"homological-algebra"},{"id":"stacks:09K6","tag":"09K6","title":"Distinguished triangles · Lemma 09K6","summary":"Let (A, d) be a differential graded algebra. Let 0 → K → L → M → 0 be an admissible short exact sequence of differential graded A-modules. The triangle K → L → M xrightarrowδ K[1] with δ as in Lemma [Tag 09JU] is, up to canonical isomorphism in K(Mod_(A, d)), independent of the choices made in Lemma [Tag 09JU].","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Let\n$0 \\to K \\to L \\to M \\to 0$ be an admissible short exact sequence\nof differential graded $A$-modules. The triangle\n\\begin{equation}\n\nK \\to L \\to M \\xrightarrow{\\delta} K[1]\n\\end{equation}\nwith $\\delta$ as in Lemma \\ref{lemma-admissible-ses} is, up to canonical\nisomorphism in $K(\\text{Mod}_{(A, \\text{d})})$, independent of the choices\nmade in Lemma \\ref{lemma-admissible-ses}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Distinguished triangles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09K6","source_file":"dga.tex","source_line":682,"source_end_line":694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L682-L694","statement_sha256":"efcc107f3933287eea1d6883c12f991568934a9e0b28b644bea50e13ff3e7eb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4689,"rank":4689,"depth":9,"x":2525.161,"y":324.712,"cluster":"homological-algebra"},{"id":"stacks:09K8","tag":"09K8","title":"Distinguished triangles · Definition 09K8","summary":"Let (A, d) be a differential graded algebra. • If 0 → K → L → M → 0 is an admissible short exact sequence of differential graded A-modules, then the triangle associated to 0 → K → L → M → 0 is the triangle ([Tag 09K7]) of K(Mod_(A, d)). • A triangle of K(Mod_(A, d)) is called a distinguished triangle if it is isomorphic to a triangle associated to an admissible short exact sequence of differential graded A-modules.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\n\\begin{enumerate}\n\\item If $0 \\to K \\to L \\to M \\to 0$ is an admissible short exact sequence\nof differential graded $A$-modules, then the {\\it triangle associated\nto $0 \\to K \\to L \\to M \\to 0$} is the triangle \n(\\ref{equation-triangle-associated-to-admissible-ses})\nof $K(\\text{Mod}_{(A, \\text{d})})$.\n\\item A triangle of $K(\\text{Mod}_{(A, \\text{d})})$ is called a\n{\\it distinguished triangle} if it is isomorphic to a triangle\nassociated to an admissible short exact sequence\nof differential graded $A$-modules.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Distinguished triangles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09K8","source_file":"dga.tex","source_line":707,"source_end_line":721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L707-L721","statement_sha256":"59133fe41a3db0ea64b5cec2a6f364b8e3e3cbd8027782c8abeac920ac49ee3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4690,"rank":4690,"depth":10,"x":2362.318,"y":168.336,"cluster":"homological-algebra"},{"id":"stacks:09KB","tag":"09KB","title":"Cones and distinguished triangles · Lemma 09KB","summary":"Let (A, d) be a differential graded algebra. Let f : K → L be a homomorphism of differential graded modules. The triangle (L, C(f), K[1], i, p, f[1]) is the triangle associated to the admissible short exact sequence 0 → L → C(f) → K[1] → 0 coming from the definition of the cone of f.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $f : K \\to L$ be a homomorphism of differential graded modules.\nThe triangle $(L, C(f), K[1], i, p, f[1])$ is\nthe triangle associated to the admissible short exact sequence\n$$\n0 \\to L \\to C(f) \\to K[1] \\to 0\n$$\ncoming from the definition of the cone of $f$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Cones and distinguished triangles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KB","source_file":"dga.tex","source_line":745,"source_end_line":755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L745-L755","statement_sha256":"a07f0a9895ebd4c2a4605079a40a280d238b726e62c6ca2f059f7c57d66e167a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4691,"rank":4691,"depth":0,"x":2608.512,"y":191.279,"cluster":"homological-algebra"},{"id":"stacks:09KC","tag":"09KC","title":"Cones and distinguished triangles · Lemma 09KC","summary":"Let (A, d) be a differential graded algebra. Let α : K → L and β : L → M define an admissible short exact sequence 0 → K → L → M → 0 of differential graded A-modules. Let (K, L, M, α, β, δ) be the associated triangle. Then the triangles (M[-1], K, L, δ[-1], α, β) and (M[-1], K, C(δ[-1]), δ[-1], i, p) are isomorphic.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $\\alpha : K \\to L$ and $\\beta : L \\to M$\ndefine an admissible short exact sequence\n$$\n0 \\to K \\to L \\to M \\to 0\n$$\nof differential graded $A$-modules.\nLet $(K, L, M, \\alpha, \\beta, \\delta)$\nbe the associated triangle. Then the triangles\n$$\n(M[-1], K, L, \\delta[-1], \\alpha, \\beta)\n\\quad\\text{and}\\quad\n(M[-1], K, C(\\delta[-1]), \\delta[-1], i, p)\n$$\nare isomorphic.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Cones and distinguished triangles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KC","source_file":"dga.tex","source_line":761,"source_end_line":778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L761-L778","statement_sha256":"1a73f59dc2d8c04fb50a9a30b68a93b842d2fbf458fc19ec6d679915281c5dbf","origin":"The Stacks Project","memory_eligible":false,"source_rank":4692,"rank":4692,"depth":0,"x":2408.229,"y":314.238,"cluster":"homological-algebra"},{"id":"stacks:09KE","tag":"09KE","title":"Cones and distinguished triangles · Lemma 09KE","summary":"Let (A, d) be a differential graded algebra. Let f_1 : K_1 → L_1 and f_2 : K_2 → L_2 be homomorphisms of differential graded A-modules. Let (a, b, c) : (K_1, L_1, C(f_1), f_1, i_1, p_1) → (K_1, L_1, C(f_1), f_2, i_2, p_2) be any morphism of triangles of K(Mod_(A, d)). If a and b are homotopy equivalences then so is c.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $f_1 : K_1 \\to L_1$ and $f_2 : K_2 \\to L_2$ be homomorphisms of\ndifferential graded $A$-modules. Let\n$$\n(a, b, c) :\n(K_1, L_1, C(f_1), f_1, i_1, p_1)\n\\longrightarrow\n(K_1, L_1, C(f_1), f_2, i_2, p_2)\n$$\nbe any morphism of triangles of $K(\\text{Mod}_{(A, \\text{d})})$.\nIf $a$ and $b$ are homotopy equivalences then so is $c$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Cones and distinguished triangles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KE","source_file":"dga.tex","source_line":798,"source_end_line":811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L798-L811","statement_sha256":"5061c60b4b0b00806e450ad3141f303efeade608618833ddcf7e9f486ffa46af","origin":"The Stacks Project","memory_eligible":false,"source_rank":4693,"rank":4693,"depth":2,"x":2457.106,"y":109.625,"cluster":"homological-algebra"},{"id":"stacks:09KF","tag":"09KF","title":"Cones and distinguished triangles · Lemma 09KF","summary":"Let (A, d) be a differential graded algebra. • Given an admissible short exact sequence 0 → K xrightarrowα L → M → 0 of differential graded A-modules there exists a homotopy equivalence C(α) → M such that the diagram xymatrix K ar[r] ar[d] & L ar[d] ar[r] & C(α) ar[r]_-p ar[d] & K[1] ar[d] K ar[r]^α & L ar[r]^β & M ar[r]^δ & K[1] defines an isomorphism of triangles in K(Mod_(A, d)). • Given a morphism of complexes f : K → L there exists an isomorphism of triangles…","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\n\\begin{enumerate}\n\\item Given an admissible short exact sequence\n$0 \\to K \\xrightarrow{\\alpha} L \\to M \\to 0$\nof differential graded $A$-modules there exists a homotopy equivalence\n$C(\\alpha) \\to M$ such that the diagram\n$$\n\\xymatrix{\nK \\ar[r] \\ar[d] & L \\ar[d] \\ar[r] &\nC(\\alpha) \\ar[r]_{-p} \\ar[d] & K[1] \\ar[d] \\\\\nK \\ar[r]^\\alpha & L \\ar[r]^\\beta &\nM \\ar[r]^\\delta & K[1]\n}\n$$\ndefines an isomorphism of triangles in $K(\\text{Mod}_{(A, \\text{d})})$.\n\\item Given a morphism of complexes $f : K \\to L$\nthere exists an isomorphism of triangles\n$$\n\\xymatrix{\nK \\ar[r] \\ar[d] & \\tilde L \\ar[d] \\ar[r] &\nM \\ar[r]_{\\delta} \\ar[d] & K[1] \\ar[d] \\\\\nK \\ar[r] & L \\ar[r] &\nC(f) \\ar[r]^{-p} & K[1]\n}\n$$\nwhere the upper triangle is the triangle associated to a\nadmissible short exact sequence $K \\to \\tilde L \\to M$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Cones and distinguished triangles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KF","source_file":"dga.tex","source_line":852,"source_end_line":882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L852-L882","statement_sha256":"3225afe0389a02652aa3640f8aaa43ed7444383ec74bcb4c8921f39a463da8c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4694,"rank":4694,"depth":3,"x":2585.797,"y":288.498,"cluster":"homological-algebra"},{"id":"stacks:09KH","tag":"09KH","title":"The homotopy category is triangulated · Lemma 09KH","summary":"Let (A, d) be a differential graded algebra. The homotopy category K(Mod_(A, d)) with its natural translation functors and distinguished triangles is a pre-triangulated category.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nThe homotopy category $K(\\text{Mod}_{(A, \\text{d})})$\nwith its natural translation functors and distinguished triangles\nis a pre-triangulated category.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The homotopy category is triangulated","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KH","source_file":"dga.tex","source_line":1007,"source_end_line":1013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1007-L1013","statement_sha256":"30c28d0084e6603960aea372b61f5b491c84ea35bba11c2b60383dbc87b251ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":4695,"rank":4695,"depth":4,"x":2346.709,"y":229.536,"cluster":"homological-algebra"},{"id":"stacks:09KI","tag":"09KI","title":"The homotopy category is triangulated · Lemma 09KI","summary":"Let (A, d) be a differential graded algebra. Suppose that α : K → L and β : L → M are admissible monomorphisms of differential graded A-modules. Then there exist distinguished triangles (K, L, Q_1, α, p_1, d_1), (K, M, Q_2, β ∘ α, p_2, d_2) and (L, M, Q_3, β, p_3, d_3) for which TR4 holds.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Suppose that\n$\\alpha : K \\to L$ and $\\beta : L \\to M$ are admissible monomorphisms\nof differential graded $A$-modules. Then there exist distinguished triangles\n$(K, L, Q_1, \\alpha, p_1, d_1)$, $(K, M, Q_2, \\beta \\circ \\alpha, p_2, d_2)$\nand $(L, M, Q_3, \\beta, p_3, d_3)$ for which TR4 holds.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The homotopy category is triangulated","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KI","source_file":"dga.tex","source_line":1068,"source_end_line":1075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1068-L1075","statement_sha256":"513276d62d0a31f1e9619e3ce10ed0b16f0a96287ed6aeec99ba511543ec4933","origin":"The Stacks Project","memory_eligible":false,"source_rank":4696,"rank":4696,"depth":0,"x":2570.75,"y":137.216,"cluster":"homological-algebra"},{"id":"stacks:09KJ","tag":"09KJ","title":"The homotopy category is triangulated · Proposition 09KJ","summary":"Let (A, d) be a differential graded algebra. The homotopy category K(Mod_(A, d)) of differential graded A-modules with its natural translation functors and distinguished triangles is a triangulated category.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. The homotopy category\n$K(\\text{Mod}_{(A, \\text{d})})$ of differential graded $A$-modules with its\nnatural translation functors and distinguished triangles is a triangulated\ncategory.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The homotopy category is triangulated","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KJ","source_file":"dga.tex","source_line":1129,"source_end_line":1135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1129-L1135","statement_sha256":"2cf94d3ba4bfaf827951bc45a6c84cf0320a71f08560c9e367200f7829c54451","origin":"The Stacks Project","memory_eligible":false,"source_rank":4697,"rank":4697,"depth":11,"x":2479.655,"y":332.7,"cluster":"homological-algebra"},{"id":"stacks:09JG","tag":"09JG","title":"Left modules · Definition 09JG","summary":"Let R be a ring. Let (A, d) be a differential graded algebra over R. The opposite differential graded algebra is the differential graded algebra (A^opp, d) over R where A^opp = A as a graded R-module, d = d, and multiplication is given by a ·_opp b = (-1)^deg(a)deg(b) b a for homogeneous elements a, b ∈ A.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded algebra\nover $R$. The {\\it opposite differential graded algebra} is the differential\ngraded algebra $(A^{opp}, \\text{d})$ over $R$ where $A^{opp} = A$\nas a graded $R$-module, $\\text{d} = \\text{d}$, and multiplication is\ngiven by\n$$\na \\cdot_{opp} b = (-1)^{\\deg(a)\\deg(b)} b a\n$$\nfor homogeneous elements $a, b \\in A$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Left modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09JG","source_file":"dga.tex","source_line":1186,"source_end_line":1197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1186-L1197","statement_sha256":"146240e55d20e3fb9694867cd432d5824d7ad09c623b3aaba391b779320f6fc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4698,"rank":4698,"depth":0,"x":2389.494,"y":136.581,"cluster":"homological-algebra"},{"id":"stacks:0FQ0","tag":"0FQ0","title":"Left modules · Lemma 0FQ0","summary":"Let (A, d) be a differential graded R-algebra. The functor M ↦ M^opp from the category of left differential graded A-modules to the category of right differential graded A^opp-modules is an equivalence.","statement_latex":"Let $(A, \\text{d})$ be a differential graded $R$-algebra.\nThe functor $M \\mapsto M^{opp}$ from the category of\nleft differential graded $A$-modules to the category of right\ndifferential graded $A^{opp}$-modules is an equivalence.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Left modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQ0","source_file":"dga.tex","source_line":1271,"source_end_line":1277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1271-L1277","statement_sha256":"3d929dec435f9761eb8bf51a843503d2ea83032a2f72f2cdac022373b43652ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":4699,"rank":4699,"depth":0,"x":2614.013,"y":230.172,"cluster":"homological-algebra"},{"id":"stacks:0FQ1","tag":"0FQ1","title":"Left modules · Definition 0FQ1","summary":"Let R be a ring. Let A be a Z-graded R-algebra. • Given a right graded A-module M we define the kth shifted A-module M[k] as the same as a right A-module but with grading (M[k])^n = M^n + k. • Given a left graded A-module M we define the kth shifted A-module M[k] as the module with grading (M[k])^n = M^n + k and multiplication A^n × (M[k])^m → (M[k])^n + m equal to (-1)^nk times the given multiplication A^n × M^m + k → M^n + m + k.","statement_latex":"Let $R$ be a ring. Let $A$ be a $\\mathbf{Z}$-graded $R$-algebra.\n\\begin{enumerate}\n\\item Given a right graded $A$-module $M$ we define the\n{\\it $k$th shifted $A$-module} $M[k]$ as the same as\na right $A$-module but with grading $(M[k])^n = M^{n + k}$.\n\\item Given a left graded $A$-module $M$ we define the\n{\\it $k$th shifted $A$-module} $M[k]$ as the module\nwith grading $(M[k])^n = M^{n + k}$ and multiplication\n$A^n \\times (M[k])^m \\to (M[k])^{n + m}$\nequal to $(-1)^{nk}$ times the given multiplication\n$A^n \\times M^{m + k} \\to M^{n + m + k}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Left modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQ1","source_file":"dga.tex","source_line":1338,"source_end_line":1352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1338-L1352","statement_sha256":"40cfed459acacd157b63ef78619cb636bb7c87ed7a56ec7b79be8be5e2f1e79b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4700,"rank":4700,"depth":0,"x":2372.848,"y":288.64,"cluster":"homological-algebra"},{"id":"stacks:0FQ3","tag":"0FQ3","title":"Hom complexes and differential graded modules · Lemma 0FQ3","summary":"In the situation above, let A be a differential graded R-algebra. To give a left A-module structure on M is the same thing as giving a homomorphism A → E of differential graded R-algebras.","statement_latex":"In the situation above, let $A$ be a differential graded $R$-algebra.\nTo give a left $A$-module structure on $M$ is the same thing as\ngiving a homomorphism $A \\to E$ of differential graded $R$-algebras.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Hom complexes and differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQ3","source_file":"dga.tex","source_line":1500,"source_end_line":1505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1500-L1505","statement_sha256":"b208906e455e4d589dafd8c23f7ab1e5c6050e40bfc0eb1e28d045d1d7ed89be","origin":"The Stacks Project","memory_eligible":false,"source_rank":4701,"rank":4701,"depth":0,"x":2503.848,"y":108.426,"cluster":"homological-algebra"},{"id":"stacks:0FQ4","tag":"0FQ4","title":"Hom complexes and differential graded modules · Lemma 0FQ4","summary":"Let R be a ring. Let (A, d) be a differential graded R-algebra. Let M' be a right differential graded A-module and let M be a left differential graded A-module. Let N^bullet be a complex of R-modules. Then we have Hom_Mod_(A, d)(M', Hom(M, N^bullet)) = Hom_Comp(R)(M' ⊗_A M, N^bullet) where M ⊗_A M is viewed as a complex of R-modules as in Section [Tag 09LL].","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded $R$-algebra.\nLet $M'$ be a right differential graded $A$-module and let\n$M$ be a left differential graded $A$-module.\nLet $N^\\bullet$ be a complex of $R$-modules. Then we have\n$$\n\\Hom_{\\text{Mod}_{(A, d)}}(M', \\Hom(M, N^\\bullet)) =\n\\Hom_{\\text{Comp}(R)}(M' \\otimes_A M, N^\\bullet)\n$$\nwhere $M \\otimes_A M$ is viewed as a complex of $R$-modules\nas in Section \\ref{section-tensor-product}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Hom complexes and differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQ4","source_file":"dga.tex","source_line":1543,"source_end_line":1555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1543-L1555","statement_sha256":"9871102d784c1a93dd1520e44eeeb7632da2855ddba6d4cf919e45dc20c0c897","origin":"The Stacks Project","memory_eligible":false,"source_rank":4702,"rank":4702,"depth":2,"x":2552.243,"y":315.942,"cluster":"homological-algebra"},{"id":"stacks:0FQ5","tag":"0FQ5","title":"Hom complexes and differential graded modules · Lemma 0FQ5","summary":"In the situation above, let A be a differential graded R-algebra. To give a right A-module structure on M is the same thing as giving a homomorphism τ : A → E^opp of differential graded R-algebras.","statement_latex":"In the situation above, let $A$ be a differential graded $R$-algebra.\nTo give a right $A$-module structure on $M$ is the same thing as\ngiving a homomorphism $\\tau : A \\to E^{opp}$\nof differential graded $R$-algebras.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Hom complexes and differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQ5","source_file":"dga.tex","source_line":1608,"source_end_line":1614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1608-L1614","statement_sha256":"ec974e1c3d9a065fb434f34d964ab55b362c173f53fa5fe15801279df94a4932","origin":"The Stacks Project","memory_eligible":false,"source_rank":4703,"rank":4703,"depth":0,"x":2349.391,"y":190.204,"cluster":"homological-algebra"},{"id":"stacks:0FQ6","tag":"0FQ6","title":"Hom complexes and differential graded modules · Lemma 0FQ6","summary":"Let R be a ring. Let (A, d) be a differential graded R-algebra. Let M be a right differential graded A-module and let M' be a left differential graded A-module. Let N^bullet be a complex of R-modules. Then we have Hom_left diff graded A-modules(M', Hom(M, N^bullet)) = Hom_Comp(R)(M ⊗_A M', N^bullet) where M ⊗_A M' is viewed as a complex of R-modules as in Section [Tag 09LL].","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ be a differential graded $R$-algebra.\nLet $M$ be a right differential graded $A$-module and let\n$M'$ be a left differential graded $A$-module.\nLet $N^\\bullet$ be a complex of $R$-modules. Then we have\n$$\n\\Hom_{\\text{left diff graded }A\\text{-modules}}(M', \\Hom(M, N^\\bullet)) =\n\\Hom_{\\text{Comp}(R)}(M \\otimes_A M', N^\\bullet)\n$$\nwhere $M \\otimes_A M'$ is viewed as a complex of $R$-modules\nas in Section \\ref{section-tensor-product}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Hom complexes and differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQ6","source_file":"dga.tex","source_line":1667,"source_end_line":1679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1667-L1679","statement_sha256":"6c610d9619c40dce9dcabbf859ad8d26dbd3ee226f8b0ed565adf7cbc6d2602e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4704,"rank":4704,"depth":2,"x":2600.441,"y":167.786,"cluster":"homological-algebra"},{"id":"stacks:09K0","tag":"09K0","title":"Projective modules and differential graded algebras · Lemma 09K0","summary":"Let (A, d) be a differential graded algebra. Let M → P be a surjective homomorphism of differential graded A-modules. If P is projective as a graded A-module, then M → P is an admissible epimorphism.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $M \\to P$ be a surjective homomorphism of differential graded\n$A$-modules. If $P$ is projective as a graded $A$-module, then\n$M \\to P$ is an admissible epimorphism.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Projective modules and differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09K0","source_file":"dga.tex","source_line":1888,"source_end_line":1894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1888-L1894","statement_sha256":"07b4158213c175b0a7ff278d38c6d1c777fc47a194ac7b3f0b09e120824ea992","origin":"The Stacks Project","memory_eligible":false,"source_rank":4705,"rank":4705,"depth":0,"x":2433.111,"y":326.993,"cluster":"homological-algebra"},{"id":"stacks:09K1","tag":"09K1","title":"Projective modules and differential graded algebras · Lemma 09K1","summary":"Let (A, d) be a differential graded algebra. Then we have Hom_Mod_(A, d)(A[k], M) = Ker(d : M^-k → M^-k + 1) and Hom_K(Mod_(A, d))(A[k], M) = H^-k(M) for any differential graded A-module M.","statement_latex":"Let $(A, d)$ be a differential graded algebra. Then we have\n$$\n\\Hom_{\\text{Mod}_{(A, \\text{d})}}(A[k], M) =\n\\Ker(\\text{d} : M^{-k} \\to M^{-k + 1})\n$$\nand\n$$\n\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(A[k], M) = H^{-k}(M)\n$$\nfor any differential graded $A$-module $M$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Projective modules and differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09K1","source_file":"dga.tex","source_line":1900,"source_end_line":1912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L1900-L1912","statement_sha256":"8f940e6bf6ca4976769aa1f820d11fbfad66e28961778f9ae363cee7b7fdfdf5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4706,"rank":4706,"depth":0,"x":2428.461,"y":114.35,"cluster":"homological-algebra"},{"id":"stacks:09K2","tag":"09K2","title":"Injective modules and differential graded algebras · Lemma 09K2","summary":"Let (A, d) be a differential graded algebra. Let I → M be an injective homomorphism of differential graded A-modules. If I is graded injective, then I → M is an admissible monomorphism.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $I \\to M$ be an injective homomorphism of differential graded\n$A$-modules. If $I$ is graded injective, then\n$I \\to M$ is an admissible monomorphism.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Injective modules and differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09K2","source_file":"dga.tex","source_line":2099,"source_end_line":2105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2099-L2105","statement_sha256":"249056af63f42cf57bac6167dda38c6d4eb82b6298d3fce6df88cb10d7112653","origin":"The Stacks Project","memory_eligible":false,"source_rank":4707,"rank":4707,"depth":0,"x":2603.137,"y":268.728,"cluster":"homological-algebra"},{"id":"stacks:09K3","tag":"09K3","title":"Injective modules and differential graded algebras · Lemma 09K3","summary":"Let (A, d) be a differential graded algebra. If M is a left differential graded A-module and N is a right differential graded A-module, then Hom_Mod_(A, d)(N, M^vee) & = Hom_Comp(Z)(N ⊗_A M, Q/Z) & = DifferentialGradedBilinear_A(N × M, Q/Z)","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. If\n$M$ is a left differential graded $A$-module and $N$ is a\nright differential graded $A$-module, then\n\\begin{align*}\n\\Hom_{\\text{Mod}_{(A, \\text{d})}}(N, M^\\vee)\n& =\n\\Hom_{\\text{Comp}(\\mathbf{Z})}(N \\otimes_A M, \\mathbf{Q}/\\mathbf{Z}) \\\\\n& =\n\\text{DifferentialGradedBilinear}_A(N \\times M, \\mathbf{Q}/\\mathbf{Z})\n\\end{align*}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Injective modules and differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09K3","source_file":"dga.tex","source_line":2129,"source_end_line":2141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2129-L2141","statement_sha256":"742203598d644d4b6137f22243010dd62c4f6fe5c3134f51638adac25e2950c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4708,"rank":4708,"depth":3,"x":2349.833,"y":253.987,"cluster":"homological-algebra"},{"id":"stacks:09K4","tag":"09K4","title":"Injective modules and differential graded algebras · Lemma 09K4","summary":"Let (A, d) be a differential graded algebra. Then we have Hom_Mod_(A, d)(M, A^vee[k]) = Ker(d : (M^vee)^k → (M^vee)^k + 1) and Hom_K(Mod_(A, d))(M, A^vee[k]) = H^k(M^vee) as functors in the differential graded A-module M.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Then we have\n$$\n\\Hom_{\\text{Mod}_{(A, \\text{d})}}(M, A^\\vee[k]) =\n\\Ker(\\text{d} : (M^\\vee)^k \\to (M^\\vee)^{k + 1})\n$$\nand\n$$\n\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(M, A^\\vee[k]) = H^k(M^\\vee)\n$$\nas functors in the differential graded $A$-module $M$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Injective modules and differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09K4","source_file":"dga.tex","source_line":2148,"source_end_line":2160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2148-L2160","statement_sha256":"5bad973ff7265f63740656ec1794ab4f82078179b5c6792f1bc39e78f59d93be","origin":"The Stacks Project","memory_eligible":false,"source_rank":4709,"rank":4709,"depth":0,"x":2548.748,"y":120.941,"cluster":"homological-algebra"},{"id":"stacks:09KL","tag":"09KL","title":"P-resolutions · Lemma 09KL","summary":"Let (A, d) be a differential graded algebra. Let P be a differential graded A-module. If F_bullet is a filtration as in property (P), then we obtain an admissible short exact sequence 0 → bigoplus F_iP → bigoplus F_iP → P → 0 of differential graded A-modules.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $P$ be a differential graded $A$-module. If $F_\\bullet$\nis a filtration as in property (P), then we obtain an\nadmissible short exact sequence\n$$\n0 \\to\n\\bigoplus\\nolimits F_iP \\to\n\\bigoplus\\nolimits F_iP \\to P \\to 0\n$$\nof differential graded $A$-modules.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"P-resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KL","source_file":"dga.tex","source_line":2207,"source_end_line":2219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2207-L2219","statement_sha256":"97bb901938da10fc1a36f25dee9cbb11598251a0fb5d4e71be0941788f02f358","origin":"The Stacks Project","memory_eligible":false,"source_rank":4710,"rank":4710,"depth":0,"x":2509.008,"y":332.21,"cluster":"homological-algebra"},{"id":"stacks:09KM","tag":"09KM","title":"P-resolutions · Lemma 09KM","summary":"Let (A, d) be a differential graded algebra. Let P be a differential graded A-module with property (P). Then Hom_K(Mod_(A, d))(P, N) = 0 for all acyclic differential graded A-modules N.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $P$ be a differential graded $A$-module with property (P).\nThen\n$$\n\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(P, N) = 0\n$$\nfor all acyclic differential graded $A$-modules $N$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"P-resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KM","source_file":"dga.tex","source_line":2239,"source_end_line":2248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2239-L2248","statement_sha256":"df2ba58f39554d8469f8c962f1fff46ae0641adb8aa5e7b708534253273b42d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4711,"rank":4711,"depth":12,"x":2368.219,"y":153.626,"cluster":"homological-algebra"},{"id":"stacks:09KN","tag":"09KN","title":"P-resolutions · Lemma 09KN","summary":"Let (A, d) be a differential graded algebra. Let M be a differential graded A-module. There exists a homomorphism P → M of differential graded A-modules with the following properties • P → M is surjective, • Ker(d_P) → Ker(d_M) is surjective, and • P sits in an admissible short exact sequence 0 → P' → P → P\" → 0 where P', P\" are direct sums of shifts of A.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $M$ be a differential graded $A$-module. There exists a homomorphism\n$P \\to M$ of differential graded $A$-modules with the following\nproperties\n\\begin{enumerate}\n\\item $P \\to M$ is surjective,\n\\item $\\Ker(\\text{d}_P) \\to \\Ker(\\text{d}_M)$ is surjective, and\n\\item $P$ sits in an admissible short exact sequence\n$0 \\to P' \\to P \\to P'' \\to 0$ where $P'$, $P''$ are direct sums\nof shifts of $A$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"P-resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KN","source_file":"dga.tex","source_line":2272,"source_end_line":2285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2272-L2285","statement_sha256":"3ee2982c3877dbbad610c0222886eecde10a21f65cfd5ae97d7a3af802280e34","origin":"The Stacks Project","memory_eligible":false,"source_rank":4712,"rank":4712,"depth":0,"x":2615.99,"y":205.496,"cluster":"homological-algebra"},{"id":"stacks:09KP","tag":"09KP","title":"P-resolutions · Lemma 09KP","summary":"Let (A, d) be a differential graded algebra. Let M be a differential graded A-module. There exists a homomorphism P → M of differential graded A-modules such that • P → M is a quasi-isomorphism, and • P has property (P).","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $M$ be a differential graded $A$-module. There exists a homomorphism\n$P \\to M$ of differential graded $A$-modules such that\n\\begin{enumerate}\n\\item $P \\to M$ is a quasi-isomorphism, and\n\\item $P$ has property (P).\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"P-resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KP","source_file":"dga.tex","source_line":2301,"source_end_line":2310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2301-L2310","statement_sha256":"a13b7438368f9ae682574800efca8994fb92f1a5437cee8229cd49c809b183c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4713,"rank":4713,"depth":1,"x":2391.264,"y":307.98,"cluster":"homological-algebra"},{"id":"stacks:09KR","tag":"09KR","title":"I-resolutions · Lemma 09KR","summary":"Let (A, d) be a differential graded algebra. Let I be a differential graded A-module. If F_bullet is a filtration as in property (I), then we obtain an admissible short exact sequence 0 → I → ∏ I/F_iI → ∏ I/F_iI → 0 of differential graded A-modules.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $I$ be a differential graded $A$-module. If $F_\\bullet$\nis a filtration as in property (I), then we obtain an\nadmissible short exact sequence\n$$\n0 \\to I \\to\n\\prod\\nolimits I/F_iI \\to\n\\prod\\nolimits I/F_iI \\to 0\n$$\nof differential graded $A$-modules.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"I-resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KR","source_file":"dga.tex","source_line":2380,"source_end_line":2392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2380-L2392","statement_sha256":"685da1b630da5c6bc587b54b119c98fc22722dd63c55159191ad8b056e7f8c7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4714,"rank":4714,"depth":1,"x":2474.674,"y":104.615,"cluster":"homological-algebra"},{"id":"stacks:09KS","tag":"09KS","title":"I-resolutions · Lemma 09KS","summary":"Let (A, d) be a differential graded algebra. Let I be a differential graded A-module with property (I). Then Hom_K(Mod_(A, d))(N, I) = 0 for all acyclic differential graded A-modules N.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $I$ be a differential graded $A$-module with property (I).\nThen\n$$\n\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(N, I) = 0\n$$\nfor all acyclic differential graded $A$-modules $N$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"I-resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KS","source_file":"dga.tex","source_line":2403,"source_end_line":2412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2403-L2412","statement_sha256":"d22e26b03595a6da2845359121b278944528898d12a0597a439fd639fca6f8ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":4715,"rank":4715,"depth":12,"x":2576.849,"y":302.173,"cluster":"homological-algebra"},{"id":"stacks:09KT","tag":"09KT","title":"I-resolutions · Lemma 09KT","summary":"Let (A, d) be a differential graded algebra. Let M be a differential graded A-module. There exists a homomorphism M → I of differential graded A-modules with the following properties • M → I is injective, • Coker(d_M) → Coker(d_I) is injective, and • I sits in an admissible short exact sequence 0 → I' → I → I\" → 0 where I', I\" are products of shifts of A^vee.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $M$ be a differential graded $A$-module. There exists a homomorphism\n$M \\to I$ of differential graded $A$-modules with the following\nproperties\n\\begin{enumerate}\n\\item $M \\to I$ is injective,\n\\item $\\Coker(\\text{d}_M) \\to \\Coker(\\text{d}_I)$ is injective,\nand\n\\item $I$ sits in an admissible short exact sequence\n$0 \\to I' \\to I \\to I'' \\to 0$ where $I'$, $I''$ are products\nof shifts of $A^\\vee$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"I-resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KT","source_file":"dga.tex","source_line":2437,"source_end_line":2451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2437-L2451","statement_sha256":"5edc933ef842ef915f314cf29237777cb6e4bc917888bb1ee86eeab26741cc41","origin":"The Stacks Project","memory_eligible":false,"source_rank":4716,"rank":4716,"depth":1,"x":2342.315,"y":214.354,"cluster":"homological-algebra"},{"id":"stacks:09KU","tag":"09KU","title":"I-resolutions · Lemma 09KU","summary":"Let (A, d) be a differential graded algebra. Let M be a differential graded A-module. There exists a homomorphism M → I of differential graded A-modules such that • M → I is a quasi-isomorphism, and • I has property (I).","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $M$ be a differential graded $A$-module. There exists a homomorphism\n$M \\to I$ of differential graded $A$-modules such that\n\\begin{enumerate}\n\\item $M \\to I$ is a quasi-isomorphism, and\n\\item $I$ has property (I).\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"I-resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KU","source_file":"dga.tex","source_line":2490,"source_end_line":2499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2490-L2499","statement_sha256":"6879eb405a276868237daf83cae8f6819862a71f1d3db856cb070e1daebd7615","origin":"The Stacks Project","memory_eligible":false,"source_rank":4717,"rank":4717,"depth":2,"x":2586.212,"y":145.937,"cluster":"homological-algebra"},{"id":"stacks:09KW","tag":"09KW","title":"The derived category · Lemma 09KW","summary":"Let (A, d) be a differential graded algebra. The full subcategory Ac of K(Mod_(A, d)) consisting of acyclic modules is a strictly full saturated triangulated subcategory of K(Mod_(A, d)). The corresponding saturated multiplicative system (see Derived Categories, Lemma [Tag 05RL]) of K(Mod_(A, d)) is the class Qis of quasi-isomorphisms. In particular, the kernel of the localization functor Q : K(Mod_(A, d)) → Qis^-1K(Mod_(A, d)) is Ac. Moreover, the functor H^0 factors…","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nThe full subcategory $\\text{Ac}$ of $K(\\text{Mod}_{(A, \\text{d})})$\nconsisting of acyclic modules is a strictly full saturated triangulated\nsubcategory of $K(\\text{Mod}_{(A, \\text{d})})$.\nThe corresponding saturated multiplicative system\n(see Derived Categories, Lemma \\ref{derived-lemma-operations})\nof $K(\\text{Mod}_{(A, \\text{d})})$ is the class $\\text{Qis}$\nof quasi-isomorphisms. In particular, the kernel of the localization\nfunctor\n$$\nQ : K(\\text{Mod}_{(A, \\text{d})}) \\to\n\\text{Qis}^{-1}K(\\text{Mod}_{(A, \\text{d})})\n$$\nis $\\text{Ac}$. Moreover, the functor $H^0$ factors through $Q$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KW","source_file":"dga.tex","source_line":2550,"source_end_line":2566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2550-L2566","statement_sha256":"0c745b9c266af981bc349128290f4e77b26be04709eec1047a6f54ffb59dd5aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4718,"rank":4718,"depth":15,"x":2461.215,"y":335.036,"cluster":"homological-algebra"},{"id":"stacks:09KX","tag":"09KX","title":"The derived category · Definition 09KX","summary":"Let (A, d) be a differential graded algebra. Let Ac and Qis be as in Lemma [Tag 09KW]. The derived category of (A, d) is the triangulated category D(A, d) = K(Mod_(A, d))/Ac = Qis^-1K(Mod_(A, d)). We denote H^0 : D(A, d) → Mod_R the unique functor whose composition with the quotient functor gives back the functor H^0 defined above.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $\\text{Ac}$ and $\\text{Qis}$ be as in Lemma \\ref{lemma-acyclic}.\nThe {\\it derived category of $(A, \\text{d})$} is the triangulated\ncategory\n$$\nD(A, \\text{d}) =\nK(\\text{Mod}_{(A, \\text{d})})/\\text{Ac} =\n\\text{Qis}^{-1}K(\\text{Mod}_{(A, \\text{d})}).\n$$\nWe denote $H^0 : D(A, \\text{d}) \\to \\text{Mod}_R$ the unique functor\nwhose composition with the quotient functor gives back the functor\n$H^0$ defined above.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KX","source_file":"dga.tex","source_line":2612,"source_end_line":2626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2612-L2626","statement_sha256":"72a5597a5906a57de8a92a17c89ad440b0b7256dff40e7b8de49e71f7be506b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4719,"rank":4719,"depth":16,"x":2401.236,"y":124.385,"cluster":"homological-algebra"},{"id":"stacks:09KY","tag":"09KY","title":"The derived category · Lemma 09KY","summary":"Let (A, d) be a differential graded algebra. Let M and N be differential graded A-modules. • Let P → M be a P-resolution as in Lemma [Tag 09KP]. Then Hom_D(A, d)(M, N) = Hom_K(Mod_(A, d))(P, N) • Let N → I be an I-resolution as in Lemma [Tag 09KU]. Then Hom_D(A, d)(M, N) = Hom_K(Mod_(A, d))(M, I)","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nLet $M$ and $N$ be differential graded $A$-modules.\n\\begin{enumerate}\n\\item Let $P \\to M$ be a P-resolution as in\nLemma \\ref{lemma-resolve}. Then\n$$\n\\Hom_{D(A, \\text{d})}(M, N) =\n\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(P, N)\n$$\n\\item Let $N \\to I$ be an I-resolution as in\nLemma \\ref{lemma-right-resolution}. Then\n$$\n\\Hom_{D(A, \\text{d})}(M, N) =\n\\Hom_{K(\\text{Mod}_{(A, \\text{d})})}(M, I)\n$$\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09KY","source_file":"dga.tex","source_line":2632,"source_end_line":2650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2632-L2650","statement_sha256":"8690c425fe2427b76ab0c29253ae21987ce31a2885b295627eef54946cc9e8ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":4720,"rank":4720,"depth":13,"x":2615.153,"y":245.847,"cluster":"homological-algebra"},{"id":"stacks:09QI","tag":"09QI","title":"The derived category · Lemma 09QI","summary":"Let (A, d) be a differential graded algebra. Then • D(A, d) has both direct sums and products, • direct sums are obtained by taking direct sums of differential graded modules, • products are obtained by taking products of differential graded modules.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Then\n\\begin{enumerate}\n\\item $D(A, \\text{d})$ has both direct sums and products,\n\\item direct sums are obtained by taking direct sums of differential graded\nmodules,\n\\item products are obtained by taking products of differential\ngraded modules.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QI","source_file":"dga.tex","source_line":2677,"source_end_line":2687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2677-L2687","statement_sha256":"10fc16babc6bfe740f74a335bcec177bb0f29ce94afe11bb8c3dcf9dc1a03648","origin":"The Stacks Project","memory_eligible":false,"source_rank":4721,"rank":4721,"depth":14,"x":2359.392,"y":277.707,"cluster":"homological-algebra"},{"id":"stacks:09L0","tag":"09L0","title":"The canonical delta-functor · Lemma 09L0","summary":"Let (A, d) be a differential graded algebra. The functor Mod_(A, d) → D(A, d) defined has the natural structure of a δ-functor, with δ_K → L → M = - p ∘ q^-1 with p and q as explained above.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. The functor\n$\\text{Mod}_{(A, \\text{d})} \\to D(A, \\text{d})$\ndefined has the natural structure of a $\\delta$-functor, with\n$$\n\\delta_{K \\to L \\to M} = - p \\circ q^{-1}\n$$\nwith $p$ and $q$ as explained above.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The canonical delta-functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09L0","source_file":"dga.tex","source_line":2804,"source_end_line":2813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2804-L2813","statement_sha256":"86fdc4d19ba2db198e687d816cb4e72159988709fc779e07323dbd4c047c79ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":4722,"rank":4722,"depth":2,"x":2522.585,"y":108.862,"cluster":"homological-algebra"},{"id":"stacks:0CRL","tag":"0CRL","title":"The canonical delta-functor · Lemma 0CRL","summary":"Let (A, d) be a differential graded algebra. Let M_n be a system of differential graded modules. Then the derived colimit hocolim M_n in D(A, d) is represented by the differential graded module colim M_n.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Let\n$M_n$ be a system of differential graded modules. Then the derived\ncolimit $\\text{hocolim} M_n$ in $D(A, \\text{d})$ is represented\nby the differential graded module $\\colim M_n$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"The canonical delta-functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRL","source_file":"dga.tex","source_line":2825,"source_end_line":2831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2825-L2831","statement_sha256":"29289cd49ee2a14827367f3cd161726f801245bc0721b60a0bf3c62dba8ff887","origin":"The Stacks Project","memory_eligible":false,"source_rank":4723,"rank":4723,"depth":15,"x":2538.05,"y":326.257,"cluster":"homological-algebra"},{"id":"stacks:09MJ","tag":"09MJ","title":"Linear categories · Definition 09MJ","summary":"Let R be a ring. An R-linear category A is a category where every morphism set is given the structure of an R-module and where for x, y, z ∈ Ob(A) composition law Hom_A(y, z) × Hom_A(x, y) → Hom_A(x, z) is R-bilinear.","statement_latex":"Let $R$ be a ring. An {\\it $R$-linear category $\\mathcal{A}$} is a category\nwhere every morphism set is given the structure of an $R$-module\nand where for $x, y, z \\in \\Ob(\\mathcal{A})$ composition law\n$$\n\\Hom_\\mathcal{A}(y, z) \\times \\Hom_\\mathcal{A}(x, y)\n\\longrightarrow\n\\Hom_\\mathcal{A}(x, z)\n$$\nis $R$-bilinear.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Linear categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MJ","source_file":"dga.tex","source_line":2863,"source_end_line":2874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2863-L2874","statement_sha256":"167dcfd870da3500872846fd3f35e0f6143cc94fe9a46e5fd07878056ced98d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4724,"rank":4724,"depth":0,"x":2351.574,"y":174.526,"cluster":"homological-algebra"},{"id":"stacks:09MK","tag":"09MK","title":"Linear categories · Definition 09MK","summary":"Let R be a ring. A functor of R-linear categories, or an R-linear functor is a functor F : A → B where for all objects x, y of A the map F : Hom_A(x, y) → Hom_B(F(x), F(y)) is a homomorphism of R-modules.","statement_latex":"Let $R$ be a ring. A {\\it functor of $R$-linear categories}, or an\n{\\it $R$-linear functor} is a functor $F : \\mathcal{A} \\to \\mathcal{B}$\nwhere for all objects $x, y$ of $\\mathcal{A}$ the map\n$F : \\Hom_\\mathcal{A}(x, y) \\to \\Hom_\\mathcal{B}(F(x), F(y))$\nis a homomorphism of $R$-modules.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Linear categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MK","source_file":"dga.tex","source_line":2886,"source_end_line":2893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2886-L2893","statement_sha256":"02d58b8b7df2d3c1d90bd06a2e85aa070b5ecb35cd01996909444dbd7ad26c47","origin":"The Stacks Project","memory_eligible":false,"source_rank":4725,"rank":4725,"depth":0,"x":2611.444,"y":180.608,"cluster":"homological-algebra"},{"id":"stacks:09L2","tag":"09L2","title":"Graded categories · Definition 09L2","summary":"Let R be a ring. A graded category A over R is a category where every morphism set is given the structure of a graded R-module and where for x, y, z ∈ Ob(A) composition is R-bilinear and induces a homomorphism Hom_A(y, z) ⊗_R Hom_A(x, y) → Hom_A(x, z) of graded R-modules (i.e., preserving degrees).","statement_latex":"Let $R$ be a ring. A {\\it graded category $\\mathcal{A}$\nover $R$} is a category where every morphism set is given the structure\nof a graded $R$-module and where for\n$x, y, z \\in \\Ob(\\mathcal{A})$ composition is $R$-bilinear and induces\na homomorphism\n$$\n\\Hom_\\mathcal{A}(y, z) \\otimes_R \\Hom_\\mathcal{A}(x, y)\n\\longrightarrow\n\\Hom_\\mathcal{A}(x, z)\n$$\nof graded $R$-modules (i.e., preserving degrees).","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Graded categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09L2","source_file":"dga.tex","source_line":2907,"source_end_line":2920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2907-L2920","statement_sha256":"56eb76589ba05c91a803c2d8a2cbfe5b03e52980899d196b121dff45c79acc3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4726,"rank":4726,"depth":0,"x":2414.667,"y":323.77,"cluster":"homological-algebra"},{"id":"stacks:09L3","tag":"09L3","title":"Graded categories · Definition 09L3","summary":"Let R be a ring. A functor of graded categories over R, or a graded functor is a functor F : A → B where for all objects x, y of A the map F : Hom_A(x, y) → Hom_A(F(x), F(y)) is a homomorphism of graded R-modules.","statement_latex":"Let $R$ be a ring. A {\\it functor of graded categories over $R$}, or a\n{\\it graded functor}\nis a functor $F : \\mathcal{A} \\to \\mathcal{B}$ where for all objects\n$x, y$ of $\\mathcal{A}$ the map\n$F : \\Hom_\\mathcal{A}(x, y) \\to \\Hom_\\mathcal{A}(F(x), F(y))$\nis a homomorphism of graded $R$-modules.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Graded categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09L3","source_file":"dga.tex","source_line":2931,"source_end_line":2939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2931-L2939","statement_sha256":"58e77af454e8ff2e2dce7800369ce5fdd4077412e7aa17d41d224ccb1804aaff","origin":"The Stacks Project","memory_eligible":false,"source_rank":4727,"rank":4727,"depth":0,"x":2444.68,"y":106.26,"cluster":"homological-algebra"},{"id":"stacks:09ML","tag":"09ML","title":"Graded categories · Definition 09ML","summary":"Let R be a ring. Let A be a graded category over R. We let A^0 be the category with the same objects as A and with Hom_A^0(x, y) = Hom^0_A(x, y) the degree 0 graded piece of the graded module of morphisms of A.","statement_latex":"Let $R$ be a ring. Let $\\mathcal{A}$ be a graded category\nover $R$. We let {\\it $\\mathcal{A}^0$} be the category with the\nsame objects as $\\mathcal{A}$ and with\n$$\n\\Hom_{\\mathcal{A}^0}(x, y) = \\Hom^0_\\mathcal{A}(x, y)\n$$\nthe degree $0$ graded piece of the graded module of morphisms of\n$\\mathcal{A}$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Graded categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ML","source_file":"dga.tex","source_line":2946,"source_end_line":2956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2946-L2956","statement_sha256":"d02538b0f884eef91b29e4b50245178c1d9f5a2acb7364ca52b4728928031e5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4728,"rank":4728,"depth":0,"x":2597.667,"y":283.914,"cluster":"homological-algebra"},{"id":"stacks:09P2","tag":"09P2","title":"Graded categories · Definition 09P2","summary":"Let R be a ring. Let A be a graded category over R. A direct sum (x, y, z, i, j, p, q) in A (notation as in Homology, Remark [Tag 0103]) is a graded direct sum if i, j, p, q are homogeneous of degree 0.","statement_latex":"Let $R$ be a ring. Let $\\mathcal{A}$ be a graded category over $R$.\nA direct sum $(x, y, z, i, j, p, q)$ in $\\mathcal{A}$ (notation as in\nHomology, Remark \\ref{homology-remark-direct-sum})\nis a {\\it graded direct sum} if $i, j, p, q$ are homogeneous\nof degree $0$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Graded categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09P2","source_file":"dga.tex","source_line":2958,"source_end_line":2965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L2958-L2965","statement_sha256":"21b8f2f1a2f3005d9c8f5d64bccf0a76b80e16326fdc46ff251c1ea5494b9987","origin":"The Stacks Project","memory_eligible":false,"source_rank":4729,"rank":4729,"depth":0,"x":2341.654,"y":239.664,"cluster":"homological-algebra"},{"id":"stacks:09L5","tag":"09L5","title":"Differential graded categories · Definition 09L5","summary":"Let R be a ring. A differential graded category A over R is a category where every morphism set is given the structure of a differential graded R-module and where for x, y, z ∈ Ob(A) composition is R-bilinear and induces a homomorphism Hom_A(y, z) ⊗_R Hom_A(x, y) → Hom_A(x, z) of differential graded R-modules.","statement_latex":"Let $R$ be a ring. A {\\it differential graded category $\\mathcal{A}$\nover $R$} is a category where every morphism set is given the structure\nof a differential graded $R$-module and where for\n$x, y, z \\in \\Ob(\\mathcal{A})$ composition is $R$-bilinear and induces\na homomorphism\n$$\n\\Hom_\\mathcal{A}(y, z) \\otimes_R \\Hom_\\mathcal{A}(x, y)\n\\longrightarrow\n\\Hom_\\mathcal{A}(x, z)\n$$\nof differential graded $R$-modules.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09L5","source_file":"dga.tex","source_line":3104,"source_end_line":3117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3104-L3117","statement_sha256":"a842660435bf5c1d22e80e67dbabe05e12778e82e0f91d9ae2ead029d5f4d09f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4730,"rank":4730,"depth":0,"x":2566.314,"y":126.877,"cluster":"homological-algebra"},{"id":"stacks:09L6","tag":"09L6","title":"Differential graded categories · Definition 09L6","summary":"Let R be a ring. A functor of differential graded categories over R is a functor F : A → B where for all objects x, y of A the map F : Hom_A(x, y) → Hom_A(F(x), F(y)) is a homomorphism of differential graded R-modules.","statement_latex":"Let $R$ be a ring. A {\\it functor of differential graded categories over $R$}\nis a functor $F : \\mathcal{A} \\to \\mathcal{B}$ where for all objects\n$x, y$ of $\\mathcal{A}$ the map\n$F : \\Hom_\\mathcal{A}(x, y) \\to \\Hom_\\mathcal{A}(F(x), F(y))$\nis a homomorphism of differential graded $R$-modules.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09L6","source_file":"dga.tex","source_line":3131,"source_end_line":3138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3131-L3138","statement_sha256":"abc1899a43a2d9f4135dcfc4b449fea51e82cce5ce1d5e90ed2ee88d861948d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4731,"rank":4731,"depth":0,"x":2491.255,"y":337.799,"cluster":"homological-algebra"},{"id":"stacks:09L7","tag":"09L7","title":"Differential graded categories · Definition 09L7","summary":"Let R be a ring. Let A be a differential graded category over R. Then we let • the category of complexes of A be the category Comp(A) whose objects are the same as the objects of A and with Hom_Comp(A)(x, y) = Ker(d : Hom^0_A(x, y) → Hom^1_A(x, y)) • the homotopy category of A be the category K(A) whose objects are the same as the objects of A and with Hom_K(A)(x, y) = H^0(Hom_A(x, y))","statement_latex":"Let $R$ be a ring. Let $\\mathcal{A}$ be a differential graded category\nover $R$. Then we let\n\\begin{enumerate}\n\\item the {\\it category of complexes of $\\mathcal{A}$}\\footnote{This may\nbe nonstandard terminology.} be the category\n$\\text{Comp}(\\mathcal{A})$ whose objects are the same as the objects\nof $\\mathcal{A}$ and with\n$$\n\\Hom_{\\text{Comp}(\\mathcal{A})}(x, y) =\n\\Ker(d : \\Hom^0_\\mathcal{A}(x, y) \\to \\Hom^1_\\mathcal{A}(x, y))\n$$\n\\item the {\\it homotopy category of $\\mathcal{A}$} be the category\n$K(\\mathcal{A})$ whose objects are the same as the objects\nof $\\mathcal{A}$ and with\n$$\n\\Hom_{K(\\mathcal{A})}(x, y) = H^0(\\Hom_\\mathcal{A}(x, y))\n$$\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09L7","source_file":"dga.tex","source_line":3145,"source_end_line":3165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3145-L3165","statement_sha256":"1c8a34e3cfce1dc055ac21155bc711cd7d0b339e4bb805fa0acad02b57a75cdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4732,"rank":4732,"depth":0,"x":2376.836,"y":139.418,"cluster":"homological-algebra"},{"id":"stacks:09P4","tag":"09P4","title":"Differential graded categories · Definition 09P4","summary":"Let R be a ring. Let A be a differential graded category over R. A direct sum (x, y, z, i, j, p, q) in A (notation as in Homology, Remark [Tag 0103]) is a differential graded direct sum if i, j, p, q are homogeneous of degree 0 and closed, i.e., d(i) = 0, etc.","statement_latex":"Let $R$ be a ring. Let $\\mathcal{A}$ be a differential graded category over\n$R$. A direct sum $(x, y, z, i, j, p, q)$ in $\\mathcal{A}$ (notation as in\nHomology, Remark \\ref{homology-remark-direct-sum})\nis a {\\it differential graded direct sum} if $i, j, p, q$ are homogeneous\nof degree $0$ and closed, i.e., $\\text{d}(i) = 0$, etc.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09P4","source_file":"dga.tex","source_line":3171,"source_end_line":3178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3171-L3178","statement_sha256":"1174edc007e1ae2b3f221530e68acb9879f18c03c0ddd44200ec7165fbe18278","origin":"The Stacks Project","memory_eligible":false,"source_rank":4733,"rank":4733,"depth":0,"x":2621.058,"y":220.883,"cluster":"homological-algebra"},{"id":"stacks:09L8","tag":"09L8","title":"Differential graded categories · Lemma 09L8","summary":"Let R be a ring. A functor F : A → B of differential graded categories over R induces functors Comp(A) → Comp(B) and K(A) → K(B).","statement_latex":"Let $R$ be a ring. A functor $F : \\mathcal{A} \\to \\mathcal{B}$\nof differential graded categories over $R$ induces functors\n$\\text{Comp}(\\mathcal{A}) \\to \\text{Comp}(\\mathcal{B})$\nand $K(\\mathcal{A}) \\to K(\\mathcal{B})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09L8","source_file":"dga.tex","source_line":3180,"source_end_line":3186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3180-L3186","statement_sha256":"62854a7d7b859f1912abfa1bdb16401f8183d3138ba1030906e0f87e865ae4a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4734,"rank":4734,"depth":0,"x":2375.14,"y":299.492,"cluster":"homological-algebra"},{"id":"stacks:09LB","tag":"09LB","title":"Differential graded categories · Lemma 09LB","summary":"Let F : B → B' be an additive functor between additive categories. Then F induces a functor of differential graded categories F : Comp^dg(B) → Comp^dg(B') of Example [Tag 09L9] inducing the usual functors on the category of complexes and the homotopy categories.","statement_latex":"Let $F : \\mathcal{B} \\to \\mathcal{B}'$ be an additive functor between\nadditive categories. Then $F$ induces a functor of differential\ngraded categories\n$$\nF : \\text{Comp}^{dg}(\\mathcal{B}) \\to \\text{Comp}^{dg}(\\mathcal{B}')\n$$\nof Example \\ref{example-category-complexes}\ninducing the usual functors on the category of complexes and the\nhomotopy categories.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LB","source_file":"dga.tex","source_line":3290,"source_end_line":3301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3290-L3301","statement_sha256":"5fa8506b107983ad8e62613bdab5c33721334b90afb3f4fa30a6f7b5f11433dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4735,"rank":4735,"depth":0,"x":2493.415,"y":101.73,"cluster":"homological-algebra"},{"id":"stacks:09LD","tag":"09LD","title":"Differential graded categories · Lemma 09LD","summary":"Let φ : (A, d) → (E, d) be a homomorphism of differential graded algebras. Then φ induces a functor of differential graded categories F : Mod^dg_(E, d) → Mod^dg_(A, d) of Example [Tag 09LC] inducing obvious restriction functors on the categories of differential graded modules and homotopy categories.","statement_latex":"Let $\\varphi : (A, \\text{d}) \\to (E, \\text{d})$ be a homomorphism of\ndifferential graded algebras. Then $\\varphi$ induces a functor of differential\ngraded categories\n$$\nF :\n\\text{Mod}^{dg}_{(E, \\text{d})}\n\\longrightarrow\n\\text{Mod}^{dg}_{(A, \\text{d})}\n$$\nof Example \\ref{example-dgm-dg-cat} inducing obvious restriction functors\non the categories of differential graded modules and homotopy categories.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LD","source_file":"dga.tex","source_line":3392,"source_end_line":3405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3392-L3405","statement_sha256":"038c8255bfee3a7e2d802878874d65671971813e3e45aaeb16a8746badfd0c21","origin":"The Stacks Project","memory_eligible":false,"source_rank":4736,"rank":4736,"depth":0,"x":2565.328,"y":314.945,"cluster":"homological-algebra"},{"id":"stacks:09LE","tag":"09LE","title":"Differential graded categories · Lemma 09LE","summary":"Let R be a ring. Let A be a differential graded category over R. Let x be an object of A. Let (E, d) = Hom_A(x, x) be the differential graded R-algebra of endomorphisms of x. We obtain a functor A → Mod^dg_(E, d), y ↦ Hom_A(x, y) of differential graded categories by letting E act on Hom_A(x, y) via composition in A. This functor induces functors Comp(A) → Mod_(A, d) and K(A) → K(Mod_(A, d)) by an application of Lemma [Tag 09L8].","statement_latex":"Let $R$ be a ring. Let $\\mathcal{A}$ be a differential graded category\nover $R$. Let $x$ be an object of $\\mathcal{A}$. Let\n$$\n(E, \\text{d}) = \\Hom_\\mathcal{A}(x, x)\n$$\nbe the differential graded $R$-algebra of endomorphisms of $x$.\nWe obtain a functor\n$$\n\\mathcal{A} \\longrightarrow \\text{Mod}^{dg}_{(E, \\text{d})},\\quad\ny \\longmapsto \\Hom_\\mathcal{A}(x, y)\n$$\nof differential graded categories by letting $E$ act on\n$\\Hom_\\mathcal{A}(x, y)$ via composition in $\\mathcal{A}$.\nThis functor induces functors\n$$\n\\text{Comp}(\\mathcal{A}) \\to \\text{Mod}_{(A, \\text{d})}\n\\quad\\text{and}\\quad\nK(\\mathcal{A}) \\to K(\\text{Mod}_{(A, \\text{d})})\n$$\nby an application of Lemma \\ref{lemma-functorial}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Differential graded categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LE","source_file":"dga.tex","source_line":3411,"source_end_line":3433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3411-L3433","statement_sha256":"d1054f100fcef51bbbc5ee813ec013d1b9f9ec9e618a9f019bdd025b76869beb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4737,"rank":4737,"depth":1,"x":2340.549,"y":198.378,"cluster":"homological-algebra"},{"id":"stacks:09P6","tag":"09P6","title":"Obtaining triangulated categories · Lemma 09P6","summary":"Let A be a differential graded category satisfying axioms (A) and (B). Given an admissible short exact sequence x → y → z we obtain (see proof) a triangle x → y → z → x[1] in Comp(A) with the property that any two compositions in z[-1] → x → y → z → x[1] are zero in K(A).","statement_latex":"Let $\\mathcal{A}$ be a differential graded category satisfying\naxioms (A) and (B). Given an admissible short exact sequence\n$x \\to y \\to z$ we obtain (see proof) a triangle\n$$\nx \\to y \\to z \\to x[1]\n$$\nin $\\text{Comp}(\\mathcal{A})$ with the property that any two compositions\nin $z[-1] \\to x \\to y \\to z \\to x[1]$ are zero in $K(\\mathcal{A})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09P6","source_file":"dga.tex","source_line":3490,"source_end_line":3500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3490-L3500","statement_sha256":"af2ee7a3a856bcfb1cc6913ad29df958acc9861fd8b41c9f2a08bca6cad1b6f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4738,"rank":4738,"depth":0,"x":2600.372,"y":156.734,"cluster":"homological-algebra"},{"id":"stacks:09P7","tag":"09P7","title":"Obtaining triangulated categories · Lemma 09P7","summary":"The homotopy category is a triangulated category. This lemma proves a part of the axioms of a triangulated category. In Situation [Tag 09QJ] suppose that xymatrix x_1 ar[r]_f_1 ar[d]_a & y_1 ar[d]^b x_2 ar[r]^f_2 & y_2 is a diagram of Comp(A) commutative up to homotopy. Then there exists a morphism c : c(f_1) → c(f_2) which gives rise to a morphism of triangles (a, b, c) : (x_1, y_1, c(f_1)) → (x_1, y_1, c(f_1)) in K(A).","statement_latex":"\\begin{slogan}\nThe homotopy category is a triangulated category.\nThis lemma proves a part of the axioms of a triangulated category.\n\\end{slogan}\nIn Situation \\ref{situation-ABC} suppose that\n$$\n\\xymatrix{\nx_1 \\ar[r]_{f_1} \\ar[d]_a & y_1 \\ar[d]^b \\\\\nx_2 \\ar[r]^{f_2} & y_2\n}\n$$\nis a diagram of $\\text{Comp}(\\mathcal{A})$ commutative up to homotopy.\nThen there exists a morphism $c : c(f_1) \\to c(f_2)$ which gives rise to\na morphism of triangles\n$$\n(a, b, c) : (x_1, y_1, c(f_1)) \\to (x_1, y_1, c(f_1))\n$$\nin $K(\\mathcal{A})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09P7","source_file":"dga.tex","source_line":3561,"source_end_line":3581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3561-L3581","statement_sha256":"d54eb09ab82e3f2a3ecc552d336f7eca6bc194af5abce9e6801d5d6d04809520","origin":"The Stacks Project","memory_eligible":false,"source_rank":4739,"rank":4739,"depth":1,"x":2442.068,"y":335.101,"cluster":"homological-algebra"},{"id":"stacks:09QK","tag":"09QK","title":"Obtaining triangulated categories · Lemma 09QK","summary":"In Situation [Tag 09QJ] given any object x of A, and the cone C(1_x) of the identity morphism 1_x : x → x, the identity morphism on C(1_x) is homotopic to zero.","statement_latex":"In Situation \\ref{situation-ABC}\ngiven any object $x$ of $\\mathcal{A}$, and the cone $C(1_x)$ of the\nidentity morphism $1_x : x \\to x$, the identity morphism on\n$C(1_x)$ is homotopic to zero.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QK","source_file":"dga.tex","source_line":3649,"source_end_line":3655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3649-L3655","statement_sha256":"1c722629a5bad8091737bdf095b0dd996b141f39d765a9ac7447be96516f09e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4740,"rank":4740,"depth":1,"x":2415.327,"y":113.466,"cluster":"homological-algebra"},{"id":"stacks:09QL","tag":"09QL","title":"Obtaining triangulated categories · Lemma 09QL","summary":"In Situation [Tag 09QJ] given a diagram xymatrixxar[r]^far[d]_a & yar[d]^b zar[r]^g & w in Comp(A) commuting up to homotopy. Then • If f is an admissible monomorphism, then b is homotopic to a morphism b' which makes the diagram commute. • If g is an admissible epimorphism, then a is homotopic to a morphism a' which makes the diagram commute.","statement_latex":"In Situation \\ref{situation-ABC} given a diagram\n$$\n\\xymatrix{x\\ar[r]^f\\ar[d]_a & y\\ar[d]^b\\\\\nz\\ar[r]^g & w}\n$$\nin $\\text{Comp}(\\mathcal{A})$ commuting up to homotopy. Then\n\\begin{enumerate}\n\\item If $f$ is an admissible monomorphism, then $b$ is homotopic\nto a morphism $b'$ which makes the diagram commute.\n\\item If $g$ is an admissible epimorphism, then $a$ is homotopic\nto a morphism $a'$ which makes the diagram commute.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QL","source_file":"dga.tex","source_line":3683,"source_end_line":3697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3683-L3697","statement_sha256":"05fd61fab4eea10b31f46c92b88a2e055630f6e93a81d4e0d63a5b716b79876e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4741,"rank":4741,"depth":0,"x":2613.531,"y":261.913,"cluster":"homological-algebra"},{"id":"stacks:09QM","tag":"09QM","title":"Obtaining triangulated categories · Lemma 09QM","summary":"In Situation [Tag 09QJ] let α : x → y be a morphism in Comp(A). Then there exists a factorization in Comp(A): xymatrix x ar[r]^tildeα & tildey ar@<0.5ex>[r]^π & yar@<0.5ex>[l]^s such that • tildeα is an admissible monomorphism, and πtildeα=α. • There exists a morphism s:y→tildey in Comp(A) such that π s=1_y and sπ is homotopic to 1_tildey.","statement_latex":"In Situation \\ref{situation-ABC} let $\\alpha : x \\to y$\nbe a morphism in $\\text{Comp}(\\mathcal{A})$. Then there exists\na factorization in $\\text{Comp}(\\mathcal{A})$:\n$$\n\\xymatrix{\nx \\ar[r]^{\\tilde{\\alpha}}  &\n\\tilde{y} \\ar@<0.5ex>[r]^{\\pi} &\ny\\ar@<0.5ex>[l]^s\n}\n$$\nsuch that\n\\begin{enumerate}\n\\item $\\tilde{\\alpha}$ is an admissible monomorphism, and\n$\\pi\\tilde{\\alpha}=\\alpha$.\n\\item There exists a morphism\n$s:y\\to\\tilde{y}$ in $\\text{Comp}(\\mathcal{A})$\nsuch that $\\pi s=1_y$ and $s\\pi$ is homotopic to $1_{\\tilde{y}}$. \n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QM","source_file":"dga.tex","source_line":3720,"source_end_line":3740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3720-L3740","statement_sha256":"3dbf011dc5fd8593c942cc7672d4a7f37222b3a2a0816ec5c94d654d4de65257","origin":"The Stacks Project","memory_eligible":false,"source_rank":4742,"rank":4742,"depth":2,"x":2347.663,"y":264.92,"cluster":"homological-algebra"},{"id":"stacks:09QN","tag":"09QN","title":"Obtaining triangulated categories · Lemma 09QN","summary":"In Situation [Tag 09QJ] let x_1 → x_2 → … → x_n be a sequence of composable morphisms in Comp(A). Then there exists a commutative diagram in Comp(A): xymatrixx_1ar[r] & x_2ar[r] & …ar[r] & x_n y_1ar[r]ar[u] & y_2ar[r]ar[u] & …ar[r] & y_nar[u] such that each y_i→ y_i+1 is an admissible monomorphism and each y_i→ x_i is a homotopy equivalence.","statement_latex":"In Situation \\ref{situation-ABC}\nlet $x_1 \\to x_2 \\to \\ldots \\to x_n$\nbe a sequence of composable morphisms in $\\text{Comp}(\\mathcal{A})$.\nThen there exists a commutative diagram in $\\text{Comp}(\\mathcal{A})$:\n$$\n\\xymatrix{x_1\\ar[r] & x_2\\ar[r] & \\ldots\\ar[r] & x_n\\\\\ny_1\\ar[r]\\ar[u] & y_2\\ar[r]\\ar[u] & \\ldots\\ar[r] & y_n\\ar[u]}\n$$\nsuch that each $y_i\\to y_{i+1}$ is an admissible monomorphism\nand each $y_i\\to x_i$ is a homotopy equivalence.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QN","source_file":"dga.tex","source_line":3804,"source_end_line":3816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3804-L3816","statement_sha256":"687885c60e9e1ab6842c9bde983ab3028312b12eb26c9db3b2733ae01af94500","origin":"The Stacks Project","memory_eligible":false,"source_rank":4743,"rank":4743,"depth":3,"x":2541.538,"y":111.648,"cluster":"homological-algebra"},{"id":"stacks:09QP","tag":"09QP","title":"Obtaining triangulated categories · Lemma 09QP","summary":"In Situation [Tag 09QJ] let x_i → y_i → z_i be morphisms in A (i=1,2,3) such that x_2 → y_2→ z_2 is an admissible short exact sequence. Let b : y_1 → y_2 and b' : y_2→ y_3 be morphisms in Comp(A) such that vcenter xymatrix x_1 ar[d]_0 ar[r] & y_1 ar[r] ar[d]_b & z_1 ar[d]_0 x_2 ar[r] & y_2 ar[r] & z_2 and vcenter xymatrix x_2 ar[d]^0 ar[r] & y_2 ar[r] ar[d]^b' & z_2 ar[d]^0 x_3 ar[r] & y_3 ar[r] & z_3 commute up to homotopy. Then b'∘ b is homotopic to 0.","statement_latex":"In Situation \\ref{situation-ABC} let $x_i \\to y_i \\to z_i$\nbe morphisms in $\\mathcal{A}$ ($i=1,2,3$) such that\n$x_2 \\to y_2\\to z_2$ is an admissible short exact sequence.\nLet $b : y_1 \\to y_2$ and $b' : y_2\\to y_3$ be morphisms\nin $\\text{Comp}(\\mathcal{A})$ such that\n$$\n\\vcenter{\n\\xymatrix{\nx_1 \\ar[d]_0 \\ar[r] &\ny_1 \\ar[r] \\ar[d]_b &\nz_1 \\ar[d]_0 \\\\\nx_2 \\ar[r] & y_2 \\ar[r] & z_2\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nx_2 \\ar[d]^0 \\ar[r] &\ny_2 \\ar[r] \\ar[d]^{b'} &\nz_2 \\ar[d]^0 \\\\\nx_3 \\ar[r] & y_3 \\ar[r] & z_3\n}\n}\n$$\ncommute up to homotopy. Then $b'\\circ b$ is homotopic to $0$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QP","source_file":"dga.tex","source_line":3832,"source_end_line":3859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3832-L3859","statement_sha256":"5bb4874cbce8c9b57c521969d9bb91c6a038b2f4e353c5fab66672673dc49abb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4744,"rank":4744,"depth":1,"x":2521.809,"y":334.961,"cluster":"homological-algebra"},{"id":"stacks:09QQ","tag":"09QQ","title":"Obtaining triangulated categories · Lemma 09QQ","summary":"In Situation [Tag 09QJ] let 0 → x → y → z → 0 be an admissible short exact sequence in Comp(A). The triangle xymatrixxar[r] & yar[r] & zar[r]^δ & x[1] with δ : z → x[1] as defined in Lemma [Tag 09P6] is up to canonical isomorphism in K(A), independent of the choices made in Lemma [Tag 09P6].","statement_latex":"In Situation \\ref{situation-ABC}\nlet $0 \\to x \\to y \\to z \\to 0$ be an admissible short\nexact sequence in $\\text{Comp}(\\mathcal{A})$. The triangle\n$$\n\\xymatrix{x\\ar[r] & y\\ar[r] & z\\ar[r]^{\\delta} & x[1]}\n$$\nwith $\\delta : z \\to x[1]$ as defined in Lemma \\ref{lemma-get-triangle}\nis up to canonical isomorphism in $K(\\mathcal{A})$, independent of the\nchoices made in Lemma \\ref{lemma-get-triangle}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QQ","source_file":"dga.tex","source_line":3887,"source_end_line":3898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3887-L3898","statement_sha256":"8866fafee7171d702b168784b944fc2f884c1a3540afd39336330082159e0d1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4745,"rank":4745,"depth":1,"x":2356.566,"y":158.876,"cluster":"homological-algebra"},{"id":"stacks:09QR","tag":"09QR","title":"Obtaining triangulated categories · Lemma 09QR","summary":"In Situation [Tag 09QJ] let f: x → y be a morphism in Comp(A). The triangle (y, c(f), x[1], i, p, f[1]) is the triangle associated to the admissible short exact sequence xymatrixyar[r] & c(f) ar[r] & x[1] where the cone c(f) is defined as in Lemma [Tag 09P6].","statement_latex":"In Situation \\ref{situation-ABC}\nlet $f: x \\to y$ be a morphism in $\\text{Comp}(\\mathcal{A})$.\nThe triangle $(y, c(f), x[1], i, p, f[1])$ is the triangle associated\nto the admissible short exact sequence \n$$\n\\xymatrix{y\\ar[r] & c(f) \\ar[r] & x[1]}\n$$\nwhere the cone $c(f)$ is defined as in Lemma \\ref{lemma-get-triangle}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QR","source_file":"dga.tex","source_line":3930,"source_end_line":3940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3930-L3940","statement_sha256":"928e56e8d4d047c890173b1b60fa377ed15b345e67583d097b62f155e9ca2d49","origin":"The Stacks Project","memory_eligible":false,"source_rank":4746,"rank":4746,"depth":1,"x":2620.349,"y":195.001,"cluster":"homological-algebra"},{"id":"stacks:09QS","tag":"09QS","title":"Obtaining triangulated categories · Lemma 09QS","summary":"In Situation [Tag 09QJ] let α : x → y and β : y → z define an admissible short exact sequence xymatrix x ar[r] & yar[r] & z in Comp(A). Let (x, y, z, α, β, δ) be the associated triangle in K(A). Then, the triangles (z[-1], x, y, δ[-1], α, β) and (z[-1], x, c(δ[-1]), δ[-1], i, p) are isomorphic.","statement_latex":"In Situation \\ref{situation-ABC} let $\\alpha : x \\to y$ and $\\beta : y \\to z$\ndefine an admissible short exact sequence\n$$\n\\xymatrix{\nx \\ar[r] &\ny\\ar[r] &\nz\n}\n$$\nin $\\text{Comp}(\\mathcal{A})$. Let $(x, y, z, \\alpha, \\beta, \\delta)$\nbe the associated triangle in $K(\\mathcal{A})$. Then, the triangles\n$$\n(z[-1], x, y, \\delta[-1], \\alpha, \\beta)\n\\quad\\text{and}\\quad\n(z[-1], x, c(\\delta[-1]), \\delta[-1], i, p)\n$$\nare isomorphic.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QS","source_file":"dga.tex","source_line":3949,"source_end_line":3968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L3949-L3968","statement_sha256":"5c06583c037f6e5a7820e878dee7697cdf16691cd1582cb3b012e338458977d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4747,"rank":4747,"depth":1,"x":2396.507,"y":318.195,"cluster":"homological-algebra"},{"id":"stacks:09QT","tag":"09QT","title":"Obtaining triangulated categories · Lemma 09QT","summary":"In Situation [Tag 09QJ] let f_1 : x_1 → y_1 and f_2 : x_2 → y_2 be morphisms in Comp(A). Let (a,b,c): (x_1,y_1,c(f_1), f_1, i_1, p_1) → (x_2,y_2, c(f_2), f_2, i_1, p_1) be any morphism of triangles in K(A). If a and b are homotopy equivalences, then so is c.","statement_latex":"In Situation \\ref{situation-ABC} let $f_1 : x_1 \\to y_1$ and\n$f_2 : x_2 \\to y_2$ be morphisms in $\\text{Comp}(\\mathcal{A})$. Let \n$$\n(a,b,c): (x_1,y_1,c(f_1), f_1, i_1, p_1) \\to (x_2,y_2, c(f_2), f_2, i_1, p_1)\n$$\nbe any morphism of triangles in $K(\\mathcal{A})$.\nIf $a$ and $b$ are homotopy equivalences, then so is $c$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QT","source_file":"dga.tex","source_line":4043,"source_end_line":4052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4043-L4052","statement_sha256":"c984afba6a7d0428057c87cd5b68b934440962d1e28484dbfeb05d2f11603c45","origin":"The Stacks Project","memory_eligible":false,"source_rank":4748,"rank":4748,"depth":2,"x":2462.579,"y":100.067,"cluster":"homological-algebra"},{"id":"stacks:09QU","tag":"09QU","title":"Obtaining triangulated categories · Lemma 09QU","summary":"In Situation [Tag 09QJ]. • Given an admissible short exact sequence xxrightarrowα yxrightarrowβ z. Then there exists a homotopy equivalence e:C(α)→ z such that the diagram vcenter xymatrix xar[r]^αar[d] & yar[r]^bar[d] & C(α)ar[r]^-car@.>[d]^e & x[1]ar[d] xar[r]^α & yar[r]^β & zar[r]^δ & x[1] defines an isomorphism of triangles in K(A). Here yxrightarrowbC(α)xrightarrowcx[1] is the admissible short exact sequence given as in axiom (C). • Given a morphism α : x → y in…","statement_latex":"In Situation \\ref{situation-ABC}.\n\\begin{enumerate}\n\\item Given an admissible short exact sequence\n$x\\xrightarrow{\\alpha} y\\xrightarrow{\\beta} z$.\nThen there exists a homotopy equivalence\n$e:C(\\alpha)\\to z$ such that the diagram\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\nx\\ar[r]^{\\alpha}\\ar[d] &\ny\\ar[r]^{b}\\ar[d] &\nC(\\alpha)\\ar[r]^{-c}\\ar@{.>}[d]^{e} &\nx[1]\\ar[d] \\\\\nx\\ar[r]^{\\alpha} &\ny\\ar[r]^{\\beta} &\nz\\ar[r]^{\\delta} & x[1]\n}\n}\n\\end{equation}\ndefines an isomorphism of triangles in $K(\\mathcal{A})$. Here\n$y\\xrightarrow{b}C(\\alpha)\\xrightarrow{c}x[1]$\nis the admissible short exact sequence given as in axiom (C).\n\\item Given a morphism\n$\\alpha : x \\to y$ in $\\text{Comp}(\\mathcal{A})$, let\n$x \\xrightarrow{\\tilde{\\alpha}} \\tilde{y} \\to y$ be the\nfactorization given as in Lemma \\ref{lemma-factor}, where the admissible\nmonomorphism $x \\xrightarrow{\\tilde{\\alpha}} y$ extends to the\nadmissible short exact sequence\n$$\n\\xymatrix{\nx \\ar[r]^{\\tilde{\\alpha}} &\n\\tilde{y} \\ar[r] & z\n}\n$$\nThen there exists an isomorphism of triangles\n$$\n\\xymatrix{\nx \\ar[r]^{\\tilde{\\alpha}} \\ar[d] &\n\\tilde{y} \\ar[r] \\ar[d] &\nz \\ar[r]^{\\delta} \\ar@{.>}[d]^{e} &\nx[1] \\ar[d] \\\\\nx \\ar[r]^{\\alpha} &\ny \\ar[r] &\nC(\\alpha) \\ar[r]^{-c} &\nx[1]\n}\n$$\nwhere the upper triangle is the triangle\nassociated to the sequence\n$x \\xrightarrow{\\tilde{\\alpha}} \\tilde{y} \\to z$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QU","source_file":"dga.tex","source_line":4108,"source_end_line":4162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4108-L4162","statement_sha256":"db1be75bdd0e20ee3a56807835997db46f9ffa10ba9d9c83217221bd7f067407","origin":"The Stacks Project","memory_eligible":false,"source_rank":4749,"rank":4749,"depth":3,"x":2589.43,"y":298.65,"cluster":"homological-algebra"},{"id":"stacks:09QW","tag":"09QW","title":"Obtaining triangulated categories · Lemma 09QW","summary":"In Situation [Tag 09QJ] the homotopy category K(A) with its natural translation functors and distinguished triangles is a pre-triangulated category.","statement_latex":"In Situation \\ref{situation-ABC} the homotopy category $K(\\mathcal{A})$\nwith its natural translation functors and distinguished triangles\nis a pre-triangulated category.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QW","source_file":"dga.tex","source_line":4278,"source_end_line":4283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4278-L4283","statement_sha256":"58308d8c064fc50c4c8ebef3486cf176e1c8dcde9cf8d6c711b2260a239253b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4750,"rank":4750,"depth":4,"x":2335.88,"y":224.103,"cluster":"homological-algebra"},{"id":"stacks:09QX","tag":"09QX","title":"Obtaining triangulated categories · Lemma 09QX","summary":"In Situation [Tag 09QJ] given admissible monomorphisms x xrightarrowα y, y xrightarrowβ z in A, there exist distinguished triangles (x,y,q_1,α,p_1,δ_1), (x,z,q_2,βα,p_2,δ_2) and (y,z,q_3,β,p_3,δ_3) for which TR4 holds.","statement_latex":"In Situation \\ref{situation-ABC} given admissible monomorphisms\n$x \\xrightarrow{\\alpha} y$, $y \\xrightarrow{\\beta} z$ in $\\mathcal{A}$,\nthere exist distinguished triangles\n$(x,y,q_1,\\alpha,p_1,\\delta_1)$, $(x,z,q_2,\\beta\\alpha,p_2,\\delta_2)$\nand $(y,z,q_3,\\beta,p_3,\\delta_3)$ for which TR4 holds.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QX","source_file":"dga.tex","source_line":4359,"source_end_line":4366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4359-L4366","statement_sha256":"b2cca20581e059e919d8fa98358c857c26cff9d4236e94e5970b4dae09e434ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":4751,"rank":4751,"depth":2,"x":2583.101,"y":135.092,"cluster":"homological-algebra"},{"id":"stacks:09QY","tag":"09QY","title":"Obtaining triangulated categories · Proposition 09QY","summary":"In Situation [Tag 09QJ] the homotopy category K(A) with its natural translation functors and distinguished triangles is a triangulated category.","statement_latex":"In Situation \\ref{situation-ABC} the homotopy category $K(\\mathcal{A})$\nwith its natural translation functors and distinguished triangles is a\ntriangulated category.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QY","source_file":"dga.tex","source_line":4477,"source_end_line":4482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4477-L4482","statement_sha256":"5a3cc1878340cc312d0dc77d846437df55d8bfabeb698e1c5e550f02cdea72e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4752,"rank":4752,"depth":11,"x":2472.246,"y":341.262,"cluster":"homological-algebra"},{"id":"stacks:0FQF","tag":"0FQF","title":"Obtaining triangulated categories · Lemma 0FQF","summary":"Let R be a ring. Let F : A → B be a functor between differential graded categories over R satisfying axioms (A), (B), and (C) such that F(x[1]) = F(x)[1]. Then F induces an exact functor K(A) → K(B) of triangulated categories.","statement_latex":"Let $R$ be a ring. Let $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor\nbetween differential graded categories over $R$ satisfying axioms\n(A), (B), and (C) such that $F(x[1]) = F(x)[1]$.\nThen $F$ induces an exact functor\n$K(\\mathcal{A}) \\to K(\\mathcal{B})$ of triangulated categories.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Obtaining triangulated categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQF","source_file":"dga.tex","source_line":4493,"source_end_line":4500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4493-L4500","statement_sha256":"c1f2b0bea4f74b4584cd2b6d38afebfca2ad7f8a997c2a06fb660e72ca504c77","origin":"The Stacks Project","memory_eligible":false,"source_rank":4753,"rank":4753,"depth":1,"x":2388.088,"y":126.067,"cluster":"homological-algebra"},{"id":"stacks:0FQH","tag":"0FQH","title":"Bimodules · Definition 0FQH","summary":"Bimodules. Let R be a ring. • Let A and B be R-algebras. An (A, B)-bimodule is an R-module M equippend with R-bilinear maps A × M → M, (a, x) ↦ ax and M × B → M, (x, b) ↦ xb such that the following hold • a'(ax) = (a'a)x and (xb)b' = x(bb'), • a(xb) = (ax)b, and • 1 x = x = x 1. • Let A and B be Z-graded R-algebras. A graded (A, B)-bimodule is an (A, B)-bimodule M which has a grading M = bigoplus M^n such that A^n M^m ⊂ M^n + m and M^n B^m ⊂ M^n + m. • Let A and B be…","statement_latex":"Bimodules. Let $R$ be a ring.\n\\begin{enumerate}\n\\item Let $A$ and $B$ be $R$-algebras. An {\\it $(A, B)$-bimodule}\nis an $R$-module $M$ equippend with $R$-bilinear maps\n$$\nA \\times M \\to M, (a, x) \\mapsto ax\n\\quad\\text{and}\\quad\nM \\times B \\to M, (x, b) \\mapsto xb\n$$\nsuch that the following hold\n\\begin{enumerate}\n\\item $a'(ax) = (a'a)x$ and $(xb)b' = x(bb')$,\n\\item $a(xb) = (ax)b$, and\n\\item $1 x = x = x 1$.\n\\end{enumerate}\n\\item Let $A$ and $B$ be $\\mathbf{Z}$-graded $R$-algebras. A\n{\\it graded $(A, B)$-bimodule} is an $(A, B)$-bimodule $M$ which\nhas a grading $M = \\bigoplus M^n$ such that\n$A^n M^m \\subset M^{n + m}$ and $M^n B^m \\subset M^{n + m}$.\n\\item Let $A$ and $B$ be differential graded $R$-algebras. A\n{\\it differential graded $(A, B)$-bimodule} is a graded $(A, B)$-bimodule\nwhich comes equipped with a differential\n$\\text{d} : M \\to M$ homogeneous of degree $1$\nsuch that $\\text{d}(ax) = \\text{d}(a)x + (-1)^{\\deg(a)}a\\text{d}(x)$ and\n$\\text{d}(xb) = \\text{d}(x)b + (-1)^{\\deg(x)}x\\text{d}(b)$\nfor homogeneous elements $a \\in A$, $x \\in M$, $b \\in B$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Bimodules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQH","source_file":"dga.tex","source_line":4523,"source_end_line":4552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4523-L4552","statement_sha256":"53f9673923120657ab2357123f703a7eec9d7f41ff00de120a8943cba5184152","origin":"The Stacks Project","memory_eligible":false,"source_rank":4754,"rank":4754,"depth":0,"x":2623.49,"y":237.133,"cluster":"homological-algebra"},{"id":"stacks:0FQI","tag":"0FQI","title":"Bimodules · Lemma 0FQI","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded algebras over R. Let M be a right differential graded B-module. There is a 1-to-1 correspondence between (A, B)-bimodule structures on M compatible with the given differential graded B-module structure and homomorphisms A → Hom_Mod^dg_(B, d)(M, M) of differential graded R-algebras.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be\ndifferential graded algebras over $R$. Let $M$ be a right differential\ngraded $B$-module. There is a $1$-to-$1$ correspondence\nbetween $(A, B)$-bimodule structures on $M$ compatible with the given\ndifferential graded $B$-module structure and homomorphisms\n$$\nA\n\\longrightarrow\n\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(M, M)\n$$\nof differential graded $R$-algebras.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Bimodules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQI","source_file":"dga.tex","source_line":4562,"source_end_line":4575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4562-L4575","statement_sha256":"2938686c358e6de4bfbc393e188037f309463f40f512eb3b781f7e2fad989b66","origin":"The Stacks Project","memory_eligible":false,"source_rank":4755,"rank":4755,"depth":1,"x":2360.268,"y":288.871,"cluster":"homological-algebra"},{"id":"stacks:0FQJ","tag":"0FQJ","title":"Bimodules · Lemma 0FQJ","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded algebras over R. The construction above defines an equivalence of categories differential graded (A, B)-bimodules longleftrightarrow right differential graded A^opp ⊗_R B-modules","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$\nbe differential graded algebras over $R$. The construction above\ndefines an equivalence of categories\n$$\n\\begin{matrix}\n\\text{differential graded}\\\\\n(A, B)\\text{-bimodules}\n\\end{matrix}\n\\longleftrightarrow\n\\begin{matrix}\n\\text{right differential graded }\\\\\nA^{opp} \\otimes_R B\\text{-modules}\n\\end{matrix}\n$$","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Bimodules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQJ","source_file":"dga.tex","source_line":4606,"source_end_line":4622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4606-L4622","statement_sha256":"f574d3627f13f67d06ec8e7fad7ce6280e409f9ca61b2d278141c4916b56b75b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4756,"rank":4756,"depth":0,"x":2512.945,"y":101.131,"cluster":"homological-algebra"},{"id":"stacks:0FQK","tag":"0FQK","title":"Bimodules · Lemma 0FQK","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Let M be a differential graded (A, B)-bimodule. There exists a homomorphism P → M of differential graded (A, B)-bimodules which is a quasi-isomorphism such that P has property (P) as defined above.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be\ndifferential graded $R$-algebras. Let $M$ be a differential graded\n$(A, B)$-bimodule. There exists a homomorphism $P \\to M$\nof differential graded $(A, B)$-bimodules which is a quasi-isomorphism\nsuch that $P$ has property (P) as defined above.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Bimodules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQK","source_file":"dga.tex","source_line":4645,"source_end_line":4652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4645-L4652","statement_sha256":"213db3291a3488ea4269cdd650a7360a28dff596c4a64f140f9e6e7ddc3ea36c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4757,"rank":4757,"depth":2,"x":2551.385,"y":326.477,"cluster":"homological-algebra"},{"id":"stacks:0FQL","tag":"0FQL","title":"Bimodules · Lemma 0FQL","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Let P be a differential graded (A, B)-bimodule having property (P) with corresponding filtration F_bullet, then we obtain a short exact sequence 0 → bigoplus F_iP → bigoplus F_iP → P → 0 of differential graded (A, B)-bimodules which is split as a sequence of graded (A, B)-bimodules.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be\ndifferential graded $R$-algebras. Let $P$ be a\ndifferential graded $(A, B)$-bimodule having property (P)\nwith corresponding filtration $F_\\bullet$, then we obtain a\nshort exact sequence\n$$\n0 \\to\n\\bigoplus\\nolimits F_iP \\to\n\\bigoplus\\nolimits F_iP \\to P \\to 0\n$$\nof differential graded $(A, B)$-bimodules which is split as a sequence\nof graded $(A, B)$-bimodules.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Bimodules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQL","source_file":"dga.tex","source_line":4659,"source_end_line":4673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4659-L4673","statement_sha256":"a403fa705144c26896eea1f89f5ee00f8df40418641320d3d45f5b971e625bbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":4758,"rank":4758,"depth":1,"x":2341.567,"y":181.945,"cluster":"homological-algebra"},{"id":"stacks:09LM","tag":"09LM","title":"Bimodules and tensor product · Lemma 09LM","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded algebras over R. Let N be a differential graded (A, B)-bimodule. Then M ↦ M ⊗_A N defines a functor - ⊗_A N : Mod^dg_(A, d) → Mod^dg_(B, d) of differential graded categories. This functor induces functors Mod_(A, d) → Mod_(B, d) and K(Mod_(A, d)) → K(Mod_(B, d)) by an application of Lemma [Tag 09L8].","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$\nbe differential graded algebras over $R$. Let $N$ be a\ndifferential graded $(A, B)$-bimodule. Then\n$M \\mapsto M \\otimes_A N$ defines a functor\n$$\n- \\otimes_A N :\n\\text{Mod}^{dg}_{(A, \\text{d})}\n\\longrightarrow\n\\text{Mod}^{dg}_{(B, \\text{d})}\n$$\nof differential graded categories. This functor induces functors\n$$\n\\text{Mod}_{(A, \\text{d})} \\to \\text{Mod}_{(B, \\text{d})}\n\\quad\\text{and}\\quad\nK(\\text{Mod}_{(A, \\text{d})}) \\to K(\\text{Mod}_{(B, \\text{d})})\n$$\nby an application of Lemma \\ref{lemma-functorial}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Bimodules and tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LM","source_file":"dga.tex","source_line":4727,"source_end_line":4746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4727-L4746","statement_sha256":"8017bf59a421b9a7ae23f3e52be44b0cf6ce574f978dc89e5e0456fb287f0c3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4759,"rank":4759,"depth":1,"x":2612.843,"y":169.449,"cluster":"homological-algebra"},{"id":"stacks:0FQQ","tag":"0FQQ","title":"Bimodules and internal hom · Lemma 0FQQ","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded algebras over R. Let N be a differential graded (A, B)-bimodule. The construction above defines a functor Hom_Mod^dg_(B, d)(N, -) : Mod^dg_(B, d) → Mod^dg_(A, d) of differential graded categories. This functor induces functors Mod_(B, d) → Mod_(A, d) and K(Mod_(B, d)) → K(Mod_(A, d)) by an application of Lemma [Tag 09L8].","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$\nbe differential graded algebras over $R$. Let $N$ be a\ndifferential graded $(A, B)$-bimodule. The construction above\ndefines a functor\n$$\n\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, -) :\n\\text{Mod}^{dg}_{(B, \\text{d})}\n\\longrightarrow\n\\text{Mod}^{dg}_{(A, \\text{d})}\n$$\nof differential graded categories. This functor induces functors\n$$\n\\text{Mod}_{(B, \\text{d})} \\to \\text{Mod}_{(A, \\text{d})}\n\\quad\\text{and}\\quad\nK(\\text{Mod}_{(B, \\text{d})}) \\to K(\\text{Mod}_{(A, \\text{d})})\n$$\nby an application of Lemma \\ref{lemma-functorial}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Bimodules and internal hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQQ","source_file":"dga.tex","source_line":4878,"source_end_line":4897,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4878-L4897","statement_sha256":"60448277e4b8b9bf244d3acbf877ee340c00b8273133b04b6e2f820eed73118e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4760,"rank":4760,"depth":1,"x":2422.625,"y":332.791,"cluster":"homological-algebra"},{"id":"stacks:09LN","tag":"09LN","title":"Bimodules and internal hom · Lemma 09LN","summary":"Let R be a ring. Let A and B be R-algebras. Let M be a right A-module, N an (A, B)-bimodule, and N' a right B-module. Then we have a canonical isomorphism Hom_B(M ⊗_A N, N') = Hom_A(M, Hom_B(N, N')) of R-modules. If A, B, M, N, N' are compatibly graded, then we have a canonical isomorphism Hom_Mod_B^gr(M ⊗_A N, N') = Hom_Mod_A^gr(M, Hom_Mod_B^gr(N, N')) of graded R-modules If A, B, M, N, N' are compatibly differential graded, then we have a canonical isomorphism…","statement_latex":"Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras.\nLet $M$ be a right $A$-module, $N$ an $(A, B)$-bimodule, and\n$N'$ a right $B$-module. Then we have a canonical isomorphism\n$$\n\\Hom_B(M \\otimes_A N, N') = \\Hom_A(M, \\Hom_B(N, N'))\n$$\nof $R$-modules.\nIf $A$, $B$, $M$, $N$, $N'$ are compatibly graded, then we have a\ncanonical isomorphism\n$$\n\\Hom_{\\text{Mod}_B^{gr}}(M \\otimes_A N, N') =\n\\Hom_{\\text{Mod}_A^{gr}}(M, \\Hom_{\\text{Mod}_B^{gr}}(N, N'))\n$$\nof graded $R$-modules\nIf $A$, $B$, $M$, $N$, $N'$ are compatibly differential graded, then\nwe have a canonical isomorphism\n$$\n\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(M \\otimes_A N, N') =\n\\Hom_{\\text{Mod}^{dg}_{(A, \\text{d})}}(M,\n\\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, N'))\n$$\nof complexes of $R$-modules.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Bimodules and internal hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LN","source_file":"dga.tex","source_line":4953,"source_end_line":4977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L4953-L4977","statement_sha256":"1625d7deb6b0bb1b567d7a278c5cafcff6a8c1bb346acc80461db61cc08e6a19","origin":"The Stacks Project","memory_eligible":false,"source_rank":4761,"rank":4761,"depth":3,"x":2431.545,"y":104.134,"cluster":"homological-algebra"},{"id":"stacks:09LH","tag":"09LH","title":"Derived Hom · Lemma 09LH","summary":"The functor ([Tag 09LG]) defines an exact functor K(Mod_(B, d)) → K(Mod_(A, d)) of triangulated categories.","statement_latex":"The functor (\\ref{equation-restriction}) defines an exact functor\n$K(\\text{Mod}_{(B, \\text{d})}) \\to K(\\text{Mod}_{(A, \\text{d})})$\nof triangulated categories.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LH","source_file":"dga.tex","source_line":5015,"source_end_line":5020,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5015-L5020","statement_sha256":"b0e83b4b57e200b69114e0d825c37d1f273a5a442442ae1ffc2f9bcae720be2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4762,"rank":4762,"depth":2,"x":2609.062,"y":278.014,"cluster":"homological-algebra"},{"id":"stacks:09LI","tag":"09LI","title":"Derived Hom · Lemma 09LI","summary":"In the situation above, the right derived functor of F exists. We denote it RHom(N, -) : D(B, d) → D(A, d).","statement_latex":"In the situation above, the right derived functor of $F$ exists.\nWe denote it $R\\Hom(N, -) : D(B, \\text{d}) \\to D(A, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LI","source_file":"dga.tex","source_line":5054,"source_end_line":5058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5054-L5058","statement_sha256":"819858ec4a0b29e5d957829c7e63d2098d83e7b29c41f7d0dd2ea38294c6e672","origin":"The Stacks Project","memory_eligible":false,"source_rank":4763,"rank":4763,"depth":14,"x":2338.007,"y":250.491,"cluster":"homological-algebra"},{"id":"stacks:0BYV","tag":"0BYV","title":"Derived Hom · Lemma 0BYV","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Let f : N → N' be a homomorphism of differential graded (A, B)-bimodules. Then f induces a morphism of functors - ∘ f : RHom(N', -) → RHom(N, -) If f is a quasi-isomorphism, then f ∘ - is an isomorphism of functors.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be\ndifferential graded $R$-algebras. Let $f : N \\to N'$ be a\nhomomorphism of differential graded $(A, B)$-bimodules.\nThen $f$ induces a morphism of functors\n$$\n- \\circ f : R\\Hom(N', -) \\longrightarrow R\\Hom(N, -)\n$$\nIf $f$ is a quasi-isomorphism, then $f \\circ -$ is an isomorphism of\nfunctors.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYV","source_file":"dga.tex","source_line":5071,"source_end_line":5082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5071-L5082","statement_sha256":"19b7c24f81940745ed5ad9ecc13e823fac3f13ef9b532fd51013fe82acaaa650","origin":"The Stacks Project","memory_eligible":false,"source_rank":4764,"rank":4764,"depth":17,"x":2560.28,"y":116.822,"cluster":"homological-algebra"},{"id":"stacks:0CS5","tag":"0CS5","title":"Derived Hom · Lemma 0CS5","summary":"Let (A, d) and (B, d) be differential graded algebras over a ring R. Let N be a differential graded (A, B)-bimodule. Then for every n ∈ Z there are isomorphisms H^n(RHom(N, M)) = Ext^n_D(B, d)(N, M) of R-modules functorial in M. It is also functorial in N with respect to the operation described in Lemma [Tag 0BYV].","statement_latex":"Let $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded algebras\nover a ring $R$. Let $N$ be a differential graded $(A, B)$-bimodule.\nThen for every $n \\in \\mathbf{Z}$ there are isomorphisms\n$$\nH^n(R\\Hom(N, M)) = \\Ext^n_{D(B, \\text{d})}(N, M)\n$$\nof $R$-modules functorial in $M$. It is also functorial in $N$\nwith respect to the operation described in\nLemma \\ref{lemma-functoriality-derived-restriction}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CS5","source_file":"dga.tex","source_line":5136,"source_end_line":5147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5136-L5147","statement_sha256":"557c4d8b237bac6294734cac2e16cf8d0ce196aedb5cb3ded94c4ffab3493317","origin":"The Stacks Project","memory_eligible":false,"source_rank":4765,"rank":4765,"depth":18,"x":2503.805,"y":341.781,"cluster":"homological-algebra"},{"id":"stacks:0BYW","tag":"0BYW","title":"Derived Hom · Lemma 0BYW","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Let N be a differential graded (A, B)-bimodule. If Hom_D(B, d)(N, N') = Hom_K(Mod_(B, d))(N, N') for all N' ∈ K(B, d), for example if N has property (P) as a differential graded B-module, then RHom(N, M) = Hom_Mod^dg_(B, d)(N, M) functorially in M in D(B, d).","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be\ndifferential graded $R$-algebras. Let $N$ be a differential\ngraded $(A, B)$-bimodule. If\n$\\Hom_{D(B, \\text{d})}(N, N') = \\Hom_{K(\\text{Mod}_{(B, \\text{d})})}(N, N')$\nfor all $N' \\in K(B, \\text{d})$, for example if $N$\nhas property (P) as a differential graded $B$-module, then\n$$\nR\\Hom(N, M) = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(N, M)\n$$\nfunctorially in $M$ in $D(B, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYW","source_file":"dga.tex","source_line":5164,"source_end_line":5176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5164-L5176","statement_sha256":"19a351cff43fa3c27594e61e96d80b3191411a9e7e82bce7f1f9e4ae78b3b849","origin":"The Stacks Project","memory_eligible":false,"source_rank":4766,"rank":4766,"depth":15,"x":2364.376,"y":143.616,"cluster":"homological-algebra"},{"id":"stacks:09LK","tag":"09LK","title":"Variant of derived Hom · Lemma 09LK","summary":"In the situation above. If the right derived functor RHom(K^bullet, -) of Hom(K^bullet, -) : K(A) → D(Ab) is everywhere defined on D(A), then we obtain a canonical exact functor RHom(K^bullet, -) : D(A) → D(E, d) of triangulated categories which reduces to the usual one on taking associated complexes of abelian groups.","statement_latex":"In the situation above. If the right derived functor $R\\Hom(K^\\bullet, -)$\nof $\\Hom(K^\\bullet, -) : K(\\mathcal{A}) \\to D(\\textit{Ab})$\nis everywhere defined on $D(\\mathcal{A})$, then we obtain a canonical exact\nfunctor\n$$\nR\\Hom(K^\\bullet, -) : D(\\mathcal{A}) \\longrightarrow D(E, \\text{d})\n$$\nof triangulated categories which reduces to the usual one  on taking\nassociated complexes of abelian groups.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Variant of derived Hom","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LK","source_file":"dga.tex","source_line":5219,"source_end_line":5230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5219-L5230","statement_sha256":"f767bb64077c38200bcbe2a022d26ff70e988ac467fa8520646ade90e54f5050","origin":"The Stacks Project","memory_eligible":false,"source_rank":4767,"rank":4767,"depth":2,"x":2626.861,"y":210.705,"cluster":"homological-algebra"},{"id":"stacks:09LR","tag":"09LR","title":"Derived tensor product · Lemma 09LR","summary":"The functor ([Tag 09LQ]) defines an exact functor of triangulated categories K(Mod_(A, d)) → K(Mod_(B, d)).","statement_latex":"The functor (\\ref{equation-bc}) defines an exact functor\nof triangulated categories\n$K(\\text{Mod}_{(A, \\text{d})}) \\to K(\\text{Mod}_{(B, \\text{d})})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LR","source_file":"dga.tex","source_line":5290,"source_end_line":5295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5290-L5295","statement_sha256":"2ec594ee1cc462fa516d88aa0db7fbc26a58e54c17c3272ee93c792b03c4f3d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4768,"rank":4768,"depth":2,"x":2379.062,"y":310.294,"cluster":"homological-algebra"},{"id":"stacks:09LS","tag":"09LS","title":"Derived tensor product · Lemma 09LS","summary":"In the situation above, the left derived functor of F exists. We denote it - ⊗_A^L N : D(A, d) → D(B, d).","statement_latex":"In the situation above, the left derived functor of $F$ exists.\nWe denote it\n$- \\otimes_A^\\mathbf{L} N : D(A, \\text{d}) \\to D(B, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LS","source_file":"dga.tex","source_line":5323,"source_end_line":5328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5323-L5328","statement_sha256":"20b38a2784916e1293fd350eb6050238ff8d2273b5e11c96b5daf7882c8b2deb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4769,"rank":4769,"depth":14,"x":2481.821,"y":95.996,"cluster":"homological-algebra"},{"id":"stacks:09S3","tag":"09S3","title":"Derived tensor product · Lemma 09S3","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Let f : N → N' be a homomorphism of differential graded (A, B)-bimodules. Then f induces a morphism of functors 1⊗ f : - ⊗_A^L N → - ⊗_A^L N' If f is a quasi-isomorphism, then 1 ⊗ f is an isomorphism of functors.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be\ndifferential graded $R$-algebras. Let $f : N \\to N'$ be a\nhomomorphism of differential graded $(A, B)$-bimodules.\nThen $f$ induces a morphism of functors\n$$\n1\\otimes f :\n- \\otimes_A^\\mathbf{L} N\n\\longrightarrow\n- \\otimes_A^\\mathbf{L} N'\n$$\nIf $f$ is a quasi-isomorphism, then $1 \\otimes f$ is an isomorphism of\nfunctors.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09S3","source_file":"dga.tex","source_line":5341,"source_end_line":5355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5341-L5355","statement_sha256":"9394aced72a854096e559e19170d84ae1600966b000416b4701f9efdc5be60d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4770,"rank":4770,"depth":15,"x":2578.495,"y":312.581,"cluster":"homological-algebra"},{"id":"stacks:0GZ2","tag":"0GZ2","title":"Derived tensor product · Lemma 0GZ2","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Let N be a differential graded (A, B)-bimodule which has property (P) as a left differential graded A-module. Then M ⊗_A^L N is computed by M ⊗_A N for all differential graded A-modules M.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be\ndifferential graded $R$-algebras. Let $N$ be a differential graded\n$(A, B)$-bimodule which has property (P) as a left differential graded\n$A$-module. Then $M \\otimes_A^\\mathbf{L} N$ is computed by\n$M \\otimes_A N$ for all differential graded $A$-modules $M$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZ2","source_file":"dga.tex","source_line":5392,"source_end_line":5399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5392-L5399","statement_sha256":"6cb23f19e83ff82d97be418cd993a984115cc7cd2e3dd02f9951e68379364810","origin":"The Stacks Project","memory_eligible":false,"source_rank":4771,"rank":4771,"depth":16,"x":2332.746,"y":207.606,"cluster":"homological-algebra"},{"id":"stacks:09LT","tag":"09LT","title":"Derived tensor product · Lemma 09LT","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Let N be a differential graded (A, B)-bimodule. Then the functor - ⊗_A^L N : D(A, d) → D(B, d) of Lemma [Tag 09LS] is a left adjoint to the functor RHom(N, -) : D(B, d) → D(A, d) of Lemma [Tag 09LI].","statement_latex":"Let $R$ be a ring.\nLet $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras.\nLet $N$ be a differential graded $(A, B)$-bimodule.\nThen the functor\n$$\n- \\otimes_A^\\mathbf{L} N : D(A, \\text{d}) \\longrightarrow D(B, \\text{d})\n$$\nof Lemma \\ref{lemma-derived-bc} is a left adjoint to the functor\n$$\nR\\Hom(N, -) : D(B, \\text{d}) \\longrightarrow D(A, \\text{d})\n$$\nof Lemma \\ref{lemma-derived-restriction}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LT","source_file":"dga.tex","source_line":5415,"source_end_line":5429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5415-L5429","statement_sha256":"d6219d49da7e0bb3d5eadacdc80ea6c358f0a00c92f829aed170c353090c9cb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4772,"rank":4772,"depth":15,"x":2598.69,"y":145.496,"cluster":"homological-algebra"},{"id":"stacks:09R9","tag":"09R9","title":"Derived tensor product · Lemma 09R9","summary":"With notation and assumptions as in Lemma [Tag 09LT]. Assume • N defines a compact object of D(B, d), and • the map H^k(A) → Hom_D(B, d)(N, N[k]) is an isomorphism for all k ∈ Z. Then the functor -⊗_A^L N is fully faithful.","statement_latex":"With notation and assumptions as in Lemma \\ref{lemma-tensor-hom-adjoint}.\nAssume\n\\begin{enumerate}\n\\item $N$ defines a compact object of $D(B, \\text{d})$, and\n\\item the map $H^k(A) \\to \\Hom_{D(B, \\text{d})}(N, N[k])$ is an\nisomorphism for all $k \\in \\mathbf{Z}$.\n\\end{enumerate}\nThen the functor $-\\otimes_A^\\mathbf{L} N$ is fully faithful.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09R9","source_file":"dga.tex","source_line":5468,"source_end_line":5478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5468-L5478","statement_sha256":"036d4b0c02edeb3405bd3e743b383f8f9e0b6f0fa9ddabe13849b441ae39abe8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4773,"rank":4773,"depth":19,"x":2452.361,"y":342.429,"cluster":"homological-algebra"},{"id":"stacks:0BYZ","tag":"0BYZ","title":"Derived tensor product · Lemma 0BYZ","summary":"Let R → R' be a ring map. Let (A, d) be a differential graded R-algebra. Let (A', d) be the base change, i.e., A' = A ⊗_R R'. If A is K-flat as a complex of R-modules, then • - ⊗_A^L A' : D(A, d) → D(A', d) is equal to the right derived functor of K(A, d) → K(A', d), M ↦ M ⊗_R R' • the diagram xymatrix D(A, d) ar[r]_- ⊗_A^L A' ar[d]_restriction & D(A', d) ar[d]^restriction D(R) ar[r]^- ⊗_R^L R' & D(R') commutes, and • if M is K-flat as a complex of R-modules, then the…","statement_latex":"Let $R \\to R'$ be a ring map. Let $(A, \\text{d})$ be a differential\ngraded $R$-algebra. Let $(A', \\text{d})$ be the base change, i.e.,\n$A' = A \\otimes_R R'$. If $A$ is K-flat as a complex of $R$-modules,\nthen\n\\begin{enumerate}\n\\item $- \\otimes_A^\\mathbf{L} A' : D(A, \\text{d}) \\to D(A', \\text{d})$\nis equal to the right derived functor of\n$$\nK(A, \\text{d}) \\longrightarrow K(A', \\text{d}),\\quad\nM \\longmapsto M \\otimes_R R'\n$$\n\\item the diagram\n$$\n\\xymatrix{\nD(A, \\text{d}) \\ar[r]_{- \\otimes_A^\\mathbf{L} A'} \\ar[d]_{restriction} &\nD(A', \\text{d}) \\ar[d]^{restriction} \\\\\nD(R) \\ar[r]^{- \\otimes_R^\\mathbf{L} R'} & D(R')\n}\n$$\ncommutes, and\n\\item if $M$ is K-flat as a complex of $R$-modules, then the\ndifferential graded $A'$-module $M \\otimes_R R'$ represents\n$M \\otimes_A^\\mathbf{L} A'$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYZ","source_file":"dga.tex","source_line":5503,"source_end_line":5529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5503-L5529","statement_sha256":"0273ccdc40c899b5cca900f90697a43cf654ea8f433c2fabb1ee57081bf42d17","origin":"The Stacks Project","memory_eligible":false,"source_rank":4774,"rank":4774,"depth":15,"x":2401.834,"y":113.915,"cluster":"homological-algebra"},{"id":"stacks:0BZ0","tag":"0BZ0","title":"Derived tensor product · Lemma 0BZ0","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Let T be a differential graded (A, B)-bimodule. Assume • T defines a compact object of D(B, d), and • S = Hom_Mod^dg_(B, d)(T, B) represents RHom(T, B) in D(A, d). Then S has a structure of a differential graded (B, A)-bimodule and there is an isomorphism N ⊗_B^L S → RHom(T, N) functorial in N in D(B, d).","statement_latex":"Let $R$ be a ring.\nLet $(A, \\text{d})$ and $(B, \\text{d})$ be differential graded $R$-algebras.\nLet $T$ be a differential graded $(A, B)$-bimodule.\nAssume\n\\begin{enumerate}\n\\item $T$ defines a compact object of $D(B, \\text{d})$, and\n\\item $S = \\Hom_{\\text{Mod}^{dg}_{(B, \\text{d})}}(T, B)$\nrepresents $R\\Hom(T, B)$ in $D(A, \\text{d})$.\n\\end{enumerate}\nThen $S$ has a structure of a differential graded $(B, A)$-bimodule\nand there is an isomorphism\n$$\nN \\otimes_B^\\mathbf{L} S \\longrightarrow R\\Hom(T, N)\n$$\nfunctorial in $N$ in $D(B, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZ0","source_file":"dga.tex","source_line":5565,"source_end_line":5582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5565-L5582","statement_sha256":"66b38ce7181cb069b425f3d1497243fde707f22b53aef70253f66d6c153ae992","origin":"The Stacks Project","memory_eligible":false,"source_rank":4775,"rank":4775,"depth":1,"x":2623.117,"y":253.912,"cluster":"homological-algebra"},{"id":"stacks:0BZ3","tag":"0BZ3","title":"Composition of derived tensor products · Lemma 0BZ3","summary":"Let R be a ring. Let (A, d), (B, d), and (C, d) be differential graded R-algebras. Let N be a differential graded (A, B)-bimodule. Let N' be a differential graded (B, C)-module. Assume ([Tag 0BZ2]) is an isomorphism. Then the composition xymatrix D(A, d) ar[rr]^- ⊗_A^L N & & D(B, d) ar[rr]^- ⊗_B^L N' & & D(C, d) is isomorphic to - ⊗_A^L N\" with N\" = N ⊗_B N' viewed as (A, C)-bimodule.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and\n$(C, \\text{d})$ be differential graded $R$-algebras.\nLet $N$ be a differential graded $(A, B)$-bimodule.\nLet $N'$ be a differential graded $(B, C)$-module.\nAssume (\\ref{equation-plain-versus-derived}) is an isomorphism.\nThen the composition\n$$\n\\xymatrix{\nD(A, \\text{d}) \\ar[rr]^{- \\otimes_A^\\mathbf{L} N} & &\nD(B, \\text{d}) \\ar[rr]^{- \\otimes_B^\\mathbf{L} N'} & &\nD(C, \\text{d})\n}\n$$\nis isomorphic to $- \\otimes_A^\\mathbf{L} N''$ with\n$N'' = N \\otimes_B N'$ viewed as $(A, C)$-bimodule.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Composition of derived tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZ3","source_file":"dga.tex","source_line":5649,"source_end_line":5666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5649-L5666","statement_sha256":"77c74288081b331d1f706034e06880fa0c609d7bf6f7be96e6e958809545be1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4776,"rank":4776,"depth":15,"x":2347.042,"y":276.267,"cluster":"homological-algebra"},{"id":"stacks:0BZ5","tag":"0BZ5","title":"Composition of derived tensor products · Lemma 0BZ5","summary":"Let R be a ring. Let (A, d), (B, d), and (C, d) be differential graded R-algebras. Assume that ([Tag 0BZ4]) is an isomorphism. Let N be a differential graded (A, B)-bimodule. Let N' be a differential graded (B, C)-bimodule. Then the composition xymatrix D(A, d) ar[rr]^- ⊗_A^L N & & D(B, d) ar[rr]^- ⊗_B^L N' & & D(C, d) is isomorphic to - ⊗_A^L N\" for a differential graded (A, C)-bimodule N\" described in the proof.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and\n$(C, \\text{d})$ be differential graded $R$-algebras. Assume\nthat (\\ref{equation-plain-versus-derived-algebras}) is an isomorphism.\nLet $N$ be a differential graded $(A, B)$-bimodule.\nLet $N'$ be a differential graded $(B, C)$-bimodule.\nThen the composition\n$$\n\\xymatrix{\nD(A, \\text{d}) \\ar[rr]^{- \\otimes_A^\\mathbf{L} N} & &\nD(B, \\text{d}) \\ar[rr]^{- \\otimes_B^\\mathbf{L} N'} & &\nD(C, \\text{d})\n}\n$$\nis isomorphic to $- \\otimes_A^\\mathbf{L} N''$ for a differential graded\n$(A, C)$-bimodule $N''$ described in the proof.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Composition of derived tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZ5","source_file":"dga.tex","source_line":5741,"source_end_line":5758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5741-L5758","statement_sha256":"930840df5a1eb038fc386f3041ea06dcdb20366bc31b65f2552b56cd85152f7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4777,"rank":4777,"depth":16,"x":2532.855,"y":102.929,"cluster":"homological-algebra"},{"id":"stacks:09S4","tag":"09S4","title":"Composition of derived tensor products · Lemma 09S4","summary":"Let R be a ring. Let (A, d), (B, d), and (C, d) be differential graded R-algebras. If C is K-flat as a complex of R-modules, then ([Tag 0BZ4]) is an isomorphism and the conclusion of Lemma [Tag 0BZ5] is valid.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and\n$(C, \\text{d})$ be differential graded $R$-algebras.\nIf $C$ is K-flat as a complex of $R$-modules, then\n(\\ref{equation-plain-versus-derived-algebras})\nis an isomorphism and the conclusion of\nLemma \\ref{lemma-compose-tensor-functors-general-algebra} is valid.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Composition of derived tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09S4","source_file":"dga.tex","source_line":5849,"source_end_line":5857,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5849-L5857","statement_sha256":"8dd4f55b976a254ab3f10840f12a92d3dd37a7f1fdb898ef32388dc5a955db40","origin":"The Stacks Project","memory_eligible":false,"source_rank":4778,"rank":4778,"depth":17,"x":2535.235,"y":336.453,"cluster":"homological-algebra"},{"id":"stacks:09LV","tag":"09LV","title":"Variant of derived tensor product · Lemma 09LV","summary":"In the situation above there is a functor - ⊗_E K^bullet : Mod^dg_(E, d) → Comp^dg(O) of differential graded categories. This functor sends E to K^bullet and commutes with direct sums.","statement_latex":"In the situation above there is a functor\n$$\n- \\otimes_E K^\\bullet :\n\\text{Mod}^{dg}_{(E, \\text{d})}\n\\longrightarrow\n\\text{Comp}^{dg}(\\mathcal{O})\n$$\nof differential graded categories. This functor sends $E$ to $K^\\bullet$\nand commutes with direct sums.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Variant of derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LV","source_file":"dga.tex","source_line":5901,"source_end_line":5912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5901-L5912","statement_sha256":"4ad71403b8eea16f80b6a1bb9009fd48a22a70fc0fc85fbccba5348d5d457e00","origin":"The Stacks Project","memory_eligible":false,"source_rank":4779,"rank":4779,"depth":2,"x":2345.466,"y":165.408,"cluster":"homological-algebra"},{"id":"stacks:09LW","tag":"09LW","title":"Variant of derived tensor product · Lemma 09LW","summary":"The functor of Lemma [Tag 09LV] defines an exact functor of triangulated categories K(Mod_(E, d)) → K(O).","statement_latex":"The functor of Lemma \\ref{lemma-tensor-with-complex} defines an exact functor\nof triangulated categories\n$K(\\text{Mod}_{(E, \\text{d})}) \\to K(\\mathcal{O})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Variant of derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LW","source_file":"dga.tex","source_line":5938,"source_end_line":5943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5938-L5943","statement_sha256":"c8a3a168ecfd33de1d3cc751f2a56d643df7d06ff134e9e8c239be3ce6c964fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4780,"rank":4780,"depth":3,"x":2623.27,"y":183.876,"cluster":"homological-algebra"},{"id":"stacks:09LX","tag":"09LX","title":"Variant of derived tensor product · Lemma 09LX","summary":"The functor K(Mod_(E, d)) → K(O) of Lemma [Tag 09LW] has a left derived version defined on all of D(E, d). We denote it - ⊗_E^L K^bullet : D(E, d) → D(O).","statement_latex":"The functor $K(\\text{Mod}_{(E, \\text{d})}) \\to K(\\mathcal{O})$\nof Lemma \\ref{lemma-tensor-with-complex-homotopy} has a left derived\nversion defined on all of $D(E, \\text{d})$. We denote it\n$- \\otimes_E^\\mathbf{L} K^\\bullet : D(E, \\text{d}) \\to D(\\mathcal{O})$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Variant of derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LX","source_file":"dga.tex","source_line":5966,"source_end_line":5972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5966-L5972","statement_sha256":"3f9a9dcf62e1c48d5d11bd7e6d77b4e1ac20fdf0bcc2494c5d3e3a6279f70e51","origin":"The Stacks Project","memory_eligible":false,"source_rank":4781,"rank":4781,"depth":14,"x":2403.316,"y":328.057,"cluster":"homological-algebra"},{"id":"stacks:0CS6","tag":"0CS6","title":"Variant of derived tensor product · Lemma 0CS6","summary":"Let R be a ring. Let C be a site. Let O be a sheaf of commutative R-algebras. Let K^bullet be a complex of O-modules. The functor of Lemma [Tag 09LX] has the following property: For every M, N in D(E, d) there is a canonical map RHom(M, N) → RHom_O(M ⊗_E^L K^bullet, N ⊗_E^L K^bullet) in D(R) which on cohomology modules gives the maps Ext^n_D(E, d)(M, N) → Ext^n_D(O) (M ⊗_E^L K^bullet, N ⊗_E^L K^bullet) induced by the functor - ⊗_E^L K^bullet.","statement_latex":"Let $R$ be a ring. Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$\nbe a sheaf of commutative $R$-algebras. Let $K^\\bullet$\nbe a complex of $\\mathcal{O}$-modules.\nThe functor\nof Lemma \\ref{lemma-tensor-with-complex-derived} has the following\nproperty: For every $M$, $N$ in $D(E, \\text{d})$ there is a\ncanonical map\n$$\nR\\Hom(M, N)\n\\longrightarrow\nR\\Hom_\\mathcal{O}(M \\otimes_E^\\mathbf{L} K^\\bullet,\nN \\otimes_E^\\mathbf{L} K^\\bullet)\n$$\nin $D(R)$ which on cohomology modules gives the maps\n$$\n\\Ext^n_{D(E, \\text{d})}(M, N) \\to\n\\Ext^n_{D(\\mathcal{O})}\n(M \\otimes_E^\\mathbf{L} K^\\bullet, N \\otimes_E^\\mathbf{L} K^\\bullet)\n$$\ninduced by the functor $- \\otimes_E^\\mathbf{L} K^\\bullet$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Variant of derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CS6","source_file":"dga.tex","source_line":5985,"source_end_line":6007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L5985-L6007","statement_sha256":"ae6e9cdeadc378f3203bfbbd19af4a7eb200e1cada9a1c00a4d9fe0af9794414","origin":"The Stacks Project","memory_eligible":false,"source_rank":4782,"rank":4782,"depth":18,"x":2449.614,"y":96.666,"cluster":"homological-algebra"},{"id":"stacks:09LY","tag":"09LY","title":"Variant of derived tensor product · Lemma 09LY","summary":"Let (C, O) be a ringed site. Let K^bullet be a complex of O-modules. Then the functor - ⊗_E^L K^bullet : D(E, d) → D(O) of Lemma [Tag 09LX] is a left adjoint of the functor RHom(K^bullet, -) : D(O) → D(E, d) of Lemma [Tag 09LK].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $K^\\bullet$ be a complex of $\\mathcal{O}$-modules.\nThen the functor\n$$\n- \\otimes_E^\\mathbf{L} K^\\bullet :\nD(E, \\text{d})\n\\longrightarrow\nD(\\mathcal{O})\n$$\nof Lemma \\ref{lemma-tensor-with-complex-derived} is a left adjoint\nof the functor\n$$\nR\\Hom(K^\\bullet, -) : D(\\mathcal{O}) \\longrightarrow D(E, \\text{d})\n$$\nof Lemma \\ref{lemma-existence-of-derived}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Variant of derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LY","source_file":"dga.tex","source_line":6059,"source_end_line":6076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6059-L6076","statement_sha256":"69dd1757560452643881e5261a3705a3c6c447b7ebc38e23ac9095078df516a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4783,"rank":4783,"depth":15,"x":2601.728,"y":293.787,"cluster":"homological-algebra"},{"id":"stacks:09LZ","tag":"09LZ","title":"Variant of derived tensor product · Lemma 09LZ","summary":"Let (C, O) be a ringed site. Let K^bullet be a complex of O-modules. Assume • K^bullet represents a compact object of D(O), and • E = Hom_Comp^dg(O)(K^bullet, K^bullet) computes the ext groups of K^bullet in D(O). Then the functor - ⊗_E^L K^bullet : D(E, d) → D(O) of Lemma [Tag 09LX] is fully faithful.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $K^\\bullet$ be a complex of $\\mathcal{O}$-modules.\nAssume\n\\begin{enumerate}\n\\item $K^\\bullet$ represents a compact object of $D(\\mathcal{O})$, and\n\\item $E = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O})}(K^\\bullet, K^\\bullet)$\ncomputes the ext groups of $K^\\bullet$ in $D(\\mathcal{O})$.\n\\end{enumerate}\nThen the functor\n$$\n- \\otimes_E^\\mathbf{L} K^\\bullet :\nD(E, \\text{d})\n\\longrightarrow\nD(\\mathcal{O})\n$$\nof Lemma \\ref{lemma-tensor-with-complex-derived} is fully faithful.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Variant of derived tensor product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09LZ","source_file":"dga.tex","source_line":6121,"source_end_line":6139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6121-L6139","statement_sha256":"90b53539f902d9a6f480079e1b7d8428cd4eb040c0da6e679e3c6a6631f0cdcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4784,"rank":4784,"depth":16,"x":2330.731,"y":234.678,"cluster":"homological-algebra"},{"id":"stacks:09R2","tag":"09R2","title":"Characterizing compact objects · Lemma 09R2","summary":"Let (A, d) be a differential graded algebra. Let E be a compact object of D(A, d). Let P be a differential graded A-module which has a finite filtration 0 = F_-1P ⊂ F_0P ⊂ F_1P ⊂ … ⊂ F_nP = P by differential graded submodules such that F_i + 1P/F_iP ≅ bigoplus_j ∈ J_i A[k_i, j] as differential graded A-modules for some sets J_i and integers k_i, j. Let E → P be a morphism of D(A, d). Then there exists a differential graded submodule P' ⊂ P such that F_i + 1P ∩ P'/(F_iP ∩…","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Let $E$ be a compact\nobject of $D(A, \\text{d})$. Let $P$ be a differential graded $A$-module\nwhich has a finite filtration\n$$\n0 = F_{-1}P \\subset F_0P \\subset F_1P \\subset \\ldots \\subset F_nP = P\n$$\nby differential graded submodules such that\n$$\nF_{i + 1}P/F_iP \\cong \\bigoplus\\nolimits_{j \\in J_i} A[k_{i, j}]\n$$\nas differential graded $A$-modules for some sets $J_i$ and integers $k_{i, j}$.\nLet $E \\to P$ be a morphism of $D(A, \\text{d})$.\nThen there exists a differential graded submodule $P' \\subset P$ such that\n$F_{i + 1}P \\cap P'/(F_iP \\cap P')$ is equal to\n$\\bigoplus_{j \\in J'_i} A[k_{i, j}]$ for some finite subsets\n$J'_i \\subset J_i$ and such that $E \\to P$ factors through $P'$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Characterizing compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09R2","source_file":"dga.tex","source_line":6216,"source_end_line":6234,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6216-L6234","statement_sha256":"1330915e94fb5fbe2aea2f94e55a0da043aaa691d62f12d36d681cc9ac56221d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4785,"rank":4785,"depth":0,"x":2578.377,"y":124.368,"cluster":"homological-algebra"},{"id":"stacks:09R3","tag":"09R3","title":"Characterizing compact objects · Proposition 09R3","summary":"Let (A, d) be a differential graded algebra. Let E be an object of D(A, d). Then the following are equivalent • E is a compact object, • E is a direct summand of an object of D(A, d) which is represented by a differential graded module P which has a finite filtration F_bullet by differential graded submodules such that F_iP/F_i - 1P are finite direct sums of shifts of A.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Let $E$ be an\nobject of $D(A, \\text{d})$. Then the following are equivalent\n\\begin{enumerate}\n\\item $E$ is a compact object,\n\\item $E$ is a direct summand of an object of $D(A, \\text{d})$\nwhich is represented by a differential graded module $P$ which\nhas a finite filtration $F_\\bullet$ by differential graded submodules\nsuch that $F_iP/F_{i - 1}P$ are finite direct sums of shifts of $A$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Characterizing compact objects","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09R3","source_file":"dga.tex","source_line":6285,"source_end_line":6296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6285-L6296","statement_sha256":"74ae80bfe238ec03db927886ec7667d5cfcbfaadc056b9a241e3934d7adfad0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4786,"rank":4786,"depth":15,"x":2484.368,"y":346.483,"cluster":"homological-algebra"},{"id":"stacks:09RA","tag":"09RA","title":"Characterizing compact objects · Lemma 09RA","summary":"Let (A, d) be a differential graded algebra. For every compact object E of D(A, d) there exist integers a ≤ b such that Hom_D(A, d)(E, M) = 0 if H^i(M) = 0 for i ∈ [a, b].","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra.\nFor every compact object $E$ of $D(A, \\text{d})$ there\nexist integers $a \\leq b$ such that $\\Hom_{D(A, \\text{d})}(E, M) = 0$\nif $H^i(M) = 0$ for $i \\in [a, b]$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Characterizing compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RA","source_file":"dga.tex","source_line":6346,"source_end_line":6352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6346-L6352","statement_sha256":"532f4fc76747d3a2ca05ec57457d269b19926106e42950682d004207997957ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":4787,"rank":4787,"depth":16,"x":2374.945,"y":129.108,"cluster":"homological-algebra"},{"id":"stacks:09RB","tag":"09RB","title":"Characterizing compact objects · Lemma 09RB","summary":"Let (A, d) be a differential graded algebra. Assume that A^n = 0 for |n| gg 0. Let E be an object of D(A, d). The following are equivalent • E is a compact object, and • E can be represented by a differential graded A-module P which is finite projective as a graded A-module and satisfies Hom_K(A, d)(P, M) = Hom_D(A, d)(P, M) for every differential graded A-module M.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra. Assume that $A^n = 0$\nfor $|n| \\gg 0$. Let $E$ be an object of $D(A, \\text{d})$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $E$ is a compact object, and\n\\item $E$ can be represented by a differential graded $A$-module $P$\nwhich is finite projective as a graded $A$-module and satisfies\n$\\Hom_{K(A, \\text{d})}(P, M) = \\Hom_{D(A, \\text{d})}(P, M)$\nfor every differential graded $A$-module $M$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Characterizing compact objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RB","source_file":"dga.tex","source_line":6373,"source_end_line":6385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6373-L6385","statement_sha256":"220855dc70303a3e8ea3f63504fc578b991d6de1b2530f9e2700ea9e5831aa9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4788,"rank":4788,"depth":17,"x":2630.73,"y":227.42,"cluster":"homological-algebra"},{"id":"stacks:09S6","tag":"09S6","title":"Equivalences of derived categories · Lemma 09S6","summary":"Let R be a ring. Let (A, d) → (B, d) be a homomorphism of differential graded algebras over R, which induces an isomorphism on cohomology algebras. Then - ⊗_A^L B : D(A, d) → D(B, d) gives an R-linear equivalence of triangulated categories with quasi-inverse the restriction functor N ↦ N_A.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d}) \\to (B, \\text{d})$ be a\nhomomorphism of differential graded algebras over $R$, which induces\nan isomorphism on cohomology algebras. Then\n$$\n- \\otimes_A^\\mathbf{L} B : D(A, \\text{d}) \\to D(B, \\text{d})\n$$\ngives an $R$-linear equivalence of triangulated categories with\nquasi-inverse the restriction functor $N \\mapsto N_A$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Equivalences of derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09S6","source_file":"dga.tex","source_line":6507,"source_end_line":6517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6507-L6517","statement_sha256":"fb17b2224b56f9feee0361df5a83709ab6f766a82c9514eb740bb181ae5a058b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4789,"rank":4789,"depth":20,"x":2362.754,"y":300.147,"cluster":"homological-algebra"},{"id":"stacks:09S7","tag":"09S7","title":"Equivalences of derived categories · Lemma 09S7","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded algebras over R. Let N be a differential graded (A, B)-bimodule. Assume that • N defines a compact object of D(B, d), • if N' ∈ D(B, d) and Hom_D(B, d)(N, N'[n]) = 0 for n ∈ Z, then N' = 0, and • the map H^k(A) → Hom_D(B, d)(N, N[k]) is an isomorphism for all k ∈ Z. Then - ⊗_A^L N : D(A, d) → D(B, d) gives an R-linear equivalence of triangulated categories.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be \ndifferential graded algebras over $R$. Let $N$ be a\ndifferential graded $(A, B)$-bimodule. Assume that\n\\begin{enumerate}\n\\item $N$ defines a compact object of $D(B, \\text{d})$,\n\\item if $N' \\in D(B, \\text{d})$ and\n$\\Hom_{D(B, \\text{d})}(N, N'[n]) = 0$ for $n \\in \\mathbf{Z}$,\nthen $N' = 0$, and\n\\item the map $H^k(A) \\to \\Hom_{D(B, \\text{d})}(N, N[k])$ is an\nisomorphism for all $k \\in \\mathbf{Z}$.\n\\end{enumerate}\nThen\n$$\n- \\otimes_A^\\mathbf{L} N : D(A, \\text{d}) \\to D(B, \\text{d})\n$$\ngives an $R$-linear equivalence of triangulated categories.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Equivalences of derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09S7","source_file":"dga.tex","source_line":6535,"source_end_line":6553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6535-L6553","statement_sha256":"dcf27515fa800152a1f374b18226c4bffe49a6b8b810961d1e1ddbb8fbffdbdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4790,"rank":4790,"depth":20,"x":2502.028,"y":94.231,"cluster":"homological-algebra"},{"id":"stacks:09S8","tag":"09S8","title":"Equivalences of derived categories · Lemma 09S8","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Assume that A = H^0(A). The following are equivalent • D(A, d) and D(B, d) are equivalent as R-linear triangulated categories, and • there exists an object P of D(B, d) such that • P is a compact object of D(B, d), • if N ∈ D(B, d) with Hom_D(B, d)(P, N[i]) = 0 for i ∈ Z, then N = 0, • Hom_D(B, d)(P, P[i]) = 0 for i not = 0 and equal to A for i = 0. The equivalence D(A, d) → D(B, d) constructed in…","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be\ndifferential graded $R$-algebras. Assume that $A = H^0(A)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $D(A, \\text{d})$ and $D(B, \\text{d})$ are equivalent as $R$-linear\ntriangulated categories, and\n\\item there exists an object $P$ of $D(B, \\text{d})$ such that\n\\begin{enumerate}\n\\item $P$ is a compact object of $D(B, \\text{d})$,\n\\item if $N \\in D(B, \\text{d})$ with $\\Hom_{D(B, \\text{d})}(P, N[i]) = 0$\nfor $i \\in \\mathbf{Z}$, then $N = 0$,\n\\item $\\Hom_{D(B, \\text{d})}(P, P[i]) = 0$ for $i \\not = 0$ and\nequal to $A$ for $i = 0$.\n\\end{enumerate}\n\\end{enumerate}\nThe equivalence $D(A, \\text{d}) \\to D(B, \\text{d})$\nconstructed in (2) sends $A$ to $P$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Equivalences of derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09S8","source_file":"dga.tex","source_line":6584,"source_end_line":6603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6584-L6603","statement_sha256":"5a6a299093491b595b8e91b8b592db07cb1cfeedf7da232ac6d2bc9c70c6f5c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4791,"rank":4791,"depth":21,"x":2564.993,"y":325.358,"cluster":"homological-algebra"},{"id":"stacks:09SA","tag":"09SA","title":"Equivalences of derived categories · Proposition 09SA","summary":"Let R be a ring. Let (A, d) and (B, d) be differential graded R-algebras. Let F : D(A, d) → D(B, d) be an R-linear equivalence of triangulated categories. Assume that • A = H^0(A), and • B is K-flat as a complex of R-modules. Then there exists an (A, B)-bimodule N as in Lemma [Tag 09S7].","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$ and $(B, \\text{d})$ be\ndifferential graded $R$-algebras. Let $F : D(A, \\text{d}) \\to D(B, \\text{d})$\nbe an $R$-linear equivalence of triangulated categories. Assume that\n\\begin{enumerate}\n\\item $A = H^0(A)$, and\n\\item $B$ is K-flat as a complex of $R$-modules.\n\\end{enumerate}\nThen there exists an $(A, B)$-bimodule $N$ as in\nLemma \\ref{lemma-tilting-equivalence}.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Equivalences of derived categories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SA","source_file":"dga.tex","source_line":6685,"source_end_line":6696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6685-L6696","statement_sha256":"94704d2720ee3d078e7706a73a80bec3c97815ceee6b92d9624b92976db660b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4792,"rank":4792,"depth":21,"x":2332.436,"y":190.504,"cluster":"homological-algebra"},{"id":"stacks:09SC","tag":"09SC","title":"Equivalences of derived categories · Lemma 09SC","summary":"Let R be a ring. Let A and B be R-algebras. The following are equivalent • there is an R-linear equivalence D(A) → D(B) of triangulated categories, • there exists an object P of D(B) such that • P can be represented by a finite complex of finite projective B-modules, • if K ∈ D(B) with Ext^i_B(P, K) = 0 for i ∈ Z, then K = 0, and • Ext^i_B(P, P) = 0 for i not = 0 and equal to A for i= 0. Moreover, if B is flat as an R-module, then this is also equivalent to • [(3)] there…","statement_latex":"Let $R$ be a ring.\nLet $A$ and $B$ be $R$-algebras. The following are equivalent\n\\begin{enumerate}\n\\item there is an $R$-linear equivalence $D(A) \\to D(B)$\nof triangulated categories,\n\\item there exists an object $P$ of $D(B)$ such that\n\\begin{enumerate}\n\\item $P$ can be represented by a finite complex\nof finite projective $B$-modules,\n\\item if $K \\in D(B)$ with $\\Ext^i_B(P, K) = 0$ for\n$i \\in \\mathbf{Z}$, then $K = 0$, and\n\\item $\\Ext^i_B(P, P) = 0$ for $i \\not = 0$ and\nequal to $A$ for $i= 0$.\n\\end{enumerate}\n\\end{enumerate}\nMoreover, if $B$ is flat as an $R$-module, then this is also\nequivalent to\n\\begin{enumerate}\n\\item[(3)] there exists an $(A, B)$-bimodule $N$ such that\n$- \\otimes_A^\\mathbf{L} N : D(A) \\to D(B)$ is an equivalence.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Equivalences of derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09SC","source_file":"dga.tex","source_line":6754,"source_end_line":6777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6754-L6777","statement_sha256":"f6a94c8a4e0ac69b839db6ca675c08abb22cbe8736a8e6aa89817f3310ed8065","origin":"The Stacks Project","memory_eligible":false,"source_rank":4793,"rank":4793,"depth":22,"x":2612.681,"y":157.949,"cluster":"homological-algebra"},{"id":"stacks:0BZ7","tag":"0BZ7","title":"Resolutions of differential graded algebras · Lemma 0BZ7","summary":"Let R be a ring. Let (B, d) be a differential graded R-algebra. There exists a quasi-isomorphism (A, d) → (B, d) of differential graded R-algebras with the following properties • A is K-flat as a complex of R-modules, • A is a free graded R-algebra.","statement_latex":"Let $R$ be a ring. Let $(B, \\text{d})$ be a differential graded $R$-algebra.\nThere exists a quasi-isomorphism $(A, \\text{d}) \\to (B, \\text{d})$ of\ndifferential graded $R$-algebras with the following properties\n\\begin{enumerate}\n\\item $A$ is K-flat as a complex of $R$-modules,\n\\item $A$ is a free graded $R$-algebra.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Resolutions of differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZ7","source_file":"dga.tex","source_line":6939,"source_end_line":6948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6939-L6948","statement_sha256":"39ff4e987956be63df4b7a944b5dad5a5cf1aaedc9f2a3fc12f94ebbbcefcc3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4794,"rank":4794,"depth":5,"x":2432.01,"y":341.177,"cluster":"homological-algebra"},{"id":"stacks:0BZ8","tag":"0BZ8","title":"Resolutions of differential graded algebras · Lemma 0BZ8","summary":"Let R be a ring. Let (A, d), (B, d), and (C, d) be differential graded R-algebras. Assume A ⊗_R C represents A ⊗^L_R C in D(R). Let N be a differential graded (A, B)-bimodule. Let N' be a differential graded (B, C)-bimodule. Then the composition xymatrix D(A, d) ar[rr]^- ⊗_A^L N & & D(B, d) ar[rr]^- ⊗_B^L N' & & D(C, d) is isomorphic to - ⊗_A^L N\" for some differential graded (A, C)-bimodule N\".","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d})$, $(B, \\text{d})$, and\n$(C, \\text{d})$ be differential graded $R$-algebras. Assume\n$A \\otimes_R C$ represents $A \\otimes^\\mathbf{L}_R C$ in $D(R)$.\nLet $N$ be a differential graded $(A, B)$-bimodule.\nLet $N'$ be a differential graded $(B, C)$-bimodule.\nThen the composition\n$$\n\\xymatrix{\nD(A, \\text{d}) \\ar[rr]^{- \\otimes_A^\\mathbf{L} N} & &\nD(B, \\text{d}) \\ar[rr]^{- \\otimes_B^\\mathbf{L} N'} & &\nD(C, \\text{d})\n}\n$$\nis isomorphic to $- \\otimes_A^\\mathbf{L} N''$ for some differential graded\n$(A, C)$-bimodule $N''$.","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Resolutions of differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZ8","source_file":"dga.tex","source_line":6993,"source_end_line":7010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L6993-L7010","statement_sha256":"3bc565969739d15c284e9509d715c9829481ba07a4a6fdeeebb597b7f4a59a64","origin":"The Stacks Project","memory_eligible":false,"source_rank":4795,"rank":4795,"depth":21,"x":2417.87,"y":103.284,"cluster":"homological-algebra"},{"id":"stacks:0CRM","tag":"0CRM","title":"Resolutions of differential graded algebras · Lemma 0CRM","summary":"Let (A, d) be a differential graded algebra with H^i(A) countable for each i. Let M be an object of D(A, d). Then the following are equivalent • M = hocolim E_n with E_n compact in D(A, d), and • H^i(M) is countable for each i.","statement_latex":"Let $(A, \\text{d})$ be a differential graded algebra with\n$H^i(A)$ countable for each $i$. Let $M$ be an object of $D(A, \\text{d})$.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $M = \\text{hocolim} E_n$ with $E_n$ compact in $D(A, \\text{d})$, and\n\\item $H^i(M)$ is countable for each $i$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Algebra","chapter_id":"dga","section":"Resolutions of differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRM","source_file":"dga.tex","source_line":7070,"source_end_line":7079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dga.tex#L7070-L7079","statement_sha256":"3bed77475e66efb9466e251dafd5c366957827cdc36a9e3ff9ef02e0c176fdd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4796,"rank":4796,"depth":21,"x":2619.829,"y":270.867,"cluster":"homological-algebra"},{"id":"stacks:07GL","tag":"07GL","title":"Divided powers · Definition 07GL","summary":"Let A be a ring. Let I be an ideal of A. A collection of maps γ_n : I → I, n > 0 is called a divided power structure on I if for all n ≥ 0, m > 0, x, y ∈ I, and a ∈ A we have • γ_1(x) = x, we also set γ_0(x) = 1, • γ_n(x)γ_m(x) = frac(n + m)!n! m! γ_n + m(x), • γ_n(ax) = a^n γ_n(x), • γ_n(x + y) = ∑_i = 0, …, n γ_i(x)γ_n - i(y), • γ_n(γ_m(x)) = frac(nm)!n! (m!)^n γ_nm(x).","statement_latex":"Let $A$ be a ring. Let $I$ be an ideal of $A$. A collection of maps\n$\\gamma_n : I \\to I$, $n > 0$ is called a {\\it divided power structure}\non $I$ if for all $n \\geq 0$, $m > 0$, $x, y \\in I$, and $a \\in A$ we have\n\\begin{enumerate}\n\\item $\\gamma_1(x) = x$, we also set $\\gamma_0(x) = 1$,\n\\item $\\gamma_n(x)\\gamma_m(x) = \\frac{(n + m)!}{n! m!} \\gamma_{n + m}(x)$,\n\\item $\\gamma_n(ax) = a^n \\gamma_n(x)$,\n\\item $\\gamma_n(x + y) = \\sum_{i = 0, \\ldots, n} \\gamma_i(x)\\gamma_{n - i}(y)$,\n\\item $\\gamma_n(\\gamma_m(x)) = \\frac{(nm)!}{n! (m!)^n} \\gamma_{nm}(x)$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided powers","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GL","source_file":"dpa.tex","source_line":35,"source_end_line":47,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L35-L47","statement_sha256":"70ac95fdc5e799ce1a112d9e6050467d4225d30515ed3924f9c2500507cedecf","origin":"The Stacks Project","memory_eligible":false,"source_rank":4797,"rank":4797,"depth":0,"x":2335.826,"y":261.88,"cluster":"homological-algebra"},{"id":"stacks:07GM","tag":"07GM","title":"Divided powers · Lemma 07GM","summary":"Let A be a ring. Let I be an ideal of A. • If γ is a divided power structure on I, then n! γ_n(x) = x^n for n ≥ 1, x ∈ I. Assume A is torsion free as a Z-module. • [(2)] A divided power structure on I, if it exists, is unique. • [(3)] If γ_n : I → I are maps then γ is a divided power structure ⇔ n! γ_n(x) = x^n ∀ x ∈ I, n ≥ 1. • [(4)] The ideal I has a divided power structure if and only if there exists a set of generators x_i of I as an ideal such that for all n ≥ 1 we…","statement_latex":"Let $A$ be a ring. Let $I$ be an ideal of $A$.\n\\begin{enumerate}\n\\item If $\\gamma$ is a divided power structure\\footnote{Here\nand in the following, $\\gamma$ stands short for a sequence\nof maps $\\gamma_1, \\gamma_2, \\gamma_3, \\ldots$ from $I$ to $I$.}\non $I$, then\n$n! \\gamma_n(x) = x^n$ for $n \\geq 1$, $x \\in I$.\n\\end{enumerate}\nAssume $A$ is torsion free as a $\\mathbf{Z}$-module.\n\\begin{enumerate}\n\\item[(2)] A divided power structure on $I$, if it exists, is unique.\n\\item[(3)] If $\\gamma_n : I \\to I$ are maps then\n$$\n\\gamma\\text{ is a divided power structure}\n\\Leftrightarrow\nn! \\gamma_n(x) = x^n\\ \\forall x \\in I, n \\geq 1.\n$$\n\\item[(4)] The ideal $I$ has a divided power structure\nif and only if there exists\na set of generators $x_i$ of $I$ as an ideal such that\nfor all $n \\geq 1$ we have $x_i^n \\in (n!)I$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GM","source_file":"dpa.tex","source_line":58,"source_end_line":82,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L58-L82","statement_sha256":"b62ebd30237db94c8d4a28b88deaea632c408eac66677bf50ae86b8f1b4a3372","origin":"The Stacks Project","memory_eligible":false,"source_rank":4798,"rank":4798,"depth":1,"x":2552.713,"y":107.185,"cluster":"homological-algebra"},{"id":"stacks:07GP","tag":"07GP","title":"Divided powers · Lemma 07GP","summary":"Let A be a ring. Let I be an ideal of A. Let γ_n : I → I, n ≥ 1 be a sequence of maps. Assume • [(a)] (1), (3), and (4) of Definition [Tag 07GL] hold for all x, y ∈ I, and • [(b)] properties (2) and (5) hold for x in some set of generators of I as an ideal. Then γ is a divided power structure on I.","statement_latex":"Let $A$ be a ring. Let $I$ be an ideal of $A$. Let $\\gamma_n : I \\to I$,\n$n \\geq 1$ be a sequence of maps. Assume\n\\begin{enumerate}\n\\item[(a)] (1), (3), and (4) of Definition \\ref{definition-divided-powers}\nhold for all $x, y \\in I$, and\n\\item[(b)] properties (2) and (5) hold for $x$ in\nsome set of generators of $I$ as an ideal.\n\\end{enumerate}\nThen $\\gamma$ is a divided power structure on $I$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GP","source_file":"dpa.tex","source_line":150,"source_end_line":161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L150-L161","statement_sha256":"7eb86dc0aa71ef4ba859776d48d7960fe9fa3e570b8bc7e13c9da3091edfee8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4799,"rank":4799,"depth":1,"x":2517.146,"y":344.585,"cluster":"homological-algebra"},{"id":"stacks:07GQ","tag":"07GQ","title":"Divided powers · Lemma 07GQ","summary":"Let A be a ring with two ideals I, J ⊂ A. Let γ be a divided power structure on I and let δ be a divided power structure on J. Then • γ and δ agree on IJ, • if γ and δ agree on I ∩ J then they are the restriction of a unique divided power structure ε on I + J.","statement_latex":"Let $A$ be a ring with two ideals $I, J \\subset A$.\nLet $\\gamma$ be a divided power structure on $I$ and let\n$\\delta$ be a divided power structure on $J$.\nThen\n\\begin{enumerate}\n\\item $\\gamma$ and $\\delta$ agree on $IJ$,\n\\item if $\\gamma$ and $\\delta$ agree on $I \\cap J$ then they are\nthe restriction of a unique divided power structure $\\epsilon$\non $I + J$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GQ","source_file":"dpa.tex","source_line":232,"source_end_line":244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L232-L244","statement_sha256":"3790d2e7cc5c6c97e44bf5b5f9fc692065148d7729bc13b87bcc25974191ed4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4800,"rank":4800,"depth":2,"x":2352.281,"y":149.132,"cluster":"homological-algebra"},{"id":"stacks:07GR","tag":"07GR","title":"Divided powers · Lemma 07GR","summary":"Let p be a prime number. Let A be a ring, let I ⊂ A be an ideal, and let γ be a divided power structure on I. Assume p is nilpotent in A/I. Then I is locally nilpotent if and only if p is nilpotent in A.","statement_latex":"Let $p$ be a prime number. Let $A$ be a ring, let $I \\subset A$ be an ideal,\nand let $\\gamma$ be a divided power structure on $I$. Assume $p$ is nilpotent\nin $A/I$. Then $I$ is locally nilpotent if and only if $p$ is nilpotent in $A$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GR","source_file":"dpa.tex","source_line":314,"source_end_line":319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L314-L319","statement_sha256":"6c3ac3be920c92acfe2da5bd7aedac4dc05c82448f1ff604998da1a6b78b663f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4801,"rank":4801,"depth":0,"x":2631.335,"y":199.764,"cluster":"homological-algebra"},{"id":"stacks:07GU","tag":"07GU","title":"Divided power rings · Definition 07GU","summary":"A divided power ring is a triple (A, I, γ) where A is a ring, I ⊂ A is an ideal, and γ = (γ_n)_n ≥ 1 is a divided power structure on I. A homomorphism of divided power rings φ : (A, I, γ) → (B, J, δ) is a ring homomorphism φ : A → B such that φ(I) ⊂ J and such that δ_n(φ(x)) = φ(γ_n(x)) for all x ∈ I and n ≥ 1.","statement_latex":"A {\\it divided power ring} is a triple $(A, I, \\gamma)$ where\n$A$ is a ring, $I \\subset A$ is an ideal, and $\\gamma = (\\gamma_n)_{n \\geq 1}$\nis a divided power structure on $I$.\nA {\\it homomorphism of divided power rings}\n$\\varphi : (A, I, \\gamma) \\to (B, J, \\delta)$ is a ring homomorphism\n$\\varphi : A \\to B$ such that $\\varphi(I) \\subset J$ and such that\n$\\delta_n(\\varphi(x)) = \\varphi(\\gamma_n(x))$ for all $x \\in I$ and\n$n \\geq 1$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided power rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GU","source_file":"dpa.tex","source_line":343,"source_end_line":353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L343-L353","statement_sha256":"d1f5651e2a97a9687cebc062961909ed05531446b87e9eb9fb126d1bd983ed47","origin":"The Stacks Project","memory_eligible":false,"source_rank":4802,"rank":4802,"depth":0,"x":2384.577,"y":320.904,"cluster":"homological-algebra"},{"id":"stacks:07GV","tag":"07GV","title":"Divided power rings · Lemma 07GV","summary":"The category of divided power rings has all limits and they agree with limits in the category of rings.","statement_latex":"The category of divided power rings has all limits and they agree with\nlimits in the category of rings.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided power rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GV","source_file":"dpa.tex","source_line":361,"source_end_line":365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L361-L365","statement_sha256":"c1edf5130e8f7461c4ff79e65e0b6ec742d02cb2b3d9bf1f146a2f35b0fcc3c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4803,"rank":4803,"depth":1,"x":2469.208,"y":91.308,"cluster":"homological-algebra"},{"id":"stacks:07GW","tag":"07GW","title":"Divided power rings · Lemma 07GW","summary":"Let C be the category of divided power rings. Let F : C → Sets be a functor. Assume that • there exists a cardinal kappa such that for every f ∈ F(A, I, γ) there exists a morphism (A', I', γ') → (A, I, γ) of C such that f is the image of f' ∈ F(A', I', γ') and |A'| ≤ kappa, and • F commutes with limits. Then F is representable, i.e., there exists an object (B, J, δ) of C such that F(A, I, γ) = Hom_C((B, J, δ), (A, I, γ)) functorially in (A, I, γ).","statement_latex":"Let $\\mathcal{C}$ be the category of divided power rings. Let\n$F : \\mathcal{C} \\to \\textit{Sets}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item there exists a cardinal $\\kappa$ such that for every\n$f \\in F(A, I, \\gamma)$ there exists a morphism\n$(A', I', \\gamma') \\to (A, I, \\gamma)$ of $\\mathcal{C}$ such that $f$\nis the image of $f' \\in F(A', I', \\gamma')$ and $|A'| \\leq \\kappa$, and\n\\item $F$ commutes with limits.\n\\end{enumerate}\nThen $F$ is representable, i.e., there exists an object $(B, J, \\delta)$\nof $\\mathcal{C}$ such that\n$$\nF(A, I, \\gamma) = \\Hom_\\mathcal{C}((B, J, \\delta), (A, I, \\gamma))\n$$\nfunctorially in $(A, I, \\gamma)$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided power rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GW","source_file":"dpa.tex","source_line":382,"source_end_line":400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L382-L400","statement_sha256":"f8503d21d9d93ed8ae86925b113f5adfd786df63c63cb1105a308dc9369479d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4804,"rank":4804,"depth":1,"x":2591.571,"y":308.869,"cluster":"homological-algebra"},{"id":"stacks:07GX","tag":"07GX","title":"Divided power rings · Lemma 07GX","summary":"The category of divided power rings has all colimits.","statement_latex":"The category of divided power rings has all colimits.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided power rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GX","source_file":"dpa.tex","source_line":407,"source_end_line":410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L407-L410","statement_sha256":"c6afb0321e7c5132af3a3b3df732b00a4238c10d980bea31abb55d2e38e6e1b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4805,"rank":4805,"depth":2,"x":2326.096,"y":217.775,"cluster":"homological-algebra"},{"id":"stacks:07H0","tag":"07H0","title":"Extending divided powers · Definition 07H0","summary":"Given a divided power ring (A, I, γ) and a ring map A → B we say γ extends to B if there exists a divided power structure bar γ on IB such that (A, I, γ) → (B, IB, barγ) is a homomorphism of divided power rings.","statement_latex":"Given a divided power ring $(A, I, \\gamma)$ and a ring map\n$A \\to B$ we say $\\gamma$ {\\it extends} to $B$ if there exists a\ndivided power structure $\\bar \\gamma$ on $IB$ such that\n$(A, I, \\gamma) \\to (B, IB, \\bar\\gamma)$ is a homomorphism of\ndivided power rings.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Extending divided powers","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07H0","source_file":"dpa.tex","source_line":493,"source_end_line":500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L493-L500","statement_sha256":"0e6eb1af93cc4bfcbe3ed3ac57936029cd4850a7fd5c70fbaf277d8225ae5a76","origin":"The Stacks Project","memory_eligible":false,"source_rank":4806,"rank":4806,"depth":0,"x":2595.402,"y":134.218,"cluster":"homological-algebra"},{"id":"stacks:07H1","tag":"07H1","title":"Extending divided powers · Lemma 07H1","summary":"Let (A, I, γ) be a divided power ring. Let A → B be a ring map. If γ extends to B then it extends uniquely. Assume (at least) one of the following conditions holds • IB = 0, • I is principal, or • A → B is flat. Then γ extends to B.","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring.\nLet $A \\to B$ be a ring map.\nIf $\\gamma$ extends to $B$ then it extends uniquely.\nAssume (at least) one of the following conditions holds\n\\begin{enumerate}\n\\item $IB = 0$,\n\\item $I$ is principal, or\n\\item $A \\to B$ is flat.\n\\end{enumerate}\nThen $\\gamma$ extends to $B$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Extending divided powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07H1","source_file":"dpa.tex","source_line":502,"source_end_line":514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L502-L514","statement_sha256":"821b96a669e6bf83b7283a3625183edefea9fbf7591b3dc16919b598d92b47d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4807,"rank":4807,"depth":4,"x":2463.87,"y":348.876,"cluster":"homological-algebra"},{"id":"stacks:07H2","tag":"07H2","title":"Extending divided powers · Lemma 07H2","summary":"Let (A, I, γ) be a divided power ring. • If φ : (A, I, γ) → (B, J, δ) is a homomorphism of divided power rings, then Ker(φ) ∩ I is preserved by γ_n for all n ≥ 1. • Let a ⊂ A be an ideal and set I' = I ∩ a. The following are equivalent • I' is preserved by γ_n for all n > 0, • γ extends to A/ a, and • there exist a set of generators x_i of I' as an ideal such that γ_n(x_i) ∈ I' for all n > 0.","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring.\n\\begin{enumerate}\n\\item If $\\varphi : (A, I, \\gamma) \\to (B, J, \\delta)$ is a\nhomomorphism of divided power rings, then $\\Ker(\\varphi) \\cap I$\nis preserved by $\\gamma_n$ for all $n \\geq 1$.\n\\item Let $\\mathfrak a \\subset A$ be an ideal and set\n$I' = I \\cap \\mathfrak a$. The following are equivalent\n\\begin{enumerate}\n\\item $I'$ is preserved by $\\gamma_n$ for all $n > 0$,\n\\item $\\gamma$ extends to $A/\\mathfrak a$, and\n\\item there exist a set of generators $x_i$ of $I'$ as an ideal\nsuch that $\\gamma_n(x_i) \\in I'$ for all $n > 0$.\n\\end{enumerate}\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Extending divided powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07H2","source_file":"dpa.tex","source_line":607,"source_end_line":623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L607-L623","statement_sha256":"6a8d6fab9d9648558d9eb38d72dc87cc694e8a11c5401f02e98994190ffeb2a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4808,"rank":4808,"depth":1,"x":2388.156,"y":115.704,"cluster":"homological-algebra"},{"id":"stacks:07H3","tag":"07H3","title":"Extending divided powers · Lemma 07H3","summary":"Let (A, I, γ) be a divided power ring. Let E ⊂ I be a subset. Then the smallest ideal J ⊂ I preserved by γ and containing all f ∈ E is the ideal J generated by γ_n(f), n ≥ 1, f ∈ E.","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring.\nLet $E \\subset I$ be a subset.\nThen the smallest ideal $J \\subset I$ preserved by $\\gamma$\nand containing all $f \\in E$ is the ideal $J$ generated by\n$\\gamma_n(f)$, $n \\geq 1$, $f \\in E$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Extending divided powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07H3","source_file":"dpa.tex","source_line":642,"source_end_line":649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L642-L649","statement_sha256":"ac340c3c216e86347cda885acd02c0f557d1e7b75e80ed39684c89f5bf2c954a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4809,"rank":4809,"depth":2,"x":2631.761,"y":244.819,"cluster":"homological-algebra"},{"id":"stacks:07KD","tag":"07KD","title":"Extending divided powers · Lemma 07KD","summary":"Let (A, I, γ) be a divided power ring. Let p be a prime. If p is nilpotent in A/I, then • the p-adic completion A^wedge = lim_e A/p^eA surjects onto A/I, • the kernel of this map is the p-adic completion I^wedge of I, and • each γ_n is continuous for the p-adic topology and extends to γ_n^wedge : I^wedge → I^wedge defining a divided power structure on I^wedge. If moreover A is a Z_(p)-algebra, then • [(4)] for e large enough the ideal p^eA ⊂ I is preserved by the divided…","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring. Let $p$ be a prime.\nIf $p$ is nilpotent in $A/I$, then\n\\begin{enumerate}\n\\item the $p$-adic completion $A^\\wedge = \\lim_e A/p^eA$ surjects onto $A/I$,\n\\item the kernel of this map is the $p$-adic completion $I^\\wedge$ of $I$, and\n\\item each $\\gamma_n$ is continuous for the $p$-adic topology and extends\nto $\\gamma_n^\\wedge : I^\\wedge \\to I^\\wedge$ defining a divided power\nstructure on $I^\\wedge$.\n\\end{enumerate}\nIf moreover $A$ is a $\\mathbf{Z}_{(p)}$-algebra, then\n\\begin{enumerate}\n\\item[(4)] for $e$ large enough the ideal $p^eA \\subset I$ is preserved by the\ndivided power structure $\\gamma$ and\n$$\n(A^\\wedge, I^\\wedge, \\gamma^\\wedge) = \\lim_e (A/p^eA, I/p^eA, \\bar\\gamma)\n$$\nin the category of divided power rings.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Extending divided powers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KD","source_file":"dpa.tex","source_line":655,"source_end_line":675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L655-L675","statement_sha256":"896d8b2618c1de935af96e1ff19118d731e996b2d0aad5479e24fd471b52b8f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4810,"rank":4810,"depth":4,"x":2347.993,"y":287.885,"cluster":"homological-algebra"},{"id":"stacks:07H5","tag":"07H5","title":"Divided power polynomial algebras · Lemma 07H5","summary":"Let (A, I, γ) be a divided power ring. There exists a unique divided power structure δ on J = IAlangle x_1, …, x_t rangle + Alangle x_1, …, x_t rangle_+ such that • δ_n(x_i) = x_i^[n], and • (A, I, γ) → (Alangle x_1, …, x_t rangle, J, δ) is a homomorphism of divided power rings. Moreover, (Alangle x_1, …, x_t rangle, J, δ) has the following universal property: A homomorphism of divided power rings φ : (Alangle x_1, …, x_t rangle, J, δ) → (C, K, ε) is the same thing as a…","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring.\nThere exists a unique divided power structure $\\delta$ on\n$$\nJ = IA\\langle x_1, \\ldots, x_t \\rangle + A\\langle x_1, \\ldots, x_t \\rangle_{+}\n$$\nsuch that\n\\begin{enumerate}\n\\item $\\delta_n(x_i) = x_i^{[n]}$, and\n\\item $(A, I, \\gamma) \\to (A\\langle x_1, \\ldots, x_t \\rangle, J, \\delta)$\nis a homomorphism of divided power rings.\n\\end{enumerate}\nMoreover, $(A\\langle x_1, \\ldots, x_t \\rangle, J, \\delta)$ has the\nfollowing universal property: A homomorphism of divided power rings\n$\\varphi : (A\\langle x_1, \\ldots, x_t \\rangle, J, \\delta) \\to\n(C, K, \\epsilon)$ is\nthe same thing as a homomorphism of divided power rings\n$A \\to C$ and elements $k_1, \\ldots, k_t \\in K$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided power polynomial algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07H5","source_file":"dpa.tex","source_line":733,"source_end_line":752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L733-L752","statement_sha256":"f8fd0907d9136699455b4be45d2227e7e515691f63dc251d04757e00b2b6e9c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4811,"rank":4811,"depth":5,"x":2522.795,"y":94.904,"cluster":"homological-algebra"},{"id":"stacks:07GS","tag":"07GS","title":"Divided power polynomial algebras · Lemma 07GS","summary":"Let p be a prime number. Let A be a ring such that every integer n not divisible by p is invertible, i.e., A is a Z_(p)-algebra. Let I ⊂ A be an ideal. Two divided power structures γ, γ' on I are equal if and only if γ_p = γ'_p. Moreover, given a map δ : I → I such that • p!δ(x) = x^p for all x ∈ I, • δ(ax) = a^pδ(x) for all a ∈ A, x ∈ I, and • δ(x + y) = δ(x) + ∑_i + j = p, i,j ≥ 1 frac1i!j! x^i y^j + δ(y) for all x, y ∈ I, then there exists a unique divided power…","statement_latex":"Let $p$ be a prime number. Let $A$ be a ring such that every integer $n$\nnot divisible by $p$ is invertible, i.e., $A$ is a $\\mathbf{Z}_{(p)}$-algebra.\nLet $I \\subset A$ be an ideal. Two divided power structures\n$\\gamma, \\gamma'$ on $I$ are equal if and only if $\\gamma_p = \\gamma'_p$.\nMoreover, given a map $\\delta : I \\to I$ such that\n\\begin{enumerate}\n\\item $p!\\delta(x) = x^p$ for all $x \\in I$\\footnote{This condition\nfollows from the other two, see Remark \\ref{remark-ryo-suzuki}.},\n\\item $\\delta(ax) = a^p\\delta(x)$ for all $a \\in A$, $x \\in I$, and\n\\item\n$\\delta(x + y) =\n\\delta(x) +\n\\sum\\nolimits_{i + j = p, i,j \\geq 1} \\frac{1}{i!j!} x^i y^j +\n\\delta(y)$ for all $x, y \\in I$,\n\\end{enumerate}\nthen there exists a unique divided power structure $\\gamma$ on $I$ such\nthat $\\gamma_p = \\delta$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Divided power polynomial algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07GS","source_file":"dpa.tex","source_line":824,"source_end_line":843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L824-L843","statement_sha256":"ab91bb65a561cb37de3ea51c6441ab415bfbf1b3cf58352d49b360c4b5ab2993","origin":"The Stacks Project","memory_eligible":false,"source_rank":4812,"rank":4812,"depth":1,"x":2549.117,"y":336.652,"cluster":"homological-algebra"},{"id":"stacks:09PG","tag":"09PG","title":"Tate resolutions · Definition 09PG","summary":"Let R be a ring. Let A = bigoplus_d ≥ 0 A_d be a graded R-algebra which is strictly graded commutative. A collection of maps γ_n : A_even, + → A_even, + defined for all n > 0 is called a divided power structure on A if we have • γ_n(x) ∈ A_2nd if x ∈ A_2d, • γ_1(x) = x for any x, we also set γ_0(x) = 1, • γ_n(x)γ_m(x) = frac(n + m)!n! m! γ_n + m(x), • γ_n(xy) = x^n γ_n(y) for all x ∈ A_even and y ∈ A_even, +, • γ_n(xy) = 0 if x, y ∈ A_odd homogeneous and n > 1 • if x, y ∈…","statement_latex":"Let $R$ be a ring. Let $A = \\bigoplus_{d \\geq 0} A_d$ be a graded\n$R$-algebra which is strictly graded commutative. A collection of maps\n$\\gamma_n : A_{even, +} \\to A_{even, +}$ defined for all $n > 0$ is called\na {\\it divided power structure} on $A$ if we have\n\\begin{enumerate}\n\\item $\\gamma_n(x) \\in A_{2nd}$ if $x \\in A_{2d}$,\n\\item $\\gamma_1(x) = x$ for any $x$, we also set $\\gamma_0(x) = 1$,\n\\item $\\gamma_n(x)\\gamma_m(x) = \\frac{(n + m)!}{n! m!} \\gamma_{n + m}(x)$,\n\\item $\\gamma_n(xy) = x^n \\gamma_n(y)$ for all $x \\in A_{even}$ and\n$y \\in A_{even, +}$,\n\\item $\\gamma_n(xy) = 0$ if $x, y \\in A_{odd}$ homogeneous and $n > 1$\n\\item if $x, y \\in A_{even, +}$ then\n$\\gamma_n(x + y) = \\sum_{i = 0, \\ldots, n} \\gamma_i(x)\\gamma_{n - i}(y)$,\n\\item $\\gamma_n(\\gamma_m(x)) =\n\\frac{(nm)!}{n! (m!)^n} \\gamma_{nm}(x)$ for $x \\in A_{even, +}$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Tate resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PG","source_file":"dpa.tex","source_line":965,"source_end_line":983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L965-L983","statement_sha256":"f458ef083389cc08b11bd8b34710a70fa70aa64d697208cf58101f17debe7409","origin":"The Stacks Project","memory_eligible":false,"source_rank":4813,"rank":4813,"depth":0,"x":2335.07,"y":173.151,"cluster":"homological-algebra"},{"id":"stacks:09PJ","tag":"09PJ","title":"Tate resolutions · Definition 09PJ","summary":"Let R be a ring. Let A = bigoplus_d ≥ 0 A_d be a differential graded R-algebra which is strictly graded commutative. A divided power structure γ on A is compatible with the differential graded structure if d(γ_n(x)) = d(x) γ_n - 1(x) for all x ∈ A_even, +.","statement_latex":"Let $R$ be a ring. Let $A = \\bigoplus_{d \\geq 0} A_d$ be a\ndifferential graded $R$-algebra which is strictly graded commutative.\nA divided power structure $\\gamma$ on $A$ is {\\it compatible with\nthe differential graded structure} if\n$\\text{d}(\\gamma_n(x)) = \\text{d}(x) \\gamma_{n - 1}(x)$ for\nall $x \\in A_{even, +}$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Tate resolutions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PJ","source_file":"dpa.tex","source_line":1082,"source_end_line":1090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1082-L1090","statement_sha256":"c6ba67027c6631f346a091b2a4e462be5b59c2d385db2f9c7d15fd2b7f17edf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4814,"rank":4814,"depth":0,"x":2624.699,"y":172.258,"cluster":"homological-algebra"},{"id":"stacks:09PK","tag":"09PK","title":"Tate resolutions · Lemma 09PK","summary":"Let (A, d, γ) and (B, d, γ) be as in Definition [Tag 09PJ]. Let f : A → B be a map of differential graded algebras compatible with divided power structures. Assume • H_k(A) = 0 for k > 0, and • f is surjective. Then γ induces a divided power structure on the graded R-algebra H(B).","statement_latex":"Let $(A, \\text{d}, \\gamma)$ and $(B, \\text{d}, \\gamma)$ be as in\nDefinition \\ref{definition-divided-powers-dga}. Let $f : A \\to B$\nbe a map of differential graded algebras compatible with divided\npower structures. Assume\n\\begin{enumerate}\n\\item $H_k(A) = 0$ for $k > 0$, and\n\\item $f$ is surjective.\n\\end{enumerate}\nThen $\\gamma$ induces a divided power structure on the graded\n$R$-algebra $H(B)$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PK","source_file":"dpa.tex","source_line":1102,"source_end_line":1114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1102-L1114","statement_sha256":"10b4c815083c104ac000f4d8c26ea26cfec4e69c7c42aafece5f057a01073a48","origin":"The Stacks Project","memory_eligible":false,"source_rank":4815,"rank":4815,"depth":1,"x":2411.621,"y":337.435,"cluster":"homological-algebra"},{"id":"stacks:09PL","tag":"09PL","title":"Tate resolutions · Lemma 09PL","summary":"Let (A, d, γ) be as in Definition [Tag 09PJ]. Let R → R' be a ring map. Then d and γ induce similar structures on A' = A ⊗_R R' such that (A', d, γ) is as in Definition [Tag 09PJ].","statement_latex":"Let $(A, \\text{d}, \\gamma)$ be as in\nDefinition \\ref{definition-divided-powers-dga}.\nLet $R \\to R'$ be a ring map.\nThen $\\text{d}$ and $\\gamma$ induce similar structures on\n$A' = A \\otimes_R R'$ such that $(A', \\text{d}, \\gamma)$ is as in\nDefinition \\ref{definition-divided-powers-dga}.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PL","source_file":"dpa.tex","source_line":1147,"source_end_line":1155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1147-L1155","statement_sha256":"c2e3743b3bd7cd7ffd528edc2ba8180fa5b35eb270f4867398471e25cfc38dcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4816,"rank":4816,"depth":5,"x":2435.937,"y":94.472,"cluster":"homological-algebra"},{"id":"stacks:09PM","tag":"09PM","title":"Tate resolutions · Lemma 09PM","summary":"Let (A, d, γ) be as in Definition [Tag 09PJ]. Let d ≥ 1 be an integer. Let Alangle T rangle be the graded divided power polynomial algebra on T with deg(T) = d constructed in Example [Tag 09PH] or [Tag 09PI]. Let f ∈ A_d - 1 be an element with d(f) = 0. There exists a unique differential d on Alangle Trangle such that d(T) = f and such that d is compatible with the divided power structure on Alangle T rangle.","statement_latex":"Let $(A, \\text{d}, \\gamma)$ be as in\nDefinition \\ref{definition-divided-powers-dga}.\nLet $d \\geq 1$ be an integer.\nLet $A\\langle T \\rangle$ be the graded divided power polynomial algebra\non $T$ with $\\deg(T) = d$\nconstructed in Example \\ref{example-adjoining-odd} or\n\\ref{example-adjoining-even}.\nLet $f \\in A_{d - 1}$ be an element with $\\text{d}(f) = 0$.\nThere exists a unique differential $\\text{d}$\non $A\\langle T\\rangle$ such that $\\text{d}(T) = f$ and\nsuch that $\\text{d}$ is compatible with the divided power\nstructure on $A\\langle T \\rangle$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PM","source_file":"dpa.tex","source_line":1180,"source_end_line":1194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1180-L1194","statement_sha256":"9ee094e17c886b0956e36df58d66801d00a583d9f85b7b044f3c98cf601fd02d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4817,"rank":4817,"depth":1,"x":2613.58,"y":287.632,"cluster":"homological-algebra"},{"id":"stacks:09PN","tag":"09PN","title":"Tate resolutions · Lemma 09PN","summary":"Let R → S be a homomorphism of commutative rings. There exists a factorization R → A → S with the following properties: • (A, d, γ) is as in Definition [Tag 09PJ], • A → S is a quasi-isomorphism (if we endow S with the zero differential), • A_0 = R[x_j: j∈ J] → S is any surjection of a polynomial ring onto S, and • A is a graded divided power polynomial algebra over R. The last condition means that A is constructed out of A_0 by successively adjoining a set of variables T…","statement_latex":"Let $R \\to S$ be a homomorphism of commutative rings.\nThere exists a factorization\n$$\nR \\to A \\to S\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item $(A, \\text{d}, \\gamma)$ is as in\nDefinition \\ref{definition-divided-powers-dga},\n\\item $A \\to S$ is a quasi-isomorphism (if we endow $S$ with\nthe zero differential),\n\\item $A_0 = R[x_j: j\\in J] \\to S$ is any surjection of a polynomial\nring onto $S$, and\n\\item $A$ is a graded divided power polynomial algebra over $R$.\n\\end{enumerate}\nThe last condition means that $A$ is constructed out of $A_0$ by\nsuccessively adjoining a set of variables $T$ in each degree $> 0$ as in\nExample \\ref{example-adjoining-odd} or \\ref{example-adjoining-even}.\nMoreover, if $R$ is Noetherian and $R\\to S$ is of finite type,\nthen $A$ can be taken to have only finitely many generators in\neach degree.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PN","source_file":"dpa.tex","source_line":1206,"source_end_line":1229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1206-L1229","statement_sha256":"4daaa6489c39348863c53ffae99ba4b00bfe5ca36da1e18338f922434ba23e75","origin":"The Stacks Project","memory_eligible":false,"source_rank":4818,"rank":4818,"depth":2,"x":2326.95,"y":245.953,"cluster":"homological-algebra"},{"id":"stacks:0BZ9","tag":"0BZ9","title":"Tate resolutions · Lemma 0BZ9","summary":"Let R → S be a pseudo-coherent ring map (More on Algebra, Definition [Tag 067H]). Then Lemma [Tag 09PN] holds, with the resolution A of S having finitely many generators in each degree.","statement_latex":"Let $R \\to S$ be a pseudo-coherent ring map (More on Algebra, Definition\n\\ref{more-algebra-definition-pseudo-coherent-perfect}). Then\nLemma \\ref{lemma-tate-resolution} holds, with the resolution $A$ of $S$\nhaving finitely many generators in each degree.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZ9","source_file":"dpa.tex","source_line":1301,"source_end_line":1307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1301-L1307","statement_sha256":"0ca0043b0d1dd1eaaededb1f5df2981125cab53bc74d1855ab3d8031eb404340","origin":"The Stacks Project","memory_eligible":false,"source_rank":4819,"rank":4819,"depth":11,"x":2572.083,"y":113.906,"cluster":"homological-algebra"},{"id":"stacks:09PP","tag":"09PP","title":"Tate resolutions · Lemma 09PP","summary":"Let R be a commutative ring. Suppose that (A, d, γ) and (B, d, γ) are as in Definition [Tag 09PJ]. Let overlineφ : H_0(A) → H_0(B) be an R-algebra map. Assume • A is a graded divided power polynomial algebra over R. • H_k(B) = 0 for k > 0. Then there exists a map φ : A → B of differential graded R-algebras compatible with divided powers that lifts overlineφ.","statement_latex":"Let $R$ be a commutative ring. Suppose that $(A, \\text{d}, \\gamma)$ and\n$(B, \\text{d}, \\gamma)$ are as in\nDefinition \\ref{definition-divided-powers-dga}.\nLet $\\overline{\\varphi} : H_0(A) \\to H_0(B)$ be an $R$-algebra map.\nAssume\n\\begin{enumerate}\n\\item $A$ is a graded divided power polynomial algebra over $R$.\n\\item $H_k(B) = 0$ for $k > 0$.\n\\end{enumerate}\nThen there exists a map $\\varphi : A \\to B$ of differential\ngraded $R$-algebras compatible with divided powers\nthat lifts $\\overline{\\varphi}$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PP","source_file":"dpa.tex","source_line":1328,"source_end_line":1342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1328-L1342","statement_sha256":"7256f97f5e54781698f687c807a165b9014cc4c3d9b4507c3b6944cf162abefa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4820,"rank":4820,"depth":6,"x":2497.435,"y":350.62,"cluster":"homological-algebra"},{"id":"stacks:09PQ","tag":"09PQ","title":"Tate resolutions · Lemma 09PQ","summary":"Let R be a commutative ring. Let S and T be commutative R-algebras. Then there is a canonical structure of a strictly graded commutative R-algebra with divided powers on Tor_*^R(S, T).","statement_latex":"Let $R$ be a commutative ring. Let $S$ and $T$ be commutative $R$-algebras.\nThen there is a canonical structure\nof a strictly graded commutative $R$-algebra with divided powers on\n$$\n\\operatorname{Tor}_*^R(S, T).\n$$","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PQ","source_file":"dpa.tex","source_line":1374,"source_end_line":1382,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1374-L1382","statement_sha256":"407fc29fcd075638a45e2c9ab50f9337ad40b77369c2aa36ec9f447b936c55a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4821,"rank":4821,"depth":7,"x":2361.978,"y":133.485,"cluster":"homological-algebra"},{"id":"stacks:09PS","tag":"09PS","title":"Application to complete intersections · Lemma 09PS","summary":"Let R be a ring. Let (A, d, γ) be as in Definition [Tag 09PJ]. Let R' → R be a surjection of rings whose kernel has square zero and is generated by one element f. If A is a graded divided power polynomial algebra over R with finitely many variables in each degree, then we obtain a derivation theta : A/IA → A/IA where I is the annihilator of f in R.","statement_latex":"Let $R$ be a ring. Let $(A, \\text{d}, \\gamma)$ be as in\nDefinition \\ref{definition-divided-powers-dga}.\nLet $R' \\to R$ be a surjection of rings whose kernel\nhas square zero and is generated by one element $f$.\nIf $A$ is a graded divided power polynomial algebra over $R$\nwith finitely many variables in each degree,\nthen we obtain a derivation\n$\\theta : A/IA \\to A/IA$ where $I$ is the annihilator\nof $f$ in $R$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Application to complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PS","source_file":"dpa.tex","source_line":1433,"source_end_line":1444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1433-L1444","statement_sha256":"c8e6e20c263eacb39e3763c88adf71ef3bdb4c5c21a77f920a3314d9a87f3215","origin":"The Stacks Project","memory_eligible":false,"source_rank":4822,"rank":4822,"depth":1,"x":2636.765,"y":216.823,"cluster":"homological-algebra"},{"id":"stacks:09PT","tag":"09PT","title":"Application to complete intersections · Lemma 09PT","summary":"Assumption and notation as in Lemma [Tag 09PS]. Suppose S = H_0(A) is isomorphic to R[x_1, …, x_n]/(f_1, …, f_m) for some n, m, and f_j ∈ R[x_1, …, x_n]. Moreover, suppose given a relation ∑ r_j f_j = 0 with r_j ∈ R[x_1, …, x_n]. Choose r'_j, f'_j ∈ R'[x_1, …, x_n] lifting r_j, f_j. Write ∑ r'_j f'_j = gf for some g ∈ R/I[x_1, …, x_n]. If H_1(A) = 0 and all the coefficients of each r_j are in I, then there exists an element xi ∈ H_2(A/IA) such that theta(xi) = g in S/IS.","statement_latex":"Assumption and notation as in Lemma \\ref{lemma-get-derivation}.\nSuppose $S = H_0(A)$ is isomorphic to\n$R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$\nfor some $n$, $m$, and $f_j \\in R[x_1, \\ldots, x_n]$.\nMoreover, suppose given a relation\n$$\n\\sum r_j f_j = 0\n$$\nwith $r_j \\in R[x_1, \\ldots, x_n]$.\nChoose $r'_j, f'_j \\in R'[x_1, \\ldots, x_n]$ lifting $r_j, f_j$.\nWrite $\\sum r'_j f'_j = gf$ for some $g \\in R/I[x_1, \\ldots, x_n]$.\nIf $H_1(A) = 0$ and all the coefficients of each $r_j$ are in $I$, then\nthere exists an element $\\xi \\in H_2(A/IA)$ such that\n$\\theta(\\xi) = g$ in $S/IS$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Application to complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PT","source_file":"dpa.tex","source_line":1466,"source_end_line":1482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1466-L1482","statement_sha256":"2c7ab6808f7aa564e7e096bc324c3b264d26946017494cfe7835cf33ece6c26f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4823,"rank":4823,"depth":2,"x":2366.84,"y":311.391,"cluster":"homological-algebra"},{"id":"stacks:09PU","tag":"09PU","title":"Application to complete intersections · Lemma 09PU","summary":"Let R' → R be a surjection of Noetherian rings whose kernel has square zero and is generated by one element f. Let S = R[x_1, …, x_n]/(f_1, …, f_m). Let ∑ r_j f_j = 0 be a relation in R[x_1, …, x_n]. Assume that • each r_j has coefficients in the annihilator I of f in R, • for some lifts r'_j, f'_j ∈ R'[x_1, …, x_n] we have ∑ r'_j f'_j = gf where g is not nilpotent in S/IS. Then S does not have finite tor dimension over R (i.e., S is not a perfect R-algebra).","statement_latex":"Let $R' \\to R$ be a surjection of Noetherian rings whose kernel has square\nzero and is generated by one element $f$. Let\n$S = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$.\nLet $\\sum r_j f_j = 0$ be a relation in $R[x_1, \\ldots, x_n]$.\nAssume that\n\\begin{enumerate}\n\\item each $r_j$ has coefficients in the annihilator $I$ of $f$ in $R$,\n\\item for some lifts $r'_j, f'_j \\in R'[x_1, \\ldots, x_n]$ we have\n$\\sum r'_j f'_j = gf$ where $g$ is not nilpotent in $S/IS$.\n\\end{enumerate}\nThen $S$ does not have finite tor dimension over $R$ (i.e., $S$ is not\na perfect $R$-algebra).","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Application to complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PU","source_file":"dpa.tex","source_line":1537,"source_end_line":1551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1537-L1551","statement_sha256":"5f790aeb84d7272b52c14282c2ac8355a76ddeb412f81372fa486cbd1d7b4169","origin":"The Stacks Project","memory_eligible":false,"source_rank":4824,"rank":4824,"depth":3,"x":2489.959,"y":88.262,"cluster":"homological-algebra"},{"id":"stacks:09PV","tag":"09PV","title":"Application to complete intersections · Lemma 09PV","summary":"Let (A, m) be a Noetherian local ring. Let I ⊂ J ⊂ A be proper ideals. If A/J has finite tor dimension over A/I, then I/ m I → J/ m J is injective.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let\n$I \\subset J \\subset A$ be proper ideals. If $A/J$ has finite\ntor dimension over $A/I$, then $I/\\mathfrak m I \\to J/\\mathfrak m J$\nis injective.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Application to complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PV","source_file":"dpa.tex","source_line":1572,"source_end_line":1578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1572-L1578","statement_sha256":"e92765e2715fe091b4874595fe9b538621a6d4a346e7019b23b5c39746376a62","origin":"The Stacks Project","memory_eligible":false,"source_rank":4825,"rank":4825,"depth":4,"x":2578.699,"y":322.9,"cluster":"homological-algebra"},{"id":"stacks:09PW","tag":"09PW","title":"Application to complete intersections · Lemma 09PW","summary":"Let (A, m) be a Noetherian local ring. Let I ⊂ J ⊂ A be proper ideals. Assume • A/J has finite tor dimension over A/I, and • J is generated by a regular sequence. Then I is generated by a regular sequence and J/I is generated by a regular sequence.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let\n$I \\subset J \\subset A$ be proper ideals. Assume\n\\begin{enumerate}\n\\item $A/J$ has finite tor dimension over $A/I$, and\n\\item $J$ is generated by a regular sequence.\n\\end{enumerate}\nThen $I$ is generated by a regular sequence and $J/I$\nis generated by a regular sequence.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Application to complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PW","source_file":"dpa.tex","source_line":1593,"source_end_line":1603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1593-L1603","statement_sha256":"694431585b8170874137e7484a309c443ade95b0add5fc0728ec7efc1c903ea0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4826,"rank":4826,"depth":6,"x":2324.309,"y":200.107,"cluster":"homological-algebra"},{"id":"stacks:09PX","tag":"09PX","title":"Application to complete intersections · Lemma 09PX","summary":"Let R → S be a local ring map of Noetherian local rings. Let I ⊂ R and J ⊂ S be ideals with IS ⊂ J. If R → S is flat and S/ m_RS is regular, then the following are equivalent • J is generated by a regular sequence and S/J has finite tor dimension as a module over R/I, • J is generated by a regular sequence and Tor^R/I_p(S/J, R/ m_R) is nonzero for only finitely many p, • I is generated by a regular sequence and J/IS is generated by a regular sequence in S/IS.","statement_latex":"Let $R \\to S$ be a local ring map of Noetherian local rings.\nLet $I \\subset R$ and $J \\subset S$ be ideals with\n$IS \\subset J$. If $R \\to S$ is flat and $S/\\mathfrak m_RS$ is\nregular, then the following are equivalent\n\\begin{enumerate}\n\\item $J$ is generated by a regular sequence and\n$S/J$ has finite tor dimension as a module over $R/I$,\n\\item $J$ is generated by a regular sequence and\n$\\text{Tor}^{R/I}_p(S/J, R/\\mathfrak m_R)$ is nonzero\nfor only finitely many $p$,\n\\item $I$ is generated by a regular sequence\nand $J/IS$ is generated by a regular sequence in $S/IS$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Application to complete intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PX","source_file":"dpa.tex","source_line":1618,"source_end_line":1633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1618-L1633","statement_sha256":"d35cddb6bb1187748e015b46c3b1faef29fb0af68b3d36f52ef15ab592fb8cd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4827,"rank":4827,"depth":17,"x":2610.938,"y":146.25,"cluster":"homological-algebra"},{"id":"stacks:09PZ","tag":"09PZ","title":"Local complete intersection rings · Lemma 09PZ","summary":"Let (A, m) be a Noetherian complete local ring. The following are equivalent • for every surjection of local rings R → A with R a regular local ring, the kernel of R → A is generated by a regular sequence, and • for some surjection of local rings R → A with R a regular local ring, the kernel of R → A is generated by a regular sequence.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian complete local ring.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every surjection of local rings $R \\to A$ with $R$\na regular local ring, the kernel of $R \\to A$ is generated\nby a regular sequence, and\n\\item for some surjection of local rings $R \\to A$ with $R$\na regular local ring, the kernel of $R \\to A$ is generated\nby a regular sequence.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09PZ","source_file":"dpa.tex","source_line":1686,"source_end_line":1698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1686-L1698","statement_sha256":"dbeba7320676729c9c4b169ee7546d90ad8d7a6e15fb2dcb4a80103d3ee14126","origin":"The Stacks Project","memory_eligible":false,"source_rank":4828,"rank":4828,"depth":19,"x":2442.718,"y":348.813,"cluster":"homological-algebra"},{"id":"stacks:09Q0","tag":"09Q0","title":"Local complete intersection rings · Lemma 09Q0","summary":"Let R be a regular ring. Let p ⊂ R be a prime. Let f_1, …, f_r ∈ p be a regular sequence. Then the completion of A = (R/(f_1, …, f_r))_ p = R_ p/(f_1, …, f_r)R_ p is a complete intersection in the sense defined above.","statement_latex":"Let $R$ be a regular ring. Let $\\mathfrak p \\subset R$ be a prime.\nLet $f_1, \\ldots, f_r \\in \\mathfrak p$ be a regular sequence.\nThen the completion of\n$$\nA = (R/(f_1, \\ldots, f_r))_\\mathfrak p =\nR_\\mathfrak p/(f_1, \\ldots, f_r)R_\\mathfrak p\n$$\nis a complete intersection in the sense defined above.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Q0","source_file":"dpa.tex","source_line":1761,"source_end_line":1771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1761-L1771","statement_sha256":"251515c54c390323c7fdf08f0ba14d135bbd70b45f0df600717f8f101856ae3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4829,"rank":4829,"depth":19,"x":2403.824,"y":103.74,"cluster":"homological-algebra"},{"id":"stacks:09Q1","tag":"09Q1","title":"Local complete intersection rings · Lemma 09Q1","summary":"Let R be a regular ring. Let p ⊂ R be a prime. Let I ⊂ p be an ideal. Set A = (R/I)_ p = R_ p/I_ p. The following are equivalent • the completion of A is a complete intersection in the sense above, • I_ p ⊂ R_ p is generated by a regular sequence, • the module (I/I^2)_ p can be generated by dim(R_ p) - dim(A) elements, • add more here.","statement_latex":"Let $R$ be a regular ring. Let $\\mathfrak p \\subset R$ be a prime.\nLet $I \\subset \\mathfrak p$ be an ideal.\nSet $A = (R/I)_\\mathfrak p = R_\\mathfrak p/I_\\mathfrak p$.\nThe following are equivalent\n\\begin{enumerate}\n\\item the completion of $A$\nis a complete intersection in the sense above,\n\\item $I_\\mathfrak p \\subset R_\\mathfrak p$ is generated\nby a regular sequence,\n\\item the module $(I/I^2)_\\mathfrak p$ can be generated by\n$\\dim(R_\\mathfrak p) - \\dim(A)$ elements,\n\\item add more here.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Q1","source_file":"dpa.tex","source_line":1792,"source_end_line":1807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1792-L1807","statement_sha256":"4acb1b475531edfddacff5f3046ff969186be046548a4c7f569e600b709040e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4830,"rank":4830,"depth":20,"x":2629.818,"y":262.548,"cluster":"homological-algebra"},{"id":"stacks:09Q2","tag":"09Q2","title":"Local complete intersection rings · Proposition 09Q2","summary":"Let A → B be a flat local homomorphism of Noetherian local rings. Then the following are equivalent • B^wedge is a complete intersection, • A^wedge and (B/ m_A B)^wedge are complete intersections.","statement_latex":"Let $A \\to B$ be a flat local homomorphism of Noetherian local rings.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $B^\\wedge$ is a complete intersection,\n\\item $A^\\wedge$ and $(B/\\mathfrak m_A B)^\\wedge$ are complete intersections.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Q2","source_file":"dpa.tex","source_line":1841,"source_end_line":1849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1841-L1849","statement_sha256":"7379fc1197b86753df73848b4901aeec1e1781b64bd9f71392c73f9cceecb100","origin":"The Stacks Project","memory_eligible":false,"source_rank":4831,"rank":4831,"depth":21,"x":2335.161,"y":273.691,"cluster":"homological-algebra"},{"id":"stacks:09Q3","tag":"09Q3","title":"Local complete intersection rings · Definition 09Q3","summary":"Let A be a Noetherian ring. • If A is local, then we say A is a complete intersection if its completion is a complete intersection in the sense above. • In general we say A is a local complete intersection if all of its local rings are complete intersections.","statement_latex":"Let $A$ be a Noetherian ring.\n\\begin{enumerate}\n\\item If $A$ is local, then we say $A$ is a {\\it complete intersection}\nif its completion is a complete intersection in the sense above.\n\\item In general we say $A$ is a {\\it local complete intersection}\nif all of its local rings are complete intersections.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Q3","source_file":"dpa.tex","source_line":1919,"source_end_line":1928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1919-L1928","statement_sha256":"44d324325f171c891200f10ce9677d564deabb19fefb60c208a924d3d5f278b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4832,"rank":4832,"depth":0,"x":2543.691,"y":98.098,"cluster":"homological-algebra"},{"id":"stacks:09Q4","tag":"09Q4","title":"Local complete intersection rings · Lemma 09Q4","summary":"Let (A, m) be a Noetherian local ring. Let p ⊂ A be a prime ideal. If A is a complete intersection, then A_ p is a complete intersection too.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let\n$\\mathfrak p \\subset A$ be a prime ideal. If $A$ is a complete\nintersection, then $A_\\mathfrak p$ is a complete intersection too.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Q4","source_file":"dpa.tex","source_line":1940,"source_end_line":1945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1940-L1945","statement_sha256":"1040e0540098ebbf0c259ea0d6a2584a83f6659da709fa50ff395104437905fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4833,"rank":4833,"depth":22,"x":2531.118,"y":346.158,"cluster":"homological-algebra"},{"id":"stacks:09Q5","tag":"09Q5","title":"Local complete intersection rings · Lemma 09Q5","summary":"Let A be a Noetherian ring. Then A is a local complete intersection if and only if A_ m is a complete intersection for every maximal ideal m of A.","statement_latex":"Let $A$ be a Noetherian ring. Then $A$ is a local complete intersection\nif and only if $A_\\mathfrak m$ is a complete intersection for every\nmaximal ideal $\\mathfrak m$ of $A$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Q5","source_file":"dpa.tex","source_line":1969,"source_end_line":1974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1969-L1974","statement_sha256":"4d03f78ce92590288ba288c365f946452987ad25760f998fc9dc0b74991853db","origin":"The Stacks Project","memory_eligible":false,"source_rank":4834,"rank":4834,"depth":23,"x":2340.711,"y":155.913,"cluster":"homological-algebra"},{"id":"stacks:09Q6","tag":"09Q6","title":"Local complete intersection rings · Lemma 09Q6","summary":"Let S be a finite type algebra over a field k. • for a prime q ⊂ S the local ring S_ q is a complete intersection in the sense of Algebra, Definition [Tag 00SD] if and only if S_ q is a complete intersection in the sense of Definition [Tag 09Q3], and • S is a local complete intersection in the sense of Algebra, Definition [Tag 00S9] if and only if S is a local complete intersection in the sense of Definition [Tag 09Q3].","statement_latex":"Let $S$ be a finite type algebra over a field $k$.\n\\begin{enumerate}\n\\item for a prime $\\mathfrak q \\subset S$ the local ring $S_\\mathfrak q$\nis a complete intersection in the sense of\nAlgebra, Definition \\ref{algebra-definition-lci-local-ring}\nif and only if $S_\\mathfrak q$ is a complete\nintersection in the sense of Definition \\ref{definition-lci}, and\n\\item $S$ is a local complete intersection in the sense of\nAlgebra, Definition \\ref{algebra-definition-lci-field}\nif and only if $S$ is a local complete\nintersection in the sense of Definition \\ref{definition-lci}.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Q6","source_file":"dpa.tex","source_line":1980,"source_end_line":1994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L1980-L1994","statement_sha256":"876d4fd90ecebd2227f97e231aa4012ecd8cd70d5936a40cac8da9f20ed29962","origin":"The Stacks Project","memory_eligible":false,"source_rank":4835,"rank":4835,"depth":28,"x":2634.404,"y":188.188,"cluster":"homological-algebra"},{"id":"stacks:09Q7","tag":"09Q7","title":"Local complete intersection rings · Lemma 09Q7","summary":"Let A → B be a flat local homomorphism of Noetherian local rings. Then the following are equivalent • B is a complete intersection, • A and B/ m_A B are complete intersections.","statement_latex":"Let $A \\to B$ be a flat local homomorphism of Noetherian local rings.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $B$ is a complete intersection,\n\\item $A$ and $B/\\mathfrak m_A B$ are complete intersections.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Q7","source_file":"dpa.tex","source_line":2015,"source_end_line":2023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2015-L2023","statement_sha256":"e230b7e833d0bbcc7d94a28be3e5b27b0d62960b6f2e19739a3c82289ca6dee2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4836,"rank":4836,"depth":22,"x":2391.637,"y":331.183,"cluster":"homological-algebra"},{"id":"stacks:09QA","tag":"09QA","title":"Local complete intersection maps · Lemma 09QA","summary":"Let A → B be a local homomorphism of Noetherian complete local rings. The following are equivalent • for some good factorization A → S → B the kernel of S → B is generated by a regular sequence, and • for every good factorization A → S → B the kernel of S → B is generated by a regular sequence.","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian complete local rings.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some good factorization $A \\to S \\to B$ the kernel of\n$S \\to B$ is generated by a regular sequence, and\n\\item for every good factorization $A \\to S \\to B$ the kernel of\n$S \\to B$ is generated by a regular sequence.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QA","source_file":"dpa.tex","source_line":2067,"source_end_line":2077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2067-L2077","statement_sha256":"9f7e16a7bfe20ff775693438b070c652e9dd43e466786cf148f640991331c533","origin":"The Stacks Project","memory_eligible":false,"source_rank":4837,"rank":4837,"depth":19,"x":2455.722,"y":87.741,"cluster":"homological-algebra"},{"id":"stacks:09QB","tag":"09QB","title":"Local complete intersection maps · Proposition 09QB","summary":"Let A → B be a local homomorphism of Noetherian local rings. Then the following are equivalent • B is a complete intersection and Tor^A_p(B, A/ m_A) is nonzero for only finitely many p, • A is a complete intersection and A^wedge → B^wedge is a complete intersection homomorphism in the sense defined above.","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian local rings.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $B$ is a complete intersection and\n$\\text{Tor}^A_p(B, A/\\mathfrak m_A)$ is nonzero for only finitely many $p$,\n\\item $A$ is a complete intersection and\n$A^\\wedge \\to B^\\wedge$ is a complete intersection homomorphism\nin the sense defined above.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection maps","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QB","source_file":"dpa.tex","source_line":2169,"source_end_line":2180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2169-L2180","statement_sha256":"b1c0184a07c29277c17fc187949c9a2415b7c1f6e6ed0a6f81ed2db85abf2ed8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4838,"rank":4838,"depth":19,"x":2604.387,"y":303.835,"cluster":"homological-algebra"},{"id":"stacks:09QD","tag":"09QD","title":"Local complete intersection maps · Lemma 09QD","summary":"Consider a commutative diagram xymatrix S ar[r] & B & A ar[lu] ar[u] of Noetherian local rings with S → B surjective, A → S flat, and S/ m_A S a regular local ring. The following are equivalent • Ker(S → B) is generated by a regular sequence, and • A^wedge → B^wedge is a complete intersection homomorphism as defined above.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nS \\ar[r] & B \\\\\n& A \\ar[lu] \\ar[u]\n}\n$$\nof Noetherian local rings with $S \\to B$ surjective, $A \\to S$ flat, and\n$S/\\mathfrak m_A S$ a regular local ring. The following are equivalent\n\\begin{enumerate}\n\\item $\\Ker(S \\to B)$ is generated by a regular sequence, and\n\\item $A^\\wedge \\to B^\\wedge$ is a complete intersection homomorphism\nas defined above.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QD","source_file":"dpa.tex","source_line":2249,"source_end_line":2265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2249-L2265","statement_sha256":"6bed8ffeab18eaf646688d5a64081faa16dd1b58b2029cb97ae903a5eae83efb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4839,"rank":4839,"depth":20,"x":2320.699,"y":228.77,"cluster":"homological-algebra"},{"id":"stacks:09QE","tag":"09QE","title":"Local complete intersection maps · Lemma 09QE","summary":"Let A be a Noetherian ring. Let A → B be a finite type ring map. The following are equivalent • A → B is a local complete intersection in the sense of More on Algebra, Definition [Tag 07D0], • for every prime q ⊂ B and with p = A ∩ q the ring map (A_ p)^wedge → (B_ q)^wedge is a complete intersection homomorphism in the sense defined above.","statement_latex":"Let $A$ be a Noetherian ring.\nLet $A \\to B$ be a finite type ring map.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A \\to B$ is a local complete intersection in the sense of\nMore on Algebra, Definition\n\\ref{more-algebra-definition-local-complete-intersection},\n\\item for every prime $\\mathfrak q \\subset B$ and with\n$\\mathfrak p = A \\cap \\mathfrak q$ the ring map\n$(A_\\mathfrak p)^\\wedge \\to (B_\\mathfrak q)^\\wedge$ is\na complete intersection homomorphism in the sense defined above.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QE","source_file":"dpa.tex","source_line":2272,"source_end_line":2286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2272-L2286","statement_sha256":"b26c9bf3524feb09899fd837bd45b3539a6e3f8f57f528d61a07aab62f62d90f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4840,"rank":4840,"depth":21,"x":2590.525,"y":123.044,"cluster":"homological-algebra"},{"id":"stacks:09QF","tag":"09QF","title":"Local complete intersection maps · Lemma 09QF","summary":"Let A be a Noetherian ring. Let A → B be a finite type ring map such that the image of Spec(B) → Spec(A) contains all closed points of Spec(A). Then the following are equivalent • B is a complete intersection and A → B has finite tor dimension, • A is a complete intersection and A → B is a local complete intersection in the sense of More on Algebra, Definition [Tag 07D0].","statement_latex":"Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map\nsuch that the image of $\\Spec(B) \\to \\Spec(A)$ contains all closed\npoints of $\\Spec(A)$. Then the following are equivalent\n\\begin{enumerate}\n\\item $B$ is a complete intersection and $A \\to B$ has finite\ntor dimension,\n\\item $A$ is a complete intersection and $A \\to B$ is a local complete\nintersection in the sense of More on Algebra, Definition\n\\ref{more-algebra-definition-local-complete-intersection}.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Local complete intersection maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09QF","source_file":"dpa.tex","source_line":2309,"source_end_line":2321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2309-L2321","statement_sha256":"5c281e7034ef48185d297895f6c71a337251ba686399daf61fee018fd930ccab","origin":"The Stacks Project","memory_eligible":false,"source_rank":4841,"rank":4841,"depth":22,"x":2476.467,"y":354.344,"cluster":"homological-algebra"},{"id":"stacks:0FCW","tag":"0FCW","title":"Smooth ring maps and diagonals · Lemma 0FCW","summary":"Let A → B be a local ring homomorphism of Noetherian local rings such that B is flat and essentially of finite type over A. If B ⊗_A B → B is a perfect ring map, i.e., if B has finite tor dimension over B ⊗_A B, then B is the localization of a smooth A-algebra.","statement_latex":"Let $A \\to B$ be a local ring homomorphism of Noetherian local rings such that\n$B$ is flat and essentially of finite type over $A$. If\n$$\nB \\otimes_A B \\longrightarrow B\n$$\nis a perfect ring map, i.e., if $B$ has finite tor dimension over\n$B \\otimes_A B$, then $B$ is the localization of a smooth $A$-algebra.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Smooth ring maps and diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCW","source_file":"dpa.tex","source_line":2343,"source_end_line":2352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2343-L2352","statement_sha256":"f702c73fae8d685e18f725a0c4bb69255b67db125164a6ffa8e43a891b57f501","origin":"The Stacks Project","memory_eligible":false,"source_rank":4842,"rank":4842,"depth":13,"x":2374.463,"y":118.83,"cluster":"homological-algebra"},{"id":"stacks:0FCX","tag":"0FCX","title":"Smooth ring maps and diagonals · Lemma 0FCX","summary":"Let A → B be a flat finite type ring map of Noetherian rings. If B ⊗_A B → B is a perfect ring map, i.e., if B has finite tor dimension over B ⊗_A B, then B is a smooth A-algebra.","statement_latex":"Let $A \\to B$ be a flat finite type ring map of Noetherian rings. If\n$$\nB \\otimes_A B \\longrightarrow B\n$$\nis a perfect ring map, i.e., if $B$ has finite tor dimension over\n$B \\otimes_A B$, then $B$ is a smooth $A$-algebra.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Smooth ring maps and diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCX","source_file":"dpa.tex","source_line":2435,"source_end_line":2443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2435-L2443","statement_sha256":"7218a3859bd3cb4b3592bf0ece8e355d30569d3edfa2eef1dfdbb321867af2b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4843,"rank":4843,"depth":14,"x":2639.339,"y":234.732,"cluster":"homological-algebra"},{"id":"stacks:0FJQ","tag":"0FJQ","title":"Freeness of the conormal module · Lemma 0FJQ","summary":"[Vasconcelos] Let R be a Noetherian local ring. Let I ⊂ R be an ideal of finite projective dimension over R. If F ⊂ I/I^2 is a direct summand isomorphic to R/I, then there exists a nonzerodivisor x ∈ I such that the image of x in I/I^2 generates F.","statement_latex":"\\begin{reference}\n\\cite{Vasconcelos}\n\\end{reference}\nLet $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal\nof finite projective dimension over $R$. If $F \\subset I/I^2$ is a\ndirect summand isomorphic to $R/I$, then there exists a nonzerodivisor\n$x \\in I$ such that the image of $x$ in $I/I^2$ generates $F$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Freeness of the conormal module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJQ","source_file":"dpa.tex","source_line":2469,"source_end_line":2478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2469-L2478","statement_sha256":"acf242aa584db75da6c0ec700e12a6a4cb65beb024cb6b750ac8544e4b539b9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4844,"rank":4844,"depth":15,"x":2350.529,"y":299.63,"cluster":"homological-algebra"},{"id":"stacks:0FJR","tag":"0FJR","title":"Freeness of the conormal module · Lemma 0FJR","summary":"Local version of [Vasconcelos] Let R be a Noetherian local ring. Let I ⊂ R be an ideal of finite projective dimension over R. If F ⊂ I/I^2 is a direct summand free of rank r, then there exists a regular sequence x_1, …, x_r ∈ I such that x_1 bmod I^2, …, x_r bmod I^2 generate F.","statement_latex":"\\begin{reference}\nLocal version of \\cite[Theorem 1.1]{Vasconcelos}\n\\end{reference}\nLet $R$ be a Noetherian local ring. Let $I \\subset R$ be an ideal\nof finite projective dimension over $R$. If $F \\subset I/I^2$\nis a direct summand free of rank $r$, then there exists a regular sequence\n$x_1, \\ldots, x_r \\in I$ such that $x_1 \\bmod I^2, \\ldots, x_r \\bmod I^2$\ngenerate $F$.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Freeness of the conormal module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJR","source_file":"dpa.tex","source_line":2505,"source_end_line":2515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2505-L2515","statement_sha256":"cff017789c8dc2ecad36f8193f7336e0c02020e8e35d6bab23809f14069cdbf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4845,"rank":4845,"depth":16,"x":2511.466,"y":87.685,"cluster":"homological-algebra"},{"id":"stacks:0FJS","tag":"0FJS","title":"Freeness of the conormal module · Proposition 0FJS","summary":"Variant of [Vasconcelos]. See also [Iyengar] and [Ferrand-lci]. Let R be a Noetherian ring. Let I ⊂ R be an ideal which has finite projective dimension and such that I/I^2 is finite locally free over R/I. Then I is a regular ideal (More on Algebra, Definition [Tag 07CV]).","statement_latex":"\\begin{reference}\nVariant of \\cite[Corollary 1]{Vasconcelos}. See also\n\\cite{Iyengar} and \\cite{Ferrand-lci}.\n\\end{reference}\nLet $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal\nwhich has finite projective dimension and such that $I/I^2$ is\nfinite locally free over $R/I$. Then $I$ is a regular ideal\n(More on Algebra, Definition \\ref{more-algebra-definition-regular-ideal}).","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Freeness of the conormal module","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJS","source_file":"dpa.tex","source_line":2543,"source_end_line":2553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2543-L2553","statement_sha256":"6e1c84a036406adb7f44cb90e3719ffb293a11dbfd955a21e9328fed3e108cf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4846,"rank":4846,"depth":17,"x":2563.285,"y":335.537,"cluster":"homological-algebra"},{"id":"stacks:0FJT","tag":"0FJT","title":"Freeness of the conormal module · Lemma 0FJT","summary":"Let A → B be a perfect (More on Algebra, Definition [Tag 067H]) ring homomorphism of Noetherian rings. Then the following are equivalent • NL_B/A has tor-amplitude in [-1, 0], • NL_B/A is a perfect object of D(B) with tor-amplitude in [-1, 0], and • A → B is a local complete intersection (More on Algebra, Definition [Tag 07D0]).","statement_latex":"Let $A \\to B$ be a perfect (More on Algebra, Definition\n\\ref{more-algebra-definition-pseudo-coherent-perfect})\nring homomorphism of Noetherian rings. Then the following are equivalent\n\\begin{enumerate}\n\\item $\\NL_{B/A}$ has tor-amplitude in $[-1, 0]$,\n\\item $\\NL_{B/A}$ is a perfect object of $D(B)$\nwith tor-amplitude in $[-1, 0]$, and\n\\item $A \\to B$ is a local complete intersection\n(More on Algebra, Definition\n\\ref{more-algebra-definition-local-complete-intersection}).\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Freeness of the conormal module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJT","source_file":"dpa.tex","source_line":2574,"source_end_line":2587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2574-L2587","statement_sha256":"1329804a5c4c0d3794c10180a17d9a98cb9a5f15fe46da261f1ab4dd4bfb8274","origin":"The Stacks Project","memory_eligible":false,"source_rank":4847,"rank":4847,"depth":18,"x":2325.522,"y":182.025,"cluster":"homological-algebra"},{"id":"stacks:0FJV","tag":"0FJV","title":"Freeness of the conormal module · Lemma 0FJV","summary":"Let A → B be a flat ring map of finite presentation. Then the following are equivalent • NL_B/A has tor-amplitude in [-1, 0], • NL_B/A is a perfect object of D(B) with tor-amplitude in [-1, 0], • A → B is syntomic (Algebra, Definition [Tag 00SL]), and • A → B is a local complete intersection (More on Algebra, Definition [Tag 07D0]).","statement_latex":"Let $A \\to B$ be a flat ring map of finite presentation.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $\\NL_{B/A}$ has tor-amplitude in $[-1, 0]$,\n\\item $\\NL_{B/A}$ is a perfect object of $D(B)$\nwith tor-amplitude in $[-1, 0]$,\n\\item $A \\to B$ is syntomic\n(Algebra, Definition \\ref{algebra-definition-lci}), and\n\\item $A \\to B$ is a local complete intersection\n(More on Algebra, Definition\n\\ref{more-algebra-definition-local-complete-intersection}).\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Freeness of the conormal module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJV","source_file":"dpa.tex","source_line":2610,"source_end_line":2624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2610-L2624","statement_sha256":"6ec1ed0cada67bc6a9ccc8a08f21c2aee411143b3f599518734ceb295bfae8cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4848,"rank":4848,"depth":36,"x":2624.592,"y":160.288,"cluster":"homological-algebra"},{"id":"stacks:0GZ4","tag":"0GZ4","title":"Koszul complexes and Tate resolutions · Lemma 0GZ4","summary":"In the situation above, if R is Noetherian, then for every n there exists an N ≥ n and maps K_N → A → R/(f_1^N, …, f_r^N) and A → K_n with the following properties • (A, d, γ) is as in Definition [Tag 09PJ], • A → R/(f_1^N, …, f_r^N) is a quasi-isomorphism, • the composition K_N → A → R/(f_1^N, …, f_r^N) is the canonical map, • the composition K_N → A → K_n is the transition map, • A_0 = R → R/(f_1^N, …, f_r^N) is the canonical surjection, • A is a graded divided power…","statement_latex":"In the situation above, if $R$ is Noetherian, then\nfor every $n$ there exists an $N \\geq n$ and maps\n$$\nK_N \\to A \\to R/(f_1^N, \\ldots, f_r^N)\\quad\\text{and}\\quad A \\to K_n\n$$\nwith the following properties\n\\begin{enumerate}\n\\item $(A, \\text{d}, \\gamma)$ is as in\nDefinition \\ref{definition-divided-powers-dga},\n\\item $A \\to R/(f_1^N, \\ldots, f_r^N)$ is a quasi-isomorphism,\n\\item the composition $K_N \\to A \\to R/(f_1^N, \\ldots, f_r^N)$\nis the canonical map,\n\\item the composition $K_N \\to A \\to K_n$ is the transition map,\n\\item $A_0 = R \\to R/(f_1^N, \\ldots, f_r^N)$ is the canonical\nsurjection,\n\\item $A$ is a graded divided power polynomial algebra over $R$\nwith finitely many generators in each degree, and\n\\item $A \\to K_n$ is a homomorphism of differential graded $R$-algebras\ncompatible with divided powers which induces the canonical map\n$R/(f_1^N, \\ldots, f_r^N) \\to R/(f_1^n, \\ldots, f_r^n)$ on\nhomology in degree $0$.\n\\end{enumerate}\nCondition (4) means that $A$ is constructed out of $A_0$ by\nsuccessively adjoining a finite set of variables $T$ in each degree\n$> 0$ as in Example \\ref{example-adjoining-odd} or \\ref{example-adjoining-even}.","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Koszul complexes and Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZ4","source_file":"dpa.tex","source_line":2690,"source_end_line":2717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2690-L2717","statement_sha256":"08f00f6b6f1dcb709ec33ca641f95b19f4c3bbdbe80d0d4aee1cace1115fee8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4849,"rank":4849,"depth":8,"x":2421.341,"y":346.201,"cluster":"homological-algebra"},{"id":"stacks:0GZ6","tag":"0GZ6","title":"Koszul complexes and Tate resolutions · Lemma 0GZ6","summary":"Let (A, d, γ), d ≥ 1, f ∈ A_d - 1, and Alangle T rangle be as in Lemma [Tag 09PM]. • If d = 1, then there is a long exact sequences … → H_0(A) xrightarrowf H_0(A) → H_0(Alangle T rangle) → 0 • For d = 2 there is a bounded spectral sequence (E_1)_i, j = H_j - i(A) · T^[i] converging to H_i + j(Alangle T rangle). The differential (d_1)_i, j : H_j - i(A) · T^[i] → H_j - i + 1(A) · T^[i - 1] sends xi · T^[i] to the class of f xi · T^[i - 1]. • Add more here for other degrees…","statement_latex":"Let $(A, \\text{d}, \\gamma)$, $d \\geq 1$, $f \\in A_{d - 1}$,\nand $A\\langle T \\rangle$ be as in Lemma \\ref{lemma-extend-differential}.\n\\begin{enumerate}\n\\item If $d = 1$, then there is a long exact sequences\n$$\n\\ldots \\to H_0(A) \\xrightarrow{f} H_0(A) \\to H_0(A\\langle T \\rangle) \\to 0\n$$\n\\item For $d = 2$ there is a bounded spectral sequence\n$(E_1)_{i, j} = H_{j - i}(A) \\cdot T^{[i]}$\nconverging to $H_{i + j}(A\\langle T \\rangle)$. The differential\n$(d_1)_{i, j} : H_{j - i}(A) \\cdot T^{[i]} \\to\nH_{j - i + 1}(A) \\cdot T^{[i - 1]}$\nsends $\\xi \\cdot T^{[i]}$ to the class of $f \\xi \\cdot T^{[i - 1]}$.\n\\item Add more here for other degrees as needed.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Koszul complexes and Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZ6","source_file":"dpa.tex","source_line":2838,"source_end_line":2855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2838-L2855","statement_sha256":"868cf45c1d9d958f47599903a009965d76dcc2e9fd968a58ad60420695600440","origin":"The Stacks Project","memory_eligible":false,"source_rank":4850,"rank":4850,"depth":2,"x":2421.709,"y":93.531,"cluster":"homological-algebra"},{"id":"stacks:0GZ7","tag":"0GZ7","title":"Koszul complexes and Tate resolutions · Lemma 0GZ7","summary":"In the situation above, for all n ≥ t ≥ 1 there exists an N > n and a map K_t → K_n ⊗_R K_t in the derived category of left differential graded K_N-modules whose composition with the multiplication map is the transition map (in either direction).","statement_latex":"In the situation above, for all $n \\geq t \\geq 1$ there exists an $N > n$\nand a map\n$$\nK_t \\longrightarrow K_n \\otimes_R K_t\n$$\nin the derived category of left differential graded $K_N$-modules\nwhose composition with the multiplication map is the transition map\n(in either direction).","area":"Homological Algebra","chapter":"Divided Power Algebra","chapter_id":"dpa","section":"Koszul complexes and Tate resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZ7","source_file":"dpa.tex","source_line":2875,"source_end_line":2885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dpa.tex#L2875-L2885","statement_sha256":"3ffc7dd3e2a24599c175df312084414527283effc35d4a764a7546e64e71b478","origin":"The Stacks Project","memory_eligible":false,"source_rank":4851,"rank":4851,"depth":3,"x":2624.828,"y":280.24,"cluster":"homological-algebra"},{"id":"stacks:0FQW","tag":"0FQW","title":"Sheaves of graded algebras · Definition 0FQW","summary":"Let (C, O) be a ringed site. A sheaf of graded O-algebras or a sheaf of graded algebras on (C, O) is given by a family A^n indexed by n ∈ Z of O-modules endowed with O-bilinear maps A^n × A^m → A^n + m, (a, b) ↦ ab called the multiplication maps with the following properties • multiplication is associative, and • there is a global section 1 of A^0 which is a two-sided identity for multiplication. We often denote such a structure A. A homomorphism of graded O-algebras f :…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A\n{\\it sheaf of graded $\\mathcal{O}$-algebras}\nor a {\\it sheaf of graded algebras} on $(\\mathcal{C}, \\mathcal{O})$\nis given by a family $\\mathcal{A}^n$ indexed by $n \\in \\mathbf{Z}$\nof $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps\n$$\n\\mathcal{A}^n \\times \\mathcal{A}^m \\to \\mathcal{A}^{n + m},\\quad\n(a, b) \\longmapsto ab\n$$\ncalled the multiplication maps with the following properties\n\\begin{enumerate}\n\\item multiplication is associative, and\n\\item there is a global section $1$ of $\\mathcal{A}^0$\nwhich is a two-sided identity for multiplication.\n\\end{enumerate}\nWe often denote such a structure $\\mathcal{A}$.\nA {\\it homomorphism of graded $\\mathcal{O}$-algebras}\n$f : \\mathcal{A} \\to \\mathcal{B}$ is a family of maps\n$f^n : \\mathcal{A}^n \\to \\mathcal{B}^n$\nof $\\mathcal{O}$-modules compatible with the multiplication maps.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of graded algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQW","source_file":"sdga.tex","source_line":53,"source_end_line":75,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L53-L75","statement_sha256":"2c715159aaa76face6c76f7be35376b1fa85ba232d6486225bc471bf679d2961","origin":"The Stacks Project","memory_eligible":false,"source_rank":4852,"rank":4852,"depth":0,"x":2324.609,"y":257.796,"cluster":"homological-algebra"},{"id":"stacks:0FQZ","tag":"0FQZ","title":"Sheaves of graded modules · Definition 0FQZ","summary":"Let (C, O) be a ringed site. Let A be a sheaf of graded algebras on (C, O). A (right) graded A-module or (right) graded module over A is given by a family M^n indexed by n ∈ Z of O-modules endowed with O-bilinear maps M^n × A^m → M^n + m, (x, a) ↦ xa called the multiplication maps with the following properties • multiplication satisfies (xa)a' = x(aa'), • the identity section 1 of A^0 acts as the identity on M^n for all n. We often say \"let M be a graded A-module\" to…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nA (right) {\\it graded $\\mathcal{A}$-module} or (right)\n{\\it graded module} over $\\mathcal{A}$\nis given by a family $\\mathcal{M}^n$ indexed by $n \\in \\mathbf{Z}$\nof $\\mathcal{O}$-modules endowed with\n$\\mathcal{O}$-bilinear maps\n$$\n\\mathcal{M}^n \\times \\mathcal{A}^m \\to \\mathcal{M}^{n + m},\\quad\n(x, a) \\longmapsto xa\n$$\ncalled the multiplication maps with the following properties\n\\begin{enumerate}\n\\item multiplication satisfies $(xa)a' = x(aa')$,\n\\item the identity section $1$ of $\\mathcal{A}^0$\nacts as the identity on $\\mathcal{M}^n$ for all $n$.\n\\end{enumerate}\nWe often say ``let $\\mathcal{M}$ be a graded $\\mathcal{A}$-module''\nto indicate this situation.\nA {\\it homomorphism of graded $\\mathcal{A}$-modules}\n$f : \\mathcal{M} \\to \\mathcal{N}$ is a family of maps\n$f^n : \\mathcal{M}^n \\to \\mathcal{N}^n$\nof $\\mathcal{O}$-modules compatible with the multiplication maps.\nThe category of (right) graded $\\mathcal{A}$-modules\nis denoted $\\textit{Mod}(\\mathcal{A})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of graded modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FQZ","source_file":"sdga.tex","source_line":124,"source_end_line":152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L124-L152","statement_sha256":"9f88ae1d8d291f9fbb512c367991954bdf7217730388d9008fdaf7371e5838d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4853,"rank":4853,"depth":0,"x":2564.271,"y":103.843,"cluster":"homological-algebra"},{"id":"stacks:0FR0","tag":"0FR0","title":"Sheaves of graded modules · Lemma 0FR0","summary":"Let (C, O) be a ringed site. Let A be a graded O-algebra. The category Mod(A) is an abelian category with the following properties • Mod(A) has arbitrary direct sums, • Mod(A) has arbitrary colimits, • filtered colimit in Mod(A) are exact, • Mod(A) has arbitrary products, • Mod(A) has arbitrary limits. The functor Mod(A) → Mod(O), M ↦ M^n sending a graded A-module to its nth term commutes with all limits and colimits.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a graded $\\mathcal{O}$-algebra.\nThe category $\\textit{Mod}(\\mathcal{A})$ is an abelian category\nwith the following properties\n\\begin{enumerate}\n\\item $\\textit{Mod}(\\mathcal{A})$ has arbitrary direct sums,\n\\item $\\textit{Mod}(\\mathcal{A})$ has arbitrary colimits,\n\\item filtered colimit in $\\textit{Mod}(\\mathcal{A})$ are exact,\n\\item $\\textit{Mod}(\\mathcal{A})$ has arbitrary products,\n\\item $\\textit{Mod}(\\mathcal{A})$ has arbitrary limits.\n\\end{enumerate}\nThe functor\n$$\n\\textit{Mod}(\\mathcal{A}) \\longrightarrow \\textit{Mod}(\\mathcal{O}),\\quad\n\\mathcal{M} \\longmapsto \\mathcal{M}^n\n$$\nsending a graded $\\mathcal{A}$-module to its $n$th term commutes\nwith all limits and colimits.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FR0","source_file":"sdga.tex","source_line":168,"source_end_line":188,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L168-L188","statement_sha256":"5de287a757b578856dc43f6d13d28bbec61b5ad65127adc58346cd5b93f84195","origin":"The Stacks Project","memory_eligible":false,"source_rank":4854,"rank":4854,"depth":0,"x":2511.301,"y":353.601,"cluster":"homological-algebra"},{"id":"stacks:0FR5","tag":"0FR5","title":"Sheaves of graded bimodules and tensor-hom adjunction · Definition 0FR5","summary":"Let (C, O) be a ringed site. Let A and B be a sheaves of graded algebras on (C, O). A graded (A, B)-bimodule is given by a family M^n indexed by n ∈ Z of O-modules endowed with O-bilinear maps M^n × B^m → M^n + m, (x, b) ↦ xb and A^n × M^m → M^n + m, (a, x) ↦ ax called the multiplication maps with the following properties • multiplication satisfies a(a'x) = (aa')x and (xb)b' = x(bb'), • (ax)b = a(xb), • the identity section 1 of A^0 acts as the identity by multiplication,…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$\nand $\\mathcal{B}$ be a sheaves of graded algebras on\n$(\\mathcal{C}, \\mathcal{O})$. A\n{\\it graded $(\\mathcal{A}, \\mathcal{B})$-bimodule}\nis given by a family $\\mathcal{M}^n$ indexed by $n \\in \\mathbf{Z}$\nof $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps\n$$\n\\mathcal{M}^n \\times \\mathcal{B}^m \\to \\mathcal{M}^{n + m},\\quad\n(x, b) \\longmapsto xb\n$$\nand\n$$\n\\mathcal{A}^n \\times \\mathcal{M}^m \\to \\mathcal{M}^{n + m},\\quad\n(a, x) \\longmapsto ax\n$$\ncalled the multiplication maps with the following properties\n\\begin{enumerate}\n\\item multiplication satisfies $a(a'x) = (aa')x$ and\n$(xb)b' = x(bb')$,\n\\item $(ax)b = a(xb)$,\n\\item the identity section $1$ of $\\mathcal{A}^0$ acts as the\nidentity by multiplication, and\n\\item the identity section $1$ of\n$\\mathcal{B}^0$ acts as the identity by multiplication.\n\\end{enumerate}\nWe often denote such a structure $\\mathcal{M}$.\nA {\\it homomorphism of graded $(\\mathcal{A}, \\mathcal{B})$-bimodules}\n$f : \\mathcal{M} \\to \\mathcal{N}$ is a family of maps\n$f^n : \\mathcal{M}^n \\to \\mathcal{N}^n$\nof $\\mathcal{O}$-modules compatible with the multiplication maps.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of graded bimodules and tensor-hom adjunction","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FR5","source_file":"sdga.tex","source_line":440,"source_end_line":472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L440-L472","statement_sha256":"47011be9b567ca8b2288d3adceb4f16119b4f7c13b1836265f770b330015cfe9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4855,"rank":4855,"depth":0,"x":2349.353,"y":139.166,"cluster":"homological-algebra"},{"id":"stacks:0FR6","tag":"0FR6","title":"Sheaves of graded bimodules and tensor-hom adjunction · Lemma 0FR6","summary":"Let (C, O) be a ringed site. Let A and B be a sheaves of graded algebras on (C, O). Let M be a right graded A-module. Let N be a graded (A, B)-bimodule. Let L be a right graded B-module. With conventions as above we have Hom_Mod^gr(B)( M ⊗_A N, L) = Hom_Mod^gr(A)( M, SheafHom_B^gr(N, L)) and SheafHom_B^gr( M ⊗_A N, L) = SheafHom_A^gr( M, SheafHom_B^gr(N, L)) functorially in M, N, L.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$\nand $\\mathcal{B}$ be a sheaves of graded algebras on\n$(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$ be a right\ngraded $\\mathcal{A}$-module. Let $\\mathcal{N}$ be a\ngraded $(\\mathcal{A}, \\mathcal{B})$-bimodule. Let $\\mathcal{L}$\nbe a right graded $\\mathcal{B}$-module. With conventions as above\nwe have\n$$\n\\Hom_{\\textit{Mod}^{gr}(\\mathcal{B})}(\n\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) =\n\\Hom_{\\textit{Mod}^{gr}(\\mathcal{A})}(\n\\mathcal{M}, \\SheafHom_\\mathcal{B}^{gr}(\\mathcal{N}, \\mathcal{L}))\n$$\nand\n$$\n\\SheafHom_\\mathcal{B}^{gr}(\n\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) =\n\\SheafHom_\\mathcal{A}^{gr}(\n\\mathcal{M}, \\SheafHom_\\mathcal{B}^{gr}(\\mathcal{N}, \\mathcal{L}))\n$$\nfunctorially in $\\mathcal{M}$, $\\mathcal{N}$, $\\mathcal{L}$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of graded bimodules and tensor-hom adjunction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FR6","source_file":"sdga.tex","source_line":560,"source_end_line":583,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L560-L583","statement_sha256":"326edaec7c14cbc60e0bdb8ea48a18b261a802271c569528ab0ed7534c708e53","origin":"The Stacks Project","memory_eligible":false,"source_rank":4856,"rank":4856,"depth":0,"x":2641.5,"y":205.459,"cluster":"homological-algebra"},{"id":"stacks:0FR8","tag":"0FR8","title":"Pull and push for sheaves of graded modules · Lemma 0FR8","summary":"In the situation above we have Hom_Mod^gr(B)( N, f_*M) = Hom_Mod^gr(A)( f^*N, M)","statement_latex":"In the situation above we have\n$$\n\\Hom_{\\textit{Mod}^{gr}(\\mathcal{B})}(\n\\mathcal{N}, f_*\\mathcal{M}) =\n\\Hom_{\\textit{Mod}^{gr}(\\mathcal{A})}(\nf^*\\mathcal{N}, \\mathcal{M})\n$$","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Pull and push for sheaves of graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FR8","source_file":"sdga.tex","source_line":772,"source_end_line":781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L772-L781","statement_sha256":"77d298008bbac80e239b39400bf034d1d30876d2d78e425512c7859f603d2aaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4857,"rank":4857,"depth":0,"x":2372.501,"y":322.461,"cluster":"homological-algebra"},{"id":"stacks:0FRA","tag":"0FRA","title":"Localization and sheaves of graded modules · Lemma 0FRA","summary":"In the situation above we have Hom_Mod^gr(A)( j_!M, N) = Hom_Mod^gr(A_U)( M, j^*N)","statement_latex":"In the situation above we have\n$$\n\\Hom_{\\textit{Mod}^{gr}(\\mathcal{A})}(\nj_!\\mathcal{M}, \\mathcal{N}) =\n\\Hom_{\\textit{Mod}^{gr}(\\mathcal{A}_U)}(\n\\mathcal{M}, j^*\\mathcal{N})\n$$","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Localization and sheaves of graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRA","source_file":"sdga.tex","source_line":871,"source_end_line":880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L871-L880","statement_sha256":"c1ecc4e448e847bef3e8264561db26aa6c7588eee25fcee1565702c31312a1ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":4858,"rank":4858,"depth":3,"x":2476.869,"y":83.316,"cluster":"homological-algebra"},{"id":"stacks:0FRB","tag":"0FRB","title":"Localization and sheaves of graded modules · Lemma 0FRB","summary":"In the situation above, let M be a right graded A_U-module and let N be a left graded A-module. Then j_!M ⊗_A N = j_!(M ⊗_A_U N|_U) as graded O-modules functorially in M and N.","statement_latex":"In the situation above, let $\\mathcal{M}$ be a right graded\n$\\mathcal{A}_U$-module and let $\\mathcal{N}$ be a left graded\n$\\mathcal{A}$-module. Then\n$$\nj_!\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N} =\nj_!(\\mathcal{M} \\otimes_{\\mathcal{A}_U} \\mathcal{N}|_U)\n$$\nas graded $\\mathcal{O}$-modules functorially in $\\mathcal{M}$\nand $\\mathcal{N}$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Localization and sheaves of graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRB","source_file":"sdga.tex","source_line":942,"source_end_line":953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L942-L953","statement_sha256":"f514248b2d8d98a99e047e493a55f217cebc38a23fa703c55ab1feffd7416d8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4859,"rank":4859,"depth":3,"x":2592.335,"y":319.108,"cluster":"homological-algebra"},{"id":"stacks:0FRD","tag":"0FRD","title":"Shift functors on sheaves of graded modules · Lemma 0FRD","summary":"Let (C, O) be a ringed site. Let A be a graded O-algebra. The category Mod(A) is a Grothendieck abelian category.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a graded $\\mathcal{O}$-algebra.\nThe category $\\textit{Mod}(\\mathcal{A})$ is a Grothendieck abelian category.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Shift functors on sheaves of graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRD","source_file":"sdga.tex","source_line":1035,"source_end_line":1040,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L1035-L1040","statement_sha256":"7d1146123f9f0edbc0328d7be45c4413db866060828f6b5d9dcce44ff7f80f5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4860,"rank":4860,"depth":1,"x":2317.308,"y":210.652,"cluster":"homological-algebra"},{"id":"stacks:0FRF","tag":"0FRF","title":"Sheaves of differential graded algebras · Definition 0FRF","summary":"Let (C, O) be a ringed site. A sheaf of differential graded O-algebras or a sheaf of differential graded algebras on (C, O) is a cochain complex A^bullet of O-modules endowed with O-bilinear maps A^n × A^m → A^n + m, (a, b) ↦ ab called the multiplication maps with the following properties • multiplication is associative, • there is a global section 1 of A^0 which is a two-sided identity for multiplication, • for U ∈ Ob(C), a ∈ A^n(U), and b ∈ A^m(U) we have d^n + m(ab) =…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. A\n{\\it sheaf of differential graded $\\mathcal{O}$-algebras}\nor a {\\it sheaf of differential graded algebras} on\n$(\\mathcal{C}, \\mathcal{O})$ is a cochain complex\n$\\mathcal{A}^\\bullet$ of $\\mathcal{O}$-modules\nendowed with $\\mathcal{O}$-bilinear maps\n$$\n\\mathcal{A}^n \\times \\mathcal{A}^m \\to \\mathcal{A}^{n + m},\\quad\n(a, b) \\longmapsto ab\n$$\ncalled the multiplication maps with the following properties\n\\begin{enumerate}\n\\item multiplication is associative,\n\\item there is a global section $1$ of $\\mathcal{A}^0$\nwhich is a two-sided identity for multiplication,\n\\item for $U \\in \\Ob(\\mathcal{C})$, $a \\in \\mathcal{A}^n(U)$, and\n$b \\in \\mathcal{A}^m(U)$ we have\n$$\n\\text{d}^{n + m}(ab) = \\text{d}^n(a)b + (-1)^n a\\text{d}^m(b)\n$$\n\\end{enumerate}\nWe often denote such a structure $(\\mathcal{A}, \\text{d})$.\nA {\\it homomorphism of differential graded $\\mathcal{O}$-algebras}\nfrom $(\\mathcal{A}, \\text{d})$ to $(\\mathcal{B}, \\text{d})$ is a map\n$f : \\mathcal{A}^\\bullet \\to \\mathcal{B}^\\bullet$ of complexes\nof $\\mathcal{O}$-modules compatible with the multiplication maps.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of differential graded algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRF","source_file":"sdga.tex","source_line":1078,"source_end_line":1106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L1078-L1106","statement_sha256":"bb1d28d5fa52004294617a1cce88ee700e104476b77ab400a13b4f2909fad2b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4861,"rank":4861,"depth":0,"x":2607.608,"y":134.495,"cluster":"homological-algebra"},{"id":"stacks:0FRI","tag":"0FRI","title":"Sheaves of differential graded modules · Definition 0FRI","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). A (right) differential graded A-module or (right) differential graded module over A is a cochain complex M^bullet endowed with O-bilinear maps M^n × A^m → M^n + m, (x, a) ↦ xa called the multiplication maps with the following properties • multiplication satisfies (xa)a' = x(aa'), • the identity section 1 of A^0 acts as the identity on M^n for all n, • for U ∈ Ob(C), x ∈ M^n(U),…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nA (right) {\\it differential graded $\\mathcal{A}$-module} or (right)\n{\\it differential graded module} over $\\mathcal{A}$\nis a cochain complex $\\mathcal{M}^\\bullet$ endowed with\n$\\mathcal{O}$-bilinear maps\n$$\n\\mathcal{M}^n \\times \\mathcal{A}^m \\to \\mathcal{M}^{n + m},\\quad\n(x, a) \\longmapsto xa\n$$\ncalled the multiplication maps with the following properties\n\\begin{enumerate}\n\\item multiplication satisfies $(xa)a' = x(aa')$,\n\\item the identity section $1$ of $\\mathcal{A}^0$\nacts as the identity on $\\mathcal{M}^n$ for all $n$,\n\\item for $U \\in \\Ob(\\mathcal{C})$, $x \\in \\mathcal{M}^n(U)$, and\n$a \\in \\mathcal{A}^m(U)$ we have\n$$\n\\text{d}^{n + m}(xa) = \\text{d}^n(x)a + (-1)^n x\\text{d}^m(a)\n$$\n\\end{enumerate}\nWe often say ``let $\\mathcal{M}$ be a differential graded\n$\\mathcal{A}$-module'' to indicate this situation.\nA {\\it homomorphism of differential graded $\\mathcal{A}$-modules}\nfrom $\\mathcal{M}$ to $\\mathcal{N}$ is a map\n$f : \\mathcal{M}^\\bullet \\to \\mathcal{N}^\\bullet$ of complexes\nof $\\mathcal{O}$-modules compatible with the multiplication maps.\nThe category of (right) differential graded $\\mathcal{A}$-modules\nis denoted $\\textit{Mod}(\\mathcal{A}, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of differential graded modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRI","source_file":"sdga.tex","source_line":1178,"source_end_line":1210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L1178-L1210","statement_sha256":"7b546156d37837e4ea7f407089da823798c773a5fd96e218096ff4ba7cfb8f6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4862,"rank":4862,"depth":0,"x":2454.64,"y":355.589,"cluster":"homological-algebra"},{"id":"stacks:0FRJ","tag":"0FRJ","title":"Sheaves of differential graded modules · Lemma 0FRJ","summary":"Let (C, O) be a ringed site. Let (A, d) be a differential graded O-algebra. The category Mod(A, d) is an abelian category with the following properties • Mod(A, d) has arbitrary direct sums, • Mod(A, d) has arbitrary colimits, • filtered colimit in Mod(A, d) are exact, • Mod(A, d) has arbitrary products, • Mod(A, d) has arbitrary limits. The forgetful functor Mod(A, d) → Mod(A) sending a differential graded A-module to its underlying graded module commutes with all limits…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$ be a differential graded $\\mathcal{O}$-algebra.\nThe category $\\textit{Mod}(\\mathcal{A}, \\text{d})$ is an abelian category\nwith the following properties\n\\begin{enumerate}\n\\item $\\textit{Mod}(\\mathcal{A}, \\text{d})$ has arbitrary direct sums,\n\\item $\\textit{Mod}(\\mathcal{A}, \\text{d})$ has arbitrary colimits,\n\\item filtered colimit in $\\textit{Mod}(\\mathcal{A}, \\text{d})$ are exact,\n\\item $\\textit{Mod}(\\mathcal{A}, \\text{d})$ has arbitrary products,\n\\item $\\textit{Mod}(\\mathcal{A}, \\text{d})$ has arbitrary limits.\n\\end{enumerate}\nThe forgetful functor\n$$\n\\textit{Mod}(\\mathcal{A}, \\text{d})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{A})\n$$\nsending a differential graded $\\mathcal{A}$-module to its underlying\ngraded module commutes with all limits and colimits.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRJ","source_file":"sdga.tex","source_line":1226,"source_end_line":1247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L1226-L1247","statement_sha256":"c068f542732df8fe86bb768f814798cf5b02c4512a53c0255a3e4ecf568bcfd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4863,"rank":4863,"depth":1,"x":2389.576,"y":105.519,"cluster":"homological-algebra"},{"id":"stacks:0FRQ","tag":"0FRQ","title":"Sheaves of differential graded bimodules and tensor-hom adjunction · Definition 0FRQ","summary":"Let (C, O) be a ringed site. Let A and B be a sheaves of differential graded algebras on (C, O). A differential graded (A, B)-bimodule is given by a complex M^bullet of O-modules endowed with O-bilinear maps M^n × B^m → M^n + m, (x, b) ↦ xb and A^n × M^m → M^n + m, (a, x) ↦ ax called the multiplication maps with the following properties • multiplication satisfies a(a'x) = (aa')x and (xb)b' = x(bb'), • (ax)b = a(xb), • d(ax) = d(a) x + (-1)^deg(a)a d(x) and d(xb) = d(x) b…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$\nand $\\mathcal{B}$ be a sheaves of differential graded algebras on\n$(\\mathcal{C}, \\mathcal{O})$. A\n{\\it differential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule}\nis given by a complex $\\mathcal{M}^\\bullet$\nof $\\mathcal{O}$-modules endowed with $\\mathcal{O}$-bilinear maps\n$$\n\\mathcal{M}^n \\times \\mathcal{B}^m \\to \\mathcal{M}^{n + m},\\quad\n(x, b) \\longmapsto xb\n$$\nand\n$$\n\\mathcal{A}^n \\times \\mathcal{M}^m \\to \\mathcal{M}^{n + m},\\quad\n(a, x) \\longmapsto ax\n$$\ncalled the multiplication maps with the following properties\n\\begin{enumerate}\n\\item multiplication satisfies $a(a'x) = (aa')x$ and\n$(xb)b' = x(bb')$,\n\\item $(ax)b = a(xb)$,\n\\item $\\text{d}(ax) = \\text{d}(a) x + (-1)^{\\deg(a)}a \\text{d}(x)$ and\n$\\text{d}(xb) = \\text{d}(x) b + (-1)^{\\deg(x)}x \\text{d}(b)$,\n\\item the identity section $1$ of $\\mathcal{A}^0$ acts as the\nidentity by multiplication, and\n\\item the identity section $1$ of\n$\\mathcal{B}^0$ acts as the identity by multiplication.\n\\end{enumerate}\nWe often denote such a structure $\\mathcal{M}$ and sometimes\nwe write ${}_\\mathcal{A}\\mathcal{M}_\\mathcal{B}$.\nA {\\it homomorphism of differential graded\n$(\\mathcal{A}, \\mathcal{B})$-bimodules}\n$f : \\mathcal{M} \\to \\mathcal{N}$ is a map of complexes\n$f : \\mathcal{M}^\\bullet \\to \\mathcal{N}^\\bullet$\nof $\\mathcal{O}$-modules compatible with the multiplication maps.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of differential graded bimodules and tensor-hom adjunction","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRQ","source_file":"sdga.tex","source_line":1661,"source_end_line":1697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L1661-L1697","statement_sha256":"0e96ad1d9870bdcced800772fbc90a1cd3060297940522a86c1d102733a03b24","origin":"The Stacks Project","memory_eligible":false,"source_rank":4864,"rank":4864,"depth":0,"x":2638.893,"y":253.14,"cluster":"homological-algebra"},{"id":"stacks:0FRR","tag":"0FRR","title":"Sheaves of differential graded bimodules and tensor-hom adjunction · Lemma 0FRR","summary":"Let (C, O) be a ringed site. Let A and B be a sheaves of differential graded algebras on (C, O). Let N be a right differential graded B-module. There is a 1-to-1 correspondence between (A, B)-bimodule structures on N compatible with the given differential graded B-module structure and homomorphisms A → SheafHom^dg_B(N, N) of differential graded O-algebras.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$\nand $\\mathcal{B}$ be a sheaves of differential graded algebras on\n$(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{N}$ be a right differential\ngraded $\\mathcal{B}$-module. There is a $1$-to-$1$ correspondence\nbetween $(\\mathcal{A}, \\mathcal{B})$-bimodule structures on\n$\\mathcal{N}$ compatible with the given\ndifferential graded $\\mathcal{B}$-module structure and homomorphisms\n$$\n\\mathcal{A}\n\\longrightarrow\n\\SheafHom^{dg}_\\mathcal{B}(\\mathcal{N}, \\mathcal{N})\n$$\nof differential graded $\\mathcal{O}$-algebras.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of differential graded bimodules and tensor-hom adjunction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRR","source_file":"sdga.tex","source_line":1717,"source_end_line":1732,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L1717-L1732","statement_sha256":"96ce07e25edef374f3f63e159464c2aa6a42bfa9242fd3f587cf283890d4cb1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4865,"rank":4865,"depth":0,"x":2336.046,"y":285.786,"cluster":"homological-algebra"},{"id":"stacks:0FRS","tag":"0FRS","title":"Sheaves of differential graded bimodules and tensor-hom adjunction · Lemma 0FRS","summary":"Let (C, O) be a ringed site. Let A and B be a sheaves of differential graded algebras on (C, O). Let M be a right differential graded A-module. Let N be a differential graded (A, B)-bimodule. Let L be a right differential graded B-module. With conventions as above we have Hom_Mod^dg(B, d)( M ⊗_A N, L) = Hom_Mod^dg(A, d)( M, SheafHom_B^dg(N, L)) and SheafHom_B^dg( M ⊗_A N, L) = SheafHom_A^dg( M, SheafHom_B^dg(N, L)) functorially in M, N, L.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$\nand $\\mathcal{B}$ be a sheaves of differential graded algebras on\n$(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$ be a right\ndifferential graded $\\mathcal{A}$-module. Let $\\mathcal{N}$ be a\ndifferential graded $(\\mathcal{A}, \\mathcal{B})$-bimodule. Let $\\mathcal{L}$\nbe a right differential graded $\\mathcal{B}$-module. With conventions as above\nwe have\n$$\n\\Hom_{\\textit{Mod}^{dg}(\\mathcal{B}, \\text{d})}(\n\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) =\n\\Hom_{\\textit{Mod}^{dg}(\\mathcal{A}, \\text{d})}(\n\\mathcal{M}, \\SheafHom_\\mathcal{B}^{dg}(\\mathcal{N}, \\mathcal{L}))\n$$\nand\n$$\n\\SheafHom_\\mathcal{B}^{dg}(\n\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}, \\mathcal{L}) =\n\\SheafHom_\\mathcal{A}^{dg}(\n\\mathcal{M}, \\SheafHom_\\mathcal{B}^{dg}(\\mathcal{N}, \\mathcal{L}))\n$$\nfunctorially in $\\mathcal{M}$, $\\mathcal{N}$, $\\mathcal{L}$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Sheaves of differential graded bimodules and tensor-hom adjunction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRS","source_file":"sdga.tex","source_line":1815,"source_end_line":1838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L1815-L1838","statement_sha256":"bc37d0022906f005ad7db14f9135a983953ef5a68cd3fcf79bcb37a0925f7474","origin":"The Stacks Project","memory_eligible":false,"source_rank":4866,"rank":4866,"depth":1,"x":2533.297,"y":89.683,"cluster":"homological-algebra"},{"id":"stacks:0FRU","tag":"0FRU","title":"Pull and push for sheaves of differential graded modules · Lemma 0FRU","summary":"In the situation above we have Hom_Mod^dg(B, d)( N, f_*M) = Hom_Mod^dg(A, d)( f^*N, M)","statement_latex":"In the situation above we have\n$$\n\\Hom_{\\textit{Mod}^{dg}(\\mathcal{B}, \\text{d})}(\n\\mathcal{N}, f_*\\mathcal{M}) =\n\\Hom_{\\textit{Mod}^{dg}(\\mathcal{A}, \\text{d})}(\nf^*\\mathcal{N}, \\mathcal{M})\n$$","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Pull and push for sheaves of differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRU","source_file":"sdga.tex","source_line":1999,"source_end_line":2008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L1999-L2008","statement_sha256":"047ae5b1e8247c8adc3ffe3774adcd9e433725576201520866a602276601cb2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4867,"rank":4867,"depth":1,"x":2545.56,"y":346.457,"cluster":"homological-algebra"},{"id":"stacks:0FRW","tag":"0FRW","title":"Localization and sheaves of differential graded modules · Lemma 0FRW","summary":"In the situation above we have Hom_Mod^dg(A, d)( j_!M, N) = Hom_Mod^dg(A_U, d)( M, j^*N)","statement_latex":"In the situation above we have\n$$\n\\Hom_{\\textit{Mod}^{dg}(\\mathcal{A}, \\text{d})}(\nj_!\\mathcal{M}, \\mathcal{N}) =\n\\Hom_{\\textit{Mod}^{dg}(\\mathcal{A}_U, \\text{d})}(\n\\mathcal{M}, j^*\\mathcal{N})\n$$","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Localization and sheaves of differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRW","source_file":"sdga.tex","source_line":2101,"source_end_line":2110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2101-L2110","statement_sha256":"2d07d9aad410a4d853b0ff0065fd6b34d0187e56f1e49071500ba27d9cb1ba4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4868,"rank":4868,"depth":4,"x":2329.82,"y":163.899,"cluster":"homological-algebra"},{"id":"stacks:0FRX","tag":"0FRX","title":"Localization and sheaves of differential graded modules · Lemma 0FRX","summary":"In the situation above, let M be a right differential graded A_U-module and let N be a left differential graded A-module. Then j_!M ⊗_A N = j_!(M ⊗_A_U N|_U) as complexes of O-modules functorially in M and N.","statement_latex":"In the situation above, let $\\mathcal{M}$ be a right differential graded\n$\\mathcal{A}_U$-module and let $\\mathcal{N}$ be a left differential graded\n$\\mathcal{A}$-module. Then\n$$\nj_!\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N} =\nj_!(\\mathcal{M} \\otimes_{\\mathcal{A}_U} \\mathcal{N}|_U)\n$$\nas complexes of $\\mathcal{O}$-modules\nfunctorially in $\\mathcal{M}$ and $\\mathcal{N}$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Localization and sheaves of differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FRX","source_file":"sdga.tex","source_line":2118,"source_end_line":2129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2118-L2129","statement_sha256":"75724fc84bc39496aec93026f02fef6be3c2cee0752db6783b71f0549e73074a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4869,"rank":4869,"depth":4,"x":2636.003,"y":176.11,"cluster":"homological-algebra"},{"id":"stacks:0FS0","tag":"0FS0","title":"The homotopy category · Definition 0FS0","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). Let f, g : M → N be homomorphisms of differential graded A-modules. A homotopy between f and g is a graded A-module map h : M → N homogeneous of degree -1 such that f - g = d_N ∘ h + h ∘ d_M If a homotopy exists, then we say f and g are homotopic.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let\n$f, g : \\mathcal{M} \\to \\mathcal{N}$\nbe homomorphisms of differential graded $\\mathcal{A}$-modules.\nA {\\it homotopy between $f$ and $g$} is a graded $\\mathcal{A}$-module\nmap $h : \\mathcal{M} \\to \\mathcal{N}$ homogeneous of degree $-1$\nsuch that\n$$\nf - g = \\text{d}_\\mathcal{N} \\circ h + h \\circ \\text{d}_\\mathcal{M}\n$$\nIf a homotopy exists, then we say $f$ and $g$ are {\\it homotopic}.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The homotopy category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FS0","source_file":"sdga.tex","source_line":2237,"source_end_line":2251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2237-L2251","statement_sha256":"964a751ec8a76d42a093fa84b681f3a64edd0aadf897384d4f6db2964f5573ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":4870,"rank":4870,"depth":0,"x":2400.185,"y":340.999,"cluster":"homological-algebra"},{"id":"stacks:0FS1","tag":"0FS1","title":"The homotopy category · Definition 0FS1","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). The homotopy category, denoted K(Mod(A, d)), is the category whose objects are the objects of Mod(A, d) and whose morphisms are homotopy classes of homomorphisms of differential graded A-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nThe {\\it homotopy category}, denoted $K(\\textit{Mod}(\\mathcal{A}, \\text{d}))$,\nis the category whose objects are the objects of\n$\\textit{Mod}(\\mathcal{A}, \\text{d})$ and whose morphisms are homotopy classes\nof homomorphisms of differential graded $\\mathcal{A}$-modules.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The homotopy category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FS1","source_file":"sdga.tex","source_line":2274,"source_end_line":2283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2274-L2283","statement_sha256":"4c7656969a48c1d310aa5f4f016cd2c61cfe11f5be7d94acf27704aeba241fcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4871,"rank":4871,"depth":0,"x":2441.514,"y":85.361,"cluster":"homological-algebra"},{"id":"stacks:0FS2","tag":"0FS2","title":"The homotopy category · Lemma 0FS2","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). The homotopy category K(Mod(A, d)) has direct sums and products.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nThe homotopy category $K(\\textit{Mod}(\\mathcal{A}, \\text{d}))$\nhas direct sums and products.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The homotopy category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FS2","source_file":"sdga.tex","source_line":2317,"source_end_line":2324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2317-L2324","statement_sha256":"ef400ca004a70ad2d692b96dfc0f2e553acfe8bbb91cc7dbeb9bcb2eccde1495","origin":"The Stacks Project","memory_eligible":false,"source_rank":4872,"rank":4872,"depth":2,"x":2616.782,"y":297.517,"cluster":"homological-algebra"},{"id":"stacks:0FS4","tag":"0FS4","title":"Cones and triangles · Lemma 0FS4","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). The differential graded category Mod^dg(A, d) satisfies axioms (A) and (B) of Differential Graded Algebra, Section [Tag 09P5].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nThe differential graded category\n$\\textit{Mod}^{dg}(\\mathcal{A}, \\text{d})$\nsatisfies axioms (A) and (B) of\nDifferential Graded Algebra, Section \\ref{dga-section-review}.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Cones and triangles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FS4","source_file":"sdga.tex","source_line":2353,"source_end_line":2362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2353-L2362","statement_sha256":"e72d4fb36498180672a01efc7fb5799a1fce38342f155151ac3b1135631df6c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4873,"rank":4873,"depth":1,"x":2316.648,"y":240.473,"cluster":"homological-algebra"},{"id":"stacks:0FS5","tag":"0FS5","title":"Cones and triangles · Definition 0FS5","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). Let f : K → L be a homomorphism of differential graded A-modules. The cone of f is the differential graded A-module C(f) defined as follows: • the underlying complex of O-modules is the cone of the corresponding map f : K^bullet → L^bullet of complexes of A-modules, i.e., we have C(f)^n = L^n ⊕ K^n + 1 and differential d_C(f) = ( d_L & f 0 & -d_K ) • the multiplication map C(f)^n ×…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nLet $f : \\mathcal{K} \\to \\mathcal{L}$\nbe a homomorphism of differential graded $\\mathcal{A}$-modules.\nThe {\\it cone} of $f$ is the differential graded $\\mathcal{A}$-module\n$C(f)$ defined as follows:\n\\begin{enumerate}\n\\item the underlying complex of $\\mathcal{O}$-modules\nis the cone of the corresponding map\n$f : \\mathcal{K}^\\bullet \\to \\mathcal{L}^\\bullet$ of\ncomplexes of $\\mathcal{A}$-modules, i.e., we have\n$C(f)^n = \\mathcal{L}^n \\oplus \\mathcal{K}^{n + 1}$ and\ndifferential\n$$\nd_{C(f)} =\n\\left(\n\\begin{matrix}\n\\text{d}_\\mathcal{L} & f \\\\\n0 & -\\text{d}_\\mathcal{K}\n\\end{matrix}\n\\right)\n$$\n\\item the multiplication map\n$$\nC(f)^n \\times \\mathcal{A}^m \\to C(f)^{n + m}\n$$\nis the direct sum of the multiplication map\n$\\mathcal{L}^n \\times \\mathcal{A}^m \\to \\mathcal{L}^{n + m}$\nand the multiplication map\n$\\mathcal{K}^{n + 1} \\times \\mathcal{A}^m \\to \\mathcal{K}^{n + 1 + m}$.\n\\end{enumerate}\nIt comes equipped with canonical hommorphisms of differential graded\n$\\mathcal{A}$-modules $i : \\mathcal{L} \\to C(f)$\nand $p : C(f) \\to \\mathcal{K}[1]$ induced by the obvious maps.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Cones and triangles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FS5","source_file":"sdga.tex","source_line":2426,"source_end_line":2463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2426-L2463","statement_sha256":"67813a0f0b509d70d19d5313a845274bc75e13b013e8cd676f0215ecbb9bd1fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4874,"rank":4874,"depth":0,"x":2584.088,"y":112.112,"cluster":"homological-algebra"},{"id":"stacks:0FS6","tag":"0FS6","title":"Cones and triangles · Lemma 0FS6","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). The differential graded category Mod^dg(A, d) satisfies axiom (C) formulated in Differential Graded Algebra, Situation [Tag 09QJ].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nThe differential graded category\n$\\textit{Mod}^{dg}(\\mathcal{A}, \\text{d})$\nsatisfies axiom (C) formulated in\nDifferential Graded Algebra, Situation \\ref{dga-situation-ABC}.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Cones and triangles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FS6","source_file":"sdga.tex","source_line":2472,"source_end_line":2481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2472-L2481","statement_sha256":"a69cfd5d10b7a4ac861f1c888ae9041da300645525c73416368e98834f1a69b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4875,"rank":4875,"depth":0,"x":2490.017,"y":358.747,"cluster":"homological-algebra"},{"id":"stacks:0FS7","tag":"0FS7","title":"Cones and triangles · Proposition 0FS7","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). The homotopy category K(Mod(A, d)) is a triangulated category where • the shift functors are those constructed in Section [Tag 0FRY], • the distinguished triangles are those triangles in K(Mod(A, d)) which are isomorphic as a triangle to a triangle K → L → N xrightarrowδ K[1], δ = π ∘ d_L ∘ s constructed from an admissible short exact sequence 0 → K → L → N → 0 in Mod(A, d) above.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nThe homotopy category $K(\\textit{Mod}(\\mathcal{A}, \\text{d}))$\nis a triangulated category where\n\\begin{enumerate}\n\\item the shift functors are those constructed in\nSection \\ref{section-shift-dg},\n\\item the distinguished triangles are those triangles\nin $K(\\textit{Mod}(\\mathcal{A}, \\text{d}))$ which are\nisomorphic as a triangle to a triangle\n$$\n\\mathcal{K} \\to \\mathcal{L} \\to \\mathcal{N}\n\\xrightarrow{\\delta} \\mathcal{K}[1],\\quad\\quad\n\\delta = \\pi \\circ \\text{d}_\\mathcal{L} \\circ s\n$$\nconstructed from an admissible short exact sequence\n$0 \\to \\mathcal{K} \\to \\mathcal{L} \\to \\mathcal{N} \\to 0$\nin $\\textit{Mod}(\\mathcal{A}, \\text{d})$ above.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Cones and triangles","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FS7","source_file":"sdga.tex","source_line":2532,"source_end_line":2554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2532-L2554","statement_sha256":"1389f837a5a11e08d273e1126035ef9d0f17ec53d14f70a36d0f1830aadaeb42","origin":"The Stacks Project","memory_eligible":false,"source_rank":4876,"rank":4876,"depth":12,"x":2360.925,"y":123.283,"cluster":"homological-algebra"},{"id":"stacks:0FS9","tag":"0FS9","title":"Cones and triangles · Lemma 0FS9","summary":"Let (C, O) be a ringed site. Let (A, d) be a differential graded O-algebra. The category Mod(A, d) is a Grothendieck abelian category.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$ be a differential graded $\\mathcal{O}$-algebra.\nThe category $\\textit{Mod}(\\mathcal{A}, \\text{d})$\nis a Grothendieck abelian category.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Cones and triangles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FS9","source_file":"sdga.tex","source_line":2589,"source_end_line":2595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2589-L2595","statement_sha256":"deee819ddc226e55f4c6bffd72076ee82fb0ef3265f52cda1c5eba2d5a21e7b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4877,"rank":4877,"depth":2,"x":2645.737,"y":223.755,"cluster":"homological-algebra"},{"id":"stacks:0FSB","tag":"0FSB","title":"Flat resolutions · Lemma 0FSB","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). Let U ∈ Ob(C). Then j_!A_U is a good differential graded A-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let $U \\in \\Ob(\\mathcal{C})$.\nThen $j_!\\mathcal{A}_U$ is a good differential graded\n$\\mathcal{A}$-module.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSB","source_file":"sdga.tex","source_line":2673,"source_end_line":2680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2673-L2680","statement_sha256":"2ef6c01182ca239c7b669ddd8bb6d7db16643e309dbbb6022e63753172140831","origin":"The Stacks Project","memory_eligible":false,"source_rank":4878,"rank":4878,"depth":10,"x":2354.651,"y":311.36,"cluster":"homological-algebra"},{"id":"stacks:0FSC","tag":"0FSC","title":"Flat resolutions · Lemma 0FSC","summary":"et (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). Let 0 → P → P' → P\" → 0 be an admissible short exact sequence of differential graded A-modules. If two-out-of-three of these modules are good, so is the third.","statement_latex":"et $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let\n$0 \\to \\mathcal{P} \\to \\mathcal{P}' \\to \\mathcal{P}'' \\to 0$\nbe an admissible short exact sequence of differential graded\n$\\mathcal{A}$-modules. If two-out-of-three of these modules\nare good, so is the third.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSC","source_file":"sdga.tex","source_line":2765,"source_end_line":2774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2765-L2774","statement_sha256":"1f33b4f23161775c421e332b65b6c0270109a7fcda5bf47f71f385821d92c3dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4879,"rank":4879,"depth":0,"x":2498.979,"y":81.377,"cluster":"homological-algebra"},{"id":"stacks:0FSD","tag":"0FSD","title":"Flat resolutions · Lemma 0FSD","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). An arbitrary direct sum of good differential graded A-modules is good. A filtered colimit of good differential graded A-modules is good.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. An arbitrary\ndirect sum of good differential graded $\\mathcal{A}$-modules\nis good. A filtered colimit of good differential graded\n$\\mathcal{A}$-modules is good.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSD","source_file":"sdga.tex","source_line":2804,"source_end_line":2812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2804-L2812","statement_sha256":"a658bd3847d8b5ad29f530284f2559ff7f24130c26ec162a159b2dff932870c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4880,"rank":4880,"depth":0,"x":2577.572,"y":333.093,"cluster":"homological-algebra"},{"id":"stacks:0FSE","tag":"0FSE","title":"Flat resolutions · Lemma 0FSE","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). Let M be a differential graded A-module. There exists a homomorphism P → M of differential graded A-modules with the following properties • P → M is surjective, • Ker(d_P) → Ker(d_M) is surjective, and • P is good.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$\nbe a differential graded $\\mathcal{A}$-module. There exists a homomorphism\n$\\mathcal{P} \\to \\mathcal{M}$ of differential graded $\\mathcal{A}$-modules\nwith the following properties\n\\begin{enumerate}\n\\item $\\mathcal{P} \\to \\mathcal{M}$ is surjective,\n\\item $\\Ker(\\text{d}_\\mathcal{P}) \\to \\Ker(\\text{d}_\\mathcal{M})$\nis surjective, and\n\\item $\\mathcal{P}$ is good.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSE","source_file":"sdga.tex","source_line":2819,"source_end_line":2833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2819-L2833","statement_sha256":"f6e997f369e8b1cf8e63dcecaf9b47110fee00e0d530786fcece09377a3b11b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4881,"rank":4881,"depth":11,"x":2316.954,"y":191.946,"cluster":"homological-algebra"},{"id":"stacks:0FSG","tag":"0FSG","title":"Flat resolutions · Lemma 0FSG","summary":"Let (C, O) be a ringed site. Let A be a differential graded A-algebra. Let S be a sheaf of graded sets on C. Then the free graded module A[S] on S endowed with differential as in Remark [Tag 0FSF] is a good differential graded A-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a differential graded $\\mathcal{A}$-algebra.\nLet $\\mathcal{S}$ be a sheaf of graded sets on $\\mathcal{C}$.\nThen the free graded module $\\mathcal{A}[\\mathcal{S}]$\non $\\mathcal{S}$ endowed with differential as in\nRemark \\ref{remark-sheaf-graded-sets}\nis a good differential graded $\\mathcal{A}$-module.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSG","source_file":"sdga.tex","source_line":2911,"source_end_line":2920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2911-L2920","statement_sha256":"f8f789dd9ef165a148681a5246051e9ecda9029bab1d918405386dd898b20a45","origin":"The Stacks Project","memory_eligible":false,"source_rank":4882,"rank":4882,"depth":0,"x":2622.921,"y":148.103,"cluster":"homological-algebra"},{"id":"stacks:0FSH","tag":"0FSH","title":"Flat resolutions · Lemma 0FSH","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). Let M be a differential graded A-module. There exists a homomorphism P → M of differential graded A-modules with the following properties • P → M is a quasi-isomorphism, and • P is good.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{M}$\nbe a differential graded $\\mathcal{A}$-module. There exists a homomorphism\n$\\mathcal{P} \\to \\mathcal{M}$ of differential graded $\\mathcal{A}$-modules\nwith the following properties\n\\begin{enumerate}\n\\item $\\mathcal{P} \\to \\mathcal{M}$ is a quasi-isomorphism, and\n\\item $\\mathcal{P}$ is good.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSH","source_file":"sdga.tex","source_line":2968,"source_end_line":2980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L2968-L2980","statement_sha256":"12c0d6fad27c867d8f2c55b996d7bf5dc7449701fa5d2fe8ddc9069362542a9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4883,"rank":4883,"depth":1,"x":2432.385,"y":354.237,"cluster":"homological-algebra"},{"id":"stacks:0FSI","tag":"0FSI","title":"Flat resolutions · Lemma 0FSI","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). Let P be a good acyclic right differential graded A-module. • for any differential graded left A-module N the tensor product P ⊗_A N is acyclic, • for any morphism (f, f^sharp) : (Sh(C'), O') → (Sh(C), O) of ringed topoi and any differential graded O'-algebra A' and any map φ : f^-1A → A' of differential graded f^-1O-algebras the pullback f^*P is acyclic and good.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{P}$ be a good\nacyclic right differential graded $\\mathcal{A}$-module.\n\\begin{enumerate}\n\\item for any differential graded left $\\mathcal{A}$-module\n$\\mathcal{N}$ the tensor product\n$\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}$ is acyclic,\n\\item for any morphism $(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')\n\\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nof ringed topoi and any differential graded $\\mathcal{O}'$-algebra\n$\\mathcal{A}'$ and any map $\\varphi : f^{-1}\\mathcal{A} \\to \\mathcal{A}'$\nof differential graded $f^{-1}\\mathcal{O}$-algebras\nthe pullback $f^*\\mathcal{P}$ is acyclic and good.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Flat resolutions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSI","source_file":"sdga.tex","source_line":3227,"source_end_line":3244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3227-L3244","statement_sha256":"41e72c71c3331830d1f7bcf810daff2b25e458148d4d1eeee47a97f08a7c9c4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4884,"rank":4884,"depth":2,"x":2407.094,"y":93.882,"cluster":"homological-algebra"},{"id":"stacks:0FSK","tag":"0FSK","title":"The differential graded hull of a graded module · Lemma 0FSK","summary":"Let (C, O) be a ringed site. Let A be a sheaf of differential graded algebras on (C, O). The forgetful functor F : Mod(A, d) → Mod(A) has a left adjoint G : Mod(A) → Mod(A, d).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. The forgetful functor\n$F : \\textit{Mod}(\\mathcal{A}, \\text{d}) \\to \\textit{Mod}(\\mathcal{A})$\nhas a left adjoint $G : \\textit{Mod}(\\mathcal{A}) \\to\n\\textit{Mod}(\\mathcal{A}, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The differential graded hull of a graded module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSK","source_file":"sdga.tex","source_line":3295,"source_end_line":3303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3295-L3303","statement_sha256":"d31784087efd25e55b029373bf51653a2ccaed9eb8f0557da3ea7728fd77068b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4885,"rank":4885,"depth":2,"x":2635.324,"y":271.677,"cluster":"homological-algebra"},{"id":"stacks:0FSL","tag":"0FSL","title":"The differential graded hull of a graded module · Lemma 0FSL","summary":"The functors F, G of Lemma [Tag 0FSK] have the following properties. Given a graded A-module N we have • the counit N → F(G(N)) is injective, • the map overlined : N → Coker(N → F(G(N)))[1] is an isomorphism, and • G(N) is an acyclic differential graded A-module.","statement_latex":"The functors $F, G$ of Lemma \\ref{lemma-dg-hull} have\nthe following properties. Given a graded $\\mathcal{A}$-module\n$\\mathcal{N}$ we have\n\\begin{enumerate}\n\\item the counit $\\mathcal{N} \\to F(G(\\mathcal{N}))$ is injective,\n\\item the map $\\overline{\\text{d}} : \\mathcal{N} \\to\n\\Coker(\\mathcal{N} \\to F(G(\\mathcal{N})))[1]$ is an isomorphism, and\n\\item $G(\\mathcal{N})$ is an acyclic differential graded $\\mathcal{A}$-module.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The differential graded hull of a graded module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSL","source_file":"sdga.tex","source_line":3355,"source_end_line":3366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3355-L3366","statement_sha256":"7ccd1f8a8062131272c5bdc80eab0a6b5afa0df9c93e627fad149fd4e49a4afa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4886,"rank":4886,"depth":3,"x":2323.765,"y":270.075,"cluster":"homological-algebra"},{"id":"stacks:0FSN","tag":"0FSN","title":"K-injective differential graded modules · Lemma 0FSN","summary":"Let (C, O) be a ringed site. Let A be a sheaf of graded algebras on (C, O). There exists a set T and for each t ∈ T an injective map N_t → N'_t of graded A-modules such that an object I of Mod(A) is injective if and only if for every solid diagram xymatrix N_t ar[r] ar[d] & I N'_t ar@..>[ru] a dotted arrow exists in Mod(A) making the diagram commute.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $\\mathcal{A}$\nbe a sheaf of graded algebras on $(\\mathcal{C}, \\mathcal{O})$.\nThere exists a set $T$ and for each $t \\in T$ an injective map\n$\\mathcal{N}_t \\to \\mathcal{N}'_t$ of graded $\\mathcal{A}$-modules\nsuch that an object $\\mathcal{I}$ of $\\textit{Mod}(\\mathcal{A})$\nis injective if and only if for every solid diagram\n$$\n\\xymatrix{\n\\mathcal{N}_t \\ar[r] \\ar[d] & \\mathcal{I} \\\\\n\\mathcal{N}'_t \\ar@{..>}[ru]\n}\n$$\na dotted arrow exists in $\\textit{Mod}(\\mathcal{A})$ making the diagram commute.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSN","source_file":"sdga.tex","source_line":3478,"source_end_line":3493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3478-L3493","statement_sha256":"2c29cc3fadada5519a0b6d53c89967f250264e42912b4168751c40a717eb43cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":4887,"rank":4887,"depth":4,"x":2555.005,"y":94.308,"cluster":"homological-algebra"},{"id":"stacks:0FSP","tag":"0FSP","title":"K-injective differential graded modules · Definition 0FSP","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). A diffential graded A-module I is said to be graded injective if M viewed as a graded A-module is an injective object of the category Mod(A) of graded A-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. A diffential graded $\\mathcal{A}$-module\n$\\mathcal{I}$ is said to be {\\it graded injective}\\footnote{This may be\nnonstandard terminology.} if $\\mathcal{M}$ viewed as a graded\n$\\mathcal{A}$-module is an injective object of the category\n$\\textit{Mod}(\\mathcal{A})$ of graded $\\mathcal{A}$-modules.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSP","source_file":"sdga.tex","source_line":3502,"source_end_line":3511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3502-L3511","statement_sha256":"a256961a983802c75d07714d97a2307a30d3f240748ebab4b29b92ebcac92af5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4888,"rank":4888,"depth":0,"x":2525.812,"y":355.367,"cluster":"homological-algebra"},{"id":"stacks:0FSR","tag":"0FSR","title":"K-injective differential graded modules · Lemma 0FSR","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Let T be a set and for each t ∈ T let I_t be a graded injective diffential graded A-module. Then ∏ I_t is a graded injective differential graded A-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let $T$ be a set and for\neach $t \\in T$ let $\\mathcal{I}_t$ be a graded injective\ndiffential graded $\\mathcal{A}$-module. Then\n$\\prod \\mathcal{I}_t$ is a graded injective differential\ngraded $\\mathcal{A}$-module.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSR","source_file":"sdga.tex","source_line":3558,"source_end_line":3567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3558-L3567","statement_sha256":"0e2738e0a64668d9c9f45c3924228fbdcef42ac28b296f744b9b1d93f46923c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4889,"rank":4889,"depth":1,"x":2337.229,"y":146.11,"cluster":"homological-algebra"},{"id":"stacks:0FSS","tag":"0FSS","title":"K-injective differential graded modules · Lemma 0FSS","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). There exists a set T and for each t ∈ T an injective map M_t → M'_t of acyclic differential graded A-modules such that for an object I of Mod(A, d) the following are equivalent • I is graded injective, and • for every solid diagram xymatrix M_t ar[r] ar[d] & I M'_t ar@..>[ru] a dotted arrow exists in Mod(A, d) making the diagram commute.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$\nbe a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.\nThere exists a set $T$ and for each $t \\in T$ an injective map\n$\\mathcal{M}_t \\to \\mathcal{M}'_t$ of\nacyclic differential graded $\\mathcal{A}$-modules\nsuch that for an object $\\mathcal{I}$ of $\\textit{Mod}(\\mathcal{A}, \\text{d})$\nthe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{I}$ is graded injective, and\n\\item for every solid diagram\n$$\n\\xymatrix{\n\\mathcal{M}_t \\ar[r] \\ar[d] & \\mathcal{I} \\\\\n\\mathcal{M}'_t \\ar@{..>}[ru]\n}\n$$\na dotted arrow exists in $\\textit{Mod}(\\mathcal{A}, \\text{d})$\nmaking the diagram commute.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSS","source_file":"sdga.tex","source_line":3577,"source_end_line":3599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3577-L3599","statement_sha256":"b83492b5873acb2ededda0a31d7f51ae41a9e0ce48b29b4b965958553bbe0bf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4890,"rank":4890,"depth":5,"x":2644.848,"y":193.45,"cluster":"homological-algebra"},{"id":"stacks:0FST","tag":"0FST","title":"K-injective differential graded modules · Lemma 0FST","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). There exists a set S and for each s an acyclic differential graded A-module M_s such that for every nonzero acyclic differential graded A-module M there is an s ∈ S and an injective map M_s → M in Mod(A, d).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. There exists a set $S$ and for each $s$\nan acyclic differential graded $\\mathcal{A}$-module $\\mathcal{M}_s$ such\nthat for every nonzero acyclic differential graded $\\mathcal{A}$-module\n$\\mathcal{M}$ there is an $s \\in S$ and an injective map\n$\\mathcal{M}_s \\to \\mathcal{M}$ in $\\textit{Mod}(\\mathcal{A}, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FST","source_file":"sdga.tex","source_line":3634,"source_end_line":3643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3634-L3643","statement_sha256":"952055391cc315a415f977533115a6580444bb0c80a930946eb1d3da04cab292","origin":"The Stacks Project","memory_eligible":false,"source_rank":4891,"rank":4891,"depth":0,"x":2379.703,"y":333.22,"cluster":"homological-algebra"},{"id":"stacks:0FSU","tag":"0FSU","title":"K-injective differential graded modules · Definition 0FSU","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). A diffential graded A-module I is K-injective if for every acyclic differential graded M we have Hom_K(Mod(A, d))(M, I) = 0","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. A diffential graded $\\mathcal{A}$-module\n$\\mathcal{I}$ is {\\it K-injective} if for every acyclic\ndifferential graded $\\mathcal{M}$ we have \n$$\n\\Hom_{K(\\textit{Mod}(\\mathcal{A}, \\text{d}))}(\\mathcal{M}, \\mathcal{I}) = 0\n$$","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSU","source_file":"sdga.tex","source_line":3734,"source_end_line":3744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3734-L3744","statement_sha256":"a5e5501bd937ba08ae4331e042c71592fd957992ebe9cec184e8d2e58cd74646","origin":"The Stacks Project","memory_eligible":false,"source_rank":4892,"rank":4892,"depth":0,"x":2462.894,"y":79.474,"cluster":"homological-algebra"},{"id":"stacks:0FSV","tag":"0FSV","title":"K-injective differential graded modules · Lemma 0FSV","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Let T be a set and for each t ∈ T let I_t be a K-injective diffential graded A-module. Then ∏ I_t is a K-injective differential graded A-module.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let $T$ be a set and for\neach $t \\in T$ let $\\mathcal{I}_t$ be a K-injective\ndiffential graded $\\mathcal{A}$-module. Then\n$\\prod \\mathcal{I}_t$ is a K-injective differential\ngraded $\\mathcal{A}$-module.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSV","source_file":"sdga.tex","source_line":3750,"source_end_line":3759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3750-L3759","statement_sha256":"73a45fd78a6bfce1dca1a989e5ee85e9684369ebb31bc6d97255669f8699cd00","origin":"The Stacks Project","memory_eligible":false,"source_rank":4893,"rank":4893,"depth":0,"x":2605.735,"y":314.0,"cluster":"homological-algebra"},{"id":"stacks:0FSW","tag":"0FSW","title":"K-injective differential graded modules · Lemma 0FSW","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Let I be a K-injective and graded injective object of Mod(A, d). For every solid diagram in Mod(A, d) xymatrix M ar[r]_a ar[d]_b & I M' ar@..>[ru] where b is injective and M is acyclic a dotted arrow exists making the diagram commute.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$\nbe a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.\nLet $\\mathcal{I}$ be a K-injective and graded injective\nobject of $\\textit{Mod}(\\mathcal{A}, \\text{d})$.\nFor every solid diagram in $\\textit{Mod}(\\mathcal{A}, \\text{d})$\n$$\n\\xymatrix{\n\\mathcal{M} \\ar[r]_a \\ar[d]_b & \\mathcal{I} \\\\\n\\mathcal{M}' \\ar@{..>}[ru]\n}\n$$\nwhere $b$ is injective and $\\mathcal{M}$ is acyclic\na dotted arrow exists making the diagram commute.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSW","source_file":"sdga.tex","source_line":3777,"source_end_line":3793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3777-L3793","statement_sha256":"37b25f01a235499025f3e815190aef568b801fe8596ae734c9a8a5b0b9484843","origin":"The Stacks Project","memory_eligible":false,"source_rank":4894,"rank":4894,"depth":0,"x":2311.54,"y":222.033,"cluster":"homological-algebra"},{"id":"stacks:0FSX","tag":"0FSX","title":"K-injective differential graded modules · Lemma 0FSX","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Let I be a K-injective and graded injective object of Mod(A, d). For every solid diagram in Mod(A, d) xymatrix M ar[r]_a ar[d]_b & I M' ar@..>[ru] where b is a quasi-isomorphism a dotted arrow exists making the diagram commute up to homotopy.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on\n$(\\mathcal{C}, \\mathcal{O})$. Let $\\mathcal{I}$ be a\nK-injective and graded injective\nobject of $\\textit{Mod}(\\mathcal{A}, \\text{d})$.\nFor every solid diagram in $\\textit{Mod}(\\mathcal{A}, \\text{d})$\n$$\n\\xymatrix{\n\\mathcal{M} \\ar[r]_a \\ar[d]_b & \\mathcal{I} \\\\\n\\mathcal{M}' \\ar@{..>}[ru]\n}\n$$\nwhere $b$ is a quasi-isomorphism a dotted arrow exists making the\ndiagram commute up to homotopy.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSX","source_file":"sdga.tex","source_line":3806,"source_end_line":3822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3806-L3822","statement_sha256":"51243e2aa7318e7ce4c5c21c9d2e68a5c774e5614bff551eb713745785d19f85","origin":"The Stacks Project","memory_eligible":false,"source_rank":4895,"rank":4895,"depth":0,"x":2602.697,"y":122.825,"cluster":"homological-algebra"},{"id":"stacks:0FSY","tag":"0FSY","title":"K-injective differential graded modules · Lemma 0FSY","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). There exists a set R and for each r ∈ R an injective map M_r → M'_r of acyclic differential graded A-modules such that for an object I of Mod(A, d) the following are equivalent • I is K-injective and graded injective, and • for every solid diagram xymatrix M_r ar[r] ar[d] & I M'_r ar@..>[ru] a dotted arrow exists in Mod(A, d) making the diagram commute.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$\nbe a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.\nThere exists a set $R$ and for each $r \\in R$ an injective map\n$\\mathcal{M}_r \\to \\mathcal{M}'_r$ of\nacyclic differential graded $\\mathcal{A}$-modules\nsuch that for an object $\\mathcal{I}$ of $\\textit{Mod}(\\mathcal{A}, \\text{d})$\nthe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{I}$ is K-injective and graded injective, and\n\\item for every solid diagram\n$$\n\\xymatrix{\n\\mathcal{M}_r \\ar[r] \\ar[d] & \\mathcal{I} \\\\\n\\mathcal{M}'_r \\ar@{..>}[ru]\n}\n$$\na dotted arrow exists in $\\textit{Mod}(\\mathcal{A}, \\text{d})$\nmaking the diagram commute.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSY","source_file":"sdga.tex","source_line":3844,"source_end_line":3866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3844-L3866","statement_sha256":"2ca845cf699c7f46c0b3ed04bf6476e4b97bad470a2578a31ce9cf8ed013b2b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4896,"rank":4896,"depth":6,"x":2467.66,"y":361.405,"cluster":"homological-algebra"},{"id":"stacks:0FSZ","tag":"0FSZ","title":"K-injective differential graded modules · Lemma 0FSZ","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Let R be a set and for each r ∈ R let an injective map M_r → M'_r of acyclic differential graded A-modules be given. There exists a functor M : Mod(A, d) → Mod(A, d) and a natural transformation j : id → M such that • j_M : M → M(M) is injective and a quasi-isomorphism, • for every solid diagram xymatrix M_r ar[r] ar[d] & M ar[d]^j_M M'_r ar@..>[r] & M(M) a dotted arrow exists in…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$\nbe a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.\nLet $R$ be a set and for each $r \\in R$ let an injective map\n$\\mathcal{M}_r \\to \\mathcal{M}'_r$ of\nacyclic differential graded $\\mathcal{A}$-modules be given.\nThere exists a functor $M : \\textit{Mod}(\\mathcal{A}, \\text{d}) \\to\n\\textit{Mod}(\\mathcal{A}, \\text{d})$ and a natural transformation\n$j : \\text{id} \\to M$ such that\n\\begin{enumerate}\n\\item $j_\\mathcal{M} : \\mathcal{M} \\to M(\\mathcal{M})$ is injective\nand a quasi-isomorphism,\n\\item for every solid diagram\n$$\n\\xymatrix{\n\\mathcal{M}_r \\ar[r] \\ar[d] & \\mathcal{M} \\ar[d]^{j_\\mathcal{M}} \\\\\n\\mathcal{M}'_r \\ar@{..>}[r] & M(\\mathcal{M})\n}\n$$\na dotted arrow exists in $\\textit{Mod}(\\mathcal{A}, \\text{d})$\nmaking the diagram commute.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FSZ","source_file":"sdga.tex","source_line":3966,"source_end_line":3990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L3966-L3990","statement_sha256":"85cc1b458e323e29133d2d624abb585448051eeae42207904cf3ff90e7d69ccc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4897,"rank":4897,"depth":0,"x":2375.291,"y":108.628,"cluster":"homological-algebra"},{"id":"stacks:0FT0","tag":"0FT0","title":"K-injective differential graded modules · Theorem 0FT0","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). For every differential graded A-module M there exists a quasi-isomorphism M → I where I is a graded injective and K-injective differential graded A-module. Moreover, the construction is functorial in M.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$\nbe a sheaf of differential graded algebras on $(\\mathcal{C}, \\mathcal{O})$.\nFor every differential graded $\\mathcal{A}$-module $\\mathcal{M}$ there\nexists a quasi-isomorphism $\\mathcal{M} \\to \\mathcal{I}$\nwhere $\\mathcal{I}$ is a graded injective and K-injective\ndifferential graded $\\mathcal{A}$-module. Moreover, the\nconstruction is functorial in $\\mathcal{M}$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"K-injective differential graded modules","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FT0","source_file":"sdga.tex","source_line":4015,"source_end_line":4025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4015-L4025","statement_sha256":"d01d31097ba782157ab5ada210ccf3958fe702e4b7409ed418eaad53a8524f0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4898,"rank":4898,"depth":7,"x":2646.923,"y":242.73,"cluster":"homological-algebra"},{"id":"stacks:0FT2","tag":"0FT2","title":"The derived category · Lemma 0FT2","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). The functor H^0 : Mod(A, d) → Mod(O) of Section [Tag 0FRH] factors through a functor H^0 : K(Mod(A, d)) → Mod(O) which is homological in the sense of Derived Categories, Definition [Tag 0147].","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. The functor \n$H^0 : \\textit{Mod}(\\mathcal{A}, \\text{d}) \\to \\textit{Mod}(\\mathcal{O})$\nof Section \\ref{section-modules} factors through a\nfunctor\n$$\nH^0 : K(\\textit{Mod}(\\mathcal{A}, \\text{d})) \\to \\textit{Mod}(\\mathcal{O})\n$$\nwhich is homological in the sense of\nDerived Categories, Definition \\ref{derived-definition-homological}.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FT2","source_file":"sdga.tex","source_line":4098,"source_end_line":4111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4098-L4111","statement_sha256":"62ac7d74c69785bc3e58ed5adbeabed69d8cb8d499fab4c2fd0ee033e24a86de","origin":"The Stacks Project","memory_eligible":false,"source_rank":4899,"rank":4899,"depth":8,"x":2338.507,"y":298.025,"cluster":"homological-algebra"},{"id":"stacks:0FT3","tag":"0FT3","title":"The derived category · Lemma 0FT3","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). The full subcategory Ac of the homotopy category K(Mod(A, d)) consisting of acyclic modules is a strictly full saturated triangulated subcategory of K(Mod(A, d)).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. The full subcategory $\\text{Ac}$\nof the homotopy category $K(\\textit{Mod}(\\mathcal{A}, \\text{d}))$\nconsisting of acyclic modules is a strictly full saturated\ntriangulated subcategory of $K(\\textit{Mod}(\\mathcal{A}, \\text{d}))$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FT3","source_file":"sdga.tex","source_line":4131,"source_end_line":4139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4131-L4139","statement_sha256":"3babb8f5ababa727037b37bb517da342af5c259e627090d3e397ea6db1f4fcf3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4900,"rank":4900,"depth":9,"x":2521.626,"y":82.055,"cluster":"homological-algebra"},{"id":"stacks:0FT4","tag":"0FT4","title":"The derived category · Lemma 0FT4","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Consider the subclass Qis ⊂ Arrows(K(Mod(A, d))) consisting of quasi-isomorphisms. This is a saturated multiplicative system compatible with the triangulated structure on K(Mod(A, d)).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nConsider the subclass\n$\\text{Qis} \\subset \\text{Arrows}(K(\\textit{Mod}(\\mathcal{A}, \\text{d})))$\nconsisting of quasi-isomorphisms. This is a saturated multiplicative\nsystem compatible with the triangulated structure on\n$K(\\textit{Mod}(\\mathcal{A}, \\text{d}))$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FT4","source_file":"sdga.tex","source_line":4152,"source_end_line":4162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4152-L4162","statement_sha256":"73dae0d33c71937d336dce2565f777aeef8c35884bd6c79d7c4b412ff92c294d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4901,"rank":4901,"depth":12,"x":2560.309,"y":345.451,"cluster":"homological-algebra"},{"id":"stacks:0FT5","tag":"0FT5","title":"The derived category · Definition 0FT5","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Let Qis be as in Lemma [Tag 0FT4]. The derived category of (A, d) is the triangulated category D(A, d) = Qis^-1K(Mod(A, d)) discussed in more detail above.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let $\\text{Qis}$ be as in\nLemma \\ref{lemma-qis}. The\n{\\it derived category of $(\\mathcal{A}, \\text{d})$} is the triangulated\ncategory\n$$\nD(\\mathcal{A}, \\text{d}) =\n\\text{Qis}^{-1}K(\\textit{Mod}(\\mathcal{A}, \\text{d}))\n$$\ndiscussed in more detail above.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FT5","source_file":"sdga.tex","source_line":4240,"source_end_line":4253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4240-L4253","statement_sha256":"4e718395c97296aff95534937a6823c7764b89785ba5806ea4969973bc2d21dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":4902,"rank":4902,"depth":13,"x":2319.754,"y":173.022,"cluster":"homological-algebra"},{"id":"stacks:0FT6","tag":"0FT6","title":"The derived category · Lemma 0FT6","summary":"In Definition [Tag 0FT5] the kernel of the localization functor Q : K(Mod(A, d)) → D(A, d) is the category Ac of Lemma [Tag 0FT3].","statement_latex":"In Definition \\ref{definition-derived-category}\nthe kernel of the localization functor\n$Q : K(\\textit{Mod}(\\mathcal{A}, \\text{d})) \\to D(\\mathcal{A}, \\text{d})$\nis the category $\\text{Ac}$ of Lemma \\ref{lemma-acyclics}.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FT6","source_file":"sdga.tex","source_line":4258,"source_end_line":4264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4258-L4264","statement_sha256":"903af685d74cabfaddb843038604522c6b4478dac5288c0c724879fe981753bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":4903,"rank":4903,"depth":14,"x":2636.081,"y":163.663,"cluster":"homological-algebra"},{"id":"stacks:0FT7","tag":"0FT7","title":"The derived category · Lemma 0FT7","summary":"In Definition [Tag 0FT5] the functor H^0 : K(Mod(A, d)) → Mod(O) factors through a homological functor H^0 : D(A, d) → Mod(O).","statement_latex":"In Definition \\ref{definition-derived-category} the functor\n$H^0 : K(\\textit{Mod}(\\mathcal{A}, \\text{d})) \\to\n\\textit{Mod}(\\mathcal{O})$ factors through a homological functor\n$H^0 : D(\\mathcal{A}, \\text{d}) \\to \\textit{Mod}(\\mathcal{O})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FT7","source_file":"sdga.tex","source_line":4273,"source_end_line":4279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4273-L4279","statement_sha256":"c7240846434489965222083e26879d1ef22f662912cfba707e799aa72af3327d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4904,"rank":4904,"depth":14,"x":2410.151,"y":350.222,"cluster":"homological-algebra"},{"id":"stacks:0FT8","tag":"0FT8","title":"The derived category · Lemma 0FT8","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Let M and N be differential graded A-modules. Let N → I be a quasi-isomorphism with I a graded injective and K-injective differential graded A-module. Then Hom_D(A, d)(M, N) = Hom_K(Mod(A, d))(M, I)","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$.\nLet $\\mathcal{M}$ and $\\mathcal{N}$ be differential graded\n$\\mathcal{A}$-modules. Let $\\mathcal{N} \\to \\mathcal{I}$ be a\nquasi-isomorphism with $\\mathcal{I}$ a graded injective and\nK-injective differential graded $\\mathcal{A}$-module. Then\n$$\n\\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{M}, \\mathcal{N}) =\n\\Hom_{K(\\textit{Mod}(\\mathcal{A}, \\text{d}))}(\\mathcal{M}, \\mathcal{I})\n$$","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FT8","source_file":"sdga.tex","source_line":4290,"source_end_line":4303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4290-L4303","statement_sha256":"d38de117fa55d15a94e91dd698429f7f732fa30ad49b1c7c7fb6be2d95632493","origin":"The Stacks Project","memory_eligible":false,"source_rank":4905,"rank":4905,"depth":14,"x":2426.738,"y":84.222,"cluster":"homological-algebra"},{"id":"stacks:0FT9","tag":"0FT9","title":"The derived category · Lemma 0FT9","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Then • D(A, d) has both direct sums and products, • direct sums are obtained by taking direct sums of differential graded A-modules, • products are obtained by taking products of K-injective differential graded modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Then\n\\begin{enumerate}\n\\item $D(\\mathcal{A}, \\text{d})$ has both direct sums and products,\n\\item direct sums are obtained by taking direct sums of differential graded\n$\\mathcal{A}$-modules,\n\\item products are obtained by taking products of\nK-injective differential graded modules.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FT9","source_file":"sdga.tex","source_line":4334,"source_end_line":4346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4334-L4346","statement_sha256":"cb708b33c07eb99ef55b493e3dca4e1ede311867bc7b03aa05e97e59c939802a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4906,"rank":4906,"depth":15,"x":2628.596,"y":289.961,"cluster":"homological-algebra"},{"id":"stacks:0FTB","tag":"0FTB","title":"The canonical delta-functor · Lemma 0FTB","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). The localization functor Mod(A, d) → D(A, d) has the natural structure of a δ-functor, with δ_K → L → M = - p ∘ q^-1 with p and q as explained above.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. The localization functor\n$\\textit{Mod}(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{A}, \\text{d})$\nhas the natural structure of a $\\delta$-functor, with\n$$\n\\delta_{\\mathcal{K} \\to \\mathcal{L} \\to \\mathcal{M}} = - p \\circ q^{-1}\n$$\nwith $p$ and $q$ as explained above.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The canonical delta-functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTB","source_file":"sdga.tex","source_line":4473,"source_end_line":4484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4473-L4484","statement_sha256":"e76dcee7865d2140241f301be45e43ba8049d1c007e895c20cb34e362ac75e4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4907,"rank":4907,"depth":2,"x":2314.02,"y":252.758,"cluster":"homological-algebra"},{"id":"stacks:0FTC","tag":"0FTC","title":"The canonical delta-functor · Lemma 0FTC","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Let M_n be a system of differential graded A-modules. Then the derived colimit hocolim M_n in D(A, d) is represented by the differential graded module colim M_n.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras\non $(\\mathcal{C}, \\mathcal{O})$. Let\n$\\mathcal{M}_n$ be a system of differential graded $\\mathcal{A}$-modules.\nThen the derived colimit $\\text{hocolim} \\mathcal{M}_n$ in\n$D(\\mathcal{A}, \\text{d})$ is represented\nby the differential graded module $\\colim \\mathcal{M}_n$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"The canonical delta-functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTC","source_file":"sdga.tex","source_line":4498,"source_end_line":4507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4498-L4507","statement_sha256":"3779aeea86a9bb3a072cf95498d2bfade3355df505bd7c6c290ea6bf928d1426","origin":"The Stacks Project","memory_eligible":false,"source_rank":4908,"rank":4908,"depth":16,"x":2576.133,"y":101.559,"cluster":"homological-algebra"},{"id":"stacks:0FTF","tag":"0FTF","title":"Derived pullback · Lemma 0FTF","summary":"In the situation above, the functor ([Tag 0FTE]) composed with the localization functor K(Mod(A', d)) → D(A', d) has a left derived extension D(B, d) → D(A', d) whose value on a good right differential graded B-module P is f^*P ⊗_A N.","statement_latex":"In the situation above, the functor (\\ref{equation-pullback})\ncomposed with the localization functor\n$K(\\textit{Mod}(\\mathcal{A}', \\text{d})) \\to D(\\mathcal{A}', \\text{d})$\nhas a left derived extension\n$D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}', \\text{d})$ whose\nvalue on a good right differential graded $\\mathcal{B}$-module\n$\\mathcal{P}$ is $f^*\\mathcal{P} \\otimes_\\mathcal{A} \\mathcal{N}$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTF","source_file":"sdga.tex","source_line":4586,"source_end_line":4595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4586-L4595","statement_sha256":"21e3577d6eac03d5e696aa30022215eff7b9c4d22f05ee54634fcb196a790564","origin":"The Stacks Project","memory_eligible":false,"source_rank":4909,"rank":4909,"depth":3,"x":2504.381,"y":362.01,"cluster":"homological-algebra"},{"id":"stacks:0FTG","tag":"0FTG","title":"Derived pullback · Definition 0FTG","summary":"Derived tensor product and derived pullback. • Let (C, O) be a ringed site. Let A, B be differential graded O-algebras. Let N be a differential graded (A, B)-bimodule. The functor D(A, d) → D(B, d) constructed in Lemma [Tag 0FTF] is called the derived tensor product and denoted - ⊗_A^L N. • Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let A be a differential graded O_C-algebra. Let B be a differential graded O_D-algebra. Let φ : B → f_*A…","statement_latex":"Derived tensor product and derived pullback.\n\\begin{enumerate}\n\\item Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras.\nLet $\\mathcal{N}$ be a  differential graded\n$(\\mathcal{A}, \\mathcal{B})$-bimodule.\nThe functor $D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{B}, \\text{d})$\nconstructed in Lemma \\ref{lemma-derived-tensor-product}\nis called the {\\it derived tensor product} and denoted\n$- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}$.\n\\item Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential\ngraded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a\ndifferential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let\n$\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism\nof differential graded $\\mathcal{O}_\\mathcal{D}$-algebras.\nThe functor $D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}, \\text{d})$\nconstructed in Lemma \\ref{lemma-derived-tensor-product}\nis called {\\it derived pullback}\nand denote $Lf^*$.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pullback","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTG","source_file":"sdga.tex","source_line":4633,"source_end_line":4657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4633-L4657","statement_sha256":"5141e33367a279b98dbcdde1e6b55e6c6707abe43fbb4ef76de4e0228d7ac92d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4910,"rank":4910,"depth":4,"x":2347.705,"y":129.039,"cluster":"homological-algebra"},{"id":"stacks:0FTH","tag":"0FTH","title":"Derived pullback · Lemma 0FTH","summary":"In Lemma [Tag 0FTF] the functor D(B, d) → D(A', d) is equal to M ↦ Lf^*M ⊗_A^L N.","statement_latex":"In Lemma \\ref{lemma-derived-tensor-product} the functor\n$D(\\mathcal{B}, \\text{d}) \\to D(\\mathcal{A}', \\text{d})$ is equal to\n$\\mathcal{M} \\mapsto\nLf^*\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTH","source_file":"sdga.tex","source_line":4662,"source_end_line":4668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4662-L4668","statement_sha256":"dbf4d4d83e2d617e1276f7686f4dc3612fa84c1389afea34f12205864f90e8b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4911,"rank":4911,"depth":4,"x":2650.851,"y":211.999,"cluster":"homological-algebra"},{"id":"stacks:0FTI","tag":"0FTI","title":"Derived pullback · Lemma 0FTI","summary":"Let (f, f^sharp) : (Sh(C), O) → (Sh(C'), O') and (g, g^sharp) : (Sh(C'), O') → (Sh(C\"), O\") be morphisms of ringed topoi. Let A, A', and A\" be a differential graded O-algebra, O'-algebra, and O\"-algebra. Let φ : A' → f_*A and φ' : A\" → g_*A' be a homomorphism of differential graded O'-algebras and O\"-algebras. Then we have L(g ∘ f)^* = Lf^* ∘ Lg^* : D(A\", d) → D(A, d).","statement_latex":"Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O})\n\\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ and\n$(g, g^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')\n\\to (\\Sh(\\mathcal{C}''), \\mathcal{O}'')$\nbe morphisms of ringed topoi. Let $\\mathcal{A}$, $\\mathcal{A}'$, and\n$\\mathcal{A}''$ be a differential graded $\\mathcal{O}$-algebra,\n$\\mathcal{O}'$-algebra, and $\\mathcal{O}''$-algebra. Let\n$\\varphi : \\mathcal{A}' \\to f_*\\mathcal{A}$ and\n$\\varphi' : \\mathcal{A}'' \\to g_*\\mathcal{A}'$\nbe a homomorphism of differential graded $\\mathcal{O}'$-algebras\nand $\\mathcal{O}''$-algebras.\nThen we have $L(g \\circ f)^* = Lf^* \\circ Lg^* :\nD(\\mathcal{A}'', \\text{d}) \\to D(\\mathcal{A}, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTI","source_file":"sdga.tex","source_line":4678,"source_end_line":4693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4678-L4693","statement_sha256":"9baefc8dabb3e0cb056d2f3798ba2c33ce8fca8b13acc1d40910b3c92f954de7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4912,"rank":4912,"depth":0,"x":2360.346,"y":322.935,"cluster":"homological-algebra"},{"id":"stacks:0FTJ","tag":"0FTJ","title":"Derived pullback · Lemma 0FTJ","summary":"In the situation above, if N → N' is an isomorphism on cohomology sheaves, then t is an isomorphism of functors (- ⊗_A^L N) → (- ⊗_A^L N').","statement_latex":"In the situation above, if $\\mathcal{N} \\to \\mathcal{N}'$ is an isomorphism\non cohomology sheaves, then $t$ is an isomorphism of functors\n$(- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}) \\to\n(- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N}')$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTJ","source_file":"sdga.tex","source_line":4728,"source_end_line":4734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4728-L4734","statement_sha256":"6b3c8e685e3e71855a7bdf0d8051c0604acfe55d558f334e5bea418586eeb567","origin":"The Stacks Project","memory_eligible":false,"source_rank":4913,"rank":4913,"depth":1,"x":2485.458,"y":76.077,"cluster":"homological-algebra"},{"id":"stacks:0FTK","tag":"0FTK","title":"Derived pullback · Lemma 0FTK","summary":"Let (C, O) be a ringed site. Let A, B be differential graded O-algebras. Let N be a differential graded (A, B)-bimodule. If N is good as a left differential graded A-module, then we have M ⊗_A^L N = M ⊗_A N for all differential graded A-modules M.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras.\nLet $\\mathcal{N}$ be a  differential graded\n$(\\mathcal{A}, \\mathcal{B})$-bimodule. If $\\mathcal{N}$ is good\nas a left differential graded $\\mathcal{A}$-module, then\nwe have $\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N} =\n\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}$ for all\ndifferential graded $\\mathcal{A}$-modules $\\mathcal{M}$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTK","source_file":"sdga.tex","source_line":4759,"source_end_line":4769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4759-L4769","statement_sha256":"a74ec777b1683c851eaa0d3f99a9c7b3c61a44b003b2b1399a8e12d74805882a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4914,"rank":4914,"depth":0,"x":2591.812,"y":329.32,"cluster":"homological-algebra"},{"id":"stacks:0FTL","tag":"0FTL","title":"Derived pullback · Lemma 0FTL","summary":"Let (C, O) be a ringed site. Let A, A', A\" be differential graded O-algebras. Let N and N' be a differential graded (A, A')-bimodule and (A', A\")-bimodule. Assume that the canonical map N ⊗_A'^L N' → N ⊗_A' N' in D(A\", d) is a quasi-isomorphism. Then we have (M ⊗_A^L N) ⊗_A'^L N' = M ⊗_A^L (N ⊗_A' N') as functors D(A, d) → D(A\", d).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$\\mathcal{A}$, $\\mathcal{A}'$, $\\mathcal{A}''$ be differential graded\n$\\mathcal{O}$-algebras. Let $\\mathcal{N}$ and $\\mathcal{N}'$ be a\ndifferential graded $(\\mathcal{A}, \\mathcal{A}')$-bimodule\nand $(\\mathcal{A}', \\mathcal{A}'')$-bimodule. Assume\nthat the canonical map\n$$\n\\mathcal{N} \\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}'\n\\longrightarrow\n\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}'\n$$\nin $D(\\mathcal{A}'', \\text{d})$ is a quasi-isomorphism.\nThen we have\n$$\n(\\mathcal{M}\n\\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N})\n\\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}'\n=\n\\mathcal{M}\n\\otimes_\\mathcal{A}^\\mathbf{L}\n(\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}')\n$$\nas functors $D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{A}'', \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTL","source_file":"sdga.tex","source_line":4788,"source_end_line":4813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4788-L4813","statement_sha256":"0310d989fdfc749d7727f3ab9b9e82195687e04f2de17145085c3d05ed7941d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":4915,"rank":4915,"depth":2,"x":2309.491,"y":202.819,"cluster":"homological-algebra"},{"id":"stacks:0FTN","tag":"0FTN","title":"Derived pushforward · Lemma 0FTN","summary":"Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). Then any exact functor T : K(Mod(A, d)) → D of triangulated categories has a right derived extension RT : D(A, d) → D whose value on a graded injective and K-injective differential graded A-module I is T(I).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{A}, \\text{d})$ be a sheaf of differential graded algebras on\n$(\\mathcal{C}, \\mathcal{O})$. Then any exact functor\n$$\nT : K(\\textit{Mod}(\\mathcal{A}, \\text{d})) \\longrightarrow \\mathcal{D}\n$$\nof triangulated categories has a right derived extension\n$RT : D(\\mathcal{A}, \\text{d}) \\to \\mathcal{D}$\nwhose value on a graded injective and K-injective\ndifferential graded $\\mathcal{A}$-module $\\mathcal{I}$\nis $T(\\mathcal{I})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTN","source_file":"sdga.tex","source_line":4879,"source_end_line":4892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4879-L4892","statement_sha256":"c5c51f7f2b38a6a63f1bc33d1769970529ff3ab62fe5450a0d0104f3787775e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4916,"rank":4916,"depth":15,"x":2619.669,"y":135.844,"cluster":"homological-algebra"},{"id":"stacks:0FTP","tag":"0FTP","title":"Derived pushforward · Definition 0FTP","summary":"Derived internal hom and derived pushforward. • Let (C, O) be a ringed site. Let A, B be differential graded O-algebras. Let N be a differential graded (A, B)-bimodule. The right derived extension RSheafHom_B(N, -) : D(B, d) → D(A, d) of the internal hom functor SheafHom_B^dg(N, -) is called derived internal hom. • Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let A be a differential graded O_C-algebra. Let B be a differential graded…","statement_latex":"Derived internal hom and derived pushforward.\n\\begin{enumerate}\n\\item Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras.\nLet $\\mathcal{N}$ be a  differential graded\n$(\\mathcal{A}, \\mathcal{B})$-bimodule. The right derived extension\n$$\nR\\SheafHom_\\mathcal{B}(\\mathcal{N}, -) :\nD(\\mathcal{B}, \\text{d})\n\\longrightarrow\nD(\\mathcal{A}, \\text{d})\n$$\nof the internal hom functor $\\SheafHom_\\mathcal{B}^{dg}(\\mathcal{N}, -)$\nis called {\\it derived internal hom}.\n\\item Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential\ngraded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a\ndifferential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let\n$\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism\nof differential graded $\\mathcal{O}_\\mathcal{D}$-algebras.\nThe right derived extension\n$$\nRf_* :\nD(\\mathcal{A}, \\text{d})\n\\longrightarrow\nD(\\mathcal{B}, \\text{d})\n$$\nof the pushforward $f_*$ is called {\\it derived pushforward}.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pushforward","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTP","source_file":"sdga.tex","source_line":4917,"source_end_line":4949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4917-L4949","statement_sha256":"003b2655154cc76a4451e259154d5794f1e0ec70dad1b36f6ec5f2cc01cec5e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":4917,"rank":4917,"depth":0,"x":2444.655,"y":361.429,"cluster":"homological-algebra"},{"id":"stacks:0FTQ","tag":"0FTQ","title":"Derived pushforward · Lemma 0FTQ","summary":"Let (C, O) be a ringed site. Let A, B be differential graded O-algebras. Let N be a differential graded (A, B)-bimodule. Then RSheafHom_B(N, -) : D(B, d) → D(A, d) is right adjoint to - ⊗_A^L N : D(A, d) → D(B, d)","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$\\mathcal{A}$, $\\mathcal{B}$ be differential graded $\\mathcal{O}$-algebras.\nLet $\\mathcal{N}$ be a  differential graded\n$(\\mathcal{A}, \\mathcal{B})$-bimodule. Then\n$$\nR\\SheafHom_\\mathcal{B}(\\mathcal{N}, -) :\nD(\\mathcal{B}, \\text{d})\n\\longrightarrow\nD(\\mathcal{A}, \\text{d})\n$$\nis right adjoint to\n$$\n- \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{N} :\nD(\\mathcal{A}, \\text{d})\n\\longrightarrow\nD(\\mathcal{B}, \\text{d})\n$$","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTQ","source_file":"sdga.tex","source_line":4961,"source_end_line":4980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4961-L4980","statement_sha256":"d3bc8f9b5328c6e01bd7e7257a8c23d239b6b3fd0884260e39daae1d6c101f09","origin":"The Stacks Project","memory_eligible":false,"source_rank":4918,"rank":4918,"depth":3,"x":2392.253,"y":95.549,"cluster":"homological-algebra"},{"id":"stacks:0FTR","tag":"0FTR","title":"Derived pushforward · Lemma 0FTR","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let A be a differential graded O_C-algebra. Let B be a differential graded O_D-algebra. Let φ : B → f_*A be a homomorphism of differential graded O_D-algebras. Then Rf_* : D(A, d) → D(B, d) is right adjoint to Lf^* : D(B, d) → D(A, d)","statement_latex":"Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential\ngraded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a\ndifferential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let\n$\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism\nof differential graded $\\mathcal{O}_\\mathcal{D}$-algebras.\nThen\n$$\nRf_* : \nD(\\mathcal{A}, \\text{d})\n\\longrightarrow\nD(\\mathcal{B}, \\text{d})\n$$\nis right adjoint to\n$$\nLf^* :\nD(\\mathcal{B}, \\text{d})\n\\longrightarrow\nD(\\mathcal{A}, \\text{d})\n$$","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTR","source_file":"sdga.tex","source_line":4988,"source_end_line":5011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L4988-L5011","statement_sha256":"85f00f05e83ecc14f45710c3a87b4891db0dbed3506974a82a888516c2b48ed4","origin":"The Stacks Project","memory_eligible":false,"source_rank":4919,"rank":4919,"depth":3,"x":2644.927,"y":262.015,"cluster":"homological-algebra"},{"id":"stacks:0FTT","tag":"0FTT","title":"Derived pushforward · Lemma 0FTT","summary":"In the situation above, denote RT : D(A', d) → D(B, d) the right derived extension of ([Tag 0FTS]). Then we have RT(M) = Rf_* RSheafHom(N, M) functorially in M.","statement_latex":"In the situation above, denote $RT : D(\\mathcal{A}', \\text{d}) \\to\nD(\\mathcal{B}, \\text{d})$ the right derived extension of\n(\\ref{equation-pushforward}). Then we have\n$$\nRT(\\mathcal{M}) = Rf_* R\\SheafHom(\\mathcal{N}, \\mathcal{M})\n$$\nfunctorially in $\\mathcal{M}$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTT","source_file":"sdga.tex","source_line":5077,"source_end_line":5086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5077-L5086","statement_sha256":"17be5545373cb5bb22de8dabc253355d88c3c2537c511ef3be1ec2a6731b195d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4920,"rank":4920,"depth":5,"x":2324.464,"y":282.656,"cluster":"homological-algebra"},{"id":"stacks:0FTU","tag":"0FTU","title":"Derived pushforward · Lemma 0FTU","summary":"Let (f, f^sharp) : (Sh(C), O) → (Sh(C'), O') and (g, g^sharp) : (Sh(C'), O') → (Sh(C\"), O\") be morphisms of ringed topoi. Let A, A', and A\" be a differential graded O-algebra, O'-algebra, and O\"-algebra. Let φ : A' → f_*A and φ' : A\" → g_*A' be a homomorphism of differential graded O'-algebras and O\"-algebras. Then we have R(g ∘ f)_* = Rg_* ∘ Rf_* : D(A, d) → D(A\", d).","statement_latex":"Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O})\n\\to (\\Sh(\\mathcal{C}'), \\mathcal{O}')$ and\n$(g, g^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')\n\\to (\\Sh(\\mathcal{C}''), \\mathcal{O}'')$\nbe morphisms of ringed topoi. Let $\\mathcal{A}$, $\\mathcal{A}'$, and\n$\\mathcal{A}''$ be a differential graded $\\mathcal{O}$-algebra,\n$\\mathcal{O}'$-algebra, and $\\mathcal{O}''$-algebra. Let\n$\\varphi : \\mathcal{A}' \\to f_*\\mathcal{A}$ and\n$\\varphi' : \\mathcal{A}'' \\to g_*\\mathcal{A}'$\nbe a homomorphism of differential graded $\\mathcal{O}'$-algebras\nand $\\mathcal{O}''$-algebras.\nThen we have $R(g \\circ f)_* = Rg_* \\circ Rf_* :\nD(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{A}'', \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTU","source_file":"sdga.tex","source_line":5106,"source_end_line":5121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5106-L5121","statement_sha256":"64af85bb2b2df152cb0c82ee23dc6b4689461ad25ffcd7b4d8e910751807d593","origin":"The Stacks Project","memory_eligible":false,"source_rank":4921,"rank":4921,"depth":4,"x":2544.359,"y":85.427,"cluster":"homological-algebra"},{"id":"stacks:0FTV","tag":"0FTV","title":"Derived pushforward · Lemma 0FTV","summary":"Let (C, O) be a ringed site. Let A, A', A\" be differential graded O-algebras. Let N and N' be a differential graded (A, A')-bimodule and (A', A\")-bimodule. Assume that the canonical map N ⊗_A'^L N' → N ⊗_A' N' in D(A\", d) is a quasi-isomorphism. Then we have RSheafHom_A\" (N ⊗_A' N', -) = RSheafHom_A'(N, RSheafHom_A\"(N', -)) as functors D(A\", d) → D(A, d).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$\\mathcal{A}$, $\\mathcal{A}'$, $\\mathcal{A}''$ be differential graded\n$\\mathcal{O}$-algebras. Let $\\mathcal{N}$ and $\\mathcal{N}'$ be a\ndifferential graded $(\\mathcal{A}, \\mathcal{A}')$-bimodule\nand $(\\mathcal{A}', \\mathcal{A}'')$-bimodule. Assume\nthat the canonical map\n$$\n\\mathcal{N} \\otimes_{\\mathcal{A}'}^\\mathbf{L} \\mathcal{N}'\n\\longrightarrow\n\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}'\n$$\nin $D(\\mathcal{A}'', \\text{d})$ is a quasi-isomorphism.\nThen we have\n$$\nR\\SheafHom_{\\mathcal{A}''}\n(\\mathcal{N} \\otimes_{\\mathcal{A}'} \\mathcal{N}', -)\n=\nR\\SheafHom_{\\mathcal{A}'}(\\mathcal{N},\nR\\SheafHom_{\\mathcal{A}''}(\\mathcal{N}', -))\n$$\nas functors $D(\\mathcal{A}'', \\text{d}) \\to D(\\mathcal{A}, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTV","source_file":"sdga.tex","source_line":5129,"source_end_line":5152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5129-L5152","statement_sha256":"a8377a0a88d7f08f459930beaeca9454111f9fad1b211ea64a108f568a5973c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4922,"rank":4922,"depth":4,"x":2540.815,"y":355.868,"cluster":"homological-algebra"},{"id":"stacks:0FTW","tag":"0FTW","title":"Derived pushforward · Lemma 0FTW","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. Let A be a differential graded O_C-algebra. Let B be a differential graded O_D-algebra. Let φ : B → f_*A be a homomorphism of differential graded O_D-algebras. The diagram xymatrix D(A, d) ar[d]_Rf_* ar[rr]_forget & & D(O_C) ar[d]^Rf_* D(B, d) ar[rr]^forget & & D(O_D) commutes.","statement_latex":"Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})\n\\to (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nbe a morphism of ringed topoi. Let $\\mathcal{A}$ be a differential\ngraded $\\mathcal{O}_\\mathcal{C}$-algebra. Let $\\mathcal{B}$ be a\ndifferential graded $\\mathcal{O}_\\mathcal{D}$-algebra. Let\n$\\varphi : \\mathcal{B} \\to f_*\\mathcal{A}$ be a homomorphism\nof differential graded $\\mathcal{O}_\\mathcal{D}$-algebras.\nThe diagram\n$$\n\\xymatrix{\nD(\\mathcal{A}, \\text{d}) \\ar[d]_{Rf_*} \\ar[rr]_{forget} & &\nD(\\mathcal{O}_\\mathcal{C}) \\ar[d]^{Rf_*} \\\\\nD(\\mathcal{B}, \\text{d}) \\ar[rr]^{forget} & &\nD(\\mathcal{O}_\\mathcal{D})\n}\n$$\ncommutes.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTW","source_file":"sdga.tex","source_line":5160,"source_end_line":5179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5160-L5179","statement_sha256":"17ab73a007ca65bebeebcb740f1ef0d8d5d3dc7f1ad6f6887756637f0c7c8b8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4923,"rank":4923,"depth":5,"x":2325.762,"y":154.264,"cluster":"homological-algebra"},{"id":"stacks:0FTX","tag":"0FTX","title":"Derived pushforward · Lemma 0FTX","summary":"Let (C, O) be a ringed site. Let A be a differential graded O-algebra. Let M be a differential graded A-module. Let n ∈ Z. We have H^n(C, M) = Hom_D(A, d)(A, M[n]) where on the left hand side we have the cohomology of M viewed as a complex of O-modules.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{A}$ be a differential graded $\\mathcal{O}$-algebra.\nLet $\\mathcal{M}$ be a differential graded $\\mathcal{A}$-module.\nLet $n \\in \\mathbf{Z}$. We have\n$$\nH^n(\\mathcal{C}, \\mathcal{M}) =\n\\Hom_{D(\\mathcal{A}, \\text{d})}(\\mathcal{A}, \\mathcal{M}[n])\n$$\nwhere on the left hand side we have the cohomology of $\\mathcal{M}$\nviewed as a complex of $\\mathcal{O}$-modules.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Derived pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTX","source_file":"sdga.tex","source_line":5233,"source_end_line":5245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5233-L5245","statement_sha256":"9f8d298d76f67680c893b75270f21cb5f0f5d85e54d77f42df95c694caca994d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4924,"rank":4924,"depth":16,"x":2646.739,"y":180.921,"cluster":"homological-algebra"},{"id":"stacks:0FTZ","tag":"0FTZ","title":"Equivalences of derived categories · Lemma 0FTZ","summary":"Let (C, O) be a ringed site. If φ : A → B is a homomorphism of differential graded O-algebras which induces an isomorphism on cohomology sheaves, then D(A, d) → D(B, d), M ↦ M ⊗_A^L B is an equivalence of categories.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nIf $\\varphi : \\mathcal{A} \\to \\mathcal{B}$ is a homomorphism of\ndifferential graded $\\mathcal{O}$-algebras which induces an\nisomorphism on cohomology sheaves, then\n$$\nD(\\mathcal{A}, \\text{d}) \\longrightarrow D(\\mathcal{B}, \\text{d}), \\quad\n\\mathcal{M}\n\\longmapsto\n\\mathcal{M} \\otimes_\\mathcal{A}^\\mathbf{L} \\mathcal{B}\n$$\nis an equivalence of categories.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Equivalences of derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FTZ","source_file":"sdga.tex","source_line":5280,"source_end_line":5293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5280-L5293","statement_sha256":"31a3a5c34c0167566aae291662449a5e392a3c1312b2ea802a28c79c96784a73","origin":"The Stacks Project","memory_eligible":false,"source_rank":4925,"rank":4925,"depth":4,"x":2388.398,"y":343.534,"cluster":"homological-algebra"},{"id":"stacks:0FU1","tag":"0FU1","title":"Resolutions of differential graded algebras · Lemma 0FU1","summary":"In the situation above the differential graded O-algebra A = colim A_i has the following property: for any morphism (f, f^sharp) : (Sh(C'), O') → (Sh(C), O) of ringed topoi, the pullback f^*A is flat as a graded O'-module and is K-flat as a complex of O'-modules.","statement_latex":"In the situation above the differential graded $\\mathcal{O}$-algebra\n$$\n\\mathcal{A} = \\colim \\mathcal{A}_i\n$$\nhas the following property: for any morphism\n$(f, f^\\sharp) : (\\Sh(\\mathcal{C}'), \\mathcal{O}')\n\\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nof ringed topoi, the pullback $f^*\\mathcal{A}$\nis flat as a graded $\\mathcal{O}'$-module and\nis K-flat as a complex of $\\mathcal{O}'$-modules.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Resolutions of differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FU1","source_file":"sdga.tex","source_line":5414,"source_end_line":5426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5414-L5426","statement_sha256":"a25fe16362e8154cce45387c8257616cbf636c3d6477d688167c55e6cf1935b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4926,"rank":4926,"depth":1,"x":2448.177,"y":76.808,"cluster":"homological-algebra"},{"id":"stacks:0FU2","tag":"0FU2","title":"Resolutions of differential graded algebras · Lemma 0FU2","summary":"Let (C, O) be a ringed site. Let (B, d) be a differential graded O-algebra. There exists a quasi-isomorphism of differential graded O-algebras (A, d) → (B, d) such that A is graded flat and K-flat as a complex of O-modules and such that the same is true after pullback by any morphism of ringed topoi.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $(\\mathcal{B}, \\text{d})$ be a differential graded $\\mathcal{O}$-algebra.\nThere exists a quasi-isomorphism of differential graded $\\mathcal{O}$-algebras\n$(\\mathcal{A}, \\text{d}) \\to (\\mathcal{B}, \\text{d})$ such that\n$\\mathcal{A}$ is graded flat and K-flat as a complex of $\\mathcal{O}$-modules\nand such that the same is true after pullback by any morphism of\nringed topoi.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Resolutions of differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FU2","source_file":"sdga.tex","source_line":5542,"source_end_line":5551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5542-L5551","statement_sha256":"380e2f852ddbfc6cd8d3521ebf546298eede23614386c45ffe43ff503ce3c82e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4927,"rank":4927,"depth":2,"x":2618.735,"y":307.605,"cluster":"homological-algebra"},{"id":"stacks:0GZ9","tag":"0GZ9","title":"Differential graded modules on a category · Definition 0GZ9","summary":"In the situation above, we denote mathitQC(A, d) the full subcategory of D(A, d) consisting of objects M such that for all U → V in C the canonical map RΓ(V, M) ⊗_A(V)^L A(U) → RΓ(U, M) is an isomorphism in D(A(U), d).","statement_latex":"In the situation above, we denote\n{\\it $\\mathit{QC}(\\mathcal{A}, \\text{d})$}\nthe full subcategory of $D(\\mathcal{A}, \\text{d})$\nconsisting of objects $M$ such that\nfor all $U \\to V$ in $\\mathcal{C}$ the canonical map\n$$\nR\\Gamma(V, M) \\otimes_{\\mathcal{A}(V)}^\\mathbf{L} \\mathcal{A}(U)\n\\longrightarrow\nR\\Gamma(U, M)\n$$\nis an isomorphism in $D(\\mathcal{A}(U), \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Differential graded modules on a category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZ9","source_file":"sdga.tex","source_line":5904,"source_end_line":5917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5904-L5917","statement_sha256":"7d989c17b56d04a3d89c8f629f7a824549449058f5049975474595195c594155","origin":"The Stacks Project","memory_eligible":false,"source_rank":4928,"rank":4928,"depth":0,"x":2307.102,"y":234.135,"cluster":"homological-algebra"},{"id":"stacks:0GZA","tag":"0GZA","title":"Differential graded modules on a category · Lemma 0GZA","summary":"In the situation above, the subcategory mathitQC(A, d) is a strictly full, saturated, triangulated subcategory of D(A, d) preserved by arbitrary direct sums.","statement_latex":"In the situation above, the subcategory $\\mathit{QC}(\\mathcal{A}, \\text{d})$\nis a strictly full, saturated, triangulated subcategory of\n$D(\\mathcal{A}, \\text{d})$ preserved by arbitrary direct sums.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Differential graded modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZA","source_file":"sdga.tex","source_line":5919,"source_end_line":5924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5919-L5924","statement_sha256":"2cc1a8ea4030cbd35c09d186e7f69dfb6823713bd93293f0faf93c48bf5cbef3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4929,"rank":4929,"depth":16,"x":2596.224,"y":111.378,"cluster":"homological-algebra"},{"id":"stacks:0GZC","tag":"0GZC","title":"Differential graded modules on a category · Lemma 0GZC","summary":"Let g : (Sh(C'), O') → (Sh(C), O) and φ : g^*A → A' be as above. Then the functor Lg^* : D(A, d) → D(A', d) maps mathitQC(A, d) into mathitQC(A', d).","statement_latex":"Let $g : (\\Sh(\\mathcal{C}'), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nand $\\varphi : g^*\\mathcal{A} \\to \\mathcal{A}'$\nbe as above. Then the functor\n$Lg^* : D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{A}', \\text{d})$\nmaps $\\mathit{QC}(\\mathcal{A}, \\text{d})$ into\n$\\mathit{QC}(\\mathcal{A}', \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Differential graded modules on a category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZC","source_file":"sdga.tex","source_line":5987,"source_end_line":5995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L5987-L5995","statement_sha256":"543ffaf21d832aaec7276773cc1c48b3a6704c9c3f3d4f731d19bf2d3f15e3ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":4930,"rank":4930,"depth":1,"x":2481.65,"y":366.169,"cluster":"homological-algebra"},{"id":"stacks:0GZE","tag":"0GZE","title":"Differential graded modules on a category, bis · Lemma 0GZE","summary":"Let C, O, A be as in Section [Tag 0GZ8]. Let C' ⊂ C be a full subcategory with the following property: for every U ∈ Ob(C) the category U/C' of arrows U → U' is cofiltered. Denote O', A' the restrictions of O, A to C'. Then restrictions induces an equivalence mathitQC(A, d) → mathitQC(A', d).","statement_latex":"Let $\\mathcal{C}, \\mathcal{O}, \\mathcal{A}$ be as in\nSection \\ref{section-modules-cohomology}. Let\n$\\mathcal{C}' \\subset \\mathcal{C}$ be a full subcategory\nwith the following property: for every $U \\in \\Ob(\\mathcal{C})$ the\ncategory $U/\\mathcal{C}'$ of arrows $U \\to U'$ is cofiltered.\nDenote $\\mathcal{O}', \\mathcal{A}'$ the restrictions\nof $\\mathcal{O}, \\mathcal{A}$ to $\\mathcal{C}'$.\nThen restrictions induces an equivalence\n$\\mathit{QC}(\\mathcal{A}, \\text{d}) \\to \\mathit{QC}(\\mathcal{A}', \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Differential graded modules on a category, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZE","source_file":"sdga.tex","source_line":6060,"source_end_line":6071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L6060-L6071","statement_sha256":"5bad97dce5b433e7565c5596af0e4ec772777cf1f5b1977c6f80caba8a536517","origin":"The Stacks Project","memory_eligible":false,"source_rank":4931,"rank":4931,"depth":2,"x":2361.137,"y":113.063,"cluster":"homological-algebra"},{"id":"stacks:0GZF","tag":"0GZF","title":"Differential graded modules on a category, bis · Lemma 0GZF","summary":"Let C, O be as in Section [Tag 0GZ8]. Let φ : A → B be a homomorphism of differential graded O-algebras which induces an isomorphism on cohomology sheaves, then the equivalence D(A, d) → D(B, d) of Lemma [Tag 0FTZ] induces an equivalence mathitQC(A, d) → mathitQC(B, d).","statement_latex":"Let $\\mathcal{C}, \\mathcal{O}$ be as in\nSection \\ref{section-modules-cohomology}. Let\n$\\varphi : \\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of\ndifferential graded $\\mathcal{O}$-algebras which induces an\nisomorphism on cohomology sheaves, then the equivalence\n$D(\\mathcal{A}, \\text{d}) \\to D(\\mathcal{B}, \\text{d})$\nof Lemma \\ref{lemma-qis-equivalence} induces an equivalence\n$\\mathit{QC}(\\mathcal{A}, \\text{d}) \\to\n\\mathit{QC}(\\mathcal{B}, \\text{d})$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Differential graded modules on a category, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZF","source_file":"sdga.tex","source_line":6140,"source_end_line":6151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L6140-L6151","statement_sha256":"bb7ed1cdf9d4154a3ce0b2df9210fd2ce2b7d1cadfdc99b0095e6a5d6b9ddb8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4932,"rank":4932,"depth":5,"x":2653.79,"y":231.418,"cluster":"homological-algebra"},{"id":"stacks:0GZH","tag":"0GZH","title":"Inverse systems of differential graded algebras · Lemma 0GZH","summary":"In the situation above, suppose that A = (A_n) and B = (B_n) are inverse systems of differential graded R-algebras. If φ : (A_n) → (B_n) is an isomorphism of pro-objects, then the functor mathitQC(A) → mathitQC(B) constructed above is an equivalence.","statement_latex":"In the situation above, suppose that $\\mathcal{A} = (A_n)$ and\n$\\mathcal{B} = (B_n)$ are inverse systems of differential\ngraded $R$-algebras. If $\\varphi : (A_n) \\to (B_n)$ is an isomorphism\nof pro-objects, then the functor $\\mathit{QC}(\\mathcal{A}) \\to\n\\mathit{QC}(\\mathcal{B})$ constructed above is an\nequivalence.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Inverse systems of differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZH","source_file":"sdga.tex","source_line":6234,"source_end_line":6242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L6234-L6242","statement_sha256":"32d2568ae235df0172e502be88f4b91d499f481a5cadb4990a3ef34b03d6357d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4933,"rank":4933,"depth":0,"x":2342.552,"y":310.27,"cluster":"homological-algebra"},{"id":"stacks:0GZI","tag":"0GZI","title":"Inverse systems of differential graded algebras · Lemma 0GZI","summary":"If (N_n) and (N'_n) are pro-isomorphic in the derived category as defined above, then for every object (M_n) of D(N, A) we have Rlim (M_n ⊗_A_n^L N_n) = Rlim (M_n ⊗_A_n^L N'_n) in D(R).","statement_latex":"If $(N_n)$ and $(N'_n)$ are pro-isomorphic in the derived category\nas defined above,\nthen for every object $(M_n)$ of $D(\\mathbf{N}, \\mathcal{A})$ we have\n$$\nR\\lim (M_n \\otimes_{A_n}^\\mathbf{L} N_n) =\nR\\lim (M_n \\otimes_{A_n}^\\mathbf{L} N'_n)\n$$\nin $D(R)$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Inverse systems of differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZI","source_file":"sdga.tex","source_line":6318,"source_end_line":6328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L6318-L6328","statement_sha256":"69bea73caae689b39bea724827121a862169032ce98ff27ff50a95676a110600","origin":"The Stacks Project","memory_eligible":false,"source_rank":4934,"rank":4934,"depth":19,"x":2508.784,"y":75.323,"cluster":"homological-algebra"},{"id":"stacks:0GZJ","tag":"0GZJ","title":"Inverse systems of differential graded algebras · Lemma 0GZJ","summary":"This is a variant of [BS] Let R be a ring. Let f_1, …, f_r ∈ R. Let K_n be the Koszul complex on f_1^n, …, f_r^n viewed as a differential graded R-algebra. Let (M_n) be an object of D(N, (K_n)). Then for any t ≥ 1 we have Rlim (M_n ⊗_R^L K_t) = Rlim (M_n ⊗_K_n^L K_t) in D(R).","statement_latex":"\\begin{reference}\nThis is a variant of \\cite[Lemma 3.5.4]{BS}\n\\end{reference}\nLet $R$ be a ring. Let $f_1, \\ldots, f_r \\in R$.\nLet $K_n$ be the Koszul complex on $f_1^n, \\ldots, f_r^n$\nviewed as a differential graded $R$-algebra.\nLet $(M_n)$ be an object of $D(\\mathbf{N}, (K_n))$.\nThen for any $t \\geq 1$ we have\n$$\nR\\lim (M_n \\otimes_R^\\mathbf{L} K_t) =\nR\\lim (M_n \\otimes_{K_n}^\\mathbf{L} K_t)\n$$\nin $D(R)$.","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Inverse systems of differential graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZJ","source_file":"sdga.tex","source_line":6339,"source_end_line":6354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L6339-L6354","statement_sha256":"c2176420d6ca446de8d2f55bea57bd32f1076d4504c1593797d5d17098100c43","origin":"The Stacks Project","memory_eligible":false,"source_rank":4935,"rank":4935,"depth":20,"x":2575.201,"y":343.119,"cluster":"homological-algebra"},{"id":"stacks:0GZK","tag":"0GZK","title":"Inverse systems of differential graded algebras · Proposition 0GZK","summary":"Let R be a Noetherian ring. Let I ⊂ R be an ideal. The following three categories are canonically equivalent: • Let A be the sheaf of R-algebras on N corresponding to the inverse system of R-algebras A_n = R/I^n. The category mathitQC(A). • Choose generators f_1, …, f_r of I. Let B be the sheaf of differential graded R-algebras on N corresponding to the inverse system of Koszul algebras on f_1^n, …, f_r^n. The category mathitQC(B). • The full subcategory D_comp(R, I) ⊂…","statement_latex":"Let $R$ be a Noetherian ring. Let $I \\subset R$ be an ideal.\nThe following three categories are canonically equivalent:\n\\begin{enumerate}\n\\item Let $\\mathcal{A}$ be the sheaf of $R$-algebras on $\\mathbf{N}$\ncorresponding to the inverse system of $R$-algebras $A_n = R/I^n$.\nThe category $\\mathit{QC}(\\mathcal{A})$.\n\\item Choose generators $f_1, \\ldots, f_r$ of $I$. Let\n$\\mathcal{B}$ be the sheaf of differential graded $R$-algebras\non $\\mathbf{N}$ corresponding to the inverse system of\nKoszul algebras on $f_1^n, \\ldots, f_r^n$.\nThe category $\\mathit{QC}(\\mathcal{B})$.\n\\item The full subcategory $D_{comp}(R, I) \\subset D(R)$\nof derived complete objects, see More on Algebra, Definition\n\\ref{more-algebra-definition-derived-complete} and text following.\n\\end{enumerate}","area":"Homological Algebra","chapter":"Differential Graded Sheaves","chapter_id":"sdga","section":"Inverse systems of differential graded algebras","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZK","source_file":"sdga.tex","source_line":6384,"source_end_line":6401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/sdga.tex#L6384-L6401","statement_sha256":"a266f31a237733a0525481644b8676b7645e69c3ba1d49aba1404a5698431bed","origin":"The Stacks Project","memory_eligible":false,"source_rank":4936,"rank":4936,"depth":22,"x":2310.649,"y":183.202,"cluster":"homological-algebra"},{"id":"stacks:01G0","tag":"01G0","title":"Semi-representable objects · Definition 01G0","summary":"Let C be a category. We denote SR(C) the category of semi-representable objects defined as follows • objects are families of objects (U_i)_i ∈ I, and • morphisms (U_i)_i ∈ I → (V_j)_j ∈ J are given by a map α : I → J and for each i ∈ I a morphism f_i : U_i → V_α(i) of C. Let X ∈ Ob(C) be an object of C. The category of semi-representable objects over X is the category SR(C, X) = SR(C/X).","statement_latex":"Let $\\mathcal{C}$ be a category. We denote $\\text{SR}(\\mathcal{C})$\nthe category of {\\it semi-representable objects} defined as follows\n\\begin{enumerate}\n\\item objects are families of objects $\\{U_i\\}_{i \\in I}$, and\n\\item morphisms $\\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$ are given by\na map $\\alpha : I \\to J$ and for each $i \\in I$\na morphism $f_i : U_i \\to V_{\\alpha(i)}$ of $\\mathcal{C}$.\n\\end{enumerate}\nLet $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$.\nThe category of {\\it semi-representable objects over $X$}\nis the category\n$\\text{SR}(\\mathcal{C}, X) = \\text{SR}(\\mathcal{C}/X)$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Semi-representable objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01G0","source_file":"hypercovering.tex","source_line":110,"source_end_line":124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L110-L124","statement_sha256":"f55e8381bd70b798013fb97198f931be3f7b96e5a7bbaec77bb0cc60f4bbeb7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4937,"rank":4937,"depth":0,"x":1140.176,"y":820.948,"cluster":"sheaf-cohomology"},{"id":"stacks:01G1","tag":"01G1","title":"Semi-representable objects · Definition 01G1","summary":"Let C be a category. We denote F the functor which associates a presheaf to a semi-representable object. In a formula F : SR(C) & → & PSh(C) (U_i)_i ∈ I & ↦ & amalg_i∈ I h_U_i where h_U denotes the representable presheaf associated to the object U.","statement_latex":"Let $\\mathcal{C}$ be a category.\nWe denote $F$ the functor {\\it which associates a presheaf to a\nsemi-representable object}. In a formula\n\\begin{eqnarray*}\nF : \\text{SR}(\\mathcal{C}) & \\longrightarrow & \\textit{PSh}(\\mathcal{C}) \\\\\n\\{U_i\\}_{i \\in I} & \\longmapsto & \\amalg_{i\\in I} h_{U_i}\n\\end{eqnarray*}\nwhere $h_U$ denotes the representable presheaf associated to\nthe object $U$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Semi-representable objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01G1","source_file":"hypercovering.tex","source_line":144,"source_end_line":155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L144-L155","statement_sha256":"7f67e5ee1ad6cb832c93f2f504ec56b980c30a8ab7785e421e11437ec4cec55d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4938,"rank":4938,"depth":0,"x":1009.036,"y":581.749,"cluster":"sheaf-cohomology"},{"id":"stacks:01G2","tag":"01G2","title":"Semi-representable objects · Lemma 01G2","summary":"Let C be a category. • the category SR(C) has coproducts and F commutes with them, • the functor F : SR(C) → PSh(C) commutes with limits, • if C has fibre products, then SR(C) has fibre products, • if C has products of pairs, then SR(C) has products of pairs, • if C has equalizers, so does SR(C), and • if C has a final object, so does SR(C). Let X ∈ Ob(C). • the category SR(C, X) has coproducts and F commutes with them, • if C has fibre products, then SR(C, X) has finite…","statement_latex":"Let $\\mathcal{C}$ be a category.\n\\begin{enumerate}\n\\item the category $\\text{SR}(\\mathcal{C})$ has coproducts\nand $F$ commutes with them,\n\\item the functor $F : \\text{SR}(\\mathcal{C}) \\to \\textit{PSh}(\\mathcal{C})$\ncommutes with limits,\n\\item if $\\mathcal{C}$ has fibre products, then $\\text{SR}(\\mathcal{C})$\nhas fibre products,\n\\item if $\\mathcal{C}$ has products of pairs, then\n$\\text{SR}(\\mathcal{C})$ has products of pairs,\n\\item if $\\mathcal{C}$ has equalizers, so does $\\text{SR}(\\mathcal{C})$, and\n\\item if $\\mathcal{C}$ has a final object, so does $\\text{SR}(\\mathcal{C})$.\n\\end{enumerate}\nLet $X \\in \\Ob(\\mathcal{C})$.\n\\begin{enumerate}\n\\item the category $\\text{SR}(\\mathcal{C}, X)$ has coproducts\nand $F$ commutes with them,\n\\item if $\\mathcal{C}$ has fibre products, then $\\text{SR}(\\mathcal{C}, X)$\nhas finite limits and\n$F : \\text{SR}(\\mathcal{C}, X) \\to \\textit{PSh}(\\mathcal{C})/h_X$\ncommutes with them.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Semi-representable objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01G2","source_file":"hypercovering.tex","source_line":173,"source_end_line":197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L173-L197","statement_sha256":"32e495dab30fa3c6cc3d7b5cae61c9f483dd60b83298985454b5c5567c05ee43","origin":"The Stacks Project","memory_eligible":false,"source_rank":4939,"rank":4939,"depth":8,"x":1298.365,"y":683.815,"cluster":"sheaf-cohomology"},{"id":"stacks:01G3","tag":"01G3","title":"Hypercoverings · Definition 01G3","summary":"Let C be a site. Let f = (α, f_i) : (U_i)_i ∈ I → (V_j)_j ∈ J be a morphism in the category SR(C). We say that f is a covering if for every j ∈ J the family of morphisms (U_i → V_j)_i ∈ I, α(i) = j is a covering for the site C. Let X be an object of C. A morphism K → L in SR(C, X) is a covering if its image in SR(C) is a covering.","statement_latex":"Let $\\mathcal{C}$ be a site. Let\n$f = (\\alpha, f_i) : \\{U_i\\}_{i \\in I} \\to \\{V_j\\}_{j \\in J}$\nbe a morphism in the category $\\text{SR}(\\mathcal{C})$.\nWe say that $f$ is a {\\it covering} if for every $j \\in J$ the\nfamily of morphisms $\\{U_i \\to V_j\\}_{i \\in I, \\alpha(i) = j}$\nis a covering for the site $\\mathcal{C}$.\nLet $X$ be an object of $\\mathcal{C}$.\nA morphism $K \\to L$ in $\\text{SR}(\\mathcal{C}, X)$ is\na {\\it covering} if its image in $\\text{SR}(\\mathcal{C})$ is\na covering.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01G3","source_file":"hypercovering.tex","source_line":292,"source_end_line":304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L292-L304","statement_sha256":"8cf8d68898194b458e3cc941c9a167c322ab04566916b03df46238a0c0de947b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4940,"rank":4940,"depth":0,"x":1002.663,"y":772.809,"cluster":"sheaf-cohomology"},{"id":"stacks:01G4","tag":"01G4","title":"Hypercoverings · Lemma 01G4","summary":"Let C be a site. • A composition of coverings in SR(C) is a covering. • If K → L is a covering in SR(C) and L' → L is a morphism, then L' ×_L K exists and L' ×_L K → L' is a covering. • If C has products of pairs, and A → B and K → L are coverings in SR(C), then A × K → B × L is a covering. Let X ∈ Ob(C). Then (1) and (2) holds for SR(C, X) and (3) holds if C has fibre products.","statement_latex":"Let $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item A composition of coverings in $\\text{SR}(\\mathcal{C})$\nis a covering.\n\\item If $K \\to L$ is a covering in $\\text{SR}(\\mathcal{C})$\nand $L' \\to L$ is a morphism, then $L' \\times_L K$ exists\nand $L' \\times_L K \\to L'$ is a covering.\n\\item If $\\mathcal{C}$ has products of pairs, and\n$A \\to B$ and $K \\to L$ are coverings in $\\text{SR}(\\mathcal{C})$,\nthen $A \\times K \\to B \\times L$ is a covering.\n\\end{enumerate}\nLet $X \\in \\Ob(\\mathcal{C})$. Then (1) and (2) holds for\n$\\text{SR}(\\mathcal{C}, X)$ and (3) holds if $\\mathcal{C}$\nhas fibre products.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01G4","source_file":"hypercovering.tex","source_line":306,"source_end_line":322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L306-L322","statement_sha256":"0e65e0b9a43f5b4351824758b5a9baef19e7b2f57ae2892ff7c209fdc385a8c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":4941,"rank":4941,"depth":9,"x":1149.28,"y":539.179,"cluster":"sheaf-cohomology"},{"id":"stacks:01G5","tag":"01G5","title":"Hypercoverings · Definition 01G5","summary":"Let C be a site. Assume C has fibre products. Let X ∈ Ob(C) be an object of C. A hypercovering of X is a simplicial object K of SR(C, X) such that • The object K_0 is a covering of X for the site C. • For every n ≥ 0 the canonical morphism K_n + 1 → (cosk_n sk_n K)_n + 1 is a covering in the sense defined above.","statement_latex":"Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has fibre products.\nLet $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$.\nA {\\it hypercovering of $X$} is a simplicial object\n$K$ of $\\text{SR}(\\mathcal{C}, X)$ such that\n\\begin{enumerate}\n\\item The object $K_0$ is a covering of $X$ for the site $\\mathcal{C}$.\n\\item For every $n \\geq 0$ the canonical morphism\n$$\nK_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n K)_{n + 1}\n$$\nis a covering in the sense defined above.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01G5","source_file":"hypercovering.tex","source_line":345,"source_end_line":359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L345-L359","statement_sha256":"aa52cc988a2a9acd43829baed92949b7dc402965565f595c5194b95b904c27fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":4942,"rank":4942,"depth":0,"x":1229.12,"y":794.887,"cluster":"sheaf-cohomology"},{"id":"stacks:01G7","tag":"01G7","title":"Hypercoverings · Lemma 01G7","summary":"Let C be a site with fibre products. Let X ∈ Ob(C) be an object of C. The collection of all hypercoverings of X forms a set.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$.\nThe collection of all hypercoverings of $X$ forms a set.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01G7","source_file":"hypercovering.tex","source_line":436,"source_end_line":441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L436-L441","statement_sha256":"fd073796652fec5e193ed95988a108bdf407b2e6aa2e4a329a91f41603c29a09","origin":"The Stacks Project","memory_eligible":false,"source_rank":4943,"rank":4943,"depth":0,"x":964.368,"y":651.501,"cluster":"sheaf-cohomology"},{"id":"stacks:01G9","tag":"01G9","title":"Hypercoverings · Lemma 01G9","summary":"Let C be a site with fibre products. Let X ∈ Ob(C) be an object of C. Let K be a hypercovering of X. Consider the simplicial object F(K) of PSh(C), endowed with its augmentation to the constant simplicial presheaf h_X. • The morphism of presheaves F(K)_0 → h_X becomes a surjection after sheafification. • The morphism (d^1_0, d^1_1) : F(K)_1 → F(K)_0 ×_h_X F(K)_0 becomes a surjection after sheafification. • For every n ≥ 1 the morphism F(K)_n + 1 → (cosk_n sk_n F(K))_n + 1…","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X \\in \\Ob(\\mathcal{C})$ be an object of $\\mathcal{C}$.\nLet $K$ be a hypercovering of $X$.\nConsider the simplicial object $F(K)$ of $\\textit{PSh}(\\mathcal{C})$,\nendowed with its augmentation to the constant simplicial presheaf $h_X$.\n\\begin{enumerate}\n\\item The morphism of presheaves $F(K)_0 \\to h_X$ becomes\na surjection after sheafification.\n\\item The morphism\n$$\n(d^1_0, d^1_1) :\nF(K)_1\n\\longrightarrow\nF(K)_0 \\times_{h_X} F(K)_0\n$$\nbecomes a surjection after sheafification.\n\\item For every $n \\geq 1$ the morphism\n$$\nF(K)_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n F(K))_{n + 1}\n$$\nturns into a surjection after sheafification.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01G9","source_file":"hypercovering.tex","source_line":467,"source_end_line":491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L467-L491","statement_sha256":"bfe84ebf9ed73ba57c1c4b45712ae9601d5286882a3d95da22e0eb144e228157","origin":"The Stacks Project","memory_eligible":false,"source_rank":4944,"rank":4944,"depth":9,"x":1275.188,"y":606.963,"cluster":"sheaf-cohomology"},{"id":"stacks:01GB","tag":"01GB","title":"Acyclicity · Definition 01GB","summary":"Let C be a site. Let K be a simplicial object of PSh(C). By the above we get a simplicial object Z_K^\\# of Ab(C). We can take its associated complex of abelian presheaves s(Z_K^\\#), see Simplicial, Section [Tag 0194]. The homology of K is the homology of the complex of abelian sheaves s(Z_K^\\#).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $K$ be a simplicial object of $\\textit{PSh}(\\mathcal{C})$.\nBy the above we get a simplicial object $\\mathbf{Z}_K^\\#$ of\n$\\textit{Ab}(\\mathcal{C})$. We can take its associated\ncomplex of abelian presheaves $s(\\mathbf{Z}_K^\\#)$, see\nSimplicial, Section \\ref{simplicial-section-complexes}.\nThe {\\it homology of $K$} is the homology of the\ncomplex of abelian sheaves $s(\\mathbf{Z}_K^\\#)$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Acyclicity","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GB","source_file":"hypercovering.tex","source_line":572,"source_end_line":582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L572-L582","statement_sha256":"c10943fd064170f677e0a8a7ddb743f227a665aafd83dc4a195b72271d2d5b1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4945,"rank":4945,"depth":0,"x":1081.63,"y":816.366,"cluster":"sheaf-cohomology"},{"id":"stacks:01GC","tag":"01GC","title":"Acyclicity · Lemma 01GC","summary":"Let C be a site. Let F → G be a morphism of presheaves of sets. Denote K the simplicial object of PSh(C) whose nth term is the (n + 1)st fibre product of F over G, see Simplicial, Example [Tag 016E]. Then, if F → G is surjective after sheafification, we have H_i(K) = ( 0 & if & i > 0 Z_G^\\# & if & i = 0 . The isomorphism in degree 0 is given by the morphism H_0(K) → Z_G^\\# coming from the map (Z_K^\\#)_0 = Z_F^\\# → Z_G^\\#.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{F} \\to \\mathcal{G}$ be a morphism\nof presheaves of sets. Denote $K$ the simplicial\nobject of $\\textit{PSh}(\\mathcal{C})$ whose $n$th\nterm is the $(n + 1)$st fibre product of $\\mathcal{F}$\nover $\\mathcal{G}$, see\nSimplicial, Example \\ref{simplicial-example-fibre-products-simplicial-object}.\nThen, if $\\mathcal{F} \\to \\mathcal{G}$ is surjective after\nsheafification, we have\n$$\nH_i(K) =\n\\left\\{\n\\begin{matrix}\n0 & \\text{if} & i > 0\\\\\n\\mathbf{Z}_\\mathcal{G}^\\# & \\text{if} & i = 0\n\\end{matrix}\n\\right.\n$$\nThe isomorphism in degree $0$ is given by the\nmorphism $H_0(K) \\to \\mathbf{Z}_\\mathcal{G}^\\#$\ncoming from the map $(\\mathbf{Z}_K^\\#)_0 =\n\\mathbf{Z}_\\mathcal{F}^\\# \\to \\mathbf{Z}_\\mathcal{G}^\\#$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Acyclicity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GC","source_file":"hypercovering.tex","source_line":590,"source_end_line":614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L590-L614","statement_sha256":"a3bdc181b65dab4553720c31926a186fc14614b6e49a85aa7b424232af1a5eae","origin":"The Stacks Project","memory_eligible":false,"source_rank":4946,"rank":4946,"depth":8,"x":1055.937,"y":551.881,"cluster":"sheaf-cohomology"},{"id":"stacks:01GD","tag":"01GD","title":"Acyclicity · Lemma 01GD","summary":"Let C be a site. Let f : L → K be a morphism of simplicial objects of PSh(C). Let n ≥ 0 be an integer. Assume that • For i < n the morphism L_i → K_i is an isomorphism. • The morphism L_n → K_n is surjective after sheafification. • The canonical map L → cosk_n sk_n L is an isomorphism. • The canonical map K → cosk_n sk_n K is an isomorphism. Then H_i(f) : H_i(L) → H_i(K) is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $f : L \\to K$ be a morphism of\nsimplicial objects of $\\textit{PSh}(\\mathcal{C})$.\nLet $n \\geq 0$ be an integer.\nAssume that\n\\begin{enumerate}\n\\item For $i < n$ the morphism $L_i \\to K_i$ is an isomorphism.\n\\item The morphism $L_n \\to K_n$ is surjective after sheafification.\n\\item The canonical map $L \\to \\text{cosk}_n \\text{sk}_n L$ is an isomorphism.\n\\item The canonical map $K \\to \\text{cosk}_n \\text{sk}_n K$ is an isomorphism.\n\\end{enumerate}\nThen $H_i(f) : H_i(L) \\to H_i(K)$ is an isomorphism.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Acyclicity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GD","source_file":"hypercovering.tex","source_line":662,"source_end_line":676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L662-L676","statement_sha256":"3760ede3b504bd223bab6e27e3127bc54f95b17119cc4c0cd9e176301e9a8167","origin":"The Stacks Project","memory_eligible":false,"source_rank":4947,"rank":4947,"depth":9,"x":1287.788,"y":732.496,"cluster":"sheaf-cohomology"},{"id":"stacks:01GE","tag":"01GE","title":"Acyclicity · Lemma 01GE","summary":"Let C be a site. Let K be a simplicial presheaf. Let G be a presheaf. Let K → G be an augmentation of K towards G. Assume that • The morphism of presheaves K_0 → G becomes a surjection after sheafification. • The morphism (d^1_0, d^1_1) : K_1 → K_0 ×_G K_0 becomes a surjection after sheafification. • For every n ≥ 1 the morphism K_n + 1 → (cosk_n sk_n K)_n + 1 turns into a surjection after sheafification. Then H_i(K) = 0 for i > 0 and H_0(K) = Z_G^\\#.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $K$ be a simplicial presheaf.\nLet $\\mathcal{G}$ be a presheaf.\nLet $K \\to \\mathcal{G}$ be an augmentation of $K$\ntowards $\\mathcal{G}$. Assume that\n\\begin{enumerate}\n\\item The morphism of presheaves $K_0 \\to \\mathcal{G}$ becomes\na surjection after sheafification.\n\\item The morphism\n$$\n(d^1_0, d^1_1) :\nK_1\n\\longrightarrow\nK_0 \\times_\\mathcal{G} K_0\n$$\nbecomes a surjection after sheafification.\n\\item For every $n \\geq 1$ the morphism\n$$\nK_{n + 1} \\longrightarrow (\\text{cosk}_n \\text{sk}_n K)_{n + 1}\n$$\nturns into a surjection after sheafification.\n\\end{enumerate}\nThen $H_i(K) = 0$ for $i > 0$ and\n$H_0(K) = \\mathbf{Z}_\\mathcal{G}^\\#$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Acyclicity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GE","source_file":"hypercovering.tex","source_line":717,"source_end_line":743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L717-L743","statement_sha256":"1f0005836109d546ba37f0e71e39d37d0aea9aa1464a00013cba52632d076d4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4948,"rank":4948,"depth":10,"x":971.287,"y":730.87,"cluster":"sheaf-cohomology"},{"id":"stacks:01GF","tag":"01GF","title":"Acyclicity · Lemma 01GF","summary":"Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. The homology of the simplicial presheaf F(K) is 0 in degrees > 0 and equal to Z_X^\\# in degree 0.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K$ be a hypercovering of $X$.\nThe homology of the simplicial presheaf $F(K)$ is\n$0$ in degrees $> 0$ and equal to $\\mathbf{Z}_X^\\#$\nin degree $0$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Acyclicity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GF","source_file":"hypercovering.tex","source_line":783,"source_end_line":791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L783-L791","statement_sha256":"6b4273fb8d639379bce588126cedce66bdbe7226abf1ba3a0d4bfab8cecaa580","origin":"The Stacks Project","memory_eligible":false,"source_rank":4949,"rank":4949,"depth":11,"x":1206.194,"y":552.315,"cluster":"sheaf-cohomology"},{"id":"stacks:01GV","tag":"01GV","title":"v Cech cohomology and hypercoverings · Lemma 01GV","summary":"Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let F be a sheaf of abelian groups on C. Then checkH^0(K, F) = F(X).","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K$ be a hypercovering of $X$.\nLet $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$.\nThen $\\check{H}^0(K, \\mathcal{F}) = \\mathcal{F}(X)$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"v Cech cohomology and hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GV","source_file":"hypercovering.tex","source_line":844,"source_end_line":851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L844-L851","statement_sha256":"5dd0c4d0c7c00287eaa8990cc92a253deef7f7b62c165be20033274d78d9c359","origin":"The Stacks Project","memory_eligible":false,"source_rank":4950,"rank":4950,"depth":0,"x":1176.539,"y":817.514,"cluster":"sheaf-cohomology"},{"id":"stacks:01GW","tag":"01GW","title":"v Cech cohomology and hypercoverings · Lemma 01GW","summary":"Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let I be an injective sheaf of abelian groups on C. Then checkH^p(K, I) = ( I(X) & if & p = 0 0 & if & p > 0 .","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K$ be a hypercovering of $X$.\nLet $\\mathcal{I}$ be an injective sheaf of abelian groups on $\\mathcal{C}$.\nThen\n$$\n\\check{H}^p(K, \\mathcal{I}) =\n\\left\\{\n\\begin{matrix}\n\\mathcal{I}(X) & \\text{if} & p = 0 \\\\\n0 & \\text{if} & p > 0\n\\end{matrix}\n\\right.\n$$","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"v Cech cohomology and hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GW","source_file":"hypercovering.tex","source_line":886,"source_end_line":902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L886-L902","statement_sha256":"5da51eb25e665cd2f14df6ea41607a2e5090846a30786da7cac6f3b7568884d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":4951,"rank":4951,"depth":12,"x":984.965,"y":604.938,"cluster":"sheaf-cohomology"},{"id":"stacks:01GY","tag":"01GY","title":"v Cech cohomology and hypercoverings · Lemma 01GY","summary":"Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let F be a sheaf of abelian groups on C. There is a map s(F(K)) → RΓ(X, F) in D^+(Ab) functorial in F, which induces natural transformations checkH^i(K, -) → H^i(X, -) as functors Ab(C) → Ab. Moreover, there is a spectral sequence (E_r, d_r)_r ≥ 0 with E_2^p, q = checkH^p(K, underlineH^q(F)) converging to H^p + q(X, F). This spectral sequence is functorial in F and in the…","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K$ be a hypercovering of $X$.\nLet $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$.\nThere is a map\n$$\ns(\\mathcal{F}(K))\n\\longrightarrow\nR\\Gamma(X, \\mathcal{F})\n$$\nin $D^{+}(\\textit{Ab})$ functorial in $\\mathcal{F}$, which induces\nnatural transformations\n$$\n\\check{H}^i(K, -) \\longrightarrow H^i(X, -)\n$$\nas functors $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$. Moreover,\nthere is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))\n$$\nconverging to $H^{p + q}(X, \\mathcal{F})$.\nThis spectral sequence is functorial in $\\mathcal{F}$ and\nin the hypercovering $K$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"v Cech cohomology and hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GY","source_file":"hypercovering.tex","source_line":954,"source_end_line":979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L954-L979","statement_sha256":"3e12f47e52df64de8d50cfd5b5f45fdb3fde6e36923bc9c427155c1b16dd826f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4952,"rank":4952,"depth":13,"x":1297.463,"y":653.029,"cluster":"sheaf-cohomology"},{"id":"stacks:09VU","tag":"09VU","title":"Hypercoverings a la Verdier · Definition 09VU","summary":"Let C be a site. Assume C has equalizers and fibre products. Let G be a presheaf of sets. A hypercovering of G is a simplicial object K of SR(C) endowed with an augmentation F(K) → G such that • F(K_0) → G becomes surjective after sheafification, • F(K_1) → F(K_0) ×_G F(K_0) becomes surjective after sheafification, and • F(K_n + 1) → F((cosk_n sk_n K)_n + 1) for n ≥ 1 becomes surjective after sheafification. We say that a simplicial object K of SR(C) is a hypercovering if…","statement_latex":"Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has equalizers\nand fibre products. Let $\\mathcal{G}$ be a presheaf of sets.\nA {\\it hypercovering of $\\mathcal{G}$} is a simplicial object\n$K$ of $\\text{SR}(\\mathcal{C})$ endowed with an augmentation\n$F(K) \\to \\mathcal{G}$ such that\n\\begin{enumerate}\n\\item $F(K_0) \\to \\mathcal{G}$ becomes surjective\nafter sheafification,\n\\item $F(K_1) \\to F(K_0) \\times_\\mathcal{G} F(K_0)$\nbecomes surjective after sheafification, and\n\\item $F(K_{n + 1}) \\longrightarrow F((\\text{cosk}_n \\text{sk}_n K)_{n + 1})$\nfor $n \\geq 1$ becomes surjective after sheafification.\n\\end{enumerate}\nWe say that a simplicial object $K$ of $\\text{SR}(\\mathcal{C})$\nis a {\\it hypercovering} if $K$ is a hypercovering of the final\nobject $*$ of $\\textit{PSh}(\\mathcal{C})$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings a la Verdier","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VU","source_file":"hypercovering.tex","source_line":1067,"source_end_line":1085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1067-L1085","statement_sha256":"2257d542588788214ce97969bb4ed35ad805b3dab24f83bb354ea9f77efac46a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4953,"rank":4953,"depth":0,"x":1028.112,"y":795.016,"cluster":"sheaf-cohomology"},{"id":"stacks:09VV","tag":"09VV","title":"Hypercoverings a la Verdier · Lemma 09VV","summary":"Let C be a site with equalizers and fibre products. Let G be a presheaf on C. Let K be a hypercovering of G. Let F be a sheaf of abelian groups on C. Then checkH^0(K, F) = H^0(G, F).","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $\\mathcal{G}$ be a presheaf on $\\mathcal{C}$.\nLet $K$ be a hypercovering of $\\mathcal{G}$.\nLet $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$.\nThen $\\check{H}^0(K, \\mathcal{F}) = H^0(\\mathcal{G}, \\mathcal{F})$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings a la Verdier","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VV","source_file":"hypercovering.tex","source_line":1125,"source_end_line":1132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1125-L1132","statement_sha256":"08b0af373f88c5ee8a37ce0fb5f03767243d69a2f301eccc4681b8dca5044d58","origin":"The Stacks Project","memory_eligible":false,"source_rank":4954,"rank":4954,"depth":0,"x":1112.622,"y":537.246,"cluster":"sheaf-cohomology"},{"id":"stacks:09VW","tag":"09VW","title":"Hypercoverings a la Verdier · Lemma 09VW","summary":"Let C be a site with equalizers and fibre products. Let G be a presheaf on C. Let K be a hypercovering of G. Let I be an injective sheaf of abelian groups on C. Then checkH^p(K, I) = ( H^0(G, I) & if & p = 0 0 & if & p > 0 .","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $\\mathcal{G}$ be a presheaf on $\\mathcal{C}$.\nLet $K$ be a hypercovering of $\\mathcal{G}$.\nLet $\\mathcal{I}$ be an injective sheaf of abelian groups on $\\mathcal{C}$.\nThen\n$$\n\\check{H}^p(K, \\mathcal{I}) =\n\\left\\{\n\\begin{matrix}\nH^0(\\mathcal{G}, \\mathcal{I}) & \\text{if} & p = 0 \\\\\n0 & \\text{if} & p > 0\n\\end{matrix}\n\\right.\n$$","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings a la Verdier","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VW","source_file":"hypercovering.tex","source_line":1146,"source_end_line":1162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1146-L1162","statement_sha256":"c5520db3bb745318ee2cba3dab1d1af8663f6f34162559bd9314d1ee85e1b4c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":4955,"rank":4955,"depth":13,"x":1257.729,"y":775.491,"cluster":"sheaf-cohomology"},{"id":"stacks:09VX","tag":"09VX","title":"Hypercoverings a la Verdier · Lemma 09VX","summary":"Let C be a site with equalizers and fibre products. Let G be a presheaf on C. Let K be a hypercovering of G. Let F be a sheaf of abelian groups on C. There is a map s(F(K)) → RΓ(G, F) in D^+(Ab) functorial in F, which induces a natural transformation checkH^i(K, -) → H^i(G, -) of functors Ab(C) → Ab. Moreover, there is a spectral sequence (E_r, d_r)_r ≥ 0 with E_2^p, q = checkH^p(K, underlineH^q(F)) converging to H^p + q(G, F). This spectral sequence is functorial in F…","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $\\mathcal{G}$ be a presheaf on $\\mathcal{C}$.\nLet $K$ be a hypercovering of $\\mathcal{G}$.\nLet $\\mathcal{F}$ be a sheaf of abelian groups on $\\mathcal{C}$.\nThere is a map\n$$\ns(\\mathcal{F}(K)) \\longrightarrow R\\Gamma(\\mathcal{G}, \\mathcal{F})\n$$\nin $D^{+}(\\textit{Ab})$ functorial in $\\mathcal{F}$, which induces\na natural transformation\n$$\n\\check{H}^i(K, -) \\longrightarrow H^i(\\mathcal{G}, -)\n$$\nof functors $\\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}$. Moreover,\nthere is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))\n$$\nconverging to $H^{p + q}(\\mathcal{G}, \\mathcal{F})$.\nThis spectral sequence is functorial in $\\mathcal{F}$ and\nin the hypercovering $K$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings a la Verdier","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VX","source_file":"hypercovering.tex","source_line":1181,"source_end_line":1204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1181-L1204","statement_sha256":"9a56eb1899a2064bf4c29373acb58daf600a714a4fbd3407b4e5941f13b1871d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4956,"rank":4956,"depth":14,"x":958.868,"y":682.065,"cluster":"sheaf-cohomology"},{"id":"stacks:09VY","tag":"09VY","title":"Hypercoverings a la Verdier · Lemma 09VY","summary":"Let C be a site with equalizers and fibre products. Let K be a hypercovering. Let F be an abelian sheaf. There is a spectral sequence (E_r, d_r)_r ≥ 0 with E_2^p, q = checkH^p(K, underlineH^q(F)) converging to the global cohomology groups H^p + q(F).","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $K$ be a hypercovering.\nLet $\\mathcal{F}$ be an abelian sheaf. There is a\nspectral sequence $(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_2^{p, q} = \\check{H}^p(K, \\underline{H}^q(\\mathcal{F}))\n$$\nconverging to the global cohomology groups $H^{p + q}(\\mathcal{F})$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings a la Verdier","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VY","source_file":"hypercovering.tex","source_line":1260,"source_end_line":1270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1260-L1270","statement_sha256":"ff83cf385b63626ab47f2023292cd07dd1242760484fe77b106dd7bec14aae1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4957,"rank":4957,"depth":15,"x":1254.643,"y":581.283,"cluster":"sheaf-cohomology"},{"id":"stacks:01GH","tag":"01GH","title":"Covering hypercoverings · Lemma 01GH","summary":"Let C be a site with fibre products. Let X be an object of C. Let K, L, M be simplicial objects of SR(C, X). Let a : K → L, b : M → L be morphisms. Assume • K is a hypercovering of X, • the morphism M_0 → L_0 is a covering, and • for all n ≥ 0 in the diagram xymatrix M_n + 1 ar[dd] ar[rr] ar[rd]^γ & & (cosk_n sk_n M)_n + 1 ar[dd] & L_n + 1 ×_(cosk_n sk_n L)_n + 1 (cosk_n sk_n M)_n + 1 ar[ld] ar[ru] & L_n + 1 ar[rr] & & (cosk_n sk_n L)_n + 1 the arrow γ is a covering. Then…","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K, L, M$ be simplicial objects of $\\text{SR}(\\mathcal{C}, X)$.\nLet $a : K \\to L$, $b : M \\to L$ be morphisms.\nAssume\n\\begin{enumerate}\n\\item $K$ is a hypercovering of $X$,\n\\item the morphism $M_0 \\to L_0$ is a covering, and\n\\item for all $n \\geq 0$ in the diagram\n$$\n\\xymatrix{\nM_{n + 1} \\ar[dd] \\ar[rr] \\ar[rd]^\\gamma &\n&\n(\\text{cosk}_n \\text{sk}_n M)_{n + 1} \\ar[dd] \\\\\n&\nL_{n + 1}\n\\times_{(\\text{cosk}_n \\text{sk}_n L)_{n + 1}}\n(\\text{cosk}_n \\text{sk}_n M)_{n + 1}\n\\ar[ld] \\ar[ru]\n& \\\\\nL_{n + 1} \\ar[rr] & & (\\text{cosk}_n \\text{sk}_n L)_{n + 1}\n}\n$$\nthe arrow $\\gamma$ is a covering.\n\\end{enumerate}\nThen the fibre product $K \\times_L M$ is a hypercovering of $X$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Covering hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GH","source_file":"hypercovering.tex","source_line":1293,"source_end_line":1321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1293-L1321","statement_sha256":"88a913c3a3b39415ea97fa171bc9fcdd2751873159a5984ec467b9252ae2c015","origin":"The Stacks Project","memory_eligible":false,"source_rank":4958,"rank":4958,"depth":10,"x":1117.464,"y":823.647,"cluster":"sheaf-cohomology"},{"id":"stacks:01GI","tag":"01GI","title":"Covering hypercoverings · Lemma 01GI","summary":"Let C be a site with fibre products. Let X be an object of C. If K, L are hypercoverings of X, then K × L is a hypercovering of X.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nIf $K, L$ are hypercoverings of $X$, then\n$K \\times L$ is a hypercovering of $X$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Covering hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GI","source_file":"hypercovering.tex","source_line":1387,"source_end_line":1393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1387-L1393","statement_sha256":"81dd4288a91c5870dc4c39bc5125179dc46944d7cc243b9a8ce6b980db9a1486","origin":"The Stacks Project","memory_eligible":false,"source_rank":4959,"rank":4959,"depth":11,"x":1023.631,"y":566.862,"cluster":"sheaf-cohomology"},{"id":"stacks:01GJ","tag":"01GJ","title":"Covering hypercoverings · Lemma 01GJ","summary":"Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let k ≥ 0 be an integer. Let u : Z → K_k be a covering in SR(C, X). Then there exists a morphism of hypercoverings f: L → K such that L_k → K_k factors through u.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K$ be a hypercovering of $X$.\nLet $k \\geq 0$ be an integer.\nLet $u : Z \\to K_k$ be a covering\nin $\\text{SR}(\\mathcal{C}, X)$.\nThen there exists a morphism of hypercoverings\n$f: L \\to K$ such that $L_k \\to K_k$\nfactors through $u$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Covering hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GJ","source_file":"hypercovering.tex","source_line":1414,"source_end_line":1425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1414-L1425","statement_sha256":"ffc00855c67e403116fb423d5adf4b8ea2ba9c1691a71c06f9f5f909100d224f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4960,"rank":4960,"depth":11,"x":1299.571,"y":703.091,"cluster":"sheaf-cohomology"},{"id":"stacks:01GK","tag":"01GK","title":"Covering hypercoverings · Lemma 01GK","summary":"Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let n ≥ 0 be an integer. Let u : F → F(K_n) be a morphism of presheaves which becomes surjective on sheafification. Then there exists a morphism of hypercoverings f: L → K such that F(f_n) : F(L_n) → F(K_n) factors through u.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K$ be a hypercovering of $X$.\nLet $n \\geq 0$ be an integer.\nLet $u : \\mathcal{F} \\to F(K_n)$ be a morphism\nof presheaves which becomes surjective\non sheafification.\nThen there exists a morphism of hypercoverings\n$f: L \\to K$ such that $F(f_n) : F(L_n) \\to F(K_n)$\nfactors through $u$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Covering hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GK","source_file":"hypercovering.tex","source_line":1558,"source_end_line":1570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1558-L1570","statement_sha256":"abe5ec8ffbb0eeee04a517debbbfb2cbfdb040cda3b9f41a49cf199dbe094e80","origin":"The Stacks Project","memory_eligible":false,"source_rank":4961,"rank":4961,"depth":12,"x":986.263,"y":759.263,"cluster":"sheaf-cohomology"},{"id":"stacks:01GM","tag":"01GM","title":"Adding simplices · Lemma 01GM","summary":"Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let U ⊂ V be simplicial sets, with U_n, V_n finite nonempty for all n. Assume that U has finitely many nondegenerate simplices. Suppose n ≥ 0 and x ∈ V_n, x not ∈ U_n are such that • V_i = U_i for i < n, • V_n = U_n ∪ (x), • any z ∈ V_j, z not ∈ U_j for j > n is degenerate. Then the morphism Hom(V, K)_0 → Hom(U, K)_0 of SR(C, X) is a covering.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K$ be a hypercovering of $X$.\nLet $U \\subset V$ be simplicial sets, with $U_n, V_n$\nfinite nonempty for all $n$.\nAssume that $U$ has finitely many nondegenerate simplices.\nSuppose $n \\geq 0$ and $x \\in V_n$,\n$x \\not \\in U_n$ are such that\n\\begin{enumerate}\n\\item $V_i = U_i$ for $i < n$,\n\\item $V_n = U_n \\cup \\{x\\}$,\n\\item any $z \\in V_j$, $z \\not \\in U_j$ for $j > n$\nis degenerate.\n\\end{enumerate}\nThen the morphism\n$$\n\\Hom(V, K)_0\n\\longrightarrow\n\\Hom(U, K)_0\n$$\nof $\\text{SR}(\\mathcal{C}, X)$ is a covering.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Adding simplices","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GM","source_file":"hypercovering.tex","source_line":1611,"source_end_line":1634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1611-L1634","statement_sha256":"d1c87eb66bf1d927d8e57c50bc1449ee4d5386f342275c4471487285ca1f506a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4962,"rank":4962,"depth":10,"x":1172.286,"y":539.867,"cluster":"sheaf-cohomology"},{"id":"stacks:01GN","tag":"01GN","title":"Adding simplices · Lemma 01GN","summary":"Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let U ⊂ V be simplicial sets, with U_n, V_n finite nonempty for all n. Assume that U and V have finitely many nondegenerate simplices. Then the morphism Hom(V, K)_0 → Hom(U, K)_0 of SR(C, X) is a covering.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K$ be a hypercovering of $X$.\nLet $U \\subset V$ be simplicial sets, with $U_n, V_n$\nfinite nonempty for all $n$.\nAssume that $U$ and $V$ have finitely many nondegenerate simplices.\nThen the morphism\n$$\n\\Hom(V, K)_0\n\\longrightarrow\n\\Hom(U, K)_0\n$$\nof $\\text{SR}(\\mathcal{C}, X)$ is a covering.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Adding simplices","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GN","source_file":"hypercovering.tex","source_line":1672,"source_end_line":1687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1672-L1687","statement_sha256":"f9809cf7bd0dd172701ed0b23a3afcda96576852be5b791e16a043600d700f63","origin":"The Stacks Project","memory_eligible":false,"source_rank":4963,"rank":4963,"depth":11,"x":1211.583,"y":807.44,"cluster":"sheaf-cohomology"},{"id":"stacks:0DEQ","tag":"0DEQ","title":"Adding simplices · Lemma 0DEQ","summary":"Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Then • K_n is a covering of X for each n ≥ 0, • d^n_i : K_n → K_n - 1 is a covering for all n ≥ 1 and 0 ≤ i ≤ n.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products. Let $X$ be an object of\n$\\mathcal{C}$. Let $K$ be a hypercovering of $X$. Then\n\\begin{enumerate}\n\\item $K_n$ is a covering of $X$ for each $n \\geq 0$,\n\\item $d^n_i : K_n \\to K_{n - 1}$ is a covering for all $n \\geq 1$\nand $0 \\leq i \\leq n$.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Adding simplices","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEQ","source_file":"hypercovering.tex","source_line":1697,"source_end_line":1706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1697-L1706","statement_sha256":"f1951d7d6735876656a57d8b80a9525855f0e87cf15add306e5fd0ccec4e5bb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4964,"rank":4964,"depth":12,"x":967.213,"y":632.277,"cluster":"sheaf-cohomology"},{"id":"stacks:01GP","tag":"01GP","title":"Homotopies · Lemma 01GP","summary":"Let C be a site with fibre products. Let X be an object of C. Let L be a simplicial object of SR(C, X). Let n ≥ 0. Consider the commutative diagram xymatrix Hom(Δ[1], L)_n + 1 ar[r] ar[d] & (cosk_n sk_n Hom(Δ[1], L))_n + 1 ar[d] (L × L)_n + 1 ar[r] & (cosk_n sk_n (L × L))_n + 1 coming from the morphism defined above. We can identify the terms in this diagram as follows, where ∂ Δ[n + 1] = i_n!sk_n Δ[n + 1] is the n-skeleton of the (n + 1)-simplex: Hom(Δ[1], L)_n + 1 & = &…","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $L$ be a simplicial object of $\\text{SR}(\\mathcal{C}, X)$.\nLet $n \\geq 0$. Consider the commutative diagram\n\\begin{equation}\n\n\\xymatrix{\n\\Hom(\\Delta[1], L)_{n + 1} \\ar[r] \\ar[d] &\n(\\text{cosk}_n \\text{sk}_n \\Hom(\\Delta[1], L))_{n + 1} \\ar[d] \\\\\n(L \\times L)_{n + 1} \\ar[r] &\n(\\text{cosk}_n \\text{sk}_n (L \\times L))_{n + 1}\n}\n\\end{equation}\ncoming from the morphism defined above.\nWe can identify the terms in this diagram as follows,\nwhere\n$\\partial \\Delta[n + 1] = i_{n!}\\text{sk}_n \\Delta[n + 1]$\nis the $n$-skeleton of the $(n + 1)$-simplex:\n\\begin{eqnarray*}\n\\Hom(\\Delta[1], L)_{n + 1}\n& = &\n\\Hom(\\Delta[1] \\times \\Delta[n + 1], L)_0 \\\\\n(\\text{cosk}_n \\text{sk}_n \\Hom(\\Delta[1], L))_{n + 1}\n& = &\n\\Hom(\\Delta[1] \\times \\partial \\Delta[n + 1], L)_0 \\\\\n(L \\times L)_{n + 1}\n& = &\n\\Hom(\n(\\Delta[n + 1] \\amalg \\Delta[n + 1], L)_0 \\\\\n(\\text{cosk}_n \\text{sk}_n (L \\times L))_{n + 1}\n& = &\n\\Hom(\n\\partial \\Delta[n + 1]\n\\amalg\n\\partial \\Delta[n + 1], L)_0\n\\end{eqnarray*}\nand the morphism between these objects of $\\text{SR}(\\mathcal{C}, X)$\ncome from the commutative diagram of simplicial sets\n\\begin{equation}\n\n\\xymatrix{\n\\Delta[1] \\times \\Delta[n + 1] &\n\\Delta[1] \\times \\partial\\Delta[n + 1] \\ar[l] \\\\\n\\Delta[n + 1] \\amalg \\Delta[n + 1] \\ar[u] &\n\\partial\\Delta[n + 1] \\amalg \\partial\\Delta[n + 1]\n\\ar[l] \\ar[u]\n}\n\\end{equation}\nMoreover the fibre product of the bottom arrow and the\nright arrow in (\\ref{equation-diagram}) is equal to\n$$\n\\Hom(U, L)_0\n$$\nwhere $U \\subset \\Delta[1] \\times \\Delta[n + 1]$\nis the smallest simplicial subset such that both\n$\\Delta[n + 1] \\amalg \\Delta[n + 1]$ and\n$\\Delta[1] \\times \\partial\\Delta[n + 1]$ map into it.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Homotopies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GP","source_file":"hypercovering.tex","source_line":1778,"source_end_line":1837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1778-L1837","statement_sha256":"b4823990423950106f283bc093f77f2ad465d171a75d09e75731a631311959ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":4965,"rank":4965,"depth":4,"x":1288.556,"y":622.77,"cluster":"sheaf-cohomology"},{"id":"stacks:01GS","tag":"01GS","title":"Homotopies · Lemma 01GS","summary":"Let C be a site with fibre products. Let X be an object of C. Let K, L be hypercoverings of X. Let a, b : K → L be morphisms of hypercoverings. There exists a morphism of hypercoverings c : K' → K such that a ∘ c is homotopic to b ∘ c.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$.\nLet $K, L$ be hypercoverings of $X$.\nLet $a, b : K \\to L$ be morphisms of hypercoverings.\nThere exists a morphism of hypercoverings\n$c : K' \\to K$ such that $a \\circ c$ is homotopic\nto $b \\circ c$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Homotopies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01GS","source_file":"hypercovering.tex","source_line":1857,"source_end_line":1866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1857-L1866","statement_sha256":"cdca84c963f557bdff57a3af8025e81b19db0f1792348e8b2b32f973888e0f9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4966,"rank":4966,"depth":12,"x":1059.043,"y":812.287,"cluster":"sheaf-cohomology"},{"id":"stacks:01H0","tag":"01H0","title":"Cohomology and hypercoverings · Theorem 01H0","summary":"Let C be a site with fibre products. Let X be an object of C. Let i ≥ 0. The functors Ab(C) & → & Ab F & ↦ & H^i(X, F) F & ↦ & checkH^i_HC(X, F) are canonically isomorphic.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $X$ be an object of $\\mathcal{C}$. Let $i \\geq 0$.\nThe functors\n\\begin{eqnarray*}\n\\textit{Ab}(\\mathcal{C}) & \\longrightarrow & \\textit{Ab} \\\\\n\\mathcal{F} & \\longmapsto & H^i(X, \\mathcal{F}) \\\\\n\\mathcal{F} & \\longmapsto & \\check{H}^i_{\\text{HC}}(X, \\mathcal{F})\n\\end{eqnarray*}\nare canonically isomorphic.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Cohomology and hypercoverings","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01H0","source_file":"hypercovering.tex","source_line":1988,"source_end_line":1999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L1988-L1999","statement_sha256":"b3b7945bc026dc9745f030cfc3e227ebc93dd09da681aa4ee917b0f17925d0ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":4967,"rank":4967,"depth":14,"x":1075.893,"y":542.068,"cluster":"sheaf-cohomology"},{"id":"stacks:09VZ","tag":"09VZ","title":"Cohomology and hypercoverings · Proposition 09VZ","summary":"Let C be a site with fibre products and products of pairs. Let F be an abelian sheaf on C. Let i ≥ 0. Then • for every xi ∈ H^i(F) there exists a hypercovering K such that xi is in the image of the canonical map checkH^i(K, F) → H^i(F), and • if K, L are hypercoverings and xi_K ∈ checkH^i(K, F), xi_L ∈ checkH^i(L, F) are elements mapping to the same element of H^i(F), then there exists a hypercovering M and morphisms M → K and M → L such that xi_K and xi_L map to the same…","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and products of pairs.\nLet $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$.\nLet $i \\geq 0$. Then\n\\begin{enumerate}\n\\item for every $\\xi \\in H^i(\\mathcal{F})$ there exists a hypercovering\n$K$ such that $\\xi$ is in the image of the canonical map\n$\\check{H}^i(K, \\mathcal{F}) \\to H^i(\\mathcal{F})$, and\n\\item if $K, L$ are hypercoverings and $\\xi_K \\in \\check{H}^i(K, \\mathcal{F})$,\n$\\xi_L \\in \\check{H}^i(L, \\mathcal{F})$ are elements mapping\nto the same element of $H^i(\\mathcal{F})$, then there exists\na hypercovering $M$ and morphisms $M \\to K$ and $M \\to L$ such\nthat $\\xi_K$ and $\\xi_L$ map to the same element of\n$\\check{H}^i(M, \\mathcal{F})$.\n\\end{enumerate}\nIn other words, modulo set theoretical issues, the cohomology\ngroups of $\\mathcal{F}$ on $\\mathcal{C}$ are the colimit of\nthe {\\v C}ech cohomology groups of $\\mathcal{F}$ over all hypercoverings.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Cohomology and hypercoverings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VZ","source_file":"hypercovering.tex","source_line":2185,"source_end_line":2204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L2185-L2204","statement_sha256":"74547a09e118d20a98f075af16e53a20f24131e7c4d238b57705a321eaeabe69","origin":"The Stacks Project","memory_eligible":false,"source_rank":4968,"rank":4968,"depth":15,"x":1280.952,"y":751.07,"cluster":"sheaf-cohomology"},{"id":"stacks:01H6","tag":"01H6","title":"Hypercoverings of spaces · Lemma 01H6","summary":"Let X be a topological space. Let B be a basis for the topology of X. There exists a hypercovering (I, (U_i)) of X such that each U_i is an element of B.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{B}$ be a basis for the topology of $X$.\nThere exists a hypercovering $(I, \\{U_i\\})$ of $X$\nsuch that each $U_i$ is an element of $\\mathcal{B}$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01H6","source_file":"hypercovering.tex","source_line":2437,"source_end_line":2443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L2437-L2443","statement_sha256":"3cb58a7e3a0242c318a2bcdb7a27dad148bc897226bafe624148197f53562c16","origin":"The Stacks Project","memory_eligible":false,"source_rank":4969,"rank":4969,"depth":0,"x":961.388,"y":713.277,"cluster":"sheaf-cohomology"},{"id":"stacks:01H7","tag":"01H7","title":"Hypercoverings of spaces · Lemma 01H7","summary":"Let X be a topological space. Let B be a basis for the topology of X. Assume that • X is quasi-compact, • each U ∈ B is quasi-compact open, and • the intersection of any two quasi-compact opens in X is quasi-compact. Then there exists a hypercovering (I, (U_i)) of X with the following properties • each U_i is an element of the basis B, • each of the I_n is a finite set, and in particular • each of the coverings ([Tag 01H2]), ([Tag 01H3]), and ([Tag 01H4]) is finite.","statement_latex":"Let $X$ be a topological space.\nLet $\\mathcal{B}$ be a basis for the topology of $X$.\nAssume that\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item each $U \\in \\mathcal{B}$ is quasi-compact open, and\n\\item the intersection of any two quasi-compact opens in\n$X$ is quasi-compact.\n\\end{enumerate}\nThen there exists a hypercovering $(I, \\{U_i\\})$ of $X$ with the\nfollowing properties\n\\begin{enumerate}\n\\item each $U_i$ is an element of the basis $\\mathcal{B}$,\n\\item each of the $I_n$ is a finite set, and in particular\n\\item each of the coverings  (\\ref{equation-covering-X}),\n(\\ref{equation-covering-two}), and (\\ref{equation-covering-general})\nis finite.\n\\end{enumerate}","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Hypercoverings of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01H7","source_file":"hypercovering.tex","source_line":2496,"source_end_line":2516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L2496-L2516","statement_sha256":"5f741bc25a50a31db4169639b385c846c0f418444876c8e716a9a4a649a80c2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4970,"rank":4970,"depth":1,"x":1227.658,"y":559.681,"cluster":"sheaf-cohomology"},{"id":"stacks:0DAU","tag":"0DAU","title":"Constructing hypercoverings · Lemma 0DAU","summary":"Let C be a site. Let K be an r-truncated simplicial object of SR(C). The following are equivalent • K is split (Simplicial, Definition [Tag 017P]), • f_φ, i : U_n, i → U_m, α(φ)(i) is an isomorphism for r ≥ n ≥ 0, φ : [m] → [n] surjective, i ∈ I_n, and • f_σ^n_j, i : U_n, i → U_n + 1, α(σ^n_j)(i) is an isomorphism for 0 ≤ j ≤ n < r, i ∈ I_n. The same holds for simplicial objects if in (2) and (3) we set r = ∞.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $K$ be an $r$-truncated simplicial object\nof $\\text{SR}(\\mathcal{C})$. The following are equivalent\n\\begin{enumerate}\n\\item $K$ is split (Simplicial, Definition \\ref{simplicial-definition-split}),\n\\item $f_{\\varphi, i} : U_{n, i} \\to U_{m, \\alpha(\\varphi)(i)}$\nis an isomorphism for $r \\geq n \\geq 0$,\n$\\varphi : [m] \\to [n]$ surjective, $i \\in I_n$, and\n\\item $f_{\\sigma^n_j, i} : U_{n, i} \\to U_{n + 1, \\alpha(\\sigma^n_j)(i)}$\nis an isomorphism for $0 \\leq j \\leq n < r$, $i \\in I_n$.\n\\end{enumerate}\nThe same holds for simplicial objects if in (2) and (3)\nwe set $r = \\infty$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Constructing hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAU","source_file":"hypercovering.tex","source_line":2549,"source_end_line":2563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L2549-L2563","statement_sha256":"e7f51cbb5fa1d3b032ee655a88c0719198d3cb0db58fa265c013d45c5cce807e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4971,"rank":4971,"depth":1,"x":1154.768,"y":824.262,"cluster":"sheaf-cohomology"},{"id":"stacks:0DAV","tag":"0DAV","title":"Constructing hypercoverings · Lemma 0DAV","summary":"Let C be a site with fibre products. Let B ⊂ Ob(C) be a subset. Assume • any object U of C has a covering (U_j → U)_j ∈ J with U_j ∈ B, and • if (U_j → U)_j ∈ J is a covering with U_j ∈ B and (U' → U) is a morphism with U' ∈ B, then (U_j → U)_j ∈ J amalg (U' → U) is a covering. Then for any X in C there is a hypercovering K of X such that K_n = (U_n, i)_i ∈ I_n with U_n, i ∈ B for all i ∈ I_n.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products.\nLet $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset.\nAssume\n\\begin{enumerate}\n\\item any object $U$ of $\\mathcal{C}$ has a covering\n$\\{U_j \\to U\\}_{j \\in J}$ with $U_j \\in \\mathcal{B}$, and\n\\item if $\\{U_j \\to U\\}_{j \\in J}$ is a covering\nwith $U_j \\in \\mathcal{B}$ and $\\{U' \\to U\\}$ is a morphism with\n$U' \\in \\mathcal{B}$, then $\\{U_j \\to U\\}_{j \\in J} \\amalg \\{U' \\to U\\}$\nis a covering.\n\\end{enumerate}\nThen for any $X$ in $\\mathcal{C}$ there is a hypercovering $K$\nof $X$ such that $K_n = \\{U_{n, i}\\}_{i \\in I_n}$\nwith $U_{n, i} \\in \\mathcal{B}$ for all $i \\in I_n$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Constructing hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAV","source_file":"hypercovering.tex","source_line":2579,"source_end_line":2595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L2579-L2595","statement_sha256":"8ff84fe75679cd45e54c1eb811b276e4b89b4d816a0294002fb06e426d44ccb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4972,"rank":4972,"depth":2,"x":995.607,"y":587.596,"cluster":"sheaf-cohomology"},{"id":"stacks:0DAX","tag":"0DAX","title":"Constructing hypercoverings · Lemma 0DAX","summary":"Let C be a site with equalizers and fibre products. Let B ⊂ Ob(C) be a subset. Assume that any object of C has a covering whose members are elements of B. Then there is a hypercovering K such that K_n = (U_i)_i ∈ I_n with U_i ∈ B for all i ∈ I_n.","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$ be a subset. Assume\nthat any object of $\\mathcal{C}$ has a covering\nwhose members are elements of $\\mathcal{B}$.\nThen there is a hypercovering $K$ such that\n$K_n = \\{U_i\\}_{i \\in I_n}$ with $U_i \\in \\mathcal{B}$\nfor all $i \\in I_n$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Constructing hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAX","source_file":"hypercovering.tex","source_line":2739,"source_end_line":2748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L2739-L2748","statement_sha256":"c448d339562cd7b528c1e4863804efe5aeb537ba768c84b45e922e2752f4333c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4973,"rank":4973,"depth":9,"x":1303.561,"y":671.872,"cluster":"sheaf-cohomology"},{"id":"stacks:0DAY","tag":"0DAY","title":"Constructing hypercoverings · Lemma 0DAY","summary":"Let f : C → D be a morphism of sites given by the functor u : D → C. Assume D and C have equalizers and fibre products and u commutes with them. If a simplicial object K of SR(D) is a hypercovering, then u(K) is a hypercovering.","statement_latex":"Let $f : \\mathcal{C} \\to \\mathcal{D}$ be a morphism of sites\ngiven by the functor $u : \\mathcal{D} \\to \\mathcal{C}$.\nAssume $\\mathcal{D}$ and $\\mathcal{C}$ have equalizers and\nfibre products and $u$ commutes with them.\nIf a simplicial object $K$ of $\\text{SR}(\\mathcal{D})$\nis a hypercovering, then $u(K)$ is a hypercovering.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Constructing hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAY","source_file":"hypercovering.tex","source_line":2820,"source_end_line":2828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L2820-L2828","statement_sha256":"8e7fad774edc2152a4451d143032c492115635a2e31615e85dfa4cb13dd12761","origin":"The Stacks Project","memory_eligible":false,"source_rank":4974,"rank":4974,"depth":3,"x":1008.448,"y":784.568,"cluster":"sheaf-cohomology"},{"id":"stacks:0DAZ","tag":"0DAZ","title":"Constructing hypercoverings · Lemma 0DAZ","summary":"Let C, D be sites. Let u : D → C be a continuous functor. Assume D and C have fibre products and u commutes with them. Let Y ∈ D and K ∈ SR(D, Y) a hypercovering of Y. Then u(K) is a hypercovering of u(Y).","statement_latex":"Let $\\mathcal{C}$, $\\mathcal{D}$ be sites. Let\n$u : \\mathcal{D} \\to \\mathcal{C}$ be a continuous functor.\nAssume $\\mathcal{D}$ and $\\mathcal{C}$ have fibre products\nand $u$ commutes with them. Let $Y \\in \\mathcal{D}$ and\n$K \\in \\text{SR}(\\mathcal{D}, Y)$ a hypercovering of $Y$.\nThen $u(K)$ is a hypercovering of $u(Y)$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Constructing hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAZ","source_file":"hypercovering.tex","source_line":2863,"source_end_line":2871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L2863-L2871","statement_sha256":"d2569136673598200c8dad9c54c2c3d37cb3e4fe2456cb3e622f659ff4360a0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4975,"rank":4975,"depth":4,"x":1135.545,"y":533.794,"cluster":"sheaf-cohomology"},{"id":"stacks:094K","tag":"094K","title":"Constructing hypercoverings · Lemma 094K","summary":"Let C be a site. Let B ⊂ Ob(C) be a subset. Assume • C has fibre products, • for all X ∈ Ob(C) there exists a finite covering (U_i → X)_i ∈ I with U_i ∈ B, • if (U_i → X)_i ∈ I is a finite covering with U_i ∈ B and U → X is a morphism with U ∈ B, then (U_i → X)_i ∈ I amalg (U → X) is a covering. Then for every X there exists a hypercovering K of X such that each K_n = (U_n, i → X)_i ∈ I_n with I_n finite and U_n, i ∈ B.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{B} \\subset \\Ob(\\mathcal{C})$\nbe a subset. Assume\n\\begin{enumerate}\n\\item $\\mathcal{C}$ has fibre products,\n\\item for all $X \\in \\Ob(\\mathcal{C})$ there exists a finite covering\n$\\{U_i \\to X\\}_{i \\in I}$ with $U_i \\in \\mathcal{B}$,\n\\item if $\\{U_i \\to X\\}_{i \\in I}$ is a finite covering with\n$U_i \\in \\mathcal{B}$ and $U \\to X$ is a morphism with $U \\in \\mathcal{B}$,\nthen $\\{U_i \\to X\\}_{i \\in I} \\amalg \\{U \\to X\\}$ is a covering.\n\\end{enumerate}\nThen for every $X$ there exists a hypercovering $K$ of $X$\nsuch that each $K_n = \\{U_{n, i} \\to X\\}_{i \\in I_n}$ with\n$I_n$ finite and $U_{n, i} \\in \\mathcal{B}$.","area":"Sheaf Cohomology","chapter":"Hypercoverings","chapter_id":"hypercovering","section":"Constructing hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094K","source_file":"hypercovering.tex","source_line":2887,"source_end_line":2902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/hypercovering.tex#L2887-L2902","statement_sha256":"024c6711d37263e56cffb30a42d19b3c2654dd66c31fddff6a6e58faff0fae2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":4976,"rank":4976,"depth":3,"x":1243.586,"y":791.053,"cluster":"sheaf-cohomology"},{"id":"stacks:01HB","tag":"01HB","title":"Locally ringed spaces · Definition 01HB","summary":"Locally ringed spaces. • A locally ringed space (X, O_X) is a pair consisting of a topological space X and a sheaf of rings O_X all of whose stalks are local rings. • Given a locally ringed space (X, O_X) we say that O_X, x is the local ring of X at x. We denote m_X, x or simply m_x the maximal ideal of O_X, x. Moreover, the residue field of X at x is the residue field kappa(x) = O_X, x/m_x. • A morphism of locally ringed spaces (f, f^sharp) : (X, O_X) → (Y, O_Y) is a…","statement_latex":"Locally ringed spaces.\n\\begin{enumerate}\n\\item A {\\it locally ringed space $(X, \\mathcal{O}_X)$}\nis a pair consisting of a\ntopological space $X$ and a sheaf of rings $\\mathcal{O}_X$ all of whose stalks\nare local rings.\n\\item Given a locally ringed space $(X, \\mathcal{O}_X)$ we say that\n$\\mathcal{O}_{X, x}$ is the {\\it local ring of $X$ at $x$}.\nWe denote $\\mathfrak{m}_{X, x}$ or simply $\\mathfrak{m}_x$\nthe maximal ideal of $\\mathcal{O}_{X, x}$. Moreover, the\n{\\it residue field of $X$ at $x$} is the residue field\n$\\kappa(x) = \\mathcal{O}_{X, x}/\\mathfrak{m}_x$.\n\\item A {\\it morphism of locally ringed spaces}\n$(f, f^\\sharp) : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$\nis a morphism of ringed spaces such that for all $x \\in X$\nthe induced ring map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is a\nlocal ring map.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Locally ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HB","source_file":"schemes.tex","source_line":92,"source_end_line":112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L92-L112","statement_sha256":"661e91021b407454c2587f1303e87086ebd52e74809da56fe1c051f76606beb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4977,"rank":4977,"depth":0,"x":1586.39,"y":680.0,"cluster":"schemes"},{"id":"stacks:01HC","tag":"01HC","title":"Locally ringed spaces · Lemma 01HC","summary":"An isomorphism of ringed spaces between locally ringed spaces is an isomorphism of locally ringed spaces. Let X, Y be locally ringed spaces. If f : X → Y is an isomorphism of ringed spaces, then f is an isomorphism of locally ringed spaces.","statement_latex":"\\begin{slogan}\nAn isomorphism of ringed spaces between locally ringed spaces is an\nisomorphism of locally ringed spaces.\n\\end{slogan}\nLet $X$, $Y$ be locally ringed spaces.\nIf $f : X \\to Y$ is an isomorphism of\nringed spaces, then $f$ is an isomorphism\nof locally ringed spaces.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Locally ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HC","source_file":"schemes.tex","source_line":172,"source_end_line":182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L172-L182","statement_sha256":"da276c504bc491f1e13328cd0f0e9a8539547923c7b9fd5edfbc5cf097045592","origin":"The Stacks Project","memory_eligible":false,"source_rank":4978,"rank":4978,"depth":0,"x":1571.839,"y":686.28,"cluster":"schemes"},{"id":"stacks:01HE","tag":"01HE","title":"Open immersions of locally ringed spaces · Definition 01HE","summary":"Let f : X → Y be a morphism of locally ringed spaces. We say that f is an open immersion if f is a homeomorphism of X onto an open subset of Y, and the map f^-1O_Y → O_X is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a morphism of locally ringed spaces.\nWe say that $f$ is an {\\it open immersion} if\n$f$ is a homeomorphism of $X$ onto an open subset\nof $Y$, and the map $f^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X$\nis an isomorphism.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Open immersions of locally ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HE","source_file":"schemes.tex","source_line":205,"source_end_line":212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L205-L212","statement_sha256":"e195d2e5908c99a328a2e4ffbdd172268c080702f2f4f1d263c422b1283157dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":4979,"rank":4979,"depth":0,"x":1581.249,"y":668.043,"cluster":"schemes"},{"id":"stacks:01HG","tag":"01HG","title":"Open immersions of locally ringed spaces · Definition 01HG","summary":"Let X be a locally ringed space. Let U ⊂ X be an open subset. The locally ringed space (U, O_U) of Example [Tag 01HF] above is the open subspace of X associated to U.","statement_latex":"Let $X$ be a locally ringed space.\nLet $U \\subset X$ be an open subset.\nThe locally ringed space $(U, \\mathcal{O}_U)$\nof Example \\ref{example-open-subspace} above\nis the {\\it open subspace of $X$ associated to $U$}.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Open immersions of locally ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HG","source_file":"schemes.tex","source_line":231,"source_end_line":238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L231-L238","statement_sha256":"22ec56245f152b2ccaba273473ea47165ea263b1d792b17c74244f070fc4b928","origin":"The Stacks Project","memory_eligible":false,"source_rank":4980,"rank":4980,"depth":0,"x":1590.287,"y":691.27,"cluster":"schemes"},{"id":"stacks:01HH","tag":"01HH","title":"Open immersions of locally ringed spaces · Lemma 01HH","summary":"Let f : X → Y be an open immersion of locally ringed spaces. Let j : V = f(X) → Y be the open subspace of Y associated to the image of f. There is a unique isomorphism f' : X ≅ V of locally ringed spaces such that f = j ∘ f'.","statement_latex":"Let $f : X \\to Y$ be an open immersion of\nlocally ringed spaces. Let $j : V = f(X) \\to Y$\nbe the open subspace of $Y$ associated to the image of $f$.\nThere is a unique isomorphism $f' : X \\cong V$ of\nlocally ringed spaces such that $f = j \\circ f'$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Open immersions of locally ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HH","source_file":"schemes.tex","source_line":240,"source_end_line":247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L240-L247","statement_sha256":"30a9129dc83aadfd7c846b1337f276fe4a4a0c6b061ccc27932e98b1d339a286","origin":"The Stacks Project","memory_eligible":false,"source_rank":4981,"rank":4981,"depth":2,"x":1561.123,"y":677.195,"cluster":"schemes"},{"id":"stacks:01HI","tag":"01HI","title":"Open immersions of locally ringed spaces · Lemma 01HI","summary":"Let f : X → Y be a morphism of locally ringed spaces. Let U ⊂ X, and V ⊂ Y be open subsets. Suppose that f(U) ⊂ V. There exists a unique morphism of locally ringed spaces f|_U : U → V such that the following diagram is a commutative square of locally ringed spaces xymatrix U ar[d]_f|_U ar[r] & X ar[d]^f V ar[r] & Y","statement_latex":"Let $f : X \\to Y$ be a morphism of locally ringed spaces.\nLet $U \\subset X$, and $V \\subset Y$ be open subsets.\nSuppose that $f(U) \\subset V$. There exists a unique\nmorphism of locally ringed spaces $f|_U : U \\to V$ such\nthat the following diagram is a commutative square of\nlocally ringed spaces\n$$\n\\xymatrix{\nU \\ar[d]_{f|_U} \\ar[r] & X \\ar[d]^f \\\\\nV \\ar[r] & Y\n}\n$$","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Open immersions of locally ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HI","source_file":"schemes.tex","source_line":298,"source_end_line":312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L298-L312","statement_sha256":"9f13b720d643c9d9208868b694504149e23f288420eb90c4058e92ed74800541","origin":"The Stacks Project","memory_eligible":false,"source_rank":4982,"rank":4982,"depth":0,"x":1597.882,"y":670.445,"cluster":"schemes"},{"id":"stacks:01HK","tag":"01HK","title":"Closed immersions of locally ringed spaces · Definition 01HK","summary":"Let i : Z → X be a morphism of locally ringed spaces. We say that i is a closed immersion if: • The map i is a homeomorphism of Z onto a closed subset of X. • The map O_X → i_*O_Z is surjective; let I denote the kernel. • The O_X-module I is locally generated by sections.","statement_latex":"Let $i : Z \\to X$ be a morphism of locally ringed spaces.\nWe say that $i$ is a {\\it closed immersion} if:\n\\begin{enumerate}\n\\item The map $i$ is a homeomorphism of $Z$ onto a closed subset of $X$.\n\\item The map $\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective;\nlet $\\mathcal{I}$ denote the kernel.\n\\item The $\\mathcal{O}_X$-module $\\mathcal{I}$\nis locally generated by sections.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Closed immersions of locally ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HK","source_file":"schemes.tex","source_line":343,"source_end_line":354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L343-L354","statement_sha256":"652166e6e5ac41b4529fd4381ded3b13f6f6ac22fa7eae944672d58528f2d09d","origin":"The Stacks Project","memory_eligible":false,"source_rank":4983,"rank":4983,"depth":0,"x":1574.019,"y":698.69,"cluster":"schemes"},{"id":"stacks:01HL","tag":"01HL","title":"Closed immersions of locally ringed spaces · Lemma 01HL","summary":"Let f : Z → X be a morphism of locally ringed spaces. In order for f to be a closed immersion it suffices that there exists an open covering X = ⋃ U_i such that each f : f^-1U_i → U_i is a closed immersion.","statement_latex":"Let $f : Z \\to X$ be a morphism of locally ringed spaces.\nIn order for $f$ to be a closed immersion it suffices\nthat there exists an open covering $X = \\bigcup U_i$ such\nthat each $f : f^{-1}U_i \\to U_i$ is a closed immersion.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Closed immersions of locally ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HL","source_file":"schemes.tex","source_line":356,"source_end_line":362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L356-L362","statement_sha256":"cb6b4f374efa497b326ca0bc942d6c105c420fe6ce2e25c13f9fd72bb137297e","origin":"The Stacks Project","memory_eligible":false,"source_rank":4984,"rank":4984,"depth":0,"x":1568.593,"y":661.551,"cluster":"schemes"},{"id":"stacks:01HN","tag":"01HN","title":"Closed immersions of locally ringed spaces · Definition 01HN","summary":"Let X be a locally ringed space. Let I be a sheaf of ideals on X which is locally generated by sections. The locally ringed space (Z, O_Z) of Example [Tag 01HM] above is the closed subspace of X associated to the sheaf of ideals I.","statement_latex":"Let $X$ be a locally ringed space.\nLet $\\mathcal{I}$ be a sheaf of ideals on $X$\nwhich is locally generated by sections.\nThe locally ringed space $(Z, \\mathcal{O}_Z)$\nof Example \\ref{example-closed-subspace} above\nis the {\\it closed subspace of $X$ associated to\nthe sheaf of ideals $\\mathcal{I}$}.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Closed immersions of locally ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HN","source_file":"schemes.tex","source_line":388,"source_end_line":397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L388-L397","statement_sha256":"07774d49d8dadc829149af5ca17022d76b4f69c60e0b52baa17a7df36b110686","origin":"The Stacks Project","memory_eligible":false,"source_rank":4985,"rank":4985,"depth":0,"x":1604.748,"y":687.592,"cluster":"schemes"},{"id":"stacks:01HO","tag":"01HO","title":"Closed immersions of locally ringed spaces · Lemma 01HO","summary":"Let f : X → Y be a closed immersion of locally ringed spaces. Let I be the kernel of the map O_Y → f_*O_X. Let i : Z → Y be the closed subspace of Y associated to I. There is a unique isomorphism f' : X ≅ Z of locally ringed spaces such that f = i ∘ f'.","statement_latex":"Let $f : X \\to Y$ be a closed immersion of\nlocally ringed spaces. Let $\\mathcal{I}$ be the\nkernel of the map $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$.\nLet $i : Z \\to Y$ be the closed subspace of $Y$\nassociated to $\\mathcal{I}$.\nThere is a unique isomorphism $f' : X \\cong Z$ of\nlocally ringed spaces such that $f = i \\circ f'$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Closed immersions of locally ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HO","source_file":"schemes.tex","source_line":399,"source_end_line":408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L399-L408","statement_sha256":"02ff941248187953c14be1cc3150a7703d6120688efdfd0dcfa85f8bf965d7c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":4986,"rank":4986,"depth":0,"x":1554.253,"y":688.927,"cluster":"schemes"},{"id":"stacks:01HP","tag":"01HP","title":"Closed immersions of locally ringed spaces · Lemma 01HP","summary":"Let X, Y be locally ringed spaces. Let I ⊂ O_X be a sheaf of ideals locally generated by sections. Let i : Z → X be the associated closed subspace. A morphism f : Y → X factors through Z if and only if the map f^*I → f^*O_X = O_Y is zero. If this is the case the morphism g : Y → Z such that f = i ∘ g is unique.","statement_latex":"Let $X$, $Y$ be locally ringed spaces. Let\n$\\mathcal{I} \\subset \\mathcal{O}_X$ be a sheaf of ideals locally generated\nby sections. Let $i : Z \\to X$ be the associated closed subspace.\nA morphism $f : Y \\to X$ factors through $Z$ if and only if the map\n$f^*\\mathcal{I} \\to f^*\\mathcal{O}_X = \\mathcal{O}_Y$\nis zero. If this is the case the morphism $g : Y \\to Z$\nsuch that $f = i \\circ g$ is unique.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Closed immersions of locally ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HP","source_file":"schemes.tex","source_line":414,"source_end_line":423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L414-L423","statement_sha256":"72c2ba553be3fc7a871db331c7e2c1c47f453608bc704bbf0893324786037696","origin":"The Stacks Project","memory_eligible":false,"source_rank":4987,"rank":4987,"depth":1,"x":1592.412,"y":657.721,"cluster":"schemes"},{"id":"stacks:01HQ","tag":"01HQ","title":"Closed immersions of locally ringed spaces · Lemma 01HQ","summary":"Let f : X → Y be a morphism of locally ringed spaces. Let I ⊂ O_Y be a sheaf of ideals which is locally generated by sections. Let i : Z → Y be the closed subspace associated to the sheaf of ideals I. Let J be the image of the map f^*I → f^*O_Y = O_X. Then this ideal is locally generated by sections. Moreover, let i' : Z' → X be the associated closed subspace of X. There exists a unique morphism of locally ringed spaces f' : Z' → Z such that the following diagram is a…","statement_latex":"Let $f : X \\to Y$ be a morphism of locally ringed spaces.\nLet $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a sheaf of\nideals which is locally generated by sections.\nLet $i : Z \\to Y$ be the closed subspace associated to the\nsheaf of ideals $\\mathcal{I}$.\nLet $\\mathcal{J}$ be the image of the map\n$f^*\\mathcal{I} \\to f^*\\mathcal{O}_Y = \\mathcal{O}_X$.\nThen this ideal is locally generated by sections.\nMoreover, let $i' : Z' \\to X$ be the associated closed\nsubspace of $X$. There exists a unique\nmorphism of locally ringed spaces $f' : Z' \\to Z$ such\nthat the following diagram is a commutative square of\nlocally ringed spaces\n$$\n\\xymatrix{\nZ' \\ar[d]_{f'} \\ar[r]_{i'} & X \\ar[d]^f \\\\\nZ \\ar[r]^{i} & Y\n}\n$$\nMoreover, this diagram is a fibre square in the category of\nlocally ringed spaces.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Closed immersions of locally ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HQ","source_file":"schemes.tex","source_line":455,"source_end_line":478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L455-L478","statement_sha256":"5f777d31eba26f8d6f29c0d6707f62a92dc7ca5dec0e0ccf5c325992e4354e9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":4988,"rank":4988,"depth":7,"x":1589.172,"y":704.563,"cluster":"schemes"},{"id":"stacks:01HS","tag":"01HS","title":"Affine schemes · Lemma 01HS","summary":"Let R be a ring. Let f ∈ R. • If g∈ R and D(g) ⊂ D(f), then • f is invertible in R_g, • g^e = af for some e ≥ 1 and a ∈ R, • there is a canonical ring map R_f → R_g, and • there is a canonical R_f-module map M_f → M_g for any R-module M. • Any open covering of D(f) can be refined to a finite open covering of the form D(f) = ⋃_i = 1^n D(g_i). • If g_1, …, g_n ∈ R, then D(f) ⊂ ⋃ D(g_i) if and only if g_1, …, g_n generate the unit ideal in R_f.","statement_latex":"Let $R$ be a ring. Let $f \\in R$.\n\\begin{enumerate}\n\\item If $g\\in R$ and $D(g) \\subset D(f)$, then\n\\begin{enumerate}\n\\item $f$ is invertible in $R_g$,\n\\item $g^e = af$ for some $e \\geq 1$ and $a \\in R$,\n\\item there is a canonical ring map $R_f \\to R_g$, and\n\\item there is a canonical $R_f$-module map\n$M_f \\to M_g$ for any $R$-module $M$.\n\\end{enumerate}\n\\item Any open covering of $D(f)$ can be refined to a finite\nopen covering of the form $D(f) = \\bigcup_{i = 1}^n D(g_i)$.\n\\item If $g_1, \\ldots, g_n \\in R$, then $D(f) \\subset \\bigcup D(g_i)$\nif and only if $g_1, \\ldots, g_n$ generate the unit ideal in $R_f$.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HS","source_file":"schemes.tex","source_line":519,"source_end_line":536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L519-L536","statement_sha256":"bb7e291da7d7cf1c6e18c4d531c2bc0dbead010f828bdd21414d462aa1ae2f09","origin":"The Stacks Project","memory_eligible":false,"source_rank":4989,"rank":4989,"depth":4,"x":1552.356,"y":666.543,"cluster":"schemes"},{"id":"stacks:01HT","tag":"01HT","title":"Affine schemes · Definition 01HT","summary":"Let R be a ring. • A standard open covering of Spec(R) is a covering Spec(R) = ⋃_i = 1^n D(f_i), where f_1, …, f_n ∈ R. • Suppose that D(f) ⊂ Spec(R) is a standard open. A standard open covering of D(f) is a covering D(f) = ⋃_i = 1^n D(g_i), where g_1, …, g_n ∈ R.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item A {\\it standard open covering} of $\\Spec(R)$\nis a covering $\\Spec(R) = \\bigcup_{i = 1}^n D(f_i)$,\nwhere $f_1, \\ldots, f_n \\in R$.\n\\item Suppose that $D(f) \\subset \\Spec(R)$ is a standard\nopen. A {\\it standard open covering} of $D(f)$\nis a covering $D(f) = \\bigcup_{i = 1}^n D(g_i)$,\nwhere $g_1, \\ldots, g_n \\in R$.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Affine schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HT","source_file":"schemes.tex","source_line":579,"source_end_line":591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L579-L591","statement_sha256":"33c100d1808fe55d45df4c7afa1458fd3e8b26e112274ec9cf9b4d548053de37","origin":"The Stacks Project","memory_eligible":false,"source_rank":4990,"rank":4990,"depth":0,"x":1612.429,"y":674.011,"cluster":"schemes"},{"id":"stacks:01HU","tag":"01HU","title":"Affine schemes · Definition 01HU","summary":"Let R be a ring. • The structure sheaf O_Spec(R) of the spectrum of R is the unique sheaf of rings O_Spec(R) which agrees with widetilde R on the basis of standard opens. • The locally ringed space (Spec(R), O_Spec(R)) is called the spectrum of R and denoted Spec(R). • The sheaf of O_Spec(R)-modules extending widetilde M to all opens of Spec(R) is called the sheaf of O_Spec(R)-modules associated to M. This sheaf is denoted widetilde M as well.","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item The {\\it structure sheaf $\\mathcal{O}_{\\Spec(R)}$ of the\nspectrum of $R$} is the unique sheaf of rings $\\mathcal{O}_{\\Spec(R)}$\nwhich agrees with $\\widetilde R$ on the basis of standard opens.\n\\item The locally ringed space\n$(\\Spec(R), \\mathcal{O}_{\\Spec(R)})$ is called\nthe {\\it spectrum} of $R$ and denoted $\\Spec(R)$.\n\\item The sheaf of $\\mathcal{O}_{\\Spec(R)}$-modules\nextending $\\widetilde M$ to all opens of $\\Spec(R)$\nis called the sheaf of $\\mathcal{O}_{\\Spec(R)}$-modules\nassociated to $M$. This sheaf is denoted $\\widetilde M$ as\nwell.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Affine schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HU","source_file":"schemes.tex","source_line":670,"source_end_line":686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L670-L686","statement_sha256":"54c057e30354647cf59d55465aa5a4b56f7593e1c2e07adf7aba4930083e8e11","origin":"The Stacks Project","memory_eligible":false,"source_rank":4991,"rank":4991,"depth":0,"x":1560.209,"y":703.647,"cluster":"schemes"},{"id":"stacks:01HV","tag":"01HV","title":"Affine schemes · Lemma 01HV","summary":"Let R be a ring. Let M be an R-module. Let widetilde M be the sheaf of O_Spec(R)-modules associated to M. • We have Γ(Spec(R), O_Spec(R)) = R. • We have Γ(Spec(R), widetilde M) = M as an R-module. • For every f ∈ R we have Γ(D(f), O_Spec(R)) = R_f. • For every f∈ R we have Γ(D(f), widetilde M) = M_f as an R_f-module. • Whenever D(g) ⊂ D(f) the restriction mappings on O_Spec(R) and widetilde M are the maps R_f → R_g and M_f → M_g from Lemma [Tag 01HS]. • Let p be a prime…","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. Let $\\widetilde M$\nbe the sheaf of $\\mathcal{O}_{\\Spec(R)}$-modules\nassociated to $M$.\n\\begin{enumerate}\n\\item We have $\\Gamma(\\Spec(R), \\mathcal{O}_{\\Spec(R)}) = R$.\n\\item We have $\\Gamma(\\Spec(R), \\widetilde M) = M$ as an $R$-module.\n\\item For every $f \\in R$ we have\n$\\Gamma(D(f), \\mathcal{O}_{\\Spec(R)}) = R_f$.\n\\item For every $f\\in R$ we have $\\Gamma(D(f), \\widetilde M) = M_f$\nas an $R_f$-module.\n\\item Whenever $D(g) \\subset D(f)$ the restriction mappings\non $\\mathcal{O}_{\\Spec(R)}$ and $\\widetilde M$\nare the maps\n$R_f \\to R_g$ and $M_f \\to M_g$ from Lemma\n\\ref{lemma-standard-open}.\n\\item Let $\\mathfrak p$ be a prime of $R$, and let $x \\in \\Spec(R)$\nbe the corresponding point. We have\n$\\mathcal{O}_{\\Spec(R), x} = R_{\\mathfrak p}$.\n\\item Let $\\mathfrak p$ be a prime of $R$, and let $x \\in \\Spec(R)$\nbe the corresponding point. We have $\\widetilde M_x = M_{\\mathfrak p}$\nas an $R_{\\mathfrak p}$-module.\n\\end{enumerate}\nMoreover, all these identifications are functorial in the $R$\nmodule $M$. In particular, the functor $M \\mapsto \\widetilde M$\nis an exact functor from the category of $R$-modules\nto the category of $\\mathcal{O}_{\\Spec(R)}$-modules.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HV","source_file":"schemes.tex","source_line":691,"source_end_line":719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L691-L719","statement_sha256":"b4d83f623aaeba3368a6af6306fe6c8537555332e3639fece3061189dac982f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":4992,"rank":4992,"depth":5,"x":1575.428,"y":650.362,"cluster":"schemes"},{"id":"stacks:01HW","tag":"01HW","title":"Affine schemes · Definition 01HW","summary":"An affine scheme is a locally ringed space isomorphic as a locally ringed space to Spec(R) for some ring R. A morphism of affine schemes is a morphism in the category of locally ringed spaces.","statement_latex":"An {\\it affine scheme} is a locally ringed space isomorphic\nas a locally ringed space to $\\Spec(R)$ for some ring $R$.\nA {\\it morphism of affine schemes} is a morphism in the category\nof locally ringed spaces.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Affine schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HW","source_file":"schemes.tex","source_line":730,"source_end_line":736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L730-L736","statement_sha256":"0510b6c7c09f483d9e7c55484a9f115573db058793bfe79d174962ea403a89c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":4993,"rank":4993,"depth":0,"x":1608.069,"y":699.872,"cluster":"schemes"},{"id":"stacks:01HY","tag":"01HY","title":"The category of affine schemes · Lemma 01HY","summary":"Let X be a locally ringed space. Let Y be an affine scheme. Let f ∈ Mor(X, Y) be a morphism of locally ringed spaces. Given a point x ∈ X consider the ring maps Γ(Y, O_Y) xrightarrowf^sharp Γ(X, O_X) → O_X, x Let p ⊂ Γ(Y, O_Y) denote the inverse image of m_x. Let y ∈ Y be the corresponding point. Then f(x) = y.","statement_latex":"Let $X$ be a locally ringed space.\nLet $Y$ be an affine scheme.\nLet $f \\in \\Mor(X, Y)$ be a morphism\nof locally ringed spaces. Given a point $x \\in X$\nconsider the ring maps\n$$\n\\Gamma(Y, \\mathcal{O}_Y) \\xrightarrow{f^\\sharp}\n\\Gamma(X, \\mathcal{O}_X) \\to \\mathcal{O}_{X, x}\n$$\nLet $\\mathfrak p \\subset \\Gamma(Y, \\mathcal{O}_Y)$ denote\nthe inverse image of $\\mathfrak m_x$. Let $y \\in Y$ be the\ncorresponding point. Then $f(x) = y$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"The category of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HY","source_file":"schemes.tex","source_line":768,"source_end_line":782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L768-L782","statement_sha256":"f60a1f031e469a19df375aefa83621ac3da6ac5c54ac89c08ecc9e772a787bf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":4994,"rank":4994,"depth":1,"x":1542.228,"y":681.312,"cluster":"schemes"},{"id":"stacks:01HZ","tag":"01HZ","title":"The category of affine schemes · Lemma 01HZ","summary":"Let X be a locally ringed space. Let f ∈ Γ(X, O_X). The set D(f) = (x ∈ X mid image f not∈ m_x) is open. Moreover f|_D(f) has an inverse.","statement_latex":"Let $X$ be a locally ringed space.\nLet $f \\in \\Gamma(X, \\mathcal{O}_X)$.\nThe set\n$$\nD(f) = \\{x \\in X \\mid \\text{image }f \\not\\in \\mathfrak m_x\\}\n$$\nis open. Moreover $f|_{D(f)}$ has an inverse.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"The category of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01HZ","source_file":"schemes.tex","source_line":802,"source_end_line":811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L802-L811","statement_sha256":"a8bd942e0f98ba62cbb9fbae2903cd78a58781183198602c37f69adee0cd4f98","origin":"The Stacks Project","memory_eligible":false,"source_rank":4995,"rank":4995,"depth":9,"x":1607.552,"y":656.969,"cluster":"schemes"},{"id":"stacks:01I0","tag":"01I0","title":"The category of affine schemes · Lemma 01I0","summary":"In Lemma [Tag 01HZ] above, if X is an affine scheme, then the open D(f) agrees with the standard open D(f) defined previously (in Algebra, Definition [Tag 00DZ]).","statement_latex":"In Lemma \\ref{lemma-f-open} above, if $X$ is an affine scheme,\nthen the open $D(f)$ agrees with the standard open $D(f)$\ndefined previously (in\nAlgebra, Definition \\ref{algebra-definition-spectrum-ring}).","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"The category of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01I0","source_file":"schemes.tex","source_line":828,"source_end_line":834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L828-L834","statement_sha256":"bfac4ef878421ed304c9d90670af526592b1b65f7a4a8e23e5d30956ca6b0f5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":4996,"rank":4996,"depth":10,"x":1578.157,"y":713.485,"cluster":"schemes"},{"id":"stacks:01I1","tag":"01I1","title":"The category of affine schemes · Lemma 01I1","summary":"A reference for this fact is [EGA] where it is attributed to J. Tate. Let X be a locally ringed space. Let Y be an affine scheme. The map Mor(X, Y) → Hom(Γ(Y, O_Y), Γ(X, O_X)) which maps f to f^sharp (on global sections) is bijective.","statement_latex":"\\begin{reference}\nA reference for this fact is \\cite[II, Err 1, Prop. 1.8.1]{EGA}\nwhere it is attributed to J. Tate.\n\\end{reference}\nLet $X$ be a locally ringed space.\nLet $Y$ be an affine scheme.\nThe map\n$$\n\\Mor(X, Y)\n\\longrightarrow\n\\Hom(\\Gamma(Y, \\mathcal{O}_Y), \\Gamma(X, \\mathcal{O}_X))\n$$\nwhich maps $f$ to $f^\\sharp$ (on global sections) is bijective.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"The category of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01I1","source_file":"schemes.tex","source_line":840,"source_end_line":855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L840-L855","statement_sha256":"2f04a12840a93b1699e0d4c18536b49fc5e95f0f0fce3f59965b19f0f7a054aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":4997,"rank":4997,"depth":10,"x":1553.784,"y":653.611,"cluster":"schemes"},{"id":"stacks:01I2","tag":"01I2","title":"The category of affine schemes · Lemma 01I2","summary":"The category of affine schemes is equivalent to the opposite of the category of rings. The equivalence is given by the functor that associates to an affine scheme the global sections of its structure sheaf.","statement_latex":"The category of affine schemes is equivalent to the opposite of the\ncategory of rings. The equivalence is given by the functor that associates\nto an affine scheme the global sections of its structure sheaf.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"The category of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01I2","source_file":"schemes.tex","source_line":953,"source_end_line":958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L953-L958","statement_sha256":"c83d09cdf2e04621b712e1fb787c2b9964326102cf27fdf2500724509c35604b","origin":"The Stacks Project","memory_eligible":false,"source_rank":4998,"rank":4998,"depth":11,"x":1621.528,"y":684.694,"cluster":"schemes"},{"id":"stacks:01I3","tag":"01I3","title":"The category of affine schemes · Lemma 01I3","summary":"Let Y be an affine scheme. Let f ∈ Γ(Y, O_Y). The open subspace D(f) is an affine scheme.","statement_latex":"Let $Y$ be an affine scheme.\nLet $f \\in \\Gamma(Y, \\mathcal{O}_Y)$.\nThe open subspace $D(f)$ is an affine scheme.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"The category of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01I3","source_file":"schemes.tex","source_line":965,"source_end_line":970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L965-L970","statement_sha256":"ba22876fb5e3be31ef67d2945d200fca55bce977e998dc0d74a1baaa8d69bd48","origin":"The Stacks Project","memory_eligible":false,"source_rank":4999,"rank":4999,"depth":3,"x":1544.813,"y":700.565,"cluster":"schemes"},{"id":"stacks:01I4","tag":"01I4","title":"The category of affine schemes · Lemma 01I4","summary":"The category of affine schemes has finite products, and fibre products. In other words, it has finite limits. Moreover, the products and fibre products in the category of affine schemes are the same as in the category of locally ringed spaces. In a formula, we have (in the category of locally ringed spaces) Spec(R) × Spec(S) = Spec(R ⊗_Z S) and given ring maps R → A, R → B we have Spec(A) ×_Spec(R) Spec(B) = Spec(A ⊗_R B).","statement_latex":"The category of affine schemes has finite products, and fibre products.\nIn other words, it has finite limits. Moreover, the products\nand fibre products in the category of affine schemes\nare the same as in the category of locally ringed spaces.\nIn a formula, we have (in the category of locally ringed spaces)\n$$\n\\Spec(R) \\times \\Spec(S) =\n\\Spec(R \\otimes_{\\mathbf{Z}} S)\n$$\nand given ring maps $R \\to A$, $R \\to B$ we have\n$$\n\\Spec(A) \\times_{\\Spec(R)} \\Spec(B)\n=\n\\Spec(A \\otimes_R B).\n$$","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"The category of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01I4","source_file":"schemes.tex","source_line":984,"source_end_line":1001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L984-L1001","statement_sha256":"7d1a0bbc8aaf48bd21c8fae9cd7aedfd7b7414fa6ba7ee5c89935108073731e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5000,"rank":5000,"depth":11,"x":1589.615,"y":644.098,"cluster":"schemes"},{"id":"stacks:01I5","tag":"01I5","title":"The category of affine schemes · Lemma 01I5","summary":"Let X be a locally ringed space. Assume X = U amalg V with U and V open and such that U, V are affine schemes. Then X is an affine scheme.","statement_latex":"Let $X$ be a locally ringed space.\nAssume $X = U \\amalg V$ with $U$ and $V$ open and\nsuch that $U$, $V$ are affine schemes. Then $X$ is an affine scheme.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"The category of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01I5","source_file":"schemes.tex","source_line":1028,"source_end_line":1033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1028-L1033","statement_sha256":"eae573d6ddf93ec2c8fe45d646f3f7db9226da62e3e25109dd0b7f60dbc47f05","origin":"The Stacks Project","memory_eligible":false,"source_rank":5001,"rank":5001,"depth":11,"x":1602.239,"y":712.601,"cluster":"schemes"},{"id":"stacks:01I7","tag":"01I7","title":"Quasi-coherent sheaves on affines · Lemma 01I7","summary":"Let (X, O_X) = (Spec(R), O_Spec(R)) be an affine scheme. Let M be an R-module. There exists a canonical isomorphism between the sheaf widetilde M associated to the R-module M (Definition [Tag 01HU]) and the sheaf F_M associated to the R-module M (Modules, Definition [Tag 01BI]). This isomorphism is functorial in M. In particular, the sheaves widetilde M are quasi-coherent. Moreover, they are characterized by the following mapping property Hom_O_X(widetilde M, F) =…","statement_latex":"Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$\nbe an affine scheme. Let $M$ be an $R$-module. There exists a canonical\nisomorphism between the sheaf $\\widetilde M$ associated to the $R$-module\n$M$ (Definition \\ref{definition-structure-sheaf}) and the sheaf\n$\\mathcal{F}_M$ associated to the $R$-module $M$\n(Modules, Definition \\ref{modules-definition-sheaf-associated}).\nThis isomorphism is functorial in $M$. In particular,\nthe sheaves $\\widetilde M$ are quasi-coherent. Moreover, they\nare characterized by the following mapping property\n$$\n\\Hom_{\\mathcal{O}_X}(\\widetilde M, \\mathcal{F})\n=\n\\Hom_R(M, \\Gamma(X, \\mathcal{F}))\n$$\nfor any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$.\nHere a map $\\alpha : \\widetilde M \\to \\mathcal{F}$ corresponds\nto its effect on global sections.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-coherent sheaves on affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01I7","source_file":"schemes.tex","source_line":1077,"source_end_line":1096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1077-L1096","statement_sha256":"ddaa68794c37b5c31c887519851c116fea07e024fed4127015535d075fe6f6d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5002,"rank":5002,"depth":8,"x":1536.524,"y":668.349,"cluster":"schemes"},{"id":"stacks:01I8","tag":"01I8","title":"Quasi-coherent sheaves on affines · Lemma 01I8","summary":"Let (X, O_X) = (Spec(R), O_Spec(R)) be an affine scheme. There are canonical isomorphisms • widetildeM ⊗_R N ≅ widetilde M ⊗_O_X widetilde N , see Modules, Section [Tag 01CA]. • widetildeT^n(M) ≅ T^n(widetilde M) , widetildeSym^n(M) ≅ Sym^n(widetilde M) , and widetildewedge^n(M) ≅ wedge^n(widetilde M) , see Modules, Section [Tag 01CF]. • if M is a finitely presented R-module, then SheafHom_O_X(widetilde M, widetilde N) ≅ widetildeHom_R(M, N) , see Modules, Section [Tag 01CM].","statement_latex":"Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$\nbe an affine scheme. There are canonical isomorphisms\n\\begin{enumerate}\n\\item\n$\n\\widetilde{M \\otimes_R N}\n\\cong\n\\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N\n$,\nsee Modules, Section \\ref{modules-section-tensor-product}.\n\\item\n$\n\\widetilde{\\text{T}^n(M)}\n\\cong\n\\text{T}^n(\\widetilde M)\n$,\n$\n\\widetilde{\\text{Sym}^n(M)}\n\\cong\n\\text{Sym}^n(\\widetilde M)\n$, and\n$\n\\widetilde{\\wedge^n(M)}\n\\cong\n\\wedge^n(\\widetilde M)\n$,\nsee\nModules, Section \\ref{modules-section-symmetric-exterior}.\n\\item if $M$ is a finitely presented $R$-module, then\n$\n\\SheafHom_{\\mathcal{O}_X}(\\widetilde M, \\widetilde N)\n\\cong\n\\widetilde{\\Hom_R(M,  N)}\n$,\nsee\nModules, Section \\ref{modules-section-internal-hom}.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-coherent sheaves on affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01I8","source_file":"schemes.tex","source_line":1112,"source_end_line":1151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1112-L1151","statement_sha256":"668a7525528d2b66ddcd0614800a3c549f94870702f05fb1dc3ba9388ce9412e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5003,"rank":5003,"depth":15,"x":1622.231,"y":663.61,"cluster":"schemes"},{"id":"stacks:01I9","tag":"01I9","title":"Quasi-coherent sheaves on affines · Lemma 01I9","summary":"Let (X, O_X) = (Spec(S), O_Spec(S)), (Y, O_Y) = (Spec(R), O_Spec(R)) be affine schemes. Let ψ : (X, O_X) → (Y, O_Y) be a morphism of affine schemes, corresponding to the ring map ψ^sharp : R → S (see Lemma [Tag 01I2]). • We have ψ^* widetilde M = widetildeS ⊗_R M functorially in the R-module M. • We have ψ_* widetilde N = widetildeN_R functorially in the S-module N.","statement_latex":"Let\n$(X, \\mathcal{O}_X) = (\\Spec(S), \\mathcal{O}_{\\Spec(S)})$,\n$(Y, \\mathcal{O}_Y) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$\nbe affine schemes.\nLet $\\psi : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a\nmorphism of affine schemes, corresponding to the ring map\n$\\psi^\\sharp : R \\to S$ (see Lemma \\ref{lemma-category-affine-schemes}).\n\\begin{enumerate}\n\\item We have $\\psi^* \\widetilde M = \\widetilde{S \\otimes_R M}$\nfunctorially in the $R$-module $M$.\n\\item We have $\\psi_* \\widetilde N = \\widetilde{N_R}$ functorially\nin the $S$-module $N$.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-coherent sheaves on affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01I9","source_file":"schemes.tex","source_line":1241,"source_end_line":1256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1241-L1256","statement_sha256":"874be8714f1524d05daf2310bd34f1d0e023e3e759d2ef937aba9992572d0067","origin":"The Stacks Project","memory_eligible":false,"source_rank":5004,"rank":5004,"depth":12,"x":1561.704,"y":716.722,"cluster":"schemes"},{"id":"stacks:01IA","tag":"01IA","title":"Quasi-coherent sheaves on affines · Lemma 01IA","summary":"Let (X, O_X) = (Spec(R), O_Spec(R)) be an affine scheme. Let F be a quasi-coherent O_X-module. Then F is isomorphic to the sheaf associated to the R-module Γ(X, F).","statement_latex":"Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$\nbe an affine scheme. Let $\\mathcal{F}$ be a\nquasi-coherent $\\mathcal{O}_X$-module. Then\n$\\mathcal{F}$ is isomorphic to the sheaf associated to\nthe $R$-module $\\Gamma(X, \\mathcal{F})$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-coherent sheaves on affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IA","source_file":"schemes.tex","source_line":1278,"source_end_line":1285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1278-L1285","statement_sha256":"0b47a7c3f50b5ad1683b5e3b4f1163c94f1543c33893587ee2807c2dc6bd5ec3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5005,"rank":5005,"depth":9,"x":1563.672,"y":641.867,"cluster":"schemes"},{"id":"stacks:01IB","tag":"01IB","title":"Quasi-coherent sheaves on affines · Lemma 01IB","summary":"Let (X, O_X) = (Spec(R), O_Spec(R)) be an affine scheme. The functors M ↦ widetilde M and F ↦ Γ(X, F) define quasi-inverse equivalences of categories xymatrix QCoh(O_X) ar@<1ex>[r] & Mod_R ar@<1ex>[l] between the category of quasi-coherent O_X-modules and the category of R-modules.","statement_latex":"Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$\nbe an affine scheme.\nThe functors $M \\mapsto \\widetilde M$ and\n$\\mathcal{F} \\mapsto \\Gamma(X, \\mathcal{F})$ define quasi-inverse\nequivalences of categories\n$$\n\\xymatrix{\n\\QCoh(\\mathcal{O}_X)\n\\ar@<1ex>[r]\n&\n\\text{Mod}_R\n\\ar@<1ex>[l]\n}\n$$\nbetween the category of quasi-coherent $\\mathcal{O}_X$-modules\nand the category of $R$-modules.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-coherent sheaves on affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IB","source_file":"schemes.tex","source_line":1389,"source_end_line":1407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1389-L1407","statement_sha256":"b5eafcc11ca1e4c258e146fd0ffcd1153839c5ca6344f5f6b2a29169aafc5201","origin":"The Stacks Project","memory_eligible":false,"source_rank":5006,"rank":5006,"depth":10,"x":1623.448,"y":699.181,"cluster":"schemes"},{"id":"stacks:01IC","tag":"01IC","title":"Quasi-coherent sheaves on affines · Lemma 01IC","summary":"Let X = Spec(R) be an affine scheme. Kernels and cokernels of maps of quasi-coherent O_X-modules are quasi-coherent.","statement_latex":"Let $X = \\Spec(R)$ be an affine scheme.\nKernels and cokernels of maps of quasi-coherent\n$\\mathcal{O}_X$-modules are quasi-coherent.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-coherent sheaves on affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IC","source_file":"schemes.tex","source_line":1418,"source_end_line":1423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1418-L1423","statement_sha256":"4896a28e439361476a1572fa454d844b126605a9f4e12ad8bc56dffedcf58f35","origin":"The Stacks Project","memory_eligible":false,"source_rank":5007,"rank":5007,"depth":9,"x":1531.74,"y":690.686,"cluster":"schemes"},{"id":"stacks:01ID","tag":"01ID","title":"Quasi-coherent sheaves on affines · Lemma 01ID","summary":"Let X = Spec(R) be an affine scheme. The direct sum of an arbitrary collection of quasi-coherent sheaves on X is quasi-coherent. The same holds for colimits.","statement_latex":"Let $X = \\Spec(R)$ be an affine scheme.\nThe direct sum of an arbitrary collection of quasi-coherent sheaves\non $X$ is quasi-coherent. The same holds for colimits.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-coherent sheaves on affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ID","source_file":"schemes.tex","source_line":1434,"source_end_line":1439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1434-L1439","statement_sha256":"b572cbf977e34a9ab73db21de4d8cbb6c631ed8c96326f8526fb302e52da0cdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5008,"rank":5008,"depth":11,"x":1607.431,"y":644.164,"cluster":"schemes"},{"id":"stacks:01IE","tag":"01IE","title":"Quasi-coherent sheaves on affines · Lemma 01IE","summary":"Let (X, O_X) = (Spec(R), O_Spec(R)) be an affine scheme. Suppose that 0 → F_1 → F_2 → F_3 → 0 is a short exact sequence of sheaves of O_X-modules. If two out of three are quasi-coherent then so is the third.","statement_latex":"Let $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$\nbe an affine scheme. Suppose that\n$$\n0 \\to\n\\mathcal{F}_1 \\to\n\\mathcal{F}_2 \\to\n\\mathcal{F}_3 \\to\n0\n$$\nis a short exact sequence of sheaves of $\\mathcal{O}_X$-modules.\nIf two out of three are quasi-coherent then so is the third.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-coherent sheaves on affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IE","source_file":"schemes.tex","source_line":1457,"source_end_line":1470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1457-L1470","statement_sha256":"0701cb8ac30928de320f0bc390612f9dc7b94f6fc28846a4b40d0f6755e2dc46","origin":"The Stacks Project","memory_eligible":false,"source_rank":5009,"rank":5009,"depth":9,"x":1588.726,"y":722.65,"cluster":"schemes"},{"id":"stacks:01IH","tag":"01IH","title":"Closed subspaces of affine schemes · Lemma 01IH","summary":"For affine schemes, closed immersions correspond to ideals. Let (X, O_X) = (Spec(R), O_Spec(R)) be an affine scheme. Let i : Z → X be any closed immersion of locally ringed spaces. Then there exists a unique ideal I ⊂ R such that the morphism i : Z → X can be identified with the closed immersion Spec(R/I) → Spec(R) constructed in Example [Tag 01IG] above.","statement_latex":"\\begin{slogan}\nFor affine schemes, closed immersions correspond to ideals.\n\\end{slogan}\nLet $(X, \\mathcal{O}_X) = (\\Spec(R), \\mathcal{O}_{\\Spec(R)})$\nbe an affine scheme. Let $i : Z \\to X$ be any closed immersion\nof locally ringed spaces. Then there exists a unique ideal\n$I \\subset R$ such that the morphism $i : Z \\to X$ can be identified\nwith the closed immersion $\\Spec(R/I) \\to \\Spec(R)$\nconstructed in Example \\ref{example-closed-immersion-affines} above.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Closed subspaces of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IH","source_file":"schemes.tex","source_line":1599,"source_end_line":1610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1599-L1610","statement_sha256":"3f58a01dc2319a5310279a17ed864f1677063feac3dd2c7db5a2dd8e46055b68","origin":"The Stacks Project","memory_eligible":false,"source_rank":5010,"rank":5010,"depth":10,"x":1538.646,"y":653.098,"cluster":"schemes"},{"id":"stacks:01IJ","tag":"01IJ","title":"Schemes · Definition 01IJ","summary":"In [EGA1] what we call a scheme was called a \"pre-schéma\" and the name \"schéma\" was reserved for what is a separated scheme in the Stacks project. In the second edition [EGA1-second] the terminology was changed to the terminology that is now standard. However, one may occasionally encounter the terminology \"prescheme\", for example in [Murre-lectures]. A scheme is a locally ringed space with the property that every point has an open neighbourhood which is an affine scheme.…","statement_latex":"\\begin{history}\nIn \\cite{EGA1} what we call a scheme was called a ``pre-sch\\'ema'' and the\nname ``sch\\'ema'' was reserved for what is a separated scheme in the\nStacks project. In the second edition \\cite{EGA1-second} the terminology\nwas changed to the terminology that is now standard. However, one may\noccasionally encounter the terminology ``prescheme'', for example in\n\\cite{Murre-lectures}.\n\\end{history}\nA {\\it scheme} is a locally ringed space with the property that\nevery point has an open neighbourhood which is an affine scheme.\nA {\\it morphism of schemes} is a morphism of locally\nringed spaces. The category of schemes will be denoted\n$\\Sch$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IJ","source_file":"schemes.tex","source_line":1647,"source_end_line":1662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1647-L1662","statement_sha256":"03536fd5d1fb4aa54faf9fed12c48497e1dd728e3908d7c5fc99d323965f377d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5011,"rank":5011,"depth":0,"x":1632.899,"y":676.319,"cluster":"schemes"},{"id":"stacks:01IK","tag":"01IK","title":"Schemes · Lemma 01IK","summary":"Let X be a scheme. Let j : U → X be an open immersion of locally ringed spaces. Then U is a scheme. In particular, any open subspace of X is a scheme.","statement_latex":"Let $X$ be a scheme. Let $j : U \\to X$ be an open immersion\nof locally ringed spaces. Then $U$ is a scheme. In particular,\nany open subspace of $X$ is a scheme.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IK","source_file":"schemes.tex","source_line":1673,"source_end_line":1678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1673-L1678","statement_sha256":"8380ee0866b2890af9ca1ae092d6c8ca5cd715f3caaaa57b3eab5067af6fccb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5012,"rank":5012,"depth":4,"x":1543.436,"y":713.201,"cluster":"schemes"},{"id":"stacks:01IN","tag":"01IN","title":"Immersions of schemes · Lemma 01IN","summary":"Let X be a scheme. Let i : Z → X be a closed immersion of locally ringed spaces. • The locally ringed space Z is a scheme, • the kernel I of the map O_X → i_*O_Z is a quasi-coherent sheaf of ideals, • for any affine open U = Spec(R) of X the morphism i^-1(U) → U can be identified with Spec(R/I) → Spec(R) for some ideal I ⊂ R, and • we have I|_U = widetilde I. In particular, any sheaf of ideals locally generated by sections is a quasi-coherent sheaf of ideals (and vice…","statement_latex":"Let $X$ be a scheme. Let $i : Z \\to X$ be a closed immersion\nof locally ringed spaces.\n\\begin{enumerate}\n\\item The locally ringed space $Z$ is a scheme,\n\\item the kernel $\\mathcal{I}$ of the map\n$\\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is a quasi-coherent\nsheaf of ideals,\n\\item for any affine open $U = \\Spec(R)$ of $X$\nthe morphism $i^{-1}(U) \\to U$ can be identified with\n$\\Spec(R/I) \\to \\Spec(R)$ for some ideal $I \\subset R$, and\n\\item we have $\\mathcal{I}|_U = \\widetilde I$.\n\\end{enumerate}\nIn particular, any sheaf of ideals locally generated by sections\nis a quasi-coherent sheaf of ideals (and vice versa),\nand any closed subspace of $X$ is a scheme.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Immersions of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IN","source_file":"schemes.tex","source_line":1743,"source_end_line":1760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1743-L1760","statement_sha256":"d6e8e35947ab9730444d355035414bbeb13232965d0824968591645bc4b7eff5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5013,"rank":5013,"depth":11,"x":1580.266,"y":634.139,"cluster":"schemes"},{"id":"stacks:01IO","tag":"01IO","title":"Immersions of schemes · Definition 01IO","summary":"Let X be a scheme. • A morphism of schemes is called an open immersion if it is an open immersion of locally ringed spaces (see Definition [Tag 01HE]). • An open subscheme of X is an open subspace of X in the sense of Definition [Tag 01HG]; an open subscheme of X is a scheme by Lemma [Tag 01IK]. • A morphism of schemes is called a closed immersion if it is a closed immersion of locally ringed spaces (see Definition [Tag 01HK]). • A closed subscheme of X is a closed…","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item A morphism of schemes is called an {\\it open immersion}\nif it is an open immersion of locally ringed spaces (see\nDefinition \\ref{definition-immersion-locally-ringed-spaces}).\n\\item An {\\it open subscheme} of $X$ is an open subspace of $X$\nin the sense of Definition \\ref{definition-open-subspace}; an open subscheme\nof $X$ is a scheme by Lemma \\ref{lemma-open-subspace-scheme}.\n\\item A morphism of schemes is called a {\\it closed immersion}\nif it is a closed immersion of locally ringed spaces (see\nDefinition \\ref{definition-closed-immersion-locally-ringed-spaces}).\n\\item A {\\it closed subscheme} of $X$ is a closed subspace of $X$\nin the sense of Definition \\ref{definition-closed-subspace}; a closed subscheme\nis a scheme by Lemma \\ref{lemma-closed-subspace-scheme}.\n\\item A morphism of schemes $f : X \\to Y$ is called an {\\it immersion},\nor a {\\it locally closed immersion} if it can be factored as\n$j \\circ i$ where $i$ is a closed immersion and $j$ is an open\nimmersion.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Immersions of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IO","source_file":"schemes.tex","source_line":1780,"source_end_line":1801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1780-L1801","statement_sha256":"063a5092bfb08f6cb44c5fce5e16445a8cb2d3b14f524a38a5811e26fc993933","origin":"The Stacks Project","memory_eligible":false,"source_rank":5014,"rank":5014,"depth":12,"x":1617.182,"y":714.43,"cluster":"schemes"},{"id":"stacks:01IQ","tag":"01IQ","title":"Immersions of schemes · Lemma 01IQ","summary":"Let f : Y → X be an immersion of schemes. Then f is a closed immersion if and only if f(Y) ⊂ X is a closed subset.","statement_latex":"Let $f : Y \\to X$ be an immersion of schemes. Then $f$ is a closed\nimmersion if and only if $f(Y) \\subset X$ is a closed subset.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Immersions of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IQ","source_file":"schemes.tex","source_line":1848,"source_end_line":1852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1848-L1852","statement_sha256":"c5246c7449d6862240fef1865ddb7c4c7f7214ebbbee361ab6cb5f0eb4f3fbdb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5015,"rank":5015,"depth":1,"x":1524.166,"y":675.653,"cluster":"schemes"},{"id":"stacks:01IS","tag":"01IS","title":"Zariski topology of schemes · Lemma 01IS","summary":"Let X be a scheme. Any irreducible closed subset of X has a unique generic point. In other words, X is a sober topological space, see Topology, Definition [Tag 004X].","statement_latex":"Let $X$ be a scheme.\nAny irreducible closed subset of $X$ has a unique generic point.\nIn other words, $X$ is a sober topological space, see\nTopology, Definition \\ref{topology-definition-generic-point}.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Zariski topology of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IS","source_file":"schemes.tex","source_line":1929,"source_end_line":1935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1929-L1935","statement_sha256":"f2b409990b040b0e33d780c6a2a5018649ec5ad12b56cd09fc92dbc179fcef16","origin":"The Stacks Project","memory_eligible":false,"source_rank":5016,"rank":5016,"depth":1,"x":1625.242,"y":651.157,"cluster":"schemes"},{"id":"stacks:01IT","tag":"01IT","title":"Zariski topology of schemes · Lemma 01IT","summary":"Let X be a scheme. The collection of affine opens of X forms a basis for the topology on X.","statement_latex":"Let $X$ be a scheme. The collection of affine opens\nof $X$ forms a basis for the topology on $X$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Zariski topology of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IT","source_file":"schemes.tex","source_line":1951,"source_end_line":1955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1951-L1955","statement_sha256":"8454a545a557b1d8d8e3522e1dc001a2db3fba03c53277367d409454fe7adeee","origin":"The Stacks Project","memory_eligible":false,"source_rank":5017,"rank":5017,"depth":0,"x":1569.706,"y":727.529,"cluster":"schemes"},{"id":"stacks:01IV","tag":"01IV","title":"Zariski topology of schemes · Lemma 01IV","summary":"The underlying topological space of any scheme is locally quasi-compact, see Topology, Definition [Tag 0068].","statement_latex":"The underlying topological space of any scheme is\nlocally quasi-compact, see\nTopology, Definition \\ref{topology-definition-locally-quasi-compact}.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Zariski topology of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IV","source_file":"schemes.tex","source_line":1968,"source_end_line":1973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1968-L1973","statement_sha256":"a75ee0b01f169316c76979f2946b65f1ecc4cba87371848efadb95fcc862b00e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5018,"rank":5018,"depth":2,"x":1548.994,"y":638.611,"cluster":"schemes"},{"id":"stacks:01IW","tag":"01IW","title":"Zariski topology of schemes · Lemma 01IW","summary":"Let X be a scheme. Let U, V be affine opens of X, and let x ∈ U ∩ V. There exists an affine open neighbourhood W of x such that W is a standard open of both U and V.","statement_latex":"Let $X$ be a scheme.\nLet $U, V$ be affine opens of $X$, and let $x \\in U \\cap V$.\nThere exists an affine open neighbourhood $W$ of $x$\nsuch that $W$ is a standard open of both $U$ and $V$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Zariski topology of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IW","source_file":"schemes.tex","source_line":1981,"source_end_line":1987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L1981-L1987","statement_sha256":"e7f508913ac75f6137c58a5b36cce7d4beac8502fa5a01aa0de9625405cd9ba9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5019,"rank":5019,"depth":0,"x":1636.819,"y":693.08,"cluster":"schemes"},{"id":"stacks:01IX","tag":"01IX","title":"Zariski topology of schemes · Lemma 01IX","summary":"Let X be a scheme. Let X = ⋃_i U_i be an affine open covering. Let V ⊂ X be an affine open. There exists a standard open covering V = ⋃_j = 1, …, m V_j (see Definition [Tag 01HT]) such that each V_j is a standard open in one of the U_i.","statement_latex":"Let $X$ be a scheme.\nLet $X = \\bigcup_i U_i$ be an affine open covering.\nLet $V \\subset X$ be an affine open.\nThere exists a standard open covering\n$V = \\bigcup_{j = 1, \\ldots, m} V_j$ (see\nDefinition \\ref{definition-standard-covering})\nsuch that each $V_j$ is a standard open in one of the $U_i$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Zariski topology of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01IX","source_file":"schemes.tex","source_line":2004,"source_end_line":2013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2004-L2013","statement_sha256":"8fa570de1a4466e071b4412b57935226086ab1cda63356f9414ab39053768251","origin":"The Stacks Project","memory_eligible":false,"source_rank":5020,"rank":5020,"depth":1,"x":1526.972,"y":702.859,"cluster":"schemes"},{"id":"stacks:0F1A","tag":"0F1A","title":"Zariski topology of schemes · Lemma 0F1A","summary":"Let X be a scheme. Let B be the set of affine opens of X. Let F be a presheaf of sets on B, see Sheaves, Definition [Tag 009I]. The following are equivalent • F is the restriction of a sheaf on X to B, • F is a sheaf on B, and • F(∅) is a singleton and whenever U = V ∪ W with U, V, W ∈ B and V, W ⊂ U standard open (Algebra, Definition [Tag 00E1]) the map F(U) → F(V) × F(W) is injective with image the set of pairs (s, t) such that s|_V ∩ W = t|_V ∩ W.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{B}$ be the set of affine opens of $X$.\nLet $\\mathcal{F}$ be a presheaf of sets on $\\mathcal{B}$, see\nSheaves, Definition \\ref{sheaves-definition-presheaf-basis}. The following\nare equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is the restriction of a sheaf on $X$ to $\\mathcal{B}$,\n\\item $\\mathcal{F}$ is a sheaf on $\\mathcal{B}$, and\n\\item $\\mathcal{F}(\\emptyset)$ is a singleton and\nwhenever $U = V \\cup W$ with $U, V, W \\in \\mathcal{B}$ and\n$V, W \\subset U$ standard open\n(Algebra, Definition \\ref{algebra-definition-Zariski-topology})\nthe map\n$$\n\\mathcal{F}(U) \\longrightarrow \\mathcal{F}(V) \\times \\mathcal{F}(W)\n$$\nis injective with image the set of pairs $(s, t)$\nsuch that $s|_{V \\cap W} = t|_{V \\cap W}$.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Zariski topology of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1A","source_file":"schemes.tex","source_line":2023,"source_end_line":2043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2023-L2043","statement_sha256":"8fe5cb89a0e3fb0639bb577b6500068b84671b7426c5f2c5831c2958e77708e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5021,"rank":5021,"depth":5,"x":1600.956,"y":632.519,"cluster":"schemes"},{"id":"stacks:02O0","tag":"02O0","title":"Zariski topology of schemes · Lemma 02O0","summary":"Let X be a scheme whose underlying topological space is a finite discrete set. Then X is affine.","statement_latex":"Let $X$ be a scheme whose underlying topological space\nis a finite discrete set.\nThen $X$ is affine.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Zariski topology of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02O0","source_file":"schemes.tex","source_line":2084,"source_end_line":2089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2084-L2089","statement_sha256":"c8964cdd4d7fec4c8e94ba87ba32a3e2bd66347f06d91fcfa6227057e37a203f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5022,"rank":5022,"depth":12,"x":1602.984,"y":727.425,"cluster":"schemes"},{"id":"stacks:01J0","tag":"01J0","title":"Reduced schemes · Definition 01J0","summary":"Let X be a scheme. We say X is reduced if every local ring O_X, x is reduced.","statement_latex":"Let $X$ be a scheme. We say $X$ is {\\it reduced} if every local ring\n$\\mathcal{O}_{X, x}$ is reduced.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Reduced schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01J0","source_file":"schemes.tex","source_line":2147,"source_end_line":2151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2147-L2151","statement_sha256":"ca46cecc616cb597dd329eee70d12f22a7f467d03289f36002430b8eeb71a95d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5023,"rank":5023,"depth":0,"x":1524.313,"y":657.832,"cluster":"schemes"},{"id":"stacks:01J1","tag":"01J1","title":"Reduced schemes · Lemma 01J1","summary":"A scheme X is reduced if and only if O_X(U) is a reduced ring for all U ⊂ X open.","statement_latex":"A scheme $X$ is reduced if and only if $\\mathcal{O}_X(U)$\nis a reduced ring for all $U \\subset X$ open.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01J1","source_file":"schemes.tex","source_line":2153,"source_end_line":2157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2153-L2157","statement_sha256":"d0261b48b7093a53920e1bad56177758d16a4e782debd920eb9fd6fe4e4f0f86","origin":"The Stacks Project","memory_eligible":false,"source_rank":5024,"rank":5024,"depth":1,"x":1639.518,"y":664.586,"cluster":"schemes"},{"id":"stacks:01J2","tag":"01J2","title":"Reduced schemes · Lemma 01J2","summary":"An affine scheme Spec(R) is reduced if and only if R is reduced.","statement_latex":"An affine scheme $\\Spec(R)$ is reduced\nif and only if $R$ is reduced.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01J2","source_file":"schemes.tex","source_line":2171,"source_end_line":2175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2171-L2175","statement_sha256":"88dc295d79b25cacd198b13d8fcc88ce221c4111c1daa2c219e8abe0ca4bafa3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5025,"rank":5025,"depth":2,"x":1548.179,"y":725.61,"cluster":"schemes"},{"id":"stacks:01J3","tag":"01J3","title":"Reduced schemes · Lemma 01J3","summary":"Let X be a scheme. Let T ⊂ X be a closed subset. There exists a unique closed subscheme Z ⊂ X with the following properties: (a) the underlying topological space of Z is equal to T, and (b) Z is reduced.","statement_latex":"Let $X$ be a scheme. Let $T \\subset X$ be a closed subset.\nThere exists a unique closed subscheme $Z \\subset X$ with\nthe following properties: (a) the underlying topological\nspace of $Z$ is equal to $T$, and (b) $Z$ is reduced.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01J3","source_file":"schemes.tex","source_line":2185,"source_end_line":2191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2185-L2191","statement_sha256":"76a7c28d5fdf3d95dd8fa3797c38f6b57d141717662dafa78bc2d1c583963562","origin":"The Stacks Project","memory_eligible":false,"source_rank":5026,"rank":5026,"depth":3,"x":1566.651,"y":627.783,"cluster":"schemes"},{"id":"stacks:01J4","tag":"01J4","title":"Reduced schemes · Definition 01J4","summary":"Let X be a scheme. Let Z ⊂ X be a closed subset. A scheme structure on Z is given by a closed subscheme Z' of X whose underlying set is equal to Z. We often say \"let (Z, O_Z) be a scheme structure on Z\" to indicate this. The reduced induced scheme structure on Z is the one constructed in Lemma [Tag 01J3]. The reduction X_red of X is the reduced induced scheme structure on X itself.","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subset.\nA {\\it scheme structure on $Z$} is given by a closed subscheme $Z'$ of\n$X$ whose underlying set is equal to $Z$. We often say\n``let $(Z, \\mathcal{O}_Z)$ be a scheme structure on $Z$'' to\nindicate this. The {\\it reduced induced scheme structure}\non $Z$ is the one constructed in Lemma \\ref{lemma-reduced-closed-subscheme}.\nThe {\\it reduction $X_{red}$ of $X$} is the reduced induced scheme\nstructure on $X$ itself.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Reduced schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01J4","source_file":"schemes.tex","source_line":2238,"source_end_line":2248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2238-L2248","statement_sha256":"af707e95f592c711228f966fc2bb9a40725142c42683cd6713e7798cf33f3bfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5027,"rank":5027,"depth":4,"x":1632.355,"y":711.24,"cluster":"schemes"},{"id":"stacks:0356","tag":"0356","title":"Reduced schemes · Lemma 0356","summary":"Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let Y be a reduced scheme. A morphism f : Y → X factors through Z if and only if f(Y) ⊂ Z (set theoretically). In particular, any morphism Y → X factors as Y → X_red → X.","statement_latex":"Let $X$ be a scheme.\nLet $Z \\subset X$ be a closed subscheme.\nLet $Y$ be a reduced scheme.\nA morphism $f : Y \\to X$ factors through $Z$ if and only if\n$f(Y) \\subset Z$ (set theoretically). In particular, any\nmorphism $Y \\to X$ factors as $Y \\to X_{red} \\to X$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0356","source_file":"schemes.tex","source_line":2268,"source_end_line":2276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2268-L2276","statement_sha256":"78166d7116eb3175c70cda3f9b191cf05169f6a4a2b79c5e2e86c57697cd002b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5028,"rank":5028,"depth":2,"x":1515.645,"y":686.737,"cluster":"schemes"},{"id":"stacks:01J6","tag":"01J6","title":"Points of schemes · Lemma 01J6","summary":"Let X be a scheme. Let R be a local ring. The construction above gives a bijective correspondence between morphisms Spec(R) → X and pairs (x, φ) consisting of a point x ∈ X and a local homomorphism of local rings φ : O_X, x → R.","statement_latex":"Let $X$ be a scheme. Let $R$ be a local ring.\nThe construction above gives a bijective correspondence\nbetween morphisms $\\Spec(R) \\to X$ and pairs\n$(x, \\varphi)$ consisting of a point $x \\in X$ and\na local homomorphism of local rings $\\varphi : \\mathcal{O}_{X, x} \\to R$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Points of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01J6","source_file":"schemes.tex","source_line":2332,"source_end_line":2339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2332-L2339","statement_sha256":"5c103a766dc115284bd9c0a304c2f42c87b24152e54c0c96b2368f85976c36ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":5029,"rank":5029,"depth":11,"x":1622.441,"y":638.116,"cluster":"schemes"},{"id":"stacks:01J7","tag":"01J7","title":"Points of schemes · Lemma 01J7","summary":"Let X be a scheme. Let x, x' ∈ X be points of X. Then x' ∈ X is a generalization of x if and only if x' is in the image of the canonical morphism Spec(O_X, x) → X.","statement_latex":"Let $X$ be a scheme.\nLet $x, x' \\in X$ be points of $X$.\nThen $x' \\in X$ is a generalization of $x$ if and only if\n$x'$ is in the image of the canonical morphism\n$\\Spec(\\mathcal{O}_{X, x}) \\to X$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Points of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01J7","source_file":"schemes.tex","source_line":2394,"source_end_line":2401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2394-L2401","statement_sha256":"48140be4a6dc9b1e1766c66ed6465fe8a288835270790301faa48b38421bd1b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5030,"rank":5030,"depth":0,"x":1582.409,"y":735.487,"cluster":"schemes"},{"id":"stacks:01J9","tag":"01J9","title":"Points of schemes · Lemma 01J9","summary":"Let X be a scheme. Points of X correspond bijectively to equivalence classes of morphisms from spectra of fields into X. Moreover, each equivalence class contains a (unique up to unique isomorphism) smallest element Spec(kappa(x)) → X.","statement_latex":"Let $X$ be a scheme. Points of $X$ correspond bijectively\nto equivalence classes of morphisms from spectra of\nfields into $X$. Moreover, each equivalence class contains\na (unique up to unique isomorphism) smallest element\n$\\Spec(\\kappa(x)) \\to X$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Points of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01J9","source_file":"schemes.tex","source_line":2464,"source_end_line":2471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2464-L2471","statement_sha256":"543a5f45012ed510ec3b959155ec65c2ad50e9c50160541d15c7c04e44f9d664","origin":"The Stacks Project","memory_eligible":false,"source_rank":5031,"rank":5031,"depth":0,"x":1533.172,"y":640.085,"cluster":"schemes"},{"id":"stacks:01JB","tag":"01JB","title":"Glueing schemes · Lemma 01JB","summary":"If you have two locally ringed spaces, and a subspace of the first one is isomorphic to a subspace of the other, then you can glue them together into one big locally ringed space. Given any glueing data of locally ringed spaces there exists a locally ringed space X and open subspaces U_i ⊂ X together with isomorphisms φ_i : X_i → U_i of locally ringed spaces such that • X=⋃_i∈ I U_i, • φ_i(U_ij) = U_i ∩ U_j, and • φ_ij = φ_j^-1|_U_i ∩ U_j ∘ φ_i|_U_ij. The locally ringed…","statement_latex":"\\begin{slogan}\nIf you have two locally ringed spaces, and a subspace of the first one\nis isomorphic to a subspace of the other, then you can glue them together\ninto one big locally ringed space.\n\\end{slogan}\nGiven any glueing data of locally ringed spaces there\nexists a locally ringed space $X$ and open subspaces\n$U_i \\subset X$ together with isomorphisms\n$\\varphi_i : X_i \\to U_i$ of locally ringed spaces such that\n\\begin{enumerate}\n\\item $X=\\bigcup_{i\\in I} U_i$,\n\\item $\\varphi_i(U_{ij}) = U_i \\cap U_j$, and\n\\item $\\varphi_{ij} =\n\\varphi_j^{-1}|_{U_i \\cap U_j} \\circ \\varphi_i|_{U_{ij}}$.\n\\end{enumerate}\nThe locally ringed space $X$ is characterized by the following\nmapping properties: Given a locally ringed space $Y$ we have\n\\begin{eqnarray*}\n\\Mor(X, Y) & = & \\{ (f_i)_{i\\in I} \\mid\nf_i : X_i \\to Y, \\ f_j \\circ \\varphi_{ij} = f_i|_{U_{ij}}\\} \\\\\nf & \\mapsto & (f|_{U_i} \\circ \\varphi_i)_{i \\in I} \\\\\n\\Mor(Y, X) & = &\n\\left\\{\n\\begin{matrix}\n\\text{open covering }Y = \\bigcup\\nolimits_{i \\in I} V_i\\text{ and }\n(g_i : V_i \\to X_i)_{i \\in I}\n\\text{ such that}\\\\\ng_i^{-1}(U_{ij}) = V_i \\cap V_j\n\\text{ and }\ng_j|_{V_i \\cap V_j} = \\varphi_{ij} \\circ g_i|_{V_i \\cap V_j}\n\\end{matrix}\n\\right\\} \\\\\ng & \\mapsto &\nV_i = g^{-1}(U_i), \\ g_i = \\varphi_i^{-1} \\circ g|_{V_i}\n\\end{eqnarray*}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Glueing schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JB","source_file":"schemes.tex","source_line":2544,"source_end_line":2581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2544-L2581","statement_sha256":"78604e08fe011dce6f322c3a5b146f4944b78b983b99a19c4c0bc17ac9006509","origin":"The Stacks Project","memory_eligible":false,"source_rank":5032,"rank":5032,"depth":0,"x":1647.236,"y":682.888,"cluster":"schemes"},{"id":"stacks:01JC","tag":"01JC","title":"Glueing schemes · Lemma 01JC","summary":"Schemes can be glued to give new schemes. In Lemma [Tag 01JB] above, assume that all X_i are schemes. Then the resulting locally ringed space X is a scheme.","statement_latex":"\\begin{slogan}\nSchemes can be glued to give new schemes.\n\\end{slogan}\nIn Lemma \\ref{lemma-glue} above, assume that all\n$X_i$ are schemes. Then the resulting locally ringed\nspace $X$ is a scheme.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Glueing schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JC","source_file":"schemes.tex","source_line":2650,"source_end_line":2658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2650-L2658","statement_sha256":"01851adb761e52a60eb3eac6165296bb221b7c546ad7efe3c4955de226770aff","origin":"The Stacks Project","memory_eligible":false,"source_rank":5033,"rank":5033,"depth":1,"x":1527.635,"y":716.344,"cluster":"schemes"},{"id":"stacks:01JG","tag":"01JG","title":"A representability criterion · Definition 01JG","summary":"(See Categories, Definition [Tag 001Q].) Let F be a contravariant functor from the category of schemes to the category of sets (as above). We say that F is representable by a scheme or representable if there exists a scheme X such that h_X ≅ F.","statement_latex":"(See Categories, Definition \\ref{categories-definition-representable-functor}.)\nLet $F$ be a contravariant functor from the category\nof schemes to the category of sets (as above).\nWe say that $F$ is {\\it representable by a scheme}\nor {\\it representable} if there exists a scheme $X$\nsuch that $h_X \\cong F$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"A representability criterion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JG","source_file":"schemes.tex","source_line":2788,"source_end_line":2796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2788-L2796","statement_sha256":"0bb845d58a8a30f32f9ab24fedee430f5910e93f9ad9f6faa74736f24038487f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5034,"rank":5034,"depth":1,"x":1589.469,"y":622.99,"cluster":"schemes"},{"id":"stacks:01JI","tag":"01JI","title":"A representability criterion · Definition 01JI","summary":"Let F be a contravariant functor on the category of schemes with values in sets. • We say that F satisfies the sheaf property for the Zariski topology if for every scheme T and every open covering T = ⋃_i ∈ I U_i, and for any collection of elements xi_i ∈ F(U_i) such that xi_i|_U_i ∩ U_j = xi_j|_U_i ∩ U_j there exists a unique element xi ∈ F(T) such that xi_i = xi|_U_i in F(U_i). • A subfunctor H ⊂ F is a rule that associates to every scheme T a subset H(T) ⊂ F(T) such…","statement_latex":"Let $F$ be a contravariant functor on the category\nof schemes with values in sets.\n\\begin{enumerate}\n\\item We say that $F$ {\\it satisfies the sheaf property for the\nZariski topology} if for every scheme $T$ and every open covering\n$T = \\bigcup_{i \\in I} U_i$, and for any collection of elements\n$\\xi_i \\in F(U_i)$ such that $\\xi_i|_{U_i \\cap U_j} =\n\\xi_j|_{U_i \\cap U_j}$ there exists a unique element\n$\\xi \\in F(T)$ such that $\\xi_i = \\xi|_{U_i}$ in $F(U_i)$.\n\\item A {\\it subfunctor $H \\subset F$} is a rule that associates\nto every scheme $T$ a subset $H(T) \\subset F(T)$ such that\nthe maps $F(f) : F(T) \\to F(T')$ maps $H(T)$ into\n$H(T')$ for all morphisms of schemes $f : T' \\to T$.\n\\item Let $H \\subset F$ be a subfunctor. We say that\n$H \\subset F$ is {\\it representable by open immersions}\nif for all pairs $(T, \\xi)$, where $T$ is a scheme and $\\xi \\in F(T)$\nthere exists an open subscheme $U_\\xi \\subset T$ with the following\nproperty:\n\\begin{itemize}\n\\item[(*)] A morphism $f : T' \\to T$ factors through $U_\\xi$ if and only\nif $f^*\\xi \\in H(T')$.\n\\end{itemize}\n\\item Let $I$ be a set. For each $i \\in I$ let $H_i \\subset F$\nbe a subfunctor. We say that the collection $(H_i)_{i \\in I}$\n{\\it covers $F$} if and only if for every $\\xi \\in F(T)$\nthere exists an open covering $T = \\bigcup U_i$ such that\n$\\xi|_{U_i} \\in H_i(U_i)$.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"A representability criterion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JI","source_file":"schemes.tex","source_line":2855,"source_end_line":2885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2855-L2885","statement_sha256":"991e51b15bb5450f7701aaaa5daade48187ba01349aa096d763bbe954c902b10","origin":"The Stacks Project","memory_eligible":false,"source_rank":5035,"rank":5035,"depth":0,"x":1619.199,"y":727.821,"cluster":"schemes"},{"id":"stacks:01JJ","tag":"01JJ","title":"A representability criterion · Lemma 01JJ","summary":"Let F be a contravariant functor on the category of schemes with values in the category of sets. Suppose that • F satisfies the sheaf property for the Zariski topology, • there exists a set I and a collection of subfunctors F_i ⊂ F such that • each F_i is representable, • each F_i ⊂ F is representable by open immersions, and • the collection (F_i)_i ∈ I covers F. Then F is representable.","statement_latex":"Let $F$ be a contravariant functor on the category of schemes\nwith values in the category of sets. Suppose that\n\\begin{enumerate}\n\\item $F$ satisfies the sheaf property for the Zariski topology,\n\\item there exists a set $I$ and a collection of subfunctors\n$F_i \\subset F$ such that\n\\begin{enumerate}\n\\item each $F_i$ is representable,\n\\item each $F_i \\subset F$ is representable by open immersions, and\n\\item the collection $(F_i)_{i \\in I}$ covers $F$.\n\\end{enumerate}\n\\end{enumerate}\nThen $F$ is representable.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"A representability criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JJ","source_file":"schemes.tex","source_line":2892,"source_end_line":2907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L2892-L2907","statement_sha256":"43659023a45cd20adb08d3fbb3328d41a4e2d4f5d9315060454aaee42a86dbc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5036,"rank":5036,"depth":2,"x":1512.067,"y":666.87,"cluster":"schemes"},{"id":"stacks:01JM","tag":"01JM","title":"Existence of fibre products of schemes · Lemma 01JM","summary":"The category of schemes has a final object, products and fibre products. In other words, the category of schemes has finite limits, see Categories, Lemma [Tag 002O].","statement_latex":"The category of schemes has a final object, products and fibre products.\nIn other words, the category of schemes has finite limits, see\nCategories, Lemma \\ref{categories-lemma-finite-limits-exist}.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Existence of fibre products of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JM","source_file":"schemes.tex","source_line":3009,"source_end_line":3014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3009-L3014","statement_sha256":"740a45c3e9a003f4b7bda82a5aa60cc8b9b3fdfd5aedeb0ceae9625a24c720b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5037,"rank":5037,"depth":12,"x":1641.158,"y":650.894,"cluster":"schemes"},{"id":"stacks:01JP","tag":"01JP","title":"Fibre products of schemes · Definition 01JP","summary":"Given morphisms of schemes f : X → S and g : Y → S the fibre product is a scheme X ×_S Y together with projection morphisms p : X ×_S Y → X and q : X ×_S Y → Y sitting into the following commutative diagram xymatrix X ×_S Y ar[r]_q ar[d]_p & Y ar[d]^g X ar[r]^f & S which is universal among all diagrams of this sort, see Categories, Definition [Tag 001V].","statement_latex":"Given morphisms of schemes $f : X \\to S$ and $g : Y \\to S$\nthe {\\it fibre product} is a scheme $X \\times_S Y$ together\nwith projection morphisms $p : X \\times_S Y \\to X$\nand $q : X \\times_S Y \\to Y$ sitting into the following\ncommutative diagram\n$$\n\\xymatrix{\nX \\times_S Y \\ar[r]_q \\ar[d]_p & Y \\ar[d]^g \\\\\nX \\ar[r]^f & S\n}\n$$\nwhich is universal among all diagrams of this sort,\nsee Categories, Definition \\ref{categories-definition-fibre-products}.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Fibre products of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JP","source_file":"schemes.tex","source_line":3138,"source_end_line":3153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3138-L3153","statement_sha256":"13f3ba5fbed84a513e4d072c08f9bf044b2a9a3ec96af89d4954cce9d2d8fa0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5038,"rank":5038,"depth":1,"x":1558.131,"y":736.625,"cluster":"schemes"},{"id":"stacks:01JQ","tag":"01JQ","title":"Fibre products of schemes · Lemma 01JQ","summary":"Let f : X → S and g : Y → S be morphisms of schemes with the same target. If X, Y, S are all affine then X ×_S Y is affine.","statement_latex":"Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes\nwith the same target. If $X, Y, S$ are all affine then\n$X \\times_S Y$ is affine.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Fibre products of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JQ","source_file":"schemes.tex","source_line":3181,"source_end_line":3186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3181-L3186","statement_sha256":"e7d9d58750890c3279fcd7c0e6f69464f8c0afa768997210bdf0ed60fb9576ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":5039,"rank":5039,"depth":12,"x":1550.352,"y":625.399,"cluster":"schemes"},{"id":"stacks:01JR","tag":"01JR","title":"Fibre products of schemes · Lemma 01JR","summary":"Let f : X → S and g : Y → S be morphisms of schemes with the same target. Let X ×_S Y, p, q be the fibre product. Suppose that U ⊂ S, V ⊂ X, W ⊂ Y are open subschemes such that f(V) ⊂ U and g(W) ⊂ U. Then the canonical morphism V ×_U W → X ×_S Y is an open immersion which identifies V ×_U W with p^-1(V) ∩ q^-1(W).","statement_latex":"Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes\nwith the same target. Let $X \\times_S Y$, $p$, $q$ be the fibre product.\nSuppose that $U \\subset S$,\n$V \\subset X$, $W \\subset Y$ are open subschemes\nsuch that $f(V) \\subset U$ and $g(W) \\subset U$.\nThen the canonical morphism\n$V \\times_U W \\to X \\times_S Y$ is an open immersion\nwhich identifies $V \\times_U W$ with $p^{-1}(V) \\cap q^{-1}(W)$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Fibre products of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JR","source_file":"schemes.tex","source_line":3197,"source_end_line":3207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3197-L3207","statement_sha256":"80705571e7044c81be16c861e04ff360d5b680861c6483e99d850c979fcf955c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5040,"rank":5040,"depth":0,"x":1646.293,"y":703.625,"cluster":"schemes"},{"id":"stacks:01JS","tag":"01JS","title":"Fibre products of schemes · Lemma 01JS","summary":"Bare-hands construction of fiber products: an affine open cover of a fiber product of schemes can be assembled from compatible affine open covers of the pieces. Let f : X → S and g : Y → S be morphisms of schemes with the same target. Let S = ⋃ U_i be any affine open covering of S. For each i ∈ I, let f^-1(U_i) = ⋃_j ∈ J_i V_j be an affine open covering of f^-1(U_i) and let g^-1(U_i) = ⋃_k ∈ K_i W_k be an affine open covering of g^-1(U_i). Then X ×_S Y = ⋃_i ∈ I ⋃_j ∈…","statement_latex":"\\begin{slogan}\nBare-hands construction of fiber products: an affine open cover of a\nfiber product of schemes can be assembled from compatible\naffine open covers of the pieces.\n\\end{slogan}\nLet $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes\nwith the same target. Let $S = \\bigcup U_i$ be any affine open\ncovering of $S$. For each $i \\in I$, let\n$f^{-1}(U_i) = \\bigcup_{j \\in J_i} V_j$ be an affine open covering\nof $f^{-1}(U_i)$ and let\n$g^{-1}(U_i) = \\bigcup_{k \\in K_i} W_k$ be an affine open covering\nof $g^{-1}(U_i)$. Then\n$$\nX \\times_S Y =\n\\bigcup\\nolimits_{i \\in I}\n\\bigcup\\nolimits_{j \\in J_i, \\ k \\in K_i}\nV_j \\times_{U_i} W_k\n$$\nis an affine open covering of $X \\times_S Y$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Fibre products of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JS","source_file":"schemes.tex","source_line":3230,"source_end_line":3251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3230-L3251","statement_sha256":"455becb952e221a58f5fb048f9726477714c27cccfbae4338f4f892b67b1b7d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5041,"rank":5041,"depth":0,"x":1511.587,"y":700.354,"cluster":"schemes"},{"id":"stacks:01JT","tag":"01JT","title":"Fibre products of schemes · Lemma 01JT","summary":"Let f : X → S and g : Y → S be morphisms of schemes with the same target. Points z of X ×_S Y are in bijective correspondence to quadruples (x, y, s, p) where x ∈ X, y ∈ Y, s ∈ S are points with f(x) = s, g(y) = s and p is a prime ideal of the ring kappa(x) ⊗_kappa(s) kappa(y). The residue field of z corresponds to the residue field of the prime p.","statement_latex":"Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes\nwith the same target. Points $z$ of $X \\times_S Y$ are in bijective\ncorrespondence to quadruples\n$$\n(x, y, s, \\mathfrak p)\n$$\nwhere $x \\in X$, $y \\in Y$, $s \\in S$ are points with\n$f(x) = s$, $g(y) = s$ and $\\mathfrak p$ is a prime ideal\nof the ring $\\kappa(x) \\otimes_{\\kappa(s)} \\kappa(y)$.\nThe residue field of $z$ corresponds to\nthe residue field of the prime $\\mathfrak p$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Fibre products of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JT","source_file":"schemes.tex","source_line":3264,"source_end_line":3277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3264-L3277","statement_sha256":"b5a0f7e4134ba301e2c2bfc0ae844854527d99f5009ef542be91d0a588da200b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5042,"rank":5042,"depth":1,"x":1614.341,"y":625.758,"cluster":"schemes"},{"id":"stacks:01JU","tag":"01JU","title":"Fibre products of schemes · Lemma 01JU","summary":"Let f : X → S and g : Y → S be morphisms of schemes with the same target. • If f : X → S is a closed immersion, then X ×_S Y → Y is a closed immersion. Moreover, if X → S corresponds to the quasi-coherent sheaf of ideals I ⊂ O_S, then X ×_S Y → Y corresponds to the sheaf of ideals Im(g^*I → O_Y). • If f : X → S is an open immersion, then X ×_S Y → Y is an open immersion. • If f : X → S is an immersion, then X ×_S Y → Y is an immersion.","statement_latex":"Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes\nwith the same target.\n\\begin{enumerate}\n\\item If $f : X \\to S$ is a closed immersion,\nthen $X \\times_S Y \\to Y$ is a closed immersion.\nMoreover, if $X \\to S$ corresponds to the quasi-coherent\nsheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_S$, then\n$X \\times_S Y \\to Y$ corresponds to the sheaf of ideals\n$\\Im(g^*\\mathcal{I} \\to \\mathcal{O}_Y)$.\n\\item If $f : X \\to S$ is an open immersion,\nthen $X \\times_S Y \\to Y$ is an open immersion.\n\\item If $f : X \\to S$ is an immersion,\nthen $X \\times_S Y \\to Y$ is an immersion.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Fibre products of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JU","source_file":"schemes.tex","source_line":3329,"source_end_line":3345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3329-L3345","statement_sha256":"82068943637d6c20372bbef8f8c9a9ee4b59e8a3b9e68df7b6bcb69d3db0aa77","origin":"The Stacks Project","memory_eligible":false,"source_rank":5043,"rank":5043,"depth":12,"x":1598.436,"y":739.935,"cluster":"schemes"},{"id":"stacks:01JV","tag":"01JV","title":"Fibre products of schemes · Definition 01JV","summary":"Let f : X → Y be a morphism of schemes. Let Z ⊂ Y be a closed subscheme of Y. The inverse image f^-1(Z) of the closed subscheme Z is the closed subscheme Z ×_Y X of X. See Lemma [Tag 01JU] above.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $Z \\subset Y$ be a\nclosed subscheme of $Y$. The {\\it inverse image $f^{-1}(Z)$ of the\nclosed subscheme $Z$} is the closed subscheme $Z \\times_Y X$ of\n$X$. See Lemma \\ref{lemma-fibre-product-immersion} above.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Fibre products of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JV","source_file":"schemes.tex","source_line":3361,"source_end_line":3367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3361-L3367","statement_sha256":"cef30eecc657cd3b3d3d13b19aadb1b93375f979cfdc8bc40a4c0bba4855a444","origin":"The Stacks Project","memory_eligible":false,"source_rank":5044,"rank":5044,"depth":13,"x":1517.746,"y":646.014,"cluster":"schemes"},{"id":"stacks:01JX","tag":"01JX","title":"Base change in algebraic geometry · Definition 01JX","summary":"Let S be a scheme. • We say X is a scheme over S to mean that X comes equipped with a morphism of schemes X → S. The morphism X → S is sometimes called the structure morphism. • If R is a ring we say X is a scheme over R instead of X is a scheme over Spec(R). • A morphism f : X → Y of schemes over S is a morphism of schemes such that the composition X → Y → S of f with the structure morphism of Y is equal to the structure morphism of X. • We denote Mor_S(X, Y) the set of…","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item We say $X$ is a {\\it scheme over $S$} to mean that $X$\ncomes equipped with a morphism of schemes $X \\to S$.\nThe morphism $X \\to S$ is sometimes called the\n{\\it structure morphism}.\n\\item If $R$ is a ring we say\n$X$ is a {\\it scheme over $R$} instead of\n$X$ is a scheme over $\\Spec(R)$.\n\\item A {\\it morphism $f : X \\to Y$ of schemes over $S$}\nis a morphism of schemes such that the composition\n$X \\to Y \\to S$ of $f$ with the structure morphism of $Y$ is\nequal to the structure morphism of $X$.\n\\item We denote $\\Mor_S(X, Y)$ the set of all morphisms\nfrom $X$ to $Y$ over $S$.\n\\item Let $X$ be a scheme over $S$. Let $S' \\to S$ be a\nmorphism of schemes. The {\\it base change} of $X$\nis the scheme $X_{S'} = S' \\times_S X$ over $S'$.\n\\item Let $f : X \\to Y$ be a morphism of schemes over $S$. Let $S' \\to S$\nbe a morphism of schemes. The {\\it base change} of $f$ is\nthe induced morphism $f' : X_{S'} \\to Y_{S'}$ (namely the\nmorphism $\\text{id}_{S'} \\times_{\\text{id}_S} f$).\n\\item Let $R$ be a ring. Let $X$ be a scheme over $R$.\nLet $R \\to R'$ be a ring map. The {\\it base change} $X_{R'}$\nis the scheme $\\Spec(R') \\times_{\\Spec(R)} X$\nover $R'$.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Base change in algebraic geometry","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JX","source_file":"schemes.tex","source_line":3410,"source_end_line":3439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3410-L3439","statement_sha256":"c27d2db2eb8935260cd3f2fd2f9c1fa0bbf2184df0ff1d522f3bd3842dbd1431","origin":"The Stacks Project","memory_eligible":false,"source_rank":5045,"rank":5045,"depth":0,"x":1653.774,"y":669.661,"cluster":"schemes"},{"id":"stacks:01JY","tag":"01JY","title":"Base change in algebraic geometry · Lemma 01JY","summary":"Let S be a scheme. Let f : X → Y be an immersion (resp. closed immersion, resp. open immersion) of schemes over S. Then any base change of f is an immersion (resp. closed immersion, resp. open immersion).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an\nimmersion (resp.\\ closed immersion, resp. open immersion)\nof schemes over $S$. Then any base change of $f$ is an\nimmersion (resp.\\ closed immersion, resp. open immersion).","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Base change in algebraic geometry","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JY","source_file":"schemes.tex","source_line":3444,"source_end_line":3450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3444-L3450","statement_sha256":"5761896fcc4397864b93b7b2f5fa3adb048dd1eb0eb978f34ab587a319cf167d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5046,"rank":5046,"depth":13,"x":1533.58,"y":729.844,"cluster":"schemes"},{"id":"stacks:01JZ","tag":"01JZ","title":"Base change in algebraic geometry · Definition 01JZ","summary":"Properties and base change. • Let P be a property of schemes over a base. We say that P is preserved under arbitrary base change, or simply that P is preserved under base change if whenever X/S has P, any base change X_S'/S' has P. • Let P be a property of morphisms of schemes over a base. We say that P is preserved under arbitrary base change, or simply that preserved under base change if whenever f : X → Y over S has P, any base change f' : X_S' → Y_S' over S' has P.","statement_latex":"Properties and base change.\n\\begin{enumerate}\n\\item Let $\\mathcal{P}$ be a property of schemes over a base.\nWe say that $\\mathcal{P}$ is {\\it preserved under arbitrary base change},\nor simply that $\\mathcal{P}$ is {\\it preserved under base change}\nif whenever $X/S$\nhas $\\mathcal{P}$, any base change $X_{S'}/S'$ has $\\mathcal{P}$.\n\\item Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.\nWe say that $\\mathcal{P}$ is {\\it preserved under arbitrary base change},\nor simply that {\\it preserved under base change} if whenever\n$f : X \\to Y$ over $S$ has $\\mathcal{P}$, any base change\n$f' : X_{S'} \\to Y_{S'}$ over $S'$ has $\\mathcal{P}$.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Base change in algebraic geometry","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01JZ","source_file":"schemes.tex","source_line":3471,"source_end_line":3486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3471-L3486","statement_sha256":"e466a52bc3435387676cc305d7adb9e1764a7e169b91b3ccc621b329d6ab19d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5047,"rank":5047,"depth":0,"x":1574.104,"y":616.455,"cluster":"schemes"},{"id":"stacks:01K0","tag":"01K0","title":"Base change in algebraic geometry · Definition 01K0","summary":"Let f : X → S be a morphism of schemes. Let s ∈ S be a point. The scheme theoretic fibre X_s of f over s, or simply the fibre of f over s, is the scheme fitting in the following fibre product diagram xymatrix X_s = Spec(kappa(s)) ×_S X ar[r] ar[d] & X ar[d] Spec(kappa(s)) ar[r] & S We think of the fibre X_s always as a scheme over kappa(s).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $s \\in S$ be a point.\nThe {\\it scheme theoretic fibre $X_s$ of $f$ over $s$},\nor simply the {\\it fibre of $f$ over $s$},\nis the scheme fitting in the following fibre product diagram\n$$\n\\xymatrix{\nX_s = \\Spec(\\kappa(s)) \\times_S X \\ar[r] \\ar[d] &\nX \\ar[d] \\\\\n\\Spec(\\kappa(s)) \\ar[r] &\nS\n}\n$$\nWe think of the fibre $X_s$ always as a scheme over $\\kappa(s)$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Base change in algebraic geometry","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01K0","source_file":"schemes.tex","source_line":3492,"source_end_line":3508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3492-L3508","statement_sha256":"fc7a734f9fb1169e7a4b0a252c579c999401604ada75f256599db5decfcf526c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5048,"rank":5048,"depth":0,"x":1635.839,"y":723.818,"cluster":"schemes"},{"id":"stacks:01K1","tag":"01K1","title":"Base change in algebraic geometry · Lemma 01K1","summary":"Let f : X → S be a morphism of schemes. Consider the diagrams vcenter xymatrix X_s ar[r] ar[d] & X ar[d] Spec(kappa(s)) ar[r] & S and vcenter xymatrix Spec(O_S, s) ×_S X ar[r] ar[d] & X ar[d] Spec(O_S, s) ar[r] & S In both cases these diagrams induce fibre squares of topological spaces and in particular, the top horizontal arrow is a homeomorphism onto its image.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nConsider the diagrams\n$$\n\\vcenter{\n\\xymatrix{\nX_s \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(\\kappa(s)) \\ar[r] & S\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\n\\Spec(\\mathcal{O}_{S, s}) \\times_S X \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(\\mathcal{O}_{S, s}) \\ar[r] & S\n}\n}\n$$\nIn both cases these diagrams induce fibre squares of topological\nspaces and in particular, the top horizontal arrow is a homeomorphism\nonto its image.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Base change in algebraic geometry","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01K1","source_file":"schemes.tex","source_line":3510,"source_end_line":3532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3510-L3532","statement_sha256":"c22e39cc6ce50973f75a7cc19a8adc1f88c06bf95b8610e83bcc71e1380bffff","origin":"The Stacks Project","memory_eligible":false,"source_rank":5049,"rank":5049,"depth":12,"x":1503.057,"y":679.369,"cluster":"schemes"},{"id":"stacks:0HA1","tag":"0HA1","title":"Base change in algebraic geometry · Lemma 0HA1","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X with image s ∈ S. Thinking of x as a point of X_s (see Lemma [Tag 01K1]) we have O_X_s, x ≅ O_X, x/ m_sO_X, x ≅ O_X, x ⊗_O_S, s kappa(s)","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$\nwith image $s \\in S$. Thinking of $x$ as a point of $X_s$\n(see Lemma \\ref{lemma-fibre-topological}) we have\n$$\n\\mathcal{O}_{X_s, x} \\cong\n\\mathcal{O}_{X, x}/\\mathfrak m_s\\mathcal{O}_{X, x} \\cong\n\\mathcal{O}_{X, x} \\otimes_{\\mathcal{O}_{S, s}} \\kappa(s)\n$$","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Base change in algebraic geometry","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HA1","source_file":"schemes.tex","source_line":3544,"source_end_line":3554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3544-L3554","statement_sha256":"88c3b2e76b61dbc4dbf86d2cadd2292b040e10d9c8897005dc174e7a96efce93","origin":"The Stacks Project","memory_eligible":false,"source_rank":5050,"rank":5050,"depth":13,"x":1637.636,"y":636.51,"cluster":"schemes"},{"id":"stacks:01K3","tag":"01K3","title":"Quasi-compact morphisms · Definition 01K3","summary":"A morphism of schemes is called quasi-compact if the underlying map of topological spaces is quasi-compact, see Topology, Definition [Tag 005A].","statement_latex":"A morphism of schemes is called {\\it quasi-compact}\nif the underlying map of topological spaces is\nquasi-compact, see\nTopology, Definition \\ref{topology-definition-quasi-compact}.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-compact morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01K3","source_file":"schemes.tex","source_line":3575,"source_end_line":3581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3575-L3581","statement_sha256":"1003d8a1832bb55b97028f075bd83d7e83acdec48681f6487d67fee44feecfa9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5051,"rank":5051,"depth":1,"x":1572.423,"y":745.211,"cluster":"schemes"},{"id":"stacks:01K4","tag":"01K4","title":"Quasi-compact morphisms · Lemma 01K4","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • f : X → S is quasi-compact, • the inverse image of every affine open is quasi-compact, and • there exists some affine open covering S = ⋃_i ∈ I U_i such that f^-1(U_i) is quasi-compact for all i.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f : X \\to S$ is quasi-compact,\n\\item the inverse image of every affine open is quasi-compact, and\n\\item there exists some affine open covering $S = \\bigcup_{i \\in I} U_i$\nsuch that $f^{-1}(U_i)$ is quasi-compact for all $i$.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01K4","source_file":"schemes.tex","source_line":3583,"source_end_line":3593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3583-L3593","statement_sha256":"eba6cb3c7e54c004a7c01a0cb4f8cbbcc9e99de4d1d2195ce4f5521d2a4fdb1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5052,"rank":5052,"depth":1,"x":1532.834,"y":627.266,"cluster":"schemes"},{"id":"stacks:01K5","tag":"01K5","title":"Quasi-compact morphisms · Lemma 01K5","summary":"Being quasi-compact is a property of morphisms of schemes over a base which is preserved under arbitrary base change.","statement_latex":"Being quasi-compact is a property of morphisms of schemes\nover a base which is preserved under arbitrary base change.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01K5","source_file":"schemes.tex","source_line":3625,"source_end_line":3629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3625-L3629","statement_sha256":"93aa775ee84f8919f30c62019d53c86dc141d7c7d1fcd14162b533e561847b8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5053,"rank":5053,"depth":0,"x":1657.695,"y":692.202,"cluster":"schemes"},{"id":"stacks:01K6","tag":"01K6","title":"Quasi-compact morphisms · Lemma 01K6","summary":"The composition of quasi-compact morphisms is quasi-compact.","statement_latex":"The composition of quasi-compact morphisms is quasi-compact.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01K6","source_file":"schemes.tex","source_line":3635,"source_end_line":3638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3635-L3638","statement_sha256":"9bda95e61d5076336b0d6592140e4e0e0965f6802b36613c4e6a90cb358707ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":5054,"rank":5054,"depth":1,"x":1512.461,"y":715.316,"cluster":"schemes"},{"id":"stacks:01K7","tag":"01K7","title":"Quasi-compact morphisms · Lemma 01K7","summary":"A closed immersion is quasi-compact.","statement_latex":"A closed immersion is quasi-compact.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01K7","source_file":"schemes.tex","source_line":3645,"source_end_line":3648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3645-L3648","statement_sha256":"42c5426696312a081f66bc8d8d7b3dbc048b097d1ef4f6c16d667750bae10c56","origin":"The Stacks Project","memory_eligible":false,"source_rank":5055,"rank":5055,"depth":1,"x":1601.539,"y":615.222,"cluster":"schemes"},{"id":"stacks:05JL","tag":"05JL","title":"Quasi-compact morphisms · Lemma 05JL","summary":"Let f : X → S be a quasi-compact morphism of schemes. The following are equivalent • f(X) ⊂ S is closed, and • f(X) ⊂ S is stable under specialization.","statement_latex":"Let $f : X \\to S$ be a quasi-compact morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f(X) \\subset S$ is closed, and\n\\item $f(X) \\subset S$ is stable under specialization.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JL","source_file":"schemes.tex","source_line":3665,"source_end_line":3673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3665-L3673","statement_sha256":"54e6b91b61174fbf9124f4a5098339168fbba888b6c5314d2c845f2465f9c8e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5056,"rank":5056,"depth":3,"x":1616.439,"y":740.367,"cluster":"schemes"},{"id":"stacks:01K9","tag":"01K9","title":"Quasi-compact morphisms · Lemma 01K9","summary":"Let f : X → S be a quasi-compact morphism of schemes. Then f is closed if and only if specializations lift along f, see Topology, Definition [Tag 0063].","statement_latex":"Let $f : X \\to S$ be a quasi-compact morphism of schemes.\nThen $f$ is closed if and only if specializations lift\nalong $f$, see\nTopology, Definition \\ref{topology-definition-lift-specializations}.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01K9","source_file":"schemes.tex","source_line":3691,"source_end_line":3697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3691-L3697","statement_sha256":"23c27591b26f3ff4ba9dd00e271dc5873903d2a7902a98f32ac4f8e0fcbe86b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5057,"rank":5057,"depth":5,"x":1504.114,"y":656.013,"cluster":"schemes"},{"id":"stacks:01KB","tag":"01KB","title":"Valuative criterion for universal closedness · Definition 01KB","summary":"A morphism of schemes f : X → S is said to be universally closed if every base change f' : X_S' → S' is closed.","statement_latex":"A morphism of schemes $f : X \\to S$ is said to be\n{\\it universally closed} if every base change\n$f' : X_{S'} \\to S'$ is closed.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Valuative criterion for universal closedness","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KB","source_file":"schemes.tex","source_line":3737,"source_end_line":3742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3737-L3742","statement_sha256":"ebe678ce5efbff00105b534b528070bae53302504164ba2718141c16c44f009a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5058,"rank":5058,"depth":0,"x":1655.711,"y":654.471,"cluster":"schemes"},{"id":"stacks:01KC","tag":"01KC","title":"Valuative criterion for universal closedness · Lemma 01KC","summary":"Let f : X → S be a morphism of schemes. • If f is universally closed then specializations lift along any base change of f, see Topology, Definition [Tag 0063]. • If f is quasi-compact and specializations lift along any base change of f, then f is universally closed.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item If $f$ is universally closed then specializations lift\nalong any base change of $f$, see\nTopology, Definition \\ref{topology-definition-lift-specializations}.\n\\item If $f$ is quasi-compact and specializations lift\nalong any base change of $f$, then $f$ is universally closed.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KC","source_file":"schemes.tex","source_line":3757,"source_end_line":3767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3757-L3767","statement_sha256":"5d7e5b573c5760a994cb824ff32862fb8a325947aa4060a84406ba45e34f0a58","origin":"The Stacks Project","memory_eligible":false,"source_rank":5059,"rank":5059,"depth":6,"x":1544.486,"y":742.161,"cluster":"schemes"},{"id":"stacks:01KD","tag":"01KD","title":"Valuative criterion for universal closedness · Definition 01KD","summary":"Let f : X → S be a morphism of schemes. We say f satisfies the existence part of the valuative criterion if given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & S where A is a valuation ring with field of fractions K, the dotted arrow exists. We say f satisfies the uniqueness part of the valuative criterion if there is at most one dotted arrow given any diagram as above (without requiring existence of course).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. We say $f$\n{\\it satisfies the existence part of the valuative criterion}\nif given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & S\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$, the\ndotted arrow exists. We say $f$ {\\it satisfies the uniqueness\npart of the valuative criterion} if there is at most one\ndotted arrow given any diagram as above (without requiring\nexistence of course).","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Valuative criterion for universal closedness","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KD","source_file":"schemes.tex","source_line":3777,"source_end_line":3793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3777-L3793","statement_sha256":"e222804e12eba37ec359a1099ae31b4b18ea6e31bd288fb6a24d98cb1a03dc71","origin":"The Stacks Project","memory_eligible":false,"source_rank":5060,"rank":5060,"depth":0,"x":1556.056,"y":613.614,"cluster":"schemes"},{"id":"stacks:01J8","tag":"01J8","title":"Valuative criterion for universal closedness · Lemma 01J8","summary":"Specializations are witnessed by valuation rings. Let S be a scheme. Let s' leadsto s be a specialization of points of S. Then • there exists a valuation ring A and a morphism f : Spec(A) → S such that the generic point eta of Spec(A) maps to s' and the special point maps to s, and • given a field extension K/kappa(s') we may arrange it so that the extension kappa(eta)/kappa(s') induced by f is isomorphic to the given extension.","statement_latex":"\\begin{slogan}\nSpecializations are witnessed by valuation rings.\n\\end{slogan}\nLet $S$ be a scheme. Let $s' \\leadsto s$ be a specialization of points of $S$.\nThen\n\\begin{enumerate}\n\\item there exists a valuation ring $A$ and a morphism\n$f : \\Spec(A) \\to S$ such that the generic point $\\eta$ of\n$\\Spec(A)$ maps to $s'$ and the special point maps to $s$, and\n\\item given a field extension $K/\\kappa(s')$\nwe may arrange it so that the extension\n$\\kappa(\\eta)/\\kappa(s')$ induced by $f$\nis isomorphic to the given extension.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01J8","source_file":"schemes.tex","source_line":3807,"source_end_line":3823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3807-L3823","statement_sha256":"7c353a78e46b6a06e23c26204d578a5c8e46bf688a5d1772d76f772b568e5dbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":5061,"rank":5061,"depth":12,"x":1651.464,"y":715.576,"cluster":"schemes"},{"id":"stacks:01KE","tag":"01KE","title":"Valuative criterion for universal closedness · Lemma 01KE","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • Specializations lift along any base change of f • The morphism f satisfies the existence part of the valuative criterion.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item Specializations lift along any base change of $f$\n\\item The morphism $f$ satisfies the existence part of the\nvaluative criterion.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KE","source_file":"schemes.tex","source_line":3841,"source_end_line":3850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3841-L3850","statement_sha256":"b7355e73b866e945a206ef09873953959f7fe6e0a5e680d6847cde42e2683ff1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5062,"rank":5062,"depth":13,"x":1498.216,"y":694.402,"cluster":"schemes"},{"id":"stacks:01KF","tag":"01KF","title":"Valuative criterion of universal closedness · Proposition 01KF","summary":"Let f be a quasi-compact morphism of schemes. Then f is universally closed if and only if f satisfies the existence part of the valuative criterion.","statement_latex":"Let $f$ be a quasi-compact morphism of schemes.\nThen $f$ is universally closed if and only if $f$\nsatisfies the existence part of the valuative criterion.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Valuative criterion for universal closedness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KF","source_file":"schemes.tex","source_line":3911,"source_end_line":3916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L3911-L3916","statement_sha256":"17796d31604c41c344c5680130e2f23665fdbb054c98f2f2625f9843fb4007de","origin":"The Stacks Project","memory_eligible":false,"source_rank":5063,"rank":5063,"depth":14,"x":1629.007,"y":622.643,"cluster":"schemes"},{"id":"stacks:01KI","tag":"01KI","title":"Separation axioms · Lemma 01KI","summary":"The diagonal morphism of a morphism between affines is closed.","statement_latex":"The diagonal morphism of a morphism between affines is closed.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KI","source_file":"schemes.tex","source_line":4021,"source_end_line":4024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4021-L4024","statement_sha256":"9a045e8fb49fb9c6fd0543e9605c2f7505313fe3d35add834e6f10ebbebf0250","origin":"The Stacks Project","memory_eligible":false,"source_rank":5064,"rank":5064,"depth":0,"x":1590.045,"y":750.505,"cluster":"schemes"},{"id":"stacks:01KJ","tag":"01KJ","title":"Separation axioms · Lemma 01KJ","summary":"The diagonal morphism for relative schemes is an immersion. Let X be a scheme over S. The diagonal morphism Δ_X/S is an immersion.","statement_latex":"\\begin{slogan}\nThe diagonal morphism for relative schemes is an immersion.\n\\end{slogan}\nLet $X$ be a scheme over $S$.\nThe diagonal morphism $\\Delta_{X/S}$ is an immersion.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KJ","source_file":"schemes.tex","source_line":4037,"source_end_line":4044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4037-L4044","statement_sha256":"9576e930f9d724847f7d3d8f584ad0227408459ab3fcda18ecdfe146ac700ad1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5065,"rank":5065,"depth":13,"x":1515.531,"y":633.448,"cluster":"schemes"},{"id":"stacks:01KK","tag":"01KK","title":"Separation axioms · Definition 01KK","summary":"Let f : X → S be a morphism of schemes. • We say f is separated if the diagonal morphism Δ_X/S is a closed immersion. • We say f is quasi-separated if the diagonal morphism Δ_X/S is a quasi-compact morphism. • We say a scheme Y is separated if the morphism Y → Spec(Z) is separated. • We say a scheme Y is quasi-separated if the morphism Y → Spec(Z) is quasi-separated.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say $f$ is {\\it separated} if the diagonal morphism $\\Delta_{X/S}$\nis a closed immersion.\n\\item We say $f$ is {\\it quasi-separated} if the diagonal morphism\n$\\Delta_{X/S}$ is a quasi-compact morphism.\n\\item We say a scheme $Y$ is {\\it separated} if the morphism\n$Y \\to \\Spec(\\mathbf{Z})$ is separated.\n\\item We say a scheme $Y$ is {\\it quasi-separated} if the morphism\n$Y \\to \\Spec(\\mathbf{Z})$ is quasi-separated.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KK","source_file":"schemes.tex","source_line":4061,"source_end_line":4074,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4061-L4074","statement_sha256":"0261245fa68901987341ac0636a1124d0466e32c9dc58dd02f0cad449cee0de0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5066,"rank":5066,"depth":0,"x":1665.451,"y":677.733,"cluster":"schemes"},{"id":"stacks:01KM","tag":"01KM","title":"Separation axioms · Lemma 01KM","summary":"Let X, Y be schemes over S. Let a, b : X → Y be morphisms of schemes over S. There exists a largest locally closed subscheme Z ⊂ X such that a|_Z = b|_Z. In fact Z is the equalizer of (a, b). Moreover, if Y is separated over S, then Z is a closed subscheme.","statement_latex":"Let $X$, $Y$ be schemes over $S$.\nLet $a, b : X \\to Y$ be morphisms of schemes over $S$.\nThere exists a largest locally closed subscheme\n$Z \\subset X$ such that $a|_Z = b|_Z$. In fact $Z$ is\nthe equalizer of $(a, b)$. Moreover, if $Y$ is separated\nover $S$, then $Z$ is a closed subscheme.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KM","source_file":"schemes.tex","source_line":4099,"source_end_line":4107,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4099-L4107","statement_sha256":"7d112e10f905ba866291b3b3d9aa583711652b9147a72829c304dc4d9997f65e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5067,"rank":5067,"depth":14,"x":1518.473,"y":730.437,"cluster":"schemes"},{"id":"stacks:01KO","tag":"01KO","title":"Separation axioms · Lemma 01KO","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is quasi-separated. • For every pair of affine opens U, V ⊂ X which map into a common affine open of S the intersection U ∩ V is a finite union of affine opens of X. • There exists an affine open covering S = ⋃_i ∈ I U_i and for each i an affine open covering f^-1U_i = ⋃_j ∈ I_i V_j such that for each i and each pair j, j' ∈ I_i the intersection V_j ∩ V_j' is a finite union of affine…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is quasi-separated.\n\\item For every pair of affine opens $U, V \\subset X$\nwhich map into a common affine open of $S$ the intersection\n$U \\cap V$ is a finite union of affine opens of $X$.\n\\item There exists an affine open covering $S = \\bigcup_{i \\in I} U_i$\nand for each $i$ an affine open covering $f^{-1}U_i = \\bigcup_{j \\in I_i} V_j$\nsuch that for each $i$ and each pair $j, j' \\in I_i$ the\nintersection $V_j \\cap V_{j'}$ is a finite union of affine\nopens of $X$.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KO","source_file":"schemes.tex","source_line":4124,"source_end_line":4139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4124-L4139","statement_sha256":"c52dd15cdbf79f03850a6aa61f19c6fbfaeb3608d599bb580f101b06083db30c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5068,"rank":5068,"depth":2,"x":1584.835,"y":607.501,"cluster":"schemes"},{"id":"stacks:01KP","tag":"01KP","title":"Separation axioms · Lemma 01KP","summary":"Let f : X → S be a morphism of schemes. • If f is separated then for every pair of affine opens (U, V) of X which map into a common affine open of S we have • the intersection U ∩ V is affine. • the ring map O_X(U) ⊗_Z O_X(V) → O_X(U ∩ V) is surjective. • If any pair of points x_1, x_2 ∈ X lying over a common point s ∈ S are contained in affine opens x_1 ∈ U, x_2 ∈ V which map into a common affine open of S such that (a), (b) hold, then f is separated.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item If $f$ is separated then for every pair of affine\nopens $(U, V)$ of $X$ which map into a\ncommon affine open of $S$ we have\n\\begin{enumerate}\n\\item the intersection $U \\cap V$ is affine.\n\\item the ring map\n$\\mathcal{O}_X(U) \\otimes_{\\mathbf{Z}} \\mathcal{O}_X(V)\n\\to \\mathcal{O}_X(U \\cap V)$\nis surjective.\n\\end{enumerate}\n\\item If any pair of points $x_1, x_2 \\in X$ lying over a common\npoint $s \\in S$ are contained in affine opens $x_1 \\in U$,\n$x_2 \\in V$ which map into a common affine open of $S$ such\nthat (a), (b) hold, then $f$ is separated.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KP","source_file":"schemes.tex","source_line":4156,"source_end_line":4175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4156-L4175","statement_sha256":"9c2a3505e5273966e77406cd1bae8bd6624fe2e4656c72471afbf072875ac844","origin":"The Stacks Project","memory_eligible":false,"source_rank":5069,"rank":5069,"depth":12,"x":1635.033,"y":736.508,"cluster":"schemes"},{"id":"stacks:01KR","tag":"01KR","title":"Separation axioms · Lemma 01KR","summary":"Let f : X → T and g : Y → T be morphisms of schemes with the same target. Let h : T → S be a morphism of schemes. Then the induced morphism i : X ×_T Y → X ×_S Y is an immersion. If T → S is separated, then i is a closed immersion. If T → S is quasi-separated, then i is a quasi-compact morphism.","statement_latex":"Let $f : X \\to T$ and $g : Y \\to T$ be morphisms of schemes\nwith the same target. Let $h : T \\to S$ be a morphism of schemes.\nThen the induced morphism $i : X \\times_T Y \\to X \\times_S Y$ is\nan immersion. If $T \\to S$ is separated, then $i$ is a closed\nimmersion. If $T \\to S$ is quasi-separated, then $i$ is a\nquasi-compact morphism.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KR","source_file":"schemes.tex","source_line":4237,"source_end_line":4245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4237-L4245","statement_sha256":"725dff5de5519ec9f95b5ff2127899c535d1851c2c94a5378e121cbe975ec298","origin":"The Stacks Project","memory_eligible":false,"source_rank":5070,"rank":5070,"depth":14,"x":1493.515,"y":669.503,"cluster":"schemes"},{"id":"stacks:01KS","tag":"01KS","title":"Separation axioms · Lemma 01KS","summary":"Let g : X → Y be a morphism of schemes over S. The morphism i : X → X ×_S Y is an immersion. If Y is separated over S it is a closed immersion. If Y is quasi-separated over S it is quasi-compact.","statement_latex":"Let $g : X \\to Y$ be a morphism of schemes over $S$.\nThe morphism $i : X \\to X \\times_S Y$ is an immersion.\nIf $Y$ is separated over $S$ it is a closed immersion.\nIf $Y$ is quasi-separated over $S$ it is quasi-compact.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KS","source_file":"schemes.tex","source_line":4261,"source_end_line":4267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4261-L4267","statement_sha256":"db9d250b6aebab8fe3f1ada504ffd4228e3560707c7f0bc3f124674d436c2da6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5071,"rank":5071,"depth":15,"x":1652.598,"y":638.447,"cluster":"schemes"},{"id":"stacks:01KT","tag":"01KT","title":"Separation axioms · Lemma 01KT","summary":"Let f : X → S be a morphism of schemes. Let s : S → X be a section of f (in a formula f ∘ s = id_S). Then s is an immersion. If f is separated then s is a closed immersion. If f is quasi-separated, then s is quasi-compact.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $s : S \\to X$ be a section of $f$ (in a formula $f \\circ s = \\text{id}_S$).\nThen $s$ is an immersion.\nIf $f$ is separated then $s$ is a closed immersion.\nIf $f$ is quasi-separated, then $s$ is quasi-compact.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KT","source_file":"schemes.tex","source_line":4274,"source_end_line":4281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4274-L4281","statement_sha256":"cb38aafbadb458b9a1585d15f20f52c5ede005a2b6a36f8d37620988553c7ab0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5072,"rank":5072,"depth":16,"x":1559.777,"y":752.212,"cluster":"schemes"},{"id":"stacks:01KU","tag":"01KU","title":"Separation axioms · Lemma 01KU","summary":"Permanence properties. • A composition of separated morphisms is separated. • A composition of quasi-separated morphisms is quasi-separated. • The base change of a separated morphism is separated. • The base change of a quasi-separated morphism is quasi-separated. • A (fibre) product of separated morphisms is separated. • A (fibre) product of quasi-separated morphisms is quasi-separated.","statement_latex":"Permanence properties.\n\\begin{enumerate}\n\\item A composition of separated morphisms is separated.\n\\item A composition of quasi-separated morphisms is quasi-separated.\n\\item The base change of a separated morphism is separated.\n\\item The base change of a quasi-separated morphism is quasi-separated.\n\\item A (fibre) product of separated morphisms is separated.\n\\item A (fibre) product of quasi-separated morphisms is quasi-separated.\n\\end{enumerate}","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KU","source_file":"schemes.tex","source_line":4288,"source_end_line":4299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4288-L4299","statement_sha256":"21f6a67d7ed1e6eb39cc7b1878846648483f203a2feb7a3d29d885d9a3692013","origin":"The Stacks Project","memory_eligible":false,"source_rank":5073,"rank":5073,"depth":15,"x":1536.617,"y":614.941,"cluster":"schemes"},{"id":"stacks:01KV","tag":"01KV","title":"Separation axioms · Lemma 01KV","summary":"Separated and quasi-separated morphisms satisfy cancellation. Let f : X → Y and g : Y → Z be morphisms of schemes. If g ∘ f is separated then so is f. If g ∘ f is quasi-separated then so is f.","statement_latex":"\\begin{slogan}\nSeparated and quasi-separated morphisms satisfy cancellation.\n\\end{slogan}\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes.\nIf $g \\circ f$ is separated then so is $f$.\nIf $g \\circ f$ is quasi-separated then so is $f$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KV","source_file":"schemes.tex","source_line":4336,"source_end_line":4344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4336-L4344","statement_sha256":"413507a8d1192b5e5d6e88b2c854af15d01eadf566bce931536c0efb36e2cbe0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5074,"rank":5074,"depth":15,"x":1664.743,"y":703.477,"cluster":"schemes"},{"id":"stacks:03GI","tag":"03GI","title":"Separation axioms · Lemma 03GI","summary":"Let f : X → Y and g : Y → Z be morphisms of schemes. If g ∘ f is quasi-compact and g is quasi-separated then f is quasi-compact.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes.\nIf $g \\circ f$ is quasi-compact and $g$ is quasi-separated\nthen $f$ is quasi-compact.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GI","source_file":"schemes.tex","source_line":4366,"source_end_line":4371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4366-L4371","statement_sha256":"904a37f9087a01e0d712e105fec4cc5aebe4f879870e45669dc00d430dc05121","origin":"The Stacks Project","memory_eligible":false,"source_rank":5075,"rank":5075,"depth":17,"x":1498.218,"y":710.93,"cluster":"schemes"},{"id":"stacks:01KN","tag":"01KN","title":"Separation axioms · Lemma 01KN","summary":"An affine scheme is separated. A morphism from an affine scheme to another scheme is separated.","statement_latex":"An affine scheme is separated. A morphism from an affine scheme\nto another scheme is separated.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KN","source_file":"schemes.tex","source_line":4386,"source_end_line":4390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4386-L4390","statement_sha256":"eb83a01c0640dda7503e6109d16dfe77c97425685bc499db109598efb62f934c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5076,"rank":5076,"depth":16,"x":1615.61,"y":610.438,"cluster":"schemes"},{"id":"stacks:01KW","tag":"01KW","title":"Separation axioms · Lemma 01KW","summary":"Let f : X → S be a morphism. Assume f is separated and S is a separated scheme. Suppose U ⊂ X and V ⊂ X are affine open. Then U ∩ V is affine (and a closed subscheme of U × V).","statement_latex":"Let $f : X \\to S$ be a morphism.\nAssume $f$ is separated and $S$ is a separated scheme.\nSuppose $U \\subset X$ and $V \\subset X$ are affine open.\nThen $U \\cap V$ is affine (and a closed subscheme of $U \\times V$).","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KW","source_file":"schemes.tex","source_line":4408,"source_end_line":4414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4408-L4414","statement_sha256":"1fc2e68edb2bc3aaca3445f64da85e1b0a56c9066b7c29578edc9486a7953221","origin":"The Stacks Project","memory_eligible":false,"source_rank":5077,"rank":5077,"depth":16,"x":1609.829,"y":751.857,"cluster":"schemes"},{"id":"stacks:01KZ","tag":"01KZ","title":"Valuative criterion of separatedness · Lemma 01KZ","summary":"Let f : X → S be a morphism of schemes. If f is separated, then f satisfies the uniqueness part of the valuative criterion.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf $f$ is separated, then $f$ satisfies the uniqueness\npart of the valuative criterion.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Valuative criterion of separatedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01KZ","source_file":"schemes.tex","source_line":4480,"source_end_line":4485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4480-L4485","statement_sha256":"a8a1575773d335626fc71dfc16f2fe041018d80cba0ce84a6601a2ce7009419b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5078,"rank":5078,"depth":15,"x":1499.825,"y":643.761,"cluster":"schemes"},{"id":"stacks:01L0","tag":"01L0","title":"Valuative criterion separatedness · Lemma 01L0","summary":"[EGA] Let f : X → S be a morphism. Assume • the morphism f is quasi-separated, and • the morphism f satisfies the uniqueness part of the valuative criterion. Then f is separated.","statement_latex":"\\begin{reference}\n\\cite[II Proposition 7.2.3]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism.\nAssume\n\\begin{enumerate}\n\\item the morphism $f$ is quasi-separated, and\n\\item the morphism $f$ satisfies the uniqueness\npart of the valuative criterion.\n\\end{enumerate}\nThen $f$ is separated.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Valuative criterion of separatedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01L0","source_file":"schemes.tex","source_line":4498,"source_end_line":4511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4498-L4511","statement_sha256":"04b6f9100a48651cbf6e516c5d3eae24bf7fec6ca923f4fa0f966fc7e226234f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5079,"rank":5079,"depth":15,"x":1668.694,"y":661.137,"cluster":"schemes"},{"id":"stacks:01L2","tag":"01L2","title":"Monomorphisms · Definition 01L2","summary":"A morphism of schemes is called a monomorphism if it is a monomorphism in the category of schemes, see Categories, Definition [Tag 003B].","statement_latex":"A morphism of schemes is called a {\\it monomorphism} if it is\na monomorphism in the category of schemes, see\nCategories, Definition \\ref{categories-definition-mono-epi}.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01L2","source_file":"schemes.tex","source_line":4540,"source_end_line":4545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4540-L4545","statement_sha256":"1fd1d922610032b43a061e4f5a20050c3b265c273b5f23d58298caf359200a5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5080,"rank":5080,"depth":1,"x":1529.525,"y":744.548,"cluster":"schemes"},{"id":"stacks:01L3","tag":"01L3","title":"Monomorphisms · Lemma 01L3","summary":"A scheme morphism is a monomorphism iff its diagonal is an isomorphism. Let j : X → Y be a morphism of schemes. Then j is a monomorphism if and only if the diagonal morphism Δ_X/Y : X → X ×_Y X is an isomorphism.","statement_latex":"\\begin{slogan}\nA scheme morphism is a monomorphism iff its diagonal is an isomorphism.\n\\end{slogan}\nLet $j : X \\to Y$ be a morphism of schemes.\nThen $j$ is a monomorphism if and only if the\ndiagonal morphism $\\Delta_{X/Y} : X \\to X \\times_Y X$ is\nan isomorphism.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01L3","source_file":"schemes.tex","source_line":4547,"source_end_line":4556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4547-L4556","statement_sha256":"3876748f1ce3659896ffe3f63b6470ed4cafc2cea1c9917de45723845ca0788a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5081,"rank":5081,"depth":0,"x":1565.242,"y":603.397,"cluster":"schemes"},{"id":"stacks:01L4","tag":"01L4","title":"Monomorphisms · Lemma 01L4","summary":"A monomorphism of schemes is separated.","statement_latex":"A monomorphism of schemes is separated.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01L4","source_file":"schemes.tex","source_line":4562,"source_end_line":4565,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4562-L4565","statement_sha256":"430f4e09873dfaa65a8db976530a250d766355883439b4b290ba1a144c89d7d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5082,"rank":5082,"depth":1,"x":1652.829,"y":728.34,"cluster":"schemes"},{"id":"stacks:01L5","tag":"01L5","title":"Monomorphisms · Lemma 01L5","summary":"A composition of monomorphisms is a monomorphism.","statement_latex":"A composition of monomorphisms is a monomorphism.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01L5","source_file":"schemes.tex","source_line":4572,"source_end_line":4575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4572-L4575","statement_sha256":"fc2d9b90d40afb899acbc38095627bdee0738275df033c611a4b80d3cc9ce849","origin":"The Stacks Project","memory_eligible":false,"source_rank":5083,"rank":5083,"depth":0,"x":1486.987,"y":685.706,"cluster":"schemes"},{"id":"stacks:02YC","tag":"02YC","title":"Monomorphisms · Lemma 02YC","summary":"The base change of a monomorphism is a monomorphism.","statement_latex":"The base change of a monomorphism is a monomorphism.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YC","source_file":"schemes.tex","source_line":4581,"source_end_line":4584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4581-L4584","statement_sha256":"6afcf6b219373aa1dd57f79208e52771db943d295642b8451bf6e8a7ab3178c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5084,"rank":5084,"depth":0,"x":1644.296,"y":622.749,"cluster":"schemes"},{"id":"stacks:0DVA","tag":"0DVA","title":"Monomorphisms · Lemma 0DVA","summary":"Let j : X → Y be a morphism of schemes. If j is injective on points, then j is separated.","statement_latex":"Let $j : X \\to Y$ be a morphism of schemes.\nIf $j$ is injective on points, then $j$ is separated.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVA","source_file":"schemes.tex","source_line":4590,"source_end_line":4594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4590-L4594","statement_sha256":"7851e2754becf6a75fe14ee83392d49a193768f456d2d87b49f3f4727551be44","origin":"The Stacks Project","memory_eligible":false,"source_rank":5085,"rank":5085,"depth":14,"x":1578.622,"y":759.062,"cluster":"schemes"},{"id":"stacks:01L6","tag":"01L6","title":"Monomorphisms · Lemma 01L6","summary":"Let j : X → Y be a morphism of schemes. If • j is injective on points, and • for any x ∈ X the ring map j^sharp_x : O_Y, j(x) → O_X, x is surjective, then j is a monomorphism.","statement_latex":"Let $j : X \\to Y$ be a morphism of schemes.\nIf\n\\begin{enumerate}\n\\item $j$ is injective on points, and\n\\item for any $x \\in X$ the ring map\n$j^\\sharp_x : \\mathcal{O}_{Y, j(x)} \\to \\mathcal{O}_{X, x}$\nis surjective,\n\\end{enumerate}\nthen $j$ is a monomorphism.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01L6","source_file":"schemes.tex","source_line":4610,"source_end_line":4621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4610-L4621","statement_sha256":"2d5fce86858840309027f20aee887c30365d0a7b2b5e5b7a89233053478d6e48","origin":"The Stacks Project","memory_eligible":false,"source_rank":5086,"rank":5086,"depth":0,"x":1517.15,"y":620.649,"cluster":"schemes"},{"id":"stacks:01L7","tag":"01L7","title":"Monomorphisms · Lemma 01L7","summary":"An immersion of schemes is a monomorphism. In particular, any immersion is separated.","statement_latex":"An immersion of schemes is a monomorphism.\nIn particular, any immersion is separated.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01L7","source_file":"schemes.tex","source_line":4638,"source_end_line":4642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4638-L4642","statement_sha256":"acd2a5e330a84ac3f64a946a9a96cf1cef9c0f3c0d9f544a864889674d3df109","origin":"The Stacks Project","memory_eligible":false,"source_rank":5087,"rank":5087,"depth":2,"x":1674.5,"y":688.139,"cluster":"schemes"},{"id":"stacks:01L8","tag":"01L8","title":"Monomorphisms · Lemma 01L8","summary":"Let f : X → S be a separated morphism. Any locally closed subscheme Z ⊂ X is separated over S.","statement_latex":"Let $f : X \\to S$ be a separated morphism.\nAny locally closed subscheme $Z \\subset X$ is separated over $S$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01L8","source_file":"schemes.tex","source_line":4655,"source_end_line":4659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4655-L4659","statement_sha256":"f741d1e6b27bffffb21dc71f262a2886891f23b1329e3446df1fc88e654cd49a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5088,"rank":5088,"depth":16,"x":1503.429,"y":727.834,"cluster":"schemes"},{"id":"stacks:03DP","tag":"03DP","title":"Monomorphisms · Lemma 03DP","summary":"Let k_1, …, k_n be fields. For any monomorphism of schemes X → Spec(k_1 × … × k_n) there exists a subset I ⊂ (1, …, n) such that X ≅ Spec(∏_i ∈ I k_i) as schemes over Spec(k_1 × … × k_n). More generally, if X = coprod_i ∈ I Spec(k_i) is a disjoint union of spectra of fields and Y → X is a monomorphism, then there exists a subset J ⊂ I such that Y = coprod_i ∈ J Spec(k_i).","statement_latex":"Let $k_1, \\ldots, k_n$ be fields.\nFor any monomorphism of schemes\n$X \\to \\Spec(k_1 \\times \\ldots \\times k_n)$\nthere exists a subset $I \\subset \\{1, \\ldots, n\\}$ such\nthat $X \\cong \\Spec(\\prod_{i \\in I} k_i)$ as\nschemes over $\\Spec(k_1 \\times \\ldots \\times k_n)$.\nMore generally, if $X = \\coprod_{i \\in I} \\Spec(k_i)$\nis a disjoint union of spectra of fields and $Y \\to X$ is a monomorphism,\nthen there exists a subset $J \\subset I$ such that\n$Y = \\coprod_{i \\in J} \\Spec(k_i)$.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DP","source_file":"schemes.tex","source_line":4680,"source_end_line":4692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4680-L4692","statement_sha256":"ce6f6fc5675a6d9c939df8d913358948898c78f2ecdd4a0c87cefae760e0c888","origin":"The Stacks Project","memory_eligible":false,"source_rank":5089,"rank":5089,"depth":3,"x":1598.075,"y":600.929,"cluster":"schemes"},{"id":"stacks:01LC","tag":"01LC","title":"Functoriality for quasi-coherent modules · Lemma 01LC","summary":"Let f : X → S be a morphism of schemes. If f is quasi-compact and quasi-separated then f_* transforms quasi-coherent O_X-modules into quasi-coherent O_S-modules.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf $f$ is quasi-compact and quasi-separated then\n$f_*$ transforms quasi-coherent $\\mathcal{O}_X$-modules\ninto quasi-coherent $\\mathcal{O}_S$-modules.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Functoriality for quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LC","source_file":"schemes.tex","source_line":4769,"source_end_line":4775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4769-L4775","statement_sha256":"5802b30faa77733d8a88f01dcb35a626b35e262055952f8b6bfae01f0a602cc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5090,"rank":5090,"depth":13,"x":1630.48,"y":748.864,"cluster":"schemes"},{"id":"stacks:01LD","tag":"01LD","title":"Functoriality for quasi-coherent modules · Lemma 01LD","summary":"Let f : X → Y be a morphism of schemes. Suppose that • f induces a homeomorphism of X with a closed subset of Y, and • f^sharp : O_Y → f_*O_X is surjective. Then f is a closed immersion of schemes.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nSuppose that\n\\begin{enumerate}\n\\item $f$ induces a homeomorphism of $X$ with a\nclosed subset of $Y$, and\n\\item $f^\\sharp : \\mathcal{O}_Y \\to f_*\\mathcal{O}_X$\nis surjective.\n\\end{enumerate}\nThen $f$ is a closed immersion of schemes.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Functoriality for quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LD","source_file":"schemes.tex","source_line":4826,"source_end_line":4837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4826-L4837","statement_sha256":"c21ece209e5a9f2b4748a6e56f46e316214ac91f50f69596a3e3f9e70830e2cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5091,"rank":5091,"depth":14,"x":1486.993,"y":657.767,"cluster":"schemes"},{"id":"stacks:02V0","tag":"02V0","title":"Functoriality for quasi-coherent modules · Lemma 02V0","summary":"A composition of immersions of schemes is an immersion, a composition of closed immersions of schemes is a closed immersion, and a composition of open immersions of schemes is an open immersion.","statement_latex":"A composition of immersions of schemes is an immersion,\na composition of closed immersions of schemes is a closed immersion, and\na composition of open immersions of schemes is an open immersion.","area":"Schemes","chapter":"Schemes","chapter_id":"schemes","section":"Functoriality for quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V0","source_file":"schemes.tex","source_line":4860,"source_end_line":4865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/schemes.tex#L4860-L4865","statement_sha256":"d3a5005148a2bb8bd780f5d25fd17e099c730915c03a86366811c4e2e0314717","origin":"The Stacks Project","memory_eligible":false,"source_rank":5092,"rank":5092,"depth":15,"x":1666.836,"y":643.462,"cluster":"schemes"},{"id":"stacks:01LH","tag":"01LH","title":"Relative glueing · Lemma 01LH","summary":"Let S be a scheme. Let B be a basis for the topology of S. Suppose given the following data: • For every U ∈ B a scheme f_U : X_U → U over U. • For U, V ∈ B with V ⊂ U a morphism ρ^U_V : X_V → X_U over U. Assume that • [(a)] each ρ^U_V induces an isomorphism X_V → f_U^-1(V) of schemes over V, • [(b)] whenever W, V, U ∈ B, with W ⊂ V ⊂ U we have ρ^U_W = ρ^U_V ∘ ρ ^V_W. Then there exists a morphism f : X → S of schemes and isomorphisms i_U : f^-1(U) → X_U over U ∈ B such…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{B}$ be a basis for the topology of $S$.\nSuppose given the following data:\n\\begin{enumerate}\n\\item For every $U \\in \\mathcal{B}$ a scheme $f_U : X_U \\to U$ over $U$.\n\\item For $U, V \\in \\mathcal{B}$ with $V \\subset U$ a morphism\n$\\rho^U_V : X_V \\to X_U$ over $U$.\n\\end{enumerate}\nAssume that\n\\begin{enumerate}\n\\item[(a)] each $\\rho^U_V$ induces an isomorphism\n$X_V \\to f_U^{-1}(V)$ of schemes over $V$,\n\\item[(b)] whenever $W, V, U \\in \\mathcal{B}$, with\n$W \\subset V \\subset U$ we have $\\rho^U_W = \\rho^U_V \\circ \\rho ^V_W$.\n\\end{enumerate}\nThen there exists a morphism $f : X \\to S$ of schemes\nand isomorphisms $i_U : f^{-1}(U) \\to X_U$ over $U \\in \\mathcal{B}$\nsuch that for $V, U \\in \\mathcal{B}$ with $V \\subset U$ the composition\n$$\n\\xymatrix{\nX_V \\ar[r]^{i_V^{-1}} &\nf^{-1}(V) \\ar[rr]^{inclusion} & &\nf^{-1}(U) \\ar[r]^{i_U} &\nX_U\n}\n$$\nis the morphism $\\rho^U_V$. Moreover $X$ is unique up to\nunique isomorphism over $S$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LH","source_file":"constructions.tex","source_line":38,"source_end_line":68,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L38-L68","statement_sha256":"d867c4abb8380336b1600dfecf7e62ed694baec0f94aa9e255ab6a8ffdb24ac4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5093,"rank":5093,"depth":3,"x":1545.203,"y":756.543,"cluster":"schemes"},{"id":"stacks:01LI","tag":"01LI","title":"Relative glueing · Lemma 01LI","summary":"Let S be a scheme. Let B be a basis for the topology of S. Suppose given the following data: • For every U ∈ B a scheme f_U : X_U → U over U. • For every U ∈ B a quasi-coherent sheaf F_U over X_U. • For every pair U, V ∈ B such that V ⊂ U a morphism ρ^U_V : X_V → X_U. • For every pair U, V ∈ B such that V ⊂ U a morphism theta^U_V : (ρ^U_V)^*F_U → F_V. Assume that • [(a)] each ρ^U_V induces an isomorphism X_V → f_U^-1(V) of schemes over V, • [(b)] each theta^U_V is an…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{B}$ be a basis for the topology of $S$.\nSuppose given the following data:\n\\begin{enumerate}\n\\item For every $U \\in \\mathcal{B}$ a scheme $f_U : X_U \\to U$ over $U$.\n\\item For every $U \\in \\mathcal{B}$ a quasi-coherent sheaf $\\mathcal{F}_U$\nover $X_U$.\n\\item For every pair $U, V \\in \\mathcal{B}$ such that\n$V \\subset U$ a morphism $\\rho^U_V : X_V \\to X_U$.\n\\item  For every pair $U, V \\in \\mathcal{B}$ such that\n$V \\subset U$ a morphism\n$\\theta^U_V : (\\rho^U_V)^*\\mathcal{F}_U \\to \\mathcal{F}_V$.\n\\end{enumerate}\nAssume that\n\\begin{enumerate}\n\\item[(a)] each $\\rho^U_V$ induces an isomorphism\n$X_V \\to f_U^{-1}(V)$ of schemes over $V$,\n\\item[(b)] each $\\theta^U_V$ is an isomorphism,\n\\item[(c)] whenever $W, V, U \\in \\mathcal{B}$, with\n$W \\subset V \\subset U$ we have $\\rho^U_W = \\rho^U_V \\circ \\rho ^V_W$,\n\\item[(d)] whenever $W, V, U \\in \\mathcal{B}$, with\n$W \\subset V \\subset U$ we have\n$\\theta^U_W = \\theta^V_W \\circ (\\rho^V_W)^*\\theta^U_V$.\n\\end{enumerate}\nThen there exists a morphism of schemes $f : X \\to S$\ntogether with a quasi-coherent sheaf $\\mathcal{F}$ on $X$\nand isomorphisms $i_U : f^{-1}(U) \\to X_U$ and\n$\\theta_U : i_U^*\\mathcal{F}_U \\to \\mathcal{F}|_{f^{-1}(U)}$\nover $U \\in \\mathcal{B}$ such that\nfor $V, U \\in \\mathcal{B}$ with $V \\subset U$ the composition\n$$\n\\xymatrix{\nX_V \\ar[r]^{i_V^{-1}} &\nf^{-1}(V) \\ar[rr]^{inclusion} & &\nf^{-1}(U) \\ar[r]^{i_U} &\nX_U\n}\n$$\nis the morphism $\\rho^U_V$, and the composition\n\\begin{equation}\n\n(\\rho^U_V)^*\\mathcal{F}_U\n=\n(i_V^{-1})^*((i_U^*\\mathcal{F}_U)|_{f^{-1}(V)})\n\\xrightarrow{\\theta_U|_{f^{-1}(V)}}\n(i_V^{-1})^*(\\mathcal{F}|_{f^{-1}(V)})\n\\xrightarrow{\\theta_V^{-1}}\n\\mathcal{F}_V\n\\end{equation}\nis equal to $\\theta^U_V$. Moreover $(X, \\mathcal{F})$ is unique\nup to unique isomorphism over $S$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LI","source_file":"constructions.tex","source_line":156,"source_end_line":209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L156-L209","statement_sha256":"c353fe2c8f6c9015e4328587dc9e6202725a27aca7811c08c1ba241ea371031c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5094,"rank":5094,"depth":4,"x":1543.952,"y":603.489,"cluster":"schemes"},{"id":"stacks:01LN","tag":"01LN","title":"Relative spectrum via glueing · Lemma 01LN","summary":"In Situation [Tag 01LM]. Suppose U ⊂ U' ⊂ S are affine opens. Let A = A(U) and A' = A(U'). The map of rings A' → A induces a morphism Spec(A) → Spec(A'), and the diagram xymatrix Spec(A) ar[r] ar[d] & Spec(A') ar[d] U ar[r] & U' is cartesian.","statement_latex":"In Situation \\ref{situation-relative-spec}.\nSuppose $U \\subset U' \\subset S$ are affine opens.\nLet $A = \\mathcal{A}(U)$ and $A' = \\mathcal{A}(U')$.\nThe map of rings $A' \\to A$ induces a morphism\n$\\Spec(A) \\to \\Spec(A')$, and the diagram\n$$\n\\xymatrix{\n\\Spec(A) \\ar[r] \\ar[d] &\n\\Spec(A') \\ar[d] \\\\\nU \\ar[r] &\nU'\n}\n$$\nis cartesian.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum via glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LN","source_file":"constructions.tex","source_line":331,"source_end_line":347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L331-L347","statement_sha256":"4f4002704a25abebe2c24d6d629658cfddfb80dd5ab9c8ffe785d16ce797c88e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5095,"rank":5095,"depth":13,"x":1668.481,"y":716.116,"cluster":"schemes"},{"id":"stacks:01LO","tag":"01LO","title":"Relative spectrum via glueing · Lemma 01LO","summary":"In Situation [Tag 01LM]. Suppose U ⊂ U' ⊂ U\" ⊂ S are affine opens. Let A = A(U), A' = A(U') and A\" = A(U\"). The composition of the morphisms Spec(A) → Spec(A'), and Spec(A') → Spec(A\") of Lemma [Tag 01LN] gives the morphism Spec(A) → Spec(A\") of Lemma [Tag 01LN].","statement_latex":"In Situation \\ref{situation-relative-spec}.\nSuppose $U \\subset U' \\subset U'' \\subset S$ are affine opens.\nLet $A = \\mathcal{A}(U)$, $A' = \\mathcal{A}(U')$ and $A'' = \\mathcal{A}(U'')$.\nThe composition of the morphisms\n$\\Spec(A) \\to \\Spec(A')$, and\n$\\Spec(A') \\to \\Spec(A'')$ of\nLemma \\ref{lemma-spec-inclusion} gives the\nmorphism $\\Spec(A) \\to \\Spec(A'')$\nof Lemma \\ref{lemma-spec-inclusion}.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum via glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LO","source_file":"constructions.tex","source_line":363,"source_end_line":374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L363-L374","statement_sha256":"0e0e0e1c968796eb55c7ad1b4a16909477c3eef85c7065569426d9c084924b53","origin":"The Stacks Project","memory_eligible":false,"source_rank":5096,"rank":5096,"depth":14,"x":1485.317,"y":703.674,"cluster":"schemes"},{"id":"stacks:01LP","tag":"01LP","title":"Relative spectrum via glueing · Lemma 01LP","summary":"In Situation [Tag 01LM]. There exists a morphism of schemes π : underlineSpec_S(A) → S with the following properties: • for every affine open U ⊂ S there exists an isomorphism i_U : π^-1(U) → Spec(A(U)) over U, and • for U ⊂ U' ⊂ S affine open the composition xymatrix Spec(A(U)) ar[r]^i_U^-1 & π^-1(U) ar[rr]^inclusion & & π^-1(U') ar[r]^i_U' & Spec(A(U')) is the open immersion of Lemma [Tag 01LN] above. Moreover, underlineSpec_S(A) is unique up to unique isomorphism over S.","statement_latex":"In Situation \\ref{situation-relative-spec}.\nThere exists a morphism of schemes\n$$\n\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\longrightarrow S\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item for every affine open $U \\subset S$ there exists an isomorphism\n$i_U : \\pi^{-1}(U) \\to \\Spec(\\mathcal{A}(U))$ over $U$, and\n\\item for $U \\subset U' \\subset S$ affine open the composition\n$$\n\\xymatrix{\n\\Spec(\\mathcal{A}(U)) \\ar[r]^{i_U^{-1}} &\n\\pi^{-1}(U) \\ar[rr]^{inclusion} & &\n\\pi^{-1}(U') \\ar[r]^{i_{U'}} &\n\\Spec(\\mathcal{A}(U'))\n}\n$$\nis the open immersion of Lemma \\ref{lemma-spec-inclusion} above.\n\\end{enumerate}\nMoreover, $\\underline{\\Spec}_S(\\mathcal{A})$\nis unique up to unique isomorphism over $S$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum via glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LP","source_file":"constructions.tex","source_line":381,"source_end_line":405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L381-L405","statement_sha256":"c8fccf9ff16f55068485dc7f6fb855de067dc1285644515803a636db873a9aef","origin":"The Stacks Project","memory_eligible":false,"source_rank":5097,"rank":5097,"depth":15,"x":1630.991,"y":608.522,"cluster":"schemes"},{"id":"stacks:01LS","tag":"01LS","title":"Relative spectrum as a functor · Lemma 01LS","summary":"In Situation [Tag 01LM]. Let F be the functor associated to (S, A) above. Let g : S' → S be a morphism of schemes. Set A' = g^*A. Let F' be the functor associated to (S', A') above. Then there is a canonical isomorphism F' ≅ h_S' ×_h_S F of functors.","statement_latex":"In Situation \\ref{situation-relative-spec}.\nLet $F$ be the functor\nassociated to $(S, \\mathcal{A})$ above.\nLet $g : S' \\to S$ be a morphism of schemes.\nSet $\\mathcal{A}' = g^*\\mathcal{A}$. Let $F'$ be the\nfunctor associated to $(S', \\mathcal{A}')$ above.\nThen there is a canonical isomorphism\n$$\nF' \\cong h_{S'} \\times_{h_S} F\n$$\nof functors.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LS","source_file":"constructions.tex","source_line":467,"source_end_line":480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L467-L480","statement_sha256":"d1b47460208085ba0840ce2979cb9522c55520333107f8f2dc4c9e4efd1c097a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5098,"rank":5098,"depth":0,"x":1599.963,"y":761.976,"cluster":"schemes"},{"id":"stacks:01LT","tag":"01LT","title":"Relative spectrum as a functor · Lemma 01LT","summary":"In Situation [Tag 01LM]. Let F be the functor associated to (S, A) above. If S is affine, then F is representable by the affine scheme Spec(Γ(S, A)).","statement_latex":"In Situation \\ref{situation-relative-spec}.\nLet $F$ be the functor associated to $(S, \\mathcal{A})$ above.\nIf $S$ is affine, then $F$ is representable by the\naffine scheme $\\Spec(\\Gamma(S, \\mathcal{A}))$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LT","source_file":"constructions.tex","source_line":491,"source_end_line":497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L491-L497","statement_sha256":"c09bc87e1cbd6cb5879e678ac63ecda7915342cffa9d882d9df3ed0980cb70d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5099,"rank":5099,"depth":13,"x":1499.027,"y":630.678,"cluster":"schemes"},{"id":"stacks:01LU","tag":"01LU","title":"Relative spectrum as a functor · Lemma 01LU","summary":"In Situation [Tag 01LM]. The functor F is representable by a scheme.","statement_latex":"In Situation \\ref{situation-relative-spec}.\nThe functor $F$ is representable by a scheme.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LU","source_file":"constructions.tex","source_line":547,"source_end_line":551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L547-L551","statement_sha256":"17742c21b2641e5e3a43bdef3a26ae40e86514428a827735668c9d91f3fd6012","origin":"The Stacks Project","memory_eligible":false,"source_rank":5100,"rank":5100,"depth":14,"x":1679.774,"y":670.384,"cluster":"schemes"},{"id":"stacks:01LV","tag":"01LV","title":"Relative spectrum as a functor · Lemma 01LV","summary":"In Situation [Tag 01LM]. The scheme π : underlineSpec_S(A) → S constructed in Lemma [Tag 01LP] and the scheme representing the functor F are canonically isomorphic as schemes over S.","statement_latex":"In Situation \\ref{situation-relative-spec}.\nThe scheme $\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\to S$\nconstructed in Lemma \\ref{lemma-glue-relative-spec}\nand the scheme representing the functor $F$ are\ncanonically isomorphic as schemes over $S$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LV","source_file":"constructions.tex","source_line":602,"source_end_line":609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L602-L609","statement_sha256":"292db25b6f43e7e0ef49f812d8c5ce2b8932b9023cf080b1f5ce3bf3e77ad459","origin":"The Stacks Project","memory_eligible":false,"source_rank":5101,"rank":5101,"depth":16,"x":1513.896,"y":743.96,"cluster":"schemes"},{"id":"stacks:01LW","tag":"01LW","title":"Relative spectrum as a functor · Definition 01LW","summary":"Let S be a scheme. Let A be a quasi-coherent sheaf of O_S-algebras. The relative spectrum of A over S, or simply the spectrum of A over S is the scheme constructed in Lemma [Tag 01LP] which represents the functor F ([Tag 01LR]), see Lemma [Tag 01LV]. We denote it π : underlineSpec_S(A) → S. The \"universal family\" is a morphism of O_S-algebras A → π_*O_underlineSpec_S(A)","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of\n$\\mathcal{O}_S$-algebras. The {\\it relative spectrum of $\\mathcal{A}$ over\n$S$}, or simply the {\\it spectrum of $\\mathcal{A}$ over $S$} is the scheme\nconstructed in Lemma \\ref{lemma-glue-relative-spec} which represents the\nfunctor $F$ (\\ref{equation-spec}), see\nLemma \\ref{lemma-glueing-gives-functor-spec}.\nWe denote it $\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\to S$.\nThe ``universal family'' is a morphism of $\\mathcal{O}_S$-algebras\n$$\n\\mathcal{A}\n\\longrightarrow\n\\pi_*\\mathcal{O}_{\\underline{\\Spec}_S(\\mathcal{A})}\n$$","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum as a functor","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LW","source_file":"constructions.tex","source_line":641,"source_end_line":656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L641-L656","statement_sha256":"b1159dce0504ddc76927042c2f4d21a5130e0080633b6984ef9288e8ed1f172d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5102,"rank":5102,"depth":17,"x":1577.298,"y":594.99,"cluster":"schemes"},{"id":"stacks:01LX","tag":"01LX","title":"Relative spectrum as a functor · Lemma 01LX","summary":"Let S be a scheme. Let A be a quasi-coherent sheaf of O_S-algebras. Let π : underlineSpec_S(A) → S be the relative spectrum of A over S. • For every affine open U ⊂ S the inverse image π^-1(U) is affine. • For every morphism g : S' → S we have S' ×_S underlineSpec_S(A) = underlineSpec_S'(g^*A). • The universal map A → π_*O_underlineSpec_S(A) is an isomorphism of O_S-algebras.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent\nsheaf of $\\mathcal{O}_S$-algebras. Let\n$\\pi : \\underline{\\Spec}_S(\\mathcal{A}) \\to S$\nbe the relative spectrum of $\\mathcal{A}$ over $S$.\n\\begin{enumerate}\n\\item For every affine open $U \\subset S$ the inverse image\n$\\pi^{-1}(U)$ is affine.\n\\item For every morphism $g : S' \\to S$ we have\n$S' \\times_S \\underline{\\Spec}_S(\\mathcal{A}) =\n\\underline{\\Spec}_{S'}(g^*\\mathcal{A})$.\n\\item\nThe universal map\n$$\n\\mathcal{A}\n\\longrightarrow\n\\pi_*\\mathcal{O}_{\\underline{\\Spec}_S(\\mathcal{A})}\n$$\nis an isomorphism of $\\mathcal{O}_S$-algebras.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LX","source_file":"constructions.tex","source_line":662,"source_end_line":683,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L662-L683","statement_sha256":"a2aa50bec8c54c15d1458a4926ac53d9e462fac2ebd263d5dea13ca339d0fdce","origin":"The Stacks Project","memory_eligible":false,"source_rank":5103,"rank":5103,"depth":16,"x":1650.633,"y":741.393,"cluster":"schemes"},{"id":"stacks:01LY","tag":"01LY","title":"Relative spectrum as a functor · Lemma 01LY","summary":"Let f : X → S be a quasi-compact and quasi-separated morphism of schemes. By Schemes, Lemma [Tag 01LC] the sheaf f_*O_X is a quasi-coherent sheaf of O_S-algebras. There is a canonical morphism can : X → underlineSpec_S(f_*O_X) of schemes over S. For any affine open U ⊂ S the restriction can|_f^-1(U) is identified with the canonical morphism f^-1(U) → Spec(Γ(f^-1(U), O_X)) coming from Schemes, Lemma [Tag 01I1].","statement_latex":"Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism\nof schemes. By Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}\nthe sheaf $f_*\\mathcal{O}_X$ is a quasi-coherent sheaf of\n$\\mathcal{O}_S$-algebras. There is a canonical morphism\n$$\ncan : X \\longrightarrow \\underline{\\Spec}_S(f_*\\mathcal{O}_X)\n$$\nof schemes over $S$.\nFor any affine open $U \\subset S$ the restriction $can|_{f^{-1}(U)}$\nis identified with the canonical morphism\n$$\nf^{-1}(U) \\longrightarrow \\Spec(\\Gamma(f^{-1}(U), \\mathcal{O}_X))\n$$\ncoming from Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative spectrum as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01LY","source_file":"constructions.tex","source_line":693,"source_end_line":709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L693-L709","statement_sha256":"be9814a95e6e5c281281937793fd4d30468c51fd94c96e365a52880e46c5387d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5104,"rank":5104,"depth":14,"x":1478.147,"y":674.788,"cluster":"schemes"},{"id":"stacks:01M0","tag":"01M0","title":"Affine n-space · Definition 01M0","summary":"Let S be a scheme and n ≥ 0. The scheme A^n_S = underlineSpec_S(O_S[T_1, …, T_n]) over S is called affine n-space over S. If S = Spec(R) is affine then we also call this affine n-space over R and we denote it A^n_R.","statement_latex":"Let $S$ be a scheme and $n \\geq 0$.\nThe scheme\n$$\n\\mathbf{A}^n_S =\n\\underline{\\Spec}_S(\\mathcal{O}_S[T_1, \\ldots, T_n])\n$$\nover $S$ is called {\\it affine $n$-space over $S$}.\nIf $S = \\Spec(R)$ is affine then we also call this\n{\\it affine $n$-space over $R$} and we denote it $\\mathbf{A}^n_R$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Affine n-space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01M0","source_file":"constructions.tex","source_line":744,"source_end_line":755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L744-L755","statement_sha256":"648460275ee63a907931fb6c4aebb14e05fb79dd02cd6b3204a4a1190b8b13d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5105,"rank":5105,"depth":0,"x":1659.605,"y":625.839,"cluster":"schemes"},{"id":"stacks:01M2","tag":"01M2","title":"Vector bundles · Definition 01M2","summary":"Let S be a scheme. Let E be a quasi-coherent O_S-module is finite locally free. We do not do so in order to be consistent with [EGA].. The vector bundle associated to E is V(E) = underlineSpec_S(Sym(E)).","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{E}$ be a quasi-coherent\n$\\mathcal{O}_S$-module\\footnote{The reader may expect here\nthe condition that $\\mathcal{E}$ is finite locally free. We do not\ndo so in order to be consistent with \\cite[II, Definition 1.7.8]{EGA}.}.\nThe {\\it vector bundle associated to $\\mathcal{E}$} is\n$$\n\\mathbf{V}(\\mathcal{E}) = \\underline{\\Spec}_S(\\text{Sym}(\\mathcal{E})).\n$$","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Vector bundles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01M2","source_file":"constructions.tex","source_line":804,"source_end_line":814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L804-L814","statement_sha256":"48f433fcd644054c4c09e0429102c32085263c87ea9a9f8db53d4102197bebc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5106,"rank":5106,"depth":0,"x":1564.797,"y":765.435,"cluster":"schemes"},{"id":"stacks:062M","tag":"062M","title":"Vector bundles · Definition 062M","summary":"Let S be a scheme. A vector bundle π : V → S over S is an affine morphism of schemes such that π_*O_V is endowed with the structure of a graded O_S-algebra π_*O_V = bigoplus_n ≥ 0 E_n such that E_0 = O_S and such that the maps Sym^n(E_1) → E_n are isomorphisms for all n ≥ 0. A morphism of vector bundles over S is a morphism f : V → V' such that the induced map f^* : π'_*O_V' → π_*O_V is compatible with the given gradings.","statement_latex":"Let $S$ be a scheme. A {\\it vector bundle $\\pi : V \\to S$ over $S$} is an\naffine morphism of schemes such that $\\pi_*\\mathcal{O}_V$ is endowed with\nthe structure of a graded $\\mathcal{O}_S$-algebra\n$\\pi_*\\mathcal{O}_V = \\bigoplus\\nolimits_{n \\geq 0} \\mathcal{E}_n$\nsuch that $\\mathcal{E}_0 = \\mathcal{O}_S$ and such that the maps\n$$\n\\text{Sym}^n(\\mathcal{E}_1) \\longrightarrow \\mathcal{E}_n\n$$\nare isomorphisms for all $n \\geq 0$. A {\\it morphism of vector bundles\nover $S$} is a morphism $f : V \\to V'$ such that the induced map\n$$\nf^* : \\pi'_*\\mathcal{O}_{V'} \\longrightarrow \\pi_*\\mathcal{O}_V\n$$\nis compatible with the given gradings.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Vector bundles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062M","source_file":"constructions.tex","source_line":828,"source_end_line":844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L828-L844","statement_sha256":"61c4659d881535e1609b514960f49e46784ba8245e71ba1ed6c5492f650c2d82","origin":"The Stacks Project","memory_eligible":false,"source_rank":5107,"rank":5107,"depth":0,"x":1522.286,"y":608.1,"cluster":"schemes"},{"id":"stacks:062N","tag":"062N","title":"Vector bundles · Lemma 062N","summary":"The category of vector bundles over a scheme S is anti-equivalent to the category of quasi-coherent O_S-modules.","statement_latex":"The category of vector bundles over a scheme $S$ is\nanti-equivalent to the category of quasi-coherent $\\mathcal{O}_S$-modules.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Vector bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062N","source_file":"constructions.tex","source_line":852,"source_end_line":856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L852-L856","statement_sha256":"c6cc0c5358b0bddff3288c3362578d7329eaf372c5d103bfc86b905e5d281d8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5108,"rank":5108,"depth":0,"x":1680.759,"y":700.346,"cluster":"schemes"},{"id":"stacks:062Q","tag":"062Q","title":"Cones · Definition 062Q","summary":"Let S be a scheme. Let A be a quasi-coherent graded O_S-algebra. Assume that O_S → A_0 is an isomorphism is generated by A_1 over O_S. We do not assume this in order to be consistent with [EGA].. The cone associated to A or the affine cone associated to A is C(A) = underlineSpec_S(A).","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent\ngraded $\\mathcal{O}_S$-algebra. Assume that $\\mathcal{O}_S \\to \\mathcal{A}_0$\nis an isomorphism\\footnote{Often one imposes the assumption that\n$\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{O}_S$. We do not\nassume this in order to be consistent with \\cite[II, (8.3.1)]{EGA}.}.\nThe {\\it cone associated to $\\mathcal{A}$} or the\n{\\it affine cone associated to $\\mathcal{A}$}\nis\n$$\nC(\\mathcal{A}) = \\underline{\\Spec}_S(\\mathcal{A}).\n$$","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Cones","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062Q","source_file":"constructions.tex","source_line":878,"source_end_line":891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L878-L891","statement_sha256":"9fb095ed23e10b66b0e8601f99a08434f5569b98b6a594676b62c8f61d44371d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5109,"rank":5109,"depth":0,"x":1488.997,"y":722.329,"cluster":"schemes"},{"id":"stacks:062R","tag":"062R","title":"Cones · Definition 062R","summary":"Let S be a scheme. A cone π : C → S over S is an affine morphism of schemes such that π_*O_C is endowed with the structure of a graded O_S-algebra π_*O_C = bigoplus_n ≥ 0 A_n such that A_0 = O_S. A morphism of cones from π : C → S to π' : C' → S is a morphism f : C → C' such that the induced map f^* : π'_*O_C' → π_*O_C is compatible with the given gradings.","statement_latex":"Let $S$ be a scheme. A {\\it cone $\\pi : C \\to S$ over $S$} is an\naffine morphism of schemes such that $\\pi_*\\mathcal{O}_C$ is endowed with\nthe structure of a graded $\\mathcal{O}_S$-algebra\n$\\pi_*\\mathcal{O}_C = \\bigoplus\\nolimits_{n \\geq 0} \\mathcal{A}_n$\nsuch that $\\mathcal{A}_0 = \\mathcal{O}_S$. A {\\it morphism of cones}\nfrom $\\pi : C \\to S$ to $\\pi' : C' \\to S$\nis a morphism $f : C \\to C'$ such that the induced map\n$$\nf^* : \\pi'_*\\mathcal{O}_{C'} \\longrightarrow \\pi_*\\mathcal{O}_C\n$$\nis compatible with the given gradings.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Cones","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062R","source_file":"constructions.tex","source_line":902,"source_end_line":915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L902-L915","statement_sha256":"3811d5c947d0557f8a178a29df5d8c6b0919c9f07895665c15532ddf28190d62","origin":"The Stacks Project","memory_eligible":false,"source_rank":5110,"rank":5110,"depth":0,"x":1613.188,"y":596.84,"cluster":"schemes"},{"id":"stacks:01M4","tag":"01M4","title":"Proj of a graded ring · Lemma 01M4","summary":"Let S be a graded ring. Let f ∈ S homogeneous of positive degree. • If g∈ S homogeneous of positive degree and D_+(g) ⊂ D_+(f), then • f is invertible in S_g, and f^deg(g)/g^deg(f) is invertible in S_(g), • g^e = af for some e ≥ 1 and a ∈ S homogeneous, • there is a canonical S-algebra map S_f → S_g, • there is a canonical S_0-algebra map S_(f) → S_(g) compatible with the map S_f → S_g, • the map S_(f) → S_(g) induces an isomorphism (S_(f))_g^deg(f)/f^deg(g) ≅ S_(g), •…","statement_latex":"Let $S$ be a graded ring. Let $f \\in S$ homogeneous of positive degree.\n\\begin{enumerate}\n\\item If $g\\in S$ homogeneous of positive degree\nand $D_{+}(g) \\subset D_{+}(f)$, then\n\\begin{enumerate}\n\\item $f$ is invertible in $S_g$, and\n$f^{\\deg(g)}/g^{\\deg(f)}$ is invertible in $S_{(g)}$,\n\\item $g^e = af$ for some $e \\geq 1$ and $a \\in S$ homogeneous,\n\\item there is a canonical $S$-algebra map $S_f \\to S_g$,\n\\item there is a canonical $S_0$-algebra map $S_{(f)} \\to S_{(g)}$\ncompatible with the map $S_f \\to S_g$,\n\\item the map $S_{(f)} \\to S_{(g)}$ induces an isomorphism\n$$\n(S_{(f)})_{g^{\\deg(f)}/f^{\\deg(g)}} \\cong S_{(g)},\n$$\n\\item these maps induce a commutative diagram of\ntopological spaces\n$$\n\\xymatrix{\nD_{+}(g) \\ar[d] &\n\\{\\mathbf{Z}\\text{-graded primes of }S_g\\} \\ar[l] \\ar[r] \\ar[d] &\n\\Spec(S_{(g)}) \\ar[d] \\\\\nD_{+}(f) &\n\\{\\mathbf{Z}\\text{-graded primes of }S_f\\} \\ar[l] \\ar[r] &\n\\Spec(S_{(f)})\n}\n$$\nwhere the horizontal maps are homeomorphisms and the vertical maps\nare open immersions,\n\\item there are compatible canonical $S_f$ and $S_{(f)}$-module\nmaps $M_f \\to M_g$ and $M_{(f)} \\to M_{(g)}$ for any graded $S$-module $M$,\nand\n\\item the map $M_{(f)} \\to M_{(g)}$ induces an isomorphism\n$$\n(M_{(f)})_{g^{\\deg(f)}/f^{\\deg(g)}} \\cong M_{(g)}.\n$$\n\\end{enumerate}\n\\item Any open covering of $D_{+}(f)$ can be refined to a finite\nopen covering of the form $D_{+}(f) = \\bigcup_{i = 1}^n D_{+}(g_i)$.\n\\item Let $g_1, \\ldots, g_n \\in S$ be homogeneous of positive degree.\nThen $D_{+}(f) \\subset \\bigcup D_{+}(g_i)$\nif and only if\n$g_1^{\\deg(f)}/f^{\\deg(g_1)}, \\ldots, g_n^{\\deg(f)}/f^{\\deg(g_n)}$\ngenerate the unit ideal in $S_{(f)}$.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01M4","source_file":"constructions.tex","source_line":954,"source_end_line":1001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L954-L1001","statement_sha256":"7331c8d2c391ae8db0555b3fbee44e29504d59a98d3d87db1d6e2c573dea1efb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5111,"rank":5111,"depth":4,"x":1622.561,"y":760.451,"cluster":"schemes"},{"id":"stacks:01M5","tag":"01M5","title":"Proj of a graded ring · Definition 01M5","summary":"Let S be a graded ring. Suppose that D_+(f) ⊂ Proj(S) is a standard open. A standard open covering of D_+(f) is a covering D_+(f) = ⋃_i = 1^n D_+(g_i), where g_1, …, g_n ∈ S are homogeneous of positive degree.","statement_latex":"Let $S$ be a graded ring.\nSuppose that $D_{+}(f) \\subset \\text{Proj}(S)$ is a standard\nopen. A {\\it standard open covering} of $D_{+}(f)$\nis a covering $D_{+}(f) = \\bigcup_{i = 1}^n D_{+}(g_i)$,\nwhere $g_1, \\ldots, g_n \\in S$ are homogeneous of positive degree.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01M5","source_file":"constructions.tex","source_line":1055,"source_end_line":1062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1055-L1062","statement_sha256":"8ade478dc9e87de2c9bd03f2c4dc5ec8ed0cb763fbd7c6cc82519792a468a65f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5112,"rank":5112,"depth":0,"x":1483.566,"y":644.697,"cluster":"schemes"},{"id":"stacks:01M6","tag":"01M6","title":"Proj of a graded ring · Definition 01M6","summary":"Let S be a graded ring. • The structure sheaf O_Proj(S) of the homogeneous spectrum of S is the unique sheaf of rings O_Proj(S) which agrees with widetilde S on the basis of standard opens. • The locally ringed space (Proj(S), O_Proj(S)) is called the homogeneous spectrum of S and denoted Proj(S). • The sheaf of O_Proj(S)-modules extending widetilde M to all opens of Proj(S) is called the sheaf of O_Proj(S)-modules associated to M. This sheaf is denoted widetilde M as well.","statement_latex":"Let $S$ be a graded ring.\n\\begin{enumerate}\n\\item The {\\it structure sheaf $\\mathcal{O}_{\\text{Proj}(S)}$ of the\nhomogeneous spectrum of $S$} is the unique sheaf of rings\n$\\mathcal{O}_{\\text{Proj}(S)}$\nwhich agrees with $\\widetilde S$ on the basis of standard opens.\n\\item The locally ringed space\n$(\\text{Proj}(S), \\mathcal{O}_{\\text{Proj}(S)})$ is called\nthe {\\it homogeneous spectrum} of $S$ and denoted $\\text{Proj}(S)$.\n\\item The sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules\nextending $\\widetilde M$ to all opens of $\\text{Proj}(S)$\nis called the sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules\nassociated to $M$. This sheaf is denoted $\\widetilde M$ as\nwell.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01M6","source_file":"constructions.tex","source_line":1160,"source_end_line":1177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1160-L1177","statement_sha256":"2778d3d4cc50e2681aff902f8c8f9833d18dff79b0cb50851a1610e49198c0e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5113,"rank":5113,"depth":0,"x":1679.863,"y":651.208,"cluster":"schemes"},{"id":"stacks:01M7","tag":"01M7","title":"Proj of a graded ring · Lemma 01M7","summary":"Let S be a graded ring. Let M be a graded S-module. Let widetilde M be the sheaf of O_Proj(S)-modules associated to M. • For every f ∈ S homogeneous of positive degree we have Γ(D_+(f), O_Proj(S)) = S_(f). • For every f∈ S homogeneous of positive degree we have Γ(D_+(f), widetilde M) = M_(f) as an S_(f)-module. • Whenever D_+(g) ⊂ D_+(f) the restriction mappings on O_Proj(S) and widetilde M are the maps S_(f) → S_(g) and M_(f) → M_(g) from Lemma [Tag 01M4]. • Let p be a…","statement_latex":"Let $S$ be a graded ring. Let $M$ be a graded $S$-module.\nLet $\\widetilde M$ be the sheaf of $\\mathcal{O}_{\\text{Proj}(S)}$-modules\nassociated to $M$.\n\\begin{enumerate}\n\\item For every $f \\in S$ homogeneous of positive degree we have\n$$\n\\Gamma(D_{+}(f), \\mathcal{O}_{\\text{Proj}(S)}) = S_{(f)}.\n$$\n\\item For every $f\\in S$ homogeneous of positive degree\nwe have $\\Gamma(D_{+}(f), \\widetilde M) = M_{(f)}$\nas an $S_{(f)}$-module.\n\\item Whenever $D_{+}(g) \\subset D_{+}(f)$ the restriction mappings\non $\\mathcal{O}_{\\text{Proj}(S)}$ and $\\widetilde M$\nare the maps\n$S_{(f)} \\to S_{(g)}$ and $M_{(f)} \\to M_{(g)}$ from Lemma\n\\ref{lemma-standard-open}.\n\\item Let $\\mathfrak p$ be a homogeneous prime of $S$ not containing\n$S_{+}$, and let $x \\in \\text{Proj}(S)$\nbe the corresponding point. We have\n$\\mathcal{O}_{\\text{Proj}(S), x} = S_{(\\mathfrak p)}$.\n\\item Let $\\mathfrak p$ be a homogeneous prime of $S$ not containing\n$S_{+}$, and let $x \\in \\text{Proj}(S)$\nbe the corresponding point. We have $(\\widetilde M)_x = M_{(\\mathfrak p)}$\nas an $S_{(\\mathfrak p)}$-module.\n\\item\n\nThere is a canonical ring map\n$\nS_0 \\longrightarrow \\Gamma(\\text{Proj}(S), \\widetilde S)\n$\nand a canonical $S_0$-module map\n$\nM_0 \\longrightarrow \\Gamma(\\text{Proj}(S), \\widetilde M)\n$\ncompatible with the descriptions of sections over standard opens\nand stalks above.\n\\end{enumerate}\nMoreover, all these identifications are functorial in the graded\n$S$-module $M$. In particular, the functor $M \\mapsto \\widetilde M$\nis an exact functor from the category of graded $S$-modules\nto the category of $\\mathcal{O}_{\\text{Proj}(S)}$-modules.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01M7","source_file":"constructions.tex","source_line":1182,"source_end_line":1225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1182-L1225","statement_sha256":"0e9f7c816db90eacabdbf9f458ef81eeba64ae5b5b0f53d17f826e0386952bb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5114,"rank":5114,"depth":5,"x":1529.333,"y":758.179,"cluster":"schemes"},{"id":"stacks:01MA","tag":"01MA","title":"Proj of a graded ring · Lemma 01MA","summary":"Let S be a graded ring. Let f ∈ S be homogeneous of positive degree. Suppose that D(g) ⊂ Spec(S_(f)) is a standard open. Then there exists an h ∈ S homogeneous of positive degree such that D(g) corresponds to D_+(h) ⊂ D_+(f) via the homeomorphism of Algebra, Lemma [Tag 00JP]. In fact we can take h such that g = h/f^n for some n.","statement_latex":"Let $S$ be a graded ring. Let $f \\in S$ be homogeneous of positive degree.\nSuppose that $D(g) \\subset \\Spec(S_{(f)})$ is a standard open.\nThen there exists an $h \\in S$ homogeneous of positive degree such that\n$D(g)$ corresponds to $D_{+}(h) \\subset D_{+}(f)$ via the homeomorphism\nof Algebra, Lemma \\ref{algebra-lemma-topology-proj}. In fact we can\ntake $h$ such that $g = h/f^n$ for some $n$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MA","source_file":"constructions.tex","source_line":1248,"source_end_line":1256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1248-L1256","statement_sha256":"4d32ee04e1386251517edb9363927901c09e0d00aeb83a61f064990a093b2cf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5115,"rank":5115,"depth":4,"x":1554.4,"y":593.291,"cluster":"schemes"},{"id":"stacks:01MB","tag":"01MB","title":"Proj of a graded ring · Lemma 01MB","summary":"Let S be a graded ring. The locally ringed space Proj(S) is a scheme. The standard opens D_+(f) are affine opens. For any graded S-module M the sheaf widetilde M is a quasi-coherent sheaf of O_Proj(S)-modules.","statement_latex":"Let $S$ be a graded ring.\nThe locally ringed space $\\text{Proj}(S)$ is a scheme.\nThe standard opens $D_{+}(f)$ are affine opens.\nFor any graded $S$-module $M$ the sheaf\n$\\widetilde M$ is a quasi-coherent sheaf of\n$\\mathcal{O}_{\\text{Proj}(S)}$-modules.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MB","source_file":"constructions.tex","source_line":1266,"source_end_line":1274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1266-L1274","statement_sha256":"3fac5422209b17b01b04ef5ea4b06862ca624c62427cc24a1bdca2ab38c7b9d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5116,"rank":5116,"depth":6,"x":1668.924,"y":729.589,"cluster":"schemes"},{"id":"stacks:01MC","tag":"01MC","title":"Proj of a graded ring · Lemma 01MC","summary":"Let S be a graded ring. The scheme Proj(S) is separated.","statement_latex":"Let $S$ be a graded ring.\nThe scheme $\\text{Proj}(S)$ is separated.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MC","source_file":"constructions.tex","source_line":1314,"source_end_line":1318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1314-L1318","statement_sha256":"e08307501e94182e53cb7f406af4f10948a1ded359b397c3f5ea428811d6ea89","origin":"The Stacks Project","memory_eligible":false,"source_rank":5117,"rank":5117,"depth":13,"x":1474.176,"y":693.941,"cluster":"schemes"},{"id":"stacks:01MD","tag":"01MD","title":"Proj of a graded ring · Lemma 01MD","summary":"Let S be a graded ring. The scheme Proj(S) is quasi-compact if and only if there exist finitely many homogeneous elements f_1, …, f_n ∈ S_+ such that S_+ ⊂ sqrt(f_1, …, f_n). In this case Proj(S) = D_+(f_1) ∪ … ∪ D_+(f_n).","statement_latex":"Let $S$ be a graded ring.\nThe scheme $\\text{Proj}(S)$ is quasi-compact if and only\nif there exist finitely many homogeneous elements\n$f_1, \\ldots, f_n \\in S_{+}$ such that\n$S_{+} \\subset \\sqrt{(f_1, \\ldots, f_n)}$. In this case\n$\\text{Proj}(S) = D_+(f_1) \\cup \\ldots \\cup D_+(f_n)$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MD","source_file":"constructions.tex","source_line":1340,"source_end_line":1348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1340-L1348","statement_sha256":"c6c078fb823ae2a060a4ea0f7917d972b18954482497cfefb7cc81ee53da15d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5118,"rank":5118,"depth":4,"x":1647.058,"y":609.425,"cluster":"schemes"},{"id":"stacks:01ME","tag":"01ME","title":"Proj of a graded ring · Lemma 01ME","summary":"Let S be a graded ring. The scheme Proj(S) has a canonical morphism towards the affine scheme Spec(S_0), agreeing with the map on topological spaces coming from Algebra, Definition [Tag 00JN].","statement_latex":"Let $S$ be a graded ring. The scheme $\\text{Proj}(S)$ has a canonical morphism\ntowards the affine scheme $\\Spec(S_0)$, agreeing with the map on\ntopological spaces coming from\nAlgebra, Definition \\ref{algebra-definition-proj}.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ME","source_file":"constructions.tex","source_line":1361,"source_end_line":1367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1361-L1367","statement_sha256":"7c5c40b0977a99a392da4bd5dac4662cd665a0f352a0296db847728856688c4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5119,"rank":5119,"depth":11,"x":1587.333,"y":770.407,"cluster":"schemes"},{"id":"stacks:01MF","tag":"01MF","title":"Proj of a graded ring · Lemma 01MF","summary":"Let S be a graded ring. If S is finitely generated as an algebra over S_0, then the morphism Proj(S) → Spec(S_0) satisfies the existence and uniqueness parts of the valuative criterion, see Schemes, Definition [Tag 01KD].","statement_latex":"Let $S$ be a graded ring. If $S$ is finitely generated as\nan algebra over $S_0$, then\nthe morphism $\\text{Proj}(S) \\to \\Spec(S_0)$ satisfies\nthe existence and uniqueness parts of the valuative criterion,\nsee Schemes, Definition \\ref{schemes-definition-valuative-criterion}.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Proj of a graded ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MF","source_file":"constructions.tex","source_line":1384,"source_end_line":1391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1384-L1391","statement_sha256":"705867846333fdd88c43d4d91b9755a3a2e2b2add72fba7b246eeb4cade83578","origin":"The Stacks Project","memory_eligible":false,"source_rank":5120,"rank":5120,"depth":16,"x":1501.618,"y":617.278,"cluster":"schemes"},{"id":"stacks:01MK","tag":"01MK","title":"Quasi-coherent sheaves on Proj · Lemma 01MK","summary":"Let S be a graded ring. Let (X, O_X) = (Proj(S), O_Proj(S)) be the scheme of Lemma [Tag 01MB]. Let f ∈ S_+ be homogeneous. Let x ∈ X be a point corresponding to the homogeneous prime p ⊂ S. Let M, N be graded S-modules. There is a canonical map of O_Proj(S)-modules widetilde M ⊗_O_X widetilde N → widetildeM ⊗_S N which induces the canonical map M_(f) ⊗_S_(f) N_(f) → (M ⊗_S N)_(f) on sections over D_+(f) and the canonical map M_( p) ⊗_S_( p) N_( p) → (M ⊗_S N)_( p) on…","statement_latex":"Let $S$ be a graded ring.\nLet $(X, \\mathcal{O}_X) = (\\text{Proj}(S), \\mathcal{O}_{\\text{Proj}(S)})$\nbe the scheme of Lemma \\ref{lemma-proj-scheme}.\nLet $f \\in S_{+}$ be homogeneous. Let $x \\in X$ be a point\ncorresponding to the homogeneous prime $\\mathfrak p \\subset S$.\nLet $M$, $N$ be graded $S$-modules.\nThere is a canonical map of $\\mathcal{O}_{\\text{Proj}(S)}$-modules\n$$\n\\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N\n\\longrightarrow\n\\widetilde{M \\otimes_S N}\n$$\nwhich induces the canonical map\n$\nM_{(f)} \\otimes_{S_{(f)}} N_{(f)}\n\\to\n(M \\otimes_S N)_{(f)}\n$\non sections over $D_{+}(f)$ and the canonical map\n$\nM_{(\\mathfrak p)} \\otimes_{S_{(\\mathfrak p)}} N_{(\\mathfrak p)}\n\\to\n(M \\otimes_S N)_{(\\mathfrak p)}\n$\non stalks at $x$. Moreover, the following diagram\n$$\n\\xymatrix{\nM_0 \\otimes_{S_0} N_0 \\ar[r] \\ar[d] &\n(M \\otimes_S N)_0 \\ar[d] \\\\\n\\Gamma(X, \\widetilde M \\otimes_{\\mathcal{O}_X} \\widetilde N) \\ar[r] &\n\\Gamma(X, \\widetilde{M \\otimes_S N})\n}\n$$\nis commutative where the vertical maps are given by\n(\\ref{equation-map-global-sections}).","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Quasi-coherent sheaves on Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MK","source_file":"constructions.tex","source_line":1592,"source_end_line":1629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1592-L1629","statement_sha256":"fae0bf943e76609604d5b98f5cbc3b2b88bdd29f484fe86beded87b507729730","origin":"The Stacks Project","memory_eligible":false,"source_rank":5121,"rank":5121,"depth":7,"x":1688.611,"y":681.78,"cluster":"schemes"},{"id":"stacks:01MN","tag":"01MN","title":"Invertible sheaves on Proj · Definition 01MN","summary":"Let S be a graded ring. Let X = Proj(S). • We define O_X(n) = widetildeS(n). This is called the nth twist of the structure sheaf of Proj(S). • For any sheaf of O_X-modules F we set F(n) = F ⊗_O_X O_X(n).","statement_latex":"Let $S$ be a graded ring. Let $X = \\text{Proj}(S)$.\n\\begin{enumerate}\n\\item We define $\\mathcal{O}_X(n) = \\widetilde{S(n)}$.\nThis is called the $n$th\n{\\it twist of the structure sheaf of $\\text{Proj}(S)$}.\n\\item For any sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ we set\n$\\mathcal{F}(n) = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(n)$.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Invertible sheaves on Proj","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MN","source_file":"constructions.tex","source_line":1695,"source_end_line":1705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1695-L1705","statement_sha256":"a4fb09e28672974e9a1ba7c42e886a44ff4a7e2cc1e47f46776cf6ee5d009b96","origin":"The Stacks Project","memory_eligible":false,"source_rank":5122,"rank":5122,"depth":0,"x":1498.2,"y":740.523,"cluster":"schemes"},{"id":"stacks:01MS","tag":"01MS","title":"Invertible sheaves on Proj · Lemma 01MS","summary":"Let S be a graded ring. Set X = Proj(S). Let f ∈ S be homogeneous of degree d > 0. The sheaves O_X(nd)|_D_+(f) are invertible, and in fact trivial for all n ∈ Z (see Modules, Definition [Tag 01CS]). The maps ([Tag 01MO]) restricted to D_+(f) O_X(nd)|_D_+(f) ⊗_O_D_+(f) O_X(m)|_D_+(f) → O_X(nd + m)|_D_+(f), the maps ([Tag 03GJ]) restricted to D_+(f) O_X(nd)|_D_+(f) ⊗_O_D_+(f) F(m)|_D_+(f) → F(nd + m)|_D_+(f), and the maps ([Tag 01MQ]) restricted to D_+(f) widetilde…","statement_latex":"Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$.\nLet $f \\in S$ be homogeneous of degree $d > 0$.\nThe sheaves $\\mathcal{O}_X(nd)|_{D_{+}(f)}$ are invertible,\nand in fact trivial for all $n \\in \\mathbf{Z}$\n(see Modules, Definition \\ref{modules-definition-invertible}).\nThe maps (\\ref{equation-multiply}) restricted to $D_{+}(f)$\n$$\n\\mathcal{O}_X(nd)|_{D_{+}(f)} \\otimes_{\\mathcal{O}_{D_{+}(f)}}\n\\mathcal{O}_X(m)|_{D_{+}(f)}\n\\longrightarrow\n\\mathcal{O}_X(nd + m)|_{D_{+}(f)},\n$$\nthe maps (\\ref{equation-multiply-on-sheaf}) restricted to $D_+(f)$\n$$\n\\mathcal{O}_X(nd)|_{D_{+}(f)} \\otimes_{\\mathcal{O}_{D_{+}(f)}}\n\\mathcal{F}(m)|_{D_{+}(f)}\n\\longrightarrow\n\\mathcal{F}(nd + m)|_{D_{+}(f)},\n$$\nand the maps (\\ref{equation-multiply-more-generally})\nrestricted to $D_{+}(f)$\n$$\n\\widetilde M(nd)|_{D_{+}(f)}\n=\n\\widetilde M|_{D_{+}(f)}\n\\otimes_{\\mathcal{O}_{D_{+}(f)}}\n\\mathcal{O}_X(nd)|_{D_{+}(f)}\n\\longrightarrow\n\\widetilde{M(nd)}|_{D_{+}(f)}\n$$\nare isomorphisms for all $n, m \\in \\mathbf{Z}$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Invertible sheaves on Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MS","source_file":"constructions.tex","source_line":1768,"source_end_line":1801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1768-L1801","statement_sha256":"c6d8a73caba0fee9c2a3fde0d991b1d548f646e599761bc85c4da32a14512cb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5123,"rank":5123,"depth":1,"x":1591.687,"y":588.646,"cluster":"schemes"},{"id":"stacks:01MT","tag":"01MT","title":"Invertible sheaves on Proj · Lemma 01MT","summary":"Let S be a graded ring. Let M be a graded S-module. Set X = Proj(S). Assume X is covered by the standard opens D_+(f) with f ∈ S_1, e.g., if S is generated by S_1 over S_0. Then the sheaves O_X(n) are invertible and the maps ([Tag 01MO]), ([Tag 03GJ]), and ([Tag 01MQ]) are isomorphisms. In particular, these maps induce isomorphisms O_X(1)^⊗ n ≅ O_X(n) and widetildeM ⊗_O_X O_X(n) = widetildeM(n) ≅ widetildeM(n) Thus ([Tag 0AG2]) becomes a map M_n → Γ(X, widetildeM(n)) and…","statement_latex":"Let $S$ be a graded ring. Let $M$ be a graded $S$-module.\nSet $X = \\text{Proj}(S)$. Assume $X$ is covered by the standard\nopens $D_+(f)$ with $f \\in S_1$, e.g., if $S$ is generated by $S_1$\nover $S_0$. Then the sheaves $\\mathcal{O}_X(n)$\nare invertible and the maps\n(\\ref{equation-multiply}), (\\ref{equation-multiply-on-sheaf}), and\n(\\ref{equation-multiply-more-generally}) are isomorphisms.\nIn particular, these maps induce isomorphisms\n$$\n\\mathcal{O}_X(1)^{\\otimes n} \\cong\n\\mathcal{O}_X(n)\n\\quad\n\\text{and}\n\\quad\n\\widetilde{M} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(n) =\n\\widetilde{M}(n) \\cong \\widetilde{M(n)}\n$$\nThus (\\ref{equation-map-global-sections-degree-n}) becomes a map\n\\begin{equation}\n\nM_n \\longrightarrow \\Gamma(X, \\widetilde{M}(n))\n\\end{equation}\nand (\\ref{equation-global-sections-more-generally}) becomes a map\n\\begin{equation}\n\nM \\longrightarrow\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\Gamma(X, \\widetilde{M}(n)).\n\\end{equation}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Invertible sheaves on Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MT","source_file":"constructions.tex","source_line":1814,"source_end_line":1844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1814-L1844","statement_sha256":"160d2ceba1753d5e3c424f9dfeb8259ced0fbf0f88261cd5014fe56513f35353","origin":"The Stacks Project","memory_eligible":false,"source_rank":5124,"rank":5124,"depth":2,"x":1645.066,"y":754.245,"cluster":"schemes"},{"id":"stacks:01MU","tag":"01MU","title":"Invertible sheaves on Proj · Lemma 01MU","summary":"Let S be a graded ring. Set X = Proj(S). Fix d ≥ 1 an integer. The following open subsets of X are equal: • The largest open subset W = W_d ⊂ X such that each O_X(dn)|_W is invertible and all the multiplication maps O_X(nd)|_W ⊗_O_W O_X(md)|_W → O_X(nd + md)|_W (see [Tag 01MO]) are isomorphisms. • The union of the open subsets D_+(fg) with f, g ∈ S homogeneous and deg(f) = deg(g) + d. Moreover, all the maps widetilde M(nd)|_W = widetilde M|_W ⊗_O_W O_X(nd)|_W →…","statement_latex":"Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Fix $d \\geq 1$ an\ninteger. The following open subsets of $X$ are equal:\n\\begin{enumerate}\n\\item The largest open subset $W = W_d \\subset X$ such that\neach $\\mathcal{O}_X(dn)|_W$ is invertible and all the\nmultiplication maps\n$\\mathcal{O}_X(nd)|_W \\otimes_{\\mathcal{O}_W} \\mathcal{O}_X(md)|_W\n\\to \\mathcal{O}_X(nd + md)|_W$\n(see \\ref{equation-multiply}) are isomorphisms.\n\\item The union of the open subsets $D_{+}(fg)$ with\n$f, g \\in S$ homogeneous and $\\deg(f) = \\deg(g) + d$.\n\\end{enumerate}\nMoreover, all the maps\n$\\widetilde M(nd)|_W = \\widetilde M|_W \\otimes_{\\mathcal{O}_W}\n\\mathcal{O}_X(nd)|_W \\to \\widetilde{M(nd)}|_W$\n(see \\ref{equation-multiply-more-generally}) are isomorphisms.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Invertible sheaves on Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MU","source_file":"constructions.tex","source_line":1852,"source_end_line":1870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1852-L1870","statement_sha256":"74611fbfd79ad87108f4e37121d8b20301b65f18c6deae24fa474c8d41ea393a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5125,"rank":5125,"depth":2,"x":1471.953,"y":662.113,"cluster":"schemes"},{"id":"stacks:01MV","tag":"01MV","title":"Invertible sheaves on Proj · Lemma 01MV","summary":"Let S be a graded ring. Set X = Proj(S). Fix d ≥ 1 an integer. Let W = W_d ⊂ X be the open subscheme defined in Lemma [Tag 01MU]. Let n ≥ 1 and f ∈ S_nd. Denote s ∈ Γ(W, O_W(nd)) the section which is the image of f via ([Tag 01MP]) restricted to W. Then W_s = D_+(f) ∩ W.","statement_latex":"Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Fix $d \\geq 1$ an\ninteger. Let $W = W_d \\subset X$ be the open subscheme defined in\nLemma \\ref{lemma-where-invertible}. Let $n \\geq 1$ and $f \\in S_{nd}$.\nDenote $s \\in \\Gamma(W, \\mathcal{O}_W(nd))$ the section which is\nthe image of $f$ via (\\ref{equation-global-sections}) restricted to $W$. Then\n$$\nW_s = D_{+}(f) \\cap W.\n$$","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Invertible sheaves on Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MV","source_file":"constructions.tex","source_line":1918,"source_end_line":1928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1918-L1928","statement_sha256":"7aef3976ba069c2975045e08cc9835306cecdd867f4f1afbe574b0daffb16e5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5126,"rank":5126,"depth":3,"x":1674.37,"y":631.721,"cluster":"schemes"},{"id":"stacks:01MW","tag":"01MW","title":"Invertible sheaves on Proj · Lemma 01MW","summary":"Let S be a graded ring. Let X = Proj(S). Let Y ⊂ X be a quasi-compact open subscheme. Denote O_Y(n) the restriction of O_X(n) to Y. There exists an integer d ≥ 1 such that • the subscheme Y is contained in the open W_d defined in Lemma [Tag 01MU], • the sheaf O_Y(dn) is invertible for all n ∈ Z, • all the maps O_Y(nd) ⊗_O_Y O_Y(m) → O_Y(nd + m) of Equation ([Tag 01MO]) are isomorphisms, • all the maps widetilde M(nd)|_Y = widetilde M|_Y ⊗_O_Y O_X(nd)|_Y →…","statement_latex":"Let $S$ be a graded ring.\nLet $X = \\text{Proj}(S)$.\nLet $Y \\subset X$ be a quasi-compact open subscheme.\nDenote $\\mathcal{O}_Y(n)$ the restriction of\n$\\mathcal{O}_X(n)$ to $Y$.\nThere exists an integer $d \\geq 1$ such that\n\\begin{enumerate}\n\\item the subscheme $Y$ is contained in the open $W_d$ defined\nin Lemma \\ref{lemma-where-invertible},\n\\item the sheaf $\\mathcal{O}_Y(dn)$ is invertible for all $n \\in \\mathbf{Z}$,\n\\item all the maps\n$\\mathcal{O}_Y(nd) \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(m)\n\\longrightarrow\n\\mathcal{O}_Y(nd + m)$\nof Equation (\\ref{equation-multiply}) are isomorphisms,\n\\item all the maps\n$\\widetilde M(nd)|_Y = \\widetilde M|_Y \\otimes_{\\mathcal{O}_Y}\n\\mathcal{O}_X(nd)|_Y \\to \\widetilde{M(nd)}|_Y$\n(see \\ref{equation-multiply-more-generally}) are isomorphisms,\n\\item given $f \\in S_{nd}$ denote $s \\in \\Gamma(Y, \\mathcal{O}_Y(nd))$\nthe image of $f$ via (\\ref{equation-global-sections})\nrestricted to $Y$, then $D_{+}(f) \\cap Y = Y_s$,\n\\item a basis for the topology on $Y$ is given\nby the collection of opens $Y_s$, where $s \\in \\Gamma(Y, \\mathcal{O}_Y(nd))$,\n$n \\geq 1$, and\n\\item a basis for the topology of $Y$ is given\nby those opens $Y_s \\subset Y$, for\n$s \\in \\Gamma(Y, \\mathcal{O}_Y(nd))$, $n \\geq 1$ which are affine.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Invertible sheaves on Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MW","source_file":"constructions.tex","source_line":1947,"source_end_line":1978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1947-L1978","statement_sha256":"9b641b30e7c6c45615bf2617b68ec3a4ec7868e071e9576d0ac192791344e695","origin":"The Stacks Project","memory_eligible":false,"source_rank":5127,"rank":5127,"depth":4,"x":1549.136,"y":769.444,"cluster":"schemes"},{"id":"stacks:0B5I","tag":"0B5I","title":"Invertible sheaves on Proj · Lemma 0B5I","summary":"Let S be a graded ring. Set X = Proj(S). Let F be a quasi-coherent O_X-module. Set M = bigoplus_n ∈ Z Γ(X, F(n)) as a graded S-module, using ([Tag 03GK]) and ([Tag 01MP]). Then there is a canonical O_X-module map widetildeM → F functorial in F such that the induced map M_0 → Γ(X, F) is the identity.","statement_latex":"Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nSet $M = \\bigoplus_{n \\in \\mathbf{Z}} \\Gamma(X, \\mathcal{F}(n))$ as\na graded $S$-module, using\n(\\ref{equation-global-sections-module}) and (\\ref{equation-global-sections}).\nThen there is a canonical $\\mathcal{O}_X$-module map\n$$\n\\widetilde{M} \\longrightarrow \\mathcal{F}\n$$\nfunctorial in $\\mathcal{F}$ such that the induced map\n$M_0 \\to \\Gamma(X, \\mathcal{F})$ is the identity.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Invertible sheaves on Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5I","source_file":"constructions.tex","source_line":1995,"source_end_line":2008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L1995-L2008","statement_sha256":"5929141977410c4537305cad7f2384b715b250d3a703a79928c3a0f039bb3020","origin":"The Stacks Project","memory_eligible":false,"source_rank":5128,"rank":5128,"depth":9,"x":1530.668,"y":596.257,"cluster":"schemes"},{"id":"stacks:01MY","tag":"01MY","title":"Functoriality of Proj · Lemma 01MY","summary":"Let A, B be two graded rings. Set X = Proj(A) and Y = Proj(B). Let ψ : A → B be a graded ring map. Set U(ψ) = ⋃_f ∈ A_+ homogeneous D_+(ψ(f)) ⊂ Y. Then there is a canonical morphism of schemes r_ψ : U(ψ) → X and a map of Z-graded O_U(ψ)-algebras theta = theta_ψ : r_ψ^*( bigoplus_d ∈ Z O_X(d) ) → bigoplus_d ∈ Z O_U(ψ)(d). The triple (U(ψ), r_ψ, theta) is characterized by the following properties: • For every d ≥ 0 the diagram xymatrix A_d ar[d] ar[rr]_ψ & & B_d ar[d] Γ(X,…","statement_latex":"Let $A$, $B$ be two graded rings.\nSet $X = \\text{Proj}(A)$ and $Y = \\text{Proj}(B)$.\nLet $\\psi : A \\to B$ be a graded ring map.\nSet\n$$\nU(\\psi)\n=\n\\bigcup\\nolimits_{f \\in A_{+}\\ \\text{homogeneous}} D_{+}(\\psi(f))\n\\subset Y.\n$$\nThen there is a canonical morphism of schemes\n$$\nr_\\psi :\nU(\\psi)\n\\longrightarrow\nX\n$$\nand a map of $\\mathbf{Z}$-graded $\\mathcal{O}_{U(\\psi)}$-algebras\n$$\n\\theta = \\theta_\\psi :\nr_\\psi^*\\left(\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_X(d)\n\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{U(\\psi)}(d).\n$$\nThe triple $(U(\\psi), r_\\psi, \\theta)$ is\ncharacterized by the following properties:\n\\begin{enumerate}\n\\item For every $d \\geq 0$ the diagram\n$$\n\\xymatrix{\nA_d \\ar[d] \\ar[rr]_{\\psi} & &\nB_d \\ar[d] \\\\\n\\Gamma(X, \\mathcal{O}_X(d)) \\ar[r]^-\\theta &\n\\Gamma(U(\\psi), \\mathcal{O}_Y(d)) &\n\\Gamma(Y, \\mathcal{O}_Y(d)) \\ar[l]\n}\n$$\nis commutative.\n\\item For any $f \\in A_{+}$ homogeneous\nwe have $r_\\psi^{-1}(D_{+}(f)) = D_{+}(\\psi(f))$ and\nthe restriction of $r_\\psi$ to $D_{+}(\\psi(f))$\ncorresponds to the ring map\n$A_{(f)} \\to B_{(\\psi(f))}$ induced by $\\psi$.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MY","source_file":"constructions.tex","source_line":2053,"source_end_line":2101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2053-L2101","statement_sha256":"114b8a11d8912568dbc1f18c4be719dfb98bdacd3e97657efe2ef92ef5612bcf","origin":"The Stacks Project","memory_eligible":false,"source_rank":5129,"rank":5129,"depth":0,"x":1684.06,"y":713.869,"cluster":"schemes"},{"id":"stacks:01MZ","tag":"01MZ","title":"Functoriality of Proj · Lemma 01MZ","summary":"Let A, B, and C be graded rings. Set X = Proj(A), Y = Proj(B) and Z = Proj(C). Let φ : A → B, ψ : B → C be graded ring maps. Then we have U(ψ ∘ φ) = r_ψ^-1(U(φ)) and r_ψ ∘ φ = r_φ ∘ r_ψ|_U(ψ ∘ φ). In addition we have theta_ψ ∘ r_ψ^*theta_φ = theta_ψ ∘ φ with obvious notation.","statement_latex":"Let $A$, $B$, and $C$ be graded rings.\nSet $X = \\text{Proj}(A)$, $Y = \\text{Proj}(B)$ and $Z = \\text{Proj}(C)$.\nLet $\\varphi : A \\to B$, $\\psi : B \\to C$ be graded ring maps.\nThen we have\n$$\nU(\\psi \\circ \\varphi) = r_\\psi^{-1}(U(\\varphi))\n\\quad\n\\text{and}\n\\quad\nr_{\\psi \\circ \\varphi}\n=\nr_\\varphi \\circ r_\\psi|_{U(\\psi \\circ \\varphi)}.\n$$\nIn addition we have\n$$\n\\theta_\\psi \\circ r_\\psi^*\\theta_\\varphi\n=\n\\theta_{\\psi \\circ \\varphi}\n$$\nwith obvious notation.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01MZ","source_file":"constructions.tex","source_line":2137,"source_end_line":2159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2137-L2159","statement_sha256":"859875642f0d0034018fe1875ea34dc7298111ee4dd90740a41d05fb076277a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5130,"rank":5130,"depth":0,"x":1475.693,"y":714.182,"cluster":"schemes"},{"id":"stacks:01N0","tag":"01N0","title":"Functoriality of Proj · Lemma 01N0","summary":"With hypotheses and notation as in Lemma [Tag 01MY] above. Assume A_d → B_d is surjective for all d gg 0. Then • U(ψ) = Y, • r_ψ : Y → X is a closed immersion, and • the maps theta : r_ψ^*O_X(n) → O_Y(n) are surjective but not isomorphisms in general (even if A → B is surjective).","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.\nAssume $A_d \\to B_d$ is surjective for all $d \\gg 0$. Then\n\\begin{enumerate}\n\\item $U(\\psi) = Y$,\n\\item $r_\\psi : Y \\to X$ is a closed immersion, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$\nare surjective but not isomorphisms in general (even if $A \\to B$ is\nsurjective).\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01N0","source_file":"constructions.tex","source_line":2165,"source_end_line":2176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2165-L2176","statement_sha256":"cfd2f6e781d87624da2068e3677a1a9659de3fface9aa9bd8bb759cda8856448","origin":"The Stacks Project","memory_eligible":false,"source_rank":5131,"rank":5131,"depth":1,"x":1629.586,"y":595.336,"cluster":"schemes"},{"id":"stacks:07ZE","tag":"07ZE","title":"Functoriality of Proj · Lemma 07ZE","summary":"With hypotheses and notation as in Lemma [Tag 01MY] above. Assume A_d → B_d is an isomorphism for all d gg 0. Then • U(ψ) = Y, • r_ψ : Y → X is an isomorphism, and • the maps theta : r_ψ^*O_X(n) → O_Y(n) are isomorphisms.","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.\nAssume $A_d \\to B_d$ is an isomorphism for all $d \\gg 0$. Then\n\\begin{enumerate}\n\\item $U(\\psi) = Y$,\n\\item $r_\\psi : Y \\to X$ is an isomorphism, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$\nare isomorphisms.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZE","source_file":"constructions.tex","source_line":2198,"source_end_line":2208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2198-L2208","statement_sha256":"86408eefefff3debf6780e55ff76b58bd037e83a80818b8b20bcb1da6e63f786","origin":"The Stacks Project","memory_eligible":false,"source_rank":5132,"rank":5132,"depth":2,"x":1611.622,"y":770.857,"cluster":"schemes"},{"id":"stacks:01N1","tag":"01N1","title":"Functoriality of Proj · Lemma 01N1","summary":"With hypotheses and notation as in Lemma [Tag 01MY] above. Assume A_d → B_d is surjective for d gg 0 and that A is generated by A_1 over A_0. Then • U(ψ) = Y, • r_ψ : Y → X is a closed immersion, and • the maps theta : r_ψ^*O_X(n) → O_Y(n) are isomorphisms.","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.\nAssume $A_d \\to B_d$ is surjective for $d \\gg 0$ and that $A$ is generated\nby $A_1$ over $A_0$. Then\n\\begin{enumerate}\n\\item $U(\\psi) = Y$,\n\\item $r_\\psi : Y \\to X$ is a closed immersion, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$\nare isomorphisms.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01N1","source_file":"constructions.tex","source_line":2218,"source_end_line":2229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2218-L2229","statement_sha256":"adb62af6d2341d5fe13509490e0cebb3c09780863fede8dfb4ad8689d5ec5cad","origin":"The Stacks Project","memory_eligible":false,"source_rank":5133,"rank":5133,"depth":3,"x":1483.311,"y":630.79,"cluster":"schemes"},{"id":"stacks:01N2","tag":"01N2","title":"Functoriality of Proj · Lemma 01N2","summary":"[EGA] With hypotheses and notation as in Lemma [Tag 01MY] above. Assume there exists a ring map R → A_0 and a ring map R → R' such that B = R' ⊗_R A. Then • U(ψ) = Y, • the diagram xymatrix Y = Proj(B) ar[r]_r_ψ ar[d] & Proj(A) = X ar[d] Spec(R') ar[r] & Spec(R) is a fibre product square, and • the maps theta : r_ψ^*O_X(n) → O_Y(n) are isomorphisms.","statement_latex":"\\begin{reference}\n\\cite[II, Proposition 2.8.10]{EGA}\n\\end{reference}\nWith hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.\nAssume there exists a ring map $R \\to A_0$ and a ring map\n$R \\to R'$ such that $B = R' \\otimes_R A$. Then\n\\begin{enumerate}\n\\item $U(\\psi) = Y$,\n\\item the diagram\n$$\n\\xymatrix{\nY = \\text{Proj}(B) \\ar[r]_{r_\\psi} \\ar[d] &\n\\text{Proj}(A) = X \\ar[d] \\\\\n\\Spec(R') \\ar[r] &\n\\Spec(R)\n}\n$$\nis a fibre product square, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$\nare isomorphisms.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01N2","source_file":"constructions.tex","source_line":2244,"source_end_line":2267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2244-L2267","statement_sha256":"b7ea1509c6338d21f960a4781a69fd603ca6c560ef91dcd39e8180e20c7809ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":5134,"rank":5134,"depth":1,"x":1691.222,"y":661.364,"cluster":"schemes"},{"id":"stacks:01N3","tag":"01N3","title":"Functoriality of Proj · Lemma 01N3","summary":"With hypotheses and notation as in Lemma [Tag 01MY] above. Assume there exists a g ∈ A_0 such that ψ induces an isomorphism A_g → B. Then U(ψ) = Y, r_ψ : Y → X is an open immersion which induces an isomorphism of Y with the inverse image of D(g) ⊂ Spec(A_0). Moreover the map theta is an isomorphism.","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-proj} above.\nAssume there exists a $g \\in A_0$ such that $\\psi$ induces an\nisomorphism $A_g \\to B$. Then\n$U(\\psi) = Y$, $r_\\psi : Y \\to X$ is an open immersion\nwhich induces an isomorphism of $Y$ with the inverse image\nof $D(g) \\subset \\Spec(A_0)$. Moreover the map $\\theta$\nis an isomorphism.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01N3","source_file":"constructions.tex","source_line":2274,"source_end_line":2283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2274-L2283","statement_sha256":"776697434e011c52dc7714ff8eb867e96d8d74ed677ea724ce72cbe42c089c18","origin":"The Stacks Project","memory_eligible":false,"source_rank":5135,"rank":5135,"depth":2,"x":1512.763,"y":757.094,"cluster":"schemes"},{"id":"stacks:0B5J","tag":"0B5J","title":"Functoriality of Proj · Lemma 0B5J","summary":"Let S be a graded ring. Let d ≥ 1. Set S' = S^(d) with notation as in Algebra, Section [Tag 00JL]. Set X = Proj(S) and X' = Proj(S'). There is a canonical isomorphism i : X → X' of schemes such that • for any graded S-module M setting M' = M^(d), we have a canonical isomorphism widetildeM → i^*widetildeM', • we have canonical isomorphisms O_X(nd) → i^*O_X'(n) and these isomorphisms are compatible with the multiplication maps of Lemma [Tag 01MK] and hence with the maps…","statement_latex":"Let $S$ be a graded ring. Let $d \\geq 1$. Set $S' = S^{(d)}$ with notation\nas in Algebra, Section \\ref{algebra-section-graded}. Set\n$X = \\text{Proj}(S)$ and $X' = \\text{Proj}(S')$. There is a canonical\nisomorphism $i : X \\to X'$ of schemes such that\n\\begin{enumerate}\n\\item for any graded $S$-module $M$ setting $M' = M^{(d)}$,\nwe have a canonical isomorphism $\\widetilde{M} \\to i^*\\widetilde{M'}$,\n\\item we have canonical isomorphisms\n$\\mathcal{O}_{X}(nd) \\to i^*\\mathcal{O}_{X'}(n)$\n\\end{enumerate}\nand these isomorphisms are compatible with the multiplication maps\nof Lemma \\ref{lemma-widetilde-tensor} and hence with the maps\n(\\ref{equation-multiply}),\n(\\ref{equation-multiply-on-sheaf}),\n(\\ref{equation-global-sections}),\n(\\ref{equation-global-sections-module}),\n(\\ref{equation-multiply-more-generally}), and\n(\\ref{equation-global-sections-more-generally}) (see proof for precise\nstatements.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5J","source_file":"constructions.tex","source_line":2289,"source_end_line":2310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2289-L2310","statement_sha256":"1eaff62b2cd9f9800bcc237955a354faa35ce4bf3dc19f7932b946371e95b133","origin":"The Stacks Project","memory_eligible":false,"source_rank":5136,"rank":5136,"depth":8,"x":1567.544,"y":584.703,"cluster":"schemes"},{"id":"stacks:01N8","tag":"01N8","title":"Morphisms into Proj · Lemma 01N8","summary":"Let S be a graded ring, and X = Proj(S). Let d ≥ 1 and U_d ⊂ X as above. Let Y be a scheme. Let L be an invertible sheaf on Y. Let ψ : S^(d) → Γ_*(Y, L) be a graded ring homomorphism such that L is generated by the sections in the image of ψ|_S_d : S_d → Γ(Y, L). Then there exist a morphism φ : Y → X such that φ(Y) ⊂ U_d and an isomorphism α : φ^*O_U_d(d) → L such that ψ_φ^d agrees with ψ via α: xymatrix Γ_*(Y, L) & Γ_*(Y, φ^*O_U_d(d)) ar[l]^-α & Γ_*(U_d, O_U_d(d))…","statement_latex":"Let $S$ be a graded ring, and $X = \\text{Proj}(S)$.\nLet $d \\geq 1$ and $U_d \\subset X$ as above.\nLet $Y$ be a scheme.\nLet $\\mathcal{L}$ be an invertible sheaf on $Y$.\nLet $\\psi : S^{(d)} \\to \\Gamma_*(Y, \\mathcal{L})$ be\na graded ring homomorphism such that $\\mathcal{L}$ is\ngenerated by the sections in the image of\n$\\psi|_{S_d} : S_d \\to \\Gamma(Y, \\mathcal{L})$.\nThen there exist a morphism\n$\\varphi : Y \\to X$ such that $\\varphi(Y) \\subset U_d$ and\nan isomorphism $\\alpha : \\varphi^*\\mathcal{O}_{U_d}(d) \\to \\mathcal{L}$\nsuch that $\\psi_\\varphi^d$ agrees with $\\psi$ via $\\alpha$:\n$$\n\\xymatrix{\n\\Gamma_*(Y, \\mathcal{L}) &\n\\Gamma_*(Y, \\varphi^*\\mathcal{O}_{U_d}(d)) \\ar[l]^-\\alpha &\n\\Gamma_*(U_d, \\mathcal{O}_{U_d}(d)) \\ar[l]^-{\\varphi^*} \\\\\nS^{(d)} \\ar[u]^\\psi & &\nS^{(d)} \\ar[u]^{\\psi^d} \\ar[ul]^{\\psi^d_\\varphi} \\ar[ll]_{\\text{id}}\n}\n$$\ncommutes. Moreover, the pair $(\\varphi, \\alpha)$ is unique.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Morphisms into Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01N8","source_file":"constructions.tex","source_line":2448,"source_end_line":2472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2448-L2472","statement_sha256":"7b9a293d9107ba9b7edd6f2b359c39f709ee2e8140b1bfa70fd5ba14a4721f40","origin":"The Stacks Project","memory_eligible":false,"source_rank":5137,"rank":5137,"depth":14,"x":1666.087,"y":743.399,"cluster":"schemes"},{"id":"stacks:01N9","tag":"01N9","title":"Morphisms into Proj · Lemma 01N9","summary":"Let S be a graded ring. Let X = Proj(S). The open subscheme U_d ⊂ X ([Tag 01N5]) represents the functor F_d and the triple (d, O_U_d(d), ψ^d) defined above is the universal family (see Schemes, Section [Tag 01JF]).","statement_latex":"Let $S$ be a graded ring.\nLet $X = \\text{Proj}(S)$.\nThe open subscheme $U_d \\subset X$ (\\ref{equation-Ud}) represents the\nfunctor $F_d$ and the triple $(d, \\mathcal{O}_{U_d}(d), \\psi^d)$\ndefined above is the universal family (see\nSchemes, Section \\ref{schemes-section-representable}).","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Morphisms into Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01N9","source_file":"constructions.tex","source_line":2597,"source_end_line":2605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2597-L2605","statement_sha256":"a2941e748cc42d5cfcdcc8952d400a210ec71f8f7f53dbe00662b2a37fd9c928","origin":"The Stacks Project","memory_eligible":false,"source_rank":5138,"rank":5138,"depth":15,"x":1465.183,"y":682.104,"cluster":"schemes"},{"id":"stacks:01NA","tag":"01NA","title":"Morphisms into Proj · Lemma 01NA","summary":"Let S be a graded ring generated as an S_0-algebra by the elements of S_1. In this case the scheme X = Proj(S) represents the functor which associates to a scheme Y the set of pairs (L, ψ), where • L is an invertible O_Y-module, and • ψ : S → Γ_*(Y, L) is a graded ring homomorphism such that L is generated by the global sections ψ(f), with f ∈ S_1 up to strict equivalence as above.","statement_latex":"Let $S$ be a graded ring generated as an $S_0$-algebra by\nthe elements of $S_1$. In this case the scheme $X = \\text{Proj}(S)$\nrepresents the functor which associates to a scheme\n$Y$ the set of pairs $(\\mathcal{L}, \\psi)$, where\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is an invertible $\\mathcal{O}_Y$-module, and\n\\item $\\psi : S \\to \\Gamma_*(Y, \\mathcal{L})$ is a graded\nring homomorphism such that $\\mathcal{L}$ is generated by\nthe global sections $\\psi(f)$, with $f \\in S_1$\n\\end{enumerate}\nup to strict equivalence as above.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Morphisms into Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NA","source_file":"constructions.tex","source_line":2611,"source_end_line":2624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2611-L2624","statement_sha256":"d03cffdbf70ce22c5a4e443e31dd54bd76b4d58e595dc4998b4d41f7f9d65c46","origin":"The Stacks Project","memory_eligible":false,"source_rank":5139,"rank":5139,"depth":16,"x":1663.226,"y":613.094,"cluster":"schemes"},{"id":"stacks:01NB","tag":"01NB","title":"Morphisms into Proj · Lemma 01NB","summary":"Let S be a graded ring. Set X = Proj(S). Let T be a scheme. Let (d, L, ψ) and (d', L', ψ') be two triples over T. The following are equivalent: • Let n = lcm(d, d'). Write n = ad = a'd'. There exists an isomorphism β : L^⊗ a → (L')^⊗ a' with the property that β ∘ ψ|_S^(n) and ψ'|_S^(n) agree as graded ring maps S^(n) → Γ_*(Y, (L')^⊗ n). • The triples (d, L, ψ) and (d', L', ψ') are equivalent. • For some positive integer n = ad = a'd' there exists an isomorphism β : L^⊗ a…","statement_latex":"Let $S$ be a graded ring. Set $X = \\text{Proj}(S)$. Let $T$ be a scheme.\nLet $(d, \\mathcal{L}, \\psi)$ and $(d', \\mathcal{L}', \\psi')$\nbe two triples over $T$. The following are equivalent:\n\\begin{enumerate}\n\\item Let $n = \\text{lcm}(d, d')$. Write $n = ad = a'd'$. There exists\nan isomorphism\n$\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$\nwith the property that\n$\\beta \\circ \\psi|_{S^{(n)}}$ and $\\psi'|_{S^{(n)}}$ agree\nas graded ring maps $S^{(n)} \\to \\Gamma_*(Y, (\\mathcal{L}')^{\\otimes n})$.\n\\item The triples $(d, \\mathcal{L}, \\psi)$ and $(d', \\mathcal{L}', \\psi')$\nare equivalent.\n\\item For some positive integer $n = ad = a'd'$ there exists\nan isomorphism\n$\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$\nwith the property that\n$\\beta \\circ \\psi|_{S^{(n)}}$ and $\\psi'|_{S^{(n)}}$ agree\nas graded ring maps $S^{(n)} \\to \\Gamma_*(Y, (\\mathcal{L}')^{\\otimes n})$.\n\\item The morphisms $\\varphi : T \\to X$ and $\\varphi' : T \\to X$\nassociated to $(d, \\mathcal{L}, \\psi)$ and $(d', \\mathcal{L}', \\psi')$\nare equal.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Morphisms into Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NB","source_file":"constructions.tex","source_line":2647,"source_end_line":2671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2647-L2671","statement_sha256":"f161c3205cd1d78a829f4846e6c2d3dd38d26d206421944c978551a3f194063d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5140,"rank":5140,"depth":15,"x":1572.411,"y":776.855,"cluster":"schemes"},{"id":"stacks:01NC","tag":"01NC","title":"Morphisms into Proj · Lemma 01NC","summary":"Let S be a graded ring. Let X = Proj(S). The functor F defined above is representable by the scheme X.","statement_latex":"Let $S$ be a graded ring.\nLet $X = \\text{Proj}(S)$.\nThe functor $F$ defined above is representable by the scheme $X$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Morphisms into Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NC","source_file":"constructions.tex","source_line":2745,"source_end_line":2750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2745-L2750","statement_sha256":"a6244864114cf75e399c9665bdf1f91557624619b5289df42ccc688fe962086a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5141,"rank":5141,"depth":0,"x":1507.489,"y":604.045,"cluster":"schemes"},{"id":"stacks:01NE","tag":"01NE","title":"Projective space · Lemma 01NE","summary":"Let S = Z[T_0, …, T_n] with deg(T_i) = 1. The scheme P^n_Z = Proj(S) represents the functor which associates to a scheme Y the pairs (L, (s_0, …, s_n)) where • L is an invertible O_Y-module, and • s_0, …, s_n are global sections of L which generate L up to the following equivalence: (L, (s_0, …, s_n)) sim (N, (t_0, …, t_n)) ⇔ there exists an isomorphism β : L → N with β(s_i) = t_i for i = 0, …, n.","statement_latex":"Let $S = \\mathbf{Z}[T_0, \\ldots, T_n]$ with $\\deg(T_i) = 1$.\nThe scheme\n$$\n\\mathbf{P}^n_{\\mathbf{Z}} = \\text{Proj}(S)\n$$\nrepresents the functor which associates to a scheme $Y$ the pairs\n$(\\mathcal{L}, (s_0, \\ldots, s_n))$ where\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is an invertible $\\mathcal{O}_Y$-module, and\n\\item $s_0, \\ldots, s_n$ are global sections of $\\mathcal{L}$\nwhich generate $\\mathcal{L}$\n\\end{enumerate}\nup to the following equivalence:\n$(\\mathcal{L}, (s_0, \\ldots, s_n)) \\sim\n(\\mathcal{N}, (t_0, \\ldots, t_n))$ $\\Leftrightarrow$ there exists\nan isomorphism $\\beta : \\mathcal{L} \\to \\mathcal{N}$\nwith $\\beta(s_i) = t_i$ for $i = 0, \\ldots, n$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NE","source_file":"constructions.tex","source_line":2784,"source_end_line":2803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2784-L2803","statement_sha256":"c1a7f83c142d93b1795e5ec4937ab00e261638af0eadb0d7523f22f9a823adf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5142,"rank":5142,"depth":17,"x":1694.895,"y":694.908,"cluster":"schemes"},{"id":"stacks:01NF","tag":"01NF","title":"Projective space · Definition 01NF","summary":"The scheme P^n_Z = Proj(Z[T_0, …, T_n]) is called projective n-space over Z. Its base change P^n_S to a scheme S is called projective n-space over S. If R is a ring the base change to Spec(R) is denoted P^n_R and called projective n-space over R.","statement_latex":"The scheme\n$\\mathbf{P}^n_{\\mathbf{Z}} = \\text{Proj}(\\mathbf{Z}[T_0, \\ldots, T_n])$\nis called {\\it projective $n$-space over $\\mathbf{Z}$}.\nIts base change $\\mathbf{P}^n_S$ to a scheme $S$ is called\n{\\it projective $n$-space over $S$}. If $R$ is a ring the base change\nto $\\Spec(R)$ is denoted $\\mathbf{P}^n_R$ and called\n{\\it projective $n$-space over $R$}.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NF","source_file":"constructions.tex","source_line":2818,"source_end_line":2827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2818-L2827","statement_sha256":"276bfa77806204f3bc8169ec6c7a2606b5065ff6475b5a2be4cf2a4bae209085","origin":"The Stacks Project","memory_eligible":false,"source_rank":5143,"rank":5143,"depth":0,"x":1483.0,"y":734.363,"cluster":"schemes"},{"id":"stacks:01NG","tag":"01NG","title":"Projective space · Lemma 01NG","summary":"Projective n-space over Z is covered by n + 1 standard opens P^n_Z = ⋃_i = 0, …, n D_+(T_i) where each D_+(T_i) is isomorphic to A^n_Z affine n-space over Z.","statement_latex":"Projective $n$-space over $\\mathbf{Z}$ is covered by\n$n + 1$ standard opens\n$$\n\\mathbf{P}^n_{\\mathbf{Z}} =\n\\bigcup\\nolimits_{i = 0, \\ldots, n} D_{+}(T_i)\n$$\nwhere each $D_{+}(T_i)$ is isomorphic to $\\mathbf{A}^n_{\\mathbf{Z}}$\naffine $n$-space over $\\mathbf{Z}$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NG","source_file":"constructions.tex","source_line":2859,"source_end_line":2869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2859-L2869","statement_sha256":"e2891da57b6cc56c8ca80e606c161f3576d137388e168e2bf8ebacfb6264b4bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5144,"rank":5144,"depth":0,"x":1607.892,"y":584.59,"cluster":"schemes"},{"id":"stacks:01NH","tag":"01NH","title":"Projective space · Lemma 01NH","summary":"Let S be a scheme. The structure morphism P^n_S → S is • separated, • quasi-compact, • satisfies the existence and uniqueness parts of the valuative criterion, and • universally closed.","statement_latex":"Let $S$ be a scheme.\nThe structure morphism $\\mathbf{P}^n_S \\to S$ is\n\\begin{enumerate}\n\\item separated,\n\\item quasi-compact,\n\\item satisfies the existence and uniqueness parts of the valuative criterion,\nand\n\\item universally closed.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NH","source_file":"constructions.tex","source_line":2888,"source_end_line":2899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2888-L2899","statement_sha256":"50fb889b3cc9cc1ea6c66a07dc2544cf8c9742dcc949f608750f766b2ff33c3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5145,"rank":5145,"depth":17,"x":1636.326,"y":766.435,"cluster":"schemes"},{"id":"stacks:01WD","tag":"01WD","title":"Segre embedding · Lemma 01WD","summary":"Let S be a scheme. There exists a closed immersion P^n_S ×_S P^m_S → P^nm + n + m_S called the Segre embedding.","statement_latex":"Let $S$ be a scheme. There exists a closed immersion\n$$\n\\mathbf{P}^n_S \\times_S \\mathbf{P}^m_S\n\\longrightarrow\n\\mathbf{P}^{nm + n + m}_S\n$$\ncalled the {\\it Segre embedding}.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WD","source_file":"constructions.tex","source_line":2926,"source_end_line":2935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2926-L2935","statement_sha256":"35e4b87f045a661c8a9a0710d8f4f62bf68bde1d904db5ddadafcd0fcb2bbf19","origin":"The Stacks Project","memory_eligible":false,"source_rank":5146,"rank":5146,"depth":18,"x":1468.631,"y":648.131,"cluster":"schemes"},{"id":"stacks:03GL","tag":"03GL","title":"Projective space · Lemma 03GL","summary":"Let R be a ring. Let Z ⊂ P^n_R be a closed subscheme. Let I_d = Ker( R[T_0, …, T_n]_d → Γ(Z, O_P^n_R(d)|_Z)) Then I = bigoplus I_d ⊂ R[T_0, …, T_n] is a graded ideal and Z = Proj(R[T_0, …, T_n]/I).","statement_latex":"Let $R$ be a ring. Let $Z \\subset \\mathbf{P}^n_R$ be a closed subscheme.\nLet\n$$\nI_d = \\Ker\\left(\nR[T_0, \\ldots, T_n]_d\n\\longrightarrow\n\\Gamma(Z, \\mathcal{O}_{\\mathbf{P}^n_R}(d)|_Z)\\right)\n$$\nThen $I = \\bigoplus I_d \\subset R[T_0, \\ldots, T_n]$ is\na graded ideal and $Z = \\text{Proj}(R[T_0, \\ldots, T_n]/I)$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GL","source_file":"constructions.tex","source_line":2988,"source_end_line":3000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L2988-L3000","statement_sha256":"cb21d588dabca506f0766462db1895cbc5b98599a0dea4a69032884b15c6ab3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5147,"rank":5147,"depth":4,"x":1688.065,"y":640.19,"cluster":"schemes"},{"id":"stacks:03GM","tag":"03GM","title":"Projective space · Lemma 03GM","summary":"Let R be a ring. Let F be a quasi-coherent sheaf on P^n_R. For d ≥ 0 set M_d = Γ(P^n_R, F ⊗_O_P^n_R O_P^n_R(d)) = Γ(P^n_R, F(d)) Then M = bigoplus_d ≥ 0 M_d is a graded R[T_0, …, R_n]-module and there is a canonical isomorphism F = widetildeM.","statement_latex":"Let $R$ be a ring.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathbf{P}^n_R$.\nFor $d \\geq 0$ set\n$$\nM_d\n=\n\\Gamma(\\mathbf{P}^n_R,\n\\mathcal{F} \\otimes_{\\mathcal{O}_{\\mathbf{P}^n_R}}\n\\mathcal{O}_{\\mathbf{P}^n_R}(d))\n=\n\\Gamma(\\mathbf{P}^n_R, \\mathcal{F}(d))\n$$\nThen $M = \\bigoplus_{d \\geq 0} M_d$ is a graded $R[T_0, \\ldots, R_n]$-module\nand there is a canonical isomorphism $\\mathcal{F} = \\widetilde{M}$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GM","source_file":"constructions.tex","source_line":3054,"source_end_line":3070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3054-L3070","statement_sha256":"48a37578a76951a77db76be4f8dcbec51a372f1885620074ac02766641e341b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5148,"rank":5148,"depth":0,"x":1532.19,"y":770.938,"cluster":"schemes"},{"id":"stacks:0B3B","tag":"0B3B","title":"Projective space · Lemma 0B3B","summary":"Let X be a scheme. Let L be an invertible sheaf and let s_0, …, s_n be global sections of L which generate it. Let F be the kernel of the induced map O_X^⊕ n + 1 → L. Then F ⊗ L is globally generated.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf\nand let $s_0, \\ldots, s_n$ be global sections of $\\mathcal{L}$\nwhich generate it. Let $\\mathcal{F}$ be the kernel of the induced\nmap $\\mathcal{O}_X^{\\oplus n + 1} \\to \\mathcal{L}$.\nThen $\\mathcal{F} \\otimes \\mathcal{L}$ is globally generated.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3B","source_file":"constructions.tex","source_line":3153,"source_end_line":3160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3153-L3160","statement_sha256":"a7ec57ed7f9a17e9a1a77f163c6078167c28627a20f2b0fb44371a991d84efa7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5149,"rank":5149,"depth":0,"x":1542.016,"y":585.543,"cluster":"schemes"},{"id":"stacks:01NK","tag":"01NK","title":"Invertible sheaves and morphisms into Proj · Lemma 01NK","summary":"Let A be a graded ring. Set X = Proj(A). Let T be a scheme. Let L be an invertible O_T-module. Let ψ : A → Γ_*(T, L) be a homomorphism of graded rings. Set U(ψ) = ⋃_f ∈ A_+ homogeneous T_ψ(f) The morphism ψ induces a canonical morphism of schemes r_L, ψ : U(ψ) → X together with a map of Z-graded O_T-algebras theta : r_L, ψ^*( bigoplus_d ∈ Z O_X(d) ) → bigoplus_d ∈ Z L^⊗ d|_U(ψ). The triple (U(ψ), r_L, ψ, theta) is characterized by the following properties: • For f ∈ A_+…","statement_latex":"Let $A$ be a graded ring.\nSet $X = \\text{Proj}(A)$.\nLet $T$ be a scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_T$-module.\nLet $\\psi : A \\to \\Gamma_*(T, \\mathcal{L})$ be a homomorphism\nof graded rings. Set\n$$\nU(\\psi) = \\bigcup\\nolimits_{f \\in A_{+}\\text{ homogeneous}} T_{\\psi(f)}\n$$\nThe morphism $\\psi$ induces a canonical morphism of schemes\n$$\nr_{\\mathcal{L}, \\psi} :\nU(\\psi) \\longrightarrow X\n$$\ntogether with a map of $\\mathbf{Z}$-graded $\\mathcal{O}_T$-algebras\n$$\n\\theta :\nr_{\\mathcal{L}, \\psi}^*\\left(\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_X(d)\n\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes d}|_{U(\\psi)}.\n$$\nThe triple $(U(\\psi), r_{\\mathcal{L}, \\psi}, \\theta)$ is\ncharacterized by the following properties:\n\\begin{enumerate}\n\\item For $f \\in A_{+}$ homogeneous we have\n$r_{\\mathcal{L}, \\psi}^{-1}(D_{+}(f)) = T_{\\psi(f)}$.\n\\item For every $d \\geq 0$ the diagram\n$$\n\\xymatrix{\nA_d \\ar[d]_{(\\ref{equation-global-sections})} \\ar[r]_{\\psi} &\n\\Gamma(T, \\mathcal{L}^{\\otimes d}) \\ar[d]^{restrict} \\\\\n\\Gamma(X, \\mathcal{O}_X(d)) \\ar[r]^{\\theta} &\n\\Gamma(U(\\psi), \\mathcal{L}^{\\otimes d})\n}\n$$\nis commutative.\n\\end{enumerate}\nMoreover, for any $d \\geq 1$ and any open subscheme $V \\subset T$\nsuch that the sections in $\\psi(A_d)$ generate $\\mathcal{L}^{\\otimes d}|_V$\nthe morphism $r_{\\mathcal{L}, \\psi}|_V$ agrees with the morphism\n$\\varphi : V \\to \\text{Proj}(A)$ and the map $\\theta|_V$ agrees with the map\n$\\alpha : \\varphi^*\\mathcal{O}_X(d) \\to \\mathcal{L}^{\\otimes d}|_V$\nwhere $(\\varphi, \\alpha)$ is the pair\nof Lemma \\ref{lemma-converse-construction}\nassociated to\n$\\psi|_{A^{(d)}} : A^{(d)} \\to \\Gamma_*(V, \\mathcal{L}^{\\otimes d})$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Invertible sheaves and morphisms into Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NK","source_file":"constructions.tex","source_line":3216,"source_end_line":3266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3216-L3266","statement_sha256":"97e9c9d469f7d6c6486e1eb68c351dcec052eba03c1efcff7b59bca9e64a836f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5150,"rank":5150,"depth":15,"x":1684.267,"y":728.236,"cluster":"schemes"},{"id":"stacks:01NO","tag":"01NO","title":"Relative Proj via glueing · Lemma 01NO","summary":"In Situation [Tag 01NN]. Suppose U ⊂ U' ⊂ S are affine opens. Let A = A(U) and A' = A(U'). The map of graded rings A' → A induces a morphism r : Proj(A) → Proj(A'), and the diagram xymatrix Proj(A) ar[r] ar[d] & Proj(A') ar[d] U ar[r] & U' is cartesian. Moreover there are canonical isomorphisms theta : r^*O_Proj(A')(n) → O_Proj(A)(n) compatible with multiplication maps.","statement_latex":"In Situation \\ref{situation-relative-proj}.\nSuppose $U \\subset U' \\subset S$ are affine opens.\nLet $A = \\mathcal{A}(U)$ and $A' = \\mathcal{A}(U')$.\nThe map of graded rings $A' \\to A$ induces a morphism\n$r : \\text{Proj}(A) \\to \\text{Proj}(A')$, and the diagram\n$$\n\\xymatrix{\n\\text{Proj}(A) \\ar[r] \\ar[d] &\n\\text{Proj}(A') \\ar[d] \\\\\nU \\ar[r] &\nU'\n}\n$$\nis cartesian. Moreover there are canonical isomorphisms\n$\\theta : r^*\\mathcal{O}_{\\text{Proj}(A')}(n) \\to\n\\mathcal{O}_{\\text{Proj}(A)}(n)$ compatible with multiplication maps.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj via glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NO","source_file":"constructions.tex","source_line":3379,"source_end_line":3397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3379-L3397","statement_sha256":"e408da9ecc7a2472e883a719e86f0f05725f13af2cfa8215844bc1f29bd87608","origin":"The Stacks Project","memory_eligible":false,"source_rank":5151,"rank":5151,"depth":13,"x":1463.994,"y":703.663,"cluster":"schemes"},{"id":"stacks:01NP","tag":"01NP","title":"Relative Proj via glueing · Lemma 01NP","summary":"In Situation [Tag 01NN]. Suppose U ⊂ U' ⊂ U\" ⊂ S are affine opens. Let A = A(U), A' = A(U') and A\" = A(U\"). The composition of the morphisms r : Proj(A) → Proj(A'), and r' : Proj(A') → Proj(A\") of Lemma [Tag 01NO] gives the morphism r\" : Proj(A) → Proj(A\") of Lemma [Tag 01NO]. A similar statement holds for the isomorphisms theta.","statement_latex":"In Situation \\ref{situation-relative-proj}.\nSuppose $U \\subset U' \\subset U'' \\subset S$ are affine opens.\nLet $A = \\mathcal{A}(U)$, $A' = \\mathcal{A}(U')$ and $A'' = \\mathcal{A}(U'')$.\nThe composition of the morphisms\n$r : \\text{Proj}(A) \\to \\text{Proj}(A')$, and\n$r' : \\text{Proj}(A') \\to \\text{Proj}(A'')$ of\nLemma \\ref{lemma-proj-inclusion} gives the\nmorphism $r'' : \\text{Proj}(A) \\to \\text{Proj}(A'')$\nof Lemma \\ref{lemma-proj-inclusion}. A similar statement\nholds for the isomorphisms $\\theta$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj via glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NP","source_file":"constructions.tex","source_line":3412,"source_end_line":3424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3412-L3424","statement_sha256":"ad64de12b845d8df45e8bf3ce69c587f6c1694cc5ace56f2e854196264387d7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5152,"rank":5152,"depth":14,"x":1646.701,"y":596.491,"cluster":"schemes"},{"id":"stacks:01NQ","tag":"01NQ","title":"Relative Proj via glueing · Lemma 01NQ","summary":"In Situation [Tag 01NN]. There exists a morphism of schemes π : underlineProj_S(A) → S with the following properties: • for every affine open U ⊂ S there exists an isomorphism i_U : π^-1(U) → Proj(A) with A = A(U), and • for U ⊂ U' ⊂ S affine open the composition xymatrix Proj(A) ar[r]^i_U^-1 & π^-1(U) ar[rr]^inclusion & & π^-1(U') ar[r]^i_U' & Proj(A') with A = A(U), A' = A(U') is the open immersion of Lemma [Tag 01NO] above.","statement_latex":"In Situation \\ref{situation-relative-proj}.\nThere exists a morphism of schemes\n$$\n\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\longrightarrow S\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item for every affine open $U \\subset S$ there exists an isomorphism\n$i_U : \\pi^{-1}(U) \\to \\text{Proj}(A)$ with $A = \\mathcal{A}(U)$, and\n\\item for $U \\subset U' \\subset S$ affine open the composition\n$$\n\\xymatrix{\n\\text{Proj}(A) \\ar[r]^{i_U^{-1}} &\n\\pi^{-1}(U) \\ar[rr]^{inclusion} & &\n\\pi^{-1}(U') \\ar[r]^{i_{U'}} &\n\\text{Proj}(A')\n}\n$$\nwith $A = \\mathcal{A}(U)$, $A' = \\mathcal{A}(U')$\nis the open immersion of Lemma \\ref{lemma-proj-inclusion} above.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj via glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NQ","source_file":"constructions.tex","source_line":3432,"source_end_line":3455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3432-L3455","statement_sha256":"5e0e2cce676fc32e2109327cb74d984cc44ae64321d72ff82d94f1bcd4a50b46","origin":"The Stacks Project","memory_eligible":false,"source_rank":5153,"rank":5153,"depth":15,"x":1598.023,"y":779.707,"cluster":"schemes"},{"id":"stacks:01NR","tag":"01NR","title":"Relative Proj via glueing · Lemma 01NR","summary":"In Situation [Tag 01NN]. The morphism π : underlineProj_S(A) → S of Lemma [Tag 01NQ] comes with the following additional structure. There exists a quasi-coherent Z-graded sheaf of O_underlineProj_S(A)-algebras bigoplus_n ∈ Z O_underlineProj_S(A)(n), and a morphism of graded O_S-algebras ψ : A → bigoplus_n ≥ 0 π_*(O_underlineProj_S(A)(n)) uniquely determined by the following property: For every affine open U ⊂ S with A = A(U) there is an isomorphism theta_U : i_U^*(…","statement_latex":"In Situation \\ref{situation-relative-proj}.\nThe morphism $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$\nof Lemma \\ref{lemma-glue-relative-proj} comes with the following\nadditional structure.\nThere exists a quasi-coherent $\\mathbf{Z}$-graded sheaf\nof $\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}$-algebras\n$\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}\n\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)$,\nand a morphism of graded $\\mathcal{O}_S$-algebras\n$$\n\\psi :\n\\mathcal{A}\n\\longrightarrow\n\\bigoplus\\nolimits_{n \\geq 0}\n\\pi_*\\left(\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)\\right)\n$$\nuniquely determined by the following property:\nFor every affine open $U \\subset S$ with $A = \\mathcal{A}(U)$\nthere is an isomorphism\n$$\n\\theta_U :\ni_U^*\\left(\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_{\\text{Proj}(A)}(n)\n\\right)\n\\longrightarrow\n\\left(\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}\n\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)\n\\right)|_{\\pi^{-1}(U)}\n$$\nof $\\mathbf{Z}$-graded $\\mathcal{O}_{\\pi^{-1}(U)}$-algebras\nsuch that\n$$\n\\xymatrix{\nA_n\n\\ar[rr]_\\psi\n\\ar[dr]_-{(\\ref{equation-global-sections})}\n& &\n\\Gamma(\\pi^{-1}(U),\n\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(n)) \\\\\n&\n\\Gamma(\\text{Proj}(A),\n\\mathcal{O}_{\\text{Proj}(A)}(n))\n\\ar[ru]_-{\\theta_U}\n&\n}\n$$\nis commutative.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj via glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NR","source_file":"constructions.tex","source_line":3464,"source_end_line":3514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3464-L3514","statement_sha256":"b13ce9b7ad6a6f771670a990a271ea53f8455daf738dc28578b31120a4c891e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5154,"rank":5154,"depth":16,"x":1486.267,"y":616.526,"cluster":"schemes"},{"id":"stacks:01NT","tag":"01NT","title":"Relative Proj as a functor · Lemma 01NT","summary":"In Situation [Tag 01NN]. Let d ≥ 1. Let F_d be the functor associated to (S, A) above. Let g : S' → S be a morphism of schemes. Set A' = g^*A. Let F_d' be the functor associated to (S', A') above. Then there is a canonical isomorphism F'_d ≅ h_S' ×_h_S F_d of functors.","statement_latex":"In Situation \\ref{situation-relative-proj}. Let $d \\geq 1$.\nLet $F_d$ be the functor\nassociated to $(S, \\mathcal{A})$ above.\nLet $g : S' \\to S$ be a morphism of schemes.\nSet $\\mathcal{A}' = g^*\\mathcal{A}$. Let $F_d'$ be the\nfunctor associated to $(S', \\mathcal{A}')$ above.\nThen there is a canonical isomorphism\n$$\nF'_d \\cong h_{S'} \\times_{h_S} F_d\n$$\nof functors.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NT","source_file":"constructions.tex","source_line":3619,"source_end_line":3632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3619-L3632","statement_sha256":"3c8c36b33b17155b7851b004b87fd703e66a2f9dcd1f7d4e5e6626f04310825d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5155,"rank":5155,"depth":0,"x":1700.497,"y":673.6,"cluster":"schemes"},{"id":"stacks:01NU","tag":"01NU","title":"Relative Proj as a functor · Lemma 01NU","summary":"In Situation [Tag 01NN]. Let F_d be the functor associated to (d, S, A) above. If S is affine, then F_d is representable by the open subscheme U_d ([Tag 01N5]) of the scheme Proj(Γ(S, A)).","statement_latex":"In Situation \\ref{situation-relative-proj}. Let $F_d$ be the functor\nassociated to $(d, S, \\mathcal{A})$ above.\nIf $S$ is affine, then $F_d$ is representable by the open subscheme\n$U_d$ (\\ref{equation-Ud})\nof the scheme $\\text{Proj}(\\Gamma(S, \\mathcal{A}))$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NU","source_file":"constructions.tex","source_line":3650,"source_end_line":3657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3650-L3657","statement_sha256":"7c7deaafc91ff0b5438ac916bb1ce85db10373519e301702baf8822e007ba70d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5156,"rank":5156,"depth":11,"x":1496.062,"y":753.295,"cluster":"schemes"},{"id":"stacks:01NV","tag":"01NV","title":"Relative Proj as a functor · Lemma 01NV","summary":"In Situation [Tag 01NN]. The functor F_d is representable by a scheme.","statement_latex":"In Situation \\ref{situation-relative-proj}.\nThe functor $F_d$ is representable by a scheme.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NV","source_file":"constructions.tex","source_line":3690,"source_end_line":3694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3690-L3694","statement_sha256":"bf91af36f897bcd2350d597de650ddbf0e2d9875e5fee020b77b20ded4f48ee6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5157,"rank":5157,"depth":12,"x":1582.961,"y":578.044,"cluster":"schemes"},{"id":"stacks:01NW","tag":"01NW","title":"Relative Proj as a functor · Lemma 01NW","summary":"In Situation [Tag 01NN]. Let T be a scheme. Let (d, f, L, ψ), (d', f', L', ψ') be two quadruples over T. The following are equivalent: • Let m = lcm(d, d'). Write m = ad = a'd'. We have f = f' and there exists an isomorphism β : L^⊗ a → (L')^⊗ a' with the property that β ∘ ψ|_f^*A^(m) and ψ'|_f^*A^(m) agree as graded ring maps f^*A^(m) → bigoplus_n ≥ 0 (L')^⊗ mn. • The quadruples (d, f, L, ψ) and (d', f', L', ψ') are equivalent. • We have f = f' and for some positive…","statement_latex":"In Situation \\ref{situation-relative-proj}.\nLet $T$ be a scheme.\nLet $(d, f, \\mathcal{L}, \\psi)$, $(d', f', \\mathcal{L}', \\psi')$\nbe two quadruples over $T$. The following are equivalent:\n\\begin{enumerate}\n\\item Let $m = \\text{lcm}(d, d')$. Write $m = ad = a'd'$.\nWe have $f = f'$ and there exists\nan isomorphism\n$\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$\nwith the property that $\\beta \\circ \\psi|_{f^*\\mathcal{A}^{(m)}}$\nand $\\psi'|_{f^*\\mathcal{A}^{(m)}}$ agree\nas graded ring maps\n$f^*\\mathcal{A}^{(m)} \\to \\bigoplus_{n \\geq 0} (\\mathcal{L}')^{\\otimes mn}$.\n\\item The quadruples $(d, f, \\mathcal{L}, \\psi)$ and\n$(d', f', \\mathcal{L}', \\psi')$ are equivalent.\n\\item We have $f = f'$ and\nfor some positive integer $m = ad = a'd'$ there exists an isomorphism\n$\\beta : \\mathcal{L}^{\\otimes a} \\to (\\mathcal{L}')^{\\otimes a'}$\nwith the property that $\\beta \\circ \\psi|_{f^*\\mathcal{A}^{(m)}}$\nand $\\psi'|_{f^*\\mathcal{A}^{(m)}}$ agree\nas graded ring maps\n$f^*\\mathcal{A}^{(m)} \\to \\bigoplus_{n \\geq 0} (\\mathcal{L}')^{\\otimes mn}$.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NW","source_file":"constructions.tex","source_line":3773,"source_end_line":3798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3773-L3798","statement_sha256":"e01e0335e212c9a93701cd5b76e7b4fc1fe957b30b002a03ec35c81022f61e0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5158,"rank":5158,"depth":16,"x":1660.025,"y":757.072,"cluster":"schemes"},{"id":"stacks:01NY","tag":"01NY","title":"Relative Proj as a functor · Lemma 01NY","summary":"In Situation [Tag 01NN]. The functor F above is representable by a scheme.","statement_latex":"In Situation \\ref{situation-relative-proj}.\nThe functor $F$ above is representable by a scheme.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NY","source_file":"constructions.tex","source_line":3856,"source_end_line":3860,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3856-L3860","statement_sha256":"948f095d5b44a83eb31b8ca9b7ecf37cd36cd6ed58368cfe2221a1a794f8c17a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5159,"rank":5159,"depth":12,"x":1458.681,"y":668.546,"cluster":"schemes"},{"id":"stacks:01NZ","tag":"01NZ","title":"Relative Proj as a functor · Lemma 01NZ","summary":"In Situation [Tag 01NN]. The scheme π : underlineProj_S(A) → S constructed in Lemma [Tag 01NQ] and the scheme representing the functor F are canonically isomorphic as schemes over S.","statement_latex":"In Situation \\ref{situation-relative-proj}.\nThe scheme $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$\nconstructed in Lemma \\ref{lemma-glue-relative-proj}\nand the scheme representing the functor $F$\nare canonically isomorphic as schemes over $S$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01NZ","source_file":"constructions.tex","source_line":3893,"source_end_line":3900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3893-L3900","statement_sha256":"d8baca712e73de36993400aaf301df838a7f91d4e1dfbfa786f9fc43c83e33c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5160,"rank":5160,"depth":17,"x":1678.938,"y":619.443,"cluster":"schemes"},{"id":"stacks:01O0","tag":"01O0","title":"Relative Proj as a functor · Definition 01O0","summary":"Let S be a scheme. Let A be a quasi-coherent sheaf of graded O_S-algebras. The relative homogeneous spectrum of A over S, or the homogeneous spectrum of A over S, or the relative Proj of A over S is the scheme constructed in Lemma [Tag 01NQ] which represents the functor F ([Tag 01NX]), see Lemma [Tag 01NZ]. We denote it π : underlineProj_S(A) → S.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf of\ngraded $\\mathcal{O}_S$-algebras. The\n{\\it relative homogeneous spectrum of $\\mathcal{A}$ over $S$},\nor the {\\it homogeneous spectrum of $\\mathcal{A}$ over $S$}, or the\n{\\it relative Proj of $\\mathcal{A}$ over $S$} is the scheme\nconstructed in Lemma \\ref{lemma-glue-relative-proj} which represents the\nfunctor $F$ (\\ref{equation-proj}), see\nLemma \\ref{lemma-glueing-gives-functor-proj}.\nWe denote it $\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01O0","source_file":"constructions.tex","source_line":3949,"source_end_line":3960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3949-L3960","statement_sha256":"0615e902bb5cdb18210d3c29c65537319986e351baf51ee68be5fcbe25b7b481","origin":"The Stacks Project","memory_eligible":false,"source_rank":5161,"rank":5161,"depth":18,"x":1555.678,"y":781.065,"cluster":"schemes"},{"id":"stacks:01O1","tag":"01O1","title":"Relative Proj as a functor · Lemma 01O1","summary":"In Situation [Tag 01NN]. Let (f : T → S, d, L, ψ) be a quadruple. Let r_d, L, ψ : T → underlineProj_S(A) be the associated S-morphism. There exists an isomorphism of Z-graded O_T-algebras theta : r_d, L, ψ^*( bigoplus_n ∈ Z O_underlineProj_S(A)(nd) ) → bigoplus_n ∈ Z L^⊗ n such that the following diagram commutes xymatrix A^(d) ar[rr]_-ψ ar[rd]_-ψ_univ & & f_*( bigoplus_n ∈ Z L^⊗ n ) & π_*( bigoplus_n ≥ 0 O_underlineProj_S(A)(nd) ) ar[ru]_theta The commutativity of this…","statement_latex":"In Situation \\ref{situation-relative-proj}.\nLet $(f : T \\to S, d, \\mathcal{L}, \\psi)$\nbe a quadruple. Let\n$r_{d, \\mathcal{L}, \\psi} : T \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$\nbe the associated $S$-morphism.\nThere exists an isomorphism\nof $\\mathbf{Z}$-graded $\\mathcal{O}_T$-algebras\n$$\n\\theta :\nr_{d, \\mathcal{L}, \\psi}^*\\left(\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}\n\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(nd)\n\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes n}\n$$\nsuch that the following diagram commutes\n$$\n\\xymatrix{\n\\mathcal{A}^{(d)} \\ar[rr]_-{\\psi}\n \\ar[rd]_-{\\psi_{univ}} & &\nf_*\\left(\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}\n\\mathcal{L}^{\\otimes n}\n\\right) \\\\\n &\n\\pi_*\\left(\n\\bigoplus\\nolimits_{n \\geq 0}\n\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(nd)\n\\right) \\ar[ru]_\\theta\n}\n$$\nThe commutativity of this diagram uniquely determines $\\theta$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01O1","source_file":"constructions.tex","source_line":3990,"source_end_line":4025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L3990-L4025","statement_sha256":"3f04443ca0d39958b361d8a49e5e235abce59b758dbfec6b707eb4d0f4459a4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5162,"rank":5162,"depth":17,"x":1516.491,"y":591.438,"cluster":"schemes"},{"id":"stacks:01O2","tag":"01O2","title":"Relative Proj as a functor · Lemma 01O2","summary":"Let S be a scheme and A be a quasi-coherent sheaf of graded O_S-algebras. The morphism π : underlineProj_S(A) → S is separated.","statement_latex":"Let $S$ be a scheme and $\\mathcal{A}$ be a quasi-coherent sheaf\nof graded $\\mathcal{O}_S$-algebras. The morphism\n$\\pi : \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$\nis separated.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01O2","source_file":"constructions.tex","source_line":4061,"source_end_line":4067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4061-L4067","statement_sha256":"b72cce7326411297aa52ecacccd35465eb2167f332173a7e4396a87bce61d207","origin":"The Stacks Project","memory_eligible":false,"source_rank":5163,"rank":5163,"depth":16,"x":1698.365,"y":709.346,"cluster":"schemes"},{"id":"stacks:01O3","tag":"01O3","title":"Relative Proj as a functor · Lemma 01O3","summary":"Let S be a scheme and A be a quasi-coherent sheaf of graded O_S-algebras. Let g : S' → S be any morphism of schemes. Then there is a canonical isomorphism r : underlineProj_S'(g^*A) → S' ×_S underlineProj_S(A) as well as a corresponding isomorphism theta : r^*pr_2^*(bigoplus_d ∈ Z O_underlineProj_S(A)(d)) → bigoplus_d ∈ Z O_underlineProj_S'(g^*A)(d) of Z-graded O_underlineProj_S'(g^*A)-algebras.","statement_latex":"Let $S$ be a scheme and $\\mathcal{A}$ be a quasi-coherent sheaf\nof graded $\\mathcal{O}_S$-algebras. Let $g : S' \\to S$ be any morphism\nof schemes. Then there is a canonical isomorphism\n$$\nr :\n\\underline{\\text{Proj}}_{S'}(g^*\\mathcal{A})\n\\longrightarrow\nS' \\times_S \\underline{\\text{Proj}}_S(\\mathcal{A})\n$$\nas well as a corresponding isomorphism\n$$\n\\theta :\nr^*\\text{pr}_2^*\\left(\\bigoplus\\nolimits_{d \\in \\mathbf{Z}}\n\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}}\n\\mathcal{O}_{\\underline{\\text{Proj}}_{S'}(g^*\\mathcal{A})}(d)\n$$\nof $\\mathbf{Z}$-graded\n$\\mathcal{O}_{\\underline{\\text{Proj}}_{S'}(g^*\\mathcal{A})}$-algebras.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01O3","source_file":"constructions.tex","source_line":4080,"source_end_line":4102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4080-L4102","statement_sha256":"25444688e6383f57626d1959fab9811ed1c79b2016fe0b0a5d1f5ca8ffc70679","origin":"The Stacks Project","memory_eligible":false,"source_rank":5164,"rank":5164,"depth":13,"x":1468.826,"y":725.645,"cluster":"schemes"},{"id":"stacks:01O4","tag":"01O4","title":"Relative Proj as a functor · Lemma 01O4","summary":"Let S be a scheme. Let A be a quasi-coherent sheaf of graded O_S-modules generated as an A_0-algebra by A_1. In this case the scheme X = underlineProj_S(A) represents the functor F_1 which associates to a scheme f : T → S over S the set of pairs (L, ψ), where • L is an invertible O_T-module, and • ψ : f^*A → bigoplus_n ≥ 0 L^⊗ n is a graded O_T-algebra homomorphism such that f^*A_1 → L is surjective up to strict equivalence as above. Moreover, in this case all the…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_S$-modules\ngenerated as an $\\mathcal{A}_0$-algebra by $\\mathcal{A}_1$.\nIn this case the scheme $X = \\underline{\\text{Proj}}_S(\\mathcal{A})$\nrepresents the functor $F_1$ which associates to a scheme\n$f : T \\to S$ over $S$ the set of pairs $(\\mathcal{L}, \\psi)$, where\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is an invertible $\\mathcal{O}_T$-module, and\n\\item $\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$\nis a graded $\\mathcal{O}_T$-algebra homomorphism such that\n$f^*\\mathcal{A}_1 \\to \\mathcal{L}$ is surjective\n\\end{enumerate}\nup to strict equivalence as above. Moreover, in this case all the\nquasi-coherent sheaves\n$\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n)$\nare invertible\n$\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}$-modules\nand the multiplication maps induce isomorphisms\n$\n\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n)\n\\otimes_{\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}}\n\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(m) =\n\\mathcal{O}_{\\underline{\\text{Proj}}(\\mathcal{A})}(n + m)$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Relative Proj as a functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01O4","source_file":"constructions.tex","source_line":4115,"source_end_line":4140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4115-L4140","statement_sha256":"31131339de4455209e17ee80ce776525ea639392f3f4f9b9e9fd7095bf1e509c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5165,"rank":5165,"depth":17,"x":1625.392,"y":583.004,"cluster":"schemes"},{"id":"stacks:01O7","tag":"01O7","title":"Quasi-coherent sheaves on relative Proj · Lemma 01O7","summary":"In Situation [Tag 01NN]. For any quasi-coherent sheaf of graded A-modules M on S, there exists a canonical associated sheaf of O_underlineProj_S(A)-modules widetildeM with the following properties: • Given a scheme T and a quadruple (T → S, d, L, ψ) over T corresponding to a morphism h : T → underlineProj_S(A) there is a canonical isomorphism widetildeM_T = h^*widetildeM where widetildeM_T is defined by ([Tag 01O6]). • The isomorphisms of (1) are compatible with…","statement_latex":"In Situation \\ref{situation-relative-proj}.\nFor any quasi-coherent sheaf of graded $\\mathcal{A}$-modules\n$\\mathcal{M}$ on $S$, there exists a canonical associated sheaf\nof $\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}$-modules\n$\\widetilde{\\mathcal{M}}$ with the following properties:\n\\begin{enumerate}\n\\item Given a scheme $T$ and a quadruple\n$(T \\to S, d, \\mathcal{L}, \\psi)$ over $T$\ncorresponding to a morphism\n$h : T \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$ there is\na canonical isomorphism\n$\\widetilde{\\mathcal{M}}_T = h^*\\widetilde{\\mathcal{M}}$\nwhere $\\widetilde{\\mathcal{M}}_T$ is defined by (\\ref{equation-widetilde-M}).\n\\item The isomorphisms of (1) are compatible with pullbacks.\n\\item There is a canonical map\n$$\n\\pi^*\\mathcal{M}_0 \\longrightarrow \\widetilde{\\mathcal{M}}.\n$$\n\\item The construction $\\mathcal{M} \\mapsto \\widetilde{\\mathcal{M}}$\nis functorial in $\\mathcal{M}$.\n\\item The construction $\\mathcal{M} \\mapsto \\widetilde{\\mathcal{M}}$\nis exact.\n\\item There are canonical maps\n$$\n\\widetilde{\\mathcal{M}}\n\\otimes_{\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}}\n\\widetilde{\\mathcal{N}}\n\\longrightarrow\n\\widetilde{\\mathcal{M} \\otimes_\\mathcal{A} \\mathcal{N}}\n$$\nas in\nLemma \\ref{lemma-widetilde-tensor}.\n\\item There exist canonical maps\n$$\n\\pi^*\\mathcal{M}\n\\longrightarrow\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}}\n\\widetilde{\\mathcal{M}(n)}\n$$\ngeneralizing (\\ref{equation-global-sections-more-generally}).\n\\item The formation of $\\widetilde{\\mathcal{M}}$ commutes with base change.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Quasi-coherent sheaves on relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01O7","source_file":"constructions.tex","source_line":4248,"source_end_line":4292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4248-L4292","statement_sha256":"64babcacd07de1714bd244305fbcde68caf6206da42503cc47a9e9dab5c9650c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5166,"rank":5166,"depth":8,"x":1624.648,"y":777.536,"cluster":"schemes"},{"id":"stacks:07ZG","tag":"07ZG","title":"Functoriality of relative Proj · Lemma 07ZG","summary":"Let S be a scheme. Let A, B be two graded quasi-coherent O_S-algebras. Set p : X = underlineProj_S(A) → S and q : Y = underlineProj_S(B) → S. Let ψ : A → B be a homomorphism of graded O_S-algebras. There is a canonical open U(ψ) ⊂ Y and a canonical morphism of schemes r_ψ : U(ψ) → X over S and a map of Z-graded O_U(ψ)-algebras theta = theta_ψ : r_ψ^*( bigoplus_d ∈ Z O_X(d) ) → bigoplus_d ∈ Z O_U(ψ)(d). The triple (U(ψ), r_ψ, theta) is characterized by the property that…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$, $\\mathcal{B}$ be two graded\nquasi-coherent $\\mathcal{O}_S$-algebras. Set\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ and\n$q : Y = \\underline{\\text{Proj}}_S(\\mathcal{B}) \\to S$. Let\n$\\psi : \\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of\ngraded $\\mathcal{O}_S$-algebras. There is a canonical open\n$U(\\psi) \\subset Y$ and a canonical morphism of schemes\n$$\nr_\\psi :\nU(\\psi)\n\\longrightarrow\nX\n$$\nover $S$ and a map of $\\mathbf{Z}$-graded $\\mathcal{O}_{U(\\psi)}$-algebras\n$$\n\\theta = \\theta_\\psi :\nr_\\psi^*\\left(\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_X(d)\n\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{U(\\psi)}(d).\n$$\nThe triple $(U(\\psi), r_\\psi, \\theta)$ is characterized by the property\nthat for any affine open $W \\subset S$ the triple\n$$\n(U(\\psi) \\cap p^{-1}W,\\quad\nr_\\psi|_{U(\\psi) \\cap p^{-1}W} : U(\\psi) \\cap p^{-1}W \\to q^{-1}W,\\quad\n\\theta|_{U(\\psi) \\cap p^{-1}W})\n$$\nis equal to the triple associated to\n$\\psi : \\mathcal{A}(W) \\to \\mathcal{B}(W)$ in\nLemma \\ref{lemma-morphism-proj} via the identifications\n$p^{-1}W = \\text{Proj}(\\mathcal{A}(W))$ and\n$q^{-1}W = \\text{Proj}(\\mathcal{B}(W))$ of\nSection \\ref{section-relative-proj-via-glueing}.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZG","source_file":"constructions.tex","source_line":4318,"source_end_line":4355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4318-L4355","statement_sha256":"2032518cd72073f3edd8c46b4de87910fd8177f61e62ace2d45ac195d267dafa","origin":"The Stacks Project","memory_eligible":false,"source_rank":5167,"rank":5167,"depth":1,"x":1468.351,"y":633.291,"cluster":"schemes"},{"id":"stacks:07ZH","tag":"07ZH","title":"Functoriality of relative Proj · Lemma 07ZH","summary":"Let S be a scheme. Let A, B, and C be quasi-coherent graded O_S-algebras. Set X = underlineProj_S(A), Y = underlineProj_S(B) and Z = underlineProj_S(C). Let φ : A → B, ψ : B → C be graded O_S-algebra maps. Then we have U(ψ ∘ φ) = r_φ^-1(U(ψ)) and r_ψ ∘ φ = r_φ ∘ r_ψ|_U(ψ ∘ φ). In addition we have theta_ψ ∘ r_ψ^*theta_φ = theta_ψ ∘ φ with obvious notation.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be\nquasi-coherent graded $\\mathcal{O}_S$-algebras.\nSet $X = \\underline{\\text{Proj}}_S(\\mathcal{A})$,\n$Y = \\underline{\\text{Proj}}_S(\\mathcal{B})$ and\n$Z = \\underline{\\text{Proj}}_S(\\mathcal{C})$.\nLet $\\varphi : \\mathcal{A} \\to \\mathcal{B}$,\n$\\psi : \\mathcal{B} \\to \\mathcal{C}$ be graded $\\mathcal{O}_S$-algebra maps.\nThen we have\n$$\nU(\\psi \\circ \\varphi) = r_\\varphi^{-1}(U(\\psi))\n\\quad\n\\text{and}\n\\quad\nr_{\\psi \\circ \\varphi}\n=\nr_\\varphi \\circ r_\\psi|_{U(\\psi \\circ \\varphi)}.\n$$\nIn addition we have\n$$\n\\theta_\\psi \\circ r_\\psi^*\\theta_\\varphi\n=\n\\theta_{\\psi \\circ \\varphi}\n$$\nwith obvious notation.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZH","source_file":"constructions.tex","source_line":4361,"source_end_line":4387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4361-L4387","statement_sha256":"3a8fb586d67957d339a17fa898c63c9f31b3eb092712dff516b3c506da8621a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5168,"rank":5168,"depth":0,"x":1700.202,"y":651.015,"cluster":"schemes"},{"id":"stacks:07ZI","tag":"07ZI","title":"Functoriality of relative Proj · Lemma 07ZI","summary":"With hypotheses and notation as in Lemma [Tag 07ZG] above. Assume A_d → B_d is surjective for d gg 0. Then • U(ψ) = Y, • r_ψ : Y → X is a closed immersion, and • the maps theta : r_ψ^*O_X(n) → O_Y(n) are surjective but not isomorphisms in general (even if A → B is surjective).","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}\nabove. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for\n$d \\gg 0$. Then\n\\begin{enumerate}\n\\item $U(\\psi) = Y$,\n\\item $r_\\psi : Y \\to X$ is a closed immersion, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$\nare surjective but not isomorphisms in general (even if\n$\\mathcal{A} \\to \\mathcal{B}$ is surjective).\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZI","source_file":"constructions.tex","source_line":4393,"source_end_line":4405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4393-L4405","statement_sha256":"cd25fae31e59b4bf99a9177e62eb86b1aa012d0bcce64b606e5da28d885e5aa4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5169,"rank":5169,"depth":2,"x":1514.505,"y":769.81,"cluster":"schemes"},{"id":"stacks:07ZJ","tag":"07ZJ","title":"Functoriality of relative Proj · Lemma 07ZJ","summary":"With hypotheses and notation as in Lemma [Tag 07ZG] above. Assume A_d → B_d is an isomorphism for all d gg 0. Then • U(ψ) = Y, • r_ψ : Y → X is an isomorphism, and • the maps theta : r_ψ^*O_X(n) → O_Y(n) are isomorphisms.","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}\nabove. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is an isomorphism for all\n$d \\gg 0$. Then\n\\begin{enumerate}\n\\item $U(\\psi) = Y$,\n\\item $r_\\psi : Y \\to X$ is an isomorphism, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$\nare isomorphisms.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZJ","source_file":"constructions.tex","source_line":4414,"source_end_line":4425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4414-L4425","statement_sha256":"313f619070ff9db9a35af5010328600227c9b0e7e10f22a42e9fc67213b80fda","origin":"The Stacks Project","memory_eligible":false,"source_rank":5170,"rank":5170,"depth":3,"x":1556.01,"y":576.346,"cluster":"schemes"},{"id":"stacks:07ZK","tag":"07ZK","title":"Functoriality of relative Proj · Lemma 07ZK","summary":"With hypotheses and notation as in Lemma [Tag 07ZG] above. Assume A_d → B_d is surjective for d gg 0 and that A is generated by A_1 over A_0. Then • U(ψ) = Y, • r_ψ : Y → X is a closed immersion, and • the maps theta : r_ψ^*O_X(n) → O_Y(n) are isomorphisms.","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}\nabove. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for $d \\gg 0$\nand that $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$.\nThen\n\\begin{enumerate}\n\\item $U(\\psi) = Y$,\n\\item $r_\\psi : Y \\to X$ is a closed immersion, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_X(n) \\to \\mathcal{O}_Y(n)$\nare isomorphisms.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Functoriality of relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZK","source_file":"constructions.tex","source_line":4434,"source_end_line":4446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4434-L4446","statement_sha256":"55ce245d9c320e3fce75fd690a6d5885e34cc91e6fb9ab3d6c75d987e7d40fd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5171,"rank":5171,"depth":4,"x":1681.304,"y":742.981,"cluster":"schemes"},{"id":"stacks:01O9","tag":"01O9","title":"Invertible sheaves and morphisms into relative Proj · Lemma 01O9","summary":"With assumptions and notation as above. The morphism ψ induces a canonical morphism of schemes over S r_L, ψ : U(ψ) → underlineProj_S(A) together with a map of graded O_U(ψ)-algebras theta : r_L, ψ^*( bigoplus_d ≥ 0 O_underlineProj_S(A)(d) ) → bigoplus_d ≥ 0 L^⊗ d|_U(ψ) characterized by the following properties: • For every open V ⊂ S and every d ≥ 0 the diagram xymatrix A_d(V) ar[d]_ψ ar[r]_ψ & Γ(f^-1(V), L^⊗ d) ar[d]^restrict Γ(π^-1(V), O_underlineProj_S(A)(d))…","statement_latex":"With assumptions and notation as above. The morphism\n$\\psi$ induces a canonical morphism of schemes over $S$\n$$\nr_{\\mathcal{L}, \\psi} :\nU(\\psi) \\longrightarrow \\underline{\\text{Proj}}_S(\\mathcal{A})\n$$\ntogether with a map of graded $\\mathcal{O}_{U(\\psi)}$-algebras\n$$\n\\theta :\nr_{\\mathcal{L}, \\psi}^*\\left(\n\\bigoplus\\nolimits_{d \\geq 0}\n\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)\n\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\geq 0} \\mathcal{L}^{\\otimes d}|_{U(\\psi)}\n$$\ncharacterized by the following properties:\n\\begin{enumerate}\n\\item For every open $V \\subset S$ and every $d \\geq 0$ the diagram\n$$\n\\xymatrix{\n\\mathcal{A}_d(V) \\ar[d]_{\\psi} \\ar[r]_{\\psi} &\n\\Gamma(f^{-1}(V), \\mathcal{L}^{\\otimes d}) \\ar[d]^{restrict} \\\\\n\\Gamma(\\pi^{-1}(V),\n\\mathcal{O}_{\\underline{\\text{Proj}}_S(\\mathcal{A})}(d)) \\ar[r]^{\\theta} &\n\\Gamma(f^{-1}(V) \\cap U(\\psi), \\mathcal{L}^{\\otimes d})\n}\n$$\nis commutative.\n\\item For any $d \\geq 1$ and any open subscheme $W \\subset X$\nsuch that $\\psi|_W : f^*\\mathcal{A}_d|_W \\to \\mathcal{L}^{\\otimes d}|_W$\nis surjective the restriction of the morphism $r_{\\mathcal{L}, \\psi}$\nagrees with the morphism $W \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$\nwhich exists by the construction of the relative homogeneous spectrum,\nsee Definition \\ref{definition-relative-proj}.\n\\item For any affine open $V \\subset S$, the restriction\n$$\n(U(\\psi) \\cap f^{-1}(V), r_{\\mathcal{L}, \\psi}|_{U(\\psi) \\cap f^{-1}(V)},\n\\theta|_{U(\\psi) \\cap f^{-1}(V)})\n$$\nagrees via $i_V$ (see Lemma \\ref{lemma-glue-relative-proj}) with the triple\n$(U(\\psi'), r_{\\mathcal{L}, \\psi'}, \\theta')$\nof Lemma \\ref{lemma-invertible-map-into-proj} associated to the map\n$\\psi' : A = \\mathcal{A}(V) \\to \\Gamma_*(f^{-1}(V), \\mathcal{L}|_{f^{-1}(V)})$\ninduced by $\\psi$.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Invertible sheaves and morphisms into relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01O9","source_file":"constructions.tex","source_line":4502,"source_end_line":4550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4502-L4550","statement_sha256":"dd8632e9fe7a33649d13ab273d4cebcfa1944d7a94a7bc267af2902a498e67ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":5172,"rank":5172,"depth":19,"x":1454.333,"y":691.069,"cluster":"schemes"},{"id":"stacks:02NC","tag":"02NC","title":"Twisting by invertible sheaves and relative Proj · Lemma 02NC","summary":"With notation S, A, L and B as above. There is a canonical isomorphism xymatrix P = underlineProj_S(A) ar[rr]_g ar[rd]_π & & underlineProj_S(B) = P' ar[ld]^π' & S & with the following properties • There are isomorphisms theta_n : g^*O_P'(n) → O_P(n) ⊗ π^*L^⊗ n which fit together to give an isomorphism of Z-graded algebras theta : g^*( bigoplus_n ∈ Z O_P'(n) ) → bigoplus_n ∈ Z O_P(n) ⊗ π^*L^⊗ n • For every open V ⊂ S the diagrams xymatrix A_n(V) ⊗ L^⊗ n(V) ar[r]_multiply…","statement_latex":"With notation $S$, $\\mathcal{A}$, $\\mathcal{L}$ and $\\mathcal{B}$ as\nabove. There is a canonical isomorphism\n$$\n\\xymatrix{\nP = \\underline{\\text{Proj}}_S(\\mathcal{A})\n\\ar[rr]_g \\ar[rd]_\\pi & &\n\\underline{\\text{Proj}}_S(\\mathcal{B}) = P'\n\\ar[ld]^{\\pi'} \\\\\n& S &\n}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item There are isomorphisms\n$\\theta_n : g^*\\mathcal{O}_{P'}(n)\n\\to\n\\mathcal{O}_P(n) \\otimes \\pi^*\\mathcal{L}^{\\otimes n}$\nwhich fit together to give an isomorphism of $\\mathbf{Z}$-graded\nalgebras\n$$\n\\theta :\ng^*\\left(\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_{P'}(n)\n\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{O}_P(n)\n\\otimes \\pi^*\\mathcal{L}^{\\otimes n}\n$$\n\\item For every open $V \\subset S$ the diagrams\n$$\n\\xymatrix{\n\\mathcal{A}_n(V) \\otimes \\mathcal{L}^{\\otimes n}(V)\n\\ar[r]_{multiply} \\ar[d]^{\\psi \\otimes \\pi^*}\n&\n\\mathcal{B}_n(V) \\ar[dd]^\\psi \\\\\n\\Gamma(\\pi^{-1}V, \\mathcal{O}_P(n)) \\otimes\n\\Gamma(\\pi^{-1}V, \\pi^*\\mathcal{L}^{\\otimes n})\n\\ar[d]^{multiply} \\\\\n\\Gamma(\\pi^{-1}V, \\mathcal{O}_P(n) \\otimes \\pi^*\\mathcal{L}^{\\otimes n})\n&\n\\Gamma(\\pi'^{-1}V, \\mathcal{O}_{P'}(n)) \\ar[l]_-{\\theta_n}\n}\n$$\nare commutative.\n\\item Add more here as necessary.\n\\end{enumerate}","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Twisting by invertible sheaves and relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NC","source_file":"constructions.tex","source_line":4584,"source_end_line":4632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4584-L4632","statement_sha256":"e8ac66f6d6a3295c694adcfd7bceed6c845b8dbf0ca430c609f2caef6c754579","origin":"The Stacks Project","memory_eligible":false,"source_rank":5173,"rank":5173,"depth":0,"x":1663.976,"y":600.33,"cluster":"schemes"},{"id":"stacks:01OB","tag":"01OB","title":"Projective bundles · Definition 01OB","summary":"Let S be a scheme. Let E be a quasi-coherent O_S-module is finite locally free. We do not do so in order to be consistent with [EGA].. We denote π : P(E) = underlineProj_S(Sym(E)) → S and we call it the projective bundle associated to E. The symbol O_P(E)(n) indicates the invertible O_P(E)-module of Lemma [Tag 01O4] and is called the nth twist of the structure sheaf.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{E}$ be a quasi-coherent\n$\\mathcal{O}_S$-module\\footnote{The reader may expect here\nthe condition that $\\mathcal{E}$ is finite locally free. We do not\ndo so in order to be consistent with\n\\cite[II, Definition 4.1.1]{EGA}.}.\nWe denote\n$$\n\\pi :\n\\mathbf{P}(\\mathcal{E}) = \\underline{\\text{Proj}}_S(\\text{Sym}(\\mathcal{E}))\n\\longrightarrow\nS\n$$\nand we call it the {\\it projective bundle associated to $\\mathcal{E}$}.\nThe symbol $\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(n)$\nindicates the invertible $\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}$-module\nof Lemma \\ref{lemma-apply-relative} and is called the $n$th\n{\\it twist of the structure sheaf}.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective bundles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OB","source_file":"constructions.tex","source_line":4671,"source_end_line":4690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4671-L4690","statement_sha256":"bc04255dcee471825482f0cab1127348388b10b5dbd6e267e0a77c8864c39340","origin":"The Stacks Project","memory_eligible":false,"source_rank":5174,"rank":5174,"depth":18,"x":1582.151,"y":786.665,"cluster":"schemes"},{"id":"stacks:01OD","tag":"01OD","title":"Projective bundles · Lemma 01OD","summary":"Let S be a scheme. The structure morphism P(E) → S of a projective bundle over S is separated.","statement_latex":"Let $S$ be a scheme.\nThe structure morphism $\\mathbf{P}(\\mathcal{E}) \\to S$ of a\nprojective bundle over $S$ is separated.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OD","source_file":"constructions.tex","source_line":4910,"source_end_line":4915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4910-L4915","statement_sha256":"c168ab911e4b2bc28a38af6c24d2da4724c592f183668771874fa587e1b144cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5175,"rank":5175,"depth":17,"x":1492.418,"y":602.373,"cluster":"schemes"},{"id":"stacks:01OE","tag":"01OE","title":"Projective bundles · Lemma 01OE","summary":"Let S be a scheme. Let n ≥ 0. Then P^n_S is a projective bundle over S.","statement_latex":"Let $S$ be a scheme. Let $n \\geq 0$. Then\n$\\mathbf{P}^n_S$ is a projective bundle over $S$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Projective bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OE","source_file":"constructions.tex","source_line":4921,"source_end_line":4925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4921-L4925","statement_sha256":"3653f9cc26a43564b995fff29a32167d76fbc26c7fa585aeeadfc8364829537a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5176,"rank":5176,"depth":14,"x":1707.324,"y":687.563,"cluster":"schemes"},{"id":"stacks:089T","tag":"089T","title":"Grassmannians · Lemma 089T","summary":"Let 0 < k < n. The functor G(k, n) of ([Tag 089S]) is representable by a scheme.","statement_latex":"Let $0 < k < n$.\nThe functor $G(k, n)$ of (\\ref{equation-gkn}) is representable by a scheme.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Grassmannians","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089T","source_file":"constructions.tex","source_line":4993,"source_end_line":4997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L4993-L4997","statement_sha256":"959767c386f920389b04efba4767df189766dc38210ba6efa9d12383f63a2f28","origin":"The Stacks Project","memory_eligible":false,"source_rank":5177,"rank":5177,"depth":6,"x":1479.783,"y":746.835,"cluster":"schemes"},{"id":"stacks:089U","tag":"089U","title":"Grassmannians · Definition 089U","summary":"Let 0 < k < n. The scheme G(k, n) representing the functor G(k, n) is called Grassmannian over Z. Its base change G(k, n)_S to a scheme S is called Grassmannian over S. If R is a ring the base change to Spec(R) is denoted G(k, n)_R and called Grassmannian over R.","statement_latex":"Let $0 < k < n$. The scheme $\\mathbf{G}(k, n)$ representing the functor\n$G(k, n)$ is called {\\it Grassmannian over $\\mathbf{Z}$}.\nIts base change $\\mathbf{G}(k, n)_S$ to a scheme $S$ is called\n{\\it Grassmannian over $S$}. If $R$ is a ring the base change\nto $\\Spec(R)$ is denoted $\\mathbf{G}(k, n)_R$ and called\n{\\it Grassmannian over $R$}.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Grassmannians","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089U","source_file":"constructions.tex","source_line":5120,"source_end_line":5128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L5120-L5128","statement_sha256":"c63de0bd4758f3fd5d2212ce31847eb6ae62cefbb8121d98c195d707733c6dfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":5178,"rank":5178,"depth":0,"x":1600.201,"y":573.589,"cluster":"schemes"},{"id":"stacks:089V","tag":"089V","title":"Grassmannians · Lemma 089V","summary":"Let n ≥ 1. There is a canonical isomorphism G(n, n + 1) = P^n_Z.","statement_latex":"Let $n \\geq 1$. There is a canonical isomorphism\n$\\mathbf{G}(n, n + 1) = \\mathbf{P}^n_\\mathbf{Z}$.","area":"Schemes","chapter":"Constructions of Schemes","chapter_id":"constructions","section":"Grassmannians","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089V","source_file":"constructions.tex","source_line":5135,"source_end_line":5139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/constructions.tex#L5135-L5139","statement_sha256":"094f792992370cc33f3979b44e2ceda6e93b3c75aa67528cc8ddd78667970147","origin":"The Stacks Project","memory_eligible":false,"source_rank":5179,"rank":5179,"depth":18,"x":1650.85,"y":770.149,"cluster":"schemes"},{"id":"stacks:054C","tag":"054C","title":"Constructible sets · Lemma 054C","summary":"Let X be a scheme. A subset E of X is locally constructible in X if and only if E ∩ U is constructible in U for every affine open U of X.","statement_latex":"Let $X$ be a scheme.\nA subset $E$ of $X$ is locally constructible in $X$ if and only if\n$E \\cap U$ is constructible in $U$ for every affine open $U$ of $X$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054C","source_file":"properties.tex","source_line":36,"source_end_line":41,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L36-L41","statement_sha256":"198ec9dfec780f838963e8e6eaf1e5b171dbc0dcb48d6134e7fcf765e753e23b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5180,"rank":5180,"depth":3,"x":1454.956,"y":653.663,"cluster":"schemes"},{"id":"stacks:0AAW","tag":"0AAW","title":"Constructible sets · Lemma 0AAW","summary":"Let X be a scheme and let E ⊂ X be a locally constructible subset. Let xi ∈ X be a generic point of an irreducible component of X. • If xi ∈ E, then an open neighbourhood of xi is contained in E. • If xi not ∈ E, then an open neighbourhood of xi is disjoint from E.","statement_latex":"Let $X$ be a scheme and let $E \\subset X$ be a locally constructible subset.\nLet $\\xi \\in X$ be a generic point of an irreducible component of $X$.\n\\begin{enumerate}\n\\item If $\\xi \\in E$, then an open neighbourhood of\n$\\xi$ is contained in $E$.\n\\item If $\\xi \\not \\in E$, then an open neighbourhood\nof $\\xi$ is disjoint from $E$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAW","source_file":"properties.tex","source_line":58,"source_end_line":68,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L58-L68","statement_sha256":"946eed8e640702cf4b1e3bd7c160587b1afec37d21aae99f2a680053d563e9bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5181,"rank":5181,"depth":5,"x":1693.661,"y":628.342,"cluster":"schemes"},{"id":"stacks:054D","tag":"054D","title":"Constructible sets · Lemma 054D","summary":"Let X be a quasi-separated scheme. The intersection of any two quasi-compact opens of X is a quasi-compact open of X. Every quasi-compact open of X is retrocompact in X.","statement_latex":"Let $X$ be a quasi-separated scheme. The intersection of any two\nquasi-compact opens of $X$ is a quasi-compact open of $X$.\nEvery quasi-compact open of $X$ is retrocompact in $X$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054D","source_file":"properties.tex","source_line":82,"source_end_line":87,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L82-L87","statement_sha256":"05ffb18c76a03377342869512663bf7a18e03b7d05e52c1f9060163bbf4cac30","origin":"The Stacks Project","memory_eligible":false,"source_rank":5182,"rank":5182,"depth":1,"x":1537.628,"y":782.834,"cluster":"schemes"},{"id":"stacks:094L","tag":"094L","title":"Constructible sets · Lemma 094L","summary":"Let X be a quasi-compact and quasi-separated scheme. Then the underlying topological space of X is a spectral space.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nThen the underlying topological space of $X$ is a spectral space.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094L","source_file":"properties.tex","source_line":101,"source_end_line":105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L101-L105","statement_sha256":"a22c82b7452a76447579b392fec0c1a643fbb5994d4c3273ce3dde00412bfb6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5183,"rank":5183,"depth":2,"x":1528.424,"y":579.888,"cluster":"schemes"},{"id":"stacks:054E","tag":"054E","title":"Constructible sets · Lemma 054E","summary":"Let X be a quasi-compact and quasi-separated scheme. Any locally constructible subset of X is constructible.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nAny locally constructible subset of $X$ is constructible.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054E","source_file":"properties.tex","source_line":119,"source_end_line":123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L119-L123","statement_sha256":"aedaae986098abe444c870f9eaebb5881e7c5bb3680eb8156500a4b655ec444c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5184,"rank":5184,"depth":4,"x":1698.823,"y":724.663,"cluster":"schemes"},{"id":"stacks:07ZL","tag":"07ZL","title":"Constructible sets · Lemma 07ZL","summary":"Let X be a scheme. A subset E of X is retrocompact in X if and only if E ∩ U is quasi-compact for every affine open U of X.","statement_latex":"Let $X$ be a scheme. A subset $E$ of $X$ is retrocompact in $X$ if and only if\n$E \\cap U$ is quasi-compact for every affine open $U$ of $X$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZL","source_file":"properties.tex","source_line":137,"source_end_line":141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L137-L141","statement_sha256":"a65bf0bddcd8fa64f712cfe10c4b26d2ff5f63f163778fedcbacfedf99e2543a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5185,"rank":5185,"depth":0,"x":1456.171,"y":714.572,"cluster":"schemes"},{"id":"stacks:0F2M","tag":"0F2M","title":"Constructible sets · Lemma 0F2M","summary":"A partition X = coprod_i ∈ I X_i of a scheme X with retrocompact parts is locally finite if and only if the parts are locally constructible.","statement_latex":"A partition $X = \\coprod_{i \\in I} X_i$ of a scheme $X$ with\nretrocompact parts is locally finite if and only if the parts\nare locally constructible.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2M","source_file":"properties.tex","source_line":148,"source_end_line":153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L148-L153","statement_sha256":"a676d2c4b5b712b63a8090564ab900ef9005371bf912b67fbc8dd6a337c586d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5186,"rank":5186,"depth":4,"x":1643.659,"y":584.017,"cluster":"schemes"},{"id":"stacks:01OK","tag":"01OK","title":"Integral, irreducible, and reduced schemes · Definition 01OK","summary":"Let X be a scheme. We say X is integral if it is nonempty and for every nonempty affine open Spec(R) = U ⊂ X the ring R is an integral domain.","statement_latex":"Let $X$ be a scheme. We say $X$ is {\\it integral} if it is nonempty and\nfor every nonempty affine open $\\Spec(R) = U \\subset X$ the ring $R$\nis an integral domain.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Integral, irreducible, and reduced schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OK","source_file":"properties.tex","source_line":189,"source_end_line":194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L189-L194","statement_sha256":"8ed48af5847061e1f0b17b5601d81c6f376dc55c4a8cb58345c60a3a74c3c865","origin":"The Stacks Project","memory_eligible":false,"source_rank":5187,"rank":5187,"depth":0,"x":1610.318,"y":787.151,"cluster":"schemes"},{"id":"stacks:01OL","tag":"01OL","title":"Integral, irreducible, and reduced schemes · Lemma 01OL","summary":"Let X be a scheme. The following are equivalent. • The scheme X is reduced, see Schemes, Definition [Tag 01J0]. • There exists an affine open covering X = ⋃ U_i such that each Γ(U_i, O_X) is reduced. • For every affine open U ⊂ X the ring O_X(U) is reduced. • For every open U ⊂ X the ring O_X(U) is reduced.","statement_latex":"Let $X$ be a scheme.\nThe following are equivalent.\n\\begin{enumerate}\n\\item The scheme $X$ is reduced, see\nSchemes, Definition \\ref{schemes-definition-reduced}.\n\\item There exists an affine open covering $X = \\bigcup U_i$\nsuch that each $\\Gamma(U_i, \\mathcal{O}_X)$ is reduced.\n\\item For every affine open $U \\subset X$ the ring\n$\\mathcal{O}_X(U)$ is reduced.\n\\item For every open $U \\subset X$ the ring $\\mathcal{O}_X(U)$ is reduced.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Integral, irreducible, and reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OL","source_file":"properties.tex","source_line":196,"source_end_line":209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L196-L209","statement_sha256":"b8c1c45f4dd7b9f4eedc757ca78feddd81edb9db7071e30920d2b2cd4bf37ff9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5188,"rank":5188,"depth":3,"x":1471.221,"y":618.046,"cluster":"schemes"},{"id":"stacks:01OM","tag":"01OM","title":"Integral, irreducible, and reduced schemes · Lemma 01OM","summary":"Let X be a scheme. The following are equivalent. • The scheme X is irreducible. • There exists an affine open covering X = ⋃_i ∈ I U_i such that I is not empty, U_i is irreducible for all i ∈ I, and U_i ∩ U_j not = ∅ for all i, j ∈ I. • The scheme X is nonempty and every nonempty affine open U ⊂ X is irreducible.","statement_latex":"Let $X$ be a scheme.\nThe following are equivalent.\n\\begin{enumerate}\n\\item The scheme $X$ is irreducible.\n\\item There exists an affine open covering $X = \\bigcup_{i \\in I} U_i$\nsuch that $I$ is not empty, $U_i$ is irreducible for all $i \\in I$, and\n$U_i \\cap U_j \\not = \\emptyset$ for all $i, j \\in I$.\n\\item The scheme $X$ is nonempty and every nonempty affine open\n$U \\subset X$ is irreducible.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Integral, irreducible, and reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OM","source_file":"properties.tex","source_line":216,"source_end_line":228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L216-L228","statement_sha256":"3cb1c215a7ce6aef8eaf725469ef91b25120741db133d705f01c45b7587abe62","origin":"The Stacks Project","memory_eligible":false,"source_rank":5189,"rank":5189,"depth":12,"x":1710.337,"y":663.923,"cluster":"schemes"},{"id":"stacks:01ON","tag":"01ON","title":"Integral, irreducible, and reduced schemes · Lemma 01ON","summary":"A scheme X is integral if and only if it is reduced and irreducible.","statement_latex":"A scheme $X$ is integral if and only if it is reduced and irreducible.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Integral, irreducible, and reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ON","source_file":"properties.tex","source_line":259,"source_end_line":262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L259-L262","statement_sha256":"3cfa6c081bad25e0287bdb1b5e40c5841b937f45e6dbe410b7821ff58d1d8231","origin":"The Stacks Project","memory_eligible":false,"source_rank":5190,"rank":5190,"depth":13,"x":1496.626,"y":766.012,"cluster":"schemes"},{"id":"stacks:01OP","tag":"01OP","title":"Types of schemes defined by properties of rings · Definition 01OP","summary":"Let P be a property of rings. We say that P is local if the following hold: • For any ring R, and any f ∈ R we have P(R) ⇒ P(R_f). • For any ring R, and f_i ∈ R such that (f_1, …, f_n) = R then ∀ i, P(R_f_i) ⇒ P(R).","statement_latex":"Let $P$ be a property of rings.\nWe say that $P$ is {\\it local} if the following hold:\n\\begin{enumerate}\n\\item For any ring $R$, and any $f \\in R$ we have\n$P(R) \\Rightarrow P(R_f)$.\n\\item For any ring $R$, and $f_i \\in R$ such that\n$(f_1, \\ldots, f_n) = R$ then\n$\\forall i, P(R_{f_i}) \\Rightarrow P(R)$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Types of schemes defined by properties of rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OP","source_file":"properties.tex","source_line":301,"source_end_line":312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L301-L312","statement_sha256":"12fb33203b81266bbf5603417ecb22e5909039f46853644bef5adf0b9ee371e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5191,"rank":5191,"depth":0,"x":1572.292,"y":569.012,"cluster":"schemes"},{"id":"stacks:01OQ","tag":"01OQ","title":"Types of schemes defined by properties of rings · Definition 01OQ","summary":"Let P be a property of rings. Let X be a scheme. We say X is locally P if for any x ∈ X there exists an affine open neighbourhood U of x in X such that O_X(U) has property P.","statement_latex":"Let $P$ be a property of rings. Let $X$ be a scheme.\nWe say $X$ is {\\it locally $P$} if for any $x \\in X$\nthere exists an affine open neighbourhood $U$ of $x$\nin $X$ such that $\\mathcal{O}_X(U)$ has property $P$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Types of schemes defined by properties of rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OQ","source_file":"properties.tex","source_line":314,"source_end_line":320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L314-L320","statement_sha256":"5be3dfc5554d749563034110778bbc2bcd00a0a26fd80bbde2a03190ea91a87b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5192,"rank":5192,"depth":0,"x":1675.156,"y":757.646,"cluster":"schemes"},{"id":"stacks:01OR","tag":"01OR","title":"Types of schemes defined by properties of rings · Lemma 01OR","summary":"Let X be a scheme. Let P be a local property of rings. The following are equivalent: • The scheme X is locally P. • For every affine open U ⊂ X the property P(O_X(U)) holds. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) satisfies P. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is locally P. Moreover, if X is locally P then every open subscheme is locally P.","statement_latex":"Let $X$ be a scheme. Let $P$ be a local property of rings.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is locally $P$.\n\\item For every affine open $U \\subset X$ the property\n$P(\\mathcal{O}_X(U))$ holds.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ satisfies $P$.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is locally $P$.\n\\end{enumerate}\nMoreover, if $X$ is locally $P$ then every open subscheme\nis locally $P$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Types of schemes defined by properties of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OR","source_file":"properties.tex","source_line":329,"source_end_line":344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L329-L344","statement_sha256":"ebd5440f96fc0475dbfd9bc34798bb3e3f645f1bf1bb2cafc9db685dbcf17e7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5193,"rank":5193,"depth":2,"x":1447.088,"y":676.731,"cluster":"schemes"},{"id":"stacks:01OS","tag":"01OS","title":"Types of schemes defined by properties of rings · Lemma 01OS","summary":"Let X be a scheme. Then X is reduced if and only if X is \"locally reduced\" in the sense of Definition [Tag 01OQ].","statement_latex":"Let $X$ be a scheme. Then $X$ is reduced if and only if $X$ is\n``locally reduced'' in the sense of Definition \\ref{definition-locally-P}.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Types of schemes defined by properties of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OS","source_file":"properties.tex","source_line":367,"source_end_line":371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L367-L371","statement_sha256":"e9ada7933b7b42c24d463976ed5851ecd7bf1132ea1f67aa5cb8ae1848210541","origin":"The Stacks Project","memory_eligible":false,"source_rank":5194,"rank":5194,"depth":4,"x":1680.866,"y":606.826,"cluster":"schemes"},{"id":"stacks:01OT","tag":"01OT","title":"Types of schemes defined by properties of rings · Lemma 01OT","summary":"The following properties of a ring R are local. • (Cohen-Macaulay.) The ring R is Noetherian and CM, see Algebra, Definition [Tag 00NC]. • (Regular.) The ring R is Noetherian and regular, see Algebra, Definition [Tag 00OD]. • (Absolutely Noetherian.) The ring R is of finite type over Z. • Add more here as needed.","statement_latex":"The following properties of a ring $R$ are local.\n\\begin{enumerate}\n\\item (Cohen-Macaulay.)\nThe ring $R$ is Noetherian and CM, see\nAlgebra, Definition \\ref{algebra-definition-ring-CM}.\n\\item (Regular.)\nThe ring $R$ is Noetherian and regular, see\nAlgebra, Definition \\ref{algebra-definition-regular}.\n\\item (Absolutely Noetherian.)\nThe ring $R$ is of finite type over $Z$.\n\\item Add more here as needed.\\footnote{But we only list those properties\nhere which we have not already dealt with separately somewhere else.}\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Types of schemes defined by properties of rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OT","source_file":"properties.tex","source_line":377,"source_end_line":392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L377-L392","statement_sha256":"3eed69545fbc4d7e32ec2a13a1ec6b8671f4879f77ba4cc053971acd318b07de","origin":"The Stacks Project","memory_eligible":false,"source_rank":5195,"rank":5195,"depth":1,"x":1564.432,"y":791.444,"cluster":"schemes"},{"id":"stacks:01OV","tag":"01OV","title":"Noetherian schemes · Definition 01OV","summary":"Let X be a scheme. • We say X is locally Noetherian if every x ∈ X has an affine open neighbourhood Spec(R) = U ⊂ X such that the ring R is Noetherian. • We say X is Noetherian if X is locally Noetherian and quasi-compact.","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item We say $X$ is {\\it locally Noetherian} if every\n$x \\in X$ has an affine open neighbourhood\n$\\Spec(R) = U \\subset X$ such that the ring $R$ is Noetherian.\n\\item We say $X$ is {\\it Noetherian} if $X$ is locally Noetherian\nand quasi-compact.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OV","source_file":"properties.tex","source_line":420,"source_end_line":430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L420-L430","statement_sha256":"ed79d95db17d764470b9f1349784442f96ebf342ec4140803b44ce737b700fb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5196,"rank":5196,"depth":0,"x":1501.682,"y":588.783,"cluster":"schemes"},{"id":"stacks:01OW","tag":"01OW","title":"Noetherian schemes · Lemma 01OW","summary":"Let X be a scheme. The following are equivalent: • The scheme X is locally Noetherian. • For every affine open U ⊂ X the ring O_X(U) is Noetherian. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is Noetherian. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is locally Noetherian. Moreover, if X is locally Noetherian then every open subscheme is locally Noetherian.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is locally Noetherian.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis Noetherian.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ is Noetherian.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is locally Noetherian.\n\\end{enumerate}\nMoreover, if $X$ is locally Noetherian then every open subscheme\nis locally Noetherian.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OW","source_file":"properties.tex","source_line":435,"source_end_line":449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L435-L449","statement_sha256":"ed269bff1717c204f65d4f3f698c39f5a2dcb35656ec12131fc7b6257e73c8a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5197,"rank":5197,"depth":3,"x":1711.4,"y":702.874,"cluster":"schemes"},{"id":"stacks:01OX","tag":"01OX","title":"Noetherian schemes · Lemma 01OX","summary":"Any immersion Z → X with X locally Noetherian is quasi-compact.","statement_latex":"Any immersion $Z \\to X$ with $X$ locally Noetherian is quasi-compact.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OX","source_file":"properties.tex","source_line":460,"source_end_line":463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L460-L463","statement_sha256":"d08b79ae38f4ee8ad6ba4603f6f5a0515c87679d1033cb1f47a9e6e123440284","origin":"The Stacks Project","memory_eligible":false,"source_rank":5198,"rank":5198,"depth":4,"x":1464.454,"y":737.822,"cluster":"schemes"},{"id":"stacks:01OY","tag":"01OY","title":"Noetherian schemes · Lemma 01OY","summary":"A locally Noetherian scheme is quasi-separated.","statement_latex":"A locally Noetherian scheme is quasi-separated.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OY","source_file":"properties.tex","source_line":480,"source_end_line":483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L480-L483","statement_sha256":"87f85c85d09537541848166067fec36a5b8a1a079534bff9e092cd9bfc6e4afe","origin":"The Stacks Project","memory_eligible":false,"source_rank":5199,"rank":5199,"depth":5,"x":1618.789,"y":571.558,"cluster":"schemes"},{"id":"stacks:01OZ","tag":"01OZ","title":"Noetherian schemes · Lemma 01OZ","summary":"A (locally) Noetherian scheme has a (locally) Noetherian underlying topological space, see Topology, Definition [Tag 0051].","statement_latex":"A (locally) Noetherian scheme has a (locally)\nNoetherian underlying topological space,\nsee Topology, Definition \\ref{topology-definition-noetherian}.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OZ","source_file":"properties.tex","source_line":495,"source_end_line":500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L495-L500","statement_sha256":"38d6abffd45b99697b2f7880c21fb03ac1017608b27d2c2c4da25805dda86f6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5200,"rank":5200,"depth":2,"x":1638.734,"y":782.2,"cluster":"schemes"},{"id":"stacks:02IK","tag":"02IK","title":"Noetherian schemes · Lemma 02IK","summary":"Any locally closed subscheme of a (locally) Noetherian scheme is (locally) Noetherian.","statement_latex":"Any locally closed subscheme of a (locally) Noetherian\nscheme is (locally) Noetherian.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IK","source_file":"properties.tex","source_line":509,"source_end_line":513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L509-L513","statement_sha256":"82a3d62cdfb37dac37b288593f7b521fcde67f0284f9494c433ecae2b01264ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":5201,"rank":5201,"depth":0,"x":1454.226,"y":637.873,"cluster":"schemes"},{"id":"stacks:0BA8","tag":"0BA8","title":"Noetherian schemes · Lemma 0BA8","summary":"A Noetherian scheme has a finite number of irreducible components.","statement_latex":"A Noetherian scheme has a finite number of irreducible components.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BA8","source_file":"properties.tex","source_line":522,"source_end_line":525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L522-L525","statement_sha256":"d23879f43f1df329660c357ce1abba4a1e20196658c48e4fb0e785925237f22f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5202,"rank":5202,"depth":3,"x":1706.9,"y":639.608,"cluster":"schemes"},{"id":"stacks:01P0","tag":"01P0","title":"Noetherian schemes · Lemma 01P0","summary":"Any morphism of schemes f : X → Y with X Noetherian is quasi-compact.","statement_latex":"Any morphism of schemes $f : X \\to Y$ with $X$ Noetherian\nis quasi-compact.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01P0","source_file":"properties.tex","source_line":535,"source_end_line":539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L535-L539","statement_sha256":"0b103c5c1bf1eef7ff6fcf6fcf422a15001e39aa00125aa15513b2fec8485161","origin":"The Stacks Project","memory_eligible":false,"source_rank":5203,"rank":5203,"depth":4,"x":1518.774,"y":782.014,"cluster":"schemes"},{"id":"stacks:02IL","tag":"02IL","title":"Noetherian schemes · Lemma 02IL","summary":"Any nonempty locally Noetherian scheme has a closed point. Any nonempty closed subset of a locally Noetherian scheme has a closed point. Equivalently, any point of a locally Noetherian scheme specializes to a closed point.","statement_latex":"Any nonempty locally Noetherian scheme has a closed point.\nAny nonempty closed subset of a locally Noetherian scheme has a closed point.\nEquivalently, any point of a locally Noetherian scheme specializes\nto a closed point.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IL","source_file":"properties.tex","source_line":554,"source_end_line":560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L554-L560","statement_sha256":"75770e5d32e4c8ed579be096101050bfc696663e8c7d86e6bdc3aa4d162028c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5204,"rank":5204,"depth":13,"x":1543.03,"y":569.795,"cluster":"schemes"},{"id":"stacks:054F","tag":"054F","title":"Noetherian schemes · Lemma 054F","summary":"Let X be a locally Noetherian scheme. Let x' leadsto x be a specialization of points of X. Then • there exists a discrete valuation ring R and a morphism f : Spec(R) → X such that the generic point eta of Spec(R) maps to x' and the special point maps to x, and • provided x not = x' given a finitely generated field extension K/kappa(x'), we may arrange it so that the extension kappa(eta)/kappa(x') induced by f is isomorphic to the given one.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $x' \\leadsto x$ be a specialization of points of $X$. Then\n\\begin{enumerate}\n\\item there exists a discrete valuation ring $R$ and a morphism\n$f : \\Spec(R) \\to X$ such that the generic point $\\eta$ of\n$\\Spec(R)$ maps to $x'$ and the special point maps to $x$, and\n\\item provided $x \\not = x'$ given a finitely generated field\nextension $K/\\kappa(x')$, we may arrange it so that the extension\n$\\kappa(\\eta)/\\kappa(x')$\ninduced by $f$ is isomorphic to the given one.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054F","source_file":"properties.tex","source_line":591,"source_end_line":604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L591-L604","statement_sha256":"77a147dde33b9e34effe49e5c17c20004b5f1a0c6c12b5a4f55249ad01c1fc78","origin":"The Stacks Project","memory_eligible":false,"source_rank":5205,"rank":5205,"depth":17,"x":1696.137,"y":740.416,"cluster":"schemes"},{"id":"stacks:0CXG","tag":"0CXG","title":"Noetherian schemes · Lemma 0CXG","summary":"Let S be a Noetherian scheme. Let T ⊂ S be an infinite subset. Then there exists an infinite subset T' ⊂ T such that there are no nontrivial specializations among the points T'.","statement_latex":"Let $S$ be a Noetherian scheme. Let $T \\subset S$ be an infinite subset.\nThen there exists an infinite subset $T' \\subset T$\nsuch that there are no nontrivial specializations among the points $T'$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXG","source_file":"properties.tex","source_line":626,"source_end_line":631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L626-L631","statement_sha256":"27dba6ba3d1da7c7d57b389d7eb20d5d35c60ca56092c7ea6a5a745eb4ba299f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5206,"rank":5206,"depth":10,"x":1445.487,"y":701.395,"cluster":"schemes"},{"id":"stacks:0G2R","tag":"0G2R","title":"Noetherian schemes · Lemma 0G2R","summary":"Let S be a Noetherian scheme. Let T ⊂ S be a subset. Let T_0 ⊂ T be the set of t ∈ T such that there is no nontrivial specialization t' leadsto t with t' ∈ T'. Then (a) there are no specializations among the points of T_0, (b) every point of T is a specialization of a point of T_0, and (c) the closures of T and T_0 are the same.","statement_latex":"Let $S$ be a Noetherian scheme. Let $T \\subset S$ be a subset. Let\n$T_0 \\subset T$ be the set of $t \\in T$ such that there is no nontrivial\nspecialization $t' \\leadsto t$ with $t' \\in T'$. Then (a) there are\nno specializations among the points of $T_0$, (b) every point of\n$T$ is a specialization of a point of $T_0$, and (c) the closures\nof $T$ and $T_0$ are the same.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2R","source_file":"properties.tex","source_line":678,"source_end_line":686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L678-L686","statement_sha256":"240aeba781bf640dfc3b848e742fcd652d2c3ba79772327de075287e0429e0b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5207,"rank":5207,"depth":10,"x":1662.159,"y":587.699,"cluster":"schemes"},{"id":"stacks:0G2F","tag":"0G2F","title":"Noetherian schemes · Lemma 0G2F","summary":"Let S be a Noetherian scheme. Let T ⊂ S be an infinite dense subset. Then there exist a countable subset E ⊂ T which is dense in S.","statement_latex":"Let $S$ be a Noetherian scheme. Let $T \\subset S$ be an infinite dense subset.\nThen there exist a countable subset $E \\subset T$ which is dense in $S$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2F","source_file":"properties.tex","source_line":699,"source_end_line":703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L699-L703","statement_sha256":"291b5f92e4df73cb54db522c78fd9277c56f7bec5aeb0e27f989fb87f16b74d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5208,"rank":5208,"depth":11,"x":1593.672,"y":794.926,"cluster":"schemes"},{"id":"stacks:01P2","tag":"01P2","title":"Jacobson schemes · Definition 01P2","summary":"A scheme S is said to be Jacobson if its underlying topological space is Jacobson.","statement_latex":"A scheme $S$ is said to be {\\it Jacobson} if its underlying topological\nspace is Jacobson.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Jacobson schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01P2","source_file":"properties.tex","source_line":750,"source_end_line":754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L750-L754","statement_sha256":"c74eaeb95f7e490f447a56a80c30a2969a54f7d70460ed2de2c1277603baee17","origin":"The Stacks Project","memory_eligible":false,"source_rank":5209,"rank":5209,"depth":0,"x":1477.279,"y":602.848,"cluster":"schemes"},{"id":"stacks:01P3","tag":"01P3","title":"Jacobson schemes · Lemma 01P3","summary":"An affine scheme Spec(R) is Jacobson if and only if the ring R is Jacobson.","statement_latex":"An affine scheme $\\Spec(R)$ is Jacobson if and only if\nthe ring $R$ is Jacobson.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01P3","source_file":"properties.tex","source_line":761,"source_end_line":765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L761-L765","statement_sha256":"4dc6ec006cf6fe54277835c4f7dd76a41cdac44d979be1000d340e970e1efc8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5210,"rank":5210,"depth":2,"x":1718.081,"y":678.601,"cluster":"schemes"},{"id":"stacks:01P4","tag":"01P4","title":"Jacobson schemes · Lemma 01P4","summary":"Let X be a scheme. The following are equivalent: • The scheme X is Jacobson. • The scheme X is \"locally Jacobson\" in the sense of Definition [Tag 01OQ]. • For every affine open U ⊂ X the ring O_X(U) is Jacobson. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is Jacobson. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is Jacobson. Moreover, if X is Jacobson then every open subscheme is Jacobson.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is Jacobson.\n\\item The scheme $X$ is ``locally Jacobson'' in the sense of\nDefinition \\ref{definition-locally-P}.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis Jacobson.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ is Jacobson.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is Jacobson.\n\\end{enumerate}\nMoreover, if $X$ is Jacobson then every open subscheme\nis Jacobson.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01P4","source_file":"properties.tex","source_line":775,"source_end_line":791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L775-L791","statement_sha256":"aee44bcfc688ebaf650bcae87fc4235355fe2b67d6a9c3fb7f514669220a4597","origin":"The Stacks Project","memory_eligible":false,"source_rank":5211,"rank":5211,"depth":3,"x":1479.093,"y":759.55,"cluster":"schemes"},{"id":"stacks:02IM","tag":"02IM","title":"Jacobson schemes · Lemma 02IM","summary":"Examples of Noetherian Jacobson schemes. • If (R, m) is a Noetherian local ring, then the punctured spectrum Spec(R) setminus ( m) is a Jacobson scheme. • If R is a Noetherian ring with Jacobson radical rad(R) then Spec(R) setminus V(rad(R)) is a Jacobson scheme. • If (R, I) is a Zariski pair (More on Algebra, Definition [Tag 0ELY]) with R Noetherian, then Spec(R) setminus V(I) is a Jacobson scheme.","statement_latex":"Examples of Noetherian Jacobson schemes.\n\\begin{enumerate}\n\\item If $(R, \\mathfrak m)$ is a Noetherian local ring, then\nthe punctured spectrum $\\Spec(R) \\setminus \\{\\mathfrak m\\}$\nis a Jacobson scheme.\n\\item If $R$ is a Noetherian ring with Jacobson radical $\\text{rad}(R)$\nthen $\\Spec(R) \\setminus V(\\text{rad}(R))$ is a Jacobson scheme.\n\\item If $(R, I)$ is a Zariski pair (More on Algebra, Definition\n\\ref{more-algebra-definition-zariski-pair})\nwith $R$ Noetherian, then $\\Spec(R) \\setminus V(I)$ is a\nJacobson scheme.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IM","source_file":"properties.tex","source_line":817,"source_end_line":831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L817-L831","statement_sha256":"51778b271f93763c18b99c9a0f037da391f0eaade27912bd6920615aa1cf4e95","origin":"The Stacks Project","memory_eligible":false,"source_rank":5212,"rank":5212,"depth":15,"x":1590.457,"y":563.839,"cluster":"schemes"},{"id":"stacks:033I","tag":"033I","title":"Normal schemes · Definition 033I","summary":"A scheme X is normal if and only if for all x ∈ X the local ring O_X, x is a normal domain.","statement_latex":"A scheme $X$ is {\\it normal} if and only if for all $x \\in X$ the local ring\n$\\mathcal{O}_{X, x}$ is a normal domain.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Normal schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033I","source_file":"properties.tex","source_line":878,"source_end_line":882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L878-L882","statement_sha256":"4eeac8fcab4538f520bcc0b50c2e4bc980e26f5ce64fd06e136309845da47a86","origin":"The Stacks Project","memory_eligible":false,"source_rank":5213,"rank":5213,"depth":0,"x":1665.882,"y":771.781,"cluster":"schemes"},{"id":"stacks:033J","tag":"033J","title":"Normal schemes · Lemma 033J","summary":"Let X be a scheme. The following are equivalent: • The scheme X is normal. • For every affine open U ⊂ X the ring O_X(U) is normal. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is normal. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is normal. Moreover, if X is normal then every open subscheme is normal.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is normal.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis normal.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ is normal.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is normal.\n\\end{enumerate}\nMoreover, if $X$ is normal then every open subscheme\nis normal.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033J","source_file":"properties.tex","source_line":891,"source_end_line":905,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L891-L905","statement_sha256":"df05b00f187302d710da7ed84ada257ac07a190cb9c79e08599c9f14f04e5e11","origin":"The Stacks Project","memory_eligible":false,"source_rank":5214,"rank":5214,"depth":0,"x":1442.577,"y":661.014,"cluster":"schemes"},{"id":"stacks:033K","tag":"033K","title":"Normal schemes · Lemma 033K","summary":"A normal scheme is reduced.","statement_latex":"A normal scheme is reduced.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033K","source_file":"properties.tex","source_line":911,"source_end_line":914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L911-L914","statement_sha256":"31d30f67343e54e5690def7473b738adfc7603de7e83635daa4e9a922a3bb207","origin":"The Stacks Project","memory_eligible":false,"source_rank":5215,"rank":5215,"depth":0,"x":1696.844,"y":615.89,"cluster":"schemes"},{"id":"stacks:033L","tag":"033L","title":"Normal schemes · Lemma 033L","summary":"Let X be an integral scheme. Then X is normal if and only if for every nonempty affine open U ⊂ X the ring O_X(U) is a normal domain.","statement_latex":"Let $X$ be an integral scheme.\nThen $X$ is normal if and only if for every nonempty affine open\n$U \\subset X$ the ring $\\mathcal{O}_X(U)$ is a normal domain.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033L","source_file":"properties.tex","source_line":920,"source_end_line":925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L920-L925","statement_sha256":"31a3c80b54d00e4dadf984e0844dee381feb427f6dc834b4d4ba694e4d0951ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":5216,"rank":5216,"depth":2,"x":1545.325,"y":793.809,"cluster":"schemes"},{"id":"stacks:0357","tag":"0357","title":"Normal schemes · Lemma 0357","summary":"Let X be a scheme such that any quasi-compact open has a finite number of irreducible components. The following are equivalent: • X is normal, and • X is a disjoint union of normal integral schemes.","statement_latex":"Let $X$ be a scheme such that any quasi-compact open has a finite number\nof irreducible components. The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is normal, and\n\\item $X$ is a disjoint union of normal integral schemes.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0357","source_file":"properties.tex","source_line":932,"source_end_line":940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L932-L940","statement_sha256":"cc7136e7314968077b35cb6f074c5c5406acd6f232bc39cb4d7ac94f6a07f6a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5217,"rank":5217,"depth":7,"x":1513.911,"y":576.189,"cluster":"schemes"},{"id":"stacks:033M","tag":"033M","title":"Normal schemes · Lemma 033M","summary":"Let X be a Noetherian scheme. The following are equivalent: • X is normal, and • X is a finite disjoint union of normal integral schemes.","statement_latex":"Let $X$ be a Noetherian scheme.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $X$ is normal, and\n\\item $X$ is a finite disjoint union of normal integral schemes.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033M","source_file":"properties.tex","source_line":959,"source_end_line":967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L959-L967","statement_sha256":"6c2cc168dad86b76abc4bf14504183f29bad48fe4f0aaff240b25b8704034150","origin":"The Stacks Project","memory_eligible":false,"source_rank":5218,"rank":5218,"depth":8,"x":1712.487,"y":719.128,"cluster":"schemes"},{"id":"stacks:033N","tag":"033N","title":"Normal schemes · Lemma 033N","summary":"Let X be a locally Noetherian scheme. The following are equivalent: • X is normal, and • X is a disjoint union of integral normal schemes.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $X$ is normal, and\n\\item $X$ is a disjoint union of integral normal schemes.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033N","source_file":"properties.tex","source_line":978,"source_end_line":986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L978-L986","statement_sha256":"00d0e0bc353cdcd3d4ae883732cb03ce56e9634b6ef19c888fbd5c3e9ddbe816","origin":"The Stacks Project","memory_eligible":false,"source_rank":5219,"rank":5219,"depth":9,"x":1450.576,"y":726.419,"cluster":"schemes"},{"id":"stacks:0358","tag":"0358","title":"Normal schemes · Lemma 0358","summary":"The ring of functions on a normal scheme is normal. Let X be an integral normal scheme. Then Γ(X, O_X) is a normal domain.","statement_latex":"\\begin{slogan}\nThe ring of functions on a normal scheme is normal.\n\\end{slogan}\nLet $X$ be an integral normal scheme.\nThen $\\Gamma(X, \\mathcal{O}_X)$ is a normal domain.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0358","source_file":"properties.tex","source_line":1005,"source_end_line":1012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1005-L1012","statement_sha256":"60803fda091a5123624c353469bb13e7fdc61a6862e6a9d02d9b6aaafc2edb84","origin":"The Stacks Project","memory_eligible":false,"source_rank":5220,"rank":5220,"depth":3,"x":1638.225,"y":572.114,"cluster":"schemes"},{"id":"stacks:02IO","tag":"02IO","title":"Cohen-Macaulay schemes · Definition 02IO","summary":"Let X be a scheme. We say X is Cohen-Macaulay if for every x ∈ X there exists an affine open neighbourhood U ⊂ X of x such that the ring O_X(U) is Noetherian and Cohen-Macaulay.","statement_latex":"Let $X$ be a scheme. We say $X$ is {\\it Cohen-Macaulay} if\nfor every $x \\in X$ there exists an affine open neighbourhood\n$U \\subset X$ of $x$ such that the ring $\\mathcal{O}_X(U)$ is\nNoetherian and Cohen-Macaulay.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Cohen-Macaulay schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IO","source_file":"properties.tex","source_line":1058,"source_end_line":1064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1058-L1064","statement_sha256":"3e257ec956cfb50b4bd0579c48b31630ca12876e928b6e246d73f4dafd62f23a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5221,"rank":5221,"depth":0,"x":1623.914,"y":792.82,"cluster":"schemes"},{"id":"stacks:02IP","tag":"02IP","title":"Cohen-Macaulay schemes · Lemma 02IP","summary":"Let X be a scheme. The following are equivalent: • X is Cohen-Macaulay, • X is locally Noetherian and all of its local rings are Cohen-Macaulay, and • X is locally Noetherian and for any closed point x ∈ X the local ring O_X, x is Cohen-Macaulay.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is Cohen-Macaulay,\n\\item $X$ is locally Noetherian and all of its local rings are Cohen-Macaulay,\nand\n\\item $X$ is locally Noetherian and for any closed point $x \\in X$\nthe local ring $\\mathcal{O}_{X, x}$ is Cohen-Macaulay.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Cohen-Macaulay schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IP","source_file":"properties.tex","source_line":1066,"source_end_line":1076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1066-L1076","statement_sha256":"3863ea769298d32147ecdc7058465302784c5096fafae3f4c99cd55911d9f2a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5222,"rank":5222,"depth":18,"x":1456.643,"y":621.608,"cluster":"schemes"},{"id":"stacks:02IQ","tag":"02IQ","title":"Cohen-Macaulay schemes · Lemma 02IQ","summary":"Let X be a scheme. The following are equivalent: • The scheme X is Cohen-Macaulay. • For every affine open U ⊂ X the ring O_X(U) is Noetherian and Cohen-Macaulay. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is Noetherian and Cohen-Macaulay. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is Cohen-Macaulay. Moreover, if X is Cohen-Macaulay then every open subscheme is Cohen-Macaulay.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is Cohen-Macaulay.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis Noetherian and Cohen-Macaulay.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ is Noetherian and Cohen-Macaulay.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is Cohen-Macaulay.\n\\end{enumerate}\nMoreover, if $X$ is Cohen-Macaulay then every open subscheme\nis Cohen-Macaulay.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Cohen-Macaulay schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IQ","source_file":"properties.tex","source_line":1088,"source_end_line":1102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1088-L1102","statement_sha256":"d2e14dc21cceb51699a8adaedf3d125020934d2b7bd87851441798cf31b22619","origin":"The Stacks Project","memory_eligible":false,"source_rank":5223,"rank":5223,"depth":19,"x":1718.196,"y":653.007,"cluster":"schemes"},{"id":"stacks:02IS","tag":"02IS","title":"Regular schemes · Definition 02IS","summary":"Let X be a scheme. We say X is regular, or nonsingular if for every x ∈ X there exists an affine open neighbourhood U ⊂ X of x such that the ring O_X(U) is Noetherian and regular.","statement_latex":"Let $X$ be a scheme. We say $X$ is {\\it regular}, or {\\it nonsingular} if\nfor every $x \\in X$ there exists an affine open neighbourhood\n$U \\subset X$ of $x$ such that the ring $\\mathcal{O}_X(U)$ is\nNoetherian and regular.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Regular schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IS","source_file":"properties.tex","source_line":1132,"source_end_line":1138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1132-L1138","statement_sha256":"ac3b2e5adac6e8864864e5c7249b72bced87d45d4f489b81c84ffdc67a3ccf7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5224,"rank":5224,"depth":0,"x":1499.642,"y":778.517,"cluster":"schemes"},{"id":"stacks:02IT","tag":"02IT","title":"Regular schemes · Lemma 02IT","summary":"Let X be a scheme. The following are equivalent: • X is regular, • X is locally Noetherian and all of its local rings are regular, and • X is locally Noetherian and for any closed point x ∈ X the local ring O_X, x is regular.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is regular,\n\\item $X$ is locally Noetherian and all of its local rings are regular,\nand\n\\item $X$ is locally Noetherian and for any closed point $x \\in X$\nthe local ring $\\mathcal{O}_{X, x}$ is regular.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Regular schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IT","source_file":"properties.tex","source_line":1140,"source_end_line":1150,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1140-L1150","statement_sha256":"5856df5b828e82d599cf8342a89c7a4d71a2d8ac91e1762545cb52f1ba6deebe","origin":"The Stacks Project","memory_eligible":false,"source_rank":5225,"rank":5225,"depth":14,"x":1559.99,"y":561.522,"cluster":"schemes"},{"id":"stacks:02IU","tag":"02IU","title":"Regular schemes · Lemma 02IU","summary":"Let X be a scheme. The following are equivalent: • The scheme X is regular. • For every affine open U ⊂ X the ring O_X(U) is Noetherian and regular. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is Noetherian and regular. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is regular. Moreover, if X is regular then every open subscheme is regular.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is regular.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis Noetherian and regular.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ is Noetherian and regular.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is regular.\n\\end{enumerate}\nMoreover, if $X$ is regular then every open subscheme is regular.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Regular schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IU","source_file":"properties.tex","source_line":1163,"source_end_line":1176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1163-L1176","statement_sha256":"68a6c2d13b01ca086a26e27a3cad642c3e506080086178b9bab5b08004310552","origin":"The Stacks Project","memory_eligible":false,"source_rank":5226,"rank":5226,"depth":15,"x":1690.25,"y":756.161,"cluster":"schemes"},{"id":"stacks:0569","tag":"0569","title":"Regular schemes · Lemma 0569","summary":"A regular scheme is normal.","statement_latex":"A regular scheme is normal.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Regular schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0569","source_file":"properties.tex","source_line":1183,"source_end_line":1186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1183-L1186","statement_sha256":"2bc901473db6bad9219cd24bdef218f2a283d5d21ac98fc8a17106eaeaf06bd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5227,"rank":5227,"depth":18,"x":1437.174,"y":686.411,"cluster":"schemes"},{"id":"stacks:04MT","tag":"04MT","title":"Dimension · Definition 04MT","summary":"Let X be a scheme. • The dimension of X is just the dimension of X as a topological spaces, see Topology, Definition [Tag 0055]. • For x ∈ X we denote dim_x(X) the dimension of the underlying topological space of X at x as in Topology, Definition [Tag 0055]. We say dim_x(X) is the dimension of X at x.","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item The {\\it dimension} of $X$ is just the dimension of $X$\nas a topological spaces, see\nTopology, Definition \\ref{topology-definition-Krull}.\n\\item For $x \\in X$ we denote $\\dim_x(X)$ the dimension of the underlying\ntopological space of $X$ at $x$ as in\nTopology, Definition \\ref{topology-definition-Krull}.\nWe say $\\dim_x(X)$ is the {\\it dimension of $X$ at $x$}.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MT","source_file":"properties.tex","source_line":1203,"source_end_line":1215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1203-L1215","statement_sha256":"0910d683b1861f95ca49e5ac180686739cde468d0ab6bf37c2d5d06e70218aaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":5228,"rank":5228,"depth":1,"x":1680.359,"y":594.06,"cluster":"schemes"},{"id":"stacks:04MU","tag":"04MU","title":"Dimension · Lemma 04MU","summary":"Let X be a scheme. The following are equal • The dimension of X. • The supremum of the dimensions of the local rings of X. • The supremum of dim_x(X) for x ∈ X.","statement_latex":"Let $X$ be a scheme. The following are equal\n\\begin{enumerate}\n\\item The dimension of $X$.\n\\item The supremum of the dimensions of the local rings of $X$.\n\\item The supremum of $\\dim_x(X)$ for $x \\in X$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MU","source_file":"properties.tex","source_line":1232,"source_end_line":1240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1232-L1240","statement_sha256":"42fc3193ce9fb272e613d76a9382bb2c5c7727d5a0474fa01bcf3fac2e130a21","origin":"The Stacks Project","memory_eligible":false,"source_rank":5229,"rank":5229,"depth":1,"x":1575.097,"y":800.553,"cluster":"schemes"},{"id":"stacks:02IZ","tag":"02IZ","title":"Dimension · Lemma 02IZ","summary":"Let X be a scheme. Let Y ⊂ X be an irreducible closed subset. Let xi ∈ Y be the generic point. Then codim(Y, X) = dim(O_X, xi) where the codimension is as defined in Topology, Definition [Tag 02I3].","statement_latex":"Let $X$ be a scheme. Let $Y \\subset X$ be an irreducible closed\nsubset. Let $\\xi \\in Y$ be the generic point. Then\n$$\n\\text{codim}(Y, X) = \\dim(\\mathcal{O}_{X, \\xi})\n$$\nwhere the codimension is as defined in\nTopology, Definition \\ref{topology-definition-codimension}.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IZ","source_file":"properties.tex","source_line":1258,"source_end_line":1267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1258-L1267","statement_sha256":"344491f732f8b138babd5b882eb069bd0c85f021b3fc251a442984acbd92a6f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5230,"rank":5230,"depth":3,"x":1486.487,"y":588.145,"cluster":"schemes"},{"id":"stacks:0BA9","tag":"0BA9","title":"Dimension · Lemma 0BA9","summary":"Let X be a scheme. Let x ∈ X. Then x is a generic point of an irreducible component of X if and only if dim(O_X, x) = 0.","statement_latex":"Let $X$ be a scheme. Let $x \\in X$. Then $x$ is a generic point of\nan irreducible component of $X$ if and only if $\\dim(\\mathcal{O}_{X, x}) = 0$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BA9","source_file":"properties.tex","source_line":1276,"source_end_line":1280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1276-L1280","statement_sha256":"76430b967f7a1f396d630373acd68fa502566cb47d036bccaabd19c0066115b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5231,"rank":5231,"depth":4,"x":1723.101,"y":694.7,"cluster":"schemes"},{"id":"stacks:0AAX","tag":"0AAX","title":"Dimension · Lemma 0AAX","summary":"A locally Noetherian scheme of dimension 0 is a disjoint union of spectra of Artinian local rings.","statement_latex":"A locally Noetherian scheme of dimension $0$ is a disjoint\nunion of spectra of Artinian local rings.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAX","source_file":"properties.tex","source_line":1286,"source_end_line":1290,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1286-L1290","statement_sha256":"28f7b6805ee275ad188a277b0a7045adb9568411bd11d2bef1113d55dcaeffb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5232,"rank":5232,"depth":9,"x":1462.43,"y":750.496,"cluster":"schemes"},{"id":"stacks:0CKV","tag":"0CKV","title":"Dimension · Lemma 0CKV","summary":"Email from Ofer Gabber dated June 4, 2016 Let X be a scheme of dimension zero. The following are equivalent • X is quasi-separated, • X is separated, • X is Hausdorff, • every affine open is closed. In this case the connected components of X are points and every quasi-compact open of X is affine. In particular, if X is quasi-compact, then X is affine.","statement_latex":"\\begin{reference}\nEmail from Ofer Gabber dated June 4, 2016\n\\end{reference}\nLet $X$ be a scheme of dimension zero. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is quasi-separated,\n\\item $X$ is separated,\n\\item $X$ is Hausdorff,\n\\item every affine open is closed.\n\\end{enumerate}\nIn this case the connected components of $X$ are points and every\nquasi-compact open of $X$ is affine. In particular, if $X$\nis quasi-compact, then $X$ is affine.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKV","source_file":"properties.tex","source_line":1302,"source_end_line":1317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1302-L1317","statement_sha256":"f133d3671e8a9b85b45726a5cc90e9b5e606a3b9faae67e49e9ee8f6d87436da","origin":"The Stacks Project","memory_eligible":false,"source_rank":5233,"rank":5233,"depth":13,"x":1610.061,"y":561.076,"cluster":"schemes"},{"id":"stacks:0H7C","tag":"0H7C","title":"Dimension · Lemma 0H7C","summary":"Let x be a point of a locally Noetherian scheme X. Then dim_x(X) = 0 if and only if x is an isolated point of X.","statement_latex":"Let $x$ be a point of a locally Noetherian scheme $X$.\nThen $\\dim_x(X) = 0$ if and only if $x$ is an isolated point of $X$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7C","source_file":"properties.tex","source_line":1372,"source_end_line":1376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1372-L1376","statement_sha256":"7a193e55e1b277d7a4414464d30b66db1c2f02f40c89cb33728e5b16d46ff54d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5234,"rank":5234,"depth":10,"x":1653.61,"y":784.952,"cluster":"schemes"},{"id":"stacks:02IW","tag":"02IW","title":"Catenary schemes · Definition 02IW","summary":"Let S be a scheme. We say S is catenary if the underlying topological space of S is catenary.","statement_latex":"Let $S$ be a scheme. We say $S$ is {\\it catenary} if the\nunderlying topological space of $S$ is catenary.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Catenary schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IW","source_file":"properties.tex","source_line":1407,"source_end_line":1411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1407-L1411","statement_sha256":"386c6c4bd8ed400e58fac3fe44249d6501ab1a3dde6bd666739451d4b20ba95f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5235,"rank":5235,"depth":0,"x":1441.055,"y":644.31,"cluster":"schemes"},{"id":"stacks:02IX","tag":"02IX","title":"Catenary schemes · Lemma 02IX","summary":"Let S be a scheme. The following are equivalent • S is catenary, • there exists an open covering of S all of whose members are catenary schemes, • for every affine open Spec(R) = U ⊂ S the ring R is catenary, and • there exists an affine open covering S = ⋃ U_i such that each U_i is the spectrum of a catenary ring. Moreover, in this case any locally closed subscheme of S is catenary as well.","statement_latex":"Let $S$ be a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $S$ is catenary,\n\\item there exists an open covering of $S$ all of whose members are\ncatenary schemes,\n\\item for every affine open $\\Spec(R) = U \\subset S$ the ring\n$R$ is catenary, and\n\\item there exists an affine open covering $S = \\bigcup U_i$ such\nthat each $U_i$ is the spectrum of a catenary ring.\n\\end{enumerate}\nMoreover, in this case any locally closed subscheme of $S$ is catenary\nas well.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Catenary schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IX","source_file":"properties.tex","source_line":1425,"source_end_line":1439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1425-L1439","statement_sha256":"b2065438249addb21fe7bf2a108fb0b501ca705c3fd63429cb6c2d6fad78691d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5236,"rank":5236,"depth":3,"x":1711.407,"y":627.376,"cluster":"schemes"},{"id":"stacks:02IY","tag":"02IY","title":"Catenary schemes · Lemma 02IY","summary":"Let S be a locally Noetherian scheme. The following are equivalent: • S is catenary, and • locally in the Zariski topology there exists a dimension function on S (see Topology, Definition [Tag 02I9]).","statement_latex":"Let $S$ be a locally Noetherian scheme.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $S$ is catenary, and\n\\item locally in the Zariski topology there exists a dimension function\non $S$ (see Topology, Definition \\ref{topology-definition-dimension-function}).\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Catenary schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02IY","source_file":"properties.tex","source_line":1446,"source_end_line":1455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1446-L1455","statement_sha256":"2a18b6b7975e10697a4c4634aa8d6fd01557e2e2b561f463ddc1fa1e19531c25","origin":"The Stacks Project","memory_eligible":false,"source_rank":5237,"rank":5237,"depth":5,"x":1525.317,"y":793.583,"cluster":"schemes"},{"id":"stacks:02J0","tag":"02J0","title":"Catenary schemes · Lemma 02J0","summary":"Let X be a scheme. The following are equivalent • X is catenary, and • for any x ∈ X the local ring O_X, x is catenary.","statement_latex":"Let $X$ be a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is catenary, and\n\\item for any $x \\in X$ the local ring $\\mathcal{O}_{X, x}$ is\ncatenary.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Catenary schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02J0","source_file":"properties.tex","source_line":1471,"source_end_line":1479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1471-L1479","statement_sha256":"377645f5f6430a31e5a0730d420daba1fd61b02850a245cf6748fb6907af19ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":5238,"rank":5238,"depth":4,"x":1528.885,"y":564.999,"cluster":"schemes"},{"id":"stacks:033Q","tag":"033Q","title":"Serre's conditions · Definition 033Q","summary":"Let X be a locally Noetherian scheme. Let k ≥ 0. • We say X is regular in codimension k, or we say X has property (R_k) if for every x ∈ X we have dim(O_X, x) ≤ k ⇒ O_X, x is regular • We say X has property (S_k) if for every x ∈ X we have depth(O_X, x) ≥ min(k, dim(O_X, x)).","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $k \\geq 0$.\n\\begin{enumerate}\n\\item We say $X$ is {\\it regular in codimension $k$},\nor we say $X$ has property {\\it $(R_k)$} if for every $x \\in X$\nwe have\n$$\n\\dim(\\mathcal{O}_{X, x}) \\leq k\n\\Rightarrow\n\\mathcal{O}_{X, x}\\text{ is regular}\n$$\n\\item We say $X$ has property {\\it $(S_k)$} if for every $x \\in X$ we have\n$\\text{depth}(\\mathcal{O}_{X, x}) \\geq \\min(k, \\dim(\\mathcal{O}_{X, x}))$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Serre's conditions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033Q","source_file":"properties.tex","source_line":1515,"source_end_line":1530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1515-L1530","statement_sha256":"4d5eb763172f613a64131056f70379df5ec86404fcb8bfb79c36215673dd62b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5239,"rank":5239,"depth":0,"x":1710.417,"y":735.901,"cluster":"schemes"},{"id":"stacks:0B3C","tag":"0B3C","title":"Serre's conditions · Lemma 0B3C","summary":"Let X be a locally Noetherian scheme. Then X is regular if and only if X has (R_k) for all k ≥ 0.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nThen $X$ is regular if and only if $X$ has $(R_k)$ for all $k \\geq 0$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Serre's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3C","source_file":"properties.tex","source_line":1542,"source_end_line":1546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1542-L1546","statement_sha256":"453762fc0c90b72880de04930868d109d2b376f0bac2f655b1d98096a2505214","origin":"The Stacks Project","memory_eligible":false,"source_rank":5240,"rank":5240,"depth":15,"x":1438.612,"y":712.843,"cluster":"schemes"},{"id":"stacks:0342","tag":"0342","title":"Serre's conditions · Lemma 0342","summary":"Let X be a locally Noetherian scheme. Then X is Cohen-Macaulay if and only if X has (S_k) for all k ≥ 0.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nThen $X$ is Cohen-Macaulay if and only if $X$ has $(S_k)$ for all $k \\geq 0$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Serre's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0342","source_file":"properties.tex","source_line":1552,"source_end_line":1556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1552-L1556","statement_sha256":"5f9d8e0caf29ae1153eb3078346e7f06e573581c36c11973dd3f306702315bdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5241,"rank":5241,"depth":19,"x":1657.992,"y":575.36,"cluster":"schemes"},{"id":"stacks:0344","tag":"0344","title":"Serre's conditions · Lemma 0344","summary":"Let X be a locally Noetherian scheme. Then X is reduced if and only if X has properties (S_1) and (R_0).","statement_latex":"Let $X$ be a locally Noetherian scheme.\nThen $X$ is reduced if and only if $X$ has properties $(S_1)$ and $(R_0)$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Serre's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0344","source_file":"properties.tex","source_line":1565,"source_end_line":1569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1565-L1569","statement_sha256":"c62acce7e4809126bdee9c3261a0986ac93fab3b832fd56bbd8f1e19a377bae6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5242,"rank":5242,"depth":11,"x":1606.689,"y":801.641,"cluster":"schemes"},{"id":"stacks:0345","tag":"0345","title":"Serre's conditions · Lemma 0345","summary":"Let X be a locally Noetherian scheme. Then X is normal if and only if X has properties (S_2) and (R_1).","statement_latex":"Let $X$ be a locally Noetherian scheme.\nThen $X$ is normal if and only if $X$ has properties $(S_2)$ and $(R_1)$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Serre's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0345","source_file":"properties.tex","source_line":1575,"source_end_line":1579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1575-L1579","statement_sha256":"8b8a450e40219b906f5b8c785a2dc0d602a01aaf9a975d7edf00f45f952495c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5243,"rank":5243,"depth":17,"x":1462.281,"y":605.309,"cluster":"schemes"},{"id":"stacks:0BX2","tag":"0BX2","title":"Serre's conditions · Lemma 0BX2","summary":"Let X be a locally Noetherian scheme which is normal and has dimension ≤ 1. Then X is regular.","statement_latex":"Let $X$ be a locally Noetherian scheme which is normal and\nhas dimension $\\leq 1$. Then $X$ is regular.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Serre's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BX2","source_file":"properties.tex","source_line":1585,"source_end_line":1589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1585-L1589","statement_sha256":"4080a3974330f3be6624c60015eefbc91dad2b0df477e53aa731ace99cffc8b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5244,"rank":5244,"depth":18,"x":1727.141,"y":668.258,"cluster":"schemes"},{"id":"stacks:0B3D","tag":"0B3D","title":"Serre's conditions · Lemma 0B3D","summary":"Let X be a locally Noetherian scheme which is normal and has dimension ≤ 2. Then X is Cohen-Macaulay.","statement_latex":"Let $X$ be a locally Noetherian scheme which is normal and\nhas dimension $\\leq 2$. Then $X$ is Cohen-Macaulay.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Serre's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3D","source_file":"properties.tex","source_line":1595,"source_end_line":1599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1595-L1599","statement_sha256":"7b5a15236ca23d52be2cec98ad5f880536e5d581d06bc9409bfad752991342c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5245,"rank":5245,"depth":18,"x":1480.761,"y":772.32,"cluster":"schemes"},{"id":"stacks:033S","tag":"033S","title":"Japanese and Nagata schemes · Definition 033S","summary":"Let X be a scheme. • Assume X integral. We say X is Japanese if for every x ∈ X there exists an affine open neighbourhood x ∈ U ⊂ X such that the ring O_X(U) is Japanese (see Algebra, Definition [Tag 032F]). • We say X is universally Japanese if for every x ∈ X there exists an affine open neighbourhood x ∈ U ⊂ X such that the ring O_X(U) is universally Japanese (see Algebra, Definition [Tag 032R]). • We say X is Nagata if for every x ∈ X there exists an affine open…","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item Assume $X$ integral. We say $X$ is {\\it Japanese}\nif for every $x \\in X$ there exists an\naffine open neighbourhood $x \\in U \\subset X$ such that the ring\n$\\mathcal{O}_X(U)$ is Japanese (see\nAlgebra, Definition \\ref{algebra-definition-N}).\n\\item We say $X$ is {\\it universally Japanese} if for every $x \\in X$\nthere exists an affine open neighbourhood $x \\in U \\subset X$ such that\nthe ring $\\mathcal{O}_X(U)$ is universally Japanese (see\nAlgebra, Definition \\ref{algebra-definition-nagata}).\n\\item We say $X$ is {\\it Nagata} if for every $x \\in X$ there exists an\naffine open neighbourhood $x \\in U \\subset X$ such that the ring\n$\\mathcal{O}_X(U)$ is Nagata (see\nAlgebra, Definition \\ref{algebra-definition-nagata}).\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Japanese and Nagata schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033S","source_file":"properties.tex","source_line":1631,"source_end_line":1649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1631-L1649","statement_sha256":"73b8bfd27ee21260bcf3dafe45513e97e2e29b197e85a5e9252cf94ed9249cb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5246,"rank":5246,"depth":1,"x":1578.935,"y":555.385,"cluster":"schemes"},{"id":"stacks:033U","tag":"033U","title":"Japanese and Nagata schemes · Lemma 033U","summary":"A Nagata scheme is locally Noetherian.","statement_latex":"A Nagata scheme is locally Noetherian.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Japanese and Nagata schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033U","source_file":"properties.tex","source_line":1671,"source_end_line":1674,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1671-L1674","statement_sha256":"f2c17e6a0ab773f5134399cb61ded747b5e2118df215f95f433fbc229f94f11e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5247,"rank":5247,"depth":0,"x":1681.182,"y":771.452,"cluster":"schemes"},{"id":"stacks:033V","tag":"033V","title":"Japanese and Nagata schemes · Lemma 033V","summary":"Let X be an integral scheme. The following are equivalent: • The scheme X is Japanese. • For every affine open U ⊂ X the domain O_X(U) is Japanese. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is Japanese. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is Japanese. Moreover, if X is Japanese then every open subscheme is Japanese.","statement_latex":"Let $X$ be an integral scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is Japanese.\n\\item For every affine open $U \\subset X$ the domain $\\mathcal{O}_X(U)$\nis Japanese.\n\\item There exists an affine open covering $X = \\bigcup U_i$\nsuch that each $\\mathcal{O}_X(U_i)$ is Japanese.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is Japanese.\n\\end{enumerate}\nMoreover, if $X$ is Japanese then every open subscheme\nis Japanese.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Japanese and Nagata schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033V","source_file":"properties.tex","source_line":1680,"source_end_line":1694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1680-L1694","statement_sha256":"4c64b8021b7d9b171330c23ed7e0a452cd0e060fc5edee0e07277b9d37455a7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5248,"rank":5248,"depth":4,"x":1431.576,"y":669.96,"cluster":"schemes"},{"id":"stacks:033W","tag":"033W","title":"Japanese and Nagata schemes · Lemma 033W","summary":"Let X be a scheme. The following are equivalent: • The scheme X is universally Japanese. • For every affine open U ⊂ X the ring O_X(U) is universally Japanese. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is universally Japanese. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is universally Japanese. Moreover, if X is universally Japanese then every open subscheme is universally Japanese.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is universally Japanese.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis universally Japanese.\n\\item There exists an affine open covering $X = \\bigcup U_i$\nsuch that each $\\mathcal{O}_X(U_i)$ is universally Japanese.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is universally Japanese.\n\\end{enumerate}\nMoreover, if $X$ is universally Japanese then every open subscheme\nis universally Japanese.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Japanese and Nagata schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033W","source_file":"properties.tex","source_line":1702,"source_end_line":1716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1702-L1716","statement_sha256":"98141e0443445905d7572d641bf3ce44895a598f0e9b93e7b608032f88e55e64","origin":"The Stacks Project","memory_eligible":false,"source_rank":5249,"rank":5249,"depth":5,"x":1697.733,"y":603.044,"cluster":"schemes"},{"id":"stacks:033X","tag":"033X","title":"Japanese and Nagata schemes · Lemma 033X","summary":"Let X be a scheme. The following are equivalent: • The scheme X is Nagata. • For every affine open U ⊂ X the ring O_X(U) is Nagata. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is Nagata. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is Nagata. Moreover, if X is Nagata then every open subscheme is Nagata.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is Nagata.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis Nagata.\n\\item There exists an affine open covering $X = \\bigcup U_i$\nsuch that each $\\mathcal{O}_X(U_i)$ is Nagata.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is Nagata.\n\\end{enumerate}\nMoreover, if $X$ is Nagata then every open subscheme is Nagata.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Japanese and Nagata schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033X","source_file":"properties.tex","source_line":1724,"source_end_line":1737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1724-L1737","statement_sha256":"7336c924b32c9be8ae584d5da8556ba01d990c85c409cfe64ad1d81d14dc554e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5250,"rank":5250,"depth":5,"x":1555.026,"y":803.774,"cluster":"schemes"},{"id":"stacks:033Y","tag":"033Y","title":"Japanese and Nagata schemes · Lemma 033Y","summary":"Let X be a locally Noetherian scheme. Then X is Nagata if and only if every integral closed subscheme Z ⊂ X is Japanese.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nThen $X$ is Nagata if and only if every integral closed subscheme\n$Z \\subset X$ is Japanese.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Japanese and Nagata schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033Y","source_file":"properties.tex","source_line":1745,"source_end_line":1750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1745-L1750","statement_sha256":"22cba4bee0b02ae6a5d590f5fc430065ae01f4d4b8fdfc3c9433a15c1cde13d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5251,"rank":5251,"depth":12,"x":1498.733,"y":574.369,"cluster":"schemes"},{"id":"stacks:033Z","tag":"033Z","title":"Japanese and Nagata schemes · Lemma 033Z","summary":"Let X be a scheme. The following are equivalent: • X is Nagata, and • X is locally Noetherian and universally Japanese.","statement_latex":"Let $X$ be a scheme.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $X$ is Nagata, and\n\\item $X$ is locally Noetherian and universally Japanese.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Japanese and Nagata schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/033Z","source_file":"properties.tex","source_line":1781,"source_end_line":1789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1781-L1789","statement_sha256":"3c0cc276d81ab9cdfa51d0482bdc0b1708c5e52d482efff50a00f064cb12a8fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":5252,"rank":5252,"depth":33,"x":1725.13,"y":711.835,"cluster":"schemes"},{"id":"stacks:0HA3","tag":"0HA3","title":"G-schemes · Definition 0HA3","summary":"Let X be a scheme. We say X is a G-scheme if for every x ∈ X there exists an affine open neighbourhood x ∈ U ⊂ X such that the ring O_X(U) is a G-ring (see More on Algebra, Definition [Tag 07GH]).","statement_latex":"Let $X$ be a scheme. We say $X$ is a {\\it G-scheme}\\footnote{This may be\nnonstandard terminology. If $G$ is a finite group or group scheme\nthen sometimes a $G$-scheme denotes a scheme equipped with an\naction of $G$.} if for every $x \\in X$ there exists an affine\nopen neighbourhood $x \\in U \\subset X$ such that the ring\n$\\mathcal{O}_X(U)$ is a G-ring (see\nMore on Algebra, Definition \\ref{more-algebra-definition-G-ring}).","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"G-schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HA3","source_file":"properties.tex","source_line":1810,"source_end_line":1819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1810-L1819","statement_sha256":"01ca4ebea9c65196d2bc80252a0585e87ee3e507dd2dfde308b65f2e8188d932","origin":"The Stacks Project","memory_eligible":false,"source_rank":5253,"rank":5253,"depth":1,"x":1447.145,"y":738.981,"cluster":"schemes"},{"id":"stacks:0HA4","tag":"0HA4","title":"G-schemes · Lemma 0HA4","summary":"Let X be a scheme. The following are equivalent: • X is a G-scheme, • X is locally Noetherian and for all x ∈ X the ring map O_X, x → O_X, x^wedge is regular, and • X is locally Noetherian and for any closed point x ∈ X the ring map O_X, x → O_X, x^wedge is regular.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is a G-scheme,\n\\item $X$ is locally Noetherian and for all $x \\in X$ the ring\nmap $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X, x}^\\wedge$ is regular, and\n\\item $X$ is locally Noetherian and for any closed point $x \\in X$\nthe ring map $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{X, x}^\\wedge$ is regular.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"G-schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HA4","source_file":"properties.tex","source_line":1821,"source_end_line":1831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1821-L1831","statement_sha256":"dedff6a44a01294ebc5d2a84dd4a4ba773048b27fd5861cbfb4afc0c10eb2eab","origin":"The Stacks Project","memory_eligible":false,"source_rank":5254,"rank":5254,"depth":44,"x":1630.624,"y":560.91,"cluster":"schemes"},{"id":"stacks:0HA5","tag":"0HA5","title":"G-schemes · Lemma 0HA5","summary":"Let X be a scheme. The following are equivalent: • The scheme X is a G-scheme. • For every affine open U ⊂ X the ring O_X(U) is a G-ring. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is a G-ring. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is a G-scheme. Moreover, if X is a G-scheme then every open subscheme is a G-scheme.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is a G-scheme.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis a G-ring.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ is a G-ring.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is a G-scheme.\n\\end{enumerate}\nMoreover, if $X$ is a G-scheme then every open subscheme is a\nG-scheme.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"G-schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HA5","source_file":"properties.tex","source_line":1849,"source_end_line":1863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1849-L1863","statement_sha256":"aa49c66bdf7f7b9430ead2ea0fef2ebfc41e42d688f4922f9f750677e8682958","origin":"The Stacks Project","memory_eligible":false,"source_rank":5255,"rank":5255,"depth":45,"x":1638.543,"y":796.748,"cluster":"schemes"},{"id":"stacks:07R1","tag":"07R1","title":"The singular locus · Definition 07R1","summary":"Let X be a locally Noetherian scheme. The regular locus Reg(X) of X is the set of x ∈ X such that O_X, x is a regular local ring. The singular locus Sing(X) is the complement X setminus Reg(X), i.e., the set of points x ∈ X such that O_X, x is not a regular local ring.","statement_latex":"Let $X$ be a locally Noetherian scheme. The {\\it regular locus}\n$\\text{Reg}(X)$ of $X$ is the set of $x \\in X$ such that $\\mathcal{O}_{X, x}$\nis a regular local ring. The {\\it singular locus} $\\text{Sing}(X)$ is the\ncomplement $X \\setminus \\text{Reg}(X)$, i.e., the set of points $x \\in X$\nsuch that $\\mathcal{O}_{X, x}$ is not a regular local ring.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"The singular locus","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07R1","source_file":"properties.tex","source_line":1879,"source_end_line":1886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1879-L1886","statement_sha256":"f3dc4076bc3fbeeddcb0946bd778019ff6c2d01e25048bc4cb095b2d8d7cf09c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5256,"rank":5256,"depth":0,"x":1442.703,"y":627.037,"cluster":"schemes"},{"id":"stacks:0BQ2","tag":"0BQ2","title":"Local irreducibility · Definition 0BQ2","summary":"[EGA4] Let X be a scheme. Let x ∈ X. We say X is unibranch at x if the local ring O_X, x is unibranch. We say X is geometrically unibranch at x if the local ring O_X, x is geometrically unibranch. We say X is unibranch if X is unibranch at all of its points. We say X is geometrically unibranch if X is geometrically unibranch at all of its points.","statement_latex":"\\begin{reference}\n\\cite[Chapter IV (6.15.1)]{EGA4}\n\\end{reference}\nLet $X$ be a scheme. Let $x \\in X$. We say $X$ is {\\it unibranch at $x$}\nif the local ring $\\mathcal{O}_{X, x}$ is unibranch. We say $X$ is\n{\\it geometrically unibranch at $x$}\nif the local ring $\\mathcal{O}_{X, x}$ is geometrically unibranch.\nWe say $X$ is {\\it unibranch} if $X$ is unibranch at all of its points.\nWe say $X$ is {\\it geometrically unibranch} if $X$ is\ngeometrically unibranch at all of its points.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Local irreducibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQ2","source_file":"properties.tex","source_line":1911,"source_end_line":1923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1911-L1923","statement_sha256":"5b893d0334bafd4e77cc9c72300516e65ab0241f77da69aa0b7ac8b0ba9bf570","origin":"The Stacks Project","memory_eligible":false,"source_rank":5257,"rank":5257,"depth":0,"x":1724.086,"y":641.08,"cluster":"schemes"},{"id":"stacks:0BQ3","tag":"0BQ3","title":"Local irreducibility · Lemma 0BQ3","summary":"A normal scheme is geometrically unibranch.","statement_latex":"A normal scheme is geometrically unibranch.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQ3","source_file":"properties.tex","source_line":1934,"source_end_line":1937,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1934-L1937","statement_sha256":"26d5f4c44df49e7d22f63194053653f689a3f26d973a8a623f0188c719c9da15","origin":"The Stacks Project","memory_eligible":false,"source_rank":5258,"rank":5258,"depth":1,"x":1504.92,"y":790.651,"cluster":"schemes"},{"id":"stacks:0BQ4","tag":"0BQ4","title":"Local irreducibility · Lemma 0BQ4","summary":"Compare with [Etale-coverings] Let X be a Noetherian scheme. The following are equivalent • X is geometrically unibranch (Definition [Tag 0BQ2]), • for every point x ∈ X which is not the generic point of an irreducible component of X, the punctured spectrum of the strict henselization O_X, x^sh is connected.","statement_latex":"\\begin{reference}\nCompare with \\cite[Proposition 2.3]{Etale-coverings}\n\\end{reference}\nLet $X$ be a Noetherian scheme. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically unibranch (Definition \\ref{definition-unibranch}),\n\\item for every point $x \\in X$ which is not the generic point of\nan irreducible component of $X$, the punctured spectrum of the\nstrict henselization $\\mathcal{O}_{X, x}^{sh}$ is connected.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQ4","source_file":"properties.tex","source_line":1946,"source_end_line":1958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1946-L1958","statement_sha256":"784b01ac57a025efde81f4426fcb4621bf4175c2f2afe5dd1c615f4a398946ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":5259,"rank":5259,"depth":53,"x":1546.322,"y":555.588,"cluster":"schemes"},{"id":"stacks:0C38","tag":"0C38","title":"Local irreducibility · Definition 0C38","summary":"Let X be a scheme. Let x ∈ X. The number of branches of X at x is the number of branches of the local ring O_X, x as defined in More on Algebra, Definition [Tag 0C26]. The number of geometric branches of X at x is the number of geometric branches of the local ring O_X, x as defined in More on Algebra, Definition [Tag 0C26].","statement_latex":"Let $X$ be a scheme. Let $x \\in X$. The {\\it number of branches of $X$\nat $x$} is the number of branches of the local ring $\\mathcal{O}_{X, x}$\nas defined in\nMore on Algebra, Definition \\ref{more-algebra-definition-number-of-branches}.\nThe {\\it number of geometric branches of $X$ at $x$} is the number of\ngeometric branches of the local ring $\\mathcal{O}_{X, x}$ as defined in\nMore on Algebra, Definition \\ref{more-algebra-definition-number-of-branches}.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Local irreducibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C38","source_file":"properties.tex","source_line":1995,"source_end_line":2004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L1995-L2004","statement_sha256":"1a9dab1462dc388a9e6cd70e1c0856c7656e2218a5a5d9e28c19ec007a690484","origin":"The Stacks Project","memory_eligible":false,"source_rank":5260,"rank":5260,"depth":1,"x":1705.101,"y":752.757,"cluster":"schemes"},{"id":"stacks:0E20","tag":"0E20","title":"Local irreducibility · Lemma 0E20","summary":"Let X be a scheme and x ∈ X. Let X_i, i ∈ I be the irreducible components of X passing through x. Then the number of (geometric) branches of X at x is the sum over i ∈ I of the number of (geometric) branches of X_i at x.","statement_latex":"Let $X$ be a scheme and $x \\in X$. Let $X_i$, $i \\in I$ be the\nirreducible components of $X$ passing through $x$.\nThen the number of (geometric) branches of $X$ at $x$\nis the sum over $i \\in I$ of the number of (geometric)\nbranches of $X_i$ at $x$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E20","source_file":"properties.tex","source_line":2012,"source_end_line":2019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2012-L2019","statement_sha256":"3ba295262f324d3967193507600bac0160cd2a2e9355d5606706040313fbe011","origin":"The Stacks Project","memory_eligible":false,"source_rank":5261,"rank":5261,"depth":53,"x":1428.981,"y":697.366,"cluster":"schemes"},{"id":"stacks:0C39","tag":"0C39","title":"Local irreducibility · Lemma 0C39","summary":"Let X be a scheme. Let x ∈ X. • The number of branches of X at x is 1 if and only if X is unibranch at x. • The number of geometric branches of X at x is 1 if and only if X is geometrically unibranch at x.","statement_latex":"Let $X$ be a scheme. Let $x \\in X$.\n\\begin{enumerate}\n\\item The number of branches of $X$ at $x$ is $1$ if and only if\n$X$ is unibranch at $x$.\n\\item The number of geometric branches of $X$ at $x$ is $1$ if and only if\n$X$ is geometrically unibranch at $x$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C39","source_file":"properties.tex","source_line":2053,"source_end_line":2062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2053-L2062","statement_sha256":"25fb65f17c0a39e9081292561026dbe8b1d1918028805599e7ddf65724968fe2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5262,"rank":5262,"depth":53,"x":1677.563,"y":581.332,"cluster":"schemes"},{"id":"stacks:01PB","tag":"01PB","title":"Characterizing modules of finite type and finite presentation · Lemma 01PB","summary":"Let X = Spec(R) be an affine scheme. The quasi-coherent sheaf of O_X-modules widetilde M is a finite type O_X-module if and only if M is a finite R-module.","statement_latex":"Let $X = \\Spec(R)$ be an affine scheme.\nThe quasi-coherent sheaf of $\\mathcal{O}_X$-modules\n$\\widetilde M$ is a finite type $\\mathcal{O}_X$-module\nif and only if $M$ is a finite $R$-module.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Characterizing modules of finite type and finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PB","source_file":"properties.tex","source_line":2092,"source_end_line":2098,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2092-L2098","statement_sha256":"b4c0811ff267c83720c535c7d6d58f3ecdf4bf3b915288223e524118a8ad919b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5263,"rank":5263,"depth":3,"x":1587.417,"y":808.337,"cluster":"schemes"},{"id":"stacks:01PC","tag":"01PC","title":"Characterizing modules of finite type and finite presentation · Lemma 01PC","summary":"Let X = Spec(R) be an affine scheme. The quasi-coherent sheaf of O_X-modules widetilde M is an O_X-module of finite presentation if and only if M is an R-module of finite presentation.","statement_latex":"Let $X = \\Spec(R)$ be an affine scheme. The quasi-coherent sheaf\nof $\\mathcal{O}_X$-modules $\\widetilde M$ is an $\\mathcal{O}_X$-module of\nfinite presentation if and only if $M$ is an $R$-module of finite presentation.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Characterizing modules of finite type and finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PC","source_file":"properties.tex","source_line":2112,"source_end_line":2117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2112-L2117","statement_sha256":"1655644a243c9b55f55a19ee0fddbcef68e79c0d2ad34e0d04fe9803db557495","origin":"The Stacks Project","memory_eligible":false,"source_rank":5264,"rank":5264,"depth":6,"x":1471.138,"y":589.419,"cluster":"schemes"},{"id":"stacks:01P7","tag":"01P7","title":"Sections over principal opens · Lemma 01P7","summary":"Sections of quasi-coherent sheaves have only meromorphic singularities at infinity. Let X be a scheme. Let f ∈ Γ(X, O_X). Denote X_f ⊂ X the open where f is invertible, see Schemes, Lemma [Tag 01HZ]. If X is quasi-compact and quasi-separated, the canonical map Γ(X, O_X)_f → Γ(X_f, O_X) is an isomorphism. Moreover, if F is a quasi-coherent sheaf of O_X-modules the map Γ(X, F)_f → Γ(X_f, F) is an isomorphism.","statement_latex":"\\begin{slogan}\nSections of quasi-coherent sheaves have only meromorphic singularities\nat infinity.\n\\end{slogan}\nLet $X$ be a scheme. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$.\nDenote $X_f \\subset X$ the open where $f$ is invertible, see\nSchemes, Lemma \\ref{schemes-lemma-f-open}.\nIf $X$ is quasi-compact and quasi-separated, the canonical map\n$$\n\\Gamma(X, \\mathcal{O}_X)_f \\longrightarrow \\Gamma(X_f, \\mathcal{O}_X)\n$$\nis an isomorphism. Moreover, if $\\mathcal{F}$ is a quasi-coherent\nsheaf of $\\mathcal{O}_X$-modules the map\n$$\n\\Gamma(X, \\mathcal{F})_f \\longrightarrow \\Gamma(X_f, \\mathcal{F})\n$$\nis an isomorphism.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections over principal opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01P7","source_file":"properties.tex","source_line":2152,"source_end_line":2171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2152-L2171","statement_sha256":"bd1df5c9ef4628b4cb75fa4ddd3bf2025d8201e520779b1bb60e42c532251fc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5265,"rank":5265,"depth":16,"x":1733.379,"y":685.031,"cluster":"schemes"},{"id":"stacks:01PW","tag":"01PW","title":"Sections over principal opens · Lemma 01PW","summary":"[EGA1-second] Let X be a scheme. Let L be an invertible sheaf on X. Let s ∈ Γ(X, L). Let F be a quasi-coherent O_X-module. • If X is quasi-compact, then ([Tag 0B5L]) is injective, and • if X is quasi-compact and quasi-separated, then ([Tag 0B5L]) is an isomorphism. In particular, the canonical map Γ_*(X, L)_(s) → Γ(X_s, O_X), a/s^n ↦ a ⊗ s^-n is an isomorphism if X is quasi-compact and quasi-separated.","statement_latex":"\\begin{reference}\n\\cite[Section 6.8]{EGA1-second}\n\\end{reference}\nLet $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf on $X$.\nLet $s \\in \\Gamma(X, \\mathcal{L})$. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If $X$ is quasi-compact, then (\\ref{equation-module-invert-s})\nis injective, and\n\\item if $X$ is quasi-compact and quasi-separated, then\n(\\ref{equation-module-invert-s}) is an isomorphism.\n\\end{enumerate}\nIn particular, the canonical map\n$$\n\\Gamma_*(X, \\mathcal{L})_{(s)}\n\\longrightarrow\n\\Gamma(X_s, \\mathcal{O}_X),\\quad\na/s^n \\longmapsto a \\otimes s^{-n}\n$$\nis an isomorphism if $X$ is quasi-compact and quasi-separated.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections over principal opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PW","source_file":"properties.tex","source_line":2226,"source_end_line":2248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2226-L2248","statement_sha256":"f1dcaa9e9c343fec8517410377171df1c7a1f844a4492830c0abee2cc47157c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5266,"rank":5266,"depth":17,"x":1462.655,"y":763.464,"cluster":"schemes"},{"id":"stacks:01XQ","tag":"01XQ","title":"Sections over principal opens · Lemma 01XQ","summary":"Let X be a scheme. Let L be an invertible O_X-module. Let s ∈ Γ(X, L) be a section. Let F, G be quasi-coherent O_X-modules. • If X is quasi-compact and F is of finite type, then ([Tag 0B5M]) is injective, and • if X is quasi-compact and quasi-separated and F is of finite presentation, then ([Tag 0B5M]) is bijective.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$ be a section.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item If $X$ is quasi-compact and $\\mathcal{F}$ is of finite type,\nthen (\\ref{equation-hom-invert-s}) is injective, and\n\\item if $X$ is quasi-compact and quasi-separated and $\\mathcal{F}$\nis of finite presentation, then\n(\\ref{equation-hom-invert-s})\nis bijective.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections over principal opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XQ","source_file":"properties.tex","source_line":2331,"source_end_line":2344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2331-L2344","statement_sha256":"85b11d807c8c4d5903c7d76dfec7a153042f2f96519e2a2827b07a9b3d3f0265","origin":"The Stacks Project","memory_eligible":false,"source_rank":5267,"rank":5267,"depth":18,"x":1599.442,"y":551.653,"cluster":"schemes"},{"id":"stacks:01P6","tag":"01P6","title":"Quasi-affine schemes · Definition 01P6","summary":"A scheme X is called quasi-affine if it is quasi-compact and isomorphic to an open subscheme of an affine scheme.","statement_latex":"A scheme $X$ is called {\\it quasi-affine} if it is quasi-compact\nand isomorphic to an open subscheme of an affine scheme.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Quasi-affine schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01P6","source_file":"properties.tex","source_line":2406,"source_end_line":2410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2406-L2410","statement_sha256":"04f0290759a87f958b5cf520172b28ca6f7379e28ddc653b942bffaba1df5af8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5268,"rank":5268,"depth":0,"x":1669.028,"y":785.853,"cluster":"schemes"},{"id":"stacks:0EHM","tag":"0EHM","title":"Quasi-affine schemes · Lemma 0EHM","summary":"Let A be a ring and let U ⊂ Spec(A) be a quasi-compact open subscheme. For F quasi-coherent on U the canonical map widetildeH^0(U, F)|_U → F is an isomorphism.","statement_latex":"Let $A$ be a ring and let $U \\subset \\Spec(A)$ be a quasi-compact\nopen subscheme. For $\\mathcal{F}$ quasi-coherent on $U$ the canonical map\n$$\n\\widetilde{H^0(U, \\mathcal{F})}|_U \\to \\mathcal{F}\n$$\nis an isomorphism.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Quasi-affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHM","source_file":"properties.tex","source_line":2412,"source_end_line":2420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2412-L2420","statement_sha256":"949ac70050fd478ee336bce1517993d382df40449a63543147fdeecbdc614745","origin":"The Stacks Project","memory_eligible":false,"source_rank":5269,"rank":5269,"depth":14,"x":1428.973,"y":652.416,"cluster":"schemes"},{"id":"stacks:01P8","tag":"01P8","title":"Quasi-affine schemes · Lemma 01P8","summary":"Let X be a scheme. Let f ∈ Γ(X, O_X). Assume X is quasi-compact and quasi-separated and assume that X_f is affine. Then the canonical morphism j : X → Spec(Γ(X, O_X)) from Schemes, Lemma [Tag 01I1] induces an isomorphism of X_f = j^-1(D(f)) onto the standard affine open D(f) ⊂ Spec(Γ(X, O_X)).","statement_latex":"Let $X$ be a scheme. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$.\nAssume $X$ is quasi-compact and quasi-separated and assume that\n$X_f$ is affine. Then the canonical morphism\n$$\nj : X \\longrightarrow \\Spec(\\Gamma(X, \\mathcal{O}_X))\n$$\nfrom Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}\ninduces an isomorphism of $X_f = j^{-1}(D(f))$ onto the standard affine\nopen $D(f) \\subset \\Spec(\\Gamma(X, \\mathcal{O}_X))$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Quasi-affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01P8","source_file":"properties.tex","source_line":2432,"source_end_line":2443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2432-L2443","statement_sha256":"246b18f351934a20dc142dc0c8c3e2dd11411f431611e80b03b1fb92121aefd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5270,"rank":5270,"depth":17,"x":1713.773,"y":614.534,"cluster":"schemes"},{"id":"stacks:01P9","tag":"01P9","title":"Quasi-affine schemes · Lemma 01P9","summary":"Let X be a scheme. Then X is quasi-affine if and only if the canonical morphism X → Spec(Γ(X, O_X)) from Schemes, Lemma [Tag 01I1] is a quasi-compact open immersion.","statement_latex":"Let $X$ be a scheme. Then $X$ is quasi-affine if and only if\nthe canonical morphism\n$$\nX \\longrightarrow \\Spec(\\Gamma(X, \\mathcal{O}_X))\n$$\nfrom Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine} is\na quasi-compact open immersion.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Quasi-affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01P9","source_file":"properties.tex","source_line":2451,"source_end_line":2460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2451-L2460","statement_sha256":"dff3745400a7609781230ef473b7071aeea86ce92883225969210c140c11c7fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5271,"rank":5271,"depth":17,"x":1533.927,"y":804.388,"cluster":"schemes"},{"id":"stacks:0ARY","tag":"0ARY","title":"Quasi-affine schemes · Lemma 0ARY","summary":"Let U → V be an open immersion of quasi-affine schemes. Then xymatrix U ar[d] ar[rr]_-j & & Spec(Γ(U, O_U)) ar[d] U ar[r] & V ar[r]^-j' & Spec(Γ(V, O_V)) is cartesian.","statement_latex":"Let $U \\to V$ be an open immersion of quasi-affine schemes. Then\n$$\n\\xymatrix{\nU \\ar[d] \\ar[rr]_-j & & \\Spec(\\Gamma(U, \\mathcal{O}_U)) \\ar[d] \\\\\nU \\ar[r] & V \\ar[r]^-{j'} & \\Spec(\\Gamma(V, \\mathcal{O}_V))\n}\n$$\nis cartesian.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Quasi-affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARY","source_file":"properties.tex","source_line":2493,"source_end_line":2503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2493-L2503","statement_sha256":"e80dfe5aca58e939e16e630e1171c112473f41c0564f09cc6c3e2c0e73acad09","origin":"The Stacks Project","memory_eligible":false,"source_rank":5272,"rank":5272,"depth":18,"x":1513.833,"y":561.938,"cluster":"schemes"},{"id":"stacks:0F82","tag":"0F82","title":"Quasi-affine schemes · Lemma 0F82","summary":"Let X be a quasi-affine scheme. There exists an integer n ≥ 0, an affine scheme T, and a morphism T → X such that for every morphism X' → X with X' affine the fibre product X' ×_X T is isomorphic to A^n_X' over X'.","statement_latex":"Let $X$ be a quasi-affine scheme. There exists an integer $n \\geq 0$,\nan affine scheme $T$, and a morphism $T \\to X$ such that for every\nmorphism $X' \\to X$ with $X'$ affine the fibre product $X' \\times_X T$\nis isomorphic to $\\mathbf{A}^n_{X'}$ over $X'$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Quasi-affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F82","source_file":"properties.tex","source_line":2521,"source_end_line":2527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2521-L2527","statement_sha256":"d6bd2c9d15d7a943556dd5f4994a3884393a21c9311333bf7e02f54c645d2375","origin":"The Stacks Project","memory_eligible":false,"source_rank":5273,"rank":5273,"depth":0,"x":1723.973,"y":729.595,"cluster":"schemes"},{"id":"stacks:05P0","tag":"05P0","title":"Flat modules · Lemma 05P0","summary":"Flatness is the same for modules and sheaves. Let X = Spec(R) be an affine scheme. Let F = widetildeM for some R-module M. The quasi-coherent sheaf F is a flat O_X-module if and only if M is a flat R-module.","statement_latex":"\\begin{slogan}\nFlatness is the same for modules and sheaves.\n\\end{slogan}\nLet $X = \\Spec(R)$ be an affine scheme.\nLet $\\mathcal{F} = \\widetilde{M}$ for some $R$-module $M$.\nThe quasi-coherent sheaf $\\mathcal{F}$ is a flat\n$\\mathcal{O}_X$-module if and only if $M$ is a flat $R$-module.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05P0","source_file":"properties.tex","source_line":2602,"source_end_line":2611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2602-L2611","statement_sha256":"70325c50b1ce4de1acd2dd749953bf2feabdf91921f893527e0cbbaa80770933","origin":"The Stacks Project","memory_eligible":false,"source_rank":5274,"rank":5274,"depth":3,"x":1433.71,"y":725.198,"cluster":"schemes"},{"id":"stacks:05JM","tag":"05JM","title":"Locally free modules · Lemma 05JM","summary":"Let X = Spec(R) be an affine scheme. Let F = widetildeM for some R-module M. The quasi-coherent sheaf F is a (finite) locally free O_X-module of if and only if M is a (finite) locally free R-module.","statement_latex":"Let $X = \\Spec(R)$ be an affine scheme.\nLet $\\mathcal{F} = \\widetilde{M}$ for some $R$-module $M$.\nThe quasi-coherent sheaf $\\mathcal{F}$ is a (finite) locally free\n$\\mathcal{O}_X$-module of if and only if $M$ is a (finite)\nlocally free $R$-module.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Locally free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JM","source_file":"properties.tex","source_line":2634,"source_end_line":2641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2634-L2641","statement_sha256":"e5e89964dfab9271e50972817d424567b738e261b3751f5a630ac719f3d515ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":5275,"rank":5275,"depth":1,"x":1651.643,"y":563.471,"cluster":"schemes"},{"id":"stacks:05P2","tag":"05P2","title":"Locally free modules · Lemma 05P2","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. The following are equivalent: • F is a flat O_X-module of finite presentation, • F is O_X-module of finite presentation and for all x ∈ X the stalk F_x is a free O_X, x-module, • F is a locally free, finite type O_X-module, • F is a finite locally free O_X-module, and • F is an O_X-module of finite type, for every x ∈ X the stalk F_x is a free O_X, x-module, and the function ρ_F : X → Z, x ↦ dim_kappa(x) F_x ⊗_O_X,…","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module of finite presentation,\n\\item $\\mathcal{F}$ is $\\mathcal{O}_X$-module of finite presentation and\nfor all $x \\in X$ the stalk $\\mathcal{F}_x$ is a free\n$\\mathcal{O}_{X, x}$-module,\n\\item $\\mathcal{F}$ is a locally free, finite type $\\mathcal{O}_X$-module,\n\\item $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module, and\n\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type,\nfor every $x \\in X$ the stalk $\\mathcal{F}_x$ is a free\n$\\mathcal{O}_{X, x}$-module, and the function\n$$\n\\rho_\\mathcal{F} : X \\to \\mathbf{Z}, \\quad\nx \\longmapsto\n\\dim_{\\kappa(x)} \\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)\n$$\nis locally constant in the Zariski topology on $X$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Locally free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05P2","source_file":"properties.tex","source_line":2653,"source_end_line":2675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2653-L2675","statement_sha256":"8d5e5ea5c8aa30e7656f62ae50c5bbc6df07c81f3ccd19d5de019cfa9330788e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5276,"rank":5276,"depth":7,"x":1620.948,"y":806.788,"cluster":"schemes"},{"id":"stacks:0FWH","tag":"0FWH","title":"Locally free modules · Lemma 0FWH","summary":"Let X be a reduced scheme. Let F be a quasi-coherent O_X-module. Then the equivalent conditions of Lemma [Tag 05P2] are also equivalent to • [(6)] F is an O_X-module of finite type and the function ρ_F : X → Z, x ↦ dim_kappa(x) F_x ⊗_O_X, x kappa(x) is locally constant in the Zariski topology on X.","statement_latex":"Let $X$ be a reduced scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Then the equivalent conditions of\nLemma \\ref{lemma-finite-locally-free} are also equivalent to\n\\begin{enumerate}\n\\item[(6)] $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type and\nthe function\n$$\n\\rho_\\mathcal{F} : X \\to \\mathbf{Z}, \\quad\nx \\longmapsto\n\\dim_{\\kappa(x)} \\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)\n$$\nis locally constant in the Zariski topology on $X$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Locally free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWH","source_file":"properties.tex","source_line":2688,"source_end_line":2703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2688-L2703","statement_sha256":"ce33f1c2aadd90e26d4904ca12c79f13f576eb6baba777938ea89cb8670112b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5277,"rank":5277,"depth":8,"x":1447.628,"y":609.628,"cluster":"schemes"},{"id":"stacks:05JP","tag":"05JP","title":"Locally projective modules · Definition 05JP","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. We say F is locally projective if for every affine open U ⊂ X the O_X(U)-module F(U) is projective.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. We say $\\mathcal{F}$ is {\\it locally projective}\nif for every affine open $U \\subset X$ the $\\mathcal{O}_X(U)$-module\n$\\mathcal{F}(U)$ is projective.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Locally projective modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JP","source_file":"properties.tex","source_line":2721,"source_end_line":2727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2721-L2727","statement_sha256":"e10800fb369af734b8efcc0f9841a5ddcf8d71606e23c311da03f024614fe562","origin":"The Stacks Project","memory_eligible":false,"source_rank":5278,"rank":5278,"depth":0,"x":1734.454,"y":656.742,"cluster":"schemes"},{"id":"stacks:05JQ","tag":"05JQ","title":"Locally projective modules · Lemma 05JQ","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. The following are equivalent • F is locally projective, and • there exists an affine open covering X = ⋃ U_i such that the O_X(U_i)-module F(U_i) is projective for every i. In particular, if X = Spec(A) and F = widetildeM then F is locally projective if and only if M is a projective A-module.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is locally projective, and\n\\item there exists an affine open covering $X = \\bigcup U_i$\nsuch that the $\\mathcal{O}_X(U_i)$-module\n$\\mathcal{F}(U_i)$ is projective for every $i$.\n\\end{enumerate}\nIn particular, if $X = \\Spec(A)$ and $\\mathcal{F} = \\widetilde{M}$\nthen $\\mathcal{F}$ is locally projective if and only if $M$ is a projective\n$A$-module.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Locally projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JQ","source_file":"properties.tex","source_line":2729,"source_end_line":2743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2729-L2743","statement_sha256":"41d5a973814ef33702f0965a358ab0058fcb8dcaecfa72a017094257745433e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5279,"rank":5279,"depth":10,"x":1484.656,"y":784.962,"cluster":"schemes"},{"id":"stacks:060M","tag":"060M","title":"Locally projective modules · Lemma 060M","summary":"Let f : X → Y be a morphism of schemes. Let G be a quasi-coherent O_Y-module. If G is locally projective on Y, then f^*G is locally projective on X.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.\nIf $\\mathcal{G}$ is locally projective on $Y$, then $f^*\\mathcal{G}$\nis locally projective on $X$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Locally projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060M","source_file":"properties.tex","source_line":2767,"source_end_line":2773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2767-L2773","statement_sha256":"cb20dd56597cb5d8f04260d92ec30157726bd5bac07de962802ae6123eddbb69","origin":"The Stacks Project","memory_eligible":false,"source_rank":5280,"rank":5280,"depth":11,"x":1565.875,"y":548.288,"cluster":"schemes"},{"id":"stacks:01PE","tag":"01PE","title":"Extending quasi-coherent sheaves · Lemma 01PE","summary":"Let j : U → X be a quasi-compact open immersion of schemes. • Any quasi-coherent sheaf on U extends to a quasi-coherent sheaf on X. • Let F be a quasi-coherent sheaf on X. Let G ⊂ F|_U be a quasi-coherent subsheaf. There exists a quasi-coherent subsheaf H of F such that H|_U = G as subsheaves of F|_U. • Let F be a quasi-coherent sheaf on X. Let G be a quasi-coherent sheaf on U. Let φ : G → F|_U be a morphism of O_U-modules. There exists a quasi-coherent sheaf H of…","statement_latex":"Let $j : U \\to X$ be a quasi-compact open immersion of schemes.\n\\begin{enumerate}\n\\item Any quasi-coherent sheaf on $U$ extends to a quasi-coherent\nsheaf on $X$.\n\\item Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $\\mathcal{G} \\subset \\mathcal{F}|_U$ be a quasi-coherent\nsubsheaf. There exists a quasi-coherent subsheaf $\\mathcal{H}$ of\n$\\mathcal{F}$ such that $\\mathcal{H}|_U = \\mathcal{G}$\nas subsheaves of $\\mathcal{F}|_U$.\n\\item Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $\\mathcal{G}$ be a quasi-coherent sheaf on $U$.\nLet $\\varphi : \\mathcal{G} \\to \\mathcal{F}|_U$ be a morphism\nof $\\mathcal{O}_U$-modules. There exists a quasi-coherent sheaf $\\mathcal{H}$\nof $\\mathcal{O}_X$-modules and a map $\\psi : \\mathcal{H} \\to \\mathcal{F}$\nsuch that $\\mathcal{H}|_U = \\mathcal{G}$ and that\n$\\psi|_U = \\varphi$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PE","source_file":"properties.tex","source_line":2793,"source_end_line":2812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2793-L2812","statement_sha256":"1ff6d96c7a6b3398e2c379ba8a20fb2397405bfc3817228bbb68e9e84a36f7f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5281,"rank":5281,"depth":14,"x":1696.524,"y":769.252,"cluster":"schemes"},{"id":"stacks:01PF","tag":"01PF","title":"Extending quasi-coherent sheaves · Lemma 01PF","summary":"Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be a quasi-compact open. Let F be a quasi-coherent O_X-module. Let G ⊂ F|_U be a quasi-coherent O_U-submodule which is of finite type. Then there exists a quasi-coherent submodule G' ⊂ F which is of finite type such that G'|_U = G.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $U \\subset X$ be a quasi-compact open.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{G} \\subset \\mathcal{F}|_U$ be a quasi-coherent\n$\\mathcal{O}_U$-submodule which is of finite type. Then\nthere exists a quasi-coherent submodule $\\mathcal{G}' \\subset \\mathcal{F}$\nwhich is of finite type such that $\\mathcal{G}'|_U = \\mathcal{G}$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PF","source_file":"properties.tex","source_line":2846,"source_end_line":2855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2846-L2855","statement_sha256":"97d52f243f4c866195d28666cc8df220ad7032befe6f3dd1cbc7f0da0eff9209","origin":"The Stacks Project","memory_eligible":false,"source_rank":5282,"rank":5282,"depth":15,"x":1422.047,"y":680.306,"cluster":"schemes"},{"id":"stacks:01PG","tag":"01PG","title":"Extending quasi-coherent sheaves · Lemma 01PG","summary":"Let X be a quasi-compact and quasi-separated scheme. Any quasi-coherent sheaf of O_X-modules is the directed colimit of its quasi-coherent O_X-submodules which are of finite type.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nAny quasi-coherent sheaf of $\\mathcal{O}_X$-modules\nis the directed colimit of its quasi-coherent\n$\\mathcal{O}_X$-submodules which are of finite type.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PG","source_file":"properties.tex","source_line":2897,"source_end_line":2903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2897-L2903","statement_sha256":"9d6c0d280cb37aed16abdfe467b049a7d9e2682f154259e2ee24d12da3d20a1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5283,"rank":5283,"depth":16,"x":1696.414,"y":590.004,"cluster":"schemes"},{"id":"stacks:01PI","tag":"01PI","title":"Extending quasi-coherent sheaves · Lemma 01PI","summary":"Let X be a quasi-compact and quasi-separated scheme. Let F be a quasi-coherent O_X-module. Let U ⊂ X be a quasi-compact open. Let G be an O_U-module which is of finite presentation. Let φ : G → F|_U be a morphism of O_U-modules. Then there exists an O_X-module G' of finite presentation, and a morphism of O_X-modules φ' : G' → F such that G'|_U = G and such that φ'|_U = φ.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $U \\subset X$ be a quasi-compact open.\nLet $\\mathcal{G}$ be an $\\mathcal{O}_U$-module which is of finite presentation.\nLet $\\varphi : \\mathcal{G} \\to \\mathcal{F}|_U$ be a morphism of\n$\\mathcal{O}_U$-modules.\nThen there exists an $\\mathcal{O}_X$-module\n$\\mathcal{G}'$ of finite presentation, and a morphism\nof $\\mathcal{O}_X$-modules $\\varphi' : \\mathcal{G}' \\to \\mathcal{F}$\nsuch that $\\mathcal{G}'|_U = \\mathcal{G}$ and such that\n$\\varphi'|_U = \\varphi$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PI","source_file":"properties.tex","source_line":2923,"source_end_line":2936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L2923-L2936","statement_sha256":"e47d675c35b309814df53eed9cd8bf734597642763ac723f4c2fa7f77b8de596","origin":"The Stacks Project","memory_eligible":false,"source_rank":5284,"rank":5284,"depth":16,"x":1566.509,"y":812.631,"cluster":"schemes"},{"id":"stacks:0G41","tag":"0G41","title":"Extending quasi-coherent sheaves · Lemma 0G41","summary":"Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be a quasi-compact open. Let G be an O_U-module. • If G is quasi-coherent and of finite type, then there exists a quasi-coherent O_X-module G' of finite type such that G'|_U = G. • If G is of finite presentation, then there exists an O_X-module G' of finite presentation such that G'|_U = G.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \\subset X$\nbe a quasi-compact open. Let $\\mathcal{G}$ be an $\\mathcal{O}_U$-module.\n\\begin{enumerate}\n\\item If $\\mathcal{G}$ is quasi-coherent and of finite type, then\nthere exists a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{G}'$\nof finite type such that $\\mathcal{G}'|_U = \\mathcal{G}$.\n\\item If $\\mathcal{G}$ is of finite presentation, then\nthere exists an $\\mathcal{O}_X$-module $\\mathcal{G}'$\nof finite presentation such that $\\mathcal{G}'|_U = \\mathcal{G}$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G41","source_file":"properties.tex","source_line":3011,"source_end_line":3023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3011-L3023","statement_sha256":"afd68cf5ccbbf84b1ec00d1aa70341cfeb9b5df4ff15b8b93404c917d458dae7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5285,"rank":5285,"depth":17,"x":1483.135,"y":574.376,"cluster":"schemes"},{"id":"stacks:01PJ","tag":"01PJ","title":"Extending quasi-coherent sheaves · Lemma 01PJ","summary":"Quasi-coherent modules on quasi-compact and quasi-separated schemes are filtered colimits of finitely presented modules. Let X be a scheme. Assume X is quasi-compact and quasi-separated. Let F be a quasi-coherent O_X-module. There exist • a filtered index category I (see Categories, Definition [Tag 002V]), • a diagram I → Mod(O_X) (see Categories, Section [Tag 002D]), i ↦ F_i, • morphisms of O_X-modules φ_i : F_i → F such that each F_i is of finite presentation and such…","statement_latex":"\\begin{slogan}\nQuasi-coherent modules on quasi-compact and quasi-separated schemes\nare filtered colimits of finitely presented modules.\n\\end{slogan}\nLet $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThere exist\n\\begin{enumerate}\n\\item a filtered index category $\\mathcal{I}$ (see\nCategories, Definition \\ref{categories-definition-directed}),\n\\item a diagram $\\mathcal{I} \\to \\textit{Mod}(\\mathcal{O}_X)$ (see\nCategories, Section \\ref{categories-section-limits}),\n$i \\mapsto \\mathcal{F}_i$,\n\\item morphisms of $\\mathcal{O}_X$-modules\n$\\varphi_i : \\mathcal{F}_i \\to \\mathcal{F}$\n\\end{enumerate}\nsuch that each $\\mathcal{F}_i$ is of finite presentation\nand such that the morphisms $\\varphi_i$ induce an isomorphism\n$$\n\\colim_i \\mathcal{F}_i\n=\n\\mathcal{F}.\n$$","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PJ","source_file":"properties.tex","source_line":3039,"source_end_line":3064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3039-L3064","statement_sha256":"442e98b3bb16f3fc694536997ad289f500e21c272c5278f19c447962520ec9e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5286,"rank":5286,"depth":17,"x":1736.617,"y":702.958,"cluster":"schemes"},{"id":"stacks:01PK","tag":"01PK","title":"Extending quasi-coherent sheaves · Lemma 01PK","summary":"Let X be a scheme. Assume X is quasi-compact and quasi-separated. Let F be a quasi-coherent O_X-module. There exist • a directed set I (see Categories, Definition [Tag 00D3]), • a system (F_i, φ_ii') over I in Mod(O_X) (see Categories, Definition [Tag 0030]) • morphisms of O_X-modules φ_i : F_i → F such that each F_i is of finite presentation and such that the morphisms φ_i induce an isomorphism colim_i F_i = F.","statement_latex":"Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThere exist\n\\begin{enumerate}\n\\item a directed set $I$ (see\nCategories, Definition \\ref{categories-definition-directed-set}),\n\\item a system $(\\mathcal{F}_i, \\varphi_{ii'})$\nover $I$ in $\\textit{Mod}(\\mathcal{O}_X)$ (see\nCategories, Definition \\ref{categories-definition-system-over-poset})\n\\item morphisms of $\\mathcal{O}_X$-modules\n$\\varphi_i : \\mathcal{F}_i \\to \\mathcal{F}$\n\\end{enumerate}\nsuch that each $\\mathcal{F}_i$ is of finite presentation\nand such that the morphisms $\\varphi_i$ induce an isomorphism\n$$\n\\colim_i \\mathcal{F}_i\n=\n\\mathcal{F}.\n$$","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PK","source_file":"properties.tex","source_line":3192,"source_end_line":3213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3192-L3213","statement_sha256":"f897b6b823c6615b0abefab8dcfe23619bbab00ba12550c09ba2ad98f0237446","origin":"The Stacks Project","memory_eligible":false,"source_rank":5287,"rank":5287,"depth":18,"x":1445.837,"y":752.054,"cluster":"schemes"},{"id":"stacks:086M","tag":"086M","title":"Extending quasi-coherent sheaves · Lemma 086M","summary":"Let X be a scheme. Assume X is quasi-compact and quasi-separated. Let F be a finite type quasi-coherent O_X-module. Then we can write F = colim F_i with F_i of finite presentation and all transition maps F_i → F_i' surjective.","statement_latex":"Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nThen we can write $\\mathcal{F} = \\colim \\mathcal{F}_i$ with $\\mathcal{F}_i$\nof finite presentation and all transition maps\n$\\mathcal{F}_i \\to \\mathcal{F}_{i'}$ surjective.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086M","source_file":"properties.tex","source_line":3224,"source_end_line":3231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3224-L3231","statement_sha256":"8314f53abfe6756b9595db833a5e02345e9695bc9aa4805ccf642888c37ee5eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5288,"rank":5288,"depth":19,"x":1621.051,"y":550.536,"cluster":"schemes"},{"id":"stacks:080V","tag":"080V","title":"Extending quasi-coherent sheaves · Lemma 080V","summary":"Let X be a quasi-compact and quasi-separated scheme. Let F be a finite type quasi-coherent O_X-module. Let U ⊂ X be a quasi-compact open such that F|_U is of finite presentation. Then there exists a map of O_X-modules φ : G → F with (a) G of finite presentation, (b) φ is surjective, and (c) φ|_U is an isomorphism.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $U \\subset X$ be a quasi-compact open such that $\\mathcal{F}|_U$\nis of finite presentation. Then there exists a map of $\\mathcal{O}_X$-modules\n$\\varphi : \\mathcal{G} \\to \\mathcal{F}$ with\n(a) $\\mathcal{G}$ of finite presentation,\n(b) $\\varphi$ is surjective, and\n(c) $\\varphi|_U$ is an isomorphism.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080V","source_file":"properties.tex","source_line":3253,"source_end_line":3263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3253-L3263","statement_sha256":"2afc46869fb0741bca2f911e20bb3ca572147b380e106afa523575bb6a940e96","origin":"The Stacks Project","memory_eligible":false,"source_rank":5289,"rank":5289,"depth":19,"x":1653.957,"y":798.946,"cluster":"schemes"},{"id":"stacks:05JS","tag":"05JS","title":"Extending quasi-coherent sheaves · Lemma 05JS","summary":"Let X be a scheme. Assume X is quasi-compact and quasi-separated. Let A be a quasi-coherent O_X-algebra. There exist • a directed set I (see Categories, Definition [Tag 00D3]), • a system (A_i, φ_ii') over I in the category of O_X-algebras, • morphisms of O_X-algebras φ_i : A_i → A such that each A_i is a quasi-coherent O_X-algebra of finite presentation and such that the morphisms φ_i induce an isomorphism colim_i A_i = A.","statement_latex":"Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.\nLet $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra.\nThere exist\n\\begin{enumerate}\n\\item a directed set $I$ (see\nCategories, Definition \\ref{categories-definition-directed-set}),\n\\item a system $(\\mathcal{A}_i, \\varphi_{ii'})$\nover $I$ in the category of $\\mathcal{O}_X$-algebras,\n\\item morphisms of $\\mathcal{O}_X$-algebras\n$\\varphi_i : \\mathcal{A}_i \\to \\mathcal{A}$\n\\end{enumerate}\nsuch that each $\\mathcal{A}_i$ is a quasi-coherent $\\mathcal{O}_X$-algebra\nof finite presentation and such that the morphisms $\\varphi_i$\ninduce an isomorphism\n$$\n\\colim_i \\mathcal{A}_i\n=\n\\mathcal{A}.\n$$","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JS","source_file":"properties.tex","source_line":3296,"source_end_line":3317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3296-L3317","statement_sha256":"408ce0743d5bf82f2cb93f04ad70aa3938eb9f4f122b03443cc0fc6a2aebb4ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":5290,"rank":5290,"depth":19,"x":1429.575,"y":634.184,"cluster":"schemes"},{"id":"stacks:05JT","tag":"05JT","title":"Extending quasi-coherent sheaves · Lemma 05JT","summary":"Let X be a scheme. Assume X is quasi-compact and quasi-separated. Let A be a quasi-coherent O_X-algebra. Then A is the directed colimit of its finite type quasi-coherent O_X-subalgebras.","statement_latex":"Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.\nLet $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra.\nThen $\\mathcal{A}$ is the directed colimit of its finite type\nquasi-coherent $\\mathcal{O}_X$-subalgebras.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JT","source_file":"properties.tex","source_line":3348,"source_end_line":3354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3348-L3354","statement_sha256":"3c0d01bd15ee41745fb51d3aba96018a95a61f3bfe43d35da49178137e6e934c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5291,"rank":5291,"depth":20,"x":1727.997,"y":628.348,"cluster":"schemes"},{"id":"stacks:086N","tag":"086N","title":"Extending quasi-coherent sheaves · Lemma 086N","summary":"Let X be a scheme. Assume X is quasi-compact and quasi-separated. Let A be a finite quasi-coherent O_X-algebra. Then A = colim A_i is a directed colimit of finite and finitely presented quasi-coherent O_X-algebras such that all transition maps A_i' → A_i are surjective.","statement_latex":"Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.\nLet $\\mathcal{A}$ be a finite quasi-coherent $\\mathcal{O}_X$-algebra.\nThen $\\mathcal{A} = \\colim \\mathcal{A}_i$ is a directed colimit of finite\nand finitely presented quasi-coherent $\\mathcal{O}_X$-algebras\nsuch that all transition maps $\\mathcal{A}_{i'} \\to \\mathcal{A}_i$\nare surjective.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086N","source_file":"properties.tex","source_line":3380,"source_end_line":3388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3380-L3388","statement_sha256":"7c19c1c7d1b9f0aa02e7f1a67ed5b50a1f766873209ff0bdc123a83e5adeef37","origin":"The Stacks Project","memory_eligible":false,"source_rank":5292,"rank":5292,"depth":20,"x":1512.3,"y":802.256,"cluster":"schemes"},{"id":"stacks:0817","tag":"0817","title":"Extending quasi-coherent sheaves · Lemma 0817","summary":"Let X be a scheme. Assume X is quasi-compact and quasi-separated. Let A be an integral quasi-coherent O_X-algebra. Then • A is the directed colimit of its finite quasi-coherent O_X-subalgebras, and • A is a direct colimit of finite and finitely presented quasi-coherent O_X-algebras.","statement_latex":"Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated.\nLet $\\mathcal{A}$ be an integral quasi-coherent $\\mathcal{O}_X$-algebra.\nThen\n\\begin{enumerate}\n\\item $\\mathcal{A}$ is the directed colimit of its finite\nquasi-coherent $\\mathcal{O}_X$-subalgebras, and\n\\item $\\mathcal{A}$ is a direct colimit of finite and finitely\npresented quasi-coherent $\\mathcal{O}_X$-algebras.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Extending quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0817","source_file":"properties.tex","source_line":3423,"source_end_line":3434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3423-L3434","statement_sha256":"6654567b3b47abf284491fc30177b2d3cb0c2e06dce50181fd2622d4ded91911","origin":"The Stacks Project","memory_eligible":false,"source_rank":5293,"rank":5293,"depth":21,"x":1531.53,"y":551.235,"cluster":"schemes"},{"id":"stacks:077L","tag":"077L","title":"Gabber's result · Definition 077L","summary":"Let (X, O_X) be a ringed space. Let kappa be an infinite cardinal. We say a sheaf of O_X-modules F is kappa-generated if there exists an open covering X = ⋃ U_i such that F|_U_i is generated by a subset R_i ⊂ F(U_i) whose cardinality is at most kappa.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\kappa$ be an infinite\ncardinal. We say a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$ is\n{\\it $\\kappa$-generated} if there exists an open covering\n$X = \\bigcup U_i$ such that $\\mathcal{F}|_{U_i}$ is generated by\na subset $R_i \\subset \\mathcal{F}(U_i)$ whose cardinality is\nat most $\\kappa$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Gabber's result","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077L","source_file":"properties.tex","source_line":3513,"source_end_line":3521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3513-L3521","statement_sha256":"c0a83d3ba1f50c037c2ad0ff420cbfef2d41c611c63d80a1fee39b4e5cbe398b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5294,"rank":5294,"depth":0,"x":1719.507,"y":747.551,"cluster":"schemes"},{"id":"stacks:077M","tag":"077M","title":"Gabber's result · Lemma 077M","summary":"Let (X, O_X) be a ringed space. Let kappa be a cardinal. There exists a set T and a family (F_t)_t ∈ T of kappa-generated O_X-modules such that every kappa-generated O_X-module is isomorphic to one of the F_t.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\kappa$ be a cardinal.\nThere exists a set $T$ and a family $(\\mathcal{F}_t)_{t \\in T}$ of\n$\\kappa$-generated $\\mathcal{O}_X$-modules such that every $\\kappa$-generated\n$\\mathcal{O}_X$-module is isomorphic to one of the $\\mathcal{F}_t$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Gabber's result","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077M","source_file":"properties.tex","source_line":3531,"source_end_line":3537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3531-L3537","statement_sha256":"d8da832cec1b4f1b684eb1658f9de4966a984d0d80838696cd7f8e2b48e5cf47","origin":"The Stacks Project","memory_eligible":false,"source_rank":5295,"rank":5295,"depth":0,"x":1422.563,"y":709.394,"cluster":"schemes"},{"id":"stacks:077N","tag":"077N","title":"Gabber's result · Lemma 077N","summary":"Let X be a scheme. There exists a cardinal kappa such that every quasi-coherent module F is the directed colimit of its quasi-coherent kappa-generated submodules.","statement_latex":"Let $X$ be a scheme. There exists a cardinal $\\kappa$ such that\nevery quasi-coherent module $\\mathcal{F}$ is the directed colimit\nof its quasi-coherent $\\kappa$-generated submodules.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Gabber's result","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077N","source_file":"properties.tex","source_line":3554,"source_end_line":3559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3554-L3559","statement_sha256":"9bf3b9f6cc92be3ffc8c3939b18d0ccfcc0a6d4ebc18d523f10b1a21bc687527","origin":"The Stacks Project","memory_eligible":false,"source_rank":5296,"rank":5296,"depth":13,"x":1672.597,"y":568.82,"cluster":"schemes"},{"id":"stacks:077P","tag":"077P","title":"Gabber's result · Proposition 077P","summary":"Let X be a scheme. • The category QCoh(O_X) is a Grothendieck abelian category. Consequently, QCoh(O_X) has enough injectives and all limits. • The inclusion functor QCoh(O_X) → Mod(O_X) has a right adjoint. Q : Mod(O_X) → QCoh(O_X) such that for every quasi-coherent sheaf F the adjunction mapping Q(F) → F is an isomorphism.","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item The category $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck\nabelian category. Consequently, $\\QCoh(\\mathcal{O}_X)$\nhas enough injectives and all limits.\n\\item The inclusion functor\n$\\QCoh(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$\nhas a right adjoint\\footnote{This functor is sometimes called\nthe {\\it coherator}.}\n$$\nQ : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\QCoh(\\mathcal{O}_X)\n$$\nsuch that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping\n$Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Gabber's result","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077P","source_file":"properties.tex","source_line":3633,"source_end_line":3650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3633-L3650","statement_sha256":"3d5ecd4847921f8fd9883d1e8f21be7ad21ce5a039c66f732f1d072133e20896","origin":"The Stacks Project","memory_eligible":false,"source_rank":5297,"rank":5297,"depth":14,"x":1601.16,"y":814.731,"cluster":"schemes"},{"id":"stacks:01PH","tag":"01PH","title":"Sections with support in a closed subset · Lemma 01PH","summary":"Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be an open subscheme. The following are equivalent: • U is retrocompact in X, • U is quasi-compact, • U is a finite union of affine opens, and • there exists a finite type quasi-coherent sheaf of ideals I ⊂ O_X such that X setminus U = V(I) (set theoretically).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $U \\subset X$ be an open subscheme. The following are equivalent:\n\\begin{enumerate}\n\\item $U$ is retrocompact in $X$,\n\\item $U$ is quasi-compact,\n\\item $U$ is a finite union of affine opens, and\n\\item there exists a finite type quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_X$ such that $X \\setminus U = V(\\mathcal{I})$\n(set theoretically).\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PH","source_file":"properties.tex","source_line":3719,"source_end_line":3731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3719-L3731","statement_sha256":"82194cd61f8624fd758669289a95ad6e11ddb9d92c85ddfb4b5dd5ad3433f783","origin":"The Stacks Project","memory_eligible":false,"source_rank":5298,"rank":5298,"depth":16,"x":1455.859,"y":592.524,"cluster":"schemes"},{"id":"stacks:01PO","tag":"01PO","title":"Sections with support in a closed subset · Lemma 01PO","summary":"Let X be a scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let F be a quasi-coherent O_X-module. Consider the sheaf of O_X-modules F' which associates to every open U ⊂ X F'(U) = (s ∈ F(U) mid Is = 0) Assume I is of finite type. Then • F' is a quasi-coherent sheaf of O_X-modules, • on any affine open U ⊂ X we have F'(U) = (s ∈ F(U) mid I(U)s = 0), and • F'_x = (s ∈ F_x mid I_x s = 0).","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nConsider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$\nwhich associates to every open $U \\subset X$\n$$\n\\mathcal{F}'(U)\n=\n\\{s \\in \\mathcal{F}(U) \\mid\n\\mathcal{I}s = 0\\}\n$$\nAssume $\\mathcal{I}$ is of finite type. Then\n\\begin{enumerate}\n\\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules,\n\\item on any affine open $U \\subset X$ we have\n$\\mathcal{F}'(U) = \\{s \\in \\mathcal{F}(U) \\mid \\mathcal{I}(U)s = 0\\}$, and\n\\item $\\mathcal{F}'_x = \\{s \\in \\mathcal{F}_x \\mid \\mathcal{I}_x s = 0\\}$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PO","source_file":"properties.tex","source_line":3754,"source_end_line":3774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3754-L3774","statement_sha256":"7da0f85bd3ae95dfdf0b5aa959425c0d59f4d40773c78cbd3f4facb4ef941942","origin":"The Stacks Project","memory_eligible":false,"source_rank":5299,"rank":5299,"depth":4,"x":1742.134,"y":674.054,"cluster":"schemes"},{"id":"stacks:01PP","tag":"01PP","title":"Sections with support in a closed subset · Definition 01PP","summary":"Let X be a scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals of finite type. Let F be a quasi-coherent O_X-module. The subsheaf F' ⊂ F defined in Lemma [Tag 01PO] above is called the subsheaf of sections annihilated by I.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals\nof finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in\nLemma \\ref{lemma-sections-annihilated-by-ideal} above is called\nthe {\\it subsheaf of sections annihilated by $\\mathcal{I}$}.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections with support in a closed subset","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PP","source_file":"properties.tex","source_line":3797,"source_end_line":3806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3797-L3806","statement_sha256":"5cba7a8d57f9431a458f241691e09405ef3cd6b523eded0be26fcd936e515aeb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5300,"rank":5300,"depth":5,"x":1465.051,"y":776.531,"cluster":"schemes"},{"id":"stacks:07ZN","tag":"07ZN","title":"Sections with support in a closed subset · Lemma 07ZN","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of schemes. Let I ⊂ O_Y be a quasi-coherent sheaf of ideals of finite type. Let F be a quasi-coherent O_X-module. Let F' ⊂ F be the subsheaf of sections annihilated by f^-1IO_X. Then f_*F' ⊂ f_*F is the subsheaf of sections annihilated by I.","statement_latex":"Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism\nof schemes. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a quasi-coherent\nsheaf of ideals of finite type. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$\nbe the subsheaf of sections annihilated by $f^{-1}\\mathcal{I}\\mathcal{O}_X$.\nThen $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf\nof sections annihilated by $\\mathcal{I}$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZN","source_file":"properties.tex","source_line":3808,"source_end_line":3817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3808-L3817","statement_sha256":"aa5f32309bb1689a4f810e448d5c84e15183fcb96fcd3583585405f257cc43b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5301,"rank":5301,"depth":5,"x":1587.145,"y":543.388,"cluster":"schemes"},{"id":"stacks:07ZP","tag":"07ZP","title":"Sections with support in a closed subset · Lemma 07ZP","summary":"Let X be a scheme. Let Z ⊂ X be a closed subset. Let F be a quasi-coherent O_X-module. Consider the sheaf of O_X-modules F' which associates to every open U ⊂ X F'(U) = (s ∈ F(U) mid the support of s is contained in Z ∩ U) If X setminus Z is a retrocompact open of X, then • for an affine open U ⊂ X there exist a finitely generated ideal I ⊂ O_X(U) such that Z ∩ U = V(I), • for U and I as in (1) we have F'(U) = (x ∈ F(U) mid I^nx = 0 for some n), • F' is a quasi-coherent…","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subset.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nConsider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$\nwhich associates to every open $U \\subset X$\n$$\n\\mathcal{F}'(U)\n=\n\\{s \\in \\mathcal{F}(U) \\mid\n\\text{the support of }s\\text{ is contained in }Z \\cap U\\}\n$$\nIf $X \\setminus Z$ is a retrocompact open of $X$, then\n\\begin{enumerate}\n\\item for an affine open $U \\subset X$ there exist a finitely generated\nideal $I \\subset \\mathcal{O}_X(U)$ such that $Z \\cap U = V(I)$,\n\\item for $U$ and $I$ as in (1) we have\n$\\mathcal{F}'(U) = \\{x \\in \\mathcal{F}(U) \\mid\nI^nx = 0 \\text{ for some } n\\}$,\n\\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZP","source_file":"properties.tex","source_line":3834,"source_end_line":3855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3834-L3855","statement_sha256":"903e12090235a896869391f8470aff09efe2dff826d41af3bd9cecf12b8f26c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5302,"rank":5302,"depth":14,"x":1684.75,"y":784.949,"cluster":"schemes"},{"id":"stacks:084L","tag":"084L","title":"Sections with support in a closed subset · Definition 084L","summary":"Let X be a scheme. Let T ⊂ X be a closed subset whose complement is retrocompact in X. Let F be a quasi-coherent O_X-module. The quasi-coherent subsheaf F' ⊂ F defined in Lemma [Tag 07ZP] is called the subsheaf of sections supported on T.","statement_latex":"Let $X$ be a scheme.\nLet $T \\subset X$ be a closed subset whose complement\nis retrocompact in $X$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe quasi-coherent subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in\nLemma \\ref{lemma-sections-supported-on-closed-subset} is called\nthe {\\it subsheaf of sections supported on $T$}.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections with support in a closed subset","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084L","source_file":"properties.tex","source_line":3893,"source_end_line":3902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3893-L3902","statement_sha256":"c807aab978a0252c8ffd54e5797ac4d79f97a29a9f58236d2e06cdb423eadfb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5303,"rank":5303,"depth":15,"x":1418.117,"y":662.023,"cluster":"schemes"},{"id":"stacks:07ZQ","tag":"07ZQ","title":"Sections with support in a closed subset · Lemma 07ZQ","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of schemes. Let Z ⊂ Y be a closed subset such that Y setminus Z is retrocompact in Y. Let F be a quasi-coherent O_X-module. Let F' ⊂ F be the subsheaf of sections supported in f^-1Z. Then f_*F' ⊂ f_*F is the subsheaf of sections supported in Z.","statement_latex":"Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism\nof schemes. Let $Z \\subset Y$ be a closed subset such that\n$Y \\setminus Z$ is retrocompact in $Y$. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$\nbe the subsheaf of sections supported in $f^{-1}Z$.\nThen $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf\nof sections supported in $Z$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZQ","source_file":"properties.tex","source_line":3904,"source_end_line":3913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3904-L3913","statement_sha256":"e5209c784e25d4062a7bb40c5ae953e7b49aafefa87da1b12d9996e9529a97e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5304,"rank":5304,"depth":15,"x":1714.029,"y":601.281,"cluster":"schemes"},{"id":"stacks:01PM","tag":"01PM","title":"Sections of quasi-coherent sheaves · Lemma 01PM","summary":"Let A be a ring. Let I ⊂ A be a finitely generated ideal. Let M be an A-module. Then there is a canonical map colim_n Hom_A(I^n, M) → Γ(Spec(A) setminus V(I), widetildeM). This map is always injective. If for all x ∈ M we have Ix = 0 ⇒ x = 0 then this map is an isomorphism. In general, set M_n = (x ∈ M mid I^nx = 0), then there is an isomorphism colim_n Hom_A(I^n, M/M_n) → Γ(Spec(A) setminus V(I), widetildeM).","statement_latex":"Let $A$ be a ring.\nLet $I \\subset A$ be a finitely generated ideal.\nLet $M$ be an $A$-module.\nThen there is a canonical map\n$$\n\\colim_n \\Hom_A(I^n, M)\n\\longrightarrow\n\\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M}).\n$$\nThis map is always injective.\nIf for all $x \\in M$ we have $Ix = 0 \\Rightarrow x = 0$\nthen this map is an isomorphism. In general, set\n$M_n = \\{x \\in M \\mid I^nx = 0\\}$, then there is an\nisomorphism\n$$\n\\colim_n \\Hom_A(I^n, M/M_n)\n\\longrightarrow\n\\Gamma(\\Spec(A) \\setminus V(I), \\widetilde{M}).\n$$","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PM","source_file":"properties.tex","source_line":3940,"source_end_line":3961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L3940-L3961","statement_sha256":"6859ab05a0b2785494ac67ca34bbded821605b5298f0fe16b25876ef1fb213c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5305,"rank":5305,"depth":0,"x":1544.419,"y":814.299,"cluster":"schemes"},{"id":"stacks:01PQ","tag":"01PQ","title":"Sections of quasi-coherent sheaves · Lemma 01PQ","summary":"Let X be a quasi-compact scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals of finite type. Let Z ⊂ X be the closed subscheme defined by I and set U = X setminus Z. Let F be a quasi-coherent O_X-module. The canonical map colim_n Hom_O_X(I^n, F) → Γ(U, F) is injective. Assume further that X is quasi-separated. Let F_n ⊂ F be subsheaf of sections annihilated by I^n. The canonical map colim_n Hom_O_X(I^n, F/F_n) → Γ(U, F) is an isomorphism.","statement_latex":"Let $X$ be a quasi-compact scheme.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a\nquasi-coherent sheaf of ideals of finite type.\nLet $Z \\subset X$ be the closed subscheme\ndefined by $\\mathcal{I}$ and set $U = X \\setminus Z$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe canonical map\n$$\n\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n,\n\\mathcal{F})\n\\longrightarrow\n\\Gamma(U, \\mathcal{F})\n$$\nis injective. Assume further that $X$ is quasi-separated.\nLet $\\mathcal{F}_n \\subset \\mathcal{F}$\nbe subsheaf of sections annihilated by $\\mathcal{I}^n$.\nThe canonical map\n$$\n\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n,\n\\mathcal{F}/\\mathcal{F}_n)\n\\longrightarrow\n\\Gamma(U, \\mathcal{F})\n$$\nis an isomorphism.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Sections of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PQ","source_file":"properties.tex","source_line":4145,"source_end_line":4171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4145-L4171","statement_sha256":"ad6a28550a86c0a9ebe6dd54b598f80e5da6e57682f1671fe388b7c939c5fe18","origin":"The Stacks Project","memory_eligible":false,"source_rank":5306,"rank":5306,"depth":1,"x":1498.115,"y":560.602,"cluster":"schemes"},{"id":"stacks:01PS","tag":"01PS","title":"Ample invertible sheaves · Definition 01PS","summary":"[EGA] Let X be a scheme. Let L be an invertible O_X-module. We say L is ample if • X is quasi-compact, and • for every x ∈ X there exists an n ≥ 1 and s ∈ Γ(X, L^⊗ n) such that x ∈ X_s and X_s is affine.","statement_latex":"\\begin{reference}\n\\cite[II Definition 4.5.3]{EGA}\n\\end{reference}\nLet $X$ be a scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nWe say $\\mathcal{L}$ is {\\it ample} if\n\\begin{enumerate}\n\\item $X$ is quasi-compact, and\n\\item for every $x \\in X$ there exists an $n \\geq 1$\nand $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such\nthat $x \\in X_s$ and $X_s$ is affine.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PS","source_file":"properties.tex","source_line":4243,"source_end_line":4257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4243-L4257","statement_sha256":"cb6d00916ab16643596189319e391a84d62cb24cb4e327015e7a40868848b511","origin":"The Stacks Project","memory_eligible":false,"source_rank":5307,"rank":5307,"depth":0,"x":1736.631,"y":721.641,"cluster":"schemes"},{"id":"stacks:01PT","tag":"01PT","title":"Ample invertible sheaves · Lemma 01PT","summary":"[EGA] Let X be a scheme. Let L be an invertible O_X-module. Let n ≥ 1. Then L is ample if and only if L^⊗ n is ample.","statement_latex":"\\begin{reference}\n\\cite[II Proposition 4.5.6(i)]{EGA}\n\\end{reference}\nLet $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $n \\geq 1$. Then $\\mathcal{L}$ is ample if and only if\n$\\mathcal{L}^{\\otimes n}$ is ample.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PT","source_file":"properties.tex","source_line":4259,"source_end_line":4267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4259-L4267","statement_sha256":"9683e2707f319b948525ae210518cb139bdf0a00ffb51340d6b3337d4b907cd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5308,"rank":5308,"depth":0,"x":1430.794,"y":738.258,"cluster":"schemes"},{"id":"stacks:01PU","tag":"01PU","title":"Ample invertible sheaves · Lemma 01PU","summary":"Let X be a scheme. Let L be an ample invertible O_X-module. For any closed subscheme Z ⊂ X the restriction of L to Z is ample.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.\nFor any closed subscheme $Z \\subset X$ the restriction of\n$\\mathcal{L}$ to $Z$ is ample.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PU","source_file":"properties.tex","source_line":4273,"source_end_line":4279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4273-L4279","statement_sha256":"11959e89ca3818fb0f9ead10cf0eb16b1676fb4ee32291329718fc2c1bc013b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5309,"rank":5309,"depth":11,"x":1643.267,"y":552.189,"cluster":"schemes"},{"id":"stacks:01PV","tag":"01PV","title":"Ample invertible sheaves · Lemma 01PV","summary":"Let X be a scheme. Let L be an invertible O_X-module. Let s ∈ Γ(X, L). For any affine U ⊂ X the intersection U ∩ X_s is affine.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$. For any affine $U \\subset X$\nthe intersection $U \\cap X_s$ is affine.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PV","source_file":"properties.tex","source_line":4287,"source_end_line":4292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4287-L4292","statement_sha256":"86e0aee71f38cb3517e75f11856bf9725bcc3c45b4c946b58bf20559729bc8a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5310,"rank":5310,"depth":0,"x":1636.213,"y":810.338,"cluster":"schemes"},{"id":"stacks:0890","tag":"0890","title":"Ample invertible sheaves · Lemma 0890","summary":"[EGA] Let X be a scheme. Let L and M be invertible O_X-modules. If • L is ample, and • the open sets X_t where t ∈ Γ(X, M^⊗ m) for m > 0 cover X, then L ⊗ M is ample.","statement_latex":"\\begin{reference}\n\\cite[II Proposition 4.5.6(ii)]{EGA}\n\\end{reference}\nLet $X$ be a scheme. Let $\\mathcal{L}$ and $\\mathcal{M}$\nbe invertible $\\mathcal{O}_X$-modules. If\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample, and\n\\item the open sets $X_t$ where $t \\in \\Gamma(X, \\mathcal{M}^{\\otimes m})$\nfor $m > 0$ cover $X$,\n\\end{enumerate}\nthen $\\mathcal{L} \\otimes \\mathcal{M}$ is ample.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0890","source_file":"properties.tex","source_line":4316,"source_end_line":4329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4316-L4329","statement_sha256":"b56bf6a3a2e835fc294ba8a07ce6840f0ba190296fe3a14ce821901f00509115","origin":"The Stacks Project","memory_eligible":false,"source_rank":5311,"rank":5311,"depth":1,"x":1433.519,"y":615.693,"cluster":"schemes"},{"id":"stacks:01PX","tag":"01PX","title":"Ample invertible sheaves · Lemma 01PX","summary":"Let X be a scheme. Let L be an invertible O_X-module. Assume the open sets X_s, where s ∈ Γ(X, L^⊗ n) and n ≥ 1, form a basis for the topology on X. Then among those opens, the open sets X_s which are affine form a basis for the topology on X.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume the open sets $X_s$, where $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$\nand $n \\geq 1$, form a basis for the topology on $X$.\nThen among those opens, the open sets $X_s$ which are affine\nform a basis for the topology on $X$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PX","source_file":"properties.tex","source_line":4343,"source_end_line":4350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4343-L4350","statement_sha256":"f40dd56020d575b81c8f8df743dab2900b02feebf2a3abdf1728eedce96a4e27","origin":"The Stacks Project","memory_eligible":false,"source_rank":5312,"rank":5312,"depth":1,"x":1739.962,"y":644.25,"cluster":"schemes"},{"id":"stacks:01PY","tag":"01PY","title":"Ample invertible sheaves · Lemma 01PY","summary":"Let X be a scheme and L be an invertible O_X-module. Assume for every point x of X there exists n ≥ 1 and s ∈ Γ(X, L^⊗ n) such that x ∈ X_s and X_s is affine. Then X is separated.","statement_latex":"Let $X$ be a scheme and $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume for every point $x$ of $X$ there exists $n \\geq 1$ and\n$s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that\n$x \\in X_s$ and $X_s$ is affine. Then $X$ is separated.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PY","source_file":"properties.tex","source_line":4362,"source_end_line":4368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4362-L4368","statement_sha256":"d9610f732440515bb7e6673ce115321946c5a560d4ad56958a69c8a8068428f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5313,"rank":5313,"depth":17,"x":1490.665,"y":797.3,"cluster":"schemes"},{"id":"stacks:09MP","tag":"09MP","title":"Ample invertible sheaves · Lemma 09MP","summary":"Let X be a scheme. If there exists an ample invertible sheaf on X then X is separated.","statement_latex":"Let $X$ be a scheme. If there exists an ample invertible sheaf on $X$\nthen $X$ is separated.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MP","source_file":"properties.tex","source_line":4425,"source_end_line":4429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4425-L4429","statement_sha256":"082598969d47157b9e573a1f0e34dd1f3eb4f9d05b49976f2add4a6a87109c61","origin":"The Stacks Project","memory_eligible":false,"source_rank":5314,"rank":5314,"depth":18,"x":1551.503,"y":542.613,"cluster":"schemes"},{"id":"stacks:01PZ","tag":"01PZ","title":"Ample invertible sheaves · Lemma 01PZ","summary":"Let X be a scheme. Let L be an invertible O_X-module. Set S = Γ_*(X, L) as a graded ring. If every point of X is contained in one of the open subschemes X_s, for some s ∈ S_+ homogeneous, then there is a canonical morphism of schemes f : X → Y = Proj(S), to the homogeneous spectrum of S (see Constructions, Section [Tag 01M3]). This morphism has the following properties • f^-1(D_+(s)) = X_s for any s ∈ S_+ homogeneous, • there are O_X-module maps f^*O_Y(n) → L^⊗ n…","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nSet $S = \\Gamma_*(X, \\mathcal{L})$ as a graded ring.\nIf every point of $X$ is contained in one of the\nopen subschemes $X_s$, for some $s \\in S_{+}$ homogeneous, then\nthere is a canonical morphism of schemes\n$$\nf : X \\longrightarrow Y = \\text{Proj}(S),\n$$\nto the homogeneous spectrum of $S$ (see\nConstructions, Section \\ref{constructions-section-proj}).\nThis morphism has the following properties\n\\begin{enumerate}\n\\item $f^{-1}(D_{+}(s)) = X_s$ for any $s \\in S_{+}$ homogeneous,\n\\item there are $\\mathcal{O}_X$-module maps\n$f^*\\mathcal{O}_Y(n) \\to \\mathcal{L}^{\\otimes n}$\ncompatible with multiplication maps, see\nConstructions, Equation (\\ref{constructions-equation-multiply}),\n\\item the composition\n$S_n \\to \\Gamma(Y, \\mathcal{O}_Y(n)) \\to \\Gamma(X, \\mathcal{L}^{\\otimes n})$\nis the identity map, and\n\\item for every $x \\in X$ there is an integer $d \\geq 1$\nand an open neighbourhood $U \\subset X$ of $x$\nsuch that $f^*\\mathcal{O}_Y(dn)|_U \\to \\mathcal{L}^{\\otimes dn}|_U$\nis an isomorphism for all $n \\in \\mathbf{Z}$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01PZ","source_file":"properties.tex","source_line":4437,"source_end_line":4465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4437-L4465","statement_sha256":"afd9697089d51d357390277e02bf7be77657891d008acd393a29520ce57a658b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5315,"rank":5315,"depth":16,"x":1711.687,"y":765.261,"cluster":"schemes"},{"id":"stacks:01Q0","tag":"01Q0","title":"Ample invertible sheaves · Lemma 01Q0","summary":"Let X be a scheme. Let L be an invertible O_X-module. Set S = Γ_*(X, L). Assume (a) every point of X is contained in one of the open subschemes X_s, for some s ∈ S_+ homogeneous, and (b) X is quasi-compact. Then the canonical morphism of schemes f : X → Proj(S) of Lemma [Tag 01PZ] above is quasi-compact with dense image.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nSet $S = \\Gamma_*(X, \\mathcal{L})$.\nAssume (a) every point of $X$ is contained in one of the\nopen subschemes $X_s$, for some $s \\in S_{+}$ homogeneous,\nand (b) $X$ is quasi-compact. Then the canonical morphism of schemes\n$f : X \\longrightarrow \\text{Proj}(S)$ of Lemma \\ref{lemma-map-into-proj}\nabove is quasi-compact with dense image.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Q0","source_file":"properties.tex","source_line":4483,"source_end_line":4492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4483-L4492","statement_sha256":"04c9e65e78d000f81c96418e01648bdece6add344953d6017601287a1dc4441e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5316,"rank":5316,"depth":18,"x":1414.09,"y":691.869,"cluster":"schemes"},{"id":"stacks:01Q1","tag":"01Q1","title":"Ample invertible sheaves · Lemma 01Q1","summary":"Let X be a scheme. Let L be an invertible O_X-module. Set S = Γ_*(X, L). Assume L is ample. Then the canonical morphism of schemes f : X → Proj(S) of Lemma [Tag 01PZ] is an open immersion with dense image.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nSet $S = \\Gamma_*(X, \\mathcal{L})$.\nAssume $\\mathcal{L}$ is ample. Then the canonical morphism of schemes\n$f : X \\longrightarrow \\text{Proj}(S)$ of Lemma \\ref{lemma-map-into-proj}\nis an open immersion with dense image.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Q1","source_file":"properties.tex","source_line":4512,"source_end_line":4519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4512-L4519","statement_sha256":"916c9a3d4d041f2e212acb11448e310644411b60aca944240317e40c83878b96","origin":"The Stacks Project","memory_eligible":false,"source_rank":5317,"rank":5317,"depth":19,"x":1692.958,"y":576.957,"cluster":"schemes"},{"id":"stacks:01Q2","tag":"01Q2","title":"Ample invertible sheaves · Lemma 01Q2","summary":"Let X be a scheme. Let S be a graded ring. Assume X is quasi-compact, and assume there exists an open immersion j : X → Y = Proj(S). Then j^*O_Y(d) is an invertible ample sheaf for some d > 0.","statement_latex":"Let $X$ be a scheme.\nLet $S$ be a graded ring. Assume $X$ is quasi-compact,\nand assume there exists an open immersion\n$$\nj : X \\longrightarrow Y = \\text{Proj}(S).\n$$\nThen $j^*\\mathcal{O}_Y(d)$ is an invertible ample sheaf\nfor some $d > 0$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Q2","source_file":"properties.tex","source_line":4540,"source_end_line":4550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4540-L4550","statement_sha256":"0545f19e7162887b1edbb0767c2d4a27530a277aba6f3e40c20b6817a3870892","origin":"The Stacks Project","memory_eligible":false,"source_rank":5318,"rank":5318,"depth":5,"x":1579.57,"y":820.28,"cluster":"schemes"},{"id":"stacks:01Q3","tag":"01Q3","title":"Ample invertible sheaves · Proposition 01Q3","summary":"Let X be a quasi-compact scheme. Let L be an invertible sheaf on X. Set S = Γ_*(X, L). The following are equivalent: • L is ample, • the open sets X_s, with s ∈ S_+ homogeneous, cover X and the associated morphism X → Proj(S) is an open immersion, • the open sets X_s, with s ∈ S_+ homogeneous, form a basis for the topology of X, • the open sets X_s, with s ∈ S_+ homogeneous, which are affine form a basis for the topology of X, • for every quasi-coherent sheaf F on X the…","statement_latex":"Let $X$ be a quasi-compact scheme.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nSet $S = \\Gamma_*(X, \\mathcal{L})$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item\n\n$\\mathcal{L}$ is ample,\n\\item\n\nthe open sets $X_s$, with $s \\in S_{+}$ homogeneous,\ncover $X$ and the associated morphism $X \\to \\text{Proj}(S)$\nis an open immersion,\n\\item\n\nthe open sets $X_s$, with $s \\in S_{+}$ homogeneous,\nform a basis for the topology of $X$,\n\\item\n\nthe open sets $X_s$, with $s \\in S_{+}$ homogeneous,\nwhich are affine form a basis for the topology of $X$,\n\\item\n\nfor every quasi-coherent sheaf $\\mathcal{F}$ on $X$\nthe sum of the images of the canonical maps\n$$\n\\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})\n\\otimes_{\\mathbf{Z}} \\mathcal{L}^{\\otimes -n}\n\\longrightarrow\n\\mathcal{F}\n$$\nwith $n \\geq 1$ equals $\\mathcal{F}$,\n\\item\n\nsame property as (\\ref{item-qc-gg}) with $\\mathcal{F}$\nranging over all quasi-coherent sheaves of ideals,\n\\item\n\n$X$ is quasi-separated and\nfor every quasi-coherent sheaf $\\mathcal{F}$ of finite type on $X$\nthere exists an integer $n_0$ such that\n$\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$\nis globally generated for all $n \\geq n_0$,\n\\item\n\n$X$ is quasi-separated and\nfor every quasi-coherent sheaf $\\mathcal{F}$ of finite type on $X$\nthere exist integers $n > 0$, $k \\geq 0$ such that\n$\\mathcal{F}$ is a quotient of a direct sum of $k$ copies of\n$\\mathcal{L}^{\\otimes - n}$, and\n\\item\n\nsame as in (\\ref{item-c-q}) with $\\mathcal{F}$ ranging over all\nsheaves of ideals of finite type on $X$.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Q3","source_file":"properties.tex","source_line":4556,"source_end_line":4613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4556-L4613","statement_sha256":"5dac0896e3bb42dd3301a04933cfad0b414241129adaa51cbcdd6ed378348767","origin":"The Stacks Project","memory_eligible":false,"source_rank":5319,"rank":5319,"depth":20,"x":1467.346,"y":576.166,"cluster":"schemes"},{"id":"stacks:0B3E","tag":"0B3E","title":"Ample invertible sheaves · Lemma 0B3E","summary":"Let X be a scheme. Let L be an ample invertible O_X-module. Let i : X' → X be a morphism of schemes. Assume at least one of the following conditions holds • i is a quasi-compact immersion, • X' is quasi-compact and i is an immersion, • i is quasi-compact and induces a homeomorphism between X' and i(X'), • X' is quasi-compact and i induces a homeomorphism between X' and i(X'). Then i^*L is ample on X'.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible\n$\\mathcal{O}_X$-module. Let $i : X' \\to X$ be a morphism of schemes.\nAssume at least one of the following conditions holds\n\\begin{enumerate}\n\\item $i$ is a quasi-compact immersion,\n\\item $X'$ is quasi-compact and $i$ is an immersion,\n\\item $i$ is quasi-compact and induces a homeomorphism\nbetween $X'$ and $i(X')$,\n\\item $X'$ is quasi-compact and $i$ induces a homeomorphism\nbetween $X'$ and $i(X')$.\n\\end{enumerate}\nThen $i^*\\mathcal{L}$ is ample on $X'$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3E","source_file":"properties.tex","source_line":4758,"source_end_line":4772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4758-L4772","statement_sha256":"06acbb58d2c44d8f843948a578d35531b1a32a462c20cb0daa9e859304b4426b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5320,"rank":5320,"depth":21,"x":1746.809,"y":692.661,"cluster":"schemes"},{"id":"stacks:0DNK","tag":"0DNK","title":"Ample invertible sheaves · Lemma 0DNK","summary":"Let S be a quasi-separated scheme. Let X, Y be schemes over S. Let L be an ample invertible O_X-module and let N be an ample invertible O_Y-module. Then M = pr_1^*L ⊗_O_X ×_S Y pr_2^*N is an ample invertible sheaf on X ×_S Y.","statement_latex":"Let $S$ be a quasi-separated scheme. Let $X$, $Y$ be schemes over $S$.\nLet $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module\nand let $\\mathcal{N}$ be an ample invertible $\\mathcal{O}_Y$-module.\nThen $\\mathcal{M} = \\text{pr}_1^*\\mathcal{L}\n\\otimes_{\\mathcal{O}_{X \\times_S Y}} \\text{pr}_2^*\\mathcal{N}$\nis an ample invertible sheaf on $X \\times_S Y$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNK","source_file":"properties.tex","source_line":4794,"source_end_line":4802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4794-L4802","statement_sha256":"3c41f046a7493ffced930bcd36508eee18528a34404aad0a2e032d5aac13b5d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5321,"rank":5321,"depth":22,"x":1446.625,"y":765.438,"cluster":"schemes"},{"id":"stacks:01QE","tag":"01QE","title":"Affine and quasi-affine schemes · Lemma 01QE","summary":"Let X be a scheme. Then X is quasi-affine if and only if O_X is ample.","statement_latex":"Let $X$ be a scheme.\nThen $X$ is quasi-affine if and only if $\\mathcal{O}_X$ is ample.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Affine and quasi-affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QE","source_file":"properties.tex","source_line":4841,"source_end_line":4845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4841-L4845","statement_sha256":"a3ccad7453844de94a3691c0cca549e5b4de3e4f24a9bb2f9f046fbfdac6b113","origin":"The Stacks Project","memory_eligible":false,"source_rank":5322,"rank":5322,"depth":20,"x":1609.684,"y":541.121,"cluster":"schemes"},{"id":"stacks:0BCK","tag":"0BCK","title":"Affine and quasi-affine schemes · Lemma 0BCK","summary":"Let X be a quasi-affine scheme. For any quasi-compact immersion i : X' → X the scheme X' is quasi-affine.","statement_latex":"Let $X$ be a quasi-affine scheme. For any quasi-compact immersion\n$i : X' \\to X$ the scheme $X'$ is quasi-affine.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Affine and quasi-affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCK","source_file":"properties.tex","source_line":4869,"source_end_line":4873,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4869-L4873","statement_sha256":"73a9da4b87639e89e231bd6256ef0462be943a27c3e7456820ae72326c2f1aa3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5323,"rank":5323,"depth":22,"x":1669.922,"y":799.421,"cluster":"schemes"},{"id":"stacks:01QF","tag":"01QF","title":"Affine and quasi-affine schemes · Lemma 01QF","summary":"Let X be a scheme. Suppose that there exist finitely many elements f_1, …, f_n ∈ Γ(X, O_X) such that • each X_f_i is an affine open of X, and • the ideal generated by f_1, …, f_n in Γ(X, O_X) is equal to the unit ideal. Then X is affine.","statement_latex":"Let $X$ be a scheme. Suppose that there exist finitely many elements\n$f_1, \\ldots, f_n \\in \\Gamma(X, \\mathcal{O}_X)$ such that\n\\begin{enumerate}\n\\item each $X_{f_i}$ is an affine open of $X$, and\n\\item the ideal generated by $f_1, \\ldots, f_n$ in\n$\\Gamma(X, \\mathcal{O}_X)$ is equal to the unit ideal.\n\\end{enumerate}\nThen $X$ is affine.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Affine and quasi-affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QF","source_file":"properties.tex","source_line":4885,"source_end_line":4895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4885-L4895","statement_sha256":"89e248629c606487a71f5b9088f6236b02f441305cb00cc2f9609f145548ffed","origin":"The Stacks Project","memory_eligible":false,"source_rank":5324,"rank":5324,"depth":21,"x":1417.428,"y":642.912,"cluster":"schemes"},{"id":"stacks:01QI","tag":"01QI","title":"Quasi-coherent sheaves and ample invertible sheaves · Lemma 01QI","summary":"In Situation [Tag 01QH]. The canonical morphism f : X → Y maps X into the open subscheme W = W_1 ⊂ Y where O_Y(1) is invertible and where all multiplication maps O_Y(n) ⊗_O_Y O_Y(m) → O_Y(n + m) are isomorphisms (see Constructions, Lemma [Tag 01MU]). Moreover, the maps f^*O_Y(n) → L^⊗ n are all isomorphisms.","statement_latex":"In Situation \\ref{situation-ample}.\nThe canonical morphism $f : X \\to Y$\nmaps $X$ into the open subscheme $W = W_1 \\subset Y$\nwhere $\\mathcal{O}_Y(1)$ is invertible and where\nall multiplication maps\n$\\mathcal{O}_Y(n) \\otimes_{\\mathcal{O}_Y} \\mathcal{O}_Y(m) \\to\n\\mathcal{O}_Y(n + m)$\nare isomorphisms (see\nConstructions, Lemma \\ref{constructions-lemma-where-invertible}).\nMoreover, the maps $f^*\\mathcal{O}_Y(n) \\to \\mathcal{L}^{\\otimes n}$\nare all isomorphisms.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Quasi-coherent sheaves and ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QI","source_file":"properties.tex","source_line":4946,"source_end_line":4959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4946-L4959","statement_sha256":"6ce10a9b6905b1aad5e693c2d2d41fe66e82529051fc0ad55369ea1aaf086120","origin":"The Stacks Project","memory_eligible":false,"source_rank":5325,"rank":5325,"depth":21,"x":1729.915,"y":615.009,"cluster":"schemes"},{"id":"stacks:01QJ","tag":"01QJ","title":"Quasi-coherent sheaves and ample invertible sheaves · Lemma 01QJ","summary":"In Situation [Tag 01QH]. Let F be a quasi-coherent sheaf on X. Set M = Γ_*(X, L, F) as a graded S-module. There are isomorphisms f^*widetildeM → F functorial in F such that M_0 → Γ(Proj(S), widetildeM) → Γ(X, F) is the identity map.","statement_latex":"In Situation \\ref{situation-ample}.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nSet $M = \\Gamma_*(X, \\mathcal{L}, \\mathcal{F})$ as a graded $S$-module.\nThere are isomorphisms\n$$\nf^*\\widetilde{M} \\longrightarrow \\mathcal{F}\n$$\nfunctorial in $\\mathcal{F}$ such that\n$M_0 \\to \\Gamma(\\text{Proj}(S), \\widetilde{M}) \\to \\Gamma(X, \\mathcal{F})$\nis the identity map.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Quasi-coherent sheaves and ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QJ","source_file":"properties.tex","source_line":4998,"source_end_line":5010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L4998-L5010","statement_sha256":"78424a72d1b3e1b4e378404d6af39b1aaef8ef2e07d2daf963d90301ef021101","origin":"The Stacks Project","memory_eligible":false,"source_rank":5326,"rank":5326,"depth":18,"x":1521.637,"y":813.177,"cluster":"schemes"},{"id":"stacks:0AG5","tag":"0AG5","title":"Quasi-coherent sheaves and ample invertible sheaves · Lemma 0AG5","summary":"Let S be a graded ring such that X = Proj(S) is quasi-compact. Let F be a quasi-coherent O_X-module. Set M = bigoplus_n ∈ Z Γ(X, F(n)) as a graded S-module, see Constructions, Section [Tag 01MM]. The map widetildeM → F of Constructions, Lemma [Tag 0B5I] is an isomorphism. If X is covered by standard opens D_+(f) where f has degree 1, then the induced maps M_n → Γ(X, F(n)) are the identity maps.","statement_latex":"Let $S$ be a graded ring such that $X = \\text{Proj}(S)$ is quasi-compact.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Set\n$M = \\bigoplus_{n \\in \\mathbf{Z}} \\Gamma(X, \\mathcal{F}(n))$ as\na graded $S$-module, see\nConstructions, Section \\ref{constructions-section-invertible-on-proj}.\nThe map\n$$\n\\widetilde{M} \\longrightarrow \\mathcal{F}\n$$\nof Constructions, Lemma\n\\ref{constructions-lemma-comparison-proj-quasi-coherent}\nis an isomorphism.\nIf $X$ is covered by standard opens $D_+(f)$ where $f$ has degree $1$,\nthen the induced maps\n$M_n \\to \\Gamma(X, \\mathcal{F}(n))$ are the identity maps.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Quasi-coherent sheaves and ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AG5","source_file":"properties.tex","source_line":5073,"source_end_line":5090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L5073-L5090","statement_sha256":"936930cdcafceda0ae7e59226926c8660e3c6558ab21c882ff182e74d971b365","origin":"The Stacks Project","memory_eligible":false,"source_rank":5327,"rank":5327,"depth":18,"x":1515.849,"y":548.496,"cluster":"schemes"},{"id":"stacks:01ZV","tag":"01ZV","title":"Finding suitable affine opens · Lemma 01ZV","summary":"Let X be a quasi-separated scheme. Let Z_1, …, Z_n be pairwise distinct irreducible components of X, see Topology, Section [Tag 004U]. Let eta_i ∈ Z_i be their generic points, see Schemes, Lemma [Tag 01IS]. There exist affine open neighbourhoods eta_i ∈ U_i such that U_i ∩ U_j = ∅ for all i not = j. In particular, U = U_1 ∪ … ∪ U_n is an affine open containing all of the points eta_1, …, eta_n.","statement_latex":"Let $X$ be a quasi-separated scheme.\nLet $Z_1, \\ldots, Z_n$ be pairwise distinct irreducible components of $X$,\nsee Topology, Section \\ref{topology-section-irreducible-components}.\nLet $\\eta_i \\in Z_i$ be their generic points, see\nSchemes, Lemma \\ref{schemes-lemma-scheme-sober}.\nThere exist affine open neighbourhoods $\\eta_i \\in U_i$\nsuch that $U_i \\cap U_j = \\emptyset$ for all $i \\not = j$.\nIn particular, $U = U_1 \\cup \\ldots \\cup U_n$ is an affine\nopen containing all of the points $\\eta_1, \\ldots, \\eta_n$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Finding suitable affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZV","source_file":"properties.tex","source_line":5140,"source_end_line":5151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L5140-L5151","statement_sha256":"341628efaec09185b2432e04b3e2c3e456dc998630d955dd5cf090cc873a9abf","origin":"The Stacks Project","memory_eligible":false,"source_rank":5328,"rank":5328,"depth":12,"x":1733.271,"y":740.653,"cluster":"schemes"},{"id":"stacks:03J1","tag":"03J1","title":"Finding suitable affine opens · Lemma 03J1","summary":"Let X be a quasi-compact scheme. There exists a dense open V ⊂ X which is separated.","statement_latex":"Let $X$ be a quasi-compact scheme.\nThere exists a dense open $V \\subset X$ which is separated.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Finding suitable affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03J1","source_file":"properties.tex","source_line":5190,"source_end_line":5194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L5190-L5194","statement_sha256":"1be6e3a92030935dd5777c8f5a7b4d13f3fb642b28d1bec8c7609f5b1c1c8228","origin":"The Stacks Project","memory_eligible":false,"source_rank":5329,"rank":5329,"depth":0,"x":1417.978,"y":722.304,"cluster":"schemes"},{"id":"stacks:01ZX","tag":"01ZX","title":"Finding suitable affine opens · Lemma 01ZX","summary":"Let X be a quasi-separated scheme. Let Z_1, …, Z_n be pairwise distinct irreducible components of X. Let eta_i ∈ Z_i be their generic points. Let x ∈ X be arbitrary. There exists an affine open U ⊂ X containing x and all the eta_i.","statement_latex":"Let $X$ be a quasi-separated scheme. Let $Z_1, \\ldots, Z_n$ be pairwise\ndistinct irreducible components of $X$. Let $\\eta_i \\in Z_i$ be their\ngeneric points. Let $x \\in X$ be arbitrary.\nThere exists an affine open $U \\subset X$ containing\n$x$ and all the $\\eta_i$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Finding suitable affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZX","source_file":"properties.tex","source_line":5214,"source_end_line":5221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L5214-L5221","statement_sha256":"1b8bd3a1c97622baf058693cd38140b0196d35000199824abe062512f39251f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5330,"rank":5330,"depth":13,"x":1665.572,"y":556.698,"cluster":"schemes"},{"id":"stacks:01ZY","tag":"01ZY","title":"Finding suitable affine opens · Lemma 01ZY","summary":"Let X be a scheme. Assume either • The scheme X is quasi-affine. • The scheme X is isomorphic to a locally closed subscheme of an affine scheme. • There exists an ample invertible sheaf on X. • The scheme X is isomorphic to a locally closed subscheme of Proj(S) for some graded ring S. Then for any finite subset E ⊂ X there exists an affine open U ⊂ X with E ⊂ U.","statement_latex":"Let $X$ be a scheme. Assume either\n\\begin{enumerate}\n\\item The scheme $X$ is quasi-affine.\n\\item The scheme $X$ is isomorphic to a locally closed subscheme\nof an affine scheme.\n\\item There exists an ample invertible sheaf on $X$.\n\\item The scheme $X$ is isomorphic to a locally closed subscheme\nof $\\text{Proj}(S)$ for some graded ring $S$.\n\\end{enumerate}\nThen for any finite subset $E \\subset X$ there exists an\naffine open $U \\subset X$ with $E \\subset U$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Finding suitable affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZY","source_file":"properties.tex","source_line":5243,"source_end_line":5256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L5243-L5256","statement_sha256":"c7d939438d295f8bf33893f4fd6fda4f05b4ddf53c64997b4ed2ff08529cf418","origin":"The Stacks Project","memory_eligible":false,"source_rank":5331,"rank":5331,"depth":21,"x":1616.106,"y":819.67,"cluster":"schemes"},{"id":"stacks:09NV","tag":"09NV","title":"Finding suitable affine opens · Lemma 09NV","summary":"Let X be a scheme. Let L be an ample invertible sheaf on X. Let E ⊂ W ⊂ X with E finite and W open in X. Then there exists an n > 0 and a section s ∈ Γ(X, L^⊗ n) such that X_s is affine and E ⊂ X_s ⊂ W.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible sheaf on $X$.\nLet\n$$\nE \\subset W \\subset X\n$$\nwith $E$ finite and $W$ open in $X$. Then there exists an $n > 0$\nand a section $s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that\n$X_s$ is affine and $E \\subset X_s \\subset W$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Finding suitable affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NV","source_file":"properties.tex","source_line":5292,"source_end_line":5302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L5292-L5302","statement_sha256":"6c85621e96dfb93951410b905486f1aeed84c3b15f047f51d6c363697b808a09","origin":"The Stacks Project","memory_eligible":false,"source_rank":5332,"rank":5332,"depth":22,"x":1440.864,"y":597.382,"cluster":"schemes"},{"id":"stacks:0F20","tag":"0F20","title":"Finding suitable affine opens · Lemma 0F20","summary":"Let X be a quasi-affine scheme. Let L be an invertible O_X-module. Let E ⊂ W ⊂ X with E finite and W open. Then there exists an s ∈ Γ(X, L) such that X_s is affine and E ⊂ X_s ⊂ W.","statement_latex":"Let $X$ be a quasi-affine scheme. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module. Let $E \\subset W \\subset X$ with $E$ finite\nand $W$ open. Then there exists an $s \\in \\Gamma(X, \\mathcal{L})$\nsuch that $X_s$ is affine and $E \\subset X_s \\subset W$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Finding suitable affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F20","source_file":"properties.tex","source_line":5326,"source_end_line":5332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L5326-L5332","statement_sha256":"dc42cfa4e8e164e96d49ef5be6c7bb06cc1a670acd790b7ca5234bd017a4d414","origin":"The Stacks Project","memory_eligible":false,"source_rank":5333,"rank":5333,"depth":23,"x":1749.27,"y":661.947,"cluster":"schemes"},{"id":"stacks:0BX3","tag":"0BX3","title":"Finding suitable affine opens · Lemma 0BX3","summary":"Let X be a scheme and x ∈ X a point. There exists an affine open neighbourhood U ⊂ X of x such that the canonical map O_X(U) → O_X, x is injective in each of the following cases: • X is integral, • X is locally Noetherian, • X is reduced and has a finite number of irreducible components.","statement_latex":"Let $X$ be a scheme and $x \\in X$ a point. There exists an affine open\nneighbourhood $U \\subset X$ of $x$ such that the canonical map\n$\\mathcal{O}_X(U) \\to \\mathcal{O}_{X, x}$ is injective in each of\nthe following cases:\n\\begin{enumerate}\n\\item $X$ is integral,\n\\item $X$ is locally Noetherian,\n\\item $X$ is reduced and has a finite number of irreducible components.\n\\end{enumerate}","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Finding suitable affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BX3","source_file":"properties.tex","source_line":5383,"source_end_line":5394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L5383-L5394","statement_sha256":"79bb51323e7f19ea2b80f875714bdccb236ceaf2cbe26d8f7a940d064dd3aab7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5334,"rank":5334,"depth":4,"x":1469.549,"y":789.511,"cluster":"schemes"},{"id":"stacks:0H9B","tag":"0H9B","title":"Finding suitable affine opens · Lemma 0H9B","summary":"Let U, V be affine schemes and let W → U and W → V be open immersions. For any w ∈ W there exists an affine open neighbourhood W' ⊂ W of w such that W' maps to a standard open of both U and V.","statement_latex":"Let $U$, $V$ be affine schemes and let $W \\to U$ and $W \\to V$\nbe open immersions. For any $w \\in W$ there exists an affine\nopen neighbourhood $W' \\subset W$ of $w$ such that $W'$\nmaps to a standard open of both $U$ and $V$.","area":"Schemes","chapter":"Properties of Schemes","chapter_id":"properties","section":"Finding suitable affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9B","source_file":"properties.tex","source_line":5401,"source_end_line":5407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/properties.tex#L5401-L5407","statement_sha256":"15e9083cb5eea4ae22d4b0ce1094ee0a170a0c9777351566c22518af2e5e59b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5335,"rank":5335,"depth":1,"x":1573.371,"y":536.378,"cluster":"schemes"},{"id":"stacks:01QO","tag":"01QO","title":"Closed immersions · Lemma 01QO","summary":"Let i : Z → X be a morphism of schemes. The following are equivalent: • The morphism i is a closed immersion. • For every affine open Spec(R) = U ⊂ X, there exists an ideal I ⊂ R such that i^-1(U) = Spec(R/I) as schemes over U = Spec(R). • There exists an affine open covering X = ⋃_j ∈ J U_j, U_j = Spec(R_j) and for every j ∈ J there exists an ideal I_j ⊂ R_j such that i^-1(U_j) = Spec(R_j/I_j) as schemes over U_j = Spec(R_j). • The morphism i induces a homeomorphism of Z…","statement_latex":"Let $i : Z \\to X$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $i$ is a closed immersion.\n\\item For every affine open $\\Spec(R) = U \\subset X$,\nthere exists an ideal $I \\subset R$ such that\n$i^{-1}(U) = \\Spec(R/I)$ as schemes over $U = \\Spec(R)$.\n\\item There exists an affine open covering $X = \\bigcup_{j \\in J} U_j$,\n$U_j = \\Spec(R_j)$ and for every $j \\in J$ there exists\nan ideal $I_j \\subset R_j$ such that\n$i^{-1}(U_j) = \\Spec(R_j/I_j)$ as schemes over $U_j = \\Spec(R_j)$.\n\\item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset\nof $X$ and $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective.\n\\item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset\nof $X$, the map $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective,\nand the kernel $\\Ker(i^\\sharp)\\subset \\mathcal{O}_X$ is a quasi-coherent\nsheaf of ideals.\n\\item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset\nof $X$, the map $i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective,\nand the kernel $\\Ker(i^\\sharp)\\subset \\mathcal{O}_X$ is a\nsheaf of ideals which is locally generated by sections.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QO","source_file":"morphisms.tex","source_line":57,"source_end_line":81,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L57-L81","statement_sha256":"6e35a75b58cb034325b79eef3c4f89ff95c922883cae20df8401452c1d659de8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5336,"rank":5336,"depth":15,"x":2033.338,"y":680.0,"cluster":"scheme-morphisms"},{"id":"stacks:01QP","tag":"01QP","title":"Closed immersions · Lemma 01QP","summary":"Let X be a scheme. Let i : Z → X and i' : Z' → X be closed immersions and consider the ideal sheaves I = Ker(i^sharp) and I' = Ker((i')^sharp) of O_X. • The morphism i : Z → X factors as Z → Z' → X for some a : Z → Z' if and only if I' ⊂ I. If this happens, then a is a closed immersion. • We have Z ≅ Z' over X if and only if I = I'.","statement_latex":"Let $X$ be a scheme. Let $i : Z \\to X$ and $i' : Z' \\to X$\nbe closed immersions and consider the ideal sheaves\n$\\mathcal{I} = \\Ker(i^\\sharp)$ and $\\mathcal{I}' = \\Ker((i')^\\sharp)$\nof $\\mathcal{O}_X$.\n\\begin{enumerate}\n\\item The morphism $i : Z \\to X$ factors as $Z \\to Z' \\to X$\nfor some $a : Z \\to Z'$ if and only if $\\mathcal{I}' \\subset \\mathcal{I}$.\nIf this happens, then $a$ is a closed immersion.\n\\item We have $Z \\cong Z'$ over $X$ if and only if\n$\\mathcal{I} = \\mathcal{I}'$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QP","source_file":"morphisms.tex","source_line":109,"source_end_line":122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L109-L122","statement_sha256":"c90898e67c079dfa4de9db5779083ab2ae4b11c1e71123262732ac7bd786f6fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5337,"rank":5337,"depth":16,"x":2025.736,"y":683.281,"cluster":"scheme-morphisms"},{"id":"stacks:01QQ","tag":"01QQ","title":"Closed immersions · Lemma 01QQ","summary":"Let X be a scheme. Let I ⊂ O_X be a sheaf of ideals. The following are equivalent: • I is locally generated by sections as a sheaf of O_X-modules, • I is quasi-coherent as a sheaf of O_X-modules, and • there exists a closed immersion i : Z → X of schemes whose corresponding sheaf of ideals Ker(i^sharp) is equal to I.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a sheaf of ideals.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{I}$ is locally generated by\nsections as a sheaf of $\\mathcal{O}_X$-modules,\n\\item $\\mathcal{I}$ is quasi-coherent as\na sheaf of $\\mathcal{O}_X$-modules, and\n\\item there exists a closed immersion $i : Z \\to X$ of schemes whose\ncorresponding sheaf of ideals $\\Ker(i^\\sharp)$ is equal to $\\mathcal{I}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QQ","source_file":"morphisms.tex","source_line":134,"source_end_line":147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L134-L147","statement_sha256":"aa08231d224058b5c8305a246848dd6339f9e7dc1cb4a34e3ba7824cc09e5e26","origin":"The Stacks Project","memory_eligible":false,"source_rank":5338,"rank":5338,"depth":12,"x":2030.653,"y":673.754,"cluster":"scheme-morphisms"},{"id":"stacks:01QR","tag":"01QR","title":"Closed immersions · Lemma 01QR","summary":"The base change of a closed immersion is a closed immersion.","statement_latex":"The base change of a closed immersion is a closed immersion.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QR","source_file":"morphisms.tex","source_line":164,"source_end_line":167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L164-L167","statement_sha256":"22cc59732a1ca511394ae770599dfff3c551284f961ba4fd4a720013d9c27517","origin":"The Stacks Project","memory_eligible":false,"source_rank":5339,"rank":5339,"depth":14,"x":2035.374,"y":685.888,"cluster":"scheme-morphisms"},{"id":"stacks:01QS","tag":"01QS","title":"Closed immersions · Lemma 01QS","summary":"A composition of closed immersions is a closed immersion.","statement_latex":"A composition of closed immersions is a closed immersion.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QS","source_file":"morphisms.tex","source_line":173,"source_end_line":176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L173-L176","statement_sha256":"4a13c3bab6113b6dff36f818268a1336ef8801d41f368a828b1cfbb426a90e3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5340,"rank":5340,"depth":16,"x":2020.138,"y":678.535,"cluster":"scheme-morphisms"},{"id":"stacks:01QT","tag":"01QT","title":"Closed immersions · Lemma 01QT","summary":"A closed immersion is quasi-compact.","statement_latex":"A closed immersion is quasi-compact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QT","source_file":"morphisms.tex","source_line":189,"source_end_line":192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L189-L192","statement_sha256":"42c5426696312a081f66bc8d8d7b3dbc048b097d1ef4f6c16d667750bae10c56","origin":"The Stacks Project","memory_eligible":false,"source_rank":5341,"rank":5341,"depth":2,"x":2039.342,"y":675.008,"cluster":"scheme-morphisms"},{"id":"stacks:01QU","tag":"01QU","title":"Closed immersions · Lemma 01QU","summary":"A closed immersion is separated.","statement_latex":"A closed immersion is separated.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QU","source_file":"morphisms.tex","source_line":199,"source_end_line":202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L199-L202","statement_sha256":"e98d6c57ea1800352033b7ee8941e6e2792718263d9f260c07040928a52bd8c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5342,"rank":5342,"depth":3,"x":2026.875,"y":689.764,"cluster":"scheme-morphisms"},{"id":"stacks:07RK","tag":"07RK","title":"Immersions · Lemma 07RK","summary":"Let Z → Y → X be morphisms of schemes. • If Z → X is an immersion, then Z → Y is an immersion. • If Z → X is a quasi-compact immersion and Y → X is quasi-separated, then Z → Y is a quasi-compact immersion. • If Z → X is a closed immersion and Y → X is separated, then Z → Y is a closed immersion.","statement_latex":"Let $Z \\to Y \\to X$ be morphisms of schemes.\n\\begin{enumerate}\n\\item If $Z \\to X$ is an immersion, then $Z \\to Y$ is an immersion.\n\\item If $Z \\to X$ is a quasi-compact immersion and $Y \\to X$ is\nquasi-separated, then $Z \\to Y$ is a quasi-compact immersion.\n\\item If $Z \\to X$ is a closed immersion and $Y \\to X$ is separated,\nthen $Z \\to Y$ is a closed immersion.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RK","source_file":"morphisms.tex","source_line":219,"source_end_line":229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L219-L229","statement_sha256":"6dd18e115f962120a724bb6bf95ab351c5cd94f60e342a1872840d2dd9dd0d71","origin":"The Stacks Project","memory_eligible":false,"source_rank":5343,"rank":5343,"depth":17,"x":2024.041,"y":670.362,"cluster":"scheme-morphisms"},{"id":"stacks:01QV","tag":"01QV","title":"Immersions · Lemma 01QV","summary":"Let h : Z → X be an immersion. If h is quasi-compact, then we can factor h = i ∘ j with j : Z → overlineZ an open immersion and i : overlineZ → X a closed immersion.","statement_latex":"Let $h : Z \\to X$ be an immersion.\nIf $h$ is quasi-compact, then we can factor\n$h = i \\circ j$ with $j : Z \\to \\overline{Z}$ an\nopen immersion and $i : \\overline{Z} \\to X$ a closed immersion.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QV","source_file":"morphisms.tex","source_line":250,"source_end_line":256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L250-L256","statement_sha256":"711c2f38303bdbb5f137b79cf77cb0f7c6fd622ba4809a04445f6f57e21e9d35","origin":"The Stacks Project","memory_eligible":false,"source_rank":5344,"rank":5344,"depth":14,"x":2042.929,"y":683.966,"cluster":"scheme-morphisms"},{"id":"stacks:03DQ","tag":"03DQ","title":"Immersions · Lemma 03DQ","summary":"Let h : Z → X be an immersion. If Z is reduced, then we can factor h = i ∘ j with j : Z → overlineZ an open immersion and i : overlineZ → X a closed immersion.","statement_latex":"Let $h : Z \\to X$ be an immersion.\nIf $Z$ is reduced, then we can factor\n$h = i \\circ j$ with $j : Z \\to \\overline{Z}$ an\nopen immersion and $i : \\overline{Z} \\to X$ a closed immersion.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DQ","source_file":"morphisms.tex","source_line":279,"source_end_line":285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L279-L285","statement_sha256":"a2aeebd3b60fb3c9af33c9c46991515ff852b6010ba214db9df81e0c3eefd1ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":5345,"rank":5345,"depth":5,"x":2016.549,"y":684.664,"cluster":"scheme-morphisms"},{"id":"stacks:0FCZ","tag":"0FCZ","title":"Immersions · Lemma 0FCZ","summary":"Let f : Y → X be a morphism of schemes. If for all y ∈ Y there is an open subscheme f(y) ∈ U ⊂ X such that f|_f^-1(U) : f^-1(U) → U is an immersion, then f is an immersion.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes. If for all $y \\in Y$\nthere is an open subscheme $f(y) \\in U \\subset X$ such that\n$f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is an immersion, then $f$ is\nan immersion.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCZ","source_file":"morphisms.tex","source_line":341,"source_end_line":347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L341-L347","statement_sha256":"c5d022fefa4e8d12ba9839a81492feb759ee9b36272028831e9c1170d2250e0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5346,"rank":5346,"depth":18,"x":2036.484,"y":668.361,"cluster":"scheme-morphisms"},{"id":"stacks:01QY","tag":"01QY","title":"Closed immersions and quasi-coherent sheaves · Lemma 01QY","summary":"Let i : Z → X be a closed immersion of schemes. Let I ⊂ O_X be the quasi-coherent sheaf of ideals cutting out Z. The functor i_* : QCoh(O_Z) → QCoh(O_X) is exact, fully faithful, with essential image those quasi-coherent O_X-modules G such that IG = 0.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes. Let\n$\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals\ncutting out $Z$. The functor\n$$\ni_* :\n\\QCoh(\\mathcal{O}_Z)\n\\longrightarrow\n\\QCoh(\\mathcal{O}_X)\n$$\nis exact, fully faithful, with essential image those quasi-coherent\n$\\mathcal{O}_X$-modules $\\mathcal{G}$ such that $\\mathcal{I}\\mathcal{G} = 0$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QY","source_file":"morphisms.tex","source_line":385,"source_end_line":398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L385-L398","statement_sha256":"5c9a4ec94a186296453725866fd2f1e1116e599e8375af52d1dabf5ff5287d1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5347,"rank":5347,"depth":16,"x":2034.792,"y":692.832,"cluster":"scheme-morphisms"},{"id":"stacks:01QZ","tag":"01QZ","title":"Closed immersions and quasi-coherent sheaves · Lemma 01QZ","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Let G ⊂ F be a O_X-submodule. There exists a unique quasi-coherent O_X-submodule G' ⊂ G with the following property: For every quasi-coherent O_X-module H the map Hom_O_X(H, G') → Hom_O_X(H, G) is bijective. In particular G' is the largest quasi-coherent O_X-submodule of F contained in G.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Let $\\mathcal{G} \\subset \\mathcal{F}$\nbe a $\\mathcal{O}_X$-submodule. There exists a unique quasi-coherent\n$\\mathcal{O}_X$-submodule $\\mathcal{G}' \\subset \\mathcal{G}$\nwith the following property: For every quasi-coherent $\\mathcal{O}_X$-module\n$\\mathcal{H}$ the map\n$$\n\\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G}')\n\\longrightarrow\n\\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G})\n$$\nis bijective. In particular $\\mathcal{G}'$ is the largest quasi-coherent\n$\\mathcal{O}_X$-submodule of $\\mathcal{F}$ contained in $\\mathcal{G}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01QZ","source_file":"morphisms.tex","source_line":441,"source_end_line":456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L441-L456","statement_sha256":"1d03525b3f332a5f610ff23f1aa0f2b0e37d272ff83cad592a5bb367105086d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5348,"rank":5348,"depth":0,"x":2015.558,"y":672.97,"cluster":"scheme-morphisms"},{"id":"stacks:01R0","tag":"01R0","title":"Closed immersions and quasi-coherent sheaves · Lemma 01R0","summary":"Let i : Z → X be a closed immersion of schemes. There is a functor i^! : QCoh(O_X) → QCoh(O_Z) which is a right adjoint to i_*. (Compare Modules, Lemma [Tag 01AZ].)","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nThere is a functor\\footnote{This is likely nonstandard notation.}\n$i^! : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Z)$\nwhich is a right adjoint to $i_*$. (Compare\nModules, Lemma \\ref{modules-lemma-i-star-right-adjoint}.)","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01R0","source_file":"morphisms.tex","source_line":482,"source_end_line":489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L482-L489","statement_sha256":"195d15565e17d2e0b3b814d18b7c6c1536f1c9979d9aa301a71f8175d7f65642","origin":"The Stacks Project","memory_eligible":false,"source_rank":5349,"rank":5349,"depth":1,"x":2046.942,"y":676.871,"cluster":"scheme-morphisms"},{"id":"stacks:0C4H","tag":"0C4H","title":"Closed immersions and quasi-coherent sheaves · Definition 0C4H","summary":"Let X be a scheme. Let Z, Y ⊂ X be closed subschemes corresponding to quasi-coherent ideal sheaves I, J ⊂ O_X. The scheme theoretic intersection of Z and Y is the closed subscheme of X cut out by I + J. The scheme theoretic union of Z and Y is the closed subscheme of X cut out by I ∩ J.","statement_latex":"Let $X$ be a scheme. Let $Z, Y \\subset X$ be closed subschemes\ncorresponding to quasi-coherent ideal sheaves\n$\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$.\nThe {\\it scheme theoretic intersection} of $Z$ and $Y$\nis the closed subscheme of $X$ cut out by $\\mathcal{I} + \\mathcal{J}$.\nThe {\\it scheme theoretic union} of $Z$ and $Y$\nis the closed subscheme of $X$ cut out by\n$\\mathcal{I} \\cap \\mathcal{J}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4H","source_file":"morphisms.tex","source_line":515,"source_end_line":525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L515-L525","statement_sha256":"b80eb20e4a6d271b8f52bf15e05b5fec802e67f1caf86467055f706d51e70759","origin":"The Stacks Project","memory_eligible":false,"source_rank":5350,"rank":5350,"depth":0,"x":2019.661,"y":692.354,"cluster":"scheme-morphisms"},{"id":"stacks:0C4I","tag":"0C4I","title":"Closed immersions and quasi-coherent sheaves · Lemma 0C4I","summary":"Let X be a scheme. Let Z, Y ⊂ X be closed subschemes. Let Z ∩ Y be the scheme theoretic intersection of Z and Y. Then Z ∩ Y → Z and Z ∩ Y → Y are closed immersions and xymatrix Z ∩ Y ar[r] ar[d] & Z ar[d] Y ar[r] & X is a cartesian diagram of schemes, i.e., Z ∩ Y = Z ×_X Y.","statement_latex":"Let $X$ be a scheme. Let $Z, Y \\subset X$ be closed subschemes.\nLet $Z \\cap Y$ be the scheme theoretic intersection of $Z$ and $Y$.\nThen $Z \\cap Y \\to Z$ and $Z \\cap Y \\to Y$ are closed immersions\nand\n$$\n\\xymatrix{\nZ \\cap Y \\ar[r] \\ar[d] & Z \\ar[d] \\\\\nY \\ar[r] & X\n}\n$$\nis a cartesian diagram of schemes, i.e., $Z \\cap Y = Z \\times_X Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4I","source_file":"morphisms.tex","source_line":527,"source_end_line":540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L527-L540","statement_sha256":"0753db15c6eec00ef6c9cd029b929e569f564c498eb70c4eb27131a46446ce72","origin":"The Stacks Project","memory_eligible":false,"source_rank":5351,"rank":5351,"depth":17,"x":2027.611,"y":664.516,"cluster":"scheme-morphisms"},{"id":"stacks:0C4J","tag":"0C4J","title":"Closed immersions and quasi-coherent sheaves · Lemma 0C4J","summary":"Let S be a scheme. Let X, Y ⊂ S be closed subschemes. Let X ∪ Y be the scheme theoretic union of X and Y. Let X ∩ Y be the scheme theoretic intersection of X and Y. Then X → X ∪ Y and Y → X ∪ Y are closed immersions, there is a short exact sequence 0 → O_X ∪ Y → O_X × O_Y → O_X ∩ Y → 0 of O_S-modules, and the diagram xymatrix X ∩ Y ar[r] ar[d] & X ar[d] Y ar[r] & X ∪ Y is cocartesian in the category of schemes, i.e., X ∪ Y = X amalg_X ∩ Y Y.","statement_latex":"Let $S$ be a scheme. Let $X, Y \\subset S$ be closed subschemes.\nLet $X \\cup Y$ be the scheme theoretic union of $X$ and $Y$.\nLet $X \\cap Y$ be the scheme theoretic intersection of $X$ and $Y$.\nThen $X \\to X \\cup Y$ and $Y \\to X \\cup Y$ are closed immersions, there is a\nshort exact sequence\n$$\n0 \\to \\mathcal{O}_{X \\cup Y} \\to \\mathcal{O}_X \\times \\mathcal{O}_Y\n\\to \\mathcal{O}_{X \\cap Y} \\to 0\n$$\nof $\\mathcal{O}_S$-modules, and the diagram\n$$\n\\xymatrix{\nX \\cap Y \\ar[r] \\ar[d] & X \\ar[d] \\\\\nY \\ar[r] & X \\cup Y\n}\n$$\nis cocartesian in the category of schemes, i.e.,\n$X \\cup Y = X \\amalg_{X \\cap Y} Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4J","source_file":"morphisms.tex","source_line":553,"source_end_line":573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L553-L573","statement_sha256":"83d27f36d297f9d11d534d3c5e49fc6601f1eb45c14a1d0aff2d28a0920bc364","origin":"The Stacks Project","memory_eligible":false,"source_rank":5352,"rank":5352,"depth":17,"x":2044.664,"y":690.381,"cluster":"scheme-morphisms"},{"id":"stacks:056I","tag":"056I","title":"Supports of modules · Lemma 056I","summary":"Let X be a scheme. Let F be a quasi-coherent sheaf on X. Let Spec(A) = U ⊂ X be an affine open, and set M = Γ(U, F). Let x ∈ U, and let p ⊂ A be the corresponding prime. The following are equivalent • p is in the support of M, and • x is in the support of F.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $\\Spec(A) = U \\subset X$ be an affine open, and set\n$M = \\Gamma(U, \\mathcal{F})$.\nLet $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathfrak p$ is in the support of $M$, and\n\\item $x$ is in the support of $\\mathcal{F}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Supports of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056I","source_file":"morphisms.tex","source_line":641,"source_end_line":652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L641-L652","statement_sha256":"54f9d90b46a6de3924b82d799193dcd01c2f6b7bb5d6fb3f6518a4c967867a4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5353,"rank":5353,"depth":6,"x":2010.267,"y":680.685,"cluster":"scheme-morphisms"},{"id":"stacks:05AC","tag":"05AC","title":"Supports of modules · Lemma 05AC","summary":"Let X be a scheme. Let F be a quasi-coherent sheaf on X. The support of F is closed under specialization.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nThe support of $\\mathcal{F}$ is closed under specialization.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Supports of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AC","source_file":"morphisms.tex","source_line":660,"source_end_line":665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L660-L665","statement_sha256":"38605bb91a852d4297c7692aea402dbb6fbf60b3fc35bb7bedd825dbaef0c638","origin":"The Stacks Project","memory_eligible":false,"source_rank":5354,"rank":5354,"depth":0,"x":2044.394,"y":667.968,"cluster":"scheme-morphisms"},{"id":"stacks:056J","tag":"056J","title":"Supports of modules · Lemma 056J","summary":"Let F be a finite type quasi-coherent module on a scheme X. Then • The support of F is closed. • For x ∈ X we have x ∈ Supp(F) ⇔ F_x not = 0 ⇔ F_x ⊗_O_X, x kappa(x) not = 0. • For any morphism of schemes f : Y → X the pullback f^*F is of finite type as well and we have Supp(f^*F) = f^-1(Supp(F)).","statement_latex":"Let $\\mathcal{F}$ be a finite type quasi-coherent module\non a scheme $X$. Then\n\\begin{enumerate}\n\\item The support of $\\mathcal{F}$ is closed.\n\\item For $x \\in X$ we have\n$$\nx \\in \\text{Supp}(\\mathcal{F})\n\\Leftrightarrow\n\\mathcal{F}_x \\not = 0\n\\Leftrightarrow\n\\mathcal{F}_x \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) \\not = 0.\n$$\n\\item For any morphism of schemes $f : Y \\to X$ the pullback\n$f^*\\mathcal{F}$ is of finite type as well and we have\n$\\text{Supp}(f^*\\mathcal{F}) = f^{-1}(\\text{Supp}(\\mathcal{F}))$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Supports of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056J","source_file":"morphisms.tex","source_line":678,"source_end_line":696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L678-L696","statement_sha256":"70fa04f2ab1b78869ce4e5626b69a049ead1cc6753f612cd6d844c77b3c850fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5355,"rank":5355,"depth":8,"x":2029.037,"y":697.494,"cluster":"scheme-morphisms"},{"id":"stacks:05JU","tag":"05JU","title":"Supports of modules · Lemma 05JU","summary":"Let F be a finite type quasi-coherent module on a scheme X. There exists a smallest closed subscheme i : Z → X such that there exists a quasi-coherent O_Z-module G with i_*G ≅ F. Moreover: • If Spec(A) ⊂ X is any affine open, and F|_Spec(A) = widetildeM then Z ∩ Spec(A) = Spec(A/I) where I = Ann_A(M). • The quasi-coherent sheaf G is unique up to unique isomorphism. • The quasi-coherent sheaf G is of finite type. • The support of G and of F is Z.","statement_latex":"Let $\\mathcal{F}$ be a finite type quasi-coherent module\non a scheme $X$. There exists a smallest closed subscheme\n$i : Z \\to X$ such that there exists a quasi-coherent\n$\\mathcal{O}_Z$-module $\\mathcal{G}$ with\n$i_*\\mathcal{G} \\cong \\mathcal{F}$. Moreover:\n\\begin{enumerate}\n\\item If $\\Spec(A) \\subset X$ is any affine open, and\n$\\mathcal{F}|_{\\Spec(A)} = \\widetilde{M}$ then\n$Z \\cap \\Spec(A) = \\Spec(A/I)$ where $I = \\text{Ann}_A(M)$.\n\\item The quasi-coherent sheaf $\\mathcal{G}$ is unique up to unique\nisomorphism.\n\\item The quasi-coherent sheaf $\\mathcal{G}$ is of finite type.\n\\item The support of $\\mathcal{G}$ and of $\\mathcal{F}$ is $Z$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Supports of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JU","source_file":"morphisms.tex","source_line":728,"source_end_line":744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L728-L744","statement_sha256":"c4c4dbae8fe07b7277cb3463929e5f02473e0f9069d747f9ae23d121ceeeb1d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5356,"rank":5356,"depth":17,"x":2016.304,"y":666.214,"cluster":"scheme-morphisms"},{"id":"stacks:05JV","tag":"05JV","title":"Supports of modules · Definition 05JV","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module of finite type. The scheme theoretic support of F is the closed subscheme Z ⊂ X constructed in Lemma [Tag 05JU].","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module of finite type. The {\\it scheme theoretic support\nof $\\mathcal{F}$} is the closed subscheme $Z \\subset X$ constructed in\nLemma \\ref{lemma-scheme-theoretic-support}.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Supports of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JV","source_file":"morphisms.tex","source_line":771,"source_end_line":777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L771-L777","statement_sha256":"5651aa2445cd65690461fa72a8d67f22e37a5da70980184697e292e2a4ce093a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5357,"rank":5357,"depth":18,"x":2051.695,"y":682.452,"cluster":"scheme-morphisms"},{"id":"stacks:01R6","tag":"01R6","title":"Scheme theoretic image · Lemma 01R6","summary":"Let f : X → Y be a morphism of schemes. There exists a closed subscheme Z ⊂ Y such that f factors through Z and such that for any other closed subscheme Z' ⊂ Y such that f factors through Z' we have Z ⊂ Z'.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. There exists a closed\nsubscheme $Z \\subset Y$ such that $f$ factors through $Z$ and such\nthat for any other closed subscheme $Z' \\subset Y$ such that $f$\nfactors through $Z'$ we have $Z \\subset Z'$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01R6","source_file":"morphisms.tex","source_line":803,"source_end_line":809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L803-L809","statement_sha256":"0666a36a6c4ae5f2edd0dd45606729af1645f291765f96aebf3fff03e54aaba8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5358,"rank":5358,"depth":17,"x":2011.618,"y":690.744,"cluster":"scheme-morphisms"},{"id":"stacks:01R7","tag":"01R7","title":"Scheme theoretic image · Definition 01R7","summary":"Let f : X → Y be a morphism of schemes. The scheme theoretic image of f is the smallest closed subscheme Z ⊂ Y through which f factors, see Lemma [Tag 01R6] above.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The {\\it scheme theoretic image}\nof $f$ is the smallest closed subscheme $Z \\subset Y$ through which $f$\nfactors, see Lemma \\ref{lemma-scheme-theoretic-image} above.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic image","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01R7","source_file":"morphisms.tex","source_line":823,"source_end_line":828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L823-L828","statement_sha256":"e8af499afb7a095ab96ca420eb62efe985d33c83c51aca5332aada2fb51d4116","origin":"The Stacks Project","memory_eligible":false,"source_rank":5359,"rank":5359,"depth":18,"x":2035.023,"y":661.244,"cluster":"scheme-morphisms"},{"id":"stacks:01R8","tag":"01R8","title":"Scheme theoretic image · Lemma 01R8","summary":"Let f : X → Y be a morphism of schemes. Let Z ⊂ Y be the scheme theoretic image of f. If f is quasi-compact then • the sheaf of ideals I = Ker(O_Y → f_*O_X) is quasi-coherent, • the scheme theoretic image Z is the closed subscheme determined by I, • for any open U ⊂ Y the scheme theoretic image of f|_f^-1(U) : f^-1(U) → U is equal to Z ∩ U, and • the image f(X) ⊂ Z is a dense subset of Z, in other words the morphism X → Z is dominant (see Definition [Tag 01RJ]).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $Z \\subset Y$ be the scheme theoretic image of $f$.\nIf $f$ is quasi-compact then\n\\begin{enumerate}\n\\item the sheaf of ideals\n$\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f_*\\mathcal{O}_X)$\nis quasi-coherent,\n\\item the scheme theoretic image $Z$ is the closed subscheme\ndetermined by $\\mathcal{I}$,\n\\item for any open $U \\subset Y$ the scheme theoretic image of\n$f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is equal to $Z \\cap U$, and\n\\item the image $f(X) \\subset Z$ is a dense subset of $Z$, in other\nwords the morphism $X \\to Z$ is dominant\n(see Definition \\ref{definition-dominant}).\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01R8","source_file":"morphisms.tex","source_line":848,"source_end_line":865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L848-L865","statement_sha256":"71e631af709c1d91ad4dd8d58bc1f05d496f6d1e047fc8bd18ebf5ef9f94823a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5360,"rank":5360,"depth":18,"x":2041.618,"y":697.031,"cluster":"scheme-morphisms"},{"id":"stacks:02JQ","tag":"02JQ","title":"Scheme theoretic image · Lemma 02JQ","summary":"Let f : X → Y be a quasi-compact morphism. Let Z be the scheme theoretic image of f. Let z ∈ Z. There exists a valuation ring A with fraction field K and a commutative diagram xymatrix Spec(K) ar[rr] ar[d] & & X ar[d] ar[ld] Spec(A) ar[r] & Z ar[r] & Y such that the closed point of Spec(A) maps to z. In particular any point of Z is the specialization of a point of f(X).","statement_latex":"Let $f : X \\to Y$ be a quasi-compact morphism.\nLet $Z$ be the scheme theoretic image of $f$.\nLet $z \\in Z$\\footnote{By\nLemma \\ref{lemma-quasi-compact-scheme-theoretic-image} set-theoretically\n$Z$ agrees with the closure of $f(X)$ in $Y$.}.\nThere exists a valuation ring $A$ with\nfraction field $K$ and a commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[rr] \\ar[d] & & X \\ar[d] \\ar[ld] \\\\\n\\Spec(A) \\ar[r] & Z \\ar[r] & Y\n}\n$$\nsuch that the closed point of $\\Spec(A)$ maps to $z$. In particular\nany point of $Z$ is the specialization of a point of $f(X)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JQ","source_file":"morphisms.tex","source_line":898,"source_end_line":915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L898-L915","statement_sha256":"c4f3f6bde4ca17c2b1f4084a54954625083d0ffbeebae295d5f21015c0d8af18","origin":"The Stacks Project","memory_eligible":false,"source_rank":5361,"rank":5361,"depth":19,"x":2007.287,"y":673.913,"cluster":"scheme-morphisms"},{"id":"stacks:01R9","tag":"01R9","title":"Scheme theoretic image · Lemma 01R9","summary":"Let xymatrix X_1 ar[d] ar[r]_f_1 & Y_1 ar[d] X_2 ar[r]^f_2 & Y_2 be a commutative diagram of schemes. Let Z_i ⊂ Y_i, i = 1, 2 be the scheme theoretic image of f_i. Then the morphism Y_1 → Y_2 induces a morphism Z_1 → Z_2 and a commutative diagram xymatrix X_1 ar[r] ar[d] & Z_1 ar[d] ar[r] & Y_1 ar[d] X_2 ar[r] & Z_2 ar[r] & Y_2","statement_latex":"Let\n$$\n\\xymatrix{\nX_1 \\ar[d] \\ar[r]_{f_1} & Y_1 \\ar[d] \\\\\nX_2 \\ar[r]^{f_2} & Y_2\n}\n$$\nbe a commutative diagram of schemes. Let $Z_i \\subset Y_i$, $i = 1, 2$ be\nthe scheme theoretic image of $f_i$. Then the morphism\n$Y_1 \\to Y_2$ induces a morphism $Z_1 \\to Z_2$ and a\ncommutative diagram\n$$\n\\xymatrix{\nX_1 \\ar[r] \\ar[d] & Z_1 \\ar[d] \\ar[r] & Y_1 \\ar[d] \\\\\nX_2 \\ar[r] & Z_2 \\ar[r] & Y_2\n}\n$$","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01R9","source_file":"morphisms.tex","source_line":948,"source_end_line":967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L948-L967","statement_sha256":"19fe6303a74c5b1547965bb0637a3be7c2f5bf009c4ef9e26aabf9015d0109fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5362,"rank":5362,"depth":0,"x":2052.062,"y":671.438,"cluster":"scheme-morphisms"},{"id":"stacks:056B","tag":"056B","title":"Scheme theoretic image · Lemma 056B","summary":"Let f : X → Y be a morphism of schemes. If X is reduced, then the scheme theoretic image of f is the reduced induced scheme structure on overlinef(X).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nIf $X$ is reduced, then the scheme theoretic image of $f$ is\nthe reduced induced scheme structure on $\\overline{f(X)}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056B","source_file":"morphisms.tex","source_line":976,"source_end_line":981,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L976-L981","statement_sha256":"d9eee10657e00a700cca0a274b3815071f32b92a7a41b32ac93d2d2961afafab","origin":"The Stacks Project","memory_eligible":false,"source_rank":5363,"rank":5363,"depth":3,"x":2020.442,"y":699.184,"cluster":"scheme-morphisms"},{"id":"stacks:0CNG","tag":"0CNG","title":"Scheme theoretic image · Lemma 0CNG","summary":"Let f : X → Y be a separated morphism of schemes. Let V ⊂ Y be a retrocompact open. Let s : V → X be a morphism such that f ∘ s = id_V. Let Y' be the scheme theoretic image of s. Then Y' → Y is an isomorphism over V.","statement_latex":"Let $f : X \\to Y$ be a separated morphism of schemes.\nLet $V \\subset Y$ be a retrocompact open. Let $s : V \\to X$\nbe a morphism such that $f \\circ s = \\text{id}_V$.\nLet $Y'$ be the scheme theoretic image of $s$.\nThen $Y' \\to Y$ is an isomorphism over $V$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNG","source_file":"morphisms.tex","source_line":990,"source_end_line":997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L990-L997","statement_sha256":"76002e349674bacb412fe2d217681721be3ac5b42d266ee264c0d66861d7e504","origin":"The Stacks Project","memory_eligible":false,"source_rank":5364,"rank":5364,"depth":19,"x":2021.47,"y":660.078,"cluster":"scheme-morphisms"},{"id":"stacks:01RB","tag":"01RB","title":"Scheme theoretic closure and density · Definition 01RB","summary":"Let X be a scheme. Let U ⊂ X be an open subscheme. • The scheme theoretic image of the morphism U → X is called the scheme theoretic closure of U in X. • We say U is scheme theoretically dense in X if for every open V ⊂ X the scheme theoretic closure of U ∩ V in V is equal to V.","statement_latex":"Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme.\n\\begin{enumerate}\n\\item The scheme theoretic image of the morphism $U \\to X$\nis called the {\\it scheme theoretic closure of $U$ in $X$}.\n\\item We say $U$ is {\\it scheme theoretically dense in $X$}\nif for every open $V \\subset X$ the scheme theoretic closure\nof $U \\cap V$ in $V$ is equal to $V$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic closure and density","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RB","source_file":"morphisms.tex","source_line":1029,"source_end_line":1039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1029-L1039","statement_sha256":"f16fc2c7c8ad0809e5b8ba1c630e397a5124ac229bb8576331b1d27bfe06130e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5365,"rank":5365,"depth":0,"x":2052.698,"y":690.021,"cluster":"scheme-morphisms"},{"id":"stacks:01RD","tag":"01RD","title":"Scheme theoretic closure and density · Lemma 01RD","summary":"Let X be a scheme. Let U ⊂ X be an open subscheme. If the inclusion morphism U → X is quasi-compact, then U is scheme theoretically dense in X if and only if the scheme theoretic closure of U in X is X.","statement_latex":"Let $X$ be a scheme.\nLet $U \\subset X$ be an open subscheme.\nIf the inclusion morphism $U \\to X$ is quasi-compact, then $U$\nis scheme theoretically dense in $X$ if and only if the scheme theoretic\nclosure of $U$ in $X$ is $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RD","source_file":"morphisms.tex","source_line":1074,"source_end_line":1081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1074-L1081","statement_sha256":"e32e825d51976768f03c19e0fc630a815cd89da425f67618f532e802ce1fe017","origin":"The Stacks Project","memory_eligible":false,"source_rank":5366,"rank":5366,"depth":19,"x":2004.788,"y":685.582,"cluster":"scheme-morphisms"},{"id":"stacks:01RE","tag":"01RE","title":"Scheme theoretic closure and density · Lemma 01RE","summary":"Let j : U → X be an open immersion of schemes. Then U is scheme theoretically dense in X if and only if O_X → j_*O_U is injective.","statement_latex":"Let $j : U \\to X$ be an open immersion of schemes.\nThen $U$ is scheme theoretically dense in $X$ if and only if\n$\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is injective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RE","source_file":"morphisms.tex","source_line":1099,"source_end_line":1104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1099-L1104","statement_sha256":"ca10aae7d2c1e9d6061077275a4f7c586b52489fd5506631b07dc62937e87f0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5367,"rank":5367,"depth":18,"x":2044.331,"y":661.278,"cluster":"scheme-morphisms"},{"id":"stacks:01RF","tag":"01RF","title":"Scheme theoretic closure and density · Lemma 01RF","summary":"Let X be a scheme. If U, V are scheme theoretically dense open subschemes of X, then so is U ∩ V.","statement_latex":"Let $X$ be a scheme. If $U$, $V$ are scheme theoretically dense\nopen subschemes of $X$, then so is $U \\cap V$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RF","source_file":"morphisms.tex","source_line":1121,"source_end_line":1125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1121-L1125","statement_sha256":"1a6dfadadb2c4942abfe1808c0803364afe1d2b2fdcaf27a0cbd23d3d843f710","origin":"The Stacks Project","memory_eligible":false,"source_rank":5368,"rank":5368,"depth":19,"x":2034.559,"y":702.282,"cluster":"scheme-morphisms"},{"id":"stacks:01RG","tag":"01RG","title":"Scheme theoretic closure and density · Lemma 01RG","summary":"Let h : Z → X be an immersion. Assume either h is quasi-compact or Z is reduced. Let overlineZ ⊂ X be the scheme theoretic image of h. Then the morphism Z → overlineZ is an open immersion which identifies Z with a scheme theoretically dense open subscheme of overlineZ. Moreover, Z is topologically dense in overlineZ.","statement_latex":"Let $h : Z \\to X$ be an immersion. Assume either $h$ is quasi-compact\nor $Z$ is reduced. Let $\\overline{Z} \\subset X$ be the scheme theoretic\nimage of $h$. Then the morphism $Z \\to \\overline{Z}$ is an open immersion\nwhich identifies $Z$ with a scheme theoretically dense open\nsubscheme of $\\overline{Z}$. Moreover, $Z$ is topologically\ndense in $\\overline{Z}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RG","source_file":"morphisms.tex","source_line":1138,"source_end_line":1146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1138-L1146","statement_sha256":"b580515e999d0ee355a3d8700bacd3c076b7b24b8356b36a52ff7dd6d5b532b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5369,"rank":5369,"depth":19,"x":2008.396,"y":665.946,"cluster":"scheme-morphisms"},{"id":"stacks:056D","tag":"056D","title":"Scheme theoretic closure and density · Lemma 056D","summary":"Let X be a reduced scheme and let U ⊂ X be an open subscheme. Then the following are equivalent • U is topologically dense in X, • the scheme theoretic closure of U in X is X, and • U is scheme theoretically dense in X.","statement_latex":"Let $X$ be a reduced scheme and let $U \\subset X$ be an open subscheme.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $U$ is topologically dense in $X$,\n\\item the scheme theoretic closure of $U$ in $X$ is $X$, and\n\\item $U$ is scheme theoretically dense in $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056D","source_file":"morphisms.tex","source_line":1172,"source_end_line":1181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1172-L1181","statement_sha256":"9f7f60f1db6d94b3b4b9f674b4064252cc55380a94ea6fb765a112d63b519038","origin":"The Stacks Project","memory_eligible":false,"source_rank":5370,"rank":5370,"depth":20,"x":2057.636,"y":678.077,"cluster":"scheme-morphisms"},{"id":"stacks:056E","tag":"056E","title":"Scheme theoretic closure and density · Lemma 056E","summary":"Let X be a scheme and let U ⊂ X be a reduced open subscheme. Then the following are equivalent • the scheme theoretic closure of U in X is X, and • U is scheme theoretically dense in X. If this holds then X is a reduced scheme.","statement_latex":"Let $X$ be a scheme and let $U \\subset X$ be a reduced open subscheme.\nThen the following are equivalent\n\\begin{enumerate}\n\\item the scheme theoretic closure of $U$ in $X$ is $X$, and\n\\item $U$ is scheme theoretically dense in $X$.\n\\end{enumerate}\nIf this holds then $X$ is a reduced scheme.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056E","source_file":"morphisms.tex","source_line":1191,"source_end_line":1200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1191-L1200","statement_sha256":"8e33fcd0a62515d6e2c97f5b47dfe7c5c4b7512f50e8cc13364a11f8d8d6018e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5371,"rank":5371,"depth":20,"x":2010.898,"y":697.345,"cluster":"scheme-morphisms"},{"id":"stacks:01RH","tag":"01RH","title":"Scheme theoretic closure and density · Lemma 01RH","summary":"Let S be a scheme. Let X, Y be schemes over S. Let f, g : X → Y be morphisms of schemes over S. Let U ⊂ X be an open subscheme such that f|_U = g|_U. If the scheme theoretic closure of U in X is X and Y → S is separated, then f = g.","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$.\nLet $f, g : X \\to Y$ be morphisms of schemes over $S$.\nLet $U \\subset X$ be an open subscheme such that\n$f|_U = g|_U$. If the scheme theoretic closure of $U$\nin $X$ is $X$ and $Y \\to S$ is separated, then $f = g$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RH","source_file":"morphisms.tex","source_line":1211,"source_end_line":1218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1211-L1218","statement_sha256":"01384f5823ee19c8ee36fd9d4e69dd9e065c19c9bfafbc00b75777c6a5924d6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5372,"rank":5372,"depth":15,"x":2030.139,"y":656.041,"cluster":"scheme-morphisms"},{"id":"stacks:01RJ","tag":"01RJ","title":"Dominant morphisms · Definition 01RJ","summary":"A morphism f : X → S of schemes is called dominant if the image of f is a dense subset of S.","statement_latex":"A morphism $f : X \\to S$ of schemes is called {\\it dominant} if the\nimage of $f$ is a dense subset of $S$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Dominant morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RJ","source_file":"morphisms.tex","source_line":1236,"source_end_line":1240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1236-L1240","statement_sha256":"c5be1072d743a76501fc8003b8fa1395d08e99dd4a0002df5cb15641a720899b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5373,"rank":5373,"depth":0,"x":2049.425,"y":697.987,"cluster":"scheme-morphisms"},{"id":"stacks:01RK","tag":"01RK","title":"Dominant morphisms · Lemma 01RK","summary":"Let f : X → S be a morphism of schemes. If every generic point of every irreducible component of S is in the image of f, then f is dominant.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf every generic point of every irreducible component of $S$\nis in the image of $f$, then $f$ is dominant.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Dominant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RK","source_file":"morphisms.tex","source_line":1252,"source_end_line":1257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1252-L1257","statement_sha256":"c0facb6170929271feebbff85e4b76f14af4089b8c7752600eed1a3f16620f8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5374,"rank":5374,"depth":2,"x":2000.831,"y":677.729,"cluster":"scheme-morphisms"},{"id":"stacks:01RL","tag":"01RL","title":"Dominant morphisms · Lemma 01RL","summary":"Morphisms whose image contains the generic points are dominant Let f : X → S be a quasi-compact morphism of schemes. Then f is dominant if and only if for every irreducible component Z ⊂ S the generic point of Z is in the image of f.","statement_latex":"\\begin{slogan}\nMorphisms whose image contains the generic points are dominant\n\\end{slogan}\nLet $f : X \\to S$ be a quasi-compact morphism of schemes.\nThen $f$ is dominant if and only if for every irreducible\ncomponent $Z \\subset S$ the generic point of $Z$ is in the\nimage of $f$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Dominant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RL","source_file":"morphisms.tex","source_line":1271,"source_end_line":1280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1271-L1280","statement_sha256":"64566a40aa04cb29d2ecab477cb95c4a018ea667d32aa18dff42bc777129d129","origin":"The Stacks Project","memory_eligible":false,"source_rank":5375,"rank":5375,"depth":12,"x":2053.635,"y":664.932,"cluster":"scheme-morphisms"},{"id":"stacks:0H3F","tag":"0H3F","title":"Dominant morphisms · Lemma 0H3F","summary":"Let f : X → S be a quasi-compact dominant morphism of schemes. Let g : S' → S be a morphism of schemes, and let f' : X' → S' be the base change of f by g. If generalizations lift along g, then f' is dominant.","statement_latex":"Let $f : X \\to S$ be a quasi-compact dominant morphism of schemes.\nLet $g : S' \\to S$ be a morphism of schemes,  and let $f' : X' \\to S'$\nbe the base change of $f$ by $g$. If generalizations lift along $g$,\nthen $f'$ is dominant.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Dominant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3F","source_file":"morphisms.tex","source_line":1304,"source_end_line":1310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1304-L1310","statement_sha256":"84c0ecf00d8df29653df5f49af9e1985daec58adcef67b5ce826a20e953db3fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5376,"rank":5376,"depth":0,"x":2024.622,"y":704.83,"cluster":"scheme-morphisms"},{"id":"stacks:0H8F","tag":"0H8F","title":"Dominant morphisms · Lemma 0H8F","summary":"Let f : X → S be a dominant morphism of schemes. Let g : S' → S be an open morphism of schemes, and let f' : X' → S' be the base change of f by g. Then f' is dominant.","statement_latex":"Let $f : X \\to S$ be a dominant morphism of schemes.\nLet $g : S' \\to S$ be an open morphism of schemes, and let $f' : X' \\to S'$\nbe the base change of $f$ by $g$. Then $f'$ is dominant.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Dominant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8F","source_file":"morphisms.tex","source_line":1334,"source_end_line":1339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1334-L1339","statement_sha256":"05ef1e86d5be5a46b732179ef528297d520fb9155c5e83852d2041c8056ebb5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5377,"rank":5377,"depth":0,"x":2013.802,"y":658.378,"cluster":"scheme-morphisms"},{"id":"stacks:02NE","tag":"02NE","title":"Dominant morphisms · Lemma 02NE","summary":"Let f : X → S be a quasi-compact morphism of schemes. Let eta ∈ S be a generic point of an irreducible component of S. If eta not ∈ f(X) then there exists an open neighbourhood V ⊂ S of eta such that f^-1(V) = ∅.","statement_latex":"Let $f : X \\to S$ be a quasi-compact morphism of schemes.\nLet $\\eta \\in S$ be a generic point of an irreducible\ncomponent of $S$. If $\\eta \\not \\in f(X)$ then there\nexists an open neighbourhood $V \\subset S$ of $\\eta$\nsuch that $f^{-1}(V) = \\emptyset$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Dominant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NE","source_file":"morphisms.tex","source_line":1354,"source_end_line":1361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1354-L1361","statement_sha256":"9679bd0954ca16921164670e4e0629d1a26636ae5c5a4b5a8f7194359744ed47","origin":"The Stacks Project","memory_eligible":false,"source_rank":5378,"rank":5378,"depth":20,"x":2059.683,"y":686.833,"cluster":"scheme-morphisms"},{"id":"stacks:01RM","tag":"01RM","title":"Dominant morphisms · Lemma 01RM","summary":"Let f : X → S be a morphism of schemes. Suppose that X has finitely many irreducible components. Then f is dominant (if and) only if for every irreducible component Z ⊂ S the generic point of Z is in the image of f. If so, then S has finitely many irreducible components as well.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nSuppose that $X$ has finitely many irreducible components.\nThen $f$ is dominant (if and) only if for every irreducible\ncomponent $Z \\subset S$ the generic point of $Z$ is in the\nimage of $f$. If so, then $S$ has finitely many irreducible\ncomponents as well.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Dominant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RM","source_file":"morphisms.tex","source_line":1382,"source_end_line":1390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1382-L1390","statement_sha256":"7dd257c0833693739b99ee5afd14fadec1b1dc439a75c5eec5f844e979e0b730","origin":"The Stacks Project","memory_eligible":false,"source_rank":5379,"rank":5379,"depth":0,"x":2002.297,"y":691.942,"cluster":"scheme-morphisms"},{"id":"stacks:0CC1","tag":"0CC1","title":"Dominant morphisms · Lemma 0CC1","summary":"Let f : X → Y be a morphism of integral schemes. The following are equivalent • f is dominant, • f maps the generic point of X to the generic point of Y, • for some nonempty affine opens U ⊂ X and V ⊂ Y with f(U) ⊂ V the ring map O_Y(V) → O_X(U) is injective, • for all nonempty affine opens U ⊂ X and V ⊂ Y with f(U) ⊂ V the ring map O_Y(V) → O_X(U) is injective, • for some x ∈ X with image y = f(x) ∈ Y the local ring map O_Y, y → O_X, x is injective, and • for all x ∈ X…","statement_latex":"Let $f : X \\to Y$ be a morphism of integral schemes. The following\nare equivalent\n\\begin{enumerate}\n\\item $f$ is dominant,\n\\item $f$ maps the generic point of $X$ to the generic point of $Y$,\n\\item for some nonempty affine opens $U \\subset X$ and $V \\subset Y$\nwith $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$\nis injective,\n\\item for all nonempty affine opens $U \\subset X$ and $V \\subset Y$\nwith $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$\nis injective,\n\\item for some $x \\in X$ with image $y = f(x) \\in Y$ the local ring\nmap $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is injective, and\n\\item for all $x \\in X$ with image $y = f(x) \\in Y$ the local ring\nmap $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is injective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Dominant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CC1","source_file":"morphisms.tex","source_line":1407,"source_end_line":1425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1407-L1425","statement_sha256":"5575641e7e4d7d31d63da08ee13af653fee3d7c84f669c9998bd9b7571ca78ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":5380,"rank":5380,"depth":1,"x":2040.948,"y":655.195,"cluster":"scheme-morphisms"},{"id":"stacks:01RZ","tag":"01RZ","title":"Surjective morphisms · Definition 01RZ","summary":"A morphism of schemes is said to be surjective if it is surjective on underlying topological spaces.","statement_latex":"A morphism of schemes is said to be {\\it surjective}\nif it is surjective on underlying topological\nspaces.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Surjective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RZ","source_file":"morphisms.tex","source_line":1460,"source_end_line":1465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1460-L1465","statement_sha256":"ec442c220b7c87a1a3e815b762de4760e257d7da408f5663975542b4df144e73","origin":"The Stacks Project","memory_eligible":false,"source_rank":5381,"rank":5381,"depth":0,"x":2042.007,"y":704.776,"cluster":"scheme-morphisms"},{"id":"stacks:01S0","tag":"01S0","title":"Surjective morphisms · Lemma 01S0","summary":"The composition of surjective morphisms is surjective.","statement_latex":"The composition of surjective morphisms is surjective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01S0","source_file":"morphisms.tex","source_line":1467,"source_end_line":1470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1467-L1470","statement_sha256":"4dd4fe35ca16f2c73e3e366b1f2912c07a5b5b2a6386e2f44bd252c2338c7567","origin":"The Stacks Project","memory_eligible":false,"source_rank":5382,"rank":5382,"depth":0,"x":2000.908,"y":668.419,"cluster":"scheme-morphisms"},{"id":"stacks:0495","tag":"0495","title":"Surjective morphisms · Lemma 0495","summary":"Let X and Y be schemes over a base scheme S. Given points x ∈ X and y ∈ Y, there is a point of X ×_S Y mapping to x and y under the projections if and only if x and y lie above the same point of S.","statement_latex":"Let $X$ and $Y$ be schemes over a base scheme $S$. Given points $x \\in X$ and\n$y \\in Y$, there is a point of $X \\times_S Y$ mapping to $x$ and $y$ under the\nprojections if and only if $x$ and $y$ lie above the same point of $S$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0495","source_file":"morphisms.tex","source_line":1476,"source_end_line":1481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1476-L1481","statement_sha256":"c7895a058f30121d14675ab59c0e0fb215f0e12706597edac7a0b9a6ce991bae","origin":"The Stacks Project","memory_eligible":false,"source_rank":5383,"rank":5383,"depth":2,"x":2061.094,"y":671.947,"cluster":"scheme-morphisms"},{"id":"stacks:01S1","tag":"01S1","title":"Surjective morphisms · Lemma 01S1","summary":"The base change of a surjective morphism is surjective.","statement_latex":"The base change of a surjective morphism is surjective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01S1","source_file":"morphisms.tex","source_line":1488,"source_end_line":1491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1488-L1491","statement_sha256":"0d5614dbadbf8741512daa6ee61694cffd32e3cab3ae71b919ab1c129a04b3fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":5384,"rank":5384,"depth":3,"x":2013.376,"y":703.828,"cluster":"scheme-morphisms"},{"id":"stacks:04ZD","tag":"04ZD","title":"Surjective morphisms · Lemma 04ZD","summary":"Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & Z be a commutative diagram of morphisms of schemes. If f is surjective and p is quasi-compact, then q is quasi-compact.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& Z\n}\n$$\nbe a commutative diagram of morphisms of schemes.\nIf $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZD","source_file":"morphisms.tex","source_line":1529,"source_end_line":1541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1529-L1541","statement_sha256":"f6d2005c74ce1ee5cb6ef8f4bf3b471772f77345d34112979548e56067b74025","origin":"The Stacks Project","memory_eligible":false,"source_rank":5385,"rank":5385,"depth":1,"x":2023.026,"y":652.72,"cluster":"scheme-morphisms"},{"id":"stacks:01S3","tag":"01S3","title":"Radicial and universally injective morphisms · Definition 01S3","summary":"Let f : X → S be a morphism. • We say that f is universally injective if and only if for any morphism of schemes S' → S the base change f' : X_S' → S' is injective (on underlying topological spaces). • We say f is radicial if f is injective as a map of topological spaces, and for every x ∈ X the field extension kappa(x)/kappa(f(x)) is purely inseparable.","statement_latex":"Let $f : X \\to S$ be a morphism.\n\\begin{enumerate}\n\\item We say that $f$ is {\\it universally injective} if and only\nif for any morphism of schemes $S' \\to S$ the base change\n$f' : X_{S'} \\to S'$ is injective (on underlying topological spaces).\n\\item We say $f$ is {\\it radicial} if $f$ is injective as a\nmap of topological spaces, and for every $x \\in X$ the field\nextension $\\kappa(x)/\\kappa(f(x))$ is purely inseparable.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Radicial and universally injective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01S3","source_file":"morphisms.tex","source_line":1564,"source_end_line":1575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1564-L1575","statement_sha256":"01c4d9ac6e22d47af0fd93e21882edd5f9ae706241b1ea405556908b61efbe28","origin":"The Stacks Project","memory_eligible":false,"source_rank":5386,"rank":5386,"depth":0,"x":2057.351,"y":696.32,"cluster":"scheme-morphisms"},{"id":"stacks:01S4","tag":"01S4","title":"Radicial and universally injective morphisms · Lemma 01S4","summary":"[EGA1-second] Let f : X → S be a morphism of schemes. The following are equivalent: • For every field K the induced map Mor(Spec(K), X) → Mor(Spec(K), S) is injective. • The morphism f is universally injective. • The morphism f is radicial. • The diagonal morphism Δ_X/S : X → X ×_S X is surjective.","statement_latex":"\\begin{reference}\n\\cite[Proposition 3.7.1]{EGA1-second}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item For every field $K$ the induced map\n$\\Mor(\\Spec(K), X) \\to \\Mor(\\Spec(K), S)$\nis injective.\n\\item The morphism $f$ is universally injective.\n\\item The morphism $f$ is radicial.\n\\item The diagonal morphism $\\Delta_{X/S} : X \\longrightarrow X \\times_S X$\nis surjective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Radicial and universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01S4","source_file":"morphisms.tex","source_line":1577,"source_end_line":1593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1577-L1593","statement_sha256":"36914dd63b889fde4aeaea3cb3d70524aee804883c4318aa1e44dd6ccb68fa15","origin":"The Stacks Project","memory_eligible":false,"source_rank":5387,"rank":5387,"depth":2,"x":1996.38,"y":683.52,"cluster":"scheme-morphisms"},{"id":"stacks:05VE","tag":"05VE","title":"Radicial and universally injective morphisms · Lemma 05VE","summary":"A universally injective morphism is separated.","statement_latex":"A universally injective morphism is separated.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Radicial and universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VE","source_file":"morphisms.tex","source_line":1669,"source_end_line":1672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1669-L1672","statement_sha256":"857396af8ab15a0c0ccf0ea30c82a949484f73ba2270972e17d7a1f81ab4a2a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5388,"rank":5388,"depth":3,"x":2052.172,"y":658.119,"cluster":"scheme-morphisms"},{"id":"stacks:0472","tag":"0472","title":"Radicial and universally injective morphisms · Lemma 0472","summary":"A base change of a universally injective morphism is universally injective.","statement_latex":"A base change of a universally injective morphism is universally injective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Radicial and universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0472","source_file":"morphisms.tex","source_line":1683,"source_end_line":1686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1683-L1686","statement_sha256":"5f65f8c18f6aa627801a99875d828b6df72ddc5f7f24fbddcede8dd37a514f12","origin":"The Stacks Project","memory_eligible":false,"source_rank":5389,"rank":5389,"depth":0,"x":2031.258,"y":708.988,"cluster":"scheme-morphisms"},{"id":"stacks:02V1","tag":"02V1","title":"Radicial and universally injective morphisms · Lemma 02V1","summary":"A composition of radicial morphisms is radicial, and so the same holds for the equivalent condition of being universally injective.","statement_latex":"A composition of radicial morphisms is radicial, and so the same holds\nfor the equivalent condition of being universally injective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Radicial and universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V1","source_file":"morphisms.tex","source_line":1692,"source_end_line":1696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1692-L1696","statement_sha256":"490297b79a5c74c844ef165f17af814f2c5a0c4a8e01af35949ad4b54aca7f1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5390,"rank":5390,"depth":0,"x":2005.536,"y":659.147,"cluster":"scheme-morphisms"},{"id":"stacks:01S6","tag":"01S6","title":"Affine morphisms · Definition 01S6","summary":"A morphism of schemes f : X → S is called affine if the inverse image of every affine open of S is an affine open of X.","statement_latex":"A morphism of schemes $f : X \\to S$ is called {\\it affine} if\nthe inverse image of every affine open of $S$ is an affine\nopen of $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01S6","source_file":"morphisms.tex","source_line":1713,"source_end_line":1718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1713-L1718","statement_sha256":"c67ef6605844e0493b50050d9a01fa552a24f52e225c8078242359c23480d92c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5391,"rank":5391,"depth":0,"x":2065.126,"y":681.509,"cluster":"scheme-morphisms"},{"id":"stacks:01S7","tag":"01S7","title":"Affine morphisms · Lemma 01S7","summary":"An affine morphism is separated and quasi-compact.","statement_latex":"An affine morphism is separated and quasi-compact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01S7","source_file":"morphisms.tex","source_line":1720,"source_end_line":1723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1720-L1723","statement_sha256":"53ad16f4abedcaf259310c235dc5731ab6d3eb05949a0e633b0c6ca59d331314","origin":"The Stacks Project","memory_eligible":false,"source_rank":5392,"rank":5392,"depth":13,"x":2002.643,"y":698.987,"cluster":"scheme-morphisms"},{"id":"stacks:01S8","tag":"01S8","title":"Affine morphisms · Lemma 01S8","summary":"[EGA] Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is affine. • There exists an affine open covering S = ⋃ W_j such that each f^-1(W_j) is affine. • There exists a quasi-coherent sheaf of O_S-algebras A and an isomorphism X ≅ underlineSpec_S(A) of schemes over S. See Constructions, Section [Tag 01LQ] for notation. Moreover, in this case X = underlineSpec_S(f_*O_X).","statement_latex":"\\begin{reference}\n\\cite[II, Corollary 1.3.2]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is affine.\n\\item There exists an affine open covering $S = \\bigcup W_j$\nsuch that each $f^{-1}(W_j)$ is affine.\n\\item There exists a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras\n$\\mathcal{A}$ and an isomorphism\n$X \\cong \\underline{\\Spec}_S(\\mathcal{A})$\nof schemes over $S$. See\nConstructions, Section \\ref{constructions-section-spec} for notation.\n\\end{enumerate}\nMoreover, in this case $X = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01S8","source_file":"morphisms.tex","source_line":1735,"source_end_line":1753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1735-L1753","statement_sha256":"e3e741dd17fc24722eea6949ad293f04ec3c758dcb9deaa5b684e9d1faea01f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5393,"rank":5393,"depth":17,"x":2034.947,"y":650.217,"cluster":"scheme-morphisms"},{"id":"stacks:01SA","tag":"01SA","title":"Affine morphisms · Lemma 01SA","summary":"Let S be a scheme. There is an anti-equivalence of categories Schemes affine over S longleftrightarrow quasi-coherent sheaves of O_S-algebras which associates to f : X → S the sheaf f_*O_X. Moreover, this equivalence is compatible with arbitrary base change.","statement_latex":"Let $S$ be a scheme. There is an anti-equivalence of categories\n$$\n\\begin{matrix}\n\\text{Schemes affine} \\\\\n\\text{over }S\n\\end{matrix}\n\\longleftrightarrow\n\\begin{matrix}\n\\text{quasi-coherent sheaves} \\\\\n\\text{of }\\mathcal{O}_S\\text{-algebras}\n\\end{matrix}\n$$\nwhich associates to $f : X \\to S$ the sheaf $f_*\\mathcal{O}_X$.\nMoreover, this equivalence is compatible with arbitrary base change.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SA","source_file":"morphisms.tex","source_line":1817,"source_end_line":1833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1817-L1833","statement_sha256":"58a34c6d90b2955b4bd9b7cfa9253a3dd716e2cf429f9ac879049f2bbd3a27e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5394,"rank":5394,"depth":18,"x":2050.478,"y":704.983,"cluster":"scheme-morphisms"},{"id":"stacks:0H88","tag":"0H88","title":"Affine morphisms · Lemma 0H88","summary":"[EGA] Let S be a scheme and let A be a quasi-coherent O_S-algebra. An A-module is quasi-coherent as an O_S-module if and only if it is quasi-coherent as an A-module.","statement_latex":"\\begin{reference}\n\\cite[I, Proposition 9.6.1]{EGA}\n\\end{reference}\nLet $S$ be a scheme and let $\\mathcal{A}$ be a quasi-coherent\n$\\mathcal{O}_S$-algebra. An $\\mathcal{A}$-module is\nquasi-coherent as an $\\mathcal{O}_S$-module if and only if\nit is quasi-coherent as an $\\mathcal{A}$-module.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H88","source_file":"morphisms.tex","source_line":1844,"source_end_line":1853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1844-L1853","statement_sha256":"18a533fd8c0893ffc1cc2d45f837ee7021935b196a96f823c4fbf51bdfcf73e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5395,"rank":5395,"depth":0,"x":1994.51,"y":673.141,"cluster":"scheme-morphisms"},{"id":"stacks:01SB","tag":"01SB","title":"Affine morphisms · Lemma 01SB","summary":"Let f : X → S be an affine morphism of schemes. Let A = f_*O_X. The functor F ↦ f_*F induces an equivalence of categories ( category of quasi-coherent O_X-modules ) → ( category of quasi-coherent A-modules ) Moreover, an A-module is quasi-coherent as an O_S-module if and only if it is quasi-coherent as an A-module.","statement_latex":"Let $f : X \\to S$ be an affine morphism of schemes.\nLet $\\mathcal{A} = f_*\\mathcal{O}_X$.\nThe functor $\\mathcal{F} \\mapsto f_*\\mathcal{F}$ induces\nan equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{category of quasi-coherent}\\\\\n\\mathcal{O}_X\\text{-modules}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{category of quasi-coherent}\\\\\n\\mathcal{A}\\text{-modules}\n\\end{matrix}\n\\right\\}\n$$\nMoreover, an $\\mathcal{A}$-module is\nquasi-coherent as an $\\mathcal{O}_S$-module if and only if\nit is quasi-coherent as an $\\mathcal{A}$-module.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SB","source_file":"morphisms.tex","source_line":1895,"source_end_line":1919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1895-L1919","statement_sha256":"36ad025b6ac2edaef9c1267a99d24dd470559e0c30dcba0047293942c128d0a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5396,"rank":5396,"depth":14,"x":2061.95,"y":664.795,"cluster":"scheme-morphisms"},{"id":"stacks:01SC","tag":"01SC","title":"Affine morphisms · Lemma 01SC","summary":"The composition of affine morphisms is affine.","statement_latex":"The composition of affine morphisms is affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SC","source_file":"morphisms.tex","source_line":1969,"source_end_line":1972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1969-L1972","statement_sha256":"3b9345b8c80d29a730c4d673b97b2bdb26ff7d431c438e8f0234ed012b727eab","origin":"The Stacks Project","memory_eligible":false,"source_rank":5397,"rank":5397,"depth":0,"x":2018.575,"y":709.582,"cluster":"scheme-morphisms"},{"id":"stacks:01SD","tag":"01SD","title":"Affine morphisms · Lemma 01SD","summary":"The base change of an affine morphism is affine.","statement_latex":"The base change of an affine morphism is affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SD","source_file":"morphisms.tex","source_line":1981,"source_end_line":1984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1981-L1984","statement_sha256":"feae9d2f56c03a76ef9a6156e304d0f427028b609f231f125ea086532b40a895","origin":"The Stacks Project","memory_eligible":false,"source_rank":5398,"rank":5398,"depth":18,"x":2014.511,"y":651.475,"cluster":"scheme-morphisms"},{"id":"stacks:01SE","tag":"01SE","title":"Affine morphisms · Lemma 01SE","summary":"A closed immersion is affine.","statement_latex":"A closed immersion is affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SE","source_file":"morphisms.tex","source_line":1999,"source_end_line":2002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L1999-L2002","statement_sha256":"685ea3d7ef77a07866af0cd475216f20344438e1b575211977861517e5b781eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5399,"rank":5399,"depth":12,"x":2064.633,"y":692.342,"cluster":"scheme-morphisms"},{"id":"stacks:01SF","tag":"01SF","title":"Affine morphisms · Lemma 01SF","summary":"Let X be a scheme. Let L be an invertible O_X-module. Let s ∈ Γ(X, L). The inclusion morphism j : X_s → X is affine.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$.\nThe inclusion morphism $j : X_s \\to X$ is affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SF","source_file":"morphisms.tex","source_line":2011,"source_end_line":2017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2011-L2017","statement_sha256":"3908940756f7fe03e91dcbe55c16a0815b2e4cd4f8cd710dbe8af50a13b5d82a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5400,"rank":5400,"depth":1,"x":1994.259,"y":690.633,"cluster":"scheme-morphisms"},{"id":"stacks:01SG","tag":"01SG","title":"Affine morphisms · Lemma 01SG","summary":"Suppose g : X → Y is a morphism of schemes over S. • If X is affine over S and Δ : Y → Y ×_S Y is affine, then g is affine. • If X is affine over S and Y is separated over S, then g is affine. • A morphism from an affine scheme to a scheme with affine diagonal is affine. • A morphism from an affine scheme to a separated scheme is affine.","statement_latex":"Suppose $g : X \\to Y$ is a morphism of schemes over $S$.\n\\begin{enumerate}\n\\item If $X$ is affine over $S$ and $\\Delta : Y \\to Y \\times_S Y$ is affine,\nthen $g$ is affine.\n\\item If $X$ is affine over $S$ and $Y$ is separated over $S$,\nthen $g$ is affine.\n\\item A morphism from an affine scheme to a scheme with affine\ndiagonal is affine.\n\\item A morphism from an affine scheme to a separated scheme is affine.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SG","source_file":"morphisms.tex","source_line":2024,"source_end_line":2036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2024-L2036","statement_sha256":"8483dc4a99783429e3bced35dda72bfdf759810cda37dcf9af8c64497ac096f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5401,"rank":5401,"depth":19,"x":2047.941,"y":651.663,"cluster":"scheme-morphisms"},{"id":"stacks:01SH","tag":"01SH","title":"Affine morphisms · Lemma 01SH","summary":"A morphism between affine schemes is affine.","statement_latex":"A morphism between affine schemes is affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SH","source_file":"morphisms.tex","source_line":2053,"source_end_line":2056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2053-L2056","statement_sha256":"281778a96a15296f09cbd4db643aa35d45508fffc8d3672ef342d3206a1b8fdb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5402,"rank":5402,"depth":20,"x":2039.631,"y":711.311,"cluster":"scheme-morphisms"},{"id":"stacks:01SI","tag":"01SI","title":"Affine morphisms · Lemma 01SI","summary":"Let S be a scheme. Let A be an Artinian ring. Any morphism Spec(A) → S is affine.","statement_latex":"Let $S$ be a scheme.\nLet $A$ be an Artinian ring.\nAny morphism $\\Spec(A) \\to S$ is affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SI","source_file":"morphisms.tex","source_line":2064,"source_end_line":2069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2064-L2069","statement_sha256":"9ac76831a8607e3f003ee02eec9c1fe005296555d2ed8e04c98f6dc06b3b5644","origin":"The Stacks Project","memory_eligible":false,"source_rank":5403,"rank":5403,"depth":0,"x":1997.477,"y":662.245,"cluster":"scheme-morphisms"},{"id":"stacks:0C3A","tag":"0C3A","title":"Affine morphisms · Lemma 0C3A","summary":"Let j : Y → X be an immersion of schemes. Assume there exists an open U ⊂ X with complement Z = X setminus U such that • U → X is affine, • j^-1(U) → U is affine, and • j(Y) ∩ Z is closed. Then j is affine. In particular, if X is affine, so is Y.","statement_latex":"Let $j : Y \\to X$ be an immersion of schemes.\nAssume there exists an open $U \\subset X$ with complement\n$Z = X \\setminus U$ such that\n\\begin{enumerate}\n\\item $U \\to X$ is affine,\n\\item $j^{-1}(U) \\to U$ is affine, and\n\\item $j(Y) \\cap Z$ is closed.\n\\end{enumerate}\nThen $j$ is affine. In particular, if $X$ is affine, so is $Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3A","source_file":"morphisms.tex","source_line":2075,"source_end_line":2086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2075-L2086","statement_sha256":"bff9353a109016e8937c666db4c1c2cd86277a7c8aa9638bdbaac869aec42418","origin":"The Stacks Project","memory_eligible":false,"source_rank":5404,"rank":5404,"depth":18,"x":2068.541,"y":674.599,"cluster":"scheme-morphisms"},{"id":"stacks:0FXR","tag":"0FXR","title":"Families of ample invertible modules · Definition 0FXR","summary":"[SGA6] Let X be a scheme. Let (L_i)_i ∈ I be a family of invertible O_X-modules. We say (L_i)_i ∈ I is an ample family of invertible modules on X if • X is quasi-compact, and • for every x ∈ X there exists an i ∈ I, an n ≥ 1, and s ∈ Γ(X, L_i^⊗ n) such that x ∈ X_s and X_s is affine.","statement_latex":"\\begin{reference}\n\\cite[II Definition 2.2.4]{SGA6}\n\\end{reference}\nLet $X$ be a scheme. Let $\\{\\mathcal{L}_i\\}_{i \\in I}$\nbe a family of invertible $\\mathcal{O}_X$-modules. We say\n$\\{\\mathcal{L}_i\\}_{i \\in I}$ is an\n{\\it ample family of invertible modules on $X$} if\n\\begin{enumerate}\n\\item $X$ is quasi-compact, and\n\\item for every $x \\in X$ there exists an $i \\in I$, an $n \\geq 1$,\nand $s \\in \\Gamma(X, \\mathcal{L}_i^{\\otimes n})$ such\nthat $x \\in X_s$ and $X_s$ is affine.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Families of ample invertible modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXR","source_file":"morphisms.tex","source_line":2117,"source_end_line":2132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2117-L2132","statement_sha256":"3eb8bb5019a75b55fb96bd19250ecef6e7da7284d1be202aed93e3c58eb728d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5405,"rank":5405,"depth":0,"x":2005.749,"y":706.04,"cluster":"scheme-morphisms"},{"id":"stacks:0FXS","tag":"0FXS","title":"Families of ample invertible modules · Lemma 0FXS","summary":"Let X be a scheme such that for every point x ∈ X there exists an invertible O_X-module L and a global section s ∈ Γ(X, L) such that x ∈ X_s and X_s is affine. Then the diagonal of X is an affine morphism.","statement_latex":"Let $X$ be a scheme such that for every point $x \\in X$ there exists\nan invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ and a global\nsection $s \\in \\Gamma(X, \\mathcal{L})$ such that $x \\in X_s$ and\n$X_s$ is affine. Then the diagonal of $X$ is an affine morphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Families of ample invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXS","source_file":"morphisms.tex","source_line":2142,"source_end_line":2148,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2142-L2148","statement_sha256":"30c87e3519701a545a126fe3f31fd0beaf347b7e7b9097f5153956ad1b99a58c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5406,"rank":5406,"depth":18,"x":2026.92,"y":646.803,"cluster":"scheme-morphisms"},{"id":"stacks:01SK","tag":"01SK","title":"Quasi-affine morphisms · Definition 01SK","summary":"A morphism of schemes f : X → S is called quasi-affine if the inverse image of every affine open of S is a quasi-affine scheme.","statement_latex":"A morphism of schemes $f : X \\to S$ is called {\\it quasi-affine} if the\ninverse image of every affine open of $S$ is a quasi-affine scheme.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-affine morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SK","source_file":"morphisms.tex","source_line":2199,"source_end_line":2203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2199-L2203","statement_sha256":"c6bced029eaa4645784a959b992b50ff7724d77e77aab01f76589887ca872f7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5407,"rank":5407,"depth":0,"x":2059.172,"y":702.892,"cluster":"scheme-morphisms"},{"id":"stacks:01SL","tag":"01SL","title":"Quasi-affine morphisms · Lemma 01SL","summary":"A quasi-affine morphism is separated and quasi-compact.","statement_latex":"A quasi-affine morphism is separated and quasi-compact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SL","source_file":"morphisms.tex","source_line":2205,"source_end_line":2208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2205-L2208","statement_sha256":"1596c282c0fb11a6e6a5c41c0863476f4c028e40346dec4d5cf7ca6dbba8064e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5408,"rank":5408,"depth":13,"x":1989.803,"y":679.671,"cluster":"scheme-morphisms"},{"id":"stacks:01SM","tag":"01SM","title":"Quasi-affine morphisms · Lemma 01SM","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is quasi-affine. • There exists an affine open covering S = ⋃ W_j such that each f^-1(W_j) is quasi-affine. • There exists a quasi-coherent sheaf of O_S-algebras A and a quasi-compact open immersion xymatrix X ar[rr] ar[rd] & & underlineSpec_S(A) ar[dl] & S & over S. • Same as in (3) but with A = f_*O_X and the horizontal arrow the canonical morphism of Constructions, Lemma [Tag 01LY].","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is quasi-affine.\n\\item There exists an affine open covering $S = \\bigcup W_j$\nsuch that each $f^{-1}(W_j)$ is quasi-affine.\n\\item There exists a quasi-coherent sheaf of $\\mathcal{O}_S$-algebras\n$\\mathcal{A}$ and a quasi-compact open immersion\n$$\n\\xymatrix{\nX \\ar[rr] \\ar[rd] & & \\underline{\\Spec}_S(\\mathcal{A}) \\ar[dl] \\\\\n& S &\n}\n$$\nover $S$.\n\\item Same as in (3) but with $\\mathcal{A} = f_*\\mathcal{O}_X$\nand the horizontal arrow the canonical morphism of\nConstructions, Lemma \\ref{constructions-lemma-canonical-morphism}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SM","source_file":"morphisms.tex","source_line":2223,"source_end_line":2244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2223-L2244","statement_sha256":"64262879c9edf16244ca016bd94769eca7d9d5dfad7f82b49c6457331c06f870","origin":"The Stacks Project","memory_eligible":false,"source_rank":5409,"rank":5409,"depth":18,"x":2060.11,"y":657.28,"cluster":"scheme-morphisms"},{"id":"stacks:01SN","tag":"01SN","title":"Quasi-affine morphisms · Lemma 01SN","summary":"The composition of quasi-affine morphisms is quasi-affine.","statement_latex":"The composition of quasi-affine morphisms is quasi-affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SN","source_file":"morphisms.tex","source_line":2280,"source_end_line":2283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2280-L2283","statement_sha256":"2e96c483637b26537f4c363f84d8762df764bcf8a27e9cf54b7f83a013366ac2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5410,"rank":5410,"depth":19,"x":2026.042,"y":714.068,"cluster":"scheme-morphisms"},{"id":"stacks:01SO","tag":"01SO","title":"Quasi-affine morphisms · Lemma 01SO","summary":"The base change of a quasi-affine morphism is quasi-affine.","statement_latex":"The base change of a quasi-affine morphism is quasi-affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SO","source_file":"morphisms.tex","source_line":2316,"source_end_line":2319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2316-L2319","statement_sha256":"3194d114c6c502c225c3e4bd26cb9c99418c399c4b4afc344ab5314e7972dea2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5411,"rank":5411,"depth":19,"x":2005.359,"y":652.45,"cluster":"scheme-morphisms"},{"id":"stacks:02JR","tag":"02JR","title":"Quasi-affine morphisms · Lemma 02JR","summary":"A quasi-compact immersion is quasi-affine.","statement_latex":"A quasi-compact immersion is quasi-affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JR","source_file":"morphisms.tex","source_line":2342,"source_end_line":2345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2342-L2345","statement_sha256":"4ae0bc9b058536a3dc7f5bec502873d432855c34d31d2e29344ec9aeda79acce","origin":"The Stacks Project","memory_eligible":false,"source_rank":5412,"rank":5412,"depth":20,"x":2070.59,"y":686.375,"cluster":"scheme-morphisms"},{"id":"stacks:01SP","tag":"01SP","title":"Quasi-affine morphisms · Lemma 01SP","summary":"Let S be a scheme. Let X be an affine scheme. A morphism f : X → S is quasi-affine if and only if it is quasi-compact. In particular any morphism from an affine scheme to a quasi-separated scheme is quasi-affine.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine scheme.\nA morphism $f : X \\to S$ is quasi-affine if and only if it is quasi-compact.\nIn particular any morphism from an affine scheme to a quasi-separated\nscheme is quasi-affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SP","source_file":"morphisms.tex","source_line":2358,"source_end_line":2364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2358-L2364","statement_sha256":"767d8f5ed210f1dfb9d37cca566377a7b38cb0f361decb77a1da1129cd906111","origin":"The Stacks Project","memory_eligible":false,"source_rank":5413,"rank":5413,"depth":18,"x":1994.716,"y":698.45,"cluster":"scheme-morphisms"},{"id":"stacks:054G","tag":"054G","title":"Quasi-affine morphisms · Lemma 054G","summary":"Suppose g : X → Y is a morphism of schemes over S. If X is quasi-affine over S and Y is quasi-separated over S, then g is quasi-affine. In particular, any morphism from a quasi-affine scheme to a quasi-separated scheme is quasi-affine.","statement_latex":"Suppose $g : X \\to Y$ is a morphism of schemes over $S$.\nIf $X$ is quasi-affine over $S$ and $Y$ is quasi-separated over $S$,\nthen $g$ is quasi-affine. In particular, any morphism from a\nquasi-affine scheme to a quasi-separated scheme is quasi-affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054G","source_file":"morphisms.tex","source_line":2375,"source_end_line":2381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2375-L2381","statement_sha256":"4e2c2c6f34f9fab860ecf52ec767424c946a77f32a211c36a15d0234ca375fad","origin":"The Stacks Project","memory_eligible":false,"source_rank":5414,"rank":5414,"depth":21,"x":2041.253,"y":646.159,"cluster":"scheme-morphisms"},{"id":"stacks:01SR","tag":"01SR","title":"Types of morphisms defined by properties of ring maps · Definition 01SR","summary":"Let P be a property of ring maps. • We say that P is local if the following hold: • For any ring map R → A, and any f ∈ R we have P(R → A) ⇒ P(R_f → A_f). • For any rings R, A, any f ∈ R, a∈ A, and any ring map R_f → A we have P(R_f → A) ⇒ P(R → A_a). • For any ring map R → A, and a_i ∈ A such that (a_1, …, a_n) = A then ∀ i, P(R → A_a_i) ⇒ P(R → A). • We say that P is stable under base change if for any ring maps R → A, R → R' we have P(R → A) ⇒ P(R' → R' ⊗_R A). • We…","statement_latex":"Let $P$ be a property of ring maps.\n\\begin{enumerate}\n\\item We say that $P$ is {\\it local} if the following hold:\n\\begin{enumerate}\n\\item For any ring map $R \\to A$, and any $f \\in R$ we have\n$P(R \\to A) \\Rightarrow P(R_f \\to A_f)$.\n\\item For any rings $R$, $A$, any $f \\in R$, $a\\in A$, and any ring map\n$R_f \\to A$ we have $P(R_f \\to A) \\Rightarrow P(R \\to A_a)$.\n\\item For any ring map $R \\to A$, and $a_i \\in A$ such that\n$(a_1, \\ldots, a_n) = A$ then\n$\\forall i, P(R \\to A_{a_i}) \\Rightarrow P(R \\to A)$.\n\\end{enumerate}\n\\item We say that $P$ is {\\it stable under base change} if for any\nring maps $R \\to A$, $R \\to R'$ we have\n$P(R \\to A) \\Rightarrow P(R' \\to R' \\otimes_R A)$.\n\\item We say that $P$ is {\\it stable under composition} if for any\nring maps $A \\to B$, $B \\to C$ we have\n$P(A \\to B) \\wedge P(B \\to C) \\Rightarrow P(A \\to C)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Types of morphisms defined by properties of ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SR","source_file":"morphisms.tex","source_line":2412,"source_end_line":2433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2412-L2433","statement_sha256":"77585398a5d75acf41c4836632b9f8b455074815b94ba4c5d425cc0945c261a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5415,"rank":5415,"depth":0,"x":2049.036,"y":711.537,"cluster":"scheme-morphisms"},{"id":"stacks:01SS","tag":"01SS","title":"Types of morphisms defined by properties of ring maps · Definition 01SS","summary":"Let P be a property of ring maps. Let f : X → S be a morphism of schemes. We say f is locally of type P if for any x ∈ X there exists an affine open neighbourhood U of x in X which maps into an affine open V ⊂ S such that the induced ring map O_S(V) → O_X(U) has property P.","statement_latex":"Let $P$ be a property of ring maps.\nLet $f : X \\to S$ be a morphism of schemes.\nWe say $f$ is {\\it locally of type $P$} if for any $x \\in X$\nthere exists an affine open neighbourhood $U$ of $x$\nin $X$ which maps into an affine open $V \\subset S$ such that\nthe induced ring map $\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$\nhas property $P$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Types of morphisms defined by properties of ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SS","source_file":"morphisms.tex","source_line":2435,"source_end_line":2444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2435-L2444","statement_sha256":"6c0d1644c40c2cce67f466e0bb6c0d90d5be5879b78667038e7ba93f151103c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5416,"rank":5416,"depth":0,"x":1990.355,"y":667.469,"cluster":"scheme-morphisms"},{"id":"stacks:01ST","tag":"01ST","title":"Types of morphisms defined by properties of ring maps · Lemma 01ST","summary":"Let f : X → S be a morphism of schemes. Let P be a property of ring maps. Let U be an affine open of X, and V an affine open of S such that f(U) ⊂ V. If f is locally of type P and P is local, then P(O_S(V) → O_X(U)) holds.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $P$ be a property of ring maps.\nLet $U$ be an affine open of $X$,\nand $V$ an affine open of $S$ such that\n$f(U) \\subset V$.\nIf $f$ is locally of type $P$ and $P$ is local,\nthen $P(\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U))$ holds.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Types of morphisms defined by properties of ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ST","source_file":"morphisms.tex","source_line":2453,"source_end_line":2462,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2453-L2462","statement_sha256":"f99653a7364c03dbc0d689956e99eb8bbb10cef248441402d0c686ab416dbe46","origin":"The Stacks Project","memory_eligible":false,"source_rank":5417,"rank":5417,"depth":1,"x":2069.553,"y":666.663,"cluster":"scheme-morphisms"},{"id":"stacks:01SU","tag":"01SU","title":"Types of morphisms defined by properties of ring maps · Lemma 01SU","summary":"Let P be a local property of ring maps. Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is locally of type P. • For every affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V we have P(O_S(V) → O_X(U)). • There exists an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is locally of type P. • There exists an affine open covering S = ⋃_j ∈ J V_j and affine open…","statement_latex":"Let $P$ be a local property of ring maps.\nLet $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is locally of type $P$.\n\\item For every affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ we have $P(\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U))$.\n\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis locally of type $P$.\n\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat $P(\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i))$ holds, for all\n$j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is locally of type $P$ then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is locally of type $P$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Types of morphisms defined by properties of ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SU","source_file":"morphisms.tex","source_line":2494,"source_end_line":2515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2494-L2515","statement_sha256":"559c2104fc2ffaea93bc03784faab43c85b03b8da3d7721853edf805ad1b3826","origin":"The Stacks Project","memory_eligible":false,"source_rank":5418,"rank":5418,"depth":2,"x":2011.447,"y":712.475,"cluster":"scheme-morphisms"},{"id":"stacks:01SV","tag":"01SV","title":"Types of morphisms defined by properties of ring maps · Lemma 01SV","summary":"Let P be a property of ring maps. Assume P is local and stable under composition. The composition of morphisms locally of type P is locally of type P.","statement_latex":"Let $P$ be a property of ring maps.\nAssume $P$ is local and stable under composition.\nThe composition of morphisms locally of type $P$ is\nlocally of type $P$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Types of morphisms defined by properties of ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SV","source_file":"morphisms.tex","source_line":2521,"source_end_line":2527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2521-L2527","statement_sha256":"70e538a1f82dcf0509f2885e2650c4aa2338b54e6a42e406d23b079589db5935","origin":"The Stacks Project","memory_eligible":false,"source_rank":5419,"rank":5419,"depth":3,"x":2017.491,"y":645.319,"cluster":"scheme-morphisms"},{"id":"stacks:01SW","tag":"01SW","title":"Types of morphisms defined by properties of ring maps · Lemma 01SW","summary":"Let P be a property of ring maps. Assume P is local and stable under base change. The base change of a morphism locally of type P is locally of type P.","statement_latex":"Let $P$ be a property of ring maps.\nAssume $P$ is local and stable under base change.\nThe base change of a morphism locally of type $P$\nis locally of type $P$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Types of morphisms defined by properties of ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SW","source_file":"morphisms.tex","source_line":2541,"source_end_line":2547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2541-L2547","statement_sha256":"d3a316af06e322c8ef2bb13cf519d2c3ef28346cbca16378a4e0df14902a49d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5420,"rank":5420,"depth":3,"x":2067.335,"y":698.585,"cluster":"scheme-morphisms"},{"id":"stacks:01SX","tag":"01SX","title":"Types of morphisms defined by properties of ring maps · Lemma 01SX","summary":"The following properties of a ring map R → A are local. • (Isomorphism on local rings.) For every prime q of A lying over p ⊂ R the ring map R → A induces an isomorphism R_ p → A_ q. • (Open immersion.) For every prime q of A there exists an f ∈ R, φ(f) not ∈ q such that the ring map φ : R → A induces an isomorphism R_f → A_f. • (Reduced fibres.) For every prime p of R the fibre ring A ⊗_R kappa( p) is reduced. • (Fibres of dimension at most n.) For every prime p of R the…","statement_latex":"The following properties of a ring map $R \\to A$ are local.\n\\begin{enumerate}\n\\item (Isomorphism on local rings.)\nFor every prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p \\subset R$\nthe ring map $R \\to A$ induces an isomorphism\n$R_{\\mathfrak p} \\to A_{\\mathfrak q}$.\n\\item (Open immersion.)\nFor every prime $\\mathfrak q$ of $A$ there exists an $f \\in R$,\n$\\varphi(f) \\not \\in \\mathfrak q$ such that the ring map $\\varphi : R \\to A$\ninduces an isomorphism $R_f \\to A_f$.\n\\item (Reduced fibres.)\nFor every prime $\\mathfrak p$ of $R$ the fibre ring\n$A \\otimes_R \\kappa(\\mathfrak p)$ is reduced.\n\\item (Fibres of dimension at most $n$.)\nFor every prime $\\mathfrak p$ of $R$ the fibre ring\n$A \\otimes_R \\kappa(\\mathfrak p)$ has Krull dimension at most $n$.\n\\item (Locally Noetherian on the target.)\nThe ring map $R \\to A$ has the property that $A$ is Noetherian.\n\\item Add more here as needed\\footnote{But only those properties\nthat are not already dealt with separately elsewhere.}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Types of morphisms defined by properties of ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SX","source_file":"morphisms.tex","source_line":2568,"source_end_line":2591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2568-L2591","statement_sha256":"26f16207147064cbd99b4f396a1fa58f7a03a493e80c4af5e3ab3f5de73c954f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5421,"rank":5421,"depth":0,"x":1987.274,"y":687.524,"cluster":"scheme-morphisms"},{"id":"stacks:01SY","tag":"01SY","title":"Types of morphisms defined by properties of ring maps · Lemma 01SY","summary":"The following properties of ring maps are stable under base change. • (Isomorphism on local rings.) For every prime q of A lying over p ⊂ R the ring map R → A induces an isomorphism R_ p → A_ q. • (Open immersion.) For every prime q of A there exists an f ∈ R, φ(f) not ∈ q such that the ring map φ : R → A induces an isomorphism R_f → A_f. • Add more here as needed.","statement_latex":"The following properties of ring maps are stable under base change.\n\\begin{enumerate}\n\\item (Isomorphism on local rings.)\nFor every prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p \\subset R$\nthe ring map $R \\to A$ induces an isomorphism\n$R_{\\mathfrak p} \\to A_{\\mathfrak q}$.\n\\item (Open immersion.)\nFor every prime $\\mathfrak q$ of $A$ there exists an $f \\in R$,\n$\\varphi(f) \\not \\in \\mathfrak q$ such that the ring map $\\varphi : R \\to A$\ninduces an isomorphism $R_f \\to A_f$.\n\\item Add more here as needed\\footnote{But only those properties\nthat are not already dealt with separately elsewhere.}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Types of morphisms defined by properties of ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SY","source_file":"morphisms.tex","source_line":2597,"source_end_line":2612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2597-L2612","statement_sha256":"6cd58974034af51d17e9c402c49755b63a33bc6f2931c829e5debe6b2d81eefd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5422,"rank":5422,"depth":0,"x":2055.603,"y":650.035,"cluster":"scheme-morphisms"},{"id":"stacks:01SZ","tag":"01SZ","title":"Types of morphisms defined by properties of ring maps · Lemma 01SZ","summary":"The following properties of ring maps are stable under composition. • (Isomorphism on local rings.) For every prime q of A lying over p ⊂ R the ring map R → A induces an isomorphism R_ p → A_ q. • (Open immersion.) For every prime q of A there exists an f ∈ R, φ(f) not ∈ q such that the ring map φ : R → A induces an isomorphism R_f → A_f. • (Locally Noetherian on the target.) The ring map R → A has the property that A is Noetherian. • Add more here as needed.","statement_latex":"The following properties of ring maps are stable under composition.\n\\begin{enumerate}\n\\item (Isomorphism on local rings.)\nFor every prime $\\mathfrak q$ of $A$ lying over $\\mathfrak p \\subset R$\nthe ring map $R \\to A$ induces an isomorphism\n$R_{\\mathfrak p} \\to A_{\\mathfrak q}$.\n\\item (Open immersion.)\nFor every prime $\\mathfrak q$ of $A$ there exists an $f \\in R$,\n$\\varphi(f) \\not \\in \\mathfrak q$ such that the ring map $\\varphi : R \\to A$\ninduces an isomorphism $R_f \\to A_f$.\n\\item (Locally Noetherian on the target.)\nThe ring map $R \\to A$ has the property that $A$ is Noetherian.\n\\item Add more here as needed\\footnote{But only those properties\nthat are not already dealt with separately elsewhere.}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Types of morphisms defined by properties of ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01SZ","source_file":"morphisms.tex","source_line":2618,"source_end_line":2635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2618-L2635","statement_sha256":"5a10e8ccd7a6c5c39ed52c29a44b2813e7654ab0258374ce334574b3d278b68d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5423,"rank":5423,"depth":0,"x":2035.248,"y":716.833,"cluster":"scheme-morphisms"},{"id":"stacks:01T1","tag":"01T1","title":"Morphisms of finite type · Definition 01T1","summary":"Let f : X → S be a morphism of schemes. • We say that f is of finite type at x ∈ X if there exists an affine open neighbourhood Spec(A) = U ⊂ X of x and an affine open Spec(R) = V ⊂ S with f(U) ⊂ V such that the induced ring map R → A is of finite type. • We say that f is locally of finite type if it is of finite type at every point of X. • We say that f is of finite type if it is locally of finite type and quasi-compact.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say that $f$ is of {\\it finite type at $x \\in X$} if\nthere exists an affine open neighbourhood $\\Spec(A) = U \\subset X$\nof $x$ and an affine open $\\Spec(R) = V \\subset S$\nwith $f(U) \\subset V$ such that the induced ring map\n$R \\to A$ is of finite type.\n\\item We say that $f$ is {\\it locally of finite type} if it is\nof finite type at every point of $X$.\n\\item We say that $f$ is of {\\it finite type} if it is locally of\nfinite type and quasi-compact.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01T1","source_file":"morphisms.tex","source_line":2656,"source_end_line":2670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2656-L2670","statement_sha256":"92c9378fb287d2047817efd320d66cca6b6539a599a900a3de204ce9a96f9bd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5424,"rank":5424,"depth":0,"x":1996.32,"y":655.68,"cluster":"scheme-morphisms"},{"id":"stacks:01T2","tag":"01T2","title":"Morphisms of finite type · Lemma 01T2","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is locally of finite type. • For all affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the ring map O_S(V) → O_X(U) is of finite type. • There exist an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is locally of finite type. • There exist an affine open covering S = ⋃_j ∈ J V_j and affine open coverings f^-1(V_j)…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is locally of finite type.\n\\item For all affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is of finite type.\n\\item There exist an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis locally of finite type.\n\\item There exist an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is\nof finite type, for all $j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is locally of finite type then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is locally of finite type.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01T2","source_file":"morphisms.tex","source_line":2672,"source_end_line":2693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2672-L2693","statement_sha256":"fe8b7d4110ea1e18e0aff8ac5f6099fd5ef8ef9fc71c270007060154b59a3385","origin":"The Stacks Project","memory_eligible":false,"source_rank":5425,"rank":5425,"depth":4,"x":2074.642,"y":678.816,"cluster":"scheme-morphisms"},{"id":"stacks:01T3","tag":"01T3","title":"Morphisms of finite type · Lemma 01T3","summary":"The composition of two morphisms which are locally of finite type is locally of finite type. The same is true for morphisms of finite type.","statement_latex":"The composition of two morphisms which are locally of finite type is\nlocally of finite type. The same is true for morphisms of finite type.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01T3","source_file":"morphisms.tex","source_line":2710,"source_end_line":2714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2710-L2714","statement_sha256":"4e264867480cd92391a10c79abf49103a89b03ea1d9c26dfee9e19c4335be329","origin":"The Stacks Project","memory_eligible":false,"source_rank":5426,"rank":5426,"depth":5,"x":1997.857,"y":706.349,"cluster":"scheme-morphisms"},{"id":"stacks:01T4","tag":"01T4","title":"Morphisms of finite type · Lemma 01T4","summary":"The base change of a morphism which is locally of finite type is locally of finite type. The same is true for morphisms of finite type.","statement_latex":"The base change of a morphism which is locally of finite type\nis locally of finite type. The same is true for morphisms of\nfinite type.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01T4","source_file":"morphisms.tex","source_line":2731,"source_end_line":2736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2731-L2736","statement_sha256":"53b6dd437813acb98f734c0991be948d007c5d5e7a9ae561fc3e17b3d213d9d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5427,"rank":5427,"depth":5,"x":2032.526,"y":642.125,"cluster":"scheme-morphisms"},{"id":"stacks:01T5","tag":"01T5","title":"Morphisms of finite type · Lemma 01T5","summary":"A closed immersion is of finite type. An immersion is locally of finite type.","statement_latex":"A closed immersion is of finite type.\nAn immersion is locally of finite type.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01T5","source_file":"morphisms.tex","source_line":2753,"source_end_line":2757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2753-L2757","statement_sha256":"923b8cdff896d77564bc62a5c024d622e11b48b5376b18dde3416fe85be243fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":5428,"rank":5428,"depth":0,"x":2058.751,"y":709.521,"cluster":"scheme-morphisms"},{"id":"stacks:01T6","tag":"01T6","title":"Morphisms of finite type · Lemma 01T6","summary":"Let f : X → S be a morphism. If S is (locally) Noetherian and f (locally) of finite type then X is (locally) Noetherian.","statement_latex":"Let $f : X \\to S$ be a morphism.\nIf $S$ is (locally) Noetherian and $f$ (locally) of finite type\nthen $X$ is (locally) Noetherian.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01T6","source_file":"morphisms.tex","source_line":2764,"source_end_line":2769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2764-L2769","statement_sha256":"e2b8b71c6234284630c4c0979fd0ae31b1a3ab27e60bd84cfd4a66882bacf471","origin":"The Stacks Project","memory_eligible":false,"source_rank":5429,"rank":5429,"depth":1,"x":1984.818,"y":674.516,"cluster":"scheme-morphisms"},{"id":"stacks:01T7","tag":"01T7","title":"Morphisms of finite type · Lemma 01T7","summary":"Let f : X → S be locally of finite type with S locally Noetherian. Then f is quasi-separated.","statement_latex":"Let $f : X \\to S$ be locally of finite type with $S$ locally Noetherian.\nThen $f$ is quasi-separated.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01T7","source_file":"morphisms.tex","source_line":2779,"source_end_line":2783,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2779-L2783","statement_sha256":"d6f16e5452bd95b1f0e1083cab62cecfedc15335c39035987189ce74382a343e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5430,"rank":5430,"depth":16,"x":2067.927,"y":658.292,"cluster":"scheme-morphisms"},{"id":"stacks:01T8","tag":"01T8","title":"Morphisms of finite type · Lemma 01T8","summary":"Let X → Y be a morphism of schemes over a base scheme S. If X is locally of finite type over S, then X → Y is locally of finite type.","statement_latex":"Let $X \\to Y$ be a morphism of schemes over a base scheme $S$.\nIf $X$ is locally of finite type over $S$, then $X \\to Y$\nis locally of finite type.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01T8","source_file":"morphisms.tex","source_line":2793,"source_end_line":2798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2793-L2798","statement_sha256":"c0b38eb0e15769b7ad7e354f1fc9e34fb04f9ce2608db809b4df650ff1eccdd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5431,"rank":5431,"depth":5,"x":2019.435,"y":717.725,"cluster":"scheme-morphisms"},{"id":"stacks:01TA","tag":"01TA","title":"Points of finite type and Jacobson schemes · Lemma 01TA","summary":"Let S be a scheme. Let k be a field. Let f : Spec(k) → S be a morphism. The following are equivalent: • The morphism f is of finite type. • The morphism f is locally of finite type. • There exists an affine open U = Spec(R) of S such that f corresponds to a finite ring map R → k. • There exists an affine open U = Spec(R) of S such that the image of f consists of a closed point u in U and the field extension k/kappa(u) is finite.","statement_latex":"Let $S$ be a scheme. Let $k$ be a field.\nLet $f : \\Spec(k) \\to S$ be a morphism.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is of finite type.\n\\item The morphism $f$ is locally of finite type.\n\\item There exists an affine open $U = \\Spec(R)$ of $S$\nsuch that $f$ corresponds to a finite ring map $R \\to k$.\n\\item There exists an affine open $U = \\Spec(R)$ of $S$\nsuch that the image of $f$ consists of a closed point $u$ in $U$\nand the field extension $k/\\kappa(u)$ is finite.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TA","source_file":"morphisms.tex","source_line":2826,"source_end_line":2840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2826-L2840","statement_sha256":"7d5c372159595f279eb5e00ce156e6f4dab0e5ddd12fff815f969f1b8221ced7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5432,"rank":5432,"depth":8,"x":2007.336,"y":646.012,"cluster":"scheme-morphisms"},{"id":"stacks:02HV","tag":"02HV","title":"Points of finite type and Jacobson schemes · Lemma 02HV","summary":"Let S be a scheme. Let A be an Artinian local ring with residue field kappa. Let f : Spec(A) → S be a morphism of schemes. Then f is of finite type if and only if the composition Spec(kappa) → Spec(A) → S is of finite type.","statement_latex":"Let $S$ be a scheme.\nLet $A$ be an Artinian local ring with residue field $\\kappa$.\nLet $f : \\Spec(A) \\to S$ be a morphism of schemes.\nThen $f$ is of finite type if and only if the composition\n$\\Spec(\\kappa) \\to \\Spec(A) \\to S$ is of finite type.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HV","source_file":"morphisms.tex","source_line":2877,"source_end_line":2884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2877-L2884","statement_sha256":"3800e9b14d15956d2e91df59190596f4d2dbfb588f734c91bee0fddec70158c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5433,"rank":5433,"depth":8,"x":2074.272,"y":692.265,"cluster":"scheme-morphisms"},{"id":"stacks:02J1","tag":"02J1","title":"Points of finite type and Jacobson schemes · Definition 02J1","summary":"Let S be a scheme. Let us say that a point s of S is a finite type point if the canonical morphism Spec(kappa(s)) → S is of finite type. We denote S_ft-pts the set of finite type points of S.","statement_latex":"Let $S$ be a scheme.\nLet us say that a point $s$ of $S$ is a {\\it finite type point}\nif the canonical morphism $\\Spec(\\kappa(s)) \\to S$ is of finite type.\nWe denote $S_{\\text{ft-pts}}$ the set of finite type points of $S$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02J1","source_file":"morphisms.tex","source_line":2902,"source_end_line":2908,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2902-L2908","statement_sha256":"fa71a4e93f8a8e5c38a8d51f159ef3163a20076c713d298820d05dc2bb99f1f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5434,"rank":5434,"depth":0,"x":1987.275,"y":696.159,"cluster":"scheme-morphisms"},{"id":"stacks:02J2","tag":"02J2","title":"Points of finite type and Jacobson schemes · Lemma 02J2","summary":"Let S be a scheme. We have S_ft-pts = ⋃_U ⊂ S open U_0 where U_0 is the set of closed points of U. Here we may let U range over all opens or over all affine opens of S.","statement_latex":"Let $S$ be a scheme. We have\n$$\nS_{\\text{ft-pts}} = \\bigcup\\nolimits_{U \\subset S\\text{ open}} U_0\n$$\nwhere $U_0$ is the set of closed points of $U$.\nHere we may let $U$ range over all opens or over all affine opens of $S$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02J2","source_file":"morphisms.tex","source_line":2913,"source_end_line":2921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2913-L2921","statement_sha256":"5341545bc47ea3f433768f25f18ad5ed8bbdeb1ab02b24e35e8f0e0479cfdc13","origin":"The Stacks Project","memory_eligible":false,"source_rank":5435,"rank":5435,"depth":9,"x":2048.604,"y":643.659,"cluster":"scheme-morphisms"},{"id":"stacks:02J3","tag":"02J3","title":"Points of finite type and Jacobson schemes · Lemma 02J3","summary":"Let f : T → S be a morphism of schemes. If f is locally of finite type, then f(T_ft-pts) ⊂ S_ft-pts.","statement_latex":"Let $f : T \\to S$ be a morphism of schemes.\nIf $f$ is locally of finite type, then\n$f(T_{\\text{ft-pts}}) \\subset S_{\\text{ft-pts}}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02J3","source_file":"morphisms.tex","source_line":2927,"source_end_line":2932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2927-L2932","statement_sha256":"fc7a09787d18cb314160fb526a004e06346d39c1cee9cf03749632a03369f4d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5436,"rank":5436,"depth":9,"x":2045.583,"y":717.54,"cluster":"scheme-morphisms"},{"id":"stacks:06EB","tag":"06EB","title":"Points of finite type and Jacobson schemes · Lemma 06EB","summary":"Let f : T → S be a morphism of schemes. If f is locally of finite type and surjective, then f(T_ft-pts) = S_ft-pts.","statement_latex":"Let $f : T \\to S$ be a morphism of schemes.\nIf $f$ is locally of finite type and surjective, then\n$f(T_{\\text{ft-pts}}) = S_{\\text{ft-pts}}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EB","source_file":"morphisms.tex","source_line":2940,"source_end_line":2945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2940-L2945","statement_sha256":"fb1f26fdcb358e83f13a19ca38736ca6d4b530da42f914e5028fb3ee3bae6a9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5437,"rank":5437,"depth":10,"x":1988.115,"y":661.068,"cluster":"scheme-morphisms"},{"id":"stacks:02J4","tag":"02J4","title":"Points of finite type and Jacobson schemes · Lemma 02J4","summary":"Let S be a scheme. For any locally closed subset T ⊂ S we have T not = ∅ ⇒ T ∩ S_ft-pts not = ∅. In particular, for any closed subset T ⊂ S we see that T ∩ S_ft-pts is dense in T.","statement_latex":"Let $S$ be a scheme.\nFor any locally closed subset $T \\subset S$ we have\n$$\nT \\not = \\emptyset\n\\Rightarrow\nT \\cap S_{\\text{ft-pts}} \\not = \\emptyset.\n$$\nIn particular, for any closed subset $T \\subset S$ we\nsee that $T \\cap S_{\\text{ft-pts}}$ is dense in $T$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02J4","source_file":"morphisms.tex","source_line":2962,"source_end_line":2973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2962-L2973","statement_sha256":"4e64bfe9e56690a19fd1bceec75c8e69047e30e78be1e098b80c9f384d42c8e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5438,"rank":5438,"depth":10,"x":2076.336,"y":670.145,"cluster":"scheme-morphisms"},{"id":"stacks:01TB","tag":"01TB","title":"Points of finite type and Jacobson schemes · Lemma 01TB","summary":"Let S be a scheme. The following are equivalent: • the scheme S is Jacobson, • S_ft-pts is the set of closed points of S, • for all T → S locally of finite type closed points map to closed points, and • for all T → S locally of finite type closed points t ∈ T map to closed points s ∈ S with kappa(s) ⊂ kappa(t) finite.","statement_latex":"Let $S$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item the scheme $S$ is Jacobson,\n\\item $S_{\\text{ft-pts}}$ is the set of closed points of $S$,\n\\item for all $T \\to S$ locally of finite type\nclosed points map to closed points, and\n\\item for all $T \\to S$ locally of finite type\nclosed points $t \\in T$ map to closed points $s \\in S$ with\n$\\kappa(s) \\subset \\kappa(t)$ finite.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TB","source_file":"morphisms.tex","source_line":2996,"source_end_line":3008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L2996-L3008","statement_sha256":"2149a16ac5df94e7a30ccd2fdfb9b8d704055d9425eec2c74c35d8805371ef7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5439,"rank":5439,"depth":11,"x":2003.631,"y":713.721,"cluster":"scheme-morphisms"},{"id":"stacks:02J5","tag":"02J5","title":"Points of finite type and Jacobson schemes · Lemma 02J5","summary":"Let S be a Jacobson scheme. Any scheme locally of finite type over S is Jacobson.","statement_latex":"Let $S$ be a Jacobson scheme.\nAny scheme locally of finite type over $S$ is Jacobson.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02J5","source_file":"morphisms.tex","source_line":3025,"source_end_line":3029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3025-L3029","statement_sha256":"2aac0b48a358022c294a879527f7b995897f030b7406479d4cc076b1652f8e21","origin":"The Stacks Project","memory_eligible":false,"source_rank":5440,"rank":5440,"depth":10,"x":2022.29,"y":639.981,"cluster":"scheme-morphisms"},{"id":"stacks:02J6","tag":"02J6","title":"Points of finite type and Jacobson schemes · Lemma 02J6","summary":"The following types of schemes are Jacobson. • Any scheme locally of finite type over a field. • Any scheme locally of finite type over Z. • Any scheme locally of finite type over a 1-dimensional Noetherian domain with infinitely many primes. • A scheme of the form Spec(R) setminus ( m) where (R, m) is a Noetherian local ring. Also any scheme locally of finite type over it.","statement_latex":"The following types of schemes are Jacobson.\n\\begin{enumerate}\n\\item Any scheme locally of finite type over a field.\n\\item Any scheme locally of finite type over $\\mathbf{Z}$.\n\\item Any scheme locally of finite type over a $1$-dimensional\nNoetherian domain with infinitely many primes.\n\\item A scheme of the form $\\Spec(R) \\setminus \\{\\mathfrak m\\}$\nwhere $(R, \\mathfrak m)$ is a Noetherian local ring.\nAlso any scheme locally of finite type over it.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Points of finite type and Jacobson schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02J6","source_file":"morphisms.tex","source_line":3038,"source_end_line":3050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3038-L3050","statement_sha256":"388f4b76783c3f361d8dc30e6d5b5c23fa991b4b75c290414a22cf253336085b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5441,"rank":5441,"depth":16,"x":2068.047,"y":705.254,"cluster":"scheme-morphisms"},{"id":"stacks:02J8","tag":"02J8","title":"Universally catenary schemes · Definition 02J8","summary":"Let S be a scheme. Assume S is locally Noetherian. We say S is universally catenary if for every morphism X → S locally of finite type the scheme X is catenary.","statement_latex":"Let $S$ be a scheme. Assume $S$ is locally Noetherian.\nWe say $S$ is {\\it universally catenary} if for every\nmorphism $X \\to S$ locally of finite type the scheme $X$ is catenary.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally catenary schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02J8","source_file":"morphisms.tex","source_line":3084,"source_end_line":3089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3084-L3089","statement_sha256":"0ce86b877b113a28670a5de8476381cce586dbb9ffd95a13319f6dddf81e56f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5442,"rank":5442,"depth":0,"x":1981.408,"y":682.981,"cluster":"scheme-morphisms"},{"id":"stacks:02J9","tag":"02J9","title":"Universally catenary schemes · Lemma 02J9","summary":"Let S be a locally Noetherian scheme. The following are equivalent • S is universally catenary, • there exists an open covering of S all of whose members are universally catenary schemes, • for every affine open Spec(R) = U ⊂ S the ring R is universally catenary, and • there exists an affine open covering S = ⋃ U_i such that each U_i is the spectrum of a universally catenary ring. Moreover, in this case any scheme locally of finite type over S is universally catenary as well.","statement_latex":"Let $S$ be a locally Noetherian scheme. The following are equivalent\n\\begin{enumerate}\n\\item $S$ is universally catenary,\n\\item there exists an open covering of $S$ all of whose members are\nuniversally catenary schemes,\n\\item for every affine open $\\Spec(R) = U \\subset S$ the ring\n$R$ is universally catenary, and\n\\item there exists an affine open covering $S = \\bigcup U_i$ such\nthat each $U_i$ is the spectrum of a universally catenary ring.\n\\end{enumerate}\nMoreover, in this case any scheme locally of finite type over $S$\nis universally catenary as well.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally catenary schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02J9","source_file":"morphisms.tex","source_line":3118,"source_end_line":3132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3118-L3132","statement_sha256":"564beca67bfc54b40d02b009d77830755b62927fb9ce1fcf4db7daad18836252","origin":"The Stacks Project","memory_eligible":false,"source_rank":5443,"rank":5443,"depth":6,"x":2063.59,"y":650.091,"cluster":"scheme-morphisms"},{"id":"stacks:02JA","tag":"02JA","title":"Universally catenary schemes · Lemma 02JA","summary":"Let S be a locally Noetherian scheme. The following are equivalent: • S is universally catenary, and • all local rings O_S, s of S are universally catenary.","statement_latex":"Let $S$ be a locally Noetherian scheme.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $S$ is universally catenary, and\n\\item all local rings $\\mathcal{O}_{S, s}$ of $S$ are universally catenary.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally catenary schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JA","source_file":"morphisms.tex","source_line":3163,"source_end_line":3171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3163-L3171","statement_sha256":"b9055e60fc2e920a4188a419e90f49de52093a8e9f82d59fd296f11124c9f950","origin":"The Stacks Project","memory_eligible":false,"source_rank":5444,"rank":5444,"depth":7,"x":2029.28,"y":721.304,"cluster":"scheme-morphisms"},{"id":"stacks:0G42","tag":"0G42","title":"Universally catenary schemes · Lemma 0G42","summary":"Let S be a locally Noetherian scheme. Then S is universally catenary if and only if the irreducible components of S are universally catenary.","statement_latex":"Let $S$ be a locally Noetherian scheme. Then $S$ is universally catenary\nif and only if the irreducible components of $S$ are universally catenary.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally catenary schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G42","source_file":"morphisms.tex","source_line":3194,"source_end_line":3198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3194-L3198","statement_sha256":"5e3d15f87a7a3063af580b093c53eec3cde39f835526ca24d02730e99441dd17","origin":"The Stacks Project","memory_eligible":false,"source_rank":5445,"rank":5445,"depth":1,"x":1997.166,"y":648.993,"cluster":"scheme-morphisms"},{"id":"stacks:02JB","tag":"02JB","title":"Universally catenary schemes · Lemma 02JB","summary":"The following types of schemes are universally catenary. • Any scheme locally of finite type over a field. • Any scheme locally of finite type over a Cohen-Macaulay scheme. • Any scheme locally of finite type over Z. • Any scheme locally of finite type over a 1-dimensional Noetherian domain. • And so on.","statement_latex":"The following types of schemes are universally catenary.\n\\begin{enumerate}\n\\item Any scheme locally of finite type over a field.\n\\item Any scheme locally of finite type over a Cohen-Macaulay scheme.\n\\item Any scheme locally of finite type over $\\mathbf{Z}$.\n\\item Any scheme locally of finite type over a $1$-dimensional\nNoetherian domain.\n\\item And so on.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally catenary schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JB","source_file":"morphisms.tex","source_line":3205,"source_end_line":3216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3205-L3216","statement_sha256":"b489cf20547db06a2b2f06bf2b5deed12c5b67d410312edba80387687d595392","origin":"The Stacks Project","memory_eligible":false,"source_rank":5446,"rank":5446,"depth":19,"x":2079.369,"y":684.252,"cluster":"scheme-morphisms"},{"id":"stacks:035A","tag":"035A","title":"Nagata schemes, reprise · Lemma 035A","summary":"Let f : X → S be a morphism. If S is Nagata and f locally of finite type then X is Nagata. If S is universally Japanese and f locally of finite type then X is universally Japanese.","statement_latex":"Let $f : X \\to S$ be a morphism.\nIf $S$ is Nagata and $f$ locally of finite type then $X$ is Nagata.\nIf $S$ is universally Japanese\nand $f$ locally of finite type then $X$ is universally Japanese.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Nagata schemes, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035A","source_file":"morphisms.tex","source_line":3256,"source_end_line":3262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3256-L3262","statement_sha256":"a37cb2ca422412217ae58dd5bccb1cd6140f1e729e33ca532b8e7bfa0cb854eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5447,"rank":5447,"depth":33,"x":1989.998,"y":704.99,"cluster":"scheme-morphisms"},{"id":"stacks:035B","tag":"035B","title":"Nagata schemes, reprise · Lemma 035B","summary":"The following types of schemes are Nagata. • Any scheme locally of finite type over a field. • Any scheme locally of finite type over a Noetherian complete local ring. • Any scheme locally of finite type over Z. • Any scheme locally of finite type over a Dedekind ring of characteristic zero. • And so on.","statement_latex":"The following types of schemes are Nagata.\n\\begin{enumerate}\n\\item Any scheme locally of finite type over a field.\n\\item Any scheme locally of finite type over a Noetherian complete local ring.\n\\item Any scheme locally of finite type over $\\mathbf{Z}$.\n\\item Any scheme locally of finite type over a Dedekind ring of\ncharacteristic zero.\n\\item And so on.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Nagata schemes, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035B","source_file":"morphisms.tex","source_line":3271,"source_end_line":3282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3271-L3282","statement_sha256":"82b39e6cb1e84052313d5ea5788151548d07970b9f85b98a430a2ea2b5075b62","origin":"The Stacks Project","memory_eligible":false,"source_rank":5448,"rank":5448,"depth":34,"x":2039.443,"y":638.692,"cluster":"scheme-morphisms"},{"id":"stacks:07R3","tag":"07R3","title":"The singular locus, reprise · Definition 07R3","summary":"Let X be a locally Noetherian scheme. We say X is J-2 if for every morphism Y → X which is locally of finite type the regular locus Reg(Y) is open in Y.","statement_latex":"Let $X$ be a locally Noetherian scheme. We say $X$ is {\\it J-2}\nif for every morphism $Y \\to X$ which is locally of finite type\nthe regular locus $\\text{Reg}(Y)$ is open in $Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"The singular locus, reprise","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07R3","source_file":"morphisms.tex","source_line":3298,"source_end_line":3303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3298-L3303","statement_sha256":"ebd7347ffa3e1280fc25454142a76cc8e438be08357f00399a627f3db3d9775d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5449,"rank":5449,"depth":0,"x":2056.372,"y":715.976,"cluster":"scheme-morphisms"},{"id":"stacks:07R4","tag":"07R4","title":"The singular locus, reprise · Lemma 07R4","summary":"Let X be a locally Noetherian scheme. The following are equivalent • X is J-2, • there exists an open covering of X all of whose members are J-2 schemes, • for every affine open Spec(R) = U ⊂ X the ring R is J-2, and • there exists an affine open covering S = ⋃ U_i such that each O(U_i) is J-2 for all i. Moreover, in this case any scheme locally of finite type over X is J-2 as well.","statement_latex":"Let $X$ be a locally Noetherian scheme. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is J-2,\n\\item there exists an open covering of $X$ all of whose members are\nJ-2 schemes,\n\\item for every affine open $\\Spec(R) = U \\subset X$ the ring\n$R$ is J-2, and\n\\item there exists an affine open covering $S = \\bigcup U_i$ such\nthat each $\\mathcal{O}(U_i)$ is J-2 for all $i$.\n\\end{enumerate}\nMoreover, in this case any scheme locally of finite type over $X$\nis J-2 as well.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"The singular locus, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07R4","source_file":"morphisms.tex","source_line":3309,"source_end_line":3323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3309-L3323","statement_sha256":"47094a364b4347a72e5f73b0db26e1eaeef4432835e076a85c6ee990a9634059","origin":"The Stacks Project","memory_eligible":false,"source_rank":5450,"rank":5450,"depth":6,"x":1981.411,"y":668.385,"cluster":"scheme-morphisms"},{"id":"stacks:07R5","tag":"07R5","title":"The singular locus, reprise · Lemma 07R5","summary":"The following types of schemes are J-2. • Any scheme locally of finite type over a field. • Any scheme locally of finite type over a Noetherian complete local ring. • Any scheme locally of finite type over Z. • Any scheme locally of finite type over a Noetherian local ring of dimension 1. • Any scheme locally of finite type over a Nagata ring of dimension 1. • Any scheme locally of finite type over a Dedekind ring of characteristic zero. • And so on.","statement_latex":"The following types of schemes are J-2.\n\\begin{enumerate}\n\\item Any scheme locally of finite type over a field.\n\\item Any scheme locally of finite type over a Noetherian complete local ring.\n\\item Any scheme locally of finite type over $\\mathbf{Z}$.\n\\item Any scheme locally of finite type over a Noetherian local ring\nof dimension $1$.\n\\item Any scheme locally of finite type over a Nagata ring of dimension $1$.\n\\item Any scheme locally of finite type over a Dedekind ring of\ncharacteristic zero.\n\\item And so on.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"The singular locus, reprise","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07R5","source_file":"morphisms.tex","source_line":3356,"source_end_line":3370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3356-L3370","statement_sha256":"2e51b058ab7d9a83170c0005695bfba63ff0d44192947d4c009f7778be82b328","origin":"The Stacks Project","memory_eligible":false,"source_rank":5451,"rank":5451,"depth":42,"x":2075.365,"y":660.912,"cluster":"scheme-morphisms"},{"id":"stacks:0HA7","tag":"0HA7","title":"Excellent schemes · Definition 0HA7","summary":"Let X be a scheme. • We say X is quasi-excellent if for every x ∈ X there exists an affine open neighbourhood x ∈ U ⊂ X such that the ring O_X(U) is quasi-excellent (see More on Algebra, Definition [Tag 07QT]). • We say X is excellent if for every x ∈ X there exists an affine open neighbourhood x ∈ U ⊂ X such that the ring O_X(U) is excellent (see More on Algebra, Definition [Tag 07QT]).","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item We say $X$ is {\\it quasi-excellent} if for every $x \\in X$ there exists\nan affine open neighbourhood $x \\in U \\subset X$ such that the ring\n$\\mathcal{O}_X(U)$ is quasi-excellent (see\nMore on Algebra, Definition \\ref{more-algebra-definition-excellent}).\n\\item We say $X$ is {\\it excellent} if for every $x \\in X$ there exists an\naffine open neighbourhood $x \\in U \\subset X$ such that the ring\n$\\mathcal{O}_X(U)$ is excellent (see\nMore on Algebra, Definition \\ref{more-algebra-definition-excellent}).\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Excellent schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HA7","source_file":"morphisms.tex","source_line":3388,"source_end_line":3401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3388-L3401","statement_sha256":"8f54b3be6444e545923f46c33c6c96446b199ee411cbbed18e487bc17dfc7fa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5452,"rank":5452,"depth":1,"x":2011.821,"y":719.988,"cluster":"scheme-morphisms"},{"id":"stacks:0HA8","tag":"0HA8","title":"Excellent schemes · Lemma 0HA8","summary":"Let X be a scheme. The following are equivalent • X is quasi-excellent, and • X is a G-scheme and J-2.","statement_latex":"Let $X$ be a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is quasi-excellent, and\n\\item $X$ is a G-scheme and J-2.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Excellent schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HA8","source_file":"morphisms.tex","source_line":3406,"source_end_line":3413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3406-L3413","statement_sha256":"5917b0e6be0bc1b281c62930722b17c1b51d92ed17c390fc55bef3ef8650a73c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5453,"rank":5453,"depth":7,"x":2011.168,"y":640.029,"cluster":"scheme-morphisms"},{"id":"stacks:0HA9","tag":"0HA9","title":"Excellent schemes · Lemma 0HA9","summary":"A quasi-excellent scheme is Nagata.","statement_latex":"A quasi-excellent scheme is Nagata.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Excellent schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HA9","source_file":"morphisms.tex","source_line":3430,"source_end_line":3433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3430-L3433","statement_sha256":"e672a50dcc0e23613393f803981ca7faf31f6f42a69ce5b684d3463c9bf0b27c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5454,"rank":5454,"depth":51,"x":2076.225,"y":698.868,"cluster":"scheme-morphisms"},{"id":"stacks:0HAA","tag":"0HAA","title":"Excellent schemes · Lemma 0HAA","summary":"Let X be a scheme. The following are equivalent • X is excellent, and • X is quasi-excellent and universally catenary.","statement_latex":"Let $X$ be a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is excellent, and\n\\item $X$ is quasi-excellent and universally catenary.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Excellent schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAA","source_file":"morphisms.tex","source_line":3439,"source_end_line":3446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3439-L3446","statement_sha256":"531e093295bdb59b0978082d4a8c6d6a6d6c50e6b1917d979d3718ff57ab641f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5455,"rank":5455,"depth":7,"x":1980.535,"y":692.368,"cluster":"scheme-morphisms"},{"id":"stacks:0HAB","tag":"0HAB","title":"Excellent schemes · Lemma 0HAB","summary":"Let X be a scheme. The following are equivalent: • The scheme X is (quasi-)excellent. • For every affine open U ⊂ X the ring O_X(U) is (quasi-)excellent. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is (quasi-)excellent. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is (quasi-)excellent. Moreover, if X is (quasi-)excellent then every open subscheme is (quasi-)excellent.","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is (quasi-)excellent.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis (quasi-)excellent.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ is (quasi-)excellent.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is (quasi-)excellent.\n\\end{enumerate}\nMoreover, if $X$ is (quasi-)excellent then every open subscheme\nis (quasi-)excellent.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Excellent schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAB","source_file":"morphisms.tex","source_line":3461,"source_end_line":3475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3461-L3475","statement_sha256":"7003b5bc9a3a2e3954f4d5483605ff96af742934260a0d72657e4ce40e8c9d2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5456,"rank":5456,"depth":46,"x":2056.639,"y":642.658,"cluster":"scheme-morphisms"},{"id":"stacks:0HAC","tag":"0HAC","title":"Excellent schemes · Lemma 0HAC","summary":"Let f : X → S be a morphism. If S is (quasi-)excellent and f locally of finite type then X is (quasi-)excellent.","statement_latex":"Let $f : X \\to S$ be a morphism.\nIf $S$ is (quasi-)excellent and $f$ locally of finite type then\n$X$ is (quasi-)excellent.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Excellent schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAC","source_file":"morphisms.tex","source_line":3487,"source_end_line":3492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3487-L3492","statement_sha256":"060ac038ee37cb34a2e08dd10689a313b1758cf5d1c8a617cc85b7f8227b2934","origin":"The Stacks Project","memory_eligible":false,"source_rank":5457,"rank":5457,"depth":50,"x":2040.429,"y":722.826,"cluster":"scheme-morphisms"},{"id":"stacks:0HAD","tag":"0HAD","title":"Excellent schemes · Lemma 0HAD","summary":"The following types of schemes are excellent. • Any scheme locally of finite type over a field. • Any scheme locally of finite type over a Noetherian complete local ring. • Any scheme locally of finite type over Z. • Any scheme locally of finite type over a Dedekind ring of characteristic zero.","statement_latex":"The following types of schemes are excellent.\n\\begin{enumerate}\n\\item Any scheme locally of finite type over a field.\n\\item Any scheme locally of finite type over a Noetherian complete local ring.\n\\item Any scheme locally of finite type over $\\mathbf{Z}$.\n\\item Any scheme locally of finite type over a Dedekind ring of\ncharacteristic zero.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Excellent schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAD","source_file":"morphisms.tex","source_line":3498,"source_end_line":3508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3498-L3508","statement_sha256":"f4d1e1589452be6edb2010da9568006e788bf65eb841844e0cd6f77861080874","origin":"The Stacks Project","memory_eligible":false,"source_rank":5458,"rank":5458,"depth":51,"x":1987.698,"y":654.233,"cluster":"scheme-morphisms"},{"id":"stacks:01TD","tag":"01TD","title":"Quasi-finite morphisms · Definition 01TD","summary":"[EGA] Let f : X → S be a morphism of schemes. • We say that f is quasi-finite at a point x ∈ X if there exist an affine neighbourhood Spec(A) = U ⊂ X of x and an affine open Spec(R) = V ⊂ S such that f(U) ⊂ V, the ring map R → A is of finite type, and R → A is quasi-finite at the prime of A corresponding to x (see above). • We say f is locally quasi-finite if f is quasi-finite at every point x of X. • We say that f is quasi-finite if f is of finite type and every point x…","statement_latex":"\\begin{reference}\n\\cite[II Definition 6.2.3]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say that $f$ is {\\it quasi-finite at a point $x \\in X$}\nif there exist an affine neighbourhood $\\Spec(A) = U \\subset X$\nof $x$ and an affine open $\\Spec(R) = V \\subset S$ such that\n$f(U) \\subset V$, the ring map $R \\to A$ is of finite type,\nand $R \\to A$ is quasi-finite at the prime of $A$ corresponding to $x$\n(see above).\n\\item We say $f$ is {\\it locally quasi-finite} if $f$ is\nquasi-finite at every point $x$ of $X$.\n\\item We say that $f$ is {\\it quasi-finite} if $f$ is of finite type\nand every point $x$ is an isolated point of its fibre.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TD","source_file":"morphisms.tex","source_line":3537,"source_end_line":3555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3537-L3555","statement_sha256":"fbfc3bb980b5d436b752bb0596ff663554ab4e16b8c502095c736d6e20b16cdb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5459,"rank":5459,"depth":0,"x":2082.124,"y":674.977,"cluster":"scheme-morphisms"},{"id":"stacks:01TE","tag":"01TE","title":"Quasi-finite morphisms · Lemma 01TE","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f(x). If kappa(x)/kappa(s) is an algebraic field extension, then • x is a closed point of its fibre, and • if in addition s is a closed point of S, then x is a closed point of X.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point. Set $s = f(x)$.\nIf $\\kappa(x)/\\kappa(s)$\nis an algebraic field extension, then\n\\begin{enumerate}\n\\item $x$ is a closed point of its fibre, and\n\\item if in addition $s$ is a closed point of $S$, then\n$x$ is a closed point of $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TE","source_file":"morphisms.tex","source_line":3564,"source_end_line":3575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3564-L3575","statement_sha256":"1c400edf567192222b9e0b920ba3c2d26c667e14c03438ff970f5bf88d0f5c30","origin":"The Stacks Project","memory_eligible":false,"source_rank":5460,"rank":5460,"depth":13,"x":1995.466,"y":713.414,"cluster":"scheme-morphisms"},{"id":"stacks:01TF","tag":"01TF","title":"Quasi-finite morphisms · Lemma 01TF","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f(x). Assume f is locally of finite type. Then x is a closed point of its fibre if and only if kappa(x)/kappa(s) is a finite field extension.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point. Set $s = f(x)$.\nAssume $f$ is locally of finite type.\nThen $x$ is a closed point of its fibre\nif and only if $\\kappa(x)/\\kappa(s)$ is\na finite field extension.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TF","source_file":"morphisms.tex","source_line":3596,"source_end_line":3604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3596-L3604","statement_sha256":"fce32d69b2bbd1127f106750083f4d24db77c7294c36418a259bdefd29d21645","origin":"The Stacks Project","memory_eligible":false,"source_rank":5461,"rank":5461,"depth":14,"x":2028.588,"y":635.589,"cluster":"scheme-morphisms"},{"id":"stacks:053M","tag":"053M","title":"Quasi-finite morphisms · Lemma 053M","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let g : S' → S be any morphism. Denote f' : X' → S' the base change. If x' ∈ X' maps to a point x ∈ X which is closed in X_f(x) then x' is closed in X'_f'(x').","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $g : S' \\to S$ be any morphism. Denote $f' : X' \\to S'$ the base change.\nIf $x' \\in X'$ maps to a point $x \\in X$ which is closed in $X_{f(x)}$\nthen $x'$ is closed in $X'_{f'(x')}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053M","source_file":"morphisms.tex","source_line":3632,"source_end_line":3638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3632-L3638","statement_sha256":"a2a526582cd76106988caf0a3ea6bf3254efea6f1d1b6357cc0f34059129abd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5462,"rank":5462,"depth":15,"x":2066.9,"y":712.073,"cluster":"scheme-morphisms"},{"id":"stacks:01TG","tag":"01TG","title":"Quasi-finite morphisms · Lemma 01TG","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f(x). If f is quasi-finite at x, then the residue field extension kappa(x)/kappa(s) is finite.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point. Set $s = f(x)$.\nIf $f$ is quasi-finite at $x$, then the residue field\nextension $\\kappa(x)/\\kappa(s)$ is finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TG","source_file":"morphisms.tex","source_line":3651,"source_end_line":3657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3651-L3657","statement_sha256":"9f431d0e668f9fca0889753e5aff1cb13dded4c8cc957808e140c4d9ce2104fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5463,"rank":5463,"depth":26,"x":1976.79,"y":677.277,"cluster":"scheme-morphisms"},{"id":"stacks:01TH","tag":"01TH","title":"Quasi-finite morphisms · Lemma 01TH","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f(x). Let X_s be the fibre of f at s. Assume f is locally of finite type. The following are equivalent: • The morphism f is quasi-finite at x. • The point x is isolated in X_s. • The point x is closed in X_s and there is no point x' ∈ X_s, x' not = x which specializes to x. • For any pair of affine opens Spec(A) = U ⊂ X, Spec(R) = V ⊂ S with f(U) ⊂ V and x ∈ U corresponding to q ⊂ A the ring map R → A…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point. Set $s = f(x)$.\nLet $X_s$ be the fibre of $f$ at $s$.\nAssume $f$ is locally of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is quasi-finite at $x$.\n\\item The point $x$ is isolated in $X_s$.\n\\item The point $x$ is closed in $X_s$\nand there is no point $x' \\in X_s$, $x' \\not = x$\nwhich specializes to $x$.\n\\item For any pair of affine opens\n$\\Spec(A) = U \\subset X$, $\\Spec(R) = V \\subset S$ with\n$f(U) \\subset V$ and $x \\in U$ corresponding to $\\mathfrak q \\subset A$\nthe ring map $R \\to A$ is quasi-finite at $\\mathfrak q$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TH","source_file":"morphisms.tex","source_line":3663,"source_end_line":3681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3663-L3681","statement_sha256":"cc016489aa684e7532c9e7df674c580f4b84fd830a3609d3108e50b7ed042b03","origin":"The Stacks Project","memory_eligible":false,"source_rank":5464,"rank":5464,"depth":26,"x":2071.587,"y":651.705,"cluster":"scheme-morphisms"},{"id":"stacks:02NG","tag":"02NG","title":"Quasi-finite morphisms · Lemma 02NG","summary":"Let f : X → S be a morphism of schemes. Let s ∈ S. Assume that • f is locally of finite type, and • f^-1((s)) is a finite set. Then X_s is a finite discrete topological space, and f is quasi-finite at each point of X lying over s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $s \\in S$. Assume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type, and\n\\item $f^{-1}(\\{s\\})$ is a finite set.\n\\end{enumerate}\nThen $X_s$ is a finite discrete topological space, and\n$f$ is quasi-finite at each point of $X$ lying over $s$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NG","source_file":"morphisms.tex","source_line":3729,"source_end_line":3739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3729-L3739","statement_sha256":"a557a44e078d96ce757479844183d371d67625dd71f39e1c9b9f09ab0f23691b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5465,"rank":5465,"depth":27,"x":2022.057,"y":724.633,"cluster":"scheme-morphisms"},{"id":"stacks:06RT","tag":"06RT","title":"Quasi-finite morphisms · Lemma 06RT","summary":"Finite type morphisms with discrete fibers are quasi-finite. Let f : X → S be a morphism of schemes. Assume f is locally of finite type. Then the following are equivalent • f is locally quasi-finite, • for every s ∈ S the fibre X_s is a discrete topological space, and • for every morphism Spec(k) → S where k is a field the base change X_k has an underlying discrete topological space.","statement_latex":"\\begin{slogan}\nFinite type morphisms with discrete fibers are quasi-finite.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nAssume $f$ is locally of finite type.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite,\n\\item for every $s \\in S$ the fibre $X_s$ is a discrete topological space, and\n\\item for every morphism $\\Spec(k) \\to S$ where $k$ is a field\nthe base change $X_k$ has an underlying discrete topological space.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RT","source_file":"morphisms.tex","source_line":3752,"source_end_line":3766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3752-L3766","statement_sha256":"9da2ba38fccb2bf69545d735ea6910c657ff1154b4acff47405319f2950dd336","origin":"The Stacks Project","memory_eligible":false,"source_rank":5466,"rank":5466,"depth":27,"x":1999.849,"y":642.438,"cluster":"scheme-morphisms"},{"id":"stacks:01TJ","tag":"01TJ","title":"Quasi-finite morphisms · Lemma 01TJ","summary":"Let f : X → S be a morphism of schemes. Then f is quasi-finite if and only if f is locally quasi-finite and quasi-compact.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThen $f$ is quasi-finite if and only if $f$ is\nlocally quasi-finite and quasi-compact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TJ","source_file":"morphisms.tex","source_line":3786,"source_end_line":3791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3786-L3791","statement_sha256":"f914c4ec9250f76708aec6f4e921eb387a6c3679ffaec0ab48388e7eb6dd88ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":5467,"rank":5467,"depth":27,"x":2082.639,"y":690.629,"cluster":"scheme-morphisms"},{"id":"stacks:02NH","tag":"02NH","title":"Quasi-finite morphisms · Lemma 02NH","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • f is quasi-finite, and • f is locally of finite type, quasi-compact, and has finite fibres.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is quasi-finite, and\n\\item $f$ is locally of finite type, quasi-compact, and has finite fibres.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NH","source_file":"morphisms.tex","source_line":3808,"source_end_line":3816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3808-L3816","statement_sha256":"968eef1d567820758bf0e75a27b5669e0b6ff5bd59756bab3652a9c5d0cef160","origin":"The Stacks Project","memory_eligible":false,"source_rank":5468,"rank":5468,"depth":28,"x":1982.458,"y":702.114,"cluster":"scheme-morphisms"},{"id":"stacks:01TK","tag":"01TK","title":"Quasi-finite morphisms · Lemma 01TK","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is locally quasi-finite. • For every pair of affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the ring map O_S(V) → O_X(U) is quasi-finite. • There exists an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is locally quasi-finite. • There exists an affine open covering S = ⋃_j ∈ J V_j and affine open coverings…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is locally quasi-finite.\n\\item For every pair of affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is quasi-finite.\n\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis locally quasi-finite.\n\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is\nquasi-finite, for all $j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is locally quasi-finite then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is locally quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TK","source_file":"morphisms.tex","source_line":3838,"source_end_line":3859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3838-L3859","statement_sha256":"7220425b1f4be515d703049c544af2ce7760df0f4e729543f37215f7315189f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5469,"rank":5469,"depth":27,"x":2047.338,"y":636.555,"cluster":"scheme-morphisms"},{"id":"stacks:01TL","tag":"01TL","title":"Quasi-finite morphisms · Lemma 01TL","summary":"The composition of two morphisms which are locally quasi-finite is locally quasi-finite. The same is true for quasi-finite morphisms.","statement_latex":"The composition of two morphisms which are locally quasi-finite is\nlocally quasi-finite. The same is true for quasi-finite morphisms.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TL","source_file":"morphisms.tex","source_line":3896,"source_end_line":3900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3896-L3900","statement_sha256":"165ac721d451ecfccc3e7b84c708e116ebd96c1d9d0c1fc406e5dff4871b335b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5470,"rank":5470,"depth":28,"x":2052.235,"y":722.029,"cluster":"scheme-morphisms"},{"id":"stacks:01TM","tag":"01TM","title":"Quasi-finite morphisms · Lemma 01TM","summary":"(Locally) quasi-finite morphisms are stable under base change. Let f : X → S be a morphism of schemes. Let g : S' → S be a morphism of schemes. Denote f' : X' → S' the base change of f by g and denote g' : X' → X the projection. Assume X is locally of finite type over S. • Let U ⊂ X (resp. U' ⊂ X') be the set of points where f (resp. f') is quasi-finite. Then U' = U ×_S S' = (g')^-1(U). • The base change of a locally quasi-finite morphism is locally quasi-finite. • The…","statement_latex":"\\begin{slogan}\n(Locally) quasi-finite morphisms are stable under base change.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nLet $g : S' \\to S$ be a morphism of schemes.\nDenote $f' : X' \\to S'$ the base change of $f$ by $g$\nand denote $g' : X' \\to X$ the projection.\nAssume $X$ is locally of finite type over $S$.\n\\begin{enumerate}\n\\item Let $U \\subset X$ (resp.\\ $U' \\subset X'$)\nbe the set of points where $f$ (resp.\\ $f'$) is quasi-finite.\nThen $U' = U \\times_S S' = (g')^{-1}(U)$.\n\\item The base change of a locally quasi-finite morphism is\nlocally quasi-finite.\n\\item The base change of a quasi-finite morphism is\nquasi-finite.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TM","source_file":"morphisms.tex","source_line":3924,"source_end_line":3943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3924-L3943","statement_sha256":"ede59121f2386df7c4c3f9da4d82f187a883ac2a0f5399e707ef1c158c2c7e04","origin":"The Stacks Project","memory_eligible":false,"source_rank":5471,"rank":5471,"depth":28,"x":1979.621,"y":661.557,"cluster":"scheme-morphisms"},{"id":"stacks:0AAY","tag":"0AAY","title":"Quasi-finite morphisms · Lemma 0AAY","summary":"Let f : X → S be a morphism of schemes of finite type. Let s ∈ S. There are at most finitely many points of X lying over s at which f is quasi-finite.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes of finite type.\nLet $s \\in S$. There are at most finitely many points\nof $X$ lying over $s$ at which $f$ is quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAY","source_file":"morphisms.tex","source_line":3959,"source_end_line":3964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3959-L3964","statement_sha256":"ad3bb50eac49e11c8e4ffe060067195e360211004686e52be5e77e5bf2f14336","origin":"The Stacks Project","memory_eligible":false,"source_rank":5472,"rank":5472,"depth":27,"x":2082.171,"y":664.958,"cluster":"scheme-morphisms"},{"id":"stacks:0CT8","tag":"0CT8","title":"Quasi-finite morphisms · Lemma 0CT8","summary":"Let f : X → Y be a morphism of schemes. If f is locally of finite type and a monomorphism, then f is separated and locally quasi-finite.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nIf $f$ is locally of finite type and a monomorphism, then $f$\nis separated and locally quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CT8","source_file":"morphisms.tex","source_line":3976,"source_end_line":3981,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3976-L3981","statement_sha256":"62698a649c3bdceec020ba2d4ddaf41e8642482e3f5777b4a622f788e326fd93","origin":"The Stacks Project","memory_eligible":false,"source_rank":5473,"rank":5473,"depth":27,"x":2003.53,"y":720.842,"cluster":"scheme-morphisms"},{"id":"stacks:01TN","tag":"01TN","title":"Quasi-finite morphisms · Lemma 01TN","summary":"Any immersion is locally quasi-finite.","statement_latex":"Any immersion is locally quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TN","source_file":"morphisms.tex","source_line":3991,"source_end_line":3994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L3991-L3994","statement_sha256":"35c014ade76408e0236109efe74fd247115683ce9053c1e16b1ac9ba9f2439a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5474,"rank":5474,"depth":0,"x":2016.626,"y":634.701,"cluster":"scheme-morphisms"},{"id":"stacks:03WR","tag":"03WR","title":"Quasi-finite morphisms · Lemma 03WR","summary":"Let X → Y be a morphism of schemes over a base scheme S. Let x ∈ X. If X → S is quasi-finite at x, then X → Y is quasi-finite at x. If X is locally quasi-finite over S, then X → Y is locally quasi-finite.","statement_latex":"Let $X \\to Y$ be a morphism of schemes over a base scheme $S$.\nLet $x \\in X$. If $X \\to S$ is quasi-finite at $x$, then\n$X \\to Y$ is quasi-finite at $x$.\nIf $X$ is locally quasi-finite over $S$, then $X \\to Y$\nis locally quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WR","source_file":"morphisms.tex","source_line":4001,"source_end_line":4008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4001-L4008","statement_sha256":"1275cb23ab50f3b5e02fbbbc8aed45efd819c1f9834ebeecb04ad22240f8039f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5475,"rank":5475,"depth":28,"x":2076.456,"y":705.906,"cluster":"scheme-morphisms"},{"id":"stacks:0GWS","tag":"0GWS","title":"Quasi-finite morphisms · Lemma 0GWS","summary":"Let f : X → Y and g : Y → S be morphisms of schemes. If f is surjective, g ∘ f locally quasi-finite, and g locally of finite type, then g : Y → S is locally quasi-finite.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes.\nIf $f$ is surjective, $g \\circ f$ locally quasi-finite, and\n$g$ locally of finite type, then $g : Y \\to S$  is locally quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWS","source_file":"morphisms.tex","source_line":4020,"source_end_line":4025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4020-L4025","statement_sha256":"11197d32b90967a9206bc3207582392c4454d020ed98f87481f8695e535beaa6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5476,"rank":5476,"depth":27,"x":1974.715,"y":687.283,"cluster":"scheme-morphisms"},{"id":"stacks:01TP","tag":"01TP","title":"Morphisms of finite presentation · Definition 01TP","summary":"Let f : X → S be a morphism of schemes. • We say that f is of finite presentation at x ∈ X if there exists an affine open neighbourhood Spec(A) = U ⊂ X of x and affine open Spec(R) = V ⊂ S with f(U) ⊂ V such that the induced ring map R → A is of finite presentation. • We say that f is locally of finite presentation if it is of finite presentation at every point of X. • We say that f is of finite presentation if it is locally of finite presentation, quasi-compact and…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say that $f$ is of {\\it finite presentation at $x \\in X$} if\nthere exists an affine open neighbourhood $\\Spec(A) = U \\subset X$\nof $x$ and affine open $\\Spec(R) = V \\subset S$\nwith $f(U) \\subset V$ such that the induced ring map\n$R \\to A$ is of finite presentation.\n\\item We say that $f$ is {\\it locally of finite presentation} if it is\nof finite presentation at every point of $X$.\n\\item We say that $f$ is of {\\it finite presentation} if it is locally of\nfinite presentation, quasi-compact and quasi-separated.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TP","source_file":"morphisms.tex","source_line":4063,"source_end_line":4077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4063-L4077","statement_sha256":"5c4794aaa59087ae5133da4486774ffcfd6759138b355fa4d30c85f5d5cd9c7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5477,"rank":5477,"depth":0,"x":2065.033,"y":643.13,"cluster":"scheme-morphisms"},{"id":"stacks:01TQ","tag":"01TQ","title":"Morphisms of finite presentation · Lemma 01TQ","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is locally of finite presentation. • For every affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the ring map O_S(V) → O_X(U) is of finite presentation. • There exist an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is locally of finite presentation. • There exist an affine open covering S = ⋃_j ∈ J V_j and affine…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is locally of finite presentation.\n\\item For every affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is of finite presentation.\n\\item There exist an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis locally of finite presentation.\n\\item There exist an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is\nof finite presentation, for all $j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is locally of finite presentation then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is locally of finite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TQ","source_file":"morphisms.tex","source_line":4095,"source_end_line":4116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4095-L4116","statement_sha256":"1027ac4b278ca4190750c616ebd65048dabf2440944dceb37fb737f755375f21","origin":"The Stacks Project","memory_eligible":false,"source_rank":5478,"rank":5478,"depth":4,"x":2033.831,"y":727.231,"cluster":"scheme-morphisms"},{"id":"stacks:01TR","tag":"01TR","title":"Morphisms of finite presentation · Lemma 01TR","summary":"The composition of two morphisms which are locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation.","statement_latex":"The composition of two morphisms which are locally of finite presentation is\nlocally of finite presentation.\nThe same is true for morphisms of finite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TR","source_file":"morphisms.tex","source_line":4133,"source_end_line":4138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4133-L4138","statement_sha256":"5deb143c66011dcba962344d8d92c0280a81f8c9b46afedf3281d1aad53aaa61","origin":"The Stacks Project","memory_eligible":false,"source_rank":5479,"rank":5479,"depth":16,"x":1989.051,"y":647.233,"cluster":"scheme-morphisms"},{"id":"stacks:01TS","tag":"01TS","title":"Morphisms of finite presentation · Lemma 01TS","summary":"The base change of a morphism which is locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation.","statement_latex":"The base change of a morphism which is locally of finite presentation\nis locally of finite presentation. The same is true for morphisms of\nfinite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TS","source_file":"morphisms.tex","source_line":4158,"source_end_line":4163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4158-L4163","statement_sha256":"0a0d80509568863320e3b51fd00957f50cc8603d66cb5f432cc011904e9a707a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5480,"rank":5480,"depth":16,"x":2086.741,"y":680.93,"cluster":"scheme-morphisms"},{"id":"stacks:01TT","tag":"01TT","title":"Morphisms of finite presentation · Lemma 01TT","summary":"Any open immersion is locally of finite presentation.","statement_latex":"Any open immersion is locally of finite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TT","source_file":"morphisms.tex","source_line":4183,"source_end_line":4186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4183-L4186","statement_sha256":"8e60863f48330dabd8715c871d4a12a92b20ba087383e00d610855036e49c087","origin":"The Stacks Project","memory_eligible":false,"source_rank":5481,"rank":5481,"depth":0,"x":1987.266,"y":711.618,"cluster":"scheme-morphisms"},{"id":"stacks:01TU","tag":"01TU","title":"Morphisms of finite presentation · Lemma 01TU","summary":"Any open immersion is of finite presentation if and only if it is quasi-compact.","statement_latex":"Any open immersion is of finite presentation if and only if\nit is quasi-compact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TU","source_file":"morphisms.tex","source_line":4192,"source_end_line":4196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4192-L4196","statement_sha256":"aee2b1a1c09d2ef6fad89be2810c63aafb8bd14c2f2777f0ddb79475c0635a7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5482,"rank":5482,"depth":3,"x":2036.106,"y":632.275,"cluster":"scheme-morphisms"},{"id":"stacks:01TV","tag":"01TV","title":"Morphisms of finite presentation · Lemma 01TV","summary":"Closed immersions of finite presentation correspond to quasi-coherent sheaves of ideals of finite type. A closed immersion i : Z → X is of finite presentation if and only if the associated quasi-coherent sheaf of ideals I = Ker(O_X → i_*O_Z) is of finite type (as an O_X-module).","statement_latex":"\\begin{slogan}\nClosed immersions of finite presentation correspond\nto quasi-coherent sheaves of ideals of finite type.\n\\end{slogan}\nA closed immersion $i : Z \\to X$ is of finite presentation if and only if\nthe associated quasi-coherent sheaf of ideals\n$\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$\nis of finite type (as an $\\mathcal{O}_X$-module).","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TV","source_file":"morphisms.tex","source_line":4206,"source_end_line":4216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4206-L4216","statement_sha256":"c875becd26bdaf119075536a380208dedeb069fa4a10fa72b835aa38338fb3c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5483,"rank":5483,"depth":4,"x":2063.992,"y":718.787,"cluster":"scheme-morphisms"},{"id":"stacks:01TW","tag":"01TW","title":"Morphisms of finite presentation · Lemma 01TW","summary":"A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type.","statement_latex":"A morphism which is locally of finite presentation is locally of finite type.\nA morphism of finite presentation is of finite type.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TW","source_file":"morphisms.tex","source_line":4229,"source_end_line":4233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4229-L4233","statement_sha256":"92df17b1f929423d789954b1427ef46ce17eb80205bc17747f44a7084dc1c0e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5484,"rank":5484,"depth":0,"x":1973.554,"y":670.655,"cluster":"scheme-morphisms"},{"id":"stacks:01TX","tag":"01TX","title":"Morphisms of finite presentation · Lemma 01TX","summary":"Over a locally Noetherian base, finite type is finite presentation. Let f : X → S be a morphism. • If S is locally Noetherian and f locally of finite type then f is locally of finite presentation. • If S is locally Noetherian and f of finite type then f is of finite presentation.","statement_latex":"\\begin{slogan}\nOver a locally Noetherian base, finite type is finite presentation.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism.\n\\begin{enumerate}\n\\item If $S$ is locally Noetherian and $f$ locally of finite type\nthen $f$ is locally of finite presentation.\n\\item If $S$ is locally Noetherian and $f$ of finite type\nthen $f$ is of finite presentation.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TX","source_file":"morphisms.tex","source_line":4239,"source_end_line":4251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4239-L4251","statement_sha256":"46b1c7dd476cfbe6c2dd18daa98929beff15c765a89990fb7bf151110be7371c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5485,"rank":5485,"depth":17,"x":2079.301,"y":654.778,"cluster":"scheme-morphisms"},{"id":"stacks:01TY","tag":"01TY","title":"Morphisms of finite presentation · Lemma 01TY","summary":"Let S be a scheme which is quasi-compact and quasi-separated. If X is of finite presentation over S, then X is quasi-compact and quasi-separated.","statement_latex":"Let $S$ be a scheme which is quasi-compact and quasi-separated.\nIf $X$ is of finite presentation over $S$, then $X$ is quasi-compact\nand quasi-separated.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TY","source_file":"morphisms.tex","source_line":4264,"source_end_line":4269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4264-L4269","statement_sha256":"a395a36e14c5429e2fd8500114b9c365f7c50b06a2ab90d7da85fd5d5bc476d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5486,"rank":5486,"depth":0,"x":2013.876,"y":726.728,"cluster":"scheme-morphisms"},{"id":"stacks:02FV","tag":"02FV","title":"Morphisms of finite presentation · Lemma 02FV","summary":"Let f : X → Y be a morphism of schemes over S. • If X is locally of finite presentation over S and Y is locally of finite type over S, then f is locally of finite presentation. • If X is of finite presentation over S and Y is quasi-separated and locally of finite type over S, then f is of finite presentation.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes over $S$.\n\\begin{enumerate}\n\\item If $X$ is locally of finite presentation over $S$ and\n$Y$ is locally of finite type over $S$, then $f$ is locally\nof finite presentation.\n\\item If $X$ is of finite presentation over $S$ and $Y$ is quasi-separated\nand locally of finite type over $S$, then $f$ is of finite presentation.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FV","source_file":"morphisms.tex","source_line":4275,"source_end_line":4285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4275-L4285","statement_sha256":"922d5d9cdd4c98170b6c31522614306609097364aff37a28b3fbde0274ec511c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5487,"rank":5487,"depth":18,"x":2004.228,"y":636.251,"cluster":"scheme-morphisms"},{"id":"stacks:0818","tag":"0818","title":"Morphisms of finite presentation · Lemma 0818","summary":"Let f : X → Y be a morphism of schemes with diagonal Δ : X → X ×_Y X. If f is locally of finite type then Δ is locally of finite presentation. If f is quasi-separated and locally of finite type, then Δ is of finite presentation.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes with diagonal\n$\\Delta : X \\to X \\times_Y X$. If $f$ is locally of finite type\nthen $\\Delta$ is locally of finite presentation. If $f$ is quasi-separated\nand locally of finite type, then $\\Delta$ is of finite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0818","source_file":"morphisms.tex","source_line":4301,"source_end_line":4307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4301-L4307","statement_sha256":"76d6abf9e50a55ba93d67eeef52eeb914edaa69bceb8d872325ec7573c19998f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5488,"rank":5488,"depth":19,"x":2084.363,"y":697.694,"cluster":"scheme-morphisms"},{"id":"stacks:054I","tag":"054I","title":"Constructible sets · Lemma 054I","summary":"Let f : X → Y be a morphism of schemes. Let E ⊂ Y be a subset. If E is (locally) constructible in Y, then f^-1(E) is (locally) constructible in X.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $E \\subset Y$ be a subset.\nIf $E$ is (locally) constructible in $Y$, then $f^{-1}(E)$ is (locally)\nconstructible in $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054I","source_file":"morphisms.tex","source_line":4339,"source_end_line":4345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4339-L4345","statement_sha256":"1e2fdf85226f95d16440e4c14a738e08fd026f156da488ce567834bb6c55cea6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5489,"rank":5489,"depth":4,"x":1975.508,"y":697.857,"cluster":"scheme-morphisms"},{"id":"stacks:054J","tag":"054J","title":"Constructible sets · Lemma 054J","summary":"Let f : X → Y be a morphism of schemes. Assume • f is quasi-compact and locally of finite presentation, and • Y is quasi-compact and quasi-separated. Then the image of every constructible subset of X is constructible in Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume\n\\begin{enumerate}\n\\item $f$ is quasi-compact and locally of finite presentation, and\n\\item $Y$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nThen the image of every constructible subset of $X$ is constructible in $Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054J","source_file":"morphisms.tex","source_line":4368,"source_end_line":4377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4368-L4377","statement_sha256":"e96848a23a03c64853ec25c27696e15e1a2260b2e9cb47e33583f374e40b19c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5490,"rank":5490,"depth":6,"x":2055.905,"y":635.769,"cluster":"scheme-morphisms"},{"id":"stacks:054K","tag":"054K","title":"Chevalley's Theorem · Theorem 054K","summary":"[EGA] Let f : X → Y be a morphism of schemes. Assume f is quasi-compact and locally of finite presentation. Then the image of every locally constructible subset is locally constructible.","statement_latex":"\\begin{reference}\n\\cite[IV, Theorem 1.8.4]{EGA}\n\\end{reference}\nLet $f : X \\to Y$ be a morphism of schemes.\nAssume $f$ is quasi-compact and locally of finite presentation.\nThen the image of every locally constructible subset is locally constructible.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Constructible sets","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054K","source_file":"morphisms.tex","source_line":4396,"source_end_line":4404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4396-L4404","statement_sha256":"0fee746324e94edf5fb2cad88bf4d5e50da362cd20ba59a01a7423584550d172","origin":"The Stacks Project","memory_eligible":false,"source_rank":5491,"rank":5491,"depth":6,"x":2046.52,"y":727.466,"cluster":"scheme-morphisms"},{"id":"stacks:05LW","tag":"05LW","title":"Constructible sets · Lemma 05LW","summary":"Let X be a scheme. Let x ∈ X. Let E ⊂ X be a locally constructible subset. If (x' mid x' leadsto x) ⊂ E, then E contains an open neighbourhood of x.","statement_latex":"Let $X$ be a scheme. Let $x \\in X$. Let $E \\subset X$ be a locally\nconstructible subset. If $\\{x' \\mid x' \\leadsto x\\} \\subset E$,\nthen $E$ contains an open neighbourhood of $x$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LW","source_file":"morphisms.tex","source_line":4422,"source_end_line":4427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4422-L4427","statement_sha256":"978723448c0704e525851ad8072ef238e71fa91230e4c13e6ba62af005b0f185","origin":"The Stacks Project","memory_eligible":false,"source_rank":5492,"rank":5492,"depth":20,"x":1979.487,"y":654.292,"cluster":"scheme-morphisms"},{"id":"stacks:01U0","tag":"01U0","title":"Open morphisms · Definition 01U0","summary":"Let f : X → S be a morphism. • We say f is open if the map on underlying topological spaces is open. • We say f is universally open if for any morphism of schemes S' → S the base change f' : X_S' → S' is open.","statement_latex":"Let $f : X \\to S$ be a morphism.\n\\begin{enumerate}\n\\item We say $f$ is {\\it open} if the map on underlying\ntopological spaces is open.\n\\item We say $f$ is {\\it universally open} if for any morphism of\nschemes $S' \\to S$ the base change $f' : X_{S'} \\to S'$ is open.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Open morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01U0","source_file":"morphisms.tex","source_line":4450,"source_end_line":4459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4450-L4459","statement_sha256":"3698c3bb5620170bc9aa2e07c85155e918bad0ecdb226aa3ed998ac3cd117dca","origin":"The Stacks Project","memory_eligible":false,"source_rank":5493,"rank":5493,"depth":0,"x":2088.105,"y":670.264,"cluster":"scheme-morphisms"},{"id":"stacks:01U1","tag":"01U1","title":"Open morphisms · Lemma 01U1","summary":"Let f : X → S be a morphism. • If f is locally of finite presentation and generalizations lift along f, then f is open. • If f is locally of finite presentation and generalizations lift along every base change of f, then f is universally open.","statement_latex":"Let $f : X \\to S$ be a morphism.\n\\begin{enumerate}\n\\item If $f$ is locally of finite presentation and generalizations lift\nalong $f$, then $f$ is open.\n\\item If $f$ is locally of finite presentation and generalizations lift\nalong every base change of $f$, then $f$ is universally open.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01U1","source_file":"morphisms.tex","source_line":4472,"source_end_line":4481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4472-L4481","statement_sha256":"1c72d65c306fbd76272bc759acb6f8d01a91bb9b9edf0fb2148a6fb8eb67e260","origin":"The Stacks Project","memory_eligible":false,"source_rank":5494,"rank":5494,"depth":8,"x":1994.874,"y":720.276,"cluster":"scheme-morphisms"},{"id":"stacks:02V2","tag":"02V2","title":"Open morphisms · Lemma 02V2","summary":"A composition of (universally) open morphisms is (universally) open.","statement_latex":"A composition of (universally) open morphisms is (universally) open.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V2","source_file":"morphisms.tex","source_line":4495,"source_end_line":4498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4495-L4498","statement_sha256":"86b1cad316c6321d21e028573edd49242b86daa1535bf131f0bed275602c7764","origin":"The Stacks Project","memory_eligible":false,"source_rank":5495,"rank":5495,"depth":0,"x":2023.493,"y":630.215,"cluster":"scheme-morphisms"},{"id":"stacks:0383","tag":"0383","title":"Open morphisms · Lemma 0383","summary":"Let k be a field. Let X be a scheme over k. The structure morphism X → Spec(k) is universally open.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$.\nThe structure morphism $X \\to \\Spec(k)$ is universally open.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0383","source_file":"morphisms.tex","source_line":4504,"source_end_line":4508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4504-L4508","statement_sha256":"59471df62e23006c0565c7d8c658855bdc2401687fa4c997fa1aac3f41527bdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5496,"rank":5496,"depth":9,"x":2074.974,"y":713.121,"cluster":"scheme-morphisms"},{"id":"stacks:040F","tag":"040F","title":"Open morphisms · Lemma 040F","summary":"Follows from the implication (a) ⇒ (b) in [EGA] Let φ : X → Y be a morphism of schemes. If φ is open, then φ is generizing (i.e., generalizations lift along φ). If φ is universally open, then φ is universally generizing.","statement_latex":"\\begin{reference}\nFollows from the implication (a) $\\Rightarrow$ (b) in\n\\cite[IV, Corollary 1.10.4]{EGA}\n\\end{reference}\nLet $\\varphi : X \\to Y$ be a morphism of schemes.\nIf $\\varphi$ is open, then $\\varphi$ is generizing\n(i.e., generalizations lift along $\\varphi$).\nIf $\\varphi$ is universally open, then $\\varphi$ is\nuniversally generizing.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040F","source_file":"morphisms.tex","source_line":4518,"source_end_line":4529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4518-L4529","statement_sha256":"3ba003d95c03a2fc8cc83f73ddef210980278a486200bbcf7aff8d9acc7b48a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5497,"rank":5497,"depth":5,"x":1970.017,"y":681.099,"cluster":"scheme-morphisms"},{"id":"stacks:04ZE","tag":"04ZE","title":"Open morphisms · Lemma 04ZE","summary":"Let f : X → Y be a morphism of schemes. Let g : Y' → Y be open and surjective such that the base change f' : X' → Y' is quasi-compact. Then f is quasi-compact.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $g : Y' \\to Y$ be open and surjective such that the base change\n$f' : X' \\to Y'$ is quasi-compact. Then $f$ is quasi-compact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZE","source_file":"morphisms.tex","source_line":4544,"source_end_line":4549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4544-L4549","statement_sha256":"939d8ebd8477b71c543f82ae1b5b592c0d1f7fd86e3926a9862a97f221bd70be","origin":"The Stacks Project","memory_eligible":false,"source_rank":5498,"rank":5498,"depth":3,"x":2073.479,"y":645.047,"cluster":"scheme-morphisms"},{"id":"stacks:040H","tag":"040H","title":"Submersive morphisms · Definition 040H","summary":"Let f : X → Y be a morphism of schemes. • We say f is submersive if the continuous map of underlying topological spaces is submersive, see Topology, Definition [Tag 0406]. • We say f is universally submersive if for every morphism of schemes Y' → Y the base change Y' ×_Y X → Y' is submersive.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say $f$ is {\\it submersive}\\footnote{This is very different\nfrom the notion of a submersion of differential manifolds.}\nif the continuous map of underlying topological spaces is submersive, see\nTopology, Definition \\ref{topology-definition-submersive}.\n\\item We say $f$ is {\\it universally submersive} if for every\nmorphism of schemes $Y' \\to Y$ the base change\n$Y' \\times_Y X \\to Y'$ is submersive.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Submersive morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040H","source_file":"morphisms.tex","source_line":4569,"source_end_line":4581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4569-L4581","statement_sha256":"5f601fe62a8cb1273faacafdfcd821c63d46e6cdac31dd78f9726d4f25b0b5a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5499,"rank":5499,"depth":1,"x":2026.035,"y":730.599,"cluster":"scheme-morphisms"},{"id":"stacks:0CES","tag":"0CES","title":"Submersive morphisms · Lemma 0CES","summary":"The base change of a universally submersive morphism of schemes by any morphism of schemes is universally submersive.","statement_latex":"The base change of a universally submersive morphism of schemes\nby any morphism of schemes is universally submersive.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Submersive morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CES","source_file":"morphisms.tex","source_line":4586,"source_end_line":4590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4586-L4590","statement_sha256":"e58f73623d1939d0a0749014791b039df2bce51a9bb9fbeb6e533904a7a5abf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5500,"rank":5500,"depth":0,"x":1992.118,"y":640.319,"cluster":"scheme-morphisms"},{"id":"stacks:0CET","tag":"0CET","title":"Submersive morphisms · Lemma 0CET","summary":"The composition of a pair of (universally) submersive morphisms of schemes is (universally) submersive.","statement_latex":"The composition of a pair of (universally) submersive morphisms of\nschemes is (universally) submersive.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Submersive morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CET","source_file":"morphisms.tex","source_line":4596,"source_end_line":4600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4596-L4600","statement_sha256":"09ade1a7e490f6e02625853e1c08a13bfada6c366a1187c0436736a3ec6c9fee","origin":"The Stacks Project","memory_eligible":false,"source_rank":5501,"rank":5501,"depth":0,"x":2090.024,"y":687.788,"cluster":"scheme-morphisms"},{"id":"stacks:01U3","tag":"01U3","title":"Flat morphisms · Definition 01U3","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf of O_X-modules. • We say f is flat at a point x ∈ X if the local ring O_X, x is flat over the local ring O_S, f(x). • We say that F is flat over S at a point x ∈ X if the stalk F_x is a flat O_S, f(x)-module. • We say f is flat if f is flat at every point of X. • We say that F is flat over S if F is flat over S at every point x of X.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item We say $f$ is {\\it flat at a point $x \\in X$} if the\nlocal ring $\\mathcal{O}_{X, x}$ is flat over the local ring\n$\\mathcal{O}_{S, f(x)}$.\n\\item We say that $\\mathcal{F}$ is {\\it flat over $S$ at a point $x \\in X$}\nif the stalk $\\mathcal{F}_x$ is a flat $\\mathcal{O}_{S, f(x)}$-module.\n\\item We say $f$ is {\\it flat} if $f$ is flat at every point of $X$.\n\\item We say that $\\mathcal{F}$ is {\\it flat over $S$} if\n$\\mathcal{F}$ is flat over $S$ at every point $x$ of $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01U3","source_file":"morphisms.tex","source_line":4639,"source_end_line":4653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4639-L4653","statement_sha256":"fd1550989dfaad8fc8d5a0e0cf5282582c4e7ee10550e61a57e355ca4b95f266","origin":"The Stacks Project","memory_eligible":false,"source_rank":5502,"rank":5502,"depth":0,"x":1979.325,"y":708.401,"cluster":"scheme-morphisms"},{"id":"stacks:01U4","tag":"01U4","title":"Flat morphisms · Lemma 01U4","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf of O_X-modules. The following are equivalent • The sheaf F is flat over S. • For every affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the O_S(V)-module F(U) is flat. • There exists an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the modules F|_U_i is flat over V_j, for all j∈ J, i∈ I_j. • There exists an affine open covering S = ⋃_j ∈ J V_j and affine open…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.\nThe following are equivalent\n\\begin{enumerate}\n\\item The sheaf $\\mathcal{F}$ is flat over $S$.\n\\item For every affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the $\\mathcal{O}_S(V)$-module $\\mathcal{F}(U)$ is flat.\n\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the modules $\\mathcal{F}|_{U_i}$ is\nflat over $V_j$, for all $j\\in J, i\\in I_j$.\n\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat $\\mathcal{F}(U_i)$ is a flat $\\mathcal{O}_S(V_j)$-module, for all\n$j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $\\mathcal{F}$ is flat over $S$ then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $\\mathcal{F}|_U$ is flat over $V$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01U4","source_file":"morphisms.tex","source_line":4659,"source_end_line":4680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4659-L4680","statement_sha256":"c33e60379e813bb3807b04aead4a7a71a91a9faab741bf9feda62b7a9cd1fb17","origin":"The Stacks Project","memory_eligible":false,"source_rank":5503,"rank":5503,"depth":3,"x":2044.571,"y":630.156,"cluster":"scheme-morphisms"},{"id":"stacks:01U5","tag":"01U5","title":"Flat morphisms · Lemma 01U5","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is flat. • For every affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the ring map O_S(V) → O_X(U) is flat. • There exists an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is flat. • There exists an affine open covering S = ⋃_j ∈ J V_j and affine open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that O_S(V_j) →…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is flat.\n\\item For every affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is flat.\n\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis flat.\n\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is flat, for all\n$j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is flat then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is flat.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01U5","source_file":"morphisms.tex","source_line":4693,"source_end_line":4714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4693-L4714","statement_sha256":"8203f69d0f6fe87261f94689d90a9594c4751b6dcca1fa4231b11733393abd74","origin":"The Stacks Project","memory_eligible":false,"source_rank":5504,"rank":5504,"depth":4,"x":2059.426,"y":725.156,"cluster":"scheme-morphisms"},{"id":"stacks:0FLM","tag":"0FLM","title":"Flat morphisms · Lemma 0FLM","summary":"Let f : X → Y be an affine morphism of schemes over a base scheme S. Let F be a quasi-coherent O_X-module. Then F is flat over S if and only if f_*F is flat over S.","statement_latex":"Let $f : X \\to Y$ be an affine morphism of schemes over a base scheme $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $\\mathcal{F}$ is flat over $S$ if and only if\n$f_*\\mathcal{F}$ is flat over $S$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLM","source_file":"morphisms.tex","source_line":4721,"source_end_line":4727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4721-L4727","statement_sha256":"bd29ab803686ba78c539c565563091567f9f662438fc148ccef63edeec541969","origin":"The Stacks Project","memory_eligible":false,"source_rank":5505,"rank":5505,"depth":13,"x":1971.818,"y":663.351,"cluster":"scheme-morphisms"},{"id":"stacks:01U6","tag":"01U6","title":"Flat morphisms · Lemma 01U6","summary":"Let X → Y → Z be morphisms of schemes. Let F be a quasi-coherent O_X-module. Let x ∈ X with image y in Y. If F is flat over Y at x, and Y is flat over Z at y, then F is flat over Z at x.","statement_latex":"Let $X \\to Y \\to Z$ be morphisms of schemes. Let $\\mathcal{F}$ be a\nquasi-coherent $\\mathcal{O}_X$-module. Let $x \\in X$ with image $y$ in $Y$.\nIf $\\mathcal{F}$ is flat over $Y$ at $x$, and $Y$ is flat over $Z$ at\n$y$, then $\\mathcal{F}$ is flat over $Z$ at $x$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01U6","source_file":"morphisms.tex","source_line":4739,"source_end_line":4745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4739-L4745","statement_sha256":"26fe632fd691ca6a5f6147b59ac1dc6e65cd4b31c60b47e09010323726941165","origin":"The Stacks Project","memory_eligible":false,"source_rank":5506,"rank":5506,"depth":3,"x":2086.456,"y":659.202,"cluster":"scheme-morphisms"},{"id":"stacks:01U7","tag":"01U7","title":"Flat morphisms · Lemma 01U7","summary":"The composition of flat morphisms is flat.","statement_latex":"The composition of flat morphisms is flat.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01U7","source_file":"morphisms.tex","source_line":4751,"source_end_line":4754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4751-L4754","statement_sha256":"1cef9b5636167763759bac6e58fd08ed733a3d6d6e3faac26fee58c85f9d5eef","origin":"The Stacks Project","memory_eligible":false,"source_rank":5507,"rank":5507,"depth":4,"x":2005.023,"y":727.508,"cluster":"scheme-morphisms"},{"id":"stacks:01U8","tag":"01U8","title":"Flat morphisms · Lemma 01U8","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf of O_X-modules. Let g : S' → S be a morphism of schemes. Denote g' : X' = X_S' → X the projection. Let x' ∈ X' be a point with image x = g'(x') ∈ X. If F is flat over S at x, then (g')^*F is flat over S' at x'. In particular, if F is flat over S, then (g')^*F is flat over S'.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.\nLet $g : S' \\to S$ be a morphism of schemes.\nDenote $g' : X' = X_{S'} \\to X$ the projection.\nLet $x' \\in X'$ be a point with image $x = g'(x') \\in X$.\nIf $\\mathcal{F}$ is flat over $S$ at $x$, then\n$(g')^*\\mathcal{F}$ is flat over $S'$ at $x'$.\nIn particular, if $\\mathcal{F}$ is flat over $S$, then\n$(g')^*\\mathcal{F}$ is flat over $S'$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01U8","source_file":"morphisms.tex","source_line":4760,"source_end_line":4771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4760-L4771","statement_sha256":"112f55b4be8b8256590cde280c860d99808831387e725c5d2269efadb6c00c61","origin":"The Stacks Project","memory_eligible":false,"source_rank":5508,"rank":5508,"depth":1,"x":2010.156,"y":630.654,"cluster":"scheme-morphisms"},{"id":"stacks:01U9","tag":"01U9","title":"Flat morphisms · Lemma 01U9","summary":"The base change of a flat morphism is flat.","statement_latex":"The base change of a flat morphism is flat.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01U9","source_file":"morphisms.tex","source_line":4777,"source_end_line":4780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4777-L4780","statement_sha256":"b92b5be92432973f4ae3a890fc14c6e019d752c9702428d20ff93e9a6e64fae3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5509,"rank":5509,"depth":2,"x":2084.472,"y":705.2,"cluster":"scheme-morphisms"},{"id":"stacks:03HV","tag":"03HV","title":"Flat morphisms · Lemma 03HV","summary":"Let f : X → S be a flat morphism of schemes. Then generalizations lift along f, see Topology, Definition [Tag 0063].","statement_latex":"Let $f : X \\to S$ be a flat morphism of schemes.\nThen generalizations lift along $f$, see\nTopology, Definition \\ref{topology-definition-lift-specializations}.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HV","source_file":"morphisms.tex","source_line":4786,"source_end_line":4791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4786-L4791","statement_sha256":"f3661a549f392c77be6049c817fd1b79645cbb0b18cc5aef87f5a9f8a4b0a7a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5510,"rank":5510,"depth":1,"x":1969.396,"y":692.362,"cluster":"scheme-morphisms"},{"id":"stacks:01UA","tag":"01UA","title":"Flat morphisms · Lemma 01UA","summary":"A flat morphism locally of finite presentation is universally open.","statement_latex":"A flat morphism locally of finite presentation is universally open.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UA","source_file":"morphisms.tex","source_line":4797,"source_end_line":4800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4797-L4800","statement_sha256":"f84ddab678262b4e472730a6cb98f1b3051792337dd22c621cf28bc54682a941","origin":"The Stacks Project","memory_eligible":false,"source_rank":5511,"rank":5511,"depth":17,"x":2064.846,"y":636.373,"cluster":"scheme-morphisms"},{"id":"stacks:0CVT","tag":"0CVT","title":"Flat morphisms · Lemma 0CVT","summary":"Let f : X → Y be a morphism of schemes. Let F be a quasi-coherent O_X-module. Assume f locally finite presentation, F of finite type, X = Supp(F), and F flat over Y. Then f is universally open.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $f$ locally finite presentation, $\\mathcal{F}$ of\nfinite type, $X = \\text{Supp}(\\mathcal{F})$, and\n$\\mathcal{F}$ flat over $Y$. Then $f$ is universally open.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVT","source_file":"morphisms.tex","source_line":4824,"source_end_line":4831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4824-L4831","statement_sha256":"9db0e316eb96dc63b6c76a5f32c19aeaaf8abd9c649f091f1d15dd9f9e423e62","origin":"The Stacks Project","memory_eligible":false,"source_rank":5512,"rank":5512,"depth":17,"x":2039.415,"y":732.089,"cluster":"scheme-morphisms"},{"id":"stacks:02JY","tag":"02JY","title":"Flat morphisms · Lemma 02JY","summary":"[SGA1] and [EGA] Let f : X → Y be a quasi-compact, surjective, flat morphism. A subset T ⊂ Y is open (resp. closed) if and only f^-1(T) is open (resp. closed). In other words, f is a submersive morphism.","statement_latex":"\\begin{reference}\n\\cite[Expose VIII, Corollaire 4.3]{SGA1} and\n\\cite[IV, Corollaire 2.3.12]{EGA}\n\\end{reference}\nLet $f : X \\to Y$ be a quasi-compact, surjective, flat morphism.\nA subset $T \\subset Y$ is open (resp.\\ closed) if and only\n$f^{-1}(T)$ is open (resp.\\ closed). In other words, $f$ is\na submersive morphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JY","source_file":"morphisms.tex","source_line":4843,"source_end_line":4853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4843-L4853","statement_sha256":"32e01902fb89082d82cac7a2fc7e689aa12ec24093146b01381f895c37bce405","origin":"The Stacks Project","memory_eligible":false,"source_rank":5513,"rank":5513,"depth":3,"x":1981.032,"y":646.84,"cluster":"scheme-morphisms"},{"id":"stacks:02JZ","tag":"02JZ","title":"Flat morphisms · Lemma 02JZ","summary":"Let h : X → Y be a morphism of schemes over S. Let G be a quasi-coherent sheaf on Y. Let x ∈ X with y = h(x) ∈ Y. If h is flat at x, then G flat over S at y ⇔ h^*G flat over S at x. In particular: If h is surjective and flat, then G is flat over S, if and only if h^*G is flat over S. If h is surjective and flat, and X is flat over S, then Y is flat over S.","statement_latex":"Let $h : X \\to Y$ be a morphism of schemes over $S$.\nLet $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$.\nLet $x \\in X$ with $y = h(x) \\in Y$. If $h$ is flat at $x$, then\n$$\n\\mathcal{G}\\text{ flat over }S\\text{ at }y\n\\Leftrightarrow\nh^*\\mathcal{G}\\text{ flat over }S\\text{ at }x.\n$$\nIn particular: If $h$ is surjective and flat, then\n$\\mathcal{G}$ is flat over $S$, if and only if\n$h^*\\mathcal{G}$ is flat over $S$. If $h$ is surjective and\nflat, and $X$ is flat over $S$, then $Y$ is flat over $S$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JZ","source_file":"morphisms.tex","source_line":4880,"source_end_line":4894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4880-L4894","statement_sha256":"21e24c5246f893b56150405b47028991519605cc0fc76fe8ac9ef4a3594186b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5514,"rank":5514,"depth":4,"x":2092.95,"y":676.657,"cluster":"scheme-morphisms"},{"id":"stacks:07T9","tag":"07T9","title":"Flat morphisms · Lemma 07T9","summary":"Let f : Y → X be a morphism of schemes. Let F be a finite type quasi-coherent O_X-module with scheme theoretic support Z ⊂ X. If f is flat, then f^-1(Z) is the scheme theoretic support of f^*F.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes. Let $\\mathcal{F}$ be\na finite type quasi-coherent $\\mathcal{O}_X$-module with scheme\ntheoretic support $Z \\subset X$. If $f$ is flat,\nthen $f^{-1}(Z)$ is the scheme theoretic support of $f^*\\mathcal{F}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07T9","source_file":"morphisms.tex","source_line":4927,"source_end_line":4933,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4927-L4933","statement_sha256":"73c6423bc4d63676aacb90d45882301db681cfc67eb90d2844be76613270d100","origin":"The Stacks Project","memory_eligible":false,"source_rank":5515,"rank":5515,"depth":18,"x":1986.149,"y":718.291,"cluster":"scheme-morphisms"},{"id":"stacks:081H","tag":"081H","title":"Flat morphisms · Lemma 081H","summary":"Let f : X → Y be a flat morphism of schemes. Let V ⊂ Y be a retrocompact open which is scheme theoretically dense. Then f^-1V is scheme theoretically dense in X.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of schemes. Let $V \\subset Y$ be\na retrocompact open which is scheme theoretically dense. Then $f^{-1}V$\nis scheme theoretically dense in $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081H","source_file":"morphisms.tex","source_line":4941,"source_end_line":4946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4941-L4946","statement_sha256":"e4e299c8703a28a50c74809f0ace1b415b0f9a5542a53c5f5146c2d7fdcd54bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5516,"rank":5516,"depth":19,"x":2031.547,"y":626.736,"cluster":"scheme-morphisms"},{"id":"stacks:081I","tag":"081I","title":"Flat morphisms · Lemma 081I","summary":"Taking scheme theoretic images commutes with flat base change in the quasi-compact case Let f : X → Y be a flat morphism of schemes. Let g : V → Y be a quasi-compact morphism of schemes. Let Z ⊂ Y be the scheme theoretic image of g and let Z' ⊂ X be the scheme theoretic image of the base change V ×_Y X → X. Then Z' = f^-1Z.","statement_latex":"\\begin{slogan}\nTaking scheme theoretic images commutes with flat base change\nin the quasi-compact case\n\\end{slogan}\nLet $f : X \\to Y$ be a flat morphism of schemes. Let $g : V \\to Y$ be a\nquasi-compact morphism of schemes. Let $Z \\subset Y$ be the scheme theoretic\nimage of $g$ and let $Z' \\subset X$ be the scheme theoretic image of the\nbase change $V \\times_Y X \\to X$. Then $Z' = f^{-1}Z$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081I","source_file":"morphisms.tex","source_line":4964,"source_end_line":4974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L4964-L4974","statement_sha256":"bc8ec5230cb2b2027f6581dffece043c5f43708f23d5639a81b19a2061dc3f5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5517,"rank":5517,"depth":19,"x":2071.807,"y":720.264,"cluster":"scheme-morphisms"},{"id":"stacks:04PW","tag":"04PW","title":"Flat closed immersions · Lemma 04PW","summary":"Let X be a scheme. The rule which associates to a closed subscheme of X its underlying closed subset defines a bijection ( closed subschemes Z ⊂ X such that Z → X is flat ) ↔ ( closed subsets Z ⊂ X closed under generalizations ) If Z ⊂ X is such a closed subscheme, every morphism of schemes g : Y → X with g(Y) ⊂ Z set theoretically factors (scheme theoretically) through Z.","statement_latex":"Let $X$ be a scheme. The rule which associates to a closed subscheme\nof $X$ its underlying closed subset defines a bijection\n$$\n\\left\\{\n\\begin{matrix}\n\\text{closed subschemes }Z \\subset X \\\\\n\\text{such that }Z \\to X\\text{ is flat}\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{closed subsets }Z \\subset X \\\\\n\\text{closed under generalizations}\n\\end{matrix}\n\\right\\}\n$$\nIf $Z \\subset X$ is such a closed subscheme, every morphism of schemes\n$g : Y \\to X$ with $g(Y) \\subset Z$ set theoretically factors (scheme\ntheoretically) through $Z$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PW","source_file":"morphisms.tex","source_line":5002,"source_end_line":5024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5002-L5024","statement_sha256":"39e91a8d1573e10322a86ab8c7e877419222508f9c589aa68a813d9e625e0812","origin":"The Stacks Project","memory_eligible":false,"source_rank":5518,"rank":5518,"depth":6,"x":1966.62,"y":674.016,"cluster":"scheme-morphisms"},{"id":"stacks:0819","tag":"0819","title":"Flat closed immersions · Lemma 0819","summary":"A flat closed immersion of finite presentation is the open immersion of an open and closed subscheme.","statement_latex":"A flat closed immersion of finite presentation\nis the open immersion of an open and closed subscheme.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0819","source_file":"morphisms.tex","source_line":5037,"source_end_line":5041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5037-L5041","statement_sha256":"df7c81cac16bb511ab4ad7c089bc7d33e066570397cce4d7c3e5f97e104aa382","origin":"The Stacks Project","memory_eligible":false,"source_rank":5519,"rank":5519,"depth":6,"x":2081.687,"y":648.364,"cluster":"scheme-morphisms"},{"id":"stacks:04PX","tag":"04PX","title":"Flat closed immersions · Definition 04PX","summary":"Let X be a scheme. Let T ⊂ X be a connected component. The canonical scheme structure on T is the unique scheme structure on T such that the closed immersion T → X is flat, see Lemma [Tag 04PW].","statement_latex":"Let $X$ be a scheme. Let $T \\subset X$ be a connected component.\nThe {\\it canonical scheme structure on $T$} is the unique\nscheme structure on $T$ such that the closed immersion $T \\to X$\nis flat, see\nLemma \\ref{lemma-characterize-flat-closed-immersions}.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat closed immersions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PX","source_file":"morphisms.tex","source_line":5055,"source_end_line":5062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5055-L5062","statement_sha256":"259f58f62d3426b1757a68a1308f00ca7260fe07a505e09e231a3d44c68b151a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5520,"rank":5520,"depth":7,"x":2017.293,"y":732.799,"cluster":"scheme-morphisms"},{"id":"stacks:053N","tag":"053N","title":"Flat closed immersions · Lemma 053N","summary":"Let X be a scheme. The following are equivalent • every finite flat quasi-coherent O_X-module is finite locally free, and • every closed subset Z ⊂ X which is closed under generalizations is open.","statement_latex":"Let $X$ be a scheme. The following are equivalent\n\\begin{enumerate}\n\\item every finite flat quasi-coherent $\\mathcal{O}_X$-module is\nfinite locally free, and\n\\item every closed subset $Z \\subset X$ which is closed under generalizations\nis open.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Flat closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053N","source_file":"morphisms.tex","source_line":5068,"source_end_line":5077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5068-L5077","statement_sha256":"9a15015b54d9d3449a9002473080b6e7c9fdee7652701d53a71b9c5fa1daeba1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5521,"rank":5521,"depth":7,"x":1996.821,"y":633.733,"cluster":"scheme-morphisms"},{"id":"stacks:052A","tag":"052A","title":"Generic flatness · Proposition 052A","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf of O_X-modules. Assume • S is integral, • f is of finite type, and • F is a finite type O_X-module. Then there exists an open dense subscheme U ⊂ S such that X_U → U is flat and of finite presentation and such that F|_X_U is flat over U and of finite presentation over O_X_U.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.\nAssume\n\\begin{enumerate}\n\\item $S$ is integral,\n\\item $f$ is of finite type, and\n\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.\n\\end{enumerate}\nThen there exists an open dense subscheme $U \\subset S$ such that\n$X_U \\to U$ is flat and of finite presentation and such that\n$\\mathcal{F}|_{X_U}$ is flat over $U$ and of finite presentation\nover $\\mathcal{O}_{X_U}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generic flatness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052A","source_file":"morphisms.tex","source_line":5171,"source_end_line":5185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5171-L5185","statement_sha256":"d1bbd1122d1f24796d4db9caee39348bd9e989a714da5fd8e0a29310734a664c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5522,"rank":5522,"depth":15,"x":2091.837,"y":695.331,"cluster":"scheme-morphisms"},{"id":"stacks:052B","tag":"052B","title":"Generic flatness, reduced case · Proposition 052B","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf of O_X-modules. Assume • S is reduced, • f is of finite type, and • F is a finite type O_X-module. Then there exists an open dense subscheme U ⊂ S such that X_U → U is flat and of finite presentation and such that F|_X_U is flat over U and of finite presentation over O_X_U.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.\nAssume\n\\begin{enumerate}\n\\item $S$ is reduced,\n\\item $f$ is of finite type, and\n\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.\n\\end{enumerate}\nThen there exists an open dense subscheme $U \\subset S$ such that\n$X_U \\to U$ is flat and of finite presentation and such that\n$\\mathcal{F}|_{X_U}$ is flat over $U$ and of finite presentation\nover $\\mathcal{O}_{X_U}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generic flatness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052B","source_file":"morphisms.tex","source_line":5246,"source_end_line":5260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5246-L5260","statement_sha256":"8fed3a18623ad5ec067d110e6ce05218ca7dea9c29526a9dd1723045e59dfba0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5523,"rank":5523,"depth":16,"x":1971.92,"y":703.846,"cluster":"scheme-morphisms"},{"id":"stacks:02FX","tag":"02FX","title":"Morphisms and dimensions of fibres · Lemma 02FX","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X and set s = f(x). Assume f is locally of finite type. Then dim_x(X_s) = dim(O_X_s, x) + trdeg_kappa(s)(kappa(x)).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ and set $s = f(x)$.\nAssume $f$ is locally of finite type.\nThen\n$$\n\\dim_x(X_s) =\n\\dim(\\mathcal{O}_{X_s, x}) + \\text{trdeg}_{\\kappa(s)}(\\kappa(x)).\n$$","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FX","source_file":"morphisms.tex","source_line":5377,"source_end_line":5387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5377-L5387","statement_sha256":"bd3c359871a158bb214b98f427390f26fb39d36bec689717851688855dc749da","origin":"The Stacks Project","memory_eligible":false,"source_rank":5524,"rank":5524,"depth":23,"x":2053.714,"y":629.327,"cluster":"scheme-morphisms"},{"id":"stacks:02JS","tag":"02JS","title":"Morphisms and dimensions of fibres · Lemma 02JS","summary":"Let f : X → Y and g : Y → S be morphisms of schemes. Let x ∈ X and set y = f(x), s = g(y). Assume f and g locally of finite type. Then dim_x(X_s) ≤ dim_x(X_y) + dim_y(Y_s). Moreover, equality holds if O_X_s, x is flat over O_Y_s, y, which holds for example if O_X, x is flat over O_Y, y.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes.\nLet $x \\in X$ and set $y = f(x)$, $s = g(y)$.\nAssume $f$ and $g$ locally of finite type.\nThen\n$$\n\\dim_x(X_s) \\leq \\dim_x(X_y) + \\dim_y(Y_s).\n$$\nMoreover, equality holds if $\\mathcal{O}_{X_s, x}$ is flat\nover $\\mathcal{O}_{Y_s, y}$, which holds for example if $\\mathcal{O}_{X, x}$\nis flat over $\\mathcal{O}_{Y, y}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JS","source_file":"morphisms.tex","source_line":5395,"source_end_line":5407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5395-L5407","statement_sha256":"eead49c106f373141a916dba49efdfb563727edfb5b543f959c07294850ab725","origin":"The Stacks Project","memory_eligible":false,"source_rank":5525,"rank":5525,"depth":24,"x":2053.325,"y":730.955,"cluster":"scheme-morphisms"},{"id":"stacks:02FY","tag":"02FY","title":"Morphisms and dimensions of fibres · Lemma 02FY","summary":"Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a fibre product diagram of schemes. Assume f locally of finite type. Suppose that x' ∈ X', x = g'(x'), s' = f'(x') and s = g(s') = f(x). Then • dim_x(X_s) = dim_x'(X'_s'), • if F is the fibre of the morphism X'_s' → X_s over x, then dim(O_F, x') = dim(O_X'_s', x') - dim(O_X_s, x) = trdeg_kappa(s)(kappa(x)) - trdeg_kappa(s')(kappa(x')) In particular dim(O_X'_s', x') ≥ dim(O_X_s, x) and trdeg_kappa(s)(kappa(x))…","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nbe a fibre product diagram of schemes. Assume $f$ locally of finite type.\nSuppose that $x' \\in X'$, $x = g'(x')$, $s' = f'(x')$ and\n$s = g(s') = f(x)$. Then\n\\begin{enumerate}\n\\item $\\dim_x(X_s) = \\dim_{x'}(X'_{s'})$,\n\\item if $F$ is the fibre of the morphism $X'_{s'} \\to X_s$\nover $x$, then\n$$\n\\dim(\\mathcal{O}_{F, x'}) =\n\\dim(\\mathcal{O}_{X'_{s'}, x'}) - \\dim(\\mathcal{O}_{X_s, x}) =\n\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) -\n\\text{trdeg}_{\\kappa(s')}(\\kappa(x'))\n$$\nIn particular $\\dim(\\mathcal{O}_{X'_{s'}, x'}) \\geq \\dim(\\mathcal{O}_{X_s, x})$\nand $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) \\geq\n\\text{trdeg}_{\\kappa(s')}(\\kappa(x'))$.\n\\item given $s', s, x$ there exists a choice of $x'$ such that\n$\\dim(\\mathcal{O}_{X'_{s'}, x'}) = \\dim(\\mathcal{O}_{X_s, x})$ and\n$\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) = \\text{trdeg}_{\\kappa(s')}(\\kappa(x'))$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FY","source_file":"morphisms.tex","source_line":5424,"source_end_line":5453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5424-L5453","statement_sha256":"bfb3bf20088ee45cf7bb3dd56c77908807ff0cde850da0992035981fde0a0c5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5526,"rank":5526,"depth":26,"x":1971.672,"y":655.598,"cluster":"scheme-morphisms"},{"id":"stacks:02FZ","tag":"02FZ","title":"Morphisms and dimensions of fibres · Lemma 02FZ","summary":"[EGA] Let f : X → S be a morphism of schemes. Let n ≥ 0. Assume f is locally of finite type. The set U_n = (x ∈ X mid dim_x X_f(x) ≤ n) is open in X.","statement_latex":"\\begin{reference}\n\\cite[IV Theorem 13.1.3]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nLet $n \\geq 0$. Assume $f$ is locally of finite type.\nThe set\n$$\nU_n = \\{x \\in X \\mid \\dim_x X_{f(x)} \\leq n\\}\n$$\nis open in $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02FZ","source_file":"morphisms.tex","source_line":5467,"source_end_line":5479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5467-L5479","statement_sha256":"670872ecf200d8a0070bd08bd076e1c79751940634e7684102bb09c96dca2e73","origin":"The Stacks Project","memory_eligible":false,"source_rank":5527,"rank":5527,"depth":31,"x":2092.796,"y":664.858,"cluster":"scheme-morphisms"},{"id":"stacks:0A3V","tag":"0A3V","title":"Morphisms and dimensions of fibres · Lemma 0A3V","summary":"Let f : X → Y be a morphism of finite type with Y quasi-compact. Then the dimension of the fibres of f is bounded.","statement_latex":"Let $f : X \\to Y$ be a morphism of finite type with $Y$ quasi-compact.\nThen the dimension of the fibres of $f$ is bounded.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3V","source_file":"morphisms.tex","source_line":5487,"source_end_line":5491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5487-L5491","statement_sha256":"d92661887823aa7e8698f8a28aa74f2e286bec9aa39251688f0c5c65492a57e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5528,"rank":5528,"depth":32,"x":1995.784,"y":726.919,"cluster":"scheme-morphisms"},{"id":"stacks:02G0","tag":"02G0","title":"Morphisms and dimensions of fibres · Lemma 02G0","summary":"Let f : X → S be a morphism of schemes. Let n ≥ 0. Assume f is locally of finite presentation. The open U_n = (x ∈ X mid dim_x X_f(x) ≤ n) of Lemma [Tag 02FZ] is retrocompact in X. (See Topology, Definition [Tag 005A].)","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $n \\geq 0$. Assume $f$ is locally of finite presentation.\nThe open\n$$\nU_n = \\{x \\in X \\mid \\dim_x X_{f(x)} \\leq n\\}\n$$\nof Lemma \\ref{lemma-openness-bounded-dimension-fibres} is retrocompact\nin $X$. (See Topology, Definition \\ref{topology-definition-quasi-compact}.)","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02G0","source_file":"morphisms.tex","source_line":5503,"source_end_line":5513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5503-L5513","statement_sha256":"38e5227b2712bc90d186686277d7c80911f0c2bae05e925b011b5a77abc198bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5529,"rank":5529,"depth":32,"x":2017.467,"y":625.849,"cluster":"scheme-morphisms"},{"id":"stacks:06RU","tag":"06RU","title":"Morphisms and dimensions of fibres · Lemma 06RU","summary":"Let f : X → S be a morphism of schemes. Let x leadsto x' be a nontrivial specialization of points in X lying over the same point s ∈ S. Assume f is locally of finite type. Then • dim_x(X_s) ≤ dim_x'(X_s), • dim(O_X_s, x) < dim(O_X_s, x'), and • trdeg_kappa(s)(kappa(x)) > trdeg_kappa(s)(kappa(x')).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\leadsto x'$ be a nontrivial specialization of points in $X$\nlying over the same point $s \\in S$. Assume $f$ is locally of finite type.\nThen\n\\begin{enumerate}\n\\item $\\dim_x(X_s) \\leq \\dim_{x'}(X_s)$,\n\\item $\\dim(\\mathcal{O}_{X_s, x}) < \\dim(\\mathcal{O}_{X_s, x'})$, and\n\\item $\\text{trdeg}_{\\kappa(s)}(\\kappa(x)) >\n\\text{trdeg}_{\\kappa(s)}(\\kappa(x'))$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RU","source_file":"morphisms.tex","source_line":5528,"source_end_line":5540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5528-L5540","statement_sha256":"d4955cc8a98b8499dd664ac0f0fc983c5f0439663e7e3da46aa0d3bc72196636","origin":"The Stacks Project","memory_eligible":false,"source_rank":5530,"rank":5530,"depth":22,"x":2082.923,"y":712.903,"cluster":"scheme-morphisms"},{"id":"stacks:02NJ","tag":"02NJ","title":"Morphisms of given relative dimension · Definition 02NJ","summary":"Let f : X → S be a morphism of schemes. Assume f is locally of finite type. • We say f is of relative dimension ≤ d at x if dim_x(X_f(x)) ≤ d. • We say f is of relative dimension ≤ d if dim_x(X_f(x)) ≤ d for all x ∈ X. • We say f is of relative dimension d if all nonempty fibres X_s are equidimensional of dimension d.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $f$ is locally of finite type.\n\\begin{enumerate}\n\\item We say $f$ is of {\\it relative dimension $\\leq d$ at $x$} if\n$\\dim_x(X_{f(x)}) \\leq d$.\n\\item We say $f$ is of {\\it relative dimension $\\leq d$} if\n$\\dim_x(X_{f(x)}) \\leq d$ for all $x \\in X$.\n\\item We say $f$ is of {\\it relative dimension $d$} if\nall nonempty fibres $X_s$ are equidimensional of dimension $d$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of given relative dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NJ","source_file":"morphisms.tex","source_line":5563,"source_end_line":5575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5563-L5575","statement_sha256":"8bcc2612b443c76bf143c52cec5b3a0e7172556c9c3055c025adf7ff2e945507","origin":"The Stacks Project","memory_eligible":false,"source_rank":5531,"rank":5531,"depth":0,"x":1964.349,"y":685.783,"cluster":"scheme-morphisms"},{"id":"stacks:02NK","tag":"02NK","title":"Morphisms of given relative dimension · Lemma 02NK","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. If f has relative dimension d, then so does any base change of f. Same for relative dimension ≤ d.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nIf $f$ has relative dimension $d$, then so does any base change of $f$.\nSame for relative dimension $\\leq d$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of given relative dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NK","source_file":"morphisms.tex","source_line":5581,"source_end_line":5586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5581-L5586","statement_sha256":"ca1005e4af63f16e7e60539b933431af9cff3110e746246fccefd05f51f29f12","origin":"The Stacks Project","memory_eligible":false,"source_rank":5532,"rank":5532,"depth":27,"x":2073.871,"y":638.379,"cluster":"scheme-morphisms"},{"id":"stacks:02NL","tag":"02NL","title":"Morphisms of given relative dimension · Lemma 02NL","summary":"Let f : X → Y, g : Y → Z be locally of finite type. If f has relative dimension ≤ d and g has relative dimension ≤ e then g ∘ f has relative dimension ≤ d + e. If • f has relative dimension d, • g has relative dimension e, and • f is flat, then g ∘ f has relative dimension d + e.","statement_latex":"Let $f : X \\to Y$, $g : Y \\to Z$ be locally of finite type.\nIf $f$ has relative dimension $\\leq d$ and $g$ has relative dimension $\\leq e$\nthen $g \\circ f$ has relative dimension $\\leq d + e$.\nIf\n\\begin{enumerate}\n\\item $f$ has relative dimension $d$,\n\\item $g$ has relative dimension $e$, and\n\\item $f$ is flat,\n\\end{enumerate}\nthen $g \\circ f$ has relative dimension $d + e$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of given relative dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NL","source_file":"morphisms.tex","source_line":5593,"source_end_line":5605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5593-L5605","statement_sha256":"f22bbe08d16c04d821e303f4959953dae988768f304ab59aa4eeb9a29bbab653","origin":"The Stacks Project","memory_eligible":false,"source_rank":5533,"rank":5533,"depth":25,"x":2031.124,"y":735.724,"cluster":"scheme-morphisms"},{"id":"stacks:02NM","tag":"02NM","title":"Morphisms of given relative dimension · Lemma 02NM","summary":"Cohen-Macaulay morphisms decompose into clopens of pure relative dimension Let f : X → S be a morphism of schemes. Assume that • f is flat, • f is locally of finite presentation, and • for all s ∈ S the fibre X_s is Cohen-Macaulay (Properties, Definition [Tag 02IO]) Then there exist open and closed subschemes X_d ⊂ X such that X = coprod_d ≥ 0 X_d and f|_X_d : X_d → S has relative dimension d.","statement_latex":"\\begin{slogan}\nCohen-Macaulay morphisms decompose into clopens of pure relative dimension\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nAssume that\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $f$ is locally of finite presentation, and\n\\item for all $s \\in S$ the fibre $X_s$ is Cohen-Macaulay\n(Properties, Definition \\ref{properties-definition-Cohen-Macaulay})\n\\end{enumerate}\nThen there exist open and closed subschemes $X_d \\subset X$\nsuch that $X = \\coprod_{d \\geq 0} X_d$ and $f|_{X_d} : X_d \\to S$\nhas relative dimension $d$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of given relative dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NM","source_file":"morphisms.tex","source_line":5617,"source_end_line":5633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5617-L5633","statement_sha256":"105e04a0962c2b7ca0cfdf48ce351106b541dc86c00eec859a9a3987a3fb1e92","origin":"The Stacks Project","memory_eligible":false,"source_rank":5534,"rank":5534,"depth":36,"x":1984.245,"y":639.446,"cluster":"scheme-morphisms"},{"id":"stacks:0397","tag":"0397","title":"Morphisms of given relative dimension · Lemma 0397","summary":"Let f : X → S be a morphism of schemes. Assume f is locally of finite type. Let x ∈ X with s = f(x). Then f is quasi-finite at x if and only if dim_x(X_s) = 0. In particular, f is locally quasi-finite if and only if f has relative dimension 0.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $f$ is locally of finite type.\nLet $x \\in X$ with $s = f(x)$.\nThen $f$ is quasi-finite at $x$ if and only if $\\dim_x(X_s) = 0$.\nIn particular, $f$ is locally quasi-finite if and only if $f$ has relative\ndimension $0$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of given relative dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0397","source_file":"morphisms.tex","source_line":5641,"source_end_line":5649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5641-L5649","statement_sha256":"9767cb16626b2e4bcabb945c5c7c7b4193155bafa17bab9e8d578ed803ee5b9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5535,"rank":5535,"depth":27,"x":2096.517,"y":683.951,"cluster":"scheme-morphisms"},{"id":"stacks:0AFE","tag":"0AFE","title":"Morphisms of given relative dimension · Lemma 0AFE","summary":"Let f : X → Y be a morphism of locally Noetherian schemes which is flat, locally of finite type and of relative dimension d. For every point x in X with image y in Y we have dim_x(X) = dim_y(Y) + d.","statement_latex":"Let $f : X \\to Y$ be a morphism of locally Noetherian schemes\nwhich is flat, locally of finite type and of relative dimension $d$.\nFor every point $x$ in $X$ with image\n$y$ in $Y$ we have $\\dim_x(X) = \\dim_y(Y) + d$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Morphisms of given relative dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFE","source_file":"morphisms.tex","source_line":5678,"source_end_line":5684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5678-L5684","statement_sha256":"1d6511824be3f1e1f2b2945d56aeece5dfbbf6624cffabd5d57e84c73f7356ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":5536,"rank":5536,"depth":18,"x":1977.644,"y":714.916,"cluster":"scheme-morphisms"},{"id":"stacks:01UC","tag":"01UC","title":"Syntomic morphisms · Definition 01UC","summary":"Let f : X → S be a morphism of schemes. • We say that f is syntomic at x ∈ X if there exists an affine open neighbourhood Spec(A) = U ⊂ X of x and affine open Spec(R) = V ⊂ S with f(U) ⊂ V such that the induced ring map R → A is syntomic. • We say that f is syntomic if it is syntomic at every point of X. • If S = Spec(k) and f is syntomic, then we say that X is a local complete intersection over k. • A morphism of affine schemes f : X → S is called standard syntomic if…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say that $f$ is {\\it syntomic at $x \\in X$} if\nthere exists an affine open neighbourhood $\\Spec(A) = U \\subset X$\nof $x$ and affine open $\\Spec(R) = V \\subset S$\nwith $f(U) \\subset V$ such that the induced ring map\n$R \\to A$ is syntomic.\n\\item We say that $f$ is {\\it syntomic} if it is syntomic\nat every point of $X$.\n\\item If $S = \\Spec(k)$ and $f$ is syntomic, then we say that\n$X$ is a {\\it local complete intersection over $k$}.\n\\item A morphism of affine schemes $f : X \\to S$\nis called {\\it standard syntomic} if there exists a\nglobal relative complete intersection\n$R \\to R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ (see\nAlgebra,\nDefinition \\ref{algebra-definition-relative-global-complete-intersection})\nsuch that $X \\to S$ is isomorphic to\n$$\n\\Spec(R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)) \\to \\Spec(R).\n$$\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UC","source_file":"morphisms.tex","source_line":5734,"source_end_line":5758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5734-L5758","statement_sha256":"de9bb585d4473e0594116f1f227642de0da57472d17d4e648ea4dfc30862dadb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5537,"rank":5537,"depth":1,"x":2040.554,"y":624.408,"cluster":"scheme-morphisms"},{"id":"stacks:01UD","tag":"01UD","title":"Syntomic morphisms · Lemma 01UD","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is syntomic. • For every affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the ring map O_S(V) → O_X(U) is syntomic. • There exists an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is syntomic. • There exists an affine open covering S = ⋃_j ∈ J V_j and affine open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that the…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is syntomic.\n\\item For every affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is syntomic.\n\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis syntomic.\n\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is\nsyntomic, for all $j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is syntomic then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is syntomic.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UD","source_file":"morphisms.tex","source_line":5785,"source_end_line":5806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5785-L5806","statement_sha256":"d5b70e98872f4ecf56e76d5c2475f08394191ab939a1e636574161567079b7c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5538,"rank":5538,"depth":36,"x":2067.014,"y":727.096,"cluster":"scheme-morphisms"},{"id":"stacks:01UH","tag":"01UH","title":"Syntomic morphisms · Lemma 01UH","summary":"The composition of two morphisms which are syntomic is syntomic.","statement_latex":"The composition of two morphisms which are syntomic is syntomic.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UH","source_file":"morphisms.tex","source_line":5823,"source_end_line":5826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5823-L5826","statement_sha256":"4a20406b1b5f0dd3c5b2a17f7179b8a9c3eb7e297f6f475cd6b4e7c5a2d59309","origin":"The Stacks Project","memory_eligible":false,"source_rank":5539,"rank":5539,"depth":37,"x":1964.674,"y":666.241,"cluster":"scheme-morphisms"},{"id":"stacks:01UI","tag":"01UI","title":"Syntomic morphisms · Lemma 01UI","summary":"The base change of a morphism which is syntomic is syntomic.","statement_latex":"The base change of a morphism which is syntomic is syntomic.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UI","source_file":"morphisms.tex","source_line":5838,"source_end_line":5841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5838-L5841","statement_sha256":"d5e58fb8e3c751be891f61f89c4b9544113e06ba61d311348366772e09944f9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5540,"rank":5540,"depth":37,"x":2089.379,"y":653.013,"cluster":"scheme-morphisms"},{"id":"stacks:01UJ","tag":"01UJ","title":"Syntomic morphisms · Lemma 01UJ","summary":"Any open immersion is syntomic.","statement_latex":"Any open immersion is syntomic.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UJ","source_file":"morphisms.tex","source_line":5853,"source_end_line":5856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5853-L5856","statement_sha256":"515e33fb07c079bee1ec14ae23ef41c1e2c545890172be44fe8b7e82511f4668","origin":"The Stacks Project","memory_eligible":false,"source_rank":5541,"rank":5541,"depth":0,"x":2007.864,"y":733.723,"cluster":"scheme-morphisms"},{"id":"stacks:01UK","tag":"01UK","title":"Syntomic morphisms · Lemma 01UK","summary":"A syntomic morphism is locally of finite presentation.","statement_latex":"A syntomic morphism is locally of finite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UK","source_file":"morphisms.tex","source_line":5862,"source_end_line":5865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5862-L5865","statement_sha256":"5500725fe7c5bede429755d81148587bbd6f7e79a7ae97f4902036586e4f5a1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5542,"rank":5542,"depth":0,"x":2003.056,"y":627.699,"cluster":"scheme-morphisms"},{"id":"stacks:01UL","tag":"01UL","title":"Syntomic morphisms · Lemma 01UL","summary":"A syntomic morphism is flat.","statement_latex":"A syntomic morphism is flat.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UL","source_file":"morphisms.tex","source_line":5872,"source_end_line":5875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5872-L5875","statement_sha256":"e4c837523da7554db0d37dd511766f900f47b318fb9ce04e2f37e6d79076fda1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5543,"rank":5543,"depth":0,"x":2092.076,"y":703.333,"cluster":"scheme-morphisms"},{"id":"stacks:056F","tag":"056F","title":"Syntomic morphisms · Lemma 056F","summary":"A syntomic morphism is universally open.","statement_latex":"A syntomic morphism is universally open.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056F","source_file":"morphisms.tex","source_line":5881,"source_end_line":5884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5881-L5884","statement_sha256":"1497b83dc1319d91a4b5f12fd32016ab26187e4b18501de3e7ec2e1c06fd5bbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":5544,"rank":5544,"depth":18,"x":1965.309,"y":698.061,"cluster":"scheme-morphisms"},{"id":"stacks:01UG","tag":"01UG","title":"Syntomic morphisms · Lemma 01UG","summary":"Let k be a field. Let X be a scheme locally of finite type over k. The following are equivalent: • X is a local complete intersection over k, • for every x ∈ X there exists an affine open U = Spec(R) ⊂ X neighbourhood of x such that R ≅ k[x_1, …, x_n]/(f_1, …, f_c) is a global complete intersection over k, and • for every x ∈ X the local ring O_X, x is a complete intersection over k.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme locally of finite type over $k$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $X$ is a local complete intersection over $k$,\n\\item for every $x \\in X$ there exists an affine open\n$U = \\Spec(R) \\subset X$ neighbourhood of $x$\nsuch that $R \\cong k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$\nis a global complete intersection over $k$, and\n\\item for every $x \\in X$ the local ring $\\mathcal{O}_{X, x}$\nis a complete intersection over $k$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UG","source_file":"morphisms.tex","source_line":5902,"source_end_line":5916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5902-L5916","statement_sha256":"9567af0738e9dcb1aa6a9ffc15282a45e6ccc993a164482312d458b8e4f1f961","origin":"The Stacks Project","memory_eligible":false,"source_rank":5545,"rank":5545,"depth":28,"x":2063.257,"y":629.856,"cluster":"scheme-morphisms"},{"id":"stacks:01UE","tag":"01UE","title":"Syntomic morphisms · Lemma 01UE","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point with image s = f(x). Let V ⊂ S be an affine open neighbourhood of s. The following are equivalent • The morphism f is syntomic at x. • There exist an affine open U ⊂ X with x ∈ U and f(U) ⊂ V such that f|_U : U → V is standard syntomic. • The morphism f is of finite presentation at x, the local ring map O_S, s → O_X, x is flat and O_X, x/ m_s O_X, x is a complete intersection over kappa(s) (see Algebra,…","statement_latex":"Let $f : X  \\to S$ be a morphism of schemes. Let $x \\in X$ be a point\nwith image $s = f(x)$. Let $V \\subset S$ be an affine open neighbourhood\nof $s$. The following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is syntomic at $x$.\n\\item There exist an affine open $U \\subset X$ with $x \\in U$ and\n$f(U) \\subset V$ such that $f|_U : U \\to V$ is standard syntomic.\n\\item The morphism $f$ is of finite presentation at $x$, the local ring map\n$\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$\nis flat and $\\mathcal{O}_{X, x}/\\mathfrak m_s \\mathcal{O}_{X, x}$\nis a complete intersection over $\\kappa(s)$ (see\nAlgebra, Definition \\ref{algebra-definition-lci-local-ring}).\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UE","source_file":"morphisms.tex","source_line":5931,"source_end_line":5946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5931-L5946","statement_sha256":"30eefd643cc871caa6bb55f792ec26c0c90edef25d0fa9a99e9867057440517f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5546,"rank":5546,"depth":35,"x":2045.839,"y":735.978,"cluster":"scheme-morphisms"},{"id":"stacks:01UF","tag":"01UF","title":"Syntomic morphisms · Lemma 01UF","summary":"Let f : X → S be a morphism of schemes. If f is flat, locally of finite presentation, and all fibres X_s are local complete intersections, then f is syntomic.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf $f$ is flat, locally of finite presentation, and all\nfibres $X_s$ are local complete intersections, then $f$\nis syntomic.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UF","source_file":"morphisms.tex","source_line":5953,"source_end_line":5959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5953-L5959","statement_sha256":"3f45ffb5fb576c234df398463cbf91afcb60bbbf722cce99b7ff4f3c185777d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5547,"rank":5547,"depth":36,"x":1973.172,"y":647.634,"cluster":"scheme-morphisms"},{"id":"stacks:02V3","tag":"02V3","title":"Syntomic morphisms · Lemma 02V3","summary":"Let f : X → S be a morphism of schemes. Assume f locally of finite type. Formation of the set T = (x ∈ X mid O_X_f(x), x is a complete intersection over kappa(f(x))) commutes with arbitrary base change: For any morphism g : S' → S, consider the base change f' : X' → S' of f and the projection g' : X' → X. Then the corresponding set T' for the morphism f' is equal to T' = (g')^-1(T). In particular, if f is assumed flat, and locally of finite presentation then the same…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $f$ locally of finite type. Formation of the set\n$$\nT = \\{x \\in X \\mid \\mathcal{O}_{X_{f(x)}, x}\n\\text{ is a complete intersection over }\\kappa(f(x))\\}\n$$\ncommutes with arbitrary base change:\nFor any morphism $g : S' \\to S$, consider\nthe base change $f' : X' \\to S'$ of $f$ and the\nprojection $g' : X' \\to X$. Then the corresponding\nset $T'$ for the morphism $f'$ is equal to $T' = (g')^{-1}(T)$.\nIn particular, if $f$ is assumed flat, and locally of finite\npresentation then the same holds for the open set of points\nwhere $f$ is syntomic.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V3","source_file":"morphisms.tex","source_line":5973,"source_end_line":5989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L5973-L5989","statement_sha256":"88c2c3dbf067addf65f96c9b5ef0e96d3ac2201b3f333a689362e1f39701b9b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5548,"rank":5548,"depth":36,"x":2098.091,"y":671.601,"cluster":"scheme-morphisms"},{"id":"stacks:02K0","tag":"02K0","title":"Syntomic morphisms · Lemma 02K0","summary":"Let R be a ring. Let R → A = R[x_1, …, x_n]/(f_1, …, f_c) be a relative global complete intersection. Set S = Spec(R) and X = Spec(A). Consider the morphism f : X → S associated to the ring map R → A. The function x ↦ dim_x(X_f(x)) is constant with value n - c.","statement_latex":"Let $R$ be a ring.\nLet $R \\to A = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ be a relative\nglobal complete intersection. Set $S = \\Spec(R)$ and\n$X = \\Spec(A)$. Consider the morphism\n$f : X \\to S$ associated to the ring map $R \\to A$.\nThe function $x \\mapsto \\dim_x(X_{f(x)})$ is constant with value $n - c$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02K0","source_file":"morphisms.tex","source_line":6005,"source_end_line":6013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6005-L6013","statement_sha256":"addea3ea8d96c146849a07435620056bf0caf502224c16469c9cf6cf6ccc0195","origin":"The Stacks Project","memory_eligible":false,"source_rank":5549,"rank":5549,"depth":1,"x":1986.444,"y":724.934,"cluster":"scheme-morphisms"},{"id":"stacks:02K1","tag":"02K1","title":"Syntomic morphisms · Lemma 02K1","summary":"Let f : X → S be a syntomic morphism. The function x ↦ dim_x(X_f(x)) is locally constant on X.","statement_latex":"Let $f : X \\to S$ be a syntomic morphism. The function\n$x \\mapsto \\dim_x(X_{f(x)})$ is locally constant on $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02K1","source_file":"morphisms.tex","source_line":6029,"source_end_line":6033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6029-L6033","statement_sha256":"9793e7cd9615658f95d4c38a5aca7da837ca525e63d8f462ebb59457ef284b01","origin":"The Stacks Project","memory_eligible":false,"source_rank":5550,"rank":5550,"depth":36,"x":2025.973,"y":622.017,"cluster":"scheme-morphisms"},{"id":"stacks:02K2","tag":"02K2","title":"Syntomic morphisms · Definition 02K2","summary":"Let d ≥ 0 be an integer. We say a morphism of schemes f : X → S is syntomic of relative dimension d if f is syntomic and the function dim_x(X_f(x)) = d for all x ∈ X.","statement_latex":"Let $d \\geq 0$ be an integer. We say a morphism of schemes $f : X \\to S$\nis {\\it syntomic of relative dimension $d$} if $f$ is syntomic and\nthe function $\\dim_x(X_{f(x)}) = d$ for all $x \\in X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02K2","source_file":"morphisms.tex","source_line":6046,"source_end_line":6051,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6046-L6051","statement_sha256":"6fb261adc9740b23f4e3de7f426942d047dc39cd96613a9216f07b73bdea9a38","origin":"The Stacks Project","memory_eligible":false,"source_rank":5551,"rank":5551,"depth":0,"x":2079.712,"y":720.564,"cluster":"scheme-morphisms"},{"id":"stacks:02K3","tag":"02K3","title":"Syntomic morphisms · Lemma 02K3","summary":"Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & S be a commutative diagram of morphisms of schemes. Assume that • f is surjective and syntomic, • p is syntomic, and • q is locally of finite presentation. Then q is syntomic.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume that\n\\begin{enumerate}\n\\item $f$ is surjective and syntomic,\n\\item $p$ is syntomic, and\n\\item $q$ is locally of finite presentation\\footnote{In fact, if $f$ is\nsurjective, flat, and locally of finite presentation and $p$ is syntomic,\nthen both $q$ and $f$ are syntomic, see\nDescent, Lemma \\ref{descent-lemma-syntomic-permanence}.}.\n\\end{enumerate}\nThen $q$ is syntomic.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02K3","source_file":"morphisms.tex","source_line":6057,"source_end_line":6077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6057-L6077","statement_sha256":"4d8457d760a90fb31f3e68c5400bb0b4bb55befe24107ab6cd238e2d12e711a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5552,"rank":5552,"depth":38,"x":1960.564,"y":678.292,"cluster":"scheme-morphisms"},{"id":"stacks:01R2","tag":"01R2","title":"Conormal sheaf of an immersion · Definition 01R2","summary":"Let i : Z → X be an immersion. The conormal sheaf C_Z/X of Z in X or the conormal sheaf of i is the quasi-coherent O_Z-module I/I^2 described above.","statement_latex":"Let $i : Z \\to X$ be an immersion. The {\\it conormal sheaf\n$\\mathcal{C}_{Z/X}$ of $Z$ in $X$} or the {\\it conormal sheaf of $i$}\nis the quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{I}/\\mathcal{I}^2$\ndescribed above.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Conormal sheaf of an immersion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01R2","source_file":"morphisms.tex","source_line":6163,"source_end_line":6169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6163-L6169","statement_sha256":"b65ba2839067ebb5eacf80822d3d143508576abedf4b5ee4f67888e852eae21c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5553,"rank":5553,"depth":0,"x":2082.695,"y":641.772,"cluster":"scheme-morphisms"},{"id":"stacks:01R3","tag":"01R3","title":"Conormal sheaf of an immersion · Lemma 01R3","summary":"Let i : Z → X be an immersion. The conormal sheaf of i has the following properties: • Let U ⊂ X be any open subscheme such that i factors as Z xrightarrowi' U → X where i' is a closed immersion. Let I = Ker((i')^sharp) ⊂ O_U. Then C_Z/X = (i')^*I and i'_*C_Z/X = I/I^2 • For any affine open Spec(R) = U ⊂ X such that Z ∩ U = Spec(R/I) there is a canonical isomorphism Γ(Z ∩ U, C_Z/X) = I/I^2.","statement_latex":"Let $i : Z \\to X$ be an immersion. The conormal sheaf\nof $i$ has the following properties:\n\\begin{enumerate}\n\\item Let $U \\subset X$ be any open subscheme such that $i$\nfactors as $Z \\xrightarrow{i'} U \\to X$ where $i'$ is a closed\nimmersion. Let $\\mathcal{I} = \\Ker((i')^\\sharp) \\subset \\mathcal{O}_U$.\nThen\n$$\n\\mathcal{C}_{Z/X} = (i')^*\\mathcal{I}\\quad\\text{and}\\quad\ni'_*\\mathcal{C}_{Z/X} = \\mathcal{I}/\\mathcal{I}^2\n$$\n\\item\nFor any affine open $\\Spec(R) = U \\subset X$\nsuch that $Z \\cap U = \\Spec(R/I)$ there is a\ncanonical isomorphism\n$\\Gamma(Z \\cap U, \\mathcal{C}_{Z/X}) = I/I^2$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Conormal sheaf of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01R3","source_file":"morphisms.tex","source_line":6185,"source_end_line":6204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6185-L6204","statement_sha256":"ea87d7d019a6e0105dac41f16f942f622be99e7f52dc4ef86710dd1ce5fd3490","origin":"The Stacks Project","memory_eligible":false,"source_rank":5554,"rank":5554,"depth":0,"x":2021.867,"y":738.221,"cluster":"scheme-morphisms"},{"id":"stacks:01R4","tag":"01R4","title":"Conormal sheaf of an immersion · Lemma 01R4","summary":"Let xymatrix Z ar[r]_i ar[d]_f & X ar[d]^g Z' ar[r]^i' & X' be a commutative diagram in the category of schemes. Assume i, i' immersions. There is a canonical map of O_Z-modules f^*C_Z'/X' → C_Z/X characterized by the following property: For every pair of affine opens (Spec(R) = U ⊂ X, Spec(R') = U' ⊂ X') with g(U) ⊂ U' such that Z ∩ U = Spec(R/I) and Z' ∩ U' = Spec(R'/I') the induced map Γ(Z' ∩ U', C_Z'/X') = I'/I'^2 → I/I^2 = Γ(Z ∩ U, C_Z/X) is the one induced by the…","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\\nZ' \\ar[r]^{i'} & X'\n}\n$$\nbe a commutative diagram in the category of schemes.\nAssume $i$, $i'$ immersions. There is a canonical map\nof $\\mathcal{O}_Z$-modules\n$$\nf^*\\mathcal{C}_{Z'/X'}\n\\longrightarrow\n\\mathcal{C}_{Z/X}\n$$\ncharacterized by the following property: For every pair of affine opens\n$(\\Spec(R) = U \\subset X, \\Spec(R') = U' \\subset X')$ with\n$g(U) \\subset U'$ such that\n$Z \\cap U = \\Spec(R/I)$ and $Z' \\cap U' = \\Spec(R'/I')$\nthe induced map\n$$\n\\Gamma(Z' \\cap U', \\mathcal{C}_{Z'/X'}) = I'/I'^2\n\\longrightarrow\nI/I^2 = \\Gamma(Z \\cap U, \\mathcal{C}_{Z/X})\n$$\nis the one induced by the ring map $g^\\sharp : R' \\to R$ which\nhas the property $g^\\sharp(I') \\subset I$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Conormal sheaf of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01R4","source_file":"morphisms.tex","source_line":6211,"source_end_line":6240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6211-L6240","statement_sha256":"22fa0e56178acc4d65a8e9b0ad3ee0506e2833ca376684878d435c1493171a0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5555,"rank":5555,"depth":8,"x":1989.085,"y":632.346,"cluster":"scheme-morphisms"},{"id":"stacks:0473","tag":"0473","title":"Conormal sheaf of an immersion · Lemma 0473","summary":"Let xymatrix Z ar[r]_i ar[d]_f & X ar[d]^g Z' ar[r]^i' & X' be a fibre product diagram in the category of schemes with i, i' immersions. Then the canonical map f^*C_Z'/X' → C_Z/X of Lemma [Tag 01R4] is surjective. If g is flat, then it is an isomorphism.","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\\nZ' \\ar[r]^{i'} & X'\n}\n$$\nbe a fibre product diagram in the category of schemes with\n$i$, $i'$ immersions. Then the canonical map\n$f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ of\nLemma \\ref{lemma-conormal-functorial}\nis surjective. If $g$ is flat, then it is an isomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Conormal sheaf of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0473","source_file":"morphisms.tex","source_line":6277,"source_end_line":6291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6277-L6291","statement_sha256":"8fb8f38cb48b640ed2fc0d2f83f4358600ae4a7380d889dd39ae843024365be3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5556,"rank":5556,"depth":9,"x":2098.646,"y":691.95,"cluster":"scheme-morphisms"},{"id":"stacks:062S","tag":"062S","title":"Conormal sheaf of an immersion · Lemma 062S","summary":"Let Z → Y → X be immersions of schemes. Then there is a canonical exact sequence i^*C_Y/X → C_Z/X → C_Z/Y → 0 where the maps come from Lemma [Tag 01R4] and i : Z → Y is the first morphism.","statement_latex":"Let $Z \\to Y \\to X$ be immersions of schemes. Then there is a canonical\nexact sequence\n$$\ni^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nwhere the maps come from\nLemma \\ref{lemma-conormal-functorial}\nand $i : Z \\to Y$ is the first morphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Conormal sheaf of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062S","source_file":"morphisms.tex","source_line":6301,"source_end_line":6313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6301-L6313","statement_sha256":"1bb70d25642f9c026e19d42838ae45891af45c8b7414e4b1c37dbcd7139e09ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":5557,"rank":5557,"depth":9,"x":1969.636,"y":710.208,"cluster":"scheme-morphisms"},{"id":"stacks:01UQ","tag":"01UQ","title":"Sheaf of differentials of a morphism · Definition 01UQ","summary":"Let f : X → S be a morphism of schemes. The sheaf of differentials Ω_X/S of X over S is the sheaf of differentials of f viewed as a morphism of ringed spaces (Modules, Definition [Tag 08RT]) equipped with its universal S-derivation d_X/S : O_X → Ω_X/S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe {\\it sheaf of differentials $\\Omega_{X/S}$ of $X$ over $S$} is\nthe sheaf of differentials of $f$ viewed as a morphism of ringed spaces\n(Modules, Definition \\ref{modules-definition-differentials})\nequipped with its {\\it universal $S$-derivation}\n$$\n\\text{d}_{X/S} : \\mathcal{O}_X \\longrightarrow \\Omega_{X/S}.\n$$","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UQ","source_file":"morphisms.tex","source_line":6347,"source_end_line":6357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6347-L6357","statement_sha256":"61cac3e71f11ef5f2230600427b4c299cdbe3a414a117fa1704c0d998c0dd50c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5558,"rank":5558,"depth":2,"x":2050.264,"y":623.347,"cluster":"scheme-morphisms"},{"id":"stacks:01UR","tag":"01UR","title":"Sheaf of differentials of a morphism · Lemma 01UR","summary":"Let f : X → S be a morphism of schemes. The map Hom_O_X(Ω_X/S, F) → Der_S(O_X, F), α ↦ α ∘ d_X/S is an isomorphism of functors Mod(O_X) → Sets.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. The map\n$$\n\\Hom_{\\mathcal{O}_X}(\\Omega_{X/S}, \\mathcal{F})\n\\longrightarrow\n\\text{Der}_S(\\mathcal{O}_X, \\mathcal{F}),\\quad\n\\alpha \\longmapsto \\alpha \\circ \\text{d}_{X/S}\n$$\nis an isomorphism of functors $\\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Sets}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UR","source_file":"morphisms.tex","source_line":6371,"source_end_line":6381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6371-L6381","statement_sha256":"063de13456c0a221d9e10ef76aab9c5292a8826fa69b6308d2249e96d144bee5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5559,"rank":5559,"depth":0,"x":2060.684,"y":733.392,"cluster":"scheme-morphisms"},{"id":"stacks:01US","tag":"01US","title":"Sheaf of differentials of a morphism · Lemma 01US","summary":"Let f : X → S be a morphism of schemes. Let U ⊂ X, V ⊂ S be open subschemes such that f(U) ⊂ V. Then there is a unique isomorphism Ω_X/S|_U = Ω_U/V of O_U-modules such that d_X/S|_U = d_U/V.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $U \\subset X$, $V \\subset S$ be open subschemes such\nthat $f(U) \\subset V$. Then there is a unique isomorphism\n$\\Omega_{X/S}|_U = \\Omega_{U/V}$ of $\\mathcal{O}_U$-modules such that\n$\\text{d}_{X/S}|_U = \\text{d}_{U/V}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01US","source_file":"morphisms.tex","source_line":6387,"source_end_line":6394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6387-L6394","statement_sha256":"9537e12360d99dd75e98f7e0ebe3089f5cd438a41a4a2b98dd13b9a2fa1d8575","origin":"The Stacks Project","memory_eligible":false,"source_rank":5560,"rank":5560,"depth":2,"x":1964.293,"y":657.992,"cluster":"scheme-morphisms"},{"id":"stacks:01UO","tag":"01UO","title":"Sheaf of differentials of a morphism · Lemma 01UO","summary":"Let R → A be a ring map. Let F be a sheaf of O_X-modules on X = Spec(A). Set S = Spec(R). The rule which associates to an S-derivation on F its action on global sections defines a bijection between the set of S-derivations of F and the set of R-derivations on M = Γ(X, F).","statement_latex":"Let $R \\to A$ be a ring map. Let $\\mathcal{F}$\nbe a sheaf of $\\mathcal{O}_X$-modules\non $X = \\Spec(A)$. Set $S = \\Spec(R)$.\nThe rule which associates to an $S$-derivation on $\\mathcal{F}$\nits action on global sections defines a bijection between\nthe set of $S$-derivations of $\\mathcal{F}$ and the set of\n$R$-derivations on $M = \\Gamma(X, \\mathcal{F})$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UO","source_file":"morphisms.tex","source_line":6408,"source_end_line":6417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6408-L6417","statement_sha256":"576865f926b155e8287cb720ad48b506db3565c9fbdedd865aebed49d65b8194","origin":"The Stacks Project","memory_eligible":false,"source_rank":5561,"rank":5561,"depth":0,"x":2096.295,"y":658.898,"cluster":"scheme-morphisms"},{"id":"stacks:01UT","tag":"01UT","title":"Sheaf of differentials of a morphism · Lemma 01UT","summary":"Let f : X → S be a morphism of schemes. For any pair of affine opens Spec(A) = U ⊂ X, Spec(R) = V ⊂ S with f(U) ⊂ V there is a unique isomorphism Γ(U, Ω_X/S) = Ω_A/R. compatible with d_X/S and d : A → Ω_A/R.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. For any pair of affine opens\n$\\Spec(A) = U \\subset X$, $\\Spec(R) = V \\subset S$ with $f(U) \\subset V$\nthere is a unique isomorphism\n$$\n\\Gamma(U, \\Omega_{X/S}) = \\Omega_{A/R}.\n$$\ncompatible with $\\text{d}_{X/S}$ and $\\text{d} : A \\to \\Omega_{A/R}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UT","source_file":"morphisms.tex","source_line":6440,"source_end_line":6449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6440-L6449","statement_sha256":"e04495f733146e9450d72dc1f7bf5ed983bb1cacd1e593809c5a0634f764fd54","origin":"The Stacks Project","memory_eligible":false,"source_rank":5562,"rank":5562,"depth":3,"x":1998.014,"y":733.294,"cluster":"scheme-morphisms"},{"id":"stacks:08S2","tag":"08S2","title":"Sheaf of differentials of a morphism · Lemma 08S2","summary":"Let f : X → S be a morphism of schemes. There is a canonical isomorphism between Ω_X/S and the conormal sheaf of the diagonal morphism Δ_X/S : X → X ×_S X.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. There is a canonical\nisomorphism between $\\Omega_{X/S}$ and the conormal sheaf of\nthe diagonal morphism $\\Delta_{X/S} : X \\longrightarrow X \\times_S X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08S2","source_file":"morphisms.tex","source_line":6506,"source_end_line":6511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6506-L6511","statement_sha256":"4770f3537bdeedda35a86c8a7eb89e740e6b7356e855d5112a1386c87a8a1237","origin":"The Stacks Project","memory_eligible":false,"source_rank":5563,"rank":5563,"depth":14,"x":2010.686,"y":622.427,"cluster":"scheme-morphisms"},{"id":"stacks:01UV","tag":"01UV","title":"Sheaf of differentials of a morphism · Lemma 01UV","summary":"Let xymatrix X' ar[d] ar[r]_f & X ar[d] S' ar[r] & S be a commutative diagram of schemes. The canonical map O_X → f_*O_X' composed with the map f_*d_X'/S' : f_*O_X' → f_*Ω_X'/S' is a S-derivation. Hence we obtain a canonical map of O_X-modules Ω_X/S → f_*Ω_X'/S', and by adjointness of f_* and f^* a canonical O_X'-module homomorphism c_f : f^*Ω_X/S → Ω_X'/S'. It is uniquely characterized by the property that f^*d_X/S(h) maps to d_X'/S'(f^* h) for any local section h of O_X.","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\\nS' \\ar[r] & S\n}\n$$\nbe a commutative diagram of schemes. The canonical map\n$\\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with the map\n$f_*\\text{d}_{X'/S'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/S'}$ is a\n$S$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_X$-modules\n$\\Omega_{X/S} \\to f_*\\Omega_{X'/S'}$, and by\nadjointness of $f_*$ and $f^*$ a\ncanonical $\\mathcal{O}_{X'}$-module homomorphism\n$$\nc_f : f^*\\Omega_{X/S} \\longrightarrow \\Omega_{X'/S'}.\n$$\nIt is uniquely characterized by the property that\n$f^*\\text{d}_{X/S}(h)$ maps to $\\text{d}_{X'/S'}(f^* h)$\nfor any local section $h$ of $\\mathcal{O}_X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UV","source_file":"morphisms.tex","source_line":6597,"source_end_line":6619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6597-L6619","statement_sha256":"22b576aa1db228c1116dad67b58154a5a364b972b82e052a21a77895623cab55","origin":"The Stacks Project","memory_eligible":false,"source_rank":5564,"rank":5564,"depth":15,"x":2090.672,"y":711.563,"cluster":"scheme-morphisms"},{"id":"stacks:01UX","tag":"01UX","title":"Sheaf of differentials of a morphism · Lemma 01UX","summary":"Let f : X → Y, g : Y → S be morphisms of schemes. Then there is a canonical exact sequence f^*Ω_Y/S → Ω_X/S → Ω_X/Y → 0 where the maps come from applications of Lemma [Tag 01UV].","statement_latex":"Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of schemes.\nThen there is a canonical exact sequence\n$$\nf^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0\n$$\nwhere the maps come from applications of\nLemma \\ref{lemma-functoriality-differentials}.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UX","source_file":"morphisms.tex","source_line":6632,"source_end_line":6641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6632-L6641","statement_sha256":"2ab07e50012093f6e3cd1825706eadf4096330ebe4bc16cc348664dd746dd365","origin":"The Stacks Project","memory_eligible":false,"source_rank":5565,"rank":5565,"depth":16,"x":1959.727,"y":691.177,"cluster":"scheme-morphisms"},{"id":"stacks:01V0","tag":"01V0","title":"Sheaf of differentials of a morphism · Lemma 01V0","summary":"Let X → S be a morphism of schemes. Let g : S' → S be a morphism of schemes. Let X' = X_S' be the base change of X. Denote g' : X' → X the projection. Then the map (g')^*Ω_X/S → Ω_X'/S' of Lemma [Tag 01UV] is an isomorphism.","statement_latex":"Let $X \\to S$ be a morphism of schemes.\nLet $g : S' \\to S$ be a morphism of schemes.\nLet $X' = X_{S'}$ be the base change of $X$.\nDenote $g' : X' \\to X$ the projection.\nThen the map\n$$\n(g')^*\\Omega_{X/S} \\to \\Omega_{X'/S'}\n$$\nof Lemma \\ref{lemma-functoriality-differentials} is an isomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01V0","source_file":"morphisms.tex","source_line":6650,"source_end_line":6661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6650-L6661","statement_sha256":"9d4b2613ff8abda381a66956e3eab3b2e5aaf9406879930f7e64756eb5458e57","origin":"The Stacks Project","memory_eligible":false,"source_rank":5566,"rank":5566,"depth":16,"x":2072.922,"y":631.78,"cluster":"scheme-morphisms"},{"id":"stacks:01V1","tag":"01V1","title":"Sheaf of differentials of a morphism · Lemma 01V1","summary":"Let f : X → S and g : Y → S be morphisms of schemes with the same target. Let p : X ×_S Y → X and q : X ×_S Y → Y be the projection morphisms. The maps from Lemma [Tag 01UV] p^*Ω_X/S ⊕ q^*Ω_Y/S → Ω_X ×_S Y/S give an isomorphism.","statement_latex":"Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes with the same\ntarget. Let $p : X \\times_S Y \\to X$ and $q : X \\times_S Y \\to Y$ be the\nprojection morphisms. The maps from\nLemma \\ref{lemma-functoriality-differentials}\n$$\np^*\\Omega_{X/S} \\oplus q^*\\Omega_{Y/S}\n\\longrightarrow\n\\Omega_{X \\times_S Y/S}\n$$\ngive an isomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01V1","source_file":"morphisms.tex","source_line":6668,"source_end_line":6680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6668-L6680","statement_sha256":"594907bffd1ec5def5627d15c9d26d691ee71d37e6f9495483adc95cc585c08e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5567,"rank":5567,"depth":17,"x":2037.142,"y":740.04,"cluster":"scheme-morphisms"},{"id":"stacks:01V2","tag":"01V2","title":"Sheaf of differentials of a morphism · Lemma 01V2","summary":"Let f : X → S be a morphism of schemes. If f is locally of finite type, then Ω_X/S is a finite type O_X-module.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf $f$ is locally of finite type, then $\\Omega_{X/S}$ is\na finite type $\\mathcal{O}_X$-module.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01V2","source_file":"morphisms.tex","source_line":6691,"source_end_line":6696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6691-L6696","statement_sha256":"5960612313b6954701986a9a5ae95e7d65a5f06efcc143d729a4f838ef51b5fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5568,"rank":5568,"depth":5,"x":1976.336,"y":639.694,"cluster":"scheme-morphisms"},{"id":"stacks:01V3","tag":"01V3","title":"Sheaf of differentials of a morphism · Lemma 01V3","summary":"Let f : X → S be a morphism of schemes. If f is locally of finite presentation, then Ω_X/S is an O_X-module of finite presentation.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf $f$ is locally of finite presentation, then $\\Omega_{X/S}$ is\nan $\\mathcal{O}_X$-module of finite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01V3","source_file":"morphisms.tex","source_line":6706,"source_end_line":6711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6706-L6711","statement_sha256":"027c9fb881433e5b2dda5b4e5f7b25b34ae13ffd65d05286b6b3a3532bfa7a68","origin":"The Stacks Project","memory_eligible":false,"source_rank":5569,"rank":5569,"depth":7,"x":2102.137,"y":679.269,"cluster":"scheme-morphisms"},{"id":"stacks:01UY","tag":"01UY","title":"Sheaf of differentials of a morphism · Lemma 01UY","summary":"If X → S is an immersion, or more generally a monomorphism, then Ω_X/S is zero.","statement_latex":"If $X \\to S$ is an immersion, or more generally a monomorphism, then\n$\\Omega_{X/S}$ is zero.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UY","source_file":"morphisms.tex","source_line":6721,"source_end_line":6725,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6721-L6725","statement_sha256":"88a006cabd81712baa3d04e943ca1c5cebbfe58d8673c49845c81d9ca07ea4eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5570,"rank":5570,"depth":15,"x":1977.284,"y":721.559,"cluster":"scheme-morphisms"},{"id":"stacks:01UZ","tag":"01UZ","title":"Sheaf of differentials of a morphism · Lemma 01UZ","summary":"Let i : Z → X be an immersion of schemes over S. There is a canonical exact sequence C_Z/X → i^*Ω_X/S → Ω_Z/S → 0 where the first arrow is induced by d_X/S and the second arrow comes from Lemma [Tag 01UV].","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes over $S$.\nThere is a canonical exact sequence\n$$\n\\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0\n$$\nwhere the first arrow is induced by $\\text{d}_{X/S}$\nand the second arrow comes from Lemma \\ref{lemma-functoriality-differentials}.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01UZ","source_file":"morphisms.tex","source_line":6734,"source_end_line":6743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6734-L6743","statement_sha256":"9c84260e87c04ed8ed707138618e217cdc32f6013c953fe2e23aa0ef054ff874","origin":"The Stacks Project","memory_eligible":false,"source_rank":5571,"rank":5571,"depth":17,"x":2035.463,"y":619.315,"cluster":"scheme-morphisms"},{"id":"stacks:0474","tag":"0474","title":"Sheaf of differentials of a morphism · Lemma 0474","summary":"Let i : Z → X be an immersion of schemes over S, and assume i (locally) has a left inverse. Then the canonical sequence 0 → C_Z/X → i^*Ω_X/S → Ω_Z/S → 0 of Lemma [Tag 01UZ] is (locally) split exact. In particular, if s : S → X is a section of the structure morphism X → S then the map C_S/X → s^*Ω_X/S induced by d_X/S is an isomorphism.","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes over $S$, and\nassume $i$ (locally) has a left inverse. Then the canonical\nsequence\n$$\n0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0\n$$\nof\nLemma \\ref{lemma-differentials-relative-immersion}\nis (locally) split exact. In particular, if $s : S \\to X$ is a section\nof the structure morphism $X \\to S$ then the map\n$\\mathcal{C}_{S/X} \\to s^*\\Omega_{X/S}$ induced by\n$\\text{d}_{X/S}$ is an isomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0474","source_file":"morphisms.tex","source_line":6762,"source_end_line":6776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6762-L6776","statement_sha256":"21d198cb0e4897a89461a54fe9d7a0124020752bb90f73a28e19ce3fab264821","origin":"The Stacks Project","memory_eligible":false,"source_rank":5572,"rank":5572,"depth":18,"x":2074.867,"y":727.949,"cluster":"scheme-morphisms"},{"id":"stacks:067L","tag":"067L","title":"Sheaf of differentials of a morphism · Lemma 067L","summary":"Let xymatrix Z ar[r]_i ar[rd]_j & X ar[d] & Y be a commutative diagram of schemes where i and j are immersions. Then there is a canonical exact sequence C_Z/Y → C_Z/X → i^*Ω_X/Y → 0 where the first arrow comes from Lemma [Tag 01R4] and the second from Lemma [Tag 01UZ].","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\\n& Y\n}\n$$\nbe a commutative diagram of schemes where $i$ and $j$ are immersions.\nThen there is a canonical exact sequence\n$$\n\\mathcal{C}_{Z/Y} \\to\n\\mathcal{C}_{Z/X} \\to\ni^*\\Omega_{X/Y} \\to 0\n$$\nwhere the first arrow comes from\nLemma \\ref{lemma-conormal-functorial}\nand the second from\nLemma \\ref{lemma-differentials-relative-immersion}.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067L","source_file":"morphisms.tex","source_line":6823,"source_end_line":6843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6823-L6843","statement_sha256":"6914b86d7eb4ff279775c74c73277716300b339c390bb7c4555feb1ee9ff6341","origin":"The Stacks Project","memory_eligible":false,"source_rank":5573,"rank":5573,"depth":18,"x":1958.207,"y":670.081,"cluster":"scheme-morphisms"},{"id":"stacks:0G44","tag":"0G44","title":"Finite order differential operators · Lemma 0G44","summary":"Let R → A be a ring map. Denote f : X → S the corresponding morphism of affine schemes. Let F and G be O_X-modules. If F is quasi-coherent then the map Diff^k_X/S(F, G) → Diff^k_A/R(Γ(X, F), Γ(X, G)) sending a differential operator to its action on global sections is bijective.","statement_latex":"Let $R \\to A$ be a ring map. Denote $f : X \\to S$ the corresponding\nmorphism of affine schemes. Let $\\mathcal{F}$ and $\\mathcal{G}$\nbe $\\mathcal{O}_X$-modules. If $\\mathcal{F}$ is quasi-coherent then\nthe map\n$$\n\\text{Diff}^k_{X/S}(\\mathcal{F}, \\mathcal{G}) \\to\n\\text{Diff}^k_{A/R}(\\Gamma(X, \\mathcal{F}), \\Gamma(X, \\mathcal{G}))\n$$\nsending a differential operator to its action on global sections\nis bijective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G44","source_file":"morphisms.tex","source_line":6869,"source_end_line":6881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6869-L6881","statement_sha256":"8364b8e8b7a74a0d60e3671d8dfafc4fed2033e82850b5847c8c0320ee5593eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5574,"rank":5574,"depth":9,"x":2091.042,"y":646.507,"cluster":"scheme-morphisms"},{"id":"stacks:0G45","tag":"0G45","title":"Finite order differential operators · Lemma 0G45","summary":"Let a : X → S and b : Y → S be morphisms of schemes. Let F and F' be quasi-coherent O_X-modules. Let D : F → F' be a differential operator of order k on X/S. Let G be a quasi-coherent O_Y-module. Then there is a unique differential operator D' : pr_1^*F ⊗_O_X ×_S Y pr_2^*G → pr_1^*F' ⊗_O_X ×_S Y pr_2^*G of order k on X ×_S Y / Y such that D'(s ⊗ t) = D(s) ⊗ t for local sections s of F and t of G.","statement_latex":"Let $a : X \\to S$ and $b : Y \\to S$ be morphisms of schemes.\nLet $\\mathcal{F}$ and $\\mathcal{F}'$ be quasi-coherent $\\mathcal{O}_X$-modules.\nLet $D : \\mathcal{F} \\to \\mathcal{F}'$ be a differential operator\nof order $k$ on $X/S$. Let $\\mathcal{G}$ be a quasi-coherent\n$\\mathcal{O}_Y$-module. Then there is a unique differential\noperator\n$$\nD' :\n\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times_S Y}}\n\\text{pr}_2^*\\mathcal{G}\n\\longrightarrow\n\\text{pr}_1^*\\mathcal{F}' \\otimes_{\\mathcal{O}_{X \\times_S Y}}\n\\text{pr}_2^*\\mathcal{G}\n$$\nof order $k$ on $X \\times_S Y / Y$ such that\n$\nD'(s \\otimes t) = D(s) \\otimes t\n$\nfor local sections $s$ of $\\mathcal{F}$ and $t$ of $\\mathcal{G}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Finite order differential operators","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G45","source_file":"morphisms.tex","source_line":6912,"source_end_line":6933,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L6912-L6933","statement_sha256":"3f3f7baf29a68bbaaf2744c5663210428b34980e7cbb7336ca9b6891f32fb8b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5575,"rank":5575,"depth":10,"x":2011.885,"y":739.457,"cluster":"scheme-morphisms"},{"id":"stacks:01V5","tag":"01V5","title":"Smooth morphisms · Definition 01V5","summary":"Let f : X → S be a morphism of schemes. • We say that f is smooth at x ∈ X if there exist an affine open neighbourhood Spec(A) = U ⊂ X of x and affine open Spec(R) = V ⊂ S with f(U) ⊂ V such that the induced ring map R → A is smooth. • We say that f is smooth if it is smooth at every point of X. • A morphism of affine schemes f : X → S is called standard smooth if there exists a standard smooth ring map R → R[x_1, …, x_n]/(f_1, …, f_c) (see Algebra, Definition [Tag 00T6])…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say that $f$ is {\\it smooth at $x \\in X$} if\nthere exist an affine open neighbourhood $\\Spec(A) = U \\subset X$\nof $x$ and affine open $\\Spec(R) = V \\subset S$\nwith $f(U) \\subset V$ such that the induced ring map\n$R \\to A$ is smooth.\n\\item We say that $f$ is {\\it smooth} if it is smooth at every point of $X$.\n\\item A morphism of affine schemes $f : X \\to S$\nis called {\\it standard smooth} if there exists a standard smooth ring\nmap $R \\to R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)$ (see\nAlgebra, Definition \\ref{algebra-definition-standard-smooth})\nsuch that $X \\to S$ is isomorphic to\n$$\n\\Spec(R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)) \\to \\Spec(R).\n$$\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01V5","source_file":"morphisms.tex","source_line":7017,"source_end_line":7036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7017-L7036","statement_sha256":"bc0494edff2455d61c0a8acdaf39cd507c9dc63dc792aa4f03463f795962eb7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5576,"rank":5576,"depth":1,"x":1995.473,"y":625.767,"cluster":"scheme-morphisms"},{"id":"stacks:01V6","tag":"01V6","title":"Smooth morphisms · Lemma 01V6","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is smooth. • For every affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the ring map O_S(V) → O_X(U) is smooth. • There exists an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is smooth. • There exists an affine open covering S = ⋃_j ∈ J V_j and affine open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that the ring…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is smooth.\n\\item For every affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is smooth.\n\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis smooth.\n\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is\nsmooth, for all $j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is smooth then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is smooth.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01V6","source_file":"morphisms.tex","source_line":7047,"source_end_line":7068,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7047-L7068","statement_sha256":"7ba8cdd8d49fbb89b65eadf6c4173c505f5e4d804c64a925f1809fc76ff1aa23","origin":"The Stacks Project","memory_eligible":false,"source_rank":5577,"rank":5577,"depth":36,"x":2099.214,"y":700.441,"cluster":"scheme-morphisms"},{"id":"stacks:01V8","tag":"01V8","title":"Smooth morphisms · Lemma 01V8","summary":"Let f : X → S be a morphism of schemes. If f is flat, locally of finite presentation, and all fibres X_s are smooth, then f is smooth.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf $f$ is flat, locally of finite presentation, and all\nfibres $X_s$ are smooth, then $f$\nis smooth.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01V8","source_file":"morphisms.tex","source_line":7091,"source_end_line":7097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7091-L7097","statement_sha256":"e38ddbad60bee30d910b62e6701c418ba8923a6ccbacc0bd1f8de5c504cb56bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":5578,"rank":5578,"depth":36,"x":1962.386,"y":704.25,"cluster":"scheme-morphisms"},{"id":"stacks:01VA","tag":"01VA","title":"Smooth morphisms · Lemma 01VA","summary":"The composition of two morphisms which are smooth is smooth.","statement_latex":"The composition of two morphisms which are smooth is smooth.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VA","source_file":"morphisms.tex","source_line":7103,"source_end_line":7106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7103-L7106","statement_sha256":"49d85593f66da3b7dc73cd5b85395d82736db1ee711f738f4ae335c9647f287f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5579,"rank":5579,"depth":37,"x":2060.418,"y":623.638,"cluster":"scheme-morphisms"},{"id":"stacks:01VB","tag":"01VB","title":"Smooth morphisms · Lemma 01VB","summary":"The base change of a morphism which is smooth is smooth.","statement_latex":"The base change of a morphism which is smooth is smooth.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VB","source_file":"morphisms.tex","source_line":7118,"source_end_line":7121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7118-L7121","statement_sha256":"5ccdb6a81ad7c0bb1ddcd842102be59c8455721ea6daf84d37c2b496d041b451","origin":"The Stacks Project","memory_eligible":false,"source_rank":5580,"rank":5580,"depth":37,"x":2052.941,"y":738.94,"cluster":"scheme-morphisms"},{"id":"stacks:01VC","tag":"01VC","title":"Smooth morphisms · Lemma 01VC","summary":"Any open immersion is smooth.","statement_latex":"Any open immersion is smooth.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VC","source_file":"morphisms.tex","source_line":7133,"source_end_line":7136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7133-L7136","statement_sha256":"d41b63bb4a5555898c8d3d2af30966508e60443af5cae58075ebdd4eb2c54d76","origin":"The Stacks Project","memory_eligible":false,"source_rank":5581,"rank":5581,"depth":0,"x":1965.556,"y":649.495,"cluster":"scheme-morphisms"},{"id":"stacks:01VD","tag":"01VD","title":"Smooth morphisms · Lemma 01VD","summary":"A smooth morphism is syntomic.","statement_latex":"A smooth morphism is syntomic.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VD","source_file":"morphisms.tex","source_line":7142,"source_end_line":7145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7142-L7145","statement_sha256":"d7c6d94430e9c8d9afbc7c48e9aa470257c51ccc7e25d5a4f1d598220752c8c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5582,"rank":5582,"depth":35,"x":2102.197,"y":665.898,"cluster":"scheme-morphisms"},{"id":"stacks:01VE","tag":"01VE","title":"Smooth morphisms · Lemma 01VE","summary":"A smooth morphism is locally of finite presentation.","statement_latex":"A smooth morphism is locally of finite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VE","source_file":"morphisms.tex","source_line":7151,"source_end_line":7154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7151-L7154","statement_sha256":"abd398acd89e7d8d5ab1eaf5c1b10738312088e9a7163a4a2cf323a03ab244a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5583,"rank":5583,"depth":0,"x":1988.019,"y":731.468,"cluster":"scheme-morphisms"},{"id":"stacks:01VF","tag":"01VF","title":"Smooth morphisms · Lemma 01VF","summary":"A smooth morphism is flat.","statement_latex":"A smooth morphism is flat.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VF","source_file":"morphisms.tex","source_line":7161,"source_end_line":7164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7161-L7164","statement_sha256":"b5aef5da56522faebc5a53dda36b256ec0cc1a6704ba0892f9b602819847ba06","origin":"The Stacks Project","memory_eligible":false,"source_rank":5584,"rank":5584,"depth":36,"x":2019.546,"y":618.104,"cluster":"scheme-morphisms"},{"id":"stacks:056G","tag":"056G","title":"Smooth morphisms · Lemma 056G","summary":"A smooth morphism is universally open.","statement_latex":"A smooth morphism is universally open.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056G","source_file":"morphisms.tex","source_line":7170,"source_end_line":7173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7170-L7173","statement_sha256":"ad7e4d40bcd8bd560e804dd721a5ca93c30ee37c6d47746fb51affea774d5bfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":5585,"rank":5585,"depth":37,"x":2087.597,"y":719.788,"cluster":"scheme-morphisms"},{"id":"stacks:01V7","tag":"01V7","title":"Smooth morphisms · Lemma 01V7","summary":"Smooth morphisms are locally standard smooth. Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Let V ⊂ S be an affine open neighbourhood of f(x). The following are equivalent • The morphism f is smooth at x. • There exists an affine open U ⊂ X, with x ∈ U and f(U) ⊂ V such that the induced morphism f|_U : U → V is standard smooth.","statement_latex":"\\begin{slogan}\nSmooth morphisms are locally standard smooth.\n\\end{slogan}\nLet $f : X  \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point.\nLet $V \\subset S$ be an affine open neighbourhood of $f(x)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is smooth at $x$.\n\\item There exists an affine open $U \\subset X$,\nwith $x \\in U$ and $f(U) \\subset V$ such that the\ninduced morphism $f|_U : U \\to V$ is standard smooth.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01V7","source_file":"morphisms.tex","source_line":7190,"source_end_line":7205,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7190-L7205","statement_sha256":"3cb5ca8d4be24be9e3d962b05be71d22ba30d65ce158119b56259ad41d07cbc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5586,"rank":5586,"depth":35,"x":1955.384,"y":683.349,"cluster":"scheme-morphisms"},{"id":"stacks:02G1","tag":"02G1","title":"Smooth morphisms · Lemma 02G1","summary":"Let f : X → S be a morphism of schemes. Assume f is smooth. Then the module of differentials Ω_X/S of X over S is finite locally free and rank_x(Ω_X/S) = dim_x(X_f(x)) for every x ∈ X.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $f$ is smooth.\nThen the module of differentials $\\Omega_{X/S}$ of $X$ over $S$\nis finite locally free and\n$$\n\\text{rank}_x(\\Omega_{X/S}) = \\dim_x(X_{f(x)})\n$$\nfor every $x \\in X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02G1","source_file":"morphisms.tex","source_line":7213,"source_end_line":7223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7213-L7223","statement_sha256":"1c4454534a20449ad4ad8977c4e50fd29523ff60988ba980fd386f6e8e4cb17f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5587,"rank":5587,"depth":36,"x":2082.43,"y":635.103,"cluster":"scheme-morphisms"},{"id":"stacks:02G2","tag":"02G2","title":"Smooth morphisms · Definition 02G2","summary":"Let d ≥ 0 be an integer. We say a morphism of schemes f : X → S is smooth of relative dimension d if f is smooth and Ω_X/S is finite locally free of constant rank d.","statement_latex":"Let $d \\geq 0$ be an integer. We say a morphism of schemes $f : X \\to S$\nis {\\it smooth of relative dimension $d$} if $f$ is smooth and\n$\\Omega_{X/S}$ is finite locally free of constant rank $d$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02G2","source_file":"morphisms.tex","source_line":7240,"source_end_line":7245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7240-L7245","statement_sha256":"abb2bc67038c71d01c726a5106ebb1a9b1c7efb3f2ab93bd1c97d1e5accadd28","origin":"The Stacks Project","memory_eligible":false,"source_rank":5588,"rank":5588,"depth":0,"x":2027.439,"y":742.98,"cluster":"scheme-morphisms"},{"id":"stacks:01V9","tag":"01V9","title":"Smooth morphisms · Lemma 01V9","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. Set s = f(x). Assume f is locally of finite presentation. The following are equivalent: • The morphism f is smooth at x. • The local ring map O_S, s → O_X, x is flat and X_s → Spec(kappa(s)) is smooth at x. • The local ring map O_S, s → O_X, x is flat and the O_X, x-module Ω_X/S, x can be generated by at most dim_x(X_f(x)) elements. • The local ring map O_S, s → O_X, x is flat and the kappa(x)-vector space Ω_X_s/s, x…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$.\nSet $s = f(x)$.\nAssume $f$ is locally of finite presentation.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is smooth at $x$.\n\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$\nis flat and $X_s \\to \\Spec(\\kappa(s))$ is smooth at $x$.\n\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$\nis flat and the $\\mathcal{O}_{X, x}$-module $\\Omega_{X/S, x}$\ncan be generated by at most $\\dim_x(X_{f(x)})$ elements.\n\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$\nis flat and the $\\kappa(x)$-vector space\n$$\n\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =\n\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)\n$$\ncan be generated by at most $\\dim_x(X_{f(x)})$ elements.\n\\item There exist affine opens $U \\subset X$,\nand $V \\subset S$ such that $x \\in U$, $f(U) \\subset V$ and the\ninduced morphism $f|_U : U \\to V$ is standard smooth.\n\\item There exist affine opens $\\Spec(A) = U \\subset X$\nand $\\Spec(R) = V \\subset S$ with $x \\in U$ corresponding\nto $\\mathfrak q \\subset A$, and $f(U) \\subset V$\nsuch that there exists a presentation\n$$\nA = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_c)\n$$\nwith\n$$\ng =\n\\det\n\\left(\n\\begin{matrix}\n\\partial f_1/\\partial x_1 &\n\\partial f_2/\\partial x_1 &\n\\ldots &\n\\partial f_c/\\partial x_1 \\\\\n\\partial f_1/\\partial x_2 &\n\\partial f_2/\\partial x_2 &\n\\ldots &\n\\partial f_c/\\partial x_2 \\\\\n\\ldots & \\ldots & \\ldots & \\ldots \\\\\n\\partial f_1/\\partial x_c &\n\\partial f_2/\\partial x_c &\n\\ldots &\n\\partial f_c/\\partial x_c\n\\end{matrix}\n\\right)\n$$\nmapping to an element of $A$ not in $\\mathfrak q$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01V9","source_file":"morphisms.tex","source_line":7263,"source_end_line":7318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7263-L7318","statement_sha256":"7ca5cae57e831ccc74bb59ce0a2b6062f6927c1c63480f9c9630800f571a894c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5589,"rank":5589,"depth":38,"x":1981.147,"y":632.013,"cluster":"scheme-morphisms"},{"id":"stacks:02V4","tag":"02V4","title":"Smooth morphisms · Lemma 02V4","summary":"Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a cartesian diagram of schemes. Let W ⊂ X, resp. W' ⊂ X' be the open subscheme of points where f, resp. f' is smooth. Then W' = (g')^-1(W) if • f is flat and locally of finite presentation, or • f is locally of finite presentation and g is flat.","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nbe a cartesian diagram of schemes. Let $W \\subset X$, resp.\\ $W' \\subset X'$\nbe the open subscheme of points where $f$, resp.\\ $f'$ is smooth.\nThen $W' = (g')^{-1}(W)$ if\n\\begin{enumerate}\n\\item $f$ is flat and locally of finite presentation, or\n\\item $f$ is locally of finite presentation and $g$ is flat.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V4","source_file":"morphisms.tex","source_line":7353,"source_end_line":7369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7353-L7369","statement_sha256":"83997f9fadbbd039a86344ed0b4246d000d07fe43b4ab6cf690014bbfa66f4a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5590,"rank":5590,"depth":39,"x":2104.759,"y":687.68,"cluster":"scheme-morphisms"},{"id":"stacks:02K4","tag":"02K4","title":"Smooth morphisms · Lemma 02K4","summary":"Let f : X → Y, g : Y → S be morphisms of schemes. Assume f is smooth. Then 0 → f^*Ω_Y/S → Ω_X/S → Ω_X/Y → 0 (see Lemma [Tag 01UX]) is short exact.","statement_latex":"Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of schemes.\nAssume $f$ is smooth. Then\n$$\n0 \\to f^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0\n$$\n(see Lemma \\ref{lemma-triangle-differentials}) is short exact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02K4","source_file":"morphisms.tex","source_line":7406,"source_end_line":7414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7406-L7414","statement_sha256":"4c230751059d6b2414766c512d10f69688f80b0461df2eb7b7ac88ce61d4b54e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5591,"rank":5591,"depth":17,"x":1968.579,"y":716.829,"cluster":"scheme-morphisms"},{"id":"stacks:06AA","tag":"06AA","title":"Smooth morphisms · Lemma 06AA","summary":"Let i : Z → X be an immersion of schemes over S. Assume that Z is smooth over S. Then the canonical exact sequence 0 → C_Z/X → i^*Ω_X/S → Ω_Z/S → 0 of Lemma [Tag 01UZ] is short exact.","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes over $S$.\nAssume that $Z$ is smooth over $S$. Then the\ncanonical exact sequence\n$$\n0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0\n$$\nof\nLemma \\ref{lemma-differentials-relative-immersion}\nis short exact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06AA","source_file":"morphisms.tex","source_line":7428,"source_end_line":7439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7428-L7439","statement_sha256":"430de8178f52fdd8819d735349ea8dc3aa365993ee56d12ccce4ec2915c7c8c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5592,"rank":5592,"depth":18,"x":2045.705,"y":617.871,"cluster":"scheme-morphisms"},{"id":"stacks:06AB","tag":"06AB","title":"Smooth morphisms · Lemma 06AB","summary":"Let xymatrix Z ar[r]_i ar[rd]_j & X ar[d] & Y be a commutative diagram of schemes where i and j are immersions and X → Y is smooth. Then the canonical exact sequence 0 → C_Z/Y → C_Z/X → i^*Ω_X/Y → 0 of Lemma [Tag 067L] is exact.","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\\n& Y\n}\n$$\nbe a commutative diagram of schemes where $i$ and $j$ are immersions\nand $X \\to Y$ is smooth.\nThen the canonical exact sequence\n$$\n0 \\to  \\mathcal{C}_{Z/Y} \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/Y} \\to 0\n$$\nof\nLemma \\ref{lemma-two-immersions}\nis exact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06AB","source_file":"morphisms.tex","source_line":7454,"source_end_line":7472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7454-L7472","statement_sha256":"dcad923304f7be8482a614c46a036462bdc5694a369d9f2a94d1291bebee9975","origin":"The Stacks Project","memory_eligible":false,"source_rank":5593,"rank":5593,"depth":19,"x":2068.456,"y":734.829,"cluster":"scheme-morphisms"},{"id":"stacks:02K5","tag":"02K5","title":"Smooth morphisms · Lemma 02K5","summary":"Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & S be a commutative diagram of morphisms of schemes. Assume that • f is surjective, and smooth, • p is smooth, and • q is locally of finite presentation. Then q is smooth.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume that\n\\begin{enumerate}\n\\item $f$ is surjective, and smooth,\n\\item $p$ is smooth, and\n\\item $q$ is locally of finite presentation\\footnote{In fact this\nis implied by (1) and (2), see\nDescent, Lemma \\ref{descent-lemma-flat-finitely-presented-permanence}.\nMoreover, it suffices to assume $f$ is surjective, flat and locally\nof finite presentation, see\nDescent, Lemma \\ref{descent-lemma-smooth-permanence}.}.\n\\end{enumerate}\nThen $q$ is smooth.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02K5","source_file":"morphisms.tex","source_line":7487,"source_end_line":7509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7487-L7509","statement_sha256":"41b1675622709a0a1f8beceb31d49d07ff68b7923997374cf9e086866b242e7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5594,"rank":5594,"depth":39,"x":1957.412,"y":661.355,"cluster":"scheme-morphisms"},{"id":"stacks:05D9","tag":"05D9","title":"Smooth morphisms · Lemma 05D9","summary":"Let f : X → S be a morphism of schemes. Let σ : S → X be a section of f. Let s ∈ S be a point such that f is smooth at x = σ(s). Then there exist affine open neighbourhoods Spec(A) = U ⊂ S of s and Spec(B) = V ⊂ X of x such that • f(V) ⊂ U and σ(U) ⊂ V, • with I = Ker(σ^\\# : B → A) the module I/I^2 is a free A-module, and • B^wedge ≅ A[[x_1, …, x_d]] as A-algebras where B^wedge denotes the completion of B with respect to I.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\sigma : S \\to X$ be a section of $f$.\nLet $s \\in S$ be a point such that $f$ is smooth at $x = \\sigma(s)$.\nThen there exist affine open neighbourhoods\n$\\Spec(A) = U \\subset S$ of $s$ and $\\Spec(B) = V \\subset X$\nof $x$ such that\n\\begin{enumerate}\n\\item $f(V) \\subset U$ and $\\sigma(U) \\subset V$,\n\\item with $I = \\Ker(\\sigma^\\# : B \\to A)$ the module $I/I^2$\nis a free $A$-module, and\n\\item $B^\\wedge \\cong A[[x_1, \\ldots, x_d]]$ as $A$-algebras where\n$B^\\wedge$ denotes the completion of $B$ with respect to $I$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05D9","source_file":"morphisms.tex","source_line":7532,"source_end_line":7547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7532-L7547","statement_sha256":"76f787c330b83e01eea59356080e40a364b9063da8c82efa030fecace3293e06","origin":"The Stacks Project","memory_eligible":false,"source_rank":5595,"rank":5595,"depth":7,"x":2098.65,"y":652.508,"cluster":"scheme-morphisms"},{"id":"stacks:0AFF","tag":"0AFF","title":"Smooth morphisms · Lemma 0AFF","summary":"Let f : X → Y be a smooth morphism of locally Noetherian schemes. For every point x in X with image y in Y, dim_x(X) = dim_y(Y) + dim_x(X_y), where X_y denotes the fiber over y.","statement_latex":"Let $f : X \\to Y$ be a smooth morphism of locally Noetherian schemes.\nFor every point $x$ in $X$ with image $y$ in $Y$,\n$$\n\\dim_x(X) = \\dim_y(Y) + \\dim_x(X_y),\n$$\nwhere $X_y$ denotes the fiber over $y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFF","source_file":"morphisms.tex","source_line":7571,"source_end_line":7579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7571-L7579","statement_sha256":"9886eed21545363758a1f2f3cf0a1c90ae29b9842286500e0ec485136263d83d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5596,"rank":5596,"depth":37,"x":2001.432,"y":739.338,"cluster":"scheme-morphisms"},{"id":"stacks:02G4","tag":"02G4","title":"Unramified morphisms · Definition 02G4","summary":"Let f : X → S be a morphism of schemes. • We say that f is unramified at x ∈ X if there exists an affine open neighbourhood Spec(A) = U ⊂ X of x and affine open Spec(R) = V ⊂ S with f(U) ⊂ V such that the induced ring map R → A is unramified. • We say that f is G-unramified at x ∈ X if there exists an affine open neighbourhood Spec(A) = U ⊂ X of x and affine open Spec(R) = V ⊂ S with f(U) ⊂ V such that the induced ring map R → A is G-unramified. • We say that f is…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say that $f$ is {\\it unramified at $x \\in X$} if\nthere exists an affine open neighbourhood $\\Spec(A) = U \\subset X$\nof $x$ and affine open $\\Spec(R) = V \\subset S$\nwith $f(U) \\subset V$ such that the induced ring map\n$R \\to A$ is unramified.\n\\item We say that $f$ is {\\it G-unramified at $x \\in X$} if\nthere exists an affine open neighbourhood $\\Spec(A) = U \\subset X$\nof $x$ and affine open $\\Spec(R) = V \\subset S$\nwith $f(U) \\subset V$ such that the induced ring map\n$R \\to A$ is G-unramified.\n\\item We say that $f$ is {\\it unramified} if it is unramified\nat every point of $X$.\n\\item We say that $f$ is {\\it G-unramified} if it is G-unramified\nat every point of $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02G4","source_file":"morphisms.tex","source_line":7615,"source_end_line":7634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7615-L7634","statement_sha256":"94f2607c7842533f555eefa0e48a28f5497678b168991618a9f9e0ee2e17bda5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5597,"rank":5597,"depth":0,"x":2003.296,"y":619.921,"cluster":"scheme-morphisms"},{"id":"stacks:02G5","tag":"02G5","title":"Unramified morphisms · Lemma 02G5","summary":"Let f : X → S be a morphism of schemes. Then • f is unramified if and only if f is locally of finite type and Ω_X/S = 0, and • f is G-unramified if and only if f is locally of finite presentation and Ω_X/S = 0.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Then\n\\begin{enumerate}\n\\item $f$ is unramified if and only if $f$ is locally of finite type\nand $\\Omega_{X/S} = 0$, and\n\\item $f$ is G-unramified if and only if $f$ is locally of finite presentation\nand $\\Omega_{X/S} = 0$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02G5","source_file":"morphisms.tex","source_line":7645,"source_end_line":7654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7645-L7654","statement_sha256":"fb6fb697f1673dcc57a3ea88805ccf9f4f612be632691a1d785dbb88b8c64fa1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5598,"rank":5598,"depth":4,"x":2098.133,"y":709.204,"cluster":"scheme-morphisms"},{"id":"stacks:02G6","tag":"02G6","title":"Unramified morphisms · Lemma 02G6","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is unramified (resp. G-unramified). • For every affine open U ⊂ X, V ⊂ S with f(U) ⊂ V the ring map O_S(V) → O_X(U) is unramified (resp. G-unramified). • There exists an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is unramified (resp. G-unramified). • There exists an affine open covering S = ⋃_j ∈ J…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is unramified (resp.\\ G-unramified).\n\\item For every affine open $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is unramified (resp.\\ G-unramified).\n\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis unramified (resp.\\ G-unramified).\n\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is\nunramified (resp.\\ G-unramified), for all $j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is unramified (resp.\\ G-unramified) then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is unramified (resp.\\ G-unramified).","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02G6","source_file":"morphisms.tex","source_line":7668,"source_end_line":7689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7668-L7689","statement_sha256":"ec051661bf5ad5ce022bb32d57d9338cd47db1df222ffd924bdd6e3c39344c48","origin":"The Stacks Project","memory_eligible":false,"source_rank":5599,"rank":5599,"depth":5,"x":1956.136,"y":697.158,"cluster":"scheme-morphisms"},{"id":"stacks:02G9","tag":"02G9","title":"Unramified morphisms · Lemma 02G9","summary":"The composition of two morphisms which are unramified is unramified. The same holds for G-unramified morphisms.","statement_latex":"The composition of two morphisms which are unramified is unramified.\nThe same holds for G-unramified morphisms.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02G9","source_file":"morphisms.tex","source_line":7700,"source_end_line":7704,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7700-L7704","statement_sha256":"be126fab8da8f10c0590db3e51a446e19d9e4610a2e97f559c48e279a11c0ab3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5600,"rank":5600,"depth":6,"x":2070.745,"y":625.333,"cluster":"scheme-morphisms"},{"id":"stacks:02GA","tag":"02GA","title":"Unramified morphisms · Lemma 02GA","summary":"The base change of a morphism which is unramified is unramified. The same holds for G-unramified morphisms.","statement_latex":"The base change of a morphism which is unramified is unramified.\nThe same holds for G-unramified morphisms.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GA","source_file":"morphisms.tex","source_line":7717,"source_end_line":7721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7717-L7721","statement_sha256":"580ab5e7e134753ca76be1328522d62b43c3f513b416f5f044875fab140bb018","origin":"The Stacks Project","memory_eligible":false,"source_rank":5601,"rank":5601,"depth":6,"x":2043.943,"y":743.548,"cluster":"scheme-morphisms"},{"id":"stacks:04EV","tag":"04EV","title":"Unramified morphisms · Lemma 04EV","summary":"Let f : X → S be a morphism of schemes. Assume S is locally Noetherian. Then f is unramified if and only if f is G-unramified.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian.\nThen $f$ is unramified if and only if $f$ is G-unramified.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EV","source_file":"morphisms.tex","source_line":7733,"source_end_line":7737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7733-L7737","statement_sha256":"7c915736aa0d3850e1d200dcc0260a28274e30e833d0b9b44b79a96fb864c34a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5602,"rank":5602,"depth":18,"x":1968.501,"y":640.98,"cluster":"scheme-morphisms"},{"id":"stacks:02GB","tag":"02GB","title":"Unramified morphisms · Lemma 02GB","summary":"Any open immersion is G-unramified.","statement_latex":"Any open immersion is G-unramified.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GB","source_file":"morphisms.tex","source_line":7745,"source_end_line":7748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7745-L7748","statement_sha256":"5cb2f77d28542a584ed8bdf3259fb03e0ec81b1d31c795fb620e071944fca4ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":5603,"rank":5603,"depth":0,"x":2106.87,"y":673.866,"cluster":"scheme-morphisms"},{"id":"stacks:02GC","tag":"02GC","title":"Unramified morphisms · Lemma 02GC","summary":"A closed immersion i : Z → X is unramified. It is G-unramified if and only if the associated quasi-coherent sheaf of ideals I = Ker(O_X → i_*O_Z) is of finite type (as an O_X-module).","statement_latex":"A closed immersion $i : Z \\to X$ is unramified.\nIt is G-unramified if and only if the associated quasi-coherent sheaf of\nideals $\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$\nis of finite type (as an $\\mathcal{O}_X$-module).","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GC","source_file":"morphisms.tex","source_line":7754,"source_end_line":7760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7754-L7760","statement_sha256":"22a142442cf476147f18d1a666de0d429f3227ce7c4308c57f427b7fc678ac11","origin":"The Stacks Project","memory_eligible":false,"source_rank":5604,"rank":5604,"depth":5,"x":1978.155,"y":728.23,"cluster":"scheme-morphisms"},{"id":"stacks:02GD","tag":"02GD","title":"Unramified morphisms · Lemma 02GD","summary":"An unramified morphism is locally of finite type. A G-unramified morphism is locally of finite presentation.","statement_latex":"An unramified morphism is locally of finite type.\nA G-unramified morphism is locally of finite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GD","source_file":"morphisms.tex","source_line":7767,"source_end_line":7771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7767-L7771","statement_sha256":"7dc7b16c0c36927dd5a6068d902d5797018793c9232e8dd7723c65c8a5a3368c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5605,"rank":5605,"depth":0,"x":2029.443,"y":614.898,"cluster":"scheme-morphisms"},{"id":"stacks:02V5","tag":"02V5","title":"Unramified morphisms · Lemma 02V5","summary":"Let f : X → S be a morphism of schemes. If f is unramified at x then f is quasi-finite at x. In particular, an unramified morphism is locally quasi-finite.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf $f$ is unramified at $x$ then $f$ is quasi-finite at $x$.\nIn particular, an unramified morphism is locally quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V5","source_file":"morphisms.tex","source_line":7778,"source_end_line":7783,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7778-L7783","statement_sha256":"fa0f39e120382ebab2ff2760a5f05468f71384332393a3189175ba31d444afae","origin":"The Stacks Project","memory_eligible":false,"source_rank":5606,"rank":5606,"depth":42,"x":2082.86,"y":727.777,"cluster":"scheme-morphisms"},{"id":"stacks:02G7","tag":"02G7","title":"Unramified morphisms · Lemma 02G7","summary":"Fibres of unramified morphisms. • Let X be a scheme over a field k. The structure morphism X → Spec(k) is unramified if and only if X is a disjoint union of spectra of finite separable field extensions of k. • If f : X → S is an unramified morphism then for every s ∈ S the fibre X_s is a disjoint union of spectra of finite separable field extensions of kappa(s).","statement_latex":"Fibres of unramified morphisms.\n\\begin{enumerate}\n\\item Let $X$ be a scheme over a field $k$.\nThe structure morphism $X \\to \\Spec(k)$ is unramified if\nand only if $X$ is a disjoint union of spectra of finite separable\nfield extensions of $k$.\n\\item If $f : X \\to S$ is an unramified morphism then for every $s \\in S$\nthe fibre $X_s$ is a disjoint union of spectra of finite separable field\nextensions of $\\kappa(s)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02G7","source_file":"morphisms.tex","source_line":7789,"source_end_line":7801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7789-L7801","statement_sha256":"758beaf99c6014467eba87b6269c56549abd99604895f4298f0e4d85aaad855b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5607,"rank":5607,"depth":43,"x":1952.46,"y":674.755,"cluster":"scheme-morphisms"},{"id":"stacks:02G8","tag":"02G8","title":"Unramified morphisms · Lemma 02G8","summary":"Let f : X → S be a morphism of schemes. • If f is unramified then for any x ∈ X the field extension kappa(x)/kappa(f(x)) is finite separable. • If f is locally of finite type, and for every s ∈ S the fibre X_s is a disjoint union of spectra of finite separable field extensions of kappa(s) then f is unramified. • If f is locally of finite presentation, and for every s ∈ S the fibre X_s is a disjoint union of spectra of finite separable field extensions of kappa(s) then f…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item If $f$ is unramified then for any $x \\in X$ the field extension\n$\\kappa(x)/\\kappa(f(x))$ is finite separable.\n\\item If $f$ is locally of finite type, and for every\n$s \\in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable\nfield extensions of $\\kappa(s)$ then $f$ is unramified.\n\\item If $f$ is locally of finite presentation, and for every\n$s \\in S$ the fibre $X_s$ is a disjoint union of spectra of finite separable\nfield extensions of $\\kappa(s)$ then $f$ is G-unramified.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02G8","source_file":"morphisms.tex","source_line":7827,"source_end_line":7840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7827-L7840","statement_sha256":"a0a407c60fa87732693030187d644aab442ee8697ae01780928f667824bd8f06","origin":"The Stacks Project","memory_eligible":false,"source_rank":5608,"rank":5608,"depth":41,"x":2091.506,"y":639.797,"cluster":"scheme-morphisms"},{"id":"stacks:02GE","tag":"02GE","title":"Unramified morphisms · Lemma 02GE","summary":"Let f : X → S be a morphism. • If f is unramified, then the diagonal morphism Δ : X → X ×_S X is an open immersion. • If f is locally of finite type and Δ is an open immersion, then f is unramified. • If f is locally of finite presentation and Δ is an open immersion, then f is G-unramified.","statement_latex":"Let $f : X \\to S$ be a morphism.\n\\begin{enumerate}\n\\item If $f$ is unramified, then the diagonal morphism\n$\\Delta : X \\to X \\times_S X$ is an open immersion.\n\\item If $f$ is locally of finite type\nand $\\Delta$ is an open immersion, then $f$ is unramified.\n\\item If $f$ is locally of finite presentation and $\\Delta$ is an open\nimmersion, then $f$ is G-unramified.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GE","source_file":"morphisms.tex","source_line":7852,"source_end_line":7863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7852-L7863","statement_sha256":"e734cdb1a0be79ff1be4d868223a5af765dddd7f8ddfa089ee4e4a216b88b97e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5609,"rank":5609,"depth":15,"x":2016.953,"y":744.663,"cluster":"scheme-morphisms"},{"id":"stacks:02GF","tag":"02GF","title":"Unramified morphisms · Lemma 02GF","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. Set s = f(x). Assume f is locally of finite type (resp. locally of finite presentation). The following are equivalent: • The morphism f is unramified (resp. G-unramified) at x. • The fibre X_s is unramified over kappa(s) at x. • The O_X, x-module Ω_X/S, x is zero. • The O_X_s, x-module Ω_X_s/s, x is zero. • The kappa(x)-vector space Ω_X_s/s, x ⊗_O_X_s, x kappa(x) = Ω_X/S, x ⊗_O_X, x kappa(x) is zero. • We have m_sO_X, x =…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$.\nSet $s = f(x)$.\nAssume $f$ is locally of finite type (resp.\\ locally of finite presentation).\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is unramified (resp.\\ G-unramified) at $x$.\n\\item The fibre $X_s$ is unramified over $\\kappa(s)$ at $x$.\n\\item The $\\mathcal{O}_{X, x}$-module $\\Omega_{X/S, x}$ is zero.\n\\item The $\\mathcal{O}_{X_s, x}$-module $\\Omega_{X_s/s, x}$ is zero.\n\\item The $\\kappa(x)$-vector space\n$$\n\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =\n\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)\n$$\nis zero.\n\\item We have $\\mathfrak m_s\\mathcal{O}_{X, x} = \\mathfrak m_x$\nand the field extension $\\kappa(x)/\\kappa(s)$ is finite\nseparable.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GF","source_file":"morphisms.tex","source_line":7874,"source_end_line":7896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7874-L7896","statement_sha256":"26e7b2f84a264c087a70abe49041c1585fe41e380a0d004aa596be06a0f46f4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5610,"rank":5610,"depth":42,"x":1987.544,"y":624.816,"cluster":"scheme-morphisms"},{"id":"stacks:0475","tag":"0475","title":"Unramified morphisms · Lemma 0475","summary":"Let f : X → S be a morphism of schemes. Assume f locally of finite type. Formation of the open set T & = (x ∈ X mid X_f(x) is unramified over kappa(f(x)) at x) & = (x ∈ X mid X is unramified over S at x) commutes with arbitrary base change: For any morphism g : S' → S, consider the base change f' : X' → S' of f and the projection g' : X' → X. Then the corresponding set T' for the morphism f' is equal to T' = (g')^-1(T). If f is assumed locally of finite presentation then…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $f$ locally of finite type. Formation of the open set\n\\begin{align*}\nT\n& =\n\\{x \\in X \\mid X_{f(x)}\\text{ is unramified over }\\kappa(f(x))\\text{ at }x\\} \\\\\n& =\n\\{x \\in X \\mid X\\text{ is unramified over }S\\text{ at }x\\}\n\\end{align*}\ncommutes with arbitrary base change:\nFor any morphism $g : S' \\to S$, consider\nthe base change $f' : X' \\to S'$ of $f$ and the\nprojection $g' : X' \\to X$. Then the corresponding\nset $T'$ for the morphism $f'$ is equal to $T' = (g')^{-1}(T)$.\nIf $f$ is assumed locally of finite presentation then the same holds\nfor the open set of points where $f$ is G-unramified.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0475","source_file":"morphisms.tex","source_line":7937,"source_end_line":7955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7937-L7955","statement_sha256":"d4754ea16acc94c350f62b99944f4fff7a050f4f16195abd4b982eace432453c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5611,"rank":5611,"depth":43,"x":2105.82,"y":696.631,"cluster":"scheme-morphisms"},{"id":"stacks:02GG","tag":"02GG","title":"Unramified morphisms · Lemma 02GG","summary":"Let f : X → Y be a morphism of schemes over S. • If X is unramified over S, then f is unramified. • If X is G-unramified over S and Y is locally of finite type over S, then f is G-unramified.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes over $S$.\n\\begin{enumerate}\n\\item If $X$ is unramified over $S$, then $f$ is unramified.\n\\item If $X$ is G-unramified over $S$ and $Y$ is locally of finite type\nover $S$, then $f$ is G-unramified.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GG","source_file":"morphisms.tex","source_line":7979,"source_end_line":7987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L7979-L7987","statement_sha256":"dcfac7eab143a2e996a80e4d98f66f9af0e9874c8c2b54a00ff31c70a0ea8d77","origin":"The Stacks Project","memory_eligible":false,"source_rank":5612,"rank":5612,"depth":19,"x":1960.593,"y":710.813,"cluster":"scheme-morphisms"},{"id":"stacks:04HB","tag":"04HB","title":"Unramified morphisms · Lemma 04HB","summary":"Let S be a scheme. Let X, Y be schemes over S. Let f, g : X → Y be morphisms over S. Let x ∈ X. Assume that • the structure morphism Y → S is unramified, • f(x) = g(x) in Y, say y = f(x) = g(x), and • the induced maps f^sharp, g^sharp : kappa(y) → kappa(x) are equal. Then there exists an open neighbourhood of x in X on which f and g are equal.","statement_latex":"Let $S$ be a scheme.\nLet $X$, $Y$ be schemes over $S$.\nLet $f, g : X \\to Y$ be morphisms over $S$. Let $x \\in X$.\nAssume that\n\\begin{enumerate}\n\\item the structure morphism $Y \\to S$ is unramified,\n\\item $f(x) = g(x)$ in $Y$, say $y = f(x) = g(x)$, and\n\\item the induced maps $f^\\sharp, g^\\sharp : \\kappa(y) \\to \\kappa(x)$\nare equal.\n\\end{enumerate}\nThen there exists an open neighbourhood of $x$ in $X$ on which\n$f$ and $g$ are equal.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HB","source_file":"morphisms.tex","source_line":8002,"source_end_line":8016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8002-L8016","statement_sha256":"81080fca5b4bbf7863ba4aa8a99096e79d370f425e1f5046965000f41543d822","origin":"The Stacks Project","memory_eligible":false,"source_rank":5613,"rank":5613,"depth":16,"x":2056.447,"y":617.785,"cluster":"scheme-morphisms"},{"id":"stacks:02GI","tag":"02GI","title":"Étale morphisms · Definition 02GI","summary":"Let f : X → S be a morphism of schemes. • We say that f is étale at x ∈ X if there exists an affine open neighbourhood Spec(A) = U ⊂ X of x and affine open Spec(R) = V ⊂ S with f(U) ⊂ V such that the induced ring map R → A is étale. • We say that f is étale if it is étale at every point of X. • A morphism of affine schemes f : X → S is called standard étale if X → S is isomorphic to Spec(R[x]_h/(g)) → Spec(R) where R → R[x]_h/(g) is a standard étale ring map, see Algebra,…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say that $f$ is {\\it \\'etale at $x \\in X$} if\nthere exists an affine open neighbourhood $\\Spec(A) = U \\subset X$\nof $x$ and affine open $\\Spec(R) = V \\subset S$\nwith $f(U) \\subset V$ such that the induced ring map\n$R \\to A$ is \\'etale.\n\\item We say that $f$ is {\\it \\'etale} if it is \\'etale at every point of $X$.\n\\item A morphism of affine schemes $f : X \\to S$ is called\n{\\it standard \\'etale} if $X \\to S$ is isomorphic to\n$$\n\\Spec(R[x]_h/(g)) \\to \\Spec(R)\n$$\nwhere $R \\to R[x]_h/(g)$ is a standard \\'etale ring map, see\nAlgebra, Definition \\ref{algebra-definition-standard-etale},\ni.e., $g$ is monic and $g'$ invertible in $R[x]_h/(g)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GI","source_file":"morphisms.tex","source_line":8057,"source_end_line":8076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8057-L8076","statement_sha256":"ea4e7bc78f26a51ddde75aead9d85c37b284b1be6556a5131759f219f26bb9dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5614,"rank":5614,"depth":1,"x":2060.584,"y":740.992,"cluster":"scheme-morphisms"},{"id":"stacks:02GJ","tag":"02GJ","title":"Étale morphisms · Lemma 02GJ","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is étale. • For every affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the ring map O_S(V) → O_X(U) is étale. • There exists an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is étale. • There exists an affine open covering S = ⋃_j ∈ J V_j and affine open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that the ring map…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is \\'etale.\n\\item For every affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is \\'etale.\n\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis \\'etale.\n\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is\n\\'etale, for all $j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is \\'etale then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is \\'etale.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GJ","source_file":"morphisms.tex","source_line":8089,"source_end_line":8110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8089-L8110","statement_sha256":"3348395e06c1d4f706ce191cfde8e9e71c583fdf3c8c06317fbbd39100479c51","origin":"The Stacks Project","memory_eligible":false,"source_rank":5615,"rank":5615,"depth":38,"x":1958.273,"y":652.331,"cluster":"scheme-morphisms"},{"id":"stacks:02GN","tag":"02GN","title":"Étale morphisms · Lemma 02GN","summary":"The composition of two morphisms which are étale is étale.","statement_latex":"The composition of two morphisms which are \\'etale is \\'etale.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GN","source_file":"morphisms.tex","source_line":8120,"source_end_line":8123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8120-L8123","statement_sha256":"3d4788e4012952dc51f4a745520959be8050d5b90fb438e57b6b2b273f877c53","origin":"The Stacks Project","memory_eligible":false,"source_rank":5616,"rank":5616,"depth":39,"x":2105.274,"y":659.667,"cluster":"scheme-morphisms"},{"id":"stacks:02GO","tag":"02GO","title":"Étale morphisms · Lemma 02GO","summary":"The base change of a morphism which is étale is étale.","statement_latex":"The base change of a morphism which is \\'etale is \\'etale.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GO","source_file":"morphisms.tex","source_line":8135,"source_end_line":8138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8135-L8138","statement_sha256":"5fd32c9e307ffb238d8249bc85974efdb4c3691ce9ed815881d0c0773d1f04f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5617,"rank":5617,"depth":39,"x":1990.776,"y":737.807,"cluster":"scheme-morphisms"},{"id":"stacks:02GK","tag":"02GK","title":"Étale morphisms · Lemma 02GK","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. Then f is étale at x if and only if f is smooth and unramified at x.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$. Then $f$ is \\'etale at $x$ if and only if $f$ is\nsmooth and unramified at $x$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GK","source_file":"morphisms.tex","source_line":8150,"source_end_line":8155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8150-L8155","statement_sha256":"97335c71d1963090e4d8a227174e81d5f51529589a804b682d8c194190a4b6e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5618,"rank":5618,"depth":0,"x":2012.406,"y":615.004,"cluster":"scheme-morphisms"},{"id":"stacks:03WS","tag":"03WS","title":"Étale morphisms · Lemma 03WS","summary":"An étale morphism is locally quasi-finite.","statement_latex":"An \\'etale morphism is locally quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WS","source_file":"morphisms.tex","source_line":8161,"source_end_line":8164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8161-L8164","statement_sha256":"921fdf2a5e0eb2c54c7fc3601b40f65eeb59eb27c9499463dc6d88c1d4e0ab70","origin":"The Stacks Project","memory_eligible":false,"source_rank":5619,"rank":5619,"depth":43,"x":2095.356,"y":718.01,"cluster":"scheme-morphisms"},{"id":"stacks:02GL","tag":"02GL","title":"Étale morphisms · Lemma 02GL","summary":"Description of the étale schemes over fields and fibres of étale morphisms. Fibres of étale morphisms. • Let X be a scheme over a field k. The structure morphism X → Spec(k) is étale if and only if X is a disjoint union of spectra of finite separable field extensions of k. • If f : X → S is an étale morphism, then for every s ∈ S the fibre X_s is a disjoint union of spectra of finite separable field extensions of kappa(s).","statement_latex":"\\begin{slogan}\nDescription of the \\'etale schemes over fields and fibres\nof \\'etale morphisms.\n\\end{slogan}\nFibres of \\'etale morphisms.\n\\begin{enumerate}\n\\item Let $X$ be a scheme over a field $k$.\nThe structure morphism $X \\to \\Spec(k)$ is \\'etale if\nand only if $X$ is a disjoint union of spectra of finite separable\nfield extensions of $k$.\n\\item If $f : X \\to S$ is an \\'etale morphism, then for every $s \\in S$ the\nfibre $X_s$ is a disjoint union of spectra of finite separable field\nextensions of $\\kappa(s)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GL","source_file":"morphisms.tex","source_line":8174,"source_end_line":8190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8174-L8190","statement_sha256":"b630ed72915ca6e8e879aea49ac48197756aec3c284de74d55676c6c5e2667f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5620,"rank":5620,"depth":44,"x":1951.104,"y":689.072,"cluster":"scheme-morphisms"},{"id":"stacks:02GM","tag":"02GM","title":"Étale morphisms · Lemma 02GM","summary":"Let f : X → S be a morphism of schemes. If f is flat, locally of finite presentation, and for every s ∈ S the fibre X_s is a disjoint union of spectra of finite separable field extensions of kappa(s), then f is étale.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf $f$ is flat, locally of finite presentation, and for every $s \\in S$\nthe fibre $X_s$ is a disjoint union of spectra of finite separable\nfield extensions of $\\kappa(s)$, then $f$ is \\'etale.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GM","source_file":"morphisms.tex","source_line":8213,"source_end_line":8219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8213-L8219","statement_sha256":"e55bd1ccd892fb475bca37f0ed7bf7121883d04c7719d2dfb9e3bd1b4dab5912","origin":"The Stacks Project","memory_eligible":false,"source_rank":5621,"rank":5621,"depth":45,"x":2080.969,"y":628.454,"cluster":"scheme-morphisms"},{"id":"stacks:02GP","tag":"02GP","title":"Étale morphisms · Lemma 02GP","summary":"Any open immersion is étale.","statement_latex":"Any open immersion is \\'etale.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GP","source_file":"morphisms.tex","source_line":8235,"source_end_line":8238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8235-L8238","statement_sha256":"2c6e98577c2208112d328c9d701d69ea6849ea0ab3a07e193f1dacaf287bc8dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5622,"rank":5622,"depth":0,"x":2033.875,"y":747.046,"cluster":"scheme-morphisms"},{"id":"stacks:02GQ","tag":"02GQ","title":"Étale morphisms · Lemma 02GQ","summary":"An étale morphism is syntomic.","statement_latex":"An \\'etale morphism is syntomic.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GQ","source_file":"morphisms.tex","source_line":8244,"source_end_line":8247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8244-L8247","statement_sha256":"f11f5a82305b9054e3898c462bdb5336f5deff692de10968852b921400306ab7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5623,"rank":5623,"depth":35,"x":1973.128,"y":632.678,"cluster":"scheme-morphisms"},{"id":"stacks:02GR","tag":"02GR","title":"Étale morphisms · Lemma 02GR","summary":"An étale morphism is locally of finite presentation.","statement_latex":"An \\'etale morphism is locally of finite presentation.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GR","source_file":"morphisms.tex","source_line":8254,"source_end_line":8257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8254-L8257","statement_sha256":"d95511c50d74a338843b60a69ac50117442d4883596e998385bd7cd1192af430","origin":"The Stacks Project","memory_eligible":false,"source_rank":5624,"rank":5624,"depth":0,"x":2110.128,"y":682.628,"cluster":"scheme-morphisms"},{"id":"stacks:02GS","tag":"02GS","title":"Étale morphisms · Lemma 02GS","summary":"An étale morphism is flat.","statement_latex":"An \\'etale morphism is flat.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GS","source_file":"morphisms.tex","source_line":8264,"source_end_line":8267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8264-L8267","statement_sha256":"d750e770627d589c477fe92c53e27b0fd6c8c792562cc691291ac3771094e002","origin":"The Stacks Project","memory_eligible":false,"source_rank":5625,"rank":5625,"depth":36,"x":1968.696,"y":723.603,"cluster":"scheme-morphisms"},{"id":"stacks:03WT","tag":"03WT","title":"Étale morphisms · Lemma 03WT","summary":"An étale morphism is open.","statement_latex":"An \\'etale morphism is open.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WT","source_file":"morphisms.tex","source_line":8273,"source_end_line":8276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8273-L8276","statement_sha256":"6fe005b8c5fff0c48652a35aa259d5374ec5dac69687502ea8edc550f2f0edfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5626,"rank":5626,"depth":37,"x":2040.157,"y":612.948,"cluster":"scheme-morphisms"},{"id":"stacks:02GT","tag":"02GT","title":"Étale morphisms · Lemma 02GT","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Let V ⊂ S be an affine open neighbourhood of f(x). The following are equivalent • The morphism f is étale at x. • There exist an affine open U ⊂ X with x ∈ U and f(U) ⊂ V such that the induced morphism f|_U : U → V is standard étale (see Definition [Tag 02GI]).","statement_latex":"Let $f : X  \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point.\nLet $V \\subset S$ be an affine open neighbourhood of $f(x)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is \\'etale at $x$.\n\\item There exist an affine open $U \\subset X$ with\n$x \\in U$ and $f(U) \\subset V$ such that the\ninduced morphism $f|_U : U \\to V$ is standard \\'etale\n(see Definition \\ref{definition-etale}).\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GT","source_file":"morphisms.tex","source_line":8291,"source_end_line":8304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8291-L8304","statement_sha256":"6ae6b8490cc6a4e3de75c08b27578f0234a8ac84915a426de5ad2184555690d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5627,"rank":5627,"depth":43,"x":2076.51,"y":735.3,"cluster":"scheme-morphisms"},{"id":"stacks:02GU","tag":"02GU","title":"Étale morphisms · Lemma 02GU","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. Set s = f(x). Assume f is locally of finite presentation. The following are equivalent: • The morphism f is étale at x. • The local ring map O_S, s → O_X, x is flat and X_s → Spec(kappa(s)) is étale at x. • The local ring map O_S, s → O_X, x is flat and X_s → Spec(kappa(s)) is unramified at x. • The local ring map O_S, s → O_X, x is flat and the O_X, x-module Ω_X/S, x is zero. • The local ring map O_S, s → O_X, x is flat…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$.\nSet $s = f(x)$.\nAssume $f$ is locally of finite presentation.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is \\'etale at $x$.\n\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$\nis flat and $X_s \\to \\Spec(\\kappa(s))$ is \\'etale at $x$.\n\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$\nis flat and $X_s \\to \\Spec(\\kappa(s))$ is unramified at $x$.\n\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$\nis flat and the $\\mathcal{O}_{X, x}$-module $\\Omega_{X/S, x}$\nis zero.\n\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$\nis flat and the $\\kappa(x)$-vector space\n$$\n\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =\n\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)\n$$\nis zero.\n\\item The local ring map $\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}$\nis flat, we have $\\mathfrak m_s\\mathcal{O}_{X, x} = \\mathfrak m_x$ and\nthe field extension $\\kappa(x)/\\kappa(s)$ is finite\nseparable.\n\\item There exist affine opens $U \\subset X$,\nand $V \\subset S$ such that $x \\in U$, $f(U) \\subset V$ and the\ninduced morphism $f|_U : U \\to V$ is standard smooth\nof relative dimension $0$.\n\\item There exist affine opens $\\Spec(A) = U \\subset X$\nand $\\Spec(R) = V \\subset S$ with $x \\in U$ corresponding\nto $\\mathfrak q \\subset A$, and $f(U) \\subset V$\nsuch that there exists a presentation\n$$\nA = R[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)\n$$\nwith\n$$\ng =\n\\det\n\\left(\n\\begin{matrix}\n\\partial f_1/\\partial x_1 &\n\\partial f_2/\\partial x_1 &\n\\ldots &\n\\partial f_n/\\partial x_1 \\\\\n\\partial f_1/\\partial x_2 &\n\\partial f_2/\\partial x_2 &\n\\ldots &\n\\partial f_n/\\partial x_2 \\\\\n\\ldots & \\ldots & \\ldots & \\ldots \\\\\n\\partial f_1/\\partial x_n &\n\\partial f_2/\\partial x_n &\n\\ldots &\n\\partial f_n/\\partial x_n\n\\end{matrix}\n\\right)\n$$\nmapping to an element of $A$ not in $\\mathfrak q$.\n\\item There exist affine opens $U \\subset X$,\nand $V \\subset S$ such that $x \\in U$, $f(U) \\subset V$ and the\ninduced morphism $f|_U : U \\to V$ is standard \\'etale.\n\\item There exist affine opens $\\Spec(A) = U \\subset X$\nand $\\Spec(R) = V \\subset S$ with $x \\in U$ corresponding\nto $\\mathfrak q \\subset A$, and $f(U) \\subset V$\nsuch that there exists a presentation\n$$\nA = R[x]_Q/(P) = R[x, 1/Q]/(P)\n$$\nwith $P, Q \\in R[x]$, $P$ monic and $P' = \\text{d}P/\\text{d}x$ mapping to\nan element of $A$ not in $\\mathfrak q$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GU","source_file":"morphisms.tex","source_line":8317,"source_end_line":8391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8317-L8391","statement_sha256":"92c2a1f54d0205c05496892e21b041f4629c8a72ef1a1e1f4b633e476e4d9b29","origin":"The Stacks Project","memory_eligible":false,"source_rank":5628,"rank":5628,"depth":44,"x":1951.1,"y":665.589,"cluster":"scheme-morphisms"},{"id":"stacks:02GV","tag":"02GV","title":"Étale morphisms · Lemma 02GV","summary":"A morphism is étale at a point if and only if it is flat and G-unramified at that point. A morphism is étale if and only if it is flat and G-unramified.","statement_latex":"A morphism is \\'etale at a point if and only if it is flat and G-unramified\nat that point.\nA morphism is \\'etale if and only if it is flat and G-unramified.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GV","source_file":"morphisms.tex","source_line":8404,"source_end_line":8409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8404-L8409","statement_sha256":"9b1ea1f11909bc529210656c30cf5e97ef0caf90df00844457f28c7034bfc033","origin":"The Stacks Project","memory_eligible":false,"source_rank":5629,"rank":5629,"depth":45,"x":2099.886,"y":645.799,"cluster":"scheme-morphisms"},{"id":"stacks:0476","tag":"0476","title":"Étale morphisms · Lemma 0476","summary":"Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a cartesian diagram of schemes. Let W ⊂ X, resp. W' ⊂ X' be the open subscheme of points where f, resp. f' is étale. Then W' = (g')^-1(W) if • f is flat and locally of finite presentation, or • f is locally of finite presentation and g is flat.","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nbe a cartesian diagram of schemes. Let $W \\subset X$, resp.\\ $W' \\subset X'$\nbe the open subscheme of points where $f$, resp.\\ $f'$ is \\'etale.\nThen $W' = (g')^{-1}(W)$ if\n\\begin{enumerate}\n\\item $f$ is flat and locally of finite presentation, or\n\\item $f$ is locally of finite presentation and $g$ is flat.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0476","source_file":"morphisms.tex","source_line":8416,"source_end_line":8432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8416-L8432","statement_sha256":"769738afeb35392c97a79275e32fbf8a43b5d80688ea4b4afd65462165531108","origin":"The Stacks Project","memory_eligible":false,"source_rank":5630,"rank":5630,"depth":46,"x":2005.93,"y":744.983,"cluster":"scheme-morphisms"},{"id":"stacks:0HB1","tag":"0HB1","title":"Étale morphisms · Lemma 0HB1","summary":"See for example [GWII]. Let f : X → Y be a morphism of schemes over S. If X is étale over S and Y is unramified over S, then f is étale.","statement_latex":"\\begin{reference}\nSee for example \\cite[Remark 18.30 (2).]{GWII}.\n\\end{reference}\nLet $f : X \\to Y$ be a morphism of schemes over $S$.\nIf $X$ is \\'etale over $S$ and $Y$ is unramified over $S$, then\n$f$ is \\'etale.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HB1","source_file":"morphisms.tex","source_line":8458,"source_end_line":8466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8458-L8466","statement_sha256":"8d6d184d5d686d8b34269637114905b2e5c3c2d2ac6e2a238bd08495ae017f79","origin":"The Stacks Project","memory_eligible":false,"source_rank":5631,"rank":5631,"depth":40,"x":1995.433,"y":618.321,"cluster":"scheme-morphisms"},{"id":"stacks:02GW","tag":"02GW","title":"Étale morphisms · Lemma 02GW","summary":"Cancellation law for étale morphisms Let f : X → Y be a morphism of schemes over S. If X and Y are étale over S, then f is étale.","statement_latex":"\\begin{slogan}\nCancellation law for \\'etale morphisms\n\\end{slogan}\nLet $f : X \\to Y$ be a morphism of schemes over $S$.\nIf $X$ and $Y$ are \\'etale over $S$, then\n$f$ is \\'etale.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02GW","source_file":"morphisms.tex","source_line":8490,"source_end_line":8498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8490-L8498","statement_sha256":"7cf321c139f965a4c76aacc7348945cc4b5b3400eb3b0de5059378ad897709ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":5632,"rank":5632,"depth":42,"x":2105.215,"y":705.91,"cluster":"scheme-morphisms"},{"id":"stacks:02K6","tag":"02K6","title":"Étale morphisms · Lemma 02K6","summary":"Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & S be a commutative diagram of morphisms of schemes. Assume that • f is surjective, and étale, • p is étale, and • q is locally of finite presentation. Then q is étale.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume that\n\\begin{enumerate}\n\\item $f$ is surjective, and \\'etale,\n\\item $p$ is \\'etale, and\n\\item $q$ is locally of finite presentation\\footnote{In fact this\nis implied by (1) and (2), see\nDescent, Lemma \\ref{descent-lemma-flat-finitely-presented-permanence}.\nMoreover, it suffices to assume that $f$ is surjective, flat and\nlocally of finite presentation, see\nDescent, Lemma \\ref{descent-lemma-smooth-permanence}.}.\n\\end{enumerate}\nThen $q$ is \\'etale.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02K6","source_file":"morphisms.tex","source_line":8506,"source_end_line":8528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8506-L8528","statement_sha256":"a8f252447b9eeeea8f19548426415fad2cf5b29c2ce5189886fc4c6367254384","origin":"The Stacks Project","memory_eligible":false,"source_rank":5633,"rank":5633,"depth":40,"x":1953.575,"y":703.612,"cluster":"scheme-morphisms"},{"id":"stacks:054L","tag":"054L","title":"Étale morphisms · Lemma 054L","summary":"Smooth schemes are étale-locally like affine spaces. Let φ : X → Y be a morphism of schemes. Let x ∈ X. Let V ⊂ Y be an affine open neighbourhood of φ(x). If φ is smooth at x, then there exists an integer d ≥ 0 and an affine open U ⊂ X with x ∈ U and φ(U) ⊂ V such that there exists a commutative diagram xymatrix X ar[d] & U ar[l] ar[d] ar[r]_-π & A^d_V ar[ld] Y & V ar[l] where π is étale.","statement_latex":"\\begin{slogan}\nSmooth schemes are \\'etale-locally like affine spaces.\n\\end{slogan}\nLet $\\varphi : X \\to Y$ be a morphism of schemes. Let $x \\in X$.\nLet $V \\subset Y$ be an affine open neighbourhood of $\\varphi(x)$.\nIf $\\varphi$ is smooth at $x$, then there exists an integer $d \\geq 0$\nand an affine open $U \\subset X$ with $x \\in U$ and\n$\\varphi(U) \\subset V$ such that there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] & U \\ar[l] \\ar[d] \\ar[r]_-\\pi & \\mathbf{A}^d_V \\ar[ld] \\\\\nY & V \\ar[l]\n}\n$$\nwhere $\\pi$ is \\'etale.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054L","source_file":"morphisms.tex","source_line":8542,"source_end_line":8559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8542-L8559","statement_sha256":"dae409dc21e4f4fbb0588565d59a247e6f1e475d1f25cee32db196c0a3ca4955","origin":"The Stacks Project","memory_eligible":false,"source_rank":5634,"rank":5634,"depth":36,"x":2067.428,"y":619.122,"cluster":"scheme-morphisms"},{"id":"stacks:01VH","tag":"01VH","title":"Relatively ample sheaves · Definition 01VH","summary":"[EGA] Let f : X → S be a morphism of schemes. Let L be an invertible O_X-module. We say L is relatively ample, or f-relatively ample, or ample on X/S, or f-ample if f : X → S is quasi-compact, and if for every affine open V ⊂ S the restriction of L to the open subscheme f^-1(V) of X is ample.","statement_latex":"\\begin{reference}\n\\cite[II Definition 4.6.1]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nWe say $\\mathcal{L}$ is {\\it relatively ample}, or {\\it $f$-relatively ample},\nor {\\it ample on $X/S$}, or {\\it $f$-ample} if $f : X \\to S$\nis quasi-compact, and if for every affine open $V \\subset S$\nthe restriction of $\\mathcal{L}$ to the open subscheme\n$f^{-1}(V)$ of $X$ is ample.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VH","source_file":"morphisms.tex","source_line":8626,"source_end_line":8638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8626-L8638","statement_sha256":"397cc149b2b9adcde1996731eb13e25a6a9ff3e8b197602051b5d3dc875589cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":5635,"rank":5635,"depth":0,"x":2051.392,"y":746.237,"cluster":"scheme-morphisms"},{"id":"stacks:02NN","tag":"02NN","title":"Relatively ample sheaves · Lemma 02NN","summary":"Let X → S be a morphism of schemes. Let L be an invertible O_X-module. Let n ≥ 1. Then L is f-ample if and only if L^⊗ n is f-ample.","statement_latex":"Let $X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $n \\geq 1$. Then $\\mathcal{L}$ is $f$-ample if and only if\n$\\mathcal{L}^{\\otimes n}$ is $f$-ample.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NN","source_file":"morphisms.tex","source_line":8644,"source_end_line":8650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8644-L8650","statement_sha256":"c2ea7945a229413a896104955b269ae5745222708e1208eab1acfb0dc9795f53","origin":"The Stacks Project","memory_eligible":false,"source_rank":5636,"rank":5636,"depth":1,"x":1960.846,"y":643.236,"cluster":"scheme-morphisms"},{"id":"stacks:01VI","tag":"01VI","title":"Relatively ample sheaves · Lemma 01VI","summary":"Let f : X → S be a morphism of schemes. If there exists an f-ample invertible sheaf, then f is separated.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf there exists an $f$-ample invertible sheaf, then\n$f$ is separated.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VI","source_file":"morphisms.tex","source_line":8656,"source_end_line":8661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8656-L8661","statement_sha256":"ac493f572e0f1cbe1282d8d37f800954783d75566b16bc14c9800aba50c48763","origin":"The Stacks Project","memory_eligible":false,"source_rank":5637,"rank":5637,"depth":19,"x":2110.69,"y":667.85,"cluster":"scheme-morphisms"},{"id":"stacks:01VJ","tag":"01VJ","title":"Relatively ample sheaves · Lemma 01VJ","summary":"[EGA] Let f : X → S be a quasi-compact morphism of schemes. Let L be an invertible sheaf on X. The following are equivalent: • The invertible sheaf L is f-ample. • There exists an open covering S = ⋃ V_i such that each L|_f^-1(V_i) is ample relative to f^-1(V_i) → V_i. • There exists an affine open covering S = ⋃ V_i such that each L|_f^-1(V_i) is ample. • There exists a quasi-coherent graded O_S-algebra A and a map of graded O_X-algebras ψ : f^*A → bigoplus_d ≥ 0 L^⊗ d…","statement_latex":"\\begin{reference}\n\\cite[II, Proposition 4.6.3]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a quasi-compact morphism of schemes.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The invertible sheaf $\\mathcal{L}$ is $f$-ample.\n\\item There exists an open covering $S = \\bigcup V_i$\nsuch that each $\\mathcal{L}|_{f^{-1}(V_i)}$ is ample\nrelative to $f^{-1}(V_i) \\to V_i$.\n\\item There exists an affine open covering $S = \\bigcup V_i$\nsuch that each $\\mathcal{L}|_{f^{-1}(V_i)}$ is ample.\n\\item There exists a quasi-coherent graded $\\mathcal{O}_S$-algebra\n$\\mathcal{A}$ and a map of graded $\\mathcal{O}_X$-algebras\n$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d}$\nsuch that $U(\\psi) = X$ and\n$$\nr_{\\mathcal{L}, \\psi} :\nX\n\\longrightarrow\n\\underline{\\text{Proj}}_S(\\mathcal{A})\n$$\nis an open immersion (see Constructions, Lemma\n\\ref{constructions-lemma-invertible-map-into-relative-proj} for notation).\n\\item The morphism $f$ is quasi-separated and\npart (4) above holds with\n$\\mathcal{A} = f_*(\\bigoplus_{d \\geq 0} \\mathcal{L}^{\\otimes d})$\nand $\\psi$ the adjunction mapping.\n\\item Same as (4) but just requiring $r_{\\mathcal{L}, \\psi}$\nto be an immersion.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VJ","source_file":"morphisms.tex","source_line":8679,"source_end_line":8713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8679-L8713","statement_sha256":"7d5e87048ad764e1cb259317104285a2020f27e69abe79fdfd3627881d7e0ba6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5638,"rank":5638,"depth":20,"x":1980.19,"y":734.834,"cluster":"scheme-morphisms"},{"id":"stacks:01VK","tag":"01VK","title":"Relatively ample sheaves · Lemma 01VK","summary":"[EGA] Let f : X → S be a morphism of schemes. Let L be an invertible O_X-module. Assume S affine. Then L is f-relatively ample if and only if L is ample on X.","statement_latex":"\\begin{reference}\n\\cite[II Corollary 4.6.6]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume $S$ affine.\nThen $\\mathcal{L}$ is $f$-relatively ample if and only\nif $\\mathcal{L}$ is ample on $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VK","source_file":"morphisms.tex","source_line":8797,"source_end_line":8807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8797-L8807","statement_sha256":"7f4952eeb590706a60ea5e81c54458592f250db7f088835a67f44f53f30afb87","origin":"The Stacks Project","memory_eligible":false,"source_rank":5639,"rank":5639,"depth":21,"x":2022.621,"y":611.191,"cluster":"scheme-morphisms"},{"id":"stacks:0891","tag":"0891","title":"Relatively ample sheaves · Lemma 0891","summary":"[EGA] Let f : X → S be a morphism of schemes. Then f is quasi-affine if and only if O_X is f-relatively ample.","statement_latex":"\\begin{reference}\n\\cite[II Proposition 5.1.6]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes. Then $f$ is quasi-affine\nif and only if $\\mathcal{O}_X$ is $f$-relatively ample.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0891","source_file":"morphisms.tex","source_line":8814,"source_end_line":8821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8814-L8821","statement_sha256":"7c0670c72ef19ddd70c89105ca4d93db8bc61e551338b14750f67f8b3a98f3ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":5640,"rank":5640,"depth":21,"x":2090.875,"y":726.628,"cluster":"scheme-morphisms"},{"id":"stacks:0892","tag":"0892","title":"Relatively ample sheaves · Lemma 0892","summary":"Let f : X → Y be a morphism of schemes, M an invertible O_Y-module, and L an invertible O_X-module. • If L is f-ample and M is ample, then L ⊗ f^*M^⊗ a is ample for a gg 0. • If M is ample and f quasi-affine, then f^*M is ample.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes, $\\mathcal{M}$\nan invertible $\\mathcal{O}_Y$-module, and $\\mathcal{L}$ an\ninvertible $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If $\\mathcal{L}$ is $f$-ample and $\\mathcal{M}$\nis ample, then $\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a}$ is ample\nfor $a \\gg 0$.\n\\item If $\\mathcal{M}$ is ample\nand $f$ quasi-affine, then $f^*\\mathcal{M}$ is ample.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0892","source_file":"morphisms.tex","source_line":8828,"source_end_line":8840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8828-L8840","statement_sha256":"9bf95bb812fdda2e7bb392da63bfa7c8f3bd4b1d002b09e9c140aa5ba1160b03","origin":"The Stacks Project","memory_eligible":false,"source_rank":5641,"rank":5641,"depth":22,"x":1947.482,"y":680.16,"cluster":"scheme-morphisms"},{"id":"stacks:0C4K","tag":"0C4K","title":"Relatively ample sheaves · Lemma 0C4K","summary":"Let g : Y → S and f : X → Y be morphisms of schemes. Let M be an invertible O_Y-module. Let L be an invertible O_X-module. If S is quasi-compact, M is g-ample, and L is f-ample, then L ⊗ f^*M^⊗ a is g ∘ f-ample for a gg 0.","statement_latex":"Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes.\nLet $\\mathcal{M}$ be an invertible $\\mathcal{O}_Y$-module.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nIf $S$ is quasi-compact, $\\mathcal{M}$ is $g$-ample, and\n$\\mathcal{L}$ is $f$-ample, then\n$\\mathcal{L} \\otimes f^*\\mathcal{M}^{\\otimes a}$\nis $g \\circ f$-ample for $a \\gg 0$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4K","source_file":"morphisms.tex","source_line":8887,"source_end_line":8896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8887-L8896","statement_sha256":"c4b22bbf6a8730d66e822cd4aa266eb904c50d2473d326d7d1b23d1d42c0bfed","origin":"The Stacks Project","memory_eligible":false,"source_rank":5642,"rank":5642,"depth":23,"x":2090.817,"y":632.984,"cluster":"scheme-morphisms"},{"id":"stacks:0893","tag":"0893","title":"Relatively ample sheaves · Lemma 0893","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible O_X-module. Let S' → S be a morphism of schemes. Let f' : X' → S' be the base change of f and denote L' the pullback of L to X'. If L is f-ample, then L' is f'-ample.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $S' \\to S$ be a morphism of schemes.\nLet $f' : X' \\to S'$ be the base change of $f$ and denote\n$\\mathcal{L}'$ the pullback of $\\mathcal{L}$ to $X'$.\nIf $\\mathcal{L}$ is $f$-ample, then $\\mathcal{L}'$ is $f'$-ample.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0893","source_file":"morphisms.tex","source_line":8908,"source_end_line":8916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8908-L8916","statement_sha256":"931d5766834ed611b32fa60c45834a6db772994716b0cf41889b34aaf548f1d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5643,"rank":5643,"depth":23,"x":2022.952,"y":749.289,"cluster":"scheme-morphisms"},{"id":"stacks:0C4L","tag":"0C4L","title":"Relatively ample sheaves · Lemma 0C4L","summary":"Let g : Y → S and f : X → Y be morphisms of schemes. Let L be an invertible O_X-module. If L is g ∘ f-ample and f is quasi-compact then L is f-ample.","statement_latex":"Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nIf $\\mathcal{L}$ is $g \\circ f$-ample and $f$ is\nquasi-compact\\footnote{This follows if $g$ is quasi-separated by\nSchemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.}\nthen $\\mathcal{L}$ is $f$-ample.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4L","source_file":"morphisms.tex","source_line":8931,"source_end_line":8939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8931-L8939","statement_sha256":"9d89f02e72c2cb5bb854efb5fc99f9b59f97c08cf578940b0b023e6536f0e79a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5644,"rank":5644,"depth":22,"x":1979.395,"y":624.82,"cluster":"scheme-morphisms"},{"id":"stacks:01VM","tag":"01VM","title":"Very ample sheaves · Definition 01VM","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible O_X-module. We say L is relatively very ample or more precisely f-relatively very ample, or very ample on X/S, or f-very ample if there exist a quasi-coherent O_S-module E and an immersion i : X → P(E) over S such that L ≅ i^*O_P(E)(1).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nWe say $\\mathcal{L}$ is {\\it relatively very ample} or more\nprecisely {\\it $f$-relatively very ample}, or\n{\\it very ample on $X/S$}, or {\\it $f$-very ample} if\nthere exist a quasi-coherent $\\mathcal{O}_S$-module\n$\\mathcal{E}$ and an immersion $i : X \\to \\mathbf{P}(\\mathcal{E})$\nover $S$ such that\n$\\mathcal{L} \\cong i^*\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(1)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Very ample sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VM","source_file":"morphisms.tex","source_line":8977,"source_end_line":8988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8977-L8988","statement_sha256":"3686c4f60e9ee8b1af83fb0ad50543f29bf2169e5626663ed19aeee611f11a60","origin":"The Stacks Project","memory_eligible":false,"source_rank":5645,"rank":5645,"depth":0,"x":2111.82,"y":691.994,"cluster":"scheme-morphisms"},{"id":"stacks:01VN","tag":"01VN","title":"Very ample sheaves · Lemma 01VN","summary":"[EGA] Let f : X → S be a morphism of schemes. Let L be an invertible O_X-module. If f is quasi-compact and L is a relatively very ample invertible sheaf, then L is a relatively ample invertible sheaf.","statement_latex":"\\begin{reference}\n\\cite[II, Proposition 4.6.2]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nIf $f$ is quasi-compact and $\\mathcal{L}$ is a relatively\nvery ample invertible sheaf, then $\\mathcal{L}$ is a relatively\nample invertible sheaf.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Very ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VN","source_file":"morphisms.tex","source_line":8997,"source_end_line":9007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L8997-L9007","statement_sha256":"35d9e2c3db3c3e8888572b3b89cdd8e5c7fc58a99bdff66d6960dce1d919512c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5646,"rank":5646,"depth":21,"x":1959.91,"y":717.642,"cluster":"scheme-morphisms"},{"id":"stacks:01VQ","tag":"01VQ","title":"Very ample sheaves · Lemma 01VQ","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. If L is relatively very ample on X/S then f is separated.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nIf $\\mathcal{L}$ is relatively very ample on $X/S$ then\n$f$ is separated.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Very ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VQ","source_file":"morphisms.tex","source_line":9127,"source_end_line":9133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9127-L9133","statement_sha256":"b13c3f77f1d98c4dfcd18660d9050995f580aa753043c9f3d1b852cdff9c0c90","origin":"The Stacks Project","memory_eligible":false,"source_rank":5647,"rank":5647,"depth":14,"x":2051.446,"y":612.365,"cluster":"scheme-morphisms"},{"id":"stacks:01VR","tag":"01VR","title":"Very ample sheaves · Lemma 01VR","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume f is quasi-compact. The following are equivalent • L is relatively very ample on X/S, • there exists an open covering S = ⋃ V_j such that L|_f^-1(V_j) is relatively very ample on f^-1(V_j)/V_j for all j, • there exists a quasi-coherent sheaf of graded O_S-algebras A generated in degree 1 over O_S and a map of graded O_X-algebras ψ : f^*A → bigoplus_n ≥ 0 L^⊗ n such that f^*A_1 → L is…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nAssume $f$ is quasi-compact. The following are\nequivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is relatively very ample on $X/S$,\n\\item there exists an open covering $S = \\bigcup V_j$ such\nthat $\\mathcal{L}|_{f^{-1}(V_j)}$ is relatively very ample\non $f^{-1}(V_j)/V_j$ for all $j$,\n\\item there exists a quasi-coherent sheaf of graded\n$\\mathcal{O}_S$-algebras $\\mathcal{A}$ generated in degree\n$1$ over $\\mathcal{O}_S$ and a map of graded $\\mathcal{O}_X$-algebras\n$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$\nsuch that $f^*\\mathcal{A}_1 \\to \\mathcal{L}$ is surjective and the\nassociated morphism\n$r_{\\mathcal{L}, \\psi} : X \\to \\underline{\\text{Proj}}_S(\\mathcal{A})$\nis an immersion, and\n\\item $f$ is quasi-separated, the canonical map\n$\\psi : f^*f_*\\mathcal{L} \\to \\mathcal{L}$ is surjective, and\nthe associated map $r_{\\mathcal{L}, \\psi} : X \\to \\mathbf{P}(f_*\\mathcal{L})$\nis an immersion.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Very ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VR","source_file":"morphisms.tex","source_line":9145,"source_end_line":9169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9145-L9169","statement_sha256":"fef3e69a7b3366c50d0e94fc5c1f2e0c3d29f83686f4bc56ca07d59d09d3d0d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5648,"rank":5648,"depth":15,"x":2068.637,"y":742.14,"cluster":"scheme-morphisms"},{"id":"stacks:0B3F","tag":"0B3F","title":"Very ample sheaves · Lemma 0B3F","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible O_X-module. Let S' → S be a morphism of schemes. Let f' : X' → S' be the base change of f and denote L' the pullback of L to X'. If L is f-very ample, then L' is f'-very ample.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $S' \\to S$ be a morphism of schemes.\nLet $f' : X' \\to S'$ be the base change of $f$ and denote\n$\\mathcal{L}'$ the pullback of $\\mathcal{L}$ to $X'$.\nIf $\\mathcal{L}$ is $f$-very ample, then $\\mathcal{L}'$ is $f'$-very ample.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Very ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3F","source_file":"morphisms.tex","source_line":9239,"source_end_line":9247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9239-L9247","statement_sha256":"b6f983587e4b09536e953ac31fc06025b850117eb070222c620eb7a6bce55c05","origin":"The Stacks Project","memory_eligible":false,"source_rank":5649,"rank":5649,"depth":14,"x":1951.415,"y":656.065,"cluster":"scheme-morphisms"},{"id":"stacks:02NP","tag":"02NP","title":"Ample and very ample sheaves relative to finite type morphisms · Lemma 02NP","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume that • the invertible sheaf L is very ample on X/S, • the morphism X → S is of finite type, and • S is affine. Then there exist an n ≥ 0 and an immersion i : X → P^n_S over S such that L ≅ i^*O_P^n_S(1).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nAssume that\n\\begin{enumerate}\n\\item the invertible sheaf $\\mathcal{L}$ is very ample on $X/S$,\n\\item the morphism $X \\to S$ is of finite type, and\n\\item $S$ is affine.\n\\end{enumerate}\nThen there exist an $n \\geq 0$ and an immersion\n$i : X \\to \\mathbf{P}^n_S$ over $S$ such that\n$\\mathcal{L} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Ample and very ample sheaves relative to finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NP","source_file":"morphisms.tex","source_line":9279,"source_end_line":9292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9279-L9292","statement_sha256":"7cd809d402b12d7869bcc3861f058077c751b6336da0b2d9e60d18df7ec0273d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5650,"rank":5650,"depth":5,"x":2107.317,"y":653.016,"cluster":"scheme-morphisms"},{"id":"stacks:04II","tag":"04II","title":"Ample and very ample sheaves relative to finite type morphisms · Lemma 04II","summary":"Let π : X → S be a morphism of schemes. Assume that X is quasi-affine and that π is locally of finite type. Then there exist n ≥ 0 and an immersion i : X → A^n_S over S.","statement_latex":"Let $\\pi : X \\to S$ be a morphism of schemes.\nAssume that $X$ is quasi-affine and that $\\pi$ is locally of finite type.\nThen there exist $n \\geq 0$ and an immersion $i : X \\to \\mathbf{A}^n_S$\nover $S$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Ample and very ample sheaves relative to finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04II","source_file":"morphisms.tex","source_line":9372,"source_end_line":9378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9372-L9378","statement_sha256":"eb66ae51438136d59d6bf507f3cea141ad41a907f79224b3607b7a82496e2c2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5651,"rank":5651,"depth":18,"x":1994.632,"y":743.869,"cluster":"scheme-morphisms"},{"id":"stacks:01VS","tag":"01VS","title":"Ample and very ample sheaves relative to finite type morphisms · Lemma 01VS","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume that • the invertible sheaf L is ample on X, and • the morphism X → S is locally of finite type. Then there exists a d_0 ≥ 1 such that for every d ≥ d_0 there exist an n ≥ 0 and an immersion i : X → P^n_S over S such that L^⊗ d ≅ i^*O_P^n_S(1).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nAssume that\n\\begin{enumerate}\n\\item the invertible sheaf $\\mathcal{L}$ is ample on $X$, and\n\\item the morphism $X \\to S$ is locally of finite type.\n\\end{enumerate}\nThen there exists a $d_0 \\geq 1$ such that for every $d \\geq d_0$\nthere exist an $n \\geq 0$ and an immersion\n$i : X \\to \\mathbf{P}^n_S$ over $S$ such that\n$\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Ample and very ample sheaves relative to finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VS","source_file":"morphisms.tex","source_line":9412,"source_end_line":9425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9412-L9425","statement_sha256":"31c73d0cba173f2f7667ec61fe49707188e3557d0759132711857614951c6e5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5652,"rank":5652,"depth":21,"x":2004.678,"y":612.73,"cluster":"scheme-morphisms"},{"id":"stacks:01VT","tag":"01VT","title":"Ample and very ample sheaves relative to finite type morphisms · Lemma 01VT","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible O_X-module. Assume S affine and f of finite type. The following are equivalent • L is ample on X, • L is f-ample, • L^⊗ d is f-very ample for some d ≥ 1, • L^⊗ d is f-very ample for all d gg 1, • for some d ≥ 1 there exist n ≥ 1 and an immersion i : X → P^n_S such that L^⊗ d ≅ i^*O_P^n_S(1), and • for all d gg 1 there exist n ≥ 1 and an immersion i : X → P^n_S such that L^⊗ d ≅ i^*O_P^n_S(1).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume $S$ affine and $f$ of finite type.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample on $X$,\n\\item $\\mathcal{L}$ is $f$-ample,\n\\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for some $d \\geq 1$,\n\\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for all $d \\gg 1$,\n\\item for some $d \\geq 1$ there exist $n \\geq 1$ and an immersion\n$i : X \\to \\mathbf{P}^n_S$ such that\n$\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$, and\n\\item for all $d \\gg 1$ there exist $n \\geq 1$ and an immersion\n$i : X \\to \\mathbf{P}^n_S$ such that\n$\\mathcal{L}^{\\otimes d} \\cong i^*\\mathcal{O}_{\\mathbf{P}^n_S}(1)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Ample and very ample sheaves relative to finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VT","source_file":"morphisms.tex","source_line":9517,"source_end_line":9535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9517-L9535","statement_sha256":"8916da3b6f0de34931ec5a2fc3f022bfcecc21c2b0545e5bb1dd46ce8679c252","origin":"The Stacks Project","memory_eligible":false,"source_rank":5653,"rank":5653,"depth":22,"x":2102.881,"y":715.29,"cluster":"scheme-morphisms"},{"id":"stacks:01VU","tag":"01VU","title":"Ample and very ample sheaves relative to finite type morphisms · Lemma 01VU","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible O_X-module. Assume S quasi-compact and f of finite type. The following are equivalent • L is f-ample, • L^⊗ d is f-very ample for some d ≥ 1, • L^⊗ d is f-very ample for all d gg 1.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume $S$ quasi-compact and $f$ of finite type.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is $f$-ample,\n\\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for some $d \\geq 1$,\n\\item $\\mathcal{L}^{\\otimes d}$ is $f$-very ample for all $d \\gg 1$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Ample and very ample sheaves relative to finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VU","source_file":"morphisms.tex","source_line":9552,"source_end_line":9563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9552-L9563","statement_sha256":"083069058522e303e73eb921c9b6195659099d8d8a2645a58323df5c561a57b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5654,"rank":5654,"depth":23,"x":1947.751,"y":695.356,"cluster":"scheme-morphisms"},{"id":"stacks:02NQ","tag":"02NQ","title":"Ample and very ample sheaves relative to finite type morphisms · Lemma 02NQ","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume f is of finite type. The following are equivalent: • L is f-relatively very ample, and • there exist an open covering S = ⋃ V_j, for each j an integer n_j, and immersions i_j : X_j = f^-1(V_j) = V_j ×_S X → P^n_j_V_j over V_j such that L|_X_j ≅ i_j^*O_P^n_j_V_j(1).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nAssume $f$ is of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is $f$-relatively very ample, and\n\\item there exist an open covering $S = \\bigcup V_j$,\nfor each $j$ an integer $n_j$, and immersions\n$$\ni_j :\nX_j = f^{-1}(V_j) = V_j \\times_S X\n\\longrightarrow\n\\mathbf{P}^{n_j}_{V_j}\n$$\nover $V_j$ such that\n$\\mathcal{L}|_{X_j} \\cong i_j^*\\mathcal{O}_{\\mathbf{P}^{n_j}_{V_j}}(1)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Ample and very ample sheaves relative to finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NQ","source_file":"morphisms.tex","source_line":9581,"source_end_line":9600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9581-L9600","statement_sha256":"4fbd5c034ca5bc2f31bf3400e2994d1d8cee7f7f599634c1abc969de90956f80","origin":"The Stacks Project","memory_eligible":false,"source_rank":5655,"rank":5655,"depth":16,"x":2078.375,"y":621.917,"cluster":"scheme-morphisms"},{"id":"stacks:02NR","tag":"02NR","title":"Ample and very ample sheaves relative to finite type morphisms · Lemma 02NR","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume f is of finite type. The following are equivalent: • L is f-relatively ample, and • there exist an open covering S = ⋃ V_j, for each j an integers d_j ≥ 1, n_j ≥ 0, and immersions i_j : X_j = f^-1(V_j) = V_j ×_S X → P^n_j_V_j over V_j such that L^⊗ d_j|_X_j ≅ i_j^*O_P^n_j_V_j(1).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nAssume $f$ is of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is $f$-relatively ample, and\n\\item there exist an open covering $S = \\bigcup V_j$,\nfor each $j$ an integers $d_j \\geq 1$,\n$n_j \\geq 0$, and immersions\n$$\ni_j :\nX_j = f^{-1}(V_j) = V_j \\times_S X\n\\longrightarrow\n\\mathbf{P}^{n_j}_{V_j}\n$$\nover $V_j$ such that\n$\\mathcal{L}^{\\otimes d_j}|_{X_j} \\cong\ni_j^*\\mathcal{O}_{\\mathbf{P}^{n_j}_{V_j}}(1)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Ample and very ample sheaves relative to finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NR","source_file":"morphisms.tex","source_line":9610,"source_end_line":9631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9610-L9631","statement_sha256":"6719b527d0f0240c4583a908bb518b203e58ade7977559f6d0d76ffa2c346743","origin":"The Stacks Project","memory_eligible":false,"source_rank":5656,"rank":5656,"depth":23,"x":2041.055,"y":750.387,"cluster":"scheme-morphisms"},{"id":"stacks:0FVC","tag":"0FVC","title":"Ample and very ample sheaves relative to finite type morphisms · Lemma 0FVC","summary":"Let f : X → S be a morphism of schemes. Let N, L be invertible O_X-modules. Assume S is quasi-compact, f is of finite type, and L is f-ample. Then N ⊗_O_X L^⊗ d is f-very ample for all d gg 1.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{N}$, $\\mathcal{L}$ be invertible $\\mathcal{O}_X$-modules.\nAssume $S$ is quasi-compact, $f$ is of finite type, and $\\mathcal{L}$\nis $f$-ample. Then\n$\\mathcal{N} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}$\nis $f$-very ample for all $d \\gg 1$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Ample and very ample sheaves relative to finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVC","source_file":"morphisms.tex","source_line":9641,"source_end_line":9649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9641-L9649","statement_sha256":"6e76a96faf22a3dae084969758ba6dd2df7f0de4a83776ec3f15dc815b043a85","origin":"The Stacks Project","memory_eligible":false,"source_rank":5657,"rank":5657,"depth":23,"x":1965.146,"y":634.3,"cluster":"scheme-morphisms"},{"id":"stacks:01VW","tag":"01VW","title":"Quasi-projective morphisms · Definition 01VW","summary":"[EGA] and [H] Let f : X → S be a morphism of schemes. • We say f is quasi-projective if f is of finite type and there exists an f-relatively ample invertible O_X-module. • We say f is H-quasi-projective if there exists a quasi-compact immersion X → P^n_S over S for some n. • We say f is locally quasi-projective if there exists an open covering S = ⋃ V_j such that each f^-1(V_j) → V_j is quasi-projective.","statement_latex":"\\begin{reference}\n\\cite[II, Definition 5.3.1]{EGA} and \\cite[page 103]{H}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say $f$ is {\\it quasi-projective} if $f$ is of finite type\nand there exists an $f$-relatively ample invertible $\\mathcal{O}_X$-module.\n\\item We say $f$ is {\\it H-quasi-projective} if there exists\na quasi-compact immersion $X \\to \\mathbf{P}^n_S$ over $S$ for some\n$n$.\\footnote{This is not exactly the same as the definition in Hartshorne.\nNamely, the definition in Hartshorne (8th corrected printing, 1997) is that\n$f$ should be the composition of an open immersion followed by a H-projective\nmorphism (see Definition \\ref{definition-projective}), which does not imply\n$f$ is quasi-compact. See\nLemma \\ref{lemma-H-quasi-projective-open-H-projective} for\nthe implication in the other direction.}\n\\item We say $f$ is {\\it locally quasi-projective} if there exists\nan open covering $S = \\bigcup V_j$ such that each $f^{-1}(V_j) \\to V_j$\nis quasi-projective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-projective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VW","source_file":"morphisms.tex","source_line":9713,"source_end_line":9735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9713-L9735","statement_sha256":"6ccf068c8fe782ab89aebd7bb75191680d3ef5b04bae844c095fa52487be9681","origin":"The Stacks Project","memory_eligible":false,"source_rank":5658,"rank":5658,"depth":0,"x":2114.702,"y":676.894,"cluster":"scheme-morphisms"},{"id":"stacks:0B3G","tag":"0B3G","title":"Quasi-projective morphisms · Lemma 0B3G","summary":"A base change of a quasi-projective morphism is quasi-projective.","statement_latex":"A base change of a quasi-projective morphism is quasi-projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3G","source_file":"morphisms.tex","source_line":9743,"source_end_line":9746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9743-L9746","statement_sha256":"3cd6d77da555825b5a178f5a76056fc9bdd58cad2fc6c110d84e3cfd0cb40f80","origin":"The Stacks Project","memory_eligible":false,"source_rank":5659,"rank":5659,"depth":24,"x":1969.948,"y":730.43,"cluster":"scheme-morphisms"},{"id":"stacks:0C4M","tag":"0C4M","title":"Quasi-projective morphisms · Lemma 0C4M","summary":"Let f : X → Y and g : Y → S be morphisms of schemes. If S is quasi-compact and f and g are quasi-projective, then g ∘ f is quasi-projective.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes.\nIf $S$ is quasi-compact and $f$ and $g$ are quasi-projective,\nthen $g \\circ f$ is quasi-projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4M","source_file":"morphisms.tex","source_line":9754,"source_end_line":9759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9754-L9759","statement_sha256":"7d12a7c2a604687401fc90bcce11b26bd94824bbe77d513509c4818957950355","origin":"The Stacks Project","memory_eligible":false,"source_rank":5660,"rank":5660,"depth":24,"x":2033.733,"y":608.631,"cluster":"scheme-morphisms"},{"id":"stacks:01VX","tag":"01VX","title":"Quasi-projective morphisms · Lemma 01VX","summary":"Let f : X → S be a morphism of schemes. If f is quasi-projective, or H-quasi-projective or locally quasi-projective, then f is separated of finite type.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. If $f$ is quasi-projective,\nor H-quasi-projective or locally quasi-projective, then $f$ is\nseparated of finite type.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VX","source_file":"morphisms.tex","source_line":9767,"source_end_line":9772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9767-L9772","statement_sha256":"7af6c7a997336e45c4226773d0017b837231cae22234031793dc1bdc73fc6f9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5661,"rank":5661,"depth":0,"x":2084.724,"y":734.828,"cluster":"scheme-morphisms"},{"id":"stacks:01VY","tag":"01VY","title":"Quasi-projective morphisms · Lemma 01VY","summary":"A H-quasi-projective morphism is quasi-projective.","statement_latex":"A H-quasi-projective morphism is quasi-projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VY","source_file":"morphisms.tex","source_line":9778,"source_end_line":9781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9778-L9781","statement_sha256":"5dcc275d4e3e7109eeb47836b00f1bc1d15ea92e14c0e07ad4cfe2745d4784e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5662,"rank":5662,"depth":0,"x":1945.429,"y":670.608,"cluster":"scheme-morphisms"},{"id":"stacks:01VZ","tag":"01VZ","title":"Quasi-projective morphisms · Lemma 01VZ","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is locally quasi-projective. • There exists an open covering S = ⋃ V_j such that each f^-1(V_j) → V_j is H-quasi-projective.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is locally quasi-projective.\n\\item There exists an open covering $S = \\bigcup V_j$ such\nthat each $f^{-1}(V_j) \\to V_j$ is H-quasi-projective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01VZ","source_file":"morphisms.tex","source_line":9787,"source_end_line":9796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9787-L9796","statement_sha256":"d456133f663f749f5f001ecb432c974fc30ba1f4a33a7a8bc01a29e0e5d79995","origin":"The Stacks Project","memory_eligible":false,"source_rank":5663,"rank":5663,"depth":23,"x":2100.02,"y":638.876,"cluster":"scheme-morphisms"},{"id":"stacks:0B3H","tag":"0B3H","title":"Quasi-projective morphisms · Lemma 0B3H","summary":"[EGA] A quasi-affine morphism of finite type is quasi-projective.","statement_latex":"\\begin{reference}\n\\cite[II, Proposition 5.3.4 (i)]{EGA}\n\\end{reference}\nA quasi-affine morphism of finite type is quasi-projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3H","source_file":"morphisms.tex","source_line":9812,"source_end_line":9818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9812-L9818","statement_sha256":"28cba0ac6159431cf6ca2fdab2f96adbe33a9a0c549b1aced8c536e0642fc3dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5664,"rank":5664,"depth":22,"x":2011.412,"y":750.161,"cluster":"scheme-morphisms"},{"id":"stacks:0C4N","tag":"0C4N","title":"Quasi-projective morphisms · Lemma 0C4N","summary":"Let g : Y → S and f : X → Y be morphisms of schemes. If g ∘ f is quasi-projective and f is quasi-compact, then f is quasi-projective.","statement_latex":"Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes.\nIf $g \\circ f$ is quasi-projective and $f$ is\nquasi-compact\\footnote{This follows if $g$ is quasi-separated by\nSchemes, Lemma \\ref{schemes-lemma-quasi-compact-permanence}.},\nthen $f$ is quasi-projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Quasi-projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4N","source_file":"morphisms.tex","source_line":9824,"source_end_line":9831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9824-L9831","statement_sha256":"f8368f04dfe98b5e7a9d66256b2db485bacdd2efe4b8fd4bd23a6621c40d5034","origin":"The Stacks Project","memory_eligible":false,"source_rank":5665,"rank":5665,"depth":23,"x":1987.221,"y":617.624,"cluster":"scheme-morphisms"},{"id":"stacks:01W1","tag":"01W1","title":"Proper morphisms · Definition 01W1","summary":"Let f : X → S be a morphism of schemes. We say f is proper if f is separated, finite type, and universally closed.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nWe say $f$ is {\\it proper} if $f$ is separated, finite type, and\nuniversally closed.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01W1","source_file":"morphisms.tex","source_line":9853,"source_end_line":9858,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9853-L9858","statement_sha256":"2ebd838666ed3c04831e2ab70002c3fcaec82e9d6bc5f3acc1af7edc5695d5ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":5666,"rank":5666,"depth":0,"x":2111.828,"y":701.754,"cluster":"scheme-morphisms"},{"id":"stacks:02K7","tag":"02K7","title":"Proper morphisms · Lemma 02K7","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is universally closed. • There exists an open covering S = ⋃ V_j such that f^-1(V_j) → V_j is universally closed for all indices j.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is universally closed.\n\\item There exists an open covering $S = \\bigcup V_j$ such\nthat $f^{-1}(V_j) \\to V_j$ is universally closed for all indices $j$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02K7","source_file":"morphisms.tex","source_line":9867,"source_end_line":9876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9867-L9876","statement_sha256":"46a2438b7c64913a250fe20e4bbf24f1ed810adc5c6ab3008731bffeda0c57b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5667,"rank":5667,"depth":0,"x":1952.051,"y":710.435,"cluster":"scheme-morphisms"},{"id":"stacks:01W2","tag":"01W2","title":"Proper morphisms · Lemma 01W2","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is proper. • There exists an open covering S = ⋃ V_j such that f^-1(V_j) → V_j is proper for all indices j.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is proper.\n\\item There exists an open covering $S = \\bigcup V_j$ such\nthat $f^{-1}(V_j) \\to V_j$ is proper for all indices $j$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01W2","source_file":"morphisms.tex","source_line":9882,"source_end_line":9891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9882-L9891","statement_sha256":"dafc6a5bbb21bb7cf3c7b198f9d75453de1e665d81477d027476833c20ec9529","origin":"The Stacks Project","memory_eligible":false,"source_rank":5668,"rank":5668,"depth":0,"x":2063.052,"y":613.229,"cluster":"scheme-morphisms"},{"id":"stacks:01W3","tag":"01W3","title":"Proper morphisms · Lemma 01W3","summary":"The composition of proper morphisms is proper. The same is true for universally closed morphisms.","statement_latex":"The composition of proper morphisms is proper.\nThe same is true for universally closed morphisms.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01W3","source_file":"morphisms.tex","source_line":9897,"source_end_line":9901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9897-L9901","statement_sha256":"44d91026b6c1ffe17fa24ef797e21c4b8948687565522e276439f95341700618","origin":"The Stacks Project","memory_eligible":false,"source_rank":5669,"rank":5669,"depth":16,"x":2059.367,"y":748.091,"cluster":"scheme-morphisms"},{"id":"stacks:01W4","tag":"01W4","title":"Proper morphisms · Lemma 01W4","summary":"The base change of a proper morphism is proper. The same is true for universally closed morphisms.","statement_latex":"The base change of a proper morphism is proper.\nThe same is true for universally closed morphisms.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01W4","source_file":"morphisms.tex","source_line":9917,"source_end_line":9921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9917-L9921","statement_sha256":"f92b1177097893db214332b5ea17ba31483229b990e67b83a2288d452a6cffa6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5670,"rank":5670,"depth":16,"x":1953.475,"y":646.404,"cluster":"scheme-morphisms"},{"id":"stacks:01W5","tag":"01W5","title":"Proper morphisms · Lemma 01W5","summary":"A closed immersion is proper, hence a fortiori universally closed.","statement_latex":"A closed immersion is proper, hence a fortiori universally closed.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01W5","source_file":"morphisms.tex","source_line":9932,"source_end_line":9935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9932-L9935","statement_sha256":"336b98f551e5c4c01e7e7ffec89e96a860f1ebc29e4c4e87772a886157f4aa33","origin":"The Stacks Project","memory_eligible":false,"source_rank":5671,"rank":5671,"depth":14,"x":2113.568,"y":661.323,"cluster":"scheme-morphisms"},{"id":"stacks:01W6","tag":"01W6","title":"Proper morphisms · Lemma 01W6","summary":"Suppose given a commutative diagram of schemes xymatrix X ar[rr] ar[rd] & & Y ar[ld] & S & with Y separated over S. • If X → S is universally closed, then the morphism X → Y is universally closed. • If X is proper over S, then the morphism X → Y is proper. In particular, in both cases the image of X in Y is closed.","statement_latex":"Suppose given a commutative diagram of schemes\n$$\n\\xymatrix{\nX \\ar[rr] \\ar[rd] & &\nY \\ar[ld] \\\\\n& S &\n}\n$$\nwith $Y$ separated over $S$.\n\\begin{enumerate}\n\\item If $X \\to S$ is universally closed, then the morphism\n$X \\to Y$ is universally closed.\n\\item If $X$ is proper over $S$, then the morphism $X \\to Y$ is proper.\n\\end{enumerate}\nIn particular, in both cases the image of $X$ in $Y$ is closed.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01W6","source_file":"morphisms.tex","source_line":9948,"source_end_line":9965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9948-L9965","statement_sha256":"000b1f57958deec4af1dd22dba097a0fd5855db84b579a3fa4e6e110fe7a9e6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5672,"rank":5672,"depth":17,"x":1983.329,"y":741.28,"cluster":"scheme-morphisms"},{"id":"stacks:04XU","tag":"04XU","title":"Proper morphisms · Lemma 04XU","summary":"Due to Bjorn Poonen. A universally closed morphism of schemes is quasi-compact.","statement_latex":"\\begin{reference}\nDue to Bjorn Poonen.\n\\end{reference}\nA universally closed morphism of schemes is quasi-compact.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XU","source_file":"morphisms.tex","source_line":9985,"source_end_line":9991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L9985-L9991","statement_sha256":"da8b09666a5e9c81f48d36eb04bccfca278a1403be7817c78a785d4f6e949abe","origin":"The Stacks Project","memory_eligible":false,"source_rank":5673,"rank":5673,"depth":2,"x":2015.113,"y":608.226,"cluster":"scheme-morphisms"},{"id":"stacks:03GN","tag":"03GN","title":"Proper morphisms · Lemma 03GN","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. If X is universally closed over S and f is surjective then Y is universally closed over S. In particular, if also Y is separated and locally of finite type over S, then Y is proper over S.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.\nIf $X$ is universally closed over $S$ and $f$ is surjective then\n$Y$ is universally closed over $S$. In particular, if also $Y$ is\nseparated and locally of finite type over $S$, then $Y$ is proper over $S$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GN","source_file":"morphisms.tex","source_line":10053,"source_end_line":10059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10053-L10059","statement_sha256":"64daba10c2ee32709a20379bcecb6830cb65dfe0ece3e85db239f1af29d97276","origin":"The Stacks Project","memory_eligible":false,"source_rank":5674,"rank":5674,"depth":4,"x":2098.797,"y":724.543,"cluster":"scheme-morphisms"},{"id":"stacks:0AH6","tag":"0AH6","title":"Proper morphisms · Lemma 0AH6","summary":"Suppose given a commutative diagram of schemes xymatrix X ar[rr]_h ar[rd]_f & & Y ar[ld]^g & S Assume • X → S is a universally closed (for example proper) morphism, and • Y → S is separated and locally of finite type. Then the scheme theoretic image Z ⊂ Y of h is proper over S and X → Z is surjective.","statement_latex":"Suppose given a commutative diagram of schemes\n$$\n\\xymatrix{\nX \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\\n& S\n}\n$$\nAssume\n\\begin{enumerate}\n\\item $X \\to S$ is a universally closed (for example proper) morphism, and\n\\item $Y \\to S$ is separated and locally of finite type.\n\\end{enumerate}\nThen the scheme theoretic image $Z \\subset Y$ of $h$\nis proper over $S$ and $X \\to Z$ is surjective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AH6","source_file":"morphisms.tex","source_line":10077,"source_end_line":10093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10077-L10093","statement_sha256":"77175c5457785e6c1fa0c88d0fbc48867b8048351f454074e3fba2de1f976d98","origin":"The Stacks Project","memory_eligible":false,"source_rank":5675,"rank":5675,"depth":19,"x":1943.325,"y":686.201,"cluster":"scheme-morphisms"},{"id":"stacks:09MQ","tag":"09MQ","title":"Proper morphisms · Lemma 09MQ","summary":"Let S be a scheme. Let f : X → Y be a surjective universally closed morphism of schemes over S. • If X is quasi-separated, then Y is quasi-separated. • If X is separated, then Y is separated. • If X is quasi-separated over S, then Y is quasi-separated over S. • If X is separated over S, then Y is separated over S.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a surjective universally closed\nmorphism of schemes over $S$.\n\\begin{enumerate}\n\\item If $X$ is quasi-separated, then $Y$ is quasi-separated.\n\\item If $X$ is separated, then $Y$ is separated.\n\\item If $X$ is quasi-separated over $S$, then $Y$ is quasi-separated over $S$.\n\\item If $X$ is separated over $S$, then $Y$ is separated over $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MQ","source_file":"morphisms.tex","source_line":10113,"source_end_line":10123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10113-L10123","statement_sha256":"e0a30682403fea542a2fb29e5f12eb163c99082ce7610b970716791cfebddd9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5676,"rank":5676,"depth":3,"x":2089.012,"y":626.168,"cluster":"scheme-morphisms"},{"id":"stacks:0BX5","tag":"0BX5","title":"Valuative criterion for properness · Lemma 0BX5","summary":"[EGA] Let S be a scheme. Let f : X → Y be a morphism of schemes over S. Assume f is of finite type and quasi-separated. Then the following are equivalent • f is proper, • f satisfies the valuative criterion (Schemes, Definition [Tag 01KD]), • given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & Y where A is a valuation ring with field of fractions K, there exists a unique dotted arrow making the diagram commute.","statement_latex":"\\begin{reference}\n\\cite[II Theorem 7.3.8]{EGA}\n\\end{reference}\nLet $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes\nover $S$. Assume $f$ is of finite type and quasi-separated.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item $f$ satisfies the valuative criterion\n(Schemes, Definition \\ref{schemes-definition-valuative-criterion}),\n\\item given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$, there exists\na unique dotted arrow making the diagram commute.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BX5","source_file":"morphisms.tex","source_line":10181,"source_end_line":10203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10181-L10203","statement_sha256":"b29d50fe266047f02e88c8619200bac646d84e6ccb709a5bd58571dd142fcb38","origin":"The Stacks Project","memory_eligible":false,"source_rank":5677,"rank":5677,"depth":16,"x":2029.775,"y":753.285,"cluster":"scheme-morphisms"},{"id":"stacks:0894","tag":"0894","title":"Valuative criteria · Lemma 0894","summary":"Let f : X → S and h : U → X be morphisms of schemes. Assume that f and h are quasi-compact and that h(U) is dense in X. If given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & U ar[r]^h & X ar[d]^f Spec(A) ar[rr] ar@-->[rru] & & S where A is a valuation ring with field of fractions K, there exists a unique dotted arrow making the diagram commute, then f is universally closed. If moreover f is quasi-separated, then f is separated.","statement_latex":"Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes.\nAssume that $f$ and $h$ are quasi-compact and that $h(U)$ is dense in $X$.\nIf given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$, there\nexists a unique dotted arrow making the diagram commute, then $f$\nis universally closed. If moreover $f$ is quasi-separated, then\n$f$ is separated.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0894","source_file":"morphisms.tex","source_line":10219,"source_end_line":10234,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10219-L10234","statement_sha256":"d52af22d3adc873156fb68e981312086dab723bdbff5eba5866c730786ddc17c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5678,"rank":5678,"depth":20,"x":1971.147,"y":625.755,"cluster":"scheme-morphisms"},{"id":"stacks:0BX6","tag":"0BX6","title":"Valuative criteria · Lemma 0BX6","summary":"Let S be a scheme. Let X, Y be schemes over S. Let s ∈ S and x ∈ X, y ∈ Y points over s. • Let f, g : X → Y be morphisms over S such that f(x) = g(x) = y and f^sharp_x = g^sharp_x : O_Y, y → O_X, x. Then there is an open neighbourhood U ⊂ X with f|_U = g|_U in the following cases • Y is locally of finite type over S, • X is integral, • X is locally Noetherian, or • X is reduced with finitely many irreducible components. • Let φ : O_Y, y → O_X, x be a local O_S, s-algebra…","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$.\nLet $s \\in S$ and $x \\in X$, $y \\in Y$ points over $s$.\n\\begin{enumerate}\n\\item Let $f, g : X \\to Y$ be morphisms over $S$ such that\n$f(x) = g(x) = y$ and\n$f^\\sharp_x = g^\\sharp_x : \\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$.\nThen there is an open neighbourhood $U \\subset X$ with\n$f|_U = g|_U$ in the following cases\n\\begin{enumerate}\n\\item $Y$ is locally of finite type over $S$,\n\\item $X$ is integral,\n\\item $X$ is locally Noetherian, or\n\\item $X$ is reduced with finitely many irreducible components.\n\\end{enumerate}\n\\item Let $\\varphi : \\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$\nbe a local $\\mathcal{O}_{S, s}$-algebra map. Then there exists\nan open neighbourhood $U \\subset X$ of $x$ and a morphism $f : U \\to Y$\nmapping $x$ to $y$ with $f^\\sharp_x = \\varphi$ in the following cases\n\\begin{enumerate}\n\\item $Y$ is locally of finite presentation over $S$,\n\\item $Y$ is locally of finite type and $X$ is integral,\n\\item $Y$ is locally of finite type and $X$ is locally Noetherian, or\n\\item $Y$ is locally of finite type and $X$ is reduced with finitely\nmany irreducible components.\n\\end{enumerate}\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BX6","source_file":"morphisms.tex","source_line":10338,"source_end_line":10366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10338-L10366","statement_sha256":"c1b6e366a742725745263f6b79b6612460e64ee997798273b8c59d556e13b4a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5679,"rank":5679,"depth":6,"x":2117.145,"y":686.614,"cluster":"scheme-morphisms"},{"id":"stacks:0BX7","tag":"0BX7","title":"Valuative criteria · Lemma 0BX7","summary":"Let S be a scheme. Let X, Y be schemes over S. Let x ∈ X. Let U ⊂ X be an open and let f : U → Y be a morphism over S. Assume • x is in the closure of U, • X is reduced with finitely many irreducible components or X is Noetherian, • O_X, x is a valuation ring, • Y → S is proper Then there exists an open U ⊂ U' ⊂ X containing x and an S-morphism f' : U' → Y extending f.","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $x \\in X$.\nLet $U \\subset X$ be an open and let $f : U \\to Y$ be a morphism over $S$.\nAssume\n\\begin{enumerate}\n\\item $x$ is in the closure of $U$,\n\\item $X$ is reduced with finitely many irreducible components or\n$X$ is Noetherian,\n\\item $\\mathcal{O}_{X, x}$ is a valuation ring,\n\\item $Y \\to S$ is proper\n\\end{enumerate}\nThen there exists an open $U \\subset U' \\subset X$ containing\n$x$ and an $S$-morphism $f' : U' \\to Y$ extending $f$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BX7","source_file":"morphisms.tex","source_line":10424,"source_end_line":10438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10424-L10438","statement_sha256":"178942ab89edb3fa125e9d28073e525bfabe5b2b42489de3b8ddadf281b86adf","origin":"The Stacks Project","memory_eligible":false,"source_rank":5680,"rank":5680,"depth":17,"x":1960.322,"y":724.635,"cluster":"scheme-morphisms"},{"id":"stacks:01W8","tag":"01W8","title":"Projective morphisms · Definition 01W8","summary":"Let f : X → S be a morphism of schemes. • We say f is projective if X is isomorphic as an S-scheme to a closed subscheme of a projective bundle P(E) for some quasi-coherent, finite type O_S-module E. • We say f is H-projective if there exists an integer n and a closed immersion X → P^n_S over S. • We say f is locally projective if there exists an open covering S = ⋃ U_i such that each f^-1(U_i) → U_i is projective.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say $f$ is {\\it projective} if $X$ is isomorphic as\nan $S$-scheme to a closed subscheme of a projective\nbundle $\\mathbf{P}(\\mathcal{E})$\nfor some quasi-coherent, finite type $\\mathcal{O}_S$-module $\\mathcal{E}$.\n\\item We say $f$ is {\\it H-projective} if there exists an integer $n$ and\na closed immersion $X \\to \\mathbf{P}^n_S$ over $S$.\n\\item We say $f$ is {\\it locally projective} if there exists an open\ncovering $S = \\bigcup U_i$ such that each $f^{-1}(U_i) \\to U_i$ is\nprojective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01W8","source_file":"morphisms.tex","source_line":10487,"source_end_line":10501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10487-L10501","statement_sha256":"c8b8465c6e9f6a401e7cfe56e5dcb608a8a284057a5ce716cdfb1afaa53db4ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":5681,"rank":5681,"depth":0,"x":2045.508,"y":607.446,"cluster":"scheme-morphisms"},{"id":"stacks:01W9","tag":"01W9","title":"Projective morphisms · Lemma 01W9","summary":"An H-projective morphism is H-quasi-projective. An H-projective morphism is projective.","statement_latex":"An H-projective morphism is H-quasi-projective.\nAn H-projective morphism is projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01W9","source_file":"morphisms.tex","source_line":10531,"source_end_line":10535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10531-L10535","statement_sha256":"895a12c4deca725d9afc3bb036694e86cebb0e5b15a2402ff032dcb388f0f034","origin":"The Stacks Project","memory_eligible":false,"source_rank":5682,"rank":5682,"depth":15,"x":2076.977,"y":742.388,"cluster":"scheme-morphisms"},{"id":"stacks:01WB","tag":"01WB","title":"Projective morphisms · Lemma 01WB","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is locally projective. • There exists an open covering S = ⋃ U_i such that each f^-1(U_i) → U_i is H-projective.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is locally projective.\n\\item There exists an open covering $S = \\bigcup U_i$ such\nthat each $f^{-1}(U_i) \\to U_i$ is H-projective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WB","source_file":"morphisms.tex","source_line":10543,"source_end_line":10552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10543-L10552","statement_sha256":"7700077063ddb5c490e79aa5257f79531431d7a5154cd53e2bb60ef4775da73b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5683,"rank":5683,"depth":16,"x":1945.068,"y":660.625,"cluster":"scheme-morphisms"},{"id":"stacks:01WC","tag":"01WC","title":"Projective morphisms · Lemma 01WC","summary":"A locally projective morphism is proper.","statement_latex":"A locally projective morphism is proper.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WC","source_file":"morphisms.tex","source_line":10579,"source_end_line":10582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10579-L10582","statement_sha256":"535dd114bd992d3239323d2f4340555205f5df81cb88b3e62f45de6c7d0a1597","origin":"The Stacks Project","memory_eligible":false,"source_rank":5684,"rank":5684,"depth":17,"x":2108.319,"y":646.047,"cluster":"scheme-morphisms"},{"id":"stacks:0B5N","tag":"0B5N","title":"Projective morphisms · Lemma 0B5N","summary":"Let f : X → S be a proper morphism of schemes. If there exists an f-ample invertible sheaf on X, then f is locally projective.","statement_latex":"Let $f : X \\to S$ be a proper morphism of schemes. If there exists\nan $f$-ample invertible sheaf on $X$, then $f$ is locally projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5N","source_file":"morphisms.tex","source_line":10616,"source_end_line":10620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10616-L10620","statement_sha256":"02b1abeeaf2e5085bb103dff89d4c2b167d1a53d96a6de00c9ba2bb42f014994","origin":"The Stacks Project","memory_eligible":false,"source_rank":5685,"rank":5685,"depth":23,"x":1999.51,"y":749.575,"cluster":"scheme-morphisms"},{"id":"stacks:01WE","tag":"01WE","title":"Projective morphisms · Lemma 01WE","summary":"A composition of H-projective morphisms is H-projective.","statement_latex":"A composition of H-projective morphisms is H-projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WE","source_file":"morphisms.tex","source_line":10630,"source_end_line":10633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10630-L10633","statement_sha256":"9100d477c950a9389ffaed996aaa338471d166fd32a188e5db3bc0eeee7d6772","origin":"The Stacks Project","memory_eligible":false,"source_rank":5686,"rank":5686,"depth":19,"x":1996.486,"y":611.299,"cluster":"scheme-morphisms"},{"id":"stacks:01WF","tag":"01WF","title":"Projective morphisms · Lemma 01WF","summary":"A base change of a H-projective morphism is H-projective.","statement_latex":"A base change of a H-projective morphism is H-projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WF","source_file":"morphisms.tex","source_line":10657,"source_end_line":10660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10657-L10660","statement_sha256":"1e25347f901894de8f55124d6609bccd7ac26703dc79d58a17d037e1e512743d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5687,"rank":5687,"depth":14,"x":2110.072,"y":711.687,"cluster":"scheme-morphisms"},{"id":"stacks:02V6","tag":"02V6","title":"Projective morphisms · Lemma 02V6","summary":"A base change of a (locally) projective morphism is (locally) projective.","statement_latex":"A base change of a (locally) projective morphism is (locally) projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V6","source_file":"morphisms.tex","source_line":10669,"source_end_line":10672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10669-L10672","statement_sha256":"858ca19a99c47cc85ed4f8271e79dc37941622831e045431a3d03664681cef8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5688,"rank":5688,"depth":14,"x":1945.356,"y":702.101,"cluster":"scheme-morphisms"},{"id":"stacks:07RL","tag":"07RL","title":"Projective morphisms · Lemma 07RL","summary":"A projective morphism is quasi-projective.","statement_latex":"A projective morphism is quasi-projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RL","source_file":"morphisms.tex","source_line":10685,"source_end_line":10688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10685-L10688","statement_sha256":"d322d9c5fea40e3b4371f9ea248ee18dd8136c3eaa17951ac9ae841a4442d2ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":5689,"rank":5689,"depth":22,"x":2074.705,"y":615.584,"cluster":"scheme-morphisms"},{"id":"stacks:01WA","tag":"01WA","title":"Projective morphisms · Lemma 01WA","summary":"Let f : X → S be a H-quasi-projective morphism. Then f factors as X → X' → S where X → X' is an open immersion and X' → S is H-projective.","statement_latex":"Let $f : X \\to S$ be a H-quasi-projective morphism.\nThen $f$ factors as $X \\to X' \\to S$ where $X \\to X'$ is an\nopen immersion and $X' \\to S$ is H-projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WA","source_file":"morphisms.tex","source_line":10702,"source_end_line":10707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10702-L10707","statement_sha256":"fb1e6b6541712a99620d1ad8d08eead2a81ba2ad717fa26eeffdec6b68e53109","origin":"The Stacks Project","memory_eligible":false,"source_rank":5690,"rank":5690,"depth":20,"x":2048.863,"y":752.967,"cluster":"scheme-morphisms"},{"id":"stacks:07RM","tag":"07RM","title":"Projective morphisms · Lemma 07RM","summary":"Let f : X → S be a quasi-projective morphism with S quasi-compact and quasi-separated. Then f factors as X → X' → S where X → X' is an open immersion and X' → S is projective.","statement_latex":"Let $f : X \\to S$ be a quasi-projective morphism with $S$ quasi-compact\nand quasi-separated. Then $f$ factors as $X \\to X' \\to S$ where $X \\to X'$\nis an open immersion and $X' \\to S$ is projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RM","source_file":"morphisms.tex","source_line":10717,"source_end_line":10722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10717-L10722","statement_sha256":"b477cfec49650a301170a4931e17c7205ec3132fd45ecd48fcf3c8be13243c52","origin":"The Stacks Project","memory_eligible":false,"source_rank":5691,"rank":5691,"depth":24,"x":1957.312,"y":636.839,"cluster":"scheme-morphisms"},{"id":"stacks:0BCL","tag":"0BCL","title":"Projective morphisms · Lemma 0BCL","summary":"Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a morphism of schemes. Then • f is projective if and only if f is quasi-projective and proper, and • f is H-projective if and only if f is H-quasi-projective and proper.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to S$ be a morphism of schemes. Then\n\\begin{enumerate}\n\\item $f$ is projective if and only if $f$ is quasi-projective and proper, and\n\\item $f$ is H-projective if and only if $f$ is H-quasi-projective and proper.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCL","source_file":"morphisms.tex","source_line":10776,"source_end_line":10784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10776-L10784","statement_sha256":"e757a47f37123080084d8d923d990245aef12400f29bb4a48f7a465346737637","origin":"The Stacks Project","memory_eligible":false,"source_rank":5692,"rank":5692,"depth":25,"x":2118.43,"y":670.569,"cluster":"scheme-morphisms"},{"id":"stacks:0C4P","tag":"0C4P","title":"Projective morphisms · Lemma 0C4P","summary":"Let f : X → Y and g : Y → S be morphisms of schemes. If S is quasi-compact and quasi-separated and f and g are projective, then g ∘ f is projective.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to S$ be morphisms of schemes.\nIf $S$ is quasi-compact and quasi-separated and $f$ and $g$ are projective,\nthen $g \\circ f$ is projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4P","source_file":"morphisms.tex","source_line":10805,"source_end_line":10810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10805-L10810","statement_sha256":"50beeaf02532a95f6ce8d7d9b63079e2dd3b6a9916663e8918a653c1c55237d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5693,"rank":5693,"depth":26,"x":1972.298,"y":737.211,"cluster":"scheme-morphisms"},{"id":"stacks:0C4Q","tag":"0C4Q","title":"Projective morphisms · Lemma 0C4Q","summary":"Let g : Y → S and f : X → Y be morphisms of schemes. If g ∘ f is projective and g is separated, then f is projective.","statement_latex":"Let $g : Y \\to S$ and $f : X \\to Y$ be morphisms of schemes.\nIf $g \\circ f$ is projective and $g$ is separated,\nthen $f$ is projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4Q","source_file":"morphisms.tex","source_line":10823,"source_end_line":10828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10823-L10828","statement_sha256":"850f1f3eb3baab05144d46fff0f55f46cd6fe61aab54761618c897db7ba95ce6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5694,"rank":5694,"depth":16,"x":2026.537,"y":604.969,"cluster":"scheme-morphisms"},{"id":"stacks:087S","tag":"087S","title":"Projective morphisms · Lemma 087S","summary":"Let S be a scheme which admits an ample invertible sheaf. Then • any projective morphism X → S is H-projective, and • any quasi-projective morphism X → S is H-quasi-projective.","statement_latex":"Let $S$ be a scheme which admits an ample invertible sheaf. Then\n\\begin{enumerate}\n\\item any projective morphism $X \\to S$ is H-projective, and\n\\item any quasi-projective morphism $X \\to S$ is H-quasi-projective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087S","source_file":"morphisms.tex","source_line":10847,"source_end_line":10854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10847-L10854","statement_sha256":"247d1aa8a4bee683f014246c222243292f1e4e5f8c9683864cf760b0793090db","origin":"The Stacks Project","memory_eligible":false,"source_rank":5695,"rank":5695,"depth":25,"x":2092.978,"y":733.435,"cluster":"scheme-morphisms"},{"id":"stacks:0C6J","tag":"0C6J","title":"Projective morphisms · Lemma 0C6J","summary":"Let f : X → S be a universally closed morphism. Let L be an f-ample invertible O_X-module. Then the canonical morphism r : X → underlineProj_S ( bigoplus_d ≥ 0 f_*L^⊗ d ) of Lemma [Tag 01VJ] is an isomorphism.","statement_latex":"Let $f : X \\to S$ be a universally closed morphism.\nLet $\\mathcal{L}$ be an $f$-ample invertible $\\mathcal{O}_X$-module.\nThen the canonical morphism\n$$\nr : X\n\\longrightarrow\n\\underline{\\text{Proj}}_S\n\\left(\n\\bigoplus\\nolimits_{d \\geq 0} f_*\\mathcal{L}^{\\otimes d}\n\\right)\n$$\nof Lemma \\ref{lemma-characterize-relatively-ample} is an isomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6J","source_file":"morphisms.tex","source_line":10884,"source_end_line":10898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10884-L10898","statement_sha256":"0291d894955abc16a32ba84598e858e939501cd9dee3824895d9cc60f6e07b38","origin":"The Stacks Project","memory_eligible":false,"source_rank":5696,"rank":5696,"depth":21,"x":1940.468,"y":676.328,"cluster":"scheme-morphisms"},{"id":"stacks:0EKE","tag":"0EKE","title":"Projective morphisms · Lemma 0EKE","summary":"Let f : X → S be a universally closed morphism. Let L be an f-ample invertible O_X-module. Let s ∈ Γ(X, L). Then X_s → S is an affine morphism.","statement_latex":"Let $f : X \\to S$ be a universally closed morphism. Let $\\mathcal{L}$\nbe an $f$-ample invertible $\\mathcal{O}_X$-module. Let\n$s \\in \\Gamma(X, \\mathcal{L})$. Then $X_s \\to S$ is an affine morphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKE","source_file":"morphisms.tex","source_line":10919,"source_end_line":10924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10919-L10924","statement_sha256":"2875a2e0389eecb1dd9bb32b2d79545e73c1bd4472b1a99c02d6cf5bd354dbfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5697,"rank":5697,"depth":22,"x":2099.067,"y":631.839,"cluster":"scheme-morphisms"},{"id":"stacks:01WH","tag":"01WH","title":"Integral and finite morphisms · Definition 01WH","summary":"Let f : X → S be a morphism of schemes. • We say that f is integral if f is affine and if for every affine open Spec(R) = V ⊂ S with inverse image Spec(A) = f^-1(V) ⊂ X the associated ring map R → A is integral. • We say that f is finite if f is affine and if for every affine open Spec(R) = V ⊂ S with inverse image Spec(A) = f^-1(V) ⊂ X the associated ring map R → A is finite.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say that $f$ is {\\it integral} if $f$ is affine\nand if for every affine open $\\Spec(R) = V \\subset S$\nwith inverse image $\\Spec(A) = f^{-1}(V) \\subset X$\nthe associated ring map $R \\to A$ is integral.\n\\item We say that $f$ is {\\it finite} if $f$ is affine\nand if for every affine open $\\Spec(R) = V \\subset S$\nwith inverse image $\\Spec(A) = f^{-1}(V) \\subset X$\nthe associated ring map $R \\to A$ is finite.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WH","source_file":"morphisms.tex","source_line":10956,"source_end_line":10969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10956-L10969","statement_sha256":"c8fa95c3aec235965ecdc7c4a236dfeedc5a54a74742bcb3325c6659e6a65929","origin":"The Stacks Project","memory_eligible":false,"source_rank":5698,"rank":5698,"depth":0,"x":2017.784,"y":754.805,"cluster":"scheme-morphisms"},{"id":"stacks:02K8","tag":"02K8","title":"Integral and finite morphisms · Lemma 02K8","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is integral. • There exists an affine open covering S = ⋃ U_i such that each f^-1(U_i) is affine and O_S(U_i) → O_X(f^-1(U_i)) is integral. • There exists an open covering S = ⋃ U_i such that each f^-1(U_i) → U_i is integral. Moreover, if f is integral then for every open subscheme U ⊂ S the morphism f : f^-1(U) → U is integral.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is integral.\n\\item There exists an affine open covering $S = \\bigcup U_i$ such that\neach $f^{-1}(U_i)$ is affine and\n$\\mathcal{O}_S(U_i) \\to \\mathcal{O}_X(f^{-1}(U_i))$ is integral.\n\\item There exists an open covering $S = \\bigcup U_i$\nsuch that each $f^{-1}(U_i) \\to U_i$ is integral.\n\\end{enumerate}\nMoreover, if $f$ is integral then for every open subscheme\n$U \\subset S$ the morphism $f : f^{-1}(U) \\to U$ is integral.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02K8","source_file":"morphisms.tex","source_line":10979,"source_end_line":10993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L10979-L10993","statement_sha256":"d0a5ee285f5401e2e4bcf97b606b564a080b7736ac8330ec3b6e3c41d7693b90","origin":"The Stacks Project","memory_eligible":false,"source_rank":5699,"rank":5699,"depth":1,"x":1978.782,"y":617.824,"cluster":"scheme-morphisms"},{"id":"stacks:01WI","tag":"01WI","title":"Integral and finite morphisms · Lemma 01WI","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is finite. • There exists an affine open covering S = ⋃ U_i such that each f^-1(U_i) is affine and O_S(U_i) → O_X(f^-1(U_i)) is finite. • There exists an open covering S = ⋃ U_i such that each f^-1(U_i) → U_i is finite. Moreover, if f is finite then for every open subscheme U ⊂ S the morphism f : f^-1(U) → U is finite.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is finite.\n\\item There exists an affine open covering $S = \\bigcup U_i$ such that\neach $f^{-1}(U_i)$ is affine and\n$\\mathcal{O}_S(U_i) \\to \\mathcal{O}_X(f^{-1}(U_i))$ is finite.\n\\item There exists an open covering $S = \\bigcup U_i$\nsuch that each $f^{-1}(U_i) \\to U_i$ is finite.\n\\end{enumerate}\nMoreover, if $f$ is finite then for every open subscheme\n$U \\subset S$ the morphism $f : f^{-1}(U) \\to U$ is finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WI","source_file":"morphisms.tex","source_line":11000,"source_end_line":11014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11000-L11014","statement_sha256":"7b3bd81231a56bdaef328cbffe274f838f51b61c130a41c6352f633544246b91","origin":"The Stacks Project","memory_eligible":false,"source_rank":5700,"rank":5700,"depth":1,"x":2117.886,"y":696.808,"cluster":"scheme-morphisms"},{"id":"stacks:01WJ","tag":"01WJ","title":"Integral and finite morphisms · Lemma 01WJ","summary":"A finite morphism is integral. An integral morphism which is locally of finite type is finite.","statement_latex":"A finite morphism is integral.\nAn integral morphism which is locally of finite type is finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WJ","source_file":"morphisms.tex","source_line":11021,"source_end_line":11025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11021-L11025","statement_sha256":"78fb550ed0932ffd3ba2b9eb115d6aefb2393e7d3b97113b34f655d73553b7f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5701,"rank":5701,"depth":6,"x":1951.572,"y":717.525,"cluster":"scheme-morphisms"},{"id":"stacks:01WK","tag":"01WK","title":"Integral and finite morphisms · Lemma 01WK","summary":"A composition of finite morphisms is finite. Same is true for integral morphisms.","statement_latex":"A composition of finite morphisms is finite.\nSame is true for integral morphisms.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WK","source_file":"morphisms.tex","source_line":11032,"source_end_line":11036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11032-L11036","statement_sha256":"d25355ec1f5ff182f41b7e0bd6f5d458ac029c9dabcbd5e9303c28424d0532e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5702,"rank":5702,"depth":6,"x":2057.692,"y":607.73,"cluster":"scheme-morphisms"},{"id":"stacks:01WL","tag":"01WL","title":"Integral and finite morphisms · Lemma 01WL","summary":"A base change of a finite morphism is finite. Same is true for integral morphisms.","statement_latex":"A base change of a finite morphism is finite.\nSame is true for integral morphisms.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WL","source_file":"morphisms.tex","source_line":11043,"source_end_line":11047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11043-L11047","statement_sha256":"ff3d288f9aab8609e0e0da8cb40894b7b4afd62ec0403f4885059dffdc6b497c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5703,"rank":5703,"depth":1,"x":2067.748,"y":749.096,"cluster":"scheme-morphisms"},{"id":"stacks:01WM","tag":"01WM","title":"Integral and finite morphisms · Lemma 01WM","summary":"integral = affine + universally closed Let f : X → S be a morphism of schemes. The following are equivalent • f is integral, and • f is affine and universally closed.","statement_latex":"\\begin{slogan}\nintegral $=$ affine $+$ universally closed\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is integral, and\n\\item $f$ is affine and universally closed.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WM","source_file":"morphisms.tex","source_line":11053,"source_end_line":11064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11053-L11064","statement_sha256":"73f768e1dc1d13ba0e1b63c04332160f25be1f81bcca261930d32106a6cf5465","origin":"The Stacks Project","memory_eligible":false,"source_rank":5704,"rank":5704,"depth":19,"x":1946.488,"y":650.429,"cluster":"scheme-morphisms"},{"id":"stacks:02NT","tag":"02NT","title":"Integral and finite morphisms · Lemma 02NT","summary":"Let f : X → S be an integral morphism. Then every point of X is closed in its fibre.","statement_latex":"Let $f : X \\to S$ be an integral morphism.\nThen every point of $X$ is closed in its fibre.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NT","source_file":"morphisms.tex","source_line":11120,"source_end_line":11124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11120-L11124","statement_sha256":"eaecbb7ebc5d2227d0ad1d5ee707bdacaf563e1c3bbbc90b490f564185bc86f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5705,"rank":5705,"depth":8,"x":2115.474,"y":654.384,"cluster":"scheme-morphisms"},{"id":"stacks:0ECG","tag":"0ECG","title":"Integral and finite morphisms · Lemma 0ECG","summary":"Let f : X → Y be an integral morphism. Then dim(X) ≤ dim(Y). If f is surjective then dim(X) = dim(Y).","statement_latex":"Let $f : X \\to Y$ be an integral morphism. Then $\\dim(X) \\leq \\dim(Y)$.\nIf $f$ is surjective then $\\dim(X) = \\dim(Y)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECG","source_file":"morphisms.tex","source_line":11130,"source_end_line":11134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11130-L11134","statement_sha256":"5644166eebfe80bee8e66d244e52ce6bff69d106e98e4c386db33adbd5bf26d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5706,"rank":5706,"depth":9,"x":1987.516,"y":747.478,"cluster":"scheme-morphisms"},{"id":"stacks:02NU","tag":"02NU","title":"Integral and finite morphisms · Lemma 02NU","summary":"A finite morphism is quasi-finite.","statement_latex":"A finite morphism is quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NU","source_file":"morphisms.tex","source_line":11146,"source_end_line":11149,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11146-L11149","statement_sha256":"f77d07df0922b280c9b493d7b66215bba15d97381675b4c527b0087f9f452a52","origin":"The Stacks Project","memory_eligible":false,"source_rank":5707,"rank":5707,"depth":28,"x":2007.033,"y":606.038,"cluster":"scheme-morphisms"},{"id":"stacks:01WN","tag":"01WN","title":"Integral and finite morphisms · Lemma 01WN","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • f is finite, and • f is affine and proper.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is finite, and\n\\item $f$ is affine and proper.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WN","source_file":"morphisms.tex","source_line":11161,"source_end_line":11168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11161-L11168","statement_sha256":"666b00369f544492ce934d7e57a691b42a05858e1aa09f0d6ba187bc8414b6c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5708,"rank":5708,"depth":20,"x":2106.515,"y":721.561,"cluster":"scheme-morphisms"},{"id":"stacks:035C","tag":"035C","title":"Integral and finite morphisms · Lemma 035C","summary":"A closed immersion is finite (and a fortiori integral).","statement_latex":"A closed immersion is finite (and a fortiori integral).","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035C","source_file":"morphisms.tex","source_line":11180,"source_end_line":11183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11180-L11183","statement_sha256":"e24b81e4748fb76e21a19ad4b2d5f1561c446f1b1a133f5bc27ee2e265740582","origin":"The Stacks Project","memory_eligible":false,"source_rank":5709,"rank":5709,"depth":13,"x":1940.038,"y":692.787,"cluster":"scheme-morphisms"},{"id":"stacks:0CYI","tag":"0CYI","title":"Integral and finite morphisms · Lemma 0CYI","summary":"Let X_i → Y, i = 1, …, n be finite morphisms of schemes. Then X_1 amalg … amalg X_n → Y is finite too.","statement_latex":"Let $X_i \\to Y$, $i = 1, \\ldots, n$ be finite morphisms of schemes.\nThen $X_1 \\amalg \\ldots \\amalg X_n \\to Y$ is finite too.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYI","source_file":"morphisms.tex","source_line":11191,"source_end_line":11195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11191-L11195","statement_sha256":"5b152187b2b05c4485440ba5ce4082050fba254f7d1a05efb171add934697922","origin":"The Stacks Project","memory_eligible":false,"source_rank":5710,"rank":5710,"depth":0,"x":2086.128,"y":619.445,"cluster":"scheme-morphisms"},{"id":"stacks:035D","tag":"035D","title":"Integral and finite morphisms · Lemma 035D","summary":"Let f : X → Y and g : Y → Z be morphisms. • If g ∘ f is finite and g separated then f is finite. • If g ∘ f is integral and g separated then f is integral.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms.\n\\begin{enumerate}\n\\item If $g \\circ f$ is finite and $g$ separated then $f$ is finite.\n\\item If $g \\circ f$ is integral and $g$ separated then $f$ is integral.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035D","source_file":"morphisms.tex","source_line":11202,"source_end_line":11209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11202-L11209","statement_sha256":"363f89c5e795183d81e7648f24c63846cb361e8acf49220b0e78e83bea225cf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5711,"rank":5711,"depth":17,"x":2037.319,"y":756.601,"cluster":"scheme-morphisms"},{"id":"stacks:03BB","tag":"03BB","title":"Integral and finite morphisms · Lemma 03BB","summary":"Let f : X → Y be a morphism of schemes. If f is finite and a monomorphism, then f is a closed immersion.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nIf $f$ is finite and a monomorphism, then $f$ is a closed immersion.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BB","source_file":"morphisms.tex","source_line":11225,"source_end_line":11229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11225-L11229","statement_sha256":"7c5dfb5aca878cc6e42afb733c4162a85bb8bf76a25025504d31236fd1b032b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5712,"rank":5712,"depth":2,"x":1962.915,"y":627.6,"cluster":"scheme-morphisms"},{"id":"stacks:0B3I","tag":"0B3I","title":"Integral and finite morphisms · Lemma 0B3I","summary":"A finite morphism is projective.","statement_latex":"A finite morphism is projective.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3I","source_file":"morphisms.tex","source_line":11236,"source_end_line":11239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11236-L11239","statement_sha256":"b8e4fdb8687af2825809903093d5943c2a06e70c1b1f3666d7a9e9aea042e308","origin":"The Stacks Project","memory_eligible":false,"source_rank":5713,"rank":5713,"depth":21,"x":2121.726,"y":680.574,"cluster":"scheme-morphisms"},{"id":"stacks:04DD","tag":"04DD","title":"Universal homeomorphisms · Definition 04DD","summary":"A morphism f : X → Y of schemes is called a universal homeomorphism if the base change f' : Y' ×_Y X → Y' is a homeomorphism for every morphism Y' → Y.","statement_latex":"A morphism $f : X \\to Y$ of schemes is called a {\\it universal homeomorphism}\nif the base change $f' : Y' \\times_Y X \\to Y'$ is a homeomorphism for\nevery morphism $Y' \\to Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DD","source_file":"morphisms.tex","source_line":11303,"source_end_line":11308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11303-L11308","statement_sha256":"064e4a6af35e7cb2d553d1604f5af11c95d36325f8009b88036a29b1da99cbfc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5714,"rank":5714,"depth":0,"x":1961.812,"y":731.691,"cluster":"scheme-morphisms"},{"id":"stacks:0CEU","tag":"0CEU","title":"Universal homeomorphisms · Lemma 0CEU","summary":"The base change of a universal homeomorphism of schemes by any morphism of schemes is a universal homeomorphism.","statement_latex":"The base change of a universal homeomorphism of schemes\nby any morphism of schemes is a universal homeomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEU","source_file":"morphisms.tex","source_line":11313,"source_end_line":11317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11313-L11317","statement_sha256":"dbaa9ec3cbda910daa5d1efbdc59306096ded3ef0d70060814e53fbf5bced32c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5715,"rank":5715,"depth":0,"x":2038.723,"y":603.093,"cluster":"scheme-morphisms"},{"id":"stacks:0CEV","tag":"0CEV","title":"Universal homeomorphisms · Lemma 0CEV","summary":"The composition of a pair of universal homeomorphisms of schemes is a universal homeomorphism.","statement_latex":"The composition of a pair of universal homeomorphisms of\nschemes is a universal homeomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEV","source_file":"morphisms.tex","source_line":11323,"source_end_line":11327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11323-L11327","statement_sha256":"0f65a7d5ca040e032ef87fa032032bfc4cb3da05b1e93c3541026625d2afdd96","origin":"The Stacks Project","memory_eligible":false,"source_rank":5716,"rank":5716,"depth":0,"x":2085.486,"y":741.74,"cluster":"scheme-morphisms"},{"id":"stacks:04DE","tag":"04DE","title":"Universal homeomorphisms · Lemma 04DE","summary":"Let f : X → Y be a morphism of schemes. If f is a homeomorphism onto a closed subset of Y then f is affine.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. If $f$ is a homeomorphism\nonto a closed subset of $Y$ then $f$ is affine.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DE","source_file":"morphisms.tex","source_line":11337,"source_end_line":11341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11337-L11341","statement_sha256":"3d9846afc4b1c7f18ff2c5718c844510f200c6b1f9138a0f975aeac5a44b6478","origin":"The Stacks Project","memory_eligible":false,"source_rank":5717,"rank":5717,"depth":18,"x":1939.319,"y":665.94,"cluster":"scheme-morphisms"},{"id":"stacks:04DF","tag":"04DF","title":"Universal homeomorphisms · Lemma 04DF","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent: • f is a universal homeomorphism, and • f is integral, universally injective and surjective.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are\nequivalent:\n\\begin{enumerate}\n\\item $f$ is a universal homeomorphism, and\n\\item $f$ is integral, universally injective and surjective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DF","source_file":"morphisms.tex","source_line":11360,"source_end_line":11368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11360-L11368","statement_sha256":"6e49145e96b24cde4d944b738e9861f33ba2d4a12729d02764efa8d51cf56e00","origin":"The Stacks Project","memory_eligible":false,"source_rank":5718,"rank":5718,"depth":20,"x":2108.274,"y":638.86,"cluster":"scheme-morphisms"},{"id":"stacks:054M","tag":"054M","title":"Universal homeomorphisms · Lemma 054M","summary":"Let X be a scheme. The canonical closed immersion X_red → X (see Schemes, Definition [Tag 01J4]) is a universal homeomorphism.","statement_latex":"Let $X$ be a scheme. The canonical closed immersion $X_{red} \\to X$ (see\nSchemes, Definition \\ref{schemes-definition-reduced-induced-scheme})\nis a universal homeomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054M","source_file":"morphisms.tex","source_line":11386,"source_end_line":11391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11386-L11391","statement_sha256":"4340e9302e5fdd2c8003e28e10b9eac9ebeb1fdbf598ca07ec859c4dff566294","origin":"The Stacks Project","memory_eligible":false,"source_rank":5719,"rank":5719,"depth":5,"x":2005.334,"y":754.847,"cluster":"scheme-morphisms"},{"id":"stacks:0896","tag":"0896","title":"Universal homeomorphisms · Lemma 0896","summary":"Let f : X → S and S' → S be morphisms of schemes. Assume • S' → S is a closed immersion, • S' → S is bijective on points, • X ×_S S' → S' is a closed immersion, and • X → S is of finite type or S' → S is of finite presentation. Then f : X → S is a closed immersion.","statement_latex":"Let $f : X \\to S$ and $S' \\to S$ be morphisms of schemes.\nAssume\n\\begin{enumerate}\n\\item $S' \\to S$ is a closed immersion,\n\\item $S' \\to S$ is bijective on points,\n\\item $X \\times_S S' \\to S'$ is a closed immersion, and\n\\item $X \\to S$ is of finite type or $S' \\to S$ is of finite presentation.\n\\end{enumerate}\nThen $f : X \\to S$ is a closed immersion.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0896","source_file":"morphisms.tex","source_line":11397,"source_end_line":11408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11397-L11408","statement_sha256":"813ca3c36dab73de8483b764f986a8a60442339ee61a8d00193a53291dee0bba","origin":"The Stacks Project","memory_eligible":false,"source_rank":5720,"rank":5720,"depth":19,"x":1987.945,"y":610.724,"cluster":"scheme-morphisms"},{"id":"stacks:0H2M","tag":"0H2M","title":"Universal homeomorphisms · Lemma 0H2M","summary":"Let f : X → Z be the composition of two morphisms g : X → Y and h : Y → Z. If two of the morphisms (f, g, h) are universal homeomorphisms, so is the third morphism.","statement_latex":"Let $f : X \\to Z$ be the composition of two morphisms \n$g : X \\to Y$ and $h : Y \\to Z$.\nIf two of the morphisms $\\{f, g, h\\}$ are universal homeomorphisms,\nso is the third morphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2M","source_file":"morphisms.tex","source_line":11433,"source_end_line":11439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11433-L11439","statement_sha256":"cf0159d9b0d55fce58002f1ce925b5624a0dcb21bfb5c42f43c8c7d15781a19c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5721,"rank":5721,"depth":21,"x":2116.831,"y":707.255,"cluster":"scheme-morphisms"},{"id":"stacks:0CN7","tag":"0CN7","title":"Universal homeomorphisms of affine schemes · Lemma 0CN7","summary":"Let A → B be a ring map such that the induced morphism of schemes f : Spec(B) → Spec(A) is a universal homeomorphism, resp. a universal homeomorphism inducing isomorphisms on residue fields, resp. universally closed, resp. universally closed and universally injective. Then for any A-subalgebra B' ⊂ B the same thing is true for f' : Spec(B') → Spec(A).","statement_latex":"Let $A \\to B$ be a ring map such that the induced morphism of\nschemes $f : \\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism,\nresp.\\ a universal homeomorphism inducing isomorphisms on residue fields,\nresp.\\ universally closed,\nresp.\\ universally closed and universally injective.\nThen for any $A$-subalgebra $B' \\subset B$\nthe same thing is true for $f' : \\Spec(B') \\to \\Spec(A)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CN7","source_file":"morphisms.tex","source_line":11499,"source_end_line":11508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11499-L11508","statement_sha256":"9b43f13461622624f49ceea2b889ef244307bc8b1631c338d13a492677ebc92d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5722,"rank":5722,"depth":21,"x":1943.945,"y":709.21,"cluster":"scheme-morphisms"},{"id":"stacks:0CN8","tag":"0CN8","title":"Universal homeomorphisms of affine schemes · Lemma 0CN8","summary":"Let A be a ring. Let B = colim B_λ be a filtered colimit of A-algebras. If each f_λ : Spec(B_λ) → Spec(A) is a universal homeomorphism, resp. a universal homeomorphism inducing isomorphisms on residue fields, resp. universally closed, resp. universally closed and universally injective, then the same thing is true for f : Spec(B) → Spec(A).","statement_latex":"Let $A$ be a ring. Let $B = \\colim B_\\lambda$ be a filtered colimit\nof $A$-algebras. If each $f_\\lambda : \\Spec(B_\\lambda) \\to \\Spec(A)$\nis a universal homeomorphism,\nresp.\\ a universal homeomorphism inducing isomorphisms on residue fields,\nresp.\\ universally closed,\nresp.\\ universally closed and universally injective,\nthen the same thing is true for $f : \\Spec(B) \\to \\Spec(A)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CN8","source_file":"morphisms.tex","source_line":11538,"source_end_line":11547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11538-L11547","statement_sha256":"7ecaeda6de459c837cd8f81e955dd8836657d3529821b3acdb411b6bf5ac4aba","origin":"The Stacks Project","memory_eligible":false,"source_rank":5723,"rank":5723,"depth":21,"x":2070.017,"y":609.542,"cluster":"scheme-morphisms"},{"id":"stacks:0CN9","tag":"0CN9","title":"Universal homeomorphisms of affine schemes · Lemma 0CN9","summary":"Let A ⊂ B be a ring extension. Let S ⊂ A be a multiplicative subset. Let n ≥ 1 and b_i ∈ B for 1 ≤ i ≤ n. Any x ∈ S^-1B such that x not ∈ S^-1A and b_i x^i ∈ S^-1A for i = 1, …, n is equal to s^-1y with s ∈ S and y ∈ B such that y not ∈ A and b_i y^i ∈ A for i = 1, …, n","statement_latex":"Let $A \\subset B$ be a ring extension. Let $S \\subset A$ be a\nmultiplicative subset. Let $n \\geq 1$ and\n$b_i \\in B$ for $1 \\leq i \\leq n$. Any $x \\in S^{-1}B$ such that\n$$\nx \\not \\in S^{-1}A\\text{ and } b_i x^i \\in S^{-1}A\\text{ for }i = 1, \\ldots, n\n$$\nis equal to $s^{-1}y$ with $s \\in S$ and $y \\in B$ such that\n$$\ny \\not \\in A\\text{ and } b_i y^i \\in A\\text{ for }i = 1, \\ldots, n\n$$","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CN9","source_file":"morphisms.tex","source_line":11578,"source_end_line":11590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11578-L11590","statement_sha256":"ecf43cde4ad2f7b6e4f4df16754f60593dc733387da78da32f96fc02b2cf7594","origin":"The Stacks Project","memory_eligible":false,"source_rank":5724,"rank":5724,"depth":0,"x":2057.187,"y":754.756,"cluster":"scheme-morphisms"},{"id":"stacks:0CNA","tag":"0CNA","title":"Universal homeomorphisms of affine schemes · Lemma 0CNA","summary":"Let A ⊂ B be a ring extension. If there exists b ∈ B, b not ∈ A and an integer n ≥ 2 with b^n ∈ A and b^n + 1 ∈ A, then there exists a b' ∈ B, b' not ∈ A with (b')^2 ∈ A and (b')^3 ∈ A.","statement_latex":"Let $A \\subset B$ be a ring extension. If there exists\n$b \\in B$, $b \\not \\in A$ and an integer $n \\geq 2$ with\n$b^n \\in A$ and $b^{n + 1} \\in A$, then there exists a\n$b' \\in B$, $b' \\not \\in A$\nwith $(b')^2 \\in A$ and $(b')^3 \\in A$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNA","source_file":"morphisms.tex","source_line":11596,"source_end_line":11603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11596-L11603","statement_sha256":"9d99b51728244cf6f12f6965007cb4b1b5907aabd6b865071a05da97e20ec260","origin":"The Stacks Project","memory_eligible":false,"source_rank":5725,"rank":5725,"depth":0,"x":1949.735,"y":640.253,"cluster":"scheme-morphisms"},{"id":"stacks:0CNB","tag":"0CNB","title":"Universal homeomorphisms of affine schemes · Lemma 0CNB","summary":"Let A ⊂ B be a ring extension such that Spec(B) → Spec(A) is a universal homeomorphism inducing isomorphisms on residue fields. If A not = B, then there exists a b ∈ B, b not ∈ A with b^2 ∈ A and b^3 ∈ A.","statement_latex":"Let $A \\subset B$ be a ring extension such that $\\Spec(B) \\to \\Spec(A)$\nis a universal homeomorphism inducing isomorphisms on residue fields.\nIf $A \\not = B$, then there exists a $b \\in B$, $b \\not \\in A$ with\n$b^2 \\in A$ and $b^3 \\in A$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNB","source_file":"morphisms.tex","source_line":11614,"source_end_line":11620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11614-L11620","statement_sha256":"19296eaad029fa6334cafcbadf9970fddedb23e750ae7e1a1143337ef743fc2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5726,"rank":5726,"depth":22,"x":2121.265,"y":663.744,"cluster":"scheme-morphisms"},{"id":"stacks:0CNC","tag":"0CNC","title":"Universal homeomorphisms of affine schemes · Lemma 0CNC","summary":"Let A ⊂ B be a ring extension such that Spec(B) → Spec(A) is a universal homeomorphism. If A not = B, then either there exists a b ∈ B, b not ∈ A with b^2 ∈ A and b^3 ∈ A or there exists a prime number p and a b ∈ B, b not ∈ A with pb ∈ A and b^p ∈ A.","statement_latex":"Let $A \\subset B$ be a ring extension such that $\\Spec(B) \\to \\Spec(A)$\nis a universal homeomorphism.\nIf $A \\not = B$, then either there exists a $b \\in B$, $b \\not \\in A$ with\n$b^2 \\in A$ and $b^3 \\in A$ or there exists a prime number $p$\nand a $b \\in B$, $b \\not \\in A$ with $pb \\in A$ and $b^p \\in A$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNC","source_file":"morphisms.tex","source_line":11656,"source_end_line":11663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11656-L11663","statement_sha256":"f57fc7e1be7f54a9926703da0ab232cbf8cda9c245256f2a7ff1e26b338f2e1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5727,"rank":5727,"depth":23,"x":1975.707,"y":743.853,"cluster":"scheme-morphisms"},{"id":"stacks:0CND","tag":"0CND","title":"Universal homeomorphisms of affine schemes · Proposition 0CND","summary":"Let A ⊂ B be a ring extension. The following are equivalent • Spec(B) → Spec(A) is a universal homeomorphism inducing isomorphisms on residue fields, and • every finite subset E ⊂ B is contained in an extension A[b_1, …, b_n] ⊂ B such that b_i^2, b_i^3 ∈ A[b_1, …, b_i - 1] for i = 1, …, n.","statement_latex":"Let $A \\subset B$ be a ring extension. The following are equivalent\n\\begin{enumerate}\n\\item $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism inducing\nisomorphisms on residue fields, and\n\\item every finite subset $E \\subset B$ is contained in an extension\n$$\nA[b_1, \\ldots, b_n] \\subset B\n$$\nsuch that $b_i^2, b_i^3 \\in A[b_1, \\ldots, b_{i - 1}]$ for $i = 1, \\ldots, n$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CND","source_file":"morphisms.tex","source_line":11712,"source_end_line":11724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11712-L11724","statement_sha256":"e41ba0d3b42764455bfe74cdee0e837c9abc1c982af3409570d29c2f9c8cad07","origin":"The Stacks Project","memory_eligible":false,"source_rank":5728,"rank":5728,"depth":23,"x":2018.672,"y":602.011,"cluster":"scheme-morphisms"},{"id":"stacks:0CNE","tag":"0CNE","title":"Universal homeomorphisms of affine schemes · Proposition 0CNE","summary":"Let A ⊂ B be a ring extension. The following are equivalent • Spec(B) → Spec(A) is a universal homeomorphism, and • every finite subset E ⊂ B is contained in an extension A[b_1, …, b_n] ⊂ B such that for i = 1, …, n we have • b_i^2, b_i^3 ∈ A[b_1, …, b_i - 1], or • there exists a prime number p with pb_i, b_i^p ∈ A[b_1, …, b_i - 1].","statement_latex":"Let $A \\subset B$ be a ring extension. The following are equivalent\n\\begin{enumerate}\n\\item $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism, and\n\\item every finite subset $E \\subset B$ is contained in an extension\n$$\nA[b_1, \\ldots, b_n] \\subset B\n$$\nsuch that for $i = 1, \\ldots, n$ we have\n\\begin{enumerate}\n\\item $b_i^2, b_i^3 \\in A[b_1, \\ldots, b_{i - 1}]$, or\n\\item there exists a prime number $p$ with\n$pb_i, b_i^p \\in A[b_1, \\ldots, b_{i - 1}]$.\n\\end{enumerate}\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNE","source_file":"morphisms.tex","source_line":11784,"source_end_line":11800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11784-L11800","statement_sha256":"d76e1c1fd2457cba316f33befaa693b53fca9256dab0b28ec4ba240c2e80527d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5729,"rank":5729,"depth":24,"x":2101.159,"y":731.144,"cluster":"scheme-morphisms"},{"id":"stacks:0CNF","tag":"0CNF","title":"Universal homeomorphisms of affine schemes · Lemma 0CNF","summary":"Let p be a prime number. Let A → B be a ring map which induces an isomorphism A[1/p] → B[1/p] (for example if p is nilpotent in A). The following are equivalent • Spec(B) → Spec(A) is a universal homeomorphism, and • the kernel of A → B is a locally nilpotent ideal and for every b ∈ B there exists a p-power q with qb and b^q in the image of A → B.","statement_latex":"Let $p$ be a prime number. Let $A \\to B$ be a ring map\nwhich induces an isomorphism $A[1/p] \\to B[1/p]$\n(for example if $p$ is nilpotent in $A$).\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\Spec(B) \\to \\Spec(A)$ is a universal homeomorphism, and\n\\item the kernel of $A \\to B$ is a locally nilpotent ideal and\nfor every $b \\in B$ there exists a $p$-power $q$ with $qb$ and $b^q$\nin the image of $A \\to B$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNF","source_file":"morphisms.tex","source_line":11825,"source_end_line":11837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11825-L11837","statement_sha256":"2f2e000e89f7636076cdb44a9697b85cc66815c5522856189a301b7cbcc23c9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5730,"rank":5730,"depth":24,"x":1936.283,"y":682.668,"cluster":"scheme-morphisms"},{"id":"stacks:0EUI","tag":"0EUI","title":"Universal homeomorphisms of affine schemes · Lemma 0EUI","summary":"Let A be a ring. Let x, y ∈ A. • If x^3 = y^2 in A, then A → B = A[t]/(t^2 - x, t^3 - y) induces bijections on residue fields and a universal homeomorphism on spectra. • If there is a prime number p such that p^px = y^p in A, then A → B = A[t]/(t^p - x, pt - y) induces a universal homeomorphism on spectra.","statement_latex":"Let $A$ be a ring. Let $x, y \\in A$.\n\\begin{enumerate}\n\\item If $x^3 = y^2$ in $A$, then $A \\to B = A[t]/(t^2 - x, t^3 - y)$\ninduces bijections on residue fields and a\nuniversal homeomorphism on spectra.\n\\item If there is a prime number $p$ such that $p^px = y^p$ in $A$, then\n$A \\to B = A[t]/(t^p - x, pt - y)$ induces a universal\nhomeomorphism on spectra.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUI","source_file":"morphisms.tex","source_line":11872,"source_end_line":11883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11872-L11883","statement_sha256":"859210a08e7b8d2fe0df2e2712a0bae8a8bc7c166b6715441734c658ff535dfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5731,"rank":5731,"depth":21,"x":2097.044,"y":624.787,"cluster":"scheme-morphisms"},{"id":"stacks:0EUJ","tag":"0EUJ","title":"Universal homeomorphisms of affine schemes · Lemma 0EUJ","summary":"Let A → B be a ring map. • If A → B induces a universal homeomorphism on spectra, then B = colim B_i is a filtered colimit of finitely presented A-algebras B_i such that A → B_i induces a universal homeomorphism on spectra. • If A → B induces isomorphisms on residue fields and a universal homeomorphism on spectra, then B = colim B_i is a filtered colimit of finitely presented A-algebras B_i such that A → B_i induces isomorphisms on residue fields and a universal…","statement_latex":"Let $A \\to B$ be a ring map.\n\\begin{enumerate}\n\\item If $A \\to B$ induces a universal homeomorphism on spectra,\nthen $B = \\colim B_i$ is a filtered colimit of finitely presented $A$-algebras\n$B_i$ such that $A \\to B_i$ induces a universal homeomorphism on spectra.\n\\item If $A \\to B$ induces isomorphisms on residue fields and\na universal homeomorphism on spectra, then $B = \\colim B_i$ is a\nfiltered colimit of finitely presented $A$-algebras $B_i$ such that\n$A \\to B_i$ induces isomorphisms on residue fields and a\nuniversal homeomorphism on spectra.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universal homeomorphisms of affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUJ","source_file":"morphisms.tex","source_line":11910,"source_end_line":11923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11910-L11923","statement_sha256":"b2a977cb327695cae8702d6a0a140072e8c2649579efa0c7f4b9b3c440468e99","origin":"The Stacks Project","memory_eligible":false,"source_rank":5732,"rank":5732,"depth":25,"x":2024.958,"y":758.853,"cluster":"scheme-morphisms"},{"id":"stacks:0EUL","tag":"0EUL","title":"Absolute weak normalization and seminormalization · Definition 0EUL","summary":"Let A be a ring. • We say A is seminormal if for all x, y ∈ A with x^3 = y^2 there is a unique a ∈ A with x = a^2 and y = a^3. • We say A is absolutely weakly normal if (a) A is seminormal and (b) for any prime number p and x, y ∈ A with p^px = y^p there is a unique a ∈ A with x = a^p and y = pa.","statement_latex":"Let $A$ be a ring.\n\\begin{enumerate}\n\\item We say $A$ is {\\it seminormal} if for all $x, y \\in A$\nwith $x^3 = y^2$ there is a unique $a \\in A$ with\n$x = a^2$ and $y = a^3$.\n\\item We say $A$ is {\\it absolutely weakly normal} if\n(a) $A$ is seminormal and\n(b) for any prime number $p$ and $x, y \\in A$ with $p^px = y^p$\nthere is a unique $a \\in A$ with $x = a^p$ and $y = pa$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUL","source_file":"morphisms.tex","source_line":11996,"source_end_line":12008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L11996-L12008","statement_sha256":"b46e5dedd0c5472465cef56fc831fd0bcb57e5d237abe927c9aa31cd1d233b87","origin":"The Stacks Project","memory_eligible":false,"source_rank":5733,"rank":5733,"depth":0,"x":1970.232,"y":618.918,"cluster":"scheme-morphisms"},{"id":"stacks:0EUM","tag":"0EUM","title":"Absolute weak normalization and seminormalization · Lemma 0EUM","summary":"Being seminormal or being absolutely weakly normal is a local property of rings, see Properties, Definition [Tag 01OP].","statement_latex":"Being seminormal or being absolutely weakly normal\nis a local property of rings, see\nProperties, Definition \\ref{properties-definition-property-local}.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUM","source_file":"morphisms.tex","source_line":12023,"source_end_line":12028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12023-L12028","statement_sha256":"f6d79f2d93148d06ce040837f7a600723bd5d2da7271c722eb5c616dfe5493fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":5734,"rank":5734,"depth":4,"x":2123.307,"y":691.141,"cluster":"scheme-morphisms"},{"id":"stacks:0EUN","tag":"0EUN","title":"Absolute weak normalization and seminormalization · Definition 0EUN","summary":"Let X be a scheme. • We say X is seminormal if every x ∈ X has an affine open neighbourhood Spec(R) = U ⊂ X such that the ring R is seminormal. • We say X is absolutely weakly normal if every x ∈ X has an affine open neighbourhood Spec(R) = U ⊂ X such that the ring R is absolutely weakly normal.","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item We say $X$ is {\\it seminormal} if every $x \\in X$ has\nan affine open neighbourhood $\\Spec(R) = U \\subset X$\nsuch that the ring $R$ is seminormal.\n\\item We say $X$ is {\\it absolutely weakly normal} if every $x \\in X$ has\nan affine open neighbourhood $\\Spec(R) = U \\subset X$\nsuch that the ring $R$ is absolutely weakly normal.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUN","source_file":"morphisms.tex","source_line":12062,"source_end_line":12073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12062-L12073","statement_sha256":"dd28b4255b521bc25a058df0cbcd4e53e074ce1fdfb0bf5a1d59440c7d33e804","origin":"The Stacks Project","memory_eligible":false,"source_rank":5735,"rank":5735,"depth":0,"x":1952.142,"y":724.785,"cluster":"scheme-morphisms"},{"id":"stacks:0EUP","tag":"0EUP","title":"Absolute weak normalization and seminormalization · Lemma 0EUP","summary":"Let X be a scheme. The following are equivalent: • The scheme X is seminormal. • For every affine open U ⊂ X the ring O_X(U) is seminormal. • There exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is seminormal. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is seminormal. Moreover, if X is seminormal then every open subscheme is seminormal. The same statements are true with \"seminormal\" replaced by \"absolutely weakly normal\".","statement_latex":"Let $X$ be a scheme. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is seminormal.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis seminormal.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ is seminormal.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is seminormal.\n\\end{enumerate}\nMoreover, if $X$ is seminormal\nthen every open subscheme is seminormal.\nThe same statements are true with ``seminormal'' replaced by\n``absolutely weakly normal''.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUP","source_file":"morphisms.tex","source_line":12078,"source_end_line":12094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12078-L12094","statement_sha256":"04f98d498c5c275148d1be03bdedc70029451c61fce840261db339d8a24adb88","origin":"The Stacks Project","memory_eligible":false,"source_rank":5736,"rank":5736,"depth":5,"x":2051.423,"y":602.703,"cluster":"scheme-morphisms"},{"id":"stacks:0EUQ","tag":"0EUQ","title":"Absolute weak normalization and seminormalization · Lemma 0EUQ","summary":"A seminormal scheme or ring is reduced. A fortiori the same is true for absolutely weakly normal schemes or rings.","statement_latex":"A seminormal scheme or ring is reduced. A fortiori the same\nis true for absolutely weakly normal schemes or rings.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUQ","source_file":"morphisms.tex","source_line":12101,"source_end_line":12105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12101-L12105","statement_sha256":"5380e59756ebbb22b6b368d533ab07191d1897e22756738372d04eab56c2494e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5737,"rank":5737,"depth":0,"x":2076.42,"y":749.239,"cluster":"scheme-morphisms"},{"id":"stacks:0EUR","tag":"0EUR","title":"Absolute weak normalization and seminormalization · Lemma 0EUR","summary":"Let A be a ring. • The category of ring maps A → B inducing a universal homeomorphism on spectra has a final object A → A^awn. • Given A → B in the category of (1) the resulting map B → A^awn is an isomorphism if and only if B is absolutely weakly normal. • The category of ring maps A → B inducing isomorphisms on residue fields and a universal homeomorphism on spectra has a final object A → A^sn. • Given A → B in the category of (3) the resulting map B → A^sn is an…","statement_latex":"Let $A$ be a ring.\n\\begin{enumerate}\n\\item The category of ring maps $A \\to B$ inducing a\nuniversal homeomorphism on spectra has a final object $A \\to A^{awn}$.\n\\item Given $A \\to B$ in the category of (1) the resulting map\n$B \\to A^{awn}$ is an isomorphism if and only if $B$ is\nabsolutely weakly normal.\n\\item The category of ring maps $A \\to B$ inducing isomorphisms on\nresidue fields and a universal homeomorphism on spectra has a final\nobject $A \\to A^{sn}$.\n\\item Given $A \\to B$ in the category of (3) the resulting map\n$B \\to A^{sn}$ is an isomorphism if and only if $B$ is seminormal.\n\\end{enumerate}\nFor any ring map $\\varphi : A \\to A'$ there are unique maps\n$\\varphi^{awn} : A^{awn} \\to (A')^{awn}$ and\n$\\varphi^{sn} : A^{sn} \\to (A')^{sn}$ compatible with $\\varphi$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUR","source_file":"morphisms.tex","source_line":12113,"source_end_line":12131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12113-L12131","statement_sha256":"894f291b4f3553e72cf503795f774c59468aef731abfaec52ee0e496682de02c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5738,"rank":5738,"depth":26,"x":1939.981,"y":655.254,"cluster":"scheme-morphisms"},{"id":"stacks:0EUS","tag":"0EUS","title":"Absolute weak normalization and seminormalization · Lemma 0EUS","summary":"Let X be a scheme. • The category of universal homeomorphisms Y → X has an initial object X^awn → X. • Given Y → X in the category of (1) the resulting morphism X^awn → Y is an isomorphism if and only if Y is absolutely weakly normal. • The category of universal homeomorphisms Y → X which induce ismomorphisms on residue fields has an initial object X^sn → X. • Given Y → X in the category of (3) the resulting morphism X^sn → Y is an isomorphism if and only if Y is…","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item The category of universal homeomorphisms $Y \\to X$ has\nan initial object $X^{awn} \\to X$.\n\\item Given $Y \\to X$ in the category of (1) the resulting morphism\n$X^{awn} \\to Y$ is an isomorphism if and only if $Y$ is\nabsolutely weakly normal.\n\\item The category of universal homeomorphisms $Y \\to X$ which\ninduce ismomorphisms on residue fields has an initial object\n$X^{sn} \\to X$.\n\\item Given $Y \\to X$ in the category of (3) the resulting morphism\n$X^{sn} \\to Y$ is an isomorphism if and only if $Y$ is seminormal.\n\\end{enumerate}\nFor any morphism $h : X' \\to X$ of schemes there are unique morphisms\n$h^{awn} : (X')^{awn} \\to X^{awn}$ and $h^{sn} : (X')^{sn} \\to X^{sn}$\ncompatible with $h$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUS","source_file":"morphisms.tex","source_line":12213,"source_end_line":12231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12213-L12231","statement_sha256":"c2b73a219bf758159ca07915b06287c9e7a3d7f9d7145adf182c8c983538f9bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5739,"rank":5739,"depth":27,"x":2116.384,"y":647.128,"cluster":"scheme-morphisms"},{"id":"stacks:0EUT","tag":"0EUT","title":"Absolute weak normalization and seminormalization · Definition 0EUT","summary":"Let X be a scheme. • The morphism X^sn → X constructed in Lemma [Tag 0EUS] is the seminormalization of X. • The morphism X^awn → X constructed in Lemma [Tag 0EUS] is the absolute weak normalization of X.","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item The morphism $X^{sn} \\to X$ constructed in\nLemma \\ref{lemma-seminormalization}\nis the {\\it seminormalization} of $X$.\n\\item The morphism $X^{awn} \\to X$ constructed in\nLemma \\ref{lemma-seminormalization}\nis the {\\it absolute weak normalization} of $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUT","source_file":"morphisms.tex","source_line":12268,"source_end_line":12279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12268-L12279","statement_sha256":"ef4824d518c14a9e970038a79729e387b61f709d167a6a9dcee87c8927fd703f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5740,"rank":5740,"depth":28,"x":1992.691,"y":753.345,"cluster":"scheme-morphisms"},{"id":"stacks:0H3G","tag":"0H3G","title":"Absolute weak normalization and seminormalization · Lemma 0H3G","summary":"Let X be a scheme. The following are equivalent • X is seminormal, • X is equal to its own seminormalization, i.e., the morphism X^sn → X is an isomorphism, • if π : Y → X is a universal homeomorphism inducing isomorphisms on residue fields with Y reduced, then π is an isomorphism.","statement_latex":"Let $X$ be a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is seminormal,\n\\item $X$ is equal to its own seminormalization, i.e., the morphism\n$X^{sn} \\to X$ is an isomorphism,\n\\item if $\\pi : Y \\to X$ is a universal homeomorphism inducing isomorphisms\non residue fields with $Y$ reduced, then $\\pi$ is an isomorphism.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3G","source_file":"morphisms.tex","source_line":12300,"source_end_line":12310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12300-L12310","statement_sha256":"8201ed6b4087782da96dadbe8afb08fadbfb14ccbf5c0588a433a25db3981b1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5741,"rank":5741,"depth":28,"x":1998.491,"y":604.655,"cluster":"scheme-morphisms"},{"id":"stacks:0H3H","tag":"0H3H","title":"Absolute weak normalization and seminormalization · Lemma 0H3H","summary":"Let X be a scheme. The following are equivalent • X is absolutely weakly normal, • X is equal to its own absolute weak normalization, i.e., the morphism X^awn → X is an isomorphism, • if π : Y → X is a universal homeomorphism with Y reduced, then π is an isomorphism.","statement_latex":"Let $X$ be a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is absolutely weakly normal,\n\\item $X$ is equal to its own absolute weak normalization, i.e., the morphism\n$X^{awn} \\to X$ is an isomorphism,\n\\item if $\\pi : Y \\to X$ is a universal homeomorphism with $Y$ reduced, then\n$\\pi$ is an isomorphism.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Absolute weak normalization and seminormalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3H","source_file":"morphisms.tex","source_line":12329,"source_end_line":12339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12329-L12339","statement_sha256":"48968024a49681db8233d95de0d536d1d3ea6486af255f7cad75bc99f0e446e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5742,"rank":5742,"depth":29,"x":2113.926,"y":717.725,"cluster":"scheme-morphisms"},{"id":"stacks:02KA","tag":"02KA","title":"Finite locally free morphisms · Definition 02KA","summary":"Let f : X → S be a morphism of schemes. We say f is finite locally free if f is affine and f_*O_X is a finite locally free O_S-module. In this case we say f is has rank or degree d if the sheaf f_*O_X is finite locally free of degree d.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nWe say $f$ is {\\it finite locally free} if $f$ is\naffine and $f_*\\mathcal{O}_X$ is a finite locally\nfree $\\mathcal{O}_S$-module. In this case we say $f$ is\nhas {\\it rank} or {\\it degree} $d$\nif the sheaf $f_*\\mathcal{O}_X$ is finite locally free of degree $d$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Finite locally free morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KA","source_file":"morphisms.tex","source_line":12359,"source_end_line":12367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12359-L12367","statement_sha256":"b632ae94a17b2c483a42a1dd883cf70524bc4c7614e4760ba8be44c45fa40ee9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5743,"rank":5743,"depth":0,"x":1937.666,"y":699.828,"cluster":"scheme-morphisms"},{"id":"stacks:02KB","tag":"02KB","title":"Finite locally free morphisms · Lemma 02KB","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • f is finite locally free, • f is finite, flat, and locally of finite presentation. If S is locally Noetherian these are also equivalent to • [(3)] f is finite and flat.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is finite locally free,\n\\item $f$ is finite, flat, and locally of finite presentation.\n\\end{enumerate}\nIf $S$ is locally Noetherian these are also equivalent to\n\\begin{enumerate}\n\\item[(3)] $f$ is finite and flat.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KB","source_file":"morphisms.tex","source_line":12374,"source_end_line":12386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12374-L12386","statement_sha256":"429d9babf227826ae41f95f9eb9fc5c939c8e458526c8840ac44e711a74cacbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":5744,"rank":5744,"depth":5,"x":2082.204,"y":612.906,"cluster":"scheme-morphisms"},{"id":"stacks:02KC","tag":"02KC","title":"Finite locally free morphisms · Lemma 02KC","summary":"A composition of finite locally free morphisms is finite locally free.","statement_latex":"A composition of finite locally free morphisms is finite locally free.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KC","source_file":"morphisms.tex","source_line":12410,"source_end_line":12413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12410-L12413","statement_sha256":"7a6d6581fb0174ac9d4023dcdefac2288b271e085575f90a107de3854795c9b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5745,"rank":5745,"depth":0,"x":2045.479,"y":759.191,"cluster":"scheme-morphisms"},{"id":"stacks:02KD","tag":"02KD","title":"Finite locally free morphisms · Lemma 02KD","summary":"A base change of a finite locally free morphism is finite locally free.","statement_latex":"A base change of a finite locally free morphism is finite locally free.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KD","source_file":"morphisms.tex","source_line":12419,"source_end_line":12422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12419-L12422","statement_sha256":"33d8c0faa5d1a5c3b4744a21806480107ec83334ae42e85732025cec44571d90","origin":"The Stacks Project","memory_eligible":false,"source_rank":5746,"rank":5746,"depth":0,"x":1954.813,"y":630.33,"cluster":"scheme-morphisms"},{"id":"stacks:04MH","tag":"04MH","title":"Finite locally free morphisms · Lemma 04MH","summary":"Let f : X → S be a finite locally free morphism of schemes. There exists a disjoint union decomposition S = coprod_d ≥ 0 S_d by open and closed subschemes such that setting X_d = f^-1(S_d) the restrictions f|_X_d are finite locally free morphisms X_d → S_d of degree d.","statement_latex":"Let $f : X \\to S$ be a finite locally free morphism of schemes.\nThere exists a disjoint union decomposition\n$S = \\coprod_{d \\geq 0} S_d$ by open and closed subschemes\nsuch that setting $X_d = f^{-1}(S_d)$ the restrictions\n$f|_{X_d}$ are finite locally free morphisms $X_d \\to S_d$\nof degree $d$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MH","source_file":"morphisms.tex","source_line":12428,"source_end_line":12436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12428-L12436","statement_sha256":"d35062c01ddfb41657ab78c14992a235383ef3481f5f8a8d1a51c62bdeb52c52","origin":"The Stacks Project","memory_eligible":false,"source_rank":5747,"rank":5747,"depth":0,"x":2125.499,"y":673.956,"cluster":"scheme-morphisms"},{"id":"stacks:03HW","tag":"03HW","title":"Finite locally free morphisms · Lemma 03HW","summary":"Let f : Y → X be a finite morphism with X affine. There exists a diagram xymatrix Z' ar[rd] & Y' ar[l]^i ar[d] ar[r] & Y ar[d] & X' ar[r] & X where • Y' → Y and X' → X are surjective finite locally free, • Y' = X' ×_X Y, • i : Y' → Z' is a closed immersion, • Z' → X' is finite locally free, and • Z' = ⋃_j = 1, …, m Z'_j is a (set theoretic) finite union of closed subschemes, each of which maps isomorphically to X'.","statement_latex":"Let $f : Y \\to X$ be a finite morphism with $X$ affine.\nThere exists a diagram\n$$\n\\xymatrix{\nZ' \\ar[rd] &\nY' \\ar[l]^i \\ar[d] \\ar[r] &\nY \\ar[d] \\\\\n & X' \\ar[r] & X\n}\n$$\nwhere\n\\begin{enumerate}\n\\item $Y' \\to Y$ and $X' \\to X$ are surjective finite locally free,\n\\item $Y' = X' \\times_X Y$,\n\\item $i : Y' \\to Z'$ is a closed immersion,\n\\item $Z' \\to X'$ is finite locally free, and\n\\item $Z' = \\bigcup_{j = 1, \\ldots, m} Z'_j$ is a (set theoretic)\nfinite union of closed subschemes, each of which maps isomorphically\nto $X'$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HW","source_file":"morphisms.tex","source_line":12443,"source_end_line":12465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12443-L12465","statement_sha256":"28d40f68526be353e69d69861fee87728b290cc68a60e60b4bef8cd75db210cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5748,"rank":5748,"depth":35,"x":1964.363,"y":738.716,"cluster":"scheme-morphisms"},{"id":"stacks:03HX","tag":"03HX","title":"Finite locally free morphisms · Lemma 03HX","summary":"Let f : Y → X be a finite morphism of schemes. Let T ⊂ Y be a closed nowhere dense subset of Y. Then f(T) ⊂ X is a closed nowhere dense subset of X.","statement_latex":"Let $f : Y \\to X$ be a finite morphism of schemes.\nLet $T \\subset Y$ be a closed nowhere dense subset of $Y$.\nThen $f(T) \\subset X$ is a closed nowhere dense subset of $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HX","source_file":"morphisms.tex","source_line":12499,"source_end_line":12504,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12499-L12504","statement_sha256":"2dc86f9326cf4d5caac297da096f008393e134f213d107394c0b43fe11cceaaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":5749,"rank":5749,"depth":36,"x":2031.184,"y":599.364,"cluster":"scheme-morphisms"},{"id":"stacks:01RS","tag":"01RS","title":"Rational maps · Definition 01RS","summary":"Let X, Y be schemes. • Let f : U → Y, g : V → Y be morphisms of schemes defined on dense open subsets U, V of X. We say that f is equivalent to g if f|_W = g|_W for some W ⊂ U ∩ V dense open in X. • A rational map from X to Y is an equivalence class for the equivalence relation defined in (1). • If X, Y are schemes over a base scheme S we say that a rational map from X to Y is an S-rational map from X to Y if there exists a representative f : U → Y of the equivalence…","statement_latex":"Let $X$, $Y$ be schemes.\n\\begin{enumerate}\n\\item Let $f : U \\to Y$, $g : V \\to Y$ be morphisms of schemes defined\non dense open subsets $U$, $V$ of $X$. We say that $f$ is\n{\\it equivalent} to $g$ if $f|_W = g|_W$ for some $W \\subset U \\cap V$\ndense open in $X$.\n\\item A {\\it rational map from $X$ to $Y$}\nis an equivalence class for the equivalence relation defined in (1).\n\\item If $X$, $Y$ are schemes over a base scheme $S$ we say that\na rational map from $X$ to $Y$ is an {\\it $S$-rational map from $X$\nto $Y$} if there exists a representative $f : U \\to Y$ of the equivalence\nclass which is an $S$-morphism.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RS","source_file":"morphisms.tex","source_line":12585,"source_end_line":12600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12585-L12600","statement_sha256":"541890725fb774125ba72310545650df7a4981de46c85b0a725a0cf46c1c7d6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5750,"rank":5750,"depth":0,"x":2094.048,"y":740.203,"cluster":"scheme-morphisms"},{"id":"stacks:0BX8","tag":"0BX8","title":"Rational maps · Lemma 0BX8","summary":"Let S be a scheme. Let X and Y be schemes over S. Assume X has finitely many irreducible components with generic points x_1, …, x_n. Let s_i ∈ S be the image of x_i. Consider the map ( S-rational maps from X to Y ) → ( (y_1, φ_1, …, y_n, φ_n) where y_i ∈ Y lies over s_i and φ_i : O_Y, y_i → O_X, x_i is a local O_S, s_i-algebra map ) which sends f : U → Y to the 2n-tuple with y_i = f(x_i) and φ_i = f^sharp_x_i. Then • If Y → S is locally of finite type, then the map is…","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Assume $X$ has\nfinitely many irreducible components with generic points\n$x_1, \\ldots, x_n$. Let $s_i \\in S$ be the image of $x_i$.\nConsider the map\n$$\n\\left\\{\n\\begin{matrix}\nS\\text{-rational maps} \\\\\n\\text{from }X\\text{ to }Y\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\n(y_1, \\varphi_1, \\ldots, y_n, \\varphi_n)\\text{ where }\ny_i \\in Y\\text{ lies over }s_i\\text{ and}\\\\\n\\varphi_i : \\mathcal{O}_{Y, y_i} \\to \\mathcal{O}_{X, x_i}\n\\text{ is a local }\\mathcal{O}_{S, s_i}\\text{-algebra map}\n\\end{matrix}\n\\right\\}\n$$\nwhich sends $f : U \\to Y$ to the $2n$-tuple with\n$y_i = f(x_i)$ and $\\varphi_i = f^\\sharp_{x_i}$. Then\n\\begin{enumerate}\n\\item If $Y \\to S$ is locally of finite type, then the map is injective.\n\\item If $Y \\to S$ is locally of finite presentation, then the map is bijective.\n\\item If $Y \\to S$ is locally of finite type and $X$ reduced,\nthen the map is bijective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BX8","source_file":"morphisms.tex","source_line":12608,"source_end_line":12639,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12608-L12639","statement_sha256":"f5136ebf5b833f63945e9589ce264f147304d2ce46f330a1a2122ea502a1f4fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5751,"rank":5751,"depth":7,"x":1934.245,"y":671.94,"cluster":"scheme-morphisms"},{"id":"stacks:01RT","tag":"01RT","title":"Rational maps · Definition 01RT","summary":"Let X be a scheme. A rational function on X is a rational map from X to A^1_Z.","statement_latex":"Let $X$ be a scheme. A {\\it rational function on $X$} is a rational map\nfrom $X$ to $\\mathbf{A}^1_{\\mathbf{Z}}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RT","source_file":"morphisms.tex","source_line":12660,"source_end_line":12664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12660-L12664","statement_sha256":"5f1090df75d8ea3639fe02df228e89b7ae962fb005ff2ffdefe7313e053b7e83","origin":"The Stacks Project","memory_eligible":false,"source_rank":5752,"rank":5752,"depth":0,"x":2107.181,"y":631.552,"cluster":"scheme-morphisms"},{"id":"stacks:01RU","tag":"01RU","title":"Rational maps · Definition 01RU","summary":"Let X be a scheme. The ring of rational functions on X is the ring R(X) whose elements are rational functions with addition and multiplication as just described.","statement_latex":"Let $X$ be a scheme. The {\\it ring of rational functions on $X$}\nis the ring $R(X)$ whose elements are rational functions with\naddition and multiplication as just described.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RU","source_file":"morphisms.tex","source_line":12694,"source_end_line":12699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12694-L12699","statement_sha256":"75a3490e1075c6fee8ee6c717d102fbddbac991d38ca47b801de7105a8a0527d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5753,"rank":5753,"depth":0,"x":2012.027,"y":759.612,"cluster":"scheme-morphisms"},{"id":"stacks:01RV","tag":"01RV","title":"Rational maps · Lemma 01RV","summary":"Let X be a scheme with finitely many irreducible components X_1, …, X_n. If eta_i ∈ X_i is the generic point, then R(X) = O_X, eta_1 × … × O_X, eta_n If X is reduced this is equal to ∏ kappa(eta_i). If X is integral then R(X) = O_X, eta = kappa(eta) is a field.","statement_latex":"Let $X$ be a scheme with finitely many irreducible components\n$X_1, \\ldots, X_n$. If $\\eta_i \\in X_i$ is the generic point, then\n$$\nR(X) = \\mathcal{O}_{X, \\eta_1} \\times \\ldots \\times \\mathcal{O}_{X, \\eta_n}\n$$\nIf $X$ is reduced this is equal to $\\prod \\kappa(\\eta_i)$.\nIf $X$ is integral then $R(X) = \\mathcal{O}_{X, \\eta} = \\kappa(\\eta)$\nis a field.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RV","source_file":"morphisms.tex","source_line":12704,"source_end_line":12714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12704-L12714","statement_sha256":"7f9c374eb504e0c4f5afeae40794b5a44bfb6b7478c3b93b92f9a337c8773711","origin":"The Stacks Project","memory_eligible":false,"source_rank":5754,"rank":5754,"depth":2,"x":1979.171,"y":611.016,"cluster":"scheme-morphisms"},{"id":"stacks:01RW","tag":"01RW","title":"Rational maps · Definition 01RW","summary":"Let X be an integral scheme. The function field, or the field of rational functions of X is the field R(X).","statement_latex":"Let $X$ be an integral scheme.\nThe {\\it function field}, or the {\\it field of rational functions}\nof $X$ is the field $R(X)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RW","source_file":"morphisms.tex","source_line":12741,"source_end_line":12746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12741-L12746","statement_sha256":"d553735cc174fb7d7cec7a14333a73fcbc681c5b8cbcbce686ac8ffe525e9d75","origin":"The Stacks Project","memory_eligible":false,"source_rank":5755,"rank":5755,"depth":0,"x":2123.065,"y":702.052,"cluster":"scheme-morphisms"},{"id":"stacks:02NF","tag":"02NF","title":"Rational maps · Lemma 02NF","summary":"Let X be an integral separated scheme. Let Z_1, Z_2 be distinct irreducible closed subsets of X. Let eta_i be the generic point of Z_i. If Z_1 not⊂ Z_2, then O_X, eta_1 not ⊂ O_X, eta_2 as subrings of R(X). In particular, if Z_1 = (x) consists of one closed point x, there exists a function regular in a neighborhood of x which is not in O_X, eta_2.","statement_latex":"Let $X$ be an integral separated scheme.\nLet $Z_1$, $Z_2$ be distinct irreducible closed subsets of $X$.\nLet $\\eta_i$ be the generic point of $Z_i$.\nIf $Z_1 \\not\\subset Z_2$, then\n$\\mathcal{O}_{X, \\eta_1} \\not \\subset \\mathcal{O}_{X, \\eta_2}$\nas subrings of $R(X)$.\nIn particular, if $Z_1 = \\{x\\}$ consists of one closed point $x$,\nthere exists a function regular in a neighborhood of $x$\nwhich is not in $\\mathcal{O}_{X, \\eta_{2}}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NF","source_file":"morphisms.tex","source_line":12756,"source_end_line":12767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12756-L12767","statement_sha256":"f596967fbdac946239fa507899864ee9c2ab6dbdbf971122004282292efe5940","origin":"The Stacks Project","memory_eligible":false,"source_rank":5756,"rank":5756,"depth":15,"x":1943.541,"y":716.589,"cluster":"scheme-morphisms"},{"id":"stacks:0A1X","tag":"0A1X","title":"Rational maps · Definition 0A1X","summary":"Let φ be a rational map between two schemes X and Y. We say φ is defined in a point x ∈ X if there exists a representative (U, f) of φ with x ∈ U. The domain of definition of φ is the set of all points where φ is defined.","statement_latex":"Let $\\varphi$ be a rational map between two schemes $X$ and $Y$. We say\n$\\varphi$ is {\\it defined in a point $x \\in X$} if there exists a\nrepresentative $(U, f)$ of $\\varphi$ with $x \\in U$. The\n{\\it domain of definition} of $\\varphi$ is the set of all points\nwhere $\\varphi$ is defined.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1X","source_file":"morphisms.tex","source_line":12806,"source_end_line":12813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12806-L12813","statement_sha256":"de1de76498a80a88bb6e1d2151dc4a9ca541635f98ae933dd121fd1397328b45","origin":"The Stacks Project","memory_eligible":false,"source_rank":5757,"rank":5757,"depth":0,"x":2064.37,"y":603.872,"cluster":"scheme-morphisms"},{"id":"stacks:0A1Y","tag":"0A1Y","title":"Rational maps · Lemma 0A1Y","summary":"Let X and Y be schemes. Assume X reduced and Y separated. Let φ be a rational map from X to Y with domain of definition U ⊂ X. Then there exists a unique morphism f : U → Y representing φ. If X and Y are schemes over a separated scheme S and if φ is an S-rational map, then f is a morphism over S.","statement_latex":"Let $X$ and $Y$ be schemes. Assume $X$ reduced and $Y$ separated. Let\n$\\varphi$ be a rational map from $X$ to $Y$ with domain of definition\n$U \\subset X$. Then there exists a unique morphism $f : U \\to Y$\nrepresenting $\\varphi$. If $X$ and $Y$ are schemes over a separated\nscheme $S$ and if $\\varphi$ is an $S$-rational map, then $f$ is a\nmorphism over $S$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1Y","source_file":"morphisms.tex","source_line":12819,"source_end_line":12827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12819-L12827","statement_sha256":"55e05a9f708db0f159ef0620c5fafc2108ffc024b61f509237bd58969ae1e162","origin":"The Stacks Project","memory_eligible":false,"source_rank":5758,"rank":5758,"depth":15,"x":2065.918,"y":755.726,"cluster":"scheme-morphisms"},{"id":"stacks:0A1Z","tag":"0A1Z","title":"Rational maps · Definition 0A1Z","summary":"Let X and Y be irreducible schemes. A rational map from X to Y is called dominant if any representative f : U → Y is a dominant morphism of schemes.","statement_latex":"Let $X$ and $Y$ be irreducible schemes. A rational map from $X$ to $Y$\nis called {\\it dominant} if any representative $f : U \\to Y$ is a dominant\nmorphism of schemes.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1Z","source_file":"morphisms.tex","source_line":12850,"source_end_line":12855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12850-L12855","statement_sha256":"f5a7eb8b043679bd92963b9dfbfd8bbe5afba4a94a4457984239982680d1d544","origin":"The Stacks Project","memory_eligible":false,"source_rank":5759,"rank":5759,"depth":0,"x":1942.517,"y":644.5,"cluster":"scheme-morphisms"},{"id":"stacks:0A20","tag":"0A20","title":"Rational maps · Definition 0A20","summary":"Let X and Y be irreducible schemes. • We say X and Y are birational if X and Y are isomorphic in the category of irreducible schemes and dominant rational maps. • Assume X and Y are schemes over a base scheme S. We say X and Y are S-birational if X and Y are isomorphic in the category of irreducible schemes over S and dominant S-rational maps.","statement_latex":"Let $X$ and $Y$ be irreducible schemes.\n\\begin{enumerate}\n\\item We say $X$ and $Y$ are {\\it birational} if $X$ and $Y$ are isomorphic\nin the category of irreducible schemes and dominant rational maps.\n\\item Assume $X$ and $Y$ are schemes over a base scheme $S$.\nWe say $X$ and $Y$ are {\\it $S$-birational} if $X$ and $Y$ are\nisomorphic in the category of irreducible schemes over $S$ and\ndominant $S$-rational maps.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A20","source_file":"morphisms.tex","source_line":12877,"source_end_line":12888,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12877-L12888","statement_sha256":"034afc0478a0e9930ba64c913cb01b245f1d0ce3ddb6cde8fb77b1bc4e890acc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5760,"rank":5760,"depth":0,"x":2123.164,"y":656.509,"cluster":"scheme-morphisms"},{"id":"stacks:0BAA","tag":"0BAA","title":"Rational maps · Lemma 0BAA","summary":"Let X and Y be irreducible schemes. • The schemes X and Y are birational if and only if they have isomorphic nonempty opens. • Assume X and Y are schemes over a base scheme S. Then X and Y are S-birational if and only if there are nonempty opens U ⊂ X and V ⊂ Y which are S-isomorphic.","statement_latex":"Let $X$ and $Y$ be irreducible schemes.\n\\begin{enumerate}\n\\item The schemes $X$ and $Y$ are birational if and only if they have\nisomorphic nonempty opens.\n\\item Assume $X$ and $Y$ are schemes over a base scheme $S$. Then\n$X$ and $Y$ are $S$-birational if and only if there are nonempty\nopens $U \\subset X$ and $V \\subset Y$ which are $S$-isomorphic.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Rational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAA","source_file":"morphisms.tex","source_line":12899,"source_end_line":12909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12899-L12909","statement_sha256":"224297e267864f3c183fb46ce6ea5ff16b022c29636476b59762c0274ece66cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5761,"rank":5761,"depth":17,"x":1980.135,"y":750.266,"cluster":"scheme-morphisms"},{"id":"stacks:01RO","tag":"01RO","title":"Birational morphisms · Definition 01RO","summary":"[EGA1] Let X, Y be schemes. Assume X and Y have finitely many irreducible components. We say a morphism f : X → Y is birational if • f induces a bijection between the set of generic points of irreducible components of X and the set of generic points of the irreducible components of Y, and • for every generic point eta ∈ X of an irreducible component of X the local ring map O_Y, f(eta) → O_X, eta is an isomorphism.","statement_latex":"\\begin{reference}\n\\cite[(2.2.9)]{EGA1}\n\\end{reference}\nLet $X$, $Y$ be schemes. Assume $X$ and $Y$ have finitely many\nirreducible components. We say a morphism $f : X \\to Y$ is\n{\\it birational} if\n\\begin{enumerate}\n\\item $f$ induces a bijection between the set of generic points\nof irreducible components of $X$ and the set of generic points\nof the irreducible components of $Y$, and\n\\item for every generic point $\\eta \\in X$ of an irreducible component\nof $X$ the local ring map\n$\\mathcal{O}_{Y, f(\\eta)} \\to \\mathcal{O}_{X, \\eta}$\nis an isomorphism.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Birational morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RO","source_file":"morphisms.tex","source_line":12981,"source_end_line":12998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L12981-L12998","statement_sha256":"b67cd3c1c10ec81b7673baa53b578b3d0c9365f25b314e34981ab3e411707e6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5762,"rank":5762,"depth":0,"x":2010.241,"y":599.8,"cluster":"scheme-morphisms"},{"id":"stacks:01RP","tag":"01RP","title":"Birational morphisms · Lemma 01RP","summary":"Let f : X → Y be a morphism of schemes having finitely many irreducible components. If f is birational then f is dominant.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes having finitely\nmany irreducible components. If $f$ is birational then\n$f$ is dominant.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01RP","source_file":"morphisms.tex","source_line":13007,"source_end_line":13012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13007-L13012","statement_sha256":"531f2da9d5c465042a0eb5424117a5d9be14d21c2dc16613032dfd0c64dd50d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5763,"rank":5763,"depth":3,"x":2109.156,"y":727.982,"cluster":"scheme-morphisms"},{"id":"stacks:0BAB","tag":"0BAB","title":"Birational morphisms · Lemma 0BAB","summary":"Let f : X → Y be a birational morphism of schemes having finitely many irreducible components. If y ∈ Y is the generic point of an irreducible component, then the base change X ×_Y Spec(O_Y, y) → Spec(O_Y, y) is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a birational morphism of schemes having finitely\nmany irreducible components. If $y \\in Y$ is the generic point of\nan irreducible component, then the base change\n$X \\times_Y \\Spec(\\mathcal{O}_{Y, y}) \\to \\Spec(\\mathcal{O}_{Y, y})$\nis an isomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAB","source_file":"morphisms.tex","source_line":13019,"source_end_line":13026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13019-L13026","statement_sha256":"3ca88ee53ef842d9656cebb547102bdb2b293cf07380f72ac1e22a2c537404f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5764,"rank":5764,"depth":0,"x":1932.935,"y":689.545,"cluster":"scheme-morphisms"},{"id":"stacks:0BAC","tag":"0BAC","title":"Birational morphisms · Lemma 0BAC","summary":"Let f : X → Y be a birational morphism of schemes having finitely many irreducible components over a base scheme S. Assume one of the following conditions is satisfied • f is locally of finite type and Y reduced, • f is locally of finite presentation. Then there exist dense opens U ⊂ X and V ⊂ Y such that f(U) ⊂ V and f|_U : U → V is an isomorphism. In particular if X and Y are irreducible, then X and Y are S-birational.","statement_latex":"Let $f : X \\to Y$ be a birational morphism of schemes having finitely\nmany irreducible components over a base scheme $S$. Assume one of the\nfollowing conditions is satisfied\n\\begin{enumerate}\n\\item $f$ is locally of finite type and $Y$ reduced,\n\\item $f$ is locally of finite presentation.\n\\end{enumerate}\nThen there exist dense opens $U \\subset X$ and $V \\subset Y$\nsuch that $f(U) \\subset V$ and $f|_U : U \\to V$ is an isomorphism.\nIn particular if $X$ and $Y$ are irreducible, then\n$X$ and $Y$ are $S$-birational.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAC","source_file":"morphisms.tex","source_line":13071,"source_end_line":13084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13071-L13084","statement_sha256":"df69247a7bd87c93587fd1ff78f93fd658eea08adb61ea1d5cd3ab224397ff5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5765,"rank":5765,"depth":1,"x":2093.972,"y":617.813,"cluster":"scheme-morphisms"},{"id":"stacks:0BAD","tag":"0BAD","title":"Birational morphisms · Lemma 0BAD","summary":"Let S be a scheme. Let X and Y be irreducible schemes locally of finite presentation over S. Let x ∈ X and y ∈ Y be the generic points. The following are equivalent • X and Y are S-birational, • there exist nonempty opens of X and Y which are S-isomorphic, and • x and y map to the same point s of S and O_X, x and O_Y, y are isomorphic as O_S, s-algebras.","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be irreducible schemes\nlocally of finite presentation over $S$. Let $x \\in X$ and $y \\in Y$\nbe the generic points. The following are equivalent\n\\begin{enumerate}\n\\item $X$ and $Y$ are $S$-birational,\n\\item there exist nonempty opens of $X$ and $Y$\nwhich are $S$-isomorphic, and\n\\item $x$ and $y$ map to the same point $s$ of $S$ and\n$\\mathcal{O}_{X, x}$ and $\\mathcal{O}_{Y, y}$ are isomorphic as\n$\\mathcal{O}_{S, s}$-algebras.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAD","source_file":"morphisms.tex","source_line":13116,"source_end_line":13129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13116-L13129","statement_sha256":"7f3f0c3279174eed5ec9650ed0e736912873a47e8dc9e87bd99f03bcb02ce584","origin":"The Stacks Project","memory_eligible":false,"source_rank":5766,"rank":5766,"depth":18,"x":2032.84,"y":762.248,"cluster":"scheme-morphisms"},{"id":"stacks:0552","tag":"0552","title":"Birational morphisms · Lemma 0552","summary":"Let S be a scheme. Let X and Y be integral schemes locally of finite type over S. Let x ∈ X and y ∈ Y be the generic points. The following are equivalent • X and Y are S-birational, • there exist nonempty opens of X and Y which are S-isomorphic, and • x and y map to the same point s ∈ S and kappa(x) ≅ kappa(y) as kappa(s)-extensions.","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be integral schemes locally\nof finite type over $S$. Let $x \\in X$ and $y \\in Y$ be the generic points.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ and $Y$ are $S$-birational,\n\\item there exist nonempty opens of $X$ and $Y$ which are $S$-isomorphic, and\n\\item $x$ and $y$ map to the same point $s \\in S$ and\n$\\kappa(x) \\cong \\kappa(y)$ as $\\kappa(s)$-extensions.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0552","source_file":"morphisms.tex","source_line":13146,"source_end_line":13157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13146-L13157","statement_sha256":"1c597f24e7c31bd25312ed6ab0abffb32d6f1271156a7651592afdb8da86e87b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5767,"rank":5767,"depth":18,"x":1961.686,"y":620.896,"cluster":"scheme-morphisms"},{"id":"stacks:02NW","tag":"02NW","title":"Generically finite morphisms · Lemma 02NW","summary":"Let X, Y be schemes. Let f : X → Y be locally of finite type. Let eta ∈ Y be a generic point of an irreducible component of Y. The following are equivalent: • the set f^-1((eta)) is finite, • there exist affine opens U_i ⊂ X, i = 1, …, n and V ⊂ Y with f(U_i) ⊂ V, eta ∈ V and f^-1((eta)) ⊂ ⋃ U_i such that each f|_U_i : U_i → V is finite. If f is quasi-separated, then these are also equivalent to • [(3)] there exist affine opens V ⊂ Y, and U ⊂ X with f(U) ⊂ V, eta ∈ V and…","statement_latex":"Let $X$, $Y$ be schemes.\nLet $f : X \\to Y$ be locally of finite type.\nLet $\\eta \\in Y$ be a generic point of an irreducible component\nof $Y$. The following are equivalent:\n\\begin{enumerate}\n\\item the set $f^{-1}(\\{\\eta\\})$ is finite,\n\\item there exist affine opens $U_i \\subset X$, $i = 1, \\ldots, n$\nand $V \\subset Y$ with $f(U_i) \\subset V$,\n$\\eta \\in V$ and $f^{-1}(\\{\\eta\\}) \\subset \\bigcup U_i$\nsuch that each $f|_{U_i} : U_i \\to V$ is finite.\n\\end{enumerate}\nIf $f$ is quasi-separated, then these are also equivalent to\n\\begin{enumerate}\n\\item[(3)] there exist affine opens $V \\subset Y$,\nand $U \\subset X$ with $f(U) \\subset V$,\n$\\eta \\in V$ and $f^{-1}(\\{\\eta\\}) \\subset U$\nsuch that $f|_U : U \\to V$ is finite.\n\\end{enumerate}\nIf $f$ is quasi-compact and quasi-separated,\nthen these are also equivalent to\n\\begin{enumerate}\n\\item[(4)] there exists an affine open $V \\subset Y$, $\\eta \\in V$\nsuch that $f^{-1}(V) \\to V$ is finite.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NW","source_file":"morphisms.tex","source_line":13191,"source_end_line":13217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13191-L13217","statement_sha256":"bceb8db51f6a48c80492ad15764d452db15b87910c3b64984cb94cd5cb351180","origin":"The Stacks Project","memory_eligible":false,"source_rank":5768,"rank":5768,"depth":28,"x":2128.015,"y":684.825,"cluster":"scheme-morphisms"},{"id":"stacks:0BAH","tag":"0BAH","title":"Generically finite morphisms · Lemma 0BAH","summary":"Let X, Y be schemes. Let f : X → Y be locally of finite type. Let X^0, resp. Y^0 denote the set of generic points of irreducible components of X, resp. Y. Let eta ∈ Y^0. The following are equivalent • f^-1((eta)) ⊂ X^0, • f is quasi-finite at all points lying over eta, • f is quasi-finite at all xi ∈ X^0 lying over eta.","statement_latex":"Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be locally of finite type.\nLet $X^0$, resp.\\ $Y^0$ denote the set of generic points of irreducible\ncomponents of $X$, resp.\\ $Y$. Let $\\eta \\in Y^0$. The following are\nequivalent\n\\begin{enumerate}\n\\item $f^{-1}(\\{\\eta\\}) \\subset X^0$,\n\\item $f$ is quasi-finite at all points lying over $\\eta$,\n\\item $f$ is quasi-finite at all $\\xi \\in X^0$ lying over $\\eta$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAH","source_file":"morphisms.tex","source_line":13303,"source_end_line":13314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13303-L13314","statement_sha256":"6f7062dbef496131cb3f77cf61e379b4c362d569a1045d6941b96df228551806","origin":"The Stacks Project","memory_eligible":false,"source_rank":5769,"rank":5769,"depth":27,"x":1953.758,"y":732.117,"cluster":"scheme-morphisms"},{"id":"stacks:0BAI","tag":"0BAI","title":"Generically finite morphisms · Lemma 0BAI","summary":"Let X, Y be schemes. Let f : X → Y be locally of finite type. Let X^0, resp. Y^0 denote the set of generic points of irreducible components of X, resp. Y. Assume • X^0 and Y^0 are finite and f^-1(Y^0) = X^0, • either f is quasi-compact or f is separated. Then there exists a dense open V ⊂ Y such that f^-1(V) → V is finite.","statement_latex":"Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be locally of finite type.\nLet $X^0$, resp.\\ $Y^0$ denote the set of generic points of irreducible\ncomponents of $X$, resp.\\ $Y$. Assume\n\\begin{enumerate}\n\\item $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$,\n\\item either $f$ is quasi-compact or $f$ is separated.\n\\end{enumerate}\nThen there exists a dense open $V \\subset Y$\nsuch that $f^{-1}(V) \\to V$ is finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAI","source_file":"morphisms.tex","source_line":13328,"source_end_line":13339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13328-L13339","statement_sha256":"518f73976d7494f9c0706809708717c6a2673b711226c734e3e16e82137331e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5770,"rank":5770,"depth":29,"x":2044.324,"y":598.216,"cluster":"scheme-morphisms"},{"id":"stacks:0BAJ","tag":"0BAJ","title":"Generically finite morphisms · Lemma 0BAJ","summary":"Let X, Y be schemes. Let f : X → Y be a birational morphism between schemes which have finitely many irreducible components. Assume • either f is quasi-compact or f is separated, and • either f is locally of finite type and Y is reduced or f is locally of finite presentation. Then there exists a dense open V ⊂ Y such that f^-1(V) → V is an isomorphism.","statement_latex":"Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be a birational morphism\nbetween schemes which have finitely many irreducible components.\nAssume\n\\begin{enumerate}\n\\item either $f$ is quasi-compact or $f$ is separated, and\n\\item either $f$ is locally of finite type and $Y$ is reduced or\n$f$ is locally of finite presentation.\n\\end{enumerate}\nThen there exists a dense open $V \\subset Y$\nsuch that $f^{-1}(V) \\to V$ is an isomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAJ","source_file":"morphisms.tex","source_line":13369,"source_end_line":13381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13369-L13381","statement_sha256":"196d6fe0a3f9cad50878647768191d4a7cea26786c67601ea1330bdb3090ed06","origin":"The Stacks Project","memory_eligible":false,"source_rank":5771,"rank":5771,"depth":30,"x":2085.268,"y":748.512,"cluster":"scheme-morphisms"},{"id":"stacks:02NX","tag":"02NX","title":"Generically finite morphisms · Lemma 02NX","summary":"Let X, Y be integral schemes. Let f : X → Y be locally of finite type. Assume f is dominant. The following are equivalent: • the extension R(Y) ⊂ R(X) has transcendence degree 0, • the extension R(Y) ⊂ R(X) is finite, • there exist nonempty affine opens U ⊂ X and V ⊂ Y such that f(U) ⊂ V and f|_U : U → V is finite, and • the generic point of X is the only point of X mapping to the generic point of Y. If f is separated or if f is quasi-compact, then these are also…","statement_latex":"Let $X$, $Y$ be integral schemes.\nLet $f : X \\to Y$ be locally of finite type.\nAssume $f$ is dominant.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the extension $R(Y) \\subset R(X)$ has\ntranscendence degree $0$,\n\\item the extension $R(Y) \\subset R(X)$ is finite,\n\\item there exist nonempty affine opens $U \\subset X$\nand $V \\subset Y$ such that $f(U) \\subset V$\nand $f|_U : U \\to V$ is finite, and\n\\item the generic point of $X$ is the only point of $X$ mapping to\nthe generic point of $Y$.\n\\end{enumerate}\nIf $f$ is separated or if $f$ is quasi-compact, then these are\nalso equivalent to\n\\begin{enumerate}\n\\item[(5)] there exists a nonempty affine open $V \\subset Y$ such\nthat $f^{-1}(V) \\to V$ is finite.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NX","source_file":"morphisms.tex","source_line":13393,"source_end_line":13415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13393-L13415","statement_sha256":"ed8fce086d46e7a36d88971b46209a85b63bdb0a57ea94ca9d82b081734d0501","origin":"The Stacks Project","memory_eligible":false,"source_rank":5772,"rank":5772,"depth":30,"x":1934.043,"y":660.82,"cluster":"scheme-morphisms"},{"id":"stacks:02NY","tag":"02NY","title":"Generically finite morphisms · Definition 02NY","summary":"Let X and Y be integral schemes. Let f : X → Y be locally of finite type and dominant. Assume [R(X) : R(Y)] < ∞, or any other of the equivalent conditions (1) -- (4) of Lemma [Tag 02NX]. Then the positive integer deg(X/Y) = [R(X) : R(Y)] is called the degree of X over Y.","statement_latex":"Let $X$ and $Y$ be integral schemes.\nLet $f : X \\to Y$ be locally of finite type and dominant.\nAssume $[R(X) : R(Y)] < \\infty$, or any other of the equivalent\nconditions (1) -- (4) of Lemma \\ref{lemma-finite-degree}.\nThen the positive integer\n$$\n\\deg(X/Y) = [R(X) : R(Y)]\n$$\nis called the {\\it degree of $X$ over $Y$}.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generically finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NY","source_file":"morphisms.tex","source_line":13461,"source_end_line":13472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13461-L13472","statement_sha256":"2f8594913d3ba842f1dc555360865e6ece91866978f6372d378332ebeb1566cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5773,"rank":5773,"depth":31,"x":2116.278,"y":639.65,"cluster":"scheme-morphisms"},{"id":"stacks:02NZ","tag":"02NZ","title":"Generically finite morphisms · Lemma 02NZ","summary":"Let X, Y, Z be integral schemes. Let f : X → Y and g : Y → Z be dominant morphisms locally of finite type. Assume that [R(X) : R(Y)] < ∞ and [R(Y) : R(Z)] < ∞. Then deg(X/Z) = deg(X/Y) deg(Y/Z).","statement_latex":"Let $X$, $Y$, $Z$ be integral schemes.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be dominant morphisms locally\nof finite type. Assume that $[R(X) : R(Y)] < \\infty$ and\n$[R(Y) : R(Z)] < \\infty$. Then\n$$\n\\deg(X/Z) = \\deg(X/Y) \\deg(Y/Z).\n$$","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NZ","source_file":"morphisms.tex","source_line":13492,"source_end_line":13501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13492-L13501","statement_sha256":"9fd173152699d9751e52587b3dd2fcb76e196cc2011eada97977bf02f9fd8fe0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5774,"rank":5774,"depth":1,"x":1998.793,"y":758.798,"cluster":"scheme-morphisms"},{"id":"stacks:0AAZ","tag":"0AAZ","title":"Generically finite morphisms · Definition 0AAZ","summary":"Let X be an integral scheme. A modification of X is a birational proper morphism f : X' → X with X' integral.","statement_latex":"Let $X$ be an integral scheme. A {\\it modification of $X$}\nis a birational proper morphism $f : X' \\to X$ with $X'$\nintegral.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generically finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAZ","source_file":"morphisms.tex","source_line":13531,"source_end_line":13536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13531-L13536","statement_sha256":"0bb515a2614743be55bb78c955174430d94e157312b61430faf3eb2c6310d19d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5775,"rank":5775,"depth":0,"x":1989.599,"y":604.103,"cluster":"scheme-morphisms"},{"id":"stacks:0AB0","tag":"0AB0","title":"Generically finite morphisms · Definition 0AB0","summary":"[alterations] Let X be an integral scheme. An alteration of X is a proper dominant morphism f : Y → X with Y integral such that f^-1(U) → U is finite for some nonempty open U ⊂ X.","statement_latex":"\\begin{reference}\n\\cite[Definition 2.20]{alterations}\n\\end{reference}\nLet $X$ be an integral scheme. An {\\it alteration of $X$}\nis a proper dominant morphism $f : Y \\to X$ with $Y$ integral such\nthat $f^{-1}(U) \\to U$ is finite for some nonempty open $U \\subset X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Generically finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AB0","source_file":"morphisms.tex","source_line":13552,"source_end_line":13560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13552-L13560","statement_sha256":"fa253df115eda6026ccd2b1e17db6ff25f795c9cda19b79717d3a52e4d72b4fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":5776,"rank":5776,"depth":0,"x":2120.927,"y":713.077,"cluster":"scheme-morphisms"},{"id":"stacks:02JU","tag":"02JU","title":"The dimension formula · Lemma 02JU","summary":"Let S be a scheme. Let f : X → S be a morphism of schemes. Let x ∈ X, and set s = f(x). Assume • S is locally Noetherian, • f is locally of finite type, • X and S integral, and • f dominant. We have dim(O_X, x) ≤ dim(O_S, s) + trdeg_R(S)R(X) - trdeg_kappa(s) kappa(x). Moreover, equality holds if S is universally catenary.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$, and set $s = f(x)$.\nAssume\n\\begin{enumerate}\n\\item $S$ is locally Noetherian,\n\\item $f$ is locally of finite type,\n\\item $X$ and $S$ integral, and\n\\item $f$ dominant.\n\\end{enumerate}\nWe have\n\\begin{equation}\n\n\\dim(\\mathcal{O}_{X, x})\n\\leq\n\\dim(\\mathcal{O}_{S, s}) + \\text{trdeg}_{R(S)}R(X)\n- \\text{trdeg}_{\\kappa(s)} \\kappa(x).\n\\end{equation}\nMoreover, equality holds if $S$ is universally catenary.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"The dimension formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JU","source_file":"morphisms.tex","source_line":13581,"source_end_line":13602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13581-L13602","statement_sha256":"255c4b44a52f23c6792270be35584724229ef7ef2fbdc16206fe9ed898f1e61c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5777,"rank":5777,"depth":13,"x":1936.248,"y":707.233,"cluster":"scheme-morphisms"},{"id":"stacks:0BAE","tag":"0BAE","title":"The dimension formula · Lemma 0BAE","summary":"Let S be a scheme. Let f : X → S be a morphism of schemes. Let x ∈ X, and set s = f(x). Assume S is locally Noetherian and f is locally of finite type, We have dim(O_X, x) ≤ dim(O_S, s) + E - trdeg_kappa(s) kappa(x). where E is the maximum of trdeg_kappa(f(xi))(kappa(xi)) where xi runs over the generic points of irreducible components of X containing x.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$, and set $s = f(x)$. Assume $S$ is locally Noetherian\nand $f$ is locally of finite type,\nWe have\n\\begin{equation}\n\n\\dim(\\mathcal{O}_{X, x})\n\\leq\n\\dim(\\mathcal{O}_{S, s}) + E - \\text{trdeg}_{\\kappa(s)} \\kappa(x).\n\\end{equation}\nwhere $E$ is the maximum of $\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi))$\nwhere $\\xi$ runs over the generic points of irreducible components\nof $X$ containing $x$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"The dimension formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAE","source_file":"morphisms.tex","source_line":13609,"source_end_line":13624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13609-L13624","statement_sha256":"0bc5ad1c99757a771fdc140bb930fb7758214642ba0a140dec52c021e0c270dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5778,"rank":5778,"depth":14,"x":2077.283,"y":606.64,"cluster":"scheme-morphisms"},{"id":"stacks:02JW","tag":"02JW","title":"The dimension formula · Lemma 02JW","summary":"Let S be a locally Noetherian and universally catenary scheme. Let δ : S → Z be a dimension function. Let f : X → S be a morphism of schemes. Assume f locally of finite type. Then the map δ = δ_X/S : X & → Z x & ↦ δ(f(x)) + trdeg_kappa(f(x)) kappa(x) is a dimension function on X.","statement_latex":"Let $S$ be a locally Noetherian and universally catenary scheme.\nLet $\\delta : S \\to \\mathbf{Z}$ be a dimension function.\nLet $f : X \\to S$ be a morphism of schemes.\nAssume $f$ locally of finite type.\nThen the map\n\\begin{align*}\n\\delta = \\delta_{X/S} : X & \\longrightarrow \\mathbf{Z} \\\\\nx & \\longmapsto \\delta(f(x)) + \\text{trdeg}_{\\kappa(f(x))} \\kappa(x)\n\\end{align*}\nis a dimension function on $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"The dimension formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JW","source_file":"morphisms.tex","source_line":13656,"source_end_line":13668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13656-L13668","statement_sha256":"16b090c8e6f8e1a77f61b6abbcba4a0799e52204781475362ffbd0d73a6bcc89","origin":"The Stacks Project","memory_eligible":false,"source_rank":5779,"rank":5779,"depth":14,"x":2054.155,"y":761.014,"cluster":"scheme-morphisms"},{"id":"stacks:02JX","tag":"02JX","title":"The dimension formula · Lemma 02JX","summary":"Let f : X → Y be a morphism of schemes. Assume that • Y is locally Noetherian, • X and Y are integral schemes, • f is dominant, and • f is locally of finite type. Then we have dim(X) ≤ dim(Y) + trdeg_R(Y) R(X). If f is closed then equality holds.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume that\n\\begin{enumerate}\n\\item $Y$ is locally Noetherian,\n\\item $X$ and $Y$ are integral schemes,\n\\item $f$ is dominant, and\n\\item $f$ is locally of finite type.\n\\end{enumerate}\nThen we have\n$$\n\\dim(X) \\leq \\dim(Y) + \\text{trdeg}_{R(Y)} R(X).\n$$\nIf $f$ is closed\\footnote{For example if $f$ is proper, see\nDefinition \\ref{definition-proper}.} then equality holds.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"The dimension formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02JX","source_file":"morphisms.tex","source_line":13744,"source_end_line":13759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13744-L13759","statement_sha256":"c22d377a38eb0125291b12d19b2eb230b0426260a5920d9611856455fe5cb208","origin":"The Stacks Project","memory_eligible":false,"source_rank":5780,"rank":5780,"depth":20,"x":1946.948,"y":633.917,"cluster":"scheme-morphisms"},{"id":"stacks:0BAG","tag":"0BAG","title":"The dimension formula · Lemma 0BAG","summary":"Let f : X → Y be a morphism of schemes. Assume that Y is locally Noetherian and f is locally of finite type. Then dim(X) ≤ dim(Y) + E where E is the supremum of trdeg_kappa(f(xi))(kappa(xi)) where xi runs through the generic points of the irreducible components of X.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume that\n$Y$ is locally Noetherian and $f$ is locally of finite type.\nThen\n$$\n\\dim(X) \\leq \\dim(Y) + E\n$$\nwhere $E$ is the supremum of $\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi))$\nwhere $\\xi$ runs through the generic points of the irreducible components\nof $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"The dimension formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAG","source_file":"morphisms.tex","source_line":13813,"source_end_line":13824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13813-L13824","statement_sha256":"90776f2a12fa5a6c11a54f3f59fc6b3f765982a46da55f209314bcb5ccbdc660","origin":"The Stacks Project","memory_eligible":false,"source_rank":5781,"rank":5781,"depth":15,"x":2128.409,"y":666.841,"cluster":"scheme-morphisms"},{"id":"stacks:035F","tag":"035F","title":"Relative normalization · Lemma 035F","summary":"Let X be a scheme. Let A be a quasi-coherent sheaf of O_X-algebras. The subsheaf A' ⊂ A defined by the rule U ↦ (f ∈ A(U) mid f_x ∈ A_x integral over O_X, x for all x ∈ U) is a quasi-coherent O_X-algebra, the stalk A'_x is the integral closure of O_X, x in A_x, and for any affine open U ⊂ X the ring A'(U) ⊂ A(U) is the integral closure of O_X(U) in A(U).","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf\nof $\\mathcal{O}_X$-algebras. The subsheaf $\\mathcal{A}' \\subset \\mathcal{A}$\ndefined by the rule\n$$\nU \\longmapsto \\{f \\in \\mathcal{A}(U) \\mid\nf_x \\in \\mathcal{A}_x \\text{ integral over } \\mathcal{O}_{X, x}\n\\text{ for all }x \\in U\\}\n$$\nis a quasi-coherent $\\mathcal{O}_X$-algebra, the stalk $\\mathcal{A}'_x$\nis the integral closure of $\\mathcal{O}_{X, x}$ in $\\mathcal{A}_x$, and\nfor any affine open $U \\subset X$ the ring\n$\\mathcal{A}'(U) \\subset \\mathcal{A}(U)$ is\nthe integral closure of $\\mathcal{O}_X(U)$ in $\\mathcal{A}(U)$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035F","source_file":"morphisms.tex","source_line":13844,"source_end_line":13859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13844-L13859","statement_sha256":"140ae48356a4c0c608a2c3ee477d8cc3513a9220e2332af586b09eff82f6a5b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5782,"rank":5782,"depth":6,"x":1967.949,"y":745.615,"cluster":"scheme-morphisms"},{"id":"stacks:035G","tag":"035G","title":"Relative normalization · Definition 035G","summary":"Let X be a scheme. Let A be a quasi-coherent sheaf of O_X-algebras. The integral closure of O_X in A is the quasi-coherent O_X-subalgebra A' ⊂ A constructed in Lemma [Tag 035F] above.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent sheaf\nof $\\mathcal{O}_X$-algebras. The {\\it integral closure of $\\mathcal{O}_X$\nin $\\mathcal{A}$} is the quasi-coherent $\\mathcal{O}_X$-subalgebra\n$\\mathcal{A}' \\subset \\mathcal{A}$ constructed in\nLemma \\ref{lemma-integral-closure} above.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035G","source_file":"morphisms.tex","source_line":13878,"source_end_line":13885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13878-L13885","statement_sha256":"e0850467f20854aac56c15b5dd5aacdab1ba83159fc6ef02b7e2b2c87410a3d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5783,"rank":5783,"depth":7,"x":2022.982,"y":596.315,"cluster":"scheme-morphisms"},{"id":"stacks:035H","tag":"035H","title":"Relative normalization · Definition 035H","summary":"Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Let O' be the integral closure of O_X in f_*O_Y. The normalization of X in Y is the scheme_X, then X' = X. ν : X' = underlineSpec_X(O') → X over X. It comes equipped with a natural factorization Y xrightarrowf' X' xrightarrowν X of the initial morphism f.","statement_latex":"Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $\\mathcal{O}'$ be the integral closure of $\\mathcal{O}_X$ in\n$f_*\\mathcal{O}_Y$. The {\\it normalization of $X$ in $Y$} is the\nscheme\\footnote{The scheme $X'$ need not be normal, for example if\n$Y = X$ and $f = \\text{id}_X$, then $X' = X$.}\n$$\n\\nu : X' = \\underline{\\Spec}_X(\\mathcal{O}') \\to X\n$$\nover $X$. It comes equipped with a natural factorization\n$$\nY \\xrightarrow{f'} X' \\xrightarrow{\\nu} X\n$$\nof the initial morphism $f$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035H","source_file":"morphisms.tex","source_line":13909,"source_end_line":13924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13909-L13924","statement_sha256":"314656150889c533c6b2df9aae984a8558eb2d92bccfd1cc17f53d6dbce92185","origin":"The Stacks Project","memory_eligible":false,"source_rank":5784,"rank":5784,"depth":0,"x":2102.551,"y":737.789,"cluster":"scheme-morphisms"},{"id":"stacks:035I","tag":"035I","title":"Relative normalization · Lemma 035I","summary":"Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. The factorization f = ν ∘ f', where ν : X' → X is the normalization of X in Y is characterized by the following two properties: • the morphism ν is integral, and • for any factorization f = π ∘ g, with π : Z → X integral, there exists a commutative diagram xymatrix Y ar[d]_f' ar[r]_g & Z ar[d]^π X' ar[ru]^h ar[r]^ν & X for some unique morphism h : X' → Z. Moreover, the morphism f' : Y → X' is…","statement_latex":"Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.\nThe factorization $f = \\nu \\circ f'$, where $\\nu : X' \\to X$ is the\nnormalization of $X$ in $Y$ is characterized by the following\ntwo properties:\n\\begin{enumerate}\n\\item the morphism $\\nu$ is integral, and\n\\item for any factorization $f = \\pi \\circ g$, with $\\pi : Z \\to X$\nintegral, there exists a commutative diagram\n$$\n\\xymatrix{\nY \\ar[d]_{f'} \\ar[r]_g & Z \\ar[d]^\\pi \\\\\nX' \\ar[ru]^h \\ar[r]^\\nu & X\n}\n$$\nfor some unique morphism $h : X' \\to Z$.\n\\end{enumerate}\nMoreover, the morphism $f' : Y \\to X'$ is dominant and in (2) the\nmorphism $h : X' \\to Z$ is the normalization of $Z$ in $Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035I","source_file":"morphisms.tex","source_line":13935,"source_end_line":13955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L13935-L13955","statement_sha256":"10c0b50d8aac1392b34358ce4e9b3a901c4c1e5c86f7cdfa0292b41d9b8c0b2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5785,"rank":5785,"depth":19,"x":1929.92,"y":678.553,"cluster":"scheme-morphisms"},{"id":"stacks:035J","tag":"035J","title":"Relative normalization · Lemma 035J","summary":"Let xymatrix Y_2 ar[d]_f_2 ar[r] & Y_1 ar[d]^f_1 X_2 ar[r] & X_1 be a commutative diagram of morphisms of schemes. Assume f_1, f_2 quasi-compact and quasi-separated. Let f_i = ν_i ∘ f_i', i = 1, 2 be the canonical factorizations, where ν_i : X_i' → X_i is the normalization of X_i in Y_i. Then there exists a unique arrow X'_2 → X'_1 fitting into a commutative diagram xymatrix Y_2 ar[d]_f_2' ar[r] & Y_1 ar[d]^f_1' X_2' ar[d]_ν_2 ar[r] & X_1' ar[d]^ν_1 X_2 ar[r] & X_1","statement_latex":"Let\n$$\n\\xymatrix{\nY_2 \\ar[d]_{f_2} \\ar[r] & Y_1 \\ar[d]^{f_1} \\\\\nX_2 \\ar[r] & X_1\n}\n$$\nbe a commutative diagram of morphisms of schemes.\nAssume $f_1$, $f_2$ quasi-compact and quasi-separated.\nLet $f_i = \\nu_i \\circ f_i'$, $i = 1, 2$\nbe the canonical factorizations, where $\\nu_i : X_i' \\to X_i$ is\nthe normalization of $X_i$ in $Y_i$. Then there exists a unique\narrow $X'_2 \\to X'_1$ fitting into a\ncommutative diagram\n$$\n\\xymatrix{\nY_2 \\ar[d]_{f_2'} \\ar[r] & Y_1 \\ar[d]^{f_1'} \\\\\nX_2' \\ar[d]_{\\nu_2} \\ar[r] & X_1' \\ar[d]^{\\nu_1} \\\\\nX_2 \\ar[r] & X_1\n}\n$$","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035J","source_file":"morphisms.tex","source_line":14001,"source_end_line":14024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14001-L14024","statement_sha256":"c04ed06f227733a414d29b013d7dabb7e7d11751aab25423e8526963dee5b005","origin":"The Stacks Project","memory_eligible":false,"source_rank":5786,"rank":5786,"depth":20,"x":2105.042,"y":624.219,"cluster":"scheme-morphisms"},{"id":"stacks:035K","tag":"035K","title":"Relative normalization · Lemma 035K","summary":"Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Let U ⊂ X be an open subscheme and set V = f^-1(U). Then the normalization of U in V is the inverse image of U in the normalization of X in Y.","statement_latex":"Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $U \\subset X$ be an open subscheme and set $V = f^{-1}(U)$.\nThen the normalization of $U$ in $V$ is the inverse image of $U$\nin the normalization of $X$ in $Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035K","source_file":"morphisms.tex","source_line":14038,"source_end_line":14044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14038-L14044","statement_sha256":"c8a0d758e634424ae9a040981d2a2dc78ab3664d61c8fa5b2acc394fdc6d6d20","origin":"The Stacks Project","memory_eligible":false,"source_rank":5787,"rank":5787,"depth":0,"x":2019.511,"y":763.804,"cluster":"scheme-morphisms"},{"id":"stacks:0BXA","tag":"0BXA","title":"Relative normalization · Lemma 0BXA","summary":"Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Let X' be the normalization of X in Y. Then the normalization of X' in Y is X'.","statement_latex":"Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $X'$ be the normalization of $X$ in $Y$. Then the normalization of\n$X'$ in $Y$ is $X'$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXA","source_file":"morphisms.tex","source_line":14050,"source_end_line":14055,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14050-L14055","statement_sha256":"0592d65bff60b104e745c570abe6ee7addd1fe1fe2c5e291b56922e0cb6cdf8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5788,"rank":5788,"depth":20,"x":1970.276,"y":612.179,"cluster":"scheme-morphisms"},{"id":"stacks:0AXN","tag":"0AXN","title":"Relative normalization · Lemma 0AXN","summary":"Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Let X' → X be the normalization of X in Y. If Y is reduced, so is X'.","statement_latex":"Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $X' \\to X$ be the normalization of $X$ in $Y$. If $Y$ is reduced, so\nis $X'$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXN","source_file":"morphisms.tex","source_line":14066,"source_end_line":14071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14066-L14071","statement_sha256":"22e33c938574e42bcf7c2e3bde2658b7af566e6c80b5f72cfac1fddc642efcb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5789,"rank":5789,"depth":0,"x":2128.686,"y":696.139,"cluster":"scheme-morphisms"},{"id":"stacks:0AXP","tag":"0AXP","title":"Relative normalization · Lemma 0AXP","summary":"Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Let X' → X be the normalization of X in Y. Every generic point of an irreducible component of X' is the image of a generic point of an irreducible component of Y.","statement_latex":"Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $X' \\to X$ be the normalization of $X$ in $Y$. Every generic point of\nan irreducible component of $X'$ is the image of a generic point of\nan irreducible component of $Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXP","source_file":"morphisms.tex","source_line":14078,"source_end_line":14084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14078-L14084","statement_sha256":"920cee2ef89fad377ee7d206f560f3564e396eebcf2e97bfd63e84ab32c2faa4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5790,"rank":5790,"depth":2,"x":1944.159,"y":724.144,"cluster":"scheme-morphisms"},{"id":"stacks:03GO","tag":"03GO","title":"Relative normalization · Lemma 03GO","summary":"Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Suppose that Y = Y_1 amalg Y_2 is a disjoint union of two schemes. Write f_i = f|_Y_i. Let X_i' be the normalization of X in Y_i. Then X_1' amalg X_2' is the normalization of X in Y.","statement_latex":"Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of schemes.\nSuppose that $Y = Y_1 \\amalg Y_2$ is a disjoint union of two schemes.\nWrite $f_i = f|_{Y_i}$. Let $X_i'$ be the normalization of $X$ in $Y_i$.\nThen $X_1' \\amalg X_2'$ is the normalization of $X$ in $Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GO","source_file":"morphisms.tex","source_line":14095,"source_end_line":14101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14095-L14101","statement_sha256":"434f97fd67a5e745e106e8485b2f63e26bbdce0d7f229a4fcb51e9d8835a2ff9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5791,"rank":5791,"depth":0,"x":2057.828,"y":598.654,"cluster":"scheme-morphisms"},{"id":"stacks:03GQ","tag":"03GQ","title":"Relative normalization · Lemma 03GQ","summary":"Let f : X → S be a quasi-compact, quasi-separated and universally closed morphisms of schemes. Then f_*O_X is integral over O_S. In other words, the normalization of S in X is equal to the factorization X → underlineSpec_S(f_*O_X) → S of Constructions, Lemma [Tag 01LY].","statement_latex":"Let $f : X \\to S$ be a quasi-compact, quasi-separated and\nuniversally closed morphisms of schemes.\nThen $f_*\\mathcal{O}_X$ is integral over $\\mathcal{O}_S$. In other\nwords, the normalization of $S$ in $X$ is equal to the factorization\n$$\nX \\longrightarrow \\underline{\\Spec}_S(f_*\\mathcal{O}_X)\n\\longrightarrow S\n$$\nof Constructions, Lemma \\ref{constructions-lemma-canonical-morphism}.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GQ","source_file":"morphisms.tex","source_line":14115,"source_end_line":14126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14115-L14126","statement_sha256":"cd2aead6a8f8fc9e9f72ff47f47fc7a8e4ae7a76228187423f55a499d1bc895e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5792,"rank":5792,"depth":20,"x":2074.945,"y":755.856,"cluster":"scheme-morphisms"},{"id":"stacks:03GP","tag":"03GP","title":"Relative normalization · Lemma 03GP","summary":"Let f : Y → X be an integral morphism. Then the normalization of X in Y is equal to Y.","statement_latex":"Let $f : Y \\to X$ be an integral morphism.\nThen the normalization of $X$ in $Y$ is equal to $Y$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GP","source_file":"morphisms.tex","source_line":14158,"source_end_line":14162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14158-L14162","statement_sha256":"14befbdbb58fed15e0a2b5a28eac306fc7bd5cd998c56bd3126500caa21683d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5793,"rank":5793,"depth":21,"x":1935.756,"y":649.535,"cluster":"scheme-morphisms"},{"id":"stacks:035L","tag":"035L","title":"Relative normalization · Lemma 035L","summary":"Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Let X' be the normalization of X in Y. Assume • Y is a normal scheme, • quasi-compact opens of Y have finitely many irreducible components. Then X' is a disjoint union of integral normal schemes. Moreover, the morphism Y → X' is dominant and induces a bijection of irreducible components.","statement_latex":"Let $f : Y \\to X$ be a quasi-compact and quasi-separated morphism\nof schemes. Let $X'$ be the normalization of $X$ in $Y$. Assume\n\\begin{enumerate}\n\\item $Y$ is a normal scheme,\n\\item quasi-compact opens of $Y$ have finitely many irreducible components.\n\\end{enumerate}\nThen $X'$ is a disjoint union of integral normal schemes. Moreover,\nthe morphism $Y \\to X'$ is dominant and induces a bijection of\nirreducible components.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035L","source_file":"morphisms.tex","source_line":14169,"source_end_line":14180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14169-L14180","statement_sha256":"e712a53109f4c2f7a97acb835c8c5c1661417900984626aad940ca41e9677dcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5794,"rank":5794,"depth":8,"x":2124.093,"y":648.955,"cluster":"scheme-morphisms"},{"id":"stacks:0AVK","tag":"0AVK","title":"Relative normalization · Lemma 0AVK","summary":"Let f : X → S be a morphism. Assume that • S is a Nagata scheme, • f is quasi-compact and quasi-separated, • quasi-compact opens of X have finitely many irreducible components, • if x ∈ X is a generic point of an irreducible component, then the field extension kappa(x)/kappa(f(x)) is finitely generated, and • X is reduced. Then the normalization ν : S' → S of S in X is finite.","statement_latex":"Let $f : X \\to S$ be a morphism. Assume that\n\\begin{enumerate}\n\\item $S$ is a Nagata scheme,\n\\item $f$ is quasi-compact and quasi-separated,\n\\item quasi-compact opens of $X$ have finitely many irreducible components,\n\\item if $x \\in X$ is a generic point of an irreducible component,\nthen the field extension $\\kappa(x)/\\kappa(f(x))$ is finitely\ngenerated, and\n\\item $X$ is reduced.\n\\end{enumerate}\nThen the normalization $\\nu : S' \\to S$ of $S$ in $X$ is finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVK","source_file":"morphisms.tex","source_line":14220,"source_end_line":14233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14220-L14233","statement_sha256":"67b2e0fb8d1fc5b474786f58a9aa8f99aba6cbe22b0e51b0bcdb4d36193d2fbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5795,"rank":5795,"depth":7,"x":1985.535,"y":756.364,"cluster":"scheme-morphisms"},{"id":"stacks:03GR","tag":"03GR","title":"Relative normalization · Lemma 03GR","summary":"Let f : X → S be a morphism. Assume that • S is a Nagata scheme, • f is of finite type, • X is reduced. Then the normalization ν : S' → S of S in X is finite.","statement_latex":"Let $f : X \\to S$ be a morphism. Assume that\n\\begin{enumerate}\n\\item $S$ is a Nagata scheme,\n\\item $f$ is of finite type,\n\\item $X$ is reduced.\n\\end{enumerate}\nThen the normalization $\\nu : S' \\to S$ of $S$ in $X$ is finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GR","source_file":"morphisms.tex","source_line":14269,"source_end_line":14278,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14269-L14278","statement_sha256":"b7860f84ba5ac8ae71544c1bcac17efbb00f1411a6b246da69447c23d1c6af52","origin":"The Stacks Project","memory_eligible":false,"source_rank":5796,"rank":5796,"depth":17,"x":2001.347,"y":598.373,"cluster":"scheme-morphisms"},{"id":"stacks:0BXB","tag":"0BXB","title":"Relative normalization · Lemma 0BXB","summary":"Let f : Y → X be a finite type morphism of schemes with Y reduced and X Nagata. Let X' be the normalization of X in Y. Let x' ∈ X' be a point such that • dim(O_X', x') = 1, and • the fibre of Y → X' over x' is empty. Then O_X', x' is a discrete valuation ring.","statement_latex":"Let $f : Y \\to X$ be a finite type morphism of schemes with $Y$ reduced\nand $X$ Nagata. Let $X'$ be the normalization of $X$ in $Y$.\nLet $x' \\in X'$ be a point such that\n\\begin{enumerate}\n\\item $\\dim(\\mathcal{O}_{X', x'}) = 1$, and\n\\item the fibre of $Y \\to X'$ over $x'$ is empty.\n\\end{enumerate}\nThen $\\mathcal{O}_{X', x'}$ is a discrete valuation ring.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Relative normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXB","source_file":"morphisms.tex","source_line":14292,"source_end_line":14302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14292-L14302","statement_sha256":"9d7202b28fed9fecc46eec5557e3994551aefef97c2c3131e6879865959b400f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5797,"rank":5797,"depth":21,"x":2116.863,"y":723.979,"cluster":"scheme-morphisms"},{"id":"stacks:035N","tag":"035N","title":"Normalization · Definition 035N","summary":"Let X be a scheme such that every quasi-compact open has finitely many irreducible components. We define the normalization of X as the morphism ν : X^ν → X which is the normalization of X in the morphism f : Y → X ([Tag 035M]) constructed above.","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has\nfinitely many irreducible components. We define the\n{\\it normalization} of $X$ as the morphism\n$$\n\\nu : X^\\nu \\longrightarrow X\n$$\nwhich is the normalization of $X$ in the morphism $f : Y \\to X$\n(\\ref{equation-generic-points}) constructed above.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035N","source_file":"morphisms.tex","source_line":14378,"source_end_line":14388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14378-L14388","statement_sha256":"4f9feb64b51048364feaadb0d6ae535ecf27567005380f2a07449e2a6247d2f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5798,"rank":5798,"depth":0,"x":1930.476,"y":696.876,"cluster":"scheme-morphisms"},{"id":"stacks:035O","tag":"035O","title":"Normalization · Lemma 035O","summary":"Let X be a scheme such that every quasi-compact open has finitely many irreducible components. The normalization morphism ν factors through the reduction X_red and X^ν → X_red is the normalization of X_red.","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has\nfinitely many irreducible components. The normalization morphism\n$\\nu$ factors through the reduction $X_{red}$ and $X^\\nu \\to X_{red}$\nis the normalization of $X_{red}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035O","source_file":"morphisms.tex","source_line":14397,"source_end_line":14403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14397-L14403","statement_sha256":"f0e1f5a83802531471e1bf7affe3dae6905c23ea0440a07a6588ff51a4251bec","origin":"The Stacks Project","memory_eligible":false,"source_rank":5799,"rank":5799,"depth":20,"x":2089.879,"y":611.01,"cluster":"scheme-morphisms"},{"id":"stacks:035P","tag":"035P","title":"Normalization · Lemma 035P","summary":"Let X be a reduced scheme such that every quasi-compact open has finitely many irreducible components. Let Spec(A) = U ⊂ X be an affine open. Then • A has finitely many minimal primes q_1, …, q_t, • the total ring of fractions Q(A) of A is Q(A/ q_1) × … × Q(A/ q_t), • the integral closure A' of A in Q(A) is the product of the integral closures of the domains A/ q_i in the fields Q(A/ q_i), and • ν^-1(U) is identified with the spectrum of A' where ν : X^ν → X is the…","statement_latex":"Let $X$ be a reduced scheme such that every quasi-compact open has\nfinitely many irreducible components. Let $\\Spec(A) = U \\subset X$\nbe an affine open. Then\n\\begin{enumerate}\n\\item $A$ has finitely many minimal primes\n$\\mathfrak q_1, \\ldots, \\mathfrak q_t$,\n\\item the total ring of fractions $Q(A)$ of $A$ is\n$Q(A/\\mathfrak q_1) \\times \\ldots \\times Q(A/\\mathfrak q_t)$,\n\\item the integral closure $A'$ of $A$ in $Q(A)$ is the product of\nthe integral closures of the domains $A/\\mathfrak q_i$\nin the fields $Q(A/\\mathfrak q_i)$, and\n\\item $\\nu^{-1}(U)$ is identified with the spectrum of $A'$ where\n$\\nu : X^\\nu \\to X$ is the normalization morphism.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035P","source_file":"morphisms.tex","source_line":14425,"source_end_line":14441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14425-L14441","statement_sha256":"fac7927a027b1d32201f74ecfc7d416812e8ff68e37afa29f2223fda4845061c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5800,"rank":5800,"depth":7,"x":2041.338,"y":764.938,"cluster":"scheme-morphisms"},{"id":"stacks:0C3B","tag":"0C3B","title":"Normalization · Lemma 0C3B","summary":"Let X be a scheme such that every quasi-compact open has a finite number of irreducible components. Let ν : X^ν → X be the normalization of X. Let x ∈ X. Then the following are canonically isomorphic as O_X, x-algebras • the stalk (ν_*O_X^ν)_x, • the integral closure of O_X, x in the total ring of fractions of (O_X, x)_red, • the integral closure of O_X, x in the product of the residue fields of the minimal primes of O_X, x (and there are finitely many of these).","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has a finite\nnumber of irreducible components. Let $\\nu : X^\\nu \\to X$\nbe the normalization of $X$. Let $x \\in X$. Then the following\nare canonically isomorphic as $\\mathcal{O}_{X, x}$-algebras\n\\begin{enumerate}\n\\item the stalk $(\\nu_*\\mathcal{O}_{X^\\nu})_x$,\n\\item the integral closure of $\\mathcal{O}_{X, x}$ in\nthe total ring of fractions of $(\\mathcal{O}_{X, x})_{red}$,\n\\item the integral closure of $\\mathcal{O}_{X, x}$ in the product\nof the residue fields of the minimal primes of $\\mathcal{O}_{X, x}$\n(and there are finitely many of these).\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3B","source_file":"morphisms.tex","source_line":14465,"source_end_line":14479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14465-L14479","statement_sha256":"ffbb1e7cac46c9635f9d515f783eccdf50f8af6f8af86eedd43d1acbf97ec764","origin":"The Stacks Project","memory_eligible":false,"source_rank":5801,"rank":5801,"depth":8,"x":1953.254,"y":623.742,"cluster":"scheme-morphisms"},{"id":"stacks:035Q","tag":"035Q","title":"Normalization · Lemma 035Q","summary":"Let X be a scheme such that every quasi-compact open has finitely many irreducible components. • The normalization X^ν is a disjoint union of integral normal schemes. • The morphism ν : X^ν → X is integral, surjective, and induces a bijection on irreducible components. • For any integral morphism α : X' → X such that for U ⊂ X quasi-compact open the inverse image α^-1(U) has finitely many irreducible components and α|_α^-1(U) : α^-1(U) → U is birational → X_red satisfies…","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has\nfinitely many irreducible components.\n\\begin{enumerate}\n\\item The normalization $X^\\nu$ is a disjoint union of integral normal schemes.\n\\item The morphism $\\nu : X^\\nu \\to X$ is integral, surjective, and\ninduces a bijection on irreducible components.\n\\item For any integral morphism $\\alpha : X' \\to X$ such that for\n$U \\subset X$ quasi-compact open the inverse image $\\alpha^{-1}(U)$ has\nfinitely many irreducible components and\n$\\alpha|_{\\alpha^{-1}(U)} : \\alpha^{-1}(U) \\to U$ is birational\\footnote{This\nawkward formulation is necessary as we've only defined what\nit means for a morphism to be birational if the source and target\nhave finitely many irreducible components. It suffices if\n$X'_{red} \\to X_{red}$ satisfies the condition.}\nthere exists a factorization\n$X^\\nu \\to X' \\to X$ and $X^\\nu \\to X'$ is the normalization of $X'$.\n\\item For any morphism $Z \\to X$ with $Z$ a normal scheme\nsuch that each irreducible component of $Z$ dominates an irreducible\ncomponent of $X$ there exists a unique factorization $Z \\to X^\\nu \\to X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035Q","source_file":"morphisms.tex","source_line":14503,"source_end_line":14525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14503-L14525","statement_sha256":"64a99749ad8ef3ccf7c2318a80ae461f58895ac01a45ed7e976392bad6583398","origin":"The Stacks Project","memory_eligible":false,"source_rank":5802,"rank":5802,"depth":21,"x":2131.94,"y":677.934,"cluster":"scheme-morphisms"},{"id":"stacks:0CDV","tag":"0CDV","title":"Normalization · Lemma 0CDV","summary":"Let X be a scheme such that every quasi-compact open has finitely many irreducible components. Let Z_i ⊂ X, i ∈ I be the irreducible components of X endowed with the reduced induced structure. Let Z_i^ν → Z_i be the normalization. Then coprod_i ∈ I Z_i^ν → X is the normalization of X.","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has\nfinitely many irreducible components. Let $Z_i \\subset X$, $i \\in I$\nbe the irreducible components of $X$ endowed with the reduced\ninduced structure. Let $Z_i^\\nu \\to Z_i$ be the normalization.\nThen $\\coprod_{i \\in I} Z_i^\\nu \\to X$ is the normalization of $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDV","source_file":"morphisms.tex","source_line":14587,"source_end_line":14594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14587-L14594","statement_sha256":"47924751722a579bac8e949fd0e2be98fd9788c0d03f260c1ff908e197be2dca","origin":"The Stacks Project","memory_eligible":false,"source_rank":5803,"rank":5803,"depth":22,"x":1956.415,"y":739.429,"cluster":"scheme-morphisms"},{"id":"stacks:0BXC","tag":"0BXC","title":"Normalization · Lemma 0BXC","summary":"Let X be a reduced scheme with finitely many irreducible components. Then the normalization morphism X^ν → X is birational.","statement_latex":"Let $X$ be a reduced scheme with finitely many irreducible components.\nThen the normalization morphism $X^\\nu \\to X$ is birational.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXC","source_file":"morphisms.tex","source_line":14605,"source_end_line":14609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14605-L14609","statement_sha256":"641ee42521008db6077c71b86c9d4667ae8ad42ca816601bf5520725b86df858","origin":"The Stacks Project","memory_eligible":false,"source_rank":5804,"rank":5804,"depth":22,"x":2036.476,"y":594.335,"cluster":"scheme-morphisms"},{"id":"stacks:0AB1","tag":"0AB1","title":"Normalization · Lemma 0AB1","summary":"A finite (or even integral) birational morphism f : X → Y of integral schemes with Y normal is an isomorphism.","statement_latex":"A finite (or even integral) birational morphism $f : X \\to Y$\nof integral schemes with $Y$ normal is an isomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AB1","source_file":"morphisms.tex","source_line":14629,"source_end_line":14633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14629-L14633","statement_sha256":"6d302aba71c94d3221067bac945b9c1834117e24d6efb68d80eba6bd2ae14b65","origin":"The Stacks Project","memory_eligible":false,"source_rank":5805,"rank":5805,"depth":0,"x":2094.181,"y":746.913,"cluster":"scheme-morphisms"},{"id":"stacks:0H7D","tag":"0H7D","title":"Normalization · Lemma 0H7D","summary":"Let X be a scheme with locally finitely many irreducible components. The normalization morphism ν : X^ν → X is an isomorphism if and only if X is normal.","statement_latex":"Let $X$ be a scheme with locally finitely many irreducible components.\nThe normalization morphism $\\nu : X^\\nu \\to X$ is an isomorphism\nif and only if $X$ is normal.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7D","source_file":"morphisms.tex","source_line":14648,"source_end_line":14653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14648-L14653","statement_sha256":"dfb7407105a8127ffd0625c57caf538c0ec7268ecea9a141aa100969b98466cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5806,"rank":5806,"depth":23,"x":1928.759,"y":667.064,"cluster":"scheme-morphisms"},{"id":"stacks:035R","tag":"035R","title":"Normalization · Lemma 035R","summary":"Let X be an integral, Japanese scheme. The normalization ν : X^ν → X is a finite morphism.","statement_latex":"Let $X$ be an integral, Japanese scheme.\nThe normalization $\\nu : X^\\nu \\to X$ is a finite morphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035R","source_file":"morphisms.tex","source_line":14669,"source_end_line":14673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14669-L14673","statement_sha256":"4db34a4e871720d8cfdbc1c7c12c1106f3356226dcc5005840a5af306dd69e56","origin":"The Stacks Project","memory_eligible":false,"source_rank":5807,"rank":5807,"depth":8,"x":2115.145,"y":632.042,"cluster":"scheme-morphisms"},{"id":"stacks:035S","tag":"035S","title":"Normalization · Lemma 035S","summary":"Let X be a Nagata scheme. The normalization ν : X^ν → X is a finite morphism.","statement_latex":"Let $X$ be a Nagata scheme.\nThe normalization $\\nu : X^\\nu \\to X$ is a finite morphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035S","source_file":"morphisms.tex","source_line":14683,"source_end_line":14687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14683-L14687","statement_sha256":"ddcc678aff5f2b0391c05a487e9075a0ef384f130de56494985aceeaa58ef09a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5808,"rank":5808,"depth":8,"x":2005.756,"y":763.764,"cluster":"scheme-morphisms"},{"id":"stacks:0GIQ","tag":"0GIQ","title":"Normalization · Lemma 0GIQ","summary":"Let X be an irreducible, geometrically unibranch scheme. The normalization morphism ν : X^ν → X is a universal homeomorphism.","statement_latex":"Let $X$ be an irreducible, geometrically unibranch scheme.\nThe normalization morphism $\\nu : X^\\nu \\to X$ is a universal\nhomeomorphism.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIQ","source_file":"morphisms.tex","source_line":14709,"source_end_line":14714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14709-L14714","statement_sha256":"6614d104a3938b218fb8db41ba4a509f8060599ab2ca3137ab1d6a3ea8316222","origin":"The Stacks Project","memory_eligible":false,"source_rank":5809,"rank":5809,"depth":22,"x":1980.465,"y":604.399,"cluster":"scheme-morphisms"},{"id":"stacks:0H3J","tag":"0H3J","title":"Weak normalization · Lemma 0H3J","summary":"Let A → B be a ring map inducing a dominant morphism Spec(B) → Spec(A) of spectra. There exists an A-subalgebra B' ⊂ B such that • Spec(B') → Spec(A) is a universal homeomorphism, • given a factorization A → C → B such that Spec(C) → Spec(A) is a universal homeomorphism, the image of C → B is contained in B'.","statement_latex":"Let $A \\to B$ be a ring map inducing a dominant morphism\n$\\Spec(B) \\to \\Spec(A)$ of spectra. There exists an $A$-subalgebra\n$B' \\subset B$ such that\n\\begin{enumerate}\n\\item $\\Spec(B') \\to \\Spec(A)$ is a universal homeomorphism,\n\\item given a factorization $A \\to C \\to B$ such that\n$\\Spec(C) \\to \\Spec(A)$ is a universal homeomorphism, the\nimage of $C \\to B$ is contained in $B'$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3J","source_file":"morphisms.tex","source_line":14747,"source_end_line":14758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14747-L14758","statement_sha256":"ae2fdb98634202877b09e46c7a55789c28674972f3c7362fead100b87f0bccc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5810,"rank":5810,"depth":25,"x":2127.423,"y":707.668,"cluster":"scheme-morphisms"},{"id":"stacks:0H3K","tag":"0H3K","title":"Weak normalization · Lemma 0H3K","summary":"Let A → B be a ring map inducing a dominant morphism Spec(B) → Spec(A) of spectra. Formation of the A-subalgebra B' ⊂ B in Lemma [Tag 0H3J] commutes with localization (see proof for explanation).","statement_latex":"Let $A \\to B$ be a ring map inducing a dominant morphism\n$\\Spec(B) \\to \\Spec(A)$ of spectra. Formation of the $A$-subalgebra\n$B' \\subset B$ in Lemma \\ref{lemma-relative-weak-normalization-algebra}\ncommutes with localization (see proof for explanation).","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3K","source_file":"morphisms.tex","source_line":14812,"source_end_line":14818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14812-L14818","statement_sha256":"2ca44e7a009281c91080234312f141646954bb9906d9a1a32e535bb6e79ffa5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5811,"rank":5811,"depth":26,"x":1935.815,"y":714.914,"cluster":"scheme-morphisms"},{"id":"stacks:0H3L","tag":"0H3L","title":"Weak normalization · Lemma 0H3L","summary":"Let A → B be a ring map inducing a dominant morphism Spec(B) → Spec(A) of spectra. There exists an A-subalgebra B' ⊂ B such that • Spec(B') → Spec(A) is a universal homeomorphism inducing isomorphisms on residue fields, • given a factorization A → C → B such that Spec(C) → Spec(A) is a universal homeomorphism inducing isomorphisms on residue fields, the image of C → B is contained in B'.","statement_latex":"Let $A \\to B$ be a ring map inducing a dominant morphism\n$\\Spec(B) \\to \\Spec(A)$ of spectra. There exists an $A$-subalgebra\n$B' \\subset B$ such that\n\\begin{enumerate}\n\\item $\\Spec(B') \\to \\Spec(A)$ is a universal homeomorphism inducing\nisomorphisms on residue fields,\n\\item given a factorization $A \\to C \\to B$ such that $\\Spec(C) \\to \\Spec(A)$\nis a universal homeomorphism inducing isomorphisms on residue fields,\nthe image of $C \\to B$ is contained in $B'$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3L","source_file":"morphisms.tex","source_line":14856,"source_end_line":14868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14856-L14868","statement_sha256":"59e6e2f9d9eec810605870faa08dc31b09906b3f636c1955efa09c1c4e5a8a9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5812,"rank":5812,"depth":26,"x":2071.416,"y":600.731,"cluster":"scheme-morphisms"},{"id":"stacks:0H3M","tag":"0H3M","title":"Weak normalization · Lemma 0H3M","summary":"Let A → B be a ring map inducing a dominant morphism Spec(B) → Spec(A) of spectra. Formation of the A-subalgebra B' ⊂ B in Lemma [Tag 0H3L] commutes with localization (see proof for explanation).","statement_latex":"Let $A \\to B$ be a ring map inducing a dominant morphism\n$\\Spec(B) \\to \\Spec(A)$ of spectra. Formation of the $A$-subalgebra\n$B' \\subset B$ in Lemma \\ref{lemma-relative-seminormalization-algebra}\ncommutes with localization (see proof for explanation).","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3M","source_file":"morphisms.tex","source_line":14877,"source_end_line":14883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14877-L14883","statement_sha256":"578fa9eb773a129cb26915da0b7e11f9250c34a8e232c8635a7695a6f66d9e35","origin":"The Stacks Project","memory_eligible":false,"source_rank":5813,"rank":5813,"depth":27,"x":2063.241,"y":762.037,"cluster":"scheme-morphisms"},{"id":"stacks:0H3N","tag":"0H3N","title":"Weak normalization · Lemma 0H3N","summary":"Let f : Y → X be a quasi-compact, quasi-separated, and dominant morphism of schemes. • The category of factorizations Y → X' → X where X' → X is a universal homeomorphism has an initial object Y → X^Y/wn → X. • The category of factorizations Y → X' → X where X' → X is a universal homeomorphism inducing isomorphisms on residue fields has an initial object Y → X^Y/sn → X. Moreover, formation of the factorization Y → X^Y/wn → X and Y → X^Y/sn → X commutes with base change to…","statement_latex":"Let $f : Y \\to X$ be a quasi-compact, quasi-separated, and dominant\nmorphism of schemes.\n\\begin{enumerate}\n\\item The category of factorizations $Y \\to X' \\to X$ where $X' \\to X$\nis a universal homeomorphism has an initial object $Y \\to X^{Y/wn} \\to X$.\n\\item The category of factorizations $Y \\to X' \\to X$ where $X' \\to X$\nis a universal homeomorphism inducing isomorphisms on residue fields\nhas an initial object $Y \\to X^{Y/sn} \\to X$.\n\\end{enumerate}\nMoreover, formation of the factorization $Y \\to X^{Y/wn} \\to X$ and\n$Y \\to X^{Y/sn} \\to X$ commutes with base change to open subschemes of $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3N","source_file":"morphisms.tex","source_line":14890,"source_end_line":14903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14890-L14903","statement_sha256":"ac11427c7106548411dc0ca6cf3cfe35e1d1ba66ac5d16cdccd00ef14b486425","origin":"The Stacks Project","memory_eligible":false,"source_rank":5814,"rank":5814,"depth":27,"x":1939.424,"y":638.326,"cluster":"scheme-morphisms"},{"id":"stacks:0H3P","tag":"0H3P","title":"Weak normalization · Definition 0H3P","summary":"Let f : Y → X be a quasi-compact, quasi-separated, and dominant morphism of schemes. • The factorization Y → X^Y/sn → X constructed in Lemma [Tag 0H3N] part (2) is the seminormalization of X in Y. • The factorization Y → X^Y/wn → X constructed in Lemma [Tag 0H3N] part (1) is the weak normalization of X in Y.","statement_latex":"Let $f : Y \\to X$ be a quasi-compact, quasi-separated, and dominant\nmorphism of schemes.\n\\begin{enumerate}\n\\item The factorization $Y \\to X^{Y/sn} \\to X$ constructed in\nLemma \\ref{lemma-relative-seminormalization} part (2)\nis the {\\it seminormalization of $X$ in $Y$}.\n\\item The factorization $Y \\to X^{Y/wn} \\to X$ constructed in\nLemma \\ref{lemma-relative-seminormalization} part (1)\nis the {\\it weak normalization of $X$ in $Y$}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3P","source_file":"morphisms.tex","source_line":14968,"source_end_line":14980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14968-L14980","statement_sha256":"623adaba098877bbf08d996103cafb2e9272ac4db821baf960c187dea1697e3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5815,"rank":5815,"depth":28,"x":2130.405,"y":659.314,"cluster":"scheme-morphisms"},{"id":"stacks:0H3Q","tag":"0H3Q","title":"Weak normalization · Lemma 0H3Q","summary":"Let X be a scheme such that every quasi-compact open has finitely many irreducible components. Let ν : X^ν → X be the normalization of X. Then the seminormalization of X in X^ν is is the seminormalization of X. In a formula: X^sn = X^X^ν/sn.","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has\nfinitely many irreducible components. Let $\\nu : X^\\nu \\to X$\nbe the normalization of $X$. Then the seminormalization of $X$ in\n$X^\\nu$ is is the seminormalization of $X$. In a formula:\n$X^{sn} = X^{X^\\nu/sn}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3Q","source_file":"morphisms.tex","source_line":14986,"source_end_line":14993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L14986-L14993","statement_sha256":"7b188520cb75f431c9819a90b6e8a6c1e4e08776892fd601bbcb492f7043088a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5816,"rank":5816,"depth":20,"x":1972.54,"y":752.299,"cluster":"scheme-morphisms"},{"id":"stacks:0H3R","tag":"0H3R","title":"Weak normalization · Definition 0H3R","summary":"Let X be a scheme such that every quasi-compact open has finitely many irreducible components. We define the weak normalization of X as the weak normalization X^ν → X^wn → X of X in the normalization X^ν of X (Definition [Tag 035N]). In a formula: X^wn = X^X^ν/wn.","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has\nfinitely many irreducible components. We define the\n{\\it weak normalization} of $X$ as the weak normalization\n$$\nX^\\nu \\longrightarrow X^{wn} \\longrightarrow X\n$$\nof $X$ in the normalization $X^\\nu$ of $X$\n(Definition \\ref{definition-normalization}).\nIn a formula: $X^{wn} = X^{X^\\nu/wn}$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3R","source_file":"morphisms.tex","source_line":15018,"source_end_line":15029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15018-L15029","statement_sha256":"d819ece8750c1d73f439f10d36256c4ab1c128d10dbedb14a40f67e252e49f43","origin":"The Stacks Project","memory_eligible":false,"source_rank":5817,"rank":5817,"depth":1,"x":2014.213,"y":593.996,"cluster":"scheme-morphisms"},{"id":"stacks:0H3S","tag":"0H3S","title":"Weak normalization · Definition 0H3S","summary":"Let X be a scheme such that every quasi-compact open has finitely many irreducible components. We say X is weakly normal if the weak normalization X^wn → X is an isomorphism (Definition [Tag 0H3R]).","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has\nfinitely many irreducible components. We say $X$ is {\\it weakly normal}\nif the weak normalization $X^{wn} \\to X$ is an isomorphism\n(Definition \\ref{definition-weak-normalization}).","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3S","source_file":"morphisms.tex","source_line":15042,"source_end_line":15048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15042-L15048","statement_sha256":"d81b987a9bb4c316a4e605f7624c5bfc3bba778f8577262cdf6cf7be10179d93","origin":"The Stacks Project","memory_eligible":false,"source_rank":5818,"rank":5818,"depth":2,"x":2110.885,"y":734.516,"cluster":"scheme-morphisms"},{"id":"stacks:0H3T","tag":"0H3T","title":"Weak normalization · Lemma 0H3T","summary":"Let X = Spec(A) be an affine scheme which has finitely many irreducible components. Then X is weakly normal if and only if • A is seminormal (Definition [Tag 0EUL]), • for a prime number p and z, w ∈ A such that (a) z is a nonzerodivisor, (b) w^p is divisible by z^p, and (c) pw is divisible by z, then w is divisible by z.","statement_latex":"Let $X = \\Spec(A)$ be an affine scheme which has finitely many\nirreducible components. Then $X$ is weakly normal if and only if\n\\begin{enumerate}\n\\item $A$ is seminormal (Definition \\ref{definition-seminormal-ring}),\n\\item for a prime number $p$ and $z, w \\in A$ such that\n(a) $z$ is a nonzerodivisor, (b) $w^p$ is divisible by $z^p$, and\n(c) $pw$ is divisible by $z$, then $w$ is divisible by $z$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3T","source_file":"morphisms.tex","source_line":15061,"source_end_line":15071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15061-L15071","statement_sha256":"a605c5a5bbc01c6d2aebcc1af194ffff027e4caa32a42a06a4d19fce06b84163","origin":"The Stacks Project","memory_eligible":false,"source_rank":5819,"rank":5819,"depth":26,"x":1926.411,"y":685.703,"cluster":"scheme-morphisms"},{"id":"stacks:0H3U","tag":"0H3U","title":"Weak normalization · Lemma 0H3U","summary":"Let X be a scheme such that every quasi-compact open has finitely many irreducible components. The following are equivalent: • The scheme X is weakly normal. • For every affine open U ⊂ X the ring O_X(U) satisfies conditions (1) and (2) of Lemma [Tag 0H3T]. • There exists an affine open covering X = ⋃ U_i such that each ring O_X(U_i) satisfies conditions (1) and (2) of Lemma [Tag 0H3T]. • There exists an open covering X = ⋃ X_j such that each open subscheme X_j is weakly…","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has\nfinitely many irreducible components. The following are equivalent:\n\\begin{enumerate}\n\\item The scheme $X$ is weakly normal.\n\\item For every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nsatisfies conditions (1) and (2) of Lemma \\ref{lemma-affine-weakly-normal}.\n\\item There exists an affine open covering $X = \\bigcup U_i$ such that\neach ring $\\mathcal{O}_X(U_i)$\nsatisfies conditions (1) and (2) of Lemma \\ref{lemma-affine-weakly-normal}.\n\\item There exists an open covering $X = \\bigcup X_j$\nsuch that each open subscheme $X_j$ is weakly normal.\n\\end{enumerate}\nMoreover, if $X$ is weakly normal then every open subscheme is weakly normal.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Weak normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3U","source_file":"morphisms.tex","source_line":15111,"source_end_line":15126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15111-L15126","statement_sha256":"bd37af2bad290821b1532aa3de2b86640d25e884f6a2bf9296b327bf22e815b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5820,"rank":5820,"depth":28,"x":2101.871,"y":616.952,"cluster":"scheme-morphisms"},{"id":"stacks:03GT","tag":"03GT","title":"Algebraic version of Zariski's Main Theorem · Theorem 03GT","summary":"Let f : Y → X be an affine morphism of schemes. Assume f is of finite type. Let X' be the normalization of X in Y. Picture: xymatrix Y ar[rd]_f ar[rr]_f' & & X' ar[ld]^ν & X & Then there exists an open subscheme U' ⊂ X' such that • (f')^-1(U') → U' is an isomorphism, and • (f')^-1(U') ⊂ Y is the set of points at which f is quasi-finite.","statement_latex":"Let $f : Y \\to X$ be an affine morphism of schemes.\nAssume $f$ is of finite type.\nLet $X'$ be the normalization of $X$ in $Y$. Picture:\n$$\n\\xymatrix{\nY \\ar[rd]_f \\ar[rr]_{f'} & & X' \\ar[ld]^\\nu \\\\\n& X &\n}\n$$\nThen there exists an open subscheme $U' \\subset X'$ such that\n\\begin{enumerate}\n\\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and\n\\item $(f')^{-1}(U') \\subset Y$ is the set of points at which\n$f$ is quasi-finite.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Zariski's Main Theorem (algebraic version)","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GT","source_file":"morphisms.tex","source_line":15158,"source_end_line":15175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15158-L15175","statement_sha256":"3052f9c1e05235c8b010cb6118e055c297ae541a6b2fac27dd92fc5d61745ae4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5821,"rank":5821,"depth":28,"x":2027.702,"y":767.36,"cluster":"scheme-morphisms"},{"id":"stacks:01TI","tag":"01TI","title":"Zariski's Main Theorem (algebraic version) · Lemma 01TI","summary":"The locally quasi-finite locus of a morphism is open Let f : X → S be a morphism of schemes. The set of points of X where f is quasi-finite is an open U ⊂ X. The induced morphism U → S is locally quasi-finite.","statement_latex":"\\begin{slogan}\nThe locally quasi-finite locus of a morphism is open\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nThe set of points of $X$ where $f$ is quasi-finite is an open\n$U \\subset X$. The induced morphism $U \\to S$ is locally quasi-finite.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Zariski's Main Theorem (algebraic version)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01TI","source_file":"morphisms.tex","source_line":15201,"source_end_line":15209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15201-L15209","statement_sha256":"56cf02caac7fd3b194008bc403bae2c17b5928746d77abe27202ab289fd6d3c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5822,"rank":5822,"depth":29,"x":1961.373,"y":614.212,"cluster":"scheme-morphisms"},{"id":"stacks:03GU","tag":"03GU","title":"Zariski's Main Theorem (algebraic version) · Lemma 03GU","summary":"Let f : Y → X be a morphism of schemes. Assume • X and Y are affine, and • f is quasi-finite. Then there exists a diagram xymatrix Y ar[rd]_f ar[rr]_j & & Z ar[ld]^π & X & with Z affine, π finite and j an open immersion.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes.\nAssume\n\\begin{enumerate}\n\\item $X$ and $Y$ are affine, and\n\\item $f$ is quasi-finite.\n\\end{enumerate}\nThen there exists a diagram\n$$\n\\xymatrix{\nY \\ar[rd]_f \\ar[rr]_j & & Z \\ar[ld]^\\pi \\\\\n& X &\n}\n$$\nwith $Z$ affine, $\\pi$ finite and $j$ an open immersion.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Zariski's Main Theorem (algebraic version)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GU","source_file":"morphisms.tex","source_line":15229,"source_end_line":15245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15229-L15245","statement_sha256":"fdb127cb6cd673ba793becb82b61d447b13d3be05096bb52d24d3470ae203d56","origin":"The Stacks Project","memory_eligible":false,"source_rank":5823,"rank":5823,"depth":29,"x":2133.614,"y":689.58,"cluster":"scheme-morphisms"},{"id":"stacks:03J2","tag":"03J2","title":"Zariski's Main Theorem (algebraic version) · Lemma 03J2","summary":"Let f : Y → X be a quasi-finite morphism of schemes. Let T ⊂ Y be a closed nowhere dense subset of Y. Then f(T) ⊂ X is a nowhere dense subset of X.","statement_latex":"Let $f : Y \\to X$ be a quasi-finite morphism of schemes.\nLet $T \\subset Y$ be a closed nowhere dense subset of $Y$.\nThen $f(T) \\subset X$ is a nowhere dense subset of $X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Zariski's Main Theorem (algebraic version)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03J2","source_file":"morphisms.tex","source_line":15253,"source_end_line":15258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15253-L15258","statement_sha256":"1e4e6936ca8f661f292985984be74fac9370f4bbcd942d1e58e90a9b4c88fb11","origin":"The Stacks Project","memory_eligible":false,"source_rank":5824,"rank":5824,"depth":37,"x":1945.809,"y":731.781,"cluster":"scheme-morphisms"},{"id":"stacks:03J4","tag":"03J4","title":"Universally bounded fibres · Definition 03J4","summary":"Let f : X → Y be a morphism of schemes. • We say the integer n bounds the degrees of the fibres of f if for all y ∈ Y the fibre X_y is a finite scheme over kappa(y) whose degree over kappa(y) is ≤ n. • We say the fibres of f are universally bounded if there exists an integer n which bounds the degrees of the fibres of f.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\n\\begin{enumerate}\n\\item We say the integer $n$ {\\it bounds the degrees of the fibres\nof $f$} if for all $y \\in Y$\nthe fibre $X_y$ is a finite scheme over $\\kappa(y)$ whose\ndegree over $\\kappa(y)$ is $\\leq n$.\n\\item We say the {\\it fibres of $f$ are universally bounded}\\footnote{This is\nprobably nonstandard notation.}\nif there exists an integer $n$ which bounds the degrees of the fibres\nof $f$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03J4","source_file":"morphisms.tex","source_line":15301,"source_end_line":15314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15301-L15314","statement_sha256":"bf40444499b2072d71eaa80e864c9fcc0a8c27438a7e0a389e1880878989f994","origin":"The Stacks Project","memory_eligible":false,"source_rank":5825,"rank":5825,"depth":0,"x":2050.461,"y":593.959,"cluster":"scheme-morphisms"},{"id":"stacks:03J5","tag":"03J5","title":"Universally bounded fibres · Lemma 03J5","summary":"Let f : X → Y be a morphism of schemes. Let n ≥ 0. The following are equivalent: • the integer n bounds the degrees of the fibres of f, and • for every morphism Spec(k) → Y, where k is a field, the fibre product X_k = Spec(k) ×_Y X is finite over k of degree ≤ n. In this case the fibres of f are universally bounded and the schemes X_k have at most n points. More precisely, if X_k = (x_1, …, x_t), then we have n ≥ ∑_i = 1, …, t [kappa(x_i) : k]","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $n \\geq 0$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the integer $n$ bounds the degrees of the fibres of $f$, and\n\\item for every morphism $\\Spec(k) \\to Y$, where $k$ is a field,\nthe fibre product $X_k = \\Spec(k) \\times_Y X$ is finite over $k$\nof degree $\\leq n$.\n\\end{enumerate}\nIn this case the fibres of $f$ are universally bounded and the schemes\n$X_k$ have at most $n$ points. More precisely, if\n$X_k = \\{x_1, \\ldots, x_t\\}$, then we have\n$$\nn \\geq \\sum\\nolimits_{i = 1, \\ldots, t} [\\kappa(x_i) : k]\n$$","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03J5","source_file":"morphisms.tex","source_line":15321,"source_end_line":15337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15321-L15337","statement_sha256":"6f52757c2d66c2dc7d99499b8576af181666544b1d41b956433bf681c208a6e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5826,"rank":5826,"depth":7,"x":2084.158,"y":755.13,"cluster":"scheme-morphisms"},{"id":"stacks:0CC2","tag":"0CC2","title":"Universally bounded fibres · Lemma 0CC2","summary":"If f is a finite locally free morphism of degree d, then d bounds the degree of the fibres of f.","statement_latex":"If $f$ is a finite locally free morphism of degree $d$, then\n$d$ bounds the degree of the fibres of $f$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CC2","source_file":"morphisms.tex","source_line":15357,"source_end_line":15361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15357-L15361","statement_sha256":"fea5035bcf65e0360bd5e9ec06bd8b9c759f0701f7cdae996bf7a20941558b3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5827,"rank":5827,"depth":1,"x":1929.547,"y":655.306,"cluster":"scheme-morphisms"},{"id":"stacks:03J6","tag":"03J6","title":"Universally bounded fibres · Lemma 03J6","summary":"A composition of morphisms with universally bounded fibres is a morphism with universally bounded fibres. More precisely, assume that n bounds the degrees of the fibres of f : X → Y and m bounds the degrees of g : Y → Z. Then nm bounds the degrees of the fibres of g ∘ f : X → Z.","statement_latex":"A composition of morphisms with universally bounded fibres\nis a morphism with universally bounded fibres. More precisely,\nassume that $n$ bounds the degrees of the fibres of $f : X \\to Y$ and\n$m$ bounds the degrees of $g : Y \\to Z$.\nThen $nm$ bounds the degrees of the fibres of $g \\circ f : X \\to Z$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03J6","source_file":"morphisms.tex","source_line":15370,"source_end_line":15377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15370-L15377","statement_sha256":"dd3066afa6209dd53fde761ba61e68e6b07e3a037a9b228d9e516a87e3c43df3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5828,"rank":5828,"depth":13,"x":2124.024,"y":641.171,"cluster":"scheme-morphisms"},{"id":"stacks:03J7","tag":"03J7","title":"Universally bounded fibres · Lemma 03J7","summary":"A base change of a morphism with universally bounded fibres is a morphism with universally bounded fibres. More precisely, if n bounds the degrees of the fibres of f : X → Y and Y' → Y is any morphism, then the degrees of the fibres of the base change f' : Y' ×_Y X → Y' is also bounded by n.","statement_latex":"A base change of a morphism with universally bounded fibres is\na morphism with universally bounded fibres. More precisely, if\n$n$ bounds the degrees of the fibres of $f : X \\to Y$ and $Y' \\to Y$\nis any morphism, then the degrees of the fibres of the base change\n$f' : Y' \\times_Y X \\to Y'$ is also bounded by $n$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03J7","source_file":"morphisms.tex","source_line":15421,"source_end_line":15428,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15421-L15428","statement_sha256":"d14f96645c8053bd100eea9b79d68f72472292becb7ed0bcabdeaf497a029e0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5829,"rank":5829,"depth":8,"x":1991.855,"y":762.065,"cluster":"scheme-morphisms"},{"id":"stacks:03J8","tag":"03J8","title":"Universally bounded fibres · Lemma 03J8","summary":"Let f : X → Y be a morphism of schemes. Let Y' → Y be a morphism of schemes, and let f' : X' = X_Y' → Y' be the base change of f. If Y' → Y is surjective and f' has universally bounded fibres, then f has universally bounded fibres. More precisely, if n bounds the degree of the fibres of f', then also n bounds the degrees of the fibres of f.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $Y' \\to Y$ be a morphism of schemes, and let\n$f' : X' = X_{Y'} \\to Y'$ be the base change of $f$.\nIf $Y' \\to Y$ is surjective and $f'$ has universally bounded fibres,\nthen $f$ has universally bounded fibres. More precisely, if $n$ bounds\nthe degree of the fibres of $f'$, then also $n$ bounds the degrees\nof the fibres of $f$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03J8","source_file":"morphisms.tex","source_line":15435,"source_end_line":15444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15435-L15444","statement_sha256":"c68bfcac23922ff1ae426a2419d81cec80f297035c70d25a9c13e77e8b603c6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5830,"rank":5830,"depth":0,"x":1992.095,"y":597.761,"cluster":"scheme-morphisms"},{"id":"stacks:03J9","tag":"03J9","title":"Universally bounded fibres · Lemma 03J9","summary":"An immersion has universally bounded fibres.","statement_latex":"An immersion has universally bounded fibres.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03J9","source_file":"morphisms.tex","source_line":15458,"source_end_line":15461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15458-L15461","statement_sha256":"6a129549ac3044c541b120314a5894dd40ad2fe57171f42f715482f3e03c951f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5831,"rank":5831,"depth":0,"x":2124.178,"y":719.173,"cluster":"scheme-morphisms"},{"id":"stacks:03WU","tag":"03WU","title":"Universally bounded fibres · Lemma 03WU","summary":"Let f : X → Y be an étale morphism of schemes. Let n ≥ 0. The following are equivalent • the integer n bounds the degrees of the fibres, • for every field k and morphism Spec(k) → Y the base change X_k = Spec(k) ×_Y X has at most n points, and • for every y ∈ Y and every separable algebraic closure kappa(y) ⊂ kappa(y)^sep the scheme X_kappa(y)^sep has at most n points.","statement_latex":"Let $f : X \\to Y$ be an \\'etale morphism of schemes.\nLet $n \\geq 0$. The following are equivalent\n\\begin{enumerate}\n\\item the integer $n$ bounds the degrees of the fibres,\n\\item for every field $k$ and morphism $\\Spec(k) \\to Y$ the\nbase change $X_k = \\Spec(k) \\times_Y X$ has at most $n$ points, and\n\\item for every $y \\in Y$ and every separable algebraic closure\n$\\kappa(y) \\subset \\kappa(y)^{sep}$ the scheme\n$X_{\\kappa(y)^{sep}}$ has at most $n$ points.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WU","source_file":"morphisms.tex","source_line":15467,"source_end_line":15479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15467-L15479","statement_sha256":"8b1357bd7fc83b3cc465dbd2b5564123b96050b89617001ce2ae8c4be915a749","origin":"The Stacks Project","memory_eligible":false,"source_rank":5832,"rank":5832,"depth":45,"x":1928.954,"y":704.578,"cluster":"scheme-morphisms"},{"id":"stacks:03JA","tag":"03JA","title":"Universally bounded fibres · Lemma 03JA","summary":"Let f : X → Y be a morphism of schemes. Assume that • f is locally quasi-finite, and • X is quasi-compact. Then f has universally bounded fibres.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite, and\n\\item $X$ is quasi-compact.\n\\end{enumerate}\nThen $f$ has universally bounded fibres.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JA","source_file":"morphisms.tex","source_line":15494,"source_end_line":15503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15494-L15503","statement_sha256":"d2aedc42cfae1915f459b6d63962bd8c3f6bda02c49e871380a3dc9b039dca12","origin":"The Stacks Project","memory_eligible":false,"source_rank":5833,"rank":5833,"depth":30,"x":2084.799,"y":604.466,"cluster":"scheme-morphisms"},{"id":"stacks:03JB","tag":"03JB","title":"Universally bounded fibres · Lemma 03JB","summary":"Consider a commutative diagram of morphisms of schemes xymatrix X ar[rd]_g ar[rr]_f & & Y ar[ld]^h & Z & If g has universally bounded fibres, and f is surjective and flat, then also h has universally bounded fibres. More precisely, if n bounds the degree of the fibres of g, then also n bounds the degree of the fibres of h.","statement_latex":"Consider a commutative diagram of morphisms of schemes\n$$\n\\xymatrix{\nX \\ar[rd]_g \\ar[rr]_f & & Y \\ar[ld]^h \\\\\n& Z &\n}\n$$\nIf $g$ has universally bounded fibres, and $f$ is surjective and flat,\nthen also $h$ has universally bounded fibres. More precisely, if $n$\nbounds the degree of the fibres of $g$, then also $n$ bounds the\ndegree of the fibres of $h$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JB","source_file":"morphisms.tex","source_line":15553,"source_end_line":15566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15553-L15566","statement_sha256":"a448516edac8c8fd5d63a507fa0bc13ca98b6dea8bbd5cabd44feff5da618a5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5834,"rank":5834,"depth":13,"x":2050.354,"y":766.877,"cluster":"scheme-morphisms"},{"id":"stacks:0H1M","tag":"0H1M","title":"Miscellany · Lemma 0H1M","summary":"Let f : Y → X be a morphism of schemes. Let x ∈ X be a point. Assume that Y is reduced and f(Y) is set-theoretically contained in (x). Then f factors through the canonical morphism x = Spec(kappa(x)) → X.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes. Let $x \\in X$ be a point.\nAssume that $Y$ is reduced and $f(Y)$ is set-theoretically contained\nin $\\{x\\}$. Then $f$ factors through the canonical morphism\n$x = \\Spec(\\kappa(x)) \\to X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1M","source_file":"morphisms.tex","source_line":15600,"source_end_line":15606,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15600-L15606","statement_sha256":"ad8263a6a26206cbfeab78008296766809fbfa832a70f0b7c3ac04af9cef08c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5835,"rank":5835,"depth":0,"x":1945.044,"y":627.436,"cluster":"scheme-morphisms"},{"id":"stacks:0H1N","tag":"0H1N","title":"Miscellany · Lemma 0H1N","summary":"Let f : Y → X be a morphism of schemes. Let E ⊂ X. Assume X is locally Noetherian, there are no nontrivial specializations among the elements of E, Y is reduced, and f(Y) ⊂ E. Then f factors through coprod_x ∈ E x → X.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes. Let $E \\subset X$.\nAssume $X$ is locally Noetherian, there are no nontrivial specializations\namong the elements of $E$, $Y$ is reduced, and $f(Y) \\subset E$.\nThen $f$ factors through $\\coprod_{x \\in E} x \\to X$.","area":"Scheme Morphisms","chapter":"Morphisms of Schemes","chapter_id":"morphisms","section":"Miscellany","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1N","source_file":"morphisms.tex","source_line":15616,"source_end_line":15622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/morphisms.tex#L15616-L15622","statement_sha256":"4dc9de887eb5e8650b525923572f20d3ca2408802009ac038289e72dcfa3a1ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":5836,"rank":5836,"depth":4,"x":2135.019,"y":670.544,"cluster":"scheme-morphisms"},{"id":"stacks:01X9","tag":"01X9","title":"v Cech cohomology of quasi-coherent sheaves · Lemma 01X9","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Let U : U = ⋃_i = 1^n D(f_i) be a standard open covering of an affine open of X. Then checkH^p(U, F) = 0 for all p > 0.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{U} : U = \\bigcup_{i = 1}^n D(f_i)$ be a standard\nopen covering of an affine open of $X$.\nThen $\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = 0$ for\nall $p > 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"v Cech cohomology of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01X9","source_file":"coherent.tex","source_line":44,"source_end_line":52,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L44-L52","statement_sha256":"0ece078d21274c501aa8b8e4ac1066785dd257b976d7e18b0021b0bfed20105e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5837,"rank":5837,"depth":3,"x":2035.291,"y":1140.0,"cluster":"duality-cohomology"},{"id":"stacks:01XB","tag":"01XB","title":"v Cech cohomology of quasi-coherent sheaves · Lemma 01XB","summary":"Serre vanishing: Higher cohomology vanishes on affine schemes for quasi-coherent modules. Let X be a scheme. Let F be a quasi-coherent O_X-module. For any affine open U ⊂ X we have H^p(U, F) = 0 for all p > 0.","statement_latex":"\\begin{slogan}\nSerre vanishing: Higher cohomology vanishes on affine schemes\nfor quasi-coherent modules.\n\\end{slogan}\nLet $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nFor any affine open $U \\subset X$ we have\n$H^p(U, \\mathcal{F}) = 0$ for all $p > 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"v Cech cohomology of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XB","source_file":"coherent.tex","source_line":145,"source_end_line":155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L145-L155","statement_sha256":"c588c043cfc7a2aa9823f834276c91136df205f2d7e6bead9bed80858a3fd81f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5838,"rank":5838,"depth":23,"x":2023.243,"y":1145.2,"cluster":"duality-cohomology"},{"id":"stacks:01XC","tag":"01XC","title":"v Cech cohomology of quasi-coherent sheaves · Lemma 01XC","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. If f is affine then R^if_*F = 0 for all i > 0.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $f$ is affine then $R^if_*\\mathcal{F} = 0$ for all $i > 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"v Cech cohomology of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XC","source_file":"coherent.tex","source_line":180,"source_end_line":185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L180-L185","statement_sha256":"7f4768fa5f915bb72d2b81e6cef70f6131ec0fa3e2d59838e27d0f9052c1677c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5839,"rank":5839,"depth":24,"x":2031.034,"y":1130.1,"cluster":"duality-cohomology"},{"id":"stacks:089W","tag":"089W","title":"v Cech cohomology of quasi-coherent sheaves · Lemma 089W","summary":"Let f : X → S be an affine morphism of schemes. Let F be a quasi-coherent O_X-module. Then H^i(X, F) = H^i(S, f_*F) for all i ≥ 0.","statement_latex":"Let $f : X \\to S$ be an affine morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $H^i(X, \\mathcal{F}) = H^i(S, f_*\\mathcal{F})$ for all $i \\geq 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"v Cech cohomology of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089W","source_file":"coherent.tex","source_line":201,"source_end_line":206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L201-L206","statement_sha256":"67375e766f2f73bb735413a23153f4f3a8b3ad734da57818e3e2063441981cf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5840,"rank":5840,"depth":25,"x":2038.517,"y":1149.332,"cluster":"duality-cohomology"},{"id":"stacks:0BDX","tag":"0BDX","title":"v Cech cohomology of quasi-coherent sheaves · Lemma 0BDX","summary":"Let X be a scheme. The following are equivalent • X has affine diagonal Δ : X → X × X, • for U, V ⊂ X affine open, the intersection U ∩ V is affine, and • there exists an open covering U : X = ⋃_i ∈ I U_i such that U_i_0 … i_p is affine open for all p ge 0 and all i_0, …, i_p ∈ I. In particular this holds if X is separated.","statement_latex":"Let $X$ be a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $X$ has affine diagonal $\\Delta : X \\to X \\times X$,\n\\item for $U, V \\subset X$ affine open, the intersection\n$U \\cap V$ is affine, and\n\\item there exists an open covering $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$\nsuch that $U_{i_0 \\ldots i_p}$ is affine open for all $p \\ge 0$ and all\n$i_0, \\ldots, i_p \\in I$.\n\\end{enumerate}\nIn particular this holds if $X$ is separated.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"v Cech cohomology of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDX","source_file":"coherent.tex","source_line":218,"source_end_line":230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L218-L230","statement_sha256":"27e5b69d69e81731672e5eee965b01b5837c54591a6167f94a668c6a57f9f856","origin":"The Stacks Project","memory_eligible":false,"source_rank":5841,"rank":5841,"depth":18,"x":2014.37,"y":1137.678,"cluster":"duality-cohomology"},{"id":"stacks:01XD","tag":"01XD","title":"v Cech cohomology of quasi-coherent sheaves · Lemma 01XD","summary":"Let X be a scheme. Let U : X = ⋃_i ∈ I U_i be an open covering such that U_i_0 … i_p is affine open for all p ge 0 and all i_0, …, i_p ∈ I. In this case for any quasi-coherent sheaf F we have checkH^p(U, F) = H^p(X, F) as Γ(X, O_X)-modules for all p.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be an open covering such that\n$U_{i_0 \\ldots i_p}$ is affine open for all $p \\ge 0$ and all\n$i_0, \\ldots, i_p \\in I$.\nIn this case for any quasi-coherent sheaf $\\mathcal{F}$ we have\n$$\n\\check{H}^p(\\mathcal{U}, \\mathcal{F}) = H^p(X, \\mathcal{F})\n$$\nas $\\Gamma(X, \\mathcal{O}_X)$-modules for all $p$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"v Cech cohomology of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XD","source_file":"coherent.tex","source_line":245,"source_end_line":256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L245-L256","statement_sha256":"59965935d7545bc5cd20c45da24e9dcc0448cbb6b5c4cd1ccd87e90dbcb81040","origin":"The Stacks Project","memory_eligible":false,"source_rank":5842,"rank":5842,"depth":24,"x":2044.806,"y":1132.088,"cluster":"duality-cohomology"},{"id":"stacks:01XF","tag":"01XF","title":"Vanishing of cohomology · Lemma 01XF","summary":"[Serre-criterion], [EGA] Serre's criterion for affineness. Let X be a scheme. Assume that • X is quasi-compact, • for every quasi-coherent sheaf of ideals I ⊂ O_X we have H^1(X, I) = 0. Then X is affine.","statement_latex":"\\begin{reference}\n\\cite{Serre-criterion}, \\cite[II, Theorem 5.2.1 (d') and IV (1.7.17)]{EGA}\n\\end{reference}\n\\begin{slogan}\nSerre's criterion for affineness.\n\\end{slogan}\nLet $X$ be a scheme.\nAssume that\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item for every quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_X$ we have $H^1(X, \\mathcal{I}) = 0$.\n\\end{enumerate}\nThen $X$ is affine.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XF","source_file":"coherent.tex","source_line":282,"source_end_line":298,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L282-L298","statement_sha256":"7383fa8c0eab774ce0dc278ebf759a26d3e36c1de7b04b721dd2b4baffcc88f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5843,"rank":5843,"depth":22,"x":2025.048,"y":1155.475,"cluster":"duality-cohomology"},{"id":"stacks:01XG","tag":"01XG","title":"Vanishing of cohomology · Lemma 01XG","summary":"[Serre-criterion], [EGA] Serre's criterion for affineness. Let X be a scheme. Assume that • X is quasi-compact, • X is quasi-separated, and • H^1(X, I) = 0 for every quasi-coherent sheaf of ideals I of finite type. Then X is affine.","statement_latex":"\\begin{reference}\n\\cite{Serre-criterion}, \\cite[II, Theorem 5.2.1]{EGA}\n\\end{reference}\n\\begin{slogan}\nSerre's criterion for affineness.\n\\end{slogan}\nLet $X$ be a scheme. Assume that\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item $X$ is quasi-separated, and\n\\item $H^1(X, \\mathcal{I}) = 0$ for every quasi-coherent sheaf\nof ideals $\\mathcal{I}$ of finite type.\n\\end{enumerate}\nThen $X$ is affine.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XG","source_file":"coherent.tex","source_line":390,"source_end_line":406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L390-L406","statement_sha256":"fb1dd570241bed3f4f88d8309cecbcd58f66d80e0f7dded1782761df2d46f365","origin":"The Stacks Project","memory_eligible":false,"source_rank":5844,"rank":5844,"depth":24,"x":2020.555,"y":1124.724,"cluster":"duality-cohomology"},{"id":"stacks:0B5P","tag":"0B5P","title":"Vanishing of cohomology · Lemma 0B5P","summary":"Let X be a scheme. Let L be an invertible O_X-module. Assume that • X is quasi-compact, • for every quasi-coherent sheaf of ideals I ⊂ O_X there exists an n ≥ 1 such that H^1(X, I ⊗_O_X L^⊗ n) = 0. Then L is ample.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume that\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item for every quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_X$\nthere exists an $n \\geq 1$ such that\n$H^1(X, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$.\n\\end{enumerate}\nThen $\\mathcal{L}$ is ample.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5P","source_file":"coherent.tex","source_line":425,"source_end_line":437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L425-L437","statement_sha256":"50edbb3f4f1bdaaa40605ec4ed7d37356c01eadc2d22b70ab591ec4e4e35f080","origin":"The Stacks Project","memory_eligible":false,"source_rank":5845,"rank":5845,"depth":23,"x":2050.491,"y":1146.286,"cluster":"duality-cohomology"},{"id":"stacks:0F83","tag":"0F83","title":"Vanishing of cohomology · Lemma 0F83","summary":"Let f : X → Y be a quasi-compact morphism with X and Y quasi-separated. If R^1f_*I = 0 for every quasi-coherent sheaf of ideals I on X, then f is affine.","statement_latex":"Let $f : X \\to Y$ be a quasi-compact morphism with $X$ and $Y$ quasi-separated.\nIf $R^1f_*\\mathcal{I} = 0$ for every quasi-coherent sheaf of ideals\n$\\mathcal{I}$ on $X$, then $f$ is affine.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F83","source_file":"coherent.tex","source_line":501,"source_end_line":506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L501-L506","statement_sha256":"c3701093389fc6f7041b168ee8205bb588acfbaafeb37994f1ea299b7d66a96d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5846,"rank":5846,"depth":25,"x":2008.682,"y":1147.392,"cluster":"duality-cohomology"},{"id":"stacks:08DR","tag":"08DR","title":"Induction Principle · Lemma 08DR","summary":"[BvdB] Let X be a quasi-compact and quasi-separated scheme. Let P be a property of the quasi-compact opens of X. Assume that • P holds for every affine open of X, • if U is quasi-compact open, V affine open, P holds for U, V, and U ∩ V, then P holds for U ∪ V. Then P holds for every quasi-compact open of X and in particular for X.","statement_latex":"\\begin{reference}\n\\cite[Proposition 3.3.1]{BvdB}\n\\end{reference}\nLet $X$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property\nof the quasi-compact opens of $X$. Assume that\n\\begin{enumerate}\n\\item $P$ holds for every affine open of $X$,\n\\item if $U$ is quasi-compact open, $V$ affine open,\n$P$ holds for $U$, $V$, and $U \\cap V$, then\n$P$ holds for $U \\cup V$.\n\\end{enumerate}\nThen $P$ holds for every quasi-compact open of $X$\nand in particular for $X$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Quasi-coherence of higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DR","source_file":"coherent.tex","source_line":556,"source_end_line":571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L556-L571","statement_sha256":"cb826e845d864bc8f2c196c6528d0ed60d32975511d0adf08388d8746dda5b0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5847,"rank":5847,"depth":13,"x":2040.277,"y":1121.553,"cluster":"duality-cohomology"},{"id":"stacks:01XI","tag":"01XI","title":"Quasi-coherence of higher direct images · Lemma 01XI","summary":"For schemes with affine diagonal, the cohomology of quasi-coherent modules vanishes in degrees bigger than the number of affine opens needed in a covering. Let X be a quasi-compact scheme with affine diagonal (for example if X is separated). Let t = t(X) be the minimal number of affine opens needed to cover X. Then H^n(X, F) = 0 for all n ≥ t and all quasi-coherent sheaves F.","statement_latex":"\\begin{slogan}\nFor schemes with affine diagonal, the cohomology of quasi-coherent\nmodules vanishes in degrees bigger than the number of affine\nopens needed in a covering.\n\\end{slogan}\nLet $X$ be a quasi-compact scheme with affine diagonal (for example\nif $X$ is separated).\nLet $t = t(X)$ be the minimal number of affine opens needed to\ncover $X$. Then $H^n(X, \\mathcal{F}) = 0$ for all $n \\geq t$ and all\nquasi-coherent sheaves $\\mathcal{F}$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Quasi-coherence of higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XI","source_file":"coherent.tex","source_line":589,"source_end_line":601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L589-L601","statement_sha256":"c55a4949beed59c5b5b2542faef792f6471a756f55f59ab06f00b9b403c90041","origin":"The Stacks Project","memory_eligible":false,"source_rank":5848,"rank":5848,"depth":25,"x":2037.594,"y":1160.337,"cluster":"duality-cohomology"},{"id":"stacks:0BDY","tag":"0BDY","title":"Quasi-coherence of higher direct images · Lemma 0BDY","summary":"Let X be a quasi-compact scheme with affine diagonal (for example if X is separated). Then • given a quasi-coherent O_X-module F there exists an embedding F → F' of quasi-coherent O_X-modules such that H^p(X, F') = 0 for all p ≥ 1, and • (H^n(X, -))_n ≥ 0 is a universal δ-functor from QCoh(O_X) to Ab.","statement_latex":"Let $X$ be a quasi-compact scheme with affine diagonal\n(for example if $X$ is separated). Then\n\\begin{enumerate}\n\\item given a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$\nthere exists an embedding $\\mathcal{F} \\to \\mathcal{F}'$ of\nquasi-coherent $\\mathcal{O}_X$-modules\nsuch that $H^p(X, \\mathcal{F}') = 0$ for all $p \\geq 1$, and\n\\item $\\{H^n(X, -)\\}_{n \\geq 0}$\nis a universal $\\delta$-functor from $\\QCoh(\\mathcal{O}_X)$ to\n$\\textit{Ab}$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Quasi-coherence of higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDY","source_file":"coherent.tex","source_line":646,"source_end_line":659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L646-L659","statement_sha256":"d1916bfd4fceb3999b141253251ca2da0ba4a2750f7b15c150b31c122c1bd16e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5849,"rank":5849,"depth":26,"x":2007.111,"y":1128.858,"cluster":"duality-cohomology"},{"id":"stacks:071L","tag":"071L","title":"Quasi-coherence of higher direct images · Lemma 071L","summary":"Let X be a quasi-compact quasi-separated scheme. Let X = U_1 ∪ … ∪ U_n be an open covering with each U_i quasi-compact and separated (for example affine). Set d = max_I ⊂ (1, …, n) (|I| + t(⋂_i ∈ I U_i) - 1) where t(U) is the minimal number of affines needed to cover the scheme U. Then H^p(X, F) = 0 for all p ≥ d and all quasi-coherent sheaves F.","statement_latex":"Let $X$ be a quasi-compact quasi-separated scheme.\nLet $X = U_1 \\cup \\ldots \\cup U_n$ be an open covering\nwith each $U_i$ quasi-compact and separated (for example affine).\nSet\n$$\nd = \\max\\nolimits_{I \\subset \\{1, \\ldots, n\\}}\n\\left(|I| + t(\\bigcap\\nolimits_{i \\in I} U_i) - 1\\right)\n$$\nwhere $t(U)$ is the minimal number of affines needed to cover\nthe scheme $U$. Then $H^p(X, \\mathcal{F}) = 0$ for all $p \\geq d$ and all\nquasi-coherent sheaves $\\mathcal{F}$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Quasi-coherence of higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/071L","source_file":"coherent.tex","source_line":690,"source_end_line":703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L690-L703","statement_sha256":"98880d04148145723d2150722ec3249297a455fcf6652353ed2494868b6b76ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":5850,"rank":5850,"depth":26,"x":2056.851,"y":1135.041,"cluster":"duality-cohomology"},{"id":"stacks:01XJ","tag":"01XJ","title":"Quasi-coherence of higher direct images · Lemma 01XJ","summary":"Let f : X → S be a morphism of schemes. Assume that f is quasi-separated and quasi-compact. • For any quasi-coherent O_X-module F the higher direct images R^pf_*F are quasi-coherent on S. • If S is quasi-compact, there exists an integer n = n(X, S, f) such that R^pf_*F = 0 for all p ≥ n and any quasi-coherent sheaf F on X. • In fact, if S is quasi-compact we can find n = n(X, S, f) such that for every morphism of schemes S' → S we have R^p(f')_*F' = 0 for p ≥ n and any…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume that $f$ is quasi-separated and quasi-compact.\n\\begin{enumerate}\n\\item For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the\nhigher direct images $R^pf_*\\mathcal{F}$ are quasi-coherent on $S$.\n\\item If $S$ is quasi-compact, there exists an integer $n = n(X, S, f)$\nsuch that $R^pf_*\\mathcal{F} = 0$ for all $p \\geq n$ and any\nquasi-coherent sheaf $\\mathcal{F}$ on $X$.\n\\item In fact, if $S$ is quasi-compact we can find $n = n(X, S, f)$\nsuch that for every\nmorphism of schemes $S' \\to S$ we have $R^p(f')_*\\mathcal{F}' = 0$\nfor $p \\geq n$ and any quasi-coherent sheaf $\\mathcal{F}'$\non $X'$. Here $f' : X' = S' \\times_S X \\to S'$ is the base change of $f$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Quasi-coherence of higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XJ","source_file":"coherent.tex","source_line":741,"source_end_line":757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L741-L757","statement_sha256":"87c79e446874d4025cee7e546a4c76f04f74e55714e39f7e4bd1fc5c78e1fbf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5851,"rank":5851,"depth":27,"x":2013.613,"y":1159.579,"cluster":"duality-cohomology"},{"id":"stacks:01XK","tag":"01XK","title":"Quasi-coherence of higher direct images · Lemma 01XK","summary":"Let f : X → S be a morphism of schemes. Assume that f is quasi-separated and quasi-compact. Assume S is affine. For any quasi-coherent O_X-module F we have H^q(X, F) = H^0(S, R^qf_*F) for all q ∈ Z.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume that $f$ is quasi-separated and quasi-compact.\nAssume $S$ is affine.\nFor any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$\nwe have\n$$\nH^q(X, \\mathcal{F}) = H^0(S, R^qf_*\\mathcal{F})\n$$\nfor all $q \\in \\mathbf{Z}$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Quasi-coherence of higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XK","source_file":"coherent.tex","source_line":843,"source_end_line":854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L843-L854","statement_sha256":"b350599513db81b6c6671c8243102ea33c381202eb4b18ec18941fafb94ee0e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5852,"rank":5852,"depth":28,"x":2026.214,"y":1115.46,"cluster":"duality-cohomology"},{"id":"stacks:02KG","tag":"02KG","title":"Cohomology and base change, I · Lemma 02KG","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. Assume f is affine. In this case f_*F ≅ Rf_*F is a quasi-coherent sheaf, and for every base change diagram ([Tag 02KF]) we have g^*f_*F = f'_*(g')^*F.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $f$ is affine.\nIn this case $f_*\\mathcal{F} \\cong Rf_*\\mathcal{F}$ is\na quasi-coherent sheaf, and for every base change diagram\n(\\ref{equation-base-change-diagram})\nwe have\n$$\ng^*f_*\\mathcal{F} = f'_*(g')^*\\mathcal{F}.\n$$","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KG","source_file":"coherent.tex","source_line":906,"source_end_line":918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L906-L918","statement_sha256":"377c807ceb0e4c6203c6d202d93cd6f8fa1c78d1ab6cf621c23add7d541f3942","origin":"The Stacks Project","memory_eligible":false,"source_rank":5853,"rank":5853,"depth":25,"x":2053.241,"y":1156.453,"cluster":"duality-cohomology"},{"id":"stacks:02KH","tag":"02KH","title":"Flat base change · Lemma 02KH","summary":"Consider a cartesian diagram of schemes xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f S' ar[r]^g & S Let F be a quasi-coherent O_X-module with pullback F' = (g')^*F. Assume that g is flat and that f is quasi-compact and quasi-separated. For any i ≥ 0 • the base change map of Cohomology, Lemma [Tag 02N7] is an isomorphism g^*R^if_*F → R^if'_*F', • if S = Spec(A) and S' = Spec(B), then H^i(X, F) ⊗_A B = H^i(X', F').","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nwith pullback $\\mathcal{F}' = (g')^*\\mathcal{F}$.\nAssume that $g$ is flat and that $f$ is quasi-compact and quasi-separated.\nFor any $i \\geq 0$\n\\begin{enumerate}\n\\item the base change map of\nCohomology, Lemma \\ref{cohomology-lemma-base-change-map-flat-case}\nis an isomorphism\n$$\ng^*R^if_*\\mathcal{F} \\longrightarrow R^if'_*\\mathcal{F}',\n$$\n\\item if $S = \\Spec(A)$ and $S' = \\Spec(B)$, then\n$H^i(X, \\mathcal{F}) \\otimes_A B = H^i(X', \\mathcal{F}')$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KH","source_file":"coherent.tex","source_line":947,"source_end_line":970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L947-L970","statement_sha256":"698f3b7f505019617e03dbc0c85953155d36bd985a963ac5360a37c4d9bbcda6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5854,"rank":5854,"depth":29,"x":1998.725,"y":1141.086,"cluster":"duality-cohomology"},{"id":"stacks:0CKW","tag":"0CKW","title":"Finite locally free base change · Lemma 0CKW","summary":"Consider a cartesian diagram of schemes xymatrix Y ar[d]_g ar[r]_h & X ar[d]^f Spec(B) ar[r] & Spec(A) Let F be a quasi-coherent O_X-module with pullback G = h^*F. If B is a finite locally free A-module, then H^i(X, F) ⊗_A B = H^i(Y, G).","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nY \\ar[d]_{g} \\ar[r]_h & X \\ar[d]^f \\\\\n\\Spec(B) \\ar[r] & \\Spec(A)\n}\n$$\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nwith pullback $\\mathcal{G} = h^*\\mathcal{F}$.\nIf $B$ is a finite locally free $A$-module, then\n$H^i(X, \\mathcal{F}) \\otimes_A B = H^i(Y, \\mathcal{G})$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKW","source_file":"coherent.tex","source_line":1052,"source_end_line":1065,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L1052-L1065","statement_sha256":"975c041f42b1883abbe7a6406f3e77edf975b9ed902b93086a2c959498ac09d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5855,"rank":5855,"depth":0,"x":2052.812,"y":1120.931,"cluster":"duality-cohomology"},{"id":"stacks:07TB","tag":"07TB","title":"Colimits and higher direct images · Lemma 07TB","summary":"Let f : X → S be a quasi-compact and quasi-separated morphism of schemes. Let F = colim F_i be a filtered colimit of abelian sheaves on X. Then for any p ≥ 0 we have R^pf_*F = colim R^pf_*F_i.","statement_latex":"Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $\\mathcal{F} = \\colim \\mathcal{F}_i$ be a filtered colimit\nof abelian sheaves on $X$. Then for any $p \\geq 0$ we have\n$$\nR^pf_*\\mathcal{F} = \\colim R^pf_*\\mathcal{F}_i.\n$$","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Colimits and higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TB","source_file":"coherent.tex","source_line":1145,"source_end_line":1153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L1145-L1153","statement_sha256":"556efb2a86a9c3bf0f0162fc0ae32d491f40d2cbd6bcfd3b178897383bd61c9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5856,"rank":5856,"depth":24,"x":2028.474,"y":1167.725,"cluster":"duality-cohomology"},{"id":"stacks:01XL","tag":"01XL","title":"Cohomology and base change, II · Lemma 01XL","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. Assume X is quasi-compact and X and S have affine diagonal (e.g., if X and S are separated). In this case we can compute Rf_*F as follows: • Choose a finite affine open covering U : X = ⋃_i = 1, …, n U_i. • For i_0, …, i_p ∈ (1, …, n) denote f_i_0 … i_p : U_i_0 … i_p → S the restriction of f to the intersection U_i_0 … i_p = U_i_0 ∩ … ∩ U_i_p. • Set F_i_0 … i_p equal to the restriction of F to…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $X$ is quasi-compact and $X$ and $S$ have affine diagonal\n(e.g., if $X$ and $S$ are separated).\nIn this case we can compute $Rf_*\\mathcal{F}$ as follows:\n\\begin{enumerate}\n\\item Choose a finite affine open covering\n$\\mathcal{U} : X = \\bigcup_{i = 1, \\ldots, n} U_i$.\n\\item For $i_0, \\ldots, i_p \\in \\{1, \\ldots, n\\}$ denote\n$f_{i_0 \\ldots i_p} : U_{i_0 \\ldots i_p} \\to S$ the restriction of $f$\nto the intersection $U_{i_0 \\ldots i_p} = U_{i_0} \\cap \\ldots \\cap U_{i_p}$.\n\\item Set $\\mathcal{F}_{i_0 \\ldots i_p}$ equal to the restriction\nof $\\mathcal{F}$ to $U_{i_0 \\ldots i_p}$.\n\\item Set\n$$\n\\check{\\mathcal{C}}^p(\\mathcal{U}, f, \\mathcal{F}) =\n\\bigoplus\\nolimits_{i_0 \\ldots i_p}\nf_{i_0 \\ldots i_p *} \\mathcal{F}_{i_0 \\ldots i_p}\n$$\nand define differentials\n$d : \\check{\\mathcal{C}}^p(\\mathcal{U}, f, \\mathcal{F})\n\\to \\check{\\mathcal{C}}^{p + 1}(\\mathcal{U}, f, \\mathcal{F})$\nas in Cohomology, Equation (\\ref{cohomology-equation-d-cech}).\n\\end{enumerate}\nThen the complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})$\nis a complex of quasi-coherent sheaves on $S$ which comes equipped with an\nisomorphism\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})\n\\longrightarrow\nRf_*\\mathcal{F}\n$$\nin $D^{+}(S)$. This isomorphism is functorial in the quasi-coherent\nsheaf $\\mathcal{F}$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology and base change, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XL","source_file":"coherent.tex","source_line":1199,"source_end_line":1235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L1199-L1235","statement_sha256":"7cbf51abe52f5aaf14c617da1f343e58ba45a85628141d5086f21656286acb52","origin":"The Stacks Project","memory_eligible":false,"source_rank":5857,"rank":5857,"depth":25,"x":2008.294,"y":1118.151,"cluster":"duality-cohomology"},{"id":"stacks:01XM","tag":"01XM","title":"Cohomology and base change, II · Lemma 01XM","summary":"With notation as in diagram ([Tag 02KF]). Assume f : X → S and F satisfy the hypotheses of Lemma [Tag 01XL]. Choose a finite affine open covering U : X = ⋃ U_i of X. There is a canonical isomorphism g^*checkC^bullet(U, f, F) → Rf'_*F' in D^+(S'). Moreover, if S' → S is affine, then in fact g^*checkC^bullet(U, f, F) = checkC^bullet(U', f', F') with U' : X' = ⋃ U_i' where U_i' = (g')^-1(U_i) = U_i, S' is also affine.","statement_latex":"With notation as in diagram (\\ref{equation-base-change-diagram}).\nAssume $f : X \\to S$ and $\\mathcal{F}$ satisfy the hypotheses of\nLemma \\ref{lemma-separated-case-relative-cech}. Choose a finite\naffine open covering $\\mathcal{U} : X = \\bigcup U_i$ of $X$.\nThere is a canonical isomorphism\n$$\ng^*\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})\n\\longrightarrow\nRf'_*\\mathcal{F}'\n$$\nin $D^{+}(S')$. Moreover, if $S' \\to S$ is affine, then in fact\n$$\ng^*\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, f, \\mathcal{F})\n=\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}', f', \\mathcal{F}')\n$$\nwith $\\mathcal{U}' : X' = \\bigcup U_i'$ where\n$U_i' = (g')^{-1}(U_i) = U_{i, S'}$ is also affine.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology and base change, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XM","source_file":"coherent.tex","source_line":1274,"source_end_line":1294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L1274-L1294","statement_sha256":"98bc09e0f8f9dd83a0937848ffbe83dfcb0dc44e1d3f02a4fd97d96bfdd578f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5858,"rank":5858,"depth":26,"x":2064.385,"y":1143.886,"cluster":"duality-cohomology"},{"id":"stacks:01XN","tag":"01XN","title":"Cohomology and base change, II · Lemma 01XN","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Assume that f is quasi-compact and quasi-separated and that S is quasi-compact and separated. There exists a bounded below complex K^bullet of quasi-coherent O_S-modules with the following property: For every morphism g : S' → S the complex g^*K^bullet is a representative for Rf'_*F' with notation as in diagram ([Tag 02KF]).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nAssume that $f$ is quasi-compact and quasi-separated and\nthat $S$ is quasi-compact and separated.\nThere exists a bounded below complex $\\mathcal{K}^\\bullet$\nof quasi-coherent $\\mathcal{O}_S$-modules with the\nfollowing property: For every morphism\n$g : S' \\to S$ the complex $g^*\\mathcal{K}^\\bullet$ is\na representative for $Rf'_*\\mathcal{F}'$ with notation as in\ndiagram (\\ref{equation-base-change-diagram}).","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology and base change, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XN","source_file":"coherent.tex","source_line":1348,"source_end_line":1360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L1348-L1360","statement_sha256":"94f5b7a6191815068945bfe38cc744f7120f8017046c3777d1cdbd845582ebc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":5859,"rank":5859,"depth":27,"x":2000.866,"y":1157.028,"cluster":"duality-cohomology"},{"id":"stacks:0GN5","tag":"0GN5","title":"Cohomology and base change, II · Lemma 0GN5","summary":"Consider a cartesian diagram of schemes xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f S' ar[r]^g & S Let F be a quasi-coherent O_X-module. Let G be a quasi-coherent O_S'-module flat over S. Assume f is quasi-compact and quasi-separated. For any i ≥ 0 there is an identification G ⊗_O_S' g^*R^if_*F = R^if'_*((f')^*G ⊗_O_X' (g')^*F)","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_{S'}$-module\nflat over $S$. Assume $f$ is quasi-compact and quasi-separated.\nFor any $i \\geq 0$ there is an identification\n$$\n\\mathcal{G} \\otimes_{\\mathcal{O}_{S'}} g^*R^if_*\\mathcal{F} =\nR^if'_*\\left((f')^*\\mathcal{G}\n\\otimes_{\\mathcal{O}_{X'}} (g')^*\\mathcal{F}\\right)\n$$","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology and base change, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GN5","source_file":"coherent.tex","source_line":1431,"source_end_line":1449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L1431-L1449","statement_sha256":"91b747939c10e3349a04cf12e0a7ad33a937fac07724979d8d685f0bf97150b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5860,"rank":5860,"depth":14,"x":2037.961,"y":1110.274,"cluster":"duality-cohomology"},{"id":"stacks:01XT","tag":"01XT","title":"Cohomology of projective space · Lemma 01XT","summary":"[EGA] Let R be a ring. Let n ≥ 0 be an integer. We have H^q(P^n, O_P^n_R(d)) = ( (R[T_0, …, T_n])_d & if & q = 0, d ≥ 0 (frac1T_0 … T_n R[frac1T_0, …, frac1T_n])_d & if & q = n, d < 0 0 & if & q, n, d not as above . as R-modules.","statement_latex":"\\begin{reference}\n\\cite[III Proposition 2.1.12]{EGA}\n\\end{reference}\nLet $R$ be a ring.\nLet $n \\geq 0$ be an integer.\nWe have\n$$\nH^q(\\mathbf{P}^n, \\mathcal{O}_{\\mathbf{P}^n_R}(d)) =\n\\left\\{\n\\begin{matrix}\n(R[T_0, \\ldots, T_n])_d & \\text{if} & q = 0,\\ d \\geq 0 \\\\\n\\left(\\frac{1}{T_0 \\ldots T_n} R[\\frac{1}{T_0}, \\ldots, \\frac{1}{T_n}]\\right)_d\n& \\text{if} & q = n,\\ d < 0 \\\\\n0 & \\text{if} & q, n, d\\text{ not as above}\n\\end{matrix}\n\\right.\n$$\nas $R$-modules.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology of projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XT","source_file":"coherent.tex","source_line":1602,"source_end_line":1622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L1602-L1622","statement_sha256":"f9cd3f2129b225babbcfb5a4047e222aea397e64da86f2593e92486319d0a126","origin":"The Stacks Project","memory_eligible":false,"source_rank":5861,"rank":5861,"depth":25,"x":2048.414,"y":1166.993,"cluster":"duality-cohomology"},{"id":"stacks:01XV","tag":"01XV","title":"Cohomology of projective space · Lemma 01XV","summary":"The identifications of Equation ([Tag 01XU]) are compatible with base change w.r.t. ring maps R → R'. Moreover, for any f ∈ R[T_0, …, T_n] homogeneous of degree m the map multiplication by f O_P^n_R(d) → O_P^n_R(d + m) induces the map on the cohomology group via the identifications of Equation ([Tag 01XU]) which is multiplication by f for H^0 and the contragredient of multiplication by f (R[T_0, …, T_n])_-n - 1 - (d + m) → (R[T_0, …, T_n])_-n - 1 - d on H^n.","statement_latex":"The identifications of Equation (\\ref{equation-identify}) are\ncompatible with base change w.r.t.\\ ring maps $R \\to R'$.\nMoreover, for any $f \\in R[T_0, \\ldots, T_n]$ homogeneous\nof degree $m$ the map multiplication by $f$\n$$\n\\mathcal{O}_{\\mathbf{P}^n_R}(d)\n\\longrightarrow\n\\mathcal{O}_{\\mathbf{P}^n_R}(d + m)\n$$\ninduces the map on the cohomology group via the identifications\nof Equation (\\ref{equation-identify}) which is multiplication by\n$f$ for $H^0$ and the contragredient of multiplication by $f$\n$$\n(R[T_0, \\ldots, T_n])_{-n - 1 - (d + m)}\n\\longrightarrow\n(R[T_0, \\ldots, T_n])_{-n - 1 - d}\n$$\non $H^n$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology of projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XV","source_file":"coherent.tex","source_line":1884,"source_end_line":1904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L1884-L1904","statement_sha256":"8192f855524f949e71eff60d388def30e9eb5aa1e0141d166d43d485e2d0a7f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5862,"rank":5862,"depth":26,"x":1994.003,"y":1130.353,"cluster":"duality-cohomology"},{"id":"stacks:01XW","tag":"01XW","title":"Cohomology of projective space · Lemma 01XW","summary":"Let S be a scheme. Let n ≥ 0 be an integer. Consider the structure morphism f : P^n_S → S. We have R^qf_*(O_P^n_S(d)) = ( (O_S[T_0, …, T_n])_d & if & q = 0 0 & if & q not = 0, n SheafHom_O_S( (O_S[T_0, …, T_n])_- n - 1 - d, O_S) & if & q = n .","statement_latex":"Let $S$ be a scheme.\nLet $n \\geq 0$ be an integer.\nConsider the structure morphism\n$$\nf : \\mathbf{P}^n_S \\longrightarrow S.\n$$\nWe have\n$$\nR^qf_*(\\mathcal{O}_{\\mathbf{P}^n_S}(d)) =\n\\left\\{\n\\begin{matrix}\n(\\mathcal{O}_S[T_0, \\ldots, T_n])_d & \\text{if} & q = 0 \\\\\n0 & \\text{if} & q \\not = 0, n \\\\\n\\SheafHom_{\\mathcal{O}_S}(\n(\\mathcal{O}_S[T_0, \\ldots, T_n])_{- n - 1 - d}, \\mathcal{O}_S)\n& \\text{if} & q = n\n\\end{matrix}\n\\right.\n$$","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology of projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XW","source_file":"coherent.tex","source_line":2004,"source_end_line":2025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2004-L2025","statement_sha256":"57cea8ceb740ba32456168c0ad8eb8d4bd319214a4a237f5d04768580a5877d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5863,"rank":5863,"depth":27,"x":2064.967,"y":1126.429,"cluster":"duality-cohomology"},{"id":"stacks:01XX","tag":"01XX","title":"Cohomology of projective space · Lemma 01XX","summary":"Let S be a scheme. Let n ≥ 1. Let E be a finite locally free O_S-module of constant rank n + 1. Consider the structure morphism π : P(E) → S. We have R^qπ_*(O_P(E)(d)) = ( Sym^d(E) & if & q = 0 0 & if & q not = 0, n SheafHom_O_S( Sym^- n - 1 - d(E) ⊗_O_S wedge^n + 1E, O_S) & if & q = n . These identifications are compatible with base change and isomorphism between locally free sheaves.","statement_latex":"Let $S$ be a scheme. Let $n \\geq 1$.\nLet $\\mathcal{E}$ be a finite locally\nfree $\\mathcal{O}_S$-module of constant rank $n + 1$.\nConsider the structure morphism\n$$\n\\pi : \\mathbf{P}(\\mathcal{E}) \\longrightarrow S.\n$$\nWe have\n$$\nR^q\\pi_*(\\mathcal{O}_{\\mathbf{P}(\\mathcal{E})}(d)) =\n\\left\\{\n\\begin{matrix}\n\\text{Sym}^d(\\mathcal{E}) & \\text{if} & q = 0 \\\\\n0 & \\text{if} & q \\not = 0, n \\\\\n\\SheafHom_{\\mathcal{O}_S}(\n\\text{Sym}^{- n - 1 - d}(\\mathcal{E})\n\\otimes_{\\mathcal{O}_S}\n\\wedge^{n + 1}\\mathcal{E},\n\\mathcal{O}_S)\n& \\text{if} & q = n\n\\end{matrix}\n\\right.\n$$\nThese identifications are compatible with base change and\nisomorphism between locally free sheaves.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology of projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XX","source_file":"coherent.tex","source_line":2054,"source_end_line":2081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2054-L2081","statement_sha256":"e1b4504c10ef781acb8979733fec686f4dbdecf9971b21036cb7d09db8e283ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":5864,"rank":5864,"depth":28,"x":2014.852,"y":1170.405,"cluster":"duality-cohomology"},{"id":"stacks:01XZ","tag":"01XZ","title":"Coherent sheaves on locally Noetherian schemes · Lemma 01XZ","summary":"Let X be a locally Noetherian scheme. Let F be an O_X-module. The following are equivalent • F is coherent, • F is a quasi-coherent, finite type O_X-module, • F is a finitely presented O_X-module, • for any affine open Spec(A) = U ⊂ X we have F|_U = widetilde M with M a finite A-module, and • there exists an affine open covering X = ⋃ U_i, U_i = Spec(A_i) such that each F|_U_i = widetilde M_i with M_i a finite A_i-module. In particular O_X is coherent, any invertible…","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is coherent,\n\\item $\\mathcal{F}$ is a quasi-coherent, finite type $\\mathcal{O}_X$-module,\n\\item $\\mathcal{F}$ is a finitely presented $\\mathcal{O}_X$-module,\n\\item for any affine open $\\Spec(A) = U \\subset X$ we have\n$\\mathcal{F}|_U = \\widetilde M$ with $M$ a finite $A$-module, and\n\\item there exists an affine open covering $X = \\bigcup U_i$,\n$U_i = \\Spec(A_i)$ such that each\n$\\mathcal{F}|_{U_i} = \\widetilde M_i$ with $M_i$ a finite $A_i$-module.\n\\end{enumerate}\nIn particular $\\mathcal{O}_X$ is coherent, any invertible\n$\\mathcal{O}_X$-module is coherent, and more generally any\nfinite locally free $\\mathcal{O}_X$-module is coherent.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XZ","source_file":"coherent.tex","source_line":2224,"source_end_line":2242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2224-L2242","statement_sha256":"f420bdcd636738856b6200477fc5c263fa6219da13bf54fcf0fe21b0fc5975de","origin":"The Stacks Project","memory_eligible":false,"source_rank":5865,"rank":5865,"depth":7,"x":2016.48,"y":1108.426,"cluster":"duality-cohomology"},{"id":"stacks:01Y0","tag":"01Y0","title":"Coherent sheaves on locally Noetherian schemes · Lemma 01Y0","summary":"Let X be a locally Noetherian scheme. The category of coherent O_X-modules is abelian. More precisely, the kernel and cokernel of a map of coherent O_X-modules are coherent. Any extension of coherent sheaves is coherent.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nThe category of coherent $\\mathcal{O}_X$-modules is abelian.\nMore precisely, the kernel and cokernel of a map of coherent\n$\\mathcal{O}_X$-modules are coherent. Any extension\nof coherent sheaves is coherent.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Y0","source_file":"coherent.tex","source_line":2309,"source_end_line":2316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2309-L2316","statement_sha256":"bd129dc1a1ca9088025709a1b6dba396eab44e93f74da4cafbd4c8d2a8c2c6dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5866,"rank":5866,"depth":2,"x":2065.974,"y":1155.882,"cluster":"duality-cohomology"},{"id":"stacks:01Y1","tag":"01Y1","title":"Coherent sheaves on locally Noetherian schemes · Lemma 01Y1","summary":"Let X be a locally Noetherian scheme. Let F be a coherent O_X-module. Any quasi-coherent submodule of F is coherent. Any quasi-coherent quotient module of F is coherent.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nAny quasi-coherent submodule of $\\mathcal{F}$ is coherent.\nAny quasi-coherent quotient module of $\\mathcal{F}$ is coherent.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Y1","source_file":"coherent.tex","source_line":2328,"source_end_line":2334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2328-L2334","statement_sha256":"7268f7d74ba9e0d66f1acb50296a872e69d0f3f33c66d8703695d1c5fb630e84","origin":"The Stacks Project","memory_eligible":false,"source_rank":5867,"rank":5867,"depth":8,"x":1990.042,"y":1148.848,"cluster":"duality-cohomology"},{"id":"stacks:01Y2","tag":"01Y2","title":"Coherent sheaves on locally Noetherian schemes · Lemma 01Y2","summary":"Let X be a locally Noetherian scheme. Let F, G be coherent O_X-modules. The O_X-modules F ⊗_O_X G and SheafHom_O_X(F, G) are coherent.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.\nThe $\\mathcal{O}_X$-modules $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$\nand $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ are\ncoherent.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Y2","source_file":"coherent.tex","source_line":2346,"source_end_line":2353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2346-L2353","statement_sha256":"cb1750cfeb84a64d7eb314a76cc274f2136d0a3a5fd5de718670ad3147342e38","origin":"The Stacks Project","memory_eligible":false,"source_rank":5868,"rank":5868,"depth":6,"x":2052.713,"y":1110.328,"cluster":"duality-cohomology"},{"id":"stacks:01Y3","tag":"01Y3","title":"Coherent sheaves on locally Noetherian schemes · Lemma 01Y3","summary":"Let X be a locally Noetherian scheme. Let F, G be coherent O_X-modules. Let φ : G → F be a homomorphism of O_X-modules. Let x ∈ X. • If F_x = 0 then there exists an open neighbourhood U ⊂ X of x such that F|_U = 0. • If φ_x : G_x → F_x is injective, then there exists an open neighbourhood U ⊂ X of x such that φ|_U is injective. • If φ_x : G_x → F_x is surjective, then there exists an open neighbourhood U ⊂ X of x such that φ|_U is surjective. • If φ_x : G_x → F_x is…","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.\nLet $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism\nof $\\mathcal{O}_X$-modules. Let $x \\in X$.\n\\begin{enumerate}\n\\item If $\\mathcal{F}_x = 0$ then there exists an open neighbourhood\n$U \\subset X$ of $x$ such that $\\mathcal{F}|_U = 0$.\n\\item If $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ is injective,\nthen there exists an open neighbourhood $U \\subset X$ of $x$ such that\n$\\varphi|_U$ is injective.\n\\item If $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ is surjective,\nthen there exists an open neighbourhood $U \\subset X$ of $x$ such that\n$\\varphi|_U$ is surjective.\n\\item If $\\varphi_x : \\mathcal{G}_x \\to \\mathcal{F}_x$ is bijective,\nthen there exists an open neighbourhood $U \\subset X$ of $x$ such that\n$\\varphi|_U$ is an isomorphism.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Y3","source_file":"coherent.tex","source_line":2363,"source_end_line":2382,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2363-L2382","statement_sha256":"83f7050ba8f508cb1e589a96b2e155f4eede7da89eafc86b84e9214490319790","origin":"The Stacks Project","memory_eligible":false,"source_rank":5869,"rank":5869,"depth":3,"x":2037.225,"y":1175.314,"cluster":"duality-cohomology"},{"id":"stacks:01Y4","tag":"01Y4","title":"Coherent sheaves on locally Noetherian schemes · Lemma 01Y4","summary":"Let X be a locally Noetherian scheme. Let F, G be coherent O_X-modules. Let x ∈ X. Suppose ψ : G_x → F_x is a map of O_X, x-modules. Then there exists an open neighbourhood U ⊂ X of x and a map φ : G|_U → F|_U such that φ_x = ψ.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.\nLet $x \\in X$.\nSuppose $\\psi : \\mathcal{G}_x \\to \\mathcal{F}_x$ is a map of\n$\\mathcal{O}_{X, x}$-modules.\nThen there exists an open neighbourhood $U \\subset X$ of $x$ and a map\n$\\varphi : \\mathcal{G}|_U \\to \\mathcal{F}|_U$ such that\n$\\varphi_x = \\psi$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Y4","source_file":"coherent.tex","source_line":2391,"source_end_line":2401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2391-L2401","statement_sha256":"7cdfe0667e99861ac2d4281cb61404b1123f97a851a64686d28d2ebc0a989857","origin":"The Stacks Project","memory_eligible":false,"source_rank":5870,"rank":5870,"depth":8,"x":1995.76,"y":1117.725,"cluster":"duality-cohomology"},{"id":"stacks:01Y5","tag":"01Y5","title":"Coherent sheaves on locally Noetherian schemes · Lemma 01Y5","summary":"Let X be a locally Noetherian scheme. Let F be a coherent O_X-module. Then Supp(F) is closed, and F comes from a coherent sheaf on the scheme theoretic support of F, see Morphisms, Definition [Tag 05JV].","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a coherent\n$\\mathcal{O}_X$-module. Then $\\text{Supp}(\\mathcal{F})$ is closed, and\n$\\mathcal{F}$ comes from a coherent sheaf on the scheme theoretic support\nof $\\mathcal{F}$, see\nMorphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support}.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Y5","source_file":"coherent.tex","source_line":2409,"source_end_line":2416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2409-L2416","statement_sha256":"d233766b6e33fce267d84d947a472daedddfeef6c5800a958d605acacf640b0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5871,"rank":5871,"depth":19,"x":2073.799,"y":1136.952,"cluster":"duality-cohomology"},{"id":"stacks:087T","tag":"087T","title":"Coherent sheaves on locally Noetherian schemes · Lemma 087T","summary":"Let i : Z → X be a closed immersion of locally Noetherian schemes. Let I ⊂ O_X be the quasi-coherent sheaf of ideals cutting out Z. The functor i_* induces an equivalence between the category of coherent O_X-modules annihilated by I and the category of coherent O_Z-modules.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of locally Noetherian schemes.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals\ncutting out $Z$. The functor $i_*$ induces an equivalence between the\ncategory of coherent $\\mathcal{O}_X$-modules annihilated by $\\mathcal{I}$\nand the category of coherent $\\mathcal{O}_Z$-modules.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087T","source_file":"coherent.tex","source_line":2430,"source_end_line":2437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2430-L2437","statement_sha256":"90ed84b39982be3830f2eac52368f1d7860327585b3c0c32436caf2435aa3b65","origin":"The Stacks Project","memory_eligible":false,"source_rank":5872,"rank":5872,"depth":17,"x":1999.725,"y":1167.49,"cluster":"duality-cohomology"},{"id":"stacks:01Y6","tag":"01Y6","title":"Coherent sheaves on locally Noetherian schemes · Lemma 01Y6","summary":"Let f : X → Y be a morphism of schemes. Let F be a quasi-coherent O_X-module. Assume f is finite and Y locally Noetherian. Then R^pf_*F = 0 for p > 0 and f_*F is coherent if F is coherent.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $f$ is finite and $Y$ locally Noetherian.\nThen $R^pf_*\\mathcal{F} = 0$ for $p > 0$ and\n$f_*\\mathcal{F}$ is coherent if $\\mathcal{F}$ is coherent.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Y6","source_file":"coherent.tex","source_line":2454,"source_end_line":2461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2454-L2461","statement_sha256":"5232f9444717286fa3acdfc5d56a8f76f812df242d0a010f49c068d14c1e144b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5873,"rank":5873,"depth":25,"x":2030.221,"y":1102.028,"cluster":"duality-cohomology"},{"id":"stacks:0B3J","tag":"0B3J","title":"Coherent sheaves on locally Noetherian schemes · Lemma 0B3J","summary":"Let X be a locally Noetherian scheme. Let F be a coherent sheaf with dim(Supp(F)) ≤ 0. Then F is generated by global sections and H^i(X, F) = 0 for i > 0.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$\nbe a coherent sheaf with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq 0$.\nThen $\\mathcal{F}$ is generated by global sections and\n$H^i(X, \\mathcal{F}) = 0$ for $i > 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3J","source_file":"coherent.tex","source_line":2487,"source_end_line":2493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2487-L2493","statement_sha256":"e7b3bfbe7dc88911ee7b1e09142b0043b2dfbdbf34adecf0eaeee14336e70a73","origin":"The Stacks Project","memory_eligible":false,"source_rank":5874,"rank":5874,"depth":26,"x":2060.786,"y":1168.507,"cluster":"duality-cohomology"},{"id":"stacks:0CYJ","tag":"0CYJ","title":"Coherent sheaves on locally Noetherian schemes · Lemma 0CYJ","summary":"Let X be a scheme. Let j : U → X be the inclusion of an open. Let T ⊂ X be a closed subset contained in U. If F is a coherent O_U-module with Supp(F) ⊂ T, then j_*F is a coherent O_X-module.","statement_latex":"Let $X$ be a scheme. Let $j : U \\to X$ be the inclusion of an open.\nLet $T \\subset X$ be a closed subset contained in $U$.\nIf $\\mathcal{F}$ is a coherent $\\mathcal{O}_U$-module\nwith $\\text{Supp}(\\mathcal{F}) \\subset T$, then\n$j_*\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYJ","source_file":"coherent.tex","source_line":2512,"source_end_line":2519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2512-L2519","statement_sha256":"cf8cb0af79f635dcdb0713abd18868330ff7dd5ab8f8182b68dcf6b5204f436e","origin":"The Stacks Project","memory_eligible":false,"source_rank":5875,"rank":5875,"depth":0,"x":1983.771,"y":1136.401,"cluster":"duality-cohomology"},{"id":"stacks:01Y8","tag":"01Y8","title":"Coherent sheaves on Noetherian schemes · Lemma 01Y8","summary":"Let X be a Noetherian scheme. Let F be a coherent O_X-module. The ascending chain condition holds for quasi-coherent submodules of F. In other words, given any sequence F_1 ⊂ F_2 ⊂ … ⊂ F of quasi-coherent submodules, then F_n = F_n + 1 = … for some n ≥ 0.","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThe ascending chain condition holds for quasi-coherent submodules\nof $\\mathcal{F}$. In other words, given any sequence\n$$\n\\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\ldots \\subset \\mathcal{F}\n$$\nof quasi-coherent submodules, then\n$\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\ldots $ for some $n \\geq 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Y8","source_file":"coherent.tex","source_line":2548,"source_end_line":2559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2548-L2559","statement_sha256":"694ca63b7519e80838477da9ffb465bf8f57626c3b6a37d96f82009856dbcee6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5876,"rank":5876,"depth":1,"x":2067.459,"y":1116.118,"cluster":"duality-cohomology"},{"id":"stacks:01Y9","tag":"01Y9","title":"Coherent sheaves on Noetherian schemes · Lemma 01Y9","summary":"Let X be a Noetherian scheme. Let F be a coherent sheaf on X. Let I ⊂ O_X be a quasi-coherent sheaf of ideals corresponding to a closed subscheme Z ⊂ X. Then there is some n ≥ 0 such that I^nF = 0 if and only if Supp(F) ⊂ Z (set theoretically).","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent\nsheaf of ideals corresponding to a closed subscheme $Z \\subset X$.\nThen there is some $n \\geq 0$ such that $\\mathcal{I}^n\\mathcal{F} = 0$\nif and only if $\\text{Supp}(\\mathcal{F}) \\subset Z$ (set theoretically).","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Y9","source_file":"coherent.tex","source_line":2568,"source_end_line":2576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2568-L2576","statement_sha256":"2546f8dab9a1f91b12f04b39a8b136a7de4e0a6ac8fbb21bc85e9039a1865e29","origin":"The Stacks Project","memory_eligible":false,"source_rank":5877,"rank":5877,"depth":4,"x":2021.477,"y":1179.353,"cluster":"duality-cohomology"},{"id":"stacks:01YA","tag":"01YA","title":"Artin-Rees · Lemma 01YA","summary":"Let X be a Noetherian scheme. Let F be a coherent sheaf on X. Let G ⊂ F be a quasi-coherent subsheaf. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Then there exists a c ≥ 0 such that for all n ≥ c we have I^n - c(I^cF ∩ G) = I^nF ∩ G","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nLet $\\mathcal{G} \\subset \\mathcal{F}$ be a quasi-coherent subsheaf.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of\nideals.\nThen there exists a $c \\geq 0$ such that for all $n \\geq c$ we\nhave\n$$\n\\mathcal{I}^{n - c}(\\mathcal{I}^c\\mathcal{F} \\cap \\mathcal{G})\n=\n\\mathcal{I}^n\\mathcal{F} \\cap \\mathcal{G}\n$$","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YA","source_file":"coherent.tex","source_line":2584,"source_end_line":2598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2584-L2598","statement_sha256":"0d718c21cb631852d9a6ded24b54098e50fe3fb12db7dd9a85b9b3e4434b907f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5878,"rank":5878,"depth":2,"x":2004.327,"y":1105.731,"cluster":"duality-cohomology"},{"id":"stacks:0GN6","tag":"0GN6","title":"Coherent sheaves on Noetherian schemes · Lemma 0GN6","summary":"Let X be a Noetherian scheme. Every quasi-coherent O_X-module is the filtered colimit of its coherent submodules.","statement_latex":"Let $X$ be a Noetherian scheme. Every quasi-coherent $\\mathcal{O}_X$-module\nis the filtered colimit of its coherent submodules.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GN6","source_file":"coherent.tex","source_line":2606,"source_end_line":2610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2606-L2610","statement_sha256":"c5f9f7c8e3a3f62fae0f6b42094ba5cf147e645289120c9d44cb42313141628f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5879,"rank":5879,"depth":17,"x":2077.045,"y":1150.83,"cluster":"duality-cohomology"},{"id":"stacks:01YB","tag":"01YB","title":"Coherent sheaves on Noetherian schemes · Lemma 01YB","summary":"Let X be a Noetherian scheme. Let F be a quasi-coherent O_X-module. Let G be a coherent O_X-module. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Denote Z ⊂ X the corresponding closed subscheme and set U = X setminus Z. There is a canonical isomorphism colim_n Hom_O_X(I^nG, F) → Hom_O_U(G|_U, F|_U). In particular we have an isomorphism colim_n Hom_O_X( I^n, F) → Γ(U, F).","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of\nideals. Denote $Z \\subset X$ the corresponding closed subscheme and\nset $U = X \\setminus Z$.\nThere is a canonical isomorphism\n$$\n\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n\\mathcal{G}, \\mathcal{F})\n\\longrightarrow\n\\Hom_{\\mathcal{O}_U}(\\mathcal{G}|_U, \\mathcal{F}|_U).\n$$\nIn particular we have an isomorphism\n$$\n\\colim_n \\Hom_{\\mathcal{O}_X}(\n\\mathcal{I}^n, \\mathcal{F})\n\\longrightarrow\n\\Gamma(U, \\mathcal{F}).\n$$","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YB","source_file":"coherent.tex","source_line":2620,"source_end_line":2641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2620-L2641","statement_sha256":"fcd21e6b48943d12280f01189bb12a1e0235f7d73337fa8f8c37c52d7512a55b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5880,"rank":5880,"depth":17,"x":1986.094,"y":1158.927,"cluster":"duality-cohomology"},{"id":"stacks:0FD0","tag":"0FD0","title":"Coherent sheaves on Noetherian schemes · Lemma 0FD0","summary":"Let X be a locally Noetherian scheme. Let F, G be coherent O_X-modules. Let U ⊂ X be open and let φ : F|_U → G|_U be an O_U-module map. Then there exists a coherent submodule F' ⊂ F agreeing with F over U such that φ extends to φ' : F' → G.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$\nbe coherent $\\mathcal{O}_X$-modules. Let $U \\subset X$ be open and\nlet $\\varphi : \\mathcal{F}|_U \\to \\mathcal{G}|_U$ be an\n$\\mathcal{O}_U$-module map. Then there exists a coherent\nsubmodule $\\mathcal{F}' \\subset \\mathcal{F}$ agreeing with\n$\\mathcal{F}$ over $U$ such that $\\varphi$ extends to\n$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FD0","source_file":"coherent.tex","source_line":2745,"source_end_line":2754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2745-L2754","statement_sha256":"709ef7dd4a877dfc0650346c4bc6335b14a65234e398d7997fd5b200dd8e08ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":5881,"rank":5881,"depth":18,"x":2047.351,"y":1100.687,"cluster":"duality-cohomology"},{"id":"stacks:0341","tag":"0341","title":"Depth · Definition 0341","summary":"Let X be a locally Noetherian scheme. Let F be a coherent O_X-module. Let k ≥ 0 be an integer. • We say F has depth k at a point x of X if depth_O_X, x(F_x) = k. • We say X has depth k at a point x of X if depth(O_X, x) = k. • We say F has property (S_k) if depth_O_X, x(F_x) ≥ min(k, dim(Supp(F_x))) for all x ∈ X. • We say X has property (S_k) if O_X has property (S_k).","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nLet $k \\geq 0$ be an integer.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ has {\\it depth $k$ at a point}\n$x$ of $X$ if $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) = k$.\n\\item We say $X$ has {\\it depth $k$ at a point} $x$ of $X$ if\n$\\text{depth}(\\mathcal{O}_{X, x}) = k$.\n\\item We say $\\mathcal{F}$ has property {\\it $(S_k)$} if\n$$\n\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x)\n\\geq \\min(k, \\dim(\\text{Supp}(\\mathcal{F}_x)))\n$$\nfor all $x \\in X$.\n\\item We say $X$ has property {\\it $(S_k)$} if $\\mathcal{O}_X$ has\nproperty $(S_k)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Depth","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0341","source_file":"coherent.tex","source_line":2810,"source_end_line":2829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2810-L2829","statement_sha256":"845a7342e2d00c3bcc6763640e6289e7bb27c7ec72825773b84ee068c5b9a397","origin":"The Stacks Project","memory_eligible":false,"source_rank":5882,"rank":5882,"depth":0,"x":2049.03,"y":1179.267,"cluster":"duality-cohomology"},{"id":"stacks:0EBC","tag":"0EBC","title":"Depth · Lemma 0EBC","summary":"Let X be a locally Noetherian scheme. Let F, G be coherent O_X-modules and x ∈ X. • If G_x has depth ≥ 1, then SheafHom_O_X(F, G)_x has depth ≥ 1. • If G_x has depth ≥ 2, then Hom_O_X(F, G)_x has depth ≥ 2.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$\nbe coherent $\\mathcal{O}_X$-modules and $x \\in X$.\n\\begin{enumerate}\n\\item If $\\mathcal{G}_x$ has depth $\\geq 1$, then\n$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x$\nhas depth $\\geq 1$.\n\\item If $\\mathcal{G}_x$ has depth $\\geq 2$, then\n$\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})_x$ has depth $\\geq 2$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBC","source_file":"coherent.tex","source_line":2836,"source_end_line":2847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2836-L2847","statement_sha256":"dadea8adee32605ae9a9251957455cd23d000658136ac6bb727f187245c42a4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5883,"rank":5883,"depth":14,"x":1983.892,"y":1121.645,"cluster":"duality-cohomology"},{"id":"stacks:0AXQ","tag":"0AXQ","title":"Depth · Lemma 0AXQ","summary":"Let X be a locally Noetherian scheme. Let F, G be coherent O_X-modules. • If G has property (S_1), then SheafHom_O_X(F, G) has property (S_1). • If G has property (S_2), then SheafHom_O_X(F, G) has property (S_2).","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$, $\\mathcal{G}$\nbe coherent $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{G}$ has property $(S_1)$, then\n$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ has property $(S_1)$.\n\\item If $\\mathcal{G}$ has property $(S_2)$, then\n$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ has property $(S_2)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXQ","source_file":"coherent.tex","source_line":2860,"source_end_line":2870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2860-L2870","statement_sha256":"344ac214544d3117c5f037b624f919a35b5f3d1f090cc173c7bbfa54e38bd243","origin":"The Stacks Project","memory_eligible":false,"source_rank":5884,"rank":5884,"depth":15,"x":2079.28,"y":1127.237,"cluster":"duality-cohomology"},{"id":"stacks:0343","tag":"0343","title":"Depth · Definition 0343","summary":"Let X be a locally Noetherian scheme. Let F be a coherent O_X-module. We say F is Cohen-Macaulay if and only if (S_k) holds for all k ≥ 0.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nWe say $\\mathcal{F}$ is {\\it Cohen-Macaulay} if and only\nif $(S_k)$ holds for all $k \\geq 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Depth","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0343","source_file":"coherent.tex","source_line":2884,"source_end_line":2890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2884-L2890","statement_sha256":"2495613e4a9de45645bd3dd5da194ea66fa761464d352b5ea55294d40c88e62f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5885,"rank":5885,"depth":0,"x":2003.652,"y":1177.764,"cluster":"duality-cohomology"},{"id":"stacks:0B3K","tag":"0B3K","title":"Depth · Lemma 0B3K","summary":"Let X be a regular scheme. Let F be a coherent O_X-module. The following are equivalent • F is Cohen-Macaulay and Supp(F) = X, • F is finite locally free of rank > 0.","statement_latex":"Let $X$ be a regular scheme. Let $\\mathcal{F}$ be a coherent\n$\\mathcal{O}_X$-module. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is Cohen-Macaulay and $\\text{Supp}(\\mathcal{F}) = X$,\n\\item $\\mathcal{F}$ is finite locally free of rank $> 0$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3K","source_file":"coherent.tex","source_line":2892,"source_end_line":2900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2892-L2900","statement_sha256":"9684e442fc932db39cec04e8ad6e512dd4352b4ec6a453dfd030b0626dc25a71","origin":"The Stacks Project","memory_eligible":false,"source_rank":5886,"rank":5886,"depth":13,"x":2018.947,"y":1096.765,"cluster":"duality-cohomology"},{"id":"stacks:01YD","tag":"01YD","title":"Devissage of coherent sheaves · Lemma 01YD","summary":"Let X be a Noetherian scheme. Let F be a coherent sheaf on X. Suppose that Supp(F) = Z ∪ Z' with Z, Z' closed. Then there exists a short exact sequence of coherent sheaves 0 → G' → F → G → 0 with Supp(G') ⊂ Z' and Supp(G) ⊂ Z.","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nSuppose that $\\text{Supp}(\\mathcal{F}) = Z \\cup Z'$ with $Z$, $Z'$ closed.\nThen there exists a short exact sequence of coherent sheaves\n$$\n0 \\to \\mathcal{G}' \\to \\mathcal{F} \\to \\mathcal{G} \\to 0\n$$\nwith $\\text{Supp}(\\mathcal{G}') \\subset Z'$ and\n$\\text{Supp}(\\mathcal{G}) \\subset Z$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YD","source_file":"coherent.tex","source_line":2943,"source_end_line":2954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2943-L2954","statement_sha256":"287d0672fce32de994aa1178cdf6b5b23d39b43a88b7bf7f9ccd73d384c827c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5887,"rank":5887,"depth":5,"x":2073.349,"y":1165.866,"cluster":"duality-cohomology"},{"id":"stacks:01YE","tag":"01YE","title":"Devissage of coherent sheaves · Lemma 01YE","summary":"Let X be a Noetherian scheme. Let i : Z → X be an integral closed subscheme. Let xi ∈ Z be the generic point. Let F be a coherent sheaf on X. Assume that F_xi is annihilated by m_xi. Then there exist an integer r ≥ 0 and a coherent sheaf of ideals I ⊂ O_Z and an injective map of coherent sheaves i_*(I^⊕ r) → F which is an isomorphism in a neighbourhood of xi.","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $i : Z \\to X$ be an integral closed subscheme.\nLet $\\xi \\in Z$ be the generic point.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nAssume that $\\mathcal{F}_\\xi$ is annihilated by\n$\\mathfrak m_\\xi$. Then there exist an integer\n$r \\geq 0$ and a coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$\nand an injective map of coherent sheaves\n$$\ni_*\\left(\\mathcal{I}^{\\oplus r}\\right) \\to \\mathcal{F}\n$$\nwhich is an isomorphism in a neighbourhood of $\\xi$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YE","source_file":"coherent.tex","source_line":2981,"source_end_line":2995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L2981-L2995","statement_sha256":"c6bae2d35396d28ff1aabde78a4ccc792f3f8ed69c5337e393a24054350986a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5888,"rank":5888,"depth":18,"x":1976.715,"y":1145.578,"cluster":"duality-cohomology"},{"id":"stacks:01YF","tag":"01YF","title":"Devissage of coherent sheaves · Lemma 01YF","summary":"Let X be a Noetherian scheme. Let F be a coherent sheaf on X. There exists a filtration 0 = F_0 ⊂ F_1 ⊂ … ⊂ F_m = F by coherent subsheaves such that for each j = 1, …, m there exist an integral closed subscheme Z_j ⊂ X and a nonzero coherent sheaf of ideals I_j ⊂ O_Z_j such that F_j/F_j - 1 ≅ (Z_j → X)_* I_j","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nThere exists a filtration\n$$\n0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset\n\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}\n$$\nby coherent subsheaves such that for each $j = 1, \\ldots, m$\nthere exist an integral closed subscheme $Z_j \\subset X$ and\na nonzero coherent sheaf of ideals $\\mathcal{I}_j \\subset \\mathcal{O}_{Z_j}$\nsuch that\n$$\n\\mathcal{F}_j/\\mathcal{F}_{j - 1}\n\\cong (Z_j \\to X)_* \\mathcal{I}_j\n$$","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YF","source_file":"coherent.tex","source_line":3050,"source_end_line":3067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3050-L3067","statement_sha256":"2df431d135118bb2c6ef00d2366bdd143e8c2b6622d0483dc86234f40d77ad70","origin":"The Stacks Project","memory_eligible":false,"source_rank":5889,"rank":5889,"depth":19,"x":2065.141,"y":1105.321,"cluster":"duality-cohomology"},{"id":"stacks:01YG","tag":"01YG","title":"Devissage of coherent sheaves · Lemma 01YG","summary":"Let X be a Noetherian scheme. Let P be a property of coherent sheaves on X. Assume • For any short exact sequence of coherent sheaves 0 → F_1 → F → F_2 → 0 if F_i, i = 1, 2 have property P then so does F. • For every integral closed subscheme Z ⊂ X and every quasi-coherent sheaf of ideals I ⊂ O_Z we have P for i_*I. Then property P holds for every coherent sheaf on X.","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume\n\\begin{enumerate}\n\\item For any short exact sequence of coherent sheaves\n$$\n0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0\n$$\nif $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$\nthen so does $\\mathcal{F}$.\n\\item For every integral closed subscheme $Z \\subset X$\nand every quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_Z$ we have\n$\\mathcal{P}$ for $i_*\\mathcal{I}$.\n\\end{enumerate}\nThen property $\\mathcal{P}$ holds for every coherent sheaf\non $X$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YG","source_file":"coherent.tex","source_line":3143,"source_end_line":3161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3143-L3161","statement_sha256":"030ba77e9dbba4fbccc5f700e1a6b9ee7858b2e3443d1ec4d9619d93f67280f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5890,"rank":5890,"depth":20,"x":2031.995,"y":1185.942,"cluster":"duality-cohomology"},{"id":"stacks:01YH","tag":"01YH","title":"Devissage of coherent sheaves · Lemma 01YH","summary":"Let X be a Noetherian scheme. Let Z_0 ⊂ X be an irreducible closed subset with generic point xi. Let P be a property of coherent sheaves on X with support contained in Z_0 such that • For any short exact sequence of coherent sheaves if two out of three of them have property P then so does the third. • For every integral closed subscheme Z ⊂ Z_0 ⊂ X, Z not = Z_0 and every quasi-coherent sheaf of ideals I ⊂ O_Z we have P for (Z → X)_*I. • There exists some coherent sheaf G…","statement_latex":"Let $X$ be a Noetherian scheme. Let $Z_0 \\subset X$ be an irreducible closed\nsubset with generic point $\\xi$. Let $\\mathcal{P}$ be a property of coherent\nsheaves on $X$ with support contained in $Z_0$ such that\n\\begin{enumerate}\n\\item For any short exact sequence of coherent sheaves if two\nout of three of them have property $\\mathcal{P}$ then so does the\nthird.\n\\item For every integral closed subscheme $Z \\subset Z_0 \\subset X$,\n$Z \\not = Z_0$ and every quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_Z$ we have\n$\\mathcal{P}$ for $(Z \\to X)_*\\mathcal{I}$.\n\\item There exists some coherent sheaf $\\mathcal{G}$ on $X$ such that\n\\begin{enumerate}\n\\item $\\text{Supp}(\\mathcal{G}) = Z_0$,\n\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$,\n\\item $\\dim_{\\kappa(\\xi)} \\mathcal{G}_\\xi = 1$, and\n\\item property $\\mathcal{P}$ holds for $\\mathcal{G}$.\n\\end{enumerate}\n\\end{enumerate}\nThen property $\\mathcal{P}$ holds for every coherent sheaf\n$\\mathcal{F}$ on $X$ whose support is contained in $Z_0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YH","source_file":"coherent.tex","source_line":3178,"source_end_line":3201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3178-L3201","statement_sha256":"86e3a512ced07b4bda686d557be9e4353754db0111fddade81b9e87b65394a91","origin":"The Stacks Project","memory_eligible":false,"source_rank":5891,"rank":5891,"depth":20,"x":1991.227,"y":1106.951,"cluster":"duality-cohomology"},{"id":"stacks:01YI","tag":"01YI","title":"Devissage of coherent sheaves · Lemma 01YI","summary":"Let X be a Noetherian scheme. Let P be a property of coherent sheaves on X such that • For any short exact sequence of coherent sheaves if two out of three of them have property P then so does the third. • For every integral closed subscheme Z ⊂ X with generic point xi there exists some coherent sheaf G such that • Supp(G) = Z, • G_xi is annihilated by m_xi, • dim_kappa(xi) G_xi = 1, and • property P holds for G. Then property P holds for every coherent sheaf on X.","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that\n\\begin{enumerate}\n\\item For any short exact sequence of coherent sheaves if two\nout of three of them have property $\\mathcal{P}$ then so does the\nthird.\n\\item For every integral closed subscheme $Z \\subset X$\nwith generic point $\\xi$ there exists\nsome coherent sheaf $\\mathcal{G}$ such that\n\\begin{enumerate}\n\\item $\\text{Supp}(\\mathcal{G}) = Z$,\n\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$,\n\\item $\\dim_{\\kappa(\\xi)} \\mathcal{G}_\\xi = 1$, and\n\\item property $\\mathcal{P}$ holds for $\\mathcal{G}$.\n\\end{enumerate}\n\\end{enumerate}\nThen property $\\mathcal{P}$ holds for every coherent sheaf\non $X$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YI","source_file":"coherent.tex","source_line":3269,"source_end_line":3289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3269-L3289","statement_sha256":"96b8c17886d521c5852c89e9775d554de6d22f7c0beaf8a5c72be0cea11b6aef","origin":"The Stacks Project","memory_eligible":false,"source_rank":5892,"rank":5892,"depth":21,"x":2085.67,"y":1142.391,"cluster":"duality-cohomology"},{"id":"stacks:01YL","tag":"01YL","title":"Devissage of coherent sheaves · Lemma 01YL","summary":"Let X be a Noetherian scheme. Let Z_0 ⊂ X be an irreducible closed subset with generic point xi. Let P be a property of coherent sheaves on X such that • For any short exact sequence of coherent sheaves 0 → F_1 → F → F_2 → 0 if F_i, i = 1, 2 have property P then so does F. • If P holds for F^⊕ r for some r ≥ 1, then it holds for F. • For every integral closed subscheme Z ⊂ Z_0 ⊂ X, Z not = Z_0 and every quasi-coherent sheaf of ideals I ⊂ O_Z we have P for (Z → X)_*I. •…","statement_latex":"Let $X$ be a Noetherian scheme. Let $Z_0 \\subset X$ be an irreducible\nclosed subset with generic point $\\xi$. Let $\\mathcal{P}$ be a property\nof coherent sheaves on $X$ such that\n\\begin{enumerate}\n\\item For any short exact sequence of coherent sheaves\n$$\n0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0\n$$\nif $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$\nthen so does $\\mathcal{F}$.\n\\item If $\\mathcal{P}$ holds for $\\mathcal{F}^{\\oplus r}$ for\nsome $r \\geq 1$, then it holds for $\\mathcal{F}$.\n\\item For every integral closed subscheme $Z \\subset Z_0 \\subset X$,\n$Z \\not = Z_0$ and every quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_Z$ we have\n$\\mathcal{P}$ for $(Z \\to X)_*\\mathcal{I}$.\n\\item There exists some coherent sheaf $\\mathcal{G}$ such that\n\\begin{enumerate}\n\\item $\\text{Supp}(\\mathcal{G}) = Z_0$,\n\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, and\n\\item for every quasi-coherent sheaf of ideals\n$\\mathcal{J} \\subset \\mathcal{O}_X$ such that\n$\\mathcal{J}_\\xi = \\mathcal{O}_{X, \\xi}$ there exists a quasi-coherent\nsubsheaf $\\mathcal{G}' \\subset \\mathcal{J}\\mathcal{G}$ with\n$\\mathcal{G}'_\\xi = \\mathcal{G}_\\xi$ and such that\n$\\mathcal{P}$ holds for $\\mathcal{G}'$.\n\\end{enumerate}\n\\end{enumerate}\nThen property $\\mathcal{P}$ holds for every coherent sheaf\n$\\mathcal{F}$ on $X$ whose support is contained in $Z_0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YL","source_file":"coherent.tex","source_line":3306,"source_end_line":3338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3306-L3338","statement_sha256":"1872eea35db5dc1f37daf736cd4151b1262beb9deecf58830622ea3aff0b6dc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5893,"rank":5893,"depth":20,"x":1986.642,"y":1170.092,"cluster":"duality-cohomology"},{"id":"stacks:01YM","tag":"01YM","title":"Devissage of coherent sheaves · Lemma 01YM","summary":"Let X be a Noetherian scheme. Let P be a property of coherent sheaves on X such that • For any short exact sequence of coherent sheaves 0 → F_1 → F → F_2 → 0 if F_i, i = 1, 2 have property P then so does F. • If P holds for F^⊕ r for some r ≥ 1, then it holds for F. • For every integral closed subscheme Z ⊂ X with generic point xi there exists some coherent sheaf G such that • Supp(G) = Z, • G_xi is annihilated by m_xi, and • for every quasi-coherent sheaf of ideals J ⊂…","statement_latex":"Let $X$ be a Noetherian scheme.\nLet $\\mathcal{P}$ be a property of coherent sheaves on $X$ such that\n\\begin{enumerate}\n\\item For any short exact sequence of coherent sheaves\n$$\n0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0\n$$\nif $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$\nthen so does $\\mathcal{F}$.\n\\item If $\\mathcal{P}$ holds  for $\\mathcal{F}^{\\oplus r}$ for\nsome $r \\geq 1$, then it holds for $\\mathcal{F}$.\n\\item For every integral closed subscheme $Z \\subset X$\nwith generic point $\\xi$ there exists\nsome coherent sheaf $\\mathcal{G}$ such that\n\\begin{enumerate}\n\\item $\\text{Supp}(\\mathcal{G}) = Z$,\n\\item $\\mathcal{G}_\\xi$ is annihilated by $\\mathfrak m_\\xi$, and\n\\item for every quasi-coherent sheaf of ideals\n$\\mathcal{J} \\subset \\mathcal{O}_X$ such that\n$\\mathcal{J}_\\xi = \\mathcal{O}_{X, \\xi}$ there exists a quasi-coherent\nsubsheaf $\\mathcal{G}' \\subset \\mathcal{J}\\mathcal{G}$ with\n$\\mathcal{G}'_\\xi = \\mathcal{G}_\\xi$ and such that\n$\\mathcal{P}$ holds for $\\mathcal{G}'$.\n\\end{enumerate}\n\\end{enumerate}\nThen property $\\mathcal{P}$ holds for every coherent sheaf\non $X$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YM","source_file":"coherent.tex","source_line":3424,"source_end_line":3453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3424-L3453","statement_sha256":"62d5e01b4b5eb9ab11c147778a7dbbe87a4d10aa56d28c09a2720091f4988f4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5894,"rank":5894,"depth":22,"x":2037.84,"y":1092.797,"cluster":"duality-cohomology"},{"id":"stacks:01YO","tag":"01YO","title":"Finite morphisms and affines · Lemma 01YO","summary":"Let f : Y → X be a morphism of schemes. Assume f is finite, surjective and X locally Noetherian. Let Z ⊂ X be an integral closed subscheme with generic point xi. Then there exists a coherent sheaf F on Y such that the support of f_*F is equal to Z and (f_*F)_xi is annihilated by m_xi.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes.\nAssume $f$ is finite, surjective and $X$ locally Noetherian.\nLet $Z \\subset X$ be an integral closed subscheme with\ngeneric point $\\xi$. Then\nthere exists a coherent sheaf $\\mathcal{F}$ on $Y$\nsuch that the support of $f_*\\mathcal{F}$ is equal to $Z$\nand $(f_*\\mathcal{F})_\\xi$ is annihilated by $\\mathfrak m_\\xi$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Finite morphisms and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YO","source_file":"coherent.tex","source_line":3478,"source_end_line":3487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3478-L3487","statement_sha256":"3f32cbda98df26deebf3c132d3d0b876034c6b041f5508e9c6e99eea99eec4c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5895,"rank":5895,"depth":26,"x":2062.456,"y":1179.595,"cluster":"duality-cohomology"},{"id":"stacks:01YP","tag":"01YP","title":"Finite morphisms and affines · Lemma 01YP","summary":"Let f : Y → X be a morphism of schemes. Let F be a quasi-coherent sheaf on Y. Let I be a quasi-coherent sheaf of ideals on X. If the morphism f is affine then If_*F = f_*(f^-1IF).","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $Y$.\nLet $\\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$.\nIf the morphism $f$ is affine then\n$\\mathcal{I}f_*\\mathcal{F} = f_*(f^{-1}\\mathcal{I}\\mathcal{F})$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Finite morphisms and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YP","source_file":"coherent.tex","source_line":3522,"source_end_line":3529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3522-L3529","statement_sha256":"f481203835d894bec2232d79a4545e9211b4e9f038e594e2b1854bf668e24d9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5896,"rank":5896,"depth":0,"x":1973.753,"y":1129.128,"cluster":"duality-cohomology"},{"id":"stacks:01YQ","tag":"01YQ","title":"Finite morphisms and affines · Lemma 01YQ","summary":"Let f : Y → X be a morphism of schemes. Assume • f finite, • f surjective, • Y affine, and • X Noetherian. Then X is affine.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes.\nAssume\n\\begin{enumerate}\n\\item $f$ finite,\n\\item $f$ surjective,\n\\item $Y$ affine, and\n\\item $X$ Noetherian.\n\\end{enumerate}\nThen $X$ is affine.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Finite morphisms and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YQ","source_file":"coherent.tex","source_line":3556,"source_end_line":3567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3556-L3567","statement_sha256":"0b879ceac21bde2efe7084bf8b40a006276240eb5e355f84827359344a149fda","origin":"The Stacks Project","memory_eligible":false,"source_rank":5897,"rank":5897,"depth":27,"x":2080.638,"y":1115.901,"cluster":"duality-cohomology"},{"id":"stacks:01YS","tag":"01YS","title":"Coherent sheaves on Proj, I · Lemma 01YS","summary":"Let R be a Noetherian ring. Let n ≥ 0 be an integer. For every coherent sheaf F on P^n_R we have the following: • There exists an r ≥ 0 and d_1, …, d_r ∈ Z and a surjection bigoplus_j = 1, …, r O_P^n_R(d_j) → F. • We have H^i(P^n_R, F) = 0 unless 0 ≤ i ≤ n. • For any i the cohomology group H^i(P^n_R, F) is a finite R-module. • If i > 0, then H^i(P^n_R, F(d)) = 0 for all d large enough. • For any k ∈ Z the graded R[T_0, …, T_n]-module bigoplus_d ≥ k H^0(P^n_R, F(d)) is a…","statement_latex":"Let $R$ be a Noetherian ring.\nLet $n \\geq 0$ be an integer.\nFor every coherent sheaf $\\mathcal{F}$ on $\\mathbf{P}^n_R$\nwe have the following:\n\\begin{enumerate}\n\\item There exists an $r \\geq 0$ and\n$d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection\n$$\n\\bigoplus\\nolimits_{j = 1, \\ldots, r}\n\\mathcal{O}_{\\mathbf{P}^n_R}(d_j)\n\\longrightarrow\n\\mathcal{F}.\n$$\n\\item We have $H^i(\\mathbf{P}^n_R, \\mathcal{F}) = 0$ unless\n$0 \\leq i \\leq n$.\n\\item For any $i$ the cohomology group $H^i(\\mathbf{P}^n_R, \\mathcal{F})$\nis a finite $R$-module.\n\\item If $i > 0$, then\n$H^i(\\mathbf{P}^n_R, \\mathcal{F}(d)) = 0$ for all $d$ large enough.\n\\item For any $k \\in \\mathbf{Z}$ the graded $R[T_0, \\ldots, T_n]$-module\n$$\n\\bigoplus\\nolimits_{d \\geq k} H^0(\\mathbf{P}^n_R, \\mathcal{F}(d))\n$$\nis a finite $R[T_0, \\ldots, T_n]$-module.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Proj, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YS","source_file":"coherent.tex","source_line":3640,"source_end_line":3667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3640-L3667","statement_sha256":"156390a3a07304929ec0108b057186499614e85639ce4481dc67d31bdd4c17c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5898,"rank":5898,"depth":26,"x":2011.893,"y":1186.885,"cluster":"duality-cohomology"},{"id":"stacks:0AG6","tag":"0AG6","title":"Coherent sheaves on Proj, I · Lemma 0AG6","summary":"Let A be a graded ring such that A_0 is Noetherian and A is generated by finitely many elements of A_1 over A_0. Set X = Proj(A). Then X is a Noetherian scheme. Let F be a coherent O_X-module. • There exists an r ≥ 0 and d_1, …, d_r ∈ Z and a surjection bigoplus_j = 1, …, r O_X(d_j) → F. • For any p the cohomology group H^p(X, F) is a finite A_0-module. • If p > 0, then H^p(X, F(d)) = 0 for all d large enough. • For any k ∈ Z the graded A-module bigoplus_d ≥ k H^0(X,…","statement_latex":"Let $A$ be a graded ring such that $A_0$ is Noetherian and\n$A$ is generated by finitely many elements of $A_1$ over $A_0$.\nSet $X = \\text{Proj}(A)$. Then $X$ is a Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item There exists an $r \\geq 0$ and\n$d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection\n$$\n\\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{O}_X(d_j)\n\\longrightarrow \\mathcal{F}.\n$$\n\\item For any $p$ the cohomology group $H^p(X, \\mathcal{F})$ is a finite\n$A_0$-module.\n\\item If $p > 0$, then $H^p(X, \\mathcal{F}(d)) = 0$ for all $d$ large enough.\n\\item For any $k \\in \\mathbf{Z}$ the graded $A$-module\n$$\n\\bigoplus\\nolimits_{d \\geq k} H^0(X, \\mathcal{F}(d))\n$$\nis a finite $A$-module.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Proj, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AG6","source_file":"coherent.tex","source_line":3809,"source_end_line":3831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3809-L3831","statement_sha256":"ff8223499788c79c62cba4350afec929e97f52f1aa1cf42d322dba258bb21aa7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5899,"rank":5899,"depth":27,"x":2005.452,"y":1094.791,"cluster":"duality-cohomology"},{"id":"stacks:0AG7","tag":"0AG7","title":"Coherent sheaves on Proj, I · Lemma 0AG7","summary":"Let A be a graded ring such that A_0 is Noetherian and A is generated by finitely many elements of A_1 over A_0. Let M be a finite graded A-module. Set X = Proj(A) and let widetildeM be the quasi-coherent O_X-module on X associated to M. The maps M_n → Γ(X, widetildeM(n)) from Constructions, Lemma [Tag 01MT] are isomorphisms for all sufficiently large n.","statement_latex":"Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated\nby finitely many elements of $A_1$ over $A_0$. Let $M$ be a\nfinite graded $A$-module. Set $X = \\text{Proj}(A)$ and let $\\widetilde{M}$\nbe the quasi-coherent $\\mathcal{O}_X$-module on $X$ associated to $M$.\nThe maps\n$$\nM_n \\longrightarrow \\Gamma(X, \\widetilde{M}(n))\n$$\nfrom Constructions, Lemma \\ref{constructions-lemma-apply-modules}\nare isomorphisms for all sufficiently large $n$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Proj, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AG7","source_file":"coherent.tex","source_line":3861,"source_end_line":3873,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3861-L3873","statement_sha256":"75be47d96bd911f153bd19744fa24180124ce07d5a8b1bc38f9f7ca732f539da","origin":"The Stacks Project","memory_eligible":false,"source_rank":5900,"rank":5900,"depth":28,"x":2084.89,"y":1159.561,"cluster":"duality-cohomology"},{"id":"stacks:0BXD","tag":"0BXD","title":"Coherent sheaves on Proj, I · Proposition 0BXD","summary":"Let A be a graded ring such that A_0 is Noetherian and A is generated by finitely many elements of A_1 over A_0. Set X = Proj(A). The functor M ↦ widetilde M induces an equivalence Mod^fg_A/Mod^fg_A, torsion → Coh(O_X) whose quasi-inverse is given by F ↦ bigoplus_n ≥ 0 Γ(X, F(n)).","statement_latex":"Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated\nby finitely many elements of $A_1$ over $A_0$.\nSet $X = \\text{Proj}(A)$. The functor $M \\mapsto \\widetilde M$\ninduces an equivalence\n$$\n\\text{Mod}^{fg}_A/\\text{Mod}^{fg}_{A, torsion}\n\\longrightarrow\n\\textit{Coh}(\\mathcal{O}_X)\n$$\nwhose quasi-inverse is given by\n$\\mathcal{F} \\longmapsto \\bigoplus_{n \\geq 0} \\Gamma(X, \\mathcal{F}(n))$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Proj, I","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXD","source_file":"coherent.tex","source_line":3950,"source_end_line":3963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L3950-L3963","statement_sha256":"50db088d728c6115d6285e6edb1503a9b51ffdfcb7137a0cf7892738f27512eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5901,"rank":5901,"depth":29,"x":1973.355,"y":1156.853,"cluster":"duality-cohomology"},{"id":"stacks:0B5Q","tag":"0B5Q","title":"Coherent sheaves on Proj, II · Lemma 0B5Q","summary":"Let A be a Noetherian graded ring. Set X = Proj(A). Then X is a Noetherian scheme. Let F be a coherent O_X-module. • There exists an r ≥ 0 and d_1, …, d_r ∈ Z and a surjection bigoplus_j = 1, …, r O_X(d_j) → F. • For any p the cohomology group H^p(X, F) is a finite A_0-module. • If p > 0, then H^p(X, F(d)) = 0 for all d large enough. • For any k ∈ Z the graded A-module bigoplus_d ≥ k H^0(X, F(d)) is a finite A-module.","statement_latex":"Let $A$ be a Noetherian graded ring. Set $X = \\text{Proj}(A)$. Then $X$\nis a Noetherian scheme. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item There exists an $r \\geq 0$ and\n$d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection\n$$\n\\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{O}_X(d_j)\n\\longrightarrow \\mathcal{F}.\n$$\n\\item For any $p$ the cohomology group $H^p(X, \\mathcal{F})$ is a finite\n$A_0$-module.\n\\item If $p > 0$, then $H^p(X, \\mathcal{F}(d)) = 0$ for all $d$ large enough.\n\\item For any $k \\in \\mathbf{Z}$ the graded $A$-module\n$$\n\\bigoplus\\nolimits_{d \\geq k} H^0(X, \\mathcal{F}(d))\n$$\nis a finite $A$-module.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Proj, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5Q","source_file":"coherent.tex","source_line":4019,"source_end_line":4039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4019-L4039","statement_sha256":"25a7ee9900098921c4ce06450bf38dfd3cba772e7292e9f66110724fa1f764df","origin":"The Stacks Project","memory_eligible":false,"source_rank":5902,"rank":5902,"depth":28,"x":2058.434,"y":1095.088,"cluster":"duality-cohomology"},{"id":"stacks:0B5R","tag":"0B5R","title":"Coherent sheaves on Proj, II · Lemma 0B5R","summary":"Let A be a Noetherian graded ring and let d be the lcm of generators of A over A_0. Let M be a finite graded A-module. Set X = Proj(A) and let widetildeM be the quasi-coherent O_X-module on X associated to M. Let k ∈ Z. • N' = bigoplus_n ≥ k H^0(X, widetildeM(n)) is a finite A-module, • N = bigoplus_n ≥ k H^0(X, widetildeM(n)) is a finite A-module, • there is a canonical map N → N', • if k is small enough there is a canonical map M → N', • the map M_n → N'_n is an…","statement_latex":"Let $A$ be a Noetherian graded ring and let $d$ be the lcm of generators\nof $A$ over $A_0$. Let $M$ be a finite graded $A$-module.\nSet $X = \\text{Proj}(A)$ and let $\\widetilde{M}$ be\nthe quasi-coherent $\\mathcal{O}_X$-module on $X$ associated to $M$.\nLet $k \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item $N' = \\bigoplus_{n \\geq k} H^0(X, \\widetilde{M(n)})$\nis a finite $A$-module,\n\\item $N = \\bigoplus_{n \\geq k} H^0(X, \\widetilde{M}(n))$\nis a finite $A$-module,\n\\item there is a canonical map $N \\to N'$,\n\\item if $k$ is small enough there is a canonical map $M \\to N'$,\n\\item the map $M_n \\to N'_n$ is an isomorphism for $n \\gg 0$,\n\\item $N_n \\to N'_n$ is an isomorphism for $d | n$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Proj, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5R","source_file":"coherent.tex","source_line":4080,"source_end_line":4097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4080-L4097","statement_sha256":"db96358a84af33984a4dedf3730cf32a597007986b02095b4f7ae2ea040126aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":5903,"rank":5903,"depth":29,"x":2045.265,"y":1189.625,"cluster":"duality-cohomology"},{"id":"stacks:0BXF","tag":"0BXF","title":"Coherent sheaves on Proj, II · Proposition 0BXF","summary":"Let A be a Noetherian graded ring. Set X = Proj(A). The functor M ↦ widetilde M induces an equivalence Mod^fg_A/Mod^fg_A, irrelevant → Coh(O_X) whose quasi-inverse is given by F ↦ bigoplus_n ≥ 0 Γ(X, F(n)).","statement_latex":"Let $A$ be a Noetherian graded ring. Set $X = \\text{Proj}(A)$. The functor\n$M \\mapsto \\widetilde M$ induces an equivalence\n$$\n\\text{Mod}^{fg}_A/\\text{Mod}^{fg}_{A, irrelevant}\n\\longrightarrow\n\\textit{Coh}(\\mathcal{O}_X)\n$$\nwhose quasi-inverse is given by\n$\\mathcal{F} \\longmapsto \\bigoplus_{n \\geq 0} \\Gamma(X, \\mathcal{F}(n))$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent sheaves on Proj, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXF","source_file":"coherent.tex","source_line":4225,"source_end_line":4236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4225-L4236","statement_sha256":"8df2bf47f42e28c961ea99f119cc13be66a88123dac47d3edfbccb8bb64f8e7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5904,"rank":5904,"depth":30,"x":1978.455,"y":1111.86,"cluster":"duality-cohomology"},{"id":"stacks:0B5T","tag":"0B5T","title":"Higher direct images along projective morphisms · Lemma 0B5T","summary":"Let R be a Noetherian ring. Let X → Spec(R) be a proper morphism. Let L be an ample invertible sheaf on X. Let F be a coherent O_X-module. • The graded ring A = bigoplus_d ≥ 0 H^0(X, L^⊗ d) is a finitely generated R-algebra. • There exists an r ≥ 0 and d_1, …, d_r ∈ Z and a surjection bigoplus_j = 1, …, r L^⊗ d_j → F. • For any p the cohomology group H^p(X, F) is a finite R-module. • If p > 0, then H^p(X, F ⊗_O_X L^⊗ d) = 0 for all d large enough. • For any k ∈ Z the…","statement_latex":"Let $R$ be a Noetherian ring. Let $X \\to \\Spec(R)$ be a proper morphism.\nLet $\\mathcal{L}$ be an ample invertible sheaf on $X$. Let $\\mathcal{F}$\nbe a coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item The graded ring\n$A = \\bigoplus_{d \\geq 0} H^0(X, \\mathcal{L}^{\\otimes d})$\nis a finitely generated $R$-algebra.\n\\item There exists an $r \\geq 0$ and\n$d_1, \\ldots, d_r \\in \\mathbf{Z}$ and a surjection\n$$\n\\bigoplus\\nolimits_{j = 1, \\ldots, r} \\mathcal{L}^{\\otimes d_j}\n\\longrightarrow \\mathcal{F}.\n$$\n\\item For any $p$ the cohomology group $H^p(X, \\mathcal{F})$ is a finite\n$R$-module.\n\\item If $p > 0$, then\n$H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0$\nfor all $d$ large enough.\n\\item For any $k \\in \\mathbf{Z}$ the graded $A$-module\n$$\n\\bigoplus\\nolimits_{d \\geq k}\nH^0(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})\n$$\nis a finite $A$-module.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Higher direct images along projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5T","source_file":"coherent.tex","source_line":4328,"source_end_line":4355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4328-L4355","statement_sha256":"9b9b6ba8499770dd4bf502325ec9298400f14a086d81d53b53576b88e3a9ae1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5905,"rank":5905,"depth":27,"x":2091.084,"y":1131.439,"cluster":"duality-cohomology"},{"id":"stacks:02O1","tag":"02O1","title":"Higher direct images along projective morphisms · Lemma 02O1","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. Let L be an invertible sheaf on X. Assume that • S is Noetherian, • f is proper, • F is coherent, and • L is relatively ample on X/S. Then there exists an n_0 such that for all n ≥ n_0 we have R^pf_*(F ⊗_O_X L^⊗ n) = 0 for all p > 0.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nAssume that\n\\begin{enumerate}\n\\item $S$ is Noetherian,\n\\item $f$ is proper,\n\\item $\\mathcal{F}$ is coherent, and\n\\item $\\mathcal{L}$ is relatively ample on $X/S$.\n\\end{enumerate}\nThen there exists an $n_0$ such that for all $n \\geq n_0$\nwe have\n$$\nR^pf_*\\left(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}\\right)\n=\n0\n$$\nfor all $p > 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Higher direct images along projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02O1","source_file":"coherent.tex","source_line":4426,"source_end_line":4446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4426-L4446","statement_sha256":"ef2b55ca152dc35144949b05f2dea1b8eba90be541a62bfce43499b22d1e01ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":5906,"rank":5906,"depth":29,"x":1991.565,"y":1181.27,"cluster":"duality-cohomology"},{"id":"stacks:02O4","tag":"02O4","title":"Higher direct images along projective morphisms · Lemma 02O4","summary":"Let S be a locally Noetherian scheme. Let f : X → S be a locally projective morphism. Let F be a coherent O_X-module. Then R^if_*F is a coherent O_S-module for all i ≥ 0.","statement_latex":"Let $S$ be a locally Noetherian scheme.\nLet $f : X \\to S$ be a locally projective morphism.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThen $R^if_*\\mathcal{F}$ is a coherent $\\mathcal{O}_S$-module\nfor all $i \\geq 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Higher direct images along projective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02O4","source_file":"coherent.tex","source_line":4465,"source_end_line":4472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4465-L4472","statement_sha256":"1bfbf2e65a935dfde8f5f4293fa0107b0592fd7e2140fc3a8f979703b91703d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5907,"rank":5907,"depth":29,"x":2025.118,"y":1087.386,"cluster":"duality-cohomology"},{"id":"stacks:0B5U","tag":"0B5U","title":"Ample invertible sheaves and cohomology · Lemma 0B5U","summary":"[EGA] Let R be a Noetherian ring. Let f : X → Spec(R) be a proper morphism. Let L be an invertible O_X-module. The following are equivalent • L is ample on X (this is equivalent to many other things, see Properties, Proposition [Tag 01Q3] and Morphisms, Lemma [Tag 01VT]), • for every coherent O_X-module F there exists an n_0 ≥ 0 such that H^p(X, F ⊗ L^⊗ n) = 0 for all n ≥ n_0 and p > 0, and • for every quasi-coherent sheaf of ideals I ⊂ O_X, there exists an n ≥ 1 such…","statement_latex":"\\begin{reference}\n\\cite[III Proposition 2.6.1]{EGA}\n\\end{reference}\nLet $R$ be a Noetherian ring. Let $f : X \\to \\Spec(R)$ be a proper morphism.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample on $X$ (this is equivalent to many other\nthings, see\nProperties, Proposition \\ref{properties-proposition-characterize-ample} and\nMorphisms, Lemma\n\\ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}),\n\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists\nan $n_0 \\geq 0$ such that\n$H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$ for all $n \\geq n_0$\nand $p > 0$, and\n\\item for every quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_X$, there exists an $n \\geq 1$\nsuch that $H^1(X, \\mathcal{I} \\otimes \\mathcal{L}^{\\otimes n}) = 0$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5U","source_file":"coherent.tex","source_line":4515,"source_end_line":4537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4515-L4537","statement_sha256":"f6f641326293575f3b42fef49ae398edc181db4089c6a55524c150a79270f6f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5908,"rank":5908,"depth":28,"x":2076.234,"y":1176.281,"cluster":"duality-cohomology"},{"id":"stacks:0B5V","tag":"0B5V","title":"Ample invertible sheaves and cohomology · Lemma 0B5V","summary":"Let R be a Noetherian ring. Let f : Y → X be a morphism of schemes proper over R. Let L be an invertible O_X-module. Assume f is finite and surjective. Then L is ample if and only if f^*L is ample.","statement_latex":"Let $R$ be a Noetherian ring. Let $f : Y \\to X$ be a morphism of\nschemes proper over $R$. Let $\\mathcal{L}$ be an\ninvertible $\\mathcal{O}_X$-module. Assume $f$ is finite and surjective.\nThen $\\mathcal{L}$ is ample if and only if $f^*\\mathcal{L}$ is ample.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5V","source_file":"coherent.tex","source_line":4547,"source_end_line":4553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4547-L4553","statement_sha256":"bd68501abe08ad8a74c9b05390a3d85f34ff2375de1379599e3f8b7a0fe1db7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5909,"rank":5909,"depth":29,"x":1966.292,"y":1139.478,"cluster":"duality-cohomology"},{"id":"stacks:09MR","tag":"09MR","title":"Ample invertible sheaves and cohomology · Lemma 09MR","summary":"Let X be a scheme. Let L be an invertible sheaf on X. Let s ∈ Γ(X, L). Let F be a quasi-coherent O_X-module. If X is quasi-compact and quasi-separated, the canonical map H^p_*(X, L, F)_(s) → H^p(X_s, F) which maps xi/s^n to s^-nxi is an isomorphism.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf on $X$.\nLet $s \\in \\Gamma(X, \\mathcal{L})$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $X$ is quasi-compact and quasi-separated, the canonical map\n$$\nH^p_*(X, \\mathcal{L}, \\mathcal{F})_{(s)}\n\\longrightarrow\nH^p(X_s, \\mathcal{F})\n$$\nwhich maps $\\xi/s^n$ to $s^{-n}\\xi$ is an isomorphism.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MR","source_file":"coherent.tex","source_line":4647,"source_end_line":4659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4647-L4659","statement_sha256":"57f10602d4d0af40d17a192079b9efda0a9bdf5c4f109cb7ea0503a093a01bec","origin":"The Stacks Project","memory_eligible":false,"source_rank":5910,"rank":5910,"depth":24,"x":2077.722,"y":1103.991,"cluster":"duality-cohomology"},{"id":"stacks:01XR","tag":"01XR","title":"Ample invertible sheaves and cohomology · Lemma 01XR","summary":"Let X be a scheme. Let L be an invertible O_X-module. Let s ∈ Γ(X, L) be a section. Assume that • X is quasi-compact and quasi-separated, and • X_s is affine. Then for every quasi-coherent O_X-module F and every p > 0 and all xi ∈ H^p(X, F) there exists an n ≥ 0 such that s^nxi = 0 in H^p(X, F ⊗_O_X L^⊗ n).","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$ be a section.\nAssume that\n\\begin{enumerate}\n\\item $X$ is quasi-compact and quasi-separated, and\n\\item $X_s$ is affine.\n\\end{enumerate}\nThen for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ and\nevery $p > 0$ and all $\\xi \\in H^p(X, \\mathcal{F})$ there exists\nan $n \\geq 0$ such that $s^n\\xi = 0$ in\n$H^p(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n})$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01XR","source_file":"coherent.tex","source_line":4708,"source_end_line":4722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4708-L4722","statement_sha256":"45881ec840a33d170df635680c67071dc909bdf4fed459a1e8d969713a1e2cc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5911,"rank":5911,"depth":25,"x":2023.726,"y":1193.994,"cluster":"duality-cohomology"},{"id":"stacks:09MS","tag":"09MS","title":"Ample invertible sheaves and cohomology · Lemma 09MS","summary":"Let i : Z → X be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Let L be an invertible sheaf on X. Then i^*L is ample on Z, if and only if L is ample on X.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of Noetherian schemes\ninducing a homeomorphism of underlying topological spaces.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nThen $i^*\\mathcal{L}$ is ample on $Z$, if and only if\n$\\mathcal{L}$ is ample on $X$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MS","source_file":"coherent.tex","source_line":4736,"source_end_line":4743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4736-L4743","statement_sha256":"bfaaf007b738b2fa02b76baa57fe990397fc6bdcee22c5e5e4fba9164f58cc90","origin":"The Stacks Project","memory_eligible":false,"source_rank":5912,"rank":5912,"depth":28,"x":1990.947,"y":1096.337,"cluster":"duality-cohomology"},{"id":"stacks:0B7K","tag":"0B7K","title":"Ample invertible sheaves and cohomology · Lemma 0B7K","summary":"Let i : Z → X be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Then X is quasi-affine if and only if Z is quasi-affine.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of Noetherian schemes\ninducing a homeomorphism of underlying topological spaces.\nThen $X$ is quasi-affine if and only if $Z$ is quasi-affine.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7K","source_file":"coherent.tex","source_line":4829,"source_end_line":4834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4829-L4834","statement_sha256":"c9d29c1eea7f017297adf096c8b6cdccc5db72f1cfa8638f09b956919ef2d86f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5913,"rank":5913,"depth":29,"x":2094.33,"y":1150.103,"cluster":"duality-cohomology"},{"id":"stacks:0EBD","tag":"0EBD","title":"Ample invertible sheaves and cohomology · Lemma 0EBD","summary":"Let X be a scheme. Let L be an ample invertible O_X-module. Let n_0 be an integer. If H^p(X, L^⊗ -n) = 0 for n ≥ n_0 and p > 0, then X is affine.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible\n$\\mathcal{O}_X$-module. Let $n_0$ be an integer.\nIf $H^p(X, \\mathcal{L}^{\\otimes -n}) = 0$ for $n \\geq n_0$ and $p > 0$,\nthen $X$ is affine.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBD","source_file":"coherent.tex","source_line":4848,"source_end_line":4854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4848-L4854","statement_sha256":"acd0fd372a2fd18ae62c8b2d7b3e4580777306f4612cf683391d8f23d61bec60","origin":"The Stacks Project","memory_eligible":false,"source_rank":5914,"rank":5914,"depth":26,"x":1974.079,"y":1169.241,"cluster":"duality-cohomology"},{"id":"stacks:0EBE","tag":"0EBE","title":"Ample invertible sheaves and cohomology · Lemma 0EBE","summary":"Let X be a quasi-affine scheme. If H^p(X, O_X) = 0 for p > 0, then X is affine.","statement_latex":"Let $X$ be a quasi-affine scheme.\nIf $H^p(X, \\mathcal{O}_X) = 0$ for $p > 0$,\nthen $X$ is affine.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBE","source_file":"coherent.tex","source_line":4893,"source_end_line":4898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4893-L4898","statement_sha256":"917adf01f47cc43307255152918111a0b5152fc021d7fbaf23ed6b12b5e8855d","origin":"The Stacks Project","memory_eligible":false,"source_rank":5915,"rank":5915,"depth":27,"x":2047.834,"y":1086.365,"cluster":"duality-cohomology"},{"id":"stacks:0200","tag":"0200","title":"Chow's Lemma · Lemma 0200","summary":"[EGA] Let S be a Noetherian scheme. Let f : X → S be a separated morphism of finite type. Then there exist an n ≥ 0 and a diagram xymatrix X ar[rd] & X' ar[d] ar[l]^π ar[r] & P^n_S ar[dl] & S & where X' → P^n_S is an immersion, and π : X' → X is proper and surjective. Moreover, we may arrange it such that there exists a dense open subscheme U ⊂ X such that π^-1(U) → U is an isomorphism.","statement_latex":"\\begin{reference}\n\\cite[II Theorem 5.6.1(a)]{EGA}\n\\end{reference}\nLet $S$ be a Noetherian scheme.\nLet $f : X \\to S$ be a separated morphism of finite type.\nThen there exist an $n \\geq 0$ and a diagram\n$$\n\\xymatrix{\nX \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\\n& S &\n}\n$$\nwhere $X' \\to \\mathbf{P}^n_S$ is an immersion, and\n$\\pi : X' \\to X$ is proper and surjective. Moreover, we may\narrange it such that there exists a dense open subscheme\n$U \\subset X$ such that $\\pi^{-1}(U) \\to U$ is an isomorphism.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Chow's Lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0200","source_file":"coherent.tex","source_line":4922,"source_end_line":4940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L4922-L4940","statement_sha256":"9d31d8e2a135bf9dcfcfeaf7bfe53aa1556881162ef9525287f7a9f8b6b4e04c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5916,"rank":5916,"depth":22,"x":2060.171,"y":1189.983,"cluster":"duality-cohomology"},{"id":"stacks:02O5","tag":"02O5","title":"Higher direct images of coherent sheaves · Proposition 02O5","summary":"[EGA] Let S be a locally Noetherian scheme. Let f : X → S be a proper morphism. Let F be a coherent O_X-module. Then R^if_*F is a coherent O_S-module for all i ≥ 0.","statement_latex":"\\begin{reference}\n\\cite[III Theorem 3.2.1]{EGA}\n\\end{reference}\nLet $S$ be a locally Noetherian scheme.\nLet $f : X \\to S$ be a proper morphism.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThen $R^if_*\\mathcal{F}$ is a coherent $\\mathcal{O}_S$-module\nfor all $i \\geq 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Higher direct images of coherent sheaves","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02O5","source_file":"coherent.tex","source_line":5107,"source_end_line":5117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5107-L5117","statement_sha256":"a42da28a945130907b8147eedb58ece080d791bc3535de420ac4fa5ea5cf3442","origin":"The Stacks Project","memory_eligible":false,"source_rank":5917,"rank":5917,"depth":30,"x":1967.167,"y":1120.139,"cluster":"duality-cohomology"},{"id":"stacks:02O6","tag":"02O6","title":"Higher direct images of coherent sheaves · Lemma 02O6","summary":"Let S = Spec(A) with A a Noetherian ring. Let f : X → S be a proper morphism. Let F be a coherent O_X-module. Then H^i(X, F) is a finite A-module for all i ≥ 0.","statement_latex":"Let $S = \\Spec(A)$ with $A$ a Noetherian ring.\nLet $f : X \\to S$ be a proper morphism.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThen $H^i(X, \\mathcal{F})$ is a finite $A$-module for all $i \\geq 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Higher direct images of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02O6","source_file":"coherent.tex","source_line":5226,"source_end_line":5232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5226-L5232","statement_sha256":"59e2607646c915a3ff30765547921af72e81f7e8b796016e885a4361d5df52ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":5918,"rank":5918,"depth":31,"x":2092.688,"y":1118.863,"cluster":"duality-cohomology"},{"id":"stacks:0897","tag":"0897","title":"Higher direct images of coherent sheaves · Lemma 0897","summary":"Let A be a Noetherian ring. Let B be a finitely generated graded A-algebra. Let f : X → Spec(A) be a proper morphism. Set B = f^*widetilde B. Let F be a quasi-coherent graded B-module of finite type. • For every p ≥ 0 the graded B-module H^p(X, F) is a finite B-module. • If L is an ample invertible O_X-module, then there exists an integer d_0 such that H^p(X, F ⊗ L^⊗ d) = 0 for all p > 0 and d ≥ d_0.","statement_latex":"Let $A$ be a Noetherian ring.\nLet $B$ be a finitely generated graded $A$-algebra.\nLet $f : X \\to \\Spec(A)$ be a proper morphism.\nSet $\\mathcal{B} = f^*\\widetilde B$.\nLet $\\mathcal{F}$ be a quasi-coherent\ngraded $\\mathcal{B}$-module of finite type.\n\\begin{enumerate}\n\\item For every $p \\geq 0$ the graded $B$-module $H^p(X, \\mathcal{F})$\nis a finite $B$-module.\n\\item If $\\mathcal{L}$ is an ample invertible $\\mathcal{O}_X$-module,\nthen there exists an integer $d_0$ such that\n$H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes d}) = 0$\nfor all $p > 0$ and $d \\geq d_0$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Higher direct images of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0897","source_file":"coherent.tex","source_line":5246,"source_end_line":5262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5246-L5262","statement_sha256":"4efb007ab9ffb4c45e2d023822280c2a35a557ebd93235c3ffb097a711392aeb","origin":"The Stacks Project","memory_eligible":false,"source_rank":5919,"rank":5919,"depth":32,"x":2000.595,"y":1191.469,"cluster":"duality-cohomology"},{"id":"stacks:02O8","tag":"02O8","title":"The theorem on formal functions · Lemma 02O8","summary":"[EGA] Let A be a Noetherian ring. Let I ⊂ A be an ideal. Set B = bigoplus_n ≥ 0 I^n. Let f : X → Spec(A) be a proper morphism. Let F be a coherent sheaf on X. Then for every p ≥ 0 the graded B-module bigoplus_n ≥ 0 H^p(X, I^nF) is a finite B-module.","statement_latex":"\\begin{reference}\n\\cite[III Cor 3.3.2]{EGA}\n\\end{reference}\nLet $A$ be a Noetherian ring.\nLet $I \\subset A$ be an ideal.\nSet $B = \\bigoplus_{n \\geq 0} I^n$.\nLet $f : X \\to \\Spec(A)$ be a proper morphism.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nThen for every $p \\geq 0$ the graded $B$-module\n$\\bigoplus_{n \\geq 0} H^p(X, I^n\\mathcal{F})$ is\na finite $B$-module.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02O8","source_file":"coherent.tex","source_line":5379,"source_end_line":5392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5379-L5392","statement_sha256":"a367f1a1cd124ead890964832461e152d065a89077d1d5bd821f5b22be7874d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5920,"rank":5920,"depth":33,"x":2010.174,"y":1085.034,"cluster":"duality-cohomology"},{"id":"stacks:02O9","tag":"02O9","title":"The theorem on formal functions · Lemma 02O9","summary":"Given a morphism of schemes f : X → Y, a quasi-coherent sheaf F on X, and a quasi-coherent sheaf of ideals I ⊂ O_Y. Assume Y locally Noetherian, f proper, and F coherent. Then M = bigoplus_n ≥ 0 R^pf_*(I^nF) is a graded A = bigoplus_n ≥ 0 I^n-module which is quasi-coherent and of finite type.","statement_latex":"Given a morphism of schemes $f : X \\to Y$, a quasi-coherent sheaf\n$\\mathcal{F}$ on $X$, and a quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_Y$. Assume $Y$ locally\nNoetherian, $f$ proper, and $\\mathcal{F}$ coherent.\nThen\n$$\n\\mathcal{M} =\n\\bigoplus\\nolimits_{n \\geq 0} R^pf_*(\\mathcal{I}^n\\mathcal{F})\n$$\nis a graded $\\mathcal{A} = \\bigoplus_{n \\geq 0} \\mathcal{I}^n$-module\nwhich is quasi-coherent and of finite type.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02O9","source_file":"coherent.tex","source_line":5401,"source_end_line":5414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5401-L5414","statement_sha256":"720eb1414b6d3bb7e2d419dc00872b6730734e9bb4af35b29b5f67801c268db9","origin":"The Stacks Project","memory_eligible":false,"source_rank":5921,"rank":5921,"depth":34,"x":2089.171,"y":1169.456,"cluster":"duality-cohomology"},{"id":"stacks:02OA","tag":"02OA","title":"The theorem on formal functions · Lemma 02OA","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let f : X → Spec(A) be a proper morphism. Let F be a coherent sheaf on X. Then for every p ≥ 0 there exists an integer c ≥ 0 such that • the multiplication map I^n - c ⊗ H^p(X, I^cF) → H^p(X, I^nF) is surjective for all n ≥ c, • the image of H^p(X, I^n + mF) → H^p(X, I^nF) is contained in the submodule I^m - e H^p(X, I^nF) where e = max(0, c - n) for n + m ≥ c, n, m ≥ 0, • we have Ker(H^p(X, I^nF) → H^p(X, F)) =…","statement_latex":"Let $A$ be a Noetherian ring.\nLet $I \\subset A$ be an ideal.\nLet $f : X \\to \\Spec(A)$ be a proper morphism.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nThen for every $p \\geq 0$ there exists an integer $c \\geq 0$\nsuch that\n\\begin{enumerate}\n\\item the multiplication map\n$I^{n - c} \\otimes H^p(X, I^c\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$\nis surjective for all $n \\geq c$,\n\\item the image of $H^p(X, I^{n + m}\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$\nis contained in the submodule $I^{m - e} H^p(X, I^n\\mathcal{F})$\nwhere $e = \\max(0, c - n)$ for $n + m \\geq c$, $n, m \\geq 0$,\n\\item we have\n$$\n\\Ker(H^p(X, I^n\\mathcal{F}) \\to H^p(X, \\mathcal{F})) =\n\\Ker(H^p(X, I^n\\mathcal{F}) \\to H^p(X, I^{n - c}\\mathcal{F}))\n$$\nfor $n \\geq c$,\n\\item there are maps $I^nH^p(X, \\mathcal{F}) \\to H^p(X, I^{n - c}\\mathcal{F})$\nfor $n \\geq c$ such that the compositions\n$$\nH^p(X, I^n\\mathcal{F}) \\to \nI^{n - c}H^p(X, \\mathcal{F}) \\to\nH^p(X, I^{n - 2c}\\mathcal{F})\n$$\nand\n$$\nI^nH^p(X, \\mathcal{F}) \\to\nH^p(X, I^{n - c}\\mathcal{F}) \\to\nI^{n - 2c}H^p(X, \\mathcal{F})\n$$\nfor $n \\geq 2c$ are the canonical ones, and\n\\item the inverse systems $(H^p(X, I^n\\mathcal{F}))$ and\n$(I^nH^p(X, \\mathcal{F}))$ are pro-isomorphic.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OA","source_file":"coherent.tex","source_line":5423,"source_end_line":5461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5423-L5461","statement_sha256":"e993d10d8908237ccdd033513bc0af65aca543932e24272fbb82babf84dfe345","origin":"The Stacks Project","memory_eligible":false,"source_rank":5922,"rank":5922,"depth":34,"x":1962.285,"y":1151.925,"cluster":"duality-cohomology"},{"id":"stacks:02OB","tag":"02OB","title":"The theorem on formal functions · Lemma 02OB","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let f : X → Spec(A) be a proper morphism. Let F be a coherent sheaf on X. Fix p ≥ 0. There exists a c ≥ 0 such that • for all n ≥ c we have Ker(H^p(X, F) → H^p(X, F/I^nF)) ⊂ I^n - cH^p(X, F). • the inverse system (H^p(X, F/I^nF))_n ∈ N satisfies the Mittag-Leffler condition (see Homology, Definition [Tag 02N0]), and • we have Im(H^p(X, F/I^kF) → H^p(X, F/I^nF)) = Im(H^p(X, F) → H^p(X, F/I^nF)) for all k ≥ n + c.","statement_latex":"Let $A$ be a Noetherian ring.\nLet $I \\subset A$ be an ideal.\nLet $f : X \\to \\Spec(A)$ be a proper morphism.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nFix $p \\geq 0$. There exists a $c \\geq 0$ such that\n\\begin{enumerate}\n\\item for all $n \\geq c$ we have\n$$\n\\Ker(H^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})) \\subset\nI^{n - c}H^p(X, \\mathcal{F}).\n$$\n\\item the inverse system\n$$\n\\left(H^p(X, \\mathcal{F}/I^n\\mathcal{F})\\right)_{n \\in \\mathbf{N}}\n$$\nsatisfies the Mittag-Leffler condition (see\nHomology, Definition \\ref{homology-definition-Mittag-Leffler}), and\n\\item we have\n$$\n\\Im(H^p(X, \\mathcal{F}/I^k\\mathcal{F})\n\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))\n=\n\\Im(H^p(X, \\mathcal{F})\n\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))\n$$\nfor all $k \\geq n + c$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OB","source_file":"coherent.tex","source_line":5552,"source_end_line":5581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5552-L5581","statement_sha256":"89fd7d03872f15cfdaf32f3c63e202e3304034888d39f3eb851583f28871f56b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5923,"rank":5923,"depth":35,"x":2070.577,"y":1092.509,"cluster":"duality-cohomology"},{"id":"stacks:02OC","tag":"02OC","title":"Theorem on formal functions · Theorem 02OC","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let f : X → Spec(A) be a proper morphism. Let F be a coherent sheaf on X. Fix p ≥ 0. The system of maps H^p(X, F)/I^nH^p(X, F) → H^p(X, F/I^nF) define an isomorphism of limits H^p(X, F)^wedge → lim_n H^p(X, F/I^nF) where the left hand side is the completion of the A-module H^p(X, F) with respect to the ideal I, see Algebra, Section [Tag 00M9]. Moreover, this is in fact a homeomorphism for the limit topologies.","statement_latex":"Let $A$ be a Noetherian ring.\nLet $I \\subset A$ be an ideal.\nLet $f : X \\to \\Spec(A)$ be a proper morphism.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nFix $p \\geq 0$.\nThe system of maps\n$$\nH^p(X, \\mathcal{F})/I^nH^p(X, \\mathcal{F})\n\\longrightarrow\nH^p(X, \\mathcal{F}/I^n\\mathcal{F})\n$$\ndefine an isomorphism of limits\n$$\nH^p(X, \\mathcal{F})^\\wedge\n\\longrightarrow\n\\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F})\n$$\nwhere the left hand side is the completion of the $A$-module\n$H^p(X, \\mathcal{F})$ with respect to the ideal $I$, see\nAlgebra, Section \\ref{algebra-section-completion}.\nMoreover, this is in fact a homeomorphism for the limit topologies.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"The theorem on formal functions","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OC","source_file":"coherent.tex","source_line":5647,"source_end_line":5670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5647-L5670","statement_sha256":"a12e80fcf5889d6611fbd8f971f702f4bbd0fd5c6837a2444d9f7d15d0ae8b58","origin":"The Stacks Project","memory_eligible":false,"source_rank":5924,"rank":5924,"depth":36,"x":2038.317,"y":1198.377,"cluster":"duality-cohomology"},{"id":"stacks:087U","tag":"087U","title":"The theorem on formal functions · Lemma 087U","summary":"Let A be a ring. Let I ⊂ A be an ideal. Assume A is Noetherian and complete with respect to I. Let f : X → Spec(A) be a proper morphism. Let F be a coherent sheaf on X. Then H^p(X, F) = lim_n H^p(X, F/I^nF) for all p ≥ 0.","statement_latex":"Let $A$ be a ring. Let $I \\subset A$ be an ideal. Assume $A$ is\nNoetherian and complete with respect to $I$.\nLet $f : X \\to \\Spec(A)$ be a proper morphism.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nThen\n$$\nH^p(X, \\mathcal{F}) = \\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F})\n$$\nfor all $p \\geq 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087U","source_file":"coherent.tex","source_line":5697,"source_end_line":5708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5697-L5708","statement_sha256":"8e65bef03e8bce150d8ddcb7d0aa1c52fe1a8f3b8340ae8e0683e758c39099bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":5925,"rank":5925,"depth":37,"x":1976.621,"y":1101.456,"cluster":"duality-cohomology"},{"id":"stacks:02OD","tag":"02OD","title":"The theorem on formal functions · Lemma 02OD","summary":"Given a morphism of schemes f : X → Y and a quasi-coherent sheaf F on X. Assume • Y locally Noetherian, • f proper, and • F coherent. Let y ∈ Y be a point. Consider the infinitesimal neighbourhoods xymatrix X_n = Spec(O_Y, y/ m_y^n) ×_Y X ar[r]_-i_n ar[d]_f_n & X ar[d]^f Spec(O_Y, y/ m_y^n) ar[r]^-c_n & Y of the fibre X_1 = X_y and set F_n = i_n^*F. Then we have (R^pf_*F)_y^wedge ≅ lim_n H^p(X_n, F_n) as O_Y, y^wedge-modules.","statement_latex":"Given a morphism of schemes $f : X \\to Y$ and a quasi-coherent sheaf\n$\\mathcal{F}$ on $X$. Assume\n\\begin{enumerate}\n\\item $Y$ locally Noetherian,\n\\item $f$ proper, and\n\\item $\\mathcal{F}$ coherent.\n\\end{enumerate}\nLet $y \\in Y$ be a point. Consider the infinitesimal neighbourhoods\n$$\n\\xymatrix{\nX_n =\n\\Spec(\\mathcal{O}_{Y, y}/\\mathfrak m_y^n) \\times_Y X\n\\ar[r]_-{i_n} \\ar[d]_{f_n} &\nX \\ar[d]^f \\\\\n\\Spec(\\mathcal{O}_{Y, y}/\\mathfrak m_y^n) \\ar[r]^-{c_n} & Y\n}\n$$\nof the fibre $X_1 = X_y$ and set $\\mathcal{F}_n = i_n^*\\mathcal{F}$.\nThen we have\n$$\n\\left(R^pf_*\\mathcal{F}\\right)_y^\\wedge\n\\cong\n\\lim_n H^p(X_n, \\mathcal{F}_n)\n$$\nas $\\mathcal{O}_{Y, y}^\\wedge$-modules.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OD","source_file":"coherent.tex","source_line":5720,"source_end_line":5747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5720-L5747","statement_sha256":"c3234a8094fa4c73ec220fb167039946d1e22f00c80d605c5aa26da7da793272","origin":"The Stacks Project","memory_eligible":false,"source_rank":5926,"rank":5926,"depth":37,"x":2100.752,"y":1138.123,"cluster":"duality-cohomology"},{"id":"stacks:02OE","tag":"02OE","title":"The theorem on formal functions · Lemma 02OE","summary":"Let f : X → Y be a morphism of schemes. Let y ∈ Y. Assume • Y locally Noetherian, • f is proper, and • f^-1((y)) is finite. Then for any coherent sheaf F on X we have (R^pf_*F)_y = 0 for all p > 0.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $y \\in Y$.\nAssume\n\\begin{enumerate}\n\\item $Y$ locally Noetherian,\n\\item $f$ is proper, and\n\\item $f^{-1}(\\{y\\})$ is finite.\n\\end{enumerate}\nThen for any coherent sheaf $\\mathcal{F}$ on $X$ we have\n$(R^pf_*\\mathcal{F})_y = 0$ for all $p > 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OE","source_file":"coherent.tex","source_line":5795,"source_end_line":5807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5795-L5807","statement_sha256":"0806512b40d8691e52e9ed776e47aadb1daf7e2c6221c3b25ca88341441ba9ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":5927,"rank":5927,"depth":38,"x":1979.057,"y":1181.761,"cluster":"duality-cohomology"},{"id":"stacks:02V7","tag":"02V7","title":"The theorem on formal functions · Lemma 02V7","summary":"Let f : X → Y be a morphism of schemes. Let y ∈ Y. Assume • Y locally Noetherian, • f is proper, and • dim(X_y) = d. Then for any coherent sheaf F on X we have (R^pf_*F)_y = 0 for all p > d.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $y \\in Y$.\nAssume\n\\begin{enumerate}\n\\item $Y$ locally Noetherian,\n\\item $f$ is proper, and\n\\item $\\dim(X_y) = d$.\n\\end{enumerate}\nThen for any coherent sheaf $\\mathcal{F}$ on $X$ we have\n$(R^pf_*\\mathcal{F})_y = 0$ for all $p > d$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V7","source_file":"coherent.tex","source_line":5829,"source_end_line":5841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5829-L5841","statement_sha256":"24471a79908580b5d56dfe1004b7fff7af8dfafe8724feeab78ad3b41ff824e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5928,"rank":5928,"depth":38,"x":2034.004,"y":1079.972,"cluster":"duality-cohomology"},{"id":"stacks:02OG","tag":"02OG","title":"Applications of the theorem on formal functions · Lemma 02OG","summary":"(For a more general version see More on Morphisms, Lemma [Tag 02LS].) Let f : X → S be a morphism of schemes. Assume S is locally Noetherian. The following are equivalent • f is finite, and • f is proper with finite fibres.","statement_latex":"(For a more general version see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-characterize-finite}.)\nLet $f : X \\to S$ be a morphism of schemes.\nAssume $S$ is locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is finite, and\n\\item $f$ is proper with finite fibres.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Applications of the theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OG","source_file":"coherent.tex","source_line":5880,"source_end_line":5891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5880-L5891","statement_sha256":"fc1a67fb398bc37782d0d5457d20f71de3e8a35959f741403c2741774e86afc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5929,"rank":5929,"depth":39,"x":2075.567,"y":1186.788,"cluster":"duality-cohomology"},{"id":"stacks:02OH","tag":"02OH","title":"Applications of the theorem on formal functions · Lemma 02OH","summary":"A proper morphism is finite in a neighbourhood of a finite fiber. (For a more general version see More on Morphisms, Lemma [Tag 02UP].) Let f : X → S be a morphism of schemes. Let s ∈ S. Assume • S is locally Noetherian, • f is proper, and • f^-1((s)) is a finite set. Then there exists an open neighbourhood V ⊂ S of s such that f|_f^-1(V) : f^-1(V) → V is finite.","statement_latex":"\\begin{slogan}\nA proper morphism is finite in a neighbourhood of a finite fiber.\n\\end{slogan}\n(For a more general version see\nMore on Morphisms,\nLemma \\ref{more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}.)\nLet $f : X \\to S$ be a morphism of schemes.\nLet $s \\in S$.\nAssume\n\\begin{enumerate}\n\\item $S$ is locally Noetherian,\n\\item $f$ is proper, and\n\\item $f^{-1}(\\{s\\})$ is a finite set.\n\\end{enumerate}\nThen there exists an open neighbourhood $V \\subset S$ of $s$\nsuch that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Applications of the theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OH","source_file":"coherent.tex","source_line":5937,"source_end_line":5955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5937-L5955","statement_sha256":"051b7a568d9406f5ac5f63b6e3222e919ac8a4268fa1f97000a81cc1a13b2308","origin":"The Stacks Project","memory_eligible":false,"source_rank":5930,"rank":5930,"depth":40,"x":1958.392,"y":1131.308,"cluster":"duality-cohomology"},{"id":"stacks:0D2M","tag":"0D2M","title":"Applications of the theorem on formal functions · Lemma 0D2M","summary":"Let f : X → Y be a proper morphism of schemes with Y Noetherian. Let L be an invertible O_X-module. Let F be a coherent O_X-module. Let y ∈ Y be a point such that L_y is ample on X_y. Then there exists a d_0 such that for all d ≥ d_0 we have R^pf_*(F ⊗_O_X L^⊗ d)_y = 0 for p > 0 and the map f_*(F ⊗_O_X L^⊗ d)_y → H^0(X_y, F_y ⊗_O_X_y L_y^⊗ d) is surjective.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes with $Y$ Noetherian.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nLet $y \\in Y$ be a point such that $\\mathcal{L}_y$ is ample on $X_y$.\nThen there exists a $d_0$ such that for all $d \\geq d_0$ we have\n$$\nR^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y = 0\n\\text{ for }p > 0\n$$\nand the map\n$$\nf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y\n\\longrightarrow\nH^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d})\n$$\nis surjective.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Applications of the theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2M","source_file":"coherent.tex","source_line":5973,"source_end_line":5991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L5973-L5991","statement_sha256":"1230598a2c4087366ba9c2d57224703fa289b6b1a53856239ee9206a9c2fa1ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":5931,"rank":5931,"depth":37,"x":2090.11,"y":1105.595,"cluster":"duality-cohomology"},{"id":"stacks:0D2N","tag":"0D2N","title":"Applications of the theorem on formal functions · Lemma 0D2N","summary":"(For a more general version see More on Morphisms, Lemma [Tag 0D2S].) Let f : X → Y be a proper morphism of schemes with Y Noetherian. Let L be an invertible O_X-module. Let y ∈ Y be a point such that L_y is ample on X_y. Then there is an open neighbourhood V ⊂ Y of y such that L|_f^-1(V) is ample on f^-1(V)/V.","statement_latex":"(For a more general version see\nMore on Morphisms,\nLemma \\ref{more-morphisms-lemma-ample-in-neighbourhood}.)\nLet $f : X \\to Y$ be a proper morphism of schemes with $Y$ Noetherian.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $y \\in Y$ be a point such that $\\mathcal{L}_y$ is ample\non $X_y$. Then there is an open neighbourhood $V \\subset Y$\nof $y$ such that $\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Applications of the theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2N","source_file":"coherent.tex","source_line":6090,"source_end_line":6100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6090-L6100","statement_sha256":"aee95f21a2a46c9a7ab1113a0f8b2447efd2aff9eb1af902980badb7135c37fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":5932,"rank":5932,"depth":41,"x":2013.256,"y":1199.79,"cluster":"duality-cohomology"},{"id":"stacks:07VK","tag":"07VK","title":"Cohomology and base change, III · Lemma 07VK","summary":"Let A be a Noetherian ring and set S = Spec(A). Let f : X → S be a proper morphism of schemes. Let F be a coherent O_X-module flat over S. Then • RΓ(X, F) is a perfect object of D(A), and • for any ring map A → A' the base change map RΓ(X, F) ⊗_A^L A' → RΓ(X_A', F_A') is an isomorphism.","statement_latex":"Let $A$ be a Noetherian ring and set $S = \\Spec(A)$. Let $f : X \\to S$ be a\nproper morphism of schemes. Let $\\mathcal{F}$ be a coherent\n$\\mathcal{O}_X$-module flat over $S$. Then\n\\begin{enumerate}\n\\item $R\\Gamma(X, \\mathcal{F})$ is a perfect object of $D(A)$, and\n\\item for any ring map $A \\to A'$ the base change map\n$$\nR\\Gamma(X, \\mathcal{F}) \\otimes_A^{\\mathbf{L}} A'\n\\longrightarrow\nR\\Gamma(X_{A'}, \\mathcal{F}_{A'})\n$$\nis an isomorphism.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Cohomology and base change, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VK","source_file":"coherent.tex","source_line":6171,"source_end_line":6186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6171-L6186","statement_sha256":"e377870baf3a835f66d5133a796298426b30f464bb27b855a1bd47af7d612ebd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5933,"rank":5933,"depth":32,"x":1994.079,"y":1086.132,"cluster":"duality-cohomology"},{"id":"stacks:087W","tag":"087W","title":"Coherent formal modules · Lemma 087W","summary":"If X = Spec(A) is the spectrum of a Noetherian ring and I is the quasi-coherent sheaf of ideals associated to the ideal I ⊂ A, then Coh(X, I) is equivalent to the category of finite A^wedge-modules where A^wedge is the completion of A with respect to I.","statement_latex":"If $X = \\Spec(A)$ is the spectrum of a Noetherian ring and\n$\\mathcal{I}$ is the quasi-coherent sheaf of ideals associated to the ideal\n$I \\subset A$, then $\\textit{Coh}(X, \\mathcal{I})$ is equivalent to the\ncategory of finite $A^\\wedge$-modules where $A^\\wedge$ is the completion\nof $A$ with respect to $I$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087W","source_file":"coherent.tex","source_line":6271,"source_end_line":6278,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6271-L6278","statement_sha256":"6e0dd9137ae2a57fa6175f7b12b515d7815ccac2b5987301ea7c0ec9b8c70496","origin":"The Stacks Project","memory_eligible":false,"source_rank":5934,"rank":5934,"depth":8,"x":2100.166,"y":1159.439,"cluster":"duality-cohomology"},{"id":"stacks:087X","tag":"087X","title":"Coherent formal modules · Lemma 087X","summary":"Let X be a Noetherian scheme and let I ⊂ O_X be a quasi-coherent sheaf of ideals. • The category Coh(X, I) is abelian. • For U ⊂ X open the restriction functor Coh(X, I) → Coh(U, I|_U) is exact. • Exactness in Coh(X, I) may be checked by restricting to the members of an open covering of X.","statement_latex":"Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals.\n\\begin{enumerate}\n\\item The category $\\textit{Coh}(X, \\mathcal{I})$ is abelian.\n\\item For $U \\subset X$ open the restriction functor\n$\\textit{Coh}(X, \\mathcal{I}) \\to \\textit{Coh}(U, \\mathcal{I}|_U)$\nis exact.\n\\item Exactness in $\\textit{Coh}(X, \\mathcal{I})$ may be checked by\nrestricting to the members of an open covering of $X$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087X","source_file":"coherent.tex","source_line":6312,"source_end_line":6324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6312-L6324","statement_sha256":"7060132471541208e91320e00937ff05ed35787d4f7845a501f782d2b64c1eb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5935,"rank":5935,"depth":9,"x":1962.286,"y":1165.61,"cluster":"duality-cohomology"},{"id":"stacks:087Y","tag":"087Y","title":"Coherent formal modules · Lemma 087Y","summary":"Let X be a Noetherian scheme and let I ⊂ O_X be a quasi-coherent sheaf of ideals. A map (F_n) → (G_n) is surjective in Coh(X, I) if and only if F_1 → G_1 is surjective.","statement_latex":"Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. A map\n$(\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ is surjective in\n$\\textit{Coh}(X, \\mathcal{I})$\nif and only if $\\mathcal{F}_1 \\to \\mathcal{G}_1$ is surjective.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087Y","source_file":"coherent.tex","source_line":6395,"source_end_line":6402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6395-L6402","statement_sha256":"8ea79d97540cf888ef4750c36bbeae9203a544d618b69d6c5edb2ad3e61728dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":5936,"rank":5936,"depth":9,"x":2059.485,"y":1082.404,"cluster":"duality-cohomology"},{"id":"stacks:0881","tag":"0881","title":"Coherent formal modules · Lemma 0881","summary":"The functor ([Tag 0880]) is exact.","statement_latex":"The functor (\\ref{equation-completion-functor}) is exact.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0881","source_file":"coherent.tex","source_line":6422,"source_end_line":6425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6422-L6425","statement_sha256":"3d84ad28c8ce1221d5e389cd6fb6d0c0c70c60cbc1f625d1ea281346a2c1ea5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5937,"rank":5937,"depth":9,"x":2054.698,"y":1199.496,"cluster":"duality-cohomology"},{"id":"stacks:0882","tag":"0882","title":"Coherent formal modules · Lemma 0882","summary":"Let X be a Noetherian scheme and let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let F, G be coherent O_X-modules. Set H = SheafHom_O_X(G, F). Then lim H^0(X, H/I^nH) = Mor_Coh(X, I) (G^wedge, F^wedge).","statement_latex":"Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. Let $\\mathcal{F}$, $\\mathcal{G}$ be\ncoherent $\\mathcal{O}_X$-modules. Set\n$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{G}, \\mathcal{F})$.\nThen\n$$\n\\lim H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}) =\n\\Mor_{\\textit{Coh}(X, \\mathcal{I})}\n(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge).\n$$","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0882","source_file":"coherent.tex","source_line":6437,"source_end_line":6449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6437-L6449","statement_sha256":"873f70c4a5f742ad1321db00c440e789fc25309b0a354ef01f11c017446d5f20","origin":"The Stacks Project","memory_eligible":false,"source_rank":5938,"rank":5938,"depth":14,"x":1963.616,"y":1109.995,"cluster":"duality-cohomology"},{"id":"stacks:0889","tag":"0889","title":"Coherent formal modules · Lemma 0889","summary":"Let X be a Noetherian scheme and let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let G be a coherent O_X-module. Let (F_n) an object of Coh(X, I). • If α : (F_n) → G^wedge is a map whose kernel and cokernel are annihilated by a power of I, then there exists a unique (up to unique isomorphism) triple (F, a, β) where • F is a coherent O_X-module, • a : F → G is an O_X-module map whose kernel and cokernel are annihilated by a power of I, • β : (F_n) → F^wedge is an…","statement_latex":"Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. Let $\\mathcal{G}$ be a coherent\n$\\mathcal{O}_X$-module. Let $(\\mathcal{F}_n)$ an object of\n$\\textit{Coh}(X, \\mathcal{I})$.\n\\begin{enumerate}\n\\item If $\\alpha : (\\mathcal{F}_n) \\to \\mathcal{G}^\\wedge$ is\na map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,\nthen there exists a unique (up to unique isomorphism) triple\n$(\\mathcal{F}, a, \\beta)$ where\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module,\n\\item $a : \\mathcal{F} \\to \\mathcal{G}$ is an $\\mathcal{O}_X$-module map\nwhose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,\n\\item $\\beta : (\\mathcal{F}_n) \\to \\mathcal{F}^\\wedge$ is an isomorphism, and\n\\item $\\alpha = a^\\wedge \\circ \\beta$.\n\\end{enumerate}\n\\item If $\\alpha : \\mathcal{G}^\\wedge \\to (\\mathcal{F}_n)$ is\na map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,\nthen there exists a unique (up to unique isomorphism) triple\n$(\\mathcal{F}, a, \\beta)$ where\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module,\n\\item $a : \\mathcal{G} \\to \\mathcal{F}$ is an $\\mathcal{O}_X$-module map\nwhose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,\n\\item $\\beta : \\mathcal{F}^\\wedge \\to (\\mathcal{F}_n)$ is an isomorphism, and\n\\item $\\alpha = \\beta \\circ a^\\wedge$.\n\\end{enumerate}\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0889","source_file":"coherent.tex","source_line":6486,"source_end_line":6516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6486-L6516","statement_sha256":"3c49d7f8bc44629fe67ef1f42ab68bdf1a247948983555f9153ae1765c4ac82a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5939,"rank":5939,"depth":21,"x":2103.437,"y":1124.382,"cluster":"duality-cohomology"},{"id":"stacks:0EHP","tag":"0EHP","title":"Coherent formal modules · Lemma 0EHP","summary":"Let X be a Noetherian scheme and let I ⊂ O_X be a quasi-coherent sheaf of ideals. Any object of Coh(X, I) which is annihilated by a power of I is in the essential image of ([Tag 0880]). Moreover, if F, G are in Coh(O_X) and either F or G is annihilated by a power of I, then the maps xymatrix Hom_X(F, G) ar[d] & Ext_X(F, G) ar[d] Hom_Coh(X, I)(F^wedge, G^wedge) & Ext_Coh(X, I)(F^wedge, G^wedge) are isomorphisms.","statement_latex":"Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. Any object of\n$\\textit{Coh}(X, \\mathcal{I})$ which is annihilated\nby a power of $\\mathcal{I}$ is in the essential image of\n(\\ref{equation-completion-functor}).\nMoreover, if $\\mathcal{F}$, $\\mathcal{G}$ are in $\\textit{Coh}(\\mathcal{O}_X)$\nand either $\\mathcal{F}$ or $\\mathcal{G}$ is annihilated by a power of\n$\\mathcal{I}$, then the maps\n$$\n\\xymatrix{\n\\Hom_X(\\mathcal{F}, \\mathcal{G}) \\ar[d] &\n\\Ext_X(\\mathcal{F}, \\mathcal{G}) \\ar[d] \\\\\n\\Hom_{\\textit{Coh}(X, \\mathcal{I})}(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge) &\n\\Ext_{\\textit{Coh}(X, \\mathcal{I})}(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge)\n}\n$$\nare isomorphisms.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHP","source_file":"coherent.tex","source_line":6548,"source_end_line":6567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6548-L6567","statement_sha256":"512f056839fd3b2edb2b0125dc5d8c6eb9aa483ad4786f39be539bd4789577b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5940,"rank":5940,"depth":22,"x":1988.207,"y":1193.444,"cluster":"duality-cohomology"},{"id":"stacks:087Z","tag":"087Z","title":"Coherent formal modules · Lemma 087Z","summary":"Let X be a Noetherian scheme and let I ⊂ O_X be a quasi-coherent sheaf of ideals. If (F_n) is an object of Coh(X, I) then bigoplus Ker(F_n + 1 → F_n) is a finite type, graded, quasi-coherent bigoplus I^n/I^n + 1-module.","statement_latex":"Let $X$ be a Noetherian scheme and let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. If $(\\mathcal{F}_n)$ is an object of\n$\\textit{Coh}(X, \\mathcal{I})$ then\n$\\bigoplus \\Ker(\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n)$ is\na finite type, graded, quasi-coherent\n$\\bigoplus \\mathcal{I}^n/\\mathcal{I}^{n + 1}$-module.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087Z","source_file":"coherent.tex","source_line":6611,"source_end_line":6619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6611-L6619","statement_sha256":"0e426038ce9d3ffeacfbd932b724ce255fa208de53ea8ac5785246c7877b9cb2","origin":"The Stacks Project","memory_eligible":false,"source_rank":5941,"rank":5941,"depth":9,"x":2017.781,"y":1076.574,"cluster":"duality-cohomology"},{"id":"stacks:0887","tag":"0887","title":"Coherent formal modules · Lemma 0887","summary":"Let f : X → Y be a morphism of Noetherian schemes. Let J ⊂ O_Y be a quasi-coherent sheaf of ideals and set I = f^-1J O_X. Then there is a right exact functor f^* : Coh(Y, J) → Coh(X, I) which sends (G_n) to (f^*G_n). If f is flat, then f^* is an exact functor.","statement_latex":"Let $f : X \\to Y$ be a morphism of Noetherian schemes.\nLet $\\mathcal{J} \\subset \\mathcal{O}_Y$ be a quasi-coherent sheaf\nof ideals and set $\\mathcal{I} = f^{-1}\\mathcal{J} \\mathcal{O}_X$.\nThen there is a right exact functor\n$$\nf^* : \\textit{Coh}(Y, \\mathcal{J}) \\longrightarrow \\textit{Coh}(X, \\mathcal{I})\n$$\nwhich sends $(\\mathcal{G}_n)$ to $(f^*\\mathcal{G}_n)$. If $f$ is flat,\nthen $f^*$ is an exact functor.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0887","source_file":"coherent.tex","source_line":6630,"source_end_line":6641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6630-L6641","statement_sha256":"82e5e8489064aab632865f7cfc036aa5592f44efb02f3b46e860f8f582e9be09","origin":"The Stacks Project","memory_eligible":false,"source_rank":5942,"rank":5942,"depth":10,"x":2090.301,"y":1180.025,"cluster":"duality-cohomology"},{"id":"stacks:0EHQ","tag":"0EHQ","title":"Coherent formal modules · Lemma 0EHQ","summary":"Let f : X' → X be a morphism of Noetherian schemes. Let Z ⊂ X be a closed subscheme and denote Z' = f^-1Z the scheme theoretic inverse image. Let I ⊂ O_X, I' ⊂ O_X' be the corresponding quasi-coherent sheaves of ideals. If f is flat and the induced morphism Z' → Z is an isomorphism, then the pullback functor f^* : Coh(X, I) → Coh(X', I') (Lemma [Tag 0887]) is an equivalence.","statement_latex":"Let $f : X' \\to X$ be a morphism of Noetherian schemes. Let $Z \\subset X$\nbe a closed subscheme and denote $Z' = f^{-1}Z$ the scheme theoretic\ninverse image. Let $\\mathcal{I} \\subset \\mathcal{O}_X$,\n$\\mathcal{I}' \\subset \\mathcal{O}_{X'}$ be the corresponding\nquasi-coherent sheaves of ideals.\nIf $f$ is flat and the induced morphism $Z' \\to Z$\nis an isomorphism, then the pullback functor\n$f^* : \\textit{Coh}(X, \\mathcal{I}) \\to \\textit{Coh}(X', \\mathcal{I}')$\n(Lemma \\ref{lemma-inverse-systems-pullback})\nis an equivalence.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHQ","source_file":"coherent.tex","source_line":6665,"source_end_line":6677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6665-L6677","statement_sha256":"7931b09ed08247dce2c3ae8a7a56e28e6a1d6f319d2c088b817b0c8f128d20fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":5943,"rank":5943,"depth":18,"x":1952.987,"y":1144.725,"cluster":"duality-cohomology"},{"id":"stacks:0EHR","tag":"0EHR","title":"Coherent formal modules · Lemma 0EHR","summary":"Let X be a Noetherian scheme. Let I, J ⊂ O_X be quasi-coherent sheaves of ideals. If V(I) = V(J) is the same closed subset of X, then Coh(X, I) and Coh(X, J) are equivalent.","statement_latex":"Let $X$ be a Noetherian scheme. Let\n$\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$\nbe quasi-coherent sheaves of ideals.\nIf $V(\\mathcal{I}) = V(\\mathcal{J})$ is the same closed subset\nof $X$, then $\\textit{Coh}(X, \\mathcal{I})$ and $\\textit{Coh}(X, \\mathcal{J})$\nare equivalent.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHR","source_file":"coherent.tex","source_line":6705,"source_end_line":6713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6705-L6713","statement_sha256":"3f014787bfc3157c51c17c160482734258a0305ff22b66098cd120176cffbc23","origin":"The Stacks Project","memory_eligible":false,"source_rank":5944,"rank":5944,"depth":9,"x":2083.236,"y":1092.597,"cluster":"duality-cohomology"},{"id":"stacks:0883","tag":"0883","title":"Grothendieck's existence theorem, I · Lemma 0883","summary":"Let A be Noetherian ring complete with respect to an ideal I. Let f : X → Spec(A) be a proper morphism. Let I = IO_X. Then the functor ([Tag 0880]) is fully faithful.","statement_latex":"Let $A$ be Noetherian ring complete with respect to an ideal $I$.\nLet $f : X \\to \\Spec(A)$ be a proper morphism. Let\n$\\mathcal{I} = I\\mathcal{O}_X$.\nThen the functor (\\ref{equation-completion-functor}) is fully faithful.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's existence theorem, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0883","source_file":"coherent.tex","source_line":6759,"source_end_line":6765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6759-L6765","statement_sha256":"f1c1b3eb286ddca43ce67beeb51fbeeac6b939fa3f8b85612a53566fe8dc2104","origin":"The Stacks Project","memory_eligible":false,"source_rank":5945,"rank":5945,"depth":38,"x":2028.859,"y":1205.462,"cluster":"duality-cohomology"},{"id":"stacks:0884","tag":"0884","title":"Grothendieck's existence theorem, I · Lemma 0884","summary":"Let A be Noetherian ring and I ⊂ A an ideal. Let f : X → Spec(A) be a proper morphism and let L be an f-ample invertible sheaf. Let I = IO_X. Let (F_n) be an object of Coh(X, I). Then there exists an integer d_0 such that H^1(X, Ker(F_n + 1 → F_n) ⊗ L^⊗ d ) = 0 for all n ≥ 0 and all d ≥ d_0.","statement_latex":"Let $A$ be Noetherian ring and $I \\subset A$ an ideal.\nLet $f : X \\to \\Spec(A)$ be a proper morphism and let\n$\\mathcal{L}$ be an $f$-ample invertible sheaf. Let\n$\\mathcal{I} = I\\mathcal{O}_X$. Let $(\\mathcal{F}_n)$ be an\nobject of $\\textit{Coh}(X, \\mathcal{I})$. Then there exists an\ninteger $d_0$ such that\n$$\nH^1(X, \\Ker(\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n)\n\\otimes \\mathcal{L}^{\\otimes d} )\n= 0\n$$\nfor all $n \\geq 0$ and all $d \\geq d_0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's existence theorem, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0884","source_file":"coherent.tex","source_line":6785,"source_end_line":6799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6785-L6799","statement_sha256":"ba50a132adf69286929970338b25810954656de86528424ad1e519c23a4421f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5946,"rank":5946,"depth":33,"x":1977.961,"y":1090.858,"cluster":"duality-cohomology"},{"id":"stacks:0885","tag":"0885","title":"Grothendieck's existence theorem, I · Lemma 0885","summary":"Let A be Noetherian ring complete with respect to an ideal I. Let f : X → Spec(A) be a projective morphism. Let I = IO_X. Then the functor ([Tag 0880]) is an equivalence.","statement_latex":"Let $A$ be Noetherian ring complete with respect to an ideal $I$.\nLet $f : X \\to \\Spec(A)$ be a projective morphism. Let\n$\\mathcal{I} = I\\mathcal{O}_X$.\nThen the functor (\\ref{equation-completion-functor}) is an equivalence.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's existence theorem, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0885","source_file":"coherent.tex","source_line":6811,"source_end_line":6817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6811-L6817","statement_sha256":"3b7959d69c8879e0e570826c912cc5c4bf0132b7fc0ec28408e9ddc249d355e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5947,"rank":5947,"depth":39,"x":2108.245,"y":1146.739,"cluster":"duality-cohomology"},{"id":"stacks:088A","tag":"088A","title":"Grothendieck's existence theorem, II · Lemma 088A","summary":"Let X be a Noetherian scheme. Let I, K ⊂ O_X be quasi-coherent sheaves of ideals. Let X_e ⊂ X be the closed subscheme cut out by K^e. Let I_e = IO_X_e. Let (F_n) be an object of Coh(X, I). Assume • the functor Coh(O_X_e) → Coh(X_e, I_e) is an equivalence for all e ≥ 1, and • there exists a coherent sheaf H on X and a map α : (F_n) → H^wedge whose kernel and cokernel are annihilated by a power of K. Then (F_n) is in the essential image of ([Tag 0880]).","statement_latex":"Let $X$ be a Noetherian scheme. Let\n$\\mathcal{I}, \\mathcal{K} \\subset \\mathcal{O}_X$\nbe quasi-coherent sheaves of ideals.\nLet $X_e \\subset X$ be the closed subscheme cut out by $\\mathcal{K}^e$.\nLet $\\mathcal{I}_e = \\mathcal{I}\\mathcal{O}_{X_e}$.\nLet $(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(X, \\mathcal{I})$.\nAssume\n\\begin{enumerate}\n\\item the functor\n$\\textit{Coh}(\\mathcal{O}_{X_e}) \\to \\textit{Coh}(X_e, \\mathcal{I}_e)$\nis an equivalence for all $e \\geq 1$, and\n\\item there exists a coherent sheaf $\\mathcal{H}$ on $X$ and a map\n$\\alpha : (\\mathcal{F}_n) \\to \\mathcal{H}^\\wedge$ whose\nkernel and cokernel are annihilated by a power of $\\mathcal{K}$.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ is in the essential image of\n(\\ref{equation-completion-functor}).","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's existence theorem, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088A","source_file":"coherent.tex","source_line":6939,"source_end_line":6958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L6939-L6958","statement_sha256":"ec06031b3a7b9b2a5bf8ecdff0501d616a9577ac3073d701d59f8f0e7629606f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5948,"rank":5948,"depth":22,"x":1966.601,"y":1179.606,"cluster":"duality-cohomology"},{"id":"stacks:088B","tag":"088B","title":"Grothendieck's existence theorem, II · Lemma 088B","summary":"Let Y be a Noetherian scheme. Let J, K ⊂ O_Y be quasi-coherent sheaves of ideals. Let f : X → Y be a proper morphism which is an isomorphism over V = Y setminus V(K). Set I = f^-1J O_X. Let (G_n) be an object of Coh(Y, J), let F be a coherent O_X-module, and let β : (f^*G_n) → F^wedge be an isomorphism in Coh(X, I). Then there exists a map α : (G_n) → (f_*F)^wedge in Coh(Y, J) whose kernel and cokernel are annihilated by a power of K.","statement_latex":"Let $Y$ be a Noetherian scheme. Let\n$\\mathcal{J}, \\mathcal{K} \\subset \\mathcal{O}_Y$\nbe quasi-coherent sheaves of ideals.\nLet $f : X \\to Y$ be a proper morphism which is an isomorphism\nover $V = Y \\setminus V(\\mathcal{K})$.\nSet $\\mathcal{I} = f^{-1}\\mathcal{J} \\mathcal{O}_X$.\nLet $(\\mathcal{G}_n)$ be an object of $\\textit{Coh}(Y, \\mathcal{J})$,\nlet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module, and let\n$\\beta : (f^*\\mathcal{G}_n)  \\to \\mathcal{F}^\\wedge$ be an isomorphism in\n$\\textit{Coh}(X, \\mathcal{I})$. Then there exists a map\n$$\n\\alpha :\n(\\mathcal{G}_n)\n\\longrightarrow\n(f_*\\mathcal{F})^\\wedge\n$$\nin $\\textit{Coh}(Y, \\mathcal{J})$ whose kernel and cokernel\nare annihilated by a power of $\\mathcal{K}$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's existence theorem, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088B","source_file":"coherent.tex","source_line":7024,"source_end_line":7044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7024-L7044","statement_sha256":"8526dae034c1d5945f5a6c727bed86f82f80a296899a6a37b2ce2bf471eefe43","origin":"The Stacks Project","memory_eligible":false,"source_rank":5949,"rank":5949,"depth":39,"x":2044.966,"y":1074.531,"cluster":"duality-cohomology"},{"id":"stacks:088C","tag":"088C","title":"Grothendieck's existence theorem, II · Proposition 088C","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Let f : X → Spec(A) be a proper morphism of schemes. Set I = IO_X. Then the functor ([Tag 0880]) is an equivalence.","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nLet $f : X \\to \\Spec(A)$ be a proper morphism of schemes.\nSet $\\mathcal{I} = I\\mathcal{O}_X$.\nThen the functor (\\ref{equation-completion-functor}) is an equivalence.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's existence theorem, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088C","source_file":"coherent.tex","source_line":7146,"source_end_line":7152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7146-L7152","statement_sha256":"7d8e4b6b887262a5fc298d5f24f6c46c29c987ff6d55633b3a4481f6e84c5399","origin":"The Stacks Project","memory_eligible":false,"source_rank":5950,"rank":5950,"depth":40,"x":2071.797,"y":1197.019,"cluster":"duality-cohomology"},{"id":"stacks:0CYL","tag":"0CYL","title":"Being proper over a base · Lemma 0CYL","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let Z ⊂ X be a closed subset. The following are equivalent • the morphism Z → S is proper if Z is endowed with the reduced induced closed subscheme structure (Schemes, Definition [Tag 01J4]), • for some closed subscheme structure on Z the morphism Z → S is proper, • for any closed subscheme structure on Z the morphism Z → S is proper.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $Z \\subset X$ be a closed subset. The following are equivalent\n\\begin{enumerate}\n\\item the morphism $Z \\to S$ is proper if $Z$ is endowed with the reduced\ninduced closed subscheme structure\n(Schemes, Definition \\ref{schemes-definition-reduced-induced-scheme}),\n\\item for some closed subscheme structure on $Z$ the morphism $Z \\to S$\nis proper,\n\\item for any closed subscheme structure on $Z$ the morphism\n$Z \\to S$ is proper.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYL","source_file":"coherent.tex","source_line":7212,"source_end_line":7225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7212-L7225","statement_sha256":"9dc8be5dd09ee9e622a8f75ca0227386624d6dd0e046e31a902f63298a2063b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5951,"rank":5951,"depth":18,"x":1952.992,"y":1121.592,"cluster":"duality-cohomology"},{"id":"stacks:0CYM","tag":"0CYM","title":"Being proper over a base · Definition 0CYM","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let Z ⊂ X be a closed subset. We say Z is proper over S if the equivalent conditions of Lemma [Tag 0CYL] are satisfied.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $Z \\subset X$ be a closed subset.\nWe say {\\it $Z$ is proper over $S$}\nif the equivalent conditions of Lemma \\ref{lemma-closed-proper-over-base}\nare satisfied.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYM","source_file":"coherent.tex","source_line":7260,"source_end_line":7267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7260-L7267","statement_sha256":"cc0be9a2f0b6631653d5782a9135bb882f357011d4305f51abf66a3d04aab70a","origin":"The Stacks Project","memory_eligible":false,"source_rank":5952,"rank":5952,"depth":19,"x":2101.899,"y":1109.747,"cluster":"duality-cohomology"},{"id":"stacks:0CYN","tag":"0CYN","title":"Being proper over a base · Lemma 0CYN","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let Y ⊂ Z ⊂ X be closed subsets. If Z is proper over S, then the same is true for Y.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $Y \\subset Z \\subset X$ be closed subsets.\nIf $Z$ is proper over $S$, then the same is true for $Y$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYN","source_file":"coherent.tex","source_line":7275,"source_end_line":7280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7275-L7280","statement_sha256":"a07e287c7f11f0e4f321185d87f98e3ac4e5e38859e72e1eba7b8dadd116a0b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5953,"rank":5953,"depth":0,"x":2001.188,"y":1203.376,"cluster":"duality-cohomology"},{"id":"stacks:0CYP","tag":"0CYP","title":"Being proper over a base · Lemma 0CYP","summary":"Consider a cartesian diagram of schemes xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f S' ar[r]^g & S with f locally of finite type. If Z is a closed subset of X proper over S, then (g')^-1(Z) is a closed subset of X' proper over S'.","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nwith $f$ locally of finite type.\nIf $Z$ is a closed subset of $X$ proper over $S$, then\n$(g')^{-1}(Z)$ is a closed subset of $X'$ proper over $S'$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYP","source_file":"coherent.tex","source_line":7286,"source_end_line":7298,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7286-L7298","statement_sha256":"a317464de2297036371a7813af2d65c1d065b416b88ac9d6757a431c63186248","origin":"The Stacks Project","memory_eligible":false,"source_rank":5954,"rank":5954,"depth":17,"x":2000.153,"y":1076.65,"cluster":"duality-cohomology"},{"id":"stacks:0CYQ","tag":"0CYQ","title":"Being proper over a base · Lemma 0CYQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes which are locally of finite type over S. • If Y is separated over S and Z ⊂ X is a closed subset proper over S, then f(Z) is a closed subset of Y proper over S. • If f is universally closed and Z ⊂ X is a closed subset proper over S, then f(Z) is a closed subset of Y proper over S. • If f is proper and Z ⊂ Y is a closed subset proper over S, then f^-1(Z) is a closed subset of X proper over S.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes which\nare locally of finite type over $S$.\n\\begin{enumerate}\n\\item If $Y$ is separated over $S$ and $Z \\subset X$ is a closed subset\nproper over $S$, then $f(Z)$ is a closed subset of $Y$ proper over $S$.\n\\item If $f$ is universally closed and $Z \\subset X$ is a\nclosed subset proper over $S$, then $f(Z)$ is a closed subset\nof $Y$ proper over $S$.\n\\item If $f$ is proper and $Z \\subset Y$ is a closed subset\nproper over $S$, then $f^{-1}(Z)$ is a closed subset of $X$ proper over $S$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYQ","source_file":"coherent.tex","source_line":7311,"source_end_line":7324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7311-L7324","statement_sha256":"6ac104722563a8c40b594f2189a59f84ce5412c2c102b7cca6bee9a424147a17","origin":"The Stacks Project","memory_eligible":false,"source_rank":5955,"rank":5955,"depth":20,"x":2103.261,"y":1169.903,"cluster":"duality-cohomology"},{"id":"stacks:0CYR","tag":"0CYR","title":"Being proper over a base · Lemma 0CYR","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let Z_i ⊂ X, i = 1, …, n be closed subsets. If Z_i, i = 1, …, n are proper over S, then the same is true for Z_1 ∪ … ∪ Z_n.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $Z_i \\subset X$, $i = 1, \\ldots, n$ be closed subsets.\nIf $Z_i$, $i = 1, \\ldots, n$ are proper over $S$, then the same is\ntrue for $Z_1 \\cup \\ldots \\cup Z_n$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYR","source_file":"coherent.tex","source_line":7373,"source_end_line":7379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7373-L7379","statement_sha256":"ce1f2f7806d04ece5049ce7f4c0372bdc62246c7a3127fc5172394e8afea39e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5956,"rank":5956,"depth":21,"x":1951.604,"y":1159.602,"cluster":"duality-cohomology"},{"id":"stacks:0CYS","tag":"0CYS","title":"Being proper over a base · Lemma 0CYS","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a finite type, quasi-coherent O_X-module. The following are equivalent • the support of F is proper over S, • the scheme theoretic support of F (Morphisms, Definition [Tag 05JV]) is proper over S, and • there exists a closed subscheme Z ⊂ X and a finite type, quasi-coherent O_Z-module G such that (a) Z → S is proper, and (b) (Z → X)_*G = F.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally\nof finite type. Let $\\mathcal{F}$ be a finite type, quasi-coherent\n$\\mathcal{O}_X$-module. The following are equivalent\n\\begin{enumerate}\n\\item the support of $\\mathcal{F}$ is proper over $S$,\n\\item the scheme theoretic support of $\\mathcal{F}$\n(Morphisms, Definition \\ref{morphisms-definition-scheme-theoretic-support})\nis proper over $S$, and\n\\item there exists a closed subscheme $Z \\subset X$ and\na finite type, quasi-coherent $\\mathcal{O}_Z$-module\n$\\mathcal{G}$ such that (a) $Z \\to S$ is proper, and (b)\n$(Z \\to X)_*\\mathcal{G} = \\mathcal{F}$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYS","source_file":"coherent.tex","source_line":7407,"source_end_line":7422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7407-L7422","statement_sha256":"700c3d030880393fb648464765c5d482fb5c36785c4b110cea6ea6c78d34e183","origin":"The Stacks Project","memory_eligible":false,"source_rank":5957,"rank":5957,"depth":20,"x":2072.22,"y":1080.817,"cluster":"duality-cohomology"},{"id":"stacks:0CYT","tag":"0CYT","title":"Being proper over a base · Lemma 0CYT","summary":"Consider a cartesian diagram of schemes xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f S' ar[r]^g & S with f locally of finite type. Let F be a finite type, quasi-coherent O_X-module. If the support of F is proper over S, then the support of (g')^*F is proper over S'.","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nwith $f$ locally of finite type. Let $\\mathcal{F}$ be a\nfinite type, quasi-coherent $\\mathcal{O}_X$-module.\nIf the support of $\\mathcal{F}$ is proper over $S$, then\nthe support of $(g')^*\\mathcal{F}$ is proper over $S'$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYT","source_file":"coherent.tex","source_line":7441,"source_end_line":7454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7441-L7454","statement_sha256":"4ba576e9bdacc8a80ed89616354eae380b13d1747fa170b692e451b37b006b71","origin":"The Stacks Project","memory_eligible":false,"source_rank":5958,"rank":5958,"depth":18,"x":2046.529,"y":1207.875,"cluster":"duality-cohomology"},{"id":"stacks:0CYU","tag":"0CYU","title":"Being proper over a base · Lemma 0CYU","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F, G be finite type, quasi-coherent O_X-module. • If the supports of F, G are proper over S, then the same is true for F ⊕ G, for any extension of G by F, for Im(u) and Coker(u) given any O_X-module map u : F → G, and for any quasi-coherent quotient of F or G. • If S is locally Noetherian, then the category of coherent O_X-modules with support proper over S is a Serre subcategory (Homology,…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally\nof finite type. Let $\\mathcal{F}$, $\\mathcal{G}$\nbe finite type, quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If the supports of $\\mathcal{F}$, $\\mathcal{G}$\nare proper over $S$, then the same is true\nfor $\\mathcal{F} \\oplus \\mathcal{G}$, for any extension\nof $\\mathcal{G}$ by $\\mathcal{F}$, for $\\Im(u)$ and $\\Coker(u)$\ngiven any $\\mathcal{O}_X$-module map $u : \\mathcal{F} \\to \\mathcal{G}$,\nand for any quasi-coherent quotient of $\\mathcal{F}$ or $\\mathcal{G}$.\n\\item If $S$ is locally Noetherian, then the category of\ncoherent $\\mathcal{O}_X$-modules with support proper over\n$S$ is a Serre subcategory (Homology, Definition\n\\ref{homology-definition-serre-subcategory})\nof the abelian category of\ncoherent $\\mathcal{O}_X$-modules.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYU","source_file":"coherent.tex","source_line":7466,"source_end_line":7485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7466-L7485","statement_sha256":"164037b263e5864907010a8995a236009f47fd11c2c57ad14c31dbbe7b07bd60","origin":"The Stacks Project","memory_eligible":false,"source_rank":5959,"rank":5959,"depth":22,"x":1962.956,"y":1099.162,"cluster":"duality-cohomology"},{"id":"stacks:08DS","tag":"08DS","title":"Being proper over a base · Lemma 08DS","summary":"Let S be a locally Noetherian scheme. Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a coherent O_X-module with support proper over S. Then R^pf_*F is a coherent O_S-module for all p ≥ 0.","statement_latex":"Let $S$ be a locally Noetherian scheme.\nLet $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module\nwith support proper over $S$. Then $R^pf_*\\mathcal{F}$\nis a coherent $\\mathcal{O}_S$-module for all $p \\geq 0$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DS","source_file":"coherent.tex","source_line":7507,"source_end_line":7514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7507-L7514","statement_sha256":"d1224f8f496271a6011947f31d84aaefae6b8d7418914ba550d326f83e781fd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":5960,"rank":5960,"depth":31,"x":2112.611,"y":1132.038,"cluster":"duality-cohomology"},{"id":"stacks:0CYV","tag":"0CYV","title":"Being proper over a base · Lemma 0CYV","summary":"Let S be a Noetherian scheme. Let f : X → S be a finite type morphism. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. The following are Serre subcategories of Coh(X, I) • the full subcategory of Coh(X, I) consisting of those objects (F_n) such that the support of F_1 is proper over S, • the full subcategory of Coh(X, I) consisting of those objects (F_n) such that there exists a closed subscheme Z ⊂ X proper over S with I_Z F_n = 0 for all n ≥ 1.","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a finite type morphism.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be\na quasi-coherent sheaf of ideals. The following are Serre subcategories\nof $\\textit{Coh}(X, \\mathcal{I})$\n\\begin{enumerate}\n\\item the full subcategory of $\\textit{Coh}(X, \\mathcal{I})$\nconsisting of those objects $(\\mathcal{F}_n)$ such that\nthe support of $\\mathcal{F}_1$ is proper over $S$,\n\\item the full subcategory of $\\textit{Coh}(X, \\mathcal{I})$\nconsisting of those objects $(\\mathcal{F}_n)$ such that\nthere exists a closed subscheme $Z \\subset X$ proper over $S$\nwith $\\mathcal{I}_Z \\mathcal{F}_n = 0$ for all $n \\geq 1$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYV","source_file":"coherent.tex","source_line":7530,"source_end_line":7545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7530-L7545","statement_sha256":"1fab082bd453721f949653592a24f845170e5df2bc7e18b511fd45f32402d8da","origin":"The Stacks Project","memory_eligible":false,"source_rank":5961,"rank":5961,"depth":22,"x":1975.267,"y":1192.958,"cluster":"duality-cohomology"},{"id":"stacks:088E","tag":"088E","title":"Grothendieck's existence theorem · Theorem 088E","summary":"[EGA] Let A be a Noetherian ring complete with respect to an ideal I. Let X be a separated, finite type scheme over A. Then the functor ([Tag 088D]) Coh_support proper over A(O_X) → Coh_support proper over A(X, I) is an equivalence.","statement_latex":"\\begin{reference}\n\\cite[III Theorem 5.1.5]{EGA}\n\\end{reference}\nLet $A$ be a Noetherian ring complete with respect to an ideal $I$.\nLet $X$ be a separated, finite type scheme over $A$. Then\nthe functor\n(\\ref{equation-completion-functor-proper-over-A})\n$$\n\\textit{Coh}_{\\text{support proper over }A}(\\mathcal{O}_X)\n\\longrightarrow\n\\textit{Coh}_{\\text{support proper over }A}(X, \\mathcal{I})\n$$\nis an equivalence.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's existence theorem, III","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088E","source_file":"coherent.tex","source_line":7639,"source_end_line":7654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7639-L7654","statement_sha256":"59dbc24c5d6d72142b74118504754fe7e50f52e0ea9322bdf6a8d7919d60fa65","origin":"The Stacks Project","memory_eligible":false,"source_rank":5962,"rank":5962,"depth":41,"x":2027.763,"y":1069.613,"cluster":"duality-cohomology"},{"id":"stacks:0899","tag":"0899","title":"Grothendieck's algebraization theorem · Lemma 0899","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Write S = Spec(A) and S_n = Spec(A/I^n). Let X → S be a separated morphism of finite type. For n ≥ 1 we set X_n = X ×_S S_n. Suppose given a commutative diagram xymatrix Z_1 ar[r] ar[d] & Z_2 ar[r] ar[d] & Z_3 ar[r] ar[d] & … X_1 ar[r]^i_1 & X_2 ar[r]^i_2 & X_3 ar[r] & … of schemes with cartesian squares. Assume that • Z_1 → X_1 is a closed immersion, and • Z_1 → S_1 is proper. Then there exists a closed…","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nWrite $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.\nLet $X \\to S$ be a separated morphism of finite type.\nFor $n \\geq 1$ we set $X_n = X \\times_S S_n$.\nSuppose given a commutative diagram\n$$\n\\xymatrix{\nZ_1 \\ar[r] \\ar[d] & Z_2 \\ar[r] \\ar[d] & Z_3 \\ar[r] \\ar[d] & \\ldots \\\\\nX_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots\n}\n$$\nof schemes with cartesian squares. Assume that\n\\begin{enumerate}\n\\item $Z_1 \\to X_1$ is a closed immersion, and\n\\item $Z_1 \\to S_1$ is proper.\n\\end{enumerate}\nThen there exists a closed immersion of schemes $Z \\to X$ such that\n$Z_n = Z \\times_S S_n$. Moreover, $Z$ is proper over $S$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's algebraization theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0899","source_file":"coherent.tex","source_line":7829,"source_end_line":7849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7829-L7849","statement_sha256":"63cd6c375c08be2181dd723d3e884cfbb51c18623f75a8a4b9cb319aa2f02a20","origin":"The Stacks Project","memory_eligible":false,"source_rank":5963,"rank":5963,"depth":20,"x":2088.483,"y":1190.833,"cluster":"duality-cohomology"},{"id":"stacks:09ZT","tag":"09ZT","title":"Grothendieck's algebraization theorem · Lemma 09ZT","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Write S = Spec(A) and S_n = Spec(A/I^n). Let X → S be a separated morphism of finite type. For n ≥ 1 we set X_n = X ×_S S_n. Suppose given a commutative diagram xymatrix Y_1 ar[r] ar[d] & Y_2 ar[r] ar[d] & Y_3 ar[r] ar[d] & … X_1 ar[r]^i_1 & X_2 ar[r]^i_2 & X_3 ar[r] & … of schemes with cartesian squares. Assume that • Y_n → X_n is a finite morphism, and • Y_1 → S_1 is proper. Then there exists a finite…","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nWrite $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.\nLet $X \\to S$ be a separated morphism of finite type.\nFor $n \\geq 1$ we set $X_n = X \\times_S S_n$.\nSuppose given a commutative diagram\n$$\n\\xymatrix{\nY_1 \\ar[r] \\ar[d] & Y_2 \\ar[r] \\ar[d] & Y_3 \\ar[r] \\ar[d] & \\ldots \\\\\nX_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots\n}\n$$\nof schemes with cartesian squares. Assume that\n\\begin{enumerate}\n\\item $Y_n \\to X_n$ is a finite morphism, and\n\\item $Y_1 \\to S_1$ is proper.\n\\end{enumerate}\nThen there exists a finite morphism of schemes $Y \\to X$ such that\n$Y_n = Y \\times_S S_n$. Moreover, $Y$ is proper over $S$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's algebraization theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZT","source_file":"coherent.tex","source_line":7886,"source_end_line":7906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7886-L7906","statement_sha256":"4b7a28bee5784be7dc93dcb7afd88c75c5562dbff29a94c2bd168d5a40111455","origin":"The Stacks Project","memory_eligible":false,"source_rank":5964,"rank":5964,"depth":42,"x":1945.667,"y":1135.685,"cluster":"duality-cohomology"},{"id":"stacks:0A42","tag":"0A42","title":"Grothendieck's algebraization theorem · Lemma 0A42","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Write S = Spec(A) and S_n = Spec(A/I^n). Let X, Y be schemes over S. For n ≥ 1 we set X_n = X ×_S S_n and Y_n = Y ×_S S_n. Suppose given a compatible system of commutative diagrams xymatrix & & X_n + 1 ar[rd] ar[rr]_g_n + 1 & & Y_n + 1 ar[ld] X_n ar[rru] ar[rd] ar[rr]_g_n & & Y_n ar[rru] ar[ld] & S_n + 1 & S_n ar[rru] Assume that • X → S is proper, and • Y → S is separated of finite type. Then there exists a…","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nWrite $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X$, $Y$ be schemes\nover $S$. For $n \\geq 1$ we set $X_n = X \\times_S S_n$ and\n$Y_n = Y \\times_S S_n$. Suppose given a compatible system of\ncommutative diagrams\n$$\n\\xymatrix{\n& & X_{n + 1} \\ar[rd] \\ar[rr]_{g_{n + 1}} & & Y_{n + 1} \\ar[ld] \\\\\nX_n \\ar[rru] \\ar[rd] \\ar[rr]_{g_n} & & Y_n \\ar[rru] \\ar[ld] & S_{n + 1} \\\\\n& S_n \\ar[rru]\n}\n$$\nAssume that\n\\begin{enumerate}\n\\item $X \\to S$ is proper, and\n\\item $Y \\to S$ is separated of finite type.\n\\end{enumerate}\nThen there exists a unique morphism of schemes $g : X \\to Y$\nover $S$ such that $g_n$ is the base change of $g$ to $S_n$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's algebraization theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A42","source_file":"coherent.tex","source_line":7928,"source_end_line":7949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7928-L7949","statement_sha256":"66618a7cd458d0c4f640379f065e0aaa70e9fe8ba05727db2ad4e8787c0004af","origin":"The Stacks Project","memory_eligible":false,"source_rank":5965,"rank":5965,"depth":41,"x":2095.911,"y":1095.156,"cluster":"duality-cohomology"},{"id":"stacks:089A","tag":"089A","title":"Grothendieck's algebraization theorem · Theorem 089A","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Set S = Spec(A) and S_n = Spec(A/I^n). Consider a commutative diagram xymatrix X_1 ar[r]_i_1 ar[d] & X_2 ar[r]_i_2 ar[d] & X_3 ar[r] ar[d] & … S_1 ar[r] & S_2 ar[r] & S_3 ar[r] & … of schemes with cartesian squares. Suppose given (L_n, φ_n) where each L_n is an invertible sheaf on X_n and φ_n : i_n^*L_n + 1 → L_n is an isomorphism. If • X_1 → S_1 is proper, and • L_1 is ample on X_1 then there exists a proper…","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nSet $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Consider a commutative\ndiagram\n$$\n\\xymatrix{\nX_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] &\n\\ldots \\\\\nS_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots\n}\n$$\nof schemes with cartesian squares. Suppose given $(\\mathcal{L}_n, \\varphi_n)$\nwhere each $\\mathcal{L}_n$ is an invertible sheaf on $X_n$ and\n$\\varphi_n : i_n^*\\mathcal{L}_{n + 1} \\to \\mathcal{L}_n$\nis an isomorphism. If\n\\begin{enumerate}\n\\item $X_1 \\to S_1$ is proper, and\n\\item $\\mathcal{L}_1$ is ample on $X_1$\n\\end{enumerate}\nthen there exists a proper morphism of schemes $X \\to S$\nand an ample invertible $\\mathcal{O}_X$-module $\\mathcal{L}$\nand isomorphisms $X_n \\cong X \\times_S S_n$ and\n$\\mathcal{L}_n \\cong \\mathcal{L}|_{X_n}$ compatible with\nthe morphisms $i_n$ and $\\varphi_n$.","area":"Duality & Cohomology","chapter":"Cohomology of Schemes","chapter_id":"coherent","section":"Grothendieck's algebraization theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089A","source_file":"coherent.tex","source_line":7979,"source_end_line":8004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/coherent.tex#L7979-L8004","statement_sha256":"3a9e3a4fed42378bbe3f5cc8ac264006f0ffd447d04c335af7f71d92bd299a71","origin":"The Stacks Project","memory_eligible":false,"source_rank":5966,"rank":5966,"depth":33,"x":2017.412,"y":1210.739,"cluster":"duality-cohomology"},{"id":"stacks:02OJ","tag":"02OJ","title":"Associated points · Definition 02OJ","summary":"Let X be a scheme. Let F be a quasi-coherent sheaf on X. • We say x ∈ X is associated to F if the maximal ideal m_x is associated to the O_X, x-module F_x. • We denote Ass(F) or Ass_X(F) the set of associated points of F. • The associated points of X are the associated points of O_X.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\n\\begin{enumerate}\n\\item We say $x \\in X$ is {\\it associated} to $\\mathcal{F}$\nif the maximal ideal\n$\\mathfrak m_x$ is associated to the $\\mathcal{O}_{X, x}$-module\n$\\mathcal{F}_x$.\n\\item We denote $\\text{Ass}(\\mathcal{F})$ or $\\text{Ass}_X(\\mathcal{F})$\nthe set of associated points of $\\mathcal{F}$.\n\\item The {\\it associated points of $X$} are the associated\npoints of $\\mathcal{O}_X$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OJ","source_file":"divisors.tex","source_line":38,"source_end_line":52,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L38-L52","statement_sha256":"9ba56d3b2bd10ac1c0809e31102eaa70b9cdfa78182bf388ede29aedc5a15e59","origin":"The Stacks Project","memory_eligible":false,"source_rank":5967,"rank":5967,"depth":0,"x":2484.749,"y":680.0,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OK","tag":"02OK","title":"Associated points · Lemma 02OK","summary":"Let X be a scheme. Let F be a quasi-coherent sheaf on X. Let Spec(A) = U ⊂ X be an affine open, and set M = Γ(U, F). Let x ∈ U, and let p ⊂ A be the corresponding prime. • If p is associated to M, then x is associated to F. • If p is finitely generated, then the converse holds as well. In particular, if X is locally Noetherian, then the equivalence p ∈ Ass(M) ⇔ x ∈ Ass(F) holds for all pairs ( p, x) as above.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $\\Spec(A) = U \\subset X$ be an affine open, and set\n$M = \\Gamma(U, \\mathcal{F})$.\nLet $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime.\n\\begin{enumerate}\n\\item If $\\mathfrak p$ is associated to $M$, then $x$ is associated\nto $\\mathcal{F}$.\n\\item If $\\mathfrak p$ is finitely generated, then the converse holds\nas well.\n\\end{enumerate}\nIn particular, if $X$ is locally Noetherian, then the equivalence\n$$\n\\mathfrak p \\in \\text{Ass}(M) \\Leftrightarrow x \\in \\text{Ass}(\\mathcal{F})\n$$\nholds for all pairs $(\\mathfrak p, x)$ as above.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OK","source_file":"divisors.tex","source_line":67,"source_end_line":84,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L67-L84","statement_sha256":"375e20ec6339285f9398ae58ddb93b6d54b83a1cc17e6d1c5311ddde0f80f4f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5968,"rank":5968,"depth":6,"x":2473.935,"y":684.667,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AD","tag":"05AD","title":"Associated points · Lemma 05AD","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Then Ass(F) ⊂ Supp(F).","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $\\text{Ass}(\\mathcal{F}) \\subset \\text{Supp}(\\mathcal{F})$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AD","source_file":"divisors.tex","source_line":119,"source_end_line":124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L119-L124","statement_sha256":"a17bbc6c4a0e9961b09e320cc1d73e04048ccc81e52fb3492f738b3c8c2b9b4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5969,"rank":5969,"depth":0,"x":2480.928,"y":671.114,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AE","tag":"05AE","title":"Associated points · Lemma 05AE","summary":"Let X be a scheme. Let 0 → F_1 → F_2 → F_3 → 0 be a short exact sequence of quasi-coherent sheaves on X. Then Ass(F_2) ⊂ Ass(F_1) ∪ Ass(F_3) and Ass(F_1) ⊂ Ass(F_2).","statement_latex":"Let $X$ be a scheme.\nLet $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nbe a short exact sequence of quasi-coherent sheaves on $X$.\nThen\n$\\text{Ass}(\\mathcal{F}_2) \\subset\n\\text{Ass}(\\mathcal{F}_1) \\cup \\text{Ass}(\\mathcal{F}_3)$\nand\n$\\text{Ass}(\\mathcal{F}_1) \\subset \\text{Ass}(\\mathcal{F}_2)$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AE","source_file":"divisors.tex","source_line":130,"source_end_line":140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L130-L140","statement_sha256":"af51cc8ddbd7ff667574fdbe174def97700e0da31b00ddeed3bfc0c43412e2e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5970,"rank":5970,"depth":1,"x":2487.645,"y":688.376,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AF","tag":"05AF","title":"Associated points · Lemma 05AF","summary":"Let X be a locally Noetherian scheme. Let F be a coherent O_X-module. Then Ass(F) ∩ U is finite for every quasi-compact open U ⊂ X.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThen $\\text{Ass}(\\mathcal{F}) \\cap U$ is finite for\nevery quasi-compact open $U \\subset X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AF","source_file":"divisors.tex","source_line":150,"source_end_line":156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L150-L156","statement_sha256":"2a3b330b176842bc24bc9796145c34e2266d483af4010a47059da96c5fe91233","origin":"The Stacks Project","memory_eligible":false,"source_rank":5971,"rank":5971,"depth":8,"x":2465.971,"y":677.916,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AG","tag":"05AG","title":"Associated points · Lemma 05AG","summary":"Let X be a locally Noetherian scheme. Let F be a quasi-coherent O_X-module. Then F = 0 ⇔ Ass(F) = ∅.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a\nquasi-coherent $\\mathcal{O}_X$-module. Then\n$$\n\\mathcal{F} = 0 \\Leftrightarrow \\text{Ass}(\\mathcal{F}) = \\emptyset.\n$$","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AG","source_file":"divisors.tex","source_line":171,"source_end_line":178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L171-L178","statement_sha256":"0ba2a6caf43a34f46695001912016230e51daac1172f037bddb534db7ad35a40","origin":"The Stacks Project","memory_eligible":false,"source_rank":5972,"rank":5972,"depth":9,"x":2493.289,"y":672.899,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B3L","tag":"0B3L","title":"Associated points · Lemma 0B3L","summary":"Let X be a locally Noetherian scheme. Let F be a quasi-coherent O_X-module. If U ⊂ X is open and Ass(F) ⊂ U, then Γ(X, F) → Γ(U, F) is injective.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. If $U \\subset X$ is open and\n$\\text{Ass}(\\mathcal{F}) \\subset U$, then\n$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$ is injective.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3L","source_file":"divisors.tex","source_line":198,"source_end_line":204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L198-L204","statement_sha256":"18a3711574c2a964ed07ecb431d0d8d76c70e3b3438468f957edefdb1b5f8c88","origin":"The Stacks Project","memory_eligible":false,"source_rank":5973,"rank":5973,"depth":10,"x":2475.555,"y":693.89,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AH","tag":"05AH","title":"Associated points · Lemma 05AH","summary":"Let X be a locally Noetherian scheme. Let F be a quasi-coherent O_X-module. Let x ∈ Supp(F) be a point in the support of F which is not a specialization of another point of Supp(F). Then x ∈ Ass(F). In particular, any generic point of an irreducible component of X is an associated point of X.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in \\text{Supp}(\\mathcal{F})$ be a point in the support\nof $\\mathcal{F}$ which is not a specialization of another point of\n$\\text{Supp}(\\mathcal{F})$. Then $x \\in \\text{Ass}(\\mathcal{F})$.\nIn particular, any generic point of an irreducible component of $X$\nis an associated point of $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AH","source_file":"divisors.tex","source_line":219,"source_end_line":228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L219-L228","statement_sha256":"42f3fefb28808e6d41203485d4c9af953ed6d7828073d096c9d0c641788d1616","origin":"The Stacks Project","memory_eligible":false,"source_rank":5974,"rank":5974,"depth":9,"x":2471.523,"y":666.289,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AVL","tag":"0AVL","title":"Associated points · Lemma 0AVL","summary":"Let X be a locally Noetherian scheme. Let φ : F → G be a map of quasi-coherent O_X-modules. Assume that for every x ∈ X at least one of the following happens • F_x → G_x is injective, or • x not ∈ Ass(F). Then φ is injective.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of\nquasi-coherent $\\mathcal{O}_X$-modules.\nAssume that for every $x \\in X$\nat least one of the following happens\n\\begin{enumerate}\n\\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is injective, or\n\\item $x \\not \\in \\text{Ass}(\\mathcal{F})$.\n\\end{enumerate}\nThen $\\varphi$ is injective.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVL","source_file":"divisors.tex","source_line":249,"source_end_line":261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L249-L261","statement_sha256":"e2d2deee0366e9a15660e89afd4eebbe3da3d3e406fd55105d6dd1e4a55e20b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":5975,"rank":5975,"depth":10,"x":2498.392,"y":685.642,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AVM","tag":"0AVM","title":"Associated points · Lemma 0AVM","summary":"Let X be a locally Noetherian scheme. Let φ : F → G be a map of quasi-coherent O_X-modules. Assume F is coherent and that for every x ∈ X one of the following happens • F_x → G_x is an isomorphism, or • depth(F_x) ≥ 2 and x not ∈ Ass(G). Then φ is an isomorphism.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of\nquasi-coherent $\\mathcal{O}_X$-modules. Assume $\\mathcal{F}$ is coherent\nand that for every $x \\in X$ one of the following happens\n\\begin{enumerate}\n\\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is an isomorphism, or\n\\item $\\text{depth}(\\mathcal{F}_x) \\geq 2$ and\n$x \\not \\in \\text{Ass}(\\mathcal{G})$.\n\\end{enumerate}\nThen $\\varphi$ is an isomorphism.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVM","source_file":"divisors.tex","source_line":268,"source_end_line":280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L268-L280","statement_sha256":"d09abdc08334ec8d5b6d7a215e891489dc455bcc0466334e39b6b5de3fadec3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":5976,"rank":5976,"depth":14,"x":2460.866,"y":686.634,"cluster":"divisors-intersection-theory"},{"id":"stacks:05DB","tag":"05DB","title":"Morphisms and associated points · Lemma 05DB","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X which is flat over S. Let G be a quasi-coherent sheaf on S. Then we have Ass_X(F ⊗_O_X f^*G) ⊃ ⋃_s ∈ Ass_S(G) Ass_X_s(F_s) and equality holds if S is locally Noetherian (for the notation F_s see above).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$ which is flat over $S$.\nLet $\\mathcal{G}$ be a quasi-coherent sheaf on $S$.\nThen we have\n$$\n\\text{Ass}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G})\n\\supset\n\\bigcup\\nolimits_{s \\in \\text{Ass}_S(\\mathcal{G})}\n\\text{Ass}_{X_s}(\\mathcal{F}_s)\n$$\nand equality holds if $S$ is locally Noetherian (for the notation\n$\\mathcal{F}_s$ see above).","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Morphisms and associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DB","source_file":"divisors.tex","source_line":310,"source_end_line":324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L310-L324","statement_sha256":"c1f6096347f8e1bcb1df6a0cff907eacf5177ee142d7ebc0164a09252b38aa53","origin":"The Stacks Project","memory_eligible":false,"source_rank":5977,"rank":5977,"depth":13,"x":2489.224,"y":663.443,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AK","tag":"05AK","title":"Embedded points · Definition 05AK","summary":"Let X be a scheme. Let F be a quasi-coherent sheaf on X. • An embedded associated point of F is an associated point which is not maximal among the associated points of F, i.e., it is the specialization of another associated point of F. • A point x of X is called an embedded point if x is an embedded associated point of O_X. • An embedded component of X is an irreducible closed subset Z = overline(x) where x is an embedded point of X.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\n\\begin{enumerate}\n\\item An {\\it embedded associated point} of $\\mathcal{F}$\nis an associated point which is not maximal among the\nassociated points of $\\mathcal{F}$, i.e., it is the specialization\nof another associated point of $\\mathcal{F}$.\n\\item A point $x$ of $X$ is called an {\\it embedded point}\nif $x$ is an embedded associated point of $\\mathcal{O}_X$.\n\\item An {\\it embedded component} of $X$ is an irreducible\nclosed subset $Z = \\overline{\\{x\\}}$ where $x$ is an embedded\npoint of $X$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Embedded points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AK","source_file":"divisors.tex","source_line":360,"source_end_line":375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L360-L375","statement_sha256":"13a77741f97a7ae803cddfc495d1b60f3c486e60ca3af79ed2962b185da07d60","origin":"The Stacks Project","memory_eligible":false,"source_rank":5978,"rank":5978,"depth":0,"x":2486.816,"y":698.254,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AL","tag":"05AL","title":"Embedded points · Lemma 05AL","summary":"Let X be a locally Noetherian scheme. Let F be a coherent O_X-module. Then • the generic points of irreducible components of Supp(F) are associated points of F, and • an associated point of F is embedded if and only if it is not a generic point of an irreducible component of Supp(F). In particular an embedded point of X is an associated point of X which is not a generic point of an irreducible component of X.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThen\n\\begin{enumerate}\n\\item the generic points of irreducible components of\n$\\text{Supp}(\\mathcal{F})$ are associated points of $\\mathcal{F}$, and\n\\item an associated point of $\\mathcal{F}$ is embedded if and only\nif it is not a generic point of an irreducible component\nof $\\text{Supp}(\\mathcal{F})$.\n\\end{enumerate}\nIn particular an embedded point of $X$ is an associated point of $X$\nwhich is not a generic point of an irreducible component of $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Embedded points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AL","source_file":"divisors.tex","source_line":381,"source_end_line":395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L381-L395","statement_sha256":"75c12f011793b81b8fd079e2d08e3bcd9836676142eeb187b1b851331c239c38","origin":"The Stacks Project","memory_eligible":false,"source_rank":5979,"rank":5979,"depth":10,"x":2459.456,"y":669.999,"cluster":"divisors-intersection-theory"},{"id":"stacks:0346","tag":"0346","title":"Embedded points · Lemma 0346","summary":"Let X be a locally Noetherian scheme. Let F be a coherent sheaf on X. Then the following are equivalent: • F has no embedded associated points, and • F has property (S_1).","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nThen the following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ has no embedded associated points, and\n\\item $\\mathcal{F}$ has property $(S_1)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Embedded points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0346","source_file":"divisors.tex","source_line":410,"source_end_line":419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L410-L419","statement_sha256":"7db1ce132d4796ac8baa783f27e9380e01cfdf172db587ef0ce55047424b548c","origin":"The Stacks Project","memory_eligible":false,"source_rank":5980,"rank":5980,"depth":12,"x":2504.1,"y":675.549,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BXG","tag":"0BXG","title":"Embedded points · Lemma 0BXG","summary":"Let X be a locally Noetherian scheme of dimension ≤ 1. The following are equivalent • X is Cohen-Macaulay, and • X has no embedded points.","statement_latex":"Let $X$ be a locally Noetherian scheme of dimension $\\leq 1$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is Cohen-Macaulay, and\n\\item $X$ has no embedded points.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Embedded points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXG","source_file":"divisors.tex","source_line":426,"source_end_line":434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L426-L434","statement_sha256":"0ee349efc46a8a928837fd869e65389ac7bb4dd0e8594ad47661bf2d654bd666","origin":"The Stacks Project","memory_eligible":false,"source_rank":5981,"rank":5981,"depth":13,"x":2465.292,"y":697.573,"cluster":"divisors-intersection-theory"},{"id":"stacks:083P","tag":"083P","title":"Embedded points · Lemma 083P","summary":"Let X be a locally Noetherian scheme. Let U ⊂ X be an open subscheme. The following are equivalent • U is scheme theoretically dense in X (Morphisms, Definition [Tag 01RB]), • U is dense in X and U contains all embedded points of X, and • U contains all associated points of X.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be an\nopen subscheme. The following are equivalent\n\\begin{enumerate}\n\\item $U$ is scheme theoretically dense in $X$\n(Morphisms, Definition \\ref{morphisms-definition-scheme-theoretically-dense}),\n\\item $U$ is dense in $X$ and $U$ contains all embedded points of $X$, and\n\\item $U$ contains all associated points of $X$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Embedded points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083P","source_file":"divisors.tex","source_line":440,"source_end_line":450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L440-L450","statement_sha256":"7b8436476fa36ac71522bfb3ec6342c38a24bfd200b0278437b132c63bcf1ac0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5982,"rank":5982,"depth":10,"x":2476.602,"y":657.974,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OL","tag":"02OL","title":"Embedded points · Lemma 02OL","summary":"Let X be a locally Noetherian scheme. Let F be a coherent sheaf on X. The set of coherent subsheaves ( K ⊂ F mid Supp(K) is nowhere dense in Supp(F) ) has a maximal element K. Setting F' = F/K we have the following • Supp(F') = Supp(F), • F' has no embedded associated points, and • there exists a dense open U ⊂ X such that U ∩ Supp(F) is dense in Supp(F) and F'|_U ≅ F|_U.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nThe set of coherent subsheaves\n$$\n\\{\n\\mathcal{K} \\subset \\mathcal{F}\n\\mid\n\\text{Supp}(\\mathcal{K})\\text{ is nowhere dense in }\\text{Supp}(\\mathcal{F})\n\\}\n$$\nhas a maximal element $\\mathcal{K}$.\nSetting $\\mathcal{F}' = \\mathcal{F}/\\mathcal{K}$ we have the\nfollowing\n\\begin{enumerate}\n\\item $\\text{Supp}(\\mathcal{F}') = \\text{Supp}(\\mathcal{F})$,\n\\item $\\mathcal{F}'$ has no embedded associated points, and\n\\item there exists a dense open $U \\subset X$ such that\n$U \\cap \\text{Supp}(\\mathcal{F})$ is dense in $\\text{Supp}(\\mathcal{F})$\nand $\\mathcal{F}'|_U \\cong \\mathcal{F}|_U$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Embedded points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OL","source_file":"divisors.tex","source_line":487,"source_end_line":509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L487-L509","statement_sha256":"af3b12c1b37b687e78fc5e542581316e00938a543af858ca221658c26e8159ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":5983,"rank":5983,"depth":10,"x":2500.86,"y":694.768,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OM","tag":"02OM","title":"Embedded points · Lemma 02OM","summary":"Let X be a locally Noetherian scheme. Let F be a coherent O_X-module without embedded associated points. Set I = Ker(O_X → SheafHom_O_X(F, F)). This is a coherent sheaf of ideals which defines a closed subscheme Z ⊂ X without embedded points. Moreover there exists a coherent sheaf G on Z such that (a) F = (Z → X)_*G, (b) G has no associated embedded points, and (c) Supp(G) = Z (as sets).","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module\nwithout embedded associated points. Set\n$$\n\\mathcal{I}\n=\n\\Ker(\\mathcal{O}_X\n\\longrightarrow\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{F})).\n$$\nThis is a coherent sheaf of ideals which defines a closed\nsubscheme $Z \\subset X$ without embedded points. Moreover\nthere exists a coherent sheaf $\\mathcal{G}$ on $Z$\nsuch that (a) $\\mathcal{F} = (Z \\to X)_*\\mathcal{G}$,\n(b) $\\mathcal{G}$ has no associated embedded points, and\n(c) $\\text{Supp}(\\mathcal{G}) = Z$ (as sets).","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Embedded points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OM","source_file":"divisors.tex","source_line":519,"source_end_line":537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L519-L537","statement_sha256":"042ff818eaff85c1f83cde78614fd02aae5b91bc9e611f3e0f61e5ce1d501832","origin":"The Stacks Project","memory_eligible":false,"source_rank":5984,"rank":5984,"depth":20,"x":2451.929,"y":680.975,"cluster":"divisors-intersection-theory"},{"id":"stacks:056L","tag":"056L","title":"Weakly associated points · Definition 056L","summary":"Let X be a scheme. Let F be a quasi-coherent sheaf on X. • We say x ∈ X is weakly associated to F if the maximal ideal m_x is weakly associated to the O_X, x-module F_x. • We denote WeakAss(F) the set of weakly associated points of F. • The weakly associated points of X are the weakly associated points of O_X.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\n\\begin{enumerate}\n\\item We say $x \\in X$ is {\\it weakly associated} to $\\mathcal{F}$\nif the maximal ideal $\\mathfrak m_x$ is weakly associated to the\n$\\mathcal{O}_{X, x}$-module $\\mathcal{F}_x$.\n\\item We denote $\\text{WeakAss}(\\mathcal{F})$ the set of weakly associated\npoints of $\\mathcal{F}$.\n\\item The {\\it weakly associated points of $X$} are the weakly associated\npoints of $\\mathcal{O}_X$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056L","source_file":"divisors.tex","source_line":562,"source_end_line":575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L562-L575","statement_sha256":"57546ad2f1c1c428610ca2b2f674450a0cac326c53d74b44c0e2096fabdc7db1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5985,"rank":5985,"depth":0,"x":2500.475,"y":662.884,"cluster":"divisors-intersection-theory"},{"id":"stacks:056M","tag":"056M","title":"Weakly associated points · Lemma 056M","summary":"Let X be a scheme. Let F be a quasi-coherent sheaf on X. Let Spec(A) = U ⊂ X be an affine open, and set M = Γ(U, F). Let x ∈ U, and let p ⊂ A be the corresponding prime. The following are equivalent • p is weakly associated to M, and • x is weakly associated to F.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $\\Spec(A) = U \\subset X$ be an affine open, and set\n$M = \\Gamma(U, \\mathcal{F})$.\nLet $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathfrak p$ is weakly associated to $M$, and\n\\item $x$ is weakly associated to $\\mathcal{F}$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056M","source_file":"divisors.tex","source_line":581,"source_end_line":592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L581-L592","statement_sha256":"e28d8f6e8c99182aad85269230e74c3b704247bcca7fb19883f7ef2a22f8e818","origin":"The Stacks Project","memory_eligible":false,"source_rank":5986,"rank":5986,"depth":1,"x":2478.63,"y":704.885,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AM","tag":"05AM","title":"Weakly associated points · Lemma 05AM","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Then Ass(F) ⊂ WeakAss(F) ⊂ Supp(F).","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen\n$$\n\\text{Ass}(\\mathcal{F}) \\subset \\text{WeakAss}(\\mathcal{F}) \\subset\n\\text{Supp}(\\mathcal{F}).\n$$","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AM","source_file":"divisors.tex","source_line":599,"source_end_line":608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L599-L608","statement_sha256":"ed40413551e0033970ecceed0a027ecd7a5d1917d6309626a15ae19b2c25aaf8","origin":"The Stacks Project","memory_eligible":false,"source_rank":5987,"rank":5987,"depth":0,"x":2460.518,"y":660.389,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AN","tag":"05AN","title":"Weakly associated points · Lemma 05AN","summary":"Let X be a scheme. Let 0 → F_1 → F_2 → F_3 → 0 be a short exact sequence of quasi-coherent sheaves on X. Then WeakAss(F_2) ⊂ WeakAss(F_1) ∪ WeakAss(F_3) and WeakAss(F_1) ⊂ WeakAss(F_2).","statement_latex":"Let $X$ be a scheme.\nLet $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nbe a short exact sequence of quasi-coherent sheaves on $X$.\nThen\n$\\text{WeakAss}(\\mathcal{F}_2) \\subset\n\\text{WeakAss}(\\mathcal{F}_1) \\cup \\text{WeakAss}(\\mathcal{F}_3)$\nand\n$\\text{WeakAss}(\\mathcal{F}_1) \\subset \\text{WeakAss}(\\mathcal{F}_2)$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AN","source_file":"divisors.tex","source_line":614,"source_end_line":624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L614-L624","statement_sha256":"5acd64d98a49c99cb15327156108f0b65ce3d55b62fdceee433a54ec7204cb9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5988,"rank":5988,"depth":2,"x":2510.862,"y":683.488,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AP","tag":"05AP","title":"Weakly associated points · Lemma 05AP","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Then F = (0) ⇔ WeakAss(F) = ∅","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen\n$$\n\\mathcal{F} = (0) \\Leftrightarrow \\text{WeakAss}(\\mathcal{F}) = \\emptyset\n$$","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AP","source_file":"divisors.tex","source_line":634,"source_end_line":642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L634-L642","statement_sha256":"3eb3ace39ae1309e80ecc4d072c933c086427c252da78f48d645efe41c1b845b","origin":"The Stacks Project","memory_eligible":false,"source_rank":5989,"rank":5989,"depth":3,"x":2453.851,"y":695.283,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B3M","tag":"0B3M","title":"Weakly associated points · Lemma 0B3M","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. If U ⊂ X is open and WeakAss(F) ⊂ U, then Γ(X, F) → Γ(U, F) is injective.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. If $U \\subset X$ is open and\n$\\text{WeakAss}(\\mathcal{F}) \\subset U$, then\n$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$\nis injective.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3M","source_file":"divisors.tex","source_line":651,"source_end_line":658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L651-L658","statement_sha256":"8852ab9125e05683f307122466fb0732229c22faa9163e002eb1c7e869ef90a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":5990,"rank":5990,"depth":4,"x":2487.146,"y":653.319,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AQ","tag":"05AQ","title":"Weakly associated points · Lemma 05AQ","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Let x ∈ Supp(F) be a point in the support of F which is not a specialization of another point of Supp(F). Then x ∈ WeakAss(F). In particular, any generic point of an irreducible component of X is weakly associated to O_X.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in \\text{Supp}(\\mathcal{F})$ be a point in the support\nof $\\mathcal{F}$ which is not a specialization of another point of\n$\\text{Supp}(\\mathcal{F})$. Then\n$x \\in \\text{WeakAss}(\\mathcal{F})$.\nIn particular, any generic point of an irreducible component of $X$\nis weakly associated to $\\mathcal{O}_X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AQ","source_file":"divisors.tex","source_line":673,"source_end_line":683,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L673-L683","statement_sha256":"7c4c8ed4a281cbeff074dc02de553153403968910ddb379b38e710d3169e7449","origin":"The Stacks Project","memory_eligible":false,"source_rank":5991,"rank":5991,"depth":3,"x":2496.527,"y":704.227,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AR","tag":"05AR","title":"Weakly associated points · Lemma 05AR","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. If m_x is a finitely generated ideal of O_X, x, then x ∈ Ass(F) ⇔ x ∈ WeakAss(F). In particular, if X is locally Noetherian, then Ass(F) = WeakAss(F).","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $\\mathfrak m_x$ is a finitely generated ideal of $\\mathcal{O}_{X, x}$,\nthen\n$$\nx \\in \\text{Ass}(\\mathcal{F}) \\Leftrightarrow\nx \\in \\text{WeakAss}(\\mathcal{F}).\n$$\nIn particular, if $X$ is locally Noetherian, then\n$\\text{Ass}(\\mathcal{F}) = \\text{WeakAss}(\\mathcal{F})$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AR","source_file":"divisors.tex","source_line":700,"source_end_line":712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L700-L712","statement_sha256":"37f7064094c53036b40f29349f245dec83b317b5729cbb6931d4be06f93c8dc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":5992,"rank":5992,"depth":3,"x":2447.691,"y":671.342,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AVN","tag":"0AVN","title":"Weakly associated points · Lemma 0AVN","summary":"Let f : X → S be a quasi-compact and quasi-separated morphism of schemes. Let F be a quasi-coherent O_X-module. Let s ∈ S be a point which is not in the image of f. Then s is not weakly associated to f_*F.","statement_latex":"Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $s \\in S$ be a point which is not in the image of $f$. Then\n$s$ is not weakly associated to $f_*\\mathcal{F}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVN","source_file":"divisors.tex","source_line":719,"source_end_line":725,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L719-L725","statement_sha256":"3e34295d3df921dff6d39c09b46aa7d08572bea95e996cf6756fb9bb08370cb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":5993,"rank":5993,"depth":30,"x":2511.384,"y":667.82,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AVP","tag":"0AVP","title":"Weakly associated points · Lemma 0AVP","summary":"Let X be a scheme. Let φ : F → G be a map of quasi-coherent O_X-modules. Assume that for every x ∈ X at least one of the following happens • F_x → G_x is injective, or • x not ∈ WeakAss(F). Then φ is injective.","statement_latex":"Let $X$ be a scheme. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of\nquasi-coherent $\\mathcal{O}_X$-modules. Assume that for every $x \\in X$\nat least one of the following happens\n\\begin{enumerate}\n\\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is injective, or\n\\item $x \\not \\in \\text{WeakAss}(\\mathcal{F})$.\n\\end{enumerate}\nThen $\\varphi$ is injective.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVP","source_file":"divisors.tex","source_line":756,"source_end_line":766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L756-L766","statement_sha256":"d6ae85f4c15d40a6f5106a82584c481eddda863e470055f8d7f8452ed659f308","origin":"The Stacks Project","memory_eligible":false,"source_rank":5994,"rank":5994,"depth":4,"x":2466.404,"y":707.29,"cluster":"divisors-intersection-theory"},{"id":"stacks:0E9I","tag":"0E9I","title":"Weakly associated points · Lemma 0E9I","summary":"Let X be a locally Noetherian scheme. Let F be a coherent O_X-module. Let j : U → X be an open subscheme such that for x ∈ X setminus U we have depth(F_x) ≥ 2. Then F → j_*(F|_U) is an isomorphism and consequently Γ(X, F) → Γ(U, F) is an isomorphism too.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{F}$\nbe a coherent $\\mathcal{O}_X$-module. Let $j : U \\to X$\nbe an open subscheme such that for $x \\in X \\setminus U$\nwe have $\\text{depth}(\\mathcal{F}_x) \\geq 2$. Then\n$$\n\\mathcal{F} \\longrightarrow j_*(\\mathcal{F}|_U)\n$$\nis an isomorphism and consequently\n$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$\nis an isomorphism too.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9I","source_file":"divisors.tex","source_line":773,"source_end_line":785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L773-L785","statement_sha256":"463d73db5968a8279474da3849dd96c6253cfd256e851408b62a6eb665795f91","origin":"The Stacks Project","memory_eligible":false,"source_rank":5995,"rank":5995,"depth":31,"x":2467.865,"y":651.661,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EME","tag":"0EME","title":"Weakly associated points · Lemma 0EME","summary":"Let X be a reduced scheme. Then the weakly associated points of X are exactly the generic points of the irreducible components of X.","statement_latex":"Let $X$ be a reduced scheme. Then the weakly associated points of $X$\nare exactly the generic points of the irreducible components of $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EME","source_file":"divisors.tex","source_line":798,"source_end_line":802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L798-L802","statement_sha256":"c703e88431f3bb73099ac4d1e470c59f8385ad94a765f9db892ac9af10504ce1","origin":"The Stacks Project","memory_eligible":false,"source_rank":5996,"rank":5996,"depth":2,"x":2512.289,"y":694.255,"cluster":"divisors-intersection-theory"},{"id":"stacks:05EX","tag":"05EX","title":"Morphisms and weakly associated points · Lemma 05EX","summary":"Let f : X → S be an affine morphism of schemes. Let F be a quasi-coherent O_X-module. Then we have WeakAss_S(f_*F) ⊂ f(WeakAss_X(F))","statement_latex":"Let $f : X \\to S$ be an affine morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen we have\n$$\n\\text{WeakAss}_S(f_*\\mathcal{F}) \\subset f(\\text{WeakAss}_X(\\mathcal{F}))\n$$","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EX","source_file":"divisors.tex","source_line":813,"source_end_line":821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L813-L821","statement_sha256":"1964a9dbe75ed7ac6263f3cb6139aeaa5a9c2380c746427f7123a5ff28426114","origin":"The Stacks Project","memory_eligible":false,"source_rank":5997,"rank":5997,"depth":2,"x":2444.135,"y":687.941,"cluster":"divisors-intersection-theory"},{"id":"stacks:05EY","tag":"05EY","title":"Morphisms and weakly associated points · Lemma 05EY","summary":"Let f : X → S be an affine morphism of schemes. Let F be a quasi-coherent O_X-module. If X is locally Noetherian, then we have f(Ass_X(F)) = Ass_S(f_*F) = WeakAss_S(f_*F) = f(WeakAss_X(F))","statement_latex":"Let $f : X \\to S$ be an affine morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $X$ is locally Noetherian, then we have\n$$\nf(\\text{Ass}_X(\\mathcal{F})) =\n\\text{Ass}_S(f_*\\mathcal{F}) =\n\\text{WeakAss}_S(f_*\\mathcal{F}) =\nf(\\text{WeakAss}_X(\\mathcal{F}))\n$$","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EY","source_file":"divisors.tex","source_line":834,"source_end_line":845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L834-L845","statement_sha256":"8a7a0238646b1bd67df69104e05322927df66d62220add665586cdf248df3f53","origin":"The Stacks Project","memory_eligible":false,"source_rank":5998,"rank":5998,"depth":9,"x":2500.386,"y":653.368,"cluster":"divisors-intersection-theory"},{"id":"stacks:05EZ","tag":"05EZ","title":"Morphisms and weakly associated points · Lemma 05EZ","summary":"Let f : X → S be a finite morphism of schemes. Let F be a quasi-coherent O_X-module. Then WeakAss(f_*F) = f(WeakAss(F)).","statement_latex":"Let $f : X \\to S$ be a finite morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $\\text{WeakAss}(f_*\\mathcal{F}) = f(\\text{WeakAss}(\\mathcal{F}))$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EZ","source_file":"divisors.tex","source_line":879,"source_end_line":884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L879-L884","statement_sha256":"7753b9380a7367ca665c2bd2c6a9f0e14a4962072b19198fb8a427df1969b168","origin":"The Stacks Project","memory_eligible":false,"source_rank":5999,"rank":5999,"depth":10,"x":2486.485,"y":711.696,"cluster":"divisors-intersection-theory"},{"id":"stacks:05F0","tag":"05F0","title":"Morphisms and weakly associated points · Lemma 05F0","summary":"Let f : X → S be a morphism of schemes. Let G be a quasi-coherent O_S-module. Let x ∈ X with s = f(x). If f is flat at x, the point x is a generic point of the fibre X_s, and s ∈ WeakAss_S(G), then x ∈ WeakAss(f^*G).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{G}$ be a\nquasi-coherent $\\mathcal{O}_S$-module. Let $x \\in X$ with $s = f(x)$.\nIf $f$ is flat at $x$, the point $x$ is a generic point of the fibre $X_s$, and\n$s \\in \\text{WeakAss}_S(\\mathcal{G})$, then\n$x \\in \\text{WeakAss}(f^*\\mathcal{G})$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05F0","source_file":"divisors.tex","source_line":897,"source_end_line":904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L897-L904","statement_sha256":"a6c919211bf17430d4bb6e561f9b38a4f75ba63b6863179d08acda520a0ce3c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6000,"rank":6000,"depth":2,"x":2449.268,"y":660.007,"cluster":"divisors-intersection-theory"},{"id":"stacks:0CUC","tag":"0CUC","title":"Morphisms and weakly associated points · Lemma 0CUC","summary":"Let K/k be a field extension. Let X be a scheme over k. Let F be a quasi-coherent O_X-module. Let y ∈ X_K with image x ∈ X. If y is a weakly associated point of the pullback F_K, then x is a weakly associated point of F.","statement_latex":"Let $K/k$ be a field extension. Let $X$ be a scheme over $k$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $y \\in X_K$ with image $x \\in X$. If $y$ is a weakly\nassociated point of the pullback $\\mathcal{F}_K$, then $x$\nis a weakly associated point of $\\mathcal{F}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUC","source_file":"divisors.tex","source_line":919,"source_end_line":926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L919-L926","statement_sha256":"57ec7369f4e1e7900dfe18d4c24fde7dcd9bbe2b4403807ed7c13580c30df42a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6001,"rank":6001,"depth":11,"x":2519.312,"y":677.264,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EY0","tag":"0EY0","title":"Morphisms and weakly associated points · Lemma 0EY0","summary":"Let f : X → S be a quasi-compact and quasi-separated morphism of schemes. Let F be a quasi-coherent O_X-module. Let s ∈ S. • If s not ∈ f(X), then s is not weakly associated to f_*F. • If s not ∈ f(X) and O_S, s is Noetherian, then s is not associated to f_*F. • If s not ∈ f(X), (f_*F)_s is a finite O_S, s-module, and O_S, s is Noetherian, then depth((f_*F)_s) ≥ 2. • If F is flat over S and a ∈ m_s is a nonzerodivisor, then a is a nonzerodivisor on (f_*F)_s. • If F is…","statement_latex":"Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $s \\in S$.\n\\begin{enumerate}\n\\item If $s \\not \\in f(X)$, then $s$ is not weakly associated\nto $f_*\\mathcal{F}$.\n\\item If $s \\not \\in f(X)$ and $\\mathcal{O}_{S, s}$ is Noetherian,\nthen $s$ is not associated to $f_*\\mathcal{F}$.\n\\item If $s \\not \\in f(X)$, $(f_*\\mathcal{F})_s$ is a finite\n$\\mathcal{O}_{S, s}$-module, and $\\mathcal{O}_{S, s}$\nis Noetherian, then $\\text{depth}((f_*\\mathcal{F})_s) \\geq 2$.\n\\item If $\\mathcal{F}$ is flat over $S$ and $a \\in \\mathfrak m_s$\nis a nonzerodivisor, then $a$ is a nonzerodivisor on $(f_*\\mathcal{F})_s$.\n\\item If $\\mathcal{F}$ is flat over $S$ and $a, b \\in \\mathfrak m_s$\nis a regular sequence, then $a$ is a nonzerodivisor on $(f_*\\mathcal{F})_s$\nand $b$ is a nonzerodivisor on $(f_*\\mathcal{F})_s/a(f_*\\mathcal{F})_s$.\n\\item If $\\mathcal{F}$ is flat over $S$ and $(f_*\\mathcal{F})_s$\nis a finite $\\mathcal{O}_{S, s}$-module, then\n$\\text{depth}((f_*\\mathcal{F})_s) \\geq\n\\min(2, \\text{depth}(\\mathcal{O}_{S, s}))$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EY0","source_file":"divisors.tex","source_line":938,"source_end_line":961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L938-L961","statement_sha256":"a3b3e5c22070cd3172b1bb1bbcf28c3ee2c16b167593f515ecc6ec1c7cfea5d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6002,"rank":6002,"depth":31,"x":2452.827,"y":704.674,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AT","tag":"05AT","title":"Relative assassin · Definition 05AT","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. The relative assassin of F in X over S is the set Ass_X/S(F) = ⋃_s ∈ S Ass_X_s(F_s) where F_s = (X_s → X)^*F is the restriction of F to the fibre of f at s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe {\\it relative assassin of $\\mathcal{F}$ in $X$ over $S$}\nis the set\n$$\n\\text{Ass}_{X/S}(\\mathcal{F}) =\n\\bigcup\\nolimits_{s \\in S} \\text{Ass}_{X_s}(\\mathcal{F}_s)\n$$\nwhere $\\mathcal{F}_s = (X_s \\to X)^*\\mathcal{F}$ is the restriction\nof $\\mathcal{F}$ to the fibre of $f$ at $s$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative assassin","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AT","source_file":"divisors.tex","source_line":1056,"source_end_line":1068,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1056-L1068","statement_sha256":"8c81df2d520babcf2f45eee760b295743eb36f7adb4009e025f0dc2ab9333dee","origin":"The Stacks Project","memory_eligible":false,"source_rank":6003,"rank":6003,"depth":0,"x":2480.198,"y":645.918,"cluster":"divisors-intersection-theory"},{"id":"stacks:0CU5","tag":"0CU5","title":"Relative assassin · Lemma 0CU5","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let U ⊂ X and V ⊂ S be affine opens with f(U) ⊂ V. Write U = Spec(A), V = Spec(R), and set M = Γ(U, F). Let x ∈ U, and let p ⊂ A be the corresponding prime. Then p ∈ Ass_A/R(M) ⇒ x ∈ Ass_X/S(F) If all fibres X_s of f are locally Noetherian, then p ∈ Ass_A/R(M) ⇔ x ∈ Ass_X/S(F) for all pairs ( p, x) as above.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $U \\subset X$ and $V \\subset S$ be affine opens\nwith $f(U) \\subset V$. Write $U = \\Spec(A)$, $V = \\Spec(R)$, and set\n$M = \\Gamma(U, \\mathcal{F})$.\nLet $x \\in U$, and let $\\mathfrak p \\subset A$ be the corresponding prime.\nThen\n$$\n\\mathfrak p \\in \\text{Ass}_{A/R}(M) \\Rightarrow\nx \\in \\text{Ass}_{X/S}(\\mathcal{F})\n$$\nIf all fibres $X_s$ of $f$ are locally Noetherian, then\n$\\mathfrak p \\in \\text{Ass}_{A/R}(M) \\Leftrightarrow\nx \\in \\text{Ass}_{X/S}(\\mathcal{F})$\nfor all pairs $(\\mathfrak p, x)$ as above.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CU5","source_file":"divisors.tex","source_line":1078,"source_end_line":1095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1078-L1095","statement_sha256":"f6ccdae12ebba54bf37dfd3459527e018e5c6f33b0be4bddfb1f6d8dc4be5189","origin":"The Stacks Project","memory_eligible":false,"source_rank":6004,"rank":6004,"depth":7,"x":2507.632,"y":705.587,"cluster":"divisors-intersection-theory"},{"id":"stacks:05DC","tag":"05DC","title":"Relative assassin · Lemma 05DC","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. Let g : S' → S be a morphism of schemes. Consider the base change diagram xymatrix X' ar[d] ar[r]_g' & X ar[d] S' ar[r]^g & S and set F' = (g')^*F. Let x' ∈ X' be a point with images x ∈ X, s' ∈ S' and s ∈ S. Assume f locally of finite type. Then x' ∈ Ass_X'/S'(F') if and only if x ∈ Ass_X/S(F) and x' corresponds to a generic point of an irreducible component of Spec(kappa(s') ⊗_kappa(s) kappa(x)).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $g : S' \\to S$ be a morphism of schemes.\nConsider the base change diagram\n$$\n\\xymatrix{\nX' \\ar[d] \\ar[r]_{g'} & X \\ar[d] \\\\\nS' \\ar[r]^g & S\n}\n$$\nand set $\\mathcal{F}' = (g')^*\\mathcal{F}$. Let $x' \\in X'$ be a point\nwith images $x \\in X$, $s' \\in S'$ and $s \\in S$.\nAssume $f$ locally of finite type.\nThen $x' \\in \\text{Ass}_{X'/S'}(\\mathcal{F}')$ if and only if\n$x \\in \\text{Ass}_{X/S}(\\mathcal{F})$ and $x'$ corresponds to\na generic point of an irreducible component of\n$\\Spec(\\kappa(s') \\otimes_{\\kappa(s)} \\kappa(x))$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DC","source_file":"divisors.tex","source_line":1119,"source_end_line":1138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1119-L1138","statement_sha256":"fa09319793c508916550ba70261798990226be168fef5b71901e1b01c66eda82","origin":"The Stacks Project","memory_eligible":false,"source_rank":6005,"rank":6005,"depth":14,"x":2438.507,"y":676.77,"cluster":"divisors-intersection-theory"},{"id":"stacks:05AV","tag":"05AV","title":"Relative weak assassin · Definition 05AV","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. The relative weak assassin of F in X over S is the set WeakAss_X/S(F) = ⋃_s ∈ S WeakAss(F_s) where F_s = (X_s → X)^*F is the restriction of F to the fibre of f at s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe {\\it relative weak assassin of $\\mathcal{F}$ in $X$ over $S$}\nis the set\n$$\n\\text{WeakAss}_{X/S}(\\mathcal{F}) =\n\\bigcup\\nolimits_{s \\in S} \\text{WeakAss}(\\mathcal{F}_s)\n$$\nwhere $\\mathcal{F}_s = (X_s \\to X)^*\\mathcal{F}$ is the restriction\nof $\\mathcal{F}$ to the fibre of $f$ at $s$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative weak assassin","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AV","source_file":"divisors.tex","source_line":1197,"source_end_line":1209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1197-L1209","statement_sha256":"a87e4905e13b3a01da508768b04fc86fc0203f3b9e4dd6b7d12df60b77fc4869","origin":"The Stacks Project","memory_eligible":false,"source_rank":6006,"rank":6006,"depth":0,"x":2513.622,"y":658.565,"cluster":"divisors-intersection-theory"},{"id":"stacks:05F2","tag":"05F2","title":"Relative weak assassin · Lemma 05F2","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent O_X-module. Then WeakAss_X/S(F) = Ass_X/S(F).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $\\text{WeakAss}_{X/S}(\\mathcal{F}) = \\text{Ass}_{X/S}(\\mathcal{F})$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05F2","source_file":"divisors.tex","source_line":1211,"source_end_line":1216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1211-L1216","statement_sha256":"a7e0d5f701ab812d6a99f4362207264e4311dd1821d18926decc58be1411bc44","origin":"The Stacks Project","memory_eligible":false,"source_rank":6007,"rank":6007,"depth":4,"x":2472.35,"y":715.322,"cluster":"divisors-intersection-theory"},{"id":"stacks:0CUD","tag":"0CUD","title":"Relative weak assassin · Lemma 0CUD","summary":"Let f : X → S be a morphism of schemes. Let i : Z → X be a finite morphism. Let F be a quasi-coherent O_Z-module. Then WeakAss_X/S(i_*F) = i(WeakAss_Z/S(F)).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $i : Z \\to X$ be a finite morphism.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_Z$-module.\nThen $\\text{WeakAss}_{X/S}(i_*\\mathcal{F}) =\ni(\\text{WeakAss}_{Z/S}(\\mathcal{F}))$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUD","source_file":"divisors.tex","source_line":1225,"source_end_line":1232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1225-L1232","statement_sha256":"f84d5ae0a5469f58b2b17a5c56721c241b70942831e665517c8c31a62471b0ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":6008,"rank":6008,"depth":26,"x":2456.957,"y":649.242,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C3D","tag":"0C3D","title":"Fitting ideals · Lemma 0C3D","summary":"Let f : T → S be a morphism of schemes. Let F be a finite type quasi-coherent O_S-module. Then f^-1Fit_i(F) · O_T = Fit_i(f^*F).","statement_latex":"Let $f : T \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_S$-module.\nThen\n$f^{-1}\\text{Fit}_i(\\mathcal{F}) \\cdot \\mathcal{O}_T =\n\\text{Fit}_i(f^*\\mathcal{F})$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3D","source_file":"divisors.tex","source_line":1283,"source_end_line":1290,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1283-L1290","statement_sha256":"f7f86c7e335ca36ae3c323503db21c79342e104e08943f2890aafd05c5a542f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6009,"rank":6009,"depth":4,"x":2522.225,"y":689.721,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C3E","tag":"0C3E","title":"Fitting ideals · Lemma 0C3E","summary":"Let S be a scheme. Let F be a finitely presented O_S-module. Then Fit_r(F) is a quasi-coherent ideal of finite type.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_S$-module.\nThen $\\text{Fit}_r(\\mathcal{F})$ is a quasi-coherent ideal of finite type.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3E","source_file":"divisors.tex","source_line":1297,"source_end_line":1302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1297-L1302","statement_sha256":"cfa28a5c80ad008eed8802dac71b11e73674a19b0943430675df537ebc9982ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":6010,"rank":6010,"depth":4,"x":2440.592,"y":696.988,"cluster":"divisors-intersection-theory"},{"id":"stacks:0CYX","tag":"0CYX","title":"Fitting ideals · Lemma 0CYX","summary":"Let S be a scheme. Let F be a finite type, quasi-coherent O_S-module. Let Z_0 ⊂ S be the closed subscheme cut out by Fit_0(F). Let Z ⊂ S be the scheme theoretic support of F. Then • Z ⊂ Z_0 ⊂ S as closed subschemes, • Z = Z_0 = Supp(F) as closed subsets, • there exists a finite type, quasi-coherent O_Z_0-module G_0 with (Z_0 → X)_*G_0 = F.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_S$-module.\nLet $Z_0 \\subset S$ be the closed subscheme cut out by\n$\\text{Fit}_0(\\mathcal{F})$.\nLet $Z \\subset S$ be the scheme theoretic support of $\\mathcal{F}$.\nThen\n\\begin{enumerate}\n\\item $Z \\subset Z_0 \\subset S$ as closed subschemes,\n\\item $Z = Z_0 = \\text{Supp}(\\mathcal{F})$ as closed subsets,\n\\item there exists a finite type, quasi-coherent $\\mathcal{O}_{Z_0}$-module\n$\\mathcal{G}_0$ with\n$$\n(Z_0 \\to X)_*\\mathcal{G}_0 = \\mathcal{F}.\n$$\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYX","source_file":"divisors.tex","source_line":1309,"source_end_line":1326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1309-L1326","statement_sha256":"bd66813c2aacb309660fd62bba4157e91e0677bb2d4bc876cc445bff893db913","origin":"The Stacks Project","memory_eligible":false,"source_rank":6011,"rank":6011,"depth":19,"x":2495.573,"y":644.714,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C3F","tag":"0C3F","title":"Fitting ideals · Lemma 0C3F","summary":"Let S be a scheme. Let F be a finite type, quasi-coherent O_S-module. Let s ∈ S. Then F can be generated by r elements in a neighbourhood of s if and only if Fit_r(F)_s = O_S, s.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent\n$\\mathcal{O}_S$-module. Let $s \\in S$. Then $\\mathcal{F}$ can be\ngenerated by $r$ elements in a neighbourhood of $s$ if and only\nif $\\text{Fit}_r(\\mathcal{F})_s = \\mathcal{O}_{S, s}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3F","source_file":"divisors.tex","source_line":1351,"source_end_line":1357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1351-L1357","statement_sha256":"0d2ba0d748fc6029190561887335c6cd920ce0233aba25291d3bc97175ca820d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6012,"rank":6012,"depth":5,"x":2497.08,"y":715.244,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C3G","tag":"0C3G","title":"Fitting ideals · Lemma 0C3G","summary":"Let S be a scheme. Let F be a finite type, quasi-coherent O_S-module. Let r ≥ 0. The following are equivalent • F is finite locally free of rank r • Fit_r - 1(F) = 0 and Fit_r(F) = O_S, and • Fit_k(F) = 0 for k < r and Fit_k(F) = O_S for k ≥ r.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent\n$\\mathcal{O}_S$-module. Let $r \\geq 0$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is finite locally free of rank $r$\n\\item $\\text{Fit}_{r - 1}(\\mathcal{F}) = 0$ and\n$\\text{Fit}_r(\\mathcal{F}) = \\mathcal{O}_S$, and\n\\item $\\text{Fit}_k(\\mathcal{F}) = 0$ for $k < r$ and\n$\\text{Fit}_k(\\mathcal{F}) = \\mathcal{O}_S$ for $k \\geq r$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3G","source_file":"divisors.tex","source_line":1364,"source_end_line":1375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1364-L1375","statement_sha256":"55e049d43cc02c2876a51bafd12824f749ae51b0a4bad0bbd85751a7f3333218","origin":"The Stacks Project","memory_eligible":false,"source_rank":6013,"rank":6013,"depth":6,"x":2438.616,"y":663.525,"cluster":"divisors-intersection-theory"},{"id":"stacks:05P8","tag":"05P8","title":"Fitting ideals · Lemma 05P8","summary":"Let S be a scheme. Let F be a finite type, quasi-coherent O_S-module. The closed subschemes S = Z_-1 ⊃ Z_0 ⊃ Z_1 ⊃ Z_2 … defined by the Fitting ideals of F have the following properties • The intersection ⋂ Z_r is empty. • The functor (Sch/S)^opp → Sets defined by the rule T ↦ ( (*) & if F_T is locally generated by ≤ r sections ∅ & otherwise . is representable by the open subscheme S setminus Z_r. • The functor F_r : (Sch/S)^opp → Sets defined by the rule T ↦ ( (*) & if…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type, quasi-coherent\n$\\mathcal{O}_S$-module. The closed subschemes\n$$\nS = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset Z_2 \\ldots\n$$\ndefined by the Fitting ideals of $\\mathcal{F}$ have the following\nproperties\n\\begin{enumerate}\n\\item The intersection $\\bigcap Z_r$ is empty.\n\\item The functor $(\\Sch/S)^{opp} \\to \\textit{Sets}$ defined by the rule\n$$\nT \\longmapsto\n\\left\\{\n\\begin{matrix}\n\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ is locally generated by }\n\\leq r\\text{ sections} \\\\\n\\emptyset & \\text{otherwise}\n\\end{matrix}\n\\right.\n$$\nis representable by the open subscheme $S \\setminus Z_r$.\n\\item The functor $F_r : (\\Sch/S)^{opp} \\to \\textit{Sets}$ defined by the rule\n$$\nT \\longmapsto\n\\left\\{\n\\begin{matrix}\n\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ locally free rank }r\\\\\n\\emptyset & \\text{otherwise}\n\\end{matrix}\n\\right.\n$$\nis representable by the locally closed subscheme $Z_{r - 1} \\setminus Z_r$\nof $S$.\n\\end{enumerate}\nIf $\\mathcal{F}$ is of finite presentation, then\n$Z_r \\to S$, $S \\setminus Z_r \\to S$, and $Z_{r - 1} \\setminus Z_r \\to S$\nare of finite presentation.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05P8","source_file":"divisors.tex","source_line":1383,"source_end_line":1422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1383-L1422","statement_sha256":"04690a86a1dd7fd4024696db6ee6fd34bf1fefd5f239efdb772a74956af9d009","origin":"The Stacks Project","memory_eligible":false,"source_rank":6014,"rank":6014,"depth":17,"x":2524.232,"y":668.545,"cluster":"divisors-intersection-theory"},{"id":"stacks:05P9","tag":"05P9","title":"Fitting ideals · Lemma 05P9","summary":"Let S be a scheme. Let F be an O_S-module of finite presentation. Let S = Z_-1 ⊃ Z_0 ⊃ Z_1 ⊃ … be as in Lemma [Tag 05P8]. Set S_r = Z_r - 1 setminus Z_r. Then S' = coprod_r ≥ 0 S_r represents the functor F_flat : Sch/S → Sets, T ↦ ( (*) & if F_T flat over T ∅ & otherwise . Moreover, F|_S_r is locally free of rank r and the morphisms S_r → S and S' → S are of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}_S$-module\nof finite presentation. Let $S = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset \\ldots$\nbe as in Lemma \\ref{lemma-locally-free-rank-r-pullback}.\nSet $S_r = Z_{r - 1} \\setminus Z_r$.\nThen $S' = \\coprod_{r \\geq 0} S_r$ represents the functor\n$$\nF_{flat} : \\Sch/S \\longrightarrow \\textit{Sets},\\quad\\quad\nT \\longmapsto\n\\left\\{\n\\begin{matrix}\n\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ flat over }T\\\\\n\\emptyset & \\text{otherwise}\n\\end{matrix}\n\\right.\n$$\nMoreover, $\\mathcal{F}|_{S_r}$ is locally free of rank $r$ and the\nmorphisms $S_r \\to S$ and $S' \\to S$ are of finite presentation.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05P9","source_file":"divisors.tex","source_line":1464,"source_end_line":1483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1464-L1483","statement_sha256":"c1721ccaa51833d8ea20318ea82022d89e685c075dcfa51f2c8be2259954a81d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6015,"rank":6015,"depth":18,"x":2456.352,"y":713.895,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C3I","tag":"0C3I","title":"The singular locus of a morphism · Lemma 0C3I","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let X = Z_-1 ⊃ Z_0 ⊃ Z_1 ⊃ … be the closed subschemes defined by the fitting ideals of Ω_X/S. Then the formation of Z_i commutes with arbitrary base change.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $X = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset \\ldots$\nbe the closed subschemes defined by the fitting ideals\nof $\\Omega_{X/S}$. Then the formation of $Z_i$ commutes\nwith arbitrary base change.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The singular locus of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3I","source_file":"divisors.tex","source_line":1569,"source_end_line":1576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1569-L1576","statement_sha256":"6ff0c80d7d098df7d7a769273ab3f649ef2998eef369fe7c5a023930d0979789","origin":"The Stacks Project","memory_eligible":false,"source_rank":6016,"rank":6016,"depth":17,"x":2470.08,"y":641.194,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C3J","tag":"0C3J","title":"The singular locus of a morphism · Lemma 0C3J","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. The closed subscheme Z ⊂ X cut out by the 0th fitting ideal of Ω_X/S is exactly the set of points where f is not unramified.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nThe closed subscheme $Z \\subset X$ cut out by the $0$th fitting ideal of\n$\\Omega_{X/S}$ is exactly the set of points where $f$ is not unramified.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The singular locus of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3J","source_file":"divisors.tex","source_line":1600,"source_end_line":1605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1600-L1605","statement_sha256":"c1196d98688c7daaf1ee7f390e3148a56f870a394a4f4848e3093307995c9030","origin":"The Stacks Project","memory_eligible":false,"source_rank":6017,"rank":6017,"depth":20,"x":2518.908,"y":703.216,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C3K","tag":"0C3K","title":"The singular locus of a morphism · Lemma 0C3K","summary":"Let f : X → S be a morphism of schemes. Let d ≥ 0 be an integer. Assume • f is flat, • f is locally of finite presentation, and • every nonempty fibre of f is equidimensional of dimension d. Let Z ⊂ X be the closed subscheme cut out by the dth fitting ideal of Ω_X/S. Then Z is exactly the set of points where f is not smooth.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $d \\geq 0$ be an integer.\nAssume\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $f$ is locally of finite presentation, and\n\\item every nonempty fibre of $f$ is equidimensional of dimension $d$.\n\\end{enumerate}\nLet $Z \\subset X$ be the closed subscheme cut out by the $d$th fitting\nideal of $\\Omega_{X/S}$. Then $Z$ is exactly the set of points\nwhere $f$ is not smooth.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The singular locus of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3K","source_file":"divisors.tex","source_line":1614,"source_end_line":1626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1614-L1626","statement_sha256":"4161472934125e18ba3d8d82a26d62716a1ebecd86aaecae47db62202d4f5650","origin":"The Stacks Project","memory_eligible":false,"source_rank":6018,"rank":6018,"depth":39,"x":2432.174,"y":685.007,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AXR","tag":"0AXR","title":"Torsion free modules · Lemma 0AXR","summary":"Let X be an integral scheme with generic point eta. Let F be a quasi-coherent O_X-module. Let U ⊂ X be nonempty open and s ∈ F(U). The following are equivalent • for some x ∈ U the image of s in F_x is torsion, • for all x ∈ U the image of s in F_x is torsion, • the image of s in F_eta is zero, • the image of s in j_*F_eta is zero, where j : eta → X is the inclusion morphism.","statement_latex":"Let $X$ be an integral scheme with generic point $\\eta$. Let $\\mathcal{F}$\nbe a quasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be nonempty\nopen and $s \\in \\mathcal{F}(U)$. The following are equivalent\n\\begin{enumerate}\n\\item for some $x \\in U$ the image of $s$ in $\\mathcal{F}_x$ is torsion,\n\\item for all $x \\in U$ the image of $s$ in $\\mathcal{F}_x$ is torsion,\n\\item the image of $s$ in $\\mathcal{F}_\\eta$ is zero,\n\\item the image of $s$ in $j_*\\mathcal{F}_\\eta$ is zero, where $j : \\eta \\to X$\nis the inclusion morphism.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXR","source_file":"divisors.tex","source_line":1649,"source_end_line":1661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1649-L1661","statement_sha256":"d5776bdd44d1796742a0cc48f0c38707965a7429aedb1ff47cd88de575648dce","origin":"The Stacks Project","memory_eligible":false,"source_rank":6019,"rank":6019,"depth":0,"x":2511.541,"y":648.874,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AVR","tag":"0AVR","title":"Torsion free modules · Definition 0AVR","summary":"Let X be an integral scheme. Let F be a quasi-coherent O_X-module. • We say a local section of F is torsion if it satisfies the equivalent conditions of Lemma [Tag 0AXR]. • We say F is torsion free if every torsion section of F is 0.","statement_latex":"Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item We say a local section of $\\mathcal{F}$ is {\\it torsion}\nif it satisfies the equivalent conditions of Lemma \\ref{lemma-torsion-sections}.\n\\item We say $\\mathcal{F}$ is {\\it torsion free} if every torsion section\nof $\\mathcal{F}$ is $0$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVR","source_file":"divisors.tex","source_line":1667,"source_end_line":1677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1667-L1677","statement_sha256":"baf2b56b3306b78601a02d1d3d3ab1854ddd459b469a8265a05e781cca4a04d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6020,"rank":6020,"depth":1,"x":2481.79,"y":721.235,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AXS","tag":"0AXS","title":"Torsion free modules · Lemma 0AXS","summary":"Let X be an integral scheme. Let F be a quasi-coherent O_X-module. The following are equivalent • F is torsion free, • for U ⊂ X affine open F(U) is a torsion free O(U)-module.","statement_latex":"Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is torsion free,\n\\item for $U \\subset X$ affine open $\\mathcal{F}(U)$\nis a torsion free $\\mathcal{O}(U)$-module.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXS","source_file":"divisors.tex","source_line":1682,"source_end_line":1691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1682-L1691","statement_sha256":"fd37e3b49e4848f7fcc2f2b35f388ee95a0f1303e54925e57b54d8e60b572bd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6021,"rank":6021,"depth":0,"x":2445.2,"y":650.337,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AXT","tag":"0AXT","title":"Torsion free modules · Lemma 0AXT","summary":"Let X be an integral scheme. Let F be a quasi-coherent O_X-module. The torsion sections of F form a quasi-coherent O_X-submodule F_tors ⊂ F. The quotient module F/F_tors is torsion free.","statement_latex":"Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. The torsion sections of $\\mathcal{F}$ form\na quasi-coherent $\\mathcal{O}_X$-submodule\n$\\mathcal{F}_{tors} \\subset \\mathcal{F}$.\nThe quotient module $\\mathcal{F}/\\mathcal{F}_{tors}$ is torsion free.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXT","source_file":"divisors.tex","source_line":1697,"source_end_line":1704,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1697-L1704","statement_sha256":"358e724db9adad8beb59f0bed164848cd25117ca3ee0a7ec73065592e3b6c408","origin":"The Stacks Project","memory_eligible":false,"source_rank":6022,"rank":6022,"depth":1,"x":2529.967,"y":682.146,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AXU","tag":"0AXU","title":"Torsion free modules · Lemma 0AXU","summary":"Let X be an integral scheme. Any flat quasi-coherent O_X-module is torsion free.","statement_latex":"Let $X$ be an integral scheme. Any flat quasi-coherent $\\mathcal{O}_X$-module\nis torsion free.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXU","source_file":"divisors.tex","source_line":1711,"source_end_line":1715,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1711-L1715","statement_sha256":"f92c21f792da7d11ffe93aa32c01fe33a9ce167d6a8b21fa66d6ffed78ec9800","origin":"The Stacks Project","memory_eligible":false,"source_rank":6023,"rank":6023,"depth":1,"x":2441.084,"y":707.009,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AXV","tag":"0AXV","title":"Torsion free modules · Lemma 0AXV","summary":"Let f : X → Y be a flat morphism of integral schemes. Let G be a torsion free quasi-coherent O_Y-module. Then f^*G is a torsion free O_X-module.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of integral schemes.\nLet $\\mathcal{G}$ be a torsion free quasi-coherent $\\mathcal{O}_Y$-module.\nThen $f^*\\mathcal{G}$ is a torsion free $\\mathcal{O}_X$-module.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXV","source_file":"divisors.tex","source_line":1721,"source_end_line":1726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1721-L1726","statement_sha256":"908bf057dfeedcb598026eda6725c8114dac6e64269305b521eb3fd7ebc39c31","origin":"The Stacks Project","memory_eligible":false,"source_rank":6024,"rank":6024,"depth":1,"x":2487.037,"y":637.633,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BCM","tag":"0BCM","title":"Torsion free modules · Lemma 0BCM","summary":"Let f : X → Y be a flat morphism of schemes. If Y is integral and the generic fibre of f is integral, then X is integral.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of schemes. If $Y$ is integral\nand the generic fibre of $f$ is integral, then $X$ is integral.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCM","source_file":"divisors.tex","source_line":1734,"source_end_line":1738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1734-L1738","statement_sha256":"fb1cefb91de65b460ad7d57fa256ae95b47723d962d0187756e0d00b5154218e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6025,"rank":6025,"depth":1,"x":2509.131,"y":715.538,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AXW","tag":"0AXW","title":"Torsion free modules · Lemma 0AXW","summary":"Let X be an integral scheme. Let F be a quasi-coherent O_X-module. Then F is torsion free if and only if F_x is a torsion free O_X, x-module for all x ∈ X.","statement_latex":"Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is torsion free if and only if\n$\\mathcal{F}_x$ is a torsion free $\\mathcal{O}_{X, x}$-module for all $x \\in X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXW","source_file":"divisors.tex","source_line":1749,"source_end_line":1754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1749-L1754","statement_sha256":"cd38d5ef5b7f27d5a7cd722f510698e2c7df9b434b83ff92708a37ca44b3b9d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6026,"rank":6026,"depth":3,"x":2429.515,"y":670.242,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AXX","tag":"0AXX","title":"Torsion free modules · Lemma 0AXX","summary":"Let X be an integral scheme. Let 0 → F → F' → F\" → 0 be a short exact sequence of quasi-coherent O_X-modules. If F and F\" are torsion free, then F' is torsion free.","statement_latex":"Let $X$ be an integral scheme. Let\n$0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to 0$\nbe a short exact sequence of quasi-coherent $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}$ and $\\mathcal{F}''$ are torsion free, then $\\mathcal{F}'$\nis torsion free.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXX","source_file":"divisors.tex","source_line":1761,"source_end_line":1768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1761-L1768","statement_sha256":"5ce3a38fd90290423ea42e822c88ea4f2be4a176ef093c17864f9413828c2545","origin":"The Stacks Project","memory_eligible":false,"source_rank":6027,"rank":6027,"depth":1,"x":2525.45,"y":658.37,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AXY","tag":"0AXY","title":"Torsion free modules · Lemma 0AXY","summary":"Let X be a locally Noetherian integral scheme with generic point eta. Let F be a nonzero coherent O_X-module. The following are equivalent • F is torsion free, • eta is the only associated prime of F, • eta is in the support of F and F has property (S_1), and • eta is in the support of F and F has no embedded associated prime.","statement_latex":"Let $X$ be a locally Noetherian integral scheme with generic point $\\eta$.\nLet $\\mathcal{F}$ be a nonzero coherent $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is torsion free,\n\\item $\\eta$ is the only associated prime of $\\mathcal{F}$,\n\\item $\\eta$ is in the support of $\\mathcal{F}$ and $\\mathcal{F}$\nhas property $(S_1)$, and\n\\item $\\eta$ is in the support of $\\mathcal{F}$ and $\\mathcal{F}$\nhas no embedded associated prime.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXY","source_file":"divisors.tex","source_line":1776,"source_end_line":1789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1776-L1789","statement_sha256":"36adac5c499d37861cdb57d68121262cf47013bfb28be12138d1121769b23061","origin":"The Stacks Project","memory_eligible":false,"source_rank":6028,"rank":6028,"depth":13,"x":2463.748,"y":722.082,"cluster":"divisors-intersection-theory"},{"id":"stacks:0CC4","tag":"0CC4","title":"Torsion free modules · Lemma 0CC4","summary":"Let X be an integral regular scheme of dimension ≤ 1. Let F be a coherent O_X-module. The following are equivalent • F is torsion free, • F is finite locally free.","statement_latex":"Let $X$ be an integral regular scheme of dimension $\\leq 1$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is torsion free,\n\\item $\\mathcal{F}$ is finite locally free.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CC4","source_file":"divisors.tex","source_line":1797,"source_end_line":1806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1797-L1806","statement_sha256":"a210f500ef39e1644868d275a10ce2e9694029dc5316f08568103a4bcfc11ffe","origin":"The Stacks Project","memory_eligible":false,"source_rank":6029,"rank":6029,"depth":18,"x":2457.967,"y":639.423,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AXZ","tag":"0AXZ","title":"Torsion free modules · Lemma 0AXZ","summary":"Let X be an integral scheme. Let F, G be quasi-coherent O_X-modules. If G is torsion free and F is of finite presentation, then SheafHom_O_X(F, G) is torsion free.","statement_latex":"Let $X$ be an integral scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be\nquasi-coherent $\\mathcal{O}_X$-modules.\nIf $\\mathcal{G}$ is torsion free and $\\mathcal{F}$ is of finite presentation,\nthen $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is torsion free.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXZ","source_file":"divisors.tex","source_line":1825,"source_end_line":1831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1825-L1831","statement_sha256":"7d658811246d44e1bfad6a666cbe97cef38c14ac7502aba21c3729ea24014849","origin":"The Stacks Project","memory_eligible":false,"source_rank":6030,"rank":6030,"depth":1,"x":2529.266,"y":697.557,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AVS","tag":"0AVS","title":"Torsion free modules · Lemma 0AVS","summary":"Let X be an integral locally Noetherian scheme. Let φ : F → G be a map of quasi-coherent O_X-modules. Assume F is coherent, G is torsion free, and that for every x ∈ X one of the following happens • F_x → G_x is an isomorphism, or • depth(F_x) ≥ 2. Then φ is an isomorphism.","statement_latex":"Let $X$ be an integral locally Noetherian scheme. Let\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of\nquasi-coherent $\\mathcal{O}_X$-modules. Assume $\\mathcal{F}$ is coherent,\n$\\mathcal{G}$ is torsion free, and that for every $x \\in X$ one of the\nfollowing happens\n\\begin{enumerate}\n\\item $\\mathcal{F}_x \\to \\mathcal{G}_x$ is an isomorphism, or\n\\item $\\text{depth}(\\mathcal{F}_x) \\geq 2$.\n\\end{enumerate}\nThen $\\varphi$ is an isomorphism.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Torsion free modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVS","source_file":"divisors.tex","source_line":1842,"source_end_line":1854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1842-L1854","statement_sha256":"c39080f3f0b94e8a6ad3afc73b04cf4d8559daa047c6dbcff2084a5e6cf9046a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6031,"rank":6031,"depth":15,"x":2429.158,"y":695.126,"cluster":"divisors-intersection-theory"},{"id":"stacks:0HAF","tag":"0HAF","title":"Ranks of modules · Definition 0HAF","summary":"Let X be an integral scheme with generic point eta. Let F be a quasi-coherent O_X-module. The rank of F is dim_kappa(eta) F_eta.","statement_latex":"Let $X$ be an integral scheme with generic point $\\eta$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe {\\it rank} of $\\mathcal{F}$ is $\\dim_{\\kappa(\\eta)} \\mathcal{F}_\\eta$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Ranks of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAF","source_file":"divisors.tex","source_line":1881,"source_end_line":1886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1881-L1886","statement_sha256":"3c222d78b20ced7990eead065b01b66f00bbefe8f4f9d621cf76a39edb9cce10","origin":"The Stacks Project","memory_eligible":false,"source_rank":6032,"rank":6032,"depth":0,"x":2505.521,"y":639.689,"cluster":"divisors-intersection-theory"},{"id":"stacks:0HAG","tag":"0HAG","title":"Ranks of modules · Lemma 0HAG","summary":"Let X be an integral scheme. Let F be a quasi-coherent O_X-module. Let r be a cardinal. The following are equivalent • F has rank r, • for all U ⊂ X nonempty affine open F(U) is an O_X(U)-module of rank r, and • for some U ⊂ X nonempty affine open F(U) is an O_X(U)-module of rank r.","statement_latex":"Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Let $r$ be a cardinal. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ has rank $r$,\n\\item for all $U \\subset X$ nonempty affine open $\\mathcal{F}(U)$\nis an $\\mathcal{O}_X(U)$-module of rank $r$, and\n\\item for some $U \\subset X$ nonempty affine open $\\mathcal{F}(U)$\nis an $\\mathcal{O}_X(U)$-module of rank $r$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAG","source_file":"divisors.tex","source_line":1898,"source_end_line":1909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1898-L1909","statement_sha256":"59e30be9a93062f8f72c234daf82d3d8bc2a1005dfacd0a2246ab0dceaed704d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6033,"rank":6033,"depth":0,"x":2493.701,"y":724.541,"cluster":"divisors-intersection-theory"},{"id":"stacks:0HAH","tag":"0HAH","title":"Ranks of modules · Lemma 0HAH","summary":"Let f : X → Y be a dominant morphism of integral schemes. Let G be a quasi-coherent O_Y-module. Then the rank of G on Y is equal to the rank of f^*G on X.","statement_latex":"Let $f : X \\to Y$ be a dominant morphism of integral schemes.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.\nThen the rank of $\\mathcal{G}$ on $Y$ is equal to the rank\nof $f^*\\mathcal{G}$ on $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAH","source_file":"divisors.tex","source_line":1915,"source_end_line":1921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1915-L1921","statement_sha256":"d5cda61dc3a55302b57787b1df7e090da750f8f508e7c7b52bca30a8154af4aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":6034,"rank":6034,"depth":2,"x":2433.736,"y":654.743,"cluster":"divisors-intersection-theory"},{"id":"stacks:0HAI","tag":"0HAI","title":"Ranks of modules · Lemma 0HAI","summary":"Let X be an integral scheme. Let 0 → F → F' → F\" → 0 be a short exact sequence of quasi-coherent O_X-modules. Then the rank of F' is the sum of the ranks of F and F\".","statement_latex":"Let $X$ be an integral scheme.\nLet $0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}'' \\to 0$\nbe a short  exact sequence of quasi-coherent $\\mathcal{O}_X$-modules.\nThen the rank of $\\mathcal{F}'$ is the sum of the ranks of\n$\\mathcal{F}$ and $\\mathcal{F}''$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAI","source_file":"divisors.tex","source_line":1930,"source_end_line":1937,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1930-L1937","statement_sha256":"197a5ca47be8069ced7fc3604c452db95c6d1046120f614c968150b12f86bb83","origin":"The Stacks Project","memory_eligible":false,"source_rank":6035,"rank":6035,"depth":1,"x":2534.826,"y":672.316,"cluster":"divisors-intersection-theory"},{"id":"stacks:0HAJ","tag":"0HAJ","title":"Ranks of modules · Lemma 0HAJ","summary":"Let X be an integral scheme. Let F and G be quasi-coherent O_X-modules. • The rank of F ⊗_O_X G is the product of the ranks of F and G. • If F is of finite presentation, then the rank of SheafHom_O_X(F, G) is the product of the ranks of F and G.","statement_latex":"Let $X$ be an integral scheme. Let\n$\\mathcal{F}$ and $\\mathcal{G}$ be\nquasi-coherent $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item The rank of $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$\nis the product of the ranks of $\\mathcal{F}$ and $\\mathcal{G}$.\n\\item If $\\mathcal{F}$ is of finite presentation, then\nthe rank of $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$\nis the product of the ranks of $\\mathcal{F}$ and $\\mathcal{G}$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAJ","source_file":"divisors.tex","source_line":1943,"source_end_line":1955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1943-L1955","statement_sha256":"1284c1e9d47263a8a693dc43513d73a4e9d976cb0415712c2d2cfb0f693c5489","origin":"The Stacks Project","memory_eligible":false,"source_rank":6036,"rank":6036,"depth":3,"x":2445.503,"y":717.042,"cluster":"divisors-intersection-theory"},{"id":"stacks:0HAK","tag":"0HAK","title":"Ranks of modules · Lemma 0HAK","summary":"Let X be an integral scheme. Let F be a finite type quasi-coherent O_X-module. Then • F has finite rank r ≥ 0, and • there exists a nonempty open U ⊂ X such that F|_U is free of rank r.","statement_latex":"Let $X$ be an integral scheme. Let $\\mathcal{F}$ be a finite type\nquasi-coherent $\\mathcal{O}_X$-module. Then\n\\begin{enumerate}\n\\item $\\mathcal{F}$ has finite rank $r \\geq 0$, and\n\\item there exists a nonempty open $U \\subset X$\nsuch that $\\mathcal{F}|_U$ is free of rank $r$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Ranks of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAK","source_file":"divisors.tex","source_line":1966,"source_end_line":1975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L1966-L1975","statement_sha256":"52610d7a9879e17647d194e5b7aa67995ae0fac6129bc0f76ea3c2ac5f7d382e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6037,"rank":6037,"depth":1,"x":2475.618,"y":632.776,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AVU","tag":"0AVU","title":"Reflexive modules · Definition 0AVU","summary":"Let X be an integral locally Noetherian scheme. Let F be a coherent O_X-module. The reflexive hull of F is the O_X-module F^** = SheafHom_O_X( SheafHom_O_X(F, O_X), O_X) We say F is reflexive if the natural map j : F → F^** is an isomorphism.","statement_latex":"Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$\nbe a coherent $\\mathcal{O}_X$-module. The {\\it reflexive hull}\nof $\\mathcal{F}$ is the $\\mathcal{O}_X$-module\n$$\n\\mathcal{F}^{**} = \\SheafHom_{\\mathcal{O}_X}(\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X), \\mathcal{O}_X)\n$$\nWe say $\\mathcal{F}$ is {\\it reflexive} if the natural map\n$j : \\mathcal{F} \\longrightarrow \\mathcal{F}^{**}$\nis an isomorphism.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVU","source_file":"divisors.tex","source_line":2004,"source_end_line":2016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2004-L2016","statement_sha256":"990545ece05b67961a071e5ee9c64b4ffa52d37b7b58ca96ce8db0266d9b9a00","origin":"The Stacks Project","memory_eligible":false,"source_rank":6038,"rank":6038,"depth":0,"x":2521.497,"y":712.564,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AY0","tag":"0AY0","title":"Reflexive modules · Lemma 0AY0","summary":"Let X be an integral locally Noetherian scheme. Let F be a coherent O_X-module. The following are equivalent • F is reflexive, • for U ⊂ X affine open F(U) is a reflexive O(U)-module.","statement_latex":"Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$ be a\ncoherent $\\mathcal{O}_X$-module. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is reflexive,\n\\item for $U \\subset X$ affine open $\\mathcal{F}(U)$\nis a reflexive $\\mathcal{O}(U)$-module.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AY0","source_file":"divisors.tex","source_line":2025,"source_end_line":2034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2025-L2034","statement_sha256":"a79b33c9f0c24359faae160cd69072de3d0bdedc85dd9ebaf0625dc348d2afdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6039,"rank":6039,"depth":0,"x":2422.819,"y":679.531,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AY2","tag":"0AY2","title":"Reflexive modules · Lemma 0AY2","summary":"Let X be an integral locally Noetherian scheme. Let F be a coherent O_X-module. • If F is reflexive, then F is torsion free. • The map j : F → F^** is injective if and only if F is torsion free.","statement_latex":"Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$\nbe a coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is reflexive, then $\\mathcal{F}$ is torsion free.\n\\item The map $j : \\mathcal{F} \\longrightarrow \\mathcal{F}^{**}$\nis injective if and only if $\\mathcal{F}$ is torsion free.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AY2","source_file":"divisors.tex","source_line":2056,"source_end_line":2065,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2056-L2065","statement_sha256":"f121a8bc7edd26870e79f613f699c99e94b39d0f10e7059c97b79446caf0cf0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6040,"rank":6040,"depth":2,"x":2522.833,"y":647.68,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AY3","tag":"0AY3","title":"Reflexive modules · Lemma 0AY3","summary":"Let X be an integral locally Noetherian scheme. Let F be a coherent O_X-module. The following are equivalent • F is reflexive, • F_x is a reflexive O_X, x-module for all x ∈ X, • F_x is a reflexive O_X, x-module for all closed points x ∈ X.","statement_latex":"Let $X$ be an integral locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is reflexive,\n\\item $\\mathcal{F}_x$ is a reflexive $\\mathcal{O}_{X, x}$-module\nfor all $x \\in X$,\n\\item $\\mathcal{F}_x$ is a reflexive $\\mathcal{O}_{X, x}$-module\nfor all closed points $x \\in X$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AY3","source_file":"divisors.tex","source_line":2072,"source_end_line":2084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2072-L2084","statement_sha256":"e81d7ba0d5d1788b8b4bd4e9442d5a9cf1f1b11dd24d7465af25bf502f7506f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6041,"rank":6041,"depth":14,"x":2474.369,"y":728.462,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EBF","tag":"0EBF","title":"Reflexive modules · Lemma 0EBF","summary":"Let f : X → Y be a flat morphism of integral locally Noetherian schemes. Let G be a coherent reflexive O_Y-module. Then f^*G is a coherent reflexive O_X-module.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of integral locally Noetherian schemes.\nLet $\\mathcal{G}$ be a coherent reflexive $\\mathcal{O}_Y$-module.\nThen $f^*\\mathcal{G}$ is a coherent reflexive $\\mathcal{O}_X$-module.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBF","source_file":"divisors.tex","source_line":2094,"source_end_line":2099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2094-L2099","statement_sha256":"1284a385250592bfc595933d15e1fb6ed271d2fa1a6458d8d236a0c93d6b1a50","origin":"The Stacks Project","memory_eligible":false,"source_rank":6042,"rank":6042,"depth":5,"x":2444.948,"y":640.81,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EBG","tag":"0EBG","title":"Reflexive modules · Lemma 0EBG","summary":"Let X be an integral locally Noetherian scheme. Let 0 → F → F' → F\" be an exact sequence of coherent O_X-modules. If F' is reflexive and F\" is torsion free, then F is reflexive.","statement_latex":"Let $X$ be an integral locally Noetherian scheme.\nLet $0 \\to \\mathcal{F} \\to \\mathcal{F}' \\to \\mathcal{F}''$ be\nan exact sequence of coherent $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F}'$ is reflexive and $\\mathcal{F}''$ is torsion free,\nthen $\\mathcal{F}$ is reflexive.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBG","source_file":"divisors.tex","source_line":2107,"source_end_line":2114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2107-L2114","statement_sha256":"93430b5afb4823940cb12986dc5da5f14809f0f2a0098779866586cad84154c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6043,"rank":6043,"depth":3,"x":2537.739,"y":689.068,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AY4","tag":"0AY4","title":"Reflexive modules · Lemma 0AY4","summary":"Let X be an integral locally Noetherian scheme. Let F, G be coherent O_X-modules. If G is reflexive, then SheafHom_O_X(F, G) is reflexive.","statement_latex":"Let $X$ be an integral locally Noetherian scheme.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be\ncoherent $\\mathcal{O}_X$-modules.\nIf $\\mathcal{G}$ is reflexive,\nthen $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is reflexive.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AY4","source_file":"divisors.tex","source_line":2120,"source_end_line":2127,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2120-L2127","statement_sha256":"20466c83a774660aa9800522d7ff484fb97e021b786e4422c36fef70ad901823","origin":"The Stacks Project","memory_eligible":false,"source_rank":6044,"rank":6044,"depth":7,"x":2429.808,"y":706.245,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AY5","tag":"0AY5","title":"Reflexive modules · Lemma 0AY5","summary":"Let X be an integral locally Noetherian scheme. Let F be a coherent O_X-module. The following are equivalent • F is reflexive, • for each x ∈ X one of the following happens • F_x is a reflexive O_X, x-module, or • depth(F_x) ≥ 2.","statement_latex":"Let $X$ be an integral locally Noetherian scheme. Let $\\mathcal{F}$\nbe a coherent $\\mathcal{O}_X$-module. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is reflexive,\n\\item for each $x \\in X$ one of the following happens\n\\begin{enumerate}\n\\item $\\mathcal{F}_x$ is a reflexive $\\mathcal{O}_{X, x}$-module, or\n\\item $\\text{depth}(\\mathcal{F}_x) \\geq 2$.\n\\end{enumerate}\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AY5","source_file":"divisors.tex","source_line":2189,"source_end_line":2201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2189-L2201","statement_sha256":"4962d1e54718a9dba1b2f2b4adb8286c7175d46d707c457bdc69ad7f06693913","origin":"The Stacks Project","memory_eligible":false,"source_rank":6045,"rank":6045,"depth":16,"x":2496.007,"y":631.86,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EBI","tag":"0EBI","title":"Reflexive modules · Lemma 0EBI","summary":"Let X be an integral locally Noetherian scheme. Let F be a coherent reflexive O_X-module. Let x ∈ X. • If depth(O_X, x) ≥ 2, then depth(F_x) ≥ 2. • If X is (S_2), then F is (S_2).","statement_latex":"Let $X$ be an integral locally Noetherian scheme.\nLet $\\mathcal{F}$ be a coherent reflexive $\\mathcal{O}_X$-module.\nLet $x \\in X$.\n\\begin{enumerate}\n\\item If $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$, then\n$\\text{depth}(\\mathcal{F}_x) \\geq 2$.\n\\item If $X$ is $(S_2)$, then $\\mathcal{F}$ is $(S_2)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBI","source_file":"divisors.tex","source_line":2207,"source_end_line":2217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2207-L2217","statement_sha256":"69d92bd985101a09353b41096916f6e0f57c9a946b3157824baf2cdd53d56718","origin":"The Stacks Project","memory_eligible":false,"source_rank":6046,"rank":6046,"depth":16,"x":2507.08,"y":724.863,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EBJ","tag":"0EBJ","title":"Reflexive modules · Lemma 0EBJ","summary":"Let X be an integral locally Noetherian scheme. Let j : U → X be an open subscheme with complement Z. Assume O_X, z has depth ≥ 2 for all z ∈ Z. Then j^* and j_* define an equivalence of categories between the category of coherent reflexive O_X-modules and the category of coherent reflexive O_U-modules.","statement_latex":"Let $X$ be an integral locally Noetherian scheme. Let $j : U \\to X$\nbe an open subscheme with complement $Z$. Assume $\\mathcal{O}_{X, z}$\nhas depth $\\geq 2$ for all $z \\in Z$. Then $j^*$ and $j_*$ define\nan equivalence of categories between the category of coherent reflexive\n$\\mathcal{O}_X$-modules and the category of coherent reflexive\n$\\mathcal{O}_U$-modules.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBJ","source_file":"divisors.tex","source_line":2223,"source_end_line":2231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2223-L2231","statement_sha256":"756b25fbd7cf086728112bfa6bbcd93e6ed16777d1671d6fde4d94469f69c8bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6047,"rank":6047,"depth":32,"x":2423.605,"y":662.174,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AY6","tag":"0AY6","title":"Reflexive modules · Lemma 0AY6","summary":"Let X be an integral locally Noetherian normal scheme. Let F be a coherent O_X-module. The following are equivalent • F is reflexive, • F is torsion free and has property (S_2), and • there exists an open subscheme j : U → X such that • every irreducible component of X setminus U has codimension ≥ 2 in X, • j^*F is finite locally free, and • F = j_*j^*F.","statement_latex":"Let $X$ be an integral locally Noetherian normal scheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is reflexive,\n\\item $\\mathcal{F}$ is torsion free and has property $(S_2)$, and\n\\item there exists an open subscheme $j : U \\to X$ such that\n\\begin{enumerate}\n\\item every irreducible component of $X \\setminus U$\nhas codimension $\\geq 2$ in $X$,\n\\item $j^*\\mathcal{F}$ is finite locally free, and\n\\item $\\mathcal{F} = j_*j^*\\mathcal{F}$.\n\\end{enumerate}\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AY6","source_file":"divisors.tex","source_line":2254,"source_end_line":2270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2254-L2270","statement_sha256":"286147a396dd984c64252251ce4289e8d1ac9f349af1090bf026afe8c74fbfc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6048,"rank":6048,"depth":33,"x":2536.265,"y":661.028,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AY7","tag":"0AY7","title":"Reflexive modules · Lemma 0AY7","summary":"Let X be an integral locally Noetherian normal scheme with generic point eta. Let F, G be coherent O_X-modules. Let T : G_eta → F_eta be a linear map. Then T extends to a map G → F^** of O_X-modules if and only if • [(*)] for every x ∈ X with dim(O_X, x) = 1 we have T(Im(G_x → G_eta)) ⊂ Im(F_x → F_eta).","statement_latex":"Let $X$ be an integral locally Noetherian normal scheme with\ngeneric point $\\eta$. Let $\\mathcal{F}$, $\\mathcal{G}$ be coherent\n$\\mathcal{O}_X$-modules. Let $T : \\mathcal{G}_\\eta \\to \\mathcal{F}_\\eta$\nbe a linear map. Then $T$ extends to a map\n$\\mathcal{G} \\to \\mathcal{F}^{**}$ of $\\mathcal{O}_X$-modules\nif and only if\n\\begin{itemize}\n\\item[(*)] for every $x \\in X$ with $\\dim(\\mathcal{O}_{X, x}) = 1$\nwe have\n$$\nT\\left(\\Im(\\mathcal{G}_x \\to \\mathcal{G}_\\eta)\\right) \\subset\n\\Im(\\mathcal{F}_x \\to \\mathcal{F}_\\eta).\n$$\n\\end{itemize}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AY7","source_file":"divisors.tex","source_line":2303,"source_end_line":2319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2303-L2319","statement_sha256":"8727672491f24501fdbb3a31dc8dc7dc77de5e959558b1de071bbde9081e9a34","origin":"The Stacks Project","memory_eligible":false,"source_rank":6049,"rank":6049,"depth":20,"x":2453.608,"y":726.196,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B3N","tag":"0B3N","title":"Reflexive modules · Lemma 0B3N","summary":"Let X be a regular scheme of dimension ≤ 2. Let F be a coherent O_X-module. The following are equivalent • F is reflexive, • F is finite locally free.","statement_latex":"Let $X$ be a regular scheme of dimension $\\leq 2$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is reflexive,\n\\item $\\mathcal{F}$ is finite locally free.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Reflexive modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3N","source_file":"divisors.tex","source_line":2336,"source_end_line":2345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2336-L2345","statement_sha256":"b35402b8ce80aec73870464926e684a10f851a61b671e887ab6c1d4be774b7df","origin":"The Stacks Project","memory_eligible":false,"source_rank":6050,"rank":6050,"depth":19,"x":2462.206,"y":630.665,"cluster":"divisors-intersection-theory"},{"id":"stacks:01WR","tag":"01WR","title":"Effective Cartier divisors · Definition 01WR","summary":"Let S be a scheme. • A locally principal closed subscheme of S is a closed subscheme whose sheaf of ideals is locally generated by a single element. • An effective Cartier divisor on S is a closed subscheme D ⊂ S whose ideal sheaf I_D ⊂ O_S is an invertible O_S-module.","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item A {\\it locally principal closed subscheme} of $S$ is a closed subscheme\nwhose sheaf of ideals is locally generated by a single element.\n\\item An {\\it effective Cartier divisor} on $S$ is a closed subscheme\n$D \\subset S$ whose ideal sheaf $\\mathcal{I}_D \\subset \\mathcal{O}_S$\nis an invertible $\\mathcal{O}_S$-module.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WR","source_file":"divisors.tex","source_line":2383,"source_end_line":2393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2383-L2393","statement_sha256":"d88ddbba9526a286427e7acb99ce2d7b90c5e23e59002321fe6061ae0fd95384","origin":"The Stacks Project","memory_eligible":false,"source_rank":6051,"rank":6051,"depth":0,"x":2533.109,"y":706.438,"cluster":"divisors-intersection-theory"},{"id":"stacks:01WS","tag":"01WS","title":"Effective Cartier divisors · Lemma 01WS","summary":"Let S be a scheme. Let D ⊂ S be a closed subscheme. The following are equivalent: • The subscheme D is an effective Cartier divisor on S. • For every x ∈ D there exists an affine open neighbourhood Spec(A) = U ⊂ S of x such that U ∩ D = Spec(A/(f)) with f ∈ A a nonzerodivisor.","statement_latex":"Let $S$ be a scheme.\nLet $D \\subset S$ be a closed subscheme.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The subscheme $D$ is an effective Cartier divisor on $S$.\n\\item For every $x \\in D$ there exists an affine open neighbourhood\n$\\Spec(A) = U \\subset S$ of $x$ such that\n$U \\cap D = \\Spec(A/(f))$ with $f \\in A$ a nonzerodivisor.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WS","source_file":"divisors.tex","source_line":2402,"source_end_line":2413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2402-L2413","statement_sha256":"13009024b6d6fc48a28fd56db356a5db78c100aa994f12f0156927b33890fc5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6052,"rank":6052,"depth":0,"x":2419.222,"y":690.703,"cluster":"divisors-intersection-theory"},{"id":"stacks:07ZT","tag":"07ZT","title":"Effective Cartier divisors · Lemma 07ZT","summary":"Let S be a scheme. Let Z ⊂ S be a locally principal closed subscheme. Let U = S setminus Z. Then U → S is an affine morphism.","statement_latex":"Let $S$ be a scheme. Let $Z \\subset S$ be a locally principal closed\nsubscheme. Let $U = S \\setminus Z$. Then $U \\to S$ is an affine morphism.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZT","source_file":"divisors.tex","source_line":2435,"source_end_line":2439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2435-L2439","statement_sha256":"53b1ae4c311e5f6ef41ead52337a9a93f6fceb404cfe7761110fdd0372382298","origin":"The Stacks Project","memory_eligible":false,"source_rank":6053,"rank":6053,"depth":18,"x":2516.42,"y":637.375,"cluster":"divisors-intersection-theory"},{"id":"stacks:07ZU","tag":"07ZU","title":"Effective Cartier divisors · Lemma 07ZU","summary":"Let S be a scheme. Let D ⊂ S be an effective Cartier divisor. Let U = S setminus D. Then U → S is an affine morphism and U is scheme theoretically dense in S.","statement_latex":"Let $S$ be a scheme. Let $D \\subset S$ be an effective Cartier divisor.\nLet $U = S \\setminus D$. Then $U \\to S$ is an affine morphism and $U$\nis scheme theoretically dense in $S$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZU","source_file":"divisors.tex","source_line":2448,"source_end_line":2453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2448-L2453","statement_sha256":"0ecf86528ba3e7578581485b66967f03d8330856b60adbaa76c782fe9f0bc4ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":6054,"rank":6054,"depth":19,"x":2487.465,"y":732.396,"cluster":"divisors-intersection-theory"},{"id":"stacks:056N","tag":"056N","title":"Effective Cartier divisors · Lemma 056N","summary":"Let S be a scheme. Let D ⊂ S be an effective Cartier divisor. Let s ∈ D. If dim_s(S) < ∞, then dim_s(D) < dim_s(S).","statement_latex":"Let $S$ be a scheme.\nLet $D \\subset S$ be an effective Cartier divisor.\nLet $s \\in D$.\nIf $\\dim_s(S) < \\infty$, then $\\dim_s(D) < \\dim_s(S)$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056N","source_file":"divisors.tex","source_line":2467,"source_end_line":2473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2467-L2473","statement_sha256":"3cd7c7b37b0153dac0445fb5a234ea1b1790b8b5bd31b56e189152c58ce063ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":6055,"rank":6055,"depth":1,"x":2432.09,"y":645.405,"cluster":"divisors-intersection-theory"},{"id":"stacks:01WT","tag":"01WT","title":"Effective Cartier divisors · Definition 01WT","summary":"Let S be a scheme. Given effective Cartier divisors D_1, D_2 on S we set D = D_1 + D_2 equal to the closed subscheme of S corresponding to the quasi-coherent sheaf of ideals I_D_1I_D_2 ⊂ O_S. We call this the sum of the effective Cartier divisors D_1 and D_2.","statement_latex":"Let $S$ be a scheme. Given effective Cartier divisors\n$D_1$, $D_2$ on $S$ we set $D = D_1 + D_2$ equal to the\nclosed subscheme of $S$ corresponding to the quasi-coherent\nsheaf of ideals\n$\\mathcal{I}_{D_1}\\mathcal{I}_{D_2} \\subset \\mathcal{O}_S$.\nWe call this the {\\it sum of the effective Cartier divisors\n$D_1$ and $D_2$}.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WT","source_file":"divisors.tex","source_line":2488,"source_end_line":2497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2488-L2497","statement_sha256":"60f3b5fdc7b5eae7d968ed8173e6e2f2a1fe2168420a8beea356771f70b28840","origin":"The Stacks Project","memory_eligible":false,"source_rank":6056,"rank":6056,"depth":0,"x":2543.504,"y":678.315,"cluster":"divisors-intersection-theory"},{"id":"stacks:01WU","tag":"01WU","title":"Effective Cartier divisors · Lemma 01WU","summary":"The sum of two effective Cartier divisors is an effective Cartier divisor.","statement_latex":"The sum of two effective Cartier divisors is an effective\nCartier divisor.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WU","source_file":"divisors.tex","source_line":2504,"source_end_line":2508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2504-L2508","statement_sha256":"1da1775ac3b845d3907ce767039e7a5cdb5d961c64577292b80a7a4d8da9fb33","origin":"The Stacks Project","memory_eligible":false,"source_rank":6057,"rank":6057,"depth":0,"x":2434.276,"y":717.483,"cluster":"divisors-intersection-theory"},{"id":"stacks:02ON","tag":"02ON","title":"Effective Cartier divisors · Lemma 02ON","summary":"Let X be a scheme. Let D, D' be two effective Cartier divisors on X. If D ⊂ D' (as closed subschemes of X), then there exists an effective Cartier divisor D\" such that D' = D + D\".","statement_latex":"Let $X$ be a scheme.\nLet $D, D'$ be two effective Cartier divisors on $X$.\nIf $D \\subset D'$ (as closed subschemes of $X$), then\nthere exists an effective Cartier divisor $D''$ such\nthat $D' = D + D''$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ON","source_file":"divisors.tex","source_line":2515,"source_end_line":2522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2515-L2522","statement_sha256":"47d5495a875189dc69e8981f168470793159cf4dd1c880c5ad303d67d517db67","origin":"The Stacks Project","memory_eligible":false,"source_rank":6058,"rank":6058,"depth":0,"x":2483.593,"y":626.122,"cluster":"divisors-intersection-theory"},{"id":"stacks:07ZV","tag":"07ZV","title":"Effective Cartier divisors · Lemma 07ZV","summary":"Let X be a scheme. Let Z, Y be two closed subschemes of X with ideal sheaves I and J. If IJ defines an effective Cartier divisor D ⊂ X, then Z and Y are effective Cartier divisors and D = Z + Y.","statement_latex":"Let $X$ be a scheme. Let $Z, Y$ be two closed subschemes of $X$\nwith ideal sheaves $\\mathcal{I}$ and $\\mathcal{J}$. If $\\mathcal{I}\\mathcal{J}$\ndefines an effective Cartier divisor $D \\subset X$, then $Z$ and $Y$\nare effective Cartier divisors and $D = Z + Y$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZV","source_file":"divisors.tex","source_line":2528,"source_end_line":2534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2528-L2534","statement_sha256":"35e75db5152d1507fa978782998b7d0cc0ddb78ab970a8254590d59aecf2b3d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6059,"rank":6059,"depth":6,"x":2520.898,"y":721.995,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C4R","tag":"0C4R","title":"Effective Cartier divisors · Lemma 0C4R","summary":"Let X be a scheme. Let D, D' ⊂ X be effective Cartier divisors such that the scheme theoretic intersection D ∩ D' is an effective Cartier divisor on D'. Then D + D' is the scheme theoretic union of D and D'.","statement_latex":"Let $X$ be a scheme. Let $D, D' \\subset X$ be effective Cartier divisors\nsuch that the scheme theoretic intersection $D \\cap D'$ is an effective\nCartier divisor on $D'$. Then $D + D'$ is the scheme theoretic\nunion of $D$ and $D'$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4R","source_file":"divisors.tex","source_line":2544,"source_end_line":2550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2544-L2550","statement_sha256":"a6e1d568a328be48818734d74aa9b393bd25c6fddcc0d0eb0790e916de804db5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6060,"rank":6060,"depth":1,"x":2415.728,"y":672.199,"cluster":"divisors-intersection-theory"},{"id":"stacks:053P","tag":"053P","title":"Effective Cartier divisors · Lemma 053P","summary":"Let f : S' → S be a morphism of schemes. Let Z ⊂ S be a locally principal closed subscheme. Then the inverse image f^-1(Z) is a locally principal closed subscheme of S'.","statement_latex":"Let $f : S' \\to S$ be a morphism of schemes. Let $Z \\subset S$\nbe a locally principal closed subscheme. Then the inverse image\n$f^{-1}(Z)$ is a locally principal closed subscheme of $S'$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053P","source_file":"divisors.tex","source_line":2569,"source_end_line":2574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2569-L2574","statement_sha256":"057e9df8f2854332ebe761006b20c7d594731d484b64479b3b5f0ad7fb497959","origin":"The Stacks Project","memory_eligible":false,"source_rank":6061,"rank":6061,"depth":0,"x":2533.952,"y":649.12,"cluster":"divisors-intersection-theory"},{"id":"stacks:01WV","tag":"01WV","title":"Effective Cartier divisors · Definition 01WV","summary":"Let f : S' → S be a morphism of schemes. Let D ⊂ S be an effective Cartier divisor. We say the pullback of D by f is defined if the closed subscheme f^-1(D) ⊂ S' is an effective Cartier divisor. In this case we denote it either f^*D or f^-1(D) and we call it the pullback of the effective Cartier divisor.","statement_latex":"Let $f : S' \\to S$ be a morphism of schemes. Let $D \\subset S$\nbe an effective Cartier divisor. We say the {\\it pullback of\n$D$ by $f$ is defined} if the closed subscheme $f^{-1}(D) \\subset S'$\nis an effective Cartier divisor. In this case we denote it either\n$f^*D$ or $f^{-1}(D)$ and we call it the\n{\\it pullback of the effective Cartier divisor}.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WV","source_file":"divisors.tex","source_line":2580,"source_end_line":2588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2580-L2588","statement_sha256":"3a2caec3cce9fd4d508f46586998228f8cf5149459de06e9408ea721191aa42e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6062,"rank":6062,"depth":0,"x":2464.971,"y":733.665,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OO","tag":"02OO","title":"Effective Cartier divisors · Lemma 02OO","summary":"Let f : X → Y be a morphism of schemes. Let D ⊂ Y be an effective Cartier divisor. The pullback of D by f is defined in each of the following cases: • f(x) not ∈ D for any weakly associated point x of X, • X, Y integral and f dominant, • X reduced and f(xi) not ∈ D for any generic point xi of any irreducible component of X, • X is locally Noetherian and f(x) not ∈ D for any associated point x of X, • X is locally Noetherian, has no embedded points, and f(xi) not ∈ D for…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $D \\subset Y$ be an effective Cartier divisor.\nThe pullback of $D$ by $f$ is defined in each of the following cases:\n\\begin{enumerate}\n\\item $f(x) \\not \\in D$ for any weakly associated point $x$ of $X$,\n\\item $X$, $Y$ integral and $f$ dominant,\n\\item $X$ reduced and $f(\\xi) \\not \\in D$ for any generic point $\\xi$ of any\nirreducible component of $X$,\n\\item $X$ is locally Noetherian and $f(x) \\not \\in D$ for any associated point\n$x$ of $X$,\n\\item $X$ is locally Noetherian, has no embedded points, and\n$f(\\xi) \\not \\in D$ for any generic point $\\xi$ of an irreducible component of\n$X$,\n\\item $f$ is flat, and\n\\item add more here as needed.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OO","source_file":"divisors.tex","source_line":2594,"source_end_line":2612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2594-L2612","statement_sha256":"e46bff9445c7281768c6665cb01708ea9dfc0d408d1b7b7df698c8e70a298a3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6063,"rank":6063,"depth":4,"x":2447.759,"y":631.651,"cluster":"divisors-intersection-theory"},{"id":"stacks:01WW","tag":"01WW","title":"Effective Cartier divisors · Lemma 01WW","summary":"Let f : S' → S be a morphism of schemes. Let D_1, D_2 be effective Cartier divisors on S. If the pullbacks of D_1 and D_2 are defined then the pullback of D = D_1 + D_2 is defined and f^*D = f^*D_1 + f^*D_2.","statement_latex":"Let $f : S' \\to S$ be a morphism of schemes.\nLet $D_1$, $D_2$ be effective Cartier divisors on $S$.\nIf the pullbacks of $D_1$ and $D_2$ are defined then the\npullback of $D = D_1 + D_2$ is defined and\n$f^*D = f^*D_1 + f^*D_2$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WW","source_file":"divisors.tex","source_line":2634,"source_end_line":2641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2634-L2641","statement_sha256":"797584b9b62e7a4f89bba7f298408700b4d533b5d03e5f147c570f31b5d879e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6064,"rank":6064,"depth":0,"x":2542.977,"y":697.447,"cluster":"divisors-intersection-theory"},{"id":"stacks:01WX","tag":"01WX","title":"Effective Cartier divisors and invertible sheaves · Definition 01WX","summary":"Let S be a scheme. Let D ⊂ S be an effective Cartier divisor with ideal sheaf I_D. • The invertible sheaf O_S(D) associated to D is defined by O_S(D) = SheafHom_O_S(I_D, O_S) = I_D^⊗ -1. • The canonical section, usually denoted 1 or 1_D, is the global section of O_S(D) corresponding to the inclusion mapping I_D → O_S. • We write O_S(-D) = O_S(D)^⊗ -1 = I_D. • Given a second effective Cartier divisor D' ⊂ S we define O_S(D - D') = O_S(D) ⊗_O_S O_S(-D').","statement_latex":"Let $S$ be a scheme. Let $D \\subset S$ be an effective Cartier divisor\nwith ideal sheaf $\\mathcal{I}_D$.\n\\begin{enumerate}\n\\item The {\\it invertible sheaf $\\mathcal{O}_S(D)$ associated to $D$}\nis defined by\n$$\n\\mathcal{O}_S(D) =\n\\SheafHom_{\\mathcal{O}_S}(\\mathcal{I}_D, \\mathcal{O}_S) =\n\\mathcal{I}_D^{\\otimes -1}.\n$$\n\\item The {\\it canonical section}, usually denoted $1$ or $1_D$, is the\nglobal section of $\\mathcal{O}_S(D)$ corresponding to\nthe inclusion mapping $\\mathcal{I}_D \\to \\mathcal{O}_S$.\n\\item We write\n$\\mathcal{O}_S(-D) = \\mathcal{O}_S(D)^{\\otimes -1} = \\mathcal{I}_D$.\n\\item Given a second effective Cartier divisor $D' \\subset S$ we define\n$\\mathcal{O}_S(D - D') =\n\\mathcal{O}_S(D) \\otimes_{\\mathcal{O}_S} \\mathcal{O}_S(-D')$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WX","source_file":"divisors.tex","source_line":2659,"source_end_line":2680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2659-L2680","statement_sha256":"a18e63f170a31c83918f4bf9cee8f0d8fc4d89d34e0d2348e9e4aa2c630c2963","origin":"The Stacks Project","memory_eligible":false,"source_rank":6065,"rank":6065,"depth":0,"x":2419.223,"y":702.986,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B3P","tag":"0B3P","title":"Effective Cartier divisors and invertible sheaves · Lemma 0B3P","summary":"Let S be a scheme and let D ⊂ S be an effective Cartier divisor. Then the conormal sheaf is C_D/S = I_D|_D = O_S(-D)|_D and the normal sheaf is N_D/S = O_S(D)|_D.","statement_latex":"Let $S$ be a scheme and let $D \\subset S$ be an effective Cartier divisor.\nThen the conormal sheaf is $\\mathcal{C}_{D/S} = \\mathcal{I}_D|_D =\n\\mathcal{O}_S(-D)|_D$ and the normal sheaf is\n$\\mathcal{N}_{D/S} = \\mathcal{O}_S(D)|_D$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3P","source_file":"divisors.tex","source_line":2701,"source_end_line":2707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2701-L2707","statement_sha256":"c6f31ed535ece81733c291b910e137372e3647b92250abdfb2db93b54c55d1b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6066,"rank":6066,"depth":1,"x":2506.464,"y":628.305,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C4T","tag":"0C4T","title":"Effective Cartier divisors and invertible sheaves · Lemma 0C4T","summary":"Let X be a scheme. Let D, D' ⊂ X be effective Cartier divisors with D ⊂ D' and let C ⊂ X be an effective Cartier divisor such that D' = D + C (which exists by Lemma [Tag 02ON]). Then there is a short exact sequence 0 → O_X(-D)|_C → O_D' → O_D → 0 of O_X-modules.","statement_latex":"Let $X$ be a scheme. Let $D, D' \\subset X$ be effective Cartier divisors\nwith $D \\subset D'$ and let $C \\subset X$ be an effective Cartier divisor\nsuch that $D' = D + C$ (which exists by\nLemma \\ref{lemma-difference-effective-Cartier-divisors}).\nThen there is a short exact sequence\n$$\n0 \\to \\mathcal{O}_X(-D)|_C \\to \\mathcal{O}_{D'} \\to \\mathcal{O}_D \\to 0\n$$\nof $\\mathcal{O}_X$-modules.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4T","source_file":"divisors.tex","source_line":2713,"source_end_line":2724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2713-L2724","statement_sha256":"98506aefedf6d26113108659f2281165e96f29fb0e1e0759ad9ae405f8180414","origin":"The Stacks Project","memory_eligible":false,"source_rank":6067,"rank":6067,"depth":17,"x":2502.168,"y":733.401,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OP","tag":"02OP","title":"Effective Cartier divisors and invertible sheaves · Lemma 02OP","summary":"Let S be a scheme. Let D_1, D_2 be effective Cartier divisors on S. Let D = D_1 + D_2. Then there is a unique isomorphism O_S(D_1) ⊗_O_S O_S(D_2) → O_S(D) which maps 1_D_1 ⊗ 1_D_2 to 1_D.","statement_latex":"Let $S$ be a scheme.\nLet $D_1$, $D_2$ be effective Cartier divisors on $S$.\nLet $D = D_1 + D_2$.\nThen there is a unique isomorphism\n$$\n\\mathcal{O}_S(D_1) \\otimes_{\\mathcal{O}_S} \\mathcal{O}_S(D_2)\n\\longrightarrow\n\\mathcal{O}_S(D)\n$$\nwhich maps $1_{D_1} \\otimes 1_{D_2}$ to $1_D$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OP","source_file":"divisors.tex","source_line":2745,"source_end_line":2757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2745-L2757","statement_sha256":"1aeabf1d092b5359a81355247e7ec7598c20323661d944d8864ea0b4a689085f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6068,"rank":6068,"depth":0,"x":2420.417,"y":653.069,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C4U","tag":"0C4U","title":"Effective Cartier divisors and invertible sheaves · Lemma 0C4U","summary":"Let f : S' → S be a morphism of schemes. Let D be an effective Cartier divisor on S. If the pullback of D is defined then f^*O_S(D) = O_S'(f^*D) and the canonical section 1_D pulls back to the canonical section 1_f^*D.","statement_latex":"Let $f : S' \\to S$ be a morphism of schemes.\nLet $D$ be an effective Cartier divisor on $S$.\nIf the pullback of $D$ is defined then\n$f^*\\mathcal{O}_S(D) = \\mathcal{O}_{S'}(f^*D)$\nand the canonical section $1_D$ pulls back to\nthe canonical section $1_{f^*D}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4U","source_file":"divisors.tex","source_line":2763,"source_end_line":2771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2763-L2771","statement_sha256":"ff9efea59258029c4960144256b8cd2a18464d1fce9f098cb59f19c34d6ad930","origin":"The Stacks Project","memory_eligible":false,"source_rank":6069,"rank":6069,"depth":0,"x":2545.913,"y":665.982,"cluster":"divisors-intersection-theory"},{"id":"stacks:01WY","tag":"01WY","title":"Effective Cartier divisors and invertible sheaves · Definition 01WY","summary":"Let (X, O_X) be a locally ringed space. Let L be an invertible sheaf on X. A global section s ∈ Γ(X, L) is called a regular section if the map O_X → L, f ↦ fs is injective.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a locally ringed space.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nA global section $s \\in \\Gamma(X, \\mathcal{L})$ is called a\n{\\it regular section} if the map $\\mathcal{O}_X \\to \\mathcal{L}$,\n$f \\mapsto fs$ is injective.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WY","source_file":"divisors.tex","source_line":2777,"source_end_line":2784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2777-L2784","statement_sha256":"c837844873a962dddb5f8d1974bd15fc73f85ab02606173663ba470d21f8b8da","origin":"The Stacks Project","memory_eligible":false,"source_rank":6070,"rank":6070,"depth":0,"x":2442.489,"y":727.969,"cluster":"divisors-intersection-theory"},{"id":"stacks:01WZ","tag":"01WZ","title":"Effective Cartier divisors and invertible sheaves · Lemma 01WZ","summary":"Let X be a locally ringed space. Let f ∈ Γ(X, O_X). The following are equivalent: • f is a regular section, and • for any x ∈ X the image f ∈ O_X, x is a nonzerodivisor. If X is a scheme these are also equivalent to • [(3)] for any affine open Spec(A) = U ⊂ X the image f ∈ A is a nonzerodivisor, • [(4)] there exists an affine open covering X = ⋃ Spec(A_i) such that the image of f in A_i is a nonzerodivisor for all i.","statement_latex":"Let $X$ be a locally ringed space. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is a regular section, and\n\\item for any $x \\in X$ the image $f \\in \\mathcal{O}_{X, x}$\nis a nonzerodivisor.\n\\end{enumerate}\nIf $X$ is a scheme these are also equivalent to\n\\begin{enumerate}\n\\item[(3)] for any affine open $\\Spec(A) = U \\subset X$\nthe image $f \\in A$ is a nonzerodivisor,\n\\item[(4)] there exists an affine open covering\n$X = \\bigcup \\Spec(A_i)$ such that\nthe image of $f$ in $A_i$ is a nonzerodivisor for all $i$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01WZ","source_file":"divisors.tex","source_line":2786,"source_end_line":2803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2786-L2803","statement_sha256":"3beffdf653cfcb43533bdc8f6166fb887f1cae6f393fbd3e358bf7c96ab15c3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6071,"rank":6071,"depth":0,"x":2469.032,"y":623.072,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OQ","tag":"02OQ","title":"Effective Cartier divisors and invertible sheaves · Definition 02OQ","summary":"Let X be a scheme. Let L be an invertible sheaf. Let s ∈ Γ(X, L) be a global section. The zero scheme of s is the closed subscheme Z(s) ⊂ X defined by the quasi-coherent sheaf of ideals I ⊂ O_X which is the image of the map s : L^⊗ -1 → O_X.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an invertible sheaf.\nLet $s \\in \\Gamma(X, \\mathcal{L})$ be a global section.\nThe {\\it zero scheme} of $s$ is the closed subscheme $Z(s) \\subset X$\ndefined by the quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_X$ which is the image of the\nmap $s : \\mathcal{L}^{\\otimes -1} \\to \\mathcal{O}_X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OQ","source_file":"divisors.tex","source_line":2817,"source_end_line":2825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2817-L2825","statement_sha256":"7a06a29f9e321b986a4f6344fc7f67c727a5a770bdd4da8330af9392bb60ca74","origin":"The Stacks Project","memory_eligible":false,"source_rank":6072,"rank":6072,"depth":0,"x":2534.123,"y":715.925,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OR","tag":"02OR","title":"Effective Cartier divisors and invertible sheaves · Lemma 02OR","summary":"Let X be a scheme. Let L be an invertible sheaf. Let s ∈ Γ(X, L). • Consider closed immersions i : Z → X such that i^*s ∈ Γ(Z, i^*L) is zero ordered by inclusion. The zero scheme Z(s) is the maximal element of this ordered set. • For any morphism of schemes f : Y → X we have f^*s = 0 in Γ(Y, f^*L) if and only if f factors through Z(s). • The zero scheme Z(s) is a locally principal closed subscheme. • The zero scheme Z(s) is an effective Cartier divisor if and only if s is…","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{L}$ be an invertible sheaf.\nLet $s \\in \\Gamma(X, \\mathcal{L})$.\n\\begin{enumerate}\n\\item Consider closed immersions $i : Z \\to X$ such that\n$i^*s \\in \\Gamma(Z, i^*\\mathcal{L})$ is zero\nordered by inclusion. The zero scheme $Z(s)$ is the\nmaximal element of this ordered set.\n\\item For any morphism of schemes $f : Y \\to X$ we have\n$f^*s = 0$ in $\\Gamma(Y, f^*\\mathcal{L})$ if and only if\n$f$ factors through $Z(s)$.\n\\item The zero scheme $Z(s)$ is a locally principal closed subscheme.\n\\item The zero scheme $Z(s)$ is an effective Cartier divisor\nif and only if $s$ is a regular section of $\\mathcal{L}$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OR","source_file":"divisors.tex","source_line":2827,"source_end_line":2844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2827-L2844","statement_sha256":"07ae3fb7cc3a0c6cda9e4b48e4250ceffd8a15c1a41f3074922f9f3cf8a8d4dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6073,"rank":6073,"depth":0,"x":2410.877,"y":684.24,"cluster":"divisors-intersection-theory"},{"id":"stacks:01X0","tag":"01X0","title":"Effective Cartier divisors and invertible sheaves · Lemma 01X0","summary":"Effective Cartier divisors on a scheme are the same as invertible sheaves with fixed regular global section. Let X be a scheme. • If D ⊂ X is an effective Cartier divisor, then the canonical section 1_D of O_X(D) is regular. • Conversely, if s is a regular section of the invertible sheaf L, then there exists a unique effective Cartier divisor D = Z(s) ⊂ X and a unique isomorphism O_X(D) → L which maps 1_D to s. The constructions D ↦ (O_X(D), 1_D) and (L, s) ↦ Z(s) give…","statement_latex":"\\begin{slogan}\nEffective Cartier divisors on a scheme are the same as invertible sheaves\nwith fixed regular global section.\n\\end{slogan}\nLet $X$ be a scheme.\n\\begin{enumerate}\n\\item If $D \\subset X$ is an effective Cartier divisor, then\nthe canonical section $1_D$ of $\\mathcal{O}_X(D)$ is regular.\n\\item Conversely, if $s$ is a regular section of the invertible\nsheaf $\\mathcal{L}$, then there exists a unique effective\nCartier divisor $D = Z(s) \\subset X$ and a unique isomorphism\n$\\mathcal{O}_X(D) \\to \\mathcal{L}$ which maps $1_D$ to $s$.\n\\end{enumerate}\nThe constructions\n$D \\mapsto (\\mathcal{O}_X(D), 1_D)$ and $(\\mathcal{L}, s) \\mapsto Z(s)$\ngive mutually inverse maps\n$$\n\\left\\{\n\\begin{matrix}\n\\text{effective Cartier divisors on }X\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{isomorphism classes of pairs }(\\mathcal{L}, s)\\\\\n\\text{consisting of an invertible }\n\\mathcal{O}_X\\text{-module}\\\\\n\\mathcal{L}\\text{ and a regular global section }s\n\\end{matrix}\n\\right\\}\n$$","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01X0","source_file":"divisors.tex","source_line":2850,"source_end_line":2884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2850-L2884","statement_sha256":"0b05e3a2425960c00fa34a529de891a31c3874642b5c303135268f6e3787edff","origin":"The Stacks Project","memory_eligible":false,"source_rank":6074,"rank":6074,"depth":0,"x":2527.782,"y":637.454,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AYL","tag":"0AYL","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0AYL","summary":"Let X be a locally Noetherian scheme. Let L be an invertible O_X-module. Let s ∈ Γ(X, L). Then s is a regular section if and only if s does not vanish in the associated points of X.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module. Let $s \\in \\Gamma(X, \\mathcal{L})$. Then $s$\nis a regular section if and only if $s$ does not vanish in the associated\npoints of $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYL","source_file":"divisors.tex","source_line":2924,"source_end_line":2930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2924-L2930","statement_sha256":"7cea496788af7ffec99e745c2d84164fb29833687829fc54e4ff22728031023f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6075,"rank":6075,"depth":10,"x":2478.976,"y":738.756,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AG8","tag":"0AG8","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0AG8","summary":"Let X be a locally Noetherian scheme. Let D ⊂ X be a closed subscheme corresponding to the quasi-coherent ideal sheaf I ⊂ O_X. • If for every x ∈ D the ideal I_x ⊂ O_X, x can be generated by one element, then D is locally principal. • If for every x ∈ D the ideal I_x ⊂ O_X, x can be generated by a single nonzerodivisor, then D is an effective Cartier divisor.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a closed subscheme\ncorresponding to the quasi-coherent ideal sheaf\n$\\mathcal{I} \\subset \\mathcal{O}_X$.\n\\begin{enumerate}\n\\item If for every $x \\in D$ the ideal\n$\\mathcal{I}_x \\subset \\mathcal{O}_{X, x}$\ncan be generated by one element, then $D$ is locally principal.\n\\item If for every $x \\in D$ the ideal\n$\\mathcal{I}_x \\subset \\mathcal{O}_{X, x}$\ncan be generated by a single nonzerodivisor, then $D$ is an\neffective Cartier divisor.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AG8","source_file":"divisors.tex","source_line":2938,"source_end_line":2952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2938-L2952","statement_sha256":"e612459a671d075c7401bfd4ced7ff3fb20ecd197d25d94cbde500d96221cd1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6076,"rank":6076,"depth":1,"x":2433.292,"y":635.893,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BCN","tag":"0BCN","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0BCN","summary":"Let X be a locally Noetherian scheme. • Let D ⊂ X be a locally principal closed subscheme. Let xi ∈ D be a generic point of an irreducible component of D. Then dim(O_X, xi) ≤ 1. • Let D ⊂ X be an effective Cartier divisor. Let xi ∈ D be a generic point of an irreducible component of D. Then dim(O_X, xi) = 1.","statement_latex":"Let $X$ be a locally Noetherian scheme.\n\\begin{enumerate}\n\\item Let $D \\subset X$ be a locally principal closed subscheme.\nLet $\\xi \\in D$ be a generic point of an irreducible component of $D$.\nThen $\\dim(\\mathcal{O}_{X, \\xi}) \\leq 1$.\n\\item Let $D \\subset X$ be an effective Cartier divisor.\nLet $\\xi \\in D$ be a generic point of an irreducible component of $D$.\nThen $\\dim(\\mathcal{O}_{X, \\xi}) = 1$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCN","source_file":"divisors.tex","source_line":2974,"source_end_line":2985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L2974-L2985","statement_sha256":"dd4a965a564f76559ca96abcb0f5a83c8f164d8f30fbbb04a290b14cd561ac8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6077,"rank":6077,"depth":10,"x":2550.229,"y":686.048,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AG9","tag":"0AG9","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0AG9","summary":"Let X be a Noetherian scheme. Let D ⊂ X be an integral closed subscheme which is also an effective Cartier divisor. Then the local ring of X at the generic point of D is a discrete valuation ring.","statement_latex":"Let $X$ be a Noetherian scheme. Let $D \\subset X$ be an\nintegral closed subscheme which is also an\neffective Cartier divisor. Then the local ring of $X$\nat the generic point of $D$ is a discrete valuation ring.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AG9","source_file":"divisors.tex","source_line":3005,"source_end_line":3011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3005-L3011","statement_sha256":"7b83568bf4cbe83c8e092ae6d9e5837143825c59b31e19682f405af910daa9f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6078,"rank":6078,"depth":16,"x":2423.096,"y":715.548,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B3R","tag":"0B3R","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0B3R","summary":"Let X be a locally Noetherian scheme. Let D ⊂ X be an effective Cartier divisor. If X is (S_k), then D is (S_k - 1).","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be an\neffective Cartier divisor. If $X$ is $(S_k)$, then $D$ is $(S_{k - 1})$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3R","source_file":"divisors.tex","source_line":3024,"source_end_line":3028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3024-L3028","statement_sha256":"c206e50ed925a35622abb9309052cc28ba0ea8eb51d4551ce5d86f8106ab50ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":6079,"rank":6079,"depth":14,"x":2493.433,"y":621.238,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B3S","tag":"0B3S","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0B3S","summary":"Let X be a locally Noetherian normal scheme. Let D ⊂ X be an effective Cartier divisor. Then D is (S_1).","statement_latex":"Let $X$ be a locally Noetherian normal scheme. Let $D \\subset X$ be an\neffective Cartier divisor. Then $D$ is $(S_1)$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3S","source_file":"divisors.tex","source_line":3043,"source_end_line":3047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3043-L3047","statement_sha256":"bdc4339e0ddfdf4827a519dba4b850c62ddb319f66948e675a4752a0e289baba","origin":"The Stacks Project","memory_eligible":false,"source_rank":6080,"rank":6080,"depth":18,"x":2517.515,"y":731.177,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AGA","tag":"0AGA","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0AGA","summary":"Let X be a locally Noetherian scheme. Let D ⊂ X be an integral closed subscheme. Assume that • D has codimension 1 in X, and • O_X, x is a UFD for all x ∈ D. Then D is an effective Cartier divisor.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be an integral\nclosed subscheme. Assume that\n\\begin{enumerate}\n\\item $D$ has codimension $1$ in $X$, and\n\\item $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in D$.\n\\end{enumerate}\nThen $D$ is an effective Cartier divisor.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGA","source_file":"divisors.tex","source_line":3055,"source_end_line":3064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3055-L3064","statement_sha256":"7a135e9deae247aac7eb886dbe5b2e70c7033c8704cdb7fbf3bf834d0620b8e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6081,"rank":6081,"depth":2,"x":2410.881,"y":663.478,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AGB","tag":"0AGB","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0AGB","summary":"Let X be a Noetherian scheme. Let Z ⊂ X be a closed subscheme. Assume there exist • a collection of integral effective Cartier divisors D_i ⊂ X, i ∈ I • a closed subset Z' ⊂ X all of whose irreducible components have codimension ≥ 2 in X (Topology, Definition [Tag 02I3]) such that Z ⊂ Z' ∪ ⋃_i ∈ I D_i set-theoretically. Then there exist integers a_i ≥ 0 with a_i = 0 for almost all i ∈ I such that the effective Cartier divisor D = ∑ a_i D_i is contained in Z and such that…","statement_latex":"Let $X$ be a Noetherian scheme. Let $Z \\subset X$ be a closed subscheme.\nAssume there exist\n\\begin{enumerate}\n\\item a collection of integral effective Cartier divisors\n$D_i \\subset X$, $i \\in I$\n\\item a closed subset $Z' \\subset X$ all of whose irreducible components\nhave codimension $\\geq 2$ in $X$\n(Topology, Definition \\ref{topology-definition-codimension})\n\\end{enumerate}\nsuch that $Z \\subset Z' \\cup \\bigcup_{i \\in I} D_i$ set-theoretically.\nThen there exist integers $a_i \\geq 0$ with $a_i = 0$ for almost all $i \\in I$\nsuch that the effective Cartier divisor\n$$\nD = \\sum a_i D_i\n$$\nis contained in $Z$ and such that the inclusion morphism $D \\to Z$\nis an isomorphism away from codimension $2$ in $X$\n(in the sense that there exists an open $U \\subset Z$ such that\n$D \\cap U \\to Z \\cap U$ is an isomorphism and such that every\nirreducible component of $Z \\setminus U$ has codimension $\\geq 2$ in $X$).\nWhen $Z$ is nowhere dense in $X$ existence of the $D_i$, $i \\in I$ and $Z'$\nis guaranteed if $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in Z$\nor if $X$ is regular.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGB","source_file":"divisors.tex","source_line":3075,"source_end_line":3100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3075-L3100","statement_sha256":"9f095d7f0a2be477f794f0316d0d7e430670361138e68312d335cf51138f6622","origin":"The Stacks Project","memory_eligible":false,"source_rank":6082,"rank":6082,"depth":19,"x":2544.533,"y":652.846,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BXH","tag":"0BXH","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0BXH","summary":"Let Z ⊂ X be a closed subscheme of a Noetherian scheme. Assume • Z has no embedded points, • every irreducible component of Z has codimension 1 in X, • every local ring O_X, x, x ∈ Z is a UFD or X is regular. Then Z is an effective Cartier divisor.","statement_latex":"Let $Z \\subset X$ be a closed subscheme of a Noetherian scheme. Assume\n\\begin{enumerate}\n\\item $Z$ has no embedded points,\n\\item every irreducible component of $Z$ has codimension $1$ in $X$,\n\\item every local ring $\\mathcal{O}_{X, x}$, $x \\in Z$ is\na UFD or $X$ is regular.\n\\end{enumerate}\nThen $Z$ is an effective Cartier divisor.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXH","source_file":"divisors.tex","source_line":3157,"source_end_line":3167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3157-L3167","statement_sha256":"45cee8e1c657cdb3e33109565302014e8816c982799e4fba4fac280827de2ce2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6083,"rank":6083,"depth":20,"x":2454.14,"y":736.883,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BXI","tag":"0BXI","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0BXI","summary":"Let R be a Noetherian UFD. Let I ⊂ R be an ideal such that R/I has no embedded primes and such that every minimal prime over I has height 1. Then I = (f) for some f ∈ R.","statement_latex":"Let $R$ be a Noetherian UFD. Let $I \\subset R$ be an ideal\nsuch that $R/I$ has no embedded primes and such that\nevery minimal prime over $I$ has height $1$.\nThen $I = (f)$ for some $f \\in R$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXI","source_file":"divisors.tex","source_line":3178,"source_end_line":3184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3178-L3184","statement_sha256":"37d567f3a07e5dcb8f97dee9c76a7a473e5054c0668827c82480d075f4a6689f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6084,"rank":6084,"depth":21,"x":2453.211,"y":623.14,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BCP","tag":"0BCP","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0BCP","summary":"Let X be a Noetherian scheme. Let D ⊂ X be an effective Cartier divisor. Assume that there exist integral effective Cartier divisors D_i ⊂ X such that D ⊂ ⋃ D_i set theoretically. Then D = ∑ a_i D_i for some a_i ≥ 0. The existence of the D_i is guaranteed if O_X, x is a UFD for all x ∈ D or if X is regular.","statement_latex":"Let $X$ be a Noetherian scheme. Let $D \\subset X$ be an effective\nCartier divisor. Assume that there exist integral effective Cartier\ndivisors $D_i \\subset X$ such that $D \\subset \\bigcup D_i$\nset theoretically. Then $D = \\sum a_i D_i$ for some $a_i \\geq 0$.\nThe existence of the $D_i$ is guaranteed if $\\mathcal{O}_{X, x}$\nis a UFD for all $x \\in D$ or if $X$ is regular.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCP","source_file":"divisors.tex","source_line":3193,"source_end_line":3201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3193-L3201","statement_sha256":"e2d38854052f4dfaa2cff5937e219f4a6d9e5acfc75c72f76ff28fc6fc807cb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6085,"rank":6085,"depth":20,"x":2545.755,"y":706.84,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AYM","tag":"0AYM","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0AYM","summary":"On a projective scheme, every line bundle has a regular meromorphic section. Let X be a Noetherian scheme which has an ample invertible sheaf. Then every invertible O_X-module is isomorphic to O_X(D - D') = O_X(D) ⊗_O_X O_X(D')^⊗ -1 for some effective Cartier divisors D, D' in X. Moreover, given a finite subset E ⊂ X we may choose D, D' such that E ∩ D = ∅ and E ∩ D' = ∅. If X is quasi-affine, then we may choose D' = ∅.","statement_latex":"\\begin{slogan}\nOn a projective scheme, every line bundle has a regular meromorphic section.\n\\end{slogan}\nLet $X$ be a Noetherian scheme which has an ample invertible sheaf.\nThen every invertible $\\mathcal{O}_X$-module is isomorphic to\n$$\n\\mathcal{O}_X(D - D') =\n\\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D')^{\\otimes -1}\n$$\nfor some effective Cartier divisors $D, D'$ in $X$. Moreover, given a\nfinite subset $E \\subset X$ we may choose $D, D'$ such that\n$E \\cap D = \\emptyset$ and $E \\cap D' = \\emptyset$. If\n$X$ is quasi-affine, then we may choose $D' = \\emptyset$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYM","source_file":"divisors.tex","source_line":3215,"source_end_line":3230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3215-L3230","statement_sha256":"24f899e18739f0fbc430457091b7626ce0fcf62edc3c49808ac64a810ef0e4b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6086,"rank":6086,"depth":24,"x":2409.636,"y":697.593,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B3T","tag":"0B3T","title":"Effective Cartier divisors on Noetherian schemes · Lemma 0B3T","summary":"Let X be an integral regular scheme of dimension 2. Let i : D → X be the immersion of an effective Cartier divisor. Let F → F' → i_*G → 0 be an exact sequence of coherent O_X-modules. Assume • F, F' are locally free of rank r on a nonempty open of X, • D is an integral scheme, • G is a finite locally free O_D-module of rank s. Then L = (wedge^rF)^** and L' = (wedge^r F')^** are invertible O_X-modules and L' ≅ L(k D) for some k ∈ (0, …, min(s, r)).","statement_latex":"Let $X$ be an integral regular scheme of dimension $2$.\nLet $i : D \\to X$ be the immersion of an effective Cartier divisor.\nLet $\\mathcal{F} \\to \\mathcal{F}' \\to i_*\\mathcal{G} \\to 0$\nbe an exact sequence of coherent $\\mathcal{O}_X$-modules.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}, \\mathcal{F}'$ are locally free of rank $r$ on a nonempty\nopen of $X$,\n\\item $D$ is an integral scheme,\n\\item $\\mathcal{G}$ is a finite locally free $\\mathcal{O}_D$-module\nof rank $s$.\n\\end{enumerate}\nThen $\\mathcal{L} = (\\wedge^r\\mathcal{F})^{**}$ and\n$\\mathcal{L}' = (\\wedge^r \\mathcal{F}')^{**}$\nare invertible $\\mathcal{O}_X$-modules and\n$\\mathcal{L}' \\cong \\mathcal{L}(k D)$ for some\n$k \\in \\{0, \\ldots, \\min(s, r)\\}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Effective Cartier divisors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3T","source_file":"divisors.tex","source_line":3276,"source_end_line":3295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3276-L3295","statement_sha256":"686adba5ba7397840556034d5db93a29f8af7f1254ede536f6dce80564615165","origin":"The Stacks Project","memory_eligible":false,"source_rank":6087,"rank":6087,"depth":20,"x":2517.894,"y":626.881,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BCR","tag":"0BCR","title":"Complements of affine opens · Lemma 0BCR","summary":"Let (A, m) be a Noetherian local ring. The punctured spectrum U = Spec(A) setminus ( m) of A is affine if and only if dim(A) ≤ 1.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nThe punctured spectrum $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$\nof $A$ is affine if and only if $\\dim(A) \\leq 1$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Complements of affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCR","source_file":"divisors.tex","source_line":3348,"source_end_line":3353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3348-L3353","statement_sha256":"439daf65ecb779b26c2e6b41fbb162be9b9ccff2b78ac17f937be7ba70c790a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6088,"rank":6088,"depth":11,"x":2494.835,"y":740.921,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BCU","tag":"0BCU","title":"Complements of affine opens · Lemma 0BCU","summary":"[EGA4] Let X be a locally Noetherian scheme. Let U ⊂ X be an open subscheme such that the inclusion morphism U → X is affine. For every generic point xi of an irreducible component of X setminus U the local ring O_X, xi has dimension ≤ 1. If U is dense or if xi is in the closure of U, then dim(O_X, xi) = 1.","statement_latex":"\\begin{reference}\n\\cite[EGA IV, Corollaire 21.12.7]{EGA4}\n\\end{reference}\nLet $X$ be a locally Noetherian scheme. Let $U \\subset X$ be an open subscheme\nsuch that the inclusion morphism $U \\to X$ is affine.\nFor every generic point $\\xi$ of an irreducible component of\n$X \\setminus U$ the local ring $\\mathcal{O}_{X, \\xi}$\nhas dimension $\\leq 1$. If $U$ is dense or if $\\xi$ is in the closure\nof $U$, then $\\dim(\\mathcal{O}_{X, \\xi}) = 1$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Complements of affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCU","source_file":"divisors.tex","source_line":3449,"source_end_line":3460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3449-L3460","statement_sha256":"71022b8dc993092372ea828a0923d613ec36bd0a299a8a0b8da81f0a9b11550e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6089,"rank":6089,"depth":19,"x":2419.825,"y":643.346,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BCV","tag":"0BCV","title":"Complements of affine opens · Lemma 0BCV","summary":"Let X be a separated locally Noetherian scheme. Let U ⊂ X be an affine open. For every generic point xi of an irreducible component of X setminus U the local ring O_X, xi has dimension ≤ 1. If U is dense or if xi is in the closure of U, then dim(O_X, xi) = 1.","statement_latex":"Let $X$ be a separated locally Noetherian scheme. Let $U \\subset X$ be an\naffine open. For every generic point $\\xi$ of an irreducible component of\n$X \\setminus U$ the local ring $\\mathcal{O}_{X, \\xi}$\nhas dimension $\\leq 1$. If $U$ is dense or if $\\xi$ is in the closure\nof $U$, then $\\dim(\\mathcal{O}_{X, \\xi}) = 1$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Complements of affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCV","source_file":"divisors.tex","source_line":3484,"source_end_line":3491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3484-L3491","statement_sha256":"d416a5785a03c43ed329d3e27c277bc811f854075a58d2a84439d706f9afd4c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6090,"rank":6090,"depth":20,"x":2554.148,"y":672.854,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BCW","tag":"0BCW","title":"Complements of affine opens · Lemma 0BCW","summary":"Let X be a Noetherian separated scheme. Let U ⊂ X be a dense affine open. If O_X, x is a UFD for all x ∈ X setminus U, then there exists an effective Cartier divisor D ⊂ X with U = X setminus D.","statement_latex":"Let $X$ be a Noetherian separated scheme. Let $U \\subset X$ be\na dense affine open. If $\\mathcal{O}_{X, x}$ is a UFD for all\n$x \\in X \\setminus U$, then there exists an effective Cartier\ndivisor $D \\subset X$ with $U = X \\setminus D$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Complements of affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCW","source_file":"divisors.tex","source_line":3503,"source_end_line":3509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3503-L3509","statement_sha256":"565ba88fb558d107adfea79d530eeb2988bfa4cef95b04f005ed2369b6b6fb51","origin":"The Stacks Project","memory_eligible":false,"source_rank":6091,"rank":6091,"depth":21,"x":2430.875,"y":727.533,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EGJ","tag":"0EGJ","title":"Complements of affine opens · Lemma 0EGJ","summary":"Let X be a Noetherian scheme with affine diagonal. Let U ⊂ X be a dense affine open. If O_X, x is a UFD for all x ∈ X setminus U, then there exists an effective Cartier divisor D ⊂ X with U = X setminus D.","statement_latex":"Let $X$ be a Noetherian scheme with affine diagonal. Let $U \\subset X$ be\na dense affine open. If $\\mathcal{O}_{X, x}$ is a UFD for all\n$x \\in X \\setminus U$, then there exists an effective Cartier\ndivisor $D \\subset X$ with $U = X \\setminus D$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Complements of affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EGJ","source_file":"divisors.tex","source_line":3524,"source_end_line":3530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3524-L3530","statement_sha256":"095769debea52df1921f55396662d9e4a464402777864bd552f93632c1157a9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6092,"rank":6092,"depth":22,"x":2477.992,"y":616.824,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GML","tag":"0GML","title":"Complements of affine opens · Lemma 0GML","summary":"Let X be a quasi-compact, regular scheme with affine diagonal. Then X has an ample family of invertible modules (Morphisms, Definition [Tag 0FXR]).","statement_latex":"Let $X$ be a quasi-compact, regular scheme with affine diagonal.\nThen $X$ has an ample family of invertible modules\n(Morphisms, Definition\n\\ref{morphisms-definition-family-ample-invertible-modules}).","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Complements of affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GML","source_file":"divisors.tex","source_line":3553,"source_end_line":3559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3553-L3559","statement_sha256":"6e319156f05419bd673fec1487059447ebf0b6df73b05174dfa961a29a77804d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6093,"rank":6093,"depth":23,"x":2532.492,"y":725.625,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BUT","tag":"0BUT","title":"Norms · Lemma 0BUT","summary":"Let π : X → Y be a finite morphism of schemes. Let L be an invertible O_X-module. Let y ∈ Y. There exists an open neighbourhood V ⊂ Y of y such that L|_π^-1(V) is trivial.","statement_latex":"Let $\\pi : X \\to Y$ be a finite morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $y \\in Y$. There exists an open neighbourhood\n$V \\subset Y$ of $y$ such that $\\mathcal{L}|_{\\pi^{-1}(V)}$ is trivial.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUT","source_file":"divisors.tex","source_line":3616,"source_end_line":3622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3616-L3622","statement_sha256":"8d96e1e56c56be307901ae4ccc53b4d4176afc5e2048f90d85f2724459be90da","origin":"The Stacks Project","memory_eligible":false,"source_rank":6094,"rank":6094,"depth":24,"x":2404.307,"y":676.127,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BCY","tag":"0BCY","title":"Norms · Lemma 0BCY","summary":"Let π : X → Y be a finite morphism of schemes. If there exists a norm of degree d for π, then there exists a homomorphism of abelian groups Norm_π : Pic(X) → Pic(Y) such that Norm_π(π^*N) ≅ N^⊗ d for all invertible O_Y-modules N.","statement_latex":"Let $\\pi : X \\to Y$ be a finite morphism of schemes. If there exists\na norm of degree $d$ for $\\pi$, then there exists a homomorphism of\nabelian groups\n$$\n\\text{Norm}_\\pi : \\Pic(X) \\to \\Pic(Y)\n$$\nsuch that $\\text{Norm}_\\pi(\\pi^*\\mathcal{N}) \\cong \\mathcal{N}^{\\otimes d}$\nfor all invertible $\\mathcal{O}_Y$-modules $\\mathcal{N}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCY","source_file":"divisors.tex","source_line":3654,"source_end_line":3664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3654-L3664","statement_sha256":"fc9cf782f6a4074a6fbc0606ee2dd6b82312d54d0c6ac372a76a26391309fe3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6095,"rank":6095,"depth":25,"x":2539.159,"y":639.75,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BCZ","tag":"0BCZ","title":"Norms · Lemma 0BCZ","summary":"Let π : X → Y be a finite morphism of schemes. Assume there exists a norm of degree d for π. For any O_X-linear map φ : L → L' of invertible O_X-modules there is an O_Y-linear map Norm_π(φ) : Norm_π(L) → Norm_π(L') with Norm_π(L), Norm_π(L') as in Lemma [Tag 0BCY]. Moreover, for y ∈ Y the following are equivalent • φ is zero at a point of x ∈ X with π(x) = y, and • Norm_π(φ) is zero at y.","statement_latex":"Let $\\pi : X \\to Y$ be a finite morphism of schemes. Assume there exists\na norm of degree $d$ for $\\pi$. For any $\\mathcal{O}_X$-linear map\n$\\varphi : \\mathcal{L} \\to \\mathcal{L}'$\nof invertible $\\mathcal{O}_X$-modules there is an $\\mathcal{O}_Y$-linear\nmap\n$$\n\\text{Norm}_\\pi(\\varphi) :\n\\text{Norm}_\\pi(\\mathcal{L})\n\\longrightarrow\n\\text{Norm}_\\pi(\\mathcal{L}')\n$$\nwith $\\text{Norm}_\\pi(\\mathcal{L})$, $\\text{Norm}_\\pi(\\mathcal{L}')$\nas in Lemma \\ref{lemma-norm-invertible}. Moreover, for\n$y \\in Y$ the following are equivalent\n\\begin{enumerate}\n\\item $\\varphi$ is zero at a point of $x \\in X$ with $\\pi(x) = y$, and\n\\item $\\text{Norm}_\\pi(\\varphi)$ is zero at $y$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BCZ","source_file":"divisors.tex","source_line":3695,"source_end_line":3715,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3695-L3715","statement_sha256":"50937b79a80e36eb0e0514f2f584a9311def561c46cdc37dad3a8f30c3657984","origin":"The Stacks Project","memory_eligible":false,"source_rank":6096,"rank":6096,"depth":26,"x":2468.702,"y":743.492,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BD0","tag":"0BD0","title":"Norms · Lemma 0BD0","summary":"Let π : X → Y be a finite morphism of schemes. Assume X has an ample invertible sheaf and there exists a norm of degree d for π. Then Y has an ample invertible sheaf.","statement_latex":"Let $\\pi : X \\to Y$ be a finite morphism of schemes. Assume $X$ has\nan ample invertible sheaf and there exists a norm of degree $d$\nfor $\\pi$. Then $Y$ has an ample invertible sheaf.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BD0","source_file":"divisors.tex","source_line":3737,"source_end_line":3742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3737-L3742","statement_sha256":"65977bc691b1e08a947dd34303c0955f85d90ca81d3976d9d9f780611feed3b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6097,"rank":6097,"depth":27,"x":2437.109,"y":626.567,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BD1","tag":"0BD1","title":"Norms · Lemma 0BD1","summary":"Let π : X → Y be a finite morphism of schemes. Assume X is quasi-affine and there exists a norm of degree d for π. Then Y is quasi-affine.","statement_latex":"Let $\\pi : X \\to Y$ be a finite morphism of schemes. Assume $X$ is quasi-affine\nand there exists a norm of degree $d$ for $\\pi$. Then $Y$ is quasi-affine.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BD1","source_file":"divisors.tex","source_line":3772,"source_end_line":3776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3772-L3776","statement_sha256":"d4e4aea653d66108bc7817bbcb8093c3f299f7c0e5c4f075e3f87cbb4b73104a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6098,"rank":6098,"depth":28,"x":2554.88,"y":695.121,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BD2","tag":"0BD2","title":"Norms · Lemma 0BD2","summary":"Let π : X → Y be a finite locally free morphism of degree d ≥ 1. Then there exists a canonical norm of degree d whose formation commutes with arbitrary base change.","statement_latex":"Let $\\pi : X \\to Y$ be a finite locally free morphism of degree $d \\geq 1$.\nThen there exists a canonical norm of degree $d$ whose formation commutes\nwith arbitrary base change.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BD2","source_file":"divisors.tex","source_line":3788,"source_end_line":3793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3788-L3793","statement_sha256":"d41d0138a637201e2481fe5613a04b7e294cb2692f8aac6d46c37782b8c2d1ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":6099,"rank":6099,"depth":0,"x":2412.371,"y":711.457,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BD3","tag":"0BD3","title":"Norms · Lemma 0BD3","summary":"Let π : X → Y be a finite surjective morphism with X and Y integral and Y normal. Then there exists a norm of degree [R(X) : R(Y)] for π.","statement_latex":"Let $\\pi : X \\to Y$ be a finite surjective morphism with $X$ and $Y$\nintegral and $Y$ normal. Then there exists a norm of degree\n$[R(X) : R(Y)]$ for $\\pi$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BD3","source_file":"divisors.tex","source_line":3825,"source_end_line":3830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3825-L3830","statement_sha256":"9bc81bbde1289f2854b9fb0b4ca1db92423d699cd8335d63e474f5f4beceea45","origin":"The Stacks Project","memory_eligible":false,"source_rank":6100,"rank":6100,"depth":8,"x":2504.664,"y":618.199,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BDZ","tag":"0BDZ","title":"Norms · Lemma 0BDZ","summary":"Let X be a Noetherian scheme. Let p be a prime number such that pO_X = 0. Then for some e > 0 there exists a norm of degree p^e for X_red → X where X_red is the reduction of X.","statement_latex":"Let $X$ be a Noetherian scheme. Let $p$ be a prime number such that\n$p\\mathcal{O}_X = 0$. Then for some $e > 0$ there exists a norm\nof degree $p^e$ for $X_{red} \\to X$ where $X_{red}$ is the reduction\nof $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDZ","source_file":"divisors.tex","source_line":3869,"source_end_line":3875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3869-L3875","statement_sha256":"4d04ca6604a1c5155295d34f15144f525c4f1217ada565066e584014ab231aaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6101,"rank":6101,"depth":1,"x":2511.629,"y":739.787,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BD4","tag":"0BD4","title":"Norms · Proposition 0BD4","summary":"Let π : X → Y be a finite surjective morphism of schemes. Assume that X has an ample invertible O_X-module. If • π is finite locally free, or • Y is an integral normal scheme, or • Y is Noetherian, pO_Y = 0, and X = Y_red, then Y has an ample invertible O_Y-module.","statement_latex":"Let $\\pi : X \\to Y$ be a finite surjective morphism of schemes.\nAssume that $X$ has an ample invertible $\\mathcal{O}_X$-module. If\n\\begin{enumerate}\n\\item $\\pi$ is finite locally free, or\n\\item $Y$ is an integral normal scheme, or\n\\item $Y$ is Noetherian, $p\\mathcal{O}_Y = 0$, and $X = Y_{red}$,\n\\end{enumerate}\nthen $Y$ has an ample invertible $\\mathcal{O}_Y$-module.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BD4","source_file":"divisors.tex","source_line":3892,"source_end_line":3902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3892-L3902","statement_sha256":"c1725bca234e861cdd7052c42c1b332c7dc0f68aa0cb7e302a02bb7d05559f5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6102,"rank":6102,"depth":28,"x":2408.334,"y":653.764,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BD5","tag":"0BD5","title":"Norms · Lemma 0BD5","summary":"Let π : X → Y be a finite surjective morphism of schemes. Assume that X is quasi-affine. If either • π is finite locally free, or • Y is an integral normal scheme then Y is quasi-affine.","statement_latex":"Let $\\pi : X \\to Y$ be a finite surjective morphism of schemes.\nAssume that $X$ is quasi-affine. If either\n\\begin{enumerate}\n\\item $\\pi$ is finite locally free, or\n\\item $Y$ is an integral normal scheme\n\\end{enumerate}\nthen $Y$ is quasi-affine.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Norms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BD5","source_file":"divisors.tex","source_line":3915,"source_end_line":3924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3915-L3924","statement_sha256":"7d18cbcc49df06e8048d4858b76a82a530dc07f15cc97aae3acc595656fc3a1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6103,"rank":6103,"depth":29,"x":2554.214,"y":658.603,"cluster":"divisors-intersection-theory"},{"id":"stacks:056Q","tag":"056Q","title":"Relative effective Cartier divisors · Lemma 056Q","summary":"Let f : X → S be a morphism of schemes. Let D ⊂ X be a closed subscheme. Assume • D is an effective Cartier divisor, and • D → S is a flat morphism. Then for every morphism of schemes g : S' → S the pullback (g')^-1D is an effective Cartier divisor on X' = S' ×_S X where g' : X' → X is the projection.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $D \\subset X$ be a closed subscheme.\nAssume\n\\begin{enumerate}\n\\item $D$ is an effective Cartier divisor, and\n\\item $D \\to S$ is a flat morphism.\n\\end{enumerate}\nThen for every morphism of schemes $g : S' \\to S$ the pullback\n$(g')^{-1}D$ is an effective Cartier divisor on $X' = S' \\times_S X$\nwhere $g' : X' \\to X$ is the projection.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056Q","source_file":"divisors.tex","source_line":3947,"source_end_line":3959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3947-L3959","statement_sha256":"6919701120bb072a4082907a3d8b105b52bd43fe6b5fb2d7f7f9aeabaaa5e92e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6104,"rank":6104,"depth":1,"x":2442.347,"y":738.099,"cluster":"divisors-intersection-theory"},{"id":"stacks:062T","tag":"062T","title":"Relative effective Cartier divisors · Definition 062T","summary":"Let f : X → S be a morphism of schemes. A relative effective Cartier divisor on X/S is an effective Cartier divisor D ⊂ X such that D → S is a flat morphism of schemes.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nA {\\it relative effective Cartier divisor} on $X/S$ is an\neffective Cartier divisor $D \\subset X$ such that $D \\to S$\nis a flat morphism of schemes.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative effective Cartier divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062T","source_file":"divisors.tex","source_line":3979,"source_end_line":3985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3979-L3985","statement_sha256":"2cb9f981219b028cb8bcbebbc6b86455affa49e9084502a0cc42b802e3a53924","origin":"The Stacks Project","memory_eligible":false,"source_rank":6105,"rank":6105,"depth":0,"x":2460.975,"y":615.561,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B8U","tag":"0B8U","title":"Relative effective Cartier divisors · Lemma 0B8U","summary":"Let f : X → S be a morphism of schemes. If D_1, D_2 ⊂ X are relative effective Cartier divisor on X/S then so is D_1 + D_2 (Definition [Tag 01WT]).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. If $D_1, D_2 \\subset X$\nare relative effective Cartier divisor on $X/S$ then so\nis $D_1 + D_2$ (Definition \\ref{definition-sum-effective-Cartier-divisors}).","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8U","source_file":"divisors.tex","source_line":3995,"source_end_line":4000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L3995-L4000","statement_sha256":"8aed25361f41820998681b980a25bb023e1f5b8d714713864dec775cb4c168d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6106,"rank":6106,"depth":4,"x":2546.084,"y":716.852,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B8V","tag":"0B8V","title":"Relative effective Cartier divisors · Lemma 0B8V","summary":"Let f : X → S be a morphism of schemes. If D_1, D_2 ⊂ X are relative effective Cartier divisor on X/S and D_1 ⊂ D_2 as closed subschemes, then the effective Cartier divisor D such that D_2 = D_1 + D (Lemma [Tag 02ON]) is a relative effective Cartier divisor on X/S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. If $D_1, D_2 \\subset X$\nare relative effective Cartier divisor on $X/S$ and $D_1 \\subset D_2$\nas closed subschemes, then the effective Cartier divisor $D$\nsuch that $D_2 = D_1 + D$\n(Lemma \\ref{lemma-difference-effective-Cartier-divisors}) is\na relative effective Cartier divisor on $X/S$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8V","source_file":"divisors.tex","source_line":4016,"source_end_line":4024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4016-L4024","statement_sha256":"c7211df2db53c40512eadc8ecaa1e3245c56842f51bbc0178a029f248625fdd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6107,"rank":6107,"depth":4,"x":2401.356,"y":690.36,"cluster":"divisors-intersection-theory"},{"id":"stacks:062U","tag":"062U","title":"Relative effective Cartier divisors · Lemma 062U","summary":"Let f : X → S be a morphism of schemes. Let D ⊂ X be a relative effective Cartier divisor on X/S. If x ∈ D and O_X, x is Noetherian, then f is flat at x.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $D \\subset X$ be a relative effective Cartier divisor on $X/S$.\nIf $x \\in D$ and $\\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062U","source_file":"divisors.tex","source_line":4040,"source_end_line":4045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4040-L4045","statement_sha256":"bff515809e7ab432dd57e7765fbe54cd13ce37adeba4fdbee2de86533ab0337c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6108,"rank":6108,"depth":3,"x":2529.835,"y":627.552,"cluster":"divisors-intersection-theory"},{"id":"stacks:062V","tag":"062V","title":"Relative effective Cartier divisors · Lemma 062V","summary":"Let f : X → S be a morphism of schemes. Let D ⊂ X be a relative effective Cartier divisor. If f is locally of finite presentation, then there exists an open subscheme U ⊂ X such that D ⊂ U and such that f|_U : U → S is flat.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $D \\subset X$ be a relative effective Cartier divisor.\nIf $f$ is locally of finite presentation, then there exists\nan open subscheme $U \\subset X$ such that $D \\subset U$ and\nsuch that $f|_U : U \\to S$ is flat.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062V","source_file":"divisors.tex","source_line":4081,"source_end_line":4088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4081-L4088","statement_sha256":"5cfbe9a60935b220c04abe19901c4950c0da5bd78bd52aa423f328dcbb9296b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6109,"rank":6109,"depth":35,"x":2485.449,"y":747.187,"cluster":"divisors-intersection-theory"},{"id":"stacks:062W","tag":"062W","title":"Relative effective Cartier divisors · Lemma 062W","summary":"Let f : X → S be a morphism of schemes. Let D ⊂ X be a relative effective Cartier divisor on X/S. If f is flat at all points of X setminus D, then f is flat.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $D \\subset X$ be a relative effective Cartier divisor on $X/S$.\nIf $f$ is flat at all points of $X \\setminus D$, then $f$ is flat.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062W","source_file":"divisors.tex","source_line":4141,"source_end_line":4146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4141-L4146","statement_sha256":"f2a58a20a99b05fd3c9e0b04784d9829f7084ae3c62da0bd9a2482ee131cb0e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6110,"rank":6110,"depth":4,"x":2421.75,"y":633.388,"cluster":"divisors-intersection-theory"},{"id":"stacks:062Y","tag":"062Y","title":"Relative effective Cartier divisors · Lemma 062Y","summary":"Let φ : X → S be a flat morphism which is locally of finite presentation. Let Z ⊂ X be a closed subscheme. Let x ∈ Z with image s ∈ S. • If Z_s ⊂ X_s is a Cartier divisor in a neighbourhood of x, then there exists an open U ⊂ X and a relative effective Cartier divisor D ⊂ U such that Z ∩ U ⊂ D and Z_s ∩ U = D_s. • If Z_s ⊂ X_s is a Cartier divisor in a neighbourhood of x, the morphism Z → X is of finite presentation, and Z → S is flat at x, then we can choose U and D such…","statement_latex":"Let $\\varphi : X \\to S$ be a flat morphism which is locally of finite\npresentation. Let $Z \\subset X$ be a closed subscheme.\nLet $x \\in Z$ with image $s \\in S$.\n\\begin{enumerate}\n\\item If $Z_s \\subset X_s$ is a Cartier divisor in a neighbourhood of $x$,\nthen there exists an open $U \\subset X$ and a\nrelative effective Cartier divisor $D \\subset U$ such that\n$Z \\cap U \\subset D$ and $Z_s \\cap U = D_s$.\n\\item If $Z_s \\subset X_s$ is a Cartier divisor in a neighbourhood of $x$,\nthe morphism $Z \\to X$ is of finite presentation, and $Z \\to S$ is flat at\n$x$, then we can choose $U$ and $D$ such that $Z \\cap U = D$.\n\\item If $Z_s \\subset X_s$ is a Cartier divisor in a neighbourhood of $x$\nand $Z$ is a locally principal closed subscheme of $X$ in a neighbourhood\nof $x$, then we can choose $U$ and $D$ such that $Z \\cap U = D$.\n\\end{enumerate}\nIn particular, if $Z \\to S$ is locally of finite presentation and flat and\nall fibres $Z_s \\subset X_s$ are effective Cartier divisors, then\n$Z$ is a relative effective Cartier divisor. Similarly, if $Z$\nis a locally principal closed subscheme of $X$ such that all fibres\n$Z_s \\subset X_s$ are effective Cartier divisors, then\n$Z$ is a relative effective Cartier divisor.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/062Y","source_file":"divisors.tex","source_line":4190,"source_end_line":4213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4190-L4213","statement_sha256":"68fd5d81118bfbc1929b54f025631dee7208cc2571efb249b7f9a3bac383eace","origin":"The Stacks Project","memory_eligible":false,"source_rank":6111,"rank":6111,"depth":35,"x":2560.715,"y":681.323,"cluster":"divisors-intersection-theory"},{"id":"stacks:0630","tag":"0630","title":"The normal cone of an immersion · Definition 0630","summary":"Let f : Z → X be an immersion. The conormal algebra C_Z/X, * of Z in X or the conormal algebra of f is the quasi-coherent sheaf of graded O_Z-algebras bigoplus_n ≥ 0 I^n/I^n + 1 described above.","statement_latex":"Let $f : Z \\to X$ be an immersion. The {\\it conormal algebra\n$\\mathcal{C}_{Z/X, *}$ of $Z$ in $X$} or the {\\it conormal algebra of $f$}\nis the quasi-coherent sheaf of graded $\\mathcal{O}_Z$-algebras\n$\\bigoplus_{n \\geq 0} \\mathcal{I}^n/\\mathcal{I}^{n + 1}$ described above.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The normal cone of an immersion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0630","source_file":"divisors.tex","source_line":4324,"source_end_line":4330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4324-L4330","statement_sha256":"0f7fe20a7f7498199417aff364135eea18d616ee90d86f6d1f0eb17a23051ed0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6112,"rank":6112,"depth":0,"x":2419.21,"y":724.978,"cluster":"divisors-intersection-theory"},{"id":"stacks:0633","tag":"0633","title":"The normal cone of an immersion · Lemma 0633","summary":"Let i : Z → X be an immersion. The conormal algebra of i has the following properties: • Let U ⊂ X be any open such that i(Z) is a closed subset of U. Let I ⊂ O_U be the sheaf of ideals corresponding to the closed subscheme i(Z) ⊂ U. Then C_Z/X, * = i^*(bigoplus_n ≥ 0 I^n) = i^-1( bigoplus_n ≥ 0 I^n/I^n + 1 ) • For any affine open Spec(R) = U ⊂ X such that Z ∩ U = Spec(R/I) there is a canonical isomorphism Γ(Z ∩ U, C_Z/X, *) = bigoplus_n ≥ 0 I^n/I^n + 1.","statement_latex":"Let $i : Z \\to X$ be an immersion. The conormal algebra\nof $i$ has the following properties:\n\\begin{enumerate}\n\\item Let $U \\subset X$ be any open such that $i(Z)$ is\na closed subset of $U$. Let $\\mathcal{I} \\subset \\mathcal{O}_U$\nbe the sheaf of ideals corresponding to the closed subscheme\n$i(Z) \\subset U$. Then\n$$\n\\mathcal{C}_{Z/X, *} =\ni^*\\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n\\right) =\ni^{-1}\\left(\n\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n/\\mathcal{I}^{n + 1}\n\\right)\n$$\n\\item\nFor any affine open $\\Spec(R) = U \\subset X$\nsuch that $Z \\cap U = \\Spec(R/I)$ there is a\ncanonical isomorphism\n$\\Gamma(Z \\cap U, \\mathcal{C}_{Z/X, *}) = \\bigoplus_{n \\geq 0} I^n/I^{n + 1}$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The normal cone of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0633","source_file":"divisors.tex","source_line":4351,"source_end_line":4373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4351-L4373","statement_sha256":"61bfefc7c1717a3c8c22f15a5544db66571f2d65e5a08bab1557e93d2a0db59c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6113,"rank":6113,"depth":0,"x":2488.685,"y":612.11,"cluster":"divisors-intersection-theory"},{"id":"stacks:0634","tag":"0634","title":"The normal cone of an immersion · Lemma 0634","summary":"Let xymatrix Z ar[r]_i ar[d]_f & X ar[d]^g Z' ar[r]^i' & X' be a commutative diagram in the category of schemes. Assume i, i' immersions. There is a canonical map of graded O_Z-algebras f^*C_Z'/X', * → C_Z/X, * characterized by the following property: For every pair of affine opens (Spec(R) = U ⊂ X, Spec(R') = U' ⊂ X') with g(U) ⊂ U' such that Z ∩ U = Spec(R/I) and Z' ∩ U' = Spec(R'/I') the induced map Γ(Z' ∩ U', C_Z'/X', *) = bigoplus (I')^n/(I')^n + 1 → bigoplus_n ≥ 0…","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\\nZ' \\ar[r]^{i'} & X'\n}\n$$\nbe a commutative diagram in the category of schemes.\nAssume $i$, $i'$ immersions. There is a canonical map\nof graded $\\mathcal{O}_Z$-algebras\n$$\nf^*\\mathcal{C}_{Z'/X', *}\n\\longrightarrow\n\\mathcal{C}_{Z/X, *}\n$$\ncharacterized by the following property: For every pair of affine opens\n$(\\Spec(R) = U \\subset X, \\Spec(R') = U' \\subset X')$ with\n$g(U) \\subset U'$ such that\n$Z \\cap U = \\Spec(R/I)$ and $Z' \\cap U' = \\Spec(R'/I')$\nthe induced map\n$$\n\\Gamma(Z' \\cap U', \\mathcal{C}_{Z'/X', *}) =\n\\bigoplus\\nolimits (I')^n/(I')^{n + 1}\n\\longrightarrow\n\\bigoplus\\nolimits_{n \\geq 0} I^n/I^{n + 1} =\n\\Gamma(Z \\cap U, \\mathcal{C}_{Z/X, *})\n$$\nis the one induced by the ring map $f^\\sharp : R' \\to R$ which\nhas the property $f^\\sharp(I') \\subset I$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The normal cone of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0634","source_file":"divisors.tex","source_line":4381,"source_end_line":4412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4381-L4412","statement_sha256":"cac44ea91604317f473b025d9d052e56c5bf71e1ab76ae562bffd753f055148e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6114,"rank":6114,"depth":8,"x":2528.354,"y":735.176,"cluster":"divisors-intersection-theory"},{"id":"stacks:0635","tag":"0635","title":"The normal cone of an immersion · Lemma 0635","summary":"Let xymatrix Z ar[r]_i ar[d]_f & X ar[d]^g Z' ar[r]^i' & X' be a fibre product diagram in the category of schemes with i, i' immersions. Then the canonical map f^*C_Z'/X', * → C_Z/X, * of Lemma [Tag 0634] is surjective. If g is flat, then it is an isomorphism.","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\\nZ' \\ar[r]^{i'} & X'\n}\n$$\nbe a fibre product diagram in the category of schemes with\n$i$, $i'$ immersions. Then the canonical map\n$f^*\\mathcal{C}_{Z'/X', *} \\to \\mathcal{C}_{Z/X, *}$ of\nLemma \\ref{lemma-conormal-algebra-functorial}\nis surjective. If $g$ is flat, then it is an isomorphism.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The normal cone of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0635","source_file":"divisors.tex","source_line":4453,"source_end_line":4467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4453-L4467","statement_sha256":"158cceea1764ad7143b69706fcff1cdd50f863480581cf51ef4faf3ab3917cfc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6115,"rank":6115,"depth":9,"x":2399.704,"y":666.707,"cluster":"divisors-intersection-theory"},{"id":"stacks:0636","tag":"0636","title":"The normal cone of an immersion · Definition 0636","summary":"Let i : Z → X be an immersion of schemes. The normal cone C_ZX of Z in X is C_ZX = underlineSpec_Z(C_Z/X, *) see Constructions, Definitions [Tag 062Q] and [Tag 062R]. The normal bundle of Z in X is the vector bundle N_ZX = underlineSpec_Z(Sym(C_Z/X)) see Constructions, Definitions [Tag 01M2] and [Tag 062M].","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes.\nThe {\\it normal cone $C_ZX$} of $Z$ in $X$ is\n$$\nC_ZX = \\underline{\\Spec}_Z(\\mathcal{C}_{Z/X, *})\n$$\nsee\nConstructions,\nDefinitions \\ref{constructions-definition-cone} and\n\\ref{constructions-definition-abstract-cone}. The {\\it normal bundle}\nof $Z$ in $X$ is the vector bundle\n$$\nN_ZX = \\underline{\\Spec}_Z(\\text{Sym}(\\mathcal{C}_{Z/X}))\n$$\nsee\nConstructions,\nDefinitions \\ref{constructions-definition-vector-bundle} and\n\\ref{constructions-definition-abstract-vector-bundle}.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The normal cone of an immersion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0636","source_file":"divisors.tex","source_line":4476,"source_end_line":4495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4476-L4495","statement_sha256":"922c78fefb8d609195b81ae6b051802b629066ef327371956f0a0fd57cbd213c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6116,"rank":6116,"depth":1,"x":2550.132,"y":644.121,"cluster":"divisors-intersection-theory"},{"id":"stacks:063C","tag":"063C","title":"Regular ideal sheaves · Lemma 063C","summary":"Let X be a ringed space. Let f_1, …, f_r ∈ Γ(X, O_X). We have the following implications f_1, …, f_r is a regular sequence ⇒ f_1, …, f_r is a Koszul-regular sequence ⇒ f_1, …, f_r is an H_1-regular sequence ⇒ f_1, …, f_r is a quasi-regular sequence.","statement_latex":"Let $X$ be a ringed space.\nLet $f_1, \\ldots, f_r \\in \\Gamma(X, \\mathcal{O}_X)$.\nWe have the following implications\n$f_1, \\ldots, f_r$ is a regular sequence $\\Rightarrow$\n$f_1, \\ldots, f_r$ is a Koszul-regular sequence $\\Rightarrow$\n$f_1, \\ldots, f_r$ is an $H_1$-regular sequence $\\Rightarrow$\n$f_1, \\ldots, f_r$ is a quasi-regular sequence.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular ideal sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063C","source_file":"divisors.tex","source_line":4582,"source_end_line":4591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4582-L4591","statement_sha256":"2355453ce6f8124229181053da48b7543f912ba41c79a4013756e34a950901b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6117,"rank":6117,"depth":5,"x":2457.063,"y":746.471,"cluster":"divisors-intersection-theory"},{"id":"stacks:063D","tag":"063D","title":"Regular ideal sheaves · Definition 063D","summary":"The concept of a Koszul-regular ideal sheaf was introduced in [SGA6] where it was called a regular ideal sheaf. Let X be a ringed space. Let J ⊂ O_X be a sheaf of ideals. • We say J is regular if for every x ∈ Supp(O_X/J) there exists an open neighbourhood x ∈ U ⊂ X and a regular sequence f_1, …, f_r ∈ O_X(U) such that J|_U is generated by f_1, …, f_r. • We say J is Koszul-regular if for every x ∈ Supp(O_X/J) there exists an open neighbourhood x ∈ U ⊂ X and a…","statement_latex":"\\begin{reference}\nThe concept of a Koszul-regular ideal sheaf was introduced in\n\\cite[Expose VII, Definition 1.4]{SGA6} where it was called a\nregular ideal sheaf.\n\\end{reference}\nLet $X$ be a ringed space. Let $\\mathcal{J} \\subset \\mathcal{O}_X$\nbe a sheaf of ideals.\n\\begin{enumerate}\n\\item We say $\\mathcal{J}$ is {\\it regular} if for every\n$x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open\nneighbourhood $x \\in U \\subset X$ and a regular sequence\n$f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$\nis generated by $f_1, \\ldots, f_r$.\n\\item We say $\\mathcal{J}$ is {\\it Koszul-regular} if for every\n$x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open\nneighbourhood $x \\in U \\subset X$ and a Koszul-regular sequence\n$f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$\nis generated by $f_1, \\ldots, f_r$.\n\\item We say $\\mathcal{J}$ is {\\it $H_1$-regular} if for every\n$x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open\nneighbourhood $x \\in U \\subset X$ and a $H_1$-regular sequence\n$f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$\nis generated by $f_1, \\ldots, f_r$.\n\\item We say $\\mathcal{J}$ is {\\it quasi-regular} if for every\n$x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists an open\nneighbourhood $x \\in U \\subset X$ and a quasi-regular sequence\n$f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$ such that $\\mathcal{J}|_U$\nis generated by $f_1, \\ldots, f_r$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular ideal sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063D","source_file":"divisors.tex","source_line":4611,"source_end_line":4642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4611-L4642","statement_sha256":"ae946e2a0da7cd8e22f41ce9a49e1ed784e22e647156e56f72474103f1b53cd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6118,"rank":6118,"depth":0,"x":2443.339,"y":617.766,"cluster":"divisors-intersection-theory"},{"id":"stacks:063E","tag":"063E","title":"Regular ideal sheaves · Lemma 063E","summary":"Let X be a ringed space. Let J be a sheaf of ideals. We have the following implications: J is regular ⇒ J is Koszul-regular ⇒ J is H_1-regular ⇒ J is quasi-regular.","statement_latex":"Let $X$ be a ringed space. Let $\\mathcal{J}$ be a sheaf of ideals.\nWe have the following implications:\n$\\mathcal{J}$ is regular $\\Rightarrow$\n$\\mathcal{J}$ is Koszul-regular $\\Rightarrow$\n$\\mathcal{J}$ is $H_1$-regular $\\Rightarrow$\n$\\mathcal{J}$ is quasi-regular.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular ideal sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063E","source_file":"divisors.tex","source_line":4649,"source_end_line":4657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4649-L4657","statement_sha256":"0a5d98c20b295ef2c3bd0c3e8316b1dc05931dd3015037b0de98f1545adebe67","origin":"The Stacks Project","memory_eligible":false,"source_rank":6119,"rank":6119,"depth":6,"x":2557.333,"y":705.17,"cluster":"divisors-intersection-theory"},{"id":"stacks:063H","tag":"063H","title":"Regular ideal sheaves · Lemma 063H","summary":"Let X be a locally ringed space. Let J ⊂ O_X be a sheaf of ideals. Then J is quasi-regular if and only if the following conditions are satisfied: • J is an O_X-module of finite type, • J/J^2 is a finite locally free O_X/J-module, and • the canonical maps Sym^n_O_X/J(J/J^2) → J^n/J^n + 1 are isomorphisms for all n ≥ 0.","statement_latex":"Let $X$ be a locally ringed space. Let $\\mathcal{J} \\subset \\mathcal{O}_X$\nbe a sheaf of ideals. Then $\\mathcal{J}$ is quasi-regular if and\nonly if the following conditions are satisfied:\n\\begin{enumerate}\n\\item $\\mathcal{J}$ is an $\\mathcal{O}_X$-module of finite type,\n\\item $\\mathcal{J}/\\mathcal{J}^2$ is a finite locally free\n$\\mathcal{O}_X/\\mathcal{J}$-module, and\n\\item the canonical maps\n$$\n\\text{Sym}^n_{\\mathcal{O}_X/\\mathcal{J}}(\\mathcal{J}/\\mathcal{J}^2)\n\\longrightarrow\n\\mathcal{J}^n/\\mathcal{J}^{n + 1}\n$$\nare isomorphisms for all $n \\geq 0$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular ideal sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063H","source_file":"divisors.tex","source_line":4664,"source_end_line":4681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4664-L4681","statement_sha256":"fd0fe45951b34b4fb965a5ca9367986df7ad7d70250445df2e6eb0cb0547ae26","origin":"The Stacks Project","memory_eligible":false,"source_rank":6120,"rank":6120,"depth":3,"x":2402.484,"y":705.403,"cluster":"divisors-intersection-theory"},{"id":"stacks:067N","tag":"067N","title":"Regular ideal sheaves · Lemma 067N","summary":"Let (X, O_X) be a locally ringed space. Let J ⊂ O_X be a sheaf of ideals. Let x ∈ X and f_1, …, f_r ∈ J_x whose images give a basis for the kappa(x)-vector space J_x/ m_xJ_x. • If J is quasi-regular, then there exists an open neighbourhood such that f_1, …, f_r ∈ O_X(U) form a quasi-regular sequence generating J|_U. • If J is H_1-regular, then there exists an open neighbourhood such that f_1, …, f_r ∈ O_X(U) form an H_1-regular sequence generating J|_U. • If J is…","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a locally ringed space.\nLet $\\mathcal{J} \\subset \\mathcal{O}_X$ be a sheaf of ideals.\nLet $x \\in X$ and $f_1, \\ldots, f_r \\in \\mathcal{J}_x$ whose images\ngive a basis for the $\\kappa(x)$-vector space\n$\\mathcal{J}_x/\\mathfrak m_x\\mathcal{J}_x$.\n\\begin{enumerate}\n\\item If $\\mathcal{J}$ is quasi-regular, then there exists an open\nneighbourhood such that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$\nform a quasi-regular sequence generating $\\mathcal{J}|_U$.\n\\item If $\\mathcal{J}$ is $H_1$-regular, then there exists an open\nneighbourhood such that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$\nform an $H_1$-regular sequence generating $\\mathcal{J}|_U$.\n\\item If $\\mathcal{J}$ is Koszul-regular, then there exists an open\nneighbourhood such that $f_1, \\ldots, f_r \\in \\mathcal{O}_X(U)$\nform an Koszul-regular sequence generating $\\mathcal{J}|_U$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular ideal sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067N","source_file":"divisors.tex","source_line":4716,"source_end_line":4734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4716-L4734","statement_sha256":"e341a494f791906ed0dc11eed235c5b23bd65554a2dd1f0c8616449e29b6ac96","origin":"The Stacks Project","memory_eligible":false,"source_rank":6121,"rank":6121,"depth":4,"x":2516.85,"y":617.081,"cluster":"divisors-intersection-theory"},{"id":"stacks:063F","tag":"063F","title":"Regular ideal sheaves · Lemma 063F","summary":"Any regular, Koszul-regular, H_1-regular, or quasi-regular sheaf of ideals on a scheme is a finite type quasi-coherent sheaf of ideals.","statement_latex":"Any regular, Koszul-regular, $H_1$-regular, or quasi-regular sheaf\nof ideals on a scheme is a finite type quasi-coherent sheaf of ideals.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular ideal sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063F","source_file":"divisors.tex","source_line":4769,"source_end_line":4773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4769-L4773","statement_sha256":"3698b4393a94b2ac76f50166bbe9b8c265548c4503d8355574c2f66b9cc30828","origin":"The Stacks Project","memory_eligible":false,"source_rank":6122,"rank":6122,"depth":12,"x":2503.5,"y":747.521,"cluster":"divisors-intersection-theory"},{"id":"stacks:063G","tag":"063G","title":"Regular ideal sheaves · Lemma 063G","summary":"Let X be a scheme. Let J be a sheaf of ideals. Then J is regular (resp. Koszul-regular, H_1-regular, quasi-regular) if and only if for every x ∈ Supp(O_X/J) there exists an affine open neighbourhood x ∈ U ⊂ X, U = Spec(A) such that J|_U = widetildeI and such that I is generated by a regular (resp. Koszul-regular, H_1-regular, quasi-regular) sequence f_1, …, f_r ∈ A.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{J}$ be a sheaf of ideals.\nThen $\\mathcal{J}$ is regular\n(resp.\\ Koszul-regular, $H_1$-regular, quasi-regular) if and only if\nfor every $x \\in \\text{Supp}(\\mathcal{O}_X/\\mathcal{J})$ there exists\nan affine open neighbourhood $x \\in U \\subset X$, $U = \\Spec(A)$\nsuch that $\\mathcal{J}|_U = \\widetilde{I}$ and such that $I$\nis generated by a regular (resp.\\ Koszul-regular, $H_1$-regular,\nquasi-regular) sequence $f_1, \\ldots, f_r \\in A$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular ideal sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063G","source_file":"divisors.tex","source_line":4782,"source_end_line":4792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4782-L4792","statement_sha256":"f4a7a09e2ce943126438861b4e2515deacf081004e26c54e066d0b8d0e8132c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6123,"rank":6123,"depth":13,"x":2408.144,"y":643.429,"cluster":"divisors-intersection-theory"},{"id":"stacks:063I","tag":"063I","title":"Regular ideal sheaves · Lemma 063I","summary":"Let X be a locally Noetherian scheme. Let J ⊂ O_X be a quasi-coherent sheaf of ideals. Let x be a point of the support of O_X/J. The following are equivalent • J_x is generated by a regular sequence in O_X, x, • J_x is generated by a Koszul-regular sequence in O_X, x, • J_x is generated by an H_1-regular sequence in O_X, x, • J_x is generated by a quasi-regular sequence in O_X, x, • there exists an affine neighbourhood U = Spec(A) of x such that J|_U = widetildeI and I is…","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{J} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. Let $x$ be a point of the support of\n$\\mathcal{O}_X/\\mathcal{J}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{J}_x$ is generated by a regular sequence in\n$\\mathcal{O}_{X, x}$,\n\\item $\\mathcal{J}_x$ is generated by a Koszul-regular sequence in\n$\\mathcal{O}_{X, x}$,\n\\item $\\mathcal{J}_x$ is generated by an $H_1$-regular sequence in\n$\\mathcal{O}_{X, x}$,\n\\item $\\mathcal{J}_x$ is generated by a quasi-regular sequence in\n$\\mathcal{O}_{X, x}$,\n\\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such\nthat $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by a\nregular sequence in $A$, and\n\\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such\nthat $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by a\nKoszul-regular sequence in $A$, and\n\\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such\nthat $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by an\n$H_1$-regular sequence in $A$, and\n\\item there exists an affine neighbourhood $U = \\Spec(A)$ of $x$ such\nthat $\\mathcal{J}|_U = \\widetilde{I}$ and $I$ is generated by a\nquasi-regular sequence in $A$,\n\\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$\nis regular, and\n\\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$\nis Koszul-regular, and\n\\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$\nis $H_1$-regular, and\n\\item there exists a neighbourhood $U$ of $x$ such that $\\mathcal{J}|_U$\nis quasi-regular.\n\\end{enumerate}\nIn particular, on a locally Noetherian scheme the notions of\nregular, Koszul-regular, $H_1$-regular, or quasi-regular ideal sheaf all agree.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular ideal sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063I","source_file":"divisors.tex","source_line":4822,"source_end_line":4859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4822-L4859","statement_sha256":"bfe2a7daf48675386d9e3997d0c39648d4f1e615b857f45dd57e000c306794a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6124,"rank":6124,"depth":14,"x":2562.655,"y":666.15,"cluster":"divisors-intersection-theory"},{"id":"stacks:063J","tag":"063J","title":"Regular immersions · Definition 063J","summary":"The concept of a Koszul-regular immersion was introduced in [SGA6] where it was called a regular immersion. Let i : Z → X be an immersion of schemes. Choose an open subscheme U ⊂ X such that i identifies Z with a closed subscheme of U and denote I ⊂ O_U the corresponding quasi-coherent sheaf of ideals. • We say i is a regular immersion if I is regular. • We say i is a Koszul-regular immersion if I is Koszul-regular. • We say i is a H_1-regular immersion if I is…","statement_latex":"\\begin{reference}\nThe concept of a Koszul-regular immersion was introduced in\n\\cite[Expose VII, Definition 1.4]{SGA6} where it was called a\nregular immersion.\n\\end{reference}\nLet $i : Z \\to X$ be an immersion of schemes. Choose an open subscheme\n$U \\subset X$ such that $i$ identifies $Z$ with a closed\nsubscheme of $U$ and denote $\\mathcal{I} \\subset \\mathcal{O}_U$\nthe corresponding quasi-coherent sheaf of ideals.\n\\begin{enumerate}\n\\item We say $i$ is a {\\it regular immersion} if\n$\\mathcal{I}$ is regular.\n\\item We say $i$ is a {\\it Koszul-regular immersion} if\n$\\mathcal{I}$ is Koszul-regular.\n\\item We say $i$ is a {\\it $H_1$-regular immersion} if\n$\\mathcal{I}$ is $H_1$-regular.\n\\item We say $i$ is a {\\it quasi-regular immersion} if\n$\\mathcal{I}$ is quasi-regular.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063J","source_file":"divisors.tex","source_line":4909,"source_end_line":4930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4909-L4930","statement_sha256":"0f9da474fc0e275d3760ed8fe0e84a5ece02bcfdf65d5e2149bf228bfae00e17","origin":"The Stacks Project","memory_eligible":false,"source_rank":6125,"rank":6125,"depth":0,"x":2430.032,"y":737.293,"cluster":"divisors-intersection-theory"},{"id":"stacks:063K","tag":"063K","title":"Regular immersions · Lemma 063K","summary":"Let i : Z → X be an immersion of schemes. We have the following implications: i is regular ⇒ i is Koszul-regular ⇒ i is H_1-regular ⇒ i is quasi-regular.","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes.\nWe have the following implications:\n$i$ is regular $\\Rightarrow$\n$i$ is Koszul-regular $\\Rightarrow$\n$i$ is $H_1$-regular $\\Rightarrow$\n$i$ is quasi-regular.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063K","source_file":"divisors.tex","source_line":4941,"source_end_line":4949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4941-L4949","statement_sha256":"900484601f088f65ce44f66b9acab695cf78e5adf37d83ce7d9e7c60dc6759dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6126,"rank":6126,"depth":7,"x":2470.743,"y":609.179,"cluster":"divisors-intersection-theory"},{"id":"stacks:063L","tag":"063L","title":"Regular immersions · Lemma 063L","summary":"Let i : Z → X be an immersion of schemes. Assume X is locally Noetherian. Then i is regular ⇔ i is Koszul-regular ⇔ i is H_1-regular ⇔ i is quasi-regular.","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes.\nAssume $X$ is locally Noetherian. Then\n$i$ is regular $\\Leftrightarrow$\n$i$ is Koszul-regular $\\Leftrightarrow$\n$i$ is $H_1$-regular $\\Leftrightarrow$\n$i$ is quasi-regular.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063L","source_file":"divisors.tex","source_line":4956,"source_end_line":4964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4956-L4964","statement_sha256":"7ee5de7f02b2a0f53b8e88d60daa7feafa86d3f2f7fc13ba4ae0a8f17861fdbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6127,"rank":6127,"depth":15,"x":2543.976,"y":727.116,"cluster":"divisors-intersection-theory"},{"id":"stacks:067P","tag":"067P","title":"Regular immersions · Lemma 067P","summary":"Let i : Z → X be a regular (resp. Koszul-regular, H_1-regular, quasi-regular) immersion. Let X' → X be a flat morphism. Then the base change i' : Z ×_X X' → X' is a regular (resp. Koszul-regular, H_1-regular, quasi-regular) immersion.","statement_latex":"Let $i : Z \\to X$ be a regular (resp.\\ Koszul-regular,\n$H_1$-regular, quasi-regular) immersion. Let $X' \\to X$ be a flat\nmorphism. Then the base change $i' : Z \\times_X X' \\to X'$\nis a regular (resp.\\ Koszul-regular,\n$H_1$-regular, quasi-regular) immersion.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067P","source_file":"divisors.tex","source_line":4973,"source_end_line":4980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4973-L4980","statement_sha256":"6a65253784a7627657f62dc2b1017d2eab1e8589c2d3e2479cb2a249b940886c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6128,"rank":6128,"depth":14,"x":2394.673,"y":681.564,"cluster":"divisors-intersection-theory"},{"id":"stacks:063M","tag":"063M","title":"Regular immersions · Lemma 063M","summary":"Let i : Z → X be an immersion of schemes. Then i is a quasi-regular immersion if and only if the following conditions are satisfied • i is locally of finite presentation, • the conormal sheaf C_Z/X is finite locally free, and • the map ([Tag 0632]) is an isomorphism.","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes. Then $i$ is a quasi-regular\nimmersion if and only if the following conditions are satisfied\n\\begin{enumerate}\n\\item $i$ is locally of finite presentation,\n\\item the conormal sheaf $\\mathcal{C}_{Z/X}$ is finite locally free, and\n\\item the map (\\ref{equation-conormal-algebra-quotient}) is an isomorphism.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063M","source_file":"divisors.tex","source_line":4993,"source_end_line":5002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L4993-L5002","statement_sha256":"ab80197887ab97413344f761fb052fc056c999e9550efb621590df2d87f4191b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6129,"rank":6129,"depth":5,"x":2541.85,"y":630.278,"cluster":"divisors-intersection-theory"},{"id":"stacks:063N","tag":"063N","title":"Regular immersions · Lemma 063N","summary":"Let Z → Y → X be immersions of schemes. Assume that Z → Y is H_1-regular. Then the canonical sequence of Morphisms, Lemma [Tag 062S] 0 → i^*C_Y/X → C_Z/X → C_Z/Y → 0 is exact and locally split.","statement_latex":"Let $Z \\to Y \\to X$ be immersions of schemes. Assume that\n$Z \\to Y$ is $H_1$-regular. Then the canonical sequence of\nMorphisms, Lemma \\ref{morphisms-lemma-transitivity-conormal}\n$$\n0 \\to i^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nis exact and locally split.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063N","source_file":"divisors.tex","source_line":5017,"source_end_line":5028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5017-L5028","statement_sha256":"6c603011208abadceb0af00103ee9400d63d6bc34298e883aa873444940f70bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6130,"rank":6130,"depth":10,"x":2474.36,"y":751.978,"cluster":"divisors-intersection-theory"},{"id":"stacks:067Q","tag":"067Q","title":"Regular immersions · Lemma 067Q","summary":"Let i : Z → Y and j : Y → X be immersions of schemes. • If i and j are regular immersions, so is j ∘ i. • If i and j are Koszul-regular immersions, so is j ∘ i. • If i and j are H_1-regular immersions, so is j ∘ i. • If i is an H_1-regular immersion and j is a quasi-regular immersion, then j ∘ i is a quasi-regular immersion.","statement_latex":"Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes.\n\\begin{enumerate}\n\\item If $i$ and $j$ are regular immersions, so is $j \\circ i$.\n\\item If $i$ and $j$ are Koszul-regular immersions, so is $j \\circ i$.\n\\item If $i$ and $j$ are $H_1$-regular immersions, so is $j \\circ i$.\n\\item If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular\nimmersion, then $j \\circ i$ is a quasi-regular immersion.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067Q","source_file":"divisors.tex","source_line":5052,"source_end_line":5062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5052-L5062","statement_sha256":"449226ccb1a0be519cc056b080f2cf309038b34a8a14721a6e7b78a5bd3bcf5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6131,"rank":6131,"depth":7,"x":2426.113,"y":623.553,"cluster":"divisors-intersection-theory"},{"id":"stacks:068Z","tag":"068Z","title":"Regular immersions · Lemma 068Z","summary":"Let i : Z → Y and j : Y → X be immersions of schemes. Assume j is locally of finite presentation and that the sequence 0 → i^*C_Y/X → C_Z/X → C_Z/Y → 0 of Morphisms, Lemma [Tag 062S] is exact and locally split. • If j ∘ i is a quasi-regular immersion, so is i. • If j ∘ i is a H_1-regular immersion, so is i. • If both j and j ∘ i are Koszul-regular immersions, so is i.","statement_latex":"Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes. Assume\n$j$ is locally of finite presentation and that the sequence\n$$\n0 \\to i^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nof\nMorphisms, Lemma \\ref{morphisms-lemma-transitivity-conormal}\nis exact and locally split.\n\\begin{enumerate}\n\\item If $j \\circ i$ is a quasi-regular immersion, so is $i$.\n\\item If $j \\circ i$ is a $H_1$-regular immersion, so is $i$.\n\\item If both $j$ and $j \\circ i$ are Koszul-regular immersions, so is $i$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068Z","source_file":"divisors.tex","source_line":5075,"source_end_line":5092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5075-L5092","statement_sha256":"3f633e68fa553ee3784d62658018b8fa1ffa68293874a75d879ce75a3761a99e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6132,"rank":6132,"depth":10,"x":2565.385,"y":691.079,"cluster":"divisors-intersection-theory"},{"id":"stacks:0690","tag":"0690","title":"Regular immersions · Lemma 0690","summary":"Let i : Z → Y and j : Y → X be immersions of schemes. Pick z ∈ Z and denote y ∈ Y, x ∈ X the corresponding points. Assume X is locally Noetherian. The following are equivalent • i is a regular immersion in a neighbourhood of z and j is a regular immersion in a neighbourhood of y, • i and j ∘ i are regular immersions in a neighbourhood of z, • j ∘ i is a regular immersion in a neighbourhood of z and the conormal sequence 0 → i^*C_Y/X → C_Z/X → C_Z/Y → 0 is split exact in a…","statement_latex":"Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes.\nPick $z \\in Z$ and denote $y \\in Y$, $x \\in X$ the corresponding points.\nAssume $X$ is locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $i$ is a regular immersion in a neighbourhood of $z$ and $j$\nis a regular immersion in a neighbourhood of $y$,\n\\item $i$ and $j \\circ i$ are regular immersions in a neighbourhood of $z$,\n\\item $j \\circ i$ is a regular immersion in a neighbourhood of $z$ and the\nconormal sequence\n$$\n0 \\to i^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nis split exact in a neighbourhood of $z$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0690","source_file":"divisors.tex","source_line":5144,"source_end_line":5163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5144-L5163","statement_sha256":"eb441225ffb539a40e03901fc06c23f36a37765afc5c092ad383ea1c1528aabd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6133,"rank":6133,"depth":15,"x":2407.914,"y":720.401,"cluster":"divisors-intersection-theory"},{"id":"stacks:0692","tag":"0692","title":"Regular immersions · Lemma 0692","summary":"Let i : Z → X be a Koszul regular closed immersion. Then there exists a surjective smooth morphism X' → X such that the base change i' : Z ×_X X' → X' of i is a regular immersion.","statement_latex":"Let $i : Z \\to X$ be a Koszul regular closed immersion.\nThen there exists a surjective smooth morphism $X' \\to X$ such\nthat the base change $i' : Z \\times_X X' \\to X'$ of $i$ is\na regular immersion.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0692","source_file":"divisors.tex","source_line":5209,"source_end_line":5215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5209-L5215","statement_sha256":"07d396280b7c6fd26ad76e5cefd758d0a089bb3a84096cc36769f3ac8fe3a4ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":6134,"rank":6134,"depth":14,"x":2500.728,"y":609.095,"cluster":"divisors-intersection-theory"},{"id":"stacks:0E9J","tag":"0E9J","title":"Regular immersions · Lemma 0E9J","summary":"Let i : Z → X be an immersion. If Z and X are regular schemes, then i is a regular immersion.","statement_latex":"Let $i : Z \\to X$ be an immersion. If $Z$ and $X$ are\nregular schemes, then $i$ is a regular immersion.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9J","source_file":"divisors.tex","source_line":5227,"source_end_line":5231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5227-L5231","statement_sha256":"cf8e42bec51d6f447059f87b16109bc5b6f7ce8c172ee3cac478a26628693f16","origin":"The Stacks Project","memory_eligible":false,"source_rank":6135,"rank":6135,"depth":15,"x":2521.859,"y":744.235,"cluster":"divisors-intersection-theory"},{"id":"stacks:063R","tag":"063R","title":"Relative regular immersions · Lemma 063R","summary":"Let f : X → S be a morphism of schemes. Let i : Z ⊂ X be an immersion. Assume • i is an H_1-regular (resp. quasi-regular) immersion, and • Z → S is a flat morphism. Then for every morphism of schemes g : S' → S the base change Z' = S' ×_S Z → X' = S' ×_S X is an H_1-regular (resp. quasi-regular) immersion.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $i : Z \\subset X$ be an immersion.\nAssume\n\\begin{enumerate}\n\\item $i$ is an $H_1$-regular (resp.\\ quasi-regular) immersion, and\n\\item $Z \\to S$ is a flat morphism.\n\\end{enumerate}\nThen for every morphism of schemes $g : S' \\to S$ the base change\n$Z' = S' \\times_S Z \\to X' = S' \\times_S X$\nis an $H_1$-regular (resp.\\ quasi-regular) immersion.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063R","source_file":"divisors.tex","source_line":5255,"source_end_line":5267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5255-L5267","statement_sha256":"5f698c8988808fe928f8aa3be2c74e73d9a481ab448edf0f820eb5cf8b86ea35","origin":"The Stacks Project","memory_eligible":false,"source_rank":6136,"rank":6136,"depth":14,"x":2397.235,"y":656.316,"cluster":"divisors-intersection-theory"},{"id":"stacks:063S","tag":"063S","title":"Relative regular immersions · Definition 063S","summary":"Let f : X → S be a morphism of schemes. Let i : Z → X be an immersion. • We say i is a relative quasi-regular immersion if Z → S is flat and i is a quasi-regular immersion. • We say i is a relative H_1-regular immersion if Z → S is flat and i is an H_1-regular immersion.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $i : Z \\to X$ be an immersion.\n\\begin{enumerate}\n\\item We say $i$ is a {\\it relative quasi-regular immersion}\nif $Z \\to S$ is flat and $i$ is a quasi-regular immersion.\n\\item We say $i$ is a {\\it relative $H_1$-regular immersion}\nif $Z \\to S$ is flat and $i$ is an $H_1$-regular immersion.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063S","source_file":"divisors.tex","source_line":5279,"source_end_line":5289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5279-L5289","statement_sha256":"1a6c576487c3f752f4dec030cfbfea679d35537f9ef84d821ceffdbcf96f20c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6137,"rank":6137,"depth":0,"x":2560.31,"y":650.415,"cluster":"divisors-intersection-theory"},{"id":"stacks:063T","tag":"063T","title":"Relative regular immersions · Lemma 063T","summary":"Let f : X → S be a morphism of schemes. Let Z → X be a relative quasi-regular immersion. If x ∈ Z and O_X, x is Noetherian, then f is flat at x.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $Z \\to X$ be a relative quasi-regular immersion.\nIf $x \\in Z$ and $\\mathcal{O}_{X, x}$ is Noetherian, then $f$ is flat at $x$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063T","source_file":"divisors.tex","source_line":5308,"source_end_line":5313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5308-L5313","statement_sha256":"a87f5cbd620ed14ef28ece0adcfdfc5b48f647d8de52a2ffad444cc488da62b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6138,"rank":6138,"depth":5,"x":2444.469,"y":747.581,"cluster":"divisors-intersection-theory"},{"id":"stacks:063U","tag":"063U","title":"Relative regular immersions · Lemma 063U","summary":"Let X → S be a morphism of schemes. Let Z → X be an immersion. Assume • X → S is flat and locally of finite presentation, • Z → X is a relative quasi-regular immersion. Then Z → X is a regular immersion and the same remains true after any base change.","statement_latex":"Let $X \\to S$ be a morphism of schemes.\nLet $Z \\to X$ be an immersion.\nAssume\n\\begin{enumerate}\n\\item $X \\to S$ is flat and locally of finite presentation,\n\\item $Z \\to X$ is a relative quasi-regular immersion.\n\\end{enumerate}\nThen $Z \\to X$ is a regular immersion and\nthe same remains true after any base change.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063U","source_file":"divisors.tex","source_line":5331,"source_end_line":5342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5331-L5342","statement_sha256":"c1ca319b0fcf13095667d419ad016a58bb7863b417e4adfed4a5e3fdfc00d513","origin":"The Stacks Project","memory_eligible":false,"source_rank":6139,"rank":6139,"depth":36,"x":2451.772,"y":609.804,"cluster":"divisors-intersection-theory"},{"id":"stacks:063V","tag":"063V","title":"Relative regular immersions · Lemma 063V","summary":"Let X → S be a morphism of schemes. Let Z → X be a relative H_1-regular immersion. Assume X → S is locally of finite presentation. Then • there exists an open subscheme U ⊂ X such that Z ⊂ U and such that U → S is flat, and • Z → X is a regular immersion and the same remains true after any base change.","statement_latex":"Let $X \\to S$ be a morphism of schemes.\nLet $Z \\to X$ be a relative $H_1$-regular immersion.\nAssume $X \\to S$ is locally of finite presentation. Then\n\\begin{enumerate}\n\\item there exists an open subscheme $U \\subset X$ such that\n$Z \\subset U$ and such that $U \\to S$ is flat, and\n\\item $Z \\to X$ is a regular immersion and the same remains\ntrue after any base change.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063V","source_file":"divisors.tex","source_line":5395,"source_end_line":5406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5395-L5406","statement_sha256":"525dd6bc332615ad840c6cf7441be6094016da5a51c01ace387ec439ccffffe8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6140,"rank":6140,"depth":37,"x":2557.487,"y":715.847,"cluster":"divisors-intersection-theory"},{"id":"stacks:063W","tag":"063W","title":"Relative regular immersions · Lemma 063W","summary":"Let φ : X → S be a flat morphism which is locally of finite presentation. Let T ⊂ X be a closed subscheme. Let x ∈ T with image s ∈ S. • If T_s ⊂ X_s is a quasi-regular immersion in a neighbourhood of x, then there exists an open U ⊂ X and a relative quasi-regular immersion Z ⊂ U such that Z_s = T_s ∩ U_s and T ∩ U ⊂ Z. • If T_s ⊂ X_s is a quasi-regular immersion in a neighbourhood of x, the morphism T → X is of finite presentation, and T → S is flat at x, then we can…","statement_latex":"Let $\\varphi : X \\to S$ be a flat morphism which is locally of finite\npresentation. Let $T \\subset X$ be a closed subscheme.\nLet $x \\in T$ with image $s \\in S$.\n\\begin{enumerate}\n\\item If $T_s \\subset X_s$ is a quasi-regular immersion\nin a neighbourhood of $x$, then there exists an open\n$U \\subset X$ and a relative quasi-regular immersion\n$Z \\subset U$ such that $Z_s = T_s \\cap U_s$ and $T \\cap U \\subset Z$.\n\\item If $T_s \\subset X_s$ is a quasi-regular immersion\nin a neighbourhood of $x$, the morphism $T \\to X$ is of finite\npresentation, and $T \\to S$ is flat at $x$, then we can choose $U$ and\n$Z$ as in (1) such that $T \\cap U = Z$.\n\\item If $T_s \\subset X_s$ is a quasi-regular immersion in a neighbourhood\nof $x$, and $T$ is cut out by $c$ equations in a neighbourhood of $x$,\nwhere $c = \\dim_x(X_s) - \\dim_x(T_s)$, then we can choose $U$ and $Z$ as in (1)\nsuch that $T \\cap U = Z$.\n\\end{enumerate}\nIn each case $Z \\to U$ is a regular immersion by\nLemma \\ref{lemma-relative-regular-immersion-flat-in-neighbourhood}.\nIn particular, if $T \\to S$ is locally of finite presentation and flat and\nall fibres $T_s \\subset X_s$ are quasi-regular immersions, then\n$T \\to X$ is a relative quasi-regular immersion.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063W","source_file":"divisors.tex","source_line":5491,"source_end_line":5515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5491-L5515","statement_sha256":"3e992114916b44f3f0a0e5b62d43e8fdd677fb64dafbf984aaae9ca9f56230d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6141,"rank":6141,"depth":37,"x":2393.79,"y":697.585,"cluster":"divisors-intersection-theory"},{"id":"stacks:067R","tag":"067R","title":"Relative regular immersions · Lemma 067R","summary":"Let f : X → S be a smooth morphism of schemes. Let σ : S → X be a section of f. Then σ is a regular immersion.","statement_latex":"Let $f : X \\to S$ be a smooth morphism of schemes.\nLet $\\sigma : S \\to X$ be a section of $f$.\nThen $\\sigma$ is a regular immersion.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067R","source_file":"divisors.tex","source_line":5600,"source_end_line":5605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5600-L5605","statement_sha256":"08b1aafeb84d359d859c2cd2cb35d0a6309c4842f288b7eca7b43d3963393ac6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6142,"rank":6142,"depth":38,"x":2529.569,"y":617.939,"cluster":"divisors-intersection-theory"},{"id":"stacks:067S","tag":"067S","title":"Relative regular immersions · Lemma 067S","summary":"Let xymatrix Y ar[rd]_j ar[rr]_i & & X ar[ld] & S be a commutative diagram of morphisms of schemes. Assume X → S smooth, and i, j immersions. If j is a regular (resp. Koszul-regular, H_1-regular, quasi-regular) immersion, then so is i.","statement_latex":"Let\n$$\n\\xymatrix{\nY \\ar[rd]_j \\ar[rr]_i & & X \\ar[ld] \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes.\nAssume $X \\to S$ smooth, and $i$, $j$ immersions.\nIf $j$ is a regular (resp.\\ Koszul-regular, $H_1$-regular, quasi-regular)\nimmersion, then so is $i$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067S","source_file":"divisors.tex","source_line":5642,"source_end_line":5655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5642-L5655","statement_sha256":"2c9bdb0ec0ce27e122c8e5eb75e7861b931ad5a811672d2ded0d847e3f964f3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6143,"rank":6143,"depth":39,"x":2493.394,"y":754.098,"cluster":"divisors-intersection-theory"},{"id":"stacks:067T","tag":"067T","title":"Relative regular immersions · Lemma 067T","summary":"Let xymatrix Y ar[rd] ar[rr]_i & & X ar[ld] & S be a commutative diagram of morphisms of schemes. Assume that Y → S is syntomic, X → S smooth, and i an immersion. Then i is a regular immersion.","statement_latex":"Let\n$$\n\\xymatrix{\nY \\ar[rd] \\ar[rr]_i & & X \\ar[ld] \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes.\nAssume that $Y \\to S$ is syntomic, $X \\to S$ smooth, and\n$i$ an immersion. Then $i$ is a regular immersion.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067T","source_file":"divisors.tex","source_line":5672,"source_end_line":5684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5672-L5684","statement_sha256":"c64b33564d56647068bcbebd8df4a88abdc5ec88c860d7d99599144ef9f1a0a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6144,"rank":6144,"depth":38,"x":2410.341,"y":632.829,"cluster":"divisors-intersection-theory"},{"id":"stacks:067U","tag":"067U","title":"Relative regular immersions · Lemma 067U","summary":"Let xymatrix Y ar[rd] ar[rr]_i & & X ar[ld] & S be a commutative diagram of morphisms of schemes. Assume that Y → S is smooth, X → S smooth, and i an immersion. Then i is a regular immersion.","statement_latex":"Let\n$$\n\\xymatrix{\nY \\ar[rd] \\ar[rr]_i & & X \\ar[ld] \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes.\nAssume that $Y \\to S$ is smooth, $X \\to S$ smooth, and\n$i$ an immersion. Then $i$ is a regular immersion.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067U","source_file":"divisors.tex","source_line":5719,"source_end_line":5731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5719-L5731","statement_sha256":"bab943802946826835fe8d567ed820e894ef3398493d4cafcf2976a3b19857ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":6145,"rank":6145,"depth":39,"x":2569.548,"y":675.244,"cluster":"divisors-intersection-theory"},{"id":"stacks:0693","tag":"0693","title":"Relative regular immersions · Lemma 0693","summary":"Let xymatrix Y ar[rd]_j ar[rr]_i & & X ar[ld] & S be a commutative diagram of morphisms of schemes. Assume X → S smooth and i and j immersions. If i is a Koszul-regular (resp. H_1-regular, quasi-regular) immersion, then so is j.","statement_latex":"Let\n$$\n\\xymatrix{\nY \\ar[rd]_j \\ar[rr]_i & & X \\ar[ld] \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes.\nAssume $X \\to S$ smooth and $i$ and $j$ immersions.\nIf $i$ is a Koszul-regular (resp.\\ $H_1$-regular, quasi-regular)\nimmersion, then so is $j$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0693","source_file":"divisors.tex","source_line":5740,"source_end_line":5753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5740-L5753","statement_sha256":"685c304f2c0283752bfb518ae25226d49360839374fe7e61322bf6270a8a40d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6146,"rank":6146,"depth":40,"x":2417.621,"y":734.47,"cluster":"divisors-intersection-theory"},{"id":"stacks:01X2","tag":"01X2","title":"Meromorphic functions and sections · Definition 01X2","summary":"Let (X, O_X) be a locally ringed space. The sheaf of meromorphic functions on X is the sheaf K_X associated to the presheaf displayed above. A meromorphic function on X is a global section of K_X.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a locally ringed space.\nThe {\\it sheaf of meromorphic functions on $X$} is\nthe sheaf {\\it $\\mathcal{K}_X$} associated to the presheaf\ndisplayed above. A {\\it meromorphic function} on $X$\nis a global section of $\\mathcal{K}_X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01X2","source_file":"divisors.tex","source_line":5882,"source_end_line":5889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5882-L5889","statement_sha256":"c58838ccaca0ef9bf2edd6e4ab46efa5cf2c87f3e4e06bef8e288c51ac524aa5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6147,"rank":6147,"depth":0,"x":2482.201,"y":604.231,"cluster":"divisors-intersection-theory"},{"id":"stacks:01X4","tag":"01X4","title":"Meromorphic functions and sections · Definition 01X4","summary":"Let X be a locally ringed space. Let F be a sheaf of O_X-modules. • We denote K_X(F) the sheaf of K_X-modules which is the sheafification of the presheaf U ↦ S(U)^-1F(U). Equivalently K_X(F) = F ⊗_O_X K_X (see above). • A meromorphic section of F is a global section of K_X(F).","statement_latex":"Let $X$ be a locally ringed space.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item We denote\n$\\mathcal{K}_X(\\mathcal{F})$\nthe sheaf of $\\mathcal{K}_X$-modules which is\nthe sheafification of the presheaf\n$U \\mapsto \\mathcal{S}(U)^{-1}\\mathcal{F}(U)$. Equivalently\n$\\mathcal{K}_X(\\mathcal{F}) =\n\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{K}_X$ (see above).\n\\item A {\\it meromorphic section of $\\mathcal{F}$}\nis a global section of $\\mathcal{K}_X(\\mathcal{F})$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01X4","source_file":"divisors.tex","source_line":5917,"source_end_line":5932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5917-L5932","statement_sha256":"d24b767e45af12539c63e57c79e7b20147df47dc178b3fb07be583f285cf0ba8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6148,"rank":6148,"depth":0,"x":2539.471,"y":737.276,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OT","tag":"02OT","title":"Meromorphic functions and sections · Definition 02OT","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of locally ringed spaces. We say that pullbacks of meromorphic functions are defined for f if for every pair of open U ⊂ X, V ⊂ Y such that f(U) ⊂ V, and any section s ∈ Γ(V, S_Y) the pullback f^sharp(s) ∈ Γ(U, O_X) is an element of Γ(U, S_X).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism\nof locally ringed spaces. We say that {\\it pullbacks of meromorphic\nfunctions are defined for $f$} if for every pair of open\n$U \\subset X$, $V \\subset Y$ such that $f(U) \\subset V$, and any\nsection $s \\in \\Gamma(V, \\mathcal{S}_Y)$ the pullback\n$f^\\sharp(s) \\in \\Gamma(U, \\mathcal{O}_X)$ is an element\nof $\\Gamma(U, \\mathcal{S}_X)$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OT","source_file":"divisors.tex","source_line":5956,"source_end_line":5965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5956-L5965","statement_sha256":"d91e838e71fa67c298c6bdc22fd5e76461596493a71bc93131166fbed86c1171","origin":"The Stacks Project","memory_eligible":false,"source_rank":6149,"rank":6149,"depth":0,"x":2389.841,"y":671.488,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OU","tag":"02OU","title":"Meromorphic functions and sections · Lemma 02OU","summary":"Let f : X → Y be a morphism of schemes. In each of the following cases pullbacks of meromorphic functions are defined. • every weakly associated point of X maps to a generic point of an irreducible component of Y, • X, Y are integral and f is dominant, • X is integral and the generic point of X maps to a generic point of an irreducible component of Y, • X is reduced and every generic point of every irreducible component of X maps to the generic point of an irreducible…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nIn each of the following cases pullbacks of meromorphic\nfunctions are defined.\n\\begin{enumerate}\n\\item every weakly associated point of $X$ maps to\na generic point of an irreducible component of $Y$,\n\\item $X$, $Y$ are integral and $f$ is dominant,\n\\item $X$ is integral and the generic point of $X$ maps\nto a generic point of an irreducible component of $Y$,\n\\item $X$ is reduced and every generic point of every irreducible\ncomponent of $X$ maps to the generic point of an irreducible component\nof $Y$,\n\\item $X$ is locally Noetherian, and any associated point of\n$X$ maps to a generic point of an irreducible component of $Y$,\n\\item $X$ is locally Noetherian, has no embedded points and\nany generic point of an irreducible component of\n$X$ maps to the generic point of an irreducible component of $Y$, and\n\\item $f$ is flat.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OU","source_file":"divisors.tex","source_line":5983,"source_end_line":6004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L5983-L6004","statement_sha256":"3ce1d662dfa3b85e80c4a5da7a19de5203bcc583285b8b6d6c8f4c1e1193d4d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6150,"rank":6150,"depth":4,"x":2553.526,"y":634.997,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EMF","tag":"0EMF","title":"Meromorphic functions and sections · Lemma 0EMF","summary":"Let X be a scheme such that • [(a)] every weakly associated point of X is a generic point of an irreducible component of X, and • [(b)] any quasi-compact open has a finite number of irreducible components. Let X^0 be the set of generic points of irreducible components of X. Then we have K_X = bigoplus_eta ∈ X^0 j_eta, *O_X, eta = ∏_eta ∈ X^0 j_eta, *O_X, eta where j_eta : Spec(O_X, eta) → X is the canonical map of Schemes, Section [Tag 01J5]. Moreover • K_X is a…","statement_latex":"Let $X$ be a scheme such that\n\\begin{enumerate}\n\\item[(a)] every weakly associated point of $X$ is a generic point of an\nirreducible component of $X$, and\n\\item[(b)] any quasi-compact open has a finite number of irreducible components.\n\\end{enumerate}\nLet $X^0$ be the set of generic points of irreducible components of $X$.\nThen we have\n$$\n\\mathcal{K}_X =\n\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} =\n\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta}\n$$\nwhere $j_\\eta : \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$ is the canonical map\nof Schemes, Section \\ref{schemes-section-points}. Moreover\n\\begin{enumerate}\n\\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of\n$\\mathcal{O}_X$-algebras,\n\\item for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf\n$$\n\\mathcal{K}_X(\\mathcal{F}) =\n\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta =\n\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta\n$$\nof meromorphic sections of $\\mathcal{F}$\nis quasi-coherent,\n\\item $\\mathcal{S}_x \\subset \\mathcal{O}_{X, x}$\nis the set of nonzerodivisors for any $x \\in X$,\n\\item $\\mathcal{K}_{X, x}$ is the total quotient ring of $\\mathcal{O}_{X, x}$\nfor any $x \\in X$,\n\\item $\\mathcal{K}_X(U)$ equals the total quotient ring of $\\mathcal{O}_X(U)$\nfor any affine open $U \\subset X$,\n\\item the ring of rational functions of $X$\n(Morphisms, Definition \\ref{morphisms-definition-rational-function})\nis the ring of meromorphic\nfunctions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMF","source_file":"divisors.tex","source_line":6034,"source_end_line":6073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6034-L6073","statement_sha256":"0e3c7e00f372fd77bb3920cab25cb8625789b24225dfafd2a37192cd57e47cee","origin":"The Stacks Project","memory_eligible":false,"source_rank":6151,"rank":6151,"depth":14,"x":2461.925,"y":755.107,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OX","tag":"02OX","title":"Meromorphic functions and sections · Definition 02OX","summary":"Let X be a locally ringed space. Let L be an invertible O_X-module. A meromorphic section s of L is said to be regular if the induced map K_X → K_X(L) is injective. In other words, s is a regular section of the invertible K_X-module K_X(L), see Definition [Tag 01WY].","statement_latex":"Let $X$ be a locally ringed space.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nA meromorphic section $s$ of $\\mathcal{L}$ is said to be {\\it regular}\nif the induced map\n$\\mathcal{K}_X \\to \\mathcal{K}_X(\\mathcal{L})$\nis injective. In other words, $s$ is a regular\nsection of the invertible $\\mathcal{K}_X$-module\n$\\mathcal{K}_X(\\mathcal{L})$, see\nDefinition \\ref{definition-regular-section}.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OX","source_file":"divisors.tex","source_line":6134,"source_end_line":6145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6134-L6145","statement_sha256":"31ca4db65f83346c235bab017c7f17bfb08355d00c4c38272dba2475932df196","origin":"The Stacks Project","memory_eligible":false,"source_rank":6152,"rank":6152,"depth":1,"x":2432.803,"y":614.184,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OY","tag":"02OY","title":"Meromorphic functions and sections · Lemma 02OY","summary":"Let f : X → Y be a morphism of locally ringed spaces. Assume that pullbacks of meromorphic functions are defined for f (see Definition [Tag 02OT]). • Let F be a sheaf of O_Y-modules. There is a canonical pullback map f^* : Γ(Y, K_Y(F)) → Γ(X, K_X(f^*F)) for meromorphic sections of F. • Let L be an invertible O_Y-module. A regular meromorphic section s of L pulls back to a regular meromorphic section f^*s of f^*L.","statement_latex":"Let $f : X \\to Y$ be a morphism of locally ringed spaces.\nAssume that pullbacks of meromorphic functions are defined\nfor $f$ (see\nDefinition \\ref{definition-pullback-meromorphic-sections}).\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules.\nThere is a canonical pullback map\n$f^* : \\Gamma(Y, \\mathcal{K}_Y(\\mathcal{F})) \\to\n\\Gamma(X, \\mathcal{K}_X(f^*\\mathcal{F}))$\nfor meromorphic sections of $\\mathcal{F}$.\n\\item Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module.\nA regular meromorphic section $s$ of $\\mathcal{L}$ pulls back\nto a regular meromorphic section $f^*s$ of $f^*\\mathcal{L}$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OY","source_file":"divisors.tex","source_line":6150,"source_end_line":6166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6150-L6166","statement_sha256":"2df3531f366c20df10a71730612fe50a8da877debc356c7591220a38fed7f910","origin":"The Stacks Project","memory_eligible":false,"source_rank":6153,"rank":6153,"depth":1,"x":2567.964,"y":701.809,"cluster":"divisors-intersection-theory"},{"id":"stacks:02P0","tag":"02P0","title":"Meromorphic functions and sections · Lemma 02P0","summary":"Let X be a scheme. Let L be an invertible O_X-module. Let s be a regular meromorphic section of L. Let us denote I ⊂ O_X the sheaf of ideals defined by the rule I(V) = (f ∈ O_X(V) mid fs ∈ L(V)). The formula makes sense since L(V) ⊂ K_X(L)(V). Then I is a quasi-coherent sheaf of ideals and we have injective maps 1 : I → O_X, s : I → L whose cokernels are supported on closed nowhere dense subsets of X.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s$ be a regular meromorphic section of $\\mathcal{L}$.\nLet us denote $\\mathcal{I} \\subset \\mathcal{O}_X$ the\nsheaf of ideals defined by the rule\n$$\n\\mathcal{I}(V)\n=\n\\{f \\in \\mathcal{O}_X(V) \\mid fs \\in \\mathcal{L}(V)\\}.\n$$\nThe formula makes sense since\n$\\mathcal{L}(V) \\subset \\mathcal{K}_X(\\mathcal{L})(V)$.\nThen $\\mathcal{I}$ is a quasi-coherent sheaf of ideals and\nwe have injective maps\n$$\n1 : \\mathcal{I} \\longrightarrow \\mathcal{O}_X,\n\\quad\ns : \\mathcal{I} \\longrightarrow \\mathcal{L}\n$$\nwhose cokernels are supported on closed nowhere dense subsets of $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02P0","source_file":"divisors.tex","source_line":6172,"source_end_line":6194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6172-L6194","statement_sha256":"e5618857a407d5d4ce2bf567c085c61c3b4dfb72189be6e669fed0bcb6330b54","origin":"The Stacks Project","memory_eligible":false,"source_rank":6154,"rank":6154,"depth":0,"x":2397.38,"y":713.921,"cluster":"divisors-intersection-theory"},{"id":"stacks:02P1","tag":"02P1","title":"Meromorphic functions and sections · Definition 02P1","summary":"Let X be a scheme. Let L be an invertible O_X-module. Let s be a regular meromorphic section of L. The sheaf of ideals I constructed in Lemma [Tag 02P0] is called the ideal sheaf of denominators of s.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s$ be a regular meromorphic section of $\\mathcal{L}$.\nThe sheaf of ideals $\\mathcal{I}$ constructed in\nLemma \\ref{lemma-regular-meromorphic-ideal-denominators}\nis called the {\\it ideal sheaf of denominators of $s$}.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02P1","source_file":"divisors.tex","source_line":6219,"source_end_line":6227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6219-L6227","statement_sha256":"015ec789f3cadc40a2a79fca2d663048a1fc5ab3202d049f119d92d804824ca0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6155,"rank":6155,"depth":1,"x":2513.733,"y":607.916,"cluster":"divisors-intersection-theory"},{"id":"stacks:08I7","tag":"08I7","title":"Meromorphic functions and sections; Noetherian case · Lemma 08I7","summary":"Let X be a quasi-compact scheme. Let h ∈ Γ(X, O_X) and f ∈ Γ(X, K_X) such that f restricts to zero on X_h. Then h^n f = 0 for some n gg 0.","statement_latex":"Let $X$ be a quasi-compact scheme. Let $h \\in \\Gamma(X, \\mathcal{O}_X)$ and\n$f \\in \\Gamma(X, \\mathcal{K}_X)$ such that $f$ restricts\nto zero on $X_h$. Then $h^n f = 0$ for some $n \\gg 0$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections; Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08I7","source_file":"divisors.tex","source_line":6241,"source_end_line":6246,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6241-L6246","statement_sha256":"095ff43500b34ed9eb76e0e9a549648a5f6d1bb719e3c49777174f8ad290ebd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6156,"rank":6156,"depth":0,"x":2513.18,"y":752.484,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OV","tag":"02OV","title":"Meromorphic functions and sections; Noetherian case · Lemma 02OV","summary":"Let X be a locally Noetherian scheme. • For any x ∈ X we have S_x ⊂ O_X, x is the set of nonzerodivisors, and hence K_X, x is the total quotient ring of O_X, x. • For any affine open U ⊂ X the ring K_X(U) equals the total quotient ring of O_X(U).","statement_latex":"Let $X$ be a locally Noetherian scheme.\n\\begin{enumerate}\n\\item For any $x \\in X$ we have $\\mathcal{S}_x \\subset \\mathcal{O}_{X, x}$\nis the set of nonzerodivisors, and hence $\\mathcal{K}_{X, x}$\nis the total quotient ring of $\\mathcal{O}_{X, x}$.\n\\item For any affine open $U \\subset X$ the ring\n$\\mathcal{K}_X(U)$ equals the total quotient ring of $\\mathcal{O}_X(U)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections; Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OV","source_file":"divisors.tex","source_line":6260,"source_end_line":6270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6260-L6270","statement_sha256":"8def270b975e7c3ee200c71b74e5c1d353585d9f4e46a6cb51cd87daf0e26e73","origin":"The Stacks Project","memory_eligible":false,"source_rank":6157,"rank":6157,"depth":10,"x":2397.027,"y":645.288,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EMH","tag":"0EMH","title":"Meromorphic functions and sections; Noetherian case · Lemma 0EMH","summary":"Let X be a locally Noetherian scheme having no embedded points. Let X^0 be the set of generic points of irreducible components of X. Then we have K_X = bigoplus_eta ∈ X^0 j_eta, *O_X, eta = ∏_eta ∈ X^0 j_eta, *O_X, eta where j_eta : Spec(O_X, eta) → X is the canonical map of Schemes, Section [Tag 01J5]. Moreover • K_X is a quasi-coherent sheaf of O_X-algebras, • for every quasi-coherent O_X-module F the sheaf K_X(F) = bigoplus_eta ∈ X^0 j_eta, *F_eta = ∏_eta ∈ X^0 j_eta,…","statement_latex":"Let $X$ be a locally Noetherian scheme having no embedded points.\nLet $X^0$ be the set of generic points of irreducible components of $X$.\nThen we have\n$$\n\\mathcal{K}_X =\n\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} =\n\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta}\n$$\nwhere $j_\\eta : \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$ is the canonical map\nof Schemes, Section \\ref{schemes-section-points}. Moreover\n\\begin{enumerate}\n\\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras,\n\\item for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf\n$$\n\\mathcal{K}_X(\\mathcal{F}) =\n\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta =\n\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta\n$$\nof meromorphic sections of $\\mathcal{F}$ is quasi-coherent, and\n\\item the ring of rational functions of $X$ is the ring of meromorphic\nfunctions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections; Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMH","source_file":"divisors.tex","source_line":6314,"source_end_line":6338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6314-L6338","statement_sha256":"6b2603fae9a997f3dc0b8dd3c0d90691e0056212b94ebc08b647ad077089eeac","origin":"The Stacks Project","memory_eligible":false,"source_rank":6158,"rank":6158,"depth":15,"x":2569.329,"y":658.459,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EMI","tag":"0EMI","title":"Meromorphic functions and sections; Noetherian case · Lemma 0EMI","summary":"Let X be a locally Noetherian scheme having no embedded points. Let L be an invertible O_X-module. Then L has a regular meromorphic section.","statement_latex":"Let $X$ be a locally Noetherian scheme having no embedded points.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen $\\mathcal{L}$ has a regular meromorphic section.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections; Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMI","source_file":"divisors.tex","source_line":6348,"source_end_line":6353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6348-L6353","statement_sha256":"8236f39b8df15028055053023aa0df4d51f5e05bd7fc65edf3fab9e029069720","origin":"The Stacks Project","memory_eligible":false,"source_rank":6159,"rank":6159,"depth":16,"x":2431.327,"y":746.743,"cluster":"divisors-intersection-theory"},{"id":"stacks:02P2","tag":"02P2","title":"Meromorphic functions and sections; Noetherian case · Lemma 02P2","summary":"Suppose given • X a locally Noetherian scheme, • L an invertible O_X-module, • s a regular meromorphic section of L, and • F coherent on X without embedded associated points and Supp(F) = X. Let I ⊂ O_X be the ideal of denominators of s. Let T ⊂ X be the union of the supports of O_X/I and L/s(I) which is a nowhere dense closed subset T ⊂ X according to Lemma [Tag 02P0]. Then there are canonical injective maps 1 : IF → F, s : IF → F ⊗_O_XL whose cokernels are supported on T.","statement_latex":"Suppose given\n\\begin{enumerate}\n\\item $X$ a locally Noetherian scheme,\n\\item $\\mathcal{L}$ an invertible $\\mathcal{O}_X$-module,\n\\item $s$ a regular meromorphic section of $\\mathcal{L}$, and\n\\item $\\mathcal{F}$ coherent on $X$\nwithout embedded associated points and $\\text{Supp}(\\mathcal{F}) = X$.\n\\end{enumerate}\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be the ideal of\ndenominators of $s$. Let $T \\subset X$ be the union\nof the supports of $\\mathcal{O}_X/\\mathcal{I}$ and\n$\\mathcal{L}/s(\\mathcal{I})$ which is a nowhere dense closed\nsubset $T \\subset X$ according to\nLemma \\ref{lemma-regular-meromorphic-ideal-denominators}.\nThen there are canonical injective maps\n$$\n1 : \\mathcal{I}\\mathcal{F} \\to \\mathcal{F}, \\quad\ns : \\mathcal{I}\\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_X}\\mathcal{L}\n$$\nwhose cokernels are supported on $T$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections; Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02P2","source_file":"divisors.tex","source_line":6365,"source_end_line":6387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6365-L6387","statement_sha256":"fd971ea02119a7232a87a0a8967a1ac6238519757076de709a29d41d235036eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6160,"rank":6160,"depth":1,"x":2462.172,"y":602.969,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OW","tag":"02OW","title":"Meromorphic functions and sections; reduced case · Lemma 02OW","summary":"Let X be a reduced scheme such that any quasi-compact open has a finite number of irreducible components. Let X^0 be the set of generic points of irreducible components of X. Then we have K_X = bigoplus_eta ∈ X^0 j_eta, *kappa(eta) = ∏_eta ∈ X^0 j_eta, *kappa(eta) where j_eta : Spec(kappa(eta)) → X is the canonical map of Schemes, Section [Tag 01J5]. Moreover • K_X is a quasi-coherent sheaf of O_X-algebras, • for every quasi-coherent O_X-module F the sheaf K_X(F) =…","statement_latex":"Let $X$ be a reduced scheme such that any quasi-compact open\nhas a finite number of irreducible components. Let $X^0$ be the set\nof generic points of irreducible components of $X$. Then we have\n$$\n\\mathcal{K}_X =\n\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\kappa(\\eta) =\n\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\kappa(\\eta)\n$$\nwhere $j_\\eta : \\Spec(\\kappa(\\eta)) \\to X$ is the canonical map\nof Schemes, Section \\ref{schemes-section-points}. Moreover\n\\begin{enumerate}\n\\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of\n$\\mathcal{O}_X$-algebras,\n\\item for every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the sheaf\n$$\n\\mathcal{K}_X(\\mathcal{F}) =\n\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta =\n\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta\n$$\nof meromorphic sections of $\\mathcal{F}$\nis quasi-coherent,\n\\item $\\mathcal{S}_x \\subset \\mathcal{O}_{X, x}$\nis the set of nonzerodivisors for any $x \\in X$,\n\\item $\\mathcal{K}_{X, x}$ is the total quotient ring of $\\mathcal{O}_{X, x}$\nfor any $x \\in X$,\n\\item $\\mathcal{K}_X(U)$ equals the total quotient ring of $\\mathcal{O}_X(U)$\nfor any affine open $U \\subset X$,\n\\item the ring of rational functions of $X$ is the ring of meromorphic\nfunctions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections; reduced case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OW","source_file":"divisors.tex","source_line":6424,"source_end_line":6456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6424-L6456","statement_sha256":"eab04d4a9b5a1da266539cf0a40b6c19ef51bfcdb450071763a15766ec2f3962","origin":"The Stacks Project","memory_eligible":false,"source_rank":6161,"rank":6161,"depth":15,"x":2555.285,"y":726.805,"cluster":"divisors-intersection-theory"},{"id":"stacks:035T","tag":"035T","title":"Meromorphic functions and sections; reduced case · Lemma 035T","summary":"Let X be a scheme. Assume X is reduced and any quasi-compact open U ⊂ X has a finite number of irreducible components. Then the normalization morphism ν : X^ν → X is the morphism underlineSpec_X(O') → X where O' ⊂ K_X is the integral closure of O_X in the sheaf of meromorphic functions.","statement_latex":"Let $X$ be a scheme.\nAssume $X$ is reduced and any quasi-compact open $U \\subset X$\nhas a finite number of irreducible components.\nThen the normalization morphism $\\nu : X^\\nu \\to X$ is the\nmorphism\n$$\n\\underline{\\Spec}_X(\\mathcal{O}') \\longrightarrow X\n$$\nwhere $\\mathcal{O}' \\subset \\mathcal{K}_X$ is the integral\nclosure of $\\mathcal{O}_X$ in the sheaf of meromorphic functions.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections; reduced case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035T","source_file":"divisors.tex","source_line":6466,"source_end_line":6478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6466-L6478","statement_sha256":"35c657689222ae932f43f3a8138962b128ca4482b9863518d82285279a7b9b04","origin":"The Stacks Project","memory_eligible":false,"source_rank":6162,"rank":6162,"depth":16,"x":2386.61,"y":688.226,"cluster":"divisors-intersection-theory"},{"id":"stacks:01X5","tag":"01X5","title":"Meromorphic functions and sections; reduced case · Lemma 01X5","summary":"Let X be an integral scheme with generic point eta. We have • the sheaf of meromorphic functions is isomorphic to the constant sheaf with value the function field (see Morphisms, Definition [Tag 01RW]) of X. • for any quasi-coherent sheaf F on X the sheaf K_X(F) is isomorphic to the constant sheaf with value F_eta.","statement_latex":"Let $X$ be an integral scheme with generic point $\\eta$. We have\n\\begin{enumerate}\n\\item the sheaf of meromorphic functions is\nisomorphic to the constant sheaf with value the\nfunction field (see\nMorphisms, Definition \\ref{morphisms-definition-function-field})\nof $X$.\n\\item for any quasi-coherent sheaf $\\mathcal{F}$ on $X$ the\nsheaf $\\mathcal{K}_X(\\mathcal{F})$ is isomorphic to the\nconstant sheaf with value $\\mathcal{F}_\\eta$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections; reduced case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01X5","source_file":"divisors.tex","source_line":6488,"source_end_line":6501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6488-L6501","statement_sha256":"f461ae01374e55936662f2c60a1a886cd9b6924b0c27a89495419fa97fac72cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6163,"rank":6163,"depth":1,"x":2542.407,"y":620.793,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OZ","tag":"02OZ","title":"Meromorphic functions and sections; reduced case · Lemma 02OZ","summary":"Let X be a scheme. Let L be an invertible O_X-module. In each of the following cases L has a regular meromorphic section: • X is integral, • X is reduced and any quasi-compact open has a finite number of irreducible components, • X is locally Noetherian and has no embedded points.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nIn each of the following cases $\\mathcal{L}$ has a regular meromorphic\nsection:\n\\begin{enumerate}\n\\item $X$ is integral,\n\\item $X$ is reduced and any quasi-compact open has a finite\nnumber of irreducible components,\n\\item $X$ is locally Noetherian and has no embedded points.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Meromorphic functions and sections; reduced case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OZ","source_file":"divisors.tex","source_line":6510,"source_end_line":6522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6510-L6522","statement_sha256":"e7e8284a2b2130d575a3d9e3d24b44019d9108de03c561f5ad989d0eb16c15a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6164,"rank":6164,"depth":17,"x":2481.599,"y":759.269,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BE1","tag":"0BE1","title":"Weil divisors · Lemma 0BE1","summary":"Let X be a locally Noetherian scheme. Let Z ⊂ X be a closed subscheme. The collection of irreducible components of Z is locally finite in X.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $Z \\subset X$ be a closed\nsubscheme. The collection of irreducible components of $Z$\nis locally finite in $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BE1","source_file":"divisors.tex","source_line":6562,"source_end_line":6567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6562-L6567","statement_sha256":"414e1adb488d6ed70822dfa5a8e2884a1a97601192fd0c4e72b3c0fb4149f37d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6165,"rank":6165,"depth":3,"x":2414.912,"y":622.311,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BE2","tag":"0BE2","title":"Weil divisors · Definition 0BE2","summary":"Let X be a locally Noetherian integral scheme. • A prime divisor is an integral closed subscheme Z ⊂ X of codimension 1. • A Weil divisor is a formal sum D = ∑ n_Z Z where the sum is over prime divisors of X and the collection (Z mid n_Z not = 0) is locally finite (Topology, Definition [Tag 0BDS]). The group of all Weil divisors on X is denoted Div(X).","statement_latex":"Let $X$ be a locally Noetherian integral scheme.\n\\begin{enumerate}\n\\item A {\\it prime divisor} is an integral closed subscheme $Z \\subset X$\nof codimension $1$.\n\\item A {\\it Weil divisor} is a formal sum $D = \\sum n_Z Z$ where\nthe sum is over prime divisors of $X$ and the collection\n$\\{Z \\mid n_Z \\not = 0\\}$ is locally finite\n(Topology, Definition \\ref{topology-definition-locally-finite}).\n\\end{enumerate}\nThe group of all Weil divisors on $X$ is denoted $\\text{Div}(X)$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BE2","source_file":"divisors.tex","source_line":6585,"source_end_line":6597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6585-L6597","statement_sha256":"9c7bf9a6f998b90e04c137e30c0b432e8b439f24715a0cd2e476cd168f3e8ead","origin":"The Stacks Project","memory_eligible":false,"source_rank":6166,"rank":6166,"depth":1,"x":2574.622,"y":685.621,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RJ","tag":"02RJ","title":"Weil divisors · Definition 02RJ","summary":"Let X be a locally Noetherian integral scheme. Let f ∈ R(X)^*. For every prime divisor Z ⊂ X we define the order of vanishing of f along Z as the integer ord_Z(f) = ord_O_X, xi(f) where the right hand side is the notion of Algebra, Definition [Tag 02MD] and xi is the generic point of Z.","statement_latex":"Let $X$ be a locally Noetherian integral scheme. Let $f \\in R(X)^*$.\nFor every prime divisor $Z \\subset X$ we define the\n{\\it order of vanishing of $f$ along $Z$} as the integer\n$$\n\\text{ord}_Z(f) = \\text{ord}_{\\mathcal{O}_{X, \\xi}}(f)\n$$\nwhere the right hand side is the notion of\nAlgebra, Definition \\ref{algebra-definition-ord}\nand $\\xi$ is the generic point of $Z$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RJ","source_file":"divisors.tex","source_line":6604,"source_end_line":6615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6604-L6615","statement_sha256":"f92da44741c996794c483affec743f7312467f87b8efd11cb314a26aedbb142b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6167,"rank":6167,"depth":1,"x":2405.523,"y":729.669,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RL","tag":"02RL","title":"Weil divisors · Lemma 02RL","summary":"Let X be a locally Noetherian integral scheme. Let f ∈ R(X)^*. Then the collections (Z ⊂ X mid Z a prime divisor with generic point xi and f not in O_X, xi) and (Z ⊂ X mid Z a prime divisor and ord_Z(f) not = 0) are locally finite in X.","statement_latex":"Let $X$ be a locally Noetherian integral scheme. Let $f \\in R(X)^*$.\nThen the collections\n$$\n\\{Z \\subset X \\mid Z\\text{ a prime divisor with generic point }\\xi\n\\text{ and }f\\text{ not in }\\mathcal{O}_{X, \\xi}\\}\n$$\nand\n$$\n\\{Z \\subset X \\mid Z \\text{ a prime divisor and }\\text{ord}_Z(f) \\not = 0\\}\n$$\nare locally finite in $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RL","source_file":"divisors.tex","source_line":6630,"source_end_line":6643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6630-L6643","statement_sha256":"aea200c65e5dc43730e75db5c867654bf6b81f570f732bc19392d6b5329ec05c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6168,"rank":6168,"depth":4,"x":2495.013,"y":600.92,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BE3","tag":"0BE3","title":"Weil divisors · Definition 0BE3","summary":"Let X be a locally Noetherian integral scheme. Let f ∈ R(X)^*. The principal Weil divisor associated to f is the Weil divisor div(f) = div_X(f) = ∑ ord_Z(f) [Z] where the sum is over prime divisors and ord_Z(f) is as in Definition [Tag 02RJ]. This makes sense by Lemma [Tag 02RL].","statement_latex":"Let $X$ be a locally Noetherian integral scheme. Let $f \\in R(X)^*$.\nThe {\\it principal Weil divisor associated to $f$} is the Weil divisor\n$$\n\\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z]\n$$\nwhere the sum is over prime divisors and $\\text{ord}_Z(f)$ is as in\nDefinition \\ref{definition-order-vanishing}. This makes sense\nby Lemma \\ref{lemma-divisor-locally-finite}.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BE3","source_file":"divisors.tex","source_line":6656,"source_end_line":6666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6656-L6666","statement_sha256":"04997a8494e7ae4edb7bdbc8dc4e3d5c0787484f1f9eab41f98f59938ed5fe16","origin":"The Stacks Project","memory_eligible":false,"source_rank":6169,"rank":6169,"depth":5,"x":2532.653,"y":746.995,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RP","tag":"02RP","title":"Weil divisors · Lemma 02RP","summary":"Let X be a locally Noetherian integral scheme. Let f, g ∈ R(X)^*. Then div_X(fg) = div_X(f) + div_X(g) as Weil divisors on X.","statement_latex":"Let $X$ be a locally Noetherian integral scheme. Let $f, g \\in R(X)^*$. Then\n$$\n\\text{div}_X(fg) = \\text{div}_X(f) + \\text{div}_X(g)\n$$\nas Weil divisors on $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RP","source_file":"divisors.tex","source_line":6668,"source_end_line":6675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6668-L6675","statement_sha256":"87cf9e9f0942f974849be327160c598db5132aa539ab50117dbc603e6196f586","origin":"The Stacks Project","memory_eligible":false,"source_rank":6170,"rank":6170,"depth":0,"x":2387.072,"y":660.427,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BE4","tag":"0BE4","title":"Weil divisors · Definition 0BE4","summary":"Let X be a locally Noetherian integral scheme. The Weil divisor class group of X is the quotient of the group of Weil divisors by the subgroup of principal Weil divisors. Notation: Cl(X).","statement_latex":"Let $X$ be a locally Noetherian integral scheme. The\n{\\it Weil divisor class group} of $X$ is the quotient of\nthe group of Weil divisors by the subgroup of principal Weil divisors.\nNotation: $\\text{Cl}(X)$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BE4","source_file":"divisors.tex","source_line":6686,"source_end_line":6692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6686-L6692","statement_sha256":"c81d7974c90422a40d0297e7e5a93ebb4506a5e299be04bd284420712c299550","origin":"The Stacks Project","memory_eligible":false,"source_rank":6171,"rank":6171,"depth":0,"x":2564.468,"y":641.61,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SF","tag":"02SF","title":"The Weil divisor class associated to an invertible module · Definition 02SF","summary":"Let X be a locally Noetherian integral scheme. Let L be an invertible O_X-module. Let s ∈ Γ(X, K_X(L)) be a regular meromorphic section of L. For every prime divisor Z ⊂ X we define the order of vanishing of s along Z as the integer ord_Z, L(s) = ord_O_X, xi(s/s_xi) where the right hand side is the notion of Algebra, Definition [Tag 02MD], xi ∈ Z is the generic point, and s_xi ∈ L_xi is a generator.","statement_latex":"Let $X$ be a locally Noetherian integral scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{K}_X(\\mathcal{L}))$\nbe a regular meromorphic section of $\\mathcal{L}$.\nFor every prime divisor $Z \\subset X$ we define the\n{\\it order of vanishing of $s$ along $Z$} as the integer\n$$\n\\text{ord}_{Z, \\mathcal{L}}(s)\n= \\text{ord}_{\\mathcal{O}_{X, \\xi}}(s/s_\\xi)\n$$\nwhere the right hand side is the notion of\nAlgebra, Definition \\ref{algebra-definition-ord},\n$\\xi \\in Z$ is the generic point,\nand $s_\\xi \\in \\mathcal{L}_\\xi$ is a generator.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The Weil divisor class associated to an invertible module","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SF","source_file":"divisors.tex","source_line":6737,"source_end_line":6753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6737-L6753","statement_sha256":"0f387349f39a8b67ab3dc320a4578c8c800f2b21c3bc2964d9ec7974d132f291","origin":"The Stacks Project","memory_eligible":false,"source_rank":6172,"rank":6172,"depth":1,"x":2448.511,"y":756.422,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SG","tag":"02SG","title":"The Weil divisor class associated to an invertible module · Lemma 02SG","summary":"Let X be a locally Noetherian integral scheme. Let L be an invertible O_X-module. Let s ∈ K_X(L) be a regular (i.e., nonzero) meromorphic section of L. Then the sets (Z ⊂ X mid Z a prime divisor with generic point xi and s not in L_xi) and (Z ⊂ X mid Z is a prime divisor and ord_Z, L(s) not = 0) are locally finite in X.","statement_latex":"Let $X$ be a locally Noetherian integral scheme. Let $\\mathcal{L}$ be an\ninvertible $\\mathcal{O}_X$-module. Let $s \\in \\mathcal{K}_X(\\mathcal{L})$ be a\nregular (i.e., nonzero) meromorphic section of $\\mathcal{L}$. Then the sets\n$$\n\\{Z \\subset X \\mid Z \\text{ a prime divisor with generic point }\\xi\n\\text{ and }s\\text{ not in }\\mathcal{L}_\\xi\\}\n$$\nand\n$$\n\\{Z \\subset X \\mid Z \\text{ is a prime divisor and }\n\\text{ord}_{Z, \\mathcal{L}}(s) \\not = 0\\}\n$$\nare locally finite in $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The Weil divisor class associated to an invertible module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SG","source_file":"divisors.tex","source_line":6758,"source_end_line":6773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6758-L6773","statement_sha256":"942d33345973cecfc07f6782338c166fb370acade569bca0666049dfdbfdcaa1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6173,"rank":6173,"depth":4,"x":2441.671,"y":605.601,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SH","tag":"02SH","title":"The Weil divisor class associated to an invertible module · Lemma 02SH","summary":"Let X be a locally Noetherian integral scheme. Let L be an invertible O_X-module. Let s, s' ∈ K_X(L) be nonzero meromorphic sections of L. Then f = s/s' is an element of R(X)^* and we have ∑ ord_Z, L(s)[Z] = ∑ ord_Z, L(s')[Z] + div(f) as Weil divisors.","statement_latex":"Let $X$ be a locally Noetherian integral scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s, s' \\in \\mathcal{K}_X(\\mathcal{L})$ be nonzero\nmeromorphic sections of $\\mathcal{L}$. Then $f = s/s'$\nis an element of $R(X)^*$ and we have\n$$\n\\sum \\text{ord}_{Z, \\mathcal{L}}(s)[Z]\n=\n\\sum \\text{ord}_{Z, \\mathcal{L}}(s')[Z]\n+\n\\text{div}(f)\n$$\nas Weil divisors.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The Weil divisor class associated to an invertible module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SH","source_file":"divisors.tex","source_line":6784,"source_end_line":6799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6784-L6799","statement_sha256":"22d8ed1cf85e59341e5d7516abc26e6e40a16722d8c15074d53084df1175b06f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6174,"rank":6174,"depth":5,"x":2568.304,"y":713.191,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BE6","tag":"0BE6","title":"The Weil divisor class associated to an invertible module · Definition 0BE6","summary":"Let X be a locally Noetherian integral scheme. Let L be an invertible O_X-module. • For any nonzero meromorphic section s of L we define the Weil divisor associated to s as div_L(s) = ∑ ord_Z, L(s) [Z] ∈ Div(X) where the sum is over prime divisors. • We define Weil divisor class associated to L as the image of div_L(s) in Cl(X) where s is any nonzero meromorphic section of L over X. This is well defined by Lemma [Tag 02SH].","statement_latex":"Let $X$ be a locally Noetherian integral scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$\nwe define the {\\it Weil divisor associated to $s$} as\n$$\n\\text{div}_\\mathcal{L}(s) =\n\\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z] \\in \\text{Div}(X)\n$$\nwhere the sum is over prime divisors.\n\\item We define {\\it Weil divisor class associated to $\\mathcal{L}$}\nas the image of $\\text{div}_\\mathcal{L}(s)$ in $\\text{Cl}(X)$\nwhere $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over\n$X$. This is well defined by\nLemma \\ref{lemma-divisor-meromorphic-well-defined}.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The Weil divisor class associated to an invertible module","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BE6","source_file":"divisors.tex","source_line":6807,"source_end_line":6825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6807-L6825","statement_sha256":"5cf803339e19100f0843e9885a9235d0068508bf3fa1cda7f18158185690a594","origin":"The Stacks Project","memory_eligible":false,"source_rank":6175,"rank":6175,"depth":6,"x":2387.976,"y":705.692,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SL","tag":"02SL","title":"The Weil divisor class associated to an invertible module · Lemma 02SL","summary":"Let X be a locally Noetherian integral scheme. Let L, N be invertible O_X-modules. Let s, resp. t be a nonzero meromorphic section of L, resp. N. Then st is a nonzero meromorphic section of L ⊗ N, and div_L ⊗ N(st) = div_L(s) + div_N(t) in Div(X). In particular, the Weil divisor class of L ⊗_O_X N is the sum of the Weil divisor classes of L and N.","statement_latex":"Let $X$ be a locally Noetherian integral scheme.\nLet $\\mathcal{L}$, $\\mathcal{N}$ be invertible $\\mathcal{O}_X$-modules.\nLet $s$, resp.\\ $t$ be a nonzero meromorphic section\nof $\\mathcal{L}$, resp.\\ $\\mathcal{N}$. Then $st$ is a nonzero\nmeromorphic section of $\\mathcal{L} \\otimes \\mathcal{N}$, and\n$$\n\\text{div}_{\\mathcal{L} \\otimes \\mathcal{N}}(st)\n=\n\\text{div}_\\mathcal{L}(s) + \\text{div}_\\mathcal{N}(t)\n$$\nin $\\text{Div}(X)$. In particular, the Weil divisor class of\n$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ is the sum\nof the Weil divisor classes of $\\mathcal{L}$ and $\\mathcal{N}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The Weil divisor class associated to an invertible module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SL","source_file":"divisors.tex","source_line":6830,"source_end_line":6845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6830-L6845","statement_sha256":"bb475f1e8a7c063d2ab1c402f6a76f098507b11681b0e8680a8038e485acca64","origin":"The Stacks Project","memory_eligible":false,"source_rank":6176,"rank":6176,"depth":0,"x":2527.309,"y":608.669,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BE8","tag":"0BE8","title":"The Weil divisor class associated to an invertible module · Lemma 0BE8","summary":"Let X be a locally Noetherian integral scheme. If X is normal, then the map ([Tag 0BE7]) Pic(X) → Cl(X) is injective.","statement_latex":"Let $X$ be a locally Noetherian integral scheme. If $X$ is normal,\nthen the map (\\ref{equation-c1}) $\\Pic(X) \\to \\text{Cl}(X)$\nis injective.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The Weil divisor class associated to an invertible module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BE8","source_file":"divisors.tex","source_line":6873,"source_end_line":6878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6873-L6878","statement_sha256":"f7a59172f73d3db8519de4a93156a62c7c37ef2c7eabe5c17088342614266adf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6177,"rank":6177,"depth":18,"x":2502.531,"y":759.63,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BE9","tag":"0BE9","title":"The Weil divisor class associated to an invertible module · Lemma 0BE9","summary":"Let X be a locally Noetherian integral scheme. Consider the map ([Tag 0BE7]) Pic(X) → Cl(X). The following are equivalent • the local rings of X are UFDs, and • X is normal and Pic(X) → Cl(X) is surjective. In this case Pic(X) → Cl(X) is an isomorphism.","statement_latex":"Let $X$ be a locally Noetherian integral scheme. Consider the map\n(\\ref{equation-c1}) $\\Pic(X) \\to \\text{Cl}(X)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item the local rings of $X$ are UFDs, and\n\\item $X$ is normal and $\\Pic(X) \\to \\text{Cl}(X)$\nis surjective.\n\\end{enumerate}\nIn this case $\\Pic(X) \\to \\text{Cl}(X)$ is an isomorphism.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"The Weil divisor class associated to an invertible module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BE9","source_file":"divisors.tex","source_line":6907,"source_end_line":6918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6907-L6918","statement_sha256":"a3a8eea9e9f3e71df9351a23ef863419843b8b1381346aff694d68fb900ed078","origin":"The Stacks Project","memory_eligible":false,"source_rank":6178,"rank":6178,"depth":19,"x":2399.16,"y":633.959,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BD7","tag":"0BD7","title":"More on invertible modules · Lemma 0BD7","summary":"Let φ : X → Y be a morphism of schemes. Let L be an invertible O_X-module. Assume that • X is locally Noetherian, • Y is locally Noetherian, integral, and normal, • φ is flat with integral (hence nonempty) fibres, • φ is either quasi-compact or locally of finite type, • L is trivial when restricted to the generic fibre of φ. Then L ≅ φ^*N for some invertible O_Y-module N.","statement_latex":"Let $\\varphi : X \\to Y$ be a morphism of schemes. Let $\\mathcal{L}$\nbe an invertible $\\mathcal{O}_X$-module. Assume that\n\\begin{enumerate}\n\\item $X$ is locally Noetherian,\n\\item $Y$ is locally Noetherian, integral, and normal,\n\\item $\\varphi$ is flat with integral (hence nonempty) fibres,\n\\item $\\varphi$ is either quasi-compact or locally of finite type,\n\\item $\\mathcal{L}$ is trivial when restricted to the generic fibre of\n$\\varphi$.\n\\end{enumerate}\nThen $\\mathcal{L} \\cong \\varphi^*\\mathcal{N}$ for some invertible\n$\\mathcal{O}_Y$-module $\\mathcal{N}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"More on invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BD7","source_file":"divisors.tex","source_line":6967,"source_end_line":6981,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L6967-L6981","statement_sha256":"ecc6eba5d2762685e7a76d226faf1a7b7b2ad78c2e12a4eefdbd6019489c5d79","origin":"The Stacks Project","memory_eligible":false,"source_rank":6179,"rank":6179,"depth":18,"x":2576.861,"y":668.052,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BD8","tag":"0BD8","title":"More on invertible modules · Lemma 0BD8","summary":"Let X be a locally Noetherian scheme. Let U ⊂ X be an open and let D ⊂ U be an effective Cartier divisor. If O_X, x is a UFD for all x ∈ X setminus U, then there exists an effective Cartier divisor D' ⊂ X with D = U ∩ D'.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be\nan open and let $D \\subset U$ be an effective Cartier divisor.\nIf $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in X \\setminus U$,\nthen there exists an effective Cartier divisor $D' \\subset X$\nwith $D = U \\cap D'$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"More on invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BD8","source_file":"divisors.tex","source_line":7092,"source_end_line":7099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7092-L7099","statement_sha256":"66ccc6740abe178fd775d416203dffe9b888431bab5a01aaaf477373d16e851f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6180,"rank":6180,"depth":21,"x":2418.04,"y":743.92,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BD9","tag":"0BD9","title":"More on invertible modules · Lemma 0BD9","summary":"Let X be a locally Noetherian scheme. Let U ⊂ X be an open and let L be an invertible O_U-module. If O_X, x is a UFD for all x ∈ X setminus U, then there exists an invertible O_X-module L' with L ≅ L'|_U.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be\nan open and let $\\mathcal{L}$ be an invertible $\\mathcal{O}_U$-module.\nIf $\\mathcal{O}_{X, x}$ is a UFD for all $x \\in X \\setminus U$,\nthen there exists an invertible $\\mathcal{O}_X$-module $\\mathcal{L}'$\nwith $\\mathcal{L} \\cong \\mathcal{L}'|_U$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"More on invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BD9","source_file":"divisors.tex","source_line":7126,"source_end_line":7133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7126-L7133","statement_sha256":"688a79e93dd5bbc2b8802cc1789f5a5d40188ad44e7a65669bb67c77c0aaaf35","origin":"The Stacks Project","memory_eligible":false,"source_rank":6181,"rank":6181,"depth":25,"x":2474.272,"y":597.518,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BDA","tag":"0BDA","title":"More on invertible modules · Lemma 0BDA","summary":"Let R be a UFD. The Picard groups of the following are trivial. • Spec(R) and any open subscheme of it. • A^n_R = Spec(R[x_1, …, x_n]) and any open subscheme of it. In particular, the Picard group of any open subscheme of affine n-space A^n_k over a field k is trivial.","statement_latex":"Let $R$ be a UFD. The Picard groups of the following are\ntrivial.\n\\begin{enumerate}\n\\item $\\Spec(R)$ and any open subscheme of it.\n\\item $\\mathbf{A}^n_R = \\Spec(R[x_1, \\ldots, x_n])$ and any open subscheme\nof it.\n\\end{enumerate}\nIn particular, the Picard group of any open subscheme of affine\n$n$-space $\\mathbf{A}^n_k$ over a field $k$ is trivial.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"More on invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDA","source_file":"divisors.tex","source_line":7186,"source_end_line":7197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7186-L7197","statement_sha256":"f5b78908be3467e2027df93b8eb1e392e7eb02fe102ba28ca10017f4bdbe3977","origin":"The Stacks Project","memory_eligible":false,"source_rank":6182,"rank":6182,"depth":26,"x":2550.716,"y":737.703,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BXJ","tag":"0BXJ","title":"More on invertible modules · Lemma 0BXJ","summary":"Let R be a UFD. The Picard group of P^n_R is Z. More precisely, there is an isomorphism Z → Pic(P^n_R), m ↦ O_P^n_R(m) In particular, the Picard group of P^n_k of projective space over a field k is Z.","statement_latex":"Let $R$ be a UFD. The Picard group of $\\mathbf{P}^n_R$\nis $\\mathbf{Z}$. More precisely, there is an isomorphism\n$$\n\\mathbf{Z} \\longrightarrow \\Pic(\\mathbf{P}^n_R),\\quad\nm \\longmapsto \\mathcal{O}_{\\mathbf{P}^n_R}(m)\n$$\nIn particular, the Picard group of $\\mathbf{P}^n_k$ of projective\nspace over a field $k$ is $\\mathbf{Z}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"More on invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXJ","source_file":"divisors.tex","source_line":7254,"source_end_line":7264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7254-L7264","statement_sha256":"34716870c8027323db3547fd7cd27ded1ec8d6fa5f6a709285cf24f033944b2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6183,"rank":6183,"depth":27,"x":2381.225,"y":677.571,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EBL","tag":"0EBL","title":"Weil divisors on normal schemes · Lemma 0EBL","summary":"Let X be an integral locally Noetherian normal scheme. For F and G coherent reflexive O_X-modules the map (SheafHom_O_X(F, O_X) ⊗_O_X G)^** → SheafHom_O_X(F, G) is an isomorphism. The rule F, G ↦ (F ⊗_O_X G)^** defines an abelian group law on the set of isomorphism classes of rank 1 coherent reflexive O_X-modules.","statement_latex":"Let $X$ be an integral locally Noetherian normal scheme.\nFor $\\mathcal{F}$ and $\\mathcal{G}$ coherent reflexive\n$\\mathcal{O}_X$-modules the map\n$$\n(\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{O}_X)\n\\otimes_{\\mathcal{O}_X} \\mathcal{G})^{**} \\to\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})\n$$\nis an isomorphism. The rule $\\mathcal{F}, \\mathcal{G} \\mapsto\n(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G})^{**}$\ndefines an abelian group law on the set of isomorphism classes of rank $1$\ncoherent reflexive $\\mathcal{O}_X$-modules.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors on normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBL","source_file":"divisors.tex","source_line":7314,"source_end_line":7328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7314-L7328","statement_sha256":"e9aedf8af1e81fd71fa66cd79dec10f35870f9b7dcd735424589f381b89c1434","origin":"The Stacks Project","memory_eligible":false,"source_rank":6184,"rank":6184,"depth":34,"x":2554.959,"y":625.62,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EBM","tag":"0EBM","title":"Weil divisors on normal schemes · Lemma 0EBM","summary":"Let X be an integral locally Noetherian normal scheme. The group of rank 1 coherent reflexive O_X-modules is isomorphic to the Weil divisor class group Cl(X) of X.","statement_latex":"Let $X$ be an integral locally Noetherian normal scheme.\nThe group of rank $1$ coherent reflexive $\\mathcal{O}_X$-modules\nis isomorphic to the Weil divisor class group $\\text{Cl}(X)$ of $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors on normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBM","source_file":"divisors.tex","source_line":7361,"source_end_line":7366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7361-L7366","statement_sha256":"4f2caaddd62b5e00b1773da6298178849c66c0286a1ea8f5ab54b1d439487298","origin":"The Stacks Project","memory_eligible":false,"source_rank":6185,"rank":6185,"depth":34,"x":2468.431,"y":762.821,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EBN","tag":"0EBN","title":"Weil divisors on normal schemes · Lemma 0EBN","summary":"Let X be an integral locally Noetherian normal scheme. Let F be a rank 1 coherent reflexive O_X-module. Let s ∈ Γ(X, F). Let U = (x ∈ X mid s : O_X, x → F_x is an isomorphism) Then j : U → X is an open subscheme of X and j_*O_U = colim (O_X xrightarrows F xrightarrows F^[2] xrightarrows F^[3] xrightarrows …) where F^[1] = F and inductively F^[n + 1] = (F ⊗_O_X F^[n])^**.","statement_latex":"Let $X$ be an integral locally Noetherian normal scheme.\nLet $\\mathcal{F}$ be a rank 1 coherent reflexive $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{F})$. Let\n$$\nU = \\{x \\in X \\mid s : \\mathcal{O}_{X, x} \\to \\mathcal{F}_x\n\\text{ is an isomorphism}\\}\n$$\nThen $j : U \\to X$ is an open subscheme of $X$ and\n$$\nj_*\\mathcal{O}_U =\n\\colim (\\mathcal{O}_X \\xrightarrow{s} \\mathcal{F}\n\\xrightarrow{s} \\mathcal{F}^{[2]}\n\\xrightarrow{s} \\mathcal{F}^{[3]}\n\\xrightarrow{s} \\ldots)\n$$\nwhere $\\mathcal{F}^{[1]} = \\mathcal{F}$ and\ninductively $\\mathcal{F}^{[n + 1]} =\n(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{F}^{[n]})^{**}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors on normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBN","source_file":"divisors.tex","source_line":7456,"source_end_line":7476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7456-L7476","statement_sha256":"de7811407130ae17e586eb876e925cfe4dfc9fdb8496fec983c4000bd487020d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6186,"rank":6186,"depth":34,"x":2421.798,"y":612.211,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EBP","tag":"0EBP","title":"Weil divisors on normal schemes · Lemma 0EBP","summary":"Assumptions and notation as in Lemma [Tag 0EBN]. If s is nonzero, then every irreducible component of X setminus U has codimension 1 in X.","statement_latex":"Assumptions and notation as in Lemma \\ref{lemma-structure-sheaf-Xs}.\nIf $s$ is nonzero, then every irreducible component of $X \\setminus U$\nhas codimension $1$ in $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors on normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBP","source_file":"divisors.tex","source_line":7520,"source_end_line":7525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7520-L7525","statement_sha256":"9f3ceb95245b542654d5aa80ed62751460dbe87c3f0b7c9ef228398331c6e0a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6187,"rank":6187,"depth":35,"x":2577.651,"y":696.999,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EBR","tag":"0EBR","title":"Weil divisors on normal schemes · Lemma 0EBR","summary":"Assumptions and notation as in Lemma [Tag 0EBN]. The following are equivalent • the inclusion morphism j : U → X is affine, and • for every x ∈ X setminus U there is an n > 0 such that s^n ∈ m_x F^[n]_x.","statement_latex":"Assumptions and notation as in Lemma \\ref{lemma-structure-sheaf-Xs}.\nThe following are equivalent\n\\begin{enumerate}\n\\item the inclusion morphism $j : U \\to X$ is affine, and\n\\item for every $x \\in X \\setminus U$ there is an $n > 0$\nsuch that $s^n \\in \\mathfrak m_x \\mathcal{F}^{[n]}_x$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Weil divisors on normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBR","source_file":"divisors.tex","source_line":7558,"source_end_line":7567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7558-L7567","statement_sha256":"064060b5aece1550e87e4501c09c6459a7396e16bbab9aaa6bd6e1cd0d952c57","origin":"The Stacks Project","memory_eligible":false,"source_rank":6188,"rank":6188,"depth":35,"x":2394.131,"y":722.971,"cluster":"divisors-intersection-theory"},{"id":"stacks:07ZX","tag":"07ZX","title":"Relative Proj · Lemma 07ZX","summary":"Let S be a scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. If one of the following holds • A is of finite type as a sheaf of A_0-algebras, • A is generated by A_1 as an A_0-algebra and A_1 is a finite type A_0-module, • there exists a finite type quasi-coherent A_0-submodule F ⊂ A_+ such that A_+/FA is a locally nilpotent sheaf of ideals of A/FA, then p is quasi-compact.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded\n$\\mathcal{O}_S$-algebra. Let\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. If one of the following holds\n\\begin{enumerate}\n\\item $\\mathcal{A}$ is of finite type as a sheaf of\n$\\mathcal{A}_0$-algebras,\n\\item $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ as an\n$\\mathcal{A}_0$-algebra and $\\mathcal{A}_1$ is a finite type\n$\\mathcal{A}_0$-module,\n\\item there exists a finite type quasi-coherent $\\mathcal{A}_0$-submodule\n$\\mathcal{F} \\subset \\mathcal{A}_{+}$ such that\n$\\mathcal{A}_{+}/\\mathcal{F}\\mathcal{A}$ is a locally nilpotent\nsheaf of ideals of $\\mathcal{A}/\\mathcal{F}\\mathcal{A}$,\n\\end{enumerate}\nthen $p$ is quasi-compact.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZX","source_file":"divisors.tex","source_line":7646,"source_end_line":7664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7646-L7664","statement_sha256":"9d7abe4836156dd615608cea3b22da9d081bf76a2b5598a7b6d91926dd3f61a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6189,"rank":6189,"depth":5,"x":2508.826,"y":599.41,"cluster":"divisors-intersection-theory"},{"id":"stacks:07ZY","tag":"07ZY","title":"Relative Proj · Lemma 07ZY","summary":"Let S be a scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. If A is of finite type as a sheaf of O_S-algebras, then p is of finite type and O_X(d) is a finite type O_X-module.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded\n$\\mathcal{O}_S$-algebra. Let\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. If $\\mathcal{A}$ is of finite type as a sheaf of\n$\\mathcal{O}_S$-algebras, then $p$ is of finite type and $\\mathcal{O}_X(d)$\nis a finite type $\\mathcal{O}_X$-module.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZY","source_file":"divisors.tex","source_line":7688,"source_end_line":7696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7688-L7696","statement_sha256":"2a466a1c3f76bf4ceeccccc471c158a0669915072dda2fb0c3be7e404f56bfd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6190,"rank":6190,"depth":6,"x":2523.649,"y":755.951,"cluster":"divisors-intersection-theory"},{"id":"stacks:07ZZ","tag":"07ZZ","title":"Relative Proj · Lemma 07ZZ","summary":"Let S be a scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. If O_S → A_0 is an integral algebra map_S in A_0, see Morphisms, Definition [Tag 035G], equals A_0. and A is of finite type as an A_0-algebra, then p is universally closed.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded\n$\\mathcal{O}_S$-algebra. Let\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. If $\\mathcal{O}_S \\to \\mathcal{A}_0$\nis an integral algebra map\\footnote{In other words, the integral\nclosure of $\\mathcal{O}_S$ in $\\mathcal{A}_0$, see\nMorphisms, Definition \\ref{morphisms-definition-integral-closure}, equals\n$\\mathcal{A}_0$.} and $\\mathcal{A}$ is of finite type as an\n$\\mathcal{A}_0$-algebra, then $p$ is universally closed.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ZZ","source_file":"divisors.tex","source_line":7715,"source_end_line":7726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7715-L7726","statement_sha256":"a6046bc4489bdc6b54426d4ec9a2d5aef1cdc4bf77729378d3f2156b9fb3e186","origin":"The Stacks Project","memory_eligible":false,"source_rank":6191,"rank":6191,"depth":20,"x":2386.53,"y":648.693,"cluster":"divisors-intersection-theory"},{"id":"stacks:0800","tag":"0800","title":"Relative Proj · Lemma 0800","summary":"Let S be a scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. The following conditions are equivalent • A_0 is a finite type O_S-module and A is of finite type as an A_0-algebra, • A_0 is a finite type O_S-module and A is of finite type as an O_S-algebra If these conditions hold, then p is locally projective and in particular proper.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded\n$\\mathcal{O}_S$-algebra. Let\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. The following conditions are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_S$-module\nand $\\mathcal{A}$ is of finite type as an $\\mathcal{A}_0$-algebra,\n\\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_S$-module\nand $\\mathcal{A}$ is of finite type as an $\\mathcal{O}_S$-algebra\n\\end{enumerate}\nIf these conditions hold, then $p$ is locally projective and in\nparticular proper.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0800","source_file":"divisors.tex","source_line":7750,"source_end_line":7764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7750-L7764","statement_sha256":"9187bdd801d9ba0c716cb3f89f566245ddd1af5a1b6b032aeddea70c48c8b955","origin":"The Stacks Project","memory_eligible":false,"source_rank":6192,"rank":6192,"depth":25,"x":2574.307,"y":649.982,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B3U","tag":"0B3U","title":"Relative Proj · Lemma 0B3U","summary":"Let S be a scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. If A is generated by A_1 over A_0 and A_1 is a finite type O_S-module, then p is projective.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded\n$\\mathcal{O}_S$-algebra. Let\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. If $\\mathcal{A}$ is generated by\n$\\mathcal{A}_1$ over $\\mathcal{A}_0$ and $\\mathcal{A}_1$\nis a finite type $\\mathcal{O}_S$-module, then $p$ is projective.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3U","source_file":"divisors.tex","source_line":7802,"source_end_line":7810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7802-L7810","statement_sha256":"eecc6549a8fa8c9159e1da980b15748c7bdfa2a34f90f246344a27683f0c8cbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6193,"rank":6193,"depth":5,"x":2434.5,"y":755.812,"cluster":"divisors-intersection-theory"},{"id":"stacks:0D4C","tag":"0D4C","title":"Relative Proj · Lemma 0D4C","summary":"Let S be a scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. If A_d is a flat O_S-module for d gg 0, then p is flat and O_X(d) is flat over S.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded\n$\\mathcal{O}_S$-algebra. Let\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. If $\\mathcal{A}_d$ is a flat $\\mathcal{O}_S$-module\nfor $d \\gg 0$, then $p$ is flat and $\\mathcal{O}_X(d)$ is\nflat over $S$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4C","source_file":"divisors.tex","source_line":7824,"source_end_line":7832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7824-L7832","statement_sha256":"d187935dcc2a648643311c51cfc973c877e42297102559815aa607b65948dfac","origin":"The Stacks Project","memory_eligible":false,"source_rank":6194,"rank":6194,"depth":3,"x":2452.525,"y":598.101,"cluster":"divisors-intersection-theory"},{"id":"stacks:0D4D","tag":"0D4D","title":"Relative Proj · Lemma 0D4D","summary":"Let S be a scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. If A is a finitely presented O_S-algebra, then p is of finite presentation and O_X(d) is an O_X-module of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded\n$\\mathcal{O}_S$-algebra. Let\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. If $\\mathcal{A}$ is a finitely presented\n$\\mathcal{O}_S$-algebra, then $p$ is of finite presentation\nand $\\mathcal{O}_X(d)$ is an $\\mathcal{O}_X$-module of finite presentation.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4D","source_file":"divisors.tex","source_line":7845,"source_end_line":7853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7845-L7853","statement_sha256":"9e28274eb08e73255b6a487dfb3e96f22cb41b086bb0acf40cb39a50ab6ebf26","origin":"The Stacks Project","memory_eligible":false,"source_rank":6195,"rank":6195,"depth":18,"x":2566.308,"y":724.899,"cluster":"divisors-intersection-theory"},{"id":"stacks:0801","tag":"0801","title":"Closed subschemes of relative proj · Lemma 0801","summary":"Let S be a scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. Let i : Z → X be a closed subscheme. Denote I ⊂ A the kernel of the canonical map A → bigoplus_d ≥ 0 p_*((i_*O_Z)(d)). If p is quasi-compact, then there is an isomorphism Z = underlineProj_S(A/I).","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{A}$ be a quasi-coherent graded\n$\\mathcal{O}_S$-algebra. Let\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subscheme. Denote\n$\\mathcal{I} \\subset \\mathcal{A}$ the kernel of the canonical map\n$$\n\\mathcal{A}\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\geq 0} p_*\\left((i_*\\mathcal{O}_Z)(d)\\right).\n$$\nIf $p$ is quasi-compact, then there is an isomorphism\n$Z = \\underline{\\text{Proj}}_S(\\mathcal{A}/\\mathcal{I})$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Closed subschemes of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0801","source_file":"divisors.tex","source_line":7903,"source_end_line":7917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L7903-L7917","statement_sha256":"720a46d6a6155980ce2860946406c9520a8b2c55f57cff255c6dbb5983ff8300","origin":"The Stacks Project","memory_eligible":false,"source_rank":6196,"rank":6196,"depth":17,"x":2380.036,"y":695.9,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BXL","tag":"0BXL","title":"Closed subschemes of relative proj · Lemma 0BXL","summary":"Let R be a Noetherian UFD. Let Z ⊂ P^n_R be a closed subscheme which has no embedded points such that every irreducible component of Z has codimension 1 in P^n_R. Then the ideal I(Z) ⊂ R[T_0, …, T_n] corresponding to Z is principal.","statement_latex":"Let $R$ be a Noetherian UFD. Let $Z \\subset \\mathbf{P}^n_R$ be a\nclosed subscheme\nwhich has no embedded points such that every irreducible component\nof $Z$ has codimension $1$ in $\\mathbf{P}^n_R$.\nThen the ideal $I(Z) \\subset R[T_0, \\ldots, T_n]$ corresponding\nto $Z$ is principal.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Closed subschemes of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXL","source_file":"divisors.tex","source_line":8041,"source_end_line":8049,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8041-L8049","statement_sha256":"c9152600f5b1ef1dbc4916fba9df2046aa125e9a8a2915debff72df02553d726","origin":"The Stacks Project","memory_eligible":false,"source_rank":6197,"rank":6197,"depth":28,"x":2541.057,"y":611.406,"cluster":"divisors-intersection-theory"},{"id":"stacks:0802","tag":"0802","title":"Closed subschemes of relative proj · Lemma 0802","summary":"Let S be a quasi-compact and quasi-separated scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. Let i : Z → X be a closed subscheme. If p is quasi-compact and i of finite presentation, then there exists a d > 0 and a quasi-coherent finite type O_S-submodule F ⊂ A_d such that Z = underlineProj_S(A/FA).","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra. Let\n$p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subscheme.\nIf $p$ is quasi-compact and $i$ of finite presentation, then there exists\na $d > 0$ and a quasi-coherent finite type $\\mathcal{O}_S$-submodule\n$\\mathcal{F} \\subset \\mathcal{A}_d$ such that\n$Z = \\underline{\\text{Proj}}_S(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Closed subschemes of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0802","source_file":"divisors.tex","source_line":8098,"source_end_line":8108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8098-L8108","statement_sha256":"2ef745abe1558a060a9067117766e935701360c06f982d33ca37a22805219a4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6198,"rank":6198,"depth":18,"x":2490.16,"y":765.408,"cluster":"divisors-intersection-theory"},{"id":"stacks:0803","tag":"0803","title":"Closed subschemes of relative proj · Lemma 0803","summary":"Let S be a quasi-compact and quasi-separated scheme. Let A be a quasi-coherent graded O_S-algebra. Let p : X = underlineProj_S(A) → S be the relative Proj of A. Let i : Z → X be a closed subscheme. Let U ⊂ S be an open. Assume that • p is quasi-compact, • i of finite presentation, • U ∩ p(i(Z)) = ∅, • U is quasi-compact, • A_n is a finite type O_S-module for all n. Then there exists a d > 0 and a quasi-coherent finite type O_S-submodule F ⊂ A_d with (a) Z =…","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_S$-algebra.\nLet $p : X = \\underline{\\text{Proj}}_S(\\mathcal{A}) \\to S$ be the relative\nProj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subscheme.\nLet $U \\subset S$ be an open. Assume that\n\\begin{enumerate}\n\\item $p$ is quasi-compact,\n\\item $i$ of finite presentation,\n\\item $U \\cap p(i(Z)) = \\emptyset$,\n\\item $U$ is quasi-compact,\n\\item $\\mathcal{A}_n$ is a finite type $\\mathcal{O}_S$-module for all $n$.\n\\end{enumerate}\nThen there exists a $d > 0$ and a quasi-coherent finite type\n$\\mathcal{O}_S$-submodule $\\mathcal{F} \\subset \\mathcal{A}_d$ with (a)\n$Z = \\underline{\\text{Proj}}_S(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$\nand (b) the support of $\\mathcal{A}_d/\\mathcal{F}$ is disjoint from $U$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Closed subschemes of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0803","source_file":"divisors.tex","source_line":8144,"source_end_line":8162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8144-L8162","statement_sha256":"aba96fec5c799608585a50b0aed75b2195481e12220d0ead2bdded5ae41fded2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6199,"rank":6199,"depth":19,"x":2403.662,"y":622.664,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B3V","tag":"0B3V","title":"Closed subschemes of relative proj · Lemma 0B3V","summary":"Let X be a scheme. Let E be a quasi-coherent O_X-module. There is a bijection ( sections σ of the morphism P(E) → X ) ↔ ( surjections E → L where L is an invertible O_X-module ) In this case σ is a closed immersion and there is a canonical isomorphism Ker(E → L) ⊗_O_X L^⊗ -1 → C_σ(X)/P(E) Both the bijection and isomorphism are compatible with base change.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{E}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. There is a bijection\n$$\n\\left\\{\n\\begin{matrix}\n\\text{sections }\\sigma\\text{ of the } \\\\\n\\text{morphism } \\mathbf{P}(\\mathcal{E}) \\to X\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{surjections }\\mathcal{E} \\to \\mathcal{L}\\text{ where} \\\\\n\\mathcal{L}\\text{ is an invertible }\\mathcal{O}_X\\text{-module}\n\\end{matrix}\n\\right\\}\n$$\nIn this case $\\sigma$ is a closed immersion and there is a canonical\nisomorphism\n$$\n\\Ker(\\mathcal{E} \\to \\mathcal{L})\n\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes -1}\n\\longrightarrow\n\\mathcal{C}_{\\sigma(X)/\\mathbf{P}(\\mathcal{E})}\n$$\nBoth the bijection and isomorphism are compatible with base change.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Closed subschemes of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3V","source_file":"divisors.tex","source_line":8189,"source_end_line":8217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8189-L8217","statement_sha256":"1760b23bff1525e7dbc877050e856ab2f5c22b7217828ed1e65e496f8895f5ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":6200,"rank":6200,"depth":19,"x":2582.616,"y":678.96,"cluster":"divisors-intersection-theory"},{"id":"stacks:01OG","tag":"01OG","title":"Blowing up · Definition 01OG","summary":"Let X be a scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals, and let Z ⊂ X be the closed subscheme corresponding to I, see Schemes, Definition [Tag 01IO]. The blowing up of X along Z, or the blowing up of X in the ideal sheaf I is the morphism b : underlineProj_X (bigoplus_n ≥ 0 I^n) → X The exceptional divisor of the blowup is the inverse image b^-1(Z). Sometimes Z is called the center of the blowup.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a\nquasi-coherent sheaf of ideals, and let $Z \\subset X$ be the closed subscheme\ncorresponding to $\\mathcal{I}$, see\nSchemes, Definition \\ref{schemes-definition-immersion}.\nThe {\\it blowing up of $X$ along $Z$}, or the\n{\\it blowing up of $X$ in the ideal sheaf $\\mathcal{I}$} is\nthe morphism\n$$\nb :\n\\underline{\\text{Proj}}_X\n\\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n\\right)\n\\longrightarrow\nX\n$$\nThe {\\it exceptional divisor} of the blowup is the inverse image\n$b^{-1}(Z)$. Sometimes $Z$ is called the {\\it center} of the blowup.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01OG","source_file":"divisors.tex","source_line":8283,"source_end_line":8301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8283-L8301","statement_sha256":"d19344066dd47d4b39fb4996cff181d9e284e4f459f8aeb19f59b0ad12061f4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6201,"rank":6201,"depth":13,"x":2405.01,"y":739.118,"cluster":"divisors-intersection-theory"},{"id":"stacks:0804","tag":"0804","title":"Blowing up · Lemma 0804","summary":"Let X be a scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let U = Spec(A) be an affine open subscheme of X and let I ⊂ A be the ideal corresponding to I|_U. If b : X' → X is the blowup of X in I, then there is a canonical isomorphism b^-1(U) = Proj(bigoplus_d ≥ 0 I^d) of b^-1(U) with the homogeneous spectrum of the Rees algebra of I in A. Moreover, b^-1(U) has an affine open covering by spectra of the affine blowup algebras A[fracIa].","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a\nquasi-coherent sheaf of ideals. Let $U = \\Spec(A)$ be an affine open\nsubscheme of $X$ and let $I \\subset A$ be the ideal corresponding to\n$\\mathcal{I}|_U$. If $b : X' \\to X$ is the blowup of $X$ in $\\mathcal{I}$,\nthen there is a canonical isomorphism\n$$\nb^{-1}(U) = \\text{Proj}(\\bigoplus\\nolimits_{d \\geq 0} I^d)\n$$\nof $b^{-1}(U)$ with the homogeneous spectrum of the Rees algebra\nof $I$ in $A$. Moreover, $b^{-1}(U)$ has an affine open covering by\nspectra of the affine blowup algebras $A[\\frac{I}{a}]$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0804","source_file":"divisors.tex","source_line":8321,"source_end_line":8334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8321-L8334","statement_sha256":"dc459bc6760e9d2c4c982238c6fb8eb579ccd1aef9b0946ebb002a9021a8cf28","origin":"The Stacks Project","memory_eligible":false,"source_rank":6202,"rank":6202,"depth":1,"x":2487.771,"y":593.674,"cluster":"divisors-intersection-theory"},{"id":"stacks:0805","tag":"0805","title":"Blowing up · Lemma 0805","summary":"Blowing up commutes with flat base change. Let X_1 → X_2 be a flat morphism of schemes. Let Z_2 ⊂ X_2 be a closed subscheme. Let Z_1 be the inverse image of Z_2 in X_1. Let X'_i be the blowup of Z_i in X_i. Then there exists a cartesian diagram xymatrix X_1' ar[r] ar[d] & X_2' ar[d] X_1 ar[r] & X_2 of schemes.","statement_latex":"\\begin{slogan}\nBlowing up commutes with flat base change.\n\\end{slogan}\nLet $X_1 \\to X_2$ be a flat morphism of schemes. Let $Z_2 \\subset X_2$ be a\nclosed subscheme. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$.\nLet $X'_i$ be the blowup of $Z_i$ in $X_i$. Then there exists a cartesian\ndiagram\n$$\n\\xymatrix{\nX_1' \\ar[r] \\ar[d] & X_2' \\ar[d] \\\\\nX_1 \\ar[r] & X_2\n}\n$$\nof schemes.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0805","source_file":"divisors.tex","source_line":8350,"source_end_line":8366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8350-L8366","statement_sha256":"401a38f5a01f4ec75a1b4e2fea850762fd6bbacb2060f260115d09b4455ffbb2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6203,"rank":6203,"depth":14,"x":2543.824,"y":748.208,"cluster":"divisors-intersection-theory"},{"id":"stacks:02OS","tag":"02OS","title":"Blowing up · Lemma 02OS","summary":"Let X be a scheme. Let Z ⊂ X be a closed subscheme. The blowing up b : X' → X of Z in X has the following properties: • b|_b^-1(X setminus Z) : b^-1(X setminus Z) → X setminus Z is an isomorphism, • the exceptional divisor E = b^-1(Z) is an effective Cartier divisor on X', • there is a canonical isomorphism O_X'(-1) = O_X'(E)","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme.\nThe blowing up $b : X' \\to X$ of $Z$ in $X$\nhas the following properties:\n\\begin{enumerate}\n\\item $b|_{b^{-1}(X \\setminus Z)} : b^{-1}(X \\setminus Z) \\to X \\setminus Z$\nis an isomorphism,\n\\item the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor\non $X'$,\n\\item there is a canonical isomorphism\n$\\mathcal{O}_{X'}(-1) = \\mathcal{O}_{X'}(E)$\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02OS","source_file":"divisors.tex","source_line":8382,"source_end_line":8395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8382-L8395","statement_sha256":"241f8605f7a1fb4276e22a35ac12633486d293c65d3e23ff832fa90119f6f20e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6204,"rank":6204,"depth":19,"x":2377.873,"y":665.89,"cluster":"divisors-intersection-theory"},{"id":"stacks:0806","tag":"0806","title":"Universal property blowing up · Lemma 0806","summary":"Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let C be the full subcategory of (Sch/X) consisting of Y → X such that the inverse image of Z is an effective Cartier divisor on Y. Then the blowing up b : X' → X of Z in X is a final object of C.","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme.\nLet $\\mathcal{C}$ be the full subcategory of $(\\Sch/X)$ consisting\nof $Y \\to X$ such that the inverse image of $Z$ is an effective\nCartier divisor on $Y$. Then the blowing up $b : X' \\to X$ of $Z$ in $X$\nis a final object of $\\mathcal{C}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0806","source_file":"divisors.tex","source_line":8433,"source_end_line":8440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8433-L8440","statement_sha256":"809f3b1410c4f0d92a680033e45c50e976b423514e501313aa470ff782440e31","origin":"The Stacks Project","memory_eligible":false,"source_rank":6205,"rank":6205,"depth":20,"x":2566.834,"y":632.356,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BFL","tag":"0BFL","title":"Blowing up · Lemma 0BFL","summary":"Let b : X' → X be the blowing up of the scheme X along a closed subscheme Z. Let U = Spec(A) be an affine open of X and let I ⊂ A be the ideal corresponding to Z ∩ U. Let a ∈ I and let x' ∈ X' be a point mapping to a point of U. Then x' is a point of the affine open U' = Spec(A[fracIa]) if and only if the image of a in O_X', x' cuts out the exceptional divisor.","statement_latex":"Let $b : X' \\to X$ be the blowing up of the scheme $X$ along a closed\nsubscheme $Z$. Let $U = \\Spec(A)$ be an affine open of $X$ and let\n$I \\subset A$ be the ideal corresponding to $Z \\cap U$.\nLet $a \\in I$ and let $x' \\in X'$ be a point mapping to a point of $U$.\nThen $x'$ is a point of the affine open $U' = \\Spec(A[\\frac{I}{a}])$\nif and only if the image of $a$ in $\\mathcal{O}_{X', x'}$ cuts\nout the exceptional divisor.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFL","source_file":"divisors.tex","source_line":8469,"source_end_line":8478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8469-L8478","statement_sha256":"52ccfe7783f6d68f09824af21b7cc7efd414548a47b640389cb2ba0cad92b47d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6206,"rank":6206,"depth":21,"x":2454.231,"y":764.578,"cluster":"divisors-intersection-theory"},{"id":"stacks:0807","tag":"0807","title":"Blowing up · Lemma 0807","summary":"Let X be a scheme. Let Z ⊂ X be an effective Cartier divisor. The blowup of X in Z is the identity morphism of X.","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be an effective Cartier divisor.\nThe blowup of $X$ in $Z$ is the identity morphism of $X$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0807","source_file":"divisors.tex","source_line":8505,"source_end_line":8509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8505-L8509","statement_sha256":"f1837224cea8db19797479350aa41f85fde95dd67d8fa328cc94b5693f706c5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6207,"rank":6207,"depth":21,"x":2430.885,"y":602.852,"cluster":"divisors-intersection-theory"},{"id":"stacks:0808","tag":"0808","title":"Blowing up · Lemma 0808","summary":"Let X be a scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. If X is reduced, then the blowup X' of X in I is reduced.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a\nquasi-coherent sheaf of ideals. If $X$ is reduced, then the\nblowup $X'$ of $X$ in $\\mathcal{I}$ is reduced.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0808","source_file":"divisors.tex","source_line":8516,"source_end_line":8521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8516-L8521","statement_sha256":"224a7bff2572ed28f0cf95179f27184a69e9c5e8794ef6c3f96b7bec13fe445d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6208,"rank":6208,"depth":2,"x":2578.458,"y":709.078,"cluster":"divisors-intersection-theory"},{"id":"stacks:02ND","tag":"02ND","title":"Blowing up · Lemma 02ND","summary":"Let X be a scheme. Let I ⊂ O_X be a nonzero quasi-coherent sheaf of ideals. If X is integral, then the blowup X' of X in I is integral.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a\nnonzero quasi-coherent sheaf of ideals. If $X$ is integral, then the\nblowup $X'$ of $X$ in $\\mathcal{I}$ is integral.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ND","source_file":"divisors.tex","source_line":8528,"source_end_line":8533,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8528-L8533","statement_sha256":"e499d097675590c267e2ac0cc3ca3c0806bc0c8f24d484cb6a7b95ad8c3fce44","origin":"The Stacks Project","memory_eligible":false,"source_rank":6209,"rank":6209,"depth":2,"x":2383.818,"y":714.496,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BFM","tag":"0BFM","title":"Blowing up · Lemma 0BFM","summary":"Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let b : X' → X be the blowing up of X along Z. Then b induces an bijective map from the set of generic points of irreducible components of X' to the set of generic points of irreducible components of X which are not in Z.","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme.\nLet $b : X' \\to X$ be the blowing up of $X$ along $Z$. Then\n$b$ induces an bijective map from the set of generic points\nof irreducible components of $X'$ to the set of generic points of\nirreducible components of $X$ which are not in $Z$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFM","source_file":"divisors.tex","source_line":8540,"source_end_line":8547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8540-L8547","statement_sha256":"07dd34555a65017d6bd2f155fade684452d6dca523ffd22c42c64855a0f8072a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6210,"rank":6210,"depth":20,"x":2523.27,"y":599.824,"cluster":"divisors-intersection-theory"},{"id":"stacks:0809","tag":"0809","title":"Blowing up · Lemma 0809","summary":"Let X be a scheme. Let b : X' → X be a blowup of X in a closed subscheme. The pullback b^-1D is defined for all effective Cartier divisors D ⊂ X and pullbacks of meromorphic functions are defined for b (Definitions [Tag 01WV] and [Tag 02OT]).","statement_latex":"Let $X$ be a scheme. Let $b : X' \\to X$ be a blowup of $X$ in a closed\nsubscheme. The pullback $b^{-1}D$ is defined\nfor all effective Cartier divisors $D \\subset X$\nand pullbacks of meromorphic functions are defined for $b$\n(Definitions\n\\ref{definition-pullback-effective-Cartier-divisor} and\n\\ref{definition-pullback-meromorphic-sections}).","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0809","source_file":"divisors.tex","source_line":8566,"source_end_line":8575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8566-L8575","statement_sha256":"381bae550f8af615700964101e5f18f84e4c1b147b0fc03e2232d95bde6ec423","origin":"The Stacks Project","memory_eligible":false,"source_rank":6211,"rank":6211,"depth":2,"x":2512.635,"y":763.843,"cluster":"divisors-intersection-theory"},{"id":"stacks:080A","tag":"080A","title":"Blowing up · Lemma 080A","summary":"Let X be a scheme. Let I, J ⊂ O_X be quasi-coherent sheaves of ideals. Let b : X' → X be the blowing up of X in I. Let b' : X\" → X' be the blowing up of X' in b^-1J O_X'. Then X\" → X is canonically isomorphic to the blowing up of X in IJ.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$\nbe quasi-coherent sheaves of ideals. Let $b : X' \\to X$\nbe the blowing up of $X$ in $\\mathcal{I}$. Let $b' : X'' \\to X'$ be the\nblowing up of $X'$ in $b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$. Then $X'' \\to X$\nis canonically isomorphic to the blowing up of $X$ in $\\mathcal{I}\\mathcal{J}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080A","source_file":"divisors.tex","source_line":8588,"source_end_line":8595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8588-L8595","statement_sha256":"899a429b24d06e2cacbc38b386887a71eeadcb6ffc70c603e8ab39da36a39a07","origin":"The Stacks Project","memory_eligible":false,"source_rank":6212,"rank":6212,"depth":21,"x":2388.326,"y":636.606,"cluster":"divisors-intersection-theory"},{"id":"stacks:02NS","tag":"02NS","title":"Blowing up · Lemma 02NS","summary":"Let X be a scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let b : X' → X be the blowing up of X in the ideal sheaf I. If I is of finite type, then • b : X' → X is a projective morphism, and • O_X'(1) is a b-relatively ample invertible sheaf.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a\nquasi-coherent sheaf of ideals. Let $b : X' \\to X$ be the blowing up of $X$\nin the ideal sheaf $\\mathcal{I}$. If $\\mathcal{I}$ is of finite type, then\n\\begin{enumerate}\n\\item $b : X' \\to X$ is a projective morphism, and\n\\item $\\mathcal{O}_{X'}(1)$ is a $b$-relatively ample invertible sheaf.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02NS","source_file":"divisors.tex","source_line":8622,"source_end_line":8631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8622-L8631","statement_sha256":"f171b1f59f87957b2a28abe26e3d94bd542d922ed6a4e4eeb6cc766516654f0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6213,"rank":6213,"depth":21,"x":2582.702,"y":659.94,"cluster":"divisors-intersection-theory"},{"id":"stacks:080B","tag":"080B","title":"Blowing up · Lemma 080B","summary":"Composition of blowing ups is a blowing up Let X be a quasi-compact and quasi-separated scheme. Let Z ⊂ X be a closed subscheme of finite presentation. Let b : X' → X be the blowing up with center Z. Let Z' ⊂ X' be a closed subscheme of finite presentation. Let X\" → X' be the blowing up with center Z'. There exists a closed subscheme Y ⊂ X of finite presentation, such that • Y = Z ∪ b(Z') set theoretically, and • the composition X\" → X is isomorphic to the blowing up of X…","statement_latex":"\\begin{slogan}\nComposition of blowing ups is a blowing up\n\\end{slogan}\nLet $X$ be a quasi-compact and quasi-separated scheme.\nLet $Z \\subset X$ be a closed subscheme of finite presentation.\nLet $b : X' \\to X$ be the blowing up with center $Z$. Let $Z' \\subset X'$ be\na closed subscheme of finite presentation.\nLet $X'' \\to X'$ be the blowing up with center $Z'$.\nThere exists a closed subscheme $Y \\subset X$ of finite presentation,\nsuch that\n\\begin{enumerate}\n\\item $Y = Z \\cup b(Z')$ set theoretically, and\n\\item the composition $X'' \\to X$ is isomorphic to the blowing up\nof $X$ in $Y$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080B","source_file":"divisors.tex","source_line":8656,"source_end_line":8673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8656-L8673","statement_sha256":"3afbd4ade144311fbcc5ab8a825d59a352ab3c1a97c22be3d0c7516be09f7076","origin":"The Stacks Project","memory_eligible":false,"source_rank":6214,"rank":6214,"depth":22,"x":2420.281,"y":753.214,"cluster":"divisors-intersection-theory"},{"id":"stacks:080D","tag":"080D","title":"Strict transform · Definition 080D","summary":"With Z ⊂ S and f : X → S as above. • Given a quasi-coherent O_X-module F the strict transform of F with respect to the blowup of S in Z is the quotient F' of pr_X^*F by the submodule of sections supported on pr_S'^-1E. • The strict transform of X is the closed subscheme X' ⊂ X ×_S S' cut out by the quasi-coherent ideal of sections of O_X ×_S S' supported on pr_S'^-1E.","statement_latex":"With $Z \\subset S$ and $f : X \\to S$ as above.\n\\begin{enumerate}\n\\item Given a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$\nthe {\\it strict transform} of $\\mathcal{F}$ with respect to the blowup\nof $S$ in $Z$ is the quotient $\\mathcal{F}'$ of $\\text{pr}_X^*\\mathcal{F}$\nby the submodule of sections supported on $\\text{pr}_{S'}^{-1}E$.\n\\item The {\\it strict transform} of $X$ is the closed subscheme\n$X' \\subset X \\times_S S'$ cut out by the quasi-coherent ideal of\nsections of $\\mathcal{O}_{X \\times_S S'}$ supported on $\\text{pr}_{S'}^{-1}E$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Strict transform","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080D","source_file":"divisors.tex","source_line":8780,"source_end_line":8792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8780-L8792","statement_sha256":"4b7de5b6b2e5561eb1ce45e80c730aefb3c253d99d59af0fb4162fca50ac7028","origin":"The Stacks Project","memory_eligible":false,"source_rank":6215,"rank":6215,"depth":0,"x":2465.13,"y":591.952,"cluster":"divisors-intersection-theory"},{"id":"stacks:080E","tag":"080E","title":"Strict transform · Lemma 080E","summary":"In the situation of Definition [Tag 080D]. • The strict transform X' of X is the blowup of X in the closed subscheme f^-1Z of X. • For a quasi-coherent O_X-module F the strict transform F' is canonically isomorphic to the pushforward along X' → X ×_S S' of the strict transform of F relative to the blowing up X' → X.","statement_latex":"In the situation of Definition \\ref{definition-strict-transform}.\n\\begin{enumerate}\n\\item The strict transform $X'$ of $X$ is the blowup of $X$ in the closed\nsubscheme $f^{-1}Z$ of $X$.\n\\item For a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the\nstrict transform $\\mathcal{F}'$ is canonically isomorphic to\nthe pushforward along $X' \\to X \\times_S S'$ of the strict transform of\n$\\mathcal{F}$ relative to the blowing up $X' \\to X$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080E","source_file":"divisors.tex","source_line":8801,"source_end_line":8812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8801-L8812","statement_sha256":"2c836581bd580e2b1d9a36807c33943896eef6a76bb3ddb6b1faf20af8f1c106","origin":"The Stacks Project","memory_eligible":false,"source_rank":6216,"rank":6216,"depth":21,"x":2561.933,"y":736.6,"cluster":"divisors-intersection-theory"},{"id":"stacks:080F","tag":"080F","title":"Strict transform · Lemma 080F","summary":"In the situation of Definition [Tag 080D]. • If X is flat over S at all points lying over Z, then the strict transform of X is equal to the base change X ×_S S'. • Let F be a quasi-coherent O_X-module. If F is flat over S at all points lying over Z, then the strict transform F' of F is equal to the pullback pr_X^*F.","statement_latex":"In the situation of Definition \\ref{definition-strict-transform}.\n\\begin{enumerate}\n\\item If $X$ is flat over $S$ at all points lying over $Z$, then\nthe strict transform of $X$ is equal to the base change $X \\times_S S'$.\n\\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $\\mathcal{F}$ is flat over $S$ at all points lying over $Z$, then\nthe strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is equal to the\npullback $\\text{pr}_X^*\\mathcal{F}$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080F","source_file":"divisors.tex","source_line":8868,"source_end_line":8879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8868-L8879","statement_sha256":"9ac5e02beb37becdf796b5d03cf939d6880c2923146ccc26e734846c18ade676","origin":"The Stacks Project","memory_eligible":false,"source_rank":6217,"rank":6217,"depth":15,"x":2373.858,"y":684.765,"cluster":"divisors-intersection-theory"},{"id":"stacks:080G","tag":"080G","title":"Strict transform · Lemma 080G","summary":"Let S be a scheme. Let Z ⊂ S be a closed subscheme. Let b : S' → S be the blowing up of Z in S. Let g : X → Y be an affine morphism of schemes over S. Let F be a quasi-coherent sheaf on X. Let g' : X ×_S S' → Y ×_S S' be the base change of g. Let F' be the strict transform of F relative to b. Then g'_*F' is the strict transform of g_*F.","statement_latex":"Let $S$ be a scheme. Let $Z \\subset S$ be a closed subscheme.\nLet $b : S' \\to S$ be the blowing up of $Z$ in $S$. Let\n$g : X \\to Y$ be an affine morphism of schemes over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $g' : X \\times_S S' \\to Y \\times_S S'$ be the base change\nof $g$. Let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$\nrelative to $b$. Then $g'_*\\mathcal{F}'$ is the strict transform\nof $g_*\\mathcal{F}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080G","source_file":"divisors.tex","source_line":8908,"source_end_line":8918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8908-L8918","statement_sha256":"d771a35762cb02830788b4f1962abedaa57a6520feee7e43e1863931df658281","origin":"The Stacks Project","memory_eligible":false,"source_rank":6218,"rank":6218,"depth":26,"x":2554.583,"y":616.133,"cluster":"divisors-intersection-theory"},{"id":"stacks:080H","tag":"080H","title":"Strict transform · Lemma 080H","summary":"Let S be a scheme. Let Z ⊂ S be a closed subscheme. Let D ⊂ S be an effective Cartier divisor. Let Z' ⊂ S be the closed subscheme cut out by the product of the ideal sheaves of Z and D. Let S' → S be the blowup of S in Z. • The blowup of S in Z' is isomorphic to S' → S. • Let f : X → S be a morphism of schemes and let F be a quasi-coherent O_X-module. If F has no nonzero local sections supported in f^-1D, then the strict transform of F relative to the blowing up in Z…","statement_latex":"Let $S$ be a scheme. Let $Z \\subset S$ be a closed subscheme.\nLet $D \\subset S$ be an effective Cartier divisor.\nLet $Z' \\subset S$ be the closed subscheme cut out by the product\nof the ideal sheaves of $Z$ and $D$.\nLet $S' \\to S$ be the blowup of $S$ in $Z$.\n\\begin{enumerate}\n\\item The blowup of $S$ in $Z'$ is isomorphic to $S' \\to S$.\n\\item Let $f : X \\to S$ be a morphism of schemes and let $\\mathcal{F}$\nbe a quasi-coherent $\\mathcal{O}_X$-module. If $\\mathcal{F}$ has\nno nonzero local sections supported in $f^{-1}D$, then the\nstrict transform of $\\mathcal{F}$ relative to the blowing up\nin $Z$ agrees with the strict transform of $\\mathcal{F}$ relative\nto the blowing up of $S$ in $Z'$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080H","source_file":"divisors.tex","source_line":8934,"source_end_line":8950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8934-L8950","statement_sha256":"3ae94e5e4d0352f6087a3d7659a7ddfa206d1f460746018bffbe2c71ed4cddfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6219,"rank":6219,"depth":22,"x":2476.357,"y":769.59,"cluster":"divisors-intersection-theory"},{"id":"stacks:080I","tag":"080I","title":"Strict transform · Lemma 080I","summary":"Let S be a scheme. Let Z ⊂ S be a closed subscheme. Let b : S' → S be the blowing up with center Z. Let Z' ⊂ S' be a closed subscheme. Let S\" → S' be the blowing up with center Z'. Let Y ⊂ S be a closed subscheme such that Y = Z ∪ b(Z') set theoretically and the composition S\" → S is isomorphic to the blowing up of S in Y. In this situation, given any scheme X over S and F ∈ QCoh(O_X) we have • the strict transform of F with respect to the blowing up of S in Y is equal to…","statement_latex":"Let $S$ be a scheme. Let $Z \\subset S$ be a closed subscheme.\nLet $b : S' \\to S$ be the blowing up with center $Z$. Let $Z' \\subset S'$ be\na closed subscheme. Let $S'' \\to S'$ be the blowing up with center $Z'$.\nLet $Y \\subset S$ be a closed subscheme such that\n$Y = Z \\cup b(Z')$ set theoretically and the composition $S'' \\to S$\nis isomorphic to the blowing up of $S$ in $Y$.\nIn this situation, given any scheme $X$ over $S$ and\n$\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have\n\\begin{enumerate}\n\\item the strict transform of $\\mathcal{F}$ with respect to the blowing\nup of $S$ in $Y$ is equal to the strict transform with respect to the\nblowup $S'' \\to S'$ in $Z'$ of the strict transform of $\\mathcal{F}$\nwith respect to the blowup $S' \\to S$ of $S$ in $Z$, and\n\\item the strict transform of $X$ with respect to the blowing\nup of $S$ in $Y$ is equal to the strict transform with respect to the\nblowup $S'' \\to S'$ in $Z'$ of the strict transform of $X$\nwith respect to the blowup $S' \\to S$ of $S$ in $Z$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080I","source_file":"divisors.tex","source_line":8969,"source_end_line":8989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L8969-L8989","statement_sha256":"bf758e340f151d136e529834102151e87119b0088d579625de37240ab59698bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6220,"rank":6220,"depth":23,"x":2410.505,"y":611.737,"cluster":"divisors-intersection-theory"},{"id":"stacks:080W","tag":"080W","title":"Strict transform · Lemma 080W","summary":"In the situation of Definition [Tag 080D]. Suppose that 0 → F_1 → F_2 → F_3 → 0 is an exact sequence of quasi-coherent sheaves on X which remains exact after any base change T → S. Then the strict transforms of F_i' relative to any blowup S' → S form a short exact sequence 0 → F'_1 → F'_2 → F'_3 → 0 too.","statement_latex":"In the situation of Definition \\ref{definition-strict-transform}.\nSuppose that\n$$\n0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0\n$$\nis an exact sequence of quasi-coherent sheaves on $X$ which remains\nexact after any base change $T \\to S$. Then the strict transforms of\n$\\mathcal{F}_i'$ relative to any blowup $S' \\to S$\nform a short exact sequence\n$0 \\to \\mathcal{F}'_1 \\to \\mathcal{F}'_2 \\to \\mathcal{F}'_3 \\to 0$ too.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080W","source_file":"divisors.tex","source_line":9036,"source_end_line":9048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9036-L9048","statement_sha256":"a4d553bde8681dcc054c360a941a20f1d4fa6b8e3e1d1f50e7e090b6fcb5f9f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6221,"rank":6221,"depth":27,"x":2586.347,"y":690.924,"cluster":"divisors-intersection-theory"},{"id":"stacks:080K","tag":"080K","title":"Admissible blowups · Definition 080K","summary":"Let X be a scheme. Let U ⊂ X be an open subscheme. A morphism X' → X is called a U-admissible blowup if there exists a closed immersion Z → X of finite presentation with Z disjoint from U such that X' is isomorphic to the blowup of X in Z.","statement_latex":"Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme. A morphism\n$X' \\to X$ is called a {\\it $U$-admissible blowup} if there exists a\nclosed immersion $Z \\to X$ of finite presentation with $Z$ disjoint from\n$U$ such that $X'$ is isomorphic to the blowup of $X$ in $Z$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Admissible blowups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080K","source_file":"divisors.tex","source_line":9088,"source_end_line":9094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9088-L9094","statement_sha256":"28edd8562632751ec53c3a7d9ebb7312c1b0914015262445736f936489df59ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":6222,"rank":6222,"depth":0,"x":2392.627,"y":732.39,"cluster":"divisors-intersection-theory"},{"id":"stacks:080L","tag":"080L","title":"Admissible blowups · Lemma 080L","summary":"Admissible blowups are stable under composition. Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be a quasi-compact open subscheme. Let b : X' → X be a U-admissible blowup. Let X\" → X' be a U-admissible blowup. Then the composition X\" → X is a U-admissible blowup.","statement_latex":"\\begin{slogan}\nAdmissible blowups are stable under composition.\n\\end{slogan}\nLet $X$ be a quasi-compact and quasi-separated scheme.\nLet $U \\subset X$ be a quasi-compact open subscheme.\nLet $b : X' \\to X$ be a $U$-admissible blowup.\nLet $X'' \\to X'$ be a $U$-admissible blowup.\nThen the composition $X'' \\to X$ is a $U$-admissible blowup.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Admissible blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080L","source_file":"divisors.tex","source_line":9109,"source_end_line":9119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9109-L9119","statement_sha256":"6bd94d71d555ba14146785f0d33d72d8da741c2aafc4632f6aac4d742f0784ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":6223,"rank":6223,"depth":23,"x":2502.34,"y":591.621,"cluster":"divisors-intersection-theory"},{"id":"stacks:080M","tag":"080M","title":"Admissible blowups · Lemma 080M","summary":"Let X be a quasi-compact and quasi-separated scheme. Let U, V ⊂ X be quasi-compact open subschemes. Let b : V' → V be a U ∩ V-admissible blowup. Then there exists a U-admissible blowup X' → X whose restriction to V is V'.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $U, V \\subset X$ be quasi-compact open subschemes.\nLet $b : V' \\to V$ be a $U \\cap V$-admissible blowup.\nThen there exists a $U$-admissible blowup $X' \\to X$\nwhose restriction to $V$ is $V'$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Admissible blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080M","source_file":"divisors.tex","source_line":9126,"source_end_line":9133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9126-L9133","statement_sha256":"03db885259d8b3bb4072385d28e1fcc0e6248ff1b0be9e681da0e9ae56149452","origin":"The Stacks Project","memory_eligible":false,"source_rank":6224,"rank":6224,"depth":16,"x":2534.704,"y":757.996,"cluster":"divisors-intersection-theory"},{"id":"stacks:080N","tag":"080N","title":"Admissible blowups · Lemma 080N","summary":"Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be a quasi-compact open subscheme. Let b_i : X_i → X, i = 1, …, n be U-admissible blowups. There exists a U-admissible blowup b : X' → X such that (a) b factors as X' → X_i → X for i = 1, …, n and (b) each of the morphisms X' → X_i is a U-admissible blowup.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $U \\subset X$ be a quasi-compact open subscheme.\nLet $b_i : X_i \\to X$, $i = 1, \\ldots, n$ be $U$-admissible blowups.\nThere exists a $U$-admissible blowup $b : X' \\to X$ such that\n(a) $b$ factors as $X' \\to X_i \\to X$ for $i = 1, \\ldots, n$ and\n(b) each of the morphisms $X' \\to X_i$ is a $U$-admissible blowup.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Admissible blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080N","source_file":"divisors.tex","source_line":9150,"source_end_line":9158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9150-L9158","statement_sha256":"5477e77e454c53f8a8cc321a312e18cc04f34edb277b2317313199d965dfe0dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6225,"rank":6225,"depth":22,"x":2376.742,"y":653.477,"cluster":"divisors-intersection-theory"},{"id":"stacks:080P","tag":"080P","title":"Admissible blowups · Lemma 080P","summary":"Separate irreducible components by blowing up. Let X be a quasi-compact and quasi-separated scheme. Let U, V be quasi-compact disjoint open subschemes of X. Then there exist a U ∪ V-admissible blowup b : X' → X such that X' is a disjoint union of open subschemes X' = X'_1 amalg X'_2 with b^-1(U) ⊂ X'_1 and b^-1(V) ⊂ X'_2.","statement_latex":"\\begin{slogan}\nSeparate irreducible components by blowing up.\n\\end{slogan}\nLet $X$ be a quasi-compact and quasi-separated scheme.\nLet $U, V$ be quasi-compact disjoint open subschemes of $X$.\nThen there exist a $U \\cup V$-admissible blowup $b : X' \\to X$\nsuch that $X'$ is a disjoint union of open subschemes\n$X' = X'_1 \\amalg X'_2$ with $b^{-1}(U) \\subset X'_1$ and\n$b^{-1}(V) \\subset X'_2$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Admissible blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080P","source_file":"divisors.tex","source_line":9170,"source_end_line":9181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9170-L9181","statement_sha256":"ac01c16fdefdea35c6a94818aa2c3f15d8d93c80ba3523d6699ebf6be217e1bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6226,"rank":6226,"depth":17,"x":2577.656,"y":640.892,"cluster":"divisors-intersection-theory"},{"id":"stacks:0ESL","tag":"0ESL","title":"Admissible blowups · Lemma 0ESL","summary":"Let X be a locally Noetherian scheme. Let L be an invertible O_X-module. Let s be a regular meromorphic section of L. Let U ⊂ X be the maximal open subscheme such that s corresponds to a section of L over U. The blowup b : X' → X in the ideal of denominators of s is U-admissible. There exists an effective Cartier divisor D ⊂ X' and an isomorphism b^*L = O_X'(D - E), where E ⊂ X' is the exceptional divisor such that the meromorphic section b^*s corresponds, via the…","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s$ be a regular meromorphic section of $\\mathcal{L}$.\nLet $U \\subset X$ be the maximal open subscheme such that\n$s$ corresponds to a section of $\\mathcal{L}$ over $U$.\nThe blowup $b : X' \\to X$ in the ideal of denominators\nof $s$ is $U$-admissible. There exists an effective Cartier divisor\n$D \\subset X'$ and an isomorphism\n$$\nb^*\\mathcal{L} = \\mathcal{O}_{X'}(D - E),\n$$\nwhere $E \\subset X'$ is the exceptional divisor such that the\nmeromorphic section $b^*s$ corresponds, via the isomorphism,\nto the meromorphic section $1_D \\otimes (1_E)^{-1}$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Admissible blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESL","source_file":"divisors.tex","source_line":9232,"source_end_line":9248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9232-L9248","statement_sha256":"a7a4621b9819e22d50207a877803d029f732ad03e7dac2eb0bd24582b36f76c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6227,"rank":6227,"depth":3,"x":2439.362,"y":764.41,"cluster":"divisors-intersection-theory"},{"id":"stacks:0CZP","tag":"0CZP","title":"Blowing up and flatness · Lemma 0CZP","summary":"Let S be a scheme. Let F be a finite type quasi-coherent O_S-module. Let Z_k ⊂ S be the closed subscheme cut out by Fit_k(F), see Section [Tag 0C3C]. Let S' → S be the blowup of S in Z_k and let F' be the strict transform of F. Then F' can locally be generated by ≤ k sections.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type\nquasi-coherent $\\mathcal{O}_S$-module. Let $Z_k \\subset S$ be the closed\nsubscheme cut out by $\\text{Fit}_k(\\mathcal{F})$, see\nSection \\ref{section-fitting-ideals}.\nLet $S' \\to S$ be the blowup of $S$ in $Z_k$ and let\n$\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$.\nThen $\\mathcal{F}'$ can locally be generated by $\\leq k$\nsections.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZP","source_file":"divisors.tex","source_line":9302,"source_end_line":9312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9302-L9312","statement_sha256":"40c7c6303be5efc72905c1edc749b75025cf833ba2e224dd82191ef81a6b9bbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6228,"rank":6228,"depth":6,"x":2442.014,"y":594.536,"cluster":"divisors-intersection-theory"},{"id":"stacks:0CZQ","tag":"0CZQ","title":"Blowing up and flatness · Lemma 0CZQ","summary":"Let S be a scheme. Let F be a finite type quasi-coherent O_S-module. Let Z_k ⊂ S be the closed subscheme cut out by Fit_k(F), see Section [Tag 0C3C]. Assume that F is locally free of rank k on S setminus Z_k. Let S' → S be the blowup of S in Z_k and let F' be the strict transform of F. Then F' is locally free of rank k.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a finite type\nquasi-coherent $\\mathcal{O}_S$-module. Let $Z_k \\subset S$ be the closed\nsubscheme cut out by $\\text{Fit}_k(\\mathcal{F})$, see\nSection \\ref{section-fitting-ideals}.\nAssume that $\\mathcal{F}$ is locally free of rank $k$ on $S \\setminus Z_k$.\nLet $S' \\to S$ be the blowup of $S$ in $Z_k$ and let\n$\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$.\nThen $\\mathcal{F}'$ is locally free of rank $k$.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZQ","source_file":"divisors.tex","source_line":9322,"source_end_line":9332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9322-L9332","statement_sha256":"8f75db2d988f58c5c9a246d8f84e37b9837dc6041a1b28c41261acad9119a519","origin":"The Stacks Project","memory_eligible":false,"source_rank":6229,"rank":6229,"depth":7,"x":2576.921,"y":721.543,"cluster":"divisors-intersection-theory"},{"id":"stacks:0ESN","tag":"0ESN","title":"Blowing up and flatness · Lemma 0ESN","summary":"Let X be a scheme. Let F be a finitely presented O_X-module. Let U ⊂ X be a scheme theoretically dense open such that F|_U is finite locally free of constant rank r. Then • the blowup b : X' → X of X in the rth Fitting ideal of F is U-admissible, • the strict transform F' of F with respect to b is locally free of rank r, • the kernel K of the surjection b^*F → F' is finitely presented and K|_U = 0, • b^*F and K are perfect O_X'-modules of tor dimension ≤ 1.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a finitely presented\n$\\mathcal{O}_X$-module. Let $U \\subset X$ be a scheme theoretically\ndense open such that $\\mathcal{F}|_U$ is finite locally free of\nconstant rank $r$. Then\n\\begin{enumerate}\n\\item the blowup $b : X' \\to X$ of $X$ in the $r$th Fitting\nideal of $\\mathcal{F}$ is $U$-admissible,\n\\item the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$\nwith respect to $b$ is locally free of rank $r$,\n\\item the kernel $\\mathcal{K}$ of the surjection\n$b^*\\mathcal{F} \\to \\mathcal{F}'$ is\nfinitely presented and $\\mathcal{K}|_U = 0$,\n\\item $b^*\\mathcal{F}$ and $\\mathcal{K}$ are perfect\n$\\mathcal{O}_{X'}$-modules of tor dimension $\\leq 1$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESN","source_file":"divisors.tex","source_line":9339,"source_end_line":9356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9339-L9356","statement_sha256":"7a0d5fe80263996c80ddd2a57a8a4d3c6bf28e1422894e43815e573f51cb923a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6230,"rank":6230,"depth":9,"x":2374.927,"y":704.408,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AYP","tag":"0AYP","title":"Modifications · Lemma 0AYP","summary":"Let X be an integral scheme. Let E be a finite locally free O_X-module. There exists a modification f : X' → X such that f^*E has a filtration whose successive quotients are invertible O_X'-modules.","statement_latex":"Let $X$ be an integral scheme. Let $\\mathcal{E}$ be a finite locally free\n$\\mathcal{O}_X$-module. There exists a modification $f : X' \\to X$\nsuch that $f^*\\mathcal{E}$ has a filtration whose successive quotients\nare invertible $\\mathcal{O}_{X'}$-modules.","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYP","source_file":"divisors.tex","source_line":9412,"source_end_line":9418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9412-L9418","statement_sha256":"ea408f893929eb46f33e052e6711f8232447b803c5271e599fb5b0c0b4aac5e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6231,"rank":6231,"depth":26,"x":2537.96,"y":602.236,"cluster":"divisors-intersection-theory"},{"id":"stacks:0C4V","tag":"0C4V","title":"Modifications · Lemma 0C4V","summary":"Let S be a scheme. Let X, Y be schemes over S. Assume X is Noetherian and Y is proper over S. Given an S-rational map f : U → Y from X to Y there exists a morphism p : X' → X and an S-morphism f' : X' → Y such that • p is proper and p^-1(U) → U is an isomorphism, • f'|_p^-1(U) is equal to f ∘ p|_p^-1(U).","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$.\nAssume $X$ is Noetherian and $Y$ is proper over $S$.\nGiven an $S$-rational map $f : U \\to Y$ from $X$ to $Y$\nthere exists a morphism $p : X' \\to X$ and an\n$S$-morphism $f' : X' \\to Y$ such that\n\\begin{enumerate}\n\\item $p$ is proper and $p^{-1}(U) \\to U$ is an isomorphism,\n\\item $f'|_{p^{-1}(U)}$ is equal to $f \\circ p|_{p^{-1}(U)}$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Divisors","chapter_id":"divisors","section":"Modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4V","source_file":"divisors.tex","source_line":9451,"source_end_line":9462,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/divisors.tex#L9451-L9462","statement_sha256":"ce7f14961835f426e5986d426271423e2bfe5384c71c1c57a3865ffade262ac6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6232,"rank":6232,"depth":20,"x":2499.834,"y":770.399,"cluster":"divisors-intersection-theory"},{"id":"stacks:01YW","tag":"01YW","title":"Directed limits of schemes with affine transition maps · Lemma 01YW","summary":"Let I be a directed set. Let (S_i, f_ii') be an inverse system of schemes over I. If all the schemes S_i are affine, then the limit S = lim_i S_i exists in the category of schemes. In fact S is affine and S = Spec(colim_i R_i) with R_i = Γ(S_i, O).","statement_latex":"Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an inverse system of\nschemes over $I$.  If all the schemes $S_i$\nare affine, then the limit $S = \\lim_i S_i$ exists\nin the category of schemes.\nIn fact $S$ is affine and $S = \\Spec(\\colim_i R_i)$\nwith $R_i = \\Gamma(S_i, \\mathcal{O})$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Directed limits of schemes with affine transition maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YW","source_file":"limits.tex","source_line":41,"source_end_line":49,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L41-L49","statement_sha256":"dfa3f79e7b7d86567eb6930b749d403e1668196c516e76013b758c7655b27753","origin":"The Stacks Project","memory_eligible":false,"source_rank":6233,"rank":6233,"depth":11,"x":1960.097,"y":746.627,"cluster":"scheme-morphisms"},{"id":"stacks:01YX","tag":"01YX","title":"Directed limits of schemes with affine transition maps · Lemma 01YX","summary":"Let I be a directed set. Let (S_i, f_ii') be an inverse system of schemes over I. If all the morphisms f_ii' : S_i → S_i' are affine, then the limit S = lim_i S_i exists in the category of schemes. Moreover, • each of the morphisms f_i : S → S_i is affine, • for an element 0 ∈ I and any open subscheme U_0 ⊂ S_0 we have f_0^-1(U_0) = lim_i ≥ 0 f_i0^-1(U_0) in the category of schemes.","statement_latex":"Let $I$ be a directed set. Let $(S_i, f_{ii'})$ be an\ninverse system of schemes over $I$. If all the morphisms\n$f_{ii'} : S_i \\to S_{i'}$ are affine, then the limit $S = \\lim_i S_i$ exists\nin the category of schemes. Moreover,\n\\begin{enumerate}\n\\item each of the morphisms $f_i : S \\to S_i$ is affine,\n\\item for an element $0 \\in I$ and any open subscheme $U_0 \\subset S_0$\nwe have\n$$\nf_0^{-1}(U_0) = \\lim_{i \\geq 0} f_{i0}^{-1}(U_0)\n$$\nin the category of schemes.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Directed limits of schemes with affine transition maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YX","source_file":"limits.tex","source_line":57,"source_end_line":72,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L57-L72","statement_sha256":"06f059cf10ba22740bbe5868c93ee5784b2f4964e35bf85cc23015b6ac8a0ed4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6234,"rank":6234,"depth":20,"x":2027.964,"y":591.118,"cluster":"scheme-morphisms"},{"id":"stacks:01YZ","tag":"01YZ","title":"Directed limits of schemes with affine transition maps · Lemma 01YZ","summary":"Let I be a directed set. Let (S_i, f_ii') be an inverse system of schemes over I. Assume all the morphisms f_ii' : S_i → S_i' are affine, Let S = lim_i S_i. Let 0 ∈ I. Suppose that T is a scheme over S_0. Then T ×_S_0 S = lim_i ≥ 0 T ×_S_0 S_i","statement_latex":"Let $I$ be a directed set.\nLet $(S_i, f_{ii'})$ be an inverse system of schemes over $I$.\nAssume all the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine,\nLet $S = \\lim_i S_i$. Let $0 \\in I$.\nSuppose that $T$ is a scheme over $S_0$.\nThen\n$$\nT \\times_{S_0} S = \\lim_{i \\geq 0} T \\times_{S_0} S_i\n$$","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Directed limits of schemes with affine transition maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YZ","source_file":"limits.tex","source_line":113,"source_end_line":124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L113-L124","statement_sha256":"dbec4383d16f1ede5048d857c01a9d9f165a8ebf73d167e50191eb12000c8a9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6235,"rank":6235,"depth":21,"x":2103.049,"y":744.447,"cluster":"scheme-morphisms"},{"id":"stacks:0CNI","tag":"0CNI","title":"Infinite products · Lemma 0CNI","summary":"Infinite products of affine schemes exist and are affine. Let S be a scheme. Let I be a set and for each i ∈ I let f_i : T_i → S be an affine morphism. Then the product T = ∏ T_i exists in the category of schemes over S. In fact, we have T = lim_(i_1, …, i_n) ⊂ I T_i_1 ×_S … ×_S T_i_n and the projection morphisms T → T_i_1 ×_S … ×_S T_i_n are affine.","statement_latex":"\\begin{slogan}\nInfinite products of affine schemes exist and are affine.\n\\end{slogan}\nLet $S$ be a scheme. Let $I$ be a set and for each $i \\in I$\nlet $f_i : T_i \\to S$ be an affine morphism. Then the\nproduct $T = \\prod T_i$ exists in the category of schemes\nover $S$. In fact, we have\n$$\nT = \\lim_{\\{i_1, \\ldots, i_n\\} \\subset I}\nT_{i_1} \\times_S \\ldots \\times_S T_{i_n}\n$$\nand the projection morphisms $T \\to T_{i_1} \\times_S \\ldots \\times_S T_{i_n}$\nare affine.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Infinite products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNI","source_file":"limits.tex","source_line":159,"source_end_line":174,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L159-L174","statement_sha256":"219e417dcd3279692bde4eb7792607a4039bea195008a0f3782ff63aa5768998","origin":"The Stacks Project","memory_eligible":false,"source_rank":6236,"rank":6236,"depth":21,"x":1924.206,"y":673.921,"cluster":"scheme-morphisms"},{"id":"stacks:0CNJ","tag":"0CNJ","title":"Infinite products · Lemma 0CNJ","summary":"Let S be a scheme. Let I be a set and for each i ∈ I let f_i : T_i → S be a surjective affine morphism. Then the product T = ∏ T_i in the category of schemes over S (Lemma [Tag 0CNI]) maps surjectively to S.","statement_latex":"Let $S$ be a scheme. Let $I$ be a set and for each $i \\in I$\nlet $f_i : T_i \\to S$ be a surjective affine morphism. Then the\nproduct $T = \\prod T_i$ in the category of schemes over $S$\n(Lemma \\ref{lemma-infinite-product})\nmaps surjectively to $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Infinite products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNJ","source_file":"limits.tex","source_line":182,"source_end_line":189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L182-L189","statement_sha256":"36ad1347298c111359bfdc780b6b1bc135a2aa14a18c10779c8a380987971343","origin":"The Stacks Project","memory_eligible":false,"source_rank":6237,"rank":6237,"depth":22,"x":2112.98,"y":624.398,"cluster":"scheme-morphisms"},{"id":"stacks:0CNK","tag":"0CNK","title":"Infinite products · Lemma 0CNK","summary":"Let S be a scheme. Let I be a set and for each i ∈ I let f_i : T_i → S be an integral morphism. Then the product T = ∏ T_i in the category of schemes over S (Lemma [Tag 0CNI]) is integral over S.","statement_latex":"Let $S$ be a scheme. Let $I$ be a set and for each $i \\in I$\nlet $f_i : T_i \\to S$ be an integral morphism. Then the\nproduct $T = \\prod T_i$ in the category of schemes over $S$\n(Lemma \\ref{lemma-infinite-product})\nis integral over $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Infinite products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNK","source_file":"limits.tex","source_line":200,"source_end_line":207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L200-L207","statement_sha256":"d7d937274eb80f3ad0975c270463819a4f8a2a758d3e41d8115ef970298f8f87","origin":"The Stacks Project","memory_eligible":false,"source_rank":6238,"rank":6238,"depth":22,"x":2013.509,"y":768.17,"cluster":"scheme-morphisms"},{"id":"stacks:0CUE","tag":"0CUE","title":"Descending properties · Lemma 0CUE","summary":"Let S = lim S_i be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma [Tag 01YX]). Then S_set = lim_i S_i, set where S_set indicates the underlying set of the scheme S.","statement_latex":"Let $S = \\lim S_i$ be the limit of a directed inverse system\nof schemes with affine transition morphisms\n(Lemma \\ref{lemma-directed-inverse-system-has-limit}). Then\n$S_{set} = \\lim_i S_{i, set}$ where $S_{set}$\nindicates the underlying set of the scheme $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUE","source_file":"limits.tex","source_line":225,"source_end_line":232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L225-L232","statement_sha256":"c9f689ea2a9c160fc76c599eb156a288d57a126ad2a71de8e41a562dfdbddac4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6239,"rank":6239,"depth":21,"x":1971.2,"y":605.556,"cluster":"scheme-morphisms"},{"id":"stacks:0CUF","tag":"0CUF","title":"Descending properties · Lemma 0CUF","summary":"[EGA] Let S = lim S_i be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma [Tag 01YX]). Then S_top = lim_i S_i, top where S_top indicates the underlying topological space of the scheme S.","statement_latex":"\\begin{reference}\n\\cite[IV, Proposition 8.2.9]{EGA}\n\\end{reference}\nLet $S = \\lim S_i$ be the limit of a directed inverse system\nof schemes with affine transition morphisms\n(Lemma \\ref{lemma-directed-inverse-system-has-limit}). Then\n$S_{top} = \\lim_i S_{i, top}$ where $S_{top}$\nindicates the underlying topological space of the scheme $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUF","source_file":"limits.tex","source_line":267,"source_end_line":277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L267-L277","statement_sha256":"c5929aa84e268ca853bdf259b43c1f424160c272b8c2b5bca285e10e626d54ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":6240,"rank":6240,"depth":22,"x":2133.324,"y":701.55,"cluster":"scheme-morphisms"},{"id":"stacks:01Z2","tag":"01Z2","title":"Descending properties · Lemma 01Z2","summary":"Let S = lim S_i be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma [Tag 01YX]). If all the schemes S_i are nonempty and quasi-compact, then the limit S = lim_i S_i is nonempty.","statement_latex":"Let $S = \\lim S_i$ be the limit of a directed inverse system\nof schemes with affine transition morphisms\n(Lemma \\ref{lemma-directed-inverse-system-has-limit}).\nIf all the schemes $S_i$ are nonempty and quasi-compact,\nthen the limit $S = \\lim_i S_i$ is nonempty.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Z2","source_file":"limits.tex","source_line":311,"source_end_line":318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L311-L318","statement_sha256":"3e24bd854e625d820624263fe6f0ab120972e941877ffd01951469058dcab6e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6241,"rank":6241,"depth":21,"x":1936.391,"y":722.779,"cluster":"scheme-morphisms"},{"id":"stacks:0CUG","tag":"0CUG","title":"Descending properties · Lemma 0CUG","summary":"Let S = lim S_i be the limit of a directed inverse system of schemes with affine transition morphisms (Lemma [Tag 01YX]). Let s ∈ S with images s_i ∈ S_i. Then • s = lim s_i as schemes, i.e., kappa(s) = colim kappa(s_i), • overline(s) = lim overline(s_i) as sets, and • overline(s) = lim overline(s_i) as schemes where overline(s) and overline(s_i) are endowed with the reduced induced scheme structure.","statement_latex":"Let $S = \\lim S_i$ be the limit of a directed inverse system\nof schemes with affine transition morphisms\n(Lemma \\ref{lemma-directed-inverse-system-has-limit}).\nLet $s \\in S$ with images $s_i \\in S_i$.\nThen\n\\begin{enumerate}\n\\item $s = \\lim s_i$ as schemes, i.e., $\\kappa(s) = \\colim \\kappa(s_i)$,\n\\item $\\overline{\\{s\\}} = \\lim \\overline{\\{s_i\\}}$ as sets, and\n\\item $\\overline{\\{s\\}} = \\lim \\overline{\\{s_i\\}}$ as schemes\nwhere $\\overline{\\{s\\}}$ and $\\overline{\\{s_i\\}}$ are\nendowed with the reduced induced scheme structure.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUG","source_file":"limits.tex","source_line":331,"source_end_line":345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L331-L345","statement_sha256":"a2379b29ffe99d6d57202f5160a55d529d20daa4e95d682cf669c38c3e06edcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6242,"rank":6242,"depth":22,"x":2064.657,"y":595.258,"cluster":"scheme-morphisms"},{"id":"stacks:01YY","tag":"01YY","title":"Descending properties · Lemma 01YY","summary":"In Situation [Tag 086P]. • We have S_set = lim_i S_i, set where S_set indicates the underlying set of the scheme S. • We have S_top = lim_i S_i, top where S_top indicates the underlying topological space of the scheme S. • If s, s' ∈ S and s' is not a specialization of s then for some i ∈ I the image s'_i ∈ S_i of s' is not a specialization of the image s_i ∈ S_i of s. • Add more easy facts on topology of S here. (Requirement: whatever is added should be easy in the…","statement_latex":"In Situation \\ref{situation-descent}.\n\\begin{enumerate}\n\\item We have $S_{set} = \\lim_i S_{i, set}$ where $S_{set}$\nindicates the underlying set of the scheme $S$.\n\\item We have $S_{top} = \\lim_i S_{i, top}$ where $S_{top}$\nindicates the underlying topological space of the scheme $S$.\n\\item If $s, s' \\in S$ and $s'$ is not a specialization of $s$\nthen for some $i \\in I$ the image $s'_i \\in S_i$ of $s'$ is not\na specialization of the image $s_i \\in S_i$ of $s$.\n\\item Add more easy facts on topology of $S$ here.\n(Requirement: whatever is added should be easy in the affine case.)\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01YY","source_file":"limits.tex","source_line":407,"source_end_line":421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L407-L421","statement_sha256":"d8a87cf8d441c3497c40f6e83b332869670f9e51aa06b1ccf9fdd678c5853e04","origin":"The Stacks Project","memory_eligible":false,"source_rank":6243,"rank":6243,"depth":23,"x":2072.632,"y":762.231,"cluster":"scheme-morphisms"},{"id":"stacks:01Z0","tag":"01Z0","title":"Descending properties · Lemma 01Z0","summary":"In Situation [Tag 086P]. Suppose that F_0 is a quasi-coherent sheaf on S_0. Set F_i = f_i0^*F_0 for i ≥ 0 and set F = f_0^*F_0. Then Γ(S, F) = colim_i ≥ 0 Γ(S_i, F_i)","statement_latex":"In Situation \\ref{situation-descent}.\nSuppose that $\\mathcal{F}_0$ is a quasi-coherent sheaf on $S_0$.\nSet $\\mathcal{F}_i = f_{i0}^*\\mathcal{F}_0$ for $i \\geq 0$ and set\n$\\mathcal{F} = f_0^*\\mathcal{F}_0$.\nThen\n$$\n\\Gamma(S, \\mathcal{F}) = \\colim_{i \\geq 0} \\Gamma(S_i, \\mathcal{F}_i)\n$$","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Z0","source_file":"limits.tex","source_line":433,"source_end_line":443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L433-L443","statement_sha256":"a3839d7ddf51c4650caec7c2b2c1b3540cfb1c8900e23c38ce6c783750717fb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6244,"rank":6244,"depth":21,"x":1932.343,"y":643.519,"cluster":"scheme-morphisms"},{"id":"stacks:01Z3","tag":"01Z3","title":"Descending properties · Lemma 01Z3","summary":"In Situation [Tag 086P]. Suppose for each i we are given a nonempty closed subset Z_i ⊂ S_i with f_i'i(Z_i') ⊂ Z_i for all i' ≥ i. Then there exists a point s ∈ S with f_i(s) ∈ Z_i for all i.","statement_latex":"In Situation \\ref{situation-descent}.\nSuppose for each $i$ we are given a nonempty closed subset\n$Z_i \\subset S_i$ with $f_{i'i}(Z_{i'}) \\subset Z_i$ for all\n$i' \\geq i$.\nThen there exists a point $s \\in S$ with $f_i(s) \\in Z_i$ for\nall $i$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Z3","source_file":"limits.tex","source_line":483,"source_end_line":491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L483-L491","statement_sha256":"51772471fb95c0318f01d2445fa9b23599d30a863692bc6ed3181f2b276991cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6245,"rank":6245,"depth":22,"x":2131.444,"y":651.46,"cluster":"scheme-morphisms"},{"id":"stacks:05F3","tag":"05F3","title":"Descending properties · Lemma 05F3","summary":"In Situation [Tag 086P]. Suppose we are given an i and a morphism T → S_i such that • T ×_S_i S = ∅, and • T is quasi-compact. Then T ×_S_i S_i' = ∅ for all sufficiently large i'.","statement_latex":"In Situation \\ref{situation-descent}.\nSuppose we are given an $i$ and a morphism $T \\to S_i$ such that\n\\begin{enumerate}\n\\item $T \\times_{S_i} S = \\emptyset$, and\n\\item $T$ is quasi-compact.\n\\end{enumerate}\nThen $T \\times_{S_i} S_{i'} = \\emptyset$ for all sufficiently large $i'$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05F3","source_file":"limits.tex","source_line":511,"source_end_line":520,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L511-L520","statement_sha256":"33231ecf790382b1897c44cb656eda3a2d8a7fe5a3b78c87990ed78f6f7a5f5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6246,"rank":6246,"depth":22,"x":1978.098,"y":758.682,"cluster":"scheme-morphisms"},{"id":"stacks:05F4","tag":"05F4","title":"Descending properties · Lemma 05F4","summary":"In Situation [Tag 086P]. Suppose we are given an i and a locally constructible subset E ⊂ S_i such that f_i(S) ⊂ E. Then f_i'i(S_i') ⊂ E for all sufficiently large i'.","statement_latex":"In Situation \\ref{situation-descent}.\nSuppose we are given an $i$ and a locally constructible subset\n$E \\subset S_i$ such that $f_i(S) \\subset E$.\nThen $f_{i'i}(S_{i'}) \\subset E$ for all sufficiently large $i'$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05F4","source_file":"limits.tex","source_line":529,"source_end_line":535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L529-L535","statement_sha256":"cc46027b0f85a49b09f45da31a54abf7115dcb76e7072a29aecd898015c0f452","origin":"The Stacks Project","memory_eligible":false,"source_rank":6247,"rank":6247,"depth":23,"x":2004.974,"y":592.448,"cluster":"scheme-morphisms"},{"id":"stacks:01Z4","tag":"01Z4","title":"Descending properties · Lemma 01Z4","summary":"In Situation [Tag 086P] we have the following: • Given any quasi-compact open V ⊂ S = lim_i S_i there exists an i ∈ I and a quasi-compact open V_i ⊂ S_i such that f_i^-1(V_i) = V. • Given V_i ⊂ S_i and V_i' ⊂ S_i' quasi-compact opens such that f_i^-1(V_i) = f_i'^-1(V_i') there exists an index i\" ≥ i, i' such that f_i\"i^-1(V_i) = f_i\"i'^-1(V_i'). • If V_1, i, …, V_n, i ⊂ S_i are quasi-compact opens and S = f_i^-1(V_1, i) ∪ … ∪ f_i^-1(V_n, i) then S_i' = f_i'i^-1(V_1, i) ∪…","statement_latex":"In Situation \\ref{situation-descent} we have the following:\n\\begin{enumerate}\n\\item Given any quasi-compact open $V \\subset S = \\lim_i S_i$\nthere exists an $i \\in I$ and a quasi-compact open $V_i \\subset S_i$\nsuch that $f_i^{-1}(V_i) = V$.\n\\item Given $V_i \\subset S_i$ and $V_{i'} \\subset S_{i'}$\nquasi-compact opens such that $f_i^{-1}(V_i) = f_{i'}^{-1}(V_{i'})$\nthere exists an index $i'' \\geq i, i'$ such that\n$f_{i''i}^{-1}(V_i) = f_{i''i'}^{-1}(V_{i'})$.\n\\item If $V_{1, i}, \\ldots, V_{n, i} \\subset S_i$ are quasi-compact\nopens and $S = f_i^{-1}(V_{1, i}) \\cup \\ldots \\cup f_i^{-1}(V_{n, i})$\nthen $S_{i'} = f_{i'i}^{-1}(V_{1, i}) \\cup \\ldots \\cup f_{i'i}^{-1}(V_{n, i})$\nfor some $i' \\geq i$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Z4","source_file":"limits.tex","source_line":553,"source_end_line":569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L553-L569","statement_sha256":"f17ac82c0b0bcaee85f39529640696154650b2f5d4560c1932948065677abb8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6248,"rank":6248,"depth":0,"x":2118.945,"y":730.407,"cluster":"scheme-morphisms"},{"id":"stacks:01Z5","tag":"01Z5","title":"Descending properties · Lemma 01Z5","summary":"In Situation [Tag 086P] if S is quasi-affine, then for some i_0 ∈ I the schemes S_i for i ≥ i_0 are quasi-affine.","statement_latex":"In Situation \\ref{situation-descent} if $S$ is quasi-affine, then\nfor some $i_0 \\in I$ the schemes $S_i$ for $i \\geq i_0$ are quasi-affine.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Z5","source_file":"limits.tex","source_line":625,"source_end_line":629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L625-L629","statement_sha256":"6f73519ac40c16e203d426d76ea8511dda6d56622293ec2cfaeb2ac38b3a96c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6249,"rank":6249,"depth":22,"x":1923.777,"y":693.313,"cluster":"scheme-morphisms"},{"id":"stacks:01Z6","tag":"01Z6","title":"Descending properties · Lemma 01Z6","summary":"In Situation [Tag 086P] if S is affine, then for some i_0 ∈ I the schemes S_i for i ≥ i_0 are affine.","statement_latex":"In Situation \\ref{situation-descent} if $S$ is affine,\nthen for some $i_0 \\in I$ the schemes $S_i$ for $i \\geq i_0$\nare affine.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Z6","source_file":"limits.tex","source_line":679,"source_end_line":684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L679-L684","statement_sha256":"fbd0795a382d8d7fdf8cc21c184773cf09b45eb95d1f609b4b5a670aad81da23","origin":"The Stacks Project","memory_eligible":false,"source_rank":6250,"rank":6250,"depth":24,"x":2097.685,"y":609.844,"cluster":"scheme-morphisms"},{"id":"stacks:086Q","tag":"086Q","title":"Descending properties · Lemma 086Q","summary":"In Situation [Tag 086P] if S is separated, then for some i_0 ∈ I the schemes S_i for i ≥ i_0 are separated.","statement_latex":"In Situation \\ref{situation-descent} if $S$ is separated,\nthen for some $i_0 \\in I$ the schemes $S_i$ for $i \\geq i_0$\nare separated.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086Q","source_file":"limits.tex","source_line":715,"source_end_line":720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L715-L720","statement_sha256":"0c9c36dd50895c8b64d7f58b0b417f08f3ffba0e4754a38104763f15888bae08","origin":"The Stacks Project","memory_eligible":false,"source_rank":6251,"rank":6251,"depth":25,"x":2036.514,"y":770.223,"cluster":"scheme-morphisms"},{"id":"stacks:09MT","tag":"09MT","title":"Descending properties · Lemma 09MT","summary":"In Situation [Tag 086P] let L_0 be an invertible sheaf of modules on S_0. If the pullback L to S is ample, then for some i ∈ I the pullback L_i to S_i is ample.","statement_latex":"In Situation \\ref{situation-descent} let $\\mathcal{L}_0$ be an invertible\nsheaf of modules on $S_0$. If the pullback $\\mathcal{L}$ to $S$ is ample,\nthen for some $i \\in I$ the pullback $\\mathcal{L}_i$ to $S_i$ is ample.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MT","source_file":"limits.tex","source_line":787,"source_end_line":792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L787-L792","statement_sha256":"a370b3fb4ad88406e6f0f41b8a76854b005dd374c586a04c482bc6a3fd975f30","origin":"The Stacks Project","memory_eligible":false,"source_rank":6252,"rank":6252,"depth":25,"x":1952.569,"y":617.108,"cluster":"scheme-morphisms"},{"id":"stacks:081B","tag":"081B","title":"Descending properties · Lemma 081B","summary":"Let S be a scheme. Let X = lim X_i be a directed limit of schemes over S with affine transition morphisms. Let Y → X be a morphism of schemes over S. • If Y → X is a closed immersion, X_i quasi-compact, and Y locally of finite type over S, then Y → X_i is a closed immersion for i large enough. • If Y → X is an immersion, X_i quasi-separated, Y → S locally of finite type, and Y quasi-compact, then Y → X_i is an immersion for i large enough. • If Y → X is an isomorphism,…","statement_latex":"Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of\nschemes over $S$ with affine transition morphisms. Let $Y \\to X$\nbe a morphism of schemes over $S$.\n\\begin{enumerate}\n\\item If $Y \\to X$ is a closed immersion, $X_i$ quasi-compact, and\n$Y$ locally of finite type over $S$, then $Y \\to X_i$ is a closed\nimmersion for $i$ large enough.\n\\item If $Y \\to X$ is an immersion, $X_i$ quasi-separated, $Y \\to S$ locally\nof finite type, and $Y$ quasi-compact, then $Y \\to X_i$ is an\nimmersion for $i$ large enough.\n\\item If $Y \\to X$ is an isomorphism, $X_i$ quasi-compact,\n$X_i \\to S$ locally of finite type, the transition morphisms\n$X_{i'} \\to X_i$ are closed immersions, and $Y \\to S$ is locally\nof finite presentation, then $Y \\to X_i$ is an isomorphism for $i$\nlarge enough.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081B","source_file":"limits.tex","source_line":808,"source_end_line":826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L808-L826","statement_sha256":"dc26b591a7908f113f85da4f3d49721b14d959bb5ee6f3f4d1f3e2f4b5827036","origin":"The Stacks Project","memory_eligible":false,"source_rank":6253,"rank":6253,"depth":1,"x":2137.774,"y":682.442,"cluster":"scheme-morphisms"},{"id":"stacks:01ZH","tag":"01ZH","title":"Descending properties · Lemma 01ZH","summary":"Let S be a scheme. Let X = lim X_i be a directed limit of schemes over S with affine transition morphisms. Assume • S quasi-separated, • X_i quasi-compact and quasi-separated, • X → S separated. Then X_i → S is separated for all i large enough.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed\nlimit of schemes over $S$ with affine transition morphisms.\nAssume\n\\begin{enumerate}\n\\item $S$ quasi-separated,\n\\item $X_i$ quasi-compact and quasi-separated,\n\\item $X \\to S$ separated.\n\\end{enumerate}\nThen $X_i \\to S$ is separated for all $i$ large enough.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZH","source_file":"limits.tex","source_line":862,"source_end_line":873,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L862-L873","statement_sha256":"cec87f96b2ddc8eae1d3483c58408466ef680c9183cf06c96a0ea9e2206b5daa","origin":"The Stacks Project","memory_eligible":false,"source_rank":6254,"rank":6254,"depth":26,"x":1948.489,"y":739.409,"cluster":"scheme-morphisms"},{"id":"stacks:09ZM","tag":"09ZM","title":"Descending properties · Lemma 09ZM","summary":"Let S be a scheme. Let X = lim X_i be a directed limit of schemes over S with affine transition morphisms. Assume • S quasi-compact and quasi-separated, • X_i quasi-compact and quasi-separated, • X → S affine. Then X_i → S is affine for i large enough.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes\nover $S$ with affine transition morphisms. Assume\n\\begin{enumerate}\n\\item $S$ quasi-compact and quasi-separated,\n\\item $X_i$ quasi-compact and quasi-separated,\n\\item $X \\to S$ affine.\n\\end{enumerate}\nThen $X_i \\to S$ is affine for $i$ large enough.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZM","source_file":"limits.tex","source_line":890,"source_end_line":900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L890-L900","statement_sha256":"28399a067a4e5bd840fd1bc81cabaa8d4fbd27d7c277e4c1814f34c7c9632669","origin":"The Stacks Project","memory_eligible":false,"source_rank":6255,"rank":6255,"depth":25,"x":2042.342,"y":589.858,"cluster":"scheme-morphisms"},{"id":"stacks:09ZN","tag":"09ZN","title":"Descending properties · Lemma 09ZN","summary":"Let S be a scheme. Let X = lim X_i be a directed limit of schemes over S with affine transition morphisms. Assume • S quasi-compact and quasi-separated, • X_i quasi-compact and quasi-separated, • the transition morphisms X_i' → X_i are finite, • X_i → S locally of finite type • X → S integral. Then X_i → S is finite for i large enough.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes\nover $S$ with affine transition morphisms. Assume\n\\begin{enumerate}\n\\item $S$ quasi-compact and quasi-separated,\n\\item $X_i$ quasi-compact and quasi-separated,\n\\item the transition morphisms $X_{i'} \\to X_i$ are finite,\n\\item $X_i \\to S$ locally of finite type\n\\item $X \\to S$ integral.\n\\end{enumerate}\nThen $X_i \\to S$ is finite for $i$ large enough.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZN","source_file":"limits.tex","source_line":913,"source_end_line":925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L913-L925","statement_sha256":"c595d75755fb02d176ddac52c9b007f92ef93ca002008e8fcc32e482e6949662","origin":"The Stacks Project","memory_eligible":false,"source_rank":6256,"rank":6256,"depth":26,"x":2093.449,"y":753.541,"cluster":"scheme-morphisms"},{"id":"stacks:0A0N","tag":"0A0N","title":"Descending properties · Lemma 0A0N","summary":"Let S be a scheme. Let X = lim X_i be a directed limit of schemes over S with affine transition morphisms. Assume • S quasi-compact and quasi-separated, • X_i quasi-compact and quasi-separated, • the transition morphisms X_i' → X_i are closed immersions, • X_i → S locally of finite type • X → S a closed immersion. Then X_i → S is a closed immersion for i large enough.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes\nover $S$ with affine transition morphisms. Assume\n\\begin{enumerate}\n\\item $S$ quasi-compact and quasi-separated,\n\\item $X_i$ quasi-compact and quasi-separated,\n\\item the transition morphisms $X_{i'} \\to X_i$ are closed immersions,\n\\item $X_i \\to S$ locally of finite type\n\\item $X \\to S$ a closed immersion.\n\\end{enumerate}\nThen $X_i \\to S$ is a closed immersion for $i$ large enough.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0N","source_file":"limits.tex","source_line":955,"source_end_line":967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L955-L967","statement_sha256":"67408eb4016d9a2fcb06177bdfa8667f331a2fd39b0ce598d651a9ae547c6343","origin":"The Stacks Project","memory_eligible":false,"source_rank":6257,"rank":6257,"depth":26,"x":1923.976,"y":661.757,"cluster":"scheme-morphisms"},{"id":"stacks:0GIH","tag":"0GIH","title":"Descending properties · Lemma 0GIH","summary":"Let S be a scheme. Let X = lim X_i be a directed limit of schemes over S with affine transition morphisms. Assume • S quasi-separated, • X_i quasi-compact and quasi-separated, • the transition morphisms X_i' → X_i are closed immersions, • X_i → S locally of finite type, and • X → S an immersion. Then X_i → S is an immersion for i large enough.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of schemes\nover $S$ with affine transition morphisms. Assume\n\\begin{enumerate}\n\\item $S$ quasi-separated,\n\\item $X_i$ quasi-compact and quasi-separated,\n\\item the transition morphisms $X_{i'} \\to X_i$ are closed immersions,\n\\item $X_i \\to S$ locally of finite type, and\n\\item $X \\to S$ an immersion.\n\\end{enumerate}\nThen $X_i \\to S$ is an immersion for $i$ large enough.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIH","source_file":"limits.tex","source_line":990,"source_end_line":1002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L990-L1002","statement_sha256":"3613a652895d7d72a128758ac30abb0efc301df0289acefb54e626cf4f84615c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6258,"rank":6258,"depth":27,"x":2122.937,"y":633.248,"cluster":"scheme-morphisms"},{"id":"stacks:01Z7","tag":"01Z7","title":"Absolute Noetherian Approximation · Lemma 01Z7","summary":"Let W be a quasi-affine scheme of finite type over Z. Suppose W → Spec(R) is an open immersion into an affine scheme. There exists a finite type Z-algebra A ⊂ R which induces an open immersion W → Spec(A). Moreover, R is the directed colimit of such subalgebras.","statement_latex":"Let $W$ be a quasi-affine scheme of finite type over\n$\\mathbf{Z}$. Suppose $W \\to \\Spec(R)$ is an\nopen immersion into an affine scheme. There exists a\nfinite type $\\mathbf{Z}$-algebra $A \\subset R$\nwhich induces an open immersion $W \\to \\Spec(A)$.\nMoreover, $R$ is the directed colimit of such subalgebras.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Absolute Noetherian Approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Z7","source_file":"limits.tex","source_line":1037,"source_end_line":1045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1037-L1045","statement_sha256":"1ceed218b800977841a6bd74c19bd1d6f5439f24d2503b68ab24b028b01183d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6259,"rank":6259,"depth":0,"x":1999.037,"y":767.29,"cluster":"scheme-morphisms"},{"id":"stacks:01Z9","tag":"01Z9","title":"Absolute Noetherian Approximation · Lemma 01Z9","summary":"Suppose given a cartesian diagram of rings xymatrix B ar[r]_s & R B'ar[u] ar[r] & R' ar[u]_t Let W' ⊂ Spec(R') be an open of the form W' = D(f_1) ∪ … ∪ D(f_n) such that t(f_i) = s(g_i) for some g_i ∈ B and B_g_i ≅ R_s(g_i). Then B' → R' induces an open immersion of W' into Spec(B').","statement_latex":"Suppose given a cartesian diagram of rings\n$$\n\\xymatrix{\nB \\ar[r]_s & R \\\\\nB'\\ar[u] \\ar[r] & R' \\ar[u]_t\n}\n$$\nLet $W' \\subset \\Spec(R')$ be an open of\nthe form $W' = D(f_1) \\cup \\ldots \\cup D(f_n)$\nsuch that $t(f_i) = s(g_i)$ for some $g_i \\in B$\nand $B_{g_i} \\cong R_{s(g_i)}$. Then $B' \\to R'$\ninduces an open immersion of $W'$ into $\\Spec(B')$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Absolute Noetherian Approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01Z9","source_file":"limits.tex","source_line":1061,"source_end_line":1075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1061-L1075","statement_sha256":"d005790333e02fb7a58fb49289d548151b9aeb338d42d98ea183055d6cef32c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6260,"rank":6260,"depth":2,"x":1982.591,"y":597.989,"cluster":"scheme-morphisms"},{"id":"stacks:07RN","tag":"07RN","title":"Absolute Noetherian Approximation · Lemma 07RN","summary":"Let S be a quasi-compact and quasi-separated scheme. Let V ⊂ S be a quasi-compact open. Let I be a directed set and let (V_i, f_ii') be an inverse system of schemes over I with affine transition maps, with each V_i of finite type over Z, and with V = lim V_i. Then there exist • a directed set J, • an inverse system of schemes (S_j, g_jj') over J, • an order preserving map α : J → I, • open subschemes V'_j ⊂ S_j, and • isomorphisms V'_j → V_α(j) such that • the transition…","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme. Let $V \\subset S$\nbe a quasi-compact open. Let $I$ be a directed set\nand let $(V_i, f_{ii'})$ be an inverse system of schemes over $I$\nwith affine transition maps, with each $V_i$ of finite type over $\\mathbf{Z}$,\nand with $V = \\lim V_i$. Then there exist\n\\begin{enumerate}\n\\item a directed set $J$,\n\\item an inverse system of schemes $(S_j, g_{jj'})$ over $J$,\n\\item an order preserving map $\\alpha : J \\to I$,\n\\item open subschemes $V'_j \\subset S_j$, and\n\\item isomorphisms $V'_j \\to V_{\\alpha(j)}$\n\\end{enumerate}\nsuch that\n\\begin{enumerate}\n\\item the transition morphisms $g_{jj'} : S_j \\to S_{j'}$ are affine,\n\\item each $S_j$ is of finite type over $\\mathbf{Z}$,\n\\item $g_{jj'}^{-1}(V'_{j'}) = V'_j$,\n\\item $S = \\lim S_j$ and $V = \\lim V'_j$, and\n\\item the diagrams\n$$\n\\vcenter{\n\\xymatrix{\nV \\ar[d] \\ar[rd] \\\\\nV'_j \\ar[r] & V_{\\alpha(j)}\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nV'_j \\ar[r] \\ar[d] & V_{\\alpha(j)} \\ar[d] \\\\\nV'_{j'} \\ar[r] & V_{\\alpha(j')}\n}\n}\n$$\nare commutative.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Absolute Noetherian Approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RN","source_file":"limits.tex","source_line":1087,"source_end_line":1125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1087-L1125","statement_sha256":"1466037ed4546cbb0500922197e9a826880f1d0b780a2ecf7df622803f7db74b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6261,"rank":6261,"depth":23,"x":2131.003,"y":713.604,"cluster":"scheme-morphisms"},{"id":"stacks:01ZA","tag":"01ZA","title":"Absolute Noetherian Approximation · Proposition 01ZA","summary":"Let S be a quasi-compact and quasi-separated scheme. There exist a directed set I and an inverse system of schemes (S_i, f_ii') over I such that • the transition morphisms f_ii' are affine • each S_i is of finite type over Z, and • S = lim_i S_i.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nThere exist a directed set $I$\nand an inverse system of schemes $(S_i, f_{ii'})$ over $I$\nsuch that\n\\begin{enumerate}\n\\item the transition morphisms $f_{ii'}$ are affine\n\\item each $S_i$ is of finite type over $\\mathbf{Z}$, and\n\\item $S = \\lim_i S_i$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Absolute Noetherian Approximation","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZA","source_file":"limits.tex","source_line":1254,"source_end_line":1265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1254-L1265","statement_sha256":"62bc9e066e76ea2dd24854530b910d86d5e1c0f38c217e748a76ccbf3c450a0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6262,"rank":6262,"depth":24,"x":1928.405,"y":712.563,"cluster":"scheme-morphisms"},{"id":"stacks:01ZC","tag":"01ZC","title":"Limits and morphisms of finite presentation · Proposition 01ZC","summary":"[EGA] Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is locally of finite presentation. • For any directed set I, and any inverse system (T_i, f_ii') of S-schemes over I with each T_i affine, we have Mor_S(lim_i T_i, X) = colim_i Mor_S(T_i, X) • For any directed set I, and any inverse system (T_i, f_ii') of S-schemes over I with each f_ii' affine and every T_i quasi-compact and quasi-separated as a scheme, we have Mor_S(lim_i T_i,…","statement_latex":"\\begin{reference}\n\\cite[IV, Proposition 8.14.2]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is locally of finite presentation.\n\\item For any directed set $I$, and any\ninverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$\nwith each $T_i$ affine, we have\n$$\n\\Mor_S(\\lim_i T_i, X) =\n\\colim_i \\Mor_S(T_i, X)\n$$\n\\item For any directed set $I$, and any\ninverse system $(T_i, f_{ii'})$ of $S$-schemes over $I$\nwith each $f_{ii'}$ affine and every $T_i$ quasi-compact and\nquasi-separated as a scheme, we have\n$$\n\\Mor_S(\\lim_i T_i, X) =\n\\colim_i \\Mor_S(T_i, X)\n$$\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Limits and morphisms of finite presentation","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZC","source_file":"limits.tex","source_line":1284,"source_end_line":1309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1284-L1309","statement_sha256":"bff78f032577dc28e00b53c0bc788fa8acd774c59f361a79a0a7af801338ba40","origin":"The Stacks Project","memory_eligible":false,"source_rank":6263,"rank":6263,"depth":25,"x":2078.774,"y":598.266,"cluster":"scheme-morphisms"},{"id":"stacks:0CM0","tag":"0CM0","title":"Limits and morphisms of finite presentation · Lemma 0CM0","summary":"Let f : X → S be a morphism of schemes. If for every directed limit T = lim_i ∈ I T_i of affine schemes over S the map colim Mor_S(T_i, X) → Mor_S(T, X) is surjective, then f is locally of finite presentation. In other words, in Proposition [Tag 01ZC] parts (2) and (3) it suffices to check surjectivity of the map.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. If for every directed limit\n$T = \\lim_{i \\in I} T_i$ of affine schemes over $S$ the map\n$$\n\\colim \\Mor_S(T_i, X) \\longrightarrow \\Mor_S(T, X)\n$$\nis surjective, then $f$ is locally of finite presentation.\nIn other words, in\nProposition \\ref{proposition-characterize-locally-finite-presentation}\nparts (2) and (3) it suffices to check surjectivity of the map.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Limits and morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CM0","source_file":"limits.tex","source_line":1462,"source_end_line":1473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1462-L1473","statement_sha256":"62a7dab98acb4aa86e8bfcd54db8ee8d66b368e9e8ee772628b6f08f3cd7eff4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6264,"rank":6264,"depth":26,"x":2059.791,"y":768.025,"cluster":"scheme-morphisms"},{"id":"stacks:0GWT","tag":"0GWT","title":"Limits and morphisms of finite presentation · Lemma 0GWT","summary":"Let S be a scheme. Let X and Y be schemes over S. Assume Y is locally of finite presentation over S. Let x ∈ X be a closed point such that U = X setminus (x) → X is quasi-compact. With V = Spec(O_X, x) setminus (x) there is a bijection ( morphisms X → Y over S ) → ( (a, b) where a : U → Y and b : Spec(O_X, x) → Y are morphisms over S which agree over V )","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$.\nAssume $Y$ is locally of finite presentation over $S$.\nLet $x \\in X$ be a closed point such that\n$U = X \\setminus \\{x\\} \\to X$ is quasi-compact. With\n$V = \\Spec(\\mathcal{O}_{X, x}) \\setminus \\{x\\}$ there is\na bijection\n$$\n\\left\\{\n\\begin{matrix}\n\\text{morphisms }X \\to Y\\text{ over }S\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\n(a, b)\\text{ where }\na : U \\to Y\\text{ and }b : \\Spec(\\mathcal{O}_{X, x}) \\to Y\\\\\n\\text{ are morphisms over }S\n\\text{ which agree over }V\n\\end{matrix}\n\\right\\}\n$$","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Limits and morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWT","source_file":"limits.tex","source_line":1529,"source_end_line":1553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1529-L1553","statement_sha256":"56f628ccb9095422674325bbc87e8149faee0d9b8267d41858b41ebe0c6e9bbf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6265,"rank":6265,"depth":26,"x":1937.16,"y":631.951,"cluster":"scheme-morphisms"},{"id":"stacks:0GS1","tag":"0GS1","title":"Relative approximation · Lemma 0GS1","summary":"Let f : X → S be a morphism of quasi-compact and quasi-separated schemes. Then there exists a direct set I and an inverse system (f_i : X_i → S_i) of morphisms schemes over I, such that the transition morphisms X_i → X_i' and S_i → S_i' are affine, such that X_i and S_i are of finite type over Z, and such that (X → S) = lim (X_i → S_i).","statement_latex":"Let $f : X \\to S$ be a morphism of quasi-compact and quasi-separated schemes.\nThen there exists a direct set $I$ and an inverse system $(f_i : X_i \\to S_i)$\nof morphisms schemes over $I$, such that the transition morphisms\n$X_i \\to X_{i'}$ and $S_i \\to S_{i'}$ are affine, such that $X_i$\nand $S_i$ are of finite type over $\\mathbf{Z}$, and such that\n$(X \\to S) = \\lim (X_i \\to S_i)$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Relative approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GS1","source_file":"limits.tex","source_line":1593,"source_end_line":1601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1593-L1601","statement_sha256":"afbeb61f8a375d375f7d8d4ed589d544ed0ea876ed8fd0dc19afa121bb320ece","origin":"The Stacks Project","memory_eligible":false,"source_rank":6266,"rank":6266,"depth":26,"x":2137.196,"y":662.735,"cluster":"scheme-morphisms"},{"id":"stacks:09MV","tag":"09MV","title":"Relative approximation · Lemma 09MV","summary":"Let f : X → S be a morphism of schemes. Assume that • X is quasi-compact and quasi-separated, and • S is quasi-separated. Then X = lim X_i is a limit of a directed system of schemes X_i of finite presentation over S with affine transition morphisms over S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume that\n\\begin{enumerate}\n\\item $X$ is quasi-compact and quasi-separated, and\n\\item $S$ is quasi-separated.\n\\end{enumerate}\nThen $X = \\lim X_i$ is a limit of a directed system of schemes\n$X_i$ of finite presentation over $S$ with affine transition morphisms\nover $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Relative approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MV","source_file":"limits.tex","source_line":1654,"source_end_line":1664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1654-L1664","statement_sha256":"79651e57ecb261aef0f0964efab55c083f2b2fd143c628c8a37d96cdb0fe4825","origin":"The Stacks Project","memory_eligible":false,"source_rank":6267,"rank":6267,"depth":27,"x":1964.78,"y":753.624,"cluster":"scheme-morphisms"},{"id":"stacks:09YZ","tag":"09YZ","title":"Relative approximation · Lemma 09YZ","summary":"Let X → S be an integral morphism with S quasi-compact and quasi-separated. Then X = lim X_i with X_i → S finite and of finite presentation.","statement_latex":"Let $X \\to S$ be an integral morphism with $S$ quasi-compact and\nquasi-separated. Then $X = \\lim X_i$ with $X_i \\to S$ finite and\nof finite presentation.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Relative approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YZ","source_file":"limits.tex","source_line":1696,"source_end_line":1701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1696-L1701","statement_sha256":"850b45938f1ac7a27f8c5c6fb5eda4355a15dcc36bea07a4faf8e9db5d86079e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6268,"rank":6268,"depth":22,"x":2018.876,"y":588.62,"cluster":"scheme-morphisms"},{"id":"stacks:01ZN","tag":"01ZN","title":"Descending properties of morphisms · Lemma 01ZN","summary":"Notation and assumptions as in Situation [Tag 081D]. If f is affine, then there exists an index i ≥ 0 such that f_i is affine.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf $f$ is affine, then there exists an index $i \\geq 0$\nsuch that $f_i$ is affine.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZN","source_file":"limits.tex","source_line":1749,"source_end_line":1754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1749-L1754","statement_sha256":"2cac885348dd68bc06f43beeb0ac3a52888f7a422f94bd352d8ef42883a230b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6269,"rank":6269,"depth":25,"x":2111.762,"y":741.126,"cluster":"scheme-morphisms"},{"id":"stacks:01ZO","tag":"01ZO","title":"Descending properties of morphisms · Lemma 01ZO","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is a finite morphism, and • f_0 is locally of finite type, then there exists an i ≥ 0 such that f_i is finite.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is a finite morphism, and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen there exists an $i \\geq 0$ such that $f_i$ is finite.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZO","source_file":"limits.tex","source_line":1771,"source_end_line":1780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1771-L1780","statement_sha256":"aa6d446418007e2d0c4ef9112450093a3720abea2b58c2906914e944a918f21b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6270,"rank":6270,"depth":26,"x":1920.455,"y":681.322,"cluster":"scheme-morphisms"},{"id":"stacks:0C4W","tag":"0C4W","title":"Descending properties of morphisms · Lemma 0C4W","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is unramified, and • f_0 is locally of finite type, then there exists an i ≥ 0 such that f_i is unramified.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is unramified, and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen there exists an $i \\geq 0$ such that $f_i$ is unramified.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4W","source_file":"limits.tex","source_line":1793,"source_end_line":1802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1793-L1802","statement_sha256":"9dd8a5e85f0ae0b30a587294c9066086a1b0eab35dd320d7539e56c3c5f90f23","origin":"The Stacks Project","memory_eligible":false,"source_rank":6271,"rank":6271,"depth":3,"x":2109.786,"y":616.81,"cluster":"scheme-morphisms"},{"id":"stacks:01ZP","tag":"01ZP","title":"Descending properties of morphisms · Lemma 01ZP","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is a closed immersion, and • f_0 is locally of finite type, then there exists an i ≥ 0 such that f_i is a closed immersion.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is a closed immersion, and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen there exists an $i \\geq 0$ such that $f_i$ is a closed immersion.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZP","source_file":"limits.tex","source_line":1827,"source_end_line":1836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1827-L1836","statement_sha256":"cea35a0e7d8b7dda188b390167f4bfe1872d1b5948c6f8256c0f77fc327b9ef2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6272,"rank":6272,"depth":26,"x":2021.976,"y":771.951,"cluster":"scheme-morphisms"},{"id":"stacks:01ZQ","tag":"01ZQ","title":"Descending properties of morphisms · Lemma 01ZQ","summary":"Notation and assumptions as in Situation [Tag 081D]. If f is separated, then f_i is separated for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf $f$ is separated, then $f_i$ is separated for some $i \\geq 0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZQ","source_file":"limits.tex","source_line":1850,"source_end_line":1854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1850-L1854","statement_sha256":"b00e7306b8ca4d76165dc43c10c3be901d45c25cde97a7d697c3705d91e6123b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6273,"rank":6273,"depth":27,"x":1961.911,"y":607.578,"cluster":"scheme-morphisms"},{"id":"stacks:04AI","tag":"04AI","title":"Descending properties of morphisms · Lemma 04AI","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is flat, • f_0 is locally of finite presentation, then f_i is flat for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is flat for some $i \\geq 0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AI","source_file":"limits.tex","source_line":1867,"source_end_line":1876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1867-L1876","statement_sha256":"d4e465317d9e73cefb22dfc22a2a1e451eb2e1075244441ec839edc2aab1c972","origin":"The Stacks Project","memory_eligible":false,"source_rank":6274,"rank":6274,"depth":35,"x":2138.545,"y":694.781,"cluster":"scheme-morphisms"},{"id":"stacks:06AC","tag":"06AC","title":"Descending properties of morphisms · Lemma 06AC","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is finite locally free (of degree d), • f_0 is locally of finite presentation, then f_i is finite locally free (of degree d) for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is finite locally free (of degree $d$),\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is finite locally free (of degree $d$) for some $i \\geq 0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06AC","source_file":"limits.tex","source_line":1901,"source_end_line":1910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1901-L1910","statement_sha256":"7d29a91a1a2f611ddf325254e9c7b4102bad8808a9ea4aef9a52ed5484679869","origin":"The Stacks Project","memory_eligible":false,"source_rank":6275,"rank":6275,"depth":36,"x":1937.991,"y":730.738,"cluster":"scheme-morphisms"},{"id":"stacks:0C0C","tag":"0C0C","title":"Descending properties of morphisms · Lemma 0C0C","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is smooth, • f_0 is locally of finite presentation, then f_i is smooth for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is smooth,\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is smooth for some $i \\geq 0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0C","source_file":"limits.tex","source_line":1930,"source_end_line":1939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1930-L1939","statement_sha256":"5bdb7ec383e9c2d4fc29c74bee7f5875af75c24904baa3cc3d764a419c72a856","origin":"The Stacks Project","memory_eligible":false,"source_rank":6276,"rank":6276,"depth":37,"x":2057.068,"y":590.298,"cluster":"scheme-morphisms"},{"id":"stacks:07RP","tag":"07RP","title":"Descending properties of morphisms · Lemma 07RP","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is étale, • f_0 is locally of finite presentation, then f_i is étale for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is \\'etale,\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is \\'etale for some $i \\geq 0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RP","source_file":"limits.tex","source_line":1948,"source_end_line":1957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1948-L1957","statement_sha256":"968336fec67c47e6653ce1a27c4e9c55f2592fb61b55d2580ab963d5e2326b76","origin":"The Stacks Project","memory_eligible":false,"source_rank":6277,"rank":6277,"depth":39,"x":2082.223,"y":761.577,"cluster":"scheme-morphisms"},{"id":"stacks:081E","tag":"081E","title":"Descending properties of morphisms · Lemma 081E","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is an isomorphism, and • f_0 is locally of finite presentation, then f_i is an isomorphism for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is an isomorphism, and\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is an isomorphism for some $i \\geq 0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081E","source_file":"limits.tex","source_line":1966,"source_end_line":1975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1966-L1975","statement_sha256":"4de058312f97e07ee3fb134a6cc0aeb08854c88142c946086e13100a48e94aff","origin":"The Stacks Project","memory_eligible":false,"source_rank":6278,"rank":6278,"depth":40,"x":1925.797,"y":649.451,"cluster":"scheme-morphisms"},{"id":"stacks:0EUU","tag":"0EUU","title":"Descending properties of morphisms · Lemma 0EUU","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is an open immersion, and • f_0 is locally of finite presentation, then f_i is an open immersion for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is an open immersion, and\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is an open immersion for some $i \\geq 0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUU","source_file":"limits.tex","source_line":1990,"source_end_line":1999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L1990-L1999","statement_sha256":"65c61b5eb8c58c71444478d341f1f1706012d7895ac654a2a60fcb3f93be7e29","origin":"The Stacks Project","memory_eligible":false,"source_rank":6279,"rank":6279,"depth":41,"x":2131.495,"y":643.366,"cluster":"scheme-morphisms"},{"id":"stacks:0GTB","tag":"0GTB","title":"Descending properties of morphisms · Lemma 0GTB","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is an immersion, and • f_0 is locally of finite type, then f_i is an immersion for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is an immersion, and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen $f_i$ is an immersion for some $i \\geq 0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTB","source_file":"limits.tex","source_line":2013,"source_end_line":2022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2013-L2022","statement_sha256":"b1f75d2873eefc729435f1fc17265b72ea38c932043aee131562cbf52babecb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6280,"rank":6280,"depth":27,"x":1984.579,"y":764.679,"cluster":"scheme-morphisms"},{"id":"stacks:07RQ","tag":"07RQ","title":"Descending properties of morphisms · Lemma 07RQ","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is a monomorphism, and • f_0 is locally of finite type, then f_i is a monomorphism for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is a monomorphism, and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen $f_i$ is a monomorphism for some $i \\geq 0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RQ","source_file":"limits.tex","source_line":2042,"source_end_line":2051,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2042-L2051","statement_sha256":"161d978675e46e71dc13f5d396a0b88d7d68a440baca47d34c95be20896087c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6281,"rank":6281,"depth":41,"x":1995.365,"y":591.707,"cluster":"scheme-morphisms"},{"id":"stacks:07RR","tag":"07RR","title":"Descending properties of morphisms · Lemma 07RR","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is surjective, and • f_0 is locally of finite presentation, then there exists an i ≥ 0 such that f_i is surjective.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is surjective, and\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen there exists an $i \\geq 0$ such that $f_i$ is surjective.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RR","source_file":"limits.tex","source_line":2068,"source_end_line":2077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2068-L2077","statement_sha256":"74bb1ebe3a9a292302661be557b0d407cfdc850ec5f6ede026e2ecc0a80e5064","origin":"The Stacks Project","memory_eligible":false,"source_rank":6282,"rank":6282,"depth":24,"x":2126.628,"y":725.493,"cluster":"scheme-morphisms"},{"id":"stacks:0C3L","tag":"0C3L","title":"Descending properties of morphisms · Lemma 0C3L","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is syntomic, and • f_0 is locally of finite presentation, then there exists an i ≥ 0 such that f_i is syntomic.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is syntomic, and\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen there exists an $i \\geq 0$ such that $f_i$ is syntomic.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3L","source_file":"limits.tex","source_line":2090,"source_end_line":2099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2090-L2099","statement_sha256":"d0127bdc975483132525d4edebe8414607502086c02e4fe431fe4cecbb7dffb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6283,"rank":6283,"depth":36,"x":1922.068,"y":701.301,"cluster":"scheme-morphisms"},{"id":"stacks:01ZE","tag":"01ZE","title":"Finite type closed in finite presentation · Lemma 01ZE","summary":"Let f : X → S be a morphism of schemes. Assume: • The morphism f is locally of finite type. • The scheme X is quasi-compact and quasi-separated. Then there exists a morphism of finite presentation f' : X' → S and an immersion X → X' of schemes over S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume:\n\\begin{enumerate}\n\\item The morphism $f$ is locally of finite type.\n\\item The scheme $X$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nThen there exists a morphism of finite presentation\n$f' : X' \\to S$ and an immersion $X \\to X'$ of schemes over $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZE","source_file":"limits.tex","source_line":2135,"source_end_line":2145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2135-L2145","statement_sha256":"83d1226870b47f18ff85ad37f5550b9d8b6f4269577f446d823a2c51f5d221a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6284,"rank":6284,"depth":25,"x":2092.512,"y":602.981,"cluster":"scheme-morphisms"},{"id":"stacks:01ZG","tag":"01ZG","title":"Finite type closed in finite presentation · Lemma 01ZG","summary":"Let f : X → S be a morphism of schemes. Assume: • The morphism f is of locally of finite type. • The scheme X is quasi-compact and quasi-separated, and • The scheme S is quasi-separated. Then there exists a morphism of finite presentation f' : X' → S and a closed immersion X → X' of schemes over S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume:\n\\begin{enumerate}\n\\item The morphism $f$ is of locally of finite type.\n\\item The scheme $X$ is quasi-compact and quasi-separated, and\n\\item The scheme $S$ is quasi-separated.\n\\end{enumerate}\nThen there exists a morphism of finite presentation\n$f' : X' \\to S$ and a closed immersion $X \\to X'$ of schemes over $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZG","source_file":"limits.tex","source_line":2206,"source_end_line":2217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2206-L2217","statement_sha256":"a31dd5d58eba9e0a3b9cbbe66aface377100c61591e8fe779d1edc5e28a32aab","origin":"The Stacks Project","memory_eligible":false,"source_rank":6285,"rank":6285,"depth":26,"x":2045.854,"y":772.345,"cluster":"scheme-morphisms"},{"id":"stacks:09ZP","tag":"09ZP","title":"Finite type closed in finite presentation · Lemma 09ZP","summary":"Let X → Y be a closed immersion of schemes. Assume Y quasi-compact and quasi-separated. Then X can be written as a directed limit X = lim X_i of schemes over Y where X_i → Y is a closed immersion of finite presentation.","statement_latex":"Let $X \\to Y$ be a closed immersion of schemes. Assume $Y$ quasi-compact and\nquasi-separated. Then $X$ can be written as a directed limit $X = \\lim X_i$\nof schemes over $Y$ where $X_i \\to Y$ is a closed immersion\nof finite presentation.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZP","source_file":"limits.tex","source_line":2244,"source_end_line":2250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2244-L2250","statement_sha256":"0bf8e552308f53e6be2caad00f384aad25e618838dcd2882d771e8d495ae4a02","origin":"The Stacks Project","memory_eligible":false,"source_rank":6286,"rank":6286,"depth":17,"x":1943.972,"y":620.85,"cluster":"scheme-morphisms"},{"id":"stacks:09ZQ","tag":"09ZQ","title":"Finite type closed in finite presentation · Lemma 09ZQ","summary":"Let f : X → S be a morphism of schemes. Assume • The morphism f is of locally of finite type. • The scheme X is quasi-compact and quasi-separated, and • The scheme S is quasi-separated. Then X = lim X_i where the X_i → S are of finite presentation, the X_i are quasi-compact and quasi-separated, and the transition morphisms X_i' → X_i are closed immersions (which implies that X → X_i are closed immersions for all i).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item The morphism $f$ is of locally of finite type.\n\\item The scheme $X$ is quasi-compact and quasi-separated, and\n\\item The scheme $S$ is quasi-separated.\n\\end{enumerate}\nThen $X = \\lim X_i$ where the $X_i \\to S$ are of\nfinite presentation, the $X_i$ are quasi-compact and quasi-separated,\nand the transition morphisms $X_{i'} \\to X_i$ are closed immersions\n(which implies that $X \\to X_i$ are closed immersions for all $i$).","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZQ","source_file":"limits.tex","source_line":2271,"source_end_line":2283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2271-L2283","statement_sha256":"beb6e3c3565454f743c54f254ac761b8ba2baad78844a139d073d8e544cb8a36","origin":"The Stacks Project","memory_eligible":false,"source_rank":6287,"rank":6287,"depth":27,"x":2141.1,"y":674.797,"cluster":"scheme-morphisms"},{"id":"stacks:01ZJ","tag":"01ZJ","title":"Finite type closed in finite presentation · Proposition 01ZJ","summary":"Let f : X → S be a morphism of schemes. Assume • f is of finite type and separated, and • S is quasi-compact and quasi-separated. Then there exists a separated morphism of finite presentation f' : X' → S and a closed immersion X → X' of schemes over S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $f$ is of finite type and separated, and\n\\item $S$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nThen there exists a separated morphism of finite presentation\n$f' : X' \\to S$ and a closed immersion $X \\to X'$ of schemes over $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Finite type closed in finite presentation","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZJ","source_file":"limits.tex","source_line":2299,"source_end_line":2308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2299-L2308","statement_sha256":"5a8bd911fbf0fb5be67dbfa84e950936380b40e4b873784c1f433975531b19e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6288,"rank":6288,"depth":28,"x":1952.192,"y":746.936,"cluster":"scheme-morphisms"},{"id":"stacks:01ZK","tag":"01ZK","title":"Finite type closed in finite presentation · Lemma 01ZK","summary":"Let f : X → S be a morphism of schemes. Assume • f is finite, and • S is quasi-compact and quasi-separated. Then there exists a morphism which is finite and of finite presentation f' : X' → S and a closed immersion X → X' of schemes over S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $f$ is finite, and\n\\item $S$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nThen there exists a morphism which is finite and of finite presentation\n$f' : X' \\to S$ and a closed immersion $X \\to X'$ of schemes over $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZK","source_file":"limits.tex","source_line":2317,"source_end_line":2326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2317-L2326","statement_sha256":"6bb38a689dcb6f6c4dc15658c23de3cdc76ded4ec6e56843b4adbe79bc44dfd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6289,"rank":6289,"depth":28,"x":2033.549,"y":586.411,"cluster":"scheme-morphisms"},{"id":"stacks:09YY","tag":"09YY","title":"Finite type closed in finite presentation · Lemma 09YY","summary":"Let f : X → S be a morphism of schemes. Assume • f is finite, and • S quasi-compact and quasi-separated. Then X is a directed limit X = lim X_i where the transition maps are closed immersions and the objects X_i are finite and of finite presentation over S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $f$ is finite, and\n\\item $S$ quasi-compact and quasi-separated.\n\\end{enumerate}\nThen $X$ is a directed limit $X = \\lim X_i$\nwhere the transition maps are closed immersions and the objects\n$X_i$ are finite and of finite presentation over $S$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YY","source_file":"limits.tex","source_line":2335,"source_end_line":2345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2335-L2345","statement_sha256":"77eefd4b842afdecd203b07b98d98b2314e425a7c5489420987f220ad3140468","origin":"The Stacks Project","memory_eligible":false,"source_rank":6290,"rank":6290,"depth":28,"x":2102.708,"y":751.087,"cluster":"scheme-morphisms"},{"id":"stacks:01ZM","tag":"01ZM","title":"Descending relative objects · Lemma 01ZM","summary":"Let I be a directed set. Let (S_i, f_ii') be an inverse system of schemes over I. Assume • the morphisms f_ii' : S_i → S_i' are affine, • the schemes S_i are quasi-compact and quasi-separated. Let S = lim_i S_i. Then we have the following: • For any morphism of finite presentation X → S there exists an index i ∈ I and a morphism of finite presentation X_i → S_i such that X ≅ X_i, S as schemes over S. • Given an index i ∈ I, schemes X_i, Y_i of finite presentation over…","statement_latex":"Let $I$ be a directed set.\nLet $(S_i, f_{ii'})$ be an inverse system of schemes over $I$.\nAssume\n\\begin{enumerate}\n\\item the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine,\n\\item the schemes $S_i$ are quasi-compact and quasi-separated.\n\\end{enumerate}\nLet $S = \\lim_i S_i$. Then we have the following:\n\\begin{enumerate}\n\\item For any morphism of finite presentation $X \\to S$\nthere exists an index $i \\in I$ and a morphism of finite\npresentation $X_i \\to S_i$ such that $X \\cong X_{i, S}$ as\nschemes over $S$.\n\\item Given an index $i \\in I$, schemes\n$X_i$, $Y_i$ of finite presentation over $S_i$, and a morphism\n$\\varphi : X_{i, S} \\to Y_{i, S}$ over $S$, there exists an index\n$i' \\geq i$ and a morphism\n$\\varphi_{i'} : X_{i, S_{i'}} \\to Y_{i, S_{i'}}$\nwhose base change to $S$ is $\\varphi$.\n\\item Given an index $i \\in I$, schemes $X_i$, $Y_i$ of finite presentation\nover $S_i$ and a pair of morphisms $\\varphi_i, \\psi_i : X_i \\to Y_i$\nwhose base changes $\\varphi_{i, S} = \\psi_{i, S}$ are equal,\nthere exists an index $i' \\geq i$ such that\n$\\varphi_{i, S_{i'}} = \\psi_{i, S_{i'}}$.\n\\end{enumerate}\nIn other words, the category of schemes of finite presentation over\n$S$ is the colimit over $I$ of the categories of schemes of finite\npresentation over $S_i$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending relative objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZM","source_file":"limits.tex","source_line":2369,"source_end_line":2399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2369-L2399","statement_sha256":"89bae67ad68d239a42489dc590942a4722afe9bbe2daf101add118d5af17611d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6291,"rank":6291,"depth":25,"x":1919.123,"y":668.828,"cluster":"scheme-morphisms"},{"id":"stacks:01ZR","tag":"01ZR","title":"Descending relative objects · Lemma 01ZR","summary":"Let I be a directed set. Let (S_i, f_ii') be an inverse system of schemes over I. Assume • all the morphisms f_ii' : S_i → S_i' are affine, • all the schemes S_i are quasi-compact and quasi-separated. Let S = lim_i S_i. Then we have the following: • For any sheaf of O_S-modules F of finite presentation there exists an index i ∈ I and a sheaf of O_S_i-modules of finite presentation F_i such that F ≅ f_i^*F_i. • Suppose given an index i ∈ I, sheaves of O_S_i-modules F_i,…","statement_latex":"Let $I$ be a directed set.\nLet $(S_i, f_{ii'})$ be an inverse system of schemes over $I$.\nAssume\n\\begin{enumerate}\n\\item all the morphisms $f_{ii'} : S_i \\to S_{i'}$ are affine,\n\\item all the schemes $S_i$ are quasi-compact and quasi-separated.\n\\end{enumerate}\nLet $S = \\lim_i S_i$. Then we have the following:\n\\begin{enumerate}\n\\item For any sheaf of $\\mathcal{O}_S$-modules\n$\\mathcal{F}$ of finite presentation there exists an index\n$i \\in I$ and a sheaf of $\\mathcal{O}_{S_i}$-modules of finite\npresentation $\\mathcal{F}_i$ such that\n$\\mathcal{F} \\cong f_i^*\\mathcal{F}_i$.\n\\item Suppose given an index $i \\in I$, sheaves\nof $\\mathcal{O}_{S_i}$-modules $\\mathcal{F}_i$, $\\mathcal{G}_i$\nof finite presentation and a morphism\n$\\varphi : f_i^*\\mathcal{F}_i \\to f_i^*\\mathcal{G}_i$ over $S$.\nThen there exists an index $i' \\geq i$ and a morphism\n$\\varphi_{i'} : f_{i'i}^*\\mathcal{F}_i \\to f_{i'i}^*\\mathcal{G}_i$\nwhose base change to $S$ is $\\varphi$.\n\\item Suppose given an index $i \\in I$, sheaves of $\\mathcal{O}_{S_i}$-modules\n$\\mathcal{F}_i$, $\\mathcal{G}_i$ of finite presentation\nand a pair of morphisms $\\varphi_i, \\psi_i : \\mathcal{F}_i \\to \\mathcal{G}_i$.\nAssume that the base changes are equal: $f_i^*\\varphi_i = f_i^*\\psi_i$.\nThen there exists an index $i' \\geq i$ such that\n$f_{i'i}^*\\varphi_i = f_{i'i}^*\\psi_i$.\n\\end{enumerate}\nIn other words, the category of modules\nof finite presentation over $S$ is the colimit over $I$\nof the categories modules of finite presentation over $S_i$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending relative objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZR","source_file":"limits.tex","source_line":2547,"source_end_line":2580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2547-L2580","statement_sha256":"4ce3a1bb7729f3ace96c9a7e8b3453f3f651b22154477bee34197a244c187ea1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6292,"rank":6292,"depth":26,"x":2120.823,"y":625.276,"cluster":"scheme-morphisms"},{"id":"stacks:0B8W","tag":"0B8W","title":"Descending relative objects · Lemma 0B8W","summary":"Let S = lim S_i be the limit of a directed system of quasi-compact and quasi-separated schemes S_i with affine transition morphisms. Then • any finite locally free O_S-module is the pullback of a finite locally free O_S_i-module for some i, • any invertible O_S-module is the pullback of an invertible O_S_i-module for some i, and • any finite type quasi-coherent ideal I ⊂ O_S is of the form I_i · O_S for some i and some finite type quasi-coherent ideal I_i ⊂ O_S_i.","statement_latex":"Let $S = \\lim S_i$ be the limit of a directed system of quasi-compact and\nquasi-separated schemes $S_i$ with affine transition morphisms. Then\n\\begin{enumerate}\n\\item any finite locally free $\\mathcal{O}_S$-module is the pullback\nof a finite locally free $\\mathcal{O}_{S_i}$-module for some $i$,\n\\item any invertible $\\mathcal{O}_S$-module is the pullback of an invertible\n$\\mathcal{O}_{S_i}$-module for some $i$, and\n\\item any finite type quasi-coherent ideal $\\mathcal{I} \\subset \\mathcal{O}_S$\nis of the form $\\mathcal{I}_i \\cdot \\mathcal{O}_S$ for some $i$ and some\nfinite type quasi-coherent ideal $\\mathcal{I}_i \\subset \\mathcal{O}_{S_i}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending relative objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8W","source_file":"limits.tex","source_line":2607,"source_end_line":2620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2607-L2620","statement_sha256":"119ef926aa57f3c33fa5bf5121054a4cae523bf5171227cca797342d4c8d4b9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6293,"rank":6293,"depth":35,"x":2007.016,"y":771.968,"cluster":"scheme-morphisms"},{"id":"stacks:05LY","tag":"05LY","title":"Descending relative objects · Lemma 05LY","summary":"With notation and assumptions as in Lemma [Tag 01ZM]. Let i ∈ I. Suppose that φ_i : X_i → Y_i is a morphism of schemes of finite presentation over S_i and that F_i is a quasi-coherent O_X_i-module of finite presentation. If the pullback of F_i to X_i ×_S_i S is flat over Y_i ×_S_i S, then there exists an index i' ≥ i such that the pullback of F_i to X_i ×_S_i S_i' is flat over Y_i ×_S_i S_i'.","statement_latex":"With notation and assumptions as in\nLemma \\ref{lemma-descend-finite-presentation}.\nLet $i \\in I$.\nSuppose that $\\varphi_i : X_i \\to Y_i$ is a morphism of schemes\nof finite presentation over $S_i$ and that $\\mathcal{F}_i$ is a\nquasi-coherent $\\mathcal{O}_{X_i}$-module of finite presentation.\nIf the pullback of $\\mathcal{F}_i$ to $X_i \\times_{S_i} S$ is flat\nover $Y_i \\times_{S_i} S$, then there exists an index $i' \\geq i$\nsuch that the pullback of $\\mathcal{F}_i$ to $X_i \\times_{S_i} S_{i'}$\nis flat over $Y_i \\times_{S_i} S_{i'}$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending relative objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LY","source_file":"limits.tex","source_line":2666,"source_end_line":2678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2666-L2678","statement_sha256":"39440f0793d4e46843634d353625e23e84778c9ef0d2e0e6b74868ae4a4903b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6294,"rank":6294,"depth":36,"x":1972.941,"y":599.073,"cluster":"scheme-morphisms"},{"id":"stacks:0EY1","tag":"0EY1","title":"Descending relative objects · Lemma 0EY1","summary":"For a scheme T denote C_T the full subcategory of schemes W over T such that W is quasi-compact and quasi-separated and such that the structure morphism W → T is locally of finite presentation. Let S = lim S_i be a directed limit of schemes with affine transition morphisms. Then there is an equivalence of categories colim C_S_i → C_S given by the base change functors.","statement_latex":"For a scheme $T$ denote $\\mathcal{C}_T$ the full subcategory of\nschemes $W$ over $T$ such that $W$ is quasi-compact and quasi-separated\nand such that the structure morphism $W \\to T$ is\nlocally of finite presentation.\nLet $S = \\lim S_i$ be a directed limit of schemes with affine\ntransition morphisms. Then there is an equivalence\nof categories\n$$\n\\colim \\mathcal{C}_{S_i} \\longrightarrow \\mathcal{C}_S\n$$\ngiven by the base change functors.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending relative objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EY1","source_file":"limits.tex","source_line":2721,"source_end_line":2734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2721-L2734","statement_sha256":"f290ac6cbb227c3ab03c75d9862005db0117734f01707ddc969088097cb53bc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6295,"rank":6295,"depth":0,"x":2137.247,"y":707.321,"cluster":"scheme-morphisms"},{"id":"stacks:01ZT","tag":"01ZT","title":"Characterizing affine schemes · Lemma 01ZT","summary":"A scheme, admitting a finite surjective map from an affine scheme, is affine. Let f : X → S be a morphism of schemes. Assume that f is surjective and finite, and assume that X is affine. Then S is affine.","statement_latex":"\\begin{slogan}\nA scheme, admitting a finite surjective map from an affine scheme, is affine.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nAssume that $f$ is surjective and finite, and assume that $X$ is affine.\nThen $S$ is affine.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Characterizing affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/01ZT","source_file":"limits.tex","source_line":2833,"source_end_line":2841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2833-L2841","statement_sha256":"7f6e84f3a28d282b5849b7c4489afa081fbb824bca37e8ba10007cc6c73a86b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6296,"rank":6296,"depth":29,"x":1928.859,"y":720.744,"cluster":"scheme-morphisms"},{"id":"stacks:05YU","tag":"05YU","title":"Characterizing affine schemes · Proposition 05YU","summary":"A scheme admitting a surjective integral map from an affine scheme is affine. Let f : X → S be a morphism of schemes. Assume X is affine and that f is surjective and universally closed. Then S is affine.","statement_latex":"\\begin{slogan}\nA scheme admitting a surjective integral map from an affine scheme is affine.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes. Assume $X$ is affine\nand that $f$ is surjective and universally closed\\footnote{An integral\nmorphism is universally closed, see\nMorphisms, Lemma \\ref{morphisms-lemma-integral-universally-closed}.}.\nThen $S$ is affine.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Characterizing affine schemes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YU","source_file":"limits.tex","source_line":2875,"source_end_line":2885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2875-L2885","statement_sha256":"4e4f8dc1008b904834c8cce28537e0e6493a77eabc5596982f55208b35d4c82a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6297,"rank":6297,"depth":30,"x":2071.851,"y":592.491,"cluster":"scheme-morphisms"},{"id":"stacks:09NL","tag":"09NL","title":"Characterizing affine schemes · Lemma 09NL","summary":"Let X be a scheme which is set theoretically the union of finitely many affine closed subschemes. Then X is affine.","statement_latex":"Let $X$ be a scheme which is set theoretically the union of\nfinitely many affine closed subschemes. Then $X$ is affine.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Characterizing affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NL","source_file":"limits.tex","source_line":2911,"source_end_line":2915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2911-L2915","statement_sha256":"d5a4806b24987f580ef98e0d1cd2d871a542f887d00744f4557709e43396f1c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6298,"rank":6298,"depth":31,"x":2069.546,"y":768.351,"cluster":"scheme-morphisms"},{"id":"stacks:09MW","tag":"09MW","title":"Characterizing affine schemes · Lemma 09MW","summary":"Let i : Z → X be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Let L be an invertible sheaf on X. Then i^*L is ample on Z, if and only if L is ample on X.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes\ninducing a homeomorphism of underlying topological spaces.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nThen $i^*\\mathcal{L}$ is ample on $Z$, if and only if $\\mathcal{L}$\nis ample on $X$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Characterizing affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MW","source_file":"limits.tex","source_line":2924,"source_end_line":2931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2924-L2931","statement_sha256":"5afa82e40711c28e6cbcae81f16d88d3c1400779f8e71dbb73a8c6b67d233bb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6299,"rank":6299,"depth":29,"x":1929.704,"y":637.254,"cluster":"scheme-morphisms"},{"id":"stacks:0B7L","tag":"0B7L","title":"Characterizing affine schemes · Lemma 0B7L","summary":"Let i : Z → X be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Then X is quasi-affine if and only if Z is quasi-affine.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes\ninducing a homeomorphism of underlying topological spaces.\nThen $X$ is quasi-affine if and only if $Z$ is quasi-affine.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Characterizing affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7L","source_file":"limits.tex","source_line":2981,"source_end_line":2986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L2981-L2986","statement_sha256":"d290d79b039592c932acd6f3facfe112c72454509a64576b61b8265cb9c7468f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6300,"rank":6300,"depth":30,"x":2138.425,"y":654.588,"cluster":"scheme-morphisms"},{"id":"stacks:0E21","tag":"0E21","title":"Characterizing affine schemes · Lemma 0E21","summary":"Let X be a scheme. Let L be an ample invertible sheaf on X. Assume we have morphisms of schemes Spec(k) ← Spec(A) → W ⊂ X where k is a field, A is an integral k-algebra, W is open in X. Then there exists an n > 0 and a section s ∈ Γ(X, L^⊗ n) such that X_s is affine, X_s ⊂ W, and Spec(A) → W factors through X_s","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{L}$ be an ample invertible sheaf on $X$.\nAssume we have morphisms of schemes\n$$\n\\Spec(k) \\leftarrow \\Spec(A) \\to W \\subset X\n$$\nwhere $k$ is a field, $A$ is an integral $k$-algebra, $W$ is open in $X$.\nThen there exists an $n > 0$ and a section\n$s \\in \\Gamma(X, \\mathcal{L}^{\\otimes n})$ such that\n$X_s$ is affine, $X_s \\subset W$, and $\\Spec(A) \\to W$ factors through $X_s$","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Characterizing affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E21","source_file":"limits.tex","source_line":3003,"source_end_line":3014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3003-L3014","statement_sha256":"01238bf3eb43e049a65674397f7647b0cfa8eb16e06665c51b931d1da5fb4b1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6301,"rank":6301,"depth":36,"x":1970.434,"y":760.33,"cluster":"scheme-morphisms"},{"id":"stacks:0202","tag":"0202","title":"Variants of Chow's Lemma · Lemma 0202","summary":"Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a separated morphism of finite type. Then there exists an n ≥ 0 and a diagram xymatrix X ar[rd] & X' ar[d] ar[l]^π ar[r] & P^n_S ar[dl] & S & where X' → P^n_S is an immersion, and π : X' → X is proper and surjective.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to S$ be a separated morphism of finite type.\nThen there exists an $n \\geq 0$ and a diagram\n$$\n\\xymatrix{\nX \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\\n& S &\n}\n$$\nwhere $X' \\to \\mathbf{P}^n_S$ is an immersion, and\n$\\pi : X' \\to X$ is proper and surjective.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Variants of Chow's Lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0202","source_file":"limits.tex","source_line":3070,"source_end_line":3083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3070-L3083","statement_sha256":"8d92ea82c76c96d4e15c5cb0aeddbe89fa7d6cd3b5cfaa2a55657323d67cbdd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6302,"rank":6302,"depth":29,"x":2009.306,"y":586.886,"cluster":"scheme-morphisms"},{"id":"stacks:0203","tag":"0203","title":"Variants of Chow's Lemma · Lemma 0203","summary":"Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a separated morphism of finite type. Assume that X has finitely many irreducible components. Then there exists an n ≥ 0 and a diagram xymatrix X ar[rd] & X' ar[d] ar[l]^π ar[r] & P^n_S ar[dl] & S & where X' → P^n_S is an immersion, and π : X' → X is proper and surjective. Moreover, there exists an open dense subscheme U ⊂ X such that π^-1(U) → U is an isomorphism of schemes.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to S$ be a separated morphism of finite type.\nAssume that $X$ has finitely many irreducible components.\nThen there exists an $n \\geq 0$ and a diagram\n$$\n\\xymatrix{\nX \\ar[rd] & X' \\ar[d] \\ar[l]^\\pi \\ar[r] & \\mathbf{P}^n_S \\ar[dl] \\\\\n& S &\n}\n$$\nwhere $X' \\to \\mathbf{P}^n_S$ is an immersion, and\n$\\pi : X' \\to X$ is proper and surjective. Moreover, there exists\nan open dense subscheme $U \\subset X$ such that $\\pi^{-1}(U) \\to U$\nis an isomorphism of schemes.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Variants of Chow's Lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0203","source_file":"limits.tex","source_line":3137,"source_end_line":3153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3137-L3153","statement_sha256":"5f992357757e0a781311e49840b153b1a10a708110192f0d3217bf84d81fe5b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6303,"rank":6303,"depth":30,"x":2120.216,"y":736.967,"cluster":"scheme-morphisms"},{"id":"stacks:081F","tag":"081F","title":"Applications of Chow's lemma · Lemma 081F","summary":"If the base change of a scheme to a limit is proper, then already the base change is proper at a finite level. Assumptions and notation as in Situation [Tag 081D]. If • f is proper, and • f_0 is locally of finite type, then there exists an i such that f_i is proper.","statement_latex":"\\begin{slogan}\nIf the base change of a scheme to a limit is proper, then\nalready the base change is proper at a finite level.\n\\end{slogan}\nAssumptions and notation as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is proper, and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen there exists an $i$ such that $f_i$ is proper.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Applications of Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081F","source_file":"limits.tex","source_line":3264,"source_end_line":3277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3264-L3277","statement_sha256":"8d4039e647f8e5df041b664d118957d9a52d4820689ce4509e8d7915610314ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":6304,"rank":6304,"depth":30,"x":1917.569,"y":689.192,"cluster":"scheme-morphisms"},{"id":"stacks:09ZR","tag":"09ZR","title":"Applications of Chow's lemma · Lemma 09ZR","summary":"Let f : X → S be a proper morphism with S quasi-compact and quasi-separated. Then X = lim X_i is a directed limit of schemes X_i proper and of finite presentation over S such that all transition morphisms and the morphisms X → X_i are closed immersions.","statement_latex":"Let $f : X \\to S$ be a proper morphism with $S$ quasi-compact and\nquasi-separated. Then $X = \\lim X_i$ is a directed limit of schemes\n$X_i$ proper and of finite presentation over $S$ such that\nall transition morphisms and the morphisms $X \\to X_i$ are closed\nimmersions.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Applications of Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZR","source_file":"limits.tex","source_line":3308,"source_end_line":3315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3308-L3315","statement_sha256":"6e71a5fbcfd07f4981c092c95c21e9ad90b8ccfcecffc868ffbe426868f8307a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6305,"rank":6305,"depth":30,"x":2105.577,"y":609.366,"cluster":"scheme-morphisms"},{"id":"stacks:0A0P","tag":"0A0P","title":"Applications of Chow's lemma · Lemma 0A0P","summary":"Let f : X → S be a proper morphism with S quasi-compact and quasi-separated. Then there exists a directed set I, an inverse system (f_i : X_i → S_i) of morphisms of schemes over I, such that the transition morphisms X_i → X_i' and S_i → S_i' are affine, such that f_i is proper, such that S_i is of finite type over Z, and such that (X → S) = lim (X_i → S_i).","statement_latex":"Let $f : X \\to S$ be a proper morphism with $S$ quasi-compact and\nquasi-separated. Then there exists a directed set $I$, an\ninverse system $(f_i : X_i \\to S_i)$ of morphisms of schemes over $I$,\nsuch that the transition morphisms $X_i \\to X_{i'}$ and $S_i \\to S_{i'}$\nare affine, such that $f_i$ is proper, such that $S_i$ is of finite\ntype over $\\mathbf{Z}$, and such that\n$(X \\to S) = \\lim (X_i \\to S_i)$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Applications of Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0P","source_file":"limits.tex","source_line":3345,"source_end_line":3354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3345-L3354","statement_sha256":"5865ebb6067579c42366049e117037aba3e2e4308735aa38f450539b92f5b353","origin":"The Stacks Project","memory_eligible":false,"source_rank":6306,"rank":6306,"depth":31,"x":2031.073,"y":775.049,"cluster":"scheme-morphisms"},{"id":"stacks:0EX1","tag":"0EX1","title":"Applications of Chow's lemma · Lemma 0EX1","summary":"Let S be a scheme. Let X = lim X_i be a directed limit of schemes over S with affine transition morphisms. Let Y → X be a morphism of schemes over S. If Y → X is proper, X_i quasi-compact and quasi-separated, and Y locally of finite type over S, then Y → X_i is proper for i large enough.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim X_i$ be a directed limit of\nschemes over $S$ with affine transition morphisms. Let $Y \\to X$\nbe a morphism of schemes over $S$.\nIf $Y \\to X$ is proper, $X_i$ quasi-compact and quasi-separated, and\n$Y$ locally of finite type over $S$, then $Y \\to X_i$ is proper\nfor $i$ large enough.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Applications of Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EX1","source_file":"limits.tex","source_line":3377,"source_end_line":3385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3377-L3385","statement_sha256":"72db430bb30ad1aea2a9188ad8d3ff147ef0399419d28482b485a9e8d514eea1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6307,"rank":6307,"depth":31,"x":1952.707,"y":610.462,"cluster":"scheme-morphisms"},{"id":"stacks:081G","tag":"081G","title":"Applications of Chow's lemma · Lemma 081G","summary":"Assumptions and notation as in Situation [Tag 081D]. Let F_0 be a quasi-coherent O_X_0-module. Denote F and F_i the pullbacks of F_0 to X and X_i. Assume • f_0 is locally of finite type, • F_0 is of finite type, • the scheme theoretic support of F is proper over Y. Then the scheme theoretic support of F_i is proper over Y_i for some i.","statement_latex":"Assumptions and notation as in Situation \\ref{situation-descent-property}.\nLet $\\mathcal{F}_0$ be a quasi-coherent $\\mathcal{O}_{X_0}$-module.\nDenote $\\mathcal{F}$ and $\\mathcal{F}_i$ the pullbacks of\n$\\mathcal{F}_0$ to $X$ and $X_i$. Assume\n\\begin{enumerate}\n\\item $f_0$ is locally of finite type,\n\\item $\\mathcal{F}_0$ is of finite type,\n\\item the scheme theoretic support of $\\mathcal{F}$ is proper over $Y$.\n\\end{enumerate}\nThen the scheme theoretic support of $\\mathcal{F}_i$ is proper over $Y_i$\nfor some $i$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Applications of Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081G","source_file":"limits.tex","source_line":3404,"source_end_line":3417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3404-L3417","statement_sha256":"27edaacebf561f7ceb44250e69e5fdbbc500c3fb83b930eeea2791e809869c27","origin":"The Stacks Project","memory_eligible":false,"source_rank":6308,"rank":6308,"depth":31,"x":2143.011,"y":687.425,"cluster":"scheme-morphisms"},{"id":"stacks:05BD","tag":"05BD","title":"Universally closed morphisms · Lemma 05BD","summary":"Let f : X → S be a quasi-compact morphism of schemes. Let g : T → S be a morphism of schemes. Let t ∈ T be a point and Z ⊂ X_T be a closed subscheme such that Z ∩ X_t = ∅. Then there exists an open neighbourhood V ⊂ T of t, a commutative diagram xymatrix V ar[d] ar[r]_a & T' ar[d]^b T ar[r]^g & S, and a closed subscheme Z' ⊂ X_T' such that • the morphism b : T' → S is locally of finite presentation, • with t' = a(t) we have Z' ∩ X_t' = ∅, and • Z ∩ X_V maps into Z' via…","statement_latex":"Let $f : X \\to S$ be a quasi-compact morphism of schemes.\nLet $g : T \\to S$ be a morphism of schemes.\nLet $t \\in T$ be a point and $Z \\subset X_T$ be a closed\nsubscheme such that $Z \\cap X_t = \\emptyset$.\nThen there exists an open neighbourhood\n$V \\subset T$ of $t$, a commutative diagram\n$$\n\\xymatrix{\nV \\ar[d] \\ar[r]_a & T' \\ar[d]^b \\\\\nT \\ar[r]^g & S,\n}\n$$\nand a closed subscheme $Z' \\subset X_{T'}$ such that\n\\begin{enumerate}\n\\item the morphism $b : T' \\to S$ is locally of finite presentation,\n\\item with $t' = a(t)$ we have $Z' \\cap X_{t'} = \\emptyset$, and\n\\item $Z \\cap X_V$ maps into $Z'$ via the morphism $X_V \\to X_{T'}$.\n\\end{enumerate}\nMoreover, we may assume $V$ and $T'$ are affine.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BD","source_file":"limits.tex","source_line":3454,"source_end_line":3475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3454-L3475","statement_sha256":"c9268f27e7efc93f80c76ba0b0b7d72576a096248ae2dc8c62635353a7e3783e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6309,"rank":6309,"depth":0,"x":1940.621,"y":738.7,"cluster":"scheme-morphisms"},{"id":"stacks:05JX","tag":"05JX","title":"Universally closed morphisms · Lemma 05JX","summary":"Let f : X → S be a quasi-compact morphism of schemes. The following are equivalent • f is universally closed, • for every morphism S' → S which is locally of finite presentation the base change X_S' → S' is closed, and • for every n the morphism A^n × X → A^n × S is closed.","statement_latex":"Let $f : X \\to S$ be a quasi-compact morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed,\n\\item for every morphism $S' \\to S$ which is locally of finite presentation\nthe base change $X_{S'} \\to S'$ is closed, and\n\\item for every $n$ the morphism\n$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$\nis closed.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JX","source_file":"limits.tex","source_line":3545,"source_end_line":3557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3545-L3557","statement_sha256":"b613a27d5f4a471952f02dbea7105905e81765214a8544f6386ccfd4f081d2b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6310,"rank":6310,"depth":6,"x":2048.717,"y":585.921,"cluster":"scheme-morphisms"},{"id":"stacks:0205","tag":"0205","title":"Universally closed morphisms · Lemma 0205","summary":"Let S be a scheme. Let f : X → S be a separated morphism of finite type. The following are equivalent: • The morphism f is proper. • For any morphism S' → S which is locally of finite type the base change X_S' → S' is closed. • For every n ≥ 0 the morphism A^n × X → A^n × S is closed.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to S$ be a separated morphism of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is proper.\n\\item For any morphism $S' \\to S$ which is locally of finite type\nthe base change $X_{S'} \\to S'$ is closed.\n\\item For every $n \\geq 0$ the morphism\n$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$ is closed.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0205","source_file":"limits.tex","source_line":3616,"source_end_line":3628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3616-L3628","statement_sha256":"2a2e1eaea5335059368ffa07682e1d9aaa380b5d951d97c04429fb82ca9eabe4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6311,"rank":6311,"depth":7,"x":2091.906,"y":760.061,"cluster":"scheme-morphisms"},{"id":"stacks:0CM2","tag":"0CM2","title":"Noetherian valuative criterion · Lemma 0CM2","summary":"Let f : X → Y be a morphism of schemes. Assume f finite type and Y locally Noetherian. Let y ∈ Y be a point in the closure of the image of f. Then there exists a commutative diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d]^f Spec(A) ar[r] & Y where A is a discrete valuation ring and K is its field of fractions mapping the closed point of Spec(A) to y. Moreover, we can assume that the image point of Spec(K) → X is a generic point eta of an irreducible component of X and that…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume $f$ finite type and $Y$ locally Noetherian.\nLet $y \\in Y$ be a point in the closure of the image of $f$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[r] & Y\n}\n$$\nwhere $A$ is a discrete valuation ring and $K$ is its field of fractions\nmapping the closed point of $\\Spec(A)$ to $y$. Moreover, we can assume\nthat the image point of $\\Spec(K) \\to X$ is a generic point $\\eta$\nof an irreducible component of $X$ and that $K = \\kappa(\\eta)$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CM2","source_file":"limits.tex","source_line":3740,"source_end_line":3756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3740-L3756","statement_sha256":"031d7650808b12ecc751051a64e548b788175322d8669631bfb60980c6dbd0d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6312,"rank":6312,"depth":20,"x":1919.876,"y":656.071,"cluster":"scheme-morphisms"},{"id":"stacks:0207","tag":"0207","title":"Noetherian valuative criterion · Lemma 0207","summary":"Let S be a locally Noetherian scheme. Let f : X → S be a morphism of schemes. Assume f is locally of finite type. The following are equivalent: • The morphism f is separated. • For any diagram ([Tag 0206]) there is at most one dotted arrow. • For all diagrams ([Tag 0206]) with A a discrete valuation ring there is at most one dotted arrow. • For any irreducible component X_0 of X with generic point eta ∈ X_0, for any discrete valuation ring A ⊂ K = kappa(eta) with fraction…","statement_latex":"Let $S$ be a locally Noetherian scheme.\nLet $f : X \\to S$ be a morphism of schemes.\nAssume $f$ is locally of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is separated.\n\\item For any diagram (\\ref{equation-valuative}) there is at most\none dotted arrow.\n\\item For all diagrams (\\ref{equation-valuative}) with $A$ a discrete\nvaluation ring there is at most one dotted arrow.\n\\item For any irreducible component $X_0$ of $X$ with\ngeneric point $\\eta \\in X_0$, for any discrete valuation ring\n$A \\subset K = \\kappa(\\eta)$ with fraction field $K$ and any\ndiagram (\\ref{equation-valuative}) such that\nthe morphism $\\Spec(K) \\to X$ is the canonical one\n(see Schemes, Section \\ref{schemes-section-points})\nthere is at most one dotted arrow.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0207","source_file":"limits.tex","source_line":3798,"source_end_line":3818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3798-L3818","statement_sha256":"3e3f246d086a76d6bc9a0f0547e52c280b2a956645cc416c813d9d0f30036873","origin":"The Stacks Project","memory_eligible":false,"source_rank":6313,"rank":6313,"depth":17,"x":2130.531,"y":635.12,"cluster":"scheme-morphisms"},{"id":"stacks:0208","tag":"0208","title":"Noetherian valuative criterion · Lemma 0208","summary":"Let S be a locally Noetherian scheme. Let f : X → S be a morphism of finite type. The following are equivalent: • The morphism f is proper. • For any diagram ([Tag 0206]) there exists exactly one dotted arrow. • For all diagrams ([Tag 0206]) with A a discrete valuation ring there exists exactly one dotted arrow. • For any irreducible component X_0 of X with generic point eta ∈ X_0, for any discrete valuation ring A ⊂ K = kappa(eta) with fraction field K and any diagram…","statement_latex":"Let $S$ be a locally Noetherian scheme.\nLet $f : X \\to S$ be a morphism of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is proper.\n\\item For any diagram (\\ref{equation-valuative}) there exists exactly\none dotted arrow.\n\\item For all diagrams (\\ref{equation-valuative}) with $A$ a discrete\nvaluation ring there exists exactly one dotted arrow.\n\\item For any irreducible component $X_0$ of $X$ with\ngeneric point $\\eta \\in X_0$, for any discrete valuation ring\n$A \\subset K = \\kappa(\\eta)$ with fraction field $K$ and any\ndiagram (\\ref{equation-valuative}) such that\nthe morphism $\\Spec(K) \\to X$ is the canonical one\n(see\nSchemes, Section \\ref{schemes-section-points})\nthere exists exactly one dotted arrow.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0208","source_file":"limits.tex","source_line":3926,"source_end_line":3946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L3926-L3946","statement_sha256":"50160e9e4aa0e422733215c1f7b3df1d29eb7f4b6080b7b486b2971aed180406","origin":"The Stacks Project","memory_eligible":false,"source_rank":6314,"rank":6314,"depth":23,"x":1991.93,"y":770.213,"cluster":"scheme-morphisms"},{"id":"stacks:05JY","tag":"05JY","title":"Noetherian valuative criterion · Lemma 05JY","summary":"Let f : X → S be a finite type morphism of schemes. Assume S is locally Noetherian. Then the following are equivalent • f is universally closed, • for every n the morphism A^n × X → A^n × S is closed, • for any diagram ([Tag 0206]) there exists some dotted arrow, • for all diagrams ([Tag 0206]) with A a discrete valuation ring there exists some dotted arrow.","statement_latex":"Let $f : X \\to S$ be a finite type morphism of schemes.\nAssume $S$ is locally Noetherian. Then the following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed,\n\\item for every $n$ the morphism\n$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$ is closed,\n\\item for any diagram (\\ref{equation-valuative}) there exists some\ndotted arrow,\n\\item for all diagrams (\\ref{equation-valuative}) with $A$ a discrete\nvaluation ring there exists some dotted arrow.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JY","source_file":"limits.tex","source_line":4008,"source_end_line":4021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4008-L4021","statement_sha256":"396f4d7fc438ee94012bb9e1a7e71aa27320f0ac1a2413d2413484bbe24452bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6315,"rank":6315,"depth":18,"x":1985.489,"y":591.803,"cluster":"scheme-morphisms"},{"id":"stacks:0CM3","tag":"0CM3","title":"Refined Noetherian valuative criteria · Lemma 0CM3","summary":"Let f : X → S and h : U → X be morphisms of schemes. Assume that S is locally Noetherian, that f and h are of finite type, that f is separated, and that h(U) is dense in X. If given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & U ar[r]^h & X ar[d]^f Spec(A) ar[rr] ar@-->[rru] & & S where A is a discrete valuation ring with field of fractions K, there exists a dotted arrow making the diagram commute, then f is proper.","statement_latex":"Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes.\nAssume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type,\nthat $f$ is separated, and that $h(U)$ is dense in $X$.\nIf given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S\n}\n$$\nwhere $A$ is a discrete valuation ring with field of fractions $K$, there\nexists a dotted arrow making the diagram commute, then $f$ is proper.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Refined Noetherian valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CM3","source_file":"limits.tex","source_line":4084,"source_end_line":4098,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4084-L4098","statement_sha256":"55585f84de6676c4354f1c0477a1f422b062e883e7b3663ddcda2412a38d9950","origin":"The Stacks Project","memory_eligible":false,"source_rank":6316,"rank":6316,"depth":25,"x":2133.834,"y":719.812,"cluster":"scheme-morphisms"},{"id":"stacks:0CM4","tag":"0CM4","title":"Refined Noetherian valuative criteria · Lemma 0CM4","summary":"Let f : X → S and h : U → X be morphisms of schemes. Assume that S is locally Noetherian, that f is locally of finite type, that h is of finite type, and that h(U) is dense in X. If given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & U ar[r]^h & X ar[d]^f Spec(A) ar[rr] ar@-->[rru] & & S where A is a discrete valuation ring with field of fractions K, there exists at most one dotted arrow making the diagram commute, then f is separated.","statement_latex":"Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes.\nAssume that $S$ is locally Noetherian, that $f$ is locally of finite type,\nthat $h$ is of finite type, and that $h(U)$ is dense in $X$.\nIf given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S\n}\n$$\nwhere $A$ is a discrete valuation ring with field of fractions $K$, there\nexists at most one dotted arrow making the diagram commute, then $f$ is\nseparated.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Refined Noetherian valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CM4","source_file":"limits.tex","source_line":4162,"source_end_line":4177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4162-L4177","statement_sha256":"8854eb04e69cc6dcd854c0c33adda4e18996b3d3aef4970db38c4dc7392714cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6317,"rank":6317,"depth":26,"x":1921.328,"y":709.586,"cluster":"scheme-morphisms"},{"id":"stacks:0CM5","tag":"0CM5","title":"Refined Noetherian valuative criteria · Lemma 0CM5","summary":"Let f : X → S and h : U → X be morphisms of schemes. Assume that S is locally Noetherian, that f and h are of finite type, and that h(U) is dense in X. If given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & U ar[r]^h & X ar[d]^f Spec(A) ar[rr] ar@-->[rru] & & S where A is a discrete valuation ring with field of fractions K, there exists a unique dotted arrow making the diagram commute, then f is proper.","statement_latex":"Let $f : X \\to S$ and $h : U \\to X$ be morphisms of schemes.\nAssume that $S$ is locally Noetherian, that $f$ and $h$ are of finite type, and\nthat $h(U)$ is dense in $X$. If given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & S\n}\n$$\nwhere $A$ is a discrete valuation ring with field of fractions $K$, there\nexists a unique dotted arrow making the diagram commute, then $f$ is proper.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Refined Noetherian valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CM5","source_file":"limits.tex","source_line":4200,"source_end_line":4213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4200-L4213","statement_sha256":"34877be1cec8755132fb143e27f467b40022af21bf3c3b3303e49d40118d3547","origin":"The Stacks Project","memory_eligible":false,"source_rank":6318,"rank":6318,"depth":27,"x":2086.388,"y":596.451,"cluster":"scheme-morphisms"},{"id":"stacks:0GWV","tag":"0GWV","title":"Valuative criteria over a Nagata base · Lemma 0GWV","summary":"Let S be a Nagata scheme (and in particular locally Noetherian). Let f : X → Y be a quasi-compact morphism of schemes locally of finite type over S. The following are equivalent • f is universally closed, • for every n the morphism A^n × X → A^n × Y is closed, • for any commutative diagram xymatrix U ar[r] ar[d] & X ar[d]^f C ar[r] ar@..>[ru] & Y of schemes over S such that • C is a normal integral scheme of finite type over S, • U = C setminus (c) for some closed point c…","statement_latex":"Let $S$ be a Nagata scheme (and in particular locally Noetherian).\nLet $f : X \\to Y$ be a quasi-compact morphism of schemes locally\nof finite type over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed,\n\\item for every $n$ the morphism\n$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times Y$ is closed,\n\\item for any commutative diagram\n$$\n\\xymatrix{\nU \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\nC \\ar[r] \\ar@{..>}[ru] & Y\n}\n$$\nof schemes over $S$ such that\n\\begin{enumerate}\n\\item $C$ is a normal integral scheme of finite type over $S$,\n\\item $U = C \\setminus \\{c\\}$ for some closed point $c \\in C$,\n\\item $A = \\mathcal{O}_{C, c}$ has dimension $1$\\footnote{It follows\nthat $A$ is a discrete valuation ring, see\nAlgebra, Lemma \\ref{algebra-lemma-characterize-dvr}.\nMoreover, $c$ maps to a finite type point $s \\in S$ and\n$A$ is essentially of finite type over $\\mathcal{O}_{S, s}$.}\n\\end{enumerate}\nthen in the commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $K = \\text{Frac}(A)$ some dotted arrow exists\\footnote{By\nLemma \\ref{lemma-morphism-glueing-near-closed-point} this is\nequivalent to asking for the existence of dotted arrow\nmaking the first commutative diagram commute.}\nmaking the diagram commute.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Valuative criteria over a Nagata base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWV","source_file":"limits.tex","source_line":4237,"source_end_line":4276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4237-L4276","statement_sha256":"88159ca16af9c79744ab5d50640d729c91a419545c738641ba8d3beb3ff70d84","origin":"The Stacks Project","memory_eligible":false,"source_rank":6319,"rank":6319,"depth":34,"x":2055.63,"y":773.681,"cluster":"scheme-morphisms"},{"id":"stacks:0GWW","tag":"0GWW","title":"Valuative criteria over a Nagata base · Lemma 0GWW","summary":"Let S be a Nagata scheme (and in particular locally Noetherian). Let f : X → Y be a morphism of schemes locally of finite type over S. The following are equivalent • f separated, • for any commutative diagram xymatrix U ar[r] ar[d] & X ar[d]^f C ar[r] ar@..>[ru] & Y of schemes over S such that • C is a normal integral scheme of finite type over S, • U = C setminus (c) for some closed point c ∈ C, • A = O_C, c has dimension 1_S, s. then in the commutative diagram xymatrix…","statement_latex":"Let $S$ be a Nagata scheme (and in particular locally Noetherian).\nLet $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ separated,\n\\item for any commutative diagram\n$$\n\\xymatrix{\nU \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\nC \\ar[r] \\ar@{..>}[ru] & Y\n}\n$$\nof schemes over $S$ such that\n\\begin{enumerate}\n\\item $C$ is a normal integral scheme of finite type over $S$,\n\\item $U = C \\setminus \\{c\\}$ for some closed point $c \\in C$,\n\\item $A = \\mathcal{O}_{C, c}$ has dimension $1$\\footnote{It follows\nthat $A$ is a discrete valuation ring, see\nAlgebra, Lemma \\ref{algebra-lemma-characterize-dvr}.\nMoreover, $c$ maps to a finite type point $s \\in S$ and\n$A$ is essentially of finite type over $\\mathcal{O}_{S, s}$.}\n\\end{enumerate}\nthen in the commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $K = \\text{Frac}(A)$ there exists at most one dotted arrow\\footnote{By\nLemma \\ref{lemma-morphism-glueing-near-closed-point} this is\nequivalent to asking there to be at most one dotted arrow\nmaking the first commutative diagram commute.}\nmaking the diagram commute.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Valuative criteria over a Nagata base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWW","source_file":"limits.tex","source_line":4366,"source_end_line":4403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4366-L4403","statement_sha256":"ea9bd3647bf1413502ce8d0e09f80c0904603e57fc206ba27370eb914fbf88f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6320,"rank":6320,"depth":35,"x":1935.687,"y":625.419,"cluster":"scheme-morphisms"},{"id":"stacks:0GWX","tag":"0GWX","title":"Valuative criteria over a Nagata base · Lemma 0GWX","summary":"Let S be a Nagata scheme (and in particular locally Noetherian). Let f : X → Y be a quasi-compact morphism of schemes locally of finite type over S. The following are equivalent • f proper, • for any commutative diagram xymatrix U ar[r] ar[d] & X ar[d]^f C ar[r] ar@..>[ru] & Y of schemes over S such that • C is a normal integral scheme of finite type over S, • U = C setminus (c) for some closed point c ∈ C, • A = O_C, c has dimension 1_S, s. then in the commutative…","statement_latex":"Let $S$ be a Nagata scheme (and in particular locally Noetherian).\nLet $f : X \\to Y$ be a quasi-compact morphism of schemes locally\nof finite type over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ proper,\n\\item for any commutative diagram\n$$\n\\xymatrix{\nU \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\nC \\ar[r] \\ar@{..>}[ru] & Y\n}\n$$\nof schemes over $S$ such that\n\\begin{enumerate}\n\\item $C$ is a normal integral scheme of finite type over $S$,\n\\item $U = C \\setminus \\{c\\}$ for some closed point $c \\in C$,\n\\item $A = \\mathcal{O}_{C, c}$ has dimension $1$\\footnote{It follows\nthat $A$ is a discrete valuation ring, see\nAlgebra, Lemma \\ref{algebra-lemma-characterize-dvr}.\nMoreover, $c$ maps to a finite type point $s \\in S$ and\n$A$ is essentially of finite type over $\\mathcal{O}_{S, s}$.}\n\\end{enumerate}\nthen in the commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $K = \\text{Frac}(A)$ there exists exactly one dotted arrow\\footnote{By\nLemma \\ref{lemma-morphism-glueing-near-closed-point} this is\nequivalent to asking for the existence and uniqueness of the dotted arrow\nmaking the first commutative diagram commute.}\nmaking the diagram commute.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Valuative criteria over a Nagata base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWX","source_file":"limits.tex","source_line":4443,"source_end_line":4480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4443-L4480","statement_sha256":"f43bd5b0ddd3b9789c35408b342b3c5f1e50bfe8e9774c98211a7173eea9c426","origin":"The Stacks Project","memory_eligible":false,"source_rank":6321,"rank":6321,"depth":36,"x":2143.532,"y":666.721,"cluster":"scheme-morphisms"},{"id":"stacks:05M5","tag":"05M5","title":"Limits and dimensions of fibres · Lemma 05M5","summary":"Let I be a directed set. Let (f_i : X_i → S_i) be an inverse system of morphisms of schemes over I. Assume • all the morphisms S_i' → S_i are affine, • all the schemes S_i are quasi-compact and quasi-separated, • the morphisms f_i are of finite type, and • the morphisms X_i' → X_i ×_S_i S_i' are closed immersions. Let f : X = lim_i X_i → S = lim_i S_i be the limit. Let d ≥ 0. If every fibre of f has dimension ≤ d, then for some i every fibre of f_i has dimension ≤ d.","statement_latex":"Let $I$ be a directed set.\nLet $(f_i : X_i \\to S_i)$ be an inverse system of morphisms of schemes\nover $I$. Assume\n\\begin{enumerate}\n\\item all the morphisms $S_{i'} \\to S_i$ are affine,\n\\item all the schemes $S_i$ are quasi-compact and quasi-separated,\n\\item the morphisms $f_i$ are of finite type, and\n\\item the morphisms $X_{i'} \\to X_i \\times_{S_i} S_{i'}$ are closed\nimmersions.\n\\end{enumerate}\nLet $f : X = \\lim_i X_i \\to S = \\lim_i S_i$ be the limit.\nLet $d \\geq 0$.\nIf every fibre of $f$ has dimension $\\leq d$, then for some $i$\nevery fibre of $f_i$ has dimension $\\leq d$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Limits and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05M5","source_file":"limits.tex","source_line":4504,"source_end_line":4520,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4504-L4520","statement_sha256":"e932b10b46008d2928cbdeb00b16b8eb55f4ea678d20d637e918938b6dd85c81","origin":"The Stacks Project","memory_eligible":false,"source_rank":6322,"rank":6322,"depth":32,"x":1956.902,"y":754.274,"cluster":"scheme-morphisms"},{"id":"stacks:094M","tag":"094M","title":"Limits and dimensions of fibres · Lemma 094M","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f is a quasi-finite morphism, and • f_0 is locally of finite type, then there exists an i ≥ 0 such that f_i is quasi-finite.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is a quasi-finite morphism, and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen there exists an $i \\geq 0$ such that $f_i$ is quasi-finite.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Limits and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094M","source_file":"limits.tex","source_line":4555,"source_end_line":4564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4555-L4564","statement_sha256":"e6b5d67dd3eafeeeeb44391e9781a341412d64947e432432c97ef9de4859d24b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6323,"rank":6323,"depth":33,"x":2024.168,"y":583.675,"cluster":"scheme-morphisms"},{"id":"stacks:0H3V","tag":"0H3V","title":"Limits and dimensions of fibres · Lemma 0H3V","summary":"Assumptions and notation as in Situation [Tag 081D]. Let d ≥ 0. If • f has relative dimension ≤ d (Morphisms, Definition [Tag 02NJ]), and • f_0 is locally of finite type, then there exists an i such that f_i has relative dimension ≤ d.","statement_latex":"Assumptions and notation as in Situation \\ref{situation-descent-property}.\nLet $d \\geq 0$. If\n\\begin{enumerate}\n\\item $f$ has relative dimension $\\leq d$\n(Morphisms, Definition \\ref{morphisms-definition-relative-dimension-d}), and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen there exists an $i$ such that $f_i$ has relative dimension $\\leq d$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Limits and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3V","source_file":"limits.tex","source_line":4570,"source_end_line":4580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4570-L4580","statement_sha256":"9d34c67a7aa3eb68ad36ef85a5eed930bdf26cd3eb4ca46652f84e2bb835078c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6324,"rank":6324,"depth":33,"x":2111.83,"y":747.775,"cluster":"scheme-morphisms"},{"id":"stacks:0EY2","tag":"0EY2","title":"Limits and dimensions of fibres · Lemma 0EY2","summary":"Notation and assumptions as in Situation [Tag 081D]. If • f has relative dimension d, and • f_0 is locally of finite presentation, then there exists an i ≥ 0 such that f_i has relative dimension d.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ has relative dimension $d$, and\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen there exists an $i \\geq 0$ such that $f_i$\nhas relative dimension $d$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Limits and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EY2","source_file":"limits.tex","source_line":4586,"source_end_line":4596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4586-L4596","statement_sha256":"c4f9196f1b6af7e15fa8b273f25ae06a472d346fafa8a34293bfa3f49e4a5e6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6325,"rank":6325,"depth":33,"x":1915.062,"y":676.453,"cluster":"scheme-morphisms"},{"id":"stacks:05M6","tag":"05M6","title":"Limits and dimensions of fibres · Lemma 05M6","summary":"Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a morphism of finite presentation. Let d ≥ 0 be an integer. If Z ⊂ X be a closed subscheme such that dim(Z_s) ≤ d for all s ∈ S, then there exists a closed subscheme Z' ⊂ X such that • Z ⊂ Z', • Z' → X is of finite presentation, and • dim(Z'_s) ≤ d for all s ∈ S.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to S$ be a morphism of finite presentation.\nLet $d \\geq 0$ be an integer.\nIf $Z \\subset X$ be a closed subscheme such that\n$\\dim(Z_s) \\leq d$ for all $s \\in S$, then there exists a\nclosed subscheme $Z' \\subset X$ such that\n\\begin{enumerate}\n\\item $Z \\subset Z'$,\n\\item $Z' \\to X$ is of finite presentation, and\n\\item $\\dim(Z'_s) \\leq d$ for all $s \\in S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Limits and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05M6","source_file":"limits.tex","source_line":4615,"source_end_line":4628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4615-L4628","statement_sha256":"46acd988175ac4af4cf104a650b50e57e21f32e7468135142f60e4da4dc9adb2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6326,"rank":6326,"depth":33,"x":2117.678,"y":617.346,"cluster":"scheme-morphisms"},{"id":"stacks:0EX3","tag":"0EX3","title":"Base change in top degree · Lemma 0EX3","summary":"Let f : X → Y be a morphism of schemes. Let d ≥ 0. Assume • X and Y are quasi-compact and quasi-separated, and • R^if_*F = 0 for i > d and every quasi-coherent O_X-module F. Then we have • [(a)] for any base change diagram xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f Y' ar[r]^g & Y we have R^if'_*F' = 0 for i > d and any quasi-coherent O_X'-module F', • [(b)] R^df'_*(F' ⊗_O_X' (f')^*G') = R^df'_*F' ⊗_O_Y' G' for any quasi-coherent O_Y'-module G', • [(c)] formation of…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $d \\geq 0$. Assume\n\\begin{enumerate}\n\\item $X$ and $Y$ are quasi-compact and quasi-separated, and\n\\item $R^if_*\\mathcal{F} = 0$ for $i > d$ and\nevery quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$.\n\\end{enumerate}\nThen we have\n\\begin{enumerate}\n\\item[(a)] for any base change diagram\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nwe have $R^if'_*\\mathcal{F}' = 0$ for $i > d$ and any quasi-coherent\n$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$,\n\\item[(b)]\n$R^df'_*(\\mathcal{F}' \\otimes_{\\mathcal{O}_{X'}} (f')^*\\mathcal{G}') =\nR^df'_*\\mathcal{F}' \\otimes_{\\mathcal{O}_{Y'}} \\mathcal{G}'$\nfor any quasi-coherent $\\mathcal{O}_{Y'}$-module $\\mathcal{G}'$,\n\\item[(c)] formation of $R^df'_*\\mathcal{F}'$ commutes with arbitrary\nfurther base change (see proof for explanation).\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Base change in top degree","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EX3","source_file":"limits.tex","source_line":4667,"source_end_line":4693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4667-L4693","statement_sha256":"777dbe5b1455bdc620893e90fc7b991ed0e4b31a60a43e7c9af7ba7ffddd46e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6327,"rank":6327,"depth":29,"x":2015.721,"y":776.029,"cluster":"scheme-morphisms"},{"id":"stacks:0E7D","tag":"0E7D","title":"Base change in top degree · Lemma 0E7D","summary":"Let f : X → Y be a morphism of schemes. Let y ∈ Y. Assume f is proper and dim(X_y) = d. Then • for F ∈ QCoh(O_X) we have (R^if_*F)_y = 0 for all i > d, • there is an affine open neighbourhood V ⊂ Y of y such that f^-1(V) → V and d satisfy the assumptions and conclusions of Lemma [Tag 0EX3].","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $y \\in Y$.\nAssume $f$ is proper and $\\dim(X_y) = d$.\nThen\n\\begin{enumerate}\n\\item for $\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$\nwe have $(R^if_*\\mathcal{F})_y = 0$ for all $i > d$,\n\\item there is an affine open neighbourhood $V \\subset Y$\nof $y$ such that $f^{-1}(V) \\to V$ and $d$ satisfy the\nassumptions and conclusions of Lemma \\ref{lemma-top-cohomology-functor}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Base change in top degree","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7D","source_file":"limits.tex","source_line":4807,"source_end_line":4819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4807-L4819","statement_sha256":"4803b971c73636b89cc0fe954953d5ab6106621e004e5506d0b74264162ca063","origin":"The Stacks Project","memory_eligible":false,"source_rank":6328,"rank":6328,"depth":39,"x":1963.251,"y":601.023,"cluster":"scheme-morphisms"},{"id":"stacks:0EX4","tag":"0EX4","title":"Base change in top degree · Lemma 0EX4","summary":"Let f : X → Y be a morphism of schemes. Let d ≥ 0. Let F be an O_X-module. Assume • f is a proper morphism all of whose fibres have dimension ≤ d, • F is a quasi-coherent O_X-module of finite type. Then R^df_*F is a quasi-coherent O_X-module of finite type.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $d \\geq 0$. Let $\\mathcal{F}$\nbe an $\\mathcal{O}_X$-module. Assume\n\\begin{enumerate}\n\\item $f$ is a proper morphism all of whose fibres have dimension $\\leq d$,\n\\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module of finite type.\n\\end{enumerate}\nThen $R^df_*\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module\nof finite type.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Base change in top degree","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EX4","source_file":"limits.tex","source_line":4888,"source_end_line":4898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4888-L4898","statement_sha256":"5a74c797856e503bed69c9f74b741d74865e4371474975bf3e507ebdbb0450a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6329,"rank":6329,"depth":40,"x":2142.823,"y":700.378,"cluster":"scheme-morphisms"},{"id":"stacks:0EX5","tag":"0EX5","title":"Base change in top degree · Lemma 0EX5","summary":"Let f : X → Y be a morphism of schemes. Let d ≥ 0. Let F be an O_X-module. Assume • f is a proper morphism of finite presentation all of whose fibres have dimension ≤ d, • F is an O_X-module of finite presentation. Then R^df_*F is an O_X-module of finite presentation.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $d \\geq 0$.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume\n\\begin{enumerate}\n\\item $f$ is a proper morphism of finite presentation\nall of whose fibres have dimension $\\leq d$,\n\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation.\n\\end{enumerate}\nThen $R^df_*\\mathcal{F}$ is an $\\mathcal{O}_X$-module\nof finite presentation.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Base change in top degree","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EX5","source_file":"limits.tex","source_line":4964,"source_end_line":4975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L4964-L4975","statement_sha256":"b32c27d4b398c92c75f46401c054852989fe31e1aa5501d5b53c919c4ddceb0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6330,"rank":6330,"depth":41,"x":1930.337,"y":729.032,"cluster":"scheme-morphisms"},{"id":"stacks:0BPA","tag":"0BPA","title":"Glueing in closed fibres · Lemma 0BPA","summary":"Let S be a scheme. Let s ∈ S be a closed point such that U = S setminus (s) → S is quasi-compact. With V = Spec(O_S, s) setminus (s) there is an equivalence of categories ( X → S of finite presentation ) → ( vcenter xymatrix X' ar[d] & Y' ar[d] ar[l] ar[r] & Y ar[d] U & V ar[l] ar[r] & Spec(O_S, s) ) where on the right hand side we consider commutative diagrams whose squares are cartesian and whose vertical arrows are of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $s \\in S$ be a closed point such that\n$U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With\n$V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ there is\nan equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\nX \\to S\\text{ of finite presentation}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\vcenter{\n\\xymatrix{\nX' \\ar[d] & Y' \\ar[d] \\ar[l] \\ar[r] & Y \\ar[d] \\\\\nU & V \\ar[l] \\ar[r] & \\Spec(\\mathcal{O}_{S, s})\n}\n}\n\\right\\}\n$$\nwhere on the right hand side we consider commutative diagrams\nwhose squares are cartesian and whose vertical arrows are\nof finite presentation.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Glueing in closed fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPA","source_file":"limits.tex","source_line":5001,"source_end_line":5026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5001-L5026","statement_sha256":"5ab714ef13032c3b338a37f02411f2e2abf76de82c740e19ce66399126c6b00a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6331,"rank":6331,"depth":26,"x":2064.087,"y":587.218,"cluster":"scheme-morphisms"},{"id":"stacks:0F21","tag":"0F21","title":"Glueing in closed fibres · Lemma 0F21","summary":"Let S be a scheme. Let s ∈ S be a closed point such that U = S setminus (s) → S is quasi-compact. With V = Spec(O_S, s) setminus (s) there is an equivalence of categories ( O_S-modules F of finite presentation ) → ( (G, H, α) ) where on the right hand side we consider triples consisting of a O_U-module G of finite presentation, a O_Spec(O_S, s)-module H of finite presentation, and an isomorphism α : G|_V → H|_V of O_V-modules.","statement_latex":"Let $S$ be a scheme. Let $s \\in S$ be a closed point such that\n$U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With\n$V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ there is\nan equivalence of categories\n$$\n\\left\\{\n\\mathcal{O}_S\\text{-modules }\\mathcal{F}\\text{ of finite presentation}\n\\right\\}\n\\longrightarrow\n\\left\\{\n(\\mathcal{G}, \\mathcal{H}, \\alpha)\n\\right\\}\n$$\nwhere on the right hand side we consider triples\nconsisting of a $\\mathcal{O}_U$-module $\\mathcal{G}$ of\nfinite presentation, a $\\mathcal{O}_{\\Spec(\\mathcal{O}_{S, s})}$-module\n$\\mathcal{H}$ of finite presentation, and an isomorphism\n$\\alpha : \\mathcal{G}|_V \\to \\mathcal{H}|_V$ of\n$\\mathcal{O}_V$-modules.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Glueing in closed fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F21","source_file":"limits.tex","source_line":5073,"source_end_line":5094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5073-L5094","statement_sha256":"f9c6e1f8ac0b5ae536bcea705a94f907dd9faae518d5e10e72352fb871411563","origin":"The Stacks Project","memory_eligible":false,"source_rank":6332,"rank":6332,"depth":27,"x":2079.518,"y":767.829,"cluster":"scheme-morphisms"},{"id":"stacks:0BQ5","tag":"0BQ5","title":"Glueing in closed fibres · Lemma 0BQ5","summary":"Let S be a scheme. Let U ⊂ S be a retrocompact open. Let s ∈ S be a point in the complement of U. With V = Spec(O_S, s) ∩ U there is an equivalence of categories colim_s ∈ U' ⊃ U open ( vcenter xymatrix X ar[d] U' ) → ( vcenter xymatrix X' ar[d] & Y' ar[d] ar[l] ar[r] & Y ar[d] U & V ar[l] ar[r] & Spec(O_S, s) ) where on the left hand side the vertical arrow is of finite presentation and on the right hand side we consider commutative diagrams whose squares are cartesian…","statement_latex":"Let $S$ be a scheme. Let $U \\subset S$ be a retrocompact open.\nLet $s \\in S$ be a point in the complement of $U$. With\n$V = \\Spec(\\mathcal{O}_{S, s}) \\cap U$ there is\nan equivalence of categories\n$$\n\\colim_{s \\in U' \\supset U\\text{ open}}\n\\left\\{\n\\vcenter{\n\\xymatrix{\nX \\ar[d] \\\\\nU'\n}\n}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\vcenter{\n\\xymatrix{\nX' \\ar[d] & Y' \\ar[d] \\ar[l] \\ar[r] & Y \\ar[d] \\\\\nU & V \\ar[l] \\ar[r] & \\Spec(\\mathcal{O}_{S, s})\n}\n}\n\\right\\}\n$$\nwhere on the left hand side the vertical arrow is of finite\npresentation and on the right hand side we consider commutative diagrams\nwhose squares are cartesian and whose vertical arrows are\nof finite presentation.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Glueing in closed fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQ5","source_file":"limits.tex","source_line":5107,"source_end_line":5137,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5107-L5137","statement_sha256":"a50a3cc8c7640a99368da8dba0612b3298af3d7ab94275411cd54109a20c7bab","origin":"The Stacks Project","memory_eligible":false,"source_rank":6333,"rank":6333,"depth":26,"x":1922.77,"y":643.304,"cluster":"scheme-morphisms"},{"id":"stacks:0EY3","tag":"0EY3","title":"Glueing in closed fibres · Lemma 0EY3","summary":"Notation and assumptions as in Lemma [Tag 0BQ5]. Let U ⊂ U' ⊂ X be an open containing s. • Let f' : X → U' correspond to f : X' → U and g : Y → Spec(O_S, s) via the equivalence. If f and g are separated, proper, finite, étale, then after possibly shrinking U' the morphism f' has the same property. • Let a : X_1 → X_2 be a morphism of schemes of finite presentation over U' with base change a' : X'_1 → X'_2 over U and b : Y_1 → Y_2 over Spec(O_S, s). If a' and b are…","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-glueing-near-point}.\nLet $U \\subset U' \\subset X$ be an open containing $s$.\n\\begin{enumerate}\n\\item  Let $f' : X \\to U'$ correspond to $f : X' \\to U$ and\n$g : Y \\to \\Spec(\\mathcal{O}_{S, s})$\nvia the equivalence. If $f$ and $g$ are separated, proper, finite, \\'etale,\nthen after possibly shrinking $U'$ the morphism $f'$ has the same property.\n\\item Let $a : X_1 \\to X_2$\nbe a morphism of schemes of finite presentation over $U'$\nwith base change $a' : X'_1 \\to X'_2$ over $U$ and\n$b : Y_1 \\to Y_2$ over $\\Spec(\\mathcal{O}_{S, s})$.\nIf $a'$ and $b$ are separated, proper, finite, \\'etale,\nthen after possibly shrinking $U'$ the morphism $a$ has the same property.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Glueing in closed fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EY3","source_file":"limits.tex","source_line":5184,"source_end_line":5200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5184-L5200","statement_sha256":"29406c4fd3935a269a56b32cf4a1250a051c087b19ef0995e6afc26ff8858a50","origin":"The Stacks Project","memory_eligible":false,"source_rank":6334,"rank":6334,"depth":40,"x":2138.668,"y":646.187,"cluster":"scheme-morphisms"},{"id":"stacks:0E8Q","tag":"0E8Q","title":"Glueing in closed fibres · Lemma 0E8Q","summary":"Let S be a scheme. Let s_1, …, s_n ∈ S be pairwise distinct closed points such that U = S setminus (s_1, …, s_n) → S is quasi-compact. With S_i = Spec(O_S, s_i) and U_i = S_i setminus (s_i) there is an equivalence of categories FP_S → FP_U ×_(FP_U_1 × … × FP_U_n) (FP_S_1 × … × FP_S_n) where FP_T is the category of schemes of finite presentation over the scheme T.","statement_latex":"Let $S$ be a scheme. Let $s_1, \\ldots, s_n \\in S$ be pairwise distinct\nclosed points such that\n$U = S \\setminus \\{s_1, \\ldots, s_n\\} \\to S$ is quasi-compact. With\n$S_i = \\Spec(\\mathcal{O}_{S, s_i})$ and $U_i = S_i \\setminus \\{s_i\\}$\nthere is an equivalence of categories\n$$\nFP_S \\longrightarrow\nFP_U \\times_{(FP_{U_1} \\times \\ldots \\times FP_{U_n})}\n(FP_{S_1} \\times \\ldots \\times FP_{S_n})\n$$\nwhere $FP_T$ is the category of schemes of finite presentation over\nthe scheme $T$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Glueing in closed fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8Q","source_file":"limits.tex","source_line":5221,"source_end_line":5235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5221-L5235","statement_sha256":"474fa79a4863bf10815c949eb5e1b044c87bd46dd2d48c5d6487e1f94b179861","origin":"The Stacks Project","memory_eligible":false,"source_rank":6335,"rank":6335,"depth":27,"x":1977.019,"y":766.664,"cluster":"scheme-morphisms"},{"id":"stacks:0B3X","tag":"0B3X","title":"Application to modifications · Lemma 0B3X","summary":"Let S be a scheme. Let s ∈ S be a closed point such that U = S setminus (s) → S is quasi-compact. With V = Spec(O_S, s) setminus (s) the base change functor ( f : X → S of finite presentation f^-1(U) → U is an isomorphism ) → ( g : Y → Spec(O_S, s) of finite presentation g^-1(V) → V is an isomorphism ) is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Let $s \\in S$ be a closed point such that\n$U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With\n$V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ the base change functor\n$$\n\\left\\{\n\\begin{matrix}\nf : X \\to S\\text{ of finite presentation} \\\\\nf^{-1}(U) \\to U\\text{ is an isomorphism}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\ng : Y \\to \\Spec(\\mathcal{O}_{S, s})\\text{ of finite presentation} \\\\\ng^{-1}(V) \\to V\\text{ is an isomorphism}\n\\end{matrix}\n\\right\\}\n$$\nis an equivalence of categories.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Application to modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3X","source_file":"limits.tex","source_line":5270,"source_end_line":5291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5270-L5291","statement_sha256":"aa1fbba335a9c8596caf81324fd180f5865a3195f482d605d4d98b33e36e9f07","origin":"The Stacks Project","memory_eligible":false,"source_rank":6336,"rank":6336,"depth":27,"x":1999.35,"y":585.957,"cluster":"scheme-morphisms"},{"id":"stacks:0BFN","tag":"0BFN","title":"Application to modifications · Lemma 0BFN","summary":"Notation and assumptions as in Lemma [Tag 0B3X]. Let f : X → S correspond to g : Y → Spec(O_S, s) via the equivalence. Then f is separated, proper, finite, étale and add more here if and only if g is so.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-modifications}.\nLet $f : X \\to S$ correspond to $g : Y \\to \\Spec(\\mathcal{O}_{S, s})$\nvia the equivalence. Then $f$ is separated, proper, finite, \\'etale\nand add more here if and only if $g$ is so.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Application to modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFN","source_file":"limits.tex","source_line":5297,"source_end_line":5303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5297-L5303","statement_sha256":"24bf6c74c642b33b6d318182ea64bb5e5175cba215cd4f18e182e6ac110cc2d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6337,"rank":6337,"depth":41,"x":2128.307,"y":731.997,"cluster":"scheme-morphisms"},{"id":"stacks:0CNN","tag":"0CNN","title":"Descending finite type schemes · Lemma 0CNN","summary":"In Situation [Tag 0CNM]. Let X → S be quasi-separated and of finite type. Then there exists an i ∈ I and a diagram vcenter xymatrix X ar[r] ar[d] & W ar[d] S ar[r] & S_i such that W → S_i is of finite type and such that the induced morphism X → S ×_S_i W is a closed immersion.","statement_latex":"In Situation \\ref{situation-limit-noetherian}.\nLet $X \\to S$ be quasi-separated and of finite type.\nThen there exists an $i \\in I$ and a diagram\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\nX \\ar[r] \\ar[d] & W \\ar[d] \\\\\nS \\ar[r] & S_i\n}\n}\n\\end{equation}\nsuch that $W \\to S_i$ is of finite type and such that\nthe induced morphism $X \\to S \\times_{S_i} W$ is a closed\nimmersion.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending finite type schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNN","source_file":"limits.tex","source_line":5371,"source_end_line":5388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5371-L5388","statement_sha256":"ce54fef06a676c04be5b0592d424e550ef3d3879890a0e0ceeda026425adf830","origin":"The Stacks Project","memory_eligible":false,"source_rank":6338,"rank":6338,"depth":27,"x":1915.604,"y":697.454,"cluster":"scheme-morphisms"},{"id":"stacks:0CNQ","tag":"0CNQ","title":"Descending finite type schemes · Lemma 0CNQ","summary":"In Situation [Tag 0CNM]. Let X → S be quasi-separated and of finite type. Given i ∈ I and a diagram vcenter xymatrix X ar[r] ar[d] & W ar[d] S ar[r] & S_i as in ([Tag 0CNP]) for i' ≥ i let X_i' be the scheme theoretic image of X → S_i' ×_S_i W. Then X = lim_i' ≥ i X_i'.","statement_latex":"In Situation \\ref{situation-limit-noetherian}.\nLet $X \\to S$ be quasi-separated and of finite type.\nGiven $i \\in I$ and a diagram\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[r] \\ar[d] & W \\ar[d] \\\\\nS \\ar[r] & S_i\n}\n}\n$$\nas in (\\ref{equation-good-diagram}) for $i' \\geq i$ let\n$X_{i'}$ be the scheme theoretic image of $X \\to S_{i'} \\times_{S_i} W$.\nThen $X = \\lim_{i' \\geq i} X_{i'}$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending finite type schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNQ","source_file":"limits.tex","source_line":5399,"source_end_line":5415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5399-L5415","statement_sha256":"152ad19f2cbbf03b72566a146125c701bb3213393439f647f98e82e589892b2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6339,"rank":6339,"depth":28,"x":2100.374,"y":602.156,"cluster":"scheme-morphisms"},{"id":"stacks:0CNR","tag":"0CNR","title":"Descending finite type schemes · Lemma 0CNR","summary":"In Situation [Tag 0CNM]. Let f : X → Y be a morphism of schemes quasi-separated and of finite type over S. Let vcenter xymatrix X ar[r] ar[d] & W ar[d] S ar[r] & S_i_1 and vcenter xymatrix Y ar[r] ar[d] & V ar[d] S ar[r] & S_i_2 be diagrams as in ([Tag 0CNP]). Let X = lim_i ≥ i_1 X_i and Y = lim_i ≥ i_2 Y_i be the corresponding limit descriptions as in Lemma [Tag 0CNQ]. Then there exists an i_0 ≥ max(i_1, i_2) and a morphism (f_i)_i ≥ i_0 : (X_i)_i ≥ i_0 → (Y_i)_i ≥ i_0…","statement_latex":"In Situation \\ref{situation-limit-noetherian}.\nLet $f : X \\to Y$ be a morphism of schemes quasi-separated\nand of finite type over $S$. Let\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[r] \\ar[d] & W \\ar[d] \\\\\nS \\ar[r] & S_{i_1}\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nY \\ar[r] \\ar[d] & V \\ar[d] \\\\\nS \\ar[r] & S_{i_2}\n}\n}\n$$\nbe diagrams as in (\\ref{equation-good-diagram}). Let\n$X = \\lim_{i \\geq i_1} X_i$ and\n$Y = \\lim_{i \\geq i_2} Y_i$ be the corresponding\nlimit descriptions as in Lemma \\ref{lemma-limit-from-good-diagram}.\nThen there exists an $i_0 \\geq \\max(i_1, i_2)$ and a morphism\n$$\n(f_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}\n$$\nof inverse systems over $(S_i)_{i \\geq i_0}$ such that\nsuch that $f = \\lim_{i \\geq i_0} f_i$.\nIf $(g_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}$\nis a second morphism of inverse systems over $(S_i)_{i \\geq i_0}$ such that\nsuch that $f = \\lim_{i \\geq i_0} g_i$\nthen $f_i = g_i$ for all $i \\gg i_0$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending finite type schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNR","source_file":"limits.tex","source_line":5446,"source_end_line":5480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5446-L5480","statement_sha256":"9560f0b136ef84c75e48832f4800184eeab8b23bbf1797c00e9c573d91d07af7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6340,"rank":6340,"depth":29,"x":2040.716,"y":777.411,"cluster":"scheme-morphisms"},{"id":"stacks:0CNT","tag":"0CNT","title":"Descending finite type schemes · Lemma 0CNT","summary":"Notation and assumptions as in Lemma [Tag 0CNR]. If f is flat and of finite presentation, then there exists an i_3 ≥ i_0 such that for i ≥ i_3 we have f_i is flat, X_i = Y_i ×_Y_i_3 X_i_3, and X = Y ×_Y_i_3 X_i_3.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.\nIf $f$ is flat and of finite presentation, then\nthere exists an $i_3 \\geq i_0$ such that for $i \\geq i_3$ we have\n$f_i$ is flat, $X_i = Y_i \\times_{Y_{i_3}} X_{i_3}$, and\n$X = Y \\times_{Y_{i_3}} X_{i_3}$.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending finite type schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNT","source_file":"limits.tex","source_line":5552,"source_end_line":5559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5552-L5559","statement_sha256":"99415dd79738265f1276f472497f984acc070de969afec6b2e893b7b394635e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6341,"rank":6341,"depth":36,"x":1943.694,"y":614.198,"cluster":"scheme-morphisms"},{"id":"stacks:0CNU","tag":"0CNU","title":"Descending finite type schemes · Lemma 0CNU","summary":"Notation and assumptions as in Lemma [Tag 0CNR]. If f is smooth, then there exists an i_3 ≥ i_0 such that for i ≥ i_3 we have f_i is smooth.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.\nIf $f$ is smooth, then there exists an $i_3 \\geq i_0$ such that for\n$i \\geq i_3$ we have $f_i$ is smooth.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending finite type schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNU","source_file":"limits.tex","source_line":5590,"source_end_line":5595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5590-L5595","statement_sha256":"7987ba2a7e3161572533f10d0fd51b5403fc9ffd728010e7810abb9cf10a0178","origin":"The Stacks Project","memory_eligible":false,"source_rank":6342,"rank":6342,"depth":38,"x":2146.65,"y":679.549,"cluster":"scheme-morphisms"},{"id":"stacks:0CNV","tag":"0CNV","title":"Descending finite type schemes · Lemma 0CNV","summary":"Notation and assumptions as in Lemma [Tag 0CNR]. If f is proper, then there exists an i_3 ≥ i_0 such that for i ≥ i_3 we have f_i is proper.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.\nIf $f$ is proper, then there exists an $i_3 \\geq i_0$ such that for\n$i \\geq i_3$ we have $f_i$ is proper.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending finite type schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNV","source_file":"limits.tex","source_line":5602,"source_end_line":5607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5602-L5607","statement_sha256":"fd618d138e0bab5ebb489a7f1130a9c5c0fe75600e6c7b6b7e12bddd1c65e6b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6343,"rank":6343,"depth":31,"x":1944.279,"y":746.576,"cluster":"scheme-morphisms"},{"id":"stacks:0CNW","tag":"0CNW","title":"Descending finite type schemes · Lemma 0CNW","summary":"In Situation [Tag 0CNM] suppose that we have a cartesian diagram xymatrix X^1 ar[r]_p ar[d]_q & X^3 ar[d]^a X^2 ar[r]^b & X^4 of schemes quasi-separated and of finite type over S. For each j = 1, 2, 3, 4 choose i_j ∈ I and a diagram xymatrix X^j ar[r] ar[d] & W^j ar[d] S ar[r] & S_i_j as in ([Tag 0CNP]). Let X^j = lim_i ≥ i_j X^j_i be the corresponding limit descriptions as in Lemma [Tag 0CNR]. Let (a_i)_i ≥ i_5, (b_i)_i ≥ i_6, (p_i)_i ≥ i_7, and (q_i)_i ≥ i_8 be the…","statement_latex":"In Situation \\ref{situation-limit-noetherian} suppose that we have a\ncartesian diagram\n$$\n\\xymatrix{\nX^1 \\ar[r]_p \\ar[d]_q & X^3 \\ar[d]^a \\\\\nX^2 \\ar[r]^b & X^4\n}\n$$\nof schemes quasi-separated and of finite type over $S$.\nFor each $j = 1, 2, 3, 4$ choose $i_j \\in I$ and a diagram\n$$\n\\xymatrix{\nX^j \\ar[r] \\ar[d] & W^j \\ar[d] \\\\\nS \\ar[r] & S_{i_j}\n}\n$$\nas in (\\ref{equation-good-diagram}). Let\n$X^j = \\lim_{i \\geq i_j} X^j_i$ be the corresponding limit descriptions\nas in Lemma \\ref{lemma-morphism-good-diagram}.\nLet $(a_i)_{i \\geq i_5}$, $(b_i)_{i \\geq i_6}$, $(p_i)_{i \\geq i_7}$, and\n$(q_i)_{i \\geq i_8}$ be the corresponding morphisms of systems constructed\nin Lemma \\ref{lemma-morphism-good-diagram}. Then there exists an\n$i_9 \\geq \\max(i_5, i_6, i_7, i_8)$ such that for $i \\geq i_9$ we have\n$a_i \\circ p_i = b_i \\circ q_i$ and such that\n$$\n(q_i, p_i) : X^1_i \\longrightarrow X^2_i \\times_{b_i, X^4_i, a_i} X^3_i\n$$\nis a closed immersion.\nIf $a$ and $b$ are flat and of finite presentation, then there exists an\n$i_{10} \\geq \\max(i_5, i_6, i_7, i_8, i_9)$ such that for $i \\geq i_{10}$\nthe last displayed morphism is an isomorphism.","area":"Scheme Morphisms","chapter":"Limits of Schemes","chapter_id":"limits","section":"Descending finite type schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNW","source_file":"limits.tex","source_line":5629,"source_end_line":5662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/limits.tex#L5629-L5662","statement_sha256":"4889ab45611aec63bcb66365f56bd5a2c8ab05c5192bc772a710be44da34363f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6344,"rank":6344,"depth":41,"x":2039.679,"y":582.19,"cluster":"scheme-morphisms"},{"id":"stacks:020D","tag":"020D","title":"Varieties · Definition 020D","summary":"Let k be a field. A variety is a scheme X over k such that X is integral and the structure morphism X → Spec(k) is separated and of finite type.","statement_latex":"Let $k$ be a field. A {\\it variety} is a scheme $X$ over $k$\nsuch that $X$ is integral and the structure morphism\n$X \\to \\Spec(k)$ is separated and of finite type.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Varieties","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020D","source_file":"varieties.tex","source_line":53,"source_end_line":58,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L53-L58","statement_sha256":"92f35ec5cf4694eceda6aee2758d70bbd7495542c2c63a5f3bf964383d91d505","origin":"The Stacks Project","memory_eligible":false,"source_rank":6345,"rank":6345,"depth":0,"x":235.188,"y":1140.0,"cluster":"varieties-curves"},{"id":"stacks:05P3","tag":"05P3","title":"Varieties · Lemma 05P3","summary":"Products of varieties are varieties over algebraically closed fields. Let k be an algebraically closed field. Let X, Y be varieties over k. Then X ×_Spec(k) Y is a variety over k.","statement_latex":"\\begin{slogan}\nProducts of varieties are varieties over algebraically closed fields.\n\\end{slogan}\nLet $k$ be an algebraically closed field.\nLet $X$, $Y$ be varieties over $k$.\nThen $X \\times_{\\Spec(k)} Y$ is a variety over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05P3","source_file":"varieties.tex","source_line":85,"source_end_line":93,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L85-L93","statement_sha256":"3de7e95931f75acd7019a29df456b52fccd36ea88330228f2b9b3154cb4e45b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6346,"rank":6346,"depth":16,"x":223.375,"y":1145.098,"cluster":"varieties-curves"},{"id":"stacks:0BXN","tag":"0BXN","title":"Varieties and rational maps · Theorem 0BXN","summary":"Let k be a field. The category of varieties and dominant rational maps is equivalent to the category of finitely generated field extensions K/k.","statement_latex":"Let $k$ be a field. The category of varieties and\ndominant rational maps is equivalent to the category of\nfinitely generated field extensions $K/k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Varieties and rational maps","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXN","source_file":"varieties.tex","source_line":164,"source_end_line":169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L164-L169","statement_sha256":"3d06f2b340adbd57e16dc4d550e4f5992ab40a3954d06a3592f536c4b9ab4305","origin":"The Stacks Project","memory_eligible":false,"source_rank":6347,"rank":6347,"depth":8,"x":231.014,"y":1130.293,"cluster":"varieties-curves"},{"id":"stacks:0BXP","tag":"0BXP","title":"Varieties and rational maps · Lemma 0BXP","summary":"Let X and Y be varieties over a field k. The following are equivalent • X and Y are birational varieties, • the function fields k(X) and k(Y) are isomorphic, • there exist nonempty opens of X and Y which are isomorphic as varieties, • there exists an open U ⊂ X and a birational morphism U → Y of varieties.","statement_latex":"Let $X$ and $Y$ be varieties over a field $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ and $Y$ are birational varieties,\n\\item the function fields $k(X)$ and $k(Y)$ are isomorphic,\n\\item there exist nonempty opens of $X$ and $Y$ which are isomorphic\nas varieties,\n\\item there exists an open $U \\subset X$ and a birational morphism\n$U \\to Y$ of varieties.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Varieties and rational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXP","source_file":"varieties.tex","source_line":205,"source_end_line":217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L205-L217","statement_sha256":"19e0a35d944a6638d339eeefac4a1d83ba191de72e2523c880910eb37e5c4966","origin":"The Stacks Project","memory_eligible":false,"source_rank":6348,"rank":6348,"depth":19,"x":238.351,"y":1149.149,"cluster":"varieties-curves"},{"id":"stacks:0C4Y","tag":"0C4Y","title":"Change of fields and local rings · Lemma 0C4Y","summary":"Let K/k be an extension of fields. Let X be scheme over k and set Y = X_K. If y ∈ Y with image x ∈ X, then • O_X, x → O_Y, y is a faithfully flat local ring homomorphism, • with p_0 = Ker(kappa(x) ⊗_k K → kappa(y)) we have kappa(y) = kappa( p_0), • O_Y, y = (O_X, x ⊗_k K)_ p where p ⊂ O_X, x ⊗_k K is the inverse image of p_0. • we have O_Y, y/ m_xO_Y, y = (kappa(x) ⊗_k K)_ p_0","statement_latex":"Let $K/k$ be an extension of fields. Let $X$ be scheme over $k$\nand set $Y = X_K$. If $y \\in Y$ with image $x \\in X$, then\n\\begin{enumerate}\n\\item $\\mathcal{O}_{X, x} \\to \\mathcal{O}_{Y, y}$ is a\nfaithfully flat local ring homomorphism,\n\\item with $\\mathfrak p_0 = \\Ker(\\kappa(x) \\otimes_k K \\to \\kappa(y))$\nwe have $\\kappa(y) = \\kappa(\\mathfrak p_0)$,\n\\item $\\mathcal{O}_{Y, y} = (\\mathcal{O}_{X, x} \\otimes_k K)_\\mathfrak p$\nwhere $\\mathfrak p \\subset \\mathcal{O}_{X, x} \\otimes_k K$ is the inverse\nimage of $\\mathfrak p_0$.\n\\item we have\n$\\mathcal{O}_{Y, y}/\\mathfrak m_x\\mathcal{O}_{Y, y} =\n(\\kappa(x) \\otimes_k K)_{\\mathfrak p_0}$\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4Y","source_file":"varieties.tex","source_line":237,"source_end_line":253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L237-L253","statement_sha256":"561d4efdc152aa09aaefea6768d49b9243e70dd154fc38152c4a5efb2a81485e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6349,"rank":6349,"depth":4,"x":214.675,"y":1137.723,"cluster":"varieties-curves"},{"id":"stacks:0C4Z","tag":"0C4Z","title":"Change of fields and local rings · Lemma 0C4Z","summary":"Notation as in Lemma [Tag 0C4Y]. Assume X is locally of finite type over k. Then dim(O_Y, y/ m_xO_Y, y) = trdeg_k(kappa(x)) - trdeg_K(kappa(y)) = dim(O_Y, y) - dim(O_X, x)","statement_latex":"Notation as in Lemma \\ref{lemma-change-fields-flat}.\nAssume $X$ is locally of finite type over $k$. Then\n$$\n\\dim(\\mathcal{O}_{Y, y}/\\mathfrak m_x\\mathcal{O}_{Y, y}) =\n\\text{trdeg}_k(\\kappa(x)) - \\text{trdeg}_K(\\kappa(y)) =\n\\dim(\\mathcal{O}_{Y, y}) - \\dim(\\mathcal{O}_{X, x})\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4Z","source_file":"varieties.tex","source_line":283,"source_end_line":292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L283-L292","statement_sha256":"65768d578bb4a446c5763ed413df956255e85e846d74526e945fb35f1870fccb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6350,"rank":6350,"depth":26,"x":244.517,"y":1132.243,"cluster":"varieties-curves"},{"id":"stacks:0C50","tag":"0C50","title":"Change of fields and local rings · Lemma 0C50","summary":"Notation as in Lemma [Tag 0C4Y]. Assume X is locally of finite type over k, that dim(O_X, x) = dim(O_Y, y) and that kappa(x) ⊗_k K is reduced (for example if kappa(x)/k is separable or K/k is separable). Then m_x O_Y, y = m_y.","statement_latex":"Notation as in Lemma \\ref{lemma-change-fields-flat}.\nAssume $X$ is locally of finite type over $k$,\nthat $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$\nand that $\\kappa(x) \\otimes_k K$ is reduced\n(for example if $\\kappa(x)/k$ is separable or $K/k$ is separable).\nThen $\\mathfrak m_x \\mathcal{O}_{Y, y} = \\mathfrak m_y$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C50","source_file":"varieties.tex","source_line":299,"source_end_line":307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L299-L307","statement_sha256":"aba94677cd5f3a400a4044f4e8e03d443a99ad6f7a9801741572f6003c0709d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6351,"rank":6351,"depth":27,"x":225.144,"y":1155.173,"cluster":"varieties-curves"},{"id":"stacks:035V","tag":"035V","title":"Geometrically reduced schemes · Definition 035V","summary":"Let k be a field. Let X be a scheme over k. • Let x ∈ X be a point. We say X is geometrically reduced at x if for any field extension k'/k and any point x' ∈ X_k' lying over x the local ring O_X_k', x' is reduced. • We say X is geometrically reduced over k if X is geometrically reduced at every point of X.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\n\\begin{enumerate}\n\\item Let $x \\in X$ be a point.\nWe say $X$ is {\\it geometrically reduced at $x$}\nif for any field extension $k'/k$\nand any point $x' \\in X_{k'}$ lying over $x$\nthe local ring $\\mathcal{O}_{X_{k'}, x'}$ is reduced.\n\\item We say $X$ is {\\it geometrically reduced} over $k$\nif $X$ is geometrically reduced at every point of $X$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035V","source_file":"varieties.tex","source_line":330,"source_end_line":343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L330-L343","statement_sha256":"6a5a3d41e022f87d4be9b7b177cdaa5239ae4d5a776e4a8a677cff72bd2cd36a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6352,"rank":6352,"depth":0,"x":220.74,"y":1125.023,"cluster":"varieties-curves"},{"id":"stacks:035W","tag":"035W","title":"Geometrically reduced schemes · Lemma 035W","summary":"Geometric reducedness can be checked on local rings. Let k be a field. Let X be a scheme over k. Let x ∈ X. The following are equivalent • X is geometrically reduced at x, and • the ring O_X, x is geometrically reduced over k (see Algebra, Definition [Tag 030S]).","statement_latex":"\\begin{slogan}\nGeometric reducedness can be checked on local rings.\n\\end{slogan}\nLet $k$ be a field.\nLet $X$ be a scheme over $k$.\nLet $x \\in X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically reduced at $x$, and\n\\item the ring $\\mathcal{O}_{X, x}$ is geometrically\nreduced over $k$ (see\nAlgebra, Definition \\ref{algebra-definition-geometrically-reduced}).\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035W","source_file":"varieties.tex","source_line":353,"source_end_line":368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L353-L368","statement_sha256":"39235639b46f6521efba6dec3c88e055ba6f14cf5eb992e79256b79dd0383b22","origin":"The Stacks Project","memory_eligible":false,"source_rank":6353,"rank":6353,"depth":9,"x":250.091,"y":1146.163,"cluster":"varieties-curves"},{"id":"stacks:020I","tag":"020I","title":"Geometrically reduced schemes · Lemma 020I","summary":"Let X be a scheme over a perfect field k (e.g. k has characteristic zero). Let x ∈ X. If O_X, x is reduced, then X is geometrically reduced at x. If X is reduced, then X is geometrically reduced over k.","statement_latex":"Let $X$ be a scheme over a perfect field $k$ (e.g.\\ $k$ has\ncharacteristic zero). Let $x \\in X$. If $\\mathcal{O}_{X, x}$ is\nreduced, then $X$ is geometrically reduced at $x$.\nIf $X$ is reduced, then $X$ is geometrically reduced over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020I","source_file":"varieties.tex","source_line":400,"source_end_line":406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L400-L406","statement_sha256":"050e518ed8993dcfadd98752d06f83fe7e4dff155727043cd2abc381d3ecaa65","origin":"The Stacks Project","memory_eligible":false,"source_rank":6354,"rank":6354,"depth":10,"x":209.098,"y":1147.247,"cluster":"varieties-curves"},{"id":"stacks:035X","tag":"035X","title":"Geometrically reduced schemes · Lemma 035X","summary":"Let k be a field of characteristic p > 0. Let X be a scheme over k. The following are equivalent • X is geometrically reduced, • X_k' is reduced for every field extension k'/k, • X_k' is reduced for every finite purely inseparable field extension k'/k, • X_k^1/p is reduced, • X_k^perf is reduced, • X_bar k is reduced, • for every affine open U ⊂ X the ring O_X(U) is geometrically reduced (see Algebra, Definition [Tag 030S]).","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically reduced,\n\\item $X_{k'}$ is reduced for every field extension $k'/k$,\n\\item $X_{k'}$ is reduced for every finite purely inseparable field extension\n$k'/k$,\n\\item $X_{k^{1/p}}$ is reduced,\n\\item $X_{k^{perf}}$ is reduced,\n\\item $X_{\\bar k}$ is reduced,\n\\item for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis geometrically reduced (see\nAlgebra, Definition \\ref{algebra-definition-geometrically-reduced}).\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035X","source_file":"varieties.tex","source_line":417,"source_end_line":433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L417-L433","statement_sha256":"eb740dbd130387d2148e810cfcfb8f1ae46470ed2081063c3e7d777e30c8af3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6355,"rank":6355,"depth":9,"x":240.076,"y":1121.913,"cluster":"varieties-curves"},{"id":"stacks:035Y","tag":"035Y","title":"Geometrically reduced schemes · Lemma 035Y","summary":"Let k be a field of characteristic p > 0. Let X be a scheme over k. Let x ∈ X. The following are equivalent • X is geometrically reduced at x, • O_X_k', x' is reduced for every finite purely inseparable field extension k' of k and x' ∈ X_k' the unique point lying over x, • O_X_k^1/p, x' is reduced for x' ∈ X_k^1/p the unique point lying over x, and • O_X_k^perf, x' is reduced for x' ∈ X_k^perf the unique point lying over x.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$.\nLet $x \\in X$. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically reduced at $x$,\n\\item $\\mathcal{O}_{X_{k'}, x'}$ is reduced for every\nfinite purely inseparable field extension $k'$ of $k$ and\n$x' \\in X_{k'}$ the unique point lying over $x$,\n\\item $\\mathcal{O}_{X_{k^{1/p}}, x'}$ is reduced for\n$x' \\in X_{k^{1/p}}$ the unique point lying over $x$, and\n\\item $\\mathcal{O}_{X_{k^{perf}}, x'}$ is reduced for\n$x' \\in X_{k^{perf}}$ the unique point lying over $x$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035Y","source_file":"varieties.tex","source_line":462,"source_end_line":476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L462-L476","statement_sha256":"22d5bcc89550e957a5e4906d065ad094c02b2c777cc9116797c358703c6ed01b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6356,"rank":6356,"depth":10,"x":237.446,"y":1159.94,"cluster":"varieties-curves"},{"id":"stacks:0384","tag":"0384","title":"Geometrically reduced schemes · Lemma 0384","summary":"Let k be a field. Let X be a scheme over k. Let k'/k be a field extension. Let x ∈ X be a point, and let x' ∈ X_k' be a point lying over x. The following are equivalent • X is geometrically reduced at x, • X_k' is geometrically reduced at x'. In particular, X is geometrically reduced over k if and only if X_k' is geometrically reduced over k'.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nLet $k'/k$ be a field extension.\nLet $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically reduced at $x$,\n\\item $X_{k'}$ is geometrically reduced at $x'$.\n\\end{enumerate}\nIn particular, $X$ is geometrically reduced over $k$ if and only if\n$X_{k'}$ is geometrically reduced over $k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0384","source_file":"varieties.tex","source_line":490,"source_end_line":503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L490-L503","statement_sha256":"ad8c8aba08ced6d5fbf4d909e3e1c9051bfbf92d03a019e7c266f79492eae7bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6357,"rank":6357,"depth":11,"x":207.558,"y":1129.075,"cluster":"varieties-curves"},{"id":"stacks:035Z","tag":"035Z","title":"Geometrically reduced schemes · Lemma 035Z","summary":"Let k be a field. Let X, Y be schemes over k. • If X is geometrically reduced at x, and Y reduced, then X ×_k Y is reduced at every point lying over x. • If X geometrically reduced over k and Y reduced, then X ×_k Y is reduced. • If X and Y are geometrically reduced over k, then X ×_k Y is geometrically reduced. • If k is perfect and X and Y are reduced, then X ×_k Y is reduced. • Add more here.","statement_latex":"Let $k$ be a field. Let $X$, $Y$ be schemes over $k$.\n\\begin{enumerate}\n\\item If $X$ is geometrically reduced at $x$, and $Y$ reduced,\nthen $X \\times_k Y$ is reduced at every point lying over $x$.\n\\item If $X$ geometrically reduced over $k$ and $Y$ reduced, then\n$X \\times_k Y$ is reduced.\n\\item If $X$ and $Y$ are geometrically reduced over $k$, then\n$X \\times_k Y$ is geometrically reduced.\n\\item If $k$ is perfect and $X$ and $Y$ are reduced, then\n$X \\times_k Y$ is reduced.\n\\item Add more here.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/035Z","source_file":"varieties.tex","source_line":524,"source_end_line":538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L524-L538","statement_sha256":"d5d9c9eaddb252deaa60c4eb74916538835c918fea55f22287c42c88caa3b68c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6358,"rank":6358,"depth":11,"x":256.327,"y":1135.138,"cluster":"varieties-curves"},{"id":"stacks:04KS","tag":"04KS","title":"Geometrically reduced schemes · Lemma 04KS","summary":"Let k be a field. Let X be a scheme over k. • If x' leadsto x is a specialization and X is geometrically reduced at x, then X is geometrically reduced at x'. • If x ∈ X such that (a) O_X, x is reduced, and (b) for each specialization x' leadsto x where x' is a generic point of an irreducible component of X the scheme X is geometrically reduced at x', then X is geometrically reduced at x. • If X is reduced and geometrically reduced at all generic points of irreducible…","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\n\\begin{enumerate}\n\\item If $x' \\leadsto x$ is a specialization and $X$ is geometrically\nreduced at $x$, then $X$ is geometrically reduced at $x'$.\n\\item If $x \\in X$ such that (a) $\\mathcal{O}_{X, x}$\nis reduced, and (b) for each specialization $x' \\leadsto x$ where\n$x'$ is a generic point of an irreducible component of $X$ the\nscheme $X$ is geometrically reduced at $x'$, then $X$ is geometrically\nreduced at $x$.\n\\item If $X$ is reduced and geometrically reduced at all generic\npoints of irreducible components of $X$, then $X$ is geometrically\nreduced.\n\\item If $X$ is a variety over $k$,\nthen $X$ is geometrically reduced if and only if its function field\nis a separable extension of $k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KS","source_file":"varieties.tex","source_line":554,"source_end_line":573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L554-L573","statement_sha256":"73802872c0aed9f46ea18a81fcc6a64cc7d39e353389fc80183be28c1f256315","origin":"The Stacks Project","memory_eligible":false,"source_rank":6359,"rank":6359,"depth":10,"x":213.933,"y":1159.197,"cluster":"varieties-curves"},{"id":"stacks:0360","tag":"0360","title":"Geometrically reduced schemes · Lemma 0360","summary":"Let k be a field. Let X be a scheme over k. Let x ∈ X. Assume X locally Noetherian and geometrically reduced at x. Then there exists an open neighbourhood U ⊂ X of x which is geometrically reduced over k.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nLet $x \\in X$.\nAssume $X$ locally Noetherian and geometrically reduced at $x$.\nThen there exists an open neighbourhood $U \\subset X$ of $x$\nwhich is geometrically reduced over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0360","source_file":"varieties.tex","source_line":598,"source_end_line":606,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L598-L606","statement_sha256":"60897d48353364c226a6fe6161a0873aceb31542dee78fa476bafd88c780b3d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6360,"rank":6360,"depth":10,"x":226.288,"y":1115.939,"cluster":"varieties-curves"},{"id":"stacks:04KT","tag":"04KT","title":"Geometrically reduced schemes · Lemma 04KT","summary":"Let k be a field. Let X → Spec(k) be locally of finite type. Assume X has finitely many irreducible components. Then there exists a finite purely inseparable extension k'/k such that (X_k')_red is geometrically reduced over k'.","statement_latex":"Let $k$ be a field.\nLet $X \\to \\Spec(k)$ be locally of finite type.\nAssume $X$ has finitely many irreducible components.\nThen there exists a finite purely inseparable extension $k'/k$\nsuch that $(X_{k'})_{red}$ is geometrically reduced over $k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically reduced schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KT","source_file":"varieties.tex","source_line":636,"source_end_line":643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L636-L643","statement_sha256":"0d49ee3aa9f384933b633ddc4138df899997e0fb6a0bf7bde5ba50a856e7db46","origin":"The Stacks Project","memory_eligible":false,"source_rank":6361,"rank":6361,"depth":11,"x":252.787,"y":1156.132,"cluster":"varieties-curves"},{"id":"stacks:0362","tag":"0362","title":"Geometrically connected schemes · Definition 0362","summary":"Let X be a scheme over the field k. We say X is geometrically connected over k if the scheme X_k' is connected for every field extension k' of k.","statement_latex":"Let $X$ be a scheme over the field $k$. We say $X$ is\n{\\it geometrically connected} over $k$ if the scheme $X_{k'}$ is connected\nfor every field extension $k'$ of $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0362","source_file":"varieties.tex","source_line":687,"source_end_line":692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L687-L692","statement_sha256":"2ca5d82d2c90ed0335fe576ded992e29b9f88e86071f493302528b63e00cff96","origin":"The Stacks Project","memory_eligible":false,"source_rank":6362,"rank":6362,"depth":0,"x":199.336,"y":1141.065,"cluster":"varieties-curves"},{"id":"stacks:054N","tag":"054N","title":"Geometrically connected schemes · Lemma 054N","summary":"Let X be a scheme over the field k. Let k'/k be a field extension. Then X is geometrically connected over k if and only if X_k' is geometrically connected over k'.","statement_latex":"Let $X$ be a scheme over the field $k$.\nLet $k'/k$ be a field extension.\nThen $X$ is geometrically connected over $k$ if and only if\n$X_{k'}$ is geometrically connected over $k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054N","source_file":"varieties.tex","source_line":710,"source_end_line":716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L710-L716","statement_sha256":"1aa12cce9d866c47331ca40d271780ab01b3781e9627d9834cd94c5240dbb3ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":6363,"rank":6363,"depth":2,"x":252.367,"y":1121.303,"cluster":"varieties-curves"},{"id":"stacks:0385","tag":"0385","title":"Geometrically connected schemes · Lemma 0385","summary":"Let k be a field. Let X, Y be schemes over k. Assume X is geometrically connected over k. Then the projection morphism p : X ×_k Y → Y induces a bijection between connected components.","statement_latex":"Let $k$ be a field.\nLet $X$, $Y$ be schemes over $k$.\nAssume $X$ is geometrically connected over $k$.\nThen the projection morphism\n$$\np : X \\times_k Y \\longrightarrow Y\n$$\ninduces a bijection between connected components.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0385","source_file":"varieties.tex","source_line":731,"source_end_line":741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L731-L741","statement_sha256":"116085e364a08d33148ec32374917387284a56ecc1907248144a52470786ca31","origin":"The Stacks Project","memory_eligible":false,"source_rank":6364,"rank":6364,"depth":13,"x":228.504,"y":1167.184,"cluster":"varieties-curves"},{"id":"stacks:0386","tag":"0386","title":"Geometrically connected schemes · Lemma 0386","summary":"Let k be a field. Let A be a k-algebra. Then X = Spec(A) is geometrically connected over k if and only if A is geometrically connected over k (see Algebra, Definition [Tag 037T]).","statement_latex":"Let $k$ be a field.\nLet $A$ be a $k$-algebra.\nThen $X = \\Spec(A)$ is geometrically connected over $k$\nif and only if $A$ is geometrically connected over $k$ (see\nAlgebra, Definition \\ref{algebra-definition-geometrically-connected}).","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0386","source_file":"varieties.tex","source_line":757,"source_end_line":764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L757-L764","statement_sha256":"ceaf68b9dadedd77a39a706324e6a0e959e141d87498275efc50aee1e1801522","origin":"The Stacks Project","memory_eligible":false,"source_rank":6365,"rank":6365,"depth":1,"x":208.718,"y":1118.577,"cluster":"varieties-curves"},{"id":"stacks:0363","tag":"0363","title":"Geometrically connected schemes · Lemma 0363","summary":"Let k'/k be an extension of fields. Let X be a scheme over k. Assume k separably algebraically closed. Then the morphism X_k' → X induces a bijection of connected components. In particular, X is geometrically connected over k if and only if X is connected.","statement_latex":"Let $k'/k$ be an extension of fields.\nLet $X$ be a scheme over $k$.\nAssume $k$ separably algebraically closed.\nThen the morphism $X_{k'} \\to X$ induces a bijection of connected\ncomponents. In particular, $X$ is geometrically connected over $k$\nif and only if $X$ is connected.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0363","source_file":"varieties.tex","source_line":770,"source_end_line":778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L770-L778","statement_sha256":"4017210d4401225cce75c22b08777e55105ef2693f15c15a882cff11c49aadc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6366,"rank":6366,"depth":14,"x":263.714,"y":1143.81,"cluster":"varieties-curves"},{"id":"stacks:0387","tag":"0387","title":"Geometrically connected schemes · Lemma 0387","summary":"Let k be a field. Let X be a scheme over k. Let overlinek be a separable algebraic closure of k. Then X is geometrically connected if and only if the base change X_overlinek is connected.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nLet $\\overline{k}$ be a separable algebraic closure of $k$.\nThen $X$ is geometrically connected if and only if the base change\n$X_{\\overline{k}}$ is connected.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0387","source_file":"varieties.tex","source_line":791,"source_end_line":798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L791-L798","statement_sha256":"c9fc0c871854fd70d1d9359ff03610112c85dbe94e6d1fe13667f92b9f469ba7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6367,"rank":6367,"depth":15,"x":201.435,"y":1156.695,"cluster":"varieties-curves"},{"id":"stacks:0388","tag":"0388","title":"Geometrically connected schemes · Lemma 0388","summary":"Let k be a field. Let X be a scheme over k. Let A be a k-algebra. Let V ⊂ X_A be a quasi-compact open. Then there exists a finitely generated k-subalgebra A' ⊂ A and a quasi-compact open V' ⊂ X_A' such that V = V'_A.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nLet $A$ be a $k$-algebra.\nLet $V \\subset X_A$ be a quasi-compact open.\nThen there exists a finitely generated $k$-subalgebra $A' \\subset A$\nand a quasi-compact open $V' \\subset X_{A'}$\nsuch that $V = V'_A$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0388","source_file":"varieties.tex","source_line":811,"source_end_line":820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L811-L820","statement_sha256":"e3d3d9923d590b245033643914ca27d324950bf1d63fe4ba697e9e46f63513a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6368,"rank":6368,"depth":1,"x":237.806,"y":1110.854,"cluster":"varieties-curves"},{"id":"stacks:04KU","tag":"04KU","title":"Geometrically connected schemes · Lemma 04KU","summary":"Let k be a field. Let X be a scheme over k. Let overlinek be a (possibly infinite) Galois extension of k. Let V ⊂ X_overlinek be a quasi-compact open. Then • there exists a finite subextension overlinek/k'/k and a quasi-compact open V' ⊂ X_k' such that V = (V')_overlinek, • there exists an open subgroup H ⊂ Gal(overlinek/k) such that σ(V) = V for all σ ∈ H.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$.\nLet $\\overline{k}$ be a (possibly infinite) Galois extension of $k$.\nLet $V \\subset X_{\\overline{k}}$ be a quasi-compact open.\nThen\n\\begin{enumerate}\n\\item there exists a finite subextension $\\overline{k}/k'/k$\nand a quasi-compact open $V' \\subset X_{k'}$ such that\n$V = (V')_{\\overline{k}}$,\n\\item there exists an open subgroup $H \\subset \\text{Gal}(\\overline{k}/k)$\nsuch that $\\sigma(V) = V$ for all $\\sigma \\in H$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KU","source_file":"varieties.tex","source_line":876,"source_end_line":889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L876-L889","statement_sha256":"ef545b2c39faf52db49d69a4b5edc7bda0a07d837913b4ddb24206d80bc82032","origin":"The Stacks Project","memory_eligible":false,"source_rank":6369,"rank":6369,"depth":2,"x":248.054,"y":1166.466,"cluster":"varieties-curves"},{"id":"stacks:038B","tag":"038B","title":"Geometrically connected schemes · Lemma 038B","summary":"Let k be a field. Let overlinek/k be a (possibly infinite) Galois extension. Let X be a scheme over k. Let overlineT ⊂ X_overlinek have the following properties • overlineT is a closed subset of X_overlinek, • for every σ ∈ Gal(overlinek/k) we have σ(overlineT) = overlineT. Then there exists a closed subset T ⊂ X whose inverse image in X_overlinek is overlineT.","statement_latex":"Let $k$ be a field. Let $\\overline{k}/k$ be a (possibly infinite)\nGalois extension. Let $X$ be a scheme over $k$. Let\n$\\overline{T} \\subset X_{\\overline{k}}$ have the following properties\n\\begin{enumerate}\n\\item $\\overline{T}$ is a closed subset of $X_{\\overline{k}}$,\n\\item for every $\\sigma \\in \\text{Gal}(\\overline{k}/k)$\nwe have $\\sigma(\\overline{T}) = \\overline{T}$.\n\\end{enumerate}\nThen there exists a closed subset $T \\subset X$ whose inverse image\nin $X_{\\overline{k}}$ is $\\overline{T}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038B","source_file":"varieties.tex","source_line":898,"source_end_line":910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L898-L910","statement_sha256":"ab2a6d1e7d454b8bfd144dd3b709dda1aab40d9a3a0575d1ef43381c0f235abf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6370,"rank":6370,"depth":0,"x":194.706,"y":1130.542,"cluster":"varieties-curves"},{"id":"stacks:0389","tag":"0389","title":"Geometrically connected schemes · Lemma 0389","summary":"Let k be a field. Let X be a scheme over k. The following are equivalent • X is geometrically connected, • for every finite separable field extension k'/k the scheme X_k' is connected.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically connected,\n\\item for every finite separable field extension $k'/k$\nthe scheme $X_{k'}$ is connected.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0389","source_file":"varieties.tex","source_line":935,"source_end_line":945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L935-L945","statement_sha256":"1ecb2ee3fb6dda3fa7321a639b35f424e4a33d826f97862983a25764bc35cab3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6371,"rank":6371,"depth":16,"x":264.284,"y":1126.694,"cluster":"varieties-curves"},{"id":"stacks:038C","tag":"038C","title":"Geometrically connected schemes · Lemma 038C","summary":"Let k be a field. Let overlinek/k be a (possibly infinite) Galois extension. Let f : T → X be a morphism of schemes over k. Assume T_overlinek connected and X_overlinek disconnected. Then X is disconnected.","statement_latex":"Let $k$ be a field. Let $\\overline{k}/k$ be a (possibly infinite)\nGalois extension. Let $f : T \\to X$ be a morphism of schemes over $k$.\nAssume $T_{\\overline{k}}$ connected and $X_{\\overline{k}}$\ndisconnected. Then $X$ is disconnected.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038C","source_file":"varieties.tex","source_line":1019,"source_end_line":1025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1019-L1025","statement_sha256":"139b13a018fa8bb231ec1e396e40bfda935475ddcdfcb7842c3ac3861bfc00e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6372,"rank":6372,"depth":1,"x":215.147,"y":1169.811,"cluster":"varieties-curves"},{"id":"stacks:056R","tag":"056R","title":"Geometrically connected schemes · Lemma 056R","summary":"[EGA] Let k be a field. Let T → X be a morphism of schemes over k. Assume T is geometrically connected and X connected. Then X is geometrically connected.","statement_latex":"\\begin{reference}\n\\cite[IV Corollary 4.5.13.1(i)]{EGA}\n\\end{reference}\nLet $k$ be a field. Let $T \\to X$ be a morphism of schemes over $k$.\nAssume $T$ is geometrically connected and $X$ connected.\nThen $X$ is geometrically connected.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056R","source_file":"varieties.tex","source_line":1066,"source_end_line":1074,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1066-L1074","statement_sha256":"fd82867306f67096b94cdc599878c8ef9752ed53d04fe9d95d619a40b4071a42","origin":"The Stacks Project","memory_eligible":false,"source_rank":6373,"rank":6373,"depth":2,"x":216.744,"y":1109.043,"cluster":"varieties-curves"},{"id":"stacks:04KV","tag":"04KV","title":"Geometrically connected schemes · Lemma 04KV","summary":"Let k be a field. Let X be a scheme over k. Assume X is connected and has a point x such that k is algebraically closed in kappa(x). Then X is geometrically connected. In particular, if X has a k-rational point and X is connected, then X is geometrically connected.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$.\nAssume $X$ is connected and has a point $x$ such that\n$k$ is algebraically closed in $\\kappa(x)$.\nThen $X$ is geometrically connected.\nIn particular, if $X$ has a $k$-rational point and $X$ is connected,\nthen $X$ is geometrically connected.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KV","source_file":"varieties.tex","source_line":1081,"source_end_line":1089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1081-L1089","statement_sha256":"f9c165dc779e58708a68c7a865ac4606cb86d844e31dcc48ba4c30ba922d46e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6374,"rank":6374,"depth":16,"x":265.272,"y":1155.572,"cluster":"varieties-curves"},{"id":"stacks:04PY","tag":"04PY","title":"Geometrically connected schemes · Lemma 04PY","summary":"Let K/k be an extension of fields. Let X be a scheme over k. For every connected component T of X the inverse image T_K ⊂ X_K is a union of connected components of X_K.","statement_latex":"Let $K/k$ be an extension of fields.\nLet $X$ be a scheme over $k$.\nFor every connected component $T$ of $X$ the inverse image\n$T_K \\subset X_K$ is a union of connected components of $X_K$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PY","source_file":"varieties.tex","source_line":1103,"source_end_line":1109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1103-L1109","statement_sha256":"ae0e6e49c59c19e76fdcd0d9b629c9d890418eddaae05dbb5124e28ce627af05","origin":"The Stacks Project","memory_eligible":false,"source_rank":6375,"rank":6375,"depth":0,"x":190.822,"y":1148.675,"cluster":"varieties-curves"},{"id":"stacks:07VM","tag":"07VM","title":"Geometrically connected schemes · Lemma 07VM","summary":"Let K/k be a finite extension of fields and let X be a scheme over k. Denote by p : X_K → X the projection morphism. For every connected component T of X_K the image p(T) is a connected component of X.","statement_latex":"Let $K/k$ be a finite extension of fields and let $X$ be a scheme over \n$k$. Denote by $p : X_K \\to X$ the projection morphism. For every connected \ncomponent $T$ of $X_K$ the image $p(T)$ is a connected component of \n$X$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VM","source_file":"varieties.tex","source_line":1124,"source_end_line":1130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1124-L1130","statement_sha256":"74f4bd422fecfc19ece9d704147e17994661b15c1882c37559eaba32b7a254da","origin":"The Stacks Project","memory_eligible":false,"source_rank":6376,"rank":6376,"depth":29,"x":252.269,"y":1110.908,"cluster":"varieties-curves"},{"id":"stacks:04PZ","tag":"04PZ","title":"Gabber · Lemma 04PZ","summary":"Email from Ofer Gabber dated June 4, 2016 Let K/k be an extension of fields. Let X be a scheme over k. Denote p : X_K → X the projection morphism. Let overlineT ⊂ X_K be a connected component. Then p(overlineT) is a connected component of X.","statement_latex":"\\begin{reference}\nEmail from Ofer Gabber dated June 4, 2016\n\\end{reference}\nLet $K/k$ be an extension of fields. Let $X$ be a scheme over $k$.\nDenote $p : X_K \\to X$ the projection morphism.\nLet $\\overline{T} \\subset X_K$ be a connected component.\nThen $p(\\overline{T})$ is a connected component of $X$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PZ","source_file":"varieties.tex","source_line":1147,"source_end_line":1156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1147-L1156","statement_sha256":"419da47e0af339402020d50009ea444a7c9d0032342183f5dce15d592eac16e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6377,"rank":6377,"depth":48,"x":237.084,"y":1174.624,"cluster":"varieties-curves"},{"id":"stacks:038D","tag":"038D","title":"Geometrically connected schemes · Lemma 038D","summary":"Let k be a field, with separable algebraic closure overlinek. Let X be a scheme over k. There is an action Gal(overlinek/k)^opp × π_0(X_overlinek) → π_0(X_overlinek) with the following properties: • An element overlineT ∈ π_0(X_overlinek) is fixed by the action if and only if there exists a connected component T ⊂ X, which is geometrically connected over k, such that T_overlinek = overlineT. • For any field extension k'/k with separable algebraic closure overlinek' the…","statement_latex":"Let $k$ be a field, with separable algebraic closure $\\overline{k}$.\nLet $X$ be a scheme over $k$.\nThere is an action\n$$\n\\text{Gal}(\\overline{k}/k)^{opp} \\times \\pi_0(X_{\\overline{k}})\n\\longrightarrow\n\\pi_0(X_{\\overline{k}})\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item An element $\\overline{T} \\in \\pi_0(X_{\\overline{k}})$\nis fixed by the action if and only if there exists a connected component\n$T \\subset X$, which is geometrically connected over $k$,\nsuch that $T_{\\overline{k}} = \\overline{T}$.\n\\item For any field extension $k'/k$ with separable\nalgebraic closure $\\overline{k}'$ the diagram\n$$\n\\xymatrix{\n\\text{Gal}(\\overline{k}'/k') \\times \\pi_0(X_{\\overline{k}'})\n\\ar[r] \\ar[d] &\n\\pi_0(X_{\\overline{k}'}) \\ar[d] \\\\\n\\text{Gal}(\\overline{k}/k) \\times \\pi_0(X_{\\overline{k}})\n\\ar[r] &\n\\pi_0(X_{\\overline{k}})\n}\n$$\nis commutative (where the right vertical arrow is a bijection\naccording to Lemma \\ref{lemma-separably-closed-field-connected-components}).\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038D","source_file":"varieties.tex","source_line":1229,"source_end_line":1260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1229-L1260","statement_sha256":"b71dd963206da77a5f13392a2a168396317a56489138c9441e81c984907fece3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6378,"rank":6378,"depth":16,"x":196.428,"y":1118.16,"cluster":"varieties-curves"},{"id":"stacks:038E","tag":"038E","title":"Geometrically connected schemes · Lemma 038E","summary":"Let k be a field, with separable algebraic closure overlinek. Let X be a scheme over k. Assume • X is quasi-compact, and • the connected components of X_overlinek are open. Then • [(a)] π_0(X_overlinek) is finite, and • [(b)] the action of Gal(overlinek/k) on π_0(X_overlinek) is continuous. Moreover, assumptions (1) and (2) are satisfied when X is of finite type over k.","statement_latex":"Let $k$ be a field, with separable algebraic closure $\\overline{k}$.\nLet $X$ be a scheme over $k$.\nAssume\n\\begin{enumerate}\n\\item $X$ is quasi-compact, and\n\\item the connected components of $X_{\\overline{k}}$ are open.\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item[(a)] $\\pi_0(X_{\\overline{k}})$ is finite, and\n\\item[(b)] the action of $\\text{Gal}(\\overline{k}/k)$ on\n$\\pi_0(X_{\\overline{k}})$ is continuous.\n\\end{enumerate}\nMoreover, assumptions (1) and (2) are satisfied when $X$ is\nof finite type over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically connected schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038E","source_file":"varieties.tex","source_line":1283,"source_end_line":1300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1283-L1300","statement_sha256":"1cd24f4fbe0639021a7c53b2eb8d4e2dfd9348147980e0a3db1be7e884a99bcc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6379,"rank":6379,"depth":6,"x":272.944,"y":1137.011,"cluster":"varieties-curves"},{"id":"stacks:0365","tag":"0365","title":"Geometrically irreducible schemes · Definition 0365","summary":"Let X be a scheme over the field k. We say X is geometrically irreducible over k if the scheme X_k' is irreducible for any field extension k' of k.","statement_latex":"Let $X$ be a scheme over the field $k$.\nWe say $X$ is {\\it geometrically irreducible} over $k$ if the scheme\n$X_{k'}$ is irreducible\\footnote{An irreducible space is nonempty.}\nfor any field extension $k'$ of $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0365","source_file":"varieties.tex","source_line":1337,"source_end_line":1343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1337-L1343","statement_sha256":"fa56e042bcc013c60812ff2d1437cdb0a52db502bcc139586c2fdf32ccfa48bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6380,"rank":6380,"depth":0,"x":200.316,"y":1166.953,"cluster":"varieties-curves"},{"id":"stacks:054P","tag":"054P","title":"Geometrically irreducible schemes · Lemma 054P","summary":"Let X be a scheme over the field k. Let k'/k be a field extension. Then X is geometrically irreducible over k if and only if X_k' is geometrically irreducible over k'.","statement_latex":"Let $X$ be a scheme over the field $k$.\nLet $k'/k$ be a field extension.\nThen $X$ is geometrically irreducible over $k$ if and only if\n$X_{k'}$ is geometrically irreducible over $k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054P","source_file":"varieties.tex","source_line":1345,"source_end_line":1351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1345-L1351","statement_sha256":"bfe031a86ac3e27aafa0a7980c1b324f1a59503ab67ac73076db80705e9b873c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6381,"rank":6381,"depth":2,"x":230.216,"y":1102.769,"cluster":"varieties-curves"},{"id":"stacks:020J","tag":"020J","title":"Geometrically irreducible schemes · Lemma 020J","summary":"Let X be a scheme over a separably closed field k. If X is irreducible, then X_K is irreducible for any field extension K/k. I.e., X is geometrically irreducible over k.","statement_latex":"Let $X$ be a scheme over a separably closed field $k$.\nIf $X$ is irreducible, then $X_K$ is irreducible for any\nfield extension $K/k$. I.e., $X$ is geometrically\nirreducible over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020J","source_file":"varieties.tex","source_line":1366,"source_end_line":1372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1366-L1372","statement_sha256":"72d3f1dd82130673cf6b2d9817bed2235292fe77b962400a7dd1b58d14964281","origin":"The Stacks Project","memory_eligible":false,"source_rank":6382,"rank":6382,"depth":13,"x":260.185,"y":1167.951,"cluster":"varieties-curves"},{"id":"stacks:038F","tag":"038F","title":"Geometrically irreducible schemes · Lemma 038F","summary":"Let k be a field. Let X, Y be schemes over k. Assume X is geometrically irreducible over k. Then the projection morphism p : X ×_k Y → Y induces a bijection between irreducible components.","statement_latex":"Let $k$ be a field.\nLet $X$, $Y$ be schemes over $k$.\nAssume $X$ is geometrically irreducible over $k$.\nThen the projection morphism\n$$\np : X \\times_k Y \\longrightarrow Y\n$$\ninduces a bijection between irreducible components.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038F","source_file":"varieties.tex","source_line":1379,"source_end_line":1389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1379-L1389","statement_sha256":"6a954b049db4723aa4d8ddd518354da9e31b16cc10290e72d355135e72bfd331","origin":"The Stacks Project","memory_eligible":false,"source_rank":6383,"rank":6383,"depth":13,"x":184.673,"y":1136.471,"cluster":"varieties-curves"},{"id":"stacks:038G","tag":"038G","title":"Geometrically irreducible schemes · Lemma 038G","summary":"Geometric irreductibility is Zariski local modulo connectedness. Let k be a field. Let X be a scheme over k. The following are equivalent • X is geometrically irreducible over k, • for every nonempty affine open U the k-algebra O_X(U) is geometrically irreducible over k (see Algebra, Definition [Tag 037L]), • X is irreducible and there exists an affine open covering X = ⋃ U_i such that each k-algebra O_X(U_i) is geometrically irreducible, and • there exists an open…","statement_latex":"\\begin{slogan}\nGeometric irreductibility is Zariski local modulo connectedness.\n\\end{slogan}\nLet $k$ be a field. Let $X$ be a scheme over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically irreducible over $k$,\n\\item for every nonempty affine open $U$ the $k$-algebra $\\mathcal{O}_X(U)$\nis geometrically irreducible over $k$ (see\nAlgebra, Definition \\ref{algebra-definition-geometrically-irreducible}),\n\\item $X$ is irreducible and there exists an affine open covering\n$X = \\bigcup U_i$ such that each $k$-algebra $\\mathcal{O}_X(U_i)$ is\ngeometrically irreducible, and\n\\item there exists an open covering $X = \\bigcup_{i \\in I} X_i$\nwith $I \\not = \\emptyset$ such\nthat $X_i$ is geometrically irreducible for each $i$ and such that\n$X_i \\cap X_j \\not = \\emptyset$ for all $i, j \\in I$.\n\\end{enumerate}\nMoreover, if $X$ is geometrically irreducible so is every nonempty\nopen subscheme of $X$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038G","source_file":"varieties.tex","source_line":1404,"source_end_line":1426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1404-L1426","statement_sha256":"0817d998affbaf08bfc02c74fd93a4705b1c5a6c640fd9c61efd48446f829e24","origin":"The Stacks Project","memory_eligible":false,"source_rank":6384,"rank":6384,"depth":13,"x":266.728,"y":1116.585,"cluster":"varieties-curves"},{"id":"stacks:054Q","tag":"054Q","title":"Geometrically irreducible schemes · Lemma 054Q","summary":"Let X be an irreducible scheme over the field k. Let xi ∈ X be its generic point. The following are equivalent: • X is geometrically irreducible over k; • kappa(xi) is geometrically irreducible over k; • the separable algebraic closure of k in kappa(xi) is equal to k.","statement_latex":"Let $X$ be an irreducible scheme over the field $k$. Let $\\xi \\in X$\nbe its generic point. The following are equivalent:\n\\begin{enumerate}\n\\item $X$ is geometrically irreducible over $k$;\n\\item $\\kappa(\\xi)$ is geometrically irreducible over $k$;\n\\item the separable algebraic closure of $k$ in $\\kappa(\\xi)$ is equal to $k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054Q","source_file":"varieties.tex","source_line":1445,"source_end_line":1454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1445-L1454","statement_sha256":"d5d80901dd4fd6f26893635b39be29156faef91e80c992bc1e2c283a03d1acb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6385,"rank":6385,"depth":14,"x":221.644,"y":1178.585,"cluster":"varieties-curves"},{"id":"stacks:0H8G","tag":"0H8G","title":"Geometrically irreducible schemes · Lemma 0H8G","summary":"A scheme X over a field k is geometrically irreducible over k if and only if X×_k X is irreducible.","statement_latex":"A scheme $X$ over a field $k$ is geometrically irreducible\nover $k$ if and only if $X\\times_k X$ is irreducible.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8G","source_file":"varieties.tex","source_line":1486,"source_end_line":1490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1486-L1490","statement_sha256":"3122cea78c59059a553b9ba397bfca7baa5310c0156b3e1c57190d6fdc379f29","origin":"The Stacks Project","memory_eligible":false,"source_rank":6386,"rank":6386,"depth":15,"x":204.828,"y":1106.4,"cluster":"varieties-curves"},{"id":"stacks:038H","tag":"038H","title":"Geometrically irreducible schemes · Lemma 038H","summary":"Let k'/k be an extension of fields. Let X be a scheme over k. Set X' = X_k'. Assume k separably algebraically closed. Then the morphism X' → X induces a bijection of irreducible components.","statement_latex":"Let $k'/k$ be an extension of fields.\nLet $X$ be a scheme over $k$. Set $X' = X_{k'}$.\nAssume $k$ separably algebraically closed.\nThen the morphism $X' \\to X$ induces a bijection of irreducible components.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038H","source_file":"varieties.tex","source_line":1523,"source_end_line":1529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1523-L1529","statement_sha256":"3c5d930f615acd5b8e710530d0e0e5f5ebac7c62b9005870daed47e0d4c1a634","origin":"The Stacks Project","memory_eligible":false,"source_rank":6387,"rank":6387,"depth":14,"x":276.126,"y":1150.619,"cluster":"varieties-curves"},{"id":"stacks:038I","tag":"038I","title":"Geometrically irreducible schemes · Lemma 038I","summary":"Geometric irreducibility can be tested over a separable algebraic closure of the base field. Let k be a field. Let X be a scheme over k. The following are equivalent: • X is geometrically irreducible over k, • for every finite separable field extension k'/k the scheme X_k' is irreducible, and • X_overlinek is irreducible, where k ⊂ overlinek is a separable algebraic closure of k.","statement_latex":"\\begin{slogan}\nGeometric irreducibility can be tested over a separable algebraic\nclosure of the base field.\n\\end{slogan}\nLet $k$ be a field. Let $X$ be a scheme over $k$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $X$ is geometrically irreducible over $k$,\n\\item for every finite separable field extension $k'/k$\nthe scheme $X_{k'}$ is irreducible, and\n\\item $X_{\\overline{k}}$ is irreducible, where $k \\subset \\overline{k}$\nis a separable algebraic closure of $k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038I","source_file":"varieties.tex","source_line":1541,"source_end_line":1556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1541-L1556","statement_sha256":"0d78e84b82a278223a5eb46c0946e02238ff77db4e03749fd01ca469d3758859","origin":"The Stacks Project","memory_eligible":false,"source_rank":6388,"rank":6388,"depth":15,"x":186.951,"y":1158.557,"cluster":"varieties-curves"},{"id":"stacks:04KW","tag":"04KW","title":"Geometrically irreducible schemes · Lemma 04KW","summary":"Let K/k be an extension of fields. Let X be a scheme over k. For every irreducible component T of X the inverse image T_K ⊂ X_K is a union of irreducible components of X_K.","statement_latex":"Let $K/k$ be an extension of fields.\nLet $X$ be a scheme over $k$.\nFor every irreducible component $T$ of $X$ the inverse image\n$T_K \\subset X_K$ is a union of irreducible components of $X_K$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KW","source_file":"varieties.tex","source_line":1595,"source_end_line":1601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1595-L1601","statement_sha256":"a9aea120300e012f5f592e0618c8e3630f3e97db59221651dcccec9ce66f668a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6389,"rank":6389,"depth":0,"x":247.012,"y":1101.454,"cluster":"varieties-curves"},{"id":"stacks:04KX","tag":"04KX","title":"Geometrically irreducible schemes · Lemma 04KX","summary":"Let K/k be an extension of fields. Let X be a scheme over k. For every irreducible component overlineT ⊂ X_K the image of overlineT in X is an irreducible component in X. This defines a canonical map IrredComp(X_K) → IrredComp(X) which is surjective.","statement_latex":"Let $K/k$ be an extension of fields.\nLet $X$ be a scheme over $k$.\nFor every irreducible component $\\overline{T} \\subset X_K$\nthe image of $\\overline{T}$ in $X$ is an irreducible component in $X$.\nThis defines a canonical map\n$$\n\\text{IrredComp}(X_K)\n\\longrightarrow\n\\text{IrredComp}(X)\n$$\nwhich is surjective.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KX","source_file":"varieties.tex","source_line":1621,"source_end_line":1634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1621-L1634","statement_sha256":"45ad2f245e4f529e8a5a783f7364c0c7b45324f978ddfafb57e65f1120522600","origin":"The Stacks Project","memory_eligible":false,"source_rank":6390,"rank":6390,"depth":20,"x":248.659,"y":1178.501,"cluster":"varieties-curves"},{"id":"stacks:0G69","tag":"0G69","title":"Geometrically irreducible schemes · Lemma 0G69","summary":"Let k be a field. Let X be a scheme over k. If X is irreducible and has a dense set of k-rational points, then X is geometrically irreducible.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$. If $X$ is irreducible\nand has a dense set of $k$-rational points, then $X$ is geometrically\nirreducible.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G69","source_file":"varieties.tex","source_line":1672,"source_end_line":1677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1672-L1677","statement_sha256":"075fc5b66e329d6cbfafb39c2775541c186525b8561a5827ff69606ff4dc3dff","origin":"The Stacks Project","memory_eligible":false,"source_rank":6391,"rank":6391,"depth":37,"x":184.792,"y":1122.003,"cluster":"varieties-curves"},{"id":"stacks:038J","tag":"038J","title":"Geometrically irreducible schemes · Lemma 038J","summary":"Let k be a field, with separable algebraic closure overlinek. Let X be a scheme over k. There is an action Gal(overlinek/k)^opp × IrredComp(X_overlinek) → IrredComp(X_overlinek) with the following properties: • An element overlineT ∈ IrredComp(X_overlinek) is fixed by the action if and only if there exists an irreducible component T ⊂ X, which is geometrically irreducible over k, such that T_overlinek = overlineT. • For any field extension k'/k with separable algebraic…","statement_latex":"Let $k$ be a field, with separable algebraic closure $\\overline{k}$.\nLet $X$ be a scheme over $k$.\nThere is an action\n$$\n\\text{Gal}(\\overline{k}/k)^{opp} \\times \\text{IrredComp}(X_{\\overline{k}})\n\\longrightarrow\n\\text{IrredComp}(X_{\\overline{k}})\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item An element $\\overline{T} \\in \\text{IrredComp}(X_{\\overline{k}})$\nis fixed by the action if and only if there exists an irreducible\ncomponent $T \\subset X$, which is geometrically irreducible over $k$,\nsuch that $T_{\\overline{k}} = \\overline{T}$.\n\\item For any field extension $k'/k$ with separable\nalgebraic closure $\\overline{k}'$ the diagram\n$$\n\\xymatrix{\n\\text{Gal}(\\overline{k}'/k') \\times \\text{IrredComp}(X_{\\overline{k}'})\n\\ar[r] \\ar[d] &\n\\text{IrredComp}(X_{\\overline{k}'}) \\ar[d] \\\\\n\\text{Gal}(\\overline{k}/k) \\times \\text{IrredComp}(X_{\\overline{k}})\n\\ar[r] &\n\\text{IrredComp}(X_{\\overline{k}})\n}\n$$\nis commutative (where the right vertical arrow is a bijection\naccording to Lemma \\ref{lemma-separably-closed-field-irreducible-components}).\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038J","source_file":"varieties.tex","source_line":1699,"source_end_line":1730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1699-L1730","statement_sha256":"66a919260b2ebad8a7704695b4e5b63dfdffd471344f9d2bdfc2c4de750e67ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":6392,"rank":6392,"depth":16,"x":278.318,"y":1127.487,"cluster":"varieties-curves"},{"id":"stacks:04KY","tag":"04KY","title":"Geometrically irreducible schemes · Lemma 04KY","summary":"Let k be a field, with separable algebraic closure overlinek. Let X be a scheme over k. The fibres of the map IrredComp(X_overlinek) → IrredComp(X) of Lemma [Tag 04KX] are exactly the orbits of Gal(overlinek/k) under the action of Lemma [Tag 038J].","statement_latex":"Let $k$ be a field, with separable algebraic closure $\\overline{k}$.\nLet $X$ be a scheme over $k$.\nThe fibres of the map\n$$\n\\text{IrredComp}(X_{\\overline{k}})\n\\longrightarrow\n\\text{IrredComp}(X)\n$$\nof\nLemma \\ref{lemma-image-irreducible}\nare exactly the orbits of $\\text{Gal}(\\overline{k}/k)$ under the action of\nLemma \\ref{lemma-galois-action-irreducible-components}.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KY","source_file":"varieties.tex","source_line":1759,"source_end_line":1773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1759-L1773","statement_sha256":"c8f35d61c99d12248fd3c212f107b37494f8c965184b9b4eb5ab4ca229340966","origin":"The Stacks Project","memory_eligible":false,"source_rank":6393,"rank":6393,"depth":21,"x":204.167,"y":1177.027,"cluster":"varieties-curves"},{"id":"stacks:04KZ","tag":"04KZ","title":"Geometrically irreducible schemes · Lemma 04KZ","summary":"Let k be a field. Assume X → Spec(k) locally of finite type. In this case • the action Gal(overlinek/k)^opp × IrredComp(X_overlinek) → IrredComp(X_overlinek) is continuous if we give IrredComp(X_overlinek) the discrete topology, • every irreducible component of X_overlinek can be defined over a finite extension of k, and • given any irreducible component T ⊂ X the scheme T_overlinek is a finite union of irreducible components of X_overlinek which are all in the same…","statement_latex":"Let $k$ be a field.\nAssume $X \\to \\Spec(k)$ locally of finite type.\nIn this case\n\\begin{enumerate}\n\\item the action\n$$\n\\text{Gal}(\\overline{k}/k)^{opp} \\times \\text{IrredComp}(X_{\\overline{k}})\n\\longrightarrow\n\\text{IrredComp}(X_{\\overline{k}})\n$$\nis continuous if we give $\\text{IrredComp}(X_{\\overline{k}})$ the discrete\ntopology,\n\\item every irreducible component of $X_{\\overline{k}}$\ncan be defined over a finite extension of $k$, and\n\\item given any irreducible component $T \\subset X$ the scheme\n$T_{\\overline{k}}$ is a finite union of irreducible components of\n$X_{\\overline{k}}$ which are all in the same\n$\\text{Gal}(\\overline{k}/k)$-orbit.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KZ","source_file":"varieties.tex","source_line":1788,"source_end_line":1809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1788-L1809","statement_sha256":"a14c8aee599098843035a1bf417d849e3aa52ce05ef4b113c3b106f05f7a2d53","origin":"The Stacks Project","memory_eligible":false,"source_rank":6394,"rank":6394,"depth":21,"x":219.163,"y":1097.609,"cluster":"varieties-curves"},{"id":"stacks:054R","tag":"054R","title":"Geometrically irreducible schemes · Lemma 054R","summary":"Let k be a field. Let X → Spec(k) be locally of finite type. Assume X has finitely many irreducible components. Then there exists a finite separable extension k'/k such that every irreducible component of X_k' is geometrically irreducible over k'.","statement_latex":"Let $k$ be a field.\nLet $X \\to \\Spec(k)$ be locally of finite type.\nAssume $X$ has finitely many irreducible components.\nThen there exists a finite separable extension $k'/k$\nsuch that every irreducible component of $X_{k'}$\nis geometrically irreducible over $k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054R","source_file":"varieties.tex","source_line":1847,"source_end_line":1855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1847-L1855","statement_sha256":"46f8aa1fb4c8bdab54ef4e0000d97e982508d31b5e9bbbd92cac91dff257e6f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6395,"rank":6395,"depth":22,"x":272.503,"y":1165.361,"cluster":"varieties-curves"},{"id":"stacks:054S","tag":"054S","title":"Geometrically irreducible schemes · Lemma 054S","summary":"Let X be a scheme over the field k. Assume X has finitely many irreducible components which are all geometrically irreducible. Then X has finitely many connected components each of which is geometrically connected.","statement_latex":"Let $X$ be a scheme over the field $k$.\nAssume $X$ has finitely many irreducible components which are\nall geometrically irreducible.\nThen $X$ has finitely many connected components each of which is\ngeometrically connected.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically irreducible schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054S","source_file":"varieties.tex","source_line":1870,"source_end_line":1877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1870-L1877","statement_sha256":"90ad4f48a5d3554f4ed72af56ddc8b1b457c0f5ac2dab49d40b0bc015e108012","origin":"The Stacks Project","memory_eligible":false,"source_rank":6396,"rank":6396,"depth":0,"x":177.756,"y":1145.469,"cluster":"varieties-curves"},{"id":"stacks:020H","tag":"020H","title":"Geometrically integral schemes · Definition 020H","summary":"Let X be a scheme over the field k. • Let x ∈ X. We say X is geometrically pointwise integral at x if for every field extension k'/k and every x' ∈ X_k' lying over x the local ring O_X_k', x' is integral. • We say X is geometrically pointwise integral if X is geometrically pointwise integral at every point. • We say X is geometrically integral over k if the scheme X_k' is integral for every field extension k' of k.","statement_latex":"Let $X$ be a scheme over the field $k$.\n\\begin{enumerate}\n\\item Let $x \\in X$. We say $X$ is\n{\\it geometrically pointwise integral at $x$} if for every\nfield extension $k'/k$ and every $x' \\in X_{k'}$ lying over $x$\nthe local ring $\\mathcal{O}_{X_{k'}, x'}$ is integral.\n\\item We say $X$ is {\\it geometrically pointwise integral} if $X$\nis geometrically pointwise integral at every point.\n\\item We say $X$ is {\\it geometrically integral} over $k$ if the scheme\n$X_{k'}$ is integral for every field extension $k'$ of $k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically integral schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020H","source_file":"varieties.tex","source_line":1898,"source_end_line":1911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1898-L1911","statement_sha256":"ab3a3b0288a49ac130e2082cca9c62803cea5bafaeb3fa1137f674c51a4ea023","origin":"The Stacks Project","memory_eligible":false,"source_rank":6397,"rank":6397,"depth":0,"x":264.455,"y":1105.998,"cluster":"varieties-curves"},{"id":"stacks:038K","tag":"038K","title":"Geometrically integral schemes · Lemma 038K","summary":"Let k be a field. Let X be a scheme over k. Then X is geometrically integral over k if and only if X is both geometrically reduced and geometrically irreducible over k.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nThen $X$ is geometrically integral over $k$ if and only if\n$X$ is both geometrically reduced and geometrically irreducible\nover $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically integral schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038K","source_file":"varieties.tex","source_line":1919,"source_end_line":1926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1919-L1926","statement_sha256":"cb41ebfcd55607bd76e48e2f42ca81623d7b924a3f9018a860d5cd8fe38a434e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6398,"rank":6398,"depth":14,"x":231.956,"y":1185.045,"cluster":"varieties-curves"},{"id":"stacks:0BUG","tag":"0BUG","title":"Geometrically integral schemes · Lemma 0BUG","summary":"Let k be a field. Let X be a proper scheme over k. • A = H^0(X, O_X) is a finite dimensional k-algebra, • A = ∏_i = 1, …, n A_i is a product of Artinian local k-algebras, one factor for each connected component of X, • if X is reduced, then A = ∏_i = 1, …, n k_i is a product of fields, each a finite extension of k, • if X is geometrically reduced, then k_i is finite separable over k, • if X is geometrically connected, then A is geometrically irreducible over k, • if X is…","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\n\\begin{enumerate}\n\\item $A = H^0(X, \\mathcal{O}_X)$ is a finite dimensional $k$-algebra,\n\\item $A = \\prod_{i = 1, \\ldots, n} A_i$ is a product of Artinian\nlocal $k$-algebras, one factor for each connected component of $X$,\n\\item if $X$ is reduced, then $A = \\prod_{i = 1, \\ldots, n} k_i$\nis a product of fields, each a finite extension of $k$,\n\\item if $X$ is geometrically reduced, then $k_i$ is finite separable\nover $k$,\n\\item if $X$ is geometrically connected, then $A$ is geometrically\nirreducible over $k$,\n\\item if $X$ is geometrically irreducible, then $A$ is geometrically\nirreducible over $k$,\n\\item if $X$ is geometrically reduced and geometrically connected, then\n$A = k$, and\n\\item if $X$ is geometrically integral, then $A = k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically integral schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUG","source_file":"varieties.tex","source_line":1932,"source_end_line":1951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L1932-L1951","statement_sha256":"f92afa60bf377dd0e233f73c76723342de316d8f4f270b1f1ece17c468350ddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6399,"rank":6399,"depth":32,"x":191.984,"y":1107.596,"cluster":"varieties-curves"},{"id":"stacks:0FD1","tag":"0FD1","title":"Geometrically integral schemes · Lemma 0FD1","summary":"Let X be a proper scheme over a field k. Set A = H^0(X, O_X). The fibres of the canonical morphism X → Spec(A) are geometrically connected.","statement_latex":"Let $X$ be a proper scheme over a field $k$. Set\n$A = H^0(X, \\mathcal{O}_X)$. The fibres of the canonical\nmorphism $X \\to \\Spec(A)$ are geometrically connected.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically integral schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FD1","source_file":"varieties.tex","source_line":2012,"source_end_line":2017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2012-L2017","statement_sha256":"6107b832e8ef02daa5d07b8844361000f2ce775efecfc3a1025ad882f10ac41b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6400,"rank":6400,"depth":33,"x":284.584,"y":1142.344,"cluster":"varieties-curves"},{"id":"stacks:0FD2","tag":"0FD2","title":"Geometrically integral schemes · Lemma 0FD2","summary":"Let k be a field. Let X be a proper geometrically reduced scheme over k. The following are equivalent • H^0(X, O_X) = k, and • X is geometrically connected.","statement_latex":"Let $k$ be a field. Let $X$ be a proper geometrically reduced scheme over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $H^0(X, \\mathcal{O}_X) = k$, and\n\\item $X$ is geometrically connected.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically integral schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FD2","source_file":"varieties.tex","source_line":2074,"source_end_line":2082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2074-L2082","statement_sha256":"d6d4b29360d9496ccec50fb663e9776ed2c5087ede36dbc33196602caf9dc9f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6401,"rank":6401,"depth":34,"x":187.489,"y":1169.505,"cluster":"varieties-curves"},{"id":"stacks:038M","tag":"038M","title":"Geometrically normal schemes · Definition 038M","summary":"Let X be a scheme over the field k. • Let x ∈ X. We say X is geometrically normal at x if for every field extension k'/k and every x' ∈ X_k' lying over x the local ring O_X_k', x' is normal. • We say X is geometrically normal over k if X is geometrically normal at every x ∈ X.","statement_latex":"Let $X$ be a scheme over the field $k$.\n\\begin{enumerate}\n\\item Let $x \\in X$. We say $X$ is\n{\\it geometrically normal at $x$} if for every\nfield extension $k'/k$ and every $x' \\in X_{k'}$ lying over $x$\nthe local ring $\\mathcal{O}_{X_{k'}, x'}$ is normal.\n\\item We say $X$ is {\\it geometrically normal} over $k$ if $X$\nis geometrically normal at every $x \\in X$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically normal schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038M","source_file":"varieties.tex","source_line":2104,"source_end_line":2115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2104-L2115","statement_sha256":"edb8c1ada56e7e1e3fa82d2e47070680ffda5b151eb1df60470908a4d0cc3b4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6402,"rank":6402,"depth":0,"x":237.687,"y":1093.718,"cluster":"varieties-curves"},{"id":"stacks:038N","tag":"038N","title":"Geometrically normal schemes · Lemma 038N","summary":"Let k be a field. Let X be a scheme over k. Let x ∈ X. The following are equivalent • X is geometrically normal at x, • for every finite purely inseparable field extension k' of k and x' ∈ X_k' lying over x the local ring O_X_k', x' is normal, and • the ring O_X, x is geometrically normal over k (see Algebra, Definition [Tag 0380]).","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nLet $x \\in X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically normal at $x$,\n\\item for every finite purely inseparable field extension $k'$ of $k$\nand $x' \\in X_{k'}$ lying over $x$ the local ring\n$\\mathcal{O}_{X_{k'}, x'}$ is normal, and\n\\item the ring $\\mathcal{O}_{X, x}$ is geometrically\nnormal over $k$ (see\nAlgebra, Definition \\ref{algebra-definition-geometrically-normal}).\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038N","source_file":"varieties.tex","source_line":2117,"source_end_line":2132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2117-L2132","statement_sha256":"8ed78315318824492ffe90aa346a61b5673c02d4b190d07cc6835c8d79e3fe8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6403,"rank":6403,"depth":42,"x":261.823,"y":1178.822,"cluster":"varieties-curves"},{"id":"stacks:038O","tag":"038O","title":"Geometrically normal schemes · Lemma 038O","summary":"Let k be a field. Let X be a scheme over k. The following are equivalent • X is geometrically normal, • X_k' is a normal scheme for every field extension k'/k, • X_k' is a normal scheme for every finitely generated field extension k'/k, • X_k' is a normal scheme for every finite purely inseparable field extension k'/k, • for every affine open U ⊂ X the ring O_X(U) is geometrically normal (see Algebra, Definition [Tag 0380]), and • X_k^perf is a normal scheme.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically normal,\n\\item $X_{k'}$ is a normal scheme for every field extension $k'/k$,\n\\item $X_{k'}$ is a normal scheme for every finitely generated field\nextension $k'/k$,\n\\item $X_{k'}$ is a normal scheme for every finite purely inseparable\nfield extension $k'/k$,\n\\item for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis geometrically normal (see\nAlgebra, Definition \\ref{algebra-definition-geometrically-normal}), and\n\\item $X_{k^{perf}}$ is a normal scheme.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038O","source_file":"varieties.tex","source_line":2159,"source_end_line":2176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2159-L2176","statement_sha256":"d800f178fda947b198ad69621f288922fd68dba15a8145e4b83d0c5b80128938","origin":"The Stacks Project","memory_eligible":false,"source_rank":6404,"rank":6404,"depth":42,"x":174.851,"y":1129.341,"cluster":"varieties-curves"},{"id":"stacks:038P","tag":"038P","title":"Geometrically normal schemes · Lemma 038P","summary":"Let k be a field. Let X be a scheme over k. Let k'/k be a field extension. Let x ∈ X be a point, and let x' ∈ X_k' be a point lying over x. The following are equivalent • X is geometrically normal at x, • X_k' is geometrically normal at x'. In particular, X is geometrically normal over k if and only if X_k' is geometrically normal over k'.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nLet $k'/k$ be a field extension.\nLet $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically normal at $x$,\n\\item $X_{k'}$ is geometrically normal at $x'$.\n\\end{enumerate}\nIn particular, $X$ is geometrically normal over $k$ if and only if\n$X_{k'}$ is geometrically normal over $k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038P","source_file":"varieties.tex","source_line":2209,"source_end_line":2222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2209-L2222","statement_sha256":"dd8d65a8b4e2a79b5260507311362a24adad1af3b8011de042fdb43689277d2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6405,"rank":6405,"depth":43,"x":279.649,"y":1116.372,"cluster":"varieties-curves"},{"id":"stacks:06DG","tag":"06DG","title":"Geometrically normal schemes · Lemma 06DG","summary":"Let k be a field. Let X be a geometrically normal scheme over k and let Y be a normal scheme over k. Then X ×_k Y is a normal scheme.","statement_latex":"Let $k$ be a field. Let $X$ be a geometrically normal scheme over $k$\nand let $Y$ be a normal scheme over $k$. Then $X \\times_k Y$ is a normal\nscheme.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DG","source_file":"varieties.tex","source_line":2243,"source_end_line":2248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2243-L2248","statement_sha256":"ff1bddf75837133b8a6de9d7e0c8a7edae7e89d6e9c13be2d7f38b00205833ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":6406,"rank":6406,"depth":43,"x":212.246,"y":1185.97,"cluster":"varieties-curves"},{"id":"stacks:0C3M","tag":"0C3M","title":"Geometrically normal schemes · Lemma 0C3M","summary":"Let k be a field. Let X be a normal scheme over k. Let K/k be a separable field extension. Then X_K is a normal scheme.","statement_latex":"Let $k$ be a field. Let $X$ be a normal scheme over $k$. Let $K/k$\nbe a separable field extension. Then $X_K$ is a normal scheme.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3M","source_file":"varieties.tex","source_line":2257,"source_end_line":2261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2257-L2261","statement_sha256":"b38fad4c43d66fa063e3fe669bb69f603f25f72ebd650188d1dadfaed88c6858","origin":"The Stacks Project","memory_eligible":false,"source_rank":6407,"rank":6407,"depth":44,"x":205.931,"y":1095.674,"cluster":"varieties-curves"},{"id":"stacks:0FD3","tag":"0FD3","title":"Geometrically normal schemes · Lemma 0FD3","summary":"Let k be a field. Let X be a proper geometrically normal scheme over k. The following are equivalent • H^0(X, O_X) = k, • X is geometrically connected, • X is geometrically irreducible, and • X is geometrically integral.","statement_latex":"Let $k$ be a field. Let $X$ be a proper geometrically normal scheme over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $H^0(X, \\mathcal{O}_X) = k$,\n\\item $X$ is geometrically connected,\n\\item $X$ is geometrically irreducible, and\n\\item $X$ is geometrically integral.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FD3","source_file":"varieties.tex","source_line":2269,"source_end_line":2279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2269-L2279","statement_sha256":"0b2cd36c9fc9d1a8b624c07784adbb987bed54c6682d6bb623a1dc90aa0e942a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6408,"rank":6408,"depth":35,"x":283.818,"y":1159.179,"cluster":"varieties-curves"},{"id":"stacks:038R","tag":"038R","title":"Change of fields and locally Noetherian schemes · Lemma 038R","summary":"Let k be a field. Let X be a scheme over k. Let k'/k be a finitely generated field extension. Then X is locally Noetherian if and only if X_k' is locally Noetherian.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nLet $k'/k$ be a finitely generated field extension.\nThen $X$ is locally Noetherian if and only if $X_{k'}$ is locally\nNoetherian.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and locally Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038R","source_file":"varieties.tex","source_line":2314,"source_end_line":2321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2314-L2321","statement_sha256":"e008277356bbdecdcf9470b6889e4f40376a18ac65675e42b143279b92955edf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6409,"rank":6409,"depth":5,"x":174.461,"y":1156.524,"cluster":"varieties-curves"},{"id":"stacks:038T","tag":"038T","title":"Geometrically regular schemes · Definition 038T","summary":"Let k be a field. Let X be a locally Noetherian scheme over k. • Let x ∈ X. We say X is geometrically regular at x over k if for every finitely generated field extension k'/k and any x' ∈ X_k' lying over x the local ring O_X_k', x' is regular. • We say X is geometrically regular over k if X is geometrically regular at all of its points.","statement_latex":"Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$.\n\\begin{enumerate}\n\\item Let $x \\in X$. We say $X$ is {\\it geometrically regular at $x$}\nover $k$ if for every finitely generated field extension $k'/k$\nand any $x' \\in X_{k'}$ lying over $x$ the local ring\n$\\mathcal{O}_{X_{k'}, x'}$ is regular.\n\\item We say $X$ is {\\it geometrically regular over $k$} if\n$X$ is geometrically regular at all of its points.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically regular schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038T","source_file":"varieties.tex","source_line":2363,"source_end_line":2374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2363-L2374","statement_sha256":"fc57402213800495df819ffb36798013650f1c39ebe02b990d007900b4d14317","origin":"The Stacks Project","memory_eligible":false,"source_rank":6410,"rank":6410,"depth":0,"x":257.879,"y":1095.965,"cluster":"varieties-curves"},{"id":"stacks:038U","tag":"038U","title":"Geometrically regular schemes · Lemma 038U","summary":"Let k be a field. Let X be a locally Noetherian scheme over k. Let x ∈ X. The following are equivalent • X is geometrically regular at x, • for every finite purely inseparable field extension k' of k and x' ∈ X_k' lying over x the local ring O_X_k', x' is regular, and • the ring O_X, x is geometrically regular over k (see Algebra, Definition [Tag 0382]).","statement_latex":"Let $k$ be a field.\nLet $X$ be a locally Noetherian scheme over $k$.\nLet $x \\in X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically regular at $x$,\n\\item for every finite purely inseparable field extension $k'$ of $k$\nand $x' \\in X_{k'}$ lying over $x$ the local ring\n$\\mathcal{O}_{X_{k'}, x'}$ is regular, and\n\\item the ring $\\mathcal{O}_{X, x}$ is geometrically\nregular over $k$ (see\nAlgebra, Definition \\ref{algebra-definition-geometrically-regular}).\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically regular schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038U","source_file":"varieties.tex","source_line":2381,"source_end_line":2396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2381-L2396","statement_sha256":"db1fea7990674d10f67c7673927c2cdd68c8303286aa570726b9ccaf8533177c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6411,"rank":6411,"depth":42,"x":244.967,"y":1188.656,"cluster":"varieties-curves"},{"id":"stacks:038V","tag":"038V","title":"Geometrically regular schemes · Lemma 038V","summary":"Let k be a field. Let X be a locally Noetherian scheme over k. The following are equivalent • X is geometrically regular, • X_k' is a regular scheme for every finitely generated field extension k'/k, • X_k' is a regular scheme for every finite purely inseparable field extension k'/k, • for every affine open U ⊂ X the ring O_X(U) is geometrically regular (see Algebra, Definition [Tag 0382]), and • there exists an affine open covering X = ⋃ U_i such that each O_X(U_i) is…","statement_latex":"Let $k$ be a field.\nLet $X$ be a locally Noetherian scheme over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically regular,\n\\item $X_{k'}$ is a regular scheme for every finitely generated field\nextension $k'/k$,\n\\item $X_{k'}$ is a regular scheme for every finite purely inseparable\nfield extension $k'/k$,\n\\item for every affine open $U \\subset X$ the ring $\\mathcal{O}_X(U)$\nis geometrically regular (see\nAlgebra, Definition \\ref{algebra-definition-geometrically-regular}), and\n\\item there exists an affine open covering $X = \\bigcup U_i$ such that\neach $\\mathcal{O}_X(U_i)$ is geometrically regular over $k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically regular schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038V","source_file":"varieties.tex","source_line":2423,"source_end_line":2440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2423-L2440","statement_sha256":"4ba373f305f63cafa5995a0d0fd0be5cb7bec1cf78e8cf570d7754d68e296868","origin":"The Stacks Project","memory_eligible":false,"source_rank":6412,"rank":6412,"depth":42,"x":179.461,"y":1112.41,"cluster":"varieties-curves"},{"id":"stacks:038W","tag":"038W","title":"Geometrically regular schemes · Lemma 038W","summary":"Let k be a field. Let X be a scheme over k. Let k'/k be a finitely generated field extension. Let x ∈ X be a point, and let x' ∈ X_k' be a point lying over x. The following are equivalent • X is geometrically regular at x, • X_k' is geometrically regular at x'. In particular, X is geometrically regular over k if and only if X_k' is geometrically regular over k'.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$.\nLet $k'/k$ be a finitely generated field extension.\nLet $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying over $x$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically regular at $x$,\n\\item $X_{k'}$ is geometrically regular at $x'$.\n\\end{enumerate}\nIn particular, $X$ is geometrically regular over $k$ if and only if\n$X_{k'}$ is geometrically regular over $k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically regular schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038W","source_file":"varieties.tex","source_line":2470,"source_end_line":2483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2470-L2483","statement_sha256":"81b24ba9601d3ebf02b543261cc11a79a8271451b66f7a8e67f830ea226eea2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6413,"rank":6413,"depth":43,"x":289.892,"y":1131.606,"cluster":"varieties-curves"},{"id":"stacks:05AW","tag":"05AW","title":"Geometrically regular schemes · Lemma 05AW","summary":"Let k be a field. Let f : X → Y be a morphism of locally Noetherian schemes over k. Let x ∈ X be a point and set y = f(x). If X is geometrically regular at x and f is flat at x then Y is geometrically regular at y. In particular, if X is geometrically regular over k and f is flat and surjective, then Y is geometrically regular over k.","statement_latex":"Let $k$ be a field.\nLet $f : X \\to Y$ be a morphism of locally Noetherian schemes over $k$.\nLet $x \\in X$ be a point and set $y = f(x)$.\nIf $X$ is geometrically regular at $x$ and\n$f$ is flat at $x$ then $Y$ is geometrically regular at $y$.\nIn particular, if $X$ is geometrically regular over $k$ and\n$f$ is flat and surjective, then $Y$ is geometrically regular over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically regular schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AW","source_file":"varieties.tex","source_line":2510,"source_end_line":2519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2510-L2519","statement_sha256":"2be0af589ff6888c7f395a0c0939c7b86b6327eacef3f37662c68c7dd8b338fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":6414,"rank":6414,"depth":43,"x":192.316,"y":1180.464,"cluster":"varieties-curves"},{"id":"stacks:038X","tag":"038X","title":"Geometrically regular schemes · Lemma 038X","summary":"Let k be a field. Let X be a scheme locally of finite type over k. Let x ∈ X. Then X is geometrically regular at x if and only if X → Spec(k) is smooth at x (Morphisms, Definition [Tag 01V5]).","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme locally of finite type over $k$.\nLet $x \\in X$.\nThen $X$ is geometrically regular at $x$ if and only if $X \\to \\Spec(k)$\nis smooth at $x$ (Morphisms, Definition \\ref{morphisms-definition-smooth}).","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Geometrically regular schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/038X","source_file":"varieties.tex","source_line":2543,"source_end_line":2550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2543-L2550","statement_sha256":"67fecd10b391b7ddb18a197cee72fc7f45b0fdb7e5fd53d3c8895dc571a47e6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6415,"rank":6415,"depth":38,"x":225.214,"y":1088.413,"cluster":"varieties-curves"},{"id":"stacks:045P","tag":"045P","title":"Change of fields and the Cohen-Macaulay property · Lemma 045P","summary":"Let X be a locally Noetherian scheme over the field k. Let k'/k be a finitely generated field extension. Let x ∈ X be a point, and let x' ∈ X_k' be a point lying over x. Then we have O_X, x is Cohen-Macaulay ⇔ O_X_k', x' is Cohen-Macaulay If X is locally of finite type over k, the same holds for any field extension k'/k.","statement_latex":"Let $X$ be a locally Noetherian scheme over the field $k$.\nLet $k'/k$ be a finitely generated field extension.\nLet $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying\nover $x$. Then we have\n$$\n\\mathcal{O}_{X, x}\\text{ is Cohen-Macaulay}\n\\Leftrightarrow\n\\mathcal{O}_{X_{k'}, x'}\\text{ is Cohen-Macaulay}\n$$\nIf $X$ is locally of finite type over $k$, the same holds for any\nfield extension $k'/k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and the Cohen-Macaulay property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045P","source_file":"varieties.tex","source_line":2633,"source_end_line":2646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2633-L2646","statement_sha256":"341c73cb9d6d05a976b87def7993d553a413d2df75607b9b9202a44334c22e05","origin":"The Stacks Project","memory_eligible":false,"source_rank":6416,"rank":6416,"depth":31,"x":275.331,"y":1175.572,"cluster":"varieties-curves"},{"id":"stacks:0478","tag":"0478","title":"Change of fields and the Jacobson property · Lemma 0478","summary":"Let X be a scheme which is locally of finite type over k. Then • for any closed point x ∈ X the extension kappa(x)/k is algebraic, and • X is a Jacobson scheme (Properties, Definition [Tag 01P2]).","statement_latex":"Let $X$ be a scheme which is locally of finite type over $k$.\nThen\n\\begin{enumerate}\n\\item for any closed point $x \\in X$ the extension $\\kappa(x)/k$\nis algebraic, and\n\\item $X$ is a Jacobson scheme\n(Properties, Definition \\ref{properties-definition-jacobson}).\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and the Jacobson property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0478","source_file":"varieties.tex","source_line":2669,"source_end_line":2679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2669-L2679","statement_sha256":"33b4a14be75ecf195faa5ba2256c39d944b6cbd75afd07bc620eebdefcac049f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6417,"rank":6417,"depth":17,"x":167.536,"y":1139.488,"cluster":"varieties-curves"},{"id":"stacks:0479","tag":"0479","title":"Change of fields and the Jacobson property · Lemma 0479","summary":"Let X be a scheme over a field k. For any field extension K/k whose cardinality is large enough we have • for any closed point x ∈ X_K the extension kappa(x)/K is algebraic, and • X_K is a Jacobson scheme (Properties, Definition [Tag 01P2]).","statement_latex":"Let $X$ be a scheme over a field $k$.\nFor any field extension $K/k$ whose cardinality is large enough\nwe have\n\\begin{enumerate}\n\\item for any closed point $x \\in X_K$ the extension $\\kappa(x)/K$\nis algebraic, and\n\\item $X_K$ is a Jacobson scheme\n(Properties, Definition \\ref{properties-definition-jacobson}).\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and the Jacobson property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0479","source_file":"varieties.tex","source_line":2699,"source_end_line":2710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2699-L2710","statement_sha256":"168c2815b4e73a538f0d9f9b5e2c525a49ffab06509f7d7c3f7bc6e5b18d4c0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6418,"rank":6418,"depth":9,"x":276.79,"y":1104.694,"cluster":"varieties-curves"},{"id":"stacks:0BDC","tag":"0BDC","title":"Change of fields and ample invertible sheaves · Lemma 0BDC","summary":"Let k be a field. Let X be a scheme over k. If there exists an ample invertible sheaf on X_K for some field extension K/k, then X has an ample invertible sheaf.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$.\nIf there exists an ample invertible sheaf on $X_K$ for some\nfield extension $K/k$, then $X$ has an ample invertible\nsheaf.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDC","source_file":"varieties.tex","source_line":2739,"source_end_line":2745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2739-L2745","statement_sha256":"e956866dd714704e81d8d6513fae9b0f7c5f30611b83295d4b43e3ffdcf39691","origin":"The Stacks Project","memory_eligible":false,"source_rank":6419,"rank":6419,"depth":36,"x":223.849,"y":1192.94,"cluster":"varieties-curves"},{"id":"stacks:0BDD","tag":"0BDD","title":"Change of fields and ample invertible sheaves · Lemma 0BDD","summary":"Let k be a field. Let X be a scheme over k. If X_K is quasi-affine for some field extension K/k, then X is quasi-affine.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is quasi-affine\nfor some field extension $K/k$, then $X$ is quasi-affine.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDD","source_file":"varieties.tex","source_line":2782,"source_end_line":2786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2782-L2786","statement_sha256":"3fae8f38589a6f133fc493dd3a53ee289badfdb795747072a2a7fcfeb5a6f4da","origin":"The Stacks Project","memory_eligible":false,"source_rank":6420,"rank":6420,"depth":30,"x":191.71,"y":1097.189,"cluster":"varieties-curves"},{"id":"stacks:0BDE","tag":"0BDE","title":"Change of fields and ample invertible sheaves · Lemma 0BDE","summary":"Let k be a field. Let X be a scheme over k. If X_K is quasi-projective over K for some field extension K/k, then X is quasi-projective over k.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is quasi-projective\nover $K$ for some field extension $K/k$, then $X$ is quasi-projective\nover $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDE","source_file":"varieties.tex","source_line":2816,"source_end_line":2821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2816-L2821","statement_sha256":"7276c8a5c42f099c91ad279e9405a1ade368966e21f58027e6c4cc1eaaa5b6a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6421,"rank":6421,"depth":37,"x":293.074,"y":1149.906,"cluster":"varieties-curves"},{"id":"stacks:0BDF","tag":"0BDF","title":"Change of fields and ample invertible sheaves · Lemma 0BDF","summary":"Let k be a field. Let X be a scheme over k. If X_K is proper over K for some field extension K/k, then X is proper over k.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is proper\nover $K$ for some field extension $K/k$, then $X$ is proper\nover $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDF","source_file":"varieties.tex","source_line":2848,"source_end_line":2853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2848-L2853","statement_sha256":"5b276e4eecd83fe3f40614f1fd037657fcd20febf46596dc52fa0ab5477394fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6422,"rank":6422,"depth":31,"x":175.17,"y":1168.67,"cluster":"varieties-curves"},{"id":"stacks:0BDG","tag":"0BDG","title":"Change of fields and ample invertible sheaves · Lemma 0BDG","summary":"Let k be a field. Let X be a scheme over k. If X_K is projective over K for some field extension K/k, then X is projective over k.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$. If $X_K$ is projective\nover $K$ for some field extension $K/k$, then $X$ is projective\nover $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Change of fields and ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDG","source_file":"varieties.tex","source_line":2892,"source_end_line":2897,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2892-L2897","statement_sha256":"cf74e6f004cdc426a93e24f6cd91bd13d22861018837523950e09f2c27148534","origin":"The Stacks Project","memory_eligible":false,"source_rank":6423,"rank":6423,"depth":38,"x":247.486,"y":1087.412,"cluster":"varieties-curves"},{"id":"stacks:0B29","tag":"0B29","title":"Tangent spaces · Definition 0B29","summary":"For any ring R the dual numbers over R is the R-algebra denoted R[ε]. As an R-module it is free with basis 1, ε and the R-algebra structure comes from setting ε^2 = 0.","statement_latex":"For any ring $R$ the {\\it dual numbers} over $R$ is the\n$R$-algebra denoted $R[\\epsilon]$. As an $R$-module it is free with\nbasis $1$, $\\epsilon$ and the $R$-algebra structure comes from setting\n$\\epsilon^2 = 0$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Tangent spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B29","source_file":"varieties.tex","source_line":2918,"source_end_line":2924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2918-L2924","statement_sha256":"543408639809ed2870eb729c9f597950ceda662716f07a0d9c044bd5f685b84c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6424,"rank":6424,"depth":0,"x":259.582,"y":1189.007,"cluster":"varieties-curves"},{"id":"stacks:0B2B","tag":"0B2B","title":"Tangent spaces · Lemma 0B2B","summary":"The set of dotted arrows making ([Tag 0B2A]) commute has a canonical kappa(x)-vector space structure.","statement_latex":"The set of dotted arrows making (\\ref{equation-tangent-space}) commute\nhas a canonical $\\kappa(x)$-vector space structure.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2B","source_file":"varieties.tex","source_line":2946,"source_end_line":2950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2946-L2950","statement_sha256":"15e739d49eabd630858f76f0b2b0086c0b12401974888f7d66fd60507357e0d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6425,"rank":6425,"depth":12,"x":168.394,"y":1120.527,"cluster":"varieties-curves"},{"id":"stacks:0B2C","tag":"0B2C","title":"Tangent spaces · Definition 0B2C","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. The set of dotted arrows making ([Tag 0B2A]) commute with its canonical kappa(x)-vector space structure is called the tangent space of X over S at x and we denote it T_X/S, x. An element of this space is called a tangent vector of X/S at x.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$. The set of\ndotted arrows making (\\ref{equation-tangent-space}) commute with\nits canonical $\\kappa(x)$-vector space structure is called\nthe {\\it tangent space of $X$ over $S$ at $x$} and we denote it $T_{X/S, x}$.\nAn element of this space is called a {\\it tangent vector} of $X/S$ at $x$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Tangent spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2C","source_file":"varieties.tex","source_line":2987,"source_end_line":2994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L2987-L2994","statement_sha256":"d1af033db4e5575e9e425d69ee0016e948602eaae95db76089280e30683d8f8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6426,"rank":6426,"depth":0,"x":291.464,"y":1119.275,"cluster":"varieties-curves"},{"id":"stacks:0B2D","tag":"0B2D","title":"Tangent spaces · Lemma 0B2D","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. There is a canonical isomorphism T_X/S, x = Hom_O_X, x(Ω_X/S, x, kappa(x)) of vector spaces over kappa(x).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$.\nThere is a canonical isomorphism\n$$\nT_{X/S, x} = \\Hom_{\\mathcal{O}_{X, x}}(\\Omega_{X/S, x}, \\kappa(x))\n$$\nof vector spaces over $\\kappa(x)$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2D","source_file":"varieties.tex","source_line":3013,"source_end_line":3021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3013-L3021","statement_sha256":"fd3f84a073ac1c26061170ed8ed8af7c6e139bfacb4e7c0072daab53e57b9ded","origin":"The Stacks Project","memory_eligible":false,"source_rank":6427,"rank":6427,"depth":17,"x":201.169,"y":1190.464,"cluster":"varieties-curves"},{"id":"stacks:0B2E","tag":"0B2E","title":"Tangent spaces · Lemma 0B2E","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point and let s = f(x) ∈ S. Assume that kappa(x) = kappa(s). Then there are canonical isomorphisms m_x/( m_x^2 + m_sO_X, x) = Ω_X/S, x ⊗_O_X, x kappa(x) and T_X/S, x = Hom_kappa(x)( m_x/( m_x^2 + m_sO_X, x), kappa(x)) This works more generally if kappa(x)/kappa(s) is a separable algebraic extension.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point and let $s = f(x) \\in S$.\nAssume that $\\kappa(x) = \\kappa(s)$. Then there are canonical isomorphisms\n$$\n\\mathfrak m_x/(\\mathfrak m_x^2 + \\mathfrak m_s\\mathcal{O}_{X, x})\n=\n\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)\n$$\nand\n$$\nT_{X/S, x} =\n\\Hom_{\\kappa(x)}(\n\\mathfrak m_x/(\\mathfrak m_x^2 + \\mathfrak m_s\\mathcal{O}_{X, x}),\n\\kappa(x))\n$$\nThis works more generally if $\\kappa(x)/\\kappa(s)$ is a separable\nalgebraic extension.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2E","source_file":"varieties.tex","source_line":3058,"source_end_line":3077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3058-L3077","statement_sha256":"edc0045dfd3f92e19dceea0de5d7fd9ca402a846d6fee05c7eabaad6d9710c11","origin":"The Stacks Project","memory_eligible":false,"source_rank":6428,"rank":6428,"depth":18,"x":210.561,"y":1086.107,"cluster":"varieties-curves"},{"id":"stacks:0B2F","tag":"0B2F","title":"Tangent spaces · Lemma 0B2F","summary":"Let f : X → Y be a morphism of schemes over a base scheme S. Let x ∈ X be a point. Set y = f(x). If kappa(y) = kappa(x), then f induces a natural linear map df : T_X/S, x → T_Y/S, y which is dual to the linear map Ω_Y/S, y ⊗ kappa(y) → Ω_X/S, x ⊗ kappa(x) via the identifications of Lemma [Tag 0B2D].","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes over a base scheme $S$.\nLet $x \\in X$ be a point. Set $y = f(x)$. If $\\kappa(y) = \\kappa(x)$,\nthen $f$ induces a natural linear map\n$$\n\\text{d}f : T_{X/S, x} \\longrightarrow T_{Y/S, y}\n$$\nwhich is dual to the linear map\n$\\Omega_{Y/S, y} \\otimes \\kappa(y) \\to \\Omega_{X/S, x} \\otimes \\kappa(x)$\nvia the identifications of Lemma \\ref{lemma-tangent-space-cotangent-space}.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2F","source_file":"varieties.tex","source_line":3107,"source_end_line":3118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3107-L3118","statement_sha256":"b670d8809b5944607539dd2221e9301492570aedb7934b1fc0a8b350a5c65647","origin":"The Stacks Project","memory_eligible":false,"source_rank":6429,"rank":6429,"depth":18,"x":288.016,"y":1168.881,"cluster":"varieties-curves"},{"id":"stacks:0BEB","tag":"0BEB","title":"Tangent spaces · Lemma 0BEB","summary":"Let X, Y be schemes over a base S. Let x ∈ X and y ∈ Y with the same image point s ∈ S such that kappa(s) = kappa(x) and kappa(s) = kappa(y). There is a canonical isomorphism T_X ×_S Y/S, (x, y) = T_X/S, x ⊕ T_Y/S, y The map from left to right is induced by the maps on tangent spaces coming from the projections X ×_S Y → X and X ×_S Y → Y. The map from right to left is induced by the maps 1 × y : X_s → X_s ×_s Y_s and x × 1 : Y_s → X_s ×_s Y_s via the identification ([Tag…","statement_latex":"Let $X$, $Y$ be schemes over a base $S$. Let $x \\in X$ and $y \\in Y$ with\nthe same image point $s \\in S$ such that $\\kappa(s) = \\kappa(x)$ and\n$\\kappa(s) = \\kappa(y)$. There is a canonical isomorphism\n$$\nT_{X \\times_S Y/S, (x, y)} = T_{X/S, x} \\oplus T_{Y/S, y}\n$$\nThe map from left to right is induced by the maps on tangent spaces coming\nfrom the projections $X \\times_S Y \\to X$ and $X \\times_S Y \\to Y$.\nThe map from right to left is induced by the maps\n$1 \\times y : X_s \\to X_s \\times_s Y_s$ and\n$x \\times 1 : Y_s \\to X_s \\times_s Y_s$ via the identification\n(\\ref{equation-tangent-space-fibre}) of\ntangent spaces with tangent spaces of fibres.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEB","source_file":"varieties.tex","source_line":3124,"source_end_line":3139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3124-L3139","statement_sha256":"2c9771986f3c16d76d712f0800c99a4de59f38b3f0481a23498beb352d0bac1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6430,"rank":6430,"depth":19,"x":163.607,"y":1151.692,"cluster":"varieties-curves"},{"id":"stacks:0B2G","tag":"0B2G","title":"Tangent spaces · Lemma 0B2G","summary":"Let f : X → Y be a morphism of schemes locally of finite type over a base scheme S. Let x ∈ X be a point. Set y = f(x) and assume that kappa(y) = kappa(x). Then the following are equivalent • df : T_X/S, x → T_Y/S, y is injective, and • f is unramified at x.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes locally of finite type over a\nbase scheme $S$. Let $x \\in X$ be a point. Set $y = f(x)$ and assume\nthat $\\kappa(y) = \\kappa(x)$. Then the following are equivalent\n\\begin{enumerate}\n\\item $\\text{d}f : T_{X/S, x} \\longrightarrow T_{Y/S, y}$ is injective, and\n\\item $f$ is unramified at $x$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2G","source_file":"varieties.tex","source_line":3148,"source_end_line":3157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3148-L3157","statement_sha256":"fd97123777ee13742822aeb909f3bac724f06af9c7935b68f9a3071fa3665a90","origin":"The Stacks Project","memory_eligible":false,"source_rank":6431,"rank":6431,"depth":43,"x":269.785,"y":1093.436,"cluster":"varieties-curves"},{"id":"stacks:0AB6","tag":"0AB6","title":"Generically finite morphisms · Lemma 0AB6","summary":"Let f : X → Y be locally of finite type. Let y ∈ Y be a point such that O_Y, y is Noetherian of dimension ≤ 1. Assume in addition one of the following conditions is satisfied • for every generic point eta of an irreducible component of X the field extension kappa(eta)/kappa(f(eta)) is finite (or algebraic), • for every generic point eta of an irreducible component of X such that f(eta) leadsto y the field extension kappa(eta)/kappa(f(eta)) is finite (or algebraic), • f is…","statement_latex":"Let $f : X \\to Y$ be locally of finite type. Let $y \\in Y$ be a point\nsuch that $\\mathcal{O}_{Y, y}$ is Noetherian of dimension $\\leq 1$.\nAssume in addition one of the following conditions is satisfied\n\\begin{enumerate}\n\\item for every generic point $\\eta$ of an irreducible component\nof $X$ the field extension $\\kappa(\\eta)/\\kappa(f(\\eta))$\nis finite (or algebraic),\n\\item for every generic point $\\eta$ of an irreducible component\nof $X$ such that $f(\\eta) \\leadsto y$ the field extension\n$\\kappa(\\eta)/\\kappa(f(\\eta))$ is finite (or algebraic),\n\\item $f$ is quasi-finite at every generic point of an\nirreducible component of $X$,\n\\item $Y$ is locally Noetherian and $f$\nis quasi-finite at a dense set of points of $X$,\n\\item add more here.\n\\end{enumerate}\nThen $f$ is quasi-finite at every point of $X$ lying over $y$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AB6","source_file":"varieties.tex","source_line":3187,"source_end_line":3206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3187-L3206","statement_sha256":"11171c30c74927315efb84a20f3c8ab0f40bc6070b59664f04bf7c645dbc4c0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6432,"rank":6432,"depth":30,"x":238.155,"y":1197.237,"cluster":"varieties-curves"},{"id":"stacks:0AB7","tag":"0AB7","title":"Generically finite morphisms · Lemma 0AB7","summary":"Let f : X → Y be a proper morphism. Let y ∈ Y be a point such that O_Y, y is Noetherian of dimension ≤ 1. Assume in addition one of the following conditions is satisfied • for every generic point eta of an irreducible component of X the field extension kappa(eta)/kappa(f(eta)) is finite (or algebraic), • for every generic point eta of an irreducible component of X such that f(eta) leadsto y the field extension kappa(eta)/kappa(f(eta)) is finite (or algebraic), • f is…","statement_latex":"Let $f : X \\to Y$ be a proper morphism. Let $y \\in Y$ be a point\nsuch that $\\mathcal{O}_{Y, y}$ is Noetherian of dimension $\\leq 1$.\nAssume in addition one of the following conditions is satisfied\n\\begin{enumerate}\n\\item for every generic point $\\eta$ of an irreducible component\nof $X$ the field extension $\\kappa(\\eta)/\\kappa(f(\\eta))$\nis finite (or algebraic),\n\\item for every generic point $\\eta$ of an irreducible component\nof $X$ such that $f(\\eta) \\leadsto y$ the field extension\n$\\kappa(\\eta)/\\kappa(f(\\eta))$ is finite (or algebraic),\n\\item $f$ is quasi-finite at every generic point of $X$,\n\\item $Y$ is locally Noetherian and $f$\nis quasi-finite at a dense set of points of $X$,\n\\item add more here.\n\\end{enumerate}\nThen there exists an open neighbourhood $V \\subset Y$ of $y$ such that\n$f^{-1}(V) \\to V$ is finite.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AB7","source_file":"varieties.tex","source_line":3267,"source_end_line":3286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3267-L3286","statement_sha256":"98a43e2201a780bb14575d5160accb4e5b07a7ff09ca7daa9813adf0886c560f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6433,"rank":6433,"depth":41,"x":177.663,"y":1102.208,"cluster":"varieties-curves"},{"id":"stacks:0BFP","tag":"0BFP","title":"Generically finite morphisms · Lemma 0BFP","summary":"Let X be a Noetherian scheme. Let f : Y → X be a birational proper morphism of schemes with Y reduced. Let U ⊂ X be the maximal open over which f is an isomorphism. Then U contains • every point of codimension 0 in X, • every x ∈ X of codimension 1 on X such that O_X, x is a discrete valuation ring, • every x ∈ X such that the fibre of Y → X over x is finite and such that O_X, x is normal, and • every x ∈ X such that f is quasi-finite at some y ∈ Y lying over x and O_X, x…","statement_latex":"Let $X$ be a Noetherian scheme. Let $f : Y \\to X$ be a birational proper\nmorphism of schemes with $Y$ reduced. Let $U \\subset X$ be the\nmaximal open over which $f$ is an isomorphism. Then $U$ contains\n\\begin{enumerate}\n\\item every point of codimension $0$ in $X$,\n\\item every $x \\in X$ of codimension $1$ on $X$ such that\n$\\mathcal{O}_{X, x}$ is a discrete valuation ring,\n\\item every $x \\in X$ such that the fibre of $Y \\to X$ over $x$ is\nfinite and such that $\\mathcal{O}_{X, x}$ is normal, and\n\\item every $x \\in X$ such that $f$ is quasi-finite at some\n$y \\in Y$ lying over $x$ and $\\mathcal{O}_{X, x}$ is normal.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFP","source_file":"varieties.tex","source_line":3299,"source_end_line":3313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3299-L3313","statement_sha256":"7e4a6dc91c1d74ad18a3bf0d50584440f569f975819fd2be86b0250fe2112caf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6434,"rank":6434,"depth":42,"x":299.371,"y":1138.16,"cluster":"varieties-curves"},{"id":"stacks:0CBH","tag":"0CBH","title":"Variants of Noether normalization · Lemma 0CBH","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X with image s ∈ S. Let V ⊂ S be an affine open neighbourhood of s. If f is locally of finite type and dim_x(X_s) = d, then there exists an affine open U ⊂ X with x ∈ U and f(U) ⊂ V and a factorization U xrightarrowπ A^d_V → V of f|_U : U → V such that π is quasi-finite.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$ with\nimage $s \\in S$. Let $V \\subset S$ be an affine open neighbourhood\nof $s$. If $f$ is locally of finite type and $\\dim_x(X_s) = d$,\nthen there exists an affine open $U \\subset X$ with\n$x \\in U$ and $f(U) \\subset V$ and a factorization\n$$\nU \\xrightarrow{\\pi} \\mathbf{A}^d_V \\to V\n$$\nof $f|_U : U \\to V$ such that $\\pi$ is quasi-finite.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Variants of Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBH","source_file":"varieties.tex","source_line":3397,"source_end_line":3408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3397-L3408","statement_sha256":"6926938fa852919f03b61b9da0d94e544e568b322348878f6e47089dfbace914","origin":"The Stacks Project","memory_eligible":false,"source_rank":6435,"rank":6435,"depth":30,"x":180.051,"y":1180.946,"cluster":"varieties-curves"},{"id":"stacks:0CBI","tag":"0CBI","title":"Variants of Noether normalization · Lemma 0CBI","summary":"Let f : X → S be a finite type morphism of affine schemes. Let s ∈ S. If dim(X_s) = d, then there exists a factorization X xrightarrowπ A^d_S → S of f such that the morphism π_s : X_s → A^d_kappa(s) of fibres over s is finite.","statement_latex":"Let $f : X \\to S$ be a finite type morphism of affine schemes.\nLet $s \\in S$. If $\\dim(X_s) = d$, then there exists a factorization\n$$\nX \\xrightarrow{\\pi} \\mathbf{A}^d_S \\to S\n$$\nof $f$ such that the morphism $\\pi_s : X_s \\to \\mathbf{A}^d_{\\kappa(s)}$\nof fibres over $s$ is finite.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Variants of Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBI","source_file":"varieties.tex","source_line":3415,"source_end_line":3424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3415-L3424","statement_sha256":"4e249f4837a0a6bd9fd581a0eef678afdeefc57cf633dc04eb5dca7d498dc058","origin":"The Stacks Project","memory_eligible":false,"source_rank":6436,"rank":6436,"depth":11,"x":233.925,"y":1081.144,"cluster":"varieties-curves"},{"id":"stacks:0CBJ","tag":"0CBJ","title":"Variants of Noether normalization · Lemma 0CBJ","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. Let V = Spec(A) be an affine open neighbourhood of f(x) in S. If f is unramified at x, then there exist exists an affine open U ⊂ X with x ∈ U and f(U) ⊂ V such that we have a commutative diagram xymatrix X ar[d] & U ar[l] ar[rd] ar[r]^-j & Spec(A[t]_g'/(g)) ar[d] ar[r] & Spec(A[t]) = A^1_V ar[ld] Y & & V ar[ll] where j is an immersion, g ∈ A[t] is a monic polynomial, and g' is the derivative of g with respect to t. If f…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $x \\in X$.\nLet $V = \\Spec(A)$ be an affine open neighbourhood of $f(x)$ in $S$.\nIf $f$ is unramified at $x$, then there exist exists an affine open\n$U \\subset X$ with $x \\in U$ and $f(U) \\subset V$\nsuch that we have a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] & U \\ar[l] \\ar[rd] \\ar[r]^-j &\n\\Spec(A[t]_{g'}/(g)) \\ar[d] \\ar[r] &\n\\Spec(A[t]) = \\mathbf{A}^1_V \\ar[ld] \\\\\nY & & V \\ar[ll]\n}\n$$\nwhere $j$ is an immersion, $g \\in A[t]$ is a monic polynomial, and\n$g'$ is the derivative of $g$ with respect to $t$. If $f$ is \\'etale\nat $x$, then we may choose the diagram such that $j$ is an open immersion.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Variants of Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBJ","source_file":"varieties.tex","source_line":3440,"source_end_line":3458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3440-L3458","statement_sha256":"e4c9812276ae39e488eea0e8ac1331bb48e931907832d326920f46df225db216","origin":"The Stacks Project","memory_eligible":false,"source_rank":6437,"rank":6437,"depth":44,"x":274.677,"y":1185.875,"cluster":"varieties-curves"},{"id":"stacks:0CBK","tag":"0CBK","title":"Variants of Noether normalization · Lemma 0CBK","summary":"Let f : X → S be a finite type morphism of affine schemes. Let x ∈ X with image s ∈ S. Let r = dim_kappa(x) Ω_X/S, x ⊗_O_X, x kappa(x) = dim_kappa(x) Ω_X_s/s, x ⊗_O_X_s, x kappa(x) = dim_kappa(x) T_X/S, x Then there exists a factorization X xrightarrowπ A^r_S → S of f such that π is unramified at x.","statement_latex":"Let $f : X \\to S$ be a finite type morphism of affine schemes.\nLet $x \\in X$ with image $s \\in S$. Let\n$$\nr =\n\\dim_{\\kappa(x)} \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) =\n\\dim_{\\kappa(x)} \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =\n\\dim_{\\kappa(x)} T_{X/S, x}\n$$\nThen there exists a factorization\n$$\nX \\xrightarrow{\\pi} \\mathbf{A}^r_S \\to S\n$$\nof $f$ such that $\\pi$ is unramified at $x$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Variants of Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBK","source_file":"varieties.tex","source_line":3470,"source_end_line":3485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3470-L3485","statement_sha256":"f61aceb58701bddb43ccff6df05b8f440cbb66060ed5293dc0ff22619eefb5d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6438,"rank":6438,"depth":18,"x":159.79,"y":1131.478,"cluster":"varieties-curves"},{"id":"stacks:0CBL","tag":"0CBL","title":"Variants of Noether normalization · Lemma 0CBL","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X with image s ∈ S. Let V ⊂ S be an affine open neighbourhood of s. If f is locally of finite type and r = dim_kappa(x) Ω_X/S, x ⊗_O_X, x kappa(x) = dim_kappa(x) Ω_X_s/s, x ⊗_O_X_s, x kappa(x) = dim_kappa(x) T_X/S, x then there exist • an affine open U ⊂ X with x ∈ U and f(U) ⊂ V and a factorization U xrightarrowj A^r + 1_V → V of f|_U such that j is an immersion, or • an affine open U ⊂ X with x ∈ U and f(U) ⊂ V and a…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ with image $s \\in S$. Let $V \\subset S$ be\nan affine open neighbourhood of $s$. If $f$ is locally of\nfinite type and\n$$\nr =\n\\dim_{\\kappa(x)} \\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x) =\n\\dim_{\\kappa(x)} \\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =\n\\dim_{\\kappa(x)} T_{X/S, x}\n$$\nthen there exist\n\\begin{enumerate}\n\\item an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ and a\nfactorization\n$$\nU \\xrightarrow{j} \\mathbf{A}^{r + 1}_V \\to V\n$$\nof $f|_U$ such that $j$ is an immersion, or\n\\item an affine open $U \\subset X$ with $x \\in U$ and $f(U) \\subset V$ and a\nfactorization\n$$\nU \\xrightarrow{j} D \\to V\n$$\nof $f|_U$ such that $j$ is a closed immersion and $D \\to V$\nis smooth of relative dimension $r$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Variants of Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBL","source_file":"varieties.tex","source_line":3520,"source_end_line":3548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3520-L3548","statement_sha256":"4fd4031b6a3b02a47b2730cf757fb3be34307a7d191f659f8e5ad8c0fa04fd5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6439,"rank":6439,"depth":45,"x":288.936,"y":1106.267,"cluster":"varieties-curves"},{"id":"stacks:0B2I","tag":"0B2I","title":"Dimension of fibres · Lemma 0B2I","summary":"Let f : X → Y be locally of finite type. Let x ∈ X be a point with image y ∈ Y such that O_Y, y is Noetherian of dimension ≤ 1. Let d ≥ 0 be an integer such that for every generic point eta of an irreducible component of X which contains x, we have dim_eta(X_f(eta)) = d. Then dim_x(X_y) = d.","statement_latex":"Let $f : X \\to Y$ be locally of finite type. Let $x \\in X$ be a point\nwith image $y \\in Y$ such that $\\mathcal{O}_{Y, y}$ is Noetherian of\ndimension $\\leq 1$. Let $d \\geq 0$ be an integer such that for every\ngeneric point $\\eta$ of an irreducible component of $X$ which contains\n$x$, we have $\\dim_\\eta(X_{f(\\eta)}) = d$. Then $\\dim_x(X_y) = d$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2I","source_file":"varieties.tex","source_line":3596,"source_end_line":3603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3596-L3603","statement_sha256":"ef8d2d2cdaed0bc6f7ca967bccde49cedbfcfe01668750d340ce1c422a0488b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6440,"rank":6440,"depth":32,"x":213.583,"y":1198.623,"cluster":"varieties-curves"},{"id":"stacks:0B2J","tag":"0B2J","title":"Dimension of fibres · Lemma 0B2J","summary":"Let f : X → Spec(R) be a morphism from an irreducible scheme to the spectrum of a valuation ring. If f is locally of finite type and surjective, then the special fibre is equidimensional of dimension equal to the dimension of the generic fibre.","statement_latex":"Let $f : X \\to \\Spec(R)$ be a morphism from an irreducible\nscheme to the spectrum of a valuation ring. If $f$ is locally\nof finite type and surjective, then the special fibre is\nequidimensional of dimension equal to the dimension of the generic fibre.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2J","source_file":"varieties.tex","source_line":3652,"source_end_line":3658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3652-L3658","statement_sha256":"cec5676007dbf227a2bd6661c22e7a72d580498142e104d9e9dc49ea6eac9d1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6441,"rank":6441,"depth":32,"x":194.781,"y":1087.184,"cluster":"varieties-curves"},{"id":"stacks:0B2K","tag":"0B2K","title":"Dimension of fibres · Lemma 0B2K","summary":"Let f : X → Y be locally of finite type. Let x ∈ X be a point with image y ∈ Y such that O_Y, y is Noetherian. Let d ≥ 0 be an integer such that for every generic point eta of an irreducible component of X which contains x, we have f(eta) not = y and dim_eta(X_f(eta)) = d. Then dim_x(X_y) ≤ d + dim(O_Y, y) - 1.","statement_latex":"Let $f : X \\to Y$ be locally of finite type. Let $x \\in X$ be a point\nwith image $y \\in Y$ such that $\\mathcal{O}_{Y, y}$ is Noetherian. Let\n$d \\geq 0$ be an integer such that for every generic point $\\eta$ of an\nirreducible component of $X$ which contains $x$, we have\n$f(\\eta) \\not = y$ and $\\dim_\\eta(X_{f(\\eta)}) = d$. Then\n$\\dim_x(X_y) \\leq d + \\dim(\\mathcal{O}_{Y, y}) - 1$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2K","source_file":"varieties.tex","source_line":3670,"source_end_line":3678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3670-L3678","statement_sha256":"e49eaa8b027e4dde360c520deb59089c091df452e432aff0f718779cc9bec135","origin":"The Stacks Project","memory_eligible":false,"source_rank":6442,"rank":6442,"depth":33,"x":298.796,"y":1159.059,"cluster":"varieties-curves"},{"id":"stacks:06LG","tag":"06LG","title":"Algebraic schemes · Definition 06LG","summary":"Let k be a field. An algebraic k-scheme is a scheme X over k such that the structure morphism X → Spec(k) is of finite type. A locally algebraic k-scheme is a scheme X over k such that the structure morphism X → Spec(k) is locally of finite type.","statement_latex":"Let $k$ be a field. An {\\it algebraic $k$-scheme} is a scheme $X$ over $k$\nsuch that the structure morphism $X \\to \\Spec(k)$ is of\nfinite type. A {\\it locally algebraic $k$-scheme} is a scheme $X$ over $k$\nsuch that the structure morphism $X \\to \\Spec(k)$ is\nlocally of finite type.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Algebraic schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LG","source_file":"varieties.tex","source_line":3719,"source_end_line":3726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3719-L3726","statement_sha256":"a87c4c421b2459073f7b287390ed66f543e3e86a2aacf39090c0f4936334c63d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6443,"rank":6443,"depth":0,"x":163.608,"y":1165.11,"cluster":"varieties-curves"},{"id":"stacks:06LH","tag":"06LH","title":"Algebraic schemes · Lemma 06LH","summary":"Let k be a field. Let X be a locally algebraic k-scheme of dimension 0. Then X is a disjoint union of spectra of local Artinian k-algebras A with dim_k(A) < ∞. If X is an algebraic k-scheme of dimension 0, then in addition X is affine and the morphism X → Spec(k) is finite.","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme of\ndimension $0$. Then $X$ is a disjoint union of spectra of local Artinian\n$k$-algebras $A$ with $\\dim_k(A) < \\infty$. If $X$ is an algebraic $k$-scheme\nof dimension $0$, then in addition $X$ is affine and the morphism\n$X \\to \\Spec(k)$ is finite.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Algebraic schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LH","source_file":"varieties.tex","source_line":3735,"source_end_line":3742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3735-L3742","statement_sha256":"9e199fbdaf74ade23b44bebd49c4c0848ad8d7950ff197b7ea0b2d1768aaebb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6444,"rank":6444,"depth":12,"x":258.909,"y":1083.529,"cluster":"varieties-curves"},{"id":"stacks:0A21","tag":"0A21","title":"Algebraic schemes · Lemma 0A21","summary":"Let k be a field. Let X be a locally algebraic k-scheme. • The topological space of X is catenary (Topology, Definition [Tag 02I1]). • For x ∈ X closed, we have dim_x(X) = dim(O_X, x). • For X irreducible we have dim(X) = dim(U) for any nonempty open U ⊂ X and dim(X) = dim_x(X) for any x ∈ X. • For X irreducible any chain of irreducible closed subsets can be extended to a maximal chain and all maximal chains of irreducible closed subsets have length equal to dim(X). • For…","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.\n\\begin{enumerate}\n\\item\n\nThe topological space of $X$ is catenary\n(Topology, Definition \\ref{topology-definition-catenary}).\n\\item\n\nFor $x \\in X$ closed, we have $\\dim_x(X) = \\dim(\\mathcal{O}_{X, x})$.\n\\item\n\nFor $X$ irreducible we have $\\dim(X) = \\dim(U)$ for any\nnonempty open $U \\subset X$ and $\\dim(X) = \\dim_x(X)$\nfor any $x \\in X$.\n\\item\n\nFor $X$ irreducible any chain of irreducible closed subsets can be\nextended to a maximal chain and all maximal chains of irreducible\nclosed subsets have length equal to $\\dim(X)$.\n\\item\n\nFor $x \\in X$ we have\n$\\dim_x(X) = \\max \\dim(Z) = \\min \\dim(\\mathcal{O}_{X, x'})$\nwhere the maximum is over irreducible components\n$Z \\subset X$ containing $x$ and the minimum is over\nspecializations $x \\leadsto x'$ with $x'$ closed in $X$.\n\\item\n\nIf $X$ is irreducible with generic point $x$, then\n$\\dim(X) = \\text{trdeg}_k(\\kappa(x))$.\n\\item\n\nIf $x \\leadsto x'$ is an immediate specialization\nof points of $X$, then we have\n$\\text{trdeg}_k(\\kappa(x)) = \\text{trdeg}_k(\\kappa(x')) + 1$.\n\\item\n\nThe dimension of $X$ is the supremum of the numbers\n$\\text{trdeg}_k(\\kappa(x))$ where $x$ runs over the\ngeneric points of the irreducible components of $X$.\n\\item\n\nIf $x \\leadsto x'$ is a nontrivial specialization of points of $X$, then\n\\begin{enumerate}\n\\item $\\dim_x(X) \\leq \\dim_{x'}(X)$,\n\\item $\\dim(\\mathcal{O}_{X, x}) < \\dim(\\mathcal{O}_{X, x'})$,\n\\item $\\text{trdeg}_k(\\kappa(x)) > \\text{trdeg}_k(\\kappa(x'))$, and\n\\item any maximal chain of nontrivial specializations\n$x = x_0 \\leadsto x_1 \\leadsto \\ldots \\leadsto x_n = x'$ has\nlength $n = \\text{trdeg}_k(\\kappa(x)) - \\text{trdeg}_k(\\kappa(x'))$.\n\\end{enumerate}\n\\item\n\nFor $x \\in X$ we have\n$\\dim_x(X) = \\text{trdeg}_k(\\kappa(x)) + \\dim(\\mathcal{O}_{X, x})$.\n\\item\n\nIf $x \\leadsto x'$ is an immediate specialization\nof points of $X$ and $X$ is irreducible or equidimensional, then\n$\\dim(\\mathcal{O}_{X, x'}) = \\dim(\\mathcal{O}_{X, x}) + 1$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Algebraic schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A21","source_file":"varieties.tex","source_line":3782,"source_end_line":3845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3782-L3845","statement_sha256":"fc6b8184f75bfd71e06664a55298b0a8ad75f596a55c683517148661699d2222","origin":"The Stacks Project","memory_eligible":false,"source_rank":6445,"rank":6445,"depth":23,"x":254.216,"y":1198.335,"cluster":"varieties-curves"},{"id":"stacks:0B2L","tag":"0B2L","title":"Algebraic schemes · Lemma 0B2L","summary":"Let k be a field. Let f : X → Y be a morphism of locally algebraic k-schemes. • For y ∈ Y, the fibre X_y is a locally algebraic scheme over kappa(y) hence all the results of Lemma [Tag 0A21] apply. • Assume X is irreducible. Set Z = overlinef(X) and d = dim(X) - dim(Z). Then • dim_x(X_f(x)) ≥ d for all x ∈ X, • the set of x ∈ X with dim_x(X_f(x)) = d is dense open, • if dim(O_Z, f(x)) ≥ 1, then dim_x(X_f(x)) ≤ d + dim(O_Z, f(x)) - 1, • if dim(O_Z, f(x)) = 1, then…","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a morphism of locally algebraic\n$k$-schemes.\n\\begin{enumerate}\n\\item For $y \\in Y$, the fibre $X_y$ is a locally\nalgebraic scheme over $\\kappa(y)$ hence all the results of\nLemma \\ref{lemma-dimension-locally-algebraic} apply.\n\\item Assume $X$ is irreducible. Set $Z = \\overline{f(X)}$ and\n$d = \\dim(X) - \\dim(Z)$. Then\n\\begin{enumerate}\n\\item $\\dim_x(X_{f(x)}) \\geq d$ for all $x \\in X$,\n\\item the set of $x \\in X$ with $\\dim_x(X_{f(x)}) = d$ is dense open,\n\\item if $\\dim(\\mathcal{O}_{Z, f(x)}) \\geq 1$, then\n$\\dim_x(X_{f(x)}) \\leq d + \\dim(\\mathcal{O}_{Z, f(x)}) - 1$,\n\\item if $\\dim(\\mathcal{O}_{Z, f(x)}) = 1$, then $\\dim_x(X_{f(x)}) = d$,\n\\end{enumerate}\n\\item For $x \\in X$ with $y = f(x)$ we have\n$\\dim_x(X_y) \\geq \\dim_x(X) - \\dim_y(Y)$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Algebraic schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2L","source_file":"varieties.tex","source_line":3952,"source_end_line":3972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L3952-L3972","statement_sha256":"359e2052d3720dbe63d3cecd77a2cf749f2695800cef37a9ee4a9db4e425b785","origin":"The Stacks Project","memory_eligible":false,"source_rank":6446,"rank":6446,"depth":34,"x":164.912,"y":1110.581,"cluster":"varieties-curves"},{"id":"stacks:0B2M","tag":"0B2M","title":"Algebraic schemes · Lemma 0B2M","summary":"The dimension of the product is the sum of the dimensions. Let k be a field. Let X, Y be locally algebraic k-schemes. • For z ∈ X × Y lying over (x, y) we have dim_z(X × Y) = dim_x(X) + dim_y(Y). • We have dim(X × Y) = dim(X) + dim(Y).","statement_latex":"\\begin{slogan}\nThe dimension of the product is the sum of the dimensions.\n\\end{slogan}\nLet $k$ be a field. Let $X$, $Y$ be locally algebraic $k$-schemes.\n\\begin{enumerate}\n\\item For $z \\in X \\times Y$ lying over $(x, y)$ we have\n$\\dim_z(X \\times Y) = \\dim_x(X) + \\dim_y(Y)$.\n\\item We have $\\dim(X \\times Y) = \\dim(X) + \\dim(Y)$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Algebraic schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2M","source_file":"varieties.tex","source_line":4004,"source_end_line":4015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4004-L4015","statement_sha256":"8f45861ebd4724fb4d32863540b1225fd72fd00dc9f89d5c0396f9ce3c4d7a8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6447,"rank":6447,"depth":27,"x":302.003,"y":1124.686,"cluster":"varieties-curves"},{"id":"stacks:0C52","tag":"0C52","title":"Complete local rings · Lemma 0C52","summary":"Let k be a field. Let X be a locally Noetherian scheme over k. Let x ∈ X be a point with residue field kappa. There is an isomorphism kappa[[x_1, …, x_n]]/I → O_X, x^wedge inducing the identity on residue fields. In general we cannot choose ([Tag 0C53]) to be a k-algebra isomorphism. However, if the extension kappa/k is separable, then we can choose ([Tag 0C53]) to be an isomorphism of k-algebras.","statement_latex":"Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$.\nLet $x \\in X$ be a point with residue field $\\kappa$.\nThere is an isomorphism\n\\begin{equation}\n\n\\kappa[[x_1, \\ldots, x_n]]/I \\longrightarrow \\mathcal{O}_{X, x}^\\wedge\n\\end{equation}\ninducing the identity on residue fields.\nIn general we cannot choose (\\ref{equation-complete-local-ring})\nto be a $k$-algebra isomorphism. However, if the extension $\\kappa/k$\nis separable, then we can choose\n(\\ref{equation-complete-local-ring}) to be an isomorphism of $k$-algebras.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Complete local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C52","source_file":"varieties.tex","source_line":4043,"source_end_line":4057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4043-L4057","statement_sha256":"30a59b383f8b0f2a56283c77785297404606304002cee79c199810006f19ffbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6448,"rank":6448,"depth":18,"x":189.023,"y":1192.401,"cluster":"varieties-curves"},{"id":"stacks:0C54","tag":"0C54","title":"Complete local rings · Lemma 0C54","summary":"Let K/k be an extension of fields. Let X be a locally algebraic k-scheme. Set Y = X_K. Let y ∈ Y be a point with image x ∈ X. Assume that dim(O_X, x) = dim(O_Y, y) and that kappa(x)/k is separable. Choose an isomorphism kappa(x)[[x_1, …, x_n]]/(g_1, …, g_m) → O_X, x^wedge of k-algebras as in ([Tag 0C53]). Then we have an isomorphism kappa(y)[[x_1, …, x_n]]/(g_1, …, g_m) → O_Y, y^wedge of K-algebras as in ([Tag 0C53]). Here we use kappa(x) → kappa(y) to view g_j as a power…","statement_latex":"Let $K/k$ be an extension of fields. Let $X$ be a locally algebraic\n$k$-scheme. Set $Y = X_K$. Let $y \\in Y$ be a point with image $x \\in X$.\nAssume that $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y})$\nand that $\\kappa(x)/k$ is separable.\nChoose an isomorphism\n$$\n\\kappa(x)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\longrightarrow\n\\mathcal{O}_{X, x}^\\wedge\n$$\nof $k$-algebras as in (\\ref{equation-complete-local-ring}).\nThen we have an isomorphism\n$$\n\\kappa(y)[[x_1, \\ldots, x_n]]/(g_1, \\ldots, g_m) \\longrightarrow\n\\mathcal{O}_{Y, y}^\\wedge\n$$\nof $K$-algebras as in (\\ref{equation-complete-local-ring}). Here we use\n$\\kappa(x) \\to \\kappa(y)$ to view $g_j$ as a power series\nover $\\kappa(y)$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Complete local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C54","source_file":"varieties.tex","source_line":4100,"source_end_line":4120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4100-L4120","statement_sha256":"83ca213613d7a3d2320fc3f3912ddef5d64f2d62af9cb44f48430d1cef94009a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6449,"rank":6449,"depth":28,"x":218.019,"y":1077.812,"cluster":"varieties-curves"},{"id":"stacks:0B57","tag":"0B57","title":"Global generation · Lemma 0B57","summary":"Let X → Spec(A) be a morphism of schemes. Let A ⊂ A' be a faithfully flat ring map. Let F be a quasi-coherent O_X-module. Then F is globally generated if and only if the base change F_A' is globally generated.","statement_latex":"Let $X \\to \\Spec(A)$ be a morphism of schemes. Let $A \\subset A'$\nbe a faithfully flat ring map. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is globally generated\nif and only if the base change $\\mathcal{F}_{A'}$ is globally generated.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Global generation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B57","source_file":"varieties.tex","source_line":4202,"source_end_line":4208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4202-L4208","statement_sha256":"10693a557ce374ad63d1ab818a33ff5b88b544b3f605afe3fdd649b40d6ee22f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6450,"rank":6450,"depth":30,"x":289.124,"y":1179.244,"cluster":"varieties-curves"},{"id":"stacks:0B58","tag":"0B58","title":"Global generation · Lemma 0B58","summary":"Let k be an infinite field. Let X be a scheme of finite type over k. Let L be a very ample invertible sheaf on X. Let n ≥ 0 and x, x_1, …, x_n ∈ X be points with x a k-rational point, i.e., kappa(x) = k, and x not = x_i for i = 1, …, n. Then there exists an s ∈ H^0(X, L) which vanishes at x but not at x_i.","statement_latex":"Let $k$ be an infinite field. Let $X$ be a scheme of finite type over $k$.\nLet $\\mathcal{L}$ be a very ample invertible sheaf on $X$.\nLet $n \\geq 0$ and $x, x_1, \\ldots, x_n \\in X$ be points with\n$x$ a $k$-rational point, i.e., $\\kappa(x) = k$, and\n$x \\not = x_i$ for $i = 1, \\ldots, n$.\nThen there exists an $s \\in H^0(X, \\mathcal{L})$ which vanishes at\n$x$ but not at $x_i$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Global generation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B58","source_file":"varieties.tex","source_line":4232,"source_end_line":4241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4232-L4241","statement_sha256":"9dded4bb8759f52440b42162c21317e244ccb81c3216abc401270fe0e1a605cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6451,"rank":6451,"depth":0,"x":154.49,"y":1144.632,"cluster":"varieties-curves"},{"id":"stacks:0B3Z","tag":"0B3Z","title":"Global generation · Lemma 0B3Z","summary":"Let k be an infinite field. Let X be an algebraic k-scheme. Let L be an invertible O_X-module. Let V → Γ(X, L) be a linear map of k-vector spaces whose image generates L. Then there exists a subspace W ⊂ V with dim_k(W) ≤ dim(X) + 1 which generates L.","statement_latex":"Let $k$ be an infinite field. Let $X$ be an algebraic $k$-scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $V \\to \\Gamma(X, \\mathcal{L})$ be a linear map of $k$-vector spaces\nwhose image generates $\\mathcal{L}$. Then there exists a subspace\n$W \\subset V$ with $\\dim_k(W) \\leq \\dim(X) + 1$ which generates $\\mathcal{L}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Global generation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B3Z","source_file":"varieties.tex","source_line":4259,"source_end_line":4266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4259-L4266","statement_sha256":"5b230410f482269f4325e3925c15a5589ea852d319e643e0f21ae97bf0973198","origin":"The Stacks Project","memory_eligible":false,"source_rank":6452,"rank":6452,"depth":13,"x":282.197,"y":1093.523,"cluster":"varieties-curves"},{"id":"stacks:0E8S","tag":"0E8S","title":"Separating points and tangent vectors · Lemma 0E8S","summary":"Let k be an algebraically closed field. Let X be a proper k-scheme. Let L be an invertible O_X-module. Let V ⊂ H^0(X, L) be a k-subvector space. If • for every pair of distinct closed points x, y ∈ X there is a section s ∈ V which vanishes at x but not at y, and • for every closed point x ∈ X and nonzero tangent vector theta ∈ T_X/k, x there exists a section s ∈ V which vanishes at x but whose pullback by theta is nonzero, then L is very ample and the canonical morphism…","statement_latex":"Let $k$ be an algebraically closed field.\nLet $X$ be a proper $k$-scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $V \\subset H^0(X, \\mathcal{L})$ be a $k$-subvector space. If\n\\begin{enumerate}\n\\item for every pair of distinct closed points $x, y \\in X$\nthere is a section $s \\in V$ which vanishes at $x$ but not at $y$, and\n\\item for every closed point $x \\in X$ and nonzero tangent vector\n$\\theta \\in T_{X/k, x}$ there exists a section $s \\in V$\nwhich vanishes at $x$ but whose pullback by $\\theta$ is nonzero,\n\\end{enumerate}\nthen $\\mathcal{L}$ is very ample and the canonical morphism\n$\\varphi_{\\mathcal{L}, V} : X \\to \\mathbf{P}(V)$\nis a closed immersion.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Separating points and tangent vectors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8S","source_file":"varieties.tex","source_line":4308,"source_end_line":4324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4308-L4324","statement_sha256":"d4afc45be27706c77258f660a83f2df73b9b3a031a315dcbde9412dca18d7d39","origin":"The Stacks Project","memory_eligible":false,"source_rank":6453,"rank":6453,"depth":41,"x":228.882,"y":1204.184,"cluster":"varieties-curves"},{"id":"stacks:0E8T","tag":"0E8T","title":"Separating points and tangent vectors · Lemma 0E8T","summary":"Let k be an algebraically closed field. Let X be a proper k-scheme. Let L be an invertible O_X-module. Suppose that for every closed subscheme Z ⊂ X of dimension 0 and degree 2 over k the map H^0(X, L) → H^0(Z, L|_Z) is surjective. Then L is very ample on X over k.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $X$ be a proper $k$-scheme.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nSuppose that for every closed subscheme $Z \\subset X$\nof dimension $0$ and degree $2$ over $k$ the map\n$$\nH^0(X, \\mathcal{L}) \\longrightarrow H^0(Z, \\mathcal{L}|_Z)\n$$\nis surjective. Then $\\mathcal{L}$ is very ample on $X$ over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Separating points and tangent vectors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8T","source_file":"varieties.tex","source_line":4385,"source_end_line":4396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4385-L4396","statement_sha256":"0ed058dc289e0aba8c3b643fb480c6ab272db68e88221fc7eb95f6f93cf8c913","origin":"The Stacks Project","memory_eligible":false,"source_rank":6454,"rank":6454,"depth":42,"x":178.977,"y":1091.817,"cluster":"varieties-curves"},{"id":"stacks:047B","tag":"047B","title":"Closures of products · Lemma 047B","summary":"Let k be a field. Let X, Y be schemes over k, and let A ⊂ X, B ⊂ Y be subsets. Set AB = (z ∈ X ×_k Y mid pr_X(z) ∈ A, pr_Y(z) ∈ B) ⊂ X ×_k Y Then set theoretically we have overlineA ×_k overlineB = overlineAB","statement_latex":"Let $k$ be a field.\nLet $X$, $Y$ be schemes over $k$, and let\n$A \\subset X$, $B \\subset Y$ be subsets.\nSet\n$$\nAB =\n\\{z \\in X \\times_k Y \\mid \\text{pr}_X(z) \\in A, \\ \\text{pr}_Y(z) \\in B\\}\n\\subset X \\times_k Y\n$$\nThen set theoretically we have\n$$\n\\overline{A} \\times_k \\overline{B} = \\overline{AB}\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Closures of products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047B","source_file":"varieties.tex","source_line":4424,"source_end_line":4439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4424-L4439","statement_sha256":"43fa1939ebe11bae07cd80f600bc0a5216573ab6cff976fcb154d51bc1764444","origin":"The Stacks Project","memory_eligible":false,"source_rank":6455,"rank":6455,"depth":10,"x":306.717,"y":1146.607,"cluster":"varieties-curves"},{"id":"stacks:04Q0","tag":"04Q0","title":"Closures of products · Lemma 04Q0","summary":"Let k be a field. Let f : A → X, g : B → Y be morphisms of schemes over k. Then set theoretically we have overlinef(A) ×_k overlineg(B) = overline(f × g)(A ×_k B)","statement_latex":"Let $k$ be a field.\nLet $f : A \\to X$, $g : B \\to Y$ be morphisms of schemes over $k$.\nThen set theoretically we have\n$$\n\\overline{f(A)} \\times_k \\overline{g(B)} =\n\\overline{(f \\times g)(A \\times_k B)}\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Closures of products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Q0","source_file":"varieties.tex","source_line":4462,"source_end_line":4471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4462-L4471","statement_sha256":"3ff1ab093af1d45af562f8a110ceeedb6b2d5097a7f108a4176988ccef495730","origin":"The Stacks Project","memory_eligible":false,"source_rank":6456,"rank":6456,"depth":11,"x":167.839,"y":1178.833,"cluster":"varieties-curves"},{"id":"stacks:04Q1","tag":"04Q1","title":"Closures of products · Lemma 04Q1","summary":"Let k be a field. Let f : A → X, g : B → Y be quasi-compact morphisms of schemes over k. Let Z ⊂ X be the scheme theoretic image of f, see Morphisms, Definition [Tag 01R7]. Similarly, let Z' ⊂ Y be the scheme theoretic image of g. Then Z ×_k Z' is the scheme theoretic image of f × g.","statement_latex":"Let $k$ be a field.\nLet $f : A \\to X$, $g : B \\to Y$ be quasi-compact morphisms of schemes\nover $k$. Let $Z \\subset X$ be the scheme theoretic image of $f$, see\nMorphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image}.\nSimilarly, let $Z' \\subset Y$ be the scheme theoretic image of $g$.\nThen $Z \\times_k Z'$ is the scheme theoretic image of $f \\times g$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Closures of products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Q1","source_file":"varieties.tex","source_line":4480,"source_end_line":4488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4480-L4488","statement_sha256":"75dcb55cea612b0f4c2b4f0b6d02536267fdccbb3565182359d2afabc5fecb16","origin":"The Stacks Project","memory_eligible":false,"source_rank":6457,"rank":6457,"depth":19,"x":244.674,"y":1075.809,"cluster":"varieties-curves"},{"id":"stacks:04QN","tag":"04QN","title":"Schemes smooth over fields · Lemma 04QN","summary":"Let k be a field. Let X be a scheme over k. Assume • X is locally of finite type over k, • Ω_X/k is locally free, and • k has characteristic zero. Then the structure morphism X → Spec(k) is smooth.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite type over $k$,\n\\item $\\Omega_{X/k}$ is locally free, and\n\\item $k$ has characteristic zero.\n\\end{enumerate}\nThen the structure morphism $X \\to \\Spec(k)$ is smooth.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QN","source_file":"varieties.tex","source_line":4530,"source_end_line":4540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4530-L4540","statement_sha256":"bcb5fb0fa7d854ec0330da169bee0645a9fdf8de0314616db1177400aa3e7169","origin":"The Stacks Project","memory_eligible":false,"source_rank":6458,"rank":6458,"depth":39,"x":270.981,"y":1195.905,"cluster":"varieties-curves"},{"id":"stacks:04QP","tag":"04QP","title":"Schemes smooth over fields · Lemma 04QP","summary":"Let k be a field. Let X be a scheme over k. Assume • X is locally of finite type over k, • Ω_X/k is locally free, • X is reduced, and • k is perfect. Then the structure morphism X → Spec(k) is smooth.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite type over $k$,\n\\item $\\Omega_{X/k}$ is locally free,\n\\item $X$ is reduced, and\n\\item $k$ is perfect.\n\\end{enumerate}\nThen the structure morphism $X \\to \\Spec(k)$ is smooth.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QP","source_file":"varieties.tex","source_line":4556,"source_end_line":4567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4556-L4567","statement_sha256":"17cd9d49405d3f7bdfa70b266bf99e6fb9dda39cdbbcd3adcabda54867534470","origin":"The Stacks Project","memory_eligible":false,"source_rank":6459,"rank":6459,"depth":37,"x":154.495,"y":1121.951,"cluster":"varieties-curves"},{"id":"stacks:056S","tag":"056S","title":"Schemes smooth over fields · Lemma 056S","summary":"Smooth over a field implies regular Let X → Spec(k) be a smooth morphism where k is a field. Then X is a regular scheme.","statement_latex":"\\begin{slogan}\nSmooth over a field implies regular\n\\end{slogan}\nLet $X \\to \\Spec(k)$ be a smooth morphism where $k$ is a field.\nThen $X$ is a regular scheme.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056S","source_file":"varieties.tex","source_line":4596,"source_end_line":4603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4596-L4603","statement_sha256":"4838f150d525acf3bb690525e43921ed2733ba6a0445352577ed8cba4bbe28f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6460,"rank":6460,"depth":39,"x":300.495,"y":1110.338,"cluster":"varieties-curves"},{"id":"stacks:056T","tag":"056T","title":"Schemes smooth over fields · Lemma 056T","summary":"Let X → Spec(k) be a smooth morphism where k is a field. Then X is geometrically regular, geometrically normal, and geometrically reduced over k.","statement_latex":"Let $X \\to \\Spec(k)$ be a smooth morphism where $k$ is a field.\nThen $X$ is geometrically regular, geometrically normal, and\ngeometrically reduced over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056T","source_file":"varieties.tex","source_line":4616,"source_end_line":4621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4616-L4621","statement_sha256":"e81be08a95406de05d0363a168ca9bc5a8de13a63102844ab4f05ce95dc1a990","origin":"The Stacks Project","memory_eligible":false,"source_rank":6461,"rank":6461,"depth":43,"x":201.751,"y":1202.139,"cluster":"varieties-curves"},{"id":"stacks:055T","tag":"055T","title":"Schemes smooth over fields · Lemma 055T","summary":"Let k be a field. Let d ≥ 0. Let W ⊂ A^d_k be nonempty open. Then there exists a closed point w ∈ W such that k ⊂ kappa(w) is finite separable.","statement_latex":"Let $k$ be a field. Let $d \\geq 0$. Let $W \\subset \\mathbf{A}^d_k$\nbe nonempty open. Then there exists a closed point $w \\in W$ such that\n$k \\subset \\kappa(w)$ is finite separable.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055T","source_file":"varieties.tex","source_line":4642,"source_end_line":4647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4642-L4647","statement_sha256":"fb8c6b78e802975154c5c6d1054f2f313d2dd6dc360121f93218c72fd2d897fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6462,"rank":6462,"depth":0,"x":200.736,"y":1077.887,"cluster":"varieties-curves"},{"id":"stacks:056U","tag":"056U","title":"Schemes smooth over fields · Lemma 056U","summary":"Let k be a field. If X is smooth over Spec(k) then the set (x ∈ X closed such that k ⊂ kappa(x) is finite separable) is dense in X.","statement_latex":"Let $k$ be a field. If $X$ is smooth over $\\Spec(k)$ then\nthe set\n$$\n\\{x \\in X\\text{ closed such that }k \\subset \\kappa(x)\n\\text{ is finite separable}\\}\n$$\nis dense in $X$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056U","source_file":"varieties.tex","source_line":4657,"source_end_line":4666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4657-L4666","statement_sha256":"52ff65d53ca5229a59fe5715be0e1a7af8a500083a0e59327f0689f736f92cce","origin":"The Stacks Project","memory_eligible":false,"source_rank":6463,"rank":6463,"depth":45,"x":301.831,"y":1169.32,"cluster":"varieties-curves"},{"id":"stacks:056V","tag":"056V","title":"Schemes smooth over fields · Lemma 056V","summary":"Let X be a reduced scheme that is locally of finite type over a field k. Then X is geometrically reduced over k if and only if X contains a dense open which is smooth over k.","statement_latex":"Let $X$ be a reduced scheme that is locally of finite type over a field $k$.\nThen $X$ is geometrically reduced over $k$ if and only if\n$X$ contains a dense open which is smooth over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056V","source_file":"varieties.tex","source_line":4693,"source_end_line":4698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4693-L4698","statement_sha256":"002a36d469d625505a4aaeb1691446641e55084e53ca560a5e09fccbbbb187ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":6464,"rank":6464,"depth":44,"x":153.134,"y":1159.219,"cluster":"varieties-curves"},{"id":"stacks:0B8X","tag":"0B8X","title":"Schemes smooth over fields · Lemma 0B8X","summary":"Let k be a perfect field. Let X be a locally algebraic reduced k-scheme, for example a variety over k. Then we have (x ∈ X mid X → Spec(k) is smooth at x) = (x ∈ X mid O_X, x is regular) and this is a dense open subscheme of X.","statement_latex":"Let $k$ be a perfect field. Let $X$ be a locally algebraic\nreduced $k$-scheme, for example a variety over $k$. Then we have\n$$\n\\{x \\in X \\mid X \\to \\Spec(k)\\text{ is smooth at }x\\} =\n\\{x \\in X \\mid \\mathcal{O}_{X, x}\\text{ is regular}\\}\n$$\nand this is a dense open subscheme of $X$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8X","source_file":"varieties.tex","source_line":4729,"source_end_line":4738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4729-L4738","statement_sha256":"2212a7d592ee0eeb206a5107138253e6999a5ea7c94f379947eace2678d0ca5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6465,"rank":6465,"depth":45,"x":271.395,"y":1081.973,"cluster":"varieties-curves"},{"id":"stacks:05AX","tag":"05AX","title":"Schemes smooth over fields · Lemma 05AX","summary":"Let k be a field. Let f : X → Y be a morphism of schemes locally of finite type over k. Let x ∈ X be a point and set y = f(x). If X → Spec(k) is smooth at x and f is flat at x then Y → Spec(k) is smooth at y. In particular, if X is smooth over k and f is flat and surjective, then Y is smooth over k.","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a morphism of schemes locally\nof finite type over $k$. Let $x \\in X$ be a point and set $y = f(x)$.\nIf $X \\to \\Spec(k)$ is smooth at $x$ and $f$ is flat at $x$\nthen $Y \\to \\Spec(k)$ is smooth at $y$. In particular, if $X$ is\nsmooth over $k$ and $f$ is flat and surjective, then $Y$ is smooth over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AX","source_file":"varieties.tex","source_line":4756,"source_end_line":4763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4756-L4763","statement_sha256":"7753f8785b02fda6667afd80f179f36e62ed6de66bb241b1114d05a15b198725","origin":"The Stacks Project","memory_eligible":false,"source_rank":6466,"rank":6466,"depth":44,"x":246.206,"y":1206.55,"cluster":"varieties-curves"},{"id":"stacks:0CDW","tag":"0CDW","title":"Schemes smooth over fields · Lemma 0CDW","summary":"Let k be a field. Let X be a variety over k which has a k-rational point x such that X is smooth at x. Then X is geometrically integral over k.","statement_latex":"Let $k$ be a field. Let $X$ be a variety over $k$ which has\na $k$-rational point $x$ such that $X$ is smooth at $x$.\nThen $X$ is geometrically integral over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDW","source_file":"varieties.tex","source_line":4775,"source_end_line":4780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4775-L4780","statement_sha256":"7a0ba7382a3199e69b82ea42e3d6bdecb4b2dbc6ba1479def8d06a628f6b688f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6467,"rank":6467,"depth":44,"x":164.265,"y":1099.96,"cluster":"varieties-curves"},{"id":"stacks:0H3W","tag":"0H3W","title":"Schemes smooth over fields · Lemma 0H3W","summary":"Let X be a scheme of finite type over a field k. There exists a finite purely inseparable extension k'/k, an integer t ≥ 0, and closed subschemes X_k' ⊃ Z_0 ⊃ Z_1 ⊃ … ⊃ Z_t = ∅ such that Z_0 = (X_k')_red and Z_i setminus Z_i + 1 is smooth over k' for all i.","statement_latex":"Let $X$ be a scheme of finite type over a field $k$.\nThere exists a finite purely inseparable extension $k'/k$,\nan integer $t \\geq 0$, and closed subschemes\n$$\nX_{k'} \\supset Z_0 \\supset Z_1 \\supset \\ldots \\supset Z_t = \\emptyset\n$$\nsuch that $Z_0 = (X_{k'})_{red}$ and $Z_i \\setminus Z_{i + 1}$\nis smooth over $k'$ for all $i$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Schemes smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3W","source_file":"varieties.tex","source_line":4797,"source_end_line":4807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4797-L4807","statement_sha256":"e29d958c93ab1525ad370e94ee89c3758cd3840008dd449250cc003669068a71","origin":"The Stacks Project","memory_eligible":false,"source_rank":6468,"rank":6468,"depth":45,"x":310.998,"y":1132.194,"cluster":"varieties-curves"},{"id":"stacks:04L1","tag":"04L1","title":"Types of varieties · Definition 04L1","summary":"Let k be a field. Let X be a variety over k. • We say X is an affine variety if X is an affine scheme. This is equivalent to requiring X to be isomorphic to a closed subscheme of A^n_k for some n. • We say X is a projective variety if the structure morphism X → Spec(k) is projective. By Morphisms, Lemma [Tag 01WB] this is true if and only if X is isomorphic to a closed subscheme of P^n_k for some n. • We say X is a quasi-projective variety if the structure morphism X →…","statement_latex":"Let $k$ be a field. Let $X$ be a variety over $k$.\n\\begin{enumerate}\n\\item We say $X$ is an {\\it affine variety} if $X$ is an affine scheme.\nThis is equivalent to requiring $X$ to be isomorphic to a closed\nsubscheme of $\\mathbf{A}^n_k$ for some $n$.\n\\item We say $X$ is a {\\it projective variety} if the\nstructure morphism $X \\to \\Spec(k)$ is projective. By\nMorphisms, Lemma \\ref{morphisms-lemma-characterize-locally-projective}\nthis is true if and only if $X$ is isomorphic to a closed\nsubscheme of $\\mathbf{P}^n_k$ for some $n$.\n\\item We say $X$ is a {\\it quasi-projective variety} if\nthe structure morphism $X \\to \\Spec(k)$ is quasi-projective. By\nMorphisms, Lemma \\ref{morphisms-lemma-characterize-locally-quasi-projective}\nthis is true if and only if $X$ is isomorphic to a\nlocally closed subscheme of $\\mathbf{P}^n_k$ for some $n$.\n\\item A {\\it proper variety} is a variety such that the\nmorphism $X \\to \\Spec(k)$ is proper.\n\\item A {\\it smooth variety} is a variety such that the\nmorphism $X \\to \\Spec(k)$ is smooth.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Types of varieties","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04L1","source_file":"varieties.tex","source_line":4843,"source_end_line":4865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4843-L4865","statement_sha256":"641ac14a24f20c3898267e2967cb7e7b92561c7a294d3ffac54c9b5282f962db","origin":"The Stacks Project","memory_eligible":false,"source_rank":6469,"rank":6469,"depth":24,"x":176.336,"y":1191.924,"cluster":"varieties-curves"},{"id":"stacks:04L2","tag":"04L2","title":"Types of varieties · Lemma 04L2","summary":"Let X be a proper variety over k. Then • K = H^0(X, O_X) is a field which is a finite extension of the field k, • if X is geometrically reduced, then K/k is separable, • if X is geometrically irreducible, then K/k is purely inseparable, • if X is geometrically integral, then K = k.","statement_latex":"Let $X$ be a proper variety over $k$. Then\n\\begin{enumerate}\n\\item $K = H^0(X, \\mathcal{O}_X)$ is a field which is\na finite extension of the field $k$,\n\\item if $X$ is geometrically reduced, then $K/k$ is separable,\n\\item if $X$ is geometrically irreducible, then $K/k$\nis purely inseparable,\n\\item if $X$ is geometrically integral, then $K = k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Types of varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04L2","source_file":"varieties.tex","source_line":4875,"source_end_line":4886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4875-L4886","statement_sha256":"1227cff0ac8711557d548ddad7004277cba11f2151451021ad623d311af680ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":6470,"rank":6470,"depth":33,"x":227.806,"y":1070.987,"cluster":"varieties-curves"},{"id":"stacks:0BXR","tag":"0BXR","title":"Normalization · Lemma 0BXR","summary":"Let k be a field. Let X be a locally algebraic scheme over k. Let ν : X^ν → X be the normalization morphism, see Morphisms, Definition [Tag 035N]. Then • ν is finite, dominant, and X^ν is a disjoint union of normal irreducible locally algebraic schemes over k, • ν factors as X^ν → X_red → X and the first morphism is the normalization morphism of X_red, • if X is a reduced algebraic scheme, then ν is birational, • if X is a variety, then X^ν is a variety and ν is a finite…","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic scheme over $k$.\nLet $\\nu : X^\\nu \\to X$ be the normalization morphism, see\nMorphisms, Definition \\ref{morphisms-definition-normalization}.\nThen\n\\begin{enumerate}\n\\item $\\nu$ is finite, dominant, and $X^\\nu$ is a disjoint\nunion of normal irreducible locally algebraic schemes over $k$,\n\\item $\\nu$ factors as $X^\\nu \\to X_{red} \\to X$ and the first\nmorphism is the normalization morphism of $X_{red}$,\n\\item if $X$ is a reduced algebraic scheme, then $\\nu$ is\nbirational,\n\\item if $X$ is a variety, then $X^\\nu$ is a variety and\n$\\nu$ is a finite birational morphism of varieties.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXR","source_file":"varieties.tex","source_line":4902,"source_end_line":4918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4902-L4918","statement_sha256":"33d77ddbfc7b132152ee9eb74aefd45c9246b50fab055f59c5d727847f0bf418","origin":"The Stacks Project","memory_eligible":false,"source_rank":6471,"rank":6471,"depth":35,"x":287.341,"y":1189.84,"cluster":"varieties-curves"},{"id":"stacks:0GK4","tag":"0GK4","title":"Normalization · Lemma 0GK4","summary":"Let k be a field. Let X be a proper scheme over k. Let ν : X^ν → X be the normalization morphism, see Morphisms, Definition [Tag 035N]. Then X^ν is proper over k. If X is projective over k, then X^ν is projective over k.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\nLet $\\nu : X^\\nu \\to X$ be the normalization morphism, see\nMorphisms, Definition \\ref{morphisms-definition-normalization}.\nThen $X^\\nu$ is proper over $k$. If $X$ is projective over $k$,\nthen $X^\\nu$ is projective over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GK4","source_file":"varieties.tex","source_line":4949,"source_end_line":4956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4949-L4956","statement_sha256":"00a966828d291661e637b1e37781bcd0f0dbdcaf6a5dae5d35379f6763d74914","origin":"The Stacks Project","memory_eligible":false,"source_rank":6472,"rank":6472,"depth":36,"x":147.314,"y":1135.769,"cluster":"varieties-curves"},{"id":"stacks:0BXS","tag":"0BXS","title":"Normalization · Lemma 0BXS","summary":"Let k be a field. Let f : Y → X be a quasi-compact morphism of locally algebraic schemes over k. Let X' be the normalization of X in Y. If Y is reduced, then X' → X is finite.","statement_latex":"Let $k$ be a field. Let $f : Y \\to X$ be a quasi-compact\nmorphism of locally algebraic schemes over $k$. Let $X'$\nbe the normalization of $X$ in $Y$. If $Y$ is reduced, then\n$X' \\to X$ is finite.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXS","source_file":"varieties.tex","source_line":4969,"source_end_line":4975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4969-L4975","statement_sha256":"fcda3996d3510f89c45c7965310c216d91184870886fdd5e6f7dd2eece03f3da","origin":"The Stacks Project","memory_eligible":false,"source_rank":6473,"rank":6473,"depth":35,"x":294.625,"y":1096.031,"cluster":"varieties-curves"},{"id":"stacks:0BXT","tag":"0BXT","title":"Normalization · Lemma 0BXT","summary":"Let k be a field. Let X be an algebraic k-scheme. Then there exists a finite purely inseparable extension k'/k such that the normalization Y of X_k' is geometrically normal over k'.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic $k$-scheme.\nThen there exists a finite purely inseparable extension $k'/k$\nsuch that the normalization $Y$ of $X_{k'}$ is geometrically normal over $k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXT","source_file":"varieties.tex","source_line":4993,"source_end_line":4998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L4993-L4998","statement_sha256":"91bb0c655221d47584174ab7989fc030c6a1825f2ef2a9ad371e05880fde438a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6474,"rank":6474,"depth":43,"x":217.658,"y":1209.358,"cluster":"varieties-curves"},{"id":"stacks:0C3N","tag":"0C3N","title":"Normalization · Lemma 0C3N","summary":"Let k be a field. Let X be a locally algebraic k-scheme. Let K/k be an extension of fields. Let ν : X^ν → X be the normalization of X and let Y^ν → X_K be the normalization of the base change. Then the canonical morphism Y^ν → X^ν ×_Spec(k) Spec(K) is an isomorphism if K/k is separable and a universal homeomorphism in general.","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.\nLet $K/k$ be an extension of fields. Let $\\nu : X^\\nu \\to X$\nbe the normalization of $X$ and let $Y^\\nu \\to X_K$ be the\nnormalization of the base change. Then the canonical morphism\n$$\nY^\\nu \\longrightarrow X^\\nu \\times_{\\Spec(k)} \\Spec(K)\n$$\nis an isomorphism if $K/k$ is separable and a universal homeomorphism\nin general.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3N","source_file":"varieties.tex","source_line":5026,"source_end_line":5037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5026-L5037","statement_sha256":"09e63bf31240eaf714baf3b9e205afc139caeeb6a84ab2ea2a2f89a7be328375","origin":"The Stacks Project","memory_eligible":false,"source_rank":6475,"rank":6475,"depth":45,"x":183.146,"y":1081.63,"cluster":"varieties-curves"},{"id":"stacks:0C3P","tag":"0C3P","title":"Normalization · Lemma 0C3P","summary":"Let k be a field. Let X be a locally algebraic k-scheme. Let ν : X^ν → X be the normalization of X. Let x ∈ X be a point such that (a) O_X, x is reduced, (b) dim(O_X, x) = 1, and (c) for every x' ∈ X^ν with ν(x') = x the extension kappa(x')/k is separable. Then X is geometrically reduced at x and X^ν is geometrically regular at x' with ν(x') = x.","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.\nLet $\\nu : X^\\nu \\to X$ be the normalization of $X$.\nLet $x \\in X$ be a point such that (a) $\\mathcal{O}_{X, x}$\nis reduced, (b) $\\dim(\\mathcal{O}_{X, x}) = 1$, and (c)\nfor every $x' \\in X^\\nu$ with $\\nu(x') = x$ the extension\n$\\kappa(x')/k$ is separable. Then $X$ is geometrically reduced at $x$\nand $X^\\nu$ is geometrically regular at $x'$ with $\\nu(x') = x$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3P","source_file":"varieties.tex","source_line":5101,"source_end_line":5110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5101-L5110","statement_sha256":"6b2ee14afa2c98cbdf087082206f16fbd1c4ddc90e165094c9312c23b06adb06","origin":"The Stacks Project","memory_eligible":false,"source_rank":6476,"rank":6476,"depth":45,"x":311.798,"y":1156.518,"cluster":"varieties-curves"},{"id":"stacks:04L4","tag":"04L4","title":"Groups of invertible functions · Lemma 04L4","summary":"Let k be an algebraically closed field. Let overlineX be a proper variety over k. Let X ⊂ overlineX be an open subscheme. Assume X is normal. Then O^*(X)/k^* is a finitely generated abelian group.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $\\overline{X}$ be a proper variety over $k$.\nLet $X \\subset \\overline{X}$ be an open subscheme.\nAssume $X$ is normal.\nThen $\\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Groups of invertible functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04L4","source_file":"varieties.tex","source_line":5156,"source_end_line":5163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5156-L5163","statement_sha256":"3c1f4eeef5fa000f66bef9cea84ad15ab2ec11171c44e97b50503b9c1ed517c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6477,"rank":6477,"depth":36,"x":156.122,"y":1174.364,"cluster":"varieties-curves"},{"id":"stacks:04L5","tag":"04L5","title":"Groups of invertible functions · Lemma 04L5","summary":"Let k be an algebraically closed field. Let X be an integral scheme locally of finite type over k. Then O^*(X)/k^* is a finitely generated abelian group.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $X$ be an integral scheme locally of finite type over $k$.\nThen $\\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Groups of invertible functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04L5","source_file":"varieties.tex","source_line":5245,"source_end_line":5250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5245-L5250","statement_sha256":"454b1d7a2bb34c47dd5ff1609b81c7cd0008ec744a636a83ad5e3c55a8b045b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6478,"rank":6478,"depth":37,"x":256.942,"y":1072.489,"cluster":"varieties-curves"},{"id":"stacks:04L6","tag":"04L6","title":"Groups of invertible functions · Lemma 04L6","summary":"Let k be an algebraically closed field. Let X be a connected reduced scheme which is locally of finite type over k with finitely many irreducible components. Then O^*(X)/k^* is a finitely generated abelian group.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $X$ be a connected reduced scheme which is locally of finite type\nover $k$ with finitely many irreducible components.\nThen $\\mathcal{O}^*(X)/k^*$ is a finitely generated abelian group.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Groups of invertible functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04L6","source_file":"varieties.tex","source_line":5288,"source_end_line":5294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5288-L5294","statement_sha256":"9ca82e32899bf1748e6972624e13e35eede7cabe1b339dac2fe89409a49cbc51","origin":"The Stacks Project","memory_eligible":false,"source_rank":6479,"rank":6479,"depth":38,"x":264.552,"y":1205.311,"cluster":"varieties-curves"},{"id":"stacks:04MI","tag":"04MI","title":"Groups of invertible functions · Lemma 04MI","summary":"Let k be a field. Let X be a scheme over k which is connected and reduced. Then the integral closure of k in Γ(X, O_X) is a field.","statement_latex":"Let $k$ be a field.\nLet $X$ be a scheme over $k$ which is connected and reduced.\nThen the integral closure of $k$ in $\\Gamma(X, \\mathcal{O}_X)$\nis a field.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Groups of invertible functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MI","source_file":"varieties.tex","source_line":5318,"source_end_line":5324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5318-L5324","statement_sha256":"6998df66fe1477a87ea81f074cbf834bee7f0a4b376311c0826e0f8ad5a79792","origin":"The Stacks Project","memory_eligible":false,"source_rank":6480,"rank":6480,"depth":11,"x":151.713,"y":1111.34,"cluster":"varieties-curves"},{"id":"stacks:04L7","tag":"04L7","title":"Groups of invertible functions · Proposition 04L7","summary":"Let k be a field. Let X be a scheme over k. Assume that X is locally of finite type over k, connected, reduced, and has finitely many irreducible components. Then O(X)^*/k^* is a finitely generated abelian group if in addition to the conditions above at least one of the following conditions is satisfied: • the integral closure of k in Γ(X, O_X) is k, • X has a k-rational point, or • X is geometrically integral.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme over $k$. Assume that $X$ is\nlocally of finite type over $k$, connected, reduced, and has finitely many\nirreducible components. Then $\\mathcal{O}(X)^*/k^*$ is a finitely generated\nabelian group if in addition to the conditions above at least\none of the following conditions is satisfied:\n\\begin{enumerate}\n\\item the integral closure of $k$ in $\\Gamma(X, \\mathcal{O}_X)$ is $k$,\n\\item $X$ has a $k$-rational point, or\n\\item $X$ is geometrically integral.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Groups of invertible functions","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04L7","source_file":"varieties.tex","source_line":5348,"source_end_line":5360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5348-L5360","statement_sha256":"4368a2c89aac958a75adf9f82b011ff8d005c4b0338c2632b701544e24a148c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6481,"rank":6481,"depth":39,"x":311.071,"y":1116.626,"cluster":"varieties-curves"},{"id":"stacks:04L8","tag":"04L8","title":"Groups of invertible functions · Lemma 04L8","summary":"Let k be a field. Let X be a variety over k. The group O(X)^*/k^* is a finitely generated abelian group provided at least one of the following conditions holds: • k is integrally closed in Γ(X, O_X), • k is algebraically closed in k(X), • X is geometrically integral over k, or • k is the \"intersection\" of the field extensions kappa(x)/k where x runs over the closed points of x.","statement_latex":"Let $k$ be a field.\nLet $X$ be a variety over $k$.\nThe group $\\mathcal{O}(X)^*/k^*$ is a finitely generated abelian group\nprovided at least one of the following conditions holds:\n\\begin{enumerate}\n\\item $k$ is integrally closed in $\\Gamma(X, \\mathcal{O}_X)$,\n\\item $k$ is algebraically closed in $k(X)$,\n\\item $X$ is geometrically integral over $k$, or\n\\item $k$ is the ``intersection'' of the field extensions\n$\\kappa(x)/k$ where $x$ runs over the closed points of $x$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Groups of invertible functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04L8","source_file":"varieties.tex","source_line":5405,"source_end_line":5418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5405-L5418","statement_sha256":"d67c5b7e3b1a5897ecd2c2f7988d1cd725e7209f1af403e4eb9977a7047239a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6482,"rank":6482,"depth":40,"x":188.868,"y":1203.467,"cluster":"varieties-curves"},{"id":"stacks:0BED","tag":"0BED","title":"K\\\"unneth formula, I · Lemma 0BED","summary":"Let k be a field. Let X and Y be schemes over k and let F, resp. G be a quasi-coherent O_X-module, resp. O_Y-module. Then we have a canonical isomorphism H^n(X ×_Spec(k) Y, pr_1^*F ⊗_O_X ×_Spec(k) Y pr_2^*G) = bigoplus_p + q = n H^p(X, F) ⊗_k H^q(Y, G) provided X and Y are quasi-compact and have affine diagonal (for example if X and Y are separated).","statement_latex":"Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$ and\nlet $\\mathcal{F}$, resp.\\ $\\mathcal{G}$ be a quasi-coherent\n$\\mathcal{O}_X$-module, resp.\\ $\\mathcal{O}_Y$-module.\nThen we have a canonical isomorphism\n$$\nH^n(X \\times_{\\Spec(k)} Y, \\text{pr}_1^*\\mathcal{F}\n\\otimes_{\\mathcal{O}_{X \\times_{\\Spec(k)} Y}} \\text{pr}_2^*\\mathcal{G}) =\n\\bigoplus\\nolimits_{p + q = n}\nH^p(X, \\mathcal{F}) \\otimes_k H^q(Y, \\mathcal{G})\n$$\nprovided $X$ and $Y$ are quasi-compact and have affine\ndiagonal\\footnote{The case where $X$ and $Y$ are quasi-separated\nwill be discussed in Lemma \\ref{lemma-kunneth-general} below.}\n(for example if $X$ and $Y$ are separated).","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"K\\\"unneth formula, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BED","source_file":"varieties.tex","source_line":5441,"source_end_line":5457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5441-L5457","statement_sha256":"64dc6224c19991166d8326874c407969498ee15a0706b0c37e6e247f3ed07f2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6483,"rank":6483,"depth":27,"x":209.217,"y":1069.608,"cluster":"varieties-curves"},{"id":"stacks:0BEF","tag":"0BEF","title":"K\\\"unneth formula, I · Lemma 0BEF","summary":"Let k be a field. Let X and Y be schemes over k and let F, resp. G be a quasi-coherent O_X-module, resp. O_Y-module. Then we have a canonical isomorphism H^n(X ×_Spec(k) Y, pr_1^*F ⊗_O_X ×_Spec(k) Y pr_2^*G) = bigoplus_p + q = n H^p(X, F) ⊗_k H^q(Y, G) provided X and Y are quasi-compact and quasi-separated.","statement_latex":"Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$ and\nlet $\\mathcal{F}$, resp.\\ $\\mathcal{G}$ be a quasi-coherent\n$\\mathcal{O}_X$-module, resp.\\ $\\mathcal{O}_Y$-module.\nThen we have a canonical isomorphism\n$$\nH^n(X \\times_{\\Spec(k)} Y, \\text{pr}_1^*\\mathcal{F}\n\\otimes_{\\mathcal{O}_{X \\times_{\\Spec(k)} Y}} \\text{pr}_2^*\\mathcal{G}) =\n\\bigoplus\\nolimits_{p + q = n}\nH^p(X, \\mathcal{F}) \\otimes_k H^q(Y, \\mathcal{G})\n$$\nprovided $X$ and $Y$ are quasi-compact and quasi-separated.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"K\\\"unneth formula, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEF","source_file":"varieties.tex","source_line":5632,"source_end_line":5645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5632-L5645","statement_sha256":"3e35beb794388f982afdf1b2ac34b59733c263f7fc037e0905e79a84f2a5e72b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6484,"rank":6484,"depth":28,"x":302.19,"y":1180.257,"cluster":"varieties-curves"},{"id":"stacks:0CDX","tag":"0CDX","title":"Picard groups of varieties · Lemma 0CDX","summary":"Let A → B be a faithfully flat ring map. Let X be a quasi-compact and quasi-separated scheme over A. Let L be an invertible O_X-module whose pullback to X_B is trivial. Then H^0(X, L) and H^0(X, L^⊗ -1) are invertible H^0(X, O_X)-modules and the multiplication map induces an isomorphism H^0(X, L) ⊗_H^0(X, O_X) H^0(X, L^⊗ -1) → H^0(X, O_X)","statement_latex":"Let $A \\to B$ be a faithfully flat ring map. Let $X$ be a quasi-compact and\nquasi-separated scheme over $A$. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module whose pullback to $X_B$ is trivial. Then\n$H^0(X, \\mathcal{L})$ and $H^0(X, \\mathcal{L}^{\\otimes -1})$ are invertible\n$H^0(X, \\mathcal{O}_X)$-modules and the\nmultiplication map induces an isomorphism\n$$\nH^0(X, \\mathcal{L}) \\otimes_{H^0(X, \\mathcal{O}_X)}\nH^0(X, \\mathcal{L}^{\\otimes -1}) \\longrightarrow\nH^0(X, \\mathcal{O}_X)\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Picard groups of varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDX","source_file":"varieties.tex","source_line":5742,"source_end_line":5755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5742-L5755","statement_sha256":"2d610b5bffbcfa3ca272fa68a9c19ddbee5d5abc49d4bff7db181f9bf0cc83fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6485,"rank":6485,"depth":30,"x":144.09,"y":1151.317,"cluster":"varieties-curves"},{"id":"stacks:0CDY","tag":"0CDY","title":"Picard groups of varieties · Lemma 0CDY","summary":"Let A → B be a faithfully flat ring map. Let X be a scheme over A such that • X is quasi-compact and quasi-separated, and • R = H^0(X, O_X) is a semi-local ring. Then the pullback map Pic(X) → Pic(X_B) is injective.","statement_latex":"Let $A \\to B$ be a faithfully flat ring map.\nLet $X$ be a scheme over $A$ such that\n\\begin{enumerate}\n\\item $X$ is quasi-compact and quasi-separated, and\n\\item $R = H^0(X, \\mathcal{O}_X)$ is a semi-local ring.\n\\end{enumerate}\nThen the pullback map $\\Pic(X) \\to \\Pic(X_B)$ is injective.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Picard groups of varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDY","source_file":"varieties.tex","source_line":5782,"source_end_line":5791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5782-L5791","statement_sha256":"481fe788002740674c5ddc6504ebbcd0086d60626866d8db911aec1365d9af13","origin":"The Stacks Project","memory_eligible":false,"source_rank":6486,"rank":6486,"depth":31,"x":284.439,"y":1082.706,"cluster":"varieties-curves"},{"id":"stacks:0CC5","tag":"0CC5","title":"Picard groups of varieties · Lemma 0CC5","summary":"Let k'/k be a field extension. Let X be a scheme over k such that • X is quasi-compact and quasi-separated, and • R = H^0(X, O_X) is semi-local, e.g., if dim_k R < ∞. Then the pullback map Pic(X) → Pic(X_k') is injective.","statement_latex":"Let $k'/k$ be a field extension.\nLet $X$ be a scheme over $k$ such that\n\\begin{enumerate}\n\\item $X$ is quasi-compact and quasi-separated, and\n\\item $R = H^0(X, \\mathcal{O}_X)$ is semi-local, e.g., if $\\dim_k R < \\infty$.\n\\end{enumerate}\nThen the pullback map $\\Pic(X) \\to \\Pic(X_{k'})$ is injective.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Picard groups of varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CC5","source_file":"varieties.tex","source_line":5807,"source_end_line":5816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5807-L5816","statement_sha256":"16bf9662027e0e52da18874b08e3e057bcf658d099b018c0fd357bfd98cd5bf9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6487,"rank":6487,"depth":32,"x":235.953,"y":1213.394,"cluster":"varieties-curves"},{"id":"stacks:0BEH","tag":"0BEH","title":"Picard groups of varieties · Lemma 0BEH","summary":"Let k be a field. Let X be a normal variety over k. Let U ⊂ A^n_k be an open subscheme with k-rational points p, q ∈ U(k). For every invertible module L on X ×_Spec(k) U the restrictions L|_X × p and L|_X × q are isomorphic.","statement_latex":"Let $k$ be a field. Let $X$ be a normal variety over $k$.\nLet $U \\subset \\mathbf{A}^n_k$ be an open subscheme with\n$k$-rational points $p, q \\in U(k)$. For every invertible\nmodule $\\mathcal{L}$ on $X \\times_{\\Spec(k)} U$ the restrictions\n$\\mathcal{L}|_{X \\times p}$ and $\\mathcal{L}|_{X \\times q}$\nare isomorphic.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Picard groups of varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEH","source_file":"varieties.tex","source_line":5866,"source_end_line":5874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5866-L5874","statement_sha256":"f46579c9c8a2e55500f1fc82641ea5ed8d9a1cac54099917a93542ee5da467ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":6488,"rank":6488,"depth":27,"x":166.368,"y":1089.081,"cluster":"varieties-curves"},{"id":"stacks:04MK","tag":"04MK","title":"Uniqueness of base field · Proposition 04MK","summary":"Let X be a scheme. Let a : X → Spec(k_1) and b : X → Spec(k_2) be morphisms from X to spectra of fields. Assume a, b are locally of finite type, and X is reduced, and connected. Then we have k_1' = k_2', where k_i' ⊂ Γ(X, O_X) is the integral closure of k_i in Γ(X, O_X).","statement_latex":"Let $X$ be a scheme. Let $a : X \\to \\Spec(k_1)$ and\n$b : X \\to \\Spec(k_2)$ be morphisms from $X$ to spectra of fields.\nAssume $a, b$ are locally of finite type, and\n$X$ is reduced, and connected. Then we have\n$k_1' = k_2'$, where $k_i' \\subset \\Gamma(X, \\mathcal{O}_X)$ is\nthe integral closure of $k_i$ in $\\Gamma(X, \\mathcal{O}_X)$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Uniqueness of base field","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MK","source_file":"varieties.tex","source_line":5964,"source_end_line":5972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L5964-L5972","statement_sha256":"dd7d905089980e31042029aace185f9f2586c8142c646ab4dd1c0d78f09c5c31","origin":"The Stacks Project","memory_eligible":false,"source_rank":6489,"rank":6489,"depth":40,"x":318.172,"y":1141.445,"cluster":"varieties-curves"},{"id":"stacks:0G05","tag":"0G05","title":"Automorphisms · Lemma 0G05","summary":"Let X be a reduced scheme of finite type over a field k. Let f : X → X be an automorphism over k which induces the identity map on the underlying topological space of X. Then • f^*F ≅ F for every coherent O_X-module, and • if dim(Z) > 0 for every irreducible component Z ⊂ X, then f is the identity.","statement_latex":"Let $X$ be a reduced scheme of finite type over a field $k$. Let $f : X \\to X$\nbe an automorphism over $k$ which induces the identity map on the underlying\ntopological space of $X$. Then\n\\begin{enumerate}\n\\item $f^*\\mathcal{F} \\cong \\mathcal{F}$ for every coherent\n$\\mathcal{O}_X$-module, and\n\\item if $\\dim(Z) > 0$ for every irreducible component $Z \\subset X$,\nthen $f$ is the identity.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Automorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G05","source_file":"varieties.tex","source_line":6072,"source_end_line":6083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6072-L6083","statement_sha256":"0113ac5090ff92ac943eae91c7b156aec1a3bb29bd1348c638be205160d36272","origin":"The Stacks Project","memory_eligible":false,"source_rank":6490,"rank":6490,"depth":0,"x":163.593,"y":1189.133,"cluster":"varieties-curves"},{"id":"stacks:0BEJ","tag":"0BEJ","title":"Euler characteristics · Definition 0BEJ","summary":"Let k be a field. Let X be a proper scheme over k. Let F be a coherent O_X-module. In this situation the Euler characteristic of F is the integer chi(X, F) = ∑_i (-1)^i dim_k H^i(X, F). For justification of the formula see below.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$\nbe a coherent $\\mathcal{O}_X$-module. In this situation the\n{\\it Euler characteristic of $\\mathcal{F}$} is the integer\n$$\n\\chi(X, \\mathcal{F}) = \\sum\\nolimits_i (-1)^i \\dim_k H^i(X, \\mathcal{F}).\n$$\nFor justification of the formula see below.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Euler characteristics","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEJ","source_file":"varieties.tex","source_line":6160,"source_end_line":6169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6160-L6169","statement_sha256":"c8ed65430172e8311ed6331eaec832b2d07df843ad8d96446ebd1272bdc86150","origin":"The Stacks Project","memory_eligible":false,"source_rank":6491,"rank":6491,"depth":0,"x":239.488,"y":1065.837,"cluster":"varieties-curves"},{"id":"stacks:08AA","tag":"08AA","title":"Euler characteristics · Lemma 08AA","summary":"Let k be a field. Let X be a proper scheme over k. Let 0 → F_1 → F_2 → F_3 → 0 be a short exact sequence of coherent modules on X. Then chi(X, F_2) = chi(X, F_1) + chi(X, F_3)","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\nLet $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nbe a short exact sequence of coherent modules on $X$. Then\n$$\n\\chi(X, \\mathcal{F}_2) = \\chi(X, \\mathcal{F}_1) + \\chi(X, \\mathcal{F}_3)\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Euler characteristics","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AA","source_file":"varieties.tex","source_line":6182,"source_end_line":6190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6182-L6190","statement_sha256":"e90b00654f4dbc915ad5c724c27262024d1b65aee5a8731e869f8771cfe2e0e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6492,"rank":6492,"depth":0,"x":282.822,"y":1200.274,"cluster":"varieties-curves"},{"id":"stacks:0AYT","tag":"0AYT","title":"Euler characteristics · Lemma 0AYT","summary":"Let k be a field. Let X be a proper scheme over k. Let F be a coherent sheaf with dim(Supp(F)) ≤ 0. Then • F is generated by global sections, • H^0(X, F) = bigoplus_x ∈ Supp(F) F_x, • H^i(X, F) = 0 for i > 0, • chi(X, F) = dim_k H^0(X, F), and • chi(X, F ⊗ E) = nchi(X, F) for every locally free module E of rank n.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$\nbe a coherent sheaf with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq 0$.\nThen\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is generated by global sections,\n\\item $H^0(X, \\mathcal{F}) =\n\\bigoplus_{x \\in \\text{Supp}(\\mathcal{F})} \\mathcal{F}_x$,\n\\item $H^i(X, \\mathcal{F}) = 0$ for $i > 0$,\n\\item $\\chi(X, \\mathcal{F}) = \\dim_k H^0(X, \\mathcal{F})$, and\n\\item\n$\\chi(X, \\mathcal{F} \\otimes \\mathcal{E}) = n\\chi(X, \\mathcal{F})$\nfor every locally free module $\\mathcal{E}$ of rank $n$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Euler characteristics","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYT","source_file":"varieties.tex","source_line":6207,"source_end_line":6222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6207-L6222","statement_sha256":"24ac3c0c7bfec457a8b567c252ece76de8a1ff74cb1d168ef942b2fb73e8b252","origin":"The Stacks Project","memory_eligible":false,"source_rank":6493,"rank":6493,"depth":26,"x":142.286,"y":1125.479,"cluster":"varieties-curves"},{"id":"stacks:08AB","tag":"08AB","title":"Euler characteristics · Lemma 08AB","summary":"Let k'/k be an extension of fields. Let X be a proper scheme over k. Let F be a coherent sheaf on X. Let F' be the pullback of F to X_k'. Then chi(X, F) = chi(X', F').","statement_latex":"Let $k'/k$ be an extension of fields. Let $X$ be a proper scheme\nover $k$. Let $\\mathcal{F}$ be a coherent sheaf on $X$.\nLet $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $X_{k'}$.\nThen $\\chi(X, \\mathcal{F}) = \\chi(X', \\mathcal{F}')$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Euler characteristics","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AB","source_file":"varieties.tex","source_line":6259,"source_end_line":6265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6259-L6265","statement_sha256":"853dc4875021e43aa300cd3e1c2b9e472e9824a95849169352102d524eb6f0cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6494,"rank":6494,"depth":30,"x":306.612,"y":1100.806,"cluster":"varieties-curves"},{"id":"stacks:0BEK","tag":"0BEK","title":"Euler characteristics · Lemma 0BEK","summary":"Let k be a field. Let f : Y → X be a morphism of proper schemes over k. Let G be a coherent O_Y-module. Then chi(Y, G) = ∑ (-1)^i chi(X, R^if_*G)","statement_latex":"Let $k$ be a field. Let $f : Y \\to X$ be a morphism of proper schemes over\n$k$. Let $\\mathcal{G}$ be a coherent $\\mathcal{O}_Y$-module. Then\n$$\n\\chi(Y, \\mathcal{G}) = \\sum (-1)^i \\chi(X, R^if_*\\mathcal{G})\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Euler characteristics","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEK","source_file":"varieties.tex","source_line":6276,"source_end_line":6283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6276-L6283","statement_sha256":"f896346c4352aa31dedb64740c2ab5f493df570af880e5a1afbfdd95a658712d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6495,"rank":6495,"depth":31,"x":204.944,"y":1212.612,"cluster":"varieties-curves"},{"id":"stacks:0B2P","tag":"0B2P","title":"Projective space · Lemma 0B2P","summary":"Projective space is smooth. Let k be a field and n ≥ 0. Then P^n_k is a smooth projective variety of dimension n over k.","statement_latex":"\\begin{slogan}\nProjective space is smooth.\n\\end{slogan}\nLet $k$ be a field and $n \\geq 0$. Then $\\mathbf{P}^n_k$ is a\nsmooth projective variety of dimension $n$ over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2P","source_file":"varieties.tex","source_line":6325,"source_end_line":6332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6325-L6332","statement_sha256":"a0fbb91ddef5ea5a73ba96e9bd92edde7bb94bec668279b3325731e39004bbea","origin":"The Stacks Project","memory_eligible":false,"source_rank":6496,"rank":6496,"depth":0,"x":189.952,"y":1072.016,"cluster":"varieties-curves"},{"id":"stacks:0B2Q","tag":"0B2Q","title":"Projective space · Lemma 0B2Q","summary":"Let k be a field and n ≥ 0. Let X, Y ⊂ A^n_k be closed subsets. Assume that X and Y are equidimensional, dim(X) = r and dim(Y) = s. Then every irreducible component of X ∩ Y has dimension ≥ r + s - n.","statement_latex":"Let $k$ be a field and $n \\geq 0$. Let $X, Y \\subset \\mathbf{A}^n_k$\nbe closed subsets. Assume that $X$ and $Y$ are equidimensional,\n$\\dim(X) = r$ and $\\dim(Y) = s$.\nThen every irreducible component of $X \\cap Y$ has dimension $\\geq r + s - n$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2Q","source_file":"varieties.tex","source_line":6338,"source_end_line":6344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6338-L6344","statement_sha256":"87aea55a963d1e840544866c38272d17d5b1c91b5c71389b41f154d3d9b87f93","origin":"The Stacks Project","memory_eligible":false,"source_rank":6497,"rank":6497,"depth":28,"x":314.478,"y":1167.496,"cluster":"varieties-curves"},{"id":"stacks:0B2R","tag":"0B2R","title":"Projective space · Lemma 0B2R","summary":"Let k be a field and n ≥ 0. Let X, Y ⊂ P^n_k be nonempty closed subsets. If dim(X) = r and dim(Y) = s and r + s ≥ n, then X ∩ Y is nonempty and dim(X ∩ Y) ≥ r + s - n.","statement_latex":"Let $k$ be a field and $n \\geq 0$. Let $X, Y \\subset \\mathbf{P}^n_k$\nbe nonempty closed subsets. If $\\dim(X) = r$ and $\\dim(Y) = s$ and\n$r + s \\geq n$, then $X \\cap Y$ is nonempty and\n$\\dim(X \\cap Y) \\geq r + s - n$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2R","source_file":"varieties.tex","source_line":6365,"source_end_line":6371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6365-L6371","statement_sha256":"6fa971ae4c6c9bfc384e4740cfd656ae124f52ffef7581cb9807bdba8f7c9cff","origin":"The Stacks Project","memory_eligible":false,"source_rank":6498,"rank":6498,"depth":35,"x":145.322,"y":1167.75,"cluster":"varieties-curves"},{"id":"stacks:0BXU","tag":"0BXU","title":"Projective space · Lemma 0BXU","summary":"Let k be a field. Let Z ⊂ P^n_k be a closed subscheme which has no embedded points such that every irreducible component of Z has dimension n - 1. Then the ideal I(Z) ⊂ k[T_0, …, T_n] corresponding to Z is principal.","statement_latex":"Let $k$ be a field. Let $Z \\subset \\mathbf{P}^n_k$ be a closed subscheme\nwhich has no embedded points such that every irreducible component\nof $Z$ has dimension $n - 1$. Then the ideal $I(Z) \\subset k[T_0, \\ldots, T_n]$\ncorresponding to $Z$ is principal.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXU","source_file":"varieties.tex","source_line":6412,"source_end_line":6418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6412-L6418","statement_sha256":"65439382502ec07276adb407f6e788caa6c4871a2b9613bf2ce84b900e71f26c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6499,"rank":6499,"depth":29,"x":270.255,"y":1071.268,"cluster":"varieties-curves"},{"id":"stacks:089Z","tag":"089Z","title":"Coherent sheaves on projective space · Lemma 089Z","summary":"Let k be a field. Let n ≥ 1. Let i : H → P^n_k be a hyperplane. Then there exists an isomorphism φ : P^n - 1_k → H such that i^*O(1) pulls back to O(1).","statement_latex":"Let $k$ be a field. Let $n \\geq 1$.\nLet $i : H \\to \\mathbf{P}^n_k$ be a hyperplane.\nThen there exists an isomorphism\n$$\n\\varphi : \\mathbf{P}^{n - 1}_k \\longrightarrow H\n$$\nsuch that $i^*\\mathcal{O}(1)$ pulls back to $\\mathcal{O}(1)$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089Z","source_file":"varieties.tex","source_line":6454,"source_end_line":6463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6454-L6463","statement_sha256":"6d39a95a381d6982df81a65252b1bb7bfa5b5d9bd12ad85a74df991da00fe74a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6500,"rank":6500,"depth":5,"x":255.671,"y":1213.759,"cluster":"varieties-curves"},{"id":"stacks:08A0","tag":"08A0","title":"Coherent sheaves on projective space · Lemma 08A0","summary":"Let k be an infinite field. Let n ≥ 1. Let F be a coherent module on P^n_k. Then there exist a nonzero section s ∈ Γ(P^n_k, O(1)) and a short exact sequence 0 → F(-1) → F → i_*G → 0 where i : H → P^n_k is the hyperplane H associated to s and G = i^*F.","statement_latex":"Let $k$ be an infinite field. Let $n \\geq 1$.\nLet $\\mathcal{F}$ be a coherent module on $\\mathbf{P}^n_k$.\nThen there exist a nonzero section\n$s \\in \\Gamma(\\mathbf{P}^n_k, \\mathcal{O}(1))$\nand a short exact sequence\n$$\n0 \\to \\mathcal{F}(-1) \\to \\mathcal{F} \\to i_*\\mathcal{G} \\to 0\n$$\nwhere $i : H \\to \\mathbf{P}^n_k$ is the hyperplane $H$ associated to $s$\nand $\\mathcal{G} = i^*\\mathcal{F}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08A0","source_file":"varieties.tex","source_line":6495,"source_end_line":6507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6495-L6507","statement_sha256":"5332fd620f029a00c9b8a962d30e598a6950c4b9461e1b40495b7ddf6df32876","origin":"The Stacks Project","memory_eligible":false,"source_rank":6501,"rank":6501,"depth":18,"x":151.506,"y":1100.051,"cluster":"varieties-curves"},{"id":"stacks:08A3","tag":"08A3","title":"Coherent sheaves on projective space · Definition 08A3","summary":"Let k be a field. Let n ≥ 0. Let F be a coherent sheaf on P^n_k. We say F is m-regular if H^i(P^n_k, F(m - i)) = 0 for i = 1, …, n.","statement_latex":"Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent\nsheaf on $\\mathbf{P}^n_k$. We say $\\mathcal{F}$ is {\\it $m$-regular}\nif\n$$\nH^i(\\mathbf{P}^n_k, \\mathcal{F}(m - i)) = 0\n$$\nfor $i = 1, \\ldots, n$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08A3","source_file":"varieties.tex","source_line":6600,"source_end_line":6609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6600-L6609","statement_sha256":"102e4c4cac5c764b9da5f5ffbd3b3a5efe759173a15c43846e662ead1fe1de6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6502,"rank":6502,"depth":0,"x":320.292,"y":1124.871,"cluster":"varieties-curves"},{"id":"stacks:08A4","tag":"08A4","title":"Coherent sheaves on projective space · Lemma 08A4","summary":"Let k'/k be an extension of fields. Let n ≥ 0. Let F be a coherent sheaf on P^n_k. Let F' be the pullback of F to P^n_k'. Then F is m-regular if and only if F' is m-regular.","statement_latex":"Let $k'/k$ be an extension of fields. Let $n \\geq 0$.\nLet $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$.\nLet $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $\\mathbf{P}^n_{k'}$.\nThen $\\mathcal{F}$ is $m$-regular if and only if $\\mathcal{F}'$ is\n$m$-regular.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08A4","source_file":"varieties.tex","source_line":6618,"source_end_line":6625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6618-L6625","statement_sha256":"48516039fff09b8312e1884a26abc8418ab5e44e71b3bc68489f4ffc23e54590","origin":"The Stacks Project","memory_eligible":false,"source_rank":6503,"rank":6503,"depth":30,"x":175.415,"y":1202.586,"cluster":"varieties-curves"},{"id":"stacks:08A5","tag":"08A5","title":"Coherent sheaves on projective space · Lemma 08A5","summary":"In the situation of Lemma [Tag 08A0], if F is m-regular, then G is m-regular on H ≅ P^n - 1_k.","statement_latex":"In the situation of Lemma \\ref{lemma-exact-sequence-induction},\nif $\\mathcal{F}$ is $m$-regular, then $\\mathcal{G}$ is $m$-regular\non $H \\cong \\mathbf{P}^{n - 1}_k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08A5","source_file":"varieties.tex","source_line":6637,"source_end_line":6642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6637-L6642","statement_sha256":"fa1aa44a3f0c9024ad971e30fc0c32c5c1c1d745d42d6f6813dfe8c08ac6ebfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6504,"rank":6504,"depth":26,"x":219.888,"y":1062.636,"cluster":"varieties-curves"},{"id":"stacks:08A6","tag":"08A6","title":"Coherent sheaves on projective space · Lemma 08A6","summary":"Let k be a field. Let n ≥ 0. Let F be a coherent sheaf on P^n_k. If F is m-regular, then F is (m + 1)-regular.","statement_latex":"Let $k$ be a field. Let $n \\geq 0$.\nLet $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$.\nIf $\\mathcal{F}$ is $m$-regular, then $\\mathcal{F}$ is\n$(m + 1)$-regular.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08A6","source_file":"varieties.tex","source_line":6657,"source_end_line":6663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6657-L6663","statement_sha256":"2b58e9dcc5549d41dde88fa85990e3d9ab8f8908b9ecddda2a63e3a372ff5daf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6505,"rank":6505,"depth":31,"x":299.887,"y":1191.469,"cluster":"varieties-curves"},{"id":"stacks:08A7","tag":"08A7","title":"Coherent sheaves on projective space · Lemma 08A7","summary":"Let k be a field. Let n ≥ 0. Let F be a coherent sheaf on P^n_k. If F is m-regular, then the multiplication map H^0(P^n_k, F(m)) ⊗_k H^0(P^n_k, O(1)) → H^0(P^n_k, F(m + 1)) is surjective.","statement_latex":"Let $k$ be a field. Let $n \\geq 0$.\nLet $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$.\nIf $\\mathcal{F}$ is $m$-regular, then the multiplication map\n$$\nH^0(\\mathbf{P}^n_k, \\mathcal{F}(m)) \\otimes_k\nH^0(\\mathbf{P}^n_k, \\mathcal{O}(1))\n\\longrightarrow\nH^0(\\mathbf{P}^n_k, \\mathcal{F}(m + 1))\n$$\nis surjective.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08A7","source_file":"varieties.tex","source_line":6686,"source_end_line":6698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6686-L6698","statement_sha256":"fb89f634570cd77669e8be51fa2a4dd6817c0a702a84aa98df27a7627f9b37ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":6506,"rank":6506,"depth":31,"x":136.79,"y":1141.708,"cluster":"varieties-curves"},{"id":"stacks:08A8","tag":"08A8","title":"Coherent sheaves on projective space · Lemma 08A8","summary":"Let k be a field. Let n ≥ 0. Let F be a coherent sheaf on P^n_k. If F is m-regular, then F(m) is globally generated.","statement_latex":"Let $k$ be a field. Let $n \\geq 0$.\nLet $\\mathcal{F}$ be a coherent sheaf on $\\mathbf{P}^n_k$.\nIf $\\mathcal{F}$ is $m$-regular, then $\\mathcal{F}(m)$ is\nglobally generated.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08A8","source_file":"varieties.tex","source_line":6741,"source_end_line":6747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6741-L6747","statement_sha256":"2e9d989c6ea372631deefff7ef2f8ce9e0d0a17e2c218dd8477b99c58ab2e949","origin":"The Stacks Project","memory_eligible":false,"source_rank":6507,"rank":6507,"depth":32,"x":297.565,"y":1085.684,"cluster":"varieties-curves"},{"id":"stacks:08AC","tag":"08AC","title":"Coherent sheaves on projective space · Lemma 08AC","summary":"Let k be a field. Let n ≥ 0. Let F be a coherent sheaf on P^n_k. The function d ↦ chi(P^n_k, F(d)) is a polynomial.","statement_latex":"Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf\non $\\mathbf{P}^n_k$. The function\n$$\nd \\longmapsto \\chi(\\mathbf{P}^n_k, \\mathcal{F}(d))\n$$\nis a polynomial.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AC","source_file":"varieties.tex","source_line":6769,"source_end_line":6777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6769-L6777","statement_sha256":"8d5f614038562a0b37c7df81f8d988311a402a503d2428847ef24cff24eb5edd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6508,"rank":6508,"depth":31,"x":223.839,"y":1218.629,"cluster":"varieties-curves"},{"id":"stacks:08AD","tag":"08AD","title":"Coherent sheaves on projective space · Definition 08AD","summary":"Let k be a field. Let n ≥ 0. Let F be a coherent sheaf on P^n_k. The function d ↦ chi(P^n_k, F(d)) is called the Hilbert polynomial of F.","statement_latex":"Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf\non $\\mathbf{P}^n_k$. The function\n$d \\mapsto \\chi(\\mathbf{P}^n_k, \\mathcal{F}(d))$ is called the\n{\\it Hilbert polynomial} of $\\mathcal{F}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AD","source_file":"varieties.tex","source_line":6806,"source_end_line":6812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6806-L6812","statement_sha256":"4966729c65d0cebfe99bf06e2d467a64313a96872d418abf85d585bcca03c65f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6509,"rank":6509,"depth":0,"x":171.134,"y":1078.338,"cluster":"varieties-curves"},{"id":"stacks:08AE","tag":"08AE","title":"Coherent sheaves on projective space · Lemma 08AE","summary":"Let k be a field. Let n ≥ 0. Let F be a coherent sheaf on P^n_k with Hilbert polynomial P ∈ Q[t]. Then P(d) = dim_k H^0(P^n_k, F(d)) for all d gg 0.","statement_latex":"Let $k$ be a field. Let $n \\geq 0$. Let $\\mathcal{F}$ be a coherent sheaf\non $\\mathbf{P}^n_k$ with Hilbert polynomial $P \\in \\mathbf{Q}[t]$.\nThen\n$$\nP(d) = \\dim_k H^0(\\mathbf{P}^n_k, \\mathcal{F}(d))\n$$\nfor all $d \\gg 0$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AE","source_file":"varieties.tex","source_line":6827,"source_end_line":6836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6827-L6836","statement_sha256":"f6dfccf4b2811dfd93518b2e4788b111ab0a9527263070b4bbe41da39952ee3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6510,"rank":6510,"depth":27,"x":323.274,"y":1152.103,"cluster":"varieties-curves"},{"id":"stacks:08AG","tag":"08AG","title":"Coherent sheaves on projective space · Lemma 08AG","summary":"Let k be a field. Let n ≥ 0. Let r ≥ 1. Let P ∈ Q[t]. There exists an integer m depending on n, r, and P with the following property: if 0 → K → O^⊕ r → F → 0 is a short exact sequence of coherent sheaves on P^n_k and F has Hilbert polynomial P, then K is m-regular.","statement_latex":"Let $k$ be a field. Let $n \\geq 0$. Let $r \\geq 1$. Let $P \\in \\mathbf{Q}[t]$.\nThere exists an integer $m$ depending on $n$, $r$, and $P$\nwith the following property: if\n$$\n0 \\to \\mathcal{K} \\to \\mathcal{O}^{\\oplus r} \\to \\mathcal{F} \\to 0\n$$\nis a short exact sequence of coherent sheaves on $\\mathbf{P}^n_k$\nand $\\mathcal{F}$ has Hilbert polynomial $P$, then\n$\\mathcal{K}$ is $m$-regular.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Coherent sheaves on projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AG","source_file":"varieties.tex","source_line":6855,"source_end_line":6866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L6855-L6866","statement_sha256":"2158da54c8698c0f90cb0ad5062fb85fa9e1a70226165dd7cb9ef833a16ed09d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6511,"rank":6511,"depth":32,"x":151.253,"y":1184.133,"cluster":"varieties-curves"},{"id":"stacks:03SM","tag":"03SM","title":"Frobenii · Definition 03SM","summary":"Let p be a prime number. Let X be a scheme in characteristic p. The absolute frobenius of X is the morphism F_X : X → X given by the identity on the underlying topological space and with F_X^sharp : O_X → O_X given by g ↦ g^p.","statement_latex":"Let $p$ be a prime number. Let $X$ be a scheme in characteristic $p$.\nThe {\\it absolute frobenius of $X$} is the morphism $F_X : X \\to X$\ngiven by the identity on the underlying topological space and\nwith $F_X^\\sharp : \\mathcal{O}_X \\to \\mathcal{O}_X$ given by $g \\mapsto g^p$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SM","source_file":"varieties.tex","source_line":7013,"source_end_line":7019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7013-L7019","statement_sha256":"8db7906d325b4fe0c86d28fa4b1962c7a3c347135612ea9907dc0b5fe3c8257c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6512,"rank":6512,"depth":0,"x":252.643,"y":1062.544,"cluster":"varieties-curves"},{"id":"stacks:0CC7","tag":"0CC7","title":"Frobenii · Lemma 0CC7","summary":"Let p > 0 be a prime number. Let f : X → Y be a morphism of schemes in characteristic p. Then the diagram xymatrix X ar[d]_f ar[r]_F_X & X ar[d]^f Y ar[r]^F_Y & Y commutes.","statement_latex":"Let $p > 0$ be a prime number.\nLet $f : X \\to Y$ be a morphism of schemes in characteristic $p$.\nThen the diagram\n$$\n\\xymatrix{\nX \\ar[d]_f \\ar[r]_{F_X} & X \\ar[d]^f \\\\\nY \\ar[r]^{F_Y} & Y\n}\n$$\ncommutes.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CC7","source_file":"varieties.tex","source_line":7031,"source_end_line":7043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7031-L7043","statement_sha256":"c429ff83a72ad3a6a7df80d0c4875899219bc7294c3db5fa3f7ed297e00bb27a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6513,"rank":6513,"depth":0,"x":275.726,"y":1210.17,"cluster":"varieties-curves"},{"id":"stacks:0CC8","tag":"0CC8","title":"Frobenii · Lemma 0CC8","summary":"Let p > 0 be a prime number. Let X be a scheme in characteristic p. Then the absolute frobenius F_X : X → X is a universal homeomorphism, is integral, and induces purely inseparable residue field extensions.","statement_latex":"Let $p > 0$ be a prime number. Let $X$ be a scheme in characteristic $p$.\nThen the absolute frobenius $F_X : X \\to X$\nis a universal homeomorphism, is integral, and\ninduces purely inseparable residue field extensions.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CC8","source_file":"varieties.tex","source_line":7051,"source_end_line":7057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7051-L7057","statement_sha256":"ffdbe95d8ede20daab6e8108cad923d62bc2a01e725a23a771a92faec742d79f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6514,"rank":6514,"depth":8,"x":139.588,"y":1114.128,"cluster":"varieties-curves"},{"id":"stacks:0CC9","tag":"0CC9","title":"Frobenii · Definition 0CC9","summary":"Let p > 0 be a prime number. Let S be a scheme in characteristic p. Let X be a scheme over S. We define X^(p) = X^(p/S) = X ×_S, F_S S viewed as a scheme over S. Applying Lemma [Tag 0CC7] we see there is a unique morphism F_X/S : X → X^(p) over S fitting into the commutative diagram xymatrix X ar[rr]_F_X/S ar[rrd] ar@/^1em/[rrrr]^F_X & & X^(p) ar[rr] ar[d] & & X ar[d] & & S ar[rr]^F_S & & S where the right square is cartesian. The morphism F_X/S is called the relative…","statement_latex":"Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.\nLet $X$ be a scheme over $S$. We define\n$$\nX^{(p)} = X^{(p/S)} = X \\times_{S, F_S} S\n$$\nviewed as a scheme over $S$. Applying\nLemma \\ref{lemma-frobenius-endomorphism-identity}\nwe see there is a unique morphism $F_{X/S} : X \\longrightarrow X^{(p)}$\nover $S$ fitting into the commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_{F_{X/S}} \\ar[rrd] \\ar@/^1em/[rrrr]^{F_X}\n& & X^{(p)} \\ar[rr] \\ar[d] & & X \\ar[d] \\\\\n& & S \\ar[rr]^{F_S} & & S\n}\n$$\nwhere the right square is cartesian. The morphism $F_{X/S}$ is called the\n{\\it relative Frobenius morphism of $X/S$}.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CC9","source_file":"varieties.tex","source_line":7070,"source_end_line":7090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7070-L7090","statement_sha256":"91608bbd263b44c422bba1c21ca08bcf0918cbd0d795201ea8718a331f1e31ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":6515,"rank":6515,"depth":1,"x":317.73,"y":1107.682,"cluster":"varieties-curves"},{"id":"stacks:0CCA","tag":"0CCA","title":"Frobenii · Lemma 0CCA","summary":"Let p > 0 be a prime number. Let S be a scheme in characteristic p. Let f : X → Y be a morphism of schemes over S . Then the diagram xymatrix X ar[d]_f ar[r]_F_X/S & X^(p) ar[d]^f^(p) Y ar[r]^F_Y/S & Y^(p) commutes.","statement_latex":"Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.\nLet $f : X \\to Y$ be a morphism of schemes over $S$ .\nThen the diagram\n$$\n\\xymatrix{\nX \\ar[d]_f \\ar[r]_{F_{X/S}} & X^{(p)} \\ar[d]^{f^{(p)}} \\\\\nY \\ar[r]^{F_{Y/S}} & Y^{(p)}\n}\n$$\ncommutes.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCA","source_file":"varieties.tex","source_line":7097,"source_end_line":7109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7097-L7109","statement_sha256":"b8084e14256bc7544fe7901d7e655119409c65652fd938a068300fca32b8c1da","origin":"The Stacks Project","memory_eligible":false,"source_rank":6516,"rank":6516,"depth":1,"x":191.187,"y":1213.825,"cluster":"varieties-curves"},{"id":"stacks:0CCB","tag":"0CCB","title":"Frobenii · Lemma 0CCB","summary":"Let p > 0 be a prime number. Let S be a scheme in characteristic p. Let X be a scheme over S. Then the relative frobenius F_X/S : X → X^(p) is a universal homeomorphism, is integral, and induces purely inseparable residue field extensions.","statement_latex":"Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.\nLet $X$ be a scheme over $S$.\nThen the relative frobenius $F_{X/S} : X \\to X^{(p)}$\nis a universal homeomorphism, is integral, and\ninduces purely inseparable residue field extensions.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCB","source_file":"varieties.tex","source_line":7116,"source_end_line":7123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7116-L7123","statement_sha256":"09b96e128bfdfac21a0144313d9b150e2eea926dd4fe03475a5ac3eb55e5b61d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6517,"rank":6517,"depth":22,"x":199.164,"y":1063.318,"cluster":"varieties-curves"},{"id":"stacks:0CCC","tag":"0CCC","title":"Frobenii · Lemma 0CCC","summary":"Let p > 0 be a prime number. Let S be a scheme in characteristic p. Let X be a scheme over S. Then Ω_X/S = Ω_X/X^(p).","statement_latex":"Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.\nLet $X$ be a scheme over $S$. Then $\\Omega_{X/S} = \\Omega_{X/X^{(p)}}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCC","source_file":"varieties.tex","source_line":7136,"source_end_line":7140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7136-L7140","statement_sha256":"af88141c9e5e16ac3ee7d90aef6aa4f5e2659fbdc371efb7709c2ed98b4b7d55","origin":"The Stacks Project","memory_eligible":false,"source_rank":6518,"rank":6518,"depth":0,"x":314.646,"y":1179.159,"cluster":"varieties-curves"},{"id":"stacks:0CCD","tag":"0CCD","title":"Frobenii · Lemma 0CCD","summary":"Let p > 0 be a prime number. Let S be a scheme in characteristic p. Let X be a scheme over S. If X → S is locally of finite type, then F_X/S is finite.","statement_latex":"Let $p > 0$ be a prime number. Let $S$ be a scheme in characteristic $p$.\nLet $X$ be a scheme over $S$. If $X \\to S$ is locally of finite type,\nthen $F_{X/S}$ is finite.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCD","source_file":"varieties.tex","source_line":7153,"source_end_line":7158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7153-L7158","statement_sha256":"d1e25603c4cd70c7f052b06c1c9578e0eb1c3d1ff92deed9c191bcf6d545eaf8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6519,"rank":6519,"depth":6,"x":135.824,"y":1159.21,"cluster":"varieties-curves"},{"id":"stacks:0CCE","tag":"0CCE","title":"Frobenii · Lemma 0CCE","summary":"Let k be a field of characteristic p > 0. Let X be a scheme over k. Then X is geometrically reduced if and only if X^(p) is reduced.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $X$ be a scheme over $k$.\nThen $X$ is geometrically reduced if and only if $X^{(p)}$ is reduced.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCE","source_file":"varieties.tex","source_line":7172,"source_end_line":7176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7172-L7176","statement_sha256":"71d43c1cb8a31d7e435f345a7d45d191bc2c707c83efbd94305d4c439d93966b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6520,"rank":6520,"depth":10,"x":284.149,"y":1072.206,"cluster":"varieties-curves"},{"id":"stacks:0CCF","tag":"0CCF","title":"Frobenii · Lemma 0CCF","summary":"Let k be a field of characteristic p > 0. Let X be a variety over k. The following are equivalent • X^(p) is reduced, • X is geometrically reduced, • there is a nonempty open U ⊂ X smooth over k. In this case X^(p) is a variety over k and F_X/k : X → X^(p) is a finite dominant morphism of degree p^dim(X).","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $X$ be a variety over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X^{(p)}$ is reduced,\n\\item $X$ is geometrically reduced,\n\\item there is a nonempty open $U \\subset X$ smooth over $k$.\n\\end{enumerate}\nIn this case $X^{(p)}$ is a variety over $k$ and $F_{X/k} : X \\to X^{(p)}$\nis a finite dominant morphism of degree $p^{\\dim(X)}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCF","source_file":"varieties.tex","source_line":7185,"source_end_line":7196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7185-L7196","statement_sha256":"3474428c1460d0265feccf7f737b5418dbe2a178796ee129615a382f2e54fc19","origin":"The Stacks Project","memory_eligible":false,"source_rank":6521,"rank":6521,"depth":45,"x":244.631,"y":1220.944,"cluster":"varieties-curves"},{"id":"stacks:09MZ","tag":"09MZ","title":"Glueing dimension one rings · Lemma 09MZ","summary":"In Situation [Tag 09MY] assume that B is a valuation ring. Then for every unit u of A either u ∈ R or u^-1 ∈ R.","statement_latex":"In Situation \\ref{situation-glue} assume that $B$ is a valuation ring.\nThen for every unit $u$ of $A$ either $u \\in R$ or $u^{-1} \\in R$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MZ","source_file":"varieties.tex","source_line":7316,"source_end_line":7320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7316-L7320","statement_sha256":"046075233bc09589f1f937e317e792f7e9a9ffaa6beba552ffce35d8ad985eaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":6522,"rank":6522,"depth":8,"x":153.905,"y":1088.471,"cluster":"varieties-curves"},{"id":"stacks:09N0","tag":"09N0","title":"Glueing dimension one rings · Lemma 09N0","summary":"In Situation [Tag 09MY] assume A is a Noetherian ring of dimension 1. The following are equivalent • A ⊗ B → K is not surjective, • there exists a discrete valuation ring O ⊂ K containing both A and B.","statement_latex":"In Situation \\ref{situation-glue} assume $A$ is a Noetherian\nring of dimension $1$. The following are equivalent\n\\begin{enumerate}\n\\item $A \\otimes B \\to K$ is not surjective,\n\\item there exists a discrete valuation ring $\\mathcal{O} \\subset K$\ncontaining both $A$ and $B$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09N0","source_file":"varieties.tex","source_line":7334,"source_end_line":7343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7334-L7343","statement_sha256":"f398cb75f4b57120e2e767c301b57c852be8315482dcf0fd9b6be9c7d49e8474","origin":"The Stacks Project","memory_eligible":false,"source_rank":6523,"rank":6523,"depth":11,"x":327.822,"y":1134.804,"cluster":"varieties-curves"},{"id":"stacks:09N1","tag":"09N1","title":"Glueing dimension one rings · Lemma 09N1","summary":"In Situation [Tag 09MY] assume • A is a Noetherian semi-local domain of dimension 1, • B is a discrete valuation ring, Then we have the following two possibilities • [(a)] If A^* is not contained in R, then Spec(A) → Spec(R) and Spec(B) → Spec(R) are open immersions covering Spec(R) and K = A ⊗_R B. • [(b)] If A^* is contained in R, then B dominates one of the local rings of A at a maximal ideal and A ⊗ B → K is not surjective.","statement_latex":"In Situation \\ref{situation-glue} assume\n\\begin{enumerate}\n\\item $A$ is a Noetherian semi-local domain of dimension $1$,\n\\item $B$ is a discrete valuation ring,\n\\end{enumerate}\nThen we have the following two possibilities\n\\begin{enumerate}\n\\item[(a)] If $A^*$ is not contained in $R$, then\n$\\Spec(A) \\to \\Spec(R)$ and $\\Spec(B) \\to \\Spec(R)$\nare open immersions covering $\\Spec(R)$ and $K = A \\otimes_R B$.\n\\item[(b)] If $A^*$ is contained in $R$, then $B$ dominates one of\nthe local rings of $A$ at a maximal ideal and $A \\otimes B \\to K$\nis not surjective.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09N1","source_file":"varieties.tex","source_line":7356,"source_end_line":7372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7356-L7372","statement_sha256":"94ec8d14e7cb0e05e4ae7b202aa0443e009070eaf0ad6e27d9bf7f7c92d4198e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6524,"rank":6524,"depth":0,"x":161.857,"y":1199.502,"cluster":"varieties-curves"},{"id":"stacks:09N2","tag":"09N2","title":"Glueing dimension one rings · Lemma 09N2","summary":"Let B be a semi-local Noetherian domain of dimension 1. Let B' be the integral closure of B in its fraction field. Then B' is a semi-local Dedekind domain. Let x be a nonzero element of the Jacobson radical of B'. Then for every y ∈ B' there exists an n such that x^n y ∈ B.","statement_latex":"Let $B$ be a semi-local Noetherian domain of dimension $1$.\nLet $B'$ be the integral closure of $B$ in its fraction field.\nThen $B'$ is a semi-local Dedekind domain.\nLet $x$ be a nonzero element of the Jacobson radical of $B'$.\nThen for every $y \\in B'$ there exists an $n$ such that\n$x^n y \\in B$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09N2","source_file":"varieties.tex","source_line":7416,"source_end_line":7424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7416-L7424","statement_sha256":"be835e40db953208d4a545db7d5cce483b4cd50592998e967a02562c4f74eba0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6525,"rank":6525,"depth":18,"x":232.404,"y":1057.23,"cluster":"varieties-curves"},{"id":"stacks:09N3","tag":"09N3","title":"Glueing dimension one rings · Lemma 09N3","summary":"In Situation [Tag 09MY] assume • A is a Noetherian semi-local domain of dimension 1, • B is a Noetherian semi-local domain of dimension 1, • A ⊗ B → K is surjective. Then Spec(A) → Spec(R) and Spec(B) → Spec(R) are open immersions covering Spec(R) and K = A ⊗_R B.","statement_latex":"In Situation \\ref{situation-glue} assume\n\\begin{enumerate}\n\\item $A$ is a Noetherian semi-local domain of dimension $1$,\n\\item $B$ is a Noetherian semi-local domain of dimension $1$,\n\\item $A \\otimes B \\to K$ is surjective.\n\\end{enumerate}\nThen $\\Spec(A) \\to \\Spec(R)$ and $\\Spec(B) \\to \\Spec(R)$\nare open immersions covering $\\Spec(R)$ and $K = A \\otimes_R B$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09N3","source_file":"varieties.tex","source_line":7446,"source_end_line":7456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7446-L7456","statement_sha256":"830bc0a390738e0b8b2cf2622a59b0abc65cd0710bc2e0da738902cae776b3d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6526,"rank":6526,"depth":19,"x":294.966,"y":1202.568,"cluster":"varieties-curves"},{"id":"stacks:09N4","tag":"09N4","title":"Glueing dimension one rings · Lemma 09N4","summary":"Let K be a field. Let A_1, …, A_r ⊂ K be Noetherian semi-local rings of dimension 1 with fraction field K. If A_i ⊗ A_j → K is surjective for all i not = j, then there exists a Noetherian semi-local domain A ⊂ K of dimension 1 contained in A_1, …, A_r such that • A → A_i induces an open immersion j_i : Spec(A_i) → Spec(A), • Spec(A) is the union of the opens j_i(Spec(A_i)), • each closed point of Spec(A) lies in exactly one of these opens.","statement_latex":"Let $K$ be a field. Let $A_1, \\ldots, A_r \\subset K$ be Noetherian\nsemi-local rings of dimension $1$ with fraction field $K$. If\n$A_i \\otimes A_j \\to K$ is surjective for all $i \\not = j$, then\nthere exists a Noetherian semi-local domain $A \\subset K$\nof dimension $1$ contained in $A_1, \\ldots, A_r$ such that\n\\begin{enumerate}\n\\item $A \\to A_i$ induces an open immersion $j_i : \\Spec(A_i) \\to \\Spec(A)$,\n\\item $\\Spec(A)$ is the union of the opens $j_i(\\Spec(A_i))$,\n\\item each closed point of $\\Spec(A)$ lies in exactly one of these\nopens.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09N4","source_file":"varieties.tex","source_line":7508,"source_end_line":7521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7508-L7521","statement_sha256":"f73627bf3a63a06d88e9a33a65a1946c64a289b3d7d9ee2e371da27f8c3dd1f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6527,"rank":6527,"depth":20,"x":131.511,"y":1130.701,"cluster":"varieties-curves"},{"id":"stacks:09N5","tag":"09N5","title":"Glueing dimension one rings · Lemma 09N5","summary":"Let A be a domain with fraction field K. Let B_1, …, B_r ⊂ K be Noetherian 1-dimensional semi-local domains whose fraction fields are K. If A ⊗ B_i → K are surjective for i = 1, …, r, then there exists an x ∈ A such that x^-1 is in the Jacobson radical of B_i for i = 1, …, r.","statement_latex":"Let $A$ be a domain with fraction field $K$. Let $B_1, \\ldots, B_r \\subset K$\nbe Noetherian $1$-dimensional semi-local domains whose fraction\nfields are $K$. If $A \\otimes B_i \\to K$ are surjective for $i = 1, \\ldots, r$,\nthen there exists an $x \\in A$ such that $x^{-1}$ is in the Jacobson radical of\n$B_i$ for $i = 1, \\ldots, r$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09N5","source_file":"varieties.tex","source_line":7536,"source_end_line":7543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7536-L7543","statement_sha256":"a1a21927ba6adbe3a5d673cdef8503b6219b2f5c7c86125f9f1479b581e478c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6528,"rank":6528,"depth":19,"x":310.319,"y":1090.839,"cluster":"varieties-curves"},{"id":"stacks:0AB2","tag":"0AB2","title":"Glueing dimension one rings · Lemma 0AB2","summary":"Let A be a Noetherian local ring of dimension 1. Let L = ∏ A_ p where the product is over the minimal primes of A. Let a_1, a_2 ∈ m_A map to the same element of L. Then a_1^n = a_2^n for some n > 0.","statement_latex":"Let $A$ be a Noetherian local ring of dimension $1$.\nLet $L = \\prod A_\\mathfrak p$ where the product is over\nthe minimal primes of $A$. Let $a_1, a_2 \\in \\mathfrak m_A$\nmap to the same element of $L$. Then $a_1^n = a_2^n$ for\nsome $n > 0$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AB2","source_file":"varieties.tex","source_line":7581,"source_end_line":7588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7581-L7588","statement_sha256":"cea43bbad11c1c756c5e982bda35c21ad8980bf7ea63a6f7b6d0111a98dde0d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6529,"rank":6529,"depth":0,"x":210.255,"y":1222.047,"cluster":"varieties-curves"},{"id":"stacks:0AB3","tag":"0AB3","title":"Glueing dimension one rings · Lemma 0AB3","summary":"Let A be a Noetherian local ring of dimension 1. Let L = ∏ A_ p and I = ⋂ p where the product and intersection are over the minimal primes of A. Let f ∈ L be an element of the form f = i + a where a ∈ m_A and i ∈ IL. Then some power of f is in the image of A → L.","statement_latex":"Let $A$ be a Noetherian local ring of dimension $1$.\nLet $L = \\prod A_\\mathfrak p$ and $I = \\bigcap \\mathfrak p$\nwhere the product and intersection are over\nthe minimal primes of $A$. Let $f \\in L$ be an element\nof the form $f = i + a$ where $a \\in \\mathfrak m_A$ and\n$i \\in IL$. Then some power of $f$ is in the image of $A \\to L$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AB3","source_file":"varieties.tex","source_line":7605,"source_end_line":7613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7605-L7613","statement_sha256":"1d20539424d7b22fa80a85f558e9af3d8ac71bd21b9d92d35490a4c95cc10933","origin":"The Stacks Project","memory_eligible":false,"source_rank":6530,"rank":6530,"depth":9,"x":178.442,"y":1068.103,"cluster":"varieties-curves"},{"id":"stacks:0AB4","tag":"0AB4","title":"Glueing dimension one rings · Lemma 0AB4","summary":"Let A be a Noetherian local ring of dimension 1. Let L = ∏ A_ p where the product is over the minimal primes of A. Let K → L be an integral ring map. Then there exist a ∈ m_A and x ∈ K which map to the same element of L such that m_A = sqrt(a).","statement_latex":"Let $A$ be a Noetherian local ring of dimension $1$.\nLet $L = \\prod A_\\mathfrak p$ where the product is over\nthe minimal primes of $A$. Let $K \\to L$ be an integral ring map.\nThen there exist $a \\in \\mathfrak m_A$ and $x \\in K$\nwhich map to the same element of $L$ such that $\\mathfrak m_A = \\sqrt{(a)}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AB4","source_file":"varieties.tex","source_line":7629,"source_end_line":7636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7629-L7636","statement_sha256":"eca674af8b7a426c1daaefaa64a673f9b93ee47daf220c9bd66ffe51690771aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":6531,"rank":6531,"depth":19,"x":326.091,"y":1163.824,"cluster":"varieties-curves"},{"id":"stacks:09N6","tag":"09N6","title":"Glueing dimension one rings · Lemma 09N6","summary":"Let A be a ring. Let p_1, …, p_r be a finite set of a primes of A. Let S = A setminus ⋃ p_i. Then S is a multiplicative system and S^-1A is a semi-local ring whose maximal ideals correspond to the maximal elements of the set ( p_i).","statement_latex":"Let $A$ be a ring. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$\nbe a finite set of a primes of $A$. Let\n$S = A \\setminus \\bigcup \\mathfrak p_i$. Then $S$ is a multiplicative\nsystem and $S^{-1}A$ is a semi-local ring whose maximal ideals\ncorrespond to the maximal elements of the set $\\{\\mathfrak p_i\\}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Glueing dimension one rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09N6","source_file":"varieties.tex","source_line":7710,"source_end_line":7717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7710-L7717","statement_sha256":"2982f6296e97bdd5b924fd274cf19b8117319c2d71c4e9e193587b3beb805bab","origin":"The Stacks Project","memory_eligible":false,"source_rank":6532,"rank":6532,"depth":2,"x":139.746,"y":1177.055,"cluster":"varieties-curves"},{"id":"stacks:09N8","tag":"09N8","title":"One dimensional Noetherian schemes · Lemma 09N8","summary":"Let X be a scheme all of whose local rings are Noetherian of dimension ≤ 1. Let U ⊂ X be a retrocompact open. Denote j : U → X the inclusion morphism. Then R^pj_*F = 0, p > 0 for every quasi-coherent O_U-module F.","statement_latex":"Let $X$ be a scheme all of whose local rings are Noetherian of dimension\n$\\leq 1$. Let $U \\subset X$ be a retrocompact open. Denote\n$j : U \\to X$ the inclusion morphism. Then $R^pj_*\\mathcal{F} = 0$, $p > 0$\nfor every quasi-coherent $\\mathcal{O}_U$-module $\\mathcal{F}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09N8","source_file":"varieties.tex","source_line":7746,"source_end_line":7752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7746-L7752","statement_sha256":"f9a169d707beca47e1e065bf0616f6b237394c1e9bf9855508c2f517429ec3dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6533,"rank":6533,"depth":30,"x":266.85,"y":1061.256,"cluster":"varieties-curves"},{"id":"stacks:09N9","tag":"09N9","title":"One dimensional Noetherian schemes · Lemma 09N9","summary":"Let X be an affine scheme all of whose local rings are Noetherian of dimension ≤ 1. Then any quasi-compact open U ⊂ X is affine.","statement_latex":"Let $X$ be an affine scheme all of whose local rings are Noetherian\nof dimension $\\leq 1$. Then any quasi-compact open $U \\subset X$ is affine.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09N9","source_file":"varieties.tex","source_line":7772,"source_end_line":7776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7772-L7776","statement_sha256":"7090bc5fe85b88f6ff39e48ac5fe143b9deca0c192552a08f019682048fca0a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6534,"rank":6534,"depth":31,"x":266.246,"y":1219.181,"cluster":"varieties-curves"},{"id":"stacks:09NA","tag":"09NA","title":"One dimensional Noetherian schemes · Lemma 09NA","summary":"Let X be a scheme. Let U ⊂ X be an open. Assume • U is a retrocompact open of X, • X setminus U is discrete, and • for x ∈ X setminus U the local ring O_X, x is Noetherian of dimension ≤ 1. Then (1) there exists an invertible O_X-module L and a section s such that U = X_s and (2) the map Pic(X) → Pic(U) is surjective.","statement_latex":"Let $X$ be a scheme. Let $U \\subset X$ be an open. Assume\n\\begin{enumerate}\n\\item $U$ is a retrocompact open of $X$,\n\\item $X \\setminus U$ is discrete, and\n\\item for $x \\in X \\setminus U$ the local ring\n$\\mathcal{O}_{X, x}$ is Noetherian of dimension $\\leq 1$.\n\\end{enumerate}\nThen (1) there exists an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$\nand a section $s$ such that $U = X_s$ and (2) the map\n$\\Pic(X) \\to \\Pic(U)$ is surjective.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NA","source_file":"varieties.tex","source_line":7796,"source_end_line":7808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7796-L7808","statement_sha256":"489bd01830fb9a746df8313fc0367198bccf2e7d00e5f6206e833f2933137372","origin":"The Stacks Project","memory_eligible":false,"source_rank":6535,"rank":6535,"depth":8,"x":139.361,"y":1102.081,"cluster":"varieties-curves"},{"id":"stacks:09NB","tag":"09NB","title":"One dimensional Noetherian schemes · Lemma 09NB","summary":"Let X be an integral separated scheme. Let U ⊂ X be a nonempty affine open such that X setminus U is a finite set of points x_1, …, x_r with O_X, x_i Noetherian of dimension 1. Then there exists a globally generated invertible O_X-module L and a section s such that U = X_s.","statement_latex":"Let $X$ be an integral separated scheme. Let $U \\subset X$ be a nonempty\naffine open such that $X \\setminus U$ is a finite set of points\n$x_1, \\ldots, x_r$ with $\\mathcal{O}_{X, x_i}$ Noetherian of dimension $1$.\nThen there exists a globally generated invertible $\\mathcal{O}_X$-module\n$\\mathcal{L}$ and a section $s$ such that $U = X_s$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NB","source_file":"varieties.tex","source_line":7855,"source_end_line":7862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7855-L7862","statement_sha256":"3268b92442e49ba4ae4704b3b5cef3baff0d664fe45b2631a881dac1a8170d19","origin":"The Stacks Project","memory_eligible":false,"source_rank":6536,"rank":6536,"depth":20,"x":327.582,"y":1116.469,"cluster":"varieties-curves"},{"id":"stacks:09NC","tag":"09NC","title":"One dimensional Noetherian schemes · Lemma 09NC","summary":"Let X be a quasi-compact scheme. If for every x ∈ X there exists a pair (L, s) consisting of a globally generated invertible sheaf L and a global section s such that x ∈ X_s and X_s is affine, then X has an ample invertible sheaf.","statement_latex":"Let $X$ be a quasi-compact scheme. If for every $x \\in X$\nthere exists a pair $(\\mathcal{L}, s)$ consisting of a globally generated\ninvertible sheaf $\\mathcal{L}$ and a global section $s$ such that\n$x \\in X_s$ and $X_s$ is affine, then $X$ has an ample invertible\nsheaf.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NC","source_file":"varieties.tex","source_line":7893,"source_end_line":7900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7893-L7900","statement_sha256":"35db9e60f5f43c6d8fdba42a912a71262b7fd55aabfb55d54d292347489422d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6537,"rank":6537,"depth":1,"x":176.83,"y":1212.91,"cluster":"varieties-curves"},{"id":"stacks:09ND","tag":"09ND","title":"One dimensional Noetherian schemes · Lemma 09ND","summary":"Let X be a Noetherian integral separated scheme of dimension 1. Then X has an ample invertible sheaf.","statement_latex":"Let $X$ be a Noetherian integral separated scheme of dimension $1$.\nThen $X$ has an ample invertible sheaf.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ND","source_file":"varieties.tex","source_line":7926,"source_end_line":7930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7926-L7930","statement_sha256":"f52f87d9607d94b67740afd17ac61625f3fc36881fa697c00aebf0b3e5fea6d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6538,"rank":6538,"depth":21,"x":210.525,"y":1055.852,"cluster":"varieties-curves"},{"id":"stacks:0C0T","tag":"0C0T","title":"One dimensional Noetherian schemes · Lemma 0C0T","summary":"Let f : X → Y be a finite morphism of schemes. Assume there exists an open V ⊂ Y such that f^-1(V) → V is an isomorphism and Y setminus V is a discrete space. Then every invertible O_X-module is the pullback of an invertible O_Y-module.","statement_latex":"Let $f : X \\to Y$ be a finite morphism of schemes. Assume there\nexists an open $V \\subset Y$ such that $f^{-1}(V) \\to V$ is an\nisomorphism and $Y \\setminus V$ is a discrete space. Then every\ninvertible $\\mathcal{O}_X$-module is the pullback of an invertible\n$\\mathcal{O}_Y$-module.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0T","source_file":"varieties.tex","source_line":7943,"source_end_line":7950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7943-L7950","statement_sha256":"88b79cbd59013527e104b2330413fd38a6a46d8685ff4a2bbeb303f221830cc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6539,"rank":6539,"depth":25,"x":312.24,"y":1191.13,"cluster":"varieties-curves"},{"id":"stacks:09NE","tag":"09NE","title":"One dimensional Noetherian schemes · Lemma 09NE","summary":"Let X be a scheme. Let Z_1, …, Z_n ⊂ X be closed subschemes. Let L_i be an invertible sheaf on Z_i. Assume that • X is reduced, • X = ⋃ Z_i set theoretically, and • Z_i ∩ Z_j is a discrete topological space for i not = j. Then there exists an invertible sheaf L on X whose restriction to Z_i is L_i. Moreover, if we are given sections s_i ∈ Γ(Z_i, L_i) which are nonvanishing at the points of Z_i ∩ Z_j, then we can choose L such that there exists a s ∈ Γ(X, L) with s|_Z_i =…","statement_latex":"Let $X$ be a scheme. Let $Z_1, \\ldots, Z_n \\subset X$ be closed\nsubschemes. Let $\\mathcal{L}_i$ be an invertible sheaf on $Z_i$.\nAssume that\n\\begin{enumerate}\n\\item $X$ is reduced,\n\\item $X = \\bigcup Z_i$ set theoretically, and\n\\item $Z_i \\cap Z_j$ is a discrete topological space for $i \\not = j$.\n\\end{enumerate}\nThen there exists an invertible sheaf $\\mathcal{L}$ on $X$ whose restriction\nto $Z_i$ is $\\mathcal{L}_i$. Moreover, if we are given sections\n$s_i \\in \\Gamma(Z_i, \\mathcal{L}_i)$ which are nonvanishing at the\npoints of $Z_i \\cap Z_j$, then we can choose $\\mathcal{L}$ such\nthat there exists a $s \\in \\Gamma(X, \\mathcal{L})$ with\n$s|_{Z_i} = s_i$ for all $i$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NE","source_file":"varieties.tex","source_line":7989,"source_end_line":8005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L7989-L8005","statement_sha256":"522836b7b37cf088db4a186eccd6adc81e4d8790dae15fad0fbab1e80806cd25","origin":"The Stacks Project","memory_eligible":false,"source_rank":6540,"rank":6540,"depth":26,"x":127.981,"y":1148.986,"cluster":"varieties-curves"},{"id":"stacks:09NX","tag":"09NX","title":"One dimensional Noetherian schemes · Lemma 09NX","summary":"Let X be a Noetherian reduced separated scheme of dimension 1. Then X has an ample invertible sheaf.","statement_latex":"Let $X$ be a Noetherian reduced separated scheme of dimension $1$.\nThen $X$ has an ample invertible sheaf.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NX","source_file":"varieties.tex","source_line":8079,"source_end_line":8083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8079-L8083","statement_sha256":"c883418b0ca0b192cea83664198a0a3d3880548cbca7346b0e36397643c9afe9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6541,"rank":6541,"depth":32,"x":298.173,"y":1075.323,"cluster":"varieties-curves"},{"id":"stacks:09NY","tag":"09NY","title":"One dimensional Noetherian schemes · Lemma 09NY","summary":"Let i : Z → X be a closed immersion of schemes. If the underlying topological space of X is Noetherian and dim(X) ≤ 1, then Pic(X) → Pic(Z) is surjective.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nIf the underlying topological space of $X$ is Noetherian and\n$\\dim(X) \\leq 1$, then $\\Pic(X) \\to \\Pic(Z)$ is surjective.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NY","source_file":"varieties.tex","source_line":8137,"source_end_line":8142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8137-L8142","statement_sha256":"d7bccc86f713932c164091c21637906d0ab264bbae80db189e38373bc144000d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6542,"rank":6542,"depth":26,"x":231.746,"y":1226.593,"cluster":"varieties-curves"},{"id":"stacks:09NZ","tag":"09NZ","title":"One dimensional Noetherian schemes · Proposition 09NZ","summary":"Let X be a Noetherian separated scheme of dimension 1. Then X has an ample invertible sheaf.","statement_latex":"Let $X$ be a Noetherian separated scheme of dimension $1$.\nThen $X$ has an ample invertible sheaf.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NZ","source_file":"varieties.tex","source_line":8163,"source_end_line":8167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8163-L8167","statement_sha256":"f08a106add2bb1578145198f8bd1eb23a29173c82eb926f4373460e4af9a14f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6543,"rank":6543,"depth":33,"x":158.899,"y":1076.981,"cluster":"varieties-curves"},{"id":"stacks:0C0U","tag":"0C0U","title":"One dimensional Noetherian schemes · Lemma 0C0U","summary":"Let f : X → Y be a morphism of schemes. Assume Y is Noetherian of dimension ≤ 1, f is finite, and there exists a dense open V ⊂ Y such that f^-1(V) → V is a closed immersion. Then every invertible O_X-module is the pullback of an invertible O_Y-module.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume\n$Y$ is Noetherian of dimension $\\leq 1$, $f$ is finite, and\nthere exists a dense open $V \\subset Y$ such that\n$f^{-1}(V) \\to V$ is a closed immersion. Then every invertible\n$\\mathcal{O}_X$-module is the pullback of an invertible $\\mathcal{O}_Y$-module.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"One dimensional Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0U","source_file":"varieties.tex","source_line":8199,"source_end_line":8206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8199-L8206","statement_sha256":"5b3c4bb3074cf3006a29198d985180b5cbf61c2a26c2197873f18a28639f33b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6544,"rank":6544,"depth":27,"x":333.364,"y":1146.14,"cluster":"varieties-curves"},{"id":"stacks:0C3R","tag":"0C3R","title":"The delta invariant · Lemma 0C3R","summary":"Let (A, m) be a Noetherian 1-dimensional local ring. Let f ∈ m. The following are equivalent • m = sqrt(f), • f is not contained in any minimal prime of A, and • A_f = ∏_ p minimal A_ p as A-algebras. Such an f ∈ m exists. If depth(A) = 1 (for example A is reduced), then (1) -- (3) are also equivalent to • [(4)] f is a nonzerodivisor, • [(5)] A_f is the total ring of fractions of A. If A is reduced, then (1) -- (5) are also equivalent to • [(6)] A_f is the product of the…","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian $1$-dimensional local ring.\nLet $f \\in \\mathfrak m$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathfrak m = \\sqrt{(f)}$,\n\\item $f$ is not contained in any minimal prime of $A$, and\n\\item $A_f = \\prod_{\\mathfrak p\\text{ minimal}} A_\\mathfrak p$ as $A$-algebras.\n\\end{enumerate}\nSuch an $f \\in \\mathfrak m$ exists. If $\\text{depth}(A) = 1$ (for example\n$A$ is reduced), then (1) -- (3) are also equivalent to\n\\begin{enumerate}\n\\item[(4)] $f$ is a nonzerodivisor,\n\\item[(5)] $A_f$ is the total ring of fractions of $A$.\n\\end{enumerate}\nIf $A$ is reduced, then (1) -- (5) are also equivalent to\n\\begin{enumerate}\n\\item[(6)] $A_f$ is the product of the residue fields at the minimal\nprimes of $A$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The delta invariant","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3R","source_file":"varieties.tex","source_line":8231,"source_end_line":8251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8231-L8251","statement_sha256":"236a21f88f01e2f34e8e1ae261a188cfb9856a9ebe052b09cf54f2c12630b45c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6545,"rank":6545,"depth":13,"x":148.642,"y":1194.258,"cluster":"varieties-curves"},{"id":"stacks:0C3S","tag":"0C3S","title":"The delta invariant · Lemma 0C3S","summary":"Let (A, m) be a reduced Nagata 1-dimensional local ring. Let A' be the integral closure of A in the total ring of fractions of A. Then A' is a normal Nagata ring, A → A' is finite, and A'/A has finite length as an A-module.","statement_latex":"Let $(A, \\mathfrak m)$ be a reduced Nagata $1$-dimensional local ring.\nLet $A'$ be the integral closure of $A$ in the total ring of fractions\nof $A$. Then $A'$ is a normal Nagata ring, $A \\to A'$ is finite, and\n$A'/A$ has finite length as an $A$-module.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The delta invariant","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3S","source_file":"varieties.tex","source_line":8293,"source_end_line":8299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8293-L8299","statement_sha256":"4defd28ce94a015c67f58328209c8b399f090e916f5d47aa13f995e11ceddd6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6546,"rank":6546,"depth":31,"x":246.4,"y":1053.614,"cluster":"varieties-curves"},{"id":"stacks:0C3T","tag":"0C3T","title":"The delta invariant · Definition 0C3T","summary":"Let A be a reduced Nagata local ring of dimension 1. The δ-invariant of A is length_A(A'/A) where A' is as in Lemma [Tag 0C3S].","statement_latex":"Let $A$ be a reduced Nagata local ring of dimension $1$.\nThe {\\it $\\delta$-invariant of $A$} is $\\text{length}_A(A'/A)$\nwhere $A'$ is as in Lemma \\ref{lemma-pre-delta-invariant}.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The delta invariant","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3T","source_file":"varieties.tex","source_line":8317,"source_end_line":8322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8317-L8322","statement_sha256":"3eec4f5516e854d49736eb0dc5e60a1049631221cfb808709acc0bd2899ff4ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":6547,"rank":6547,"depth":32,"x":287.518,"y":1213.185,"cluster":"varieties-curves"},{"id":"stacks:0C3U","tag":"0C3U","title":"The delta invariant · Lemma 0C3U","summary":"Let A be a reduced Nagata local ring of dimension 1. The δ-invariant of A is 0 if and only if A is a discrete valuation ring.","statement_latex":"Let $A$ be a reduced Nagata local ring of dimension $1$.\nThe $\\delta$-invariant of $A$ is $0$ if and only if\n$A$ is a discrete valuation ring.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The delta invariant","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3U","source_file":"varieties.tex","source_line":8327,"source_end_line":8332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8327-L8332","statement_sha256":"a4ffe273e3f79c6ca588a3f754619667423d11a2026a8a95820c30e408d71d68","origin":"The Stacks Project","memory_eligible":false,"source_rank":6548,"rank":6548,"depth":16,"x":128.486,"y":1118.619,"cluster":"varieties-curves"},{"id":"stacks:0C3V","tag":"0C3V","title":"The delta invariant · Lemma 0C3V","summary":"Let A be a reduced Nagata local ring of dimension 1. Let A → A' be as in Lemma [Tag 0C3S]. Let A^h, A^sh, resp. A^wedge be the henselization, strict henselization, resp. completion of A. Then A^h, A^sh, resp. A^wedge is a reduced Nagata local ring of dimension 1 and A' ⊗_A A^h, A' ⊗_A A^sh, resp. A' ⊗_A A^wedge is the integral closure of A^h, A^sh, resp. A^wedge in its total ring of fractions.","statement_latex":"Let $A$ be a reduced Nagata local ring of dimension $1$.\nLet $A \\to A'$ be as in Lemma \\ref{lemma-pre-delta-invariant}.\nLet $A^h$, $A^{sh}$, resp.\\ $A^\\wedge$\nbe the henselization, strict henselization, resp.\\ completion of $A$.\nThen $A^h$, $A^{sh}$, resp. $A^\\wedge$ is a reduced Nagata local\nring of dimension $1$ and\n$A' \\otimes_A A^h$, $A' \\otimes_A A^{sh}$, resp. $A' \\otimes_A A^\\wedge$\nis the integral closure of $A^h$, $A^{sh}$, resp.\\ $A^\\wedge$\nin its total ring of fractions.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The delta invariant","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3V","source_file":"varieties.tex","source_line":8345,"source_end_line":8356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8345-L8356","statement_sha256":"3972afa3286f2bed025cfda00785f9ff8ed0f1252288228e226cf06cb4965492","origin":"The Stacks Project","memory_eligible":false,"source_rank":6549,"rank":6549,"depth":53,"x":322.272,"y":1098.063,"cluster":"varieties-curves"},{"id":"stacks:0C3W","tag":"0C3W","title":"The delta invariant · Lemma 0C3W","summary":"Let A be a reduced Nagata local ring of dimension 1. The δ-invariant of A is the same as the δ-invariant of the henselization, strict henselization, or the completion of A.","statement_latex":"Let $A$ be a reduced Nagata local ring of dimension $1$.\nThe $\\delta$-invariant of $A$ is the same as the\n$\\delta$-invariant of the henselization, strict henselization,\nor the completion of $A$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The delta invariant","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3W","source_file":"varieties.tex","source_line":8425,"source_end_line":8431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8425-L8431","statement_sha256":"faa4d7652780174538bb3a12c71b11c846788120eb6863c78c83e1e3f5bcf62b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6550,"rank":6550,"depth":54,"x":195.601,"y":1223.483,"cluster":"varieties-curves"},{"id":"stacks:0C1T","tag":"0C1T","title":"The delta invariant · Definition 0C1T","summary":"Let k be a field. Let X be a locally algebraic k-scheme. Let x ∈ X be a point such that O_X, x is reduced and dim(O_X, x) = 1. The δ-invariant of X at x is the δ-invariant of O_X, x as defined in Definition [Tag 0C3T].","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.\nLet $x \\in X$ be a point such that $\\mathcal{O}_{X, x}$\nis reduced and $\\dim(\\mathcal{O}_{X, x}) = 1$.\nThe {\\it $\\delta$-invariant of $X$ at $x$} is the\n$\\delta$-invariant of $\\mathcal{O}_{X, x}$ as defined in\nDefinition \\ref{definition-delta-invariant-algebra}.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The delta invariant","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1T","source_file":"varieties.tex","source_line":8446,"source_end_line":8454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8446-L8454","statement_sha256":"dc318cd019c16343a67ac894a796fda789b3633915a552db0cf8bd372217c500","origin":"The Stacks Project","memory_eligible":false,"source_rank":6551,"rank":6551,"depth":33,"x":188.13,"y":1058.727,"cluster":"varieties-curves"},{"id":"stacks:0C3X","tag":"0C3X","title":"The delta invariant · Lemma 0C3X","summary":"Let k be a field. Let X be a locally algebraic k-scheme. Let K/k be a field extension and set Y = X_K. Let y ∈ Y with image x ∈ X. Assume X is geometrically reduced at x and dim(O_X, x) = dim(O_Y, y) = 1. Then δ-invariant of X at x ≤ δ-invariant of Y at y","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.\nLet $K/k$ be a field extension and set $Y = X_K$.\nLet $y \\in Y$ with image $x \\in X$.\nAssume $X$ is geometrically reduced at $x$ and\n$\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y}) = 1$.\nThen\n$$\n\\delta\\text{-invariant of }X\\text{ at }x \\leq\n\\delta\\text{-invariant of }Y\\text{ at }y\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The delta invariant","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3X","source_file":"varieties.tex","source_line":8473,"source_end_line":8485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8473-L8485","statement_sha256":"61d85bcded4f0aa07b84eda3dba1c3b73d57e78a51cc6be814bd8fba127bd294","origin":"The Stacks Project","memory_eligible":false,"source_rank":6552,"rank":6552,"depth":32,"x":326.463,"y":1176.258,"cluster":"varieties-curves"},{"id":"stacks:0C3Y","tag":"0C3Y","title":"The delta invariant · Lemma 0C3Y","summary":"Let k be a field. Let X be a locally algebraic k-scheme. Let K/k be a field extension and set Y = X_K. Let y ∈ Y with image x ∈ X. Assume assumptions (a), (b), (c) of Lemma [Tag 0C3P] hold for x ∈ X and that dim(O_Y, y) = 1. Then the δ-invariant of X at x is δ-invariant of Y at y.","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.\nLet $K/k$ be a field extension and set $Y = X_K$.\nLet $y \\in Y$ with image $x \\in X$.\nAssume assumptions (a), (b), (c) of\nLemma \\ref{lemma-geometrically-normal-in-codim-1}\nhold for $x \\in X$ and that $\\dim(\\mathcal{O}_{Y, y}) = 1$.\nThen the $\\delta$-invariant\nof $X$ at $x$ is $\\delta$-invariant of $Y$ at $y$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The delta invariant","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C3Y","source_file":"varieties.tex","source_line":8522,"source_end_line":8532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8522-L8532","statement_sha256":"5959aa3ee90da8cfd998d365b51c62fdbaa6120157369f43f80b46a326d73416","origin":"The Stacks Project","memory_eligible":false,"source_rank":6553,"rank":6553,"depth":46,"x":129.473,"y":1168.066,"cluster":"varieties-curves"},{"id":"stacks:0C1S","tag":"0C1S","title":"The number of branches · Lemma 0C1S","summary":"Let X be a scheme. Assume every quasi-compact open of X has finitely many irreducible components. Let ν : X^ν → X be the normalization of X. Let x ∈ X. • The number of branches of X at x is the number of inverse images of x in X^ν. • The number of geometric branches of X at x is ∑_ν(x^ν) = x [kappa(x^ν) : kappa(x)]_s.","statement_latex":"Let $X$ be a scheme. Assume every quasi-compact open of $X$ has\nfinitely many irreducible components. Let $\\nu : X^\\nu \\to X$\nbe the normalization of $X$. Let $x \\in X$.\n\\begin{enumerate}\n\\item The number of branches of $X$ at $x$ is the number of\ninverse images of $x$ in $X^\\nu$.\n\\item The number of geometric branches of $X$ at $x$ is\n$\\sum_{\\nu(x^\\nu) = x} [\\kappa(x^\\nu) : \\kappa(x)]_s$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The number of branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1S","source_file":"varieties.tex","source_line":8574,"source_end_line":8585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8574-L8585","statement_sha256":"8626ff68268bc34e6bf6f8c74cbfe2fb29ec4e397fe814f6b8fe67989ed574fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":6554,"rank":6554,"depth":53,"x":281.68,"y":1062.079,"cluster":"varieties-curves"},{"id":"stacks:0C40","tag":"0C40","title":"The number of branches · Lemma 0C40","summary":"Let k be a field. Let X be a locally algebraic k-scheme. Let K/k be an extension of fields. Let y ∈ X_K be a point with image x in X. Then the number of geometric branches of X at x is the number of geometric branches of X_K at y.","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.\nLet $K/k$ be an extension of fields. Let $y \\in X_K$ be a\npoint with image $x$ in $X$. Then the number of\ngeometric branches of $X$ at $x$ is the number of geometric\nbranches of $X_K$ at $y$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The number of branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C40","source_file":"varieties.tex","source_line":8607,"source_end_line":8614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8607-L8614","statement_sha256":"969038b318904c2439f7efa961253d4f5809cc9ecdf87a8b702c9160144b56d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6555,"rank":6555,"depth":54,"x":254.612,"y":1226.987,"cluster":"varieties-curves"},{"id":"stacks:0C55","tag":"0C55","title":"The number of branches · Lemma 0C55","summary":"Let k be a field. Let X be a locally algebraic k-scheme. Let K/k be an extension of fields. Let y ∈ X_K be a point with image x in X. Then X is geometrically unibranch at x if and only if X_K is geometrically unibranch at y.","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.\nLet $K/k$ be an extension of fields. Let $y \\in X_K$ be a\npoint with image $x$ in $X$. Then $X$ is geometrically unibranch\nat $x$ if and only if $X_K$ is geometrically unibranch at $y$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The number of branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C55","source_file":"varieties.tex","source_line":8658,"source_end_line":8664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8658-L8664","statement_sha256":"783803d0fdd71c0638eaafcfa7f7af4d21e18808e57f584909d8d3d8c1b9f72f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6556,"rank":6556,"depth":55,"x":141.691,"y":1089.705,"cluster":"varieties-curves"},{"id":"stacks:0C41","tag":"0C41","title":"The number of branches · Definition 0C41","summary":"Let A and A_i, 1 ≤ i ≤ n be local rings. We say A is a wedge of A_1, …, A_n if there exist isomorphisms kappa_A_1 → kappa_A_2 → … → kappa_A_n and A is isomorphic to the ring consisting of n-tuples (a_1, …, a_n) ∈ A_1 × … × A_n which map to the same element of kappa_A_n.","statement_latex":"Let $A$ and $A_i$, $1 \\leq i \\leq n$ be local rings. We say\n{\\it $A$ is a wedge of $A_1, \\ldots, A_n$}\nif there exist isomorphisms\n$$\n\\kappa_{A_1} \\to \\kappa_{A_2} \\to \\ldots \\to \\kappa_{A_n}\n$$\nand $A$ is isomorphic to the ring consisting of $n$-tuples\n$(a_1, \\ldots, a_n) \\in A_1 \\times \\ldots \\times A_n$ which map to the\nsame element of $\\kappa_{A_n}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The number of branches","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C41","source_file":"varieties.tex","source_line":8672,"source_end_line":8683,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8672-L8683","statement_sha256":"1ce2a165d3c0fce2dd6bc2526b158f4ce9c751baec77b51ae7e3439706bd22de","origin":"The Stacks Project","memory_eligible":false,"source_rank":6557,"rank":6557,"depth":0,"x":335.81,"y":1126.948,"cluster":"varieties-curves"},{"id":"stacks:0C42","tag":"0C42","title":"The number of branches · Lemma 0C42","summary":"Let (A, m) be a strictly henselian 1-dimensional reduced Nagata local ring. Then δ-invariant of A ≥ number of geometric branches of A - 1 If equality holds, then A is a wedge of n ≥ 1 strictly henselian discrete valuation rings.","statement_latex":"Let $(A, \\mathfrak m)$ be a strictly henselian\n$1$-dimensional reduced Nagata local ring. Then\n$$\n\\delta\\text{-invariant of }A \\geq \\text{number of geometric branches of }A - 1\n$$\nIf equality holds, then $A$ is a wedge of $n \\geq 1$ strictly henselian\ndiscrete valuation rings.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The number of branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C42","source_file":"varieties.tex","source_line":8698,"source_end_line":8707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8698-L8707","statement_sha256":"4c4ffee61bd82017477f2091a75c55a879b008ce7b657aad297693acf02b67d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6558,"rank":6558,"depth":53,"x":162.316,"y":1209.826,"cluster":"varieties-curves"},{"id":"stacks:0C43","tag":"0C43","title":"The number of branches · Lemma 0C43","summary":"Let (A, m) be a 1-dimensional reduced Nagata local ring. Then δ-invariant of A ≥ number of geometric branches of A - 1","statement_latex":"Let $(A, \\mathfrak m)$ be a $1$-dimensional reduced Nagata local ring. Then\n$$\n\\delta\\text{-invariant of }A \\geq \\text{number of geometric branches of }A - 1\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"The number of branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C43","source_file":"varieties.tex","source_line":8739,"source_end_line":8745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8739-L8745","statement_sha256":"242ec567e02330055502e78f3d413393566da7a757764530f73eda9a27087d73","origin":"The Stacks Project","memory_eligible":false,"source_rank":6559,"rank":6559,"depth":55,"x":223.743,"y":1049.897,"cluster":"varieties-curves"},{"id":"stacks:0C45","tag":"0C45","title":"Normalization of one dimensional schemes · Lemma 0C45","summary":"Let X be a locally Noetherian scheme of dimension 1. Let ν : X^ν → X be the normalization. Then • ν is integral, surjective, and induces a bijection on irreducible components, • there is a factorization X^ν → X_red → X and the morphism X^ν → X_red is the normalization of X_red, • X^ν → X_red is birational, • for every closed point x ∈ X the stalk (ν_*O_X^ν)_x is the integral closure of O_X, x in the total ring of fractions of (O_X, x)_red = O_X_red, x, • the fibres of ν…","statement_latex":"Let $X$ be a locally Noetherian scheme of dimension $1$.\nLet $\\nu : X^\\nu \\to X$ be the normalization. Then\n\\begin{enumerate}\n\\item $\\nu$ is integral, surjective, and induces a bijection\non irreducible components,\n\\item there is a factorization $X^\\nu \\to X_{red} \\to X$\nand the morphism $X^\\nu \\to X_{red}$ is the normalization\nof $X_{red}$,\n\\item $X^\\nu \\to X_{red}$ is birational,\n\\item for every closed point $x \\in X$ the stalk\n$(\\nu_*\\mathcal{O}_{X^\\nu})_x$ is the integral closure\nof $\\mathcal{O}_{X, x}$ in the total ring of fractions\nof $(\\mathcal{O}_{X, x})_{red} = \\mathcal{O}_{X_{red}, x}$,\n\\item the fibres of $\\nu$ are finite and the residue\nfield extensions are finite,\n\\item $X^\\nu$ is a disjoint union of integral normal locally Noetherian\nschemes and each affine open is the spectrum of a finite\nproduct of Dedekind domains.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Normalization of one dimensional schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C45","source_file":"varieties.tex","source_line":8770,"source_end_line":8791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8770-L8791","statement_sha256":"d3536fe5dcef74a597a7d7085039813612b895f264ad3ec28698bec496eab2f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6560,"rank":6560,"depth":23,"x":307.25,"y":1203.035,"cluster":"varieties-curves"},{"id":"stacks:0C1R","tag":"0C1R","title":"Normalization of one dimensional schemes · Lemma 0C1R","summary":"Let X be a reduced Nagata scheme of dimension 1. Let ν : X^ν → X be the normalization. Let x ∈ X denote a closed point. Then • ν : X^ν → X is finite, surjective, and birational, • O_X ⊂ ν_*O_X^ν and ν_*O_X^ν/O_X is a direct sum of skyscraper sheaves with value Q_x at x which is nonzero if and only if x is a singular point of X, • A' = (ν_*O_X^ν)_x is the integral closure of A = O_X, x in its total ring of fractions, • Q_x = A'/A has finite length equal to the δ-invariant…","statement_latex":"Let $X$ be a reduced Nagata scheme of dimension $1$. Let $\\nu : X^\\nu \\to X$\nbe the normalization. Let $x \\in X$ denote a closed point. Then\n\\begin{enumerate}\n\\item $\\nu : X^\\nu \\to X$ is finite, surjective, and birational,\n\\item $\\mathcal{O}_X \\subset \\nu_*\\mathcal{O}_{X^\\nu}$ and\n$\\nu_*\\mathcal{O}_{X^\\nu}/\\mathcal{O}_X$ is a direct sum of\nskyscraper sheaves with value $\\mathcal{Q}_x$ at $x$ which is nonzero\nif and only if $x$ is a singular point of $X$,\n\\item $A' = (\\nu_*\\mathcal{O}_{X^\\nu})_x$ is the integral closure\nof $A = \\mathcal{O}_{X, x}$ in its total ring of fractions,\n\\item $\\mathcal{Q}_x = A'/A$ has finite length equal to the\n$\\delta$-invariant of $X$ at $x$,\n\\item $A'$ is a semi-local ring which is a finite product of\nDedekind domains,\n\\item $A^\\wedge$ is a reduced Noetherian complete\nlocal ring of dimension $1$,\n\\item $(A')^\\wedge$ is the integral closure of $A^\\wedge$\nin its total ring of fractions,\n\\item $(A')^\\wedge$ is a finite product of\ncomplete discrete valuation rings, and\n\\item $A'/A \\cong (A')^\\wedge/A^\\wedge$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Normalization of one dimensional schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1R","source_file":"varieties.tex","source_line":8818,"source_end_line":8842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8818-L8842","statement_sha256":"b597b1a5c02a64738518f19391a5c32f2aa82b67e3ae14f12aa603df43208022","origin":"The Stacks Project","memory_eligible":false,"source_rank":6561,"rank":6561,"depth":54,"x":122.099,"y":1137.346,"cluster":"varieties-curves"},{"id":"stacks:09NG","tag":"09NG","title":"Finding affine opens · Lemma 09NG","summary":"Let f : X → Y be a morphism of schemes. Let X^0 denote the set of generic points of irreducible components of X. If • f is separated, • there is an open covering X = ⋃ U_i such that f|_U_i : U_i → Y is an open immersion, and • if xi, xi' ∈ X^0, xi not = xi', then f(xi) not = f(xi'), then f is an open immersion.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $X^0$ denote the set\nof generic points of irreducible components of $X$. If\n\\begin{enumerate}\n\\item $f$ is separated,\n\\item there is an open covering $X = \\bigcup U_i$ such that\n$f|_{U_i} : U_i \\to Y$ is an open immersion, and\n\\item if $\\xi, \\xi' \\in X^0$, $\\xi \\not = \\xi'$, then $f(\\xi) \\not = f(\\xi')$,\n\\end{enumerate}\nthen $f$ is an open immersion.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Finding affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NG","source_file":"varieties.tex","source_line":8890,"source_end_line":8901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8890-L8901","statement_sha256":"7cd884f8be60e8594e7a298ef952c8d6b8de09092930ba402b61e540b7ddff8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6562,"rank":6562,"depth":0,"x":311.885,"y":1080.596,"cluster":"varieties-curves"},{"id":"stacks:09NH","tag":"09NH","title":"Finding affine opens · Lemma 09NH","summary":"Let X → S be a morphism of schemes. Let x ∈ X be a point with image s ∈ S. If • O_X, x = O_S, s, • S is reduced, • X → S is of finite type, and • S has finitely many irreducible components, then there exists an open neighbourhood U of x such that f|_U is an open immersion.","statement_latex":"Let $X \\to S$ be a morphism of schemes. Let $x \\in X$ be a point with\nimage $s \\in S$. If\n\\begin{enumerate}\n\\item $\\mathcal{O}_{X, x} = \\mathcal{O}_{S, s}$,\n\\item $S$ is reduced,\n\\item $X \\to S$ is of finite type, and\n\\item $S$ has finitely many irreducible components,\n\\end{enumerate}\nthen there exists an open neighbourhood $U$\nof $x$ such that $f|_U$ is an open immersion.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Finding affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NH","source_file":"varieties.tex","source_line":8912,"source_end_line":8924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8912-L8924","statement_sha256":"c62a90d4d96e2e0f3a827b04bbb388fd356a30ccc93c55ef6682430baf39a434","origin":"The Stacks Project","memory_eligible":false,"source_rank":6563,"rank":6563,"depth":0,"x":217.362,"y":1230.473,"cluster":"varieties-curves"},{"id":"stacks:09NI","tag":"09NI","title":"Finding affine opens · Lemma 09NI","summary":"Let f : T → X be a morphism of schemes. Let X^0, resp. T^0 denote the sets of generic points of irreducible components. Let t_1, …, t_m ∈ T be a finite set of points with images x_j = f(t_j). If • T is affine, • X is quasi-separated, • X^0 is finite • f(T^0) ⊂ X^0 and f : T^0 → X^0 is injective, and • O_X, x_j = O_T, t_j, then there exists an affine open of X containing x_1, …, x_r.","statement_latex":"Let $f : T \\to X$ be a morphism of schemes. Let $X^0$, resp.\\ $T^0$\ndenote the sets of generic points of irreducible components.\nLet $t_1, \\ldots, t_m \\in T$ be a finite set of points\nwith images $x_j = f(t_j)$. If\n\\begin{enumerate}\n\\item $T$ is affine,\n\\item $X$ is quasi-separated,\n\\item $X^0$ is finite\n\\item $f(T^0) \\subset X^0$ and $f : T^0 \\to X^0$ is injective, and\n\\item $\\mathcal{O}_{X, x_j} = \\mathcal{O}_{T, t_j}$,\n\\end{enumerate}\nthen there exists an affine open of $X$ containing $x_1, \\ldots, x_r$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Finding affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NI","source_file":"varieties.tex","source_line":8951,"source_end_line":8965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L8951-L8965","statement_sha256":"991dcb216d942294acba43cfc0efa0e487b9c55e541a3084c19f7275e9dafa7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6564,"rank":6564,"depth":31,"x":166.42,"y":1065.948,"cluster":"varieties-curves"},{"id":"stacks:09NJ","tag":"09NJ","title":"Finding affine opens · Lemma 09NJ","summary":"Let X be an integral separated scheme. Let x_1, …, x_r ∈ X be a finite set of points such that O_X, x_i is Noetherian of dimension ≤ 1. Then there exists an affine open subscheme of X containing all of x_1, …, x_r.","statement_latex":"Let $X$ be an integral separated scheme. Let $x_1, \\ldots, x_r \\in X$\nbe a finite set of points such that $\\mathcal{O}_{X, x_i}$\nis Noetherian of dimension $\\leq 1$. Then there exists an affine\nopen subscheme of $X$ containing all of $x_1, \\ldots, x_r$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Finding affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NJ","source_file":"varieties.tex","source_line":9016,"source_end_line":9022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9016-L9022","statement_sha256":"cb269e2ddedd97de52f7099f6fd1c7770a738c311e58d8f6bce1875936d8cf4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6565,"rank":6565,"depth":32,"x":336.673,"y":1158.57,"cluster":"varieties-curves"},{"id":"stacks:09NK","tag":"09NK","title":"Finding affine opens · Lemma 09NK","summary":"Let A be a ring, I ⊂ A an ideal, p_1, …, p_r primes of A, and overlinef ∈ A/I an element. If I not ⊂ p_i for all i, then there exists an f ∈ A, f not ∈ p_i which maps to overlinef in A/I.","statement_latex":"Let $A$ be a ring, $I \\subset A$ an ideal,\n$\\mathfrak p_1, \\ldots, \\mathfrak p_r$ primes of $A$, and\n$\\overline{f} \\in A/I$ an element. If $I \\not \\subset \\mathfrak p_i$\nfor all $i$, then there exists an $f \\in A$, $f \\not \\in \\mathfrak p_i$\nwhich maps to $\\overline{f}$ in $A/I$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Finding affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NK","source_file":"varieties.tex","source_line":9060,"source_end_line":9067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9060-L9067","statement_sha256":"ecc29476d0fad67af30912cfd0b8e73be6ffee0838539321b5a47b3a9e662c79","origin":"The Stacks Project","memory_eligible":false,"source_rank":6566,"rank":6566,"depth":2,"x":136.198,"y":1186.941,"cluster":"varieties-curves"},{"id":"stacks:09NM","tag":"09NM","title":"Finding affine opens · Lemma 09NM","summary":"Let X be a scheme. Let T ⊂ X be finite set of points. Assume • X has finitely many irreducible components Z_1, …, Z_t, and • Z_i ∩ T is contained in an affine open of the reduced induced subscheme corresponding to Z_i. Then there exists an affine open subscheme of X containing T.","statement_latex":"Let $X$ be a scheme. Let $T \\subset X$ be finite set of points. Assume\n\\begin{enumerate}\n\\item $X$ has finitely many irreducible components $Z_1, \\ldots, Z_t$, and\n\\item $Z_i \\cap T$ is contained in an affine open of the reduced\ninduced subscheme corresponding to $Z_i$.\n\\end{enumerate}\nThen there exists an affine open subscheme of $X$ containing $T$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Finding affine opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NM","source_file":"varieties.tex","source_line":9083,"source_end_line":9092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9083-L9092","statement_sha256":"5c4c4ba3ed915eab1cd610e5212469ab4c64f1bed617726b1228628c0ba98a22","origin":"The Stacks Project","memory_eligible":false,"source_rank":6567,"rank":6567,"depth":32,"x":261.49,"y":1051.965,"cluster":"varieties-curves"},{"id":"stacks:09NN","tag":"09NN","title":"Finding affine opens · Proposition 09NN","summary":"Let X be a separated scheme such that every quasi-compact open has a finite number of irreducible components. Let x_1, …, x_r ∈ X be points such that O_X, x_i is Noetherian of dimension ≤ 1. Then there exists an affine open subscheme of X containing all of x_1, …, x_r.","statement_latex":"Let $X$ be a separated scheme such that every quasi-compact open\nhas a finite number of irreducible components. Let $x_1, \\ldots, x_r \\in X$\nbe points such that $\\mathcal{O}_{X, x_i}$ is Noetherian\nof dimension $\\leq 1$. Then there exists an affine open subscheme\nof $X$ containing all of $x_1, \\ldots, x_r$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Finding affine opens","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NN","source_file":"varieties.tex","source_line":9140,"source_end_line":9147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9140-L9147","statement_sha256":"a1903209b554a84af11907255dad45e5588491f067a04ba0031a66e3bccf3ac1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6568,"rank":6568,"depth":33,"x":277.681,"y":1222.968,"cluster":"varieties-curves"},{"id":"stacks:0A23","tag":"0A23","title":"Curves · Definition 0A23","summary":"Let k be a field. A curve is a variety of dimension 1 over k.","statement_latex":"Let $k$ be a field. A {\\it curve} is a variety of dimension $1$ over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A23","source_file":"varieties.tex","source_line":9168,"source_end_line":9171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9168-L9171","statement_sha256":"3000ee20349dcb4c2204c241e23f5e5cba6a7ff9cec8f5afee8160b9481ec5e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6569,"rank":6569,"depth":0,"x":127.894,"y":1105.801,"cluster":"varieties-curves"},{"id":"stacks:0A24","tag":"0A24","title":"Curves · Lemma 0A24","summary":"Let X be a separated, irreducible scheme of dimension > 0 over a field k. Let x ∈ X be a closed point. The open subscheme X setminus (x) is not proper over k.","statement_latex":"Let $X$ be a separated, irreducible scheme of dimension $> 0$ over a field $k$.\nLet $x \\in X$ be a closed point. The open subscheme $X \\setminus \\{x\\}$\nis not proper over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A24","source_file":"varieties.tex","source_line":9187,"source_end_line":9192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9187-L9192","statement_sha256":"4c88eda831c0baeaacfdc2e82a8d65b510adc75123dbd1241c1a200635e39590","origin":"The Stacks Project","memory_eligible":false,"source_rank":6570,"rank":6570,"depth":18,"x":333.02,"y":1107.209,"cluster":"varieties-curves"},{"id":"stacks:0A25","tag":"0A25","title":"Curves · Lemma 0A25","summary":"Let X be a separated finite type scheme over a field k. If dim(X) ≤ 1 then X is H-quasi-projective over k.","statement_latex":"Let $X$ be a separated finite type scheme over a field $k$.\nIf $\\dim(X) \\leq 1$ then $X$ is H-quasi-projective over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A25","source_file":"varieties.tex","source_line":9201,"source_end_line":9205,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9201-L9205","statement_sha256":"309bab79542294fc99e697e1c1d27e8bfc8fef543f9d1efa4eaa15af5a06174c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6571,"rank":6571,"depth":34,"x":180.296,"y":1222.817,"cluster":"varieties-curves"},{"id":"stacks:0A26","tag":"0A26","title":"Curves · Lemma 0A26","summary":"Let X be a proper scheme over a field k. If dim(X) ≤ 1 then X is H-projective over k.","statement_latex":"Let $X$ be a proper scheme over a field $k$.\nIf $\\dim(X) \\leq 1$ then $X$ is H-projective over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A26","source_file":"varieties.tex","source_line":9217,"source_end_line":9221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9217-L9221","statement_sha256":"c1237a4cda31f948324b60d0be6d97a005574c263c1c045c39292b759898795e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6572,"rank":6572,"depth":35,"x":199.987,"y":1050.534,"cluster":"varieties-curves"},{"id":"stacks:0BXV","tag":"0BXV","title":"Curves · Lemma 0BXV","summary":"Let X be a separated scheme of finite type over k. If dim(X) ≤ 1, then there exists an open immersion j : X → overlineX with the following properties • overlineX is H-projective over k, i.e., overlineX is a closed subscheme of P^d_k for some d, • j(X) ⊂ overlineX is dense and scheme theoretically dense, • overlineX setminus X = (x_1, …, x_n) for some closed points x_i ∈ overlineX.","statement_latex":"Let $X$ be a separated scheme of finite type over $k$.\nIf $\\dim(X) \\leq 1$, then there exists an open immersion\n$j : X \\to \\overline{X}$ with the following properties\n\\begin{enumerate}\n\\item $\\overline{X}$ is H-projective over $k$, i.e., $\\overline{X}$\nis a closed subscheme of $\\mathbf{P}^d_k$ for some $d$,\n\\item $j(X) \\subset \\overline{X}$ is dense and scheme\ntheoretically dense,\n\\item $\\overline{X} \\setminus X = \\{x_1, \\ldots, x_n\\}$\nfor some closed points $x_i \\in \\overline{X}$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXV","source_file":"varieties.tex","source_line":9231,"source_end_line":9244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9231-L9244","statement_sha256":"df8024343cac56b002c5796387371c9ef61164bf4c4a3076aeae66a58d7db55b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6573,"rank":6573,"depth":35,"x":324.282,"y":1189.047,"cluster":"varieties-curves"},{"id":"stacks:0BXW","tag":"0BXW","title":"Curves · Lemma 0BXW","summary":"Let X be a separated scheme of finite type over k. If X is reduced and dim(X) ≤ 1, then there exists an open immersion j : X → overlineX such that • overlineX is H-projective over k, i.e., overlineX is a closed subscheme of P^d_k for some d, • j(X) ⊂ overlineX is dense and scheme theoretically dense, • overlineX setminus X = (x_1, …, x_n) for some closed points x_i ∈ overlineX, • the local rings O_overlineX, x_i are discrete valuation rings for i = 1, …, n.","statement_latex":"Let $X$ be a separated scheme of finite type over $k$.\nIf $X$ is reduced and $\\dim(X) \\leq 1$, then there exists\nan open immersion $j : X \\to \\overline{X}$ such that\n\\begin{enumerate}\n\\item $\\overline{X}$ is H-projective over $k$, i.e., $\\overline{X}$\nis a closed subscheme of $\\mathbf{P}^d_k$ for some $d$,\n\\item $j(X) \\subset \\overline{X}$ is dense and scheme\ntheoretically dense,\n\\item $\\overline{X} \\setminus X = \\{x_1, \\ldots, x_n\\}$\nfor some closed points $x_i \\in \\overline{X}$,\n\\item the local rings $\\mathcal{O}_{\\overline{X}, x_i}$\nare discrete valuation rings for $i = 1, \\ldots, n$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXW","source_file":"varieties.tex","source_line":9263,"source_end_line":9278,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9263-L9278","statement_sha256":"743408446464de36041707e92114e178365100191497bc8f8776f90faa8eb801","origin":"The Stacks Project","memory_eligible":false,"source_rank":6574,"rank":6574,"depth":36,"x":120.8,"y":1157.369,"cluster":"varieties-curves"},{"id":"stacks:0GK5","tag":"0GK5","title":"Curves · Lemma 0GK5","summary":"Let X be a separated scheme of finite type over k with dim(X) ≤ 1. Then there exists a commutative diagram xymatrix overlineY_1 amalg … amalg overlineY_n ar[rd] & Y_1 amalg … amalg Y_n ar[r]_-ν ar[d] ar[l]^j & X_k' ar[r] ar[d] & X ar[d]^f & Spec(k'_1) amalg … amalg Spec(k'_n) ar[r] & Spec(k') ar[r] & Spec(k) of schemes with the following properties: • k'/k is a finite purely inseparable extension of fields, • ν is the normalization of X_k', • j is an open immersion with…","statement_latex":"Let $X$ be a separated scheme of finite type over $k$ with $\\dim(X) \\leq 1$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n\\overline{Y}_1 \\amalg \\ldots \\amalg \\overline{Y}_n \\ar[rd] &\nY_1 \\amalg \\ldots \\amalg Y_n \\ar[r]_-\\nu \\ar[d] \\ar[l]^j &\nX_{k'} \\ar[r] \\ar[d] &\nX \\ar[d]^f \\\\\n& \\Spec(k'_1) \\amalg \\ldots \\amalg \\Spec(k'_n) \\ar[r] &\n\\Spec(k') \\ar[r] &\n\\Spec(k)\n}\n$$\nof schemes with the following properties:\n\\begin{enumerate}\n\\item $k'/k$ is a finite purely inseparable extension of fields,\n\\item $\\nu$ is the normalization of $X_{k'}$,\n\\item $j$ is an open immersion with dense image,\n\\item $k'_i/k'$ is a finite separable extension for $i = 1, \\ldots, n$,\n\\item $\\overline{Y}_i$ is smooth, projective, geometrically irreducible\ndimension $\\leq 1$ over $k'_i$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GK5","source_file":"varieties.tex","source_line":9302,"source_end_line":9326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9302-L9326","statement_sha256":"c3befadbdee2ccc3c56a46eaf97b5ddfcab9f59d810efc1dbb136b5de71ec790","origin":"The Stacks Project","memory_eligible":false,"source_rank":6575,"rank":6575,"depth":44,"x":296.699,"y":1065.069,"cluster":"varieties-curves"},{"id":"stacks:0B8Y","tag":"0B8Y","title":"Curves · Lemma 0B8Y","summary":"Let k be a field. Let X be a curve over k. Let x ∈ X be a closed point. We think of x as a (reduced) closed subscheme of X with sheaf of ideals I. The following are equivalent • O_X, x is regular, • O_X, x is normal, • O_X, x is a discrete valuation ring, • I is an invertible O_X-module, • x is an effective Cartier divisor on X. If k is perfect or if kappa(x) is separable over k, these are also equivalent to • [(6)] X → Spec(k) is smooth at x.","statement_latex":"Let $k$ be a field. Let $X$ be a curve over $k$. Let $x \\in X$ be a closed\npoint. We think of $x$ as a (reduced) closed subscheme of $X$ with sheaf\nof ideals $\\mathcal{I}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{O}_{X, x}$ is regular,\n\\item $\\mathcal{O}_{X, x}$ is normal,\n\\item $\\mathcal{O}_{X, x}$ is a discrete valuation ring,\n\\item $\\mathcal{I}$ is an invertible $\\mathcal{O}_X$-module,\n\\item $x$ is an effective Cartier divisor on $X$.\n\\end{enumerate}\nIf $k$ is perfect or if $\\kappa(x)$ is separable over $k$,\nthese are also equivalent to\n\\begin{enumerate}\n\\item[(6)] $X \\to \\Spec(k)$ is smooth at $x$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8Y","source_file":"varieties.tex","source_line":9355,"source_end_line":9372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9355-L9372","statement_sha256":"2103f03f1f044aa4cac5244e7917a7603d8170a76bdd5dde6cadc80430a4cafb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6576,"rank":6576,"depth":46,"x":241.099,"y":1233.299,"cluster":"varieties-curves"},{"id":"stacks:0A27","tag":"0A27","title":"Curves · Lemma 0A27","summary":"Let X be a curve over k. Then either X is an affine scheme or X is H-projective over k.","statement_latex":"Let $X$ be a curve over $k$. Then either $X$ is an affine scheme or $X$\nis H-projective over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A27","source_file":"varieties.tex","source_line":9411,"source_end_line":9415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9411-L9415","statement_sha256":"3f1ab2fc53b6eaafdc567bcdb069b50613f1f92b1c0ee6622e141c908391042e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6577,"rank":6577,"depth":47,"x":146.609,"y":1077.367,"cluster":"varieties-curves"},{"id":"stacks:0A28","tag":"0A28","title":"Curves · Lemma 0A28","summary":"Let X be a separated scheme of finite type over k. If dim(X) ≤ 1 and no irreducible component of X is proper of dimension 1, then X is affine.","statement_latex":"Let $X$ be a separated scheme of finite type over $k$.\nIf $\\dim(X) \\leq 1$ and no irreducible component of $X$\nis proper of dimension $1$, then $X$ is affine.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A28","source_file":"varieties.tex","source_line":9501,"source_end_line":9506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9501-L9506","statement_sha256":"a304fff562c9d369a796d027c779e56281f94222db6b6c6f16546b6c43c4e031","origin":"The Stacks Project","memory_eligible":false,"source_rank":6578,"rank":6578,"depth":48,"x":342.097,"y":1138.864,"cluster":"varieties-curves"},{"id":"stacks:0AYR","tag":"0AYR","title":"Degrees on curves · Definition 0AYR","summary":"Let k be a field, let X be a proper scheme of dimension ≤ 1 over k, and let L be an invertible O_X-module. The degree of L is defined by deg(L) = chi(X, L) - chi(X, O_X) More generally, if E is a locally free sheaf of rank n we define the degree of E by deg(E) = chi(X, E) - nchi(X, O_X)","statement_latex":"Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$\nover $k$, and let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThe {\\it degree} of $\\mathcal{L}$ is defined by\n$$\n\\deg(\\mathcal{L}) = \\chi(X, \\mathcal{L}) - \\chi(X, \\mathcal{O}_X)\n$$\nMore generally, if $\\mathcal{E}$ is a locally free sheaf of rank $n$\nwe define the {\\it degree} of $\\mathcal{E}$ by\n$$\n\\deg(\\mathcal{E}) = \\chi(X, \\mathcal{E}) - n\\chi(X, \\mathcal{O}_X)\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYR","source_file":"varieties.tex","source_line":9540,"source_end_line":9553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9540-L9553","statement_sha256":"83e23b13df98f4fa637ae18a6d50fe2fdcafe32cd21ef4458572c9f1420323f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6579,"rank":6579,"depth":0,"x":148.082,"y":1204.58,"cluster":"varieties-curves"},{"id":"stacks:0B59","tag":"0B59","title":"Degrees on curves · Lemma 0B59","summary":"Let k'/k be an extension of fields. Let X be a proper scheme of dimension ≤ 1 over k. Let E be a locally free O_X-module of constant rank n. Then the degree of E/X/k is equal to the degree of E_k'/X_k'/k'.","statement_latex":"Let $k'/k$ be an extension of fields. Let $X$ be a proper scheme of\ndimension $\\leq 1$ over $k$. Let $\\mathcal{E}$ be a locally free\n$\\mathcal{O}_X$-module of constant rank $n$. Then the degree of\n$\\mathcal{E}/X/k$ is equal to the degree of\n$\\mathcal{E}_{k'}/X_{k'}/k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B59","source_file":"varieties.tex","source_line":9569,"source_end_line":9576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9569-L9576","statement_sha256":"81388e84889f8d114f688e98c1e0dd3493c3494b509da8a7c030c53a42fef07b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6580,"rank":6580,"depth":30,"x":238.49,"y":1045.699,"cluster":"varieties-curves"},{"id":"stacks:0AYS","tag":"0AYS","title":"Degrees on curves · Lemma 0AYS","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. Let 0 → E_1 → E_2 → E_3 → 0 be a short exact sequence of locally free O_X-modules each of finite constant rank. Then deg(E_2) = deg(E_1) + deg(E_3)","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$\nover $k$. Let $0 \\to \\mathcal{E}_1 \\to \\mathcal{E}_2 \\to \\mathcal{E}_3 \\to 0$\nbe a short exact sequence of locally free $\\mathcal{O}_X$-modules\neach of finite constant rank. Then\n$$\n\\deg(\\mathcal{E}_2) = \\deg(\\mathcal{E}_1) + \\deg(\\mathcal{E}_3)\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYS","source_file":"varieties.tex","source_line":9591,"source_end_line":9600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9591-L9600","statement_sha256":"0f07eb9eccd08e440709c62818a8e9df31e3ace0a486caadfa88d24aad8165de","origin":"The Stacks Project","memory_eligible":false,"source_rank":6581,"rank":6581,"depth":1,"x":299.721,"y":1214.51,"cluster":"varieties-curves"},{"id":"stacks:0AYU","tag":"0AYU","title":"Degrees on curves · Lemma 0AYU","summary":"Let k be a field. Let f : X' → X be a birational morphism of proper schemes of dimension ≤ 1 over k. Then deg(f^*E) = deg(E) for every finite locally free sheaf of constant rank. More generally it suffices if f induces a bijection between irreducible components of dimension 1 and isomorphisms of local rings at the corresponding generic points.","statement_latex":"Let $k$ be a field. Let $f : X' \\to X$ be a birational morphism of\nproper schemes of dimension $\\leq 1$ over $k$. Then\n$$\n\\deg(f^*\\mathcal{E}) = \\deg(\\mathcal{E})\n$$\nfor every finite locally free sheaf of constant rank. More generally\nit suffices if $f$ induces a bijection between irreducible components\nof dimension $1$ and isomorphisms of local rings at the corresponding\ngeneric points.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYU","source_file":"varieties.tex","source_line":9608,"source_end_line":9619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9608-L9619","statement_sha256":"8af26641023bd79629f03ba7f3e08bcc1fb4de3eeca4e51c5881d76d28146c54","origin":"The Stacks Project","memory_eligible":false,"source_rank":6582,"rank":6582,"depth":41,"x":118.438,"y":1124.587,"cluster":"varieties-curves"},{"id":"stacks:0AYV","tag":"0AYV","title":"Degrees on curves · Lemma 0AYV","summary":"Let k be a field. Let X be a proper curve over k with generic point xi. Let E be a locally free O_X-module of rank n and let F be a coherent O_X-module. Then chi(X, E ⊗ F) = r deg(E) + n chi(X, F) where r = dim_kappa(xi) F_xi is the rank of F.","statement_latex":"Let $k$ be a field. Let $X$ be a proper curve over $k$ with generic point\n$\\xi$. Let $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $n$\nand let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Then\n$$\n\\chi(X, \\mathcal{E} \\otimes \\mathcal{F}) =\nr \\deg(\\mathcal{E}) + n \\chi(X, \\mathcal{F})\n$$\nwhere $r = \\dim_{\\kappa(\\xi)} \\mathcal{F}_\\xi$ is the rank of $\\mathcal{F}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYV","source_file":"varieties.tex","source_line":9668,"source_end_line":9678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9668-L9678","statement_sha256":"01e97e1dfdccc3fc57b8e88c5409d59badc0f207cf676e245f5113940852472f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6583,"rank":6583,"depth":27,"x":324.856,"y":1087.954,"cluster":"varieties-curves"},{"id":"stacks:0AYW","tag":"0AYW","title":"Degrees on curves · Lemma 0AYW","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. Let E be a locally free O_X-module of rank n. Then deg(E) = ∑ m_i deg(E|_C_i) where C_i ⊂ X, i = 1, …, t are the irreducible components of dimension 1 with reduced induced scheme structure and m_i is the multiplicity of C_i in X.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.\nLet $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $n$.\nThen\n$$\n\\deg(\\mathcal{E}) = \\sum m_i \\deg(\\mathcal{E}|_{C_i})\n$$\nwhere $C_i \\subset X$, $i = 1, \\ldots, t$ are the irreducible components\nof dimension $1$ with reduced induced scheme structure and $m_i$ is the\nmultiplicity of $C_i$ in $X$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYW","source_file":"varieties.tex","source_line":9712,"source_end_line":9723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9712-L9723","statement_sha256":"f7898d434bb90c5bc4b34aa23b9ecab888fc344656c6ba83a2289fffcb9a56a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6584,"rank":6584,"depth":42,"x":201.85,"y":1232.393,"cluster":"varieties-curves"},{"id":"stacks:0AYX","tag":"0AYX","title":"Degrees on curves · Lemma 0AYX","summary":"Let k be a field, let X be a proper scheme of dimension ≤ 1 over k, and let E, V be locally free O_X-modules of constant finite rank. Then deg(E ⊗ V) = rank(E) deg(V) + rank(V) deg(E)","statement_latex":"Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$\nover $k$, and let $\\mathcal{E}$, $\\mathcal{V}$ be locally free\n$\\mathcal{O}_X$-modules of constant finite rank. Then\n$$\n\\deg(\\mathcal{E} \\otimes \\mathcal{V}) =\n\\text{rank}(\\mathcal{E}) \\deg(\\mathcal{V}) +\n\\text{rank}(\\mathcal{V}) \\deg(\\mathcal{E})\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYX","source_file":"varieties.tex","source_line":9783,"source_end_line":9793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9783-L9793","statement_sha256":"2bf1d1545b08a376c9a2733e4aef9b7455c73d59d98bdc941a740be5d6127227","origin":"The Stacks Project","memory_eligible":false,"source_rank":6585,"rank":6585,"depth":43,"x":176.347,"y":1055.724,"cluster":"varieties-curves"},{"id":"stacks:0DJ5","tag":"0DJ5","title":"Degrees on curves · Lemma 0DJ5","summary":"Let k be a field, let X be a proper scheme of dimension ≤ 1 over k, and let E be a locally free O_X-module of rank n. Then deg(E) = deg(wedge^n(E)) = deg(det(E))","statement_latex":"Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$\nover $k$, and let $\\mathcal{E}$ be a locally free\n$\\mathcal{O}_X$-module of rank $n$. Then\n$$\n\\deg(\\mathcal{E}) = \\deg(\\wedge^n(\\mathcal{E})) = \\deg(\\det(\\mathcal{E}))\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJ5","source_file":"varieties.tex","source_line":9801,"source_end_line":9809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9801-L9809","statement_sha256":"e722e03d9de2905bdbe94a90f1fdffbfac1152d7ca56bfddb21f959455e27ee9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6586,"rank":6586,"depth":44,"x":337.555,"y":1171.765,"cluster":"varieties-curves"},{"id":"stacks:0AYY","tag":"0AYY","title":"Degrees on curves · Lemma 0AYY","summary":"Let k be a field, let X be a proper scheme of dimension ≤ 1 over k. Let D be an effective Cartier divisor on X. Then D is finite over Spec(k) of degree deg(D) = dim_k Γ(D, O_D). For a locally free sheaf E of rank n we have deg(E(D)) = ndeg(D) + deg(E) where E(D) = E ⊗_O_X O_X(D).","statement_latex":"Let $k$ be a field, let $X$ be a proper scheme of dimension $\\leq 1$\nover $k$. Let $D$ be an effective Cartier divisor on $X$.\nThen $D$ is finite over $\\Spec(k)$ of degree\n$\\deg(D) = \\dim_k \\Gamma(D, \\mathcal{O}_D)$. For a locally free sheaf\n$\\mathcal{E}$ of rank $n$ we have\n$$\n\\deg(\\mathcal{E}(D)) = n\\deg(D) + \\deg(\\mathcal{E})\n$$\nwhere $\\mathcal{E}(D) = \\mathcal{E} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D)$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYY","source_file":"varieties.tex","source_line":9832,"source_end_line":9843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9832-L9843","statement_sha256":"51a5555b86d23256ac35e98783b31e3a4e869dd37c7ff9ac9e22457c89eb1475","origin":"The Stacks Project","memory_eligible":false,"source_rank":6587,"rank":6587,"depth":27,"x":124.931,"y":1177.684,"cluster":"varieties-curves"},{"id":"stacks:0C6P","tag":"0C6P","title":"Degrees on curves · Lemma 0C6P","summary":"Let k be a field. Let X be a proper scheme over k which is reduced and connected. Let kappa = H^0(X, O_X). Then kappa/k is a finite extension of fields and w = [kappa : k] divides • deg(E) for all locally free O_X-modules E, • [kappa(x) : k] for all closed points x ∈ X, and • deg(D) for all closed subschemes D ⊂ X of dimension zero.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$\nwhich is reduced and connected.\nLet $\\kappa = H^0(X, \\mathcal{O}_X)$. Then $\\kappa/k$ is a\nfinite extension of fields and $w = [\\kappa : k]$ divides\n\\begin{enumerate}\n\\item $\\deg(\\mathcal{E})$ for all locally free $\\mathcal{O}_X$-modules\n$\\mathcal{E}$,\n\\item $[\\kappa(x) : k]$ for all closed points $x \\in X$, and\n\\item $\\deg(D)$ for all closed subschemes $D \\subset X$\nof dimension zero.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6P","source_file":"varieties.tex","source_line":9871,"source_end_line":9884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9871-L9884","statement_sha256":"bec8bdf3947409d45ccd17d7d79d1b063d7c89fde0db95b793817328c05aeffe","origin":"The Stacks Project","memory_eligible":false,"source_rank":6588,"rank":6588,"depth":33,"x":277.268,"y":1052.416,"cluster":"varieties-curves"},{"id":"stacks:0AYZ","tag":"0AYZ","title":"Degrees on curves · Lemma 0AYZ","summary":"Let k be a field. Let f : X → Y be a nonconstant morphism of proper curves over k. Let E be a locally free O_Y-module. Then deg(f^*E) = deg(X/Y) deg(E)","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a nonconstant morphism of\nproper curves over $k$. Let $\\mathcal{E}$ be a locally free\n$\\mathcal{O}_Y$-module. Then\n$$\n\\deg(f^*\\mathcal{E}) = \\deg(X/Y) \\deg(\\mathcal{E})\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYZ","source_file":"varieties.tex","source_line":9895,"source_end_line":9903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9895-L9903","statement_sha256":"6d5f7fe75f543792308d257f7c524fc02df51e1999ca8d5712048a4c2c228e14","origin":"The Stacks Project","memory_eligible":false,"source_rank":6589,"rank":6589,"depth":32,"x":265.65,"y":1231.59,"cluster":"varieties-curves"},{"id":"stacks:0B40","tag":"0B40","title":"Degrees on curves · Lemma 0B40","summary":"Let k be a field. Let X be a proper curve over k. Let L be an invertible O_X-module. • If L has a nonzero section, then deg(L) ≥ 0. • If L has a nonzero section s which vanishes at a point, then deg(L) > 0. • If L and L^-1 have nonzero sections, then L ≅ O_X. • If deg(L) ≤ 0 and L has a nonzero section, then L ≅ O_X. • If N → L is a nonzero map of invertible O_X-modules, then deg(L) ≥ deg(N) and if equality holds then it is an isomorphism.","statement_latex":"Let $k$ be a field. Let $X$ be a proper curve over $k$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If $\\mathcal{L}$ has a nonzero section, then\n$\\deg(\\mathcal{L}) \\geq 0$.\n\\item If $\\mathcal{L}$ has a nonzero section $s$ which vanishes\nat a point, then $\\deg(\\mathcal{L}) > 0$.\n\\item If $\\mathcal{L}$ and $\\mathcal{L}^{-1}$ have nonzero sections, then\n$\\mathcal{L} \\cong \\mathcal{O}_X$.\n\\item If $\\deg(\\mathcal{L}) \\leq 0$ and $\\mathcal{L}$ has a nonzero\nsection, then $\\mathcal{L} \\cong \\mathcal{O}_X$.\n\\item If $\\mathcal{N} \\to \\mathcal{L}$ is a nonzero map of invertible\n$\\mathcal{O}_X$-modules, then $\\deg(\\mathcal{L}) \\geq \\deg(\\mathcal{N})$\nand if equality holds then it is an isomorphism.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B40","source_file":"varieties.tex","source_line":9934,"source_end_line":9951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9934-L9951","statement_sha256":"e6bf5f85b070dd5e95a4970cbf6fec006f77743e70939c34f0786b51609af18f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6590,"rank":6590,"depth":44,"x":129.856,"y":1092.596,"cluster":"varieties-curves"},{"id":"stacks:0E22","tag":"0E22","title":"Degrees on curves · Lemma 0E22","summary":"Let k be a field. Let X be a proper scheme over k which is reduced, connected, and equidimensional of dimension 1. Let L be an invertible O_X-module. If deg(L|_C) ≤ 0 for all irreducible components C of X, then either H^0(X, L) = 0 or L ≅ O_X.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$\nwhich is reduced, connected, and equidimensional of dimension $1$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nIf $\\deg(\\mathcal{L}|_C) \\leq 0$ for all irreducible components $C$\nof $X$, then either $H^0(X, \\mathcal{L}) = 0$ or\n$\\mathcal{L} \\cong \\mathcal{O}_X$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E22","source_file":"varieties.tex","source_line":9975,"source_end_line":9983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9975-L9983","statement_sha256":"42b10e61685c6d61a8fa9afbed29e44cd87ba7dbf66f8c3a33065fe8f47eb01e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6591,"rank":6591,"depth":45,"x":342.191,"y":1118.087,"cluster":"varieties-curves"},{"id":"stacks:0B5X","tag":"0B5X","title":"Degrees on curves · Lemma 0B5X","summary":"Let k be a field. Let X be a proper curve over k. Let L be an invertible O_X-module. Then L is ample if and only if deg(L) > 0.","statement_latex":"Let $k$ be a field. Let $X$ be a proper curve over $k$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen $\\mathcal{L}$ is ample if and only if $\\deg(\\mathcal{L}) > 0$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5X","source_file":"varieties.tex","source_line":9997,"source_end_line":10002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L9997-L10002","statement_sha256":"c0c1dffd27fe3f5f4492a4f4614b66d4d011faacec4d7db2f744eb176c736f26","origin":"The Stacks Project","memory_eligible":false,"source_rank":6592,"rank":6592,"depth":45,"x":164.764,"y":1219.978,"cluster":"varieties-curves"},{"id":"stacks:0B5Y","tag":"0B5Y","title":"Degrees on curves · Lemma 0B5Y","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. Let L be an invertible O_X-module. Let C_i ⊂ X, i = 1, …, t be the irreducible components of dimension 1. The following are equivalent: • L is ample, and • deg(L|_C_i) > 0 for i = 1, …, t.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$\nover $k$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $C_i \\subset X$, $i = 1, \\ldots, t$ be the irreducible components\nof dimension $1$. The following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample, and\n\\item $\\deg(\\mathcal{L}|_{C_i}) > 0$ for $i = 1, \\ldots, t$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5Y","source_file":"varieties.tex","source_line":10069,"source_end_line":10079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10069-L10079","statement_sha256":"eba53a5a07770030a1c1ae102f0ada8edfd502e0633ef44f6e071f9d6b996b27","origin":"The Stacks Project","memory_eligible":false,"source_rank":6593,"rank":6593,"depth":46,"x":213.756,"y":1043.817,"cluster":"varieties-curves"},{"id":"stacks:0B8Z","tag":"0B8Z","title":"Degrees on curves · Lemma 0B8Z","summary":"Let k be an algebraically closed field. Let X be a proper curve over k. Then there exist • an invertible O_X-module L with dim_k H^0(X, L) = 1 and H^1(X, L) = 0, and • an invertible O_X-module N with dim_k H^0(X, N) = 0 and H^1(X, N) = 0.","statement_latex":"Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$.\nThen there exist\n\\begin{enumerate}\n\\item an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ with\n$\\dim_k H^0(X, \\mathcal{L}) = 1$ and $H^1(X, \\mathcal{L}) = 0$, and\n\\item an invertible $\\mathcal{O}_X$-module $\\mathcal{N}$ with\n$\\dim_k H^0(X, \\mathcal{N}) = 0$ and $H^1(X, \\mathcal{N}) = 0$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8Z","source_file":"varieties.tex","source_line":10096,"source_end_line":10106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10096-L10106","statement_sha256":"02119ce24d6da99533892b65819e9f4c1302d9bc5040424dfb33f349160e0743","origin":"The Stacks Project","memory_eligible":false,"source_rank":6594,"rank":6594,"depth":47,"x":319.503,"y":1201.829,"cluster":"varieties-curves"},{"id":"stacks:0B90","tag":"0B90","title":"Degrees on curves · Lemma 0B90","summary":"Let k be an algebraically closed field. Let X be a proper curve over k. Set g = dim_k H^1(X, O_X). For every invertible O_X-module L with deg(L) ≥ 2g - 1 we have H^1(X, L) = 0.","statement_latex":"Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$.\nSet $g = \\dim_k H^1(X, \\mathcal{O}_X)$. For every invertible\n$\\mathcal{O}_X$-module $\\mathcal{L}$ with $\\deg(\\mathcal{L}) \\geq 2g - 1$\nwe have $H^1(X, \\mathcal{L}) = 0$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Degrees on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B90","source_file":"varieties.tex","source_line":10151,"source_end_line":10157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10151-L10157","statement_sha256":"e29fa4827192a41e5159b862cae03a82e15d751a12b9fd0d0f32530ff9acce90","origin":"The Stacks Project","memory_eligible":false,"source_rank":6595,"rank":6595,"depth":48,"x":114.051,"y":1145.205,"cluster":"varieties-curves"},{"id":"stacks:0BEM","tag":"0BEM","title":"Numerical intersections · Lemma 0BEM","summary":"Let k be a field. Let X be a proper scheme over k. Let F be a coherent O_X-module. Let L_1, …, L_r be invertible O_X-modules. The map (n_1, …, n_r) ↦ chi(X, F ⊗ L_1^⊗ n_1 ⊗ … ⊗ L_r^⊗ n_r) is a numerical polynomial in n_1, …, n_r of total degree at most the dimension of the support of F.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{F}$\nbe a coherent $\\mathcal{O}_X$-module. Let\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules.\nThe map\n$$\n(n_1, \\ldots, n_r) \\longmapsto\n\\chi(X, \\mathcal{F} \\otimes\n\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes\n\\mathcal{L}_r^{\\otimes n_r})\n$$\nis a numerical polynomial in $n_1, \\ldots, n_r$ of total degree at\nmost the dimension of the support of $\\mathcal{F}$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEM","source_file":"varieties.tex","source_line":10194,"source_end_line":10208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10194-L10208","statement_sha256":"3339ddb9960ace5e2c0ea90c0e1ed365510e97c5ffa0abe9b94ecbb0badc5b7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6596,"rank":6596,"depth":27,"x":311.474,"y":1070.232,"cluster":"varieties-curves"},{"id":"stacks:0BEN","tag":"0BEN","title":"Numerical intersections · Lemma 0BEN","summary":"Let k be a field. Let X be a proper scheme over k. Let F be a coherent O_X-module. Let L_1, …, L_r be invertible O_X-modules. Let d = dim(Supp(F)). Let Z_i ⊂ X be the irreducible components of Supp(F) of dimension d. Let xi_i ∈ Z_i be the generic point and set m_i = length_O_X, xi_i(F_xi_i). Then chi(X, F ⊗ L_1^⊗ n_1 ⊗ … ⊗ L_r^⊗ n_r) - ∑_i m_i chi(Z_i, L_1^⊗ n_1 ⊗ … ⊗ L_r^⊗ n_r|_Z_i) is a numerical polynomial in n_1, …, n_r of total degree < d.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let\n$\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules.\nLet $d = \\dim(\\text{Supp}(\\mathcal{F}))$.\nLet $Z_i \\subset X$ be the irreducible components\nof $\\text{Supp}(\\mathcal{F})$ of dimension $d$. Let $\\xi_i \\in Z_i$\nbe the generic point and set\n$m_i = \\text{length}_{\\mathcal{O}_{X, \\xi_i}}(\\mathcal{F}_{\\xi_i})$.\nThen\n$$\n\\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes\n\\mathcal{L}_r^{\\otimes n_r}) -\n\\sum\\nolimits_i\nm_i\\ \\chi(Z_i, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes\n\\mathcal{L}_r^{\\otimes n_r}|_{Z_i})\n$$\nis a numerical polynomial in $n_1, \\ldots, n_r$ of total degree $< d$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEN","source_file":"varieties.tex","source_line":10289,"source_end_line":10308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10289-L10308","statement_sha256":"c2b20931ebc98f331cc708624ab50fce8da3028deef0c848d22a4773a7c5722f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6597,"rank":6597,"depth":28,"x":226.02,"y":1237.868,"cluster":"varieties-curves"},{"id":"stacks:0BEP","tag":"0BEP","title":"Numerical intersections · Definition 0BEP","summary":"Let k be a field. Let X be a proper scheme over k. Let i : Z → X be a closed subscheme of dimension d. Let L_1, …, L_d be invertible O_X-modules. We define the intersection number (L_1 … L_d · Z) as the coefficient of n_1 … n_d in the numerical polynomial chi(X, i_*O_Z ⊗ L_1^⊗ n_1 ⊗ … ⊗ L_d^⊗ n_d) = chi(Z, L_1^⊗ n_1 ⊗ … ⊗ L_d^⊗ n_d|_Z) In the special case that L_1 = … = L_d = L we write (L^d · Z).","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let\n$i : Z \\to X$ be a closed subscheme of dimension $d$. Let\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible\n$\\mathcal{O}_X$-modules. We define the {\\it intersection number}\n$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$\nas the coefficient of $n_1 \\ldots n_d$ in the numerical polynomial\n$$\n\\chi(X, i_*\\mathcal{O}_Z \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes\n\\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}) =\n\\chi(Z, \\mathcal{L}_1^{\\otimes n_1} \\otimes\n\\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}|_Z)\n$$\nIn the special\ncase that $\\mathcal{L}_1 = \\ldots = \\mathcal{L}_d = \\mathcal{L}$\nwe write $(\\mathcal{L}^d \\cdot Z)$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEP","source_file":"varieties.tex","source_line":10388,"source_end_line":10405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10388-L10405","statement_sha256":"80fc18da8fc1daf0fb264e1984de86d0ee1ba8b8f39378a61de52c74c0daf2f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6598,"rank":6598,"depth":0,"x":154.084,"y":1065.43,"cluster":"varieties-curves"},{"id":"stacks:0BEQ","tag":"0BEQ","title":"Numerical intersections · Lemma 0BEQ","summary":"In the situation of Definition [Tag 0BEP] the intersection number (L_1 … L_d · Z) is an integer.","statement_latex":"In the situation of Definition \\ref{definition-intersection-number}\nthe intersection number\n$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$\nis an integer.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEQ","source_file":"varieties.tex","source_line":10415,"source_end_line":10421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10415-L10421","statement_sha256":"6d13db06b5785650261fc58d22116b8dd7b2f7a0946763372b23663730d44722","origin":"The Stacks Project","memory_eligible":false,"source_rank":6599,"rank":6599,"depth":1,"x":346.172,"y":1151.934,"cluster":"varieties-curves"},{"id":"stacks:0BER","tag":"0BER","title":"Numerical intersections · Lemma 0BER","summary":"In the situation of Definition [Tag 0BEP] the intersection number (L_1 … L_d · Z) is additive: if L_i = L_i' ⊗ L_i\", then we have (L_1 … L_i … L_d · Z) = (L_1 … L_i' … L_d · Z) + (L_1 … L_i\" … L_d · Z)","statement_latex":"In the situation of Definition \\ref{definition-intersection-number}\nthe intersection number\n$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$\nis additive: if $\\mathcal{L}_i = \\mathcal{L}_i' \\otimes \\mathcal{L}_i''$,\nthen we have\n$$\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_i \\cdots \\mathcal{L}_d \\cdot Z) =\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_i' \\cdots \\mathcal{L}_d \\cdot Z) +\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_i'' \\cdots \\mathcal{L}_d \\cdot Z)\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BER","source_file":"varieties.tex","source_line":10431,"source_end_line":10443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10431-L10443","statement_sha256":"0223f05c50f05970e9ab2c26e340e714ea336e966c856ce12225d9c16c5cf554","origin":"The Stacks Project","memory_eligible":false,"source_rank":6600,"rank":6600,"depth":28,"x":134.555,"y":1197.23,"cluster":"varieties-curves"},{"id":"stacks:0BES","tag":"0BES","title":"Numerical intersections · Lemma 0BES","summary":"In the situation of Definition [Tag 0BEP] let Z_i ⊂ Z be the irreducible components of dimension d. Let m_i = length_O_X, xi_i(O_Z, xi_i) where xi_i ∈ Z_i is the generic point. Then (L_1 … L_d · Z) = ∑ m_i(L_1 … L_d · Z_i)","statement_latex":"In the situation of Definition \\ref{definition-intersection-number}\nlet $Z_i \\subset Z$ be the irreducible components of dimension $d$. Let\n$m_i = \\text{length}_{\\mathcal{O}_{X, \\xi_i}}(\\mathcal{O}_{Z, \\xi_i})$\nwhere $\\xi_i \\in Z_i$ is the generic point. Then\n$$\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =\n\\sum m_i(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z_i)\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BES","source_file":"varieties.tex","source_line":10459,"source_end_line":10469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10459-L10469","statement_sha256":"2c99af842f07e16a2668d6912c7e72c087942827a381ce58a2f13a0ee2837fd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6601,"rank":6601,"depth":29,"x":254.404,"y":1043.455,"cluster":"varieties-curves"},{"id":"stacks:0BET","tag":"0BET","title":"Numerical intersections · Lemma 0BET","summary":"Let k be a field. Let f : Y → X be a morphism of proper schemes over k. Let Z ⊂ Y be an integral closed subscheme of dimension d and let L_1, …, L_d be invertible O_X-modules. Then (f^*L_1 … f^*L_d · Z) = deg(f|_Z : Z → f(Z)) (L_1 … L_d · f(Z)) where deg(Z → f(Z)) is as in Morphisms, Definition [Tag 02NY] or 0 if dim(f(Z)) < d.","statement_latex":"Let $k$ be a field. Let $f : Y \\to X$ be a morphism of proper schemes over $k$.\nLet $Z \\subset Y$ be an integral closed subscheme of dimension $d$ and let\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules.\nThen\n$$\n(f^*\\mathcal{L}_1 \\cdots f^*\\mathcal{L}_d \\cdot Z) =\n\\deg(f|_Z : Z \\to f(Z)) (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot f(Z))\n$$\nwhere $\\deg(Z \\to f(Z))$ is as in\nMorphisms, Definition \\ref{morphisms-definition-degree}\nor $0$ if $\\dim(f(Z)) < d$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BET","source_file":"varieties.tex","source_line":10476,"source_end_line":10489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10476-L10489","statement_sha256":"e5be1eedd4ac54bb9cee46bcd531129f2026d4a0aaa4f0dd17e7f717a79b7b56","origin":"The Stacks Project","memory_eligible":false,"source_rank":6602,"rank":6602,"depth":35,"x":289.758,"y":1225.202,"cluster":"varieties-curves"},{"id":"stacks:0BEU","tag":"0BEU","title":"Numerical intersections · Lemma 0BEU","summary":"Let k be a field. Let X be proper over k. Let Z ⊂ X be a closed subscheme of dimension d. Let L_1, …, L_d be invertible O_X-modules. Assume there exists an effective Cartier divisor D ⊂ Z such that L_1|_Z ≅ O_Z(D). Then (L_1 … L_d · Z) = (L_2 … L_d · D)","statement_latex":"Let $k$ be a field. Let $X$ be proper over $k$. Let $Z \\subset X$ be\na closed subscheme of dimension $d$. Let $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$\nbe invertible $\\mathcal{O}_X$-modules. Assume there exists an\neffective Cartier divisor $D \\subset Z$ such that\n$\\mathcal{L}_1|_Z \\cong \\mathcal{O}_Z(D)$. Then\n$$\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =\n(\\mathcal{L}_2 \\cdots \\mathcal{L}_d \\cdot D)\n$$","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEU","source_file":"varieties.tex","source_line":10540,"source_end_line":10551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10540-L10551","statement_sha256":"79603ccc504b7fa0b10fd2fe67324ff78376f97c16e4adc7a07bcd0695444d62","origin":"The Stacks Project","memory_eligible":false,"source_rank":6603,"rank":6603,"depth":0,"x":117.202,"y":1111.026,"cluster":"varieties-curves"},{"id":"stacks:0BEV","tag":"0BEV","title":"Numerical intersections · Lemma 0BEV","summary":"Let k be a field. Let X be proper over k. Let Z ⊂ X be a closed subscheme of dimension d. If L_1, …, L_d are ample, then (L_1 … L_d · Z) is positive.","statement_latex":"Let $k$ be a field. Let $X$ be proper over $k$. Let $Z \\subset X$ be\na closed subscheme of dimension $d$. If $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$\nare ample, then $(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$ is positive.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEV","source_file":"varieties.tex","source_line":10579,"source_end_line":10584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10579-L10584","statement_sha256":"8fee6912cab220f277a7f73c94f321a92a2b12b6f729fd888aee17fc0040cbd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6604,"rank":6604,"depth":30,"x":336.679,"y":1097.279,"cluster":"varieties-curves"},{"id":"stacks:0BEW","tag":"0BEW","title":"Numerical intersections · Definition 0BEW","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module. For any closed subscheme the degree of Z with respect to L, denoted deg_L(Z), is the intersection number (L^d · Z) where d = dim(Z).","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let\n$\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.\nFor any closed subscheme the {\\it degree of $Z$ with respect to\n$\\mathcal{L}$}, denoted $\\deg_\\mathcal{L}(Z)$, is\nthe intersection number $(\\mathcal{L}^d \\cdot Z)$\nwhere $d = \\dim(Z)$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEW","source_file":"varieties.tex","source_line":10621,"source_end_line":10629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10621-L10629","statement_sha256":"cb6993b478567a6d6aa2404ca81f7bae3c533f7a4d2aa22eb6f797202dab20d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6605,"rank":6605,"depth":0,"x":185.607,"y":1232.209,"cluster":"varieties-curves"},{"id":"stacks:0BEX","tag":"0BEX","title":"Numerical intersections · Lemma 0BEX","summary":"Let k be a field. Let f : Y → X be a finite dominant morphism of proper varieties over k. Let L be an ample invertible O_X-module. Then deg_f^*L(Y) = deg(f) deg_L(X) where deg(f) is as in Morphisms, Definition [Tag 02NY].","statement_latex":"Let $k$ be a field. Let $f : Y \\to X$ be a finite\ndominant morphism of proper varieties over $k$. Let $\\mathcal{L}$\nbe an ample invertible $\\mathcal{O}_X$-module.\nThen\n$$\n\\deg_{f^*\\mathcal{L}}(Y) = \\deg(f) \\deg_\\mathcal{L}(X)\n$$\nwhere $\\deg(f)$ is as in\nMorphisms, Definition \\ref{morphisms-definition-degree}.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEX","source_file":"varieties.tex","source_line":10643,"source_end_line":10654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10643-L10654","statement_sha256":"a23680140b6c054231a405bc0046d1dc0490bc6b51849812db4f5b27c12c908a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6606,"rank":6606,"depth":36,"x":188.504,"y":1046.64,"cluster":"varieties-curves"},{"id":"stacks:0BEY","tag":"0BEY","title":"Numerical intersections · Lemma 0BEY","summary":"Let k be a field. Let X be a proper scheme over k. Let Z ⊂ X be a closed subscheme of dimension ≤ 1. Let L be an invertible O_X-module. Then (L · Z) = deg(L|_Z) where deg(L|_Z) is as in Definition [Tag 0AYR]. If L is ample, then deg_L(Z) = deg(L|_Z).","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\nLet $Z \\subset X$ be a closed subscheme of dimension $\\leq 1$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen\n$$\n(\\mathcal{L} \\cdot Z) = \\deg(\\mathcal{L}|_Z)\n$$\nwhere $\\deg(\\mathcal{L}|_Z)$ is as in\nDefinition \\ref{definition-degree-invertible-sheaf}.\nIf $\\mathcal{L}$ is ample, then\n$\\deg_\\mathcal{L}(Z) = \\deg(\\mathcal{L}|_Z)$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BEY","source_file":"varieties.tex","source_line":10668,"source_end_line":10681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10668-L10681","statement_sha256":"c7318d5659244e52bfe6529cb5132ab7b3b8f14a9946d4189f46c9e3e56d5eb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6607,"rank":6607,"depth":1,"x":335.875,"y":1185.382,"cluster":"varieties-curves"},{"id":"stacks:0BJ8","tag":"0BJ8","title":"Asymptotic Riemann-Roch · Proposition 0BJ8","summary":"Let k be a field. Let X be a proper scheme over k of dimension d. Let L be an ample invertible O_X-module. Then dim_k Γ(X, L^⊗ n) sim c n^d + l.o.t. where c = deg_L(X)/d! is a positive constant.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $d$.\nLet $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.\nThen\n$$\n\\dim_k \\Gamma(X, \\mathcal{L}^{\\otimes n}) \\sim c n^d + l.o.t.\n$$\nwhere $c = \\deg_\\mathcal{L}(X)/d!$ is a positive constant.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Numerical intersections","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJ8","source_file":"varieties.tex","source_line":10689,"source_end_line":10698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10689-L10698","statement_sha256":"111567b56f5a9fdf6a17625b4436b7007ea02eb134dec51c015e160f50920725","origin":"The Stacks Project","memory_eligible":false,"source_rank":6608,"rank":6608,"depth":31,"x":115.219,"y":1166.663,"cluster":"varieties-curves"},{"id":"stacks:0C1Q","tag":"0C1Q","title":"Embedding dimension · Definition 0C1Q","summary":"Let k be an algebraically closed field. Let X be a locally algebraic k-scheme and let x ∈ X be a closed point. The embedding dimension of X at x is dim_k m_x/ m_x^2.","statement_latex":"Let $k$ be an algebraically closed field. Let $X$ be a locally algebraic\n$k$-scheme and let $x \\in X$ be a closed point. The\n{\\it embedding dimension of $X$ at $x$} is\n$\\dim_k \\mathfrak m_x/\\mathfrak m_x^2$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Embedding dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1Q","source_file":"varieties.tex","source_line":10720,"source_end_line":10726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10720-L10726","statement_sha256":"8156b8e01a3cdd8f1adcac66f60338993ed8d8d1f906d70c7b923440541fbff1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6609,"rank":6609,"depth":0,"x":293.315,"y":1055.051,"cluster":"varieties-curves"},{"id":"stacks:0C2H","tag":"0C2H","title":"Embedding dimension · Definition 0C2H","summary":"Let k be a field. Let X be a locally algebraic k-scheme. Let x ∈ X be a point. The embedding dimension of X/k at x is dim_kappa(x)(T_X/k, x).","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme.\nLet $x \\in X$ be a point. The {\\it embedding dimension of $X/k$ at $x$}\nis $\\dim_{\\kappa(x)}(T_{X/k, x})$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Embedding dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2H","source_file":"varieties.tex","source_line":10785,"source_end_line":10790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10785-L10790","statement_sha256":"1d9c2daed8e3542a20205a6f51cddb82760064eba5a6399b350d3014b92f645c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6610,"rank":6610,"depth":0,"x":251.667,"y":1238.751,"cluster":"varieties-curves"},{"id":"stacks:0FD5","tag":"0FD5","title":"Bertini theorems · Lemma 0FD5","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module. Let Z ⊂ X be a closed subscheme. Then there exists an integer n_0 such that for all n ≥ n_0 the kernel V_n of Γ(X, L^⊗ n) → Γ(Z, L^⊗ n|_Z) generates L^⊗ n|_X setminus Z and the canonical morphism X setminus Z → P(V_n) is an immersion of schemes over k.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$\nbe an ample invertible $\\mathcal{O}_X$-module. Let $Z \\subset X$ be a\nclosed subscheme. Then there exists an integer $n_0$ such that for all\n$n \\geq n_0$ the kernel $V_n$ of\n$\\Gamma(X, \\mathcal{L}^{\\otimes n}) \\to \\Gamma(Z, \\mathcal{L}^{\\otimes n}|_Z)$\ngenerates $\\mathcal{L}^{\\otimes n}|_{X \\setminus Z}$ and\nthe canonical morphism\n$$\nX \\setminus Z \\longrightarrow \\mathbf{P}(V_n)\n$$\nis an immersion of schemes over $k$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Bertini theorems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FD5","source_file":"varieties.tex","source_line":10820,"source_end_line":10833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10820-L10833","statement_sha256":"ea23c44470aecc882e90828bdff60f01080cb9213991ba37abd1c1c903eee97c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6611,"rank":6611,"depth":21,"x":134.433,"y":1079.364,"cluster":"varieties-curves"},{"id":"stacks:0FD6","tag":"0FD6","title":"Bertini theorems · Lemma 0FD6","summary":"In Situation [Tag 0G47] assume • X is smooth over k, • the image of ψ : V → Γ(X, L) generates L, • the corresponding morphism φ_L, ψ : X → P(V) is an immersion. Then for general v ∈ V ⊗_k k' the scheme H_v is smooth over k'.","statement_latex":"In Situation \\ref{situation-family-divisors} assume\n\\begin{enumerate}\n\\item $X$ is smooth over $k$,\n\\item the image of $\\psi : V \\to \\Gamma(X, \\mathcal{L})$\ngenerates $\\mathcal{L}$,\n\\item the corresponding morphism\n$\\varphi_{\\mathcal{L}, \\psi} : X \\to \\mathbf{P}(V)$ is\nan immersion.\n\\end{enumerate}\nThen for general $v \\in V \\otimes_k k'$\nthe scheme $H_v$ is smooth over $k'$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Bertini theorems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FD6","source_file":"varieties.tex","source_line":10951,"source_end_line":10964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L10951-L10964","statement_sha256":"b4ea9641386605ac3a4d60d405a57f407b261d2975bf07d9c106f17cc7e7e76d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6612,"rank":6612,"depth":39,"x":349.452,"y":1130.467,"cluster":"varieties-curves"},{"id":"stacks:0FD7","tag":"0FD7","title":"Enriques-Severi-Zariski · Lemma 0FD7","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module. Let F be a coherent O_X-module. If Ass(F) does not contain any closed points, then Γ(X, F ⊗_O_X L^⊗ n) = 0 for n ll 0.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$\nbe an ample invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a\ncoherent $\\mathcal{O}_X$-module. If $\\text{Ass}(\\mathcal{F})$ does not\ncontain any closed points, then\n$\\Gamma(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$\nfor $n \\ll 0$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Enriques-Severi-Zariski","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FD7","source_file":"varieties.tex","source_line":11074,"source_end_line":11082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L11074-L11082","statement_sha256":"dbe90bb1b79fde9195ba6a0a0de0627cc8105d881c0de260f4141f6ddb34a145","origin":"The Stacks Project","memory_eligible":false,"source_rank":6613,"rank":6613,"depth":33,"x":149.435,"y":1214.947,"cluster":"varieties-curves"},{"id":"stacks:0FD8","tag":"0FD8","title":"Enriques-Severi-Zariski · Lemma 0FD8","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module. Let F be a coherent O_X-module. Assume that for x ∈ X closed we have depth(F_x) ≥ 2. Then H^1(X, F ⊗_O_X L^⊗ m) = 0 for m ll 0.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$\nbe an ample invertible $\\mathcal{O}_X$-module. Let $\\mathcal{F}$ be a\ncoherent $\\mathcal{O}_X$-module. Assume that\nfor $x \\in X$ closed we have $\\text{depth}(\\mathcal{F}_x) \\geq 2$.\nThen\n$H^1(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes m}) = 0$\nfor $m \\ll 0$.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Enriques-Severi-Zariski","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FD8","source_file":"varieties.tex","source_line":11157,"source_end_line":11166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L11157-L11166","statement_sha256":"ef7842cbb7debcb61598794221323cd5253796c5b2d3413bde0681a61f0983f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6614,"rank":6614,"depth":34,"x":229.135,"y":1038.835,"cluster":"varieties-curves"},{"id":"stacks:0FD9","tag":"0FD9","title":"Enriques-Severi-Zariski · Lemma 0FD9","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module. Let s ∈ Γ(X, L). Assume • s is a regular section (Divisors, Definition [Tag 01WY]), • for every closed point x ∈ X we have depth(O_X, x) ≥ 2, and • X is connected. Then the zero scheme Z(s) of s is connected.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\\mathcal{L}$\nbe an ample invertible $\\mathcal{O}_X$-module. Let\n$s \\in \\Gamma(X, \\mathcal{L})$. Assume\n\\begin{enumerate}\n\\item $s$ is a regular section\n(Divisors, Definition \\ref{divisors-definition-regular-section}),\n\\item for every closed point $x \\in X$ we have\n$\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$, and\n\\item $X$ is connected.\n\\end{enumerate}\nThen the zero scheme $Z(s)$ of $s$ is connected.","area":"Varieties & Curves","chapter":"Varieties","chapter_id":"varieties","section":"Enriques-Severi-Zariski","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FD9","source_file":"varieties.tex","source_line":11292,"source_end_line":11305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/varieties.tex#L11292-L11305","statement_sha256":"ea7d507e6ac67045eac51cea3e795cb63a5d089d922dfef05eca7b485f73e91a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6615,"rank":6615,"depth":35,"x":312.142,"y":1214.242,"cluster":"varieties-curves"},{"id":"stacks:020O","tag":"020O","title":"The Zariski topology · Definition 020O","summary":"Let T be a scheme. A Zariski covering of T is a family of morphisms (f_i : T_i → T)_i ∈ I of schemes such that each f_i is an open immersion and such that T = ⋃ f_i(T_i).","statement_latex":"Let $T$ be a scheme. A {\\it Zariski covering of $T$} is a family\nof morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes\nsuch that each $f_i$ is an open immersion and such\nthat $T = \\bigcup f_i(T_i)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020O","source_file":"topologies.tex","source_line":80,"source_end_line":86,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L80-L86","statement_sha256":"4d2e6bd0822f2cf7c315afa0b61cb5fa958efd1faf065c067145446b5adad59d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6616,"rank":6616,"depth":0,"x":686.833,"y":1140.0,"cluster":"descent"},{"id":"stacks:020P","tag":"020P","title":"The Zariski topology · Lemma 020P","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is a Zariski covering of T. • If (T_i → T)_i∈ I is a Zariski covering and for each i we have a Zariski covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is a Zariski covering. • If (T_i → T)_i∈ I is a Zariski covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is a Zariski covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis a Zariski covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a Zariski covering and for each\n$i$ we have a Zariski covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a Zariski covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a Zariski covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a Zariski covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020P","source_file":"topologies.tex","source_line":93,"source_end_line":106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L93-L106","statement_sha256":"d1c6b357c73611efca45de068463511ff1244d76fe6ca15576a611d5b712a20c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6617,"rank":6617,"depth":0,"x":671.274,"y":1146.715,"cluster":"descent"},{"id":"stacks:020Q","tag":"020Q","title":"The Zariski topology · Lemma 020Q","summary":"Let T be an affine scheme. Let (T_i → T)_i ∈ I be a Zariski covering of T. Then there exists a Zariski covering (U_j → T)_j = 1, …, m which is a refinement of (T_i → T)_i ∈ I such that each U_j is a standard open of T, see Schemes, Definition [Tag 01HT]. Moreover, we may choose each U_j to be an open of one of the T_i.","statement_latex":"Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be a\nZariski covering of $T$. Then there exists a Zariski covering\n$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement\nof $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is a standard\nopen of $T$, see\nSchemes, Definition \\ref{schemes-definition-standard-covering}.\nMoreover, we may choose each $U_j$ to be an open of one of the $T_i$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020Q","source_file":"topologies.tex","source_line":112,"source_end_line":121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L112-L121","statement_sha256":"175b30ff22af4a862779ee3dea2fea6bb27bc281cf8c1c63dd461074c0586f4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6618,"rank":6618,"depth":5,"x":681.336,"y":1127.215,"cluster":"descent"},{"id":"stacks:020R","tag":"020R","title":"The Zariski topology · Definition 020R","summary":"Compare Schemes, Definition [Tag 01HT]. Let T be an affine scheme. A standard Zariski covering of T is a Zariski covering (U_j → T)_j = 1, …, m with each U_j → T inducing an isomorphism with a standard affine open of T.","statement_latex":"Compare Schemes, Definition \\ref{schemes-definition-standard-covering}.\nLet $T$ be an affine scheme. A {\\it standard Zariski covering}\nof $T$ is a Zariski covering $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$\nwith each $U_j \\to T$ inducing an isomorphism with a standard affine open\nof $T$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020R","source_file":"topologies.tex","source_line":132,"source_end_line":139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L132-L139","statement_sha256":"5a93693ff743aef875564519727a2c838c5c9cf75ad444c18ef87da8f1444df2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6619,"rank":6619,"depth":1,"x":690.999,"y":1152.051,"cluster":"descent"},{"id":"stacks:020S","tag":"020S","title":"The Zariski topology · Definition 020S","summary":"A big Zariski site is any site Sch_Zar as in Sites, Definition [Tag 00VH] constructed as follows: • Choose any set of schemes S_0, and any set of Zariski coverings Cov_0 among these schemes. • As underlying category of Sch_Zar take any category Sch_α constructed as in Sets, Lemma [Tag 000J] starting with the set S_0. • As coverings of Sch_Zar choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Sch_α and the class of Zariski coverings, and…","statement_latex":"A {\\it big Zariski site} is any site $\\Sch_{Zar}$ as in\nSites, Definition \\ref{sites-definition-site} constructed as follows:\n\\begin{enumerate}\n\\item Choose any set of schemes $S_0$, and any set of Zariski coverings\n$\\text{Cov}_0$ among these schemes.\n\\item As underlying category of $\\Sch_{Zar}$\ntake any category $\\Sch_\\alpha$ constructed as in\nSets, Lemma \\ref{sets-lemma-construct-category} starting with the set $S_0$.\n\\item As coverings of $\\Sch_{Zar}$ choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\Sch_\\alpha$ and the class of Zariski coverings,\nand the set $\\text{Cov}_0$ chosen above.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020S","source_file":"topologies.tex","source_line":141,"source_end_line":156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L141-L156","statement_sha256":"28f5a5ac1e3a6fce5e47ec389b8d16e7369ec9838a9761563574c1ea228475a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6620,"rank":6620,"depth":2,"x":659.816,"y":1137.001,"cluster":"descent"},{"id":"stacks:03WV","tag":"03WV","title":"The Zariski topology · Lemma 03WV","summary":"Let Sch_Zar be a big Zariski site as in Definition [Tag 020S]. Let T ∈ Ob(Sch_Zar). Let (T_i → T)_i ∈ I be an arbitrary Zariski covering of T. There exists a covering (U_j → T)_j ∈ J of T in the site Sch_Zar which is tautologically equivalent (see Sites, Definition [Tag 00VU]) to (T_i → T)_i ∈ I.","statement_latex":"Let $\\Sch_{Zar}$ be a big Zariski site as in\nDefinition \\ref{definition-big-zariski-site}.\nLet $T \\in \\Ob(\\Sch_{Zar})$.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary Zariski covering of $T$.\nThere exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site\n$\\Sch_{Zar}$ which is tautologically equivalent (see\nSites, Definition \\ref{sites-definition-combinatorial-tautological})\nto $\\{T_i \\to T\\}_{i \\in I}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WV","source_file":"topologies.tex","source_line":184,"source_end_line":194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L184-L194","statement_sha256":"857ad854b51ddb8eab5d514c97ee84c9681514ffc045d049e58720b3f7c420d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6621,"rank":6621,"depth":3,"x":699.12,"y":1129.783,"cluster":"descent"},{"id":"stacks:020T","tag":"020T","title":"The Zariski topology · Definition 020T","summary":"Let S be a scheme. Let Sch_Zar be a big Zariski site containing S. • The big Zariski site of S, denoted (Sch/S)_Zar, is the site Sch_Zar/S introduced in Sites, Section [Tag 00XZ]. • The small Zariski site of S, which we denote S_Zar, is the full subcategory of (Sch/S)_Zar whose objects are those U/S such that U → S is an open immersion. A covering of S_Zar is any covering (U_i → U) of (Sch/S)_Zar with U ∈ Ob(S_Zar). • The big affine Zariski site of S, denoted (Aff/S)_Zar,…","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski\nsite containing $S$.\n\\begin{enumerate}\n\\item The {\\it big Zariski site of $S$}, denoted\n$(\\Sch/S)_{Zar}$, is the site $\\Sch_{Zar}/S$\nintroduced in Sites, Section \\ref{sites-section-localize}.\n\\item The {\\it small Zariski site of $S$}, which we denote\n$S_{Zar}$, is the full subcategory of $(\\Sch/S)_{Zar}$\nwhose objects are those $U/S$ such that $U \\to S$ is an open immersion.\nA covering of $S_{Zar}$ is any covering $\\{U_i \\to U\\}$ of\n$(\\Sch/S)_{Zar}$ with $U \\in \\Ob(S_{Zar})$.\n\\item The {\\it big affine Zariski site of $S$}, denoted\n$(\\textit{Aff}/S)_{Zar}$, is the full subcategory of\n$(\\Sch/S)_{Zar}$ consisting of objects $U/S$ such that $U$ is an\naffine scheme. A covering of $(\\textit{Aff}/S)_{Zar}$ is any covering\n$\\{U_i \\to U\\}$ of $(\\Sch/S)_{Zar}$ with $U \\in \\Ob((\\textit{Aff}/S)_{Zar})$\nwhich is a standard Zariski covering.\n\\item The {\\it small affine Zariski site of $S$}, denoted\n$S_{affine, Zar}$, is the full subcategory of $S_{Zar}$\nwhose objects are those $U/S$ such that $U$ is an affine scheme.\nA covering of $S_{affine, Zar}$ is any covering $\\{U_i \\to U\\}$ of\n$S_{Zar}$ with $U \\in \\Ob(S_{affine, Zar})$\nwhich is a standard Zariski covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020T","source_file":"topologies.tex","source_line":207,"source_end_line":233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L207-L233","statement_sha256":"ba7d2a9db29e0faf7cd5932cd6b2bafe530b68882bdcaca6b99ff35e7808db11","origin":"The Stacks Project","memory_eligible":false,"source_rank":6622,"rank":6622,"depth":0,"x":673.605,"y":1159.984,"cluster":"descent"},{"id":"stacks:020U","tag":"020U","title":"The Zariski topology · Lemma 020U","summary":"Let S be a scheme. Let Sch_Zar be a big Zariski site containing S. The structures S_Zar, (Aff/S)_Zar, and S_affine, Zar defined above are sites.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski\nsite containing $S$. The structures $S_{Zar}$, $(\\textit{Aff}/S)_{Zar}$,\nand $S_{affine, Zar}$ defined above are sites.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020U","source_file":"topologies.tex","source_line":240,"source_end_line":245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L240-L245","statement_sha256":"7aaff95d4aa3e0013d1f6e9688d9de170c6f3216bdceab4bd5fda06b7e38045f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6623,"rank":6623,"depth":1,"x":667.803,"y":1120.273,"cluster":"descent"},{"id":"stacks:020V","tag":"020V","title":"The Zariski topology · Lemma 020V","summary":"Let S be a scheme. Let Sch_Zar be a big Zariski site containing S. The underlying categories of the sites Sch_Zar, (Sch/S)_Zar, S_Zar, (Aff/S)_Zar, and S_affine, Zar have fibre products. In each case the obvious functor into the category Sch of all schemes commutes with taking fibre products. The categories (Sch/S)_Zar, and S_Zar both have a final object, namely S/S.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski\nsite containing $S$. The underlying categories of the sites\n$\\Sch_{Zar}$, $(\\Sch/S)_{Zar}$,\n$S_{Zar}$, $(\\textit{Aff}/S)_{Zar}$, and $S_{affine, Zar}$ have fibre products.\nIn each case the obvious functor into the category $\\Sch$ of\nall schemes commutes with taking fibre products. The categories\n$(\\Sch/S)_{Zar}$, and $S_{Zar}$ both have a final object,\nnamely $S/S$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020V","source_file":"topologies.tex","source_line":280,"source_end_line":290,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L280-L290","statement_sha256":"4eb75ef0e3a6b366e93784544900a74e24b39c8f36798ead767bd350855fa485","origin":"The Stacks Project","memory_eligible":false,"source_rank":6624,"rank":6624,"depth":3,"x":706.462,"y":1148.118,"cluster":"descent"},{"id":"stacks:020W","tag":"020W","title":"The Zariski topology · Lemma 020W","summary":"Let S be a scheme. Let Sch_Zar be a big Zariski site containing S. The functor (Aff/S)_Zar → (Sch/S)_Zar is a special cocontinuous functor. Hence it induces an equivalence of topoi from Sh((Aff/S)_Zar) to Sh((Sch/S)_Zar).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski\nsite containing $S$.\nThe functor $(\\textit{Aff}/S)_{Zar} \\to (\\Sch/S)_{Zar}$\nis a special cocontinuous functor. Hence it induces an equivalence\nof topoi from $\\Sh((\\textit{Aff}/S)_{Zar})$ to\n$\\Sh((\\Sch/S)_{Zar})$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020W","source_file":"topologies.tex","source_line":313,"source_end_line":321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L313-L321","statement_sha256":"7abb0074cef7b9ab85eb758600ebc06b18e2e6b7f4e678845e066afc530c9983","origin":"The Stacks Project","memory_eligible":false,"source_rank":6625,"rank":6625,"depth":8,"x":652.471,"y":1149.546,"cluster":"descent"},{"id":"stacks:0F1B","tag":"0F1B","title":"The Zariski topology · Lemma 0F1B","summary":"Let S be a scheme. Let Sch_Zar be a big Zariski site containing S. The functor S_affine, Zar → S_Zar is a special cocontinuous functor. Hence it induces an equivalence of topoi from Sh(S_affine, Zar) to Sh(S_Zar).","statement_latex":"Let $S$ be a scheme.  Let $\\Sch_{Zar}$ be a big Zariski\nsite containing $S$. The functor $S_{affine, Zar} \\to S_{Zar}$\nis a special cocontinuous functor. Hence it induces an equivalence\nof topoi from $\\Sh(S_{affine, Zar})$ to $\\Sh(S_{Zar})$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1B","source_file":"topologies.tex","source_line":341,"source_end_line":347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L341-L347","statement_sha256":"2b546e874e9733a22b83c9a40dba06129f5d7ada71ad3fc66168bc656da4ef43","origin":"The Stacks Project","memory_eligible":false,"source_rank":6626,"rank":6626,"depth":9,"x":693.271,"y":1116.178,"cluster":"descent"},{"id":"stacks:020X","tag":"020X","title":"The Zariski topology · Lemma 020X","summary":"The category of sheaves on S_Zar is equivalent to the category of sheaves on the underlying topological space of S.","statement_latex":"The category of sheaves on $S_{Zar}$ is equivalent to the\ncategory of sheaves on the underlying topological space of $S$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020X","source_file":"topologies.tex","source_line":358,"source_end_line":362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L358-L362","statement_sha256":"4142636716a11569bb013972eba3196a47ca25f35085a3bd73607e9237d24ad5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6627,"rank":6627,"depth":3,"x":689.807,"y":1166.263,"cluster":"descent"},{"id":"stacks:020Y","tag":"020Y","title":"The Zariski topology · Lemma 020Y","summary":"Let Sch_Zar be a big Zariski site. Let f : T → S be a morphism in Sch_Zar. The functor T_Zar → (Sch/S)_Zar is cocontinuous and induces a morphism of topoi i_f : Sh(T_Zar) → Sh((Sch/S)_Zar) For a sheaf G on (Sch/S)_Zar we have the formula (i_f^-1G)(U/T) = G(U/S). The functor i_f^-1 also has a left adjoint i_f, ! which commutes with fibre products and equalizers.","statement_latex":"Let $\\Sch_{Zar}$ be a big Zariski site.\nLet $f : T \\to S$ be a morphism in $\\Sch_{Zar}$.\nThe functor $T_{Zar} \\to (\\Sch/S)_{Zar}$\nis cocontinuous and induces a morphism of topoi\n$$\ni_f :\n\\Sh(T_{Zar})\n\\longrightarrow\n\\Sh((\\Sch/S)_{Zar})\n$$\nFor a sheaf $\\mathcal{G}$ on $(\\Sch/S)_{Zar}$\nwe have the formula $(i_f^{-1}\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nThe functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes\nwith fibre products and equalizers.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020Y","source_file":"topologies.tex","source_line":390,"source_end_line":406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L390-L406","statement_sha256":"8c4dcb8bd3ec617c1d164528f7908b598a6b7e94f182f893802c76b70402aa3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6628,"rank":6628,"depth":6,"x":650.442,"y":1125.611,"cluster":"descent"},{"id":"stacks:020Z","tag":"020Z","title":"The Zariski topology · Lemma 020Z","summary":"Let S be a scheme. Let Sch_Zar be a big Zariski site containing S. The inclusion functor S_Zar → (Sch/S)_Zar satisfies the hypotheses of Sites, Lemma [Tag 00XU] and hence induces a morphism of sites π_S : (Sch/S)_Zar → S_Zar and a morphism of topoi i_S : Sh(S_Zar) → Sh((Sch/S)_Zar) such that π_S ∘ i_S = id. Moreover, i_S = i_id_S with i_id_S as in Lemma [Tag 020Y]. In particular the functor i_S^-1 = π_S, * is described by the rule i_S^-1(G)(U/S) = G(U/S).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{Zar}$ be a big Zariski\nsite containing $S$.\nThe inclusion functor $S_{Zar} \\to (\\Sch/S)_{Zar}$\nsatisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site}\nand hence induces a morphism of sites\n$$\n\\pi_S : (\\Sch/S)_{Zar} \\longrightarrow S_{Zar}\n$$\nand a morphism of topoi\n$$\ni_S : \\Sh(S_{Zar}) \\longrightarrow \\Sh((\\Sch/S)_{Zar})\n$$\nsuch that $\\pi_S \\circ i_S = \\text{id}$. Moreover, $i_S = i_{\\text{id}_S}$\nwith $i_{\\text{id}_S}$ as in Lemma \\ref{lemma-put-in-T}. In particular the\nfunctor $i_S^{-1} = \\pi_{S, *}$ is described by the rule\n$i_S^{-1}(\\mathcal{G})(U/S) = \\mathcal{G}(U/S)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/020Z","source_file":"topologies.tex","source_line":423,"source_end_line":441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L423-L441","statement_sha256":"0f047fc690743d13924e71c5664a37344d78eab6c51439db6fa8756a0c0b1c61","origin":"The Stacks Project","memory_eligible":false,"source_rank":6629,"rank":6629,"depth":8,"x":714.675,"y":1133.596,"cluster":"descent"},{"id":"stacks:04BS","tag":"04BS","title":"The Zariski topology · Definition 04BS","summary":"In the situation of Lemma [Tag 020Z] the functor i_S^-1 = π_S, * is often called the restriction to the small Zariski site, and for a sheaf F on the big Zariski site we denote F|_S_Zar this restriction.","statement_latex":"In the situation of\nLemma \\ref{lemma-at-the-bottom}\nthe functor $i_S^{-1} = \\pi_{S, *}$ is often\ncalled the {\\it restriction to the small Zariski site}, and for a sheaf\n$\\mathcal{F}$ on the big Zariski site we denote $\\mathcal{F}|_{S_{Zar}}$\nthis restriction.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BS","source_file":"topologies.tex","source_line":451,"source_end_line":459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L451-L459","statement_sha256":"10009efe8c6632995971a375caab867de6d65420643e327cf0ec627514297752","origin":"The Stacks Project","memory_eligible":false,"source_rank":6630,"rank":6630,"depth":9,"x":658.838,"y":1165.284,"cluster":"descent"},{"id":"stacks:0210","tag":"0210","title":"The Zariski topology · Lemma 0210","summary":"Let Sch_Zar be a big Zariski site. Let f : T → S be a morphism in Sch_Zar. The functor u : (Sch/T)_Zar → (Sch/S)_Zar, V/T ↦ V/S is cocontinuous, and has a continuous right adjoint v : (Sch/S)_Zar → (Sch/T)_Zar, (U → S) ↦ (U ×_S T → T). They induce the same morphism of topoi f_big : Sh((Sch/T)_Zar) → Sh((Sch/S)_Zar) We have f_big^-1(G)(U/T) = G(U/S). We have f_big, *(F)(U/S) = F(U ×_S T/T). Also, f_big^-1 has a left adjoint f_big! which commutes with fibre products and…","statement_latex":"Let $\\Sch_{Zar}$ be a big Zariski site.\nLet $f : T \\to S$ be a morphism in $\\Sch_{Zar}$.\nThe functor\n$$\nu : (\\Sch/T)_{Zar} \\longrightarrow (\\Sch/S)_{Zar},\n\\quad\nV/T \\longmapsto V/S\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv : (\\Sch/S)_{Zar} \\longrightarrow (\\Sch/T)_{Zar},\n\\quad\n(U \\to S) \\longmapsto (U \\times_S T \\to T).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\Sch/T)_{Zar})\n\\longrightarrow\n\\Sh((\\Sch/S)_{Zar})\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nWe have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0210","source_file":"topologies.tex","source_line":477,"source_end_line":504,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L477-L504","statement_sha256":"d75f0756a4f6d5bc8bb37adee1e0a55b1b5fcb9235208615dbc9716d7d247b90","origin":"The Stacks Project","memory_eligible":false,"source_rank":6631,"rank":6631,"depth":8,"x":675.111,"y":1108.309,"cluster":"descent"},{"id":"stacks:0211","tag":"0211","title":"The Zariski topology · Lemma 0211","summary":"Let Sch_Zar be a big Zariski site. Let f : T → S be a morphism in Sch_Zar. • We have i_f = f_big ∘ i_T with i_f as in Lemma [Tag 020Y] and i_T as in Lemma [Tag 020Z]. • The functor S_Zar → T_Zar, (U → S) ↦ (U ×_S T → T) is continuous and induces a morphism of sites f_small : T_Zar → S_Zar The functors f_small^-1 and f_small, * agree with the usual notions f^-1 and f_* if we identify sheaves on T_Zar, resp. S_Zar with sheaves on T, resp. S via Lemma [Tag 020X]. • We have a…","statement_latex":"Let $\\Sch_{Zar}$ be a big Zariski site.\nLet $f : T \\to S$ be a morphism in $\\Sch_{Zar}$.\n\\begin{enumerate}\n\\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in\nLemma \\ref{lemma-put-in-T} and $i_T$ as in\nLemma \\ref{lemma-at-the-bottom}.\n\\item The functor $S_{Zar} \\to T_{Zar}$,\n$(U \\to S) \\mapsto (U \\times_S T \\to T)$ is continuous and induces\na morphism of sites\n$$\nf_{small} :\nT_{Zar}\n\\longrightarrow\nS_{Zar}\n$$\nThe functors $f_{small}^{-1}$ and $f_{small, *}$ agree with\nthe usual notions $f^{-1}$ and $f_*$ if we identify sheaves\non $T_{Zar}$, resp.\\ $S_{Zar}$ with sheaves on $T$, resp.\\ $S$\nvia Lemma \\ref{lemma-Zariski-usual}.\n\\item We have a commutative diagram of morphisms of sites\n$$\n\\xymatrix{\nT_{Zar} \\ar[d]_{f_{small}} &\n(\\Sch/T)_{Zar} \\ar[d]^{f_{big}} \\ar[l]^{\\pi_T} \\\\\nS_{Zar} &\n(\\Sch/S)_{Zar} \\ar[l]_{\\pi_S}\n}\n$$\nso that $f_{small} \\circ \\pi_T = \\pi_S \\circ f_{big}$ as morphisms of topoi.\n\\item We have $f_{small} = \\pi_S \\circ f_{big} \\circ i_T = \\pi_S \\circ i_f$.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0211","source_file":"topologies.tex","source_line":523,"source_end_line":556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L523-L556","statement_sha256":"473e98d129f767258067fa357c04521b4f94483d51ff3d6bc06a37464962a348","origin":"The Stacks Project","memory_eligible":false,"source_rank":6632,"rank":6632,"depth":9,"x":710.013,"y":1161.248,"cluster":"descent"},{"id":"stacks:0212","tag":"0212","title":"The Zariski topology · Lemma 0212","summary":"Given schemes X, Y, Z in (Sch/S)_Zar and morphisms f : X → Y, g : Y → Z we have g_big ∘ f_big = (g ∘ f)_big and g_small ∘ f_small = (g ∘ f)_small.","statement_latex":"Given schemes $X$, $Y$, $Z$ in $(\\Sch/S)_{Zar}$\nand morphisms $f : X \\to Y$, $g : Y \\to Z$ we have\n$g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and\n$g_{small} \\circ f_{small} = (g \\circ f)_{small}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0212","source_file":"topologies.tex","source_line":589,"source_end_line":595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L589-L595","statement_sha256":"f56eff9c7a4e01aada77f49127f006d791ef756d9901d9d9f1eb1e595952a4f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6633,"rank":6633,"depth":9,"x":639.612,"y":1141.403,"cluster":"descent"},{"id":"stacks:0DD9","tag":"0DD9","title":"The Zariski topology · Lemma 0DD9","summary":"Let Sch_Zar be a big Zariski site. Consider a cartesian diagram xymatrix T' ar[r]_g' ar[d]_f' & T ar[d]^f S' ar[r]^g & S in Sch_Zar. Then i_g^-1 ∘ f_big, * = f'_small, * ∘ (i_g')^-1 and g_big^-1 ∘ f_big, * = f'_big, * ∘ (g'_big)^-1.","statement_latex":"Let $\\Sch_{Zar}$ be a big Zariski site. Consider a cartesian diagram\n$$\n\\xymatrix{\nT' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nin $\\Sch_{Zar}$. Then\n$i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$\nand $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DD9","source_file":"topologies.tex","source_line":606,"source_end_line":618,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L606-L618","statement_sha256":"8174b69d4f4c2b192b820282258d28989184f0eae65b6adb2927b74f543c9de5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6634,"rank":6634,"depth":10,"x":709.46,"y":1115.374,"cluster":"descent"},{"id":"stacks:0213","tag":"0213","title":"The Zariski topology · Lemma 0213","summary":"Let S be a scheme contained in a big Zariski site Sch_Zar. A sheaf F on the big Zariski site (Sch/S)_Zar is given by the following data: • for every T/S ∈ Ob((Sch/S)_Zar) a sheaf F_T on T, • for every f : T' → T in (Sch/S)_Zar a map c_f : f^-1F_T → F_T'. These data are subject to the following conditions: • [(a)] given any f : T' → T and g : T\" → T' in (Sch/S)_Zar the composition c_g ∘ g^-1c_f is equal to c_f ∘ g, and • [(b)] if f : T' → T in (Sch/S)_Zar is an open…","statement_latex":"Let $S$ be a scheme contained in a big Zariski site $\\Sch_{Zar}$.\nA sheaf $\\mathcal{F}$ on the big Zariski site $(\\Sch/S)_{Zar}$\nis given by the following data:\n\\begin{enumerate}\n\\item for every $T/S \\in \\Ob((\\Sch/S)_{Zar})$ a sheaf\n$\\mathcal{F}_T$ on $T$,\n\\item for every $f : T' \\to T$ in\n$(\\Sch/S)_{Zar}$ a map\n$c_f : f^{-1}\\mathcal{F}_T \\to \\mathcal{F}_{T'}$.\n\\end{enumerate}\nThese data are subject to the following conditions:\n\\begin{enumerate}\n\\item[(a)] given any $f : T' \\to T$ and $g : T'' \\to T'$ in\n$(\\Sch/S)_{Zar}$ the composition $c_g \\circ g^{-1}c_f$\nis equal to $c_{f \\circ g}$, and\n\\item[(b)] if $f : T' \\to T$ in $(\\Sch/S)_{Zar}$ is an\nopen immersion then $c_f$ is an isomorphism.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0213","source_file":"topologies.tex","source_line":636,"source_end_line":656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L636-L656","statement_sha256":"53cdc2b2f7702633b81e4e4fe265048602c8ab13abd4000d6c45271b8883d17e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6635,"rank":6635,"depth":7,"x":678.029,"y":1175.804,"cluster":"descent"},{"id":"stacks:0215","tag":"0215","title":"The étale topology · Definition 0215","summary":"Let T be a scheme. An étale covering of T is a family of morphisms (f_i : T_i → T)_i ∈ I of schemes such that each f_i is étale and such that T = ⋃ f_i(T_i).","statement_latex":"Let $T$ be a scheme. An {\\it \\'etale covering of $T$} is a family\nof morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes\nsuch that each $f_i$ is \\'etale and such that $T = \\bigcup f_i(T_i)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0215","source_file":"topologies.tex","source_line":724,"source_end_line":729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L724-L729","statement_sha256":"9ab851be2d88af4d9a3f581b0f714953e6bf6a2579a7dbfcebb93363b9d0cb8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6636,"rank":6636,"depth":0,"x":651.969,"y":1111.784,"cluster":"descent"},{"id":"stacks:0216","tag":"0216","title":"The étale topology · Lemma 0216","summary":"Any Zariski covering is an étale covering.","statement_latex":"Any Zariski covering is an \\'etale covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0216","source_file":"topologies.tex","source_line":731,"source_end_line":734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L731-L734","statement_sha256":"47a8295bd1ca74773f6bbc6724a7cfcc6634e1a6b91d97ef7542ee73b10ce144","origin":"The Stacks Project","memory_eligible":false,"source_rank":6637,"rank":6637,"depth":1,"x":724.404,"y":1145.019,"cluster":"descent"},{"id":"stacks:0217","tag":"0217","title":"The étale topology · Lemma 0217","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is an étale covering of T. • If (T_i → T)_i∈ I is an étale covering and for each i we have an étale covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is an étale covering. • If (T_i → T)_i∈ I is an étale covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is an étale covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis an \\'etale covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an \\'etale covering and for each\n$i$ we have an \\'etale covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an \\'etale covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an \\'etale covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an \\'etale covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0217","source_file":"topologies.tex","source_line":746,"source_end_line":759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L746-L759","statement_sha256":"1b831eeb9d4015fee5ed7b017ba130ab8add978987c84beb50279fb33fc5df08","origin":"The Stacks Project","memory_eligible":false,"source_rank":6638,"rank":6638,"depth":0,"x":642.376,"y":1161.989,"cluster":"descent"},{"id":"stacks:0218","tag":"0218","title":"The étale topology · Lemma 0218","summary":"Let T be an affine scheme. Let (T_i → T)_i ∈ I be an étale covering of T. Then there exists an étale covering (U_j → T)_j = 1, …, m which is a refinement of (T_i → T)_i ∈ I such that each U_j is an affine scheme. Moreover, we may choose each U_j to be open affine in one of the T_i.","statement_latex":"Let $T$ be an affine scheme.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an \\'etale covering of $T$.\nThen there exists an \\'etale covering\n$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement\nof $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine\nscheme. Moreover, we may choose each $U_j$ to be open affine\nin one of the $T_i$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0218","source_file":"topologies.tex","source_line":765,"source_end_line":774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L765-L774","statement_sha256":"77e05b6c76d8d7eac3aa08c9d780898697f426ad33e2d475afb061f950be7177","origin":"The Stacks Project","memory_eligible":false,"source_rank":6639,"rank":6639,"depth":0,"x":690.281,"y":1101.612,"cluster":"descent"},{"id":"stacks:0219","tag":"0219","title":"The étale topology · Definition 0219","summary":"Let T be an affine scheme. A standard étale covering of T is a family (f_j : U_j → T)_j = 1, …, m with each U_j is affine and étale over T and T = ⋃ f_j(U_j).","statement_latex":"Let $T$ be an affine scheme. A {\\it standard \\'etale covering}\nof $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$\nwith each $U_j$ is affine and \\'etale over $T$ and\n$T = \\bigcup f_j(U_j)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0219","source_file":"topologies.tex","source_line":783,"source_end_line":789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L783-L789","statement_sha256":"865b6bf6687e86254af2a47a2597f25d3b4eae4455bb407dd3e27095b79ad4cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6640,"rank":6640,"depth":0,"x":703.779,"y":1174.858,"cluster":"descent"},{"id":"stacks:021A","tag":"021A","title":"The étale topology · Definition 021A","summary":"A big étale site is any site Sch_etale as in Sites, Definition [Tag 00VH] constructed as follows: • Choose any set of schemes S_0, and any set of étale coverings Cov_0 among these schemes. • As underlying category take any category Sch_α constructed as in Sets, Lemma [Tag 000J] starting with the set S_0. • Choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Sch_α and the class of étale coverings, and the set Cov_0 chosen above.","statement_latex":"A {\\it big \\'etale site} is any site $\\Sch_\\etale$ as in\nSites, Definition \\ref{sites-definition-site} constructed as follows:\n\\begin{enumerate}\n\\item Choose any set of schemes $S_0$, and any set of \\'etale coverings\n$\\text{Cov}_0$ among these schemes.\n\\item As underlying category take any category $\\Sch_\\alpha$\nconstructed as in Sets, Lemma \\ref{sets-lemma-construct-category}\nstarting with the set $S_0$.\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\Sch_\\alpha$ and the class of \\'etale coverings,\nand the set $\\text{Cov}_0$ chosen above.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021A","source_file":"topologies.tex","source_line":802,"source_end_line":817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L802-L817","statement_sha256":"9064347c7e0c30f67a1bfe4b4e98abfc2d6ffd2143363a5d9ba065689547188f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6641,"rank":6641,"depth":2,"x":633.514,"y":1127.542,"cluster":"descent"},{"id":"stacks:03WW","tag":"03WW","title":"The étale topology · Lemma 03WW","summary":"Let Sch_etale be a big étale site as in Definition [Tag 021A]. Let T ∈ Ob(Sch_etale). Let (T_i → T)_i ∈ I be an arbitrary étale covering of T. • There exists a covering (U_j → T)_j ∈ J of T in the site Sch_etale which refines (T_i → T)_i ∈ I. • If (T_i → T)_i ∈ I is a standard étale covering, then it is tautologically equivalent to a covering in Sch_etale. • If (T_i → T)_i ∈ I is a Zariski covering, then it is tautologically equivalent to a covering in Sch_etale.","statement_latex":"Let $\\Sch_\\etale$ be a big \\'etale site as in\nDefinition \\ref{definition-big-etale-site}.\nLet $T \\in \\Ob(\\Sch_\\etale)$.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary \\'etale covering of $T$.\n\\begin{enumerate}\n\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site\n$\\Sch_\\etale$ which refines $\\{T_i \\to T\\}_{i \\in I}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard \\'etale covering, then\nit is tautologically equivalent to a covering in $\\Sch_\\etale$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then\nit is tautologically equivalent to a covering in $\\Sch_\\etale$.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WW","source_file":"topologies.tex","source_line":829,"source_end_line":843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L829-L843","statement_sha256":"8937df414f92a648a1b921a5a1bb80b1fbef3f86e0a8e1aa9d067152a0361535","origin":"The Stacks Project","memory_eligible":false,"source_rank":6642,"rank":6642,"depth":3,"x":725.155,"y":1122.475,"cluster":"descent"},{"id":"stacks:021B","tag":"021B","title":"The étale topology · Definition 021B","summary":"Let S be a scheme. Let Sch_etale be a big étale site containing S. • The big étale site of S, denoted (Sch/S)_etale, is the site Sch_etale/S introduced in Sites, Section [Tag 00XZ]. • The small étale site of S, which we denote S_etale, is the full subcategory of (Sch/S)_etale whose objects are those U/S such that U → S is étale. A covering of S_etale is any covering (U_i → U) of (Sch/S)_etale with U ∈ Ob(S_etale). • The big affine étale site of S, denoted (Aff/S)_etale,…","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale\nsite containing $S$.\n\\begin{enumerate}\n\\item The {\\it big \\'etale site of $S$}, denoted\n$(\\Sch/S)_\\etale$, is the site\n$\\Sch_\\etale/S$ introduced in\nSites, Section \\ref{sites-section-localize}.\n\\item The {\\it small \\'etale site of $S$}, which we denote\n$S_\\etale$, is the full subcategory of\n$(\\Sch/S)_\\etale$\nwhose objects are those $U/S$ such that $U \\to S$ is \\'etale.\nA covering of $S_\\etale$ is any covering $\\{U_i \\to U\\}$ of\n$(\\Sch/S)_\\etale$ with $U \\in \\Ob(S_\\etale)$.\n\\item The {\\it big affine \\'etale site of $S$}, denoted\n$(\\textit{Aff}/S)_\\etale$, is the full subcategory of\n$(\\Sch/S)_\\etale$ whose objects are those $U/S$\nsuch that $U$ is an affine scheme.\nA covering of $(\\textit{Aff}/S)_\\etale$ is any covering\n$\\{U_i \\to U\\}$ of $(\\Sch/S)_\\etale$ with $U \\in \\Ob((\\textit{Aff}/S)_\\etale)$\nwhich is a standard \\'etale covering.\n\\item The {\\it small affine \\'etale site of $S$}, denoted\n$S_{affine, \\etale}$, is the full subcategory of $S_\\etale$\nwhose objects are those $U/S$ such that $U$ is an affine scheme.\nA covering of $S_{affine, \\etale}$ is any covering $\\{U_i \\to U\\}$\nof $S_\\etale$ with $U \\in \\Ob(S_{affine, \\etale})$ which is a standard\n\\'etale covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021B","source_file":"topologies.tex","source_line":873,"source_end_line":902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L873-L902","statement_sha256":"3909b0d4640b66771d08429feb4bd1e824f9dfb1313633575d239ff1718a7082","origin":"The Stacks Project","memory_eligible":false,"source_rank":6643,"rank":6643,"depth":0,"x":660.438,"y":1179.265,"cluster":"descent"},{"id":"stacks:021C","tag":"021C","title":"The étale topology · Lemma 021C","summary":"Let S be a scheme. Let Sch_etale be a big étale site containing S. The structures S_etale, (Aff/S)_etale, and S_affine, etale are sites.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale\nsite containing $S$.\nThe structures $S_\\etale$, $(\\textit{Aff}/S)_\\etale$, and $S_{affine, \\etale}$\nare sites.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021C","source_file":"topologies.tex","source_line":909,"source_end_line":915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L909-L915","statement_sha256":"b84a899675dccd969e1ad4f07f26d2edc8868d30b4193d30f61d4b219173c604","origin":"The Stacks Project","memory_eligible":false,"source_rank":6644,"rank":6644,"depth":1,"x":662.541,"y":1099.226,"cluster":"descent"},{"id":"stacks:021D","tag":"021D","title":"The étale topology · Lemma 021D","summary":"Let S be a scheme. Let Sch_etale be a big étale site containing S. The underlying categories of the sites Sch_etale, (Sch/S)_etale, S_etale, (Aff/S)_etale, and S_affine, etale have fibre products. In each case the obvious functor into the category Sch of all schemes commutes with taking fibre products. The categories (Sch/S)_etale, and S_etale both have a final object, namely S/S.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale\nsite containing $S$. The underlying categories of the sites\n$\\Sch_\\etale$, $(\\Sch/S)_\\etale$, $S_\\etale$, $(\\textit{Aff}/S)_\\etale$,\nand $S_{affine, \\etale}$ have fibre products.\nIn each case the obvious functor into the category $\\Sch$ of\nall schemes commutes with taking fibre products. The categories\n$(\\Sch/S)_\\etale$, and $S_\\etale$ both have a\nfinal object, namely $S/S$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021D","source_file":"topologies.tex","source_line":945,"source_end_line":955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L945-L955","statement_sha256":"bac0713c4863e32c1dc24b111cdcfdb2cf3b0ea28fb27b19e653bfc0ffc3e9f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6645,"rank":6645,"depth":3,"x":726.457,"y":1160.51,"cluster":"descent"},{"id":"stacks:021E","tag":"021E","title":"The étale topology · Lemma 021E","summary":"Let S be a scheme. Let Sch_etale be a big étale site containing S. The functor (Aff/S)_etale → (Sch/S)_etale is special cocontinuous and induces an equivalence of topoi from Sh((Aff/S)_etale) to Sh((Sch/S)_etale).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale\nsite containing $S$.\nThe functor\n$(\\textit{Aff}/S)_\\etale \\to (\\Sch/S)_\\etale$\nis special cocontinuous and induces an equivalence of topoi from\n$\\Sh((\\textit{Aff}/S)_\\etale)$ to\n$\\Sh((\\Sch/S)_\\etale)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021E","source_file":"topologies.tex","source_line":978,"source_end_line":987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L978-L987","statement_sha256":"75c7a3da5b20ca8aa444eb200da55a99c02dcd54ef02a49b5870da053398d6b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6646,"rank":6646,"depth":8,"x":628.398,"y":1151.426,"cluster":"descent"},{"id":"stacks:04HR","tag":"04HR","title":"The étale topology · Lemma 04HR","summary":"Let S be a scheme. Let Sch_etale be a big étale site containing S. The functor S_affine, etale → S_etale is special cocontinuous and induces an equivalence of topoi from Sh(S_affine, etale) to Sh(S_etale).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale\nsite containing $S$. The functor $S_{affine, \\etale} \\to S_\\etale$\nis special cocontinuous and induces an equivalence of topoi from\n$\\Sh(S_{affine, \\etale})$ to $\\Sh(S_\\etale)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HR","source_file":"topologies.tex","source_line":1007,"source_end_line":1013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1007-L1013","statement_sha256":"9fcfbdafedb257ac2c9a611880a859e542d65d3c484204122371c5705108ffe4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6647,"rank":6647,"depth":9,"x":709.331,"y":1101.682,"cluster":"descent"},{"id":"stacks:021F","tag":"021F","title":"The étale topology · Lemma 021F","summary":"Let Sch_etale be a big étale site. Let f : T → S be a morphism in Sch_etale. The functor T_etale → (Sch/S)_etale is cocontinuous and induces a morphism of topoi i_f : Sh(T_etale) → Sh((Sch/S)_etale) For a sheaf G on (Sch/S)_etale we have the formula (i_f^-1G)(U/T) = G(U/S). The functor i_f^-1 also has a left adjoint i_f, ! which commutes with fibre products and equalizers.","statement_latex":"Let $\\Sch_\\etale$ be a big \\'etale site.\nLet $f : T \\to S$ be a morphism in $\\Sch_\\etale$.\nThe functor $T_\\etale \\to (\\Sch/S)_\\etale$\nis cocontinuous and induces a morphism of topoi\n$$\ni_f :\n\\Sh(T_\\etale)\n\\longrightarrow\n\\Sh((\\Sch/S)_\\etale)\n$$\nFor a sheaf $\\mathcal{G}$ on $(\\Sch/S)_\\etale$\nwe have the formula $(i_f^{-1}\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nThe functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes\nwith fibre products and equalizers.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021F","source_file":"topologies.tex","source_line":1024,"source_end_line":1040,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1024-L1040","statement_sha256":"61188bb8012d86e9bf7ca7ea928a8bc44950a240d703ee7e2c9891a84cc439f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6648,"rank":6648,"depth":6,"x":689.33,"y":1185.604,"cluster":"descent"},{"id":"stacks:021G","tag":"021G","title":"The étale topology · Lemma 021G","summary":"Let S be a scheme. Let Sch_etale be a big étale site containing S. The inclusion functor S_etale → (Sch/S)_etale satisfies the hypotheses of Sites, Lemma [Tag 00XU] and hence induces a morphism of sites π_S : (Sch/S)_etale → S_etale and a morphism of topoi i_S : Sh(S_etale) → Sh((Sch/S)_etale) such that π_S ∘ i_S = id. Moreover, i_S = i_id_S with i_id_S as in Lemma [Tag 021F]. In particular the functor i_S^-1 = π_S, * is described by the rule i_S^-1(G)(U/S) = G(U/S).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\etale$ be a big \\'etale\nsite containing $S$.\nThe inclusion functor $S_\\etale \\to (\\Sch/S)_\\etale$\nsatisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site}\nand hence induces a morphism of sites\n$$\n\\pi_S : (\\Sch/S)_\\etale \\longrightarrow S_\\etale\n$$\nand a morphism of topoi\n$$\ni_S : \\Sh(S_\\etale) \\longrightarrow \\Sh((\\Sch/S)_\\etale)\n$$\nsuch that $\\pi_S \\circ i_S = \\text{id}$. Moreover, $i_S = i_{\\text{id}_S}$\nwith $i_{\\text{id}_S}$ as in Lemma \\ref{lemma-put-in-T-etale}.\nIn particular the functor $i_S^{-1} = \\pi_{S, *}$ is described by the rule\n$i_S^{-1}(\\mathcal{G})(U/S) = \\mathcal{G}(U/S)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021G","source_file":"topologies.tex","source_line":1062,"source_end_line":1080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1062-L1080","statement_sha256":"3e7702ad6290b1876a27103a0b08535e44ce04194afa905fce821416b06fcb50","origin":"The Stacks Project","memory_eligible":false,"source_rank":6649,"rank":6649,"depth":8,"x":635.783,"y":1111.235,"cluster":"descent"},{"id":"stacks:04BT","tag":"04BT","title":"The étale topology · Definition 04BT","summary":"In the situation of Lemma [Tag 021G] the functor i_S^-1 = π_S, * is often called the restriction to the small étale site, and for a sheaf F on the big étale site we denote F|_S_etale this restriction.","statement_latex":"In the situation of\nLemma \\ref{lemma-at-the-bottom-etale}\nthe functor $i_S^{-1} = \\pi_{S, *}$ is often\ncalled the {\\it restriction to the small \\'etale site}, and for a sheaf\n$\\mathcal{F}$ on the big \\'etale site we denote\n$\\mathcal{F}|_{S_\\etale}$ this restriction.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BT","source_file":"topologies.tex","source_line":1091,"source_end_line":1099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1091-L1099","statement_sha256":"217150ea4555ac0f7cbabd30804e9582f1aff63a00d1701af840695ec917bc23","origin":"The Stacks Project","memory_eligible":false,"source_rank":6650,"rank":6650,"depth":9,"x":736.562,"y":1136.064,"cluster":"descent"},{"id":"stacks:021H","tag":"021H","title":"The étale topology · Lemma 021H","summary":"Let Sch_etale be a big étale site. Let f : T → S be a morphism in Sch_etale. The functor u : (Sch/T)_etale → (Sch/S)_etale, V/T ↦ V/S is cocontinuous, and has a continuous right adjoint v : (Sch/S)_etale → (Sch/T)_etale, (U → S) ↦ (U ×_S T → T). They induce the same morphism of topoi f_big : Sh((Sch/T)_etale) → Sh((Sch/S)_etale) We have f_big^-1(G)(U/T) = G(U/S). We have f_big, *(F)(U/S) = F(U ×_S T/T). Also, f_big^-1 has a left adjoint f_big! which commutes with fibre…","statement_latex":"Let $\\Sch_\\etale$ be a big \\'etale site.\nLet $f : T \\to S$ be a morphism in $\\Sch_\\etale$.\nThe functor\n$$\nu :\n(\\Sch/T)_\\etale\n\\longrightarrow\n(\\Sch/S)_\\etale,\n\\quad\nV/T \\longmapsto V/S\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv :\n(\\Sch/S)_\\etale\n\\longrightarrow\n(\\Sch/T)_\\etale,\n\\quad\n(U \\to S) \\longmapsto (U \\times_S T \\to T).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\Sch/T)_\\etale)\n\\longrightarrow\n\\Sh((\\Sch/S)_\\etale)\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nWe have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021H","source_file":"topologies.tex","source_line":1123,"source_end_line":1156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1123-L1156","statement_sha256":"554978dd4d04fa7db678436b136cd410797e755761228d8d0045c37a707ecec9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6651,"rank":6651,"depth":8,"x":640.904,"y":1175.5,"cluster":"descent"},{"id":"stacks:021I","tag":"021I","title":"The étale topology · Lemma 021I","summary":"Let Sch_etale be a big étale site. Let f : T → S be a morphism in Sch_etale. • We have i_f = f_big ∘ i_T with i_f as in Lemma [Tag 021F] and i_T as in Lemma [Tag 021G]. • The functor S_etale → T_etale, (U → S) ↦ (U ×_S T → T) is continuous and induces a morphism of sites f_small : T_etale → S_etale We have f_small, *(F)(U/S) = F(U ×_S T/T). • We have a commutative diagram of morphisms of sites xymatrix T_etale ar[d]_f_small & (Sch/T)_etale ar[d]^f_big ar[l]^π_T S_etale &…","statement_latex":"Let $\\Sch_\\etale$ be a big \\'etale site.\nLet $f : T \\to S$ be a morphism in $\\Sch_\\etale$.\n\\begin{enumerate}\n\\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in\nLemma \\ref{lemma-put-in-T-etale} and $i_T$ as in\nLemma \\ref{lemma-at-the-bottom-etale}.\n\\item The functor $S_\\etale \\to T_\\etale$,\n$(U \\to S) \\mapsto (U \\times_S T \\to T)$ is continuous and induces\na morphism of sites\n$$\nf_{small} : T_\\etale \\longrightarrow S_\\etale\n$$\nWe have $f_{small, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\n\\item We have a commutative diagram of morphisms of sites\n$$\n\\xymatrix{\nT_\\etale \\ar[d]_{f_{small}} &\n(\\Sch/T)_\\etale \\ar[d]^{f_{big}} \\ar[l]^{\\pi_T}\\\\\nS_\\etale &\n(\\Sch/S)_\\etale \\ar[l]_{\\pi_S}\n}\n$$\nso that $f_{small} \\circ \\pi_T = \\pi_S \\circ f_{big}$ as morphisms of topoi.\n\\item We have $f_{small} = \\pi_S \\circ f_{big} \\circ i_T = \\pi_S \\circ i_f$.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021I","source_file":"topologies.tex","source_line":1175,"source_end_line":1202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1175-L1202","statement_sha256":"0d205997167a07f61bbf7dc6914049a6a37b84444f738f89ae270f048e465835","origin":"The Stacks Project","memory_eligible":false,"source_rank":6652,"rank":6652,"depth":9,"x":680.285,"y":1090.963,"cluster":"descent"},{"id":"stacks:021J","tag":"021J","title":"The étale topology · Lemma 021J","summary":"Given schemes X, Y, Y in Sch_etale and morphisms f : X → Y, g : Y → Z we have g_big ∘ f_big = (g ∘ f)_big and g_small ∘ f_small = (g ∘ f)_small.","statement_latex":"Given schemes $X$, $Y$, $Y$ in $\\Sch_\\etale$\nand morphisms $f : X \\to Y$, $g : Y \\to Z$ we have\n$g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and\n$g_{small} \\circ f_{small} = (g \\circ f)_{small}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021J","source_file":"topologies.tex","source_line":1249,"source_end_line":1255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1249-L1255","statement_sha256":"906b5822d5c14c71f0624067b6d38ccbd6cd16faef69549c1ec0ad5556f61d44","origin":"The Stacks Project","memory_eligible":false,"source_rank":6653,"rank":6653,"depth":10,"x":719.757,"y":1176.814,"cluster":"descent"},{"id":"stacks:0DDA","tag":"0DDA","title":"The étale topology · Lemma 0DDA","summary":"Let Sch_etale be a big étale site. Consider a cartesian diagram xymatrix T' ar[r]_g' ar[d]_f' & T ar[d]^f S' ar[r]^g & S in Sch_etale. Then i_g^-1 ∘ f_big, * = f'_small, * ∘ (i_g')^-1 and g_big^-1 ∘ f_big, * = f'_big, * ∘ (g'_big)^-1.","statement_latex":"Let $\\Sch_\\etale$ be a big \\'etale site. Consider a cartesian diagram\n$$\n\\xymatrix{\nT' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nin $\\Sch_\\etale$. Then\n$i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$\nand $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDA","source_file":"topologies.tex","source_line":1265,"source_end_line":1277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1265-L1277","statement_sha256":"cde56efffd62be833ecd125aac6bd262f8265c4269bd329dd12ae9ae11135093","origin":"The Stacks Project","memory_eligible":false,"source_rank":6654,"rank":6654,"depth":9,"x":620.3,"y":1135.352,"cluster":"descent"},{"id":"stacks:021K","tag":"021K","title":"The étale topology · Lemma 021K","summary":"Let S be a scheme contained in a big étale site Sch_etale. A sheaf F on the big étale site (Sch/S)_etale is given by the following data: • for every T/S ∈ Ob((Sch/S)_etale) a sheaf F_T on T_etale, • for every f : T' → T in (Sch/S)_etale a map c_f : f_small^-1F_T → F_T'. These data are subject to the following conditions: • [(a)] given any f : T' → T and g : T\" → T' in (Sch/S)_etale the composition c_g ∘ g_small^-1c_f is equal to c_f ∘ g, and • [(b)] if f : T' → T in…","statement_latex":"Let $S$ be a scheme contained in a big \\'etale site\n$\\Sch_\\etale$.\nA sheaf $\\mathcal{F}$ on the big \\'etale site\n$(\\Sch/S)_\\etale$ is given by the following data:\n\\begin{enumerate}\n\\item for every $T/S \\in \\Ob((\\Sch/S)_\\etale)$ a sheaf\n$\\mathcal{F}_T$ on $T_\\etale$,\n\\item for every $f : T' \\to T$ in\n$(\\Sch/S)_\\etale$ a map\n$c_f : f_{small}^{-1}\\mathcal{F}_T \\to \\mathcal{F}_{T'}$.\n\\end{enumerate}\nThese data are subject to the following conditions:\n\\begin{enumerate}\n\\item[(a)] given any $f : T' \\to T$ and $g : T'' \\to T'$ in\n$(\\Sch/S)_\\etale$ the composition\n$c_g \\circ g_{small}^{-1}c_f$ is equal to $c_{f \\circ g}$, and\n\\item[(b)] if $f : T' \\to T$ in $(\\Sch/S)_\\etale$\nis \\'etale then $c_f$ is an isomorphism.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021K","source_file":"topologies.tex","source_line":1295,"source_end_line":1316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1295-L1316","statement_sha256":"458242d426f01e948fb2d322e3b759048b3866ea7ade9aab8850f8338730b06d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6655,"rank":6655,"depth":7,"x":728.375,"y":1109.16,"cluster":"descent"},{"id":"stacks:021Z","tag":"021Z","title":"The smooth topology · Definition 021Z","summary":"Let T be a scheme. A smooth covering of T is a family of morphisms (f_i : T_i → T)_i ∈ I of schemes such that each f_i is smooth and such that T = ⋃ f_i(T_i).","statement_latex":"Let $T$ be a scheme. A {\\it smooth covering of $T$} is a family\nof morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes\nsuch that each $f_i$ is smooth and such\nthat $T = \\bigcup f_i(T_i)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021Z","source_file":"topologies.tex","source_line":1393,"source_end_line":1399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1393-L1399","statement_sha256":"f83daf228e70fc6dacdf0b5895a3ca7c399cf8ea3fbbed5bfe2a4379454527de","origin":"The Stacks Project","memory_eligible":false,"source_rank":6656,"rank":6656,"depth":0,"x":668.994,"y":1190.82,"cluster":"descent"},{"id":"stacks:0220","tag":"0220","title":"The smooth topology · Lemma 0220","summary":"Any étale covering is a smooth covering, and a fortiori, any Zariski covering is a smooth covering.","statement_latex":"Any \\'etale covering is a smooth covering, and a fortiori,\nany Zariski covering is a smooth covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0220","source_file":"topologies.tex","source_line":1401,"source_end_line":1405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1401-L1405","statement_sha256":"2d2d62ba9e8d349f7ab0f12d7f22cdc1ddba522c9e5ec88ed7bbf94c700fb34d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6657,"rank":6657,"depth":2,"x":646.846,"y":1095.745,"cluster":"descent"},{"id":"stacks:0221","tag":"0221","title":"The smooth topology · Lemma 0221","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is a smooth covering of T. • If (T_i → T)_i∈ I is a smooth covering and for each i we have a smooth covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is a smooth covering. • If (T_i → T)_i∈ I is a smooth covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is a smooth covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis a smooth covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a smooth covering and for each\n$i$ we have a smooth covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a smooth covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a smooth covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a smooth covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0221","source_file":"topologies.tex","source_line":1418,"source_end_line":1431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1418-L1431","statement_sha256":"6927eef2b07aa8ded22c79143f0f5104a9f04d4bce9658535d22637ea949a64f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6658,"rank":6658,"depth":0,"x":740.753,"y":1153.986,"cluster":"descent"},{"id":"stacks:0222","tag":"0222","title":"The smooth topology · Lemma 0222","summary":"Let T be an affine scheme. Let (T_i → T)_i ∈ I be a smooth covering of T. Then there exists a smooth covering (U_j → T)_j = 1, …, m which is a refinement of (T_i → T)_i ∈ I such that each U_j is an affine scheme, and such that each morphism U_j → T is standard smooth, see Morphisms, Definition [Tag 01V5]. Moreover, we may choose each U_j to be open affine in one of the T_i.","statement_latex":"Let $T$ be an affine scheme.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be a smooth covering of $T$.\nThen there exists a smooth covering\n$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement\nof $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine\nscheme, and such that each morphism $U_j \\to T$ is standard\nsmooth, see Morphisms, Definition \\ref{morphisms-definition-smooth}.\nMoreover, we may choose each $U_j$ to be open affine in one of the $T_i$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0222","source_file":"topologies.tex","source_line":1437,"source_end_line":1447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1437-L1447","statement_sha256":"de7f707860f8c3e3d1a584d07ccecb34746a0b33502d66fc1fed12ac4c16c000","origin":"The Stacks Project","memory_eligible":false,"source_rank":6659,"rank":6659,"depth":35,"x":623.3,"y":1164.442,"cluster":"descent"},{"id":"stacks:0223","tag":"0223","title":"The smooth topology · Definition 0223","summary":"Let T be an affine scheme. A standard smooth covering of T is a family (f_j : U_j → T)_j = 1, …, m with each U_j is affine, U_j → T standard smooth and T = ⋃ f_j(U_j).","statement_latex":"Let $T$ be an affine scheme. A {\\it standard smooth covering}\nof $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$\nwith each $U_j$ is affine, $U_j \\to T$ standard smooth\nand $T = \\bigcup f_j(U_j)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0223","source_file":"topologies.tex","source_line":1456,"source_end_line":1462,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1456-L1462","statement_sha256":"c65b15be171b2951d640da2d4b147816b09259f6ba45465981c3d1263f3ec228","origin":"The Stacks Project","memory_eligible":false,"source_rank":6660,"rank":6660,"depth":0,"x":702.407,"y":1089.231,"cluster":"descent"},{"id":"stacks:03WY","tag":"03WY","title":"The smooth topology · Definition 03WY","summary":"A big smooth site is any site Sch_smooth as in Sites, Definition [Tag 00VH] constructed as follows: • Choose any set of schemes S_0, and any set of smooth coverings Cov_0 among these schemes. • As underlying category take any category Sch_α constructed as in Sets, Lemma [Tag 000J] starting with the set S_0. • Choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Sch_α and the class of smooth coverings, and the set Cov_0 chosen above.","statement_latex":"A {\\it big smooth site} is any site $\\Sch_{smooth}$ as in\nSites, Definition \\ref{sites-definition-site} constructed as follows:\n\\begin{enumerate}\n\\item Choose any set of schemes $S_0$, and any set of smooth coverings\n$\\text{Cov}_0$ among these schemes.\n\\item As underlying category take any category $\\Sch_\\alpha$\nconstructed as in Sets, Lemma \\ref{sets-lemma-construct-category}\nstarting with the set $S_0$.\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\Sch_\\alpha$ and the class of smooth coverings,\nand the set $\\text{Cov}_0$ chosen above.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WY","source_file":"topologies.tex","source_line":1464,"source_end_line":1479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1464-L1479","statement_sha256":"57a3e8c80506c9128015098aaf6685677a465028d5d74c7f73a3526e7c61209b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6661,"rank":6661,"depth":2,"x":704.575,"y":1190.709,"cluster":"descent"},{"id":"stacks:03WZ","tag":"03WZ","title":"The smooth topology · Lemma 03WZ","summary":"Let Sch_smooth be a big smooth site as in Definition [Tag 03WY]. Let T ∈ Ob(Sch_smooth). Let (T_i → T)_i ∈ I be an arbitrary smooth covering of T. • There exists a covering (U_j → T)_j ∈ J of T in the site Sch_smooth which refines (T_i → T)_i ∈ I. • If (T_i → T)_i ∈ I is a standard smooth covering, then it is tautologically equivalent to a covering of Sch_smooth. • If (T_i → T)_i ∈ I is a Zariski covering, then it is tautologically equivalent to a covering of Sch_smooth.","statement_latex":"Let $\\Sch_{smooth}$ be a big smooth site as in\nDefinition \\ref{definition-big-smooth-site}.\nLet $T \\in \\Ob(\\Sch_{smooth})$.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary smooth covering of $T$.\n\\begin{enumerate}\n\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site\n$\\Sch_{smooth}$ which refines $\\{T_i \\to T\\}_{i \\in I}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard smooth covering, then\nit is tautologically equivalent to a covering of $\\Sch_{smooth}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then\nit is tautologically equivalent to a covering of $\\Sch_{smooth}$.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WZ","source_file":"topologies.tex","source_line":1491,"source_end_line":1505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1491-L1505","statement_sha256":"aeae9c495c3f7f50d7008484e5921d0c305debdd9d549ff38f921304a64e7edf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6662,"rank":6662,"depth":3,"x":620.457,"y":1116.296,"cluster":"descent"},{"id":"stacks:03X0","tag":"03X0","title":"The smooth topology · Definition 03X0","summary":"Let S be a scheme. Let Sch_smooth be a big smooth site containing S. • The big smooth site of S, denoted (Sch/S)_smooth, is the site Sch_smooth/S introduced in Sites, Section [Tag 00XZ]. • The big affine smooth site of S, denoted (Aff/S)_smooth, is the full subcategory of (Sch/S)_smooth whose objects are affine U/S. A covering of (Aff/S)_smooth is any covering (U_i → U) of (Sch/S)_smooth which is a standard smooth covering.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{smooth}$ be a big smooth\nsite containing $S$.\n\\begin{enumerate}\n\\item The {\\it big smooth site of $S$}, denoted\n$(\\Sch/S)_{smooth}$, is the site $\\Sch_{smooth}/S$\nintroduced in Sites, Section \\ref{sites-section-localize}.\n\\item The {\\it big affine smooth site of $S$}, denoted\n$(\\textit{Aff}/S)_{smooth}$, is the full subcategory of\n$(\\Sch/S)_{smooth}$ whose objects are affine $U/S$.\nA covering of $(\\textit{Aff}/S)_{smooth}$ is any covering\n$\\{U_i \\to U\\}$ of $(\\Sch/S)_{smooth}$ which is a\nstandard smooth covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03X0","source_file":"topologies.tex","source_line":1535,"source_end_line":1550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1535-L1550","statement_sha256":"489dc459005fdcf36d50f0dcf4ff7e6afb944527bac6b815693b908c5bf638c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6663,"rank":6663,"depth":0,"x":743.64,"y":1123.518,"cluster":"descent"},{"id":"stacks:06VC","tag":"06VC","title":"The smooth topology · Lemma 06VC","summary":"Let S be a scheme. Let Sch_smooth be a big smooth site containing S. The functor (Aff/S)_smooth → (Sch/S)_smooth is special cocontinuous and induces an equivalence of topoi from Sh((Aff/S)_smooth) to Sh((Sch/S)_smooth).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{smooth}$ be a big smooth\nsite containing $S$.\nThe functor\n$(\\textit{Aff}/S)_{smooth} \\to (\\Sch/S)_{smooth}$\nis special cocontinuous and induces an equivalence of topoi from\n$\\Sh((\\textit{Aff}/S)_{smooth})$ to\n$\\Sh((\\Sch/S)_{smooth})$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VC","source_file":"topologies.tex","source_line":1556,"source_end_line":1565,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1556-L1565","statement_sha256":"16ad6317928a475556a403e1c04ce84d180f368e19633ed3218f41075c7ec3b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6664,"rank":6664,"depth":36,"x":645.975,"y":1188.768,"cluster":"descent"},{"id":"stacks:04HC","tag":"04HC","title":"The smooth topology · Lemma 04HC","summary":"Let Sch_smooth be a big smooth site. Let f : T → S be a morphism in Sch_smooth. The functor u : (Sch/T)_smooth → (Sch/S)_smooth, V/T ↦ V/S is cocontinuous, and has a continuous right adjoint v : (Sch/S)_smooth → (Sch/T)_smooth, (U → S) ↦ (U ×_S T → T). They induce the same morphism of topoi f_big : Sh((Sch/T)_smooth) → Sh((Sch/S)_smooth) We have f_big^-1(G)(U/T) = G(U/S). We have f_big, *(F)(U/S) = F(U ×_S T/T). Also, f_big^-1 has a left adjoint f_big! which commutes with…","statement_latex":"Let $\\Sch_{smooth}$ be a big smooth site.\nLet $f : T \\to S$ be a morphism in $\\Sch_{smooth}$.\nThe functor\n$$\nu : (\\Sch/T)_{smooth} \\longrightarrow (\\Sch/S)_{smooth},\n\\quad\nV/T \\longmapsto V/S\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv : (\\Sch/S)_{smooth} \\longrightarrow (\\Sch/T)_{smooth},\n\\quad\n(U \\to S) \\longmapsto (U \\times_S T \\to T).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\Sch/T)_{smooth})\n\\longrightarrow\n\\Sh((\\Sch/S)_{smooth})\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nWe have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HC","source_file":"topologies.tex","source_line":1588,"source_end_line":1615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1588-L1615","statement_sha256":"54b80ad01dfdbbeaa89247fb2262d1ec0ca517d967fd8f0b491ae0d55dfa3fdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6665,"rank":6665,"depth":8,"x":665.727,"y":1084.167,"cluster":"descent"},{"id":"stacks:0225","tag":"0225","title":"The syntomic topology · Definition 0225","summary":"Let T be a scheme. An syntomic covering of T is a family of morphisms (f_i : T_i → T)_i ∈ I of schemes such that each f_i is syntomic and such that T = ⋃ f_i(T_i).","statement_latex":"Let $T$ be a scheme. An {\\it syntomic covering of $T$} is a family\nof morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes\nsuch that each $f_i$ is syntomic and such\nthat $T = \\bigcup f_i(T_i)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0225","source_file":"topologies.tex","source_line":1651,"source_end_line":1657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1651-L1657","statement_sha256":"d49807424a14d07791b6727db44a505808f97a907e5280f5006e4974c5b09a45","origin":"The Stacks Project","memory_eligible":false,"source_rank":6666,"rank":6666,"depth":0,"x":735.98,"y":1173.403,"cluster":"descent"},{"id":"stacks:0226","tag":"0226","title":"The syntomic topology · Lemma 0226","summary":"Any smooth covering is a syntomic covering, and a fortiori, any étale or Zariski covering is a syntomic covering.","statement_latex":"Any smooth covering is a syntomic covering, and a fortiori,\nany \\'etale or Zariski covering is a syntomic covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0226","source_file":"topologies.tex","source_line":1659,"source_end_line":1663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1659-L1663","statement_sha256":"9f4e6825cface159b8a24ff0c03cdda6028bcccc1054e4f3a32b07a594e80891","origin":"The Stacks Project","memory_eligible":false,"source_rank":6667,"rank":6667,"depth":36,"x":611.189,"y":1147.204,"cluster":"descent"},{"id":"stacks:0227","tag":"0227","title":"The syntomic topology · Lemma 0227","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is a syntomic covering of T. • If (T_i → T)_i∈ I is a syntomic covering and for each i we have a syntomic covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is a syntomic covering. • If (T_i → T)_i∈ I is a syntomic covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is a syntomic covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis a syntomic covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a syntomic covering and for each\n$i$ we have a syntomic covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a syntomic covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a syntomic covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a syntomic covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0227","source_file":"topologies.tex","source_line":1676,"source_end_line":1689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1676-L1689","statement_sha256":"91bd10ac2095847955ddf57887923028f0c19ddf20a01308798fbd3c8ebeac0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6668,"rank":6668,"depth":0,"x":725.381,"y":1095.215,"cluster":"descent"},{"id":"stacks:0228","tag":"0228","title":"The syntomic topology · Lemma 0228","summary":"Let T be an affine scheme. Let (T_i → T)_i ∈ I be a syntomic covering of T. Then there exists a syntomic covering (U_j → T)_j = 1, …, m which is a refinement of (T_i → T)_i ∈ I such that each U_j is an affine scheme, and such that each morphism U_j → T is standard syntomic, see Morphisms, Definition [Tag 01UC]. Moreover, we may choose each U_j to be open affine in one of the T_i.","statement_latex":"Let $T$ be an affine scheme.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be a syntomic covering of $T$.\nThen there exists a syntomic covering\n$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement\nof $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine\nscheme, and such that each morphism $U_j \\to T$ is standard\nsyntomic, see Morphisms, Definition \\ref{morphisms-definition-syntomic}.\nMoreover, we may choose each $U_j$ to be open affine in one of the $T_i$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0228","source_file":"topologies.tex","source_line":1695,"source_end_line":1705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1695-L1705","statement_sha256":"0abba284b74490a4d848e157f2f5a93653571bfb430c50849bad7f618ed78364","origin":"The Stacks Project","memory_eligible":false,"source_rank":6669,"rank":6669,"depth":35,"x":682.576,"y":1199.329,"cluster":"descent"},{"id":"stacks:0229","tag":"0229","title":"The syntomic topology · Definition 0229","summary":"Let T be an affine scheme. A standard syntomic covering of T is a family (f_j : U_j → T)_j = 1, …, m with each U_j is affine, U_j → T standard syntomic and T = ⋃ f_j(U_j).","statement_latex":"Let $T$ be an affine scheme. A {\\it standard syntomic covering} of $T$ is\na family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j$ is\naffine, $U_j \\to T$ standard syntomic and $T = \\bigcup f_j(U_j)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0229","source_file":"topologies.tex","source_line":1714,"source_end_line":1719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1714-L1719","statement_sha256":"dc0d679f2fefe96deefac2ff3574d7da13444a27dcb62271b518ca034bb30a07","origin":"The Stacks Project","memory_eligible":false,"source_rank":6670,"rank":6670,"depth":0,"x":629.929,"y":1097.321,"cluster":"descent"},{"id":"stacks:03X1","tag":"03X1","title":"The syntomic topology · Definition 03X1","summary":"A big syntomic site is any site Sch_syntomic as in Sites, Definition [Tag 00VH] constructed as follows: • Choose any set of schemes S_0, and any set of syntomic coverings Cov_0 among these schemes. • As underlying category take any category Sch_α constructed as in Sets, Lemma [Tag 000J] starting with the set S_0. • Choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Sch_α and the class of syntomic coverings, and the set Cov_0 chosen above.","statement_latex":"A {\\it big syntomic site} is any site $\\Sch_{syntomic}$ as in\nSites, Definition \\ref{sites-definition-site} constructed as follows:\n\\begin{enumerate}\n\\item Choose any set of schemes $S_0$, and any set of syntomic coverings\n$\\text{Cov}_0$ among these schemes.\n\\item As underlying category take any category $\\Sch_\\alpha$\nconstructed as in Sets, Lemma \\ref{sets-lemma-construct-category}\nstarting with the set $S_0$.\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\Sch_\\alpha$ and the class of syntomic coverings,\nand the set $\\text{Cov}_0$ chosen above.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03X1","source_file":"topologies.tex","source_line":1721,"source_end_line":1736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1721-L1736","statement_sha256":"91284b4c95ccc64fc97c762cb9dc7b86151a936b2eac97071aae32d234d07f60","origin":"The Stacks Project","memory_eligible":false,"source_rank":6671,"rank":6671,"depth":2,"x":751.892,"y":1143.088,"cluster":"descent"},{"id":"stacks:03X2","tag":"03X2","title":"The syntomic topology · Lemma 03X2","summary":"Let Sch_syntomic be a big syntomic site as in Definition [Tag 03X1]. Let T ∈ Ob(Sch_syntomic). Let (T_i → T)_i ∈ I be an arbitrary syntomic covering of T. • There exists a covering (U_j → T)_j ∈ J of T in the site Sch_syntomic which refines (T_i → T)_i ∈ I. • If (T_i → T)_i ∈ I is a standard syntomic covering, then it is tautologically equivalent to a covering in Sch_syntomic. • If (T_i → T)_i ∈ I is a Zariski covering, then it is tautologically equivalent to a covering…","statement_latex":"Let $\\Sch_{syntomic}$ be a big syntomic site as in\nDefinition \\ref{definition-big-syntomic-site}.\nLet $T \\in \\Ob(\\Sch_{syntomic})$.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary syntomic covering of $T$.\n\\begin{enumerate}\n\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site\n$\\Sch_{syntomic}$ which refines $\\{T_i \\to T\\}_{i \\in I}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard syntomic covering, then\nit is tautologically equivalent to a covering in $\\Sch_{syntomic}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then\nit is tautologically equivalent to a covering in $\\Sch_{syntomic}$.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03X2","source_file":"topologies.tex","source_line":1748,"source_end_line":1762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1748-L1762","statement_sha256":"81956b2955ccd82ad3597769f5c75a1a35a258b6127fdacd62b7dec8410952c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6672,"rank":6672,"depth":3,"x":624.008,"y":1178.861,"cluster":"descent"},{"id":"stacks:03X3","tag":"03X3","title":"The syntomic topology · Definition 03X3","summary":"Let S be a scheme. Let Sch_syntomic be a big syntomic site containing S. • The big syntomic site of S, denoted (Sch/S)_syntomic, is the site Sch_syntomic/S introduced in Sites, Section [Tag 00XZ]. • The big affine syntomic site of S, denoted (Aff/S)_syntomic, is the full subcategory of (Sch/S)_syntomic whose objects are affine U/S. A covering of (Aff/S)_syntomic is any covering (U_i → U) of (Sch/S)_syntomic which is a standard syntomic covering.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{syntomic}$ be a big syntomic\nsite containing $S$.\n\\begin{enumerate}\n\\item The {\\it big syntomic site of $S$}, denoted\n$(\\Sch/S)_{syntomic}$, is the site $\\Sch_{syntomic}/S$\nintroduced in Sites, Section \\ref{sites-section-localize}.\n\\item The {\\it big affine syntomic site of $S$}, denoted\n$(\\textit{Aff}/S)_{syntomic}$, is the full subcategory of\n$(\\Sch/S)_{syntomic}$ whose objects are affine $U/S$.\nA covering of $(\\textit{Aff}/S)_{syntomic}$ is any covering\n$\\{U_i \\to U\\}$ of $(\\Sch/S)_{syntomic}$ which is a\nstandard syntomic covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03X3","source_file":"topologies.tex","source_line":1792,"source_end_line":1807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1792-L1807","statement_sha256":"609178b8f46f0b2c819b69645f6a5f440490ef5d83c8f1e07e511256add644ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":6673,"rank":6673,"depth":0,"x":690.125,"y":1079.042,"cluster":"descent"},{"id":"stacks:06VD","tag":"06VD","title":"The syntomic topology · Lemma 06VD","summary":"Let S be a scheme. Let Sch_syntomic be a big syntomic site containing S. The functor (Aff/S)_syntomic → (Sch/S)_syntomic is special cocontinuous and induces an equivalence of topoi from Sh((Aff/S)_syntomic) to Sh((Sch/S)_syntomic).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{syntomic}$ be a big syntomic\nsite containing $S$.\nThe functor\n$(\\textit{Aff}/S)_{syntomic} \\to (\\Sch/S)_{syntomic}$\nis special cocontinuous and induces an equivalence of topoi from\n$\\Sh((\\textit{Aff}/S)_{syntomic})$ to\n$\\Sh((\\Sch/S)_{syntomic})$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VD","source_file":"topologies.tex","source_line":1813,"source_end_line":1822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1813-L1822","statement_sha256":"171de1f1b37fa0041cf7075ba4e65d2319cddf45383517d7dfbf6d801947bb01","origin":"The Stacks Project","memory_eligible":false,"source_rank":6674,"rank":6674,"depth":36,"x":721.914,"y":1191.132,"cluster":"descent"},{"id":"stacks:04HD","tag":"04HD","title":"The syntomic topology · Lemma 04HD","summary":"Let Sch_syntomic be a big syntomic site. Let f : T → S be a morphism in Sch_syntomic. The functor u : (Sch/T)_syntomic → (Sch/S)_syntomic, V/T ↦ V/S is cocontinuous, and has a continuous right adjoint v : (Sch/S)_syntomic → (Sch/T)_syntomic, (U → S) ↦ (U ×_S T → T). They induce the same morphism of topoi f_big : Sh((Sch/T)_syntomic) → Sh((Sch/S)_syntomic) We have f_big^-1(G)(U/T) = G(U/S). We have f_big, *(F)(U/S) = F(U ×_S T/T). Also, f_big^-1 has a left adjoint f_big!…","statement_latex":"Let $\\Sch_{syntomic}$ be a big syntomic site.\nLet $f : T \\to S$ be a morphism in $\\Sch_{syntomic}$.\nThe functor\n$$\nu : (\\Sch/T)_{syntomic} \\longrightarrow (\\Sch/S)_{syntomic},\n\\quad\nV/T \\longmapsto V/S\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv : (\\Sch/S)_{syntomic} \\longrightarrow (\\Sch/T)_{syntomic},\n\\quad\n(U \\to S) \\longmapsto (U \\times_S T \\to T).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\Sch/T)_{syntomic})\n\\longrightarrow\n\\Sh((\\Sch/S)_{syntomic})\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nWe have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HD","source_file":"topologies.tex","source_line":1845,"source_end_line":1872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1845-L1872","statement_sha256":"c032f2cbd8371961575efcf55ccd8351c7c6ef0f6c89bb8e694a41d5700c5125","origin":"The Stacks Project","memory_eligible":false,"source_rank":6675,"rank":6675,"depth":8,"x":607.363,"y":1125.961,"cluster":"descent"},{"id":"stacks:021M","tag":"021M","title":"The fppf topology · Definition 021M","summary":"Let T be a scheme. An fppf covering of T is a family of morphisms (f_i : T_i → T)_i ∈ I of schemes such that each f_i is flat, locally of finite presentation and such that T = ⋃ f_i(T_i).","statement_latex":"Let $T$ be a scheme. An {\\it fppf covering of $T$} is a family\nof morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes\nsuch that each $f_i$ is flat, locally of finite presentation and such\nthat $T = \\bigcup f_i(T_i)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021M","source_file":"topologies.tex","source_line":1910,"source_end_line":1916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1910-L1916","statement_sha256":"664af346b4fdbf5271df584ced107ff2ab34c3aa329515ae84899c39580aaf59","origin":"The Stacks Project","memory_eligible":false,"source_rank":6676,"rank":6676,"depth":0,"x":745.393,"y":1108.879,"cluster":"descent"},{"id":"stacks:021N","tag":"021N","title":"The fppf topology · Lemma 021N","summary":"Any syntomic covering is an fppf covering, and a fortiori, any smooth, étale, or Zariski covering is an fppf covering.","statement_latex":"Any syntomic covering is an fppf covering, and a fortiori,\nany smooth, \\'etale, or Zariski covering is an fppf covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021N","source_file":"topologies.tex","source_line":1918,"source_end_line":1922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1918-L1922","statement_sha256":"1cbb5fdf87cfa8ffea86d7a97776a7db2502567f79b82673af57aa4bb317bab3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6677,"rank":6677,"depth":37,"x":656.617,"y":1200.547,"cluster":"descent"},{"id":"stacks:021O","tag":"021O","title":"The fppf topology · Lemma 021O","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is an fppf covering of T. • If (T_i → T)_i∈ I is an fppf covering and for each i we have an fppf covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is an fppf covering. • If (T_i → T)_i∈ I is an fppf covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is an fppf covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis an fppf covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fppf covering and for each\n$i$ we have an fppf covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an fppf covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fppf covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an fppf covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021O","source_file":"topologies.tex","source_line":1938,"source_end_line":1951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1938-L1951","statement_sha256":"25d54087c965b7727c7dfc175a5cf537296368210c589df9145d755dabfd3bd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6678,"rank":6678,"depth":17,"x":648.299,"y":1081.618,"cluster":"descent"},{"id":"stacks:021P","tag":"021P","title":"The fppf topology · Lemma 021P","summary":"Let T be an affine scheme. Let (T_i → T)_i ∈ I be an fppf covering of T. Then there exists an fppf covering (U_j → T)_j = 1, …, m which is a refinement of (T_i → T)_i ∈ I such that each U_j is an affine scheme. Moreover, we may choose each U_j to be open affine in one of the T_i.","statement_latex":"Let $T$ be an affine scheme.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an fppf covering of $T$.\nThen there exists an fppf covering\n$\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ which is a refinement\nof $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine\nscheme. Moreover, we may choose each $U_j$ to be open affine\nin one of the $T_i$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021P","source_file":"topologies.tex","source_line":1969,"source_end_line":1978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1969-L1978","statement_sha256":"cb81fa6380fac37f736557b8736b492dbb02484f1eed121430a677e98d8684df","origin":"The Stacks Project","memory_eligible":false,"source_rank":6679,"rank":6679,"depth":18,"x":750.884,"y":1165.261,"cluster":"descent"},{"id":"stacks:021Q","tag":"021Q","title":"The fppf topology · Definition 021Q","summary":"Let T be an affine scheme. A standard fppf covering of T is a family (f_j : U_j → T)_j = 1, …, m with each U_j is affine, flat and of finite presentation over T and T = ⋃ f_j(U_j).","statement_latex":"Let $T$ be an affine scheme. A {\\it standard fppf covering}\nof $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$\nwith each $U_j$ is affine, flat and of finite presentation over $T$\nand $T = \\bigcup f_j(U_j)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021Q","source_file":"topologies.tex","source_line":1989,"source_end_line":1995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1989-L1995","statement_sha256":"d5e5a95cec74c4ff0275d3c6fff6a41a30143efd7e0075fc66495963659c7f1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6680,"rank":6680,"depth":0,"x":606.849,"y":1161.763,"cluster":"descent"},{"id":"stacks:021R","tag":"021R","title":"The fppf topology · Definition 021R","summary":"A big fppf site is any site Sch_fppf as in Sites, Definition [Tag 00VH] constructed as follows: • Choose any set of schemes S_0, and any set of fppf coverings Cov_0 among these schemes. • As underlying category take any category Sch_α constructed as in Sets, Lemma [Tag 000J] starting with the set S_0. • Choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Sch_α and the class of fppf coverings, and the set Cov_0 chosen above.","statement_latex":"A {\\it big fppf site} is any site $\\Sch_{fppf}$ as in\nSites, Definition \\ref{sites-definition-site} constructed as follows:\n\\begin{enumerate}\n\\item Choose any set of schemes $S_0$, and any set of fppf coverings\n$\\text{Cov}_0$ among these schemes.\n\\item As underlying category take any category $\\Sch_\\alpha$\nconstructed as in Sets, Lemma \\ref{sets-lemma-construct-category}\nstarting with the set $S_0$.\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\Sch_\\alpha$ and the class of fppf coverings,\nand the set $\\text{Cov}_0$ chosen above.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021R","source_file":"topologies.tex","source_line":1997,"source_end_line":2012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L1997-L2012","statement_sha256":"93fd2b1c53cb4b6f7b4336aa3412d8228c9021615efd1b67997da0ec6c5fc6fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6681,"rank":6681,"depth":2,"x":716.72,"y":1082.001,"cluster":"descent"},{"id":"stacks:03WX","tag":"03WX","title":"The fppf topology · Lemma 03WX","summary":"Let Sch_fppf be a big fppf site as in Definition [Tag 021R]. Let T ∈ Ob(Sch_fppf). Let (T_i → T)_i ∈ I be an arbitrary fppf covering of T. • There exists a covering (U_j → T)_j ∈ J of T in the site Sch_fppf which refines (T_i → T)_i ∈ I. • If (T_i → T)_i ∈ I is a standard fppf covering, then it is tautologically equivalent to a covering of Sch_fppf. • If (T_i → T)_i ∈ I is a Zariski covering, then it is tautologically equivalent to a covering of Sch_fppf.","statement_latex":"Let $\\Sch_{fppf}$ be a big fppf site as in\nDefinition \\ref{definition-big-fppf-site}.\nLet $T \\in \\Ob(\\Sch_{fppf})$.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary fppf covering of $T$.\n\\begin{enumerate}\n\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site\n$\\Sch_{fppf}$ which refines $\\{T_i \\to T\\}_{i \\in I}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard fppf covering, then\nit is tautologically equivalent to a covering of $\\Sch_{fppf}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then\nit is tautologically equivalent to a covering of $\\Sch_{fppf}$.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WX","source_file":"topologies.tex","source_line":2024,"source_end_line":2038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2024-L2038","statement_sha256":"a79e382fa67f0236a42aaf14ae51402597f5ba1c0c9ba904c515966aefb4b607","origin":"The Stacks Project","memory_eligible":false,"source_rank":6682,"rank":6682,"depth":18,"x":699.713,"y":1204.085,"cluster":"descent"},{"id":"stacks:021S","tag":"021S","title":"The fppf topology · Definition 021S","summary":"Let S be a scheme. Let Sch_fppf be a big fppf site containing S. • The big fppf site of S, denoted (Sch/S)_fppf, is the site Sch_fppf/S introduced in Sites, Section [Tag 00XZ]. • The big affine fppf site of S, denoted (Aff/S)_fppf, is the full subcategory of (Sch/S)_fppf whose objects are affine U/S. A covering of (Aff/S)_fppf is any covering (U_i → U) of (Sch/S)_fppf which is a standard fppf covering.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf\nsite containing $S$.\n\\begin{enumerate}\n\\item The {\\it big fppf site of $S$}, denoted\n$(\\Sch/S)_{fppf}$, is the site $\\Sch_{fppf}/S$\nintroduced in Sites, Section \\ref{sites-section-localize}.\n\\item The {\\it big affine fppf site of $S$}, denoted\n$(\\textit{Aff}/S)_{fppf}$, is the full subcategory of\n$(\\Sch/S)_{fppf}$ whose objects are affine $U/S$.\nA covering of $(\\textit{Aff}/S)_{fppf}$ is any covering\n$\\{U_i \\to U\\}$ of $(\\Sch/S)_{fppf}$ which is a\nstandard fppf covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021S","source_file":"topologies.tex","source_line":2068,"source_end_line":2083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2068-L2083","statement_sha256":"497d15b477107576c80e8d7611ef91b1bab9e164aa09f5f3adbb395e6ffcc332","origin":"The Stacks Project","memory_eligible":false,"source_rank":6683,"rank":6683,"depth":0,"x":613.435,"y":1103.661,"cluster":"descent"},{"id":"stacks:021T","tag":"021T","title":"The fppf topology · Lemma 021T","summary":"Let S be a scheme. Let Sch_fppf be a big fppf site containing S. Then (Aff/S)_fppf is a site.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf\nsite containing $S$. Then $(\\textit{Aff}/S)_{fppf}$ is a site.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021T","source_file":"topologies.tex","source_line":2089,"source_end_line":2093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2089-L2093","statement_sha256":"07ca83d34cac6e20a8193f99e96c22f8d26a185613be2f2365b6a9d38186cc3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6684,"rank":6684,"depth":2,"x":758.883,"y":1128.945,"cluster":"descent"},{"id":"stacks:021U","tag":"021U","title":"The fppf topology · Lemma 021U","summary":"Let S be a scheme. Let Sch_fppf be a big fppf site containing S. The underlying categories of the sites Sch_fppf, (Sch/S)_fppf, and (Aff/S)_fppf have fibre products. In each case the obvious functor into the category Sch of all schemes commutes with taking fibre products. The category (Sch/S)_fppf has a final object, namely S/S.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf\nsite containing $S$. The underlying categories of the sites\n$\\Sch_{fppf}$, $(\\Sch/S)_{fppf}$,\nand $(\\textit{Aff}/S)_{fppf}$ have fibre products.\nIn each case the obvious functor into the category $\\Sch$ of\nall schemes commutes with taking fibre products. The category\n$(\\Sch/S)_{fppf}$ has a final object, namely $S/S$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021U","source_file":"topologies.tex","source_line":2109,"source_end_line":2118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2109-L2118","statement_sha256":"6b7e3ac4c3b8b7bcc84e1c9a77830036a59a8554b4b86ed0febe9940a5a16daf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6685,"rank":6685,"depth":3,"x":630.366,"y":1193.296,"cluster":"descent"},{"id":"stacks:021V","tag":"021V","title":"The fppf topology · Lemma 021V","summary":"Let S be a scheme. Let Sch_fppf be a big fppf site containing S. The functor (Aff/S)_fppf → (Sch/S)_fppf is cocontinuous and induces an equivalence of topoi from Sh((Aff/S)_fppf) to Sh((Sch/S)_fppf).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{fppf}$ be a big fppf\nsite containing $S$.\nThe functor $(\\textit{Aff}/S)_{fppf} \\to (\\Sch/S)_{fppf}$\nis cocontinuous and induces an equivalence of topoi from\n$\\Sh((\\textit{Aff}/S)_{fppf})$ to\n$\\Sh((\\Sch/S)_{fppf})$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021V","source_file":"topologies.tex","source_line":2139,"source_end_line":2147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2139-L2147","statement_sha256":"813b96b5bf64ea0cbbf7078bfadc6abc5be61a263fce48f0550a4fbe346c87bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6686,"rank":6686,"depth":19,"x":673.696,"y":1072.055,"cluster":"descent"},{"id":"stacks:021W","tag":"021W","title":"The fppf topology · Lemma 021W","summary":"Let Sch_fppf be a big fppf site. Let f : T → S be a morphism in Sch_fppf. The functor u : (Sch/T)_fppf → (Sch/S)_fppf, V/T ↦ V/S is cocontinuous, and has a continuous right adjoint v : (Sch/S)_fppf → (Sch/T)_fppf, (U → S) ↦ (U ×_S T → T). They induce the same morphism of topoi f_big : Sh((Sch/T)_fppf) → Sh((Sch/S)_fppf) We have f_big^-1(G)(U/T) = G(U/S). We have f_big, *(F)(U/S) = F(U ×_S T/T). Also, f_big^-1 has a left adjoint f_big! which commutes with fibre products…","statement_latex":"Let $\\Sch_{fppf}$ be a big fppf site.\nLet $f : T \\to S$ be a morphism in $\\Sch_{fppf}$.\nThe functor\n$$\nu : (\\Sch/T)_{fppf} \\longrightarrow (\\Sch/S)_{fppf},\n\\quad\nV/T \\longmapsto V/S\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv : (\\Sch/S)_{fppf} \\longrightarrow (\\Sch/T)_{fppf},\n\\quad\n(U \\to S) \\longmapsto (U \\times_S T \\to T).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\Sch/T)_{fppf})\n\\longrightarrow\n\\Sh((\\Sch/S)_{fppf})\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nWe have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021W","source_file":"topologies.tex","source_line":2171,"source_end_line":2198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2171-L2198","statement_sha256":"768533f6871ea1fb4b55edca63a57ac88ef5b907dd5eda19586d2fcb932976c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6687,"rank":6687,"depth":8,"x":739.706,"y":1186.853,"cluster":"descent"},{"id":"stacks:021X","tag":"021X","title":"The fppf topology · Lemma 021X","summary":"Given schemes X, Y, Z in (Sch/S)_fppf and morphisms f : X → Y, g : Y → Z we have g_big ∘ f_big = (g ∘ f)_big.","statement_latex":"Given schemes $X$, $Y$, $Z$ in $(\\Sch/S)_{fppf}$\nand morphisms $f : X \\to Y$, $g : Y \\to Z$ we have\n$g_{big} \\circ f_{big} = (g \\circ f)_{big}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/021X","source_file":"topologies.tex","source_line":2215,"source_end_line":2220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2215-L2220","statement_sha256":"04ded5857196dda746ee0e6a3d97b98268500dc618c16ad1aea5242c8d00e6bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6688,"rank":6688,"depth":9,"x":597.728,"y":1139.326,"cluster":"descent"},{"id":"stacks:0DBD","tag":"0DBD","title":"The ph topology · Definition 0DBD","summary":"Let T be an affine scheme. A standard ph covering is a family (f_j : U_j → T)_j = 1, …, m constructed from a proper surjective morphism f : U → T and an affine open covering U = ⋃_j = 1, …, m U_j by setting f_j = f|_U_j.","statement_latex":"Let $T$ be an affine scheme. A {\\it standard ph covering} is a family\n$\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ constructed from a\nproper surjective morphism $f : U \\to T$ and an affine open covering\n$U = \\bigcup_{j = 1, \\ldots, m} U_j$ by setting $f_j = f|_{U_j}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBD","source_file":"topologies.tex","source_line":2303,"source_end_line":2309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2303-L2309","statement_sha256":"f8e92782232757dda199e00513e7f2e9904985c55865d4199bbc1a489214f4c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6689,"rank":6689,"depth":0,"x":741.627,"y":1093.498,"cluster":"descent"},{"id":"stacks:0DBE","tag":"0DBE","title":"The ph topology · Lemma 0DBE","summary":"Let (f_j : U_j → T)_j = 1, …, m be a standard ph covering. Let T' → T be a morphism of affine schemes. Then (U_j ×_T T' → T')_j = 1, …, m is a standard ph covering.","statement_latex":"Let $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ be a standard ph covering.\nLet $T' \\to T$ be a morphism of affine schemes.\nThen $\\{U_j \\times_T T' \\to T'\\}_{j = 1, \\ldots, m}$ is a\nstandard ph covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBE","source_file":"topologies.tex","source_line":2316,"source_end_line":2322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2316-L2322","statement_sha256":"1183d3e0c87417e497f502ced9915ddd78c3496a55f1e27f112965d0b1167c48","origin":"The Stacks Project","memory_eligible":false,"source_rank":6690,"rank":6690,"depth":17,"x":671.898,"y":1209.727,"cluster":"descent"},{"id":"stacks:0DBF","tag":"0DBF","title":"The ph topology · Lemma 0DBF","summary":"Let T be an affine scheme. Each of the following types of families of maps with target T has a refinement by a standard ph covering: • any Zariski open covering of T, • (W_ji → T)_j = 1, …, m, i = 1, … n_j where (W_ji → U_j)_i = 1, …, n_j and (U_j → T)_j = 1, …, m are standard ph coverings.","statement_latex":"Let $T$ be an affine scheme. Each of the following types of families\nof maps with target $T$ has a refinement by a standard ph covering:\n\\begin{enumerate}\n\\item any Zariski open covering of $T$,\n\\item $\\{W_{ji} \\to T\\}_{j = 1, \\ldots, m, i = 1, \\ldots n_j}$\nwhere $\\{W_{ji} \\to U_j\\}_{i = 1, \\ldots, n_j}$\nand $\\{U_j \\to T\\}_{j = 1, \\ldots, m}$ are standard ph coverings.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBF","source_file":"topologies.tex","source_line":2336,"source_end_line":2346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2336-L2346","statement_sha256":"89133ebd63207301814098992975b70b1f49d84a99748122d37ae3390e288a7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6691,"rank":6691,"depth":30,"x":629.568,"y":1083.614,"cluster":"descent"},{"id":"stacks:0DBG","tag":"0DBG","title":"The ph topology · Definition 0DBG","summary":"Let T be a scheme. A ph covering of T is a family of morphisms (f_i : T_i → T)_i ∈ I of schemes such that f_i is locally of finite type and such that for every affine open U ⊂ T there exists a standard ph covering (U_j → U)_j = 1, …, m refining the family (T_i ×_T U → U)_i ∈ I.","statement_latex":"Let $T$ be a scheme. A {\\it ph covering of $T$} is a family\nof morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes such\nthat $f_i$ is locally of finite type and such that for every\naffine open $U \\subset T$ there exists a standard ph covering\n$\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ refining the family\n$\\{T_i \\times_T U \\to U\\}_{i \\in I}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBG","source_file":"topologies.tex","source_line":2408,"source_end_line":2416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2408-L2416","statement_sha256":"c4fcc1aef308fd5c7945e5532cf1b4844bc9f250e2c1c1173110616380869997","origin":"The Stacks Project","memory_eligible":false,"source_rank":6692,"rank":6692,"depth":0,"x":763.075,"y":1153.047,"cluster":"descent"},{"id":"stacks:0DBH","tag":"0DBH","title":"The ph topology · Lemma 0DBH","summary":"A Zariski covering is a ph covering.","statement_latex":"A Zariski covering is a ph covering\\footnote{We will see\nin More on Morphisms, Lemma \\ref{more-morphisms-lemma-fppf-ph} that\nfppf coverings (and hence syntomic, smooth, or \\'etale coverings)\nare ph coverings as well.}.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBH","source_file":"topologies.tex","source_line":2422,"source_end_line":2428,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2422-L2428","statement_sha256":"4a528d8bdc997cdc13b84f23506b5d9d82fbb4df70606187a344b906e1f01648","origin":"The Stacks Project","memory_eligible":false,"source_rank":6693,"rank":6693,"depth":31,"x":607.784,"y":1177.762,"cluster":"descent"},{"id":"stacks:0DES","tag":"0DES","title":"The ph topology · Lemma 0DES","summary":"Let f : Y → X be a surjective proper morphism of schemes. Then (Y → X) is a ph covering.","statement_latex":"Let $f : Y \\to X$ be a surjective proper morphism of schemes.\nThen $\\{Y \\to X\\}$ is a ph covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DES","source_file":"topologies.tex","source_line":2436,"source_end_line":2440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2436-L2440","statement_sha256":"500fb2358c92357e54471aec096838d4aa5987fd39976aea2348990637978661","origin":"The Stacks Project","memory_eligible":false,"source_rank":6694,"rank":6694,"depth":0,"x":703.031,"y":1070.737,"cluster":"descent"},{"id":"stacks:0ET9","tag":"0ET9","title":"The ph topology · Lemma 0ET9","summary":"Let T be a scheme. Let (f_i : T_i → T)_i ∈ I be a family of morphisms such that f_i is locally of finite type for all i. The following are equivalent • (T_i → T)_i ∈ I is a ph covering, • there is a ph covering which refines (T_i → T)_i ∈ I, and • (coprod_i ∈ I T_i → T) is a ph covering.","statement_latex":"Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family\nof morphisms such that $f_i$ is locally of finite type for all $i$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\{T_i \\to T\\}_{i \\in I}$ is a ph covering,\n\\item there is a ph covering which refines $\\{T_i \\to T\\}_{i \\in I}$, and\n\\item $\\{\\coprod_{i \\in I} T_i \\to T\\}$ is a ph covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET9","source_file":"topologies.tex","source_line":2446,"source_end_line":2456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2446-L2456","statement_sha256":"b16a779f808dc61c7876c6b41893a91d3d3ee4805e4d81166673e1cdd184fe0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6695,"rank":6695,"depth":1,"x":718.962,"y":1204.548,"cluster":"descent"},{"id":"stacks:0DBI","tag":"0DBI","title":"The ph topology · Lemma 0DBI","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is a ph covering of T. • If (T_i → T)_i∈ I is a ph covering and for each i we have a ph covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is a ph covering. • If (T_i → T)_i∈ I is a ph covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is a ph covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis a ph covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a ph covering and for each\n$i$ we have a ph covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a ph covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a ph covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a ph covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBI","source_file":"topologies.tex","source_line":2492,"source_end_line":2505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2492-L2505","statement_sha256":"d14a8d15b48c2504429c68fc1512e8af23a382af6fd1a031392e594d19fe3566","origin":"The Stacks Project","memory_eligible":false,"source_rank":6696,"rank":6696,"depth":31,"x":598.859,"y":1114.352,"cluster":"descent"},{"id":"stacks:0DBJ","tag":"0DBJ","title":"The ph topology · Definition 0DBJ","summary":"A big ph site is any site Sch_ph as in Sites, Definition [Tag 00VH] constructed as follows: • Choose any set of schemes S_0, and any set of ph coverings Cov_0 among these schemes. • As underlying category take any category Sch_α constructed as in Sets, Lemma [Tag 000J] starting with the set S_0. • Choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Sch_α and the class of ph coverings, and the set Cov_0 chosen above.","statement_latex":"A {\\it big ph site} is any site $\\Sch_{ph}$ as in\nSites, Definition \\ref{sites-definition-site} constructed as follows:\n\\begin{enumerate}\n\\item Choose any set of schemes $S_0$, and any set of ph coverings\n$\\text{Cov}_0$ among these schemes.\n\\item As underlying category take any category $\\Sch_\\alpha$\nconstructed as in Sets, Lemma \\ref{sets-lemma-construct-category}\nstarting with the set $S_0$.\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\Sch_\\alpha$ and the class of ph coverings,\nand the set $\\text{Cov}_0$ chosen above.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBJ","source_file":"topologies.tex","source_line":2552,"source_end_line":2567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2552-L2567","statement_sha256":"8d7c3062ca606499e15aa96d8a94a5d99b43d92e065c22a235cf41e14463cac3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6697,"rank":6697,"depth":2,"x":760.954,"y":1112.703,"cluster":"descent"},{"id":"stacks:0DBK","tag":"0DBK","title":"The ph topology · Lemma 0DBK","summary":"Let Sch_ph be a big ph site as in Definition [Tag 0DBJ]. Let T ∈ Ob(Sch_ph). Let (T_i → T)_i ∈ I be an arbitrary ph covering of T. • There exists a covering (U_j → T)_j ∈ J of T in the site Sch_ph which refines (T_i → T)_i ∈ I. • If (T_i → T)_i ∈ I is a standard ph covering, then it is tautologically equivalent to a covering of Sch_ph. • If (T_i → T)_i ∈ I is a Zariski covering, then it is tautologically equivalent to a covering of Sch_ph.","statement_latex":"Let $\\Sch_{ph}$ be a big ph site as in\nDefinition \\ref{definition-big-ph-site}.\nLet $T \\in \\Ob(\\Sch_{ph})$.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary ph covering of $T$.\n\\begin{enumerate}\n\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site\n$\\Sch_{ph}$ which refines $\\{T_i \\to T\\}_{i \\in I}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard ph covering, then\nit is tautologically equivalent to a covering of $\\Sch_{ph}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then\nit is tautologically equivalent to a covering of $\\Sch_{ph}$.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBK","source_file":"topologies.tex","source_line":2579,"source_end_line":2593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2579-L2593","statement_sha256":"2ca13b8795e66b9a5a5238b4156c7a91e25958097bb6d83da38a8f39df5ed60b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6698,"rank":6698,"depth":32,"x":642.027,"y":1206.466,"cluster":"descent"},{"id":"stacks:0DBL","tag":"0DBL","title":"The ph topology · Definition 0DBL","summary":"Let S be a scheme. Let Sch_ph be a big ph site containing S. • The big ph site of S, denoted (Sch/S)_ph, is the site Sch_ph/S introduced in Sites, Section [Tag 00XZ]. • The big affine ph site of S, denoted (Aff/S)_ph, is the full subcategory of (Sch/S)_ph whose objects are affine U/S. A covering of (Aff/S)_ph is any finite covering (U_i → U) of (Sch/S)_ph with U_i and U affine.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph site containing $S$.\n\\begin{enumerate}\n\\item The {\\it big ph site of $S$}, denoted\n$(\\Sch/S)_{ph}$, is the site $\\Sch_{ph}/S$\nintroduced in Sites, Section \\ref{sites-section-localize}.\n\\item The {\\it big affine ph site of $S$}, denoted\n$(\\textit{Aff}/S)_{ph}$, is the full subcategory of\n$(\\Sch/S)_{ph}$ whose objects are affine $U/S$.\nA covering of $(\\textit{Aff}/S)_{ph}$ is any finite covering\n$\\{U_i \\to U\\}$ of $(\\Sch/S)_{ph}$ with $U_i$ and $U$ affine.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBL","source_file":"topologies.tex","source_line":2623,"source_end_line":2636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2623-L2636","statement_sha256":"5bcae10287c80b0835ec10c1d5806adf320ae7fadc5a80812f30b43692be1474","origin":"The Stacks Project","memory_eligible":false,"source_rank":6699,"rank":6699,"depth":0,"x":654.397,"y":1069.017,"cluster":"descent"},{"id":"stacks:0DBM","tag":"0DBM","title":"The ph topology · Lemma 0DBM","summary":"Let S be a scheme. Let Sch_ph be a big ph site containing S. Then (Aff/S)_ph is a site.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph\nsite containing $S$. Then $(\\textit{Aff}/S)_{ph}$ is a site.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBM","source_file":"topologies.tex","source_line":2647,"source_end_line":2651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2647-L2651","statement_sha256":"1975de4cbae34f7c5205d7eff446bd048e8e5f1e1181a080f5ef7a0d22ce516f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6700,"rank":6700,"depth":32,"x":756.413,"y":1178.039,"cluster":"descent"},{"id":"stacks:0DBN","tag":"0DBN","title":"The ph topology · Lemma 0DBN","summary":"Let S be a scheme. Let Sch_ph be a big ph site containing S. The underlying categories of the sites Sch_ph, (Sch/S)_ph, and (Aff/S)_ph have fibre products. In each case the obvious functor into the category Sch of all schemes commutes with taking fibre products. The category (Sch/S)_ph has a final object, namely S/S.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph\nsite containing $S$. The underlying categories of the sites\n$\\Sch_{ph}$, $(\\Sch/S)_{ph}$, and $(\\textit{Aff}/S)_{ph}$ have fibre products.\nIn each case the obvious functor into the category $\\Sch$ of\nall schemes commutes with taking fibre products. The category\n$(\\Sch/S)_{ph}$ has a final object, namely $S/S$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBN","source_file":"topologies.tex","source_line":2667,"source_end_line":2675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2667-L2675","statement_sha256":"06024e4f50c45fe0dbba73be0b6e6ac8e8c0dcb8b6632eb31dc4e63c034f5eb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6701,"rank":6701,"depth":3,"x":592.553,"y":1155.399,"cluster":"descent"},{"id":"stacks:0DBP","tag":"0DBP","title":"The ph topology · Lemma 0DBP","summary":"Let S be a scheme. Let Sch_ph be a big ph site containing S. The functor (Aff/S)_ph → (Sch/S)_ph is cocontinuous and induces an equivalence of topoi from Sh((Aff/S)_ph) to Sh((Sch/S)_ph).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_{ph}$ be a big ph\nsite containing $S$.\nThe functor $(\\textit{Aff}/S)_{ph} \\to (\\Sch/S)_{ph}$\nis cocontinuous and induces an equivalence of topoi from\n$\\Sh((\\textit{Aff}/S)_{ph})$ to\n$\\Sh((\\Sch/S)_{ph})$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBP","source_file":"topologies.tex","source_line":2696,"source_end_line":2704,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2696-L2704","statement_sha256":"5ce37f553a7853f19a1e0dd84a825a1f190fdd562f38dfe8f76be6065d345ba8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6702,"rank":6702,"depth":8,"x":732.401,"y":1078.671,"cluster":"descent"},{"id":"stacks:0DBQ","tag":"0DBQ","title":"The ph topology · Lemma 0DBQ","summary":"Let F be a presheaf on (Sch/S)_ph. Then F is a sheaf if and only if • F satisfies the sheaf condition for Zariski coverings, and • if f : V → U is proper surjective, then F(U) maps bijectively to the equalizer of the two maps F(V) → F(V ×_U V). Moreover, in the presence of (1) property (2) is equivalent to property • [(2')] the sheaf property for (V → U) as in (2) with U affine.","statement_latex":"Let $\\mathcal{F}$ be a presheaf on $(\\Sch/S)_{ph}$.\nThen $\\mathcal{F}$ is a sheaf if and only if\n\\begin{enumerate}\n\\item $\\mathcal{F}$ satisfies the sheaf condition for\nZariski coverings, and\n\\item if $f : V \\to U$ is proper surjective, then\n$\\mathcal{F}(U)$ maps bijectively to the equalizer\nof the two maps $\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$.\n\\end{enumerate}\nMoreover, in the presence of (1) property (2) is equivalent to property\n\\begin{enumerate}\n\\item[(2')] the sheaf property for $\\{V \\to U\\}$ as in (2) with $U$ affine.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBQ","source_file":"topologies.tex","source_line":2724,"source_end_line":2739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2724-L2739","statement_sha256":"0e2130d5c0801f1f71bf88b9634c3e7a0c736e677808abd8a60ad40fde5aea90","origin":"The Stacks Project","memory_eligible":false,"source_rank":6703,"rank":6703,"depth":0,"x":690.741,"y":1215.387,"cluster":"descent"},{"id":"stacks:0DBR","tag":"0DBR","title":"The ph topology · Lemma 0DBR","summary":"Let Sch_ph be a big ph site. Let f : T → S be a morphism in Sch_ph. The functor u : (Sch/T)_ph → (Sch/S)_ph, V/T ↦ V/S is cocontinuous, and has a continuous right adjoint v : (Sch/S)_ph → (Sch/T)_ph, (U → S) ↦ (U ×_S T → T). They induce the same morphism of topoi f_big : Sh((Sch/T)_ph) → Sh((Sch/S)_ph) We have f_big^-1(G)(U/T) = G(U/S). We have f_big, *(F)(U/S) = F(U ×_S T/T). Also, f_big^-1 has a left adjoint f_big! which commutes with fibre products and equalizers.","statement_latex":"Let $\\Sch_{ph}$ be a big ph site.\nLet $f : T \\to S$ be a morphism in $\\Sch_{ph}$.\nThe functor\n$$\nu : (\\Sch/T)_{ph} \\longrightarrow (\\Sch/S)_{ph},\n\\quad\nV/T \\longmapsto V/S\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv : (\\Sch/S)_{ph} \\longrightarrow (\\Sch/T)_{ph},\n\\quad\n(U \\to S) \\longmapsto (U \\times_S T \\to T).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\Sch/T)_{ph})\n\\longrightarrow\n\\Sh((\\Sch/S)_{ph})\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nWe have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBR","source_file":"topologies.tex","source_line":2793,"source_end_line":2820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2793-L2820","statement_sha256":"d2765daa8efcc84969792cf2d457064410ba8949df5f9ac747dc32dbdda2d2cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6704,"rank":6704,"depth":8,"x":611.067,"y":1090.224,"cluster":"descent"},{"id":"stacks:0DBS","tag":"0DBS","title":"The ph topology · Lemma 0DBS","summary":"Given schemes X, Y, Y in (Sch/S)_ph and morphisms f : X → Y, g : Y → Z we have g_big ∘ f_big = (g ∘ f)_big.","statement_latex":"Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_{ph}$\nand morphisms $f : X \\to Y$, $g : Y \\to Z$ we have\n$g_{big} \\circ f_{big} = (g \\circ f)_{big}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBS","source_file":"topologies.tex","source_line":2837,"source_end_line":2842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2837-L2842","statement_sha256":"a151ad71fb17067335bed98ef661fc2ef812516b48adc26669d96c6c54d0a41a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6705,"rank":6705,"depth":9,"x":771.368,"y":1137.576,"cluster":"descent"},{"id":"stacks:022B","tag":"022B","title":"The fpqc topology · Definition 022B","summary":"Let T be a scheme. An fpqc covering of T is a family of morphisms (f_i : T_i → T)_i ∈ I of schemes such that each f_i is flat and such that for every affine open U ⊂ T there exists n ≥ 0, a map a : (1, …, n) → I and affine opens V_j ⊂ T_a(j), j = 1, …, n with ⋃_j = 1^n f_a(j)(V_j) = U.","statement_latex":"Let $T$ be a scheme. An {\\it fpqc covering of $T$} is a family\nof morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes\nsuch that each $f_i$ is flat and such that for every affine open\n$U \\subset T$ there exists $n \\geq 0$, a map\n$a : \\{1, \\ldots, n\\} \\to I$ and affine opens\n$V_j \\subset T_{a(j)}$, $j = 1, \\ldots, n$\nwith $\\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022B","source_file":"topologies.tex","source_line":2887,"source_end_line":2896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2887-L2896","statement_sha256":"a94f9fbe6a434a33d2d4bb9ff1dbaae163013a5f84fd070630a270763bd4a511","origin":"The Stacks Project","memory_eligible":false,"source_rank":6706,"rank":6706,"depth":0,"x":614.213,"y":1193.93,"cluster":"descent"},{"id":"stacks:03L7","tag":"03L7","title":"The fpqc topology · Lemma 03L7","summary":"Let T be a scheme. Let (f_i : T_i → T)_i ∈ I be a family of morphisms of schemes with target T. The following are equivalent • (f_i : T_i → T)_i ∈ I is an fpqc covering, • each f_i is flat and for every affine open U ⊂ T there exist quasi-compact opens U_i ⊂ T_i which are almost all empty, such that U = ⋃ f_i(U_i), • each f_i is flat and there exists an affine open covering T = ⋃_α ∈ A U_α and for each α ∈ A there exist i_α, 1, …, i_α, n(α) ∈ I and quasi-compact opens…","statement_latex":"Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of\nmorphisms of schemes with target $T$. The following are equivalent\n\\begin{enumerate}\n\\item $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering,\n\\item each $f_i$ is flat and for every affine open $U \\subset T$\nthere exist quasi-compact opens\n$U_i \\subset T_i$ which are almost all empty,\nsuch that $U = \\bigcup f_i(U_i)$,\n\\item each $f_i$ is flat and there exists an affine open covering\n$T = \\bigcup_{\\alpha \\in A} U_\\alpha$ and for each $\\alpha \\in A$\nthere exist $i_{\\alpha, 1}, \\ldots, i_{\\alpha, n(\\alpha)} \\in I$\nand quasi-compact opens $U_{\\alpha, j} \\subset T_{i_{\\alpha, j}}$ such that\n$U_\\alpha =\n\\bigcup_{j = 1, \\ldots, n(\\alpha)} f_{i_{\\alpha, j}}(U_{\\alpha, j})$.\n\\end{enumerate}\nIf $T$ is quasi-separated, these are also equivalent to\n\\begin{enumerate}\n\\item[(4)] each $f_i$ is flat, and for every $t \\in T$ there exist\n$i_1, \\ldots, i_n \\in I$ and quasi-compact opens $U_j \\subset T_{i_j}$\nsuch that $\\bigcup_{j = 1, \\ldots, n} f_{i_j}(U_j)$ is a\n(not necessarily open) neighbourhood of $t$ in $T$.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03L7","source_file":"topologies.tex","source_line":2903,"source_end_line":2927,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2903-L2927","statement_sha256":"6ead52acc00e91eb9105cb624ade1b14fc5cb189383c197b6ec0b7311d504c6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6707,"rank":6707,"depth":18,"x":685.17,"y":1062.481,"cluster":"descent"},{"id":"stacks:040I","tag":"040I","title":"The fpqc topology · Lemma 040I","summary":"Let T be a scheme. Let (f_i : T_i → T)_i ∈ I be a family of morphisms of schemes with target T. The following are equivalent • (f_i : T_i → T)_i ∈ I is an fpqc covering, and • setting T' = coprod_i ∈ I T_i, and f = coprod_i ∈ I f_i the family (f : T' → T) is an fpqc covering.","statement_latex":"Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of\nmorphisms of schemes with target $T$. The following are equivalent\n\\begin{enumerate}\n\\item $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering, and\n\\item setting $T' = \\coprod_{i \\in I} T_i$, and $f = \\coprod_{i \\in I} f_i$\nthe family $\\{f : T' \\to T\\}$ is an fpqc covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040I","source_file":"topologies.tex","source_line":2969,"source_end_line":2978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2969-L2978","statement_sha256":"12ff2c346447f0702786a8f128f6301e0751cd40640bc3b957aa0581cce54357","origin":"The Stacks Project","memory_eligible":false,"source_rank":6708,"rank":6708,"depth":19,"x":738.844,"y":1200.421,"cluster":"descent"},{"id":"stacks:03L8","tag":"03L8","title":"The fpqc topology · Lemma 03L8","summary":"Let T be a scheme. Let (f_i : T_i → T)_i ∈ I be a family of morphisms of schemes with target T. Assume that • each f_i is flat, and • the family (f_i : T_i → T)_i ∈ I can be refined by an fpqc covering of T. Then (f_i : T_i → T)_i ∈ I is an fpqc covering of T.","statement_latex":"Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of\nmorphisms of schemes with target $T$. Assume that\n\\begin{enumerate}\n\\item each $f_i$ is flat, and\n\\item the family $\\{f_i : T_i \\to T\\}_{i \\in I}$ can be refined by an\nfpqc covering of $T$.\n\\end{enumerate}\nThen $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering of $T$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03L8","source_file":"topologies.tex","source_line":2993,"source_end_line":3003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L2993-L3003","statement_sha256":"32f004a367c6129530123ca769cd966b070569fcff3200ee7fa6a1bfd2e833b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6709,"rank":6709,"depth":19,"x":587.526,"y":1128.776,"cluster":"descent"},{"id":"stacks:03L9","tag":"03L9","title":"The fpqc topology · Lemma 03L9","summary":"Let T be a scheme. Let (f_i : T_i → T)_i ∈ I be a family of morphisms of schemes with target T. Assume that • each f_i is flat, and • there exists an fpqc covering (g_j : S_j → T)_j ∈ J such that each (S_j ×_T T_i → S_j)_i ∈ I is an fpqc covering. Then (f_i : T_i → T)_i ∈ I is an fpqc covering of T.","statement_latex":"Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of\nmorphisms of schemes with target $T$. Assume that\n\\begin{enumerate}\n\\item each $f_i$ is flat, and\n\\item there exists an fpqc covering\n$\\{g_j : S_j \\to T\\}_{j \\in J}$ such that each\n$\\{S_j \\times_T T_i \\to S_j\\}_{i \\in I}$ is an fpqc covering.\n\\end{enumerate}\nThen $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering of $T$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03L9","source_file":"topologies.tex","source_line":3018,"source_end_line":3029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3018-L3029","statement_sha256":"c8311a9053af09fc38f8b3792e9e51c31e1ab80b677d836ad0a63f5c9ac77a27","origin":"The Stacks Project","memory_eligible":false,"source_rank":6710,"rank":6710,"depth":20,"x":757.625,"y":1095.57,"cluster":"descent"},{"id":"stacks:022C","tag":"022C","title":"The fpqc topology · Lemma 022C","summary":"Any fppf covering is an fpqc covering, and a fortiori, any syntomic, smooth, étale or Zariski covering is an fpqc covering.","statement_latex":"Any fppf covering is an fpqc covering, and a fortiori,\nany syntomic, smooth, \\'etale or Zariski covering is an fpqc covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022C","source_file":"topologies.tex","source_line":3043,"source_end_line":3047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3043-L3047","statement_sha256":"e0f0782f45db2d061347be14fe60e7c0ae1e725e3ff285df26deb4e369189698","origin":"The Stacks Project","memory_eligible":false,"source_rank":6711,"rank":6711,"depth":38,"x":658.377,"y":1217.212,"cluster":"descent"},{"id":"stacks:022D","tag":"022D","title":"The fpqc topology · Lemma 022D","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is an fpqc covering of T. • If (T_i → T)_i∈ I is an fpqc covering and for each i we have an fpqc covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is an fpqc covering. • If (T_i → T)_i∈ I is an fpqc covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is an fpqc covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis an fpqc covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering and for each\n$i$ we have an fpqc covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an fpqc covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an fpqc covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022D","source_file":"topologies.tex","source_line":3104,"source_end_line":3117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3104-L3117","statement_sha256":"141ba5bac469ba348a55ac3dd923614a389d7e9edd4524fc07889cb7adb1c690","origin":"The Stacks Project","memory_eligible":false,"source_rank":6712,"rank":6712,"depth":19,"x":633.613,"y":1070.435,"cluster":"descent"},{"id":"stacks:022E","tag":"022E","title":"The fpqc topology · Lemma 022E","summary":"Let T be an affine scheme. Let (T_i → T)_i ∈ I be an fpqc covering of T. Then there exists an fpqc covering (U_j → T)_j = 1, …, n which is a refinement of (T_i → T)_i ∈ I such that each U_j is an affine scheme. Moreover, we may choose each U_j to be open affine in one of the T_i.","statement_latex":"Let $T$ be an affine scheme.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an fpqc covering of $T$.\nThen there exists an fpqc covering\n$\\{U_j \\to T\\}_{j = 1, \\ldots, n}$ which is a refinement\nof $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine\nscheme. Moreover, we may choose each $U_j$ to be open affine\nin one of the $T_i$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022E","source_file":"topologies.tex","source_line":3150,"source_end_line":3159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3150-L3159","statement_sha256":"a27ea250d23848d0f9c87dbe334e80c29d722829dc22b89055dadb3384e6cd7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6713,"rank":6713,"depth":0,"x":770.611,"y":1165.103,"cluster":"descent"},{"id":"stacks:022F","tag":"022F","title":"The fpqc topology · Definition 022F","summary":"Let T be an affine scheme. A standard fpqc covering of T is a family (f_j : U_j → T)_j = 1, …, n with each U_j is affine, flat over T and T = ⋃ f_j(U_j).","statement_latex":"Let $T$ be an affine scheme. A {\\it standard fpqc covering}\nof $T$ is a family $\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, n}$\nwith each $U_j$ is affine, flat over $T$ and $T = \\bigcup f_j(U_j)$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022F","source_file":"topologies.tex","source_line":3165,"source_end_line":3170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3165-L3170","statement_sha256":"d38c70c8ee45e2120f022bcb6b6d3ab2f0a402f7ac80d42c2ccea00a8dc5b4ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":6714,"rank":6714,"depth":0,"x":592.554,"y":1173.072,"cluster":"descent"},{"id":"stacks:03LA","tag":"03LA","title":"The fpqc topology · Lemma 03LA","summary":"Let T be an affine scheme. • If T' → T is an isomorphism then (T' → T) is a standard fpqc covering of T. • If (T_i → T)_i∈ I is a standard fpqc covering and for each i we have a standard fpqc covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is a standard fpqc covering. • If (T_i → T)_i∈ I is a standard fpqc covering and T' → T is a morphism of affine schemes then (T' ×_T T_i → T')_i∈ I is a standard fpqc covering.","statement_latex":"Let $T$ be an affine scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis a standard fpqc covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a standard fpqc covering and for each\n$i$ we have a standard fpqc covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a standard fpqc covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a standard fpqc covering\nand $T' \\to T$ is a morphism of affine schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a standard fpqc covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LA","source_file":"topologies.tex","source_line":3177,"source_end_line":3190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3177-L3190","statement_sha256":"19b8314d7b4125371851767fee2622f49f7f21a7f4f1baa0f0f28bb533b1df6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6715,"rank":6715,"depth":13,"x":718.076,"y":1065.621,"cluster":"descent"},{"id":"stacks:03LB","tag":"03LB","title":"The fpqc topology · Lemma 03LB","summary":"Let T be a scheme. Let (f_i : T_i → T)_i ∈ I be a family of morphisms of schemes with target T. Assume that • each f_i is flat, and • every affine scheme Z and morphism h : Z → T there exists a standard fpqc covering (Z_j → Z)_j = 1, …, n which refines the family (T_i ×_T Z → Z)_i ∈ I. Then (f_i : T_i → T)_i ∈ I is an fpqc covering of T.","statement_latex":"Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of\nmorphisms of schemes with target $T$. Assume that\n\\begin{enumerate}\n\\item each $f_i$ is flat, and\n\\item every affine scheme\n$Z$ and morphism $h : Z \\to T$ there exists a standard fpqc covering\n$\\{Z_j \\to Z\\}_{j = 1, \\ldots, n}$ which refines the family\n$\\{T_i \\times_T Z \\to Z\\}_{i \\in I}$.\n\\end{enumerate}\nThen $\\{f_i : T_i \\to T\\}_{i \\in I}$ is an fpqc covering of $T$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LB","source_file":"topologies.tex","source_line":3201,"source_end_line":3213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3201-L3213","statement_sha256":"4976411f610c1790328b0b9a5e55dcdf476c5acc50916d941f9d2e9610d617f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6716,"rank":6716,"depth":21,"x":711.895,"y":1216.833,"cluster":"descent"},{"id":"stacks:022G","tag":"022G","title":"The fpqc topology · Definition 022G","summary":"Let F be a contravariant functor on the category of schemes with values in sets. • Let (U_i → T)_i ∈ I be a family of morphisms of schemes with fixed target. We say that F satisfies the sheaf property for the given family if for any collection of elements xi_i ∈ F(U_i) such that xi_i|_U_i ×_T U_j = xi_j|_U_i ×_T U_j there exists a unique element xi ∈ F(T) such that xi_i = xi|_U_i in F(U_i). • We say that F satisfies the sheaf property for the fpqc topology if it satisfies…","statement_latex":"Let $F$ be a contravariant functor on the category\nof schemes with values in sets.\n\\begin{enumerate}\n\\item Let $\\{U_i \\to T\\}_{i \\in I}$ be a family of morphisms\nof schemes with fixed target.\nWe say that $F$ {\\it satisfies the sheaf property for the given family}\nif for any collection of elements $\\xi_i \\in F(U_i)$ such that\n$\\xi_i|_{U_i \\times_T U_j} = \\xi_j|_{U_i \\times_T U_j}$\nthere exists a unique element\n$\\xi \\in F(T)$ such that $\\xi_i = \\xi|_{U_i}$ in $F(U_i)$.\n\\item We say that $F$ {\\it satisfies the sheaf property for the\nfpqc topology} if it satisfies the sheaf property for any\nfpqc covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022G","source_file":"topologies.tex","source_line":3226,"source_end_line":3242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3226-L3242","statement_sha256":"e859bcf77c7d26406c22de9ad4fd49f8edd959463c4a51d5d7f2b96cead49a0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6717,"rank":6717,"depth":0,"x":594.273,"y":1101.252,"cluster":"descent"},{"id":"stacks:022H","tag":"022H","title":"The fpqc topology · Lemma 022H","summary":"Let F be a contravariant functor on the category of schemes with values in sets. Then F satisfies the sheaf property for the fpqc topology if and only if it satisfies • the sheaf property for every Zariski covering, and • the sheaf property for any standard fpqc covering. Moreover, in the presence of (1) property (2) is equivalent to property • [(2')] the sheaf property for (V → U) with V, U affine and V → U faithfully flat.","statement_latex":"Let $F$ be a contravariant functor on the category\nof schemes with values in sets. Then $F$ satisfies\nthe sheaf property for the fpqc topology if and only\nif it satisfies\n\\begin{enumerate}\n\\item the sheaf property for every Zariski covering, and\n\\item the sheaf property for any standard fpqc covering.\n\\end{enumerate}\nMoreover, in the presence of (1) property (2) is equivalent to\nproperty\n\\begin{enumerate}\n\\item[(2')] the sheaf property for $\\{V \\to U\\}$\nwith $V$, $U$ affine and $V \\to U$ faithfully flat.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022H","source_file":"topologies.tex","source_line":3249,"source_end_line":3265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3249-L3265","statement_sha256":"ca4f1aef669b5506fadbc53513d64ac195b71acf08af1f2577022f22c122d93c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6718,"rank":6718,"depth":1,"x":774.835,"y":1119.831,"cluster":"descent"},{"id":"stacks:0BBK","tag":"0BBK","title":"The fpqc topology · Lemma 0BBK","summary":"Let R be a nonzero ring. There does not exist a set A of fpqc-coverings of Spec(R) such that every fpqc-covering can be refined by an element of A.","statement_latex":"Let $R$ be a nonzero ring. There does not exist a set $A$ of\nfpqc-coverings of $\\Spec(R)$ such that every fpqc-covering can\nbe refined by an element of $A$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBK","source_file":"topologies.tex","source_line":3313,"source_end_line":3318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3313-L3318","statement_sha256":"cc89668727cb9cf690bba1cfc0320e0f080bd2c8af627a1551cfe1cc2fa14bd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6719,"rank":6719,"depth":0,"x":626.029,"y":1209.017,"cluster":"descent"},{"id":"stacks:0ETB","tag":"0ETB","title":"The V topology · Definition 0ETB","summary":"Let T be an affine scheme. A standard V covering is a finite family (T_j → T)_j = 1, …, m with T_j affine such that for every morphism g : Spec(V) → T where V is a valuation ring, there is an extension V ⊂ W of valuation rings (More on Algebra, Definition [Tag 0ASG]), an index 1 ≤ j ≤ m, and a commutative diagram xymatrix Spec(W) ar[r] ar[d] & T_j ar[d] Spec(V) ar[r]^g & T","statement_latex":"Let $T$ be an affine scheme. A {\\it standard V covering} is a finite family\n$\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ with $T_j$ affine\nsuch that for every morphism $g : \\Spec(V) \\to T$ where $V$\nis a valuation ring, there is an extension $V \\subset W$ of valuation rings\n(More on Algebra, Definition\n\\ref{more-algebra-definition-extension-valuation-rings}),\nan index $1 \\leq j \\leq m$, and a commutative diagram\n$$\n\\xymatrix{\n\\Spec(W) \\ar[r] \\ar[d] & T_j \\ar[d] \\\\\n\\Spec(V) \\ar[r]^g & T\n}\n$$","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETB","source_file":"topologies.tex","source_line":3388,"source_end_line":3403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3388-L3403","statement_sha256":"537c85e0093426e30c0b2d91fa2cd08655511bf2482ec155e026b4c6835e4bbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":6720,"rank":6720,"depth":1,"x":664.22,"y":1058.092,"cluster":"descent"},{"id":"stacks:0ETC","tag":"0ETC","title":"The V topology · Lemma 0ETC","summary":"A standard fpqc covering is a standard V covering.","statement_latex":"A standard fpqc covering is a standard V covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETC","source_file":"topologies.tex","source_line":3408,"source_end_line":3411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3408-L3411","statement_sha256":"4a37a3209af5bbfa875a4282a6a7c7f3e013de050d4474b79775936aa08ec642","origin":"The Stacks Project","memory_eligible":false,"source_rank":6721,"rank":6721,"depth":13,"x":757.872,"y":1191.688,"cluster":"descent"},{"id":"stacks:0ETD","tag":"0ETD","title":"The V topology · Lemma 0ETD","summary":"A standard ph covering is a standard V covering.","statement_latex":"A standard ph covering is a standard V covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETD","source_file":"topologies.tex","source_line":3441,"source_end_line":3444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3441-L3444","statement_sha256":"e0ece91e8df2994e63ed4477191da54175f34314e81c31f260218f9fd7c1c7e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6722,"rank":6722,"depth":17,"x":580.546,"y":1146.101,"cluster":"descent"},{"id":"stacks:0ETE","tag":"0ETE","title":"The V topology · Lemma 0ETE","summary":"Let (T_j → T)_j = 1, …, m be a standard V covering. Let T' → T be a morphism of affine schemes. Then (T_j ×_T T' → T')_j = 1, …, m is a standard V covering.","statement_latex":"Let $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ be a standard V covering.\nLet $T' \\to T$ be a morphism of affine schemes.\nThen $\\{T_j \\times_T T' \\to T'\\}_{j = 1, \\ldots, m}$ is a\nstandard V covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETE","source_file":"topologies.tex","source_line":3478,"source_end_line":3484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3478-L3484","statement_sha256":"ccd7da87370416d7352d50f8a6e94e09bbd31661792331624316e47db29c556e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6723,"rank":6723,"depth":0,"x":748.748,"y":1078.785,"cluster":"descent"},{"id":"stacks:0ETF","tag":"0ETF","title":"The V topology · Lemma 0ETF","summary":"Let T be an affine scheme. Let (T_j → T)_j = 1, …, m be a standard V covering. Let (T_ji → T_j)_i = 1, … n_j be a standard V covering. Then (T_ji → T)_i, j is a standard V covering.","statement_latex":"Let $T$ be an affine scheme. Let $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$\nbe a standard V covering. Let $\\{T_{ji} \\to T_j\\}_{i = 1, \\ldots n_j}$\nbe a standard V covering. Then $\\{T_{ji} \\to T\\}_{i, j}$\nis a standard V covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETF","source_file":"topologies.tex","source_line":3501,"source_end_line":3507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3501-L3507","statement_sha256":"47b02602495dccfe287baecfebfb6915b630119324249f13098e40f59fc1499e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6724,"rank":6724,"depth":0,"x":678.527,"y":1224.537,"cluster":"descent"},{"id":"stacks:0ETG","tag":"0ETG","title":"The V topology · Lemma 0ETG","summary":"Let T be an affine scheme. Let (T_j → T)_j = 1, …, m be a family of morphisms with T_j affine for all j. The following are equivalent • (T_j → T)_j = 1, …, m is a standard V covering, • there is a standard V covering which refines (T_j → T)_j = 1, …, m, and • (coprod_j = 1, …, m T_j → T) is a standard V covering.","statement_latex":"Let $T$ be an affine scheme. Let $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$\nbe a family of morphisms with $T_j$ affine for all $j$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\{T_j \\to T\\}_{j = 1, \\ldots, m}$ is a standard V covering,\n\\item there is a standard V covering which refines\n$\\{T_j \\to T\\}_{j = 1, \\ldots, m}$, and\n\\item $\\{\\coprod_{j = 1, \\ldots, m} T_j \\to T\\}$ is a standard\nV covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETG","source_file":"topologies.tex","source_line":3516,"source_end_line":3528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3516-L3528","statement_sha256":"62221e3cf9ae693817e1ef5fdcc8929eaf5adce7b82504ad03350f60e883c498","origin":"The Stacks Project","memory_eligible":false,"source_rank":6725,"rank":6725,"depth":0,"x":612.797,"y":1076.539,"cluster":"descent"},{"id":"stacks:0ETH","tag":"0ETH","title":"The V topology · Definition 0ETH","summary":"Let T be a scheme. A V covering of T is a family of morphisms (T_i → T)_i ∈ I of schemes such that for every affine open U ⊂ T there exists a standard V covering (U_j → U)_j = 1, …, m refining the family (T_i ×_T U → U)_i ∈ I.","statement_latex":"Let $T$ be a scheme. A {\\it V covering of $T$} is a family\nof morphisms $\\{T_i \\to T\\}_{i \\in I}$ of schemes such that for every\naffine open $U \\subset T$ there exists a standard V covering\n$\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ refining the family\n$\\{T_i \\times_T U \\to U\\}_{i \\in I}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETH","source_file":"topologies.tex","source_line":3537,"source_end_line":3544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3537-L3544","statement_sha256":"a635ff1861c63c82988327299f71f843ca1b8990fb6768d04a1c18a67ec957db","origin":"The Stacks Project","memory_eligible":false,"source_rank":6726,"rank":6726,"depth":0,"x":781.044,"y":1148.702,"cluster":"descent"},{"id":"stacks:0ETI","tag":"0ETI","title":"The V topology · Lemma 0ETI","summary":"Let T be a scheme. Let (f_i : T_i → T)_i ∈ I be a family of morphisms. The following are equivalent • (T_i → T)_i ∈ I is a V covering, • there is a V covering which refines (T_i → T)_i ∈ I, and • (coprod_i ∈ I T_i → T) is a V covering.","statement_latex":"Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family\nof morphisms. The following are equivalent\n\\begin{enumerate}\n\\item $\\{T_i \\to T\\}_{i \\in I}$ is a V covering,\n\\item there is a V covering which refines $\\{T_i \\to T\\}_{i \\in I}$, and\n\\item $\\{\\coprod_{i \\in I} T_i \\to T\\}$ is a V covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETI","source_file":"topologies.tex","source_line":3557,"source_end_line":3566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3557-L3566","statement_sha256":"19afe68e12dff9da9a0a794016f04b1cb7c3a69942a220988a379053c094d46a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6727,"rank":6727,"depth":2,"x":598.127,"y":1191.147,"cluster":"descent"},{"id":"stacks:0ETJ","tag":"0ETJ","title":"The V topology · Lemma 0ETJ","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is a V covering of T. • If (T_i → T)_i∈ I is a V covering and for each i we have a V covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is a V covering. • If (T_i → T)_i∈ I is a V covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is a V covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis a V covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a V covering and for each\n$i$ we have a V covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a V covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a V covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a V covering.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETJ","source_file":"topologies.tex","source_line":3573,"source_end_line":3586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3573-L3586","statement_sha256":"e9a7fdb2ba018227073287424b5f20b17a0619692f91549eb6a253a81c29bfcf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6728,"rank":6728,"depth":14,"x":699.327,"y":1055.454,"cluster":"descent"},{"id":"stacks:0ETK","tag":"0ETK","title":"The V topology · Lemma 0ETK","summary":"Any fpqc covering is a V covering. A fortiori, any fppf, syntomic, smooth, étale or Zariski covering is a V covering. Also, a ph covering is a V covering.","statement_latex":"Any fpqc covering is a V covering. A fortiori,\nany fppf, syntomic, smooth, \\'etale or Zariski covering is a V covering.\nAlso, a ph covering is a V covering.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETK","source_file":"topologies.tex","source_line":3628,"source_end_line":3633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3628-L3633","statement_sha256":"35cf4cd2d33858c049057f4569ea5722b9e6f2a7446557f83389badd30c3d460","origin":"The Stacks Project","memory_eligible":false,"source_rank":6729,"rank":6729,"depth":39,"x":733.976,"y":1213.633,"cluster":"descent"},{"id":"stacks:0ETL","tag":"0ETL","title":"The V topology · Definition 0ETL","summary":"Let F be a contravariant functor on the category of schemes with values in sets. We say that F satisfies the sheaf property for the V topology if it satisfies the sheaf property for any V covering (see Definition [Tag 022G]).","statement_latex":"Let $F$ be a contravariant functor on the category\nof schemes with values in sets. We say that\n$F$ {\\it satisfies the sheaf property for the V topology}\nif it satisfies the sheaf property for any V covering\n(see Definition \\ref{definition-sheaf-property-fpqc}).","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETL","source_file":"topologies.tex","source_line":3650,"source_end_line":3657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3650-L3657","statement_sha256":"0b5be2ee87869450f3fff677d8a19d9720ab19539b0e506ffbc3a6b21c62a33b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6730,"rank":6730,"depth":1,"x":580.552,"y":1116.228,"cluster":"descent"},{"id":"stacks:0ETM","tag":"0ETM","title":"The V topology · Lemma 0ETM","summary":"Let F be a contravariant functor on the category of schemes with values in sets. Then F satisfies the sheaf property for the V topology if and only if it satisfies • the sheaf property for every Zariski covering, and • the sheaf property for any standard V covering. Moreover, in the presence of (1) property (2) is equivalent to property • [(2')] the sheaf property for a standard V covering of the form (V → U), i.e., consisting of a single arrow.","statement_latex":"Let $F$ be a contravariant functor on the category\nof schemes with values in sets. Then $F$ satisfies\nthe sheaf property for the V topology if and only\nif it satisfies\n\\begin{enumerate}\n\\item the sheaf property for every Zariski covering, and\n\\item the sheaf property for any standard V covering.\n\\end{enumerate}\nMoreover, in the presence of (1) property (2) is equivalent to\nproperty\n\\begin{enumerate}\n\\item[(2')] the sheaf property for a standard V covering\nof the form $\\{V \\to U\\}$, i.e., consisting of a single arrow.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETM","source_file":"topologies.tex","source_line":3664,"source_end_line":3680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3664-L3680","statement_sha256":"c7823c1db154adb709effd1dbb452b02ac763dd1b02313c60240982ed7d8ec50","origin":"The Stacks Project","memory_eligible":false,"source_rank":6731,"rank":6731,"depth":1,"x":772.849,"y":1100.932,"cluster":"descent"},{"id":"stacks:0ETN","tag":"0ETN","title":"The V topology · Lemma 0ETN","summary":"Let X → Y be a quasi-compact morphism of schemes. The following are equivalent • (X → Y) is a V covering, • for any valuation ring V and morphism g : Spec(V) → Y there exists an extension of valuation rings V ⊂ W and a commutative diagram xymatrix Spec(W) ar[r] ar[d] & X ar[d] Spec(V) ar[r] & Y • for any morphism Z → Y and specialization z' leadsto z of points in Z, there is a specialization w' leadsto w of points in Z ×_Y X mapping to z' leadsto z.","statement_latex":"Let $X \\to Y$ be a quasi-compact morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\{X \\to Y\\}$ is a V covering,\n\\item for any valuation ring $V$ and morphism $g : \\Spec(V) \\to Y$\nthere exists an extension of valuation rings $V \\subset W$\nand a commutative diagram\n$$\n\\xymatrix{\n\\Spec(W) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(V) \\ar[r] & Y\n}\n$$\n\\item for any morphism $Z \\to Y$ and specialization $z' \\leadsto z$\nof points in $Z$, there is a specialization $w' \\leadsto w$\nof points in $Z \\times_Y X$ mapping to $z' \\leadsto z$.\n\\end{enumerate}","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETN","source_file":"topologies.tex","source_line":3730,"source_end_line":3749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3730-L3749","statement_sha256":"eb04cc0b1df85f16e9b1fa0a08dfbaf4b43313b2887a61ea0dfeefd37ec565ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":6732,"rank":6732,"depth":13,"x":642.793,"y":1221.843,"cluster":"descent"},{"id":"stacks:0ETP","tag":"0ETP","title":"The V topology · Lemma 0ETP","summary":"Let (f_i : X_i → X)_i ∈ I be a V covering. Then coprod_i ∈ I f_i : coprod_i ∈ I X_i → X is a universally submersive morphism of schemes (Morphisms, Definition [Tag 040H]).","statement_latex":"Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a V covering.\nThen\n$$\n\\coprod\\nolimits_{i \\in I} f_i :\n\\coprod\\nolimits_{i \\in I} X_i\n\\longrightarrow\nX\n$$\nis a universally submersive morphism of schemes (Morphisms, Definition\n\\ref{morphisms-definition-submersive}).","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"The V topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETP","source_file":"topologies.tex","source_line":3828,"source_end_line":3840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L3828-L3840","statement_sha256":"7b3936670328ec28067c97b3b1ef1eb03bb200fb03257b86873f3b47286a4b95","origin":"The Stacks Project","memory_eligible":false,"source_rank":6733,"rank":6733,"depth":15,"x":641.456,"y":1058.191,"cluster":"descent"},{"id":"stacks:022J","tag":"022J","title":"Change of big sites · Lemma 022J","summary":"Any set of big Zariski sites is contained in a common big Zariski site. The same is true, mutatis mutandis, for big fppf and big étale sites.","statement_latex":"Any set of big Zariski sites is contained in a common big Zariski site.\nThe same is true, mutatis mutandis, for big fppf and big \\'etale sites.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"Change of big sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022J","source_file":"topologies.tex","source_line":4003,"source_end_line":4007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L4003-L4007","statement_sha256":"815aa09784a3148b81a4bee4b177ed68c44ba28dfe06f96d5a94084c3374e616","origin":"The Stacks Project","memory_eligible":false,"source_rank":6734,"rank":6734,"depth":2,"x":774.608,"y":1178.617,"cluster":"descent"},{"id":"stacks:022K","tag":"022K","title":"Change of big sites · Lemma 022K","summary":"Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Suppose given big sites Sch_τ and Sch'_τ. Assume that Sch_τ is contained in Sch'_τ. The inclusion functor Sch_τ → Sch'_τ satisfies the assumptions of Sites, Lemma [Tag 00XU]. There are morphisms of topoi g : Sh(Sch_τ) & → & Sh(Sch'_τ) f : Sh(Sch'_τ) & → & Sh(Sch_τ) such that f ∘ g ≅ id. For any object S of Sch_τ the inclusion functor (Sch/S)_τ → (Sch'/S)_τ satisfies the assumptions of Sites, Lemma [Tag 00XU] also. Hence…","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nSuppose given big sites $\\Sch_\\tau$ and $\\Sch'_\\tau$.\nAssume that $\\Sch_\\tau$ is contained in $\\Sch'_\\tau$.\nThe inclusion functor $\\Sch_\\tau \\to \\Sch'_\\tau$ satisfies\nthe assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}.\nThere are morphisms of topoi\n\\begin{eqnarray*}\ng : \\Sh(\\Sch_\\tau) &\n\\longrightarrow &\n\\Sh(\\Sch'_\\tau) \\\\\nf : \\Sh(\\Sch'_\\tau) &\n\\longrightarrow &\n\\Sh(\\Sch_\\tau)\n\\end{eqnarray*}\nsuch that $f \\circ g \\cong \\text{id}$. For any object $S$\nof $\\Sch_\\tau$ the inclusion functor\n$(\\Sch/S)_\\tau \\to (\\Sch'/S)_\\tau$ satisfies\nthe assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}\nalso. Hence similarly we obtain morphisms\n\\begin{eqnarray*}\ng : \\Sh((\\Sch/S)_\\tau) &\n\\longrightarrow &\n\\Sh((\\Sch'/S)_\\tau) \\\\\nf : \\Sh((\\Sch'/S)_\\tau) &\n\\longrightarrow &\n\\Sh((\\Sch/S)_\\tau)\n\\end{eqnarray*}\nwith $f \\circ g \\cong \\text{id}$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"Change of big sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022K","source_file":"topologies.tex","source_line":4017,"source_end_line":4047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L4017-L4047","statement_sha256":"ae8d0cff9ea6ff74b3cd931faa182fa75767704e8be99286b6b9d6ab4660a13f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6735,"rank":6735,"depth":19,"x":578.76,"y":1165.313,"cluster":"descent"},{"id":"stacks:0EUW","tag":"0EUW","title":"Extending functors · Lemma 0EUW","summary":"Let S be a scheme. Let C be a full subcategory of the category Sch/S of all schemes over S. Assume • if X → S is an object of C and U ⊂ X is an affine open, then U → S is isomorphic to an object of C, • if V is an affine scheme lying over an affine open U ⊂ S such that V → U is of finite presentation, then V → S is isomorphic to an object of C. Let F : C^opp → Sets be a functor. Assume • [(a)] for any Zariski covering (f_i : X_i → X)_i ∈ I with X, X_i objects of C we have…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{C}$ be a full subcategory\nof the category $\\Sch/S$ of all schemes over $S$. Assume\n\\begin{enumerate}\n\\item if $X \\to S$ is an object of $\\mathcal{C}$ and\n$U \\subset X$ is an affine open, then $U \\to S$ is isomorphic\nto an object of $\\mathcal{C}$,\n\\item if $V$ is an affine scheme lying over an affine open $U \\subset S$\nsuch that $V \\to U$ is of finite presentation, then $V \\to S$ is isomorphic\nto an object of $\\mathcal{C}$.\n\\end{enumerate}\nLet $F : \\mathcal{C}^{opp} \\to \\textit{Sets}$ be a functor.\nAssume\n\\begin{enumerate}\n\\item[(a)] for any Zariski covering $\\{f_i : X_i \\to X\\}_{i \\in I}$\nwith $X, X_i$ objects of $\\mathcal{C}$ we have\nthe sheaf condition for $F$ and this family\\footnote{As we\ndo not know that $X_i \\times_X X_j$ is in $\\mathcal{C}$\nthis has to be interpreted as follows: by property (1)\nthere exist Zariski coverings $\\{U_{ijk} \\to X_i \\times_X X_j\\}_{k \\in K_{ij}}$\nwith $U_{ijk}$ an object of $\\mathcal{C}$. Then the sheaf condition\nsays that $F(X)$ is the equalizer of the two maps\nfrom $\\prod F(X_i)$ to $\\prod F(U_{ijk})$.},\n\\item[(b)] if $X = \\lim X_i$ is a directed limit of affine schemes\nover $S$ with $X, X_i$ objects of $\\mathcal{C}$, then\n$F(X) = \\colim F(X_i)$.\n\\end{enumerate}\nThen there is a unique way to extend $F$ to a functor\n$F' : (\\Sch/S)^{opp} \\to \\textit{Sets}$ satisfying the analogues of\n(a) and (b), i.e., $F'$ satisfies the sheaf condition for any\nZariski covering and $F'(X) = \\colim F'(X_i)$ whenever $X = \\lim X_i$\nis a directed limit of affine schemes over $S$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"Extending functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUW","source_file":"topologies.tex","source_line":4108,"source_end_line":4141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L4108-L4141","statement_sha256":"6f4ba4a613c09a9aace17802861ff608669dfbdf44c0ced08e14c50e1ef02795","origin":"The Stacks Project","memory_eligible":false,"source_rank":6736,"rank":6736,"depth":26,"x":734.522,"y":1063.572,"cluster":"descent"},{"id":"stacks:049N","tag":"049N","title":"Extending functors · Lemma 049N","summary":"Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Let T be an affine scheme which is written as a limit T = lim_i ∈ I T_i of a directed inverse system of affine schemes. • Let V = (V_j → T)_j = 1, …, m be a standard τ-covering of T, see Definitions [Tag 020R], [Tag 0219], [Tag 0223], [Tag 0229], and [Tag 021Q]. Then there exists an index i and a standard τ-covering V_i = (V_i, j → T_i)_j = 1, …, m whose base change T ×_T_i V_i to T is isomorphic to V. • Let V_i, V'_i be a…","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nLet $T$ be an affine scheme which is written as a limit\n$T = \\lim_{i \\in I} T_i$ of a directed inverse system of affine schemes.\n\\begin{enumerate}\n\\item Let $\\mathcal{V} = \\{V_j \\to T\\}_{j = 1, \\ldots, m}$ be a\nstandard $\\tau$-covering of $T$, see Definitions\n\\ref{definition-standard-Zariski},\n\\ref{definition-standard-etale},\n\\ref{definition-standard-smooth},\n\\ref{definition-standard-syntomic}, and\n\\ref{definition-standard-fppf}.\nThen there exists an index $i$ and a standard $\\tau$-covering\n$\\mathcal{V}_i = \\{V_{i, j} \\to T_i\\}_{j = 1, \\ldots, m}$\nwhose base change $T \\times_{T_i} \\mathcal{V}_i$ to $T$\nis isomorphic to $\\mathcal{V}$.\n\\item Let $\\mathcal{V}_i$, $\\mathcal{V}'_i$ be a pair of standard\n$\\tau$-coverings of $T_i$. If\n$f : T \\times_{T_i} \\mathcal{V}_i \\to T \\times_{T_i} \\mathcal{V}'_i$ is\na morphism of coverings of $T$, then there exists an index\n$i' \\geq i$ and a morphism\n$f_{i'} : T_{i'} \\times_{T_i} \\mathcal{V} \\to\nT_{i'} \\times_{T_i} \\mathcal{V}'_i$\nwhose base change to $T$ is $f$.\n\\item If\n$f, g : \\mathcal{V} \\to \\mathcal{V}'_i$\nare morphisms of standard $\\tau$-coverings of $T_i$ whose\nbase changes $f_T, g_T$ to $T$ are equal then there exists an\nindex $i' \\geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$.\n\\end{enumerate}\nIn other words, the category of standard $\\tau$-coverings of $T$ is\nthe colimit over $I$ of the categories of standard $\\tau$-coverings of $T_i$.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"Extending functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049N","source_file":"topologies.tex","source_line":4291,"source_end_line":4324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L4291-L4324","statement_sha256":"a9f4f26ce389c96b1ce04b2ff265f547572f6ea0429091d5bf955f3d5c280750","origin":"The Stacks Project","memory_eligible":false,"source_rank":6737,"rank":6737,"depth":42,"x":701.345,"y":1227.653,"cluster":"descent"},{"id":"stacks:0GDW","tag":"0GDW","title":"Extending functors · Lemma 0GDW","summary":"Let S, C, F satisfy conditions (1), (2), (a), and (b) of Lemma [Tag 0EUW] and denote F' : (Sch/S)^opp → Sets the unique extension constructed in the lemma. Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Assume • [(c)] for any standard τ-covering (V_i → V)_i = 1, …, n of affines in Sch/S such that V → S factors through an affine open U ⊂ S and V → U is of finite presentation, the sheaf condition hold for F and (V_i → V)_i = 1, …, n by (2).. Then F' satisfies the sheaf…","statement_latex":"Let $S$, $\\mathcal{C}$, $F$ satisfy conditions (1), (2), (a), and (b) of\nLemma \\ref{lemma-extend} and denote $F' : (\\Sch/S)^{opp} \\to \\textit{Sets}$\nthe unique extension constructed in the lemma. Let\n$\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. Assume\n\\begin{enumerate}\n\\item[(c)] for any standard $\\tau$-covering $\\{V_i \\to V\\}_{i = 1, \\ldots, n}$\nof affines in $\\Sch/S$ such that $V \\to S$ factors through an affine open\n$U \\subset S$ and $V \\to U$ is of finite presentation, the sheaf condition\nhold for $F$ and $\\{V_i \\to V\\}_{i = 1, \\ldots, n}$\\footnote{This makes\nsense as $V$, $V_i$, and $V_i \\times_V V_j$ are isomorphic to objects\nof $\\mathcal{C}$ by (2).}.\n\\end{enumerate}\nThen $F'$ satisfies the sheaf condition for all $\\tau$-coverings.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"Extending functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDW","source_file":"topologies.tex","source_line":4366,"source_end_line":4381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L4366-L4381","statement_sha256":"9970ef4ebae23f2ced211b4e5a1442f04a99ca55d319f1e337f3e38e6e74406c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6738,"rank":6738,"depth":43,"x":593.42,"y":1087.263,"cluster":"descent"},{"id":"stacks:0EUX","tag":"0EUX","title":"Extending functors · Lemma 0EUX","summary":"Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Let S be a scheme contained in a big site Sch_τ. Let F : (Sch/S)_τ^opp → Sets be a τ-sheaf satisfying property (b) of Lemma [Tag 0EUW] with C = (Sch/S)_τ. Then the extension F' of F to the category of all schemes over S satisfies the sheaf condition for all τ-coverings.","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nLet $S$ be a scheme contained in a big site $\\Sch_\\tau$.\nLet $F : (\\Sch/S)_\\tau^{opp} \\to \\textit{Sets}$ be a $\\tau$-sheaf\nsatisfying property (b) of Lemma \\ref{lemma-extend} with\n$\\mathcal{C} = (\\Sch/S)_\\tau$. Then the extension\n$F'$ of $F$ to the category of all schemes over $S$\nsatisfies the sheaf condition for all $\\tau$-coverings.","area":"Descent","chapter":"Topologies on Schemes","chapter_id":"topologies","section":"Extending functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUX","source_file":"topologies.tex","source_line":4472,"source_end_line":4481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/topologies.tex#L4472-L4481","statement_sha256":"c3082116b6457ea1812305820b65761fe115ca28547a19934d2890ccb21f221b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6739,"rank":6739,"depth":44,"x":786.683,"y":1129.718,"cluster":"descent"},{"id":"stacks:023B","tag":"023B","title":"Descent data for quasi-coherent sheaves · Definition 023B","summary":"Let S be a scheme. Let (f_i : S_i → S)_i ∈ I be a family of morphisms with target S. • A descent datum (F_i, φ_ij) for quasi-coherent sheaves with respect to the given family is given by a quasi-coherent sheaf F_i on S_i for each i ∈ I, an isomorphism of quasi-coherent O_S_i ×_S S_j-modules φ_ij : pr_0^*F_i → pr_1^*F_j for each pair (i, j) ∈ I^2 such that for every triple of indices (i, j, k) ∈ I^3 the diagram xymatrix pr_0^*F_i ar[rd]_pr_01^*φ_ij ar[rr]_pr_02^*φ_ik & &…","statement_latex":"Let $S$ be a scheme. Let $\\{f_i : S_i \\to S\\}_{i \\in I}$ be a family\nof morphisms with target $S$.\n\\begin{enumerate}\n\\item A {\\it descent datum $(\\mathcal{F}_i, \\varphi_{ij})$\nfor quasi-coherent sheaves} with respect to the given family\nis given by a quasi-coherent sheaf $\\mathcal{F}_i$ on $S_i$ for\neach $i \\in I$, an isomorphism of quasi-coherent\n$\\mathcal{O}_{S_i \\times_S S_j}$-modules\n$\\varphi_{ij} : \\text{pr}_0^*\\mathcal{F}_i \\to \\text{pr}_1^*\\mathcal{F}_j$\nfor each pair $(i, j) \\in I^2$\nsuch that for every triple of indices $(i, j, k) \\in I^3$ the\ndiagram\n$$\n\\xymatrix{\n\\text{pr}_0^*\\mathcal{F}_i \\ar[rd]_{\\text{pr}_{01}^*\\varphi_{ij}}\n\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & &\n\\text{pr}_2^*\\mathcal{F}_k \\\\\n& \\text{pr}_1^*\\mathcal{F}_j \\ar[ru]_{\\text{pr}_{12}^*\\varphi_{jk}} &\n}\n$$\nof $\\mathcal{O}_{S_i \\times_S S_j \\times_S S_k}$-modules\ncommutes. This is called the {\\it cocycle condition}.\n\\item A {\\it morphism $\\psi : (\\mathcal{F}_i, \\varphi_{ij}) \\to\n(\\mathcal{F}'_i, \\varphi'_{ij})$ of descent data} is given\nby a family $\\psi = (\\psi_i)_{i\\in I}$ of morphisms of\n$\\mathcal{O}_{S_i}$-modules $\\psi_i : \\mathcal{F}_i \\to \\mathcal{F}'_i$\nsuch that all the diagrams\n$$\n\\xymatrix{\n\\text{pr}_0^*\\mathcal{F}_i \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\text{pr}_0^*\\psi_i}\n& \\text{pr}_1^*\\mathcal{F}_j \\ar[d]^{\\text{pr}_1^*\\psi_j} \\\\\n\\text{pr}_0^*\\mathcal{F}'_i \\ar[r]^{\\varphi'_{ij}} &\n\\text{pr}_1^*\\mathcal{F}'_j \\\\\n}\n$$\ncommute.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for quasi-coherent sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023B","source_file":"descent.tex","source_line":45,"source_end_line":84,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L45-L84","statement_sha256":"41f855894b8cfdb7a3f05ee8d7f6fd6f9e16769707043a689014b6b0e50408f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6740,"rank":6740,"depth":0,"x":609.319,"y":1208.39,"cluster":"descent"},{"id":"stacks:023C","tag":"023C","title":"Descent data for quasi-coherent sheaves · Lemma 023C","summary":"Let U = (U_i → U)_i ∈ I and V = (V_j → V)_j ∈ J be families of morphisms of schemes with fixed target. Let (g, α : I → J, (g_i)) : U → V be a morphism of families of maps with fixed target, see Sites, Definition [Tag 00VT]. Let (F_j, φ_jj') be a descent datum for quasi-coherent sheaves with respect to the family (V_j → V)_j ∈ J. Then • The system (g_i^*F_α(i), (g_i × g_i')^*φ_α(i)α(i')) is a descent datum with respect to the family (U_i → U)_i ∈ I. • This construction is…","statement_latex":"Let $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ and\n$\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$\nbe families of morphisms of schemes with fixed target.\nLet $(g, \\alpha : I \\to J, (g_i)) : \\mathcal{U} \\to \\mathcal{V}$\nbe a morphism of families of maps with fixed target, see\nSites, Definition \\ref{sites-definition-morphism-coverings}.\nLet $(\\mathcal{F}_j, \\varphi_{jj'})$ be a descent\ndatum for quasi-coherent sheaves with respect to the\nfamily $\\{V_j \\to V\\}_{j \\in J}$. Then\n\\begin{enumerate}\n\\item The system\n$$\n\\left(g_i^*\\mathcal{F}_{\\alpha(i)},\n(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}\\right)\n$$\nis a descent datum with respect to the family $\\{U_i \\to U\\}_{i \\in I}$.\n\\item This construction is functorial in the descent datum\n$(\\mathcal{F}_j, \\varphi_{jj'})$.\n\\item Given a second morphism $(g', \\alpha' : I \\to J, (g'_i))$\nof families of maps with fixed target with $g = g'$\nthere exists a functorial isomorphism of descent data\n$$\n(g_i^*\\mathcal{F}_{\\alpha(i)},\n(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')})\n\\cong\n((g'_i)^*\\mathcal{F}_{\\alpha'(i)},\n(g'_i \\times g'_{i'})^*\\varphi_{\\alpha'(i)\\alpha'(i')}).\n$$\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023C","source_file":"descent.tex","source_line":107,"source_end_line":138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L107-L138","statement_sha256":"d94a36123698ddbabaab27e97206b0b9e2b4fd271e0cfafbb02e7c8a23b4ddb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6741,"rank":6741,"depth":1,"x":677.111,"y":1049.103,"cluster":"descent"},{"id":"stacks:023D","tag":"023D","title":"Descent data for quasi-coherent sheaves · Definition 023D","summary":"Let S be a scheme. Let (S_i → S)_i ∈ I be a family of morphisms with target S. • Let F be a quasi-coherent O_S-module. We call the unique descent on F datum with respect to the covering (S → S) the trivial descent datum. • The pullback of the trivial descent datum to (S_i → S) is called the canonical descent datum. Notation: (F|_S_i, can). • A descent datum (F_i, φ_ij) for quasi-coherent sheaves with respect to the given covering is said to be effective if there exists a…","statement_latex":"Let $S$ be a scheme.\nLet $\\{S_i \\to S\\}_{i \\in I}$ be a family of morphisms\nwith target $S$.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.\nWe call the unique descent on $\\mathcal{F}$ datum with respect to the covering\n$\\{S \\to S\\}$ the {\\it trivial descent datum}.\n\\item The pullback of the trivial descent datum to\n$\\{S_i \\to S\\}$ is called the {\\it canonical descent datum}.\nNotation: $(\\mathcal{F}|_{S_i}, can)$.\n\\item A descent datum $(\\mathcal{F}_i, \\varphi_{ij})$\nfor quasi-coherent sheaves with respect to the given covering\nis said to be {\\it effective} if there exists a quasi-coherent\nsheaf $\\mathcal{F}$ on $S$ such that $(\\mathcal{F}_i, \\varphi_{ij})$\nis isomorphic to $(\\mathcal{F}|_{S_i}, can)$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for quasi-coherent sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023D","source_file":"descent.tex","source_line":155,"source_end_line":173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L155-L173","statement_sha256":"927eb32cc16d6270a55dff9657dd1558b5aca59a91ee9983579612d83c3ba43e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6742,"rank":6742,"depth":0,"x":755.525,"y":1205.645,"cluster":"descent"},{"id":"stacks:023E","tag":"023E","title":"Descent data for quasi-coherent sheaves · Lemma 023E","summary":"Let S be a scheme. Let S = ⋃ U_i be an open covering. Any descent datum on quasi-coherent sheaves for the family U = (U_i → S) is effective. Moreover, the functor from the category of quasi-coherent O_S-modules to the category of descent data with respect to U is fully faithful.","statement_latex":"Let $S$ be a scheme.\nLet $S = \\bigcup U_i$ be an open covering.\nAny descent datum on quasi-coherent sheaves\nfor the family $\\mathcal{U} = \\{U_i \\to S\\}$ is\neffective. Moreover, the functor from the category of\nquasi-coherent $\\mathcal{O}_S$-modules to the category\nof descent data with respect to $\\mathcal{U}$ is fully faithful.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023E","source_file":"descent.tex","source_line":175,"source_end_line":184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L175-L184","statement_sha256":"b4a279f97a497eda01df065b1e7389f275d38d3d472d920e64345adc69487135","origin":"The Stacks Project","memory_eligible":false,"source_rank":6743,"rank":6743,"depth":1,"x":571.094,"y":1134.427,"cluster":"descent"},{"id":"stacks:023G","tag":"023G","title":"Descent for modules · Definition 023G","summary":"Let R → A be a ring map. • A descent datum (N, φ) for modules with respect to R → A is given by an A-module N and an isomorphism of A ⊗_R A-modules φ : N ⊗_R A → A ⊗_R N such that the cocycle condition holds: the diagram of A ⊗_R A ⊗_R A-module maps xymatrix N ⊗_R A ⊗_R A ar[rr]_φ_02 ar[rd]_φ_01 & & A ⊗_R A ⊗_R N & A ⊗_R N ⊗_R A ar[ru]_φ_12 & commutes (see below for notation). • A morphism (N, φ) → (N', φ') of descent data is a morphism of A-modules ψ : N → N' such that…","statement_latex":"Let $R \\to A$ be a ring map.\n\\begin{enumerate}\n\\item A {\\it descent datum $(N, \\varphi)$ for modules\nwith respect to $R \\to A$}\nis given by an $A$-module $N$ and an isomorphism of\n$A \\otimes_R A$-modules\n$$\n\\varphi : N \\otimes_R A \\to A \\otimes_R N\n$$\nsuch that the {\\it cocycle condition} holds: the diagram\nof $A \\otimes_R A \\otimes_R A$-module maps\n$$\n\\xymatrix{\nN \\otimes_R A \\otimes_R A \\ar[rr]_{\\varphi_{02}}\n\\ar[rd]_{\\varphi_{01}}\n& &\nA \\otimes_R A \\otimes_R N \\\\\n& A \\otimes_R N \\otimes_R A \\ar[ru]_{\\varphi_{12}} &\n}\n$$\ncommutes (see below for notation).\n\\item A {\\it morphism $(N, \\varphi) \\to (N', \\varphi')$ of descent data}\nis a morphism of $A$-modules $\\psi : N \\to N'$ such that\nthe diagram\n$$\n\\xymatrix{\nN \\otimes_R A \\ar[r]_\\varphi \\ar[d]_{\\psi \\otimes \\text{id}_A} &\nA \\otimes_R N \\ar[d]^{\\text{id}_A \\otimes \\psi} \\\\\nN' \\otimes_R A \\ar[r]^{\\varphi'} &\nA \\otimes_R N'\n}\n$$\nis commutative.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023G","source_file":"descent.tex","source_line":270,"source_end_line":306,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L270-L306","statement_sha256":"1866eaeea6550a708b4bf951c724ee3ac67834791a985b753c3898c26e1fcfe1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6744,"rank":6744,"depth":0,"x":765.117,"y":1082.089,"cluster":"descent"},{"id":"stacks:023H","tag":"023H","title":"Descent for modules · Lemma 023H","summary":"Let R → A be a ring map. Given a descent datum (N, φ) we can associate to it a cosimplicial (A/R)_bullet-module N_bullet by the rules N_n = N_n, n and given β : [n] → [m] setting we define N_bullet(β) = (φ^m_β(n)m) ∘ N_β, n : N_n, n → N_m, m. This procedure is functorial in the descent datum.","statement_latex":"Let $R \\to A$ be a ring map.\nGiven a descent datum $(N, \\varphi)$ we can associate to it a\ncosimplicial $(A/R)_\\bullet$-module $N_\\bullet$\\footnote{We should really\nwrite $(N, \\varphi)_\\bullet$.} by the\nrules $N_n = N_{n, n}$ and given $\\beta : [n] \\to [m]$\nsetting we define\n$$\nN_\\bullet(\\beta) = (\\varphi^m_{\\beta(n)m}) \\circ N_{\\beta, n} :\nN_{n, n} \\longrightarrow N_{m, m}.\n$$\nThis procedure is functorial in the descent datum.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023H","source_file":"descent.tex","source_line":373,"source_end_line":386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L373-L386","statement_sha256":"3ec472e3206dbb3bff6673b135633622d5f6b0f0fc146babbe71fbc6b1cc5538","origin":"The Stacks Project","memory_eligible":false,"source_rank":6745,"rank":6745,"depth":0,"x":663.744,"y":1231.352,"cluster":"descent"},{"id":"stacks:023I","tag":"023I","title":"Descent for modules · Lemma 023I","summary":"Let R → A be a ring map. Let M be an R-module. The cosimplicial (A/R)_bullet-module associated to the canonical descent datum is isomorphic to the cosimplicial module (A/R)_bullet ⊗_R M.","statement_latex":"Let $R \\to A$ be a ring map.\nLet $M$ be an $R$-module. The cosimplicial\n$(A/R)_\\bullet$-module associated to the canonical descent\ndatum is isomorphic to the cosimplicial module $(A/R)_\\bullet \\otimes_R M$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023I","source_file":"descent.tex","source_line":462,"source_end_line":468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L462-L468","statement_sha256":"f53e9884abe7e294c334a88966843569801d08c5805869757893e1fe18fabda0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6746,"rank":6746,"depth":0,"x":618.289,"y":1063.121,"cluster":"descent"},{"id":"stacks:023J","tag":"023J","title":"Descent for modules · Definition 023J","summary":"Let R → A be a ring map. We say a descent datum (N, φ) is effective if there exists an R-module M and an isomorphism of descent data from (M ⊗_R A, can) to (N, φ).","statement_latex":"Let $R \\to A$ be a ring map.\nWe say a descent datum $(N, \\varphi)$ is {\\it effective}\nif there exists an $R$-module $M$ and an isomorphism\nof descent data from $(M \\otimes_R A, can)$ to\n$(N, \\varphi)$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023J","source_file":"descent.tex","source_line":474,"source_end_line":481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L474-L481","statement_sha256":"2aa7a639f794c424fc88e04450b6111e6118cd8c2bdacd9e507d6f51d7ca9394","origin":"The Stacks Project","memory_eligible":false,"source_rank":6747,"rank":6747,"depth":0,"x":787.737,"y":1161.755,"cluster":"descent"},{"id":"stacks:023L","tag":"023L","title":"Descent for modules · Lemma 023L","summary":"Suppose that R → A has a section. Then for any R-module M the extended cochain complex ([Tag 023K]) is exact.","statement_latex":"Suppose that $R \\to A$ has a section.\nThen for any $R$-module $M$ the extended cochain complex\n(\\ref{equation-extended-complex}) is exact.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023L","source_file":"descent.tex","source_line":523,"source_end_line":528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L523-L528","statement_sha256":"4feac02af7c2af74a32d512d13d54fcd1f26842d9e82d6eb62d060f6c8c31a60","origin":"The Stacks Project","memory_eligible":false,"source_rank":6748,"rank":6748,"depth":6,"x":582.696,"y":1185.261,"cluster":"descent"},{"id":"stacks:023M","tag":"023M","title":"Descent for modules · Lemma 023M","summary":"Suppose that R → A is faithfully flat, see Algebra, Definition [Tag 00HB]. Then for any R-module M the extended cochain complex ([Tag 023K]) is exact.","statement_latex":"Suppose that $R \\to A$ is faithfully flat, see\nAlgebra, Definition \\ref{algebra-definition-flat}.\nThen for any $R$-module $M$ the extended cochain complex\n(\\ref{equation-extended-complex}) is exact.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023M","source_file":"descent.tex","source_line":556,"source_end_line":562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L556-L562","statement_sha256":"ff42b59310abec0201d1bf5d78b74c5020bcad78827bc3d52a6a94c282917f04","origin":"The Stacks Project","memory_eligible":false,"source_rank":6749,"rank":6749,"depth":7,"x":715.486,"y":1051.081,"cluster":"descent"},{"id":"stacks:039W","tag":"039W","title":"Descent for modules · Lemma 039W","summary":"Let R → A be a faithfully flat ring map. Let (N, φ) be a descent datum. Then (N, φ) is effective if and only if the canonical map A ⊗_R H^0(s(N_bullet)) → N is an isomorphism.","statement_latex":"Let $R \\to A$ be a faithfully flat ring map.\nLet $(N, \\varphi)$ be a descent datum.\nThen $(N, \\varphi)$ is effective if and only if the canonical\nmap\n$$\nA \\otimes_R H^0(s(N_\\bullet)) \\longrightarrow N\n$$\nis an isomorphism.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039W","source_file":"descent.tex","source_line":589,"source_end_line":599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L589-L599","statement_sha256":"1745dc6ded4dc2c34360b8902a3e06cdd6cc056a91a76a6a6104cf54cb2e918e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6750,"rank":6750,"depth":8,"x":725.508,"y":1226.022,"cluster":"descent"},{"id":"stacks:039X","tag":"039X","title":"Descent for modules · Lemma 039X","summary":"Let R → A be a faithfully flat ring map, and let R → R' be faithfully flat. Set A' = R' ⊗_R A. If all descent data for R' → A' are effective, then so are all descent data for R → A.","statement_latex":"Let $R \\to A$ be a faithfully flat ring map, and let $R \\to R'$\nbe faithfully flat. Set $A' = R' \\otimes_R A$. If all descent data\nfor $R' \\to A'$ are effective, then so are all descent data for $R \\to A$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039X","source_file":"descent.tex","source_line":615,"source_end_line":620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L615-L620","statement_sha256":"313d287ef5a64e6ef4cdb3a87fd75c3df84c050baf630c9f315c668940b53e31","origin":"The Stacks Project","memory_eligible":false,"source_rank":6751,"rank":6751,"depth":9,"x":576.888,"y":1102.253,"cluster":"descent"},{"id":"stacks:023N","tag":"023N","title":"Descent for modules · Proposition 023N","summary":"Effective descent for modules along faithfully flat ring maps. Let R → A be a faithfully flat ring map. Then • any descent datum on modules with respect to R → A is effective, • the functor M ↦ (A ⊗_R M, can) from R-modules to the category of descent data is an equivalence, and • the inverse functor is given by (N, φ) ↦ H^0(s(N_bullet)).","statement_latex":"\\begin{slogan}\nEffective descent for modules along faithfully flat ring maps.\n\\end{slogan}\nLet $R \\to A$ be a faithfully flat ring map.\nThen\n\\begin{enumerate}\n\\item any descent datum on modules with respect to $R \\to A$\nis effective,\n\\item the functor $M \\mapsto (A \\otimes_R M, can)$ from $R$-modules\nto the category of descent data is an equivalence, and\n\\item the inverse functor is given by $(N, \\varphi) \\mapsto H^0(s(N_\\bullet))$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023N","source_file":"descent.tex","source_line":641,"source_end_line":655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L641-L655","statement_sha256":"f33145470d6d101b1dd6bae157c622c30dd7567bb7353b6e1cdd0a6f55d2fbef","origin":"The Stacks Project","memory_eligible":false,"source_rank":6752,"rank":6752,"depth":10,"x":786.778,"y":1109.214,"cluster":"descent"},{"id":"stacks:08WH","tag":"08WH","title":"Descent for universally injective morphisms · Definition 08WH","summary":"A split equalizer is a diagram ([Tag 08WG]) with g_1 ∘ f = g_2 ∘ f for which there exist auxiliary morphisms h : B → A and i : C → B such that h ∘ f = 1_A, f ∘ h = i ∘ g_1, i ∘ g_2 = 1_B.","statement_latex":"A {\\it split equalizer} is a diagram (\\ref{equation-equalizer}) with\n$g_1 \\circ f = g_2 \\circ f$ for which there exist auxiliary morphisms\n$h : B \\to A$ and $i : C \\to B$ such that\n\\begin{equation}\n\nh \\circ f = 1_A, \\quad f \\circ h = i \\circ g_1, \\quad i \\circ g_2 = 1_B.\n\\end{equation}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WH","source_file":"descent.tex","source_line":865,"source_end_line":874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L865-L874","statement_sha256":"d3321bf6498acbba489fa3c76aa56422d8b5ccb53b0543b43834b5c2eb8e4d5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6753,"rank":6753,"depth":0,"x":625.824,"y":1223.593,"cluster":"descent"},{"id":"stacks:08WL","tag":"08WL","title":"Descent for universally injective morphisms · Definition 08WL","summary":"A ring map f: R → S is universally injective if it is universally injective as a morphism in Mod_R.","statement_latex":"A ring map $f: R \\to S$ is {\\it universally injective}\nif it is universally injective as a morphism in $\\text{Mod}_R$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WL","source_file":"descent.tex","source_line":909,"source_end_line":913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L909-L913","statement_sha256":"9280ee1f96394df5a1c5a296561d76ad00f6e1b70aedad9bcb78745f27282b48","origin":"The Stacks Project","memory_eligible":false,"source_rank":6754,"rank":6754,"depth":0,"x":652.627,"y":1047.286,"cluster":"descent"},{"id":"stacks:08WP","tag":"08WP","title":"Descent for universally injective morphisms · Lemma 08WP","summary":"Any faithfully flat ring map is universally injective.","statement_latex":"Any faithfully flat ring map is universally injective.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WP","source_file":"descent.tex","source_line":932,"source_end_line":935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L932-L935","statement_sha256":"6eeee1f01b8cf7e73aed202f60bf8235bd710668a3cc65138934606a317dd065","origin":"The Stacks Project","memory_eligible":false,"source_rank":6755,"rank":6755,"depth":1,"x":775.082,"y":1193.023,"cluster":"descent"},{"id":"stacks:08WQ","tag":"08WQ","title":"Descent for universally injective morphisms · Definition 08WQ","summary":"Let R be a ring. Define the contravariant functor C : Mod_R → Mod_R by setting C(M) = Hom_Ab(M, Q/Z), with the R-action on C(M) given by rf(s) = f(rs).","statement_latex":"Let $R$ be a ring. Define the contravariant functor\n{\\it $C$} $ : \\text{Mod}_R \\to \\text{Mod}_R$ by setting\n$$\nC(M) = \\Hom_{\\textit{Ab}}(M, \\mathbf{Q}/\\mathbf{Z}),\n$$\nwith the $R$-action on $C(M)$ given by $rf(s) = f(rs)$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WQ","source_file":"descent.tex","source_line":947,"source_end_line":955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L947-L955","statement_sha256":"5aa9684e9dc01e47f2ee954b11b163299547946634e203177369eaba2c87e147","origin":"The Stacks Project","memory_eligible":false,"source_rank":6756,"rank":6756,"depth":0,"x":566.848,"y":1154.906,"cluster":"descent"},{"id":"stacks:08WR","tag":"08WR","title":"Descent for universally injective morphisms · Lemma 08WR","summary":"For a ring R, the functor C : Mod_R → Mod_R is exact and reflects injections and surjections.","statement_latex":"For a ring $R$, the functor $C : \\text{Mod}_R \\to \\text{Mod}_R$ is\nexact and reflects injections and surjections.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WR","source_file":"descent.tex","source_line":961,"source_end_line":965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L961-L965","statement_sha256":"3fe561b1383a0c5b25a3e5cd465033a0cf1c8c27775d0b5293680abaec7348d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6757,"rank":6757,"depth":2,"x":751.702,"y":1064.538,"cluster":"descent"},{"id":"stacks:08WU","tag":"08WU","title":"Descent for universally injective morphisms · Lemma 08WU","summary":"Let R be a ring. A morphism f: M → N in Mod_R is universally injective if and only if C(f): C(N) → C(M) is a split surjection.","statement_latex":"Let $R$ be a ring. A morphism $f: M \\to N$ in $\\text{Mod}_R$ is universally\ninjective if and only if $C(f): C(N) \\to C(M)$ is a split surjection.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WU","source_file":"descent.tex","source_line":1001,"source_end_line":1005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1001-L1005","statement_sha256":"f852bf97198bf1595d48a5c68ef403ed2d9a3fe4c67e00f4347c4e1c7df0aa6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6758,"rank":6758,"depth":3,"x":687.84,"y":1236.668,"cluster":"descent"},{"id":"stacks:08WY","tag":"08WY","title":"Descent for universally injective morphisms · Definition 08WY","summary":"The functor f^*: Mod_R → DD_S/R is called base extension along f. We say that f is a descent morphism for modules if f^* is fully faithful. We say that f is an effective descent morphism for modules if f^* is an equivalence of categories.","statement_latex":"The functor $f^*: \\text{Mod}_R \\to DD_{S/R}$\nis called {\\it base extension along $f$}. We say that $f$ is a\n{\\it descent morphism for modules} if $f^*$ is fully\nfaithful. We say that $f$ is an {\\it effective descent morphism for modules}\nif $f^*$ is an equivalence of categories.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WY","source_file":"descent.tex","source_line":1097,"source_end_line":1104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1097-L1104","statement_sha256":"b0a000f25e1d77fa6fae1deac2bbc50b09bf42524f86d8f7a8d54db7b1437183","origin":"The Stacks Project","memory_eligible":false,"source_rank":6759,"rank":6759,"depth":0,"x":596.19,"y":1072.935,"cluster":"descent"},{"id":"stacks:08WZ","tag":"08WZ","title":"Descent for universally injective morphisms · Lemma 08WZ","summary":"For (M,theta) ∈ DD_S/R, the diagram xymatrix@C=8pc M ar[r]^theta ∘ (1_M ⊗ δ_0^1) & M ⊗_S, δ_1^1 S_2 ar@<1ex>[r]^(theta ⊗ δ_2^2) ∘ (1_M ⊗ δ^2_0) ar@<-1ex>[r]_1_M ⊗ S_2 ⊗ δ^2_1 & M ⊗_S, δ_12^1 S_3 is a split equalizer.","statement_latex":"For $(M,\\theta) \\in DD_{S/R}$, the diagram\n\\begin{equation}\n\n\\xymatrix@C=8pc{\nM \\ar[r]^{\\theta \\circ (1_M \\otimes \\delta_0^1)} &\nM \\otimes_{S, \\delta_1^1} S_2\n\\ar@<1ex>[r]^{(\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0)}\n\\ar@<-1ex>[r]_{1_{M \\otimes S_2} \\otimes \\delta^2_1} & \nM \\otimes_{S, \\delta_{12}^1} S_3\n}\n\\end{equation}\nis a split equalizer.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WZ","source_file":"descent.tex","source_line":1111,"source_end_line":1125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1111-L1125","statement_sha256":"8132f80109abec8f30b234ff7a423da1d6ec1a87aeab9032f32cba2232887109","origin":"The Stacks Project","memory_eligible":false,"source_rank":6760,"rank":6760,"depth":1,"x":796.132,"y":1141.904,"cluster":"descent"},{"id":"stacks:08X1","tag":"08X1","title":"Descent for universally injective morphisms · Lemma 08X1","summary":"For (M, theta) ∈ DD_S/R, the diagram xymatrix@C=8pc C(M ⊗_S, δ_12^1 S_3) ar@<1ex>[r]^C((theta ⊗ δ_2^2) ∘ (1_M ⊗ δ^2_0)) ar@<-1ex>[r]_C(1_M ⊗ S_2 ⊗ δ^2_1) & C(M ⊗_S, δ_1^1 S_2 ) ar[r]^C(theta ∘ (1_M ⊗ δ_0^1)) & C(M). obtained by applying C to ([Tag 08X0]) is a split coequalizer.","statement_latex":"For $(M, \\theta) \\in DD_{S/R}$, the diagram\n\\begin{equation}\n\n\\xymatrix@C=8pc{\nC(M \\otimes_{S, \\delta_{12}^1} S_3)\n\\ar@<1ex>[r]^{C((\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0))}\n\\ar@<-1ex>[r]_{C(1_{M \\otimes S_2} \\otimes \\delta^2_1)} &\nC(M \\otimes_{S, \\delta_1^1} S_2 )\n\\ar[r]^{C(\\theta \\circ (1_M \\otimes \\delta_0^1))} & C(M).\n}\n\\end{equation}\nobtained by applying $C$ to (\\ref{equation-equalizer-M}) is a split\ncoequalizer.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08X1","source_file":"descent.tex","source_line":1145,"source_end_line":1160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1145-L1160","statement_sha256":"429a997a8b3e0952698e771788903e025902ae8d70511b6eeec6acbf0ca642a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6761,"rank":6761,"depth":2,"x":592.536,"y":1204.714,"cluster":"descent"},{"id":"stacks:08X3","tag":"08X3","title":"Descent for universally injective morphisms · Lemma 08X3","summary":"The diagram xymatrix@C=8pc S_1 ar[r]^δ^1_1 & S_2 ar@<1ex>[r]^δ^2_2 ar@<-1ex>[r]_δ^2_1 & S_3 is a split equalizer.","statement_latex":"The diagram\n\\begin{equation}\n\n\\xymatrix@C=8pc{\nS_1 \\ar[r]^{\\delta^1_1} &\nS_2 \\ar@<1ex>[r]^{\\delta^2_2} \\ar@<-1ex>[r]_{\\delta^2_1} & \nS_3\n}\n\\end{equation}\nis a split equalizer.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08X3","source_file":"descent.tex","source_line":1166,"source_end_line":1178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1166-L1178","statement_sha256":"c70f3f95ffeae35f62110ddd587ec0ba2ff6ccc3b02b58858e35f4fc1dfab603","origin":"The Stacks Project","memory_eligible":false,"source_rank":6762,"rank":6762,"depth":2,"x":692.496,"y":1042.32,"cluster":"descent"},{"id":"stacks:08X5","tag":"08X5","title":"Descent for universally injective morphisms · Definition 08X5","summary":"Define the functor f_* : DD_S/R → Mod_R by taking f_*(M, theta) to be the R-submodule of M for which the diagram xymatrix@C=8pcf_*(M,theta) ar[r] & M ar@<1ex>^theta ∘ (1_M ⊗ δ_0^1)[r] ar@<-1ex>_1_M ⊗ δ_1^1[r] & M ⊗_S, δ_1^1 S_2 is an equalizer.","statement_latex":"Define the functor {\\it $f_*$} $: DD_{S/R} \\to \\text{Mod}_R$ by taking\n$f_*(M, \\theta)$ to be the $R$-submodule of $M$ for which the diagram\n\\begin{equation}\n\n\\xymatrix@C=8pc{f_*(M,\\theta) \\ar[r] & M \\ar@<1ex>^{\\theta \\circ (1_M \\otimes \n\\delta_0^1)}[r] \\ar@<-1ex>_{1_M \\otimes \\delta_1^1}[r] & \nM \\otimes_{S, \\delta_1^1} S_2 \n}\n\\end{equation}\nis an equalizer.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08X5","source_file":"descent.tex","source_line":1187,"source_end_line":1199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1187-L1199","statement_sha256":"126b612fb78b8a89e11ec240e330c5ddc8cdde616c1cc5fe9126444d0ccfe52a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6763,"rank":6763,"depth":0,"x":749.572,"y":1219.386,"cluster":"descent"},{"id":"stacks:08X7","tag":"08X7","title":"Descent for universally injective morphisms · Lemma 08X7","summary":"If f is universally injective, then the diagram xymatrix@C=8pc f_*(M, theta) ⊗_R S ar[r]^theta ∘ (1_M ⊗ δ_0^1) & M ⊗_S, δ_1^1 S_2 ar@<1ex>[r]^(theta ⊗ δ_2^2) ∘ (1_M ⊗ δ^2_0) ar@<-1ex>[r]_1_M ⊗ S_2 ⊗ δ^2_1 & M ⊗_S, δ_12^1 S_3 obtained by tensoring ([Tag 08X6]) over R with S is an equalizer.","statement_latex":"If $f$ is universally injective, then the diagram\n\\begin{equation}\n\n\\xymatrix@C=8pc{\nf_*(M, \\theta) \\otimes_R S\n\\ar[r]^{\\theta \\circ (1_M \\otimes \\delta_0^1)} &\nM \\otimes_{S, \\delta_1^1} S_2 \n\\ar@<1ex>[r]^{(\\theta \\otimes \\delta_2^2) \\circ (1_M \\otimes \\delta^2_0)}\n\\ar@<-1ex>[r]_{1_{M \\otimes S_2} \\otimes \\delta^2_1} &\nM \\otimes_{S, \\delta_{12}^1} S_3\n}\n\\end{equation}\nobtained by tensoring (\\ref{equation-equalizer-f}) over $R$ with $S$ is an \nequalizer.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08X7","source_file":"descent.tex","source_line":1212,"source_end_line":1228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1212-L1228","statement_sha256":"285cfea8763db9b7963440364f39c7e1cf04dbecf5f6ab64b8f9206757d21dd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6764,"rank":6764,"depth":4,"x":564.471,"y":1120.874,"cluster":"descent"},{"id":"stacks:08XA","tag":"08XA","title":"Descent for universally injective morphisms · Theorem 08XA","summary":"The following conditions are equivalent. • [(a)] The morphism f is a descent morphism for modules. • [(b)] The morphism f is an effective descent morphism for modules. • [(c)] The morphism f is universally injective.","statement_latex":"The following conditions are equivalent.\n\\begin{enumerate}\n\\item[(a)] The morphism $f$ is a descent morphism for modules.\n\\item[(b)] The morphism $f$ is an effective descent morphism for modules.\n\\item[(c)] The morphism $f$ is universally injective.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XA","source_file":"descent.tex","source_line":1289,"source_end_line":1297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1289-L1297","statement_sha256":"8bb235bad70cc6c553039ba33e2c48b7d89d0f3e179788eb5193c00aebaf075d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6765,"rank":6765,"depth":5,"x":780.905,"y":1088.377,"cluster":"descent"},{"id":"stacks:08XC","tag":"08XC","title":"Descent for universally injective morphisms · Lemma 08XC","summary":"If M ∈ Mod_R is flat, then C(M) is an injective R-module.","statement_latex":"If $M \\in \\text{Mod}_R$ is flat, then $C(M)$ is an injective $R$-module.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XC","source_file":"descent.tex","source_line":1332,"source_end_line":1335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1332-L1335","statement_sha256":"f1168a4822297fb089825b4c9ccc3ceed83432befb99a57a650434164e0791af","origin":"The Stacks Project","memory_eligible":false,"source_rank":6766,"rank":6766,"depth":3,"x":646.998,"y":1235.638,"cluster":"descent"},{"id":"stacks:08XD","tag":"08XD","title":"Descent for universally injective morphisms · Theorem 08XD","summary":"If M ⊗_R S has one of the following properties as an S-module • [(a)] finitely generated; • [(b)] finitely presented; • [(c)] flat; • [(d)] faithfully flat; • [(e)] finite projective; then so does M as an R-module (and conversely).","statement_latex":"If $M \\otimes_R S$ has one of the following properties as an $S$-module\n\\begin{enumerate}\n\\item[(a)]\nfinitely generated;\n\\item[(b)]\nfinitely presented;\n\\item[(c)]\nflat;\n\\item[(d)]\nfaithfully flat;\n\\item[(e)]\nfinite projective;\n\\end{enumerate}\nthen so does $M$ as an $R$-module (and conversely).","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XD","source_file":"descent.tex","source_line":1356,"source_end_line":1372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1356-L1372","statement_sha256":"d4a626f8e4fa15c0fd4be00a2b7cc2d51314d28c380737b573c57df007cefb0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6767,"rank":6767,"depth":5,"x":627.252,"y":1050.458,"cluster":"descent"},{"id":"stacks:08XE","tag":"08XE","title":"Descent for universally injective morphisms · Theorem 08XE","summary":"If A ⊗_R S has one of the following properties as an S-algebra • [(a)] of finite type; • [(b)] of finite presentation; • [(c)] formally unramified; • [(d)] unramified; • [(e)] étale; then so does A as an R-algebra (and of course conversely).","statement_latex":"If $A \\otimes_R S$ has one of the following properties as an $S$-algebra\n\\begin{enumerate}\n\\item[(a)]\nof finite type;\n\\item[(b)]\nof finite presentation;\n\\item[(c)]\nformally unramified;\n\\item[(d)]\nunramified;\n\\item[(e)]\n\\'etale;\n\\end{enumerate}\nthen so does $A$ as an $R$-algebra (and of course conversely).","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent for universally injective morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XE","source_file":"descent.tex","source_line":1445,"source_end_line":1461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1445-L1461","statement_sha256":"b96f51c11325e2844c5a5aea7eec9e10e754930cbd57a3a2edf1fce87dd4ee47","origin":"The Stacks Project","memory_eligible":false,"source_rank":6768,"rank":6768,"depth":42,"x":791.266,"y":1176.215,"cluster":"descent"},{"id":"stacks:023S","tag":"023S","title":"Fpqc descent of quasi-coherent sheaves · Lemma 023S","summary":"Let S be an affine scheme. Let U = (f_i : U_i → S)_i = 1, …, n be a standard fpqc covering of S, see Topologies, Definition [Tag 022F]. Any descent datum on quasi-coherent sheaves for U = (U_i → S) is effective. Moreover, the functor from the category of quasi-coherent O_S-modules to the category of descent data with respect to U is fully faithful.","statement_latex":"Let $S$ be an affine scheme.\nLet $\\mathcal{U} = \\{f_i : U_i \\to S\\}_{i = 1, \\ldots, n}$\nbe a standard fpqc covering of $S$, see\nTopologies, Definition \\ref{topologies-definition-standard-fpqc}.\nAny descent datum on quasi-coherent sheaves\nfor $\\mathcal{U} = \\{U_i \\to S\\}$ is effective.\nMoreover, the functor from the category of\nquasi-coherent $\\mathcal{O}_S$-modules to the category\nof descent data with respect to $\\mathcal{U}$ is fully faithful.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc descent of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023S","source_file":"descent.tex","source_line":1549,"source_end_line":1560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1549-L1560","statement_sha256":"1e6d73b953e78a7225e42fb72f16cde9f4b870d7709efa75396f64d07f3c2c97","origin":"The Stacks Project","memory_eligible":false,"source_rank":6769,"rank":6769,"depth":11,"x":568.47,"y":1176.549,"cluster":"descent"},{"id":"stacks:023T","tag":"023T","title":"Fpqc descent of quasi-coherent sheaves · Proposition 023T","summary":"Let S be a scheme. Let U = (φ_i : U_i → S) be an fpqc covering, see Topologies, Definition [Tag 022B]. Any descent datum on quasi-coherent sheaves for U = (U_i → S) is effective. Moreover, the functor from the category of quasi-coherent O_S-modules to the category of descent data with respect to U is fully faithful.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{U} = \\{\\varphi_i : U_i \\to S\\}$ be an fpqc covering, see\nTopologies, Definition \\ref{topologies-definition-fpqc-covering}.\nAny descent datum on quasi-coherent sheaves\nfor $\\mathcal{U} = \\{U_i \\to S\\}$ is effective.\nMoreover, the functor from the category of\nquasi-coherent $\\mathcal{O}_S$-modules to the category\nof descent data with respect to $\\mathcal{U}$ is fully faithful.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc descent of quasi-coherent sheaves","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023T","source_file":"descent.tex","source_line":1583,"source_end_line":1593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1583-L1593","statement_sha256":"4b914d84bb2afbf40c214df2bee4666da3391c2e9e45ce584180439cd4267b1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6770,"rank":6770,"depth":20,"x":733.019,"y":1049.473,"cluster":"descent"},{"id":"stacks:0CDR","tag":"0CDR","title":"Galois descent for quasi-coherent sheaves · Lemma 0CDR","summary":"Let k'/k be a (finite) Galois extension with Galois group G. Let X be a scheme over k. The category of quasi-coherent O_X-modules is equivalent to the category of systems (F, (φ_σ)_σ ∈ G) where • F is a quasi-coherent module on X_k', • φ_σ : F → f_σ^*F is an isomorphism of modules, • φ_στ = f_σ^*φ_τ ∘ φ_σ for all σ, τ ∈ G. Here f_σ = id_X × Spec(σ) : X_k' → X_k'.","statement_latex":"Let $k'/k$ be a (finite) Galois extension with Galois group $G$.\nLet $X$ be a scheme over $k$. The category of quasi-coherent\n$\\mathcal{O}_X$-modules is equivalent to the category of systems\n$(\\mathcal{F}, (\\varphi_\\sigma)_{\\sigma \\in G})$ where\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a quasi-coherent module on $X_{k'}$,\n\\item $\\varphi_\\sigma : \\mathcal{F} \\to f_\\sigma^*\\mathcal{F}$\nis an isomorphism of modules,\n\\item $\\varphi_{\\sigma\\tau} = f_\\sigma^*\\varphi_\\tau \\circ \\varphi_\\sigma$\nfor all $\\sigma, \\tau \\in G$.\n\\end{enumerate}\nHere $f_\\sigma = \\text{id}_X \\times \\Spec(\\sigma) : X_{k'} \\to X_{k'}$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Galois descent for quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDR","source_file":"descent.tex","source_line":1815,"source_end_line":1829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1815-L1829","statement_sha256":"bae2b1202a21b5afddfea78a05836d79f6509c2ee636fca6446574ab290910d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6771,"rank":6771,"depth":21,"x":713.812,"y":1237.149,"cluster":"descent"},{"id":"stacks:0D1V","tag":"0D1V","title":"Galois descent for quasi-coherent sheaves · Lemma 0D1V","summary":"Let X → Y, G, and f_σ : X → X be as above. The category of quasi-coherent O_Y-modules is equivalent to the category of systems (F, (φ_σ)_σ ∈ G) where • F is a quasi-coherent O_X-module, • φ_σ : F → f_σ^*F is an isomorphism of modules, • φ_στ = f_σ^*φ_τ ∘ φ_σ for all σ, τ ∈ G.","statement_latex":"Let $X \\to Y$, $G$, and $f_\\sigma : X \\to X$ be as above.\nThe category of quasi-coherent\n$\\mathcal{O}_Y$-modules is equivalent to the category of systems\n$(\\mathcal{F}, (\\varphi_\\sigma)_{\\sigma \\in G})$ where\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module,\n\\item $\\varphi_\\sigma : \\mathcal{F} \\to f_\\sigma^*\\mathcal{F}$\nis an isomorphism of modules,\n\\item $\\varphi_{\\sigma\\tau} = f_\\sigma^*\\varphi_\\tau \\circ \\varphi_\\sigma$\nfor all $\\sigma, \\tau \\in G$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Galois descent for quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1V","source_file":"descent.tex","source_line":1850,"source_end_line":1863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1850-L1863","statement_sha256":"418b2eeaf739eb0a84d04c33444bf7aa3efaf3b19fcbe9f41437c6bff67b7cf8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6772,"rank":6772,"depth":21,"x":576.615,"y":1087.383,"cluster":"descent"},{"id":"stacks:05AZ","tag":"05AZ","title":"Descent of finiteness properties of modules · Lemma 05AZ","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is a finite type O_X_i-module. Then F is a finite type O_X-module.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is a finite type $\\mathcal{O}_{X_i}$-module.\nThen $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05AZ","source_file":"descent.tex","source_line":1895,"source_end_line":1902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1895-L1902","statement_sha256":"7343743f3c58080b6823873603d8fdc14b7336175cddd27db5ff2d049e3fa860","origin":"The Stacks Project","memory_eligible":false,"source_rank":6773,"rank":6773,"depth":3,"x":798.924,"y":1120.073,"cluster":"descent"},{"id":"stacks:09UB","tag":"09UB","title":"Descent of finiteness properties of modules · Lemma 09UB","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of locally ringed spaces. Let F be a sheaf of O_Y-modules. If • f is open as a map of topological spaces, • f is surjective and flat, and • f^*F is of finite type, then F is of finite type.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of\nlocally ringed spaces. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules.\nIf\n\\begin{enumerate}\n\\item $f$ is open as a map of topological spaces,\n\\item $f$ is surjective and flat, and\n\\item $f^*\\mathcal{F}$ is of finite type,\n\\end{enumerate}\nthen $\\mathcal{F}$ is of finite type.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UB","source_file":"descent.tex","source_line":1909,"source_end_line":1920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1909-L1920","statement_sha256":"95196116c18c7b64608b6007e7e76a441f23e802178f58757352cdc89a9f374c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6774,"rank":6774,"depth":4,"x":608.106,"y":1222.433,"cluster":"descent"},{"id":"stacks:05B0","tag":"05B0","title":"Descent of finiteness properties of modules · Lemma 05B0","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is an O_X_i-module of finite presentation. Then F is an O_X-module of finite presentation.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is an $\\mathcal{O}_{X_i}$-module of finite\npresentation. Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module\nof finite presentation.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05B0","source_file":"descent.tex","source_line":1952,"source_end_line":1960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1952-L1960","statement_sha256":"65019377b3d0e8343afc13ddd9f296a94920add871b8f66dc3b8949f6f6d4d87","origin":"The Stacks Project","memory_eligible":false,"source_rank":6775,"rank":6775,"depth":3,"x":666.682,"y":1038.104,"cluster":"descent"},{"id":"stacks:082U","tag":"082U","title":"Descent of finiteness properties of modules · Lemma 082U","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is locally generated by r sections as an O_X_i-module. Then F is locally generated by r sections as an O_X-module.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is locally generated by $r$ sections as an\n$\\mathcal{O}_{X_i}$-module. Then $\\mathcal{F}$ is locally generated by\n$r$ sections as an $\\mathcal{O}_X$-module.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082U","source_file":"descent.tex","source_line":1967,"source_end_line":1975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1967-L1975","statement_sha256":"3ccf0ed27e101bf6d7bc64a3d6f788e48edbd482d6076d60725baae60b9e1a6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6776,"rank":6776,"depth":4,"x":772.048,"y":1207.79,"cluster":"descent"},{"id":"stacks:05B1","tag":"05B1","title":"Descent of finiteness properties of modules · Lemma 05B1","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is a flat O_X_i-module. Then F is a flat O_X-module.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is a flat $\\mathcal{O}_{X_i}$-module.\nThen $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05B1","source_file":"descent.tex","source_line":1990,"source_end_line":1997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L1990-L1997","statement_sha256":"2db4755635cf8966665361fc53a5808a7f95b1c748ea8520f817dba212eb88da","origin":"The Stacks Project","memory_eligible":false,"source_rank":6777,"rank":6777,"depth":3,"x":557.232,"y":1142.25,"cluster":"descent"},{"id":"stacks:05B2","tag":"05B2","title":"Descent of finiteness properties of modules · Lemma 05B2","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is a finite locally free O_X_i-module. Then F is a finite locally free O_X-module.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is a finite locally free $\\mathcal{O}_{X_i}$-module.\nThen $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05B2","source_file":"descent.tex","source_line":2004,"source_end_line":2011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2004-L2011","statement_sha256":"7262313fe4a6649e8671f45a185ce8e9f41dd260b2f2d6ae152fe45742e90cfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6778,"rank":6778,"depth":5,"x":768.99,"y":1068.461,"cluster":"descent"},{"id":"stacks:05JZ","tag":"05JZ","title":"Descent of finiteness properties of modules · Lemma 05JZ","summary":"Let X be a scheme. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is a locally projective O_X_i-module. Then F is a locally projective O_X-module.","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is a locally projective $\\mathcal{O}_{X_i}$-module.\nThen $\\mathcal{F}$ is a locally projective $\\mathcal{O}_X$-module.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JZ","source_file":"descent.tex","source_line":2027,"source_end_line":2034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2027-L2034","statement_sha256":"e7d29d62bf8a8fec87539ddbc7e5a7f4c81d3d52cb0c1d143f6c6ca41f94df28","origin":"The Stacks Project","memory_eligible":false,"source_rank":6779,"rank":6779,"depth":11,"x":671.886,"y":1243.562,"cluster":"descent"},{"id":"stacks:05B3","tag":"05B3","title":"Descent of finiteness properties of modules · Lemma 05B3","summary":"Let f : X → Y be a morphism of schemes. Let F be a quasi-coherent O_X-module. Assume f is a finite morphism. Then F is an O_X-module of finite type if and only if f_*F is an O_Y-module of finite type.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $f$ is a finite morphism.\nThen $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type\nif and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite\ntype.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05B3","source_file":"descent.tex","source_line":2074,"source_end_line":2082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2074-L2082","statement_sha256":"3d0575476b83eb8a27fe7275adf9d35a6b6aec9ddde6259593da08a101d34a06","origin":"The Stacks Project","memory_eligible":false,"source_rank":6780,"rank":6780,"depth":4,"x":602.467,"y":1058.785,"cluster":"descent"},{"id":"stacks:05B4","tag":"05B4","title":"Descent of finiteness properties of modules · Lemma 05B4","summary":"Let f : X → Y be a morphism of schemes. Let F be a quasi-coherent O_X-module. Assume f is finite and of finite presentation. Then F is an O_X-module of finite presentation if and only if f_*F is an O_Y-module of finite presentation.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $f$ is finite and of finite presentation.\nThen $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation\nif and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite\npresentation.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05B4","source_file":"descent.tex","source_line":2098,"source_end_line":2106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2098-L2106","statement_sha256":"25c6cc74b8186dbd17839e7bea562457de4c2a302d5776c08d69bc83f7022f45","origin":"The Stacks Project","memory_eligible":false,"source_rank":6781,"rank":6781,"depth":7,"x":802.851,"y":1155.941,"cluster":"descent"},{"id":"stacks:03DT","tag":"03DT","title":"Quasi-coherent sheaves and topologies, I · Lemma 03DT","summary":"Let S be a scheme. Let F be a quasi-coherent O_S-module. Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf, linebreak[0] fpqc). The functor defined in ([Tag 03DS]) satisfies the sheaf condition with respect to any τ-covering (T_i → T)_i ∈ I of any scheme T over S.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0] smooth,\n\\linebreak[0] syntomic, \\linebreak[0] fppf, \\linebreak[0] fpqc\\}$.\nThe functor defined in (\\ref{equation-quasi-coherent-presheaf})\nsatisfies the sheaf condition with respect to any $\\tau$-covering\n$\\{T_i \\to T\\}_{i \\in I}$ of any scheme $T$ over $S$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DT","source_file":"descent.tex","source_line":2157,"source_end_line":2166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2157-L2166","statement_sha256":"f2581454270b141a70d357635b89e1e5ef34f9dc3b0dd3c93b67502d243f6f56","origin":"The Stacks Project","memory_eligible":false,"source_rank":6782,"rank":6782,"depth":39,"x":576.283,"y":1198.128,"cluster":"descent"},{"id":"stacks:03DU","tag":"03DU","title":"Quasi-coherent sheaves and topologies, I · Definition 03DU","summary":"Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf). Let S be a scheme. Let Sch_τ be a big site containing S. Let F be a quasi-coherent O_S-module. • The structure sheaf of the big site (Sch/S)_τ is the sheaf of rings T/S ↦ Γ(T, O_T) which is denoted O or O_S. • If τ = Zariski or τ = etale the structure sheaf of the small site S_Zar or S_etale is the sheaf of rings T/S ↦ Γ(T, O_T) which is denoted O or O_S. • The sheaf of…","statement_latex":"Let $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic, \\linebreak[0] fppf\\}$.\nLet $S$ be a scheme.\nLet $\\Sch_\\tau$ be a big site containing $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.\n\\begin{enumerate}\n\\item The {\\it structure sheaf of the big site $(\\Sch/S)_\\tau$}\nis the sheaf of rings $T/S \\mapsto \\Gamma(T, \\mathcal{O}_T)$ which is\ndenoted $\\mathcal{O}$ or $\\mathcal{O}_S$.\n\\item If $\\tau = Zariski$ or $\\tau = \\etale$ the\n{\\it structure sheaf of the small site} $S_{Zar}$ or $S_\\etale$\nis the sheaf of rings $T/S \\mapsto \\Gamma(T, \\mathcal{O}_T)$\nwhich is denoted $\\mathcal{O}$ or $\\mathcal{O}_S$.\n\\item The {\\it sheaf of $\\mathcal{O}$-modules associated to\n$\\mathcal{F}$} on the big site $(\\Sch/S)_\\tau$\nis the sheaf of $\\mathcal{O}$-modules\n$(f : T \\to S) \\mapsto \\Gamma(T, f^*\\mathcal{F})$\nwhich is denoted $\\mathcal{F}^a$ (and often simply $\\mathcal{F}$).\n\\item If $\\tau = Zariski$ or $\\tau = \\etale$ the\n{\\it sheaf of $\\mathcal{O}$-modules associated to $\\mathcal{F}$}\non the small site $S_{Zar}$ or $S_\\etale$ is the sheaf of\n$\\mathcal{O}$-modules $(f : T \\to S) \\mapsto \\Gamma(T, f^*\\mathcal{F})$\nwhich is denoted $\\mathcal{F}^a$ (and often simply $\\mathcal{F}$).\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, I","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DU","source_file":"descent.tex","source_line":2197,"source_end_line":2223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2197-L2223","statement_sha256":"bf6406b4ed274ccb9cbada25f497068b25a896afd02ab4a1ab55416b4e00f118","origin":"The Stacks Project","memory_eligible":false,"source_rank":6783,"rank":6783,"depth":0,"x":709.823,"y":1037.983,"cluster":"descent"},{"id":"stacks:070S","tag":"070S","title":"Quasi-coherent sheaves and topologies, I · Lemma 070S","summary":"Let S be a scheme. Denote id_τ, Zar & : & (Sch/S)_τ → S_Zar, & τ ∈ (Zar, etale, smooth, syntomic, fppf) id_τ, etale & : & (Sch/S)_τ → S_etale, & τ ∈ (etale, smooth, syntomic, fppf) id_small, etale, Zar & : & S_etale → S_Zar, the morphisms of ringed sites of Remark [Tag 070R]. Let F be a sheaf of O_S-modules which we view a sheaf of O-modules on S_Zar. Then • (id_τ, Zar)^*F is the τ-sheafification of the Zariski sheaf (f : T → S) ↦ Γ(T, f^*F) on (Sch/S)_τ, and • (id_small,…","statement_latex":"Let $S$ be a scheme. Denote\n$$\n\\begin{matrix}\n\\text{id}_{\\tau, Zar} & : & (\\Sch/S)_\\tau \\to S_{Zar}, &\n\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\} \\\\\n\\text{id}_{\\tau, \\etale} & : &\n(\\Sch/S)_\\tau \\to S_\\etale, &\n\\tau \\in \\{\\etale, smooth, syntomic, fppf\\} \\\\\n\\text{id}_{small, \\etale, Zar} & : & S_\\etale \\to S_{Zar},\n\\end{matrix}\n$$\nthe morphisms of ringed sites of\nRemark \\ref{remark-change-topologies-ringed}.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_S$-modules\nwhich we view a sheaf of $\\mathcal{O}$-modules on $S_{Zar}$. Then\n\\begin{enumerate}\n\\item $(\\text{id}_{\\tau, Zar})^*\\mathcal{F}$ is the $\\tau$-sheafification\nof the Zariski sheaf\n$$\n(f : T \\to S) \\longmapsto \\Gamma(T, f^*\\mathcal{F})\n$$\non $(\\Sch/S)_\\tau$, and\n\\item $(\\text{id}_{small, \\etale, Zar})^*\\mathcal{F}$ is the\n\\'etale sheafification of the Zariski sheaf\n$$\n(f : T \\to S) \\longmapsto \\Gamma(T, f^*\\mathcal{F})\n$$\non $S_\\etale$.\n\\end{enumerate}\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules\non $S_\\etale$. Then\n\\begin{enumerate}\n\\item[(3)] $(\\text{id}_{\\tau, \\etale})^*\\mathcal{G}$ is the\n$\\tau$-sheafification of the \\'etale sheaf\n$$\n(f : T \\to S) \\longmapsto \\Gamma(T, f_{small}^*\\mathcal{G})\n$$\nwhere $f_{small} : T_\\etale \\to S_\\etale$\nis the morphism of ringed small \\'etale sites of\nRemark \\ref{remark-change-topologies-ringed}.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/070S","source_file":"descent.tex","source_line":2299,"source_end_line":2342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2299-L2342","statement_sha256":"c1d3e810d9a5f7f15d699b3904c71b15da2820ddb8977ba9feaf1a816f7837a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6784,"rank":6784,"depth":10,"x":740.226,"y":1232.421,"cluster":"descent"},{"id":"stacks:03DV","tag":"03DV","title":"Quasi-coherent sheaves and topologies, I · Lemma 03DV","summary":"Let S be a scheme. Let F be a quasi-coherent O_S-module. Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf). • The sheaf F^a is a quasi-coherent O-module on (Sch/S)_τ, as defined in Modules on Sites, Definition [Tag 03DL]. • If τ = Zariski or τ = etale, then the sheaf F^a is a quasi-coherent O-module on S_Zar or S_etale as defined in Modules on Sites, Definition [Tag 03DL].","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic, \\linebreak[0] fppf\\}$.\n\\begin{enumerate}\n\\item The sheaf $\\mathcal{F}^a$ is a quasi-coherent\n$\\mathcal{O}$-module on $(\\Sch/S)_\\tau$, as defined in\nModules on Sites, Definition \\ref{sites-modules-definition-site-local}.\n\\item If $\\tau = Zariski$ or $\\tau = \\etale$, then the sheaf\n$\\mathcal{F}^a$ is a quasi-coherent $\\mathcal{O}$-module on\n$S_{Zar}$ or $S_\\etale$ as defined in\nModules on Sites, Definition \\ref{sites-modules-definition-site-local}.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DV","source_file":"descent.tex","source_line":2398,"source_end_line":2413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2398-L2413","statement_sha256":"9a1544cd8f8a38618011754a501b677569a86c0c59aa9f16b34eb845c3d2011f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6785,"rank":6785,"depth":7,"x":560.919,"y":1105.924,"cluster":"descent"},{"id":"stacks:0GN7","tag":"0GN7","title":"Quasi-coherent sheaves and topologies, I · Lemma 0GN7","summary":"Let S be a scheme. Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf). Each of the functors F ↦ F^a of Definition [Tag 03DU] QCoh(O_S) → QCoh((Sch/S)_τ, O) or QCoh(O_S) → QCoh(S_τ, O) is fully faithful.","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic, \\linebreak[0] fppf\\}$.\nEach of the functors $\\mathcal{F} \\mapsto \\mathcal{F}^a$\nof Definition \\ref{definition-structure-sheaf}\n$$\n\\QCoh(\\mathcal{O}_S) \\to \\QCoh((\\Sch/S)_\\tau, \\mathcal{O})\n\\quad\\text{or}\\quad\n\\QCoh(\\mathcal{O}_S) \\to \\QCoh(S_\\tau, \\mathcal{O})\n$$\nis fully faithful.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GN7","source_file":"descent.tex","source_line":2458,"source_end_line":2471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2458-L2471","statement_sha256":"3843620b6fe472c3a18a11a77e6f53f3bf702c7f43b6ce34fb297f1a833c4242","origin":"The Stacks Project","memory_eligible":false,"source_rank":6786,"rank":6786,"depth":8,"x":795.549,"y":1097.433,"cluster":"descent"},{"id":"stacks:03DX","tag":"03DX","title":"Quasi-coherent sheaves and topologies, I · Proposition 03DX","summary":"Let S be a scheme. Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf). • The functor F ↦ F^a defines an equivalence of categories QCoh(O_S) → QCoh((Sch/S)_τ, O) between the category of quasi-coherent sheaves on S and the category of quasi-coherent O-modules on the big τ site of S. • Let τ = Zariski or τ = etale. The functor F ↦ F^a defines an equivalence of categories QCoh(O_S) → QCoh(S_τ, O) between the category of…","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic, \\linebreak[0] fppf\\}$.\n\\begin{enumerate}\n\\item The functor $\\mathcal{F} \\mapsto \\mathcal{F}^a$\ndefines an equivalence of categories\n$$\n\\QCoh(\\mathcal{O}_S)\n\\longrightarrow\n\\QCoh((\\Sch/S)_\\tau, \\mathcal{O})\n$$\nbetween the category of quasi-coherent sheaves on $S$ and the category\nof quasi-coherent $\\mathcal{O}$-modules on the big $\\tau$ site of $S$.\n\\item Let $\\tau = Zariski$ or $\\tau = \\etale$.\nThe functor $\\mathcal{F} \\mapsto \\mathcal{F}^a$\ndefines an equivalence of categories\n$$\n\\QCoh(\\mathcal{O}_S)\n\\longrightarrow\n\\QCoh(S_\\tau, \\mathcal{O})\n$$\nbetween the category of quasi-coherent sheaves on $S$ and the category\nof quasi-coherent $\\mathcal{O}$-modules on the small $\\tau$ site of $S$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, I","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DX","source_file":"descent.tex","source_line":2498,"source_end_line":2524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2498-L2524","statement_sha256":"47704309cc89f68441e9a617816b3ee159678a94be26499f8f9a0333c682d5c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6787,"rank":6787,"depth":21,"x":628.879,"y":1237.235,"cluster":"descent"},{"id":"stacks:05VG","tag":"05VG","title":"Quasi-coherent sheaves and topologies, I · Lemma 05VG","summary":"Let S be a scheme. Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf). Let P be one of the properties of modules defined in Modules on Sites, Definitions [Tag 03DE], [Tag 03DL], and [Tag 03ER]. The equivalences of categories QCoh(O_S) → QCoh((Sch/S)_τ, O) and QCoh(O_S) → QCoh(S_τ, O) defined by the rule F ↦ F^a seen in Proposition [Tag 03DX] have the property F has P ⇔ F^a has P as an O-module except (possibly) when P is…","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic, \\linebreak[0] fppf\\}$.\nLet $\\mathcal{P}$ be one of the properties of modules\\footnote{The list is:\nfree, finite free, generated by global sections,\ngenerated by $r$ global sections, generated by finitely many global sections,\nhaving a global presentation, having a global finite presentation,\nlocally free, finite locally free, locally generated by sections,\nlocally generated by $r$ sections, finite type, of finite presentation,\ncoherent, or flat.} defined in\nModules on Sites, Definitions \\ref{sites-modules-definition-global},\n\\ref{sites-modules-definition-site-local}, and\n\\ref{sites-modules-definition-flat}.\nThe equivalences of categories\n$$\n\\QCoh(\\mathcal{O}_S)\n\\longrightarrow\n\\QCoh((\\Sch/S)_\\tau, \\mathcal{O})\n\\quad\\text{and}\\quad\n\\QCoh(\\mathcal{O}_S)\n\\longrightarrow\n\\QCoh(S_\\tau, \\mathcal{O})\n$$\ndefined by the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ seen in\nProposition \\ref{proposition-equivalence-quasi-coherent}\nhave the property\n$$\n\\mathcal{F}\\text{ has }\\mathcal{P}\n\\Leftrightarrow\n\\mathcal{F}^a\\text{ has }\\mathcal{P}\\text{ as an }\\mathcal{O}\\text{-module}\n$$\nexcept (possibly) when $\\mathcal{P}$ is ``locally free'' or ``coherent''.\nIf $\\mathcal{P}=$``coherent'' the equivalence\nholds for $\\QCoh(\\mathcal{O}_S) \\to \\QCoh(S_\\tau, \\mathcal{O})$\nwhen $S$ is locally Noetherian and $\\tau$ is Zariski or \\'etale.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VG","source_file":"descent.tex","source_line":2612,"source_end_line":2649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2612-L2649","statement_sha256":"4ccd44dff3ff80d2a651998687ed0f7cf6d8a1cb0641b66223b51c85f3712502","origin":"The Stacks Project","memory_eligible":false,"source_rank":6788,"rank":6788,"depth":22,"x":639.385,"y":1039.002,"cluster":"descent"},{"id":"stacks:03FI","tag":"03FI","title":"Cohomology of quasi-coherent modules and topologies · Lemma 03FI","summary":"Let S be a scheme. Let • [(a)] τ ∈ (Zariski, linebreak[0] fppf, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic) and C = (Sch/S)_τ, or • [(b)] let τ = etale and C = S_etale, or • [(c)] let τ = Zariski and C = S_Zar. Let F be an abelian sheaf on C. Let U ∈ Ob(C) be affine. Let U = (U_i → U)_i = 1, …, n be a standard affine τ-covering in C. Then • V = (coprod_i = 1, …, n U_i → U) is a τ-covering of U, • U is a refinement of V, and • the induced map on v Cech…","statement_latex":"Let $S$ be a scheme. Let\n\\begin{enumerate}\n\\item[(a)] $\\tau \\in \\{Zariski, \\linebreak[0] fppf, \\linebreak[0]\n\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$\nand $\\mathcal{C} = (\\Sch/S)_\\tau$, or\n\\item[(b)] let $\\tau = \\etale$ and $\\mathcal{C} = S_\\etale$, or\n\\item[(c)] let $\\tau = Zariski$ and $\\mathcal{C} = S_{Zar}$.\n\\end{enumerate}\nLet $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{C}$.\nLet $U \\in \\Ob(\\mathcal{C})$ be affine.\nLet $\\mathcal{U} = \\{U_i \\to U\\}_{i = 1, \\ldots, n}$ be a standard affine\n$\\tau$-covering in $\\mathcal{C}$. Then\n\\begin{enumerate}\n\\item $\\mathcal{V} = \\{\\coprod_{i = 1, \\ldots, n} U_i \\to U\\}$ is a\n$\\tau$-covering of $U$,\n\\item $\\mathcal{U}$ is a refinement of $\\mathcal{V}$, and\n\\item the induced map on {\\v C}ech complexes\n(Cohomology on Sites,\nEquation (\\ref{sites-cohomology-equation-map-cech-complexes}))\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{V}, \\mathcal{F})\n\\longrightarrow\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nis an isomorphism of complexes.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Cohomology of quasi-coherent modules and topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FI","source_file":"descent.tex","source_line":2686,"source_end_line":2714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2686-L2714","statement_sha256":"b83c13034347ccad3259d0a09e7d769741b950440a86895547a1d35a6d533b29","origin":"The Stacks Project","memory_eligible":false,"source_rank":6789,"rank":6789,"depth":0,"x":791.488,"y":1191.576,"cluster":"descent"},{"id":"stacks:03FJ","tag":"03FJ","title":"Cohomology of quasi-coherent modules and topologies · Lemma 03FJ","summary":"Let S be a scheme. Let F be a quasi-coherent sheaf on S. Let τ, C, U, U be as in Lemma [Tag 03FI]. Then there is an isomorphism of complexes checkC^bullet(U, F^a) ≅ s((A/R)_bullet ⊗_R M) (see Section [Tag 023F]) where R = Γ(U, O_U), M = Γ(U, F^a) and R → A is a faithfully flat ring map. In particular checkH^p(U, F^a) = 0 for all p ≥ 1.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $S$.\nLet $\\tau$, $\\mathcal{C}$, $U$, $\\mathcal{U}$ be as in\nLemma \\ref{lemma-standard-covering-Cech}. Then there is an isomorphism\nof complexes\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^a)\n\\cong\ns((A/R)_\\bullet \\otimes_R M)\n$$\n(see Section \\ref{section-descent-modules})\nwhere $R = \\Gamma(U, \\mathcal{O}_U)$, $M = \\Gamma(U, \\mathcal{F}^a)$\nand $R \\to A$ is a faithfully flat ring map. In particular\n$$\n\\check{H}^p(\\mathcal{U}, \\mathcal{F}^a) = 0\n$$\nfor all $p \\geq 1$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Cohomology of quasi-coherent modules and topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FJ","source_file":"descent.tex","source_line":2730,"source_end_line":2748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2730-L2748","statement_sha256":"4b661177f7946bd50d2b5c229dbf25413fa5bf5d1b4423c384a30cf68bd58df0","origin":"The Stacks Project","memory_eligible":false,"source_rank":6790,"rank":6790,"depth":8,"x":555.961,"y":1165.301,"cluster":"descent"},{"id":"stacks:03DW","tag":"03DW","title":"Cohomology of quasi-coherent modules and topologies · Proposition 03DW","summary":"Cohomology of quasi-coherent sheaves is the same no matter which topology you use. Let S be a scheme. Let F be a quasi-coherent sheaf on S. Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf). • There is a canonical isomorphism H^q(S, F) = H^q((Sch/S)_τ, F^a). • There are canonical isomorphisms H^q(S, F) = H^q(S_Zar, F^a) = H^q(S_etale, F^a).","statement_latex":"\\begin{slogan}\nCohomology of quasi-coherent sheaves is the same no matter which\ntopology you use.\n\\end{slogan}\nLet $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $S$.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic, \\linebreak[0] fppf\\}$.\n\\begin{enumerate}\n\\item There is a canonical isomorphism\n$$\nH^q(S, \\mathcal{F}) = H^q((\\Sch/S)_\\tau, \\mathcal{F}^a).\n$$\n\\item There are canonical isomorphisms\n$$\nH^q(S, \\mathcal{F}) =\nH^q(S_{Zar}, \\mathcal{F}^a) =\nH^q(S_\\etale, \\mathcal{F}^a).\n$$\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Cohomology of quasi-coherent modules and topologies","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DW","source_file":"descent.tex","source_line":2772,"source_end_line":2793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2772-L2793","statement_sha256":"6ad04b57f0e54c44e1c1781646976c769abbdddfd385048c00a8050d3dd565fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":6791,"rank":6791,"depth":25,"x":751.32,"y":1050.708,"cluster":"descent"},{"id":"stacks:03LC","tag":"03LC","title":"Cohomology of quasi-coherent modules and topologies · Proposition 03LC","summary":"Let f : T → S be a morphism of schemes. • The equivalences of categories of Proposition [Tag 03DX] are compatible with pullback. More precisely, we have f^*(G^a) = (f^*G)^a for any quasi-coherent sheaf G on S. • The equivalences of categories of Proposition [Tag 03DX] part (1) are not compatible with pushforward in general. • If f is quasi-compact and quasi-separated, and τ ∈ (Zariski, etale) then f_* and f_small, * preserve quasi-coherent sheaves and the diagram xymatrix…","statement_latex":"Let $f : T \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item The equivalences of categories of\nProposition \\ref{proposition-equivalence-quasi-coherent}\nare compatible with pullback.\nMore precisely, we have $f^*(\\mathcal{G}^a) = (f^*\\mathcal{G})^a$\nfor any quasi-coherent sheaf $\\mathcal{G}$ on $S$.\n\\item The equivalences of categories of\nProposition \\ref{proposition-equivalence-quasi-coherent} part (1)\nare {\\bf not} compatible with pushforward in general.\n\\item If $f$ is quasi-compact and quasi-separated, and\n$\\tau \\in \\{Zariski, \\etale\\}$ then $f_*$ and $f_{small, *}$\npreserve quasi-coherent sheaves and the diagram\n$$\n\\xymatrix{\n\\QCoh(\\mathcal{O}_T)\n\\ar[rr]_{f_*} \\ar[d]_{\\mathcal{F} \\mapsto \\mathcal{F}^a} & &\n\\QCoh(\\mathcal{O}_S)\n\\ar[d]^{\\mathcal{G} \\mapsto \\mathcal{G}^a} \\\\\n\\QCoh(T_\\tau, \\mathcal{O}) \\ar[rr]^{f_{small, *}} & &\n\\QCoh(S_\\tau, \\mathcal{O})\n}\n$$\nis commutative, i.e., $f_{small, *}(\\mathcal{F}^a) = (f_*\\mathcal{F})^a$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Cohomology of quasi-coherent modules and topologies","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LC","source_file":"descent.tex","source_line":2891,"source_end_line":2918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2891-L2918","statement_sha256":"6a3756879428107f2354f2df26809b41c92ceb8e25d064c21f7f5853c8dc342b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6792,"rank":6792,"depth":30,"x":699.271,"y":1246.611,"cluster":"descent"},{"id":"stacks:071N","tag":"071N","title":"Cohomology of quasi-coherent modules and topologies · Lemma 071N","summary":"Let f : T → S be a quasi-compact and quasi-separated morphism of schemes. Let F be a quasi-coherent sheaf on T. For either the étale or Zariski topology, there are canonical isomorphisms R^if_small, *(F^a) = (R^if_*F)^a.","statement_latex":"Let $f : T \\to S$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the \\'etale\nor Zariski topology, there are canonical isomorphisms\n$R^if_{small, *}(\\mathcal{F}^a) = (R^if_*\\mathcal{F})^a$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Cohomology of quasi-coherent modules and topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/071N","source_file":"descent.tex","source_line":2947,"source_end_line":2953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L2947-L2953","statement_sha256":"ba4a0792778a31157f4900b7433063faa8df1646b3a122cac1cbede182d677d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6793,"rank":6793,"depth":30,"x":579.776,"y":1072.131,"cluster":"descent"},{"id":"stacks:0GNA","tag":"0GNA","title":"Quasi-coherent sheaves and topologies, II · Lemma 0GNA","summary":"In Lemma [Tag 070S] the morphism of ringed sites id_small, etale, Zar : S_etale → S_Zar is flat.","statement_latex":"In Lemma \\ref{lemma-compare-sites} the morphism of ringed\nsites $\\text{id}_{small, \\etale, Zar} : S_\\etale \\to S_{Zar}$ is flat.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNA","source_file":"descent.tex","source_line":3003,"source_end_line":3007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3003-L3007","statement_sha256":"b2807a0e18b66c3ac66771501fbd35770afea9faa3f3e6c8a99ae64978eb33c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6794,"rank":6794,"depth":38,"x":808.841,"y":1133.157,"cluster":"descent"},{"id":"stacks:06VE","tag":"06VE","title":"Quasi-coherent sheaves and topologies, II · Lemma 06VE","summary":"Let S be a scheme. Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf). The functors QCoh(O_S) → Mod((Sch/S)_τ, O) and QCoh(O_S) → Mod(S_τ, O) defined by the rule F ↦ F^a seen in Proposition [Tag 03DX] are • fully faithful, • commmute with direct sums, • commmute with colimits, • right exact, • exact as a functor QCoh(O_S) → Mod(S_etale, O), • not exact as a functor QCoh(O_S) → Mod((Sch/S)_τ, O) in general, • given two…","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] \\etale,\n\\linebreak[0] smooth, \\linebreak[0] syntomic, \\linebreak[0] fppf\\}$.\nThe functors\n$$\n\\QCoh(\\mathcal{O}_S)\n\\longrightarrow\n\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})\n\\quad\\text{and}\\quad\n\\QCoh(\\mathcal{O}_S)\n\\longrightarrow\n\\textit{Mod}(S_\\tau, \\mathcal{O})\n$$\ndefined by the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ seen in\nProposition \\ref{proposition-equivalence-quasi-coherent}\nare\n\\begin{enumerate}\n\\item fully faithful,\n\\item commmute with direct sums,\n\\item commmute with colimits,\n\\item right exact,\n\\item exact as a functor\n$\\QCoh(\\mathcal{O}_S) \\to \\textit{Mod}(S_\\etale, \\mathcal{O})$,\n\\item {\\bf not} exact as a functor\n$\\QCoh(\\mathcal{O}_S) \\to\n\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$\nin general,\n\\item given two quasi-coherent $\\mathcal{O}_S$-modules\n$\\mathcal{F}$, $\\mathcal{G}$ we have\n$(\\mathcal{F} \\otimes_{\\mathcal{O}_S} \\mathcal{G})^a =\n\\mathcal{F}^a \\otimes_\\mathcal{O} \\mathcal{G}^a$,\n\\item if $\\tau = \\etale$ or $\\tau = Zariski$,\ngiven two quasi-coherent $\\mathcal{O}_S$-modules\n$\\mathcal{F}$, $\\mathcal{G}$ such that $\\mathcal{F}$\nis of finite presentation we have\n$(\\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, \\mathcal{G}))^a =\n\\SheafHom_\\mathcal{O}(\\mathcal{F}^a, \\mathcal{G}^a)$ in\n$\\textit{Mod}(S_\\tau, \\mathcal{O})$,\n\\item given two quasi-coherent $\\mathcal{O}_S$-modules\n$\\mathcal{F}$, $\\mathcal{G}$ we do {\\bf not} have\n$(\\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, \\mathcal{G}))^a =\n\\SheafHom_\\mathcal{O}(\\mathcal{F}^a, \\mathcal{G}^a)$\nin $\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$ in general\neven if $\\mathcal{F}$ is of finite presentation, and\n\\item given a short exact sequence\n$0 \\to \\mathcal{F}_1^a \\to \\mathcal{E} \\to \\mathcal{F}_2^a \\to 0$\nof $\\mathcal{O}$-modules then $\\mathcal{E}$ is\nquasi-coherent\\footnote{Warning: This is misleading. See part (6).}, i.e.,\n$\\mathcal{E}$ is in the essential image of the functor.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VE","source_file":"descent.tex","source_line":3038,"source_end_line":3090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3038-L3090","statement_sha256":"bab3882830f17ac3f15973d4cf81e48200e7663ee7a4360f4f77b195b9ba8856","origin":"The Stacks Project","memory_eligible":false,"source_rank":6795,"rank":6795,"depth":39,"x":590.249,"y":1218.37,"cluster":"descent"},{"id":"stacks:0GNB","tag":"0GNB","title":"Quasi-coherent sheaves and topologies, II · Lemma 0GNB","summary":"Let S be a scheme. The category QCoh(S_etale, O) of quasi-coherent modules on S_etale has the following properties: • Any direct sum of quasi-coherent sheaves is quasi-coherent. • Any colimit of quasi-coherent sheaves is quasi-coherent. • The kernel and cokernel of a morphism of quasi-coherent sheaves is quasi-coherent. • Given a short exact sequence of O-modules 0 → F_1 → F_2 → F_3 → 0 if two out of three are quasi-coherent so is the third. • Given two quasi-coherent…","statement_latex":"Let $S$ be a scheme. The category $\\QCoh(S_\\etale, \\mathcal{O})$\nof quasi-coherent modules on $S_\\etale$\nhas the following properties:\n\\begin{enumerate}\n\\item Any direct sum of quasi-coherent sheaves is quasi-coherent.\n\\item Any colimit of quasi-coherent sheaves is quasi-coherent.\n\\item The kernel and cokernel of a morphism of quasi-coherent sheaves\nis quasi-coherent.\n\\item Given a short exact sequence of $\\mathcal{O}$-modules\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nif two out of three are quasi-coherent so is the third.\n\\item Given two quasi-coherent $\\mathcal{O}$-modules\nthe tensor product is quasi-coherent.\n\\item Given two quasi-coherent $\\mathcal{O}$-modules\n$\\mathcal{F}$, $\\mathcal{G}$ such that $\\mathcal{F}$\nis of finite presentation.\nthen the internal hom\n$\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$\nis quasi-coherent.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNB","source_file":"descent.tex","source_line":3209,"source_end_line":3231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3209-L3231","statement_sha256":"1fb32b92ca43b2125c5e0d98d8c04ebb3ede2d9449997dd6db42f2062b2bc296","origin":"The Stacks Project","memory_eligible":false,"source_rank":6796,"rank":6796,"depth":40,"x":683.166,"y":1030.984,"cluster":"descent"},{"id":"stacks:0GNC","tag":"0GNC","title":"Quasi-coherent sheaves and topologies, II · Lemma 0GNC","summary":"Let S be a scheme. Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf). The category QCoh((Sch/S)_τ, O) of quasi-coherent modules on (Sch/S)_τ has the following properties: • Any direct sum of quasi-coherent sheaves is quasi-coherent. • Any colimit of quasi-coherent sheaves is quasi-coherent. • The cokernel of a morphism of quasi-coherent sheaves is quasi-coherent. • Given a short exact sequence of O-modules 0 → F_1 → F_2 →…","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic, \\linebreak[0] fppf\\}$.\nThe category $\\QCoh((\\Sch/S)_\\tau, \\mathcal{O})$\nof quasi-coherent modules on $(\\Sch/S)_\\tau$\nhas the following properties:\n\\begin{enumerate}\n\\item Any direct sum of quasi-coherent sheaves is quasi-coherent.\n\\item Any colimit of quasi-coherent sheaves is quasi-coherent.\n\\item The cokernel of a morphism of quasi-coherent sheaves\nis quasi-coherent.\n\\item Given a short exact sequence of $\\mathcal{O}$-modules\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nif $\\mathcal{F}_1$ and $\\mathcal{F}_3$ are quasi-coherent so is\n$\\mathcal{F}_2$.\n\\item Given two quasi-coherent $\\mathcal{O}$-modules\nthe tensor product is quasi-coherent.\n\\item Given two quasi-coherent $\\mathcal{O}$-modules\n$\\mathcal{F}$, $\\mathcal{G}$ such that $\\mathcal{F}$\nis finite locally free, the internal hom\n$\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$\nis quasi-coherent.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNC","source_file":"descent.tex","source_line":3297,"source_end_line":3322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3297-L3322","statement_sha256":"26e0b672ddf61090b9a47e7dd5cd7fcdce03e3ecd2378ac38d1d5ba205637ba4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6797,"rank":6797,"depth":40,"x":765.567,"y":1222.409,"cluster":"descent"},{"id":"stacks:0GNE","tag":"0GNE","title":"Quasi-coherent sheaves and topologies, II · Lemma 0GNE","summary":"Let S be a scheme. • The category QCoh((Sch/S)_fppf, O) has colimits and they agree with colimits in the categories Mod((Sch/S)_Zar, O), Mod((Sch/S)_etale, O), and Mod((Sch/S)_fppf, O). • Given F, G in QCoh((Sch/S)_fppf, O) the tensor products F ⊗_O G computed in Mod((Sch/S)_Zar, O), Mod((Sch/S)_etale, O), or Mod((Sch/S)_fppf, O) agree and the common value is an object of QCoh((Sch/S)_fppf, O). • Given F, G in QCoh((Sch/S)_fppf, O) with F finite locally free (in fppf, or…","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item The category $\\QCoh((\\Sch/S)_{fppf}, \\mathcal{O})$\nhas colimits and they agree with colimits in the categories\n$\\textit{Mod}((\\Sch/S)_{Zar}, \\mathcal{O})$,\n$\\textit{Mod}((\\Sch/S)_\\etale, \\mathcal{O})$, and\n$\\textit{Mod}((\\Sch/S)_{fppf}, \\mathcal{O})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\QCoh((\\Sch/S)_{fppf}, \\mathcal{O})$\nthe tensor products $\\mathcal{F} \\otimes_\\mathcal{O} \\mathcal{G}$\ncomputed in $\\textit{Mod}((\\Sch/S)_{Zar}, \\mathcal{O})$,\n$\\textit{Mod}((\\Sch/S)_\\etale, \\mathcal{O})$, or\n$\\textit{Mod}((\\Sch/S)_{fppf}, \\mathcal{O})$ agree and the common value\nis an object of $\\QCoh((\\Sch/S)_{fppf}, \\mathcal{O})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\QCoh((\\Sch/S)_{fppf}, \\mathcal{O})$\nwith $\\mathcal{F}$ finite locally free (in fppf, or equivalently \\'etale, or\nequivalently Zariski topology) the internal homs\n$\\SheafHom_\\mathcal{O}(\\mathcal{F}, \\mathcal{G})$\ncomputed in $\\textit{Mod}((\\Sch/S)_{Zar}, \\mathcal{O})$,\n$\\textit{Mod}((\\Sch/S)_\\etale, \\mathcal{O})$, or\n$\\textit{Mod}((\\Sch/S)_{fppf}, \\mathcal{O})$ agree and the common value\nis an object of $\\QCoh((\\Sch/S)_{fppf}, \\mathcal{O})$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent sheaves and topologies, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNE","source_file":"descent.tex","source_line":3397,"source_end_line":3421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3397-L3421","statement_sha256":"00fbe18889da18a7a1952eb520af4a6df4436395580363b0bd2897d5bf49a7c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6798,"rank":6798,"depth":41,"x":550.28,"y":1127.753,"cluster":"descent"},{"id":"stacks:0GZV","tag":"0GZV","title":"Quasi-coherent modules and affines · Lemma 0GZV","summary":"Let S be a scheme. Let F be a presheaf of O_Aff-modules on (Aff/S)_fppf. The following are equivalent • for every morphism U → U' of (Aff/S)_fppf the map F(U') ⊗_O(U') O(U) → F(U) is an isomorphism, • F is a sheaf on (Aff/S)_Zar and a quasi-coherent module on the ringed site ((Aff/S)_Zar, O_Aff) in the sense of Modules on Sites, Definition [Tag 03DL], • same as in (2) for the étale topology, • same as in (2) for the smooth topology, • same as in (2) for the syntomic…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a presheaf\nof $\\mathcal{O}_{\\textit{Aff}}$-modules on $(\\textit{Aff}/S)_{fppf}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every morphism $U \\to U'$ of $(\\textit{Aff}/S)_{fppf}$ the map\n$\\mathcal{F}(U') \\otimes_{\\mathcal{O}(U')} \\mathcal{O}(U) \\to \\mathcal{F}(U)$\nis an isomorphism,\n\\item $\\mathcal{F}$ is a sheaf on $(\\textit{Aff}/S)_{Zar}$ and\na quasi-coherent module on the ringed site\n$((\\textit{Aff}/S)_{Zar}, \\mathcal{O}_{\\textit{Aff}})$ in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local},\n\\item same as in (2) for the \\'etale topology,\n\\item same as in (2) for the smooth topology,\n\\item same as in (2) for the syntomic topology,\n\\item same as in (2) for the fppf topology,\n\\item $\\mathcal{F}$ corresponds to a quasi-coherent module on\n$(\\Sch/S)_{Zar}$,\n$(\\Sch/S)_\\etale$,\n$(\\Sch/S)_{smooth}$,\n$(\\Sch/S)_{syntomic}$, or\n$(\\Sch/S)_{fppf}$\nvia the equivalence (\\ref{equation-alternative-ringed}),\n\\item $\\mathcal{F}$ comes from a unique quasi-coherent\n$\\mathcal{O}_S$-module $\\mathcal{G}$ by the procedure\ndescribed in Section \\ref{section-quasi-coherent-sheaves}.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent modules and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZV","source_file":"descent.tex","source_line":3531,"source_end_line":3559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3531-L3559","statement_sha256":"8f37f81c79ec30247ecd80447fcbbef66b243dddd985545480411f3597764f52","origin":"The Stacks Project","memory_eligible":false,"source_rank":6799,"rank":6799,"depth":22,"x":785.789,"y":1075.25,"cluster":"descent"},{"id":"stacks:0GZX","tag":"0GZX","title":"Quasi-coherent modules and affines · Lemma 0GZX","summary":"Let S be a scheme. Let τ ∈ (Zariski, etale). Let F be a presheaf of O_affine-modules on S_affine, τ. The following are equivalent • for every morphism U → U' of S_affine, τ the map F(U') ⊗_O(U') O(U) → F(U) is an isomorphism, • F is a sheaf on S_affine, τ and a quasi-coherent module on the ringed site (S_affine, τ, O_affine) in the sense of Modules on Sites, Definition [Tag 03DL], • F corresponds to a quasi-coherent module on S_τ via the equivalence ([Tag 0GZW]), • F…","statement_latex":"Let $S$ be a scheme. Let $\\tau \\in \\{Zariski, \\etale\\}$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_{affine}$-modules\non $S_{affine, \\tau}$. The following are equivalent\n\\begin{enumerate}\n\\item for every morphism $U \\to U'$ of $S_{affine, \\tau}$ the map\n$\\mathcal{F}(U') \\otimes_{\\mathcal{O}(U')} \\mathcal{O}(U) \\to \\mathcal{F}(U)$\nis an isomorphism,\n\\item $\\mathcal{F}$ is a sheaf on $S_{affine, \\tau}$ and\na quasi-coherent module on the ringed site\n$(S_{affine, \\tau}, \\mathcal{O}_{affine})$ in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local},\n\\item $\\mathcal{F}$ corresponds to a quasi-coherent module on\n$S_\\tau$ via the equivalence (\\ref{equation-alternative-small-ringed}),\n\\item $\\mathcal{F}$ comes from a unique quasi-coherent\n$\\mathcal{O}_S$-module $\\mathcal{G}$ by the procedure\ndescribed in Section \\ref{section-quasi-coherent-sheaves}.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Quasi-coherent modules and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZX","source_file":"descent.tex","source_line":3657,"source_end_line":3676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3657-L3676","statement_sha256":"a8ed14c8e631b71781d344fd1baa1e9fac31ffde10a8834039e0b0565511abb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6800,"rank":6800,"depth":22,"x":653.993,"y":1248.064,"cluster":"descent"},{"id":"stacks:06ZL","tag":"06ZL","title":"Parasitic modules · Definition 06ZL","summary":"Let S be a scheme. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). Let F be a presheaf of O-modules on (Sch/S)_τ. • F is called parasitic if for every flat morphism U → S we have F(U) = 0. • F is called parasitic for the τ-topology if for every τ-covering (U_i → S)_i ∈ I we have F(U_i) = 0 for all i.","statement_latex":"Let $S$ be a scheme. Let $\\tau \\in \\{Zar, \\etale,\nsmooth, syntomic, fppf\\}$. Let $\\mathcal{F}$ be a presheaf\nof $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$.\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is called\n{\\it parasitic}\\footnote{This may be nonstandard notation.}\nif for every flat morphism $U \\to S$ we have $\\mathcal{F}(U) = 0$.\n\\item $\\mathcal{F}$ is called {\\it parasitic for the $\\tau$-topology}\nif for every $\\tau$-covering $\\{U_i \\to S\\}_{i \\in I}$ we have\n$\\mathcal{F}(U_i) = 0$ for all $i$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Parasitic modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZL","source_file":"descent.tex","source_line":3767,"source_end_line":3780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3767-L3780","statement_sha256":"2e81a75e4df5fb5d00b23c4d152343efee613ba44ad6a11ed43a46fd7036086e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6801,"rank":6801,"depth":0,"x":612.093,"y":1045.305,"cluster":"descent"},{"id":"stacks:0755","tag":"0755","title":"Parasitic modules · Lemma 0755","summary":"Let S be a scheme. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). Let G be a presheaf of O-modules on (Sch/S)_τ. • If G is parasitic for the τ-topology, then H^p_τ(U, G) = 0 for every U open in S, resp. étale over S, resp. smooth over S, resp. syntomic over S, resp. flat and locally of finite presentation over S. • If G is parasitic then H^p_τ(U, G) = 0 for every U flat over S.","statement_latex":"Let $S$ be a scheme. Let $\\tau \\in \\{Zar, \\etale, smooth,\nsyntomic, fppf\\}$. Let $\\mathcal{G}$ be a presheaf of\n$\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$.\n\\begin{enumerate}\n\\item If $\\mathcal{G}$ is parasitic for the $\\tau$-topology, then\n$H^p_\\tau(U, \\mathcal{G}) = 0$ for every $U$ open in $S$,\nresp.\\ \\'etale over $S$,\nresp.\\ smooth over $S$,\nresp.\\ syntomic over $S$,\nresp.\\ flat and locally of finite presentation over $S$.\n\\item If $\\mathcal{G}$ is parasitic then $H^p_\\tau(U, \\mathcal{G}) = 0$\nfor every $U$ flat over $S$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Parasitic modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0755","source_file":"descent.tex","source_line":3787,"source_end_line":3802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3787-L3802","statement_sha256":"4f4774388cd7ba6a7c96ce1e7a7e03eece2ccca83bc8acfb34ae7b1b52630017","origin":"The Stacks Project","memory_eligible":false,"source_rank":6802,"rank":6802,"depth":23,"x":806.562,"y":1171.378,"cluster":"descent"},{"id":"stacks:07AG","tag":"07AG","title":"Parasitic modules · Lemma 07AG","summary":"Let f : T → S be a morphism of schemes. For any parasitic O-module on (Sch/T)_τ the pushforward f_*F and the higher direct images R^if_*F are parasitic O-modules on (Sch/S)_τ.","statement_latex":"Let $f : T \\to S$ be a morphism of schemes. For any parasitic\n$\\mathcal{O}$-module on $(\\Sch/T)_\\tau$ the pushforward\n$f_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$\nare parasitic $\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Parasitic modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AG","source_file":"descent.tex","source_line":3814,"source_end_line":3820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3814-L3820","statement_sha256":"34deb7eec543a827ce8e6f2f871ad56104eb3bcc6de1d2ad5954c850b9b2917a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6803,"rank":6803,"depth":24,"x":561.127,"y":1188.806,"cluster":"descent"},{"id":"stacks:0756","tag":"0756","title":"Parasitic modules · Lemma 0756","summary":"Let S be a scheme. Let τ ∈ (Zar, etale). Let G be a sheaf of O-modules on (Sch/S)_fppf such that • G|_S_τ is quasi-coherent, and • for every flat, locally finitely presented morphism g : U → S the canonical map g_τ, small^*(G|_S_τ) → G|_U_τ is an isomorphism. Then H^p(U, G) = H^p(U, G|_U_τ) for every U flat and locally of finite presentation over S.","statement_latex":"Let $S$ be a scheme. Let $\\tau \\in \\{Zar, \\etale\\}$.\nLet $\\mathcal{G}$ be a sheaf of $\\mathcal{O}$-modules on\n$(\\Sch/S)_{fppf}$ such that\n\\begin{enumerate}\n\\item $\\mathcal{G}|_{S_\\tau}$ is quasi-coherent, and\n\\item for every flat, locally finitely presented morphism\n$g : U \\to S$ the canonical map\n$g_{\\tau, small}^*(\\mathcal{G}|_{S_\\tau}) \\to \\mathcal{G}|_{U_\\tau}$\nis an isomorphism.\n\\end{enumerate}\nThen $H^p(U, \\mathcal{G}) = H^p(U, \\mathcal{G}|_{U_\\tau})$\nfor every $U$ flat and locally of finite presentation over $S$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Parasitic modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0756","source_file":"descent.tex","source_line":3839,"source_end_line":3853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3839-L3853","statement_sha256":"2e5f03b343419534f98187cbcdd997c1284b6e2385eb1fcdae5c64bd751100ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":6804,"rank":6804,"depth":26,"x":728.535,"y":1036.287,"cluster":"descent"},{"id":"stacks:02KI","tag":"02KI","title":"Fpqc coverings are universal effective epimorphisms · Lemma 02KI","summary":"For a scheme X denote |X| the underlying set. Let f : X → S be a morphism of schemes. Then |X ×_S X| → |X| ×_|S| |X| is surjective.","statement_latex":"For a scheme $X$ denote $|X|$ the underlying set.\nLet $f : X \\to S$ be a morphism of schemes.\nThen\n$$\n|X \\times_S X| \\to |X| \\times_{|S|} |X|\n$$\nis surjective.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc coverings are universal effective epimorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KI","source_file":"descent.tex","source_line":3905,"source_end_line":3914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3905-L3914","statement_sha256":"79f4b7b5df04ec33faac8e8369818442ec9184d4bf8020637fd8e431fd38de3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6805,"rank":6805,"depth":2,"x":727.74,"y":1244.29,"cluster":"descent"},{"id":"stacks:0EUA","tag":"0EUA","title":"Fpqc coverings are universal effective epimorphisms · Lemma 0EUA","summary":"Let (f_i : X_i → X)_i ∈ I be a family of morphisms of affine schemes. The following are equivalent • for any quasi-coherent O_X-module F we have Γ(X, F) = Equalizer( xymatrix ∏_i ∈ I Γ(X_i, f_i^*F) ar@<1ex>[r] ar@<-1ex>[r] & ∏_i, j ∈ I Γ(X_i ×_X X_j, (f_i × f_j)^*F) ) • (f_i : X_i → X)_i ∈ I is a universal effective epimorphism (Sites, Definition [Tag 00WP]) in the category of affine schemes.","statement_latex":"Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of affine schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ we have\n$$\n\\Gamma(X, \\mathcal{F}) =\n\\text{Equalizer}\\left(\n\\xymatrix{\n\\prod\\nolimits_{i \\in I} \\Gamma(X_i, f_i^*\\mathcal{F})\n\\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\prod\\nolimits_{i, j \\in I}\n\\Gamma(X_i \\times_X X_j, (f_i \\times f_j)^*\\mathcal{F})\n}\n\\right)\n$$\n\\item $\\{f_i : X_i \\to X\\}_{i \\in I}$ is a universal effective epimorphism\n(Sites, Definition \\ref{sites-definition-universal-effective-epimorphisms})\nin the category of affine schemes.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc coverings are universal effective epimorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUA","source_file":"descent.tex","source_line":3922,"source_end_line":3943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L3922-L3943","statement_sha256":"5cc13046f21e4445df519865f7750f5553d57f12d20bc8d29e7f2df3cf80d725","origin":"The Stacks Project","memory_eligible":false,"source_rank":6806,"rank":6806,"depth":26,"x":560.619,"y":1090.057,"cluster":"descent"},{"id":"stacks:0EUB","tag":"0EUB","title":"Fpqc coverings are universal effective epimorphisms · Lemma 0EUB","summary":"Let (f_i : X_i → X)_i ∈ I be a family of morphisms of schemes. • If the family is universal effective epimorphism in the category of schemes, then coprod f_i is surjective. • If X and X_i are affine and the family is a universal effective epimorphism in the category of affine schemes, then coprod f_i is surjective.","statement_latex":"Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of schemes.\n\\begin{enumerate}\n\\item If the family is universal effective\nepimorphism in the category of schemes, then $\\coprod f_i$ is surjective.\n\\item If $X$ and $X_i$ are affine and the family is a universal effective\nepimorphism in the category of affine schemes, then\n$\\coprod f_i$ is surjective.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc coverings are universal effective epimorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUB","source_file":"descent.tex","source_line":4014,"source_end_line":4024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4014-L4024","statement_sha256":"29d7015612cffea35397d8e2eae3141b23719aabc56cdac2bc53de5d1fcacfe2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6807,"rank":6807,"depth":0,"x":808.526,"y":1109.008,"cluster":"descent"},{"id":"stacks:0EUC","tag":"0EUC","title":"Fpqc coverings are universal effective epimorphisms · Lemma 0EUC","summary":"Let (f_i : X_i → X)_i ∈ I be a family of morphisms of schemes. If for every morphism Y → X with Y affine the family of base changes g_i : Y_i → Y forms an effective epimorphism, then the family of f_i forms a universal effective epimorphism in the category of schemes.","statement_latex":"Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of schemes.\nIf for every morphism $Y \\to X$ with $Y$ affine the family of base changes\n$g_i : Y_i \\to Y$ forms an effective epimorphism, then\nthe family of $f_i$ forms a universal effective epimorphism\nin the category of schemes.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc coverings are universal effective epimorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUC","source_file":"descent.tex","source_line":4031,"source_end_line":4038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4031-L4038","statement_sha256":"c35b869c1d8167f94387c1a6eb759c4f36c750ca0b5a3449ad760fcfeea46478","origin":"The Stacks Project","memory_eligible":false,"source_rank":6808,"rank":6808,"depth":0,"x":609.97,"y":1236.03,"cluster":"descent"},{"id":"stacks:0EUD","tag":"0EUD","title":"Fpqc coverings are universal effective epimorphisms · Lemma 0EUD","summary":"Let (f_i : X_i → X)_i ∈ I be a family of morphisms of affine schemes. Assume the equivalent assumption of Lemma [Tag 0EUA] hold and that moreover for any morphism of affines Y → X the map coprod X_i ×_X Y → Y is a submersive map of topological spaces (Topology, Definition [Tag 0406]). Then our family of morphisms is a universal effective epimorphism in the category of schemes.","statement_latex":"Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms of affine\nschemes. Assume the equivalent assumption of\nLemma \\ref{lemma-universal-effective-epimorphism-affine} hold\nand that moreover for any morphism of affines $Y \\to X$ the map\n$$\n\\coprod X_i \\times_X Y \\longrightarrow Y\n$$\nis a submersive map of topological spaces\n(Topology, Definition \\ref{topology-definition-submersive}).\nThen our family of morphisms is a universal effective epimorphism\nin the category of schemes.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc coverings are universal effective epimorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUD","source_file":"descent.tex","source_line":4058,"source_end_line":4071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4058-L4071","statement_sha256":"4d68c0ad2ce7814c4337c73f62c48939a14786f409c47e44cce36bc995dec5b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6809,"rank":6809,"depth":27,"x":654.349,"y":1029.168,"cluster":"descent"},{"id":"stacks:03N0","tag":"03N0","title":"Fpqc coverings are universal effective epimorphisms · Lemma 03N0","summary":"Let (f_i : T_i → T)_i ∈ I be a fpqc covering. Suppose that for each i we have an open subset W_i ⊂ T_i such that for all i, j ∈ I we have pr_0^-1(W_i) = pr_1^-1(W_j) as open subsets of T_i ×_T T_j. Then there exists a unique open subset W ⊂ T such that W_i = f_i^-1(W) for each i.","statement_latex":"Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a fpqc covering.\nSuppose that for each $i$ we have an open subset $W_i \\subset T_i$\nsuch that for all $i, j \\in I$ we have\n$\\text{pr}_0^{-1}(W_i) = \\text{pr}_1^{-1}(W_j)$ as open\nsubsets of $T_i \\times_T T_j$. Then there exists a unique open subset\n$W \\subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc coverings are universal effective epimorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03N0","source_file":"descent.tex","source_line":4100,"source_end_line":4108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4100-L4108","statement_sha256":"079a15abaa638a8869629087488b99f594cfbce4087141e3b8e0624eba94b49d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6810,"rank":6810,"depth":4,"x":788.319,"y":1207.343,"cluster":"descent"},{"id":"stacks:023Q","tag":"023Q","title":"Fpqc coverings are universal effective epimorphisms · Lemma 023Q","summary":"Let (T_i → T) be an fpqc covering, see Topologies, Definition [Tag 022B]. Then (T_i → T) is a universal effective epimorphism in the category of schemes, see Sites, Definition [Tag 00WP]. In other words, every representable functor on the category of schemes satisfies the sheaf condition for the fpqc topology, see Topologies, Definition [Tag 022G].","statement_latex":"Let $\\{T_i \\to T\\}$ be an fpqc covering, see\nTopologies, Definition \\ref{topologies-definition-fpqc-covering}.\nThen $\\{T_i \\to T\\}$ is a universal effective epimorphism\nin the category of schemes, see\nSites, Definition \\ref{sites-definition-universal-effective-epimorphisms}.\nIn other words, every representable functor on the category of schemes\nsatisfies the sheaf condition for the fpqc topology, see\nTopologies, Definition \\ref{topologies-definition-sheaf-property-fpqc}.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc coverings are universal effective epimorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023Q","source_file":"descent.tex","source_line":4130,"source_end_line":4140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4130-L4140","statement_sha256":"3336d9f1901ef04d8206f6e2ddc7cc3849a24dc5364508ed07934f5926b38fbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6811,"rank":6811,"depth":40,"x":545.631,"y":1151.835,"cluster":"descent"},{"id":"stacks:0BMN","tag":"0BMN","title":"Fpqc coverings are universal effective epimorphisms · Lemma 0BMN","summary":"Consider schemes X, Y, Z and morphisms a, b : X → Y and a morphism c : Y → Z with c ∘ a = c ∘ b. Set d = c ∘ a = c ∘ b. If there exists an fpqc covering (Z_i → Z) such that • for all i the morphism Y ×_c, Z Z_i → Z_i is the coequalizer of (a, 1) : X ×_d, Z Z_i → Y ×_c, Z Z_i and (b, 1) : X ×_d, Z Z_i → Y ×_c, Z Z_i, and • for all i and i' the morphism Y ×_c, Z (Z_i ×_Z Z_i') → (Z_i ×_Z Z_i') is the coequalizer of (a, 1) : X ×_d, Z (Z_i ×_Z Z_i') → Y ×_c, Z (Z_i ×_Z Z_i')…","statement_latex":"Consider schemes $X, Y, Z$ and morphisms $a, b : X \\to Y$ and\na morphism $c : Y \\to Z$ with $c \\circ a = c \\circ b$. Set\n$d = c \\circ a = c \\circ b$. If there exists an\nfpqc covering $\\{Z_i \\to Z\\}$ such that\n\\begin{enumerate}\n\\item for all $i$ the morphism $Y \\times_{c, Z} Z_i \\to Z_i$\nis the coequalizer of $(a, 1) : X \\times_{d, Z} Z_i \\to Y \\times_{c, Z} Z_i$\nand $(b, 1) : X \\times_{d, Z} Z_i \\to Y \\times_{c, Z} Z_i$, and\n\\item for all $i$ and $i'$ the morphism\n$Y \\times_{c, Z} (Z_i \\times_Z Z_{i'}) \\to (Z_i \\times_Z Z_{i'})$\nis the coequalizer of\n$(a, 1) : X \\times_{d, Z} (Z_i \\times_Z Z_{i'}) \\to\nY \\times_{c, Z} (Z_i \\times_Z Z_{i'})$ and\n$(b, 1) : X \\times_{d, Z} (Z_i \\times_Z Z_{i'}) \\to\nY \\times_{c, Z} (Z_i \\times_Z Z_{i'})$\n\\end{enumerate}\nthen $c$ is the coequalizer of $a$ and $b$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fpqc coverings are universal effective epimorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMN","source_file":"descent.tex","source_line":4208,"source_end_line":4227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4208-L4227","statement_sha256":"9dc94d99a36e1b61f2596d776bebd26039b69a44dc78654225b8d9f78094f433","origin":"The Stacks Project","memory_eligible":false,"source_rank":6812,"rank":6812,"depth":41,"x":769.791,"y":1054.813,"cluster":"descent"},{"id":"stacks:02KK","tag":"02KK","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 02KK","summary":"Let R → A → B be ring maps. Assume R → B is of finite presentation and A → B faithfully flat and of finite presentation. Then R → A is of finite presentation.","statement_latex":"Let $R \\to A \\to B$ be ring maps.\nAssume $R \\to B$ is of finite presentation and\n$A \\to B$ faithfully flat and of finite presentation.\nThen $R \\to A$ is of finite presentation.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KK","source_file":"descent.tex","source_line":4260,"source_end_line":4266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4260-L4266","statement_sha256":"144d930d9fbce898d67703b468e44b1c44f5e85e1386362530125382c0fb0808","origin":"The Stacks Project","memory_eligible":false,"source_rank":6813,"rank":6813,"depth":35,"x":682.3,"y":1254.052,"cluster":"descent"},{"id":"stacks:0367","tag":"0367","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 0367","summary":"Let R → A → B be ring maps. Assume R → B is of finite type and A → B faithfully flat and of finite presentation. Then R → A is of finite type.","statement_latex":"Let $R \\to A \\to B$ be ring maps.\nAssume $R \\to B$ is of finite type and\n$A \\to B$ faithfully flat and of finite presentation.\nThen $R \\to A$ is of finite type.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0367","source_file":"descent.tex","source_line":4351,"source_end_line":4357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4351-L4357","statement_sha256":"953cc314c1bc00fdb2aa33ff6b08268f417c4ddef705e668df2ad599cc46610c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6814,"rank":6814,"depth":36,"x":586.353,"y":1056.998,"cluster":"descent"},{"id":"stacks:02KL","tag":"02KL","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 02KL","summary":"[EGA]. Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & S be a commutative diagram of morphisms of schemes. Assume that f is surjective, flat and locally of finite presentation and assume that p is locally of finite presentation (resp. locally of finite type). Then q is locally of finite presentation (resp. locally of finite type).","statement_latex":"\\begin{reference}\n\\cite[IV, 17.7.5 (i) and (ii)]{EGA}.\n\\end{reference}\nLet\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume that $f$ is\nsurjective, flat and locally of finite presentation and assume\nthat $p$ is locally of finite presentation (resp.\\ locally of finite type).\nThen $q$ is locally of finite presentation (resp.\\ locally of finite type).","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KL","source_file":"descent.tex","source_line":4384,"source_end_line":4401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4384-L4401","statement_sha256":"0eb628ad5cc424d520aa3bf1173b2cc2b242d01fd661895b12b567b8eb37874e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6815,"rank":6815,"depth":37,"x":816.141,"y":1148.087,"cluster":"descent"},{"id":"stacks:02KM","tag":"02KM","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 02KM","summary":"Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & S be a commutative diagram of morphisms of schemes. Assume that • f is surjective, and syntomic (resp. smooth, resp. étale), • p is syntomic (resp. smooth, resp. étale). Then q is syntomic (resp. smooth, resp. étale).","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume that\n\\begin{enumerate}\n\\item $f$ is surjective, and syntomic (resp.\\ smooth, resp.\\ \\'etale),\n\\item $p$ is syntomic (resp.\\ smooth, resp.\\ \\'etale).\n\\end{enumerate}\nThen $q$ is syntomic (resp.\\ smooth, resp.\\ \\'etale).","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KM","source_file":"descent.tex","source_line":4422,"source_end_line":4438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4422-L4438","statement_sha256":"c7962c2d058de03c631f192da918a50878ff8ac268c7eeae993117785322fbd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6816,"rank":6816,"depth":41,"x":572.843,"y":1211.463,"cluster":"descent"},{"id":"stacks:05B5","tag":"05B5","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 05B5","summary":"Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & S be a commutative diagram of morphisms of schemes. Assume that • f is surjective, flat, and locally of finite presentation, • p is smooth (resp. étale). Then q is smooth (resp. étale).","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume that\n\\begin{enumerate}\n\\item $f$ is surjective, flat, and locally of finite presentation,\n\\item $p$ is smooth (resp.\\ \\'etale).\n\\end{enumerate}\nThen $q$ is smooth (resp.\\ \\'etale).","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05B5","source_file":"descent.tex","source_line":4451,"source_end_line":4467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4451-L4467","statement_sha256":"8ddff6c34f4efe07de5c4269eba707e1b1626d352ea5efb7e42d683fc7214c92","origin":"The Stacks Project","memory_eligible":false,"source_rank":6817,"rank":6817,"depth":45,"x":701.6,"y":1026.22,"cluster":"descent"},{"id":"stacks:05B7","tag":"05B7","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 05B7","summary":"Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & S be a commutative diagram of morphisms of schemes. Assume that • f is surjective, flat, and locally of finite presentation, • p is syntomic. Then both q and f are syntomic.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume that\n\\begin{enumerate}\n\\item $f$ is surjective, flat, and locally of finite presentation,\n\\item $p$ is syntomic.\n\\end{enumerate}\nThen both $q$ and $f$ are syntomic.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05B7","source_file":"descent.tex","source_line":4495,"source_end_line":4511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4495-L4511","statement_sha256":"299839d86b201b5c42e0e45c098874a744f9e58360a875a19a3d13a2a6a2439b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6818,"rank":6818,"depth":38,"x":755.756,"y":1236.392,"cluster":"descent"},{"id":"stacks:06NB","tag":"06NB","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 06NB","summary":"Let X → Y → Z be morphism of schemes. Let P be one of the following properties of morphisms of schemes: flat, locally finite type, locally finite presentation. Assume that X → Z has P and that (X → Y) can be refined by an fppf covering of Y. Then Y → Z is P.","statement_latex":"Let $X \\to Y \\to Z$ be morphism of schemes.\nLet $P$ be one of the following properties of morphisms of schemes:\nflat, locally finite type, locally finite presentation.\nAssume that $X \\to Z$ has $P$ and that $\\{X \\to Y\\}$\ncan be refined by an fppf covering of $Y$. Then $Y \\to Z$ is $P$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NB","source_file":"descent.tex","source_line":4540,"source_end_line":4547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4540-L4547","statement_sha256":"b777fec387aa56711aa88662e9eba09ec8ce87b3dfe9b29f3f6608626d00a8d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6819,"rank":6819,"depth":37,"x":546.296,"y":1111.839,"cluster":"descent"},{"id":"stacks:0348","tag":"0348","title":"Local properties of schemes · Definition 0348","summary":"Let P be a property of schemes. Let τ ∈ (fpqc, linebreak[0] fppf, linebreak[0] syntomic, linebreak[0] smooth, linebreak[0] etale, linebreak[0] Zariski). We say P is local in the τ-topology if for any τ-covering (S_i → S)_i ∈ I (see Topologies, Section [Tag 020M]) we have S has P ⇔ each S_i has P.","statement_latex":"Let $\\mathcal{P}$ be a property of schemes. Let\n$\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0]\nsmooth, \\linebreak[0] \\etale, \\linebreak[0] Zariski\\}$.\nWe say $\\mathcal{P}$ is {\\it local in the $\\tau$-topology} if for any\n$\\tau$-covering $\\{S_i \\to S\\}_{i \\in I}$ (see\nTopologies, Section \\ref{topologies-section-procedure})\nwe have\n$$\nS \\text{ has }\\mathcal{P}\n\\Leftrightarrow\n\\text{each }S_i \\text{ has }\\mathcal{P}.\n$$","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Local properties of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0348","source_file":"descent.tex","source_line":4652,"source_end_line":4666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4652-L4666","statement_sha256":"b3c510ef9e874d64ee319d6562481fa9fb53dc1515f0f72d7352126da51adab9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6820,"rank":6820,"depth":0,"x":801.532,"y":1084.765,"cluster":"descent"},{"id":"stacks:0349","tag":"0349","title":"Local properties of schemes · Lemma 0349","summary":"Let P be a property of schemes. Let τ ∈ (fpqc, linebreak[0] fppf, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic). Assume that • the property is local in the Zariski topology, • for any morphism of affine schemes S' → S which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether τ is fpqc, fppf, étale, smooth, or syntomic, property P holds for S' if property P holds for S, and • for any surjective morphism of affine schemes…","statement_latex":"Let $\\mathcal{P}$ be a property of schemes.\nLet $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0]\n\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.\nAssume that\n\\begin{enumerate}\n\\item the property is local in the Zariski topology,\n\\item for any morphism of affine schemes $S' \\to S$\nwhich is flat, flat of finite presentation,\n\\'etale, smooth or syntomic depending on whether $\\tau$ is\nfpqc, fppf, \\'etale, smooth, or syntomic,\nproperty $\\mathcal{P}$ holds for $S'$ if property $\\mathcal{P}$\nholds for $S$, and\n\\item for any surjective morphism of affine schemes $S' \\to S$\nwhich is flat, flat of finite presentation,\n\\'etale, smooth or syntomic depending on whether $\\tau$ is\nfpqc, fppf, \\'etale, smooth, or syntomic,\nproperty $\\mathcal{P}$ holds for $S$ if property $\\mathcal{P}$\nholds for $S'$.\n\\end{enumerate}\nThen $\\mathcal{P}$ is $\\tau$ local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Local properties of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0349","source_file":"descent.tex","source_line":4700,"source_end_line":4722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4700-L4722","statement_sha256":"c16adbb05f9801a4dfae5873c2206fa3dfa3a70acef8f892368293fde551c1c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6821,"rank":6821,"depth":36,"x":634.693,"y":1249.955,"cluster":"descent"},{"id":"stacks:034C","tag":"034C","title":"Properties of schemes local in the fppf topology · Lemma 034C","summary":"The property P(S) =\"S is locally Noetherian\" is local in the fppf topology.","statement_latex":"The property $\\mathcal{P}(S) =$``$S$ is locally Noetherian'' is local\nin the fppf topology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034C","source_file":"descent.tex","source_line":4761,"source_end_line":4765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4761-L4765","statement_sha256":"4673330127b2f1c62bc907c438c7fcc4e345329e953bab13a01e77b61c7435d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6822,"rank":6822,"depth":37,"x":624.853,"y":1032.955,"cluster":"descent"},{"id":"stacks:0368","tag":"0368","title":"Properties of schemes local in the fppf topology · Lemma 0368","summary":"The property P(S) =\"S is Jacobson\" is local in the fppf topology.","statement_latex":"The property $\\mathcal{P}(S) =$``$S$ is Jacobson'' is local\nin the fppf topology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0368","source_file":"descent.tex","source_line":4784,"source_end_line":4788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4784-L4788","statement_sha256":"33fedc2a69dd9fad3963a80cd34b37bd88b0836be3548bd7966f815861603052","origin":"The Stacks Project","memory_eligible":false,"source_rank":6823,"rank":6823,"depth":37,"x":807.051,"y":1187.755,"cluster":"descent"},{"id":"stacks:0BAL","tag":"0BAL","title":"Properties of schemes local in the fppf topology · Lemma 0BAL","summary":"The property P(S) =\"every quasi-compact open of S has a finite number of irreducible components\" is local in the fppf topology.","statement_latex":"The property $\\mathcal{P}(S) =$``every quasi-compact open of $S$\nhas a finite number of irreducible components'' is local\nin the fppf topology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAL","source_file":"descent.tex","source_line":4852,"source_end_line":4857,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4852-L4857","statement_sha256":"efb9edafa64d5f200be971a5323481a536f84381ba0bf21ffbc4cc06a7c14a90","origin":"The Stacks Project","memory_eligible":false,"source_rank":6824,"rank":6824,"depth":37,"x":547.596,"y":1176.966,"cluster":"descent"},{"id":"stacks:036A","tag":"036A","title":"Properties of schemes local in the syntomic topology · Lemma 036A","summary":"The property P(S) =\"S is locally Noetherian and (S_k)\" is local in the syntomic topology.","statement_latex":"The property $\\mathcal{P}(S) =$``$S$ is locally Noetherian and $(S_k)$''\nis local in the syntomic topology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036A","source_file":"descent.tex","source_line":4892,"source_end_line":4896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4892-L4896","statement_sha256":"cd916310d023b53f4f126cdb0de0e8b63a2529d062f71f866c710f863992065e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6825,"rank":6825,"depth":38,"x":748.067,"y":1037.37,"cluster":"descent"},{"id":"stacks:036B","tag":"036B","title":"Properties of schemes local in the syntomic topology · Lemma 036B","summary":"The property P(S) =\"S is Cohen-Macaulay\" is local in the syntomic topology.","statement_latex":"The property $\\mathcal{P}(S) =$``$S$ is Cohen-Macaulay''\nis local in the syntomic topology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036B","source_file":"descent.tex","source_line":4925,"source_end_line":4929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4925-L4929","statement_sha256":"8a4e2ef822fc65e24995204b1fda4aa431dc6e5fe3c53f6496f74ecc9b93b2cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6826,"rank":6826,"depth":39,"x":712.417,"y":1254.571,"cluster":"descent"},{"id":"stacks:034E","tag":"034E","title":"Properties of schemes local in the smooth topology · Lemma 034E","summary":"The property P(S) =\"S is reduced\" is local in the smooth topology.","statement_latex":"The property $\\mathcal{P}(S) =$``$S$ is reduced'' is local in the smooth\ntopology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034E","source_file":"descent.tex","source_line":4951,"source_end_line":4955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4951-L4955","statement_sha256":"367f0594d66705ba7c5be94a76efae458204e0fdbe313547424c77cd19fa38cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":6827,"rank":6827,"depth":37,"x":563.689,"y":1073.756,"cluster":"descent"},{"id":"stacks:034F","tag":"034F","title":"Properties of schemes local in the smooth topology · Lemma 034F","summary":"Normality is local in the smooth topology. The property P(S) =\"S is normal\" is local in the smooth topology.","statement_latex":"\\begin{slogan}\nNormality is local in the smooth topology.\n\\end{slogan}\nThe property $\\mathcal{P}(S) =$``$S$ is normal'' is local in the smooth\ntopology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/034F","source_file":"descent.tex","source_line":4973,"source_end_line":4980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4973-L4980","statement_sha256":"a3335eb867fe7b70ffcd86fc6fa6fd00a1edd66375448abdeaf72433155bb73e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6828,"rank":6828,"depth":37,"x":819.363,"y":1122.809,"cluster":"descent"},{"id":"stacks:036C","tag":"036C","title":"Properties of schemes local in the smooth topology · Lemma 036C","summary":"The property P(S) =\"S is locally Noetherian and (R_k)\" is local in the smooth topology.","statement_latex":"The property $\\mathcal{P}(S) =$``$S$ is locally Noetherian and $(R_k)$''\nis local in the smooth topology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036C","source_file":"descent.tex","source_line":4998,"source_end_line":5002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L4998-L5002","statement_sha256":"c9be4b723f5d71049faf891e595db8fb85acb100521c033bace979bf37557ff5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6829,"rank":6829,"depth":38,"x":590.853,"y":1231.968,"cluster":"descent"},{"id":"stacks:036D","tag":"036D","title":"Properties of schemes local in the smooth topology · Lemma 036D","summary":"The property P(S) =\"S is regular\" is local in the smooth topology.","statement_latex":"The property $\\mathcal{P}(S) =$``$S$ is regular''\nis local in the smooth topology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036D","source_file":"descent.tex","source_line":5029,"source_end_line":5033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5029-L5033","statement_sha256":"3a6c0e1f018d63680da7489d8f88c726efc2c2900a790f163e02756c66b11c3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6830,"rank":6830,"depth":39,"x":671.759,"y":1021.326,"cluster":"descent"},{"id":"stacks:036E","tag":"036E","title":"Properties of schemes local in the smooth topology · Lemma 036E","summary":"The property P(S) =\"S is Nagata\" is local in the smooth topology.","statement_latex":"The property $\\mathcal{P}(S) =$``$S$ is Nagata''\nis local in the smooth topology.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of schemes local in the smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036E","source_file":"descent.tex","source_line":5041,"source_end_line":5045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5041-L5045","statement_sha256":"fd767503e6f94323ad83701e827ef454bb65312d1b7eaa1678afbd0b2f535567","origin":"The Stacks Project","memory_eligible":false,"source_rank":6831,"rank":6831,"depth":37,"x":781.746,"y":1223.023,"cluster":"descent"},{"id":"stacks:06QM","tag":"06QM","title":"Variants on descending properties · Lemma 06QM","summary":"If f : X → Y is a flat and surjective morphism of schemes and X is reduced, then Y is reduced.","statement_latex":"If $f : X \\to Y$ is a flat and surjective morphism of schemes\nand $X$ is reduced, then $Y$ is reduced.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Variants on descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QM","source_file":"descent.tex","source_line":5076,"source_end_line":5080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5076-L5080","statement_sha256":"4470647ccadbb6afa3e0e095b966016ada7df785325227bc008e73088602c101","origin":"The Stacks Project","memory_eligible":false,"source_rank":6832,"rank":6832,"depth":2,"x":537.884,"y":1136.505,"cluster":"descent"},{"id":"stacks:06QN","tag":"06QN","title":"Variants on descending properties · Lemma 06QN","summary":"Let f : X → Y be a morphism of algebraic spaces. If f is locally of finite presentation, flat, and surjective and X is regular, then Y is regular.","statement_latex":"Let $f : X \\to Y$ be a morphism of algebraic spaces.\nIf $f$ is locally of finite presentation, flat, and surjective and\n$X$ is regular, then $Y$ is regular.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Variants on descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QN","source_file":"descent.tex","source_line":5089,"source_end_line":5094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5089-L5094","statement_sha256":"c1d8ec0e180cf55ded5a2bf8dd756a5b71af04dd8a70942a92219ca83b733249","origin":"The Stacks Project","memory_eligible":false,"source_rank":6833,"rank":6833,"depth":18,"x":787.851,"y":1061.759,"cluster":"descent"},{"id":"stacks:04QR","tag":"04QR","title":"Germs of schemes · Definition 04QR","summary":"Germs of schemes. • A pair (X, x) consisting of a scheme X and a point x ∈ X is called the germ of X at x. • A morphism of germs f : (X, x) → (S, s) is an equivalence class of morphisms of schemes f : U → S with f(x) = s where U ⊂ X is an open neighbourhood of x. Two such f, f' are said to be equivalent if and only if f and f' agree in some open neighbourhood of x. • We define the composition of morphisms of germs by composing representatives (this is well defined).","statement_latex":"Germs of schemes.\n\\begin{enumerate}\n\\item A pair $(X, x)$ consisting of a scheme $X$ and a point $x \\in X$ is\ncalled the {\\it germ of $X$ at $x$}.\n\\item A {\\it morphism of germs} $f : (X, x) \\to (S, s)$\nis an equivalence class of morphisms of schemes $f : U \\to S$ with $f(x) = s$\nwhere $U \\subset X$ is an open neighbourhood of $x$. Two such\n$f$, $f'$ are said to be equivalent if and only if $f$ and $f'$\nagree in some open neighbourhood of $x$.\n\\item We define the {\\it composition of morphisms of germs}\nby composing representatives (this is well defined).\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Germs of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QR","source_file":"descent.tex","source_line":5117,"source_end_line":5131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5117-L5131","statement_sha256":"d988602feb2e08315cb00a1741c2bc0972b6a1d42078943454820c88ba90770e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6834,"rank":6834,"depth":0,"x":663.354,"y":1259.162,"cluster":"descent"},{"id":"stacks:04QS","tag":"04QS","title":"Germs of schemes · Definition 04QS","summary":"Let f : (X, x) → (S, s) be a morphism of germs. We say f is étale (resp. smooth) if there exists a representative f : U → S of f which is an étale morphism (resp. a smooth morphism) of schemes.","statement_latex":"Let $f : (X, x) \\to (S, s)$ be a morphism of germs.\nWe say $f$ is {\\it \\'etale} (resp.\\ {\\it smooth}) if there exists a\nrepresentative $f : U \\to S$ of $f$ which is an \\'etale morphism\n(resp.\\ a smooth morphism) of schemes.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Germs of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QS","source_file":"descent.tex","source_line":5136,"source_end_line":5142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5136-L5142","statement_sha256":"de0f9b2da2066ea576f7ff5d3e679827be65530804cc2942d0e96dcd7cfc2416","origin":"The Stacks Project","memory_eligible":false,"source_rank":6835,"rank":6835,"depth":0,"x":596.259,"y":1042.466,"cluster":"descent"},{"id":"stacks:04N1","tag":"04N1","title":"Local properties of germs · Definition 04N1","summary":"Let P be a property of germs of schemes. We say that P is étale local (resp. smooth local) if for any étale (resp. smooth) morphism of germs (U', u') → (U, u) we have P(U, u) ⇔ P(U', u').","statement_latex":"Let $\\mathcal{P}$ be a property of germs of schemes.\nWe say that $\\mathcal{P}$ is {\\it \\'etale local}\n(resp.\\ {\\it smooth local}) if for any\n\\'etale (resp.\\ smooth) morphism of germs $(U', u') \\to (U, u)$\nwe have $\\mathcal{P}(U, u) \\Leftrightarrow \\mathcal{P}(U', u')$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Local properties of germs","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04N1","source_file":"descent.tex","source_line":5154,"source_end_line":5161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5154-L5161","statement_sha256":"bc013575b02a4dc08741350a4dc2cdf47a68994b753a4c2106e169ef3670f8d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6836,"rank":6836,"depth":0,"x":820.499,"y":1164.458,"cluster":"descent"},{"id":"stacks:04N4","tag":"04N4","title":"Local properties of germs · Lemma 04N4","summary":"Let f : U → V be an étale morphism of schemes. Let u ∈ U and v = f(u). Then dim_u(U) = dim_v(V).","statement_latex":"Let $f : U \\to V$ be an \\'etale morphism of schemes.\nLet $u \\in U$ and $v = f(u)$. Then $\\dim_u(U) = \\dim_v(V)$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Local properties of germs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04N4","source_file":"descent.tex","source_line":5172,"source_end_line":5176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5172-L5176","statement_sha256":"f8ebda2497d018721f58fa2672ea4c318b1797b626d5d076b7c645b3aa802f1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6837,"rank":6837,"depth":45,"x":556.453,"y":1201.826,"cluster":"descent"},{"id":"stacks:04N8","tag":"04N8","title":"Local properties of germs · Lemma 04N8","summary":"Let f : U → V be an étale morphism of schemes. Let u ∈ U and v = f(u). Then dim(O_U, u) = dim(O_V, v).","statement_latex":"Let $f : U \\to V$ be an \\'etale morphism of schemes.\nLet $u \\in U$ and $v = f(u)$. Then\n$\\dim(\\mathcal{O}_{U, u}) = \\dim(\\mathcal{O}_{V, v})$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Local properties of germs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04N8","source_file":"descent.tex","source_line":5236,"source_end_line":5241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5236-L5241","statement_sha256":"e3acbe1d36ab33c8e93f1f8b9bd791c616e59c0c1341232f7da3f479b16f6fc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6838,"rank":6838,"depth":14,"x":721.475,"y":1024.048,"cluster":"descent"},{"id":"stacks:0AH7","tag":"0AH7","title":"Local properties of germs · Lemma 0AH7","summary":"Let f : U → V be an étale morphism of schemes. Let u ∈ U and v = f(u). Then O_U, u is a regular local ring if and only if O_V, v is a regular local ring.","statement_latex":"Let $f : U \\to V$ be an \\'etale morphism of schemes.\nLet $u \\in U$ and $v = f(u)$. Then\n$\\mathcal{O}_{U, u}$ is a regular local ring if and only if\n$\\mathcal{O}_{V, v}$ is a regular local ring.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Local properties of germs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AH7","source_file":"descent.tex","source_line":5259,"source_end_line":5265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5259-L5265","statement_sha256":"7c1bd07492aefad13a029450268bbb60ed4d0b97b4efe3d97c17f02bf711f7fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":6839,"rank":6839,"depth":42,"x":742.801,"y":1249.277,"cluster":"descent"},{"id":"stacks:02KO","tag":"02KO","title":"Properties of morphisms local on the target · Definition 02KO","summary":"Let P be a property of morphisms of schemes over a base. Let τ ∈ (fpqc, fppf, syntomic, smooth, etale, Zariski). We say P is τ local on the base, or τ local on the target, or local on the base for the τ-topology if for any τ-covering (Y_i → Y)_i ∈ I (see Topologies, Section [Tag 020M]) and any morphism of schemes f : X → Y over S we have f has P ⇔ each Y_i ×_Y X → Y_i has P.","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.\nLet $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale, Zariski\\}$.\nWe say $\\mathcal{P}$ is {\\it $\\tau$ local on the base}, or\n{\\it $\\tau$ local on the target}, or\n{\\it local on the base for the $\\tau$-topology} if for any\n$\\tau$-covering $\\{Y_i \\to Y\\}_{i \\in I}$ (see\nTopologies, Section \\ref{topologies-section-procedure})\nand any morphism of schemes $f : X \\to Y$ over $S$ we\nhave\n$$\nf \\text{ has }\\mathcal{P}\n\\Leftrightarrow\n\\text{each }Y_i \\times_Y X \\to Y_i\\text{ has }\\mathcal{P}.\n$$","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local on the target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KO","source_file":"descent.tex","source_line":5309,"source_end_line":5325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5309-L5325","statement_sha256":"950381acfb1c6d8fd2f74c9ef758314d7d4e3e1e6f92c689e6d848b0c9921a59","origin":"The Stacks Project","memory_eligible":false,"source_rank":6840,"rank":6840,"depth":0,"x":545.517,"y":1094.956,"cluster":"descent"},{"id":"stacks:04QU","tag":"04QU","title":"Properties of morphisms local on the target · Lemma 04QU","summary":"Let τ ∈ (fpqc, fppf, syntomic, smooth, etale, Zariski). Let P be a property of morphisms which is τ local on the target. Let f : X → Y have property P. For any morphism Y' → Y which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. étale, resp. an open immersion, the base change f' : Y' ×_Y X → Y' of f has property P.","statement_latex":"Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale, Zariski\\}$.\nLet $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local\non the target. Let $f : X \\to Y$ have property $\\mathcal{P}$.\nFor any morphism $Y' \\to Y$ which is\nflat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic,\nresp.\\ \\'etale, resp.\\ an open immersion, the base change\n$f' : Y' \\times_Y X \\to Y'$ of $f$ has property $\\mathcal{P}$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QU","source_file":"descent.tex","source_line":5335,"source_end_line":5344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5335-L5344","statement_sha256":"b21447d3976d28bc26b9b11f7a762e6a7ed50f1f46011d63e67268fb6bf84278","origin":"The Stacks Project","memory_eligible":false,"source_rank":6841,"rank":6841,"depth":0,"x":815.688,"y":1096.811,"cluster":"descent"},{"id":"stacks:06QP","tag":"06QP","title":"Properties of morphisms local on the target · Lemma 06QP","summary":"Let τ ∈ (fppf, syntomic, smooth, etale). Let P be a property of morphisms which is τ local on the target. For any morphism of schemes f : X → Y there exists a largest open W(f) ⊂ Y such that the restriction X_W(f) → W(f) has P. Moreover, • if g : Y' → Y is flat and locally of finite presentation, syntomic, smooth, or étale and the base change f' : X_Y' → Y' has P, then g(Y') ⊂ W(f), • if g : Y' → Y is flat and locally of finite presentation, syntomic, smooth, or étale,…","statement_latex":"Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale\\}$.\nLet $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local\non the target. For any morphism of schemes $f : X \\to Y$ there exists\na largest open $W(f) \\subset Y$ such that the restriction\n$X_{W(f)} \\to W(f)$ has $\\mathcal{P}$. Moreover,\n\\begin{enumerate}\n\\item if $g : Y' \\to Y$ is flat and locally of finite presentation,\nsyntomic, smooth, or \\'etale and the base change $f' : X_{Y'} \\to Y'$\nhas $\\mathcal{P}$, then $g(Y') \\subset W(f)$,\n\\item if $g : Y' \\to Y$ is flat and locally of finite presentation,\nsyntomic, smooth, or \\'etale, then $W(f') = g^{-1}(W(f))$, and\n\\item if $\\{g_i : Y_i \\to Y\\}$ is a $\\tau$-covering, then\n$g_i^{-1}(W(f)) = W(f_i)$, where $f_i$ is the base change of $f$\nby $Y_i \\to Y$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QP","source_file":"descent.tex","source_line":5359,"source_end_line":5376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5359-L5376","statement_sha256":"57c460662b426c632692a0c4b049ece2d4e8e14f01a8273a87baf401251eba36","origin":"The Stacks Project","memory_eligible":false,"source_rank":6842,"rank":6842,"depth":18,"x":614.534,"y":1249.078,"cluster":"descent"},{"id":"stacks:02KP","tag":"02KP","title":"Properties of morphisms local on the target · Lemma 02KP","summary":"Let P be a property of morphisms of schemes over a base. Let τ ∈ (fpqc, fppf, etale, smooth, syntomic). Assume that • the property is preserved under flat, flat and locally of finite presentation, étale, smooth, or syntomic base change depending on whether τ is fpqc, fppf, étale, smooth, or syntomic (compare with Schemes, Definition [Tag 01JZ]), • the property is Zariski local on the base. • for any surjective morphism of affine schemes S' → S which is flat, flat of…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.\nLet $\\tau \\in \\{fpqc, fppf, \\etale, smooth, syntomic\\}$.\nAssume that\n\\begin{enumerate}\n\\item the property is preserved under\nflat, flat and locally of finite presentation, \\'etale, smooth, or syntomic\nbase change depending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or\nsyntomic (compare with\nSchemes, Definition \\ref{schemes-definition-preserved-by-base-change}),\n\\item the property is Zariski local on the base.\n\\item for any surjective morphism of affine schemes $S' \\to S$\nwhich is flat, flat of finite presentation,\n\\'etale, smooth or syntomic depending on whether $\\tau$ is\nfpqc, fppf, \\'etale, smooth, or syntomic,\nand any morphism of schemes $f : X \\to S$ property\n$\\mathcal{P}$ holds for $f$ if property $\\mathcal{P}$\nholds for the base change $f' : X' = S' \\times_S X \\to S'$.\n\\end{enumerate}\nThen $\\mathcal{P}$ is $\\tau$ local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KP","source_file":"descent.tex","source_line":5406,"source_end_line":5427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5406-L5427","statement_sha256":"26a041b6a7be6d7d9c70e1e4f4fdaba5c9b498204e786ce68f8ded0529eae162","origin":"The Stacks Project","memory_eligible":false,"source_rank":6843,"rank":6843,"depth":36,"x":640.47,"y":1022.164,"cluster":"descent"},{"id":"stacks:02KQ","tag":"02KQ","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KQ","summary":"The property P(f) =\"f is quasi-compact\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is quasi-compact''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KQ","source_file":"descent.tex","source_line":5470,"source_end_line":5474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5470-L5474","statement_sha256":"59061219405bfcf9191402e259d3ce9c3efa9f4b7fab7681c81a026071cd9e73","origin":"The Stacks Project","memory_eligible":false,"source_rank":6844,"rank":6844,"depth":37,"x":804.179,"y":1204.6,"cluster":"descent"},{"id":"stacks:02KR","tag":"02KR","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KR","summary":"The property P(f) =\"f is quasi-separated\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is quasi-separated''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KR","source_file":"descent.tex","source_line":5490,"source_end_line":5494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5490-L5494","statement_sha256":"c5ea0cdca23d40d22cafb02fbfb0b756fdd6768fd6e4cc5d1dca1ef4b274432b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6845,"rank":6845,"depth":38,"x":536.172,"y":1162.876,"cluster":"descent"},{"id":"stacks:02KS","tag":"02KS","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KS","summary":"The property P(f) =\"f is universally closed\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is universally closed''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KS","source_file":"descent.tex","source_line":5514,"source_end_line":5518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5514-L5518","statement_sha256":"4334f44a11d3f10807a73b4de5307f3b6ae3f69c21870d69a5c088fed59ba7cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6846,"rank":6846,"depth":37,"x":767.849,"y":1041.308,"cluster":"descent"},{"id":"stacks:02KT","tag":"02KT","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KT","summary":"The property P(f) =\"f is universally open\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is universally open''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KT","source_file":"descent.tex","source_line":5555,"source_end_line":5559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5555-L5559","statement_sha256":"7f2368f4497d441d30ef06b01c0c3e7c3e9ba28f37a1c59f5cf458190d481581","origin":"The Stacks Project","memory_eligible":false,"source_rank":6847,"rank":6847,"depth":38,"x":694.619,"y":1262.885,"cluster":"descent"},{"id":"stacks:0CEW","tag":"0CEW","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 0CEW","summary":"The property P(f) =\"f is universally submersive\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is universally submersive''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEW","source_file":"descent.tex","source_line":5566,"source_end_line":5570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5566-L5570","statement_sha256":"96fdb5e7c220df003df94ef12262d8740e34fccd9ee547b9092b117ddb968732","origin":"The Stacks Project","memory_eligible":false,"source_rank":6848,"rank":6848,"depth":38,"x":570.166,"y":1057.506,"cluster":"descent"},{"id":"stacks:02KU","tag":"02KU","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KU","summary":"The property P(f) =\"f is separated\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is separated''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KU","source_file":"descent.tex","source_line":5580,"source_end_line":5584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5580-L5584","statement_sha256":"97c027685279b12307797d4294039b28bcd17764757362b3047abcc2b3aad1d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6849,"rank":6849,"depth":38,"x":827.643,"y":1138.504,"cluster":"descent"},{"id":"stacks:02KV","tag":"02KV","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KV","summary":"The property P(f) =\"f is surjective\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is surjective''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KV","source_file":"descent.tex","source_line":5608,"source_end_line":5612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5608-L5612","statement_sha256":"3e15938fd4adf118c868b89240c680f1ad264443e154f43d8eb3fdfae7565b11","origin":"The Stacks Project","memory_eligible":false,"source_rank":6850,"rank":6850,"depth":0,"x":572.105,"y":1225.059,"cluster":"descent"},{"id":"stacks:0H8H","tag":"0H8H","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 0H8H","summary":"The property P(f) =\"f is quasi-compact and dominant\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is quasi-compact and dominant''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8H","source_file":"descent.tex","source_line":5618,"source_end_line":5622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5618-L5622","statement_sha256":"4929a7dc298120d62142b8f3eab1568f731e171e027d4ca5630c7bc95ae1d9df","origin":"The Stacks Project","memory_eligible":false,"source_rank":6851,"rank":6851,"depth":38,"x":691.182,"y":1015.795,"cluster":"descent"},{"id":"stacks:0H8I","tag":"0H8I","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 0H8I","summary":"Let E/k be a field extension. Then a morphism X → Y over k is dominant if and only if the pullback X_E → Y_E is dominant.","statement_latex":"Let $E/k$ be a field extension. Then a morphism\n$X \\to Y$ over $k$ is dominant if and only if\nthe pullback $X_E \\to Y_E$ is dominant.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8I","source_file":"descent.tex","source_line":5644,"source_end_line":5649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5644-L5649","statement_sha256":"7141c23e3b856aa06e5dc168d3fff0163bc254c689c6d1cf38afae7599d8be3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6852,"rank":6852,"depth":10,"x":771.829,"y":1238.137,"cluster":"descent"},{"id":"stacks:02KW","tag":"02KW","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KW","summary":"The property P(f) =\"f is universally injective\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is universally injective''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KW","source_file":"descent.tex","source_line":5666,"source_end_line":5670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5666-L5670","statement_sha256":"545b119c25a7022b4c0600163bf79756c521a5e569655859a8f56cbc62727020","origin":"The Stacks Project","memory_eligible":false,"source_rank":6853,"rank":6853,"depth":37,"x":533.061,"y":1119.699,"cluster":"descent"},{"id":"stacks:0CEX","tag":"0CEX","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 0CEX","summary":"The property P(f) =\"f is a universal homeomorphism\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a universal homeomorphism''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEX","source_file":"descent.tex","source_line":5716,"source_end_line":5720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5716-L5720","statement_sha256":"9a15771b46f44c805fd97d06884f04a5095479f836d15f7cd2a0bf62842aeb93","origin":"The Stacks Project","memory_eligible":false,"source_rank":6854,"rank":6854,"depth":39,"x":804.935,"y":1071.451,"cluster":"descent"},{"id":"stacks:02KX","tag":"02KX","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KX","summary":"The property P(f) =\"f is locally of finite type\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is locally of finite type''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KX","source_file":"descent.tex","source_line":5736,"source_end_line":5740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5736-L5740","statement_sha256":"e6d8618cd13ceeae8b842a0f5ed2db263ee3a12c1eeac36e8d870f0b1d07e135","origin":"The Stacks Project","memory_eligible":false,"source_rank":6855,"rank":6855,"depth":37,"x":642.923,"y":1261.69,"cluster":"descent"},{"id":"stacks:02KY","tag":"02KY","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KY","summary":"The property P(f) =\"f is locally of finite presentation\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is locally of finite presentation''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KY","source_file":"descent.tex","source_line":5766,"source_end_line":5770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5766-L5770","statement_sha256":"748eb336627c26fed014da30630d2168a63b50bfffb1264d3bcc76b8c66e27e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6856,"rank":6856,"depth":37,"x":609.334,"y":1029.001,"cluster":"descent"},{"id":"stacks:02KZ","tag":"02KZ","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02KZ","summary":"The property P(f) =\"f is of finite type\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is of finite type''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02KZ","source_file":"descent.tex","source_line":5796,"source_end_line":5800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5796-L5800","statement_sha256":"cc476dc1ead67397ba86401264147629e9af75fcf4d9a77130973107df003a50","origin":"The Stacks Project","memory_eligible":false,"source_rank":6857,"rank":6857,"depth":38,"x":821.661,"y":1181.838,"cluster":"descent"},{"id":"stacks:02L0","tag":"02L0","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L0","summary":"The property P(f) =\"f is of finite presentation\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is of finite presentation''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L0","source_file":"descent.tex","source_line":5807,"source_end_line":5811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5807-L5811","statement_sha256":"6f163446313d85218781f742b32fdaf24dc8c5fd28d0a8fb52147aaa500e8ba1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6858,"rank":6858,"depth":39,"x":541.614,"y":1189.633,"cluster":"descent"},{"id":"stacks:02L1","tag":"02L1","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L1","summary":"The property P(f) =\"f is proper\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is proper''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L1","source_file":"descent.tex","source_line":5819,"source_end_line":5823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5819-L5823","statement_sha256":"8fd6c941df459479d8528bc6500071620cc7b42f013a6b217946769545670682","origin":"The Stacks Project","memory_eligible":false,"source_rank":6859,"rank":6859,"depth":39,"x":742.257,"y":1024.643,"cluster":"descent"},{"id":"stacks:02L2","tag":"02L2","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L2","summary":"The property P(f) =\"f is flat\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is flat''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L2","source_file":"descent.tex","source_line":5832,"source_end_line":5836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5832-L5836","statement_sha256":"ae3bb5793167093d0bb67a8c829f197b195db25ca36ce28c4c0c1251abf2474a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6860,"rank":6860,"depth":37,"x":726.955,"y":1260.633,"cluster":"descent"},{"id":"stacks:02L3","tag":"02L3","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L3","summary":"The property P(f) =\"f is an open immersion\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is an open immersion''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L3","source_file":"descent.tex","source_line":5861,"source_end_line":5865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5861-L5865","statement_sha256":"043281993c402db64118b4e3e04749413800e5c0fab81e113cfd0f03340d4896","origin":"The Stacks Project","memory_eligible":false,"source_rank":6861,"rank":6861,"depth":39,"x":548.101,"y":1077.564,"cluster":"descent"},{"id":"stacks:02L4","tag":"02L4","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L4","summary":"The property P(f) =\"f is an isomorphism\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is an isomorphism''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L4","source_file":"descent.tex","source_line":5914,"source_end_line":5918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5914-L5918","statement_sha256":"b95c624baa3a7cf8751946208ad9999eb8f600aa34918af5811a8d50afc8f7be","origin":"The Stacks Project","memory_eligible":false,"source_rank":6862,"rank":6862,"depth":40,"x":827.766,"y":1111.138,"cluster":"descent"},{"id":"stacks:02L5","tag":"02L5","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L5","summary":"The property P(f) =\"f is affine\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is affine''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L5","source_file":"descent.tex","source_line":5925,"source_end_line":5929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5925-L5929","statement_sha256":"939661cde60c5ee60ad3b70d29d46f5fb213f7c0af2da1db6274c2e779ff0943","origin":"The Stacks Project","memory_eligible":false,"source_rank":6863,"rank":6863,"depth":41,"x":594.077,"y":1245.339,"cluster":"descent"},{"id":"stacks:02L6","tag":"02L6","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L6","summary":"The property P(f) =\"f is a closed immersion\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a closed immersion''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L6","source_file":"descent.tex","source_line":5967,"source_end_line":5971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L5967-L5971","statement_sha256":"a7ac62d40c48a3defa81f2d71d5ea6d4de7800ad399237d6367d3eeb8a07fa49","origin":"The Stacks Project","memory_eligible":false,"source_rank":6864,"rank":6864,"depth":42,"x":658.605,"y":1013.317,"cluster":"descent"},{"id":"stacks:02L7","tag":"02L7","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L7","summary":"The property P(f) =\"f is quasi-affine\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is quasi-affine''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L7","source_file":"descent.tex","source_line":6002,"source_end_line":6006,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6002-L6006","statement_sha256":"c48ab1474052b839b4ee47386a2a5e2eb8a6f95e4f86478568a86dadbf09a809","origin":"The Stacks Project","memory_eligible":false,"source_rank":6865,"rank":6865,"depth":40,"x":797.885,"y":1221.435,"cluster":"descent"},{"id":"stacks:02L8","tag":"02L8","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L8","summary":"The property P(f) =\"f is a quasi-compact immersion\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a quasi-compact immersion''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L8","source_file":"descent.tex","source_line":6045,"source_end_line":6049,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6045-L6049","statement_sha256":"70f8ab0149083ef1e2d90820453c421f6c131839bab864c3710e59952efa2e90","origin":"The Stacks Project","memory_eligible":false,"source_rank":6866,"rank":6866,"depth":40,"x":527.284,"y":1146.855,"cluster":"descent"},{"id":"stacks:02L9","tag":"02L9","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02L9","summary":"The property P(f) =\"f is integral\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is integral''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02L9","source_file":"descent.tex","source_line":6086,"source_end_line":6090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6086-L6090","statement_sha256":"39fcce1122ccba9dcb993fa69faad528be9ad30e97e9a432de8a3e1b6eb3b438","origin":"The Stacks Project","memory_eligible":false,"source_rank":6867,"rank":6867,"depth":42,"x":787.309,"y":1048.109,"cluster":"descent"},{"id":"stacks:02LA","tag":"02LA","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02LA","summary":"The property P(f) =\"f is finite\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is finite''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LA","source_file":"descent.tex","source_line":6101,"source_end_line":6105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6101-L6105","statement_sha256":"d865de4420fc7cb74376474f1ecc7ff14f5484acff58654f61f21722f4bebe51","origin":"The Stacks Project","memory_eligible":false,"source_rank":6868,"rank":6868,"depth":43,"x":674.758,"y":1268.901,"cluster":"descent"},{"id":"stacks:02VI","tag":"02VI","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02VI","summary":"The properties P(f) =\"f is locally quasi-finite\" and P(f) =\"f is quasi-finite\" are fpqc local on the base.","statement_latex":"The properties\n$\\mathcal{P}(f) =$``$f$ is locally quasi-finite''\nand\n$\\mathcal{P}(f) =$``$f$ is quasi-finite''\nare fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VI","source_file":"descent.tex","source_line":6116,"source_end_line":6123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6116-L6123","statement_sha256":"1052253810aa83233ef4fa4eecf46ecd3a27bb3ea7e602098691b8b910afa73c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6869,"rank":6869,"depth":38,"x":580.011,"y":1041.784,"cluster":"descent"},{"id":"stacks:02VJ","tag":"02VJ","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02VJ","summary":"The property P(f) =\"f is locally of finite type of relative dimension d\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is locally of finite type\nof relative dimension $d$'' is fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VJ","source_file":"descent.tex","source_line":6140,"source_end_line":6144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6140-L6144","statement_sha256":"44803463a71518f0d112ed5fa080741cd7f6c56d67addb6f7adbdf9757d70a7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6870,"rank":6870,"depth":27,"x":833.011,"y":1155.718,"cluster":"descent"},{"id":"stacks:02VK","tag":"02VK","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02VK","summary":"The property P(f) =\"f is syntomic\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is syntomic''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VK","source_file":"descent.tex","source_line":6152,"source_end_line":6156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6152-L6156","statement_sha256":"653b3b3e36bf6d6cccf7f1ed62482c7c762411b98926b425ee9fdde7fc07eb3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6871,"rank":6871,"depth":38,"x":554.289,"y":1215.378,"cluster":"descent"},{"id":"stacks:02VL","tag":"02VL","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02VL","summary":"The property P(f) =\"f is smooth\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is smooth''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VL","source_file":"descent.tex","source_line":6169,"source_end_line":6173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6169-L6173","statement_sha256":"2f19af09181afd64e555f0277fd9a5987daf5c69de8a177848dee18beccb7c0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6872,"rank":6872,"depth":40,"x":712.143,"y":1012.841,"cluster":"descent"},{"id":"stacks:02VM","tag":"02VM","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02VM","summary":"The property P(f) =\"f is unramified\" is fpqc local on the base. The property P(f) =\"f is G-unramified\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is unramified''\nis fpqc local on the base.\nThe property $\\mathcal{P}(f) =$``$f$ is G-unramified''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VM","source_file":"descent.tex","source_line":6186,"source_end_line":6192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6186-L6192","statement_sha256":"77d938335af2e837203741dbb50bed15bbb613f4b6d5fe2735667d92e52dfa36","origin":"The Stacks Project","memory_eligible":false,"source_rank":6873,"rank":6873,"depth":40,"x":758.708,"y":1252.22,"cluster":"descent"},{"id":"stacks:02VN","tag":"02VN","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02VN","summary":"The property P(f) =\"f is étale\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is \\'etale''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VN","source_file":"descent.tex","source_line":6208,"source_end_line":6212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6208-L6212","statement_sha256":"01cf90a9fd8b4a49b70f1f0ca7779a5d32b45bdafa4d0e346accef2b0f6fe22b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6874,"rank":6874,"depth":46,"x":531.434,"y":1101.839,"cluster":"descent"},{"id":"stacks:02VO","tag":"02VO","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02VO","summary":"The property P(f) =\"f is finite locally free\" is fpqc local on the base. Let d ≥ 0. The property P(f) =\"f is finite locally free of degree d\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is finite locally free''\nis fpqc local on the base.\nLet $d \\geq 0$.\nThe property $\\mathcal{P}(f) =$``$f$ is finite locally free of degree $d$''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VO","source_file":"descent.tex","source_line":6224,"source_end_line":6231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6224-L6231","statement_sha256":"537725466d4cc62bffdc94398982a387cc7b93f343df2cff23303f1f16a57ca4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6875,"rank":6875,"depth":44,"x":820.507,"y":1083.732,"cluster":"descent"},{"id":"stacks:02YK","tag":"02YK","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 02YK","summary":"The property P(f) =\"f is a monomorphism\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a monomorphism''\nis fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YK","source_file":"descent.tex","source_line":6248,"source_end_line":6252,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6248-L6252","statement_sha256":"fd73e68bbabba7d4f91b64914a6c339dd3f0552869985dfa5ab698aef1319a70","origin":"The Stacks Project","memory_eligible":false,"source_rank":6876,"rank":6876,"depth":41,"x":621.53,"y":1261.449,"cluster":"descent"},{"id":"stacks:0694","tag":"0694","title":"Properties of morphisms local in the fpqc topology on the target · Lemma 0694","summary":"The properties • [] P(f) =\"f is a Koszul-regular immersion\", • [] P(f) =\"f is an H_1-regular immersion\", and • [] P(f) =\"f is a quasi-regular immersion\" are fpqc local on the base.","statement_latex":"The properties\n\\begin{enumerate}\n\\item[] $\\mathcal{P}(f) =$``$f$ is a Koszul-regular immersion'',\n\\item[] $\\mathcal{P}(f) =$``$f$ is an $H_1$-regular immersion'', and\n\\item[] $\\mathcal{P}(f) =$``$f$ is a quasi-regular immersion''\n\\end{enumerate}\nare fpqc local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0694","source_file":"descent.tex","source_line":6270,"source_end_line":6279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6270-L6279","statement_sha256":"633f1d711a26f59b0b6d161a9775d96bc7e458dd22c4c2ea885962a1699be479","origin":"The Stacks Project","memory_eligible":false,"source_rank":6877,"rank":6877,"depth":37,"x":625.345,"y":1017.036,"cluster":"descent"},{"id":"stacks:02YM","tag":"02YM","title":"Properties of morphisms local in the fppf topology on the target · Lemma 02YM","summary":"The property P(f) =\"f is an immersion\" is fppf local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is an immersion''\nis fppf local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fppf topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YM","source_file":"descent.tex","source_line":6315,"source_end_line":6319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6315-L6319","statement_sha256":"6a0d06d8a1ded3398a9cfb2d856ab1063eac535f99a912ceb32945d63e90371e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6878,"rank":6878,"depth":43,"x":819.449,"y":1199.772,"cluster":"descent"},{"id":"stacks:0H8J","tag":"0H8J","title":"Properties of morphisms local in the fppf topology on the target · Lemma 0H8J","summary":"The property P(f) =\"f is dominant\" is fppf local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is dominant''\nis fppf local on the base.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fppf topology on the target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8J","source_file":"descent.tex","source_line":6346,"source_end_line":6350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6346-L6350","statement_sha256":"5616320ac2339c25a0f4cfc5d6d752cdb21df3668cd38b109d0036afd9cf2f3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6879,"rank":6879,"depth":18,"x":528.822,"y":1175.117,"cluster":"descent"},{"id":"stacks:06NC","tag":"06NC","title":"Application of fpqc descent of properties of morphisms · Lemma 06NC","summary":"Let f : X → Y be a flat, quasi-compact, surjective monomorphism. Then f is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a flat, quasi-compact, surjective monomorphism.\nThen f is an isomorphism.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Application of fpqc descent of properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NC","source_file":"descent.tex","source_line":6381,"source_end_line":6385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6381-L6385","statement_sha256":"d72d19355526870d8d1274d2ee5ae7c64639b008d2ed11784bbd85b9c749568c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6880,"rank":6880,"depth":41,"x":763.392,"y":1028.113,"cluster":"descent"},{"id":"stacks:02LC","tag":"02LC","title":"Application of fpqc descent of properties of morphisms · Lemma 02LC","summary":"A universally injective étale morphism is an open immersion.","statement_latex":"A universally injective \\'etale morphism is an open immersion.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Application of fpqc descent of properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LC","source_file":"descent.tex","source_line":6403,"source_end_line":6406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6403-L6406","statement_sha256":"20c4611ff1972fb6f28b253167abfee4aa0b06782390cb4e12700bc96afb8634","origin":"The Stacks Project","memory_eligible":false,"source_rank":6881,"rank":6881,"depth":46,"x":708.537,"y":1270.065,"cluster":"descent"},{"id":"stacks:09NP","tag":"09NP","title":"Application of fpqc descent of properties of morphisms · Lemma 09NP","summary":"Let f : X → Y be a morphism of schemes. Let X^0 denote the set of generic points of irreducible components of X. If • f is flat and separated, • for xi ∈ X^0 we have kappa(f(xi)) = kappa(xi), and • if xi, xi' ∈ X^0, xi not = xi', then f(xi) not = f(xi'), then f is universally injective.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $X^0$ denote the set\nof generic points of irreducible components of $X$. If\n\\begin{enumerate}\n\\item $f$ is flat and separated,\n\\item for $\\xi \\in X^0$ we have $\\kappa(f(\\xi)) = \\kappa(\\xi)$, and\n\\item if $\\xi, \\xi' \\in X^0$, $\\xi \\not = \\xi'$, then $f(\\xi) \\not = f(\\xi')$,\n\\end{enumerate}\nthen $f$ is universally injective.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Application of fpqc descent of properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NP","source_file":"descent.tex","source_line":6458,"source_end_line":6468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6458-L6468","statement_sha256":"d80013d092917c53c1e094e23b68d70aec3ba535df6615ff7de75a7e6296a0e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6882,"rank":6882,"depth":3,"x":554.129,"y":1060.137,"cluster":"descent"},{"id":"stacks:09NQ","tag":"09NQ","title":"Application of fpqc descent of properties of morphisms · Lemma 09NQ","summary":"Let f : X → Y be a morphism of schemes. Let X^0 denote the set of generic points of irreducible components of X. If • f is étale and separated, • for xi ∈ X^0 we have kappa(f(xi)) = kappa(xi), and • if xi, xi' ∈ X^0, xi not = xi', then f(xi) not = f(xi'), then f is an open immersion.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $X^0$ denote the set\nof generic points of irreducible components of $X$. If\n\\begin{enumerate}\n\\item $f$ is \\'etale and separated,\n\\item for $\\xi \\in X^0$ we have $\\kappa(f(\\xi)) = \\kappa(\\xi)$, and\n\\item if $\\xi, \\xi' \\in X^0$, $\\xi \\not = \\xi'$, then $f(\\xi) \\not = f(\\xi')$,\n\\end{enumerate}\nthen $f$ is an open immersion.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Application of fpqc descent of properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NQ","source_file":"descent.tex","source_line":6492,"source_end_line":6502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6492-L6502","statement_sha256":"d349127f61f65baa24dbb9c92d8d95d02a188f949e00ddd25eaea177e141c5a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6883,"rank":6883,"depth":47,"x":837.33,"y":1127.445,"cluster":"descent"},{"id":"stacks:0F4J","tag":"0F4J","title":"Application of fpqc descent of properties of morphisms · Lemma 0F4J","summary":"Let f : X → Y be a morphism of schemes which is locally of finite type. Let Z be a closed subset of X. If there exists an fpqc covering (Y_i → Y) such that the inverse image Z_i ⊂ Y_i ×_Y X is proper over Y_i (Cohomology of Schemes, Definition [Tag 0CYM]) then Z is proper over Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type.\nLet $Z$ be a closed subset of $X$. If there exists an fpqc covering\n$\\{Y_i \\to Y\\}$ such that the inverse image $Z_i \\subset Y_i \\times_Y X$\nis proper over $Y_i$\n(Cohomology of Schemes, Definition \\ref{coherent-definition-proper-over-base})\nthen $Z$ is proper over $Y$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Application of fpqc descent of properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4J","source_file":"descent.tex","source_line":6509,"source_end_line":6517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6509-L6517","statement_sha256":"550f4420bd15541ea0b31243c635fe05c3497b2052a5d5fa49040193ca83cb2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6884,"rank":6884,"depth":40,"x":573.888,"y":1238.713,"cluster":"descent"},{"id":"stacks:0D2P","tag":"0D2P","title":"Application of fpqc descent of properties of morphisms · Lemma 0D2P","summary":"Let f : X → S be a morphism of schemes. Let L be an invertible O_X-module. Let (g_i : S_i → S)_i ∈ I be an fpqc covering. Let f_i : X_i → S_i be the base change of f and let L_i be the pullback of L to X_i. The following are equivalent • L is ample on X/S, and • L_i is ample on X_i/S_i for every i ∈ I.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $\\{g_i : S_i \\to S\\}_{i \\in I}$ be an fpqc covering.\nLet $f_i : X_i \\to S_i$ be the base change of $f$ and let $\\mathcal{L}_i$\nbe the pullback of $\\mathcal{L}$ to $X_i$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample on $X/S$, and\n\\item $\\mathcal{L}_i$ is ample on $X_i/S_i$\nfor every $i \\in I$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Application of fpqc descent of properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2P","source_file":"descent.tex","source_line":6532,"source_end_line":6545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6532-L6545","statement_sha256":"95b765728cccd1369c793ced3bce8df3f973b4462184aac465f1001cc6408fdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6885,"rank":6885,"depth":40,"x":678.861,"y":1006.756,"cluster":"descent"},{"id":"stacks:036G","tag":"036G","title":"Properties of morphisms local on the source · Definition 036G","summary":"Let P be a property of morphisms of schemes. Let τ ∈ (Zariski, linebreak[0] fpqc, linebreak[0] fppf, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic). We say P is τ local on the source, or local on the source for the τ-topology if for any morphism of schemes f : X → Y over S, and any τ-covering (X_i → X)_i ∈ I we have f has P ⇔ each X_i → Y has P.","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] fpqc, \\linebreak[0] fppf, \\linebreak[0]\n\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.\nWe say $\\mathcal{P}$ is\n{\\it $\\tau$ local on the source}, or\n{\\it local on the source for the $\\tau$-topology} if for\nany morphism of schemes $f : X \\to Y$ over $S$, and any\n$\\tau$-covering $\\{X_i \\to X\\}_{i \\in I}$ we\nhave\n$$\nf \\text{ has }\\mathcal{P}\n\\Leftrightarrow\n\\text{each }X_i \\to Y\\text{ has }\\mathcal{P}.\n$$","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local on the source","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036G","source_file":"descent.tex","source_line":6630,"source_end_line":6646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6630-L6646","statement_sha256":"204086ee7ef163ecf014e1e68c9a2aef831b45842182c696109f1533b1f4f11a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6886,"rank":6886,"depth":0,"x":788.189,"y":1237.785,"cluster":"descent"},{"id":"stacks:04QV","tag":"04QV","title":"Properties of morphisms local on the source · Lemma 04QV","summary":"Let τ ∈ (fpqc, fppf, syntomic, smooth, etale, Zariski). Let P be a property of morphisms which is τ local on the source. Let f : X → Y have property P. For any morphism a : X' → X which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. étale, resp. an open immersion, the composition f ∘ a : X' → Y has property P.","statement_latex":"Let $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale, Zariski\\}$.\nLet $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local\non the source. Let $f : X \\to Y$ have property $\\mathcal{P}$.\nFor any morphism $a : X' \\to X$ which is\nflat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic,\nresp.\\ \\'etale, resp.\\ an open immersion, the composition\n$f \\circ a : X' \\to Y$ has property $\\mathcal{P}$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QV","source_file":"descent.tex","source_line":6656,"source_end_line":6665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6656-L6665","statement_sha256":"c0c70ee21394dc5c1309f2c328bc48ede214812e20d63afddea7f65c7f3be9e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":6887,"rank":6887,"depth":0,"x":521.298,"y":1129.265,"cluster":"descent"},{"id":"stacks:0CEY","tag":"0CEY","title":"Properties of morphisms local on the source · Lemma 0CEY","summary":"Let τ ∈ (fppf, syntomic, smooth, etale). Let P be a property of morphisms which is τ local on the source. For any morphism of schemes f : X → Y there exists a largest open W(f) ⊂ X such that the restriction f|_W(f) : W(f) → Y has P. Moreover, if g : X' → X is flat and locally of finite presentation, syntomic, smooth, or étale and f' = f ∘ g : X' → Y, then g^-1(W(f)) = W(f').","statement_latex":"Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale\\}$.\nLet $\\mathcal{P}$ be a property of morphisms which is $\\tau$ local\non the source. For any morphism of schemes $f : X \\to Y$ there exists\na largest open $W(f) \\subset X$ such that the restriction\n$f|_{W(f)} : W(f) \\to Y$ has $\\mathcal{P}$. Moreover,\nif $g : X' \\to X$ is flat and locally of finite presentation,\nsyntomic, smooth, or \\'etale and $f' = f \\circ g : X' \\to Y$, then\n$g^{-1}(W(f)) = W(f')$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEY","source_file":"descent.tex","source_line":6672,"source_end_line":6682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6672-L6682","statement_sha256":"1bfbdfd1371c492412d865d0c402a250bf614043bf9b99906297ebdda3059f8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6888,"rank":6888,"depth":18,"x":805.886,"y":1057.715,"cluster":"descent"},{"id":"stacks:036H","tag":"036H","title":"Properties of morphisms local on the source · Lemma 036H","summary":"Let P be a property of morphisms of schemes. Let τ ∈ (fpqc, linebreak[0] fppf, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic). Assume that • the property is preserved under precomposing with flat, flat locally of finite presentation, étale, smooth or syntomic morphisms depending on whether τ is fpqc, fppf, étale, smooth, or syntomic, • the property is Zariski local on the source, • the property is Zariski local on the target, • for any morphism of affine…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes.\nLet $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0]\n\\etale, \\linebreak[0] smooth, \\linebreak[0] syntomic\\}$.\nAssume that\n\\begin{enumerate}\n\\item the property is preserved under precomposing with\nflat, flat locally of finite presentation, \\'etale, smooth or syntomic morphisms\ndepending on whether $\\tau$ is fpqc, fppf, \\'etale, smooth, or syntomic,\n\\item the property is Zariski local on the source,\n\\item the property is Zariski local on the target,\n\\item for any morphism of affine schemes $f : X \\to Y$, and\nany surjective morphism of affine schemes $X' \\to X$\nwhich is flat, flat of finite presentation,\n\\'etale, smooth or syntomic depending on whether $\\tau$ is\nfpqc, fppf, \\'etale, smooth, or syntomic, property\n$\\mathcal{P}$ holds for $f$ if property $\\mathcal{P}$\nholds for the composition $f' : X' \\to Y$.\n\\end{enumerate}\nThen $\\mathcal{P}$ is $\\tau$ local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036H","source_file":"descent.tex","source_line":6702,"source_end_line":6723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6702-L6723","statement_sha256":"7cd86d1581f523676e5587f996bb9bbf5a285510676c0c60566ef095c3ebfca4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6889,"rank":6889,"depth":36,"x":653.296,"y":1272.345,"cluster":"descent"},{"id":"stacks:036K","tag":"036K","title":"Properties of morphisms local in the fpqc topology on the source · Lemma 036K","summary":"The property P(f)=\"f is flat\" is fpqc local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is flat'' is fpqc local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036K","source_file":"descent.tex","source_line":6764,"source_end_line":6767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6764-L6767","statement_sha256":"e4d2afff810aacef13a02a4587dbcf293c471d182edbf795b908b606d5066287","origin":"The Stacks Project","memory_eligible":false,"source_rank":6890,"rank":6890,"depth":5,"x":593.106,"y":1027.055,"cluster":"descent"},{"id":"stacks:036L","tag":"036L","title":"Properties of morphisms local in the fpqc topology on the source · Lemma 036L","summary":"Then property P(f : X → Y)=\"for every x ∈ X the map of local rings O_Y, f(x) → O_X, x is injective\" is fpqc local on the source.","statement_latex":"Then property\n$\\mathcal{P}(f : X \\to Y)=$``for every $x \\in X$ the map of local\nrings $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is injective''\nis fpqc local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fpqc topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036L","source_file":"descent.tex","source_line":6788,"source_end_line":6794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6788-L6794","statement_sha256":"5e4a7bbffe6bd60c05fe1d70ce1d6bfc6c313455619aad6b86c9e5e5fb719f85","origin":"The Stacks Project","memory_eligible":false,"source_rank":6891,"rank":6891,"depth":0,"x":835.181,"y":1174.039,"cluster":"descent"},{"id":"stacks:036N","tag":"036N","title":"Properties of morphisms local in the fppf topology on the source · Lemma 036N","summary":"The property P(f)=\"f is locally of finite presentation\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is locally of finite presentation''\nis fppf local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fppf topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036N","source_file":"descent.tex","source_line":6810,"source_end_line":6814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6810-L6814","statement_sha256":"364c47c877e8745394e6833ee79a416b51899399917e3dae84e90bef0d1bfe92","origin":"The Stacks Project","memory_eligible":false,"source_rank":6892,"rank":6892,"depth":37,"x":537.945,"y":1203.066,"cluster":"descent"},{"id":"stacks:036O","tag":"036O","title":"Properties of morphisms local in the fppf topology on the source · Lemma 036O","summary":"The property P(f)=\"f is locally of finite type\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is locally of finite type''\nis fppf local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fppf topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036O","source_file":"descent.tex","source_line":6828,"source_end_line":6832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6828-L6832","statement_sha256":"b2107398f2bbfc7a852bc596680e83c5a5a00bacf6f1cc93e27b815d7ebfe93e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6893,"rank":6893,"depth":37,"x":734.13,"y":1012.664,"cluster":"descent"},{"id":"stacks:036P","tag":"036P","title":"Properties of morphisms local in the fppf topology on the source · Lemma 036P","summary":"The property P(f)=\"f is open\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is open''\nis fppf local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fppf topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036P","source_file":"descent.tex","source_line":6848,"source_end_line":6852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6848-L6852","statement_sha256":"04bad7246fa67b43820259ed3e28d78c5ce5a5c7e95b662a1fabb6b56797779a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6894,"rank":6894,"depth":37,"x":742.597,"y":1264.832,"cluster":"descent"},{"id":"stacks:036Q","tag":"036Q","title":"Properties of morphisms local in the fppf topology on the source · Lemma 036Q","summary":"The property P(f)=\"f is universally open\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is universally open''\nis fppf local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the fppf topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036Q","source_file":"descent.tex","source_line":6867,"source_end_line":6871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6867-L6871","statement_sha256":"c5579e9331b059ed8085050d3f43c0ceeb274eeb7cdbb116f78dbc0c020edf6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6895,"rank":6895,"depth":38,"x":533.196,"y":1083.369,"cluster":"descent"},{"id":"stacks:036S","tag":"036S","title":"Properties of morphisms local in the syntomic topology on the source · Lemma 036S","summary":"The property P(f)=\"f is syntomic\" is syntomic local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is syntomic''\nis syntomic local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the syntomic topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036S","source_file":"descent.tex","source_line":6901,"source_end_line":6905,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6901-L6905","statement_sha256":"faac328264a93af43121a9b5bb48fbde7a0cab54cc027b0809abbe335f3e02ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":6896,"rank":6896,"depth":42,"x":834.064,"y":1098.385,"cluster":"descent"},{"id":"stacks:036U","tag":"036U","title":"Properties of morphisms local in the smooth topology on the source · Lemma 036U","summary":"The property P(f)=\"f is smooth\" is smooth local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is smooth''\nis smooth local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the smooth topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036U","source_file":"descent.tex","source_line":6927,"source_end_line":6931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6927-L6931","statement_sha256":"602b67e004a79b5229ec1bc89fcccdf05d00f0678190e33428d46affbe100fd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6897,"rank":6897,"depth":42,"x":599.721,"y":1258.313,"cluster":"descent"},{"id":"stacks:036W","tag":"036W","title":"Properties of morphisms local in the étale topology on the source · Lemma 036W","summary":"The property P(f)=\"f is étale\" is étale local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is \\'etale''\nis \\'etale local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the étale topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/036W","source_file":"descent.tex","source_line":6949,"source_end_line":6953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6949-L6953","statement_sha256":"25d4615ae4106b28cbb32ae528c96a1455fdb6c92cf7d022e20f6bf85a1e1263","origin":"The Stacks Project","memory_eligible":false,"source_rank":6898,"rank":6898,"depth":42,"x":643.989,"y":1006.972,"cluster":"descent"},{"id":"stacks:03X4","tag":"03X4","title":"Properties of morphisms local in the étale topology on the source · Lemma 03X4","summary":"The property P(f)=\"f is locally quasi-finite\" is étale local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is locally quasi-finite''\nis \\'etale local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the étale topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03X4","source_file":"descent.tex","source_line":6963,"source_end_line":6967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6963-L6967","statement_sha256":"c6ccf25d56e4c6b0f91c610c797ee9a52cd051bd325e9ec7276fb67f6d63c84a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6899,"rank":6899,"depth":44,"x":813.764,"y":1217.795,"cluster":"descent"},{"id":"stacks:03YV","tag":"03YV","title":"Properties of morphisms local in the étale topology on the source · Lemma 03YV","summary":"The property P(f)=\"f is unramified\" is étale local on the source. The property P(f)=\"f is G-unramified\" is étale local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is unramified''\nis \\'etale local on the source.\nThe property $\\mathcal{P}(f)=$``$f$ is G-unramified''\nis \\'etale local on the source.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms local in the étale topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YV","source_file":"descent.tex","source_line":6996,"source_end_line":7002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L6996-L7002","statement_sha256":"8505aa5bab0d76d60cfef94e8d97babfa30a6409b44a13561409b61fb111c7a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6900,"rank":6900,"depth":37,"x":518.523,"y":1158.568,"cluster":"descent"},{"id":"stacks:04QZ","tag":"04QZ","title":"Properties of morphisms étale local on source-and-target · Definition 04QZ","summary":"Let P be a property of morphisms of schemes. We say P is étale local on source-and-target if • (stable under precomposing with étale maps) if f : X → Y is étale and g : Y → Z has P, then g ∘ f has P, • (stable under étale base change) if f : X → Y has P and Y' → Y is étale, then the base change f' : Y' ×_Y X → Y' has P, and • (locality) given a morphism f : X → Y the following are equivalent • f has P, • for every x ∈ X there exists a commutative diagram xymatrix U…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes.\nWe say $\\mathcal{P}$ is {\\it \\'etale local on source-and-target} if\n\\begin{enumerate}\n\\item (stable under precomposing with \\'etale maps)\nif $f : X \\to Y$ is \\'etale and $g : Y \\to Z$ has $\\mathcal{P}$,\nthen $g \\circ f$ has $\\mathcal{P}$,\n\\item (stable under \\'etale base change)\nif $f : X \\to Y$ has $\\mathcal{P}$ and $Y' \\to Y$ is \\'etale, then\nthe base change $f' : Y' \\times_Y X \\to Y'$ has $\\mathcal{P}$, and\n\\item (locality) given a morphism $f : X \\to Y$ the following are equivalent\n\\begin{enumerate}\n\\item $f$ has $\\mathcal{P}$,\n\\item for every $x \\in X$ there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwith \\'etale vertical arrows and $u \\in U$ with $a(u) = x$ such that\n$h$ has $\\mathcal{P}$.\n\\end{enumerate}\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms étale local on source-and-target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QZ","source_file":"descent.tex","source_line":7143,"source_end_line":7168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7143-L7168","statement_sha256":"9397eed46479e27cdfd43745e4d29a2b9810551a08a7f4e0e51becd77aa55b6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6901,"rank":6901,"depth":0,"x":784.319,"y":1034.499,"cluster":"descent"},{"id":"stacks:04R0","tag":"04R0","title":"Properties of morphisms étale local on source-and-target · Lemma 04R0","summary":"Let P be a property of morphisms of schemes which is étale local on source-and-target. Then • P is étale local on the source, • P is étale local on the target, • P is stable under postcomposing with étale morphisms: if f : X → Y has P and g : Y → Z is étale, then g ∘ f has P, and • P has a permanence property: given f : X → Y and g : Y → Z étale such that g ∘ f has P, then f has P.","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes which is\n\\'etale local on source-and-target. Then\n\\begin{enumerate}\n\\item $\\mathcal{P}$ is \\'etale local on the source,\n\\item $\\mathcal{P}$ is \\'etale local on the target,\n\\item $\\mathcal{P}$ is stable under postcomposing with \\'etale morphisms:\nif $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is \\'etale, then\n$g \\circ f$ has $\\mathcal{P}$, and\n\\item $\\mathcal{P}$ has a permanence property: given $f : X \\to Y$ and\n$g : Y \\to Z$ \\'etale such that $g \\circ f$ has $\\mathcal{P}$, then\n$f$ has $\\mathcal{P}$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms étale local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04R0","source_file":"descent.tex","source_line":7181,"source_end_line":7195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7181-L7195","statement_sha256":"0ea1717cdd219bc226e059be16429ed0e5a397aa9e4314eca39e9f3b7fadc0af","origin":"The Stacks Project","memory_eligible":false,"source_rank":6902,"rank":6902,"depth":43,"x":687.931,"y":1277.224,"cluster":"descent"},{"id":"stacks:04R1","tag":"04R1","title":"Properties of morphisms étale local on source-and-target · Lemma 04R1","summary":"Let P be a property of morphisms of schemes which is étale local on source-and-target. Let f : X → Y be a morphism of schemes. The following are equivalent: • [(a)] f has property P, • [(b)] for every x ∈ X there exists an étale morphism of germs a : (U, u) → (X, x), an étale morphism b : V → Y, and a morphism h : U → V such that f ∘ a = b ∘ h and h has P, • [(c)] for any commutative diagram xymatrix U ar[d]_a ar[r]_h & V ar[d]^b X ar[r]^f & Y with a, b étale the morphism…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes which is\n\\'etale local on source-and-target. Let $f : X \\to Y$ be a morphism\nof schemes. The following are equivalent:\n\\begin{enumerate}\n\\item[(a)] $f$ has property $\\mathcal{P}$,\n\\item[(b)] for every $x \\in X$ there exists an \\'etale morphism of germs\n$a : (U, u) \\to (X, x)$, an \\'etale morphism $b : V \\to Y$, and\na morphism $h : U \\to V$ such that $f \\circ a = b \\circ h$ and\n$h$ has $\\mathcal{P}$,\n\\item[(c)]\nfor any commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwith $a$, $b$ \\'etale the morphism $h$ has $\\mathcal{P}$,\n\\item[(d)] for some diagram as in (c)\nwith $a : U \\to X$ surjective $h$ has $\\mathcal{P}$,\n\\item[(e)] there exists an \\'etale covering $\\{Y_i \\to Y\\}_{i \\in I}$ such\nthat each base change $Y_i \\times_Y X \\to Y_i$ has $\\mathcal{P}$,\n\\item[(f)] there exists an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$ such\nthat each composition $X_i \\to Y$ has $\\mathcal{P}$,\n\\item[(g)] there exists an \\'etale covering $\\{Y_i \\to Y\\}_{i \\in I}$ and\nfor each $i \\in I$ an \\'etale covering\n$\\{X_{ij} \\to Y_i \\times_Y X\\}_{j \\in J_i}$ such that each morphism\n$X_{ij} \\to Y_i$ has $\\mathcal{P}$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms étale local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04R1","source_file":"descent.tex","source_line":7251,"source_end_line":7282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7251-L7282","statement_sha256":"4aa37a91fcd3e2486ce5e4dca008f8afe07fea3746aabb807d61d0c3e528bfd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6903,"rank":6903,"depth":44,"x":563.599,"y":1043.147,"cluster":"descent"},{"id":"stacks:04R2","tag":"04R2","title":"Properties of morphisms étale local on source-and-target · Lemma 04R2","summary":"Let P be a property of morphisms of schemes. Assume • P is étale local on the source, • P is étale local on the target, and • P is stable under postcomposing with open immersions: if f : X → Y has P and Y ⊂ Z is an open subscheme then X → Z has P. Then P is étale local on the source-and-target.","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{P}$ is \\'etale local on the source,\n\\item $\\mathcal{P}$ is \\'etale local on the target, and\n\\item $\\mathcal{P}$ is stable under postcomposing with open immersions:\nif $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\subset Z$ is an open\nsubscheme then $X \\to Z$ has $\\mathcal{P}$.\n\\end{enumerate}\nThen $\\mathcal{P}$ is \\'etale local on the source-and-target.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms étale local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04R2","source_file":"descent.tex","source_line":7314,"source_end_line":7326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7314-L7326","statement_sha256":"b1301124ae3241d2f792ce8716a3336ffd01dad1927a7c23d672d5d07ccb261d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6904,"rank":6904,"depth":47,"x":844.0,"y":1145.379,"cluster":"descent"},{"id":"stacks:0CEZ","tag":"0CEZ","title":"Properties of morphisms étale local on source-and-target · Lemma 0CEZ","summary":"Let P be a property of morphisms of schemes which is étale local on the source-and-target. Given a commutative diagram of schemes vcenter xymatrix X' ar[d]_g' ar[r]_f' & Y' ar[d]^g X ar[r]^f & Y with points vcenter xymatrix x' ar[d] ar[r] & y' ar[d] x ar[r] & y such that g' is étale at x' and g is étale at y', then x ∈ W(f) ⇔ x' ∈ W(f') where W(-) is as in Lemma [Tag 0CEY].","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes which\nis \\'etale local on the source-and-target.\nGiven a commutative diagram of schemes\n$$\n\\vcenter{\n\\xymatrix{\nX' \\ar[d]_{g'} \\ar[r]_{f'} & Y' \\ar[d]^g \\\\\nX \\ar[r]^f & Y\n}\n}\n\\quad\\text{with points}\\quad\n\\vcenter{\n\\xymatrix{\nx' \\ar[d] \\ar[r] & y' \\ar[d] \\\\\nx \\ar[r] & y\n}\n}\n$$\nsuch that $g'$ is \\'etale at $x'$ and $g$ is \\'etale at $y'$, then\n$x \\in W(f) \\Leftrightarrow x' \\in W(f')$\nwhere $W(-)$ is as in Lemma \\ref{lemma-largest-open-of-the-source}.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms étale local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEZ","source_file":"descent.tex","source_line":7569,"source_end_line":7592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7569-L7592","statement_sha256":"98b485d7bfc4721cb8cec5497c54554f004adff0c428c3e6dd8e0c8555740cee","origin":"The Stacks Project","memory_eligible":false,"source_rank":6905,"rank":6905,"depth":44,"x":554.529,"y":1229.243,"cluster":"descent"},{"id":"stacks:0CF0","tag":"0CF0","title":"Properties of morphisms étale local on source-and-target · Lemma 0CF0","summary":"Let k be a field. Let n ≥ 2. For 1 ≤ i, j ≤ n with i not = j and d ≥ 0 denote T_i, j, d the automorphism of A^n_k given in coordinates by (x_1, …, x_n) ↦ (x_1, …, x_i - 1, x_i + x_j^d, x_i + 1, …, x_n) Let W ⊂ A^n_k be a nonempty open subscheme such that T_i, j, d(W) = W for all i, j, d as above. Then either W = A^n_k or the characteristic of k is p > 0 and A^n_k setminus W is a finite set of closed points whose coordinates are algebraic over F_p.","statement_latex":"Let $k$ be a field. Let $n \\geq 2$. For $1 \\leq i, j \\leq n$ with\n$i \\not = j$ and $d \\geq 0$ denote $T_{i, j, d}$ the automorphism\nof $\\mathbf{A}^n_k$ given in coordinates by\n$$\n(x_1, \\ldots, x_n) \\longmapsto\n(x_1, \\ldots, x_{i - 1}, x_i + x_j^d, x_{i + 1}, \\ldots, x_n)\n$$\nLet $W \\subset \\mathbf{A}^n_k$ be a nonempty open subscheme\nsuch that $T_{i, j, d}(W) = W$ for all $i, j, d$ as above.\nThen either $W = \\mathbf{A}^n_k$ or the characteristic of $k$\nis $p > 0$ and $\\mathbf{A}^n_k \\setminus W$ is a finite set\nof closed points whose coordinates are algebraic over $\\mathbf{F}_p$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms étale local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CF0","source_file":"descent.tex","source_line":7629,"source_end_line":7643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7629-L7643","statement_sha256":"7a4dcea8646e6af029a4df037ae35c9a19b3434817fc4eab3b3fdf7b5eceea7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6906,"rank":6906,"depth":0,"x":700.789,"y":1002.765,"cluster":"descent"},{"id":"stacks:0CF1","tag":"0CF1","title":"Properties of morphisms étale local on source-and-target · Lemma 0CF1","summary":"Let P be a property of morphisms of schemes. Assume • P is étale local on the source, • P is smooth local on the target, • P is stable under postcomposing with open immersions: if f : X → Y has P and Y ⊂ Z is an open subscheme then X → Z has P. Given a commutative diagram of schemes vcenter xymatrix X' ar[d]_g' ar[r]_f' & Y' ar[d]^g X ar[r]^f & Y with points vcenter xymatrix x' ar[d] ar[r] & y' ar[d] x ar[r] & y such that g is smooth y' and X' → X ×_Y Y' is étale at x',…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes. Assume\n\\begin{enumerate}\n\\item $\\mathcal{P}$ is \\'etale local on the source,\n\\item $\\mathcal{P}$ is smooth local on the target,\n\\item $\\mathcal{P}$ is stable under postcomposing with open immersions:\nif $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\subset Z$ is an open\nsubscheme then $X \\to Z$ has $\\mathcal{P}$.\n\\end{enumerate}\nGiven a commutative diagram of schemes\n$$\n\\vcenter{\n\\xymatrix{\nX' \\ar[d]_{g'} \\ar[r]_{f'} & Y' \\ar[d]^g \\\\\nX \\ar[r]^f & Y\n}\n}\n\\quad\\text{with points}\\quad\n\\vcenter{\n\\xymatrix{\nx' \\ar[d] \\ar[r] & y' \\ar[d] \\\\\nx \\ar[r] & y\n}\n}\n$$\nsuch that $g$ is smooth $y'$ and $X' \\to X \\times_Y Y'$ is \\'etale\nat $x'$, then $x \\in W(f) \\Leftrightarrow x' \\in W(f')$\nwhere $W(-)$ is as in Lemma \\ref{lemma-largest-open-of-the-source}.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms étale local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CF1","source_file":"descent.tex","source_line":7686,"source_end_line":7715,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7686-L7715","statement_sha256":"ddd0be6478f3aafe646cc37d13fe741c9453e20df504ded6e5042384dd8e5a0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6907,"rank":6907,"depth":48,"x":775.193,"y":1253.183,"cluster":"descent"},{"id":"stacks:04NB","tag":"04NB","title":"Properties of morphisms of germs local on source-and-target · Definition 04NB","summary":"Let Q be a property of morphisms of germs of schemes. We say Q is étale local on the source-and-target if for any commutative diagram xymatrix (U', u') ar[d]_a ar[r]_h' & (V', v') ar[d]^b (U, u) ar[r]^h & (V, v) of germs with étale vertical arrows we have Q(h) ⇔ Q(h').","statement_latex":"Let $\\mathcal{Q}$ be a property of morphisms of germs of schemes.\nWe say $\\mathcal{Q}$ is  {\\it \\'etale local on the source-and-target}\nif for any commutative diagram\n$$\n\\xymatrix{\n(U', u') \\ar[d]_a \\ar[r]_{h'} & (V', v') \\ar[d]^b \\\\\n(U, u) \\ar[r]^h & (V, v)\n}\n$$\nof germs with \\'etale vertical arrows we have\n$\\mathcal{Q}(h) \\Leftrightarrow \\mathcal{Q}(h')$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms of germs local on source-and-target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NB","source_file":"descent.tex","source_line":7837,"source_end_line":7850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7837-L7850","statement_sha256":"98cc6e285437b9eee2b4b6fdf4ace3bd6d23b093b8d7c507ed9b9820c881a7e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6908,"rank":6908,"depth":0,"x":518.515,"y":1110.506,"cluster":"descent"},{"id":"stacks:04R6","tag":"04R6","title":"Properties of morphisms of germs local on source-and-target · Lemma 04R6","summary":"Let P be a property of morphisms of schemes which is étale local on the source-and-target. Consider the property Q of morphisms of germs defined by the rule Q((X, x) → (S, s)) ⇔ there exists a representative U → S which has P Then Q is étale local on the source-and-target as in Definition [Tag 04NB].","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes\nwhich is \\'etale local on the source-and-target.\nConsider the property $\\mathcal{Q}$ of\nmorphisms of germs defined by the rule\n$$\n\\mathcal{Q}((X, x) \\to (S, s))\n\\Leftrightarrow\n\\text{there exists a representative }U \\to S\n\\text{ which has }\\mathcal{P}\n$$\nThen $\\mathcal{Q}$ is \\'etale local on the source-and-target as in\nDefinition \\ref{definition-local-source-target-at-point}.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms of germs local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04R6","source_file":"descent.tex","source_line":7852,"source_end_line":7866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7852-L7866","statement_sha256":"8d6311b2c13e55c646af04edb13b55ed2cdb7b9509b615dc46e404c1ce252737","origin":"The Stacks Project","memory_eligible":false,"source_rank":6909,"rank":6909,"depth":45,"x":823.036,"y":1070.0,"cluster":"descent"},{"id":"stacks:04R7","tag":"04R7","title":"Properties of morphisms of germs local on source-and-target · Lemma 04R7","summary":"Let P be a property of morphisms of schemes which is étale local on source-and-target. Let Q be the associated property of morphisms of germs, see Lemma [Tag 04R6]. Let f : X → Y be a morphism of schemes. The following are equivalent: • f has property P, and • for every x ∈ X the morphism of germs (X, x) → (Y, f(x)) has property Q.","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes which is\n\\'etale local on source-and-target. Let $Q$ be the associated property\nof morphisms of germs, see\nLemma \\ref{lemma-local-source-target-global-implies-local}.\nLet $f : X \\to Y$ be a morphism\nof schemes. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ has property $\\mathcal{P}$, and\n\\item for every $x \\in X$ the morphism of germs $(X, x) \\to (Y, f(x))$\nhas property $\\mathcal{Q}$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms of germs local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04R7","source_file":"descent.tex","source_line":7893,"source_end_line":7906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7893-L7906","statement_sha256":"fe4b29997b67c4de4dcf0db285682252f9fd994854583324dab40beee7b3782a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6910,"rank":6910,"depth":46,"x":630.737,"y":1273.002,"cluster":"descent"},{"id":"stacks:04ND","tag":"04ND","title":"Properties of morphisms of germs local on source-and-target · Lemma 04ND","summary":"The property of morphisms of germs P((X, x) → (S, s)) = O_S, s → O_X, x is flat is étale local on the source-and-target.","statement_latex":"The property of morphisms of germs\n$$\n\\mathcal{P}((X, x) \\to (S, s)) =\n\\mathcal{O}_{S, s} \\to \\mathcal{O}_{X, x}\\text{ is flat}\n$$\nis \\'etale local on the source-and-target.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms of germs local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ND","source_file":"descent.tex","source_line":7918,"source_end_line":7926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7918-L7926","statement_sha256":"98e6bb95d1cf514f2a1ea65e650e5e228a925d9553b31af01e692893a0b4600a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6911,"rank":6911,"depth":42,"x":609.251,"y":1013.762,"cluster":"descent"},{"id":"stacks:04NI","tag":"04NI","title":"Properties of morphisms of germs local on source-and-target · Lemma 04NI","summary":"Consider a commutative diagram of morphisms of schemes xymatrix U' ar[r] ar[d] & V' ar[d] U ar[r] & V with étale vertical arrows and a point v' ∈ V' mapping to v ∈ V. Then the morphism of fibres U'_v' → U_v is étale.","statement_latex":"Consider a commutative diagram of morphisms of schemes\n$$\n\\xymatrix{\nU' \\ar[r] \\ar[d] & V' \\ar[d] \\\\\nU \\ar[r] & V\n}\n$$\nwith \\'etale vertical arrows and a point $v' \\in V'$ mapping to $v \\in V$.\nThen the morphism of fibres $U'_{v'} \\to U_v$ is \\'etale.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms of germs local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NI","source_file":"descent.tex","source_line":7968,"source_end_line":7979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L7968-L7979","statement_sha256":"5f3392736610ddee54874467ebf0121a454648c5d21c6bae20075ada6ff448c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6912,"rank":6912,"depth":45,"x":833.943,"y":1193.03,"cluster":"descent"},{"id":"stacks:04NJ","tag":"04NJ","title":"Properties of morphisms of germs local on source-and-target · Lemma 04NJ","summary":"Let d ∈ (0, 1, 2, …, ∞). The property of morphisms of germs P_d((X, x) → (S, s)) = the local ring O_X_s, x of the fibre has dimension d is étale local on the source-and-target.","statement_latex":"Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.\nThe property of morphisms of germs\n$$\n\\mathcal{P}_d((X, x) \\to (S, s)) =\n\\text{the local ring }\n\\mathcal{O}_{X_s, x}\n\\text{ of the fibre has dimension }d\n$$\nis \\'etale local on the source-and-target.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms of germs local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NJ","source_file":"descent.tex","source_line":8000,"source_end_line":8011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8000-L8011","statement_sha256":"f5dc12da40517c2667715194c468bd784ea29b0de50bc0c7d9ac3c118ae00b37","origin":"The Stacks Project","memory_eligible":false,"source_rank":6913,"rank":6913,"depth":46,"x":523.58,"y":1188.328,"cluster":"descent"},{"id":"stacks:04NK","tag":"04NK","title":"Properties of morphisms of germs local on source-and-target · Lemma 04NK","summary":"Let r ∈ (0, 1, 2, …, ∞). The property of morphisms of germs P_r((X, x) → (S, s)) ⇔ trdeg_kappa(s) kappa(x) = r is étale local on the source-and-target.","statement_latex":"Let $r \\in \\{0, 1, 2, \\ldots, \\infty\\}$.\nThe property of morphisms of germs\n$$\n\\mathcal{P}_r((X, x) \\to (S, s))\n\\Leftrightarrow\n\\text{trdeg}_{\\kappa(s)} \\kappa(x) = r\n$$\nis \\'etale local on the source-and-target.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms of germs local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NK","source_file":"descent.tex","source_line":8023,"source_end_line":8033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8023-L8033","statement_sha256":"985df1875c83259945f760120c5b534b6d845e5141ce6f3a0ef6c3ae70bcd783","origin":"The Stacks Project","memory_eligible":false,"source_rank":6914,"rank":6914,"depth":1,"x":756.605,"y":1015.401,"cluster":"descent"},{"id":"stacks:04NL","tag":"04NL","title":"Properties of morphisms of germs local on source-and-target · Lemma 04NL","summary":"Let d ∈ (0, 1, 2, …, ∞). The property of morphisms of germs P_d((X, x) → (S, s)) ⇔ dim_x (X_s) = d is étale local on the source-and-target.","statement_latex":"Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.\nThe property of morphisms of germs\n$$\n\\mathcal{P}_d((X, x) \\to (S, s))\n\\Leftrightarrow\n\\dim_x (X_s) = d\n$$\nis \\'etale local on the source-and-target.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Properties of morphisms of germs local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NL","source_file":"descent.tex","source_line":8062,"source_end_line":8072,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8062-L8072","statement_sha256":"f3042dad570d1e3ac9a743bd764ecd3169a38bf6b7daa6ae9f11f6b6da4e5bbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":6915,"rank":6915,"depth":46,"x":723.784,"y":1275.568,"cluster":"descent"},{"id":"stacks:023V","tag":"023V","title":"Descent data for schemes over schemes · Definition 023V","summary":"Let f : X → S be a morphism of schemes. • Let V → X be a scheme over X. A descent datum for V/X/S is an isomorphism φ : V ×_S X → X ×_S V of schemes over X ×_S X satisfying the cocycle condition that the diagram xymatrix V ×_S X ×_S X ar[rd]^φ_01 ar[rr]_φ_02 & & X ×_S X ×_S V & X ×_S V ×_S X ar[ru]^φ_12 commutes (with obvious notation). • We also say that the pair (V/X, φ) is a descent datum relative to X → S. • A morphism g : (V/X, φ) → (V'/X, φ') of descent data…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item Let $V \\to X$ be a scheme over $X$.\nA {\\it descent datum for $V/X/S$} is an isomorphism\n$\\varphi : V \\times_S X \\to X \\times_S V$ of schemes over\n$X \\times_S X$ satisfying the {\\it cocycle condition}\nthat the diagram\n$$\n\\xymatrix{\nV \\times_S X \\times_S X \\ar[rd]^{\\varphi_{01}} \\ar[rr]_{\\varphi_{02}} &\n&\nX \\times_S X \\times_S V\\\\\n&\nX \\times_S V \\times_S X \\ar[ru]^{\\varphi_{12}}\n}\n$$\ncommutes (with obvious notation).\n\\item We also say that the pair $(V/X, \\varphi)$ is\na {\\it descent datum relative to $X \\to S$}.\n\\item A {\\it morphism $g : (V/X, \\varphi) \\to (V'/X, \\varphi')$ of\ndescent data relative to $X \\to S$} is a morphism\n$g : V \\to V'$ of schemes over $X$ such that\nthe diagram\n$$\n\\xymatrix{\nV \\times_S X \\ar[r]_{\\varphi} \\ar[d]_{g \\times \\text{id}_X} &\nX \\times_S V \\ar[d]^{\\text{id}_X \\times g} \\\\\nV' \\times_S X \\ar[r]^{\\varphi'} & X \\times_S V'\n}\n$$\ncommutes.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for schemes over schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023V","source_file":"descent.tex","source_line":8100,"source_end_line":8134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8100-L8134","statement_sha256":"b4394ea14fe1322e7eaf14e270a3cc12e5002eb8ae1695edc314c8422ef147c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6916,"rank":6916,"depth":0,"x":538.462,"y":1064.755,"cluster":"descent"},{"id":"stacks:023W","tag":"023W","title":"Descent data for schemes over schemes · Definition 023W","summary":"Let S be a scheme. Let (X_i → S)_i ∈ I be a family of morphisms with target S. • A descent datum (V_i, φ_ij) relative to the family (X_i → S) is given by a scheme V_i over X_i for each i ∈ I, an isomorphism φ_ij : V_i ×_S X_j → X_i ×_S V_j of schemes over X_i ×_S X_j for each pair (i, j) ∈ I^2 such that for every triple of indices (i, j, k) ∈ I^3 the diagram xymatrix V_i ×_S X_j ×_S X_k ar[rd]^pr_01^*φ_ij ar[rr]_pr_02^*φ_ik & & X_i ×_S X_j ×_S V_k & X_i ×_S V_j ×_S X_k…","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$.\n\\begin{enumerate}\n\\item A {\\it descent datum $(V_i, \\varphi_{ij})$ relative to the\nfamily $\\{X_i \\to S\\}$} is given by a scheme $V_i$ over $X_i$\nfor each $i \\in I$, an isomorphism\n$\\varphi_{ij} : V_i \\times_S X_j \\to X_i \\times_S V_j$\nof schemes over $X_i \\times_S X_j$ for each pair $(i, j) \\in I^2$\nsuch that for every triple of indices $(i, j, k) \\in I^3$\nthe diagram\n$$\n\\xymatrix{\nV_i \\times_S X_j \\times_S X_k\n\\ar[rd]^{\\text{pr}_{01}^*\\varphi_{ij}}\n\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} &\n&\nX_i \\times_S X_j \\times_S V_k\\\\\n&\nX_i \\times_S V_j \\times_S X_k\n\\ar[ru]^{\\text{pr}_{12}^*\\varphi_{jk}}\n}\n$$\nof schemes over $X_i \\times_S X_j \\times_S X_k$ commutes\n(with obvious notation).\n\\item A {\\it morphism\n$\\psi : (V_i, \\varphi_{ij}) \\to (V'_i, \\varphi'_{ij})$\nof descent data} is given by a family\n$\\psi = (\\psi_i)_{i \\in I}$ of morphisms of\n$X_i$-schemes $\\psi_i : V_i \\to V'_i$ such that all the diagrams\n$$\n\\xymatrix{\nV_i \\times_S X_j \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\psi_i \\times \\text{id}} &\nX_i \\times_S V_j \\ar[d]^{\\text{id} \\times \\psi_j} \\\\\nV'_i \\times_S X_j \\ar[r]^{\\varphi'_{ij}} & X_i \\times_S V'_j\n}\n$$\ncommute.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for schemes over schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023W","source_file":"descent.tex","source_line":8176,"source_end_line":8216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8176-L8216","statement_sha256":"007bf57fd6a906ad244c4391b9d810ec112826430874e6bb1f9ed46d77b54cb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6917,"rank":6917,"depth":0,"x":845.149,"y":1115.132,"cluster":"descent"},{"id":"stacks:023X","tag":"023X","title":"Descent data for schemes over schemes · Lemma 023X","summary":"Let S be a scheme. Let (X_i → S)_i ∈ I be a family of morphisms with target S. Set X = coprod_i ∈ I X_i, and consider it as an S-scheme. There is a canonical equivalence of categories category of descent data relative to the family (X_i → S)_i ∈ I → category of descent data relative to X/S which maps (V_i, φ_ij) to (V, φ) with V = coprod_i∈ I V_i and φ = coprod φ_ij.","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to S\\}_{i \\in I}$ be a family of morphisms with target $S$.\nSet $X = \\coprod_{i \\in I} X_i$, and consider it as an $S$-scheme.\nThere is a canonical equivalence of categories\n$$\n\\begin{matrix}\n\\text{category of descent data } \\\\\n\\text{relative to the family } \\{X_i \\to S\\}_{i \\in I}\n\\end{matrix}\n\\longrightarrow\n\\begin{matrix}\n\\text{ category of descent data} \\\\\n\\text{ relative to } X/S\n\\end{matrix}\n$$\nwhich maps $(V_i, \\varphi_{ij})$ to $(V, \\varphi)$ with\n$V = \\coprod_{i\\in I} V_i$ and $\\varphi = \\coprod \\varphi_{ij}$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for schemes over schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023X","source_file":"descent.tex","source_line":8249,"source_end_line":8268,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8249-L8268","statement_sha256":"93fdef1b6020d91f74436ecc8f63860f3b3836cec110f8911193d431040d6415","origin":"The Stacks Project","memory_eligible":false,"source_rank":6918,"rank":6918,"depth":0,"x":578.053,"y":1252.23,"cluster":"descent"},{"id":"stacks:023Y","tag":"023Y","title":"Descent data for schemes over schemes · Lemma 023Y","summary":"Pullback of descent data for schemes over schemes. • Let xymatrix X' ar[r]_f ar[d]_a' & X ar[d]^a S' ar[r]^h & S be a commutative diagram of morphisms of schemes. The construction (V → X, φ) ↦ f^*(V → X, φ) = (V' → X', φ') where V' = X' ×_X V and where φ' is defined as the composition xymatrix V' ×_S' X' ar@=[r] & (X' ×_X V) ×_S' X' ar@=[r] & (X' ×_S' X') ×_X ×_S X (V ×_S X) ar[d]^id × φ X' ×_S' V' ar@=[r] & X' ×_S' (X' ×_X V) & (X' ×_S' X') ×_X ×_S X (X ×_S V) ar@=[l]…","statement_latex":"Pullback of descent data for schemes over schemes.\n\\begin{enumerate}\n\\item Let\n$$\n\\xymatrix{\nX' \\ar[r]_f \\ar[d]_{a'} & X \\ar[d]^a \\\\\nS' \\ar[r]^h & S\n}\n$$\nbe a commutative diagram of morphisms of schemes.\nThe construction\n$$\n(V \\to X, \\varphi) \\longmapsto f^*(V \\to X, \\varphi) = (V' \\to X', \\varphi')\n$$\nwhere $V' = X' \\times_X V$ and where\n$\\varphi'$ is defined as the composition\n$$\n\\xymatrix{\nV' \\times_{S'} X' \\ar@{=}[r] &\n(X' \\times_X V) \\times_{S'} X' \\ar@{=}[r] &\n(X' \\times_{S'} X') \\times_{X \\times_S X} (V \\times_S X)\n\\ar[d]^{\\text{id} \\times \\varphi} \\\\\nX' \\times_{S'} V' \\ar@{=}[r] &\nX' \\times_{S'} (X' \\times_X V) &\n(X' \\times_{S'} X') \\times_{X \\times_S X} (X \\times_S V) \\ar@{=}[l]\n}\n$$\ndefines a functor from the category of descent data\nrelative to $X \\to S$ to the category of descent data\nrelative to $X' \\to S'$.\n\\item Given two morphisms $f_i : X' \\to X$, $i = 0, 1$ making the\ndiagram commute the functors $f_0^*$ and $f_1^*$ are\ncanonically isomorphic.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for schemes over schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023Y","source_file":"descent.tex","source_line":8278,"source_end_line":8314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8278-L8314","statement_sha256":"95ce61e2ef23d3ccc896891646c437701e2e3aa4ebf9ba7f5384b8cc6a657de1","origin":"The Stacks Project","memory_eligible":false,"source_rank":6919,"rank":6919,"depth":0,"x":664.897,"y":999.167,"cluster":"descent"},{"id":"stacks:02VR","tag":"02VR","title":"Descent data for schemes over schemes · Definition 02VR","summary":"With S, S', X, X', f, a, a', h as in Lemma [Tag 023Y] the functor (V, φ) ↦ f^*(V, φ) constructed in that lemma is called the pullback functor on descent data.","statement_latex":"With $S, S', X, X', f, a, a', h$ as in Lemma \\ref{lemma-pullback} the functor\n$$\n(V, \\varphi) \\longmapsto f^*(V, \\varphi)\n$$\nconstructed in that lemma is called the {\\it pullback functor} on descent data.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for schemes over schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VR","source_file":"descent.tex","source_line":8343,"source_end_line":8350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8343-L8350","statement_sha256":"d20310e7178fe518c3596298c1c7db9c3b05d18af9c47720cf1fc5ca7e0ff727","origin":"The Stacks Project","memory_eligible":false,"source_rank":6920,"rank":6920,"depth":1,"x":804.593,"y":1235.433,"cluster":"descent"},{"id":"stacks:02VS","tag":"02VS","title":"Pullback of descent data for schemes over families · Lemma 02VS","summary":"Let U = (U_i → S')_i ∈ I and V = (V_j → S)_j ∈ J be families of morphisms with fixed target. Let α : I → J, h : S' → S and g_i : U_i → V_α(i) be a morphism of families of maps with fixed target, see Sites, Definition [Tag 00VT]. • Let (Y_j, φ_jj') be a descent datum relative to the family (V_j → S'). The system ( g_i^*Y_α(i), (g_i × g_i')^*φ_α(i)α(i') ) (with notation as in Remark [Tag 02VQ]) is a descent datum relative to V. • This construction defines a functor between…","statement_latex":"Let $\\mathcal{U} = \\{U_i \\to S'\\}_{i \\in I}$ and\n$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$ be families of morphisms with\nfixed target. Let $\\alpha : I \\to J$, $h : S' \\to S$ and\n$g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families\nof maps with fixed target, see\nSites, Definition \\ref{sites-definition-morphism-coverings}.\n\\begin{enumerate}\n\\item Let $(Y_j, \\varphi_{jj'})$ be a descent datum relative to the\nfamily $\\{V_j \\to S'\\}$. The system\n$$\n\\left(\ng_i^*Y_{\\alpha(i)},\n(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}\n\\right)\n$$\n(with notation as in Remark \\ref{remark-easier-family})\nis a descent datum relative to $\\mathcal{V}$.\n\\item This construction defines a functor between descent data relative\nto $\\mathcal{U}$ and descent data relative to $\\mathcal{V}$.\n\\item Given a second $\\alpha' : I \\to J$, $h' : S' \\to S$ and\n$g'_i : U_i \\to V_{\\alpha'(i)}$ morphism of families\nof maps with fixed target, then if $h = h'$ the two resulting functors\nbetween descent data are canonically isomorphic.\n\\item These functors agree, via Lemma \\ref{lemma-family-is-one},\nwith the pullback functors constructed in Lemma \\ref{lemma-pullback}.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for schemes over schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VS","source_file":"descent.tex","source_line":8352,"source_end_line":8380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8352-L8380","statement_sha256":"b33db12062ab3d1cc1d9d9e86f71e02e6a3ba2641148749a618f0a96c92d9a19","origin":"The Stacks Project","memory_eligible":false,"source_rank":6921,"rank":6921,"depth":1,"x":511.109,"y":1140.327,"cluster":"descent"},{"id":"stacks:02VT","tag":"02VT","title":"Descent data for schemes over schemes · Definition 02VT","summary":"With U = (U_i → S')_i ∈ I, V = (V_j → S)_j ∈ J, α : I → J, h : S' → S, and g_i : U_i → V_α(i) as in Lemma [Tag 02VS] the functor (Y_j, φ_jj') ↦ (g_i^*Y_α(i), (g_i × g_i')^*φ_α(i)α(i')) constructed in that lemma is called the pullback functor on descent data.","statement_latex":"With $\\mathcal{U} = \\{U_i \\to S'\\}_{i \\in I}$,\n$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$, $\\alpha : I \\to J$, $h : S' \\to S$,\nand $g_i : U_i \\to V_{\\alpha(i)}$ as in Lemma \\ref{lemma-pullback-family}\nthe functor\n$$\n(Y_j, \\varphi_{jj'}) \\longmapsto\n(g_i^*Y_{\\alpha(i)}, (g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')})\n$$\nconstructed in that lemma\nis called the {\\it pullback functor} on descent data.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for schemes over schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VT","source_file":"descent.tex","source_line":8387,"source_end_line":8399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8387-L8399","statement_sha256":"75cdf02cef9cf70c4c058b2abdf419d37b32764ba78aeb12a14c927bfe36e43e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6922,"rank":6922,"depth":2,"x":804.475,"y":1043.771,"cluster":"descent"},{"id":"stacks:023Z","tag":"023Z","title":"Descent data for schemes over schemes · Definition 023Z","summary":"Let S be a scheme. Let f : X → S be a morphism of schemes. • Given a scheme U over S we have the trivial descent datum of U relative to id : S → S, namely the identity morphism on U. • By Lemma [Tag 023Y] we get a canonical descent datum on X ×_S U relative to X → S by pulling back the trivial descent datum via f. We often denote (X ×_S U, can) this descent datum. • A descent datum (V, φ) relative to X/S is called effective if (V, φ) is isomorphic to the canonical descent…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item  Given a scheme $U$ over $S$ we have the\n{\\it trivial descent datum} of $U$ relative to\n$\\text{id} : S \\to S$, namely the identity morphism on $U$.\n\\item By Lemma \\ref{lemma-pullback} we get a\n{\\it canonical descent datum} on $X \\times_S U$\nrelative to $X \\to S$ by pulling back the trivial\ndescent datum via $f$. We often\ndenote $(X \\times_S U, can)$ this descent datum.\n\\item A descent datum $(V, \\varphi)$ relative to $X/S$ is\ncalled {\\it effective} if $(V, \\varphi)$\nis isomorphic to the canonical descent datum\n$(X \\times_S U, can)$ for some scheme $U$ over $S$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for schemes over schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/023Z","source_file":"descent.tex","source_line":8411,"source_end_line":8429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8411-L8429","statement_sha256":"de53810a35b6d6ef55c270ecf7fd1aa5fdbe911cc576d112af598db2425ae03c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6923,"rank":6923,"depth":1,"x":665.575,"y":1281.815,"cluster":"descent"},{"id":"stacks:02VU","tag":"02VU","title":"Descent data for schemes over schemes · Definition 02VU","summary":"Let S be a scheme. Let (X_i → S) be a family of morphisms with target S. • Given a scheme U over S we have a canonical descent datum on the family of schemes X_i ×_S U by pulling back the trivial descent datum for U relative to (id : S → S). We denote this descent datum (X_i ×_S U, can). • A descent datum (V_i, φ_ij) relative to (X_i → S) is called effective if there exists a scheme U over S such that (V_i, φ_ij) is isomorphic to (X_i ×_S U, can).","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to S\\}$ be a family of morphisms\nwith target $S$.\n\\begin{enumerate}\n\\item  Given a scheme $U$ over $S$\nwe have a {\\it canonical descent datum} on the family of\nschemes $X_i \\times_S U$ by pulling back the trivial\ndescent datum for $U$ relative to $\\{\\text{id} : S \\to S\\}$.\nWe denote this descent datum $(X_i \\times_S U, can)$.\n\\item A descent datum $(V_i, \\varphi_{ij})$\nrelative to $\\{X_i \\to S\\}$ is called {\\it effective}\nif there exists a scheme $U$ over $S$ such that\n$(V_i, \\varphi_{ij})$ is isomorphic to $(X_i \\times_S U, can)$.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data for schemes over schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VU","source_file":"descent.tex","source_line":8443,"source_end_line":8459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8443-L8459","statement_sha256":"1277091119e56352ac78b81f1fecaa6dcf27492abac39866292d6ecd7f34bc37","origin":"The Stacks Project","memory_eligible":false,"source_rank":6924,"rank":6924,"depth":0,"x":576.427,"y":1027.062,"cluster":"descent"},{"id":"stacks:02VW","tag":"02VW","title":"Fully faithfulness of the pullback functors · Lemma 02VW","summary":"A surjective and flat morphism is an epimorphism in the category of schemes.","statement_latex":"A surjective and flat morphism is an epimorphism in the\ncategory of schemes.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VW","source_file":"descent.tex","source_line":8482,"source_end_line":8486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8482-L8486","statement_sha256":"15a65d3070c10bf033365bf1b9628cae409c8f92b3b142534ff44e6fb49e6dd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6925,"rank":6925,"depth":4,"x":847.462,"y":1164.548,"cluster":"descent"},{"id":"stacks:02VX","tag":"02VX","title":"Fully faithfulness of the pullback functors · Lemma 02VX","summary":"Let h : S' → S be a surjective, flat morphism of schemes. The base change functor Sch/S → Sch/S', X ↦ S' ×_S X is faithful.","statement_latex":"Let $h : S' \\to S$ be a surjective, flat morphism of\nschemes. The base change functor\n$$\n\\Sch/S \\longrightarrow \\Sch/S', \\quad\nX \\longmapsto S' \\times_S X\n$$\nis faithful.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VX","source_file":"descent.tex","source_line":8506,"source_end_line":8515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8506-L8515","statement_sha256":"91f6f8972a3bb526f7992de4e72da43ea8ed947acefd0d1839049b83e63f26f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6926,"rank":6926,"depth":5,"x":536.547,"y":1217.043,"cluster":"descent"},{"id":"stacks:0240","tag":"0240","title":"Fully faithfulness of the pullback functors · Lemma 0240","summary":"In the situation of Lemma [Tag 023Y] assume that f : X' → X is surjective and flat. Then the pullback functor is faithful.","statement_latex":"In the situation of Lemma \\ref{lemma-pullback}\nassume that $f : X' \\to X$ is surjective\nand flat. Then the pullback functor is faithful.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0240","source_file":"descent.tex","source_line":8540,"source_end_line":8545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8540-L8545","statement_sha256":"9702f3f52e3feba9f6a6e3be20105abebfa0800eadd8c5ad12b56de41a5f86cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6927,"rank":6927,"depth":6,"x":723.894,"y":1001.571,"cluster":"descent"},{"id":"stacks:0241","tag":"0241","title":"Fully faithfulness of the pullback functors · Lemma 0241","summary":"In the situation of Lemma [Tag 023Y] assume • (f : X' → X) is an fpqc covering (for example if f is surjective, flat, and quasi-compact), and • S = S'. Then the pullback functor is fully faithful.","statement_latex":"In the situation of Lemma \\ref{lemma-pullback}\nassume\n\\begin{enumerate}\n\\item $\\{f : X' \\to X\\}$ is an fpqc covering (for example if $f$ is\nsurjective, flat, and quasi-compact), and\n\\item $S = S'$.\n\\end{enumerate}\nThen the pullback functor is fully faithful.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0241","source_file":"descent.tex","source_line":8560,"source_end_line":8570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8560-L8570","statement_sha256":"5725a3e095e8f60180d8bd822fa0b1b2ce96c3edb5066bb2ac2a04f33c721ea7","origin":"The Stacks Project","memory_eligible":false,"source_rank":6928,"rank":6928,"depth":41,"x":759.079,"y":1267.183,"cluster":"descent"},{"id":"stacks:0242","tag":"0242","title":"Fully faithfulness of the pullback functors · Lemma 0242","summary":"Let X → S be a morphism of schemes. Let f : X → X be a selfmap of X over S. In this case pullback by f is isomorphic to the identity functor on the category of descent data relative to X → S.","statement_latex":"Let $X \\to S$ be a morphism of schemes.\nLet $f : X \\to X$ be a selfmap of $X$ over $S$.\nIn this case pullback by $f$ is isomorphic to the\nidentity functor on the category of descent data\nrelative to $X \\to S$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0242","source_file":"descent.tex","source_line":8665,"source_end_line":8672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8665-L8672","statement_sha256":"b4813a83001fe4fd98a69a0f899262306cdb3cd31f2a7b487a1439beddea3364","origin":"The Stacks Project","memory_eligible":false,"source_rank":6929,"rank":6929,"depth":1,"x":519.159,"y":1091.012,"cluster":"descent"},{"id":"stacks:0243","tag":"0243","title":"Fully faithfulness of the pullback functors · Lemma 0243","summary":"Let f : X' → X be a morphism of schemes over a base scheme S. Assume there exists a morphism g : X → X' over S, for example if f has a section. Then the pullback functor of Lemma [Tag 023Y] defines an equivalence of categories between the category of descent data relative to X/S and X'/S.","statement_latex":"Let $f : X' \\to X$ be a morphism of schemes over a base scheme $S$.\nAssume there exists a morphism $g : X \\to X'$ over $S$, for example\nif $f$ has a section. Then the pullback functor\nof Lemma \\ref{lemma-pullback} defines an equivalence of\ncategories between the category of descent data relative to\n$X/S$ and $X'/S$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0243","source_file":"descent.tex","source_line":8679,"source_end_line":8687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8679-L8687","statement_sha256":"02d9e48cc4404c9a122db96a8f36a074d7acdb51755a7d5188086f97cf2a666c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6930,"rank":6930,"depth":2,"x":838.246,"y":1084.771,"cluster":"descent"},{"id":"stacks:040J","tag":"040J","title":"Fully faithfulness of the pullback functors · Lemma 040J","summary":"Let f : X → X' be a morphism of schemes over a base scheme S. Assume X → S is surjective and flat. Then the pullback functor of Lemma [Tag 023Y] is a faithful functor from the category of descent data relative to X'/S to the category of descent data relative to X/S.","statement_latex":"Let $f : X \\to X'$ be a morphism of schemes over a base scheme $S$.\nAssume $X \\to S$ is surjective and flat. Then the pullback functor\nof Lemma \\ref{lemma-pullback} is a faithful functor\nfrom the category of descent data relative to $X'/S$ to the\ncategory of descent data relative to $X/S$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040J","source_file":"descent.tex","source_line":8697,"source_end_line":8704,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8697-L8704","statement_sha256":"66652ec3d6e9751fd1fde1457b7a25bf3e13a72ed29826b914f4211670593083","origin":"The Stacks Project","memory_eligible":false,"source_rank":6931,"rank":6931,"depth":7,"x":607.612,"y":1270.722,"cluster":"descent"},{"id":"stacks:040K","tag":"040K","title":"Fully faithfulness of the pullback functors · Lemma 040K","summary":"Let f : X → X' be a morphism of schemes over a base scheme S. Assume (X → S) is an fpqc covering (for example if f is surjective, flat and quasi-compact). Then the pullback functor of Lemma [Tag 023Y] is a fully faithful functor from the category of descent data relative to X'/S to the category of descent data relative to X/S.","statement_latex":"Let $f : X \\to X'$ be a morphism of schemes over a base scheme $S$.\nAssume $\\{X \\to S\\}$ is an fpqc covering (for example if $f$ is\nsurjective, flat and quasi-compact).\nThen the pullback functor of Lemma \\ref{lemma-pullback} is a\nfully faithful functor from the category of descent data relative\nto $X'/S$ to the category of descent data relative to $X/S$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040K","source_file":"descent.tex","source_line":8715,"source_end_line":8723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8715-L8723","statement_sha256":"a34103ecfca754c49509f0865e9a28ed8da0398f4a24e9bc1b578b448828226e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6932,"rank":6932,"depth":42,"x":628.174,"y":1002.319,"cluster":"descent"},{"id":"stacks:02VZ","tag":"02VZ","title":"Fully faithfulness of the pullback functors · Lemma 02VZ","summary":"Let S be a scheme. Let U = (U_i → S)_i ∈ I, and V = (V_j → S)_j ∈ J, be families of morphisms with target S. Let α : I → J, id : S → S and g_i : U_i → V_α(i) be a morphism of families of maps with fixed target, see Sites, Definition [Tag 00VT]. Assume that for each j ∈ J the family (g_i : U_i → V_j)_α(i) = j is an fpqc covering of V_j. Then the pullback functor descent data relative to V → descent data relative to U of Lemma [Tag 02VS] is fully faithful.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{U} = \\{U_i \\to S\\}_{i \\in I}$, and\n$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$,\nbe families of morphisms with target $S$.\nLet $\\alpha : I \\to J$, $\\text{id} : S \\to S$ and\n$g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families\nof maps with fixed target, see\nSites, Definition \\ref{sites-definition-morphism-coverings}.\nAssume that for each $j \\in J$ the family\n$\\{g_i : U_i \\to V_j\\}_{\\alpha(i) = j}$ is an fpqc\ncovering of $V_j$. Then the pullback functor\n$$\n\\text{descent data relative to }\n\\mathcal{V}\n\\longrightarrow\n\\text{descent data relative to }\n\\mathcal{U}\n$$\nof Lemma \\ref{lemma-pullback-family} is fully faithful.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VZ","source_file":"descent.tex","source_line":8734,"source_end_line":8755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8734-L8755","statement_sha256":"137e8ba6343acc9e97e011589a7ee6b612a77856b67c0bb42e8ebe162a34cb5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6933,"rank":6933,"depth":42,"x":829.167,"y":1212.229,"cluster":"descent"},{"id":"stacks:02VY","tag":"02VY","title":"Fully faithfulness of the pullback functors · Lemma 02VY","summary":"Let S be a scheme. Let U = (U_i → S)_i ∈ I, and V = (V_j → S)_j ∈ J, be families of morphisms with target S. Let α : I → J, id : S → S and g_i : U_i → V_α(i) be a morphism of families of maps with fixed target, see Sites, Definition [Tag 00VT]. Assume that for each j ∈ J the family (g_i : U_i → V_j)_α(i) = j is a Zariski covering (see Topologies, Definition [Tag 020O]) of V_j. Then the pullback functor descent data relative to V → descent data relative to U of Lemma [Tag…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{U} = \\{U_i \\to S\\}_{i \\in I}$, and\n$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$,\nbe families of morphisms with target $S$.\nLet $\\alpha : I \\to J$, $\\text{id} : S \\to S$ and\n$g_i : U_i \\to V_{\\alpha(i)}$ be a morphism of families\nof maps with fixed target, see\nSites, Definition \\ref{sites-definition-morphism-coverings}.\nAssume that for each $j \\in J$ the family\n$\\{g_i : U_i \\to V_j\\}_{\\alpha(i) = j}$ is a Zariski covering (see\nTopologies, Definition \\ref{topologies-definition-zariski-covering})\nof $V_j$. Then the pullback functor\n$$\n\\text{descent data relative to }\n\\mathcal{V}\n\\longrightarrow\n\\text{descent data relative to }\n\\mathcal{U}\n$$\nof Lemma \\ref{lemma-pullback-family} is an equivalence of categories.\nIn particular, the category of schemes over $S$\nis equivalent to the category\nof descent data relative to any Zariski covering of $S$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VY","source_file":"descent.tex","source_line":8802,"source_end_line":8827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8802-L8827","statement_sha256":"3a35131be23e15d3fa8ae5574e6990b1ee64efafcdff18b44f9bc62fa47da54e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6934,"rank":6934,"depth":43,"x":511.661,"y":1171.429,"cluster":"descent"},{"id":"stacks:02W0","tag":"02W0","title":"Fully faithfulness of the pullback functors · Lemma 02W0","summary":"Let S be a scheme. Let U = (U_i → S)_i ∈ I, and V = (V_j → S)_j ∈ J, be fpqc-coverings of S. If U is a refinement of V, then the pullback functor descent data relative to V → descent data relative to U is fully faithful. In particular, the category of schemes over S is identified with a full subcategory of the category of descent data relative to any fpqc-covering of S.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{U} = \\{U_i \\to S\\}_{i \\in I}$, and\n$\\mathcal{V} = \\{V_j \\to S\\}_{j \\in J}$,\nbe fpqc-coverings of $S$.\nIf $\\mathcal{U}$ is a refinement of $\\mathcal{V}$,\nthen the pullback functor\n$$\n\\text{descent data relative to }\n\\mathcal{V}\n\\longrightarrow\n\\text{descent data relative to }\n\\mathcal{U}\n$$\nis fully faithful.\nIn particular, the category of schemes over $S$\nis identified with a full subcategory of the category\nof descent data relative to any fpqc-covering of $S$.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02W0","source_file":"descent.tex","source_line":8857,"source_end_line":8876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8857-L8876","statement_sha256":"a9299f563984f8c023acc7ae003ce1bca3f8acb1cff85f1e6bd51d5f72503b11","origin":"The Stacks Project","memory_eligible":false,"source_rank":6935,"rank":6935,"depth":43,"x":779.009,"y":1021.121,"cluster":"descent"},{"id":"stacks:0AP4","tag":"0AP4","title":"Fully faithfulness of the pullback functors · Lemma 0AP4","summary":"Let X → S be a surjective, quasi-compact, flat morphism of schemes. Let (V, φ) be a descent datum relative to X/S. Suppose that for all v ∈ V there exists an open subscheme v ∈ W ⊂ V such that φ(W ×_S X) ⊂ X ×_S W and such that the descent datum (W, φ|_W ×_S X) is effective. Then (V, φ) is effective.","statement_latex":"Let $X \\to S$ be a surjective, quasi-compact, flat morphism of\nschemes. Let $(V, \\varphi)$ be a descent datum relative to $X/S$.\nSuppose that for all $v \\in V$ there exists an open subscheme\n$v \\in W \\subset V$ such that $\\varphi(W \\times_S X) \\subset X \\times_S W$\nand such that the descent datum $(W, \\varphi|_{W \\times_S X})$\nis effective. Then $(V, \\varphi)$ is effective.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Fully faithfulness of the pullback functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AP4","source_file":"descent.tex","source_line":8927,"source_end_line":8935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8927-L8935","statement_sha256":"e9b156a1ba6c63e3526c28d0e479a1578eb6620cc0d296ec606d5da3d9258349","origin":"The Stacks Project","memory_eligible":false,"source_rank":6936,"rank":6936,"depth":5,"x":702.626,"y":1284.061,"cluster":"descent"},{"id":"stacks:02W2","tag":"02W2","title":"Descending types of morphisms · Definition 02W2","summary":"Let P be a property of morphisms of schemes over a base. Let τ ∈ (Zariski, fpqc, fppf, etale, smooth, syntomic). We say morphisms of type P satisfy descent for τ-coverings if for any τ-covering U : (U_i → S)_i ∈ I (see Topologies, Section [Tag 020M]), any descent datum (X_i, φ_ij) relative to U such that each morphism X_i → U_i has property P is effective.","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.\nLet $\\tau \\in \\{Zariski, fpqc, fppf, \\etale, smooth, syntomic\\}$.\nWe say\n{\\it morphisms of type $\\mathcal{P}$ satisfy descent for $\\tau$-coverings}\nif for\nany $\\tau$-covering $\\mathcal{U} : \\{U_i \\to S\\}_{i \\in I}$\n(see Topologies, Section \\ref{topologies-section-procedure}),\nany descent datum $(X_i, \\varphi_{ij})$ relative to $\\mathcal{U}$\nsuch that each morphism $X_i \\to U_i$ has property $\\mathcal{P}$\nis effective.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descending types of morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02W2","source_file":"descent.tex","source_line":8992,"source_end_line":9004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L8992-L9004","statement_sha256":"2e2bba3c97d5d5ace2a09c8f19522f4b05b59425a7ce61247565d5ca59a55527","origin":"The Stacks Project","memory_eligible":false,"source_rank":6937,"rank":6937,"depth":0,"x":547.262,"y":1046.466,"cluster":"descent"},{"id":"stacks:02W3","tag":"02W3","title":"Descending types of morphisms · Lemma 02W3","summary":"Let P be a property of morphisms of schemes over a base. Let τ ∈ (fpqc, fppf, etale, smooth, syntomic). Suppose that • P is stable under any base change (see Schemes, Definition [Tag 01JZ]), • if Y_j → V_j, j = 1, …, m have P, then so does coprod Y_j → coprod V_j, and • for any surjective morphism of affines X → S which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether τ is fpqc, fppf, étale, smooth, or syntomic, any descent datum (V, φ)…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes over a base.\nLet $\\tau \\in \\{fpqc, fppf, \\etale, smooth, syntomic\\}$.\nSuppose that\n\\begin{enumerate}\n\\item $\\mathcal{P}$ is stable under any base change\n(see Schemes, Definition \\ref{schemes-definition-preserved-by-base-change}),\n\\item if $Y_j \\to V_j$, $j = 1, \\ldots, m$ have $\\mathcal{P}$,\nthen so does $\\coprod Y_j \\to \\coprod V_j$, and\n\\item for any surjective morphism of affines\n$X \\to S$ which is flat, flat of finite presentation,\n\\'etale, smooth or syntomic depending on whether $\\tau$ is\nfpqc, fppf, \\'etale, smooth, or syntomic,\nany descent datum $(V, \\varphi)$ relative\nto $X$ over $S$ such that $\\mathcal{P}$ holds for\n$V \\to X$ is effective.\n\\end{enumerate}\nThen morphisms of type $\\mathcal{P}$ satisfy descent for $\\tau$-coverings.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descending types of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02W3","source_file":"descent.tex","source_line":9027,"source_end_line":9046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L9027-L9046","statement_sha256":"fad09d11e0e90f368f2d409dcc2cebe51304ef3a475444ffee25f04e86073c58","origin":"The Stacks Project","memory_eligible":false,"source_rank":6938,"rank":6938,"depth":44,"x":853.361,"y":1133.642,"cluster":"descent"},{"id":"stacks:0245","tag":"0245","title":"Descending affine morphisms · Lemma 0245","summary":"Let S be a scheme. Let (X_i → S)_i∈ I be an fpqc covering, see Topologies, Definition [Tag 022B]. Let (V_i/X_i, φ_ij) be a descent datum relative to (X_i → S). If each morphism V_i → X_i is affine, then the descent datum is effective.","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering, see\nTopologies, Definition \\ref{topologies-definition-fpqc-covering}.\nLet $(V_i/X_i, \\varphi_{ij})$ be a descent datum\nrelative to $\\{X_i \\to S\\}$. If each morphism\n$V_i \\to X_i$ is affine, then the descent datum is\neffective.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descending affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0245","source_file":"descent.tex","source_line":9147,"source_end_line":9156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L9147-L9156","statement_sha256":"b3c7c08f17030ca64df9dd21e3b22835d95adfbfe005007fe2b6d9ea5728307a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6939,"rank":6939,"depth":45,"x":557.091,"y":1243.215,"cluster":"descent"},{"id":"stacks:03I0","tag":"03I0","title":"Descending affine morphisms · Lemma 03I0","summary":"Let S be a scheme. Let (X_i → S)_i∈ I be an fpqc covering, see Topologies, Definition [Tag 022B]. Let (V_i/X_i, φ_ij) be a descent datum relative to (X_i → S). If each morphism V_i → X_i is a closed immersion, then the descent datum is effective.","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering, see\nTopologies, Definition \\ref{topologies-definition-fpqc-covering}.\nLet $(V_i/X_i, \\varphi_{ij})$ be a descent datum\nrelative to $\\{X_i \\to S\\}$. If each morphism\n$V_i \\to X_i$ is a closed immersion, then the descent datum is\neffective.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descending affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03I0","source_file":"descent.tex","source_line":9206,"source_end_line":9215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L9206-L9215","statement_sha256":"49b5bbd1dd4930a18bde6c18a95b8a6a7e136234cb290180ca4c33f9ba8e850d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6940,"rank":6940,"depth":46,"x":687.64,"y":993.927,"cluster":"descent"},{"id":"stacks:0247","tag":"0247","title":"Descending quasi-affine morphisms · Lemma 0247","summary":"Let S be a scheme. Let (X_i → S)_i∈ I be an fpqc covering, see Topologies, Definition [Tag 022B]. Let (V_i/X_i, φ_ij) be a descent datum relative to (X_i → S). If each morphism V_i → X_i is quasi-affine, then the descent datum is effective.","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering, see\nTopologies, Definition \\ref{topologies-definition-fpqc-covering}.\nLet $(V_i/X_i, \\varphi_{ij})$ be a descent datum\nrelative to $\\{X_i \\to S\\}$. If each morphism\n$V_i \\to X_i$ is quasi-affine, then the descent datum is\neffective.","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descending quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0247","source_file":"descent.tex","source_line":9232,"source_end_line":9241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L9232-L9241","statement_sha256":"816033e2a423b35456ccfc0673b123c36e6d98384f3d1f4cdfd8de7446a63b8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":6941,"rank":6941,"depth":46,"x":792.004,"y":1252.217,"cluster":"descent"},{"id":"stacks:02W5","tag":"02W5","title":"Descent data in terms of sheaves · Lemma 02W5","summary":"Let τ ∈ (Zariski, fppf, etale, smooth, syntomic). Let Sch_τ be a big τ-site. Let S ∈ Ob(Sch_τ). Let (S_i → S)_i ∈ I be a covering in the site (Sch/S)_τ. There is an equivalence of categories ( descent data (X_i, φ_ii') such that each X_i ∈ Ob((Sch/S)_τ) ) ↔ ( sheaves F on (Sch/S)_τ such that each h_S_i × F is representable ). Moreover, • the objects representing h_S_i × F on the right hand side correspond to the schemes X_i on the left hand side, and • the sheaf F is…","statement_latex":"Let $\\tau \\in \\{Zariski, fppf, \\etale, smooth, syntomic\\}$\\footnote{The\nfact that fpqc is missing is not a typo. See discussion\nin Topologies, Section \\ref{topologies-section-fpqc}.}.\nLet $\\Sch_\\tau$ be a big $\\tau$-site.\nLet $S \\in \\Ob(\\Sch_\\tau)$.\nLet $\\{S_i \\to S\\}_{i \\in I}$ be a covering in the\nsite $(\\Sch/S)_\\tau$. There is an equivalence of\ncategories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{descent data }(X_i, \\varphi_{ii'})\\text{ such that}\\\\\n\\text{each }X_i \\in \\Ob((\\Sch/S)_\\tau)\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{sheaves }F\\text{ on }(\\Sch/S)_\\tau\\text{ such that}\\\\\n\\text{each }h_{S_i} \\times F\\text{ is representable}\n\\end{matrix}\n\\right\\}.\n$$\nMoreover,\n\\begin{enumerate}\n\\item the objects representing $h_{S_i} \\times F$ on the right hand side\ncorrespond to the schemes $X_i$ on the left hand side, and\n\\item the sheaf $F$ is representable if and only if the\ncorresponding descent datum $(X_i, \\varphi_{ii'})$ is effective.\n\\end{enumerate}","area":"Descent","chapter":"Descent","chapter_id":"descent","section":"Descent data in terms of sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02W5","source_file":"descent.tex","source_line":9334,"source_end_line":9366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/descent.tex#L9334-L9366","statement_sha256":"bec337316d309119b05e6b2c093e16a9b1f71be85ee0804d11452a9587ff9bea","origin":"The Stacks Project","memory_eligible":false,"source_rank":6942,"rank":6942,"depth":1,"x":506.907,"y":1120.778,"cluster":"descent"},{"id":"stacks:08DT","tag":"08DT","title":"Derived category of quasi-coherent modules · Lemma 08DT","summary":"Let X be a scheme. Then D_QCoh(O_X) has direct sums.","statement_latex":"Let $X$ be a scheme. Then $D_\\QCoh(\\mathcal{O}_X)$\nhas direct sums.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DT","source_file":"perfect.tex","source_line":79,"source_end_line":83,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L79-L83","statement_sha256":"72143b5c2a2085755d8ee37cbd99ebd00c0bbc195e73935f3f3e4096ac320c1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6943,"rank":6943,"depth":8,"x":189.362,"y":764.41,"cluster":"derived-categories"},{"id":"stacks:0A0J","tag":"0A0J","title":"Derived category of quasi-coherent modules · Lemma 0A0J","summary":"Let X be a scheme. Let (K_n) be an inverse system of D_QCoh(O_X) with derived limit K = Rlim K_n in D(O_X). Assume H^q(K_n + 1) → H^q(K_n) is surjective for all q ∈ Z and n ≥ 1. Then • H^q(K) = lim H^q(K_n), • Rlim H^q(K_n) = lim H^q(K_n), and • for every affine open U ⊂ X we have H^p(U, lim H^q(K_n)) = 0 for p > 0.","statement_latex":"Let $X$ be a scheme. Let $(K_n)$ be an inverse system of\n$D_\\QCoh(\\mathcal{O}_X)$ with derived limit\n$K = R\\lim K_n$ in $D(\\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \\to H^q(K_n)$\nis surjective for all $q \\in \\mathbf{Z}$ and $n \\geq 1$.\nThen\n\\begin{enumerate}\n\\item $H^q(K) = \\lim H^q(K_n)$,\n\\item $R\\lim H^q(K_n) = \\lim H^q(K_n)$, and\n\\item for every affine open $U \\subset X$ we have\n$H^p(U, \\lim H^q(K_n)) = 0$ for $p > 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0J","source_file":"perfect.tex","source_line":101,"source_end_line":114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L101-L114","statement_sha256":"5ab5aca0dff89504164569c7431c9396c7d1c2c4badbdfca9c687c1032697063","origin":"The Stacks Project","memory_eligible":false,"source_rank":6944,"rank":6944,"depth":24,"x":192.014,"y":594.536,"cluster":"derived-categories"},{"id":"stacks:08D3","tag":"08D3","title":"Derived category of quasi-coherent modules · Lemma 08D3","summary":"Let X be a scheme. Let E be an object of D_QCoh(O_X). Then the map E → Rlim τ_≥ -nE of Derived Categories, Remark [Tag 0H72] is an isomorphism.","statement_latex":"Let $X$ be a scheme. Let $E$ be an object of\n$D_\\QCoh(\\mathcal{O}_X)$. Then the map $E \\to R\\lim \\tau_{\\geq -n}E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism\\footnote{In particular,\n$E$ has a K-injective representative by\nDerived Categories, Lemma \\ref{derived-lemma-difficulty-K-injectives}.}.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08D3","source_file":"perfect.tex","source_line":135,"source_end_line":144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L135-L144","statement_sha256":"e93004c59d1d2e059618cc190b14cb1787086fbed8eba1bb883ebb4daaac3621","origin":"The Stacks Project","memory_eligible":false,"source_rank":6945,"rank":6945,"depth":24,"x":326.921,"y":721.543,"cluster":"derived-categories"},{"id":"stacks:08D4","tag":"08D4","title":"Derived category of quasi-coherent modules · Lemma 08D4","summary":"Let X be a scheme. Let F : Mod(O_X) → Ab be an additive functor and N ≥ 0 an integer. Assume that • F commutes with countable direct products, • R^pF(F) = 0 for all p ≥ N and F quasi-coherent. Then for E ∈ D_QCoh(O_X) • H^i(RF(τ_≤ aE)) → H^i(RF(E)) is an isomorphism for i ≤ a, • H^i(RF(E)) → H^i(RF(τ_≥ b - N + 1E)) is an isomorphism for i ≥ b, • if H^i(E) = 0 for i not ∈ [a, b] for some -∞ ≤ a ≤ b ≤ ∞, then H^i(RF(E)) = 0 for i not ∈ [a, b + N - 1].","statement_latex":"Let $X$ be a scheme. Let $F : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Ab}$\nbe an additive functor and $N \\geq 0$ an integer. Assume that\n\\begin{enumerate}\n\\item $F$ commutes with countable direct products,\n\\item $R^pF(\\mathcal{F}) = 0$ for all $p \\geq N$ and $\\mathcal{F}$\nquasi-coherent.\n\\end{enumerate}\nThen for $E \\in D_\\QCoh(\\mathcal{O}_X)$\n\\begin{enumerate}\n\\item $H^i(RF(\\tau_{\\leq a}E)) \\to H^i(RF(E))$ is an isomorphism\nfor $i \\leq a$,\n\\item $H^i(RF(E)) \\to H^i(RF(\\tau_{\\geq b - N + 1}E))$ is an isomorphism\nfor $i \\geq b$,\n\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some\n$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(RF(E)) = 0$\nfor $i \\not \\in [a, b + N - 1]$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08D4","source_file":"perfect.tex","source_line":157,"source_end_line":176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L157-L176","statement_sha256":"89314ed4b3d218ca40b95a65bec722b11227f3c76c9df2e3110d8793ff35c2dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":6946,"rank":6946,"depth":25,"x":124.927,"y":704.408,"cluster":"derived-categories"},{"id":"stacks:06Z0","tag":"06Z0","title":"Derived category of quasi-coherent modules · Lemma 06Z0","summary":"Let X = Spec(A) be an affine scheme. All the functors in the diagram xymatrix D(QCoh(O_X)) ar[rr]_([Tag 06VT]) & & D_QCoh(O_X) ar[ld]^RΓ(X, -) & D(A) ar[lu]^widetilde are equivalences of triangulated categories. Moreover, for E in D_QCoh(O_X) we have H^0(X, E) = H^0(X, H^0(E)).","statement_latex":"Let $X = \\Spec(A)$ be an affine scheme. All the functors in the diagram\n$$\n\\xymatrix{\nD(\\QCoh(\\mathcal{O}_X)) \\ar[rr]_{(\\ref{equation-compare})}\n& &\nD_\\QCoh(\\mathcal{O}_X) \\ar[ld]^{R\\Gamma(X, -)} \\\\\n& D(A) \\ar[lu]^{\\widetilde{\\ \\ }}\n}\n$$\nare equivalences of triangulated categories. Moreover, for $E$ in\n$D_\\QCoh(\\mathcal{O}_X)$ we have $H^0(X, E) = H^0(X, H^0(E))$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Z0","source_file":"perfect.tex","source_line":236,"source_end_line":249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L236-L249","statement_sha256":"5f402ae4aaee0076737f248402470cb278d4fe61e604e9605fd9809b923358b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6947,"rank":6947,"depth":26,"x":287.96,"y":602.236,"cluster":"derived-categories"},{"id":"stacks:08DV","tag":"08DV","title":"Derived category of quasi-coherent modules · Lemma 08DV","summary":"Let X = Spec(A) be an affine scheme. If K^bullet is a K-flat complex of A-modules, then widetildeK^bullet is a K-flat complex of O_X-modules.","statement_latex":"Let $X = \\Spec(A)$ be an affine scheme. If $K^\\bullet$ is a K-flat\ncomplex of $A$-modules, then $\\widetilde{K^\\bullet}$ is a K-flat\ncomplex of $\\mathcal{O}_X$-modules.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DV","source_file":"perfect.tex","source_line":326,"source_end_line":331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L326-L331","statement_sha256":"972d1b8557e23f800e780c547f05dcb186224ff40e196089b7eca34de11f609a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6948,"rank":6948,"depth":6,"x":249.834,"y":770.399,"cluster":"derived-categories"},{"id":"stacks:0DJK","tag":"0DJK","title":"Derived category of quasi-coherent modules · Lemma 0DJK","summary":"If f : X → Y is a morphism of affine schemes given by the ring map A → B, then the diagram xymatrix D(B) ar[d] ar[r] & D_QCoh(O_X) ar[d]^Rf_* D(A) ar[r] & D_QCoh(O_Y) commutes.","statement_latex":"If $f : X \\to Y$ is a morphism of affine schemes given by the ring map\n$A \\to B$, then the diagram\n$$\n\\xymatrix{\nD(B) \\ar[d] \\ar[r] & D_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\\nD(A) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y)\n}\n$$\ncommutes.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJK","source_file":"perfect.tex","source_line":344,"source_end_line":355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L344-L355","statement_sha256":"58f32f230bd60d9ef125e01f81f219519307dce1d88efc3101dbcaa6b5cf7b57","origin":"The Stacks Project","memory_eligible":false,"source_rank":6949,"rank":6949,"depth":27,"x":142.516,"y":624.493,"cluster":"derived-categories"},{"id":"stacks:08DW","tag":"08DW","title":"Derived category of quasi-coherent modules · Lemma 08DW","summary":"Let f : Y → X be a morphism of schemes. • The functor Lf^* sends D_QCoh(O_X) into D_QCoh(O_Y). • If X and Y are affine and f is given by the ring map A → B, then the diagram xymatrix D(B) ar[r] & D_QCoh(O_Y) D(A) ar[r] ar[u]^- ⊗_A^L B & D_QCoh(O_X) ar[u]_Lf^* commutes.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes.\n\\begin{enumerate}\n\\item The functor $Lf^*$ sends $D_\\QCoh(\\mathcal{O}_X)$\ninto $D_\\QCoh(\\mathcal{O}_Y)$.\n\\item If $X$ and $Y$ are affine and $f$ is given by the ring map\n$A \\to B$, then the diagram\n$$\n\\xymatrix{\nD(B) \\ar[r] & D_\\QCoh(\\mathcal{O}_Y) \\\\\nD(A) \\ar[r] \\ar[u]^{- \\otimes_A^\\mathbf{L} B} &\nD_\\QCoh(\\mathcal{O}_X) \\ar[u]_{Lf^*}\n}\n$$\ncommutes.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DW","source_file":"perfect.tex","source_line":363,"source_end_line":380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L363-L380","statement_sha256":"2a0986ad1bed86eafe94c31ec1441a36c55800a70fe5aa5cf8e75d2ccf8c6e20","origin":"The Stacks Project","memory_eligible":false,"source_rank":6950,"rank":6950,"depth":27,"x":339.349,"y":671.274,"cluster":"derived-categories"},{"id":"stacks:08DX","tag":"08DX","title":"Derived category of quasi-coherent modules · Lemma 08DX","summary":"Let X be a scheme. • For objects K, L of D_QCoh(O_X) the derived tensor product K ⊗^L_O_X L is in D_QCoh(O_X). • If X = Spec(A) is affine then widetildeM^bullet ⊗_O_X^L widetildeK^bullet = widetildeM^bullet ⊗_A^L K^bullet for any pair of complexes of A-modules K^bullet, M^bullet.","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item For objects $K, L$ of $D_\\QCoh(\\mathcal{O}_X)$\nthe derived tensor product $K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} L$ is in\n$D_\\QCoh(\\mathcal{O}_X)$.\n\\item If $X = \\Spec(A)$ is affine then\n$$\n\\widetilde{M^\\bullet} \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\widetilde{K^\\bullet}\n=\n\\widetilde{M^\\bullet \\otimes_A^\\mathbf{L} K^\\bullet}\n$$\nfor any pair of complexes of $A$-modules $K^\\bullet$, $M^\\bullet$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DX","source_file":"perfect.tex","source_line":402,"source_end_line":417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L402-L417","statement_sha256":"cc95d40142181ce8cbacec44c6f542e5d73af255a453270ac56bb70477b2812a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6951,"rank":6951,"depth":27,"x":156.249,"y":748.608,"cluster":"derived-categories"},{"id":"stacks:08D5","tag":"08D5","title":"Total direct image · Lemma 08D5","summary":"Let f : X → S be a morphism of schemes. Assume that f is quasi-separated and quasi-compact. • The functor Rf_* sends D_QCoh(O_X) into D_QCoh(O_S). • If S is quasi-compact, there exists an integer N = N(X, S, f) such that for an object E of D_QCoh(O_X) with H^m(E) = 0 for m > 0 we have H^m(Rf_*E) = 0 for m ≥ N. • In fact, if S is quasi-compact we can find N = N(X, S, f) such that for every morphism of schemes S' → S the same conclusion holds for the functor R(f')_* where…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume that $f$ is quasi-separated and quasi-compact.\n\\begin{enumerate}\n\\item The functor $Rf_*$ sends $D_\\QCoh(\\mathcal{O}_X)$\ninto $D_\\QCoh(\\mathcal{O}_S)$.\n\\item If $S$ is quasi-compact, there exists an integer $N = N(X, S, f)$\nsuch that for an object $E$ of $D_\\QCoh(\\mathcal{O}_X)$\nwith $H^m(E) = 0$ for $m > 0$ we have\n$H^m(Rf_*E) = 0$ for $m \\geq N$.\n\\item In fact, if $S$ is quasi-compact we can find $N = N(X, S, f)$\nsuch that for every morphism of schemes $S' \\to S$\nthe same conclusion holds for the functor $R(f')_*$\nwhere $f' : X' \\to S'$ is the base change of $f$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Total direct image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08D5","source_file":"perfect.tex","source_line":443,"source_end_line":459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L443-L459","statement_sha256":"f78995c7fe3616a105bf1d50d02492efe019776dc6dfb6a5a06a5a193f7bd12e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6952,"rank":6952,"depth":29,"x":229.208,"y":587.392,"cluster":"derived-categories"},{"id":"stacks:0G9N","tag":"0G9N","title":"Total direct image · Lemma 0G9N","summary":"Let f : X → S be a quasi-separated and quasi-compact morphism of schemes. Let F^bullet be a complex of quasi-coherent O_X-modules each of which is right acyclic for f_*. Then f_*F^bullet represents Rf_*F^bullet in D(O_S).","statement_latex":"Let $f : X \\to S$ be a quasi-separated and quasi-compact morphism\nof schemes. Let $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent\n$\\mathcal{O}_X$-modules each of which is right acyclic for $f_*$.\nThen $f_*\\mathcal{F}^\\bullet$ represents $Rf_*\\mathcal{F}^\\bullet$\nin $D(\\mathcal{O}_S)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Total direct image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9N","source_file":"perfect.tex","source_line":513,"source_end_line":520,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L513-L520","statement_sha256":"a078e077936cf240883d0045b3d923bdf55f5980e0ba0ee1eed2e9511a2be603","origin":"The Stacks Project","memory_eligible":false,"source_rank":6953,"rank":6953,"depth":30,"x":305.195,"y":747.963,"cluster":"derived-categories"},{"id":"stacks:0G9P","tag":"0G9P","title":"Total direct image · Lemma 0G9P","summary":"Let X be a quasi-separated and quasi-compact scheme. Let F^bullet be a complex of quasi-coherent O_X-modules each of which is right acyclic for Γ(X, -). Then Γ(X, F^bullet) represents RΓ(X, F^bullet) in D(Γ(X, O_X).","statement_latex":"Let $X$ be a quasi-separated and quasi-compact scheme.\nLet $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent\n$\\mathcal{O}_X$-modules each of which is right acyclic for $\\Gamma(X, -)$.\nThen $\\Gamma(X, \\mathcal{F}^\\bullet)$ represents\n$R\\Gamma(X, \\mathcal{F}^\\bullet)$ in $D(\\Gamma(X, \\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Total direct image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9P","source_file":"perfect.tex","source_line":539,"source_end_line":546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L539-L546","statement_sha256":"7240b6a67ea37f07abe7c377ef0d9606b6c707f02adb26cc0d5e206451d12cc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6954,"rank":6954,"depth":31,"x":119.698,"y":672.539,"cluster":"derived-categories"},{"id":"stacks:0G9Q","tag":"0G9Q","title":"Total direct image · Lemma 0G9Q","summary":"Let X be a quasi-separated and quasi-compact scheme. For any object K of D_QCoh(O_X) the spectral sequence E_2^i, j = H^i(X, H^j(K)) ⇒ H^i + j(X, K) of Cohomology, Example [Tag 0BKM] is bounded and converges.","statement_latex":"Let $X$ be a quasi-separated and quasi-compact scheme. For any object\n$K$ of $D_\\QCoh(\\mathcal{O}_X)$ the spectral sequence\n$$\nE_2^{i, j} = H^i(X, H^j(K)) \\Rightarrow H^{i + j}(X, K)\n$$\nof Cohomology, Example \\ref{cohomology-example-spectral-sequence}\nis bounded and converges.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Total direct image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9Q","source_file":"perfect.tex","source_line":553,"source_end_line":562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L553-L562","statement_sha256":"777cdaeff11ffc41118fba4c54fe42dfa608ab4e19b5e4d403fb2ef860813502","origin":"The Stacks Project","memory_eligible":false,"source_rank":6955,"rank":6955,"depth":28,"x":317.494,"y":622.81,"cluster":"derived-categories"},{"id":"stacks:08DZ","tag":"08DZ","title":"Total direct image · Lemma 08DZ","summary":"Let f : X → S be a quasi-separated and quasi-compact morphism of schemes. Then Rf_* : D_QCoh(O_X) → D_QCoh(O_S) commutes with direct sums.","statement_latex":"Let $f : X \\to S$ be a quasi-separated and quasi-compact morphism of\nschemes. Then\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$\ncommutes with direct sums.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Total direct image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DZ","source_file":"perfect.tex","source_line":585,"source_end_line":591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L585-L591","statement_sha256":"86c5c51999fdd59f6300ffba0ab4827786d013048ef639a18cfdcce3801e8ad9","origin":"The Stacks Project","memory_eligible":false,"source_rank":6956,"rank":6956,"depth":30,"x":211.44,"y":771.984,"cluster":"derived-categories"},{"id":"stacks:0G9R","tag":"0G9R","title":"Affine morphisms · Lemma 0G9R","summary":"Let f : X → S be an affine morphism of schemes. Let F^bullet be a complex of quasi-coherent O_X-modules. Then f_*F^bullet = Rf_*F^bullet.","statement_latex":"Let $f : X \\to S$ be an affine morphism of schemes. Let $\\mathcal{F}^\\bullet$\nbe a complex of quasi-coherent $\\mathcal{O}_X$-modules. Then\n$f_*\\mathcal{F}^\\bullet = Rf_*\\mathcal{F}^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9R","source_file":"perfect.tex","source_line":635,"source_end_line":640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L635-L640","statement_sha256":"12994bd799c9572ce77e1cdc1beb7d5f744743a817e9240b8932470e20989b6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6957,"rank":6957,"depth":31,"x":169.606,"y":601.5,"cluster":"derived-categories"},{"id":"stacks:08I8","tag":"08I8","title":"Affine morphisms · Lemma 08I8","summary":"Let f : X → S be an affine morphism of schemes. Then Rf_* : D_QCoh(O_X) → D_QCoh(O_S) reflects isomorphisms.","statement_latex":"Let $f : X \\to S$ be an affine morphism of schemes.\nThen\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$\nreflects isomorphisms.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08I8","source_file":"perfect.tex","source_line":649,"source_end_line":655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L649-L655","statement_sha256":"caa78160d6adb703d81b222b0b48edca3936021bc62f2830bcc689b78d536f9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6958,"rank":6958,"depth":27,"x":337.855,"y":703.658,"cluster":"derived-categories"},{"id":"stacks:08I9","tag":"08I9","title":"Affine morphisms · Lemma 08I9","summary":"Let f : X → S be an affine morphism of schemes. For E in D_QCoh(O_S) we have Rf_* Lf^* E = E ⊗^L_O_S f_*O_X.","statement_latex":"Let $f : X \\to S$ be an affine morphism of schemes.\nFor $E$ in $D_\\QCoh(\\mathcal{O}_S)$ we have\n$Rf_* Lf^* E = E \\otimes^\\mathbf{L}_{\\mathcal{O}_S} f_*\\mathcal{O}_X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08I9","source_file":"perfect.tex","source_line":670,"source_end_line":675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L670-L675","statement_sha256":"c10b9e0c28e159be9ebbce388dc3565445ab79a3b6fa042617da1e9ab8889c6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6959,"rank":6959,"depth":28,"x":131.268,"y":723.833,"cluster":"derived-categories"},{"id":"stacks:0AVW","tag":"0AVW","title":"Affine morphisms · Lemma 0AVW","summary":"Let f : X → Y be an affine morphism of schemes. Then f_* induces an equivalence Φ : D_QCoh(O_X) → D_QCoh(f_*O_X) whose composition with D_QCoh(f_*O_X) → D_QCoh(O_Y) is Rf_* : D_QCoh(O_X) → D_QCoh(O_Y).","statement_latex":"Let $f : X \\to Y$ be an affine morphism of schemes. Then $f_*$ induces\nan equivalence\n$$\n\\Phi : D_\\QCoh(\\mathcal{O}_X) \\longrightarrow D_\\QCoh(f_*\\mathcal{O}_X)\n$$\nwhose composition with $D_\\QCoh(f_*\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$\nis $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVW","source_file":"perfect.tex","source_line":712,"source_end_line":721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L712-L721","statement_sha256":"a45164dd9f6bcd8485956a1ac8b80047faa38373612c4520ecdde3965f799f3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6960,"rank":6960,"depth":26,"x":267.622,"y":591.498,"cluster":"derived-categories"},{"id":"stacks:08DA","tag":"08DA","title":"Cohomology with support in a closed subset · Definition 08DA","summary":"Let X be a scheme. Let E be an object of D(O_X). Let T ⊂ X be a closed subset. We say E is supported on T if the cohomology sheaves H^i(E) are supported on T.","statement_latex":"Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$.\nLet $T \\subset X$ be a closed subset.\nWe say $E$ is {\\it supported on $T$} if the\ncohomology sheaves $H^i(E)$ are supported on $T$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology with support in a closed subset","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DA","source_file":"perfect.tex","source_line":787,"source_end_line":793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L787-L793","statement_sha256":"a57671e24adc0d91368b775f6e4bce76e6deb2d2aa0088d46dafa1ef496c6a29","origin":"The Stacks Project","memory_eligible":false,"source_rank":6961,"rank":6961,"depth":0,"x":273.507,"y":766.762,"cluster":"derived-categories"},{"id":"stacks:0G7G","tag":"0G7G","title":"Cohomology with support in a closed subset · Lemma 0G7G","summary":"Let X be a scheme. Let T ⊂ X be a closed subset such that X setminus T is a retrocompact open of X. Let i : T → X be the inclusion. • For E in D_QCoh(O_X) we have i_*RH_T(E) in D_QCoh, T(O_X). • The functor i_* ∘ RH_T : D_QCoh(O_X) → D_QCoh, T(O_X) is right adjoint to the inclusion functor D_QCoh, T(O_X) → D_QCoh(O_X).","statement_latex":"Let $X$ be a scheme. Let $T \\subset X$ be a closed subset such that\n$X \\setminus T$ is a retrocompact open of $X$. Let $i : T \\to X$ be\nthe inclusion.\n\\begin{enumerate}\n\\item For $E$ in $D_\\QCoh(\\mathcal{O}_X)$ we have\n$i_*R\\mathcal{H}_T(E)$ in $D_{\\QCoh, T}(\\mathcal{O}_X)$.\n\\item The functor\n$i_* \\circ R\\mathcal{H}_T : D_\\QCoh(\\mathcal{O}_X) \\to\nD_{\\QCoh, T}(\\mathcal{O}_X)$ is right adjoint to the inclusion functor\n$D_{\\QCoh, T}(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_X)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7G","source_file":"perfect.tex","source_line":820,"source_end_line":833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L820-L833","statement_sha256":"9268669d55494b0f37bcb70a9188f56ec2f4ef99f148dbe249d59fc9ec314091","origin":"The Stacks Project","memory_eligible":false,"source_rank":6962,"rank":6962,"depth":30,"x":127.967,"y":640.64,"cluster":"derived-categories"},{"id":"stacks:0G7H","tag":"0G7H","title":"Cohomology with support in a closed subset · Lemma 0G7H","summary":"Let X be a scheme. Let T ⊂ X be a closed subset such that X setminus T is a retrocompact open of X. Then for a family of objects E_i, i ∈ I of D_QCoh(O_X) we have RH_T(bigoplus E_i) = bigoplus RH_T(E_i).","statement_latex":"Let $X$ be a scheme. Let $T \\subset X$ be a closed subset such that\n$X \\setminus T$ is a retrocompact open of $X$. Then for a family of\nobjects $E_i$, $i \\in I$ of $D_\\QCoh(\\mathcal{O}_X)$ we have\n$R\\mathcal{H}_T(\\bigoplus E_i) = \\bigoplus R\\mathcal{H}_T(E_i)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7H","source_file":"perfect.tex","source_line":850,"source_end_line":856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L850-L856","statement_sha256":"ff6860fe7c7b8634c099b5bb320ca6eef0d0d0fb91565ca982e55cbc2068f08a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6963,"rank":6963,"depth":31,"x":337.078,"y":651.076,"cluster":"derived-categories"},{"id":"stacks:0G7K","tag":"0G7K","title":"Cohomology with support in a closed subset · Lemma 0G7K","summary":"With X, f_1, …, f_c ∈ Γ(X, O_X), and F as in Remark [Tag 0G7I] the complex ([Tag 0G7J]) restricts to an acyclic complex over X setminus Z.","statement_latex":"With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and\n$\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}\nthe complex (\\ref{equation-extended-alternating}) restricts to an acyclic\ncomplex over $X \\setminus Z$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7K","source_file":"perfect.tex","source_line":911,"source_end_line":917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L911-L917","statement_sha256":"d4b5815b10c2e3bbc9bcd6a8de687ea4c7cf0d45314333c99ab0af06c40234a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":6964,"rank":6964,"depth":0,"x":174.204,"y":762.231,"cluster":"derived-categories"},{"id":"stacks:0G7M","tag":"0G7M","title":"Cohomology with support in a closed subset · Lemma 0G7M","summary":"With X, f_1, …, f_c ∈ Γ(X, O_X), and F as in Remark [Tag 0G7I]. If F is quasi-coherent, then the complex ([Tag 0G7J]) represents i_* RH_Z(F) in D_Z(O_X).","statement_latex":"With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and\n$\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}.\nIf $\\mathcal{F}$ is quasi-coherent, then the complex\n(\\ref{equation-extended-alternating}) represents\n$i_* R\\mathcal{H}_Z(\\mathcal{F})$ in $D_Z(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7M","source_file":"perfect.tex","source_line":967,"source_end_line":974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L967-L974","statement_sha256":"82ca16851683ac04358fe4e60ef7ad32b944d5e44c7e399bdd01f44108a2f206","origin":"The Stacks Project","memory_eligible":false,"source_rank":6965,"rank":6965,"depth":25,"x":204.972,"y":587.542,"cluster":"derived-categories"},{"id":"stacks:0G7N","tag":"0G7N","title":"Cohomology with support in a closed subset · Lemma 0G7N","summary":"Let X be a scheme. Let T ⊂ X be a closed subset which can locally be cut out by at most c elements of the structure sheaf. Then H^i_Z(F) = 0 for i > c and any quasi-coherent O_X-module F.","statement_latex":"Let $X$ be a scheme. Let $T \\subset X$ be a closed subset which can\nlocally be cut out by at most $c$ elements of the structure sheaf.\nThen $\\mathcal{H}^i_Z(\\mathcal{F}) = 0$ for $i > c$ and any\nquasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7N","source_file":"perfect.tex","source_line":1007,"source_end_line":1013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1007-L1013","statement_sha256":"ff9fc663df1c446c405e77805e698371912ddb1e00b7a78ca7f9e3ee7a65508e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6966,"rank":6966,"depth":26,"x":322.97,"y":734.07,"cluster":"derived-categories"},{"id":"stacks:0G7P","tag":"0G7P","title":"Cohomology with support in a closed subset · Lemma 0G7P","summary":"Let X be a scheme. Let Z ⊂ X be a closed subset which can locally be cut out by a Koszul regular sequence having c elements. Then H^i_Z(F) = 0 for i not = c for every flat, quasi-coherent O_X-module F.","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subset which can\nlocally be cut out by a Koszul regular sequence having $c$ elements.\nThen $\\mathcal{H}^i_Z(\\mathcal{F}) = 0$ for $i \\not = c$ for every\nflat, quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7P","source_file":"perfect.tex","source_line":1021,"source_end_line":1027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1021-L1027","statement_sha256":"1bac139b6c6a7a801ff8344d5e1105674e6964fa9dfa8a9ec6a5a151297a126d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6967,"rank":6967,"depth":26,"x":117.769,"y":692.906,"cluster":"derived-categories"},{"id":"stacks:0G7S","tag":"0G7S","title":"Cohomology with support in a closed subset · Lemma 0G7S","summary":"With X, f_1, …, f_c ∈ Γ(X, O_X), and F as in Remark [Tag 0G7I]. Let a_ji ∈ Γ(X, O_X) for 1 ≤ i, j ≤ c and set g_j = ∑_i = 1, …, c a_jif_i. Assume g_1, …, g_c scheme theoretically cut out Z. If F is quasi-coherent, then c_f_1, …, f_c = det(a_ji) c_g_1, …, g_c where c_f_1, …, f_c and c_g_1, …, g_c are as in Remark [Tag 0G7Q].","statement_latex":"With $X$, $f_1, \\ldots, f_c \\in \\Gamma(X, \\mathcal{O}_X)$, and\n$\\mathcal{F}$ as in Remark \\ref{remark-support-c-equations}.\nLet $a_{ji} \\in \\Gamma(X, \\mathcal{O}_X)$ for $1 \\leq i, j \\leq c$\nand set $g_j = \\sum_{i = 1, \\ldots, c} a_{ji}f_i$. Assume $g_1, \\ldots, g_c$\nscheme theoretically cut out $Z$. If $\\mathcal{F}$ is quasi-coherent, then\n$$\nc_{f_1, \\ldots, f_c} = \\det(a_{ji}) c_{g_1, \\ldots, g_c}\n$$\nwhere $c_{f_1, \\ldots, f_c}$ and $c_{g_1, \\ldots, g_c}$ are as in\nRemark \\ref{remark-supported-map-c-equations}.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7S","source_file":"perfect.tex","source_line":1105,"source_end_line":1117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1105-L1117","statement_sha256":"d597f0c316af742609204abb1fa804d8fca48f20dbabeb7cf37a764a81b5cf1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":6968,"rank":6968,"depth":26,"x":302.505,"y":606.674,"cluster":"derived-categories"},{"id":"stacks:0G7T","tag":"0G7T","title":"Cohomology with support in a closed subset · Lemma 0G7T","summary":"Let X be a scheme. Let Z → X be a closed immersion of finite presentation whose conormal sheaf C_Z/X is locally free of rank c. Then there is a canonical map c : wedge^c(C_Z/X)^vee ⊗_O_Z i^*F → H_Z^c(F) functorial in the quasi-coherent module F.","statement_latex":"Let $X$ be a scheme. Let $Z \\to X$ be a closed immersion of finite presentation\nwhose conormal sheaf $\\mathcal{C}_{Z/X}$ is locally free of rank $c$.\nThen there is a canonical map\n$$\nc :\n\\wedge^c(\\mathcal{C}_{Z/X})^\\vee \\otimes_{\\mathcal{O}_Z} i^*\\mathcal{F}\n\\longrightarrow\n\\mathcal{H}_Z^c(\\mathcal{F})\n$$\nfunctorial in the quasi-coherent module $\\mathcal{F}$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7T","source_file":"perfect.tex","source_line":1228,"source_end_line":1240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1228-L1240","statement_sha256":"9b8d6ec8de8568da4b62b42a3750bc0426ccd7f9273828da41e1f708d74f7bdb","origin":"The Stacks Project","memory_eligible":false,"source_rank":6969,"rank":6969,"depth":27,"x":235.512,"y":775.375,"cluster":"derived-categories"},{"id":"stacks:08D7","tag":"08D7","title":"The coherator · Lemma 08D7","summary":"Let f : X → Y be an affine morphism of schemes. Then f_* defines a derived functor f_* : D(QCoh(O_X)) → D(QCoh(O_Y)). This functor has the property that xymatrix D(QCoh(O_X)) ar[d]_f_* ar[r] & D_QCoh(O_X) ar[d]^Rf_* D(QCoh(O_Y)) ar[r] & D_QCoh(O_Y) commutes.","statement_latex":"Let $f : X \\to Y$ be an affine morphism of schemes.\nThen $f_*$ defines a derived functor\n$f_* : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$.\nThis functor has the property that\n$$\n\\xymatrix{\nD(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{f_*} \\ar[r] &\nD_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\\nD(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &\nD_\\QCoh(\\mathcal{O}_Y)\n}\n$$\ncommutes.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08D7","source_file":"perfect.tex","source_line":1362,"source_end_line":1377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1362-L1377","statement_sha256":"1d8d246266cef0f71b97730786a9ef1787ded864e67ae49cf913e629caf42856","origin":"The Stacks Project","memory_eligible":false,"source_rank":6970,"rank":6970,"depth":32,"x":149.098,"y":612.684,"cluster":"derived-categories"},{"id":"stacks:08D8","tag":"08D8","title":"The coherator · Lemma 08D8","summary":"Let f : X → Y be a morphism of schemes. Assume f is quasi-compact, quasi-separated, and flat. Then, denoting Φ : D(QCoh(O_X)) → D(QCoh(O_Y)) the right derived functor of f_* : QCoh(O_X) → QCoh(O_Y) we have RQ_Y ∘ Rf_* = Φ ∘ RQ_X.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ is\nquasi-compact, quasi-separated, and flat. Then, denoting\n$$\n\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))\n$$\nthe right derived functor of\n$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$\nwe have $RQ_Y \\circ Rf_* = \\Phi \\circ RQ_X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08D8","source_file":"perfect.tex","source_line":1391,"source_end_line":1401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1391-L1401","statement_sha256":"ce604c01022db1f995d6e5a795cdffec623682937efd78949b62e7876f38b68b","origin":"The Stacks Project","memory_eligible":false,"source_rank":6971,"rank":6971,"depth":19,"x":343.984,"y":683.739,"cluster":"derived-categories"},{"id":"stacks:08D9","tag":"08D9","title":"The coherator · Lemma 08D9","summary":"Let X = Spec(A) be an affine scheme. Then • Q_X : Mod(O_X) → QCoh(O_X) is the functor which sends F to the quasi-coherent O_X-module associated to the A-module Γ(X, F), • RQ_X : D(O_X) → D(QCoh(O_X)) is the functor which sends E to the complex of quasi-coherent O_X-modules associated to the object RΓ(X, E) of D(A), • restricted to D_QCoh(O_X) the functor RQ_X defines a quasi-inverse to ([Tag 06VT]).","statement_latex":"Let $X = \\Spec(A)$ be an affine scheme. Then\n\\begin{enumerate}\n\\item $Q_X : \\textit{Mod}(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_X)$\nis the functor\nwhich sends $\\mathcal{F}$ to the quasi-coherent $\\mathcal{O}_X$-module\nassociated to the $A$-module $\\Gamma(X, \\mathcal{F})$,\n\\item $RQ_X : D(\\mathcal{O}_X) \\to D(\\QCoh(\\mathcal{O}_X))$\nis the functor which sends $E$ to the complex of quasi-coherent\n$\\mathcal{O}_X$-modules associated to the object $R\\Gamma(X, E)$ of $D(A)$,\n\\item restricted to $D_\\QCoh(\\mathcal{O}_X)$ the functor\n$RQ_X$ defines a quasi-inverse to (\\ref{equation-compare}).\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08D9","source_file":"perfect.tex","source_line":1452,"source_end_line":1466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1452-L1466","statement_sha256":"50d55d81334d161014100883b1adb1416f397e7718b723ea980f4462610ad451","origin":"The Stacks Project","memory_eligible":false,"source_rank":6972,"rank":6972,"depth":27,"x":142.794,"y":742.027,"cluster":"derived-categories"},{"id":"stacks:09T6","tag":"09T6","title":"The coherator · Lemma 09T6","summary":"Let X be a quasi-compact and quasi-separated scheme. Suppose that for every affine open U ⊂ X the right derived functor Φ : D(QCoh(O_U)) → D(QCoh(O_X)) of the left exact functor j_* : QCoh(O_U) → QCoh(O_X) fits into a commutative diagram xymatrix D(QCoh(O_U)) ar[d]_Φ ar[r]_i_U & D_QCoh(O_U) ar[d]^Rj_* D(QCoh(O_X)) ar[r]^i_X & D_QCoh(O_X) Then the functor ([Tag 06VT]) D(QCoh(O_X)) → D_QCoh(O_X) is an equivalence with quasi-inverse given by RQ_X.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Suppose that\nfor every affine open $U \\subset X$ the right derived functor\n$$\n\\Phi : D(\\QCoh(\\mathcal{O}_U)) \\to D(\\QCoh(\\mathcal{O}_X))\n$$\nof the left exact functor\n$j_* : \\QCoh(\\mathcal{O}_U) \\to \\QCoh(\\mathcal{O}_X)$\nfits into a commutative diagram\n$$\n\\xymatrix{\nD(\\QCoh(\\mathcal{O}_U)) \\ar[d]_\\Phi \\ar[r]_{i_U} &\nD_\\QCoh(\\mathcal{O}_U) \\ar[d]^{Rj_*} \\\\\nD(\\QCoh(\\mathcal{O}_X)) \\ar[r]^{i_X} &\nD_\\QCoh(\\mathcal{O}_X)\n}\n$$\nThen the functor (\\ref{equation-compare})\n$$\nD(\\QCoh(\\mathcal{O}_X))\n\\longrightarrow\nD_\\QCoh(\\mathcal{O}_X)\n$$\nis an equivalence with quasi-inverse given by $RQ_X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09T6","source_file":"perfect.tex","source_line":1484,"source_end_line":1509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1484-L1509","statement_sha256":"10433d246ff05e983fb1f86dd791884f4e040423906049c26a6726214f5e6a0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":6973,"rank":6973,"depth":28,"x":244.449,"y":584.618,"cluster":"derived-categories"},{"id":"stacks:08DB","tag":"08DB","title":"The coherator · Proposition 08DB","summary":"Let X be a quasi-compact scheme with affine diagonal. Then the functor ([Tag 06VT]) D(QCoh(O_X)) → D_QCoh(O_X) is an equivalence with quasi-inverse given by RQ_X.","statement_latex":"Let $X$ be a quasi-compact scheme with affine diagonal.\nThen the functor (\\ref{equation-compare})\n$$\nD(\\QCoh(\\mathcal{O}_X))\n\\longrightarrow\nD_\\QCoh(\\mathcal{O}_X)\n$$\nis an equivalence with quasi-inverse given by $RQ_X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DB","source_file":"perfect.tex","source_line":1583,"source_end_line":1593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1583-L1593","statement_sha256":"7813d5e8d95ad7d774782d03ad4bc7a15c16798b73a4b287b72f83a0831eed17","origin":"The Stacks Project","memory_eligible":false,"source_rank":6974,"rank":6974,"depth":33,"x":296.162,"y":758.665,"cluster":"derived-categories"},{"id":"stacks:0CRX","tag":"0CRX","title":"The coherator · Lemma 0CRX","summary":"Let f : X → Y be a morphism of schemes. Assume X and Y are quasi-compact and have affine diagonal. Then, denoting Φ : D(QCoh(O_X)) → D(QCoh(O_Y)) the right derived functor of f_* : QCoh(O_X) → QCoh(O_Y) the diagram xymatrix D(QCoh(O_X)) ar[d]_Φ ar[r] & D_QCoh(O_X) ar[d]^Rf_* D(QCoh(O_Y)) ar[r] & D_QCoh(O_Y) is commutative.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume $X$ and $Y$ are quasi-compact and have affine diagonal.\nThen, denoting\n$$\n\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))\n$$\nthe right derived functor of\n$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$\nthe diagram\n$$\n\\xymatrix{\nD(\\QCoh(\\mathcal{O}_X)) \\ar[d]_\\Phi \\ar[r] &\nD_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\\nD(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &\nD_\\QCoh(\\mathcal{O}_Y)\n}\n$$\nis commutative.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRX","source_file":"perfect.tex","source_line":1603,"source_end_line":1623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1603-L1623","statement_sha256":"dde4aa81de6bf1421fe8b4367294e4a7572c0f8fda06e30fac4a791e03c34a3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6975,"rank":6975,"depth":34,"x":117.763,"y":659.501,"cluster":"derived-categories"},{"id":"stacks:09T2","tag":"09T2","title":"The coherator for Noetherian schemes · Lemma 09T2","summary":"Let X be a Noetherian scheme. Let J be an injective object of QCoh(O_X). Then J is a flasque sheaf of O_X-modules.","statement_latex":"Let $X$ be a Noetherian scheme. Let $\\mathcal{J}$ be an injective\nobject of $\\QCoh(\\mathcal{O}_X)$. Then $\\mathcal{J}$\nis a flasque sheaf of $\\mathcal{O}_X$-modules.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator for Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09T2","source_file":"perfect.tex","source_line":1734,"source_end_line":1739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1734-L1739","statement_sha256":"79284961bd374329c205d92e38bd8a1774193a9fa16550f30413dfa1c22758ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":6976,"rank":6976,"depth":18,"x":329.414,"y":631.348,"cluster":"derived-categories"},{"id":"stacks:09T3","tag":"09T3","title":"The coherator for Noetherian schemes · Lemma 09T3","summary":"Let f : X → Y be a morphism of Noetherian schemes. Then f_* on quasi-coherent sheaves has a right derived extension Φ : D(QCoh(O_X)) → D(QCoh(O_Y)) such that the diagram xymatrix D(QCoh(O_X)) ar[d]_Φ ar[r] & D_QCoh(O_X) ar[d]^Rf_* D(QCoh(O_Y)) ar[r] & D_QCoh(O_Y) commutes.","statement_latex":"Let $f : X \\to Y$ be a morphism of Noetherian schemes.\nThen $f_*$ on quasi-coherent sheaves has a right derived\nextension\n$\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$\nsuch that the diagram\n$$\n\\xymatrix{\nD(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{\\Phi} \\ar[r] &\nD_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\\nD(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &\nD_\\QCoh(\\mathcal{O}_Y)\n}\n$$\ncommutes.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator for Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09T3","source_file":"perfect.tex","source_line":1754,"source_end_line":1770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1754-L1770","statement_sha256":"d6864293dffbf915f2c4ded3a21eb22b3ecce0fdd3d542b796f3879847634100","origin":"The Stacks Project","memory_eligible":false,"source_rank":6977,"rank":6977,"depth":31,"x":195.761,"y":772.44,"cluster":"derived-categories"},{"id":"stacks:09T4","tag":"09T4","title":"The coherator for Noetherian schemes · Proposition 09T4","summary":"Let X be a Noetherian scheme. Then the functor ([Tag 06VT]) D(QCoh(O_X)) → D_QCoh(O_X) is an equivalence with quasi-inverse given by RQ_X.","statement_latex":"Let $X$ be a Noetherian scheme. Then the functor (\\ref{equation-compare})\n$$\nD(\\QCoh(\\mathcal{O}_X))\n\\longrightarrow\nD_\\QCoh(\\mathcal{O}_X)\n$$\nis an equivalence with quasi-inverse given by $RQ_X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator for Noetherian schemes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09T4","source_file":"perfect.tex","source_line":1808,"source_end_line":1817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1808-L1817","statement_sha256":"0f3dd6b571a23a10522c068f7ad8526bb0063b5d4b86790b2a2cf50347a74c84","origin":"The Stacks Project","memory_eligible":false,"source_rank":6978,"rank":6978,"depth":32,"x":180.828,"y":592.261,"cluster":"derived-categories"},{"id":"stacks:08D0","tag":"08D0","title":"Koszul complexes · Lemma 08D0","summary":"In Situation [Tag 08CZ]. Let M be an A-module and denote F the associated O_X-module. Then there is a canonical isomorphism of complexes Ψ : colim_e Hom_A(I^bullet(f_1^e, …, f_r^e), M) → checkC_alt^bullet(U, F) functorial in M where the differentials on the Hom-complex are the contragredients of the differentials on I^bullet(f_1^e, …, f_r^e).","statement_latex":"In Situation \\ref{situation-complex}. Let $M$ be an $A$-module and\ndenote $\\mathcal{F}$ the associated $\\mathcal{O}_X$-module. Then\nthere is a canonical isomorphism of complexes\n$$\n\\Psi : \\colim_e \\Hom_A(I^\\bullet(f_1^e, \\ldots, f_r^e), M)\n\\longrightarrow\n\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F})\n$$\nfunctorial in $M$ where the differentials on the $\\Hom$-complex\nare the contragredients of the differentials on\n$I^\\bullet(f_1^e, \\ldots, f_r^e)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Koszul complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08D0","source_file":"perfect.tex","source_line":1910,"source_end_line":1923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L1910-L1923","statement_sha256":"c0f4940f096fbb7f5f6a4ecbf9f541592eb8d07c801458af45e688fa507dca11","origin":"The Stacks Project","memory_eligible":false,"source_rank":6979,"rank":6979,"depth":2,"x":336.994,"y":716.857,"cluster":"derived-categories"},{"id":"stacks:08D1","tag":"08D1","title":"Koszul complexes · Lemma 08D1","summary":"In Situation [Tag 08CZ]. Let M^bullet be a complex of A-modules and denote F^bullet the associated complex of O_X-modules. Then there is a canonical isomorphism of complexes colim_e Hom^bullet(I^bullet(f_1^e, …, f_r^e), M^bullet) → Tot(checkC_alt^bullet(U, F^bullet)) functorial in M^bullet.","statement_latex":"In Situation \\ref{situation-complex}. Let $M^\\bullet$ be a\ncomplex of $A$-modules and\ndenote $\\mathcal{F}^\\bullet$ the associated complex of\n$\\mathcal{O}_X$-modules. Then\nthere is a canonical isomorphism of complexes\n$$\n\\colim_e \\Hom^\\bullet(I^\\bullet(f_1^e, \\ldots, f_r^e), M^\\bullet)\n\\longrightarrow\n\\text{Tot}(\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))\n$$\nfunctorial in $M^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Koszul complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08D1","source_file":"perfect.tex","source_line":2013,"source_end_line":2026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2013-L2026","statement_sha256":"4e245bd22b647e5e73077df4f1f2f79f0f898f40f447504847c288b45af215e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":6980,"rank":6980,"depth":3,"x":121.284,"y":713.589,"cluster":"derived-categories"},{"id":"stacks:08D2","tag":"08D2","title":"Koszul complexes · Lemma 08D2","summary":"In Situation [Tag 08CZ]. Let F^bullet be a complex of quasi-coherent O_X-modules. Then there is a canonical isomorphism Tot(checkC_alt^bullet(U, F^bullet)) → RΓ(U, F^bullet) in D(A) functorial in F^bullet.","statement_latex":"In Situation \\ref{situation-complex}. Let $\\mathcal{F}^\\bullet$\nbe a complex of quasi-coherent $\\mathcal{O}_X$-modules. Then\nthere is a canonical isomorphism\n$$\n\\text{Tot}(\\check{\\mathcal{C}}_{alt}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))\n\\longrightarrow\nR\\Gamma(U, \\mathcal{F}^\\bullet)\n$$\nin $D(A)$ functorial in $\\mathcal{F}^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Koszul complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08D2","source_file":"perfect.tex","source_line":2054,"source_end_line":2065,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2054-L2065","statement_sha256":"4cb25844feb57187f39c8e8ab7466b89a96d3c2be11c289cb598cf7a8f6b8997","origin":"The Stacks Project","memory_eligible":false,"source_rank":6981,"rank":6981,"depth":24,"x":283.242,"y":593.4,"cluster":"derived-categories"},{"id":"stacks:08DD","tag":"08DD","title":"Koszul complexes · Proposition 08DD","summary":"In Situation [Tag 08CZ]. For every object E of D_QCoh(O_X) the map ([Tag 08DC]) is an isomorphism.","statement_latex":"In Situation \\ref{situation-complex}. For every object $E$\nof $D_\\QCoh(\\mathcal{O}_X)$ the map\n(\\ref{equation-comparison}) is an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Koszul complexes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08DD","source_file":"perfect.tex","source_line":2098,"source_end_line":2103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2098-L2103","statement_sha256":"4821b5bb08903b275ef69199a4c9d38708a0381704d32540b2d1944e7a56b623","origin":"The Stacks Project","memory_eligible":false,"source_rank":6982,"rank":6982,"depth":27,"x":260.431,"y":774.224,"cluster":"derived-categories"},{"id":"stacks:08E3","tag":"08E3","title":"Koszul complexes · Lemma 08E3","summary":"In Situation [Tag 08CZ]. Let E be an object of D_QCoh(O_X). Assume that H^i(E)|_U = 0 for i = - r + 1, …, 0. Then given s ∈ H^0(X, E) there exists an e ≥ 0 and a morphism K_e → E such that s is in the image of H^0(X, K_e) → H^0(X, E).","statement_latex":"In Situation \\ref{situation-complex}. Let $E$ be an object of\n$D_\\QCoh(\\mathcal{O}_X)$.\nAssume that $H^i(E)|_U = 0$ for $i = - r + 1, \\ldots, 0$.\nThen given $s \\in H^0(X, E)$ there exists an $e \\geq 0$ and\na morphism $K_e \\to E$ such that $s$ is in the image of\n$H^0(X, K_e) \\to H^0(X, E)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Koszul complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08E3","source_file":"perfect.tex","source_line":2156,"source_end_line":2164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2156-L2164","statement_sha256":"c5eb7dc085f262750a90e2713740966d91c52f8b80e8800b4fe7fd4ad48d231e","origin":"The Stacks Project","memory_eligible":false,"source_rank":6983,"rank":6983,"depth":28,"x":131.627,"y":627.702,"cluster":"derived-categories"},{"id":"stacks:08E5","tag":"08E5","title":"Pseudo-coherent and perfect complexes · Lemma 08E5","summary":"Let X be a scheme. If E is an m-pseudo-coherent object of D(O_X), then H^i(E) is a quasi-coherent O_X-module for i > m and H^m(E) is a quotient of a quasi-coherent O_X-module. If E is pseudo-coherent, then E is an object of D_QCoh(O_X).","statement_latex":"Let $X$ be a scheme. If $E$ is an $m$-pseudo-coherent\nobject of $D(\\mathcal{O}_X)$, then $H^i(E)$ is a quasi-coherent\n$\\mathcal{O}_X$-module for $i > m$ and $H^m(E)$ is a quotient\nof a quasi-coherent $\\mathcal{O}_X$-module.\nIf $E$ is pseudo-coherent, then $E$ is an object of\n$D_\\QCoh(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08E5","source_file":"perfect.tex","source_line":2204,"source_end_line":2212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2204-L2212","statement_sha256":"e0bc298068dcb6dd172889e266bd566ce6dab3286af644ad2239acfa654190db","origin":"The Stacks Project","memory_eligible":false,"source_rank":6984,"rank":6984,"depth":0,"x":344.783,"y":662.716,"cluster":"derived-categories"},{"id":"stacks:08E7","tag":"08E7","title":"Pseudo-coherent and perfect complexes · Lemma 08E7","summary":"Let X = Spec(A) be an affine scheme. Let M^bullet be a complex of A-modules and let E be the corresponding object of D(O_X). Then E is an m-pseudo-coherent (resp. pseudo-coherent) as an object of D(O_X) if and only if M^bullet is m-pseudo-coherent (resp. pseudo-coherent) as a complex of A-modules.","statement_latex":"Let $X = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a\ncomplex of $A$-modules and let $E$ be the corresponding object\nof $D(\\mathcal{O}_X)$. Then $E$ is an $m$-pseudo-coherent\n(resp.\\ pseudo-coherent) as an object of $D(\\mathcal{O}_X)$\nif and only if $M^\\bullet$ is $m$-pseudo-coherent (resp.\\ pseudo-coherent)\nas a complex of $A$-modules.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08E7","source_file":"perfect.tex","source_line":2222,"source_end_line":2230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2222-L2230","statement_sha256":"652d3a07cd8e4b0baea78d9b3163a3aa21d58e60cb343e19df6eb2aa0aed3765","origin":"The Stacks Project","memory_eligible":false,"source_rank":6985,"rank":6985,"depth":12,"x":159.144,"y":758.003,"cluster":"derived-categories"},{"id":"stacks:08E8","tag":"08E8","title":"Pseudo-coherent and perfect complexes · Lemma 08E8","summary":"Let X be a Noetherian scheme. Let E be an object of D_QCoh(O_X). For m ∈ Z the following are equivalent • H^i(E) is coherent for i ≥ m and zero for i gg 0, and • E is m-pseudo-coherent. In particular, E is pseudo-coherent if and only if E is an object of D^-_Coh(O_X).","statement_latex":"Let $X$ be a Noetherian scheme. Let $E$ be an object of\n$D_\\QCoh(\\mathcal{O}_X)$. For $m \\in \\mathbf{Z}$ the\nfollowing are equivalent\n\\begin{enumerate}\n\\item $H^i(E)$ is coherent for $i \\geq m$ and zero for $i \\gg 0$, and\n\\item $E$ is $m$-pseudo-coherent.\n\\end{enumerate}\nIn particular, $E$ is pseudo-coherent if and only if $E$ is an object\nof $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08E8","source_file":"perfect.tex","source_line":2262,"source_end_line":2273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2262-L2273","statement_sha256":"4a2a71b01abb6b815f4a5fecf882f0c7e427a1df82a40d1eb86fa01e840802c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6986,"rank":6986,"depth":27,"x":219.503,"y":582.117,"cluster":"derived-categories"},{"id":"stacks:08E9","tag":"08E9","title":"Pseudo-coherent and perfect complexes · Lemma 08E9","summary":"Let X = Spec(A) be an affine scheme. Let M^bullet be a complex of A-modules and let E be the corresponding object of D(O_X). Then • E has tor amplitude in [a, b] if and only if M^bullet has tor amplitude in [a, b]. • E has finite tor dimension if and only if M^bullet has finite tor dimension.","statement_latex":"Let $X = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a\ncomplex of $A$-modules and let $E$ be the corresponding object\nof $D(\\mathcal{O}_X)$. Then\n\\begin{enumerate}\n\\item $E$ has tor amplitude in $[a, b]$ if and only if $M^\\bullet$\nhas tor amplitude in $[a, b]$.\n\\item $E$ has finite tor dimension if and only if $M^\\bullet$\nhas finite tor dimension.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08E9","source_file":"perfect.tex","source_line":2290,"source_end_line":2301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2290-L2301","statement_sha256":"9fd88666ced36bf62d4f907d12094e9b23bbe8c45014ff9b3d55367f203fa1ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":6987,"rank":6987,"depth":27,"x":316.596,"y":746.329,"cluster":"derived-categories"},{"id":"stacks:0DHY","tag":"0DHY","title":"Pseudo-coherent and perfect complexes · Lemma 0DHY","summary":"Let f : X → S be a morphism of affine schemes corresponding to the ring map R → A. Let M^bullet be a complex of A-modules and let E be the corresponding object of D(O_X). Then • E as an object of D(f^-1O_S) has tor amplitude in [a, b] if and only if M^bullet has tor amplitude in [a, b] as an object of D(R). • E locally has finite tor dimension as an object of D(f^-1O_S) if and only if M^bullet has finite tor dimension as an object of D(R).","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes corresponding\nto the ring map $R \\to A$. Let $M^\\bullet$ be a\ncomplex of $A$-modules and let $E$ be the corresponding object\nof $D(\\mathcal{O}_X)$. Then\n\\begin{enumerate}\n\\item $E$ as an object of $D(f^{-1}\\mathcal{O}_S)$ has tor amplitude in\n$[a, b]$ if and only if $M^\\bullet$ has tor amplitude in $[a, b]$\nas an object of $D(R)$.\n\\item $E$ locally has finite tor dimension as an object of\n$D(f^{-1}\\mathcal{O}_S)$ if and only if $M^\\bullet$\nhas finite tor dimension as an object of $D(R)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHY","source_file":"perfect.tex","source_line":2334,"source_end_line":2348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2334-L2348","statement_sha256":"68829039d29c381159de24b76b2193e45655ba3cb81d3854df48bb0e714c0857","origin":"The Stacks Project","memory_eligible":false,"source_rank":6988,"rank":6988,"depth":8,"x":112.616,"y":680.227,"cluster":"derived-categories"},{"id":"stacks:08EA","tag":"08EA","title":"Pseudo-coherent and perfect complexes · Lemma 08EA","summary":"Let X be a quasi-separated scheme. Let E be an object of D_QCoh(O_X). Let a ≤ b. The following are equivalent • E has tor amplitude in [a, b], and • for all F in QCoh(O_X) we have H^i(E ⊗_O_X^L F) = 0 for i not ∈ [a, b].","statement_latex":"Let $X$ be a quasi-separated scheme. Let $E$ be an object\nof $D_\\QCoh(\\mathcal{O}_X)$. Let $a \\leq b$. The\nfollowing are equivalent\n\\begin{enumerate}\n\\item $E$ has tor amplitude in $[a, b]$, and\n\\item for all $\\mathcal{F}$ in $\\QCoh(\\mathcal{O}_X)$\nwe have $H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}) = 0$\nfor $i \\not \\in [a, b]$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EA","source_file":"perfect.tex","source_line":2382,"source_end_line":2393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2382-L2393","statement_sha256":"402fe3ecfae1d33e793384464fef15ffeacd13647d8e8b69996836cfe8f82f0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":6989,"rank":6989,"depth":28,"x":316.514,"y":613.118,"cluster":"derived-categories"},{"id":"stacks:08EB","tag":"08EB","title":"Pseudo-coherent and perfect complexes · Lemma 08EB","summary":"Let X = Spec(A) be an affine scheme. Let M^bullet be a complex of A-modules and let E be the corresponding object of D(O_X). Then E is a perfect object of D(O_X) if and only if M^bullet is perfect as an object of D(A).","statement_latex":"Let $X = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a\ncomplex of $A$-modules and let $E$ be the corresponding object\nof $D(\\mathcal{O}_X)$. Then $E$ is a perfect object of $D(\\mathcal{O}_X)$\nif and only if $M^\\bullet$ is perfect as an object of $D(A)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EB","source_file":"perfect.tex","source_line":2418,"source_end_line":2424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2418-L2424","statement_sha256":"a19dbd455b86094ca442d84f98b2619f27f00a4621b36622ebedce1449066132","origin":"The Stacks Project","memory_eligible":false,"source_rank":6990,"rank":6990,"depth":28,"x":219.974,"y":778.566,"cluster":"derived-categories"},{"id":"stacks:0A6H","tag":"0A6H","title":"Pseudo-coherent and perfect complexes · Lemma 0A6H","summary":"Let X be a scheme. • If L is in D^+_QCoh(O_X) and K in D(O_X) is pseudo-coherent, then RSheafHom(K, L) is in D_QCoh(O_X) and locally bounded below. • If L is in D_QCoh(O_X) and K in D(O_X) is perfect, then RSheafHom(K, L) is in D_QCoh(O_X). • If X = Spec(A) is affine and K, L ∈ D(A) then RSheafHom(widetildeK, widetildeL) = widetildeRHom_A(K, L) in the following two cases • K is pseudo-coherent and L is bounded below, • K is perfect and L arbitrary. • If X = Spec(A) and K,…","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item If $L$ is in $D^+_\\QCoh(\\mathcal{O}_X)$ and\n$K$ in $D(\\mathcal{O}_X)$ is pseudo-coherent, then\n$R\\SheafHom(K, L)$ is in $D_\\QCoh(\\mathcal{O}_X)$\nand locally bounded below.\n\\item If $L$ is in $D_\\QCoh(\\mathcal{O}_X)$ and\n$K$ in $D(\\mathcal{O}_X)$ is perfect, then\n$R\\SheafHom(K, L)$ is in $D_\\QCoh(\\mathcal{O}_X)$.\n\\item If $X = \\Spec(A)$ is affine and $K, L \\in D(A)$ then\n$$\nR\\SheafHom(\\widetilde{K}, \\widetilde{L}) = \\widetilde{R\\Hom_A(K, L)}\n$$\nin the following two cases\n\\begin{enumerate}\n\\item $K$ is pseudo-coherent and $L$ is bounded below,\n\\item $K$ is perfect and $L$ arbitrary.\n\\end{enumerate}\n\\item If $X = \\Spec(A)$ and $K, L$ are in $D(A)$, then the $n$th\ncohomology sheaf of $R\\SheafHom(\\widetilde{K}, \\widetilde{L})$\nis the sheaf associated to the presheaf\n$$\nX \\supset D(f) \\longmapsto \\Ext^n_{A_f}(K \\otimes_A A_f, L \\otimes_A A_f)\n$$\nfor $f \\in A$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6H","source_file":"perfect.tex","source_line":2439,"source_end_line":2467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2439-L2467","statement_sha256":"e6c4948058271ab98ffa47be29b3021d340fb7b6889728235e27c9efd94004c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":6991,"rank":6991,"depth":29,"x":158.014,"y":601.505,"cluster":"derived-categories"},{"id":"stacks:0ATN","tag":"0ATN","title":"Pseudo-coherent and perfect complexes · Lemma 0ATN","summary":"Let X be a scheme. Let K, L, M be objects of D_QCoh(O_X). The map K ⊗_O_X^L RSheafHom(M, L) → RSheafHom(M, K ⊗_O_X^L L) of Cohomology, Lemma [Tag 0BYS] is an isomorphism in the following cases • M perfect, or • K is perfect, or • M is pseudo-coherent, L ∈ D^+(O_X), and K has finite tor dimension.","statement_latex":"Let $X$ be a scheme. Let $K, L, M$ be objects of $D_\\QCoh(\\mathcal{O}_X)$.\nThe map\n$$\nK \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, L)\n\\longrightarrow\nR\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)\n$$\nof Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-diagonal-better}\nis an isomorphism in the following cases\n\\begin{enumerate}\n\\item $M$ perfect, or\n\\item $K$ is perfect, or\n\\item $M$ is pseudo-coherent, $L \\in D^+(\\mathcal{O}_X)$, and $K$ has finite\ntor dimension.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATN","source_file":"perfect.tex","source_line":2501,"source_end_line":2518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2501-L2518","statement_sha256":"91c315831070a559a08bab8fd44f7121c4283c95e6def5a4884d0b41ced97ff5","origin":"The Stacks Project","memory_eligible":false,"source_rank":6992,"rank":6992,"depth":30,"x":346.391,"y":697.062,"cluster":"derived-categories"},{"id":"stacks:0FDA","tag":"0FDA","title":"Derived category of coherent modules · Lemma 0FDA","summary":"Let X be a Noetherian scheme. Then the functor D^-(Coh(O_X)) → D^-_Coh(O_X)(QCoh(O_X)) is an equivalence.","statement_latex":"Let $X$ be a Noetherian scheme. Then the functor\n$$\nD^-(\\textit{Coh}(\\mathcal{O}_X))\n\\longrightarrow\nD^-_{\\textit{Coh}(\\mathcal{O}_X)}(\\QCoh(\\mathcal{O}_X))\n$$\nis an equivalence.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDA","source_file":"perfect.tex","source_line":2564,"source_end_line":2573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2564-L2573","statement_sha256":"6033bd5d39e487a99bef979ed4495db4a85e3f1fb0b3e4a5f2550daa9003da04","origin":"The Stacks Project","memory_eligible":false,"source_rank":6993,"rank":6993,"depth":17,"x":130.296,"y":733.547,"cluster":"derived-categories"},{"id":"stacks:0FDB","tag":"0FDB","title":"Derived category of coherent modules · Proposition 0FDB","summary":"Let X be a Noetherian scheme. Then the functors D^-(Coh(O_X)) → D^-_Coh(O_X) and D^b(Coh(O_X)) → D^b_Coh(O_X) are equivalences.","statement_latex":"Let $X$ be a Noetherian scheme. Then the functors\n$$\nD^-(\\textit{Coh}(\\mathcal{O}_X))\n\\longrightarrow\nD^-_{\\textit{Coh}}(\\mathcal{O}_X)\n\\quad\\text{and}\\quad\nD^b(\\textit{Coh}(\\mathcal{O}_X))\n\\longrightarrow\nD^b_{\\textit{Coh}}(\\mathcal{O}_X)\n$$\nare equivalences.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of coherent modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDB","source_file":"perfect.tex","source_line":2595,"source_end_line":2608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2595-L2608","statement_sha256":"f0f67d6ca77d1f2af476f97fce03e9192cd837028fc9c1bcb405a4939bbb4420","origin":"The Stacks Project","memory_eligible":false,"source_rank":6994,"rank":6994,"depth":33,"x":260.507,"y":583.788,"cluster":"derived-categories"},{"id":"stacks:08E2","tag":"08E2","title":"Derived category of coherent modules · Lemma 08E2","summary":"Let S be a Noetherian scheme. Let f : X → S be a morphism of schemes which is locally of finite type. Let E be an object of D^b_Coh(O_X) such that the support of H^i(E) is proper over S for all i. Then Rf_*E is an object of D^b_Coh(O_S).","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes\nwhich is locally of finite type. Let $E$ be an object of\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$\nis proper over $S$ for all $i$.\nThen $Rf_*E$ is an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_S)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08E2","source_file":"perfect.tex","source_line":2634,"source_end_line":2641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2634-L2641","statement_sha256":"b7f14cdf069db5977fb4a2ac4dbb8b296c4d06714f82369fb836ad2792b69289","origin":"The Stacks Project","memory_eligible":false,"source_rank":6995,"rank":6995,"depth":32,"x":284.962,"y":768.395,"cluster":"derived-categories"},{"id":"stacks:0D0B","tag":"0D0B","title":"Derived category of coherent modules · Lemma 0D0B","summary":"Let S be a Noetherian scheme. Let f : X → S be a morphism of schemes which is locally of finite type. Let E be an object of D^+_Coh(O_X) such that the support of H^i(E) is proper over S for all i. Then Rf_*E is an object of D^+_Coh(O_S).","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes\nwhich is locally of finite type. Let $E$ be an object of\n$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$\nis proper over $S$ for all $i$.\nThen $Rf_*E$ is an object of $D^+_{\\textit{Coh}}(\\mathcal{O}_S)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0B","source_file":"perfect.tex","source_line":2662,"source_end_line":2669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2662-L2669","statement_sha256":"c63e9570401f60c2afb472df07bfe2c3b7bf0bc98f1635050b45007a10b62f07","origin":"The Stacks Project","memory_eligible":false,"source_rank":6996,"rank":6996,"depth":33,"x":118.211,"y":645.952,"cluster":"derived-categories"},{"id":"stacks:0D0C","tag":"0D0C","title":"Derived category of coherent modules · Lemma 0D0C","summary":"Let X be a locally Noetherian scheme. If L is in D^+_Coh(O_X) and K in D^-_Coh(O_X), then RSheafHom(K, L) is in D^+_Coh(O_X).","statement_latex":"Let $X$ be a locally Noetherian scheme. If $L$ is in\n$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in\n$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then\n$R\\SheafHom(K, L)$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0C","source_file":"perfect.tex","source_line":2681,"source_end_line":2687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2681-L2687","statement_sha256":"0bdd78ed408bffc60dc608882452efbb64c7f3eef1fef56ef8ae4133357caff6","origin":"The Stacks Project","memory_eligible":false,"source_rank":6997,"rank":6997,"depth":30,"x":339.985,"y":641.615,"cluster":"derived-categories"},{"id":"stacks:0FXU","tag":"0FXU","title":"Derived category of coherent modules · Lemma 0FXU","summary":"Let X be a Noetherian scheme. Let E in D(O_X) be perfect. Then • E is in D^b_Coh(O_X), • if L is in D_Coh(O_X) then E ⊗_O_X^L L and RSheafHom_O_X(E, L) are in D_Coh(O_X), • if L is in D^b_Coh(O_X) then E ⊗_O_X^L L and RSheafHom_O_X(E, L) are in D^b_Coh(O_X), • if L is in D^+_Coh(O_X) then E ⊗_O_X^L L and RSheafHom_O_X(E, L) are in D^+_Coh(O_X), • if L is in D^-_Coh(O_X) then E ⊗_O_X^L L and RSheafHom_O_X(E, L) are in D^-_Coh(O_X).","statement_latex":"Let $X$ be a Noetherian scheme. Let $E$ in $D(\\mathcal{O}_X)$ be perfect.\nThen\n\\begin{enumerate}\n\\item $E$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$,\n\\item if $L$ is in $D_{\\textit{Coh}}(\\mathcal{O}_X)$ then\n$E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and\n$R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in\n$D_{\\textit{Coh}}(\\mathcal{O}_X)$,\n\\item if $L$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ then\n$E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and\n$R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$,\n\\item if $L$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ then\n$E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and\n$R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in\n$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$,\n\\item if $L$ is in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ then\n$E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$ and\n$R\\SheafHom_{\\mathcal{O}_X}(E, L)$ are in\n$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXU","source_file":"perfect.tex","source_line":2714,"source_end_line":2737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2714-L2737","statement_sha256":"505491c9e37612279b087fbcffe64c4821d90003cf81c9c53f70def907043003","origin":"The Stacks Project","memory_eligible":false,"source_rank":6998,"rank":6998,"depth":7,"x":179.689,"y":770.855,"cluster":"derived-categories"},{"id":"stacks:0D0D","tag":"0D0D","title":"Derived category of coherent modules · Lemma 0D0D","summary":"Let A be a Noetherian ring. Let X be a proper scheme over A. For L in D^+_Coh(O_X) and K in D^-_Coh(O_X), the A-modules Ext_O_X^n(K, L) are finite.","statement_latex":"Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $A$.\nFor $L$ in\n$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in\n$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, the $A$-modules\n$\\Ext_{\\mathcal{O}_X}^n(K, L)$ are finite.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0D","source_file":"perfect.tex","source_line":2753,"source_end_line":2760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2753-L2760","statement_sha256":"183f9b5110ac666f5fcab573b46a5768aa18f4f8d3f2da8afacd61f0ee568074","origin":"The Stacks Project","memory_eligible":false,"source_rank":6999,"rank":6999,"depth":34,"x":193.98,"y":584.308,"cluster":"derived-categories"},{"id":"stacks:0FDC","tag":"0FDC","title":"Derived category of coherent modules · Lemma 0FDC","summary":"Let X be a regular scheme. Then every object of D^b_Coh(O_X) is perfect. If X is quasi-compact, i.e., Noetherian and regular, then conversely every perfect object of D(O_X) is in D^b_Coh(O_X).","statement_latex":"Let $X$ be a regular scheme. Then\nevery object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ is perfect.\nIf $X$ is quasi-compact, i.e., Noetherian and regular,\nthen conversely every perfect object of\n$D(\\mathcal{O}_X)$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDC","source_file":"perfect.tex","source_line":2776,"source_end_line":2783,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2776-L2783","statement_sha256":"cb74848360f94e626ff3acccb16bd89c53fe6c5519834cb7fd1dc3aa8f95117e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7000,"rank":7000,"depth":29,"x":333.675,"y":730.201,"cluster":"derived-categories"},{"id":"stacks:09UD","tag":"09UD","title":"Descent finiteness properties of complexes · Lemma 09UD","summary":"Let f : X → Y be a surjective flat morphism of schemes (or more generally locally ringed spaces). Let E ∈ D(O_Y). Let a, b ∈ Z. Then E has tor-amplitude in [a, b] if and only if Lf^*E has tor-amplitude in [a, b].","statement_latex":"Let $f : X \\to Y$ be a surjective flat morphism of schemes\n(or more generally locally ringed spaces).\nLet $E \\in D(\\mathcal{O}_Y)$. Let $a, b \\in \\mathbf{Z}$.\nThen $E$ has tor-amplitude in $[a, b]$ if and only if\n$Lf^*E$ has tor-amplitude in $[a, b]$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Descent finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UD","source_file":"perfect.tex","source_line":2806,"source_end_line":2813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2806-L2813","statement_sha256":"b876df82c939f86eb282136c969fe487094039b8e0985a8537f801236b1d4495","origin":"The Stacks Project","memory_eligible":false,"source_rank":7001,"rank":7001,"depth":8,"x":113.0,"y":701.844,"cluster":"derived-categories"},{"id":"stacks:09UE","tag":"09UE","title":"Descent finiteness properties of complexes · Lemma 09UE","summary":"Let (f_i : X_i → X) be an fpqc covering of schemes. Let E ∈ D_QCoh(O_X). Let m ∈ Z. Then E is m-pseudo-coherent if and only if each Lf_i^*E is m-pseudo-coherent.","statement_latex":"Let $\\{f_i : X_i \\to X\\}$ be an fpqc covering of schemes. Let\n$E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.\nThen $E$ is $m$-pseudo-coherent if and only if each\n$Lf_i^*E$ is $m$-pseudo-coherent.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Descent finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UE","source_file":"perfect.tex","source_line":2827,"source_end_line":2833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2827-L2833","statement_sha256":"1d32c5f73c8629a6c3432efb28237f51ad4dbf8f16b96aa45f207a0a1bd2bf71","origin":"The Stacks Project","memory_eligible":false,"source_rank":7002,"rank":7002,"depth":27,"x":298.814,"y":597.376,"cluster":"derived-categories"},{"id":"stacks:09UF","tag":"09UF","title":"Descent finiteness properties of complexes · Lemma 09UF","summary":"Let (f_i : X_i → X) be an fppf covering of schemes. Let E ∈ D(O_X). Let m ∈ Z. Then E is m-pseudo-coherent if and only if each Lf_i^*E is m-pseudo-coherent.","statement_latex":"Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of schemes. Let\n$E \\in D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.\nThen $E$ is $m$-pseudo-coherent if and only if each\n$Lf_i^*E$ is $m$-pseudo-coherent.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Descent finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UF","source_file":"perfect.tex","source_line":2853,"source_end_line":2859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2853-L2859","statement_sha256":"8f41ed7b3542a8fa7df77eb1581ce6844967ed91a1122de3fa73336302bcaa40","origin":"The Stacks Project","memory_eligible":false,"source_rank":7003,"rank":7003,"depth":11,"x":245.726,"y":780.127,"cluster":"derived-categories"},{"id":"stacks:09UG","tag":"09UG","title":"Descent finiteness properties of complexes · Lemma 09UG","summary":"Let (f_i : X_i → X) be an fpqc covering of schemes. Let E ∈ D(O_X). Then E is perfect if and only if each Lf_i^*E is perfect.","statement_latex":"Let $\\{f_i : X_i \\to X\\}$ be an fpqc covering of schemes. Let\n$E \\in D(\\mathcal{O}_X)$. Then $E$ is perfect\nif and only if each $Lf_i^*E$ is perfect.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Descent finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UG","source_file":"perfect.tex","source_line":2900,"source_end_line":2905,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2900-L2905","statement_sha256":"dc367fdf95da638d1c2b9da9ee495e7996533b174c7dba3f5f4a3f54f83f9c74","origin":"The Stacks Project","memory_eligible":false,"source_rank":7004,"rank":7004,"depth":28,"x":137.744,"y":614.991,"cluster":"derived-categories"},{"id":"stacks:09VA","tag":"09VA","title":"Descent finiteness properties of complexes · Lemma 09VA","summary":"Let i : Z → X be a morphism of ringed spaces such that i is a closed immersion of underlying topological spaces and such that i_*O_Z is pseudo-coherent as an O_X-module. Let E ∈ D(O_Z). Then E is m-pseudo-coherent if and only if Ri_*E is m-pseudo-coherent.","statement_latex":"Let $i : Z \\to X$ be a morphism of ringed spaces such that\n$i$ is a closed immersion of underlying topological spaces and such that\n$i_*\\mathcal{O}_Z$ is pseudo-coherent as an $\\mathcal{O}_X$-module.\nLet $E \\in D(\\mathcal{O}_Z)$. Then $E$ is $m$-pseudo-coherent\nif and only if $Ri_*E$ is $m$-pseudo-coherent.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Descent finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VA","source_file":"perfect.tex","source_line":2925,"source_end_line":2932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L2925-L2932","statement_sha256":"e3287298169b8f835e8c6bf9f328a1abdf0da031da14656f4f18eda1e4a1f72d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7005,"rank":7005,"depth":9,"x":350.491,"y":675.581,"cluster":"derived-categories"},{"id":"stacks:09VB","tag":"09VB","title":"Descent finiteness properties of complexes · Lemma 09VB","summary":"Let f : X → Y be a finite morphism of schemes such that f_*O_X is pseudo-coherent as an O_Y-module. Let E ∈ D_QCoh(O_X). Then E is m-pseudo-coherent if and only if Rf_*E is m-pseudo-coherent.","statement_latex":"Let $f : X \\to Y$ be a finite morphism of schemes such that\n$f_*\\mathcal{O}_X$ is pseudo-coherent as an\n$\\mathcal{O}_Y$-module\\footnote{This means that $f$ is pseudo-coherent, see\nMore on Morphisms, Lemma\n\\ref{more-morphisms-lemma-finite-pseudo-coherent}.}.\nLet $E \\in D_\\QCoh(\\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent\nif and only if $Rf_*E$ is $m$-pseudo-coherent.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Descent finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VB","source_file":"perfect.tex","source_line":3006,"source_end_line":3015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3006-L3015","statement_sha256":"7d68b182ff624e6d04007538b1d72bc210ec0bf43db05ff0aeeeaed5dd33fa01","origin":"The Stacks Project","memory_eligible":false,"source_rank":7006,"rank":7006,"depth":27,"x":144.575,"y":751.737,"cluster":"derived-categories"},{"id":"stacks:08ED","tag":"08ED","title":"Lifting complexes · Lemma 08ED","summary":"Let X be a scheme and let j : U → X be a quasi-compact open immersion. The functors D_QCoh(O_X) → D_QCoh(O_U) and D^+_QCoh(O_X) → D^+_QCoh(O_U) are essentially surjective. If X is quasi-compact, then the functors D^-_QCoh(O_X) → D^-_QCoh(O_U) and D^b_QCoh(O_X) → D^b_QCoh(O_U) are essentially surjective.","statement_latex":"Let $X$ be a scheme and let $j : U \\to X$ be a quasi-compact\nopen immersion. The functors\n$$\nD_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_U)\n\\quad\\text{and}\\quad\nD^+_\\QCoh(\\mathcal{O}_X) \\to D^+_\\QCoh(\\mathcal{O}_U)\n$$\nare essentially surjective. If $X$ is quasi-compact, then the functors\n$$\nD^-_\\QCoh(\\mathcal{O}_X) \\to D^-_\\QCoh(\\mathcal{O}_U)\n\\quad\\text{and}\\quad\nD^b_\\QCoh(\\mathcal{O}_X) \\to D^b_\\QCoh(\\mathcal{O}_U)\n$$\nare essentially surjective.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ED","source_file":"perfect.tex","source_line":3037,"source_end_line":3053,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3037-L3053","statement_sha256":"d800f68a04c1a3aa8e3cb0ca5c3a673b236bbe39675b6fd599063254ba80e292","origin":"The Stacks Project","memory_eligible":false,"source_rank":7007,"rank":7007,"depth":30,"x":235.31,"y":578.475,"cluster":"derived-categories"},{"id":"stacks:0G48","tag":"0G48","title":"Lifting complexes · Lemma 0G48","summary":"Let X be a Noetherian scheme and let j : U → X be an open immersion. The functor D^b_Coh(O_X) → D^b_Coh(O_U) is essentially surjective.","statement_latex":"Let $X$ be a Noetherian scheme and let $j : U \\to X$ be an open immersion.\nThe functor\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_U)$\nis essentially surjective.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G48","source_file":"perfect.tex","source_line":3070,"source_end_line":3076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3070-L3076","statement_sha256":"615cb22e7eebf755f3c43fcb4e16da0f8194943fcf12079f54268613fbd9e2b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7008,"rank":7008,"depth":34,"x":307.846,"y":757.994,"cluster":"derived-categories"},{"id":"stacks:08EE","tag":"08EE","title":"Lifting complexes · Lemma 08EE","summary":"Let X be an affine scheme and let U ⊂ X be a quasi-compact open subscheme. For any pseudo-coherent object E of D(O_U) there exists a bounded above complex of finite free O_X-modules whose restriction to U is isomorphic to E.","statement_latex":"Let $X$ be an affine scheme and let $U \\subset X$ be a quasi-compact\nopen subscheme. For any pseudo-coherent object $E$ of $D(\\mathcal{O}_U)$\nthere exists a bounded above complex of finite free $\\mathcal{O}_X$-modules \nwhose restriction to $U$ is isomorphic to $E$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EE","source_file":"perfect.tex","source_line":3103,"source_end_line":3109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3103-L3109","statement_sha256":"e8848fdc75cd72a788011e9c6124ad03e4cd3c3d205c3380a5cb3a8c06a3c1d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7009,"rank":7009,"depth":31,"x":109.695,"y":666.64,"cluster":"derived-categories"},{"id":"stacks:08EF","tag":"08EF","title":"Lifting complexes · Lemma 08EF","summary":"Let X be a quasi-compact and quasi-separated scheme. Let E ∈ D^b_QCoh(O_X). There exists an integer n_0 > 0 such that Ext^n_D(O_X)(E, E) = 0 for every finite locally free O_X-module E and every n ≥ n_0.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $E \\in D^b_\\QCoh(\\mathcal{O}_X)$.\nThere exists an integer $n_0 > 0$ such that\n$\\Ext^n_{D(\\mathcal{O}_X)}(\\mathcal{E}, E) = 0$\nfor every finite locally free\n$\\mathcal{O}_X$-module $\\mathcal{E}$ and every $n \\geq n_0$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EF","source_file":"perfect.tex","source_line":3196,"source_end_line":3204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3196-L3204","statement_sha256":"194e941295ed98ec29fcb24b689a6d88ca8e6f0df7789e3c417c4a794c425862","origin":"The Stacks Project","memory_eligible":false,"source_rank":7010,"rank":7010,"depth":27,"x":329.605,"y":621.5,"cluster":"derived-categories"},{"id":"stacks:09M4","tag":"09M4","title":"Lifting complexes · Lemma 09M4","summary":"Let X be a quasi-compact and quasi-separated scheme. Let K be a perfect object of D(O_X). Then • there exist integers a ≤ b such that for any L ∈ D_QCoh(O_X) with H^i(L) = 0 for i ∈ [a, b] we have Hom_D(O_X)(K, L) = 0, and • if L is bounded, then Ext^n_D(O_X)(K, L) is zero for all but finitely many n.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then\n\\begin{enumerate}\n\\item there exist integers $a \\leq b$ such that for any\n$L \\in D_\\QCoh(\\mathcal{O}_X)$ with $H^i(L) = 0$ for $i \\in [a, b]$\nwe have $\\Hom_{D(\\mathcal{O}_X)}(K, L) = 0$, and\n\\item if $L$ is bounded, then $\\Ext^n_{D(\\mathcal{O}_X)}(K, L)$\nis zero for all but finitely many $n$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09M4","source_file":"perfect.tex","source_line":3247,"source_end_line":3258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3247-L3258","statement_sha256":"81c515569ec5d9ebd717516f8bc4ae29ab13e9c0c3a38cd29ea5979227cf5d9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7011,"rank":7011,"depth":28,"x":203.558,"y":779.805,"cluster":"derived-categories"},{"id":"stacks:08EG","tag":"08EG","title":"Lifting complexes · Lemma 08EG","summary":"Let X be an affine scheme. Let U ⊂ X be a quasi-compact open. For every perfect object E of D(O_U) there exists an integer r and a finite locally free sheaf F on U such that F[-r] ⊕ E is the restriction of a perfect object of D(O_X).","statement_latex":"Let $X$ be an affine scheme. Let $U \\subset X$ be a quasi-compact open.\nFor every perfect object $E$ of $D(\\mathcal{O}_U)$ there exists an integer\n$r$ and a finite locally free sheaf $\\mathcal{F}$ on $U$ such that\n$\\mathcal{F}[-r] \\oplus E$ is the restriction of a perfect object of\n$D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EG","source_file":"perfect.tex","source_line":3307,"source_end_line":3314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3307-L3314","statement_sha256":"b8492c9651a0dc04a3801d6d4ac035c89307b5c8cb32fe3e12cfec9a052b1b35","origin":"The Stacks Project","memory_eligible":false,"source_rank":7012,"rank":7012,"depth":32,"x":169.146,"y":591.268,"cluster":"derived-categories"},{"id":"stacks:08EH","tag":"08EH","title":"Lifting complexes · Lemma 08EH","summary":"Let X be an affine scheme. Let U ⊂ X be a quasi-compact open. Let E, E' be objects of D_QCoh(O_X) with E perfect. For every map α : E|_U → E'|_U there exist maps E xleftarrowβ E_1 xrightarrowγ E' of complexes on X with E_1 perfect such that β : E_1 → E restricts to an isomorphism on U and such that α = γ|_U ∘ β|_U^-1. Moreover we can assume E_1 = E ⊗_O_X^L I for some perfect complex I on X.","statement_latex":"Let $X$ be an affine scheme. Let $U \\subset X$ be a quasi-compact open.\nLet $E, E'$ be objects of $D_\\QCoh(\\mathcal{O}_X)$ with $E$ perfect.\nFor every map $\\alpha : E|_U \\to E'|_U$ there exist maps\n$$\nE \\xleftarrow{\\beta} E_1 \\xrightarrow{\\gamma} E'\n$$\nof complexes on $X$ with $E_1$ perfect such that $\\beta : E_1 \\to E$\nrestricts to an isomorphism on $U$ and such that\n$\\alpha = \\gamma|_U \\circ \\beta|_U^{-1}$.\nMoreover we can assume $E_1 = E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} I$\nfor some perfect complex $I$ on $X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EH","source_file":"perfect.tex","source_line":3353,"source_end_line":3366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3353-L3366","statement_sha256":"520297615c8d34ec633e82b84a40b5d0e6dfd98f38fc126e6c0db5293788d9f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7013,"rank":7013,"depth":29,"x":346.402,"y":710.946,"cluster":"derived-categories"},{"id":"stacks:08EI","tag":"08EI","title":"Lifting complexes · Lemma 08EI","summary":"Let X be an affine scheme. Let U ⊂ X be a quasi-compact open. For every perfect object F of D(O_U) the object F ⊕ F[1] is the restriction of a perfect object of D(O_X).","statement_latex":"Let $X$ be an affine scheme. Let $U \\subset X$ be a quasi-compact open.\nFor every perfect object $F$ of $D(\\mathcal{O}_U)$\nthe object $F \\oplus F[1]$ is the restriction of\na perfect object of $D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EI","source_file":"perfect.tex","source_line":3397,"source_end_line":3403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3397-L3403","statement_sha256":"d65d7ce731a16190a076c709439c82ad9470a55c2d066c341119beba473948a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7014,"rank":7014,"depth":33,"x":119.116,"y":723.295,"cluster":"derived-categories"},{"id":"stacks:08EJ","tag":"08EJ","title":"Lifting complexes · Lemma 08EJ","summary":"Let X be a quasi-compact and quasi-separated scheme. Let f ∈ Γ(X, O_X). For any morphism α : E → E' in D_QCoh(O_X) such that • E is perfect, and • E' is supported on T = V(f) there exists an n ≥ 0 such that f^n α = 0.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $f \\in \\Gamma(X, \\mathcal{O}_X)$.\nFor any morphism $\\alpha : E \\to E'$ in\n$D_\\QCoh(\\mathcal{O}_X)$ such that\n\\begin{enumerate}\n\\item $E$ is perfect, and\n\\item $E'$ is supported on $T = V(f)$\n\\end{enumerate}\nthere exists an $n \\geq 0$ such that $f^n \\alpha  = 0$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EJ","source_file":"perfect.tex","source_line":3426,"source_end_line":3437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3426-L3437","statement_sha256":"9b514d923cabb9257b7a155239adac7b956de685a00455108be14feea50b3d50","origin":"The Stacks Project","memory_eligible":false,"source_rank":7015,"rank":7015,"depth":29,"x":277.017,"y":585.016,"cluster":"derived-categories"},{"id":"stacks:08EK","tag":"08EK","title":"Lifting complexes · Lemma 08EK","summary":"Let X be an affine scheme. Let T ⊂ X be a closed subset such that X setminus T is quasi-compact. Let U ⊂ X be a quasi-compact open. For every perfect object F of D(O_U) supported on T ∩ U the object F ⊕ F[1] is the restriction of a perfect object E of D(O_X) supported in T.","statement_latex":"Let $X$ be an affine scheme. Let $T \\subset X$ be a closed subset\nsuch that $X \\setminus T$ is quasi-compact. Let $U \\subset X$ be a\nquasi-compact open. For every perfect object $F$ of $D(\\mathcal{O}_U)$\nsupported on $T \\cap U$ the object $F \\oplus F[1]$ is the restriction of\na perfect object $E$ of $D(\\mathcal{O}_X)$ supported in $T$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EK","source_file":"perfect.tex","source_line":3471,"source_end_line":3478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3471-L3478","statement_sha256":"f1e7118addd270bd9a7e7c171cfa2b802b4d248890f49f7cf73667359a183a9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7016,"rank":7016,"depth":34,"x":271.775,"y":776.862,"cluster":"derived-categories"},{"id":"stacks:09IM","tag":"09IM","title":"Lifting complexes · Lemma 09IM","summary":"Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be a quasi-compact open. Let T ⊂ X be a closed subset with X setminus T retro-compact in X. Let E be an object of D_QCoh(O_X). Let α : P → E|_U be a map where P is a perfect object of D(O_U) supported on T ∩ U. Then there exists a map β : R → E where R is a perfect object of D(O_X) supported on T such that P is a direct summand of R|_U in D(O_U) compatible α and β|_U.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $U \\subset X$ be a quasi-compact open. Let $T \\subset X$\nbe a closed subset with $X \\setminus T$ retro-compact in $X$.\nLet $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$.\nLet $\\alpha : P \\to E|_U$ be a map where $P$ is a perfect object of\n$D(\\mathcal{O}_U)$ supported on $T \\cap U$. Then there exists a map\n$\\beta : R \\to E$ where $R$ is a perfect object of $D(\\mathcal{O}_X)$\nsupported on $T$ such that $P$ is a direct summand of $R|_U$ in\n$D(\\mathcal{O}_U)$ compatible $\\alpha$ and $\\beta|_U$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Lifting complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IM","source_file":"perfect.tex","source_line":3534,"source_end_line":3545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3534-L3545","statement_sha256":"6fb30ea09fdace4d9ef684402af271281f100b78c62783ed6c90b40bcd7d90f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7017,"rank":7017,"depth":35,"x":121.142,"y":632.21,"cluster":"derived-categories"},{"id":"stacks:08EM","tag":"08EM","title":"Approximation by perfect complexes · Definition 08EM","summary":"Let X be a scheme. Consider triples (T, E, m) where • T ⊂ X is a closed subset, • E is an object of D_QCoh(O_X), and • m ∈ Z. We say approximation holds for the triple (T, E, m) if there exists a perfect object P of D(O_X) supported on T and a map α : P → E which induces isomorphisms H^i(P) → H^i(E) for i > m and a surjection H^m(P) → H^m(E).","statement_latex":"Let $X$ be a scheme. Consider triples $(T, E, m)$ where\n\\begin{enumerate}\n\\item $T \\subset X$ is a closed subset,\n\\item $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$, and\n\\item $m \\in \\mathbf{Z}$.\n\\end{enumerate}\nWe say {\\it approximation holds for the triple} $(T, E, m)$ if\nthere exists a perfect object $P$ of $D(\\mathcal{O}_X)$ supported on $T$\nand a map $\\alpha : P \\to E$ which induces isomorphisms $H^i(P) \\to H^i(E)$\nfor $i > m$ and a surjection $H^m(P) \\to H^m(E)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Approximation by perfect complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EM","source_file":"perfect.tex","source_line":3596,"source_end_line":3608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3596-L3608","statement_sha256":"c8d064b2149e8665c91065f91982d4ef3dde2136f29929bdc41e227048e92dae","origin":"The Stacks Project","memory_eligible":false,"source_rank":7018,"rank":7018,"depth":0,"x":348.877,"y":653.432,"cluster":"derived-categories"},{"id":"stacks:08EN","tag":"08EN","title":"Approximation by perfect complexes · Definition 08EN","summary":"Let X be a scheme. We say approximation by perfect complexes holds on X if for any closed subset T ⊂ X with X setminus T retro-compact in X there exists an integer r such that for every triple (T, E, m) as in Definition [Tag 08EM] with • E is (m - r)-pseudo-coherent, and • H^i(E) is supported on T for i ≥ m - r approximation holds.","statement_latex":"Let $X$ be a scheme. We say {\\it approximation by perfect complexes holds}\non $X$ if for any closed subset $T \\subset X$ with $X \\setminus T$\nretro-compact in $X$ there exists an integer $r$ such that\nfor every triple $(T, E, m)$ as in\nDefinition \\ref{definition-approximation-holds} with\n\\begin{enumerate}\n\\item $E$ is $(m - r)$-pseudo-coherent, and\n\\item $H^i(E)$ is supported on $T$ for $i \\geq m - r$\n\\end{enumerate}\napproximation holds.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Approximation by perfect complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EN","source_file":"perfect.tex","source_line":3623,"source_end_line":3635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3623-L3635","statement_sha256":"e20da1726a8de45b2c051703d150105a742ea1d60e497aa46dfde234b62dff98","origin":"The Stacks Project","memory_eligible":false,"source_rank":7019,"rank":7019,"depth":1,"x":163.61,"y":767.172,"cluster":"derived-categories"},{"id":"stacks:08EP","tag":"08EP","title":"Approximation by perfect complexes · Lemma 08EP","summary":"Let X be a scheme. Let U ⊂ X be an open subscheme. Let (T, E, m) be a triple as in Definition [Tag 08EM]. If • T ⊂ U, • approximation holds for (T, E|_U, m), and • the sheaves H^i(E) for i ≥ m are supported on T, then approximation holds for (T, E, m).","statement_latex":"Let $X$ be a scheme. Let $U \\subset X$ be an open subscheme.\nLet $(T, E, m)$ be a triple as in\nDefinition \\ref{definition-approximation-holds}.\nIf\n\\begin{enumerate}\n\\item $T \\subset U$,\n\\item approximation holds for $(T, E|_U, m)$, and\n\\item the sheaves $H^i(E)$ for $i \\geq m$ are supported on $T$,\n\\end{enumerate}\nthen approximation holds for $(T, E, m)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Approximation by perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EP","source_file":"perfect.tex","source_line":3644,"source_end_line":3656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3644-L3656","statement_sha256":"c7efd56cd30be5054783266d0d6fcf5a5730b4ee02d75ce32a4f2b69f6e12f4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7020,"rank":7020,"depth":6,"x":208.823,"y":577.9,"cluster":"derived-categories"},{"id":"stacks:08EQ","tag":"08EQ","title":"Approximation by perfect complexes · Lemma 08EQ","summary":"Let X be an affine scheme. Then approximation holds for every triple (T, E, m) as in Definition [Tag 08EM] such that there exists an integer r ≥ 0 with • E is m-pseudo-coherent, • H^i(E) is supported on T for i ≥ m - r + 1, • X setminus T is the union of r affine opens. In particular, approximation by perfect complexes holds for affine schemes.","statement_latex":"Let $X$ be an affine scheme. Then approximation holds for every\ntriple $(T, E, m)$ as in Definition \\ref{definition-approximation-holds}\nsuch that there exists an integer $r \\geq 0$ with\n\\begin{enumerate}\n\\item $E$ is $m$-pseudo-coherent,\n\\item $H^i(E)$ is supported on $T$ for $i \\geq m - r + 1$,\n\\item $X \\setminus T$ is the union of $r$ affine opens.\n\\end{enumerate}\nIn particular, approximation by perfect complexes holds for affine schemes.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Approximation by perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EQ","source_file":"perfect.tex","source_line":3667,"source_end_line":3678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3667-L3678","statement_sha256":"afed75da08bba214f9feaf46ac15182d8f47df374691a90a2ff3578da0ca48ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":7021,"rank":7021,"depth":29,"x":327.864,"y":743.363,"cluster":"derived-categories"},{"id":"stacks:08ER","tag":"08ER","title":"Approximation by perfect complexes · Lemma 08ER","summary":"Let X be a scheme. Let X = U ∪ V be an open covering with U quasi-compact, V affine, and U ∩ V quasi-compact. If approximation by perfect complexes holds on U, then approximation holds on X.","statement_latex":"Let $X$ be a scheme. Let $X = U \\cup V$ be an open covering\nwith $U$ quasi-compact, $V$ affine, and $U \\cap V$ quasi-compact.\nIf approximation by perfect complexes holds on $U$,\nthen approximation holds on $X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Approximation by perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ER","source_file":"perfect.tex","source_line":3727,"source_end_line":3733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3727-L3733","statement_sha256":"680a2628ffd63cd6815c399f9872f968255cae2c036212dc7bc657bae26069ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":7022,"rank":7022,"depth":35,"x":106.703,"y":688.821,"cluster":"derived-categories"},{"id":"stacks:08ES","tag":"08ES","title":"Approximation by perfect complexes · Theorem 08ES","summary":"Let X be a quasi-compact and quasi-separated scheme. Then approximation by perfect complexes holds on X.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nThen approximation by perfect complexes holds on $X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Approximation by perfect complexes","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ES","source_file":"perfect.tex","source_line":3813,"source_end_line":3817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3813-L3817","statement_sha256":"f109f02d296921866d081d7c72d80991556b06ab7ee965c14c5d2736b8683004","origin":"The Stacks Project","memory_eligible":false,"source_rank":7023,"rank":7023,"depth":36,"x":313.946,"y":603.423,"cluster":"derived-categories"},{"id":"stacks:09IQ","tag":"09IQ","title":"Generating derived categories · Lemma 09IQ","summary":"Let X be a quasi-compact and quasi-separated scheme. Let U be a quasi-compact open subscheme. Let P be a perfect object of D(O_U). Then P is a direct summand of the restriction of a perfect object of D(O_X).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $U$ be a quasi-compact open subscheme.\nLet $P$ be a perfect object of $D(\\mathcal{O}_U)$.\nThen $P$ is a direct summand of the restriction of a perfect\nobject of $D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Generating derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IQ","source_file":"perfect.tex","source_line":3840,"source_end_line":3847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3840-L3847","statement_sha256":"3c7469a71d1b37590b0437baaabaaea5f3442176b6699fea47d13ba2660c91fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7024,"rank":7024,"depth":36,"x":229.68,"y":784.25,"cluster":"derived-categories"},{"id":"stacks:09IR","tag":"09IR","title":"Generating derived categories · Lemma 09IR","summary":"[Bokstedt-Neeman] In Situation [Tag 08CZ] denote j : U → X the open immersion and let K be the perfect object of D(O_X) corresponding to the Koszul complex on f_1, …, f_r over A. For E ∈ D_QCoh(O_X) the following are equivalent • E = Rj_*(E|_U), and • Hom_D(O_X)(K[n], E) = 0 for all n ∈ Z.","statement_latex":"\\begin{reference}\n\\cite[Proposition 6.1]{Bokstedt-Neeman}\n\\end{reference}\nIn Situation \\ref{situation-complex} denote $j : U \\to X$ the open\nimmersion and let $K$ be the perfect object of $D(\\mathcal{O}_X)$\ncorresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.\nFor $E \\in D_\\QCoh(\\mathcal{O}_X)$ the following are equivalent\n\\begin{enumerate}\n\\item $E = Rj_*(E|_U)$, and\n\\item $\\Hom_{D(\\mathcal{O}_X)}(K[n], E) = 0$ for all $n \\in \\mathbf{Z}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Generating derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IR","source_file":"perfect.tex","source_line":3853,"source_end_line":3866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3853-L3866","statement_sha256":"12e597a6f8b5c2fd49478a0ed83664297ccd1f6519021162e3ff9b9dd0230bba","origin":"The Stacks Project","memory_eligible":false,"source_rank":7025,"rank":7025,"depth":29,"x":146.28,"y":602.835,"cluster":"derived-categories"},{"id":"stacks:09IS","tag":"09IS","title":"Generating derived categories · Theorem 09IS","summary":"Let X be a quasi-compact and quasi-separated scheme. The category D_QCoh(O_X) can be generated by a single perfect object. More precisely, there exists a perfect object P of D(O_X) such that for E ∈ D_QCoh(O_X) the following are equivalent • E = 0, and • Hom_D(O_X)(P[n], E) = 0 for all n ∈ Z.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. The category\n$D_\\QCoh(\\mathcal{O}_X)$ can be generated by a single\nperfect object. More precisely, there exists a perfect object\n$P$ of $D(\\mathcal{O}_X)$ such that for \n$E \\in D_\\QCoh(\\mathcal{O}_X)$ the following are equivalent\n\\begin{enumerate}\n\\item $E = 0$, and\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[n], E) = 0$ for all $n \\in \\mathbf{Z}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Generating derived categories","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IS","source_file":"perfect.tex","source_line":3893,"source_end_line":3904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3893-L3904","statement_sha256":"34e3b3729fa1b8d22ac0920109ef71c07ae9c1b177573563c3162631bc0f0d0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7026,"rank":7026,"depth":37,"x":353.966,"y":689.409,"cluster":"derived-categories"},{"id":"stacks:0A9A","tag":"0A9A","title":"Generating derived categories · Lemma 0A9A","summary":"[Rouquier-dimensions] Let X be a quasi-compact and quasi-separated scheme. Let T ⊂ X be a closed subset such that X setminus T is quasi-compact. With notation as above, the category D_QCoh, T(O_X) is generated by a single perfect object.","statement_latex":"\\begin{reference}\n\\cite[Theorem 6.8]{Rouquier-dimensions}\n\\end{reference}\nLet $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$ be a\nclosed subset such that $X \\setminus T$ is quasi-compact. With notation\nas above, the category $D_{\\QCoh, T}(\\mathcal{O}_X)$ is generated by a\nsingle perfect object.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Generating derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9A","source_file":"perfect.tex","source_line":3967,"source_end_line":3976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L3967-L3976","statement_sha256":"bb5c4d94dd654ea0135d1b1374b87c46f116c39145ad8a70826f7123a44c669a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7027,"rank":7027,"depth":36,"x":130.881,"y":743.494,"cluster":"derived-categories"},{"id":"stacks:0BQR","tag":"0BQR","title":"An example generator · Lemma 0BQR","summary":"Let X be a scheme and L an ample invertible O_X-module. If K is a nonzero object of D_QCoh(O_X), then for some n ≥ 0 and p ∈ Z the cohomology group H^p(X, K ⊗_O_X^L L^⊗ n) is nonzero.","statement_latex":"Let $X$ be a scheme and $\\mathcal{L}$ an ample invertible\n$\\mathcal{O}_X$-module. If $K$ is a nonzero object of\n$D_\\QCoh(\\mathcal{O}_X)$, then for some $n \\geq 0$ and $p \\in \\mathbf{Z}$\nthe cohomology group\n$H^p(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{L}^{\\otimes n})$\nis nonzero.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"An example generator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQR","source_file":"perfect.tex","source_line":4067,"source_end_line":4075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4067-L4075","statement_sha256":"56e8d3a09cfdebd30c8c16bd790e52152db00033230ffd0a6c40ca4968797517","origin":"The Stacks Project","memory_eligible":false,"source_rank":7028,"rank":7028,"depth":34,"x":252.06,"y":576.791,"cluster":"derived-categories"},{"id":"stacks:0BQS","tag":"0BQS","title":"An example generator · Lemma 0BQS","summary":"Let A be a ring. Let X = P^n_A. For every a ∈ Z there exists an exact complex 0 → O_X(a) → … → O_X(a + i)^⊕ n + 1 choose i → … → O_X(a + n + 1) → 0 of vector bundles on X.","statement_latex":"Let $A$ be a ring. Let $X = \\mathbf{P}^n_A$. For every $a \\in \\mathbf{Z}$\nthere exists an exact complex\n$$\n0 \\to \\mathcal{O}_X(a) \\to \\ldots\n\\to \\mathcal{O}_X(a + i)^{\\oplus {n + 1 \\choose i}} \\to\n\\ldots \\to \\mathcal{O}_X(a + n + 1) \\to 0\n$$\nof vector bundles on $X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"An example generator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQS","source_file":"perfect.tex","source_line":4109,"source_end_line":4119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4109-L4119","statement_sha256":"98839d30ea1dc662658ae20c4e0e33201b1d791d82f51af5d28940f4daea5ba3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7029,"rank":7029,"depth":6,"x":296.826,"y":768.748,"cluster":"derived-categories"},{"id":"stacks:0A9V","tag":"0A9V","title":"An example generator · Lemma 0A9V","summary":"Let A be a ring. Let X = P^n_A. Then E = O_X ⊕ O_X(-1) ⊕ … ⊕ O_X(-n) is a generator (Derived Categories, Definition [Tag 09SJ]) of D_QCoh(X).","statement_latex":"Let $A$ be a ring. Let $X = \\mathbf{P}^n_A$. Then\n$$\nE =\n\\mathcal{O}_X \\oplus \\mathcal{O}_X(-1) \\oplus \\ldots \\oplus \\mathcal{O}_X(-n)\n$$\nis a generator\n(Derived Categories, Definition \\ref{derived-definition-generators})\nof $D_\\QCoh(X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"An example generator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9V","source_file":"perfect.tex","source_line":4154,"source_end_line":4164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4154-L4164","statement_sha256":"d21601270bd03a956a5129531f94dbd52c0b5f0fd69306861e08943e81f133aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7030,"rank":7030,"depth":35,"x":109.183,"y":652.438,"cluster":"derived-categories"},{"id":"stacks:09M1","tag":"09M1","title":"Compact and perfect objects · Proposition 09M1","summary":"Let X be a quasi-compact and quasi-separated scheme. An object of D_QCoh(O_X) is compact if and only if it is perfect.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nAn object of $D_\\QCoh(\\mathcal{O}_X)$ is compact\nif and only if it is perfect.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Compact and perfect objects","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09M1","source_file":"perfect.tex","source_line":4246,"source_end_line":4251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4246-L4251","statement_sha256":"92a89b97c4d2a9f1ca9d836a9d3c33ea53c57d721949515c42706b12516297ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":7031,"rank":7031,"depth":31,"x":341.411,"y":631.701,"cluster":"derived-categories"},{"id":"stacks:0A9B","tag":"0A9B","title":"Compact and perfect objects · Lemma 0A9B","summary":"Let X be a quasi-compact and quasi-separated scheme. Let T ⊂ X be a closed subset such that X setminus T is quasi-compact. An object of D_QCoh, T(O_X) is compact if and only if it is perfect as an object of D(O_X).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $T \\subset X$ be a closed subset such that $X \\setminus T$\nis quasi-compact. An object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ is compact\nif and only if it is perfect as an object of $D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Compact and perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9B","source_file":"perfect.tex","source_line":4312,"source_end_line":4318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4312-L4318","statement_sha256":"09c414dae9045bbb32f0a69d3ac0114613a8772639409d180b7d0905efab0860","origin":"The Stacks Project","memory_eligible":false,"source_rank":7032,"rank":7032,"depth":37,"x":186.627,"y":778.972,"cluster":"derived-categories"},{"id":"stacks:0A9C","tag":"0A9C","title":"Compact and perfect objects · Lemma 0A9C","summary":"Let X be a quasi-compact and quasi-separated scheme. Let T ⊂ X be a closed subset such that U = X setminus T is quasi-compact. Let α : P → E be a morphism of D_QCoh(O_X) with either • P is perfect and E supported on T, or • P pseudo-coherent, E supported on T, and E bounded below. Then there exists a perfect complex of O_X-modules I and a map I → O_X[0] such that I ⊗^L P → E is zero and such that I|_U → O_U[0] is an isomorphism.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \\subset X$\nbe a closed subset such that $U = X \\setminus T$ is quasi-compact.\nLet $\\alpha : P \\to E$ be a morphism of $D_\\QCoh(\\mathcal{O}_X)$ with\neither\n\\begin{enumerate}\n\\item $P$ is perfect and $E$ supported on $T$, or\n\\item $P$ pseudo-coherent, $E$ supported on $T$, and $E$ bounded below.\n\\end{enumerate}\nThen there exists a perfect complex of $\\mathcal{O}_X$-modules $I$\nand a map $I \\to \\mathcal{O}_X[0]$ such that\n$I \\otimes^\\mathbf{L} P \\to E$ is zero and such that\n$I|_U \\to \\mathcal{O}_U[0]$ is an\nisomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Compact and perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9C","source_file":"perfect.tex","source_line":4361,"source_end_line":4376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4361-L4376","statement_sha256":"cf0b2fa1b6ce8aaf84a64996ea147cbf836967845516eecce2e45e4fb452b327","origin":"The Stacks Project","memory_eligible":false,"source_rank":7033,"rank":7033,"depth":37,"x":182.325,"y":582.272,"cluster":"derived-categories"},{"id":"stacks:09M3","tag":"09M3","title":"Derived categories as module categories · Lemma 09M3","summary":"Let X be a scheme. Let K^bullet be a complex of O_X-modules whose cohomology sheaves are quasi-coherent. Let (E, d) = Hom_Comp^dg(O_X)(K^bullet, K^bullet) be the endomorphism differential graded algebra. Then the functor - ⊗_E^L K^bullet : D(E, d) → D(O_X) of Differential Graded Algebra, Lemma [Tag 09LX] has image contained in D_QCoh(O_X).","statement_latex":"Let $X$ be a scheme. Let $K^\\bullet$ be a complex of $\\mathcal{O}_X$-modules\nwhose cohomology sheaves are quasi-coherent. Let\n$(E, d) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)$\nbe the endomorphism differential graded algebra. Then the functor\n$$\n- \\otimes_E^\\mathbf{L} K^\\bullet :\nD(E, \\text{d}) \\longrightarrow D(\\mathcal{O}_X)\n$$\nof\nDifferential Graded Algebra, Lemma\n\\ref{dga-lemma-tensor-with-complex-derived}\nhas image contained in $D_\\QCoh(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived categories as module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09M3","source_file":"perfect.tex","source_line":4432,"source_end_line":4446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4432-L4446","statement_sha256":"23be37bbfbaf8ee63a6ed2485a287dd8327b5743caa26b369885b3e6070a75d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7034,"rank":7034,"depth":15,"x":343.905,"y":725.075,"cluster":"derived-categories"},{"id":"stacks:09M5","tag":"09M5","title":"Derived categories as module categories · Theorem 09M5","summary":"Let X be a quasi-compact and quasi-separated scheme. Then there exist a differential graded algebra (E, d) with only a finite number of nonzero cohomology groups H^i(E) such that D_QCoh(O_X) is equivalent to D(E, d).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nThen there exist a differential graded algebra $(E, \\text{d})$\nwith only a finite number of nonzero cohomology groups $H^i(E)$\nsuch that $D_\\QCoh(\\mathcal{O}_X)$ is equivalent\nto $D(E, \\text{d})$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Derived categories as module categories","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09M5","source_file":"perfect.tex","source_line":4463,"source_end_line":4470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4463-L4470","statement_sha256":"e31074c6a845ba9d71dcabae02244394c8cecfff03f41f2aac193e1a87952680","origin":"The Stacks Project","memory_eligible":false,"source_rank":7035,"rank":7035,"depth":38,"x":109.592,"y":711.439,"cluster":"derived-categories"},{"id":"stacks:0DJN","tag":"0DJN","title":"Characterizing pseudo-coherent complexes, I · Lemma 0DJN","summary":"Let X be a quasi-compact and quasi-separated scheme. Let K ∈ D(O_X). The following are equivalent • K is pseudo-coherent, and • K = hocolim K_n where K_n is perfect and τ_≥ -nK_n → τ_≥ -nK is an isomorphism for all n.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $K \\in D(\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item $K$ is pseudo-coherent, and\n\\item $K = \\text{hocolim} K_n$ where\n$K_n$ is perfect and $\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$\nis an isomorphism for all $n$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Characterizing pseudo-coherent complexes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJN","source_file":"perfect.tex","source_line":4596,"source_end_line":4606,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4596-L4606","statement_sha256":"af8066660080013604efef673ea096e935887c7c871d9a74544473275839f91f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7036,"rank":7036,"depth":37,"x":293.594,"y":588.368,"cluster":"derived-categories"},{"id":"stacks:0DJP","tag":"0DJP","title":"Characterizing pseudo-coherent complexes, I · Lemma 0DJP","summary":"Let X be a quasi-compact and quasi-separated scheme. Let T ⊂ X be a closed subset such that X setminus T is quasi-compact. Let K ∈ D(O_X) supported on T. The following are equivalent • K is pseudo-coherent, and • K = hocolim K_n where K_n is perfect, supported on T, and τ_≥ -nK_n → τ_≥ -nK is an isomorphism for all n.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $T \\subset X$ be a closed subset such that $X \\setminus T$\nis quasi-compact. Let $K \\in D(\\mathcal{O}_X)$ supported on $T$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ is pseudo-coherent, and\n\\item $K = \\text{hocolim} K_n$ where\n$K_n$ is perfect, supported on $T$, and\n$\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$ is an isomorphism for all $n$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Characterizing pseudo-coherent complexes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJP","source_file":"perfect.tex","source_line":4657,"source_end_line":4669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4657-L4669","statement_sha256":"77b40fb1d2d6f253675c13c1857db1543d219e5e6cd97f9873d1ca0107299ad7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7037,"rank":7037,"depth":38,"x":256.833,"y":783.797,"cluster":"derived-categories"},{"id":"stacks:0BQU","tag":"0BQU","title":"An example equivalence · Lemma 0BQU","summary":"[Beilinson] Let A be a ring. Let X = P^n_A = Proj(S) where S = A[X_0, …, X_n]. With P as in ([Tag 0CS8]) and R as in ([Tag 0CS9]) the functor - ⊗_R^L P : D(R) → D_QCoh(O_X) is an A-linear equivalence of triangulated categories sending R to P.","statement_latex":"\\begin{reference}\n\\cite{Beilinson}\n\\end{reference}\nLet $A$ be a ring. Let $X = \\mathbf{P}^n_A = \\text{Proj}(S)$\nwhere $S = A[X_0, \\ldots, X_n]$. With\n$P$ as in (\\ref{equation-generator-Pn}) and\n$R$ as in (\\ref{equation-algebra-for-Pn})\nthe functor\n$$\n- \\otimes_R^\\mathbf{L} P : D(R) \\longrightarrow D_\\QCoh(\\mathcal{O}_X)\n$$\nis an $A$-linear equivalence of triangulated categories sending $R$\nto $P$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"An example equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQU","source_file":"perfect.tex","source_line":4743,"source_end_line":4758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4743-L4758","statement_sha256":"eb525af3506ee3fbef81e17aa3ae77f9de788115b1ff6008534bdc786e7bf82b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7038,"rank":7038,"depth":0,"x":126.6,"y":618.602,"cluster":"derived-categories"},{"id":"stacks:0CR0","tag":"0CR0","title":"The coherator revisited · Lemma 0CR0","summary":"Let X be a quasi-compact and quasi-separated scheme. The inclusion functor D_QCoh(O_X) → D(O_X) has a right adjoint DQ_X.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nThe inclusion functor $D_\\QCoh(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$\nhas a right adjoint $DQ_X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CR0","source_file":"perfect.tex","source_line":4827,"source_end_line":4832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4827-L4832","statement_sha256":"0a5c37dd5da278af471107624b1ee5f815bc98fc519c459d18ce22b63de6e386","origin":"The Stacks Project","memory_eligible":false,"source_rank":7039,"rank":7039,"depth":30,"x":355.794,"y":666.584,"cluster":"derived-categories"},{"id":"stacks:0CR1","tag":"0CR1","title":"The coherator revisited · Lemma 0CR1","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of schemes. If the right adjoints DQ_X and DQ_Y of the inclusion functors D_QCoh → D exist for X and Y, then Rf_* ∘ DQ_X = DQ_Y ∘ Rf_*","statement_latex":"Let $f : X \\to Y$ be a quasi-compact and quasi-separated\nmorphism of schemes. If the right adjoints $DQ_X$ and $DQ_Y$\nof the inclusion functors $D_\\QCoh \\to D$ exist for $X$ and $Y$, then\n$$\nRf_* \\circ DQ_X = DQ_Y \\circ Rf_*\n$$","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CR1","source_file":"perfect.tex","source_line":4917,"source_end_line":4925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4917-L4925","statement_sha256":"479ddb727aa41562b3c9c68d9031ec9aae6fbe368e76723556f07e0ba676f9ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":7040,"rank":7040,"depth":30,"x":147.917,"y":761.384,"cluster":"derived-categories"},{"id":"stacks:0CSA","tag":"0CSA","title":"The coherator revisited · Lemma 0CSA","summary":"Let X be a quasi-compact and quasi-separated scheme. The functor DQ_X of Lemma [Tag 0CR0] has the following boundedness property: there exists an integer N = N(X) such that, if K in D(O_X) with H^i(U, K) = 0 for U affine open in X and i not ∈ [a, b], then the cohomology sheaves H^i(DQ_X(K)) are zero for i not ∈ [a, b + N].","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. The functor\n$DQ_X$ of Lemma \\ref{lemma-better-coherator}\nhas the following boundedness property:\nthere exists an integer $N = N(X)$ such that, if\n$K$ in $D(\\mathcal{O}_X)$ with\n$H^i(U, K) = 0$ for $U$ affine open in $X$ and $i \\not \\in [a, b]$, then\nthe cohomology sheaves $H^i(DQ_X(K))$ are zero for\n$i \\not \\in [a, b + N]$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The coherator revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSA","source_file":"perfect.tex","source_line":4955,"source_end_line":4965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L4955-L4965","statement_sha256":"c2ebc397ee622b6247634173f72b3740297019ffef433a5f54679a5be28f8ace","origin":"The Stacks Project","memory_eligible":false,"source_rank":7041,"rank":7041,"depth":31,"x":225.073,"y":573.266,"cluster":"derived-categories"},{"id":"stacks:08EU","tag":"08EU","title":"Cohomology and base change, IV · Lemma 08EU","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of schemes. For E in D_QCoh(O_X) and K in D_QCoh(O_Y) the map Rf_*(E) ⊗_O_Y^L K → Rf_*(E ⊗_O_X^L Lf^*K) defined in Cohomology, Equation ([Tag 0B53]) is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism\nof schemes. For $E$ in $D_\\QCoh(\\mathcal{O}_X)$ and\n$K$ in $D_\\QCoh(\\mathcal{O}_Y)$ the map\n$$\nRf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} K\n\\longrightarrow\nRf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*K)\n$$\ndefined in\nCohomology, Equation (\\ref{cohomology-equation-projection-formula-map})\nis an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EU","source_file":"perfect.tex","source_line":5051,"source_end_line":5064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5051-L5064","statement_sha256":"272c3c2af2063dac9e802da4068dae5a0c12e75ae097b2d31af33ee2d36000cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7042,"rank":7042,"depth":31,"x":319.588,"y":756.012,"cluster":"derived-categories"},{"id":"stacks:08IA","tag":"08IA","title":"Cohomology and base change, IV · Definition 08IA","summary":"Let S be a scheme. Let X, Y be schemes over S. We say X and Y are Tor independent over S if for every x ∈ X and y ∈ Y mapping to the same point s ∈ S the rings O_X, x and O_Y, y are Tor independent over O_S, s (see More on Algebra, Definition [Tag 0660]).","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. We say $X$ and\n$Y$ are {\\it Tor independent over $S$} if for every $x \\in X$ and\n$y \\in Y$ mapping to the same point $s \\in S$ the rings\n$\\mathcal{O}_{X, x}$ and $\\mathcal{O}_{Y, y}$ are Tor independent\nover $\\mathcal{O}_{S, s}$ (see\nMore on Algebra, Definition \\ref{more-algebra-definition-tor-independent}).","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, IV","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IA","source_file":"perfect.tex","source_line":5096,"source_end_line":5104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5096-L5104","statement_sha256":"7660ef58575278bf736bf368c8df797101425806e0cfa0238a9bab9baf9034aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7043,"rank":7043,"depth":1,"x":102.638,"y":674.777,"cluster":"derived-categories"},{"id":"stacks:0FXV","tag":"0FXV","title":"Cohomology and base change, IV · Lemma 0FXV","summary":"Let f : X → S and g : Y → S be morphisms of schemes. The following are equivalent • X and Y are tor independent over S, and • for every affine opens U ⊂ X, V ⊂ Y, W ⊂ S with f(U) ⊂ W and g(V) ⊂ W the rings O_X(U) and O_Y(V) are tor independent over O_S(W). • there exists an affine open overing S = ⋃ W_i and for each i affine open coverings f^-1(W_i) = ⋃ U_ij and g^-1(W_i) = ⋃ V_ik such that the rings O_X(U_ij) and O_Y(V_ik) are tor independent over O_S(W_i) for all i, j, k.","statement_latex":"Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ and $Y$ are tor independent over $S$, and\n\\item for every affine opens $U \\subset X$, $V \\subset Y$, $W \\subset S$\nwith $f(U) \\subset W$ and $g(V) \\subset W$ the rings\n$\\mathcal{O}_X(U)$ and $\\mathcal{O}_Y(V)$ are tor independent over\n$\\mathcal{O}_S(W)$.\n\\item there exists an affine open overing $S = \\bigcup W_i$ and\nfor each $i$ affine open coverings $f^{-1}(W_i) = \\bigcup U_{ij}$\nand $g^{-1}(W_i) = \\bigcup V_{ik}$ such that the rings\n$\\mathcal{O}_X(U_{ij})$ and $\\mathcal{O}_Y(V_{ik})$ are tor independent over\n$\\mathcal{O}_S(W_i)$ for all $i, j, k$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXV","source_file":"perfect.tex","source_line":5106,"source_end_line":5122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5106-L5122","statement_sha256":"545ff9d5d3ea4567cef53e995c3fd97775872b7d782adc87974831d785f9e8d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7044,"rank":7044,"depth":6,"x":328.249,"y":611.49,"cluster":"derived-categories"},{"id":"stacks:0FXW","tag":"0FXW","title":"Cohomology and base change, IV · Lemma 0FXW","summary":"Let X → S and Y → S be morphisms of schemes. Let S' → S be a morphism of schemes and denote X' = X ×_S S' and Y' = Y ×_S S'. If X and Y are tor independent over S and S' → S is flat, then X' and Y' are tor independent over S'.","statement_latex":"Let $X \\to S$ and $Y \\to S$ be morphisms of schemes. Let $S' \\to S$ be a\nmorphism of schemes and denote $X' = X \\times_S S'$\nand $Y' = Y \\times_S S'$.\nIf $X$ and $Y$ are tor independent over $S$ and $S' \\to S$ is flat,\nthen $X'$ and $Y'$ are tor independent over $S'$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXW","source_file":"perfect.tex","source_line":5129,"source_end_line":5136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5129-L5136","statement_sha256":"f98bd83dfedb54a0f955ed6b4f0a0df79b65f3e6d4b9b02ebe10462f510a214b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7045,"rank":7045,"depth":7,"x":212.623,"y":786.412,"cluster":"derived-categories"},{"id":"stacks:08IB","tag":"08IB","title":"Cohomology and base change, IV · Lemma 08IB","summary":"Let g : S' → S be a morphism of schemes. Let f : X → S be quasi-compact and quasi-separated. Consider the base change diagram xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S If X and S' are Tor independent over S, then for all E ∈ D_QCoh(O_X) the canonical arrow Lg^*Rf_*E → Rf'_*L(g')^*E is an isomorphism.","statement_latex":"Let $g : S' \\to S$ be a morphism of schemes.\nLet $f : X \\to S$ be quasi-compact and quasi-separated.\nConsider the base change diagram\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nS' \\ar[r]^g &\nS\n}\n$$\nIf $X$ and $S'$ are Tor independent over $S$, then for all\n$E \\in D_\\QCoh(\\mathcal{O}_X)$ the canonical arrow\n$Lg^*Rf_*E \\to Rf'_*L(g')^*E$ is an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IB","source_file":"perfect.tex","source_line":5144,"source_end_line":5160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5144-L5160","statement_sha256":"27c1e01ac8d95804ca6a79812506c2f62cf7cc4ef871c75449682de80bef58a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7046,"rank":7046,"depth":32,"x":157.142,"y":591.554,"cluster":"derived-categories"},{"id":"stacks:0AA7","tag":"0AA7","title":"Cohomology and base change, IV · Lemma 0AA7","summary":"Consider a cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S of quasi-compact and quasi-separated schemes. Assume g and f Tor independent and S = Spec(R), S' = Spec(R') affine. For M, K ∈ D(O_X) the canonical map RHom_X(M, K) ⊗^L_R R' → RHom_X'(L(g')^*M, L(g')^*K) in D(R') is an isomorphism in the following two cases • M ∈ D(O_X) is perfect and K ∈ D_QCoh(X), or • M ∈ D(O_X) is pseudo-coherent, K ∈ D_QCoh^+(X), and R' has finite tor dimension over R.","statement_latex":"Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nof quasi-compact and quasi-separated schemes. Assume $g$ and $f$\nTor independent and $S = \\Spec(R)$, $S' = \\Spec(R')$ affine. For\n$M, K \\in D(\\mathcal{O}_X)$ the canonical map\n$$\nR\\Hom_X(M, K) \\otimes^\\mathbf{L}_R R'\n\\longrightarrow\nR\\Hom_{X'}(L(g')^*M, L(g')^*K)\n$$\nin $D(R')$ is an isomorphism in the following two cases\n\\begin{enumerate}\n\\item $M \\in D(\\mathcal{O}_X)$ is perfect and $K \\in D_\\QCoh(X)$, or\n\\item $M \\in D(\\mathcal{O}_X)$ is pseudo-coherent,\n$K \\in D_\\QCoh^+(X)$, and $R'$ has finite tor dimension over $R$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AA7","source_file":"perfect.tex","source_line":5205,"source_end_line":5228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5205-L5228","statement_sha256":"baa19c15218a8f9c563339e2d137880981bbc1ddb3fa4e24167b5879a37f79f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7047,"rank":7047,"depth":33,"x":355.02,"y":703.909,"cluster":"derived-categories"},{"id":"stacks:0C0V","tag":"0C0V","title":"Cohomology and base change, IV · Lemma 0C0V","summary":"Consider a cartesian square of schemes xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S Assume g and f Tor independent. • If E ∈ D(O_X) has tor amplitude in [a, b] as a complex of f^-1O_S-modules, then L(g')^*E has tor amplitude in [a, b] as a complex of f^-1O_S'-modules. • If G is an O_X-module flat over S, then L(g')^*G = (g')^*G.","statement_latex":"Consider a cartesian square of schemes\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nAssume $g$ and $f$ Tor independent.\n\\begin{enumerate}\n\\item If $E \\in D(\\mathcal{O}_X)$ has tor amplitude\nin $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_S$-modules,\nthen $L(g')^*E$ has tor amplitude\nin $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_{S'}$-modules.\n\\item If $\\mathcal{G}$ is an $\\mathcal{O}_X$-module flat\nover $S$, then $L(g')^*\\mathcal{G} = (g')^*\\mathcal{G}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0V","source_file":"perfect.tex","source_line":5295,"source_end_line":5313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5295-L5313","statement_sha256":"c808d8d457f56b43aa7ad1507d9d78c85c6c2f1b898d565ec1544d6e455793a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7048,"rank":7048,"depth":8,"x":118.435,"y":733.381,"cluster":"derived-categories"},{"id":"stacks:0E23","tag":"0E23","title":"Cohomology and base change, IV · Lemma 0E23","summary":"Consider a cartesian diagram of schemes xymatrix Z' ar[r]_i' ar[d]_g & X' ar[d]^f Z ar[r]^i & X where i is a closed immersion. If Z and X' are tor independent over X, then Ri'_* ∘ Lg^* = Lf^* ∘ Ri_* as functors D(O_Z) → D(O_X').","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nZ' \\ar[r]_{i'} \\ar[d]_g & X' \\ar[d]^f \\\\\nZ \\ar[r]^i & X\n}\n$$\nwhere $i$ is a closed immersion. If $Z$ and $X'$ are\ntor independent over $X$, then $Ri'_* \\circ Lg^* = Lf^* \\circ Ri_*$\nas functors $D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_{X'})$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E23","source_file":"perfect.tex","source_line":5336,"source_end_line":5348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5336-L5348","statement_sha256":"7176a5211a6e29a25c3c4acb39fd5f411f99ad4c11fbbe669a78fba67d85244f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7049,"rank":7049,"depth":1,"x":269.392,"y":577.195,"cluster":"derived-categories"},{"id":"stacks:0FLQ","tag":"0FLQ","title":"K\\\"unneth formula, II · Lemma 0FLQ","summary":"In the situation above, if a and b are quasi-compact and quasi-separated and X and Y are tor-independent over S, then ([Tag 0FLP]) is an isomorphism for K ∈ D_QCoh(O_X) and M ∈ D_QCoh(O_Y). If in addition S = Spec(A) is affine, then the map ([Tag 0G7V]) is an isomorphism.","statement_latex":"In the situation above, if $a$ and $b$ are quasi-compact and quasi-separated\nand $X$ and $Y$ are tor-independent over $S$, then (\\ref{equation-kunneth})\nis an isomorphism for $K \\in D_\\QCoh(\\mathcal{O}_X)$ and\n$M \\in D_\\QCoh(\\mathcal{O}_Y)$. If in addition $S = \\Spec(A)$ is affine,\nthen the map (\\ref{equation-kunneth-global}) is an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K\\\"unneth formula, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLQ","source_file":"perfect.tex","source_line":5418,"source_end_line":5425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5418-L5425","statement_sha256":"4c5fb6c0fb4b7f671806c276b9a6ce281f7bb4bff172b8eb8fba3da465df84a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7050,"rank":7050,"depth":33,"x":283.698,"y":778.291,"cluster":"derived-categories"},{"id":"stacks:0FML","tag":"0FML","title":"K\\\"unneth formula, II · Lemma 0FML","summary":"Let a : X → S be a quasi-compact and quasi-separated morphism of schemes. Let F^bullet be a locally bounded complex of a^-1O_S-modules. Assume for all n ∈ Z the sheaf F^n is a flat a^-1O_S-module and F^n has the structure of a quasi-coherent O_X-module compatible with the given a^-1O_S-module structure (but the differentials in the complex F^bullet need not be O_X-linear). Then the following hold • Ra_*F^bullet is locally bounded, • Ra_*F^bullet is in D_QCoh(O_S), •…","statement_latex":"Let $a : X \\to S$ be a quasi-compact and quasi-separated morphism\nof schemes. Let $\\mathcal{F}^\\bullet$ be a locally bounded\ncomplex of $a^{-1}\\mathcal{O}_S$-modules. Assume for all $n \\in \\mathbf{Z}$\nthe sheaf $\\mathcal{F}^n$ is a flat $a^{-1}\\mathcal{O}_S$-module and\n$\\mathcal{F}^n$ has the structure of a quasi-coherent $\\mathcal{O}_X$-module\ncompatible with the given $a^{-1}\\mathcal{O}_S$-module structure (but the\ndifferentials in the complex $\\mathcal{F}^\\bullet$ need not\nbe $\\mathcal{O}_X$-linear). Then the following hold\n\\begin{enumerate}\n\\item $Ra_*\\mathcal{F}^\\bullet$ is locally bounded,\n\\item $Ra_*\\mathcal{F}^\\bullet$ is in $D_\\QCoh(\\mathcal{O}_S)$,\n\\item $Ra_*\\mathcal{F}^\\bullet$ locally has finite tor dimension,\n\\item $\\mathcal{G} \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Ra_*\\mathcal{F}^\\bullet =\nRa_*(a^{-1}\\mathcal{G} \\otimes_{a^{-1}\\mathcal{O}_S} \\mathcal{F}^\\bullet)$\nfor $\\mathcal{G} \\in \\QCoh(\\mathcal{O}_S)$, and\n\\item $K \\otimes_{\\mathcal{O}_S}^\\mathbf{L} Ra_*\\mathcal{F}^\\bullet =\nRa_*(a^{-1}K \\otimes_{a^{-1}\\mathcal{O}_S}^\\mathbf{L} \\mathcal{F}^\\bullet)$\nfor $K \\in D_\\QCoh(\\mathcal{O}_S)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K\\\"unneth formula, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FML","source_file":"perfect.tex","source_line":5553,"source_end_line":5574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5553-L5574","statement_sha256":"b91149197a96385c52dc3b15abf73cb0dbbcc7824fa4fe4b4a3c6d9b1b68d074","origin":"The Stacks Project","memory_eligible":false,"source_rank":7051,"rank":7051,"depth":32,"x":111.203,"y":637.935,"cluster":"derived-categories"},{"id":"stacks:0FMQ","tag":"0FMQ","title":"K\\\"unneth formula, II · Lemma 0FMQ","summary":"Let f : X → Y be a morphism of schemes with Y = Spec(A) affine. Let U : X = ⋃_i ∈ I U_i be a finite affine open covering such that all the finite intersections U_i_0 … i_p = U_i_0 ∩ … ∩ U_i_p are affine. Let F^bullet be a bounded complex of f^-1O_Y-modules. Assume for all n ∈ Z the sheaf F^n is a flat f^-1O_Y-module and F^n has the structure of a quasi-coherent O_X-module compatible with the given p^-1O_Y-module structure (but the differentials in the complex F^bullet…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes with $Y = \\Spec(A)$ affine.\nLet $\\mathcal{U} : X = \\bigcup_{i \\in I} U_i$ be a finite affine open covering\nsuch that all the finite intersections\n$U_{i_0 \\ldots i_p} = U_{i_0} \\cap \\ldots \\cap U_{i_p}$\nare affine. Let $\\mathcal{F}^\\bullet$ be a bounded complex of\n$f^{-1}\\mathcal{O}_Y$-modules. Assume for all $n \\in \\mathbf{Z}$\nthe sheaf $\\mathcal{F}^n$ is a flat $f^{-1}\\mathcal{O}_Y$-module and\n$\\mathcal{F}^n$ has the structure of a quasi-coherent $\\mathcal{O}_X$-module\ncompatible with the given $p^{-1}\\mathcal{O}_Y$-module structure (but the\ndifferentials in the complex $\\mathcal{F}^\\bullet$ need not\nbe $\\mathcal{O}_X$-linear). Then the complex\n$\\text{Tot}(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F}^\\bullet))$\nis K-flat as a complex of $A$-modules.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K\\\"unneth formula, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMQ","source_file":"perfect.tex","source_line":5680,"source_end_line":5695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5680-L5695","statement_sha256":"7e2d4af8ba1bc4b941a42b91f942f090633c27200ede922d07ddacd3e481ce61","origin":"The Stacks Project","memory_eligible":false,"source_rank":7052,"rank":7052,"depth":6,"x":351.589,"y":643.561,"cluster":"derived-categories"},{"id":"stacks:0FU4","tag":"0FU4","title":"K\\\"unneth formula, II · Lemma 0FU4","summary":"In the situation above the map ([Tag 0G49]) is an isomorphism if S is affine, F and G are S-flat and quasi-coherent and X and Y are quasi-compact with affine diagonal.","statement_latex":"In the situation above the map (\\ref{equation-kunneth-single-sheaves}) is an\nisomorphism if $S$ is affine, $\\mathcal{F}$ and $\\mathcal{G}$ are $S$-flat and\nquasi-coherent and $X$ and $Y$ are quasi-compact with affine diagonal.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K\\\"unneth formula, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FU4","source_file":"perfect.tex","source_line":5756,"source_end_line":5761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L5756-L5761","statement_sha256":"f58743b6068e7520830e9185ca674611891590ebd107e9d69281f963b4484f3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7053,"rank":7053,"depth":32,"x":169.565,"y":775.989,"cluster":"derived-categories"},{"id":"stacks:0FLT","tag":"0FLT","title":"K\\\"unneth formula, III · Lemma 0FLT","summary":"In the situation above the cup product ([Tag 0FLR]) is an isomorphism in D(A) if the following assumptions hold • S = Spec(A) is affine, • X and Y are quasi-compact with affine diagonal, • F^bullet is bounded, • G^bullet is bounded below, • F^n is S-flat, and • G^m is S-flat.","statement_latex":"In the situation above the cup product (\\ref{equation-de-rham-kunneth})\nis an isomorphism in $D(A)$ if the following assumptions hold\n\\begin{enumerate}\n\\item $S = \\Spec(A)$ is affine,\n\\item $X$ and $Y$ are quasi-compact with affine diagonal,\n\\item $\\mathcal{F}^\\bullet$ is bounded,\n\\item $\\mathcal{G}^\\bullet$ is bounded below,\n\\item $\\mathcal{F}^n$ is $S$-flat, and\n\\item $\\mathcal{G}^m$ is $S$-flat.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K\\\"unneth formula, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLT","source_file":"perfect.tex","source_line":6163,"source_end_line":6175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6163-L6175","statement_sha256":"165cd70442c53271fb519729bd8866689eb42938270e998e86b84f31038be8f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7054,"rank":7054,"depth":33,"x":197.328,"y":574.787,"cluster":"derived-categories"},{"id":"stacks:0FXZ","tag":"0FXZ","title":"K\\\"unneth formula for Ext · Lemma 0FXZ","summary":"In the situation above, assume a and b are quasi-compact and quasi-separated and X and Y are tor independent over S. If K is perfect, K' ∈ D_QCoh(O_X), M is perfect, and M' ∈ D_QCoh(O_Y), then ([Tag 0FXY]) is an isomorphism.","statement_latex":"In the situation above, assume $a$ and $b$ are quasi-compact and\nquasi-separated and $X$ and $Y$ are tor independent over $S$.\nIf $K$ is perfect, $K' \\in D_\\QCoh(\\mathcal{O}_X)$, $M$ is perfect, and\n$M' \\in D_\\QCoh(\\mathcal{O}_Y)$, then (\\ref{equation-kunneth-ext})\nis an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K\\\"unneth formula for Ext","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FXZ","source_file":"perfect.tex","source_line":6350,"source_end_line":6357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6350-L6357","statement_sha256":"193ec64aae082798faccdcb0bb2bb0d31035ab986b9d80442d7c4f820a0cc55e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7055,"rank":7055,"depth":34,"x":338.845,"y":739.122,"cluster":"derived-categories"},{"id":"stacks:0DJ8","tag":"0DJ8","title":"Cohomology and base change, V · Lemma 0DJ8","summary":"Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a cartesian diagram of schemes. Let K ∈ D_QCoh(O_X) and let L(g')^*K → K' be a map in D_QCoh(O_X'). The following are equivalent • for any x' ∈ X' and i ∈ Z the map ([Tag 0DJ7]) is an isomorphism, • for U ⊂ X, V' ⊂ S' affine open both mapping into the affine open V ⊂ S with U' = V' ×_V U the composition RΓ(U, K) ⊗_O_S(U)^L O_S'(V') → RΓ(U, K) ⊗_O_X(U)^L O_X'(U') → RΓ(U', K') is an isomorphism in D(O_S'(V')),…","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nS' \\ar[r]^g &\nS\n}\n$$\nbe a cartesian diagram of schemes. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$\nand let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any $x' \\in X'$ and $i \\in \\mathbf{Z}$ the map (\\ref{equation-bc})\nis an isomorphism,\n\\item for $U \\subset X$, $V' \\subset S'$ affine open both mapping\ninto the affine open $V \\subset S$ with $U' = V' \\times_V U$\nthe composition\n$$\nR\\Gamma(U, K) \\otimes_{\\mathcal{O}_S(U)}^\\mathbf{L} \\mathcal{O}_{S'}(V')\n\\to\nR\\Gamma(U, K) \\otimes_{\\mathcal{O}_X(U)}^\\mathbf{L} \\mathcal{O}_{X'}(U')\n\\to\nR\\Gamma(U', K')\n$$\nis an isomorphism in $D(\\mathcal{O}_{S'}(V'))$, and\n\\item there is a set $I$ of quadruples $U_i, V_i', V_i, U_i'$, $i \\in I$\nas in (2) with $X' = \\bigcup U'_i$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJ8","source_file":"perfect.tex","source_line":6436,"source_end_line":6467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6436-L6467","statement_sha256":"8ee271487e69b0b7eefa4af618753c55e553bdd5410c7be704e63137be479623","origin":"The Stacks Project","memory_eligible":false,"source_rank":7056,"rank":7056,"depth":28,"x":102.027,"y":698.19,"cluster":"derived-categories"},{"id":"stacks:0DJ9","tag":"0DJ9","title":"Cohomology and base change, V · Lemma 0DJ9","summary":"Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a cartesian diagram of schemes. Let K ∈ D_QCoh(O_X) and let L(g')^*K → K' be a map in D_QCoh(O_X'). If • the equivalent conditions of Lemma [Tag 0DJ8] hold, and • f is quasi-compact and quasi-separated, then the composition Lg^*Rf_*K → Rf'_*L(g')^*K → Rf'_*K' is an isomorphism.","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nS' \\ar[r]^g &\nS\n}\n$$\nbe a cartesian diagram of schemes. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$\nand let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$.\nIf\n\\begin{enumerate}\n\\item the equivalent conditions of\nLemma \\ref{lemma-single-complex-base-change-condition} hold, and\n\\item $f$ is quasi-compact and quasi-separated,\n\\end{enumerate}\nthen the composition $Lg^*Rf_*K \\to Rf'_*L(g')^*K \\to Rf'_*K'$\nis an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJ9","source_file":"perfect.tex","source_line":6531,"source_end_line":6552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6531-L6552","statement_sha256":"0ddf4f27ac2f32e3b9cb0a20a9b70dbf189a08b262e63ea39d40fb29f0999ec9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7057,"rank":7057,"depth":33,"x":309.843,"y":593.859,"cluster":"derived-categories"},{"id":"stacks:0DJA","tag":"0DJA","title":"Cohomology and base change, V · Lemma 0DJA","summary":"Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a cartesian diagram of schemes. Let K ∈ D_QCoh(O_X) and let L(g')^*K → K' be a map in D_QCoh(O_X'). If the equivalent conditions of Lemma [Tag 0DJ8] hold, then • for E ∈ D_QCoh(O_X) the equivalent conditions of Lemma [Tag 0DJ8] hold for L(g')^*(E ⊗^L K) → L(g')^*E ⊗^L K', • if E in D(O_X) is perfect the equivalent conditions of Lemma [Tag 0DJ8] hold for L(g')^*RSheafHom(E, K) → RSheafHom(L(g')^*E, K'), and •…","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nS' \\ar[r]^g &\nS\n}\n$$\nbe a cartesian diagram of schemes. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$\nand let $L(g')^*K \\to K'$ be a map in $D_\\QCoh(\\mathcal{O}_{X'})$.\nIf the equivalent conditions of\nLemma \\ref{lemma-single-complex-base-change-condition} hold, then\n\\begin{enumerate}\n\\item for $E \\in D_\\QCoh(\\mathcal{O}_X)$ the equivalent\nconditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold\nfor $L(g')^*(E \\otimes^\\mathbf{L} K) \\to L(g')^*E \\otimes^\\mathbf{L} K'$,\n\\item if $E$ in $D(\\mathcal{O}_X)$ is perfect the equivalent conditions of\nLemma \\ref{lemma-single-complex-base-change-condition} hold for\n$L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$, and\n\\item if $K$ is bounded below and $E$ in $D(\\mathcal{O}_X)$\npseudo-coherent the equivalent conditions of\nLemma \\ref{lemma-single-complex-base-change-condition} hold for\n$L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJA","source_file":"perfect.tex","source_line":6606,"source_end_line":6633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6606-L6633","statement_sha256":"d90eb469ca7c20ecc6e905da59a24a7b0d943fe976ce77e2480532ef6bd19998","origin":"The Stacks Project","memory_eligible":false,"source_rank":7058,"rank":7058,"depth":30,"x":240.411,"y":788.967,"cluster":"derived-categories"},{"id":"stacks:0A1D","tag":"0A1D","title":"Cohomology and base change, V · Lemma 0A1D","summary":"Let f : X → S be a quasi-compact and quasi-separated morphism of schemes. Let E ∈ D_QCoh(O_X). Let G^bullet be a bounded above complex of quasi-coherent O_X-modules flat over S. Then formation of Rf_*(E ⊗^L_O_X G^bullet) commutes with arbitrary base change (see proof for precise statement).","statement_latex":"Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of\nschemes. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $\\mathcal{G}^\\bullet$\nbe a bounded above complex of quasi-coherent\n$\\mathcal{O}_X$-modules flat over $S$. Then formation of\n$$\nRf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)\n$$\ncommutes with arbitrary base change (see proof for precise statement).","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1D","source_file":"perfect.tex","source_line":6662,"source_end_line":6672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6662-L6672","statement_sha256":"ed9a421af9ed8905ae3930b9a54eeb809c7ebd6f1bf3d6fb04a92140adb2b221","origin":"The Stacks Project","memory_eligible":false,"source_rank":7059,"rank":7059,"depth":34,"x":134.57,"y":605.46,"cluster":"derived-categories"},{"id":"stacks:08IE","tag":"08IE","title":"Cohomology and base change, V · Lemma 08IE","summary":"Let f : X → S be a quasi-compact and quasi-separated morphism of schemes. Let E be an object of D(O_X). Let G^bullet be a complex of quasi-coherent O_X-modules. If • E is perfect, G^bullet is a bounded above, and G^n is flat over S, or • E is pseudo-coherent, G^bullet is bounded, and G^n is flat over S, then formation of Rf_*RSheafHom(E, G^bullet) commutes with arbitrary base change (see proof for precise statement).","statement_latex":"Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\nLet $\\mathcal{G}^\\bullet$ be a complex of\nquasi-coherent $\\mathcal{O}_X$-modules. If\n\\begin{enumerate}\n\\item $E$ is perfect, $\\mathcal{G}^\\bullet$ is a bounded above,\nand $\\mathcal{G}^n$ is flat over $S$, or\n\\item $E$ is pseudo-coherent, $\\mathcal{G}^\\bullet$ is bounded,\nand $\\mathcal{G}^n$ is flat over $S$,\n\\end{enumerate}\nthen formation of\n$$\nRf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)\n$$\ncommutes with arbitrary base change (see proof for precise statement).","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IE","source_file":"perfect.tex","source_line":6710,"source_end_line":6727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6710-L6727","statement_sha256":"3b4aac7cae1f0aec0811c91c818a1f0231ec2b4eed051fb7c61f46558eeb81fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":7060,"rank":7060,"depth":35,"x":360.482,"y":680.817,"cluster":"derived-categories"},{"id":"stacks:08EV","tag":"08EV","title":"Producing perfect complexes · Lemma 08EV","summary":"Let S be a Noetherian scheme. Let f : X → S be a morphism of schemes which is locally of finite type. Let E ∈ D(O_X) such that • E ∈ D^b_Coh(O_X), • the support of H^i(E) is proper over S for all i, and • E has finite tor dimension as an object of D(f^-1O_S). Then Rf_*E is a perfect object of D(O_S).","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes\nwhich is locally of finite type. Let $E \\in D(\\mathcal{O}_X)$ such that\n\\begin{enumerate}\n\\item $E \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$,\n\\item the support of $H^i(E)$ is proper over $S$ for all $i$, and\n\\item $E$ has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_S)$.\n\\end{enumerate}\nThen $Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Producing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EV","source_file":"perfect.tex","source_line":6772,"source_end_line":6782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6772-L6782","statement_sha256":"2289fc73c6444d1a6ef84589c2fe2544e8074838b1858b0a53dd032c5dd894db","origin":"The Stacks Project","memory_eligible":false,"source_rank":7061,"rank":7061,"depth":33,"x":133.002,"y":753.532,"cluster":"derived-categories"},{"id":"stacks:0DJQ","tag":"0DJQ","title":"Producing perfect complexes · Lemma 0DJQ","summary":"Let S be a Noetherian scheme. Let f : X → S be a morphism of schemes which is locally of finite type. Let E ∈ D(O_X) be perfect. Let G^bullet be a bounded complex of coherent O_X-modules flat over S with support proper over S. Then K = Rf_*(E ⊗_O_X^L G^bullet) is a perfect object of D(O_S).","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes\nwhich is locally of finite type. Let $E \\in D(\\mathcal{O}_X)$ be perfect.\nLet $\\mathcal{G}^\\bullet$ be a bounded complex of coherent\n$\\mathcal{O}_X$-modules flat over $S$ with support proper over $S$.\nThen $K = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)$\nis a perfect object of $D(\\mathcal{O}_S)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Producing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJQ","source_file":"perfect.tex","source_line":6820,"source_end_line":6828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6820-L6828","statement_sha256":"5049533b24465039843c8cc5857c1687e9d6bd228eb6a430bdd09bbdd750b626","origin":"The Stacks Project","memory_eligible":false,"source_rank":7062,"rank":7062,"depth":34,"x":242.409,"y":570.598,"cluster":"derived-categories"},{"id":"stacks:0DJR","tag":"0DJR","title":"Producing perfect complexes · Lemma 0DJR","summary":"Let S be a Noetherian scheme. Let f : X → S be a morphism of schemes which is locally of finite type. Let E ∈ D(O_X) be perfect. Let G^bullet be a bounded complex of coherent O_X-modules flat over S with support proper over S. Then K = Rf_*RSheafHom(E, G^bullet) is a perfect object of D(O_S).","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of schemes\nwhich is locally of finite type. Let $E \\in D(\\mathcal{O}_X)$ be perfect.\nLet $\\mathcal{G}^\\bullet$ be a bounded complex of coherent\n$\\mathcal{O}_X$-modules flat over $S$ with support proper over $S$.\nThen $K = Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)$ is a perfect object of\n$D(\\mathcal{O}_S)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Producing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJR","source_file":"perfect.tex","source_line":6845,"source_end_line":6853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6845-L6853","statement_sha256":"37aff4256890af2cc366c2d489c674c4cff1d37bf231bf9819eb7fd9f5e8b9cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7063,"rank":7063,"depth":35,"x":308.93,"y":767.826,"cluster":"derived-categories"},{"id":"stacks:0B6F","tag":"0B6F","title":"Producing perfect complexes · Lemma 0B6F","summary":"Let S be a Noetherian scheme. Let f : X → S be a flat proper morphism of schemes. Let E ∈ D(O_X) be perfect. Then Rf_*E is a perfect object of D(O_S).","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a flat proper\nmorphism of schemes. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then\n$Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Producing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6F","source_file":"perfect.tex","source_line":6871,"source_end_line":6876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6871-L6876","statement_sha256":"b50c8249ad1e7b3c1b627f0da3516a6f66c8870bf70cc3df90f7eb9fe6b468b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7064,"rank":7064,"depth":34,"x":101.004,"y":659.999,"cluster":"derived-categories"},{"id":"stacks:0A1F","tag":"0A1F","title":"A projection formula for Ext · Lemma 0A1F","summary":"Assumptions and notation as in Lemma [Tag 0DJQ]. Then there are functorial isomorphisms H^i(S, K ⊗^L_O_S F) → H^i(X, E ⊗_O_X^L (G^bullet ⊗_O_X f^*F)) for F quasi-coherent on S compatible with boundary maps (see proof).","statement_latex":"Assumptions and notation as in Lemma \\ref{lemma-tensor-perfect}.\nThen there are functorial isomorphisms\n$$\nH^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F})\n\\longrightarrow\nH^i(X, E \\otimes_{\\mathcal{O}_X}^\\mathbf{L}\n(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}))\n$$\nfor $\\mathcal{F}$ quasi-coherent on $S$\ncompatible with boundary maps (see proof).","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"A projection formula for Ext","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1F","source_file":"perfect.tex","source_line":6911,"source_end_line":6923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6911-L6923","statement_sha256":"a80e8ecca9610e5f58b94987d019a39fda3156fe88afff9964ea807509017917","origin":"The Stacks Project","memory_eligible":false,"source_rank":7065,"rank":7065,"depth":35,"x":341.347,"y":621.478,"cluster":"derived-categories"},{"id":"stacks:08ID","tag":"08ID","title":"A projection formula for Ext · Lemma 08ID","summary":"Assumptions and notation as in Lemma [Tag 0DJR]. Then there are functorial isomorphisms H^i(S, K ⊗^L_O_S F) → Ext^i_O_X(E, G^bullet ⊗_O_X f^*F) for F quasi-coherent on S compatible with boundary maps (see proof).","statement_latex":"Assumptions and notation as in Lemma \\ref{lemma-ext-perfect}.\nThen there are functorial isomorphisms\n$$\nH^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F})\n\\longrightarrow\n\\Ext^i_{\\mathcal{O}_X}(E,\n\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})\n$$\nfor $\\mathcal{F}$ quasi-coherent on $S$\ncompatible with boundary maps (see proof).","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"A projection formula for Ext","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ID","source_file":"perfect.tex","source_line":6993,"source_end_line":7005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L6993-L7005","statement_sha256":"c72c3690c7a64d7234b5702d4a3516d9090069340e21330a9903ab533ad5c7db","origin":"The Stacks Project","memory_eligible":false,"source_rank":7066,"rank":7066,"depth":36,"x":194.912,"y":786.471,"cluster":"derived-categories"},{"id":"stacks:08IF","tag":"08IF","title":"A projection formula for Ext · Lemma 08IF","summary":"Let f : X → S be a morphism of schemes, E ∈ D(O_X) and G^bullet a complex of O_X-modules. Assume • S is Noetherian, • f is locally of finite type, • E ∈ D^-_Coh(O_X), • G^bullet is a bounded complex of coherent O_X-modules flat over S with support proper over S. Then the following two statements are true • [(A)] for every m ∈ Z there exists a perfect object K of D(O_S) and functorial maps α^i_F : Ext^i_O_X(E, G^bullet ⊗_O_X f^*F) → H^i(S, K ⊗^L_O_S F) for F quasi-coherent…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes, $E \\in D(\\mathcal{O}_X)$\nand $\\mathcal{G}^\\bullet$ a complex of $\\mathcal{O}_X$-modules.\nAssume\n\\begin{enumerate}\n\\item $S$ is Noetherian,\n\\item $f$ is locally of finite type,\n\\item $E \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$,\n\\item $\\mathcal{G}^\\bullet$ is a bounded complex of\ncoherent $\\mathcal{O}_X$-modules flat over $S$ with support proper over $S$.\n\\end{enumerate}\nThen the following two statements are true\n\\begin{enumerate}\n\\item[(A)] for every $m \\in \\mathbf{Z}$ there exists a perfect object $K$\nof $D(\\mathcal{O}_S)$ and functorial maps\n$$\n\\alpha^i_\\mathcal{F} :\n\\Ext^i_{\\mathcal{O}_X}(E,\n\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})\n\\longrightarrow\nH^i(S, K \\otimes^\\mathbf{L}_{\\mathcal{O}_S} \\mathcal{F})\n$$\nfor $\\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps\n(see proof) such that $\\alpha^i_\\mathcal{F}$ is an isomorphism for $i \\leq m$\n\\item[(B)] there exists a pseudo-coherent $L \\in D(\\mathcal{O}_S)$\nand functorial isomorphisms\n$$\n\\Ext^i_{\\mathcal{O}_S}(L, \\mathcal{F}) \\longrightarrow\n\\Ext^i_{\\mathcal{O}_X}(E,\n\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})\n$$\nfor $\\mathcal{F}$ quasi-coherent on $S$ compatible with boundary maps.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"A projection formula for Ext","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IF","source_file":"perfect.tex","source_line":7057,"source_end_line":7091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7057-L7091","statement_sha256":"f07e45765132750454c484ca8cfcaa580f9be953637b05948e1c724b24f5cd10","origin":"The Stacks Project","memory_eligible":false,"source_rank":7067,"rank":7067,"depth":38,"x":170.176,"y":581.454,"cluster":"derived-categories"},{"id":"stacks:09RE","tag":"09RE","title":"Limits and derived categories · Lemma 09RE","summary":"In Situation [Tag 09RD]. Let E_0 and K_0 be objects of D(O_S_0). Set E_i = Lf_i0^*E_0 and K_i = Lf_i0^*K_0 for i ≥ 0 and set E = Lf_0^*E_0 and K = Lf_0^*K_0. Then the map colim_i ≥ 0 Hom_D(O_S_i)(E_i, K_i) → Hom_D(O_S)(E, K) is an isomorphism if either • E_0 is perfect and K_0 ∈ D_QCoh(O_S_0), or • E_0 is pseudo-coherent and K_0 ∈ D_QCoh(O_S_0) has finite tor dimension.","statement_latex":"In Situation \\ref{situation-descent}.\nLet $E_0$ and $K_0$ be objects of\n$D(\\mathcal{O}_{S_0})$.\nSet $E_i = Lf_{i0}^*E_0$ and $K_i = Lf_{i0}^*K_0$ for $i \\geq 0$\nand set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map\n$$\n\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{S_i})}(E_i, K_i)\n\\longrightarrow\n\\Hom_{D(\\mathcal{O}_S)}(E, K)\n$$\nis an isomorphism if either\n\\begin{enumerate}\n\\item $E_0$ is perfect and $K_0 \\in D_\\QCoh(\\mathcal{O}_{S_0})$, or\n\\item $E_0$ is pseudo-coherent and\n$K_0 \\in D_\\QCoh(\\mathcal{O}_{S_0})$ has finite tor dimension.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Limits and derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RE","source_file":"perfect.tex","source_line":7229,"source_end_line":7247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7229-L7247","statement_sha256":"7f17810218f5ed1ed63dd55e096b4ba5b0677c484f75d4207e1a187f6d23d3fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7068,"rank":7068,"depth":29,"x":353.519,"y":718.77,"cluster":"derived-categories"},{"id":"stacks:09RF","tag":"09RF","title":"Limits and derived categories · Lemma 09RF","summary":"In Situation [Tag 09RD] the category of perfect objects of D(O_S) is the colimit of the categories of perfect objects of D(O_S_i).","statement_latex":"In Situation \\ref{situation-descent} the category of perfect\nobjects of $D(\\mathcal{O}_S)$ is the colimit of the categories\nof perfect objects of $D(\\mathcal{O}_{S_i})$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Limits and derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RF","source_file":"perfect.tex","source_line":7314,"source_end_line":7319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7314-L7319","statement_sha256":"a35beddfa937ff1944fae5eee6466d5ae108946cc351133526c9ee40d5055291","origin":"The Stacks Project","memory_eligible":false,"source_rank":7069,"rank":7069,"depth":30,"x":107.585,"y":721.552,"cluster":"derived-categories"},{"id":"stacks:0A1H","tag":"0A1H","title":"Cohomology and base change, VI · Lemma 0A1H","summary":"Let f : X → S be a morphism of finite presentation. Let E ∈ D(O_X) be a perfect object. Let G^bullet be a bounded complex of finitely presented O_X-modules, flat over S, with support proper over S. Then K = Rf_*(E ⊗_O_X^L G^bullet) is a perfect object of D(O_S) and its formation commutes with arbitrary base change.","statement_latex":"Let $f : X \\to S$ be a morphism of finite presentation.\nLet $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$\nbe a bounded complex of finitely presented $\\mathcal{O}_X$-modules,\nflat over $S$, with support proper over $S$. Then\n$$\nK = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)\n$$\nis a perfect object of $D(\\mathcal{O}_S)$ and its formation\ncommutes with arbitrary base change.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1H","source_file":"perfect.tex","source_line":7396,"source_end_line":7407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7396-L7407","statement_sha256":"5bf106ee4cb6961875476867fffbbb7cc4dbdf3dedde42cf4f9b74e909515752","origin":"The Stacks Project","memory_eligible":false,"source_rank":7070,"rank":7070,"depth":37,"x":286.924,"y":579.772,"cluster":"derived-categories"},{"id":"stacks:0CSC","tag":"0CSC","title":"Cohomology and base change, VI · Lemma 0CSC","summary":"Let f : X → S be a morphism of finite presentation. Let E ∈ D(O_X) be a pseudo-coherent object. Let G^bullet be a bounded above complex of finitely presented O_X-modules, flat over S, with support proper over S. Then K = Rf_*(E ⊗_O_X^L G^bullet) is a pseudo-coherent object of D(O_S) and its formation commutes with arbitrary base change.","statement_latex":"Let $f : X \\to S$ be a morphism of finite presentation.\nLet $E \\in D(\\mathcal{O}_X)$ be a pseudo-coherent object.\nLet $\\mathcal{G}^\\bullet$ be a bounded above complex of\nfinitely presented $\\mathcal{O}_X$-modules, flat over $S$,\nwith support proper over $S$. Then\n$$\nK = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)\n$$\nis a pseudo-coherent object of $D(\\mathcal{O}_S)$ and its formation\ncommutes with arbitrary base change.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSC","source_file":"perfect.tex","source_line":7457,"source_end_line":7469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7457-L7469","statement_sha256":"e0e99b45bad55eb1c82da962e6159b4cdb82be88ff1c224cbac50a1b1dd6c37a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7071,"rank":7071,"depth":38,"x":268.675,"y":786.342,"cluster":"derived-categories"},{"id":"stacks:0B91","tag":"0B91","title":"Cohomology and base change, VI · Lemma 0B91","summary":"Let S be a scheme. Let f : X → S be a proper morphism of finite presentation. • Let E ∈ D(O_X) be perfect and f flat. Then Rf_*E is a perfect object of D(O_S) and its formation commutes with arbitrary base change. • Let G be an O_X-module of finite presentation, flat over S. Then Rf_*G is a perfect object of D(O_S) and its formation commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to S$ be a proper\nmorphism of finite presentation.\n\\begin{enumerate}\n\\item Let $E \\in D(\\mathcal{O}_X)$ be perfect and $f$ flat. Then\n$Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$ and its formation\ncommutes with arbitrary base change.\n\\item Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation,\nflat over $S$. Then $Rf_*\\mathcal{G}$ is a perfect object of\n$D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B91","source_file":"perfect.tex","source_line":7524,"source_end_line":7536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7524-L7536","statement_sha256":"431f33cebb3552e13cb1d9efe320a12618679010181d670840715f29a9ae7334","origin":"The Stacks Project","memory_eligible":false,"source_rank":7072,"rank":7072,"depth":38,"x":115.821,"y":623.459,"cluster":"derived-categories"},{"id":"stacks:0CSD","tag":"0CSD","title":"Cohomology and base change, VI · Lemma 0CSD","summary":"Let S be a scheme. Let f : X → S be a flat proper morphism of finite presentation. Let E ∈ D(O_X) be pseudo-coherent. Then Rf_*E is a pseudo-coherent object of D(O_S) and its formation commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to S$ be a flat proper\nmorphism of finite presentation. Let $E \\in D(\\mathcal{O}_X)$\nbe pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of\n$D(\\mathcal{O}_S)$ and its formation commutes with arbitrary base change.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSD","source_file":"perfect.tex","source_line":7546,"source_end_line":7552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7546-L7552","statement_sha256":"e22dfd1442fd0e03082ad15e9557bb06e163ce76acf2f99627ca654e180cab25","origin":"The Stacks Project","memory_eligible":false,"source_rank":7073,"rank":7073,"depth":0,"x":359.826,"y":656.875,"cluster":"derived-categories"},{"id":"stacks:0D2Q","tag":"0D2Q","title":"Cohomology and base change, VI · Lemma 0D2Q","summary":"Let R be a ring. Let X be a scheme and let f : X → Spec(R) be proper, flat, and of finite presentation. Let (M_n) be an inverse system of R-modules with surjective transition maps. Then the canonical map O_X ⊗_R (lim M_n) → lim O_X ⊗_R M_n induces an isomorphism from the source to DQ_X applied to the target.","statement_latex":"Let $R$ be a ring. Let $X$ be a scheme and let\n$f : X \\to \\Spec(R)$ be proper, flat, and\nof finite presentation. Let $(M_n)$ be an inverse\nsystem of $R$-modules with surjective transition maps.\nThen the canonical map\n$$\n\\mathcal{O}_X \\otimes_R (\\lim M_n)\n\\longrightarrow\n\\lim \\mathcal{O}_X \\otimes_R M_n\n$$\ninduces an isomorphism from the source to $DQ_X$ applied to the target.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2Q","source_file":"perfect.tex","source_line":7568,"source_end_line":7581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7568-L7581","statement_sha256":"23498fd2e8da0706004be27cdef2d917e2aa74e5f09f3766d9bd3274d7c6089c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7074,"rank":7074,"depth":39,"x":152.767,"y":770.832,"cluster":"derived-categories"},{"id":"stacks:0A1J","tag":"0A1J","title":"Cohomology and base change, VI · Lemma 0A1J","summary":"Let f : X → S be a morphism of finite presentation. Let E ∈ D(O_X) be a perfect object. Let G^bullet be a bounded complex of finitely presented O_X-modules, flat over S, with support proper over S. Then K = Rf_*RSheafHom(E, G^bullet) is a perfect object of D(O_S) and its formation commutes with arbitrary base change.","statement_latex":"Let $f : X \\to S$ be a morphism of finite presentation.\nLet $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$\nbe a bounded complex of finitely presented $\\mathcal{O}_X$-modules,\nflat over $S$, with support proper over $S$. Then\n$$\nK = Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)\n$$\nis a perfect object of $D(\\mathcal{O}_S)$ and its formation\ncommutes with arbitrary base change.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1J","source_file":"perfect.tex","source_line":7625,"source_end_line":7636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7625-L7636","statement_sha256":"49d41ddbc5a51955bed2500216cbcde21b2c747caccb8d5df87c8f862859854d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7075,"rank":7075,"depth":37,"x":213.885,"y":569.059,"cluster":"derived-categories"},{"id":"stacks:0BDI","tag":"0BDI","title":"Perfect complexes · Lemma 0BDI","summary":"Let X be a scheme. Let E ∈ D(O_X) be pseudo-coherent (for example perfect). For any i ∈ Z consider the function β_i : X → (0, 1, 2, …), x ↦ dim_kappa(x) H^i(E ⊗_O_X^L kappa(x)) Then we have • formation of β_i commutes with arbitrary base change, • the functions β_i are upper semi-continuous, and • the level sets of β_i are locally constructible in X.","statement_latex":"Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent\n(for example perfect). For any $i \\in \\mathbf{Z}$ consider the function\n$$\n\\beta_i : X \\longrightarrow \\{0, 1, 2, \\ldots\\},\\quad\nx \\longmapsto\n\\dim_{\\kappa(x)}\nH^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\kappa(x))\n$$\nThen we have\n\\begin{enumerate}\n\\item formation of $\\beta_i$ commutes with arbitrary base change,\n\\item the functions $\\beta_i$ are upper semi-continuous, and\n\\item the level sets of $\\beta_i$ are locally constructible in $X$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDI","source_file":"perfect.tex","source_line":7689,"source_end_line":7705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7689-L7705","statement_sha256":"b530e9508bbeb41844ae6b59ebd0a6e4f1e01480230d59a692c4cfb316b79c3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7076,"rank":7076,"depth":32,"x":331.225,"y":752.753,"cluster":"derived-categories"},{"id":"stacks:0BDJ","tag":"0BDJ","title":"Perfect complexes · Lemma 0BDJ","summary":"Let X be a scheme. Let E ∈ D(O_X) be perfect. The function chi_E : X → Z, x ↦ ∑ (-1)^i dim_kappa(x) H^i(E ⊗_O_X^L kappa(x)) is locally constant on X.","statement_latex":"Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect.\nThe function\n$$\n\\chi_E : X \\longrightarrow \\mathbf{Z},\\quad\nx \\longmapsto \\sum (-1)^i\n\\dim_{\\kappa(x)} H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\kappa(x))\n$$\nis locally constant on $X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDJ","source_file":"perfect.tex","source_line":7777,"source_end_line":7787,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7777-L7787","statement_sha256":"4ca60016278f05bbaa6a5c41231aed5a0c9808180fcf7f9cb3a2e81dc17884dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7077,"rank":7077,"depth":7,"x":96.685,"y":683.795,"cluster":"derived-categories"},{"id":"stacks:0BDK","tag":"0BDK","title":"Perfect complexes · Lemma 0BDK","summary":"Let X be a scheme. Let E ∈ D(O_X) be perfect. Given i, r ∈ Z, there exists an open subscheme U ⊂ X characterized by the following • E|_U ≅ H^i(E|_U)[-i] and H^i(E|_U) is a locally free O_U-module of rank r, • a morphism f : Y → X factors through U if and only if Lf^*E is isomorphic to a locally free module of rank r placed in degree i.","statement_latex":"Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect.\nGiven $i, r \\in \\mathbf{Z}$, there exists an\nopen subscheme $U \\subset X$ characterized by the following\n\\begin{enumerate}\n\\item $E|_U \\cong H^i(E|_U)[-i]$ and $H^i(E|_U)$ is a locally free\n$\\mathcal{O}_U$-module of rank $r$,\n\\item a morphism $f : Y \\to X$ factors through $U$ if and only if\n$Lf^*E$ is isomorphic to a locally free module of rank $r$\nplaced in degree $i$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDK","source_file":"perfect.tex","source_line":7798,"source_end_line":7810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7798-L7810","statement_sha256":"892e073436e05cde5cbf92c06cef979d8aa73ff46cc3bebc471f155d9f6bd628","origin":"The Stacks Project","memory_eligible":false,"source_rank":7078,"rank":7078,"depth":33,"x":325.371,"y":601.458,"cluster":"derived-categories"},{"id":"stacks:0BDL","tag":"0BDL","title":"Perfect complexes · Lemma 0BDL","summary":"Let X be a scheme. Let E ∈ D(O_X) be perfect of tor-amplitude in [a, b] for some a, b ∈ Z. Let r ≥ 0. Then there exists a locally closed subscheme j : Z → X characterized by the following • H^a(Lj^*E) is a locally free O_Z-module of rank r, and • a morphism f : Y → X factors through Z if and only if for all morphisms g : Y' → Y the O_Y'-module H^a(L(f ∘ g)^*E) is locally free of rank r. Moreover, j : Z → X is of finite presentation and we have • [(3)] if f : Y → X factors…","statement_latex":"Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect\nof tor-amplitude in $[a, b]$ for some $a, b \\in \\mathbf{Z}$.\nLet $r \\geq 0$.\nThen there exists a locally closed subscheme $j : Z \\to X$\ncharacterized by the following\n\\begin{enumerate}\n\\item $H^a(Lj^*E)$ is a locally free $\\mathcal{O}_Z$-module of rank $r$, and\n\\item a morphism $f : Y \\to X$ factors through $Z$\nif and only if for all morphisms $g : Y' \\to Y$ the\n$\\mathcal{O}_{Y'}$-module $H^a(L(f \\circ g)^*E)$ is locally free\nof rank $r$.\n\\end{enumerate}\nMoreover, $j : Z \\to X$ is of finite presentation and we have\n\\begin{enumerate}\n\\item[(3)] if $f : Y \\to X$ factors as $Y \\xrightarrow{g} Z \\to X$, then\n$H^a(Lf^*E) = g^*H^a(Lj^*E)$,\n\\item[(4)] if $\\beta_a(x) \\leq r$ for all $x \\in X$, then\n$j$ is a closed immersion and given $f : Y \\to X$ the following\nare equivalent\n\\begin{enumerate}\n\\item $f : Y \\to X$ factors through $Z$,\n\\item $H^a(Lf^*E)$ is a locally free $\\mathcal{O}_Y$-module of rank $r$,\n\\end{enumerate}\nand if $r = 1$ these are also equivalent to\n\\begin{enumerate}\n\\item[(c)] $\\mathcal{O}_Y \\to \\SheafHom_{\\mathcal{O}_Y}(H^a(Lf^*E), H^a(Lf^*E))$\nis injective.\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDL","source_file":"perfect.tex","source_line":7831,"source_end_line":7862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7831-L7862","statement_sha256":"e74155d0d6e951112a96435f6219d4d8c7d49c0407ae3da635925f04573ea323","origin":"The Stacks Project","memory_eligible":false,"source_rank":7079,"rank":7079,"depth":33,"x":222.827,"y":792.17,"cluster":"derived-categories"},{"id":"stacks:0BDN","tag":"0BDN","title":"Applications · Lemma 0BDN","summary":"Let f : X → S be a proper morphism of finite presentation. Let F be an O_X-module of finite presentation, flat over S. For fixed i ∈ Z consider the function β_i : S → (0, 1, 2, …), s ↦ dim_kappa(s) H^i(X_s, F_s) Then we have • formation of β_i commutes with arbitrary base change, • the functions β_i are upper semi-continuous, and • the level sets of β_i are locally constructible in S.","statement_latex":"Let $f : X \\to S$ be a proper morphism of finite presentation.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,\nflat over $S$. For fixed $i \\in \\mathbf{Z}$ consider the function\n$$\n\\beta_i : S \\to \\{0, 1, 2, \\ldots\\},\\quad\ns \\longmapsto \\dim_{\\kappa(s)} H^i(X_s, \\mathcal{F}_s)\n$$\nThen we have\n\\begin{enumerate}\n\\item formation of $\\beta_i$ commutes with arbitrary base change,\n\\item the functions $\\beta_i$ are upper semi-continuous, and\n\\item the level sets of $\\beta_i$ are locally constructible in $S$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDN","source_file":"perfect.tex","source_line":7931,"source_end_line":7946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7931-L7946","statement_sha256":"61d4112f0389e26c771f8f6b63cf3ce71f3eed4e7317efbea461ff4add929e45","origin":"The Stacks Project","memory_eligible":false,"source_rank":7080,"rank":7080,"depth":39,"x":144.979,"y":593.11,"cluster":"derived-categories"},{"id":"stacks:0B9T","tag":"0B9T","title":"Applications · Lemma 0B9T","summary":"Let f : X → S be a proper morphism of finite presentation. Let F be an O_X-module of finite presentation, flat over S. The function s ↦ chi(X_s, F_s) is locally constant on S. Formation of this function commutes with base change.","statement_latex":"Let $f : X \\to S$ be a proper morphism of finite presentation.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,\nflat over $S$. The function\n$$\ns \\longmapsto \\chi(X_s, \\mathcal{F}_s)\n$$\nis locally constant on $S$. Formation of this function commutes with\nbase change.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9T","source_file":"perfect.tex","source_line":7961,"source_end_line":7971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7961-L7971","statement_sha256":"cf76199b0ec664612fa3a4fd38f02dab2f56b2c66989ab0a4d0017d868b8bf3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7081,"rank":7081,"depth":39,"x":362.731,"y":695.848,"cluster":"derived-categories"},{"id":"stacks:0B9S","tag":"0B9S","title":"Applications · Lemma 0B9S","summary":"Let f : X → S be a proper morphism of finite presentation. Let F be an O_X-module of finite presentation, flat over S. Fix i, r ∈ Z. Then there exists an open subscheme U ⊂ S with the following property: A morphism T → S factors through U if and only if Rf_T, *F_T is isomorphic to a finite locally free module of rank r placed in degree i.","statement_latex":"Let $f : X \\to S$ be a proper morphism of finite presentation.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,\nflat over $S$. Fix $i, r \\in \\mathbf{Z}$.\nThen there exists an open subscheme\n$U \\subset S$ with the following property:\nA morphism $T \\to S$ factors through $U$ if and only if\n$Rf_{T, *}\\mathcal{F}_T$ is isomorphic to a\nfinite locally free module of rank $r$ placed in degree $i$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9S","source_file":"perfect.tex","source_line":7986,"source_end_line":7996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L7986-L7996","statement_sha256":"3f8f8b31e55b1aff84c67a5d3d3c3343818889b5611efba21a4ba6872753c8be","origin":"The Stacks Project","memory_eligible":false,"source_rank":7082,"rank":7082,"depth":39,"x":119.245,"y":743.707,"cluster":"derived-categories"},{"id":"stacks:0D4E","tag":"0D4E","title":"Applications · Lemma 0D4E","summary":"Let f : X → S be a morphism of finite presentation. Let F be an O_X-module of finite presentation, flat over S with support proper over S. If R^if_*F = 0 for i > 0, then f_*F is locally free and its formation commutes with arbitrary base change (see proof for explanation).","statement_latex":"Let $f : X \\to S$ be a morphism of finite presentation.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,\nflat over $S$ with support proper over $S$. If $R^if_*\\mathcal{F} = 0$\nfor $i > 0$, then $f_*\\mathcal{F}$ is locally free and its formation\ncommutes with arbitrary base change (see proof for explanation).","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4E","source_file":"perfect.tex","source_line":8007,"source_end_line":8014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8007-L8014","statement_sha256":"8f91ef33ce911e360ec80d48e22b6135b96f4073986bf263f709aabc1632dffb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7083,"rank":7083,"depth":38,"x":260.475,"y":570.043,"cluster":"derived-categories"},{"id":"stacks:0E62","tag":"0E62","title":"Applications · Lemma 0E62","summary":"Let f : X → S be a morphism of schemes. Assume • f is proper, flat, and of finite presentation, and • for all s ∈ S we have kappa(s) = H^0(X_s, O_X_s). Then we have • [(a)] f_*O_X = O_S and this holds after any base change, • [(b)] locally on S we have Rf_*O_X = O_S ⊕ P in D(O_S) where P is perfect of tor amplitude in [1, ∞).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $f$ is proper, flat, and of finite presentation, and\n\\item for all $s \\in S$ we have $\\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$.\n\\end{enumerate}\nThen we have\n\\begin{enumerate}\n\\item[(a)] $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and\nthis holds after any base change,\n\\item[(b)] locally on $S$ we have\n$$\nRf_*\\mathcal{O}_X = \\mathcal{O}_S \\oplus P\n$$\nin $D(\\mathcal{O}_S)$\nwhere $P$ is perfect of tor amplitude in $[1, \\infty)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E62","source_file":"perfect.tex","source_line":8046,"source_end_line":8064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8046-L8064","statement_sha256":"5b1913bb117c044cdecf772dbdeff62c649d4ca5096bad12621b8f9dd82a28e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7084,"rank":7084,"depth":39,"x":296.033,"y":778.493,"cluster":"derived-categories"},{"id":"stacks:0E0L","tag":"0E0L","title":"Applications · Lemma 0E0L","summary":"Let f : X → S be a morphism of schemes. Assume • f is proper, flat, and of finite presentation, and • the geometric fibres of f are reduced and connected. Then f_*O_X = O_S and this holds after any base change.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $f$ is proper, flat, and of finite presentation, and\n\\item the geometric fibres of $f$ are reduced and connected.\n\\end{enumerate}\nThen $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds\nafter any base change.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0L","source_file":"perfect.tex","source_line":8096,"source_end_line":8105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8096-L8105","statement_sha256":"4efa9d3e947e707ba7c4b32e3773297b2252df6fe18a41a4e6d9b610f6310764","origin":"The Stacks Project","memory_eligible":false,"source_rank":7085,"rank":7085,"depth":40,"x":101.946,"y":644.798,"cluster":"derived-categories"},{"id":"stacks:0G7X","tag":"0G7X","title":"Applications · Lemma 0G7X","summary":"Let f : X → S be a proper morphism of schemes. Let s ∈ S and let e ∈ H^0(X_s, O_X_s) be an idempotent. Then e is in the image of the map (f_*O_X)_s → H^0(X_s, O_X_s).","statement_latex":"Let $f : X \\to S$ be a proper morphism of schemes. Let $s \\in S$\nand let $e \\in H^0(X_s, \\mathcal{O}_{X_s})$ be an idempotent.\nThen $e$ is in the image of the map\n$(f_*\\mathcal{O}_X)_s \\to H^0(X_s, \\mathcal{O}_{X_s})$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7X","source_file":"perfect.tex","source_line":8116,"source_end_line":8122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8116-L8122","statement_sha256":"34a2dcd07f5fdfa6aee7f898f74019857227f66167151fd7e26e91f2f3c4f165","origin":"The Stacks Project","memory_eligible":false,"source_rank":7086,"rank":7086,"depth":38,"x":352.883,"y":633.239,"cluster":"derived-categories"},{"id":"stacks:0G7Y","tag":"0G7Y","title":"Applications · Lemma 0G7Y","summary":"Let f : X → S be a morphism of schemes. Let s ∈ S. Assume • f is proper, flat, and of finite presentation, and • the fibre X_s is geometrically reduced. Then, after replacing S by an open neighbourhood of s, there exists a direct sum decomposition Rf_*O_X = f_*O_X ⊕ P in D(O_S) where f_*O_X is a finite étale O_S-algebra and P is a perfect of tor amplitude in [1, ∞).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $s \\in S$. Assume\n\\begin{enumerate}\n\\item $f$ is proper, flat, and of finite presentation, and\n\\item the fibre $X_s$ is geometrically reduced.\n\\end{enumerate}\nThen, after replacing $S$ by an open neighbourhood of $s$, there\nexists a direct sum decomposition\n$Rf_*\\mathcal{O}_X = f_*\\mathcal{O}_X \\oplus P$\nin $D(\\mathcal{O}_S)$ where $f_*\\mathcal{O}_X$ is a finite \\'etale\n$\\mathcal{O}_S$-algebra and\n$P$ is a perfect of tor amplitude in $[1, \\infty)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7Y","source_file":"perfect.tex","source_line":8182,"source_end_line":8195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8182-L8195","statement_sha256":"4f0ba103c9e1d8432a1eea0b939a5960cae9d66375e9745fa52ed235c35832ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":7087,"rank":7087,"depth":47,"x":176.927,"y":784.334,"cluster":"derived-categories"},{"id":"stacks:0EX6","tag":"0EX6","title":"Other applications · Lemma 0EX6","summary":"Let R be a coherent ring. Let X be a scheme of finite presentation over R. Let G be an O_X-module of finite presentation, flat over R, with support proper over R. Then H^i(X, G) is a coherent R-module.","statement_latex":"Let $R$ be a coherent ring. Let $X$ be a scheme of finite presentation over $R$.\nLet $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation,\nflat over $R$, with support proper over $R$. Then\n$H^i(X, \\mathcal{G})$ is a coherent $R$-module.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Other applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EX6","source_file":"perfect.tex","source_line":8276,"source_end_line":8282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8276-L8282","statement_sha256":"08545a239e2f482665682bfbb22f93c08a1e1fbcaa17a1728dc0d135fc463288","origin":"The Stacks Project","memory_eligible":false,"source_rank":7088,"rank":7088,"depth":38,"x":185.178,"y":572.821,"cluster":"derived-categories"},{"id":"stacks:0CRP","tag":"0CRP","title":"Other applications · Lemma 0CRP","summary":"Let X be a quasi-compact and quasi-separated scheme. Let K be an object of D_QCoh(O_X) such that the cohomology sheaves H^i(K) have countable sets of sections over affine opens. Then for any quasi-compact open U ⊂ X and any perfect object E in D(O_X) the sets H^i(U, K ⊗^L E), Ext^i(E|_U, K|_U) are countable.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$\nsuch that the cohomology sheaves $H^i(K)$ have countable\nsets of sections over affine opens. Then for any quasi-compact open\n$U \\subset X$ and any perfect object $E$ in $D(\\mathcal{O}_X)$\nthe sets\n$$\nH^i(U, K \\otimes^\\mathbf{L} E),\\quad \\Ext^i(E|_U, K|_U)\n$$\nare countable.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Other applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRP","source_file":"perfect.tex","source_line":8290,"source_end_line":8302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8290-L8302","statement_sha256":"2b51815feee59ba1bb79c53cc5cfc32167526557d1343f27ac14048156046711","origin":"The Stacks Project","memory_eligible":false,"source_rank":7089,"rank":7089,"depth":29,"x":349.386,"y":733.664,"cluster":"derived-categories"},{"id":"stacks:0CRQ","tag":"0CRQ","title":"Other applications · Lemma 0CRQ","summary":"Let X be a quasi-compact and quasi-separated scheme such that the sets of sections of O_X over affine opens are countable. Let K be an object of D_QCoh(O_X). The following are equivalent • K = hocolim E_n with E_n a perfect object of D(O_X), and • the cohomology sheaves H^i(K) have countable sets of sections over affine opens.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme such that\nthe sets of sections of $\\mathcal{O}_X$ over affine opens are countable.\nLet $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The\nfollowing are equivalent\n\\begin{enumerate}\n\\item $K = \\text{hocolim} E_n$ with $E_n$ a perfect object of\n$D(\\mathcal{O}_X)$, and\n\\item the cohomology sheaves $H^i(K)$ have countable\nsets of sections over affine opens.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Other applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRQ","source_file":"perfect.tex","source_line":8339,"source_end_line":8351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8339-L8351","statement_sha256":"a71f723e99cef70acea711f5c4efae2c199767a21fa50c11c20e627fafdd5d79","origin":"The Stacks Project","memory_eligible":false,"source_rank":7090,"rank":7090,"depth":39,"x":98.653,"y":708.205,"cluster":"derived-categories"},{"id":"stacks:0CRR","tag":"0CRR","title":"Other applications · Lemma 0CRR","summary":"Let A be a ring. Let X be a scheme of finite presentation over A. Let f : U → X be a flat morphism of finite presentation. Then • there exists an inverse system of perfect objects L_n of D(O_X) such that RΓ(U, Lf^*K) = hocolim RHom_X(L_n, K) in D(A) functorially in K in D_QCoh(O_X), and • there exists a system of perfect objects E_n of D(O_X) such that RΓ(U, Lf^*K) = hocolim RΓ(X, E_n ⊗^L K) in D(A) functorially in K in D_QCoh(O_X).","statement_latex":"Let $A$ be a ring. Let $X$ be a scheme of finite presentation over $A$.\nLet $f : U \\to X$ be a flat morphism of finite presentation. Then\n\\begin{enumerate}\n\\item there exists an inverse system of perfect objects $L_n$ of\n$D(\\mathcal{O}_X)$ such that\n$$\nR\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Hom_X(L_n, K)\n$$\nin $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$, and\n\\item there exists a system of perfect objects $E_n$ of\n$D(\\mathcal{O}_X)$ such that\n$$\nR\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Gamma(X, E_n \\otimes^\\mathbf{L} K)\n$$\nin $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Other applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRR","source_file":"perfect.tex","source_line":8392,"source_end_line":8410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8392-L8410","statement_sha256":"12d0d43ba201f4b9adfa54dab5c25a1f57c468723a41e56ca79ee668292178bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7091,"rank":7091,"depth":40,"x":304.261,"y":584.558,"cluster":"derived-categories"},{"id":"stacks:0CSF","tag":"0CSF","title":"Characterizing pseudo-coherent complexes, II · Lemma 0CSF","summary":"Let A be a ring. Let R be a (possibly noncommutative) A-algebra which is finite free as an A-module. Then any object M of D(R) which is pseudo-coherent in D(A) can be represented by a bounded above complex of finite free (right) R-modules.","statement_latex":"Let $A$ be a ring. Let $R$ be a (possibly noncommutative) $A$-algebra\nwhich is finite free as an $A$-module. Then any object $M$ of $D(R)$\nwhich is pseudo-coherent in $D(A)$ can be represented by a\nbounded above complex of finite free (right) $R$-modules.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Characterizing pseudo-coherent complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSF","source_file":"perfect.tex","source_line":8469,"source_end_line":8475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8469-L8475","statement_sha256":"259455d19810596162601e89f6bf9a992a563ed3b2899df1a00b4bbbe16c74ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":7092,"rank":7092,"depth":9,"x":252.02,"y":792.65,"cluster":"derived-categories"},{"id":"stacks:0CSG","tag":"0CSG","title":"Characterizing pseudo-coherent complexes, II · Lemma 0CSG","summary":"Let A be a ring. Let n ≥ 0. Let K ∈ D_QCoh(O_P^n_A). The following are equivalent • K is pseudo-coherent, • RΓ(P^n_A, E ⊗^L K) is a pseudo-coherent object of D(A) for each pseudo-coherent object E of D(O_P^n_A), • RΓ(P^n_A, E ⊗^L K) is a pseudo-coherent object of D(A) for each perfect object E of D(O_P^n_A), • RHom_P^n_A(E, K) is a pseudo-coherent object of D(A) for each perfect object E of D(O_P^n_A), • RΓ(P^n_A, K ⊗^L O_P^n_A(d)) is pseudo-coherent object of D(A) for d…","statement_latex":"Let $A$ be a ring. Let $n \\geq 0$. Let\n$K \\in D_\\QCoh(\\mathcal{O}_{\\mathbf{P}^n_A})$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ is pseudo-coherent,\n\\item $R\\Gamma(\\mathbf{P}^n_A, E \\otimes^\\mathbf{L} K)$ is a pseudo-coherent\nobject of $D(A)$ for each pseudo-coherent object $E$ of\n$D(\\mathcal{O}_{\\mathbf{P}^n_A})$,\n\\item $R\\Gamma(\\mathbf{P}^n_A, E \\otimes^\\mathbf{L} K)$ is a pseudo-coherent\nobject of $D(A)$ for each perfect object $E$ of\n$D(\\mathcal{O}_{\\mathbf{P}^n_A})$,\n\\item $R\\Hom_{\\mathbf{P}^n_A}(E, K)$ is a pseudo-coherent\nobject of $D(A)$ for each perfect object $E$ of\n$D(\\mathcal{O}_{\\mathbf{P}^n_A})$,\n\\item $R\\Gamma(\\mathbf{P}^n_A,\nK \\otimes^\\mathbf{L} \\mathcal{O}_{\\mathbf{P}^n_A}(d))$ is pseudo-coherent\nobject of $D(A)$ for $d = 0, 1, \\ldots, n$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Characterizing pseudo-coherent complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSG","source_file":"perfect.tex","source_line":8555,"source_end_line":8575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8555-L8575","statement_sha256":"60ed9e4685991f899aa905d48b2889d7dcb808b4bb4a4f9002e78bb4eb58c89e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7093,"rank":7093,"depth":19,"x":123.045,"y":609.343,"cluster":"derived-categories"},{"id":"stacks:0CSH","tag":"0CSH","title":"Characterizing pseudo-coherent complexes, II · Lemma 0CSH","summary":"Let A be a ring. Let X be a scheme over A which is quasi-compact and quasi-separated. Let K ∈ D^-_QCoh(O_X). If RΓ(X, E ⊗^L K) is pseudo-coherent in D(A) for every perfect E in D(O_X), then RΓ(X, E ⊗^L K) is pseudo-coherent in D(A) for every pseudo-coherent E in D(O_X).","statement_latex":"Let $A$ be a ring. Let $X$ be a scheme over $A$ which is quasi-compact\nand quasi-separated. Let $K \\in D^-_\\QCoh(\\mathcal{O}_X)$.\nIf $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent\nin $D(A)$ for every perfect $E$ in $D(\\mathcal{O}_X)$,\nthen $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent\nin $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Characterizing pseudo-coherent complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSH","source_file":"perfect.tex","source_line":8647,"source_end_line":8655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8647-L8655","statement_sha256":"de641e7509bb6b78df0484e0648d3be27a01d8bc8defa0964f176a4dbaa2ecd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7094,"rank":7094,"depth":37,"x":365.85,"y":671.402,"cluster":"derived-categories"},{"id":"stacks:0DI0","tag":"0DI0","title":"Relatively perfect objects · Definition 0DI0","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. An object E of D(O_X) is perfect relative to S or S-perfect if E is pseudo-coherent (Cohomology, Definition [Tag 08CB]) and E locally has finite tor dimension as an object of D(f^-1O_S) (Cohomology, Definition [Tag 08CG]).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation. An object $E$ of $D(\\mathcal{O}_X)$ is\n{\\it perfect relative to $S$} or\n{\\it $S$-perfect} if $E$ is pseudo-coherent\n(Cohomology, Definition \\ref{cohomology-definition-pseudo-coherent}) and\n$E$ locally has finite tor dimension as an object of\n$D(f^{-1}\\mathcal{O}_S)$\n(Cohomology, Definition \\ref{cohomology-definition-tor-amplitude}).","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DI0","source_file":"perfect.tex","source_line":8702,"source_end_line":8712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8702-L8712","statement_sha256":"152a7eb9aa2679d920bcd55d9f88416641c31ebd1ba6ef3d2a6c5ef7c77ace28","origin":"The Stacks Project","memory_eligible":false,"source_rank":7095,"rank":7095,"depth":1,"x":136.63,"y":763.524,"cluster":"derived-categories"},{"id":"stacks:0DI2","tag":"0DI2","title":"Relatively perfect objects · Lemma 0DI2","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. Let E be an object of D_QCoh(O_X). The following are equivalent • E is S-perfect, • for any affine open U ⊂ X mapping into an affine open V ⊂ S the complex RΓ(U, E) is O_S(V)-perfect. • there exists an affine open covering S = ⋃ V_i and for each i an affine open covering f^-1(V_i) = ⋃ U_ij such that the complex RΓ(U_ij, E) is O_S(V_i)-perfect.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation. Let $E$ be an object of\n$D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item $E$ is $S$-perfect,\n\\item for any affine open $U \\subset X$ mapping into an affine open\n$V \\subset S$ the complex $R\\Gamma(U, E)$ is $\\mathcal{O}_S(V)$-perfect.\n\\item there exists an affine open covering $S = \\bigcup V_i$\nand for each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$\nsuch that the complex $R\\Gamma(U_{ij}, E)$ is $\\mathcal{O}_S(V_i)$-perfect.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DI2","source_file":"perfect.tex","source_line":8733,"source_end_line":8746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8733-L8746","statement_sha256":"f0a6a12ed26a91c38c55394e1a02c11e0509b8855ad8abdaa3093723d8762f04","origin":"The Stacks Project","memory_eligible":false,"source_rank":7096,"rank":7096,"depth":27,"x":231.684,"y":565.294,"cluster":"derived-categories"},{"id":"stacks:0DI3","tag":"0DI3","title":"Relatively perfect objects · Lemma 0DI3","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. The full subcategory of D(O_X) consisting of S-perfect objects is a saturated triangulated subcategory.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is\nflat and locally of finite presentation. The full subcategory\nof $D(\\mathcal{O}_X)$ consisting of $S$-perfect objects is\na saturated\\footnote{Derived Categories, Definition\n\\ref{derived-definition-saturated}.} triangulated subcategory.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DI3","source_file":"perfect.tex","source_line":8772,"source_end_line":8779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8772-L8779","statement_sha256":"1cb739f5e65972f2a737765fc914665a5907c859336c120fdd8095429da7f613","origin":"The Stacks Project","memory_eligible":false,"source_rank":7097,"rank":7097,"depth":10,"x":321.11,"y":765.64,"cluster":"derived-categories"},{"id":"stacks:0DI4","tag":"0DI4","title":"Relatively perfect objects · Lemma 0DI4","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. A perfect object of D(O_X) is S-perfect. If K, M ∈ D(O_X), then K ⊗_O_X^L M is S-perfect if K is perfect and M is S-perfect.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and locally\nof finite presentation. A perfect object of $D(\\mathcal{O}_X)$ is $S$-perfect.\nIf $K, M \\in D(\\mathcal{O}_X)$, then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$\nis $S$-perfect if $K$ is perfect and $M$ is $S$-perfect.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DI4","source_file":"perfect.tex","source_line":8789,"source_end_line":8795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8789-L8795","statement_sha256":"c4315402efe674f3c47ff2727ea85aa0b54e0638d924abafd936a966689eb207","origin":"The Stacks Project","memory_eligible":false,"source_rank":7098,"rank":7098,"depth":28,"x":93.787,"y":668.535,"cluster":"derived-categories"},{"id":"stacks:0DI5","tag":"0DI5","title":"Relatively perfect objects · Lemma 0DI5","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. Let g : S' → S be a morphism of schemes. Set X' = S' ×_S X and denote g' : X' → X the projection. If K ∈ D(O_X) is S-perfect, then L(g')^*K is S'-perfect.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation.\nLet $g : S' \\to S$ be a morphism of schemes. Set $X' = S' \\times_S X$\nand denote $g' : X' \\to X$ the projection.\nIf $K \\in D(\\mathcal{O}_X)$ is $S$-perfect, then $L(g')^*K$\nis $S'$-perfect.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DI5","source_file":"perfect.tex","source_line":8804,"source_end_line":8812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8804-L8812","statement_sha256":"cb6f5dc84dff595e8e6da4fd5d7be22e9648b2a1706afbc7027e9e1d4ee570b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7099,"rank":7099,"depth":28,"x":339.791,"y":611.082,"cluster":"derived-categories"},{"id":"stacks:0DI7","tag":"0DI7","title":"Relatively perfect objects · Lemma 0DI7","summary":"In Situation [Tag 0DI6]. Let K_0 and L_0 be objects of D(O_X_0). Set K_i = Lf_i0^*K_0 and L_i = Lf_i0^*L_0 for i ≥ 0 and set K = Lf_0^*K_0 and L = Lf_0^*L_0. Then the map colim_i ≥ 0 Hom_D(O_X_i)(K_i, L_i) → Hom_D(O_X)(K, L) is an isomorphism if K_0 is pseudo-coherent and L_0 ∈ D_QCoh(O_X_0) has (locally) finite tor dimension as an object of D((X_0 → S_0)^-1O_S_0)","statement_latex":"In Situation \\ref{situation-relative-descent}.\nLet $K_0$ and $L_0$ be objects of $D(\\mathcal{O}_{X_0})$.\nSet $K_i = Lf_{i0}^*K_0$ and $L_i = Lf_{i0}^*L_0$ for $i \\geq 0$\nand set $K = Lf_0^*K_0$ and $L = Lf_0^*L_0$. Then the map\n$$\n\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{X_i})}(K_i, L_i)\n\\longrightarrow\n\\Hom_{D(\\mathcal{O}_X)}(K, L)\n$$\nis an isomorphism if $K_0$ is pseudo-coherent and\n$L_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$ has (locally)\nfinite tor dimension as an object of\n$D((X_0 \\to S_0)^{-1}\\mathcal{O}_{S_0})$","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DI7","source_file":"perfect.tex","source_line":8839,"source_end_line":8854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8839-L8854","statement_sha256":"8ff2044058e23b959e3a2d0cc0b04e0638bcdc15113c804dd22af1d0298a7804","origin":"The Stacks Project","memory_eligible":false,"source_rank":7100,"rank":7100,"depth":37,"x":204.434,"y":793.25,"cluster":"derived-categories"},{"id":"stacks:0DI8","tag":"0DI8","title":"Relatively perfect objects · Lemma 0DI8","summary":"In Situation [Tag 0DI6] the category of S-perfect objects of D(O_X) is the colimit of the categories of S_i-perfect objects of D(O_X_i).","statement_latex":"In Situation \\ref{situation-relative-descent} the category of\n$S$-perfect objects of $D(\\mathcal{O}_X)$ is the colimit of the categories\nof $S_i$-perfect objects of $D(\\mathcal{O}_{X_i})$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DI8","source_file":"perfect.tex","source_line":8901,"source_end_line":8906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8901-L8906","statement_sha256":"ed4e10d5dae94b46c594e15bca8db679585b3c0ce5dd8b021b36156751d2a0ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":7101,"rank":7101,"depth":38,"x":157.695,"y":581.869,"cluster":"derived-categories"},{"id":"stacks:0DJT","tag":"0DJT","title":"Relatively perfect objects · Lemma 0DJT","summary":"Let f : X → S be a morphism of schemes which is flat, proper, and of finite presentation. Let E ∈ D(O_X) be S-perfect. Then Rf_*E is a perfect object of D(O_S) and its formation commutes with arbitrary base change.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat, proper, and\nof finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be $S$-perfect.\nThen $Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$\nand its formation commutes with arbitrary base change.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJT","source_file":"perfect.tex","source_line":8970,"source_end_line":8976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L8970-L8976","statement_sha256":"1b8bf9ec9dec757f5f02f0943d4def0a518b827cc641be875c93a445f099bd61","origin":"The Stacks Project","memory_eligible":false,"source_rank":7102,"rank":7102,"depth":39,"x":362.386,"y":711.369,"cluster":"derived-categories"},{"id":"stacks:0DJU","tag":"0DJU","title":"Relatively perfect objects · Lemma 0DJU","summary":"Let f : X → S be a morphism of schemes. Let E, K ∈ D(O_X). Assume • S is quasi-compact and quasi-separated, • f is proper, flat, and of finite presentation, • E is S-perfect, • K is pseudo-coherent. Then there exists a pseudo-coherent L ∈ D(O_S) such that Rf_*RSheafHom(K, E) = RSheafHom(L, O_S) and the same is true after arbitrary base change: given vcenter xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S cartesian, then we have Rf'_*RSheafHom(L(g')^*K, L(g')^*E)…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $E, K \\in D(\\mathcal{O}_X)$.\nAssume\n\\begin{enumerate}\n\\item $S$ is quasi-compact and quasi-separated,\n\\item $f$ is proper, flat, and of finite presentation,\n\\item $E$ is $S$-perfect,\n\\item $K$ is pseudo-coherent.\n\\end{enumerate}\nThen there exists a pseudo-coherent $L \\in D(\\mathcal{O}_S)$ such that\n$$\nRf_*R\\SheafHom(K, E) = R\\SheafHom(L, \\mathcal{O}_S)\n$$\nand the same is true after arbitrary base change: given\n$$\n\\vcenter{\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nS' \\ar[r]^g &\nS\n}\n}\n\\quad\\quad\n\\begin{matrix}\n\\text{cartesian, then we have } \\\\\nRf'_*R\\SheafHom(L(g')^*K, L(g')^*E) \\\\\n= R\\SheafHom(Lg^*L, \\mathcal{O}_{S'})\n\\end{matrix}\n$$","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJU","source_file":"perfect.tex","source_line":9001,"source_end_line":9032,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9001-L9032","statement_sha256":"c05abd35153f9f8ecbdc56a15333a4970c5c5bfc18d29cdfbd69cf2f3b904365","origin":"The Stacks Project","memory_eligible":false,"source_rank":7103,"rank":7103,"depth":40,"x":107.01,"y":732.049,"cluster":"derived-categories"},{"id":"stacks:0GEH","tag":"0GEH","title":"Relatively perfect objects · Lemma 0GEH","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. Let E be a pseudo-coherent object of D(O_X). The following are equivalent • E is S-perfect, and • E is locally bounded below and for every point s ∈ S the object L(X_s → X)^*E of D(O_X_s) is locally bounded below.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation. Let $E$ be a pseudo-coherent\nobject of $D(\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item $E$ is $S$-perfect, and\n\\item $E$ is locally bounded below and for every point $s \\in S$\nthe object $L(X_s \\to X)^*E$ of $D(\\mathcal{O}_{X_s})$\nis locally bounded below.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEH","source_file":"perfect.tex","source_line":9133,"source_end_line":9144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9133-L9144","statement_sha256":"a87227878c9aff3e21dce0bef4428dfdb981a776ac7aff2e15eaf3c5769800d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7104,"rank":7104,"depth":38,"x":278.892,"y":571.707,"cluster":"derived-categories"},{"id":"stacks:0F86","tag":"0F86","title":"The resolution property · Definition 0F86","summary":"Let X be a scheme. We say X has the resolution property if every quasi-coherent O_X-module of finite type is the quotient of a finite locally free O_X-module.","statement_latex":"Let $X$ be a scheme. We say $X$ has the {\\it resolution property}\nif every quasi-coherent $\\mathcal{O}_X$-module of finite type\nis the quotient of a finite locally free $\\mathcal{O}_X$-module.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F86","source_file":"perfect.tex","source_line":9217,"source_end_line":9222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9217-L9222","statement_sha256":"5581d184804a2cbfe2e53004fdcd837b92a4420e9b68768bc33106af77c3c13c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7105,"rank":7105,"depth":0,"x":281.094,"y":787.721,"cluster":"derived-categories"},{"id":"stacks:0F87","tag":"0F87","title":"The resolution property · Lemma 0F87","summary":"Let X be a scheme. If X has an ample invertible O_X-module, then X has the resolution property.","statement_latex":"Let $X$ be a scheme. If $X$ has an ample invertible $\\mathcal{O}_X$-module,\nthen $X$ has the resolution property.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F87","source_file":"perfect.tex","source_line":9234,"source_end_line":9238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9234-L9238","statement_sha256":"08aeb21e30c50b87436c54cb8e9aebc5fe11ced263d4f7d96066247f2990753c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7106,"rank":7106,"depth":21,"x":105.553,"y":629.501,"cluster":"derived-categories"},{"id":"stacks:0FDD","tag":"0FDD","title":"The resolution property · Lemma 0FDD","summary":"Let f : X → Y be a morphism of schemes. Assume • Y is quasi-compact and quasi-separated and has the resolution property, • there exists an f-ample invertible module on X. Then X has the resolution property.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $Y$ is quasi-compact and quasi-separated and has the resolution property,\n\\item there exists an $f$-ample invertible module on $X$.\n\\end{enumerate}\nThen $X$ has the resolution property.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDD","source_file":"perfect.tex","source_line":9245,"source_end_line":9253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9245-L9253","statement_sha256":"ad9133ed5a157a6cf2eaf7fb3c8c1e5844d4bbf4bf9ac09be666876c9433f936","origin":"The Stacks Project","memory_eligible":false,"source_rank":7107,"rank":7107,"depth":21,"x":362.528,"y":646.584,"cluster":"derived-categories"},{"id":"stacks:0F88","tag":"0F88","title":"The resolution property · Lemma 0F88","summary":"Let f : X → Y be an affine or quasi-affine morphism of schemes with Y quasi-compact and quasi-separated. If Y has the resolution property, so does X.","statement_latex":"Let $f : X \\to Y$ be an affine or quasi-affine morphism of schemes with\n$Y$ quasi-compact and quasi-separated.\nIf $Y$ has the resolution property, so does $X$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F88","source_file":"perfect.tex","source_line":9289,"source_end_line":9294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9289-L9294","statement_sha256":"81ee59ad844de6171e63c97c67ea5e5b25900082c6556d914fb365af10d890ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":7108,"rank":7108,"depth":22,"x":159.066,"y":779.955,"cluster":"derived-categories"},{"id":"stacks:0GTC","tag":"0GTC","title":"The resolution property · Lemma 0GTC","summary":"Let f : X → Y be a surjective finite locally free morphism of schemes. If X has the resolution property, so does Y.","statement_latex":"Let $f : X \\to Y$ be a surjective finite locally free morphism of schemes.\nIf $X$ has the resolution property, so does $Y$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTC","source_file":"perfect.tex","source_line":9305,"source_end_line":9309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9305-L9309","statement_sha256":"620b0e2c18ebd992f43055055f21071c624af5deffa2405df4e1bd5d3270beac","origin":"The Stacks Project","memory_eligible":false,"source_rank":7109,"rank":7109,"depth":25,"x":201.892,"y":565.913,"cluster":"derived-categories"},{"id":"stacks:0F89","tag":"0F89","title":"The resolution property · Lemma 0F89","summary":"Let X be a scheme. Suppose given • a finite affine open covering X = U_1 ∪ … ∪ U_m • finite type quasi-coherent ideals I_j with V(I_j) = X setminus U_j Then X has the resolution property if and only if I_j is the quotient of a finite locally free O_X-module for j = 1, …, m.","statement_latex":"Let $X$ be a scheme. Suppose given\n\\begin{enumerate}\n\\item a finite affine open covering $X = U_1 \\cup \\ldots \\cup U_m$\n\\item finite type quasi-coherent ideals $\\mathcal{I}_j$\nwith $V(\\mathcal{I}_j) = X \\setminus U_j$\n\\end{enumerate}\nThen $X$ has the resolution property if and only if $\\mathcal{I}_j$\nis the quotient of a finite locally free $\\mathcal{O}_X$-module\nfor $j = 1, \\ldots, m$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F89","source_file":"perfect.tex","source_line":9349,"source_end_line":9360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9349-L9360","statement_sha256":"7e9f0309fce347bf8a91d176a3232d17a1c4a77940327d75fb4d021a83f1cd15","origin":"The Stacks Project","memory_eligible":false,"source_rank":7110,"rank":7110,"depth":36,"x":342.601,"y":748.255,"cluster":"derived-categories"},{"id":"stacks:0GMM","tag":"0GMM","title":"The resolution property · Lemma 0GMM","summary":"Let X be a scheme. If X has an ample family of invertible modules (Morphisms, Definition [Tag 0FXR]), then X has the resolution property.","statement_latex":"Let $X$ be a scheme. If $X$ has an ample family of invertible modules\n(Morphisms, Definition\n\\ref{morphisms-definition-family-ample-invertible-modules}),\nthen $X$ has the resolution property.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMM","source_file":"perfect.tex","source_line":9411,"source_end_line":9417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9411-L9417","statement_sha256":"f461bef4fd07f774a7dcdfbe7933a1c1963d9db7ce6ebc21b1da7201bc84f422","origin":"The Stacks Project","memory_eligible":false,"source_rank":7111,"rank":7111,"depth":37,"x":91.923,"y":693.578,"cluster":"derived-categories"},{"id":"stacks:0F8A","tag":"0F8A","title":"The resolution property · Lemma 0F8A","summary":"Let X be a quasi-compact, regular scheme with affine diagonal. Then X has the resolution property.","statement_latex":"Let $X$ be a quasi-compact, regular scheme with affine diagonal.\nThen $X$ has the resolution property.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8A","source_file":"perfect.tex","source_line":9432,"source_end_line":9436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9432-L9436","statement_sha256":"196cf0796ebacb0a37bafc8ffac24ff7f6e86b320fe64cbe0054d318a473dfbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7112,"rank":7112,"depth":38,"x":321.001,"y":591.538,"cluster":"derived-categories"},{"id":"stacks:0F8B","tag":"0F8B","title":"The resolution property · Lemma 0F8B","summary":"Let X = lim X_i be a limit of a direct system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then X has the resolution property if and only if X_i has the resolution properties for some i.","statement_latex":"Let $X = \\lim X_i$ be a limit of a direct system of quasi-compact\nand quasi-separated schemes with affine transition morphisms.\nThen $X$ has the resolution property if and only if $X_i$ has\nthe resolution properties for some $i$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8B","source_file":"perfect.tex","source_line":9443,"source_end_line":9449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9443-L9449","statement_sha256":"dd312b2aa87414668f392aa4f573a092cbcbd36c17debad9496027add3523cee","origin":"The Stacks Project","memory_eligible":false,"source_rank":7113,"rank":7113,"depth":37,"x":234.04,"y":797.0,"cluster":"derived-categories"},{"id":"stacks:0F8C","tag":"0F8C","title":"The resolution property · Lemma 0F8C","summary":"Special case of [totaro_resolution]. Let X be a quasi-compact and quasi-separated scheme with the resolution property. Then X has affine diagonal.","statement_latex":"\\begin{reference}\nSpecial case of \\cite[Proposition 1.3]{totaro_resolution}.\n\\end{reference}\nLet $X$ be a quasi-compact and quasi-separated scheme with\nthe resolution property. Then $X$ has affine diagonal.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8C","source_file":"perfect.tex","source_line":9487,"source_end_line":9494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9487-L9494","statement_sha256":"20c11a0f5c132ee84f240883fb01eaf1647a8431a8eda5273492559f7ed4892c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7114,"rank":7114,"depth":38,"x":132.822,"y":595.923,"cluster":"derived-categories"},{"id":"stacks:0F8E","tag":"0F8E","title":"The resolution property and perfect complexes · Lemma 0F8E","summary":"Let X be a quasi-compact and quasi-separated scheme with the resolution property. Let F^bullet be a bounded below complex of quasi-coherent O_X-modules representing a perfect object of D(O_X). Then there exists a bounded complex E^bullet of finite locally free O_X-modules and a quasi-isomorphism E^bullet → F^bullet.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme with the\nresolution property.\nLet $\\mathcal{F}^\\bullet$ be a bounded below complex of quasi-coherent\n$\\mathcal{O}_X$-modules representing a perfect object of\n$D(\\mathcal{O}_X)$. Then there exists a bounded complex\n$\\mathcal{E}^\\bullet$ of finite locally free $\\mathcal{O}_X$-modules\nand a quasi-isomorphism $\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8E","source_file":"perfect.tex","source_line":9572,"source_end_line":9581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9572-L9581","statement_sha256":"50de878ac27863da56d66a78618b89a046edc220225cafac5f5a5d2cff429064","origin":"The Stacks Project","memory_eligible":false,"source_rank":7115,"rank":7115,"depth":17,"x":369.429,"y":686.864,"cluster":"derived-categories"},{"id":"stacks:0F8F","tag":"0F8F","title":"The resolution property and perfect complexes · Lemma 0F8F","summary":"Let X be a quasi-compact and quasi-separated scheme with the resolution property. Then every perfect object of D(O_X) can be represented by a bounded complex of finite locally free O_X-modules.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme with the\nresolution property. Then every perfect object of $D(\\mathcal{O}_X)$\ncan be represented by a bounded complex of finite locally free\n$\\mathcal{O}_X$-modules.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8F","source_file":"perfect.tex","source_line":9672,"source_end_line":9678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9672-L9678","statement_sha256":"dcafdb6fa959c1a6f9b2828e01791cc005a36f56e51e634371368d84ec7bd559","origin":"The Stacks Project","memory_eligible":false,"source_rank":7116,"rank":7116,"depth":39,"x":121.545,"y":754.138,"cluster":"derived-categories"},{"id":"stacks:0F8G","tag":"0F8G","title":"The resolution property and perfect complexes · Lemma 0F8G","summary":"Let X be a quasi-compact and quasi-separated scheme with the resolution property. Let E^bullet and F^bullet be finite complexes of finite locally free O_X-modules. Then any α ∈ Hom_D(O_X)(E^bullet, F^bullet) can be represented by a diagram E^bullet ← G^bullet → F^bullet where G^bullet is a bounded complex of finite locally free O_X-modules and where G^bullet → E^bullet is a quasi-isomorphism.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme with the\nresolution property. Let $\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$\nbe finite complexes of finite locally free $\\mathcal{O}_X$-modules.\nThen any\n$\\alpha \\in \\Hom_{D(\\mathcal{O}_X)}(\\mathcal{E}^\\bullet, \\mathcal{F}^\\bullet)$\ncan be represented by a diagram\n$$\n\\mathcal{E}^\\bullet \\leftarrow \\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet\n$$\nwhere $\\mathcal{G}^\\bullet$ is a bounded complex of finite locally free\n$\\mathcal{O}_X$-modules and where $\\mathcal{G}^\\bullet \\to \\mathcal{E}^\\bullet$\nis a quasi-isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8G","source_file":"perfect.tex","source_line":9694,"source_end_line":9708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9694-L9708","statement_sha256":"0bab14b6b415922df8cd3c4aca94aafd3094f07d1377af0f8f531e63967581d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7117,"rank":7117,"depth":39,"x":250.377,"y":563.661,"cluster":"derived-categories"},{"id":"stacks:0F8H","tag":"0F8H","title":"The resolution property and perfect complexes · Lemma 0F8H","summary":"Let X be a quasi-compact and quasi-separated scheme with the resolution property. Let E^bullet and F^bullet be finite complexes of finite locally free O_X-modules. Let α^bullet, β^bullet :E^bullet → F^bullet be two maps of complexes defining the same map in D(O_X). Then there exists a quasi-isomorphism γ^bullet : G^bullet → E^bullet where G^bullet is a bounded complex of finite locally free O_X-modules such that α^bullet ∘ γ^bullet and β^bullet ∘ γ^bullet are homotopic…","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme with the\nresolution property. Let $\\mathcal{E}^\\bullet$ and $\\mathcal{F}^\\bullet$\nbe finite complexes of finite locally free $\\mathcal{O}_X$-modules.\nLet $\\alpha^\\bullet, \\beta^\\bullet :\\mathcal{E}^\\bullet \\to \\mathcal{F}^\\bullet$\nbe two maps of complexes defining the same map in $D(\\mathcal{O}_X)$.\nThen there exists a quasi-isomorphism\n$\\gamma^\\bullet : \\mathcal{G}^\\bullet \\to \\mathcal{E}^\\bullet$\nwhere $\\mathcal{G}^\\bullet$ is a bounded complex of finite locally free\n$\\mathcal{O}_X$-modules\nsuch that $\\alpha^\\bullet \\circ \\gamma^\\bullet$ and\n$\\beta^\\bullet \\circ \\gamma^\\bullet$ are homotopic maps of complexes.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8H","source_file":"perfect.tex","source_line":9734,"source_end_line":9747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9734-L9747","statement_sha256":"8c29514de7de0dbfcf86f3de41c50e8eed8f747433e2949ae9ce0d826b967965","origin":"The Stacks Project","memory_eligible":false,"source_rank":7118,"rank":7118,"depth":39,"x":308.62,"y":777.459,"cluster":"derived-categories"},{"id":"stacks:0F8I","tag":"0F8I","title":"The resolution property and perfect complexes · Proposition 0F8I","summary":"Let X be a quasi-compact and quasi-separated scheme with the resolution property. Denote • A the additive category of finite locally free O_X-modules, • K^b(A) the homotopy category of bounded complexes in A, see Derived Categories, Section [Tag 05RN], and • D_perf(O_X) the strictly full, saturated, triangulated subcategory of D(O_X) consisting of perfect objects. With this notation the obvious functor K^b(A) → D_perf(O_X) is an exact functor of trianglated categories…","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme with the\nresolution property. Denote\n\\begin{enumerate}\n\\item $\\mathcal{A}$ the additive category of finite locally free\n$\\mathcal{O}_X$-modules,\n\\item $K^b(\\mathcal{A})$ the homotopy category of bounded complexes\nin $\\mathcal{A}$, see\nDerived Categories, Section \\ref{derived-section-homotopy}, and\n\\item $D_{perf}(\\mathcal{O}_X)$ the strictly full, saturated,\ntriangulated subcategory of $D(\\mathcal{O}_X)$ consisting of\nperfect objects.\n\\end{enumerate}\nWith this notation the obvious functor\n$$\nK^b(\\mathcal{A}) \\longrightarrow D_{perf}(\\mathcal{O}_X)\n$$\nis an exact functor of trianglated categories which factors through an\nequivalence $S^{-1}K^b(\\mathcal{A}) \\to D_{perf}(\\mathcal{O}_X)$\nof triangulated categories\nwhere $S$ is the saturated multiplicative system of quasi-isomorphisms\nin $K^b(\\mathcal{A})$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"The resolution property and perfect complexes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8I","source_file":"perfect.tex","source_line":9772,"source_end_line":9795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9772-L9795","statement_sha256":"5d0576c0b80e5e1543f332170ff33645f713f0c95a5e0d3c5992e74e4504421e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7119,"rank":7119,"depth":40,"x":93.499,"y":652.715,"cluster":"derived-categories"},{"id":"stacks:0FDF","tag":"0FDF","title":"K-groups · Lemma 0FDF","summary":"Let X be a Noetherian scheme. Then K_0(Coh(O_X)) = K_0(D^b(Coh(O_X)) = K_0(D^b_Coh(O_X))","statement_latex":"Let $X$ be a Noetherian scheme. Then\n$$\nK_0(\\textit{Coh}(\\mathcal{O}_X)) =\nK_0(D^b(\\textit{Coh}(\\mathcal{O}_X)) =\nK_0(D^b_{\\textit{Coh}}(\\mathcal{O}_X))\n$$","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDF","source_file":"perfect.tex","source_line":9864,"source_end_line":9872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9864-L9872","statement_sha256":"a096dd0ee5196294c14f7c9227c68c9a64ae6af9132f040eb2f6674e93965b79","origin":"The Stacks Project","memory_eligible":false,"source_rank":7120,"rank":7120,"depth":34,"x":352.733,"y":622.601,"cluster":"derived-categories"},{"id":"stacks:0FDG","tag":"0FDG","title":"K-groups · Definition 0FDG","summary":"Let X be a scheme. • We denote K_0(X) the Grothendieck group of X. It is the zeroth K-group of the strictly full, saturated, triangulated subcategory D_perf(O_X) of D(O_X) consisting of perfect objects. In a formula K_0(X) = K_0(D_perf(O_X)) • If X is locally Noetherian, then we denote K'_0(X) the Grothendieck group of coherent sheaves on X. It is the is the zeroth K-group of the abelian category of coherent O_X-modules. In a formula K'_0(X) = K_0(Coh(O_X))","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item We denote $K_0(X)$ the {\\it Grothendieck group of $X$}. It is the\nzeroth K-group of the strictly full, saturated, triangulated subcategory\n$D_{perf}(\\mathcal{O}_X)$ of $D(\\mathcal{O}_X)$ consisting of perfect objects.\nIn a formula\n$$\nK_0(X) = K_0(D_{perf}(\\mathcal{O}_X))\n$$\n\\item If $X$ is locally Noetherian, then we denote $K'_0(X)$ the\n{\\it Grothendieck group of coherent sheaves on $X$}. It is the\nis the zeroth $K$-group of the abelian category\nof coherent $\\mathcal{O}_X$-modules. In a formula\n$$\nK'_0(X) = K_0(\\textit{Coh}(\\mathcal{O}_X))\n$$\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K-groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDG","source_file":"perfect.tex","source_line":9890,"source_end_line":9909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9890-L9909","statement_sha256":"32925c48164341f715fd5bd0878f5088634d01b3af20a436c799018bad253540","origin":"The Stacks Project","memory_eligible":false,"source_rank":7121,"rank":7121,"depth":0,"x":185.608,"y":792.092,"cluster":"derived-categories"},{"id":"stacks:0FDH","tag":"0FDH","title":"K-groups · Lemma 0FDH","summary":"Let X = Spec(R) be an affine scheme. Then K_0(X) = K_0(R) and if R is Noetherian then K'_0(X) = K'_0(R).","statement_latex":"Let $X = \\Spec(R)$ be an affine scheme. Then $K_0(X) = K_0(R)$\nand if $R$ is Noetherian then $K'_0(X) = K'_0(R)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDH","source_file":"perfect.tex","source_line":9917,"source_end_line":9921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9917-L9921","statement_sha256":"f4ba1891e70bdfe4654afae91a2aab8443fd7a6a52185da69e7bc8e06658d203","origin":"The Stacks Project","memory_eligible":false,"source_rank":7122,"rank":7122,"depth":29,"x":172.529,"y":572.035,"cluster":"derived-categories"},{"id":"stacks:0FDI","tag":"0FDI","title":"K-groups · Lemma 0FDI","summary":"Let X be a Noetherian regular scheme. Then the map K_0(X) → K'_0(X) is an isomorphism.","statement_latex":"Let $X$ be a Noetherian regular scheme. Then\nthe map $K_0(X) \\to K'_0(X)$ is an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDI","source_file":"perfect.tex","source_line":9960,"source_end_line":9964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L9960-L9964","statement_sha256":"b9dad0bbb20f29c7f86d2d21eaecb2c6d5ae9279d43e902d6c8abb5d5a75289b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7123,"rank":7123,"depth":30,"x":359.345,"y":727.053,"cluster":"derived-categories"},{"id":"stacks:0FDJ","tag":"0FDJ","title":"K-groups · Lemma 0FDJ","summary":"Let X be a quasi-compact and quasi-separated scheme with the resolution property. Then the map K_0(Vect(X)) → K_0(X) is an isomorphism.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme with the\nresolution property. Then the map $K_0(\\textit{Vect}(X)) \\to K_0(X)$\nis an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDJ","source_file":"perfect.tex","source_line":10011,"source_end_line":10016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10011-L10016","statement_sha256":"20320c88465c3db83f37b28d85d9a8f49b7ab65cceaaade51c3750c42cd40977","origin":"The Stacks Project","memory_eligible":false,"source_rank":7124,"rank":7124,"depth":40,"x":96.636,"y":718.74,"cluster":"derived-categories"},{"id":"stacks:0FDM","tag":"0FDM","title":"K-groups · Lemma 0FDM","summary":"Let f : X → Y be a proper morphism of locally Noetherian schemes. Then we have f_*(α · f^*β) = f_*α · β for α ∈ K'_0(X) and β ∈ K_0(Y).","statement_latex":"Let $f : X \\to Y$ be a proper morphism of locally Noetherian schemes.\nThen we have $f_*(\\alpha \\cdot f^*\\beta) = f_*\\alpha \\cdot \\beta$\nfor $\\alpha \\in K'_0(X)$ and $\\beta \\in K_0(Y)$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDM","source_file":"perfect.tex","source_line":10214,"source_end_line":10219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10214-L10219","statement_sha256":"361b289c2b7ed28255a34724c864601cda9ab7a9eab2ae3122641ca3fe5ff80b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7125,"rank":7125,"depth":32,"x":297.262,"y":575.644,"cluster":"derived-categories"},{"id":"stacks:0FJX","tag":"0FJX","title":"Determinants of complexes · Lemma 0FJX","summary":"Let X be a scheme. There is a functor det : ( category of perfect complexes with tor amplitude in [-1, 0] morphisms are isomorphisms ) → ( category of invertible modules morphisms are isomorphisms ) In addition, given a rank 0 perfect object L of D(O_X) with tor-amplitude in [-1, 0] there is a canonical element δ(L) ∈ Γ(X, det(L)) such that for any isomorphism a : L → K in D(O_X) we have det(a)(δ(L)) = δ(K). Moreover, the construction is affine locally given by the…","statement_latex":"Let $X$ be a scheme. There is a functor\n$$\n\\det :\n\\left\\{\n\\begin{matrix}\n\\text{category of perfect complexes} \\\\\n\\text{with tor amplitude in }[-1, 0] \\\\\n\\text{morphisms are isomorphisms}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{category of invertible modules} \\\\\n\\text{morphisms are isomorphisms}\n\\end{matrix}\n\\right\\}\n$$\nIn addition, given a rank $0$ perfect object $L$ of $D(\\mathcal{O}_X)$ with\ntor-amplitude in $[-1, 0]$ there is a canonical element\n$\\delta(L) \\in \\Gamma(X, \\det(L))$ such that for any isomorphism\n$a : L \\to K$ in $D(\\mathcal{O}_X)$ we have $\\det(a)(\\delta(L)) = \\delta(K)$.\nMoreover, the construction is affine locally given by the construction\nof More on Algebra, Section \\ref{more-algebra-section-determinants-complexes}.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Determinants of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJX","source_file":"perfect.tex","source_line":10290,"source_end_line":10316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10290-L10316","statement_sha256":"bfa5ff853f27521c9595b0a28acb05bca4125d75e4993c156f28e1881003f93b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7126,"rank":7126,"depth":29,"x":264.361,"y":795.244,"cluster":"derived-categories"},{"id":"stacks:0GEJ","tag":"0GEJ","title":"Detecting Boundedness · Lemma 0GEJ","summary":"In Situation [Tag 08CZ] denote j : U → X the open immersion and let K be the perfect object of D(O_X) corresponding to the Koszul complex on f_1, …, f_r over A. Let E ∈ D_QCoh(O_X) and a ∈ Z. Consider the following conditions • The canonical map τ_≥ aE → τ_≥ a Rj_*(E|_U) is an isomorphism. • We have Hom_D(O_X)(K[-n], E) = 0 for all n ≥ a. Then (2) implies (1) and (1) implies (2) with a replaced by a + 1.","statement_latex":"In Situation \\ref{situation-complex} denote $j : U \\to X$ the open\nimmersion and let $K$ be the perfect object of $D(\\mathcal{O}_X)$\ncorresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.\nLet $E \\in D_\\QCoh(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$.\nConsider the following conditions\n\\begin{enumerate}\n\\item The canonical map $\\tau_{\\geq a}E \\to \\tau_{\\geq a} Rj_*(E|_U)$\nis an isomorphism.\n\\item We have $\\Hom_{D(\\mathcal{O}_X)}(K[-n], E) = 0$ for all $n \\geq a$.\n\\end{enumerate}\nThen (2) implies (1) and (1) implies (2) with $a$ replaced by $a + 1$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Detecting Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEJ","source_file":"perfect.tex","source_line":10366,"source_end_line":10379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10366-L10379","statement_sha256":"260af50e8aea741fcad8a83f90ccff83ded8d18124005a25f4aed15423ade46c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7127,"rank":7127,"depth":29,"x":111.856,"y":614.445,"cluster":"derived-categories"},{"id":"stacks:0GEK","tag":"0GEK","title":"Detecting Boundedness · Lemma 0GEK","summary":"In Situation [Tag 08CZ] denote j : U → X the open immersion and let K be the perfect object of D(O_X) corresponding to the Koszul complex on f_1, …, f_r over A. Let E ∈ D_QCoh(O_X) and a ∈ Z. Consider the following conditions • The canonical map τ_≤ aE → τ_≤ a Rj_*(E|_U) is an isomorphism, and • Hom_D(O_X)(K[-n], E) = 0 for all n ≤ a. Then (2) implies (1) and (1) implies (2) with a replaced by a - 1.","statement_latex":"In Situation \\ref{situation-complex} denote $j : U \\to X$ the open\nimmersion and let $K$ be the perfect object of $D(\\mathcal{O}_X)$\ncorresponding to the Koszul complex on $f_1, \\ldots, f_r$ over $A$.\nLet $E \\in D_\\QCoh(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$. Consider\nthe following conditions\n\\begin{enumerate}\n\\item The canonical map $\\tau_{\\leq a}E \\to \\tau_{\\leq a} Rj_*(E|_U)$\nis an isomorphism, and\n\\item $\\Hom_{D(\\mathcal{O}_X)}(K[-n], E) = 0$ for all $n \\leq a$.\n\\end{enumerate}\nThen (2) implies (1) and (1) implies (2) with $a$ replaced by $a - 1$.","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Detecting Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEK","source_file":"perfect.tex","source_line":10411,"source_end_line":10424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10411-L10424","statement_sha256":"d4a64f65c0068fbe5713e725716c2a571108092f8bae79304b727fb7b8bef5a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7128,"rank":7128,"depth":29,"x":369.989,"y":661.281,"cluster":"derived-categories"},{"id":"stacks:0GEL","tag":"0GEL","title":"Detecting Boundedness · Lemma 0GEL","summary":"Let X be a quasi-compact and quasi-separated scheme. Let P ∈ D_perf(O_X) and E ∈ D_QCoh(O_X). Let a ∈ Z. The following are equivalent • Hom_D(O_X)(P[-i], E) = 0 for i gg 0, and • Hom_D(O_X)(P[-i], τ_≥ a E) = 0 for i gg 0.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$. \nLet $a \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\gg 0$, and\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\geq a} E) = 0$ for $i \\gg 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Detecting Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEL","source_file":"perfect.tex","source_line":10455,"source_end_line":10464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10455-L10464","statement_sha256":"24545b3a2e75d9171d1e48be205a100cc6696d815f286249883821723d8cf96f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7129,"rank":7129,"depth":29,"x":141.731,"y":773.339,"cluster":"derived-categories"},{"id":"stacks:0GEM","tag":"0GEM","title":"Detecting Boundedness · Lemma 0GEM","summary":"Let X be a quasi-compact and quasi-separated scheme. Let P ∈ D_perf(O_X) and E ∈ D_QCoh(O_X). Let a ∈ Z. The following are equivalent • Hom_D(O_X)(P[-i], E) = 0 for i ll 0, and • Hom_D(O_X)(P[-i], τ_≤ a E) = 0 for i ll 0.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let\n$P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$.\nLet $a \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\ll 0$, and\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\leq a} E) = 0$ for $i \\ll 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Detecting Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEM","source_file":"perfect.tex","source_line":10477,"source_end_line":10486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10477-L10486","statement_sha256":"e71dbce09711e64e9a0a7893f505657b57e9a3957a4d5c084a71df04064f9d02","origin":"The Stacks Project","memory_eligible":false,"source_rank":7130,"rank":7130,"depth":29,"x":220.017,"y":560.957,"cluster":"derived-categories"},{"id":"stacks:0GEN","tag":"0GEN","title":"Detecting Boundedness · Proposition 0GEN","summary":"Let X be a quasi-compact and quasi-separated scheme. Let G ∈ D_perf(O_X) be a perfect complex which generates D_QCoh (O_X). Let E ∈ D_QCoh (O_X). The following are equivalent • E ∈ D^-_QCoh (O_X), • Hom_D(O_X)(G[-i], E) = 0 for i gg 0, • Ext^i_X(G, E) = 0 for i gg 0, • RHom_X(G, E) is in D^-(Z), • H^i(X, G^vee ⊗_O_X^L E) = 0 for i gg 0, • RΓ(X, G^vee ⊗_O_X^L E) is in D^-(Z), • for every perfect object P of D(O_X) • the assertions (2), (3), (4) hold with G replaced by P,…","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let\n$G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates \n$D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $E \\in D^-_\\QCoh (\\mathcal{O}_X)$,\n\\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\gg 0$,\n\\item $\\Ext^i_X(G, E) = 0$ for $i \\gg 0$,\n\\item $R\\Hom_X(G, E)$ is in $D^-(\\mathbf{Z})$,\n\\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$\nfor $i \\gg 0$,\n\\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$\nis in $D^-(\\mathbf{Z})$,\n\\item for every perfect object $P$ of $D(\\mathcal{O}_X)$\n\\begin{enumerate}\n\\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and\n\\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\gg 0$,\n\\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$\nis in $D^-(\\mathbf{Z})$.\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Detecting Boundedness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEN","source_file":"perfect.tex","source_line":10499,"source_end_line":10522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10499-L10522","statement_sha256":"cb4fd57b78b6e9a0f28cc800f4a1d88f8a2ae202db6859a9eb946093c94dc106","origin":"The Stacks Project","memory_eligible":false,"source_rank":7131,"rank":7131,"depth":38,"x":333.205,"y":762.206,"cluster":"derived-categories"},{"id":"stacks:0GEQ","tag":"0GEQ","title":"Detecting Boundedness · Proposition 0GEQ","summary":"Let X be a quasi-compact and quasi-separated scheme. Let G ∈ D_perf(O_X) be a perfect complex which generates D_QCoh (O_X). Let E ∈ D_QCoh (O_X). The following are equivalent • E ∈ D^+_QCoh (O_X), • Hom_D(O_X)(G[-i], E) = 0 for i ll 0, • Ext^i_X(G, E) = 0 for i ll 0, • RHom_X(G, E) is in D^+(Z), • H^i(X, G^vee ⊗_O_X^L E) = 0 for i ll 0, • RΓ(X, G^vee ⊗_O_X^L E) is in D^+(Z), • for every perfect object P of D(O_X) • the assertions (2), (3), (4) hold with G replaced by P,…","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $G \\in D_{perf}(\\mathcal{O}_X)$\nbe a perfect complex which generates $D_\\QCoh (\\mathcal{O}_X)$. Let\n$E \\in D_\\QCoh (\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item $E \\in D^+_\\QCoh (\\mathcal{O}_X)$,\n\\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\ll 0$,\n\\item $\\Ext^i_X(G, E) = 0$ for $i \\ll 0$,\n\\item $R\\Hom_X(G, E)$ is in $D^+(\\mathbf{Z})$,\n\\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$\nfor $i \\ll 0$,\n\\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$\nis in $D^+(\\mathbf{Z})$,\n\\item for every perfect object $P$ of $D(\\mathcal{O}_X)$\n\\begin{enumerate}\n\\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and\n\\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\ll 0$,\n\\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$\nis in $D^+(\\mathbf{Z})$.\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Detecting Boundedness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEQ","source_file":"perfect.tex","source_line":10617,"source_end_line":10640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10617-L10640","statement_sha256":"dd4fda718de4ea91bbb56d346d6affe148869b0e185a5d9bb3d5b6b4de0fbbb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7132,"rank":7132,"depth":38,"x":87.635,"y":677.942,"cluster":"derived-categories"},{"id":"stacks:0GZZ","tag":"0GZZ","title":"Quasi-coherent objects in the derived category · Lemma 0GZZ","summary":"In the situation above there are canonical exact equivalences between the following triangulated categories • D_QCoh(O_X), • D_QCoh(X_Zar, O), • D_QCoh(X_affine, Zar, O), • D_QCoh(X_affine, O_X), and • mathitQC(X_affine, O).","statement_latex":"In the situation above there are canonical exact equivalences between\nthe following triangulated categories\n\\begin{enumerate}\n\\item $D_\\QCoh(\\mathcal{O}_X)$,\n\\item $D_\\QCoh(X_{Zar}, \\mathcal{O})$,\n\\item $D_\\QCoh(X_{affine, Zar}, \\mathcal{O})$,\n\\item $D_\\QCoh(X_{affine}, \\mathcal{O}_X)$, and\n\\item $\\mathit{QC}(X_{affine}, \\mathcal{O})$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Schemes","chapter_id":"perfect","section":"Quasi-coherent objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZZ","source_file":"perfect.tex","source_line":10784,"source_end_line":10795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/perfect.tex#L10784-L10795","statement_sha256":"496b6829d90c21bbf8965da27cf8a3ca08d4f5e8bea7eb806e903ec789eab9cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7133,"rank":7133,"depth":25,"x":336.749,"y":600.65,"cluster":"derived-categories"},{"id":"stacks:04EX","tag":"04EX","title":"Thickenings · Definition 04EX","summary":"Thickenings. • We say a scheme X' is a thickening of a scheme X if X is a closed subscheme of X' and the underlying topological spaces are equal. • We say a scheme X' is a first order thickening of a scheme X if X is a closed subscheme of X' and the quasi-coherent sheaf of ideals I ⊂ O_X' defining X has square zero. • We say a scheme X' is a finite order thickening of a scheme X if X is a closed subscheme of X' and the quasi-coherent sheaf of ideals I ⊂ O_X' defining X is…","statement_latex":"Thickenings.\n\\begin{enumerate}\n\\item We say a scheme $X'$ is a {\\it thickening} of a scheme $X$ if\n$X$ is a closed subscheme of $X'$ and the underlying topological spaces\nare equal.\n\\item We say a scheme $X'$ is a {\\it first order thickening} of a scheme $X$ if\n$X$ is a closed subscheme of $X'$ and the quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_{X'}$ defining $X$ has square zero.\n\\item We say a scheme $X'$ is a {\\it finite order thickening} of a scheme $X$\nif $X$ is a closed subscheme of $X'$ and the quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_{X'}$ defining $X$ is nilpotent, i.e.,\nthere exists an integer $n \\geq 0$ such that $\\mathcal{I}^{n + 1} = 0$.\n\\item We say a scheme $X'$ is an {\\it $n$th order thickening} of a scheme $X$\nif $X$ is a closed subscheme of $X$ and $\\mathcal{I}^{n + 1} = 0$\nwhere $\\mathcal{I} \\subset \\mathcal{O}_{X'}$ is the\nquasi-coherent sheaf of ideals defining $X$.\n\\item Given two thickenings $X \\subset X'$ and $Y \\subset Y'$ a\n{\\it morphism of thickenings} is a morphism $f' : X' \\to Y'$ such that\n$f'(X) \\subset Y$, i.e., such that $f'|_X$ factors through the closed\nsubscheme $Y$. In this situation we set $f = f'|_X : X \\to Y$ and we say\nthat $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ is a morphism of\nthickenings.\n\\item Let $S$ be a scheme. We similarly define {\\it thickenings over $S$}, and\n{\\it morphisms of thickenings over $S$}. This means that the schemes\n$X, X', Y, Y'$ above are schemes over $S$, and that the morphisms\n$X \\to X'$, $Y \\to Y'$ and $f' : X' \\to Y'$ are morphisms over $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Thickenings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EX","source_file":"more-morphisms.tex","source_line":35,"source_end_line":64,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L35-L64","statement_sha256":"e354bfdbccb21b9eecbb62d4857eb6e04ed573b6883ea1a3a21d80ae85871f7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7134,"rank":7134,"depth":0,"x":2101.576,"y":757.677,"cluster":"scheme-morphisms"},{"id":"stacks:05YV","tag":"05YV","title":"Thickenings · Lemma 05YV","summary":"Let X be a scheme over a base S. Consider a short exact sequence 0 → I → A → O_X → 0 of sheaves on X where A is a sheaf of f^-1O_S-algebras, A → O_X is a surjection of sheaves of f^-1O_S-algebras, and I is its kernel. If • I is an ideal of square zero in A, and • I is quasi-coherent as an O_X-module then X' = (X, A) is a scheme and X → X' is a first order thickening over S. Moreover, any first order thickening over S is of this form.","statement_latex":"Let $X$ be a scheme over a base $S$. Consider a short exact sequence\n$$\n0 \\to \\mathcal{I} \\to \\mathcal{A} \\to \\mathcal{O}_X \\to 0\n$$\nof sheaves on $X$ where $\\mathcal{A}$ is a sheaf of\n$f^{-1}\\mathcal{O}_S$-algebras,\n$\\mathcal{A} \\to \\mathcal{O}_X$ is a surjection\nof sheaves of $f^{-1}\\mathcal{O}_S$-algebras, and $\\mathcal{I}$ is its kernel.\nIf\n\\begin{enumerate}\n\\item $\\mathcal{I}$ is an ideal of square zero in $\\mathcal{A}$, and\n\\item $\\mathcal{I}$ is quasi-coherent as an $\\mathcal{O}_X$-module\n\\end{enumerate}\nthen $X' = (X, \\mathcal{A})$ is a scheme and $X \\to X'$ is a first\norder thickening over $S$. Moreover, any first order thickening over\n$S$ is of this form.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YV","source_file":"more-morphisms.tex","source_line":91,"source_end_line":109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L91-L109","statement_sha256":"af18b6c8b11099610e819a4087b202a45c7f526c2343fae4301d0bc8e5f20587","origin":"The Stacks Project","memory_eligible":false,"source_rank":7135,"rank":7135,"depth":24,"x":1914.664,"y":663.323,"cluster":"scheme-morphisms"},{"id":"stacks:06AD","tag":"06AD","title":"Thickenings · Lemma 06AD","summary":"Affineness is insensitive to thickenings The case of a finite order thickening is [EGA1]. Any thickening of an affine scheme is affine.","statement_latex":"\\begin{slogan}\nAffineness is insensitive to thickenings\n\\end{slogan}\n\\begin{reference}\nThe case of a finite order thickening is \\cite[Proposition 5.1.9]{EGA1}.\n\\end{reference}\nAny thickening of an affine scheme is affine.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06AD","source_file":"more-morphisms.tex","source_line":156,"source_end_line":165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L156-L165","statement_sha256":"b05dd1dafb07a946ceaebe06d0146220d5b9506ced5dd98c37aaa2bd19dae3d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7136,"rank":7136,"depth":31,"x":2128.536,"y":626.811,"cluster":"scheme-morphisms"},{"id":"stacks:09ZU","tag":"09ZU","title":"Thickenings · Lemma 09ZU","summary":"Let S ⊂ S' be a thickening of schemes. Let X' → S' be a morphism and set X = S ×_S' X'. Then (X ⊂ X') → (S ⊂ S') is a morphism of thickenings. If S ⊂ S' is a first (resp. finite order) thickening, then X ⊂ X' is a first (resp. finite order) thickening.","statement_latex":"Let $S \\subset S'$ be a thickening of schemes. Let $X' \\to S'$ be a morphism\nand set $X = S \\times_{S'} X'$. Then $(X \\subset X') \\to (S \\subset S')$\nis a morphism of thickenings. If $S \\subset S'$ is a first\n(resp.\\ finite order) thickening, then $X \\subset X'$ is a first\n(resp.\\ finite order) thickening.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZU","source_file":"more-morphisms.tex","source_line":208,"source_end_line":215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L208-L215","statement_sha256":"27ec1722152b13b429d6e7a3553c5091ec006bac7bd6d6c6a4c944f7270eeb6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7137,"rank":7137,"depth":0,"x":2000.09,"y":775.208,"cluster":"scheme-morphisms"},{"id":"stacks:0BPE","tag":"0BPE","title":"Thickenings · Lemma 0BPE","summary":"Compositions of thickenings are thickenings If S ⊂ S' and S' ⊂ S\" are thickenings, then so is S ⊂ S\".","statement_latex":"\\begin{slogan}\nCompositions of thickenings are thickenings\n\\end{slogan}\nIf $S \\subset S'$ and $S' \\subset S''$ are thickenings, then so is\n$S \\subset S''$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPE","source_file":"more-morphisms.tex","source_line":221,"source_end_line":228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L221-L228","statement_sha256":"0621b36490111dbc7941b129574be08faa9142c0bdf3fff6851eb0fe1f7f4b70","origin":"The Stacks Project","memory_eligible":false,"source_rank":7138,"rank":7138,"depth":0,"x":1975.448,"y":592.755,"cluster":"scheme-morphisms"},{"id":"stacks:0BPF","tag":"0BPF","title":"Thickenings · Lemma 0BPF","summary":"The property of being a thickening is fpqc local. Similarly for first order thickenings.","statement_latex":"The property of being a thickening is fpqc local.\nSimilarly for first order thickenings.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPF","source_file":"more-morphisms.tex","source_line":234,"source_end_line":238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L234-L238","statement_sha256":"8d5f04e731cbd3c9274dc55ccbb72a192c785df2f2ad9e1e970110c337b8a4c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7139,"rank":7139,"depth":43,"x":2140.472,"y":713.406,"cluster":"scheme-morphisms"},{"id":"stacks:09ZV","tag":"09ZV","title":"Morphisms of thickenings · Lemma 09ZV","summary":"Let (f, f') : (X ⊂ X') → (S ⊂ S') be a morphism of thickenings. Then • f is an affine morphism if and only if f' is an affine morphism, • f is a surjective morphism if and only if f' is a surjective morphism, • f is quasi-compact if and only if f' quasi-compact, • f is universally closed if and only if f' is universally closed, • f is integral if and only if f' is integral, • f is (quasi-)separated if and only if f' is (quasi-)separated, • f is universally injective if…","statement_latex":"Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism\nof thickenings. Then\n\\begin{enumerate}\n\\item $f$ is an affine morphism if and only if $f'$ is an affine morphism,\n\\item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism,\n\\item $f$ is quasi-compact if and only if $f'$ quasi-compact,\n\\item $f$ is universally closed if and only if $f'$ is universally closed,\n\\item $f$ is integral if and only if $f'$ is integral,\n\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,\n\\item $f$ is universally injective if and only if $f'$ is universally injective,\n\\item $f$ is universally open if and only if $f'$ is universally open,\n\\item $f$ is quasi-affine if and only if $f'$ is quasi-affine, and\n\\item add more here.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Morphisms of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZV","source_file":"more-morphisms.tex","source_line":269,"source_end_line":285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L269-L285","statement_sha256":"9b81e15cff7e4d7f51561d7c29ae1832183e973c2290cce3cc73e405b056b31d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7140,"rank":7140,"depth":32,"x":1921.59,"y":718.082,"cluster":"scheme-morphisms"},{"id":"stacks:0D2R","tag":"0D2R","title":"Morphisms of thickenings · Lemma 0D2R","summary":"Let (f, f') : (X ⊂ X') → (S ⊂ S') be a morphism of thickenings. Let L' be an invertible sheaf on X' and denote L the restriction to X. Then L' is f'-ample if and only if L is f-ample.","statement_latex":"Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism\nof thickenings. Let $\\mathcal{L}'$ be an invertible sheaf on $X'$\nand denote $\\mathcal{L}$ the restriction to $X$.\nThen $\\mathcal{L}'$ is $f'$-ample if and only if\n$\\mathcal{L}$ is $f$-ample.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Morphisms of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2R","source_file":"more-morphisms.tex","source_line":311,"source_end_line":318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L311-L318","statement_sha256":"e701af795009e46115f80aa9ac676b54e868ed4d84e5770ccc64d940a10d5da9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7141,"rank":7141,"depth":32,"x":2079.354,"y":590.334,"cluster":"scheme-morphisms"},{"id":"stacks:09ZW","tag":"09ZW","title":"Morphisms of thickenings · Lemma 09ZW","summary":"Let (f, f') : (X ⊂ X') → (S ⊂ S') be a morphism of thickenings such that X = S ×_S' X'. If S ⊂ S' is a finite order thickening, then • f is a closed immersion if and only if f' is a closed immersion, • f is locally of finite type if and only if f' is locally of finite type, • f is locally quasi-finite if and only if f' is locally quasi-finite, • f is locally of finite type of relative dimension d if and only if f' is locally of finite type of relative dimension d, • Ω_X/S…","statement_latex":"Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism\nof thickenings such that $X = S \\times_{S'} X'$. If $S \\subset S'$\nis a finite order thickening, then\n\\begin{enumerate}\n\\item $f$ is a closed immersion if and only if $f'$ is a closed immersion,\n\\item $f$ is locally of finite type if and only if $f'$ is\nlocally of finite type,\n\\item $f$ is locally quasi-finite if and only if $f'$ is locally\nquasi-finite,\n\\item $f$ is locally of finite type of relative dimension $d$ if and\nonly if $f'$ is locally of finite type of relative dimension $d$,\n\\item $\\Omega_{X/S} = 0$ if and only if $\\Omega_{X'/S'} = 0$,\n\\item $f$ is unramified if and only if $f'$ is unramified,\n\\item $f$ is proper if and only if $f'$ is proper,\n\\item $f$ is finite if and only if $f'$ is finite,\n\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,\n\\item $f$ is an immersion if and only if $f'$ is an immersion, and\n\\item add more here.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Morphisms of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZW","source_file":"more-morphisms.tex","source_line":334,"source_end_line":355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L334-L355","statement_sha256":"4226abdc1c849ef095677b653225f794e23643def7b97e105ed0e1cdb664fb8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7142,"rank":7142,"depth":33,"x":2065.742,"y":774.197,"cluster":"scheme-morphisms"},{"id":"stacks:0BPG","tag":"0BPG","title":"Morphisms of thickenings · Lemma 0BPG","summary":"Let (f, f') : (X ⊂ X') → (Y → Y') be a morphism of thickenings. Assume f and f' are locally of finite type and X = Y ×_Y' X'. Then • f is locally quasi-finite if and only if f' is locally quasi-finite, • f is finite if and only if f' is finite, • f is a closed immersion if and only if f' is a closed immersion, • Ω_X/Y = 0 if and only if Ω_X'/Y' = 0, • f is unramified if and only if f' is unramified, • f is a monomorphism if and only if f' is a monomorphism, • f is an…","statement_latex":"Let $(f, f') : (X \\subset X') \\to (Y \\to Y')$ be a morphism\nof thickenings. Assume $f$ and $f'$ are locally of finite type\nand $X = Y \\times_{Y'} X'$. Then\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,\n\\item $f$ is finite if and only if $f'$ is finite,\n\\item $f$ is a closed immersion if and only if $f'$ is a closed immersion,\n\\item $\\Omega_{X/Y} = 0$ if and only if $\\Omega_{X'/Y'} = 0$,\n\\item $f$ is unramified if and only if $f'$ is unramified,\n\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,\n\\item $f$ is an immersion if and only if $f'$ is an immersion,\n\\item $f$ is proper if and only if $f'$ is proper, and\n\\item add more here.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Morphisms of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPG","source_file":"more-morphisms.tex","source_line":483,"source_end_line":499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L483-L499","statement_sha256":"8b0cea183351537725307c5180eba606fc3b06b18a7b1199e89c500fffce418b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7143,"rank":7143,"depth":33,"x":1927.814,"y":630.783,"cluster":"scheme-morphisms"},{"id":"stacks:0C6R","tag":"0C6R","title":"Picard groups of thickenings · Lemma 0C6R","summary":"Let X ⊂ X' be a first order thickening with ideal sheaf I. Then there is a canonical exact sequence xymatrix 0 ar[r] & H^0(X, I) ar[r] & H^0(X', O_X'^*) ar[r] & H^0(X, O^*_X) ar `r[d] `d[l] `l[llld] `d[dll] [dll] & H^1(X, I) ar[r] & Pic(X') ar[r] & Pic(X) ar `r[d] `d[l] `l[llld] `d[dll] [dll] & H^2(X, I) ar[r] & … ar[r] & … of abelian groups.","statement_latex":"Let $X \\subset X'$ be a first order thickening\nwith ideal sheaf $\\mathcal{I}$. Then there is a canonical\nexact sequence\n$$\n\\xymatrix{\n0 \\ar[r] &\nH^0(X, \\mathcal{I}) \\ar[r] &\nH^0(X', \\mathcal{O}_{X'}^*) \\ar[r] &\nH^0(X, \\mathcal{O}^*_X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\\n& H^1(X, \\mathcal{I}) \\ar[r] &\n\\Pic(X') \\ar[r] &\n\\Pic(X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\\n& H^2(X, \\mathcal{I}) \\ar[r] & \\ldots \\ar[r] & \\ldots\n}\n$$\nof abelian groups.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Picard groups of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6R","source_file":"more-morphisms.tex","source_line":598,"source_end_line":616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L598-L616","statement_sha256":"df51b9bbb0326855f91e1da4fa7756a60c9e5e7546b9f1c61c54380d02eac806","origin":"The Stacks Project","memory_eligible":false,"source_rank":7144,"rank":7144,"depth":21,"x":2145.019,"y":658.292,"cluster":"scheme-morphisms"},{"id":"stacks:0C6S","tag":"0C6S","title":"Picard groups of thickenings · Lemma 0C6S","summary":"Let X ⊂ X' be a thickening. Let n be an integer invertible in O_X. Then the map Pic(X')[n] → Pic(X)[n] is bijective.","statement_latex":"Let $X \\subset X'$ be a thickening. Let $n$ be an integer\ninvertible in $\\mathcal{O}_X$. Then the map\n$\\Pic(X')[n] \\to \\Pic(X)[n]$ is bijective.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Picard groups of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6S","source_file":"more-morphisms.tex","source_line":632,"source_end_line":637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L632-L637","statement_sha256":"683e8be97b9f55728f6b05808765f889c94a9df3b1d17ff8b623a4ebfacdf21d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7145,"rank":7145,"depth":22,"x":1962.591,"y":761.335,"cluster":"scheme-morphisms"},{"id":"stacks:04EY","tag":"04EY","title":"Infinitesimal neighbourhoods · Definition 04EY","summary":"Let i : Z → X be an immersion of schemes. • The first order infinitesimal neighbourhood of Z in X is the first order thickening Z ⊂ Z_1 over X described above. • The nth order infinitesimal neighbourhood of Z in X is the nth order thickening Z ⊂ Z_n over X described above.","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes.\n\\begin{enumerate}\n\\item The {\\it first order infinitesimal neighbourhood} of $Z$ in $X$ is\nthe first order thickening $Z \\subset Z_1$ over $X$ described above.\n\\item The {\\it $n$th order infinitesimal neighbourhood} of $Z$ in $X$ is\nthe $n$th order thickening $Z \\subset Z_n$ over $X$ described above.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal neighbourhoods","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EY","source_file":"more-morphisms.tex","source_line":683,"source_end_line":692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L683-L692","statement_sha256":"6246c4b7e36f6ce35816bc4dd3966730296b1ebf3b7e76fcf8c34b101ce91d71","origin":"The Stacks Project","memory_eligible":false,"source_rank":7146,"rank":7146,"depth":0,"x":2014.287,"y":581.699,"cluster":"scheme-morphisms"},{"id":"stacks:04EZ","tag":"04EZ","title":"Infinitesimal neighbourhoods · Lemma 04EZ","summary":"Let i : Z → X be an immersion of schemes. • The first order infinitesimal neighbourhood Z' of Z in X has the following universal property: Given any commutative diagram xymatrix Z ar[d]_i & T ar[l]^a ar[d] X & T' ar[l]_b where T ⊂ T' is a first order thickening over X, there exists a unique morphism (a', a) : (T ⊂ T') → (Z ⊂ Z') of thickenings over X. • For n ≥ 1 the nth order infinitesimal neighbourhood Z_n of Z in X has the following universal property: Given any…","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes.\n\\begin{enumerate}\n\\item The first order infinitesimal neighbourhood $Z'$ of $Z$ in $X$\nhas the following universal property: Given any commutative diagram\n$$\n\\xymatrix{\nZ \\ar[d]_i & T \\ar[l]^a \\ar[d] \\\\\nX & T' \\ar[l]_b\n}\n$$\nwhere $T \\subset T'$ is a first order thickening over $X$, there exists\na unique morphism $(a', a) : (T \\subset T') \\to (Z \\subset Z')$ of\nthickenings over $X$.\n\\item For $n \\geq 1$ the $n$th order infinitesimal neighbourhood $Z_n$\nof $Z$ in $X$ has the following universal property: Given any commutative\ndiagram\n$$\n\\xymatrix{\nZ \\ar[d]_i & T \\ar[l]^a \\ar[d] \\\\\nX & T' \\ar[l]_b\n}\n$$\nwhere $T \\subset T'$ is an $n$th order thickening over $X$, there exists\na unique morphism $(a', a) : (T \\subset T') \\to (Z \\subset Z_n)$ of\nthickenings over $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal neighbourhoods","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04EZ","source_file":"more-morphisms.tex","source_line":699,"source_end_line":727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L699-L727","statement_sha256":"92deed784f46bce23c75e1682491f6abbe73275d452753cd5f613352c022292f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7147,"rank":7147,"depth":2,"x":2120.708,"y":743.619,"cluster":"scheme-morphisms"},{"id":"stacks:04F0","tag":"04F0","title":"Infinitesimal neighbourhoods · Lemma 04F0","summary":"Let i : Z → X be an immersion of schemes. Let Z ⊂ Z' be the first order infinitesimal neighbourhood of Z in X. Then the diagram xymatrix Z ar[r] ar[d] & Z' ar[d] Z ar[r] & X induces a map of conormal sheaves C_Z/X → C_Z/Z' by Morphisms, Lemma [Tag 01R4]. This map is an isomorphism.","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes. Let $Z \\subset Z'$ be\nthe first order infinitesimal neighbourhood of $Z$ in $X$.\nThen the diagram\n$$\n\\xymatrix{\nZ \\ar[r] \\ar[d] & Z' \\ar[d] \\\\\nZ \\ar[r] & X\n}\n$$\ninduces a map of conormal sheaves $\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Z'}$ by\nMorphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}.\nThis map is an isomorphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal neighbourhoods","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04F0","source_file":"more-morphisms.tex","source_line":745,"source_end_line":759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L745-L759","statement_sha256":"43b86dc5deb582469c6c51fc931c2f43602d541978b6b5c523a40caa3cdcbef4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7148,"rank":7148,"depth":9,"x":1911.861,"y":684.562,"cluster":"scheme-morphisms"},{"id":"stacks:02H8","tag":"02H8","title":"Formally unramified morphisms · Definition 02H8","summary":"Let f : X → S be a morphism of schemes. We say f is formally unramified if given any solid commutative diagram xymatrix X ar[d]_f & T ar[d]^i ar[l] S & T' ar[l] ar@-->[lu] where T ⊂ T' is a first order thickening of affine schemes over S there exists at most one dotted arrow making the diagram commute.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nWe say $f$ is {\\it formally unramified} if given any solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\\nS & T' \\ar[l] \\ar@{-->}[lu]\n}\n$$\nwhere $T \\subset T'$ is a first order thickening of affine schemes over $S$\nthere exists at most one dotted arrow making the diagram commute.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally unramified morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02H8","source_file":"more-morphisms.tex","source_line":797,"source_end_line":809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L797-L809","statement_sha256":"9cc4a367858726b4eec6431fa8c41acc33c201a65c4528aee604cfd9b1c3abbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7149,"rank":7149,"depth":0,"x":2113.51,"y":609.547,"cluster":"scheme-morphisms"},{"id":"stacks:04F1","tag":"04F1","title":"Formally unramified morphisms · Lemma 04F1","summary":"If f : X → S is a formally unramified morphism, then given any solid commutative diagram xymatrix X ar[d]_f & T ar[d]^i ar[l] S & T' ar[l] ar@-->[lu] where T ⊂ T' is a first order thickening of schemes over S there exists at most one dotted arrow making the diagram commute. In other words, in Definition [Tag 02H8] the condition that T be affine may be dropped.","statement_latex":"If $f : X \\to S$ is a formally unramified morphism, then given\nany solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\\nS & T' \\ar[l] \\ar@{-->}[lu]\n}\n$$\nwhere $T \\subset T'$ is a first order thickening of schemes over $S$\nthere exists at most one dotted arrow making the diagram commute.\nIn other words, in\nDefinition \\ref{definition-formally-unramified}\nthe condition that $T$ be affine may be dropped.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04F1","source_file":"more-morphisms.tex","source_line":815,"source_end_line":830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L815-L830","statement_sha256":"1f6a2447602e7b0fc5444b5786d765df83d4e72adaf0b2e7552fb0004d2dfd51","origin":"The Stacks Project","memory_eligible":false,"source_rank":7150,"rank":7150,"depth":1,"x":2025.074,"y":779.414,"cluster":"scheme-morphisms"},{"id":"stacks:02HA","tag":"02HA","title":"Formally unramified morphisms · Lemma 02HA","summary":"A composition of formally unramified morphisms is formally unramified.","statement_latex":"A composition of formally unramified morphisms is formally unramified.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HA","source_file":"more-morphisms.tex","source_line":837,"source_end_line":840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L837-L840","statement_sha256":"614aceced4d470e43169fec0aca479e1c7d6f8a43e123bf816fa05e7b0098151","origin":"The Stacks Project","memory_eligible":false,"source_rank":7151,"rank":7151,"depth":0,"x":1953.628,"y":603.84,"cluster":"scheme-morphisms"},{"id":"stacks:02HB","tag":"02HB","title":"Formally unramified morphisms · Lemma 02HB","summary":"A base change of a formally unramified morphism is formally unramified.","statement_latex":"A base change of a formally unramified morphism is formally unramified.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HB","source_file":"more-morphisms.tex","source_line":846,"source_end_line":849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L846-L849","statement_sha256":"3844a93bbe7fa5f67f213b19add0972e29d48b7a9c27c9d7b8740e139ce35282","origin":"The Stacks Project","memory_eligible":false,"source_rank":7152,"rank":7152,"depth":0,"x":2147.652,"y":692.834,"cluster":"scheme-morphisms"},{"id":"stacks:02HC","tag":"02HC","title":"Formally unramified morphisms · Lemma 02HC","summary":"Let f : X → S be a morphism of schemes. Let U ⊂ X and V ⊂ S be open such that f(U) ⊂ V. If f is formally unramified, so is f|_U : U → V.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $U \\subset X$ and $V \\subset S$ be open such that\n$f(U) \\subset V$. If $f$ is formally unramified, so is $f|_U : U \\to V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HC","source_file":"more-morphisms.tex","source_line":855,"source_end_line":860,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L855-L860","statement_sha256":"f65858a380c24e1c1e085a0f72a66dc63d1df67d58b3f777fca6d7edd0e49d67","origin":"The Stacks Project","memory_eligible":false,"source_rank":7153,"rank":7153,"depth":1,"x":1932.85,"y":737.339,"cluster":"scheme-morphisms"},{"id":"stacks:02HD","tag":"02HD","title":"Formally unramified morphisms · Lemma 02HD","summary":"Let f : X → S be a morphism of schemes. Assume X and S are affine. Then f is formally unramified if and only if O_S(S) → O_X(X) is a formally unramified ring map.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $X$ and $S$ are affine.\nThen $f$ is formally unramified if and only if\n$\\mathcal{O}_S(S) \\to \\mathcal{O}_X(X)$ is a formally unramified\nring map.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HD","source_file":"more-morphisms.tex","source_line":876,"source_end_line":883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L876-L883","statement_sha256":"d81050a529713b14c94d4e3dd36699d7e3f8e54edf51caa77a506f0b5e243f86","origin":"The Stacks Project","memory_eligible":false,"source_rank":7154,"rank":7154,"depth":12,"x":2055.546,"y":582.519,"cluster":"scheme-morphisms"},{"id":"stacks:02H9","tag":"02H9","title":"Formally unramified morphisms · Lemma 02H9","summary":"Let f : X → S be a morphism of schemes. Then f is formally unramified if and only if Ω_X/S = 0.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThen $f$ is formally unramified if and only if $\\Omega_{X/S} = 0$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02H9","source_file":"more-morphisms.tex","source_line":897,"source_end_line":901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L897-L901","statement_sha256":"ab797037d3be538b74a738e8be22440627b4e11f9826d6a9e9a8da708f89a14f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7155,"rank":7155,"depth":4,"x":2089.6,"y":766.443,"cluster":"scheme-morphisms"},{"id":"stacks:0HAL","tag":"0HAL","title":"Formally unramified morphisms · Lemma 0HAL","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent: • f is formally unramified, • for every x ∈ X there exist opens x ∈ U ⊂ X and f(x) ∈ V ⊂ Y with f(U) ⊂ V such that f|_U : U → V is formally unramified, • for every pair of affine opens U ⊂ X and V ⊂ Y with f(U) ⊂ V the ring map O_Y(V) → O_X(U) is formally unramified, and • there exists an affine open covering Y = ⋃ V_j and for each j an affine open covering f^-1(V_j) = ⋃ U_ji such that O_Y(V) → O_X(U)…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is formally unramified,\n\\item for every $x \\in X$ there exist opens $x \\in U \\subset X$ and\n$f(x) \\in V \\subset Y$ with $f(U) \\subset V$ such that\n$f|_U : U \\to V$ is formally unramified,\n\\item for every pair of affine opens $U \\subset X$ and $V \\subset Y$\nwith $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$\nis formally unramified, and\n\\item there exists an affine open covering $Y = \\bigcup V_j$ and\nfor each $j$ an affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$\nsuch that $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ is a formally unramified\nring map for all $j$ and $i$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAL","source_file":"more-morphisms.tex","source_line":943,"source_end_line":959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L943-L959","statement_sha256":"fc187b7de6733a41c6c82d040155b5d108da86e8ef7efef5704dd053c8a636b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7156,"rank":7156,"depth":13,"x":1916.45,"y":650.054,"cluster":"scheme-morphisms"},{"id":"stacks:02HE","tag":"02HE","title":"Formally unramified morphisms · Lemma 02HE","summary":"Unramified morphisms are the same as formally unramified morphism that are locally of finite type. Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is unramified (resp. G-unramified), and • the morphism f is locally of finite type (resp. locally of finite presentation) and formally unramified.","statement_latex":"\\begin{slogan}\nUnramified morphisms are the same as formally unramified morphism that\nare locally of finite type.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is unramified (resp.\\ G-unramified), and\n\\item the morphism $f$ is locally of finite type (resp.\\ locally of finite\npresentation) and formally unramified.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HE","source_file":"more-morphisms.tex","source_line":974,"source_end_line":987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L974-L987","statement_sha256":"84e0e625c775ab6f20bd3cc704bac52d877352ba8f41499ef0bfd15de3d7a0a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7157,"rank":7157,"depth":5,"x":2137.895,"y":637.618,"cluster":"scheme-morphisms"},{"id":"stacks:04F3","tag":"04F3","title":"Universal first order thickenings · Lemma 04F3","summary":"Let h : Z → X be a formally unramified morphism of schemes. There exists a universal first order thickening Z ⊂ Z' of Z over X.","statement_latex":"Let $h : Z \\to X$ be a formally unramified morphism of schemes.\nThere exists a universal first order thickening $Z \\subset Z'$ of\n$Z$ over $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04F3","source_file":"more-morphisms.tex","source_line":1024,"source_end_line":1029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1024-L1029","statement_sha256":"b83b646bd2e20b3831cefc01cf978aad780ff5db67905c4c8472d2ff7d71e240","origin":"The Stacks Project","memory_eligible":false,"source_rank":7158,"rank":7158,"depth":6,"x":1984.488,"y":772.544,"cluster":"scheme-morphisms"},{"id":"stacks:04F4","tag":"04F4","title":"Universal first order thickenings · Definition 04F4","summary":"Let h : Z → X be a formally unramified morphism of schemes. • The universal first order thickening of Z over X is the thickening Z ⊂ Z' constructed in Lemma [Tag 04F3]. • The conormal sheaf of Z over X is the conormal sheaf of Z in its universal first order thickening Z' over X. We often denote the conormal sheaf C_Z/X in this situation.","statement_latex":"Let $h : Z \\to X$ be a formally unramified morphism of schemes.\n\\begin{enumerate}\n\\item The {\\it universal first order thickening} of $Z$ over $X$\nis the thickening $Z \\subset Z'$ constructed in\nLemma \\ref{lemma-universal-thickening}.\n\\item The {\\it conormal sheaf of $Z$ over $X$} is the conormal sheaf\nof $Z$ in its universal first order thickening $Z'$ over $X$.\n\\end{enumerate}\nWe often denote the conormal sheaf $\\mathcal{C}_{Z/X}$ in this situation.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04F4","source_file":"more-morphisms.tex","source_line":1129,"source_end_line":1140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1129-L1140","statement_sha256":"10ea7123c0d46b6cb563cb212af9dd9a22d6bfc69bdbed2454e5273e60ec5539","origin":"The Stacks Project","memory_eligible":false,"source_rank":7159,"rank":7159,"depth":7,"x":1989.107,"y":585.863,"cluster":"scheme-morphisms"},{"id":"stacks:04F5","tag":"04F5","title":"Universal first order thickenings · Lemma 04F5","summary":"Let i : Z → X be an immersion of schemes. Then • i is formally unramified, • the universal first order thickening of Z over X is the first order infinitesimal neighbourhood of Z in X of Definition [Tag 04EY], and • the conormal sheaf of i in the sense of Morphisms, Definition [Tag 01R2] agrees with the conormal sheaf of i in the sense of Definition [Tag 04F4].","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes. Then\n\\begin{enumerate}\n\\item $i$ is formally unramified,\n\\item the universal first order thickening of $Z$ over $X$ is the first order\ninfinitesimal neighbourhood of $Z$ in $X$ of\nDefinition \\ref{definition-first-order-infinitesimal-neighbourhood}, and\n\\item the conormal sheaf of $i$ in the sense of\nMorphisms, Definition \\ref{morphisms-definition-conormal-sheaf}\nagrees with the conormal sheaf of $i$ in the sense of\nDefinition \\ref{definition-universal-thickening}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04F5","source_file":"more-morphisms.tex","source_line":1151,"source_end_line":1164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1151-L1164","statement_sha256":"7e7fcd6491a4703ce631fe3be5cf2f29a48bea5a42778d866437b43b601e87f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7160,"rank":7160,"depth":10,"x":2135.937,"y":726.247,"cluster":"scheme-morphisms"},{"id":"stacks:04F6","tag":"04F6","title":"Universal first order thickenings · Lemma 04F6","summary":"Let Z → X be a formally unramified morphism of schemes. Then the universal first order thickening Z' is formally unramified over X.","statement_latex":"Let $Z \\to X$ be a formally unramified morphism of schemes.\nThen the universal first order thickening $Z'$ is formally\nunramified over $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04F6","source_file":"more-morphisms.tex","source_line":1178,"source_end_line":1183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1178-L1183","statement_sha256":"228dd976c3f7e4047a78cb33509ac1e7c9a4fd9ac205f306ccc239462a4866cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7161,"rank":7161,"depth":7,"x":1914.606,"y":706.029,"cluster":"scheme-morphisms"},{"id":"stacks:04F7","tag":"04F7","title":"Universal first order thickenings · Lemma 04F7","summary":"Consider a commutative diagram of schemes xymatrix Z ar[r]_h ar[d]_f & X ar[d]^g W ar[r]^h' & Y with h and h' formally unramified. Let Z ⊂ Z' be the universal first order thickening of Z over X. Let W ⊂ W' be the universal first order thickening of W over Y. There exists a canonical morphism (f, f') : (Z, Z') → (W, W') of thickenings over Y which fits into the following commutative diagram xymatrix & & & Z' ar[ld] ar[d]^f' Z ar[rr] ar[d]_f ar[rrru] & & X ar[d] & W' ar[ld]…","statement_latex":"Consider a commutative diagram of schemes\n$$\n\\xymatrix{\nZ \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\\nW \\ar[r]^{h'} & Y\n}\n$$\nwith $h$ and $h'$ formally unramified. Let $Z \\subset Z'$ be the universal\nfirst order thickening of $Z$ over $X$. Let $W \\subset W'$ be the universal\nfirst order thickening of $W$ over $Y$. There exists a canonical morphism\n$(f, f') : (Z, Z') \\to (W, W')$ of thickenings over $Y$ which fits into\nthe following commutative diagram\n$$\n\\xymatrix{\n& & & Z' \\ar[ld] \\ar[d]^{f'} \\\\\nZ \\ar[rr] \\ar[d]_f \\ar[rrru] & & X \\ar[d] & W' \\ar[ld] \\\\\nW \\ar[rrru]|!{[rr];[rruu]}\\hole \\ar[rr] & & Y\n}\n$$\nIn particular the morphism $(f, f')$ of thickenings induces a morphism\nof conormal sheaves $f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04F7","source_file":"more-morphisms.tex","source_line":1219,"source_end_line":1242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1219-L1242","statement_sha256":"de1a7557934ebb2ea6ac84bf4ca8a7d2a3b854dcc5a85b04ffed3cae53988bd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7162,"rank":7162,"depth":9,"x":2094.207,"y":595.265,"cluster":"scheme-morphisms"},{"id":"stacks:04F8","tag":"04F8","title":"Universal first order thickenings · Lemma 04F8","summary":"Let xymatrix Z ar[r]_h ar[d]_f & X ar[d]^g W ar[r]^h' & Y be a fibre product diagram in the category of schemes with h' formally unramified. Then h is formally unramified and if W ⊂ W' is the universal first order thickening of W over Y, then Z = X ×_Y W ⊂ X ×_Y W' is the universal first order thickening of Z over X. In particular the canonical map f^*C_W/Y → C_Z/X of Lemma [Tag 04F7] is surjective.","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\\nW \\ar[r]^{h'} & Y\n}\n$$\nbe a fibre product diagram in the category of schemes with\n$h'$ formally unramified. Then $h$ is formally unramified and if\n$W \\subset W'$ is the universal first order thickening of $W$ over $Y$,\nthen $Z = X \\times_Y W \\subset X \\times_Y W'$ is the universal\nfirst order thickening of $Z$ over $X$. In particular the canonical map\n$f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ of\nLemma \\ref{lemma-universal-thickening-functorial}\nis surjective.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04F8","source_file":"more-morphisms.tex","source_line":1251,"source_end_line":1268,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1251-L1268","statement_sha256":"c601317c057eb4aebfeccb98377118d4d6fc8bb2652f948c21f6d430508a46d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7163,"rank":7163,"depth":10,"x":2050.812,"y":778.99,"cluster":"scheme-morphisms"},{"id":"stacks:04F9","tag":"04F9","title":"Universal first order thickenings · Lemma 04F9","summary":"Let xymatrix Z ar[r]_h ar[d]_f & X ar[d]^g W ar[r]^h' & Y be a fibre product diagram in the category of schemes with h' formally unramified and g flat. In this case the corresponding map Z' → W' of universal first order thickenings is flat, and f^*C_W/Y → C_Z/X is an isomorphism.","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\\nW \\ar[r]^{h'} & Y\n}\n$$\nbe a fibre product diagram in the category of schemes with\n$h'$ formally unramified and $g$ flat. In this case the corresponding\nmap $Z' \\to W'$ of universal first order thickenings is flat, and\n$f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ is an isomorphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04F9","source_file":"more-morphisms.tex","source_line":1281,"source_end_line":1294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1281-L1294","statement_sha256":"42a6d853ce341b9df05940201efa788c76f86aa06616083881216c704ef59792","origin":"The Stacks Project","memory_eligible":false,"source_rank":7164,"rank":7164,"depth":11,"x":1934.977,"y":618.769,"cluster":"scheme-morphisms"},{"id":"stacks:04FA","tag":"04FA","title":"Universal first order thickenings · Lemma 04FA","summary":"Taking the universal first order thickenings commutes with taking opens. More precisely, let h : Z → X be a formally unramified morphism of schemes. Let V ⊂ Z, U ⊂ X be opens such that h(V) ⊂ U. Let Z' be the universal first order thickening of Z over X. Then h|_V : V → U is formally unramified and the universal first order thickening of V over U is the open subscheme V' ⊂ Z' such that V = Z ∩ V'. In particular, C_Z/X|_V = C_V/U.","statement_latex":"Taking the universal first order thickenings commutes with taking opens.\nMore precisely, let $h : Z \\to X$ be a formally unramified morphism of schemes.\nLet $V \\subset Z$, $U \\subset X$ be opens such that $h(V) \\subset U$.\nLet $Z'$ be the universal first order thickening of $Z$ over $X$.\nThen $h|_V : V \\to U$ is formally unramified and the universal first\norder thickening of $V$ over $U$ is the open subscheme $V' \\subset Z'$\nsuch that $V = Z \\cap V'$. In particular,\n$\\mathcal{C}_{Z/X}|_V = \\mathcal{C}_{V/U}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FA","source_file":"more-morphisms.tex","source_line":1309,"source_end_line":1319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1309-L1319","statement_sha256":"b437421bee031a64d20c59b10092f2cb20641cd245a0b62b8752db604dfea358","origin":"The Stacks Project","memory_eligible":false,"source_rank":7165,"rank":7165,"depth":12,"x":2149.399,"y":671.225,"cluster":"scheme-morphisms"},{"id":"stacks:04FB","tag":"04FB","title":"Universal first order thickenings · Lemma 04FB","summary":"Let h : Z → X be a formally unramified morphism of schemes over S. Let Z ⊂ Z' be the universal first order thickening of Z over X with structure morphism h' : Z' → X. The canonical map c_h' : (h')^*Ω_X/S → Ω_Z'/S induces an isomorphism h^*Ω_X/S → Ω_Z'/S ⊗ O_Z.","statement_latex":"Let $h : Z \\to X$ be a formally unramified morphism of schemes over $S$.\nLet $Z \\subset Z'$ be the universal first order thickening of $Z$\nover $X$ with structure morphism $h' : Z' \\to X$. The canonical map\n$$\nc_{h'} : (h')^*\\Omega_{X/S} \\longrightarrow \\Omega_{Z'/S}\n$$\ninduces an isomorphism\n$h^*\\Omega_{X/S} \\to \\Omega_{Z'/S} \\otimes \\mathcal{O}_Z$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FB","source_file":"more-morphisms.tex","source_line":1332,"source_end_line":1342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1332-L1342","statement_sha256":"bfbaf67d0a5b654b6114d5c7869b1b1501af6175e9602cd3444c370d18b52827","origin":"The Stacks Project","memory_eligible":false,"source_rank":7166,"rank":7166,"depth":16,"x":1948.952,"y":754.276,"cluster":"scheme-morphisms"},{"id":"stacks:04FC","tag":"04FC","title":"Universal first order thickenings · Lemma 04FC","summary":"Let h : Z → X be a formally unramified morphism of schemes over S. There is a canonical exact sequence C_Z/X → h^*Ω_X/S → Ω_Z/S → 0. The first arrow is induced by d_Z'/S where Z' is the universal first order neighbourhood of Z over X.","statement_latex":"Let $h : Z \\to X$ be a formally unramified morphism of schemes over $S$.\nThere is a canonical exact sequence\n$$\n\\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0.\n$$\nThe first arrow is induced by $\\text{d}_{Z'/S}$ where\n$Z'$ is the universal first order neighbourhood of $Z$ over $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FC","source_file":"more-morphisms.tex","source_line":1359,"source_end_line":1368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1359-L1368","statement_sha256":"2b745f0f016efd93f94c149864bd7079c0fb10d99a68a2466fdfb7fd1673752a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7167,"rank":7167,"depth":18,"x":2030.033,"y":579.166,"cluster":"scheme-morphisms"},{"id":"stacks:067V","tag":"067V","title":"Universal first order thickenings · Lemma 067V","summary":"Let xymatrix Z ar[r]_i ar[rd]_j & X ar[d] & Y be a commutative diagram of schemes where i and j are formally unramified. Then there is a canonical exact sequence C_Z/Y → C_Z/X → i^*Ω_X/Y → 0 where the first arrow comes from Lemma [Tag 04F7] and the second from Lemma [Tag 04FC].","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\\n& Y\n}\n$$\nbe a commutative diagram of schemes where $i$ and $j$ are formally\nunramified. Then there is a canonical exact sequence\n$$\n\\mathcal{C}_{Z/Y} \\to\n\\mathcal{C}_{Z/X} \\to\ni^*\\Omega_{X/Y} \\to 0\n$$\nwhere the first arrow comes from\nLemma \\ref{lemma-universal-thickening-functorial}\nand the second from\nLemma \\ref{lemma-universally-unramified-differentials-sequence}.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067V","source_file":"more-morphisms.tex","source_line":1383,"source_end_line":1403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1383-L1403","statement_sha256":"394fe069a4fca376143578425a92c1d72498025079d59836133b56b8d591337a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7168,"rank":7168,"depth":19,"x":2111.125,"y":754.428,"cluster":"scheme-morphisms"},{"id":"stacks:06AE","tag":"06AE","title":"Universal first order thickenings · Lemma 06AE","summary":"Let Z → Y → X be formally unramified morphisms of schemes. • If Z ⊂ Z' is the universal first order thickening of Z over X and Y ⊂ Y' is the universal first order thickening of Y over X, then there is a morphism Z' → Y' and Y ×_Y' Z' is the universal first order thickening of Z over Y. • There is a canonical exact sequence i^*C_Y/X → C_Z/X → C_Z/Y → 0 where the maps come from Lemma [Tag 04F7] and i : Z → Y is the first morphism.","statement_latex":"Let $Z \\to Y \\to X$ be formally unramified morphisms of schemes.\n\\begin{enumerate}\n\\item If $Z \\subset Z'$ is the universal first order thickening of $Z$\nover $X$ and $Y \\subset Y'$ is the universal first order thickening of $Y$\nover $X$, then there is a morphism $Z' \\to Y'$ and $Y \\times_{Y'} Z'$ is\nthe universal first order thickening of $Z$ over $Y$.\n\\item There is a canonical exact sequence\n$$\ni^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nwhere the maps come from\nLemma \\ref{lemma-universal-thickening-functorial}\nand $i : Z \\to Y$ is the first morphism.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06AE","source_file":"more-morphisms.tex","source_line":1437,"source_end_line":1455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1437-L1455","statement_sha256":"49a2ed7ab3304dc70df012b3f7b83f5cc9953b2f45da7509dcfa1bd4784d9ef5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7169,"rank":7169,"depth":10,"x":1910.237,"y":671.144,"cluster":"scheme-morphisms"},{"id":"stacks:02HG","tag":"02HG","title":"Formally étale morphisms · Definition 02HG","summary":"Let f : X → S be a morphism of schemes. We say f is formally étale if given any solid commutative diagram xymatrix X ar[d]_f & T ar[d]^i ar[l] S & T' ar[l] ar@-->[lu] where T ⊂ T' is a first order thickening of affine schemes over S there exists exactly one dotted arrow making the diagram commute.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nWe say $f$ is {\\it formally \\'etale} if given any solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\\nS & T' \\ar[l] \\ar@{-->}[lu]\n}\n$$\nwhere $T \\subset T'$ is a first order thickening of affine schemes over $S$\nthere exists exactly one dotted arrow making the diagram commute.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HG","source_file":"more-morphisms.tex","source_line":1508,"source_end_line":1520,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1508-L1520","statement_sha256":"d9ef972b3678e62d11d32f59b2a8b5a9baf0157519eff8f13a7b830734b1ee36","origin":"The Stacks Project","memory_eligible":false,"source_rank":7170,"rank":7170,"depth":0,"x":2125.505,"y":618.528,"cluster":"scheme-morphisms"},{"id":"stacks:04FD","tag":"04FD","title":"Formally étale morphisms · Lemma 04FD","summary":"If f : X → S is a formally étale morphism, then given any solid commutative diagram xymatrix X ar[d]_f & T ar[d]^i ar[l] S & T' ar[l] ar@-->[lu] where T ⊂ T' is a first order thickening of schemes over S there exists exactly one dotted arrow making the diagram commute. In other words, in Definition [Tag 02HG] the condition that T be affine may be dropped.","statement_latex":"If $f : X \\to S$ is a formally \\'etale morphism, then given\nany solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\\nS & T' \\ar[l] \\ar@{-->}[lu]\n}\n$$\nwhere $T \\subset T'$ is a first order thickening of schemes over $S$\nthere exists exactly one dotted arrow making the diagram commute.\nIn other words, in\nDefinition \\ref{definition-formally-etale}\nthe condition that $T$ be affine may be dropped.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FD","source_file":"more-morphisms.tex","source_line":1527,"source_end_line":1542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1527-L1542","statement_sha256":"5087039095b74eb68eab8d0590f32e0b5e3418fdd63256ff9c86f2a77948ee3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7171,"rank":7171,"depth":2,"x":2008.994,"y":779.595,"cluster":"scheme-morphisms"},{"id":"stacks:02HI","tag":"02HI","title":"Formally étale morphisms · Lemma 02HI","summary":"A composition of formally étale morphisms is formally étale.","statement_latex":"A composition of formally \\'etale morphisms is formally \\'etale.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HI","source_file":"more-morphisms.tex","source_line":1555,"source_end_line":1558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1555-L1558","statement_sha256":"97e4898911c64421df2522e9a3bd38ce2fc5ff0024401b1b868c447141a76e96","origin":"The Stacks Project","memory_eligible":false,"source_rank":7172,"rank":7172,"depth":0,"x":1965.35,"y":594.577,"cluster":"scheme-morphisms"},{"id":"stacks:02HJ","tag":"02HJ","title":"Formally étale morphisms · Lemma 02HJ","summary":"A base change of a formally étale morphism is formally étale.","statement_latex":"A base change of a formally \\'etale morphism is formally \\'etale.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HJ","source_file":"more-morphisms.tex","source_line":1564,"source_end_line":1567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1564-L1567","statement_sha256":"120023d88b68c4f8199602d927a051b5242f3ceb43b1a701d42b2a641658a485","origin":"The Stacks Project","memory_eligible":false,"source_rank":7173,"rank":7173,"depth":0,"x":2146.453,"y":706.325,"cluster":"scheme-morphisms"},{"id":"stacks:02HK","tag":"02HK","title":"Formally étale morphisms · Lemma 02HK","summary":"Let f : X → S be a morphism of schemes. Let U ⊂ X and V ⊂ S be open subschemes such that f(U) ⊂ V. If f is formally étale, so is f|_U : U → V.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $U \\subset X$ and $V \\subset S$ be open subschemes such that\n$f(U) \\subset V$. If $f$ is formally \\'etale, so is $f|_U : U \\to V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HK","source_file":"more-morphisms.tex","source_line":1573,"source_end_line":1578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1573-L1578","statement_sha256":"9d7cf0f26844342257063b4b56f6e47cbc97be2956098da58955a9d100c51f2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7174,"rank":7174,"depth":1,"x":1922.88,"y":726.702,"cluster":"scheme-morphisms"},{"id":"stacks:04FE","tag":"04FE","title":"Formally étale morphisms · Lemma 04FE","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • f is formally étale, • f is formally unramified and the universal first order thickening of X over S is equal to X, • f is formally unramified and C_X/S = 0, and • Ω_X/S = 0 and C_X/S = 0.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is formally \\'etale,\n\\item $f$ is formally unramified and the universal first order thickening\nof $X$ over $S$ is equal to $X$,\n\\item $f$ is formally unramified and $\\mathcal{C}_{X/S} = 0$, and\n\\item $\\Omega_{X/S} = 0$ and $\\mathcal{C}_{X/S} = 0$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FE","source_file":"more-morphisms.tex","source_line":1594,"source_end_line":1605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1594-L1605","statement_sha256":"fbf3b494df6db87db47d5c125f1822d95f88044b9ee0d3d834a9892d4487e2ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":7175,"rank":7175,"depth":8,"x":2071.463,"y":584.709,"cluster":"scheme-morphisms"},{"id":"stacks:04FF","tag":"04FF","title":"Formally étale morphisms · Lemma 04FF","summary":"An unramified flat morphism is formally étale.","statement_latex":"An unramified flat morphism is formally \\'etale.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FF","source_file":"more-morphisms.tex","source_line":1625,"source_end_line":1628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1625-L1628","statement_sha256":"de9d7886ca617fb74c3cd68b36c06cdce8b43c4fbfa7b0419e26588d273d18ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":7176,"rank":7176,"depth":16,"x":2076.09,"y":773.863,"cluster":"scheme-morphisms"},{"id":"stacks:02HL","tag":"02HL","title":"Formally étale morphisms · Lemma 02HL","summary":"Let f : X → S be a morphism of schemes. Assume X and S are affine. Then f is formally étale if and only if O_S(S) → O_X(X) is a formally étale ring map.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $X$ and $S$ are affine.\nThen $f$ is formally \\'etale if and only if\n$\\mathcal{O}_S(S) \\to \\mathcal{O}_X(X)$ is a formally \\'etale\nring map.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HL","source_file":"more-morphisms.tex","source_line":1654,"source_end_line":1661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1654-L1661","statement_sha256":"ef37e548f89c0bef05323166bcd4523adc2a962a8022740e1eb8f59cd0ae35a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7177,"rank":7177,"depth":12,"x":1920.451,"y":636.908,"cluster":"scheme-morphisms"},{"id":"stacks:0HAM","tag":"0HAM","title":"Formally étale morphisms · Lemma 0HAM","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent: • f is formally étale, • for every x ∈ X there exist opens x ∈ U ⊂ X and f(x) ∈ V ⊂ Y with f(U) ⊂ V such that f|_U : U → V is formally étale, • for every pair of affine opens U ⊂ X and V ⊂ Y with f(U) ⊂ V the ring map O_Y(V) → O_X(U) is formally étale, and • there exists an affine open covering Y = ⋃ V_j and for each j an affine open covering f^-1(V_j) = ⋃ U_ji such that O_Y(V) → O_X(U) is a formally…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is formally \\'etale,\n\\item for every $x \\in X$ there exist opens $x \\in U \\subset X$ and\n$f(x) \\in V \\subset Y$ with $f(U) \\subset V$ such that\n$f|_U : U \\to V$ is formally \\'etale,\n\\item for every pair of affine opens $U \\subset X$ and $V \\subset Y$\nwith $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$\nis formally \\'etale, and\n\\item there exists an affine open covering $Y = \\bigcup V_j$ and\nfor each $j$ an affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$\nsuch that $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ is a formally \\'etale\nring map for all $j$ and $i$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAM","source_file":"more-morphisms.tex","source_line":1672,"source_end_line":1688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1672-L1688","statement_sha256":"86030934776e420415577e4ac3c20fd619bb1b74222c222feefb6a54e0328c5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7178,"rank":7178,"depth":13,"x":2145.519,"y":649.593,"cluster":"scheme-morphisms"},{"id":"stacks:02HM","tag":"02HM","title":"Formally étale morphisms · Lemma 02HM","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is étale, and • the morphism f is locally of finite presentation and formally étale.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is \\'etale, and\n\\item the morphism $f$ is locally of finite presentation and\nformally \\'etale.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HM","source_file":"more-morphisms.tex","source_line":1705,"source_end_line":1714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1705-L1714","statement_sha256":"0c86781cc6fce4552761162c56b90754e6617a8a851376650e6fb245bba0937b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7179,"rank":7179,"depth":39,"x":1969.226,"y":768.035,"cluster":"scheme-morphisms"},{"id":"stacks:04FG","tag":"04FG","title":"Infinitesimal deformations of maps · Lemma 04FG","summary":"Let S be a scheme. Let X ⊂ X' and Y ⊂ Y' be two first order thickenings over S. Let (a, a'), (b, b') : (X ⊂ X') → (Y ⊂ Y') be two morphisms of thickenings over S. Assume that • a = b, and • the two maps a^*C_Y/Y' → C_X/X' (Morphisms, Lemma [Tag 01R4]) are equal. Then the map (a')^sharp - (b')^sharp factors as O_Y' → O_Y xrightarrowD a_*C_X/X' → a_*O_X' where D is an O_S-derivation.","statement_latex":"Let $S$ be a scheme.\nLet $X \\subset X'$ and $Y \\subset Y'$ be two first order thickenings\nover $S$. Let $(a, a'), (b, b') : (X \\subset X') \\to (Y \\subset Y')$\nbe two morphisms of thickenings over $S$. Assume that\n\\begin{enumerate}\n\\item $a = b$, and\n\\item the two maps $a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$\n(Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial})\nare equal.\n\\end{enumerate}\nThen the map $(a')^\\sharp - (b')^\\sharp$ factors as\n$$\n\\mathcal{O}_{Y'} \\to \\mathcal{O}_Y \\xrightarrow{D}\na_*\\mathcal{C}_{X/X'} \\to a_*\\mathcal{O}_{X'}\n$$\nwhere $D$ is an $\\mathcal{O}_S$-derivation.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FG","source_file":"more-morphisms.tex","source_line":1764,"source_end_line":1782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1764-L1782","statement_sha256":"dbefbab9041bc71936a4d1177d7efdd2cfa8e65bbadeef0f82e018f8cbfaebcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7180,"rank":7180,"depth":9,"x":2003.999,"y":580.526,"cluster":"scheme-morphisms"},{"id":"stacks:02H5","tag":"02H5","title":"Infinitesimal deformations of maps · Lemma 02H5","summary":"Let S be a scheme. Let (a, a') : (X ⊂ X') → (Y ⊂ Y') be a morphism of first order thickenings over S. Let theta : a^*Ω_Y/S → C_X/X' be an O_X-linear map. Then there exists a unique morphism of pairs (b, b') : (X ⊂ X') → (Y ⊂ Y') such that (1) and (2) of Lemma [Tag 04FG] hold and the derivation D and theta are related by Equation ([Tag 04BV]).","statement_latex":"Let $S$ be a scheme.\nLet $(a, a') : (X \\subset X') \\to (Y \\subset Y')$\nbe a morphism of first order thickenings over $S$.\nLet\n$$\n\\theta : a^*\\Omega_{Y/S} \\to \\mathcal{C}_{X/X'}\n$$\nbe an $\\mathcal{O}_X$-linear map. Then there exists a unique morphism of pairs\n$(b, b') : (X \\subset X') \\to (Y \\subset Y')$ such that\n(1) and (2) of\nLemma \\ref{lemma-difference-derivation}\nhold and the derivation $D$ and $\\theta$ are related by\nEquation (\\ref{equation-D}).","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02H5","source_file":"more-morphisms.tex","source_line":1822,"source_end_line":1837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1822-L1837","statement_sha256":"d6f53b918cdbcc604abcefe8343f07170bc1062dd910dcde33070c0015e14f42","origin":"The Stacks Project","memory_eligible":false,"source_rank":7181,"rank":7181,"depth":10,"x":2129.239,"y":738.64,"cluster":"scheme-morphisms"},{"id":"stacks:04FH","tag":"04FH","title":"Infinitesimal deformations of maps · Lemma 04FH","summary":"Let S be a scheme. Let X ⊂ X' and Y ⊂ Y' be first order thickenings over S. Assume given a morphism a : X → Y and a map A : a^*C_Y/Y' → C_X/X' of O_X-modules. For an open subscheme U' ⊂ X' consider morphisms a' : U' → Y' such that • a' is a morphism over S, • a'|_U = a|_U, and • the induced map a^*C_Y/Y'|_U → C_X/X'|_U is the restriction of A to U. Here U = X ∩ U'. Then the rule U' ↦ (a' : U' → Y' such that (1), (2), (3) hold.) defines a sheaf of sets on X'.","statement_latex":"Let $S$ be a scheme.\nLet $X \\subset X'$ and $Y \\subset Y'$ be first order thickenings\nover $S$. Assume given a morphism $a : X \\to Y$ and a map\n$A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ of\n$\\mathcal{O}_X$-modules. For an open subscheme $U' \\subset X'$\nconsider morphisms $a' : U' \\to Y'$ such that\n\\begin{enumerate}\n\\item $a'$ is a morphism over $S$,\n\\item $a'|_U = a|_U$, and\n\\item the induced map\n$a^*\\mathcal{C}_{Y/Y'}|_U \\to \\mathcal{C}_{X/X'}|_U$\nis the restriction of $A$ to $U$.\n\\end{enumerate}\nHere $U = X \\cap U'$. Then the rule\n\\begin{equation}\n\nU' \\mapsto\n\\{a' : U' \\to Y'\\text{ such that (1), (2), (3) hold.}\\}\n\\end{equation}\ndefines a sheaf of sets on $X'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FH","source_file":"more-morphisms.tex","source_line":1860,"source_end_line":1882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1860-L1882","statement_sha256":"342984193fc3c6b637f5ec9d02aa7e3b6b171dad268529ef008fb844ae16aad6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7182,"rank":7182,"depth":0,"x":1909.577,"y":693.08,"cluster":"scheme-morphisms"},{"id":"stacks:04FJ","tag":"04FJ","title":"Infinitesimal deformations of maps · Lemma 04FJ","summary":"Same notation and assumptions as in Lemma [Tag 04FH]. There is an action of the sheaf SheafHom_O_X(a^*Ω_Y/S, C_X/X') on the sheaf ([Tag 04FI]). Moreover, the action is simply transitive for any open U' ⊂ X' over which the sheaf ([Tag 04FI]) has a section.","statement_latex":"Same notation and assumptions as in Lemma \\ref{lemma-sheaf}.\nThere is an action of the sheaf\n$$\n\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/S}, \\mathcal{C}_{X/X'})\n$$\non the sheaf (\\ref{equation-sheaf}). Moreover, the action\nis simply transitive for any open $U' \\subset X'$ over which the sheaf\n(\\ref{equation-sheaf}) has a section.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FJ","source_file":"more-morphisms.tex","source_line":1897,"source_end_line":1907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1897-L1907","statement_sha256":"82e74aa4b293534da201c2c97d389539017de90150f43250beeb033e8356fc13","origin":"The Stacks Project","memory_eligible":false,"source_rank":7183,"rank":7183,"depth":11,"x":2108.337,"y":601.967,"cluster":"scheme-morphisms"},{"id":"stacks:04FL","tag":"04FL","title":"Infinitesimal deformations of maps · Lemma 04FL","summary":"Let S be a scheme. Let X ⊂ X' be a first order thickening over S. Let Y be a scheme over S. Let a', b' : X' → Y be two morphisms over S with a = a'|_X = b'|_X. This gives rise to a commutative diagram xymatrix X ar[r] ar[d]_a & X' ar[d]^(b', a') Y ar[r]^-Δ_Y/S & Y ×_S Y Since the horizontal arrows are immersions with conormal sheaves C_X/X' and Ω_Y/S, by Morphisms, Lemma [Tag 01R4], we obtain a map theta : a^*Ω_Y/S → C_X/X'. Then this theta and the derivation D of Lemma…","statement_latex":"Let $S$ be a scheme. Let $X \\subset X'$ be a first order thickening over\n$S$. Let $Y$ be a scheme over $S$. Let\n$a', b' : X' \\to Y$ be two morphisms over $S$ with\n$a = a'|_X = b'|_X$. This gives rise to a commutative diagram\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d]_a & X' \\ar[d]^{(b', a')} \\\\\nY \\ar[r]^-{\\Delta_{Y/S}} & Y \\times_S Y\n}\n$$\nSince the horizontal arrows are immersions with conormal sheaves\n$\\mathcal{C}_{X/X'}$ and $\\Omega_{Y/S}$, by\nMorphisms, Lemma \\ref{morphisms-lemma-conormal-functorial},\nwe obtain a map $\\theta : a^*\\Omega_{Y/S} \\to \\mathcal{C}_{X/X'}$.\nThen this $\\theta$ and the derivation $D$ of\nLemma \\ref{lemma-difference-derivation}\nare related by Equation (\\ref{equation-D}).","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FL","source_file":"more-morphisms.tex","source_line":1969,"source_end_line":1988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L1969-L1988","statement_sha256":"e40cc19bb4c62bbdc381af9bac8a237202516367bccaf57235c84904c166be58","origin":"The Stacks Project","memory_eligible":false,"source_rank":7184,"rank":7184,"depth":10,"x":2034.992,"y":782.065,"cluster":"scheme-morphisms"},{"id":"stacks:04BX","tag":"04BX","title":"Infinitesimal deformations of maps · Lemma 04BX","summary":"Let xymatrix X_1 ar[d] & X_2 ar[l]^f ar[d] S_1 & S_2 ar[l] be a commutative diagram of schemes with X_2 → X_1 and S_2 → S_1 étale. Then the map c_f : f^*Ω_X_1/S_1 → Ω_X_2/S_2 of Morphisms, Lemma [Tag 01UV] is an isomorphism.","statement_latex":"Let\n$$\n\\xymatrix{\nX_1 \\ar[d] & X_2 \\ar[l]^f \\ar[d] \\\\\nS_1 & S_2 \\ar[l]\n}\n$$\nbe a commutative diagram of schemes with $X_2 \\to X_1$ and $S_2 \\to S_1$\n\\'etale. Then the map $c_f : f^*\\Omega_{X_1/S_1} \\to \\Omega_{X_2/S_2}$ of\nMorphisms, Lemma \\ref{morphisms-lemma-functoriality-differentials}\nis an isomorphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BX","source_file":"more-morphisms.tex","source_line":2005,"source_end_line":2018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L2005-L2018","statement_sha256":"4a04b9fc3881a582a19413f70a4f0e55736d3f5cb743170f1e1d84c05dd312f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7185,"rank":7185,"depth":18,"x":1944.178,"y":607.518,"cluster":"scheme-morphisms"},{"id":"stacks:04BY","tag":"04BY","title":"Infinitesimal deformations of maps · Lemma 04BY","summary":"Consider a commutative diagram of first order thickenings vcenter xymatrix (T_2 ⊂ T_2') ar[d]_(h, h') ar[rr]_(a_2, a_2') & & (X_2 ⊂ X_2') ar[d]^(f, f') (T_1 ⊂ T_1') ar[rr]^(a_1, a_1') & & (X_1 ⊂ X_1') and a commutative diagram of schemes vcenter xymatrix X_2' ar[r] ar[d] & S_2 ar[d] X_1' ar[r] & S_1 with X_2 → X_1 and S_2 → S_1 étale. For any O_T_1-linear map theta_1 : a_1^*Ω_X_1/S_1 → C_T_1/T'_1 let theta_2 be the composition xymatrix a_2^*Ω_X_2/S_2 ar@=[r] &…","statement_latex":"Consider a commutative diagram of first order thickenings\n$$\n\\vcenter{\n\\xymatrix{\n(T_2 \\subset T_2') \\ar[d]_{(h, h')} \\ar[rr]_{(a_2, a_2')} & &\n(X_2 \\subset X_2') \\ar[d]^{(f, f')} \\\\\n(T_1 \\subset T_1') \\ar[rr]^{(a_1, a_1')} & &\n(X_1 \\subset X_1')\n}\n}\n\\quad\n\\begin{matrix}\n\\text{and a commutative} \\\\\n\\text{diagram of schemes}\n\\end{matrix}\n\\quad\n\\vcenter{\n\\xymatrix{\nX_2' \\ar[r] \\ar[d] & S_2 \\ar[d] \\\\\nX_1' \\ar[r] & S_1\n}\n}\n$$\nwith $X_2 \\to X_1$ and $S_2 \\to S_1$ \\'etale.\nFor any $\\mathcal{O}_{T_1}$-linear map\n$\\theta_1 : a_1^*\\Omega_{X_1/S_1} \\to \\mathcal{C}_{T_1/T'_1}$ let\n$\\theta_2$ be the composition\n$$\n\\xymatrix{\na_2^*\\Omega_{X_2/S_2} \\ar@{=}[r] &\nh^*a_1^*\\Omega_{X_1/S_1} \\ar[r]^-{h^*\\theta_1} &\nh^*\\mathcal{C}_{T_1/T'_1} \\ar[r] &\n\\mathcal{C}_{T_2/T'_2}\n}\n$$\n(equality sign is explained in the proof). Then the diagram\n$$\n\\xymatrix{\nT_2' \\ar[rr]_{\\theta_2 \\cdot a_2'} \\ar[d] & & X'_2 \\ar[d] \\\\\nT_1' \\ar[rr]^{\\theta_1 \\cdot a_1'} & & X'_1\n}\n$$\ncommutes where the actions $\\theta_2 \\cdot a_2'$ and $\\theta_1 \\cdot a_1'$\nare as in Remark \\ref{remark-action-by-derivations}.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04BY","source_file":"more-morphisms.tex","source_line":2031,"source_end_line":2077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L2031-L2077","statement_sha256":"0d3e2505e48398b33e3c5732299990760adc2065ce3418b4c2a688761fd47512","origin":"The Stacks Project","memory_eligible":false,"source_rank":7186,"rank":7186,"depth":19,"x":2151.66,"y":684.753,"cluster":"scheme-morphisms"},{"id":"stacks:063Y","tag":"063Y","title":"Infinitesimal deformations of schemes · Lemma 063Y","summary":"Let (f, f') : (X ⊂ X') → (S ⊂ S') be a morphism of first order thickenings. Assume that f is flat. Then the following are equivalent • f' is flat and X = S ×_S' X', and • the canonical map f^*C_S/S' → C_X/X' is an isomorphism.","statement_latex":"Let $(f, f') : (X \\subset X') \\to (S \\subset S')$ be a morphism\nof first order thickenings. Assume that $f$ is flat.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $f'$ is flat and $X = S \\times_{S'} X'$, and\n\\item the canonical map $f^*\\mathcal{C}_{S/S'} \\to \\mathcal{C}_{X/X'}$\nis an isomorphism.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/063Y","source_file":"more-morphisms.tex","source_line":2181,"source_end_line":2191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L2181-L2191","statement_sha256":"411023fc7ec36753eda72f1c3a043ab7f80cffcf762b8bd39ba1c9819d83f1ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":7187,"rank":7187,"depth":4,"x":1936.399,"y":745.576,"cluster":"scheme-morphisms"},{"id":"stacks:06AF","tag":"06AF","title":"Infinitesimal deformations of schemes · Lemma 06AF","summary":"Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (S ⊂ S') of thickenings. Assume • X' is flat over S', • f is flat, • S ⊂ S' is a finite order thickening, and • X = S ×_S' X' and Y = S ×_S' Y'. Then f' is flat and Y' is flat over S' at all points in the image of f'.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\\n& (S \\subset S')\n}\n$$\nof thickenings. Assume\n\\begin{enumerate}\n\\item $X'$ is flat over $S'$,\n\\item $f$ is flat,\n\\item $S \\subset S'$ is a finite order thickening, and\n\\item $X = S \\times_{S'} X'$ and $Y = S \\times_{S'} Y'$.\n\\end{enumerate}\nThen $f'$ is flat and $Y'$ is flat over $S'$ at all points in\nthe image of $f'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06AF","source_file":"more-morphisms.tex","source_line":2237,"source_end_line":2255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L2237-L2255","statement_sha256":"c776a07db482e3ecff9f70402bec8cdf5c88ab1a29df502ebde04cdeab85b022","origin":"The Stacks Project","memory_eligible":false,"source_rank":7188,"rank":7188,"depth":3,"x":2046.298,"y":578.46,"cluster":"scheme-morphisms"},{"id":"stacks:06AG","tag":"06AG","title":"Infinitesimal deformations of schemes · Lemma 06AG","summary":"Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (S ⊂ S') of thickenings. Assume S ⊂ S' is a finite order thickening, X' flat over S', X = S ×_S' X', and Y = S ×_S' Y'. Then • f is flat if and only if f' is flat, • f is an isomorphism if and only if f' is an isomorphism, • f is an open immersion if and only if f' is an open immersion, • f is quasi-compact if and only if f' is quasi-compact, • f is universally closed if and only…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\\n& (S \\subset S')\n}\n$$\nof thickenings. Assume $S \\subset S'$ is a finite order thickening,\n$X'$ flat over $S'$, $X = S \\times_{S'} X'$, and\n$Y = S \\times_{S'} Y'$. Then\n\\begin{enumerate}\n\\item $f$ is flat if and only if $f'$ is flat,\n\n\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,\n\n\\item $f$ is an open immersion if and only if $f'$ is an open immersion,\n\n\\item $f$ is quasi-compact if and only if $f'$ is quasi-compact,\n\n\\item $f$ is universally closed if and only if $f'$ is universally closed,\n\n\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,\n\n\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,\n\n\\item $f$ is surjective if and only if $f'$ is surjective,\n\n\\item $f$ is universally injective if and only if $f'$ is universally injective,\n\n\\item $f$ is affine if and only if $f'$ is affine,\n\n\\item\n\n$f$ is locally of finite type if and only if $f'$ is locally of finite type,\n\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,\n\n\\item\n\n$f$ is locally of finite presentation if and only if $f'$ is locally of\nfinite presentation,\n\\item\n\n$f$ is locally of finite type of relative dimension $d$ if and only if\n$f'$ is locally of finite type of relative dimension $d$,\n\\item $f$ is universally open if and only if $f'$ is universally open,\n\n\\item $f$ is syntomic if and only if $f'$ is syntomic,\n\n\\item $f$ is smooth if and only if $f'$ is smooth,\n\n\\item $f$ is unramified if and only if $f'$ is unramified,\n\n\\item $f$ is \\'etale if and only if $f'$ is \\'etale,\n\n\\item $f$ is proper if and only if $f'$ is proper,\n\n\\item $f$ is integral if and only if $f'$ is integral,\n\n\\item $f$ is finite if and only if $f'$ is finite,\n\n\\item\n\n$f$ is finite locally free (of rank $d$) if and only if $f'$\nis finite locally free (of rank $d$), and\n\\item add more here.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06AG","source_file":"more-morphisms.tex","source_line":2266,"source_end_line":2334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L2266-L2334","statement_sha256":"a4b784e2b372e2e3b77a19e2ca7340856f6240a13a379b8182ad75887fa434eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7189,"rank":7189,"depth":46,"x":2099.689,"y":764.183,"cluster":"scheme-morphisms"},{"id":"stacks:0CF3","tag":"0CF3","title":"Infinitesimal deformations of schemes · Lemma 0CF3","summary":"Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (S ⊂ S') of thickenings. Assume • Y' → S' is locally of finite type, • X' → S' is flat and locally of finite presentation, • f is flat, and • X = S ×_S' X' and Y = S ×_S' Y'. Then f' is flat and for all y' ∈ Y' in the image of f' the local ring O_Y', y' is flat and essentially of finite presentation over O_S', s'.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\\n& (S \\subset S')\n}\n$$\nof thickenings. Assume\n\\begin{enumerate}\n\\item $Y' \\to S'$ is locally of finite type,\n\\item $X' \\to S'$ is flat and locally of finite presentation,\n\\item $f$ is flat, and\n\\item $X = S \\times_{S'} X'$ and $Y = S \\times_{S'} Y'$.\n\\end{enumerate}\nThen $f'$ is flat and for all $y' \\in Y'$ in the image of $f'$\nthe local ring $\\mathcal{O}_{Y', y'}$ is\nflat and essentially of finite presentation over $\\mathcal{O}_{S', s'}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CF3","source_file":"more-morphisms.tex","source_line":2567,"source_end_line":2586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L2567-L2586","statement_sha256":"85d02d1f0336129a0d8f1e9b5bb5f76e469dbfd09f57af05746bf171837aaafd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7190,"rank":7190,"depth":14,"x":1910.828,"y":657.451,"cluster":"scheme-morphisms"},{"id":"stacks:0CF4","tag":"0CF4","title":"Infinitesimal deformations of schemes · Lemma 0CF4","summary":"Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (S ⊂ S') of thickenings. Assume Y' → S' locally of finite type, X' → S' flat and locally of finite presentation, X = S ×_S' X', and Y = S ×_S' Y'. Then • f is flat if and only if f' is flat, • f is an isomorphism if and only if f' is an isomorphism, • f is an open immersion if and only if f' is an open immersion, • f is quasi-compact if and only if f' is quasi-compact, • f is…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\\n& (S \\subset S')\n}\n$$\nof thickenings. Assume $Y' \\to S'$ locally of finite type,\n$X' \\to S'$ flat and locally of finite presentation,\n$X = S \\times_{S'} X'$, and $Y = S \\times_{S'} Y'$. Then\n\\begin{enumerate}\n\\item $f$ is flat if and only if $f'$ is flat,\n\n\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,\n\n\\item $f$ is an open immersion if and only if $f'$ is an open immersion,\n\n\\item $f$ is quasi-compact if and only if $f'$ is quasi-compact,\n\n\\item $f$ is universally closed if and only if $f'$ is universally closed,\n\n\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,\n\n\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,\n\n\\item $f$ is surjective if and only if $f'$ is surjective,\n\n\\item $f$ is universally injective if and only if $f'$ is universally injective,\n\n\\item $f$ is affine if and only if $f'$ is affine,\n\n\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,\n\n\\item\n\n$f$ is locally of finite type of relative dimension $d$ if and only if\n$f'$ is locally of finite type of relative dimension $d$,\n\\item $f$ is universally open if and only if $f'$ is universally open,\n\n\\item $f$ is syntomic if and only if $f'$ is syntomic,\n\n\\item $f$ is smooth if and only if $f'$ is smooth,\n\n\\item $f$ is unramified if and only if $f'$ is unramified,\n\n\\item $f$ is \\'etale if and only if $f'$ is \\'etale,\n\n\\item $f$ is proper if and only if $f'$ is proper,\n\n\\item $f$ is finite if and only if $f'$ is finite,\n\n\\item\n\n$f$ is finite locally free (of rank $d$) if and only if $f'$\nis finite locally free (of rank $d$), and\n\\item add more here.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CF4","source_file":"more-morphisms.tex","source_line":2597,"source_end_line":2656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L2597-L2656","statement_sha256":"ec860039744150c2a71dd80a0f47ccd1f5283dfae76e273d3c3ae963daa8b99e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7191,"rank":7191,"depth":46,"x":2136.085,"y":628.969,"cluster":"scheme-morphisms"},{"id":"stacks:0D4F","tag":"0D4F","title":"Deformations of projective schemes · Lemma 0D4F","summary":"Let f : X → S be a morphism of schemes which is proper, flat, and of finite presentation. Let L be f-ample. Assume S is quasi-compact. There exists a d_0 ≥ 0 such that for every cartesian diagram vcenter xymatrix X ar[r]_i' ar[d]_f & X' ar[d]^f' S ar[r]^i & S' and invertible O_X'-module L' with L ≅ (i')^*L' where S ⊂ S' is a thickening and f' is proper, flat, of finite presentation we have • R^p(f')_*(L')^⊗ d = 0 for all p > 0 and d ≥ d_0, • A'_d = (f')_*(L')^⊗ d is…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is proper, flat, and\nof finite presentation. Let $\\mathcal{L}$ be $f$-ample. Assume\n$S$ is quasi-compact. There exists a $d_0 \\geq 0$ such that\nfor every cartesian diagram\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[r]_{i'} \\ar[d]_f & X' \\ar[d]^{f'} \\\\\nS \\ar[r]^i & S'\n}\n}\n\\quad\\text{and}\\quad\n\\begin{matrix}\n\\text{invertible }\\mathcal{O}_{X'}\\text{-module}\\\\\n\\mathcal{L}'\\text{ with }\\mathcal{L} \\cong (i')^*\\mathcal{L}'\n\\end{matrix}\n$$\nwhere $S \\subset S'$ is a thickening and $f'$ is\nproper, flat, of finite presentation we have\n\\begin{enumerate}\n\\item $R^p(f')_*(\\mathcal{L}')^{\\otimes d} = 0$\nfor all $p > 0$ and $d \\geq d_0$,\n\\item $\\mathcal{A}'_d = (f')_*(\\mathcal{L}')^{\\otimes d}$\nis finite locally free for $d \\geq d_0$,\n\\item $\\mathcal{A}' =\n\\mathcal{O}_{S'} \\oplus \\bigoplus_{d \\geq d_0} \\mathcal{A}'_d$\nis a quasi-coherent $\\mathcal{O}_{S'}$-algebra of finite presentation,\n\\item there is a canonical isomorphism\n$r' : X' \\to \\underline{\\text{Proj}}_{S'}(\\mathcal{A}')$, and\n\\item there is a canonical isomorphism\n$\\theta' : (r')^*\\mathcal{O}_{\\underline{\\text{Proj}}_{S'}(\\mathcal{A}')}(1)\n\\to \\mathcal{L}'$.\n\\end{enumerate}\nThe construction of $\\mathcal{A}'$, $r'$, $\\theta'$\nis functorial in the data $(X', S', i, i', f', \\mathcal{L}')$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Infinitesimal deformations of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4F","source_file":"more-morphisms.tex","source_line":2852,"source_end_line":2889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L2852-L2889","statement_sha256":"90741ef41d9ee9561640fae55c76def151b04102d5d69fadc0d052b6691818f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7192,"rank":7192,"depth":39,"x":1992.785,"y":777.895,"cluster":"scheme-morphisms"},{"id":"stacks:02H0","tag":"02H0","title":"Formally smooth morphisms · Definition 02H0","summary":"Let f : X → S be a morphism of schemes. We say f is formally smooth if given any solid commutative diagram xymatrix X ar[d]_f & T ar[d]^i ar[l] S & T' ar[l] ar@-->[lu] where T ⊂ T' is a first order thickening of affine schemes over S there exists a dotted arrow making the diagram commute.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nWe say $f$ is {\\it formally smooth} if given any solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\\nS & T' \\ar[l] \\ar@{-->}[lu]\n}\n$$\nwhere $T \\subset T'$ is a first order thickening of affine schemes over $S$\nthere exists a dotted arrow making the diagram commute.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02H0","source_file":"more-morphisms.tex","source_line":3081,"source_end_line":3093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3081-L3093","statement_sha256":"f70ef332d733d2722975cfc2958b74b6018dc8729366f5ee257e70507fc7dc8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7193,"rank":7193,"depth":0,"x":1978.68,"y":586.629,"cluster":"scheme-morphisms"},{"id":"stacks:02H1","tag":"02H1","title":"Formally smooth morphisms · Lemma 02H1","summary":"A composition of formally smooth morphisms is formally smooth.","statement_latex":"A composition of formally smooth morphisms is formally smooth.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02H1","source_file":"more-morphisms.tex","source_line":3110,"source_end_line":3113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3110-L3113","statement_sha256":"eeb2f93084340d9b800c583c6b4dc9f4ffe91ffb67ca174042638bbe3f3e88eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7194,"rank":7194,"depth":0,"x":2143.01,"y":719.759,"cluster":"scheme-morphisms"},{"id":"stacks:02H2","tag":"02H2","title":"Formally smooth morphisms · Lemma 02H2","summary":"A base change of a formally smooth morphism is formally smooth.","statement_latex":"A base change of a formally smooth morphism is formally smooth.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02H2","source_file":"more-morphisms.tex","source_line":3119,"source_end_line":3122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3119-L3122","statement_sha256":"772e0042117a12f8e84c911fe9f5eda6db78b586794db10b65ae09215ceb5a2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7195,"rank":7195,"depth":2,"x":1914.612,"y":714.832,"cluster":"scheme-morphisms"},{"id":"stacks:02HH","tag":"02HH","title":"Formally smooth morphisms · Lemma 02HH","summary":"Let f : X → S be a morphism of schemes. Then f is formally étale if and only if f is formally smooth and formally unramified.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThen $f$ is formally \\'etale if and only if\n$f$ is formally smooth and formally unramified.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HH","source_file":"more-morphisms.tex","source_line":3129,"source_end_line":3134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3129-L3134","statement_sha256":"b09728323f018acb37507ec49dc318d2ea8ac8e1049f7caaa02dcda5bf628c41","origin":"The Stacks Project","memory_eligible":false,"source_rank":7196,"rank":7196,"depth":0,"x":2087.115,"y":588.776,"cluster":"scheme-morphisms"},{"id":"stacks:02H3","tag":"02H3","title":"Formally smooth morphisms · Lemma 02H3","summary":"Let f : X → S be a morphism of schemes. Let U ⊂ X and V ⊂ S be open subschemes such that f(U) ⊂ V. If f is formally smooth, so is f|_U : U → V.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $U \\subset X$ and $V \\subset S$ be open subschemes such that\n$f(U) \\subset V$. If $f$ is formally smooth, so is $f|_U : U \\to V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02H3","source_file":"more-morphisms.tex","source_line":3140,"source_end_line":3145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3140-L3145","statement_sha256":"522e5dbfe6bc7209e70dc9bf4260746e7195a764b17a3deece31e478c30c7c89","origin":"The Stacks Project","memory_eligible":false,"source_rank":7197,"rank":7197,"depth":1,"x":2061.267,"y":779.747,"cluster":"scheme-morphisms"},{"id":"stacks:02H4","tag":"02H4","title":"Formally smooth morphisms · Lemma 02H4","summary":"Let f : X → S be a morphism of schemes. Assume X and S are affine. Then f is formally smooth if and only if O_S(S) → O_X(X) is a formally smooth ring map.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $X$ and $S$ are affine.\nThen $f$ is formally smooth if and only if\n$\\mathcal{O}_S(S) \\to \\mathcal{O}_X(X)$ is a formally smooth\nring map.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02H4","source_file":"more-morphisms.tex","source_line":3163,"source_end_line":3170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3163-L3170","statement_sha256":"52ea12b849ee94be936b3799414930db0101ac06b0c7e222b8daf9a5e68127d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7198,"rank":7198,"depth":12,"x":1926.656,"y":624.149,"cluster":"scheme-morphisms"},{"id":"stacks:02H6","tag":"02H6","title":"Infinitesimal lifting criterion · Lemma 02H6","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • The morphism f is smooth, and • the morphism f is locally of finite presentation and formally smooth.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is smooth, and\n\\item the morphism $f$ is locally of finite presentation and\nformally smooth.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02H6","source_file":"more-morphisms.tex","source_line":3189,"source_end_line":3198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3189-L3198","statement_sha256":"7a32dba73364fcc24d71ab7532284f9193591daf1d8c766c1ac005b9e95e11f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7199,"rank":7199,"depth":37,"x":2151.205,"y":662.531,"cluster":"scheme-morphisms"},{"id":"stacks:06B5","tag":"06B5","title":"Formally smooth morphisms · Lemma 06B5","summary":"Let f : X → Y be a formally smooth morphism of schemes. Then Ω_X/Y is locally projective on X.","statement_latex":"Let $f : X \\to Y$ be a formally smooth morphism of schemes.\nThen $\\Omega_{X/Y}$ is locally projective on $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06B5","source_file":"more-morphisms.tex","source_line":3288,"source_end_line":3292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3288-L3292","statement_sha256":"c5dd7bc230e6339dd82e44d8806190f80ea7dc283f831a9e38a7811fd86b0a83","origin":"The Stacks Project","memory_eligible":false,"source_rank":7200,"rank":7200,"depth":13,"x":1954.62,"y":761.714,"cluster":"scheme-morphisms"},{"id":"stacks:0D0E","tag":"0D0E","title":"Formally smooth morphisms · Lemma 0D0E","summary":"Let T be an affine scheme. Let F, G be quasi-coherent O_T-modules. Consider H = SheafHom_O_T(F, G). If F is locally projective, then H^1(T, H) = 0.","statement_latex":"Let $T$ be an affine scheme. Let $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent\n$\\mathcal{O}_T$-modules. Consider\n$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_T}(\\mathcal{F}, \\mathcal{G})$.\nIf $\\mathcal{F}$ is locally projective, then $H^1(T, \\mathcal{H}) = 0$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0E","source_file":"more-morphisms.tex","source_line":3310,"source_end_line":3316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3310-L3316","statement_sha256":"c0f4f8d4d0ed75fd3b210e9d922cb1ca7b6aab0bf69f7446c7f02380d2a8853e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7201,"rank":7201,"depth":24,"x":2019.865,"y":576.899,"cluster":"scheme-morphisms"},{"id":"stacks:0D0F","tag":"0D0F","title":"Formally smooth morphisms · Lemma 0D0F","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent: • f is formally smooth, • for every x ∈ X there exist opens x ∈ U ⊂ X and f(x) ∈ V ⊂ Y with f(U) ⊂ V such that f|_U : U → V is formally smooth, • for every pair of affine opens U ⊂ X and V ⊂ Y with f(U) ⊂ V the ring map O_Y(V) → O_X(U) is formally smooth, and • there exists an affine open covering Y = ⋃ V_j and for each j an affine open covering f^-1(V_j) = ⋃ U_ji such that O_Y(V) → O_X(U) is a formally…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is formally smooth,\n\\item for every $x \\in X$ there exist opens $x \\in U \\subset X$ and\n$f(x) \\in V \\subset Y$ with $f(U) \\subset V$ such that\n$f|_U : U \\to V$ is formally smooth,\n\\item for every pair of affine opens $U \\subset X$ and $V \\subset Y$\nwith $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$\nis formally smooth, and\n\\item there exists an affine open covering $Y = \\bigcup V_j$ and\nfor each $j$ an affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$\nsuch that $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$ is a formally smooth\nring map for all $j$ and $i$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0F","source_file":"more-morphisms.tex","source_line":3332,"source_end_line":3348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3332-L3348","statement_sha256":"95a2699f7143e77efe1906fbb4b733ea1ea3093c52b64af5a748a84fd6012695","origin":"The Stacks Project","memory_eligible":false,"source_rank":7202,"rank":7202,"depth":38,"x":2120.449,"y":750.324,"cluster":"scheme-morphisms"},{"id":"stacks:06B6","tag":"06B6","title":"Formally smooth morphisms · Lemma 06B6","summary":"Let f : X → Y, g : Y → S be morphisms of schemes. Assume f is formally smooth. Then 0 → f^*Ω_Y/S → Ω_X/S → Ω_X/Y → 0 (see Morphisms, Lemma [Tag 01UX]) is short exact.","statement_latex":"Let $f : X \\to Y$, $g : Y \\to S$ be morphisms of schemes.\nAssume $f$ is formally smooth. Then\n$$\n0 \\to f^*\\Omega_{Y/S} \\to \\Omega_{X/S} \\to \\Omega_{X/Y} \\to 0\n$$\n(see\nMorphisms, Lemma \\ref{morphisms-lemma-triangle-differentials})\nis short exact.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06B6","source_file":"more-morphisms.tex","source_line":3417,"source_end_line":3427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3417-L3427","statement_sha256":"b8b2f98304275a1a062aaf915c3c47d1d1e75bdbf7d484612eecfc13db8d8f80","origin":"The Stacks Project","memory_eligible":false,"source_rank":7203,"rank":7203,"depth":17,"x":1906.664,"y":679.466,"cluster":"scheme-morphisms"},{"id":"stacks:06B7","tag":"06B7","title":"Formally smooth morphisms · Lemma 06B7","summary":"Let h : Z → X be a formally unramified morphism of schemes over S. Assume that Z is formally smooth over S. Then the canonical exact sequence 0 → C_Z/X → h^*Ω_X/S → Ω_Z/S → 0 of Lemma [Tag 04FC] is short exact.","statement_latex":"Let $h : Z \\to X$ be a formally unramified morphism of schemes over $S$.\nAssume that $Z$ is formally smooth over $S$. Then the\ncanonical exact sequence\n$$\n0 \\to \\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/S} \\to \\Omega_{Z/S} \\to 0\n$$\nof\nLemma \\ref{lemma-universally-unramified-differentials-sequence}\nis short exact.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06B7","source_file":"more-morphisms.tex","source_line":3442,"source_end_line":3453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3442-L3453","statement_sha256":"436e15ffba351841a1e8f4a6160c913d854c8b5e92f1c6ea7c4ae7fac3f87974","origin":"The Stacks Project","memory_eligible":false,"source_rank":7204,"rank":7204,"depth":19,"x":2121.44,"y":610.36,"cluster":"scheme-morphisms"},{"id":"stacks:067W","tag":"067W","title":"Formally smooth morphisms · Lemma 067W","summary":"Let xymatrix Z ar[r]_i ar[rd]_j & X ar[d]^f & Y be a commutative diagram of schemes where i and j are formally unramified and f is formally smooth. Then the canonical exact sequence 0 → C_Z/Y → C_Z/X → i^*Ω_X/Y → 0 of Lemma [Tag 067V] is exact and locally split.","statement_latex":"Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[rd]_j & X \\ar[d]^f \\\\\n& Y\n}\n$$\nbe a commutative diagram of schemes where $i$ and $j$ are formally\nunramified and $f$ is formally smooth. Then the canonical exact sequence\n$$\n0 \\to\n\\mathcal{C}_{Z/Y} \\to\n\\mathcal{C}_{Z/X} \\to\ni^*\\Omega_{X/Y} \\to 0\n$$\nof\nLemma \\ref{lemma-two-unramified-morphisms}\nis exact and locally split.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067W","source_file":"more-morphisms.tex","source_line":3470,"source_end_line":3490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3470-L3490","statement_sha256":"2d8feea8529a20b86ca3647fb8122b7f11e9d808accb9f5010debed619c30103","origin":"The Stacks Project","memory_eligible":false,"source_rank":7205,"rank":7205,"depth":20,"x":2018.568,"y":783.31,"cluster":"scheme-morphisms"},{"id":"stacks:02HX","tag":"02HX","title":"Smoothness over a Noetherian base · Lemma 02HX","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. Assume that S is locally Noetherian and f locally of finite type. The following are equivalent: • f is smooth at x, • for every solid commutative diagram xymatrix X ar[d]_f & Spec(B) ar[d]^i ar[l]^-α S & Spec(B') ar[l]_-β ar@-->[lu] where B' → B is a surjection of local rings with Ker(B' → B) of square zero, and α mapping the closed point of Spec(B) to x there exists a dotted arrow making the diagram commute, • same as in…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$.\nAssume that $S$ is locally Noetherian and $f$ locally of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is smooth at $x$,\n\\item for every solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\\nS & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu]\n}\n$$\nwhere $B' \\to B$ is a surjection of local rings with\n$\\Ker(B' \\to B)$ of square zero, and $\\alpha$ mapping the\nclosed point of $\\Spec(B)$ to $x$ there exists\na dotted arrow making the diagram commute,\n\\item same as in (2) but with $B' \\to B$ ranging over small\nextensions (see Algebra, Definition \\ref{algebra-definition-small-extension}),\nand\n\\item same as in (2) but with $B' \\to B$ ranging over small\nextensions such that $\\alpha$ induces an isomorphism\n$\\kappa(x) \\to \\kappa(\\mathfrak m)$ where $\\mathfrak m \\subset B$\nis the maximal ideal.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Smoothness over a Noetherian base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HX","source_file":"more-morphisms.tex","source_line":3574,"source_end_line":3601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3574-L3601","statement_sha256":"15f012273b135cfc35c78bbfec0dbe8448d12eff9d5ac2da53274e408aa7629c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7206,"rank":7206,"depth":7,"x":1955.298,"y":597.275,"cluster":"scheme-morphisms"},{"id":"stacks:02HY","tag":"02HY","title":"Smoothness over a Noetherian base · Lemma 02HY","summary":"Let f : X → S be a morphism of schemes. Assume that S is locally Noetherian and f locally of finite type. The following are equivalent: • f is smooth, • for every solid commutative diagram xymatrix X ar[d]_f & Spec(B) ar[d]^i ar[l]^-α S & Spec(B') ar[l]_-β ar@-->[lu] where B' → B is a small extension of Artinian local rings and β of finite type (!) there exists a dotted arrow making the diagram commute.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume that $S$ is locally Noetherian and $f$ locally of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is smooth,\n\\item for every solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\\nS & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu]\n}\n$$\nwhere $B' \\to B$ is a small extension of Artinian local rings\nand $\\beta$ of finite type (!) there exists a dotted arrow making\nthe diagram commute.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Smoothness over a Noetherian base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02HY","source_file":"more-morphisms.tex","source_line":3623,"source_end_line":3641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3623-L3641","statement_sha256":"692a53cbabfd2f6e3b77925e21594385740611b0bbbfdf7a4b39bf75447bc6e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7207,"rank":7207,"depth":38,"x":2151.695,"y":698.625,"cluster":"scheme-morphisms"},{"id":"stacks:0A43","tag":"0A43","title":"Smoothness over a Noetherian base · Lemma 0A43","summary":"Let f : X → S be a finite type morphism of locally Noetherian schemes. Let Z ⊂ S be a closed subscheme with nth infinitesimal neighbourhood Z_n ⊂ S. Set X_n = Z_n ×_S X. • If X_n → Z_n is smooth for all n, then f is smooth at every point of f^-1(Z). • If X_n → Z_n is étale for all n, then f is étale at every point of f^-1(Z).","statement_latex":"Let $f : X \\to S$ be a finite type morphism of locally Noetherian schemes.\nLet $Z \\subset S$ be a closed subscheme with $n$th infinitesimal\nneighbourhood $Z_n \\subset S$. Set $X_n = Z_n \\times_S X$.\n\\begin{enumerate}\n\\item If $X_n \\to Z_n$ is smooth for all $n$, then $f$\nis smooth at every point of $f^{-1}(Z)$.\n\\item If $X_n \\to Z_n$ is \\'etale for all $n$, then $f$\nis \\'etale at every point of $f^{-1}(Z)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Smoothness over a Noetherian base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A43","source_file":"more-morphisms.tex","source_line":3670,"source_end_line":3681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3670-L3681","statement_sha256":"e1b19fc293a8b9cfcca29c937bfa2daf1364998e24b0849723229d6a905725b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7208,"rank":7208,"depth":8,"x":1925.212,"y":735.358,"cluster":"scheme-morphisms"},{"id":"stacks:0D4G","tag":"0D4G","title":"Smoothness over a Noetherian base · Lemma 0D4G","summary":"Let f : X → S be a morphism of locally Noetherian schemes. Let Z ⊂ S be a closed subscheme with nth infinitesimal neighbourhood Z_n ⊂ S. Set X_n = Z_n ×_S X. If X_n → Z_n is flat for all n, then f is flat at every point of f^-1(Z).","statement_latex":"Let $f : X \\to S$ be a morphism of locally Noetherian schemes.\nLet $Z \\subset S$ be a closed subscheme with $n$th infinitesimal\nneighbourhood $Z_n \\subset S$. Set $X_n = Z_n \\times_S X$.\nIf $X_n \\to Z_n$ is flat for all $n$, then $f$\nis flat at every point of $f^{-1}(Z)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Smoothness over a Noetherian base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4G","source_file":"more-morphisms.tex","source_line":3696,"source_end_line":3703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3696-L3703","statement_sha256":"54ff001341d56d01b3face7cda2bed402a1d3728c8d857afaab784a999d13adc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7209,"rank":7209,"depth":10,"x":2062.775,"y":579.65,"cluster":"scheme-morphisms"},{"id":"stacks:0D0H","tag":"0D0H","title":"The naive cotangent complex · Definition 0D0H","summary":"Let f : X → Y be a morphism of schemes. The naive cotangent complex of f is the complex defined in Modules, Definition [Tag 08TN]. Notation: NL_f or NL_X/Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nThe {\\it naive cotangent complex of $f$}\nis the complex defined in Modules, Definition\n\\ref{modules-definition-cotangent-complex-morphism-ringed-topoi}.\nNotation: $\\NL_f$ or $\\NL_{X/Y}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0H","source_file":"more-morphisms.tex","source_line":3726,"source_end_line":3733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3726-L3733","statement_sha256":"22ed4477cf4dbc50056e5f349ab2d18cb21a65b8d284ef022ae9c9e060feb173","origin":"The Stacks Project","memory_eligible":false,"source_rank":7210,"rank":7210,"depth":1,"x":2086.571,"y":772.659,"cluster":"scheme-morphisms"},{"id":"stacks:0D0I","tag":"0D0I","title":"The naive cotangent complex · Lemma 0D0I","summary":"Let f : X → Y be a morphism of schemes. Let Spec(A) = U ⊂ X and Spec(R) = V ⊂ S be affine opens with f(U) ⊂ V. There is a canonical map widetildeNL_A/R → NL_X/Y|_U of complexes which is an isomorphism in D(O_U).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let\n$\\Spec(A) = U \\subset X$ and $\\Spec(R) = V \\subset S$\nbe affine opens with $f(U) \\subset V$.\nThere is a canonical map\n$$\n\\widetilde{\\NL_{A/R}} \\longrightarrow \\NL_{X/Y}|_U\n$$\nof complexes which is an isomorphism in $D(\\mathcal{O}_U)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0I","source_file":"more-morphisms.tex","source_line":3735,"source_end_line":3745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3735-L3745","statement_sha256":"cbfe8825880c67d30713a984472f6f0d4d823991d4d0f3302463726daeb565d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7211,"rank":7211,"depth":9,"x":1913.69,"y":643.749,"cluster":"scheme-morphisms"},{"id":"stacks:0D0J","tag":"0D0J","title":"The naive cotangent complex · Lemma 0D0J","summary":"Let f : X → Y be a morphism of schemes. The cohomology sheaves of the complex NL_X/Y are quasi-coherent, zero outside degrees -1, 0 and equal to Ω_X/Y in degree 0.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The cohomology sheaves\nof the complex $\\NL_{X/Y}$ are quasi-coherent, zero outside\ndegrees $-1$, $0$ and equal to $\\Omega_{X/Y}$ in degree $0$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0J","source_file":"more-morphisms.tex","source_line":3776,"source_end_line":3781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3776-L3781","statement_sha256":"f1b00efb33fc653cd90af8439f1b35cd5530296cd859d8ea79b1b67b65029708","origin":"The Stacks Project","memory_eligible":false,"source_rank":7212,"rank":7212,"depth":15,"x":2144.998,"y":640.707,"cluster":"scheme-morphisms"},{"id":"stacks:0D0K","tag":"0D0K","title":"The naive cotangent complex · Lemma 0D0K","summary":"Let f : X → Y be a morphism of schemes. If f is locally of finite presentation, then NL_X/Y is locally on X quasi-isomorphic to a complex … → 0 → F^-1 → F^0 → 0 → … of quasi-coherent O_X-modules with F^0 of finite presentation and F^-1 of finite type.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. If $f$ is locally of finite\npresentation, then $\\NL_{X/Y}$ is locally on $X$ quasi-isomorphic to\na complex\n$$\n\\ldots \\to 0 \\to \\mathcal{F}^{-1} \\to \\mathcal{F}^0 \\to 0 \\to \\ldots\n$$\nof quasi-coherent $\\mathcal{O}_X$-modules\nwith $\\mathcal{F}^0$ of finite presentation\nand $\\mathcal{F}^{-1}$ of finite type.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0K","source_file":"more-morphisms.tex","source_line":3795,"source_end_line":3806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3795-L3806","statement_sha256":"3344330051a4db8d699bbc5fa77824fb1cac9f84eef03327be8ac52e0bf06d12","origin":"The Stacks Project","memory_eligible":false,"source_rank":7213,"rank":7213,"depth":10,"x":1976.763,"y":774.293,"cluster":"scheme-morphisms"},{"id":"stacks:0D0L","tag":"0D0L","title":"The naive cotangent complex · Lemma 0D0L","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent • f is formally smooth, • H^-1(NL_X/Y) = 0 and H^0(NL_X/Y) = Ω_X/Y is locally projective.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is formally smooth,\n\\item $H^{-1}(\\NL_{X/Y}) = 0$ and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$\nis locally projective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0L","source_file":"more-morphisms.tex","source_line":3823,"source_end_line":3831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3823-L3831","statement_sha256":"b3cd37f3758b6a70bc397439ad25d2461d3a3f012a23849838544b52f3e83643","origin":"The Stacks Project","memory_eligible":false,"source_rank":7214,"rank":7214,"depth":39,"x":1993.402,"y":580.192,"cluster":"scheme-morphisms"},{"id":"stacks:0D0M","tag":"0D0M","title":"The naive cotangent complex · Lemma 0D0M","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent • f is formally étale, • H^-1(NL_X/Y) = H^0(NL_X/Y) = 0.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is formally \\'etale,\n\\item $H^{-1}(\\NL_{X/Y}) = H^0(\\NL_{X/Y}) = 0$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0M","source_file":"more-morphisms.tex","source_line":3839,"source_end_line":3846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3839-L3846","statement_sha256":"b573ba341023d30111ba0e110ec764e3b6ba17c880f6a8a858e030ae25d3ac58","origin":"The Stacks Project","memory_eligible":false,"source_rank":7215,"rank":7215,"depth":40,"x":2137.325,"y":732.867,"cluster":"scheme-morphisms"},{"id":"stacks:0D0N","tag":"0D0N","title":"The naive cotangent complex · Lemma 0D0N","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent • f is smooth, and • f is locally of finite presentation, H^-1(NL_X/Y) = 0, and H^0(NL_X/Y) = Ω_X/Y is finite locally free.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is smooth, and\n\\item $f$ is locally of finite presentation,\n$H^{-1}(\\NL_{X/Y}) = 0$, and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$\nis finite locally free.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0N","source_file":"more-morphisms.tex","source_line":3861,"source_end_line":3870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3861-L3870","statement_sha256":"b544269a7e01eee4438a309518f1935cac6e1bf73a7c9311dc39702ff58a86ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":7216,"rank":7216,"depth":10,"x":1908.261,"y":701.93,"cluster":"scheme-morphisms"},{"id":"stacks:0G7Z","tag":"0G7Z","title":"The naive cotangent complex · Lemma 0G7Z","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent • f is étale, and • f is locally of finite presentation and H^-1(NL_X/Y) = H^0(NL_X/Y) = 0.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is \\'etale, and\n\\item $f$ is locally of finite presentation and\n$H^{-1}(\\NL_{X/Y}) = H^0(\\NL_{X/Y}) = 0$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G7Z","source_file":"more-morphisms.tex","source_line":3883,"source_end_line":3891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3883-L3891","statement_sha256":"5486e548ac8816b002863c9044f45b600026865dd294bfd2e2569f3dbee098e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7217,"rank":7217,"depth":10,"x":2102.183,"y":594.692,"cluster":"scheme-morphisms"},{"id":"stacks:0FV2","tag":"0FV2","title":"The naive cotangent complex · Lemma 0FV2","summary":"Let i : Z → X be an immersion of schemes. Then NL_Z/X is isomorphic to C_Z/X[1] in D(O_Z) where C_Z/X is the conormal sheaf of Z in X.","statement_latex":"Let $i : Z \\to X$ be an immersion of schemes. Then $\\NL_{Z/X}$\nis isomorphic to $\\mathcal{C}_{Z/X}[1]$ in $D(\\mathcal{O}_Z)$\nwhere $\\mathcal{C}_{Z/X}$ is the conormal sheaf of $Z$ in $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV2","source_file":"more-morphisms.tex","source_line":3902,"source_end_line":3907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3902-L3907","statement_sha256":"7348d1c463578aa46adc3adb6274a5b73adb3a5da28ec1e7fa10f420ca140a65","origin":"The Stacks Project","memory_eligible":false,"source_rank":7218,"rank":7218,"depth":10,"x":2045.386,"y":783.936,"cluster":"scheme-morphisms"},{"id":"stacks:0E44","tag":"0E44","title":"The naive cotangent complex · Lemma 0E44","summary":"Let f : X → Y and g : Y → Z be morphisms of schemes. There is a canonical six term exact sequence H^-1(f^*NL_Y/Z) → H^-1(NL_X/Z) → H^-1(NL_X/Y) → f^*Ω_Y/Z → Ω_X/Z → Ω_X/Y → 0 of cohomology sheaves.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes.\nThere is a canonical six term exact sequence\n$$\nH^{-1}(f^*\\NL_{Y/Z}) \\to\nH^{-1}(\\NL_{X/Z}) \\to\nH^{-1}(\\NL_{X/Y}) \\to\nf^*\\Omega_{Y/Z} \\to \\Omega_{X/Z} \\to \\Omega_{X/Y} \\to 0\n$$\nof cohomology sheaves.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E44","source_file":"more-morphisms.tex","source_line":3915,"source_end_line":3926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3915-L3926","statement_sha256":"15c336504184801895d0edc1c4a151722fe1a9ccba24831d870bc00d13bd8e5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7219,"rank":7219,"depth":6,"x":1935.006,"y":612.043,"cluster":"scheme-morphisms"},{"id":"stacks:0FV3","tag":"0FV3","title":"The naive cotangent complex · Lemma 0FV3","summary":"Let f : X → Y and Y → Z be morphisms of schemes. Assume X → Y is a complete intersection morphism. Then there is a canonical distinguished triangle f^*NL_Y/Z → NL_X/Z → NL_X/Y → f^*NL_Y/Z[1] in D(O_X) which recovers the 6-term exact sequence of Lemma [Tag 0E44].","statement_latex":"Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of schemes. Assume\n$X \\to Y$ is a complete intersection morphism. Then there is\na canonical distinguished triangle\n$$\nf^*\\NL_{Y/Z} \\to \\NL_{X/Z} \\to \\NL_{X/Y} \\to f^*\\NL_{Y/Z}[1]\n$$\nin $D(\\mathcal{O}_X)$ which recovers the $6$-term exact sequence of\nLemma \\ref{lemma-exact-sequence-NL}.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV3","source_file":"more-morphisms.tex","source_line":3933,"source_end_line":3943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3933-L3943","statement_sha256":"85a51898fefebaa98168073aee4cefbf5da436a07832aea5cde9f5012331b811","origin":"The Stacks Project","memory_eligible":false,"source_rank":7220,"rank":7220,"depth":10,"x":2154.783,"y":676.206,"cluster":"scheme-morphisms"},{"id":"stacks:0G80","tag":"0G80","title":"The naive cotangent complex · Lemma 0G80","summary":"Let X → Y → Z be morphisms of schemes. Assume X → Z smooth and Y → Z étale. Then X → Y is smooth.","statement_latex":"Let $X \\to Y \\to Z$ be morphisms of schemes. Assume $X \\to Z$ smooth\nand $Y \\to Z$ \\'etale. Then $X \\to Y$ is smooth.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G80","source_file":"more-morphisms.tex","source_line":3960,"source_end_line":3964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3960-L3964","statement_sha256":"507457c19aa6905a5763208be6555836961ee3968776bd7a48acbf7a42b9e2da","origin":"The Stacks Project","memory_eligible":false,"source_rank":7221,"rank":7221,"depth":19,"x":1940.976,"y":753.653,"cluster":"scheme-morphisms"},{"id":"stacks:0FV4","tag":"0FV4","title":"The naive cotangent complex · Lemma 0FV4","summary":"Let f : X → Y be a morphism of schemes which factors as f = g ∘ i with i an immersion and g : P → Y formally smooth (for example smooth). Then there is a canonical isomorphism NL_X/Y ≅ (C_X/P → i^*Ω_P/Y) in D(O_X) where the conormal sheaf C_X/P is placed in degree -1.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which factors\nas $f = g \\circ i$ with $i$ an immersion and $g : P \\to Y$\nformally smooth (for example smooth). Then there is a canonical isomorphism\n$$\n\\NL_{X/Y} \\cong \\left(\\mathcal{C}_{X/P} \\to i^*\\Omega_{P/Y}\\right)\n$$\nin $D(\\mathcal{O}_X)$ where the conormal sheaf $\\mathcal{C}_{X/P}$\nis placed in degree $-1$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV4","source_file":"more-morphisms.tex","source_line":3979,"source_end_line":3989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L3979-L3989","statement_sha256":"b56add3a999499ff3c11c0fc89c0c99a6515ff2a26c8b34a5c9221f4b6745b97","origin":"The Stacks Project","memory_eligible":false,"source_rank":7222,"rank":7222,"depth":40,"x":2036.419,"y":575.103,"cluster":"scheme-morphisms"},{"id":"stacks:0FV5","tag":"0FV5","title":"The naive cotangent complex · Lemma 0FV5","summary":"Consider a cartesian diagram of schemes xymatrix X' ar[r]_g' ar[d] & X ar[d] Y' ar[r] & Y The canonical map (g')^*NL_X/Y → NL_X'/Y' induces an isomorphism on H^0 and a surjection on H^-1.","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d] & X \\ar[d] \\\\\nY' \\ar[r] & Y\n}\n$$\nThe canonical map $(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ induces\nan isomorphism on $H^0$ and a surjection on $H^{-1}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV5","source_file":"more-morphisms.tex","source_line":4017,"source_end_line":4028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4017-L4028","statement_sha256":"d79d8aed7d16ebad7f64b06500f02286a203981b0deccc8b9381288bb5516b2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7223,"rank":7223,"depth":10,"x":2109.677,"y":761.048,"cluster":"scheme-morphisms"},{"id":"stacks:0FJZ","tag":"0FJZ","title":"The naive cotangent complex · Lemma 0FJZ","summary":"Consider a cartesian diagram of schemes xymatrix X' ar[d] ar[r]_g' & X ar[d] Y' ar[r] & Y If Y' → Y is flat, then the canonical map (g')^*NL_X/Y → NL_X'/Y' is a quasi-isomorphism.","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[d] \\ar[r]_{g'} & X \\ar[d] \\\\\nY' \\ar[r] & Y\n}\n$$\nIf $Y' \\to Y$ is flat, then the canonical map\n$(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ is a quasi-isomorphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJZ","source_file":"more-morphisms.tex","source_line":4036,"source_end_line":4047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4036-L4047","statement_sha256":"c37be88056b1dfb5866c64fa4caf87215723e6bf60d77f8e215e67b3e8aac77a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7224,"rank":7224,"depth":10,"x":1905.985,"y":665.437,"cluster":"scheme-morphisms"},{"id":"stacks:0FK0","tag":"0FK0","title":"The naive cotangent complex · Lemma 0FK0","summary":"Consider a cartesian diagram of schemes xymatrix X' ar[r]_g' ar[d] & X ar[d] Y' ar[r] & Y If X → Y is flat, then the canonical map (g')^*NL_X/Y → NL_X'/Y' is a quasi-isomorphism. If in addition NL_X/Y has tor-amplitude in [-1, 0] then L(g')^*NL_X/Y → NL_X'/Y' is a quasi-isomorphism too.","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d] & X \\ar[d] \\\\\nY' \\ar[r] & Y\n}\n$$\nIf $X \\to Y$ is flat, then the canonical map\n$(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ is a quasi-isomorphism.\nIf in addition $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$\nthen $L(g')^*\\NL_{X/Y} \\to \\NL_{X'/Y'}$ is a quasi-isomorphism too.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FK0","source_file":"more-morphisms.tex","source_line":4054,"source_end_line":4067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4054-L4067","statement_sha256":"078ea94ea15ce221d061833b5f317638e8d70347cf398f350991a1ac1b8f6a63","origin":"The Stacks Project","memory_eligible":false,"source_rank":7225,"rank":7225,"depth":28,"x":2133.229,"y":620.328,"cluster":"scheme-morphisms"},{"id":"stacks:0ET0","tag":"0ET0","title":"Pushouts in the category of schemes, I · Lemma 0ET0","summary":"Let A' → A be a surjection of rings and let B → A be a ring map. Let B' = B ×_A A' be the fibre product of rings. Set S = Spec(A), S' = Spec(A'), T = Spec(B), and T' = Spec(B'). Then vcenter xymatrix S ar[r]_i ar[d]_f & S' ar[d]^f' T ar[r]^i' & T' corresponding to vcenter xymatrix A & A' ar[l] B ar[u] & B' ar[l] ar[u] is a pushout of schemes.","statement_latex":"Let $A' \\to A$ be a surjection of rings and let $B \\to A$ be a ring map.\nLet $B' = B \\times_A A'$ be the fibre product of rings. Set\n$S = \\Spec(A)$, $S' = \\Spec(A')$, $T = \\Spec(B)$, and $T' = \\Spec(B')$.\nThen\n$$\n\\vcenter{\n\\xymatrix{\nS \\ar[r]_i \\ar[d]_f & S' \\ar[d]^{f'} \\\\\nT \\ar[r]^{i'} & T'\n}\n}\n\\quad\\text{corresponding to}\\quad\n\\vcenter{\n\\xymatrix{\nA & A' \\ar[l] \\\\\nB \\ar[u] & B' \\ar[l] \\ar[u]\n}\n}\n$$\nis a pushout of schemes.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET0","source_file":"more-morphisms.tex","source_line":4100,"source_end_line":4122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4100-L4122","statement_sha256":"4da08a42c2b76a1d9b0ff07febf89485113759b325ef8b27decbbbbc9bc7ec9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7226,"rank":7226,"depth":14,"x":2001.848,"y":782.646,"cluster":"scheme-morphisms"},{"id":"stacks:0BMP","tag":"0BMP","title":"Pushouts in the category of schemes, I · Lemma 0BMP","summary":"Let I → (Sch/S)_fppf, i ↦ X_i be a diagram of schemes. Let (W, X_i → W) be a cocone for the diagram in the category of schemes (Categories, Remark [Tag 0G2U]). If there exists a fpqc covering (W_a → W)_a ∈ A of schemes such that • for all a ∈ A we have W_a = colim X_i ×_W W_a in the category of schemes, and • for all a, b ∈ A we have W_a ×_W W_b = colim X_i ×_W W_a ×_W W_b in the category of schemes, then W = colim X_i in the category of schemes.","statement_latex":"Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$ be a diagram of schemes.\nLet $(W, X_i \\to W)$ be a cocone for the diagram in the category of schemes\n(Categories, Remark \\ref{categories-remark-cones-and-cocones}).\nIf there exists a fpqc covering $\\{W_a \\to W\\}_{a \\in A}$ of schemes such that\n\\begin{enumerate}\n\\item for all $a \\in A$ we have\n$W_a = \\colim X_i \\times_W W_a$\nin the category of schemes, and\n\\item for all $a, b \\in A$ we have\n$W_a \\times_W W_b = \\colim X_i \\times_W W_a \\times_W W_b$\nin the category of schemes,\n\\end{enumerate}\nthen $W = \\colim X_i$ in the category of schemes.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMP","source_file":"more-morphisms.tex","source_line":4182,"source_end_line":4197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4182-L4197","statement_sha256":"b544f0b0a44083abb3f0d61b73850d949a7326a8f2b202199f14862a542cb023","origin":"The Stacks Project","memory_eligible":false,"source_rank":7227,"rank":7227,"depth":41,"x":1968.172,"y":588.273,"cluster":"scheme-morphisms"},{"id":"stacks:07RT","tag":"07RT","title":"Pushouts in the category of schemes, I · Lemma 07RT","summary":"Let X → X' be a thickening of schemes and let X → Y be an affine morphism of schemes. Then there exists a pushout xymatrix X ar[r] ar[d]_f & X' ar[d]^f' Y ar[r] & Y' in the category of schemes. Moreover, Y ⊂ Y' is a thickening, X = Y ×_Y' X', and O_Y' = O_Y ×_f_*O_X f'_*O_X' as sheaves on |Y| = |Y'|.","statement_latex":"Let $X \\to X'$ be a thickening of schemes and let $X \\to Y$ be an affine\nmorphism of schemes. Then there exists a pushout\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d]_f\n&\nX' \\ar[d]^{f'}\n\\\\\nY \\ar[r]\n&\nY'\n}\n$$\nin the category of schemes. Moreover, $Y \\subset Y'$ is a\nthickening, $X = Y \\times_{Y'} X'$, and\n$$\n\\mathcal{O}_{Y'} = \\mathcal{O}_Y \\times_{f_*\\mathcal{O}_X} f'_*\\mathcal{O}_{X'}\n$$\nas sheaves on $|Y| = |Y'|$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RT","source_file":"more-morphisms.tex","source_line":4206,"source_end_line":4227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4206-L4227","statement_sha256":"78d26f81fe561c557a983c12b615717a7c9e26267393ccae9eb717671c31df1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7228,"rank":7228,"depth":42,"x":2149.438,"y":712.577,"cluster":"scheme-morphisms"},{"id":"stacks:07RV","tag":"07RV","title":"Pushouts in the category of schemes, I · Lemma 07RV","summary":"Let X → X' be a thickening of schemes and let X → Y be an affine morphism of schemes. Let Y' = Y amalg_X X' be the pushout (see Lemma [Tag 07RT]). Base change gives a functor F : (Sch/Y') → (Sch/Y) ×_(Sch/X) (Sch/X') given by V' ↦ (V' ×_Y' Y, V' ×_Y' X', 1) which has a left adjoint G : (Sch/Y) ×_(Sch/X) (Sch/X') → (Sch/Y') which sends the triple (V, U', φ) to the pushout V amalg_(V ×_Y X) U'. Finally, F ∘ G is isomorphic to the identity functor.","statement_latex":"Let $X \\to X'$ be a thickening of schemes and let $X \\to Y$ be an\naffine morphism of schemes. Let $Y' = Y \\amalg_X X'$ be the pushout\n(see Lemma \\ref{lemma-pushout-along-thickening}). Base change gives\na functor\n$$\nF : (\\Sch/Y') \\longrightarrow (\\Sch/Y) \\times_{(\\Sch/X)} (\\Sch/X')\n$$\ngiven by $V' \\longmapsto (V' \\times_{Y'} Y, V' \\times_{Y'} X', 1)$\nwhich has a left adjoint\n$$\nG : (\\Sch/Y) \\times_{(\\Sch/X)} (\\Sch/X') \\longrightarrow (\\Sch/Y')\n$$\nwhich sends the triple $(V, U', \\varphi)$ to the pushout\n$V \\amalg_{(V \\times_Y X)} U'$. Finally, $F \\circ G$ is isomorphic to the\nidentity functor.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RV","source_file":"more-morphisms.tex","source_line":4276,"source_end_line":4293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4276-L4293","statement_sha256":"8fe5d8fa52c6995e829f5d9c1a1f009219f4be9e79c4bce97793fe30288ae2b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7229,"rank":7229,"depth":43,"x":1915.652,"y":723.78,"cluster":"scheme-morphisms"},{"id":"stacks:08KU","tag":"08KU","title":"Pushouts in the category of schemes, I · Lemma 08KU","summary":"Let X → X' be a thickening of schemes and let X → Y be an affine morphism of schemes. Let Y' = Y amalg_X X' be the pushout (see Lemma [Tag 07RT]). Let V' → Y' be a morphism of schemes. Set V = Y ×_Y' V', U' = X' ×_Y' V', and U = X ×_Y' V'. There is an equivalence of categories between • quasi-coherent O_V'-modules flat over Y', and • the category of triples (G, F', φ) where • G is a quasi-coherent O_V-module flat over Y, • F' is a quasi-coherent O_U'-module flat over X',…","statement_latex":"Let $X \\to X'$ be a thickening of schemes and let $X \\to Y$ be an\naffine morphism of schemes. Let $Y' = Y \\amalg_X X'$ be the pushout\n(see Lemma \\ref{lemma-pushout-along-thickening}). Let $V' \\to Y'$\nbe a morphism of schemes. Set\n$V = Y \\times_{Y'} V'$, $U' = X' \\times_{Y'} V'$, and $U = X \\times_{Y'} V'$.\nThere is an equivalence of categories between\n\\begin{enumerate}\n\\item quasi-coherent $\\mathcal{O}_{V'}$-modules flat over $Y'$, and\n\\item the category of triples $(\\mathcal{G}, \\mathcal{F}', \\varphi)$ where\n\\begin{enumerate}\n\\item $\\mathcal{G}$ is a quasi-coherent $\\mathcal{O}_V$-module flat over $Y$,\n\\item $\\mathcal{F}'$ is a quasi-coherent $\\mathcal{O}_{U'}$-module flat\nover $X'$, and\n\\item $\\varphi : (U \\to V)^*\\mathcal{G} \\to (U \\to U')^*\\mathcal{F}'$\nis an isomorphism of $\\mathcal{O}_U$-modules.\n\\end{enumerate}\n\\end{enumerate}\nThe equivalence maps $\\mathcal{G}'$ to\n$((V \\to V')^*\\mathcal{G}', (U' \\to V')^*\\mathcal{G}', can)$.\nSuppose $\\mathcal{G}'$ corresponds to the triple\n$(\\mathcal{G}, \\mathcal{F}', \\varphi)$. Then\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{G}'$ is a finite type $\\mathcal{O}_{V'}$-module if and\nonly if $\\mathcal{G}$ and $\\mathcal{F}'$ are finite type\n$\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules.\n\\item[(b)] if $V' \\to Y'$ is locally of finite presentation, then\n$\\mathcal{G}'$ is an $\\mathcal{O}_{V'}$-module of finite\npresentation if and only if $\\mathcal{G}$ and $\\mathcal{F}'$ are\n$\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules of finite presentation.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KU","source_file":"more-morphisms.tex","source_line":4348,"source_end_line":4380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4348-L4380","statement_sha256":"f7372efaa71b2a9f55cbe94cb2147c6d0566e156bc38f4327e2374e6d2ac2c81","origin":"The Stacks Project","memory_eligible":false,"source_rank":7230,"rank":7230,"depth":43,"x":2079.145,"y":582.768,"cluster":"scheme-morphisms"},{"id":"stacks:07RX","tag":"07RX","title":"Pushouts in the category of schemes, I · Lemma 07RX","summary":"In the situation of Lemma [Tag 07RV]. If V' = G(V, U', φ) for some triple (V, U', φ), then • V' → Y' is locally of finite type if and only if V → Y and U' → X' are locally of finite type, • V' → Y' is flat if and only if V → Y and U' → X' are flat, • V' → Y' is flat and locally of finite presentation if and only if V → Y and U' → X' are flat and locally of finite presentation, • V' → Y' is smooth if and only if V → Y and U' → X' are smooth, • V' → Y' is étale if and only…","statement_latex":"In the situation of\nLemma \\ref{lemma-equivalence-categories-schemes-over-pushout}.\nIf $V' = G(V, U', \\varphi)$ for some triple $(V, U', \\varphi)$, then\n\\begin{enumerate}\n\\item $V' \\to Y'$ is locally of finite type if and only if $V \\to Y$ and\n$U' \\to X'$ are locally of finite type,\n\\item $V' \\to Y'$ is flat if and only if $V \\to Y$ and $U' \\to X'$ are flat,\n\\item $V' \\to Y'$ is flat and locally of finite presentation if and only if\n$V \\to Y$ and $U' \\to X'$ are flat and locally of finite presentation,\n\\item $V' \\to Y'$ is smooth if and only if $V \\to Y$ and $U' \\to X'$ are smooth,\n\\item $V' \\to Y'$ is \\'etale if and only if $V \\to Y$ and $U' \\to X'$\nare \\'etale, and\n\\item add more here as needed.\n\\end{enumerate}\nIf $W'$ is flat over $Y'$, then the adjunction mapping\n$G(F(W')) \\to W'$ is an isomorphism. Hence $F$ and $G$ define mutually\nquasi-inverse functors between the category of schemes flat over $Y'$\nand the category of triples $(V, U', \\varphi)$ with $V \\to Y$\nand $U' \\to X'$ flat.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RX","source_file":"more-morphisms.tex","source_line":4402,"source_end_line":4423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4402-L4423","statement_sha256":"7c79048de8c6f6a982cc3ac15af13849fe2ea4727be536e3bed5e6de18060f9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7231,"rank":7231,"depth":44,"x":2071.981,"y":779.652,"cluster":"scheme-morphisms"},{"id":"stacks:0399","tag":"0399","title":"Openness of the flat locus · Theorem 0399","summary":"[EGA] Let S be a scheme. Let f : X → S be a morphism which is locally of finite presentation. Let F be a quasi-coherent O_X-module which is locally of finite presentation. Then U = (x ∈ X mid F is flat over S at x) is open in X.","statement_latex":"\\begin{reference}\n\\cite[IV Theorem 11.3.1]{EGA}\n\\end{reference}\nLet $S$ be a scheme.\nLet $f : X \\to S$ be a morphism which is locally of finite presentation.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module which is\nlocally of finite presentation. Then\n$$\nU = \\{x \\in X \\mid \\mathcal{F}\\text{ is flat over }S\\text{ at }x\\}\n$$\nis open in $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Openness of the flat locus","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0399","source_file":"more-morphisms.tex","source_line":4445,"source_end_line":4458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4445-L4458","statement_sha256":"164dba3a355488f37a48e0c01e03e9829a9dd7cb511adc2a78d46ad95f7ef0f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7232,"rank":7232,"depth":34,"x":1918.831,"y":630.305,"cluster":"scheme-morphisms"},{"id":"stacks:047C","tag":"047C","title":"Openness of the flat locus · Lemma 047C","summary":"Let S be a scheme. Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a cartesian diagram of schemes. Let F be a quasi-coherent O_X-module. Let x' ∈ X' with images x = g'(x') and s' = f'(x'). • If F is flat over S at x, then (g')^*F is flat over S' at x'. • If g is flat at s' and (g')^*F is flat over S' at x', then F is flat over S at x. In particular, if g is flat, f is locally of finite presentation, and F is locally of finite presentation, then formation…","statement_latex":"Let $S$ be a scheme.\nLet\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nbe a cartesian diagram of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x' \\in X'$ with images\n$x = g'(x')$ and $s' = f'(x')$.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is flat over $S$ at $x$, then\n$(g')^*\\mathcal{F}$ is flat over $S'$ at $x'$.\n\\item If $g$ is flat at $s'$ and $(g')^*\\mathcal{F}$ is flat over $S'$ at\n$x'$, then $\\mathcal{F}$ is flat over $S$ at $x$.\n\\end{enumerate}\nIn particular, if $g$ is flat, $f$ is locally of finite presentation,\nand $\\mathcal{F}$ is locally of finite presentation,\nthen formation of the open subset of\nTheorem \\ref{theorem-openness-flatness}\ncommutes with base change.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Openness of the flat locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047C","source_file":"more-morphisms.tex","source_line":4467,"source_end_line":4492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4467-L4492","statement_sha256":"86a412b5898a8d3454294f8418ee68652593f1ad9eaa916efe0a3e7c081b83ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":7233,"rank":7233,"depth":35,"x":2152.021,"y":653.546,"cluster":"scheme-morphisms"},{"id":"stacks:039B","tag":"039B","title":"Crit\\`ere de platitude par fibres · Theorem 039B","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. Let F be a quasi-coherent O_X-module. Let x ∈ X. Set y = f(x) and s ∈ S the image of x in S. Assume S, X, Y locally Noetherian, F coherent, and F_x not = 0. Then the following are equivalent: • F is flat over S at x, and F_s is flat over Y_s at x, and • Y is flat over S at y and F is flat over Y at x.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of schemes over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in X$. Set $y = f(x)$ and $s \\in S$ the image of $x$ in $S$.\nAssume $S$, $X$, $Y$ locally Noetherian,\n$\\mathcal{F}$ coherent, and $\\mathcal{F}_x \\not = 0$.\nThen the following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat over $S$ at $x$, and\n$\\mathcal{F}_s$ is flat over $Y_s$ at $x$, and\n\\item $Y$ is flat over $S$ at $y$ and $\\mathcal{F}$ is\nflat over $Y$ at $x$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039B","source_file":"more-morphisms.tex","source_line":4546,"source_end_line":4561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4546-L4561","statement_sha256":"caf7cabf732006e05dc07f5560c9c8da956bae0b9fa4343e873b811427ca04d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7234,"rank":7234,"depth":10,"x":1961.251,"y":768.805,"cluster":"scheme-morphisms"},{"id":"stacks:039C","tag":"039C","title":"Crit\\`ere de platitude par fibres · Theorem 039C","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. Let F be a quasi-coherent O_X-module. Assume • X is locally of finite presentation over S, • F an O_X-module of finite presentation, and • Y is locally of finite type over S. Let x ∈ X. Set y = f(x) and let s ∈ S be the image of x in S. If F_x not = 0, then the following are equivalent: • F is flat over S at x, and F_s is flat over Y_s at x, and • Y is flat over S at y and F is flat over Y at x. Moreover,…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of schemes over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite presentation over $S$,\n\\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation, and\n\\item $Y$ is locally of finite type over $S$.\n\\end{enumerate}\nLet $x \\in X$. Set $y = f(x)$ and let $s \\in S$ be the image of $x$ in $S$.\nIf $\\mathcal{F}_x \\not = 0$, then the following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat over $S$ at $x$, and\n$\\mathcal{F}_s$ is flat over $Y_s$ at $x$, and\n\\item $Y$ is flat over $S$ at $y$ and $\\mathcal{F}$ is\nflat over $Y$ at $x$.\n\\end{enumerate}\nMoreover, the set of points $x$ where (1) and (2) hold is open in\n$\\text{Supp}(\\mathcal{F})$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039C","source_file":"more-morphisms.tex","source_line":4589,"source_end_line":4610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4589-L4610","statement_sha256":"45fbc00306f8a44244ccc2de5524eda9ae23dec6a683af47291963278f1c2292","origin":"The Stacks Project","memory_eligible":false,"source_rank":7235,"rank":7235,"depth":35,"x":2009.266,"y":575.436,"cluster":"scheme-morphisms"},{"id":"stacks:039D","tag":"039D","title":"Crit\\`ere de platitude par fibres · Lemma 039D","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. Assume • S, X, Y are locally Noetherian, • X is flat over S, • for every s ∈ S the morphism f_s : X_s → Y_s is flat. Then f is flat. If f is also surjective, then Y is flat over S.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of schemes over $S$.\nAssume\n\\begin{enumerate}\n\\item $S$, $X$, $Y$ are locally Noetherian,\n\\item $X$ is flat over $S$,\n\\item for every $s \\in S$ the morphism\n$f_s : X_s \\to Y_s$ is flat.\n\\end{enumerate}\nThen $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039D","source_file":"more-morphisms.tex","source_line":4662,"source_end_line":4674,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4662-L4674","statement_sha256":"2b007bc579957bc6cdbc9040b959dfbdd6134cf315d66d492bfd27eea9662ae7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7236,"rank":7236,"depth":11,"x":2129.444,"y":745.383,"cluster":"scheme-morphisms"},{"id":"stacks:039E","tag":"039E","title":"Crit\\`ere de platitude par fibres · Lemma 039E","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. Assume • X is locally of finite presentation over S, • X is flat over S, • for every s ∈ S the morphism f_s : X_s → Y_s is flat, and • Y is locally of finite type over S. Then f is flat. If f is also surjective, then Y is flat over S.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of schemes over $S$.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite presentation over $S$,\n\\item $X$ is flat over $S$,\n\\item for every $s \\in S$ the morphism\n$f_s : X_s \\to Y_s$ is flat, and\n\\item $Y$ is locally of finite type over $S$.\n\\end{enumerate}\nThen $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039E","source_file":"more-morphisms.tex","source_line":4681,"source_end_line":4694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4681-L4694","statement_sha256":"84a9b4eeb20b6cfd6720dcf3ecc78c5dc0c9a1ad1d495424b7045de5035c62e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7237,"rank":7237,"depth":36,"x":1904.007,"y":688.22,"cluster":"scheme-morphisms"},{"id":"stacks:05VJ","tag":"05VJ","title":"Crit\\`ere de platitude par fibres · Lemma 05VJ","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. Let F be a quasi-coherent O_X-module. Assume • X is locally of finite presentation over S, • F an O_X-module of finite presentation, • F is flat over S, and • Y is locally of finite type over S. Then the set U = (x ∈ X mid F flat at x over Y). is open in X and its formation commutes with arbitrary base change: If S' → S is a morphism of schemes, and U' is the set of points of X' = X ×_S S' where F' = F ×_S…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite presentation over $S$,\n\\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation,\n\\item $\\mathcal{F}$ is flat over $S$, and\n\\item $Y$ is locally of finite type over $S$.\n\\end{enumerate}\nThen the set\n$$\nU = \\{x \\in X \\mid \\mathcal{F} \\text{ flat at }x \\text{ over }Y\\}.\n$$\nis open in $X$ and its formation commutes with arbitrary base change:\nIf $S' \\to S$ is a morphism of schemes, and $U'$ is the set of points\nof $X' = X \\times_S S'$ where $\\mathcal{F}' = \\mathcal{F} \\times_S S'$\nis flat over $Y' = Y \\times_S S'$, then $U' = U \\times_S S'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VJ","source_file":"more-morphisms.tex","source_line":4701,"source_end_line":4720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4701-L4720","statement_sha256":"627528d83a94fd6829514136e2ca7a9a6a86ffb984f1da6c6997d4db1e9036b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7238,"rank":7238,"depth":36,"x":2116.354,"y":602.395,"cluster":"scheme-morphisms"},{"id":"stacks:05VK","tag":"05VK","title":"Crit\\`ere de platitude par fibres · Lemma 05VK","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. Assume • X is locally of finite presentation over S, • X is flat over S, and • Y is locally of finite type over S. Then the set U = (x ∈ X mid X flat at x over Y). is open in X and its formation commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite presentation over $S$,\n\\item $X$ is flat over $S$, and\n\\item $Y$ is locally of finite type over $S$.\n\\end{enumerate}\nThen the set\n$$\nU = \\{x \\in X \\mid X\\text{ flat at }x \\text{ over }Y\\}.\n$$\nis open in $X$ and its formation commutes with arbitrary base change.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VK","source_file":"more-morphisms.tex","source_line":4758,"source_end_line":4772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4758-L4772","statement_sha256":"48d46b2956ff5ad13b8ebec47d777e5f8c915cf211b2a34b7cfb14aff99d24e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7239,"rank":7239,"depth":37,"x":2028.731,"y":786.296,"cluster":"scheme-morphisms"},{"id":"stacks:080Q","tag":"080Q","title":"Crit\\`ere de platitude par fibres · Lemma 080Q","summary":"Let f : X → S be a morphism of schemes of finite presentation. Let F be a finitely presented O_X-module. Let x ∈ X with image s ∈ S. If F is flat at x over S and (F_s)_x is a flat O_X_s, x-module, then F is finite free in a neighbourhood of x.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes of finite presentation.\nLet $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module.\nLet $x \\in X$ with image $s \\in S$.\nIf $\\mathcal{F}$ is flat at $x$ over $S$ and $(\\mathcal{F}_s)_x$ is a flat\n$\\mathcal{O}_{X_s, x}$-module, then $\\mathcal{F}$\nis finite free in a neighbourhood of $x$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/080Q","source_file":"more-morphisms.tex","source_line":4790,"source_end_line":4798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4790-L4798","statement_sha256":"c4ce4cc11500d99765ac73dda82efcffb8f5049433bafe383b68c43b4e28e2da","origin":"The Stacks Project","memory_eligible":false,"source_rank":7240,"rank":7240,"depth":36,"x":1945.399,"y":600.846,"cluster":"scheme-morphisms"},{"id":"stacks:0CZR","tag":"0CZR","title":"Crit\\`ere de platitude par fibres · Lemma 0CZR","summary":"Let f : X → S be a morphism of schemes which is locally of finite presentation. Let F be a finitely presented O_X-module flat over S. Then the set (x ∈ X : F free in a neighbourhood of x) is open in X and its formation commutes with arbitrary base change S' → S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is\nlocally of finite presentation.\nLet $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module\nflat over $S$. Then the set\n$$\n\\{x \\in X : \\mathcal{F}\\text{ free in a neighbourhood of }x\\}\n$$\nis open in $X$ and its formation commutes with arbitrary base change\n$S' \\to S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZR","source_file":"more-morphisms.tex","source_line":4822,"source_end_line":4833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4822-L4833","statement_sha256":"0f6c50915f38f0f2e7a3f062e6419badbd0196447bd73a809ab092ea4af033ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":7241,"rank":7241,"depth":37,"x":2156.122,"y":690.369,"cluster":"scheme-morphisms"},{"id":"stacks:0FUE","tag":"0FUE","title":"Closed immersions between smooth schemes · Lemma 0FUE","summary":"Let S be a scheme. Let Y → X be a closed immersion of schemes smooth over S. For every y ∈ Y there exist integers 0 ≤ m, n and a commutative diagram xymatrix Y ar[d] & V ar[l] ar[d] ar[r] & A^m_S ar[d]^(a_1, …, a_m) ↦ (a_1, …, a_m, 0 …, 0) X & U ar[l] ar[r]^-π & A^m + n_S where U ⊂ X is open, V = Y ∩ U, π is étale, V = π^-1(A^m_S), and y ∈ V.","statement_latex":"Let $S$ be a scheme. Let $Y \\to X$ be a closed immersion of schemes\nsmooth over $S$. For every $y \\in Y$ there exist integers\n$0 \\leq m, n$ and a commutative diagram\n$$\n\\xymatrix{\nY \\ar[d] &\nV \\ar[l] \\ar[d] \\ar[r] &\n\\mathbf{A}^m_S\n\\ar[d]^{(a_1, \\ldots, a_m) \\mapsto (a_1, \\ldots, a_m, 0 \\ldots, 0)} \\\\\nX &\nU \\ar[l] \\ar[r]^-\\pi &\n\\mathbf{A}^{m + n}_S\n}\n$$\nwhere $U \\subset X$ is open, $V = Y \\cap U$,\n$\\pi$ is \\'etale, $V = \\pi^{-1}(\\mathbf{A}^m_S)$, and $y \\in V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Closed immersions between smooth schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUE","source_file":"more-morphisms.tex","source_line":4860,"source_end_line":4878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4860-L4878","statement_sha256":"aa3fee0b60ecd6289f3f671a113a493159c4b497119c327b28eee55b84318c24","origin":"The Stacks Project","memory_eligible":false,"source_rank":7242,"rank":7242,"depth":45,"x":1928.593,"y":743.962,"cluster":"scheme-morphisms"},{"id":"stacks:0FUT","tag":"0FUT","title":"Closed immersions between smooth schemes · Lemma 0FUT","summary":"Let S be a scheme. Let Z → X be a closed immersion of schemes smooth over S. Let b : X' → X be the blowing up of Z with exceptional divisor E ⊂ X'. Then X' and E are smooth over S. The morphism p : E → Z is canonically isomorphic to the projective space bundle P(I/I^2) → Z where I ⊂ O_X is the ideal sheaf of Z. The relative O_E(1) coming from the projective space bundle structure is isomorphic to the restriction of O_X'(-E) to E.","statement_latex":"Let $S$ be a scheme. Let $Z \\to X$ be a closed immersion of schemes\nsmooth over $S$. Let $b : X' \\to X$ be the blowing up of $Z$ with\nexceptional divisor $E \\subset X'$. Then $X'$ and $E$ are smooth\nover $S$. The morphism $p : E \\to Z$ is canonically isomorphic\nto the projective space bundle\n$$\n\\mathbf{P}(\\mathcal{I}/\\mathcal{I}^2) \\longrightarrow Z\n$$\nwhere $\\mathcal{I} \\subset \\mathcal{O}_X$ is the ideal sheaf\nof $Z$. The relative $\\mathcal{O}_E(1)$ coming from the projective\nspace bundle structure is isomorphic to the restriction of\n$\\mathcal{O}_{X'}(-E)$ to $E$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Closed immersions between smooth schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUT","source_file":"more-morphisms.tex","source_line":4974,"source_end_line":4988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L4974-L4988","statement_sha256":"0b1c196879cc4e9a5e631759f1ce792118961d4c8fe82146a2df931b8453da7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7243,"rank":7243,"depth":46,"x":2053.356,"y":575.224,"cluster":"scheme-morphisms"},{"id":"stacks:0GSG","tag":"0GSG","title":"Flat modules and relative assassins · Lemma 0GSG","summary":"Let A be a valuation ring. Let A → B is a local homomorphism of local rings which is essentially of finite type. Let u : N → M be a map of finite B-modules. Assume M is flat over A and overlineu : N/ m_A N → M/ m_A M is injective. Then u is injective and M/u(N) is flat over A.","statement_latex":"Let $A$ be a valuation ring. Let $A \\to B$ is a local homomorphism of\nlocal rings which is essentially of finite type.\nLet $u : N \\to M$ be a map of finite $B$-modules.\nAssume $M$ is flat over $A$ and\n$\\overline{u} : N/\\mathfrak m_A N \\to M/\\mathfrak m_A M$ is injective.\nThen $u$ is injective and $M/u(N)$ is flat over $A$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Flat modules and relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSG","source_file":"more-morphisms.tex","source_line":5043,"source_end_line":5051,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5043-L5051","statement_sha256":"dd65cea82b90384ed0b228d4ed2436251a16b0f82ad44914a27c7c91908a4daa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7244,"rank":7244,"depth":38,"x":2097.08,"y":770.573,"cluster":"scheme-morphisms"},{"id":"stacks:0GSH","tag":"0GSH","title":"Flat modules and relative assassins · Lemma 0GSH","summary":"This can be found in the proof of [EGA] Let f : X → S be a morphism of schemes. Let y ∈ X be a point with image t ∈ S. Denote Y ⊂ X the closure of (y) viewed as an integral closed subscheme of X. Let s ∈ S and let x ∈ Y_s be a generic point of an irreducible component of Y_s. There exists a cartesian diagram xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S with the following properties: • S' is the spectrum of a valuation ring with generic point t' and closed…","statement_latex":"\\begin{reference}\nThis can be found in the proof of\n\\cite[IV Proposition 12.1.1.5]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes. Let $y \\in X$ be a point\nwith image $t \\in S$. Denote $Y \\subset X$ the closure of $\\{y\\}$\nviewed as an integral closed subscheme of $X$. Let $s \\in S$ and let\n$x \\in Y_s$ be a generic point of an irreducible component of $Y_s$.\nThere exists a cartesian diagram\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item $S'$ is the spectrum of a valuation ring\nwith generic point $t'$ and closed point $s'$,\n\\item $g(t') = t$ and $g(s') = s$,\n\\item there exists a point $y' \\in X'_{t'}$ which is\na generic point of an irreducible component of\n$(S' \\times_S Y)_{t'} = Y_t \\times_t t'$\nand satisfies $g'(y') = y$,\n\\item denoting $Y' \\subset X'$ the closure of $\\{y'\\}$\nviewed as an integral closed subscheme of $X'$\nthere exists a point $x' \\in Y'_{s'}$ which is a generic\npoint of an irreducible component of $Y'_{s'}$\nwith $g'(x') = x$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Flat modules and relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSH","source_file":"more-morphisms.tex","source_line":5093,"source_end_line":5125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5093-L5125","statement_sha256":"96d906deeddbca4d0ec8d854fed8cbf1ceef1f06c91f626e7963b649b8ca0c21","origin":"The Stacks Project","memory_eligible":false,"source_rank":7245,"rank":7245,"depth":13,"x":1907.618,"y":651.256,"cluster":"scheme-morphisms"},{"id":"stacks:0GSI","tag":"0GSI","title":"Flat modules and relative assassins · Lemma 0GSI","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent finite type O_X-module. Let y ∈ Ass_X/S(F) with image t ∈ S. Denote Y ⊂ X the closure of (y) in X viewed as an integral closed subscheme. Let s ∈ S and let x ∈ Y_s be a generic point of an irreducible component of Y_s. If F is flat over S at x, then x ∈ Ass_X/S(F) and dim_x(Y_s) = dim(Y_t).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent finite type $\\mathcal{O}_X$-module.\nLet $y \\in \\text{Ass}_{X/S}(\\mathcal{F})$ with image $t \\in S$.\nDenote $Y \\subset X$ the closure of $\\{y\\}$ in $X$ viewed\nas an integral closed subscheme. Let $s \\in S$\nand let $x \\in Y_s$ be a generic point of an irreducible component\nof $Y_s$. If $\\mathcal{F}$ is flat over $S$ at $x$, then\n$x \\in \\text{Ass}_{X/S}(\\mathcal{F})$ and\n$\\dim_x(Y_s) = \\dim(Y_t)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Flat modules and relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSI","source_file":"more-morphisms.tex","source_line":5166,"source_end_line":5177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5166-L5177","statement_sha256":"b9a643ad8581487bfc11586892c423b313515a83aafbfd4d1b1cdfe7a2d9a37c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7246,"rank":7246,"depth":39,"x":2143.433,"y":631.72,"cluster":"scheme-morphisms"},{"id":"stacks:0H3X","tag":"0H3X","title":"Flat modules and relative assassins · Lemma 0H3X","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a finite type, quasi-coherent O_X-module flat over S. Assume S is irreducible with generic point eta. If dim(Supp(F_eta)) ≤ r then for all s ∈ S we have dim(Supp(F_s)) ≤ r.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module\nflat over $S$. Assume $S$ is irreducible with generic point $\\eta$.\nIf $\\dim(\\text{Supp}(\\mathcal{F}_\\eta)) \\leq r$ then for all $s \\in S$\nwe have $\\dim(\\text{Supp}(\\mathcal{F}_s)) \\leq r$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Flat modules and relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3X","source_file":"more-morphisms.tex","source_line":5255,"source_end_line":5262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5255-L5262","statement_sha256":"3aa38361a9a50ab4534f1446cea3a1cfea5be3353355a29795f5452ccdf2935d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7247,"rank":7247,"depth":40,"x":1985.151,"y":780.032,"cluster":"scheme-morphisms"},{"id":"stacks:0GSJ","tag":"0GSJ","title":"Flat modules and relative assassins · Lemma 0GSJ","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let y ∈ Ass_X/S(F). Denote Y ⊂ X the closure of (y) in X viewed as an integral closed subscheme. Denote T ⊂ S the closure of (f(y)) viewed as an integral closed subscheme. We obtain a commutative diagram xymatrix Y ar[r] ar[d] & X ar[d] T ar[r] & S where Y → T is dominant. Assume F is flat over S at all generic points of irreducible components of…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $y \\in \\text{Ass}_{X/S}(\\mathcal{F})$.\nDenote $Y \\subset X$ the closure of $\\{y\\}$ in $X$ viewed\nas an integral closed subscheme. Denote $T \\subset S$ the closure\nof $\\{f(y)\\}$ viewed as an integral closed subscheme. We obtain\na commutative diagram\n$$\n\\xymatrix{\nY \\ar[r] \\ar[d] & X \\ar[d] \\\\\nT \\ar[r] & S\n}\n$$\nwhere $Y \\to T$ is dominant. Assume $\\mathcal{F}$ is flat over $S$\nat all generic points of irreducible components of fibres of $Y \\to T$\n(for example if $\\mathcal{F}$ is flat over $S$). Then\n\\begin{enumerate}\n\\item if $s \\in S$ and $x \\in Y_s$ is the generic point of an\nirreducible component of $Y_s$, then $x \\in \\text{Ass}_{X/S}(\\mathcal{F})$, and\n\\item there is an integer $d \\geq 0$ such that\n$Y \\to T$ is of relative dimension $d$, see\nMorphisms, Definition \\ref{morphisms-definition-relative-dimension-d}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Flat modules and relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSJ","source_file":"more-morphisms.tex","source_line":5275,"source_end_line":5300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5275-L5300","statement_sha256":"47c7ef5f7c9b4551fb9033bbf733a2224bb215d830b2d42135d6e1dc3153f7b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7248,"rank":7248,"depth":40,"x":1982.596,"y":580.723,"cluster":"scheme-morphisms"},{"id":"stacks:081K","tag":"081K","title":"Normalization revisited · Lemma 081K","summary":"Let f : Y → X be a smooth morphism of schemes. Let A be a quasi-coherent sheaf of O_X-algebras. The integral closure of O_Y in f^*A is equal to f^*A' where A' ⊂ A is the integral closure of O_X in A.","statement_latex":"Let $f : Y \\to X$ be a smooth morphism of schemes. Let $\\mathcal{A}$ be a\nquasi-coherent sheaf of $\\mathcal{O}_X$-algebras. The integral closure\nof $\\mathcal{O}_Y$ in $f^*\\mathcal{A}$ is equal to $f^*\\mathcal{A}'$\nwhere $\\mathcal{A}' \\subset \\mathcal{A}$ is the integral closure of\n$\\mathcal{O}_X$ in $\\mathcal{A}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Normalization revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081K","source_file":"more-morphisms.tex","source_line":5358,"source_end_line":5365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5358-L5365","statement_sha256":"794c49a772e70c37098e6bbfbb88eac80c001a97e65187b626f70b71532427f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7249,"rank":7249,"depth":45,"x":2144.867,"y":726.338,"cluster":"scheme-morphisms"},{"id":"stacks:03GV","tag":"03GV","title":"Normalization commutes with smooth base change · Lemma 03GV","summary":"Let xymatrix Y_2 ar[r] ar[d]_f_2 & Y_1 ar[d]^f_1 X_2 ar[r]^φ & X_1 be a fibre square in the category of schemes. Assume f_1 is quasi-compact and quasi-separated, and φ is smooth. Let Y_i → X_i' → X_i be the normalization of X_i in Y_i. Then X_2' ≅ X_2 ×_X_1 X_1'.","statement_latex":"Let\n$$\n\\xymatrix{\nY_2 \\ar[r] \\ar[d]_{f_2} & Y_1 \\ar[d]^{f_1} \\\\\nX_2 \\ar[r]^\\varphi & X_1\n}\n$$\nbe a fibre square in the category of schemes. Assume $f_1$ is quasi-compact\nand quasi-separated, and $\\varphi$ is smooth.\nLet $Y_i \\to X_i' \\to X_i$ be the normalization of $X_i$ in $Y_i$.\nThen $X_2' \\cong X_2 \\times_{X_1} X_1'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Normalization revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GV","source_file":"more-morphisms.tex","source_line":5373,"source_end_line":5386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5373-L5386","statement_sha256":"339b58c594c6fd4e1f570af52ed6bd3d37887f3be88e40a29fb35c121c19e4b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7250,"rank":7250,"depth":46,"x":1907.954,"y":711.03,"cluster":"scheme-morphisms"},{"id":"stacks:07TD","tag":"07TD","title":"Normalization and smooth morphisms · Lemma 07TD","summary":"Let X → Y be a smooth morphism of schemes. Assume every quasi-compact open of Y has finitely many irreducible components. Then the same is true for X and there is a unique isomorphism X^ν = X ×_Y Y^ν over X where X^ν, Y^ν are the normalizations of X, Y.","statement_latex":"Let $X \\to Y$ be a smooth morphism of schemes. Assume every quasi-compact\nopen of $Y$ has finitely many irreducible components. Then the same\nis true for $X$ and there is a unique isomorphism $X^\\nu = X \\times_Y Y^\\nu$\nover $X$ where $X^\\nu$, $Y^\\nu$ are the normalizations of $X$, $Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Normalization revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TD","source_file":"more-morphisms.tex","source_line":5409,"source_end_line":5415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5409-L5415","statement_sha256":"dee54234ad9bcad857da9d9d608f25b2c504c677e1e2f187400a0e0ca20b16cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7251,"rank":7251,"depth":38,"x":2095.084,"y":587.806,"cluster":"scheme-morphisms"},{"id":"stacks:0CBM","tag":"0CBM","title":"Normalization and henselization · Lemma 0CBM","summary":"Let X be a locally Noetherian scheme. Let ν : X^ν → X be the normalization morphism. Then for any point x ∈ X the base change X^ν ×_X Spec(O_X, x^h) → Spec(O_X, x^h), resp. X^ν ×_X Spec(O_X, x^sh) → Spec(O_X, x^sh) is the normalization of Spec(O_X, x^h), resp. Spec(O_X, x^sh).","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\nu : X^\\nu \\to X$\nbe the normalization morphism. Then for any point $x \\in X$\nthe base change\n$$\nX^\\nu \\times_X \\Spec(\\mathcal{O}_{X, x}^h) \\to \\Spec(\\mathcal{O}_{X, x}^h),\n\\quad\\text{resp.}\\quad\nX^\\nu \\times_X \\Spec(\\mathcal{O}_{X, x}^{sh}) \\to \\Spec(\\mathcal{O}_{X, x}^{sh})\n$$\nis the normalization of $\\Spec(\\mathcal{O}_{X, x}^h)$,\nresp.\\ $\\Spec(\\mathcal{O}_{X, x}^{sh})$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Normalization revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBM","source_file":"more-morphisms.tex","source_line":5438,"source_end_line":5450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5438-L5450","statement_sha256":"cf8358c8ed23e527aa11b75412334af894a12cceb5b0ceba6f36645e97599c99","origin":"The Stacks Project","memory_eligible":false,"source_rank":7252,"rank":7252,"depth":52,"x":2056.165,"y":784.982,"cluster":"scheme-morphisms"},{"id":"stacks:0390","tag":"0390","title":"Normal morphisms · Definition 0390","summary":"Let f : X → Y be a morphism of schemes. Assume that all the fibres X_y are locally Noetherian schemes. • Let x ∈ X, and y = f(x). We say that f is normal at x if f is flat at x, and the scheme X_y is geometrically normal at x over kappa(y) (see Varieties, Definition [Tag 038M]). • We say f is a normal morphism if f is normal at every point of X.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume that all the fibres $X_y$ are locally Noetherian schemes.\n\\begin{enumerate}\n\\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is {\\it normal at $x$}\nif $f$ is flat at $x$, and the scheme $X_y$ is geometrically\nnormal at $x$ over $\\kappa(y)$ (see\nVarieties, Definition \\ref{varieties-definition-geometrically-normal}).\n\\item We say $f$ is a {\\it normal morphism} if $f$ is normal\nat every point of $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Normal morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0390","source_file":"more-morphisms.tex","source_line":5518,"source_end_line":5530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5518-L5530","statement_sha256":"c78f573b92580b9782c834bf391248bb24048e586498736523d777b904dc6b24","origin":"The Stacks Project","memory_eligible":false,"source_rank":7253,"rank":7253,"depth":1,"x":1926.214,"y":617.393,"cluster":"scheme-morphisms"},{"id":"stacks:0391","tag":"0391","title":"Normal morphisms · Lemma 0391","summary":"Let f : X → Y be a morphism of schemes. Assume all fibres of f are locally Noetherian. The following are equivalent • f is normal, and • f is flat and its fibres are geometrically normal schemes.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume all fibres of $f$ are locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is normal, and\n\\item $f$ is flat and its fibres are geometrically normal schemes.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Normal morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0391","source_file":"more-morphisms.tex","source_line":5537,"source_end_line":5546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5537-L5546","statement_sha256":"475ff17ba2f4571ed1b1ef411326aad6402a5202b0cdc32961239c8cdded3d50","origin":"The Stacks Project","memory_eligible":false,"source_rank":7254,"rank":7254,"depth":0,"x":2156.961,"y":667.266,"cluster":"scheme-morphisms"},{"id":"stacks:056W","tag":"056W","title":"Normal morphisms · Lemma 056W","summary":"A smooth morphism is normal.","statement_latex":"A smooth morphism is normal.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Normal morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056W","source_file":"more-morphisms.tex","source_line":5552,"source_end_line":5555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5552-L5555","statement_sha256":"6d64b242365106437004b3e5a52d78f965afc99cafc4787bd47e2dc15b70b6fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7255,"rank":7255,"depth":44,"x":1946.566,"y":761.484,"cluster":"scheme-morphisms"},{"id":"stacks:0392","tag":"0392","title":"Normal morphisms · Lemma 0392","summary":"The property P(f)=\"the fibres of f are locally Noetherian\" is local in the fppf topology on the source and the target.","statement_latex":"The property $\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian''\nis local in the fppf topology on the source and the target.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Normal morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0392","source_file":"more-morphisms.tex","source_line":5577,"source_end_line":5581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5577-L5581","statement_sha256":"4cc1bc4423c4064c3d38c6a6cc556d39d295ecfae2a20455412de8b16c5e6de1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7256,"rank":7256,"depth":38,"x":2025.993,"y":572.501,"cluster":"scheme-morphisms"},{"id":"stacks:0393","tag":"0393","title":"Normal morphisms · Lemma 0393","summary":"The property P(f)=\"the fibres of f are locally Noetherian and f is normal\" is local in the fppf topology on the target and local in the smooth topology on the source.","statement_latex":"The property\n$\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is normal''\nis local in the fppf topology on the target and\nlocal in the smooth topology on the source.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Normal morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0393","source_file":"more-morphisms.tex","source_line":5606,"source_end_line":5612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5606-L5612","statement_sha256":"637115f64ed6fb007e15a9a2b8ec808a378f52d325187533a938eb31e0c73dad","origin":"The Stacks Project","memory_eligible":false,"source_rank":7257,"rank":7257,"depth":44,"x":2119.461,"y":757.045,"cluster":"scheme-morphisms"},{"id":"stacks:07R7","tag":"07R7","title":"Regular morphisms · Definition 07R7","summary":"Let f : X → Y be a morphism of schemes. Assume that all the fibres X_y are locally Noetherian schemes. • Let x ∈ X, and y = f(x). We say that f is regular at x if f is flat at x, and the scheme X_y is geometrically regular at x over kappa(y) (see Varieties, Definition [Tag 038T]). • We say f is a regular morphism if f is regular at every point of X.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume that all the fibres $X_y$ are locally Noetherian schemes.\n\\begin{enumerate}\n\\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is {\\it regular at $x$}\nif $f$ is flat at $x$, and the scheme $X_y$ is geometrically\nregular at $x$ over $\\kappa(y)$ (see\nVarieties, Definition \\ref{varieties-definition-geometrically-regular}).\n\\item We say $f$ is a {\\it regular morphism} if $f$ is regular\nat every point of $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Regular morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07R7","source_file":"more-morphisms.tex","source_line":5662,"source_end_line":5674,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5662-L5674","statement_sha256":"b342cb962f9cb2d07758958a2bd8c67f3acf88308ea498e079403935163347e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7258,"rank":7258,"depth":1,"x":1901.992,"y":673.947,"cluster":"scheme-morphisms"},{"id":"stacks:07R8","tag":"07R8","title":"Regular morphisms · Lemma 07R8","summary":"Let f : X → Y be a morphism of schemes. Assume all fibres of f are locally Noetherian. The following are equivalent • f is regular, • f is flat and its fibres are geometrically regular schemes, • for every pair of affine opens U ⊂ X, V ⊂ Y with f(U) ⊂ V the ring map O_Y(V) → O_X(U) is regular, • there exists an open covering Y = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j is regular, and • there exists an affine open…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume all fibres of $f$ are locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is regular,\n\\item $f$ is flat and its fibres are geometrically regular schemes,\n\\item for every pair of affine opens $U \\subset X$, $V \\subset Y$\nwith $f(U) \\subset V$ the ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(U)$\nis regular,\n\\item there exists an open covering $Y = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$ is regular, and\n\\item there exists an affine open covering $Y = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat the ring maps $\\mathcal{O}_Y(V_j) \\to \\mathcal{O}_X(U_i)$ are regular.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Regular morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07R8","source_file":"more-morphisms.tex","source_line":5681,"source_end_line":5699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5681-L5699","statement_sha256":"834b29932502e8f7b6071ae53e39c04a488a5d07517946fd43ef97caded63db5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7259,"rank":7259,"depth":43,"x":2129.325,"y":611.784,"cluster":"scheme-morphisms"},{"id":"stacks:07R9","tag":"07R9","title":"Regular morphisms · Lemma 07R9","summary":"A smooth morphism is regular.","statement_latex":"A smooth morphism is regular.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Regular morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07R9","source_file":"more-morphisms.tex","source_line":5735,"source_end_line":5738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5735-L5738","statement_sha256":"07cec3313e01ab9f16eef9be13d46d16a73d8ade28ae2e8920aafef095feb69c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7260,"rank":7260,"depth":44,"x":2011.605,"y":786.731,"cluster":"scheme-morphisms"},{"id":"stacks:07RA","tag":"07RA","title":"Regular morphisms · Lemma 07RA","summary":"The property P(f)=\"the fibres of f are locally Noetherian and f is regular\" is local in the fppf topology on the target and local in the smooth topology on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``the fibres of $f$ are\nlocally Noetherian and $f$ is regular''\nis local in the fppf topology on the target and\nlocal in the smooth topology on the source.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Regular morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RA","source_file":"more-morphisms.tex","source_line":5755,"source_end_line":5761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5755-L5761","statement_sha256":"2dfc1b8bb42faa10e97af7236c38ad5b53ac645b42a30d05c2a12999a663992d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7261,"rank":7261,"depth":44,"x":1957.687,"y":590.802,"cluster":"scheme-morphisms"},{"id":"stacks:045R","tag":"045R","title":"Cohen-Macaulay morphisms · Definition 045R","summary":"Let f : X → Y be a morphism of schemes. Assume that all the fibres X_y are locally Noetherian schemes. • Let x ∈ X, and y = f(x). We say that f is Cohen-Macaulay at x if f is flat at x, and the local ring of the scheme X_y at x is Cohen-Macaulay. • We say f is a Cohen-Macaulay morphism if f is Cohen-Macaulay at every point of X.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume that all the fibres $X_y$ are locally Noetherian schemes.\n\\begin{enumerate}\n\\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is\n{\\it Cohen-Macaulay at $x$} if $f$ is flat at $x$, and the\nlocal ring of the scheme $X_y$ at $x$ is Cohen-Macaulay.\n\\item We say $f$ is a {\\it Cohen-Macaulay morphism} if $f$ is\nCohen-Macaulay at every point of $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045R","source_file":"more-morphisms.tex","source_line":5815,"source_end_line":5826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5815-L5826","statement_sha256":"6b133eb33d2e857dc9a4b9446d8f6d6114c11fa77f21ca879bac97e4f4311561","origin":"The Stacks Project","memory_eligible":false,"source_rank":7262,"rank":7262,"depth":0,"x":2155.134,"y":704.757,"cluster":"scheme-morphisms"},{"id":"stacks:045S","tag":"045S","title":"Cohen-Macaulay morphisms · Lemma 045S","summary":"Let f : X → Y be a morphism of schemes. Assume all fibres of f are locally Noetherian. The following are equivalent • f is Cohen-Macaulay, and • f is flat and its fibres are Cohen-Macaulay schemes.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume all fibres of $f$ are locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is Cohen-Macaulay, and\n\\item $f$ is flat and its fibres are Cohen-Macaulay schemes.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045S","source_file":"more-morphisms.tex","source_line":5831,"source_end_line":5840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5831-L5840","statement_sha256":"a658dd62d3a1387313eae24838f69584d5b5c9d769d613e54a8496a147cec4c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7263,"rank":7263,"depth":0,"x":1917.746,"y":732.783,"cluster":"scheme-morphisms"},{"id":"stacks:0AFG","tag":"0AFG","title":"Cohen-Macaulay morphisms · Lemma 0AFG","summary":"Let f : X → Y be a morphism of locally Noetherian schemes which is locally of finite type and Cohen-Macaulay. For every point x in X with image y in Y, dim_x(X) = dim_y(Y) + dim_x(X_y), where X_y denotes the fiber over y.","statement_latex":"Let $f : X \\to Y$ be a morphism of locally Noetherian schemes\nwhich is locally of finite type and Cohen-Macaulay.\nFor every point $x$ in $X$ with image $y$ in $Y$,\n$$\n\\dim_x(X) = \\dim_y(Y) + \\dim_x(X_y),\n$$\nwhere $X_y$ denotes the fiber over $y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFG","source_file":"more-morphisms.tex","source_line":5846,"source_end_line":5855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5846-L5855","statement_sha256":"e930619daa25f999d809305e41fdaa7ae10983287d217af3d54ceb0b39c53873","origin":"The Stacks Project","memory_eligible":false,"source_rank":7264,"rank":7264,"depth":37,"x":2070.354,"y":577.316,"cluster":"scheme-morphisms"},{"id":"stacks:0C0W","tag":"0C0W","title":"Cohen-Macaulay morphisms · Lemma 0C0W","summary":"Let f : X → Y and g : Y → Z be morphisms of schemes. Assume that the fibres of f, g, and g ∘ f are locally Noetherian. Let x ∈ X with images y ∈ Y and z ∈ Z. • If f is Cohen-Macaulay at x and g is Cohen-Macaulay at f(x), then g ∘ f is Cohen-Macaulay at x. • If f and g are Cohen-Macaulay, then g ∘ f is Cohen-Macaulay. • If g ∘ f is Cohen-Macaulay at x and f is flat at x, then f is Cohen-Macaulay at x and g is Cohen-Macaulay at f(x). • If g ∘ f is Cohen-Macaulay and f is…","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes. Assume that the\nfibres of $f$, $g$, and $g \\circ f$ are locally Noetherian.\nLet $x \\in X$ with images $y \\in Y$ and $z \\in Z$.\n\\begin{enumerate}\n\\item If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay\nat $f(x)$, then $g \\circ f$ is Cohen-Macaulay at $x$.\n\\item If $f$ and $g$ are Cohen-Macaulay, then $g \\circ f$ is Cohen-Macaulay.\n\\item If $g \\circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$,\nthen $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$.\n\\item If $g \\circ f$ is Cohen-Macaulay and $f$ is flat, then\n$f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in\nthe image of $f$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0W","source_file":"more-morphisms.tex","source_line":5868,"source_end_line":5883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5868-L5883","statement_sha256":"1c2b815bc23e90cf707cf1928137836a6acb42411e53b28ca0c1b94096ff8a8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7265,"rank":7265,"depth":17,"x":2082.854,"y":778.68,"cluster":"scheme-morphisms"},{"id":"stacks:0C0X","tag":"0C0X","title":"Cohen-Macaulay morphisms · Lemma 0C0X","summary":"Let f : X → Y be a flat morphism of locally Noetherian schemes. If X is Cohen-Macaulay, then f is Cohen-Macaulay and O_Y, f(x) is Cohen-Macaulay for all x ∈ X.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of locally Noetherian schemes.\nIf $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and\n$\\mathcal{O}_{Y, f(x)}$ is Cohen-Macaulay for all $x \\in X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0X","source_file":"more-morphisms.tex","source_line":5899,"source_end_line":5904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5899-L5904","statement_sha256":"c73a84c2c092684ed3cc94ab58e67fb9c298195aca142a1b7edefb33426eedcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7266,"rank":7266,"depth":17,"x":1911.594,"y":637.198,"cluster":"scheme-morphisms"},{"id":"stacks:045T","tag":"045T","title":"Cohen-Macaulay morphisms · Lemma 045T","summary":"Let f : X → Y be a morphism of schemes. Assume that all the fibres X_y are locally Noetherian schemes. Let Y' → Y be locally of finite type. Let f' : X' → Y' be the base change of f. Let x' ∈ X' be a point with image x ∈ X. • If f is Cohen-Macaulay at x, then f' : X' → Y' is Cohen-Macaulay at x'. • If f is flat at x and f' is Cohen-Macaulay at x', then f is Cohen-Macaulay at x. • If Y' → Y is flat at f'(x') and f' is Cohen-Macaulay at x', then f is Cohen-Macaulay at x.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume that all the fibres $X_y$ are locally Noetherian schemes.\nLet $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$\nbe the base change of $f$.\nLet $x' \\in X'$ be a point with image $x \\in X$.\n\\begin{enumerate}\n\\item If $f$ is Cohen-Macaulay at $x$, then\n$f' : X' \\to Y'$ is Cohen-Macaulay at $x'$.\n\\item If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$\nis Cohen-Macaulay at $x$.\n\\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at\n$x'$, then $f$ is Cohen-Macaulay at $x$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045T","source_file":"more-morphisms.tex","source_line":5911,"source_end_line":5926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5911-L5926","statement_sha256":"f078eaa68202108771c3039e816318a79c4ed905d22c7be679d70b8b9705c9fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7267,"rank":7267,"depth":32,"x":2151.81,"y":644.352,"cluster":"scheme-morphisms"},{"id":"stacks:045U","tag":"045U","title":"Cohen-Macaulay morphisms · Lemma 045U","summary":"[EGA] Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. Let W = (x ∈ X mid f is Cohen-Macaulay at x) Then • W = (x ∈ X mid O_X_f(x), x is Cohen-Macaulay), • W is open in X, • W is dense in every fibre of X → S, • the formation of W commutes with arbitrary base change of f: For any morphism g : S' → S, consider the base change f' : X' → S' of f and the projection g' : X' → X. Then the corresponding set W' for the morphism f' is equal…","statement_latex":"\\begin{reference}\n\\cite[IV Corollary 12.1.7(iii)]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes which is flat and locally\nof finite presentation. Let\n$$\nW = \\{x \\in X \\mid f\\text{ is Cohen-Macaulay at }x\\}\n$$\nThen\n\\begin{enumerate}\n\\item $W = \\{x \\in X \\mid \\mathcal{O}_{X_{f(x)}, x}\\text{ is Cohen-Macaulay}\\}$,\n\\item $W$ is open in $X$,\n\\item $W$ is dense in every fibre of $X \\to S$,\n\\item the formation of $W$ commutes with arbitrary base change of $f$:\nFor any morphism $g : S' \\to S$, consider\nthe base change $f' : X' \\to S'$ of $f$ and the\nprojection $g' : X' \\to X$. Then the corresponding\nset $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045U","source_file":"more-morphisms.tex","source_line":5980,"source_end_line":6001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L5980-L6001","statement_sha256":"eb6cb75ff0735b181a976b53524c78766f6a91ee3fd8b79386a6fc9bf7824abf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7268,"rank":7268,"depth":36,"x":1968.806,"y":775.466,"cluster":"scheme-morphisms"},{"id":"stacks:0BUU","tag":"0BUU","title":"Cohen-Macaulay morphisms · Lemma 0BUU","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. Let x ∈ X with image s ∈ S. Set d = dim_x(X_s). The following are equivalent • f is Cohen-Macaulay at x, • there exists an open neighbourhood U ⊂ X of x and a locally quasi-finite morphism U → A^d_S over S which is flat at x, • there exists an open neighbourhood U ⊂ X of x and a locally quasi-finite flat morphism U → A^d_S over S, • for any S-morphism g : U → A^d_S of an open…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and locally\nof finite presentation. Let $x \\in X$ with image $s \\in S$.\nSet $d = \\dim_x(X_s)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is Cohen-Macaulay at $x$,\n\\item there exists an open neighbourhood $U \\subset X$ of $x$\nand a locally quasi-finite morphism $U \\to \\mathbf{A}^d_S$ over $S$\nwhich is flat at $x$,\n\\item there exists an open neighbourhood $U \\subset X$ of $x$\nand a locally quasi-finite flat morphism $U \\to \\mathbf{A}^d_S$ over $S$,\n\\item for any $S$-morphism $g : U \\to \\mathbf{A}^d_S$\nof an open neighbourhood $U \\subset X$ of $x$ we have:\n$g$ is quasi-finite at $x$ $\\Rightarrow$ $g$ is flat at $x$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUU","source_file":"more-morphisms.tex","source_line":6013,"source_end_line":6030,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6013-L6030","statement_sha256":"85dac1b13b0a77d5aec2bd893458ac8a4b9fcf6083f2a6f021b06e6788d04b0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7269,"rank":7269,"depth":36,"x":1998.332,"y":574.814,"cluster":"scheme-morphisms"},{"id":"stacks:054T","tag":"054T","title":"Cohen-Macaulay morphisms · Lemma 054T","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. For d ≥ 0 there exist opens U_d ⊂ X with the following properties • W = ⋃_d ≥ 0 U_d is dense in every fibre of f, and • U_d → S is of relative dimension d (see Morphisms, Definition [Tag 02NJ]).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and locally\nof finite presentation. For $d \\geq 0$ there exist opens $U_d \\subset X$\nwith the following properties\n\\begin{enumerate}\n\\item $W = \\bigcup_{d \\geq 0} U_d$ is dense in every fibre of $f$, and\n\\item $U_d \\to S$ is of relative dimension $d$ (see\nMorphisms, Definition \\ref{morphisms-definition-relative-dimension-d}).\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054T","source_file":"more-morphisms.tex","source_line":6067,"source_end_line":6077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6067-L6077","statement_sha256":"8a79915aa735e35feab918544da7e7545810a5eb5ac52c2dce823598cc5bf4a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7270,"rank":7270,"depth":37,"x":2138.009,"y":739.632,"cluster":"scheme-morphisms"},{"id":"stacks:054U","tag":"054U","title":"Cohen-Macaulay morphisms · Lemma 054U","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. Suppose x' leadsto x is a specialization of points of X with image s' leadsto s in S. If x is a generic point of an irreducible component of X_s then dim_x'(X_s') = dim_x(X_s).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and locally\nof finite presentation.\nSuppose $x' \\leadsto x$ is a specialization of points of $X$\nwith image $s' \\leadsto s$ in $S$. If $x$ is a generic point of an\nirreducible component of $X_s$ then $\\dim_{x'}(X_{s'}) = \\dim_x(X_s)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054U","source_file":"more-morphisms.tex","source_line":6087,"source_end_line":6094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6087-L6094","statement_sha256":"ff17cc84817b82ba1147e58ee1d673c937bd61e2491d493df55ae873c0416fed","origin":"The Stacks Project","memory_eligible":false,"source_rank":7271,"rank":7271,"depth":38,"x":1902.319,"y":697.327,"cluster":"scheme-morphisms"},{"id":"stacks:045V","tag":"045V","title":"Cohen-Macaulay morphisms · Lemma 045V","summary":"The property P(f)=\"the fibres of f are locally Noetherian and f is Cohen-Macaulay\" is local in the fppf topology on the target and local in the syntomic topology on the source.","statement_latex":"The property\n$\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is\nCohen-Macaulay'' is local in the fppf topology on the target and\nlocal in the syntomic topology on the source.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045V","source_file":"more-morphisms.tex","source_line":6101,"source_end_line":6107,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6101-L6107","statement_sha256":"decad6b85c2ee25f17ffcb416c22cbe75348b053867fe49749c02ddcb3fc4e93","origin":"The Stacks Project","memory_eligible":false,"source_rank":7272,"rank":7272,"depth":40,"x":2110.268,"y":594.719,"cluster":"scheme-morphisms"},{"id":"stacks:056Y","tag":"056Y","title":"Slicing Cohen-Macaulay morphisms · Lemma 056Y","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point with image s ∈ S. Let h ∈ m_x ⊂ O_X, x. Assume • f is locally of finite presentation, • f is flat at x, and • the image overlineh of h in O_X_s, x = O_X, x/ m_sO_X, x is a nonzerodivisor. Then there exists an affine open neighbourhood U ⊂ X of x such that h comes from h ∈ Γ(U, O_U) and such that D = V(h) is an effective Cartier divisor in U with x ∈ D and D → S flat and locally of finite presentation.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point with image $s \\in S$.\nLet $h \\in \\mathfrak m_x \\subset \\mathcal{O}_{X, x}$.\nAssume\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $f$ is flat at $x$, and\n\\item the image $\\overline{h}$ of $h$ in\n$\\mathcal{O}_{X_s, x} = \\mathcal{O}_{X, x}/\\mathfrak m_s\\mathcal{O}_{X, x}$\nis a nonzerodivisor.\n\\end{enumerate}\nThen there exists an affine open neighbourhood $U \\subset X$ of $x$\nsuch that $h$ comes from $h \\in \\Gamma(U, \\mathcal{O}_U)$ and such\nthat $D = V(h)$ is an effective Cartier divisor in $U$ with $x \\in D$ and\n$D \\to S$ flat and locally of finite presentation.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056Y","source_file":"more-morphisms.tex","source_line":6165,"source_end_line":6182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6165-L6182","statement_sha256":"63e387b6e3ddb57975229b5f55f73e92deaa8753f018c2fbaabd5d588e8a94f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7273,"rank":7273,"depth":35,"x":2039.398,"y":788.5,"cluster":"scheme-morphisms"},{"id":"stacks:06LI","tag":"06LI","title":"Slicing Cohen-Macaulay morphisms · Lemma 06LI","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point with image s ∈ S. Let h_1, …, h_r ∈ O_X, x. Assume • f is locally of finite presentation, • f is flat at x, and • the images of h_1, …, h_r in O_X_s, x = O_X, x/ m_sO_X, x form a regular sequence. Then there exists an affine open neighbourhood U ⊂ X of x such that h_1, …, h_r come from h_1, …, h_r ∈ Γ(U, O_U) and such that Z = V(h_1, …, h_r) → U is a regular immersion with x ∈ Z and Z → S flat and locally of…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point with image $s \\in S$.\nLet $h_1, \\ldots, h_r \\in \\mathcal{O}_{X, x}$.\nAssume\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $f$ is flat at $x$, and\n\\item the images of $h_1, \\ldots, h_r$ in\n$\\mathcal{O}_{X_s, x} = \\mathcal{O}_{X, x}/\\mathfrak m_s\\mathcal{O}_{X, x}$\nform a regular sequence.\n\\end{enumerate}\nThen there exists an affine open neighbourhood $U \\subset X$ of $x$\nsuch that $h_1, \\ldots, h_r$ come from\n$h_1, \\ldots, h_r \\in \\Gamma(U, \\mathcal{O}_U)$ and such\nthat $Z = V(h_1, \\ldots, h_r) \\to U$ is a regular immersion with\n$x \\in Z$ and $Z \\to S$ flat and locally of finite presentation.\nMoreover, the base change $Z_{S'} \\to U_{S'}$ is a regular immersion\nfor any scheme $S'$ over $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LI","source_file":"more-morphisms.tex","source_line":6263,"source_end_line":6283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6263-L6283","statement_sha256":"6b3f48553e4967517b346130e6502d7c0ad057e43b34babb3a4d3475697ea71b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7274,"rank":7274,"depth":38,"x":1935.757,"y":605.278,"cluster":"scheme-morphisms"},{"id":"stacks:056Z","tag":"056Z","title":"Slicing Cohen-Macaulay morphisms · Lemma 056Z","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point with image s ∈ S. Assume • f is locally of finite presentation, • f is flat at x, and • O_X_s, x has depth ≥ 1. Then there exists an affine open neighbourhood U ⊂ X of x and an effective Cartier divisor D ⊂ U containing x such that D → S is flat and of finite presentation.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point with image $s \\in S$.\nAssume\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $f$ is flat at $x$, and\n\\item $\\mathcal{O}_{X_s, x}$ has $\\text{depth} \\geq 1$.\n\\end{enumerate}\nThen there exists an affine open neighbourhood $U \\subset X$ of $x$\nand an effective Cartier divisor $D \\subset U$ containing $x$ such that\n$D \\to S$ is flat and of finite presentation.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/056Z","source_file":"more-morphisms.tex","source_line":6336,"source_end_line":6349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6336-L6349","statement_sha256":"c9b0663c0620360caf00a81ff778fe7c59ab523445dfde3db0964c4e55476a59","origin":"The Stacks Project","memory_eligible":false,"source_rank":7275,"rank":7275,"depth":36,"x":2159.666,"y":681.624,"cluster":"scheme-morphisms"},{"id":"stacks:0570","tag":"0570","title":"Slicing Cohen-Macaulay morphisms · Lemma 0570","summary":"[EGA] Let f : X → S be a morphism of schemes. Let x ∈ X be a point with image s ∈ S. Assume • f is locally of finite presentation, • f is Cohen-Macaulay at x, and • x is a closed point of X_s. Then there exists a regular immersion Z → X containing x such that • [(a)] Z → S is flat and locally of finite presentation, • [(b)] Z → S is locally quasi-finite, and • [(c)] Z_s = (x) set theoretically.","statement_latex":"\\begin{reference}\n\\cite[IV Proposition 17.16.1]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point with image $s \\in S$.\nAssume\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $f$ is Cohen-Macaulay at $x$, and\n\\item $x$ is a closed point of $X_s$.\n\\end{enumerate}\nThen there exists a regular immersion $Z \\to X$ containing $x$ such that\n\\begin{enumerate}\n\\item[(a)] $Z \\to S$ is flat and locally of finite presentation,\n\\item[(b)] $Z \\to S$ is locally quasi-finite, and\n\\item[(c)] $Z_s = \\{x\\}$ set theoretically.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0570","source_file":"more-morphisms.tex","source_line":6357,"source_end_line":6376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6357-L6376","statement_sha256":"ffc540472cd692cdfe9321bdf88f7cf3428f8956117afbf3e7687066b7236e24","origin":"The Stacks Project","memory_eligible":false,"source_rank":7276,"rank":7276,"depth":37,"x":1933.018,"y":752.425,"cluster":"scheme-morphisms"},{"id":"stacks:0571","tag":"0571","title":"Slicing Cohen-Macaulay morphisms · Lemma 0571","summary":"Let f : X → S be a flat morphism of schemes which is locally of finite presentation. Let s ∈ S be a point in the image of f. Then there exists a commutative diagram xymatrix S' ar[rr] ar[rd]_g & & X ar[ld]^f & S where g : S' → S is flat, locally of finite presentation, locally quasi-finite, and s ∈ g(S').","statement_latex":"Let $f : X \\to S$ be a flat morphism of schemes which is\nlocally of finite presentation. Let $s \\in S$ be a point in the image of $f$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\nS' \\ar[rr] \\ar[rd]_g & & X \\ar[ld]^f \\\\\n& S\n}\n$$\nwhere $g : S' \\to S$ is flat, locally of finite presentation,\nlocally quasi-finite, and $s \\in g(S')$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0571","source_file":"more-morphisms.tex","source_line":6436,"source_end_line":6449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6436-L6449","statement_sha256":"6af8b50312f0890826fdb28c68582596ccdae7bb65be3580a7380f977dbf55d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7277,"rank":7277,"depth":38,"x":2043.279,"y":571.496,"cluster":"scheme-morphisms"},{"id":"stacks:0572","tag":"0572","title":"Slicing Cohen-Macaulay morphisms · Lemma 0572","summary":"Let S be a scheme. Let U = (S_i → S)_i ∈ I be an fppf covering of S, see Topologies, Definition [Tag 021M]. Then there exists an fppf covering V = (T_j → S)_j ∈ J which refines (see Sites, Definition [Tag 00VT]) U such that each T_j → S is locally quasi-finite.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{U} = \\{S_i \\to S\\}_{i \\in I}$ be an fppf\ncovering of $S$, see\nTopologies, Definition \\ref{topologies-definition-fppf-covering}.\nThen there exists an fppf covering $\\mathcal{V} = \\{T_j \\to S\\}_{j \\in J}$\nwhich refines (see\nSites, Definition \\ref{sites-definition-morphism-coverings})\n$\\mathcal{U}$ such that each $T_j \\to S$ is locally quasi-finite.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0572","source_file":"more-morphisms.tex","source_line":6465,"source_end_line":6474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6465-L6474","statement_sha256":"0b2a50a8261ca96530cdd7382d229ee44a2074d01e2c9d0bcfb9f6a3e4cfd672","origin":"The Stacks Project","memory_eligible":false,"source_rank":7278,"rank":7278,"depth":39,"x":2107.513,"y":767.6,"cluster":"scheme-morphisms"},{"id":"stacks:054W","tag":"054W","title":"Generic fibres · Lemma 054W","summary":"Let f : X → Y be a finite type morphism of schemes. Assume Y irreducible with generic point eta. If X_eta = ∅ then there exists a nonempty open V ⊂ Y such that X_V = V ×_Y X = ∅.","statement_latex":"Let $f : X \\to Y$ be a finite type morphism of schemes. Assume\n$Y$ irreducible with generic point $\\eta$. If $X_\\eta = \\emptyset$\nthen there exists a nonempty open $V \\subset Y$ such that\n$X_V = V \\times_Y X = \\emptyset$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054W","source_file":"more-morphisms.tex","source_line":6495,"source_end_line":6501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6495-L6501","statement_sha256":"221c255aacbde9d17bcb61cb6cd58d974ef2760f8e6b2ca0d63d9ac9b8c81486","origin":"The Stacks Project","memory_eligible":false,"source_rank":7279,"rank":7279,"depth":21,"x":1902.316,"y":659.375,"cluster":"scheme-morphisms"},{"id":"stacks:05F5","tag":"05F5","title":"Generic fibres · Lemma 05F5","summary":"Let f : X → Y be a finite type morphism of schemes. Assume Y irreducible with generic point eta. If X_eta not = ∅ then there exists a nonempty open V ⊂ Y such that X_V = V ×_Y X → V is surjective.","statement_latex":"Let $f : X \\to Y$ be a finite type morphism of schemes. Assume\n$Y$ irreducible with generic point $\\eta$. If $X_\\eta \\not = \\emptyset$\nthen there exists a nonempty open $V \\subset Y$ such that\n$X_V = V \\times_Y X \\to V$ is surjective.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05F5","source_file":"more-morphisms.tex","source_line":6509,"source_end_line":6515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6509-L6515","statement_sha256":"250b5740ecba5d07e546790a572b3bd60afbac714d556c35472d6f5805aec132","origin":"The Stacks Project","memory_eligible":false,"source_rank":7280,"rank":7280,"depth":7,"x":2140.809,"y":622.721,"cluster":"scheme-morphisms"},{"id":"stacks:054X","tag":"054X","title":"Generic fibres · Lemma 054X","summary":"Let f : X → Y be a finite type morphism of schemes. Assume Y irreducible with generic point eta. If Z ⊂ X is a closed subset with Z_eta nowhere dense in X_eta, then there exists a nonempty open V ⊂ Y such that Z_y is nowhere dense in X_y for all y ∈ V.","statement_latex":"Let $f : X \\to Y$ be a finite type morphism of schemes. Assume\n$Y$ irreducible with generic point $\\eta$.\nIf $Z \\subset X$ is a closed subset with $Z_\\eta$ nowhere dense\nin $X_\\eta$, then there exists a nonempty open $V \\subset Y$ such\nthat $Z_y$ is nowhere dense in $X_y$ for all $y \\in V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054X","source_file":"more-morphisms.tex","source_line":6523,"source_end_line":6530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6523-L6530","statement_sha256":"fe6a1d941b8a6892ca5c8d4387f72d2f0380e07850fa5b9bf42fe0d5d0ba12ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":7281,"rank":7281,"depth":39,"x":1994.33,"y":785.179,"cluster":"scheme-morphisms"},{"id":"stacks:0573","tag":"0573","title":"Generic fibres · Lemma 0573","summary":"Let f : X → Y be a finite type morphism of schemes. Assume Y irreducible with generic point eta. Let U ⊂ X be an open subscheme such that U_eta is scheme theoretically dense in X_eta. Then there exists a nonempty open V ⊂ Y such that U_y is scheme theoretically dense in X_y for all y ∈ V.","statement_latex":"Let $f : X \\to Y$ be a finite type morphism of schemes.\nAssume $Y$ irreducible with generic point $\\eta$.\nLet $U \\subset X$ be an open subscheme such that $U_\\eta$ is\nscheme theoretically dense in $X_\\eta$.\nThen there exists a nonempty open $V \\subset Y$ such\nthat $U_y$ is scheme theoretically dense in $X_y$ for all $y \\in V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0573","source_file":"more-morphisms.tex","source_line":6562,"source_end_line":6570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6562-L6570","statement_sha256":"c16f7b64f5a7003fba2a1465bdfcca9e37d05292e0b040d501e93d30de768e86","origin":"The Stacks Project","memory_eligible":false,"source_rank":7282,"rank":7282,"depth":15,"x":1971.683,"y":582.14,"cluster":"scheme-morphisms"},{"id":"stacks:054Y","tag":"054Y","title":"Generic fibres · Lemma 054Y","summary":"Let f : X → Y be a finite type morphism of schemes. Assume Y irreducible with generic point eta. Let X_eta = Z_1, eta ∪ … ∪ Z_n, eta be a covering of the generic fibre by closed subsets of X_eta. Let Z_i be the closure of Z_i, eta in X (see discussion above). Then there exists a nonempty open V ⊂ Y such that X_y = Z_1, y ∪ … ∪ Z_n, y for all y ∈ V.","statement_latex":"Let $f : X \\to Y$ be a finite type morphism of schemes. Assume\n$Y$ irreducible with generic point $\\eta$. Let\n$X_\\eta = Z_{1, \\eta} \\cup \\ldots \\cup Z_{n, \\eta}$ be a covering of\nthe generic fibre by closed subsets of $X_\\eta$.\nLet $Z_i$ be the closure of $Z_{i, \\eta}$ in $X$ (see discussion above).\nThen there exists a nonempty open $V \\subset Y$ such\nthat $X_y = Z_{1, y} \\cup \\ldots \\cup Z_{n, y}$ for all $y \\in V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054Y","source_file":"more-morphisms.tex","source_line":6657,"source_end_line":6666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6657-L6666","statement_sha256":"60d742c80f11ba41272a0964fc396088b90b55b3f1a7e5da901827e35be89f5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7283,"rank":7283,"depth":22,"x":2151.775,"y":719.094,"cluster":"scheme-morphisms"},{"id":"stacks:054Z","tag":"054Z","title":"Generic fibres · Lemma 054Z","summary":"Let f : X → Y be a morphism of schemes. Let eta ∈ Y be a generic point of an irreducible component of Y. Then (X_eta)_red = (X_red)_eta.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $\\eta \\in Y$ be a generic\npoint of an irreducible component of $Y$. Then\n$(X_\\eta)_{red} = (X_{red})_\\eta$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/054Z","source_file":"more-morphisms.tex","source_line":6689,"source_end_line":6694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6689-L6694","statement_sha256":"570eb067de51b549233b696ddb6e246a66fbbcef1d684e93cc66bbbd59369d59","origin":"The Stacks Project","memory_eligible":false,"source_rank":7284,"rank":7284,"depth":2,"x":1908.689,"y":720.296,"cluster":"scheme-morphisms"},{"id":"stacks:0550","tag":"0550","title":"Generic fibres · Lemma 0550","summary":"Let f : X → Y be a morphism of schemes. Assume that Y is irreducible and f is of finite type. There exists a diagram xymatrix X' ar[d]_f' ar[r]_g' & X_V ar[r] ar[d] & X ar[d]^f Y' ar[r]^g & V ar[r] & Y where • V is a nonempty open of Y, • X_V = V ×_Y X, • g : Y' → V is a finite universal homeomorphism, • X' = (Y' ×_Y X)_red = (Y' ×_V X_V)_red, • g' is a finite universal homeomorphism, • Y' is an integral affine scheme, • f' is flat and of finite presentation, and • the…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume that $Y$ is irreducible and $f$ is of finite type.\nThere exists a diagram\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X_V \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\nY' \\ar[r]^g & V \\ar[r] & Y\n}\n$$\nwhere\n\\begin{enumerate}\n\\item $V$ is a nonempty open of $Y$,\n\\item $X_V = V \\times_Y X$,\n\\item $g : Y' \\to V$ is a finite universal homeomorphism,\n\\item $X' = (Y' \\times_Y X)_{red} = (Y' \\times_V X_V)_{red}$,\n\\item $g'$ is a finite universal homeomorphism,\n\\item $Y'$ is an integral affine scheme,\n\\item $f'$ is flat and of finite presentation, and\n\\item the generic fibre of $f'$ is geometrically reduced.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0550","source_file":"more-morphisms.tex","source_line":6718,"source_end_line":6740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6718-L6740","statement_sha256":"5097440c6d0fbc0e3c03360dcd35201cb88e5479795f507b8a379afd7fe07249","origin":"The Stacks Project","memory_eligible":false,"source_rank":7285,"rank":7285,"depth":16,"x":2087.084,"y":581.389,"cluster":"scheme-morphisms"},{"id":"stacks:0551","tag":"0551","title":"Generic fibres · Lemma 0551","summary":"Let f : X → Y be a morphism of schemes. Assume that Y is irreducible and f is of finite type. There exists a diagram xymatrix X' ar[d]_f' ar[r]_g' & X_V ar[r] ar[d] & X ar[d]^f Y' ar[r]^g & V ar[r] & Y where • V is a nonempty open of Y, • X_V = V ×_Y X, • g : Y' → V is surjective finite étale, • X' = Y' ×_Y X = Y' ×_V X_V, • g' is surjective finite étale, • Y' is an irreducible affine scheme, and • all irreducible components of the generic fibre of f' are geometrically…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume that $Y$ is irreducible and $f$ is of finite type.\nThere exists a diagram\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X_V \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\nY' \\ar[r]^g & V \\ar[r] & Y\n}\n$$\nwhere\n\\begin{enumerate}\n\\item $V$ is a nonempty open of $Y$,\n\\item $X_V = V \\times_Y X$,\n\\item $g : Y' \\to V$ is surjective finite \\'etale,\n\\item $X' = Y' \\times_Y X = Y' \\times_V X_V$,\n\\item $g'$ is surjective finite \\'etale,\n\\item $Y'$ is an irreducible affine scheme, and\n\\item all irreducible components of the generic fibre of $f'$\nare geometrically irreducible.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0551","source_file":"more-morphisms.tex","source_line":6771,"source_end_line":6793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6771-L6793","statement_sha256":"7a560ac9eed72f17ef4723d36599d73af5b64542445591c1c32cae9d3a4c1e77","origin":"The Stacks Project","memory_eligible":false,"source_rank":7286,"rank":7286,"depth":37,"x":2067.231,"y":785.171,"cluster":"scheme-morphisms"},{"id":"stacks:05F1","tag":"05F1","title":"Relative assassins · Lemma 05F1","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. Let xi ∈ Ass_X/S(F) and set Z = overline(xi) ⊂ X. If f is locally of finite type and F is a finite type O_X-module, then there exists a nonempty open V ⊂ Z such that for every s ∈ f(V) the generic points of V_s are elements of Ass_X/S(F).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\xi \\in \\text{Ass}_{X/S}(\\mathcal{F})$ and set\n$Z = \\overline{\\{\\xi\\}} \\subset X$.\nIf $f$ is locally of finite type and $\\mathcal{F}$ is a\nfinite type $\\mathcal{O}_X$-module, then there exists a nonempty\nopen $V \\subset Z$ such that for every $s \\in f(V)$ the generic\npoints of $V_s$ are elements of $\\text{Ass}_{X/S}(\\mathcal{F})$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05F1","source_file":"more-morphisms.tex","source_line":6840,"source_end_line":6850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6840-L6850","statement_sha256":"9bc0ab6d07290fbe2b5a79d5905040a25d9ae0574d83aa201763bca0bce75304","origin":"The Stacks Project","memory_eligible":false,"source_rank":7287,"rank":7287,"depth":15,"x":1917.9,"y":623.538,"cluster":"scheme-morphisms"},{"id":"stacks:05KN","tag":"05KN","title":"Relative assassins · Lemma 05KN","summary":"Let f : X → Y be a morphism of schemes. Let F be a quasi-coherent O_X-module. Let U ⊂ X be an open subscheme. Assume • f is of finite type, • F is of finite type, • Y is irreducible with generic point eta, and • Ass_X_eta(F_eta) is not contained in U_eta. Then there exists a nonempty open subscheme V ⊂ Y such that for all y ∈ V the set Ass_X_y(F_y) is not contained in U_y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a\nquasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be an open\nsubscheme. Assume\n\\begin{enumerate}\n\\item $f$ is of finite type,\n\\item $\\mathcal{F}$ is of finite type,\n\\item $Y$ is irreducible with generic point $\\eta$, and\n\\item $\\text{Ass}_{X_\\eta}(\\mathcal{F}_\\eta)$ is not contained in $U_\\eta$.\n\\end{enumerate}\nThen there exists a nonempty open subscheme $V \\subset Y$ such that\nfor all $y \\in V$ the set $\\text{Ass}_{X_y}(\\mathcal{F}_y)$ is not\ncontained in $U_y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KN","source_file":"more-morphisms.tex","source_line":6907,"source_end_line":6921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6907-L6921","statement_sha256":"d4ab1e01955f1ef761ab7ea09f86b42d569309d8b52589b2b41cddf250b42193","origin":"The Stacks Project","memory_eligible":false,"source_rank":7288,"rank":7288,"depth":22,"x":2158.147,"y":658.012,"cluster":"scheme-morphisms"},{"id":"stacks:05KP","tag":"05KP","title":"Relative assassins · Lemma 05KP","summary":"Let f : X → Y be a morphism of schemes. Let F be a quasi-coherent O_X-module. Let U ⊂ X be an open subscheme. Assume • f is of finite type, • F is of finite type, • Y is irreducible with generic point eta, and • Ass_X_eta(F_eta) ⊂ U_eta. Then there exists a nonempty open subscheme V ⊂ Y such that for all y ∈ V we have Ass_X_y(F_y) ⊂ U_y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a\nquasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset X$ be an open\nsubscheme. Assume\n\\begin{enumerate}\n\\item $f$ is of finite type,\n\\item $\\mathcal{F}$ is of finite type,\n\\item $Y$ is irreducible with generic point $\\eta$, and\n\\item $\\text{Ass}_{X_\\eta}(\\mathcal{F}_\\eta) \\subset U_\\eta$.\n\\end{enumerate}\nThen there exists a nonempty open subscheme $V \\subset Y$ such that\nfor all $y \\in V$ we have $\\text{Ass}_{X_y}(\\mathcal{F}_y) \\subset U_y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KP","source_file":"more-morphisms.tex","source_line":6940,"source_end_line":6953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L6940-L6953","statement_sha256":"e2dc4a189f83f7333ed39c2704233f0fff2fe26b8ee494577c394862ecfe9c12","origin":"The Stacks Project","memory_eligible":false,"source_rank":7289,"rank":7289,"depth":16,"x":1953.141,"y":768.983,"cluster":"scheme-morphisms"},{"id":"stacks:05KQ","tag":"05KQ","title":"Relative assassins · Lemma 05KQ","summary":"Let f : X → S be a morphism which is locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let U ⊂ X be an open subscheme. Let g : S' → S be a morphism of schemes, let f' : X' = X_S' → S' be the base change of f, let g' : X' → X be the projection, set F' = (g')^*F, and set U' = (g')^-1(U). Finally, let s' ∈ S' with image s = g(s'). In this case Ass_X_s(F_s) ⊂ U_s ⇔ Ass_X'_s'(F'_s') ⊂ U'_s'.","statement_latex":"Let $f : X \\to S$ be a morphism which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nof finite type. Let $U \\subset X$ be an open subscheme.\nLet $g : S' \\to S$ be a morphism of schemes, let\n$f' : X' = X_{S'} \\to S'$ be the base change of $f$,\nlet $g' : X' \\to X$ be the projection, set\n$\\mathcal{F}' = (g')^*\\mathcal{F}$, and set\n$U' = (g')^{-1}(U)$. Finally, let $s' \\in S'$ with image $s = g(s')$.\nIn this case\n$$\n\\text{Ass}_{X_s}(\\mathcal{F}_s) \\subset U_s\n\\Leftrightarrow\n\\text{Ass}_{X'_{s'}}(\\mathcal{F}'_{s'}) \\subset U'_{s'}.\n$$","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KQ","source_file":"more-morphisms.tex","source_line":7025,"source_end_line":7041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7025-L7041","statement_sha256":"79910425948240f61dc86e2f0e69a829e14a3dcec0bc3d1443fd5975c0304533","origin":"The Stacks Project","memory_eligible":false,"source_rank":7290,"rank":7290,"depth":15,"x":2015.108,"y":570.705,"cluster":"scheme-morphisms"},{"id":"stacks:05KR","tag":"05KR","title":"Relative assassins · Lemma 05KR","summary":"Let f : X → Y be a morphism of finite presentation. Let F be a quasi-coherent O_X-module of finite presentation. Let U ⊂ X be an open subscheme such that U → Y is quasi-compact. Then the set E = (y ∈ Y mid Ass_X_y(F_y) ⊂ U_y) is locally constructible in Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of finite presentation.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nof finite presentation. Let $U \\subset X$ be an open subscheme\nsuch that $U \\to Y$ is quasi-compact. Then the set\n$$\nE = \\{y \\in Y \\mid \\text{Ass}_{X_y}(\\mathcal{F}_y) \\subset U_y\\}\n$$\nis locally constructible in $Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative assassins","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KR","source_file":"more-morphisms.tex","source_line":7050,"source_end_line":7060,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7050-L7060","statement_sha256":"1feb83cb4aef95a4288748d0adcade75be175fc344b93970b51f6ffea290f6fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":7291,"rank":7291,"depth":27,"x":2128.936,"y":752.188,"cluster":"scheme-morphisms"},{"id":"stacks:0575","tag":"0575","title":"Reduced fibres · Lemma 0575","summary":"Let f : X → Y be a morphism of schemes. Assume Y irreducible with generic point eta and f of finite type. If X_eta is nonreduced, then there exists a nonempty open V ⊂ Y such that for all y ∈ V the fibre X_y is nonreduced.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with\ngeneric point $\\eta$ and $f$ of finite type. If $X_\\eta$ is nonreduced,\nthen there exists a nonempty open $V \\subset Y$\nsuch that for all $y \\in V$ the fibre $X_y$ is nonreduced.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0575","source_file":"more-morphisms.tex","source_line":7109,"source_end_line":7115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7109-L7115","statement_sha256":"827d696eff67d74f86c0ac2b11e21db009d1ae51c18190fe6900dc5dc60861c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7292,"rank":7292,"depth":15,"x":1898.913,"y":682.91,"cluster":"scheme-morphisms"},{"id":"stacks:0576","tag":"0576","title":"Reduced fibres · Lemma 0576","summary":"Let f : X → Y be a morphism of schemes. Let g : Y' → Y be any morphism, and denote f' : X' → Y' the base change of f. Then (y' ∈ Y' mid X'_y' is geometrically reduced) = g^-1((y ∈ Y mid X_y is geometrically reduced)).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $g : Y' \\to Y$ be any morphism, and denote\n$f' : X' \\to Y'$ the base change of $f$.\nThen\n\\begin{align*}\n\\{y' \\in Y' \\mid X'_{y'}\\text{ is geometrically reduced}\\} \\\\\n= g^{-1}(\\{y \\in Y \\mid X_y\\text{ is geometrically reduced}\\}).\n\\end{align*}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0576","source_file":"more-morphisms.tex","source_line":7160,"source_end_line":7170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7160-L7170","statement_sha256":"ab42c9a494c5fe2547e6935cc2c535dda366d9cb8ccd35dd03e228c5f276d781","origin":"The Stacks Project","memory_eligible":false,"source_rank":7293,"rank":7293,"depth":12,"x":2124.381,"y":603.424,"cluster":"scheme-morphisms"},{"id":"stacks:0577","tag":"0577","title":"Reduced fibres · Lemma 0577","summary":"Let f : X → Y be a morphism of schemes. Assume Y irreducible with generic point eta and f of finite type. If X_eta is not geometrically reduced, then there exists a nonempty open V ⊂ Y such that for all y ∈ V the fibre X_y is not geometrically reduced.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with\ngeneric point $\\eta$ and $f$ of finite type. If $X_\\eta$ is not\ngeometrically reduced, then there exists a nonempty open $V \\subset Y$\nsuch that for all $y \\in V$ the fibre $X_y$ is not geometrically reduced.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0577","source_file":"more-morphisms.tex","source_line":7180,"source_end_line":7186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7180-L7186","statement_sha256":"10dec33e1ed1aa26c5e29161f5e12e7d6fa172800713b84a1566dea5bd77116f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7294,"rank":7294,"depth":17,"x":2021.98,"y":790.089,"cluster":"scheme-morphisms"},{"id":"stacks:0578","tag":"0578","title":"Reduced fibres · Lemma 0578","summary":"Let f : X → Y be a morphism of schemes. Assume • Y is irreducible with generic point eta, • X_eta is geometrically reduced, and • f is of finite type. Then there exists a nonempty open subscheme V ⊂ Y such that X_V → V has geometrically reduced fibres.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume\n\\begin{enumerate}\n\\item $Y$ is irreducible with generic point $\\eta$,\n\\item $X_\\eta$ is geometrically reduced, and\n\\item $f$ is of finite type.\n\\end{enumerate}\nThen there exists a nonempty open subscheme $V \\subset Y$\nsuch that $X_V \\to V$ has geometrically reduced fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0578","source_file":"more-morphisms.tex","source_line":7214,"source_end_line":7225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7214-L7225","statement_sha256":"94d71aec39c731db4003037118af6cb1da06e8b666af71b6555c29f18080f92f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7295,"rank":7295,"depth":45,"x":1947.332,"y":594.218,"cluster":"scheme-morphisms"},{"id":"stacks:0579","tag":"0579","title":"Reduced fibres · Lemma 0579","summary":"Let f : X → Y be a morphism which is quasi-compact and locally of finite presentation. Then the set E = (y ∈ Y mid X_y is geometrically reduced) is locally constructible in Y.","statement_latex":"Let $f : X \\to Y$ be a morphism which is quasi-compact and\nlocally of finite presentation. Then the set\n$$\nE = \\{y \\in Y \\mid X_y\\text{ is geometrically reduced}\\}\n$$\nis locally constructible in $Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0579","source_file":"more-morphisms.tex","source_line":7258,"source_end_line":7266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7258-L7266","statement_sha256":"a0799e3ea8050c333f2ad3ae3c963203ac6ca9d1a60b9e5475dc00ac77a43886","origin":"The Stacks Project","memory_eligible":false,"source_rank":7296,"rank":7296,"depth":46,"x":2160.023,"y":696.356,"cluster":"scheme-morphisms"},{"id":"stacks:0C0D","tag":"0C0D","title":"Reduced fibres · Lemma 0C0D","summary":"Let f : X → Spec(R) be a proper morphism where R is a discrete valuation ring. Assume every irreducible component of X dominates Spec(R) (for example if f is flat) and assume the special fibre is reduced. Then both X and the generic fibre X_eta are reduced.","statement_latex":"Let $f : X \\to \\Spec(R)$ be a proper morphism where $R$ is a\ndiscrete valuation ring. Assume every irreducible component\nof $X$ dominates $\\Spec(R)$ (for example if $f$ is flat) and\nassume the special fibre is reduced. Then\nboth $X$ and the generic fibre $X_\\eta$ are reduced.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0D","source_file":"more-morphisms.tex","source_line":7304,"source_end_line":7311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7304-L7311","statement_sha256":"763b23525998e5af97daac18b831936717ea86d47e5d93c06226539cd25a6749","origin":"The Stacks Project","memory_eligible":false,"source_rank":7297,"rank":7297,"depth":14,"x":1920.902,"y":741.756,"cluster":"scheme-morphisms"},{"id":"stacks:0C0E","tag":"0C0E","title":"Reduced fibres · Lemma 0C0E","summary":"Let f : X → Y be a flat proper morphism of finite presentation. Then the set (y ∈ Y mid X_y is geometrically reduced) is open in Y.","statement_latex":"Let $f : X \\to Y$ be a flat proper morphism of finite presentation.\nThen the set $\\{y \\in Y \\mid X_y\\text{ is geometrically reduced}\\}$\nis open in $Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0E","source_file":"more-morphisms.tex","source_line":7341,"source_end_line":7346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7341-L7346","statement_sha256":"5f8f9f64cb88168e47349114eb1e0e5f83704a97b1ba8da014c2babf34aa6884","origin":"The Stacks Project","memory_eligible":false,"source_rank":7298,"rank":7298,"depth":47,"x":2060.804,"y":572.491,"cluster":"scheme-morphisms"},{"id":"stacks:0554","tag":"0554","title":"Irreducible components of fibres · Lemma 0554","summary":"Let f : X → Y be a morphism of schemes. Assume Y irreducible with generic point eta and f of finite type. If X_eta has n irreducible components, then there exists a nonempty open V ⊂ Y such that for all y ∈ V the fibre X_y has at least n irreducible components.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with\ngeneric point $\\eta$ and $f$ of finite type. If $X_\\eta$ has $n$\nirreducible components, then there exists a nonempty open $V \\subset Y$\nsuch that for all $y \\in V$ the fibre $X_y$ has at least $n$\nirreducible components.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Irreducible components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0554","source_file":"more-morphisms.tex","source_line":7392,"source_end_line":7399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7392-L7399","statement_sha256":"9b745add5bc5ce2f2376cd80145fad61ae37d4cf29787d616d4da28071455978","origin":"The Stacks Project","memory_eligible":false,"source_rank":7299,"rank":7299,"depth":40,"x":2093.781,"y":776.815,"cluster":"scheme-morphisms"},{"id":"stacks:0555","tag":"0555","title":"Irreducible components of fibres · Lemma 0555","summary":"Let f : X → Y be a morphism of schemes. Let g : Y' → Y be any morphism, and denote f' : X' → Y' the base change of f. Then (y' ∈ Y' mid X'_y' is geometrically irreducible) = g^-1((y ∈ Y mid X_y is geometrically irreducible)).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $g : Y' \\to Y$ be any morphism, and denote\n$f' : X' \\to Y'$ the base change of $f$.\nThen\n\\begin{align*}\n\\{y' \\in Y' \\mid X'_{y'}\\text{ is geometrically irreducible}\\} \\\\\n= g^{-1}(\\{y \\in Y \\mid X_y\\text{ is geometrically irreducible}\\}).\n\\end{align*}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Irreducible components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0555","source_file":"more-morphisms.tex","source_line":7431,"source_end_line":7441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7431-L7441","statement_sha256":"883843b168249faab34b0a5d138d4844887c359da7050bf20cf608222c523f3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7300,"rank":7300,"depth":3,"x":1905.035,"y":644.779,"cluster":"scheme-morphisms"},{"id":"stacks:0556","tag":"0556","title":"Irreducible components of fibres · Lemma 0556","summary":"Let f : X → Y be a morphism of schemes. Let n_X/Y : Y → (0, 1, 2, 3, …, ∞) be the function which associates to y ∈ Y the number of irreducible components of (X_y)_K where K is a separably closed extension of kappa(y). This is well defined and if g : Y' → Y is a morphism then n_X'/Y' = n_X/Y ∘ g where X' → Y' is the base change of f.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let\n$$\nn_{X/Y} : Y \\to \\{0, 1, 2, 3, \\ldots, \\infty\\}\n$$\nbe the function which associates to $y \\in Y$ the number of irreducible\ncomponents of $(X_y)_K$ where $K$ is a separably closed extension\nof $\\kappa(y)$. This is well defined and if $g : Y' \\to Y$ is a morphism\nthen\n$$\nn_{X'/Y'} = n_{X/Y} \\circ g\n$$\nwhere $X' \\to Y'$ is the base change of $f$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Irreducible components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0556","source_file":"more-morphisms.tex","source_line":7452,"source_end_line":7466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7452-L7466","statement_sha256":"4c56bb57ef8add6586541931cdb0d3a091bdab159fc257f1b9f66f3af3b7ae6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7301,"rank":7301,"depth":15,"x":2150.545,"y":635.035,"cluster":"scheme-morphisms"},{"id":"stacks:0557","tag":"0557","title":"Irreducible components of fibres · Lemma 0557","summary":"Let A be a domain with fraction field K. Let P ∈ A[x_1, …, x_n]. Denote overlineK the algebraic closure of K. Assume P is irreducible in overlineK[x_1, …, x_n]. Then there exists a f ∈ A such that P^φ ∈ kappa[x_1, …, x_n] is irreducible for all homomorphisms φ : A_f → kappa into fields.","statement_latex":"Let $A$ be a domain with fraction field $K$.\nLet $P \\in A[x_1, \\ldots, x_n]$.\nDenote $\\overline{K}$ the algebraic closure of $K$.\nAssume $P$ is irreducible in $\\overline{K}[x_1, \\ldots, x_n]$.\nThen there exists a $f \\in A$ such that\n$P^\\varphi \\in \\kappa[x_1, \\ldots, x_n]$ is irreducible for all\nhomomorphisms $\\varphi : A_f \\to \\kappa$ into fields.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Irreducible components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0557","source_file":"more-morphisms.tex","source_line":7491,"source_end_line":7500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7491-L7500","statement_sha256":"3f3e1d18f6f1d8763e6a0b1a6fc4e75cc14a01a8b413f3d4a39bb432f8f48fec","origin":"The Stacks Project","memory_eligible":false,"source_rank":7302,"rank":7302,"depth":7,"x":1977.238,"y":781.619,"cluster":"scheme-morphisms"},{"id":"stacks:0559","tag":"0559","title":"Irreducible components of fibres · Lemma 0559","summary":"Let f : X → Y be a morphism of schemes. Assume • Y is irreducible with generic point eta, • X_eta is geometrically irreducible, and • f is of finite type. Then there exists a nonempty open subscheme V ⊂ Y such that X_V → V has geometrically irreducible fibres.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume\n\\begin{enumerate}\n\\item $Y$ is irreducible with generic point $\\eta$,\n\\item $X_\\eta$ is geometrically irreducible, and\n\\item $f$ is of finite type.\n\\end{enumerate}\nThen there exists a nonempty open subscheme $V \\subset Y$\nsuch that $X_V \\to V$ has geometrically irreducible fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Irreducible components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0559","source_file":"more-morphisms.tex","source_line":7556,"source_end_line":7567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7556-L7567","statement_sha256":"9fc776e5b0159465027ed0fa3ebd8d9fdcacfac067a4204c5a3dbfc08b169a1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7303,"rank":7303,"depth":40,"x":1987.16,"y":575.065,"cluster":"scheme-morphisms"},{"id":"stacks:055A","tag":"055A","title":"Irreducible components of fibres · Lemma 055A","summary":"Let f : X → Y be a morphism of schemes. Let n_X/Y be the function on Y counting the numbers of geometrically irreducible components of fibres of f introduced in Lemma [Tag 0556]. Assume f of finite type. Let y ∈ Y be a point. Then there exists a nonempty open V ⊂ overline(y) such that n_X/Y|_V is constant.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let\n$n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically\nirreducible components of fibres of $f$ introduced in\nLemma \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}.\nAssume $f$ of finite type.\nLet $y \\in Y$ be a point. Then there exists a nonempty open\n$V \\subset \\overline{\\{y\\}}$ such that $n_{X/Y}|_V$ is constant.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Irreducible components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055A","source_file":"more-morphisms.tex","source_line":7740,"source_end_line":7749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7740-L7749","statement_sha256":"8c7caeaa11449546da56f1fd0d91bc20dbb60dc3586b4b0308cf417925ef7972","origin":"The Stacks Project","memory_eligible":false,"source_rank":7304,"rank":7304,"depth":41,"x":2146.048,"y":733.102,"cluster":"scheme-morphisms"},{"id":"stacks:055B","tag":"055B","title":"Irreducible components of fibres · Lemma 055B","summary":"Let f : X → Y be a morphism of schemes. Let n_X/Y be the function on Y counting the numbers of geometrically irreducible components of fibres of f introduced in Lemma [Tag 0556]. Assume f of finite presentation. Then the level sets E_n = (y ∈ Y mid n_X/Y(y) = n) of n_X/Y are locally constructible in Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let\n$n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically\nirreducible components of fibres of $f$ introduced in\nLemma \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}.\nAssume $f$ of finite presentation. Then the level sets\n$$\nE_n = \\{y \\in Y \\mid n_{X/Y}(y) = n\\}\n$$\nof $n_{X/Y}$ are locally constructible in $Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Irreducible components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055B","source_file":"more-morphisms.tex","source_line":7793,"source_end_line":7804,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7793-L7804","statement_sha256":"fb29b90c97ab8ab3aff6b86a907ba4449c3e6d4becbf0b3201db65bcb326db51","origin":"The Stacks Project","memory_eligible":false,"source_rank":7305,"rank":7305,"depth":42,"x":1901.646,"y":706.708,"cluster":"scheme-morphisms"},{"id":"stacks:055D","tag":"055D","title":"Connected components of fibres · Lemma 055D","summary":"Let f : X → Y be a morphism of schemes. Assume Y irreducible with generic point eta and f of finite type. If X_eta has n connected components, then there exists a nonempty open V ⊂ Y such that for all y ∈ V the fibre X_y has at least n connected components.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with\ngeneric point $\\eta$ and $f$ of finite type. If $X_\\eta$ has $n$\nconnected components, then there exists a nonempty open $V \\subset Y$\nsuch that for all $y \\in V$ the fibre $X_y$ has at least $n$\nconnected components.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055D","source_file":"more-morphisms.tex","source_line":7846,"source_end_line":7853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7846-L7853","statement_sha256":"5f556d28d396fee6c440492665df44d014f67dc33e0706bd7cfb5015037e9982","origin":"The Stacks Project","memory_eligible":false,"source_rank":7306,"rank":7306,"depth":23,"x":2103.213,"y":587.418,"cluster":"scheme-morphisms"},{"id":"stacks:055E","tag":"055E","title":"Connected components of fibres · Lemma 055E","summary":"Let f : X → Y be a morphism of schemes. Let g : Y' → Y be any morphism, and denote f' : X' → Y' the base change of f. Then (y' ∈ Y' mid X'_y' is geometrically connected) = g^-1((y ∈ Y mid X_y is geometrically connected)).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $g : Y' \\to Y$ be any morphism, and denote\n$f' : X' \\to Y'$ the base change of $f$.\nThen\n\\begin{align*}\n\\{y' \\in Y' \\mid X'_{y'}\\text{ is geometrically connected}\\} \\\\\n= g^{-1}(\\{y \\in Y \\mid X_y\\text{ is geometrically connected}\\}).\n\\end{align*}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055E","source_file":"more-morphisms.tex","source_line":7883,"source_end_line":7893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7883-L7893","statement_sha256":"029a5bed68465d2c4a345134897727811786f31cb815057c9ca1a1d30045866f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7307,"rank":7307,"depth":3,"x":2050.478,"y":789.879,"cluster":"scheme-morphisms"},{"id":"stacks:055F","tag":"055F","title":"Connected components of fibres · Lemma 055F","summary":"Let f : X → Y be a morphism of schemes. Let n_X/Y : Y → (0, 1, 2, 3, …, ∞) be the function which associates to y ∈ Y the number of connected components of (X_y)_K where K is a separably closed extension of kappa(y). This is well defined and if g : Y' → Y is a morphism then n_X'/Y' = n_X/Y ∘ g where X' → Y' is the base change of f.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let\n$$\nn_{X/Y} : Y \\to \\{0, 1, 2, 3, \\ldots, \\infty\\}\n$$\nbe the function which associates to $y \\in Y$ the number of connected\ncomponents of $(X_y)_K$ where $K$ is a separably closed extension\nof $\\kappa(y)$. This is well defined and if $g : Y' \\to Y$ is a morphism\nthen\n$$\nn_{X'/Y'} = n_{X/Y} \\circ g\n$$\nwhere $X' \\to Y'$ is the base change of $f$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055F","source_file":"more-morphisms.tex","source_line":7904,"source_end_line":7918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7904-L7918","statement_sha256":"0a78c7764f190f3e3d3f6cded1c30294ac45f229b20b88a3c376b62285fc3d8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7308,"rank":7308,"depth":15,"x":1926.474,"y":610.554,"cluster":"scheme-morphisms"},{"id":"stacks:055G","tag":"055G","title":"Connected components of fibres · Lemma 055G","summary":"Let f : X → Y be a morphism of schemes. Assume • Y is irreducible with generic point eta, • X_eta is geometrically connected, and • f is of finite type. Then there exists a nonempty open subscheme V ⊂ Y such that X_V → V has geometrically connected fibres.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume\n\\begin{enumerate}\n\\item $Y$ is irreducible with generic point $\\eta$,\n\\item $X_\\eta$ is geometrically connected, and\n\\item $f$ is of finite type.\n\\end{enumerate}\nThen there exists a nonempty open subscheme $V \\subset Y$\nsuch that $X_V \\to V$ has geometrically connected fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055G","source_file":"more-morphisms.tex","source_line":7943,"source_end_line":7954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L7943-L7954","statement_sha256":"1bb5f10c001d6d33e917a6af8fa5a5f45ca7aec8ba9385feed5435bb91ad0a4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7309,"rank":7309,"depth":41,"x":2162.266,"y":672.461,"cluster":"scheme-morphisms"},{"id":"stacks:055H","tag":"055H","title":"Connected components of fibres · Lemma 055H","summary":"Let f : X → Y be a morphism of schemes. Let n_X/Y be the function on Y counting the numbers of geometrically connected components of fibres of f introduced in Lemma [Tag 055F]. Assume f of finite type. Let y ∈ Y be a point. Then there exists a nonempty open V ⊂ overline(y) such that n_X/Y|_V is constant.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let\n$n_{X/Y}$ be the function on $Y$ counting the numbers of geometrically\nconnected components of fibres of $f$ introduced in\nLemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}.\nAssume $f$ of finite type.\nLet $y \\in Y$ be a point. Then there exists a nonempty open\n$V \\subset \\overline{\\{y\\}}$ such that $n_{X/Y}|_V$ is constant.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055H","source_file":"more-morphisms.tex","source_line":8035,"source_end_line":8044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8035-L8044","statement_sha256":"850d834d03a28a66080076af5a7543d0bc8ad17197af7b137c5d78e03490dd81","origin":"The Stacks Project","memory_eligible":false,"source_rank":7310,"rank":7310,"depth":42,"x":1938.476,"y":760.659,"cluster":"scheme-morphisms"},{"id":"stacks:055I","tag":"055I","title":"Connected components of fibres · Lemma 055I","summary":"Let f : X → Y be a morphism of schemes. Let n_X/Y be the function on Y counting the numbers of geometric connected components of fibres of f introduced in Lemma [Tag 055F]. Assume f of finite presentation. Then the level sets E_n = (y ∈ Y mid n_X/Y(y) = n) of n_X/Y are locally constructible in Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let\n$n_{X/Y}$ be the function on $Y$ counting the numbers of geometric\nconnected components of fibres of $f$ introduced in\nLemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}.\nAssume $f$ of finite presentation. Then the level sets\n$$\nE_n = \\{y \\in Y \\mid n_{X/Y}(y) = n\\}\n$$\nof $n_{X/Y}$ are locally constructible in $Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055I","source_file":"more-morphisms.tex","source_line":8093,"source_end_line":8104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8093-L8104","statement_sha256":"0609a48ca81cb5d3d582e18fc149bfbba07917001de671dc057e879ce74aaed0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7311,"rank":7311,"depth":43,"x":2032.626,"y":568.522,"cluster":"scheme-morphisms"},{"id":"stacks:055J","tag":"055J","title":"Connected components of fibres · Lemma 055J","summary":"A flat degeneration of a disconnected scheme is either disconnected or nonreduced. Let f : X → S be a morphism of schemes. Assume that • S is the spectrum of a discrete valuation ring, • f is flat, • X is connected, • the closed fibre X_s is reduced. Then the generic fibre X_eta is connected.","statement_latex":"\\begin{slogan}\nA flat degeneration of a disconnected scheme is either disconnected\nor nonreduced.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nAssume that\n\\begin{enumerate}\n\\item $S$ is the spectrum of a discrete valuation ring,\n\\item $f$ is flat,\n\\item $X$ is connected,\n\\item the closed fibre $X_s$ is reduced.\n\\end{enumerate}\nThen the generic fibre $X_\\eta$ is connected.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055J","source_file":"more-morphisms.tex","source_line":8131,"source_end_line":8146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8131-L8146","statement_sha256":"f185db88cbd698ea58da3df1b7ac317df04c82cafca8285e0db8637f425b87d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7312,"rank":7312,"depth":1,"x":2117.765,"y":763.744,"cluster":"scheme-morphisms"},{"id":"stacks:055M","tag":"055M","title":"Connected components meeting a section · Lemma 055M","summary":"Let f : X → Y, s : Y → X be as in Situation [Tag 055L]. If g : Y' → Y is any morphism, consider the base change diagram xymatrix X' ar[r]_g' ar[d]^f' & X ar[d]_f Y' ar@/^1pc/[u]^s' ar[r]^g & Y ar@/_1pc/[u]_s so that we obtain (X')^0 ⊂ X'. Then (X')^0 = (g')^-1(X^0).","statement_latex":"Let $f : X \\to Y$, $s : Y \\to X$ be as in\nSituation \\ref{situation-connected-along-section}.\nIf $g : Y' \\to Y$ is any morphism, consider the base change diagram\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]^{f'} & X \\ar[d]_f \\\\\nY' \\ar@/^1pc/[u]^{s'} \\ar[r]^g & Y \\ar@/_1pc/[u]_s\n}\n$$\nso that we obtain $(X')^0 \\subset X'$.\nThen $(X')^0 = (g')^{-1}(X^0)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components meeting a section","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055M","source_file":"more-morphisms.tex","source_line":8186,"source_end_line":8199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8186-L8199","statement_sha256":"073889a3210fd7184a184a538d02dbd598a4c34faae15cefa46c0a368a947a22","origin":"The Stacks Project","memory_eligible":false,"source_rank":7313,"rank":7313,"depth":17,"x":1897.859,"y":668.041,"cluster":"scheme-morphisms"},{"id":"stacks:055N","tag":"055N","title":"Connected components meeting a section · Lemma 055N","summary":"Let f : X → Y, s : Y → X be as in Situation [Tag 055L]. Assume f of finite type. Let y ∈ Y be a point. Then there exists a nonempty open V ⊂ overline(y) such that the inverse image of X^0 in the base change X_V is open and closed in X_V.","statement_latex":"Let $f : X \\to Y$, $s : Y \\to X$ be as in\nSituation \\ref{situation-connected-along-section}.\nAssume $f$ of finite type. Let $y \\in Y$ be a point.\nThen there exists a nonempty open $V \\subset \\overline{\\{y\\}}$ such that\nthe inverse image of $X^0$ in the base change $X_V$ is open and closed in\n$X_V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components meeting a section","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055N","source_file":"more-morphisms.tex","source_line":8217,"source_end_line":8225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8217-L8225","statement_sha256":"09f2c97203dee33dcb572558948a924a2d94d3fc2fd8c430732e71f604ac9a1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7314,"rank":7314,"depth":42,"x":2137.121,"y":613.798,"cluster":"scheme-morphisms"},{"id":"stacks:055P","tag":"055P","title":"Connected components meeting a section · Lemma 055P","summary":"Let f : X → Y, s : Y → X be as in Situation [Tag 055L]. If f is of finite presentation then X^0 is locally constructible in X.","statement_latex":"Let $f : X \\to Y$, $s : Y \\to X$ be as in\nSituation \\ref{situation-connected-along-section}.\nIf $f$ is of finite presentation then $X^0$ is locally constructible\nin $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components meeting a section","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055P","source_file":"more-morphisms.tex","source_line":8267,"source_end_line":8273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8267-L8273","statement_sha256":"05f44fe8b30c8eb019ea21b9dad8cd4d7518f0b64a26f864013c57e3675d7637","origin":"The Stacks Project","memory_eligible":false,"source_rank":7315,"rank":7315,"depth":43,"x":2004.233,"y":789.666,"cluster":"scheme-morphisms"},{"id":"stacks:055Q","tag":"055Q","title":"Connected components meeting a section · Lemma 055Q","summary":"Let f : X → Y, s : Y → X be as in Situation [Tag 055L]. Let y ∈ Y be a point. Assume • f is of finite presentation and flat, and • the fibre X_y is geometrically reduced. Then X^0 is a neighbourhood of X^0_y in X.","statement_latex":"Let $f : X \\to Y$, $s : Y \\to X$ be as in\nSituation \\ref{situation-connected-along-section}.\nLet $y \\in Y$ be a point.\nAssume\n\\begin{enumerate}\n\\item $f$ is of finite presentation and flat, and\n\\item the fibre $X_y$ is geometrically reduced.\n\\end{enumerate}\nThen $X^0$ is a neighbourhood of $X^0_y$ in $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components meeting a section","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055Q","source_file":"more-morphisms.tex","source_line":8309,"source_end_line":8320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8309-L8320","statement_sha256":"d090381785d857022abea7611ce42aaa4d3046e59691639937e3c0a817d089d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7316,"rank":7316,"depth":36,"x":1960.768,"y":584.455,"cluster":"scheme-morphisms"},{"id":"stacks:055R","tag":"055R","title":"Connected components meeting a section · Lemma 055R","summary":"Let f : X → Y, s : Y → X be as in Situation [Tag 055L]. Assume • f is of finite presentation and flat, and • all fibres of f are geometrically reduced. Then X^0 is open in X.","statement_latex":"Let $f : X \\to Y$, $s : Y \\to X$ be as in\nSituation \\ref{situation-connected-along-section}.\nAssume\n\\begin{enumerate}\n\\item $f$ is of finite presentation and flat, and\n\\item all fibres of $f$ are geometrically reduced.\n\\end{enumerate}\nThen $X^0$ is open in $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Connected components meeting a section","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055R","source_file":"more-morphisms.tex","source_line":8386,"source_end_line":8396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8386-L8396","statement_sha256":"1b772104bb01a6772b0ea4dfb17ff783f50a40c3380066a5d306c4b3983cbad7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7317,"rank":7317,"depth":37,"x":2157.963,"y":711.188,"cluster":"scheme-morphisms"},{"id":"stacks:05F7","tag":"05F7","title":"Dimension of fibres · Lemma 05F7","summary":"Let f : X → Y be a morphism of schemes. Assume Y irreducible with generic point eta and f of finite type. If X_eta has dimension n, then there exists a nonempty open V ⊂ Y such that for all y ∈ V the fibre X_y has dimension n.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume $Y$ irreducible with\ngeneric point $\\eta$ and $f$ of finite type. If $X_\\eta$ has dimension $n$,\nthen there exists a nonempty open $V \\subset Y$\nsuch that for all $y \\in V$ the fibre $X_y$ has dimension $n$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05F7","source_file":"more-morphisms.tex","source_line":8411,"source_end_line":8417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8411-L8417","statement_sha256":"f2589a3515e8296ae207eba12b04e9923a6c1566b6501762abb13a1625364ccf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7318,"rank":7318,"depth":32,"x":1910.489,"y":729.642,"cluster":"scheme-morphisms"},{"id":"stacks:05F8","tag":"05F8","title":"Dimension of fibres · Lemma 05F8","summary":"Let f : X → Y be a morphism of finite type. Let n_X/Y : Y → (0, 1, 2, 3, …, ∞) be the function which associates to y ∈ Y the dimension of X_y. If g : Y' → Y is a morphism then n_X'/Y' = n_X/Y ∘ g where X' → Y' is the base change of f.","statement_latex":"Let $f : X \\to Y$ be a morphism of finite type. Let\n$$\nn_{X/Y} : Y \\to \\{0, 1, 2, 3, \\ldots, \\infty\\}\n$$\nbe the function which associates to $y \\in Y$ the dimension of $X_y$.\nIf $g : Y' \\to Y$ is a morphism then\n$$\nn_{X'/Y'} = n_{X/Y} \\circ g\n$$\nwhere $X' \\to Y'$ is the base change of $f$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05F8","source_file":"more-morphisms.tex","source_line":8434,"source_end_line":8446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8434-L8446","statement_sha256":"9d0de53c6f2c94e4d2c66a275e94745e15054677a213bc954f12fcb7ec4b4808","origin":"The Stacks Project","memory_eligible":false,"source_rank":7319,"rank":7319,"depth":27,"x":2078.234,"y":575.518,"cluster":"scheme-morphisms"},{"id":"stacks:05F9","tag":"05F9","title":"Dimension of fibres · Lemma 05F9","summary":"Let f : X → Y be a morphism of schemes. Let n_X/Y be the function on Y giving the dimension of fibres of f introduced in Lemma [Tag 05F8]. Assume f of finite presentation. Then the level sets E_n = (y ∈ Y mid n_X/Y(y) = n) of n_X/Y are locally constructible in Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let\n$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$\nintroduced in\nLemma \\ref{lemma-base-change-dimension-fibres}.\nAssume $f$ of finite presentation. Then the level sets\n$$\nE_n = \\{y \\in Y \\mid n_{X/Y}(y) = n\\}\n$$\nof $n_{X/Y}$ are locally constructible in $Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05F9","source_file":"more-morphisms.tex","source_line":8453,"source_end_line":8464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8453-L8464","statement_sha256":"cb34a41e39e23e4d943d5eed85528c8822acfb1e18cb0096bfcfc43d067796c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7320,"rank":7320,"depth":33,"x":2078.483,"y":784.476,"cluster":"scheme-morphisms"},{"id":"stacks:0D4H","tag":"0D4H","title":"Dimension of fibres · Lemma 0D4H","summary":"Let f : X → Y be a flat morphism of schemes of finite presentation. Let n_X/Y be the function on Y giving the dimension of fibres of f introduced in Lemma [Tag 05F8]. Then n_X/Y is lower semi-continuous.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of schemes of finite presentation. Let\n$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$\nintroduced in Lemma \\ref{lemma-base-change-dimension-fibres}.\nThen $n_{X/Y}$ is lower semi-continuous.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4H","source_file":"more-morphisms.tex","source_line":8491,"source_end_line":8497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8491-L8497","statement_sha256":"583099c673d00d9c0ae377ce2298d27d4876e23a6c51f608317c5bbca8ee8209","origin":"The Stacks Project","memory_eligible":false,"source_rank":7321,"rank":7321,"depth":38,"x":1910.16,"y":630.442,"cluster":"scheme-morphisms"},{"id":"stacks:0D4I","tag":"0D4I","title":"Dimension of fibres · Lemma 0D4I","summary":"Let f : X → Y be a proper morphism of schemes. Let n_X/Y be the function on Y giving the dimension of fibres of f introduced in Lemma [Tag 05F8]. Then n_X/Y is upper semi-continuous.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let\n$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$\nintroduced in Lemma \\ref{lemma-base-change-dimension-fibres}.\nThen $n_{X/Y}$ is upper semi-continuous.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4I","source_file":"more-morphisms.tex","source_line":8511,"source_end_line":8517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8511-L8517","statement_sha256":"1eb3a16d81aeb3820d3917f83ac4c4f345eb3fb72247115f725a0ce5f4d529ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":7322,"rank":7322,"depth":32,"x":2158.299,"y":648.524,"cluster":"scheme-morphisms"},{"id":"stacks:0D4J","tag":"0D4J","title":"Dimension of fibres · Lemma 0D4J","summary":"Let f : X → Y be a proper, flat morphism of schemes of finite presentation. Let n_X/Y be the function on Y giving the dimension of fibres of f introduced in Lemma [Tag 05F8]. Then n_X/Y is locally constant.","statement_latex":"Let $f : X \\to Y$ be a proper, flat morphism of schemes of finite presentation.\nLet $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$\nintroduced in Lemma \\ref{lemma-base-change-dimension-fibres}.\nThen $n_{X/Y}$ is locally constant.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4J","source_file":"more-morphisms.tex","source_line":8528,"source_end_line":8534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8528-L8534","statement_sha256":"0f2dfe113ff5b330226b993ff9233aa6542d830bc497a507e9563592d6750681","origin":"The Stacks Project","memory_eligible":false,"source_rank":7323,"rank":7323,"depth":39,"x":1960.665,"y":776.067,"cluster":"scheme-morphisms"},{"id":"stacks:0GTE","tag":"0GTE","title":"Weak relative Noether normalization · Lemma 0GTE","summary":"Let R be a ring. Let p_1, …, p_r be prime ideals of R with p_i not ⊂ p_j if i not = j. Let k_i ⊂ kappa( p_i) be subfields such that the extensions kappa( p_i)/k_i are not algebraic. Let J ⊂ R be an ideal not contained in any of the p_i. Then there exists an element x ∈ J such that the image of x in kappa( p_i) is transcendental over k_i for i = 1, …, r.","statement_latex":"Let $R$ be a ring. Let $\\mathfrak p_1, \\ldots, \\mathfrak p_r$\nbe prime ideals of $R$ with $\\mathfrak p_i \\not \\subset \\mathfrak p_j$\nif $i \\not = j$. Let $k_i \\subset \\kappa(\\mathfrak p_i)$ be\nsubfields such that the extensions $\\kappa(\\mathfrak p_i)/k_i$\nare not algebraic. Let $J \\subset R$ be an ideal not contained\nin any of the $\\mathfrak p_i$. Then there exists an element $x \\in J$\nsuch that the image of $x$ in $\\kappa(\\mathfrak p_i)$\nis transcendental over $k_i$ for $i = 1, \\ldots, r$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weak relative Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTE","source_file":"more-morphisms.tex","source_line":8565,"source_end_line":8575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8565-L8575","statement_sha256":"8b753f446cfbb1b9968ef46c96e33b5d904082b2750be5914a494df7579b8b8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7324,"rank":7324,"depth":1,"x":2003.857,"y":569.753,"cluster":"scheme-morphisms"},{"id":"stacks:0GTF","tag":"0GTF","title":"Weak relative Noether normalization · Lemma 0GTF","summary":"Let R → S be a finite type ring map. Let d ≥ 0. Let a, b ∈ S. Assume that the fibres of f_a : Spec(S) → A^1_R given by the R-algebra map R[x] → S sending x to a have dimension ≤ d. Then there exists an n_0 such that for n ≥ n_0 the fibres of f_a^n + b : Spec(S) → A^1_R given by the R-algebra map R[x] → S sending x to a^n + b have dimension ≤ d.","statement_latex":"Let $R \\to S$ be a finite type ring map. Let $d \\geq 0$. Let $a, b \\in S$.\nAssume that the fibres of\n$$\nf_a : \\Spec(S) \\longrightarrow \\mathbf{A}^1_R\n$$\ngiven by the $R$-algebra map $R[x] \\to S$ sending $x$ to $a$\nhave dimension $\\leq d$. Then there exists an $n_0$ such that for\n$n \\geq n_0$ the fibres of\n$$\nf_{a^n + b} : \\Spec(S) \\longrightarrow \\mathbf{A}^1_R\n$$\ngiven by the $R$-algebra map $R[x] \\to S$ sending $x$ to $a^n + b$\nhave dimension $\\leq d$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weak relative Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTF","source_file":"more-morphisms.tex","source_line":8595,"source_end_line":8610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8595-L8610","statement_sha256":"99dc3a8e65f24eea418aea3b0daceacffaf205c2e9b4aaf92b0682eaeb0b18bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7325,"rank":7325,"depth":40,"x":2138.0,"y":746.5,"cluster":"scheme-morphisms"},{"id":"stacks:0GTG","tag":"0GTG","title":"Weak relative Noether normalization · Lemma 0GTG","summary":"Let R → S be a finite type ring map. Let d be the maximum of the dimensions of fibres of Spec(S) → Spec(R). Then there exists a quasi-finite ring map R[t_1, …, t_d] → S.","statement_latex":"Let $R \\to S$ be a finite type ring map. Let $d$ be the maximum\nof the dimensions of fibres of $\\Spec(S) \\to \\Spec(R)$.\nThen there exists a quasi-finite ring map\n$R[t_1, \\ldots, t_d] \\to S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weak relative Noether normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTG","source_file":"more-morphisms.tex","source_line":8779,"source_end_line":8785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8779-L8785","statement_sha256":"f1d14ba6f0f6b164d098dabacce9ce45641bf7627e9c3818306020db0f8758aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7326,"rank":7326,"depth":41,"x":1896.805,"y":692.253,"cluster":"scheme-morphisms"},{"id":"stacks:0G4D","tag":"0G4D","title":"Bertini theorems · Lemma 0G4D","summary":"See pages 71 and 72 of [Jou] Let K/k be a geometrically irreducible and finitely generated field extension. Let n ≥ 1. Let g_1, …, g_n ∈ K be elements such that there exist c_1, …, c_n ∈ k such that the elements x_1, …, x_n, ∑ g_ix_i, ∑ c_ig_i ∈ K(x_1, …, x_n) are algebraically independent over k. Then K(x_1, …, x_n) is geometrically irreducible over k(x_1, …, x_n, ∑ g_ix_i).","statement_latex":"\\begin{reference}\nSee pages 71 and 72 of \\cite{Jou}\n\\end{reference}\nLet $K/k$ be a geometrically irreducible and finitely generated\nfield extension. Let $n \\geq 1$.\nLet $g_1, \\ldots, g_n \\in K$ be elements such that there\nexist $c_1, \\ldots, c_n \\in k$ such that the elements\n$$\nx_1, \\ldots, x_n, \\sum g_ix_i, \\sum c_ig_i \\in K(x_1, \\ldots, x_n)\n$$\nare algebraically independent over $k$. Then\n$K(x_1, \\ldots, x_n)$ is geometrically irreducible over\n$k(x_1, \\ldots, x_n, \\sum g_ix_i)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Bertini theorems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4D","source_file":"more-morphisms.tex","source_line":8906,"source_end_line":8921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L8906-L8921","statement_sha256":"5052f004e6f565e11f536c315cb7c17334189c24dd1f0923e2607fab1f99182b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7327,"rank":7327,"depth":15,"x":2118.415,"y":595.336,"cluster":"scheme-morphisms"},{"id":"stacks:0G4E","tag":"0G4E","title":"Bertini theorems · Lemma 0G4E","summary":"Let A be a domain of finite type over a field k. Let n ≥ 2. Let g_1, …, g_n ∈ A be elements such that V(g_1, g_2) has an irreducible component of dimension dim(A) - 2. Then there exist c_1, …, c_n ∈ k such that the elements x_1, …, x_n, ∑ g_ix_i, ∑ c_ig_i ∈ Frac(A)(x_1, …, x_n) are algebraically independent over k.","statement_latex":"Let $A$ be a domain of finite type over a field $k$. Let $n \\geq 2$.\nLet $g_1, \\ldots, g_n \\in A$ be elements such that $V(g_1, g_2)$\nhas an irreducible component of dimension $\\dim(A) - 2$.\nThen there exist $c_1, \\ldots, c_n \\in k$ such that the elements\n$$\nx_1, \\ldots, x_n, \\sum g_ix_i, \\sum c_ig_i \\in\n\\text{Frac}(A)(x_1, \\ldots, x_n)\n$$\nare algebraically independent over $k$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Bertini theorems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4E","source_file":"more-morphisms.tex","source_line":9011,"source_end_line":9022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9011-L9022","statement_sha256":"0cfd9266f23c6fcef21c0d0f479a604fe8493376a3eddfc4ab61c0ba3a7dabf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7328,"rank":7328,"depth":35,"x":2032.89,"y":792.666,"cluster":"scheme-morphisms"},{"id":"stacks:0G4F","tag":"0G4F","title":"Bertini theorems · Lemma 0G4F","summary":"[Jou] In Varieties, Situation [Tag 0G47] assume • X is of finite type over k, • X is geometrically irreducible over k, • there exist v_1, v_2, v_3 ∈ V and an irreducible component Z of H_v_2 ∩ H_v_3 such that Z not ⊂ H_v_1 and codim(Z, X) = 2, and • every irreducible component Y of ⋂_v ∈ V H_v has codim(Y, X) ≥ 2. Then for general v ∈ V ⊗_k k' the scheme H_v is geometrically irreducible over k'.","statement_latex":"\\begin{reference}\n\\cite[Theorem 6.3 part 4)]{Jou}\n\\end{reference}\nIn Varieties, Situation \\ref{varieties-situation-family-divisors} assume\n\\begin{enumerate}\n\\item $X$ is of finite type over $k$,\n\\item $X$ is geometrically irreducible over $k$,\n\\item there exist $v_1, v_2, v_3 \\in V$ and an irreducible component\n$Z$ of $H_{v_2} \\cap H_{v_3}$ such that $Z \\not \\subset H_{v_1}$ and\n$\\text{codim}(Z, X) = 2$, and\n\\item every irreducible component $Y$ of $\\bigcap_{v \\in V} H_v$\nhas $\\text{codim}(Y, X) \\geq 2$.\n\\end{enumerate}\nThen for general $v \\in V \\otimes_k k'$\nthe scheme $H_v$ is geometrically irreducible over $k'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Bertini theorems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4F","source_file":"more-morphisms.tex","source_line":9054,"source_end_line":9071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9054-L9071","statement_sha256":"a91124cdf4df64e726eba3f30dc09853fc0c8a7ddb81d36710e8c88cfc6ce984","origin":"The Stacks Project","memory_eligible":false,"source_rank":7329,"rank":7329,"depth":41,"x":1937.21,"y":598.513,"cluster":"scheme-morphisms"},{"id":"stacks:0BDP","tag":"0BDP","title":"Theorem of the cube · Lemma 0BDP","summary":"Let f : X → S be a flat, proper morphism of finite presentation. Let E be a finite locally free O_X-module. For a morphism g : T → S consider the base change diagram xymatrix X_T ar[d]_p ar[r]_q & X ar[d]^f T ar[r]^g & S Assume O_T → p_*O_X_T is an isomorphism for all g : T → S. Then there exists an immersion j : Z → S of finite presentation such that a morphism g : T → S factors through Z if and only if there exists a finite locally free O_T-module N with p^*N ≅ q^*E.","statement_latex":"Let $f : X \\to S$ be a flat, proper morphism of finite presentation.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.\nFor a morphism $g : T \\to S$ consider the base change diagram\n$$\n\\xymatrix{\nX_T \\ar[d]_p \\ar[r]_q & X \\ar[d]^f \\\\\nT \\ar[r]^g & S\n}\n$$\nAssume $\\mathcal{O}_T \\to p_*\\mathcal{O}_{X_T}$ is an\nisomorphism for all $g : T \\to S$. Then there exists an\nimmersion $j : Z \\to S$ of finite presentation such that\na morphism $g : T \\to S$ factors through $Z$ if and only if\nthere exists a finite locally free $\\mathcal{O}_T$-module $\\mathcal{N}$\nwith $p^*\\mathcal{N} \\cong q^*\\mathcal{E}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Theorem of the cube","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BDP","source_file":"more-morphisms.tex","source_line":9198,"source_end_line":9215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9198-L9215","statement_sha256":"6e7fad5d9dd6827a63355a87e4d8f10288a2e305579a9c93f46f04b37103dadf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7330,"rank":7330,"depth":39,"x":2164.031,"y":687.441,"cluster":"scheme-morphisms"},{"id":"stacks:0EX7","tag":"0EX7","title":"Theorem of the cube · Lemma 0EX7","summary":"Let f : X → S be a flat, proper morphism of finite presentation such that f_*O_X = O_S and this remains true after arbitrary base change. Let E be a finite locally free O_X-module. Assume • E|_X_s is isomorphic to O_X_s^⊕ r_s for all s ∈ S, and • S is reduced. Then E = f^*N for some finite locally free O_S-module N.","statement_latex":"Let $f : X \\to S$ be a flat, proper morphism of finite presentation\nsuch that $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this remains\ntrue after arbitrary base change. Let $\\mathcal{E}$ be a finite\nlocally free $\\mathcal{O}_X$-module. Assume\n\\begin{enumerate}\n\\item $\\mathcal{E}|_{X_s}$ is isomorphic to\n$\\mathcal{O}_{X_s}^{\\oplus r_s}$ for all $s \\in S$, and\n\\item $S$ is reduced.\n\\end{enumerate}\nThen $\\mathcal{E} = f^*\\mathcal{N}$ for some finite locally free\n$\\mathcal{O}_S$-module $\\mathcal{N}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Theorem of the cube","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EX7","source_file":"more-morphisms.tex","source_line":9311,"source_end_line":9324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9311-L9324","statement_sha256":"08954ed3a0203e6888cec2c09240db6ab23e92739fc96eb729e723c5cd4b52b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7331,"rank":7331,"depth":40,"x":1925.121,"y":750.608,"cluster":"scheme-morphisms"},{"id":"stacks:0EX8","tag":"0EX8","title":"Theorem of the cube · Lemma 0EX8","summary":"Let f : X → S be a proper flat morphism of finite presentation. Let L be an invertible O_X-module. Assume • S is the spectrum of a valuation ring, • L is trivial on the generic fibre X_eta of f, • the closed fibre X_0 of f is integral, • H^0(X_eta, O_X_eta) is equal to the function field of S. Then L is trivial.","statement_latex":"Let $f : X \\to S$ be a proper flat morphism of finite presentation.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $S$ is the spectrum of a valuation ring,\n\\item $\\mathcal{L}$ is trivial on the generic fibre $X_\\eta$ of $f$,\n\\item the closed fibre $X_0$ of $f$ is integral,\n\\item $H^0(X_\\eta, \\mathcal{O}_{X_\\eta})$ is equal to the function field of $S$.\n\\end{enumerate}\nThen $\\mathcal{L}$ is trivial.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Theorem of the cube","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EX8","source_file":"more-morphisms.tex","source_line":9333,"source_end_line":9345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9333-L9345","statement_sha256":"17dd00b73af76cf4c9b7e76a2ebb43b402a0f98ac4c8335782f5ad0142500133","origin":"The Stacks Project","memory_eligible":false,"source_rank":7332,"rank":7332,"depth":39,"x":2050.567,"y":568.358,"cluster":"scheme-morphisms"},{"id":"stacks:0BF0","tag":"0BF0","title":"Theorem of the cube · Lemma 0BF0","summary":"Let f : X → S and E be as in Lemma [Tag 0BDP] and in addition assume E is an invertible O_X-module. If moreover the geometric fibres of f are integral, then Z is closed in S.","statement_latex":"Let $f : X \\to S$ and $\\mathcal{E}$ be as in\nLemma \\ref{lemma-diagonal-picard-flat-proper}\nand in addition assume $\\mathcal{E}$ is an invertible $\\mathcal{O}_X$-module.\nIf moreover the geometric fibres of $f$ are\nintegral, then $Z$ is closed in $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Theorem of the cube","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BF0","source_file":"more-morphisms.tex","source_line":9409,"source_end_line":9416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9409-L9416","statement_sha256":"dfa937066ec0599ed89553eb2ff8ce122daebc96ec16d74167db53e9809575af","origin":"The Stacks Project","memory_eligible":false,"source_rank":7333,"rank":7333,"depth":40,"x":2104.658,"y":774.05,"cluster":"scheme-morphisms"},{"id":"stacks:0BF1","tag":"0BF1","title":"Theorem of the cube · Lemma 0BF1","summary":"Consider a commutative diagram of schemes xymatrix X' ar[rr] ar[dr]_f' & & X ar[dl]^f & S with f' : X' → S and f : X → S satisfying the hypotheses of Lemma [Tag 0BDP]. Let L be an invertible O_X-module and let L' be the pullback to X'. Let Z ⊂ S, resp. Z' ⊂ S be the locally closed subscheme constructed in Lemma [Tag 0BDP] for (f, L), resp. (f', L') so that Z ⊂ Z'. If s ∈ Z and H^1(X_s, O) → H^1(X'_s, O) is injective, then Z ∩ U = Z' ∩ U for some open neighbourhood U of s.","statement_latex":"Consider a commutative diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[rr] \\ar[dr]_{f'} & & X \\ar[dl]^f \\\\\n& S\n}\n$$\nwith $f' : X' \\to S$ and $f : X \\to S$ satisfying the hypotheses of\nLemma \\ref{lemma-diagonal-picard-flat-proper}.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module\nand let $\\mathcal{L}'$ be the pullback to $X'$. Let $Z \\subset S$,\nresp.\\ $Z' \\subset S$ be the locally closed subscheme constructed\nin Lemma \\ref{lemma-diagonal-picard-flat-proper}\nfor $(f, \\mathcal{L})$, resp.\\ $(f', \\mathcal{L}')$\nso that $Z \\subset Z'$. If $s \\in Z$ and\n$$\nH^1(X_s, \\mathcal{O}) \\longrightarrow H^1(X'_s, \\mathcal{O})\n$$\nis injective, then $Z \\cap U = Z' \\cap U$ for some open neighbourhood\n$U$ of $s$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Theorem of the cube","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BF1","source_file":"more-morphisms.tex","source_line":9429,"source_end_line":9451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9429-L9451","statement_sha256":"bc406002d7afb149ae3bf39a00b98ef5d8f836013cfefb047e92ba14df947732","origin":"The Stacks Project","memory_eligible":false,"source_rank":7334,"rank":7334,"depth":40,"x":1899.238,"y":652.996,"cluster":"scheme-morphisms"},{"id":"stacks:0BF2","tag":"0BF2","title":"Theorem of the cube · Lemma 0BF2","summary":"Consider n commutative diagrams of schemes xymatrix X_i ar[rr] ar[dr]_f_i & & X ar[dl]^f & S with f_i : X_i → S and f : X → S satisfying the hypotheses of Lemma [Tag 0BDP]. Let L be an invertible O_X-module and let L_i be the pullback to X_i. Let Z ⊂ S, resp. Z_i ⊂ S be the locally closed subscheme constructed in Lemma [Tag 0BDP] for (f, L), resp. (f_i, L_i) so that Z ⊂ ⋂_i = 1, …, n Z_i. If s ∈ Z and H^1(X_s, O) → bigoplus_i = 1, …, n H^1(X_i, s, O) is injective, then Z…","statement_latex":"Consider $n$ commutative diagrams of schemes\n$$\n\\xymatrix{\nX_i \\ar[rr] \\ar[dr]_{f_i} & & X \\ar[dl]^f \\\\\n& S\n}\n$$\nwith $f_i : X_i \\to S$ and $f : X \\to S$ satisfying the hypotheses of\nLemma \\ref{lemma-diagonal-picard-flat-proper}.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module\nand let $\\mathcal{L}_i$ be the pullback to $X_i$. Let $Z \\subset S$,\nresp.\\ $Z_i \\subset S$ be the locally closed subscheme constructed\nin Lemma \\ref{lemma-diagonal-picard-flat-proper}\nfor $(f, \\mathcal{L})$, resp.\\ $(f_i, \\mathcal{L}_i)$\nso that $Z \\subset \\bigcap_{i = 1, \\ldots, n} Z_i$. If $s \\in Z$ and\n$$\nH^1(X_s, \\mathcal{O}) \\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} H^1(X_{i, s}, \\mathcal{O})\n$$\nis injective, then $Z \\cap U = (\\bigcap_{i = 1, \\ldots, n} Z_i) \\cap U$\n(scheme theoretic intersection) for some open neighbourhood $U$ of $s$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Theorem of the cube","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BF2","source_file":"more-morphisms.tex","source_line":9518,"source_end_line":9541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9518-L9541","statement_sha256":"49801ef6027d8271309d5a0fbc4fbdbbeeb9c84c63dd78072eb08541ca0d7f1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7335,"rank":7335,"depth":41,"x":2148.207,"y":625.683,"cluster":"scheme-morphisms"},{"id":"stacks:0BF3","tag":"0BF3","title":"Theorem of the cube · Lemma 0BF3","summary":"Let f : X → S and g : Y → S be morphisms of schemes satisfying the hypotheses of Lemma [Tag 0BDP]. Let σ : S → X and τ : S → Y be sections of f and g. Let s ∈ S. Let L be an invertible sheaf on X ×_S Y. If (1 × τ)^*L on X, (σ × 1)^*L on Y, and L|_(X ×_S Y)_s are trivial, then there is an open neighbourhood U of s such that L is trivial over (X ×_S Y)_U.","statement_latex":"Let $f : X \\to S$ and $g : Y \\to S$ be morphisms of schemes\nsatisfying the hypotheses of Lemma \\ref{lemma-diagonal-picard-flat-proper}.\nLet $\\sigma : S \\to X$ and $\\tau : S \\to Y$ be sections of\n$f$ and $g$. Let $s \\in S$.\nLet $\\mathcal{L}$ be an invertible sheaf on $X \\times_S Y$.\nIf $(1 \\times \\tau)^*\\mathcal{L}$ on $X$, $(\\sigma \\times 1)^*\\mathcal{L}$\non $Y$, and $\\mathcal{L}|_{(X \\times_S Y)_s}$ are trivial, then\nthere is an open neighbourhood $U$ of $s$ such that\n$\\mathcal{L}$ is trivial over $(X \\times_S Y)_U$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Theorem of the cube","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BF3","source_file":"more-morphisms.tex","source_line":9621,"source_end_line":9632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9621-L9632","statement_sha256":"c0b695758cd5003fa671c0e50e35b25360465301ceacfe9ada7ac72eac58fcba","origin":"The Stacks Project","memory_eligible":false,"source_rank":7336,"rank":7336,"depth":42,"x":1986.49,"y":787.19,"cluster":"scheme-morphisms"},{"id":"stacks:0BF4","tag":"0BF4","title":"Theorem of the cube · Theorem 0BF4","summary":"Let S be a scheme. Let X, Y, and Z be schemes over S. Let x : S → X and y : S → Y be sections of the structure morphisms. Let L be an invertible module on X ×_S Y ×_S Z. If • X → S and Y → S are flat, proper morphisms of finite presentation with geometrically integral fibres, • the pullbacks of L by x × id_Y × id_Z and id_X × y × id_Z are trivial over Y ×_S Z and X ×_S Z, • there is a point z ∈ Z such that L restricted to X ×_S Y ×_S z is trivial, and • Z is connected,…","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$, and $Z$ be schemes over $S$.\nLet $x : S \\to X$ and $y : S \\to Y$ be sections of the structure morphisms.\nLet $\\mathcal{L}$ be an invertible module on $X \\times_S Y \\times_S Z$. If\n\\begin{enumerate}\n\\item $X \\to S$ and $Y \\to S$ are flat, proper morphisms\nof finite presentation with geometrically integral fibres,\n\\item the pullbacks of $\\mathcal{L}$ by\n$x \\times \\text{id}_Y \\times \\text{id}_Z$ and\n$\\text{id}_X \\times y \\times \\text{id}_Z$\nare trivial over $Y \\times_S Z$ and $X \\times_S Z$,\n\\item there is a point $z \\in Z$ such that $\\mathcal{L}$\nrestricted to $X \\times_S Y \\times_S z$ is trivial, and\n\\item $Z$ is connected,\n\\end{enumerate}\nthen $\\mathcal{L}$ is trivial.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Theorem of the cube","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BF4","source_file":"more-morphisms.tex","source_line":9652,"source_end_line":9669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9652-L9669","statement_sha256":"17aff5c0a7e20ffd042c90ad4e37d053d56cf9cee199c80f6c191d854c357053","origin":"The Stacks Project","memory_eligible":false,"source_rank":7337,"rank":7337,"depth":0,"x":1975.852,"y":576.21,"cluster":"scheme-morphisms"},{"id":"stacks:05FB","tag":"05FB","title":"Limit arguments · Lemma 05FB","summary":"Let f : X → S be a morphism of affine schemes, which is of finite presentation. Then there exists a cartesian diagram xymatrix X_0 ar[d]_f_0 & X ar[l]^g ar[d]^f S_0 & S ar[l] such that • X_0, S_0 are affine schemes, • S_0 is of finite type over Z, • f_0 is of finite type.","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes, which is of finite\npresentation. Then there exists a cartesian diagram\n$$\n\\xymatrix{\nX_0 \\ar[d]_{f_0} & X \\ar[l]^g \\ar[d]^f \\\\\nS_0 & S \\ar[l]\n}\n$$\nsuch that\n\\begin{enumerate}\n\\item $X_0$, $S_0$ are affine schemes,\n\\item $S_0$ is of finite type over $\\mathbf{Z}$,\n\\item $f_0$ is of finite type.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FB","source_file":"more-morphisms.tex","source_line":9738,"source_end_line":9754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9738-L9754","statement_sha256":"9aaeb39ed361d2907d67bbfe535c265032475466c21177c2aeac0c781ab3b87b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7338,"rank":7338,"depth":5,"x":2153.466,"y":725.836,"cluster":"scheme-morphisms"},{"id":"stacks:05FC","tag":"05FC","title":"Limit arguments · Lemma 05FC","summary":"Let f : X → S be a morphism of affine schemes, which is of finite presentation. Let F be a quasi-coherent O_X-module of finite presentation. Then there exists a diagram as in Lemma [Tag 05FB] such that there exists a coherent O_X_0-module F_0 with g^*F_0 = F.","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes, which is of finite\npresentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nof finite presentation. Then there exists a diagram as in\nLemma \\ref{lemma-Noetherian-approximation}\nsuch that there exists a coherent $\\mathcal{O}_{X_0}$-module $\\mathcal{F}_0$\nwith $g^*\\mathcal{F}_0 = \\mathcal{F}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FC","source_file":"more-morphisms.tex","source_line":9766,"source_end_line":9774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9766-L9774","statement_sha256":"569ba0d45ae857a44db3015916d397d153958c4e14c6126cd793d682f9aba60a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7339,"rank":7339,"depth":7,"x":1902.023,"y":716.28,"cluster":"scheme-morphisms"},{"id":"stacks:05FD","tag":"05FD","title":"Limit arguments · Lemma 05FD","summary":"Let f : X → S be a morphism of affine schemes, which is of finite presentation. Let F be a quasi-coherent O_X-module of finite presentation and flat over S. Then we may choose a diagram as in Lemma [Tag 05FC] and sheaf F_0 such that in addition F_0 is flat over S_0.","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes, which is of finite\npresentation. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nof finite presentation and flat over $S$. Then we may choose a diagram as in\nLemma \\ref{lemma-Noetherian-approximation-module}\nand sheaf $\\mathcal{F}_0$ such that in addition $\\mathcal{F}_0$\nis flat over $S_0$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FD","source_file":"more-morphisms.tex","source_line":9789,"source_end_line":9797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9789-L9797","statement_sha256":"9e4479cbadf7bd1733a1a6b0663cc390ff86c66bc1af1e09ec43ee5293aeae0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7340,"rank":7340,"depth":35,"x":2095.231,"y":580.572,"cluster":"scheme-morphisms"},{"id":"stacks:05FE","tag":"05FE","title":"Limit arguments · Lemma 05FE","summary":"Let f : X → S be a morphism of affine schemes, which is of finite presentation and flat. Then there exists a diagram as in Lemma [Tag 05FB] such that in addition f_0 is flat.","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes, which is of finite\npresentation and flat. Then there exists a diagram as in\nLemma \\ref{lemma-Noetherian-approximation}\nsuch that in addition $f_0$ is flat.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FE","source_file":"more-morphisms.tex","source_line":9814,"source_end_line":9820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9814-L9820","statement_sha256":"e340b5be7081a7c1146f7c8395bd84552fc94b0445240138fdd757e3df062d9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7341,"rank":7341,"depth":36,"x":2061.876,"y":790.395,"cluster":"scheme-morphisms"},{"id":"stacks:05FF","tag":"05FF","title":"Limit arguments · Lemma 05FF","summary":"Let f : X → S be a morphism of affine schemes, which is smooth. Then there exists a diagram as in Lemma [Tag 05FB] such that in addition f_0 is smooth.","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes, which is smooth.\nThen there exists a diagram as in\nLemma \\ref{lemma-Noetherian-approximation}\nsuch that in addition $f_0$ is smooth.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FF","source_file":"more-morphisms.tex","source_line":9827,"source_end_line":9833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9827-L9833","statement_sha256":"7af9710f6c558e5773022d217f8f32d20c09b9e2f3f2cc0b73135105a38c4470","origin":"The Stacks Project","memory_eligible":false,"source_rank":7342,"rank":7342,"depth":37,"x":1917.652,"y":616.646,"cluster":"scheme-morphisms"},{"id":"stacks:05FG","tag":"05FG","title":"Limit arguments · Lemma 05FG","summary":"Let f : X → S be a morphism of affine schemes, which is of finite presentation with geometrically reduced fibres. Then there exists a diagram as in Lemma [Tag 05FB] such that in addition f_0 has geometrically reduced fibres.","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes, which is\nof finite presentation with geometrically reduced fibres.\nThen there exists a diagram as in\nLemma \\ref{lemma-Noetherian-approximation}\nsuch that in addition $f_0$ has geometrically reduced fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FG","source_file":"more-morphisms.tex","source_line":9844,"source_end_line":9851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9844-L9851","statement_sha256":"a2c88214cfee2a751db94e80b583514509c11a6a06c3d6d53689127e94be4492","origin":"The Stacks Project","memory_eligible":false,"source_rank":7343,"rank":7343,"depth":47,"x":2163.869,"y":662.957,"cluster":"scheme-morphisms"},{"id":"stacks:05FH","tag":"05FH","title":"Limit arguments · Lemma 05FH","summary":"Let f : X → S be a morphism of affine schemes, which is of finite presentation with geometrically irreducible fibres. Then there exists a diagram as in Lemma [Tag 05FB] such that in addition f_0 has geometrically irreducible fibres.","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes, which is\nof finite presentation with geometrically irreducible fibres.\nThen there exists a diagram as in\nLemma \\ref{lemma-Noetherian-approximation}\nsuch that in addition $f_0$ has geometrically irreducible fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FH","source_file":"more-morphisms.tex","source_line":9879,"source_end_line":9886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9879-L9886","statement_sha256":"a7992ceed626e67ce85e5a2ea9a590bcea176bb6d9551341df748f253ffe7390","origin":"The Stacks Project","memory_eligible":false,"source_rank":7344,"rank":7344,"depth":43,"x":1944.942,"y":768.58,"cluster":"scheme-morphisms"},{"id":"stacks:05FI","tag":"05FI","title":"Limit arguments · Lemma 05FI","summary":"Let f : X → S be a morphism of affine schemes, which is of finite presentation with geometrically connected fibres. Then there exists a diagram as in Lemma [Tag 05FB] such that in addition f_0 has geometrically connected fibres.","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes, which is\nof finite presentation with geometrically connected fibres.\nThen there exists a diagram as in\nLemma \\ref{lemma-Noetherian-approximation}\nsuch that in addition $f_0$ has geometrically connected fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FI","source_file":"more-morphisms.tex","source_line":9914,"source_end_line":9921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9914-L9921","statement_sha256":"981ebeafdd4dd31b67cb64a777d3d18aba23b11faa399777d1cf0504ba94b015","origin":"The Stacks Project","memory_eligible":false,"source_rank":7345,"rank":7345,"depth":44,"x":2021.482,"y":566.352,"cluster":"scheme-morphisms"},{"id":"stacks:05FJ","tag":"05FJ","title":"Limit arguments · Lemma 05FJ","summary":"Let d ≥ 0 be an integer. Let f : X → S be a morphism of affine schemes, which is of finite presentation all of whose fibres have dimension d. Then there exists a diagram as in Lemma [Tag 05FB] such that in addition all fibres of f_0 have dimension d.","statement_latex":"Let $d \\geq 0$ be an integer.\nLet $f : X \\to S$ be a morphism of affine schemes, which is\nof finite presentation all of whose fibres have dimension $d$.\nThen there exists a diagram as in\nLemma \\ref{lemma-Noetherian-approximation}\nsuch that in addition all fibres of $f_0$ have dimension $d$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FJ","source_file":"more-morphisms.tex","source_line":9949,"source_end_line":9957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9949-L9957","statement_sha256":"4b544dea1033e90a9a4b81b2f0d9393bd365c1a80b7b6e4499f94cf1aa1ef48c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7346,"rank":7346,"depth":34,"x":2127.73,"y":759.015,"cluster":"scheme-morphisms"},{"id":"stacks:05FK","tag":"05FK","title":"Limit arguments · Lemma 05FK","summary":"Let f : X → S be a morphism of affine schemes, which is standard syntomic (see Morphisms, Definition [Tag 01UC]). Then there exists a diagram as in Lemma [Tag 05FB] such that in addition f_0 is standard syntomic.","statement_latex":"Let $f : X \\to S$ be a morphism of affine schemes, which is\nstandard syntomic (see\nMorphisms, Definition \\ref{morphisms-definition-syntomic}).\nThen there exists a diagram as in\nLemma \\ref{lemma-Noetherian-approximation}\nsuch that in addition $f_0$ is standard syntomic.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FK","source_file":"more-morphisms.tex","source_line":9985,"source_end_line":9993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L9985-L9993","statement_sha256":"8d4ccaedf92fce38e274a7ced9767077bd983e4b61a1e061cbdc2a9dd53581d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7347,"rank":7347,"depth":32,"x":1894.314,"y":677.188,"cluster":"scheme-morphisms"},{"id":"stacks:05FL","tag":"05FL","title":"Limit arguments · Lemma 05FL","summary":"(Noetherian approximation and combining properties.) Let P, Q be properties of morphisms of schemes which are stable under base change. Let f : X → S be a morphism of finite presentation of affine schemes. Assume we can find cartesian diagrams vcenter xymatrix X_1 ar[d]_f_1 & X ar[l] ar[d]^f S_1 & S ar[l] and vcenter xymatrix X_2 ar[d]_f_2 & X ar[l] ar[d]^f S_2 & S ar[l] of affine schemes, with S_1, S_2 of finite type over Z and f_1, f_2 of finite type such that f_1 has…","statement_latex":"(Noetherian approximation and combining properties.)\nLet $P$, $Q$ be properties of morphisms of schemes which are stable\nunder base change. Let $f : X \\to S$ be a morphism of finite presentation\nof affine schemes. Assume we can find cartesian diagrams\n$$\n\\vcenter{\n\\xymatrix{\nX_1 \\ar[d]_{f_1} & X \\ar[l] \\ar[d]^f \\\\\nS_1 & S \\ar[l]\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nX_2 \\ar[d]_{f_2} & X \\ar[l] \\ar[d]^f \\\\\nS_2 & S \\ar[l]\n}\n}\n$$\nof affine schemes, with $S_1$, $S_2$ of finite type over $\\mathbf{Z}$\nand $f_1$, $f_2$ of finite type such that $f_1$ has property $P$\nand $f_2$ has property $Q$. Then we can find a cartesian diagram\n$$\n\\xymatrix{\nX_0 \\ar[d]_{f_0} & X \\ar[l] \\ar[d]^f \\\\\nS_0 & S \\ar[l]\n}\n$$\nof affine schemes with $S_0$ of finite type over $\\mathbf{Z}$\nand $f_0$ of finite type such that $f_0$ has both property $P$ and\nproperty $Q$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Limit arguments","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FL","source_file":"more-morphisms.tex","source_line":10001,"source_end_line":10034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10001-L10034","statement_sha256":"99c7f70c88e0b4c47ac8a2ee310fe506739a2502fa99dfa8e8cdaa9be8fc5d5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7348,"rank":7348,"depth":26,"x":2132.373,"y":605.038,"cluster":"scheme-morphisms"},{"id":"stacks:02LE","tag":"02LE","title":"Étale neighbourhoods · Definition 02LE","summary":"Let S be a scheme. Let s ∈ S be a point. • An étale neighbourhood of (S, s) is a pair (U, u) together with an étale morphism of schemes φ : U → S such that φ(u) = s. • A morphism of étale neighbourhoods f : (V, v) → (U, u) of (S, s) is simply a morphism of S-schemes f : V → U such that f(v) = u. • An elementary étale neighbourhood is an étale neighbourhood φ : (U, u) → (S, s) such that kappa(s) = kappa(u).","statement_latex":"Let $S$ be a scheme. Let $s \\in S$ be a point.\n\\begin{enumerate}\n\\item An {\\it \\'etale neighbourhood of $(S, s)$} is a\npair $(U, u)$ together with an \\'etale morphism\nof schemes $\\varphi : U \\to S$ such that $\\varphi(u) = s$.\n\\item A {\\it morphism of \\'etale neighbourhoods} $f : (V, v) \\to (U, u)$\nof $(S, s)$ is simply a morphism of $S$-schemes $f : V \\to U$ such\nthat $f(v) = u$.\n\\item An {\\it elementary \\'etale neighbourhood} is an \\'etale neighbourhood\n$\\varphi : (U, u) \\to (S, s)$ such that $\\kappa(s) = \\kappa(u)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LE","source_file":"more-morphisms.tex","source_line":10093,"source_end_line":10106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10093-L10106","statement_sha256":"e95a7ec033858b4031e175273737833e8841e3c68735846fe6fcfffc9282870d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7349,"rank":7349,"depth":0,"x":2014.785,"y":793.431,"cluster":"scheme-morphisms"},{"id":"stacks:02LF","tag":"02LF","title":"Étale neighbourhoods · Lemma 02LF","summary":"Let S be a scheme. Let s ∈ S. Let k/kappa(s) be a finite separable field extension. Then there exists an étale neighbourhood (U, u) → (S, s) such that the field extension kappa(u)/kappa(s) is isomorphic to k/kappa(s).","statement_latex":"Let $S$ be a scheme.\nLet $s \\in S$.\nLet $k/\\kappa(s)$ be a finite separable field extension.\nThen there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$\nsuch that the field extension $\\kappa(u)/\\kappa(s)$ is\nisomorphic to $k/\\kappa(s)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LF","source_file":"more-morphisms.tex","source_line":10137,"source_end_line":10145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10137-L10145","statement_sha256":"69582d695fe200dbc4f137a1c55347d8448312cc109c7641e4fad9a7988517e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7350,"rank":7350,"depth":1,"x":1949.955,"y":587.671,"cluster":"scheme-morphisms"},{"id":"stacks:057A","tag":"057A","title":"Étale neighbourhoods · Lemma 057A","summary":"Let S be a scheme, and let s be a point of S. The category of étale neighborhoods has the following properties: • Let (U_i, u_i)_i=1, 2 be two étale neighborhoods of s in S. Then there exists a third étale neighborhood (U, u) and morphisms (U, u) → (U_i, u_i), i = 1, 2. • Let h_1, h_2: (U, u) → (U', u') be two morphisms between étale neighborhoods of s. Assume h_1, h_2 induce the same map kappa(u') → kappa(u) of residue fields. Then there exist an étale neighborhood (U\",…","statement_latex":"Let $S$ be a scheme, and let $s$ be a point of $S$.\nThe category of \\'etale neighborhoods has the following properties:\n\\begin{enumerate}\n\\item Let $(U_i, u_i)_{i=1, 2}$ be two \\'etale neighborhoods of\n$s$ in $S$. Then there exists a third \\'etale neighborhood\n$(U, u)$ and morphisms\n$(U, u) \\to (U_i, u_i)$, $i = 1, 2$.\n\\item Let $h_1, h_2: (U, u) \\to (U', u')$ be two\nmorphisms between \\'etale neighborhoods of $s$.\nAssume $h_1$, $h_2$ induce the same map $\\kappa(u') \\to \\kappa(u)$ of residue\nfields. Then there exist an \\'etale neighborhood $(U'', u'')$ and a morphism\n$h : (U'', u'') \\to (U, u)$\nwhich equalizes $h_1$ and $h_2$, i.e., such that\n$h_1 \\circ h = h_2 \\circ h$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057A","source_file":"more-morphisms.tex","source_line":10153,"source_end_line":10170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10153-L10170","statement_sha256":"9c3cd5284337ddd80c38f64ca3c28757f6ab9372aedad8b37878ff9d0768cf52","origin":"The Stacks Project","memory_eligible":false,"source_rank":7351,"rank":7351,"depth":43,"x":2163.35,"y":702.676,"cluster":"scheme-morphisms"},{"id":"stacks:057B","tag":"057B","title":"Étale neighbourhoods · Lemma 057B","summary":"Let S be a scheme, and let s be a point of S. The category of elementary étale neighborhoods of (S, s) is cofiltered (see Categories, Definition [Tag 04AZ]).","statement_latex":"Let $S$ be a scheme, and let $s$ be a point of $S$.\nThe category of elementary \\'etale neighborhoods of $(S, s)$\nis cofiltered (see\nCategories, Definition \\ref{categories-definition-codirected}).","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057B","source_file":"more-morphisms.tex","source_line":10202,"source_end_line":10208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10202-L10208","statement_sha256":"ce9c1e6168f09a3e050866e76a2efc0f3508270825b59d785952af94d0012557","origin":"The Stacks Project","memory_eligible":false,"source_rank":7352,"rank":7352,"depth":44,"x":1913.367,"y":738.979,"cluster":"scheme-morphisms"},{"id":"stacks:05KS","tag":"05KS","title":"Étale neighbourhoods · Lemma 05KS","summary":"Let S be a scheme. Let s ∈ S. Then we have O_S, s^h = colim_(U, u) O(U) where the colimit is over the filtered category which is opposite to the category of elementary étale neighbourhoods (U, u) of (S, s).","statement_latex":"Let $S$ be a scheme. Let $s \\in S$. Then we have\n$$\n\\mathcal{O}_{S, s}^h =\n\\colim_{(U, u)} \\mathcal{O}(U)\n$$\nwhere the colimit is over the filtered category which is opposite to the\ncategory of elementary \\'etale neighbourhoods $(U, u)$ of $(S, s)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KS","source_file":"more-morphisms.tex","source_line":10215,"source_end_line":10224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10215-L10224","statement_sha256":"b0f0096c5e0d4e43997c3dfae5c5dcea0a20779c4b9db06f70bda89fcf2046ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":7353,"rank":7353,"depth":44,"x":2068.596,"y":570.265,"cluster":"scheme-morphisms"},{"id":"stacks:0CAS","tag":"0CAS","title":"Étale neighbourhoods · Lemma 0CAS","summary":"Lift étale neighbourhood of point on fibre to total space. Let X → S be a morphism of schemes. Let x ∈ X with image s ∈ S. Let (V, v) → (X_s, x) be an étale neighbourhood. Then there exists an étale neighbourhood (U, u) → (X, x) such that there exists a morphism (U_s, u) → (V, v) of étale neighbourhoods of (X_s, x) which is an open immersion.","statement_latex":"\\begin{slogan}\nLift \\'etale neighbourhood of point on fibre to total space.\n\\end{slogan}\nLet $X \\to S$ be a morphism of schemes. Let $x \\in X$ with image $s \\in S$.\nLet $(V, v) \\to (X_s, x)$ be an \\'etale neighbourhood.\nThen there exists an \\'etale neighbourhood $(U, u) \\to (X, x)$\nsuch that there exists a morphism $(U_s, u) \\to (V, v)$\nof \\'etale neighbourhoods of $(X_s, x)$ which is an open immersion.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAS","source_file":"more-morphisms.tex","source_line":10246,"source_end_line":10256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10246-L10256","statement_sha256":"9cfb94777d9b20d5fdfc4a532b3e1d6b670070a3217acdace1a917f480bce9aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7354,"rank":7354,"depth":38,"x":2089.82,"y":782.877,"cluster":"scheme-morphisms"},{"id":"stacks:0CB3","tag":"0CB3","title":"Étale neighbourhoods and branches · Lemma 0CB3","summary":"Let R = colim R_i be colimit of a directed system of rings whose transition maps are faithfully flat. Then the number of minimal primes of R taken as an element of (0, 1, 2, …, ∞) is the supremum of the numbers of minimal primes of the R_i.","statement_latex":"Let $R = \\colim R_i$ be colimit of a directed system of rings\nwhose transition maps are faithfully flat.\nThen the number of minimal primes of $R$\ntaken as an element of $\\{0, 1, 2, \\ldots, \\infty\\}$\nis the supremum of the numbers of minimal primes of the $R_i$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CB3","source_file":"more-morphisms.tex","source_line":10302,"source_end_line":10309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10302-L10309","statement_sha256":"03482c18d900f57426f297c562ccd921b07bcadd5a0d4e7b7008faeb6c4f85ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":7355,"rank":7355,"depth":5,"x":1903.086,"y":638.059,"cluster":"scheme-morphisms"},{"id":"stacks:0CB4","tag":"0CB4","title":"Étale neighbourhoods and branches · Lemma 0CB4","summary":"Let X be a scheme and x ∈ X a point. Then • the number of branches of X at x is equal to the supremum of the number of irreducible components of U passing through u taken over elementary étale neighbourhoods (U, u) → (X, x), • the number of geometric branches of X at x is equal to the supremum of the number of irreducible components of U passing through u taken over étale neighbourhoods (U, u) → (X, x), • X is unibranch at x if and only if for every elementary étale…","statement_latex":"Let $X$ be a scheme and $x \\in X$ a point. Then\n\\begin{enumerate}\n\\item the number of branches of $X$ at $x$ is equal to\nthe supremum of the number of irreducible components of $U$\npassing through $u$ taken over elementary \\'etale neighbourhoods\n$(U, u) \\to (X, x)$,\n\\item the number of geometric branches of $X$ at $x$ is equal to\nthe supremum of the number of irreducible components of $U$\npassing through $u$ taken over \\'etale neighbourhoods\n$(U, u) \\to (X, x)$,\n\\item $X$ is unibranch at $x$ if and only if for every\nelementary \\'etale neighbourhood $(U, u) \\to (X, x)$ there\nis exactly one irreducible component of $U$ passing through $u$, and\n\\item $X$ is geometrically unibranch at $x$ if and only if for every\n\\'etale neighbourhood $(U, u) \\to (X, x)$ there\nis exactly one irreducible component of $U$ passing through $u$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CB4","source_file":"more-morphisms.tex","source_line":10333,"source_end_line":10352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10333-L10352","statement_sha256":"fbf443c6bd73fb8acb8f09b6429a877cfd4c7c9051c033d6786c4c71cfb7af73","origin":"The Stacks Project","memory_eligible":false,"source_rank":7356,"rank":7356,"depth":54,"x":2157.386,"y":638.889,"cluster":"scheme-morphisms"},{"id":"stacks:0CB5","tag":"0CB5","title":"Étale neighbourhoods and branches · Lemma 0CB5","summary":"Let X → S be a morphism of schemes and x ∈ X a point with image s. Then • the number of branches of the fibre X_s at x is equal to the supremum of the number of irreducible components of the fibre U_s passing through u taken over elementary étale neighbourhoods (U, u) → (X, x), • the number of geometric branches of the fibre X_s at x is equal to the supremum of the number of irreducible components of the fibre U_s passing through u taken over étale neighbourhoods (U, u) →…","statement_latex":"Let $X \\to S$ be a morphism of schemes and $x \\in X$ a point with image $s$.\nThen\n\\begin{enumerate}\n\\item the number of branches of the fibre $X_s$ at $x$ is equal to\nthe supremum of the number of irreducible components of the fibre $U_s$\npassing through $u$ taken over elementary \\'etale neighbourhoods\n$(U, u) \\to (X, x)$,\n\\item the number of geometric branches of the fibre $X_s$ at $x$ is equal to\nthe supremum of the number of irreducible components of the fibre $U_s$\npassing through $u$ taken over \\'etale neighbourhoods\n$(U, u) \\to (X, x)$,\n\\item the fibre $X_s$ is unibranch at $x$ if and only if for every\nelementary \\'etale neighbourhood $(U, u) \\to (X, x)$ there is\nexactly one irreducible component of the fibre $U_s$ passing through $u$, and\n\\item $X$ is geometrically unibranch at $x$ if and only if for every\n\\'etale neighbourhood $(U, u) \\to (X, x)$ there\nis exactly one irreducible component of $U_s$ passing through $u$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CB5","source_file":"more-morphisms.tex","source_line":10407,"source_end_line":10427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10407-L10427","statement_sha256":"6a71c9055890a497dd7b828d04a36d3f5e663eff50f9dcc1ce3a587954fa0991","origin":"The Stacks Project","memory_eligible":false,"source_rank":7357,"rank":7357,"depth":55,"x":1969.093,"y":782.656,"cluster":"scheme-morphisms"},{"id":"stacks:0DQ2","tag":"0DQ2","title":"Étale neighbourhoods and branches · Lemma 0DQ2","summary":"Let X → S be a smooth morphism of schemes. Let x ∈ X with image s ∈ S. Then • The number of geometric branches of X at x is equal to the number of geometric branches of S at s. • If kappa(x)/kappa(s) is a purely inseparable extension of fields, then number of branches of X at x is equal to the number of branches of S at s.","statement_latex":"Let $X \\to S$ be a smooth morphism of schemes.\nLet $x \\in X$ with image $s \\in S$.\nThen\n\\begin{enumerate}\n\\item The number of geometric branches of $X$ at $x$\nis equal to the number of geometric branches of $S$ at $s$.\n\\item If $\\kappa(x)/\\kappa(s)$ is a purely inseparable\\footnote{In fact,\nit would suffice if $\\kappa(x)$ is geometrically irreducible over\n$\\kappa(s)$. If we ever need this we will add a detailed proof.}\nextension of fields, then number of branches of $X$ at $x$\nis equal to the number of branches of $S$ at $s$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and branches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQ2","source_file":"more-morphisms.tex","source_line":10434,"source_end_line":10448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10434-L10448","statement_sha256":"a7cd704bae35660b3bb04930a0e62f16ab532c49b296e73024d89e1093a97164","origin":"The Stacks Project","memory_eligible":false,"source_rank":7358,"rank":7358,"depth":54,"x":1992.337,"y":569.68,"cluster":"scheme-morphisms"},{"id":"stacks:0GS8","tag":"0GS8","title":"Unramified and étale morphisms · Lemma 0GS8","summary":"Let f : X → Y be a morphism of schemes. Let x ∈ X with image y ∈ Y. Assume • Y is integral and geometrically unibranch at y, • f is locally of finite type, • there is a specialization x' leadsto x such that f(x') is the generic point of Y, • f is unramified at x. Then f is étale at x.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $x \\in X$ with\nimage $y \\in Y$. Assume\n\\begin{enumerate}\n\\item $Y$ is integral and geometrically unibranch at $y$,\n\\item $f$ is locally of finite type,\n\\item there is a specialization $x' \\leadsto x$ such that $f(x')$\nis the generic point of $Y$,\n\\item $f$ is unramified at $x$.\n\\end{enumerate}\nThen $f$ is \\'etale at $x$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Unramified and étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GS8","source_file":"more-morphisms.tex","source_line":10465,"source_end_line":10477,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10465-L10477","statement_sha256":"73fd39b72121aeb64bb3eaa3c42f852802896a503f7d3d756b9aab2e7073b366","origin":"The Stacks Project","memory_eligible":false,"source_rank":7359,"rank":7359,"depth":55,"x":2146.556,"y":740.012,"cluster":"scheme-morphisms"},{"id":"stacks:0GS9","tag":"0GS9","title":"Unramified and étale morphisms · Lemma 0GS9","summary":"[SGA1] Let f : X → Y be a morphism of schemes. Assume • Y is integral and geometrically unibranch, • at least one irreducible component of X dominates Y, • f is unramified, and • X is connected. Then f is étale and X is irreducible.","statement_latex":"\\begin{reference}\n\\cite[Expose I, Corollary 9.11]{SGA1}\n\\end{reference}\nLet $f : X \\to Y$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $Y$ is integral and geometrically unibranch,\n\\item at least one irreducible component of $X$ dominates $Y$,\n\\item $f$ is unramified, and\n\\item $X$ is connected.\n\\end{enumerate}\nThen $f$ is \\'etale and $X$ is irreducible.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Unramified and étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GS9","source_file":"more-morphisms.tex","source_line":10495,"source_end_line":10508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10495-L10508","statement_sha256":"60baaadfbd966565e8fdbc34bc9237df34b264b088af5d2622f521e0ac6dcf6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7360,"rank":7360,"depth":56,"x":1895.717,"y":701.897,"cluster":"scheme-morphisms"},{"id":"stacks:0GSA","tag":"0GSA","title":"Unramified and étale morphisms · Lemma 0GSA","summary":"Let f : X → Y and g : Y → Z be morphisms of schemes. Let x ∈ X with image y ∈ Y. Assume • Y is integral and geometrically unibranch at y, • g ∘ f is étale at x, • there is a specialization x' leadsto x such that f(x') is the generic point of Y. Then f is étale at x and g is étale at y.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes.\nLet $x \\in X$ with image $y \\in Y$. Assume\n\\begin{enumerate}\n\\item $Y$ is integral and geometrically unibranch at $y$,\n\\item $g \\circ f$ is \\'etale at $x$,\n\\item there is a specialization $x' \\leadsto x$ such that $f(x')$\nis the generic point of $Y$.\n\\end{enumerate}\nThen $f$ is \\'etale at $x$ and $g$ is \\'etale at $y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Unramified and étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSA","source_file":"more-morphisms.tex","source_line":10529,"source_end_line":10540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10529-L10540","statement_sha256":"d269b81a99a045577d5a544f9358c7a7270924b9249a784b7948fefae5b158ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":7361,"rank":7361,"depth":56,"x":2111.456,"y":587.605,"cluster":"scheme-morphisms"},{"id":"stacks:0GSB","tag":"0GSB","title":"Unramified and étale morphisms · Lemma 0GSB","summary":"Let f : X → Y and g : Y → Z be morphisms of schemes. Assume • Y is integral and geometrically unibranch, • g ∘ f is étale, • every irreducible component of X dominates Y. Then f is étale and g is étale at every point in the image of f.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes.\nAssume\n\\begin{enumerate}\n\\item $Y$ is integral and geometrically unibranch,\n\\item $g \\circ f$ is \\'etale,\n\\item every irreducible component of $X$ dominates $Y$.\n\\end{enumerate}\nThen $f$ is \\'etale and $g$ is \\'etale at every point in\nthe image of $f$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Unramified and étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GSB","source_file":"more-morphisms.tex","source_line":10555,"source_end_line":10566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10555-L10566","statement_sha256":"a3c19ebdf650e0f1a665f0dc23bd29c5fc7630f2dd156170fa01f0b043187180","origin":"The Stacks Project","memory_eligible":false,"source_rank":7362,"rank":7362,"depth":57,"x":2044.245,"y":794.417,"cluster":"scheme-morphisms"},{"id":"stacks:057C","tag":"057C","title":"Slicing smooth morphisms · Lemma 057C","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point with image s ∈ S. Let h ∈ m_x ⊂ O_X, x. Assume • f is smooth at x, and • the image doverlineh of dh in Ω_X_s/s, x ⊗_O_X_s, x kappa(x) = Ω_X/S, x ⊗_O_X, x kappa(x) is nonzero. Then there exists an affine open neighbourhood U ⊂ X of x such that h comes from h ∈ Γ(U, O_U) and such that D = V(h) is an effective Cartier divisor in U with x ∈ D and D → S smooth.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point with image $s \\in S$.\nLet $h \\in \\mathfrak m_x \\subset \\mathcal{O}_{X, x}$.\nAssume\n\\begin{enumerate}\n\\item $f$ is smooth at $x$, and\n\\item the image $\\text{d}\\overline{h}$ of $\\text{d}h$ in\n$$\n\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x) =\n\\Omega_{X/S, x} \\otimes_{\\mathcal{O}_{X, x}} \\kappa(x)\n$$\nis nonzero.\n\\end{enumerate}\nThen there exists an affine open neighbourhood $U \\subset X$ of $x$\nsuch that $h$ comes from $h \\in \\Gamma(U, \\mathcal{O}_U)$ and such\nthat $D = V(h)$ is an effective Cartier divisor in $U$ with $x \\in D$ and\n$D \\to S$ smooth.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057C","source_file":"more-morphisms.tex","source_line":10590,"source_end_line":10609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10590-L10609","statement_sha256":"9013bda58d1dfa23a6ab3e333460f71fe66487b8fa377016f6f24805db4ec4d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7363,"rank":7363,"depth":39,"x":1927.426,"y":603.67,"cluster":"scheme-morphisms"},{"id":"stacks:057D","tag":"057D","title":"Slicing smooth morphisms · Lemma 057D","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point with image s ∈ S. Assume • f is smooth at x, and • the map Ω_X_s/s, x ⊗_O_X_s, x kappa(x) → Ω_kappa(x)/kappa(s) has a nonzero kernel. Then there exists an affine open neighbourhood U ⊂ X of x and an effective Cartier divisor D ⊂ U containing x such that D → S is smooth.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point with image $s \\in S$.\nAssume\n\\begin{enumerate}\n\\item $f$ is smooth at $x$, and\n\\item the map\n$$\n\\Omega_{X_s/s, x} \\otimes_{\\mathcal{O}_{X_s, x}} \\kappa(x)\n\\longrightarrow\n\\Omega_{\\kappa(x)/\\kappa(s)}\n$$\nhas a nonzero kernel.\n\\end{enumerate}\nThen there exists an affine open neighbourhood $U \\subset X$ of $x$\nand an effective Cartier divisor $D \\subset U$ containing $x$ such that\n$D \\to S$ is smooth.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057D","source_file":"more-morphisms.tex","source_line":10660,"source_end_line":10678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10660-L10678","statement_sha256":"dcfeeac79a1cb449ef0794c94a15172f648fb880f86515533de223d43515e703","origin":"The Stacks Project","memory_eligible":false,"source_rank":7364,"rank":7364,"depth":40,"x":2167.097,"y":678.081,"cluster":"scheme-morphisms"},{"id":"stacks:057F","tag":"057F","title":"Slicing smooth morphisms · Lemma 057F","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point with image s ∈ S. Assume • f is smooth at x, • the residue field extension kappa(x)/kappa(s) is separable, and • x is not a generic point of X_s. Then there exists an affine open neighbourhood U ⊂ X of x and an effective Cartier divisor D ⊂ U containing x such that D → S is smooth.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point with image $s \\in S$.\nAssume\n\\begin{enumerate}\n\\item $f$ is smooth at $x$,\n\\item the residue field extension $\\kappa(x)/\\kappa(s)$\nis separable, and\n\\item $x$ is not a generic point of $X_s$.\n\\end{enumerate}\nThen there exists an affine open neighbourhood $U \\subset X$ of $x$\nand an effective Cartier divisor $D \\subset U$ containing $x$ such that\n$D \\to S$ is smooth.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057F","source_file":"more-morphisms.tex","source_line":10730,"source_end_line":10744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10730-L10744","statement_sha256":"be504754bbb9b2347a511fe5ed2f5130bf7665f6595898950edadd05970be73c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7365,"rank":7365,"depth":40,"x":1930.394,"y":759.253,"cluster":"scheme-morphisms"},{"id":"stacks:057G","tag":"057G","title":"Slicing smooth morphisms · Lemma 057G","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X be a point with image s ∈ S. Assume • f is smooth at x, and • x is a closed point of X_s and kappa(s) ⊂ kappa(x) is separable. Then there exists an immersion Z → X containing x such that • Z → S is étale, and • Z_s = (x) set theoretically.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ be a point with image $s \\in S$.\nAssume\n\\begin{enumerate}\n\\item $f$ is smooth at $x$, and\n\\item $x$ is a closed point of $X_s$ and $\\kappa(s) \\subset \\kappa(x)$\nis separable.\n\\end{enumerate}\nThen there exists an immersion $Z \\to X$ containing $x$ such that\n\\begin{enumerate}\n\\item $Z \\to S$ is \\'etale, and\n\\item $Z_s = \\{x\\}$ set theoretically.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057G","source_file":"more-morphisms.tex","source_line":10774,"source_end_line":10789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10774-L10789","statement_sha256":"9dc0dfdf5c912f2988e76b04b0804393b83686ad313943561c9765bcad1d5b2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7366,"rank":7366,"depth":41,"x":2039.721,"y":564.976,"cluster":"scheme-morphisms"},{"id":"stacks:055U","tag":"055U","title":"Slicing smooth morphisms · Lemma 055U","summary":"Smooth morphisms admit étale local sections. See [EGA4]. Let f : X → S be a smooth morphism of schemes. Let s ∈ S be a point in the image of f. Then there exists an étale neighbourhood (S', s') → (S, s) and a S-morphism S' → X.","statement_latex":"\\begin{slogan}\nSmooth morphisms admit \\'etale local sections.\n\\end{slogan}\n\\begin{reference}\nSee \\cite[Corollaire 17.16.3 (ii)]{EGA4}.\n\\end{reference}\nLet $f : X \\to S$ be a smooth morphism of schemes.\nLet $s \\in S$ be a point in the image of $f$.\nThen there exists an \\'etale neighbourhood $(S', s') \\to (S, s)$\nand a $S$-morphism $S' \\to X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055U","source_file":"more-morphisms.tex","source_line":10845,"source_end_line":10857,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10845-L10857","statement_sha256":"3a90f967f485d728d2985e520ca9f4402c44c5bed4135fb97308e26bfea9d353","origin":"The Stacks Project","memory_eligible":false,"source_rank":7367,"rank":7367,"depth":46,"x":2115.38,"y":770.384,"cluster":"scheme-morphisms"},{"id":"stacks:055V","tag":"055V","title":"Slicing smooth morphisms · Lemma 055V","summary":"Let S be a scheme. Let U = (S_i → S)_i ∈ I be a smooth covering of S, see Topologies, Definition [Tag 021Z]. Then there exists an étale covering V = (T_j → S)_j ∈ J (see Topologies, Definition [Tag 0215]) which refines (see Sites, Definition [Tag 00VT]) U.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{U} = \\{S_i \\to S\\}_{i \\in I}$ be a smooth\ncovering of $S$, see\nTopologies, Definition \\ref{topologies-definition-smooth-covering}.\nThen there exists an \\'etale covering $\\mathcal{V} = \\{T_j \\to S\\}_{j \\in J}$\n(see\nTopologies, Definition \\ref{topologies-definition-etale-covering})\nwhich refines (see\nSites, Definition \\ref{sites-definition-morphism-coverings})\n$\\mathcal{U}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055V","source_file":"more-morphisms.tex","source_line":10934,"source_end_line":10945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10934-L10945","statement_sha256":"b2efbc2f8e6bf28d42ce99bd3b5bfe48dd95b4faea1b979737dca70324dd0ad5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7368,"rank":7368,"depth":47,"x":1894.28,"y":661.788,"cluster":"scheme-morphisms"},{"id":"stacks:0EY4","tag":"0EY4","title":"Slicing smooth morphisms · Lemma 0EY4","summary":"Let f : X → S be a smooth morphism of schemes. Then there exists an étale covering (U_i → X)_i ∈ I such that U_i → S factors as U_i → V_i → S where V_i → S is étale and U_i → V_i is a smooth morphism of affine schemes, which has a section, and has geometrically connected fibres.","statement_latex":"Let $f : X \\to S$ be a smooth morphism of schemes. Then there exists an\n\\'etale covering $\\{U_i \\to X\\}_{i \\in I}$ such that $U_i \\to S$\nfactors as $U_i \\to V_i \\to S$ where $V_i \\to S$ is \\'etale and\n$U_i \\to V_i$ is a smooth morphism of affine schemes, which\nhas a section, and has geometrically connected fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Slicing smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EY4","source_file":"more-morphisms.tex","source_line":10956,"source_end_line":10963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L10956-L10963","statement_sha256":"edc351f4f4fa8c78f83d7f060185b6e87043ffa42d9e2d7534080ad4884090c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7369,"rank":7369,"depth":46,"x":2144.788,"y":616.382,"cluster":"scheme-morphisms"},{"id":"stacks:0CAU","tag":"0CAU","title":"Étale neighbourhoods and Artin approximation · Lemma 0CAU","summary":"Let S be a locally Noetherian scheme. Let X, Y be schemes locally of finite type over S. Let x ∈ X and y ∈ Y be points lying over the same point s ∈ S. Assume O_S, s is a G-ring. Assume further we are given a local O_S, s-algebra map φ : O_Y, y → O_X, x^wedge For every N ≥ 1 there exists an elementary étale neighbourhood (U, u) → (X, x) and an S-morphism f : U → Y mapping u to y such that the diagram xymatrix O_X, x^wedge ar[r] & O_U, u^wedge O_Y, y ar[r]^f^sharp_u…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be\nschemes locally of finite\ntype over $S$. Let $x \\in X$ and $y \\in Y$ be points lying over the\nsame point $s \\in S$. Assume $\\mathcal{O}_{S, s}$ is a G-ring.\nAssume further we are given a local $\\mathcal{O}_{S, s}$-algebra map\n$$\n\\varphi : \\mathcal{O}_{Y, y} \\longrightarrow \\mathcal{O}_{X, x}^\\wedge\n$$\nFor every $N \\geq 1$\nthere exists an elementary \\'etale neighbourhood\n$(U, u) \\to (X, x)$ and an $S$-morphism\n$f : U \\to Y$ mapping $u$ to $y$ such that the diagram\n$$\n\\xymatrix{\n\\mathcal{O}_{X, x}^\\wedge \\ar[r] &\n\\mathcal{O}_{U, u}^\\wedge \\\\\n\\mathcal{O}_{Y, y} \\ar[r]^{f^\\sharp_u} \\ar[u]^\\varphi &\n\\mathcal{O}_{U, u} \\ar[u]\n}\n$$\ncommutes modulo $\\mathfrak m_u^N$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and Artin approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAU","source_file":"more-morphisms.tex","source_line":11016,"source_end_line":11039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11016-L11039","statement_sha256":"7c126621df7d94bf43b474f61b6bfb8b80ec38fbe8f56c7d8d77dd1249907d55","origin":"The Stacks Project","memory_eligible":false,"source_rank":7370,"rank":7370,"depth":55,"x":1996.497,"y":792.108,"cluster":"scheme-morphisms"},{"id":"stacks:0CAV","tag":"0CAV","title":"Étale neighbourhoods and Artin approximation · Lemma 0CAV","summary":"Let S be a locally Noetherian scheme. Let X, Y be schemes locally of finite type over S. Let x ∈ X and y ∈ Y be points lying over the same point s ∈ S. Assume O_S, s is a G-ring. Assume we have an O_S, s-algebra isomorphism φ : O_Y, y^wedge → O_X, x^wedge between the complete local rings. Then for every N ≥ 1 there exists morphisms (X, x) ← (U, u) → (Y, y) of pointed schemes over S such that both arrows define elementary étale neighbourhoods and such that the diagram…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $X$, $Y$ be\nschemes locally of finite\ntype over $S$. Let $x \\in X$ and $y \\in Y$ be points lying over the\nsame point $s \\in S$. Assume $\\mathcal{O}_{S, s}$ is a G-ring.\nAssume we have an $\\mathcal{O}_{S, s}$-algebra isomorphism\n$$\n\\varphi : \\mathcal{O}_{Y, y}^\\wedge \\longrightarrow \\mathcal{O}_{X, x}^\\wedge\n$$\nbetween the complete local rings. Then for every $N \\geq 1$\nthere exists morphisms\n$$\n(X, x) \\leftarrow (U, u) \\rightarrow (Y, y)\n$$\nof pointed schemes over $S$ such that both arrows define elementary\n\\'etale neighbourhoods and such that the diagram\n$$\n\\xymatrix{\n& \\mathcal{O}_{U, u}^\\wedge \\\\\n\\mathcal{O}_{Y, y}^\\wedge \\ar[rr]^\\varphi \\ar[ru] & &\n\\mathcal{O}_{X, x}^\\wedge \\ar[lu]\n}\n$$\ncommutes modulo $\\mathfrak m_u^N$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and Artin approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAV","source_file":"more-morphisms.tex","source_line":11070,"source_end_line":11095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11070-L11095","statement_sha256":"801b73a2ec03a3026f14e080b2fb44c0d7c8e46bce2e234c7206940c5f80c82b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7371,"rank":7371,"depth":56,"x":1964.513,"y":578.266,"cluster":"scheme-morphisms"},{"id":"stacks:0GDX","tag":"0GDX","title":"Étale neighbourhoods and Artin approximation · Lemma 0GDX","summary":"Let X → S, Y → T, x, s, y, t, σ, y_σ, and φ be given as follows: we have morphisms of schemes vcenter xymatrix X ar[d] & Y ar[d] S & T with points vcenter xymatrix x ar[d] & y ar[d] s & t Here S is locally Noetherian and T is of finite type over Z. The morphisms X → S and Y → T are locally of finite type. The local ring O_S, s is a G-ring. The map σ : O_T, t → O_S, s^wedge is a local homomorphism. Set Y_σ = Y ×_T, σ Spec(O_S, s^wedge). Next, y_σ is a point of Y_σ mapping…","statement_latex":"Let $X \\to S$, $Y \\to T$, $x$, $s$, $y$, $t$, $\\sigma$, $y_\\sigma$, and\n$\\varphi$ be given as follows: we have morphisms of schemes\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[d] & Y \\ar[d] \\\\\nS & T\n}\n}\n\\quad\\text{with points}\\quad\n\\vcenter{\n\\xymatrix{\nx \\ar[d] & y \\ar[d] \\\\\ns & t\n}\n}\n$$\nHere $S$ is locally Noetherian and $T$ is of finite type over $\\mathbf{Z}$.\nThe morphisms $X \\to S$ and $Y \\to T$ are locally of finite type.\nThe local ring $\\mathcal{O}_{S, s}$ is a G-ring. The map\n$$\n\\sigma : \\mathcal{O}_{T, t} \\longrightarrow \\mathcal{O}_{S, s}^\\wedge\n$$\nis a local homomorphism. Set\n$Y_\\sigma = Y \\times_{T, \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge)$.\nNext, $y_\\sigma$ is a point of $Y_\\sigma$ mapping to $y$ and\nthe closed point of $\\Spec(\\mathcal{O}_{S, s}^\\wedge)$. Finally\n$$\n\\varphi :\n\\mathcal{O}_{X, x}^\\wedge\n\\longrightarrow\n\\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge\n$$\nis an isomorphism of $\\mathcal{O}_{S, s}^\\wedge$-algebras.\nIn this situation there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] &\nW \\ar[l] \\ar[rd] \\ar[rr] & &\nY \\times_{T, \\tau} V \\ar[r] \\ar[ld] & Y \\ar[d] \\\\\nS & &\nV \\ar[ll] \\ar[rr]^\\tau & &\nT\n}\n$$\nof schemes and points $w \\in W$, $v \\in V$ such that\n\\begin{enumerate}\n\\item $(V, v) \\to (S, s)$ is an elementary \\'etale neighbourhood,\n\\item $(W, w) \\to (X, x)$ is an elementary \\'etale neighbourhood, and\n\\item $\\tau(v) = t$.\n\\end{enumerate}\nLet $y_\\tau \\in Y \\times_T V$ correspond to $y_\\sigma$\nvia the identification $(Y_\\sigma)_s = (Y \\times_T V)_v$.\nThen\n\\begin{enumerate}\n\\item[(4)] $(W, w) \\to (Y \\times_{T, \\tau} V, y_\\tau)$ is an elementary\n\\'etale neighbourhood.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and Artin approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDX","source_file":"more-morphisms.tex","source_line":11143,"source_end_line":11203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11143-L11203","statement_sha256":"05144e535de9ef2105fadbf918829f46f8ca1105f66f45c3e965596ef6eabe2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7372,"rank":7372,"depth":57,"x":2160.175,"y":717.88,"cluster":"scheme-morphisms"},{"id":"stacks:0CAW","tag":"0CAW","title":"Étale neighbourhoods and Artin approximation · Lemma 0CAW","summary":"Consider a diagram vcenter xymatrix X ar[d] & Y ar[d] S & T ar[l] with points vcenter xymatrix x ar[d] & y ar[d] s & t ar[l] where S be a locally Noetherian scheme and the morphisms are locally of finite type. Assume O_S, s is a G-ring. Assume further we are given a local O_S, s-algebra map σ : O_T, t → O_S, s^wedge and a local O_S, s-algebra map φ : O_X, x → O_Y_σ, y_σ^wedge where Y_σ = Y ×_T, σ Spec(O_S, s^wedge) and y_σ is the unique point of Y_σ lying over y. For…","statement_latex":"Consider a diagram\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[d] & Y \\ar[d] \\\\\nS & T \\ar[l]\n}\n}\n\\quad\\text{with points}\\quad\n\\vcenter{\n\\xymatrix{\nx \\ar[d] & y \\ar[d] \\\\\ns & t \\ar[l]\n}\n}\n$$\nwhere $S$ be a locally Noetherian scheme and the morphisms are\nlocally of finite type. Assume $\\mathcal{O}_{S, s}$ is a G-ring.\nAssume further we are given a local $\\mathcal{O}_{S, s}$-algebra map\n$$\n\\sigma : \\mathcal{O}_{T, t} \\longrightarrow \\mathcal{O}_{S, s}^\\wedge\n$$\nand a local $\\mathcal{O}_{S, s}$-algebra map\n$$\n\\varphi :\n\\mathcal{O}_{X, x}\n\\longrightarrow\n\\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge\n$$\nwhere $Y_\\sigma = Y \\times_{T, \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge)$\nand $y_\\sigma$ is the unique point of $Y_\\sigma$ lying over $y$.\nFor every $N \\geq 1$ there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] & X \\times_S V \\ar[l] \\ar[rd] &\nW \\ar[l]^-f \\ar[r] \\ar[d] &\nY \\times_{T, \\tau} V \\ar[r] \\ar[ld] & Y \\ar[d] \\\\\nS & & V \\ar[ll] \\ar[rr]^\\tau & & T\n}\n$$\nof schemes over $S$ and points $w \\in W$, $v \\in V$ such that\n\\begin{enumerate}\n\\item $v \\mapsto s$, $\\tau(v) = t$, $f(w) = (x, v)$, and $w \\mapsto (y, v)$,\n\\item $(V, v) \\to (S, s)$ is an elementary \\'etale neighbourhood,\n\\item the diagram\n$$\n\\xymatrix{\n\\mathcal{O}_{S, s}^\\wedge \\ar[r] & \\mathcal{O}_{V, v}^\\wedge \\\\\n\\mathcal{O}_{T, t} \\ar[r]^{\\tau^\\sharp_v} \\ar[u]_\\sigma &\n\\mathcal{O}_{V, v} \\ar[u]\n}\n$$\ncommutes module $\\mathfrak m_v^N$,\n\\item $(W, w) \\to (Y \\times_{T, \\tau} V, (y, v))$ is an\nelementary \\'etale neighbourhood,\n\\item the diagram\n$$\n\\xymatrix{\n\\mathcal{O}_{X, x} \\ar[r]_\\varphi &\n\\mathcal{O}_{Y_\\sigma, y_\\sigma}^\\wedge \\ar[r] &\n\\mathcal{O}_{Y_\\sigma, y_\\sigma}/\\mathfrak m_{y_\\sigma}^N \\ar@{=}[r] &\n\\mathcal{O}_{Y \\times_{T, \\tau} V, (y, v)}/\\mathfrak m_{(y, v)}^N\n\\ar[d]_{\\cong} \\\\\n\\mathcal{O}_{X, x} \\ar[r] \\ar@{=}[u] &\n\\mathcal{O}_{X \\times_S V, (x, v)} \\ar[r]^{f^\\sharp_w} &\n\\mathcal{O}_{W, w} \\ar[r] &\n\\mathcal{O}_{W, w}/\\mathfrak m_w^N\n}\n$$\ncommutes. The equality comes from the fact that\n$Y_\\sigma$ and $Y \\times_{T, \\tau} V$ are canonically isomorphic over\n$\\mathcal{O}_{V, v}/\\mathfrak m_v^N = \\mathcal{O}_{S, s}/\\mathfrak m_s^N$\nby parts (2) and (3).\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and Artin approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAW","source_file":"more-morphisms.tex","source_line":11288,"source_end_line":11364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11288-L11364","statement_sha256":"41ff2c977effe678eef06926b326db5fe0493f425cfb55e084b78e85e45f1e60","origin":"The Stacks Project","memory_eligible":false,"source_rank":7373,"rank":7373,"depth":56,"x":1903.478,"y":725.958,"cluster":"scheme-morphisms"},{"id":"stacks:0CAX","tag":"0CAX","title":"Étale neighbourhoods and Artin approximation · Lemma 0CAX","summary":"Let T → S be finite type morphisms of Noetherian schemes. Let t ∈ T map to s ∈ S and let σ : O_T, t → O_S, s^wedge be a local O_S, s-algebra map. For every N ≥ 1 there exists a finite type morphism (T', t') → (T, t) such that σ factors through O_T, t → O_T', t' and such that for every local O_S, s-algebra map σ' : O_T, t → O_S, s^wedge which factors through O_T, t → O_T', t' the maps σ and σ' agree modulo m_s^N.","statement_latex":"Let $T \\to S$ be finite type morphisms of Noetherian schemes.\nLet $t \\in T$ map to $s \\in S$ and let\n$\\sigma : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$\nbe a local $\\mathcal{O}_{S, s}$-algebra map. For every $N \\geq 1$\nthere exists a finite type morphism $(T', t') \\to (T, t)$\nsuch that $\\sigma$ factors through\n$\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$\nand such that for every local $\\mathcal{O}_{S, s}$-algebra map\n$\\sigma' : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$\nwhich factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$\nthe maps $\\sigma$ and $\\sigma'$ agree modulo $\\mathfrak m_s^N$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and Artin approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAX","source_file":"more-morphisms.tex","source_line":11529,"source_end_line":11542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11529-L11542","statement_sha256":"c7476dd8727d5ec35e1f3b8575e6c43a4112b658f6b8a93455aaacfcbe23eaae","origin":"The Stacks Project","memory_eligible":false,"source_rank":7374,"rank":7374,"depth":0,"x":2086.369,"y":574.26,"cluster":"scheme-morphisms"},{"id":"stacks:0CAY","tag":"0CAY","title":"Étale neighbourhoods and Artin approximation · Lemma 0CAY","summary":"Let Y → T → S be finite type morphisms of Noetherian schemes. Let t ∈ T map to s ∈ S and let σ : O_T, t → O_S, s^wedge be a local O_S, s-algebra map. There exists a finite type morphism (T', t') → (T, t) such that σ factors through O_T, t → O_T', t' and such that for every local O_S, s-algebra map σ' : O_T, t → O_S, s^wedge which factors through O_T, t → O_T', t' the closed immersions Y ×_T, σ Spec(O_S, s^wedge) = Y_σ longleftarrow Y_t → Y_σ' = Y ×_T, σ' Spec(O_S,…","statement_latex":"Let $Y \\to T \\to S$ be finite type morphisms of Noetherian schemes.\nLet $t \\in T$ map to $s \\in S$ and let\n$\\sigma : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$\nbe a local $\\mathcal{O}_{S, s}$-algebra map.\nThere exists a finite type morphism $(T', t') \\to (T, t)$\nsuch that $\\sigma$ factors through\n$\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$\nand such that for every local $\\mathcal{O}_{S, s}$-algebra map\n$\\sigma' : \\mathcal{O}_{T, t} \\to \\mathcal{O}_{S, s}^\\wedge$\nwhich factors through $\\mathcal{O}_{T, t} \\to \\mathcal{O}_{T', t'}$\nthe closed immersions\n$$\nY \\times_{T, \\sigma} \\Spec(\\mathcal{O}_{S, s}^\\wedge) = Y_\\sigma\n\\longleftarrow Y_t \\longrightarrow\nY_{\\sigma'} =\nY \\times_{T, \\sigma'} \\Spec(\\mathcal{O}_{S, s}^\\wedge)\n$$\nhave isomorphic conormal algebras.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and Artin approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAY","source_file":"more-morphisms.tex","source_line":11570,"source_end_line":11590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11570-L11590","statement_sha256":"16e6ba7a3986246eba84a3ac112c946a2dfbb4996f3a3c9e98c2a81579273170","origin":"The Stacks Project","memory_eligible":false,"source_rank":7375,"rank":7375,"depth":7,"x":2073.492,"y":790.019,"cluster":"scheme-morphisms"},{"id":"stacks:0CB1","tag":"0CB1","title":"Étale neighbourhoods and Artin approximation · Lemma 0CB1","summary":"With notation an assumptions as in Lemma [Tag 0CAW] assume that φ induces an isomorphism on completions. Then we can choose our diagram such that f is étale.","statement_latex":"With notation an assumptions as in Lemma \\ref{lemma-relative-map-approximation}\nassume that $\\varphi$ induces an isomorphism on completions.\nThen we can choose our diagram such that $f$ is \\'etale.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale neighbourhoods and Artin approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CB1","source_file":"more-morphisms.tex","source_line":11720,"source_end_line":11725,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11720-L11725","statement_sha256":"7713ab7fcc549a4c6048b84c67adea22c5cf337224e69f7abdd7ae1a3f5a1d02","origin":"The Stacks Project","memory_eligible":false,"source_rank":7376,"rank":7376,"depth":57,"x":1909.388,"y":623.521,"cluster":"scheme-morphisms"},{"id":"stacks:02LG","tag":"02LG","title":"Finite free locally dominates étale · Lemma 02LG","summary":"Let S be a scheme. Let s ∈ S. Let f : (U, u) → (S, s) be an étale neighbourhood. There exists an affine open neighbourhood s ∈ V ⊂ S and a surjective, finite locally free morphism π : T → V such that for every t ∈ π^-1(s) there exists an open neighbourhood t ∈ W_t ⊂ T and a commutative diagram xymatrix T ar[d]^π & W_t ar[l] ar[rr]_h_t ar[rd] & & U ar[dl] V ar[rr] & & S with h_t(t) = u.","statement_latex":"Let $S$ be a scheme. Let $s \\in S$.\nLet $f : (U, u) \\to (S, s)$ be an \\'etale neighbourhood.\nThere exists an affine open neighbourhood $s \\in V \\subset S$\nand a surjective, finite locally free morphism $\\pi : T \\to V$\nsuch that for every $t \\in \\pi^{-1}(s)$ there exists an\nopen neighbourhood $t \\in W_t \\subset T$ and a commutative\ndiagram\n$$\n\\xymatrix{\nT \\ar[d]^\\pi & W_t \\ar[l] \\ar[rr]_{h_t} \\ar[rd] & & U \\ar[dl] \\\\\nV \\ar[rr] & & S\n}\n$$\nwith $h_t(t) = u$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Finite free locally dominates étale","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LG","source_file":"more-morphisms.tex","source_line":11822,"source_end_line":11838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11822-L11838","statement_sha256":"8dde421f18d1927994f8f3486138ae6970750148efcab14d5ae0208c3c9e3597","origin":"The Stacks Project","memory_eligible":false,"source_rank":7377,"rank":7377,"depth":44,"x":2164.432,"y":653.194,"cluster":"scheme-morphisms"},{"id":"stacks:02LH","tag":"02LH","title":"Finite free locally dominates étale · Lemma 02LH","summary":"Let f : U → S be a surjective étale morphism of affine schemes. There exists a surjective, finite locally free morphism π : T → S and a finite open covering T = T_1 ∪ … ∪ T_n such that each T_i → S factors through U → S. Diagram: xymatrix & coprod T_i ar[rd] ar[ld] & T ar[rd]^π & & U ar[ld]_f & S & where the south-west arrow is a Zariski-covering.","statement_latex":"Let $f : U \\to S$ be a surjective \\'etale morphism of affine schemes.\nThere exists a surjective, finite locally free morphism\n$\\pi : T \\to S$ and a finite open covering\n$T = T_1 \\cup \\ldots \\cup T_n$ such that each\n$T_i \\to S$ factors through $U \\to S$. Diagram:\n$$\n\\xymatrix{\n& \\coprod T_i  \\ar[rd] \\ar[ld] & \\\\\nT \\ar[rd]^\\pi & & U \\ar[ld]_f \\\\\n& S &\n}\n$$\nwhere the south-west arrow is a Zariski-covering.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Finite free locally dominates étale","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LH","source_file":"more-morphisms.tex","source_line":11851,"source_end_line":11866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11851-L11866","statement_sha256":"1aadd6e106552f618748120640d48b19e9eb83aa019e19a706bef77af45eac68","origin":"The Stacks Project","memory_eligible":false,"source_rank":7378,"rank":7378,"depth":44,"x":1952.385,"y":776.101,"cluster":"scheme-morphisms"},{"id":"stacks:02LK","tag":"02LK","title":"Étale localization of quasi-finite morphisms · Lemma 02LK","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. Set s = f(x). Assume that • f is locally of finite type, and • x ∈ X_s is isolated. Then there exist • [(a)] an elementary étale neighbourhood (U, u) → (S, s), • [(b)] an open subscheme V ⊂ X_U (see [Tag 02LJ]) such that • [(romannumeral1)] V → U is a finite morphism, • [(romannumeral2)] there is a unique point v of V mapping to u in U, and • [(romannumeral3)] the point v maps to x under the morphism X_U → X, inducing…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$. Set $s = f(x)$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type, and\n\\item $x \\in X_s$ is isolated\\footnote{In the presence of (1)\nthis means that $f$ is\nquasi-finite at $x$, see\nMorphisms, Lemma \\ref{morphisms-lemma-quasi-finite-at-point-characterize}.}.\n\\end{enumerate}\nThen there exist\n\\begin{enumerate}\n\\item[(a)] an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$,\n\\item[(b)] an open subscheme $V \\subset X_U$\n(see \\ref{equation-basic-diagram})\n\\end{enumerate}\nsuch that\n\\begin{enumerate}\n\\item[(\\romannumeral1)] $V \\to U$ is a finite morphism,\n\\item[(\\romannumeral2)] there is a unique point $v$ of $V$\nmapping to $u$ in $U$, and\n\\item[(\\romannumeral3)] the point $v$ maps to $x$\nunder the morphism $X_U \\to X$, inducing $\\kappa(x) = \\kappa(v)$.\n\\end{enumerate}\nMoreover, for any elementary \\'etale neighbourhood $(U', u') \\to (U, u)$\nsetting $V' = U' \\times_U V \\subset X_{U'}$ the triple $(U', u', V')$\nsatisfies the properties\n(\\romannumeral1), (\\romannumeral2), and (\\romannumeral3) as well.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale localization of quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LK","source_file":"more-morphisms.tex","source_line":11923,"source_end_line":11953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11923-L11953","statement_sha256":"6df7e131e1a0a8df0c45aacc75644b4e3295c739c4e435836432b13867c41a5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7379,"rank":7379,"depth":29,"x":2009.939,"y":565.032,"cluster":"scheme-morphisms"},{"id":"stacks:02LL","tag":"02LL","title":"Étale localization of quasi-finite morphisms · Lemma 02LL","summary":"Let f : X → S be a morphism of schemes. Let x_1, …, x_n ∈ X be points having the same image s in S. Assume that • f is locally of finite type, and • x_i ∈ X_s is isolated for i = 1, …, n. Then there exist • [(a)] an elementary étale neighbourhood (U, u) → (S, s), • [(b)] for each i an open subscheme V_i ⊂ X_U, such that for each i we have • [(romannumeral1)] V_i → U is a finite morphism, • [(romannumeral2)] there is a unique point v_i of V_i mapping to u in U, and •…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type, and\n\\item $x_i \\in X_s$ is isolated for $i = 1, \\ldots, n$.\n\\end{enumerate}\nThen there exist\n\\begin{enumerate}\n\\item[(a)] an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$,\n\\item[(b)] for each $i$ an open subscheme $V_i \\subset X_U$,\n\\end{enumerate}\nsuch that for each $i$ we have\n\\begin{enumerate}\n\\item[(\\romannumeral1)] $V_i \\to U$ is a finite morphism,\n\\item[(\\romannumeral2)] there is a unique point $v_i$ of $V_i$\nmapping to $u$ in $U$, and\n\\item[(\\romannumeral3)] the point $v_i$ maps to $x_i$ in $X$ and\n$\\kappa(x_i) = \\kappa(v_i)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale localization of quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LL","source_file":"more-morphisms.tex","source_line":11973,"source_end_line":11995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L11973-L11995","statement_sha256":"8af0a14d8ea0549b46c7b4e075e22f265c750af51905afcf4af4268658b773fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7380,"rank":7380,"depth":30,"x":2137.307,"y":753.434,"cluster":"scheme-morphisms"},{"id":"stacks:02LM","tag":"02LM","title":"Étale localization of quasi-finite morphisms · Lemma 02LM","summary":"Let f : X → S be a morphism of schemes. Let x_1, …, x_n ∈ X be points having the same image s in S. Assume that • f is locally of finite type, and • x_i ∈ X_s is isolated for i = 1, …, n. Then there exist • [(a)] an étale neighbourhood (U, u) → (S, s), • [(b)] for each i an integer m_i and open subschemes V_i, j ⊂ X_U, j = 1, …, m_i such that we have • [(romannumeral1)] each V_i, j → U is a finite morphism, • [(romannumeral2)] there is a unique point v_i, j of V_i, j…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type, and\n\\item $x_i \\in X_s$ is isolated for $i = 1, \\ldots, n$.\n\\end{enumerate}\nThen there exist\n\\begin{enumerate}\n\\item[(a)] an \\'etale neighbourhood $(U, u) \\to (S, s)$,\n\\item[(b)] for each $i$ an integer $m_i$ and\nopen subschemes $V_{i, j} \\subset X_U$, $j = 1, \\ldots, m_i$\n\\end{enumerate}\nsuch that we have\n\\begin{enumerate}\n\\item[(\\romannumeral1)] each $V_{i, j} \\to U$ is a finite morphism,\n\\item[(\\romannumeral2)] there is a unique point $v_{i, j}$ of $V_{i, j}$\nmapping to $u$ in $U$ with $\\kappa(u) \\subset \\kappa(v_{i, j})$\nfinite purely inseparable,\n\\item[(\\romannumeral4)] if $v_{i, j} = v_{i', j'}$, then $i = i'$ and\n$j = j'$, and\n\\item[(\\romannumeral3)] the points $v_{i, j}$ map to $x_i$ in $X$ and\nno other points of $(X_U)_u$ map to $x_i$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale localization of quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LM","source_file":"more-morphisms.tex","source_line":12018,"source_end_line":12044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12018-L12044","statement_sha256":"7aa6531e2c3a97e4b264d9b4ecedacc56257ef25d1bef7acfea3a14de3805637","origin":"The Stacks Project","memory_eligible":false,"source_rank":7381,"rank":7381,"depth":31,"x":1891.742,"y":686.743,"cluster":"scheme-morphisms"},{"id":"stacks:02LN","tag":"02LN","title":"Étale localization of quasi-finite morphisms · Lemma 02LN","summary":"Let f : X → S be a morphism of schemes. Let s ∈ S. Let x_1, …, x_n ∈ X_s. Assume that • f is locally of finite type, • f is separated, and • x_1, …, x_n are pairwise distinct isolated points of X_s. Then there exists an elementary étale neighbourhood (U, u) → (S, s) and a decomposition U ×_S X = W amalg V_1 amalg … amalg V_n into open and closed subschemes such that the morphisms V_i → U are finite, the fibres of V_i → U over u are singletons (v_i), each v_i maps to x_i…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $s \\in S$. Let $x_1, \\ldots, x_n \\in X_s$. Assume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item $f$ is separated, and\n\\item $x_1, \\ldots, x_n$ are pairwise distinct isolated points of $X_s$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$\nand a decomposition\n$$\nU \\times_S X = W \\amalg V_1 \\amalg \\ldots \\amalg V_n\n$$\ninto open and closed subschemes such that the morphisms\n$V_i \\to U$ are finite, the fibres of $V_i \\to U$ over $u$ are\nsingletons $\\{v_i\\}$, each $v_i$ maps to $x_i$ with\n$\\kappa(x_i) = \\kappa(v_i)$, and the fibre of $W \\to U$\nover $u$ contains no points mapping to any of the $x_i$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale localization of quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LN","source_file":"more-morphisms.tex","source_line":12093,"source_end_line":12112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12093-L12112","statement_sha256":"e7ad7f398ca7d9b001db3aef9f463cd50818ed5abe48c26011408a4b08c2fb69","origin":"The Stacks Project","memory_eligible":false,"source_rank":7382,"rank":7382,"depth":31,"x":2126.581,"y":596.53,"cluster":"scheme-morphisms"},{"id":"stacks:02LO","tag":"02LO","title":"Étale localization of quasi-finite morphisms · Lemma 02LO","summary":"Let f : X → S be a morphism of schemes. Let s ∈ S. Let x_1, …, x_n ∈ X_s. Assume that • f is locally of finite type, • f is separated, and • x_1, …, x_n are pairwise distinct isolated points of X_s. Then there exists an étale neighbourhood (U, u) → (S, s) and a decomposition U ×_S X = W amalg coprod_i = 1, …, n coprod_j = 1, …, m_i V_i, j into open and closed subschemes such that the morphisms V_i, j → U are finite, the fibres of V_i, j → U over u are singletons (v_i, j),…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $s \\in S$. Let $x_1, \\ldots, x_n \\in X_s$. Assume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item $f$ is separated, and\n\\item $x_1, \\ldots, x_n$ are pairwise distinct isolated points of $X_s$.\n\\end{enumerate}\nThen there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$\nand a decomposition\n$$\nU \\times_S X =\nW \\amalg\n\\ \\coprod\\nolimits_{i = 1, \\ldots, n}\n\\ \\coprod\\nolimits_{j = 1, \\ldots, m_i}\nV_{i, j}\n$$\ninto open and closed subschemes such that the morphisms\n$V_{i, j} \\to U$ are finite, the fibres of $V_{i, j} \\to U$ over $u$ are\nsingletons $\\{v_{i, j}\\}$, each $v_{i, j}$ maps to $x_i$,\n$\\kappa(u) \\subset \\kappa(v_{i, j})$ is purely inseparable,\nand the fibre of $W \\to U$ over $u$ contains no points mapping\nto any of the $x_i$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale localization of quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LO","source_file":"more-morphisms.tex","source_line":12135,"source_end_line":12159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12135-L12159","statement_sha256":"3b54a3a5b0c3a2f022756e9970baabac3312fbe122d7009d5c328a9c20197a90","origin":"The Stacks Project","memory_eligible":false,"source_rank":7383,"rank":7383,"depth":32,"x":2025.905,"y":796.416,"cluster":"scheme-morphisms"},{"id":"stacks:02LP","tag":"02LP","title":"Étale localization of quasi-finite morphisms · Lemma 02LP","summary":"Let f : X → S be a morphism of schemes. Let s ∈ S. Assume that • f is locally of finite type, • f is separated, and • X_s has at most finitely many isolated points. Then there exists an elementary étale neighbourhood (U, u) → (S, s) and a decomposition U ×_S X = W amalg V into open and closed subschemes such that the morphism V → U is finite, and the fibre W_u of the morphism W → U contains no isolated points. In particular, if f^-1(s) is a finite set, then W_u = ∅.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $s \\in S$. Assume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item $f$ is separated, and\n\\item $X_s$ has at most finitely many isolated points.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(U, u) \\to (S, s)$\nand a decomposition\n$$\nU \\times_S X = W \\amalg V\n$$\ninto open and closed subschemes such that the morphism\n$V \\to U$ is finite, and the fibre $W_u$ of the\nmorphism $W \\to U$ contains no isolated points.\nIn particular, if $f^{-1}(s)$ is a finite set, then $W_u = \\emptyset$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale localization of quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LP","source_file":"more-morphisms.tex","source_line":12171,"source_end_line":12189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12171-L12189","statement_sha256":"cab5bbacefb281ceb44c0035c42e0aa29d93f465578fe299c5bd4172c655a47c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7384,"rank":7384,"depth":32,"x":1939.351,"y":591.783,"cluster":"scheme-morphisms"},{"id":"stacks:0BSR","tag":"0BSR","title":"Étale localization of integral morphisms · Lemma 0BSR","summary":"Let R → S be an integral ring map. Let p ⊂ R be a prime ideal. Assume • there are finitely many primes q_1, …, q_n lying over p, and • for each i the maximal separable subextension kappa( q)/kappa( q_i)_sep/kappa( p) (Fields, Lemma [Tag 030K]) is finite over kappa( p). Then there exists an étale ring map R → R' and a prime p' lying over p such that S ⊗_R R' = A_1 × … × A_m with R' → A_j integral having a unique prime r_j over p' such that kappa( r_j)/kappa( p') is purely…","statement_latex":"Let $R \\to S$ be an integral ring map. Let $\\mathfrak p \\subset R$ be a prime\nideal. Assume\n\\begin{enumerate}\n\\item there are finitely many primes $\\mathfrak q_1, \\ldots, \\mathfrak q_n$\nlying over $\\mathfrak p$, and\n\\item for each $i$ the maximal separable subextension\n$\\kappa(\\mathfrak q)/\\kappa(\\mathfrak q_i)_{sep}/\\kappa(\\mathfrak p)$\n(Fields, Lemma \\ref{fields-lemma-separable-first})\nis finite over $\\kappa(\\mathfrak p)$.\n\\end{enumerate}\nThen there exists an \\'etale ring map $R \\to R'$ and a prime\n$\\mathfrak p'$ lying over $\\mathfrak p$ such that\n$$\nS \\otimes_R R' = A_1 \\times \\ldots \\times A_m\n$$\nwith $R' \\to A_j$ integral having a unique prime $\\mathfrak r_j$\nover $\\mathfrak p'$ such that $\\kappa(\\mathfrak r_j)/\\kappa(\\mathfrak p')$\nis purely inseparable.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Étale localization of integral morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSR","source_file":"more-morphisms.tex","source_line":12210,"source_end_line":12230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12210-L12230","statement_sha256":"23b92d85e74aea1e1ca557c807aae30f42c3a340e311bf061e53e6a6b8013754","origin":"The Stacks Project","memory_eligible":false,"source_rank":7385,"rank":7385,"depth":31,"x":2167.862,"y":693.621,"cluster":"scheme-morphisms"},{"id":"stacks:03GW","tag":"03GW","title":"Zariski's Main Theorem · Lemma 03GW","summary":"Let f : X → S be a morphism of schemes. Assume f is of finite type and separated. Let S' be the normalization of S in X, see Morphisms, Definition [Tag 035H]. Picture: xymatrix X ar[rd]_f ar[rr]_f' & & S' ar[ld]^ν & S & Then there exists an open subscheme U' ⊂ S' such that • (f')^-1(U') → U' is an isomorphism, and • (f')^-1(U') ⊂ X is the set of points at which f is quasi-finite.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $f$ is of finite type and separated.\nLet $S'$ be the normalization of $S$ in $X$, see\nMorphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.\nPicture:\n$$\n\\xymatrix{\nX \\ar[rd]_f \\ar[rr]_{f'} & & S' \\ar[ld]^\\nu \\\\\n& S &\n}\n$$\nThen there exists an open subscheme $U' \\subset S'$ such that\n\\begin{enumerate}\n\\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and\n\\item $(f')^{-1}(U') \\subset X$ is the set of points at which\n$f$ is quasi-finite.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GW","source_file":"more-morphisms.tex","source_line":12327,"source_end_line":12346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12327-L12346","statement_sha256":"db727277f7e5aa283d3f7cbbbe4ea384e14b85b3b6a6782ddfe0d68eabbabb74","origin":"The Stacks Project","memory_eligible":false,"source_rank":7386,"rank":7386,"depth":47,"x":1917.327,"y":748.22,"cluster":"scheme-morphisms"},{"id":"stacks:02LR","tag":"02LR","title":"Zariski's Main Theorem · Lemma 02LR","summary":"Quasi-finite, separated morphisms are quasi-affine Let f : X → S be a morphism of schemes. Assume f is quasi-finite and separated. Let S' be the normalization of S in X, see Morphisms, Definition [Tag 035H]. Picture: xymatrix X ar[rd]_f ar[rr]_f' & & S' ar[ld]^ν & S & Then f' is a quasi-compact open immersion and ν is integral. In particular f is quasi-affine.","statement_latex":"\\begin{slogan}\nQuasi-finite, separated morphisms are quasi-affine\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nAssume $f$ is quasi-finite and separated.\nLet $S'$ be the normalization of $S$ in $X$, see\nMorphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.\nPicture:\n$$\n\\xymatrix{\nX \\ar[rd]_f \\ar[rr]_{f'} & & S' \\ar[ld]^\\nu \\\\\n& S &\n}\n$$\nThen $f'$ is a quasi-compact open immersion and $\\nu$ is integral.\nIn particular $f$ is quasi-affine.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LR","source_file":"more-morphisms.tex","source_line":12425,"source_end_line":12443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12425-L12443","statement_sha256":"60ce9dbb654f25ce61c5b092f515217f5d38b247394192f84045387b63370bf5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7387,"rank":7387,"depth":48,"x":2058.238,"y":565.698,"cluster":"scheme-morphisms"},{"id":"stacks:05K0","tag":"05K0","title":"Zariski's Main Theorem · Lemma 05K0","summary":"[EGA] Let f : X → S be a morphism of schemes. Assume f is quasi-finite and separated and assume that S is quasi-compact and quasi-separated. Then there exists a factorization xymatrix X ar[rd]_f ar[rr]_j & & T ar[ld]^π & S & where j is a quasi-compact open immersion and π is finite.","statement_latex":"\\begin{reference}\n\\cite[IV Corollary 18.12.13]{EGA}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nAssume $f$ is quasi-finite and separated and assume that\n$S$ is quasi-compact and quasi-separated. Then there exists\na factorization\n$$\n\\xymatrix{\nX \\ar[rd]_f \\ar[rr]_j & & T \\ar[ld]^\\pi \\\\\n& S &\n}\n$$\nwhere $j$ is a quasi-compact open immersion and $\\pi$ is finite.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05K0","source_file":"more-morphisms.tex","source_line":12456,"source_end_line":12472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12456-L12472","statement_sha256":"f56cfea3a75ac7e86e97fe11d64aefbf95227f66ee9dccb6e5322ff956e06ff7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7388,"rank":7388,"depth":49,"x":2101.135,"y":780.363,"cluster":"scheme-morphisms"},{"id":"stacks:0F2N","tag":"0F2N","title":"Zariski's Main Theorem · Lemma 0F2N","summary":"With notation and hypotheses as in Lemma [Tag 05K0]. Assume moreover that f is locally of finite presentation. Then we can choose the factorization such that T is finite and of finite presentation over S.","statement_latex":"With notation and hypotheses as in\nLemma \\ref{lemma-quasi-finite-separated-pass-through-finite}.\nAssume moreover that $f$ is locally of finite presentation. Then we can\nchoose the factorization such that $T$ is finite and of\nfinite presentation over $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2N","source_file":"more-morphisms.tex","source_line":12505,"source_end_line":12512,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12505-L12512","statement_sha256":"66af12a429f5a51ab0cf5f3aadc5dd672c851cbd81dfd4f69e896b21b4e1446f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7389,"rank":7389,"depth":50,"x":1896.763,"y":646.339,"cluster":"scheme-morphisms"},{"id":"stacks:02LS","tag":"02LS","title":"Applications of Zariski's Main Theorem, I · Lemma 02LS","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • f is finite, • f is proper with finite fibres, • f is proper and locally quasi-finite, • f is universally closed, separated, locally of finite type and has finite fibres.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is finite,\n\\item $f$ is proper with finite fibres,\n\\item $f$ is proper and locally quasi-finite,\n\\item $f$ is universally closed, separated, locally of finite type\nand has finite fibres.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02LS","source_file":"more-morphisms.tex","source_line":12543,"source_end_line":12554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12543-L12554","statement_sha256":"aa4d2101df4b25582a4ceffe60a086ca1f98411204b8e4cd98f86550c8477e86","origin":"The Stacks Project","memory_eligible":false,"source_rank":7390,"rank":7390,"depth":44,"x":2155.386,"y":629.191,"cluster":"scheme-morphisms"},{"id":"stacks:02UP","tag":"02UP","title":"Applications of Zariski's Main Theorem, I · Lemma 02UP","summary":"Let f : X → S be a morphism of schemes. Let s ∈ S. Assume that f is proper and f^-1((s)) is a finite set. Then there exists an open neighbourhood V ⊂ S of s such that f|_f^-1(V) : f^-1(V) → V is finite.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $s \\in S$.\nAssume that $f$ is proper and $f^{-1}(\\{s\\})$ is a finite set.\nThen there exists an open neighbourhood $V \\subset S$ of $s$\nsuch that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UP","source_file":"more-morphisms.tex","source_line":12594,"source_end_line":12601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12594-L12601","statement_sha256":"81dac61bd3cc5af3a4a484e2c0c54a060b9c9fde33c5868b4a11f344dd17b3b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7391,"rank":7391,"depth":45,"x":1978.373,"y":788.673,"cluster":"scheme-morphisms"},{"id":"stacks:0AH8","tag":"0AH8","title":"Applications of Zariski's Main Theorem, I · Lemma 0AH8","summary":"Consider a commutative diagram of schemes xymatrix X ar[rr]_h ar[rd]_f & & Y ar[ld]^g & S Let s ∈ S. Assume • X → S is a proper morphism, • Y → S is separated and locally of finite type, and • the image of X_s → Y_s is finite. Then there is an open subspace U ⊂ S containing s such that X_U → Y_U factors through a closed subscheme Z ⊂ Y_U finite over U.","statement_latex":"Consider a commutative diagram of schemes\n$$\n\\xymatrix{\nX \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\\n& S\n}\n$$\nLet $s \\in S$. Assume\n\\begin{enumerate}\n\\item $X \\to S$ is a proper morphism,\n\\item $Y \\to S$ is separated and locally of finite type, and\n\\item the image of $X_s \\to Y_s$ is finite.\n\\end{enumerate}\nThen there is an open\nsubspace $U \\subset S$ containing $s$ such that $X_U \\to Y_U$\nfactors through a closed subscheme $Z \\subset Y_U$ finite over $U$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AH8","source_file":"more-morphisms.tex","source_line":12619,"source_end_line":12637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12619-L12637","statement_sha256":"4c8812cbdc3dcb61f7291728085d1c34c7247dcb98ca0ccf10dc724048ce13d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7392,"rank":7392,"depth":46,"x":1980.65,"y":570.512,"cluster":"scheme-morphisms"},{"id":"stacks:07S0","tag":"07S0","title":"Applications of Zariski's Main Theorem, II · Lemma 07S0","summary":"Let f : X → Y be a separated, locally quasi-finite morphism with Y affine. Then every finite set of points of X is contained in an open affine of X.","statement_latex":"Let $f : X \\to Y$ be a separated, locally quasi-finite morphism\nwith $Y$ affine. Then every finite set of points of $X$ is contained\nin an open affine of $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07S0","source_file":"more-morphisms.tex","source_line":12661,"source_end_line":12666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12661-L12666","statement_sha256":"09d408f694c7b5b5c2544a7c0bf313106e2a84a30abb8c5aad2f0bb51e98cc3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7393,"rank":7393,"depth":49,"x":2154.506,"y":732.762,"cluster":"scheme-morphisms"},{"id":"stacks:03I1","tag":"03I1","title":"Applications of Zariski's Main Theorem, II · Lemma 03I1","summary":"Let f : Y → X be a quasi-finite morphism. There exists a dense open U ⊂ X such that f|_f^-1(U) : f^-1(U) → U is finite.","statement_latex":"Let $f : Y \\to X$ be a quasi-finite morphism.\nThere exists a dense open $U \\subset X$ such that\n$f|_{f^{-1}(U)} : f^{-1}(U) \\to U$ is finite.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03I1","source_file":"more-morphisms.tex","source_line":12677,"source_end_line":12682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12677-L12682","statement_sha256":"0d96861a9e7afc6a2959c29afe1d12da314c8441540fda819cf658351aa6327c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7394,"rank":7394,"depth":45,"x":1895.687,"y":711.76,"cluster":"scheme-morphisms"},{"id":"stacks:07RY","tag":"07RY","title":"Applications of Zariski's Main Theorem, II · Lemma 07RY","summary":"Let f : X → S be flat, locally of finite presentation, separated, locally quasi-finite with universally bounded fibres. Then there exist closed subsets ∅ = Z_-1 ⊂ Z_0 ⊂ Z_1 ⊂ Z_2 ⊂ … ⊂ Z_n = S such that with S_r = Z_r setminus Z_r - 1 the stratification S = coprod_r = 0, …, n S_r is characterized by the following universal property: Given g : T → S the projection X ×_S T → T is finite locally free of degree r if and only if g(T) ⊂ S_r (set theoretically).","statement_latex":"Let $f : X \\to S$ be flat, locally of finite presentation, separated,\nlocally quasi-finite with universally bounded fibres. Then there exist\nclosed subsets\n$$\n\\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset\n\\ldots \\subset Z_n = S\n$$\nsuch that with $S_r = Z_r \\setminus Z_{r - 1}$ the stratification\n$S = \\coprod_{r = 0, \\ldots, n} S_r$ is characterized by the following\nuniversal property: Given $g : T \\to S$ the projection\n$X \\times_S T \\to T$ is finite locally\nfree of degree $r$ if and only if $g(T) \\subset S_r$ (set theoretically).","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RY","source_file":"more-morphisms.tex","source_line":12760,"source_end_line":12774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12760-L12774","statement_sha256":"5736cb3ded737f056e2f30d96b4ba458d01b390e92166d96ff5e25c7ee8857fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7395,"rank":7395,"depth":45,"x":2103.54,"y":580.314,"cluster":"scheme-morphisms"},{"id":"stacks:07RZ","tag":"07RZ","title":"Applications of Zariski's Main Theorem, II · Lemma 07RZ","summary":"Let f : X → S be a morphism of schemes which is flat, locally of finite presentation, separated, and quasi-finite. Then there exist closed subsets ∅ = Z_-1 ⊂ Z_0 ⊂ Z_1 ⊂ Z_2 ⊂ … ⊂ S such that with S_r = Z_r setminus Z_r - 1 the stratification S = coprod S_r is characterized by the following universal property: Given a morphism g : T → S the projection X ×_S T → T is finite locally free of degree r if and only if g(T) ⊂ S_r (set theoretically). Moreover, the inclusion maps…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat, locally of\nfinite presentation, separated, and quasi-finite. Then there exist\nclosed subsets\n$$\n\\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset\n\\ldots \\subset S\n$$\nsuch that with $S_r = Z_r \\setminus Z_{r - 1}$ the stratification\n$S = \\coprod S_r$ is characterized by the following universal property:\nGiven a morphism $g : T \\to S$ the projection $X \\times_S T \\to T$ is\nfinite locally free of degree $r$ if and only if $g(T) \\subset S_r$\n(set theoretically). Moreover, the inclusion maps $S_r \\to S$ are\nquasi-compact.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07RZ","source_file":"more-morphisms.tex","source_line":12844,"source_end_line":12859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12844-L12859","statement_sha256":"2e57ec4d3aa1766a08d5bd7486bdfc031b63fa724d626992314a61020203cf16","origin":"The Stacks Project","memory_eligible":false,"source_rank":7396,"rank":7396,"depth":46,"x":2055.952,"y":795.299,"cluster":"scheme-morphisms"},{"id":"stacks:086R","tag":"086R","title":"Applications of Zariski's Main Theorem, II · Lemma 086R","summary":"Let f : X → S be a flat, locally of finite presentation, separated, and locally quasi-finite morphism of schemes. Then there exist open subschemes S = U_0 ⊃ U_1 ⊃ U_2 ⊃ … such that a morphism Spec(k) → S where k is a field factors through U_d if and only if X ×_S Spec(k) has degree ≥ d over k.","statement_latex":"Let $f : X \\to S$ be a flat, locally of finite presentation, separated, and\nlocally quasi-finite morphism of schemes. Then there\nexist open subschemes\n$$\nS = U_0 \\supset U_1 \\supset U_2 \\supset \\ldots\n$$\nsuch that a morphism $\\Spec(k) \\to S$ where $k$ is a field\nfactors through $U_d$ if and\nonly if $X \\times_S \\Spec(k)$ has degree $\\geq d$ over $k$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086R","source_file":"more-morphisms.tex","source_line":12891,"source_end_line":12902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12891-L12902","statement_sha256":"52c148c82e1f59c0b9d90bd09f894a0fabb18d8304a534faafc746d4026ce1e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7397,"rank":7397,"depth":47,"x":1918.082,"y":609.667,"cluster":"scheme-morphisms"},{"id":"stacks:082V","tag":"082V","title":"Applications of Zariski's Main Theorem, II · Lemma 082V","summary":"Let f : X → S be a morphism of schemes which is flat, locally of finite presentation, and locally quasi-finite. Let g ∈ Γ(X, O_X) nonzero. Then there exist an open V ⊂ X such that g|_V not = 0, an open U ⊂ S fitting into a commutative diagram xymatrix V ar[r] ar[d]_π & X ar[d]^f U ar[r] & S, a quasi-coherent subsheaf F ⊂ O_U, an integer r > 0, and an injective O_U-module map F^⊕ r → π_*O_V whose image contains g|_V.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat, locally of\nfinite presentation, and locally quasi-finite. Let\n$g \\in \\Gamma(X, \\mathcal{O}_X)$ nonzero. Then there exist\nan open $V \\subset X$ such that $g|_V \\not = 0$, an open\n$U \\subset S$ fitting into a commutative diagram\n$$\n\\xymatrix{\nV \\ar[r] \\ar[d]_\\pi & X \\ar[d]^f \\\\\nU \\ar[r] & S,\n}\n$$\na quasi-coherent subsheaf $\\mathcal{F} \\subset \\mathcal{O}_U$, an integer\n$r > 0$, and an injective $\\mathcal{O}_U$-module map\n$\\mathcal{F}^{\\oplus r} \\to \\pi_*\\mathcal{O}_V$\nwhose image contains $g|_V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082V","source_file":"more-morphisms.tex","source_line":12913,"source_end_line":12930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12913-L12930","statement_sha256":"04b8076f665363de5b4e0429d9f132f62a13c4f326316510590e02dfc8699ddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7398,"rank":7398,"depth":47,"x":2169.163,"y":668.351,"cluster":"scheme-morphisms"},{"id":"stacks:09Z0","tag":"09Z0","title":"Applications of Zariski's Main Theorem, II · Lemma 09Z0","summary":"Let U → X be a surjective étale morphism of schemes. Assume X is quasi-compact and quasi-separated. Then there exists a surjective integral morphism Y → X, such that for every y ∈ Y there is an open neighbourhood V ⊂ Y such that V → X factors through U. In fact, we may assume Y → X is finite and of finite presentation.","statement_latex":"Let $U \\to X$ be a surjective \\'etale morphism of schemes. Assume $X$\nis quasi-compact and quasi-separated. Then there exists a surjective\nintegral morphism $Y \\to X$, such that for\nevery $y \\in Y$ there is an open neighbourhood $V \\subset Y$\nsuch that $V \\to X$ factors through $U$. In fact, we may assume\n$Y \\to X$ is finite and of finite presentation.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Applications of Zariski's Main Theorem, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Z0","source_file":"more-morphisms.tex","source_line":12978,"source_end_line":12986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L12978-L12986","statement_sha256":"3a8111e1c774781a2f96fa8f4c12b04f37478481c4a7bc3884d6d22dc9ad1527","origin":"The Stacks Project","memory_eligible":false,"source_rank":7399,"rank":7399,"depth":49,"x":1936.7,"y":767.602,"cluster":"scheme-morphisms"},{"id":"stacks:057I","tag":"057I","title":"Application to morphisms with connected fibres · Lemma 057I","summary":"Consider a diagram of morphisms of schemes xymatrix Z ar[r]_σ ar[rd] & X ar[d] & Y an a point y ∈ Y. Assume • X → Y is of finite presentation and flat, • Z → Y is finite locally free, • Z_y not = ∅, • all fibres of X → Y are geometrically reduced, and • X_y is geometrically connected over kappa(y). Then there exists a quasi-compact open X^0 ⊂ X such that X^0_y = X_y and such that all nonempty fibres of X^0 → Y are geometrically connected.","statement_latex":"Consider a diagram of morphisms of schemes\n$$\n\\xymatrix{\nZ \\ar[r]_{\\sigma} \\ar[rd] & X \\ar[d] \\\\\n& Y\n}\n$$\nan a point $y \\in Y$. Assume\n\\begin{enumerate}\n\\item $X \\to Y$ is of finite presentation and flat,\n\\item $Z \\to Y$ is finite locally free,\n\\item $Z_y \\not = \\emptyset$,\n\\item all fibres of $X \\to Y$ are geometrically reduced, and\n\\item $X_y$ is geometrically connected over $\\kappa(y)$.\n\\end{enumerate}\nThen there exists a quasi-compact open $X^0 \\subset X$ such that $X^0_y = X_y$\nand such that all nonempty fibres of $X^0 \\to Y$ are geometrically connected.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to morphisms with connected fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057I","source_file":"more-morphisms.tex","source_line":13087,"source_end_line":13106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13087-L13106","statement_sha256":"3b4251105b4608337a6f3deb6edb4074acba5f1297f1a3afd94829e0fbe2f3a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7400,"rank":7400,"depth":44,"x":2028.35,"y":562.399,"cluster":"scheme-morphisms"},{"id":"stacks:055W","tag":"055W","title":"Application to morphisms with connected fibres · Lemma 055W","summary":"Let h : Y → S be a morphism of schemes. Let s ∈ S be a point. Let T ⊂ Y_s be an open subscheme. Assume • h is flat and of finite presentation, • all fibres of h are geometrically reduced, and • T is geometrically connected over kappa(s). Then we can find an affine elementary étale neighbourhood (S', s') → (S, s) and a quasi-compact open V ⊂ Y_S' such that • [(a)] all fibres of V → S' are geometrically connected, • [(b)] V_s' = T ×_s s'.","statement_latex":"Let $h : Y \\to S$ be a morphism of schemes.\nLet $s \\in S$ be a point.\nLet $T \\subset Y_s$ be an open subscheme.\nAssume\n\\begin{enumerate}\n\\item $h$ is flat and of finite presentation,\n\\item all fibres of $h$ are geometrically reduced, and\n\\item $T$ is geometrically connected over $\\kappa(s)$.\n\\end{enumerate}\nThen we can find an affine elementary \\'etale neighbourhood\n$(S', s') \\to (S, s)$\nand a quasi-compact open $V \\subset Y_{S'}$ such that\n\\begin{enumerate}\n\\item[(a)] all fibres of $V \\to S'$ are geometrically connected,\n\\item[(b)] $V_{s'} = T \\times_s s'$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to morphisms with connected fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/055W","source_file":"more-morphisms.tex","source_line":13192,"source_end_line":13210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13192-L13210","statement_sha256":"4fec7b96a4f43dcc452332de4efd2c27244a333ed33f3acdfbc9d22fe5a094ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":7401,"rank":7401,"depth":45,"x":2125.841,"y":765.828,"cluster":"scheme-morphisms"},{"id":"stacks:0EY5","tag":"0EY5","title":"Application to morphisms with connected fibres · Lemma 0EY5","summary":"Let f : X → S be a morphism of schemes which is locally of finite presentation and flat with geometrically reduced fibres. Then there exists an étale covering (X_i → X)_i ∈ I such that X_i → S factors as X_i → S_i → S where S_i → S is étale and X_i → S_i is flat of finite presentation with geometrically connected and geometrically reduced fibres.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is\nlocally of finite presentation and flat with geometrically\nreduced fibres. Then there\nexists an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$\nsuch that $X_i \\to S$ factors as $X_i \\to S_i \\to S$\nwhere $S_i \\to S$ is \\'etale and $X_i \\to S_i$ is\nflat of finite presentation with geometrically connected\nand geometrically reduced fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to morphisms with connected fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EY5","source_file":"more-morphisms.tex","source_line":13257,"source_end_line":13267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13257-L13267","statement_sha256":"d55529133a8faee9873c0efae8f82668da715b04636f0b90a604ccae1a1e01b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7402,"rank":7402,"depth":46,"x":1890.232,"y":671.09,"cluster":"scheme-morphisms"},{"id":"stacks:057J","tag":"057J","title":"Application to morphisms with connected fibres · Lemma 057J","summary":"Let h : Y → S be a morphism of schemes. Let s ∈ S be a point. Let T ⊂ Y_s be an open subscheme. Assume • h is of finite presentation, • h is normal, and • T is geometrically irreducible over kappa(s). Then we can find an affine elementary étale neighbourhood (S', s') → (S, s) and a quasi-compact open V ⊂ Y_S' such that • [(a)] all fibres of V → S' are geometrically integral, • [(b)] V_s' = T ×_s s'.","statement_latex":"Let $h : Y \\to S$ be a morphism of schemes.\nLet $s \\in S$ be a point.\nLet $T \\subset Y_s$ be an open subscheme.\nAssume\n\\begin{enumerate}\n\\item $h$ is of finite presentation,\n\\item $h$ is normal, and\n\\item $T$ is geometrically irreducible over $\\kappa(s)$.\n\\end{enumerate}\nThen we can find an affine elementary \\'etale neighbourhood\n$(S', s') \\to (S, s)$ and a quasi-compact open $V \\subset Y_{S'}$ such that\n\\begin{enumerate}\n\\item[(a)] all fibres of $V \\to S'$ are geometrically integral,\n\\item[(b)] $V_{s'} = T \\times_s s'$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to morphisms with connected fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057J","source_file":"more-morphisms.tex","source_line":13326,"source_end_line":13343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13326-L13343","statement_sha256":"b7532922ec29d56197132176eb2734965f570eb8b65dd0398a94f6d0fdf07292","origin":"The Stacks Project","memory_eligible":false,"source_rank":7403,"rank":7403,"depth":46,"x":2140.289,"y":607.223,"cluster":"scheme-morphisms"},{"id":"stacks:052E","tag":"052E","title":"Application to the structure of finite type morphisms · Lemma 052E","summary":"Let f : X → S be a morphism. Let x ∈ X and set s = f(x). Assume that f is locally of finite type and that n = dim_x(X_s). Then there exists a commutative diagram xymatrix X ar[dd] & X' ar[l]^g ar[d]^π & x ar@|->[dd] & x' ar@|->[l] ar@|->[d] & Y ar[d]^h & & y ar@|->[d] S ar@=[r] & S & s & s ar@=[l] and a point x' ∈ X' with g(x') = x such that with y = π(x') we have • h : Y → S is smooth of relative dimension n, • g : (X', x') → (X, x) is an elementary étale neighbourhood,…","statement_latex":"Let $f : X \\to S$ be a morphism. Let $x \\in X$ and set $s = f(x)$.\nAssume that $f$ is locally of finite type and that $n = \\dim_x(X_s)$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[dd] & X' \\ar[l]^g \\ar[d]^\\pi & x \\ar@{|->}[dd] &\nx' \\ar@{|->}[l]  \\ar@{|->}[d] \\\\\n& Y \\ar[d]^h & & y \\ar@{|->}[d] \\\\\nS \\ar@{=}[r] & S & s & s \\ar@{=}[l]\n}\n$$\nand a point $x' \\in X'$ with $g(x') = x$ such that with $y = \\pi(x')$\nwe have\n\\begin{enumerate}\n\\item $h : Y \\to S$ is smooth of relative dimension $n$,\n\\item $g : (X', x') \\to (X, x)$ is an elementary \\'etale neighbourhood,\n\\item $\\pi$ is finite, and $\\pi^{-1}(\\{y\\}) = \\{x'\\}$, and\n\\item $\\kappa(y)$ is a purely transcendental extension of $\\kappa(s)$.\n\\end{enumerate}\nMoreover, if $f$ is locally of finite presentation then $\\pi$ is\nof finite presentation.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the structure of finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/052E","source_file":"more-morphisms.tex","source_line":13382,"source_end_line":13405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13382-L13405","statement_sha256":"89e0dd714194104fa39baeeabc82449fe421154af60a0801f47a4348d017de27","origin":"The Stacks Project","memory_eligible":false,"source_rank":7404,"rank":7404,"depth":31,"x":2007.188,"y":796.309,"cluster":"scheme-morphisms"},{"id":"stacks:057K","tag":"057K","title":"Application to the structure of finite type morphisms · Lemma 057K","summary":"A morphism of finite type is, in étale neighbourhoods, finite over a smooth morphism. Let f : X → S be a morphism. Let x ∈ X and set s = f(x). Assume that f is locally of finite type and that n = dim_x(X_s). Then there exists a commutative diagram xymatrix X ar[dd] & X' ar[l]^g ar[d]^π & x ar@|->[dd] & x' ar@|->[l] ar@|->[d] & Y' ar[d]^h & & y' ar@|->[d] S & S' ar[l]_e & s & s' ar@|->[l] and a point x' ∈ X' with g(x') = x such that with y' = π(x'), s' = h(y') we have • h…","statement_latex":"\\begin{slogan}\nA morphism of finite type is, in \\'etale neighbourhoods, finite over a\nsmooth morphism.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism. Let $x \\in X$ and set $s = f(x)$.\nAssume that $f$ is locally of finite type and that $n = \\dim_x(X_s)$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[dd] & X' \\ar[l]^g \\ar[d]^\\pi & x \\ar@{|->}[dd] &\nx' \\ar@{|->}[l] \\ar@{|->}[d] \\\\\n& Y' \\ar[d]^h & & y' \\ar@{|->}[d] \\\\\nS & S' \\ar[l]_e & s & s' \\ar@{|->}[l]\n}\n$$\nand a point $x' \\in X'$ with $g(x') = x$ such that with $y' = \\pi(x')$,\n$s' = h(y')$ we have\n\\begin{enumerate}\n\\item $h : Y' \\to S'$ is smooth of relative dimension $n$,\n\\item all fibres of $Y' \\to S'$ are geometrically integral,\n\\item $g : (X', x') \\to (X, x)$ is an elementary \\'etale neighbourhood,\n\\item $\\pi$ is finite, and $\\pi^{-1}(\\{y'\\}) = \\{x'\\}$,\n\\item $\\kappa(y')$ is a purely transcendental extension of $\\kappa(s')$, and\n\\item $e : (S', s') \\to (S, s)$ is an elementary \\'etale neighbourhood.\n\\end{enumerate}\nMoreover, if $f$ is locally of finite presentation, then $\\pi$ is\nof finite presentation.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the structure of finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057K","source_file":"more-morphisms.tex","source_line":13431,"source_end_line":13460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13431-L13460","statement_sha256":"bf15d0616d25ab8f2c799f74e6c155514fb67d7ab181d5513296da4c10e7d40a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7405,"rank":7405,"depth":47,"x":1953.248,"y":581.238,"cluster":"scheme-morphisms"},{"id":"stacks:057L","tag":"057L","title":"Application to the structure of finite type morphisms · Lemma 057L","summary":"Assumption and notation as in Lemma [Tag 057K]. In addition to properties (1) -- (6) we may also arrange it so that • [(7)] S', Y', X' are affine.","statement_latex":"Assumption and notation as in\nLemma \\ref{lemma-local-local-structure-finite-type}.\nIn addition to properties (1) -- (6) we may also arrange it so that\n\\begin{enumerate}\n\\item[(7)] $S'$, $Y'$, $X'$ are affine.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the structure of finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057L","source_file":"more-morphisms.tex","source_line":13522,"source_end_line":13530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13522-L13530","statement_sha256":"2658b07efe90169d885a9e1ed0b766d73d1f3705540f11bef3112440ffac1121","origin":"The Stacks Project","memory_eligible":false,"source_rank":7406,"rank":7406,"depth":48,"x":2166.092,"y":709.29,"cluster":"scheme-morphisms"},{"id":"stacks:05B8","tag":"05B8","title":"Application to the structure of finite type morphisms · Lemma 05B8","summary":"Let π : X → Y be a finite morphism. Let x ∈ X with y = π(x) such that π^-1((y)) = (x). Then • For every neighbourhood U ⊂ X of x in X, there exists a neighbourhood V ⊂ Y of y such that π^-1(V) ⊂ U. • The ring map O_Y, y → O_X, x is finite. • If π is of finite presentation, then O_Y, y → O_X, x is of finite presentation. • For any quasi-coherent O_X-module F we have F_x = π_*F_y as O_Y, y-modules.","statement_latex":"Let $\\pi : X \\to Y$ be a finite morphism.\nLet $x \\in X$ with $y = \\pi(x)$ such that $\\pi^{-1}(\\{y\\}) = \\{x\\}$.\nThen\n\\begin{enumerate}\n\\item For every neighbourhood $U \\subset X$ of $x$ in $X$, there\nexists a neighbourhood $V \\subset Y$ of $y$ such that\n$\\pi^{-1}(V) \\subset U$.\n\\item The ring map $\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$\nis finite.\n\\item If $\\pi$ is of finite presentation, then\n$\\mathcal{O}_{Y, y} \\to \\mathcal{O}_{X, x}$ is of finite presentation.\n\\item For any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$\nwe have $\\mathcal{F}_x = \\pi_*\\mathcal{F}_y$ as\n$\\mathcal{O}_{Y, y}$-modules.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the structure of finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05B8","source_file":"more-morphisms.tex","source_line":13551,"source_end_line":13568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13551-L13568","statement_sha256":"59e52392a613d9d55f2afb589e7c2f47360dfac2b5b38fcb6b9b299ecc1aa96c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7407,"rank":7407,"depth":1,"x":1906.026,"y":735.654,"cluster":"scheme-morphisms"},{"id":"stacks:0DET","tag":"0DET","title":"Application to the fppf topology · Lemma 0DET","summary":"Let S be a scheme. Let (S_i → S)_i ∈ I be an fppf covering. Then there exist • an étale covering (S'_a → S), • surjective finite locally free morphisms V_a → S'_a, such that the fppf covering (V_a → S) refines the given covering (S_i → S).","statement_latex":"Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering.\nThen there exist\n\\begin{enumerate}\n\\item an \\'etale covering $\\{S'_a \\to S\\}$,\n\\item surjective finite locally free morphisms $V_a \\to S'_a$,\n\\end{enumerate}\nsuch that the fppf covering $\\{V_a \\to S\\}$ refines the given\ncovering $\\{S_i \\to S\\}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DET","source_file":"more-morphisms.tex","source_line":13607,"source_end_line":13617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13607-L13617","statement_sha256":"aca559af229d30478e72fbad13b9967362a0d578eeaf17f8b5f5847777900d83","origin":"The Stacks Project","memory_eligible":false,"source_rank":7408,"rank":7408,"depth":40,"x":2076.687,"y":568.556,"cluster":"scheme-morphisms"},{"id":"stacks:05WN","tag":"05WN","title":"Application to the fppf topology · Lemma 05WN","summary":"Let S be a scheme. Let (S_i → S)_i ∈ I be an fppf covering. Then there exist • a Zariski open covering S = ⋃ U_j, • surjective finite locally free morphisms W_j → U_j, • Zariski open coverings W_j = ⋃_k W_j, k, • surjective finite locally free morphisms T_j, k → W_j, k such that the fppf covering (T_j, k → S) refines the given covering (S_i → S).","statement_latex":"Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering.\nThen there exist\n\\begin{enumerate}\n\\item a Zariski open covering $S = \\bigcup U_j$,\n\\item surjective finite locally free morphisms $W_j \\to U_j$,\n\\item Zariski open coverings $W_j = \\bigcup_k W_{j, k}$,\n\\item surjective finite locally free morphisms $T_{j, k} \\to W_{j, k}$\n\\end{enumerate}\nsuch that the fppf covering $\\{T_{j, k} \\to S\\}$ refines the given\ncovering $\\{S_i \\to S\\}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WN","source_file":"more-morphisms.tex","source_line":13648,"source_end_line":13660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13648-L13660","statement_sha256":"53375b8a3404483984afbb9419c07f69d9eb9aa2d12aed68ad5349bb8f42f2b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7409,"rank":7409,"depth":41,"x":2085.224,"y":788.727,"cluster":"scheme-morphisms"},{"id":"stacks:0CNX","tag":"0CNX","title":"Application to the fppf topology · Lemma 0CNX","summary":"Let S be a scheme. If U ⊂ S is open and V → U is a surjective integral morphism, then there exists a surjective integral morphism overlineV → S with overlineV ×_S U isomorphic to V as schemes over U.","statement_latex":"Let $S$ be a scheme. If $U \\subset S$ is open and $V \\to U$ is a surjective\nintegral morphism, then there exists a surjective integral\nmorphism $\\overline{V} \\to S$ with $\\overline{V} \\times_S U$\nisomorphic to $V$ as schemes over $U$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNX","source_file":"more-morphisms.tex","source_line":13685,"source_end_line":13691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13685-L13691","statement_sha256":"de39b295d784d3d24333c62e8284cbe3b4a9c16fb1b9ffcd8cddd19f6eda7fbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7410,"rank":7410,"depth":22,"x":1901.774,"y":631.135,"cluster":"scheme-morphisms"},{"id":"stacks:0CNY","tag":"0CNY","title":"Application to the fppf topology · Lemma 0CNY","summary":"Let S be a quasi-compact and quasi-separated scheme. If U ⊂ S is a quasi-compact open and V → U is a surjective finite morphism, then there exists a surjective finite morphism overlineV → S with overlineV ×_S U isomorphic to V as schemes over U.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nIf $U \\subset S$ is a quasi-compact open\nand $V \\to U$ is a surjective finite morphism, then there exists a\nsurjective finite morphism $\\overline{V} \\to S$ with $\\overline{V} \\times_S U$\nisomorphic to $V$ as schemes over $U$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNY","source_file":"more-morphisms.tex","source_line":13704,"source_end_line":13711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13704-L13711","statement_sha256":"531d16d92d336bda28ab20d01088e5086b96c13b1bc66d0447b192b7e3e0082e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7411,"rank":7411,"depth":50,"x":2163.92,"y":643.253,"cluster":"scheme-morphisms"},{"id":"stacks:0CNZ","tag":"0CNZ","title":"Application to the fppf topology · Lemma 0CNZ","summary":"Let S be a scheme. Let (S_i → S)_i ∈ I be an fppf covering. Then there exists a surjective integral morphism S' → S and an open covering S' = ⋃ U'_α such that for each α the morphism U'_α → S factors through S_i → S for some i.","statement_latex":"Let $S$ be a scheme. Let $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering.\nThen there exists a surjective integral morphism $S' \\to S$ and an\nopen covering $S' = \\bigcup U'_\\alpha$ such that for each $\\alpha$ the\nmorphism $U'_\\alpha \\to S$ factors through $S_i \\to S$ for some $i$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CNZ","source_file":"more-morphisms.tex","source_line":13725,"source_end_line":13731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13725-L13731","statement_sha256":"fc94d5164fe2706880da99e236732de8637efe34b1e902ba0173c173c1381518","origin":"The Stacks Project","memory_eligible":false,"source_rank":7412,"rank":7412,"depth":42,"x":1960.763,"y":783.142,"cluster":"scheme-morphisms"},{"id":"stacks:0CP0","tag":"0CP0","title":"Application to the fppf topology · Lemma 0CP0","summary":"Let S be a quasi-compact and quasi-separated scheme. Let (S_i → S)_i ∈ I be an fppf covering. Then there exists a surjective finite morphism S' → S of finite presentation and an open covering S' = ⋃ U'_α such that for each α the morphism U'_α → S factors through S_i → S for some i.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $\\{S_i \\to S\\}_{i \\in I}$ be an fppf covering.\nThen there exists a surjective finite morphism $S' \\to S$\nof finite presentation and an\nopen covering $S' = \\bigcup U'_\\alpha$ such that for each $\\alpha$ the\nmorphism $U'_\\alpha \\to S$ factors through $S_i \\to S$ for some $i$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CP0","source_file":"more-morphisms.tex","source_line":13757,"source_end_line":13765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13757-L13765","statement_sha256":"f98a3ded3d52aa882dee0118c78e4a37cd67261d3aa033091686c7e43fcc12a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7413,"rank":7413,"depth":43,"x":1998.094,"y":564.596,"cluster":"scheme-morphisms"},{"id":"stacks:0DBT","tag":"0DBT","title":"Application to the fppf topology · Lemma 0DBT","summary":"An fppf covering of schemes is a ph covering.","statement_latex":"An fppf covering of schemes is a ph covering.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Application to the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBT","source_file":"more-morphisms.tex","source_line":13787,"source_end_line":13790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13787-L13790","statement_sha256":"54e7ad55f93a51764a44cad6b1d635a19c8b7c080978ca266bf7690bdea6c3fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7414,"rank":7414,"depth":44,"x":2146.395,"y":747.029,"cluster":"scheme-morphisms"},{"id":"stacks:0B42","tag":"0B42","title":"Quasi-projective schemes · Lemma 0B42","summary":"Let S be a scheme which has an ample invertible sheaf. Let f : X → S be a morphism of schemes. The following are equivalent • X → S is quasi-projective, • X → S is H-quasi-projective, • there exists a quasi-compact open immersion X → X' of schemes over S with X' → S projective, • X → S is of finite type and X has an ample invertible sheaf, and • X → S is of finite type and there exists an f-very ample invertible sheaf.","statement_latex":"Let $S$ be a scheme which has an ample invertible sheaf.\nLet $f : X \\to S$ be a morphism of schemes. The following are\nequivalent\n\\begin{enumerate}\n\\item $X \\to S$ is quasi-projective,\n\\item $X \\to S$ is H-quasi-projective,\n\\item there exists a quasi-compact open immersion $X \\to X'$ of schemes\nover $S$ with $X' \\to S$ projective,\n\\item $X \\to S$ is of finite type and $X$ has an ample invertible\nsheaf, and\n\\item $X \\to S$ is of finite type and there exists an\n$f$-very ample invertible sheaf.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Quasi-projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B42","source_file":"more-morphisms.tex","source_line":13839,"source_end_line":13854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13839-L13854","statement_sha256":"bb5b795b43218c5832dda79518b9e6ce741aa084f03a78483e72d0a8ba3dc36e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7415,"rank":7415,"depth":26,"x":1890.194,"y":696.628,"cluster":"scheme-morphisms"},{"id":"stacks:0B43","tag":"0B43","title":"Quasi-projective schemes · Lemma 0B43","summary":"Let S be a scheme which has an ample invertible sheaf. Let QP_S be the full subcategory of the category of schemes over S satisfying the equivalent conditions of Lemma [Tag 0B42]. • if S' → S is a morphism of schemes and S' has an ample invertible sheaf, then base change determines a functor QP_S → QP_S', • if X ∈ QP_S and Y ∈ QP_X, then Y ∈ QP_S, • the category QP_S is closed under fibre products, • the category QP_S is closed under finite disjoint unions, • if X → S is…","statement_latex":"Let $S$ be a scheme which has an ample invertible sheaf.\nLet $\\text{QP}_S$ be the full subcategory of the\ncategory of schemes over $S$ satisfying the equivalent\nconditions of Lemma \\ref{lemma-quasi-projective}.\n\\begin{enumerate}\n\\item if $S' \\to S$ is a morphism of schemes and $S'$ has\nan ample invertible sheaf, then base change determines\na functor $\\text{QP}_S \\to \\text{QP}_{S'}$,\n\\item if $X \\in \\text{QP}_S$ and $Y \\in \\text{QP}_X$, then $Y \\in \\text{QP}_S$,\n\\item the category $\\text{QP}_S$ is closed under fibre products,\n\\item the category $\\text{QP}_S$ is closed under\nfinite disjoint unions,\n\\item if $X \\to S$ is projective, then $X \\in \\text{QP}_S$,\n\\item if $X \\to S$ is quasi-affine of finite type, then\n$X$ is in $\\text{QP}_S$,\n\\item if $X \\to S$ is quasi-finite and separated, then\n$X \\in \\text{QP}_S$,\n\\item if $X \\to S$ is a quasi-compact immersion, then\n$X \\in \\text{QP}_S$,\n\\item add more here.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Quasi-projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B43","source_file":"more-morphisms.tex","source_line":13891,"source_end_line":13914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13891-L13914","statement_sha256":"69af5fd1413b6930e4674f97960511bb0e0162c68f70aecbc477a5e1857f6971","origin":"The Stacks Project","memory_eligible":false,"source_rank":7416,"rank":7416,"depth":49,"x":2119.767,"y":588.361,"cluster":"scheme-morphisms"},{"id":"stacks:0EJY","tag":"0EJY","title":"Quasi-projective schemes · Lemma 0EJY","summary":"Let X be a quasi-affine scheme. Let f : U → X be an integral morphism. Then U is quasi-affine and the diagram xymatrix U ar[r] ar[d] & Spec(Γ(U, O_U)) ar[d] X ar[r] & Spec(Γ(X, O_X)) is cartesian.","statement_latex":"Let $X$ be a quasi-affine scheme. Let $f : U \\to X$ be an integral\nmorphism. Then $U$ is quasi-affine and the diagram\n$$\n\\xymatrix{\nU \\ar[r] \\ar[d] & \\Spec(\\Gamma(U, \\mathcal{O}_U)) \\ar[d] \\\\\nX \\ar[r] & \\Spec(\\Gamma(X, \\mathcal{O}_X))\n}\n$$\nis cartesian.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Quasi-projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJY","source_file":"more-morphisms.tex","source_line":13960,"source_end_line":13971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L13960-L13971","statement_sha256":"3467d8b7f2a36231f1f4e3f8fa773c67d48aa2a193c5ebcedd185a72d9520be7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7417,"rank":7417,"depth":20,"x":2037.505,"y":798.573,"cluster":"scheme-morphisms"},{"id":"stacks:0B45","tag":"0B45","title":"Projective schemes · Lemma 0B45","summary":"Let S be a scheme which has an ample invertible sheaf. Let f : X → S be a morphism of schemes. The following are equivalent • X → S is projective, • X → S is H-projective, • X → S is quasi-projective and proper, • X → S is H-quasi-projective and proper, • X → S is proper and X has an ample invertible sheaf, • X → S is proper and there exists an f-ample invertible sheaf, • X → S is proper and there exists an f-very ample invertible sheaf, • there is a quasi-coherent graded…","statement_latex":"Let $S$ be a scheme which has an ample invertible sheaf.\nLet $f : X \\to S$ be a morphism of schemes. The following are\nequivalent\n\\begin{enumerate}\n\\item $X \\to S$ is projective,\n\\item $X \\to S$ is H-projective,\n\\item $X \\to S$ is quasi-projective and proper,\n\\item $X \\to S$ is H-quasi-projective and proper,\n\\item $X \\to S$ is proper and $X$ has an ample invertible sheaf,\n\\item $X \\to S$ is proper and there exists an $f$-ample invertible sheaf,\n\\item $X \\to S$ is proper and there exists an $f$-very ample invertible sheaf,\n\\item there is a quasi-coherent graded $\\mathcal{O}_S$-algebra $\\mathcal{A}$\ngenerated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$ with $\\mathcal{A}_1$ a\nfinite type $\\mathcal{O}_S$-module such that\n$X = \\underline{\\text{Proj}}_S(\\mathcal{A})$. \n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B45","source_file":"more-morphisms.tex","source_line":14027,"source_end_line":14045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14027-L14045","statement_sha256":"f75f0efa12d985d66b3ed693b0051cb5f4df687f8498542947f3ff1c5f688bb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7418,"rank":7418,"depth":27,"x":1929.059,"y":596.78,"cluster":"scheme-morphisms"},{"id":"stacks:0B46","tag":"0B46","title":"Projective schemes · Lemma 0B46","summary":"Let S be a scheme which has an ample invertible sheaf. Let P_S be the full subcategory of the category of schemes over S satisfying the equivalent conditions of Lemma [Tag 0B45]. • if S' → S is a morphism of schemes and S' has an ample invertible sheaf, then base change determines a functor P_S → P_S', • if X ∈ P_S and Y ∈ P_X, then Y ∈ P_S, • the category P_S is closed under fibre products, • the category P_S is closed under finite disjoint unions, • if X → S is finite,…","statement_latex":"Let $S$ be a scheme which has an ample invertible sheaf.\nLet $\\text{P}_S$ be the full subcategory of the\ncategory of schemes over $S$ satisfying the equivalent\nconditions of Lemma \\ref{lemma-projective}.\n\\begin{enumerate}\n\\item if $S' \\to S$ is a morphism of schemes and $S'$ has\nan ample invertible sheaf, then base change determines\na functor $\\text{P}_S \\to \\text{P}_{S'}$,\n\\item if $X \\in \\text{P}_S$ and $Y \\in \\text{P}_X$, then $Y \\in \\text{P}_S$,\n\\item the category $\\text{P}_S$ is closed under fibre products,\n\\item the category $\\text{P}_S$ is closed under\nfinite disjoint unions,\n\\item if $X \\to S$ is finite, then $X$ is in $\\text{P}_S$,\n\\item add more here.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B46","source_file":"more-morphisms.tex","source_line":14102,"source_end_line":14119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14102-L14119","statement_sha256":"26665d8a6075044f967360524e34009551461bd628f65d1e9e59d5fcdd0f8912","origin":"The Stacks Project","memory_eligible":false,"source_rank":7419,"rank":7419,"depth":28,"x":2171.432,"y":684.091,"cluster":"scheme-morphisms"},{"id":"stacks:0D2S","tag":"0D2S","title":"Projective schemes · Lemma 0D2S","summary":"[EGA] Let f : X → Y be a proper morphism of schemes. Let L be an invertible O_X-module. Let y ∈ Y be a point such that L_y is ample on X_y. Then there is an open neighbourhood V ⊂ Y of y such that L|_f^-1(V) is ample on f^-1(V)/V.","statement_latex":"\\begin{reference}\n\\cite[IV Corollary 9.6.4]{EGA}\n\\end{reference}\nLet $f : X \\to Y$ be a proper morphism of schemes.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $y \\in Y$ be a point such that $\\mathcal{L}_y$ is ample\non $X_y$. Then there is an open neighbourhood $V \\subset Y$\nof $y$ such that $\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2S","source_file":"more-morphisms.tex","source_line":14153,"source_end_line":14163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14153-L14163","statement_sha256":"1610e496216f233ed38cc894e6a7f9558a27e274394e38ddcfcdb02728545557","origin":"The Stacks Project","memory_eligible":false,"source_rank":7420,"rank":7420,"depth":42,"x":1922.363,"y":757.277,"cluster":"scheme-morphisms"},{"id":"stacks:0EKI","tag":"0EKI","title":"Proj and Spec · Lemma 0EKI","summary":"Let R be a ring. Let P be a proper scheme over R and let L be an ample invertible O_P-module. Set A = bigoplus_m ≥ 0 Γ(P, L^⊗ m). Then P = Proj(A) and diagram ([Tag 0EKG]) becomes the diagram xymatrix underlineSpec_P ( bigoplus_m ∈ Z L^⊗ m ) ar[r] ar@=[d] & L = underlineSpec_P ( bigoplus_m ≥ 0 L^⊗ m ) ar[d]^σ ar[r]_-π & P ar[d] U ar[r] & X ar[r] & Z having the properties explained above.","statement_latex":"Let $R$ be a ring. Let $P$ be a proper scheme over $R$ and let\n$\\mathcal{L}$ be an ample invertible $\\mathcal{O}_P$-module.\nSet $A = \\bigoplus_{m \\geq 0} \\Gamma(P, \\mathcal{L}^{\\otimes m})$.\nThen $P = \\text{Proj}(A)$ and diagram (\\ref{equation-proj-and-spec})\nbecomes the diagram\n$$\n\\xymatrix{\n\\underline{\\Spec}_P \\left(\n\\bigoplus\\nolimits_{m \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes m}\n\\right)\n\\ar[r] \\ar@{=}[d] &\nL =\n\\underline{\\Spec}_P \\left(\n\\bigoplus\\nolimits_{m \\geq 0} \\mathcal{L}^{\\otimes m}\n\\right) \\ar[d]^\\sigma \\ar[r]_-\\pi & P \\ar[d] \\\\\nU \\ar[r] & X \\ar[r] & Z\n}\n$$\nhaving the properties explained above.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Proj and Spec","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKI","source_file":"more-morphisms.tex","source_line":14491,"source_end_line":14512,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14491-L14512","statement_sha256":"c386f7d52d0e97abe4ed961852f2f3a4594dbd7503f6229293f7e058901b0df6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7421,"rank":7421,"depth":22,"x":2047.235,"y":561.879,"cluster":"scheme-morphisms"},{"id":"stacks:053R","tag":"053R","title":"Closed points in fibres · Lemma 053R","summary":"Let f : X → S be a morphism of schemes. Let Z ⊂ X be a closed subscheme. Let s ∈ S. Assume • S is irreducible with generic point eta, • X is irreducible, • f is dominant, • f is locally of finite type, • dim(X_s) ≤ dim(X_eta), • Z is locally principal in X, and • Z_eta = ∅. Then the fibre Z_s is (set theoretically) a union of irreducible components of X_s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $Z \\subset X$ be a closed subscheme.\nLet $s \\in S$.\nAssume\n\\begin{enumerate}\n\\item $S$ is irreducible with generic point $\\eta$,\n\\item $X$ is irreducible,\n\\item $f$ is dominant,\n\\item $f$ is locally of finite type,\n\\item $\\dim(X_s) \\leq \\dim(X_\\eta)$,\n\\item $Z$ is locally principal in $X$, and\n\\item $Z_\\eta = \\emptyset$.\n\\end{enumerate}\nThen the fibre $Z_s$ is (set theoretically) a union of\nirreducible components of $X_s$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Closed points in fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053R","source_file":"more-morphisms.tex","source_line":14537,"source_end_line":14554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14537-L14554","statement_sha256":"1cf4027f360ad9449d3c9b73ba3949f2e0790f7c903fb4ca3fb80aad1d3923f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7422,"rank":7422,"depth":32,"x":2112.325,"y":776.932,"cluster":"scheme-morphisms"},{"id":"stacks:053S","tag":"053S","title":"Closed points in fibres · Lemma 053S","summary":"Let A → B be a local homomorphism of local rings, and g ∈ m_B. Assume • A and B are domains and A ⊂ B, • B is essentially of finite type over A, • g is not contained in any minimal prime over m_AB, and • dim(B/ m_AB) + trdeg_kappa( m_A)(kappa( m_B)) = trdeg_A(B). Then A ⊂ B/gB, i.e., the generic point of Spec(A) is in the image of the morphism Spec(B/gB) → Spec(A).","statement_latex":"Let $A \\to B$ be a local homomorphism of local rings, and\n$g \\in \\mathfrak m_B$. Assume\n\\begin{enumerate}\n\\item $A$ and $B$ are domains and $A \\subset B$,\n\\item $B$ is essentially of finite type over $A$,\n\\item $g$ is not contained in any minimal prime over $\\mathfrak m_AB$, and\n\\item $\\dim(B/\\mathfrak m_AB) +\n\\text{trdeg}_{\\kappa(\\mathfrak m_A)}(\\kappa(\\mathfrak m_B)) =\n\\text{trdeg}_A(B)$.\n\\end{enumerate}\nThen $A \\subset B/gB$, i.e., the generic point of $\\Spec(A)$\nis in the image of the morphism $\\Spec(B/gB) \\to \\Spec(A)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Closed points in fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053S","source_file":"more-morphisms.tex","source_line":14701,"source_end_line":14715,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14701-L14715","statement_sha256":"47bca8431a71d56b761eee910dc5bf4f7ae71cd19b8ee887e626ff328f3439c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7423,"rank":7423,"depth":33,"x":1891.271,"y":655.224,"cluster":"scheme-morphisms"},{"id":"stacks:053T","tag":"053T","title":"Closed points in fibres · Lemma 053T","summary":"Let A → B be a local homomorphism of local rings. Assume • A and B are domains and A ⊂ B, • B is essentially of finite type over A, and • B is flat over A. Then we have dim(B/ m_AB) + trdeg_kappa( m_A)(kappa( m_B)) = trdeg_A(B).","statement_latex":"Let $A \\to B$ be a local homomorphism of local rings. Assume\n\\begin{enumerate}\n\\item $A$ and $B$ are domains and $A \\subset B$,\n\\item $B$ is essentially of finite type over $A$, and\n\\item $B$ is flat over $A$.\n\\end{enumerate}\nThen we have\n$$\n\\dim(B/\\mathfrak m_AB) +\n\\text{trdeg}_{\\kappa(\\mathfrak m_A)}(\\kappa(\\mathfrak m_B)) =\n\\text{trdeg}_A(B).\n$$","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Closed points in fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053T","source_file":"more-morphisms.tex","source_line":14754,"source_end_line":14768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14754-L14768","statement_sha256":"c6646c8f36e3b1cbb96f3acd4c261a782d41313d63986482192ad33bb7dca8e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7424,"rank":7424,"depth":32,"x":2152.286,"y":619.519,"cluster":"scheme-morphisms"},{"id":"stacks:053U","tag":"053U","title":"Closed points in fibres · Lemma 053U","summary":"Let f : X → S be a morphism of schemes. Let x leadsto x' be a specialization of points in X. Set s = f(x) and s' = f(x'). Assume • x' is a closed point of X_s', and • f is locally of finite type. Then the set (x_1 ∈ X such that f(x_1) = s and x_1 is closed in X_s and x leadsto x_1 leadsto x' ) is dense in the closure of x in X_s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\leadsto x'$ be a specialization of points in $X$.\nSet $s = f(x)$ and $s' = f(x')$.\nAssume\n\\begin{enumerate}\n\\item $x'$ is a closed point of $X_{s'}$, and\n\\item $f$ is locally of finite type.\n\\end{enumerate}\nThen the set\n$$\n\\{x_1 \\in X\n\\text{ such that }\nf(x_1) = s\n\\text{ and }\nx_1\\text{ is closed in }X_s\n\\text{ and }\nx \\leadsto x_1 \\leadsto x'\n\\}\n$$\nis dense in the closure of $x$ in $X_s$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Closed points in fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053U","source_file":"more-morphisms.tex","source_line":14816,"source_end_line":14838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14816-L14838","statement_sha256":"a4b7c1c015e781a00e119b3f46d2d38aa1626db9c10c6816299f4159abd012b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7425,"rank":7425,"depth":33,"x":1988.443,"y":794.046,"cluster":"scheme-morphisms"},{"id":"stacks:05GT","tag":"05GT","title":"Closed points in fibres · Lemma 05GT","summary":"Let φ : A → B be a local ring map of local rings. Let V ⊂ Spec(B) be an open subscheme which contains at least one prime not lying over m_A. Assume A is Noetherian, φ essentially of finite type, and A/ m_A ⊂ B/ m_B is finite. Then there exists a q ∈ V, m_A not = q ∩ A such that A → B/ q is the localization of a quasi-finite ring map.","statement_latex":"Let $\\varphi : A \\to B$ be a local ring map of local rings.\nLet $V \\subset \\Spec(B)$ be an open subscheme\nwhich contains at least one prime not lying over $\\mathfrak m_A$.\nAssume $A$ is Noetherian, $\\varphi$ essentially of finite type, and\n$A/\\mathfrak m_A \\subset B/\\mathfrak m_B$ is finite.\nThen there exists a $\\mathfrak q \\in V$,\n$\\mathfrak m_A \\not = \\mathfrak q \\cap A$ such that\n$A \\to B/\\mathfrak q$ is the localization of a quasi-finite ring map.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Closed points in fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GT","source_file":"more-morphisms.tex","source_line":14957,"source_end_line":14967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14957-L14967","statement_sha256":"eb3fc17cb4413002b131c459325e91ed75070f5f9fd989d718f3e909d2f26cf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7426,"rank":7426,"depth":29,"x":1968.898,"y":572.266,"cluster":"scheme-morphisms"},{"id":"stacks:05GU","tag":"05GU","title":"Closed points in fibres · Lemma 05GU","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X with image s ∈ S. Let U ⊂ X be an open subscheme. Assume f locally of finite type, S locally Noetherian, x a closed point of X_s, and assume there exists a point x' ∈ U with x' leadsto x and f(x') not = s. Then there exists a closed subscheme Z ⊂ X such that (a) x ∈ Z, (b) f|_Z : Z → S is quasi-finite at x, and (c) there exists a z ∈ Z, z ∈ U, z leadsto x and f(z) not = s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$ with image $s \\in S$.\nLet $U \\subset X$ be an open subscheme.\nAssume $f$ locally of finite type, $S$ locally Noetherian, $x$ a closed\npoint of $X_s$, and assume there exists a point $x' \\in U$ with\n$x' \\leadsto x$ and $f(x') \\not = s$. Then there exists a closed\nsubscheme $Z \\subset X$ such that (a) $x \\in Z$, (b) $f|_Z : Z \\to S$ is\nquasi-finite at $x$, and (c) there exists a $z \\in Z$, $z \\in U$,\n$z \\leadsto x$ and $f(z) \\not = s$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Closed points in fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GU","source_file":"more-morphisms.tex","source_line":14993,"source_end_line":15004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L14993-L15004","statement_sha256":"e36899d162e031c7292e0d451d5b4088271c08443a3caea831c8028c0dd98706","origin":"The Stacks Project","memory_eligible":false,"source_rank":7427,"rank":7427,"depth":30,"x":2161.762,"y":724.795,"cluster":"scheme-morphisms"},{"id":"stacks:05GW","tag":"05GW","title":"Closed points in fibres · Lemma 05GW","summary":"Suppose that f : X → S is locally of finite type, S locally Noetherian, x ∈ X a closed point of its fibre X_s, and U ⊂ X an open subscheme such that U ∩ X_s = ∅ and x ∈ overlineU, then the conclusions of Lemma [Tag 05GU] hold.","statement_latex":"Suppose that $f : X \\to S$ is locally of finite type, $S$ locally Noetherian,\n$x \\in X$ a closed point of its fibre $X_s$, and $U \\subset X$ an open\nsubscheme such that $U \\cap X_s = \\emptyset$ and $x \\in \\overline{U}$, then\nthe conclusions of\nLemma \\ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X}\nhold.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Closed points in fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05GW","source_file":"more-morphisms.tex","source_line":15053,"source_end_line":15061,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15053-L15061","statement_sha256":"2436d497b4b3a46eb6b2802a30c5dc22eba696af70d537204519aa7873cbc63d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7428,"rank":7428,"depth":31,"x":1896.748,"y":721.756,"cluster":"scheme-morphisms"},{"id":"stacks:03GY","tag":"03GY","title":"Stein factorization · Lemma 03GY","summary":"Let S be a scheme. Let f : X → S be a universally closed and quasi-separated morphism. There exists a factorization xymatrix X ar[rr]_f' ar[rd]_f & & S' ar[dl]^π & S & with the following properties: • the morphism f' is universally closed, quasi-compact, quasi-separated, and surjective, • the morphism π : S' → S is integral, • we have f'_*O_X = O_S', • we have S' = underlineSpec_S(f_*O_X), and • S' is the normalization of S in X, see Morphisms, Definition [Tag 035H].…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to S$ be a universally closed and\nquasi-separated morphism. There exists a factorization\n$$\n\\xymatrix{\nX \\ar[rr]_{f'} \\ar[rd]_f & & S' \\ar[dl]^\\pi \\\\\n& S &\n}\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item the morphism $f'$ is universally closed, quasi-compact, quasi-separated,\nand surjective,\n\\item the morphism $\\pi : S' \\to S$ is integral,\n\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{S'}$,\n\\item we have $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$, and\n\\item $S'$ is the normalization of $S$ in $X$, see\nMorphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.\n\\end{enumerate}\nFormation of the factorization $f = \\pi \\circ f'$ commutes\nwith flat base change.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GY","source_file":"more-morphisms.tex","source_line":15086,"source_end_line":15108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15086-L15108","statement_sha256":"223f0d4d930b4fe98acb09c9c8f19bf9cc6fd13e8e4ffcef6b63ab1bd98fe66f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7429,"rank":7429,"depth":30,"x":2094.713,"y":573.543,"cluster":"scheme-morphisms"},{"id":"stacks:0E0M","tag":"0E0M","title":"Stein factorization · Lemma 0E0M","summary":"In Lemma [Tag 03GY] assume in addition that f is locally of finite type. Then for s ∈ S the fibre π^-1((s)) = (s_1, …, s_n) is finite and the field extensions kappa(s_i)/kappa(s) are finite.","statement_latex":"In Lemma \\ref{lemma-stein-universally-closed} assume in addition that\n$f$ is locally of finite type. Then for $s \\in S$ the fibre\n$\\pi^{-1}(\\{s\\}) = \\{s_1, \\ldots, s_n\\}$ is finite and the field extensions\n$\\kappa(s_i)/\\kappa(s)$ are finite.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0M","source_file":"more-morphisms.tex","source_line":15153,"source_end_line":15159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15153-L15159","statement_sha256":"3396e624bc388b81d616381b3799bc6623f70d0b9c740f23310906380a068fbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7430,"rank":7430,"depth":31,"x":2067.911,"y":795.281,"cluster":"scheme-morphisms"},{"id":"stacks:03GZ","tag":"03GZ","title":"Stein factorization · Lemma 03GZ","summary":"Let f : X → S be a morphism of schemes. Let s ∈ S. Then X_s is geometrically connected, if and only if for every étale neighbourhood (U, u) → (S, s) the base change X_U → U has connected fibre X_u.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $s \\in S$. Then $X_s$ is geometrically connected, if and\nonly if for every \\'etale neighbourhood $(U, u) \\to (S, s)$\nthe base change $X_U \\to U$ has connected fibre $X_u$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GZ","source_file":"more-morphisms.tex","source_line":15181,"source_end_line":15187,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15181-L15187","statement_sha256":"e945194ac9844a5dfc03e03dab5d7095bd5f088bbffcc4b1346c353744b71bb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7431,"rank":7431,"depth":17,"x":1909.275,"y":616.472,"cluster":"scheme-morphisms"},{"id":"stacks:03H0","tag":"03H0","title":"Stein factorization; Noetherian case · Theorem 03H0","summary":"Let S be a locally Noetherian scheme. Let f : X → S be a proper morphism. There exists a factorization xymatrix X ar[rr]_f' ar[rd]_f & & S' ar[dl]^π & S & with the following properties: • the morphism f' is proper with geometrically connected fibres, • the morphism π : S' → S is finite, • we have f'_*O_X = O_S', • we have S' = underlineSpec_S(f_*O_X), and • S' is the normalization of S in X, see Morphisms, Definition [Tag 035H].","statement_latex":"Let $S$ be a locally Noetherian scheme.\nLet $f : X \\to S$ be a proper morphism.\nThere exists a factorization\n$$\n\\xymatrix{\nX \\ar[rr]_{f'} \\ar[rd]_f & & S' \\ar[dl]^\\pi \\\\\n& S &\n}\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item the morphism $f'$ is proper with geometrically connected fibres,\n\\item the morphism $\\pi : S' \\to S$ is finite,\n\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{S'}$,\n\\item we have $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$, and\n\\item $S'$ is the normalization of $S$ in $X$, see\nMorphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stein factorization","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03H0","source_file":"more-morphisms.tex","source_line":15204,"source_end_line":15224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15204-L15224","statement_sha256":"51faf5bed4e08c55971cb212f4b712f81547ffea50a6d2ad0e2baf7343341d46","origin":"The Stacks Project","memory_eligible":false,"source_rank":7432,"rank":7432,"depth":39,"x":2170.182,"y":658.331,"cluster":"scheme-morphisms"},{"id":"stacks:03H2","tag":"03H2","title":"Stein factorization; general case · Theorem 03H2","summary":"Let S be a scheme. Let f : X → S be a proper morphism. There exists a factorization xymatrix X ar[rr]_f' ar[rd]_f & & S' ar[dl]^π & S & with the following properties: • the morphism f' is proper with geometrically connected fibres, • the morphism π : S' → S is integral, • we have f'_*O_X = O_S', • we have S' = underlineSpec_S(f_*O_X), and • S' is the normalization of S in X, see Morphisms, Definition [Tag 035H].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to S$ be a proper morphism.\nThere exists a factorization\n$$\n\\xymatrix{\nX \\ar[rr]_{f'} \\ar[rd]_f & & S' \\ar[dl]^\\pi \\\\\n& S &\n}\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item the morphism $f'$ is proper with geometrically connected fibres,\n\\item the morphism $\\pi : S' \\to S$ is integral,\n\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{S'}$,\n\\item we have $S' = \\underline{\\Spec}_S(f_*\\mathcal{O}_X)$, and\n\\item $S'$ is the normalization of $S$ in $X$, see\nMorphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stein factorization","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03H2","source_file":"more-morphisms.tex","source_line":15289,"source_end_line":15309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15289-L15309","statement_sha256":"9f9c927a05311c053bed006268af0a5780988258b3dc718cf281de20243c9735","origin":"The Stacks Project","memory_eligible":false,"source_rank":7433,"rank":7433,"depth":39,"x":1944.012,"y":775.571,"cluster":"scheme-morphisms"},{"id":"stacks:0AY8","tag":"0AY8","title":"Stein factorization · Lemma 0AY8","summary":"Let f : X → S be a morphism of schemes. Assume • f is proper, • S is integral with generic point xi, • S is normal, • X is reduced, • every generic point of an irreducible component of X maps to xi, • we have H^0(X_xi, O) = kappa(xi). Then f_*O_X = O_S and f has geometrically connected fibres.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item $S$ is integral with generic point $\\xi$,\n\\item $S$ is normal,\n\\item $X$ is reduced,\n\\item every generic point of an irreducible component of $X$ maps to $\\xi$,\n\\item we have $H^0(X_\\xi, \\mathcal{O}) = \\kappa(\\xi)$.\n\\end{enumerate}\nThen $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and $f$\nhas geometrically connected fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AY8","source_file":"more-morphisms.tex","source_line":15357,"source_end_line":15370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15357-L15370","statement_sha256":"4b661bb2e846943cce505a0c559a1365616206872ec0f33181f3fad285608b8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7434,"rank":7434,"depth":40,"x":2016.543,"y":560.673,"cluster":"scheme-morphisms"},{"id":"stacks:0BUI","tag":"0BUI","title":"Stein factorization · Lemma 0BUI","summary":"Let X → S be a flat proper morphism of finite presentation. Let n_X/S be the function on S counting the numbers of geometric connected components of fibres of f introduced in Lemma [Tag 055F]. Then n_X/S is lower semi-continuous.","statement_latex":"Let $X \\to S$ be a flat proper morphism of finite presentation. Let\n$n_{X/S}$ be the function on $S$ counting the numbers of geometric\nconnected components of fibres of $f$ introduced in\nLemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}.\nThen $n_{X/S}$ is lower semi-continuous.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUI","source_file":"more-morphisms.tex","source_line":15393,"source_end_line":15400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15393-L15400","statement_sha256":"f29f3ffe3fa310bd63d5954dd2da86e2ee5ccefb2e35c6ed65105c676e641539","origin":"The Stacks Project","memory_eligible":false,"source_rank":7435,"rank":7435,"depth":40,"x":2135.938,"y":760.396,"cluster":"scheme-morphisms"},{"id":"stacks:0E0N","tag":"0E0N","title":"Stein factorization · Lemma 0E0N","summary":"Let f : X → S be a morphism of schemes. Assume • f is proper, flat, and of finite presentation, and • the geometric fibres of f are reduced. Then the function n_X/S : S → Z counting the numbers of geometric connected components of fibres of f is locally constant.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $f$ is proper, flat, and of finite presentation, and\n\\item the geometric fibres of $f$ are reduced.\n\\end{enumerate}\nThen the function $n_{X/S} : S \\to \\mathbf{Z}$\ncounting the numbers of geometric connected components\nof fibres of $f$ is locally constant.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0N","source_file":"more-morphisms.tex","source_line":15425,"source_end_line":15435,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15425-L15435","statement_sha256":"f16e0dc612a57f070ea583eca7ab183abfcdc0b037e428e53abe214ee1ebe6b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7436,"rank":7436,"depth":41,"x":1887.156,"y":680.829,"cluster":"scheme-morphisms"},{"id":"stacks:0CT9","tag":"0CT9","title":"Stein factorization · Lemma 0CT9","summary":"A reference for the case of an adic Noetherian base is [EGA] Let (A, I) be a henselian pair. Let X → Spec(A) be separated and of finite type. Set X_0 = X ×_Spec(A) Spec(A/I). Let Y ⊂ X_0 be an open and closed subscheme such that Y → Spec(A/I) is proper. Then there exists an open and closed subscheme W ⊂ X which is proper over A with W ×_Spec(A) Spec(A/I) = Y.","statement_latex":"\\begin{reference}\nA reference for the case of an adic Noetherian base is\n\\cite[III, Proposition 5.5.1]{EGA}\n\\end{reference}\nLet $(A, I)$ be a henselian pair. Let $X \\to \\Spec(A)$\nbe separated and of finite type. Set $X_0 = X \\times_{\\Spec(A)} \\Spec(A/I)$.\nLet $Y \\subset X_0$ be an open and closed subscheme such that\n$Y \\to \\Spec(A/I)$ is proper. Then there exists an open and closed\nsubscheme $W \\subset X$ which is proper over $A$ with\n$W \\times_{\\Spec(A)} \\Spec(A/I) = Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CT9","source_file":"more-morphisms.tex","source_line":15462,"source_end_line":15474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15462-L15474","statement_sha256":"172c46657ae5db8e8088bd75e277dc839695ddea3f4100e4d78dc43f33f074bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7437,"rank":7437,"depth":47,"x":2134.72,"y":598.292,"cluster":"scheme-morphisms"},{"id":"stacks:0ASY","tag":"0ASY","title":"Generic flatness stratification · Lemma 0ASY","summary":"Let f : X → S be a morphism of finite presentation between quasi-compact and quasi-separated schemes. Let F be an O_X-module of finite presentation. Then there exists a t ≥ 0 and closed subschemes S ⊃ S_0 ⊃ S_1 ⊃ … ⊃ S_t = ∅ such that S_i → S is defined by a finite type ideal sheaf, S_0 ⊂ S is a thickening, and F pulled back to X ×_S (S_i setminus S_i + 1) is flat over S_i setminus S_i + 1.","statement_latex":"Let $f : X \\to S$ be a morphism of finite presentation between quasi-compact\nand quasi-separated schemes. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module\nof finite presentation. Then there exists a $t \\geq 0$ and closed\nsubschemes\n$$\nS \\supset S_0 \\supset S_1 \\supset \\ldots \\supset S_t = \\emptyset\n$$\nsuch that $S_i \\to S$ is defined by a finite type ideal sheaf,\n$S_0 \\subset S$ is a thickening, and $\\mathcal{F}$ pulled back to\n$X \\times_S (S_i \\setminus S_{i + 1})$ is flat over $S_i \\setminus S_{i + 1}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic flatness stratification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASY","source_file":"more-morphisms.tex","source_line":15565,"source_end_line":15577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15565-L15577","statement_sha256":"16a23c0f1535d32a2e527e4cd7630e68493dec78ad6c072419d7d690d0f5cd45","origin":"The Stacks Project","memory_eligible":false,"source_rank":7438,"rank":7438,"depth":27,"x":2018.482,"y":799.733,"cluster":"scheme-morphisms"},{"id":"stacks:0H3Z","tag":"0H3Z","title":"Generic flatness stratification · Lemma 0H3Z","summary":"Let f : X → S be a morphism of finite presentation between quasi-compact and quasi-separated schemes. Then there exists a t ≥ 0 and closed subschemes S ⊃ S_0 ⊃ S_1 ⊃ … ⊃ S_t = ∅ such that S_i → S is defined by a finite type ideal sheaf, S_0 ⊂ S is a thickening, and X ×_S (S_i setminus S_i + 1) is flat over S_i setminus S_i + 1.","statement_latex":"Let $f : X \\to S$ be a morphism of finite presentation between quasi-compact\nand quasi-separated schemes. Then there exists a $t \\geq 0$ and closed\nsubschemes\n$$\nS \\supset S_0 \\supset S_1 \\supset \\ldots \\supset S_t = \\emptyset\n$$\nsuch that $S_i \\to S$ is defined by a finite type ideal sheaf,\n$S_0 \\subset S$ is a thickening, and\n$X \\times_S (S_i \\setminus S_{i + 1})$ is flat over $S_i \\setminus S_{i + 1}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic flatness stratification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H3Z","source_file":"more-morphisms.tex","source_line":15614,"source_end_line":15625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15614-L15625","statement_sha256":"05e28bc64090874e285552b755d1b57050e34bbb459fa3e89538e14b8f9c11b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7439,"rank":7439,"depth":28,"x":1942.161,"y":585.126,"cluster":"scheme-morphisms"},{"id":"stacks:0H89","tag":"0H89","title":"Longke Tang · Lemma 0H89","summary":"Let f : X → S be a surjective morphism of finite presentation. Let M be an object of D_QCoh(O_S) such that H^i(M) = 0 for i > 0. The following are equivalent • M is isomorphic to a flat O_S-module placed in degree 0, • Lf^*M is isomorphic to a flat O_X-module placed in degree 0.","statement_latex":"Let $f : X \\to S$ be a surjective morphism of finite presentation.\nLet $M$ be an object of $D_\\QCoh(\\mathcal{O}_S)$ such that\n$H^i(M) = 0$ for $i > 0$. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is isomorphic to a flat $\\mathcal{O}_S$-module\nplaced in degree $0$,\n\\item $Lf^*M$ is isomorphic to a flat $\\mathcal{O}_X$-module\nplaced in degree $0$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic flatness stratification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H89","source_file":"more-morphisms.tex","source_line":15632,"source_end_line":15643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15632-L15643","statement_sha256":"4add5bb1f060a185bb607f4a21ea7682972320cff7b463a6c5a3d6d8384f6f1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7440,"rank":7440,"depth":29,"x":2171.14,"y":700.127,"cluster":"scheme-morphisms"},{"id":"stacks:0H40","tag":"0H40","title":"Generic flatness stratification · Lemma 0H40","summary":"Let R be a Noetherian domain. Let R → A → B be finite type ring maps. Let M be a finite A-module and let N a finite B-module. Let M → N be an A-linear map. There exists an nonzero f ∈ R such that the cokernel of M_f → N_f is a flat R_f-module.","statement_latex":"Let $R$ be a Noetherian domain. Let $R \\to A \\to B$ be finite type ring maps.\nLet $M$ be a finite $A$-module and let $N$ a finite $B$-module.\nLet $M \\to N$ be an $A$-linear map. There exists an nonzero $f \\in R$\nsuch that the cokernel of $M_f \\to N_f$ is a flat $R_f$-module.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic flatness stratification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H40","source_file":"more-morphisms.tex","source_line":15700,"source_end_line":15706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15700-L15706","statement_sha256":"7cdff84f6cf93ab752bb140ffd4282d71d5a76728e16b7658e6876856f96b651","origin":"The Stacks Project","memory_eligible":false,"source_rank":7441,"rank":7441,"depth":13,"x":1909.677,"y":745.279,"cluster":"scheme-morphisms"},{"id":"stacks:0H41","tag":"0H41","title":"Generic flatness stratification · Lemma 0H41","summary":"Let S be a quasi-compact and quasi-separated scheme. Let f : X → Y be a morphism of schemes over S with both X and Y of finite presentation over S. Then there exists a t ≥ 0 and closed subschemes S ⊃ S_0 ⊃ S_1 ⊃ … ⊃ S_t = ∅ with the following properties: • S_i → S is defined by a finite type ideal sheaf, • S_0 ⊂ S is a thickening, and • with T_i = S_i setminus S_i + 1 and f_i the base change of f to T_i we have: formation of the scheme theoretic image of f_i/T_i commutes…","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to Y$ be a morphism of schemes over $S$ with\nboth $X$ and $Y$ of finite presentation over $S$.\nThen there exists a $t \\geq 0$ and closed subschemes\n$$\nS \\supset S_0 \\supset S_1 \\supset \\ldots \\supset S_t = \\emptyset\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item $S_i \\to S$ is defined by a finite type ideal sheaf,\n\\item $S_0 \\subset S$ is a thickening, and\n\\item with $T_i = S_i \\setminus S_{i + 1}$ and $f_i$ the base\nchange of $f$ to $T_i$ we have:\nformation of the scheme theoretic image of $f_i/T_i$\ncommutes with arbitrary base change (see discussion above the lemma).\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Generic flatness stratification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H41","source_file":"more-morphisms.tex","source_line":15775,"source_end_line":15793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15775-L15793","statement_sha256":"c0431e70c7b7e5721e2d4c9f9e3813a40786461c95e6d8fd4f20647c6c0165e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7442,"rank":7442,"depth":26,"x":2066.248,"y":563.529,"cluster":"scheme-morphisms"},{"id":"stacks:0H43","tag":"0H43","title":"Stratifying a morphism · Lemma 0H43","summary":"Let f : X → S be a morphism of schemes of finite presentation. Let eta ∈ S be a generic point of an irreducible component of S. Assume S is reduced. Then there exist • an open subscheme U ⊂ S containing eta, • a surjective, universally injective, finite locally free morphism V → U, • a t ≥ 0 and closed subschemes X ×_S V ⊃ Z_0 ⊃ Z_1 ⊃ … ⊃ Z_t = ∅ such that Z_i → X ×_S V is defined by a finite type ideal sheaf, Z_0 ⊂ X ×_S V is a thickening, and such that the morphism Z_i…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes of finite presentation.\nLet $\\eta \\in S$ be a generic point of an irreducible component of $S$.\nAssume $S$ is reduced. Then there exist\n\\begin{enumerate}\n\\item an open subscheme $U \\subset S$ containing $\\eta$,\n\\item a surjective, universally injective, finite locally free\nmorphism $V \\to U$,\n\\item a $t \\geq 0$ and closed subschemes\n$$\nX \\times_S V \\supset Z_0 \\supset Z_1 \\supset \\ldots \\supset Z_t = \\emptyset\n$$\nsuch that $Z_i \\to X \\times_S V$ is defined by a finite type ideal sheaf,\n$Z_0 \\subset X \\times_S V$ is a thickening, and such that the morphism\n$Z_i \\setminus Z_{i + 1} \\to V$ is smooth.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stratifying a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H43","source_file":"more-morphisms.tex","source_line":15885,"source_end_line":15902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15885-L15902","statement_sha256":"56e9c2d2f7285576b05565769216de36b4989804078add80810c9cb91c27b8e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7443,"rank":7443,"depth":46,"x":2096.969,"y":786.507,"cluster":"scheme-morphisms"},{"id":"stacks:0H44","tag":"0H44","title":"Stratifying a morphism · Lemma 0H44","summary":"Let f : X → S be a morphism of finite presentation between quasi-compact and quasi-separated schemes. Then there exists a t ≥ 0 and closed subschemes S ⊃ S_0 ⊃ S_1 ⊃ … ⊃ S_t = ∅ such that • S_i → S is defined by a finite type ideal sheaf, • S_0 ⊂ S is a thickening, • for each i there exists a surjective finite locally free morphism T_i → S_i setminus S_i + 1, • for each i there exists a t_i ≥ 0 and closed subschemes X_i = X ×_S T_i ⊃ Z_i, 0 ⊃ Z_i, 1 ⊃ … ⊃ Z_i, t_i = ∅…","statement_latex":"Let $f : X \\to S$ be a morphism of finite presentation between quasi-compact\nand quasi-separated schemes. Then there exists a $t \\geq 0$ and closed\nsubschemes\n$$\nS \\supset S_0 \\supset S_1 \\supset \\ldots \\supset S_t = \\emptyset\n$$\nsuch that\n\\begin{enumerate}\n\\item $S_i \\to S$ is defined by a finite type ideal sheaf,\n\\item $S_0 \\subset S$ is a thickening,\n\\item for each $i$ there exists a surjective finite locally free\nmorphism $T_i \\to S_i \\setminus S_{i + 1}$,\n\\item for each $i$ there exists a $t_i \\geq 0$ and closed subschemes\n$$\nX_i = X \\times_S T_i \\supset Z_{i, 0}\n\\supset Z_{i, 1} \\supset \\ldots \\supset Z_{i, t_i} = \\emptyset\n$$\nsuch that $Z_{i, j} \\to X_i$ is defined by a finite type ideal sheaf,\n$Z_{i, 0} \\subset X_i$ is a thickening, and such that the morphism\n$Z_{i, j} \\setminus Z_{i, j + 1} \\to T_i$ is smooth.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Stratifying a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H44","source_file":"more-morphisms.tex","source_line":15986,"source_end_line":16009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L15986-L16009","statement_sha256":"eb1bcb736ffd1ff49931939e0c9be88fc520a597c02cfac92eb540e3faa45b7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7444,"rank":7444,"depth":47,"x":1894.898,"y":639.442,"cluster":"scheme-morphisms"},{"id":"stacks:0GK7","tag":"0GK7","title":"Improving morphisms of relative dimension one · Lemma 0GK7","summary":"Let f : X → S be a morphism of schemes. Let eta ∈ S be a generic point of an irreducible component of S. Assume f is separated, of finite presentation, and dim(X_eta) ≤ 1. Then there exists a commutative diagram xymatrix overlineY_1 amalg … amalg overlineY_n ar[rd] & Y_1 amalg … amalg Y_n ar[r]_-ν ar[d] ar[l]^j & X_V ar[r] ar[d] & X_U ar[r] ar[d] & X ar[d]^f & T_1 amalg … amalg T_n ar[r] & V ar[r] & U ar[r] & S of schemes with the following properties: • U ⊂ S is an open…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $\\eta \\in S$ be a\ngeneric point of an irreducible component of $S$. Assume $f$ is\nseparated, of finite presentation, and $\\dim(X_\\eta) \\leq 1$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n\\overline{Y}_1 \\amalg \\ldots \\amalg \\overline{Y}_n \\ar[rd] &\nY_1 \\amalg \\ldots \\amalg Y_n \\ar[r]_-\\nu \\ar[d] \\ar[l]^j &\nX_V \\ar[r] \\ar[d] &\nX_U \\ar[r] \\ar[d] &\nX \\ar[d]^f \\\\\n& T_1 \\amalg \\ldots \\amalg T_n \\ar[r] &\nV \\ar[r] &\nU \\ar[r] &\nS\n}\n$$\nof schemes with the following properties:\n\\begin{enumerate}\n\\item $U \\subset S$ is an open neighbourhood of $\\eta$,\n\\item $V \\to U$ is a finite, surjective, universally injective morphism,\n\\item $X_U = U \\times_S X$ and $X_V = V \\times_S X$ are the base changes,\n\\item $\\nu$ is finite, surjective, and there is an open $W \\subset X_V$\nsuch that\n\\begin{enumerate}\n\\item $W$ is dense in all fibres of $X_V \\to V$,\n\\item $\\nu^{-1}(W) \\cap Y_i$ is dense in all fibres of $Y_i \\to T_i$, and\n\\item $\\nu^{-1}(W) \\to W$ is a thickening,\n\\end{enumerate}\n\\item $j$ is an open immersion,\n\\item $T_i \\to V$ is finite \\'etale,\n\\item $Y_i \\to T_i$ is surjective and smooth,\n\\item $\\overline{Y}_i \\to T_i$ is smooth, proper, with geometrically\nconnected fibres of dimension $\\leq 1$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Improving morphisms of relative dimension one","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GK7","source_file":"more-morphisms.tex","source_line":16079,"source_end_line":16116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16079-L16116","statement_sha256":"fa1d7624b422fc51dc8047cfa9686ca2347389a293873143008189bf764e7014","origin":"The Stacks Project","memory_eligible":false,"source_rank":7445,"rank":7445,"depth":45,"x":2162.307,"y":633.223,"cluster":"scheme-morphisms"},{"id":"stacks:02W8","tag":"02W8","title":"Descending separated locally quasi-finite morphisms · Lemma 02W8","summary":"Let S be a scheme. Let (X_i → S)_i∈ I be an fppf covering, see Topologies, Definition [Tag 021M]. Let (V_i/X_i, φ_ij) be a descent datum relative to (X_i → S). If each morphism V_i → X_i is separated and locally quasi-finite, then the descent datum is effective.","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to S\\}_{i\\in I}$ be an fppf covering, see\nTopologies, Definition \\ref{topologies-definition-fppf-covering}.\nLet $(V_i/X_i, \\varphi_{ij})$ be a descent datum\nrelative to $\\{X_i \\to S\\}$. If each morphism\n$V_i \\to X_i$ is separated and locally quasi-finite,\nthen the descent datum is effective.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Descending separated locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02W8","source_file":"more-morphisms.tex","source_line":16278,"source_end_line":16287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16278-L16287","statement_sha256":"4a1375522e3bda53712bc547047cbfa434627b6ec68bc672d354da9dc440e192","origin":"The Stacks Project","memory_eligible":false,"source_rank":7446,"rank":7446,"depth":49,"x":1970.025,"y":789.624,"cluster":"scheme-morphisms"},{"id":"stacks:05H1","tag":"05H1","title":"Relative finite presentation · Definition 05H1","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent O_X-module. We say F is finitely presented relative to S or of finite presentation relative to S if there exists an affine open covering S = ⋃ V_i and for every i an affine open covering f^-1(V_i) = ⋃_j U_ij such that F(U_ij) is a O_X(U_ij)-module of finite presentation relative to O_S(V_i).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. We say\n$\\mathcal{F}$ is {\\it finitely presented relative to $S$} or\n{\\it of finite presentation relative to $S$}\nif there exists an affine open covering $S = \\bigcup V_i$ and\nfor every $i$ an affine open covering\n$f^{-1}(V_i) = \\bigcup_j U_{ij}$ such that $\\mathcal{F}(U_{ij})$\nis a $\\mathcal{O}_X(U_{ij})$-module of finite presentation relative\nto $\\mathcal{O}_S(V_i)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative finite presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05H1","source_file":"more-morphisms.tex","source_line":16374,"source_end_line":16385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16374-L16385","statement_sha256":"0b8c14ef7137e84af21feeba84346dc8521071f4cd9f87f2d7ad1bfae84bb7d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7447,"rank":7447,"depth":0,"x":1986.045,"y":565.074,"cluster":"scheme-morphisms"},{"id":"stacks:09T7","tag":"09T7","title":"Relative finite presentation · Lemma 09T7","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent O_X-module. The following are equivalent • F is of finite presentation relative to S, • for every affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the O_X(U)-module F(U) is finitely presented relative to O_S(V). Moreover, if this is true, then for every open subschemes U ⊂ X and V ⊂ S with f(U) ⊂ V the restriction F|_U is of finite presentation relative to V.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. The following\nare equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is of finite presentation relative to $S$,\n\\item for every affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the $\\mathcal{O}_X(U)$-module $\\mathcal{F}(U)$\nis finitely presented relative to $\\mathcal{O}_S(V)$.\n\\end{enumerate}\nMoreover, if this is true, then for every open subschemes\n$U \\subset X$ and $V \\subset S$ with $f(U) \\subset V$\nthe restriction $\\mathcal{F}|_U$ is of finite presentation relative to $V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09T7","source_file":"more-morphisms.tex","source_line":16395,"source_end_line":16409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16395-L16409","statement_sha256":"a7502d5c5b9f9731cb257bdee311f611fc029b2d860269676b91eaee76ba105b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7448,"rank":7448,"depth":6,"x":2154.896,"y":739.835,"cluster":"scheme-morphisms"},{"id":"stacks:09T8","tag":"09T8","title":"Relative finite presentation · Lemma 09T8","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent O_X-module. • If f is locally of finite presentation, then F is of finite presentation relative to S if and only if F is of finite presentation. • The morphism f is locally of finite presentation if and only if O_X is of finite presentation relative to S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite\ntype. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If $f$ is locally of finite presentation, then $\\mathcal{F}$\nis of finite presentation relative to $S$ if and only if $\\mathcal{F}$\nis of finite presentation.\n\\item The morphism $f$ is locally of finite presentation if and only\nif $\\mathcal{O}_X$ is of finite presentation relative to $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09T8","source_file":"more-morphisms.tex","source_line":16452,"source_end_line":16463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16452-L16463","statement_sha256":"8e6ec0f0d113f41f54409edc19e7278c0cd00a551968923247cfad8443306f6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7449,"rank":7449,"depth":6,"x":1889.714,"y":706.762,"cluster":"scheme-morphisms"},{"id":"stacks:09T9","tag":"09T9","title":"Relative finite presentation · Lemma 09T9","summary":"Let π : X → Y be a finite morphism of schemes locally of finite type over a base scheme S. Let F be a quasi-coherent O_X-module. Then F is of finite presentation relative to S if and only if π_*F is of finite presentation relative to S.","statement_latex":"Let $\\pi : X \\to Y$ be a finite morphism of schemes locally of finite\ntype over a base scheme $S$. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Then $\\mathcal{F}$ is of finite presentation\nrelative to $S$ if and only if $\\pi_*\\mathcal{F}$ is of finite presentation\nrelative to $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09T9","source_file":"more-morphisms.tex","source_line":16472,"source_end_line":16479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16472-L16479","statement_sha256":"74937bc5594ec4fdd9081cbc84f9979bc16d4edc9cb4037044ce4e5dd8e9d544","origin":"The Stacks Project","memory_eligible":false,"source_rank":7450,"rank":7450,"depth":5,"x":2111.966,"y":580.613,"cluster":"scheme-morphisms"},{"id":"stacks:09TA","tag":"09TA","title":"Relative finite presentation · Lemma 09TA","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent O_X-module. Let S' → S be a morphism of schemes, set X' = X ×_S S' and denote F' the pullback of F to X'. If F is of finite presentation relative to S, then F' is of finite presentation relative to S'.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite\ntype. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $S' \\to S$ be a morphism of schemes, set $X' = X \\times_S S'$\nand denote $\\mathcal{F}'$ the pullback of $\\mathcal{F}$ to $X'$.\nIf $\\mathcal{F}$ is of finite presentation relative to $S$, then\n$\\mathcal{F}'$ is of finite presentation relative to $S'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TA","source_file":"more-morphisms.tex","source_line":16487,"source_end_line":16495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16487-L16495","statement_sha256":"fea58279a47ba2d64425e524a4fd2dd09da2141d3998e3ed30dfc83192d640e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7451,"rank":7451,"depth":1,"x":2049.494,"y":799.858,"cluster":"scheme-morphisms"},{"id":"stacks:09TB","tag":"09TB","title":"Relative finite presentation · Lemma 09TB","summary":"Let X → Y → S be morphisms of schemes which are locally of finite type. Let G be a quasi-coherent O_Y-module. If f : X → Y is locally of finite presentation and G of finite presentation relative to S, then f^*G is of finite presentation relative to S.","statement_latex":"Let $X \\to Y \\to S$ be morphisms of schemes which are locally of finite\ntype. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.\nIf $f : X \\to Y$ is locally of finite presentation and\n$\\mathcal{G}$ of finite presentation relative to $S$, then\n$f^*\\mathcal{G}$ is of finite presentation relative to $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TB","source_file":"more-morphisms.tex","source_line":16504,"source_end_line":16511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16504-L16511","statement_sha256":"1642b76c0ee8a333a814f3e26daf4b3b5527107071eb861e67a8a7cffd12ca05","origin":"The Stacks Project","memory_eligible":false,"source_rank":7452,"rank":7452,"depth":1,"x":1919.182,"y":602.64,"cluster":"scheme-morphisms"},{"id":"stacks:09TC","tag":"09TC","title":"Relative finite presentation · Lemma 09TC","summary":"Let X → Y → S be morphisms of schemes which are locally of finite type. Let F be a quasi-coherent O_X-module. If Y → S is locally of finite presentation and F is of finite presentation relative to Y, then F is of finite presentation relative to S.","statement_latex":"Let $X \\to Y \\to S$ be morphisms of schemes which are locally of finite\ntype. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $Y \\to S$ is locally of finite presentation and $\\mathcal{F}$\nis of finite presentation relative to $Y$, then $\\mathcal{F}$\nis of finite presentation relative to $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TC","source_file":"more-morphisms.tex","source_line":16520,"source_end_line":16527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16520-L16527","statement_sha256":"256029ebe1b60c73a545f4255425beb80e26a955b24442e3192deb2d447c5863","origin":"The Stacks Project","memory_eligible":false,"source_rank":7453,"rank":7453,"depth":1,"x":2174.0,"y":674.16,"cluster":"scheme-morphisms"},{"id":"stacks:09TD","tag":"09TD","title":"Relative finite presentation · Lemma 09TD","summary":"Let X → S be a morphism of schemes which is locally of finite type. Let 0 → F' → F → F\" → 0 be a short exact sequence of quasi-coherent O_X-modules. • If F', F\" are finitely presented relative to S, then so is F. • If F' is a finite type O_X-module and F is finitely presented relative to S, then F\" is finitely presented relative to S.","statement_latex":"Let $X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to \\mathcal{F}'' \\to 0$\nbe a short exact sequence of quasi-coherent $\\mathcal{O}_X$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}', \\mathcal{F}''$ are finitely presented relative to\n$S$, then so is $\\mathcal{F}$.\n\\item If $\\mathcal{F}'$ is a finite type $\\mathcal{O}_X$-module\nand $\\mathcal{F}$ is finitely presented relative to $S$, then\n$\\mathcal{F}''$ is finitely presented relative to $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TD","source_file":"more-morphisms.tex","source_line":16536,"source_end_line":16548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16536-L16548","statement_sha256":"cee099b9c161f65a3dcf3cd33866538780b173452e0e1012ab761d0c00751ffb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7454,"rank":7454,"depth":3,"x":1928.46,"y":766.06,"cluster":"scheme-morphisms"},{"id":"stacks:09TE","tag":"09TE","title":"Relative finite presentation · Lemma 09TE","summary":"A direct summand of a module inherits the property of being finitely presented relative to a base. Let X → S be a morphism of schemes which is locally of finite type. Let F, F' be quasi-coherent O_X-modules. If F ⊕ F' is finitely presented relative to S, then so are F and F'.","statement_latex":"\\begin{slogan}\nA direct summand of a module inherits the property of being finitely\npresented relative to a base.\n\\end{slogan}\nLet $X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}, \\mathcal{F}'$ be quasi-coherent $\\mathcal{O}_X$-modules.\nIf $\\mathcal{F} \\oplus \\mathcal{F}'$ is finitely presented relative to $S$,\nthen so are $\\mathcal{F}$ and $\\mathcal{F}'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TE","source_file":"more-morphisms.tex","source_line":16557,"source_end_line":16567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16557-L16567","statement_sha256":"2b2571244f3ccf2e18457a675a8e6ae17bfc0d9d327ae20deb8447109ae95574","origin":"The Stacks Project","memory_eligible":false,"source_rank":7455,"rank":7455,"depth":1,"x":2035.67,"y":558.863,"cluster":"scheme-morphisms"},{"id":"stacks:09VC","tag":"09VC","title":"Relative pseudo-coherence · Lemma 09VC","summary":"Let X → S be a finite type morphism of affine schemes. Let E be an object of D(O_X). Let m ∈ Z. The following are equivalent • for some closed immersion i : X → A^n_S the object Ri_*E of D(O_A^n_S) is m-pseudo-coherent, and • for all closed immersions i : X → A^n_S the object Ri_*E of D(O_A^n_S) is m-pseudo-coherent.","statement_latex":"Let $X \\to S$ be a finite type morphism of affine schemes.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\nLet $m \\in \\mathbf{Z}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some closed immersion $i : X \\to \\mathbf{A}^n_S$\nthe object $Ri_*E$ of $D(\\mathcal{O}_{\\mathbf{A}^n_S})$\nis $m$-pseudo-coherent, and\n\\item for all closed immersions $i : X \\to \\mathbf{A}^n_S$\nthe object $Ri_*E$ of $D(\\mathcal{O}_{\\mathbf{A}^n_S})$\nis $m$-pseudo-coherent.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VC","source_file":"more-morphisms.tex","source_line":16609,"source_end_line":16623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16609-L16623","statement_sha256":"7f8bf153b97ddb229f818e2aa71cfa9a00813fbe5734a71c74467edc1bbc7825","origin":"The Stacks Project","memory_eligible":false,"source_rank":7456,"rank":7456,"depth":18,"x":2123.282,"y":772.59,"cluster":"scheme-morphisms"},{"id":"stacks:09UI","tag":"09UI","title":"Relative pseudo-coherence · Definition 09UI","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let E be an object of D(O_X). Let F be an O_X-module. Fix m ∈ Z. • We say E is m-pseudo-coherent relative to S if there exists an affine open covering S = ⋃ V_i and for each i an affine open covering f^-1(V_i) = ⋃ U_ij such that the equivalent conditions of Lemma [Tag 09VC] are satisfied for each of the pairs (U_ij → V_i, E|_U_ij). • We say E is pseudo-coherent relative to S if E is m-pseudo-coherent…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $E$ be an object of $D(\\mathcal{O}_X)$. Let $\\mathcal{F}$ be an\n$\\mathcal{O}_X$-module. Fix $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item We say $E$ is {\\it $m$-pseudo-coherent relative to $S$}\nif there exists an affine open covering $S = \\bigcup V_i$ and\nfor each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$\nsuch that the equivalent conditions of\nLemma \\ref{lemma-relatively-pseudo-coherent}\nare satisfied for each of the pairs $(U_{ij} \\to V_i, E|_{U_{ij}})$.\n\\item We say $E$ is {\\it pseudo-coherent relative to $S$}\nif $E$ is $m$-pseudo-coherent relative to $S$ for all $m \\in \\mathbf{Z}$.\n\\item We say $\\mathcal{F}$ is {\\it $m$-pseudo-coherent relative to $S$} if\n$\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is\n$m$-pseudo-coherent relative to $S$.\n\\item We say $\\mathcal{F}$ is {\\it pseudo-coherent relative to $S$} if\n$\\mathcal{F}$ viewed as an object of $D(\\mathcal{O}_X)$ is\npseudo-coherent relative to $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UI","source_file":"more-morphisms.tex","source_line":16674,"source_end_line":16695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16674-L16695","statement_sha256":"fe072f868656812a3ba2230d25082f4d6a3f6c734f2b763392e7079e481ea032","origin":"The Stacks Project","memory_eligible":false,"source_rank":7457,"rank":7457,"depth":19,"x":1886.684,"y":664.648,"cluster":"scheme-morphisms"},{"id":"stacks:0CSU","tag":"0CSU","title":"Relative pseudo-coherence · Lemma 0CSU","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. If E in D(O_X) is m-pseudo-coherent relative to S, then H^i(E) is a quasi-coherent O_X-module for i > m. If E is pseudo-coherent relative to S, then E is an object of D_QCoh(O_X).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nIf $E$ in $D(\\mathcal{O}_X)$ is $m$-pseudo-coherent relative to $S$,\nthen $H^i(E)$ is a quasi-coherent $\\mathcal{O}_X$-module for $i > m$.\nIf $E$ is pseudo-coherent relative to $S$, then $E$ is an object of\n$D_\\QCoh(\\mathcal{O}_X)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSU","source_file":"more-morphisms.tex","source_line":16703,"source_end_line":16710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16703-L16710","statement_sha256":"026ba5a77cbd6ae7bb3c29c26ae1fab4764de74201601a8616fc233fb02620c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7458,"rank":7458,"depth":19,"x":2148.085,"y":609.963,"cluster":"scheme-morphisms"},{"id":"stacks:09VD","tag":"09VD","title":"Relative pseudo-coherence · Lemma 09VD","summary":"Let S be an affine scheme. Let V ⊂ S be a standard open. Let X → V be a finite type morphism of affine schemes. Let U ⊂ X be an affine open. Let E be an object of D(O_X). If the equivalent conditions of Lemma [Tag 09VC] are satisfied for the pair (X → V, E), then the equivalent conditions of Lemma [Tag 09VC] are satisfied for the pair (U → S, E|_U).","statement_latex":"Let $S$ be an affine scheme. Let $V \\subset S$ be a standard open.\nLet $X \\to V$ be a finite type morphism of affine schemes.\nLet $U \\subset X$ be an affine open. Let $E$ be an object of\n$D(\\mathcal{O}_X)$. If the equivalent conditions of\nLemma \\ref{lemma-relatively-pseudo-coherent}\nare satisfied for the pair $(X \\to V, E)$, then\nthe equivalent conditions of\nLemma \\ref{lemma-relatively-pseudo-coherent}\nare satisfied for the pair $(U \\to S, E|_U)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VD","source_file":"more-morphisms.tex","source_line":16737,"source_end_line":16748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16737-L16748","statement_sha256":"b2ac0d943d193cc38b8f7b19d170584eabfcc46fab5c636fa6fb3c00340d1b2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7459,"rank":7459,"depth":19,"x":1999.232,"y":798.709,"cluster":"scheme-morphisms"},{"id":"stacks:09VE","tag":"09VE","title":"Relative pseudo-coherence · Lemma 09VE","summary":"Let X → S be a finite type morphism of affine schemes. Let E be an object of D(O_X). Let m ∈ Z. Let X = ⋃ U_i be a standard affine open covering. The following are equivalent • the equivalent conditions of Lemma [Tag 09VC] hold for the pairs (U_i → S, E|_U_i), • the equivalent conditions of Lemma [Tag 09VC] hold for the pair (X → S, E).","statement_latex":"Let $X \\to S$ be a finite type morphism of affine schemes. Let $E$ be an\nobject of $D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.\nLet $X = \\bigcup U_i$ be a standard affine open covering.\nThe following are equivalent\n\\begin{enumerate}\n\\item the equivalent conditions of\nLemma \\ref{lemma-relatively-pseudo-coherent}\nhold for the pairs $(U_i \\to S, E|_{U_i})$,\n\\item the equivalent conditions of\nLemma \\ref{lemma-relatively-pseudo-coherent}\nhold for the pair $(X \\to S, E)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VE","source_file":"more-morphisms.tex","source_line":16788,"source_end_line":16802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16788-L16802","statement_sha256":"6d23c03177d75cc1b3e36c73ef5af70b7c99ffcef22daf418e2fb2087f65147c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7460,"rank":7460,"depth":20,"x":1957.188,"y":574.953,"cluster":"scheme-morphisms"},{"id":"stacks:09UJ","tag":"09UJ","title":"Relative pseudo-coherence · Lemma 09UJ","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let E be an object of D(O_X). Fix m ∈ Z. The following are equivalent • E is m-pseudo-coherent relative to S, • for every affine opens U ⊂ X and V ⊂ S with f(U) ⊂ V the equivalent conditions of Lemma [Tag 09VC] are satisfied for the pair (U → V, E|_U). Moreover, if this is true, then for every open subschemes U ⊂ X and V ⊂ S with f(U) ⊂ V the restriction E|_U is m-pseudo-coherent relative to V.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\nFix $m \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $E$ is $m$-pseudo-coherent relative to $S$,\n\\item for every affine opens $U \\subset X$ and $V \\subset S$\nwith $f(U) \\subset V$ the equivalent conditions of\nLemma \\ref{lemma-relatively-pseudo-coherent}\nare satisfied for the pair $(U \\to V, E|_U)$.\n\\end{enumerate}\nMoreover, if this is true, then for every open subschemes\n$U \\subset X$ and $V \\subset S$ with $f(U) \\subset V$\nthe restriction $E|_U$ is $m$-pseudo-coherent relative to $V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UJ","source_file":"more-morphisms.tex","source_line":16836,"source_end_line":16851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16836-L16851","statement_sha256":"65cfd6ccfe239af76f699c1a94fe1adecdd935d5adc0bac4e43977a796e56f67","origin":"The Stacks Project","memory_eligible":false,"source_rank":7461,"rank":7461,"depth":21,"x":2168.237,"y":716.163,"cluster":"scheme-morphisms"},{"id":"stacks:09VF","tag":"09VF","title":"Relative pseudo-coherence · Lemma 09VF","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let E be an object of D_QCoh(O_X). Fix m ∈ Z. The following are equivalent • E is m-pseudo-coherent relative to S, • there exists an affine open covering S = ⋃ V_i and for each i an affine open covering f^-1(V_i) = ⋃ U_ij such that the complex of O_X(U_ij)-modules RΓ(U_ij, E) is m-pseudo-coherent relative to O_S(V_i), and • for every affine opens U ⊂ X and V ⊂ S with f(U) ⊂ V the complex of…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$.\nFix $m \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $E$ is $m$-pseudo-coherent relative to $S$,\n\\item there exists an affine open covering $S = \\bigcup V_i$ and\nfor each $i$ an affine open covering $f^{-1}(V_i) = \\bigcup U_{ij}$\nsuch that the complex of $\\mathcal{O}_X(U_{ij})$-modules\n$R\\Gamma(U_{ij}, E)$ is $m$-pseudo-coherent relative to\n$\\mathcal{O}_S(V_i)$, and\n\\item for every affine opens $U \\subset X$ and $V \\subset S$\nwith $f(U) \\subset V$ the complex of $\\mathcal{O}_X(U)$-modules\n$R\\Gamma(U, E)$ is $m$-pseudo-coherent relative to $\\mathcal{O}_S(V)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VF","source_file":"more-morphisms.tex","source_line":16907,"source_end_line":16923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16907-L16923","statement_sha256":"a0e0cd6e39d8e97c9fea67c51127d5dc517a8937de04adceb73f155c1aa6786e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7462,"rank":7462,"depth":22,"x":1898.918,"y":731.799,"cluster":"scheme-morphisms"},{"id":"stacks:09VG","tag":"09VG","title":"Relative pseudo-coherence · Lemma 09VG","summary":"Let i : X → Y morphism of schemes locally of finite type over a base scheme S. Assume that i induces a homeomorphism of X with a closed subset of Y. Let E be an object of D(O_X). Then E is m-pseudo-coherent relative to S if and only if Ri_*E is m-pseudo-coherent relative to S.","statement_latex":"Let $i : X \\to Y$ morphism of schemes locally of finite type over a\nbase scheme $S$. Assume that $i$ induces a homeomorphism of $X$ with a closed\nsubset of $Y$. Let $E$ be an object of $D(\\mathcal{O}_X)$.\nThen $E$ is $m$-pseudo-coherent relative to $S$ if and only if\n$Ri_*E$ is $m$-pseudo-coherent relative to $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VG","source_file":"more-morphisms.tex","source_line":16948,"source_end_line":16955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16948-L16955","statement_sha256":"b7b73408b574146569509ca8d97329bf35b33cd3206d46b6b6cfcbeed9281c71","origin":"The Stacks Project","memory_eligible":false,"source_rank":7463,"rank":7463,"depth":22,"x":2085.03,"y":567.368,"cluster":"scheme-morphisms"},{"id":"stacks:09UK","tag":"09UK","title":"Relative pseudo-coherence · Lemma 09UK","summary":"Let π : X → Y be a finite morphism of schemes locally of finite type over a base scheme S. Let E be an object of D_QCoh(O_X). Then E is m-pseudo-coherent relative to S if and only if Rπ_*E is m-pseudo-coherent relative to S.","statement_latex":"Let $\\pi : X \\to Y$ be a finite morphism of schemes locally of finite\ntype over a base scheme $S$. Let $E$ be an object of\n$D_\\QCoh(\\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent\nrelative to $S$ if and only if $R\\pi_*E$ is $m$-pseudo-coherent\nrelative to $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UK","source_file":"more-morphisms.tex","source_line":16994,"source_end_line":17001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L16994-L17001","statement_sha256":"3e91a028ef48cd3ca8d875c52cfccc8c511662066fca833efda25a92e8741a03","origin":"The Stacks Project","memory_eligible":false,"source_rank":7464,"rank":7464,"depth":30,"x":2080.023,"y":794.337,"cluster":"scheme-morphisms"},{"id":"stacks:09UL","tag":"09UL","title":"Relative pseudo-coherence · Lemma 09UL","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let (E, E', E\") be a distinguished triangle of D(O_X). Let m ∈ Z. • If E is (m + 1)-pseudo-coherent relative to S and E' is m-pseudo-coherent relative to S then E\" is m-pseudo-coherent relative to S. • If E, E\" are m-pseudo-coherent relative to S, then E' is m-pseudo-coherent relative to S. • If E' is (m + 1)-pseudo-coherent relative to S and E\" is m-pseudo-coherent relative to S, then E is (m +…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $(E, E', E'')$ be a distinguished triangle of\n$D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $E$ is $(m + 1)$-pseudo-coherent relative to $S$ and\n$E'$ is $m$-pseudo-coherent relative to $S$ then $E''$ is\n$m$-pseudo-coherent relative to $S$.\n\\item If $E, E''$ are $m$-pseudo-coherent relative to $S$,\nthen $E'$ is $m$-pseudo-coherent relative to $S$.\n\\item If $E'$ is $(m + 1)$-pseudo-coherent relative to $S$\nand $E''$ is $m$-pseudo-coherent relative to $S$, then\n$E$ is $(m + 1)$-pseudo-coherent relative to $S$.\n\\end{enumerate}\nMoreover, if two out of three of $E, E', E''$ are pseudo-coherent\nrelative to $S$, the so is the third.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UL","source_file":"more-morphisms.tex","source_line":17015,"source_end_line":17032,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17015-L17032","statement_sha256":"f3e2c7b841b55ea282edcbf43dcdcf95ea501f5d5879abf6036eb5926807233e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7465,"rank":7465,"depth":22,"x":1901.102,"y":624.045,"cluster":"scheme-morphisms"},{"id":"stacks:09UM","tag":"09UM","title":"Relative pseudo-coherence · Lemma 09UM","summary":"Let X → S be a morphism of schemes which is locally of finite type. Let F be an O_X-module. Then • F is m-pseudo-coherent relative to S for all m > 0, • F is 0-pseudo-coherent relative to S if and only if F is a finite type O_X-module, • F is (-1)-pseudo-coherent relative to S if and only if F is quasi-coherent and finitely presented relative to S.","statement_latex":"Let $X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Then\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is $m$-pseudo-coherent relative to $S$ for all $m > 0$,\n\\item $\\mathcal{F}$ is $0$-pseudo-coherent relative to $S$ if and only if\n$\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module,\n\\item $\\mathcal{F}$ is $(-1)$-pseudo-coherent relative to $S$ if and only if\n$\\mathcal{F}$ is quasi-coherent and finitely presented relative to $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UM","source_file":"more-morphisms.tex","source_line":17039,"source_end_line":17050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17039-L17050","statement_sha256":"0092124e603e096141223dc22c4ebe2b19ad589bb806ae185e3178acdab5e0f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7466,"rank":7466,"depth":17,"x":2170.116,"y":648.104,"cluster":"scheme-morphisms"},{"id":"stacks:09UN","tag":"09UN","title":"Relative pseudo-coherence · Lemma 09UN","summary":"Let X → S be a morphism of schemes which is locally of finite type. Let m ∈ Z. Let E, K be objects of D(O_X). If E ⊕ K is m-pseudo-coherent relative to S so are E and K.","statement_latex":"Let $X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $m \\in \\mathbf{Z}$. Let $E, K$ be objects of $D(\\mathcal{O}_X)$.\nIf $E \\oplus K$ is $m$-pseudo-coherent relative to $S$ so are $E$ and $K$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UN","source_file":"more-morphisms.tex","source_line":17069,"source_end_line":17074,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17069-L17074","statement_sha256":"c77159f08b11b2fc2cb826f54fbec9596f5004c7521cfc843210378c5a3a10a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7467,"rank":7467,"depth":10,"x":1952.291,"y":783.077,"cluster":"scheme-morphisms"},{"id":"stacks:09UP","tag":"09UP","title":"Relative pseudo-coherence · Lemma 09UP","summary":"Let X → S be a morphism of schemes which is locally of finite type. Let m ∈ Z. Let F^bullet be a (locally) bounded above complex of O_X-modules such that F^i is (m - i)-pseudo-coherent relative to S for all i. Then F^bullet is m-pseudo-coherent relative to S.","statement_latex":"Let $X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $m \\in \\mathbf{Z}$. Let $\\mathcal{F}^\\bullet$ be a (locally) bounded\nabove complex of $\\mathcal{O}_X$-modules such that $\\mathcal{F}^i$ is\n$(m - i)$-pseudo-coherent relative to $S$ for all $i$. Then\n$\\mathcal{F}^\\bullet$ is $m$-pseudo-coherent relative to $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UP","source_file":"more-morphisms.tex","source_line":17082,"source_end_line":17089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17082-L17089","statement_sha256":"4826e9715b91f74f2b17a990f7384ada5495e660288de7412b69d71e7af8044c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7468,"rank":7468,"depth":12,"x":2004.397,"y":559.838,"cluster":"scheme-morphisms"},{"id":"stacks:09UQ","tag":"09UQ","title":"Relative pseudo-coherence · Lemma 09UQ","summary":"Let X → S be a morphism of schemes which is locally of finite type. Let m ∈ Z. Let E be an object of D(O_X). If E is (locally) bounded above and H^i(E) is (m - i)-pseudo-coherent relative to S for all i, then E is m-pseudo-coherent relative to S.","statement_latex":"Let $X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $m \\in \\mathbf{Z}$. Let $E$ be an object of $D(\\mathcal{O}_X)$.\nIf $E$ is (locally) bounded above and $H^i(E)$ is $(m - i)$-pseudo-coherent\nrelative to $S$ for all $i$, then $E$ is $m$-pseudo-coherent relative to $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UQ","source_file":"more-morphisms.tex","source_line":17097,"source_end_line":17103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17097-L17103","statement_sha256":"feb0d1622c411f4bbe9922d3c6ed04559a92ad627a03db0dc5bcd4fafe56803d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7469,"rank":7469,"depth":13,"x":2145.569,"y":754.115,"cluster":"scheme-morphisms"},{"id":"stacks:09UR","tag":"09UR","title":"Relative pseudo-coherence · Lemma 09UR","summary":"Let X → S be a morphism of schemes which is locally of finite type. Let m ∈ Z. Let E be an object of D(O_X) which is m-pseudo-coherent relative to S. Let S' → S be a morphism of schemes. Set X' = X ×_S S' and denote E' the derived pullback of E to X'. If S' and X are Tor independent over S, then E' is m-pseudo-coherent relative to S'.","statement_latex":"Let $X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $m \\in \\mathbf{Z}$. Let $E$ be an object of $D(\\mathcal{O}_X)$\nwhich is $m$-pseudo-coherent relative to $S$. Let $S' \\to S$ be a\nmorphism of schemes. Set $X' = X \\times_S S'$ and denote $E'$\nthe derived pullback of $E$ to $X'$. If $S'$ and $X$ are\nTor independent over $S$, then $E'$\nis $m$-pseudo-coherent relative to $S'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UR","source_file":"more-morphisms.tex","source_line":17111,"source_end_line":17120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17111-L17120","statement_sha256":"27fb514e90e3a82e007f309ae8c0e3f82ad8388fe13c79dbd0a948e3ae0e4249","origin":"The Stacks Project","memory_eligible":false,"source_rank":7470,"rank":7470,"depth":11,"x":1885.107,"y":690.931,"cluster":"scheme-morphisms"},{"id":"stacks:09US","tag":"09US","title":"Relative pseudo-coherence · Lemma 09US","summary":"Let f : X → Y be a morphism of schemes locally of finite type over a base S. Let m ∈ Z. Let E be an object of D(O_Y). Assume • O_X is pseudo-coherent relative to Y, and • E is m-pseudo-coherent relative to S. Then Lf^*E is m-pseudo-coherent relative to S.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes locally of finite type\nover a base $S$. Let $m \\in \\mathbf{Z}$. Let $E$ be an object of\n$D(\\mathcal{O}_Y)$. Assume\n\\begin{enumerate}\n\\item $\\mathcal{O}_X$ is pseudo-coherent relative to $Y$\\footnote{This\nmeans $f$ is pseudo-coherent, see\nDefinition \\ref{definition-pseudo-coherent}.}, and\n\\item $E$ is $m$-pseudo-coherent relative to $S$.\n\\end{enumerate}\nThen $Lf^*E$ is $m$-pseudo-coherent relative to $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09US","source_file":"more-morphisms.tex","source_line":17154,"source_end_line":17166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17154-L17166","statement_sha256":"bc299eb33b757e3b2debb84c93b4482704dc206f529b9e06860731178e17b072","origin":"The Stacks Project","memory_eligible":false,"source_rank":7471,"rank":7471,"depth":23,"x":2128.101,"y":589.678,"cluster":"scheme-morphisms"},{"id":"stacks:09UT","tag":"09UT","title":"Relative pseudo-coherence · Lemma 09UT","summary":"Let f : X → Y be a morphism of schemes locally of finite type over a base S. Let m ∈ Z. Let E be an object of D(O_X). Assume O_Y is pseudo-coherent relative to S. Then the following are equivalent • E is m-pseudo-coherent relative to Y, and • E is m-pseudo-coherent relative to S.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes locally of finite type\nover a base $S$. Let $m \\in \\mathbf{Z}$. Let $E$ be an object of\n$D(\\mathcal{O}_X)$. Assume $\\mathcal{O}_Y$ is pseudo-coherent relative\nto $S$\\footnote{This means $Y \\to S$ is pseudo-coherent, see\nDefinition \\ref{definition-pseudo-coherent}.}.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $E$ is $m$-pseudo-coherent relative to $Y$, and\n\\item $E$ is $m$-pseudo-coherent relative to $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UT","source_file":"more-morphisms.tex","source_line":17234,"source_end_line":17246,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17234-L17246","statement_sha256":"88608b151bbf519a935dfd5d0c0499f40b3ca1e72479d609dd6f9d313c17ae17","origin":"The Stacks Project","memory_eligible":false,"source_rank":7472,"rank":7472,"depth":13,"x":2030.296,"y":802.328,"cluster":"scheme-morphisms"},{"id":"stacks:09UU","tag":"09UU","title":"Relative pseudo-coherence · Lemma 09UU","summary":"Let xymatrix X ar[rd] ar[rr]_i & & P ar[ld] & S be a commutative diagram of schemes. Assume i is a closed immersion and P → S flat and locally of finite presentation. Let E be an object of D(O_X). Then the following are equivalent • E is m-pseudo-coherent relative to S, • Ri_*E is m-pseudo-coherent relative to S, and • Ri_*E is m-pseudo-coherent on P.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rd] \\ar[rr]_i & & P \\ar[ld] \\\\\n& S\n}\n$$\nbe a commutative diagram of schemes. Assume $i$ is a closed immersion\nand $P \\to S$ flat and locally of finite presentation. Let $E$\nbe an object of $D(\\mathcal{O}_X)$. Then the following\nare equivalent\n\\begin{enumerate}\n\\item $E$ is $m$-pseudo-coherent relative to $S$,\n\\item $Ri_*E$ is $m$-pseudo-coherent relative to $S$, and\n\\item $Ri_*E$ is $m$-pseudo-coherent on $P$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09UU","source_file":"more-morphisms.tex","source_line":17274,"source_end_line":17292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17274-L17292","statement_sha256":"e7abb53c6697f0edcefa9ea12b4e49d56e19a282dfed3f681713df34a8a1aed9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7473,"rank":7473,"depth":36,"x":1931.359,"y":589.919,"cluster":"scheme-morphisms"},{"id":"stacks:067Y","tag":"067Y","title":"Pseudo-coherent morphisms · Lemma 067Y","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • there exist an affine open covering S = ⋃ V_j and for each j an affine open covering f^-1(V_j) = ⋃ U_ji such that O_S(V_j) → O_X(U_ij) is a pseudo-coherent ring map, • for every pair of affine opens U ⊂ X, V ⊂ S such that f(U) ⊂ V the ring map O_S(V) → O_X(U) is pseudo-coherent, and • f is locally of finite type and O_X is pseudo-coherent relative to S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item there exist an affine open covering $S = \\bigcup V_j$ and for each $j$\nan affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$ such that\n$\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_{ij})$ is a pseudo-coherent\nring map,\n\\item for every pair of affine opens $U \\subset X$, $V \\subset S$\nsuch that $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is pseudo-coherent, and\n\\item $f$ is locally of finite type and $\\mathcal{O}_X$\nis pseudo-coherent relative to $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067Y","source_file":"more-morphisms.tex","source_line":17331,"source_end_line":17345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17331-L17345","statement_sha256":"478de72ca4b2f17cdc35c5950a2fad5d7c19a187c32088a8200603dea693b0f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7474,"rank":7474,"depth":23,"x":2175.25,"y":690.458,"cluster":"scheme-morphisms"},{"id":"stacks:067Z","tag":"067Z","title":"Pseudo-coherent morphisms · Definition 067Z","summary":"A morphism of schemes f : X → S is called pseudo-coherent if the equivalent conditions of Lemma [Tag 067Y] are satisfied. In this case we also say that X is pseudo-coherent over S.","statement_latex":"A morphism of schemes $f : X \\to S$ is called {\\it pseudo-coherent}\nif the equivalent conditions of\nLemma \\ref{lemma-pseudo-coherent}\nare satisfied. In this case we also say that $X$ is pseudo-coherent\nover $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/067Z","source_file":"more-morphisms.tex","source_line":17374,"source_end_line":17381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17374-L17381","statement_sha256":"171ede8e85850591fd270d842897d467711c37d6584a2c52d874088048204c6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7475,"rank":7475,"depth":24,"x":1914.427,"y":754.744,"cluster":"scheme-morphisms"},{"id":"stacks:0680","tag":"0680","title":"Pseudo-coherent morphisms · Lemma 0680","summary":"A flat base change of a pseudo-coherent morphism is pseudo-coherent.","statement_latex":"A flat base change of a pseudo-coherent morphism is pseudo-coherent.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0680","source_file":"more-morphisms.tex","source_line":17387,"source_end_line":17390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17387-L17390","statement_sha256":"e2655591472dc15ff273008f78e93acbd65b35010e7c0cb0e0a423f7b4a156b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7476,"rank":7476,"depth":12,"x":2055.127,"y":559.245,"cluster":"scheme-morphisms"},{"id":"stacks:0681","tag":"0681","title":"Pseudo-coherent morphisms · Lemma 0681","summary":"A composition of pseudo-coherent morphisms of schemes is pseudo-coherent.","statement_latex":"A composition of pseudo-coherent morphisms of schemes is\npseudo-coherent.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0681","source_file":"more-morphisms.tex","source_line":17401,"source_end_line":17405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17401-L17405","statement_sha256":"70433d668f36624fb69be4dfce023803ad6945224e884cc1cb07b9655f553b7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7477,"rank":7477,"depth":13,"x":2108.619,"y":783.352,"cluster":"scheme-morphisms"},{"id":"stacks:0682","tag":"0682","title":"Pseudo-coherent morphisms · Lemma 0682","summary":"A pseudo-coherent morphism is locally of finite presentation.","statement_latex":"A pseudo-coherent morphism is locally of finite presentation.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0682","source_file":"more-morphisms.tex","source_line":17418,"source_end_line":17421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17418-L17421","statement_sha256":"6c1b6570e88dc9c4a15994774025e848d5c7aaf43b168827173378c7cbd75ee1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7478,"rank":7478,"depth":0,"x":1888.844,"y":648.384,"cluster":"scheme-morphisms"},{"id":"stacks:0695","tag":"0695","title":"Pseudo-coherent morphisms · Lemma 0695","summary":"A flat morphism which is locally of finite presentation is pseudo-coherent.","statement_latex":"A flat morphism which is locally of finite presentation is pseudo-coherent.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0695","source_file":"more-morphisms.tex","source_line":17427,"source_end_line":17430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17427-L17430","statement_sha256":"9ea99dfec6791cdb392e352a671a3d2cf267d7c770dce2e5a653813fff50ad19","origin":"The Stacks Project","memory_eligible":false,"source_rank":7479,"rank":7479,"depth":36,"x":2159.576,"y":623.19,"cluster":"scheme-morphisms"},{"id":"stacks:0683","tag":"0683","title":"Pseudo-coherent morphisms · Lemma 0683","summary":"Let f : X → Y be a morphism of schemes pseudo-coherent over a base scheme S. Then f is pseudo-coherent.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes pseudo-coherent\nover a base scheme $S$. Then $f$ is pseudo-coherent.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0683","source_file":"more-morphisms.tex","source_line":17439,"source_end_line":17443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17439-L17443","statement_sha256":"06ac1cf772dad90c4ef386c15d6f45a5eda42cec229f9e6526f82a522b68e247","origin":"The Stacks Project","memory_eligible":false,"source_rank":7480,"rank":7480,"depth":13,"x":1980.113,"y":795.473,"cluster":"scheme-morphisms"},{"id":"stacks:0AVX","tag":"0AVX","title":"Pseudo-coherent morphisms · Lemma 0AVX","summary":"Let f : X → S be a finite morphism of schemes. Then f is pseudo-coherent if and only if f_*O_X is pseudo-coherent as an O_S-module.","statement_latex":"Let $f : X \\to S$ be a finite morphism of schemes.\nThen $f$ is pseudo-coherent if and only if $f_*\\mathcal{O}_X$\nis pseudo-coherent as an $\\mathcal{O}_S$-module.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVX","source_file":"more-morphisms.tex","source_line":17454,"source_end_line":17459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17454-L17459","statement_sha256":"46b3b1673022cd66c7cc1a38c76c6bbea224ebc186c737a2f5eef7bd5f1b44e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7481,"rank":7481,"depth":13,"x":1973.897,"y":566.488,"cluster":"scheme-morphisms"},{"id":"stacks:0684","tag":"0684","title":"Pseudo-coherent morphisms · Lemma 0684","summary":"Let f : X → S be a morphism of schemes. If S is locally Noetherian, then f is pseudo-coherent if and only if f is locally of finite type.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nIf $S$ is locally Noetherian, then $f$ is pseudo-coherent if\nand only if $f$ is locally of finite type.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0684","source_file":"more-morphisms.tex","source_line":17472,"source_end_line":17477,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17472-L17477","statement_sha256":"39aab635f405e5dda55caeaa6767b5c9525e9ef43602d0dda1d91ee32124e3e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7482,"rank":7482,"depth":13,"x":2162.719,"y":731.894,"cluster":"scheme-morphisms"},{"id":"stacks:0696","tag":"0696","title":"Pseudo-coherent morphisms · Lemma 0696","summary":"The property P(f) =\"f is pseudo-coherent\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is pseudo-coherent''\nis fpqc local on the base.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0696","source_file":"more-morphisms.tex","source_line":17492,"source_end_line":17496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17492-L17496","statement_sha256":"ee38401648557ea58156fd6a3720d7115347646cf821796f1a31def26f697313","origin":"The Stacks Project","memory_eligible":false,"source_rank":7483,"rank":7483,"depth":37,"x":1890.334,"y":717.06,"cluster":"scheme-morphisms"},{"id":"stacks:0697","tag":"0697","title":"Pseudo-coherent morphisms · Lemma 0697","summary":"Let A → B be a flat ring map of finite presentation. Let I ⊂ B be an ideal. Then A → B/I is pseudo-coherent if and only if I is pseudo-coherent as a B-module.","statement_latex":"Let $A \\to B$ be a flat ring map of finite presentation.\nLet $I \\subset B$ be an ideal. Then $A \\to B/I$ is pseudo-coherent\nif and only if $I$ is pseudo-coherent as a $B$-module.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0697","source_file":"more-morphisms.tex","source_line":17517,"source_end_line":17522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17517-L17522","statement_sha256":"d7d430a6f950000b7ed0d9be071acbb7faec8cc5932edd92b596f803a843d940","origin":"The Stacks Project","memory_eligible":false,"source_rank":7484,"rank":7484,"depth":37,"x":2103.221,"y":573.369,"cluster":"scheme-morphisms"},{"id":"stacks:0698","tag":"0698","title":"Pseudo-coherent morphisms · Lemma 0698","summary":"The property P(f) =\"f is pseudo-coherent\" is syntomic local on the source.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is pseudo-coherent''\nis syntomic local on the source.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0698","source_file":"more-morphisms.tex","source_line":17544,"source_end_line":17548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17544-L17548","statement_sha256":"b2cb87560ed6e93bf649f9e2a73379d5186dbf8e97842919c81b5e9640a6ea82","origin":"The Stacks Project","memory_eligible":false,"source_rank":7485,"rank":7485,"depth":38,"x":2061.773,"y":800.235,"cluster":"scheme-morphisms"},{"id":"stacks:0699","tag":"0699","title":"Pseudo-coherent morphisms · Lemma 0699","summary":"The property P(f) =\"f is pseudo-coherent\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is pseudo-coherent''\nis fppf local on the source.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0699","source_file":"more-morphisms.tex","source_line":17620,"source_end_line":17624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17620-L17624","statement_sha256":"22d8a8c084ebe3f586d5847fe64b9a59f3cebc1deaa79c08ac4ec1fc27b58cef","origin":"The Stacks Project","memory_eligible":false,"source_rank":7486,"rank":7486,"depth":42,"x":1909.822,"y":609.334,"cluster":"scheme-morphisms"},{"id":"stacks:0686","tag":"0686","title":"Perfect morphisms · Lemma 0686","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. The following are equivalent • there exist an affine open covering S = ⋃ V_j and for each j an affine open covering f^-1(V_j) = ⋃ U_ji such that O_S(V_j) → O_X(U_ij) is a perfect ring map, and • for every pair of affine opens U ⊂ X, V ⊂ S such that f(U) ⊂ V the ring map O_S(V) → O_X(U) is perfect.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exist an affine open covering $S = \\bigcup V_j$ and for each $j$\nan affine open covering $f^{-1}(V_j) = \\bigcup U_{ji}$ such that\n$\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_{ij})$ is a perfect\nring map, and\n\\item for every pair of affine opens $U \\subset X$, $V \\subset S$\nsuch that $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is perfect.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0686","source_file":"more-morphisms.tex","source_line":17697,"source_end_line":17710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17697-L17710","statement_sha256":"038afc8c4efd19d6102c5817928a1a435d287cc3091a013597f04c3e4f4429ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":7487,"rank":7487,"depth":24,"x":2175.517,"y":663.908,"cluster":"scheme-morphisms"},{"id":"stacks:0687","tag":"0687","title":"Perfect morphisms · Definition 0687","summary":"A morphism of schemes f : X → S is called perfect if the equivalent conditions of Lemma [Tag 0686] are satisfied. In this case we also say that X is perfect over S.","statement_latex":"A morphism of schemes $f : X \\to S$ is called {\\it perfect}\nif the equivalent conditions of\nLemma \\ref{lemma-perfect}\nare satisfied. In this case we also say that $X$ is perfect\nover $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0687","source_file":"more-morphisms.tex","source_line":17729,"source_end_line":17736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17729-L17736","statement_sha256":"071428b6cc6bae6feed0ba06caf3a59a7492fb587129b0276d96121ccb511295","origin":"The Stacks Project","memory_eligible":false,"source_rank":7488,"rank":7488,"depth":25,"x":1935.592,"y":774.483,"cluster":"scheme-morphisms"},{"id":"stacks:0688","tag":"0688","title":"Perfect morphisms · Lemma 0688","summary":"A flat base change of a perfect morphism is perfect.","statement_latex":"A flat base change of a perfect morphism is perfect.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0688","source_file":"more-morphisms.tex","source_line":17743,"source_end_line":17746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17743-L17746","statement_sha256":"5c36f0a840c7e616f858c0e90e07205247f5659a225749218abbe4b55fafaa23","origin":"The Stacks Project","memory_eligible":false,"source_rank":7489,"rank":7489,"depth":13,"x":2023.631,"y":556.699,"cluster":"scheme-morphisms"},{"id":"stacks:0689","tag":"0689","title":"Perfect morphisms · Lemma 0689","summary":"A composition of perfect morphisms of schemes is perfect.","statement_latex":"A composition of perfect morphisms of schemes is perfect.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0689","source_file":"more-morphisms.tex","source_line":17759,"source_end_line":17762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17759-L17762","statement_sha256":"3677e66bdd01e9ddb961346003b663cf2b2446de11d5b30715de3e2e69d8c124","origin":"The Stacks Project","memory_eligible":false,"source_rank":7490,"rank":7490,"depth":14,"x":2133.903,"y":767.349,"cluster":"scheme-morphisms"},{"id":"stacks:068A","tag":"068A","title":"Perfect morphisms · Lemma 068A","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • f is flat and perfect, and • f is flat and locally of finite presentation.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is flat and perfect, and\n\\item $f$ is flat and locally of finite presentation.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068A","source_file":"more-morphisms.tex","source_line":17782,"source_end_line":17790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17782-L17790","statement_sha256":"9fb6d8b99ea5bc348bce9e760bfaa40e136a963f67606115589e6d6188434f8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7491,"rank":7491,"depth":36,"x":1883.067,"y":674.544,"cluster":"scheme-morphisms"},{"id":"stacks:068B","tag":"068B","title":"Perfect morphisms · Lemma 068B","summary":"Let f : X → S be a morphism of schemes. Assume S is regular and f is locally of finite type. Then f is perfect.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $S$ is regular and $f$ is locally of finite type.\nThen $f$ is perfect.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068B","source_file":"more-morphisms.tex","source_line":17801,"source_end_line":17806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17801-L17806","statement_sha256":"b96adc7a8cadb0d64b56a9b62915e0082e8bfe00fdcd3373aed3cbe9f16d6341","origin":"The Stacks Project","memory_eligible":false,"source_rank":7492,"rank":7492,"depth":19,"x":2142.789,"y":600.611,"cluster":"scheme-morphisms"},{"id":"stacks:068C","tag":"068C","title":"Perfect morphisms · Lemma 068C","summary":"A regular immersion of schemes is perfect. A Koszul-regular immersion of schemes is perfect.","statement_latex":"A regular immersion of schemes is perfect.\nA Koszul-regular immersion of schemes is perfect.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068C","source_file":"more-morphisms.tex","source_line":17813,"source_end_line":17817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17813-L17817","statement_sha256":"d4ec650ecd5e8fbb0c3669ad8705a952ea80a131616c1457bf1365c81aadbac8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7493,"rank":7493,"depth":8,"x":2010.664,"y":802.6,"cluster":"scheme-morphisms"},{"id":"stacks:068D","tag":"068D","title":"Perfect morphisms · Lemma 068D","summary":"Let xymatrix X ar[rr]_f ar[rd] & & Y ar[ld] & S be a commutative diagram of morphisms of schemes. Assume Y → S smooth and X → S perfect. Then f : X → Y is perfect.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume $Y \\to S$\nsmooth and $X \\to S$ perfect. Then $f : X \\to Y$ is perfect.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/068D","source_file":"more-morphisms.tex","source_line":17833,"source_end_line":17844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17833-L17844","statement_sha256":"3176234cf88dc89b3265a33513c24cf2f2b502e15f03d07bca73181731a2e592","origin":"The Stacks Project","memory_eligible":false,"source_rank":7494,"rank":7494,"depth":39,"x":1945.625,"y":578.575,"cluster":"scheme-morphisms"},{"id":"stacks:069B","tag":"069B","title":"Perfect morphisms · Lemma 069B","summary":"The property P(f) =\"f is perfect\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is perfect''\nis fpqc local on the base.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069B","source_file":"more-morphisms.tex","source_line":17875,"source_end_line":17879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17875-L17879","statement_sha256":"f6440ecb4906da1a9f57377484dafc7e2b5c80864cd66c54b42adebf25956bc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7495,"rank":7495,"depth":37,"x":2173.851,"y":706.926,"cluster":"scheme-morphisms"},{"id":"stacks:069C","tag":"069C","title":"Perfect morphisms · Lemma 069C","summary":"Let f : X → S be a pseudo-coherent morphism of schemes. The following are equivalent • f is perfect, • O_X locally has finite tor dimension as a sheaf of f^-1O_S-modules, and • for all x ∈ X the ring O_X, x has finite tor dimension as an O_S, f(x)-module.","statement_latex":"Let $f : X \\to S$ be a pseudo-coherent morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is perfect,\n\\item $\\mathcal{O}_X$ locally has finite tor dimension as a\nsheaf of $f^{-1}\\mathcal{O}_S$-modules, and\n\\item for all $x \\in X$ the ring $\\mathcal{O}_{X, x}$ has finite tor\ndimension as an $\\mathcal{O}_{S, f(x)}$-module.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069C","source_file":"more-morphisms.tex","source_line":17900,"source_end_line":17911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17900-L17911","statement_sha256":"74b0fbac9adf72cdd5d77989984428da7f3f768a05b6d8311b38ea9c27a8d54f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7496,"rank":7496,"depth":19,"x":1902.211,"y":741.8,"cluster":"scheme-morphisms"},{"id":"stacks:0G2E","tag":"0G2E","title":"Perfect morphisms · Lemma 0G2E","summary":"Let i : Z → X be a perfect closed immersion of schemes. Then i_*O_Z is a perfect O_X-module, i.e., it is a perfect object of D(O_X).","statement_latex":"Let $i : Z \\to X$ be a perfect closed immersion of schemes.\nThen $i_*\\mathcal{O}_Z$ is a perfect $\\mathcal{O}_X$-module, i.e.,\nit is a perfect object of $D(\\mathcal{O}_X)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2E","source_file":"more-morphisms.tex","source_line":17953,"source_end_line":17958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17953-L17958","statement_sha256":"fb97a39a3b7f410f06c62b2ded922e2757d3ecadcab6d85262c19a266965c54d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7497,"rank":7497,"depth":17,"x":2074.554,"y":561.861,"cluster":"scheme-morphisms"},{"id":"stacks:0B6G","tag":"0B6G","title":"Perfect morphisms · Lemma 0B6G","summary":"Let S be a Noetherian scheme. Let f : X → S be a perfect proper morphism of schemes. Let E ∈ D(O_X) be perfect. Then Rf_*E is a perfect object of D(O_S).","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a perfect proper\nmorphism of schemes. Let $E \\in D(\\mathcal{O}_X)$ be perfect. Then\n$Rf_*E$ is a perfect object of $D(\\mathcal{O}_S)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6G","source_file":"more-morphisms.tex","source_line":17972,"source_end_line":17977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17972-L17977","statement_sha256":"a817b77ff2a692c46e20361400dc459ea024b36ec8565c1c3ce47530aaa2c618","origin":"The Stacks Project","memory_eligible":false,"source_rank":7498,"rank":7498,"depth":34,"x":2092.181,"y":792.45,"cluster":"scheme-morphisms"},{"id":"stacks:069D","tag":"069D","title":"Perfect morphisms · Lemma 069D","summary":"The property P(f) =\"f is perfect\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is perfect''\nis fppf local on the source.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069D","source_file":"more-morphisms.tex","source_line":17991,"source_end_line":17995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L17991-L17995","statement_sha256":"2f2dae0631cfc51e682586678f543ad527cdc300fb7f0d14b1c0f94693f8b56f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7499,"rank":7499,"depth":43,"x":1893.653,"y":632.341,"cluster":"scheme-morphisms"},{"id":"stacks:09RK","tag":"09RK","title":"Perfect morphisms · Lemma 09RK","summary":"Let i : Z → Y and j : Y → X be immersions of schemes. Assume • X is locally Noetherian, • j ∘ i is a regular immersion, and • i is perfect. Then i and j are regular immersions.","statement_latex":"Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of schemes.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally Noetherian,\n\\item $j \\circ i$ is a regular immersion, and\n\\item $i$ is perfect.\n\\end{enumerate}\nThen $i$ and $j$ are regular immersions.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RK","source_file":"more-morphisms.tex","source_line":18015,"source_end_line":18025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18015-L18025","statement_sha256":"93a651202e0f6ddf75fb1d73353b2cb0ebc983ff76164c14a94a8d5b5a356aa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7500,"rank":7500,"depth":15,"x":2168.935,"y":637.756,"cluster":"scheme-morphisms"},{"id":"stacks:069E","tag":"069E","title":"Local complete intersection morphisms · Lemma 069E","summary":"Let S be a scheme. Let U, P, P' be schemes over S. Let u ∈ U. Let i : U → P, i' : U → P' be immersions over S. Assume P and P' smooth over S. Then the following are equivalent • i is a Koszul-regular immersion in a neighbourhood of u, and • i' is a Koszul-regular immersion in a neighbourhood of u.","statement_latex":"Let $S$ be a scheme. Let $U$, $P$, $P'$ be schemes over $S$.\nLet $u \\in U$. Let $i : U \\to P$, $i' : U \\to P'$ be immersions over $S$.\nAssume $P$ and $P'$ smooth over $S$. Then the following are equivalent\n\\begin{enumerate}\n\\item $i$ is a Koszul-regular immersion in a neighbourhood of $u$, and\n\\item $i'$ is a Koszul-regular immersion in a neighbourhood of $u$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069E","source_file":"more-morphisms.tex","source_line":18072,"source_end_line":18081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18072-L18081","statement_sha256":"6d3f97d315f54975bd89fb9f5bd315fb363c9ea863d8b3c5b3f5e087d54dee61","origin":"The Stacks Project","memory_eligible":false,"source_rank":7501,"rank":7501,"depth":41,"x":1961.49,"y":790.039,"cluster":"scheme-morphisms"},{"id":"stacks:069F","tag":"069F","title":"Local complete intersection morphisms · Definition 069F","summary":"Let f : X → S be a morphism of schemes. • Let x ∈ X. We say that f is Koszul at x if f is of finite type at x and there exists an open neighbourhood and a factorization of f|_U as π ∘ i where i : U → P is a Koszul-regular immersion and π : P → S is smooth. • We say f is a Koszul morphism, or that f is a local complete intersection morphism if f is Koszul at every point.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item Let $x \\in X$. We say that $f$ is {\\it Koszul at $x$} if $f$\nis of finite type at $x$ and there exists an open neighbourhood\nand a factorization of $f|_U$ as $\\pi \\circ i$ where $i : U \\to P$\nis a Koszul-regular immersion and $\\pi : P \\to S$ is smooth.\n\\item We say $f$ is a {\\it Koszul morphism}, or that\n$f$ is a {\\it local complete intersection morphism}\nif $f$ is Koszul at every point.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069F","source_file":"more-morphisms.tex","source_line":18110,"source_end_line":18122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18110-L18122","statement_sha256":"c35ab40b0cf568de26d3de47c9b7562059e371b9b762c98c683c49e8f14a8a74","origin":"The Stacks Project","memory_eligible":false,"source_rank":7502,"rank":7502,"depth":0,"x":1992.009,"y":559.926,"cluster":"scheme-morphisms"},{"id":"stacks:069G","tag":"069G","title":"Local complete intersection morphisms · Lemma 069G","summary":"Let f : X → S be a local complete intersection morphism. Let P be a scheme smooth over S. Let U ⊂ X be an open subscheme and i : U → P an immersion of schemes over S. Then i is a Koszul-regular immersion.","statement_latex":"Let $f : X \\to S$ be a local complete intersection morphism.\nLet $P$ be a scheme smooth over $S$. Let $U \\subset X$ be an open subscheme\nand $i : U \\to P$ an immersion of schemes over $S$.\nThen $i$ is a Koszul-regular immersion.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069G","source_file":"more-morphisms.tex","source_line":18132,"source_end_line":18138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18132-L18138","statement_sha256":"a04d1a57cf459ee7c2068d9ab81e800e5b7756f63e8f26549133bb58d19a771a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7503,"rank":7503,"depth":0,"x":2154.635,"y":747.017,"cluster":"scheme-morphisms"},{"id":"stacks:069H","tag":"069H","title":"Local complete intersection morphisms · Lemma 069H","summary":"Let f : X → S be a local complete intersection morphism. Then • f is locally of finite presentation, • f is pseudo-coherent, and • f is perfect.","statement_latex":"Let $f : X \\to S$ be a local complete intersection morphism.\nThen\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $f$ is pseudo-coherent, and\n\\item $f$ is perfect.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069H","source_file":"more-morphisms.tex","source_line":18149,"source_end_line":18158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18149-L18158","statement_sha256":"f6824b594de01ffeafef0d6a4f0162edf8bd326fec705cfedf05fa64b25bb157","origin":"The Stacks Project","memory_eligible":false,"source_rank":7504,"rank":7504,"depth":37,"x":1884.132,"y":701.314,"cluster":"scheme-morphisms"},{"id":"stacks:07DB","tag":"07DB","title":"Local complete intersection morphisms · Lemma 07DB","summary":"Let f : X = Spec(B) → S = Spec(A) be a morphism of affine schemes. Then f is a local complete intersection morphism if and only if A → B is a local complete intersection homomorphism, see More on Algebra, Definition [Tag 07D0].","statement_latex":"Let $f : X = \\Spec(B) \\to S = \\Spec(A)$ be a morphism of affine schemes.\nThen $f$ is a local complete intersection morphism if and only if\n$A \\to B$ is a local complete intersection homomorphism, see\nMore on Algebra, Definition\n\\ref{more-algebra-definition-local-complete-intersection}.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DB","source_file":"more-morphisms.tex","source_line":18177,"source_end_line":18184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18177-L18184","statement_sha256":"dd17eb37e77e7609f6f20c065f3a0fe2030da8099bb744c23ee9233d1877e3a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7505,"rank":7505,"depth":1,"x":2120.464,"y":581.467,"cluster":"scheme-morphisms"},{"id":"stacks:069I","tag":"069I","title":"Local complete intersection morphisms · Lemma 069I","summary":"A flat base change of a local complete intersection morphism is a local complete intersection morphism.","statement_latex":"A flat base change of a local complete intersection morphism is a\nlocal complete intersection morphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069I","source_file":"more-morphisms.tex","source_line":18194,"source_end_line":18198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18194-L18198","statement_sha256":"4939fc818834b6cbedb15f33ca90b78fd08862c8bca0b3a1e103a45fc1ba6036","origin":"The Stacks Project","memory_eligible":false,"source_rank":7506,"rank":7506,"depth":16,"x":2042.537,"y":804.049,"cluster":"scheme-morphisms"},{"id":"stacks:069J","tag":"069J","title":"Local complete intersection morphisms · Lemma 069J","summary":"A composition of local complete intersection morphisms is a local complete intersection morphism.","statement_latex":"A composition of local complete intersection morphisms\nis a local complete intersection morphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069J","source_file":"more-morphisms.tex","source_line":18207,"source_end_line":18211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18207-L18211","statement_sha256":"9e8fa45acd62f2d5816140be8a0dbc8998302a97c90f5a082cf38c7b7312ad39","origin":"The Stacks Project","memory_eligible":false,"source_rank":7507,"rank":7507,"depth":38,"x":1920.946,"y":595.601,"cluster":"scheme-morphisms"},{"id":"stacks:069K","tag":"069K","title":"Local complete intersection morphisms · Lemma 069K","summary":"A morphism is flat and lci if and only if it is syntomic. Let f : X → S be a morphism of schemes. The following are equivalent • f is flat and a local complete intersection morphism, and • f is syntomic.","statement_latex":"\\begin{slogan}\nA morphism is flat and lci if and only if it is syntomic.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is flat and a local complete intersection morphism, and\n\\item $f$ is syntomic.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069K","source_file":"more-morphisms.tex","source_line":18244,"source_end_line":18255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18244-L18255","statement_sha256":"059314c7ecb6fef068bf8c396d7c9ec4709038a1ae57a68a5f76e95412e4e8ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":7508,"rank":7508,"depth":38,"x":2178.358,"y":680.355,"cluster":"scheme-morphisms"},{"id":"stacks:069L","tag":"069L","title":"Local complete intersection morphisms · Lemma 069L","summary":"A regular immersion of schemes is a local complete intersection morphism. A Koszul-regular immersion of schemes is a local complete intersection morphism.","statement_latex":"A regular immersion of schemes is a local complete intersection morphism.\nA Koszul-regular immersion of schemes is a local complete intersection\nmorphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069L","source_file":"more-morphisms.tex","source_line":18305,"source_end_line":18310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18305-L18310","statement_sha256":"a33b78ebab7e38f39dad40de1ddd270defc3bfc9b8aaa7d43ce3dcf09a2678b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7509,"rank":7509,"depth":8,"x":1920.264,"y":763.961,"cluster":"scheme-morphisms"},{"id":"stacks:069M","tag":"069M","title":"Local complete intersection morphisms · Lemma 069M","summary":"Let xymatrix X ar[rr]_f ar[rd] & & Y ar[ld] & S be a commutative diagram of morphisms of schemes. Assume Y → S smooth and X → S is a local complete intersection morphism. Then f : X → Y is a local complete intersection morphism.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume $Y \\to S$\nsmooth and $X \\to S$ is a local complete intersection morphism.\nThen $f : X \\to Y$ is a local complete intersection morphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069M","source_file":"more-morphisms.tex","source_line":18319,"source_end_line":18331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18319-L18331","statement_sha256":"dfe3d4a345f00fabbd623a1874308443dc2a15a577953ce9b4a327eb988b3de2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7510,"rank":7510,"depth":0,"x":2043.405,"y":555.762,"cluster":"scheme-morphisms"},{"id":"stacks:0E9K","tag":"0E9K","title":"Local complete intersection morphisms · Lemma 0E9K","summary":"Let f : X → Y be a morphism of schemes. If f is locally of finite type and X and Y are regular, then f is a local complete intersection morphism.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. If $f$ is locally\nof finite type and $X$ and $Y$ are regular, then\n$f$ is a local complete intersection morphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9K","source_file":"more-morphisms.tex","source_line":18337,"source_end_line":18342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18337-L18342","statement_sha256":"c4dde9c90b88e81b17a55b4cb9940834582ee1768dcce18b16fb7fe64b289f2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7511,"rank":7511,"depth":38,"x":2120.068,"y":779.266,"cluster":"scheme-morphisms"},{"id":"stacks:09RL","tag":"09RL","title":"Local complete intersection morphisms · Lemma 09RL","summary":"Let xymatrix X ar[rr]_f ar[rd] & & Y ar[ld] & S be a commutative diagram of morphisms of schemes. Assume • S is locally Noetherian, • Y → S is locally of finite type, • f : X → Y is perfect, • X → S is a local complete intersection morphism. Then X → Y is a local complete intersection morphism and Y → S is Koszul at f(x) for all x ∈ X.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume\n\\begin{enumerate}\n\\item $S$ is locally Noetherian,\n\\item $Y \\to S$ is locally of finite type,\n\\item $f : X \\to Y$ is perfect,\n\\item $X \\to S$ is a local complete intersection morphism.\n\\end{enumerate}\nThen $X \\to Y$ is a local complete intersection morphism\nand $Y \\to S$ is Koszul at $f(x)$ for all $x \\in X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RL","source_file":"more-morphisms.tex","source_line":18356,"source_end_line":18374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18356-L18374","statement_sha256":"d4fd0c03ea1b4f30457726700a40eb61014add3c20976b7b9322a00e03914a29","origin":"The Stacks Project","memory_eligible":false,"source_rank":7512,"rank":7512,"depth":18,"x":1883.688,"y":657.899,"cluster":"scheme-morphisms"},{"id":"stacks:0FJ2","tag":"0FJ2","title":"Local complete intersection morphisms · Lemma 0FJ2","summary":"Let xymatrix X ar[rr]_f ar[rd] & & Y ar[ld] & S be a commutative diagram of morphisms of schemes. Assume S is locally Noetherian, Y → S is locally of finite type, Y is regular, and X → S is a local complete intersection morphism. Then f : X → Y is a local complete intersection morphism and Y → S is Koszul at f(x) for all x ∈ X.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\\n& S\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume\n$S$ is locally Noetherian, $Y \\to S$ is locally of finite type,\n$Y$ is regular, and $X \\to S$ is a local complete intersection morphism.\nThen $f : X \\to Y$ is a local complete intersection morphism\nand $Y \\to S$ is Koszul at $f(x)$ for all $x \\in X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FJ2","source_file":"more-morphisms.tex","source_line":18420,"source_end_line":18434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18420-L18434","statement_sha256":"3838d4429471e3d0fca4e626ce1c12a585aed072075169e01bc209ef1c9bbdef","origin":"The Stacks Project","memory_eligible":false,"source_rank":7513,"rank":7513,"depth":20,"x":2155.722,"y":613.243,"cluster":"scheme-morphisms"},{"id":"stacks:0FK1","tag":"0FK1","title":"Local complete intersection morphisms · Lemma 0FK1","summary":"Let i : X → Y be an immersion. If • i is perfect, • Y is locally Noetherian, and • the conormal sheaf C_X/Y is finite locally free, then i is a regular immersion.","statement_latex":"Let $i : X \\to Y$ be an immersion. If\n\\begin{enumerate}\n\\item $i$ is perfect,\n\\item $Y$ is locally Noetherian, and\n\\item the conormal sheaf $\\mathcal{C}_{X/Y}$ is finite locally free,\n\\end{enumerate}\nthen $i$ is a regular immersion.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FK1","source_file":"more-morphisms.tex","source_line":18442,"source_end_line":18451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18442-L18451","statement_sha256":"97d01c62b181dd9ac8910a2a70a2ee161f3615b2b9f2e72ca01eb30d03d2e6bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7514,"rank":7514,"depth":18,"x":1990.96,"y":800.621,"cluster":"scheme-morphisms"},{"id":"stacks:0FV6","tag":"0FV6","title":"Local complete intersection morphisms · Lemma 0FV6","summary":"Let f : X → Y be a local complete intersection homomorphism. Then the naive cotangent complex NL_X/Y is a perfect object of D(O_X) of tor-amplitude in [-1, 0].","statement_latex":"Let $f : X \\to Y$ be a local complete intersection homomorphism.\nThen the naive cotangent complex $\\NL_{X/Y}$ is a perfect object\nof $D(\\mathcal{O}_X)$ of tor-amplitude in $[-1, 0]$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV6","source_file":"more-morphisms.tex","source_line":18458,"source_end_line":18463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18458-L18463","statement_sha256":"fa405b6d7b396a711e2ee166dc3742ada693f3a5a7a4515dba6e5684f662310c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7515,"rank":7515,"depth":29,"x":1961.755,"y":568.85,"cluster":"scheme-morphisms"},{"id":"stacks:0FK2","tag":"0FK2","title":"Local complete intersection morphisms · Lemma 0FK2","summary":"Let f : X → Y be a perfect morphism of locally Noetherian schemes. The following are equivalent • f is a local complete intersection morphism, • NL_X/Y has tor-amplitude in [-1, 0], and • NL_X/Y is perfect with tor-amplitude in [-1, 0].","statement_latex":"Let $f : X \\to Y$ be a perfect morphism of locally Noetherian schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is a local complete intersection morphism,\n\\item $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and\n\\item $\\NL_{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FK2","source_file":"more-morphisms.tex","source_line":18477,"source_end_line":18486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18477-L18486","statement_sha256":"d47d1db69eebaa7f1a83c5ef4ecb7575006b68e954cc84eaccd49b180db8fed4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7516,"rank":7516,"depth":29,"x":2169.774,"y":723.257,"cluster":"scheme-morphisms"},{"id":"stacks:0FK3","tag":"0FK3","title":"Local complete intersection morphisms · Lemma 0FK3","summary":"Let f : X → Y be a flat morphism of finite presentation. The following are equivalent • f is a local complete intersection morphism, • f is syntomic, • NL_X/Y has tor-amplitude in [-1, 0], and • NL_X/Y is perfect with tor-amplitude in [-1, 0].","statement_latex":"Let $f : X \\to Y$ be a flat morphism of finite presentation.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is a local complete intersection morphism,\n\\item $f$ is syntomic,\n\\item $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and\n\\item $\\NL_{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FK3","source_file":"more-morphisms.tex","source_line":18500,"source_end_line":18510,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18500-L18510","statement_sha256":"011e3809920c5d83e98b5f326b6f611db823bbfb1e1911d1ab6e8c8d00f71ef1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7517,"rank":7517,"depth":37,"x":1892.08,"y":727.437,"cluster":"scheme-morphisms"},{"id":"stacks:0FDP","tag":"0FDP","title":"Local complete intersection morphisms · Lemma 0FDP","summary":"Let f : X → Y be a finite type morphism of locally Noetherian schemes. Denote Δ : X → X ×_Y X the diagonal morphism. The following are equivalent • f is smooth, • f is flat and Δ : X → X ×_Y X is a regular immersion, • f is flat and Δ : X → X ×_Y X is a local complete intersection morphism, • f is flat and Δ : X → X ×_Y X is perfect.","statement_latex":"Let $f : X \\to Y$ be a finite type morphism of locally Noetherian schemes.\nDenote $\\Delta : X \\to X \\times_Y X$ the diagonal morphism.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is smooth,\n\\item $f$ is flat and $\\Delta : X \\to X \\times_Y X$ is a regular immersion,\n\\item $f$ is flat and $\\Delta : X \\to X \\times_Y X$ is a\nlocal complete intersection morphism,\n\\item $f$ is flat and $\\Delta : X \\to X \\times_Y X$ is perfect.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDP","source_file":"more-morphisms.tex","source_line":18528,"source_end_line":18540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18528-L18540","statement_sha256":"56a889c098850b79848741c963941527425c1c04b6830cd48b6925cd02a293aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7518,"rank":7518,"depth":39,"x":2093.583,"y":566.708,"cluster":"scheme-morphisms"},{"id":"stacks:069N","tag":"069N","title":"Local complete intersection morphisms · Lemma 069N","summary":"The property P(f) =\"f is a local complete intersection morphism\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection\nmorphism'' is fpqc local on the base.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069N","source_file":"more-morphisms.tex","source_line":18558,"source_end_line":18562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18558-L18562","statement_sha256":"5f34d21469d19928941013d81445ed615cc6b3e5314be16bb4834b2a3a852d06","origin":"The Stacks Project","memory_eligible":false,"source_rank":7519,"rank":7519,"depth":38,"x":2074.243,"y":799.676,"cluster":"scheme-morphisms"},{"id":"stacks:069P","tag":"069P","title":"Local complete intersection morphisms · Lemma 069P","summary":"The property P(f) =\"f is a local complete intersection morphism\" is syntomic local on the source.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection\nmorphism'' is syntomic local on the source.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/069P","source_file":"more-morphisms.tex","source_line":18589,"source_end_line":18593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18589-L18593","statement_sha256":"ce4e46da168bdbd136158335dbd39aa3f1b3b74fcf604b753e610e632e2c24bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7520,"rank":7520,"depth":39,"x":1901.074,"y":616.827,"cluster":"scheme-morphisms"},{"id":"stacks:06B8","tag":"06B8","title":"Local complete intersection morphisms · Lemma 06B8","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. Assume both X and Y are flat and locally of finite presentation over S. Then the set (x ∈ X mid f Koszul at x). is open in X and its formation commutes with arbitrary base change S' → S.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.\nAssume both $X$ and $Y$ are flat and locally of finite presentation over $S$.\nThen the set\n$$\n\\{x \\in X \\mid f\\text{ Koszul at }x\\}.\n$$\nis open in $X$ and its formation commutes with arbitrary base change\n$S' \\to S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06B8","source_file":"more-morphisms.tex","source_line":18656,"source_end_line":18666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18656-L18666","statement_sha256":"f5fa76bb5f994e9c1c8d633a173f76b386a189a8f6619785548d41d54d8e7525","origin":"The Stacks Project","memory_eligible":false,"source_rank":7521,"rank":7521,"depth":38,"x":2175.94,"y":653.415,"cluster":"scheme-morphisms"},{"id":"stacks:06B9","tag":"06B9","title":"Local complete intersection morphisms · Lemma 06B9","summary":"Let f : X → Y be a local complete intersection morphism of schemes. Then f is unramified if and only if f is formally unramified and in this case the conormal sheaf C_X/Y is finite locally free on X.","statement_latex":"Let $f : X \\to Y$ be a local complete intersection morphism of schemes.\nThen $f$ is unramified if and only if $f$ is formally unramified and in\nthis case the conormal sheaf $\\mathcal{C}_{X/Y}$ is finite locally free\non $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06B9","source_file":"more-morphisms.tex","source_line":18708,"source_end_line":18714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18708-L18714","statement_sha256":"a75759e19bbcaec6e902a4b3e41c94a3d1b1e1fb01755b4618b46a6bd9d0226c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7522,"rank":7522,"depth":21,"x":1943.724,"y":782.462,"cluster":"scheme-morphisms"},{"id":"stacks:06BA","tag":"06BA","title":"Local complete intersection morphisms · Lemma 06BA","summary":"Let Z → Y → X be formally unramified morphisms of schemes. Assume that Z → Y is a local complete intersection morphism. The exact sequence 0 → i^*C_Y/X → C_Z/X → C_Z/Y → 0 of Lemma [Tag 06AE] is short exact.","statement_latex":"Let $Z \\to Y \\to X$ be formally unramified morphisms of schemes.\nAssume that $Z \\to Y$ is a local complete intersection morphism.\nThe exact sequence\n$$\n0 \\to i^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nof\nLemma \\ref{lemma-transitivity-conormal}\nis short exact.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BA","source_file":"more-morphisms.tex","source_line":18730,"source_end_line":18743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18730-L18743","statement_sha256":"f41f9d1632495644d65a846924a7e1b74bff5bfa362f84c76d569e7348495d02","origin":"The Stacks Project","memory_eligible":false,"source_rank":7523,"rank":7523,"depth":12,"x":2011.212,"y":555.431,"cluster":"scheme-morphisms"},{"id":"stacks:094P","tag":"094P","title":"Weakly étale morphisms · Definition 094P","summary":"A morphism of schemes X → Y is weakly étale or absolutely flat if both X → Y and the diagonal morphism X → X ×_Y X are flat.","statement_latex":"A morphism of schemes $X \\to Y$ is {\\it weakly \\'etale} or\n{\\it absolutely flat} if both $X \\to Y$ and the diagonal\nmorphism $X \\to X \\times_Y X$ are flat.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094P","source_file":"more-morphisms.tex","source_line":18873,"source_end_line":18878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18873-L18878","statement_sha256":"da358f117be0dce872f332a35df2acc6e264a02973a9d1a1392e3cfc41dd3fdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7524,"rank":7524,"depth":0,"x":2144.083,"y":761.233,"cluster":"scheme-morphisms"},{"id":"stacks:094Q","tag":"094Q","title":"Weakly étale morphisms · Lemma 094Q","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent • X → Y is weakly étale, and • for every x ∈ X the ring map O_Y, f(x) → O_X, x is weakly étale.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $X \\to Y$ is weakly \\'etale, and\n\\item for every $x \\in X$ the ring map\n$\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is weakly \\'etale.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094Q","source_file":"more-morphisms.tex","source_line":18891,"source_end_line":18899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18891-L18899","statement_sha256":"9f88856931e03a65cdce6ec54dbc10adaabdc43e8c21484b850976020b8e0c70","origin":"The Stacks Project","memory_eligible":false,"source_rank":7525,"rank":7525,"depth":3,"x":1880.48,"y":684.836,"cluster":"scheme-morphisms"},{"id":"stacks:094R","tag":"094R","title":"Weakly étale morphisms · Lemma 094R","summary":"Let X → Y be a morphism of schemes such that X → X ×_Y X is flat. Let F be an O_X-module. If F is flat over Y, then F is flat over X.","statement_latex":"Let $X \\to Y$ be a morphism of schemes such that\n$X \\to X \\times_Y X$ is flat. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nIf $\\mathcal{F}$ is flat over $Y$, then $\\mathcal{F}$ is flat over $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094R","source_file":"more-morphisms.tex","source_line":18918,"source_end_line":18923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18918-L18923","statement_sha256":"56f1ab79299514bed1bdf108d89f6b4c8b164c130cd7583fbcd1eb2d438557d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7526,"rank":7526,"depth":1,"x":2136.415,"y":591.551,"cluster":"scheme-morphisms"},{"id":"stacks:094S","tag":"094S","title":"Weakly étale morphisms · Lemma 094S","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • The morphism f is weakly étale. • For every affine opens U ⊂ X, V ⊂ S with f(U) ⊂ V the ring map O_S(V) → O_X(U) is weakly étale. • There exists an open covering S = ⋃_j ∈ J V_j and open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i such that each of the morphisms U_i → V_j, j∈ J, i∈ I_j is weakly étale. • There exists an affine open covering S = ⋃_j ∈ J V_j and affine open coverings f^-1(V_j) = ⋃_i ∈ I_j U_i…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is weakly \\'etale.\n\\item For every affine opens $U \\subset X$, $V \\subset S$\nwith $f(U) \\subset V$ the ring map\n$\\mathcal{O}_S(V) \\to \\mathcal{O}_X(U)$ is weakly \\'etale.\n\\item There exists an open covering $S = \\bigcup_{j \\in J} V_j$\nand open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat each of the morphisms $U_i \\to V_j$, $j\\in J, i\\in I_j$\nis weakly \\'etale.\n\\item There exists an affine open covering $S = \\bigcup_{j \\in J} V_j$\nand affine open coverings $f^{-1}(V_j) = \\bigcup_{i \\in I_j} U_i$ such\nthat the ring map $\\mathcal{O}_S(V_j) \\to \\mathcal{O}_X(U_i)$ is\nof weakly \\'etale, for all $j\\in J, i\\in I_j$.\n\\end{enumerate}\nMoreover, if $f$ is weakly \\'etale then for\nany open subschemes $U \\subset X$, $V \\subset S$ with $f(U) \\subset V$\nthe restriction $f|_U : U \\to V$ is weakly-\\'etale.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094S","source_file":"more-morphisms.tex","source_line":18934,"source_end_line":18954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18934-L18954","statement_sha256":"2d0163b08e3d56e83f3a6d795829c4f56534f746944f6703e68c876454528b41","origin":"The Stacks Project","memory_eligible":false,"source_rank":7527,"rank":7527,"depth":5,"x":2022.656,"y":805.663,"cluster":"scheme-morphisms"},{"id":"stacks:094T","tag":"094T","title":"Weakly étale morphisms · Lemma 094T","summary":"Let X → Y → Z be morphisms of schemes. • If X → X ×_Y X and Y → Y ×_Z Y are flat, then X → X ×_Z X is flat. • If X → Y and Y → Z are weakly étale, then X → Z is weakly étale.","statement_latex":"Let $X \\to Y \\to Z$ be morphisms of schemes.\n\\begin{enumerate}\n\\item If $X \\to X \\times_Y X$ and $Y \\to Y \\times_Z Y$ are flat,\nthen $X \\to X \\times_Z X$ is flat.\n\\item If $X \\to Y$ and $Y \\to Z$ are weakly \\'etale, then\n$X \\to Z$ is weakly \\'etale.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094T","source_file":"more-morphisms.tex","source_line":18973,"source_end_line":18982,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L18973-L18982","statement_sha256":"0a60ad2b7053a9e630bb70264130e18c2f66592d3104caf1b0dfca93fb417f15","origin":"The Stacks Project","memory_eligible":false,"source_rank":7528,"rank":7528,"depth":5,"x":1934.315,"y":583.124,"cluster":"scheme-morphisms"},{"id":"stacks:094U","tag":"094U","title":"Weakly étale morphisms · Lemma 094U","summary":"Let X → Y and Y' → Y be morphisms of schemes and let X' = Y' ×_Y X be the base change of X. • If X → X ×_Y X is flat, then X' → X' ×_Y' X' is flat. • If X → Y is weakly étale, then X' → Y' is weakly étale.","statement_latex":"Let $X \\to Y$ and $Y' \\to Y$ be morphisms of schemes and let\n$X' = Y' \\times_Y X$ be the base change of $X$.\n\\begin{enumerate}\n\\item If $X \\to X \\times_Y X$ is flat, then $X' \\to X' \\times_{Y'} X'$\nis flat.\n\\item If $X \\to Y$ is weakly \\'etale, then $X' \\to Y'$ is weakly \\'etale.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094U","source_file":"more-morphisms.tex","source_line":19001,"source_end_line":19010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19001-L19010","statement_sha256":"3be6dccab455b58b809dbea235d7b4ffbe86cd7005d91a9796b9d87c19247777","origin":"The Stacks Project","memory_eligible":false,"source_rank":7529,"rank":7529,"depth":3,"x":2178.531,"y":697.149,"cluster":"scheme-morphisms"},{"id":"stacks:094V","tag":"094V","title":"Weakly étale morphisms · Lemma 094V","summary":"Let X → Y → Z be morphisms of schemes. Assume that X → Y is flat and surjective and that X → X ×_Z X is flat. Then Y → Y ×_Z Y is flat.","statement_latex":"Let $X \\to Y \\to Z$ be morphisms of schemes. Assume that $X \\to Y$ is\nflat and surjective and that $X \\to X \\times_Z X$ is flat.\nThen $Y \\to Y \\times_Z Y$ is flat.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094V","source_file":"more-morphisms.tex","source_line":19020,"source_end_line":19025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19020-L19025","statement_sha256":"983cc868129011b2413aee29cb9ffacc6a74493dd1d51e3560815e7282fe5921","origin":"The Stacks Project","memory_eligible":false,"source_rank":7530,"rank":7530,"depth":5,"x":1906.626,"y":751.669,"cluster":"scheme-morphisms"},{"id":"stacks:094W","tag":"094W","title":"Weakly étale morphisms · Lemma 094W","summary":"Let f : X → Y be a weakly étale morphism of schemes. Then f is formally unramified, i.e., Ω_X/Y = 0.","statement_latex":"Let $f : X \\to Y$ be a weakly \\'etale morphism of schemes.\nThen $f$ is formally unramified, i.e., $\\Omega_{X/Y} = 0$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094W","source_file":"more-morphisms.tex","source_line":19041,"source_end_line":19045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19041-L19045","statement_sha256":"c755a7f73a7f826145020dba79d4e7fe818561dec36f11c17ac3703a8e68049d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7531,"rank":7531,"depth":6,"x":2063.356,"y":557.089,"cluster":"scheme-morphisms"},{"id":"stacks:094X","tag":"094X","title":"Weakly étale morphisms · Lemma 094X","summary":"Let f : X → Y be a morphism of schemes. Then X → Y is weakly étale in each of the following cases • X → Y is a flat monomorphism, • X → Y is an open immersion, • X → Y is flat and unramified, • X → Y is étale.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Then $X \\to Y$ is weakly \\'etale\nin each of the following cases\n\\begin{enumerate}\n\\item $X \\to Y$ is a flat monomorphism,\n\\item $X \\to Y$ is an open immersion,\n\\item $X \\to Y$ is flat and unramified,\n\\item $X \\to Y$ is \\'etale.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094X","source_file":"more-morphisms.tex","source_line":19056,"source_end_line":19066,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19056-L19066","statement_sha256":"e5122bc0a865b21dca812ffe9e7b04530ba0a11140d1ff84db9f27b8195b42f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7532,"rank":7532,"depth":16,"x":2104.281,"y":789.611,"cluster":"scheme-morphisms"},{"id":"stacks:094Y","tag":"094Y","title":"Weakly étale morphisms · Lemma 094Y","summary":"Let f : X → Y be a morphism of schemes. If Y is reduced and f weakly étale, then X is reduced.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nIf $Y$ is reduced and $f$ weakly \\'etale, then $X$ is reduced.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094Y","source_file":"more-morphisms.tex","source_line":19080,"source_end_line":19084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19080-L19084","statement_sha256":"37c81d85891b8a736438eaefcc0730c78aacff48dbd8eff978bc3684367ed4b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7533,"rank":7533,"depth":9,"x":1887.012,"y":641.305,"cluster":"scheme-morphisms"},{"id":"stacks:094Z","tag":"094Z","title":"Weakly étale morphisms · Lemma 094Z","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent • f is weakly étale, and • for x ∈ X the local ring map O_Y, f(x) → O_X, x induces an isomorphism on strict henselizations.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is weakly \\'etale, and\n\\item for $x \\in X$ the local ring map\n$\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ induces an isomorphism\non strict henselizations.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/094Z","source_file":"more-morphisms.tex","source_line":19097,"source_end_line":19107,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19097-L19107","statement_sha256":"5337383da1d310363b5007ba1224fb397d7baec08219f6d65a7fd3b50da8edbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7534,"rank":7534,"depth":48,"x":2166.619,"y":627.374,"cluster":"scheme-morphisms"},{"id":"stacks:0950","tag":"0950","title":"Weakly étale morphisms · Lemma 0950","summary":"Let f : X → Y be a morphism of schemes. If Y is a normal scheme and f weakly étale, then X is a normal scheme.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. If $Y$ is a normal scheme\nand $f$ weakly \\'etale, then $X$ is a normal scheme.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0950","source_file":"more-morphisms.tex","source_line":19150,"source_end_line":19154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19150-L19154","statement_sha256":"f68afe5131e6e5f5b01bbce65a61cf9442ed34b8576cc45b8829d14281c10c27","origin":"The Stacks Project","memory_eligible":false,"source_rank":7535,"rank":7535,"depth":49,"x":1971.552,"y":796.382,"cluster":"scheme-morphisms"},{"id":"stacks:0951","tag":"0951","title":"Weakly étale morphisms · Lemma 0951","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. If X, Y are weakly étale over S, then f is weakly étale.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.\nIf $X$, $Y$ are weakly \\'etale over $S$, then $f$ is weakly \\'etale.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0951","source_file":"more-morphisms.tex","source_line":19164,"source_end_line":19168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19164-L19168","statement_sha256":"51135b417970ba54a3fec4a3de9a29d7ac9442af2455b9f1fe89049a3cb1fbea","origin":"The Stacks Project","memory_eligible":false,"source_rank":7536,"rank":7536,"depth":5,"x":1979.484,"y":560.961,"cluster":"scheme-morphisms"},{"id":"stacks:0F6V","tag":"0F6V","title":"Weakly étale morphisms · Lemma 0F6V","summary":"Let f : X → Y be a morphism of schemes. If f is weakly étale and a universal homeomorphism, it is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. If $f$ is weakly \\'etale and\na universal homeomorphism, it is an isomorphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6V","source_file":"more-morphisms.tex","source_line":19189,"source_end_line":19193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19189-L19193","statement_sha256":"48f9089663624ef3980eef81f2180e3e16fd5dcb55c625dbba5f7fc0137276ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":7537,"rank":7537,"depth":42,"x":2163.04,"y":739.141,"cluster":"scheme-morphisms"},{"id":"stacks:0F6W","tag":"0F6W","title":"Weakly étale morphisms · Lemma 0F6W","summary":"Let U → X be a weakly étale morphism of schemes where X is a scheme in characteristic p. Then the relative Frobenius F_U/X : U → U ×_X, F_X X is an isomorphism.","statement_latex":"Let $U \\to X$ be a weakly \\'etale morphism of schemes where $X$ is a scheme\nin characteristic $p$. Then the relative Frobenius\n$F_{U/X} : U \\to U \\times_{X, F_X} X$ is an isomorphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weakly étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6W","source_file":"more-morphisms.tex","source_line":19214,"source_end_line":19219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19214-L19219","statement_sha256":"2353e5c1817c3358666f03fd7d9a33d8ee2c4b260363234b84136c486108c4a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7538,"rank":7538,"depth":43,"x":1884.27,"y":711.895,"cluster":"scheme-morphisms"},{"id":"stacks:09IL","tag":"09IL","title":"Reduced fibre theorem · Theorem 09IL","summary":"Let A be a Dedekind ring with fraction field K. Let X be a scheme flat and of finite type over A. Assume A is a Nagata ring. There exists a finite extension L/K such that the normalized base change Y is smooth over Spec(B) at all generic points of all fibres.","statement_latex":"Let $A$ be a Dedekind ring with fraction field $K$.\nLet $X$ be a scheme flat and of finite type over $A$.\nAssume $A$ is a Nagata ring.\nThere exists a finite extension $L/K$ such that\nthe normalized base change $Y$ is smooth over $\\Spec(B)$\nat all generic points of all fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibre theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IL","source_file":"more-morphisms.tex","source_line":19281,"source_end_line":19289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19281-L19289","statement_sha256":"dcb330329a1ad29d4a4910a86ba902fd7d92898ebed3ef7c93ddf2bbffe6f354","origin":"The Stacks Project","memory_eligible":false,"source_rank":7539,"rank":7539,"depth":52,"x":2111.848,"y":573.74,"cluster":"scheme-morphisms"},{"id":"stacks:0BRQ","tag":"0BRQ","title":"Variant over curves · Lemma 0BRQ","summary":"Let f : X → S be a flat, finite type morphism of schemes. Assume S is Nagata, integral with function field K, and regular of dimension 1. Then there exists a finite extension L/K such that in the diagram xymatrix Y ar[rd]_g ar[r]_-ν & X ×_S T ar[d] ar[r] & X ar[d]_f & T ar[r] & S the morphism g is smooth at all generic points of fibres. Here T is the normalization of S in Spec(L) and ν : Y → X ×_S T is the normalization.","statement_latex":"Let $f : X \\to S$ be a flat, finite type morphism of schemes.\nAssume $S$ is Nagata, integral with function field $K$, and\nregular of dimension $1$. Then there exists a finite extension $L/K$\nsuch that in the diagram\n$$\n\\xymatrix{\nY \\ar[rd]_g \\ar[r]_-\\nu & X \\times_S T \\ar[d] \\ar[r] & X \\ar[d]_f \\\\\n& T \\ar[r] & S\n}\n$$\nthe morphism $g$ is smooth at all generic points of fibres. Here\n$T$ is the normalization of $S$ in $\\Spec(L)$ and $\\nu : Y \\to X \\times_S T$\nis the normalization.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibre theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRQ","source_file":"more-morphisms.tex","source_line":19357,"source_end_line":19372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19357-L19372","statement_sha256":"f0eb488b1dc296d3c91d42de48a85960e7a3f3177ba0652ab3f7def9ed95be97","origin":"The Stacks Project","memory_eligible":false,"source_rank":7540,"rank":7540,"depth":53,"x":2055.109,"y":804.855,"cluster":"scheme-morphisms"},{"id":"stacks:0BRR","tag":"0BRR","title":"Variant with separable extension · Lemma 0BRR","summary":"Let A be a Dedekind ring with fraction field K. Let X be a scheme flat and of finite type over A. Assume A is a Nagata ring and that for every generic point eta of an irreducible component of X the field extension kappa(eta)/K is separable. Then there exists a finite separable extension L/K such that the normalized base change Y is smooth over Spec(B) at all generic points of all fibres.","statement_latex":"Let $A$ be a Dedekind ring with fraction field $K$.\nLet $X$ be a scheme flat and of finite type over $A$.\nAssume $A$ is a Nagata ring and that for every generic point\n$\\eta$ of an irreducible component of $X$ the field\nextension $\\kappa(\\eta)/K$ is separable.\nThen there exists a finite separable extension $L/K$ such that\nthe normalized base change $Y$ is smooth over $\\Spec(B)$\nat all generic points of all fibres.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibre theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRR","source_file":"more-morphisms.tex","source_line":19400,"source_end_line":19410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19400-L19410","statement_sha256":"779571ac860196cb4f67762c4f72f9dcf904d871067dadc546b80c50d7f41b9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7541,"rank":7541,"depth":53,"x":1911.024,"y":602.145,"cluster":"scheme-morphisms"},{"id":"stacks:0BRS","tag":"0BRS","title":"Variant with separable extensions over curves · Lemma 0BRS","summary":"Let f : X → S be a flat, finite type morphism of schemes. Assume S is Nagata, integral with function field K, and regular of dimension 1. Assume the field extensions kappa(eta)/K are separable for every generic point eta of an irreducible component of X. Then there exists a finite separable extension L/K such that in the diagram xymatrix Y ar[rd]_g ar[r]_-ν & X ×_S T ar[d] ar[r] & X ar[d]_f & T ar[r] & S the morphism g is smooth at all generic points of fibres. Here T is…","statement_latex":"Let $f : X \\to S$ be a flat, finite type morphism of schemes.\nAssume $S$ is Nagata, integral with function field $K$, and\nregular of dimension $1$. Assume the field extensions $\\kappa(\\eta)/K$\nare separable for every generic point $\\eta$ of an irreducible\ncomponent of $X$. Then there exists a finite separable extension $L/K$\nsuch that in the diagram\n$$\n\\xymatrix{\nY \\ar[rd]_g \\ar[r]_-\\nu & X \\times_S T \\ar[d] \\ar[r] & X \\ar[d]_f \\\\\n& T \\ar[r] & S\n}\n$$\nthe morphism $g$ is smooth at all generic points of fibres. Here\n$T$ is the normalization of $S$ in $\\Spec(L)$ and $\\nu : Y \\to X \\times_S T$\nis the normalization.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Reduced fibre theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRS","source_file":"more-morphisms.tex","source_line":19480,"source_end_line":19497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19480-L19497","statement_sha256":"a2ec4efe7f47d680f47b09fe961f72fbf3e7d938632ec82ee27e39906ca07c6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7542,"rank":7542,"depth":54,"x":2180.411,"y":669.895,"cluster":"scheme-morphisms"},{"id":"stacks:0AP6","tag":"0AP6","title":"Ind-quasi-affine morphisms · Definition 0AP6","summary":"A scheme X is ind-quasi-affine if every quasi-compact open of X is quasi-affine. Similarly, a morphism of schemes X → Y is ind-quasi-affine if f^-1(V) is ind-quasi-affine for each affine open V in Y.","statement_latex":"A scheme $X$ is {\\it ind-quasi-affine} if every quasi-compact open of\n$X$ is quasi-affine. Similarly, a morphism of schemes $X \\to Y$\nis {\\it ind-quasi-affine} if $f^{-1}(V)$ is ind-quasi-affine\nfor each affine open $V$ in $Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Ind-quasi-affine morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AP6","source_file":"more-morphisms.tex","source_line":19521,"source_end_line":19527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19521-L19527","statement_sha256":"47c9c4d2baf70aad5cc8a1c5b2895fbd32b9d1ccdbe98e374f38092d0dc01c34","origin":"The Stacks Project","memory_eligible":false,"source_rank":7543,"rank":7543,"depth":0,"x":1927.168,"y":772.842,"cluster":"scheme-morphisms"},{"id":"stacks:0F1U","tag":"0F1U","title":"Ind-quasi-affine morphisms · Lemma 0F1U","summary":"For a morphism of schemes f : X → Y, the following are equivalent: • f is ind-quasi-affine, • for every affine open subscheme V ⊂ Y and every quasi-compact open subscheme U ⊂ f^-1(V), the induced morphism U → V is quasi-affine. • for some cover ( V_j )_j ∈ J of Y by quasi-compact and quasi-separated open subschemes V_j ⊂ Y, every j ∈ J, and every quasi-compact open subscheme U ⊂ f^-1(V_j), the induced morphism U → V_j is quasi-affine. • for every quasi-compact and…","statement_latex":"For a morphism of schemes $f : X \\to Y$, the following are equivalent:\n\\begin{enumerate}\n\\item $f$ is ind-quasi-affine,\n\\item for every affine open subscheme $V \\subset Y$ and\nevery quasi-compact open subscheme $U \\subset f^{-1}(V)$,\nthe induced morphism $U \\to V$ is quasi-affine. \n\\item\nfor some cover $\\{ V_j \\}_{j \\in J}$ of $Y$ by\nquasi-compact and quasi-separated open subschemes\n$V_j \\subset Y$, every $j \\in J$, and every quasi-compact\nopen subscheme $U \\subset f^{-1}(V_j)$, the induced morphism\n$U \\to V_j$ is quasi-affine.\n\\item for every quasi-compact and quasi-separated open subscheme\n$V \\subset Y$ and every quasi-compact open subscheme\n$U \\subset f^{-1}(V)$, the induced morphism $U \\to V$ is quasi-affine.\n\\end{enumerate}\nIn particular, the property of being an ind-quasi-affine morphism\nis Zariski local on the base.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1U","source_file":"more-morphisms.tex","source_line":19537,"source_end_line":19557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19537-L19557","statement_sha256":"d9347bd68200816d8cf4e9abab934d029c78dd838c071e3dda24d2579a69929c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7544,"rank":7544,"depth":22,"x":2031.167,"y":553.131,"cluster":"scheme-morphisms"},{"id":"stacks:0F1V","tag":"0F1V","title":"Ind-quasi-affine morphisms · Lemma 0F1V","summary":"The property of being an ind-quasi-affine morphism is stable under composition.","statement_latex":"The property of being an ind-quasi-affine morphism is stable under composition.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1V","source_file":"more-morphisms.tex","source_line":19584,"source_end_line":19587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19584-L19587","statement_sha256":"b2adf0924f98f2c089af8672790cf7674b1eae034ddc19e113db3ab453eaa9eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7545,"rank":7545,"depth":23,"x":2131.211,"y":774.257,"cluster":"scheme-morphisms"},{"id":"stacks:0F1W","tag":"0F1W","title":"Ind-quasi-affine morphisms · Lemma 0F1W","summary":"Any quasi-affine morphism is ind-quasi-affine. Any immersion is ind-quasi-affine.","statement_latex":"Any quasi-affine morphism is ind-quasi-affine.\nAny immersion is ind-quasi-affine.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1W","source_file":"more-morphisms.tex","source_line":19605,"source_end_line":19609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19605-L19609","statement_sha256":"8b3de938c9f45a6bdbe02f45cfa1e45f8ce0befca9593b160b5431d16f43b742","origin":"The Stacks Project","memory_eligible":false,"source_rank":7546,"rank":7546,"depth":24,"x":1879.499,"y":667.92,"cluster":"scheme-morphisms"},{"id":"stacks:0F1X","tag":"0F1X","title":"Ind-quasi-affine morphisms · Lemma 0F1X","summary":"If f : X → Y and g : Y → Z are morphisms of schemes such that g ∘ f is ind-quasi-affine, then f is ind-quasi-affine.","statement_latex":"If $f : X \\to Y$ and $g : Y \\to Z$ are morphisms of schemes\nsuch that $g \\circ f$ is ind-quasi-affine, then $f$ is ind-quasi-affine.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1X","source_file":"more-morphisms.tex","source_line":19620,"source_end_line":19624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19620-L19624","statement_sha256":"6270931d167214b9f74b480a87b5ce259b7e8af1aacbe2695c2f331681e18327","origin":"The Stacks Project","memory_eligible":false,"source_rank":7547,"rank":7547,"depth":23,"x":2150.747,"y":603.474,"cluster":"scheme-morphisms"},{"id":"stacks:0AP7","tag":"0AP7","title":"Ind-quasi-affine morphisms · Lemma 0AP7","summary":"The property of being ind-quasi-affine is stable under base change.","statement_latex":"The property of being ind-quasi-affine is stable under base change.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AP7","source_file":"more-morphisms.tex","source_line":19634,"source_end_line":19637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19634-L19637","statement_sha256":"193ad3f318aa21b469848d84ceb472700a81bc06a8c7c3f5864147532d819d3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7548,"rank":7548,"depth":25,"x":2002.49,"y":805.002,"cluster":"scheme-morphisms"},{"id":"stacks:0AP8","tag":"0AP8","title":"Ind-quasi-affine morphisms · Lemma 0AP8","summary":"The property of being ind-quasi-affine is fpqc local on the base.","statement_latex":"The property of being ind-quasi-affine is fpqc local on the base.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AP8","source_file":"more-morphisms.tex","source_line":19654,"source_end_line":19657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19654-L19657","statement_sha256":"2bedb0a26fadc85f8b3d25def54aaf3e00bd7f0e4e4ed6fddace2a255c0cc889","origin":"The Stacks Project","memory_eligible":false,"source_rank":7549,"rank":7549,"depth":41,"x":1949.725,"y":572.165,"cluster":"scheme-morphisms"},{"id":"stacks:0AP9","tag":"0AP9","title":"Ind-quasi-affine morphisms · Lemma 0AP9","summary":"A separated locally quasi-finite morphism of schemes is ind-quasi-affine.","statement_latex":"A separated locally quasi-finite morphism of schemes is ind-quasi-affine.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AP9","source_file":"more-morphisms.tex","source_line":19680,"source_end_line":19683,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19680-L19683","statement_sha256":"f25724e3e8e0fc1679cb7fd60c103dbbad0eaaad657b3e43f45c1a970e4a8eb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7550,"rank":7550,"depth":49,"x":2175.979,"y":713.981,"cluster":"scheme-morphisms"},{"id":"stacks:0ECJ","tag":"0ECJ","title":"Pushouts in the category of schemes, II · Lemma 0ECJ","summary":"In Situation [Tag 0ECI] then for y ∈ Y there exist affine opens U ⊂ X and V ⊂ Y with i^-1(U) = j^-1(V) and y ∈ V.","statement_latex":"In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}\nthen for $y \\in Y$ there exist affine opens $U \\subset X$ and\n$V \\subset Y$ with $i^{-1}(U) = j^{-1}(V)$ and $y \\in V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECJ","source_file":"more-morphisms.tex","source_line":19719,"source_end_line":19724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19719-L19724","statement_sha256":"a3da698ea58b87ed620377fce71838fe99073f102c6ae7b23cc929fe7895f091","origin":"The Stacks Project","memory_eligible":false,"source_rank":7551,"rank":7551,"depth":37,"x":1894.968,"y":737.802,"cluster":"scheme-morphisms"},{"id":"stacks:0E25","tag":"0E25","title":"Pushouts in the category of schemes, II · Proposition 0E25","summary":"[Ferrand-Conducteur] In Situation [Tag 0ECI] the pushout Y amalg_Z X exists in the category of schemes. Picture xymatrix Z ar[r]_i ar[d]_j & X ar[d]^a Y ar[r]^-b & Y amalg_Z X The diagram is a fibre square, the morphism a is integral, the morphism b is a closed immersion, and O_Y amalg_Z X = b_*O_Y ×_c_*O_Z a_*O_X as sheaves of rings where c = a ∘ i = b ∘ j.","statement_latex":"\\begin{reference}\n\\cite[Theorem 7.1 part iii]{Ferrand-Conducteur}\n\\end{reference}\nIn Situation \\ref{situation-pushout-along-closed-immersion-and-integral}\nthe pushout $Y \\amalg_Z X$ exists in the category of schemes. Picture\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_j & X \\ar[d]^a \\\\\nY \\ar[r]^-b & Y \\amalg_Z X\n}\n$$\nThe diagram is a fibre square, the morphism $a$ is integral,\nthe morphism $b$ is a closed immersion, and\n$$\n\\mathcal{O}_{Y \\amalg_Z X} =\nb_*\\mathcal{O}_Y \\times_{c_*\\mathcal{O}_Z} a_*\\mathcal{O}_X\n$$\nas sheaves of rings where $c = a \\circ i = b \\circ j$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E25","source_file":"more-morphisms.tex","source_line":19757,"source_end_line":19777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19757-L19777","statement_sha256":"73aa17a377495c67de42dfdf23f3a385ada28e6827735b5b2c5095ef62bd4662","origin":"The Stacks Project","memory_eligible":false,"source_rank":7552,"rank":7552,"depth":42,"x":2083.113,"y":560.702,"cluster":"scheme-morphisms"},{"id":"stacks:0E26","tag":"0E26","title":"Pushouts in the category of schemes, II · Lemma 0E26","summary":"In Situation [Tag 0ECI]. If X and Y are separated, then the pushout Y amalg_Z X (Proposition [Tag 0E25]) is separated. Same with \"separated over S\", \"quasi-separated\", and \"quasi-separated over S\".","statement_latex":"In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}.\nIf $X$ and $Y$ are separated, then the pushout $Y \\amalg_Z X$\n(Proposition \\ref{proposition-pushout-along-closed-immersion-and-integral})\nis separated. Same with ``separated over $S$'', ``quasi-separated'', and\n``quasi-separated over $S$''.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E26","source_file":"more-morphisms.tex","source_line":19846,"source_end_line":19853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19846-L19853","statement_sha256":"23ba3ee4a9771b9328472e7b6ccbaf6ffbfc7d888bb5a98be3fa215c53342985","origin":"The Stacks Project","memory_eligible":false,"source_rank":7553,"rank":7553,"depth":43,"x":2086.798,"y":798.161,"cluster":"scheme-morphisms"},{"id":"stacks:0E27","tag":"0E27","title":"Pushouts in the category of schemes, II · Lemma 0E27","summary":"In Situation [Tag 0ECI] assume S is a locally Noetherian scheme and X, Y, and Z are locally of finite type over S. Then the pushout Y amalg_Z X (Proposition [Tag 0E25]) is locally of finite type over S.","statement_latex":"In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}\nassume $S$ is a locally Noetherian scheme and $X$, $Y$, and $Z$\nare locally of finite type over $S$. Then the pushout $Y \\amalg_Z X$\n(Proposition \\ref{proposition-pushout-along-closed-immersion-and-integral})\nis locally of finite type over $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E27","source_file":"more-morphisms.tex","source_line":19861,"source_end_line":19868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19861-L19868","statement_sha256":"c9c93ea42e9ce944074150a6415e76315bc92854ed8e5ecbeb4c8212c4070345","origin":"The Stacks Project","memory_eligible":false,"source_rank":7554,"rank":7554,"depth":43,"x":1893.033,"y":625.073,"cluster":"scheme-morphisms"},{"id":"stacks:0ECK","tag":"0ECK","title":"Pushouts in the category of schemes, II · Lemma 0ECK","summary":"In Situation [Tag 0ECI] suppose given a commutative diagram xymatrix Y' ar[d]^g & Z' ar[l]^j' ar[r]_i' ar[d]^h & X' ar[d]^f Y & Z ar[l] ar[r] & X with cartesian squares and f, g, h separated and locally quasi-finite. Then • the pushouts Y amalg_Z X and Y' amalg_Z' X' exist, • Y' amalg_Z' X' → Y amalg_Z X is separated and locally quasi-finite, and • the squares xymatrix Y' ar[r] ar[d] & Y' amalg_Z' X' ar[d] & X' ar[l] ar[d] Y ar[r] & Y amalg_Z X & X ar[l] are cartesian.","statement_latex":"In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}\nsuppose given a commutative diagram\n$$\n\\xymatrix{\nY' \\ar[d]^g & Z' \\ar[l]^{j'} \\ar[r]_{i'} \\ar[d]^h & X' \\ar[d]^f \\\\\nY & Z \\ar[l] \\ar[r] & X\n}\n$$\nwith cartesian squares and $f, g, h$ separated and locally quasi-finite. Then\n\\begin{enumerate}\n\\item the pushouts $Y \\amalg_Z X$ and $Y' \\amalg_{Z'} X'$ exist,\n\\item $Y' \\amalg_{Z'} X' \\to Y \\amalg_Z X$ is\nseparated and locally quasi-finite, and\n\\item the squares\n$$\n\\xymatrix{\nY' \\ar[r] \\ar[d] & Y' \\amalg_{Z'} X' \\ar[d] & X' \\ar[l] \\ar[d] \\\\\nY \\ar[r] & Y \\amalg_Z X & X \\ar[l]\n}\n$$\nare cartesian.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECK","source_file":"more-morphisms.tex","source_line":19875,"source_end_line":19899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19875-L19899","statement_sha256":"bc7054f8d1c176da16b162cda918639b5371ca4aeff1e23b22d3366e5ad5a0a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7555,"rank":7555,"depth":50,"x":2175.235,"y":642.767,"cluster":"scheme-morphisms"},{"id":"stacks:0ECL","tag":"0ECL","title":"Pushouts in the category of schemes, II · Lemma 0ECL","summary":"In Situation [Tag 0ECI] the category of schemes flat, separated, and locally quasi-finite over the pushout Y amalg_Z X is equivalent to the category of (X', Y', Z', i', j', f, g, h) as in Lemma [Tag 0ECK] with f, g, h flat. Similarly with \"flat\" replaced with \"étale\".","statement_latex":"In Situation \\ref{situation-pushout-along-closed-immersion-and-integral}\nthe category of schemes flat, separated, and locally quasi-finite\nover the pushout $Y \\amalg_Z X$ is equivalent to the category of\n$(X', Y', Z', i', j', f, g, h)$ as in Lemma \\ref{lemma-pushout-functor}\nwith $f, g, h$ flat. Similarly with ``flat'' replaced with\n``\\'etale''.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECL","source_file":"more-morphisms.tex","source_line":19945,"source_end_line":19953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19945-L19953","statement_sha256":"8ef6c272e2c248b023c43703b8c58bd92c86880f1e947d9b2f90304a221572e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7556,"rank":7556,"depth":51,"x":1952.812,"y":789.916,"cluster":"scheme-morphisms"},{"id":"stacks:0B7M","tag":"0B7M","title":"Pushouts in the category of schemes, II · Lemma 0B7M","summary":"Let i : Z → X and j : Z → Y be closed immersions of schemes. Then the pushout Y amalg_Z X exists in the category of schemes. Picture xymatrix Z ar[r]_i ar[d]_j & X ar[d]^a Y ar[r]^-b & Y amalg_Z X The diagram is a fibre square, the morphisms a and b are closed immersions, and there is a short exact sequence 0 → O_Y amalg_Z X → a_*O_X ⊕ b_*O_Y → c_*O_Z → 0 where c = a ∘ i = b ∘ j.","statement_latex":"Let $i : Z \\to X$ and $j : Z \\to Y$ be closed immersions of schemes.\nThen the pushout $Y \\amalg_Z X$ exists in the category of schemes. Picture\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_j & X \\ar[d]^a \\\\\nY \\ar[r]^-b & Y \\amalg_Z X\n}\n$$\nThe diagram is a fibre square, the morphisms $a$ and $b$\nare closed immersions, and there is a short exact sequence\n$$\n0 \\to \\mathcal{O}_{Y \\amalg_Z X} \\to\na_*\\mathcal{O}_X \\oplus b_*\\mathcal{O}_Y \\to\nc_*\\mathcal{O}_Z \\to 0\n$$\nwhere $c = a \\circ i = b \\circ j$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7M","source_file":"more-morphisms.tex","source_line":19986,"source_end_line":20004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L19986-L20004","statement_sha256":"eb4c49603085eb68e5877e58915ae35b66815d9a6017ceb9bc2965d1299772bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7557,"rank":7557,"depth":43,"x":1998.511,"y":555.094,"cluster":"scheme-morphisms"},{"id":"stacks:0CYY","tag":"0CYY","title":"Pushouts in the category of schemes, II · Lemma 0CYY","summary":"Let i : Z → X and j : Z → Y be closed immersions of schemes. Let f : X' → X and g : Y' → Y be morphisms of schemes and let φ : X' ×_X, i Z → Y' ×_Y, j Z be an isomorphism of schemes over Z. Consider the morphism h : X' amalg_X' ×_X, i Z, φ Y' → X amalg_Z Y Then we have • h is locally of finite type if and only if f and g are locally of finite type, • h is flat if and only if f and g are flat, • h is flat and locally of finite presentation if and only if f and g are flat…","statement_latex":"Let $i : Z \\to X$ and $j : Z \\to Y$ be closed immersions of schemes.\nLet $f : X' \\to X$ and $g : Y' \\to Y$ be morphisms of schemes and let\n$\\varphi : X' \\times_{X, i} Z \\to Y' \\times_{Y, j} Z$\nbe an isomorphism of schemes over $Z$. Consider the morphism\n$$\nh :\nX' \\amalg_{X' \\times_{X, i} Z, \\varphi} Y'\n\\longrightarrow\nX \\amalg_Z Y\n$$\nThen we have\n\\begin{enumerate}\n\\item $h$ is locally of finite type if and only if $f$ and $g$ are\nlocally of finite type,\n\\item $h$ is flat if and only if $f$ and $g$ are flat,\n\\item $h$ is flat and locally of finite presentation if and only if\n$f$ and $g$ are flat and locally of finite presentation,\n\\item $h$ is smooth if and only if $f$ and $g$ are smooth,\n\\item $h$ is \\'etale if and only if $f$ and $g$ are \\'etale, and\n\\item add more here as needed.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Pushouts in the category of schemes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYY","source_file":"more-morphisms.tex","source_line":20016,"source_end_line":20039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20016-L20039","statement_sha256":"6b6a2930b2438e37b3086a49e9313d411fa905c786aec2c838cc5ed961133749","origin":"The Stacks Project","memory_eligible":false,"source_rank":7558,"rank":7558,"depth":44,"x":2153.722,"y":754.271,"cluster":"scheme-morphisms"},{"id":"stacks:05Y6","tag":"05Y6","title":"Relative morphisms · Lemma 05Y6","summary":"Let Z → S and X → S be morphisms of affine schemes. Assume Γ(Z, O_Z) is a finite free Γ(S, O_S)-module. Then mathitMor_S(Z, X) is representable by an affine scheme over S.","statement_latex":"Let $Z \\to S$ and $X \\to S$ be morphisms of affine schemes.\nAssume $\\Gamma(Z, \\mathcal{O}_Z)$ is a finite free\n$\\Gamma(S, \\mathcal{O}_S)$-module. Then $\\mathit{Mor}_S(Z, X)$\nis representable by an affine scheme over $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Y6","source_file":"more-morphisms.tex","source_line":20099,"source_end_line":20105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20099-L20105","statement_sha256":"5983646cd59d7d684e593c7bb58ea193fc44d1a35fefaf2901c6034e64ae0453","origin":"The Stacks Project","memory_eligible":false,"source_rank":7559,"rank":7559,"depth":0,"x":1878.973,"y":695.444,"cluster":"scheme-morphisms"},{"id":"stacks:0BL3","tag":"0BL3","title":"Relative morphisms · Lemma 0BL3","summary":"Let Z → S and X → S be morphisms of schemes. If Z → S is finite locally free and X → S is affine, then mathitMor_S(Z, X) is representable by a scheme affine over S.","statement_latex":"Let $Z \\to S$ and $X \\to S$ be morphisms of schemes.\nIf $Z \\to S$ is finite locally free and $X \\to S$ is affine,\nthen $\\mathit{Mor}_S(Z, X)$ is representable by a scheme\naffine over $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BL3","source_file":"more-morphisms.tex","source_line":20170,"source_end_line":20176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20170-L20176","statement_sha256":"9d6689d33587858258051d8747cc83208fd8f9a54b2e5fa15d17fd3d803039bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7560,"rank":7560,"depth":3,"x":2128.991,"y":582.871,"cluster":"scheme-morphisms"},{"id":"stacks:0BL4","tag":"0BL4","title":"Relative morphisms · Lemma 0BL4","summary":"Let Z → S and X → S be morphisms of schemes. Assume • Z → S is finite locally free, and • for all (s, x_1, …, x_d) where s ∈ S and x_1, …, x_d ∈ X_s there exists an affine open U ⊂ X with x_1, …, x_d ∈ U. Then mathitMor_S(Z, X) is representable by a scheme.","statement_latex":"Let $Z \\to S$ and $X \\to S$ be morphisms of schemes.\nAssume\n\\begin{enumerate}\n\\item $Z \\to S$ is finite locally free, and\n\\item for all $(s, x_1, \\ldots, x_d)$ where $s \\in S$ and\n$x_1, \\ldots, x_d \\in X_s$ there exists an affine open $U \\subset X$\nwith $x_1, \\ldots, x_d \\in U$.\n\\end{enumerate}\nThen $\\mathit{Mor}_S(Z, X)$ is representable by a scheme.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BL4","source_file":"more-morphisms.tex","source_line":20200,"source_end_line":20211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20200-L20211","statement_sha256":"286545ea80e95d239069eb2b7bea0e6eac4ba37a9458d461cb07e7450e07d6ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":7561,"rank":7561,"depth":4,"x":2035.117,"y":807.85,"cluster":"scheme-morphisms"},{"id":"stacks:0BL5","tag":"0BL5","title":"Relative morphisms · Lemma 0BL5","summary":"Let Z → S and X → S be morphisms of schemes. Assume Z → S is finite locally free and X → S is separated and locally quasi-finite. Then mathitMor_S(Z, X) is representable by a scheme.","statement_latex":"Let $Z \\to S$ and $X \\to S$ be morphisms of schemes.\nAssume $Z \\to S$ is finite locally free and $X \\to S$\nis separated and locally quasi-finite.\nThen $\\mathit{Mor}_S(Z, X)$ is representable by a scheme.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BL5","source_file":"more-morphisms.tex","source_line":20256,"source_end_line":20262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20256-L20262","statement_sha256":"57b8812d6928764ff730a9104f659638a9e47a9fa9a1fbfdb701d09e1f85239f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7562,"rank":7562,"depth":50,"x":1923.365,"y":588.587,"cluster":"scheme-morphisms"},{"id":"stacks:0CTA","tag":"0CTA","title":"Characterizing pseudo-coherent complexes, III · Lemma 0CTA","summary":"Consider a commutative diagram of schemes xymatrix Z' ar[d] ar[r] & Y' ar[d] X' ar[r] & S' Let S → S' be a morphism. Denote by X and Y the base changes of X' and Y' to S. Assume Y' → S' and Z' → X' are flat. Then X ×_S Y and Z' are Tor independent over X' ×_S' Y'.","statement_latex":"Consider a commutative diagram of schemes\n$$\n\\xymatrix{\nZ' \\ar[d] \\ar[r] & Y' \\ar[d] \\\\\nX' \\ar[r] & S'\n}\n$$\nLet $S \\to S'$ be a morphism. Denote by $X$ and $Y$ the base\nchanges of $X'$ and $Y'$ to $S$.\nAssume $Y' \\to S'$ and $Z' \\to X'$ are flat.\nThen $X \\times_S Y$ and $Z'$ are Tor independent over $X' \\times_{S'} Y'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Characterizing pseudo-coherent complexes, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTA","source_file":"more-morphisms.tex","source_line":20289,"source_end_line":20302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20289-L20302","statement_sha256":"4e2419d4ecafc21c11ac582bfb37d148e29600dbc4b5da7de2138c19a73a16f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7563,"rank":7563,"depth":18,"x":2182.213,"y":686.903,"cluster":"scheme-morphisms"},{"id":"stacks:0CSJ","tag":"0CSJ","title":"Derived Chow's lemma · Lemma 0CSJ","summary":"Let A be a ring. Let X be a separated scheme of finite presentation over A. Let x ∈ X. Then there exist an open neighbourhood U ⊂ X of x, an n ≥ 0, an open V ⊂ P^n_A, a closed subscheme Z ⊂ X ×_A P^n_A, a point z ∈ Z, and an object E in D(O_X ×_A P^n_A) such that • Z → X ×_A P^n_A is of finite presentation, • b : Z → X is an isomorphism over U and b(z) = x, • c : Z → P^n_A is a closed immersion over V, • b^-1(U) = c^-1(V), in particular c(z) ∈ V, • E|_X ×_A V ≅ (b,…","statement_latex":"Let $A$ be a ring. Let $X$ be a separated scheme of finite presentation\nover $A$. Let $x \\in X$. Then there exist\nan open neighbourhood $U \\subset X$ of $x$,\nan $n \\geq 0$,\nan open $V \\subset \\mathbf{P}^n_A$,\na closed subscheme $Z \\subset X \\times_A \\mathbf{P}^n_A$,\na point $z \\in Z$, and\nan object $E$ in $D(\\mathcal{O}_{X \\times_A \\mathbf{P}^n_A})$ such that\n\\begin{enumerate}\n\\item $Z \\to X \\times_A \\mathbf{P}^n_A$ is of finite presentation,\n\\item $b : Z \\to X$ is an isomorphism over $U$ and $b(z) = x$,\n\\item $c : Z \\to \\mathbf{P}^n_A$ is a closed immersion over $V$,\n\\item $b^{-1}(U) = c^{-1}(V)$, in particular $c(z) \\in V$,\n\\item $E|_{X \\times_A V} \\cong\n(b, c)_*\\mathcal{O}_Z|_{X \\times_A V}$,\n\\item $E$ is pseudo-coherent and supported on $Z$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Characterizing pseudo-coherent complexes, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSJ","source_file":"more-morphisms.tex","source_line":20336,"source_end_line":20355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20336-L20355","statement_sha256":"8d86d2a042422cdf6fba04118c9b08a541ab635d95cca8f5f4b87694c103a78e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7564,"rank":7564,"depth":33,"x":1912.156,"y":761.317,"cluster":"scheme-morphisms"},{"id":"stacks:0CSK","tag":"0CSK","title":"Characterizing pseudo-coherent complexes, III · Lemma 0CSK","summary":"Let A, x ∈ X, and U, n, V, Z, z, E be as in Lemma [Tag 0CSJ]. For any K ∈ D_QCoh(O_X) we have Rq_*(Lp^*K ⊗^L E)|_V = R(U → V)_*K|_U where p : X ×_A P^n_A → X and q : X ×_A P^n_A → P^n_A are the projections and where the morphism U → V is the finitely presented closed immersion c ∘ (b|_U)^-1.","statement_latex":"Let $A$, $x \\in X$, and\n$U, n, V, Z, z, E$ be as in Lemma \\ref{lemma-derived-chow}.\nFor any $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have\n$$\nRq_*(Lp^*K \\otimes^\\mathbf{L} E)|_V = R(U \\to V)_*K|_U\n$$\nwhere $p : X \\times_A \\mathbf{P}^n_A \\to X$ and\n$q : X \\times_A \\mathbf{P}^n_A \\to \\mathbf{P}^n_A$ are\nthe projections and where the morphism $U \\to V$ is\nthe finitely presented closed immersion $c \\circ (b|_U)^{-1}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Characterizing pseudo-coherent complexes, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSK","source_file":"more-morphisms.tex","source_line":20440,"source_end_line":20452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20440-L20452","statement_sha256":"6dbe3a872b88dd994393278b454696aee1b979d6ef18316a7ed5ac4ec0b01729","origin":"The Stacks Project","memory_eligible":false,"source_rank":7565,"rank":7565,"depth":34,"x":2051.514,"y":553.113,"cluster":"scheme-morphisms"},{"id":"stacks:0CSL","tag":"0CSL","title":"Characterizing pseudo-coherent complexes, III · Lemma 0CSL","summary":"Let A be a ring. Let X be a scheme separated and of finite presentation over A. Let K ∈ D_QCoh(O_X). If RΓ(X, E ⊗^L K) is pseudo-coherent in D(A) for every pseudo-coherent E in D(O_X), then K is pseudo-coherent relative to A.","statement_latex":"Let $A$ be a ring. Let $X$ be a scheme separated and\nof finite presentation over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$.\nIf $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent\nin $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$,\nthen $K$ is pseudo-coherent relative to $A$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Characterizing pseudo-coherent complexes, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSL","source_file":"more-morphisms.tex","source_line":20487,"source_end_line":20494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20487-L20494","statement_sha256":"83c20578e4167c14c07be1300c665d643cd9dace4262d34a8534e8da9053b2da","origin":"The Stacks Project","memory_eligible":false,"source_rank":7566,"rank":7566,"depth":37,"x":2116.215,"y":785.82,"cluster":"scheme-morphisms"},{"id":"stacks:0GES","tag":"0GES","title":"Characterizing pseudo-coherent complexes, III · Lemma 0GES","summary":"Let A be a ring. Let X be a scheme separated and of finite presentation over A. Let K ∈ D_QCoh(O_X). If R Γ (X, E ⊗ ^L K) is pseudo-coherent in D(A) for every perfect E ∈ D(O_X), then K is pseudo-coherent relative to A.","statement_latex":"Let $A$ be a ring. Let $X$ be a scheme separated and\nof finite presentation over $A$. Let \n$K \\in D_\\QCoh(\\mathcal{O}_X).$ If \n$R \\Gamma (X, E \\otimes ^{\\mathbf{L}} K)$ is\npseudo-coherent in $D(A)$ for every perfect \n$E \\in D(\\mathcal{O}_X)$, then $K$ is pseudo-coherent\nrelative to $A$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Characterizing pseudo-coherent complexes, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GES","source_file":"more-morphisms.tex","source_line":20548,"source_end_line":20557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20548-L20557","statement_sha256":"92396016a48281213f3f068a66184affff7ec56ffcb7ef1b0f9cbb34f7f7b94b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7567,"rank":7567,"depth":39,"x":1881.261,"y":650.877,"cluster":"scheme-morphisms"},{"id":"stacks:0GET","tag":"0GET","title":"Characterizing pseudo-coherent complexes, III · Lemma 0GET","summary":"Let A be a ring. Let X be a scheme separated, of finite presentation, and flat over A. Let K ∈ D_QCoh(O_X). If R Γ (X, E ⊗^L K) is perfect in D(A) for every perfect E ∈ D(O_X), then K is Spec(A)-perfect.","statement_latex":"Let $A$ be a ring. Let $X$ be a scheme separated, of\nfinite presentation, and flat over $A$. Let \n$K \\in D_\\QCoh(\\mathcal{O}_X).$ If \n$R \\Gamma (X, E \\otimes^\\mathbf{L} K)$ is perfect in\n$D(A)$ for every perfect $E \\in D(\\mathcal{O}_X)$, then $K$ is\n$\\Spec(A)$-perfect.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Characterizing pseudo-coherent complexes, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GET","source_file":"more-morphisms.tex","source_line":20570,"source_end_line":20578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20570-L20578","statement_sha256":"6cfbea2a98900787182917f3851aa518376b274fe7bb5c4767d7f8db57c93a25","origin":"The Stacks Project","memory_eligible":false,"source_rank":7568,"rank":7568,"depth":40,"x":2163.159,"y":617.048,"cluster":"scheme-morphisms"},{"id":"stacks:0CSN","tag":"0CSN","title":"Descent finiteness properties of complexes · Lemma 0CSN","summary":"Let X → S be locally of finite type. Let (f_i : X_i → X) be an fppf covering of schemes. Let E ∈ D_QCoh(O_X). Let m ∈ Z. Then E is m-pseudo-coherent relative to S if and only if each Lf_i^*E is m-pseudo-coherent relative to S.","statement_latex":"Let $X \\to S$ be locally of finite type.\nLet $\\{f_i : X_i \\to X\\}$ be an fppf covering of schemes.\nLet $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.\nThen $E$ is $m$-pseudo-coherent relative to $S$\nif and only if each $Lf_i^*E$ is $m$-pseudo-coherent relative to $S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Descent finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSN","source_file":"more-morphisms.tex","source_line":20630,"source_end_line":20637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20630-L20637","statement_sha256":"2515a97b48409c7aa8b6e83d33eec0cccbe8b26897c7eb8e9facd358815f08b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7569,"rank":7569,"depth":42,"x":1982.413,"y":802.033,"cluster":"scheme-morphisms"},{"id":"stacks:0CSP","tag":"0CSP","title":"Descent finiteness properties of complexes · Lemma 0CSP","summary":"Let X → T → S be morphisms of schemes. Assume T → S is flat and locally of finite presentation and X → T locally of finite type. Let E ∈ D(O_X). Let m ∈ Z. Then E is m-pseudo-coherent relative to S if and only if E is m-pseudo-coherent relative to T.","statement_latex":"Let $X \\to T \\to S$ be morphisms of schemes. Assume $T \\to S$\nis flat and locally of finite presentation and $X \\to T$\nlocally of finite type. Let $E \\in D(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.\nThen $E$ is $m$-pseudo-coherent relative to $S$\nif and only if $E$ is $m$-pseudo-coherent relative to $T$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Descent finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSP","source_file":"more-morphisms.tex","source_line":20682,"source_end_line":20689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20682-L20689","statement_sha256":"47d665f54193d6e7836464232a623fb2a422801f8e7e7763431957e434d2388b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7570,"rank":7570,"depth":14,"x":1966.926,"y":562.96,"cluster":"scheme-morphisms"},{"id":"stacks:0CSQ","tag":"0CSQ","title":"Descent finiteness properties of complexes · Lemma 0CSQ","summary":"Let f : X → S be locally of finite type. Let (S_i → S) be an fppf covering of schemes. Denote f_i : X_i → S_i the base change of f and g_i : X_i → X the projection. Let E ∈ D_QCoh(O_X). Let m ∈ Z. Then E is m-pseudo-coherent relative to S if and only if each Lg_i^*E is m-pseudo-coherent relative to S_i.","statement_latex":"Let $f : X \\to S$ be locally of finite type.\nLet $\\{S_i \\to S\\}$ be an fppf covering of schemes.\nDenote $f_i : X_i \\to S_i$ the base change of $f$\nand $g_i : X_i \\to X$ the projection.\nLet $E \\in D_\\QCoh(\\mathcal{O}_X)$. Let $m \\in \\mathbf{Z}$.\nThen $E$ is $m$-pseudo-coherent relative to $S$\nif and only if each $Lg_i^*E$ is $m$-pseudo-coherent relative to $S_i$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Descent finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSQ","source_file":"more-morphisms.tex","source_line":20699,"source_end_line":20708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20699-L20708","statement_sha256":"4ae7b4e813dcb5f004fd0442b84e5fe96374795a2c1b78835a2496ac4cb4acde","origin":"The Stacks Project","memory_eligible":false,"source_rank":7571,"rank":7571,"depth":43,"x":2170.694,"y":730.537,"cluster":"scheme-morphisms"},{"id":"stacks:0DJX","tag":"0DJX","title":"Relatively perfect objects · Lemma 0DJX","summary":"Let i : X → X' be a finite order thickening of schemes. Let K' ∈ D(O_X') be an object such that K = Li^*K' is pseudo-coherent. Then K' is pseudo-coherent.","statement_latex":"Let $i : X \\to X'$ be a finite order thickening of schemes. Let\n$K' \\in D(\\mathcal{O}_{X'})$ be an object such that\n$K = Li^*K'$ is pseudo-coherent. Then $K'$ is pseudo-coherent.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJX","source_file":"more-morphisms.tex","source_line":20736,"source_end_line":20741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20736-L20741","statement_sha256":"95dd81b50d5e0ad74705d25e2b84a131f947051fbfb957c7413a86db5515a850","origin":"The Stacks Project","memory_eligible":false,"source_rank":7572,"rank":7572,"depth":30,"x":1885.548,"y":722.588,"cluster":"scheme-morphisms"},{"id":"stacks:0DJY","tag":"0DJY","title":"Relatively perfect objects · Lemma 0DJY","summary":"Consider a cartesian diagram xymatrix X ar[r]_i ar[d]_f & X' ar[d]^f' Y ar[r]^j & Y' of schemes. Assume X' → Y' is flat and locally of finite presentation and Y → Y' is a finite order thickening. Let E' ∈ D(O_X'). If E = Li^*(E') is Y-perfect, then E' is Y'-perfect.","statement_latex":"Consider a cartesian diagram\n$$\n\\xymatrix{\nX \\ar[r]_i \\ar[d]_f & X' \\ar[d]^{f'} \\\\\nY \\ar[r]^j & Y'\n}\n$$\nof schemes. Assume $X' \\to Y'$ is flat and locally\nof finite presentation and $Y \\to Y'$ is a finite order thickening.\nLet $E' \\in D(\\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect,\nthen $E'$ is $Y'$-perfect.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJY","source_file":"more-morphisms.tex","source_line":20795,"source_end_line":20808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20795-L20808","statement_sha256":"6b4a44365b741beb370c589e5ce5c566d0a7527c9a5243b3aaabca88a6f71ca1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7573,"rank":7573,"depth":31,"x":2102.302,"y":566.58,"cluster":"scheme-morphisms"},{"id":"stacks:0DJZ","tag":"0DJZ","title":"Relatively perfect objects · Lemma 0DJZ","summary":"Let (R, I) be a pair consisting of a ring and an ideal I contained in the Jacobson radical. Set S = Spec(R) and S_0 = Spec(R/I). Let f : X → S be proper, flat, and of finite presentation. Denote X_0 = S_0 ×_S X. Let E ∈ D(O_X) be pseudo-coherent. If the derived restriction E_0 of E to X_0 is S_0-perfect, then E is S-perfect.","statement_latex":"Let $(R, I)$ be a pair consisting of a ring and an ideal $I$\ncontained in the Jacobson radical. Set $S = \\Spec(R)$ and $S_0 = \\Spec(R/I)$.\nLet $f : X \\to S$ be proper, flat, and of finite presentation.\nDenote $X_0 = S_0 \\times_S X$. Let $E \\in D(\\mathcal{O}_X)$\nbe pseudo-coherent. If the derived restriction $E_0$ of $E$\nto $X_0$ is $S_0$-perfect, then $E$ is $S$-perfect.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJZ","source_file":"more-morphisms.tex","source_line":20831,"source_end_line":20839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20831-L20839","statement_sha256":"a22b2a17166e30159e237d8491832d136c87e73bda3dceeb125246c28211cfaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7574,"rank":7574,"depth":38,"x":2067.913,"y":804.717,"cluster":"scheme-morphisms"},{"id":"stacks:0E7F","tag":"0E7F","title":"Contracting rational curves · Lemma 0E7F","summary":"Let f : X → Y be a proper morphism of schemes. Let y ∈ Y be a point with dim(X_y) ≤ 1. If • R^1f_*O_X = 0, or more generally • there is a morphism g : Y' → Y such that y is in the image of g and such that R^1f'_*O_X' = 0 where f' : X' → Y' is the base change of f by g. Then H^1(X_y, O_X_y) = 0.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $y \\in Y$\nbe a point with $\\dim(X_y) \\leq 1$. If\n\\begin{enumerate}\n\\item $R^1f_*\\mathcal{O}_X = 0$, or more generally\n\\item there is a morphism $g : Y' \\to Y$ such that $y$ is in the image\nof $g$ and such that $R^1f'_*\\mathcal{O}_{X'} = 0$ where $f' : X' \\to Y'$\nis the base change of $f$ by $g$.\n\\end{enumerate}\nThen $H^1(X_y, \\mathcal{O}_{X_y}) = 0$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Contracting rational curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7F","source_file":"more-morphisms.tex","source_line":20917,"source_end_line":20928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20917-L20928","statement_sha256":"62f623180cca8eb746c64655798a78cdb556cbeb9552d36170a792ad5ed735ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":7575,"rank":7575,"depth":41,"x":1901.692,"y":609.516,"cluster":"scheme-morphisms"},{"id":"stacks:0E7G","tag":"0E7G","title":"Contracting rational curves · Lemma 0E7G","summary":"Let f : X → Y be a proper morphism of schemes. Let y ∈ Y be a point with dim(X_y) ≤ 1 and H^1(X_y, O_X_y) = 0. Then there is an open neighbourhood V ⊂ Y of y such that R^1f_*O_X|_V = 0 and the same is true after base change by any Y' → V.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $y \\in Y$\nbe a point with $\\dim(X_y) \\leq 1$ and $H^1(X_y, \\mathcal{O}_{X_y}) = 0$.\nThen there is an open neighbourhood $V \\subset Y$ of $y$ such that\n$R^1f_*\\mathcal{O}_X|_V = 0$\nand the same is true after base change by any $Y' \\to V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Contracting rational curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7G","source_file":"more-morphisms.tex","source_line":20942,"source_end_line":20949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20942-L20949","statement_sha256":"641fba8d3b2396fbaa5b8554103c0d05df3da9f8677dfc5adf998598b8609111","origin":"The Stacks Project","memory_eligible":false,"source_rank":7576,"rank":7576,"depth":41,"x":2181.362,"y":659.159,"cluster":"scheme-morphisms"},{"id":"stacks:0E7H","tag":"0E7H","title":"Contracting rational curves · Lemma 0E7H","summary":"Let f : X → Y be a proper morphism of schemes such that dim(X_y) ≤ 1 and H^1(X_y, O_X_y) = 0 for all y ∈ Y. Let F be quasi-coherent on X. Then • R^pf_*F = 0 for p > 1, and • R^1f_*F = 0 if there is a surjection f^*G → F with G quasi-coherent on Y. If Y is affine, then we also have • [(3)] H^p(X, F) = 0 for p not ∈ (0, 1), and • [(4)] H^1(X, F) = 0 if F is globally generated.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes such\nthat $\\dim(X_y) \\leq 1$ and $H^1(X_y, \\mathcal{O}_{X_y}) = 0$\nfor all $y \\in Y$. Let $\\mathcal{F}$ be quasi-coherent on $X$. Then\n\\begin{enumerate}\n\\item $R^pf_*\\mathcal{F} = 0$ for $p > 1$, and\n\\item $R^1f_*\\mathcal{F} = 0$ if there is a surjection\n$f^*\\mathcal{G} \\to \\mathcal{F}$ with $\\mathcal{G}$ quasi-coherent\non $Y$.\n\\end{enumerate}\nIf $Y$ is affine, then we also have\n\\begin{enumerate}\n\\item[(3)] $H^p(X, \\mathcal{F}) = 0$ for $p \\not \\in \\{0, 1\\}$, and\n\\item[(4)] $H^1(X, \\mathcal{F}) = 0$ if $\\mathcal{F}$ is globally generated.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Contracting rational curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7H","source_file":"more-morphisms.tex","source_line":20970,"source_end_line":20986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L20970-L20986","statement_sha256":"623d248c0131a9526d39e95d607372818a8980a802dad0f71e417fb295ed8968","origin":"The Stacks Project","memory_eligible":false,"source_rank":7577,"rank":7577,"depth":42,"x":1935.105,"y":781.3,"cluster":"scheme-morphisms"},{"id":"stacks:0E7I","tag":"0E7I","title":"Contracting rational curves · Lemma 0E7I","summary":"Let f : X → Y be a proper morphism of schemes. Assume • for all y ∈ Y we have dim(X_y) ≤ 1 and H^1(X_y, O_X_y) = 0, and • O_Y → f_*O_X is surjective. Then O_Y' → f'_*O_X' is surjective for any base change f' : X' → Y' of f.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Assume\n\\begin{enumerate}\n\\item for all $y \\in Y$ we have $\\dim(X_y) \\leq 1$ and\n$H^1(X_y, \\mathcal{O}_{X_y}) = 0$, and\n\\item $\\mathcal{O}_Y \\to f_*\\mathcal{O}_X$ is surjective.\n\\end{enumerate}\nThen $\\mathcal{O}_{Y'} \\to f'_*\\mathcal{O}_{X'}$ is surjective\nfor any base change $f' : X' \\to Y'$ of $f$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Contracting rational curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7I","source_file":"more-morphisms.tex","source_line":21017,"source_end_line":21027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21017-L21027","statement_sha256":"d11048b0faeadac7e53115ef0ef341de6a130ccda9a5a084863d50866aa01f1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7578,"rank":7578,"depth":43,"x":2018.506,"y":551.399,"cluster":"scheme-morphisms"},{"id":"stacks:0E7J","tag":"0E7J","title":"Contracting rational curves · Lemma 0E7J","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd] & & Y ar[ld] & S of morphisms of schemes. Let s ∈ S be a point. Assume • X → S is locally of finite presentation and flat at points of X_s, • f is proper, • the fibres of f_s : X_s → Y_s have dimension ≤ 1 and R^1f_s, *O_X_s = 0, • O_Y_s → f_s, *O_X_s is surjective. Then there is an open Y_s ⊂ V ⊂ Y such that (a) f^-1(V) is flat over S, (b) dim(X_y) ≤ 1 for y ∈ V, (c) R^1f_*O_X|_V = 0, (d) O_V → f_*O_X|_V is…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\\n& S\n}\n$$\nof morphisms of schemes. Let $s \\in S$ be a point. Assume\n\\begin{enumerate}\n\\item $X \\to S$ is locally of finite presentation and flat at\npoints of $X_s$,\n\\item $f$ is proper,\n\\item the fibres of $f_s : X_s \\to Y_s$ have dimension $\\leq 1$\nand $R^1f_{s, *}\\mathcal{O}_{X_s} = 0$,\n\\item $\\mathcal{O}_{Y_s} \\to f_{s, *}\\mathcal{O}_{X_s}$ is surjective.\n\\end{enumerate}\nThen there is an open $Y_s \\subset V \\subset Y$ such that\n(a) $f^{-1}(V)$ is flat over $S$,\n(b) $\\dim(X_y) \\leq 1$ for $y \\in V$,\n(c) $R^1f_*\\mathcal{O}_X|_V = 0$,\n(d) $\\mathcal{O}_V \\to f_*\\mathcal{O}_X|_V$\nis surjective,\nand (b), (c), and (d) remain true after base change by any $Y' \\to V$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Contracting rational curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7J","source_file":"more-morphisms.tex","source_line":21058,"source_end_line":21083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21058-L21083","statement_sha256":"2db0802f92525d5c755b94bafd424e93c2e998d30e321cc250f5bcef751c443d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7579,"rank":7579,"depth":44,"x":2141.943,"y":768.346,"cluster":"scheme-morphisms"},{"id":"stacks:0E7K","tag":"0E7K","title":"Contracting rational curves · Lemma 0E7K","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd] & & Y ar[ld] & S of morphisms of schemes. Assume X → S is flat, f is proper, dim(X_y) ≤ 1 for y ∈ Y, and R^1f_*O_X = 0. Then f_*O_X is S-flat and formation of f_*O_X commutes with arbitrary base change S' → S.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\\n& S\n}\n$$\nof morphisms of schemes. Assume $X \\to S$\nis flat, $f$ is proper, $\\dim(X_y) \\leq 1$ for $y \\in Y$, and\n$R^1f_*\\mathcal{O}_X = 0$. Then $f_*\\mathcal{O}_X$\nis $S$-flat and formation of $f_*\\mathcal{O}_X$ commutes\nwith arbitrary base change $S' \\to S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Contracting rational curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7K","source_file":"more-morphisms.tex","source_line":21192,"source_end_line":21206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21192-L21206","statement_sha256":"31dea591a01ad8c06f536666178c54e763095dcfa8aa31d95830c9776546bf64","origin":"The Stacks Project","memory_eligible":false,"source_rank":7580,"rank":7580,"depth":43,"x":1876.34,"y":678.373,"cluster":"scheme-morphisms"},{"id":"stacks:0E7L","tag":"0E7L","title":"Contracting rational curves · Lemma 0E7L","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd] & & Y ar[ld] & S of morphisms of schemes. Let s ∈ S be a point. Assume • X → S is locally of finite presentation and flat at points of X_s, • Y → S is locally of finite presentation, • f is proper, • the fibres of f_s : X_s → Y_s have dimension ≤ 1 and R^1f_s, *O_X_s = 0, • O_Y_s → f_s, *O_X_s is an isomorphism. Then there is an open Y_s ⊂ V ⊂ Y such that (a) V is flat over S, (b) f^-1(V) is flat over S, (c)…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\\n& S\n}\n$$\nof morphisms of schemes. Let $s \\in S$ be a point. Assume\n\\begin{enumerate}\n\\item $X \\to S$ is locally of finite presentation and flat at\npoints of $X_s$,\n\\item $Y \\to S$ is locally of finite presentation,\n\\item $f$ is proper,\n\\item the fibres of $f_s : X_s \\to Y_s$ have dimension $\\leq 1$\nand $R^1f_{s, *}\\mathcal{O}_{X_s} = 0$,\n\\item $\\mathcal{O}_{Y_s} \\to f_{s, *}\\mathcal{O}_{X_s}$ is an isomorphism.\n\\end{enumerate}\nThen there is an open $Y_s \\subset V \\subset Y$ such that\n(a) $V$ is flat over $S$,\n(b) $f^{-1}(V)$ is flat over $S$,\n(c) $\\dim(X_y) \\leq 1$ for $y \\in V$,\n(d) $R^1f_*\\mathcal{O}_X|_V = 0$,\n(e) $\\mathcal{O}_V \\to f_*\\mathcal{O}_X|_V$\nis an isomorphism, and (a) -- (e)\nremain true after base change of $f^{-1}(V) \\to V$ by any $S' \\to S$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Contracting rational curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7L","source_file":"more-morphisms.tex","source_line":21234,"source_end_line":21261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21234-L21261","statement_sha256":"4ee02db7b92aecae91beb6be615e40ab2110a558ee4e34a955d93136c029180e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7581,"rank":7581,"depth":45,"x":2144.666,"y":593.971,"cluster":"scheme-morphisms"},{"id":"stacks:0E24","tag":"0E24","title":"Contracting rational curves · Lemma 0E24","summary":"Let f : X → Y be a proper morphism of Noetherian schemes such that f_*O_X = O_Y, such that the fibres of f have dimension ≤ 1, and such that H^1(X_y, O_X_y) = 0 for y ∈ Y. Then f^* : Pic(Y) → Pic(X) is a bijection onto the subgroup of L ∈ Pic(X) with L|_X_y ≅ O_X_y for all y ∈ Y.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of Noetherian schemes\nsuch that $f_*\\mathcal{O}_X = \\mathcal{O}_Y$, such that\nthe fibres of $f$ have dimension $\\leq 1$, and such that\n$H^1(X_y, \\mathcal{O}_{X_y}) = 0$ for $y \\in Y$.\nThen $f^* : \\Pic(Y) \\to \\Pic(X)$ is a bijection onto\nthe subgroup of $\\mathcal{L} \\in \\Pic(X)$ with\n$\\mathcal{L}|_{X_y} \\cong \\mathcal{O}_{X_y}$\nfor all $y \\in Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Contracting rational curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E24","source_file":"more-morphisms.tex","source_line":21345,"source_end_line":21355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21345-L21355","statement_sha256":"b5a13838beffddbc8d7b896e6e359a65546011f00cf8ff8669512dd928e90cd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7582,"rank":7582,"depth":38,"x":2014.622,"y":808.559,"cluster":"scheme-morphisms"},{"id":"stacks:0F2S","tag":"0F2S","title":"Affine stratifications · Definition 0F2S","summary":"Let X be a scheme. An affine stratification is a locally finite stratification X = coprod_i ∈ I X_i whose strata X_i are affine and such that the inclusion morphisms X_i → X are affine.","statement_latex":"Let $X$ be a scheme. An {\\it affine stratification} is a\nlocally finite stratification $X = \\coprod_{i \\in I} X_i$\nwhose strata $X_i$ are affine and such that\nthe inclusion morphisms $X_i \\to X$ are affine.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Affine stratifications","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2S","source_file":"more-morphisms.tex","source_line":21437,"source_end_line":21443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21437-L21443","statement_sha256":"a33586cb4b73b5b41c3045a7c588edf74f343b47400d44ce2fcf1ab657409ad8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7583,"rank":7583,"depth":0,"x":1937.914,"y":576.43,"cluster":"scheme-morphisms"},{"id":"stacks:0F2T","tag":"0F2T","title":"Affine stratifications · Lemma 0F2T","summary":"Let X be a scheme. Let X = coprod_i ∈ I X_i be a finite affine stratification. There exists an affine stratification with index set (0, …, n) where n is the length of I.","statement_latex":"Let $X$ be a scheme. Let $X = \\coprod_{i \\in I} X_i$ be a finite\naffine stratification. There exists an affine stratification\nwith index set $\\{0, \\ldots, n\\}$ where $n$ is the length of $I$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Affine stratifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2T","source_file":"more-morphisms.tex","source_line":21469,"source_end_line":21474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21469-L21474","statement_sha256":"5a3cc9421c98adaee28c03662611b8dfdda3a47d3108c3c98c128fb0f215bab0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7584,"rank":7584,"depth":33,"x":2181.258,"y":704.13,"cluster":"scheme-morphisms"},{"id":"stacks:0F2U","tag":"0F2U","title":"Affine stratifications · Lemma 0F2U","summary":"Let X be a scheme. The following are equivalent • X has a finite affine stratification, and • X is quasi-compact and quasi-separated.","statement_latex":"Let $X$ be a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $X$ has a finite affine stratification, and\n\\item $X$ is quasi-compact and quasi-separated.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Affine stratifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2U","source_file":"more-morphisms.tex","source_line":21511,"source_end_line":21518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21511-L21518","statement_sha256":"fd15c70e87f7784026271e16ad70d8c40d372751a0dff43abe8580fcb1eaa52c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7585,"rank":7585,"depth":18,"x":1899.001,"y":748.065,"cluster":"scheme-morphisms"},{"id":"stacks:0F2V","tag":"0F2V","title":"Affine stratifications · Definition 0F2V","summary":"Let X be a nonempty quasi-compact and quasi-separated scheme. The affine stratification number is the smallest integer n ≥ 0 such that the following equivalent conditions are satisfied • there exists a finite affine stratification X = coprod_i ∈ I X_i where I has length n, • there exists an affine stratification X = X_0 amalg X_1 amalg … amalg X_n with index set (0, …, n).","statement_latex":"Let $X$ be a nonempty quasi-compact and quasi-separated scheme. The\n{\\it affine stratification number} is the smallest integer $n \\geq 0$\nsuch that the following equivalent conditions are satisfied\n\\begin{enumerate}\n\\item there exists a finite affine stratification\n$X = \\coprod_{i \\in I} X_i$ where $I$ has length $n$,\n\\item there exists an affine stratification\n$X = X_0 \\amalg X_1 \\amalg \\ldots \\amalg X_n$ with\nindex set $\\{0, \\ldots, n\\}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Affine stratifications","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2V","source_file":"more-morphisms.tex","source_line":21568,"source_end_line":21580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21568-L21580","statement_sha256":"42b6749dd35b71b3c70e029144e9978ce158209a84b4fc5a5e167e8bd5cff0bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7586,"rank":7586,"depth":0,"x":2071.879,"y":555.422,"cluster":"scheme-morphisms"},{"id":"stacks:0F2W","tag":"0F2W","title":"Affine stratifications · Lemma 0F2W","summary":"Let X be a separated scheme which has an open covering by n + 1 affines. Then the affine stratification number of X is at most n.","statement_latex":"Let $X$ be a separated scheme which has an open covering by\n$n + 1$ affines. Then the affine stratification number of $X$\nis at most $n$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Affine stratifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2W","source_file":"more-morphisms.tex","source_line":21588,"source_end_line":21593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21588-L21593","statement_sha256":"7db438fd130d2c10ca9aabcb6c37d2b72c3e10bb8b5a51ef08889d28935c4266","origin":"The Stacks Project","memory_eligible":false,"source_rank":7587,"rank":7587,"depth":20,"x":2099.332,"y":795.677,"cluster":"scheme-morphisms"},{"id":"stacks:0F2X","tag":"0F2X","title":"Affine stratifications · Lemma 0F2X","summary":"Let X be a Noetherian scheme of dimension ∞ > d ≥ 0. Then the affine stratification number of X is at most d.","statement_latex":"Let $X$ be a Noetherian scheme of dimension $\\infty > d \\geq 0$.\nThen the affine stratification number of $X$ is at most $d$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Affine stratifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2X","source_file":"more-morphisms.tex","source_line":21607,"source_end_line":21611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21607-L21611","statement_sha256":"1d8238f5fb6271935206234b127d0660edc2ec5a8717570dd0db1e65fd070403","origin":"The Stacks Project","memory_eligible":false,"source_rank":7588,"rank":7588,"depth":18,"x":1885.787,"y":634.022,"cluster":"scheme-morphisms"},{"id":"stacks:0F2Y","tag":"0F2Y","title":"Affine stratifications · Proposition 0F2Y","summary":"Let X be a nonempty quasi-compact and quasi-separated scheme with affine stratification number n. Then H^p(X, F) = 0, p > n for every quasi-coherent O_X-module F.","statement_latex":"Let $X$ be a nonempty quasi-compact and quasi-separated scheme with\naffine stratification number $n$. Then $H^p(X, \\mathcal{F}) = 0$, $p > n$\nfor every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Affine stratifications","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2Y","source_file":"more-morphisms.tex","source_line":21638,"source_end_line":21643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21638-L21643","statement_sha256":"9ca085d0107394c9a19ffd49ac7cb5a82cf3523d6869177645ee97a84d4349dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7589,"rank":7589,"depth":26,"x":2173.379,"y":632.052,"cluster":"scheme-morphisms"},{"id":"stacks:0F31","tag":"0F31","title":"Universally open morphisms · Lemma 0F31","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • f is universally open, • for every morphism S' → S which is locally of finite presentation the base change X_S' → S' is open, and • for every n the morphism A^n × X → A^n × S is open.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally open,\n\\item for every morphism $S' \\to S$ which is locally of finite presentation\nthe base change $X_{S'} \\to S'$ is open, and\n\\item for every $n$ the morphism\n$\\mathbf{A}^n \\times X \\to \\mathbf{A}^n \\times S$\nis open.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universally open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F31","source_file":"more-morphisms.tex","source_line":21725,"source_end_line":21737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21725-L21737","statement_sha256":"ce9274f6fd6c3889db37a58ad0034b71b5e9060c2b740f1bf0e822492ecffa43","origin":"The Stacks Project","memory_eligible":false,"source_rank":7590,"rank":7590,"depth":0,"x":1962.803,"y":796.765,"cluster":"scheme-morphisms"},{"id":"stacks:0F32","tag":"0F32","title":"Universally open morphisms · Lemma 0F32","summary":"Let f : X → Y be a morphism of schemes. If • f is locally quasi-finite, • Y is geometrically unibranch and locally Noetherian, and • every irreducible component of X dominates an irreducible component of Y, then f is universally open.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. If\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite,\n\\item $Y$ is geometrically unibranch and locally Noetherian, and\n\\item every irreducible component of $X$ dominates\nan irreducible component of $Y$,\n\\end{enumerate}\nthen $f$ is universally open.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universally open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F32","source_file":"more-morphisms.tex","source_line":21787,"source_end_line":21797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21787-L21797","statement_sha256":"7a6942409d4bf58e912c9982086c044bb26f7afea0f5012329f51f6f37dae1c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7591,"rank":7591,"depth":55,"x":1985.632,"y":555.715,"cluster":"scheme-morphisms"},{"id":"stacks:0F33","tag":"0F33","title":"Universally open morphisms · Lemma 0F33","summary":"Let A → B be a ring map. Say B is generated as an A-module by b_1, …, b_d ∈ B. Set h = ∑ x_ib_i ∈ B[x_1, …, x_d]. Then Spec(B) → Spec(A) is universally open if and only if the image of D(h) in Spec(A[x_1, …, x_d]) is open.","statement_latex":"Let $A \\to B$ be a ring map. Say $B$ is generated as an $A$-module by\n$b_1, \\ldots, b_d \\in B$. Set $h = \\sum x_ib_i \\in B[x_1, \\ldots, x_d]$.\nThen $\\Spec(B) \\to \\Spec(A)$ is universally open if and only if the image of\n$D(h)$ in $\\Spec(A[x_1, \\ldots, x_d])$ is open.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universally open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F33","source_file":"more-morphisms.tex","source_line":21846,"source_end_line":21852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21846-L21852","statement_sha256":"5ebb53053b96ac05a05d43a735ccea6b91e09b6f9678033ba819b28f5fe6cbfc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7592,"rank":7592,"depth":0,"x":2162.722,"y":746.5,"cluster":"scheme-morphisms"},{"id":"stacks:0F34","tag":"0F34","title":"Universally open morphisms · Lemma 0F34","summary":"Let S = lim S_i be a limit of a directed system of schemes with affine transition morphisms. Let 0 ∈ I and let f_0 : X_0 → Y_0 be a morphism of schemes over S_0. Assume S_0, X_0, Y_0 are quasi-compact and quasi-separated. Let f_i : X_i → Y_i be the base change of f_0 to S_i and let f : X → Y be the base change of f_0 to S. If • f is locally quasi-finite and universally open, and • f_0 is locally of finite presentation, then there exists an i ≥ 0 such that f_i is locally…","statement_latex":"Let $S = \\lim S_i$ be a limit of a directed system of schemes\nwith affine transition morphisms.\nLet $0 \\in I$ and let $f_0 : X_0 \\to Y_0$ be a morphism of schemes over $S_0$.\nAssume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated.\nLet $f_i : X_i \\to Y_i$ be the base change of $f_0$ to $S_i$ and\nlet $f : X \\to Y$ be the base change of $f_0$ to $S$.\nIf\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite and universally open, and\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen there exists an $i \\geq 0$ such that $f_i$ is locally quasi-finite\nand universally open.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universally open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F34","source_file":"more-morphisms.tex","source_line":21877,"source_end_line":21892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21877-L21892","statement_sha256":"f4a5b0d2380e001aeaf45313bc7bbd31471efa97df81d6f103deb30c1da6f281","origin":"The Stacks Project","memory_eligible":false,"source_rank":7593,"rank":7593,"depth":40,"x":1878.588,"y":706.286,"cluster":"scheme-morphisms"},{"id":"stacks:0F35","tag":"0F35","title":"Universally open morphisms · Lemma 0F35","summary":"Let f : X → Y be a locally quasi-finite morphism. Then • the functions n_X/Y of Lemmas [Tag 0556] and [Tag 055F] agree, • if X is quasi-compact, then n_X/Y attains a maximum d < ∞.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism. Then\n\\begin{enumerate}\n\\item the functions $n_{X/Y}$ of\nLemmas \\ref{lemma-base-change-fibres-nr-geometrically-irreducible-components}\nand \\ref{lemma-base-change-fibres-nr-geometrically-connected-components}\nagree,\n\\item if $X$ is quasi-compact, then $n_{X/Y}$ attains a maximum $d < \\infty$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universally open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F35","source_file":"more-morphisms.tex","source_line":21959,"source_end_line":21969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21959-L21969","statement_sha256":"c2d997768b78d8578efcf6010074fe10db66039ae6fc73f4a1f6b94841298666","origin":"The Stacks Project","memory_eligible":false,"source_rank":7594,"rank":7594,"depth":31,"x":2120.551,"y":574.656,"cluster":"scheme-morphisms"},{"id":"stacks:0F36","tag":"0F36","title":"Universally open morphisms · Lemma 0F36","summary":"Let f : X → Y be a separated, locally quasi-finite, and universally open morphism of schemes. Let n_X/Y be as in Lemma [Tag 0F35]. If n_X/Y(y) ≥ d for some y ∈ Y and d ≥ 0, then n_X/Y ≥ d in an open neighbourhood of y.","statement_latex":"Let $f : X \\to Y$ be a separated, locally quasi-finite, and universally open\nmorphism of schemes. Let $n_{X/Y}$ be as in\nLemma \\ref{lemma-count-geometric-fibres}.\nIf $n_{X/Y}(y) \\geq d$ for some $y \\in Y$ and $d \\geq 0$,\nthen $n_{X/Y} \\geq d$ in an open neighbourhood of $y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universally open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F36","source_file":"more-morphisms.tex","source_line":21980,"source_end_line":21987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L21980-L21987","statement_sha256":"6b9ea944ebd74ff274d9784c7b396280635a2eb31e58514a707d74b4270967d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7595,"rank":7595,"depth":33,"x":2047.952,"y":809.118,"cluster":"scheme-morphisms"},{"id":"stacks:0F37","tag":"0F37","title":"Universally open morphisms · Lemma 0F37","summary":"Let f : X → Y be a separated, locally quasi-finite, and universally open morphism of schemes. Let n_X/Y be as in Lemma [Tag 0F35]. If n_X/Y attains a maximum d < ∞, then the set Y_d = (y ∈ Y mid n_X/Y(y) = d) is open in Y and the morphism f^-1(Y_d) → Y_d is finite.","statement_latex":"Let $f : X \\to Y$ be a separated, locally quasi-finite, and universally open\nmorphism of schemes. Let $n_{X/Y}$ be as in\nLemma \\ref{lemma-count-geometric-fibres}.\nIf $n_{X/Y}$ attains a maximum $d < \\infty$, then the set\n$$\nY_d = \\{y \\in Y \\mid n_{X/Y}(y) = d\\}\n$$\nis open in $Y$ and the morphism $f^{-1}(Y_d) \\to Y_d$ is finite.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Universally open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F37","source_file":"more-morphisms.tex","source_line":22018,"source_end_line":22028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22018-L22028","statement_sha256":"7cf0f41513bc4bf923661aaefc39a382773cc66a55df25c011939c378fa3b8e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7596,"rank":7596,"depth":44,"x":1912.878,"y":594.939,"cluster":"scheme-morphisms"},{"id":"stacks:0F39","tag":"0F39","title":"Weightings · Lemma 0F39","summary":"Given a cartesian square xymatrix U ar[d]_π & U' ar[l]^h ar[d]^π' V & V' ar[l]_g with π locally quasi-finite with finite fibres and a function w : U → Z we have (int_π w) ∘ g = int_π' (w ∘ h).","statement_latex":"Given a cartesian square\n$$\n\\xymatrix{\nU \\ar[d]_\\pi & U' \\ar[l]^h \\ar[d]^{\\pi'} \\\\\nV & V' \\ar[l]_g\n}\n$$\nwith $\\pi$ locally quasi-finite with finite fibres\nand a function $w : U \\to \\mathbf{Z}$\nwe have $(\\int_\\pi w) \\circ g = \\int_{\\pi'} (w \\circ h)$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F39","source_file":"more-morphisms.tex","source_line":22110,"source_end_line":22122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22110-L22122","statement_sha256":"efa3e775693c8e1ee26b36a82f7d82bc94d3d4657c56c8b4cd6902678be65f72","origin":"The Stacks Project","memory_eligible":false,"source_rank":7597,"rank":7597,"depth":6,"x":2184.836,"y":676.263,"cluster":"scheme-morphisms"},{"id":"stacks:0F3A","tag":"0F3A","title":"Weightings · Definition 0F3A","summary":"Let f : X → Y be a locally quasi-finite morphism. A weighting or a pondération of f is a map w : X → Z such that for any diagram xymatrix X ar[d]_f & U ar[l]^h ar[d]^π Y & V ar[l]_g where V → Y is étale, U ⊂ X_V is open, and U → V finite, the function int_π (w ∘ h) is locally constant.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism. A\n{\\it weighting} or a {\\it pond\\'eration} of $f$ is a map\n$w : X \\to \\mathbf{Z}$ such that for any diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & U \\ar[l]^h \\ar[d]^\\pi \\\\\nY & V \\ar[l]_g\n}\n$$\nwhere $V \\to Y$ is \\'etale, $U \\subset X_V$ is open, and $U \\to V$ finite,\nthe function $\\int_\\pi (w \\circ h)$ is locally constant.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3A","source_file":"more-morphisms.tex","source_line":22151,"source_end_line":22164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22151-L22164","statement_sha256":"455ef3c12bf3a82e4df1d987343a4f025ab60c3e602dfee68aaea479d77536bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7598,"rank":7598,"depth":0,"x":1918.782,"y":770.654,"cluster":"scheme-morphisms"},{"id":"stacks:0F3B","tag":"0F3B","title":"Weightings · Lemma 0F3B","summary":"Let f : X → Y be a locally quasi-finite morphism. Let w : X → Z be a weighting. Let f' : X' → Y' be the base change of f by a morphism Y' → Y. Then the composition w' : X' → Z of w and the projection X' → X is a weighting of f'.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism.\nLet $w : X \\to \\mathbf{Z}$ be a weighting. Let $f' : X' \\to Y'$\nbe the base change of $f$ by a morphism $Y' \\to Y$. Then the\ncomposition $w' : X' \\to \\mathbf{Z}$ of $w$ and the projection $X' \\to X$\nis a weighting of $f'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3B","source_file":"more-morphisms.tex","source_line":22172,"source_end_line":22179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22172-L22179","statement_sha256":"de19ebd1c11534cae1db3392d6236a2fdcddd0cdf948dd75efdf596eecc25ad3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7599,"rank":7599,"depth":33,"x":2039.113,"y":549.988,"cluster":"scheme-morphisms"},{"id":"stacks:0GK8","tag":"0GK8","title":"Weightings · Lemma 0GK8","summary":"Let f : X → Y be a locally quasi-finite morphism. Let w : X → Z be a weighting of f. If X' ⊂ X is open, then w|_X' is a weighting of f|_X' : X' → Y.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism. Let\n$w : X \\to \\mathbf{Z}$ be a weighting of $f$. If $X' \\subset X$ is open,\nthen $w|_{X'}$ is a weighting of $f|_{X'} : X' \\to Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GK8","source_file":"more-morphisms.tex","source_line":22238,"source_end_line":22243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22238-L22243","statement_sha256":"9794c79381641ce1cafb0ba6eec088f22c44d380322a8bde6d0f07219274add8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7600,"rank":7600,"depth":0,"x":2127.875,"y":781.084,"cluster":"scheme-morphisms"},{"id":"stacks:0GK9","tag":"0GK9","title":"Weightings · Lemma 0GK9","summary":"Let f : X → Y and g : Y → Z be locally quasi-finite morphisms. Let w_f : X → Z be a weighting of f and let w_g : Y → Z be a weighting of g. Then the function X → Z, x ↦ w_f(x) w_g(f(x)) is a weighting of g ∘ f.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be locally quasi-finite morphisms.\nLet $w_f : X \\to \\mathbf{Z}$ be a weighting of $f$ and let\n$w_g : Y \\to \\mathbf{Z}$ be a weighting of $g$. Then the function\n$$\nX \\longrightarrow \\mathbf{Z},\\quad\nx \\longmapsto w_f(x) w_g(f(x))\n$$\nis a weighting of $g \\circ f$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GK9","source_file":"more-morphisms.tex","source_line":22249,"source_end_line":22259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22249-L22259","statement_sha256":"bea16abd58e2e4b854d5318a41a122cf404bf2fbe5e0de3ed45786fc29473558","origin":"The Stacks Project","memory_eligible":false,"source_rank":7601,"rank":7601,"depth":33,"x":1876.471,"y":660.99,"cluster":"scheme-morphisms"},{"id":"stacks:0F3C","tag":"0F3C","title":"Weightings · Lemma 0F3C","summary":"Let f : X → Y be a locally quasi-finite morphism. Let w : X → Z be a weighting. If w(x) > 0 for all x ∈ X, then f is universally open.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism.\nLet $w : X \\to \\mathbf{Z}$ be a weighting. If $w(x) > 0$\nfor all $x \\in X$, then $f$ is universally open.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3C","source_file":"more-morphisms.tex","source_line":22311,"source_end_line":22316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22311-L22316","statement_sha256":"13a40f0464db1525130cb02ecdf005d9ebd119aa68431864080a14ffd85dbfa8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7602,"rank":7602,"depth":34,"x":2158.553,"y":606.869,"cluster":"scheme-morphisms"},{"id":"stacks:0F3D","tag":"0F3D","title":"Weightings · Lemma 0F3D","summary":"Let f : X → Y be a morphism of schemes. Assume f is locally quasi-finite, locally of finite presentation, and flat. Then there is a positive weighting w : X → Z_> 0 of f given by the rule that sends x ∈ X lying over y ∈ Y to w(x) = length_O_X, x (O_X, x/ m_y O_X, x) [kappa(x) : kappa(y)]_i where [kappa' : kappa]_i is the inseparable degree (Fields, Definition [Tag 030L]).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume $f$ is\nlocally quasi-finite, locally of finite presentation, and flat.\nThen there is a positive weighting $w : X \\to \\mathbf{Z}_{> 0}$ of $f$\ngiven by the rule that sends $x \\in X$ lying over $y \\in Y$ to\n$$\nw(x) =\n\\text{length}_{\\mathcal{O}_{X, x}}\n(\\mathcal{O}_{X, x}/\\mathfrak m_y \\mathcal{O}_{X, x})\n[\\kappa(x) : \\kappa(y)]_i\n$$\nwhere $[\\kappa' : \\kappa]_i$ is the inseparable degree\n(Fields, Definition \\ref{fields-definition-insep-degree}).","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3D","source_file":"more-morphisms.tex","source_line":22336,"source_end_line":22350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22336-L22350","statement_sha256":"a31a97f749386b21c7f190f643bc8fa15a1ff775f610f947ae34a78cc418d3eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7603,"rank":7603,"depth":6,"x":1994.0,"y":806.926,"cluster":"scheme-morphisms"},{"id":"stacks:0F3E","tag":"0F3E","title":"Weightings · Lemma 0F3E","summary":"Let f : X → Y be a morphism of schemes. Assume • f is locally quasi-finite, and • Y is geometrically unibranch and locally Noetherian. Then there is a weighting w : X → Z_≥ 0 given by the rule that sends x ∈ X lying over y ∈ Y to the \"generic separable degree\" of O_X, x^sh over O_Y, y^sh.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite, and\n\\item $Y$ is geometrically unibranch and locally Noetherian.\n\\end{enumerate}\nThen there is a weighting $w : X \\to \\mathbf{Z}_{\\geq 0}$ given by\nthe rule that sends $x \\in X$ lying over $y \\in Y$ to the\n``generic separable degree''\nof $\\mathcal{O}_{X, x}^{sh}$ over $\\mathcal{O}_{Y, y}^{sh}$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3E","source_file":"more-morphisms.tex","source_line":22409,"source_end_line":22420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22409-L22420","statement_sha256":"77895da61f456f320b5e7f2f0419e3d6683c961b556179daf9c3d5543e1950d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7604,"rank":7604,"depth":53,"x":1954.442,"y":565.93,"cluster":"scheme-morphisms"},{"id":"stacks:0F3G","tag":"0F3G","title":"More on weightings · Lemma 0F3G","summary":"Let f : X → Y be a locally quasi-finite morphism. Let w : X → Z be a weighting of f. Then the level sets of the function w are locally constructible in X.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism.\nLet $w : X \\to \\mathbf{Z}$ be a weighting of $f$. Then\nthe level sets of the function $w$ are locally constructible in $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"More on weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3G","source_file":"more-morphisms.tex","source_line":22513,"source_end_line":22518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22513-L22518","statement_sha256":"d092d316568db4932b52ab557c28bfa412ac23b80dfb07c6e5822b71e1d3c46b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7605,"rank":7605,"depth":50,"x":2177.512,"y":721.258,"cluster":"scheme-morphisms"},{"id":"stacks:0F3H","tag":"0F3H","title":"More on weightings · Lemma 0F3H","summary":"Let f : X → Y be a locally quasi-finite morphism of finite presentation. Let w : X → Z be a weighting of f. Then the level sets of the function int_f w are locally constructible in Y.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism of finite\npresentation. Let $w : X \\to \\mathbf{Z}$ be a weighting of $f$. Then\nthe level sets of the function $\\int_f w$ are locally constructible in $Y$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"More on weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3H","source_file":"more-morphisms.tex","source_line":22618,"source_end_line":22623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22618-L22623","statement_sha256":"8411dfb7d753c6476073f781fda08e229fc4fb230ad787d7e903d5b7d0b495a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7606,"rank":7606,"depth":51,"x":1887.985,"y":733.302,"cluster":"scheme-morphisms"},{"id":"stacks:0F3I","tag":"0F3I","title":"More on weightings · Lemma 0F3I","summary":"Let f : X → Y be a locally quasi-finite morphism. Let w : X → Z_> 0 be a positive weighting of f. Then w is upper semi-continuous.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism.\nLet $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$.\nThen $w$ is upper semi-continuous.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"More on weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3I","source_file":"more-morphisms.tex","source_line":22671,"source_end_line":22676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22671-L22676","statement_sha256":"cfc07419bd7ee19c966b378f3ebe1a87d3722a7116c2d1536fd41d263f7c477c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7607,"rank":7607,"depth":32,"x":2091.882,"y":560.061,"cluster":"scheme-morphisms"},{"id":"stacks:0F3J","tag":"0F3J","title":"More on weightings · Lemma 0F3J","summary":"Let f : X → Y be a separated, locally quasi-finite morphism with finite fibres. Let w : X → Z_> 0 be a positive weighting of f. Then int_f w is lower semi-continuous.","statement_latex":"Let $f : X \\to Y$ be a separated, locally quasi-finite morphism\nwith finite fibres.\nLet $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$.\nThen $\\int_f w$ is lower semi-continuous.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"More on weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3J","source_file":"more-morphisms.tex","source_line":22694,"source_end_line":22700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22694-L22700","statement_sha256":"69960666ee304ba1201f076f78009a650ee9112d3edff2a2f242accf90de30d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7608,"rank":7608,"depth":33,"x":2080.843,"y":803.609,"cluster":"scheme-morphisms"},{"id":"stacks:0F3K","tag":"0F3K","title":"More on weightings · Lemma 0F3K","summary":"Let f : X → Y be a locally quasi-finite morphism with X quasi-compact. Let w : X → Z be a weighting of f. Then int_f w attains its maximum.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism\nwith $X$ quasi-compact. Let $w : X \\to \\mathbf{Z}$ be a weighting of $f$.\nThen $\\int_f w$ attains its maximum.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"More on weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3K","source_file":"more-morphisms.tex","source_line":22730,"source_end_line":22735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22730-L22735","statement_sha256":"c450ce7b41bddc3d8dc7bbb1fdb9752a0dea664716577995f179e373708e7953","origin":"The Stacks Project","memory_eligible":false,"source_rank":7609,"rank":7609,"depth":51,"x":1893.046,"y":617.675,"cluster":"scheme-morphisms"},{"id":"stacks:0F3L","tag":"0F3L","title":"More on weightings · Lemma 0F3L","summary":"Let f : X → Y be a separated, locally quasi-finite morphism. Let w : X → Z_> 0 be a positive weighting of f. Assume int_w f attains its maximum d and let Y_d ⊂ Y be the open set of points y with (int_f w)(y) = d. Then the morphism f^-1(Y_d) → Y_d is finite.","statement_latex":"Let $f : X \\to Y$ be a separated, locally quasi-finite morphism.\nLet $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$.\nAssume $\\int_w f$ attains its maximum $d$ and let $Y_d \\subset Y$\nbe the open set of points $y$ with $(\\int_f w)(y) = d$. Then\nthe morphism $f^{-1}(Y_d) \\to Y_d$ is finite.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"More on weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3L","source_file":"more-morphisms.tex","source_line":22748,"source_end_line":22755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22748-L22755","statement_sha256":"e939e647e27f65cea2e81a46f43359039bba44a2bfb6f3117f702a2f15470f44","origin":"The Stacks Project","memory_eligible":false,"source_rank":7610,"rank":7610,"depth":44,"x":2181.174,"y":648.233,"cluster":"scheme-morphisms"},{"id":"stacks:0F3M","tag":"0F3M","title":"More on weightings · Lemma 0F3M","summary":"Let A → B be a ring map which is finite and of finite presentation. There exists a finitely presented ring map A → A_univ and an idempotent e_univ ∈ B ⊗_A A_univ such that for any ring map A → A' and idempotent e ∈ B ⊗_A A' there is a ring map A_univ → A' mapping e_univ to e.","statement_latex":"Let $A \\to B$ be a ring map which is finite and of finite presentation.\nThere exists a finitely presented ring map $A \\to A_{univ}$\nand an idempotent $e_{univ} \\in B \\otimes_A A_{univ}$\nsuch that for any ring map $A \\to A'$ and idempotent $e \\in B \\otimes_A A'$\nthere is a ring map $A_{univ} \\to A'$ mapping $e_{univ}$ to $e$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"More on weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3M","source_file":"more-morphisms.tex","source_line":22790,"source_end_line":22797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22790-L22797","statement_sha256":"227bb37ff8dde9e206794e58aa9f9089083e47ac6e963628259fefc4b2019069","origin":"The Stacks Project","memory_eligible":false,"source_rank":7611,"rank":7611,"depth":4,"x":1944.035,"y":789.252,"cluster":"scheme-morphisms"},{"id":"stacks:0F3N","tag":"0F3N","title":"More on weightings · Lemma 0F3N","summary":"Let X → Y be a morphism of affine schemes which is quasi-finite and of finite presentation. There exists a morphism Y_univ → Y of finite presentation and an open subscheme U_univ ⊂ Y_univ ×_Y X such that U_univ → Y_univ is finite with the following property: given any morphism Y' → Y of affine schemes and an open subscheme U' ⊂ Y' ×_Y X such that U' → Y' is finite, there exists a morphism Y' → Y_univ such that the inverse image of U_univ is U'.","statement_latex":"Let $X \\to Y$ be a morphism of affine schemes which is quasi-finite and\nof finite presentation. There exists a morphism $Y_{univ} \\to Y$\nof finite presentation and an open subscheme\n$U_{univ} \\subset Y_{univ} \\times_Y X$ such that\n$U_{univ} \\to Y_{univ}$ is finite with the following property:\ngiven any morphism $Y' \\to Y$ of affine schemes\nand an open subscheme $U' \\subset Y' \\times_Y X$\nsuch that $U' \\to Y'$ is finite, there exists a morphism\n$Y' \\to Y_{univ}$ such that the inverse image of $U_{univ}$ is $U'$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"More on weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3N","source_file":"more-morphisms.tex","source_line":22834,"source_end_line":22845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22834-L22845","statement_sha256":"78a2c8e91470ff690141c9c4fb9fdc9f98becae3272242a87357788c771206c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7612,"rank":7612,"depth":35,"x":2005.521,"y":550.604,"cluster":"scheme-morphisms"},{"id":"stacks:0F3P","tag":"0F3P","title":"More on weightings · Lemma 0F3P","summary":"Let Y = lim Y_i be a directed limit of affine schemes. Let 0 ∈ I and let f_0 : X_0 → Y_0 be a morphism of affine schemes which is quasi-finite and of finite presentation. Let f : X → Y and f_i : X_i → Y_i for i ≥ 0 be the base changes of f_0. If w : X → Z is a weighting of f, then for sufficiently large i there exists a weighting w_i : X_i → Z of f_i whose pullback to X is w.","statement_latex":"Let $Y = \\lim Y_i$ be a directed limit of affine schemes. Let $0 \\in I$\nand let $f_0 : X_0 \\to Y_0$ be a morphism of affine schemes which is\nquasi-finite and of finite presentation. Let $f : X  \\to Y$\nand $f_i : X_i \\to Y_i$ for $i \\geq 0$ be the base changes of $f_0$.\nIf $w : X \\to \\mathbf{Z}$ is a weighting of $f$, then for sufficiently\nlarge $i$ there exists a weighting $w_i : X_i \\to \\mathbf{Z}$\nof $f_i$ whose pullback to $X$ is $w$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"More on weightings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3P","source_file":"more-morphisms.tex","source_line":22890,"source_end_line":22899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22890-L22899","statement_sha256":"05cb9f75a049fbd7141b23a7bdbc7c8552727d76cce22322eda5a768d7177177","origin":"The Stacks Project","memory_eligible":false,"source_rank":7613,"rank":7613,"depth":52,"x":2152.16,"y":761.56,"cluster":"scheme-morphisms"},{"id":"stacks:0F3R","tag":"0F3R","title":"Weightings and affine stratification numbers · Lemma 0F3R","summary":"Let f : X → Y be a morphism of affine schemes which is quasi-finite and of finite presentation. Let w : X → Z_> 0 be a positive weighting of f. Let d < ∞ be the maximum value of int_f w. The open Y_d = (y ∈ Y mid (textstyleint_f w)(y) = d ) of Y is affine.","statement_latex":"Let $f : X \\to Y$ be a morphism of affine schemes which is\nquasi-finite and of finite presentation.\nLet $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$.\nLet $d < \\infty$ be the maximum value of $\\int_f w$. The open\n$$\nY_d = \\{y \\in Y \\mid (\\textstyle{\\int}_f w)(y) = d \\}\n$$\nof $Y$ is affine.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings and affine stratification numbers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3R","source_file":"more-morphisms.tex","source_line":22965,"source_end_line":22975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L22965-L22975","statement_sha256":"1eb49ee1f390c9bf97d614f77b042e40d1c7104189d70cf8da85c046062046d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7614,"rank":7614,"depth":53,"x":1874.265,"y":689.18,"cluster":"scheme-morphisms"},{"id":"stacks:0F3S","tag":"0F3S","title":"Weightings and affine stratification numbers · Proposition 0F3S","summary":"Let f : X → Y be a surjective quasi-finite morphism of schemes. Let w : X → Z_> 0 be a positive weighting of f. Assume X affine and Y separated and nonempty. Then the affine stratification number of Y is at most the number of distinct values of int_f w minus 1.","statement_latex":"Let $f : X \\to Y$ be a surjective quasi-finite morphism of schemes.\nLet $w : X \\to \\mathbf{Z}_{> 0}$ be a positive weighting of $f$.\nAssume $X$ affine and $Y$ separated\\footnote{It suffices if the\ndiagonal of $Y$ is affine.} and nonempty. Then the affine stratification\nnumber of $Y$ is at most the number of distinct values of $\\int_f w$\nminus $1$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Weightings and affine stratification numbers","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3S","source_file":"more-morphisms.tex","source_line":23079,"source_end_line":23087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23079-L23087","statement_sha256":"5ad1d79e856b8af1e73db9801a64a1372bf03e07ce84a4afc236d23e6e86d241","origin":"The Stacks Project","memory_eligible":false,"source_rank":7615,"rank":7615,"depth":54,"x":2137.502,"y":584.822,"cluster":"scheme-morphisms"},{"id":"stacks:0GTI","tag":"0GTI","title":"Completely decomposed morphisms · Definition 0GTI","summary":"A morphism f : X → Y of schemes is said to be completely decomposed if for all points y ∈ Y there is a point x ∈ X with f(x) = y such that the field extension kappa(x)/kappa(y) is trivial. A family of morphisms (f_i : X_i → Y)_i ∈ I of schemes with fixed target is said to be completely decomposed if coprod f_i : coprod Y_i → X is completely decomposed.","statement_latex":"A morphism $f : X \\to Y$ of schemes is said to be\n{\\it completely decomposed}\\footnote{This may be nonstandard terminology.}\nif for all points $y \\in Y$ there\nis a point $x \\in X$ with $f(x) = y$ such that the field\nextension $\\kappa(x)/\\kappa(y)$ is trivial.\nA family of morphisms $\\{f_i : X_i \\to Y\\}_{i \\in I}$ of\nschemes with fixed target is said to be {\\it completely decomposed}\nif $\\coprod f_i : \\coprod Y_i \\to X$ is completely decomposed.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Completely decomposed morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTI","source_file":"more-morphisms.tex","source_line":23136,"source_end_line":23146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23136-L23146","statement_sha256":"907ff089bd4aa182697cf1b1eff33e47bff2e9a16a111450ab1ec0a97eaa8b84","origin":"The Stacks Project","memory_eligible":false,"source_rank":7616,"rank":7616,"depth":0,"x":2027.268,"y":811.239,"cluster":"scheme-morphisms"},{"id":"stacks:0GTJ","tag":"0GTJ","title":"Completely decomposed morphisms · Lemma 0GTJ","summary":"The composition of two completely decomposed morphisms of schemes is completely decomposed. If (f_i : X_i → Y)_i ∈ I is completely decomposed and for each i we have a family (X_ij → X_i)_j ∈ J_i which is completely decomposed, then the family (X_ij → Y)_i ∈ I, j ∈ J_i is completely decomposed.","statement_latex":"The composition of two completely decomposed morphisms of schemes\nis completely decomposed.\nIf $\\{f_i : X_i \\to Y\\}_{i \\in I}$ is completely decomposed\nand for each $i$ we have a family $\\{X_{ij} \\to X_i\\}_{j \\in J_i}$\nwhich is completely decomposed, then the family\n$\\{X_{ij} \\to Y\\}_{i \\in I, j \\in J_i}$ is completely decomposed.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Completely decomposed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTJ","source_file":"more-morphisms.tex","source_line":23151,"source_end_line":23159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23151-L23159","statement_sha256":"1a254377c1a0687ce90db98c68b235699aa3cb65222abc0926be95e7445cd536","origin":"The Stacks Project","memory_eligible":false,"source_rank":7617,"rank":7617,"depth":0,"x":1926.43,"y":581.633,"cluster":"scheme-morphisms"},{"id":"stacks:0GTK","tag":"0GTK","title":"Completely decomposed morphisms · Lemma 0GTK","summary":"The base change of a completely decomposed morphism of schemes is completely decomposed. If (f_i : X_i → Y)_i ∈ I is completely decomposed and Y' → Y is a morphism of schemes, then (X_i ×_Y Y' → Y')_i ∈ I is completely decomposed.","statement_latex":"The base change of a completely decomposed morphism of schemes\nis completely decomposed.\nIf $\\{f_i : X_i \\to Y\\}_{i \\in I}$ is completely decomposed\nand $Y' \\to Y$ is a morphism of schemes, then\n$\\{X_i \\times_Y Y' \\to Y'\\}_{i \\in I}$ is completely\ndecomposed.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Completely decomposed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTK","source_file":"more-morphisms.tex","source_line":23165,"source_end_line":23173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23165-L23173","statement_sha256":"88436b0e41af7b04575af6b9092e94f052579d02821b4910fca1388849886867","origin":"The Stacks Project","memory_eligible":false,"source_rank":7618,"rank":7618,"depth":2,"x":2185.542,"y":693.772,"cluster":"scheme-morphisms"},{"id":"stacks:0GTL","tag":"0GTL","title":"Completely decomposed morphisms · Lemma 0GTL","summary":"[EHIK] Let f : X → Y be a morphism of schemes. Assume f is completely decomposed, f is locally of finite presentation, and Y is quasi-compact and quasi-separated. Then there exist n ≥ 0 and morphisms Z_i → Y, i = 1, …, n with the following properties • coprod Z_i → Y is surjective, • Z_i → Y is an immersion for all i, • Z_i → Y is of finite presentation for all i, and • the base change X ×_Y Z_i → Z_i has a section for all i.","statement_latex":"\\begin{reference}\n\\cite[Lemma 2.1.2]{EHIK}\n\\end{reference}\nLet $f : X \\to Y$ be a morphism of schemes. Assume\n$f$ is completely decomposed,\n$f$ is locally of finite presentation, and\n$Y$ is quasi-compact and quasi-separated.\nThen there exist $n \\geq 0$ and morphisms\n$Z_i \\to Y$, $i = 1, \\ldots, n$ with the following properties\n\\begin{enumerate}\n\\item $\\coprod Z_i \\to Y$ is surjective,\n\\item $Z_i \\to Y$ is an immersion for all $i$,\n\\item $Z_i \\to Y$ is of finite presentation for all $i$, and\n\\item the base change $X \\times_Y Z_i \\to Z_i$ has a section\nfor all $i$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Completely decomposed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTL","source_file":"more-morphisms.tex","source_line":23186,"source_end_line":23204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23186-L23204","statement_sha256":"19cf1631e61914bb87c414884565eb3f3127d47760b979aa958769215701c5e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7619,"rank":7619,"depth":26,"x":1904.176,"y":758.137,"cluster":"scheme-morphisms"},{"id":"stacks:0GTM","tag":"0GTM","title":"Completely decomposed morphisms · Lemma 0GTM","summary":"Let S = lim_λ ∈ Lambda S_λ be a limit of a directed system of schemes with affine transition morphisms. Let 0 ∈ Lambda and let f_0 : X_0 → Y_0 be a morphism of schemes over S_0. For λ ≥ 0 let f_λ : X_λ → Y_λ be the base change of f_0 to S_λ and let f : X → Y be the base change of f_0 to S. If • f is completely decomposed, • Y_0 is quasi-compact and quasi-separated, and • f_0 is locally of finite presentation, then there exists an λ ≥ 0 such that f_λ is completely decomposed.","statement_latex":"Let $S = \\lim_{\\lambda \\in \\Lambda} S_\\lambda$\nbe a limit of a directed system of schemes with affine transition morphisms.\nLet $0 \\in \\Lambda$ and let $f_0 : X_0 \\to Y_0$\nbe a morphism of schemes over $S_0$.\nFor $\\lambda \\geq 0$ let $f_\\lambda : X_\\lambda \\to Y_\\lambda$\nbe the base change of $f_0$ to $S_\\lambda$ and\nlet $f : X \\to Y$ be the base change of $f_0$ to $S$. If\n\\begin{enumerate}\n\\item $f$ is completely decomposed,\n\\item $Y_0$ is quasi-compact and quasi-separated, and\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen there exists an $\\lambda \\geq 0$ such that $f_\\lambda$\nis completely decomposed.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Completely decomposed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTM","source_file":"more-morphisms.tex","source_line":23239,"source_end_line":23255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23239-L23255","statement_sha256":"ea83257a1b64095ca809466ae8d605e2f94ba6286a762c16bfc2d8d447548a7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7620,"rank":7620,"depth":28,"x":2059.958,"y":550.932,"cluster":"scheme-morphisms"},{"id":"stacks:0GTP","tag":"0GTP","title":"Families of ample invertible modules · Lemma 0GTP","summary":"Let f : X → Y be a morphism of schemes. Assume • Y has an ample family of invertible modules, • there exists an f-ample invertible module on X. Then X has an ample family of invertible modules.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $Y$ has an ample family of invertible modules,\n\\item there exists an $f$-ample invertible module on $X$.\n\\end{enumerate}\nThen $X$ has an ample family of invertible modules.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Families of ample invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTP","source_file":"more-morphisms.tex","source_line":23302,"source_end_line":23310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23302-L23310","statement_sha256":"12c5ebeafac44c829e87b1423f093800b0b061849cdb3aeb61a1f10377548b28","origin":"The Stacks Project","memory_eligible":false,"source_rank":7621,"rank":7621,"depth":18,"x":2111.738,"y":792.221,"cluster":"scheme-morphisms"},{"id":"stacks:0GTQ","tag":"0GTQ","title":"Families of ample invertible modules · Lemma 0GTQ","summary":"Let f : X → Y be an affine or quasi-affine morphism of schemes. If Y has an ample family of invertible modules, so does X.","statement_latex":"Let $f : X \\to Y$ be an affine or quasi-affine morphism of schemes.\nIf $Y$ has an ample family of invertible modules, so does $X$.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Families of ample invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTQ","source_file":"more-morphisms.tex","source_line":23337,"source_end_line":23341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23337-L23341","statement_sha256":"fc9f7d4ebd85e22e4c6bda6af4a420e6e47a0e70d0a9f63b9a340a342327ab3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7622,"rank":7622,"depth":22,"x":1879.418,"y":643.615,"cluster":"scheme-morphisms"},{"id":"stacks:0GTS","tag":"0GTS","title":"Blowing up and ample families of invertible modules · Lemma 0GTS","summary":"Let X be a scheme. Suppose given effective Cartier divisors D_1, …, D_m on X and invertible modules L_1, …, L_m such that ⋂ D_i = ∅ and L_i|_X setminus D_i is ample. Then X has an ample family of invertible modules.","statement_latex":"Let $X$ be a scheme. Suppose given effective Cartier divisors\n$D_1, \\ldots, D_m$ on $X$ and invertible modules\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_m$ such that\n$\\bigcap D_i = \\emptyset$ and $\\mathcal{L}_i|_{X \\setminus D_i}$\nis ample. Then $X$ has an ample family of invertible modules.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Blowing up and ample families of invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTS","source_file":"more-morphisms.tex","source_line":23360,"source_end_line":23367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23360-L23367","statement_sha256":"5e94d858b64d7bb3c27c7f57fb8c5774e6f992555eb4b77fda5194f43e798702","origin":"The Stacks Project","memory_eligible":false,"source_rank":7623,"rank":7623,"depth":18,"x":2170.358,"y":621.361,"cluster":"scheme-morphisms"},{"id":"stacks:0GTT","tag":"0GTT","title":"Blowing up and ample families of invertible modules · Lemma 0GTT","summary":"[Gross-thesis] Let X be a quasi-compact and quasi-separated scheme with finitely many irreducible components. There exists a quasi-compact dense open U ⊂ X and a U-admissible blowing up X' → X such that the scheme X' has an ample family of invertible modules.","statement_latex":"\\begin{reference}\n\\cite[Proposition 1.3.1]{Gross-thesis}\n\\end{reference}\nLet $X$ be a quasi-compact and quasi-separated scheme with finitely\nmany irreducible components. There exists a quasi-compact dense open\n$U \\subset X$ and a $U$-admissible blowing up $X' \\to X$ such that the\nscheme $X'$ has an ample family of invertible modules.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Blowing up and ample families of invertible modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTT","source_file":"more-morphisms.tex","source_line":23391,"source_end_line":23400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23391-L23400","statement_sha256":"b39bf42676ec22880a695020a0a14b8f80f526971213de5b0906a9d828684436","origin":"The Stacks Project","memory_eligible":false,"source_rank":7624,"rank":7624,"depth":22,"x":1973.634,"y":802.935,"cluster":"scheme-morphisms"},{"id":"stacks:0GTU","tag":"0GTU","title":"Blowing up and ample families of invertible modules · Proposition 0GTU","summary":"Let X be a quasi-compact and quasi-separated scheme. There exists a morphism f : Y → X which is of finite presentation, proper, and completely decomposed (Definition [Tag 0GTI]) such that the scheme Y has an ample family of invertible modules.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. There exists a\nmorphism $f : Y \\to X$ which is of finite presentation, proper, and\ncompletely decomposed (Definition \\ref{definition-cd-morphism})\nsuch that the scheme $Y$ has an ample family of invertible modules.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"Blowing up and ample families of invertible modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTU","source_file":"more-morphisms.tex","source_line":23440,"source_end_line":23446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23440-L23446","statement_sha256":"531010f27b5534a76ea9ce59d660731d12f90c5883dc80c4db195809f1a5324b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7625,"rank":7625,"depth":25,"x":1972.678,"y":557.313,"cluster":"scheme-morphisms"},{"id":"stacks:0H2P","tag":"0H2P","title":"The extensive criterion for closed immersions · Lemma 0H2P","summary":"A morphism f : X → Y of affine schemes is a closed immersion if and only if for every injective ring map A → B and commutative square xymatrix Spec(B) ar[d] ar[r] & X ar[d]^f Spec(A) ar[r] ar@..>[ur] & Y there exists a lift Spec(A) → X making the two triangles commute.","statement_latex":"A morphism $f : X \\to Y$ of affine schemes is a closed immersion \nif and only if for every injective ring map $A \\to B$ and commutative \nsquare\n$$\n\\xymatrix{\n\\Spec(B) \\ar[d] \\ar[r] & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[r] \\ar@{..>}[ur] & Y\n}\n$$\nthere exists a lift $\\Spec(A) \\to X$ making the two triangles commute.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The extensive criterion for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2P","source_file":"more-morphisms.tex","source_line":23495,"source_end_line":23507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23495-L23507","statement_sha256":"e86d00e526bc91ab80745b160732c43c65256a7f55cd181c57af212c8662bf7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7626,"rank":7626,"depth":0,"x":2170.991,"y":737.966,"cluster":"scheme-morphisms"},{"id":"stacks:0H2Q","tag":"0H2Q","title":"The extensive criterion for closed immersions · Lemma 0H2Q","summary":"Let X be a scheme. If the canonical morphism X → Spec(Γ(X, O_X)) of Schemes, Lemma [Tag 01I1] has a retraction, then X is an affine scheme.","statement_latex":"Let $X$ be a scheme.\nIf the canonical morphism $X \\to \\Spec(\\Gamma(X, \\mathcal{O}_X))$\nof Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}\nhas a retraction, then $X$ is an affine scheme.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The extensive criterion for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2Q","source_file":"more-morphisms.tex","source_line":23546,"source_end_line":23552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23546-L23552","statement_sha256":"820aa0eb2bcbb004212ad8f85718376d200afc1b40fde52cd233c812a7818209","origin":"The Stacks Project","memory_eligible":false,"source_rank":7627,"rank":7627,"depth":11,"x":1879.356,"y":717.274,"cluster":"scheme-morphisms"},{"id":"stacks:0H2R","tag":"0H2R","title":"The extensive criterion for closed immersions · Lemma 0H2R","summary":"Let X be a scheme. Let f : X → S = Spec(Γ(X, O_X)) be the canonical morphism of Schemes, Lemma [Tag 01I1]. The largest quasi-coherent O_S-module contained in the kernel of f^sharp : O_S → f_*O_X is zero. If X is quasi-compact, then f^sharp is injective. In particular, if X is quasi-compact, then f is a dominant morphism.","statement_latex":"Let $X$ be a scheme. Let $f : X \\to S = \\Spec(\\Gamma(X, \\mathcal{O}_X))$\nbe the canonical morphism of\nSchemes, Lemma \\ref{schemes-lemma-morphism-into-affine}.\nThe largest quasi-coherent $\\mathcal{O}_S$-module contained\nin the kernel of $f^\\sharp : \\mathcal{O}_S \\to f_*\\mathcal{O}_X$\nis zero. If $X$ is quasi-compact, then $f^\\sharp$ is injective.\nIn particular, if $X$ is quasi-compact, then $f$ is a \ndominant morphism.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The extensive criterion for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2R","source_file":"more-morphisms.tex","source_line":23565,"source_end_line":23575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23565-L23575","statement_sha256":"668c1d76e2155691a1c279f36d02f6a9e520d00c76129936c83de20aa4c2f314","origin":"The Stacks Project","memory_eligible":false,"source_rank":7628,"rank":7628,"depth":19,"x":2111.143,"y":566.987,"cluster":"scheme-morphisms"},{"id":"stacks:0H2S","tag":"0H2S","title":"The extensive criterion for closed immersions · Lemma 0H2S","summary":"Let f: X → Y be a quasi-compact morphism of schemes. Then f is a closed immersion if and only if for every injective ring map A → B and commutative square xymatrix Spec(B) ar[d] ar[r] & X ar[d]^f Spec(A) ar[r] ar@..>[ur] & Y there exists a lift Spec A → X making the diagram commute.","statement_latex":"Let $f: X \\to Y$ be a quasi-compact morphism of schemes.\nThen $f$ is a closed immersion if and only if for every injective \nring map $A \\to B$ and commutative square\n$$\n\\xymatrix{\n\\Spec(B) \\ar[d] \\ar[r] & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[r] \\ar@{..>}[ur] & Y\n}\n$$\nthere exists a lift $\\Spec A \\to X$ making the diagram commute.","area":"Scheme Morphisms","chapter":"More on Morphisms","chapter_id":"more-morphisms","section":"The extensive criterion for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2S","source_file":"more-morphisms.tex","source_line":23598,"source_end_line":23610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-morphisms.tex#L23598-L23610","statement_sha256":"929be112875703375de915054de25361c9c7948c8842c59093f0cad7f241ac48","origin":"The Stacks Project","memory_eligible":false,"source_rank":7629,"rank":7629,"depth":12,"x":2061.062,"y":809.432,"cluster":"scheme-morphisms"},{"id":"stacks:057R","tag":"057R","title":"Lemmas on étale localization · Lemma 057R","summary":"Let i : Z → X be a closed immersion of affine schemes. Let Z' → Z be an étale morphism with Z' affine. Then there exists an étale morphism X' → X with X' affine such that Z' ≅ Z ×_X X' as schemes over Z.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of affine schemes.\nLet $Z' \\to Z$ be an \\'etale morphism with $Z'$ affine.\nThen there exists an \\'etale morphism $X' \\to X$ with $X'$\naffine such that $Z' \\cong Z \\times_X X'$ as schemes over $Z$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Lemmas on étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057R","source_file":"flat.tex","source_line":83,"source_end_line":89,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L83-L89","statement_sha256":"cf181ed3f29d94e21b8ff7ae139337f6e9fd930687150e88de67c42c8c5cb35c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7630,"rank":7630,"depth":7,"x":1902.955,"y":602.151,"cluster":"scheme-morphisms"},{"id":"stacks:05H2","tag":"05H2","title":"Lemmas on étale localization · Lemma 05H2","summary":"Let xymatrix X ar[d] & X' ar[l] ar[d] S & S' ar[l] be a commutative diagram of schemes with X' → X and S' → S étale. Let s' ∈ S' be a point. Then X' ×_S' Spec(O_S', s') → X ×_S Spec(O_S', s') is étale.","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l] \\ar[d] \\\\\nS & S' \\ar[l]\n}\n$$\nbe a commutative diagram of schemes with $X' \\to X$ and $S' \\to S$ \\'etale.\nLet $s' \\in S'$ be a point. Then\n$$\nX' \\times_{S'} \\Spec(\\mathcal{O}_{S', s'})\n\\longrightarrow\nX \\times_S \\Spec(\\mathcal{O}_{S', s'})\n$$\nis \\'etale.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Lemmas on étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05H2","source_file":"flat.tex","source_line":96,"source_end_line":113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L96-L113","statement_sha256":"d7f9e1d3648d51f40de34fd860b6ee6a57f4909a0f9f147f74d02bd58d3c6fa9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7631,"rank":7631,"depth":43,"x":2186.352,"y":665.31,"cluster":"scheme-morphisms"},{"id":"stacks:05B9","tag":"05B9","title":"Lemmas on étale localization · Lemma 05B9","summary":"Let X → T → S be morphisms of schemes with T → S étale. Let F be a quasi-coherent O_X-module. Let x ∈ X be a point. Then F flat over S at x ⇔ F flat over T at x In particular F is flat over S if and only if F is flat over T.","statement_latex":"Let $X \\to T \\to S$ be morphisms of schemes with $T \\to S$ \\'etale.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in X$ be a point. Then\n$$\n\\mathcal{F}\\text{ flat over }S\\text{ at }x\n\\Leftrightarrow\n\\mathcal{F}\\text{ flat over }T\\text{ at }x\n$$\nIn particular $\\mathcal{F}$ is flat over $S$ if and only if $\\mathcal{F}$\nis flat over $T$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Lemmas on étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05B9","source_file":"flat.tex","source_line":123,"source_end_line":135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L123-L135","statement_sha256":"7c1885e1acd67bb860a62c75d6c792b3b769d59110945c2675bdacf01f3d3891","origin":"The Stacks Project","memory_eligible":false,"source_rank":7632,"rank":7632,"depth":37,"x":1926.477,"y":779.593,"cluster":"scheme-morphisms"},{"id":"stacks:05BA","tag":"05BA","title":"Lemmas on étale localization · Lemma 05BA","summary":"Let T → S be an étale morphism. Let t ∈ T with image s ∈ S. Let M be a O_T, t-module. Then M flat over O_S, s ⇔ M flat over O_T, t.","statement_latex":"Let $T \\to S$ be an \\'etale morphism. Let $t \\in T$ with image $s \\in S$.\nLet $M$ be a $\\mathcal{O}_{T, t}$-module. Then\n$$\nM\\text{ flat over }\\mathcal{O}_{S, s}\n\\Leftrightarrow\nM\\text{ flat over }\\mathcal{O}_{T, t}.\n$$","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Lemmas on étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05BA","source_file":"flat.tex","source_line":159,"source_end_line":168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L159-L168","statement_sha256":"e2f1d74ed1ef519c40db49856e450b0196eef392b9117c7712be9e24998510a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7633,"rank":7633,"depth":38,"x":2026.245,"y":547.764,"cluster":"scheme-morphisms"},{"id":"stacks:05VL","tag":"05VL","title":"Lemmas on étale localization · Lemma 05VL","summary":"Let S be a scheme and s ∈ S a point. Denote O_S, s^h (resp. O_S, s^sh) the henselization (resp. strict henselization), see Algebra, Definition [Tag 04GQ]. Let M^sh be a O_S, s^sh-module. The following are equivalent • M^sh is flat over O_S, s, • M^sh is flat over O_S, s^h, and • M^sh is flat over O_S, s^sh. If M^sh = M^h ⊗_O_S, s^h O_S, s^sh this is also equivalent to • [(4)] M^h is flat over O_S, s, and • [(5)] M^h is flat over O_S, s^h. If M^h = M ⊗_O_S, s O_S, s^h this…","statement_latex":"Let $S$ be a scheme and $s \\in S$ a point. Denote $\\mathcal{O}_{S, s}^h$\n(resp.\\ $\\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\\ strict\nhenselization), see\nAlgebra, Definition \\ref{algebra-definition-henselization}.\nLet $M^{sh}$ be a $\\mathcal{O}_{S, s}^{sh}$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M^{sh}$ is flat over $\\mathcal{O}_{S, s}$,\n\\item $M^{sh}$ is flat over $\\mathcal{O}_{S, s}^h$, and\n\\item $M^{sh}$ is flat over $\\mathcal{O}_{S, s}^{sh}$.\n\\end{enumerate}\nIf $M^{sh} = M^h \\otimes_{\\mathcal{O}_{S, s}^h} \\mathcal{O}_{S, s}^{sh}$\nthis is also equivalent to\n\\begin{enumerate}\n\\item[(4)] $M^h$ is flat over $\\mathcal{O}_{S, s}$, and\n\\item[(5)] $M^h$ is flat over $\\mathcal{O}_{S, s}^h$.\n\\end{enumerate}\nIf $M^h = M \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^h$\nthis is also equivalent to\n\\begin{enumerate}\n\\item[(6)] $M$ is flat over $\\mathcal{O}_{S, s}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Lemmas on étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VL","source_file":"flat.tex","source_line":182,"source_end_line":206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L182-L206","statement_sha256":"16a02cda060032dde41ace3793cad50b9fb0dc6d426dc6993f6a23e718ec7888","origin":"The Stacks Project","memory_eligible":false,"source_rank":7634,"rank":7634,"depth":46,"x":2139.156,"y":775.419,"cluster":"scheme-morphisms"},{"id":"stacks:0DK0","tag":"0DK0","title":"Lemmas on étale localization · Lemma 0DK0","summary":"Let S be a scheme and s ∈ S a point. Denote O_S, s^h (resp. O_S, s^sh) the henselization (resp. strict henselization), see Algebra, Definition [Tag 04GQ]. Let M^sh be an object of D(O_S, s^sh). Let a, b ∈ Z. The following are equivalent • M^sh has tor amplitude in [a, b] over O_S, s, • M^sh has tor amplitude in [a, b] over O_S, s^h, and • M^sh has tor amplitude in [a, b] over O_S, s^sh. If M^sh = M^h ⊗_O_S, s^h^L O_S, s^sh for M^h ∈ D(O_S, s^h) this is also equivalent to…","statement_latex":"Let $S$ be a scheme and $s \\in S$ a point. Denote $\\mathcal{O}_{S, s}^h$\n(resp.\\ $\\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\\ strict\nhenselization), see\nAlgebra, Definition \\ref{algebra-definition-henselization}.\nLet $M^{sh}$ be an object of $D(\\mathcal{O}_{S, s}^{sh})$.\nLet $a, b \\in \\mathbf{Z}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $M^{sh}$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}$,\n\\item $M^{sh}$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}^h$, and\n\\item $M^{sh}$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}^{sh}$.\n\\end{enumerate}\nIf $M^{sh} =\nM^h \\otimes_{\\mathcal{O}_{S, s}^h}^\\mathbf{L} \\mathcal{O}_{S, s}^{sh}$\nfor $M^h \\in D(\\mathcal{O}_{S, s}^h)$ this is also equivalent to\n\\begin{enumerate}\n\\item[(4)] $M^h$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}$, and\n\\item[(5)] $M^h$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}^h$.\n\\end{enumerate}\nIf $M^h = M \\otimes_{\\mathcal{O}_{S, s}}^\\mathbf{L} \\mathcal{O}_{S, s}^h$\nfor $M \\in D(\\mathcal{O}_{S, s})$\nthis is also equivalent to\n\\begin{enumerate}\n\\item[(6)] $M$ has tor amplitude in $[a, b]$ over $\\mathcal{O}_{S, s}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Lemmas on étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DK0","source_file":"flat.tex","source_line":243,"source_end_line":270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L243-L270","statement_sha256":"9815e57d1b63afe2b742e2582990a3b4ee232cb15bb2a864b288ea2cc9344404","origin":"The Stacks Project","memory_eligible":false,"source_rank":7635,"rank":7635,"depth":47,"x":1872.709,"y":671.574,"cluster":"scheme-morphisms"},{"id":"stacks:05FN","tag":"05FN","title":"Lemmas on étale localization · Lemma 05FN","summary":"Let g : T → S be a finite flat morphism of schemes. Let G be a quasi-coherent O_S-module. Let t ∈ T be a point with image s ∈ S. Then t ∈ WeakAss(g^*G) ⇔ s ∈ WeakAss(G)","statement_latex":"Let $g : T \\to S$ be a finite flat morphism of schemes.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_S$-module.\nLet $t \\in T$ be a point with image $s \\in S$. Then\n$$\nt \\in \\text{WeakAss}(g^*\\mathcal{G})\n\\Leftrightarrow\ns \\in \\text{WeakAss}(\\mathcal{G})\n$$","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Lemmas on étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FN","source_file":"flat.tex","source_line":311,"source_end_line":321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L311-L321","statement_sha256":"2797b0ebfa707d29e2b4d4a9b34c06328c6a7770bb4fc7a5a0be9d800466816e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7636,"rank":7636,"depth":11,"x":2152.812,"y":596.927,"cluster":"scheme-morphisms"},{"id":"stacks:05FP","tag":"05FP","title":"Lemmas on étale localization · Lemma 05FP","summary":"Let h : U → S be an étale morphism of schemes. Let G be a quasi-coherent O_S-module. Let u ∈ U be a point with image s ∈ S. Then u ∈ WeakAss(h^*G) ⇔ s ∈ WeakAss(G)","statement_latex":"Let $h : U \\to S$ be an \\'etale morphism of schemes.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_S$-module.\nLet $u \\in U$ be a point with image $s \\in S$. Then\n$$\nu \\in \\text{WeakAss}(h^*\\mathcal{G})\n\\Leftrightarrow\ns \\in \\text{WeakAss}(\\mathcal{G})\n$$","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Lemmas on étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FP","source_file":"flat.tex","source_line":352,"source_end_line":362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L352-L362","statement_sha256":"671abb0677d61468f8354aa03d10d293dbeb985efe968845e80362d1b31e0ccc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7637,"rank":7637,"depth":44,"x":2006.234,"y":810.999,"cluster":"scheme-morphisms"},{"id":"stacks:0CTU","tag":"0CTU","title":"Lemmas on étale localization · Lemma 0CTU","summary":"Let S be a scheme and s ∈ S a point. Denote O_S, s^h (resp. O_S, s^sh) the henselization (resp. strict henselization), see Algebra, Definition [Tag 04GQ]. Let F be a quasi-coherent O_S-module. The following are equivalent • s is a weakly associated point of F, • m_s is a weakly associated prime of F_s, • m_s^h is a weakly associated prime of F_s ⊗_O_S, s O_S, s^h, and • m_s^sh is a weakly associated prime of F_s ⊗_O_S, s O_S, s^sh.","statement_latex":"Let $S$ be a scheme and $s \\in S$ a point. Denote $\\mathcal{O}_{S, s}^h$\n(resp.\\ $\\mathcal{O}_{S, s}^{sh}$) the henselization (resp.\\ strict\nhenselization), see\nAlgebra, Definition \\ref{algebra-definition-henselization}.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_S$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $s$ is a weakly associated point of $\\mathcal{F}$,\n\\item $\\mathfrak m_s$ is a weakly associated prime of $\\mathcal{F}_s$,\n\\item $\\mathfrak m_s^h$ is a weakly associated prime of\n$\\mathcal{F}_s \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^h$, and\n\\item $\\mathfrak m_s^{sh}$ is a weakly associated prime of\n$\\mathcal{F}_s \\otimes_{\\mathcal{O}_{S, s}} \\mathcal{O}_{S, s}^{sh}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Lemmas on étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTU","source_file":"flat.tex","source_line":379,"source_end_line":395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L379-L395","statement_sha256":"eb236389a7da0f72dc52c725c3c6ee5201a8f3e8036540777129dc84404fec4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7638,"rank":7638,"depth":46,"x":1942.141,"y":569.871,"cluster":"scheme-morphisms"},{"id":"stacks:057Q","tag":"057Q","title":"The local structure of a finite type module · Lemma 057Q","summary":"Let f : X → S be a finite type morphism of affine schemes. Let F be a finite type quasi-coherent O_X-module. Let x ∈ X with image s = f(x) in S. Set F_s = F|_X_s. Then there exist a closed immersion i : Z → X of finite presentation, and a quasi-coherent finite type O_Z-module G such that i_*G = F and Z_s = Supp(F_s).","statement_latex":"Let $f : X \\to S$ be a finite type morphism of affine schemes.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in X$ with image $s = f(x)$ in $S$.\nSet $\\mathcal{F}_s = \\mathcal{F}|_{X_s}$.\nThen there exist a closed immersion $i : Z \\to X$ of finite presentation,\nand a quasi-coherent finite type $\\mathcal{O}_Z$-module $\\mathcal{G}$\nsuch that $i_*\\mathcal{G} = \\mathcal{F}$ and\n$Z_s = \\text{Supp}(\\mathcal{F}_s)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The local structure of a finite type module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057Q","source_file":"flat.tex","source_line":445,"source_end_line":455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L445-L455","statement_sha256":"8667c1aa3b046093ee9bc97f700e568ca817bc342666abda178fc6c1f7ce4435","origin":"The Stacks Project","memory_eligible":false,"source_rank":7639,"rank":7639,"depth":9,"x":2183.414,"y":711.367,"cluster":"scheme-morphisms"},{"id":"stacks:057S","tag":"057S","title":"The local structure of a finite type module · Lemma 057S","summary":"Let f : X → S be morphism of schemes which is locally of finite type. Let F be a finite type quasi-coherent O_X-module. Let x ∈ X with image s = f(x) in S. Set F_s = F|_X_s and n = dim_x(Supp(F_s)). Then we can construct • elementary étale neighbourhoods g : (X', x') → (X, x), e : (S', s') → (S, s), • a commutative diagram xymatrix X ar[dd]_f & X' ar[dd] ar[l]^g & Z' ar[l]^i ar[d]^π & & Y' ar[d]^h S & S' ar[l]_e & S' ar@=[l] • a point z' ∈ Z' with i(z') = x', y' = π(z'),…","statement_latex":"Let $f : X \\to S$ be morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in X$ with image $s = f(x)$ in $S$.\nSet $\\mathcal{F}_s = \\mathcal{F}|_{X_s}$ and\n$n = \\dim_x(\\text{Supp}(\\mathcal{F}_s))$.\nThen we can construct\n\\begin{enumerate}\n\\item elementary \\'etale neighbourhoods $g : (X', x') \\to (X, x)$,\n$e : (S', s') \\to (S, s)$,\n\\item a commutative diagram\n$$\n\\xymatrix{\nX \\ar[dd]_f & X' \\ar[dd] \\ar[l]^g & Z' \\ar[l]^i \\ar[d]^\\pi \\\\\n& & Y' \\ar[d]^h \\\\\nS & S' \\ar[l]_e & S' \\ar@{=}[l]\n}\n$$\n\\item a point $z' \\in Z'$ with $i(z') = x'$, $y' = \\pi(z')$, $h(y') = s'$,\n\\item a finite type quasi-coherent $\\mathcal{O}_{Z'}$-module $\\mathcal{G}$,\n\\end{enumerate}\nsuch that the following properties hold\n\\begin{enumerate}\n\\item $X'$, $Z'$, $Y'$, $S'$ are affine schemes,\n\\item $i$ is a closed immersion of finite presentation,\n\\item $i_*(\\mathcal{G}) \\cong g^*\\mathcal{F}$,\n\\item $\\pi$ is finite and $\\pi^{-1}(\\{y'\\}) = \\{z'\\}$,\n\\item the extension $\\kappa(y')/\\kappa(s')$ is purely transcendental,\n\\item $h$ is smooth of relative dimension $n$\nwith geometrically integral fibres.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The local structure of a finite type module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057S","source_file":"flat.tex","source_line":493,"source_end_line":525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L493-L525","statement_sha256":"2839fa6b57b260cf681d8c5f9d93dd37b0f013711a998282f1acba49b852a458","origin":"The Stacks Project","memory_eligible":false,"source_rank":7640,"rank":7640,"depth":49,"x":1891.592,"y":743.948,"cluster":"scheme-morphisms"},{"id":"stacks:057T","tag":"057T","title":"The local structure of a finite type module · Lemma 057T","summary":"Assumptions and notation as in Lemma [Tag 057S]. If f is locally of finite presentation then π is of finite presentation. In this case the following are equivalent • F is an O_X-module of finite presentation in a neighbourhood of x, • G is an O_Z'-module of finite presentation in a neighbourhood of z', and • π_*G is an O_Y'-module of finite presentation in a neighbourhood of y'. Still assuming f locally of finite presentation the following are equivalent to each other •…","statement_latex":"Assumptions and notation as in\nLemma \\ref{lemma-elementary-devissage}.\nIf $f$ is locally of finite presentation\nthen $\\pi$ is of finite presentation.\nIn this case the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation\nin a neighbourhood of $x$,\n\\item $\\mathcal{G}$ is an $\\mathcal{O}_{Z'}$-module of finite presentation\nin a neighbourhood of $z'$, and\n\\item $\\pi_*\\mathcal{G}$ is an $\\mathcal{O}_{Y'}$-module of\nfinite presentation in a neighbourhood of $y'$.\n\\end{enumerate}\nStill assuming $f$ locally of finite presentation the following are\nequivalent to each other\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{F}_x$ is an $\\mathcal{O}_{X, x}$-module of finite\npresentation,\n\\item[(b)] $\\mathcal{G}_{z'}$ is an $\\mathcal{O}_{Z', z'}$-module of\nfinite presentation, and\n\\item[(c)] $(\\pi_*\\mathcal{G})_{y'}$ is an $\\mathcal{O}_{Y', y'}$-module\nof finite presentation.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The local structure of a finite type module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057T","source_file":"flat.tex","source_line":597,"source_end_line":622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L597-L622","statement_sha256":"1afd5b7a717cb1f5c2c2047e41a8d3c78427abfe71ee0bf966ee7699d326b32b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7641,"rank":7641,"depth":50,"x":2080.657,"y":554.256,"cluster":"scheme-morphisms"},{"id":"stacks:057U","tag":"057U","title":"The local structure of a finite type module · Lemma 057U","summary":"Assumptions and notation as in Lemma [Tag 057S]. The following are equivalent • F is flat over S in a neighbourhood of x, • G is flat over S' in a neighbourhood of z', and • π_*G is flat over S' in a neighbourhood of y'. The following are equivalent also • [(a)] F_x is flat over O_S, s, • [(b)] G_z' is flat over O_S', s', and • [(c)] (π_*G)_y' is flat over O_S', s'.","statement_latex":"Assumptions and notation as in\nLemma \\ref{lemma-elementary-devissage}.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat over $S$ in a neighbourhood of $x$,\n\\item $\\mathcal{G}$ is flat over $S'$ in a neighbourhood of $z'$, and\n\\item $\\pi_*\\mathcal{G}$ is flat over $S'$ in a neighbourhood of $y'$.\n\\end{enumerate}\nThe following are equivalent also\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{F}_x$ is flat over $\\mathcal{O}_{S, s}$,\n\\item[(b)] $\\mathcal{G}_{z'}$ is flat over $\\mathcal{O}_{S', s'}$, and\n\\item[(c)] $(\\pi_*\\mathcal{G})_{y'}$ is flat over $\\mathcal{O}_{S', s'}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The local structure of a finite type module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057U","source_file":"flat.tex","source_line":687,"source_end_line":703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L687-L703","statement_sha256":"652fd836723cfeafe1e8e4c313411ccbe1d310b59d4022de26289b1eebc39f30","origin":"The Stacks Project","memory_eligible":false,"source_rank":7642,"rank":7642,"depth":50,"x":2093.794,"y":801.517,"cluster":"scheme-morphisms"},{"id":"stacks:05H4","tag":"05H4","title":"One step dévissage · Definition 05H4","summary":"Let S be a scheme. Let X be locally of finite type over S. Let F be a quasi-coherent O_X-module of finite type. Let s ∈ S be a point. A one step dévissage of F/X/S over s is given by morphisms of schemes over S xymatrix X & Z ar[l]_i ar[r]^π & Y and a quasi-coherent O_Z-module G of finite type such that • X, S, Z and Y are affine, • i is a closed immersion of finite presentation, • F ≅ i_*G, • π is finite, and • the structure morphism Y → S is smooth with geometrically…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $s \\in S$ be a point.\nA {\\it one step d\\'evissage of $\\mathcal{F}/X/S$ over $s$}\nis given by morphisms of schemes over $S$\n$$\n\\xymatrix{\nX & Z \\ar[l]_i \\ar[r]^\\pi & Y\n}\n$$\nand a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{G}$ of finite type\nsuch that\n\\begin{enumerate}\n\\item $X$, $S$, $Z$ and $Y$ are affine,\n\\item $i$ is a closed immersion of finite presentation,\n\\item $\\mathcal{F} \\cong i_*\\mathcal{G}$,\n\\item $\\pi$ is finite, and\n\\item the structure morphism $Y \\to S$ is smooth with\ngeometrically irreducible fibres of\ndimension $\\dim(\\text{Supp}(\\mathcal{F}_s))$.\n\\end{enumerate}\nIn this case we say $(Z, Y, i, \\pi, \\mathcal{G})$ is a one step\nd\\'evissage of $\\mathcal{F}/X/S$ over $s$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"One step dévissage","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05H4","source_file":"flat.tex","source_line":779,"source_end_line":805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L779-L805","statement_sha256":"8b8bb0d9a22e8264082645c3a4522e5e36fb1a368d7a7734878318d5a53096b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7643,"rank":7643,"depth":0,"x":1885.178,"y":626.57,"cluster":"scheme-morphisms"},{"id":"stacks:05H5","tag":"05H5","title":"One step dévissage · Definition 05H5","summary":"Let S be a scheme. Let X be locally of finite type over S. Let F be a quasi-coherent O_X-module of finite type. Let x ∈ X be a point with image s in S. A one step dévissage of F/X/S at x is a system (Z, Y, i, π, G, z, y), where (Z, Y, i, π, G) is a one step dévissage of F/X/S over s and • dim_x(Supp(F_s)) = dim(Supp(F_s)), • z ∈ Z is a point with i(z) = x and π(z) = y, • we have π^-1((y)) = (z), • the extension kappa(y)/kappa(s) is purely transcendental.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $x \\in X$ be a point with image $s$ in $S$.\nA {\\it one step d\\'evissage of $\\mathcal{F}/X/S$ at $x$}\nis a system $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$, where\n$(Z, Y, i, \\pi, \\mathcal{G})$ is a one step d\\'evissage of\n$\\mathcal{F}/X/S$ over $s$ and\n\\begin{enumerate}\n\\item $\\dim_x(\\text{Supp}(\\mathcal{F}_s)) = \\dim(\\text{Supp}(\\mathcal{F}_s))$,\n\\item $z \\in Z$ is a point with $i(z) = x$ and $\\pi(z) = y$,\n\\item we have $\\pi^{-1}(\\{y\\}) = \\{z\\}$,\n\\item the extension $\\kappa(y)/\\kappa(s)$ is purely\ntranscendental.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"One step dévissage","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05H5","source_file":"flat.tex","source_line":828,"source_end_line":845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L828-L845","statement_sha256":"596597ebad20cdbf7e5ff55c9967e739a4e1aa1cc36968ebe157628d61835c5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7644,"rank":7644,"depth":0,"x":2179.82,"y":637.204,"cluster":"scheme-morphisms"},{"id":"stacks:05H6","tag":"05H6","title":"One step dévissage · Lemma 05H6","summary":"Let f : X → S be morphism of schemes which is locally of finite type. Let F be a finite type quasi-coherent O_X-module. Let x ∈ X with image s = f(x) in S. Then there exists a commutative diagram of pointed schemes xymatrix (X, x) ar[d]_f & (X', x') ar[l]^g ar[d] (S, s) & (S', s') ar[l] such that (S', s') → (S, s) and (X', x') → (X, x) are elementary étale neighbourhoods, and such that g^*F/X'/S' has a one step dévissage at x'.","statement_latex":"Let $f : X \\to S$ be morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in X$ with image $s = f(x)$ in $S$.\nThen there exists a commutative diagram of pointed schemes\n$$\n\\xymatrix{\n(X, x) \\ar[d]_f & (X', x') \\ar[l]^g \\ar[d] \\\\\n(S, s) & (S', s') \\ar[l] \\\\\n}\n$$\nsuch that $(S', s') \\to (S, s)$ and $(X', x') \\to (X, x)$\nare elementary \\'etale neighbourhoods, and such that\n$g^*\\mathcal{F}/X'/S'$ has a one step d\\'evissage at $x'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"One step dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05H6","source_file":"flat.tex","source_line":854,"source_end_line":869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L854-L869","statement_sha256":"431196e3486615ec3f8d431694f396541ad905eda0d856353bc21f58ef6cb598","origin":"The Stacks Project","memory_eligible":false,"source_rank":7645,"rank":7645,"depth":50,"x":1953.908,"y":796.618,"cluster":"scheme-morphisms"},{"id":"stacks:05H7","tag":"05H7","title":"One step dévissage · Lemma 05H7","summary":"Let S, X, F, s be as in Definition [Tag 05H4]. Let (Z, Y, i, π, G) be a one step dévissage of F/X/S over s. Let (S', s') → (S, s) be any morphism of pointed schemes. Given this data let X', Z', Y', i', π' be the base changes of X, Z, Y, i, π via S' → S. Let F' be the pullback of F to X' and let G' be the pullback of G to Z'. If S' is affine, then (Z', Y', i', π', G') is a one step dévissage of F'/X'/S' over s'.","statement_latex":"Let $S$, $X$, $\\mathcal{F}$, $s$ be as in\nDefinition \\ref{definition-one-step-devissage}.\nLet $(Z, Y, i, \\pi, \\mathcal{G})$ be a one step d\\'evissage\nof $\\mathcal{F}/X/S$ over $s$.\nLet $(S', s') \\to (S, s)$ be any morphism of pointed schemes.\nGiven this data let $X', Z', Y', i', \\pi'$ be the base\nchanges of $X, Z, Y, i, \\pi$ via $S' \\to S$.\nLet $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $X'$\nand let $\\mathcal{G}'$ be the pullback of $\\mathcal{G}$ to $Z'$.\nIf $S'$ is affine, then $(Z', Y', i', \\pi', \\mathcal{G}')$\nis a one step d\\'evissage of $\\mathcal{F}'/X'/S'$ over $s'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"One step dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05H7","source_file":"flat.tex","source_line":878,"source_end_line":891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L878-L891","statement_sha256":"a663e786f0f41e488cef2c44eca8a7f14084722c624a0068fbe53c278c785901","origin":"The Stacks Project","memory_eligible":false,"source_rank":7646,"rank":7646,"depth":38,"x":1992.312,"y":550.777,"cluster":"scheme-morphisms"},{"id":"stacks:05H8","tag":"05H8","title":"One step dévissage · Lemma 05H8","summary":"Let S, X, F, x, s be as in Definition [Tag 05H5]. Let (Z, Y, i, π, G, z, y) be a one step dévissage of F/X/S at x. Let (S', s') → (S, s) be a morphism of pointed schemes which induces an isomorphism kappa(s) = kappa(s'). Let (Z', Y', i', π', G') be as constructed in Lemma [Tag 05H7] and let x' ∈ X' (resp. z' ∈ Z', y' ∈ Y') be the unique point mapping to both x ∈ X (resp. z ∈ Z, y ∈ Y) and s' ∈ S'. If S' is affine, then (Z', Y', i', π', G', z', y') is a one step dévissage…","statement_latex":"Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in\nDefinition \\ref{definition-one-step-devissage-at-x}.\nLet $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$ be a one step d\\'evissage\nof $\\mathcal{F}/X/S$ at $x$.\nLet $(S', s') \\to (S, s)$ be a morphism of pointed schemes\nwhich induces an isomorphism $\\kappa(s) = \\kappa(s')$.\nLet $(Z', Y', i', \\pi', \\mathcal{G}')$ be as constructed in\nLemma \\ref{lemma-base-change-one-step}\nand let $x' \\in X'$ (resp.\\ $z' \\in Z'$, $y' \\in Y'$) be the\nunique point mapping to both $x \\in X$ (resp.\\ $z \\in Z$, $y \\in Y$)\nand $s' \\in S'$.\nIf $S'$ is affine, then $(Z', Y', i', \\pi', \\mathcal{G}', z', y')$\nis a one step d\\'evissage of $\\mathcal{F}'/X'/S'$ at $x'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"One step dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05H8","source_file":"flat.tex","source_line":924,"source_end_line":939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L924-L939","statement_sha256":"b6c2dc3489ea7e1883371b6e3b9cb90aed4456d336e50f3dc84496b0c0ffd198","origin":"The Stacks Project","memory_eligible":false,"source_rank":7647,"rank":7647,"depth":39,"x":2161.764,"y":753.934,"cluster":"scheme-morphisms"},{"id":"stacks:05H9","tag":"05H9","title":"One step dévissage · Definition 05H9","summary":"Let S, X, F, x, s be as in Definition [Tag 05H5]. Let (Z, Y, i, π, G, z, y) be a one step dévissage of F/X/S at x. Let us define a standard shrinking of this situation to be given by standard opens S' ⊂ S, X' ⊂ X, Z' ⊂ Z, and Y' ⊂ Y such that s ∈ S', x ∈ X', z ∈ Z', and y ∈ Y' and such that (Z', Y', i|_Z', π|_Z', G|_Z', z, y) is a one step dévissage of F|_X'/X'/S' at x.","statement_latex":"Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in\nDefinition \\ref{definition-one-step-devissage-at-x}.\nLet $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$ be a one step d\\'evissage\nof $\\mathcal{F}/X/S$ at $x$. Let us define a\n{\\it standard shrinking} of this situation to be\ngiven by standard opens $S' \\subset S$, $X' \\subset X$, $Z' \\subset Z$,\nand $Y' \\subset Y$ such that $s \\in S'$, $x \\in X'$, $z \\in Z'$, and\n$y \\in Y'$ and such that\n$$\n(Z', Y', i|_{Z'}, \\pi|_{Z'}, \\mathcal{G}|_{Z'}, z, y)\n$$\nis a one step d\\'evissage of $\\mathcal{F}|_{X'}/X'/S'$ at $x$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"One step dévissage","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05H9","source_file":"flat.tex","source_line":953,"source_end_line":967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L953-L967","statement_sha256":"41a3bd1edee6ddf6be9158cbe422010181c259ef16310758f73e95f837fc5dee","origin":"The Stacks Project","memory_eligible":false,"source_rank":7648,"rank":7648,"depth":1,"x":1873.318,"y":700.257,"cluster":"scheme-morphisms"},{"id":"stacks:05HA","tag":"05HA","title":"One step dévissage · Lemma 05HA","summary":"With assumption and notation as in Definition [Tag 05H9] we have: • If S' ⊂ S is a standard open neighbourhood of s, then setting X' = X_S', Z' = Z_S' and Y' = Y_S' we obtain a standard shrinking. • Let W ⊂ Y be a standard open neighbourhood of y. Then there exists a standard shrinking with Y' = W ×_S S'. • Let U ⊂ X be an open neighbourhood of x. Then there exists a standard shrinking with X' ⊂ U.","statement_latex":"With assumption and notation as in\nDefinition \\ref{definition-shrink}\nwe have:\n\\begin{enumerate}\n\\item\n\nIf $S' \\subset S$ is a standard open neighbourhood of $s$, then\nsetting $X' = X_{S'}$, $Z' = Z_{S'}$ and $Y' = Y_{S'}$ we obtain a\nstandard shrinking.\n\\item\n\nLet $W \\subset Y$ be a standard open neighbourhood of $y$.\nThen there exists a standard shrinking with $Y' = W \\times_S S'$.\n\\item\n\nLet $U \\subset X$ be an open neighbourhood of $x$.\nThen there exists a standard shrinking with $X' \\subset U$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"One step dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HA","source_file":"flat.tex","source_line":969,"source_end_line":989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L969-L989","statement_sha256":"b14f61ee04f663ea1ef86e402a3bd8faf5ac2715a448693b13f20f78b7b3b468","origin":"The Stacks Project","memory_eligible":false,"source_rank":7649,"rank":7649,"depth":40,"x":2129.287,"y":576.114,"cluster":"scheme-morphisms"},{"id":"stacks:05HE","tag":"05HE","title":"One step dévissage · Lemma 05HE","summary":"Let S, X, F, x, s be as in Definition [Tag 05H5]. Let (Z, Y, i, π, G, z, y) be a one step dévissage of F/X/S at x. Let xymatrix (Y, y) ar[d] & (Y', y') ar[l] ar[d] (S, s) & (S', s') ar[l] be a commutative diagram of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods. Then there exists a commutative diagram xymatrix & & (X\", x\") ar[lld] ar[d] & (Z\", z\") ar[l] ar[lld] ar[d] (X, x) ar[d] & (Z, z) ar[l] ar[d] & (S\", s\") ar[lld] & (Y\", y\")…","statement_latex":"Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in\nDefinition \\ref{definition-one-step-devissage-at-x}.\nLet $(Z, Y, i, \\pi, \\mathcal{G}, z, y)$ be a one step d\\'evissage\nof $\\mathcal{F}/X/S$ at $x$. Let\n$$\n\\xymatrix{\n(Y, y) \\ar[d] & (Y', y') \\ar[l] \\ar[d] \\\\\n(S, s) & (S', s') \\ar[l]\n}\n$$\nbe a commutative diagram of pointed schemes such that the horizontal\narrows are elementary \\'etale neighbourhoods. Then there exists\na commutative diagram\n$$\n\\xymatrix{\n& & (X'', x'') \\ar[lld] \\ar[d] & (Z'', z'') \\ar[l] \\ar[lld] \\ar[d] \\\\\n(X, x) \\ar[d] & (Z, z) \\ar[l] \\ar[d] &\n(S'', s'') \\ar[lld] & (Y'', y'') \\ar[lld] \\ar[l] \\\\\n(S, s) & (Y, y) \\ar[l]\n}\n$$\nof pointed schemes with the following properties:\n\\begin{enumerate}\n\\item $(S'', s'') \\to (S', s')$ is an elementary \\'etale neighbourhood and\nthe morphism $S'' \\to S$ is the composition $S'' \\to S' \\to S$,\n\\item $Y''$ is an open subscheme of $Y' \\times_{S'} S''$,\n\\item $Z'' = Z \\times_Y Y''$,\n\\item $(X'', x'') \\to (X, x)$ is an elementary \\'etale neighbourhood, and\n\\item $(Z'', Y'', i'', \\pi'', \\mathcal{G}'', z'', y'')$ is a one step\nd\\'evissage at $x''$ of the sheaf $\\mathcal{F}''$.\n\\end{enumerate}\nHere $\\mathcal{F}''$ (resp.\\ $\\mathcal{G}''$) is the pullback of\n$\\mathcal{F}$ (resp.\\ $\\mathcal{G}$) via the morphism $X'' \\to X$\n(resp.\\ $Z'' \\to Z$) and $i'' : Z'' \\to X''$ and $\\pi'' : Z'' \\to Y''$\nare as in the diagram.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"One step dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HE","source_file":"flat.tex","source_line":1026,"source_end_line":1063,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1026-L1063","statement_sha256":"81746f1bce47e36455ca909ff9f9733c5f0fe5c1ad532ce9e700f2f4a18a3a93","origin":"The Stacks Project","memory_eligible":false,"source_rank":7650,"rank":7650,"depth":47,"x":2040.334,"y":812.998,"cluster":"scheme-morphisms"},{"id":"stacks:05HF","tag":"05HF","title":"One step dévissage · Lemma 05HF","summary":"Let S, X, F, s be as in Definition [Tag 05H4]. Let (Z, Y, i, π, G) be a one step dévissage of F/X/S over s. Let xi ∈ Y_s be the (unique) generic point. Then there exists an integer r > 0 and an O_Y-module map α : O_Y^⊕ r → π_*G such that α : kappa(xi)^⊕ r → (π_*G)_xi ⊗_O_Y, xi kappa(xi) is an isomorphism. Moreover, in this case we have dim(Supp(Coker(α)_s)) < dim(Supp(F_s)).","statement_latex":"Let $S$, $X$, $\\mathcal{F}$, $s$ be as in\nDefinition \\ref{definition-one-step-devissage}.\nLet $(Z, Y, i, \\pi, \\mathcal{G})$ be a one step d\\'evissage\nof $\\mathcal{F}/X/S$ over $s$.\nLet $\\xi \\in Y_s$ be the (unique) generic point.\nThen there exists an integer $r > 0$ and an $\\mathcal{O}_Y$-module map\n$$\n\\alpha : \\mathcal{O}_Y^{\\oplus r} \\longrightarrow \\pi_*\\mathcal{G}\n$$\nsuch that\n$$\n\\alpha :\n\\kappa(\\xi)^{\\oplus r}\n\\longrightarrow\n(\\pi_*\\mathcal{G})_\\xi \\otimes_{\\mathcal{O}_{Y, \\xi}} \\kappa(\\xi)\n$$\nis an isomorphism. Moreover, in this case we have\n$$\n\\dim(\\text{Supp}(\\Coker(\\alpha)_s)) < \\dim(\\text{Supp}(\\mathcal{F}_s)).\n$$","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"One step dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HF","source_file":"flat.tex","source_line":1137,"source_end_line":1159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1137-L1159","statement_sha256":"c8399cf6da82e6a31b7675610cc588ccb1c79d6fb5b651c0a662369cf259da0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7651,"rank":7651,"depth":3,"x":1915.378,"y":587.755,"cluster":"scheme-morphisms"},{"id":"stacks:05HH","tag":"05HH","title":"Complete dévissage · Definition 05HH","summary":"Let S be a scheme. Let X be locally of finite type over S. Let F be a quasi-coherent O_X-module of finite type. Let s ∈ S be a point. A complete dévissage of F/X/S over s is given by a diagram xymatrix X & Z_1 ar[l]^i_1 ar[d]^π_1 & Y_1 & Z_2 ar[l]^i_2 ar[d]^π_2 & & Y_2 & Z_3 ar[l] ar[d] & & & ... & ... ar[l] ar[d] & & & & Y_n of schemes over S, finite type quasi-coherent O_Z_k-modules G_k, and O_Y_k-module maps α_k : O_Y_k^⊕ r_k → π_k, *G_k, k = 1, …, n satisfying the…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $s \\in S$ be a point.\nA {\\it complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$} is given by a\ndiagram\n$$\n\\xymatrix{\nX & Z_1 \\ar[l]^{i_1} \\ar[d]^{\\pi_1} \\\\\n& Y_1 & Z_2 \\ar[l]^{i_2} \\ar[d]^{\\pi_2} \\\\\n& & Y_2 & Z_3 \\ar[l] \\ar[d] \\\\\n& & & ... & ... \\ar[l] \\ar[d] \\\\\n& & & & Y_n\n}\n$$\nof schemes over $S$, finite type quasi-coherent $\\mathcal{O}_{Z_k}$-modules\n$\\mathcal{G}_k$, and $\\mathcal{O}_{Y_k}$-module maps\n$$\n\\alpha_k :\n\\mathcal{O}_{Y_k}^{\\oplus r_k}\n\\longrightarrow\n\\pi_{k, *}\\mathcal{G}_k,\n\\quad\nk = 1, \\ldots, n\n$$\nsatisfying the following properties:\n\\begin{enumerate}\n\\item $(Z_1, Y_1, i_1, \\pi_1, \\mathcal{G}_1)$ is a one step\nd\\'evissage of $\\mathcal{F}/X/S$ over $s$,\n\\item the map $\\alpha_k$ induces an isomorphism\n$$\n\\kappa(\\xi_k)^{\\oplus r_k} \\longrightarrow\n(\\pi_{k, *}\\mathcal{G}_k)_{\\xi_k}\n\\otimes_{\\mathcal{O}_{Y_k, \\xi_k}} \\kappa(\\xi_k)\n$$\nwhere $\\xi_k \\in (Y_k)_s$ is the unique generic point,\n\\item for $k = 2, \\ldots, n$ the system\n$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k)$\nis a one step d\\'evissage of $\\Coker(\\alpha_{k - 1})/Y_{k - 1}/S$\nover $s$,\n\\item $\\Coker(\\alpha_n) = 0$.\n\\end{enumerate}\nIn this case we say that\n$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)_{k = 1, \\ldots, n}$\nis a complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Complete dévissage","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HH","source_file":"flat.tex","source_line":1194,"source_end_line":1241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1194-L1241","statement_sha256":"6b0a04128bc816ce96ed5e76a7f294b4455e31b846d6f078c0864b71d0899f53","origin":"The Stacks Project","memory_eligible":false,"source_rank":7652,"rank":7652,"depth":0,"x":2188.768,"y":682.983,"cluster":"scheme-morphisms"},{"id":"stacks:05HI","tag":"05HI","title":"Complete dévissage · Definition 05HI","summary":"Let S be a scheme. Let X be locally of finite type over S. Let F be a quasi-coherent O_X-module of finite type. Let x ∈ X be a point with image s ∈ S. A complete dévissage of F/X/S at x is given by a system (Z_k, Y_k, i_k, π_k, G_k, α_k, z_k, y_k)_k = 1, …, n such that (Z_k, Y_k, i_k, π_k, G_k, α_k) is a complete dévissage of F/X/S over s, and such that • (Z_1, Y_1, i_1, π_1, G_1, z_1, y_1) is a one step dévissage of F/X/S at x, • for k = 2, …, n the system (Z_k, Y_k,…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $x \\in X$ be a point with image $s \\in S$.\nA {\\it complete d\\'evissage of $\\mathcal{F}/X/S$ at $x$} is given by a\nsystem\n$$\n(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 1, \\ldots, n}\n$$\nsuch that $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)$ is a\ncomplete d\\'evissage of $\\mathcal{F}/X/S$ over $s$, and such that\n\\begin{enumerate}\n\\item $(Z_1, Y_1, i_1, \\pi_1, \\mathcal{G}_1, z_1, y_1)$ is a one step\nd\\'evissage of $\\mathcal{F}/X/S$ at $x$,\n\\item for $k = 2, \\ldots, n$ the system\n$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, z_k, y_k)$\nis a one step d\\'evissage of $\\Coker(\\alpha_{k - 1})/Y_{k - 1}/S$\nat $y_{k - 1}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Complete dévissage","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HI","source_file":"flat.tex","source_line":1243,"source_end_line":1264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1243-L1264","statement_sha256":"f681c8574a6aa9ebfaa301a38fae944ebe6dd430746665c7fa0c35c45bb12dc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7653,"rank":7653,"depth":0,"x":1910.478,"y":767.927,"cluster":"scheme-morphisms"},{"id":"stacks:05HJ","tag":"05HJ","title":"Complete dévissage · Lemma 05HJ","summary":"Let S, X, F, s be as in Definition [Tag 05HH]. Let (S', s') → (S, s) be any morphism of pointed schemes. Let (Z_k, Y_k, i_k, π_k, G_k, α_k)_k = 1, …, n be a complete dévissage of F/X/S over s. Given this data let X', Z'_k, Y'_k, i'_k, π'_k be the base changes of X, Z_k, Y_k, i_k, π_k via S' → S. Let F' be the pullback of F to X' and let G'_k be the pullback of G_k to Z'_k. Let α'_k be the pullback of α_k to Y'_k. If S' is affine, then (Z'_k, Y'_k, i'_k, π'_k, G'_k,…","statement_latex":"Let $S$, $X$, $\\mathcal{F}$, $s$ be as in\nDefinition \\ref{definition-complete-devissage}.\nLet $(S', s') \\to (S, s)$ be any morphism of pointed schemes.\nLet $(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k)_{k = 1, \\ldots, n}$\nbe a complete d\\'evissage of $\\mathcal{F}/X/S$ over $s$.\nGiven this data let $X', Z'_k, Y'_k, i'_k, \\pi'_k$ be the base\nchanges of $X, Z_k, Y_k, i_k, \\pi_k$ via $S' \\to S$.\nLet $\\mathcal{F}'$ be the pullback of $\\mathcal{F}$ to $X'$\nand let $\\mathcal{G}'_k$ be the pullback of $\\mathcal{G}_k$ to $Z'_k$.\nLet $\\alpha'_k$ be the pullback of $\\alpha_k$ to $Y'_k$.\nIf $S'$ is affine, then\n$(Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k)_{k = 1, \\ldots, n}$\nis a complete d\\'evissage of $\\mathcal{F}'/X'/S'$ over $s'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Complete dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HJ","source_file":"flat.tex","source_line":1270,"source_end_line":1285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1270-L1285","statement_sha256":"ce0892630edb3d73d7b70b4a5f1faee240860407b50f70ed9b1c78f907bda941","origin":"The Stacks Project","memory_eligible":false,"source_rank":7654,"rank":7654,"depth":39,"x":2047.433,"y":547.289,"cluster":"scheme-morphisms"},{"id":"stacks:05HK","tag":"05HK","title":"Complete dévissage · Lemma 05HK","summary":"Let S, X, F, x, s be as in Definition [Tag 05HI]. Let (S', s') → (S, s) be a morphism of pointed schemes which induces an isomorphism kappa(s) = kappa(s'). Let (Z_k, Y_k, i_k, π_k, G_k, α_k, z_k, y_k)_k = 1, …, n be a complete dévissage of F/X/S at x. Let (Z'_k, Y'_k, i'_k, π'_k, G'_k, α'_k)_k = 1, …, n be as constructed in Lemma [Tag 05HJ] and let x' ∈ X' (resp. z'_k ∈ Z', y'_k ∈ Y') be the unique point mapping to both x ∈ X (resp. z_k ∈ Z_k, y_k ∈ Y_k) and s' ∈ S'. If…","statement_latex":"Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in\nDefinition \\ref{definition-complete-devissage-at-x}.\nLet $(S', s') \\to (S, s)$ be a morphism of pointed schemes\nwhich induces an isomorphism $\\kappa(s) = \\kappa(s')$. Let\n$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 1, \\ldots, n}$\nbe a complete d\\'evissage of $\\mathcal{F}/X/S$ at $x$.\nLet\n$(Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k)_{k = 1, \\ldots, n}$\nbe as constructed in\nLemma \\ref{lemma-base-change-complete}\nand let $x' \\in X'$ (resp.\\ $z'_k \\in Z'$, $y'_k \\in Y'$) be the\nunique point mapping to both $x \\in X$ (resp.\\ $z_k \\in Z_k$, $y_k \\in Y_k$)\nand $s' \\in S'$.\nIf $S'$ is affine, then\n$(Z'_k, Y'_k, i'_k, \\pi'_k, \\mathcal{G}'_k, \\alpha'_k,\nz'_k, y'_k)_{k = 1, \\ldots, n}$\nis a complete d\\'evissage of $\\mathcal{F}'/X'/S'$ at $x'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Complete dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HK","source_file":"flat.tex","source_line":1312,"source_end_line":1331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1312-L1331","statement_sha256":"6d48c5fe92516c963afeacfe416695e022e00d28055aa170088a49286aca87ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":7655,"rank":7655,"depth":40,"x":2123.907,"y":787.796,"cluster":"scheme-morphisms"},{"id":"stacks:05HL","tag":"05HL","title":"Complete dévissage · Definition 05HL","summary":"Let S, X, F, x, s be as in Definition [Tag 05HI]. Consider a complete dévissage (Z_k, Y_k, i_k, π_k, G_k, α_k, z_k, y_k)_k = 1, …, n of F/X/S at x. Let us define a standard shrinking of this situation to be given by standard opens S' ⊂ S, X' ⊂ X, Z'_k ⊂ Z_k, and Y'_k ⊂ Y_k such that s_k ∈ S', x_k ∈ X', z_k ∈ Z', and y_k ∈ Y' and such that (Z'_k, Y'_k, i'_k, π'_k, G'_k, α'_k, z_k, y_k)_k = 1, …, n is a one step dévissage of F'/X'/S' at x where G'_k = G_k|_Z'_k and F' = F|_X'.","statement_latex":"Let $S$, $X$, $\\mathcal{F}$, $x$, $s$ be as in\nDefinition \\ref{definition-complete-devissage-at-x}.\nConsider a complete d\\'evissage\n$(Z_k, Y_k, i_k, \\pi_k, \\mathcal{G}_k, \\alpha_k, z_k, y_k)_{k = 1, \\ldots, n}$\nof $\\mathcal{F}/X/S$ at $x$. Let us define a\n{\\it standard shrinking} of this situation to be\ngiven by standard opens $S' \\subset S$, $X' \\subset X$,\n$Z'_k \\subset Z_k$, and $Y'_k \\subset Y_k$ such that $s_k \\in S'$,\n$x_k \\in X'$, $z_k \\in Z'$, and $y_k \\in Y'$ and such that\n$$\n(Z'_k, Y'_k, i'_k, \\pi'_k,\n\\mathcal{G}'_k, \\alpha'_k, z_k, y_k)_{k = 1, \\ldots, n}\n$$\nis a one step d\\'evissage of $\\mathcal{F}'/X'/S'$ at $x$ where\n$\\mathcal{G}'_k = \\mathcal{G}_k|_{Z'_k}$ and\n$\\mathcal{F}' = \\mathcal{F}|_{X'}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Complete dévissage","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HL","source_file":"flat.tex","source_line":1340,"source_end_line":1358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1340-L1358","statement_sha256":"a21caa072dd2c404d3dc51c6e0d30d0de1151c949c7e8307897c3cf906724c9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7656,"rank":7656,"depth":1,"x":1874.002,"y":653.787,"cluster":"scheme-morphisms"},{"id":"stacks:05HM","tag":"05HM","title":"Complete dévissage · Lemma 05HM","summary":"With assumption and notation as in Definition [Tag 05HL] we have: • If S' ⊂ S is a standard open neighbourhood of s, then setting X' = X_S', Z'_k = Z_S' and Y'_k = Y_S' we obtain a standard shrinking. • Let W ⊂ Y_n be a standard open neighbourhood of y. Then there exists a standard shrinking with Y'_n = W ×_S S'. • Let U ⊂ X be an open neighbourhood of x. Then there exists a standard shrinking with X' ⊂ U.","statement_latex":"With assumption and notation as in\nDefinition \\ref{definition-shrink-complete}\nwe have:\n\\begin{enumerate}\n\\item\n\nIf $S' \\subset S$ is a standard open neighbourhood of $s$, then\nsetting $X' = X_{S'}$, $Z'_k = Z_{S'}$ and $Y'_k = Y_{S'}$ we obtain a\nstandard shrinking.\n\\item\n\nLet $W \\subset Y_n$ be a standard open neighbourhood of $y$.\nThen there exists a standard shrinking with $Y'_n = W \\times_S S'$.\n\\item\n\nLet $U \\subset X$ be an open neighbourhood of $x$.\nThen there exists a standard shrinking with $X' \\subset U$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Complete dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HM","source_file":"flat.tex","source_line":1360,"source_end_line":1380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1360-L1380","statement_sha256":"7b07bf429c324168a290bc542721cfc4643ea26eead28bed8b782e69d2ac0d95","origin":"The Stacks Project","memory_eligible":false,"source_rank":7657,"rank":7657,"depth":41,"x":2166.167,"y":610.783,"cluster":"scheme-morphisms"},{"id":"stacks:05HR","tag":"05HR","title":"Complete dévissage · Proposition 05HR","summary":"Let S be a scheme. Let X be locally of finite type over S. Let x ∈ X be a point with image s ∈ S. There exists a commutative diagram xymatrix (X, x) ar[d] & (X', x') ar[l]^g ar[d] (S, s) & (S', s') ar[l] of pointed schemes such that the horizontal arrows are elementary étale neighbourhoods and such that g^*F/X'/S' has a complete dévissage at x.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be locally of finite type over $S$.\nLet $x \\in X$ be a point with image $s \\in S$.\nThere exists a commutative diagram\n$$\n\\xymatrix{\n(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\\n(S, s) & (S', s') \\ar[l]\n}\n$$\nof pointed schemes such that the horizontal\narrows are elementary \\'etale neighbourhoods\nand such that $g^*\\mathcal{F}/X'/S'$ has a complete\nd\\'evissage at $x$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Complete dévissage","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HR","source_file":"flat.tex","source_line":1428,"source_end_line":1444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1428-L1444","statement_sha256":"f89ec0f0f1cf6f03311601c2e9682533ee71c42ec953a05a54dd9fc46460beaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7658,"rank":7658,"depth":51,"x":1985.236,"y":808.358,"cluster":"scheme-morphisms"},{"id":"stacks:05HU","tag":"05HU","title":"Complete dévissage · Lemma 05HU","summary":"Let X → S be a finite type morphism of schemes. Let F be a finite type quasi-coherent O_X-module. Let s ∈ S be a point. There exists an elementary étale neighbourhood (S', s') → (S, s) and étale morphisms h_i : Y_i → X_S', i = 1, …, n such that for each i there exists a complete dévissage of F_i/Y_i/S' over s', where F_i is the pullback of F to Y_i and such that X_s = (X_S')_s' ⊂ ⋃ h_i(Y_i).","statement_latex":"Let $X \\to S$ be a finite type morphism of schemes.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $s \\in S$ be a point.\nThere exists an elementary \\'etale neighbourhood\n$(S', s') \\to (S, s)$ and \\'etale morphisms\n$h_i : Y_i \\to X_{S'}$, $i = 1, \\ldots, n$ such that for each\n$i$ there exists a complete d\\'evissage of $\\mathcal{F}_i/Y_i/S'$ over $s'$,\nwhere $\\mathcal{F}_i$ is the pullback of $\\mathcal{F}$ to $Y_i$\nand such that $X_s = (X_{S'})_{s'} \\subset \\bigcup h_i(Y_i)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Complete dévissage","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HU","source_file":"flat.tex","source_line":1575,"source_end_line":1586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1575-L1586","statement_sha256":"084d199e66cf3bf57bcfa5e3e16471f38ad53260434b6fa2df95b1325c7fa70c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7659,"rank":7659,"depth":40,"x":1959.757,"y":559.901,"cluster":"scheme-morphisms"},{"id":"stacks:05HW","tag":"05HW","title":"Translation into algebra · Definition 05HW","summary":"Let R → S be a ring map. Let q be a prime of S lying over the prime p of R. A elementary étale localization of the ring map R → S at q is given by a commutative diagram of rings and accompanying primes xymatrix S ar[r] & S' R ar[u] ar[r] & R' ar[u] xymatrix q ar@-[r] & q' p ar@-[u] ar@-[r] & p' ar@-[u] such that R → R' and S → S' are étale ring maps and kappa( p) = kappa( p') and kappa( q) = kappa( q').","statement_latex":"Let $R \\to S$ be a ring map. Let $\\mathfrak q$ be a prime of $S$ lying over\nthe prime $\\mathfrak p$ of $R$. A {\\it elementary \\'etale localization of\nthe ring map $R \\to S$ at $\\mathfrak q$} is given by a commutative diagram\nof rings and accompanying primes\n$$\n\\xymatrix{\nS \\ar[r] & S' \\\\\nR \\ar[u] \\ar[r] & R' \\ar[u]\n}\n\\quad\\quad\n\\xymatrix{\n\\mathfrak q \\ar@{-}[r] & \\mathfrak q' \\\\\n\\mathfrak p \\ar@{-}[u] \\ar@{-}[r] & \\mathfrak p' \\ar@{-}[u]\n}\n$$\nsuch that $R \\to R'$ and $S \\to S'$ are \\'etale ring maps and\n$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$ and\n$\\kappa(\\mathfrak q) = \\kappa(\\mathfrak q')$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Translation into algebra","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HW","source_file":"flat.tex","source_line":1629,"source_end_line":1649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1629-L1649","statement_sha256":"11c5f9dce766783745a8a07f28259116905c7c844ff7b7108368712cf0c5a478","origin":"The Stacks Project","memory_eligible":false,"source_rank":7660,"rank":7660,"depth":0,"x":2178.438,"y":728.722,"cluster":"scheme-morphisms"},{"id":"stacks:05HX","tag":"05HX","title":"Translation into algebra · Definition 05HX","summary":"Let R → S be a finite type ring map. Let r be a prime of R. Let N be a finite S-module. A complete dévissage of N/S/R over r is given by R-algebra maps xymatrix & A_1 & & A_2 & & ... & & A_n S ar[ru] & & B_1 ar[lu] ar[ru] & & ... ar[lu] ar[ru] & & ... ar[lu] ar[ru] & & B_n ar[lu] finite A_i-modules M_i and B_i-module maps α_i : B_i^⊕ r_i → M_i such that • S → A_1 is surjective and of finite presentation, • B_i → A_i + 1 is surjective and of finite presentation, • B_i →…","statement_latex":"Let $R \\to S$ be a finite type ring map.\nLet $\\mathfrak r$ be a prime of $R$.\nLet $N$ be a finite $S$-module.\nA {\\it complete d\\'evissage of $N/S/R$ over $\\mathfrak r$}\nis given by $R$-algebra maps\n$$\n\\xymatrix{\n& A_1 & & A_2 & & ... & & A_n \\\\\nS \\ar[ru] & & B_1 \\ar[lu] \\ar[ru] & & ... \\ar[lu] \\ar[ru] & &\n... \\ar[lu] \\ar[ru] & & B_n \\ar[lu]\n}\n$$\nfinite $A_i$-modules $M_i$ and $B_i$-module maps\n$\\alpha_i : B_i^{\\oplus r_i} \\to M_i$ such that\n\\begin{enumerate}\n\\item $S \\to A_1$ is surjective and of finite presentation,\n\\item $B_i \\to A_{i + 1}$ is surjective and of finite presentation,\n\\item $B_i \\to A_i$ is finite,\n\\item $R \\to B_i$ is smooth with geometrically irreducible fibres,\n\\item $N \\cong M_1$ as $S$-modules,\n\\item $\\Coker(\\alpha_i) \\cong M_{i + 1}$ as $B_i$-modules,\n\\item $\\alpha_i : \\kappa(\\mathfrak p_i)^{\\oplus r_i}\n\\to M_i \\otimes_{B_i} \\kappa(\\mathfrak p_i)$ is an isomorphism\nwhere $\\mathfrak p_i = \\mathfrak rB_i$, and\n\\item $\\Coker(\\alpha_n) = 0$.\n\\end{enumerate}\nIn this situation we say that\n$(A_i, B_i, M_i, \\alpha_i)_{i = 1, \\ldots, n}$\nis a complete d\\'evissage of $N/S/R$ over $\\mathfrak r$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Translation into algebra","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HX","source_file":"flat.tex","source_line":1651,"source_end_line":1682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1651-L1682","statement_sha256":"e3de481c4a4413dd8b16c19c6d3697ba0ab04ee9dfa4c438e0bb25d5e28938f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7661,"rank":7661,"depth":0,"x":1881.301,"y":728.32,"cluster":"scheme-morphisms"},{"id":"stacks:05HZ","tag":"05HZ","title":"Translation into algebra · Definition 05HZ","summary":"Let R → S be a finite type ring map. Let q be a prime of S lying over the prime r of R. Let N be a finite S-module. A complete dévissage of N/S/R at q is given by a complete dévissage (A_i, B_i, M_i, α_i)_i = 1, …, n of N/S/R over r and prime ideals q_i ⊂ B_i lying over r such that • kappa( r) ⊂ kappa( q_i) is purely transcendental, • there is a unique prime q'_i ⊂ A_i lying over q_i ⊂ B_i, • q = q'_1 ∩ S and q_i = q'_i + 1 ∩ A_i, • R → B_i has relative dimension dim_…","statement_latex":"Let $R \\to S$ be a finite type ring map.\nLet $\\mathfrak q$ be a prime of $S$ lying over the prime $\\mathfrak r$ of $R$.\nLet $N$ be a finite $S$-module.\nA {\\it complete d\\'evissage of $N/S/R$ at $\\mathfrak q$} is given by a\ncomplete d\\'evissage $(A_i, B_i, M_i, \\alpha_i)_{i = 1, \\ldots, n}$\nof $N/S/R$ over $\\mathfrak r$ and prime ideals $\\mathfrak q_i \\subset B_i$\nlying over $\\mathfrak r$ such that\n\\begin{enumerate}\n\\item $\\kappa(\\mathfrak r) \\subset \\kappa(\\mathfrak q_i)$ is purely\ntranscendental,\n\\item there is a unique prime $\\mathfrak q'_i \\subset A_i$\nlying over $\\mathfrak q_i \\subset B_i$,\n\\item $\\mathfrak q = \\mathfrak q'_1 \\cap S$ and\n$\\mathfrak q_i = \\mathfrak q'_{i + 1} \\cap A_i$,\n\\item $R \\to B_i$ has relative dimension\n$\\dim_{\\mathfrak q_i}(\\text{Supp}(M_i \\otimes_R \\kappa(\\mathfrak r)))$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Translation into algebra","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05HZ","source_file":"flat.tex","source_line":1709,"source_end_line":1728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1709-L1728","statement_sha256":"ef892f6c3e9f1178cf0720666cad5d7d0e8691a32eee334ee59d3a736bd6dcf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7662,"rank":7662,"depth":0,"x":2100.82,"y":559.944,"cluster":"scheme-morphisms"},{"id":"stacks:05I1","tag":"05I1","title":"Translation into algebra · Lemma 05I1","summary":"Let R → S be a finite type ring map. Let M be a finite S-module. Let q be a prime ideal of S. There exists an elementary étale localization R' → S', q', p' of the ring map R → S at q such that there exists a complete dévissage of (M ⊗_S S')/S'/R' at q'.","statement_latex":"Let $R \\to S$ be a finite type ring map.\nLet $M$ be a finite $S$-module.\nLet $\\mathfrak q$ be a prime ideal of $S$.\nThere exists an elementary \\'etale localization\n$R' \\to S', \\mathfrak q', \\mathfrak p'$ of\nthe ring map $R \\to S$ at $\\mathfrak q$ such that\nthere exists a complete d\\'evissage of\n$(M \\otimes_S S')/S'/R'$ at $\\mathfrak q'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Translation into algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05I1","source_file":"flat.tex","source_line":1756,"source_end_line":1766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1756-L1766","statement_sha256":"48cf95fdd20a0844c348d1f9b48df86319cecc35189baf169771737a74c2b3b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7663,"rank":7663,"depth":52,"x":2074.343,"y":808.766,"cluster":"scheme-morphisms"},{"id":"stacks:05DE","tag":"05DE","title":"Localization and universally injective maps · Lemma 05DE","summary":"Let R → S be a ring map. Let N be a S-module. Assume • R is a local ring with maximal ideal m, • overlineS = S/ m S is Noetherian, and • overlineN = N/ m_R N is a finite overlineS-module. Let Sigma ⊂ S be the multiplicative subset of elements which are not a zerodivisor on overlineN. Then Sigma^-1S is a semi-local ring whose spectrum consists of primes q ⊂ S contained in an element of Ass_S(overlineN). Moreover, any maximal ideal of Sigma^-1S corresponds to an associated…","statement_latex":"Let $R \\to S$ be a ring map.\nLet $N$ be a $S$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is a local ring with maximal ideal $\\mathfrak m$,\n\\item $\\overline{S} = S/\\mathfrak m S$ is Noetherian, and\n\\item $\\overline{N} = N/\\mathfrak m_R N$ is a finite $\\overline{S}$-module.\n\\end{enumerate}\nLet $\\Sigma \\subset S$ be the multiplicative subset of elements which are not\na zerodivisor on $\\overline{N}$. Then $\\Sigma^{-1}S$ is a semi-local ring\nwhose spectrum consists of primes $\\mathfrak q \\subset S$ contained in an\nelement of $\\text{Ass}_S(\\overline{N})$. Moreover, any maximal\nideal of $\\Sigma^{-1}S$ corresponds to an associated prime of\n$\\overline{N}$ over $\\overline{S}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Localization and universally injective maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DE","source_file":"flat.tex","source_line":1782,"source_end_line":1798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1782-L1798","statement_sha256":"5d2d5200143ae718e18fbc45eee0ba3c60ec5e3bf5a52c2b964dddf3c57d4203","origin":"The Stacks Project","memory_eligible":false,"source_rank":7664,"rank":7664,"depth":10,"x":1893.695,"y":610.182,"cluster":"scheme-morphisms"},{"id":"stacks:05DF","tag":"05DF","title":"Localization and universally injective maps · Lemma 05DF","summary":"Assumption and notation as in Lemma [Tag 05DE]. Assume moreover that • S is local and R → S is a local homomorphism, • S is essentially of finite presentation over R, • N is finitely presented over S, and • N is flat over R. Then each s ∈ Sigma defines a universally injective R-module map s : N → N, and the map N → Sigma^-1N is R-universally injective.","statement_latex":"Assumption and notation as in\nLemma \\ref{lemma-homothety-spectrum}.\nAssume moreover that\n\\begin{enumerate}\n\\item $S$ is local and $R \\to S$ is a local homomorphism,\n\\item $S$ is essentially of finite presentation over $R$,\n\\item $N$ is finitely presented over $S$, and\n\\item $N$ is flat over $R$.\n\\end{enumerate}\nThen each $s \\in \\Sigma$ defines a\nuniversally injective $R$-module map $s : N \\to N$, and the\nmap $N \\to \\Sigma^{-1}N$ is $R$-universally injective.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Localization and universally injective maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DF","source_file":"flat.tex","source_line":1820,"source_end_line":1834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1820-L1834","statement_sha256":"1ded09746bfd42d08e86ce4051860357b9857bcf1e5f26cdae96c4df8c03f4ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":7665,"rank":7665,"depth":12,"x":2186.72,"y":654.13,"cluster":"scheme-morphisms"},{"id":"stacks:05DG","tag":"05DG","title":"Localization and universally injective maps · Lemma 05DG","summary":"Let R → S be a ring map. Let N be an S-module. Let S → S' be a ring map. Assume • R → S is a local homomorphism of local rings • S is essentially of finite presentation over R, • N is of finite presentation over S, • N is flat over R, • S → S' is flat, and • the image of Spec(S') → Spec(S) contains all primes q of S lying over m_R such that q is an associated prime of N/ m_R N. Then N → N ⊗_S S' is R-universally injective.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $N$ be an $S$-module.\nLet $S \\to S'$ be a ring map.\nAssume\n\\begin{enumerate}\n\\item $R \\to S$ is a local homomorphism of local rings\n\\item $S$ is essentially of finite presentation over $R$,\n\\item $N$ is of finite presentation over $S$,\n\\item $N$ is flat over $R$,\n\\item $S \\to S'$ is flat, and\n\\item the image of $\\Spec(S') \\to \\Spec(S)$ contains\nall primes $\\mathfrak q$ of $S$ lying over $\\mathfrak m_R$\nsuch that $\\mathfrak q$ is an associated prime of $N/\\mathfrak m_R N$.\n\\end{enumerate}\nThen $N \\to N \\otimes_S S'$ is $R$-universally injective.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Localization and universally injective maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DG","source_file":"flat.tex","source_line":1847,"source_end_line":1864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1847-L1864","statement_sha256":"276185b987c4bbf69f953a7585b4204b88e4a0a4fcbbb71723e495ff87482585","origin":"The Stacks Project","memory_eligible":false,"source_rank":7666,"rank":7666,"depth":13,"x":1935.202,"y":788.048,"cluster":"scheme-morphisms"},{"id":"stacks:05DH","tag":"05DH","title":"Localization and universally injective maps · Lemma 05DH","summary":"Let R → S be a ring map. Let N be an S-module. Let S → S' be a ring map. Assume • R → S is of finite presentation and N is of finite presentation over S, • N is flat over R, • S → S' is flat, and • the image of Spec(S') → Spec(S) contains all primes q such that q is an associated prime of N ⊗_R kappa( p) where p is the inverse image of q in R. Then N → N ⊗_S S' is R-universally injective.","statement_latex":"Let $R \\to S$ be a ring map.\nLet $N$ be an $S$-module.\nLet $S \\to S'$ be a ring map.\nAssume\n\\begin{enumerate}\n\\item $R \\to S$ is of finite presentation and $N$ is of finite presentation\nover $S$,\n\\item $N$ is flat over $R$,\n\\item $S \\to S'$ is flat, and\n\\item the image of $\\Spec(S') \\to \\Spec(S)$ contains\nall primes $\\mathfrak q$ such that $\\mathfrak q$ is an associated prime\nof $N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p$ is the inverse\nimage of $\\mathfrak q$ in $R$.\n\\end{enumerate}\nThen $N \\to N \\otimes_S S'$ is $R$-universally injective.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Localization and universally injective maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DH","source_file":"flat.tex","source_line":1906,"source_end_line":1923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1906-L1923","statement_sha256":"bcf24a445f0d0e644b2e6a21cf8e3a7b4ccaabae4918594b32030f194b540cd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7667,"rank":7667,"depth":14,"x":2013.006,"y":546.481,"cluster":"scheme-morphisms"},{"id":"stacks:05FQ","tag":"05FQ","title":"Localization and universally injective maps · Lemma 05FQ","summary":"Let (R, m) be a local ring. Let u : M → N be an R-module map. If M is a projective R-module, N is a flat R-module, and overlineu : M/ mM → N/ mN is injective then u is universally injective.","statement_latex":"Let $(R, \\mathfrak m)$ be a local ring. Let $u : M \\to N$ be an $R$-module map.\nIf $M$ is a projective $R$-module, $N$ is a flat $R$-module, and\n$\\overline{u} : M/\\mathfrak mM \\to N/\\mathfrak mN$ is injective\nthen $u$ is universally injective.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Localization and universally injective maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FQ","source_file":"flat.tex","source_line":1945,"source_end_line":1951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1945-L1951","statement_sha256":"334bacde1422d3865b7aabacb7e25af8e80725b831e980881e0b4029346f9c5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7668,"rank":7668,"depth":4,"x":2149.953,"y":768.849,"cluster":"scheme-morphisms"},{"id":"stacks:05FR","tag":"05FR","title":"Localization and universally injective maps · Lemma 05FR","summary":"Assumption and notation as in Lemma [Tag 05DE]. Assume moreover that N is projective as an R-module. Then each s ∈ Sigma defines a universally injective R-module map s : N → N, and the map N → Sigma^-1N is R-universally injective.","statement_latex":"Assumption and notation as in\nLemma \\ref{lemma-homothety-spectrum}.\nAssume moreover that $N$ is projective as an $R$-module.\nThen each $s \\in \\Sigma$ defines a\nuniversally injective $R$-module map $s : N \\to N$, and the\nmap $N \\to \\Sigma^{-1}N$ is $R$-universally injective.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Localization and universally injective maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FR","source_file":"flat.tex","source_line":1975,"source_end_line":1983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L1975-L1983","statement_sha256":"2675af6c1b06212cdcef52d3f9bec87c2abb387eac107850cc02ddabb2da36f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7669,"rank":7669,"depth":11,"x":1870.032,"y":682.549,"cluster":"scheme-morphisms"},{"id":"stacks:05DJ","tag":"05DJ","title":"Completion and Mittag-Leffler modules · Lemma 05DJ","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let A be a set. Assume R is Noetherian and complete with respect to I. The completion (bigoplus_α ∈ A R)^wedge is flat and Mittag-Leffler.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal. Let $A$ be a set.\nAssume $R$ is Noetherian and complete with respect to $I$. The completion\n$(\\bigoplus\\nolimits_{\\alpha \\in A} R)^\\wedge$\nis flat and Mittag-Leffler.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Completion and Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DJ","source_file":"flat.tex","source_line":2002,"source_end_line":2008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2002-L2008","statement_sha256":"87e0f5a20744cb94f5326f5c4f915c972f7b20113c1ea95d27883f71ad4e67fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7670,"rank":7670,"depth":9,"x":2145.956,"y":587.312,"cluster":"scheme-morphisms"},{"id":"stacks:05DK","tag":"05DK","title":"Completion and Mittag-Leffler modules · Lemma 05DK","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let M be an R-module. Assume • R is Noetherian and I-adically complete, • M is flat over R, and • M/IM is a projective R/I-module. Then the I-adic completion M^wedge is a flat Mittag-Leffler R-module.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nLet $M$ be an $R$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is Noetherian and $I$-adically complete,\n\\item $M$ is flat over $R$, and\n\\item $M/IM$ is a projective $R/I$-module.\n\\end{enumerate}\nThen the $I$-adic completion $M^\\wedge$ is a flat Mittag-Leffler\n$R$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Completion and Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DK","source_file":"flat.tex","source_line":2029,"source_end_line":2041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2029-L2041","statement_sha256":"4d61716a23667094071e98515e8dc6c66a23655cd48b5e8ffb2e8dcb3528e387","origin":"The Stacks Project","memory_eligible":false,"source_rank":7671,"rank":7671,"depth":10,"x":2019.028,"y":814.198,"cluster":"scheme-morphisms"},{"id":"stacks:05DL","tag":"05DL","title":"Completion and Mittag-Leffler modules · Lemma 05DL","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let R → S be a ring map, and N an S-module. Assume • R is a Noetherian ring, • S is a Noetherian ring, • N is a finite S-module, and • for any finite R-module Q, any q ∈ Ass_S(Q ⊗_R N) satisfies IS + q not = S. Then the map N → N^wedge of N into the I-adic completion of N is universally injective as a map of R-modules.","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\nLet $R \\to S$ be a ring map, and $N$ an $S$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is a Noetherian ring,\n\\item $S$ is a Noetherian ring,\n\\item $N$ is a finite $S$-module, and\n\\item for any finite $R$-module $Q$, any\n$\\mathfrak q \\in \\text{Ass}_S(Q \\otimes_R N)$\nsatisfies $IS + \\mathfrak q \\not = S$.\n\\end{enumerate}\nThen the map $N \\to N^\\wedge$ of $N$ into the $I$-adic completion of $N$\nis universally injective as a map of $R$-modules.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Completion and Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DL","source_file":"flat.tex","source_line":2059,"source_end_line":2075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2059-L2075","statement_sha256":"89a103c2dba0198910d73c4f833ff903a5d27aa3dc9888f351b371a040eeb70f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7672,"rank":7672,"depth":10,"x":1930.131,"y":574.776,"cluster":"scheme-morphisms"},{"id":"stacks:05DM","tag":"05DM","title":"Completion and Mittag-Leffler modules · Lemma 05DM","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let R → S be a ring map, and N an S-module. Assume • R is a Noetherian ring, • S is a Noetherian ring, • N is a finite S-module, • N is flat over R, and • for any prime q ⊂ S which is an associated prime of N ⊗_R kappa( p) where p = R ∩ q we have IS + q not = S. Then the map N → N^wedge of N into the I-adic completion of N is universally injective as a map of R-modules.","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\nLet $R \\to S$ be a ring map, and $N$ an $S$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is a Noetherian ring,\n\\item $S$ is a Noetherian ring,\n\\item $N$ is a finite $S$-module,\n\\item $N$ is flat over $R$, and\n\\item for any prime $\\mathfrak q \\subset S$ which is an associated prime of\n$N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p = R \\cap \\mathfrak q$\nwe have $IS + \\mathfrak q \\not = S$.\n\\end{enumerate}\nThen the map $N \\to N^\\wedge$ of $N$ into the $I$-adic completion of $N$\nis universally injective as a map of $R$-modules.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Completion and Mittag-Leffler modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DM","source_file":"flat.tex","source_line":2102,"source_end_line":2119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2102-L2119","statement_sha256":"c77e084e77c074b9fd468c444e8f1df7552c43f1fa9f07249c3c7e7796c09e0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7673,"rank":7673,"depth":13,"x":2188.326,"y":700.931,"cluster":"scheme-morphisms"},{"id":"stacks:05DP","tag":"05DP","title":"Projective modules · Lemma 05DP","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let R → S be a ring map, and N an S-module. Assume • R is Noetherian and I-adically complete, • R → S is of finite type, • N is a finite S-module, • N is flat over R, • N/IN is projective as a R/I-module, and • for any prime q ⊂ S which is an associated prime of N ⊗_R kappa( p) where p = R ∩ q we have IS + q not = S. Then N is projective as an R-module.","statement_latex":"Let $R$ be a ring.\nLet $I \\subset R$ be an ideal.\nLet $R \\to S$ be a ring map, and $N$ an $S$-module.\nAssume\n\\begin{enumerate}\n\\item $R$ is Noetherian and $I$-adically complete,\n\\item $R \\to S$ is of finite type,\n\\item $N$ is a finite $S$-module,\n\\item $N$ is flat over $R$,\n\\item $N/IN$ is projective as a $R/I$-module, and\n\\item for any prime $\\mathfrak q \\subset S$ which is an associated prime of\n$N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p = R \\cap \\mathfrak q$\nwe have $IS + \\mathfrak q \\not = S$.\n\\end{enumerate}\nThen $N$ is projective as an $R$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05DP","source_file":"flat.tex","source_line":2144,"source_end_line":2161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2144-L2161","statement_sha256":"033bf9eac6e7ed8a1be63279c77578798133050c3a5c85634c3faef7d8a6f882","origin":"The Stacks Project","memory_eligible":false,"source_rank":7674,"rank":7674,"depth":14,"x":1896.365,"y":754.434,"cluster":"scheme-morphisms"},{"id":"stacks:05FS","tag":"05FS","title":"Projective modules · Lemma 05FS","summary":"Let R be a ring. Let R → S be a ring map. Assume • R is Noetherian, • R → S is of finite type and flat, and • every fibre ring S ⊗_R kappa( p) is geometrically integral over kappa( p). Then S is projective as an R-module.","statement_latex":"Let $R$ be a ring.\nLet $R \\to S$ be a ring map.\nAssume\n\\begin{enumerate}\n\\item $R$ is Noetherian,\n\\item $R \\to S$ is of finite type and flat, and\n\\item every fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$ is\ngeometrically integral over $\\kappa(\\mathfrak p)$.\n\\end{enumerate}\nThen $S$ is projective as an $R$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FS","source_file":"flat.tex","source_line":2178,"source_end_line":2190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2178-L2190","statement_sha256":"9af0762c4606a1c12fe55e51900bce4233b9e2a5a1fc329e5ce5c69e5a62451e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7675,"rank":7675,"depth":15,"x":2068.698,"y":549.232,"cluster":"scheme-morphisms"},{"id":"stacks:05FT","tag":"05FT","title":"Projective modules · Lemma 05FT","summary":"Let R be a ring. Let R → S be a ring map. Assume • R → S is of finite presentation and flat, and • every fibre ring S ⊗_R kappa( p) is geometrically integral over kappa( p). Then S is projective as an R-module.","statement_latex":"Let $R$ be a ring. Let $R \\to S$ be a ring map.\nAssume\n\\begin{enumerate}\n\\item $R \\to S$ is of finite presentation and flat, and\n\\item every fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$ is\ngeometrically integral over $\\kappa(\\mathfrak p)$.\n\\end{enumerate}\nThen $S$ is projective as an $R$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FT","source_file":"flat.tex","source_line":2246,"source_end_line":2256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2246-L2256","statement_sha256":"bef213abb6fbd369777addf3f1f8b6b87e274a942dcc3a7cfa7f23e9709ca9ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":7676,"rank":7676,"depth":48,"x":2106.656,"y":798.433,"cluster":"scheme-morphisms"},{"id":"stacks:05I3","tag":"05I3","title":"Flat finite type modules, Part I · Lemma 05I3","summary":"Let (R, m) be a local ring. Let R → S be a finitely presented flat ring map with geometrically integral fibres. Write p = mS. Let q ⊂ S be a prime ideal lying over m. Let N be a finite S-module. There exist r ≥ 0 and an S-module map α : S^⊕ r → N such that α : kappa( p)^⊕ r → N ⊗_S kappa( p) is an isomorphism. For any such α the following are equivalent: • N_ q is R-flat, • α is R-universally injective and Coker(α)_ q is R-flat, • α is injective and Coker(α)_ q is R-flat,…","statement_latex":"Let $(R, \\mathfrak m)$ be a local ring. Let $R \\to S$ be a finitely presented\nflat ring map with geometrically integral fibres. Write\n$\\mathfrak p = \\mathfrak mS$. Let $\\mathfrak q \\subset S$ be a prime ideal\nlying over $\\mathfrak m$. Let $N$ be a finite $S$-module.\nThere exist $r \\geq 0$ and an $S$-module map\n$$\n\\alpha : S^{\\oplus r} \\longrightarrow N\n$$\nsuch that\n$\\alpha : \\kappa(\\mathfrak p)^{\\oplus r} \\to N \\otimes_S \\kappa(\\mathfrak p)$\nis an isomorphism. For any such $\\alpha$ the following are equivalent:\n\\begin{enumerate}\n\\item $N_{\\mathfrak q}$ is $R$-flat,\n\\item $\\alpha$ is $R$-universally injective and\n$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat,\n\\item $\\alpha$ is injective and\n$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat,\n\\item $\\alpha_{\\mathfrak p}$ is an isomorphism and\n$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat, and\n\\item $\\alpha_{\\mathfrak q}$ is injective and\n$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05I3","source_file":"flat.tex","source_line":2355,"source_end_line":2379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2355-L2379","statement_sha256":"eeb0ac114c94801c3522a8b88278b54cc53906d1fdcf6643113869588ac8eabc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7677,"rank":7677,"depth":49,"x":1878.172,"y":636.148,"cluster":"scheme-morphisms"},{"id":"stacks:05I4","tag":"05I4","title":"Flat finite type modules, Part I · Lemma 05I4","summary":"Let (R, m) be a local ring. Let R → S be a ring map of finite presentation. Let N be a finite S-module. Let q be a prime of S lying over m. Assume that N_ q is flat over R, and assume there exists a complete dévissage of N/S/R at q. Then N is a finitely presented S-module, free as an R-module, and there exists an isomorphism N ≅ B_1^⊕ r_1 ⊕ … ⊕ B_n^⊕ r_n as R-modules where each B_i is a smooth R-algebra with geometrically irreducible fibres.","statement_latex":"Let $(R, \\mathfrak m)$ be a local ring.\nLet $R \\to S$ be a ring map of finite presentation.\nLet $N$ be a finite $S$-module.\nLet $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak m$.\nAssume that $N_{\\mathfrak q}$ is flat over $R$, and\nassume there exists a complete d\\'evissage of $N/S/R$ at $\\mathfrak q$.\nThen $N$ is a finitely presented $S$-module, free as an $R$-module,\nand there exists an isomorphism\n$$\nN \\cong B_1^{\\oplus r_1} \\oplus \\ldots \\oplus B_n^{\\oplus r_n}\n$$\nas $R$-modules where each $B_i$ is a smooth $R$-algebra with geometrically\nirreducible fibres.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05I4","source_file":"flat.tex","source_line":2425,"source_end_line":2440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2425-L2440","statement_sha256":"6e51baa9b8ed9febea796f511795acb9fc7d1757077de9f61592e75fe9dbb612","origin":"The Stacks Project","memory_eligible":false,"source_rank":7678,"rank":7678,"depth":50,"x":2177.281,"y":626.163,"cluster":"scheme-morphisms"},{"id":"stacks:05I5","tag":"05I5","title":"Flat finite type modules, Part I · Proposition 05I5","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let x ∈ X with image s ∈ S. Assume that • f is locally of finite presentation, • F is of finite type, and • F is flat at x over S. Then there exists an elementary étale neighbourhood (S', s') → (S, s) and an open subscheme V ⊂ X ×_S Spec(O_S', s') which contains the unique point of X ×_S Spec(O_S', s') mapping to x such that the pullback of F to V is an O_V-module of finite presentation and flat…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $x \\in X$ with image $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $\\mathcal{F}$ is of finite type, and\n\\item $\\mathcal{F}$ is flat at $x$ over $S$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$\nand an open subscheme\n$$\nV \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})\n$$\nwhich contains the unique point of\n$X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ mapping to $x$\nsuch that the pullback of $\\mathcal{F}$ to $V$ is an $\\mathcal{O}_V$-module\nof finite presentation and flat over $\\mathcal{O}_{S', s'}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05I5","source_file":"flat.tex","source_line":2476,"source_end_line":2496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2476-L2496","statement_sha256":"594e462af3423423bdf82ddb19cee4694b8933c6191be3420da96237231eb4ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":7679,"rank":7679,"depth":51,"x":1964.665,"y":803.32,"cluster":"scheme-morphisms"},{"id":"stacks:05M9","tag":"05M9","title":"Flat finite type modules, Part I · Lemma 05M9","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let s ∈ S. Then the set (x ∈ X_s mid F flat over S at x) is open in the fibre X_s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $s \\in S$. Then the set\n$$\n\\{x \\in X_s \\mid \\mathcal{F} \\text{ flat over }S\\text{ at }x\\}\n$$\nis open in the fibre $X_s$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05M9","source_file":"flat.tex","source_line":2675,"source_end_line":2684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2675-L2684","statement_sha256":"c553156eb7b421202e0f98ace15d9672765826e4d0f5e92ae27d4d2fa65210ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":7680,"rank":7680,"depth":52,"x":1978.986,"y":551.94,"cluster":"scheme-morphisms"},{"id":"stacks:05KT","tag":"05KT","title":"Flat finite type modules, Part I · Lemma 05KT","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let x ∈ X with image s ∈ S. Assume that • f is locally of finite type, • F is of finite type, and • F is flat at x over S. Then there exists an elementary étale neighbourhood (S', s') → (S, s) and an open subscheme V ⊂ X ×_S Spec(O_S', s') which contains the unique point of X ×_S Spec(O_S', s') mapping to x such that the pullback of F to V is flat over O_S', s'.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $x \\in X$ with image $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item $\\mathcal{F}$ is of finite type, and\n\\item $\\mathcal{F}$ is flat at $x$ over $S$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$\nand an open subscheme\n$$\nV \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})\n$$\nwhich contains the unique point of\n$X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ mapping to $x$\nsuch that the pullback of $\\mathcal{F}$ to $V$ is flat over\n$\\mathcal{O}_{S', s'}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KT","source_file":"flat.tex","source_line":2700,"source_end_line":2720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2700-L2720","statement_sha256":"9f4c3c639bcf8949554b7508df12371bef7a8aa2bc37f8e5100076a6eec255e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7681,"rank":7681,"depth":52,"x":2170.657,"y":745.509,"cluster":"scheme-morphisms"},{"id":"stacks:05KU","tag":"05KU","title":"Flat finite type modules, Part I · Lemma 05KU","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let s ∈ S. Assume that • f is of finite presentation, • F is of finite type, and • F is flat over S at every point of the fibre X_s. Then there exists an elementary étale neighbourhood (S', s') → (S, s) and an open subscheme V ⊂ X ×_S Spec(O_S', s') which contains the fibre X_s = X ×_S s' such that the pullback of F to V is an O_V-module of finite presentation and flat over O_S', s'.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is of finite presentation,\n\\item $\\mathcal{F}$ is of finite type, and\n\\item $\\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$\nand an open subscheme\n$$\nV \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})\n$$\nwhich contains the fibre $X_s = X \\times_S s'$ such that the pullback\nof $\\mathcal{F}$ to $V$ is an $\\mathcal{O}_V$-module\nof finite presentation and flat over $\\mathcal{O}_{S', s'}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KU","source_file":"flat.tex","source_line":2744,"source_end_line":2763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2744-L2763","statement_sha256":"f6f7f65c6e8ebfbc505cb1ab9ee4f1010494851bba08de1cb2506906e5eece82","origin":"The Stacks Project","memory_eligible":false,"source_rank":7682,"rank":7682,"depth":52,"x":1873.537,"y":711.519,"cluster":"scheme-morphisms"},{"id":"stacks:05KV","tag":"05KV","title":"Flat finite type modules, Part I · Lemma 05KV","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let s ∈ S. Assume that • f is of finite type, • F is of finite type, and • F is flat over S at every point of the fibre X_s. Then there exists an elementary étale neighbourhood (S', s') → (S, s) and an open subscheme V ⊂ X ×_S Spec(O_S', s') which contains the fibre X_s = X ×_S s' such that the pullback of F to V is flat over O_S', s'.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is of finite type,\n\\item $\\mathcal{F}$ is of finite type, and\n\\item $\\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$\nand an open subscheme\n$$\nV \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})\n$$\nwhich contains the fibre $X_s = X \\times_S s'$ such that the pullback\nof $\\mathcal{F}$ to $V$ is flat over $\\mathcal{O}_{S', s'}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KV","source_file":"flat.tex","source_line":2794,"source_end_line":2812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2794-L2812","statement_sha256":"514d37750e23144285fb85dba6e66abbfcb2eba6d971dd8a09900bebbc1c69c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7683,"rank":7683,"depth":53,"x":2120.063,"y":567.932,"cluster":"scheme-morphisms"},{"id":"stacks:05I6","tag":"05I6","title":"Flat finite type modules, Part I · Lemma 05I6","summary":"Let S be a scheme. Let X be locally of finite type over S. Let x ∈ X with image s ∈ S. If X is flat at x over S, then there exists an elementary étale neighbourhood (S', s') → (S, s) and an open subscheme V ⊂ X ×_S Spec(O_S', s') which contains the unique point of X ×_S Spec(O_S', s') mapping to x such that V → Spec(O_S', s') is flat and of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $X$ be locally of finite type over $S$.\nLet $x \\in X$ with image $s \\in S$.\nIf $X$ is flat at $x$ over $S$, then there exists an elementary\n\\'etale neighbourhood $(S', s') \\to (S, s)$ and an open subscheme\n$$\nV \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})\n$$\nwhich contains the unique point of\n$X \\times_S \\Spec(\\mathcal{O}_{S', s'})$ mapping to $x$\nsuch that $V \\to \\Spec(\\mathcal{O}_{S', s'})$\nis flat and of finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05I6","source_file":"flat.tex","source_line":2844,"source_end_line":2857,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2844-L2857","statement_sha256":"0012b95a296b1ddd296043c5a872b7d316dab0c107f305efb8bc3e75051a571a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7684,"rank":7684,"depth":52,"x":2053.721,"y":813.796,"cluster":"scheme-morphisms"},{"id":"stacks:05I7","tag":"05I7","title":"Flat finite type modules, Part I · Lemma 05I7","summary":"Let f : X → S be a morphism which is locally of finite presentation. Let F be a quasi-coherent O_X-module of finite type. If x ∈ X and F is flat at x over S, then F_x is an O_X, x-module of finite presentation.","statement_latex":"Let $f : X \\to S$ be a morphism which is locally of finite presentation.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nIf $x \\in X$ and $\\mathcal{F}$ is flat at $x$ over $S$, then\n$\\mathcal{F}_x$ is an $\\mathcal{O}_{X, x}$-module of finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05I7","source_file":"flat.tex","source_line":2878,"source_end_line":2884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2878-L2884","statement_sha256":"68af15a5624df0bc8bb2c19488fb0d259ccabb259271644b81628c15c8c913f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7685,"rank":7685,"depth":52,"x":1904.862,"y":594.766,"cluster":"scheme-morphisms"},{"id":"stacks:05I8","tag":"05I8","title":"Flat finite type modules, Part I · Lemma 05I8","summary":"Let f : X → S be a morphism which is locally of finite type. Let x ∈ X with image s ∈ S. If f is flat at x over S, then O_X, x is essentially of finite presentation over O_S, s.","statement_latex":"Let $f : X \\to S$ be a morphism which is locally of finite type.\nLet $x \\in X$ with image $s \\in S$. If $f$ is flat at $x$ over $S$, then\n$\\mathcal{O}_{X, x}$ is essentially of finite presentation over\n$\\mathcal{O}_{S, s}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05I8","source_file":"flat.tex","source_line":2908,"source_end_line":2914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2908-L2914","statement_sha256":"443d5cdeaf53fad5df24cf45146122fc5360f3fc225ed76cd1538a8bc915bbe8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7686,"rank":7686,"depth":53,"x":2190.883,"y":671.841,"cluster":"scheme-morphisms"},{"id":"stacks:081N","tag":"081N","title":"Extending properties from an open · Lemma 081N","summary":"S-flat and finite type extensions of finitely presented modules on a (good) open are also X-finitely presented. Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. Let U ⊂ S be open. Assume • f is locally of finite presentation, • F is of finite type and flat over S, • U ⊂ S is retrocompact and scheme theoretically dense, • F|_f^-1U is of finite presentation. Then F is of finite presentation.","statement_latex":"\\begin{slogan}\n$S$-flat and finite type extensions of finitely presented modules\non a (good) open are also $X$-finitely presented.\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a\nquasi-coherent $\\mathcal{O}_X$-module. Let $U \\subset S$ be open.\nAssume\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $\\mathcal{F}$ is of finite type and flat over $S$,\n\\item $U \\subset S$ is retrocompact and scheme theoretically dense,\n\\item $\\mathcal{F}|_{f^{-1}U}$ is of finite presentation.\n\\end{enumerate}\nThen $\\mathcal{F}$ is of finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081N","source_file":"flat.tex","source_line":2942,"source_end_line":2958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2942-L2958","statement_sha256":"2cd7e8773cdd9c2169ca3ba5acf2e649fa7d56fca060b9da0257d4aceb5baf41","origin":"The Stacks Project","memory_eligible":false,"source_rank":7687,"rank":7687,"depth":53,"x":1917.883,"y":777.345,"cluster":"scheme-morphisms"},{"id":"stacks:081P","tag":"081P","title":"Extending properties from an open · Lemma 081P","summary":"Let f : X → S be a morphism of schemes. Let U ⊂ S be open. Assume • f is locally of finite type and flat, • U ⊂ S is retrocompact and scheme theoretically dense, • f|_f^-1U : f^-1U → U is locally of finite presentation. Then f is of locally of finite presentation.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset S$ be open.\nAssume\n\\begin{enumerate}\n\\item $f$ is locally of finite type and flat,\n\\item $U \\subset S$ is retrocompact and scheme theoretically dense,\n\\item $f|_{f^{-1}U} : f^{-1}U \\to U$ is locally of finite presentation.\n\\end{enumerate}\nThen $f$ is of locally of finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081P","source_file":"flat.tex","source_line":2999,"source_end_line":3009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L2999-L3009","statement_sha256":"bbfb401868c92c022b06da944d1359012ead67ea1632641c9fc787434fdffd70","origin":"The Stacks Project","memory_eligible":false,"source_rank":7688,"rank":7688,"depth":54,"x":2034.393,"y":544.546,"cluster":"scheme-morphisms"},{"id":"stacks:081L","tag":"081L","title":"Extending properties from an open · Lemma 081L","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite type. Let U ⊂ S be a dense open such that X_U → U has relative dimension ≤ e, see Morphisms, Definition [Tag 02NJ]. If also either • f is locally of finite presentation, or • U ⊂ S is retrocompact, then f has relative dimension ≤ e.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and locally\nof finite type. Let $U \\subset S$ be a dense open such that\n$X_U \\to U$ has relative dimension $\\leq e$, see\nMorphisms, Definition \\ref{morphisms-definition-relative-dimension-d}.\nIf also either\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation, or\n\\item $U \\subset S$ is retrocompact,\n\\end{enumerate}\nthen $f$ has relative dimension $\\leq e$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081L","source_file":"flat.tex","source_line":3019,"source_end_line":3031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3019-L3031","statement_sha256":"c28b46a7d575eb3fad410436cda1e75029532b1d94e5446d1df85fb26753910b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7689,"rank":7689,"depth":55,"x":2135.732,"y":782.416,"cluster":"scheme-morphisms"},{"id":"stacks:0B48","tag":"0B48","title":"Extending properties from an open · Lemma 0B48","summary":"Let f : X → S be a morphism of schemes which is flat and proper. Let U ⊂ S be a dense open such that X_U → U is finite. If also either f is locally of finite presentation or U ⊂ S is retrocompact, then f is finite.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and proper.\nLet $U \\subset S$ be a dense open such that $X_U \\to U$ is finite.\nIf also either $f$ is locally of finite presentation or\n$U \\subset S$ is retrocompact, then $f$ is finite.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B48","source_file":"flat.tex","source_line":3053,"source_end_line":3059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3053-L3059","statement_sha256":"aa53fb3f9d2018962fbccc188c596c1b4e9d728cefe0429e628db57bcbcc5583","origin":"The Stacks Project","memory_eligible":false,"source_rank":7690,"rank":7690,"depth":56,"x":1869.61,"y":664.468,"cluster":"scheme-morphisms"},{"id":"stacks:081M","tag":"081M","title":"Extending properties from an open · Lemma 081M","summary":"Let f : X → S be a morphism of schemes and U ⊂ S an open. If • f is separated, locally of finite type, and flat, • f^-1(U) → U is an isomorphism, and • U ⊂ S is retrocompact and scheme theoretically dense, then f is an open immersion.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes and $U \\subset S$ an open. If\n\\begin{enumerate}\n\\item $f$ is separated, locally of finite type, and flat,\n\\item $f^{-1}(U) \\to U$ is an isomorphism, and\n\\item $U \\subset S$ is retrocompact and scheme theoretically dense,\n\\end{enumerate}\nthen $f$ is an open immersion.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081M","source_file":"flat.tex","source_line":3072,"source_end_line":3081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3072-L3081","statement_sha256":"4c10696992815d587857c5f7d307b966d8445b2440b29f9fb8730b9203a7176b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7691,"rank":7691,"depth":56,"x":2160.813,"y":600.411,"cluster":"scheme-morphisms"},{"id":"stacks:05IA","tag":"05IA","title":"Flat finitely presented modules · Lemma 05IA","summary":"Let R be a ring. Let R → S be a finitely presented flat ring map with geometrically integral fibres. Let q ⊂ S be a prime ideal lying over the prime r ⊂ R. Set p = r S. Let N be a finitely presented S-module. There exists r ≥ 0 and an S-module map α : S^⊕ r → N such that α : kappa( p)^⊕ r → N ⊗_S kappa( p) is an isomorphism. For any such α the following are equivalent: • N_ q is R-flat, • there exists an f ∈ R, f not ∈ r such that α_f : S_f^⊕ r → N_f is R_f-universally…","statement_latex":"Let $R$ be a ring. Let $R \\to S$ be a finitely presented\nflat ring map with geometrically integral fibres. Let\n$\\mathfrak q \\subset S$ be a prime ideal lying over the prime\n$\\mathfrak r \\subset R$. Set $\\mathfrak p = \\mathfrak r S$.\nLet $N$ be a finitely presented $S$-module.\nThere exists $r \\geq 0$ and an $S$-module map\n$$\n\\alpha : S^{\\oplus r} \\longrightarrow N\n$$\nsuch that\n$\\alpha : \\kappa(\\mathfrak p)^{\\oplus r} \\to N \\otimes_S \\kappa(\\mathfrak p)$\nis an isomorphism. For any such $\\alpha$ the following are equivalent:\n\\begin{enumerate}\n\\item $N_{\\mathfrak q}$ is $R$-flat,\n\\item there exists an $f \\in R$, $f \\not \\in \\mathfrak r$ such that\n$\\alpha_f : S_f^{\\oplus r} \\to N_f$ is $R_f$-universally injective and\na $g \\in S$, $g \\not \\in \\mathfrak q$ such that $\\Coker(\\alpha)_g$\nis $R$-flat,\n\\item $\\alpha_{\\mathfrak r}$ is $R_{\\mathfrak r}$-universally injective and\n$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat\n\\item $\\alpha_{\\mathfrak r}$ is injective and\n$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat,\n\\item $\\alpha_{\\mathfrak p}$ is an isomorphism and\n$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat, and\n\\item $\\alpha_{\\mathfrak q}$ is injective and\n$\\Coker(\\alpha)_{\\mathfrak q}$ is $R$-flat.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IA","source_file":"flat.tex","source_line":3140,"source_end_line":3169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3140-L3169","statement_sha256":"a161bcdc4c464eb70142b7ee665f3f3ab5d7a1d7ad4afbfb2f96527667d708b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7692,"rank":7692,"depth":50,"x":1997.531,"y":812.968,"cluster":"scheme-morphisms"},{"id":"stacks:05IB","tag":"05IB","title":"Flat finitely presented modules · Lemma 05IB","summary":"Let R → S be a ring map of finite presentation. Let N be a finitely presented S-module flat over R. Let r ⊂ R be a prime ideal. Assume there exists a complete dévissage of N/S/R over r. Then there exists an f ∈ R, f not ∈ r such that N_f ≅ B_1^⊕ r_1 ⊕ … ⊕ B_n^⊕ r_n as R-modules where each B_i is a smooth R_f-algebra with geometrically irreducible fibres. Moreover, N_f is projective as an R_f-module.","statement_latex":"Let $R \\to S$ be a ring map of finite presentation.\nLet $N$ be a finitely presented $S$-module flat over $R$.\nLet $\\mathfrak r \\subset R$ be a prime ideal.\nAssume there exists a complete d\\'evissage of $N/S/R$ over $\\mathfrak r$.\nThen there exists an $f \\in R$, $f \\not \\in \\mathfrak r$\nsuch that\n$$\nN_f \\cong B_1^{\\oplus r_1} \\oplus \\ldots \\oplus B_n^{\\oplus r_n}\n$$\nas $R$-modules where each $B_i$ is a smooth $R_f$-algebra with geometrically\nirreducible fibres. Moreover, $N_f$ is projective as an $R_f$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IB","source_file":"flat.tex","source_line":3228,"source_end_line":3241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3228-L3241","statement_sha256":"51f2ee1d8a9bd4aba3611433ed5b48b9665770647058f05c50798b2bd1524835","origin":"The Stacks Project","memory_eligible":false,"source_rank":7693,"rank":7693,"depth":51,"x":1946.98,"y":563.481,"cluster":"scheme-morphisms"},{"id":"stacks:05ID","tag":"05ID","title":"Flat finitely presented modules · Proposition 05ID","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let x ∈ X with image s ∈ S. Assume that • f is locally of finite presentation, • F is of finite presentation, and • F is flat at x over S. Then there exists a commutative diagram of pointed schemes xymatrix (X, x) ar[d] & (X', x') ar[l]^g ar[d] (S, s) & (S', s') ar[l] whose horizontal arrows are elementary étale neighbourhoods such that X', S' are affine and such that Γ(X', g^*F) is a projective…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $x \\in X$ with image $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $\\mathcal{F}$ is of finite presentation, and\n\\item $\\mathcal{F}$ is flat at $x$ over $S$.\n\\end{enumerate}\nThen there exists a commutative diagram of pointed schemes\n$$\n\\xymatrix{\n(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\\n(S, s) & (S', s') \\ar[l]\n}\n$$\nwhose horizontal arrows are elementary \\'etale neighbourhoods\nsuch that $X'$, $S'$ are affine and such that\n$\\Gamma(X', g^*\\mathcal{F})$ is a projective\n$\\Gamma(S', \\mathcal{O}_{S'})$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ID","source_file":"flat.tex","source_line":3286,"source_end_line":3308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3286-L3308","statement_sha256":"2cb2dd0f421069afd2bd00bf7d30de75c67392541ad021d358c1c9840b398b15","origin":"The Stacks Project","memory_eligible":false,"source_rank":7694,"rank":7694,"depth":52,"x":2184.982,"y":718.828,"cluster":"scheme-morphisms"},{"id":"stacks:05KW","tag":"05KW","title":"Flat finitely presented modules · Lemma 05KW","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let s ∈ S. Assume that • f is of finite presentation, • F is of finite presentation, and • F is flat over S at every point of the fibre X_s. Then there exists an elementary étale neighbourhood (S', s') → (S, s) and a commutative diagram of schemes xymatrix X ar[d] & X' ar[l]^g ar[d] S & S' ar[l] such that g is étale, X_s ⊂ g(X'), the schemes X', S' are affine, and such that Γ(X', g^*F) is a…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is of finite presentation,\n\\item $\\mathcal{F}$ is of finite presentation, and\n\\item $\\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood\n$(S', s') \\to (S, s)$ and a commutative diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\\nS & S' \\ar[l]\n}\n$$\nsuch that $g$ is \\'etale, $X_s \\subset g(X')$, the schemes\n$X'$, $S'$ are affine, and such that\n$\\Gamma(X', g^*\\mathcal{F})$ is a projective\n$\\Gamma(S', \\mathcal{O}_{S'})$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KW","source_file":"flat.tex","source_line":3341,"source_end_line":3364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3341-L3364","statement_sha256":"58219ecc416815447502fab25f3dc53032b3450a87f4e80820425b07add6417a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7695,"rank":7695,"depth":53,"x":1884.436,"y":739.334,"cluster":"scheme-morphisms"},{"id":"stacks:05IE","tag":"05IE","title":"Flat finitely presented modules · Lemma 05IE","summary":"Let f : X → S be locally of finite presentation. Let x ∈ X with image s ∈ S. If f is flat at x over S, then there exists a commutative diagram of pointed schemes xymatrix (X, x) ar[d] & (X', x') ar[l]^g ar[d] (S, s) & (S', s') ar[l] whose horizontal arrows are elementary étale neighbourhoods such that X', S' are affine and such that Γ(X', O_X') is a projective Γ(S', O_S')-module.","statement_latex":"Let $f : X \\to S$ be locally of finite presentation.\nLet $x \\in X$ with image $s \\in S$.\nIf $f$ is flat at $x$ over $S$, then there exists a commutative\ndiagram of pointed schemes\n$$\n\\xymatrix{\n(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\\n(S, s) & (S', s') \\ar[l]\n}\n$$\nwhose horizontal arrows are elementary \\'etale neighbourhoods\nsuch that $X'$, $S'$ are affine and such that\n$\\Gamma(X', \\mathcal{O}_{X'})$ is a projective\n$\\Gamma(S', \\mathcal{O}_{S'})$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IE","source_file":"flat.tex","source_line":3407,"source_end_line":3423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3407-L3423","statement_sha256":"95efb4cd25e3c60562cb0756971094c0a8f92d13af76e06e0425fa2232dfe368","origin":"The Stacks Project","memory_eligible":false,"source_rank":7696,"rank":7696,"depth":53,"x":2089.647,"y":553.6,"cluster":"scheme-morphisms"},{"id":"stacks:05KX","tag":"05KX","title":"Flat finitely presented modules · Lemma 05KX","summary":"Let f : X → S be of finite presentation. Let s ∈ S. If X is flat over S at all points of X_s, then there exists an elementary étale neighbourhood (S', s') → (S, s) and a commutative diagram of schemes xymatrix X ar[d] & X' ar[l]^g ar[d] S & S' ar[l] with g étale, X_s ⊂ g(X'), such that X', S' are affine, and such that Γ(X', O_X') is a projective Γ(S', O_S')-module.","statement_latex":"Let $f : X \\to S$ be of finite presentation.\nLet $s \\in S$.\nIf $X$ is flat over $S$ at all points of $X_s$, then\nthere exists an elementary \\'etale neighbourhood\n$(S', s') \\to (S, s)$ and a commutative diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\\nS & S' \\ar[l]\n}\n$$\nwith $g$ \\'etale, $X_s \\subset g(X')$, such that $X'$, $S'$\nare affine, and such that\n$\\Gamma(X', \\mathcal{O}_{X'})$ is a projective\n$\\Gamma(S', \\mathcal{O}_{S'})$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KX","source_file":"flat.tex","source_line":3430,"source_end_line":3447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3430-L3447","statement_sha256":"bc661019346a5c30251496f8d706896af758697af616291bb6753398e3df94f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7697,"rank":7697,"depth":54,"x":2087.688,"y":807.101,"cluster":"scheme-morphisms"},{"id":"stacks:05KY","tag":"05KY","title":"Flat finitely presented modules · Lemma 05KY","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let x ∈ X with image s ∈ S. Assume that • f is locally of finite presentation, • F is of finite type, and • F is flat at x over S. Then there exists an elementary étale neighbourhood (S', s') → (S, s) and a commutative diagram of pointed schemes xymatrix (X, x) ar[d] & (X', x') ar[l]^g ar[d] (S, s) & (Spec(O_S', s'), s') ar[l] such that X' → X ×_S Spec(O_S', s') is étale, kappa(x) = kappa(x'),…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $x \\in X$ with image $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $\\mathcal{F}$ is of finite type, and\n\\item $\\mathcal{F}$ is flat at $x$ over $S$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$\nand a commutative diagram of pointed schemes\n$$\n\\xymatrix{\n(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\\n(S, s) & (\\Spec(\\mathcal{O}_{S', s'}), s') \\ar[l]\n}\n$$\nsuch that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$\nis \\'etale, $\\kappa(x) = \\kappa(x')$, the scheme $X'$ is\naffine of finite presentation over $\\mathcal{O}_{S', s'}$,\nthe sheaf $g^*\\mathcal{F}$ is of finite presentation over $\\mathcal{O}_{X'}$,\nand such that $\\Gamma(X', g^*\\mathcal{F})$ is a free\n$\\mathcal{O}_{S', s'}$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KY","source_file":"flat.tex","source_line":3460,"source_end_line":3485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3460-L3485","statement_sha256":"c7eba0c728832e11babe4994a658e78da5b2c794d11865fbb3d8520c99adce8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7698,"rank":7698,"depth":53,"x":1885.192,"y":618.986,"cluster":"scheme-morphisms"},{"id":"stacks:05KZ","tag":"05KZ","title":"Flat finitely presented modules · Lemma 05KZ","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let x ∈ X with image s ∈ S. Assume that • f is locally of finite type, • F is of finite type, and • F is flat at x over S. Then there exists an elementary étale neighbourhood (S', s') → (S, s) and a commutative diagram of pointed schemes xymatrix (X, x) ar[d] & (X', x') ar[l]^g ar[d] (S, s) & (Spec(O_S', s'), s') ar[l] such that X' → X ×_S Spec(O_S', s') is étale, kappa(x) = kappa(x'), the…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $x \\in X$ with image $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item $\\mathcal{F}$ is of finite type, and\n\\item $\\mathcal{F}$ is flat at $x$ over $S$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$\nand a commutative diagram of pointed schemes\n$$\n\\xymatrix{\n(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\\n(S, s) & (\\Spec(\\mathcal{O}_{S', s'}), s') \\ar[l]\n}\n$$\nsuch that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$\nis \\'etale, $\\kappa(x) = \\kappa(x')$, the scheme $X'$ is\naffine, and such that $\\Gamma(X', g^*\\mathcal{F})$ is a free\n$\\mathcal{O}_{S', s'}$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05KZ","source_file":"flat.tex","source_line":3500,"source_end_line":3523,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3500-L3523","statement_sha256":"1e6478881ecaf04954145f9bef44764cf7d4662efb25365cf98d69503fe7b4be","origin":"The Stacks Project","memory_eligible":false,"source_rank":7699,"rank":7699,"depth":54,"x":2185.908,"y":642.808,"cluster":"scheme-morphisms"},{"id":"stacks:05L0","tag":"05L0","title":"Flat finitely presented modules · Lemma 05L0","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let s ∈ S. Assume that • f is of finite presentation, • F is of finite type, and • F is flat over S at all points of X_s. Then there exists an elementary étale neighbourhood (S', s') → (S, s) and a commutative diagram of schemes xymatrix X ar[d] & X' ar[l]^g ar[d] S & Spec(O_S', s') ar[l] such that X' → X ×_S Spec(O_S', s') is étale, X_s = g((X')_s'), the scheme X' is affine of finite…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is of finite presentation,\n\\item $\\mathcal{F}$ is of finite type, and\n\\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$\nand a commutative diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\\nS & \\Spec(\\mathcal{O}_{S', s'}) \\ar[l]\n}\n$$\nsuch that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$\nis \\'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is\naffine of finite presentation over $\\mathcal{O}_{S', s'}$,\nthe sheaf $g^*\\mathcal{F}$ is of finite presentation over $\\mathcal{O}_{X'}$,\nand such that $\\Gamma(X', g^*\\mathcal{F})$ is a free\n$\\mathcal{O}_{S', s'}$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05L0","source_file":"flat.tex","source_line":3556,"source_end_line":3581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3556-L3581","statement_sha256":"0ab9c6675b4550553be6e3181452589131622cfc47d94db6f86b007632e24270","origin":"The Stacks Project","memory_eligible":false,"source_rank":7700,"rank":7700,"depth":54,"x":1944.911,"y":795.937,"cluster":"scheme-morphisms"},{"id":"stacks:05L1","tag":"05L1","title":"Flat finitely presented modules · Lemma 05L1","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Let s ∈ S. Assume that • f is of finite type, • F is of finite type, and • F is flat over S at all points of X_s. Then there exists an elementary étale neighbourhood (S', s') → (S, s) and a commutative diagram of schemes xymatrix X ar[d] & X' ar[l]^g ar[d] S & Spec(O_S', s') ar[l] such that X' → X ×_S Spec(O_S', s') is étale, X_s = g((X')_s'), the scheme X' is affine, and such that Γ(X', g^*F)…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is of finite type,\n\\item $\\mathcal{F}$ is of finite type, and\n\\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$.\n\\end{enumerate}\nThen there exists an elementary \\'etale neighbourhood $(S', s') \\to (S, s)$\nand a commutative diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\\nS & \\Spec(\\mathcal{O}_{S', s'}) \\ar[l]\n}\n$$\nsuch that $X' \\to X \\times_S \\Spec(\\mathcal{O}_{S', s'})$\nis \\'etale, $X_s = g((X')_{s'})$, the scheme $X'$ is affine,\nand such that $\\Gamma(X', g^*\\mathcal{F})$ is a free\n$\\mathcal{O}_{S', s'}$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05L1","source_file":"flat.tex","source_line":3626,"source_end_line":3649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3626-L3649","statement_sha256":"5a128eba9083d3a1a4632e5d79071d3e64551227475ec540e4a914ece1e01792","origin":"The Stacks Project","memory_eligible":false,"source_rank":7701,"rank":7701,"depth":55,"x":1999.498,"y":546.174,"cluster":"scheme-morphisms"},{"id":"stacks:0CU6","tag":"0CU6","title":"Flat finite type modules, Part II · Lemma 0CU6","summary":"Let R → S be a ring map of finite presentation. Let N be a finitely presented S-module. Let q ⊂ S be a prime ideal lying over p ⊂ R. Set overlineS = S ⊗_R kappa( p), overline q = q overlineS, and overlineN = N ⊗_R kappa( p). Then we can find a g ∈ S with g not ∈ q such that overlineg ∈ r for all r ∈ Ass_overlineS(overlineN) such that r not ⊂ overline q.","statement_latex":"Let $R \\to S$ be a ring map of finite presentation. Let $N$ be a\nfinitely presented $S$-module. Let $\\mathfrak q \\subset S$ be a prime ideal\nlying over $\\mathfrak p \\subset R$. Set\n$\\overline{S} = S \\otimes_R \\kappa(\\mathfrak p)$,\n$\\overline{\\mathfrak q} = \\mathfrak q \\overline{S}$, and\n$\\overline{N} = N \\otimes_R \\kappa(\\mathfrak p)$. Then\nwe can find a $g \\in S$ with\n$g \\not \\in \\mathfrak q$ such that\n$\\overline{g} \\in \\mathfrak r$ for all\n$\\mathfrak r \\in \\text{Ass}_{\\overline{S}}(\\overline{N})$\nsuch that $\\mathfrak r \\not \\subset \\overline{\\mathfrak q}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CU6","source_file":"flat.tex","source_line":3704,"source_end_line":3717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3704-L3717","statement_sha256":"aa1e88c203fcc93fc9a5239bf00b10b2583b86cacda33ea277b0c182da99a7fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7702,"rank":7702,"depth":3,"x":2160.163,"y":761.406,"cluster":"scheme-morphisms"},{"id":"stacks:05IG","tag":"05IG","title":"Flat finite type modules, Part II · Lemma 05IG","summary":"Let R → S be a ring map of finite presentation. Let N be a finitely presented S-module which is flat as an R-module. Let M be an R-module. Let q be a prime of S lying over p ⊂ R. Then q ∈ WeakAss_S(M ⊗_R N) ⇔ Big( p ∈ WeakAss_R(M) and overline q ∈ Ass_overlineS(overlineN) Big) Here overlineS = S ⊗_R kappa( p), overline q = q overlineS, and overlineN = N ⊗_R kappa( p).","statement_latex":"Let $R \\to S$ be a ring map of finite presentation.\nLet $N$ be a finitely presented $S$-module\nwhich is flat as an $R$-module. Let $M$ be an $R$-module.\nLet $\\mathfrak q$ be a prime of $S$ lying over $\\mathfrak p \\subset R$.\nThen\n$$\n\\mathfrak q \\in \\text{WeakAss}_S(M \\otimes_R N)\n\\Leftrightarrow\n\\Big(\n\\mathfrak p \\in \\text{WeakAss}_R(M)\n\\text{ and }\n\\overline{\\mathfrak q} \\in \\text{Ass}_{\\overline{S}}(\\overline{N})\n\\Big)\n$$\nHere $\\overline{S} = S \\otimes_R \\kappa(\\mathfrak p)$,\n$\\overline{\\mathfrak q} = \\mathfrak q \\overline{S}$, and\n$\\overline{N} = N \\otimes_R \\kappa(\\mathfrak p)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IG","source_file":"flat.tex","source_line":3749,"source_end_line":3768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3749-L3768","statement_sha256":"d9d5e2ffff8b0c7526c7cba865f520765bdb8dd20e909317027b056ddc8184bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7703,"rank":7703,"depth":53,"x":1868.49,"y":693.835,"cluster":"scheme-morphisms"},{"id":"stacks:05IH","tag":"05IH","title":"Flat finite type modules, Part II · Lemma 05IH","summary":"Let S be a scheme. Let f : X → S be locally of finite type. Let x ∈ X with image s ∈ S. Let F be a finite type quasi-coherent sheaf on X. Let G be a quasi-coherent sheaf on S. If F is flat at x over S, then x ∈ WeakAss_X(F ⊗_O_X f^*G) ⇔ s ∈ WeakAss_S(G) and x ∈ Ass_X_s(F_s).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to S$ be locally of finite type.\nLet $x \\in X$ with image $s \\in S$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$.\nLet $\\mathcal{G}$ be a quasi-coherent sheaf on $S$.\nIf $\\mathcal{F}$ is flat at $x$ over $S$, then\n$$\nx \\in \\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G})\n\\Leftrightarrow\ns \\in \\text{WeakAss}_S(\\mathcal{G})\n\\text{ and }\nx \\in \\text{Ass}_{X_s}(\\mathcal{F}_s).\n$$","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IH","source_file":"flat.tex","source_line":3983,"source_end_line":3998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L3983-L3998","statement_sha256":"e91d24853375d20cd1c5172af702bc3ebab55869d74731df0075040584fba9af","origin":"The Stacks Project","memory_eligible":false,"source_rank":7704,"rank":7704,"depth":54,"x":2138.012,"y":578.113,"cluster":"scheme-morphisms"},{"id":"stacks:05II","tag":"05II","title":"Flat finite type modules, Part II · Lemma 05II","summary":"Let R → S be a ring map which is essentially of finite type. Let N be a localization of a finite S-module flat over R. Let M be an R-module. Then WeakAss_S(M ⊗_R N) = ⋃_ p ∈ WeakAss_R(M) Ass_S ⊗_R kappa( p)(N ⊗_R kappa( p))","statement_latex":"Let $R \\to S$ be a ring map which is essentially of finite type.\nLet $N$ be a localization of a finite $S$-module flat over $R$.\nLet $M$ be an $R$-module. Then\n$$\n\\text{WeakAss}_S(M \\otimes_R N)\n=\n\\bigcup\\nolimits_{\\mathfrak p \\in \\text{WeakAss}_R(M)}\n\\text{Ass}_{S \\otimes_R \\kappa(\\mathfrak p)}(N \\otimes_R \\kappa(\\mathfrak p))\n$$","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05II","source_file":"flat.tex","source_line":4059,"source_end_line":4070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4059-L4070","statement_sha256":"22b6225ba84f72da0dc940f1a36b92192c16fcff62716d2af34ad5ef853a25fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7705,"rank":7705,"depth":55,"x":2032.29,"y":816.474,"cluster":"scheme-morphisms"},{"id":"stacks:05IJ","tag":"05IJ","title":"Flat finite type modules, Part II · Lemma 05IJ","summary":"Let f : X → S be a morphism which is locally of finite type. Let F be a finite type quasi-coherent sheaf on X which is flat over S. Let G be a quasi-coherent sheaf on S. Then we have WeakAss_X(F ⊗_O_X f^*G) = ⋃_s ∈ WeakAss_S(G) Ass_X_s(F_s)","statement_latex":"Let $f : X \\to S$ be a morphism which is locally of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$\nwhich is flat over $S$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $S$.\nThen we have\n$$\n\\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) =\n\\bigcup\\nolimits_{s \\in \\text{WeakAss}_S(\\mathcal{G})}\n\\text{Ass}_{X_s}(\\mathcal{F}_s)\n$$","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IJ","source_file":"flat.tex","source_line":4078,"source_end_line":4089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4078-L4089","statement_sha256":"4560a87ebf614b99a88e3b5b93af109d3b4621d4b7a26ee1a888eb112bf95c3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7706,"rank":7706,"depth":55,"x":1918.519,"y":580.626,"cluster":"scheme-morphisms"},{"id":"stacks:05IK","tag":"05IK","title":"Flat finite type modules, Part II · Theorem 05IK","summary":"The flat locus is open (non-Noetherian version). Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. Assume • X → S is locally of finite presentation, • F is an O_X-module of finite type, and • the set of weakly associated points of S is locally finite in S. Then U = (x ∈ X mid F flat at x over S) is open in X and F|_U is an O_U-module of finite presentation and flat over S.","statement_latex":"\\begin{slogan}\nThe flat locus is open (non-Noetherian version).\n\\end{slogan}\nLet $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $X \\to S$ is locally of finite presentation,\n\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, and\n\\item the set of weakly associated points of $S$ is locally finite in $S$.\n\\end{enumerate}\nThen $U = \\{x \\in X \\mid \\mathcal{F}\\text{ flat at }x\\text{ over }S\\}$\nis open in $X$ and $\\mathcal{F}|_U$ is an $\\mathcal{O}_U$-module\nof finite presentation and flat over $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IK","source_file":"flat.tex","source_line":4096,"source_end_line":4112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4096-L4112","statement_sha256":"b02cdc6efe54a4a86f0380d85f5151ee03f2ef0e689032d7e60d1f85f3703f29","origin":"The Stacks Project","memory_eligible":false,"source_rank":7707,"rank":7707,"depth":56,"x":2192.183,"y":690.024,"cluster":"scheme-morphisms"},{"id":"stacks:05IL","tag":"05IL","title":"Flat finite type modules, Part II · Lemma 05IL","summary":"Let R → S be a ring map of finite presentation. Let M be a finite S-module. Assume WeakAss_R(R) is finite. Then U = ( q ⊂ S mid M_ q flat over R) is open in Spec(S) and for every g ∈ S such that D(g) ⊂ U the localization M_g is a finitely presented S_g-module flat over R.","statement_latex":"Let $R \\to S$ be a ring map of finite presentation.\nLet $M$ be a finite $S$-module. Assume $\\text{WeakAss}_R(R)$ is finite.\nThen\n$$\nU = \\{\\mathfrak q \\subset S \\mid M_{\\mathfrak q}\\text{ flat over }R\\}\n$$\nis open in $\\Spec(S)$ and for every $g \\in S$ such that\n$D(g) \\subset U$ the localization $M_g$ is a finitely presented\n$S_g$-module flat over $R$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IL","source_file":"flat.tex","source_line":4161,"source_end_line":4172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4161-L4172","statement_sha256":"a1d05c6e1a35f8fdd56d4388db2ab84c12cb20e2e6916c332e06bdf675291d2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7708,"rank":7708,"depth":57,"x":1902.297,"y":764.669,"cluster":"scheme-morphisms"},{"id":"stacks:05IM","tag":"05IM","title":"Flat finite type modules, Part II · Lemma 05IM","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Assume the set of weakly associated points of S is locally finite in S. Then the set of points x ∈ X where f is flat is an open subscheme U ⊂ X and U → S is flat and locally of finite presentation.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite\ntype. Assume the set of weakly associated points of $S$ is locally finite\nin $S$. Then the set of points $x \\in X$ where $f$ is flat is an open\nsubscheme $U \\subset X$ and $U \\to S$ is flat and locally of finite\npresentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IM","source_file":"flat.tex","source_line":4179,"source_end_line":4186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4179-L4186","statement_sha256":"70a9cd653b074cb15b571e68f4552d137c8a08fc14075270f41eab2a8070e329","origin":"The Stacks Project","memory_eligible":false,"source_rank":7709,"rank":7709,"depth":58,"x":2056.088,"y":545.05,"cluster":"scheme-morphisms"},{"id":"stacks:05IN","tag":"05IN","title":"Flat finite type modules, Part II · Lemma 05IN","summary":"Let f : X → S be a morphism of schemes which is locally of finite type and flat. If S is integral, then f is locally of finite presentation.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is\nlocally of finite type and flat. If $S$ is integral, then $f$\nis locally of finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IN","source_file":"flat.tex","source_line":4197,"source_end_line":4202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4197-L4202","statement_sha256":"145d382ba38fcd3431b432a053f79b434e86f86026bf86aa58f4a8990bc7a658","origin":"The Stacks Project","memory_eligible":false,"source_rank":7710,"rank":7710,"depth":59,"x":2119.321,"y":794.359,"cluster":"scheme-morphisms"},{"id":"stacks:053G","tag":"053G","title":"Flat finite type modules, Part II · Proposition 053G","summary":"Let R be a domain. Let R → S be a ring map of finite type. Let M be a finite S-module. • If S is flat over R, then S is a finitely presented R-algebra. • If M is flat as an R-module, then M is finitely presented as an S-module.","statement_latex":"Let $R$ be a domain. Let $R \\to S$ be a ring map of finite type.\nLet $M$ be a finite $S$-module.\n\\begin{enumerate}\n\\item If $S$ is flat over $R$, then $S$ is a finitely presented $R$-algebra.\n\\item If $M$ is flat as an $R$-module, then $M$ is finitely presented\nas an $S$-module.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/053G","source_file":"flat.tex","source_line":4209,"source_end_line":4218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4209-L4218","statement_sha256":"b58fcbe5bb563bd585c59fac6a18b2e51ebb2e10cd6d4d0430058c4ea431b6d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7711,"rank":7711,"depth":60,"x":1872.109,"y":646.343,"cluster":"scheme-morphisms"},{"id":"stacks:05IQ","tag":"05IQ","title":"Finite type version of Theorem [Tag 05IK] · Lemma 05IQ","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. Assume • X → S is locally of finite type, • F is an O_X-module of finite type, and • the set of weakly associated points of S is locally finite in S. Then U = (x ∈ X mid F flat at x over S) is open in X and F|_U is flat over S and locally finitely presented relative to S (see More on Morphisms, Definition [Tag 05H1]).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $X \\to S$ is locally of finite type,\n\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, and\n\\item the set of weakly associated points of $S$ is locally finite in $S$.\n\\end{enumerate}\nThen $U = \\{x \\in X \\mid \\mathcal{F}\\text{ flat at }x\\text{ over }S\\}$\nis open in $X$ and $\\mathcal{F}|_U$ is flat over $S$ and locally\nfinitely presented relative to $S$ (see\nMore on Morphisms, Definition\n\\ref{more-morphisms-definition-relatively-finitely-presented-sheaf}).","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IQ","source_file":"flat.tex","source_line":4230,"source_end_line":4245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4230-L4245","statement_sha256":"b703418c70a599e5a1ea1f123913fd3e5301d2a6e1af971d3fe0f1e43538e380","origin":"The Stacks Project","memory_eligible":false,"source_rank":7712,"rank":7712,"depth":57,"x":2173.549,"y":615.201,"cluster":"scheme-morphisms"},{"id":"stacks:05IR","tag":"05IR","title":"Flat finite type modules, Part II · Lemma 05IR","summary":"Let R → S be a ring map of finite type. Let M be a finite S-module. Assume WeakAss_R(R) is finite. Then U = ( q ⊂ S mid M_ q flat over R) is open in Spec(S) and for every g ∈ S such that D(g) ⊂ U the localization M_g is flat over R and an S_g-module finitely presented relative to R (see More on Algebra, Definition [Tag 05GZ]).","statement_latex":"Let $R \\to S$ be a ring map of finite type.\nLet $M$ be a finite $S$-module.\nAssume $\\text{WeakAss}_R(R)$ is finite.\nThen\n$$\nU = \\{\\mathfrak q \\subset S \\mid M_{\\mathfrak q}\\text{ flat over }R\\}\n$$\nis open in $\\Spec(S)$ and for every $g \\in S$ such that\n$D(g) \\subset U$ the localization $M_g$ is flat over $R$ and\nan $S_g$-module finitely presented relative to $R$ (see\nMore on Algebra, Definition\n\\ref{more-algebra-definition-relatively-finitely-presented}).","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IR","source_file":"flat.tex","source_line":4270,"source_end_line":4284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4270-L4284","statement_sha256":"212fb19057bbd7dccb6eef351d099388da6218f811d54a5656d255dc76a4e9c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7713,"rank":7713,"depth":58,"x":1976.237,"y":809.287,"cluster":"scheme-morphisms"},{"id":"stacks:05FV","tag":"05FV","title":"Examples of relatively pure modules · Lemma 05FV","summary":"Let R be a local ring with maximal ideal m. Let R → S be a ring map. Let N be an S-module. Assume • N is projective as an R-module, and • S/ mS is Noetherian and N/ mN is a finite S/ mS-module. Then for any prime q ⊂ S which is an associated prime of N ⊗_R kappa( p) where p = R ∩ q we have q + m S not = S.","statement_latex":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$.\nLet $R \\to S$ be a ring map. Let $N$ be an $S$-module.\nAssume\n\\begin{enumerate}\n\\item $N$ is projective as an $R$-module, and\n\\item $S/\\mathfrak mS$ is Noetherian and $N/\\mathfrak mN$ is a finite\n$S/\\mathfrak mS$-module.\n\\end{enumerate}\nThen for any prime $\\mathfrak q \\subset S$ which is an associated prime of\n$N \\otimes_R \\kappa(\\mathfrak p)$ where $\\mathfrak p = R \\cap \\mathfrak q$\nwe have $\\mathfrak q + \\mathfrak m S \\not = S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Examples of relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FV","source_file":"flat.tex","source_line":4314,"source_end_line":4327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4314-L4327","statement_sha256":"463d06471fb6e56b9886bcc4fef8c559041e9e143d39db70a236af0144d503c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7714,"rank":7714,"depth":12,"x":1965.65,"y":554.11,"cluster":"scheme-morphisms"},{"id":"stacks:05IT","tag":"05IT","title":"Examples of relatively pure modules · Lemma 05IT","summary":"Let R be a ring. Let I ⊂ R be an ideal. Let R → S be a ring map. Let N be an S-module. If N is I-adically complete, then for any R-module M and for any prime q ⊂ S which is an associated prime of N ⊗_R M we have q + I S not = S.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal.\nLet $R \\to S$ be a ring map. Let $N$ be an $S$-module.\nIf $N$ is $I$-adically complete, then for any $R$-module $M$ and\nfor any prime $\\mathfrak q \\subset S$ which is an associated prime of\n$N \\otimes_R M$ we have $\\mathfrak q + I S \\not = S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Examples of relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IT","source_file":"flat.tex","source_line":4348,"source_end_line":4355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4348-L4355","statement_sha256":"6361837fab14f507bbfd7796bce791cc525e371ccb2a35a8afa730d62c52af42","origin":"The Stacks Project","memory_eligible":false,"source_rank":7715,"rank":7715,"depth":3,"x":2178.748,"y":736.338,"cluster":"scheme-morphisms"},{"id":"stacks:05IU","tag":"05IU","title":"Examples of relatively pure modules · Lemma 05IU","summary":"Let R be a local ring with maximal ideal m. Let R → S be a ring map. Let N be an S-module. Assume N is isomorphic as an R-module to a direct sum of finite R-modules. Then for any R-module M and for any prime q ⊂ S which is an associated prime of N ⊗_R M we have q + m S not = S.","statement_latex":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$.\nLet $R \\to S$ be a ring map. Let $N$ be an $S$-module.\nAssume $N$ is isomorphic as an $R$-module to a direct\nsum of finite $R$-modules. Then for any $R$-module $M$ and\nfor any prime $\\mathfrak q \\subset S$ which is an associated prime of\n$N \\otimes_R M$ we have $\\mathfrak q + \\mathfrak m S \\not = S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Examples of relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IU","source_file":"flat.tex","source_line":4378,"source_end_line":4386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4378-L4386","statement_sha256":"c464b6bd25c8062aec34a3d25783bfb384fa28b2fb286ce7137a41ae95d05c2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7716,"rank":7716,"depth":3,"x":1874.949,"y":722.877,"cluster":"scheme-morphisms"},{"id":"stacks:05IV","tag":"05IV","title":"Examples of relatively pure modules · Lemma 05IV","summary":"Let R be a henselian local ring with maximal ideal m. Let R → S be a ring map. Let N be an S-module. Assume N is countably generated and Mittag-Leffler as an R-module. Then for any R-module M and for any prime q ⊂ S which is an associated prime of N ⊗_R M we have q + m S not = S.","statement_latex":"Let $R$ be a henselian local ring with maximal ideal $\\mathfrak m$.\nLet $R \\to S$ be a ring map. Let $N$ be an $S$-module.\nAssume $N$ is countably generated and Mittag-Leffler as an $R$-module.\nThen for any $R$-module $M$ and for any prime $\\mathfrak q \\subset S$\nwhich is an associated prime of $N \\otimes_R M$ we have\n$\\mathfrak q + \\mathfrak m S \\not = S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Examples of relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IV","source_file":"flat.tex","source_line":4418,"source_end_line":4426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4418-L4426","statement_sha256":"943fb67ac6f7a6c81e8df748522df6c4d9d5bb15b4c2a56842c3882a4b5a19bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7717,"rank":7717,"depth":45,"x":2109.884,"y":560.356,"cluster":"scheme-morphisms"},{"id":"stacks:05IY","tag":"05IY","title":"Impurities · Definition 05IY","summary":"In Situation [Tag 05FW] we say a diagram ([Tag 05IX]) defines an impurity of F above s if xi ∈ Ass_X_T/T(F_T) and overline(xi) ∩ X_t = ∅. We will indicate this by saying \"let (g : T → S, t' leadsto t, xi) be an impurity of F above s\".","statement_latex":"In\nSituation \\ref{situation-pre-pure}\nwe say a diagram (\\ref{equation-impurity}) defines an\n{\\it impurity of $\\mathcal{F}$ above $s$}\nif $\\xi \\in \\text{Ass}_{X_T/T}(\\mathcal{F}_T)$ and\n$\\overline{\\{\\xi\\}} \\cap X_t = \\emptyset$. We will indicate\nthis by saying ``let $(g : T \\to S, t' \\leadsto t, \\xi)$ be\nan impurity of $\\mathcal{F}$ above $s$''.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Impurities","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IY","source_file":"flat.tex","source_line":4501,"source_end_line":4511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4501-L4511","statement_sha256":"b51a8f8c5a749c09e82dd038b18332e493b815b7e947284ea11a97cc222f41a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7718,"rank":7718,"depth":0,"x":2067.324,"y":813.605,"cluster":"scheme-morphisms"},{"id":"stacks:05FX","tag":"05FX","title":"Impurities · Lemma 05FX","summary":"In Situation [Tag 05FW]. If there exists an impurity of F above s, then there exists an impurity (g : T → S, t' leadsto t, xi) of F above s such that g is locally of finite presentation and t a closed point of the fibre of g above s.","statement_latex":"In Situation \\ref{situation-pre-pure}.\nIf there exists an impurity of $\\mathcal{F}$ above $s$, then\nthere exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$\nof $\\mathcal{F}$ above $s$ such that $g$ is locally of finite\npresentation and $t$ a closed point of the fibre of $g$ above $s$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Impurities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FX","source_file":"flat.tex","source_line":4513,"source_end_line":4520,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4513-L4520","statement_sha256":"b431ed51eb75f7410b09bd5fc8b72838e1c7c81dae55ccf022623fdfff04ebd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7719,"rank":7719,"depth":22,"x":1894.983,"y":602.63,"cluster":"scheme-morphisms"},{"id":"stacks:05IZ","tag":"05IZ","title":"Impurities · Lemma 05IZ","summary":"In Situation [Tag 05FW]. Let (g : T → S, t' leadsto t, xi) be an impurity of F above s. Assume T = lim_i ∈ I T_i is a directed limit of affine schemes over S. Then for some i the triple (T_i → S, t'_i leadsto t_i, xi_i) is an impurity of F above s.","statement_latex":"In Situation \\ref{situation-pre-pure}.\nLet $(g : T \\to S, t' \\leadsto t, \\xi)$ be an impurity of\n$\\mathcal{F}$ above $s$. Assume $T = \\lim_{i \\in I} T_i$\nis a directed limit of affine schemes over $S$. Then for\nsome $i$ the triple $(T_i \\to S, t'_i \\leadsto t_i, \\xi_i)$\nis an impurity of $\\mathcal{F}$ above $s$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Impurities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05IZ","source_file":"flat.tex","source_line":4572,"source_end_line":4580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4572-L4580","statement_sha256":"457e2b93dd79dd97201e42d0124e05900944263314325ba11c2475736d4e1cdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7720,"rank":7720,"depth":26,"x":2191.842,"y":660.432,"cluster":"scheme-morphisms"},{"id":"stacks:05J0","tag":"05J0","title":"Impurities · Lemma 05J0","summary":"In Situation [Tag 05FW]. If there exists an impurity (g : T → S, t' leadsto t, xi) of F above s with g quasi-finite at t, then there exists an impurity (g : T → S, t' leadsto t, xi) such that (T, t) → (S, s) is an elementary étale neighbourhood.","statement_latex":"In Situation \\ref{situation-pre-pure}.\nIf there exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$\nof $\\mathcal{F}$ above $s$ with $g$ quasi-finite at $t$, then there\nexists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$ such that\n$(T, t) \\to (S, s)$ is an elementary \\'etale neighbourhood.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Impurities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05J0","source_file":"flat.tex","source_line":4644,"source_end_line":4651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4644-L4651","statement_sha256":"223996e7c9fe98bfb5fd8852b9bad4cc9cc2a3c97bdcd2a34f63162d19ec9db6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7721,"rank":7721,"depth":30,"x":1926.355,"y":786.305,"cluster":"scheme-morphisms"},{"id":"stacks:05J1","tag":"05J1","title":"Impurities · Lemma 05J1","summary":"In Situation [Tag 05FW]. Assume that S is locally Noetherian. If there exists an impurity of F above s, then there exists an impurity (g : T → S, t' leadsto t, xi) of F above s such that g is quasi-finite at t.","statement_latex":"In Situation \\ref{situation-pre-pure}.\nAssume that $S$ is locally Noetherian.\nIf there exists an impurity of $\\mathcal{F}$ above $s$, then\nthere exists an impurity $(g : T \\to S, t' \\leadsto t, \\xi)$\nof $\\mathcal{F}$ above $s$ such that $g$ is quasi-finite at $t$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Impurities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05J1","source_file":"flat.tex","source_line":4698,"source_end_line":4705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4698-L4705","statement_sha256":"1700160941ec139743ad9634f6a251eb07348fcfc95f2e577bb7ccce1bb8d64a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7722,"rank":7722,"depth":31,"x":2020.935,"y":542.748,"cluster":"scheme-morphisms"},{"id":"stacks:05J2","tag":"05J2","title":"Impurities · Lemma 05J2","summary":"In Situation [Tag 05FW]. If there exists an impurity (S^h → S, s' leadsto s, xi) of F above s then there exists an impurity (T → S, t' leadsto t, xi) of F above s where (T, t) → (S, s) is an elementary étale neighbourhood.","statement_latex":"In Situation \\ref{situation-pre-pure}.\nIf there exists an impurity $(S^h \\to S, s' \\leadsto s, \\xi)$\nof $\\mathcal{F}$ above $s$ then there exists an impurity\n$(T \\to S, t' \\leadsto t, \\xi)$ of $\\mathcal{F}$ above $s$\nwhere $(T, t) \\to (S, s)$ is an elementary \\'etale neighbourhood.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Impurities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05J2","source_file":"flat.tex","source_line":4743,"source_end_line":4750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4743-L4750","statement_sha256":"397e01d181ba6f002f6c04f03fa5d9a729103af687b4d965ccaf19951e04779d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7723,"rank":7723,"depth":45,"x":2147.105,"y":776.102,"cluster":"scheme-morphisms"},{"id":"stacks:05J3","tag":"05J3","title":"Impurities · Lemma 05J3","summary":"In Situation [Tag 05FW] the following are equivalent • there exists an impurity (S^h → S, s' leadsto s, xi) of F above s where S^h is the henselization of S at s, • there exists an impurity (T → S, t' leadsto t, xi) of F above s such that (T, t) → (S, s) is an elementary étale neighbourhood, and • there exists an impurity (T → S, t' leadsto t, xi) of F above s such that T → S is quasi-finite at t.","statement_latex":"In Situation \\ref{situation-pre-pure} the following\nare equivalent\n\\begin{enumerate}\n\\item there exists an impurity $(S^h \\to S, s' \\leadsto s, \\xi)$\nof $\\mathcal{F}$ above $s$ where $S^h$ is the henselization of $S$ at $s$,\n\\item there exists an impurity $(T \\to S, t' \\leadsto t, \\xi)$\nof $\\mathcal{F}$ above $s$ such that $(T, t) \\to (S, s)$ is an\nelementary \\'etale neighbourhood, and\n\\item there exists an impurity $(T \\to S, t' \\leadsto t, \\xi)$\nof $\\mathcal{F}$ above $s$ such that $T \\to S$ is quasi-finite at $t$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Impurities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05J3","source_file":"flat.tex","source_line":4766,"source_end_line":4779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4766-L4779","statement_sha256":"fbbab2a07ec06ff6cdc40e72dcc4a0cb5acd8a7ad1514f5a127b11dd08bd132d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7724,"rank":7724,"depth":46,"x":1866.302,"y":675.582,"cluster":"scheme-morphisms"},{"id":"stacks:05J4","tag":"05J4","title":"Relatively pure modules · Definition 05J4","summary":"Let f : X → S be a morphism of schemes which is of finite type. Let F be a finite type quasi-coherent O_X-module. • Let s ∈ S. We say F is pure along X_s if there is no impurity (g : T → S, t' leadsto t, xi) of F above s with (T, t) → (S, s) an elementary étale neighbourhood. • We say F is universally pure along X_s if there does not exist any impurity of F above s. • We say that X is pure along X_s if O_X is pure along X_s. • We say F is universally S-pure, or…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item Let $s \\in S$. We say $\\mathcal{F}$ is {\\it pure along $X_s$}\nif there is no impurity $(g : T \\to S, t' \\leadsto t, \\xi)$\nof $\\mathcal{F}$ above $s$ with $(T, t) \\to (S, s)$ an\nelementary \\'etale neighbourhood.\n\\item We say $\\mathcal{F}$ is {\\it universally pure along $X_s$}\nif there does not exist any impurity of $\\mathcal{F}$ above $s$.\n\\item We say that $X$ is {\\it pure along $X_s$} if $\\mathcal{O}_X$\nis pure along $X_s$.\n\\item We say $\\mathcal{F}$ is {\\it universally $S$-pure}, or\n{\\it universally pure relative to $S$} if $\\mathcal{F}$ is universally\npure along $X_s$ for every $s \\in S$.\n\\item We say $\\mathcal{F}$ is {\\it $S$-pure}, or\n{\\it pure relative to $S$} if $\\mathcal{F}$ is pure along $X_s$\nfor every $s \\in S$.\n\\item We say that $X$ is {\\it $S$-pure} or {\\it pure relative to $S$}\nif $\\mathcal{O}_X$ is pure relative to $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Relatively pure modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05J4","source_file":"flat.tex","source_line":4807,"source_end_line":4829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4807-L4829","statement_sha256":"21e8584ec0cd870fbf45fdeb53982c0e8a66af6a36d8cb5357748abd269a52a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7725,"rank":7725,"depth":0,"x":2154.31,"y":590.336,"cluster":"scheme-morphisms"},{"id":"stacks:05J6","tag":"05J6","title":"Relatively pure modules · Lemma 05J6","summary":"Let f : X → S be a morphism of schemes which is of finite type. Let F be a finite type quasi-coherent O_X-module. Let s ∈ S. The following are equivalent • F is universally pure along X_s, and • for every morphism of pointed schemes (S', s') → (S, s) the pullback F_S' is pure along X_s'. In particular, F is universally pure relative to S if and only if every base change F_S' of F is pure relative to S'.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $s \\in S$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is universally pure along $X_s$, and\n\\item for every morphism of pointed schemes $(S', s') \\to (S, s)$\nthe pullback $\\mathcal{F}_{S'}$ is pure along $X_{s'}$.\n\\end{enumerate}\nIn particular, $\\mathcal{F}$ is universally pure relative to $S$ if and\nonly if every base change $\\mathcal{F}_{S'}$ of $\\mathcal{F}$ is\npure relative to $S'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05J6","source_file":"flat.tex","source_line":4888,"source_end_line":4901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4888-L4901","statement_sha256":"21a43edb19ec1a3cf4cff633766f6f953e2e54377a4faddd321ef46485f985b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7726,"rank":7726,"depth":0,"x":2010.433,"y":816.707,"cluster":"scheme-morphisms"},{"id":"stacks:05J7","tag":"05J7","title":"Relatively pure modules · Lemma 05J7","summary":"Let f : X → S be a morphism of schemes which is of finite type. Let F be a finite type quasi-coherent O_X-module. Let s ∈ S. Let (S', s') → (S, s) be a morphism of pointed schemes. If S' → S is quasi-finite at s' and F is pure along X_s, then F_S' is pure along X_s'.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $s \\in S$. Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes.\nIf $S' \\to S$ is quasi-finite at $s'$ and $\\mathcal{F}$ is pure along $X_s$,\nthen $\\mathcal{F}_{S'}$ is pure along $X_{s'}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05J7","source_file":"flat.tex","source_line":4907,"source_end_line":4914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4907-L4914","statement_sha256":"a27010a133b2af42f23f79160d89e58ea3a1b6e9416f60b2bc69b2cf12ab3291","origin":"The Stacks Project","memory_eligible":false,"source_rank":7727,"rank":7727,"depth":29,"x":1934.455,"y":568.048,"cluster":"scheme-morphisms"},{"id":"stacks:05J8","tag":"05J8","title":"Relatively pure modules · Lemma 05J8","summary":"Let f : X → S be a morphism of schemes which is of finite type. Let F be a finite type quasi-coherent O_X-module. Let s ∈ S. If O_S, s is Noetherian then F is pure along X_s if and only if F is universally pure along X_s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $s \\in S$. If $\\mathcal{O}_{S, s}$ is Noetherian then\n$\\mathcal{F}$ is pure along $X_s$ if and only if $\\mathcal{F}$\nis universally pure along $X_s$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05J8","source_file":"flat.tex","source_line":4927,"source_end_line":4934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4927-L4934","statement_sha256":"ec03a62d6784cbd5c8efa949f66c218864d54ce68ff2c04f7f36ad00e8e351ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":7728,"rank":7728,"depth":32,"x":2190.546,"y":708.348,"cluster":"scheme-morphisms"},{"id":"stacks:05J9","tag":"05J9","title":"Relatively pure modules · Lemma 05J9","summary":"Let f : X → S be a morphism of schemes which is of finite type. Let F be a finite type quasi-coherent O_X-module. Let s ∈ S. Let (S', s') → (S, s) be a morphism of pointed schemes. Assume S' → S is flat at s'. • If F_S' is pure along X_s', then F is pure along X_s. • If F_S' is universally pure along X_s', then F is universally pure along X_s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $s \\in S$. Let $(S', s') \\to (S, s)$ be a morphism of pointed schemes.\nAssume $S' \\to S$ is flat at $s'$.\n\\begin{enumerate}\n\\item If $\\mathcal{F}_{S'}$ is pure along $X_{s'}$,\nthen $\\mathcal{F}$ is pure along $X_s$.\n\\item If $\\mathcal{F}_{S'}$ is universally pure along $X_{s'}$,\nthen $\\mathcal{F}$ is universally pure along $X_s$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05J9","source_file":"flat.tex","source_line":4948,"source_end_line":4960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4948-L4960","statement_sha256":"aaf3222bada4a0fec4c04e6af6b5c88734e2b35199e9bb3e18a61cd0ea2c4950","origin":"The Stacks Project","memory_eligible":false,"source_rank":7729,"rank":7729,"depth":15,"x":1888.764,"y":750.221,"cluster":"scheme-morphisms"},{"id":"stacks:05K1","tag":"05K1","title":"Relatively pure modules · Lemma 05K1","summary":"Let i : Z → X be a closed immersion of schemes of finite type over a scheme S. Let s ∈ S. Let F be a finite type, quasi-coherent sheaf on Z. Then F is (universally) pure along Z_s if and only if i_*F is (universally) pure along X_s.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes of finite type over\na scheme $S$. Let $s \\in S$. Let $\\mathcal{F}$ be a\nfinite type, quasi-coherent sheaf on $Z$. Then $\\mathcal{F}$ is\n(universally) pure along $Z_s$ if and only if $i_*\\mathcal{F}$\nis (universally) pure along $X_s$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05K1","source_file":"flat.tex","source_line":4988,"source_end_line":4995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L4988-L4995","statement_sha256":"a0627ac4709186a0a0118c01fcb04dea77bf90667997b48150b606011c01e06a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7730,"rank":7730,"depth":27,"x":2077.694,"y":548.027,"cluster":"scheme-morphisms"},{"id":"stacks:05K3","tag":"05K3","title":"Examples of relatively pure sheaves · Lemma 05K3","summary":"Let f : X → S be a morphism of schemes which is of finite type. Let F be a finite type quasi-coherent O_X-module. • If the support of F is proper over S, then F is universally pure relative to S. • If f is proper, then F is universally pure relative to S. • If f is proper, then X is universally pure relative to S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If the support of $\\mathcal{F}$ is proper over $S$, then\n$\\mathcal{F}$ is universally pure relative to $S$.\n\\item If $f$ is proper, then\n$\\mathcal{F}$ is universally pure relative to $S$.\n\\item If $f$ is proper, then $X$ is universally pure relative to $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Examples of relatively pure sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05K3","source_file":"flat.tex","source_line":5015,"source_end_line":5026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5015-L5026","statement_sha256":"39dc681cee7a6e8dfdb1079a1f6740fa84e06282c84abc019feb6c828b402fb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7731,"rank":7731,"depth":28,"x":2100.988,"y":804.426,"cluster":"scheme-morphisms"},{"id":"stacks:05K4","tag":"05K4","title":"Examples of relatively pure sheaves · Lemma 05K4","summary":"Let f : X → S be a separated, finite type morphism of schemes. Let F be a finite type, quasi-coherent O_X-module. Assume that Supp(F_s) is finite for every s ∈ S. Then the following are equivalent • F is pure relative to S, • the scheme theoretic support of F is finite over S, and • F is universally pure relative to S. In particular, given a quasi-finite separated morphism X → S we see that X is pure relative to S if and only if X → S is finite.","statement_latex":"Let $f : X \\to S$ be a separated, finite type morphism of schemes.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module.\nAssume that $\\text{Supp}(\\mathcal{F}_s)$ is finite for every $s \\in S$.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is pure relative to $S$,\n\\item the scheme theoretic support of $\\mathcal{F}$ is finite over $S$, and\n\\item $\\mathcal{F}$ is universally pure relative to $S$.\n\\end{enumerate}\nIn particular, given a quasi-finite separated morphism $X \\to S$ we see\nthat $X$ is pure relative to $S$ if and only if $X \\to S$ is finite.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Examples of relatively pure sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05K4","source_file":"flat.tex","source_line":5046,"source_end_line":5059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5046-L5059","statement_sha256":"022c49ed71b936af67c63b247f767058f4f3c41d5e37b35bb41c1d227e213d4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7732,"rank":7732,"depth":50,"x":1877.535,"y":628.51,"cluster":"scheme-morphisms"},{"id":"stacks:05K5","tag":"05K5","title":"Examples of relatively pure sheaves · Lemma 05K5","summary":"Let f : X → S be a finite type, flat morphism of schemes with geometrically integral fibres. Then X is universally pure over S.","statement_latex":"Let $f : X \\to S$ be a finite type, flat morphism of schemes\nwith geometrically integral fibres. Then $X$ is universally pure\nover $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Examples of relatively pure sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05K5","source_file":"flat.tex","source_line":5123,"source_end_line":5128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5123-L5128","statement_sha256":"797e99b35984750544076731db763bcf8f7fef00ab207158fa98e61bc63c41b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7733,"rank":7733,"depth":2,"x":2183.893,"y":631.437,"cluster":"scheme-morphisms"},{"id":"stacks:05K6","tag":"05K6","title":"Examples of relatively pure sheaves · Lemma 05K6","summary":"Let f : X → S be a finite type, affine morphism of schemes. Let F be a finite type quasi-coherent O_X-module such that f_*F is locally projective on S, see Properties, Definition [Tag 05JP]. Then F is universally pure over S.","statement_latex":"Let $f : X \\to S$ be a finite type, affine morphism of schemes.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module\nsuch that $f_*\\mathcal{F}$ is locally projective on $S$, see\nProperties, Definition \\ref{properties-definition-locally-projective}.\nThen $\\mathcal{F}$ is universally pure over $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Examples of relatively pure sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05K6","source_file":"flat.tex","source_line":5147,"source_end_line":5154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5147-L5154","statement_sha256":"11cfb14089b16efcc418ab2468cb37bb8b813f063a82300ea8bdcbc63dc247c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7734,"rank":7734,"depth":13,"x":1955.546,"y":803.18,"cluster":"scheme-morphisms"},{"id":"stacks:05L3","tag":"05L3","title":"A criterion for purity · Lemma 05L3","summary":"Let f : X → S be a morphism of schemes of finite type. Let F be a quasi-coherent O_X-module of finite type. Let s ∈ S. Assume that F is flat over S at all points of X_s. Let x' ∈ Ass_X/S(F) with f(x') = s' such that s' leadsto s is a specialization in S. If x' specializes to a point of X_s, then x' leadsto x with x ∈ Ass_X_s(F_s).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $s \\in S$.\nAssume that $\\mathcal{F}$ is flat over $S$ at all points of $X_s$.\nLet $x' \\in \\text{Ass}_{X/S}(\\mathcal{F})$ with $f(x') = s'$\nsuch that $s' \\leadsto s$ is a specialization in $S$. If\n$x'$ specializes to a point of $X_s$, then $x' \\leadsto x$\nwith $x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05L3","source_file":"flat.tex","source_line":5174,"source_end_line":5184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5174-L5184","statement_sha256":"d6792097741389c962382b35e40ec22e38cf5bbfac851422ca0940fef5a34421","origin":"The Stacks Project","memory_eligible":false,"source_rank":7735,"rank":7735,"depth":40,"x":1985.825,"y":546.87,"cluster":"scheme-morphisms"},{"id":"stacks:05L4","tag":"05L4","title":"A criterion for purity · Lemma 05L4","summary":"Let f : X → S be a morphism of schemes of finite type. Let F be a quasi-coherent O_X-module of finite type. Let s ∈ S. Let (S', s') → (S, s) be an elementary étale neighbourhood and let xymatrix X ar[d] & X' ar[l]^g ar[d] S & S' ar[l] be a commutative diagram of morphisms of schemes. Assume • F is flat over S at all points of X_s, • X' → S' is of finite type, • g^*F is pure along X'_s', • g : X' → X is étale, and • g(X') contains Ass_X_s(F_s). In this situation F is pure…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nof finite type. Let $s \\in S$. Let $(S', s') \\to (S, s)$ be an\nelementary \\'etale neighbourhood and let\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\\nS & S' \\ar[l]\n}\n$$\nbe a commutative diagram of morphisms of schemes. Assume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$,\n\\item $X' \\to S'$ is of finite type,\n\\item $g^*\\mathcal{F}$ is pure along $X'_{s'}$,\n\\item $g : X' \\to X$ is \\'etale, and\n\\item $g(X')$ contains $\\text{Ass}_{X_s}(\\mathcal{F}_s)$.\n\\end{enumerate}\nIn this situation $\\mathcal{F}$ is pure along $X_s$ if and only\nif the image of $X' \\to X \\times_S S'$ contains the points of\n$\\text{Ass}_{X \\times_S S'/S'}(\\mathcal{F} \\times_S S')$\nlying over points in $S'$ which specialize to $s'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05L4","source_file":"flat.tex","source_line":5196,"source_end_line":5220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5196-L5220","statement_sha256":"a8179861d76f76cbe9f8a1b001f2750d6ce8c857878e33d4b6072b06a2db8093","origin":"The Stacks Project","memory_eligible":false,"source_rank":7736,"rank":7736,"depth":45,"x":2169.688,"y":753.131,"cluster":"scheme-morphisms"},{"id":"stacks:05L5","tag":"05L5","title":"A criterion for purity · Lemma 05L5","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. Let s ∈ S. Assume • f is of finite type, • F is of finite type, • F is flat over S at all points of X_s, and • F is pure along X_s. Then F is universally pure along X_s.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $s \\in S$.\nAssume\n\\begin{enumerate}\n\\item $f$ is of finite type,\n\\item $\\mathcal{F}$ is of finite type,\n\\item $\\mathcal{F}$ is flat over $S$ at all points of $X_s$, and\n\\item $\\mathcal{F}$ is pure along $X_s$.\n\\end{enumerate}\nThen $\\mathcal{F}$ is universally pure along $X_s$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05L5","source_file":"flat.tex","source_line":5275,"source_end_line":5288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5275-L5288","statement_sha256":"ceedabc2c4b06f5ed391bb48f61bb185b838d2e7009bcccf5f1b0d7ea33a4502","origin":"The Stacks Project","memory_eligible":false,"source_rank":7737,"rank":7737,"depth":56,"x":1868.123,"y":705.346,"cluster":"scheme-morphisms"},{"id":"stacks:05L6","tag":"05L6","title":"A criterion for purity · Lemma 05L6","summary":"Let f : X → S be a finite type morphism of schemes. Let F be a finite type quasi-coherent O_X-module. Assume F is flat over S. In this case F is pure relative to S if and only if F is universally pure relative to S.","statement_latex":"Let $f : X \\to S$ be a finite type morphism of schemes.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nAssume $\\mathcal{F}$ is flat over $S$. In this case\n$\\mathcal{F}$ is pure relative to $S$ if and only if $\\mathcal{F}$\nis universally pure relative to $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05L6","source_file":"flat.tex","source_line":5362,"source_end_line":5369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5362-L5369","statement_sha256":"08e863de4a067d984733106168390754b19877fac2621ff1eb5880f2426e57d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7738,"rank":7738,"depth":57,"x":2129.021,"y":569.414,"cluster":"scheme-morphisms"},{"id":"stacks:05MA","tag":"05MA","title":"A criterion for purity · Lemma 05MA","summary":"Let I be a directed set. Let (S_i, g_ii') be an inverse system of affine schemes over I. Set S = lim_i S_i and s ∈ S. Denote g_i : S → S_i the projections and set s_i = g_i(s). Suppose that f : X → S is a morphism of finite presentation, F a quasi-coherent O_X-module of finite presentation which is pure along X_s and flat over S at all points of X_s. Then there exists an i ∈ I, a morphism of finite presentation X_i → S_i, a quasi-coherent O_X_i-module F_i of finite…","statement_latex":"Let $I$ be a directed set.\nLet $(S_i, g_{ii'})$ be an inverse system of affine schemes over $I$.\nSet $S = \\lim_i S_i$ and $s \\in S$.\nDenote $g_i : S \\to S_i$ the projections and set $s_i = g_i(s)$.\nSuppose that $f : X \\to S$ is a morphism of finite presentation,\n$\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module of finite presentation\nwhich is pure along $X_s$ and flat over $S$ at all points of $X_s$.\nThen there exists an $i \\in I$, a morphism of finite presentation\n$X_i \\to S_i$, a quasi-coherent $\\mathcal{O}_{X_i}$-module $\\mathcal{F}_i$\nof finite presentation which is pure along $(X_i)_{s_i}$ and flat over $S_i$\nat all points of $(X_i)_{s_i}$ such that $X \\cong X_i \\times_{S_i} S$\nand such that the pullback of $\\mathcal{F}_i$ to $X$ is isomorphic\nto $\\mathcal{F}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05MA","source_file":"flat.tex","source_line":5377,"source_end_line":5392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5377-L5392","statement_sha256":"3d70c15dcad9d5967bbe21f74d921de3ed6cbbcb745c2608ffaacfb1a2e50a9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7739,"rank":7739,"depth":54,"x":2045.92,"y":817.785,"cluster":"scheme-morphisms"},{"id":"stacks:05MC","tag":"05MC","title":"A criterion for purity · Lemma 05MC","summary":"Let f : X → S be a morphism of finite presentation. Let F be a quasi-coherent O_X-module of finite presentation flat over S. Then the set U = (s ∈ S mid F is pure along X_s) is open in S.","statement_latex":"Let $f : X \\to S$ be a morphism of finite presentation.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nof finite presentation flat over $S$. Then the set\n$$\nU = \\{s \\in S \\mid \\mathcal{F}\\text{ is pure along }X_s\\}\n$$\nis open in $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05MC","source_file":"flat.tex","source_line":5489,"source_end_line":5498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5489-L5498","statement_sha256":"06963cf5cb7d79f6b81cebbc4905370015da692b0bd37699fa0b69095807fd71","origin":"The Stacks Project","memory_eligible":false,"source_rank":7740,"rank":7740,"depth":54,"x":1907.411,"y":587.397,"cluster":"scheme-morphisms"},{"id":"stacks:05L8","tag":"05L8","title":"How purity is used · Lemma 05L8","summary":"Let f : X → S be a morphism of finite type. Let F be a quasi-coherent sheaf of finite type on X. Assume S is local with closed point s. Assume F is pure along X_s and that F is flat over S. Let φ : F → G of quasi-coherent O_X-modules. Then the following are equivalent • the map on stalks φ_x is injective for all x ∈ Ass_X_s(F_s), and • φ is injective.","statement_latex":"Let $f : X \\to S$ be a morphism of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf of finite type on $X$.\nAssume $S$ is local with closed point $s$.\nAssume $\\mathcal{F}$ is pure along $X_s$ and\nthat $\\mathcal{F}$ is flat over $S$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ of quasi-coherent\n$\\mathcal{O}_X$-modules. Then the following are equivalent\n\\begin{enumerate}\n\\item the map on stalks $\\varphi_x$ is injective for all\n$x \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$, and\n\\item $\\varphi$ is injective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"How purity is used","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05L8","source_file":"flat.tex","source_line":5546,"source_end_line":5560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5546-L5560","statement_sha256":"27ed4710190690dbf056c377ee865b0598be99f03b5d4d85b896a637c34107ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":7741,"rank":7741,"depth":56,"x":2194.928,"y":678.724,"cluster":"scheme-morphisms"},{"id":"stacks:05MD","tag":"05MD","title":"How purity is used · Proposition 05MD","summary":"Let f : X → S be an affine, finitely presented morphism of schemes. Let F be a quasi-coherent O_X-module of finite presentation, flat over S. Then the following are equivalent • f_*F is locally projective on S, and • F is pure relative to S. In particular, given a ring map A → B of finite presentation and a finitely presented B-module N flat over A we have: N is projective as an A-module if and only if widetildeN on Spec(B) is pure relative to Spec(A).","statement_latex":"Let $f : X \\to S$ be an affine, finitely presented morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of\nfinite presentation, flat over $S$. Then the following\nare equivalent\n\\begin{enumerate}\n\\item $f_*\\mathcal{F}$ is locally projective on $S$, and\n\\item $\\mathcal{F}$ is pure relative to $S$.\n\\end{enumerate}\nIn particular, given a ring map $A \\to B$ of finite presentation and\na finitely presented $B$-module $N$ flat over $A$ we have:\n$N$ is projective as an $A$-module if and only if $\\widetilde{N}$\non $\\Spec(B)$ is pure relative to $\\Spec(A)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"How purity is used","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05MD","source_file":"flat.tex","source_line":5581,"source_end_line":5595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5581-L5595","statement_sha256":"2cb9afe59cca0f0964279e7d8aa2ee0a9c7be542211e36952616a3b0d256c740","origin":"The Stacks Project","memory_eligible":false,"source_rank":7742,"rank":7742,"depth":57,"x":1909.364,"y":774.562,"cluster":"scheme-morphisms"},{"id":"stacks:05ME","tag":"05ME","title":"How purity is used · Lemma 05ME","summary":"Let f : X → S be a morphism which is locally of finite presentation. Let F be a quasi-coherent O_X-module which is of finite presentation. Let x ∈ X with s = f(x) ∈ S. If F is flat at x over S there exists an affine elementary étale neighbourhood (S', s') → (S, s) and an affine open U' ⊂ X ×_S S' which contains x' = (x, s') such that Γ(U', F|_U') is a projective Γ(S', O_S')-module.","statement_latex":"Let $f : X \\to S$ be a morphism which is locally of finite presentation.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module which is\nof finite presentation. Let $x \\in X$ with $s = f(x) \\in S$.\nIf $\\mathcal{F}$ is flat at $x$ over $S$ there exists an affine\nelementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and\nan affine open $U' \\subset X \\times_S S'$ which contains $x' = (x, s')$\nsuch that $\\Gamma(U', \\mathcal{F}|_{U'})$ is a projective\n$\\Gamma(S', \\mathcal{O}_{S'})$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"How purity is used","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ME","source_file":"flat.tex","source_line":5661,"source_end_line":5671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5661-L5671","statement_sha256":"35b739c90ebffdc643c3e0ab9c6daff55dabe5c7848f056d0a565d5ea09995c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7743,"rank":7743,"depth":58,"x":2042.916,"y":541.766,"cluster":"scheme-morphisms"},{"id":"stacks:05MF","tag":"05MF","title":"How purity is used · Lemma 05MF","summary":"Let f : X → S be a morphism which is locally of finite type. Let F be a quasi-coherent O_X-module which is of finite type. Let x ∈ X with s = f(x) ∈ S. If F is flat at x over S there exists an affine elementary étale neighbourhood (S', s') → (S, s) and an affine open U' ⊂ X ×_S Spec(O_S', s') which contains x' = (x, s') such that Γ(U', F|_U') is a free O_S', s'-module.","statement_latex":"Let $f : X \\to S$ be a morphism which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module which is\nof finite type. Let $x \\in X$ with $s = f(x) \\in S$.\nIf $\\mathcal{F}$ is flat at $x$ over $S$ there exists an affine\nelementary \\'etale neighbourhood $(S', s') \\to (S, s)$ and\nan affine open $U' \\subset X \\times_S \\Spec(\\mathcal{O}_{S', s'})$\nwhich contains $x' = (x, s')$ such that\n$\\Gamma(U', \\mathcal{F}|_{U'})$ is a free\n$\\mathcal{O}_{S', s'}$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"How purity is used","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05MF","source_file":"flat.tex","source_line":5727,"source_end_line":5738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5727-L5738","statement_sha256":"c13c47440c7b1ebc98d7e36aee3303c8c9b66392297c85d13a4f01723e0c6bd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7744,"rank":7744,"depth":59,"x":2131.679,"y":789.303,"cluster":"scheme-morphisms"},{"id":"stacks:05U7","tag":"05U7","title":"How purity is used · Lemma 05U7","summary":"Let A → B be a local ring map of local rings which is essentially of finite type. Let N be a finite B-module which is flat as an A-module. If A is henselian, then N is a filtered colimit N = colim_i F_i of free A-modules F_i such that all transition maps u_i : F_i → F_i' of the system induce injective maps overlineu_i : F_i/ m_AF_i → F_i'/ m_AF_i'. Also, N is a Mittag-Leffler A-module.","statement_latex":"Let $A \\to B$ be a local ring map of local rings which is essentially of\nfinite type. Let $N$ be a finite $B$-module which is flat as an $A$-module.\nIf $A$ is henselian, then $N$ is a filtered colimit\n$$\nN = \\colim_i F_i\n$$\nof free $A$-modules $F_i$ such that all transition maps\n$u_i : F_i \\to F_{i'}$ of the system induce injective maps\n$\\overline{u}_i : F_i/\\mathfrak m_AF_i \\to F_{i'}/\\mathfrak m_AF_{i'}$.\nAlso, $N$ is a Mittag-Leffler $A$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"How purity is used","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05U7","source_file":"flat.tex","source_line":5758,"source_end_line":5770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5758-L5770","statement_sha256":"13181fbc383c3073b9f863bc2b328e8e4ee362f31faeea6d07e20cd5244879a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7745,"rank":7745,"depth":60,"x":1867.062,"y":657.088,"cluster":"scheme-morphisms"},{"id":"stacks:0ASX","tag":"0ASX","title":"How purity is used · Lemma 0ASX","summary":"Let A → B be a local ring map of local rings which is essentially of finite type. Let N be a finite B-module which is flat as an A-module. If A is a valuation ring, then any element of N has a content ideal I ⊂ A (More on Algebra, Definition [Tag 0ASA]). Also, I is a principal ideal.","statement_latex":"Let $A \\to B$ be a local ring map of local rings which is essentially of\nfinite type. Let $N$ be a finite $B$-module which is flat as an $A$-module.\nIf $A$ is a valuation ring, then any element of $N$ has a content ideal\n$I \\subset A$ (More on Algebra, Definition\n\\ref{more-algebra-definition-content-ideal}). Also, $I$ is a\nprincipal ideal.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"How purity is used","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ASX","source_file":"flat.tex","source_line":5800,"source_end_line":5808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5800-L5808","statement_sha256":"1e0294975434b4d320d4ea68498d1e9d5557bc3e61794d2a04e6942140b7c398","origin":"The Stacks Project","memory_eligible":false,"source_rank":7746,"rank":7746,"depth":61,"x":2168.627,"y":604.411,"cluster":"scheme-morphisms"},{"id":"stacks:0H2T","tag":"0H2T","title":"How purity is used · Lemma 0H2T","summary":"Let X → Spec(R) be a proper flat morphism where R is a valuation ring. If the special fibre is reduced, then X and every fibre of X → Spec(R) is reduced.","statement_latex":"Let $X \\to \\Spec(R)$ be a proper flat morphism where $R$ is a valuation ring.\nIf the special fibre is reduced, then $X$ and every fibre of $X \\to \\Spec(R)$\nis reduced.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"How purity is used","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2T","source_file":"flat.tex","source_line":5842,"source_end_line":5847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5842-L5847","statement_sha256":"6000cccbf0ef84f4a4634466e61d4733e236334ebaf0f84e7b762f8263da1358","origin":"The Stacks Project","memory_eligible":false,"source_rank":7747,"rank":7747,"depth":62,"x":1988.55,"y":814.451,"cluster":"scheme-morphisms"},{"id":"stacks:05MJ","tag":"05MJ","title":"Flattening functors · Lemma 05MJ","summary":"In Situation [Tag 05MH]. • Each of the functors F_iso, F_inj, F_surj, F_zero satisfies the sheaf property for the fpqc topology. • If f is quasi-compact and G is of finite type, then F_surj is limit preserving. • If f is quasi-compact and F of finite type, then F_zero is limit preserving. • If f is quasi-compact, F is of finite type, and G is of finite presentation, then F_iso is limit preserving.","statement_latex":"In Situation \\ref{situation-iso}.\n\\begin{enumerate}\n\\item Each of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$\nsatisfies the sheaf property for the fpqc topology.\n\\item If $f$ is quasi-compact and $\\mathcal{G}$ is of finite type,\nthen $F_{surj}$ is limit preserving.\n\\item If $f$ is quasi-compact and $\\mathcal{F}$ of finite type, then\n$F_{zero}$ is limit preserving.\n\\item If $f$ is quasi-compact, $\\mathcal{F}$ is of finite type, and\n$\\mathcal{G}$ is of finite presentation, then $F_{iso}$ is limit preserving.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05MJ","source_file":"flat.tex","source_line":5918,"source_end_line":5931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L5918-L5931","statement_sha256":"8b3f7383efeddc76f37fb08671f2c8d088ea7b88ac7817eb6a546f3d0e31ce4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7748,"rank":7748,"depth":20,"x":1952.413,"y":557.291,"cluster":"scheme-morphisms"},{"id":"stacks:05MN","tag":"05MN","title":"Flattening functors · Lemma 05MN","summary":"In Situation [Tag 05MK]. • If A' → A\" is a flat morphism in C then F_lf(A') = F_lf(A\"). • If A → B is essentially of finite presentation and M is a B-module of finite presentation, then F_lf is limit preserving: If (A_i)_i ∈ I is a directed system of objects of C, then F_lf(colim_i A_i) = colim_i F_lf(A_i).","statement_latex":"In Situation \\ref{situation-flat-at-point}.\n\\begin{enumerate}\n\\item If $A' \\to A''$ is a flat morphism in $\\mathcal{C}$\nthen $F_{lf}(A') = F_{lf}(A'')$.\n\\item If $A \\to B$ is essentially of finite presentation and\n$M$ is a $B$-module of finite presentation, then $F_{lf}$ is limit\npreserving: If $\\{A_i\\}_{i \\in I}$ is a\ndirected system of objects of $\\mathcal{C}$, then\n$F_{lf}(\\colim_i A_i) = \\colim_i F_{lf}(A_i)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05MN","source_file":"flat.tex","source_line":6044,"source_end_line":6056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6044-L6056","statement_sha256":"da3bde8f93fba5458da41024998e39fb19ef7fdfa7f49ed9558ed0e65cbef7e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7749,"rank":7749,"depth":36,"x":2185.951,"y":726.477,"cluster":"scheme-morphisms"},{"id":"stacks:05P4","tag":"05P4","title":"Flattening functors · Lemma 05P4","summary":"In Situation [Tag 05MK]. Let B → C is a local map of local A-algebras and N a C-module. Denote F'_lf : C → Sets the functor associated to the pair (C, N). If M ≅ N as B-modules and B → C is finite, then F_lf = F'_lf.","statement_latex":"In Situation \\ref{situation-flat-at-point}. Let $B \\to C$ is a local map of\nlocal $A$-algebras and $N$ a $C$-module. Denote\n$F'_{lf} : \\mathcal{C} \\to \\textit{Sets}$ the functor associated to the pair\n$(C, N)$. If $M \\cong N$ as $B$-modules and $B \\to C$ is finite, then\n$F_{lf} = F'_{lf}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05P4","source_file":"flat.tex","source_line":6067,"source_end_line":6074,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6067-L6074","statement_sha256":"eb17b372117086a4e3d4c1672c461a31a65a51abec3ea9a148fa7429621839f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7750,"rank":7750,"depth":7,"x":1877.57,"y":734.239,"cluster":"scheme-morphisms"},{"id":"stacks:05P5","tag":"05P5","title":"Flattening functors · Lemma 05P5","summary":"In Situation [Tag 05MK] suppose that B → C is a flat local homomorphism of local rings. Set N = M ⊗_B C. Denote F'_lf : C → Sets the functor associated to the pair (C, N). Then F_lf = F'_lf.","statement_latex":"In\nSituation \\ref{situation-flat-at-point}\nsuppose that $B \\to C$ is a flat local homomorphism of local\nrings. Set $N = M \\otimes_B C$. Denote\n$F'_{lf} : \\mathcal{C} \\to \\textit{Sets}$ the functor associated\nto the pair $(C, N)$. Then $F_{lf} = F'_{lf}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05P5","source_file":"flat.tex","source_line":6107,"source_end_line":6115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6107-L6115","statement_sha256":"c586cb6d554ba22415c6d5d4794bd50982e347193feaf1a2b6392cf411ff7179","origin":"The Stacks Project","memory_eligible":false,"source_rank":7751,"rank":7751,"depth":4,"x":2098.808,"y":553.464,"cluster":"scheme-morphisms"},{"id":"stacks:05MS","tag":"05MS","title":"Flattening functors · Lemma 05MS","summary":"In Situation [Tag 05MP]. • The functor H_p satisfies the sheaf property for the fpqc topology. • If F is of finite presentation, then functor H_p is limit preserving.","statement_latex":"In Situation \\ref{situation-free-at-generic-points}.\n\\begin{enumerate}\n\\item The functor $H_p$ satisfies the sheaf property for the fpqc topology.\n\\item If $\\mathcal{F}$ is of finite presentation, then functor $H_p$ is\nlimit preserving.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05MS","source_file":"flat.tex","source_line":6178,"source_end_line":6186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6178-L6186","statement_sha256":"23cb4c7165139c1eb3432d2681d77bd1e0ca84090128de63fec8d690929ab71d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7752,"rank":7752,"depth":50,"x":2081.039,"y":812.4,"cluster":"scheme-morphisms"},{"id":"stacks:0CWF","tag":"0CWF","title":"Flattening functors · Lemma 0CWF","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let n ≥ 0. The following are equivalent • for s ∈ S the closed subset Z ⊂ X_s of points where F is not flat over S (see Lemma [Tag 05M9]) satisfies dim(Z) < n, and • for x ∈ X such that F is not flat at x over S we have trdeg_kappa(f(x))(kappa(x)) < n. If this is true, then it remains true after any base change.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite\ntype. Let $n \\geq 0$. The following are equivalent\n\\begin{enumerate}\n\\item for $s \\in S$ the closed subset $Z \\subset X_s$ of points\nwhere $\\mathcal{F}$ is not flat over $S$ (see\nLemma \\ref{lemma-open-in-fibre-where-flat})\nsatisfies $\\dim(Z) < n$, and\n\\item for $x \\in X$ such that $\\mathcal{F}$ is not flat at $x$\nover $S$ we have $\\text{trdeg}_{\\kappa(f(x))}(\\kappa(x)) < n$.\n\\end{enumerate}\nIf this is true, then it remains true after any base change.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWF","source_file":"flat.tex","source_line":6270,"source_end_line":6284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6270-L6284","statement_sha256":"1c4a7b7b50429b3ffd5506ce0988d47846f23e406f76a52ccda0e155c7937ec0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7753,"rank":7753,"depth":53,"x":1885.837,"y":611.304,"cluster":"scheme-morphisms"},{"id":"stacks:0CWG","tag":"0CWG","title":"Flattening functors · Definition 0CWG","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let n ≥ 0. We say F is flat over S in dimensions ≥ n if the equivalent conditions of Lemma [Tag 0CWF] are satisfied.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $n \\geq 0$.\nWe say {\\it $\\mathcal{F}$ is flat over $S$ in dimensions $\\geq n$}\nif the equivalent conditions of Lemma \\ref{lemma-pre-flat-dimension-n}\nare satisfied.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWG","source_file":"flat.tex","source_line":6300,"source_end_line":6308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6300-L6308","statement_sha256":"0250645bfee447d32ee222bc4e0c3e4738700e0ff190cb9eccacdb690308b2ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":7754,"rank":7754,"depth":54,"x":2191.609,"y":648.842,"cluster":"scheme-morphisms"},{"id":"stacks:05MV","tag":"05MV","title":"Flattening functors · Lemma 05MV","summary":"In Situation [Tag 05MT]. • The functor F_n satisfies the sheaf property for the fpqc topology. • If f is quasi-compact and locally of finite presentation and F is of finite presentation, then the functor F_n is limit preserving.","statement_latex":"In Situation \\ref{situation-flat-dimension-n}.\n\\begin{enumerate}\n\\item The functor $F_n$ satisfies the sheaf property for the fpqc topology.\n\\item If $f$ is quasi-compact and locally of finite presentation\nand $\\mathcal{F}$ is of finite presentation, then the functor $F_n$ is\nlimit preserving.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05MV","source_file":"flat.tex","source_line":6336,"source_end_line":6345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6336-L6345","statement_sha256":"44c38a946f8ed466dbf217a90bd372241667166476f8d925fbc8dd10de2120bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7755,"rank":7755,"depth":37,"x":1935.853,"y":794.721,"cluster":"scheme-morphisms"},{"id":"stacks:05MY","tag":"05MY","title":"Flattening functors · Lemma 05MY","summary":"In Situation [Tag 05MW]. • The functor F_flat satisfies the sheaf property for the fpqc topology. • If f is quasi-compact and locally of finite presentation and F is of finite presentation, then the functor F_flat is limit preserving.","statement_latex":"In Situation \\ref{situation-flat}.\n\\begin{enumerate}\n\\item The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology.\n\\item If $f$ is quasi-compact and locally of finite presentation\nand $\\mathcal{F}$ is of finite presentation, then the functor\n$F_{flat}$ is limit preserving.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05MY","source_file":"flat.tex","source_line":6441,"source_end_line":6450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6441-L6450","statement_sha256":"c1c71af5251b3e2db81917859dc1458adfbcc3ead92fdec6832d60afee613358","origin":"The Stacks Project","memory_eligible":false,"source_rank":7756,"rank":7756,"depth":38,"x":2007.159,"y":541.932,"cluster":"scheme-morphisms"},{"id":"stacks:05P6","tag":"05P6","title":"Flattening stratifications · Definition 05P6","summary":"Let X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. We say that the universal flattening of F exists if the functor F_flat defined in Situation [Tag 05MW] is representable by a scheme S' over S. We say that the universal flattening of X exists if the universal flattening of O_X exists.","statement_latex":"Let $X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nWe say that the {\\it universal flattening of $\\mathcal{F}$ exists}\nif the functor $F_{flat}$ defined in Situation \\ref{situation-flat}\nis representable by a scheme $S'$ over $S$.\nWe say that the {\\it universal flattening of $X$ exists}\nif the universal flattening of $\\mathcal{O}_X$ exists.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening stratifications","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05P6","source_file":"flat.tex","source_line":6476,"source_end_line":6485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6476-L6485","statement_sha256":"eb6339ba3660691ec98205a69e5a1b16406efa404a3e61f059a0ab8ab3830d7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7757,"rank":7757,"depth":0,"x":2157.922,"y":768.882,"cluster":"scheme-morphisms"},{"id":"stacks:05P7","tag":"05P7","title":"Flattening stratifications · Definition 05P7","summary":"Let X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. We say that F has a flattening stratification if the functor F_flat defined in Situation [Tag 05MW] is representable by a monomorphism S' → S associated to a stratification of S by locally closed subschemes. We say that X has a flattening stratification if O_X has a flattening stratification.","statement_latex":"Let $X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nWe say that $\\mathcal{F}$ has a {\\it flattening stratification}\nif the functor $F_{flat}$ defined in Situation \\ref{situation-flat}\nis representable by a monomorphism $S' \\to S$ associated\nto a stratification of $S$ by locally closed subschemes.\nWe say that $X$ has a {\\it flattening stratification}\nif $\\mathcal{O}_X$ has a flattening stratification.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening stratifications","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05P7","source_file":"flat.tex","source_line":6534,"source_end_line":6544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6534-L6544","statement_sha256":"2104561df9ab009740ed2ad4758f6de00be85dc3c6657eb6e7b1915d337fb1a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7758,"rank":7758,"depth":0,"x":1864.132,"y":687.048,"cluster":"scheme-morphisms"},{"id":"stacks:05PB","tag":"05PB","title":"Flattening stratification over an Artinian ring · Lemma 05PB","summary":"Let S be the spectrum of an Artinian ring. For any scheme X over S, and any quasi-coherent O_X-module there exists a universal flattening. In fact the universal flattening is given by a closed immersion S' → S, and hence is a flattening stratification for F as well.","statement_latex":"Let $S$ be the spectrum of an Artinian ring.\nFor any scheme $X$ over $S$, and any quasi-coherent $\\mathcal{O}_X$-module\nthere exists a universal flattening. In fact the universal flattening\nis given by a closed immersion $S' \\to S$, and hence is a flattening\nstratification for $\\mathcal{F}$ as well.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening stratification over an Artinian ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PB","source_file":"flat.tex","source_line":6565,"source_end_line":6572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6565-L6572","statement_sha256":"36ba07a7dc424e058ec7adfe28333582c8a21b7162170e40de14e9fdced19a11","origin":"The Stacks Project","memory_eligible":false,"source_rank":7759,"rank":7759,"depth":9,"x":2146.685,"y":580.647,"cluster":"scheme-morphisms"},{"id":"stacks:05PD","tag":"05PD","title":"Flattening a map · Lemma 05PD","summary":"Let S be a scheme. Let g : X' → X be a flat morphism of schemes over S with X locally of finite type over S. Let F be a finite type quasi-coherent O_X-module which is flat over S. If Ass_X/S(F) ⊂ g(X') then the canonical map F → g_*g^*F is injective, and remains injective after any base change.","statement_latex":"Let $S$ be a scheme.\nLet $g : X' \\to X$ be a flat morphism of schemes over $S$\nwith $X$ locally of finite type over $S$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module\nwhich is flat over $S$. If $\\text{Ass}_{X/S}(\\mathcal{F}) \\subset g(X')$\nthen the canonical map\n$$\n\\mathcal{F} \\longrightarrow g_*g^*\\mathcal{F}\n$$\nis injective, and remains injective after any base change.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening a map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PD","source_file":"flat.tex","source_line":6597,"source_end_line":6609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6597-L6609","statement_sha256":"32856a23991684a3759c9259712e175600db6b3631166e847b8157d2dc6b1bb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7760,"rank":7760,"depth":56,"x":2023.852,"y":819.524,"cluster":"scheme-morphisms"},{"id":"stacks:05PE","tag":"05PE","title":"Flattening a map · Lemma 05PE","summary":"Let A be a ring. Let u : M → N be a surjective map of A-modules. If M is projective as an A-module, then there exists an ideal I ⊂ A such that for any ring map φ : A → B the following are equivalent • u ⊗ 1 : M ⊗_A B → N ⊗_A B is an isomorphism, and • φ(I) = 0.","statement_latex":"Let $A$ be a ring. Let $u : M \\to N$ be a surjective map of $A$-modules.\nIf $M$ is projective as an $A$-module, then there exists an ideal\n$I \\subset A$ such that for any ring map $\\varphi : A \\to B$\nthe following are equivalent\n\\begin{enumerate}\n\\item $u \\otimes 1 : M \\otimes_A B \\to N \\otimes_A B$ is an\nisomorphism, and\n\\item $\\varphi(I) = 0$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening a map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PE","source_file":"flat.tex","source_line":6645,"source_end_line":6656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6645-L6656","statement_sha256":"51865250beb807ce40dd357c41e35067ce0f692ace8c30e869d94369c93270a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7761,"rank":7761,"depth":0,"x":1922.29,"y":573.588,"cluster":"scheme-morphisms"},{"id":"stacks:05PF","tag":"05PF","title":"Flattening a map · Theorem 05PF","summary":"In Situation [Tag 05MH] assume • f is of finite presentation, • F is of finite presentation, flat over S, and pure relative to S, and • u is surjective. Then F_iso is representable by a closed immersion Z → S. Moreover Z → S is of finite presentation if G is of finite presentation.","statement_latex":"In\nSituation \\ref{situation-iso}\nassume\n\\begin{enumerate}\n\\item $f$ is of finite presentation,\n\\item $\\mathcal{F}$ is of finite presentation, flat over $S$, and\npure relative to $S$, and\n\\item $u$ is surjective.\n\\end{enumerate}\nThen $F_{iso}$ is representable by a closed immersion $Z \\to S$.\nMoreover $Z \\to S$ is of finite presentation if $\\mathcal{G}$ is\nof finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening a map","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PF","source_file":"flat.tex","source_line":6670,"source_end_line":6684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6670-L6684","statement_sha256":"eb782018892a10c7b3e47b4f86a11040947dc6a8b435196ae5cf41857dad68a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7762,"rank":7762,"depth":57,"x":2195.06,"y":697.356,"cluster":"scheme-morphisms"},{"id":"stacks:07AI","tag":"07AI","title":"Flattening a map · Lemma 07AI","summary":"Let f:X→ S be a morphism of schemes which is of finite presentation, flat, and pure. Let Y be a closed subscheme of X. Let F=f_*Y be the Weil restriction functor of Y along f, defined by F : (Sch/S)^opp → Sets, T ↦ ( (*) & if & Y_T→ X_T is an isomorphism, ∅ & else. & . Then F is representable by a closed immersion Z→ S. Moreover Z→ S is of finite presentation if Y→ S is.","statement_latex":"Let $f:X\\to S$ be a morphism of schemes which is of finite presentation,\nflat, and pure. Let $Y$ be a closed subscheme of $X$. Let $F=f_*Y$ be the\nWeil restriction functor of $Y$ along $f$, defined by\n$$\nF : (\\Sch/S)^{opp} \\to \\textit{Sets}, \\quad\nT \\mapsto\n\\left\\{\n\\begin{matrix}\n\\{*\\} & \\text{if} & Y_T\\to X_T \\text{ is an isomorphism, }\\\\\n\\emptyset & \\text{else.} &\n\\end{matrix}\n\\right.\n$$\nThen $F$ is representable by a closed immersion $Z\\to S$. Moreover\n$Z\\to S$ is of finite presentation if $Y\\to S$ is.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening a map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AI","source_file":"flat.tex","source_line":6752,"source_end_line":6769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6752-L6769","statement_sha256":"c9059124095904de683eb8c9bf5b4ade203eb01011ca996d859f5f03616a1da5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7763,"rank":7763,"depth":58,"x":1894.278,"y":760.892,"cluster":"scheme-morphisms"},{"id":"stacks:05PG","tag":"05PG","title":"Flattening in the local case · Theorem 05PG","summary":"In Situation [Tag 05MK] assume A is henselian, B is essentially of finite type over A, and M is a finite B-module. Then there exists an ideal I ⊂ A such that A/I corepresents the functor F_lf on the category C. In other words given a local homomorphism of local rings φ : A → A' with B' = B ⊗_A A' and M' = M ⊗_A A' the following are equivalent: • ∀ q ∈ V( m_A'B' + m_B B') ⊂ Spec(B') : M'_ q is flat over A', and • φ(I) = 0. If B is essentially of finite presentation over A…","statement_latex":"In\nSituation \\ref{situation-flat-at-point}\nassume $A$ is henselian, $B$ is essentially of finite type over $A$, and\n$M$ is a finite $B$-module. Then there exists an ideal\n$I \\subset A$ such that $A/I$ corepresents the functor $F_{lf}$ on the category\n$\\mathcal{C}$. In other words given a local homomorphism of local rings\n$\\varphi : A \\to A'$ with $B' = B \\otimes_A A'$ and $M' = M \\otimes_A A'$\nthe following are equivalent:\n\\begin{enumerate}\n\\item $\\forall \\mathfrak q \\in V(\\mathfrak m_{A'}B' + \\mathfrak m_B B')\n\\subset \\Spec(B') :\nM'_{\\mathfrak q}\\text{ is flat over }A'$, and\n\\item $\\varphi(I) = 0$.\n\\end{enumerate}\nIf $B$ is essentially of finite presentation over $A$ and $M$\nof finite presentation over $B$, then $I$ is a finitely generated ideal.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening in the local case","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PG","source_file":"flat.tex","source_line":6789,"source_end_line":6807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6789-L6807","statement_sha256":"de57a2a3b583710d3bfe87f78fc4c95a23ee00a568ec52d535b133bfd0647cc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7764,"rank":7764,"depth":58,"x":2065.042,"y":543.288,"cluster":"scheme-morphisms"},{"id":"stacks:05PI","tag":"05PI","title":"Flattening in the local case · Lemma 05PI","summary":"Let S be the spectrum of a henselian local ring with closed point s. Let X → S be a morphism of schemes which is locally of finite type. Let F be a finite type quasi-coherent O_X-module. Let E ⊂ X_s be a subset. There exists a closed subscheme Z ⊂ S with the following property: for any morphism of pointed schemes (T, t) → (S, s) the following are equivalent • F_T is flat over T at all points of the fibre X_t which map to a point of E ⊂ X_s, and • Spec(O_T, t) → S factors…","statement_latex":"Let $S$ be the spectrum of a henselian local ring with closed point $s$.\nLet $X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $E \\subset X_s$ be a subset. There exists a closed subscheme\n$Z \\subset S$ with the following property: for any morphism of pointed\nschemes $(T, t) \\to (S, s)$ the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}_T$ is flat over $T$ at all points of the fibre\n$X_t$ which map to a point of $E \\subset X_s$, and\n\\item $\\Spec(\\mathcal{O}_{T, t}) \\to S$ factors through $Z$.\n\\end{enumerate}\nMoreover, if $X \\to S$ is locally of finite presentation,\n$\\mathcal{F}$ is of finite presentation, and $E \\subset X_s$ is\nclosed and quasi-compact, then $Z \\to S$ is of finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flattening in the local case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05PI","source_file":"flat.tex","source_line":6930,"source_end_line":6946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L6930-L6946","statement_sha256":"fa51677d60502b798ec46ef630d5b73ca31e17f4dae7fa35e4ca34566907ec07","origin":"The Stacks Project","memory_eligible":false,"source_rank":7765,"rank":7765,"depth":59,"x":2114.133,"y":800.739,"cluster":"scheme-morphisms"},{"id":"stacks:0AT1","tag":"0AT1","title":"Variants of a lemma · Lemma 0AT1","summary":"If in Situation [Tag 0AT0] the ring A is Noetherian then the lemma holds.","statement_latex":"If in Situation \\ref{situation-mod-injective} the ring $A$ is Noetherian\nthen the lemma holds.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AT1","source_file":"flat.tex","source_line":7029,"source_end_line":7033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7029-L7033","statement_sha256":"fe0bccab96b8faf65a710931d72ed83fac039d868f54e1b854d144ed382daf55","origin":"The Stacks Project","memory_eligible":false,"source_rank":7766,"rank":7766,"depth":5,"x":1870.806,"y":638.692,"cluster":"scheme-morphisms"},{"id":"stacks:0AT2","tag":"0AT2","title":"Variants of a lemma · Lemma 0AT2","summary":"Let A_0 be a local ring. If the lemma holds for every Situation [Tag 0AT0] with A = A_0, with B a localization of a polynomial algebra over A, and N of finite presentation over B, then the lemma holds for every Situation [Tag 0AT0] with A = A_0.","statement_latex":"Let $A_0$ be a local ring. If the lemma holds for every\nSituation \\ref{situation-mod-injective} with $A = A_0$, with $B$ a\nlocalization of a polynomial algebra over $A$, and $N$ of finite presentation\nover $B$, then the lemma holds for every\nSituation \\ref{situation-mod-injective} with $A = A_0$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AT2","source_file":"flat.tex","source_line":7044,"source_end_line":7051,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7044-L7051","statement_sha256":"3e55b3934b8ec41cf80205c1bd19c332b99506be97a70bc5c4d4d2ca200cbd07","origin":"The Stacks Project","memory_eligible":false,"source_rank":7767,"rank":7767,"depth":2,"x":2180.663,"y":620.107,"cluster":"scheme-morphisms"},{"id":"stacks:0AT3","tag":"0AT3","title":"Variants of a lemma · Lemma 0AT3","summary":"If in Situation [Tag 0AT0] the ring A is henselian then the lemma holds.","statement_latex":"If in Situation \\ref{situation-mod-injective} the ring $A$ is henselian\nthen the lemma holds.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AT3","source_file":"flat.tex","source_line":7072,"source_end_line":7076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7072-L7076","statement_sha256":"a26482fee54c3e2d6552ebbe4524edb50222ecfc32d05de2474a47c510cc7f48","origin":"The Stacks Project","memory_eligible":false,"source_rank":7768,"rank":7768,"depth":61,"x":1967.044,"y":809.703,"cluster":"scheme-morphisms"},{"id":"stacks:0AT4","tag":"0AT4","title":"Variants of a lemma · Lemma 0AT4","summary":"Let A → B be a local ring homomorphism of local rings which is essentially of finite type. Let u : N → M be a B-module map. If N is a finite B-module, M is flat over A, and overlineu : N/ m_A N → M/ m_A M is injective, then u is A-universally injective, N is of finite presentation over B, and N is flat over A.","statement_latex":"Let $A \\to B$ be a local ring homomorphism of local rings which is\nessentially of finite type. Let $u : N \\to M$ be a $B$-module map.\nIf $N$ is a finite $B$-module, $M$ is flat over $A$, and\n$\\overline{u} : N/\\mathfrak m_A N \\to M/\\mathfrak m_A M$ is injective,\nthen $u$ is $A$-universally injective, $N$ is of finite presentation over\n$B$, and $N$ is flat over $A$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AT4","source_file":"flat.tex","source_line":7131,"source_end_line":7139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7131-L7139","statement_sha256":"cff59b1dcf03e2c978d7f8ec71d4f5b695177caa3e6b3e2959f22a3ae98ced1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7769,"rank":7769,"depth":62,"x":1972.097,"y":548.586,"cluster":"scheme-morphisms"},{"id":"stacks:0AT5","tag":"0AT5","title":"Variants of a lemma · Lemma 0AT5","summary":"If in Situation [Tag 0AT0] the ring A is a valuation ring then the lemma holds.","statement_latex":"If in Situation \\ref{situation-mod-injective} the ring $A$ is a\nvaluation ring then the lemma holds.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AT5","source_file":"flat.tex","source_line":7157,"source_end_line":7161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7157-L7161","statement_sha256":"c4c998052878b292393c551ecd5c6a852161c7cfc9ab484efd39e056d9a195d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7770,"rank":7770,"depth":62,"x":2178.432,"y":744.071,"cluster":"scheme-morphisms"},{"id":"stacks:0AT7","tag":"0AT7","title":"Variants of a lemma · Lemma 0AT7","summary":"In ([Tag 0AT6]) if there exists a pure spreadout, then • elements of N have content ideals in A, and • if u : N → M is a morphism to a flat A-module M such that N/ m N → M/ m M is injective for all maximal ideals m of A, then u is A-universally injective.","statement_latex":"In (\\ref{equation-star}) if there exists a pure spreadout, then\n\\begin{enumerate}\n\\item elements of $N$ have content ideals in $A$, and\n\\item if $u : N \\to M$ is a morphism to a flat $A$-module $M$\nsuch that $N/\\mathfrak m N \\to M/\\mathfrak m M$ is injective\nfor all maximal ideals $\\mathfrak m$ of $A$, then $u$ is\n$A$-universally injective.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AT7","source_file":"flat.tex","source_line":7228,"source_end_line":7238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7228-L7238","statement_sha256":"e3ccb606fc2802a3bc0701c895efb8e7e5e88428ad985c62bd0c25cb8f00205c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7771,"rank":7771,"depth":9,"x":1868.963,"y":716.993,"cluster":"scheme-morphisms"},{"id":"stacks:0AT8","tag":"0AT8","title":"Variants of a lemma · Lemma 0AT8","summary":"In ([Tag 0AT6]) for every p ∈ Spec(A) there is a finitely generated ideal I ⊂ pA_ p such that over A_ p/I we have a pure spreadout.","statement_latex":"In (\\ref{equation-star}) for every $\\mathfrak p \\in \\Spec(A)$\nthere is a finitely generated ideal $I \\subset \\mathfrak pA_\\mathfrak p$\nsuch that over $A_\\mathfrak p/I$ we have a pure spreadout.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AT8","source_file":"flat.tex","source_line":7269,"source_end_line":7274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7269-L7274","statement_sha256":"16069e5f804e5bfe9e28e6e0d39f27afaa3ee2c2e94c170adb9f834744d9eb3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7772,"rank":7772,"depth":12,"x":2119.031,"y":561.301,"cluster":"scheme-morphisms"},{"id":"stacks:0AT9","tag":"0AT9","title":"Variants of a lemma · Lemma 0AT9","summary":"In ([Tag 0AT6]) assume N is A-flat, M is a flat A-module, and u : N → M is an A-module map such that u ⊗ id_kappa( p) is injective for all p ∈ Spec(A). Then u is A-universally injective.","statement_latex":"In (\\ref{equation-star}) assume $N$ is $A$-flat, $M$ is a flat $A$-module,\nand $u : N \\to M$ is an $A$-module map such that\n$u \\otimes \\text{id}_{\\kappa(\\mathfrak p)}$ is injective for all\n$\\mathfrak p \\in \\Spec(A)$. Then $u$ is $A$-universally injective.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AT9","source_file":"flat.tex","source_line":7302,"source_end_line":7308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7302-L7308","statement_sha256":"fd117f2c99a69d25cb952e488bdcae9636d0a33c9146f45c603559fc9b363efe","origin":"The Stacks Project","memory_eligible":false,"source_rank":7773,"rank":7773,"depth":13,"x":2059.816,"y":818.097,"cluster":"scheme-morphisms"},{"id":"stacks:0ATA","tag":"0ATA","title":"Variants of a lemma · Lemma 0ATA","summary":"Let A be a local domain which is not a field. Let S be a set of finitely generated ideals of A. Assume that S is closed under products and such that ⋃_I ∈ S V(I) is the complement of the generic point of Spec(A). Then ⋂_I ∈ S I = (0).","statement_latex":"Let $A$ be a local domain which is not a field.\nLet $S$ be a set of finitely generated ideals of $A$.\nAssume that $S$ is closed under products and such that\n$\\bigcup_{I \\in S} V(I)$ is the complement of the generic point of $\\Spec(A)$.\nThen $\\bigcap_{I \\in S} I = (0)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATA","source_file":"flat.tex","source_line":7343,"source_end_line":7350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7343-L7350","statement_sha256":"e665f8567b3ad56fe2315d2601b91ec90db6e2f4c20db70844613829a4d4ce40","origin":"The Stacks Project","memory_eligible":false,"source_rank":7774,"rank":7774,"depth":5,"x":1896.91,"y":595.055,"cluster":"scheme-morphisms"},{"id":"stacks:0ATB","tag":"0ATB","title":"Variants of a lemma · Lemma 0ATB","summary":"Let A be a local ring. Let I, J ⊂ A be ideals. If J is finitely generated and I ⊂ J^n for all n ≥ 1, then V(I) contains the closed points of Spec(A) setminus V(J).","statement_latex":"Let $A$ be a local ring. Let $I, J \\subset A$ be ideals.\nIf $J$ is finitely generated and $I \\subset J^n$ for all $n \\geq 1$,\nthen $V(I)$ contains the closed points of $\\Spec(A) \\setminus V(J)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATB","source_file":"flat.tex","source_line":7370,"source_end_line":7375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7370-L7375","statement_sha256":"d3f44dfa71b8c2fd5f812834be0bcea4918bcad4cc4ef4b8b2827bed8de1d53b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7775,"rank":7775,"depth":6,"x":2196.512,"y":667.114,"cluster":"scheme-morphisms"},{"id":"stacks:0ATC","tag":"0ATC","title":"Variants of a lemma · Lemma 0ATC","summary":"Let A be a local ring. Let I ⊂ A be an ideal. Let U ⊂ Spec(A) be quasi-compact open. Let M be an A-module. Assume that • M/IM is flat over A/I, • M is flat over U, Then M/I_2M is flat over A/I_2 where I_2 = Ker(I → Γ(U, I/I^2)).","statement_latex":"Let $A$ be a local ring. Let $I \\subset A$ be an ideal.\nLet $U \\subset \\Spec(A)$ be quasi-compact open.\nLet $M$ be an $A$-module. Assume that\n\\begin{enumerate}\n\\item $M/IM$ is flat over $A/I$,\n\\item $M$ is flat over $U$,\n\\end{enumerate}\nThen $M/I_2M$ is flat over $A/I_2$ where\n$I_2 = \\Ker(I \\to \\Gamma(U, I/I^2))$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATC","source_file":"flat.tex","source_line":7387,"source_end_line":7398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7387-L7398","statement_sha256":"0db7a5044d1108afdea7a398d8f36647d3892d846d139f207d58b86ab31e57f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7776,"rank":7776,"depth":4,"x":1917.537,"y":784.024,"cluster":"scheme-morphisms"},{"id":"stacks:05U9","tag":"05U9","title":"Variants of a lemma · Proposition 05U9","summary":"Let A → B be a local ring homomorphism of local rings which is essentially of finite type. Let M be a flat A-module, N a finite B-module and u : N → M an A-module map such that overlineu : N/ m_AN → M/ m_AM is injective. Then u is A-universally injective, N is of finite presentation over B, and N is flat over A.","statement_latex":"Let $A \\to B$ be a local ring homomorphism of local rings\nwhich is essentially of finite type. Let $M$ be a flat $A$-module,\n$N$ a finite $B$-module and $u : N \\to M$ an $A$-module map such that\n$\\overline{u} : N/\\mathfrak m_AN \\to M/\\mathfrak m_AM$ is injective.\nThen $u$ is $A$-universally injective, $N$ is of finite presentation over\n$B$, and $N$ is flat over $A$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Variants of a lemma","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05U9","source_file":"flat.tex","source_line":7413,"source_end_line":7421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7413-L7421","statement_sha256":"5d8f52b03ddcec495d9b2e644894d2a2273b70a2aada4447a7407e4033fc68a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7777,"rank":7777,"depth":61,"x":2029.275,"y":539.427,"cluster":"scheme-morphisms"},{"id":"stacks:05UA","tag":"05UA","title":"Flat finite type modules, Part III · Theorem 05UA","summary":"Let f : X → S be locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let x ∈ X with image s ∈ S. The following are equivalent • F is flat at x over S, and • for every x' ∈ Ass_X_s(F_s) which specializes to x we have that F is flat at x' over S.","statement_latex":"Let $f : X \\to S$ be locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $x \\in X$ with image $s \\in S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat at $x$ over $S$, and\n\\item for every $x' \\in \\text{Ass}_{X_s}(\\mathcal{F}_s)$ which\nspecializes to $x$ we have that $\\mathcal{F}$ is flat at $x'$ over $S$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part III","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UA","source_file":"flat.tex","source_line":7558,"source_end_line":7569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7558-L7569","statement_sha256":"b8a7fbc09c631a2185faa835d6326579e956c10762707cb5a4a816a5c07492ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":7778,"rank":7778,"depth":63,"x":2143.622,"y":783.283,"cluster":"scheme-morphisms"},{"id":"stacks:05UB","tag":"05UB","title":"Flat finite type modules, Part III · Lemma 05UB","summary":"Let S be a local scheme with closed point s. Let f : X → S be locally of finite type. Let F be a finite type quasi-coherent O_X-module. Assume that • every point of Ass_X/S(F) specializes to a point of the closed fibre X_s is pure along X_s, or if f is proper., • F is flat over S at every point of X_s. Then F is flat over S.","statement_latex":"Let $S$ be a local scheme with closed point $s$.\nLet $f : X \\to S$ be locally of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nAssume that\n\\begin{enumerate}\n\\item every point of $\\text{Ass}_{X/S}(\\mathcal{F})$ specializes\nto a point of the closed fibre $X_s$\\footnote{For example this holds if\n$f$ is finite type and $\\mathcal{F}$ is pure along $X_s$, or\nif $f$ is proper.},\n\\item $\\mathcal{F}$ is flat over $S$ at every point of $X_s$.\n\\end{enumerate}\nThen $\\mathcal{F}$ is flat over $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Flat finite type modules, Part III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UB","source_file":"flat.tex","source_line":7589,"source_end_line":7603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7589-L7603","statement_sha256":"6878c0da5a7ca67648952b4e31f06366a050aad2d7b7b8395d6997548f88e3d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7779,"rank":7779,"depth":64,"x":1863.096,"y":668.308,"cluster":"scheme-morphisms"},{"id":"stacks:05UC","tag":"05UC","title":"Universal flattening · Lemma 05UC","summary":"In Situation [Tag 05MP]. For each p ≥ 0 the functor H_p ([Tag 05MR]) is representable by a locally closed immersion S_p → S. If F is of finite presentation, then S_p → S is of finite presentation.","statement_latex":"In Situation \\ref{situation-free-at-generic-points}.\nFor each $p \\geq 0$ the functor $H_p$\n(\\ref{equation-free-at-generic-points}) is representable\nby a locally closed immersion $S_p \\to S$. If $\\mathcal{F}$\nis of finite presentation, then $S_p \\to S$ is of finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Universal flattening","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UC","source_file":"flat.tex","source_line":7630,"source_end_line":7637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7630-L7637","statement_sha256":"9bfa5e96673a376f06603f16ec19e760f7231d57093ed12c6012d89eedf78e8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7780,"rank":7780,"depth":58,"x":2162.524,"y":593.884,"cluster":"scheme-morphisms"},{"id":"stacks:05UD","tag":"05UD","title":"Universal flattening · Lemma 05UD","summary":"In Situation [Tag 05MT]. Let h : X' → X be an étale morphism. Set F' = h^*F and f' = f ∘ h. Let F_n' be ([Tag 05MU]) associated to (f' : X' → S, F'). Then F_n is a subfunctor of F_n' and if h(X') ⊃ Ass_X/S(F), then F_n = F'_n.","statement_latex":"In Situation \\ref{situation-flat-dimension-n}.\nLet $h : X' \\to X$ be an \\'etale morphism.\nSet $\\mathcal{F}' = h^*\\mathcal{F}$ and $f' = f \\circ h$.\nLet $F_n'$ be (\\ref{equation-flat-dimension-n})\nassociated to $(f' : X' \\to S, \\mathcal{F}')$.\nThen $F_n$ is a subfunctor of $F_n'$ and if\n$h(X') \\supset \\text{Ass}_{X/S}(\\mathcal{F})$, then $F_n = F'_n$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Universal flattening","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UD","source_file":"flat.tex","source_line":7727,"source_end_line":7736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7727-L7736","statement_sha256":"481471147a21d3cc01fe8b3a289ed719a59932431e4d7578303e2a74621917d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7781,"rank":7781,"depth":64,"x":2001.52,"y":818.75,"cluster":"scheme-morphisms"},{"id":"stacks:05UE","tag":"05UE","title":"Universal flattening · Lemma 05UE","summary":"Assume that X → S is a smooth morphism of affine schemes with geometrically irreducible fibres of dimension d and that F is a quasi-coherent O_X-module of finite presentation. Then F_d = coprod_p = 0, …, c H_p for some c ≥ 0 with F_d as in ([Tag 05MU]) and H_p as in ([Tag 05MR]).","statement_latex":"Assume that $X \\to S$ is a smooth morphism of affine schemes\nwith geometrically irreducible fibres of dimension $d$ and that\n$\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module of finite\npresentation. Then $F_d = \\coprod_{p = 0, \\ldots, c} H_p$\nfor some $c \\geq 0$ with $F_d$ as in\n(\\ref{equation-flat-dimension-n}) and $H_p$ as in\n(\\ref{equation-free-at-generic-points}).","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Universal flattening","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UE","source_file":"flat.tex","source_line":7777,"source_end_line":7786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7777-L7786","statement_sha256":"eb9ece693bfde9a45cad24074a1c5024c4a00b930d636396b1f1ab4d32bab7c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7782,"rank":7782,"depth":37,"x":1939.388,"y":561.483,"cluster":"scheme-morphisms"},{"id":"stacks:05UF","tag":"05UF","title":"Universal flattening · Lemma 05UF","summary":"In Situation [Tag 05MT]. Let s ∈ S let d ≥ 0. Assume • there exists a complete dévissage of F/X/S over some point s ∈ S, • X is of finite presentation over S, • F is an O_X-module of finite presentation, and • F is flat in dimensions ≥ d + 1 over S. Then after possibly replacing S by an open neighbourhood of s the functor F_d ([Tag 05MU]) is representable by a monomorphism Z_d → S of finite presentation.","statement_latex":"In Situation \\ref{situation-flat-dimension-n}.\nLet $s \\in S$ let $d \\geq 0$. Assume\n\\begin{enumerate}\n\\item there exists a complete d\\'evissage\nof $\\mathcal{F}/X/S$ over some point $s \\in S$,\n\\item $X$ is of finite presentation over $S$,\n\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation, and\n\\item $\\mathcal{F}$ is flat in dimensions $\\geq d + 1$ over $S$.\n\\end{enumerate}\nThen after possibly replacing $S$ by an open neighbourhood\nof $s$ the functor $F_d$ (\\ref{equation-flat-dimension-n})\nis representable by a monomorphism $Z_d \\to S$ of finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Universal flattening","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UF","source_file":"flat.tex","source_line":7805,"source_end_line":7819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7805-L7819","statement_sha256":"aad4f1d357104cd4a023d1fa02a6f7ae25bf053dfa4e86890cd3a54f74166de5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7783,"rank":7783,"depth":59,"x":2192.185,"y":715.99,"cluster":"scheme-morphisms"},{"id":"stacks:05UG","tag":"05UG","title":"Universal flattening · Theorem 05UG","summary":"In Situation [Tag 05MT]. Assume moreover that f is of finite presentation, that F is an O_X-module of finite presentation, and that F is pure relative to S. Then F_n is representable by a monomorphism Z_n → S of finite presentation.","statement_latex":"In Situation \\ref{situation-flat-dimension-n}.\nAssume moreover that $f$ is of finite presentation, that\n$\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation,\nand that $\\mathcal{F}$ is pure relative to $S$.\nThen $F_n$ is representable by a monomorphism\n$Z_n \\to S$ of finite presentation.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Universal flattening","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UG","source_file":"flat.tex","source_line":7894,"source_end_line":7902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7894-L7902","statement_sha256":"ac1643840efc706c96a0799144dc57b00d04273747d52dbf9c2922475a375b66","origin":"The Stacks Project","memory_eligible":false,"source_rank":7784,"rank":7784,"depth":65,"x":1881.409,"y":745.513,"cluster":"scheme-morphisms"},{"id":"stacks:05UH","tag":"05UH","title":"Universal flattening · Lemma 05UH","summary":"Let f : X → S be a morphism of schemes. Let F be a quasi-coherent O_X-module. • If f is of finite presentation, F is an O_X-module of finite presentation, and F is pure relative to S, then there exists a universal flattening S' → S of F. Moreover S' → S is a monomorphism of finite presentation. • If f is of finite presentation and X is pure relative to S, then there exists a universal flattening S' → S of X. Moreover S' → S is a monomorphism of finite presentation. • If f…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If $f$ is of finite presentation, $\\mathcal{F}$ is an\n$\\mathcal{O}_X$-module of finite presentation, and $\\mathcal{F}$ is\npure relative to $S$, then there exists a universal flattening\n$S' \\to S$ of $\\mathcal{F}$. Moreover $S' \\to S$ is a monomorphism\nof finite presentation.\n\\item If $f$ is of finite presentation and $X$ is pure relative to $S$,\nthen there exists a universal flattening $S' \\to S$ of $X$.\nMoreover $S' \\to S$ is a monomorphism of finite presentation.\n\\item If $f$ is proper and of finite presentation and $\\mathcal{F}$ is an\n$\\mathcal{O}_X$-module of finite presentation, then there exists a\nuniversal flattening $S' \\to S$ of $\\mathcal{F}$. Moreover $S' \\to S$ is\na monomorphism of finite presentation.\n\\item If $f$ is proper and of finite presentation\nthen there exists a universal flattening $S' \\to S$ of $X$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Universal flattening","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UH","source_file":"flat.tex","source_line":7988,"source_end_line":8008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L7988-L8008","statement_sha256":"fcd781085b6b706f0850f08cdf1ebb539475f47a03d305f0b3417d9588e95609","origin":"The Stacks Project","memory_eligible":false,"source_rank":7785,"rank":7785,"depth":66,"x":2086.906,"y":547.328,"cluster":"scheme-morphisms"},{"id":"stacks:0CTD","tag":"0CTD","title":"Grothendieck's Existence Theorem, IV · Lemma 0CTD","summary":"In Situation [Tag 0CTC] consider K = Rlim_D_QCoh(O_X)(F_n) = DQ_X(Rlim_D(O_X)F_n) Then K is in D^b_QCoh(O_X) and in fact K has nonzero cohomology sheaves only in degrees ≥ 0.","statement_latex":"In Situation \\ref{situation-existence} consider\n$$\nK = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(\\mathcal{F}_n) =\nDQ_X(R\\lim_{D(\\mathcal{O}_X)}\\mathcal{F}_n)\n$$\nThen $K$ is in $D^b_{\\QCoh}(\\mathcal{O}_X)$ and in fact\n$K$ has nonzero cohomology sheaves only in degrees $\\geq 0$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTD","source_file":"flat.tex","source_line":8061,"source_end_line":8070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8061-L8070","statement_sha256":"bfc446ef1d1213acdb9d46359f3598a38e53ed4662a458eae3215451e21758cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7786,"rank":7786,"depth":0,"x":2094.754,"y":810.169,"cluster":"scheme-morphisms"},{"id":"stacks:0CTE","tag":"0CTE","title":"Grothendieck's Existence Theorem, IV · Lemma 0CTE","summary":"In Situation [Tag 0CTC] let K be as in Lemma [Tag 0CTD]. For any perfect object E of D(O_X) we have • M = RΓ(X, K ⊗^L E) is a perfect object of D(A) and there is a canonical isomorphism RΓ(X_n, F_n ⊗^L E|_X_n) = M ⊗_A^L A_n in D(A_n), • N = RHom_X(E, K) is a perfect object of D(A) and there is a canonical isomorphism RHom_X_n(E|_X_n, F_n) = N ⊗_A^L A_n in D(A_n). In both statements E|_X_n denotes the derived pullback of E to X_n.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}. For any perfect\nobject $E$ of $D(\\mathcal{O}_X)$ we have\n\\begin{enumerate}\n\\item $M = R\\Gamma(X, K \\otimes^\\mathbf{L} E)$ is a perfect object of $D(A)$\nand there is a canonical isomorphism\n$R\\Gamma(X_n, \\mathcal{F}_n \\otimes^\\mathbf{L} E|_{X_n}) =\nM \\otimes_A^\\mathbf{L} A_n$\nin $D(A_n)$,\n\\item $N = R\\Hom_X(E, K)$ is a perfect object of $D(A)$\nand there is a canonical isomorphism\n$R\\Hom_{X_n}(E|_{X_n}, \\mathcal{F}_n) = N \\otimes_A^\\mathbf{L} A_n$\nin $D(A_n)$.\n\\end{enumerate}\nIn both statements $E|_{X_n}$ denotes the derived pullback\nof $E$ to $X_n$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTE","source_file":"flat.tex","source_line":8078,"source_end_line":8096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8078-L8096","statement_sha256":"5422291292a943c30c77c29f1d5b1aa71fd319627ee9af54127069deb5e8736b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7787,"rank":7787,"depth":38,"x":1877.516,"y":620.736,"cluster":"scheme-morphisms"},{"id":"stacks:0CTF","tag":"0CTF","title":"Grothendieck's Existence Theorem, IV · Lemma 0CTF","summary":"In Situation [Tag 0CTC] let K be as in Lemma [Tag 0CTD]. Then K is pseudo-coherent relative to A.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}. Then $K$\nis pseudo-coherent relative to $A$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTF","source_file":"flat.tex","source_line":8127,"source_end_line":8132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8127-L8132","statement_sha256":"f18ccde9cc5bbedd6d3a748219a851a3df7112b1d709d5ebd35258a36e5f8b6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7788,"rank":7788,"depth":39,"x":2190.157,"y":637.161,"cluster":"scheme-morphisms"},{"id":"stacks:0CTG","tag":"0CTG","title":"Grothendieck's Existence Theorem, IV · Lemma 0CTG","summary":"In Situation [Tag 0CTC] let K be as in Lemma [Tag 0CTD]. For any quasi-compact open U ⊂ X we have RΓ(U, K) ⊗_A^L A_n = RΓ(U_n, F_n) in D(A_n) where U_n = U ∩ X_n.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}. For any quasi-compact\nopen $U \\subset X$ we have\n$$\nR\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n =\nR\\Gamma(U_n, \\mathcal{F}_n)\n$$\nin $D(A_n)$ where $U_n = U \\cap X_n$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTG","source_file":"flat.tex","source_line":8144,"source_end_line":8154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8144-L8154","statement_sha256":"10b00fb5c94309dcbce62d3b8e5407e9a94b30a5db3588e8eb41c9920b99a115","origin":"The Stacks Project","memory_eligible":false,"source_rank":7789,"rank":7789,"depth":41,"x":1946.321,"y":802.511,"cluster":"scheme-morphisms"},{"id":"stacks:0CTH","tag":"0CTH","title":"Grothendieck's Existence Theorem, IV · Lemma 0CTH","summary":"In Situation [Tag 0CTC] let K be as in Lemma [Tag 0CTD]. Denote X_0 ⊂ X the closed subset consisting of points lying over the closed subset Spec(A_1) = Spec(A_2) = … of Spec(A). There exists an open W ⊂ X containing X_0 such that • H^i(K)|_W is zero unless i = 0, • F = H^0(K)|_W is of finite presentation, and • F_n = F ⊗_O_X O_X_n.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}.\nDenote $X_0 \\subset X$ the closed subset\nconsisting of points lying over the closed subset\n$\\Spec(A_1) = \\Spec(A_2) = \\ldots$ of $\\Spec(A)$.\nThere exists an open $W \\subset X$ containing $X_0$\nsuch that\n\\begin{enumerate}\n\\item $H^i(K)|_W$ is zero unless $i = 0$,\n\\item $\\mathcal{F} = H^0(K)|_W$ is of finite presentation, and\n\\item $\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTH","source_file":"flat.tex","source_line":8180,"source_end_line":8194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8180-L8194","statement_sha256":"f8da7721663fa1efffd513ecf4d44c1d7b33da4091d230c0c4a710bf5d6ccc15","origin":"The Stacks Project","memory_eligible":false,"source_rank":7790,"rank":7790,"depth":42,"x":1993.169,"y":542.129,"cluster":"scheme-morphisms"},{"id":"stacks:0CTI","tag":"0CTI","title":"Grothendieck's Existence Theorem, IV · Lemma 0CTI","summary":"In Situation [Tag 0CTC] let K be as in Lemma [Tag 0CTD]. Let W ⊂ X be as in Lemma [Tag 0CTH]. Set F = H^0(K)|_W. Then, after possibly shrinking the open W, the support of F is proper over A.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}. Let $W \\subset X$\nbe as in Lemma \\ref{lemma-finitely-presented}.\nSet $\\mathcal{F} = H^0(K)|_W$. Then, after possibly shrinking the open $W$,\nthe support of $\\mathcal{F}$ is proper over $A$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTI","source_file":"flat.tex","source_line":8244,"source_end_line":8251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8244-L8251","statement_sha256":"f2df6c91cdcae8c430223dcd42e6a12e7c299912a2d6bd7e7ee9321c6c12539c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7791,"rank":7791,"depth":48,"x":2168.082,"y":760.795,"cluster":"scheme-morphisms"},{"id":"stacks:0CTJ","tag":"0CTJ","title":"Grothendieck's Existence Theorem, IV · Lemma 0CTJ","summary":"Let A = lim A_n be a limit of a system of rings whose transition maps are surjective and with locally nilpotent kernels. Let S = Spec(A). Let T → S be a monomorphism which is locally of finite type. If Spec(A_n) → S factors through T for all n, then T = S.","statement_latex":"Let $A = \\lim A_n$ be a limit of a system of rings\nwhose transition maps are surjective and with locally nilpotent\nkernels. Let $S = \\Spec(A)$. Let $T \\to S$ be a monomorphism\nwhich is locally of finite type. If $\\Spec(A_n) \\to S$\nfactors through $T$ for all $n$, then $T = S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTJ","source_file":"flat.tex","source_line":8273,"source_end_line":8280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8273-L8280","statement_sha256":"da06e3b34fc6fbcf89e991c9d4a864feec1d3176cf1a58f353e383b296ecc1c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7792,"rank":7792,"depth":50,"x":1863.145,"y":698.78,"cluster":"scheme-morphisms"},{"id":"stacks:0CTK","tag":"0CTK","title":"Grothendieck Existence Theorem · Theorem 0CTK","summary":"In Situation [Tag 0CTC] there exists a finitely presented O_X-module F, flat over A, with support proper over A, such that F_n = F ⊗_O_X O_X_n for all n compatibly with the maps φ_n.","statement_latex":"In Situation \\ref{situation-existence}\nthere exists a finitely presented $\\mathcal{O}_X$-module\n$\\mathcal{F}$, flat over $A$, with support proper over $A$,\nsuch that\n$\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$\nfor all $n$ compatibly with the maps $\\varphi_n$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, IV","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTK","source_file":"flat.tex","source_line":8318,"source_end_line":8326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8318-L8326","statement_sha256":"78e804dc6a303fe2cf05352875db642b640ec4dd15de3088e7c29df0317129d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7793,"rank":7793,"depth":67,"x":2137.974,"y":571.434,"cluster":"scheme-morphisms"},{"id":"stacks:0DIC","tag":"0DIC","title":"Grothendieck's Existence Theorem, V · Lemma 0DIC","summary":"In Situation [Tag 0DIB] consider K = Rlim_D_QCoh(O_X)(K_n) = DQ_X(Rlim_D(O_X) K_n) Then K is in D^-_QCoh(O_X).","statement_latex":"In Situation \\ref{situation-existence-derived} consider\n$$\nK = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(K_n) =\nDQ_X(R\\lim_{D(\\mathcal{O}_X)} K_n)\n$$\nThen $K$ is in $D^-_{\\QCoh}(\\mathcal{O}_X)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIC","source_file":"flat.tex","source_line":8405,"source_end_line":8413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8405-L8413","statement_sha256":"48415e60a796a77757247839754afb7e76a7e0de48a70804318d3653b843c290","origin":"The Stacks Project","memory_eligible":false,"source_rank":7794,"rank":7794,"depth":32,"x":2037.691,"y":821.374,"cluster":"scheme-morphisms"},{"id":"stacks:0DID","tag":"0DID","title":"Grothendieck's Existence Theorem, V · Lemma 0DID","summary":"In Situation [Tag 0DIB] let K be as in Lemma [Tag 0DIC]. For any perfect object E of D(O_X) the cohomology M = RΓ(X, K ⊗^L E) is a pseudo-coherent object of D(A) and there is a canonical isomorphism RΓ(X_n, K_n ⊗^L E|_X_n) = M ⊗_A^L A_n in D(A_n). Here E|_X_n denotes the derived pullback of E to X_n.","statement_latex":"In Situation \\ref{situation-existence-derived} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be-derived}. For any perfect\nobject $E$ of $D(\\mathcal{O}_X)$ the cohomology\n$$\nM = R\\Gamma(X, K \\otimes^\\mathbf{L} E)\n$$\nis a pseudo-coherent object of $D(A)$ and there is a canonical isomorphism\n$$\nR\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E|_{X_n}) = M \\otimes_A^\\mathbf{L} A_n\n$$\nin $D(A_n)$. Here $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DID","source_file":"flat.tex","source_line":8461,"source_end_line":8474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8461-L8474","statement_sha256":"82112153cf4b6f8a3db00729eac278bdd792be9328abb30d04d248cdb1c0c852","origin":"The Stacks Project","memory_eligible":false,"source_rank":7795,"rank":7795,"depth":33,"x":1910.595,"y":580.08,"cluster":"scheme-morphisms"},{"id":"stacks:0DIE","tag":"0DIE","title":"Grothendieck's Existence Theorem, V · Lemma 0DIE","summary":"In Situation [Tag 0DIB] let K be as in Lemma [Tag 0DIC]. Then K is pseudo-coherent on X.","statement_latex":"In Situation \\ref{situation-existence-derived} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be-derived}. Then $K$\nis pseudo-coherent on $X$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIE","source_file":"flat.tex","source_line":8502,"source_end_line":8507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8502-L8507","statement_sha256":"69642dcddcaa5b1cfaab4a6c9a118d748a1c5aa8ab02938ae8723e240e061ac5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7796,"rank":7796,"depth":38,"x":2198.463,"y":685.929,"cluster":"scheme-morphisms"},{"id":"stacks:0DIF","tag":"0DIF","title":"Grothendieck's Existence Theorem, V · Lemma 0DIF","summary":"In Situation [Tag 0DIB] let K be as in Lemma [Tag 0DIC]. For any quasi-compact open U ⊂ X we have RΓ(U, K) ⊗_A^L A_n = RΓ(U_n, K_n) in D(A_n) where U_n = U ∩ X_n.","statement_latex":"In Situation \\ref{situation-existence-derived} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be-derived}. For any quasi-compact\nopen $U \\subset X$ we have\n$$\nR\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n =\nR\\Gamma(U_n, K_n)\n$$\nin $D(A_n)$ where $U_n = U \\cap X_n$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIF","source_file":"flat.tex","source_line":8525,"source_end_line":8535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8525-L8535","statement_sha256":"6d74baf82a525f8963fbd6ec09f82c47ec86887e715ed55f38c3dd3d5e30cd2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7797,"rank":7797,"depth":41,"x":1900.962,"y":771.252,"cluster":"scheme-morphisms"},{"id":"stacks:0DIG","tag":"0DIG","title":"Derived Grothendieck Existence Theorem · Theorem 0DIG","summary":"In Situation [Tag 0DIB] there exists a pseudo-coherent K in D(O_X) such that K_n = K ⊗_O_X^L O_X_n for all n compatibly with the maps φ_n.","statement_latex":"In Situation \\ref{situation-existence-derived}\nthere exists a pseudo-coherent $K$ in $D(\\mathcal{O}_X)$\nsuch that $K_n = K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_{X_n}$\nfor all $n$ compatibly with the maps $\\varphi_n$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Grothendieck's Existence Theorem, V","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIG","source_file":"flat.tex","source_line":8561,"source_end_line":8567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8561-L8567","statement_sha256":"53bda896e354ad77fc31f6da8df4706510971116ebfba9d51b1799f4d824b54f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7798,"rank":7798,"depth":42,"x":2051.776,"y":539.441,"cluster":"scheme-morphisms"},{"id":"stacks:0810","tag":"0810","title":"Blowing up and flatness · Lemma 0810","summary":"Let R be a ring and let f ∈ R. Let r≥ 0 be an integer. Let R → S be a ring map and let M be an S-module. Assume • R → S is of finite presentation and flat, • every fibre ring S ⊗_R kappa( p) is geometrically integral over R, • M is a finite S-module, • M_f is a finitely presented S_f-module, • for all p ∈ R, f not ∈ p with q = pS the module M_ q is free of rank r over S_ q. Then there exists a finitely generated ideal I ⊂ R with V(f) = V(I) such that for all a ∈ I with R'…","statement_latex":"Let $R$ be a ring and let $f \\in R$. Let $r\\geq 0$ be an integer.\nLet $R \\to S$ be a ring map and let $M$ be an $S$-module. Assume\n\\begin{enumerate}\n\\item $R \\to S$ is of finite presentation and flat,\n\\item every fibre ring $S \\otimes_R \\kappa(\\mathfrak p)$ is\ngeometrically integral over $R$,\n\\item $M$ is a finite $S$-module,\n\\item $M_f$ is a finitely presented $S_f$-module,\n\\item for all $\\mathfrak p \\in R$, $f \\not \\in \\mathfrak p$ with\n$\\mathfrak q = \\mathfrak pS$ the module $M_{\\mathfrak q}$ is free\nof rank $r$ over $S_\\mathfrak q$.\n\\end{enumerate}\nThen there exists a finitely generated ideal $I \\subset R$ with\n$V(f) = V(I)$ such that for all $a \\in I$ with $R' = R[\\frac{I}{a}]$\nthe quotient\n$$\nM' = (M \\otimes_R R')/a\\text{-power torsion}\n$$\nover $S' = S \\otimes_R R'$ satisfies the following: for every prime\n$\\mathfrak p' \\subset R'$ there exists a $g \\in S'$,\n$g \\not \\in \\mathfrak p'S'$ such that $M'_g$ is a free $S'_g$-module\nof rank $r$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0810","source_file":"flat.tex","source_line":8667,"source_end_line":8691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8667-L8691","statement_sha256":"99862ed2acc77602e96df2a10150f9e82d429f1e61ac9028dfebde6caf1d2e6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7799,"rank":7799,"depth":49,"x":2127.012,"y":796.045,"cluster":"scheme-morphisms"},{"id":"stacks:0811","tag":"0811","title":"Blowing up and flatness · Lemma 0811","summary":"Let S be a quasi-compact and quasi-separated scheme. Let X → S be a morphism of schemes. Let F be a quasi-coherent module on X. Let U ⊂ S be a quasi-compact open. Assume • X → S is affine, of finite presentation, flat, geometrically integral fibres, • F is a module of finite type, • F_U is of finite presentation, • F is flat over S at all generic points of fibres lying over points of U. Then there exists a U-admissible blowup S' → S and an open subscheme V ⊂ X_S' such…","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent module on $X$.\nLet $U \\subset S$ be a quasi-compact open. Assume\n\\begin{enumerate}\n\\item $X \\to S$ is affine, of finite presentation, flat,\ngeometrically integral fibres,\n\\item $\\mathcal{F}$ is a module of finite type,\n\\item $\\mathcal{F}_U$ is of finite presentation,\n\\item $\\mathcal{F}$ is flat over $S$ at all generic points of\nfibres lying over points of $U$.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $S' \\to S$\nand an open subscheme $V \\subset X_{S'}$\nsuch that (a) the strict transform $\\mathcal{F}'$ of $\\mathcal{F}$\nrestricts to a finitely locally free $\\mathcal{O}_V$-module and\n(b) $V \\to S'$ is surjective.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0811","source_file":"flat.tex","source_line":8781,"source_end_line":8800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8781-L8800","statement_sha256":"30f24e6ee4d83ecc2f5b2745558714d5427e895de56f7718f69647246dbe2924","origin":"The Stacks Project","memory_eligible":false,"source_rank":7800,"rank":7800,"depth":50,"x":1865.084,"y":649.466,"cluster":"scheme-morphisms"},{"id":"stacks:0812","tag":"0812","title":"Blowing up and flatness · Lemma 0812","summary":"Let A → C be a finite locally free ring map of rank d. Let h ∈ C be an element such that C_h is étale over A. Let J ⊂ C be an ideal. Set I = Fit_0(C/J) where we think of C/J as a finite A-module. Then IC_h = JJ' for some ideal J' ⊂ C_h. If J is finitely generated so are I and J'.","statement_latex":"Let $A \\to C$ be a finite locally free ring map of rank $d$.\nLet $h \\in C$ be an element such that $C_h$ is \\'etale over $A$.\nLet $J \\subset C$ be an ideal. Set $I = \\text{Fit}_0(C/J)$ where we\nthink of $C/J$ as a finite $A$-module. Then $IC_h = JJ'$ for some ideal\n$J' \\subset C_h$. If $J$ is finitely generated so are $I$ and $J'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0812","source_file":"flat.tex","source_line":8848,"source_end_line":8855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8848-L8855","statement_sha256":"512031168e040754cda55fea907ebb1652184aae7196bc44278626215fe93b2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7801,"rank":7801,"depth":42,"x":2176.215,"y":608.912,"cluster":"scheme-morphisms"},{"id":"stacks:0813","tag":"0813","title":"Blowing up and flatness · Lemma 0813","summary":"Let A → B be an étale ring map. Let a ∈ A be a nonzerodivisor. Let J ⊂ B be a finite type ideal with V(J) ⊂ V(aB). For every q ⊂ B there exists a finite type ideal I ⊂ A with V(I) ⊂ V(a) and g ∈ B, g not ∈ q such that IB_g = JJ' for some finite type ideal J' ⊂ B_g.","statement_latex":"Let $A \\to B$ be an \\'etale ring map. Let $a \\in A$ be a nonzerodivisor.\nLet $J \\subset B$ be a finite type ideal with $V(J) \\subset V(aB)$.\nFor every $\\mathfrak q \\subset B$ there exists a finite type ideal\n$I \\subset A$ with $V(I) \\subset V(a)$ and\n$g \\in B$, $g \\not \\in \\mathfrak q$ such that\n$IB_g = JJ'$ for some finite type ideal $J' \\subset B_g$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0813","source_file":"flat.tex","source_line":8875,"source_end_line":8883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8875-L8883","statement_sha256":"1b28681ce5398c67f6ee7646ed7987460d3ea3730689063d6e3d973af0763376","origin":"The Stacks Project","memory_eligible":false,"source_rank":7802,"rank":7802,"depth":43,"x":1979.331,"y":815.435,"cluster":"scheme-morphisms"},{"id":"stacks:0814","tag":"0814","title":"Blowing up and flatness · Lemma 0814","summary":"Let S be a quasi-compact and quasi-separated scheme. Let X → S be a morphism of schemes. Let F be a quasi-coherent module on X. Let U ⊂ S be a quasi-compact open. Assume there exist finitely many commutative diagrams xymatrix & X_i ar[r]_j_i ar[d] & X ar[d] S_i^* ar[r] & S_i ar[r]^e_i & S where • e_i : S_i → S are quasi-compact étale morphisms and S = ⋃ e_i(S_i), • j_i : X_i → X are étale morphisms and X = ⋃ j_i(X_i), • S^*_i → S_i is an e_i^-1(U)-admissible blowup such…","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $X \\to S$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent module on $X$.\nLet $U \\subset S$ be a quasi-compact open.\nAssume there exist finitely many commutative diagrams\n$$\n\\xymatrix{\n& X_i \\ar[r]_{j_i} \\ar[d] & X \\ar[d] \\\\\nS_i^* \\ar[r] & S_i \\ar[r]^{e_i} & S\n}\n$$\nwhere\n\\begin{enumerate}\n\\item $e_i : S_i \\to S$ are quasi-compact \\'etale morphisms and\n$S = \\bigcup e_i(S_i)$,\n\\item $j_i : X_i \\to X$ are \\'etale morphisms and\n$X = \\bigcup j_i(X_i)$,\n\\item $S^*_i \\to S_i$ is an $e_i^{-1}(U)$-admissible blowup\nsuch that the strict transform $\\mathcal{F}_i^*$ of $j_i^*\\mathcal{F}$\nis flat over $S^*_i$.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $S' \\to S$ such that\nthe strict transform of $\\mathcal{F}$ is flat over $S'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0814","source_file":"flat.tex","source_line":8903,"source_end_line":8928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L8903-L8928","statement_sha256":"da548a8d49aecc1885446b0dbc69960ea24a80450b0db2f04ce1e52885018c82","origin":"The Stacks Project","memory_eligible":false,"source_rank":7803,"rank":7803,"depth":44,"x":1958.423,"y":551.334,"cluster":"scheme-morphisms"},{"id":"stacks:0815","tag":"0815","title":"Blowing up and flatness · Theorem 0815","summary":"Let S be a quasi-compact and quasi-separated scheme. Let X be a scheme over S. Let F be a quasi-coherent module on X. Let U ⊂ S be a quasi-compact open. Assume • X is quasi-compact, • X is locally of finite presentation over S, • F is a module of finite type, • F_U is of finite presentation, and • F_U is flat over U. Then there exists a U-admissible blowup S' → S such that the strict transform F' of F is an O_X ×_S S'-module of finite presentation and flat over S'.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $X$ be a scheme over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent module on $X$.\nLet $U \\subset S$ be a quasi-compact open. Assume\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item $X$ is locally of finite presentation over $S$,\n\\item $\\mathcal{F}$ is a module of finite type,\n\\item $\\mathcal{F}_U$ is of finite presentation, and\n\\item $\\mathcal{F}_U$ is flat over $U$.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $S' \\to S$ such that the\nstrict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is an\n$\\mathcal{O}_{X \\times_S S'}$-module of finite presentation and\nflat over $S'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up and flatness","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0815","source_file":"flat.tex","source_line":9016,"source_end_line":9033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9016-L9033","statement_sha256":"aa2506b98d32596e06ec798471c826655d68fae9b457f17f1f4423edc7869b7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7804,"rank":7804,"depth":54,"x":2186.307,"y":734.282,"cluster":"scheme-morphisms"},{"id":"stacks:081R","tag":"081R","title":"Applications · Lemma 081R","summary":"Let S be a quasi-compact and quasi-separated scheme. Let X be a scheme over S. Let U ⊂ S be a quasi-compact open. Assume • X → S is of finite type and quasi-separated, and • X_U → U is flat and locally of finite presentation. Then there exists a U-admissible blowup S' → S such that the strict transform of X is flat and of finite presentation over S'.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $X$ be a scheme over $S$.\nLet $U \\subset S$ be a quasi-compact open.\nAssume\n\\begin{enumerate}\n\\item $X \\to S$ is of finite type and quasi-separated, and\n\\item $X_U \\to U$ is flat and locally of finite presentation.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $S' \\to S$ such that\nthe strict transform of $X$ is flat and of finite presentation\nover $S'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081R","source_file":"flat.tex","source_line":9122,"source_end_line":9135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9122-L9135","statement_sha256":"1b7495b37105588437e850fc129952c5371ff198b7e3be3beb8897b93c78afaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7805,"rank":7805,"depth":55,"x":1871.031,"y":728.684,"cluster":"scheme-morphisms"},{"id":"stacks:0B49","tag":"0B49","title":"Applications · Lemma 0B49","summary":"Let S be a quasi-compact and quasi-separated scheme. Let X be a scheme over S. Let U ⊂ S be a quasi-compact open. Assume • X → S is proper, and • X_U → U is finite locally free. Then there exists a U-admissible blowup S' → S such that the strict transform of X is finite locally free over S'.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $X$ be a scheme over $S$.\nLet $U \\subset S$ be a quasi-compact open.\nAssume\n\\begin{enumerate}\n\\item $X \\to S$ is proper, and\n\\item $X_U \\to U$ is finite locally free.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $S' \\to S$ such that\nthe strict transform of $X$ is finite locally free over $S'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B49","source_file":"flat.tex","source_line":9172,"source_end_line":9184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9172-L9184","statement_sha256":"b3482a9a465725bd6191287cc9888494dc4032394d0630af0396141ce46f7c61","origin":"The Stacks Project","memory_eligible":false,"source_rank":7806,"rank":7806,"depth":57,"x":2108.1,"y":553.852,"cluster":"scheme-morphisms"},{"id":"stacks:081S","tag":"081S","title":"Applications · Lemma 081S","summary":"Let φ : X → S be a separated morphism of finite type with S quasi-compact and quasi-separated. Let U ⊂ S be a quasi-compact open such that φ^-1U → U is an isomorphism. Then there exists a U-admissible blowup S' → S such that the strict transform X' of X is isomorphic to an open subscheme of S'.","statement_latex":"Let $\\varphi : X \\to S$ be a separated morphism of finite type with\n$S$ quasi-compact and quasi-separated. Let $U \\subset S$ be a\nquasi-compact open such that $\\varphi^{-1}U \\to U$ is an isomorphism.\nThen there exists a $U$-admissible blowup $S' \\to S$ such that\nthe strict transform $X'$ of $X$ is isomorphic to an open subscheme\nof $S'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081S","source_file":"flat.tex","source_line":9196,"source_end_line":9204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9196-L9204","statement_sha256":"f7a87761810993cb90bc80dde0823bcd11656f87313f97115850ea89ab957702","origin":"The Stacks Project","memory_eligible":false,"source_rank":7807,"rank":7807,"depth":57,"x":2073.871,"y":817.386,"cluster":"scheme-morphisms"},{"id":"stacks:081T","tag":"081T","title":"Applications · Lemma 081T","summary":"Let φ : X → S be a proper morphism with S quasi-compact and quasi-separated. Let U ⊂ S be a quasi-compact open such that φ^-1U → U is an isomorphism. Then there exists a U-admissible blowup S' → S which dominates X, i.e., such that there exists a factorization S' → X → S of the blowup morphism.","statement_latex":"Let $\\varphi : X \\to S$ be a proper morphism with\n$S$ quasi-compact and quasi-separated. Let $U \\subset S$ be a\nquasi-compact open such that $\\varphi^{-1}U \\to U$ is an isomorphism.\nThen there exists a $U$-admissible blowup $S' \\to S$\nwhich dominates $X$, i.e., such that there exists a factorization\n$S' \\to X \\to S$ of the blowup morphism.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081T","source_file":"flat.tex","source_line":9221,"source_end_line":9229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9221-L9229","statement_sha256":"dc2d2a6af8d25519e4004d0304d1a45236524c404a0f2d41d9fe7a51ba75e893","origin":"The Stacks Project","memory_eligible":false,"source_rank":7808,"rank":7808,"depth":58,"x":1887.116,"y":603.56,"cluster":"scheme-morphisms"},{"id":"stacks:0CP1","tag":"0CP1","title":"Applications · Lemma 0CP1","summary":"Let S be a scheme. Let U ⊂ W ⊂ S be open subschemes. Let f : X → W be a morphism and let s : U → X be a morphism such that f ∘ s = id_U. Assume • f is proper, • S is quasi-compact and quasi-separated, and • U and W are quasi-compact. Then there exists a U-admissible blowup b : S' → S and a morphism s' : b^-1(W) → X extending s with f ∘ s' = b|_b^-1(W).","statement_latex":"Let $S$ be a scheme. Let $U \\subset W \\subset S$ be open subschemes.\nLet $f : X \\to W$ be a morphism and let $s : U \\to X$ be a\nmorphism such that $f \\circ s = \\text{id}_U$. Assume\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item $S$ is quasi-compact and quasi-separated, and\n\\item $U$ and $W$ are quasi-compact.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $b : S' \\to S$ and a morphism\n$s' : b^{-1}(W) \\to X$ extending $s$ with $f \\circ s' = b|_{b^{-1}(W)}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CP1","source_file":"flat.tex","source_line":9243,"source_end_line":9255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9243-L9255","statement_sha256":"d6c298796f9e79e2f6920435665446c433895c74f2e793b0a23f32963ded410b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7809,"rank":7809,"depth":59,"x":2196.893,"y":655.281,"cluster":"scheme-morphisms"},{"id":"stacks:0ATU","tag":"0ATU","title":"Compactifications · Lemma 0ATU","summary":"Let S be a quasi-compact and quasi-separated scheme. Let X be a compactifyable scheme over S. • [(a)] The category of compactifications of X over S is cofiltered. • [(b)] The full subcategory consisting of compactifications j : X → overlineX such that j(X) is dense and scheme theoretically dense in overlineX is initial (Categories, Definition [Tag 09WP]). • [(c)] If f : overlineX' → overlineX is a morphism of compactifications of X such that j'(X) is dense in overlineX',…","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $X$ be a compactifyable scheme over $S$.\n\\begin{enumerate}\n\\item[(a)] The category of compactifications of $X$ over $S$ is\ncofiltered.\n\\item[(b)] The full subcategory consisting of compactifications\n$j : X \\to \\overline{X}$ such that $j(X)$ is dense and\nscheme theoretically dense in $\\overline{X}$ is initial\n(Categories, Definition \\ref{categories-definition-initial}).\n\\item[(c)] If $f : \\overline{X}' \\to \\overline{X}$ is a morphism\nof compactifications of $X$ such that $j'(X)$ is dense in $\\overline{X}'$,\nthen $f^{-1}(j(X)) = j'(X)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATU","source_file":"flat.tex","source_line":9315,"source_end_line":9330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9315-L9330","statement_sha256":"e9b17d41d614308daf2a28261d646d73025cf236c136e9b05103dc64b586df73","origin":"The Stacks Project","memory_eligible":false,"source_rank":7810,"rank":7810,"depth":20,"x":1926.776,"y":792.968,"cluster":"scheme-morphisms"},{"id":"stacks:0A9Z","tag":"0A9Z","title":"Compactifications · Lemma 0A9Z","summary":"Let S be a quasi-compact and quasi-separated scheme. Let f : X → Y be a morphism of schemes over S with Y separated and of finite type over S and X compactifyable over S. Then X has a compactification over Y.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to Y$\nbe a morphism of schemes over $S$ with $Y$ separated and of finite type\nover $S$ and $X$ compactifyable over $S$. Then $X$ has a compactification\nover $Y$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9Z","source_file":"flat.tex","source_line":9386,"source_end_line":9392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9386-L9392","statement_sha256":"ceb8e286da34a115269d2a632a21d1fde917205ae83c6856ebef8a63c41eb97e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7811,"rank":7811,"depth":20,"x":2015.265,"y":538.075,"cluster":"scheme-morphisms"},{"id":"stacks:0ATV","tag":"0ATV","title":"Compactifications · Lemma 0ATV","summary":"Let S be a quasi-compact and quasi-separated scheme. The collection of morphisms (u, overlineu) : (X', overlineX') → (X, overlineX) such that u is an isomorphism forms a right multiplicative system (Categories, Definition [Tag 04VC]) of arrows in the category of compactifications.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nThe collection of morphisms\n$(u, \\overline{u}) : (X', \\overline{X}') \\to (X, \\overline{X})$\nsuch that $u$ is an isomorphism forms a right multiplicative system\n(Categories, Definition \\ref{categories-definition-multiplicative-system})\nof arrows in the category of compactifications.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATV","source_file":"flat.tex","source_line":9424,"source_end_line":9432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9424-L9432","statement_sha256":"b4408a689f7645513eeea5f7db8acf6e392ab9fb9b2621237f1e113b08e6d65e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7812,"rank":7812,"depth":1,"x":2155.043,"y":776.327,"cluster":"scheme-morphisms"},{"id":"stacks:0ATW","tag":"0ATW","title":"Compactifications · Lemma 0ATW","summary":"Let S be a quasi-compact and quasi-separated scheme. The functor (X, overlineX) ↦ X defines an equivalence from the category of compactifications localized (Categories, Lemma [Tag 04VH]) at the right multiplicative system of Lemma [Tag 0ATV] to the category of compactifyable schemes over S.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nThe functor $(X, \\overline{X}) \\mapsto X$ defines an\nequivalence from the category of compactifications localized\n(Categories, Lemma \\ref{categories-lemma-right-localization})\nat the right\nmultiplicative system of Lemma \\ref{lemma-right-multiplicative-system}\nto the category of compactifyable schemes over $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATW","source_file":"flat.tex","source_line":9475,"source_end_line":9484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9475-L9484","statement_sha256":"a13d1fca303fbba58e8f5696880b4d26c0f22434e45f05e543a215d253a4adce","origin":"The Stacks Project","memory_eligible":false,"source_rank":7813,"rank":7813,"depth":21,"x":1860.269,"y":679.922,"cluster":"scheme-morphisms"},{"id":"stacks:0F3U","tag":"0F3U","title":"Nagata compactification · Lemma 0F3U","summary":"Let X → S be a morphism of schemes. If X = U ∪ V is an open cover such that U → S and V → S are separated and U ∩ V → U ×_S V is closed, then X → S is separated.","statement_latex":"Let $X \\to S$ be a morphism of schemes. If $X = U \\cup V$\nis an open cover such that $U \\to S$ and $V \\to S$ are separated\nand $U \\cap V \\to U \\times_S V$ is closed, then\n$X \\to S$ is separated.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Nagata compactification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3U","source_file":"flat.tex","source_line":9539,"source_end_line":9545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9539-L9545","statement_sha256":"2351c8ac3f8346c7e4903182c038aa06dad125aedfe871a96d2b3f3042c3b871","origin":"The Stacks Project","memory_eligible":false,"source_rank":7814,"rank":7814,"depth":0,"x":2155.265,"y":583.713,"cluster":"scheme-morphisms"},{"id":"stacks:0F3V","tag":"0F3V","title":"Nagata compactification · Lemma 0F3V","summary":"Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be a quasi-compact open. • If Z_1, Z_2 ⊂ X are closed subschemes of finite presentation such that Z_1 ∩ Z_2 ∩ U = ∅, then there exists a U-admissible blowing up X' → X such that the strict transforms of Z_1 and Z_2 are disjoint. • If T_1, T_2 ⊂ U are disjoint constructible closed subsets, then there is a U-admissible blowing up X' → X such that the closures of T_1 and T_2 are disjoint.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $U \\subset X$ be a quasi-compact open.\n\\begin{enumerate}\n\\item If $Z_1, Z_2 \\subset X$ are closed subschemes of finite\npresentation such that $Z_1 \\cap Z_2 \\cap U = \\emptyset$, then\nthere exists a $U$-admissible blowing up $X' \\to X$\nsuch that the strict transforms of $Z_1$ and $Z_2$ are disjoint.\n\\item If $T_1, T_2 \\subset U$ are disjoint constructible closed subsets, then\nthere is a $U$-admissible blowing up $X' \\to X$ such that the closures of\n$T_1$ and $T_2$ are disjoint.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Nagata compactification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3V","source_file":"flat.tex","source_line":9553,"source_end_line":9566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9553-L9566","statement_sha256":"d6ba5bf62ffecdef983e25c959121d1ff2553ba071086bc192e53d86720de7d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7815,"rank":7815,"depth":22,"x":2015.058,"y":822.131,"cluster":"scheme-morphisms"},{"id":"stacks:0F3W","tag":"0F3W","title":"Nagata compactification · Lemma 0F3W","summary":"Let f : X → Y be a proper morphism of quasi-compact and quasi-separated schemes. Let V ⊂ Y be a quasi-compact open and U = f^-1(V). Let T ⊂ V be a closed subset such that f|_U : U → V is an isomorphism over an open neighbourhood of T in V. Then there exists a V-admissible blowing up Y' → Y such that the strict transform f' : X' → Y' of f is an isomorphism over an open neighbourhood of the closure of T in Y'.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of quasi-compact and\nquasi-separated schemes. Let $V \\subset Y$ be a quasi-compact open\nand $U = f^{-1}(V)$. Let $T \\subset V$ be a closed subset such that\n$f|_U : U \\to V$ is an isomorphism over an open neighbourhood of $T$\nin $V$. Then there exists a $V$-admissible blowing up $Y' \\to Y$\nsuch that the strict transform $f' : X' \\to Y'$ of $f$\nis an isomorphism over an open neighbourhood of the closure\nof $T$ in $Y'$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Nagata compactification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3W","source_file":"flat.tex","source_line":9607,"source_end_line":9617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9607-L9617","statement_sha256":"a29803460a8bdc3c3e479b738e5e44869de70cedc8345875404f6820f72af5ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":7816,"rank":7816,"depth":58,"x":1926.683,"y":566.675,"cluster":"scheme-morphisms"},{"id":"stacks:0F3X","tag":"0F3X","title":"Nagata compactification · Lemma 0F3X","summary":"Let S be a quasi-compact and quasi-separated scheme. Let U → X_1 and U → X_2 be open immersions of schemes over S and assume U, X_1, X_2 of finite type and separated over S. Then there exists a commutative diagram xymatrix X_1' ar[d] ar[r] & X & X_2' ar[l] ar[d] X_1 & U ar[l] ar[lu] ar[u] ar[ru] ar[r] & X_2 of schemes over S where X_i' → X_i is a U-admissible blowup, X_i' → X is an open immersion, and X is separated and finite type over S.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $U \\to X_1$ and $U \\to X_2$ be open immersions\nof schemes over $S$ and assume $U$, $X_1$, $X_2$ of finite\ntype and separated over $S$. Then there exists a commutative diagram\n$$\n\\xymatrix{\nX_1' \\ar[d] \\ar[r] & X & X_2' \\ar[l] \\ar[d] \\\\\nX_1 & U \\ar[l] \\ar[lu] \\ar[u] \\ar[ru] \\ar[r] & X_2\n}\n$$\nof schemes over $S$ where $X_i' \\to X_i$ is a $U$-admissible\nblowup, $X_i' \\to X$ is an open immersion, and $X$ is separated and finite\ntype over $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Nagata compactification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3X","source_file":"flat.tex","source_line":9646,"source_end_line":9661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9646-L9661","statement_sha256":"db2198f6b9465445bbbf5d7b73266808fad786ce61487b482bcc193a57a2c2eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7817,"rank":7817,"depth":58,"x":2197.378,"y":704.949,"cluster":"scheme-morphisms"},{"id":"stacks:0F3Y","tag":"0F3Y","title":"Nagata compactification · Lemma 0F3Y","summary":"Let X → S and Y → S be morphisms of schemes. Let U ⊂ X be an open subscheme. Let V → X ×_S Y be a quasi-compact morphism whose composition with the first projection maps into U. Let Z ⊂ X ×_S Y be the scheme theoretic image of V → X ×_S Y. Let X' → X be a U-admissible blowup. Then the scheme theoretic image of V → X' ×_S Y is the strict transform of Z with respect to the blowing up.","statement_latex":"Let $X \\to S$ and $Y \\to S$ be morphisms of schemes.\nLet $U \\subset X$ be an open subscheme.\nLet $V \\to X \\times_S Y$ be a quasi-compact morphism\nwhose composition with the first projection maps into $U$.\nLet $Z \\subset X \\times_S Y$ be the scheme theoretic image of\n$V \\to X \\times_S Y$. Let $X' \\to X$ be a $U$-admissible blowup.\nThen the scheme theoretic image of $V \\to X' \\times_S Y$ is the\nstrict transform of $Z$ with respect to the blowing up.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Nagata compactification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3Y","source_file":"flat.tex","source_line":9727,"source_end_line":9737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9727-L9737","statement_sha256":"1c54cc353b3b08edbea578940e706c375bb096dd2d75f7281c78189bb2ab4cd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7818,"rank":7818,"depth":19,"x":1886.463,"y":756.606,"cluster":"scheme-morphisms"},{"id":"stacks:0F3Z","tag":"0F3Z","title":"Nagata compactification · Lemma 0F3Z","summary":"Let S be a quasi-compact and quasi-separated scheme. Let U be a scheme of finite type and separated over S. Let V ⊂ U be a quasi-compact open. If V has a compactification V ⊂ Y over S, then there exists a V-admissible blowing up Y' → Y and an open V ⊂ V' ⊂ Y' such that V → U extends to a proper morphism V' → U.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme. Let $U$ be a\nscheme of finite type and separated over $S$. Let $V \\subset U$ be a\nquasi-compact open. If $V$ has a compactification $V \\subset Y$\nover $S$, then there exists a $V$-admissible blowing up $Y' \\to Y$ and an\nopen $V \\subset V' \\subset Y'$ such that $V \\to U$\nextends to a proper morphism $V' \\to U$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Nagata compactification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F3Z","source_file":"flat.tex","source_line":9761,"source_end_line":9769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9761-L9769","statement_sha256":"4f9ca65f78fc0b2a347f80ae1de41036f2914e01ac80d53b357b1c96eced77a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7819,"rank":7819,"depth":58,"x":2074.254,"y":542.015,"cluster":"scheme-morphisms"},{"id":"stacks:0F40","tag":"0F40","title":"Nagata compactification · Lemma 0F40","summary":"Let S be a Noetherian scheme. Let U be a scheme of finite type and separated over S. Let U = U_1 ∪ U_2 be opens such that U_1 and U_2 have compactifications over S and such that U_1 ∩ U_2 is dense in U. Then U has a compactification over S.","statement_latex":"Let $S$ be a Noetherian scheme. Let $U$ be a scheme of finite type\nand separated over $S$. Let $U = U_1 \\cup U_2$ be opens such that\n$U_1$ and $U_2$ have compactifications over $S$ and such that\n$U_1 \\cap U_2$ is dense in $U$. Then $U$ has a compactification over $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Nagata compactification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F40","source_file":"flat.tex","source_line":9790,"source_end_line":9796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9790-L9796","statement_sha256":"ed5ea4be4dd60d7905b68db11284799eb3d155ad78a501c3ebbf87ad26a62e4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7820,"rank":7820,"depth":59,"x":2108.359,"y":806.905,"cluster":"scheme-morphisms"},{"id":"stacks:0F41","tag":"0F41","title":"Nagata compactification · Theorem 0F41","summary":"See [Lutkebohmert], [Conrad-Nagata], [Nagata-1], [Nagata-2], [Nagata-3], and [Nagata-4] Let S be a quasi-compact and quasi-separated scheme. Let X → S be a separated, finite type morphism. Then X has a compactification over S.","statement_latex":"\\begin{reference}\nSee \\cite{Lutkebohmert}, \\cite{Conrad-Nagata}, \\cite{Nagata-1},\n\\cite{Nagata-2}, \\cite{Nagata-3}, and \\cite{Nagata-4}\n\\end{reference}\nLet $S$ be a quasi-compact and quasi-separated scheme. Let\n$X \\to S$ be a separated, finite type morphism.\nThen $X$ has a compactification over $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Nagata compactification","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F41","source_file":"flat.tex","source_line":9949,"source_end_line":9958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L9949-L9958","statement_sha256":"a3490135074a9f2b6445be6dac20439066c415e807fb2fe87d8f184bf9e9c409","origin":"The Stacks Project","memory_eligible":false,"source_rank":7821,"rank":7821,"depth":60,"x":1870.108,"y":630.868,"cluster":"scheme-morphisms"},{"id":"stacks:0ETR","tag":"0ETR","title":"The h topology · Lemma 0ETR","summary":"Let (f_i : X_i → X)_i ∈ I be a family of morphisms of schemes with fixed target with f_i locally of finite presentation for all i. The following are equivalent • (X_i → X) is a ph covering, and • (X_i → X) is a V covering.","statement_latex":"Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family of morphisms\nof schemes with fixed target with $f_i$ locally of finite\npresentation for all $i$. The following are equivalent\n\\begin{enumerate}\n\\item $\\{X_i \\to X\\}$ is a ph covering, and\n\\item $\\{X_i \\to X\\}$ is a V covering.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETR","source_file":"flat.tex","source_line":10048,"source_end_line":10057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10048-L10057","statement_sha256":"6e437530ec3182ae77fbd81358ceaccd2f29fe4d2abaf7153cd49b432ae727d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7822,"rank":7822,"depth":55,"x":2187.47,"y":625.483,"cluster":"scheme-morphisms"},{"id":"stacks:0ETS","tag":"0ETS","title":"The h topology · Definition 0ETS","summary":"Let T be a scheme. A h covering of T is a family of morphisms (f_i : T_i → T)_i ∈ I such that each f_i is locally of finite presentation and one of the equivalent conditions of Lemma [Tag 0ETR] is satisfied.","statement_latex":"Let $T$ be a scheme. A {\\it h covering of $T$} is a family of morphisms\n$\\{f_i : T_i \\to T\\}_{i \\in I}$ such that each $f_i$ is\nlocally of finite presentation and one of the equivalent conditions of\nLemma \\ref{lemma-equivalence-h-v-locally-finite-presentation} is satisfied.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETS","source_file":"flat.tex","source_line":10170,"source_end_line":10176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10170-L10176","statement_sha256":"4ddeda393031f8c342285e2d113519cbbda4cdbe4157e42715f71d4bcc2e6abb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7823,"rank":7823,"depth":56,"x":1957.699,"y":809.599,"cluster":"scheme-morphisms"},{"id":"stacks:0ETT","tag":"0ETT","title":"The h topology · Lemma 0ETT","summary":"Let X be a Noetherian scheme. Let (X_i → X)_i ∈ I be a finite family of finite type morphisms. The following are equivalent • coprod_i ∈ I X_i → X is universally submersive (Morphisms, Definition [Tag 040H]), and • (X_i → X)_i ∈ I is an h covering.","statement_latex":"Let $X$ be a Noetherian scheme. Let $\\{X_i \\to X\\}_{i \\in I}$\nbe a finite family of finite type morphisms. The following are equivalent\n\\begin{enumerate}\n\\item $\\coprod_{i \\in I} X_i \\to X$ is universally submersive\n(Morphisms, Definition\n\\ref{morphisms-definition-submersive}), and\n\\item $\\{X_i \\to X\\}_{i \\in I}$ is an h covering.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETT","source_file":"flat.tex","source_line":10183,"source_end_line":10193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10183-L10193","statement_sha256":"48bd1cb8831c9c4e6de2b51517e3703873509352b655d4a31b50ad7794d84ff4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7824,"rank":7824,"depth":56,"x":1979.075,"y":543.36,"cluster":"scheme-morphisms"},{"id":"stacks:0H45","tag":"0H45","title":"The h topology · Lemma 0H45","summary":"Let X be a locally Noetherian scheme. A family of morphisms (f_i : X_i → X)_i ∈ I with target X is an h covering if and only if it is a ph covering.","statement_latex":"Let $X$ be a locally Noetherian scheme. A family of morphisms\n$\\{f_i : X_i \\to X\\}_{i \\in I}$ with target $X$ is an h covering\nif and only if it is a ph covering.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H45","source_file":"flat.tex","source_line":10280,"source_end_line":10285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10280-L10285","statement_sha256":"49f0996c4bcf47a1f04b3956391f0a187f6b020c18dfa43a506148d69a61b7dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7825,"rank":7825,"depth":57,"x":2177.486,"y":751.886,"cluster":"scheme-morphisms"},{"id":"stacks:0ETU","tag":"0ETU","title":"The h topology · Lemma 0ETU","summary":"Let X be an affine scheme. Let (X_i → X)_i ∈ I be an h covering. Then there exists a surjective proper morphism Y → X of finite presentation (!) and a finite affine open covering Y = ⋃_j = 1, …, m Y_j such that (Y_j → X)_j = 1, …, m refines (X_i → X)_i ∈ I.","statement_latex":"Let $X$ be an affine scheme. Let $\\{X_i \\to X\\}_{i \\in I}$\nbe an h covering. Then there exists a surjective proper morphism\n$$\nY \\longrightarrow X\n$$\nof finite presentation (!) and a finite affine open covering\n$Y = \\bigcup_{j = 1, \\ldots, m} Y_j$ such that\n$\\{Y_j \\to X\\}_{j = 1, \\ldots, m}$ refines $\\{X_i \\to X\\}_{i \\in I}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETU","source_file":"flat.tex","source_line":10302,"source_end_line":10312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10302-L10312","statement_sha256":"b0a63b31b91a267e8293335d57c1898cfa7aa7bc28570617b39f4ab33b6e3bd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":7826,"rank":7826,"depth":57,"x":1863.377,"y":710.691,"cluster":"scheme-morphisms"},{"id":"stacks:0ETV","tag":"0ETV","title":"The h topology · Lemma 0ETV","summary":"An fppf covering is a h covering. Hence syntomic, smooth, étale, and Zariski coverings are h coverings as well.","statement_latex":"An fppf covering is a h covering. Hence syntomic, smooth, \\'etale,\nand Zariski coverings are h coverings as well.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETV","source_file":"flat.tex","source_line":10358,"source_end_line":10362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10358-L10362","statement_sha256":"67d2ef0116f0035bac13663c8d6ece70f6661b1e53d5855af4164beb44e58b18","origin":"The Stacks Project","memory_eligible":false,"source_rank":7827,"rank":7827,"depth":45,"x":2128.22,"y":562.781,"cluster":"scheme-morphisms"},{"id":"stacks:0ETW","tag":"0ETW","title":"The h topology · Lemma 0ETW","summary":"Let f : Y → X be a surjective proper morphism of schemes which is of finite presentation. Then (Y → X) is an h covering.","statement_latex":"Let $f : Y \\to X$ be a surjective proper morphism of schemes\nwhich is of finite presentation. Then $\\{Y \\to X\\}$ is an h covering.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETW","source_file":"flat.tex","source_line":10373,"source_end_line":10377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10373-L10377","statement_sha256":"5357d51e079470320ff96214135c4a806632a888fd327a82ed3d2b9c5c16f192","origin":"The Stacks Project","memory_eligible":false,"source_rank":7828,"rank":7828,"depth":40,"x":2051.846,"y":822.22,"cluster":"scheme-morphisms"},{"id":"stacks:0ETX","tag":"0ETX","title":"The h topology · Lemma 0ETX","summary":"Let T be a scheme. Let (f_i : T_i → T)_i ∈ I be a family of morphisms such that f_i is locally of finite presentation for all i. The following are equivalent • (T_i → T)_i ∈ I is an h covering, • there is an h covering which refines (T_i → T)_i ∈ I, and • (coprod_i ∈ I T_i → T) is an h covering.","statement_latex":"Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family\nof morphisms such that $f_i$ is locally of finite presentation for all $i$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\{T_i \\to T\\}_{i \\in I}$ is an h covering,\n\\item there is an h covering which refines $\\{T_i \\to T\\}_{i \\in I}$, and\n\\item $\\{\\coprod_{i \\in I} T_i \\to T\\}$ is an h covering.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETX","source_file":"flat.tex","source_line":10385,"source_end_line":10395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10385-L10395","statement_sha256":"e1d9cab473927de82cd90556d7da35321a9f5a51dc08dc1f14e047493fc881ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":7829,"rank":7829,"depth":3,"x":1899.475,"y":587.492,"cluster":"scheme-morphisms"},{"id":"stacks:0ETY","tag":"0ETY","title":"The h topology · Lemma 0ETY","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is an h covering of T. • If (T_i → T)_i∈ I is an h covering and for each i we have an h covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is an h covering. • If (T_i → T)_i∈ I is an h covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is an h covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis an h covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an h covering and for each\n$i$ we have an h covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is an h covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is an h covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is an h covering.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETY","source_file":"flat.tex","source_line":10408,"source_end_line":10421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10408-L10421","statement_sha256":"bbfbfb800b336c2bdbf9df35d016b48a820b961257d0ca731b58d89f2b4230c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7830,"rank":7830,"depth":32,"x":2200.701,"y":674.149,"cluster":"scheme-morphisms"},{"id":"stacks:0ETZ","tag":"0ETZ","title":"The h topology · Definition 0ETZ","summary":"A big h site is any site Sch_h as in Sites, Definition [Tag 00VH] constructed as follows: • Choose any set of schemes S_0, and any set of h coverings Cov_0 among these schemes. • As underlying category take any category Sch_α constructed as in Sets, Lemma [Tag 000J] starting with the set S_0. • Choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Sch_α and the class of h coverings, and the set Cov_0 chosen above.","statement_latex":"A {\\it big h site} is any site $\\Sch_h$ as in\nSites, Definition \\ref{sites-definition-site} constructed as follows:\n\\begin{enumerate}\n\\item Choose any set of schemes $S_0$, and any set of h coverings\n$\\text{Cov}_0$ among these schemes.\n\\item As underlying category take any category $\\Sch_\\alpha$\nconstructed as in Sets, Lemma \\ref{sets-lemma-construct-category}\nstarting with the set $S_0$.\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\Sch_\\alpha$ and the class of h coverings,\nand the set $\\text{Cov}_0$ chosen above.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ETZ","source_file":"flat.tex","source_line":10439,"source_end_line":10454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10439-L10454","statement_sha256":"ecb8b227538b983f7921e1a986fb111b17e466eb98621760f17f903598a818e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7831,"rank":7831,"depth":2,"x":1908.79,"y":781.211,"cluster":"scheme-morphisms"},{"id":"stacks:0EUY","tag":"0EUY","title":"The h topology · Definition 0EUY","summary":"Let T be an affine scheme. A standard h covering of T is a family (f_i : T_i → T)_i = 1, …, n with each T_i affine, with f_i of finite presentation satisfying either of the following equivalent conditions: (1) (T_i → T) can be refined by a standard ph covering or (2) (T_i → T) is a V covering.","statement_latex":"Let $T$ be an affine scheme. A {\\it standard h covering} of $T$\nis a family $\\{f_i : T_i \\to T\\}_{i = 1, \\ldots, n}$ with each $T_i$\naffine, with $f_i$ of finite presentation satisfying either of the\nfollowing equivalent conditions: (1) $\\{T_i \\to T\\}$ can be refined by\na standard ph covering or (2) $\\{T_i \\to T\\}$ is a V covering.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUY","source_file":"flat.tex","source_line":10461,"source_end_line":10468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10461-L10468","statement_sha256":"ed80bb25f82ffb5c4f60a46732a93bd9a638e54eecc86f51274977982e010e89","origin":"The Stacks Project","memory_eligible":false,"source_rank":7832,"rank":7832,"depth":0,"x":2037.99,"y":536.539,"cluster":"scheme-morphisms"},{"id":"stacks:0EU0","tag":"0EU0","title":"The h topology · Lemma 0EU0","summary":"Let Sch_h be a big h site as in Definition [Tag 0ETZ]. Let T ∈ Ob(Sch_h). Let (T_i → T)_i ∈ I be an arbitrary h covering of T. • There exists a covering (U_j → T)_j ∈ J of T in the site Sch_h which refines (T_i → T)_i ∈ I. • If (T_i → T)_i ∈ I is a standard h covering, then it is tautologically equivalent to a covering of Sch_h. • If (T_i → T)_i ∈ I is a Zariski covering, then it is tautologically equivalent to a covering of Sch_h.","statement_latex":"Let $\\Sch_h$ be a big h site as in\nDefinition \\ref{definition-big-h-site}.\nLet $T \\in \\Ob(\\Sch_h)$.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary h covering of $T$.\n\\begin{enumerate}\n\\item There exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site\n$\\Sch_h$ which refines $\\{T_i \\to T\\}_{i \\in I}$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a standard h covering, then\nit is tautologically equivalent to a covering of $\\Sch_h$.\n\\item If $\\{T_i \\to T\\}_{i \\in I}$ is a Zariski covering, then\nit is tautologically equivalent to a covering of $\\Sch_h$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EU0","source_file":"flat.tex","source_line":10482,"source_end_line":10496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10482-L10496","statement_sha256":"9e1b54203a8a13f380bd23ddb5f773f4806d472c378c8b2caef17a12769f78b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7833,"rank":7833,"depth":33,"x":2139.515,"y":790.359,"cluster":"scheme-morphisms"},{"id":"stacks:0EU1","tag":"0EU1","title":"The h topology · Definition 0EU1","summary":"Let S be a scheme. Let Sch_h be a big h site containing S. • The big h site of S, denoted (Sch/S)_h, is the site Sch_h/S introduced in Sites, Section [Tag 00XZ]. • The big affine h site of S, denoted (Aff/S)_h, is the full subcategory of (Sch/S)_h whose objects are affine U/S. A covering of (Aff/S)_h is any covering (U_i → U) of (Sch/S)_h which is a standard h covering.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_h$ be a big h site containing $S$.\n\\begin{enumerate}\n\\item The {\\it big h site of $S$}, denoted\n$(\\Sch/S)_h$, is the site $\\Sch_h/S$\nintroduced in Sites, Section \\ref{sites-section-localize}.\n\\item The {\\it big affine h site of $S$}, denoted\n$(\\textit{Aff}/S)_h$, is the full subcategory of\n$(\\Sch/S)_h$ whose objects are affine $U/S$.\nA covering of $(\\textit{Aff}/S)_h$ is any covering\n$\\{U_i \\to U\\}$ of $(\\Sch/S)_h$ which is a standard h covering.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EU1","source_file":"flat.tex","source_line":10503,"source_end_line":10516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10503-L10516","statement_sha256":"8d93e58cabe8e855709cd676ba72b6720e44f924b18fffc68d655f6aee906109","origin":"The Stacks Project","memory_eligible":false,"source_rank":7834,"rank":7834,"depth":0,"x":1860.437,"y":660.757,"cluster":"scheme-morphisms"},{"id":"stacks:0EU2","tag":"0EU2","title":"The h topology · Lemma 0EU2","summary":"Let S be a scheme. Let Sch_h be a big h site containing S. Then (Aff/S)_h is a site.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_h$ be a big h\nsite containing $S$. Then $(\\textit{Aff}/S)_h$ is a site.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EU2","source_file":"flat.tex","source_line":10521,"source_end_line":10525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10521-L10525","statement_sha256":"8370b7abd46d79c8190debb82e8cd41e1a1cd1bddd908fda019ac1c7ae0c48b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7835,"rank":7835,"depth":33,"x":2170.559,"y":597.946,"cluster":"scheme-morphisms"},{"id":"stacks:0EU3","tag":"0EU3","title":"The h topology · Lemma 0EU3","summary":"Let S be a scheme. Let Sch_h be a big h site containing S. The underlying categories of the sites Sch_h, (Sch/S)_h, and (Aff/S)_h have fibre products. In each case the obvious functor into the category Sch of all schemes commutes with taking fibre products. The category (Sch/S)_h has a final object, namely S/S.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_h$ be a big h\nsite containing $S$. The underlying categories of the sites\n$\\Sch_h$, $(\\Sch/S)_h$, and $(\\textit{Aff}/S)_h$ have fibre products.\nIn each case the obvious functor into the category $\\Sch$ of\nall schemes commutes with taking fibre products. The category\n$(\\Sch/S)_h$ has a final object, namely $S/S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EU3","source_file":"flat.tex","source_line":10542,"source_end_line":10550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10542-L10550","statement_sha256":"7e22ee9270beb2ce5da46a87c4ed58d3612bd82b92ac2d00be8cf43d1076134f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7836,"rank":7836,"depth":3,"x":1992.326,"y":820.312,"cluster":"scheme-morphisms"},{"id":"stacks:0EU4","tag":"0EU4","title":"The h topology · Lemma 0EU4","summary":"Let S be a scheme. Let Sch_h be a big h site containing S. The functor (Aff/S)_h → (Sch/S)_h is cocontinuous and induces an equivalence of topoi from Sh((Aff/S)_h) to Sh((Sch/S)_h).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_h$ be a big h\nsite containing $S$.\nThe functor $(\\textit{Aff}/S)_h \\to (\\Sch/S)_h$\nis cocontinuous and induces an equivalence of topoi from\n$\\Sh((\\textit{Aff}/S)_h)$ to\n$\\Sh((\\Sch/S)_h)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EU4","source_file":"flat.tex","source_line":10571,"source_end_line":10579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10571-L10579","statement_sha256":"b574d747cae7e6f1411d7c994ebc2e1b5bfc6d587faccea99b1f60143c02b596","origin":"The Stacks Project","memory_eligible":false,"source_rank":7837,"rank":7837,"depth":58,"x":1944.914,"y":555.114,"cluster":"scheme-morphisms"},{"id":"stacks:0EU5","tag":"0EU5","title":"The h topology · Lemma 0EU5","summary":"Let F be a presheaf on (Sch/S)_h. Then F is a sheaf if and only if • F satisfies the sheaf condition for Zariski coverings, and • if f : V → U is proper, surjective, and of finite presentation, then F(U) maps bijectively to the equalizer of the two maps F(V) → F(V ×_U V). Moreover, in the presence of (1) property (2) is equivalent to property • [(2')] the sheaf property for (V → U) as in (2) with U affine.","statement_latex":"Let $\\mathcal{F}$ be a presheaf on $(\\Sch/S)_h$.\nThen $\\mathcal{F}$ is a sheaf if and only if\n\\begin{enumerate}\n\\item $\\mathcal{F}$ satisfies the sheaf condition for\nZariski coverings, and\n\\item if $f : V \\to U$ is proper, surjective, and of finite presentation, then\n$\\mathcal{F}(U)$ maps bijectively to the equalizer\nof the two maps $\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$.\n\\end{enumerate}\nMoreover, in the presence of (1) property (2) is equivalent to\nproperty\n\\begin{enumerate}\n\\item[(2')] the sheaf property for $\\{V \\to U\\}$ as in (2) with $U$ affine.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EU5","source_file":"flat.tex","source_line":10599,"source_end_line":10615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10599-L10615","statement_sha256":"bd8f03ae1410af4de393ad4364698fd07b1c2d4af56d2fb13a19c82492930077","origin":"The Stacks Project","memory_eligible":false,"source_rank":7838,"rank":7838,"depth":58,"x":2193.229,"y":723.825,"cluster":"scheme-morphisms"},{"id":"stacks:0EU6","tag":"0EU6","title":"The h topology · Lemma 0EU6","summary":"Let Sch_h be a big h site. Let f : T → S be a morphism in Sch_h. The functor u : (Sch/T)_h → (Sch/S)_h, V/T ↦ V/S is cocontinuous, and has a continuous right adjoint v : (Sch/S)_h → (Sch/T)_h, (U → S) ↦ (U ×_S T → T). They induce the same morphism of topoi f_big : Sh((Sch/T)_h) → Sh((Sch/S)_h) We have f_big^-1(G)(U/T) = G(U/S). We have f_big, *(F)(U/S) = F(U ×_S T/T). Also, f_big^-1 has a left adjoint f_big! which commutes with fibre products and equalizers.","statement_latex":"Let $\\Sch_h$ be a big h site.\nLet $f : T \\to S$ be a morphism in $\\Sch_h$.\nThe functor\n$$\nu : (\\Sch/T)_h \\longrightarrow (\\Sch/S)_h,\n\\quad\nV/T \\longmapsto V/S\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv : (\\Sch/S)_h \\longrightarrow (\\Sch/T)_h,\n\\quad\n(U \\to S) \\longmapsto (U \\times_S T \\to T).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\Sch/T)_h)\n\\longrightarrow\n\\Sh((\\Sch/S)_h)\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nWe have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EU6","source_file":"flat.tex","source_line":10670,"source_end_line":10697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10670-L10697","statement_sha256":"213efb02d386c759882cde2c2ced72a5bb0733a23df26ea9725d695b4aa84c8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7839,"rank":7839,"depth":8,"x":1874.339,"y":740.326,"cluster":"scheme-morphisms"},{"id":"stacks:0EU7","tag":"0EU7","title":"The h topology · Lemma 0EU7","summary":"Given schemes X, Y, Y in (Sch/S)_h and morphisms f : X → Y, g : Y → Z we have g_big ∘ f_big = (g ∘ f)_big.","statement_latex":"Given schemes $X$, $Y$, $Y$ in $(\\Sch/S)_h$\nand morphisms $f : X \\to Y$, $g : Y \\to Z$ we have\n$g_{big} \\circ f_{big} = (g \\circ f)_{big}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"The h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EU7","source_file":"flat.tex","source_line":10714,"source_end_line":10719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10714-L10719","statement_sha256":"527af8c6268f21218433268cb695129565b688d6958a053c9f6662159d4f6de4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7840,"rank":7840,"depth":9,"x":2096.293,"y":547.143,"cluster":"scheme-morphisms"},{"id":"stacks:0EV2","tag":"0EV2","title":"More on the h topology · Lemma 0EV2","summary":"Let T be an affine scheme which is written as a limit T = lim_i ∈ I T_i of a directed inverse system of affine schemes. • Let V = (V_j → T)_j = 1, …, m be a standard h covering of T, see Definition [Tag 0EUY]. Then there exists an index i and a standard h covering V_i = (V_i, j → T_i)_j = 1, …, m whose base change T ×_T_i V_i to T is isomorphic to V. • Let V_i, V'_i be a pair of standard h coverings of T_i. If f : T ×_T_i V_i → T ×_T_i V'_i is a morphism of coverings of…","statement_latex":"Let $T$ be an affine scheme which is written as a limit\n$T = \\lim_{i \\in I} T_i$ of a directed inverse system of affine schemes.\n\\begin{enumerate}\n\\item Let $\\mathcal{V} = \\{V_j \\to T\\}_{j = 1, \\ldots, m}$ be a\nstandard h covering of $T$, see Definition\n\\ref{definition-standard-h}.\nThen there exists an index $i$ and a standard h covering\n$\\mathcal{V}_i = \\{V_{i, j} \\to T_i\\}_{j = 1, \\ldots, m}$\nwhose base change $T \\times_{T_i} \\mathcal{V}_i$ to $T$\nis isomorphic to $\\mathcal{V}$.\n\\item Let $\\mathcal{V}_i$, $\\mathcal{V}'_i$ be a pair of standard\nh coverings of $T_i$. If\n$f : T \\times_{T_i} \\mathcal{V}_i \\to T \\times_{T_i} \\mathcal{V}'_i$ is\na morphism of coverings of $T$, then there exists an index\n$i' \\geq i$ and a morphism\n$f_{i'} : T_{i'} \\times_{T_i} \\mathcal{V} \\to\nT_{i'} \\times_{T_i} \\mathcal{V}'_i$\nwhose base change to $T$ is $f$.\n\\item If\n$f, g : \\mathcal{V} \\to \\mathcal{V}'_i$\nare morphisms of standard h coverings of $T_i$ whose\nbase changes $f_T, g_T$ to $T$ are equal then there exists an\nindex $i' \\geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$.\n\\end{enumerate}\nIn other words, the category of standard h coverings of $T$ is\nthe colimit over $I$ of the categories of standard h coverings of $T_i$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"More on the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EV2","source_file":"flat.tex","source_line":10767,"source_end_line":10795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10767-L10795","statement_sha256":"1a7c3d86f9f7cc2b4c90685399ce91d11c9b52f8968eb951165306dd43115623","origin":"The Stacks Project","memory_eligible":false,"source_rank":7841,"rank":7841,"depth":58,"x":2087.977,"y":815.631,"cluster":"scheme-morphisms"},{"id":"stacks:0EV3","tag":"0EV3","title":"More on the h topology · Lemma 0EV3","summary":"Let S be a scheme contained in a big site Sch_h. Let F : (Sch/S)_h^opp → Sets be an h sheaf satisfying property (b) of Topologies, Lemma [Tag 0EUW] with C = (Sch/S)_h. Then the extension F' of F to the category of all schemes over S satisfies the sheaf condition for all h coverings and is limit preserving (Limits, Remark [Tag 05LX]).","statement_latex":"Let $S$ be a scheme contained in a big site $\\Sch_h$.\nLet $F : (\\Sch/S)_h^{opp} \\to \\textit{Sets}$ be an h sheaf satisfying\nproperty (b) of Topologies, Lemma \\ref{topologies-lemma-extend}\nwith $\\mathcal{C} = (\\Sch/S)_h$.\nThen the extension $F'$ of $F$ to the category of all\nschemes over $S$ satisfies the sheaf condition for all h coverings\nand is limit preserving (Limits, Remark \\ref{limits-remark-limit-preserving}).","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"More on the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EV3","source_file":"flat.tex","source_line":10829,"source_end_line":10838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10829-L10838","statement_sha256":"1d4fc80aabf5cb6cc82df9439ad58c40e9dc9d55e147947bdff5a7e6ed61db36","origin":"The Stacks Project","memory_eligible":false,"source_rank":7842,"rank":7842,"depth":59,"x":1878.124,"y":612.861,"cluster":"scheme-morphisms"},{"id":"stacks:0EW1","tag":"0EW1","title":"Blow up squares and the ph topology · Lemma 0EW1","summary":"Let F be a sheaf on a site (Sch/S)_ph, see Topologies, Definition [Tag 0DBL]. Then for any blow up square ([Tag 0EV5]) in the category (Sch/S)_ph the diagram xymatrix F(E) & F(X') ar[l] F(Z) ar[u] & F(X) ar[u] ar[l] is cartesian in the category of sets.","statement_latex":"Let $\\mathcal{F}$ be a sheaf on a site $(\\Sch/S)_{ph}$, see\nTopologies, Definition \\ref{topologies-definition-big-small-ph}.\nThen for any blow up square (\\ref{equation-blow-up-square})\nin the category $(\\Sch/S)_{ph}$ the diagram\n$$\n\\xymatrix{\n\\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\\n\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]\n}\n$$\nis cartesian in the category of sets.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blow up squares and the ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EW1","source_file":"flat.tex","source_line":10877,"source_end_line":10890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10877-L10890","statement_sha256":"e1baefe43f9e8c08c8225969691f363d1ac3bd028d7bef2c40a090358abb0487","origin":"The Stacks Project","memory_eligible":false,"source_rank":7843,"rank":7843,"depth":1,"x":2196.042,"y":643.316,"cluster":"scheme-morphisms"},{"id":"stacks:0EW2","tag":"0EW2","title":"Blow up squares and the ph topology · Lemma 0EW2","summary":"Let F be a sheaf on a site (Sch/S)_ph as in Topologies, Definition [Tag 0DBL]. Let X → X' be a morphism of (Sch/S)_ph which is a thickening. Then F(X') → F(X) is bijective.","statement_latex":"Let $\\mathcal{F}$ be a sheaf on a site $(\\Sch/S)_{ph}$\nas in Topologies, Definition \\ref{topologies-definition-big-small-ph}.\nLet $X \\to X'$ be a morphism of $(\\Sch/S)_{ph}$ which is\na thickening. Then\n$\\mathcal{F}(X') \\to \\mathcal{F}(X)$ is bijective.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blow up squares and the ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EW2","source_file":"flat.tex","source_line":10920,"source_end_line":10927,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10920-L10927","statement_sha256":"9d228bb999485048d3dbcb2ec1bd956b2228aa45cc275cc657d654ce9001e9a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7844,"rank":7844,"depth":1,"x":1937.03,"y":801.31,"cluster":"scheme-morphisms"},{"id":"stacks:0EV7","tag":"0EV7","title":"Almost blow up squares and the h topology · Lemma 0EV7","summary":"Consider an almost blow up square ([Tag 0EV6]). Let Y → X be any morphism. Then the base change xymatrix Y ×_X E ar[d] ar[r] & Y ×_X X' ar[d] Y ×_X Z ar[r] & Y is an almost blow up square too.","statement_latex":"Consider an almost blow up square (\\ref{equation-almost-blow-up-square}).\nLet $Y \\to X$ be any morphism. Then the base change\n$$\n\\xymatrix{\nY \\times_X E \\ar[d] \\ar[r] & Y \\times_X X' \\ar[d] \\\\\nY \\times_X Z \\ar[r] & Y\n}\n$$\nis an almost blow up square too.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EV7","source_file":"flat.tex","source_line":10986,"source_end_line":10997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L10986-L10997","statement_sha256":"b99c7ac555df97d487116beeb18f3d256897fc584fab40c018d1e1534aff3953","origin":"The Stacks Project","memory_eligible":false,"source_rank":7845,"rank":7845,"depth":22,"x":2000.99,"y":537.744,"cluster":"scheme-morphisms"},{"id":"stacks:0EV8","tag":"0EV8","title":"Almost blow up squares and the h topology · Lemma 0EV8","summary":"Consider an almost blow up square ([Tag 0EV6]). Let W → X' be a closed immersion of finite presentation. The following are equivalent • X' setminus E is scheme theoretically contained in W, • the blowup X\" of X in Z is scheme theoretically contained in W, • the diagram xymatrix E ∩ W ar[d] ar[r] & W ar[d] Z ar[r] & X is an almost blow up square. Here E ∩ W is the scheme theoretic intersection.","statement_latex":"Consider an almost blow up square (\\ref{equation-almost-blow-up-square}).\nLet $W \\to X'$ be a closed immersion of finite presentation.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X' \\setminus E$ is scheme theoretically contained in $W$,\n\\item the blowup $X''$ of $X$ in $Z$ is scheme theoretically contained in $W$,\n\\item the diagram\n$$\n\\xymatrix{\nE \\cap W \\ar[d] \\ar[r] & W \\ar[d] \\\\\nZ \\ar[r] & X\n}\n$$\nis an almost blow up square. Here $E \\cap W$ is the\nscheme theoretic intersection.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EV8","source_file":"flat.tex","source_line":11026,"source_end_line":11044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11026-L11044","statement_sha256":"04160d7eaedec0b03a9bfa741d07a8cb7f122b71f5a89b7f885025c2b14c1e96","origin":"The Stacks Project","memory_eligible":false,"source_rank":7846,"rank":7846,"depth":0,"x":2165.838,"y":768.468,"cluster":"scheme-morphisms"},{"id":"stacks:0EV9","tag":"0EV9","title":"Almost blow up squares and the h topology · Lemma 0EV9","summary":"Consider an almost blow up square ([Tag 0EV6]) with X quasi-compact and quasi-separated. Then the blowup X\" of X in Z can be written as X\" = lim X'_i where the limit is over the directed system of closed subschemes X'_i ⊂ X' of finite presentation satisfying the equivalent conditions of Lemma [Tag 0EV8].","statement_latex":"Consider an almost blow up square (\\ref{equation-almost-blow-up-square})\nwith $X$ quasi-compact and quasi-separated. Then the blowup $X''$ of $X$\nin $Z$ can be written as\n$$\nX'' = \\lim X'_i\n$$\nwhere the limit is over the directed system of closed subschemes\n$X'_i \\subset X'$ of finite presentation satisfying the equivalent\nconditions of Lemma \\ref{lemma-shrink-almost-blow-up}.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EV9","source_file":"flat.tex","source_line":11060,"source_end_line":11071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11060-L11071","statement_sha256":"96c5fcaa4f3c57ab6bce04745704f90d75d20d86de14480d7d27a6aebe9452d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7847,"rank":7847,"depth":17,"x":1858.631,"y":691.847,"cluster":"scheme-morphisms"},{"id":"stacks:0EVA","tag":"0EVA","title":"Almost blow up squares and the h topology · Lemma 0EVA","summary":"Let X be a quasi-compact and quasi-separated scheme. Let Z ⊂ X be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals. Then there exists an almost blow up square as in ([Tag 0EV6]).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let $Z \\subset X$\nbe a closed subscheme cut out by a finite type quasi-coherent\nsheaf of ideals. Then there exists an almost blow up square as in\n(\\ref{equation-almost-blow-up-square}).","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVA","source_file":"flat.tex","source_line":11086,"source_end_line":11092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11086-L11092","statement_sha256":"087586bd73671fb41094166c481b9950102f641723246bd2396584b86d24701a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7848,"rank":7848,"depth":27,"x":2146.879,"y":573.987,"cluster":"scheme-morphisms"},{"id":"stacks:0EVB","tag":"0EVB","title":"Almost blow up squares and the h topology · Lemma 0EVB","summary":"Let X be a quasi-compact and quasi-separated scheme and let Z ⊂ X be a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals. Suppose given almost blow up squares ([Tag 0EV6]) xymatrix E_k ar[r] ar[d] & X_k' ar[d] Z ar[r] & X for k = 1, 2, then there exists an almost blow up square xymatrix E ar[r] ar[d] & X' ar[d] Z ar[r] & X and closed immersions i_k : X' → X'_k over X with E = i_k^-1(E_k).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme and let $Z \\subset X$\nbe a closed subscheme cut out by a finite type quasi-coherent sheaf of ideals.\nSuppose given almost blow up squares (\\ref{equation-almost-blow-up-square})\n$$\n\\xymatrix{\nE_k \\ar[r] \\ar[d] & X_k' \\ar[d] \\\\\nZ \\ar[r] & X\n}\n$$\nfor $k = 1, 2$, then there exists an almost blow up square\n$$\n\\xymatrix{\nE \\ar[r] \\ar[d] & X' \\ar[d] \\\\\nZ \\ar[r] & X\n}\n$$\nand closed immersions $i_k : X' \\to X'_k$ over $X$\nwith $E = i_k^{-1}(E_k)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVB","source_file":"flat.tex","source_line":11121,"source_end_line":11141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11121-L11141","statement_sha256":"20538f5218e4cda314e8fa8d211e298b2a8c182f0a017c7d5b37fa0cc4ebbcca","origin":"The Stacks Project","memory_eligible":false,"source_rank":7849,"rank":7849,"depth":27,"x":2029.067,"y":824.543,"cluster":"scheme-morphisms"},{"id":"stacks:0EVC","tag":"0EVC","title":"Almost blow up squares and the h topology · Lemma 0EVC","summary":"Let Y be a quasi-compact and quasi-separated scheme. Let X be a scheme of finite presentation over Y. Let V ⊂ Y be a quasi-compact open such that X_V → V is flat. Then there exist a commutative diagram xymatrix E ar[ddd] ar[rd] & & & D ar[lll] ar[ddd] ar[ld] & Y' ar[d] & X' ar[l] ar[d] & Y & X ar[l] Z ar[ru] & & & T ar[lll] ar[lu] whose right and left hand squares are almost blow up squares, whose lower and top squares are cartesian, such that Z ∩ V = ∅, and such that X'…","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated scheme.\nLet $X$ be a scheme of finite presentation over $Y$.\nLet $V \\subset Y$ be a quasi-compact open such that\n$X_V \\to V$ is flat. Then there exist a commutative diagram\n$$\n\\xymatrix{\nE \\ar[ddd] \\ar[rd] & & & D \\ar[lll] \\ar[ddd] \\ar[ld] \\\\\n& Y' \\ar[d] & X' \\ar[l] \\ar[d] \\\\\n& Y & X \\ar[l] \\\\\nZ \\ar[ru] & & & T \\ar[lll] \\ar[lu]\n}\n$$\nwhose right and left hand squares are almost blow up squares,\nwhose lower and top squares are cartesian, such that\n$Z \\cap V = \\emptyset$, and\nsuch that $X' \\to Y'$ is flat (and of finite presentation).","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVC","source_file":"flat.tex","source_line":11161,"source_end_line":11179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11161-L11179","statement_sha256":"db5958cb56851e863f9c050501745f63ee103e3b5845b512b12c037a1f59f813","origin":"The Stacks Project","memory_eligible":false,"source_rank":7850,"rank":7850,"depth":56,"x":1914.409,"y":572.848,"cluster":"scheme-morphisms"},{"id":"stacks:0EVD","tag":"0EVD","title":"Almost blow up squares and the h topology · Lemma 0EVD","summary":"Let F be a sheaf on one of the sites (Sch/S)_h constructed in Definition [Tag 0EU1]. Then for any almost blow up square ([Tag 0EV6]) in the category (Sch/S)_h the diagram xymatrix F(E) & F(X') ar[l] F(Z) ar[u] & F(X) ar[u] ar[l] is cartesian in the category of sets.","statement_latex":"Let $\\mathcal{F}$ be a sheaf on one of the sites $(\\Sch/S)_h$\nconstructed in Definition \\ref{definition-big-small-h}.\nThen for any almost blow up square (\\ref{equation-almost-blow-up-square})\nin the category $(\\Sch/S)_h$ the diagram\n$$\n\\xymatrix{\n\\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\\n\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]\n}\n$$\nis cartesian in the category of sets.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVD","source_file":"flat.tex","source_line":11215,"source_end_line":11228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11215-L11228","statement_sha256":"d13140bca4627b13a41fff0f3322178d58dc681fa1ffc8c5ba04acac1d4f2b9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7851,"rank":7851,"depth":41,"x":2201.464,"y":693.428,"cluster":"scheme-morphisms"},{"id":"stacks:0EVE","tag":"0EVE","title":"Almost blow up squares and the h topology · Lemma 0EVE","summary":"Let F be a sheaf on one of the sites (Sch/S)_h constructed in Definition [Tag 0EU1]. Let X → X' be a morphism of (Sch/S)_h which is a thickening and of finite presentation. Then F(X') → F(X) is bijective.","statement_latex":"Let $\\mathcal{F}$ be a sheaf on one of the sites $(\\Sch/S)_h$\nconstructed in Definition \\ref{definition-big-small-h}.\nLet $X \\to X'$ be a morphism of $(\\Sch/S)_h$ which is\na thickening and of finite presentation. Then\n$\\mathcal{F}(X') \\to \\mathcal{F}(X)$ is bijective.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVE","source_file":"flat.tex","source_line":11258,"source_end_line":11265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11258-L11265","statement_sha256":"0c0289f13755cfc4d04d6fa708fb873eac0335e88b739e3dd7c62fde6e994032","origin":"The Stacks Project","memory_eligible":false,"source_rank":7852,"rank":7852,"depth":42,"x":1892.718,"y":767.422,"cluster":"scheme-morphisms"},{"id":"stacks:0EVF","tag":"0EVF","title":"Almost blow up squares and the h topology · Proposition 0EVF","summary":"Let F be a presheaf on one of the sites (Sch/S)_h constructed in Definition [Tag 0EU1]. Then F is a sheaf if and only if the following conditions are satisfied • F is a sheaf for the Zariski topology, • given a morphism f : X → Y of (Sch/S)_h with Y affine and f surjective, flat, proper, and of finite presentation, then F(Y) is the equalizer of the two maps F(X) → F(X ×_Y X), • given an almost blow up square ([Tag 0EV6]) with X affine in the category (Sch/S)_h the diagram…","statement_latex":"Let $\\mathcal{F}$ be a presheaf on one of the sites $(\\Sch/S)_h$\nconstructed in Definition \\ref{definition-big-small-h}.\nThen $\\mathcal{F}$ is a sheaf if and only if the following\nconditions are satisfied\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a sheaf for the Zariski topology,\n\\item given a morphism $f : X \\to Y$ of $(\\Sch/S)_h$ with $Y$ affine\nand $f$ surjective, flat, proper, and of finite presentation, then\n$\\mathcal{F}(Y)$ is the equalizer of the two maps\n$\\mathcal{F}(X) \\to \\mathcal{F}(X \\times_Y X)$,\n\\item given an almost blow up square (\\ref{equation-almost-blow-up-square})\nwith $X$ affine in the category $(\\Sch/S)_h$ the diagram\n$$\n\\xymatrix{\n\\mathcal{F}(E) & \\mathcal{F}(X') \\ar[l] \\\\\n\\mathcal{F}(Z) \\ar[u] & \\mathcal{F}(X) \\ar[u] \\ar[l]\n}\n$$\nis cartesian in the category of sets.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVF","source_file":"flat.tex","source_line":11291,"source_end_line":11313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11291-L11313","statement_sha256":"fef3a5b9dffc03b5d01b5febd2fcc12e452d2a59611c7381a81e4807c7ad7987","origin":"The Stacks Project","memory_eligible":false,"source_rank":7853,"rank":7853,"depth":59,"x":2060.938,"y":537.589,"cluster":"scheme-morphisms"},{"id":"stacks:0EVI","tag":"0EVI","title":"Almost blow up squares and the h topology · Lemma 0EVI","summary":"Let F be a presheaf on one of the sites (Sch/S)_h constructed in Definition [Tag 0EU1]. Then F is a sheaf if and only if the following conditions are satisfied • F is a sheaf for the Zariski topology, • given a morphism f : X → Y of (Sch/S)_h with Y affine and f surjective, flat, proper, and of finite presentation, then F(Y) is the equalizer of the two maps F(X) → F(X ×_Y X), • F turns an almost blow up square as in Example [Tag 0EVG] in the category (Sch/S)_h into a…","statement_latex":"Let $\\mathcal{F}$ be a presheaf on one of the sites $(\\Sch/S)_h$\nconstructed in Definition \\ref{definition-big-small-h}.\nThen $\\mathcal{F}$ is a sheaf if and only if the following\nconditions are satisfied\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a sheaf for the Zariski topology,\n\\item given a morphism $f : X \\to Y$ of $(\\Sch/S)_h$ with $Y$ affine\nand $f$ surjective, flat, proper, and of finite presentation, then\n$\\mathcal{F}(Y)$ is the equalizer of the two maps\n$\\mathcal{F}(X) \\to \\mathcal{F}(X \\times_Y X)$,\n\\item $\\mathcal{F}$ turns an almost blow up square as in\nExample \\ref{example-one-generator} in the category $(\\Sch/S)_h$\ninto a cartesian diagram of sets, and\n\\item $\\mathcal{F}$ turns an almost blow up square as in\nExample \\ref{example-two-generators} in the category $(\\Sch/S)_h$\ninto a cartesian diagram of sets.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVI","source_file":"flat.tex","source_line":11500,"source_end_line":11519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11500-L11519","statement_sha256":"23d1967f75e604a116f3ecd289758d78bd2b05c2314aec0508388e0b511ba0bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7854,"rank":7854,"depth":60,"x":2121.743,"y":802.61,"cluster":"scheme-morphisms"},{"id":"stacks:0EX9","tag":"0EX9","title":"Almost blow up squares and the h topology · Lemma 0EX9","summary":"Let p : S → (Sch/S)_h be a category fibred in groupoids. Then S is a stack in groupoids if and only if the following conditions are satisfied • S is a stack in groupoids for the Zariski topology, • given a morphism f : X → Y of (Sch/S)_h with Y affine and f surjective, flat, proper, and of finite presentation, then S_Y → S_X ×_S_X ×_Y X S_X is an equivalence of categories, • for an almost blow up square as in Example [Tag 0EVG] or [Tag 0EVH] in the category (Sch/S)_h the…","statement_latex":"Let $p : \\mathcal{S} \\to (\\Sch/S)_h$ be a category fibred in groupoids.\nThen $\\mathcal{S}$ is a stack in groupoids if and only if the following\nconditions are satisfied\n\\begin{enumerate}\n\\item $\\mathcal{S}$ is a stack in groupoids for the Zariski topology,\n\\item given a morphism $f : X \\to Y$ of $(\\Sch/S)_h$ with $Y$ affine\nand $f$ surjective, flat, proper, and of finite presentation, then\n$$\n\\mathcal{S}_Y \\longrightarrow\n\\mathcal{S}_X \\times_{\\mathcal{S}_{X \\times_Y X}} \\mathcal{S}_X\n$$\nis an equivalence of categories,\n\\item for an almost blow up square as in\nExample \\ref{example-one-generator} or \\ref{example-two-generators}\nin the category $(\\Sch/S)_h$ the functor\n$$\n\\mathcal{S}_X \\longrightarrow\n\\mathcal{S}_Z \\times_{\\mathcal{S}_E} \\mathcal{S}_{X'}\n$$\nis an equivalence of categories.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EX9","source_file":"flat.tex","source_line":11695,"source_end_line":11718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11695-L11718","statement_sha256":"894e65380b7c373c5744184cff96e7522c0019fed28eaf220d85982c9cb399e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7855,"rank":7855,"depth":61,"x":1863.692,"y":641.633,"cluster":"scheme-morphisms"},{"id":"stacks:0EVJ","tag":"0EVJ","title":"Absolute weak normalization and h coverings · Lemma 0EVJ","summary":"Let Z, X, X', E be an almost blow up square as in Example [Tag 0EVH]. Then H^p(X', O_X') = 0 for p > 0 and Γ(X, O_X) → Γ(X', O_X') is a surjective map of rings whose kernel is an ideal of square zero.","statement_latex":"Let $Z, X, X', E$ be an almost blow up square as in\nExample \\ref{example-two-generators}.\nThen $H^p(X', \\mathcal{O}_{X'}) = 0$ for $p > 0$ and\n$\\Gamma(X, \\mathcal{O}_X) \\to \\Gamma(X', \\mathcal{O}_{X'})$\nis a surjective map of rings whose kernel is an ideal of square zero.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Absolute weak normalization and h coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVJ","source_file":"flat.tex","source_line":11781,"source_end_line":11788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11781-L11788","statement_sha256":"58d87108368ce20bd68456acba4910a5d46e2aa5dc7ef6ec555ed68498faf729","origin":"The Stacks Project","memory_eligible":false,"source_rank":7856,"rank":7856,"depth":44,"x":2183.541,"y":613.901,"cluster":"scheme-morphisms"},{"id":"stacks:0EVK","tag":"0EVK","title":"Absolute weak normalization and h coverings · Lemma 0EVK","summary":"Let p be a prime number. Let S be a scheme over F_p. Let (Sch/S)_h be a site as in Definition [Tag 0EU1]. There is a unique sheaf F on (Sch/S)_h such that F(X) = colim_F Γ(X, O_X) for any quasi-compact and quasi-separated object X of (Sch/S)_h.","statement_latex":"Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$.\nLet $(\\Sch/S)_h$ be a site as in Definition \\ref{definition-big-small-h}.\nThere is a unique sheaf $\\mathcal{F}$ on $(\\Sch/S)_h$ such that\n$$\n\\mathcal{F}(X) = \\colim_F \\Gamma(X, \\mathcal{O}_X)\n$$\nfor any quasi-compact and quasi-separated object $X$ of $(\\Sch/S)_h$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Absolute weak normalization and h coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVK","source_file":"flat.tex","source_line":11854,"source_end_line":11863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11854-L11863","statement_sha256":"f87faeecc2cd64ed95bb746b950d39ce040870e4db129c4300ec8bfe3469a5c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7857,"rank":7857,"depth":61,"x":1969.915,"y":815.911,"cluster":"scheme-morphisms"},{"id":"stacks:0EVL","tag":"0EVL","title":"Absolute weak normalization and h coverings · Lemma 0EVL","summary":"Let p be a prime number. Let S be a scheme over F_p. Let (Sch/S)_h be a site as in Definition [Tag 0EU1]. The rule F(X) = lim_F Γ(X, O_X) defines a sheaf on (Sch/S)_h.","statement_latex":"Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$.\nLet $(\\Sch/S)_h$ be a site as in Definition \\ref{definition-big-small-h}.\nThe rule\n$$\n\\mathcal{F}(X) = \\lim_F \\Gamma(X, \\mathcal{O}_X)\n$$\ndefines a sheaf on $(\\Sch/S)_h$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Absolute weak normalization and h coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVL","source_file":"flat.tex","source_line":11930,"source_end_line":11939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11930-L11939","statement_sha256":"eee1d1e2fdb05a059206db2b4e3c6cce82f214087c551598ef7759012b19130c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7858,"rank":7858,"depth":61,"x":1964.987,"y":545.64,"cluster":"scheme-morphisms"},{"id":"stacks:0EVT","tag":"0EVT","title":"Absolute weak normalization and h coverings · Lemma 0EVT","summary":"Let (Sch/S)_ph be a site as in Topologies, Definition [Tag 0DBL]. The rule X ↦ Γ(X^awn, O_X^awn) is a sheaf on (Sch/S)_ph.","statement_latex":"Let $(\\Sch/S)_{ph}$ be a site as in\nTopologies, Definition \\ref{topologies-definition-big-small-ph}.\nThe rule\n$$\nX \\longmapsto \\Gamma(X^{awn}, \\mathcal{O}_{X^{awn}})\n$$\nis a sheaf on $(\\Sch/S)_{ph}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Absolute weak normalization and h coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVT","source_file":"flat.tex","source_line":11990,"source_end_line":11999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L11990-L11999","statement_sha256":"90a77d154c94c0022b279a782836b52792fcdd83baab9f404e610c7cd1ac1206","origin":"The Stacks Project","memory_eligible":false,"source_rank":7859,"rank":7859,"depth":28,"x":2186.043,"y":742.207,"cluster":"scheme-morphisms"},{"id":"stacks:0EVU","tag":"0EVU","title":"Absolute weak normalization and h coverings · Lemma 0EVU","summary":"Let S be a scheme. Choose a site (Sch/S)_h as in Definition [Tag 0EU1]. The rule X ↦ Γ(X^awn, O_X^awn) is the sheafification of the \"structure sheaf\" O on (Sch/S)_h. Similarly for the ph topology.","statement_latex":"Let $S$ be a scheme. Choose a site $(\\Sch/S)_h$\nas in Definition \\ref{definition-big-small-h}.\nThe rule\n$$\nX \\longmapsto \\Gamma(X^{awn}, \\mathcal{O}_{X^{awn}})\n$$\nis the sheafification of the ``structure sheaf'' $\\mathcal{O}$\non $(\\Sch/S)_h$. Similarly for the ph topology.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Absolute weak normalization and h coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVU","source_file":"flat.tex","source_line":12076,"source_end_line":12086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12076-L12086","statement_sha256":"17ce111cb2942f8b06656486721bb7adc5cdf6ede87fada1eabc6704f89c56fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7860,"rank":7860,"depth":29,"x":1864.853,"y":722.687,"cluster":"scheme-morphisms"},{"id":"stacks:0EVV","tag":"0EVV","title":"Absolute weak normalization and h coverings · Lemma 0EVV","summary":"Let p be a prime number. An F_p-algebra A is absolutely weakly normal if and only if it is perfect.","statement_latex":"Let $p$ be a prime number. An $\\mathbf{F}_p$-algebra $A$ is\nabsolutely weakly normal if and only if it is perfect.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Absolute weak normalization and h coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVV","source_file":"flat.tex","source_line":12124,"source_end_line":12128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12124-L12128","statement_sha256":"b1db81afc37f41b3c965344a338e0f706bca83390f4f07aba8f2453820f2c116","origin":"The Stacks Project","memory_eligible":false,"source_rank":7861,"rank":7861,"depth":1,"x":2117.48,"y":554.771,"cluster":"scheme-morphisms"},{"id":"stacks:0EVW","tag":"0EVW","title":"Absolute weak normalization and h coverings · Lemma 0EVW","summary":"Let p be a prime number. • If A is an F_p-algebra, then colim_F A = A^awn. • If S is a scheme over F_p, then the h sheafification of O sends a quasi-compact and quasi-separated X to colim_F Γ(X, O_X).","statement_latex":"Let $p$ be a prime number.\n\\begin{enumerate}\n\\item If $A$ is an $\\mathbf{F}_p$-algebra, then $\\colim_F A = A^{awn}$.\n\\item If $S$ is a scheme over $\\mathbf{F}_p$, then the\nh sheafification of $\\mathcal{O}$ sends a quasi-compact\nand quasi-separated $X$ to $\\colim_F \\Gamma(X, \\mathcal{O}_X)$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Absolute weak normalization and h coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVW","source_file":"flat.tex","source_line":12155,"source_end_line":12164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12155-L12164","statement_sha256":"2151a261cb5235b0dc23da4eb0e4abf5309c9e073ea7f4770ab82eb6f91ab969","origin":"The Stacks Project","memory_eligible":false,"source_rank":7862,"rank":7862,"depth":62,"x":2066.213,"y":822.03,"cluster":"scheme-morphisms"},{"id":"stacks:0EXB","tag":"0EXB","title":"Descent vector bundles in positive characteristic · Lemma 0EXB","summary":"Let p be a prime number. Let S be a quasi-compact and quasi-separated scheme over F_p. The category colim_F Vect(S) is equivalent to the category of finite locally free modules over the sheaf of rings colim_F O_S on S.","statement_latex":"Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated\nscheme over $\\mathbf{F}_p$. The category $\\colim_F \\textit{Vect}(S)$\nis equivalent to the category of finite locally free modules\nover the sheaf of rings $\\colim_F \\mathcal{O}_S$ on $S$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Descent vector bundles in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXB","source_file":"flat.tex","source_line":12222,"source_end_line":12228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12222-L12228","statement_sha256":"46ee9b53eaeaa4fb7b80a5284c86d93dcae6e797c02990ce70c5fc9d289f5873","origin":"The Stacks Project","memory_eligible":false,"source_rank":7863,"rank":7863,"depth":0,"x":1889.031,"y":595.787,"cluster":"scheme-morphisms"},{"id":"stacks:0EXC","tag":"0EXC","title":"Descent vector bundles in positive characteristic · Lemma 0EXC","summary":"Let p be a prime number. Consider an almost blowup square X, X', Z, E in characteristic p as in Example [Tag 0EVG]. Then the functor colim_F Vect(X) → colim_F Vect(Z) ×_colim_F Vect(E) colim_F Vect(X') is an equivalence.","statement_latex":"Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$\nin characteristic $p$ as in Example \\ref{example-one-generator}.\nThen the functor\n$$\n\\colim_F \\textit{Vect}(X)\n\\longrightarrow\n\\colim_F \\textit{Vect}(Z)\n\\times_{\\colim_F \\textit{Vect}(E)}\n\\colim_F \\textit{Vect}(X')\n$$\nis an equivalence.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Descent vector bundles in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXC","source_file":"flat.tex","source_line":12234,"source_end_line":12247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12234-L12247","statement_sha256":"e1f4f70433006a391d39adfc874e423c78016ac56b66cea04d7c3ddd1a74ebb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":7864,"rank":7864,"depth":62,"x":2201.73,"y":662.102,"cluster":"scheme-morphisms"},{"id":"stacks:0EXD","tag":"0EXD","title":"Descent vector bundles in positive characteristic · Lemma 0EXD","summary":"Let p be a prime number. Consider an almost blowup square X, X', Z, E in characteristic p as in Example [Tag 0EVH]. Then the functor G : colim_F Vect(X) → colim_F Vect(Z) ×_colim_F Vect(E) colim_F Vect(X') is an equivalence.","statement_latex":"Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$\nin characteristic $p$ as in Example \\ref{example-two-generators}.\nThen the functor\n$$\nG :\n\\colim_F \\textit{Vect}(X)\n\\longrightarrow\n\\colim_F \\textit{Vect}(Z)\n\\times_{\\colim_F \\textit{Vect}(E)}\n\\colim_F \\textit{Vect}(X')\n$$\nis an equivalence.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Descent vector bundles in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXD","source_file":"flat.tex","source_line":12302,"source_end_line":12316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12302-L12316","statement_sha256":"18d15d65022485e3189327cc1cdddffb0cfd2932f46f6d13e83ccab096e09211","origin":"The Stacks Project","memory_eligible":false,"source_rank":7865,"rank":7865,"depth":62,"x":1917.723,"y":790.68,"cluster":"scheme-morphisms"},{"id":"stacks:0EXE","tag":"0EXE","title":"Descent vector bundles in positive characteristic · Proposition 0EXE","summary":"Let p be a prime number. Let S be a scheme in characteristic p. Then the category fibred in groupoids p : S → (Sch/S)_h whose fibre category over U is the category of finite locally free colim_F O_U-modules over U is a stack in groupoids. Moreover, if U is quasi-compact and quasi-separated, then S_U is colim_F Vect(U).","statement_latex":"Let $p$ be a prime number. Let $S$ be a scheme in characteristic $p$.\nThen the category fibred in groupoids\n$$\np : \\mathcal{S} \\longrightarrow (\\Sch/S)_h\n$$\nwhose fibre category over $U$ is the category\nof finite locally free $\\colim_F \\mathcal{O}_U$-modules over $U$\nis a stack in groupoids. Moreover, if $U$ is quasi-compact\nand quasi-separated, then $\\mathcal{S}_U$ is $\\colim_F \\textit{Vect}(U)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Descent vector bundles in positive characteristic","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXE","source_file":"flat.tex","source_line":12439,"source_end_line":12450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12439-L12450","statement_sha256":"c7ff58f39798fa8779bed38c48ed8c0cafe4a701853b22347aa34ff4075464b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7866,"rank":7866,"depth":63,"x":2023.783,"y":534.626,"cluster":"scheme-morphisms"},{"id":"stacks:0EXF","tag":"0EXF","title":"Descent vector bundles in positive characteristic · Lemma 0EXF","summary":"Let f : X → S be a proper morphism with geometrically connected fibres where S is the spectrum of a discrete valuation ring. Denote eta ∈ S the generic point and denote X_n ⊂ X the closed subscheme cutout by the nth power of a uniformizer on S. Then there exists an integer n such that the following is true: any finite locally free O_X-module E such that E|_X_eta and E|_X_n are free, is free.","statement_latex":"Let $f : X \\to S$ be a proper morphism with geometrically connected fibres\nwhere $S$ is the spectrum of a discrete valuation ring. Denote $\\eta \\in S$\nthe generic point and denote $X_n \\subset X$ the closed subscheme\ncutout by the $n$th power of a uniformizer on $S$.\nThen there exists\nan integer $n$ such that the following is true: any finite\nlocally free $\\mathcal{O}_X$-module $\\mathcal{E}$\nsuch that $\\mathcal{E}|_{X_\\eta}$ and $\\mathcal{E}|_{X_n}$\nare free, is free.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Descent vector bundles in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXF","source_file":"flat.tex","source_line":12485,"source_end_line":12496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12485-L12496","statement_sha256":"c4d9397ddbbe51475989a86bf8baa2e5cf0e5cefdfff0b999ff9d5c03c5e0914","origin":"The Stacks Project","memory_eligible":false,"source_rank":7867,"rank":7867,"depth":33,"x":2151.532,"y":783.705,"cluster":"scheme-morphisms"},{"id":"stacks:0EXG","tag":"0EXG","title":"Descent vector bundles in positive characteristic · Lemma 0EXG","summary":"Let f : X → S be a morphism of schemes. Let E be a finite locally free O_X-module. Assume • f is flat and proper and O_S = f_*O_X, • S is a normal Noetherian scheme, • the pullback of E to X ×_S Spec(O_S, s) is free for every codimension 1 point s ∈ S. Then E is isomorphic to the pullback of a finite locally free O_S-module.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $f$ is flat and proper and $\\mathcal{O}_S = f_*\\mathcal{O}_X$,\n\\item $S$ is a normal Noetherian scheme,\n\\item the pullback of $\\mathcal{E}$ to $X \\times_S \\Spec(\\mathcal{O}_{S, s})$\nis free for every codimension $1$ point $s \\in S$.\n\\end{enumerate}\nThen $\\mathcal{E}$ is isomorphic to the pullback of a finite\nlocally free $\\mathcal{O}_S$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Descent vector bundles in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXG","source_file":"flat.tex","source_line":12640,"source_end_line":12653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12640-L12653","statement_sha256":"73a82c413decfa9dce151b1d026591090cad9dae49898f944145fd5df4dd4eda","origin":"The Stacks Project","memory_eligible":false,"source_rank":7868,"rank":7868,"depth":32,"x":1856.927,"y":672.487,"cluster":"scheme-morphisms"},{"id":"stacks:0EXH","tag":"0EXH","title":"Descent vector bundles in positive characteristic · Theorem 0EXH","summary":"Let p be a prime number. Let Y be a quasi-compact and quasi-separated scheme over F_p. Let f : X → Y be a proper, surjective morphism of finite presentation with geometrically connected fibres. Then the functor colim_F Vect(Y) → colim_F Vect(X) is fully faithful with essential image described as follows. Let E be a finite locally free O_X-module. Assume for all y ∈ Y there exists integers n_y, r_y ≥ 0 such that F^n_y, *E|_X_y, red ≅ O_X_y, red^⊕ r_y Then for some n ≥ 0…","statement_latex":"Let $p$ be a prime number. Let $Y$ be a quasi-compact and quasi-separated\nscheme over $\\mathbf{F}_p$.\nLet $f : X \\to Y$ be a proper, surjective morphism of finite presentation\nwith geometrically connected fibres.\nThen the functor\n$$\n\\colim_F \\textit{Vect}(Y) \\longrightarrow \\colim_F \\textit{Vect}(X)\n$$\nis fully faithful with essential image described as follows.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.\nAssume for all $y \\in Y$ there exists integers $n_y, r_y \\geq 0$\nsuch that\n$$\nF^{n_y, *}\\mathcal{E}|_{X_{y, red}}\n\\cong\n\\mathcal{O}_{X_{y, red}}^{\\oplus r_y}\n$$\nThen for some $n \\geq 0$ the $n$th Frobenius power pullback\n$F^{n, *}\\mathcal{E}$ is the pullback of a finite locally free\n$\\mathcal{O}_Y$-module.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Descent vector bundles in positive characteristic","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EXH","source_file":"flat.tex","source_line":12685,"source_end_line":12707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12685-L12707","statement_sha256":"8a90931423b30bce188f5a7cc95fc7727eb7971fef555d83207c3bdfd3a85d8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7869,"rank":7869,"depth":64,"x":2163.71,"y":587.301,"cluster":"scheme-morphisms"},{"id":"stacks:0ESP","tag":"0ESP","title":"Blowing up complexes · Lemma 0ESP","summary":"Let X be a scheme. Let E ∈ D(O_X) be pseudo-coherent. For every p, k ∈ Z there is an finite type quasi-coherent sheaf of ideals Fit_p, k(E) ⊂ O_X with the following property: for U ⊂ X open such that E|_U is isomorphic to … → O_U^⊕ n_b - 2 xrightarrowd_b - 2 O_U^⊕ n_b - 1 xrightarrowd_b - 1 O_U^⊕ n_b → 0 → … the restriction Fit_p, k(E)|_U is generated by the minors of the matrix of d_p of size - k + n_p + 1 - n_p + 2 + … + (-1)^b - p + 1 n_b Convention: the ideal…","statement_latex":"Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent.\nFor every $p, k \\in \\mathbf{Z}$ there is an finite type quasi-coherent\nsheaf of ideals $\\text{Fit}_{p, k}(E) \\subset \\mathcal{O}_X$\nwith the following property: for $U \\subset X$ open\nsuch that $E|_U$ is isomorphic to\n$$\n\\ldots \\to\n\\mathcal{O}_U^{\\oplus n_{b - 2}}\n\\xrightarrow{d_{b - 2}}\n\\mathcal{O}_U^{\\oplus n_{b - 1}}\n\\xrightarrow{d_{b - 1}}\n\\mathcal{O}_U^{\\oplus n_b} \\to 0 \\to \\ldots\n$$\nthe restriction $\\text{Fit}_{p, k}(E)|_U$ is generated by the\nminors of the matrix of $d_p$ of size\n$$\n- k + n_{p + 1} - n_{p + 2} + \\ldots + (-1)^{b - p + 1} n_b\n$$\nConvention: the ideal generated by $r \\times r$-minors\nis $\\mathcal{O}_U$ if $r \\leq 0$ and the ideal generated by\n$r \\times r$-minors where $r > \\min(n_p, n_{p + 1})$ is zero.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESP","source_file":"flat.tex","source_line":12962,"source_end_line":12985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L12962-L12985","statement_sha256":"255c3fb8c122b006103c6d7fde6d39d11f350c114a7e582877fd33ba95fd2661","origin":"The Stacks Project","memory_eligible":false,"source_rank":7870,"rank":7870,"depth":12,"x":2005.942,"y":824.275,"cluster":"scheme-morphisms"},{"id":"stacks:0ESQ","tag":"0ESQ","title":"Blowing up complexes · Lemma 0ESQ","summary":"Let X be a scheme. Let E ∈ D(O_X) be perfect. Let U ⊂ X be a scheme theoretically dense open subscheme such that H^i(E|_U) is finite locally free of constant rank r_i for all i ∈ Z. Then there exists a U-admissible blowup b : X' → X such that H^i(Lb^*E) is a perfect O_X'-module of tor dimension ≤ 1 for all i ∈ Z.","statement_latex":"Let $X$ be a scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect.\nLet $U \\subset X$ be a scheme theoretically dense open subscheme\nsuch that $H^i(E|_U)$ is finite locally free of constant rank $r_i$\nfor all $i \\in \\mathbf{Z}$.\nThen there exists a $U$-admissible blowup $b : X' \\to X$ such that\n$H^i(Lb^*E)$ is a perfect $\\mathcal{O}_{X'}$-module\nof tor dimension $\\leq 1$ for all $i \\in \\mathbf{Z}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESQ","source_file":"flat.tex","source_line":13035,"source_end_line":13044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13035-L13044","statement_sha256":"e6a6e915f5860e433ec2af6c97593094ff2f679302b90befc11a56bd24c3ac88","origin":"The Stacks Project","memory_eligible":false,"source_rank":7871,"rank":7871,"depth":23,"x":1931.684,"y":559.921,"cluster":"scheme-morphisms"},{"id":"stacks:0ESR","tag":"0ESR","title":"Blowing up complexes · Lemma 0ESR","summary":"Let X be an integral scheme. Let E ∈ D(O_X) be perfect. Then there exists a nonempty open U ⊂ X such that H^i(E|_U) is finite locally free of constant rank r_i for all i ∈ Z and there exists a U-admissible blowup b : X' → X such that H^i(Lb^*E) is a perfect O_X'-module of tor dimension ≤ 1 for all i ∈ Z.","statement_latex":"Let $X$ be an integral scheme. Let $E \\in D(\\mathcal{O}_X)$ be perfect.\nThen there exists a nonempty open $U \\subset X$\nsuch that $H^i(E|_U)$ is finite locally free of constant rank $r_i$\nfor all $i \\in \\mathbf{Z}$ and there exists a $U$-admissible blowup\n$b : X' \\to X$ such that $H^i(Lb^*E)$ is a perfect\n$\\mathcal{O}_{X'}$-module of tor dimension $\\leq 1$ for all $i \\in \\mathbf{Z}$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESR","source_file":"flat.tex","source_line":13105,"source_end_line":13113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13105-L13113","statement_sha256":"b791801becd8a1afec6e4374d67aa27e3bbe508fca5c5380a99b6b6245269ed5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7872,"rank":7872,"depth":39,"x":2199.12,"y":712.769,"cluster":"scheme-morphisms"},{"id":"stacks:0ESS","tag":"0ESS","title":"Blowing up perfect modules · Lemma 0ESS","summary":"Let X be a scheme. Let F be a perfect O_X-module of tor dimension ≤ 1. For any blowup b : X' → X we have Lb^*F = b^*F and b^*F is a perfect O_X-module of tor dimension ≤ 1.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a perfect\n$\\mathcal{O}_X$-module of tor dimension $\\leq 1$. For any\nblowup $b : X' \\to X$ we have $Lb^*\\mathcal{F} = b^*\\mathcal{F}$\nand $b^*\\mathcal{F}$ is a perfect $\\mathcal{O}_X$-module\nof tor dimension $\\leq 1$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESS","source_file":"flat.tex","source_line":13223,"source_end_line":13230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13223-L13230","statement_sha256":"13461757f479fd0f0547575180ded44b41d391d1c325be79bba35859056acf48","origin":"The Stacks Project","memory_eligible":false,"source_rank":7873,"rank":7873,"depth":0,"x":1878.889,"y":751.824,"cluster":"scheme-morphisms"},{"id":"stacks:0EST","tag":"0EST","title":"Blowing up perfect modules · Lemma 0EST","summary":"Let X be a scheme. Let F be a perfect O_X-module of tor dimension ≤ 1. Let U ⊂ X be a scheme theoretically dense open such that F|_U is finite locally free of constant rank r. Then there exists a U-admissible blowup b : X' → X such that there is a canonical short exact sequence 0 → K → b^*F → Q → 0 where Q is finite locally free of rank r and K is a perfect O_X-module of tor dimension ≤ 1 whose restriction to U is zero.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a perfect $\\mathcal{O}_X$-module\nof tor dimension $\\leq 1$. Let $U \\subset X$ be a scheme theoretically\ndense open such that $\\mathcal{F}|_U$ is finite locally free of constant\nrank $r$. Then there exists a $U$-admissible blowup $b : X' \\to X$ such that\nthere is a canonical short exact sequence\n$$\n0 \\to \\mathcal{K} \\to b^*\\mathcal{F} \\to \\mathcal{Q} \\to 0\n$$\nwhere $\\mathcal{Q}$ is finite locally free of rank $r$ and\n$\\mathcal{K}$ is a perfect $\\mathcal{O}_X$-module\nof tor dimension $\\leq 1$ whose restriction to $U$ is zero.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EST","source_file":"flat.tex","source_line":13245,"source_end_line":13258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13245-L13258","statement_sha256":"0097b71cf2c73a82c631e0b00f287b116b09df884c25e05455ca22247f4b690c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7874,"rank":7874,"depth":10,"x":2083.687,"y":541.245,"cluster":"scheme-morphisms"},{"id":"stacks:0ESU","tag":"0ESU","title":"Blowing up perfect modules · Lemma 0ESU","summary":"Let X be a scheme. Let F be a perfect O_X-module of tor dimension ≤ 1. Let U ⊂ X be an open such that F|_U = 0. Then there is a U-admissible blowup b : X' → X such that F' = b^*F is equipped with two canonical locally finite filtrations 0 = F^0 ⊂ F^1 ⊂ F^2 ⊂ … ⊂ F' and F' = F_1 ⊃ F_2 ⊃ F_3 ⊃ … ⊃ 0 such that for each n ≥ 1 there is an effective Cartier divisor D_n ⊂ X' with the property that F^i/F^i - 1 and F_i/F_i + 1 are finite locally free of rank i on D_i.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a perfect $\\mathcal{O}_X$-module\nof tor dimension $\\leq 1$. Let $U \\subset X$ be an open such that\n$\\mathcal{F}|_U = 0$. Then there is a $U$-admissible blowup\n$$\nb : X' \\to X\n$$\nsuch that $\\mathcal{F}' = b^*\\mathcal{F}$ is equipped with two canonical\nlocally finite filtrations\n$$\n0 = F^0 \\subset F^1 \\subset F^2 \\subset \\ldots \\subset \\mathcal{F}'\n\\quad\\text{and}\\quad\n\\mathcal{F}' = F_1 \\supset F_2 \\supset F_3 \\supset \\ldots \\supset 0\n$$\nsuch that for each $n \\geq 1$ there is an effective Cartier divisor\n$D_n \\subset X'$ with the property that\n$$\nF^i/F^{i - 1}\n\\quad\\text{and}\\quad\nF_i/F_{i + 1}\n$$\nare finite locally free of rank $i$ on $D_i$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESU","source_file":"flat.tex","source_line":13265,"source_end_line":13288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13265-L13288","statement_sha256":"1139456898c3b0705341fd77cdf0ded4f66582a42c48a22e337939c16f0069ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":7875,"rank":7875,"depth":22,"x":2102.02,"y":812.825,"cluster":"scheme-morphisms"},{"id":"stacks:0ESV","tag":"0ESV","title":"Blowing up perfect modules · Lemma 0ESV","summary":"Let X be a scheme. Let φ : F → G be a homorphism of perfect O_X-modules of tor dimension ≤ 1. Let U ⊂ X be a scheme theoretically dense open such that F|_U = 0 and G|_U = 0. Then there is a U-admissible blowup b : X' → X such that the kernel, image, and cokernel of b^*φ are perfect O_X'-modules of tor dimension ≤ 1.","statement_latex":"Let $X$ be a scheme. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nbe a homorphism of perfect $\\mathcal{O}_X$-modules of tor dimension $\\leq 1$.\nLet $U \\subset X$ be a scheme theoretically dense open\nsuch that $\\mathcal{F}|_U = 0$ and $\\mathcal{G}|_U = 0$.\nThen there is a $U$-admissible blowup $b : X' \\to X$ such that\nthe kernel, image, and cokernel of $b^*\\varphi$ are\nperfect $\\mathcal{O}_{X'}$-modules of tor dimension $\\leq 1$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up perfect modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESV","source_file":"flat.tex","source_line":13366,"source_end_line":13375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13366-L13375","statement_sha256":"5bc6d52d761cdb66a36433aff6366e649f0afa5779b9e131e54d9b0669666f8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7876,"rank":7876,"depth":24,"x":1870.024,"y":622.903,"cluster":"scheme-morphisms"},{"id":"stacks:0F8M","tag":"0F8M","title":"An operator introduced by Berthelot and Ogus · Lemma 0F8M","summary":"Let X be a scheme. Let D ⊂ X be an effective Cartier divisor with ideal sheaf I ⊂ O_X. Let F^bullet be a complex of quasi-coherent O_X-modules such that F^i is I-torsion free for all i. Then eta_IF^bullet is a complex of quasi-coherent O_X-modules. Moreover, if U = Spec(A) ⊂ X is affine open and D ∩ U = V(f), then eta_f(F^bullet(U)) is canonically isomorphic to (eta_IF^bullet)(U).","statement_latex":"Let $X$ be a scheme. Let $D \\subset X$ be an effective\nCartier divisor with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.\nLet $\\mathcal{F}^\\bullet$ be a complex of quasi-coherent\n$\\mathcal{O}_X$-modules such that $\\mathcal{F}^i$ is\n$\\mathcal{I}$-torsion free for all $i$. Then\n$\\eta_\\mathcal{I}\\mathcal{F}^\\bullet$ is a complex of\nquasi-coherent $\\mathcal{O}_X$-modules. Moreover,\nif $U = \\Spec(A) \\subset X$ is affine open and $D \\cap U = V(f)$,\nthen $\\eta_f(\\mathcal{F}^\\bullet(U))$ is canonically isomorphic\nto $(\\eta_\\mathcal{I}\\mathcal{F}^\\bullet)(U)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8M","source_file":"flat.tex","source_line":13411,"source_end_line":13423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13411-L13423","statement_sha256":"03a1356b131b278ee17d77c80db46df3409d6aa329a6f348eda43005889911fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7877,"rank":7877,"depth":0,"x":2193.937,"y":631.311,"cluster":"scheme-morphisms"},{"id":"stacks:0GTV","tag":"0GTV","title":"An operator introduced by Berthelot and Ogus · Lemma 0GTV","summary":"Let X be a scheme. Let D ⊂ X be an effective Cartier divisor with ideal sheaf I ⊂ O_X. The functor Leta_I : D(O_X) → D(O_X) of Cohomology, Lemma [Tag 0F8Q] sends D_QCoh(O_X) into itself. Moreover, if X = Spec(A) is affine and D = V(f), then the functor Leta_f on D(A) defined in More on Algebra, Lemma [Tag 0F7R] and the functor Leta_I on D_QCoh(O_X) correspond via the equivalence of Derived Categories of Schemes, Lemma [Tag 06Z0].","statement_latex":"Let $X$ be a scheme. Let $D \\subset X$ be an effective\nCartier divisor with ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.\nThe functor $L\\eta_\\mathcal{I} : D(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$ of\nCohomology, Lemma \\ref{cohomology-lemma-Leta} sends\n$D_\\QCoh(\\mathcal{O}_X)$ into itself.\nMoreover, if $X = \\Spec(A)$ is affine and $D = V(f)$,\nthen the functor $L\\eta_f$ on $D(A)$ defined in\nMore on Algebra, Lemma \\ref{more-algebra-lemma-Leta}\nand the functor $L\\eta_\\mathcal{I}$ on $D_\\QCoh(\\mathcal{O}_X)$\ncorrespond via the equivalence of Derived Categories of Schemes, Lemma\n\\ref{perfect-lemma-affine-compare-bounded}.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"An operator introduced by Berthelot and Ogus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTV","source_file":"flat.tex","source_line":13429,"source_end_line":13442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13429-L13442","statement_sha256":"2eedfe820dfabf05f1ff7b8b310f05ae9020f8a5ac12abbc7d3b1e1a3cf800ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":7878,"rank":7878,"depth":27,"x":1948.241,"y":808.969,"cluster":"scheme-morphisms"},{"id":"stacks:0F8T","tag":"0F8T","title":"Blowing up complexes, II · Lemma 0F8T","summary":"In Situation [Tag 0F8S] let h : Y → X be a morphism of schemes such that the pullback E = h^-1D of D is defined (Divisors, Definition [Tag 01WV]). Let (U, A, f, M^bullet) is an affine chart for (X, D, M). Let V = Spec(B) ⊂ Y is an affine open with h(V) ⊂ U. Denote g ∈ B the image of f ∈ A. Then • (V, B, g, M^bullet ⊗_A B) is an affine chart for (Y, E, Lh^*M), • I_i(M^bullet, f)B = I_i(M^bullet ⊗_A B, g) in B, and • if (X, D, M) is a good triple, then (Y, E, Lh^*M) is a…","statement_latex":"In Situation \\ref{situation-complex-and-divisor} let $h : Y \\to X$ be a\nmorphism of schemes such that the pullback $E = h^{-1}D$ of $D$\nis defined (Divisors, Definition\n\\ref{divisors-definition-pullback-effective-Cartier-divisor}).\nLet $(U, A, f, M^\\bullet)$ is an affine chart for $(X, D, M)$.\nLet $V = \\Spec(B) \\subset Y$ is an affine open with $h(V) \\subset U$.\nDenote $g \\in B$ the image of $f \\in A$.\nThen\n\\begin{enumerate}\n\\item $(V, B, g, M^\\bullet \\otimes_A B)$ is an affine chart for $(Y, E, Lh^*M)$,\n\\item $I_i(M^\\bullet, f)B = I_i(M^\\bullet \\otimes_A B, g)$ in $B$, and\n\\item if $(X, D, M)$ is a good triple, then\n$(Y, E, Lh^*M)$ is a good triple.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8T","source_file":"flat.tex","source_line":13487,"source_end_line":13503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13487-L13503","statement_sha256":"516c9a48f308afe696ae77aa83918f40ef224ad78f1d8aacf2b67dcacdd6d9df","origin":"The Stacks Project","memory_eligible":false,"source_rank":7879,"rank":7879,"depth":28,"x":1986.559,"y":538.46,"cluster":"scheme-morphisms"},{"id":"stacks:0GTW","tag":"0GTW","title":"Blowing up complexes, II · Lemma 0GTW","summary":"Let X, D, I, M be as in Situation [Tag 0F8S]. If (X, D, M) is a good triple, then Leta_IM is a perfect object of D(O_X).","statement_latex":"Let $X, D, \\mathcal{I}, M$ be as in\nSituation \\ref{situation-complex-and-divisor}.\nIf $(X, D, M)$ is a good triple, then\n$L\\eta_\\mathcal{I}M$ is a perfect object\nof $D(\\mathcal{O}_X)$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTW","source_file":"flat.tex","source_line":13519,"source_end_line":13526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13519-L13526","statement_sha256":"fb7c300bf7b3b89fe0ccad4d673708aec727ae7332c27d5dabc13fa0d5264ad9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7880,"rank":7880,"depth":28,"x":2175.906,"y":759.748,"cluster":"scheme-morphisms"},{"id":"stacks:0GTX","tag":"0GTX","title":"Blowing up complexes, II · Lemma 0GTX","summary":"Let X, D, I, M be as in Situation [Tag 0F8S]. Assume (X, D, M) is a good triple. If there exists a locally bounded complex M^bullet of finite locally free O_X-modules representing M, then there exists a locally bounded complex Q^bullet of finite locally free O_X'-modules representing Leta_IM.","statement_latex":"Let $X, D, \\mathcal{I}, M$ be as in\nSituation \\ref{situation-complex-and-divisor}.\nAssume $(X, D, M)$ is a good triple.\nIf there exists a locally bounded complex $\\mathcal{M}^\\bullet$\nof finite locally free $\\mathcal{O}_X$-modules representing $M$,\nthen there exists a locally bounded complex $\\mathcal{Q}^\\bullet$\nof finite locally free $\\mathcal{O}_{X'}$-modules representing\n$L\\eta_\\mathcal{I}M$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTX","source_file":"flat.tex","source_line":13534,"source_end_line":13544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13534-L13544","statement_sha256":"dbeeb71eb359d083afe55c7a1d0e0b98e4f4d96f340235908fca33a0588c42f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7881,"rank":7881,"depth":27,"x":1858.22,"y":703.994,"cluster":"scheme-morphisms"},{"id":"stacks:0F9X","tag":"0F9X","title":"Blowing up complexes, II · Lemma 0F9X","summary":"In Situation [Tag 0F8S] let h : Y → X be a morphism of schemes such that the pullback E = h^-1D is defined. If (X, D, M) is a good triple, then Lh^*(Leta_IM) = Leta_J(Lh^*M) in D(O_Y) where J is the ideal sheaf of E.","statement_latex":"In Situation \\ref{situation-complex-and-divisor} let $h : Y \\to X$\nbe a morphism of schemes such that the pullback $E = h^{-1}D$\nis defined. If $(X, D, M)$ is a good triple, then\n$$\nLh^*(L\\eta_\\mathcal{I}M) = L\\eta_\\mathcal{J}(Lh^*M)\n$$\nin $D(\\mathcal{O}_Y)$ where $\\mathcal{J}$ is the ideal sheaf of $E$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9X","source_file":"flat.tex","source_line":13569,"source_end_line":13578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13569-L13578","statement_sha256":"e983462f13eb5edc9566c843d115039c53ca30934974557f0e8e11f64f1f87a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7882,"rank":7882,"depth":28,"x":2137.41,"y":564.796,"cluster":"scheme-morphisms"},{"id":"stacks:0GTY","tag":"0GTY","title":"Blowing up complexes, II · Lemma 0GTY","summary":"In Situation [Tag 0F8S] there is a unique morphism b : X' → X such that • the pullback D' = b^-1D is defined and (X', D', M') is a good triple where M' = Lb^*M, and • for any morphism of schemes h : Y → X such that the pullback E = h^-1D is defined and (Y, E, Lh^*M) is a good triple, there is a unique factorization of h through b. Moreover, for any affine chart (U, A, f, M^bullet) the restriction b^-1(U) → U is the blowing up in the product of the ideals I_i(M^bullet, f)…","statement_latex":"In Situation \\ref{situation-complex-and-divisor}\nthere is a unique morphism $b : X' \\to X$ such that\n\\begin{enumerate}\n\\item the pullback $D' = b^{-1}D$ is defined and\n$(X', D', M')$ is a good triple where $M' = Lb^*M$, and\n\\item for any morphism of schemes $h : Y \\to X$ such that\nthe pullback $E = h^{-1}D$ is defined and $(Y, E, Lh^*M)$\nis a good triple, there is a unique factorization of $h$ through $b$.\n\\end{enumerate}\nMoreover, for any affine chart $(U, A, f, M^\\bullet)$ the restriction\n$b^{-1}(U) \\to U$ is the blowing up in the product of the ideals\n$I_i(M^\\bullet, f)$ and for any quasi-compact open $W \\subset X$ the\nrestriction $b|_{b^{-1}(W)} : b^{-1}(W) \\to W$ is a $W \\setminus D$-admissible\nblowing up.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTY","source_file":"flat.tex","source_line":13587,"source_end_line":13603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13587-L13603","statement_sha256":"7aad149d32079be497d43be56749ae9fac0dc4a3bc4704c684737dab6616e369","origin":"The Stacks Project","memory_eligible":false,"source_rank":7883,"rank":7883,"depth":29,"x":2043.446,"y":825.947,"cluster":"scheme-morphisms"},{"id":"stacks:0F8U","tag":"0F8U","title":"Blowing up complexes, II · Lemma 0F8U","summary":"In Situation [Tag 0F8S] let b : X' → X be the morphism of Lemma [Tag 0GTY]. Consider the effective Cartier divisor D' = b^-1D with ideal sheaf I' ⊂ O_X'. Then Q = Leta_I'Lb^*M is a perfect object of D(O_X').","statement_latex":"In Situation \\ref{situation-complex-and-divisor} let $b : X' \\to X$\nbe the morphism of Lemma \\ref{lemma-complex-and-divisor-blowup-pre}.\nConsider the effective Cartier divisor $D' = b^{-1}D$ with ideal sheaf\n$\\mathcal{I}' \\subset \\mathcal{O}_{X'}$. Then $Q = L\\eta_{\\mathcal{I}'}Lb^*M$\nis a perfect object of $D(\\mathcal{O}_{X'})$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8U","source_file":"flat.tex","source_line":13716,"source_end_line":13723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13716-L13723","statement_sha256":"14d29b302041285efc470156213dd3df8c91d4992a441517c6510847b14d4b5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7884,"rank":7884,"depth":30,"x":1902.674,"y":579.976,"cluster":"scheme-morphisms"},{"id":"stacks:0F8V","tag":"0F8V","title":"Blowing up complexes, II · Lemma 0F8V","summary":"In Situation [Tag 0F8S] let h : Y → X be a morphism of schemes such that the pullback E = h^-1D is defined. Let b : X' → X, resp. c : Y' → Y be as constructed in Lemma [Tag 0GTY] for D ⊂ X and M, resp. E ⊂ Y and Lh^*M. Then Y' is the strict transform of Y with respect to b : X' → X (see proof for a precise formulation of this) and Leta_J'L(h ∘ c)^*M = L(Y' → X')^*Q where Q = Leta_I'Lb^*M as in Lemma [Tag 0F8U]. In particular, if (Y, E, Lh^*M) is a good triple and k : Y →…","statement_latex":"In Situation \\ref{situation-complex-and-divisor} let $h : Y \\to X$\nbe a morphism of schemes such that the pullback $E = h^{-1}D$\nis defined. Let $b : X' \\to X$, resp.\\ $c : Y' \\to Y$ be as constructed in\nLemma \\ref{lemma-complex-and-divisor-blowup-pre} for\n$D \\subset X$ and $M$, resp.\\ $E \\subset Y$ and $Lh^*M$.\nThen $Y'$ is the strict transform of $Y$ with respect to $b : X' \\to X$\n(see proof for a precise formulation of this) and\n$$\nL\\eta_{\\mathcal{J}'}L(h \\circ c)^*M = L(Y' \\to X')^*Q\n$$\nwhere $Q = L\\eta_{\\mathcal{I}'}Lb^*M$ as in\nLemma \\ref{lemma-complex-and-divisor-blowup}.\nIn particular, if $(Y, E, Lh^*M)$ is a good triple and\n$k : Y \\to X'$ is the unique morphism such that\n$h = b \\circ k$, then $L\\eta_\\mathcal{J}Lh^*M = Lk^*Q$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8V","source_file":"flat.tex","source_line":13730,"source_end_line":13747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13730-L13747","statement_sha256":"02986f59776c1e5316a9474b7126954c7ae5c50b9e073a86f925c9791b19bfdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":7885,"rank":7885,"depth":31,"x":2204.385,"y":681.509,"cluster":"scheme-morphisms"},{"id":"stacks:0F8W","tag":"0F8W","title":"Blowing up complexes, II · Lemma 0F8W","summary":"In Situation [Tag 0F8S] let W ⊂ X be the maximal open subscheme over which the cohomology sheaves of M are locally free. Then the morphism b : X' → X of Lemma [Tag 0GTY] is an isomorphism over W.","statement_latex":"In Situation \\ref{situation-complex-and-divisor} let $W \\subset X$\nbe the maximal open subscheme over which the cohomology sheaves\nof $M$ are locally free. Then the morphism $b : X' \\to X$\nof Lemma \\ref{lemma-complex-and-divisor-blowup-pre} is an isomorphism\nover $W$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8W","source_file":"flat.tex","source_line":13791,"source_end_line":13798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13791-L13798","statement_sha256":"178e809f16fa68022aff89d0c8f381437cd0694b3ddd34b6d11fdb02d01fc755","origin":"The Stacks Project","memory_eligible":false,"source_rank":7886,"rank":7886,"depth":30,"x":1900.153,"y":777.871,"cluster":"scheme-morphisms"},{"id":"stacks:0GTZ","tag":"0GTZ","title":"Blowing up complexes, II · Lemma 0GTZ","summary":"Let X, D, I, M be as in Situation [Tag 0F8S]. If (X, D, M) is a good triple, then there exists a closed immersion i : T → D of finite presentation with the following properties • T scheme theoretically contains D ∩ W where W ⊂ X is the maximal open over which the cohomology sheaves of M are locally free, • the cohomology sheaves of Li^*Leta_IM are locally free, and • for any point t ∈ T with image x = i(t) ∈ W the rank of H^i(M)_x over O_X, x and the rank of…","statement_latex":"Let $X, D, \\mathcal{I}, M$ be as in\nSituation \\ref{situation-complex-and-divisor}.\nIf $(X, D, M)$ is a good triple, then there exists a\nclosed immersion\n$$\ni : T \\longrightarrow D\n$$\nof finite presentation with the following properties\n\\begin{enumerate}\n\\item $T$ scheme theoretically contains $D \\cap W$\nwhere $W \\subset X$ is the maximal open over which the\ncohomology sheaves of $M$ are locally free,\n\\item the cohomology sheaves of $Li^*L\\eta_\\mathcal{I}M$\nare locally free, and\n\\item for any point $t \\in T$ with image $x = i(t) \\in W$ the rank\nof $H^i(M)_x$ over $\\mathcal{O}_{X, x}$ and the rank\nof $H^i(Li^*L\\eta_\\mathcal{I}M)_t$ over $\\mathcal{O}_{T, t}$ agree.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GTZ","source_file":"flat.tex","source_line":13809,"source_end_line":13829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13809-L13829","statement_sha256":"2893775924ba125863d28c8874513550795eff0ca2584a639270378ec978bfac","origin":"The Stacks Project","memory_eligible":false,"source_rank":7887,"rank":7887,"depth":15,"x":2047.048,"y":534.103,"cluster":"scheme-morphisms"},{"id":"stacks:0F8X","tag":"0F8X","title":"Blowing up complexes, II · Lemma 0F8X","summary":"In Situation [Tag 0F8S]. Let b : X' → X and D' be as in Lemma [Tag 0GTY]. Let Q = Leta_I'Lb^*M be as in Lemma [Tag 0F8U]. Let W ⊂ X be the maximal open where M has locally free cohomology modules. Then there exists a closed immersion i : T → D' of finite presentation such that • D' ∩ b^-1(W) ⊂ T scheme theoretically, • Li^*Q has locally free cohomology sheaves, and • for t ∈ T mapping to w ∈ W the rank of H^i(Li^*Q)_t over O_T, t is equal to the rank of H^i(M)_x over O_X, x.","statement_latex":"In Situation \\ref{situation-complex-and-divisor}. Let $b : X' \\to X$\nand $D'$ be as in Lemma \\ref{lemma-complex-and-divisor-blowup-pre}. Let\n$Q = L\\eta_{\\mathcal{I}'}Lb^*M$ be as in\nLemma \\ref{lemma-complex-and-divisor-blowup}.\nLet $W \\subset X$ be the maximal open where $M$ has\nlocally free cohomology modules.\nThen there exists a closed immersion $i : T \\to D'$ of finite presentation\nsuch that\n\\begin{enumerate}\n\\item $D' \\cap b^{-1}(W) \\subset T$ scheme theoretically,\n\\item $Li^*Q$ has locally free cohomology sheaves, and\n\\item for $t \\in T$ mapping to $w \\in W$ the rank\nof $H^i(Li^*Q)_t$ over $\\mathcal{O}_{T, t}$ is equal to the\nrank of $H^i(M)_x$ over $\\mathcal{O}_{X, x}$.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8X","source_file":"flat.tex","source_line":13859,"source_end_line":13876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13859-L13876","statement_sha256":"9801471365e6e408931178640e595f44ea6ec4f76a34a7d87f3198d91152e51e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7888,"rank":7888,"depth":31,"x":2134.792,"y":797.296,"cluster":"scheme-morphisms"},{"id":"stacks:0F8Y","tag":"0F8Y","title":"Blowing up complexes, II · Lemma 0F8Y","summary":"In Situation [Tag 0F8S]. Let b : X' → X, D' ⊂ X', and Q be as in Lemma [Tag 0F8U]. Let ρ = (ρ_i)_i ∈ Z be integers. Let W(ρ) ⊂ X be the maximal open subscheme where H^i(M) is locally free of rank ρ_i for all i. Let i : T → D' be as in Lemma [Tag 0F8X]. Then there exists an open and closed subscheme T(ρ) ⊂ T containing D' ∩ b^-1(W(ρ)) scheme theoretically such that H^i(Li^*Q|_T(ρ)) is locally free of rank ρ_i for all i.","statement_latex":"In Situation \\ref{situation-complex-and-divisor}. Let $b : X' \\to X$,\n$D' \\subset X'$, and $Q$ be as in\nLemma \\ref{lemma-complex-and-divisor-blowup}.\nLet $\\rho = (\\rho_i)_{i \\in \\mathbf{Z}}$ be integers.\nLet $W(\\rho) \\subset X$ be the maximal open subscheme where\n$H^i(M)$ is locally free of rank $\\rho_i$ for all $i$.\nLet $i : T \\to D'$ be as in Lemma \\ref{lemma-complex-and-divisor-blowup-T}.\nThen there exists an open and closed subscheme $T(\\rho) \\subset T$\ncontaining $D' \\cap b^{-1}(W(\\rho))$ scheme theoretically\nsuch that $H^i(Li^*Q|_{T(\\rho)})$ is locally free of rank $\\rho_i$\nfor all $i$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F8Y","source_file":"flat.tex","source_line":13888,"source_end_line":13901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13888-L13901","statement_sha256":"61916681b3751b5b2c1f5fc3ea91d6c4ab8014abcf8e012922a130a89d52090a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7889,"rank":7889,"depth":32,"x":1858.343,"y":652.96,"cluster":"scheme-morphisms"},{"id":"stacks:0GU0","tag":"0GU0","title":"Blowing up complexes, II · Lemma 0GU0","summary":"In Situation [Tag 0F8S]. Let b : X' → X, D' ⊂ X', and Q be as in Lemma [Tag 0F8U]. If there exists a locally bounded complex M^bullet of finite locally free O_X-modules representing M, then there exists a locally bounded complex Q^bullet of finite locally free O_X'-modules representing Q.","statement_latex":"In Situation \\ref{situation-complex-and-divisor}. Let $b : X' \\to X$,\n$D' \\subset X'$, and $Q$ be as in\nLemma \\ref{lemma-complex-and-divisor-blowup}.\nIf there exists a locally bounded complex $\\mathcal{M}^\\bullet$\nof finite locally free $\\mathcal{O}_X$-modules representing $M$,\nthen there exists a locally bounded complex $\\mathcal{Q}^\\bullet$\nof finite locally free $\\mathcal{O}_{X'}$-modules representing $Q$.","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GU0","source_file":"flat.tex","source_line":13911,"source_end_line":13920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13911-L13920","statement_sha256":"132cc68a94ce8979cf120d0188be414381683d0164fbc232c812e7245f1ba984","origin":"The Stacks Project","memory_eligible":false,"source_rank":7890,"rank":7890,"depth":31,"x":2178.373,"y":602.51,"cluster":"scheme-morphisms"},{"id":"stacks:0F9Y","tag":"0F9Y","title":"Blowing up complexes, II · Lemma 0F9Y","summary":"Let X be a scheme and let D ⊂ X be an effective Cartier divisor. Let M ∈ D(O_X) be a perfect object. Let W ⊂ X be the maximal open over which the cohomology sheaves H^i(M) are locally free. There exists a proper morphism b : X' → X and an object Q in D(O_X') with the following properties • b : X' → X is an isomorphism over X setminus D, • b : X' → X is an isomorphism over W, • D' = b^-1D is an effective Cartier divisor, • Q = Leta_I'Lb^*M where I' is the ideal sheaf of…","statement_latex":"Let $X$ be a scheme and let $D \\subset X$ be an effective Cartier divisor. Let\n$M \\in D(\\mathcal{O}_X)$ be a perfect object. Let $W \\subset X$ be the maximal\nopen over which the cohomology sheaves $H^i(M)$ are locally free.\nThere exists a proper morphism $b : X' \\longrightarrow X$\nand an object $Q$ in $D(\\mathcal{O}_{X'})$ with the following properties\n\\begin{enumerate}\n\\item $b : X' \\to X$ is an isomorphism over $X \\setminus D$,\n\\item $b : X' \\to X$ is an isomorphism over $W$,\n\\item $D' = b^{-1}D$ is an effective Cartier divisor,\n\\item $Q = L\\eta_{\\mathcal{I}'}Lb^*M$ where $\\mathcal{I}'$\nis the ideal sheaf of $D'$,\n\\item $Q$ is a perfect object of $D(\\mathcal{O}_{X'})$,\n\\item there exists a closed immersion $i : T \\to D'$ of finite presentation\nsuch that\n\\begin{enumerate}\n\\item $D' \\cap b^{-1}(W) \\subset T$ scheme theoretically,\n\\item $Li^*Q$ has finite locally free cohomology sheaves,\n\\item for $t \\in T$ with image $w \\in W$ the rank\nof $H^i(Li^*Q)_t$ over $\\mathcal{O}_{T, t}$ is equal to the\nrank of $H^i(M)_x$ over $\\mathcal{O}_{X, x}$,\n\\end{enumerate}\n\\item for any affine chart $(U, A, f, M^\\bullet)$ for $(X, D, M)$\nthe restriction of $b$ to $U$ is the blowing up of $U = \\Spec(A)$\nin the ideal $I = \\prod I_i(M^\\bullet, f)$, and\n\\item for any affine chart $(V, B, g, N^\\bullet)$ for $(X', D', Lb^*N)$\nsuch that $I_i(N^\\bullet, g)$ is principal, we have\n\\begin{enumerate}\n\\item $Q|_V$ corresponds to $\\eta_gN^\\bullet$,\n\\item $T \\subset V \\cap D'$ corresponds to the ideal\n$J(N^\\bullet, g) = \\sum J_i(N^\\bullet, g) \\subset B/gB$\nstudied in\nMore on Algebra, Lemma \\ref{more-algebra-lemma-eta-vanishing-beta-plus}.\n\\end{enumerate}\n\\item If $M$ can be represented by a locally bounded complex\nof finite locally free $\\mathcal{O}_X$-modules, then $Q$ can\nbe represented by a bounded complex of finite locally free\n$\\mathcal{O}_{X'}$-modules.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9Y","source_file":"flat.tex","source_line":13932,"source_end_line":13972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L13932-L13972","statement_sha256":"73e9dfec0cb50b540a147165b27aa53b9724f9bcadafd1c0e441125b6437e3f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7891,"rank":7891,"depth":32,"x":1982.891,"y":821.379,"cluster":"scheme-morphisms"},{"id":"stacks:0F90","tag":"0F90","title":"Blowing up complexes, III · Lemma 0F90","summary":"The construction above has the following properties: • b is an isomorphism over P^1_U ∪ A^1_X, • the restriction of Q to A^1_X is equal to the pullback of E, • there exists a closed immersion i : T → W_∞ of finite presentation such that (W_∞ → X)^-1U ⊂ T scheme theoretically and such that Li^*Q has locally free cohomology sheaves, • for t ∈ T with image u ∈ U we have that the rank H^i(Li^*Q)_t over O_T, t is equal to the rank of H^i(M)_u over O_U, u, • if E can be…","statement_latex":"The construction above has the following properties:\n\\begin{enumerate}\n\\item $b$ is an isomorphism over $\\mathbf{P}^1_U \\cup \\mathbf{A}^1_X$,\n\\item the restriction of $Q$ to $\\mathbf{A}^1_X$\nis equal to the pullback of $E$,\n\\item there exists a closed immersion $i : T \\to W_\\infty$\nof finite presentation such that $(W_\\infty \\to X)^{-1}U \\subset T$\nscheme theoretically and such that $Li^*Q$ has locally free cohomology\nsheaves,\n\\item for $t \\in T$ with image $u \\in U$ we have that the\nrank $H^i(Li^*Q)_t$ over $\\mathcal{O}_{T, t}$ is equal to the rank\nof $H^i(M)_u$ over $\\mathcal{O}_{U, u}$,\n\\item if $E$ can be represented by a locally bounded complex of\nfinite locally free $\\mathcal{O}_X$-modules, then $Q$ can be represented\nby a locally bounded complex of finite locally free $\\mathcal{O}_W$-modules.\n\\end{enumerate}","area":"Scheme Morphisms","chapter":"More on Flatness","chapter_id":"flat","section":"Blowing up complexes, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F90","source_file":"flat.tex","source_line":14047,"source_end_line":14065,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/flat.tex#L14047-L14065","statement_sha256":"8bce1a254a77739364a08e833672f11eba30fb1966d9d46ad3507a3c3f316668","origin":"The Stacks Project","memory_eligible":false,"source_rank":7892,"rank":7892,"depth":33,"x":1951.017,"y":548.973,"cluster":"scheme-morphisms"},{"id":"stacks:022P","tag":"022P","title":"Equivalence relations · Definition 022P","summary":"Let S be a scheme. Let U be a scheme over S. • A pre-relation on U over S is any morphism of schemes j : R → U ×_S U. In this case we set t = pr_0 ∘ j and s = pr_1 ∘ j, so that j = (t, s). • A relation on U over S is a monomorphism of schemes j : R → U ×_S U. • A pre-equivalence relation is a pre-relation j : R → U ×_S U such that the image of j : R(T) → U(T) × U(T) is an equivalence relation for all T/S. • We say a morphism R → U ×_S U of schemes is an equivalence…","statement_latex":"Let $S$ be a scheme. Let $U$ be a scheme over $S$.\n\\begin{enumerate}\n\\item A {\\it pre-relation} on $U$ over $S$ is any morphism\nof schemes $j : R \\to U \\times_S U$. In this case we set\n$t = \\text{pr}_0 \\circ j$ and $s = \\text{pr}_1 \\circ j$, so\nthat $j = (t, s)$.\n\\item A {\\it relation} on $U$ over $S$ is a monomorphism\nof schemes $j : R \\to U \\times_S U$.\n\\item A {\\it pre-equivalence relation} is a pre-relation\n$j : R \\to U \\times_S U$ such that the image of\n$j : R(T) \\to U(T) \\times U(T)$ is an equivalence relation for\nall $T/S$.\n\\item We say a morphism $R \\to U \\times_S U$ of schemes is\nan {\\it equivalence relation on $U$ over $S$}\nif and only if for every scheme $T$ over $S$ the $T$-valued\npoints of $R$ define an equivalence relation\non the set of $T$-valued points of $U$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Equivalence relations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022P","source_file":"groupoids.tex","source_line":85,"source_end_line":105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L85-L105","statement_sha256":"bfd76738d2db608fcca9547ae55b4b449d45f3d97d985545802934994d94ef43","origin":"The Stacks Project","memory_eligible":false,"source_rank":7893,"rank":7893,"depth":0,"x":1136.939,"y":1140.0,"cluster":"groupoids-quotients"},{"id":"stacks:02V8","tag":"02V8","title":"Equivalence relations · Lemma 02V8","summary":"Let S be a scheme. Let U be a scheme over S. Let j : R → U ×_S U be a pre-relation. Let g : U' → U be a morphism of schemes. Finally, set R' = (U' ×_S U')×_U ×_S U R xrightarrowj' U' ×_S U' Then j' is a pre-relation on U' over S. If j is a relation, then j' is a relation. If j is a pre-equivalence relation, then j' is a pre-equivalence relation. If j is an equivalence relation, then j' is an equivalence relation.","statement_latex":"Let $S$ be a scheme.\nLet $U$ be a scheme over $S$.\nLet $j : R \\to U \\times_S U$ be a pre-relation.\nLet $g : U' \\to U$ be a morphism of schemes.\nFinally, set\n$$\nR' = (U' \\times_S U')\\times_{U \\times_S U} R\n\\xrightarrow{j'}\nU' \\times_S U'\n$$\nThen $j'$ is a pre-relation on $U'$ over $S$.\nIf $j$ is a relation, then $j'$ is a relation.\nIf $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation.\nIf $j$ is an equivalence relation, then $j'$ is an equivalence relation.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V8","source_file":"groupoids.tex","source_line":111,"source_end_line":127,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L111-L127","statement_sha256":"a978c452b3efaddb454ef6c5340c7ec70d9e38189aa0b1b4f7d1bba6e4102fdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7894,"rank":7894,"depth":0,"x":1121.138,"y":1146.82,"cluster":"groupoids-quotients"},{"id":"stacks:02V9","tag":"02V9","title":"Equivalence relations · Definition 02V9","summary":"Let S be a scheme. Let U be a scheme over S. Let j : R → U ×_S U be a pre-relation. Let g : U' → U be a morphism of schemes. The pre-relation j' : R' → U' ×_S U' is called the restriction, or pullback of the pre-relation j to U'. In this situation we sometimes write R' = R|_U'.","statement_latex":"Let $S$ be a scheme.\nLet $U$ be a scheme over $S$.\nLet $j : R \\to U \\times_S U$ be a pre-relation.\nLet $g : U' \\to U$ be a morphism of schemes.\nThe pre-relation $j' : R' \\to U' \\times_S U'$ is called\nthe {\\it restriction}, or {\\it pullback} of the pre-relation $j$ to $U'$.\nIn this situation we sometimes write $R' = R|_{U'}$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Equivalence relations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02V9","source_file":"groupoids.tex","source_line":133,"source_end_line":142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L133-L142","statement_sha256":"c41357fd5f1ff5d86992b7d194c366b410ceb8729a5b6df344196effc56e2b5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7895,"rank":7895,"depth":0,"x":1131.357,"y":1127.016,"cluster":"groupoids-quotients"},{"id":"stacks:022Q","tag":"022Q","title":"Equivalence relations · Lemma 022Q","summary":"Let j : R → U ×_S U be a pre-relation. Consider the relation on points of the scheme U defined by the rule x sim y ⇔ ∃ r ∈ R : t(r) = x, s(r) = y. If j is a pre-equivalence relation then this is an equivalence relation.","statement_latex":"Let $j : R \\to U \\times_S U$ be a pre-relation.\nConsider the relation on points of the scheme $U$ defined by\nthe rule\n$$\nx \\sim y\n\\Leftrightarrow\n\\exists\\ r \\in R :\nt(r) = x,\ns(r) = y.\n$$\nIf $j$ is a pre-equivalence relation then this is an\nequivalence relation.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022Q","source_file":"groupoids.tex","source_line":144,"source_end_line":158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L144-L158","statement_sha256":"3bf03858b69b07e46e0226fe16556e6ab39d9f0358d5d59a507b4f14bddca507","origin":"The Stacks Project","memory_eligible":false,"source_rank":7896,"rank":7896,"depth":0,"x":1141.171,"y":1152.239,"cluster":"groupoids-quotients"},{"id":"stacks:0DT7","tag":"0DT7","title":"Equivalence relations · Lemma 0DT7","summary":"Let j : R → U ×_S U be a pre-relation. Assume • s, t are unramified, • for any algebraically closed field k over S the map R(k) → U(k) × U(k) is an equivalence relation, • there are morphisms e : U → R, i : R → R, c : R ×_s, U, t R → R such that xymatrix U ar[r]_e ar[d]_Δ & R ar[d]_j & R ar[d]^j ar[r]_i & R ar[d]^j & R ×_s, U, t R ar[d]^j × j ar[r]_c & R ar[d]^j U ×_S U ar[r] & U ×_S U & U ×_S U ar[r]^flip & U ×_S U & U ×_S U ×_S U ar[r]^pr_02 & U ×_S U are commutative.…","statement_latex":"Let $j : R \\to U \\times_S U$ be a pre-relation. Assume\n\\begin{enumerate}\n\\item $s, t$ are unramified,\n\\item for any algebraically closed field $k$ over $S$\nthe map $R(k) \\to U(k) \\times U(k)$ is an equivalence relation,\n\\item there are morphisms $e : U \\to R$, $i : R \\to R$,\n$c : R \\times_{s, U, t} R \\to R$ such that\n$$\n\\xymatrix{\nU \\ar[r]_e \\ar[d]_\\Delta &\nR \\ar[d]_j &\nR \\ar[d]^j \\ar[r]_i &\nR \\ar[d]^j &\nR \\times_{s, U, t} R \\ar[d]^{j \\times j} \\ar[r]_c &\nR \\ar[d]^j \\\\\nU \\times_S U \\ar[r] &\nU \\times_S U &\nU \\times_S U \\ar[r]^{flip} &\nU \\times_S U &\nU \\times_S U \\times_S U \\ar[r]^{\\text{pr}_{02}} &\nU \\times_S U\n}\n$$\nare commutative.\n\\end{enumerate}\nThen $j$ is an equivalence relation.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DT7","source_file":"groupoids.tex","source_line":196,"source_end_line":224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L196-L224","statement_sha256":"ccdb31dc802a8d85e448ffcc0f16edcc55c8dbc991bbd8739c1a9ac8e2e0609b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7897,"rank":7897,"depth":20,"x":1109.501,"y":1136.954,"cluster":"groupoids-quotients"},{"id":"stacks:022S","tag":"022S","title":"Group schemes · Definition 022S","summary":"Let S be a scheme. • A group scheme over S is a pair (G, m), where G is a scheme over S and m : G ×_S G → G is a morphism of schemes over S with the following property: For every scheme T over S the pair (G(T), m) is a group. • A morphism ψ : (G, m) → (G', m') of group schemes over S is a morphism ψ : G → G' of schemes over S such that for every T/S the induced map ψ : G(T) → G'(T) is a homomorphism of groups.","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item A {\\it group scheme over $S$} is a pair $(G, m)$, where\n$G$ is a scheme over $S$ and $m : G \\times_S G \\to G$ is\na morphism of schemes over $S$ with the following property:\nFor every scheme $T$ over $S$ the pair $(G(T), m)$\nis a group.\n\\item A {\\it morphism $\\psi : (G, m) \\to (G', m')$ of group schemes over $S$}\nis a morphism $\\psi : G \\to G'$ of schemes over $S$ such that for\nevery $T/S$ the induced map $\\psi : G(T) \\to G'(T)$ is a homomorphism\nof groups.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Group schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022S","source_file":"groupoids.tex","source_line":279,"source_end_line":293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L279-L293","statement_sha256":"1911258d1f944bdba3ff868602eecbf6a79ed76f0fceb818a9e6855e58bd5031","origin":"The Stacks Project","memory_eligible":false,"source_rank":7898,"rank":7898,"depth":0,"x":1149.419,"y":1129.624,"cluster":"groupoids-quotients"},{"id":"stacks:022T","tag":"022T","title":"Group schemes · Lemma 022T","summary":"Let (G, m) be a group scheme over S. Let S' → S be a morphism of schemes. The pullback (G_S', m_S') is a group scheme over S'.","statement_latex":"Let $(G, m)$ be a group scheme over $S$.\nLet $S' \\to S$ be a morphism of schemes.\nThe pullback $(G_{S'}, m_{S'})$ is a group scheme over $S'$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022T","source_file":"groupoids.tex","source_line":317,"source_end_line":322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L317-L322","statement_sha256":"f2c2185b90d21a11195d6c9b55d45ddc823aa3e017a28d5a4649a94e81db8f11","origin":"The Stacks Project","memory_eligible":false,"source_rank":7899,"rank":7899,"depth":0,"x":1123.505,"y":1160.296,"cluster":"groupoids-quotients"},{"id":"stacks:047D","tag":"047D","title":"Group schemes · Definition 047D","summary":"Let S be a scheme. Let (G, m) be a group scheme over S. • A closed subgroup scheme of G is a closed subscheme H ⊂ G such that m|_H ×_S H factors through H and induces a group scheme structure on H over S. • An open subgroup scheme of G is an open subscheme G' ⊂ G such that m|_G' ×_S G' factors through G' and induces a group scheme structure on G' over S.","statement_latex":"Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.\n\\begin{enumerate}\n\\item A {\\it closed subgroup scheme} of $G$ is a closed subscheme\n$H \\subset G$ such that $m|_{H \\times_S H}$ factors through $H$ and induces a\ngroup scheme structure on $H$ over $S$.\n\\item An {\\it open subgroup scheme} of $G$ is an open subscheme\n$G' \\subset G$ such that $m|_{G' \\times_S G'}$ factors through $G'$\nand induces a group scheme structure on $G'$ over $S$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Group schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047D","source_file":"groupoids.tex","source_line":328,"source_end_line":339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L328-L339","statement_sha256":"060e4b618efdebf4f4c897eaafbf7eda51479bf8a279af93a8cdd34c4a98d59b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7900,"rank":7900,"depth":0,"x":1117.613,"y":1119.966,"cluster":"groupoids-quotients"},{"id":"stacks:0G8L","tag":"0G8L","title":"Group schemes · Lemma 0G8L","summary":"Let S be a scheme. Let (G, m, e, i) be a group scheme over S. • A closed subscheme H ⊂ G is a closed subgroup scheme if and only if e : S → G, m|_H ×_S H : H ×_S H → G, and i|_H : H → G factor through H. • An open subscheme H ⊂ G is an open subgroup scheme if and only if e : S → G, m|_H ×_S H : H ×_S H → G, and i|_H : H → G factor through H.","statement_latex":"Let $S$ be a scheme. Let $(G, m, e, i)$ be a group scheme over $S$.\n\\begin{enumerate}\n\\item A closed subscheme $H \\subset G$ is a closed subgroup scheme\nif and only if $e : S \\to G$, $m|_{H \\times_S H} : H \\times_S H \\to G$,\nand $i|_H : H \\to G$ factor through $H$.\n\\item An open subscheme $H \\subset G$ is an open subgroup scheme\nif and only if $e : S \\to G$, $m|_{H \\times_S H} : H \\times_S H \\to G$,\nand $i|_H : H \\to G$ factor through $H$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8L","source_file":"groupoids.tex","source_line":346,"source_end_line":357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L346-L357","statement_sha256":"24ffc00e5515d85d5956e7c7976202d24391a6d6a336b43008ffef609418d9eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7901,"rank":7901,"depth":0,"x":1156.875,"y":1148.244,"cluster":"groupoids-quotients"},{"id":"stacks:047E","tag":"047E","title":"Group schemes · Definition 047E","summary":"Let S be a scheme. Let (G, m) be a group scheme over S. • We say G is a smooth group scheme if the structure morphism G → S is smooth. • We say G is a flat group scheme if the structure morphism G → S is flat. • We say G is a separated group scheme if the structure morphism G → S is separated. Add more as needed.","statement_latex":"Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.\n\\begin{enumerate}\n\\item We say $G$ is a {\\it smooth group scheme} if the structure\nmorphism $G \\to S$ is smooth.\n\\item We say $G$ is a {\\it flat group scheme} if the structure\nmorphism $G \\to S$ is flat.\n\\item We say $G$ is a {\\it separated group scheme} if the structure\nmorphism $G \\to S$ is separated.\n\\end{enumerate}\nAdd more as needed.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Group schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047E","source_file":"groupoids.tex","source_line":364,"source_end_line":376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L364-L376","statement_sha256":"4fde50d07291527031e5b45fe3eac062801911dc8ee759f515bd0bf9b9ee33d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":7902,"rank":7902,"depth":0,"x":1102.041,"y":1149.694,"cluster":"groupoids-quotients"},{"id":"stacks:047G","tag":"047G","title":"Properties of group schemes · Lemma 047G","summary":"Let S be a scheme. Let G be a group scheme over S. Then G → S is separated (resp. quasi-separated) if and only if the identity morphism e : S → G is a closed immersion (resp. quasi-compact).","statement_latex":"Let $S$ be a scheme.\nLet $G$ be a group scheme over $S$.\nThen $G \\to S$ is separated (resp.\\ quasi-separated) if and only if\nthe identity morphism $e : S \\to G$ is a closed immersion\n(resp.\\ quasi-compact).","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047G","source_file":"groupoids.tex","source_line":574,"source_end_line":581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L574-L581","statement_sha256":"5528a42c293711169a84122e04cac446cdfacde3b0b329e99eae65171ed17568","origin":"The Stacks Project","memory_eligible":false,"source_rank":7903,"rank":7903,"depth":17,"x":1143.478,"y":1115.806,"cluster":"groupoids-quotients"},{"id":"stacks:047H","tag":"047H","title":"Properties of group schemes · Lemma 047H","summary":"Let S be a scheme. Let G be a group scheme over S. Let T be a scheme over S and let ψ : T → G be a morphism over S. If T is flat over S, then the morphism T ×_S G → G, (t, g) ↦ m(ψ(t), g) is flat. In particular, if G is flat over S, then m : G ×_S G → G is flat.","statement_latex":"Let $S$ be a scheme.\nLet $G$ be a group scheme over $S$.\nLet $T$ be a scheme over $S$ and let $\\psi : T \\to G$ be a morphism over $S$.\nIf $T$ is flat over $S$, then the morphism\n$$\nT \\times_S G \\longrightarrow G, \\quad\n(t, g) \\longmapsto m(\\psi(t), g)\n$$\nis flat. In particular, if $G$ is flat over $S$, then\n$m : G \\times_S G \\to G$ is flat.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047H","source_file":"groupoids.tex","source_line":605,"source_end_line":617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L605-L617","statement_sha256":"510ad0ad2cf4ccd6096f990bb676c443506bb269394cd2f949039f9dd832eefd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7904,"rank":7904,"depth":3,"x":1139.96,"y":1166.673,"cluster":"groupoids-quotients"},{"id":"stacks:047I","tag":"047I","title":"Properties of group schemes · Lemma 047I","summary":"[BookAV] Let (G, m, e, i) be a group scheme over the scheme S. Denote f : G → S the structure morphism. Then there exist canonical isomorphisms Ω_G/S ≅ f^*C_S/G ≅ f^*e^*Ω_G/S where C_S/G denotes the conormal sheaf of the immersion e. In particular, if S is the spectrum of a field, then Ω_G/S is a free O_G-module.","statement_latex":"\\begin{reference}\n\\cite[Proposition 3.15]{BookAV}\n\\end{reference}\nLet $(G, m, e, i)$ be a group scheme over the scheme $S$.\nDenote $f : G \\to S$ the structure morphism.\nThen there exist canonical isomorphisms\n$$\n\\Omega_{G/S} \\cong f^*\\mathcal{C}_{S/G} \\cong f^*e^*\\Omega_{G/S}\n$$\nwhere $\\mathcal{C}_{S/G}$ denotes the conormal sheaf of the\nimmersion $e$. In particular, if $S$ is the spectrum of a field, then\n$\\Omega_{G/S}$ is a free $\\mathcal{O}_G$-module.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047I","source_file":"groupoids.tex","source_line":636,"source_end_line":650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L636-L650","statement_sha256":"73bda5e9106f9fc11227f5a18459d94ff1dbce5d773418d54150b7264e05e65e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7905,"rank":7905,"depth":19,"x":1099.981,"y":1125.387,"cluster":"groupoids-quotients"},{"id":"stacks:0BF5","tag":"0BF5","title":"Properties of group schemes · Lemma 0BF5","summary":"Let S be a scheme. Let G be a group scheme over S. Let s ∈ S. Then the composition T_G/S, e(s) ⊕ T_G/S, e(s) = T_G ×_S G/S, (e(s), e(s)) → T_G/S, e(s) is addition of tangent vectors. Here the = comes from Varieties, Lemma [Tag 0BEB] and the right arrow is induced from m : G ×_S G → G via Varieties, Lemma [Tag 0B2F].","statement_latex":"Let $S$ be a scheme. Let $G$ be a group scheme over $S$.\nLet $s \\in S$. Then the composition\n$$\nT_{G/S, e(s)} \\oplus T_{G/S, e(s)} = T_{G \\times_S G/S, (e(s), e(s))}\n\\rightarrow T_{G/S, e(s)}\n$$\nis addition of tangent vectors. Here the $=$ comes from\nVarieties, Lemma \\ref{varieties-lemma-tangent-space-product}\nand the right arrow is induced from $m : G \\times_S G \\to G$ via\nVarieties, Lemma \\ref{varieties-lemma-map-tangent-spaces}.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BF5","source_file":"groupoids.tex","source_line":686,"source_end_line":698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L686-L698","statement_sha256":"55d76c8e3765b4f13c1a831f365dc59dbb7eba97995c081327b57bea1732d873","origin":"The Stacks Project","memory_eligible":false,"source_rank":7906,"rank":7906,"depth":20,"x":1165.216,"y":1133.497,"cluster":"groupoids-quotients"},{"id":"stacks:047K","tag":"047K","title":"Properties of group schemes over a field · Lemma 047K","summary":"If (G, m) is a group scheme over a field k, then the multiplication map m : G ×_k G → G is open.","statement_latex":"If $(G, m)$ is a group scheme over a field $k$, then the\nmultiplication map $m : G \\times_k G \\to G$ is open.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047K","source_file":"groupoids.tex","source_line":725,"source_end_line":729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L725-L729","statement_sha256":"6eacf4ba05266c72ecd05830c336110d3cc8dbf09444153ddeac9c7ad8d7dc47","origin":"The Stacks Project","memory_eligible":false,"source_rank":7907,"rank":7907,"depth":10,"x":1108.508,"y":1165.679,"cluster":"groupoids-quotients"},{"id":"stacks:0B7N","tag":"0B7N","title":"Properties of group schemes over a field · Lemma 0B7N","summary":"If (G, m) is a group scheme over a field k. Let U ⊂ G open and T → G a morphism of schemes. Then the image of the composition T ×_k U → G ×_k G → G is open.","statement_latex":"If $(G, m)$ is a group scheme over a field $k$. Let $U \\subset G$\nopen and $T \\to G$ a morphism of schemes. Then the image of the\ncomposition $T \\times_k U \\to G \\times_k G \\to G$ is open.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7N","source_file":"groupoids.tex","source_line":747,"source_end_line":752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L747-L752","statement_sha256":"1c60fd3563209cf7c0be64c769885b29f11f5c41946504fa33e8e495c7021b6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7908,"rank":7908,"depth":10,"x":1125.035,"y":1107.815,"cluster":"groupoids-quotients"},{"id":"stacks:047L","tag":"047L","title":"Properties of group schemes over a field · Lemma 047L","summary":"Let G be a group scheme over a field. Then G is a separated scheme.","statement_latex":"Let $G$ be a group scheme over a field.\nThen $G$ is a separated scheme.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047L","source_file":"groupoids.tex","source_line":765,"source_end_line":769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L765-L769","statement_sha256":"5fc862a3c493698ad50d22893e2f18448c5f2acf337145285e0203a80bde0b0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7909,"rank":7909,"depth":18,"x":1160.481,"y":1161.579,"cluster":"groupoids-quotients"},{"id":"stacks:047M","tag":"047M","title":"Properties of group schemes over a field · Lemma 047M","summary":"Let G be a group scheme over a field k. Then • every local ring O_G, g of G has a unique minimal prime ideal, • there is exactly one irreducible component Z of G passing through e, and • Z is geometrically irreducible over k.","statement_latex":"Let $G$ be a group scheme over a field $k$.\nThen\n\\begin{enumerate}\n\\item every local ring $\\mathcal{O}_{G, g}$ of $G$ has a unique\nminimal prime ideal,\n\\item there is exactly one irreducible component $Z$ of $G$\npassing through $e$, and\n\\item $Z$ is geometrically irreducible over $k$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047M","source_file":"groupoids.tex","source_line":785,"source_end_line":796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L785-L796","statement_sha256":"425047e3685a79c2cf0c4e54c10fc9fa99ad655e87d145941824bd1fecb481c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7910,"rank":7910,"depth":14,"x":1088.982,"y":1141.425,"cluster":"groupoids-quotients"},{"id":"stacks:047R","tag":"047R","title":"Properties of group schemes over a field · Lemma 047R","summary":"Let G be a group scheme over a perfect field k. Then the reduction G_red of G is a closed subgroup scheme of G.","statement_latex":"Let $G$ be a group scheme over a perfect field $k$.\nThen the reduction $G_{red}$ of $G$ is a closed subgroup scheme of $G$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047R","source_file":"groupoids.tex","source_line":845,"source_end_line":849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L845-L849","statement_sha256":"2839a4cb03ea4fc00991f824e797a40a62ebf7cf31121231cc43c97fb85089e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7911,"rank":7911,"depth":12,"x":1159.919,"y":1114.99,"cluster":"groupoids-quotients"},{"id":"stacks:047S","tag":"047S","title":"Properties of group schemes over a field · Lemma 047S","summary":"Let k be a field. Let ψ : G' → G be a morphism of group schemes over k. If ψ(G') is open in G, then ψ(G') is closed in G.","statement_latex":"Let $k$ be a field. Let $\\psi : G' \\to G$ be a morphism of group schemes\nover $k$. If $\\psi(G')$ is open in $G$, then $\\psi(G')$ is closed in $G$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047S","source_file":"groupoids.tex","source_line":857,"source_end_line":861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L857-L861","statement_sha256":"664ead0bc5528a9ddd9fa470f3821084a606659937a75e449d46ef1bd07dbab3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7912,"rank":7912,"depth":11,"x":1127.998,"y":1176.363,"cluster":"groupoids-quotients"},{"id":"stacks:047T","tag":"047T","title":"Properties of group schemes over a field · Lemma 047T","summary":"Let i : G' → G be an immersion of group schemes over a field k. Then i is a closed immersion, i.e., i(G') is a closed subgroup scheme of G.","statement_latex":"Let $i : G' \\to G$ be an immersion of group schemes over a field $k$.\nThen $i$ is a closed immersion, i.e., $i(G')$ is a closed subgroup scheme\nof $G$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047T","source_file":"groupoids.tex","source_line":880,"source_end_line":885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L880-L885","statement_sha256":"3c720dd12478d7ac85e89d9e5988de0b4382bebe2de9b483d34c5cd0de2d9250","origin":"The Stacks Project","memory_eligible":false,"source_rank":7913,"rank":7913,"depth":13,"x":1101.532,"y":1111.344,"cluster":"groupoids-quotients"},{"id":"stacks:0B7P","tag":"0B7P","title":"Properties of group schemes over a field · Lemma 0B7P","summary":"Let G be a group scheme over a field k. If G is irreducible, then G is quasi-compact.","statement_latex":"Let $G$ be a group scheme over a field $k$. If $G$ is irreducible,\nthen $G$ is quasi-compact.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7P","source_file":"groupoids.tex","source_line":914,"source_end_line":918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L914-L918","statement_sha256":"ab349613697aaf34eefdd318747c3c5a05768347e31f0d1ba9604b6f802771c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7914,"rank":7914,"depth":15,"x":1175.097,"y":1145.097,"cluster":"groupoids-quotients"},{"id":"stacks:0B7Q","tag":"0B7Q","title":"Properties of group schemes over a field · Lemma 0B7Q","summary":"Let G be a group scheme over a field k. If G is connected, then G is irreducible.","statement_latex":"Let $G$ be a group scheme over a field $k$. If $G$ is connected,\nthen $G$ is irreducible.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7Q","source_file":"groupoids.tex","source_line":948,"source_end_line":952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L948-L952","statement_sha256":"db9c518f58b922ed8f38a5d362da590862c6cad5144a0f7eaf2b94fc90dd41b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7915,"rank":7915,"depth":20,"x":1091.789,"y":1162.332,"cluster":"groupoids-quotients"},{"id":"stacks:0B7R","tag":"0B7R","title":"Properties of group schemes over a field · Proposition 0B7R","summary":"Let G be a group scheme over a field k. There exists a canonical closed subgroup scheme G^0 ⊂ G with the following properties • G^0 → G is a flat closed immersion, • G^0 ⊂ G is the connected component of the identity, • G^0 is geometrically irreducible, and • G^0 is quasi-compact.","statement_latex":"Let $G$ be a group scheme over a field $k$. There exists a canonical closed\nsubgroup scheme $G^0 \\subset G$ with the following properties\n\\begin{enumerate}\n\\item $G^0 \\to G$ is a flat closed immersion,\n\\item $G^0 \\subset G$ is the connected component of the identity,\n\\item $G^0$ is geometrically irreducible, and\n\\item $G^0$ is quasi-compact.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7R","source_file":"groupoids.tex","source_line":1012,"source_end_line":1022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1012-L1022","statement_sha256":"fbb678e9bb0e02b9c1018edbf80e9b8ef492e72a3a90a2c8e268b6b1ec003e48","origin":"The Stacks Project","memory_eligible":false,"source_rank":7916,"rank":7916,"depth":21,"x":1140.441,"y":1101.013,"cluster":"groupoids-quotients"},{"id":"stacks:0B7T","tag":"0B7T","title":"Properties of group schemes over a field · Lemma 0B7T","summary":"Let k be a field. Let T = Spec(A) where A is a directed colimit of algebras which are finite products of copies of k. For any scheme X over k we have |T ×_k X| = |T| × |X| as topological spaces.","statement_latex":"Let $k$ be a field. Let $T = \\Spec(A)$ where $A$ is a directed colimit of\nalgebras which are finite products of copies of $k$. For any scheme $X$\nover $k$ we have $|T \\times_k X| = |T| \\times |X|$ as topological spaces.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7T","source_file":"groupoids.tex","source_line":1051,"source_end_line":1056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1051-L1056","statement_sha256":"9c83dbb9d40de5f540294c819cc7df210e4bc4c2180176b63ed0a4b588039692","origin":"The Stacks Project","memory_eligible":false,"source_rank":7917,"rank":7917,"depth":24,"x":1154.15,"y":1175.402,"cluster":"groupoids-quotients"},{"id":"stacks:0B7U","tag":"0B7U","title":"Properties of group schemes over a field · Lemma 0B7U","summary":"Let k be an algebraically closed field. Let G be a group scheme over k. Assume that G is Jacobson and that all closed points are k-rational. Let T = Spec(A) where A is a directed colimit of algebras which are finite products of copies of k. For any morphism f : T → G there exists an affine open U ⊂ G containing f(T).","statement_latex":"Let $k$ be an algebraically closed field. Let $G$ be a group scheme over $k$.\nAssume that $G$ is Jacobson and that all closed points are $k$-rational.\nLet $T = \\Spec(A)$ where $A$ is a directed colimit of algebras which\nare finite products of copies of $k$. For any morphism $f : T \\to G$\nthere exists an affine open $U \\subset G$ containing $f(T)$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7U","source_file":"groupoids.tex","source_line":1090,"source_end_line":1097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1090-L1097","statement_sha256":"7d17f077059522d36a1e530b4143e164afb4cf727f721bbfc31ae3eb75e05065","origin":"The Stacks Project","memory_eligible":false,"source_rank":7918,"rank":7918,"depth":25,"x":1082.789,"y":1127.348,"cluster":"groupoids-quotients"},{"id":"stacks:047U","tag":"047U","title":"Properties of group schemes over a field · Lemma 047U","summary":"Let G be a group scheme over a field. There exists an open and closed subscheme G' ⊂ G which is a countable union of affines.","statement_latex":"Let $G$ be a group scheme over a field.\nThere exists an open and closed subscheme $G' \\subset G$\nwhich is a countable union of affines.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of group schemes over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047U","source_file":"groupoids.tex","source_line":1195,"source_end_line":1200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1195-L1200","statement_sha256":"d3aa2bb4ca10a3a66f9d87184fdf38c475eb88ee85ede7e415372138833526f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7919,"rank":7919,"depth":12,"x":1175.86,"y":1122.202,"cluster":"groupoids-quotients"},{"id":"stacks:045X","tag":"045X","title":"Properties of algebraic group schemes · Lemma 045X","summary":"Let k be a field. Let G be a locally algebraic group scheme over k. Then G is equidimensional and dim(G) = dim_g(G) for all g ∈ G. For any closed point g ∈ G we have dim(G) = dim(O_G, g).","statement_latex":"Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$.\nThen $G$ is equidimensional and $\\dim(G) = \\dim_g(G)$ for all $g \\in G$.\nFor any closed point $g \\in G$ we have $\\dim(G) = \\dim(\\mathcal{O}_{G, g})$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of algebraic group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045X","source_file":"groupoids.tex","source_line":1234,"source_end_line":1239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1234-L1239","statement_sha256":"afe665cc0199e90cd9b02af2b9b13e2eeef0b54d75ff0cbe962f49f7d4b16331","origin":"The Stacks Project","memory_eligible":false,"source_rank":7920,"rank":7920,"depth":27,"x":1110.132,"y":1179.877,"cluster":"groupoids-quotients"},{"id":"stacks:047N","tag":"047N","title":"Properties of algebraic group schemes · Lemma 047N","summary":"Let k be a field of characteristic 0. Let G be a locally algebraic group scheme over k. Then the structure morphism G → Spec(k) is smooth, i.e., G is a smooth group scheme.","statement_latex":"Let $k$ be a field of characteristic $0$. Let $G$ be a\nlocally algebraic group scheme over $k$. Then the structure\nmorphism $G \\to \\Spec(k)$ is smooth, i.e., $G$ is a smooth\ngroup scheme.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of algebraic group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047N","source_file":"groupoids.tex","source_line":1265,"source_end_line":1271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1265-L1271","statement_sha256":"4916876c78b1c97901be35170a30201e6202e334d144990d10da425cbc9f3572","origin":"The Stacks Project","memory_eligible":false,"source_rank":7921,"rank":7921,"depth":40,"x":1112.269,"y":1098.59,"cluster":"groupoids-quotients"},{"id":"stacks:047P","tag":"047P","title":"Properties of algebraic group schemes · Lemma 047P","summary":"Let k be a perfect field of characteristic p > 0 (see Lemma [Tag 047N] for the characteristic zero case). Let G be a locally algebraic group scheme over k. If G is reduced then the structure morphism G → Spec(k) is smooth, i.e., G is a smooth group scheme.","statement_latex":"Let $k$ be a perfect field of characteristic $p > 0$ (see\nLemma \\ref{lemma-group-scheme-characteristic-zero-smooth}\nfor the characteristic zero case).\nLet $G$ be a locally algebraic group scheme over $k$.\nIf $G$ is reduced then the structure\nmorphism $G \\to \\Spec(k)$ is smooth, i.e., $G$ is a smooth\ngroup scheme.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of algebraic group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047P","source_file":"groupoids.tex","source_line":1294,"source_end_line":1303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1294-L1303","statement_sha256":"18d44062c5e7a16a6b6579df127fee99aa274aee4ed18ddb9e1ec9cb24dbea35","origin":"The Stacks Project","memory_eligible":false,"source_rank":7922,"rank":7922,"depth":41,"x":1177.181,"y":1160.83,"cluster":"groupoids-quotients"},{"id":"stacks:0B7S","tag":"0B7S","title":"Properties of algebraic group schemes · Lemma 0B7S","summary":"Let k be an algebraically closed field. Let G be a locally algebraic group scheme over k. Let g_1, …, g_n ∈ G(k) be k-rational points. Then there exists an affine open U ⊂ G containing g_1, …, g_n.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $G$ be a locally algebraic group scheme over $k$.\nLet $g_1, \\ldots, g_n \\in G(k)$ be $k$-rational points.\nThen there exists an affine open $U \\subset G$ containing $g_1, \\ldots, g_n$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of algebraic group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7S","source_file":"groupoids.tex","source_line":1328,"source_end_line":1334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1328-L1334","statement_sha256":"c85155b88cbacecc8ba652e443448f5a331ceb653f38078d4078d945926bca8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7923,"rank":7923,"depth":22,"x":1077.593,"y":1151.604,"cluster":"groupoids-quotients"},{"id":"stacks:0BF7","tag":"0BF7","title":"Properties of algebraic group schemes · Lemma 0BF7","summary":"Let k be a field. Let G be an algebraic group scheme over k. Then G is quasi-projective over k.","statement_latex":"Let $k$ be a field. Let $G$ be an algebraic group scheme over $k$.\nThen $G$ is quasi-projective over $k$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of algebraic group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BF7","source_file":"groupoids.tex","source_line":1364,"source_end_line":1368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1364-L1368","statement_sha256":"c1b82ff47a6fe10723f65c96db46dd3a0f355a98bc05ace58834fe10fdd01c6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7924,"rank":7924,"depth":42,"x":1159.788,"y":1101.085,"cluster":"groupoids-quotients"},{"id":"stacks:0BF8","tag":"0BF8","title":"Properties of algebraic group schemes · Lemma 0BF8","summary":"Let k be a field. Let G be a locally algebraic group scheme over k. Then the center of G is a closed subgroup scheme of G.","statement_latex":"Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$.\nThen the center of $G$ is a closed subgroup scheme of $G$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Properties of algebraic group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BF8","source_file":"groupoids.tex","source_line":1482,"source_end_line":1486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1482-L1486","statement_sha256":"09cde950169be12552481fc6ca775e8b33b00a115f18e9bf3968ae315c0d8aca","origin":"The Stacks Project","memory_eligible":false,"source_rank":7925,"rank":7925,"depth":47,"x":1139.476,"y":1186.315,"cluster":"groupoids-quotients"},{"id":"stacks:03RO","tag":"03RO","title":"Abelian varieties · Definition 03RO","summary":"Let k be a field. An abelian variety is a group scheme over k which is also a proper, geometrically integral variety over k.","statement_latex":"Let $k$ be a field. An {\\it abelian variety} is a group scheme over\n$k$ which is also a proper, geometrically integral variety over\n$k$\\footnote{For equivalent definitions see Remark \\ref{remark-16-equivalent}.}.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RO","source_file":"groupoids.tex","source_line":1595,"source_end_line":1600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1595-L1600","statement_sha256":"977eb02291a11eeab44dafcb43a0ae2c50ca3256ef9c8b88ea192f066b44384e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7926,"rank":7926,"depth":0,"x":1085.093,"y":1110.786,"cluster":"groupoids-quotients"},{"id":"stacks:0BFA","tag":"0BFA","title":"Abelian varieties · Lemma 0BFA","summary":"Let k be a field. Let A be an abelian variety over k. Then A is projective.","statement_latex":"Let $k$ be a field. Let $A$ be an abelian variety over $k$.\nThen $A$ is projective.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFA","source_file":"groupoids.tex","source_line":1607,"source_end_line":1611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1607-L1611","statement_sha256":"bca7b968be417628d1565d16e21c3052fb4e8d3e8737f3e48ae0d84568ac0f18","origin":"The Stacks Project","memory_eligible":false,"source_rank":7927,"rank":7927,"depth":43,"x":1187.445,"y":1136.002,"cluster":"groupoids-quotients"},{"id":"stacks:0BFB","tag":"0BFB","title":"Abelian varieties · Lemma 0BFB","summary":"Let k be a field. Let A be an abelian variety over k. For any field extension K/k the base change A_K is an abelian variety over K.","statement_latex":"Let $k$ be a field. Let $A$ be an abelian variety over $k$.\nFor any field extension $K/k$ the base change $A_K$ is an\nabelian variety over $K$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFB","source_file":"groupoids.tex","source_line":1619,"source_end_line":1624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1619-L1624","statement_sha256":"2bff7df1b49fa22790a36c00cb09faa1d1ebc660c2c28ddf4737a8299bb4d481","origin":"The Stacks Project","memory_eligible":false,"source_rank":7928,"rank":7928,"depth":0,"x":1090.294,"y":1176.054,"cluster":"groupoids-quotients"},{"id":"stacks:0BFC","tag":"0BFC","title":"Abelian varieties · Lemma 0BFC","summary":"Let k be a field. Let A be an abelian variety over k. Then A is smooth over k.","statement_latex":"Let $k$ be a field. Let $A$ be an abelian variety over $k$.\nThen $A$ is smooth over $k$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFC","source_file":"groupoids.tex","source_line":1632,"source_end_line":1636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1632-L1636","statement_sha256":"ef4aa03215a1b4f497eef996f531cd8c41e34c2de61b56f4960c2b2cb58d660c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7929,"rank":7929,"depth":42,"x":1130.289,"y":1090.198,"cluster":"groupoids-quotients"},{"id":"stacks:0BFD","tag":"0BFD","title":"Abelian varieties · Lemma 0BFD","summary":"An abelian variety is an abelian group scheme, i.e., the group law is commutative.","statement_latex":"An abelian variety is an abelian group scheme, i.e., the group\nlaw is commutative.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFD","source_file":"groupoids.tex","source_line":1650,"source_end_line":1654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1650-L1654","statement_sha256":"10e73916719f821a13ec43c9b7b7375539db4738a1cb42467b794c901e4cb53a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7930,"rank":7930,"depth":47,"x":1170.377,"y":1177.388,"cluster":"groupoids-quotients"},{"id":"stacks:0BFE","tag":"0BFE","title":"Abelian varieties · Lemma 0BFE","summary":"Let k be a field. Let A be an abelian variety over k. Let L be an invertible O_A-module. Then there is an isomorphism m_1, 2, 3^*L ⊗ m_1^*L ⊗ m_2^*L ⊗ m_3^*L ≅ m_1, 2^*L ⊗ m_1, 3^*L ⊗ m_2, 3^*L of invertible modules on A ×_k A ×_k A where m_i_1, …, i_t : A ×_k A ×_k A → A is the morphism (x_1, x_2, x_3) ↦ ∑ x_i_j.","statement_latex":"Let $k$ be a field. Let $A$ be an abelian variety over $k$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_A$-module.\nThen there is an isomorphism\n$$\nm_{1, 2, 3}^*\\mathcal{L} \\otimes\nm_1^*\\mathcal{L} \\otimes\nm_2^*\\mathcal{L} \\otimes\nm_3^*\\mathcal{L} \\cong\nm_{1, 2}^*\\mathcal{L} \\otimes\nm_{1, 3}^*\\mathcal{L} \\otimes\nm_{2, 3}^*\\mathcal{L}\n$$\nof invertible modules on $A \\times_k A \\times_k A$\nwhere $m_{i_1, \\ldots, i_t} : A \\times_k A \\times_k A \\to A$\nis the morphism $(x_1, x_2, x_3) \\mapsto \\sum x_{i_j}$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFE","source_file":"groupoids.tex","source_line":1678,"source_end_line":1695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1678-L1695","statement_sha256":"c205e006112461ff771cab78d6dd34291b2ff47a1648b380a36d2ffa87b0e4e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":7931,"rank":7931,"depth":1,"x":1069.369,"y":1135.28,"cluster":"groupoids-quotients"},{"id":"stacks:0BFF","tag":"0BFF","title":"Abelian varieties · Lemma 0BFF","summary":"Let k be a field. Let A be an abelian variety over k. Let L be an invertible O_A-module. Then [n]^*L ≅ L^⊗ n(n + 1)/2 ⊗ ([-1]^*L)^⊗ n(n - 1)/2 where [n] : A → A sends x to x + x + … + x with n summands and where [-1] : A → A is the inverse of A.","statement_latex":"Let $k$ be a field. Let $A$ be an abelian variety over $k$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_A$-module.\nThen\n$$\n[n]^*\\mathcal{L} \\cong\n\\mathcal{L}^{\\otimes n(n + 1)/2} \\otimes\n([-1]^*\\mathcal{L})^{\\otimes n(n - 1)/2}\n$$\nwhere $[n] : A \\to A$ sends $x$ to $x + x + \\ldots + x$ with $n$ summands\nand where $[-1] : A \\to A$ is the inverse of $A$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFF","source_file":"groupoids.tex","source_line":1726,"source_end_line":1738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1726-L1738","statement_sha256":"77a386e97354c7f3dd8b9a5b25fa7bada72b5512f39c18bbdfe0dd804f9ad6f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7932,"rank":7932,"depth":2,"x":1179.129,"y":1108.679,"cluster":"groupoids-quotients"},{"id":"stacks:0BFG","tag":"0BFG","title":"Abelian varieties · Lemma 0BFG","summary":"Let k be a field. Let A be an abelian variety over k. Let [d] : A → A be the multiplication by d. Then [d] is finite locally free of degree d^2dim(A).","statement_latex":"Let $k$ be a field. Let $A$ be an abelian variety over $k$.\nLet $[d] : A \\to A$ be the multiplication by $d$.\nThen $[d]$ is finite locally free of degree $d^{2\\dim(A)}$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFG","source_file":"groupoids.tex","source_line":1768,"source_end_line":1773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1768-L1773","statement_sha256":"ea4a69848ffc20945a070563065629ddaa75fe00ccc0d12fa28066217fffbf0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7933,"rank":7933,"depth":46,"x":1118.822,"y":1191.613,"cluster":"groupoids-quotients"},{"id":"stacks:0BFH","tag":"0BFH","title":"Abelian varieties · Lemma 0BFH","summary":"Multiplication by an integer on an abelian variety is an etale morphism if and only if the integer is invertible in the base field. Let k be a field. Let A be a nonzero abelian variety over k. Then [d] : A → A is étale if and only if d is invertible in k.","statement_latex":"\\begin{slogan}\nMultiplication by an integer on an abelian variety is an etale morphism\nif and only if the integer is invertible in the base field.\n\\end{slogan}\nLet $k$ be a field. Let $A$ be a nonzero abelian variety over $k$.\nThen $[d] : A \\to A$ is \\'etale if and only if $d$ is invertible in $k$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFH","source_file":"groupoids.tex","source_line":1825,"source_end_line":1833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1825-L1833","statement_sha256":"811eb0bbd19e29b20a99c615b0da178af1e0a34a14a49531e6979c26fb835a41","origin":"The Stacks Project","memory_eligible":false,"source_rank":7934,"rank":7934,"depth":47,"x":1096.329,"y":1095.055,"cluster":"groupoids-quotients"},{"id":"stacks:0C0Y","tag":"0C0Y","title":"Abelian varieties · Lemma 0C0Y","summary":"Let k be a field of characteristic p > 0. Let A be an abelian variety of dimension g over k. The fibre of [p] : A → A over 0 has at most p^g distinct points.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $A$ be an abelian variety\nof dimension $g$\nover $k$. The fibre of $[p] : A \\to A$ over $0$ has at most\n$p^g$ distinct points.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0Y","source_file":"groupoids.tex","source_line":1850,"source_end_line":1856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1850-L1856","statement_sha256":"e718b74ee0f4fe848a0eb6f5e6b3bc2b298a5877adef1e7723377169b000abea","origin":"The Stacks Project","memory_eligible":false,"source_rank":7935,"rank":7935,"depth":48,"x":1191.701,"y":1154.204,"cluster":"groupoids-quotients"},{"id":"stacks:03RP","tag":"03RP","title":"Abelian varieties · Proposition 03RP","summary":"Wonderfully explained in [AVar]. Let A be an abelian variety over a field k. Then • A is projective over k, • A is a commutative group scheme, • the morphism [n] : A → A is surjective for all n ≥ 1, • if k is algebraically closed, then A(k) is a divisible abelian group, • A[n] = Ker([n] : A → A) is a finite group scheme of degree n^2dim A over k, • A[n] is étale over k if and only if n ∈ k^*, • if n ∈ k^* and k is algebraically closed, then A(k)[n] ≅ (Z/nZ)^⊕ 2dim(A), •…","statement_latex":"\\begin{reference}\nWonderfully explained in \\cite{AVar}.\n\\end{reference}\nLet $A$ be an abelian variety over a field $k$. Then\n\\begin{enumerate}\n\\item $A$ is projective over $k$,\n\\item $A$ is a commutative group scheme,\n\\item the morphism $[n] : A \\to A$ is surjective for all $n \\geq 1$,\n\\item if $k$ is algebraically closed, then $A(k)$ is a divisible abelian group,\n\\item $A[n] = \\Ker([n] : A \\to A)$ is a finite group scheme of degree\n$n^{2\\dim A}$ over $k$,\n\\item $A[n]$ is \\'etale over $k$ if and only if $n \\in k^*$,\n\\item if $n \\in k^*$ and $k$ is algebraically closed,\nthen $A(k)[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2\\dim(A)}$,\n\\item if $k$ is algebraically closed of characteristic $p > 0$, then\nthere exists an integer $0 \\leq f \\leq \\dim(A)$ such that\n$A(k)[p^m] \\cong (\\mathbf{Z}/p^m\\mathbf{Z})^{\\oplus f}$\nfor all $m \\geq 1$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Abelian varieties","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RP","source_file":"groupoids.tex","source_line":1902,"source_end_line":1923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L1902-L1923","statement_sha256":"5781012c8f8e41a020caa35202f50e2c8397ae6b1d19594aec785a199b8e8f68","origin":"The Stacks Project","memory_eligible":false,"source_rank":7936,"rank":7936,"depth":49,"x":1072.415,"y":1164.823,"cluster":"groupoids-quotients"},{"id":"stacks:022Z","tag":"022Z","title":"Actions of group schemes · Definition 022Z","summary":"Let S be a scheme. Let (G, m) be a group scheme over S. • An action of G on the scheme X/S is a morphism a : G ×_S X → X over S such that for every T/S the map a : G(T) × X(T) → X(T) defines the structure of a G(T)-set on X(T). • Suppose that X, Y are schemes over S each endowed with an action of G. An equivariant or more precisely a G-equivariant morphism ψ : X → Y is a morphism of schemes over S such that for every T/S the map ψ : X(T) → Y(T) is a morphism of G(T)-sets.","statement_latex":"Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.\n\\begin{enumerate}\n\\item An {\\it action of $G$ on the scheme $X/S$} is\na morphism $a : G \\times_S X \\to X$ over $S$ such that\nfor every $T/S$ the map $a : G(T) \\times X(T) \\to X(T)$\ndefines the structure of a $G(T)$-set on $X(T)$.\n\\item Suppose that $X$, $Y$ are schemes over $S$ each endowed\nwith an action of $G$. An {\\it equivariant} or more precisely\na {\\it $G$-equivariant} morphism $\\psi : X \\to Y$\nis a morphism of schemes over $S$ such\nthat for every $T/S$ the map $\\psi : X(T) \\to Y(T)$ is\na morphism of $G(T)$-sets.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Actions of group schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/022Z","source_file":"groupoids.tex","source_line":2025,"source_end_line":2040,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2025-L2040","statement_sha256":"84c083fe401874a2a65f2fca6f1fe79a2060194473b6ee597b1e599d628fc498","origin":"The Stacks Project","memory_eligible":false,"source_rank":7937,"rank":7937,"depth":0,"x":1152.756,"y":1088.439,"cluster":"groupoids-quotients"},{"id":"stacks:07S1","tag":"07S1","title":"Actions of group schemes · Definition 07S1","summary":"Let S, G → S, and X → S as in Definition [Tag 022Z]. Let a : G ×_S X → X be an action of G on X/S. We say the action is free if for every scheme T over S the action a : G(T) × X(T) → X(T) is a free action of the group G(T) on the set X(T).","statement_latex":"Let $S$, $G \\to S$, and $X \\to S$ as in\nDefinition \\ref{definition-action-group-scheme}.\nLet $a : G \\times_S X \\to X$ be an action of $G$ on $X/S$.\nWe say the action is {\\it free} if for every scheme $T$ over $S$\nthe action $a : G(T) \\times X(T) \\to X(T)$ is a free action of\nthe group $G(T)$ on the set $X(T)$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Actions of group schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07S1","source_file":"groupoids.tex","source_line":2071,"source_end_line":2079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2071-L2079","statement_sha256":"69fb0e40a1691dea473a077d42427492d21608ea609c28ad255d3fe45968e707","origin":"The Stacks Project","memory_eligible":false,"source_rank":7938,"rank":7938,"depth":1,"x":1154.959,"y":1191.501,"cluster":"groupoids-quotients"},{"id":"stacks:07S2","tag":"07S2","title":"Actions of group schemes · Lemma 07S2","summary":"Situation as in Definition [Tag 07S1], The action a is free if and only if G ×_S X → X ×_S X, (g, x) ↦ (a(g, x), x) is a monomorphism.","statement_latex":"Situation as in Definition \\ref{definition-free-action},\nThe action $a$ is free if and only if\n$$\nG \\times_S X \\to X \\times_S X, \\quad (g, x) \\mapsto (a(g, x), x)\n$$\nis a monomorphism.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Actions of group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07S2","source_file":"groupoids.tex","source_line":2081,"source_end_line":2089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2081-L2089","statement_sha256":"ba9bf1f6978cbadbfb1e43ffa28d3c8de98ea239c1af2cd7acbeede15c6403f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7939,"rank":7939,"depth":2,"x":1069.528,"y":1115.927,"cluster":"groupoids-quotients"},{"id":"stacks:0498","tag":"0498","title":"Principal homogeneous spaces · Definition 0498","summary":"Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a scheme over S, and let a : G ×_S X → X be an action of G on X. • We say X is a pseudo G-torsor or that X is formally principally homogeneous under G if the induced morphism of schemes G ×_S X → X ×_S X, (g, x) ↦ (a(g, x), x) is an isomorphism of schemes over S. • A pseudo G-torsor X is called trivial if there exists an G-equivariant isomorphism G → X over S where G acts on G by left multiplication.","statement_latex":"Let $S$ be a scheme.\nLet $(G, m)$ be a group scheme over $S$.\nLet $X$ be a scheme over $S$, and let\n$a : G \\times_S X \\to X$ be an action of $G$ on $X$.\n\\begin{enumerate}\n\\item We say $X$ is a {\\it pseudo $G$-torsor} or that $X$ is\n{\\it formally principally homogeneous under $G$} if the induced\nmorphism of schemes $G \\times_S X \\to X \\times_S X$,\n$(g, x) \\mapsto (a(g, x), x)$ is an isomorphism of schemes over $S$.\n\\item A pseudo $G$-torsor $X$ is called {\\it trivial} if there exists\nan $G$-equivariant isomorphism $G \\to X$ over $S$ where $G$ acts on\n$G$ by left multiplication.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Principal homogeneous spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0498","source_file":"groupoids.tex","source_line":2120,"source_end_line":2135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2120-L2135","statement_sha256":"8a63e573fc67cf5259a7c86b612cc1a57a872839d2f5e4b8b371b4a4661cc0fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7940,"rank":7940,"depth":0,"x":1194.633,"y":1123.261,"cluster":"groupoids-quotients"},{"id":"stacks:0499","tag":"0499","title":"Principal homogeneous spaces · Lemma 0499","summary":"In the situation of Definition [Tag 0498]. • The scheme X is a pseudo G-torsor if and only if for every scheme T over S the set X(T) is either empty or the action of the group G(T) on X(T) is simply transitive. • A pseudo G-torsor X is trivial if and only if the morphism X → S has a section.","statement_latex":"In the situation of\nDefinition \\ref{definition-pseudo-torsor}.\n\\begin{enumerate}\n\\item The scheme $X$ is a pseudo $G$-torsor if and only if for every scheme\n$T$ over $S$ the set $X(T)$ is either empty or the action of the group $G(T)$\non $X(T)$ is simply transitive.\n\\item A pseudo $G$-torsor $X$ is trivial if and only if the morphism\n$X \\to S$ has a section.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Principal homogeneous spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0499","source_file":"groupoids.tex","source_line":2142,"source_end_line":2153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2142-L2153","statement_sha256":"b2650a2d3d072904d1fb847fb3f9661d252f84bc3a8f034bb9837890d782bc75","origin":"The Stacks Project","memory_eligible":false,"source_rank":7941,"rank":7941,"depth":1,"x":1095.444,"y":1189.529,"cluster":"groupoids-quotients"},{"id":"stacks:049A","tag":"049A","title":"Principal homogeneous spaces · Definition 049A","summary":"Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a pseudo G-torsor over S. • We say X is a principal homogeneous space or a G-torsor if there exists a fpqc covering (S_i → S)_i ∈ I such that each X_S_i → S_i has a section (i.e., is a trivial pseudo G_S_i-torsor). • Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). We say X is a G-torsor in the τ topology, or a τ G-torsor, or simply a τ torsor if there exists a τ covering (S_i → S)_i ∈ I such that each…","statement_latex":"Let $S$ be a scheme.\nLet $(G, m)$ be a group scheme over $S$.\nLet $X$ be a pseudo $G$-torsor over $S$.\n\\begin{enumerate}\n\\item We say $X$ is a {\\it principal homogeneous space}\nor a {\\it $G$-torsor} if there exists a fpqc covering\\footnote{This means\nthat the default type of torsor is a pseudo torsor which is trivial on an\nfpqc covering. This is the definition in \\cite[Expos\\'e IV, 6.5]{SGA3}.\nIt is a little bit inconvenient for us as we most often work in the fppf\ntopology.}\n$\\{S_i \\to S\\}_{i \\in I}$ such that each\n$X_{S_i} \\to S_i$ has a section (i.e., is a trivial pseudo $G_{S_i}$-torsor).\n\\item Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nWe say $X$ is a {\\it $G$-torsor in the $\\tau$ topology}, or a\n{\\it $\\tau$ $G$-torsor}, or simply a {\\it $\\tau$ torsor}\nif there exists a $\\tau$ covering $\\{S_i \\to S\\}_{i \\in I}$\nsuch that each $X_{S_i} \\to S_i$ has a section.\n\\item If $X$ is a $G$-torsor, then we say that it is\n{\\it quasi-isotrivial} if it is a torsor for the \\'etale topology.\n\\item If $X$ is a $G$-torsor, then we say that it is\n{\\it locally trivial} if it is a torsor for the Zariski topology.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Principal homogeneous spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049A","source_file":"groupoids.tex","source_line":2159,"source_end_line":2183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2159-L2183","statement_sha256":"8cded131f08c14ef95508493e7c354fed2e165a69f362d80831c89ec26e5b99e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7942,"rank":7942,"depth":0,"x":1115.504,"y":1083.296,"cluster":"groupoids-quotients"},{"id":"stacks:049B","tag":"049B","title":"Principal homogeneous spaces · Lemma 049B","summary":"Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a scheme over S, and let a : G ×_S X → X be an action of G on X. Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Then X is a G-torsor in the τ-topology if and only if underlineX is a underlineG-torsor on (Sch/S)_τ.","statement_latex":"Let $S$ be a scheme.\nLet $(G, m)$ be a group scheme over $S$.\nLet $X$ be a scheme over $S$, and let\n$a : G \\times_S X \\to X$ be an action of $G$ on $X$.\nLet $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nThen $X$ is a $G$-torsor in the $\\tau$-topology if and only if\n$\\underline{X}$ is a $\\underline{G}$-torsor on $(\\Sch/S)_\\tau$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Principal homogeneous spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049B","source_file":"groupoids.tex","source_line":2192,"source_end_line":2201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2192-L2201","statement_sha256":"6904b6d143c4571ac57fa4f0999a5913f467fdf664cd3dafe27022e0419a19a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":7943,"rank":7943,"depth":0,"x":1186.854,"y":1173.924,"cluster":"groupoids-quotients"},{"id":"stacks:03LF","tag":"03LF","title":"Equivariant quasi-coherent sheaves · Definition 03LF","summary":"Let S be a scheme, let (G, m) be a group scheme over S, and let a : G ×_S X → X be an action of the group scheme G on X/S. A G-equivariant quasi-coherent O_X-module, or simply an equivariant quasi-coherent O_X-module, is a pair (F, α), where F is a quasi-coherent O_X-module, and α is a O_G ×_S X-module map α : a^*F → pr_1^*F where pr_1 : G ×_S X → X is the projection such that • the diagram xymatrix (1_G × a)^*pr_1^*F ar[r]_-pr_12^*α & pr_2^*F (1_G × a)^*a^*F ar[u]^(1_G ×…","statement_latex":"Let $S$ be a scheme, let $(G, m)$ be a group scheme over $S$, and\nlet $a : G \\times_S X \\to X$ be an action of the group scheme $G$\non $X/S$. A {\\it $G$-equivariant quasi-coherent $\\mathcal{O}_X$-module},\nor simply an {\\it equivariant quasi-coherent $\\mathcal{O}_X$-module},\nis a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent\n$\\mathcal{O}_X$-module, and $\\alpha$ is a $\\mathcal{O}_{G \\times_S X}$-module\nmap\n$$\n\\alpha : a^*\\mathcal{F} \\longrightarrow \\text{pr}_1^*\\mathcal{F}\n$$\nwhere $\\text{pr}_1 : G \\times_S X \\to X$ is the projection\nsuch that\n\\begin{enumerate}\n\\item the diagram\n$$\n\\xymatrix{\n(1_G \\times a)^*\\text{pr}_1^*\\mathcal{F} \\ar[r]_-{\\text{pr}_{12}^*\\alpha} &\n\\text{pr}_2^*\\mathcal{F} \\\\\n(1_G \\times a)^*a^*\\mathcal{F} \\ar[u]^{(1_G \\times a)^*\\alpha} \\ar@{=}[r] &\n(m \\times 1_X)^*a^*\\mathcal{F} \\ar[u]_{(m \\times 1_X)^*\\alpha}\n}\n$$\nis commutative in the category of\n$\\mathcal{O}_{G \\times_S G \\times_S X}$-modules, and\n\\item the pullback\n$$\n(e \\times 1_X)^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F}\n$$\nis the identity map.\n\\end{enumerate}\nFor explanation compare with the relevant diagrams of\nEquation (\\ref{equation-action}).","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Equivariant quasi-coherent sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LF","source_file":"groupoids.tex","source_line":2244,"source_end_line":2278,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2244-L2278","statement_sha256":"8740546f0b6141472738b2eb0df13b9027020a0fef560974c7692cdbbdf3db93","origin":"The Stacks Project","memory_eligible":false,"source_rank":7944,"rank":7944,"depth":0,"x":1060.115,"y":1147.316,"cluster":"groupoids-quotients"},{"id":"stacks:03LG","tag":"03LG","title":"Equivariant quasi-coherent sheaves · Lemma 03LG","summary":"Let S be a scheme. Let G be a group scheme over S. Let f : Y → X be a G-equivariant morphism between S-schemes endowed with G-actions. The rule (F, α) ↦ (f^*F, (1_G × f)^*α) defines a functor from the category of G-equivariant quasi-coherent O_X-modules to the category of G-equivariant quasi-coherent O_Y-modules.","statement_latex":"Let $S$ be a scheme. Let $G$ be a group scheme over $S$.\nLet $f : Y \\to X$ be a $G$-equivariant morphism between $S$-schemes\nendowed with $G$-actions. The rule\n$(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, (1_G \\times f)^*\\alpha)$\ndefines a functor from the category of $G$-equivariant quasi-coherent\n$\\mathcal{O}_X$-modules  to the category of\n$G$-equivariant quasi-coherent $\\mathcal{O}_Y$-modules.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Equivariant quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LG","source_file":"groupoids.tex","source_line":2285,"source_end_line":2294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2285-L2294","statement_sha256":"4654b9009c4da5b35c646f3c98a9239a223875d085d67dd9ebc775c163d3529f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7945,"rank":7945,"depth":0,"x":1176.089,"y":1094.517,"cluster":"groupoids-quotients"},{"id":"stacks:0EKK","tag":"0EKK","title":"Equivariant quasi-coherent sheaves · Lemma 0EKK","summary":"Let a : G_m × X → X be an action on an affine scheme. Then X is the spectrum of a Z-graded ring and the action is as in Example [Tag 0EKJ].","statement_latex":"Let $a : \\mathbf{G}_m \\times X \\to X$ be an action on an affine scheme.\nThen $X$ is the spectrum of a $\\mathbf{Z}$-graded ring\nand the action is as in Example \\ref{example-Gm-on-affine}.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Equivariant quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKK","source_file":"groupoids.tex","source_line":2339,"source_end_line":2344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2339-L2344","statement_sha256":"03003199cb9540d6a03ca4597ffd9dae98c040c2a6296fbf1846825a1c60e415","origin":"The Stacks Project","memory_eligible":false,"source_rank":7946,"rank":7946,"depth":0,"x":1132.616,"y":1200.255,"cluster":"groupoids-quotients"},{"id":"stacks:0EKL","tag":"0EKL","title":"Equivariant quasi-coherent sheaves · Lemma 0EKL","summary":"Let A be a graded ring. Let X = Spec(A) with action a : G_m × X → X as in Example [Tag 0EKJ]. Let F be a G_m-equivariant quasi-coherent O_X-module. Then M = Γ(X, F) has a canonical grading such that it is a graded A-module and such that the isomorphism widetildeM → F (Schemes, Lemma [Tag 01IA]) is an isomorphism of G_m-equivariant modules where the G_m-equivariant structure on widetildeM is the one from Example [Tag 0EKJ].","statement_latex":"Let $A$ be a graded ring. Let $X = \\Spec(A)$ with action\n$a : \\mathbf{G}_m \\times X \\to X$ as in Example \\ref{example-Gm-on-affine}.\nLet $\\mathcal{F}$ be a $\\mathbf{G}_m$-equivariant quasi-coherent\n$\\mathcal{O}_X$-module. Then $M = \\Gamma(X, \\mathcal{F})$\nhas a canonical grading such that it is a graded $A$-module\nand such that the isomorphism $\\widetilde{M} \\to \\mathcal{F}$\n(Schemes, Lemma \\ref{schemes-lemma-quasi-coherent-affine})\nis an isomorphism of $\\mathbf{G}_m$-equivariant modules where\nthe $\\mathbf{G}_m$-equivariant structure on $\\widetilde{M}$\nis the one from Example \\ref{example-Gm-on-affine}.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Equivariant quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKL","source_file":"groupoids.tex","source_line":2372,"source_end_line":2384,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2372-L2384","statement_sha256":"77a67b381550c3812d187ff8a66cc5b4c36847b172830afdc2a389fdb463cbfc","origin":"The Stacks Project","memory_eligible":false,"source_rank":7947,"rank":7947,"depth":10,"x":1079.148,"y":1096.655,"cluster":"groupoids-quotients"},{"id":"stacks:0231","tag":"0231","title":"Groupoids · Definition 0231","summary":"Let S be a scheme. • A groupoid scheme over S, or simply a groupoid over S is a quintuple (U, R, s, t, c) where U and R are schemes over S, and s, t : R → U and c : R ×_s, U, t R → R are morphisms of schemes over S with the following property: For any scheme T over S the quintuple (U(T), R(T), s, t, c) is a groupoid category in the sense described above. • A morphism f : (U, R, s, t, c) → (U', R', s', t', c') of groupoid schemes over S is given by morphisms of schemes f :…","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item A {\\it groupoid scheme over $S$}, or simply a\n{\\it groupoid over $S$} is a\nquintuple $(U, R, s, t, c)$ where\n$U$ and $R$ are schemes over $S$, and\n$s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$\nare morphisms of schemes over $S$ with the\nfollowing property: For any scheme\n$T$ over $S$ the quintuple\n$$\n(U(T), R(T), s, t, c)\n$$\nis a groupoid category in the sense described above.\n\\item A {\\it morphism\n$f : (U, R, s, t, c) \\to (U', R', s', t', c')$\nof groupoid schemes over $S$} is given by morphisms\nof schemes $f : U \\to U'$ and $f : R \\to R'$ with the\nfollowing property:  For any scheme\n$T$ over $S$ the maps $f$ define a functor from the\ngroupoid category $(U(T), R(T), s, t, c)$ to the\ngroupoid category $(U'(T), R'(T), s', t', c')$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0231","source_file":"groupoids.tex","source_line":2450,"source_end_line":2475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2450-L2475","statement_sha256":"8703776ff572df8f1625dcc1f24bdfd69fd444f26473c655fe5142a57e2b83aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":7948,"rank":7948,"depth":0,"x":1203.014,"y":1143.136,"cluster":"groupoids-quotients"},{"id":"stacks:0232","tag":"0232","title":"Groupoids · Lemma 0232","summary":"Given a groupoid scheme (U, R, s, t, c) over S the morphism j : R → U ×_S U is a pre-equivalence relation.","statement_latex":"Given a groupoid scheme $(U, R, s, t, c)$ over $S$\nthe morphism $j : R \\to U \\times_S U$ is a pre-equivalence\nrelation.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0232","source_file":"groupoids.tex","source_line":2501,"source_end_line":2506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2501-L2506","statement_sha256":"9f824623bf1079127e075f6c9fb7d393a5f36ddcc17c72414e5db3e70bd89a88","origin":"The Stacks Project","memory_eligible":false,"source_rank":7949,"rank":7949,"depth":0,"x":1073.135,"y":1179.467,"cluster":"groupoids-quotients"},{"id":"stacks:0233","tag":"0233","title":"Groupoids · Lemma 0233","summary":"Given an equivalence relation j : R → U ×_S U over S there is a unique way to extend it to a groupoid (U, R, s, t, c) over S.","statement_latex":"Given an equivalence relation $j : R \\to U \\times_S U$ over $S$\nthere is a unique way to extend it to a groupoid\n$(U, R, s, t, c)$ over $S$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0233","source_file":"groupoids.tex","source_line":2513,"source_end_line":2518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2513-L2518","statement_sha256":"0e5775add6f12b7694d66f9cda2a86f5743a6fc47c69233972c249c99c27c58e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7950,"rank":7950,"depth":0,"x":1140.283,"y":1078.091,"cluster":"groupoids-quotients"},{"id":"stacks:02YE","tag":"02YE","title":"Groupoids · Lemma 02YE","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. In the commutative diagram xymatrix & U & R ar[d]_s ar[ru]^t & R ×_s, U, t R ar[l]^-pr_0 ar[d]^pr_1 ar[r]_-c & R ar[d]^s ar[lu]_t U & R ar[l]_t ar[r]^s & U the two lower squares are fibre product squares. Moreover, the triangle on top (which is really a square) is also cartesian.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$.\nIn the commutative diagram\n$$\n\\xymatrix{\n& U & \\\\\nR \\ar[d]_s \\ar[ru]^t &\nR \\times_{s, U, t} R\n\\ar[l]^-{\\text{pr}_0} \\ar[d]^{\\text{pr}_1} \\ar[r]_-c &\nR \\ar[d]^s \\ar[lu]_t \\\\\nU & R \\ar[l]_t \\ar[r]^s & U\n}\n$$\nthe two lower squares are fibre product squares.\nMoreover, the triangle on top (which is really a square)\nis also cartesian.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YE","source_file":"groupoids.tex","source_line":2525,"source_end_line":2543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2525-L2543","statement_sha256":"4befea6700e61dc8a8da565465da37857fe5a3b88536f62ad7c9f94e5b0b0048","origin":"The Stacks Project","memory_eligible":false,"source_rank":7951,"rank":7951,"depth":0,"x":1172.567,"y":1191.93,"cluster":"groupoids-quotients"},{"id":"stacks:03C6","tag":"03C6","title":"Groupoids · Lemma 03C6","summary":"Let S be a scheme. Let (U, R, s, t, c, e, i) be a groupoid over S. The diagram xymatrix R ×_t, U, t R ar@<1ex>[r]^-pr_1 ar@<-1ex>[r]_-pr_0 ar[d]_(pr_0, c ∘ (i, 1)) & R ar[r]^t ar[d]^id_R & U ar[d]^id_U R ×_s, U, t R ar@<1ex>[r]^-c ar@<-1ex>[r]_-pr_0 ar[d]_pr_1 & R ar[r]^t ar[d]^s & U R ar@<1ex>[r]^s ar@<-1ex>[r]_t & U is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c, e, i)$ be a groupoid over $S$.\nThe diagram\n\\begin{equation}\n\n\\xymatrix{\nR \\times_{t, U, t} R\n\\ar@<1ex>[r]^-{\\text{pr}_1} \\ar@<-1ex>[r]_-{\\text{pr}_0}\n\\ar[d]_{(\\text{pr}_0, c \\circ (i, 1))} &\nR \\ar[r]^t \\ar[d]^{\\text{id}_R} &\nU \\ar[d]^{\\text{id}_U} \\\\\nR \\times_{s, U, t} R\n\\ar@<1ex>[r]^-c \\ar@<-1ex>[r]_-{\\text{pr}_0} \\ar[d]_{\\text{pr}_1} &\nR \\ar[r]^t \\ar[d]^s &\nU \\\\\nR \\ar@<1ex>[r]^s \\ar@<-1ex>[r]_t &\nU\n}\n\\end{equation}\nis commutative. The two top rows are isomorphic via the vertical maps given.\nThe two lower left squares are cartesian.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03C6","source_file":"groupoids.tex","source_line":2551,"source_end_line":2574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2551-L2574","statement_sha256":"c5dd489202271f3b0708fb67bbaa0eec1844f06bc57a8cbad7239ad634528e04","origin":"The Stacks Project","memory_eligible":false,"source_rank":7952,"rank":7952,"depth":1,"x":1056.23,"y":1125.742,"cluster":"groupoids-quotients"},{"id":"stacks:0DT8","tag":"0DT8","title":"Groupoids · Lemma 0DT8","summary":"Let (U, R, s, t, c) be a groupoid over a scheme S. Let S' → S be a morphism. Then the base changes U' = S' ×_S U, R' = S' ×_S R endowed with the base changes s', t', c' of the morphisms s, t, c form a groupoid scheme (U', R', s', t', c') over S' and the projections determine a morphism (U', R', s', t', c') → (U, R, s, t, c) of groupoid schemes over S.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid over a scheme $S$.\nLet $S' \\to S$ be a morphism. Then the base changes $U' = S' \\times_S U$,\n$R' = S' \\times_S R$ endowed with the base changes $s'$, $t'$, $c'$\nof the morphisms $s, t, c$ form a groupoid scheme\n$(U', R', s', t', c')$ over $S'$ and the projections\ndetermine a morphism\n$(U', R', s', t', c') \\to (U, R, s, t, c)$\nof groupoid schemes over $S$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DT8","source_file":"groupoids.tex","source_line":2589,"source_end_line":2599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2589-L2599","statement_sha256":"c3d5481b69afc63bd37f09d4fc4d1ec086e1e7dcb27402d61d292cec2443f633","origin":"The Stacks Project","memory_eligible":false,"source_rank":7953,"rank":7953,"depth":0,"x":1196.413,"y":1108.393,"cluster":"groupoids-quotients"},{"id":"stacks:03LI","tag":"03LI","title":"Quasi-coherent sheaves on groupoids · Definition 03LI","summary":"Let S be a scheme, let (U, R, s, t, c) be a groupoid scheme over S. A quasi-coherent module on (U, R, s, t, c) is a pair (F, α), where F is a quasi-coherent O_U-module, and α is a O_R-module map α : t^*F → s^*F such that • the diagram xymatrix & pr_1^*t^*F ar[r]_-pr_1^*α & pr_1^*s^*F ar@=[rd] & pr_0^*s^*F ar@=[ru] & & & c^*s^*F & pr_0^*t^*F ar[lu]^pr_0^*α ar@=[r] & c^*t^*F ar[ru]_c^*α is a commutative in the category of O_R ×_s, U, t R-modules, and • the pullback e^*α : F…","statement_latex":"Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nA {\\it quasi-coherent module on $(U, R, s, t, c)$}\nis a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent\n$\\mathcal{O}_U$-module, and $\\alpha$ is a $\\mathcal{O}_R$-module\nmap\n$$\n\\alpha : t^*\\mathcal{F} \\longrightarrow s^*\\mathcal{F}\n$$\nsuch that\n\\begin{enumerate}\n\\item the diagram\n$$\n\\xymatrix{\n& \\text{pr}_1^*t^*\\mathcal{F} \\ar[r]_-{\\text{pr}_1^*\\alpha} &\n\\text{pr}_1^*s^*\\mathcal{F} \\ar@{=}[rd] & \\\\\n\\text{pr}_0^*s^*\\mathcal{F} \\ar@{=}[ru] & & & c^*s^*\\mathcal{F} \\\\\n& \\text{pr}_0^*t^*\\mathcal{F} \\ar[lu]^{\\text{pr}_0^*\\alpha} \\ar@{=}[r] &\nc^*t^*\\mathcal{F} \\ar[ru]_{c^*\\alpha}\n}\n$$\nis a commutative in the category of\n$\\mathcal{O}_{R \\times_{s, U, t} R}$-modules, and\n\\item the pullback\n$$\ne^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F}\n$$\nis the identity map.\n\\end{enumerate}\nCompare with the commutative diagrams of Lemma \\ref{lemma-diagram}.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LI","source_file":"groupoids.tex","source_line":2621,"source_end_line":2652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2621-L2652","statement_sha256":"768fac2897dcbf567a671cce98d11c5d5a20d72c10bb920d9320f67fff134107","origin":"The Stacks Project","memory_eligible":false,"source_rank":7954,"rank":7954,"depth":1,"x":1106.252,"y":1201.491,"cluster":"groupoids-quotients"},{"id":"stacks:077Q","tag":"077Q","title":"Quasi-coherent sheaves on groupoids · Lemma 077Q","summary":"Let S be a scheme, let (U, R, s, t, c) be a groupoid scheme over S. If (F, α) is a quasi-coherent module on (U, R, s, t, c) then α is an isomorphism.","statement_latex":"Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nIf $(\\mathcal{F}, \\alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$\nthen $\\alpha$ is an isomorphism.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077Q","source_file":"groupoids.tex","source_line":2660,"source_end_line":2665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2660-L2665","statement_sha256":"d73a760799ad4b0fcf8bd049c642aabe766d84397dd14949cf444e774f81683c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7955,"rank":7955,"depth":2,"x":1097.804,"y":1080.707,"cluster":"groupoids-quotients"},{"id":"stacks:03LJ","tag":"03LJ","title":"Quasi-coherent sheaves on groupoids · Lemma 03LJ","summary":"Let S be a scheme. Consider a morphism f : (U, R, s, t, c) → (U', R', s', t', c') of groupoid schemes over S. Then pullback f^* given by (F, α) ↦ (f^*F, f^*α) defines a functor from the category of quasi-coherent sheaves on (U', R', s', t', c') to the category of quasi-coherent sheaves on (U, R, s, t, c).","statement_latex":"Let $S$ be a scheme. Consider a morphism\n$f : (U, R, s, t, c) \\to (U', R', s', t', c')$\nof groupoid schemes over $S$. Then pullback $f^*$ given by\n$$\n(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, f^*\\alpha)\n$$\ndefines a functor from the category of quasi-coherent sheaves on\n$(U', R', s', t', c')$ to the category of quasi-coherent sheaves on\n$(U, R, s, t, c)$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LJ","source_file":"groupoids.tex","source_line":2678,"source_end_line":2689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2678-L2689","statement_sha256":"8aed51c2f4100b12ae57ef33a5d45cd6969dec5c5c2ebb7bc288ac3b3c9af3d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":7956,"rank":7956,"depth":0,"x":1201.99,"y":1165.655,"cluster":"groupoids-quotients"},{"id":"stacks:09VH","tag":"09VH","title":"Quasi-coherent sheaves on groupoids · Lemma 09VH","summary":"Let S be a scheme. Consider a morphism f : (U, R, s, t, c) → (U', R', s', t', c') of groupoid schemes over S. Assume that • f : U → U' is quasi-compact and quasi-separated, • the square xymatrix R ar[d]_t ar[r]_f & R' ar[d]^t' U ar[r]^f & U' is cartesian, and • s' and t' are flat. Then pushforward f_* given by (F, α) ↦ (f_*F, f_*α) defines a functor from the category of quasi-coherent sheaves on (U, R, s, t, c) to the category of quasi-coherent sheaves on (U', R', s', t',…","statement_latex":"Let $S$ be a scheme. Consider a morphism\n$f : (U, R, s, t, c) \\to (U', R', s', t', c')$\nof groupoid schemes over $S$. Assume that\n\\begin{enumerate}\n\\item $f : U \\to U'$ is quasi-compact and quasi-separated,\n\\item the square\n$$\n\\xymatrix{\nR \\ar[d]_t \\ar[r]_f & R' \\ar[d]^{t'} \\\\\nU \\ar[r]^f & U'\n}\n$$\nis cartesian, and\n\\item $s'$ and $t'$ are flat.\n\\end{enumerate}\nThen pushforward $f_*$ given by\n$$\n(\\mathcal{F}, \\alpha) \\mapsto (f_*\\mathcal{F}, f_*\\alpha)\n$$\ndefines a functor from the category of quasi-coherent sheaves on\n$(U, R, s, t, c)$ to the category of quasi-coherent sheaves on\n$(U', R', s', t', c')$ which is right adjoint to pullback as defined in\nLemma \\ref{lemma-pullback}.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VH","source_file":"groupoids.tex","source_line":2695,"source_end_line":2720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2695-L2720","statement_sha256":"e8895ebcfcd92915e660c930d434060b217924fe3adf70b376b3cb952a5e7bf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":7957,"rank":7957,"depth":30,"x":1055.708,"y":1162.103,"cluster":"groupoids-quotients"},{"id":"stacks:077R","tag":"077R","title":"Quasi-coherent sheaves on groupoids · Lemma 077R","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. The category of quasi-coherent modules on (U, R, s, t, c) has colimits.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nThe category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077R","source_file":"groupoids.tex","source_line":2754,"source_end_line":2758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2754-L2758","statement_sha256":"80d71552971c1d4718cdfff585f5793918e3a082014b830d73e3288d4a63ae76","origin":"The Stacks Project","memory_eligible":false,"source_rank":7958,"rank":7958,"depth":0,"x":1167.292,"y":1081.097,"cluster":"groupoids-quotients"},{"id":"stacks:077S","tag":"077S","title":"Quasi-coherent sheaves on groupoids · Lemma 077S","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. If s, t are flat, then the category of quasi-coherent modules on (U, R, s, t, c) is abelian.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nIf $s$, $t$ are flat, then the category of quasi-coherent modules on\n$(U, R, s, t, c)$ is abelian.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077S","source_file":"groupoids.tex","source_line":2774,"source_end_line":2780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2774-L2780","statement_sha256":"8e775f638afea8bf55b9fe05ad6e01306b7cdd5c709502c18978d3b904703015","origin":"The Stacks Project","memory_eligible":false,"source_rank":7959,"rank":7959,"depth":0,"x":1150.02,"y":1205.085,"cluster":"groupoids-quotients"},{"id":"stacks:07TR","tag":"07TR","title":"Colimits of quasi-coherent modules · Lemma 07TR","summary":"Let (U, R, s, t, c) be a groupoid scheme over S. Assume s, t are flat, quasi-compact, and quasi-separated. For any quasi-coherent module G on U, there exists a canonical isomorphism α : t^*s_*t^*G → s^*s_*t^*G which turns (s_*t^*G, α) into a quasi-coherent module on (U, R, s, t, c). This construction defines a functor QCoh(O_U) → QCoh(U, R, s, t, c) which is a right adjoint to the forgetful functor (F, β) ↦ F.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume $s, t$ are flat, quasi-compact, and quasi-separated.\nFor any quasi-coherent module $\\mathcal{G}$ on $U$, there exists\na canonical isomorphism\n$\\alpha : t^*s_*t^*\\mathcal{G} \\to s^*s_*t^*\\mathcal{G}$\nwhich turns $(s_*t^*\\mathcal{G}, \\alpha)$ into a quasi-coherent module\non $(U, R, s, t, c)$. This construction defines a functor\n$$\n\\QCoh(\\mathcal{O}_U) \\longrightarrow \\QCoh(U, R, s, t, c)\n$$\nwhich is a right adjoint to the forgetful functor\n$(\\mathcal{F}, \\beta) \\mapsto \\mathcal{F}$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Colimits of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TR","source_file":"groupoids.tex","source_line":2822,"source_end_line":2836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2822-L2836","statement_sha256":"fb2d79710ef45f6750c622dfdb03b2bcb2b230c91e1114f904b401d745882c6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7960,"rank":7960,"depth":31,"x":1062.397,"y":1103.094,"cluster":"groupoids-quotients"},{"id":"stacks:07TT","tag":"07TT","title":"Colimits of quasi-coherent modules · Lemma 07TT","summary":"Let f : Y → X be a morphism of schemes. Let F be a quasi-coherent O_X-module, let G be a quasi-coherent O_Y-module, and let φ : G → f^*F be a module map. Assume • φ is injective, • f is quasi-compact, quasi-separated, flat, and surjective, • X, Y are locally Noetherian, and • G is a coherent O_Y-module. Then F ∩ f_*G defined as the pullback xymatrix F ar[r] & f_*f^*F F ∩ f_*G ar[u] ar[r] & f_*G ar[u] is a coherent O_X-module.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes. Let $\\mathcal{F}$\nbe a quasi-coherent $\\mathcal{O}_X$-module, let $\\mathcal{G}$\nbe a quasi-coherent $\\mathcal{O}_Y$-module, and let\n$\\varphi : \\mathcal{G} \\to f^*\\mathcal{F}$ be a module map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is injective,\n\\item $f$ is quasi-compact, quasi-separated, flat, and surjective,\n\\item $X$, $Y$ are locally Noetherian, and\n\\item $\\mathcal{G}$ is a coherent $\\mathcal{O}_Y$-module.\n\\end{enumerate}\nThen $\\mathcal{F} \\cap f_*\\mathcal{G}$ defined as the pullback\n$$\n\\xymatrix{\n\\mathcal{F} \\ar[r] & f_*f^*\\mathcal{F} \\\\\n\\mathcal{F} \\cap f_*\\mathcal{G} \\ar[u] \\ar[r] &\nf_*\\mathcal{G} \\ar[u]\n}\n$$\nis a coherent $\\mathcal{O}_X$-module.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Colimits of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TT","source_file":"groupoids.tex","source_line":2957,"source_end_line":2978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L2957-L2978","statement_sha256":"349a07f7a868907d4ce56de142ca9def9e2a60c16c256a619a96ae24ad773469","origin":"The Stacks Project","memory_eligible":false,"source_rank":7961,"rank":7961,"depth":30,"x":1210.114,"y":1128.772,"cluster":"groupoids-quotients"},{"id":"stacks:07TU","tag":"07TU","title":"Colimits of quasi-coherent modules · Lemma 07TU","summary":"Let (U, R, s, t, c) be a groupoid scheme over S. Assume that • U, R are Noetherian, • s, t are flat, quasi-compact, and quasi-separated. Then every quasi-coherent module (F, β) on (U, R, s, t, c) is a filtered colimit of coherent modules.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume that\n\\begin{enumerate}\n\\item $U$, $R$ are Noetherian,\n\\item $s, t$ are flat, quasi-compact, and quasi-separated.\n\\end{enumerate}\nThen every quasi-coherent module $(\\mathcal{F}, \\beta)$ on $(U, R, s, t, c)$\nis a filtered colimit of coherent modules.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Colimits of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TU","source_file":"groupoids.tex","source_line":3008,"source_end_line":3018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3008-L3018","statement_sha256":"71b14f9e8557bc1e9baf95d8c2b9d3cde2a690969b761a582f3b95133e41900e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7962,"rank":7962,"depth":32,"x":1079.591,"y":1194.127,"cluster":"groupoids-quotients"},{"id":"stacks:07TV","tag":"07TV","title":"Colimits of quasi-coherent modules · Lemma 07TV","summary":"Let (U, R, s, t, c) be a groupoid scheme over S. Assume that • U, R are affine, • there exist e_i ∈ O_R(R) such that every element g ∈ O_R(R) can be uniquely written as ∑ s^*(f_i)e_i for some f_i ∈ O_U(U). Then every quasi-coherent module (F, α) on (U, R, s, t, c) is a filtered colimit of finite type quasi-coherent modules.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume that\n\\begin{enumerate}\n\\item $U$, $R$ are affine,\n\\item there exist $e_i \\in \\mathcal{O}_R(R)$ such that\nevery element $g \\in \\mathcal{O}_R(R)$ can be uniquely written as\n$\\sum s^*(f_i)e_i$ for some $f_i \\in \\mathcal{O}_U(U)$.\n\\end{enumerate}\nThen every quasi-coherent module $(\\mathcal{F}, \\alpha)$ on $(U, R, s, t, c)$\nis a filtered colimit of finite type quasi-coherent modules.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Colimits of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TV","source_file":"groupoids.tex","source_line":3076,"source_end_line":3088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3076-L3088","statement_sha256":"a832a7067a4cf60c4601138c83ae8a2861d7845bbf068a4fc77189ae4cc5533d","origin":"The Stacks Project","memory_eligible":false,"source_rank":7963,"rank":7963,"depth":2,"x":1123.597,"y":1070.995,"cluster":"groupoids-quotients"},{"id":"stacks:077T","tag":"077T","title":"Colimits of quasi-coherent modules · Lemma 077T","summary":"Let (U, R, s, t, c) be a groupoid scheme over S. Let kappa be a cardinal. There exists a set T and a family (F_t, α_t)_t ∈ T of kappa-generated quasi-coherent modules on (U, R, s, t, c) such that every kappa-generated quasi-coherent module on (U, R, s, t, c) is isomorphic to one of the (F_t, α_t).","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $\\kappa$ be a cardinal.\nThere exists a set $T$ and a family $(\\mathcal{F}_t, \\alpha_t)_{t \\in T}$ of\n$\\kappa$-generated quasi-coherent modules on $(U, R, s, t, c)$\nsuch that every $\\kappa$-generated quasi-coherent module on\n$(U, R, s, t, c)$ is isomorphic to one of the $(\\mathcal{F}_t, \\alpha_t)$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Colimits of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077T","source_file":"groupoids.tex","source_line":3132,"source_end_line":3140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3132-L3140","statement_sha256":"78361b5a6a42e58f779c818a0107d53d53f751518f54e327b8d52b95f6929153","origin":"The Stacks Project","memory_eligible":false,"source_rank":7964,"rank":7964,"depth":1,"x":1190.638,"y":1187.583,"cluster":"groupoids-quotients"},{"id":"stacks:077U","tag":"077U","title":"Colimits of quasi-coherent modules · Lemma 077U","summary":"Let (U, R, s, t, c) be a groupoid scheme over S. Assume that s, t are flat. There exists a cardinal kappa such that every quasi-coherent module (F, α) on (U, R, s, t, c) is the directed colimit of its kappa-generated quasi-coherent submodules.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume that $s, t$ are flat. There exists a\ncardinal $\\kappa$ such that every quasi-coherent module\n$(\\mathcal{F}, \\alpha)$ on $(U, R, s, t, c)$\nis the directed colimit of its $\\kappa$-generated\nquasi-coherent submodules.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Colimits of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077U","source_file":"groupoids.tex","source_line":3152,"source_end_line":3160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3152-L3160","statement_sha256":"a6fb2c67bcf061cc9634e0a4202c541e32ab0ddc83912119c4e85e309a303cfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7965,"rank":7965,"depth":14,"x":1046.445,"y":1139.315,"cluster":"groupoids-quotients"},{"id":"stacks:0234","tag":"0234","title":"Groupoids and group schemes · Lemma 0234","summary":"Let S be a scheme. Let Y be a scheme over S. Let (G, m) be a group scheme over Y with identity e_G and inverse i_G. Let X/Y be a scheme over Y and let a : G ×_Y X → X be an action of G on X/Y. Then we get a groupoid scheme (U, R, s, t, c, e, i) over S in the following manner: • We set U = X, and R = G ×_Y X. • We set s : R → U equal to (g, x) ↦ x. • We set t : R → U equal to (g, x) ↦ a(g, x). • We set c : R ×_s, U, t R → R equal to ((g, x), (g', x')) ↦ (m(g, g'), x'). •…","statement_latex":"Let $S$ be a scheme.\nLet $Y$ be a scheme over $S$.\nLet $(G, m)$ be a group scheme over $Y$ with\nidentity $e_G$ and inverse $i_G$.\nLet $X/Y$ be a scheme over $Y$ and let $a : G \\times_Y X \\to X$\nbe an action of $G$ on $X/Y$.\nThen we get a groupoid scheme $(U, R, s, t, c, e, i)$ over $S$\nin the following manner:\n\\begin{enumerate}\n\\item We set $U = X$, and $R = G \\times_Y X$.\n\\item We set $s : R \\to U$ equal to $(g, x) \\mapsto x$.\n\\item We set $t : R \\to U$ equal to $(g, x) \\mapsto a(g, x)$.\n\\item We set $c : R \\times_{s, U, t} R \\to R$ equal to\n$((g, x), (g', x')) \\mapsto (m(g, g'), x')$.\n\\item We set $e : U \\to R$ equal to $x \\mapsto (e_G(x), x)$.\n\\item We set $i : R \\to R$ equal to $(g, x) \\mapsto (i_G(g), a(g, x))$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Groupoids and group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0234","source_file":"groupoids.tex","source_line":3282,"source_end_line":3301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3282-L3301","statement_sha256":"b45f82e2def5890c0ca6db1d322d9b8090ab47397439605cf05d1767e2d86d69","origin":"The Stacks Project","memory_eligible":false,"source_rank":7966,"rank":7966,"depth":0,"x":1192.589,"y":1092.772,"cluster":"groupoids-quotients"},{"id":"stacks:03LL","tag":"03LL","title":"Groupoids and group schemes · Lemma 03LL","summary":"Let S be a scheme. Let Y be a scheme over S. Let (G, m) be a group scheme over Y. Let X be a scheme over Y and let a : G ×_Y X → X be an action of G on X over Y. Let (U, R, s, t, c) be the groupoid scheme constructed in Lemma [Tag 0234]. The rule (F, α) ↦ (F, α) defines an equivalence of categories between G-equivariant O_X-modules and the category of quasi-coherent modules on (U, R, s, t, c).","statement_latex":"Let $S$ be a scheme.\nLet $Y$ be a scheme over $S$.\nLet $(G, m)$ be a group scheme over $Y$.\nLet $X$ be a scheme over $Y$ and let $a : G \\times_Y X \\to X$\nbe an action of $G$ on $X$ over $Y$. Let $(U, R, s, t, c)$ be\nthe groupoid scheme constructed in Lemma \\ref{lemma-groupoid-from-action}.\nThe rule\n$(\\mathcal{F}, \\alpha) \\mapsto (\\mathcal{F}, \\alpha)$ defines\nan equivalence of categories between $G$-equivariant\n$\\mathcal{O}_X$-modules and the category of quasi-coherent\nmodules on $(U, R, s, t, c)$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Groupoids and group schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LL","source_file":"groupoids.tex","source_line":3309,"source_end_line":3322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3309-L3322","statement_sha256":"90e2bb2d5dcd86407ecc6871f20137340c81da7d0090a5c40923536ba5d3d2f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":7967,"rank":7967,"depth":2,"x":1121.772,"y":1210.815,"cluster":"groupoids-quotients"},{"id":"stacks:0235","tag":"0235","title":"The stabilizer group scheme · Lemma 0235","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. The scheme G defined by the cartesian square xymatrix G ar[r] ar[d] & R ar[d]^j = (t, s) U ar[r]^-Δ & U ×_S U is a group scheme over U with composition law m induced by the composition law c.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$.\nThe scheme $G$ defined by the cartesian square\n$$\n\\xymatrix{\nG \\ar[r] \\ar[d] & R \\ar[d]^{j = (t, s)} \\\\\nU \\ar[r]^-{\\Delta} & U \\times_S U\n}\n$$\nis a group scheme over $U$ with composition law\n$m$ induced by the composition law $c$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"The stabilizer group scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0235","source_file":"groupoids.tex","source_line":3344,"source_end_line":3357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3344-L3357","statement_sha256":"cd967ed820f90515b1c3be190eb6f14f3608417d52ca723111fc9e8e7c23d561","origin":"The Stacks Project","memory_eligible":false,"source_rank":7968,"rank":7968,"depth":0,"x":1078.781,"y":1082.734,"cluster":"groupoids-quotients"},{"id":"stacks:0236","tag":"0236","title":"The stabilizer group scheme · Definition 0236","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. The group scheme j^-1(Δ_U/S)→ U is called the stabilizer of the groupoid scheme (U, R, s, t, c).","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$.\nThe group scheme $j^{-1}(\\Delta_{U/S})\\to U$\nis called the {\\it stabilizer of the groupoid scheme\n$(U, R, s, t, c)$}.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"The stabilizer group scheme","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0236","source_file":"groupoids.tex","source_line":3370,"source_end_line":3377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3370-L3377","statement_sha256":"75ea5540952c93102fe910d89940130254e7e0b06d468b3e29bdab6efdb6c535","origin":"The Stacks Project","memory_eligible":false,"source_rank":7969,"rank":7969,"depth":0,"x":1214.371,"y":1153.251,"cluster":"groupoids-quotients"},{"id":"stacks:0237","tag":"0237","title":"The stabilizer group scheme · Lemma 0237","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S, and let G/U be its stabilizer. Denote R_t/U the scheme R seen as a scheme over U via the morphism t : R → U. There is a canonical left action a : G ×_U R_t → R_t induced by the composition law c.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$, and let $G/U$ be its stabilizer.\nDenote $R_t/U$ the scheme $R$ seen as a scheme over $U$ via the\nmorphism $t : R \\to U$.\nThere is a canonical left action\n$$\na : G \\times_U R_t \\longrightarrow R_t\n$$\ninduced by the composition law $c$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"The stabilizer group scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0237","source_file":"groupoids.tex","source_line":3384,"source_end_line":3395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3384-L3395","statement_sha256":"cfb2b66cb1bf05f88302b29cd92c9d815c88447309bb5e758198acc9c47473ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":7970,"rank":7970,"depth":0,"x":1056.657,"y":1178.351,"cluster":"groupoids-quotients"},{"id":"stacks:04Q2","tag":"04Q2","title":"The stabilizer group scheme · Lemma 04Q2","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let G be the stabilizer group scheme of R. Let G_0 = G ×_U, pr_0 (U ×_S U) = G ×_S U as a group scheme over U ×_S U. The action of G on R of Lemma [Tag 0237] induces an action of G_0 on R over U ×_S U which turns R into a pseudo G_0-torsor over U ×_S U.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Let $G$ be the stabilizer group scheme of $R$.\nLet\n$$\nG_0 = G \\times_{U, \\text{pr}_0} (U \\times_S U) = G \\times_S U\n$$\nas a group scheme over $U \\times_S U$. The action of $G$ on $R$ of\nLemma \\ref{lemma-groupoid-action-stabilizer}\ninduces an action of $G_0$ on $R$ over $U \\times_S U$\nwhich turns $R$ into a pseudo $G_0$-torsor over $U \\times_S U$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"The stabilizer group scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Q2","source_file":"groupoids.tex","source_line":3401,"source_end_line":3413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3401-L3413","statement_sha256":"169e839ff620dcf12c1e797670fdeccd8f31f5bd97ecbf0e15e83ff76d49c54e","origin":"The Stacks Project","memory_eligible":false,"source_rank":7971,"rank":7971,"depth":1,"x":1153.39,"y":1069.656,"cluster":"groupoids-quotients"},{"id":"stacks:04Q3","tag":"04Q3","title":"The stabilizer group scheme · Lemma 04Q3","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let p ∈ U ×_S U be a point. Denote R_p the scheme theoretic fibre of j = (t, s) : R → U ×_S U. If R_p not = ∅, then the action G_0, kappa(p) ×_kappa(p) R_p → R_p (see Lemma [Tag 04Q2]) which turns R_p into a G_kappa(p)-torsor over kappa(p).","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $p \\in U \\times_S U$ be a point. Denote\n$R_p$ the scheme theoretic fibre of $j = (t, s) : R \\to U \\times_S U$.\nIf $R_p \\not = \\emptyset$, then the action\n$$\nG_{0, \\kappa(p)} \\times_{\\kappa(p)} R_p \\longrightarrow R_p\n$$\n(see\nLemma \\ref{lemma-groupoid-action-stabilizer-pseudo-torsor})\nwhich turns $R_p$ into a $G_{\\kappa(p)}$-torsor over $\\kappa(p)$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"The stabilizer group scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Q3","source_file":"groupoids.tex","source_line":3421,"source_end_line":3433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3421-L3433","statement_sha256":"6287388e42d57c5dd05df53283f1f5df09ec70ae54931fdb60800994212b3033","origin":"The Stacks Project","memory_eligible":false,"source_rank":7972,"rank":7972,"depth":2,"x":1169.57,"y":1205.555,"cluster":"groupoids-quotients"},{"id":"stacks:02VB","tag":"02VB","title":"Restricting groupoids · Lemma 02VB","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' → U be a morphism of schemes. Consider the following diagram xymatrix R' ar[d] ar[r] ar@/_3pc/[dd]_t' ar@/^1pc/[rr]^s'& R ×_s, U U' ar[r] ar[d] & U' ar[d]^g U' ×_U, t R ar[d] ar[r] & R ar[r]^s ar[d]_t & U U' ar[r]^g & U where all the squares are fibre product squares. Then there is a canonical composition law c' : R' ×_s', U', t' R' → R' such that (U', R', s', t', c') is a groupoid scheme over…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $g : U' \\to U$ be a morphism of schemes.\nConsider the following diagram\n$$\n\\xymatrix{\nR' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'}&\nR \\times_{s, U} U' \\ar[r] \\ar[d] &\nU' \\ar[d]^g \\\\\nU' \\times_{U, t} R \\ar[d] \\ar[r] &\nR \\ar[r]^s \\ar[d]_t &\nU \\\\\nU' \\ar[r]^g &\nU\n}\n$$\nwhere all the squares are fibre product squares. Then there is a\ncanonical composition law $c' : R' \\times_{s', U', t'} R' \\to R'$\nsuch that $(U', R', s', t', c')$ is a groupoid scheme over\n$S$ and such that $U' \\to U$, $R' \\to R$ defines a morphism\n$(U', R', s', t', c') \\to (U, R, s, t, c)$ of groupoid schemes over $S$.\nMoreover, for any scheme $T$ over $S$ the functor of groupoids\n$$\n(U'(T), R'(T), s', t', c') \\to (U(T), R(T), s, t, c)\n$$\nis the restriction (see above) of $(U(T), R(T), s, t, c)$ via the map\n$U'(T) \\to U(T)$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VB","source_file":"groupoids.tex","source_line":3474,"source_end_line":3503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3474-L3503","statement_sha256":"45ea3b4ed39c77a383f1a3aa291cf499da1c20fd41edf6191c53a37656f32407","origin":"The Stacks Project","memory_eligible":false,"source_rank":7973,"rank":7973,"depth":0,"x":1047.593,"y":1113.952,"cluster":"groupoids-quotients"},{"id":"stacks:02VC","tag":"02VC","title":"Restricting groupoids · Definition 02VC","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' → U be a morphism of schemes. The morphism of groupoids (U', R', s', t', c') → (U, R, s, t, c) constructed in Lemma [Tag 02VB] is called the restriction of (U, R, s, t, c) to U'. We sometime use the notation R' = R|_U' in this case.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $g : U' \\to U$ be a morphism of schemes.\nThe morphism of groupoids\n$(U', R', s', t', c') \\to (U, R, s, t, c)$\nconstructed in Lemma \\ref{lemma-restrict-groupoid} is called\nthe {\\it restriction of $(U, R, s, t, c)$ to $U'$}.\nWe sometime use the notation $R' = R|_{U'}$ in this case.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Restricting groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VC","source_file":"groupoids.tex","source_line":3509,"source_end_line":3519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3509-L3519","statement_sha256":"dcc853847c821ae923b29b540e3465bc22a2f5c31d7f8f65177b4357c5bae69c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7974,"rank":7974,"depth":1,"x":1212.217,"y":1112.278,"cluster":"groupoids-quotients"},{"id":"stacks:02VD","tag":"02VD","title":"Restricting groupoids · Lemma 02VD","summary":"The notions of restricting groupoids and (pre-)equivalence relations defined in Definitions [Tag 02VC] and [Tag 02V9] agree via the constructions of Lemmas [Tag 0232] and [Tag 0233].","statement_latex":"The notions of restricting groupoids and\n(pre-)equivalence relations defined in Definitions\n\\ref{definition-restrict-groupoid} and \\ref{definition-restrict-relation}\nagree via the constructions of\nLemmas \\ref{lemma-groupoid-pre-equivalence} and\n\\ref{lemma-equivalence-groupoid}.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VD","source_file":"groupoids.tex","source_line":3521,"source_end_line":3529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3521-L3529","statement_sha256":"2978cdbbb53407d68e006e6dbdce4ce944befe062e6e95aa9383ac0c713de09f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7975,"rank":7975,"depth":2,"x":1091.434,"y":1207.503,"cluster":"groupoids-quotients"},{"id":"stacks:04ML","tag":"04ML","title":"Restricting groupoids · Lemma 04ML","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' → U be a morphism of schemes. Let (U', R', s', t', c') be the restriction of (U, R, s, t, c) via g. Let G be the stabilizer of (U, R, s, t, c) and let G' be the stabilizer of (U', R', s', t', c'). Then G' is the base change of G by g, i.e., there is a canonical identification G' = U' ×_g, U G.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $g : U' \\to U$ be a morphism of schemes.\nLet $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$.\nLet $G$ be the stabilizer of $(U, R, s, t, c)$ and let\n$G'$ be the stabilizer of $(U', R', s', t', c')$.\nThen $G'$ is the base change of $G$ by $g$, i.e.,\nthere is a canonical identification $G' = U' \\times_{g, U} G$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ML","source_file":"groupoids.tex","source_line":3543,"source_end_line":3553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3543-L3553","statement_sha256":"73c765fb0bddefb597b6117d99ddb38bb76634068167718088ff26a4582d99ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":7976,"rank":7976,"depth":0,"x":1103.998,"y":1067.91,"cluster":"groupoids-quotients"},{"id":"stacks:03BC","tag":"03BC","title":"Invariant subschemes · Definition 03BC","summary":"Let (U, R, s, t, c) be a groupoid scheme over the base scheme S. • A subset W ⊂ U is set-theoretically R-invariant if t(s^-1(W)) ⊂ W. • An open W ⊂ U is R-invariant if t(s^-1(W)) ⊂ W. • A closed subscheme Z ⊂ U is called R-invariant if t^-1(Z) = s^-1(Z). Here we use the scheme theoretic inverse image, see Schemes, Definition [Tag 01JV]. • A monomorphism of schemes T → U is R-invariant if T ×_U, t R = R ×_s, U T as schemes over R.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over the base scheme $S$.\n\\begin{enumerate}\n\\item A subset $W \\subset U$ is {\\it set-theoretically $R$-invariant}\nif $t(s^{-1}(W)) \\subset W$.\n\\item An open $W \\subset U$ is {\\it $R$-invariant} if\n$t(s^{-1}(W)) \\subset W$.\n\\item A closed subscheme $Z \\subset U$ is called {\\it $R$-invariant}\nif $t^{-1}(Z) = s^{-1}(Z)$. Here we use the scheme theoretic inverse image, see\nSchemes, Definition \\ref{schemes-definition-inverse-image-closed-subscheme}.\n\\item A monomorphism of schemes $T \\to U$ is {\\it $R$-invariant} if\n$T \\times_{U, t} R = R \\times_{s, U} T$ as schemes over $R$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Invariant subschemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BC","source_file":"groupoids.tex","source_line":3570,"source_end_line":3584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3570-L3584","statement_sha256":"132430f6488661da3ea67fc0ef2d93c5f2e7848667b9c93cd266e1d0bef0ce31","origin":"The Stacks Project","memory_eligible":false,"source_rank":7977,"rank":7977,"depth":14,"x":1207.605,"y":1178.633,"cluster":"groupoids-quotients"},{"id":"stacks:03LO","tag":"03LO","title":"Invariant subschemes · Lemma 03LO","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. • For any subset W ⊂ U the subset t(s^-1(W)) is set-theoretically R-invariant. • If s and t are open, then for every open W ⊂ U the open t(s^-1(W)) is an R-invariant open subscheme. • If s and t are open and quasi-compact, then U has an open covering consisting of R-invariant quasi-compact open subschemes.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\n\\begin{enumerate}\n\\item For any subset $W \\subset U$ the subset $t(s^{-1}(W))$\nis set-theoretically $R$-invariant.\n\\item If $s$ and $t$ are open, then for every open $W \\subset U$\nthe open $t(s^{-1}(W))$ is an $R$-invariant open subscheme.\n\\item If $s$ and $t$ are open and quasi-compact, then $U$ has an open\ncovering consisting of $R$-invariant quasi-compact open subschemes.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Invariant subschemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LO","source_file":"groupoids.tex","source_line":3594,"source_end_line":3606,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3594-L3606","statement_sha256":"36331839d5a786bc08c139cc7f69ddbe8572962b1ea08c384c05681de2ee30d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7978,"rank":7978,"depth":1,"x":1041.189,"y":1155.639,"cluster":"groupoids-quotients"},{"id":"stacks:0APA","tag":"0APA","title":"Invariant subschemes · Lemma 0APA","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume s and t quasi-compact and flat and U quasi-separated. Let W ⊂ U be quasi-compact open. Then t(s^-1(W)) is an intersection of a nonempty family of quasi-compact open subsets of U.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume $s$ and $t$ quasi-compact and flat and $U$ quasi-separated.\nLet $W \\subset U$ be quasi-compact open. Then $t(s^{-1}(W))$\nis an intersection of a nonempty family of quasi-compact open subsets of $U$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Invariant subschemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APA","source_file":"groupoids.tex","source_line":3622,"source_end_line":3628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3622-L3628","statement_sha256":"8815665e235f5892f3fc97ceac8e57bc84a98ab65fd114dbaf46ba8ffd572a36","origin":"The Stacks Project","memory_eligible":false,"source_rank":7979,"rank":7979,"depth":6,"x":1183.219,"y":1077.714,"cluster":"groupoids-quotients"},{"id":"stacks:0APB","tag":"0APB","title":"Invariant subschemes · Lemma 0APB","summary":"Assumptions and notation as in Lemma [Tag 0APA]. There exists an R-invariant open V ⊂ U and a quasi-compact open W' such that W ⊂ V ⊂ W' ⊂ U.","statement_latex":"Assumptions and notation as in Lemma \\ref{lemma-first-observation}.\nThere exists an $R$-invariant open $V \\subset U$ and a quasi-compact\nopen $W'$ such that $W \\subset V \\subset W' \\subset U$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Invariant subschemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APB","source_file":"groupoids.tex","source_line":3646,"source_end_line":3651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3646-L3651","statement_sha256":"a5a312501e407792dd640c52e708fc2b8346696fb26f20c0d9032f704ca1738a","origin":"The Stacks Project","memory_eligible":false,"source_rank":7980,"rank":7980,"depth":20,"x":1140.908,"y":1216.563,"cluster":"groupoids-quotients"},{"id":"stacks:02VG","tag":"02VG","title":"Quotient sheaves · Definition 02VG","summary":"Let τ, S, and the pre-relation j : R → U ×_S U be as above. In this setting the quotient sheaf U/R associated to j is the sheafification of the presheaf ([Tag 02VF]) in the τ-topology. If j : R → U ×_S U comes from the action of a group scheme G/S on U as in Lemma [Tag 0234] then we sometimes denote the quotient sheaf U/G.","statement_latex":"Let $\\tau$, $S$, and the pre-relation $j : R \\to U \\times_S U$ be as above.\nIn this setting the {\\it quotient sheaf $U/R$} associated\nto $j$ is the sheafification of the presheaf\n(\\ref{equation-quotient-presheaf}) in the $\\tau$-topology.\nIf $j : R \\to U \\times_S U$ comes from the action of a group scheme\n$G/S$ on $U$ as in Lemma \\ref{lemma-groupoid-from-action} then we\nsometimes denote the quotient sheaf $U/G$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quotient sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VG","source_file":"groupoids.tex","source_line":3713,"source_end_line":3722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3713-L3722","statement_sha256":"8f5ed82230787efa60304fd19b39e9ef59306ae23f13eaa381fe3d04402f0315","origin":"The Stacks Project","memory_eligible":false,"source_rank":7981,"rank":7981,"depth":1,"x":1059.991,"y":1089.448,"cluster":"groupoids-quotients"},{"id":"stacks:03BD","tag":"03BD","title":"Quotient sheaves · Definition 03BD","summary":"In the situation of Definition [Tag 02VG]. We say that the pre-relation j has a representable quotient if the sheaf U/R is representable. We will say a groupoid (U, R, s, t, c) has a representable quotient if the quotient U/R with j = (t, s) is representable.","statement_latex":"In the situation of Definition \\ref{definition-quotient-sheaf}.\nWe say that the pre-relation $j$ has a\n{\\it representable quotient} if the sheaf $U/R$ is representable.\nWe will say a groupoid $(U, R, s, t, c)$ has a\n{\\it representable quotient}\nif the quotient $U/R$ with $j = (t, s)$ is representable.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quotient sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BD","source_file":"groupoids.tex","source_line":3749,"source_end_line":3757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3749-L3757","statement_sha256":"48b4a5e1c3bf6643eaae490be07e5174d6e7c067cb6846dfa0fcb140d7ed4c15","origin":"The Stacks Project","memory_eligible":false,"source_rank":7982,"rank":7982,"depth":2,"x":1222.794,"y":1137.538,"cluster":"groupoids-quotients"},{"id":"stacks:03C5","tag":"03C5","title":"Quotient sheaves · Lemma 03C5","summary":"In the situation of Definition [Tag 02VG]. Assume there is a scheme M, and a morphism U → M such that • the morphism U → M equalizes s, t, • the morphism U → M induces a surjection of sheaves h_U → h_M in the τ-topology, and • the induced map (t, s) : R → U ×_M U induces a surjection of sheaves h_R → h_U ×_M U in the τ-topology. In this case M represents the quotient sheaf U/R.","statement_latex":"In the situation of Definition \\ref{definition-quotient-sheaf}.\nAssume there is a scheme $M$, and a morphism $U \\to M$ such that\n\\begin{enumerate}\n\\item the morphism $U \\to M$ equalizes $s, t$,\n\\item the morphism $U \\to M$ induces a surjection of sheaves\n$h_U \\to h_M$ in the $\\tau$-topology, and\n\\item the induced map $(t, s) : R \\to U \\times_M U$ induces a\nsurjection of sheaves $h_R \\to h_{U \\times_M U}$ in the $\\tau$-topology.\n\\end{enumerate}\nIn this case $M$ represents the quotient sheaf $U/R$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quotient sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03C5","source_file":"groupoids.tex","source_line":3764,"source_end_line":3776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3764-L3776","statement_sha256":"48761b960eaf26683d76529f2a01a13ba5f54b013ead04ff1d7c939465368b7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7983,"rank":7983,"depth":2,"x":1063.186,"y":1194.771,"cluster":"groupoids-quotients"},{"id":"stacks:045Y","tag":"045Y","title":"Quotient sheaves · Lemma 045Y","summary":"Let τ ∈ (Zariski, etale, fppf, smooth, syntomic). Let S be a scheme. Let j : R → U ×_S U be a pre-equivalence relation over S. Assume U, R, S are objects of a τ-site Sch_τ. For T ∈ Ob((Sch/S)_τ) and a, b ∈ U(T) the following are equivalent: • a and b map to the same element of (U/R)(T), and • there exists a τ-covering (f_i : T_i → T) of T and morphisms r_i : T_i → R such that a ∘ f_i = s ∘ r_i and b ∘ f_i = t ∘ r_i. In other words, in this case the map of τ-sheaves h_R →…","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$.\nLet $S$ be a scheme.\nLet $j : R \\to U \\times_S U$ be a pre-equivalence relation over $S$.\nAssume $U, R, S$ are objects of a $\\tau$-site $\\Sch_\\tau$.\nFor $T \\in \\Ob((\\Sch/S)_\\tau)$ and\n$a, b \\in U(T)$ the following are equivalent:\n\\begin{enumerate}\n\\item $a$ and $b$ map to the same element of $(U/R)(T)$, and\n\\item there exists a $\\tau$-covering $\\{f_i : T_i \\to T\\}$ of $T$\nand morphisms $r_i : T_i \\to R$ such that\n$a \\circ f_i = s \\circ r_i$ and $b \\circ f_i = t \\circ r_i$.\n\\end{enumerate}\nIn other words, in this case the map of $\\tau$-sheaves\n$$\nh_R \\longrightarrow h_U \\times_{U/R} h_U\n$$\nis surjective.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quotient sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045Y","source_file":"groupoids.tex","source_line":3789,"source_end_line":3808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3789-L3808","statement_sha256":"0a988ee78b14d61f9093e2d55f01c4e9104c63c80798f19d84e7fe2f070a0ac3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7984,"rank":7984,"depth":1,"x":1135.251,"y":1061.271,"cluster":"groupoids-quotients"},{"id":"stacks:045Z","tag":"045Z","title":"Quotient sheaves · Lemma 045Z","summary":"Let τ ∈ (Zariski, etale, fppf, smooth, syntomic). Let S be a scheme. Let j : R → U ×_S U be a pre-equivalence relation over S and g : U' → U a morphism of schemes over S. Let j' : R' → U' ×_S U' be the restriction of j to U'. Assume U, U', R, S are objects of a τ-site Sch_τ. The map of quotient sheaves U'/R' → U/R is injective. If g defines a surjection h_U' → h_U of sheaves in the τ-topology (for example if (g : U' → U) is a τ-covering), then U'/R' → U/R is an isomorphism.","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$.\nLet $S$ be a scheme.\nLet $j : R \\to U \\times_S U$ be a pre-equivalence relation over $S$\nand $g : U' \\to U$ a morphism of schemes over $S$.\nLet $j' : R' \\to U' \\times_S U'$ be the restriction of $j$ to $U'$.\nAssume  $U, U', R, S$ are objects of a $\\tau$-site $\\Sch_\\tau$.\nThe map of quotient sheaves\n$$\nU'/R' \\longrightarrow U/R\n$$\nis injective. If $g$ defines a surjection $h_{U'} \\to h_U$ of sheaves\nin the $\\tau$-topology (for example if $\\{g : U' \\to U\\}$ is a\n$\\tau$-covering), then $U'/R' \\to U/R$ is an isomorphism.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quotient sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045Z","source_file":"groupoids.tex","source_line":3818,"source_end_line":3833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3818-L3833","statement_sha256":"6f49aa9644a196ec315470ead322bd126ad043025b50bd9ddc9099cf4e6b6f16","origin":"The Stacks Project","memory_eligible":false,"source_rank":7985,"rank":7985,"depth":2,"x":1189.762,"y":1201.364,"cluster":"groupoids-quotients"},{"id":"stacks:02VH","tag":"02VH","title":"Quotient sheaves · Lemma 02VH","summary":"Let τ ∈ (Zariski, etale, fppf, smooth, syntomic). Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' → U a morphism of schemes over S. Let (U', R', s', t', c') be the restriction of (U, R, s, t, c) to U'. Assume U, U', R, S are objects of a τ-site Sch_τ. The map of quotient sheaves U'/R' → U/R is injective. If the composition xymatrix U' ×_g, U, t R ar[r]_-pr_1 ar@/^3ex/[rr]^h & R ar[r]_s & U defines a surjection of sheaves in the τ-topology…","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, fppf, smooth, syntomic\\}$.\nLet $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $g : U' \\to U$ a morphism of schemes over $S$.\nLet $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ to $U'$.\nAssume  $U, U', R, S$ are objects of a $\\tau$-site $\\Sch_\\tau$.\nThe map of quotient sheaves\n$$\nU'/R' \\longrightarrow U/R\n$$\nis injective. If the composition\n$$\n\\xymatrix{\nU' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h\n& R \\ar[r]_s & U\n}\n$$\ndefines a surjection of sheaves in the $\\tau$-topology  then\nthe map is bijective. This holds for example if\n$\\{h : U' \\times_{g, U, t} R \\to U\\}$ is a $\\tau$-covering, or\nif $U' \\to U$ defines a surjection of sheaves in the $\\tau$-topology, or if\n$\\{g : U' \\to U\\}$ is a covering in the $\\tau$-topology.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quotient sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02VH","source_file":"groupoids.tex","source_line":3860,"source_end_line":3884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3860-L3884","statement_sha256":"0529e6353e99e945e1dae0ac37c6ff5f4216f71b6b68845be325d9be2585940b","origin":"The Stacks Project","memory_eligible":false,"source_rank":7986,"rank":7986,"depth":3,"x":1036.083,"y":1128.601,"cluster":"groupoids-quotients"},{"id":"stacks:07S3","tag":"07S3","title":"Quotient sheaves · Lemma 07S3","summary":"Let S be a scheme. Let f : (U, R, j) → (U', R', j') be a morphism between equivalence relations over S. Assume that xymatrix R ar[d]_s ar[r]_f & R' ar[d]^s' U ar[r]^f & U' is cartesian. For any τ ∈ (Zariski, etale, fppf, smooth, syntomic) the diagram xymatrix U ar[d] ar[r] & U/R ar[d]^f U' ar[r] & U'/R' is a fibre product square of τ-sheaves.","statement_latex":"Let $S$ be a scheme. Let $f : (U, R, j) \\to (U', R', j')$ be a morphism\nbetween equivalence relations over $S$. Assume that\n$$\n\\xymatrix{\nR \\ar[d]_s \\ar[r]_f & R' \\ar[d]^{s'} \\\\\nU \\ar[r]^f & U'\n}\n$$\nis cartesian. For any\n$\\tau \\in  \\{Zariski, \\etale, fppf, smooth, syntomic\\}$\nthe diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & U/R \\ar[d]^f \\\\\nU' \\ar[r] & U'/R'\n}\n$$\nis a fibre product square of $\\tau$-sheaves.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Quotient sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07S3","source_file":"groupoids.tex","source_line":3917,"source_end_line":3937,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3917-L3937","statement_sha256":"705ba6190c000e178c3d80fc846b4614987aab8c8775733c64f2613ddc98841c","origin":"The Stacks Project","memory_eligible":false,"source_rank":7987,"rank":7987,"depth":2,"x":1208.836,"y":1094.877,"cluster":"groupoids-quotients"},{"id":"stacks:0APD","tag":"0APD","title":"Descent in terms of groupoids · Definition 0APD","summary":"Let S be a scheme. Let f : (U', R', s', t', c') → (U, R, s, t, c) be a morphism of groupoid schemes over S. We say f is cartesian, or that (U', R', s', t', c') is cartesian over (U, R, s, t, c), if the diagram xymatrix R' ar[r]_f ar[d]_s' & R ar[d]^s U' ar[r]^f & U is a fibre square in the category of schemes. A morphism of groupoid schemes cartesian over (U, R, s, t, c) is a morphism of groupoid schemes compatible with the structure morphisms towards (U, R, s, t, c).","statement_latex":"Let $S$ be a scheme. Let $f : (U', R', s', t', c') \\to (U, R, s, t, c)$ be\na morphism of groupoid schemes over $S$. We say $f$ is {\\it cartesian}, or\nthat {\\it $(U', R', s', t', c')$ is cartesian over $(U, R, s, t, c)$},\nif the diagram\n$$\n\\xymatrix{\nR' \\ar[r]_f \\ar[d]_{s'} & R \\ar[d]^s \\\\\nU' \\ar[r]^f & U\n}\n$$\nis a fibre square in the category of schemes. A {\\it morphism of groupoid\nschemes cartesian over $(U, R, s, t, c)$} is a morphism of groupoid\nschemes compatible with the structure morphisms towards $(U, R, s, t, c)$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Descent in terms of groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APD","source_file":"groupoids.tex","source_line":3983,"source_end_line":3998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L3983-L3998","statement_sha256":"c36a44c280d080d2d593f3f9b7cea18c0cfd734056c4e46a1f0244ed0ba296ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":7988,"rank":7988,"depth":0,"x":1108.04,"y":1218.417,"cluster":"groupoids-quotients"},{"id":"stacks:0APE","tag":"0APE","title":"Descent in terms of groupoids · Lemma 0APE","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. The category of groupoid schemes cartesian over (U, R, s, t, c) is equivalent to the category of pairs (V, φ) where V is a scheme over U and φ : V ×_U, t R → R ×_s, U V is an isomorphism over R such that e^*φ = id_V and such that c^*φ = pr_1^*φ ∘ pr_0^*φ as morphisms of schemes over R ×_s, U, t R.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nThe category of groupoid schemes cartesian over $(U, R, s, t, c)$\nis equivalent to the category of pairs $(V, \\varphi)$ where $V$ is a\nscheme over $U$ and\n$$\n\\varphi :\nV \\times_{U, t} R\n\\longrightarrow\nR \\times_{s, U} V\n$$\nis an isomorphism over $R$ such that $e^*\\varphi = \\text{id}_V$ and such that\n$$\nc^*\\varphi = \\text{pr}_1^*\\varphi \\circ \\text{pr}_0^*\\varphi\n$$\nas morphisms of schemes over $R \\times_{s, U, t} R$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Descent in terms of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APE","source_file":"groupoids.tex","source_line":4005,"source_end_line":4022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4005-L4022","statement_sha256":"a3ecb505972c69f45bc36a2dc62b8c3b0c5390aaba5ddd9fab63dee25a225410","origin":"The Stacks Project","memory_eligible":false,"source_rank":7989,"rank":7989,"depth":0,"x":1082.889,"y":1069.35,"cluster":"groupoids-quotients"},{"id":"stacks:0APF","tag":"0APF","title":"Descent in terms of groupoids · Lemma 0APF","summary":"Let S be a scheme. Let f : X → Y be a morphism of schemes over S. The construction of Lemma [Tag 0APE] determines an equivalence category of groupoid schemes cartesian over (X, X ×_Y X, …) → category of descent data relative to X/Y","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of schemes over $S$.\nThe construction of Lemma \\ref{lemma-characterize-cartesian-schemes}\ndetermines an equivalence\n$$\n\\begin{matrix}\n\\text{category of groupoid schemes} \\\\\n\\text{cartesian over } (X, X \\times_Y X, \\ldots)\n\\end{matrix}\n\\longrightarrow\n\\begin{matrix}\n\\text{ category of descent data} \\\\\n\\text{ relative to } X/Y\n\\end{matrix}\n$$","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Descent in terms of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APF","source_file":"groupoids.tex","source_line":4073,"source_end_line":4089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4073-L4089","statement_sha256":"e66afb494bbdb542a22b4bb6decf959bd1019e73cdf1a198fa9d935108485295","origin":"The Stacks Project","memory_eligible":false,"source_rank":7990,"rank":7990,"depth":1,"x":1222.025,"y":1165.495,"cluster":"groupoids-quotients"},{"id":"stacks:02YH","tag":"02YH","title":"Separation conditions · Lemma 02YH","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let G → U be the stabilizer group scheme. The commutative diagram xymatrix R ar[d]^Δ_R/U ×_S U ar[rrr]_f ↦ (f, s(f)) & & & R ×_s, U U ar[d] ar[r] & U ar[d] R ×_(U ×_S U) R ar[rrr]^(f, g) ↦ (f, f^-1 ∘ g) & & & R ×_s, U G ar[r] & G the two left horizontal arrows are isomorphisms and the right square is a fibre product square.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$.\nLet $G \\to U$ be the stabilizer group scheme.\nThe commutative diagram\n$$\n\\xymatrix{\nR \\ar[d]^{\\Delta_{R/U \\times_S U}} \\ar[rrr]_{f \\mapsto (f, s(f))} & & &\nR \\times_{s, U} U \\ar[d] \\ar[r] & U \\ar[d] \\\\\nR \\times_{(U \\times_S U)} R \\ar[rrr]^{(f, g) \\mapsto (f, f^{-1} \\circ g)} & & &\nR \\times_{s, U} G \\ar[r] & G\n}\n$$\nthe two left horizontal arrows are isomorphisms\nand the right square is a fibre product square.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Separation conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YH","source_file":"groupoids.tex","source_line":4112,"source_end_line":4128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4112-L4128","statement_sha256":"9de8a758041ab619dc4ad247c497a2e653a10bbdafd8cd096da537be4fb7217f","origin":"The Stacks Project","memory_eligible":false,"source_rank":7991,"rank":7991,"depth":0,"x":1041.19,"y":1173.588,"cluster":"groupoids-quotients"},{"id":"stacks:02YI","tag":"02YI","title":"Separation conditions · Lemma 02YI","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let G → U be the stabilizer group scheme. • The following are equivalent • j : R → U ×_S U is separated, • G → U is separated, and • e : U → G is a closed immersion. • The following are equivalent • j : R → U ×_S U is quasi-separated, • G → U is quasi-separated, and • e : U → G is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$.\nLet $G \\to U$ be the stabilizer group scheme.\n\\begin{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $j : R \\to U \\times_S U$ is separated,\n\\item $G \\to U$ is separated, and\n\\item $e : U \\to G$ is a closed immersion.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $j : R \\to U \\times_S U$ is quasi-separated,\n\\item $G \\to U$ is quasi-separated, and\n\\item $e : U \\to G$ is quasi-compact.\n\\end{enumerate}\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Separation conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YI","source_file":"groupoids.tex","source_line":4136,"source_end_line":4155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4136-L4155","statement_sha256":"87ff2afecdee0e2e11068d31e07de7db4142153b2c4ecd809cb76ab250e0e448","origin":"The Stacks Project","memory_eligible":false,"source_rank":7992,"rank":7992,"depth":17,"x":1168.671,"y":1064.461,"cluster":"groupoids-quotients"},{"id":"stacks:03BH","tag":"03BH","title":"Finite flat groupoids, affine case · Lemma 03BH","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(A) and R = Spec(B) are affine and s, t : R → U finite locally free. Let C be as in ([Tag 03BF]). Let f ∈ A. Then Norm_s^sharp(t^sharp(f)) ∈ C.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and\n$s, t : R \\to U$ finite locally free.\nLet $C$ be as in (\\ref{equation-invariants}).\nLet $f \\in A$. Then $\\text{Norm}_{s^\\sharp}(t^\\sharp(f)) \\in C$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Finite flat groupoids, affine case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BH","source_file":"groupoids.tex","source_line":4251,"source_end_line":4258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4251-L4258","statement_sha256":"04e4a426127c9b6364c34022c4cc937f018cc96520eb9104c036e22dcd179cda","origin":"The Stacks Project","memory_eligible":false,"source_rank":7993,"rank":7993,"depth":1,"x":1162.392,"y":1218.031,"cluster":"groupoids-quotients"},{"id":"stacks:03BI","tag":"03BI","title":"Finite flat groupoids, affine case · Lemma 03BI","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume s, t : R → U finite locally free. Then U = coprod_r ≥ 1 U_r is a disjoint union of R-invariant opens such that the restriction R_r of R to U_r has the property that s, t : R_r → U_r are finite locally free of rank r.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume $s, t : R \\to U$ finite locally free.\nThen\n$$\nU = \\coprod\\nolimits_{r \\geq 1} U_r\n$$\nis a disjoint union of $R$-invariant opens such that the restriction $R_r$ of\n$R$ to $U_r$ has the property that $s, t : R_r \\to U_r$ are finite locally\nfree of rank $r$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Finite flat groupoids, affine case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BI","source_file":"groupoids.tex","source_line":4289,"source_end_line":4300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4289-L4300","statement_sha256":"d7c11bbdbce937ad8ef598d8e43ffed2d7f11c743959a17fe0023dd2b56d88c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":7994,"rank":7994,"depth":1,"x":1042.936,"y":1100.647,"cluster":"groupoids-quotients"},{"id":"stacks:03BJ","tag":"03BJ","title":"Finite flat groupoids, affine case · Lemma 03BJ","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(A) and R = Spec(B) are affine and s, t : R → U finite locally free. Let C ⊂ A be as in ([Tag 03BF]). Then A is integral over C.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and\n$s, t : R \\to U$ finite locally free.\nLet $C \\subset A$ be as in (\\ref{equation-invariants}).\nThen $A$ is integral over $C$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Finite flat groupoids, affine case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BJ","source_file":"groupoids.tex","source_line":4325,"source_end_line":4332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4325-L4332","statement_sha256":"d8127072b19311cc1aed1a52ff7af0a8cc45faa567222e7e884230442de7aa21","origin":"The Stacks Project","memory_eligible":false,"source_rank":7995,"rank":7995,"depth":2,"x":1226.315,"y":1119.516,"cluster":"groupoids-quotients"},{"id":"stacks:03BK","tag":"03BK","title":"Finite flat groupoids, affine case · Lemma 03BK","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(A) and R = Spec(B) are affine and s, t : R → U finite locally free. Let C ⊂ A be as in ([Tag 03BF]). Let C → C' be a ring map, and set U' = Spec(A ⊗_C C'), R' = Spec(B ⊗_C C'). Then • The maps s, t, c induce maps s', t', c' such that (U', R', s', t', c') is a groupoid scheme. Let C^1 ⊂ A' be the R'-invariant functions on U'. • The canonical map φ : C' → C^1 satisfies • for every f ∈ C^1…","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and\n$s, t : R \\to U$ finite locally free. Let $C \\subset A$ be as in\n(\\ref{equation-invariants}). Let $C \\to C'$ be a ring map, and set\n$U' = \\Spec(A \\otimes_C C')$,\n$R' = \\Spec(B \\otimes_C C')$.\nThen\n\\begin{enumerate}\n\\item The maps $s, t, c$ induce maps $s', t', c'$ such that\n$(U', R', s', t', c')$ is a groupoid scheme. Let $C^1 \\subset A'$\nbe the $R'$-invariant functions on $U'$.\n\\item The canonical map $\\varphi : C' \\to C^1$ satisfies\n\\begin{enumerate}\n\\item for every $f \\in C^1$ there exists an $n > 0$ and a\npolynomial $P \\in C'[x]$ whose image in $C^1[x]$ is\n$(x - f)^n$, and\n\\item for every $f \\in \\Ker(\\varphi)$ there exists\nan $n > 0$ such that $f^n = 0$.\n\\end{enumerate}\n\\item If $C \\to C'$ is flat then $\\varphi$ is an isomorphism.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Finite flat groupoids, affine case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BK","source_file":"groupoids.tex","source_line":4367,"source_end_line":4390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4367-L4390","statement_sha256":"c09abfb8c50419094f439cd940e947a8c2ecf597077c8e35a5013469b108daeb","origin":"The Stacks Project","memory_eligible":false,"source_rank":7996,"rank":7996,"depth":6,"x":1075.187,"y":1210.094,"cluster":"groupoids-quotients"},{"id":"stacks:03BL","tag":"03BL","title":"Finite flat groupoids, affine case · Lemma 03BL","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(A) and R = Spec(B) are affine and s, t : R → U finite locally free. Let C ⊂ A be as in ([Tag 03BF]). Then U → M = Spec(C) has the following properties: • the map on points |U| → |M| is surjective and u_0, u_1 ∈ |U| map to the same point if and only if there exists a r ∈ |R| with t(r) = u_0 and s(r) = u_1, in a formula |M| = |U|/|R| • for any algebraically closed field k we have M(k) =…","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume $U = \\Spec(A)$ and $R = \\Spec(B)$ are affine and\n$s, t : R \\to U$ finite locally free. Let $C \\subset A$ be as in\n(\\ref{equation-invariants}). Then $U \\to M = \\Spec(C)$ has\nthe following properties:\n\\begin{enumerate}\n\\item the map on points $|U| \\to |M|$ is surjective and\n$u_0, u_1 \\in |U|$ map to the same point if and only if\nthere exists a $r \\in |R|$ with $t(r) = u_0$ and $s(r) = u_1$, in\na formula\n$$\n|M| = |U|/|R|\n$$\n\\item for any algebraically closed field $k$ we have\n$$\nM(k) = U(k)/R(k)\n$$\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Finite flat groupoids, affine case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BL","source_file":"groupoids.tex","source_line":4452,"source_end_line":4472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4452-L4472","statement_sha256":"2c2b5483f900edcabd8e753a9b276fcd6433dfca482a0bf9f54cb3f2f16dcb83","origin":"The Stacks Project","memory_eligible":false,"source_rank":7997,"rank":7997,"depth":7,"x":1113.974,"y":1056.814,"cluster":"groupoids-quotients"},{"id":"stacks:0DT9","tag":"0DT9","title":"Finite flat groupoids, affine case · Lemma 0DT9","summary":"Let S be a scheme. Let f : (U', R', s', t') → (U, R, s, t, c) be a morphism of groupoid schemes over S. • U, R, U', R' are affine, • s, t, s', t' are finite locally free, • the diagrams xymatrix R' ar[d]_s' ar[r]_f & R ar[d]^s U' ar[r]^f & U xymatrix R' ar[d]_t' ar[r]_f & R ar[d]^t U' ar[r]^f & U xymatrix G' ar[d] ar[r]_f & G ar[d] U' ar[r]^f & U are cartesian where G and G' are the stabilizer group schemes, and • f : U' → U is étale. Then the map C → C' from the…","statement_latex":"Let $S$ be a scheme. Let $f : (U', R', s', t') \\to (U, R, s, t, c)$ be a\nmorphism of groupoid schemes over $S$.\n\\begin{enumerate}\n\\item $U$, $R$, $U'$, $R'$ are affine,\n\\item $s, t, s', t'$ are finite locally free,\n\\item the diagrams\n$$\n\\xymatrix{\nR' \\ar[d]_{s'} \\ar[r]_f & R \\ar[d]^s \\\\\nU' \\ar[r]^f & U\n}\n\\quad\n\\quad\n\\xymatrix{\nR' \\ar[d]_{t'} \\ar[r]_f & R \\ar[d]^t \\\\\nU' \\ar[r]^f & U\n}\n\\quad\n\\quad\n\\xymatrix{\nG' \\ar[d] \\ar[r]_f & G \\ar[d] \\\\\nU' \\ar[r]^f & U\n}\n$$\nare cartesian where $G$ and $G'$ are the stabilizer group schemes, and\n\\item $f : U' \\to U$ is \\'etale.\n\\end{enumerate}\nThen the map $C \\to C'$ from the $R$-invariant functions on $U$\nto the $R'$-invariant functions on $U'$ is \\'etale and\n$U' = \\Spec(C') \\times_{\\Spec(C)} U$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Finite flat groupoids, affine case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DT9","source_file":"groupoids.tex","source_line":4568,"source_end_line":4600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4568-L4600","statement_sha256":"fd87b37bda850535fcadeb8df59fbe8b7eb3e44a391d61df4b279496ab7277a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":7998,"rank":7998,"depth":48,"x":1209.087,"y":1192.494,"cluster":"groupoids-quotients"},{"id":"stacks:03C8","tag":"03C8","title":"Finite flat groupoids, affine case · Lemma 03C8","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume • U = Spec(A), and R = Spec(B) are affine, and • there exist elements x_i ∈ A, i ∈ I such that B = bigoplus_i ∈ I s^sharp(A)t^sharp(x_i). Then A = bigoplus_i∈ I Cx_i, and B ≅ A ⊗_C A where C ⊂ A is the R-invariant functions on U as in ([Tag 03BF]).","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume\n\\begin{enumerate}\n\\item $U = \\Spec(A)$, and $R = \\Spec(B)$ are affine, and\n\\item there exist elements $x_i \\in A$, $i \\in I$ such that\n$B = \\bigoplus_{i \\in I} s^\\sharp(A)t^\\sharp(x_i)$.\n\\end{enumerate}\nThen $A = \\bigoplus_{i\\in I} Cx_i$, and $B \\cong A \\otimes_C A$\nwhere $C \\subset A$ is the $R$-invariant\nfunctions on $U$ as in (\\ref{equation-invariants}).","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Finite flat groupoids, affine case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03C8","source_file":"groupoids.tex","source_line":4744,"source_end_line":4757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4744-L4757","statement_sha256":"fc8333072ae2a2b7f926ef906a4d7a0f7ea4f8addf27efe5215db6c61ac838dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":7999,"rank":7999,"depth":8,"x":1028.995,"y":1146.196,"cluster":"groupoids-quotients"},{"id":"stacks:03BM","tag":"03BM","title":"Finite flat groupoids, affine case · Proposition 03BM","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume • U = Spec(A), and R = Spec(B) are affine, • s, t : R → U finite locally free, and • j = (t, s) is an equivalence relation. In this case, let C ⊂ A be as in ([Tag 03BF]). Then U → M = Spec(C) is finite locally free and R = U ×_M U. Moreover, M represents the quotient sheaf U/R in the fppf topology (see Definition [Tag 02VG]).","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume\n\\begin{enumerate}\n\\item $U = \\Spec(A)$, and $R = \\Spec(B)$ are affine,\n\\item $s, t : R \\to U$ finite locally free, and\n\\item $j = (t, s)$ is an equivalence relation.\n\\end{enumerate}\nIn this case, let $C \\subset A$ be as in\n(\\ref{equation-invariants}). Then $U \\to M = \\Spec(C)$\nis finite locally free and $R = U \\times_M U$.\nMoreover, $M$ represents the quotient sheaf $U/R$\nin the fppf topology (see Definition \\ref{definition-quotient-sheaf}).","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Finite flat groupoids, affine case","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BM","source_file":"groupoids.tex","source_line":4815,"source_end_line":4830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4815-L4830","statement_sha256":"57860b4d12971b6b61c48e6297ac672f8c725b8edf261188dcdae62362dd7cf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8000,"rank":8000,"depth":9,"x":1199.821,"y":1077.83,"cluster":"groupoids-quotients"},{"id":"stacks:03JE","tag":"03JE","title":"Finite flat groupoids · Lemma 03JE","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume s, t are finite locally free. Let u ∈ U be a point such that t(s^-1((u))) is contained in an affine open of U. Then there exists an R-invariant affine open neighbourhood of u in U.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume $s$, $t$ are finite locally free.\nLet $u \\in U$ be a point such that $t(s^{-1}(\\{u\\}))$\nis contained in an affine open of $U$.\nThen there exists an $R$-invariant affine open neighbourhood\nof $u$ in $U$.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Finite flat groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JE","source_file":"groupoids.tex","source_line":4924,"source_end_line":4933,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4924-L4933","statement_sha256":"f94c18576ea3e45c0f404478833ceb3dbdc5a7174b49497ec1671fa1ea5b2e1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8001,"rank":8001,"depth":2,"x":1128.504,"y":1225.856,"cluster":"groupoids-quotients"},{"id":"stacks:0H8K","tag":"0H8K","title":"Descending quasi-projective schemes · Lemma 0H8K","summary":"Let X → Y be a surjective finite locally free morphism. Let d be a positive integer. Assume that for every geometric point bary : Spec(k) → Y the fiber X_bary has at most d points. Let V be a scheme over X such that for all (y, v_1, …, v_d) where y ∈ Y and v_1, …, v_d ∈ V_y there exists an affine open U ⊂ V with v_1, …, v_d ∈ U. Then any descent datum on V/X/Y is effective.","statement_latex":"Let $X \\to Y$ be a surjective finite locally free morphism.\nLet $d$ be a positive integer. Assume that for every geometric\npoint $\\bar{y} : \\mathrm{Spec}(k) \\to Y$ the fiber $X_{\\bar{y}}$\nhas at most $d$ points.\nLet $V$ be a scheme over $X$ such that for all\n$(y, v_1, \\ldots, v_d)$ where $y \\in Y$ and\n$v_1, \\ldots, v_d \\in V_y$ there exists an affine open\n$U \\subset V$ with $v_1, \\ldots, v_d \\in U$.\nThen any descent datum on $V/X/Y$ is effective.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Descending quasi-projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8K","source_file":"groupoids.tex","source_line":4990,"source_end_line":5001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L4990-L5001","statement_sha256":"f1d5ab1bb3d55fa67fb1dd1bd01ed8844d646075028d2891a6d93f3b80b16023","origin":"The Stacks Project","memory_eligible":false,"source_rank":8002,"rank":8002,"depth":46,"x":1061.749,"y":1075.548,"cluster":"groupoids-quotients"},{"id":"stacks:0CCI","tag":"0CCI","title":"Descending quasi-projective schemes · Lemma 0CCI","summary":"Let X → Y be a surjective finite locally free morphism. Let V be a scheme over X such that for all (y, v_1, …, v_d) where y ∈ Y and v_1, …, v_d ∈ V_y there exists an affine open U ⊂ V with v_1, …, v_d ∈ U. Then any descent datum on V/X/Y is effective.","statement_latex":"Let $X \\to Y$ be a surjective finite locally free morphism. \nLet $V$ be a scheme over $X$ such that for all \n$(y, v_1, \\ldots, v_d)$ where $y \\in Y$ and\n$v_1, \\ldots, v_d \\in V_y$ there exists an affine open\n$U \\subset V$ with $v_1, \\ldots, v_d \\in U$.\nThen any descent datum on $V/X/Y$ is effective.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Descending quasi-projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCI","source_file":"groupoids.tex","source_line":5046,"source_end_line":5054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L5046-L5054","statement_sha256":"6fed255e0569b8ef19f9dfc1f7fb9e785f525a8d7c6101006d4a07c99927011d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8003,"rank":8003,"depth":47,"x":1232.621,"y":1148.838,"cluster":"groupoids-quotients"},{"id":"stacks:0CCJ","tag":"0CCJ","title":"Descending quasi-projective schemes · Lemma 0CCJ","summary":"Let X → Y be a surjective finite locally free morphism. Let V be a scheme over X such that one of the following holds • V → X is projective, • V → X is quasi-projective, • there exists an ample invertible sheaf on V, • there exists an X-ample invertible sheaf on V, • there exists an X-very ample invertible sheaf on V. Then any descent datum on V/X/Y is effective.","statement_latex":"Let $X \\to Y$ be a surjective finite locally free morphism.\nLet $V$ be a scheme over $X$ such that one of the following holds\n\\begin{enumerate}\n\\item $V \\to X$ is projective,\n\\item $V \\to X$ is quasi-projective,\n\\item there exists an ample invertible sheaf on $V$,\n\\item there exists an $X$-ample invertible sheaf on $V$,\n\\item there exists an $X$-very ample invertible sheaf on $V$.\n\\end{enumerate}\nThen any descent datum on $V/X/Y$ is effective.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Descending quasi-projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCJ","source_file":"groupoids.tex","source_line":5066,"source_end_line":5078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L5066-L5078","statement_sha256":"5bd34e62ca3135bdaa791b8a8e9bf2ec2e0aea4ceadcd28e3f64672f897b73ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":8004,"rank":8004,"depth":48,"x":1046.85,"y":1191.945,"cluster":"groupoids-quotients"},{"id":"stacks:0H8L","tag":"0H8L","title":"Descending quasi-projective schemes · Lemma 0H8L","summary":"Let X → Y be a surjective finite locally free morphism which is radicial. Let V be a scheme over X. Then any descent datum on V/X/Y is effective.","statement_latex":"Let $X \\to Y$ be a surjective finite locally free morphism which is radicial. \nLet $V$ be a scheme over $X$. Then any descent datum on $V/X/Y$ is effective.","area":"Groupoids & Quotients","chapter":"Groupoid Schemes","chapter_id":"groupoids","section":"Descending quasi-projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8L","source_file":"groupoids.tex","source_line":5102,"source_end_line":5106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids.tex#L5102-L5106","statement_sha256":"b79e8b7e2c914e7e5cb79315a977e4cabf34eee414ad2dc490c18ca2af59600b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8005,"rank":8005,"depth":47,"x":1149.628,"y":1054.135,"cluster":"groupoids-quotients"},{"id":"stacks:04R9","tag":"04R9","title":"Sheaf of differentials · Lemma 04R9","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. The sheaf of differentials of R seen as a scheme over U via t is a quotient of the pullback via t of the conormal sheaf of the immersion e : U → R. In a formula: there is a canonical surjection t^*C_U/R → Ω_R/U. If s is flat, then this map is an isomorphism.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nThe sheaf of differentials of $R$ seen as a scheme over\n$U$ via $t$ is a quotient of the pullback via $t$ of the conormal sheaf of\nthe immersion $e : U \\to R$. In a formula: there is a canonical surjection\n$t^*\\mathcal{C}_{U/R} \\to \\Omega_{R/U}$. If $s$ is flat, then\nthis map is an isomorphism.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Sheaf of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04R9","source_file":"more-groupoids.tex","source_line":116,"source_end_line":125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L116-L125","statement_sha256":"0f4479d8fe77bfca36b62c1b3dd11c348d34a959e73be5ccab2d69c7fc7327c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8006,"rank":8006,"depth":17,"x":1184.818,"y":1214.782,"cluster":"groupoids-quotients"},{"id":"stacks:0CK4","tag":"0CK4","title":"Local structure · Lemma 0CK4","summary":"The map I/I^2 → J/J^2 induced by c is the composition I/I^2 xrightarrow(1, 1) I/I^2 ⊕ I/I^2 → J/J^2 where the second arrow comes from the equality J = (I ⊗ B + B ⊗ I)C. The map i : B → B induces the map -1 : I/I^2 → I/I^2.","statement_latex":"The map $I/I^2 \\to J/J^2$ induced by $c$ is the composition\n$$\nI/I^2 \\xrightarrow{(1, 1)} I/I^2 \\oplus I/I^2 \\to J/J^2\n$$\nwhere the second arrow comes from the equality\n$J = (I \\otimes B + B \\otimes I)C$.\nThe map $i : B \\to B$ induces the map $-1 : I/I^2 \\to I/I^2$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Local structure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CK4","source_file":"more-groupoids.tex","source_line":204,"source_end_line":213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L204-L213","statement_sha256":"9247651f3f3a0e217bbbfe7d0ea7984036451d1130b94ca6546790f507dc179a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8007,"rank":8007,"depth":0,"x":1029.001,"y":1115.857,"cluster":"groupoids-quotients"},{"id":"stacks:04LH","tag":"04LH","title":"Properties of groupoids · Lemma 04LH","summary":"Let S be a scheme. Let (U, R, s, t, c, e, i) be a groupoid over S. Let g : U' → U be a morphism of schemes. Denote h the composition xymatrix h : U' ×_g, U, t R ar[r]_-pr_1 & R ar[r]_s & U. Let P, Q, R be properties of morphisms of schemes. Assume • R ⇒ Q, • Q is preserved under base change and composition, • for any morphism f : X → Y which has Q there exists a largest open W(P, f) ⊂ X such that f|_W(P, f) has P, and • for any morphism f : X → Y which has Q, and any…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c, e, i)$ be a groupoid over $S$.\nLet $g : U' \\to U$ be a morphism of schemes.\nDenote $h$ the composition\n$$\n\\xymatrix{\nh : U' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} & R \\ar[r]_s & U.\n}\n$$\nLet $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ be properties of morphisms\nof schemes. Assume\n\\begin{enumerate}\n\\item $\\mathcal{R} \\Rightarrow \\mathcal{Q}$,\n\\item $\\mathcal{Q}$ is preserved under base change and composition,\n\\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$ there exists a\nlargest open $W(\\mathcal{P}, f) \\subset X$ such that $f|_{W(\\mathcal{P}, f)}$\nhas $\\mathcal{P}$, and\n\\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$,\nand any morphism $Y' \\to Y$ which has $\\mathcal{R}$ we have\n$Y' \\times_Y W(\\mathcal{P}, f) = W(\\mathcal{P}, f')$, where\n$f' : X_{Y'} \\to Y'$ is the base change of $f$.\n\\end{enumerate}\nIf $s, t$ have $\\mathcal{R}$ and $g$ has $\\mathcal{Q}$, then\nthere exists an open subscheme $W \\subset U'$ such that\n$W \\times_{g, U, t} R = W(\\mathcal{P}, h)$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LH","source_file":"more-groupoids.tex","source_line":264,"source_end_line":291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L264-L291","statement_sha256":"b92505eda211137349b7adf88f608c4be37f0ef03a7672a4736a5f833f0938ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":8008,"rank":8008,"depth":0,"x":1224.298,"y":1100.322,"cluster":"groupoids-quotients"},{"id":"stacks:03JC","tag":"03JC","title":"Properties of groupoids · Lemma 03JC","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let τ ∈ (Zariski, linebreak[0] fppf, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic). Let P be a property of morphisms of schemes which is τ-local on the target (Descent, Definition [Tag 02KO]). Assume (s : R → U) and (t : R → U) are coverings for the τ-topology. Let W ⊂ U be the maximal open subscheme such that s|_s^-1(W) : s^-1(W) → W has property P. Then W is R-invariant, see Groupoids,…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] fppf,\n\\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic\\}$\\footnote{The fact that $fpqc$ is missing\nis not a typo.}. Let $\\mathcal{P}$ be a property of morphisms of schemes\nwhich is $\\tau$-local on the target\n(Descent, Definition \\ref{descent-definition-property-morphisms-local}).\nAssume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings for the\n$\\tau$-topology. Let $W \\subset U$ be the maximal open subscheme such that\n$s|_{s^{-1}(W)} : s^{-1}(W) \\to W$ has property $\\mathcal{P}$.\nThen $W$ is $R$-invariant, see\nGroupoids, Definition \\ref{groupoids-definition-invariant-open}.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JC","source_file":"more-groupoids.tex","source_line":430,"source_end_line":445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L430-L445","statement_sha256":"14d8a8acee557bc519b20f1ef5a693e896ab2b355e9992f621cb06d26dabb8ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":8009,"rank":8009,"depth":19,"x":1092.212,"y":1223.12,"cluster":"groupoids-quotients"},{"id":"stacks:06QQ","tag":"06QQ","title":"Properties of groupoids · Lemma 06QQ","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let G → U be its stabilizer group scheme. Let τ ∈ (fppf, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic). Let P be a property of morphisms which is τ-local on the target. Assume (s : R → U) and (t : R → U) are coverings for the τ-topology. Let W ⊂ U be the maximal open subscheme such that G_W → W has property P. Then W is R-invariant (see Groupoids, Definition [Tag 03BC]).","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$.\nLet $G \\to U$ be its stabilizer group scheme.\nLet $\\tau \\in \\{fppf, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic\\}$.\nLet $\\mathcal{P}$ be a property of morphisms which is $\\tau$-local\non the target. Assume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings\nfor the $\\tau$-topology. Let $W \\subset U$ be the maximal open subscheme\nsuch that $G_W \\to W$ has property $\\mathcal{P}$. Then $W$ is $R$-invariant\n(see\nGroupoids, Definition\n\\ref{groupoids-definition-invariant-open}).","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QQ","source_file":"more-groupoids.tex","source_line":460,"source_end_line":474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L460-L474","statement_sha256":"cc5bcd1663a9256543234038dcb1aeae304f72ea33fdcd6fbbc0c02b75dffbb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8010,"rank":8010,"depth":19,"x":1090.855,"y":1056.914,"cluster":"groupoids-quotients"},{"id":"stacks:02YF","tag":"02YF","title":"Comparing fibres · Lemma 02YF","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let r, r' ∈ R with t(r) = t(r') in U. Set u = s(r), u' = s(r'). Denote F_u = s^-1(u) and F_u' = s^-1(u') the scheme theoretic fibres. • There exists a common field extension kappa(u) ⊂ k, kappa(u') ⊂ k and an isomorphism (F_u)_k ≅ (F_u')_k. • We may choose the isomorphism of (1) such that a point lying over r maps to a point lying over r'. • If the morphisms s, t are flat then the morphisms of germs s : (R, r) →…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$.\nLet $r, r' \\in R$ with $t(r) = t(r')$ in $U$.\nSet $u = s(r)$, $u' = s(r')$.\nDenote $F_u = s^{-1}(u)$ and $F_{u'} = s^{-1}(u')$ the scheme\ntheoretic fibres.\n\\begin{enumerate}\n\\item There exists a common field extension\n$\\kappa(u) \\subset k$, $\\kappa(u') \\subset k$ and\nan isomorphism $(F_u)_k \\cong (F_{u'})_k$.\n\\item We may choose the isomorphism of (1) such that a point\nlying over $r$ maps to a point lying over $r'$.\n\\item If the morphisms $s$, $t$ are flat then the morphisms of germs\n$s : (R, r) \\to (U, u)$ and $s : (R, r') \\to (U, u')$ are flat\nlocally on the base isomorphic.\n\\item If the morphisms $s$, $t$ are \\'etale\n(resp.\\ smooth, syntomic, or flat and locally of finite presentation)\nthen the morphisms of germs $s : (R, r) \\to (U, u)$ and\n$s : (R, r') \\to (U, u')$ are locally on the base isomorphic\nin the \\'etale (resp.\\ smooth, syntomic, or fppf) topology.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Comparing fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YF","source_file":"more-groupoids.tex","source_line":543,"source_end_line":566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L543-L566","statement_sha256":"bb1252cbd5455e3ae4c1bc5be31b5d5e9caee909fe1f6e6d279e8677b3b317d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8011,"rank":8011,"depth":18,"x":1226.084,"y":1179.219,"cluster":"groupoids-quotients"},{"id":"stacks:0460","tag":"0460","title":"Cohen-Macaulay presentations · Lemma 0460","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Assume s and t are flat and locally of finite presentation. Then there exists an open U' ⊂ U such that • t^-1(U') ⊂ R is the largest open subscheme of R on which the morphism s is Cohen-Macaulay, • s^-1(U') ⊂ R is the largest open subscheme of R on which the morphism t is Cohen-Macaulay, • the morphism t|_s^-1(U') : s^-1(U') → U is surjective, • the morphism s|_t^-1(U') : t^-1(U') → U is surjective, and • the…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid over $S$.\nAssume $s$ and $t$ are flat and locally of finite presentation.\nThen there exists an open $U' \\subset U$ such that\n\\begin{enumerate}\n\\item $t^{-1}(U') \\subset R$ is the largest open subscheme of\n$R$ on which the morphism $s$ is Cohen-Macaulay,\n\\item $s^{-1}(U') \\subset R$ is the largest open subscheme of\n$R$ on which the morphism $t$ is Cohen-Macaulay,\n\\item the morphism $t|_{s^{-1}(U')} : s^{-1}(U') \\to U$ is\nsurjective,\n\\item the morphism $s|_{t^{-1}(U')} : t^{-1}(U') \\to U$ is\nsurjective, and\n\\item the restriction $R' = s^{-1}(U') \\cap t^{-1}(U')$\nof $R$ to $U'$ defines a groupoid $(U', R', s', t', c')$ which has the property\nthat the morphisms $s'$ and $t'$ are Cohen-Macaulay and locally of\nfinite presentation.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Cohen-Macaulay presentations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0460","source_file":"more-groupoids.tex","source_line":623,"source_end_line":643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L623-L643","statement_sha256":"901e224b867a0feaade6a86f6dbbc7b248010cf5720742883daf079b49bc0ff3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8012,"rank":8012,"depth":37,"x":1027.181,"y":1165.708,"cluster":"groupoids-quotients"},{"id":"stacks:04MP","tag":"04MP","title":"Restricting groupoids · Lemma 04MP","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' → U be a morphism of schemes. Let (U', R', s', t', c') be the restriction of (U, R, s, t, c) via g. • If s, t are locally of finite type and g is locally of finite type, then s', t' are locally of finite type. • If s, t are locally of finite presentation and g is locally of finite presentation, then s', t' are locally of finite presentation. • If s, t are flat and g is flat, then s', t' are…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $g : U' \\to U$ be a morphism of schemes.\nLet $(U', R', s', t', c')$ be the restriction of\n$(U, R, s, t, c)$ via $g$.\n\\begin{enumerate}\n\\item If $s, t$ are locally of finite type and $g$ is locally of finite\ntype, then $s', t'$ are locally of finite type.\n\\item If $s, t$ are locally of finite presentation and $g$ is locally of finite\npresentation, then $s', t'$ are locally of finite presentation.\n\\item If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat.\n\\item Add more here.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MP","source_file":"more-groupoids.tex","source_line":705,"source_end_line":720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L705-L720","statement_sha256":"63cc16b57fe241fb5154afa0d7a469b29c8d1fad1d067fdc158d0af600286bb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8013,"rank":8013,"depth":17,"x":1185.373,"y":1062.38,"cluster":"groupoids-quotients"},{"id":"stacks:04MV","tag":"04MV","title":"Restricting groupoids · Lemma 04MV","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' → U be a morphism of schemes. Let (U', R', s', t', c') be the restriction of (U, R, s, t, c) via g, and let h = s ∘ pr_1 : U' ×_g, U, t R → U. If P is a property of morphisms of schemes such that • h has property P, and • P is preserved under base change, then s', t' have property P.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $g : U' \\to U$ be a morphism of schemes.\nLet $(U', R', s', t', c')$ be the restriction of\n$(U, R, s, t, c)$ via $g$, and let\n$h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\to U$. If\n$\\mathcal{P}$ is a property of morphisms of schemes such that\n\\begin{enumerate}\n\\item $h$ has property $\\mathcal{P}$, and\n\\item $\\mathcal{P}$ is preserved under base change,\n\\end{enumerate}\nthen $s', t'$ have property $\\mathcal{P}$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MV","source_file":"more-groupoids.tex","source_line":739,"source_end_line":753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L739-L753","statement_sha256":"50df36ba3c218e6645fb96418e02e3b9b1976b75ab9f9a4aa9c0c01375b0519a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8014,"rank":8014,"depth":0,"x":1151.678,"y":1229.021,"cluster":"groupoids-quotients"},{"id":"stacks:04MW","tag":"04MW","title":"Restricting groupoids · Lemma 04MW","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' → U and g' : U\" → U' be morphisms of schemes. Set g\" = g ∘ g'. Let (U', R', s', t', c') be the restriction of R to U'. Let h = s ∘ pr_1 : U' ×_g, U, t R → U, let h' = s' ∘ pr_1 : U\" ×_g', U', t R → U', and let h\" = s ∘ pr_1 : U\" ×_g\", U, t R → U. The following diagram is commutative xymatrix U\" ×_g', U', t R' ar[d]^h' & (U' ×_g, U, t R) ×_U (U\" ×_g\", U, t R) ar[l] ar[r] ar[d] & U\" ×_g\", U, t R…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $g : U' \\to U$ and $g' : U'' \\to U'$ be morphisms of schemes.\nSet $g'' = g \\circ g'$.\nLet $(U', R', s', t', c')$ be the restriction of $R$ to $U'$.\nLet $h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\to U$,\nlet $h' = s' \\circ \\text{pr}_1 : U'' \\times_{g', U', t} R \\to U'$, and\nlet $h'' = s \\circ \\text{pr}_1 : U'' \\times_{g'', U, t} R \\to U$.\nThe following diagram is commutative\n$$\n\\xymatrix{\nU'' \\times_{g', U', t} R' \\ar[d]^{h'} &\n(U' \\times_{g, U, t} R) \\times_U (U'' \\times_{g'', U, t} R)\n\\ar[l] \\ar[r] \\ar[d] &\nU'' \\times_{g'', U, t} R \\ar[d]_{h''} \\\\\nU' &\nU' \\times_{g, U, t} R \\ar[l]_{\\text{pr}_0} \\ar[r]^h &\nU\n}\n$$\nwith both squares cartesian where the left upper horizontal arrow\nis given by the rule\n$$\n\\begin{matrix}\n(U' \\times_{g, U, t} R) \\times_U (U'' \\times_{g'', U, t} R) &\n\\longrightarrow &\nU'' \\times_{g', U', t} R' \\\\\n((u', r_0), (u'', r_1)) &\n\\longmapsto &\n(u'', (c(r_1, i(r_0)), (g'(u''), u')))\n\\end{matrix}\n$$\nwith notation as explained in the proof.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MW","source_file":"more-groupoids.tex","source_line":761,"source_end_line":796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L761-L796","statement_sha256":"53a23aa9e58e2b481b493bac7525a1a8ef9feb9f870b0834fbd2fe6a50c6bf42","origin":"The Stacks Project","memory_eligible":false,"source_rank":8015,"rank":8015,"depth":0,"x":1042.069,"y":1086.44,"cluster":"groupoids-quotients"},{"id":"stacks:04MX","tag":"04MX","title":"Restricting groupoids · Lemma 04MX","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' → U and g' : U\" → U' be morphisms of schemes. Set g\" = g ∘ g'. Let (U', R', s', t', c') be the restriction of R to U'. Let h = s ∘ pr_1 : U' ×_g, U, t R → U, let h' = s' ∘ pr_1 : U\" ×_g', U', t R → U', and let h\" = s ∘ pr_1 : U\" ×_g\", U, t R → U. Let τ ∈ (Zariski, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic, linebreak[0] fppf, linebreak[0] fpqc). Let P be a property of…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $g : U' \\to U$ and $g' : U'' \\to U'$ be morphisms of schemes.\nSet $g'' = g \\circ g'$.\nLet $(U', R', s', t', c')$ be the restriction of $R$ to $U'$.\nLet $h = s \\circ \\text{pr}_1 : U' \\times_{g, U, t} R \\to U$,\nlet $h' = s' \\circ \\text{pr}_1 : U'' \\times_{g', U', t} R \\to U'$, and\nlet $h'' = s \\circ \\text{pr}_1 : U'' \\times_{g'', U, t} R \\to U$.\nLet $\\tau \\in \\{Zariski, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic, \\linebreak[0] fppf, \\linebreak[0] fpqc\\}$. Let\n$\\mathcal{P}$ be a property of morphisms of schemes\nwhich is preserved under base change, and which\nis local on the target for the $\\tau$-topology. If\n\\begin{enumerate}\n\\item $h(U' \\times_U R)$ is open in $U$,\n\\item $\\{h : U' \\times_U R \\to h(U' \\times_U R)\\}$ is a $\\tau$-covering,\n\\item $h'$ has property $\\mathcal{P}$,\n\\end{enumerate}\nthen $h''$ has property $\\mathcal{P}$. Conversely, if\n\\begin{enumerate}\n\\item[(a)] $\\{t : R \\to U\\}$ is a $\\tau$-covering,\n\\item[(d)] $h''$ has property $\\mathcal{P}$,\n\\end{enumerate}\nthen $h'$ has property $\\mathcal{P}$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MX","source_file":"more-groupoids.tex","source_line":859,"source_end_line":885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L859-L885","statement_sha256":"e6d39e25753effaecf00562388b6a1e3b7330270d0e211291bee98f24925f2d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8016,"rank":8016,"depth":1,"x":1238.347,"y":1129.558,"cluster":"groupoids-quotients"},{"id":"stacks:04LM","tag":"04LM","title":"Properties of groupoids on fields · Lemma 04LM","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. If U is the spectrum of a field, then the composition morphism c : R ×_s, U, t R → R is open.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. If $U$ is the spectrum of a field, then the composition\nmorphism $c : R \\times_{s, U, t} R \\to R$ is open.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LM","source_file":"more-groupoids.tex","source_line":937,"source_end_line":942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L937-L942","statement_sha256":"cf19f55f92d2d7b9c9d477dd0fa09d41b22a8e768df9ad3a5996f78c262c1374","origin":"The Stacks Project","memory_eligible":false,"source_rank":8017,"rank":8017,"depth":10,"x":1058.216,"y":1209.457,"cluster":"groupoids-quotients"},{"id":"stacks:04LN","tag":"04LN","title":"Properties of groupoids on fields · Lemma 04LN","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. If U is the spectrum of a field, then R is a separated scheme.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. If $U$ is the spectrum of a field,\nthen $R$ is a separated scheme.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LN","source_file":"more-groupoids.tex","source_line":952,"source_end_line":957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L952-L957","statement_sha256":"1c02681cdd4f819f092287063bcda7f3673c8ee60798b8eff86f68720315929c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8018,"rank":8018,"depth":19,"x":1127.066,"y":1047.685,"cluster":"groupoids-quotients"},{"id":"stacks:04LP","tag":"04LP","title":"Properties of groupoids on fields · Lemma 04LP","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(k) with k a field. For any points r, r' ∈ R there exists a field extension k'/k and points r_1, r_2 ∈ R ×_s, Spec(k) Spec(k') and a diagram xymatrix R & R ×_s, Spec(k) Spec(k') ar[l]_-pr_0 ar[r]^φ & R ×_s, Spec(k) Spec(k') ar[r]^-pr_0 & R such that φ is an isomorphism of schemes over Spec(k'), we have φ(r_1) = r_2, pr_0(r_1) = r, and pr_0(r_2) = r'.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U = \\Spec(k)$ with $k$ a field.\nFor any points $r, r' \\in R$ there exists a field extension\n$k'/k$ and points\n$r_1, r_2 \\in R \\times_{s, \\Spec(k)} \\Spec(k')$\nand a diagram\n$$\n\\xymatrix{\nR &\nR \\times_{s, \\Spec(k)} \\Spec(k')\n\\ar[l]_-{\\text{pr}_0} \\ar[r]^\\varphi &\nR \\times_{s, \\Spec(k)} \\Spec(k')\n\\ar[r]^-{\\text{pr}_0} &\nR\n}\n$$\nsuch that $\\varphi$ is an isomorphism of schemes over $\\Spec(k')$,\nwe have $\\varphi(r_1) = r_2$, $\\text{pr}_0(r_1) = r$, and\n$\\text{pr}_0(r_2) = r'$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LP","source_file":"more-groupoids.tex","source_line":972,"source_end_line":993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L972-L993","statement_sha256":"68b2e7ecad06c192d8efd02f0364492fcd9c27759d69fff88f2426a372ee1dae","origin":"The Stacks Project","memory_eligible":false,"source_rank":8019,"rank":8019,"depth":19,"x":1206.703,"y":1206.669,"cluster":"groupoids-quotients"},{"id":"stacks:04LQ","tag":"04LQ","title":"Properties of groupoids on fields · Lemma 04LQ","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(k) with k a field. Let k'/k be a field extension, U' = Spec(k') and let (U', R', s', t', c') be the restriction of (U, R, s, t, c) via U' → U. In the defining diagram xymatrix R' ar[d] ar[r] ar@/_3pc/[dd]_t' ar@/^1pc/[rr]^s' ar@..>[rd] & R ×_s, U U' ar[r] ar[d] & U' ar[d] U' ×_U, t R ar[d] ar[r] & R ar[r]^s ar[d]_t & U U' ar[r] & U all the morphisms are surjective, flat, and universally…","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U = \\Spec(k)$ with $k$ a field.\nLet $k'/k$ be a field extension, $U' = \\Spec(k')$\nand let $(U', R', s', t', c')$ be the restriction of\n$(U, R, s, t, c)$ via $U' \\to U$. In the defining diagram\n$$\n\\xymatrix{\nR' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'} \\ar@{..>}[rd] &\nR \\times_{s, U} U' \\ar[r] \\ar[d] &\nU' \\ar[d] \\\\\nU' \\times_{U, t} R \\ar[d] \\ar[r] &\nR \\ar[r]^s \\ar[d]_t &\nU \\\\\nU' \\ar[r] &\nU\n}\n$$\nall the morphisms are surjective, flat, and universally open.\nThe dotted arrow $R' \\to R$ is in addition affine.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LQ","source_file":"more-groupoids.tex","source_line":1001,"source_end_line":1022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1001-L1022","statement_sha256":"1527f71bb5d1c0ce032642dd8d82094ceb91d7fa8fc5305cf6987b71190cd075","origin":"The Stacks Project","memory_eligible":false,"source_rank":8020,"rank":8020,"depth":19,"x":1019.395,"y":1134.34,"cluster":"groupoids-quotients"},{"id":"stacks:04LR","tag":"04LR","title":"Properties of groupoids on fields · Lemma 04LR","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(k) with k a field. For any point r ∈ R there exist • a field extension k'/k with k' algebraically closed, • a point r' ∈ R' where (U', R', s', t', c') is the restriction of (U, R, s, t, c) via Spec(k') → Spec(k) such that • the point r' maps to r under the morphism R' → R, and • the maps s', t' : R' → Spec(k') induce isomorphisms k' → kappa(r').","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U = \\Spec(k)$ with $k$ a field.\nFor any point $r \\in R$ there exist\n\\begin{enumerate}\n\\item a field extension $k'/k$ with $k'$ algebraically closed,\n\\item a point $r' \\in R'$ where $(U', R', s', t', c')$ is the\nrestriction of $(U, R, s, t, c)$ via $\\Spec(k') \\to \\Spec(k)$\n\\end{enumerate}\nsuch that\n\\begin{enumerate}\n\\item the point $r'$ maps to $r$ under the morphism $R' \\to R$, and\n\\item the maps $s', t' : R' \\to \\Spec(k')$ induce isomorphisms\n$k' \\to \\kappa(r')$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LR","source_file":"more-groupoids.tex","source_line":1040,"source_end_line":1056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1040-L1056","statement_sha256":"9bb8d0d84b79dc83921ede941ab544e3e4214d8dff011f33abe3243b0cd01540","origin":"The Stacks Project","memory_eligible":false,"source_rank":8021,"rank":8021,"depth":0,"x":1216.445,"y":1081.185,"cluster":"groupoids-quotients"},{"id":"stacks:04LS","tag":"04LS","title":"Properties of groupoids on fields · Lemma 04LS","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(k) with k a field. If r ∈ R is a point such that s, t induce isomorphisms k → kappa(r), then the map R → R, x ↦ c(r, x) (see proof for precise notation) is an automorphism R → R which maps e to r.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U = \\Spec(k)$ with $k$ a field.\nIf $r \\in R$ is a point such that $s, t$ induce\nisomorphisms $k \\to \\kappa(r)$, then the map\n$$\nR \\longrightarrow R, \\quad\nx \\longmapsto c(r, x)\n$$\n(see proof for precise notation) is an automorphism $R \\to R$\nwhich maps $e$ to $r$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LS","source_file":"more-groupoids.tex","source_line":1111,"source_end_line":1123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1111-L1123","statement_sha256":"4307b29a796ffe2b561192d6fdc97e1a4051425059e82a920371211f17603ff2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8022,"rank":8022,"depth":0,"x":1113.49,"y":1232.777,"cluster":"groupoids-quotients"},{"id":"stacks:0B7V","tag":"0B7V","title":"Properties of groupoids on fields · Lemma 0B7V","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. If U is the spectrum of a field, W ⊂ R is open, and Z → R is a morphism of schemes, then the image of the composition Z ×_s, U, t W → R ×_s, U, t R → R is open.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. If $U$ is the spectrum of a field, $W \\subset R$ is open,\nand $Z \\to R$ is a morphism of schemes, then the image of the\ncomposition $Z \\times_{s, U, t} W \\to R \\times_{s, U, t} R \\to R$ is open.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7V","source_file":"more-groupoids.tex","source_line":1191,"source_end_line":1197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1191-L1197","statement_sha256":"2ac33e96037d42591369689e48128082b38ee83634c31650308a248ef41b37f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8023,"rank":8023,"depth":20,"x":1067.326,"y":1061.922,"cluster":"groupoids-quotients"},{"id":"stacks:04LT","tag":"04LT","title":"Properties of groupoids on fields · Lemma 04LT","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(k) with k a field. By abuse of notation denote e ∈ R the image of the identity morphism e : U → R. Then • every local ring O_R, r of R has a unique minimal prime ideal, • there is exactly one irreducible component Z of R passing through e, and • Z is geometrically irreducible over k via either s or t.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U = \\Spec(k)$ with $k$ a field.\nBy abuse of notation denote $e \\in R$ the image of the identity\nmorphism $e : U \\to R$. Then\n\\begin{enumerate}\n\\item every local ring $\\mathcal{O}_{R, r}$ of $R$ has a unique\nminimal prime ideal,\n\\item there is exactly one irreducible component $Z$ of $R$\npassing through $e$, and\n\\item $Z$ is geometrically irreducible over $k$ via either\n$s$ or $t$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LT","source_file":"more-groupoids.tex","source_line":1221,"source_end_line":1235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1221-L1235","statement_sha256":"ba404475929e27da4183cfb1209116f9b40d76bbcd5f8a670a3e079eb2befcf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8024,"rank":8024,"depth":22,"x":1239.418,"y":1162.095,"cluster":"groupoids-quotients"},{"id":"stacks:04LU","tag":"04LU","title":"Properties of groupoids on fields · Lemma 04LU","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(k) with k a field. Assume s, t are locally of finite type. Then • R is equidimensional, • dim(R) = dim_r(R) for all r ∈ R, • for any r ∈ R we have trdeg_s(k)(kappa(r)) = trdeg_t(k)(kappa(r)), and • for any closed point r ∈ R we have dim(R) = dim(O_R, r).","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U = \\Spec(k)$ with $k$ a field.\nAssume $s, t$ are locally of finite type.\nThen\n\\begin{enumerate}\n\\item $R$ is equidimensional,\n\\item $\\dim(R) = \\dim_r(R)$ for all $r \\in R$,\n\\item for any $r \\in R$ we have\n$\\text{trdeg}_{s(k)}(\\kappa(r)) = \\text{trdeg}_{t(k)}(\\kappa(r))$, and\n\\item for any closed point $r \\in R$ we have\n$\\dim(R) = \\dim(\\mathcal{O}_{R, r})$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LU","source_file":"more-groupoids.tex","source_line":1297,"source_end_line":1311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1297-L1311","statement_sha256":"64d5b6c6cfc0d604b2ac013d0d47a4ad9bfc838417bcd12fa3fff125c4c9c15d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8025,"rank":8025,"depth":27,"x":1031.177,"y":1185.967,"cluster":"groupoids-quotients"},{"id":"stacks:04MQ","tag":"04MQ","title":"Properties of groupoids on fields · Lemma 04MQ","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(k) with k a field. Assume s, t are locally of finite type. Then dim(R) = dim(G) where G is the stabilizer group scheme of R.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U = \\Spec(k)$ with $k$ a field.\nAssume $s, t$ are locally of finite type.\nThen $\\dim(R) = \\dim(G)$ where $G$ is the stabilizer group scheme of $R$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MQ","source_file":"more-groupoids.tex","source_line":1336,"source_end_line":1342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1336-L1342","statement_sha256":"bbdbce1f3db7af5abab73367ddd1b6b176305b7a1bab13ae07323973a7327701","origin":"The Stacks Project","memory_eligible":false,"source_rank":8026,"rank":8026,"depth":41,"x":1166.04,"y":1049.694,"cluster":"groupoids-quotients"},{"id":"stacks:04RA","tag":"04RA","title":"Properties of groupoids on fields · Lemma 04RA","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume • U = Spec(k) with k a field, • s, t are locally of finite type, and • the characteristic of k is zero. Then s, t : R → U are smooth.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume\n\\begin{enumerate}\n\\item $U = \\Spec(k)$ with $k$ a field,\n\\item $s, t$ are locally of finite type, and\n\\item the characteristic of $k$ is zero.\n\\end{enumerate}\nThen $s, t : R \\to U$ are smooth.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RA","source_file":"more-groupoids.tex","source_line":1401,"source_end_line":1411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1401-L1411","statement_sha256":"d38d440da1b00cf7afd398b35d64858ad1dfbeeff79077e340e4e201a04321d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8027,"rank":8027,"depth":40,"x":1176.218,"y":1227.364,"cluster":"groupoids-quotients"},{"id":"stacks:04RB","tag":"04RB","title":"Properties of groupoids on fields · Lemma 04RB","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume • U = Spec(k) with k a field, • s, t are locally of finite type, • R is reduced, and • k is perfect. Then s, t : R → U are smooth.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume\n\\begin{enumerate}\n\\item $U = \\Spec(k)$ with $k$ a field,\n\\item $s, t$ are locally of finite type,\n\\item $R$ is reduced, and\n\\item $k$ is perfect.\n\\end{enumerate}\nThen $s, t : R \\to U$ are smooth.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Properties of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RB","source_file":"more-groupoids.tex","source_line":1421,"source_end_line":1432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1421-L1432","statement_sha256":"13be60c982989eff0d8bf9e1dc7ebb32d7da6f2c8a632c387ea2acd2bac4e39a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8028,"rank":8028,"depth":38,"x":1025.279,"y":1101.664,"cluster":"groupoids-quotients"},{"id":"stacks:04Q6","tag":"04Q6","title":"Morphisms of groupoids on fields · Lemma 04Q6","summary":"Notation and assumptions as in Situation [Tag 04Q5]. If a(R_1) is open in R_2, then a(R_1) is closed in R_2.","statement_latex":"Notation and assumptions as in\nSituation \\ref{situation-morphism-groupoids-on-field}.\nIf $a(R_1)$ is open in $R_2$, then $a(R_1)$ is closed in $R_2$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Morphisms of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Q6","source_file":"more-groupoids.tex","source_line":1492,"source_end_line":1497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1492-L1497","statement_sha256":"fc2a340fef33a9941f77871760ead1e4245a6a76d5238543fe7f57665fcd8f91","origin":"The Stacks Project","memory_eligible":false,"source_rank":8029,"rank":8029,"depth":20,"x":1238.444,"y":1108.733,"cluster":"groupoids-quotients"},{"id":"stacks:04Q7","tag":"04Q7","title":"Morphisms of groupoids on fields · Lemma 04Q7","summary":"Notation and assumptions as in Situation [Tag 04Q5]. Let Z ⊂ R_2 be the reduced closed subscheme (see Schemes, Definition [Tag 01J4]) whose underlying topological space is the closure of the image of a : R_1 → R_2. Then c_2(Z ×_s_2, U, t_2 Z) ⊂ Z set theoretically.","statement_latex":"Notation and assumptions as in\nSituation \\ref{situation-morphism-groupoids-on-field}.\nLet $Z \\subset R_2$ be the reduced closed subscheme (see\nSchemes, Definition \\ref{schemes-definition-reduced-induced-scheme})\nwhose underlying topological space is the closure of the image of\n$a : R_1 \\to R_2$. Then\n$c_2(Z \\times_{s_2, U, t_2} Z) \\subset Z$\nset theoretically.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Morphisms of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Q7","source_file":"more-groupoids.tex","source_line":1535,"source_end_line":1545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1535-L1545","statement_sha256":"c531c2444c5afe150f961adc2e2c7a9f2920adf4919280405e7bb42bb9d943ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":8030,"rank":8030,"depth":12,"x":1074.979,"y":1224.897,"cluster":"groupoids-quotients"},{"id":"stacks:04Q8","tag":"04Q8","title":"Morphisms of groupoids on fields · Lemma 04Q8","summary":"Notation and assumptions as in Situation [Tag 04Q5]. Assume that k is perfect. Let Z ⊂ R_2 be the reduced closed subscheme (see Schemes, Definition [Tag 01J4]) whose underlying topological space is the closure of the image of a : R_1 → R_2. Then (U, Z, s_2|_Z, t_2|_Z, c_2|_Z) is a groupoid scheme over S.","statement_latex":"Notation and assumptions as in\nSituation \\ref{situation-morphism-groupoids-on-field}.\nAssume that $k$ is perfect.\nLet $Z \\subset R_2$ be the reduced closed subscheme (see\nSchemes, Definition \\ref{schemes-definition-reduced-induced-scheme})\nwhose underlying topological space is the closure of the image of\n$a : R_1 \\to R_2$. Then\n$$\n(U, Z, s_2|_Z, t_2|_Z, c_2|_Z)\n$$\nis a groupoid scheme over $S$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Morphisms of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Q8","source_file":"more-groupoids.tex","source_line":1562,"source_end_line":1575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1562-L1575","statement_sha256":"cef206640ddace9f62d914c56ff05c601962d6aed63beee24d9b57f1fd94122d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8031,"rank":8031,"depth":13,"x":1102.2,"y":1045.84,"cluster":"groupoids-quotients"},{"id":"stacks:04Q9","tag":"04Q9","title":"Morphisms of groupoids on fields · Lemma 04Q9","summary":"Notation and assumptions as in Situation [Tag 04Q5]. If the image a(R_1) is a locally closed subset of R_2 then it is a closed subset.","statement_latex":"Notation and assumptions as in\nSituation \\ref{situation-morphism-groupoids-on-field}.\nIf the image $a(R_1)$ is a locally closed subset of $R_2$\nthen it is a closed subset.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Morphisms of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Q9","source_file":"more-groupoids.tex","source_line":1597,"source_end_line":1603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1597-L1603","statement_sha256":"e32e5c198db0390123c244a79b24477f9b91f574f538d1675196a4b228faeb87","origin":"The Stacks Project","memory_eligible":false,"source_rank":8032,"rank":8032,"depth":21,"x":1226.565,"y":1193.85,"cluster":"groupoids-quotients"},{"id":"stacks:04QA","tag":"04QA","title":"Morphisms of groupoids on fields · Lemma 04QA","summary":"Notation and assumptions as in Situation [Tag 04Q5]. Assume that a : R_1 → R_2 is a quasi-compact morphism. Let Z ⊂ R_2 be the scheme theoretic image (see Morphisms, Definition [Tag 01R7]) of a : R_1 → R_2. Then (U, Z, s_2|_Z, t_2|_Z, c_2|_Z) is a groupoid scheme over S.","statement_latex":"Notation and assumptions as in\nSituation \\ref{situation-morphism-groupoids-on-field}.\nAssume that $a : R_1 \\to R_2$ is a quasi-compact morphism.\nLet $Z \\subset R_2$ be the scheme theoretic image (see\nMorphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image})\nof $a : R_1 \\to R_2$. Then\n$$\n(U, Z, s_2|_Z, t_2|_Z, c_2|_Z)\n$$\nis a groupoid scheme over $S$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Morphisms of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QA","source_file":"more-groupoids.tex","source_line":1630,"source_end_line":1642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1630-L1642","statement_sha256":"d052dc00672cfae6089206b479184481171d50b77fbfe97de0913f9db6f33c88","origin":"The Stacks Project","memory_eligible":false,"source_rank":8033,"rank":8033,"depth":20,"x":1015.082,"y":1155.139,"cluster":"groupoids-quotients"},{"id":"stacks:04QB","tag":"04QB","title":"Morphisms of groupoids on fields · Lemma 04QB","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U is the spectrum of a field. Let Z ⊂ U ×_S U be the reduced closed subscheme (see Schemes, Definition [Tag 01J4]) whose underlying topological space is the closure of the image of j = (t, s) : R → U ×_S U. Then pr_02(Z ×_pr_1, U, pr_0 Z) ⊂ Z set theoretically.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U$ is the spectrum of a field.\nLet $Z \\subset U \\times_S U$ be the reduced closed subscheme (see\nSchemes, Definition \\ref{schemes-definition-reduced-induced-scheme})\nwhose underlying topological space is the closure of the image of\n$j = (t, s) : R \\to U \\times_S U$. Then\n$\\text{pr}_{02}(Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z) \\subset Z$\nset theoretically.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Morphisms of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QB","source_file":"more-groupoids.tex","source_line":1665,"source_end_line":1675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1665-L1675","statement_sha256":"33cd3061edd1be7efb600a771739173347ed923113ea4e22caefe5935da68f8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8034,"rank":8034,"depth":13,"x":1202.82,"y":1063.36,"cluster":"groupoids-quotients"},{"id":"stacks:04QC","tag":"04QC","title":"Morphisms of groupoids on fields · Lemma 04QC","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U is the spectrum of a perfect field. Let Z ⊂ U ×_S U be the reduced closed subscheme (see Schemes, Definition [Tag 01J4]) whose underlying topological space is the closure of the image of j = (t, s) : R → U ×_S U. Then (U, Z, pr_0|_Z, pr_1|_Z, pr_02|_Z ×_pr_1, U, pr_0 Z) is a groupoid scheme over S.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U$ is the spectrum of a perfect field.\nLet $Z \\subset U \\times_S U$ be the reduced closed subscheme (see\nSchemes, Definition \\ref{schemes-definition-reduced-induced-scheme})\nwhose underlying topological space is the closure of the image of\n$j = (t, s) : R \\to U \\times_S U$.\nThen\n$$\n(U, Z, \\text{pr}_0|_Z, \\text{pr}_1|_Z,\n\\text{pr}_{02}|_{Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z})\n$$\nis a groupoid scheme over $S$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Morphisms of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QC","source_file":"more-groupoids.tex","source_line":1709,"source_end_line":1723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1709-L1723","statement_sha256":"d864189f770dc2c563ff2ae64f5b24fa5b44770063ce6823f4de6dd2fce2387b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8035,"rank":8035,"depth":14,"x":1137.963,"y":1238.176,"cluster":"groupoids-quotients"},{"id":"stacks:04QD","tag":"04QD","title":"Morphisms of groupoids on fields · Lemma 04QD","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U is the spectrum of a field and assume R is quasi-compact (equivalently s, t are quasi-compact). Let Z ⊂ U ×_S U be the scheme theoretic image (see Morphisms, Definition [Tag 01R7]) of j = (t, s) : R → U ×_S U. Then (U, Z, pr_0|_Z, pr_1|_Z, pr_02|_Z ×_pr_1, U, pr_0 Z) is a groupoid scheme over S.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Assume $U$ is the spectrum of a field and\nassume $R$ is quasi-compact (equivalently $s, t$ are quasi-compact).\nLet $Z \\subset U \\times_S U$ be the scheme theoretic image (see\nMorphisms, Definition \\ref{morphisms-definition-scheme-theoretic-image})\nof $j = (t, s) : R \\to U \\times_S U$.\nThen\n$$\n(U, Z, \\text{pr}_0|_Z, \\text{pr}_1|_Z,\n\\text{pr}_{02}|_{Z \\times_{\\text{pr}_1, U, \\text{pr}_0} Z})\n$$\nis a groupoid scheme over $S$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Morphisms of groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QD","source_file":"more-groupoids.tex","source_line":1753,"source_end_line":1767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1753-L1767","statement_sha256":"25aa8da3b699f30f95a66d36e6d3e202555d5d54cfa0fd3ca57ef99cdd2a2113","origin":"The Stacks Project","memory_eligible":false,"source_rank":8036,"rank":8036,"depth":21,"x":1044.883,"y":1071.888,"cluster":"groupoids-quotients"},{"id":"stacks:0461","tag":"0461","title":"Slicing groupoids · Lemma 0461","summary":"Let S be a scheme. Let (U, R, s, t, c, e, i) be a groupoid scheme over S. Let G → U be the stabilizer group scheme. Assume s and t are Cohen-Macaulay and locally of finite presentation. Let u ∈ U be a finite type point of the scheme U, see Morphisms, Definition [Tag 02J1]. With notation as in Situation [Tag 04MY], set d_1 = dim(G_u), d_2 = dim_e(u)(F_u). If d_2 > d_1, then there exist an affine scheme U' and a morphism g : U' → U such that (with notation as in Situation…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$.\nLet $G \\to U$ be the stabilizer group scheme.\nAssume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation.\nLet $u \\in U$ be a finite type point of the scheme $U$, see\nMorphisms, Definition \\ref{morphisms-definition-finite-type-point}.\nWith notation as in\nSituation \\ref{situation-slice},\nset\n$$\nd_1 = \\dim(G_u), \\quad\nd_2 = \\dim_{e(u)}(F_u).\n$$\nIf $d_2 > d_1$, then there exist an affine scheme $U'$\nand a morphism $g : U' \\to U$ such that (with notation as in\nSituation \\ref{situation-slice})\n\\begin{enumerate}\n\\item $g$ is an immersion\n\\item $u \\in U'$,\n\\item $g$ is locally of finite presentation,\n\\item the morphism $h : U' \\times_{g, U, t} R \\longrightarrow U$\nis Cohen-Macaulay at $(u, e(u))$, and\n\\item we have $\\dim_{e'(u)}(F'_u) = d_2 - 1$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Slicing groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0461","source_file":"more-groupoids.tex","source_line":1849,"source_end_line":1875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L1849-L1875","statement_sha256":"32cc11c57dc02fbe86f5f963664443d78fcb417f988d340b082c85f56441a6db","origin":"The Stacks Project","memory_eligible":false,"source_rank":8037,"rank":8037,"depth":42,"x":1247.944,"y":1141.933,"cluster":"groupoids-quotients"},{"id":"stacks:04MZ","tag":"04MZ","title":"Slicing groupoids · Lemma 04MZ","summary":"Let S be a scheme. Let (U, R, s, t, c, e, i) be a groupoid scheme over S. Let G → U be the stabilizer group scheme. Assume s and t are Cohen-Macaulay and locally of finite presentation. Let u ∈ U be a finite type point of the scheme U, see Morphisms, Definition [Tag 02J1]. With notation as in Situation [Tag 04MY] there exist an affine scheme U' and a morphism g : U' → U such that • g is an immersion, • u ∈ U', • g is locally of finite presentation, • the morphism h : U'…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$.\nLet $G \\to U$ be the stabilizer group scheme.\nAssume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation.\nLet $u \\in U$ be a finite type point of the scheme $U$, see\nMorphisms, Definition \\ref{morphisms-definition-finite-type-point}.\nWith notation as in\nSituation \\ref{situation-slice}\nthere exist an affine scheme $U'$ and a morphism $g : U' \\to U$ such that\n\\begin{enumerate}\n\\item $g$ is an immersion,\n\\item $u \\in U'$,\n\\item $g$ is locally of finite presentation,\n\\item the morphism $h : U' \\times_{g, U, t} R \\longrightarrow U$\nis Cohen-Macaulay and locally of finite presentation,\n\\item the morphisms $s', t' : R' \\to U'$ are Cohen-Macaulay and\nlocally of finite presentation, and\n\\item $\\dim_{e(u)}(F'_u) = \\dim(G'_u)$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Slicing groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04MZ","source_file":"more-groupoids.tex","source_line":2028,"source_end_line":2049,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2028-L2049","statement_sha256":"6d09c9e4f52d55461f70285a0f83350acac9180b8f493497589398126ff95200","origin":"The Stacks Project","memory_eligible":false,"source_rank":8038,"rank":8038,"depth":43,"x":1041.171,"y":1205.723,"cluster":"groupoids-quotients"},{"id":"stacks:04N0","tag":"04N0","title":"Slicing groupoids · Lemma 04N0","summary":"Let S be a scheme. Let (U, R, s, t, c, e, i) be a groupoid scheme over S. Let G → U be the stabilizer group scheme. Assume s and t are Cohen-Macaulay and locally of finite presentation. Let u ∈ U be a finite type point of the scheme U, see Morphisms, Definition [Tag 02J1]. Assume that G → U is locally quasi-finite. With notation as in Situation [Tag 04MY] there exist an affine scheme U' and a morphism g : U' → U such that • g is an immersion, • u ∈ U', • g is locally of…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$.\nLet $G \\to U$ be the stabilizer group scheme.\nAssume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation.\nLet $u \\in U$ be a finite type point of the scheme $U$, see\nMorphisms, Definition \\ref{morphisms-definition-finite-type-point}.\nAssume that $G \\to U$ is locally quasi-finite.\nWith notation as in\nSituation \\ref{situation-slice}\nthere exist an affine scheme $U'$ and a morphism $g : U' \\to U$ such that\n\\begin{enumerate}\n\\item $g$ is an immersion,\n\\item $u \\in U'$,\n\\item $g$ is locally of finite presentation,\n\\item the morphism $h : U' \\times_{g, U, t} R \\longrightarrow U$\nis flat, locally of finite presentation, and locally quasi-finite, and\n\\item the morphisms $s', t' : R' \\to U'$ are flat,\nlocally of finite presentation, and locally quasi-finite.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Slicing groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04N0","source_file":"more-groupoids.tex","source_line":2112,"source_end_line":2133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2112-L2133","statement_sha256":"70e0124b053812f89cbe235a045a781a0a129f345d4de8b66ff12485ceb41f46","origin":"The Stacks Project","memory_eligible":false,"source_rank":8039,"rank":8039,"depth":44,"x":1142.691,"y":1040.796,"cluster":"groupoids-quotients"},{"id":"stacks:03FL","tag":"03FL","title":"Étale localization of groupoids · Lemma 03FL","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let p ∈ S be a point, and let u ∈ U be a point lying over p. Assume that • U → S is locally of finite type, • U → S is quasi-finite at u, • U → S is separated, • R → S is separated, • s, t are flat and locally of finite presentation, and • s^-1((u)) is finite. Then there exists an étale neighbourhood (S', p') → (S, p) with kappa(p) = kappa(p') and a base change diagram xymatrix R' amalg W' ar@=[r] & S'…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $p \\in S$ be a point, and let $u \\in U$ be a point lying over $p$.\nAssume that\n\\begin{enumerate}\n\\item $U \\to S$ is locally of finite type,\n\\item $U \\to S$ is quasi-finite at $u$,\n\\item $U \\to S$ is separated,\n\\item $R \\to S$ is separated,\n\\item $s$, $t$ are flat and locally of finite presentation, and\n\\item $s^{-1}(\\{u\\})$ is finite.\n\\end{enumerate}\nThen there exists an \\'etale neighbourhood $(S', p') \\to (S, p)$ with\n$\\kappa(p) = \\kappa(p')$ and a base change diagram\n$$\n\\xymatrix{\nR' \\amalg W'\n\\ar@{=}[r] &\nS' \\times_S R\n\\ar[r] \\ar@<2ex>[d]^{s'} \\ar@<-2ex>[d]_{t'} &\nR \\ar@<1ex>[d]^s \\ar@<-1ex>[d]_t \\\\\nU' \\amalg W\n\\ar@{=}[r] &\nS' \\times_S U\n\\ar[r] \\ar[d] &\nU \\ar[d] \\\\\n &\nS' \\ar[r] &\nS\n}\n$$\nwhere the equal signs are decompositions into open and closed\nsubschemes such that\n\\begin{enumerate}\n\\item[(a)] there exists a point $u'$ of $U'$ mapping to $u$ in $U$,\n\\item[(b)] the fibre $(U')_{p'}$ equals $t'\\big((s')^{-1}(\\{u'\\})\\big)$\nset theoretically,\n\\item[(c)] the fibre $(R')_{p'}$ equals $(s')^{-1}\\big((U')_{p'}\\big)$\nset theoretically,\n\\item[(d)] the schemes $U'$ and $R'$ are finite over $S'$,\n\\item[(e)] we have $s'(R') \\subset U'$ and $t'(R') \\subset U'$,\n\\item[(f)] we have\n$c'(R' \\times_{s', U', t'} R') \\subset R'$\nwhere $c'$ is the base change of $c$, and\n\\item[(g)] the morphisms $s', t', c'$ determine a groupoid structure\nby taking the system\n$(U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \\times_{s', U', t'} R'})$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Étale localization of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FL","source_file":"more-groupoids.tex","source_line":2178,"source_end_line":2228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2178-L2228","statement_sha256":"76220d04c63e4588f6928054ef524649862ce7e96f5d2de5e9e3b47a2aa384ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":8040,"rank":8040,"depth":32,"x":1200.657,"y":1220.625,"cluster":"groupoids-quotients"},{"id":"stacks:03X5","tag":"03X5","title":"Étale localization of groupoids · Lemma 03X5","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let p ∈ S be a point, and let u ∈ U be a point lying over p. Assume assumptions (1) -- (6) of Lemma [Tag 03FL] hold as well as • [(7)] j : R → U ×_S U is universally closed. Then we can choose (S', p') → (S, p) and decompositions S' ×_S U = U' amalg W and S' ×_S R = R' amalg W' and u' ∈ U' such that (a) -- (g) of Lemma [Tag 03FL] hold as well as • [(h)] R' is the restriction of S' ×_S R to U'.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $p \\in S$ be a point, and let $u \\in U$ be a point lying over $p$.\nAssume assumptions (1) -- (6) of\nLemma \\ref{lemma-quasi-finite-over-base}\nhold as well as\n\\begin{enumerate}\n\\item[(7)] $j : R \\to U \\times_S U$ is universally closed\\footnote{In view of\nthe other conditions this is equivalent to requiring $j$ to be proper.}.\n\\end{enumerate}\nThen we can choose $(S', p') \\to (S, p)$ and decompositions\n$S' \\times_S U = U' \\amalg W$ and $S' \\times_S R = R' \\amalg W'$\nand $u' \\in U'$ such that (a) -- (g) of\nLemma \\ref{lemma-quasi-finite-over-base}\nhold as well as\n\\begin{enumerate}\n\\item[(h)] $R'$ is the restriction of $S' \\times_S R$ to $U'$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Étale localization of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03X5","source_file":"more-groupoids.tex","source_line":2346,"source_end_line":2366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2346-L2366","statement_sha256":"c6224af83dd8dccf78254c5075e53e3601f93c1ea4072ddec52bba3a8b207c24","origin":"The Stacks Project","memory_eligible":false,"source_rank":8041,"rank":8041,"depth":33,"x":1012.669,"y":1120.576,"cluster":"groupoids-quotients"},{"id":"stacks:0AB9","tag":"0AB9","title":"Finite groupoids · Lemma 0AB9","summary":"Let (U, R, s, t, c) be a groupoid scheme over a scheme S. Assume s, t are finite. There exists a sequence of R-invariant closed subschemes U = Z_0 ⊃ Z_1 ⊃ Z_2 ⊃ … such that ⋂ Z_r = ∅ and such that s^-1(Z_r - 1) setminus s^-1(Z_r) → Z_r - 1 setminus Z_r is finite locally free of rank r.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$\nare finite. There exists a sequence of $R$-invariant closed subschemes\n$$\nU = Z_0 \\supset Z_1 \\supset Z_2 \\supset \\ldots\n$$\nsuch that $\\bigcap Z_r = \\emptyset$ and such that\n$s^{-1}(Z_{r - 1}) \\setminus s^{-1}(Z_r) \\to Z_{r - 1} \\setminus Z_r$\nis finite locally free of rank $r$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AB9","source_file":"more-groupoids.tex","source_line":2420,"source_end_line":2430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2420-L2430","statement_sha256":"d4a13a77f8ece2e186c85fb715fff1ebc436f6725bcf277fbb802578b482d565","origin":"The Stacks Project","memory_eligible":false,"source_rank":8042,"rank":8042,"depth":18,"x":1232.48,"y":1087.572,"cluster":"groupoids-quotients"},{"id":"stacks:0ABA","tag":"0ABA","title":"Finite groupoids · Lemma 0ABA","summary":"Let (U, R, s, t, c) be a groupoid scheme over a scheme S. Assume s, t are finite. There exists an open subscheme W ⊂ U and a closed subscheme W' ⊂ W such that • W and W' are R-invariant, • U = t(s^-1(overlineW)) set theoretically, • W is a thickening of W', and • the maps s', t' of the restriction (W', R', s', t', c') are finite locally free.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$\nare finite. There exists an open subscheme $W \\subset U$ and a closed\nsubscheme $W' \\subset W$ such that\n\\begin{enumerate}\n\\item $W$ and $W'$ are $R$-invariant,\n\\item $U = t(s^{-1}(\\overline{W}))$ set theoretically,\n\\item $W$ is a thickening of $W'$, and\n\\item the maps $s'$, $t'$ of the restriction $(W', R', s', t', c')$\nare finite locally free.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABA","source_file":"more-groupoids.tex","source_line":2454,"source_end_line":2466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2454-L2466","statement_sha256":"cf326855a0e0178656d860e0df8024173339943e8ec23ee073caa0d0ea75c505","origin":"The Stacks Project","memory_eligible":false,"source_rank":8043,"rank":8043,"depth":19,"x":1096.484,"y":1237.13,"cluster":"groupoids-quotients"},{"id":"stacks:0ABB","tag":"0ABB","title":"Finite groupoids · Lemma 0ABB","summary":"In Lemma [Tag 0ABA] assume in addition that s and t are of finite presentation. Then • the morphism W' → W is of finite presentation, and • if u ∈ U is a point whose R-orbit consists of generic points of irreducible components of U, then u ∈ W.","statement_latex":"In Lemma \\ref{lemma-finite-flat-over-almost-dense-subscheme}\nassume in addition that $s$ and $t$ are of finite presentation.\nThen\n\\begin{enumerate}\n\\item the morphism $W' \\to W$ is of finite presentation, and\n\\item if $u \\in U$ is a point whose $R$-orbit consists of\ngeneric points of irreducible components of $U$, then $u \\in W$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABB","source_file":"more-groupoids.tex","source_line":2534,"source_end_line":2544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2534-L2544","statement_sha256":"d0dcf09f53ee6e6c1c2ec57e45de0a6e472a20d7fb506ac817e18ff2b2a008e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8044,"rank":8044,"depth":20,"x":1076.429,"y":1049.061,"cluster":"groupoids-quotients"},{"id":"stacks:0ABC","tag":"0ABC","title":"Finite groupoids · Lemma 0ABC","summary":"Let (U, R, s, t, c) be a groupoid scheme over a scheme S. Assume s, t are finite and of finite presentation and U quasi-separated. Let u_1, …, u_m ∈ U be points whose orbits consist of generic points of irreducible components of U. Then there exist R-invariant subschemes V' ⊂ V ⊂ U such that • u_1, …, u_m ∈ V', • V is open in U, • V' and V are affine, • V' ⊂ V is a thickening of finite presentation, • the morphisms s', t' of the restriction (V', R', s', t', c') are finite…","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$\nare finite and of finite presentation and $U$ quasi-separated. Let\n$u_1, \\ldots, u_m \\in U$ be points whose orbits consist of generic points\nof irreducible components of $U$. Then there exist $R$-invariant subschemes\n$V' \\subset V \\subset U$ such that\n\\begin{enumerate}\n\\item $u_1, \\ldots, u_m \\in V'$,\n\\item $V$ is open in $U$,\n\\item $V'$ and $V$ are affine,\n\\item $V' \\subset V$ is a thickening of finite presentation,\n\\item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$\nare finite locally free.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABC","source_file":"more-groupoids.tex","source_line":2581,"source_end_line":2596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2581-L2596","statement_sha256":"04463668d7f431d0217d1abd8311baae8d7676760f55f2311a651cf4f06f5db0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8045,"rank":8045,"depth":32,"x":1243.002,"y":1176.78,"cluster":"groupoids-quotients"},{"id":"stacks:0ABD","tag":"0ABD","title":"Finite groupoids · Lemma 0ABD","summary":"Let (U, R, s, t, c) be a groupoid scheme over a scheme S. Assume s, t finite, U is locally Noetherian, and u_1, …, u_m ∈ U points whose orbits consist of generic points of irreducible components of U. Then there exist R-invariant subschemes V' ⊂ V ⊂ U such that • u_1, …, u_m ∈ V', • V is open in U, • V' and V are affine, • V' ⊂ V is a thickening, • the morphisms s', t' of the restriction (V', R', s', t', c') are finite locally free.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.\nAssume $s, t$ finite, $U$ is locally Noetherian, and $u_1, \\ldots, u_m \\in U$\npoints whose orbits consist of generic points of irreducible\ncomponents of $U$. Then there exist $R$-invariant subschemes\n$V' \\subset V \\subset U$ such that\n\\begin{enumerate}\n\\item $u_1, \\ldots, u_m \\in V'$,\n\\item $V$ is open in $U$,\n\\item $V'$ and $V$ are affine,\n\\item $V' \\subset V$ is a thickening,\n\\item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$\nare finite locally free.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABD","source_file":"more-groupoids.tex","source_line":2625,"source_end_line":2640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2625-L2640","statement_sha256":"9a0df0ac5ea1e2ba1b0327d28c15823dac336811a01e47061867e5b64da4002e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8046,"rank":8046,"depth":28,"x":1016.73,"y":1177.119,"cluster":"groupoids-quotients"},{"id":"stacks:0ABE","tag":"0ABE","title":"Finite groupoids · Lemma 0ABE","summary":"Let (U, R, s, t, c) be a groupoid scheme over a scheme S with s, t integral. Let g : U' → U be an integral morphism such that every R-orbit in U meets g(U'). Let (U', R', s', t', c') be the restriction of R to U'. If u' ∈ U' is contained in an R'-invariant affine open, then the image u ∈ U is contained in an R-invariant affine open of U.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$\nwith $s, t$ integral. Let $g : U' \\to U$ be an integral morphism\nsuch that every $R$-orbit in $U$ meets $g(U')$. Let $(U', R', s', t', c')$\nbe the restriction of $R$ to $U'$. If $u' \\in U'$ is contained in an\n$R'$-invariant affine open, then the image $u \\in U$ is contained\nin an $R$-invariant affine open of $U$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABE","source_file":"more-groupoids.tex","source_line":2669,"source_end_line":2677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2669-L2677","statement_sha256":"561cdb8c41a64d34fdf7fc62fa0b897449690d67009f9a76aee410288541072a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8047,"rank":8047,"depth":31,"x":1183.847,"y":1048.061,"cluster":"groupoids-quotients"},{"id":"stacks:0ABF","tag":"0ABF","title":"Finite groupoids · Lemma 0ABF","summary":"Let (U, R, s, t, c) be a groupoid scheme with s, t finite and of finite presentation. Let u_1, …, u_m ∈ U be points whose R-orbits consist of generic points of irreducible components of U. Let j : U → Spec(A) be an immersion. Let I ⊂ A be an ideal such that j(U) ∩ V(I) = ∅ and V(I) ∪ j(U) is closed in Spec(A). Then there exists an h ∈ I such that j^-1D(h) is an R-invariant affine open subscheme of U containing u_1, …, u_m.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme with $s, t$ finite and of\nfinite presentation. Let $u_1, \\ldots, u_m \\in U$ be points whose $R$-orbits\nconsist of generic points of irreducible components of $U$.\nLet $j : U \\to \\Spec(A)$ be an immersion.\nLet $I \\subset A$ be an ideal such that $j(U) \\cap V(I) = \\emptyset$\nand $V(I) \\cup j(U)$ is closed in $\\Spec(A)$.\nThen there exists an $h \\in I$ such that $j^{-1}D(h)$\nis an $R$-invariant affine open subscheme of $U$ containing\n$u_1, \\ldots, u_m$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABF","source_file":"more-groupoids.tex","source_line":2705,"source_end_line":2716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2705-L2716","statement_sha256":"6486e88033defc3537dddedeb98358ffaad2a75de058d4a38f41779145d0accf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8048,"rank":8048,"depth":33,"x":1164.339,"y":1238.664,"cluster":"groupoids-quotients"},{"id":"stacks:0ABG","tag":"0ABG","title":"Finite groupoids · Lemma 0ABG","summary":"Let (U, R, s, t, c) be a groupoid scheme. If s, t are finite, and u, u' ∈ R are distinct points in the same orbit, then u' is not a specialization of u.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme. If $s, t$ are finite,\nand $u, u' \\in R$ are distinct points in the same orbit,\nthen $u'$ is not a specialization of $u$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABG","source_file":"more-groupoids.tex","source_line":2747,"source_end_line":2752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2747-L2752","statement_sha256":"fa6e8b163c9ba3337dc7e7ecd6e703890cdc699f986f7b1bdef52998973a5c2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8049,"rank":8049,"depth":6,"x":1025.002,"y":1086.562,"cluster":"groupoids-quotients"},{"id":"stacks:0ABH","tag":"0ABH","title":"Finite groupoids · Lemma 0ABH","summary":"Let j : V → Spec(A) be a quasi-compact immersion of schemes. Let f ∈ A be such that j^-1D(f) is affine and j(V) ∩ V(f) is closed. Then V is affine.","statement_latex":"Let $j : V \\to \\Spec(A)$ be a quasi-compact immersion of schemes.\nLet $f \\in A$ be such that $j^{-1}D(f)$ is affine and $j(V) \\cap V(f)$\nis closed. Then $V$ is affine.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABH","source_file":"more-groupoids.tex","source_line":2769,"source_end_line":2774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2769-L2774","statement_sha256":"387c5a75215021c9fcb839a0a897189a20ba1a96fb463502047b121c864caa3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8050,"rank":8050,"depth":19,"x":1250.779,"y":1119.762,"cluster":"groupoids-quotients"},{"id":"stacks:0ABI","tag":"0ABI","title":"Finite groupoids · Lemma 0ABI","summary":"Let (U, R, s, t, c) be a groupoid scheme. Let u ∈ U. Assume • s, t are finite morphisms, • U is separated and locally Noetherian, • dim(O_U, u') ≤ 1 for every point u' in the orbit of u. Then u is contained in an R-invariant affine open of U.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme. Let $u \\in U$. Assume\n\\begin{enumerate}\n\\item $s, t$ are finite morphisms,\n\\item $U$ is separated and locally Noetherian,\n\\item $\\dim(\\mathcal{O}_{U, u'}) \\leq 1$ for every point $u'$\nin the orbit of $u$.\n\\end{enumerate}\nThen $u$ is contained in an $R$-invariant affine open of $U$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Finite groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABI","source_file":"more-groupoids.tex","source_line":2805,"source_end_line":2815,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2805-L2815","statement_sha256":"346ff5af9c8e609c72077b87455ed6d8a1a3ba8a551662bfed6800ae9d726c68","origin":"The Stacks Project","memory_eligible":false,"source_rank":8051,"rank":8051,"depth":34,"x":1056.985,"y":1223.719,"cluster":"groupoids-quotients"},{"id":"stacks:0API","tag":"0API","title":"Descending ind-quasi-affine morphisms · Lemma 0API","summary":"Let X be an ind-quasi-affine scheme. Let E ⊂ X be an intersection of a nonempty family of quasi-compact opens of X. Set A = Γ(E, O_X|_E) and Y = Spec(A). Then the canonical morphism j : (E, O_X|_E) → (Y, O_Y) of Schemes, Lemma [Tag 01I1] determines an isomorphism (E, O_X|_E) → (E', O_Y|_E') where E' ⊂ Y is an intersection of quasi-compact opens. If W ⊂ E is open in X, then j(W) is open in Y.","statement_latex":"Let $X$ be an ind-quasi-affine scheme. Let $E \\subset X$ be an\nintersection of a nonempty family of quasi-compact opens of $X$.\nSet $A = \\Gamma(E, \\mathcal{O}_X|_E)$ and $Y = \\Spec(A)$.\nThen the canonical morphism\n$$\nj : (E, \\mathcal{O}_X|_E) \\longrightarrow (Y, \\mathcal{O}_Y)\n$$\nof Schemes, Lemma \\ref{schemes-lemma-morphism-into-affine}\ndetermines an isomorphism\n$(E, \\mathcal{O}_X|_E) \\to (E', \\mathcal{O}_Y|_{E'})$\nwhere $E' \\subset Y$ is an intersection of quasi-compact opens.\nIf $W \\subset E$ is open in $X$, then $j(W)$ is open in $Y$.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Descending ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0API","source_file":"more-groupoids.tex","source_line":2993,"source_end_line":3007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L2993-L3007","statement_sha256":"c1b16e5d8bf39618d97c3390292b2970968d2863b9ae7e319bedcf6d46adcf87","origin":"The Stacks Project","memory_eligible":false,"source_rank":8052,"rank":8052,"depth":24,"x":1116.474,"y":1036.514,"cluster":"groupoids-quotients"},{"id":"stacks:0APJ","tag":"0APJ","title":"Descending ind-quasi-affine morphisms · Lemma 0APJ","summary":"Suppose given a cartesian diagram xymatrix X ar[d]_f ar[r] & Spec(B) ar[d] Y ar[r] & Spec(A) of schemes. Let E ⊂ Y be an intersection of a nonempty family of quasi-compact opens of Y. Then Γ(f^-1(E), O_X|_f^-1(E)) = Γ(E, O_Y|_E) ⊗_A B provided Y is quasi-separated and A → B is flat.","statement_latex":"Suppose given a cartesian diagram\n$$\n\\xymatrix{\nX \\ar[d]_f \\ar[r] & \\Spec(B) \\ar[d] \\\\\nY \\ar[r] & \\Spec(A)\n}\n$$\nof schemes. Let $E \\subset Y$ be an intersection of a nonempty family\nof quasi-compact opens of $Y$. Then\n$$\n\\Gamma(f^{-1}(E), \\mathcal{O}_X|_{f^{-1}(E)}) =\n\\Gamma(E, \\mathcal{O}_Y|_E) \\otimes_A B\n$$\nprovided $Y$ is quasi-separated and $A \\to B$ is flat.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Descending ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APJ","source_file":"more-groupoids.tex","source_line":3063,"source_end_line":3079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L3063-L3079","statement_sha256":"7ca5c5e37bc86126345536eb3de41152e06e9f383525371e0a546e417696912c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8053,"rank":8053,"depth":30,"x":1223.484,"y":1208.847,"cluster":"groupoids-quotients"},{"id":"stacks:0APK","tag":"0APK","title":"Gabber · Lemma 0APK","summary":"Let S be a scheme. Let (X_i → S)_i∈ I be an fpqc covering. Let (V_i/X_i, φ_ij) be a descent datum relative to (X_i → S), see Descent, Definition [Tag 023W]. If each morphism V_i → X_i is ind-quasi-affine, then the descent datum is effective.","statement_latex":"Let $S$ be a scheme. Let $\\{X_i \\to S\\}_{i\\in I}$ be an fpqc covering.\nLet $(V_i/X_i, \\varphi_{ij})$ be a descent datum relative to\n$\\{X_i \\to S\\}$, see Descent, Definition\n\\ref{descent-definition-descent-datum-for-family-of-morphisms}. \nIf each morphism $V_i \\to X_i$ is ind-quasi-affine, then the descent datum\nis effective.","area":"Groupoids & Quotients","chapter":"More on Groupoid Schemes","chapter_id":"more-groupoids","section":"Descending ind-quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APK","source_file":"more-groupoids.tex","source_line":3091,"source_end_line":3099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-groupoids.tex#L3091-L3099","statement_sha256":"8a1b0d7502ea2fbc7e35510fc578b7a6b9119ab76f7a33bbf95477da03d643ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":8054,"rank":8054,"depth":45,"x":1005.317,"y":1142.285,"cluster":"groupoids-quotients"},{"id":"stacks:024M","tag":"024M","title":"Unramified morphisms · Definition 024M","summary":"Let A, B be Noetherian local rings. A local homomorphism A → B is said to be unramified homomorphism of local rings if • m_AB = m_B, • kappa( m_B) is a finite separable extension of kappa( m_A), and • B is essentially of finite type over A (this means that B is the localization of a finite type A-algebra at a prime).","statement_latex":"Let $A$, $B$ be Noetherian local rings. A local homomorphism $A \\to B$\nis said to be {\\it unramified homomorphism of local rings} if\n\\begin{enumerate}\n\\item $\\mathfrak m_AB = \\mathfrak m_B$,\n\\item $\\kappa(\\mathfrak m_B)$ is a finite separable extension of\n$\\kappa(\\mathfrak m_A)$, and\n\\item $B$ is essentially of finite type over $A$ (this means\nthat $B$ is the localization of a finite type $A$-algebra at a prime).\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Unramified morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024M","source_file":"etale.tex","source_line":73,"source_end_line":84,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L73-L84","statement_sha256":"988aa296b26b47bb5e69cb607e8ea2340c339474534ce7f6f0d688af88f5aeb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8055,"rank":8055,"depth":0,"x":1584.093,"y":1140.0,"cluster":"tale-geometry"},{"id":"stacks:039G","tag":"039G","title":"Unramified morphisms · Lemma 039G","summary":"Unramifiedness is a stalk local condition. Let A → B be of finite type with A a Noetherian ring. Let q be a prime of B lying over p ⊂ A. Then A → B is unramified at q if and only if A_ p → B_ q is an unramified homomorphism of local rings.","statement_latex":"\\begin{slogan}\nUnramifiedness is a stalk local condition.\n\\end{slogan}\nLet $A \\to B$ be of finite type with $A$ a Noetherian ring.\nLet $\\mathfrak q$ be a prime of $B$ lying over $\\mathfrak p \\subset A$.\nThen $A \\to B$ is unramified at $\\mathfrak q$ if and only if\n$A_{\\mathfrak p} \\to B_{\\mathfrak q}$ is an unramified homomorphism\nof local rings.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039G","source_file":"etale.tex","source_line":109,"source_end_line":119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L109-L119","statement_sha256":"a1e196461daba2dc3539603e8e96d0a5aa58c9b970a16d9348604b428689c5d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8056,"rank":8056,"depth":0,"x":1574.772,"y":1144.023,"cluster":"tale-geometry"},{"id":"stacks:039H","tag":"039H","title":"Unramified morphisms · Lemma 039H","summary":"Let A, B be Noetherian local rings. Let A → B be a local homomorphism. • if A → B is an unramified homomorphism of local rings, then B^wedge is a finite A^wedge module, • if A → B is an unramified homomorphism of local rings and kappa( m_A) = kappa( m_B), then A^wedge → B^wedge is surjective, • if A → B is an unramified homomorphism of local rings and kappa( m_A) is separably closed, then A^wedge → B^wedge is surjective, • if A and B are complete discrete valuation rings,…","statement_latex":"Let $A$, $B$ be Noetherian local rings.\nLet $A \\to B$ be a local homomorphism.\n\\begin{enumerate}\n\\item if $A \\to B$ is an unramified homomorphism of local rings,\nthen $B^\\wedge$ is a finite $A^\\wedge$ module,\n\\item if $A \\to B$ is an unramified homomorphism of local rings and\n$\\kappa(\\mathfrak m_A) = \\kappa(\\mathfrak m_B)$,\nthen $A^\\wedge \\to B^\\wedge$ is surjective,\n\\item if $A \\to B$ is an unramified homomorphism of local rings\nand $\\kappa(\\mathfrak m_A)$\nis separably closed, then $A^\\wedge \\to B^\\wedge$ is surjective,\n\\item if $A$ and $B$ are complete discrete valuation rings, then\n$A \\to B$ is an unramified homomorphism of local rings\nif and only if the uniformizer for $A$ maps to a uniformizer for $B$,\nand the residue field extension is finite separable (and $B$ is\nessentially of finite type over $A$).\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039H","source_file":"etale.tex","source_line":132,"source_end_line":151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L132-L151","statement_sha256":"cab72aca91522516bb1fa93e6fb0e0ddd7f1a52ce5f6554a8db64e1d45ee2708","origin":"The Stacks Project","memory_eligible":false,"source_rank":8057,"rank":8057,"depth":9,"x":1580.8,"y":1132.341,"cluster":"tale-geometry"},{"id":"stacks:039I","tag":"039I","title":"Unramified morphisms · Lemma 039I","summary":"Let A, B be Noetherian local rings. Let A → B be a local homomorphism such that B is essentially of finite type over A. The following are equivalent • A → B is an unramified homomorphism of local rings • A^wedge → B^wedge is an unramified homomorphism of local rings, and • A^wedge → B^wedge is unramified.","statement_latex":"Let $A$, $B$ be Noetherian local rings.\nLet $A \\to B$ be a local homomorphism such that $B$ is\nessentially of finite type over $A$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A \\to B$ is an unramified homomorphism of local rings\n\\item $A^\\wedge \\to B^\\wedge$ is an unramified homomorphism of local rings, and\n\\item $A^\\wedge \\to B^\\wedge$ is unramified.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039I","source_file":"etale.tex","source_line":165,"source_end_line":176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L165-L176","statement_sha256":"874d06c5972db326e6c65759452f0ea5d56ab217a4968163c306ed4f3d0e635c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8058,"rank":8058,"depth":41,"x":1586.589,"y":1147.219,"cluster":"tale-geometry"},{"id":"stacks:024N","tag":"024N","title":"Unramified morphisms · Definition 024N","summary":"(See Morphisms, Definition [Tag 02G4] for the definition in the general case.) Let Y be a locally Noetherian scheme. Let f : X → Y be locally of finite type. Let x ∈ X. • We say f is unramified at x if O_Y, f(x) → O_X, x is an unramified homomorphism of local rings. • The morphism f : X → Y is said to be unramified if it is unramified at all points of X.","statement_latex":"(See Morphisms, Definition \\ref{morphisms-definition-unramified}\nfor the definition in the general case.)\nLet $Y$ be a locally Noetherian scheme.\nLet $f : X \\to Y$ be locally of finite type.\nLet $x \\in X$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it unramified at $x$} if\n$\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$\nis an unramified homomorphism of local rings.\n\\item The morphism $f : X \\to Y$ is said to be {\\it unramified}\nif it is unramified at all points of $X$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Unramified morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024N","source_file":"etale.tex","source_line":202,"source_end_line":216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L202-L216","statement_sha256":"56726ee8a4e1d9a090f291ad169088aa3e58c844e0651bfed0415e8c1783c354","origin":"The Stacks Project","memory_eligible":false,"source_rank":8059,"rank":8059,"depth":1,"x":1567.908,"y":1138.203,"cluster":"tale-geometry"},{"id":"stacks:039J","tag":"039J","title":"Unramified morphisms · Lemma 039J","summary":"Let Y be a locally Noetherian scheme. Let f : X → Y be locally of finite type. Let x ∈ X. The morphism f is unramified at x in the sense of Definition [Tag 024N] if and only if it is unramified in the sense of Morphisms, Definition [Tag 02G4].","statement_latex":"Let $Y$ be a locally Noetherian scheme.\nLet $f : X \\to Y$ be locally of finite type.\nLet $x \\in X$. The morphism $f$ is unramified at $x$ in\nthe sense of Definition \\ref{definition-unramified-schemes}\nif and only if it is unramified in\nthe sense of Morphisms, Definition \\ref{morphisms-definition-unramified}.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039J","source_file":"etale.tex","source_line":223,"source_end_line":231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L223-L231","statement_sha256":"e0b4b26ddf4a10194a033bbb2b05e86c331af1614931f16f48e632f2eab11637","origin":"The Stacks Project","memory_eligible":false,"source_rank":8060,"rank":8060,"depth":2,"x":1591.454,"y":1133.88,"cluster":"tale-geometry"},{"id":"stacks:024P","tag":"024P","title":"Three other characterizations of unramified morphisms · Theorem 024P","summary":"Let Y be a locally Noetherian scheme. Let f : X → Y be a morphism of schemes which is locally of finite type. Let x be a point of X. The following are equivalent • f is unramified at x, • the stalk Ω_X/Y, x of the module of relative differentials at x is trivial, • there exist open neighbourhoods U of x and V of f(x), and a commutative diagram xymatrix U ar[rr]_i ar[rd] & & A^n_V ar[ld] & V where i is a closed immersion defined by a quasi-coherent sheaf of ideals I such…","statement_latex":"Let $Y$ be a locally Noetherian scheme.\nLet $f : X \\to Y$ be a morphism of schemes which is locally of finite type.\nLet $x$ be a point of $X$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is unramified at $x$,\n\\item the stalk $\\Omega_{X/Y, x}$ of the module of relative differentials\nat $x$ is trivial,\n\\item there exist open neighbourhoods $U$ of $x$ and $V$ of $f(x)$, and a\ncommutative diagram\n$$\n\\xymatrix{\nU \\ar[rr]_i \\ar[rd] & & \\mathbf{A}^n_V \\ar[ld] \\\\\n& V\n}\n$$\nwhere $i$ is a closed immersion defined by a\nquasi-coherent sheaf of ideals $\\mathcal{I}$ such that the differentials\n$\\text{d}g$ for $g \\in \\mathcal{I}_{i(x)}$ generate\n$\\Omega_{\\mathbf{A}^n_V/V, i(x)}$, and\n\\item the diagonal $\\Delta_{X/Y} : X \\to X \\times_Y X$\nis a local isomorphism at $x$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Three other characterizations of unramified morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024P","source_file":"etale.tex","source_line":276,"source_end_line":300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L276-L300","statement_sha256":"46b3cb0ef3f8af6c1943e5a8ec065cac6aa2fb5e629e5a41e808f558a04f2adb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8061,"rank":8061,"depth":43,"x":1576.169,"y":1151.972,"cluster":"tale-geometry"},{"id":"stacks:024R","tag":"024R","title":"The functorial characterization of unramified morphisms · Theorem 024R","summary":"Let f : X → S be a morphism of schemes. Assume S is a locally Noetherian scheme, and f is locally of finite type. Then the following are equivalent: • f is unramified, • the morphism f is formally unramified: for any affine S-scheme T and subscheme T_0 of T defined by a square-zero ideal, the natural map Hom_S(T, X) → Hom_S(T_0, X) is injective.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nAssume $S$ is a locally Noetherian scheme, and $f$ is locally of finite type.\nThen the following are equivalent:\n\\begin{enumerate}\n\\item $f$ is unramified,\n\\item the morphism $f$ is formally unramified:\nfor any affine $S$-scheme $T$ and subscheme $T_0$ of $T$\ndefined by a square-zero ideal,\nthe natural map\n$$\n\\Hom_S(T, X) \\longrightarrow \\Hom_S(T_0, X)\n$$\nis injective.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"The functorial characterization of unramified morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024R","source_file":"etale.tex","source_line":401,"source_end_line":417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L401-L417","statement_sha256":"d89b8a0b2c441fc66880a46581c04fc831896715a0ecad3662ea6a48196f90a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8062,"rank":8062,"depth":44,"x":1572.694,"y":1128.183,"cluster":"tale-geometry"},{"id":"stacks:024T","tag":"024T","title":"Topological properties of unramified morphisms · Proposition 024T","summary":"Sections of unramified morphisms. • Any section of an unramified morphism is an open immersion. • Any section of a separated morphism is a closed immersion. • Any section of an unramified separated morphism is open and closed.","statement_latex":"Sections of unramified morphisms.\n\\begin{enumerate}\n\\item Any section of an unramified morphism is an open immersion.\n\\item Any section of a separated morphism is a closed immersion.\n\\item Any section of an unramified separated morphism is open and closed.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of unramified morphisms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024T","source_file":"etale.tex","source_line":466,"source_end_line":474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L466-L474","statement_sha256":"e9d0c55599608143773f240cdf95df11056d89223c170f6e1af55466c2c03d91","origin":"The Stacks Project","memory_eligible":false,"source_rank":8063,"rank":8063,"depth":16,"x":1595.852,"y":1144.863,"cluster":"tale-geometry"},{"id":"stacks:024U","tag":"024U","title":"Topological properties of unramified morphisms · Theorem 024U","summary":"Let Y be a connected scheme. Let f : X → Y be unramified and separated. Every section of f is an isomorphism onto a connected component. There exists a bijective correspondence sections of f ↔ ( connected components X' of X such that the induced map X' → Y is an isomorphism ) In particular, given x ∈ X there is at most one section passing through x.","statement_latex":"Let $Y$ be a connected scheme.\nLet $f : X \\to Y$ be unramified and separated.\nEvery section of $f$ is an isomorphism onto a connected component.\nThere exists a bijective correspondence\n$$\n\\text{sections of }f\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{connected components }X'\\text{ of }X\\text{ such that}\\\\\n\\text{the induced map }X' \\to Y\\text{ is an isomorphism}\n\\end{matrix}\n\\right\\}\n$$\nIn particular, given $x \\in X$ there is at most one\nsection passing through $x$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of unramified morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024U","source_file":"etale.tex","source_line":496,"source_end_line":514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L496-L514","statement_sha256":"bc6f58372d9cd7abc72ee2b6ebbe79855508ba22076ddc22872830dc780a8a26","origin":"The Stacks Project","memory_eligible":false,"source_rank":8064,"rank":8064,"depth":17,"x":1563.508,"y":1145.718,"cluster":"tale-geometry"},{"id":"stacks:024V","tag":"024V","title":"Topological properties of unramified morphisms · Proposition 024V","summary":"Let S is be a scheme. Let π : X → S be unramified and separated. Let Y be an S-scheme and y ∈ Y a point. Let f, g : Y → X be two S-morphisms. Assume • Y is connected • x = f(y) = g(y), and • the induced maps f^sharp, g^sharp : kappa(x) → kappa(y) on residue fields are equal. Then f = g.","statement_latex":"Let $S$ is be a scheme.\nLet $\\pi : X \\to S$ be unramified and separated.\nLet $Y$ be an $S$-scheme and $y \\in Y$ a point.\nLet $f, g : Y \\to X$ be two $S$-morphisms. Assume\n\\begin{enumerate}\n\\item $Y$ is connected\n\\item $x = f(y) = g(y)$, and\n\\item the induced maps $f^\\sharp, g^\\sharp : \\kappa(x) \\to \\kappa(y)$\non residue fields are equal.\n\\end{enumerate}\nThen $f = g$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of unramified morphisms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024V","source_file":"etale.tex","source_line":528,"source_end_line":541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L528-L541","statement_sha256":"f2ee4448debc9b5d5b5862c90d6c25adf7881a1dd7cf7639be33adeb46038923","origin":"The Stacks Project","memory_eligible":false,"source_rank":8065,"rank":8065,"depth":18,"x":1587.95,"y":1125.729,"cluster":"tale-geometry"},{"id":"stacks:0AKI","tag":"0AKI","title":"Topological properties of unramified morphisms · Lemma 0AKI","summary":"Let S be a Noetherian scheme. Let X → S be a quasi-compact unramified morphism. Let Y → S be a morphism with Y Noetherian. Then Mor_S(Y, X) is a finite set.","statement_latex":"Let $S$ be a Noetherian scheme. Let $X \\to S$ be a quasi-compact unramified\nmorphism. Let $Y \\to S$ be a morphism with $Y$ Noetherian. Then\n$\\Mor_S(Y, X)$ is a finite set.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKI","source_file":"etale.tex","source_line":557,"source_end_line":562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L557-L562","statement_sha256":"97b94a7c05e1e28b168008c33335674811b3ea210027feeb8bffe1417ef2c350","origin":"The Stacks Project","memory_eligible":false,"source_rank":8066,"rank":8066,"depth":43,"x":1585.875,"y":1155.733,"cluster":"tale-geometry"},{"id":"stacks:05VH","tag":"05VH","title":"Universally injective, unramified morphisms · Lemma 05VH","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • f is unramified and a monomorphism, • f is unramified and universally injective, • f is locally of finite type and a monomorphism, • f is universally injective, locally of finite type, and formally unramified, • f is locally of finite type and X_s is either empty or X_s → s is an isomorphism for all s ∈ S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is unramified and a monomorphism,\n\\item $f$ is unramified and universally injective,\n\\item $f$ is locally of finite type and a monomorphism,\n\\item $f$ is universally injective, locally of finite type, and\nformally unramified,\n\\item $f$ is locally of finite type and $X_s$ is either empty\nor $X_s \\to s$ is an isomorphism for all $s \\in S$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Universally injective, unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VH","source_file":"etale.tex","source_line":617,"source_end_line":630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L617-L630","statement_sha256":"237aabbade7b40e763a13c6827b02cb0375feb412cecba1ef9540d485f0fe1ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":8067,"rank":8067,"depth":42,"x":1562.293,"y":1131.38,"cluster":"tale-geometry"},{"id":"stacks:04XV","tag":"04XV","title":"Universally injective, unramified morphisms · Lemma 04XV","summary":"Let f : X → S be a morphism of schemes. The following are equivalent: • f is a closed immersion, • f is a proper monomorphism, • f is proper, unramified, and universally injective, • f is universally closed, unramified, and a monomorphism, • f is universally closed, unramified, and universally injective, • f is universally closed, locally of finite type, and a monomorphism, • f is universally closed, universally injective, locally of finite type, and formally unramified.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is a closed immersion,\n\\item $f$ is a proper monomorphism,\n\\item $f$ is proper, unramified, and universally injective,\n\\item $f$ is universally closed, unramified, and a monomorphism,\n\\item $f$ is universally closed, unramified, and universally injective,\n\\item $f$ is universally closed, locally of finite type, and a monomorphism,\n\\item $f$ is universally closed, universally injective, locally of\nfinite type, and formally unramified.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Universally injective, unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XV","source_file":"etale.tex","source_line":663,"source_end_line":677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L663-L677","statement_sha256":"54ed8e287e91a9e97393ca508e894585a18eb6f4f4b302e876449e92d450e0b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8068,"rank":8068,"depth":45,"x":1600.772,"y":1136.164,"cluster":"tale-geometry"},{"id":"stacks:04DG","tag":"04DG","title":"Universally injective, unramified morphisms · Lemma 04DG","summary":"Let π : X → S be a morphism of schemes. Let s ∈ S. Assume that • π is finite, • π is unramified, • π^-1((s)) = (x), and • kappa(s) ⊂ kappa(x) is purely inseparable. Then there exists an open neighbourhood U of s such that π|_π^-1(U) : π^-1(U) → U is a closed immersion.","statement_latex":"Let $\\pi : X \\to S$ be a morphism of schemes. Let $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $\\pi$ is finite,\n\\item $\\pi$ is unramified,\n\\item $\\pi^{-1}(\\{s\\}) = \\{x\\}$, and\n\\item $\\kappa(s) \\subset \\kappa(x)$ is purely\ninseparable\\footnote{In view of condition (2)\nthis is equivalent to $\\kappa(s) = \\kappa(x)$.}.\n\\end{enumerate}\nThen there exists an open neighbourhood $U$ of $s$ such that\n$\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ is a closed immersion.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Universally injective, unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DG","source_file":"etale.tex","source_line":706,"source_end_line":720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L706-L720","statement_sha256":"df2a3cd84803fed8987b8874e8d63c3fa5aaaf6c5de57e505c0e6b68523d4965","origin":"The Stacks Project","memory_eligible":false,"source_rank":8069,"rank":8069,"depth":3,"x":1567.323,"y":1155.147,"cluster":"tale-geometry"},{"id":"stacks:0251","tag":"0251","title":"Flat morphisms · Definition 0251","summary":"Flatness of modules and rings. • A module N over a ring A is said to be flat if the functor M ↦ M ⊗_A N is exact. • If this functor is also faithful, we say that N is faithfully flat over A. • A morphism of rings f : A → B is said to be flat (resp. faithfully flat) if the functor M ↦ M ⊗_A B is exact (resp. faithful and exact).","statement_latex":"Flatness of modules and rings.\n\\begin{enumerate}\n\\item A module $N$ over a ring $A$ is said to be {\\it flat}\nif the functor $M \\mapsto M \\otimes_A N$ is exact.\n\\item If this functor is also faithful, we say that\n$N$ is {\\it faithfully flat} over $A$.\n\\item A morphism of rings $f : A \\to B$ is said to be\n{\\it flat (resp. faithfully flat)}\nif the functor $M \\mapsto M \\otimes_A B$ is exact\n(resp. faithful and exact).\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0251","source_file":"etale.tex","source_line":850,"source_end_line":863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L850-L863","statement_sha256":"3b8f5502dae4e3ce518782d4abcdee62aa8ff687a955edb58a9b0f17104e03fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8070,"rank":8070,"depth":0,"x":1577.071,"y":1121.016,"cluster":"tale-geometry"},{"id":"stacks:0252","tag":"0252","title":"Flat morphisms · Theorem 0252","summary":"Let A, B be Noetherian local rings. Let f : A → B be a local homomorphism. If M is a finite B-module that is flat as an A-module, and t ∈ m_B is an element such that multiplication by t is injective on M/ m_AM, then M/tM is also A-flat.","statement_latex":"Let $A$, $B$ be Noetherian local rings.\nLet $f : A \\to B$ be a local homomorphism.\nIf $M$ is a finite $B$-module that is flat as an $A$-module,\nand $t \\in \\mathfrak m_B$ is an element such that multiplication\nby $t$ is injective on $M/\\mathfrak m_AM$, then $M/tM$ is also $A$-flat.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Flat morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0252","source_file":"etale.tex","source_line":903,"source_end_line":910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L903-L910","statement_sha256":"7ded252bd4d2519f30dd484c81c2a22c2c36ac44646e657e01b6004fc1c30348","origin":"The Stacks Project","memory_eligible":false,"source_rank":8071,"rank":8071,"depth":5,"x":1597.979,"y":1152.728,"cluster":"tale-geometry"},{"id":"stacks:0253","tag":"0253","title":"Flat morphisms · Definition 0253","summary":"(See Morphisms, Definition [Tag 01U3]). Let f : X → Y be a morphism of schemes. Let F be a quasi-coherent O_X-module. • Let x ∈ X. We say F is flat over Y at x ∈ X if F_x is a flat O_Y, f(x)-module. This uses the map O_Y, f(x) → O_X, x to think of F_x as a O_Y, f(x)-module. • Let x ∈ X. We say f is flat at x ∈ X if O_Y, f(x) → O_X, x is flat. • We say f is flat if it is flat at all points of X. • A morphism f : X → Y that is flat and surjective is sometimes said to be…","statement_latex":"(See Morphisms, Definition \\ref{morphisms-definition-flat}).\nLet $f : X \\to Y$ be a morphism of schemes.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item Let $x \\in X$. We say $\\mathcal{F}$ is\n{\\it flat over $Y$ at $x \\in X$} if $\\mathcal{F}_x$\nis a flat $\\mathcal{O}_{Y, f(x)}$-module.\nThis uses the map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ to\nthink of $\\mathcal{F}_x$ as a $\\mathcal{O}_{Y, f(x)}$-module.\n\\item Let $x \\in X$. We say $f$ is {\\it flat at $x \\in X$}\nif $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is flat.\n\\item We say $f$ is {\\it flat} if it is flat at all points of $X$.\n\\item A morphism $f : X \\to Y$ that is flat and surjective is sometimes\nsaid to be {\\it faithfully flat}.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0253","source_file":"etale.tex","source_line":917,"source_end_line":934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L917-L934","statement_sha256":"882365c75d8683ef6b4101e98d9cfa57b74aec1c2bd17928f3e2a8e855b0a48c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8072,"rank":8072,"depth":1,"x":1555.806,"y":1140.84,"cluster":"tale-geometry"},{"id":"stacks:0255","tag":"0255","title":"Topological properties of flat morphisms · Theorem 0255","summary":"Let Y be a locally Noetherian scheme. Let f : X → Y be a morphism which is locally of finite type. Let F be a coherent O_X-module. The set of points in X where F is flat over Y is an open set. In particular the set of points where f is flat is open in X.","statement_latex":"Let $Y$ be a locally Noetherian scheme.\nLet $f : X \\to Y$ be a morphism which is locally of finite type.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThe set of points in $X$ where $\\mathcal{F}$ is flat over $Y$ is an open set.\nIn particular the set of points where $f$ is flat is open in $X$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of flat morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0255","source_file":"etale.tex","source_line":966,"source_end_line":973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L966-L973","statement_sha256":"f358716c3a04340988f1638860cfd3acd5afcaa14ef84f57e7f8b9e63bbcc87b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8073,"rank":8073,"depth":35,"x":1597.648,"y":1125.248,"cluster":"tale-geometry"},{"id":"stacks:039K","tag":"039K","title":"Topological properties of flat morphisms · Theorem 039K","summary":"Let Y be a locally Noetherian scheme. Let f : X → Y be a morphism which is flat and locally of finite type. Then f is (universally) open.","statement_latex":"Let $Y$ be a locally Noetherian scheme.\nLet $f : X \\to Y$ be a morphism which is flat and locally of finite type.\nThen $f$ is (universally) open.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of flat morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039K","source_file":"etale.tex","source_line":979,"source_end_line":984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L979-L984","statement_sha256":"68b056deb368fdd02093fa1d6e46f07a7998880b2ee5530201566c871b0fa732","origin":"The Stacks Project","memory_eligible":false,"source_rank":8074,"rank":8074,"depth":18,"x":1578.819,"y":1161.449,"cluster":"tale-geometry"},{"id":"stacks:0256","tag":"0256","title":"Topological properties of flat morphisms · Theorem 0256","summary":"A faithfully flat quasi-compact morphism is a quotient map for the Zariski topology.","statement_latex":"A faithfully flat quasi-compact morphism is a quotient map for\nthe Zariski topology.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of flat morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0256","source_file":"etale.tex","source_line":990,"source_end_line":994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L990-L994","statement_sha256":"97eab318780d662597d9ac619f97e9ad847e4173bb48d6e09e0dec3834f3b40a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8075,"rank":8075,"depth":4,"x":1563.208,"y":1123.097,"cluster":"tale-geometry"},{"id":"stacks:0258","tag":"0258","title":"Étale morphisms · Definition 0258","summary":"Let A, B be Noetherian local rings. A local homomorphism f : A → B is said to be an étale homomorphism of local rings if it is flat and an unramified homomorphism of local rings (please see Definition [Tag 024M]).","statement_latex":"Let $A$, $B$ be Noetherian local rings.\nA local homomorphism $f : A \\to B$ is said to be an\n{\\it \\'etale homomorphism of local rings}\nif it is flat and an unramified homomorphism of local rings\n(please see Definition \\ref{definition-unramified-rings}).","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0258","source_file":"etale.tex","source_line":1048,"source_end_line":1055,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1048-L1055","statement_sha256":"6c9377c94137575641bc4248f9a2d0a4d5d829c1ea3f9427c5681fa347f32747","origin":"The Stacks Project","memory_eligible":false,"source_rank":8076,"rank":8076,"depth":1,"x":1606.6,"y":1143.006,"cluster":"tale-geometry"},{"id":"stacks:039L","tag":"039L","title":"Étale morphisms · Lemma 039L","summary":"Let A → B be of finite type with A a Noetherian ring. Let q be a prime of B lying over p ⊂ A. Then A → B is étale at q if and only if A_ p → B_ q is an étale homomorphism of local rings.","statement_latex":"Let $A \\to B$ be of finite type with $A$ a Noetherian ring.\nLet $\\mathfrak q$ be a prime of $B$ lying over $\\mathfrak p \\subset A$.\nThen $A \\to B$ is \\'etale at $\\mathfrak q$ if and only if\n$A_{\\mathfrak p} \\to B_{\\mathfrak q}$ is an \\'etale homomorphism\nof local rings.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039L","source_file":"etale.tex","source_line":1093,"source_end_line":1100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1093-L1100","statement_sha256":"13863cf031f670013f518e0b0ae0a6e6b0f20aa4321e07716fad73480d90555e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8077,"rank":8077,"depth":41,"x":1557.462,"y":1153.173,"cluster":"tale-geometry"},{"id":"stacks:039M","tag":"039M","title":"Étale morphisms · Lemma 039M","summary":"Let A, B be Noetherian local rings. Let A → B be a local homomorphism such that B is essentially of finite type over A. The following are equivalent • A → B is an étale homomorphism of local rings • A^wedge → B^wedge is an étale homomorphism of local rings, and • A^wedge → B^wedge is étale. Moreover, in this case B^wedge ≅ (A^wedge)^⊕ n as A^wedge-modules for some n ≥ 1.","statement_latex":"Let $A$, $B$ be Noetherian local rings.\nLet $A \\to B$ be a local homomorphism such that $B$ is essentially of\nfinite type over $A$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A \\to B$ is an \\'etale homomorphism of local rings\n\\item $A^\\wedge \\to B^\\wedge$ is an \\'etale homomorphism of local rings, and\n\\item $A^\\wedge \\to B^\\wedge$ is \\'etale.\n\\end{enumerate}\nMoreover, in this case $B^\\wedge \\cong (A^\\wedge)^{\\oplus n}$ as\n$A^\\wedge$-modules for some $n \\geq 1$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039M","source_file":"etale.tex","source_line":1109,"source_end_line":1122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1109-L1122","statement_sha256":"0c3f5c9b2458c41c29d125100aae593508b56ccaa9048f4c7784f50339b71b4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8078,"rank":8078,"depth":42,"x":1586.159,"y":1117.004,"cluster":"tale-geometry"},{"id":"stacks:0259","tag":"0259","title":"Étale morphisms · Definition 0259","summary":"(See Morphisms, Definition [Tag 02GI].) Let Y be a locally Noetherian scheme. Let f : X → Y be a morphism of schemes which is locally of finite type. • Let x ∈ X. We say f is étale at x ∈ X if O_Y, f(x) → O_X, x is an étale homomorphism of local rings. • The morphism is said to be étale if it is étale at all its points.","statement_latex":"(See Morphisms, Definition \\ref{morphisms-definition-etale}.)\nLet $Y$ be a locally Noetherian scheme.\nLet $f : X \\to Y$ be a morphism of schemes which is locally of finite type.\n\\begin{enumerate}\n\\item Let $x \\in X$. We say $f$ is {\\it \\'etale at $x \\in X$} if\n$\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$ is an\n\\'etale homomorphism of local rings.\n\\item The morphism is said to be {\\it \\'etale} if it is \\'etale at all its\npoints.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0259","source_file":"etale.tex","source_line":1146,"source_end_line":1158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1146-L1158","statement_sha256":"95a124a1751a1a2307b77e0c4d9f5ec19179ad108bb8c0fb58b1c24c459195e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8079,"rank":8079,"depth":2,"x":1594.245,"y":1160.882,"cluster":"tale-geometry"},{"id":"stacks:039N","tag":"039N","title":"Étale morphisms · Lemma 039N","summary":"Let Y be a locally Noetherian scheme. Let f : X → Y be locally of finite type. Let x ∈ X. The morphism f is étale at x in the sense of Definition [Tag 0259] if and only if it is étale at x in the sense of Morphisms, Definition [Tag 02GI].","statement_latex":"Let $Y$ be a locally Noetherian scheme.\nLet $f : X \\to Y$ be locally of finite type.\nLet $x \\in X$. The morphism $f$ is \\'etale at $x$ in\nthe sense of Definition \\ref{definition-etale-schemes-1}\nif and only if it is \\'etale at $x$ in\nthe sense of Morphisms, Definition \\ref{morphisms-definition-etale}.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039N","source_file":"etale.tex","source_line":1165,"source_end_line":1173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1165-L1173","statement_sha256":"0aa06a3da9f172d4e3f56bbecfc13a4b6157a55fcd227b4f257721378f9a1965","origin":"The Stacks Project","memory_eligible":false,"source_rank":8080,"rank":8080,"depth":42,"x":1552.152,"y":1132.537,"cluster":"tale-geometry"},{"id":"stacks:025B","tag":"025B","title":"The structure theorem · Theorem 025B","summary":"Let f : A → B be an étale homomorphism of local rings. Then there exist f, g ∈ A[t] such that • B' = A[t]_g/(f) is standard étale -- see (a) and (b) above, and • B is isomorphic to a localization of B' at a prime.","statement_latex":"Let $f : A \\to B$ be an \\'etale homomorphism of local rings.\nThen there exist $f, g \\in A[t]$ such that\n\\begin{enumerate}\n\\item $B' = A[t]_g/(f)$ is standard \\'etale -- see (a) and (b) above, and\n\\item $B$ is isomorphic to a localization of $B'$ at a prime.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"The structure theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025B","source_file":"etale.tex","source_line":1251,"source_end_line":1259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1251-L1259","statement_sha256":"79daf3bae9abf6cbea85921eb119088b168316660d02dead0bd78194586f3c53","origin":"The Stacks Project","memory_eligible":false,"source_rank":8081,"rank":8081,"depth":43,"x":1607.05,"y":1129.502,"cluster":"tale-geometry"},{"id":"stacks:039O","tag":"039O","title":"The structure theorem · Theorem 039O","summary":"Let f : A → B be an unramified morphism of local rings. Then there exist f, g ∈ A[t] such that • B' = A[t]_g/(f) is standard étale -- see (a) and (b) above, and • B is isomorphic to a quotient of a localization of B' at a prime.","statement_latex":"Let $f : A \\to B$ be an unramified morphism of local rings.\nThen there exist $f, g \\in A[t]$ such that\n\\begin{enumerate}\n\\item $B' = A[t]_g/(f)$ is standard \\'etale -- see (a) and (b) above, and\n\\item $B$ is isomorphic to a quotient of a localization of $B'$ at a prime.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"The structure theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039O","source_file":"etale.tex","source_line":1274,"source_end_line":1282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1274-L1282","statement_sha256":"d92b6c6a6c296f53396cbfdbda35bf62bfedf399572e8c3e4e163893409fd7cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8082,"rank":8082,"depth":44,"x":1568.281,"y":1163.521,"cluster":"tale-geometry"},{"id":"stacks:025C","tag":"025C","title":"The structure theorem · Theorem 025C","summary":"Let φ : X → Y be a morphism of schemes. Let x ∈ X. Let V ⊂ Y be an affine open neighbourhood of φ(x). If φ is étale at x, then there exist exists an affine open U ⊂ X with x ∈ U and φ(U) ⊂ V such that we have the following diagram xymatrix X ar[d] & U ar[l] ar[d] ar[r]_-j & Spec(R[t]_f'/(f)) ar[d] Y & V ar[l] ar@=[r] & Spec(R) where j is an open immersion, and f ∈ R[t] is monic.","statement_latex":"Let $\\varphi : X \\to Y$ be a morphism of schemes. Let $x \\in X$.\nLet $V \\subset Y$ be an affine open neighbourhood of $\\varphi(x)$.\nIf $\\varphi$ is \\'etale at $x$, then there exist exists an affine open\n$U \\subset X$ with $x \\in U$ and $\\varphi(U) \\subset V$\nsuch that we have the following diagram\n$$\n\\xymatrix{\nX \\ar[d] & U \\ar[l] \\ar[d] \\ar[r]_-j & \\Spec(R[t]_{f'}/(f)) \\ar[d] \\\\\nY & V \\ar[l] \\ar@{=}[r] & \\Spec(R)\n}\n$$\nwhere $j$ is an open immersion, and $f \\in R[t]$ is monic.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"The structure theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025C","source_file":"etale.tex","source_line":1299,"source_end_line":1313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1299-L1313","statement_sha256":"702066fc8dd8a34c238f5f1fc06f8496c4c934d1e0b8af932af0cc6218c5b648","origin":"The Stacks Project","memory_eligible":false,"source_rank":8083,"rank":8083,"depth":45,"x":1569.541,"y":1115.574,"cluster":"tale-geometry"},{"id":"stacks:039Q","tag":"039Q","title":"Étale and smooth morphisms · Theorem 039Q","summary":"Let φ : X → Y be a morphism of schemes. Let x ∈ X. If φ is smooth at x, then there exist an integer n ≥ 0 and affine opens V ⊂ Y and U ⊂ X with x ∈ U and φ(U) ⊂ V such that there exists a commutative diagram xymatrix X ar[d] & U ar[l] ar[d] ar[r]_-π & A^n_R ar[d] ar@=[r] & Spec(R[x_1, …, x_n]) ar[dl] Y & V ar[l] ar@=[r] & Spec(R) where π is étale.","statement_latex":"Let $\\varphi : X \\to Y$ be a morphism of schemes.\nLet $x \\in X$.\nIf $\\varphi$ is smooth at $x$, then\nthere exist an integer $n \\geq 0$ and affine opens\n$V \\subset Y$ and $U \\subset X$ with $x \\in U$ and $\\varphi(U) \\subset V$\nsuch that there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] & U \\ar[l] \\ar[d] \\ar[r]_-\\pi &\n\\mathbf{A}^n_R \\ar[d] \\ar@{=}[r] &  \\Spec(R[x_1, \\ldots, x_n]) \\ar[dl] \\\\\nY & V \\ar[l] \\ar@{=}[r] & \\Spec(R)\n}\n$$\nwhere $\\pi$ is \\'etale.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale and smooth morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039Q","source_file":"etale.tex","source_line":1334,"source_end_line":1350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1334-L1350","statement_sha256":"5f00aac3f225a7f16bc2e5adc8bde7c367ed5ccd91829e504277f71478eccd33","origin":"The Stacks Project","memory_eligible":false,"source_rank":8084,"rank":8084,"depth":37,"x":1607.83,"y":1152.286,"cluster":"tale-geometry"},{"id":"stacks:025G","tag":"025G","title":"Topological properties of étale morphisms · Theorem 025G","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent: • f is an open immersion, • f is universally injective and étale, and • f is a flat monomorphism, locally of finite presentation.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is an open immersion,\n\\item $f$ is universally injective and \\'etale, and\n\\item $f$ is a flat monomorphism, locally of finite presentation.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of étale morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025G","source_file":"etale.tex","source_line":1369,"source_end_line":1378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1369-L1378","statement_sha256":"8a14f4aba59f5f2457a68ee5a6cacaa25cc6907b977fbc51583a314ff7094ca4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8085,"rank":8085,"depth":43,"x":1549.088,"y":1146.845,"cluster":"tale-geometry"},{"id":"stacks:04DH","tag":"04DH","title":"Topological properties of étale morphisms · Lemma 04DH","summary":"Let π : X → S be a morphism of schemes. Let s ∈ S. Assume that • π is finite, • π is étale, • π^-1((s)) = (x), and • kappa(s) ⊂ kappa(x) is purely inseparable. Then there exists an open neighbourhood U of s such that π|_π^-1(U) : π^-1(U) → U is an isomorphism.","statement_latex":"Let $\\pi : X \\to S$ be a morphism of schemes. Let $s \\in S$.\nAssume that\n\\begin{enumerate}\n\\item $\\pi$ is finite,\n\\item $\\pi$ is \\'etale,\n\\item $\\pi^{-1}(\\{s\\}) = \\{x\\}$, and\n\\item $\\kappa(s) \\subset \\kappa(x)$ is purely\ninseparable\\footnote{In view of condition (2)\nthis is equivalent to $\\kappa(s) = \\kappa(x)$.}.\n\\end{enumerate}\nThen there exists an open neighbourhood $U$ of $s$ such that\n$\\pi|_{\\pi^{-1}(U)} : \\pi^{-1}(U) \\to U$ is an isomorphism.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DH","source_file":"etale.tex","source_line":1411,"source_end_line":1425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1411-L1425","statement_sha256":"f895eb8438709342fafb231db0d3189d7c9c7d1a17f18b6d443d9eb26e1c57e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8086,"rank":8086,"depth":44,"x":1597.571,"y":1117.046,"cluster":"tale-geometry"},{"id":"stacks:0EBS","tag":"0EBS","title":"Topological properties of étale morphisms · Lemma 0EBS","summary":"Let U → X be an étale morphism of schemes where X is a scheme in characteristic p. Then the relative Frobenius F_U/X : U → U ×_X, F_X X is an isomorphism.","statement_latex":"Let $U \\to X$ be an \\'etale morphism of schemes\nwhere $X$ is a scheme in characteristic $p$.\nThen the relative Frobenius $F_{U/X} : U \\to U \\times_{X, F_X} X$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological properties of étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBS","source_file":"etale.tex","source_line":1438,"source_end_line":1444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1438-L1444","statement_sha256":"5800263761d3b33ecbe79b3e8ecc6a607440570e6f975a1acd0169a58dc5f08a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8087,"rank":8087,"depth":44,"x":1585.589,"y":1167.319,"cluster":"tale-geometry"},{"id":"stacks:025H","tag":"025H","title":"Topological invariance of the étale topology · Theorem 025H","summary":"Let X and Y be two schemes over a base scheme S. Let S_0 be a closed subscheme of S with the same underlying topological space (for example if the ideal sheaf of S_0 in S has square zero). Denote X_0 (resp. Y_0) the base change S_0 ×_S X (resp. S_0 ×_S Y). If X is étale over S, then the map Mor_S(Y, X) → Mor_S_0(Y_0, X_0) is bijective.","statement_latex":"Let $X$ and $Y$ be two schemes over a base scheme $S$. Let $S_0$ be a closed\nsubscheme of $S$ with the same underlying topological space\n(for example if the ideal sheaf of $S_0$ in $S$ has square zero).\nDenote $X_0$ (resp.\\ $Y_0$) the base change $S_0 \\times_S X$\n(resp.\\ $S_0 \\times_S Y$).\nIf $X$ is \\'etale over $S$, then the map\n$$\n\\Mor_S(Y, X) \\longrightarrow \\Mor_{S_0}(Y_0, X_0)\n$$\nis bijective.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological invariance of the étale topology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025H","source_file":"etale.tex","source_line":1468,"source_end_line":1480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1468-L1480","statement_sha256":"987ffdb4fc99d9831b42131586f665a260cf0e920317f013a4b0057a23fdf7f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8088,"rank":8088,"depth":16,"x":1553.512,"y":1122.768,"cluster":"tale-geometry"},{"id":"stacks:039R","tag":"039R","title":"Une equivalence remarquable de catégories · Theorem 039R","summary":"[EGA] Let S be a scheme. Let S_0 ⊂ S be a closed subscheme with the same underlying topological space (for example if the ideal sheaf of S_0 in S has square zero). The functor X ↦ X_0 = S_0 ×_S X defines an equivalence of categories ( schemes X étale over S ) ↔ ( schemes X_0 étale over S_0 )","statement_latex":"\\begin{reference}\n\\cite[IV, Theorem 18.1.2]{EGA}\n\\end{reference}\nLet $S$ be a scheme.\nLet $S_0 \\subset S$ be a closed subscheme with the same underlying\ntopological space (for example if the ideal sheaf of $S_0$ in $S$\nhas square zero). The functor\n$$\nX \\longmapsto X_0 = S_0 \\times_S X\n$$\ndefines an equivalence of categories\n$$\n\\{\n\\text{schemes }X\\text{ \\'etale over }S\n\\}\n\\leftrightarrow\n\\{\n\\text{schemes }X_0\\text{ \\'etale over }S_0\n\\}\n$$","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Topological invariance of the étale topology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039R","source_file":"etale.tex","source_line":1529,"source_end_line":1551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1529-L1551","statement_sha256":"f45ef94ea481fe3a915f2dab54f0a0618d522b61c9682da5c8c39ad3ce2dd9f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8089,"rank":8089,"depth":17,"x":1613.884,"y":1137.642,"cluster":"tale-geometry"},{"id":"stacks:025K","tag":"025K","title":"The functorial characterization · Theorem 025K","summary":"Let f : X → S be a morphism that is locally of finite presentation. The following are equivalent • f is étale, • for all affine S-schemes Y, and closed subschemes Y_0 ⊂ Y defined by square-zero ideals, the natural map Mor_S(Y, X) → Mor_S(Y_0, X) is bijective.","statement_latex":"Let $f : X \\to S$ be a morphism that is locally of finite presentation.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is \\'etale,\n\\item for all affine $S$-schemes $Y$, and closed subschemes $Y_0 \\subset Y$\ndefined by square-zero ideals, the natural map\n$$\n\\Mor_S(Y, X) \\longrightarrow \\Mor_S(Y_0, X)\n$$\nis bijective.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"The functorial characterization","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025K","source_file":"etale.tex","source_line":1640,"source_end_line":1653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1640-L1653","statement_sha256":"c2caa6a8fdf2e2f18298934ce61aef988ce6ce614ecdb1da7144c0060a0952e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8090,"rank":8090,"depth":40,"x":1556.579,"y":1161.266,"cluster":"tale-geometry"},{"id":"stacks:04HH","tag":"04HH","title":"Étale local structure of unramified morphisms · Lemma 04HH","summary":"Let f : X → S be a morphism of schemes. Let x_1, …, x_n ∈ X be points having the same image s in S. Assume f is unramified at each x_i. Then there exists an étale neighbourhood (U, u) → (S, s) and opens V_i, j ⊂ X_U, i = 1, …, n, j = 1, …, m_i such that • V_i, j → U is a closed immersion passing through u, • u is not in the image of V_i, j ∩ V_i', j' unless i = i' and j = j', and • any point of (X_U)_u mapping to x_i is in some V_i, j.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.\nAssume $f$ is unramified at each $x_i$.\nThen there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$\nand opens $V_{i, j} \\subset X_U$, $i = 1, \\ldots, n$, $j = 1, \\ldots, m_i$\nsuch that\n\\begin{enumerate}\n\\item $V_{i, j} \\to U$ is a closed immersion passing through $u$,\n\\item $u$ is not in the image of $V_{i, j} \\cap V_{i', j'}$ unless\n$i = i'$ and $j = j'$, and\n\\item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale local structure of unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HH","source_file":"etale.tex","source_line":1686,"source_end_line":1700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1686-L1700","statement_sha256":"e5102d48b96b15cfb540ccb855e2f84cd881f01072ca238d2ffa5d967a07d9f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8091,"rank":8091,"depth":32,"x":1580.171,"y":1110.624,"cluster":"tale-geometry"},{"id":"stacks:04HI","tag":"04HI","title":"Étale local structure of unramified morphisms · Lemma 04HI","summary":"Let f : X → S be a morphism of schemes. Let x_1, …, x_n ∈ X be points having the same image s in S. Assume f is separated and f is unramified at each x_i. Then there exists an étale neighbourhood (U, u) → (S, s) and a disjoint union decomposition X_U = W amalg coprod_i, j V_i, j such that • V_i, j → U is a closed immersion passing through u, • the fibre W_u contains no point mapping to any x_i. In particular, if f^-1((s)) = (x_1, …, x_n), then the fibre W_u is empty.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.\nAssume $f$ is separated and $f$ is unramified at each $x_i$.\nThen there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$\nand a disjoint union decomposition\n$$\nX_U =\nW \\amalg \\coprod\\nolimits_{i, j} V_{i, j}\n$$\nsuch that\n\\begin{enumerate}\n\\item $V_{i, j} \\to U$ is a closed immersion passing through $u$,\n\\item the fibre $W_u$ contains no point mapping to any $x_i$.\n\\end{enumerate}\nIn particular, if $f^{-1}(\\{s\\}) = \\{x_1, \\ldots, x_n\\}$, then\nthe fibre $W_u$ is empty.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale local structure of unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HI","source_file":"etale.tex","source_line":1717,"source_end_line":1735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1717-L1735","statement_sha256":"d533ba1ef4cd9011933669845230dd2c37a82772485059a4021766d36b89b605","origin":"The Stacks Project","memory_eligible":false,"source_rank":8092,"rank":8092,"depth":33,"x":1603.816,"y":1162.053,"cluster":"tale-geometry"},{"id":"stacks:04HJ","tag":"04HJ","title":"Étale local structure of unramified morphisms · Lemma 04HJ","summary":"Let f : X → S be a finite unramified morphism of schemes. Let s ∈ S. There exists an étale neighbourhood (U, u) → (S, s) and a finite disjoint union decomposition X_U = coprod_j V_j such that each V_j → U is a closed immersion.","statement_latex":"Let $f : X \\to S$ be a finite unramified morphism of schemes.\nLet $s \\in S$.\nThere exists an \\'etale neighbourhood $(U, u) \\to (S, s)$\nand a finite disjoint union decomposition\n$$\nX_U = \\coprod\\nolimits_j V_j\n$$\nsuch that each $V_j \\to U$ is a closed immersion.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale local structure of unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HJ","source_file":"etale.tex","source_line":1757,"source_end_line":1767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1757-L1767","statement_sha256":"e981dcad2f056308009235df5230c4c41cde2892ae954fac11b14bc2f526a481","origin":"The Stacks Project","memory_eligible":false,"source_rank":8093,"rank":8093,"depth":34,"x":1544.237,"y":1137.216,"cluster":"tale-geometry"},{"id":"stacks:04HL","tag":"04HL","title":"Étale local structure of étale morphisms · Lemma 04HL","summary":"Let f : X → S be a morphism of schemes. Let x_1, …, x_n ∈ X be points having the same image s in S. Assume f is étale at each x_i. Then there exists an étale neighbourhood (U, u) → (S, s) and opens V_i, j ⊂ X_U, i = 1, …, n, j = 1, …, m_i such that • V_i, j → U is an isomorphism, • u is not in the image of V_i, j ∩ V_i', j' unless i = i' and j = j', and • any point of (X_U)_u mapping to x_i is in some V_i, j.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.\nAssume $f$ is \\'etale at each $x_i$.\nThen there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$\nand opens $V_{i, j} \\subset X_U$, $i = 1, \\ldots, n$, $j = 1, \\ldots, m_i$\nsuch that\n\\begin{enumerate}\n\\item $V_{i, j} \\to U$ is an isomorphism,\n\\item $u$ is not in the image of $V_{i, j} \\cap V_{i', j'}$ unless\n$i = i'$ and $j = j'$, and\n\\item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale local structure of étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HL","source_file":"etale.tex","source_line":1793,"source_end_line":1807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1793-L1807","statement_sha256":"170eb05c2f177b18141d0360a1b14852348c69e7ba56d4e7ac8e6fab5b6afa44","origin":"The Stacks Project","memory_eligible":false,"source_rank":8094,"rank":8094,"depth":44,"x":1608.979,"y":1121.525,"cluster":"tale-geometry"},{"id":"stacks:04HM","tag":"04HM","title":"Étale local structure of étale morphisms · Lemma 04HM","summary":"Let f : X → S be a morphism of schemes. Let x_1, …, x_n ∈ X be points having the same image s in S. Assume f is separated and f is étale at each x_i. Then there exists an étale neighbourhood (U, u) → (S, s) and a finite disjoint union decomposition X_U = W amalg coprod_i, j V_i, j of schemes such that • V_i, j → U is an isomorphism, • the fibre W_u contains no point mapping to any x_i. In particular, if f^-1((s)) = (x_1, …, x_n), then the fibre W_u is empty.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x_1, \\ldots, x_n \\in X$ be points having the same image $s$ in $S$.\nAssume $f$ is separated and $f$ is \\'etale at each $x_i$.\nThen there exists an \\'etale neighbourhood $(U, u) \\to (S, s)$\nand a finite disjoint union decomposition\n$$\nX_U =\nW \\amalg \\coprod\\nolimits_{i, j} V_{i, j}\n$$\nof schemes such that\n\\begin{enumerate}\n\\item $V_{i, j} \\to U$ is an isomorphism,\n\\item the fibre $W_u$ contains no point mapping to any $x_i$.\n\\end{enumerate}\nIn particular, if $f^{-1}(\\{s\\}) = \\{x_1, \\ldots, x_n\\}$, then\nthe fibre $W_u$ is empty.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale local structure of étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HM","source_file":"etale.tex","source_line":1819,"source_end_line":1837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1819-L1837","statement_sha256":"bb6bdc6ecddd74436a449612c9b63284dab428401f62d5f246da80bcc3a082de","origin":"The Stacks Project","memory_eligible":false,"source_rank":8095,"rank":8095,"depth":45,"x":1573.407,"y":1170.444,"cluster":"tale-geometry"},{"id":"stacks:04HN","tag":"04HN","title":"Étale local structure of étale morphisms · Lemma 04HN","summary":"Let f : X → S be a finite étale morphism of schemes. Let s ∈ S. There exists an étale neighbourhood (U, u) → (S, s) and a finite disjoint union decomposition X_U = coprod_j V_j of schemes such that each V_j → U is an isomorphism.","statement_latex":"Let $f : X \\to S$ be a finite \\'etale morphism of schemes.\nLet $s \\in S$. There exists an \\'etale neighbourhood $(U, u) \\to (S, s)$\nand a finite disjoint union decomposition\n$$\nX_U = \\coprod\\nolimits_j V_j\n$$\nof schemes such that each $V_j \\to U$ is an isomorphism.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Étale local structure of étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HN","source_file":"etale.tex","source_line":1855,"source_end_line":1864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1855-L1864","statement_sha256":"93e2adf51865a17da1d916225aab4d84a631959201b1a49b0b9e52cde0fbbf20","origin":"The Stacks Project","memory_eligible":false,"source_rank":8096,"rank":8096,"depth":45,"x":1560.139,"y":1113.489,"cluster":"tale-geometry"},{"id":"stacks:039S","tag":"039S","title":"Permanence properties · Lemma 039S","summary":"Let A, B be Noetherian local rings. Let A → B be a étale homomorphism of local rings. Then dim(A) = dim(B).","statement_latex":"Let $A$, $B$ be Noetherian local rings.\nLet $A \\to B$ be a \\'etale homomorphism of local rings.\nThen $\\dim(A) = \\dim(B)$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Permanence properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039S","source_file":"etale.tex","source_line":1890,"source_end_line":1895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1890-L1895","statement_sha256":"b77b52c5c4a41dae97cc40cc9e134acba98bc557f9622a4ed3a8545660145c80","origin":"The Stacks Project","memory_eligible":false,"source_rank":8097,"rank":8097,"depth":12,"x":1616.394,"y":1148.378,"cluster":"tale-geometry"},{"id":"stacks:039T","tag":"039T","title":"Permanence properties · Proposition 039T","summary":"Let A, B be Noetherian local rings. Let f : A → B be an étale homomorphism of local rings. Then depth(A) = depth(B)","statement_latex":"Let $A$, $B$ be Noetherian local rings.\nLet $f : A \\to B$ be an \\'etale homomorphism of local rings.\nThen $\\text{depth}(A) = \\text{depth}(B)$","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Permanence properties","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/039T","source_file":"etale.tex","source_line":1902,"source_end_line":1907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1902-L1907","statement_sha256":"9939d94665d926b1d283d21f3245bb1ab1cb20fc9d767bb8dfd1b272ce62e1c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8098,"rank":8098,"depth":16,"x":1546.034,"y":1154.642,"cluster":"tale-geometry"},{"id":"stacks:025Q","tag":"025Q","title":"Permanence properties · Proposition 025Q","summary":"Being Cohen-Macaulay ascends and descends along étale maps. Let A, B be Noetherian local rings. Let f : A → B be an étale homomorphism of local rings. Then A is Cohen-Macaulay if and only if B is so.","statement_latex":"\\begin{slogan}\nBeing Cohen-Macaulay ascends and descends along \\'etale maps.\n\\end{slogan}\nLet $A$, $B$ be Noetherian local rings.\nLet $f : A \\to B$ be an \\'etale homomorphism of local rings.\nThen $A$ is Cohen-Macaulay if and only if $B$ is so.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Permanence properties","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025Q","source_file":"etale.tex","source_line":1913,"source_end_line":1921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1913-L1921","statement_sha256":"0e43934ff2c06cd092658c0df7f769aec1d52cf921217c5d487a81f171218c99","origin":"The Stacks Project","memory_eligible":false,"source_rank":8099,"rank":8099,"depth":0,"x":1593.423,"y":1109.587,"cluster":"tale-geometry"},{"id":"stacks:025N","tag":"025N","title":"Permanence properties · Proposition 025N","summary":"Let A, B be Noetherian local rings. Let f : A → B be an étale homomorphism of local rings. Then A is regular if and only if B is so.","statement_latex":"Let $A$, $B$ be Noetherian local rings.\nLet $f : A \\to B$ be an \\'etale homomorphism of local rings.\nThen $A$ is regular if and only if $B$ is so.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Permanence properties","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025N","source_file":"etale.tex","source_line":1929,"source_end_line":1934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1929-L1934","statement_sha256":"167850f9f12bde54fc0b0e65a63f858d4967a09d2e1cf7e07dcea024c3d9258e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8100,"rank":8100,"depth":18,"x":1594.722,"y":1170.378,"cluster":"tale-geometry"},{"id":"stacks:025O","tag":"025O","title":"Permanence properties · Proposition 025O","summary":"Let A, B be Noetherian local rings. Let f : A → B be an étale homomorphism of local rings. Then A is reduced if and only if B is so.","statement_latex":"Let $A$, $B$ be Noetherian local rings.\nLet $f : A \\to B$ be an \\'etale homomorphism of local rings.\nThen $A$ is reduced if and only if $B$ is so.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Permanence properties","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025O","source_file":"etale.tex","source_line":1950,"source_end_line":1955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1950-L1955","statement_sha256":"0da3ea45989f1aaa6e0440764641ba123a2e94a1c4f52e25f1f955869929acd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8101,"rank":8101,"depth":42,"x":1544.331,"y":1125.8,"cluster":"tale-geometry"},{"id":"stacks:025P","tag":"025P","title":"Permanence properties · Proposition 025P","summary":"[SGA1] Let A, B be Noetherian local rings. Let f : A → B be an étale homomorphism of local rings. Then A is a normal domain if and only if B is so.","statement_latex":"\\begin{reference}\n\\cite[Expose I, Theorem 9.5 part (i)]{SGA1}\n\\end{reference}\nLet $A$, $B$ be Noetherian local rings.\nLet $f : A \\to B$ be an \\'etale homomorphism of local rings.\nThen $A$ is a normal domain if and only if $B$ is so.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Permanence properties","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025P","source_file":"etale.tex","source_line":1983,"source_end_line":1991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L1983-L1991","statement_sha256":"82901ffa1280278f8fe2d7716b0123709cd1ac96524368d58b4879fe91b9fc52","origin":"The Stacks Project","memory_eligible":false,"source_rank":8102,"rank":8102,"depth":42,"x":1618.124,"y":1130.127,"cluster":"tale-geometry"},{"id":"stacks:0BTJ","tag":"0BTJ","title":"Descending étale morphisms · Lemma 0BTJ","summary":"If f : X → S is surjective, then the functor ([Tag 0BTI]) is faithful.","statement_latex":"If $f : X \\to S$ is surjective, then the functor\n(\\ref{equation-descent-etale}) is faithful.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Descending étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTJ","source_file":"etale.tex","source_line":2044,"source_end_line":2048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2044-L2048","statement_sha256":"7e91fdc5e93a25ceeec0a4ee719e07928384e1be4fb3c1fb3b9eed2a9f7261b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8103,"rank":8103,"depth":19,"x":1559.617,"y":1169.215,"cluster":"tale-geometry"},{"id":"stacks:0BTK","tag":"0BTK","title":"Descending étale morphisms · Lemma 0BTK","summary":"Assume f : X → S is submersive and any étale base change of f is submersive. Then the functor ([Tag 0BTI]) is fully faithful.","statement_latex":"Assume $f : X \\to S$ is submersive and any \\'etale base change\nof $f$ is submersive. Then the functor\n(\\ref{equation-descent-etale}) is fully faithful.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Descending étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTK","source_file":"etale.tex","source_line":2060,"source_end_line":2065,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2060-L2065","statement_sha256":"aa120f1cd7716f3ac37b4d18568db5d0906ec3d3760a6a3d754f5f2228244731","origin":"The Stacks Project","memory_eligible":false,"source_rank":8104,"rank":8104,"depth":44,"x":1571.45,"y":1106.553,"cluster":"tale-geometry"},{"id":"stacks:0BTL","tag":"0BTL","title":"Descending étale morphisms · Lemma 0BTL","summary":"Let f : X → S be a morphism of schemes. In the following cases the functor ([Tag 0BTI]) is fully faithful: • f is surjective and universally closed (e.g., finite, integral, or proper), • f is surjective and universally open (e.g., locally of finite presentation and flat, smooth, or etale), • f is surjective, quasi-compact, and flat.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. In the following\ncases the functor (\\ref{equation-descent-etale}) is fully faithful:\n\\begin{enumerate}\n\\item $f$ is surjective and universally closed\n(e.g., finite, integral, or proper),\n\\item $f$ is surjective and universally open\n(e.g., locally of finite presentation and flat, smooth, or etale),\n\\item $f$ is surjective, quasi-compact, and flat.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Descending étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTL","source_file":"etale.tex","source_line":2123,"source_end_line":2134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2123-L2134","statement_sha256":"0e1b944a789afb1b305e158e6dd52089b5990a79e5007f9db7410bd11cdc1731","origin":"The Stacks Project","memory_eligible":false,"source_rank":8105,"rank":8105,"depth":45,"x":1613.535,"y":1160.01,"cluster":"tale-geometry"},{"id":"stacks:0BTM","tag":"0BTM","title":"Descending étale morphisms · Lemma 0BTM","summary":"Let f : X → S be a morphism of schemes. Let (V, φ) be a descent datum relative to X/S with V → X étale. Let S = ⋃ S_i be an open covering. Assume that • the pullback of the descent datum (V, φ) to X ×_S S_i/S_i is effective, • the functor ([Tag 0BTI]) for X ×_S (S_i ∩ S_j) → (S_i ∩ S_j) is fully faithful, and • the functor ([Tag 0BTI]) for X ×_S (S_i ∩ S_j ∩ S_k) → (S_i ∩ S_j ∩ S_k) is faithful. Then (V, φ) is effective.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $(V, \\varphi)$ be a descent datum relative to $X/S$\nwith $V \\to X$ \\'etale. Let $S = \\bigcup S_i$ be an\nopen covering. Assume that\n\\begin{enumerate}\n\\item the pullback of the descent datum $(V, \\varphi)$\nto $X \\times_S S_i/S_i$ is effective,\n\\item the functor (\\ref{equation-descent-etale})\nfor $X \\times_S (S_i \\cap S_j) \\to (S_i \\cap S_j)$ is fully faithful, and\n\\item the functor (\\ref{equation-descent-etale})\nfor $X \\times_S (S_i \\cap S_j \\cap S_k) \\to (S_i \\cap S_j \\cap S_k)$\nis faithful.\n\\end{enumerate}\nThen $(V, \\varphi)$ is effective.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Descending étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTM","source_file":"etale.tex","source_line":2157,"source_end_line":2173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2157-L2173","statement_sha256":"bcd975c521bce222064ed0b56335f9d39b8c7415ee5eccae4094a637a2243922","origin":"The Stacks Project","memory_eligible":false,"source_rank":8106,"rank":8106,"depth":44,"x":1538.779,"y":1144.315,"cluster":"tale-geometry"},{"id":"stacks:0BTN","tag":"0BTN","title":"Descending étale morphisms · Lemma 0BTN","summary":"Let (A, I) be a henselian pair. Let U → Spec(A) be a quasi-compact, separated, étale morphism such that U ×_Spec(A) Spec(A/I) → Spec(A/I) is finite. Then U = U_fin amalg U_away where U_fin → Spec(A) is finite and U_away has no points lying over Z.","statement_latex":"Let $(A, I)$ be a henselian pair. Let $U \\to \\Spec(A)$ be a\nquasi-compact, separated, \\'etale morphism such that\n$U \\times_{\\Spec(A)} \\Spec(A/I) \\to \\Spec(A/I)$ is finite.\nThen\n$$\nU = U_{fin} \\amalg U_{away}\n$$\nwhere $U_{fin} \\to \\Spec(A)$ is finite and $U_{away}$ has\nno points lying over $Z$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Descending étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTN","source_file":"etale.tex","source_line":2199,"source_end_line":2210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2199-L2210","statement_sha256":"6dcb6b38f0c13b8ab6dbd68fc2932943d124d1eb3d0445e30d48631741163435","origin":"The Stacks Project","memory_eligible":false,"source_rank":8107,"rank":8107,"depth":50,"x":1607.185,"y":1113.172,"cluster":"tale-geometry"},{"id":"stacks:0BTP","tag":"0BTP","title":"Descending étale morphisms · Proposition 0BTP","summary":"Let f : X → S be a surjective integral morphism. The functor ([Tag 0BTI]) induces an equivalence schemes quasi-compact, separated, étale over S → descent data (V, φ) relative to X/S with V quasi-compact, separated, étale over X","statement_latex":"Let $f : X \\to S$ be a surjective integral morphism.\nThe functor (\\ref{equation-descent-etale}) induces an equivalence\n$$\n\\begin{matrix}\n\\text{schemes quasi-compact,}\\\\\n\\text{separated, \\'etale over }S\n\\end{matrix}\n\\longrightarrow\n\\begin{matrix}\n\\text{descent data }(V, \\varphi)\\text{ relative to }X/S\\text{ with}\\\\\nV\\text{ quasi-compact, separated, \\'etale over }X\n\\end{matrix}\n$$","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Descending étale morphisms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTP","source_file":"etale.tex","source_line":2229,"source_end_line":2244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2229-L2244","statement_sha256":"d3512a0c8e77fa88b34b58ef957245f79bbc5351be21479c8f5d00093046142a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8108,"rank":8108,"depth":51,"x":1581.543,"y":1175.541,"cluster":"tale-geometry"},{"id":"stacks:0BI9","tag":"0BI9","title":"Normal crossings divisors · Definition 0BI9","summary":"Let X be a locally Noetherian scheme. A strict normal crossings divisor on X is an effective Cartier divisor D ⊂ X such that for every p ∈ D the local ring O_X, p is regular and there exists a regular system of parameters x_1, …, x_d ∈ m_p and 1 ≤ r ≤ d such that D is cut out by x_1 … x_r in O_X, p.","statement_latex":"Let $X$ be a locally Noetherian scheme. A\n{\\it strict normal crossings divisor}\non $X$ is an effective Cartier divisor $D \\subset X$ such that\nfor every $p \\in D$ the local ring $\\mathcal{O}_{X, p}$ is regular\nand there exists a regular system of parameters\n$x_1, \\ldots, x_d \\in \\mathfrak m_p$ and $1 \\leq r \\leq d$\nsuch that $D$ is cut out by $x_1 \\ldots x_r$ in $\\mathcal{O}_{X, p}$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Normal crossings divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BI9","source_file":"etale.tex","source_line":2395,"source_end_line":2404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2395-L2404","statement_sha256":"ccfc869ff18e528d87a7d715b765fdb94a0d2582595ac8347ba0c0117fb6c316","origin":"The Stacks Project","memory_eligible":false,"source_rank":8109,"rank":8109,"depth":0,"x":1550.005,"y":1114.433,"cluster":"tale-geometry"},{"id":"stacks:0BIA","tag":"0BIA","title":"Normal crossings divisors · Lemma 0BIA","summary":"Let X be a locally Noetherian scheme. Let D ⊂ X be an effective Cartier divisor. Let D_i ⊂ D, i ∈ I be its irreducible components viewed as reduced closed subschemes of X. The following are equivalent • D is a strict normal crossings divisor, and • D is reduced, each D_i is an effective Cartier divisor, and for J ⊂ I finite the scheme theoretic intersection D_J = ⋂_j ∈ J D_j is a regular scheme each of whose irreducible components has codimension |J| in X.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be an\neffective Cartier divisor. Let $D_i \\subset D$, $i \\in I$ be its\nirreducible components viewed as reduced closed subschemes of $X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $D$ is a strict normal crossings divisor, and\n\\item $D$ is reduced, each $D_i$ is an effective Cartier divisor, and\nfor $J \\subset I$ finite the scheme theoretic\nintersection $D_J = \\bigcap_{j \\in J} D_j$ is a\nregular scheme each of whose irreducible components has\ncodimension $|J|$ in $X$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Normal crossings divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIA","source_file":"etale.tex","source_line":2418,"source_end_line":2432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2418-L2432","statement_sha256":"e831cdd369b6869239fcc7e79f6a6cd56a66134e1170982e6a2973c7fd643a51","origin":"The Stacks Project","memory_eligible":false,"source_rank":8110,"rank":8110,"depth":19,"x":1623.067,"y":1141.85,"cluster":"tale-geometry"},{"id":"stacks:0CBP","tag":"0CBP","title":"Normal crossings divisors · Lemma 0CBP","summary":"Pullback of a strict normal crossings divisor by a smooth morphism is a strict normal crossings divisor. Let X be a locally Noetherian scheme. Let D ⊂ X be a strict normal crossings divisor. If f : Y → X is a smooth morphism of schemes, then the pullback f^*D is a strict normal crossings divisor on Y.","statement_latex":"\\begin{slogan}\nPullback of a strict normal crossings divisor by a smooth\nmorphism is a strict normal crossings divisor.\n\\end{slogan}\nLet $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a\nstrict normal crossings divisor. If $f : Y \\to X$ is a smooth\nmorphism of schemes, then the pullback $f^*D$ is a\nstrict normal crossings divisor on $Y$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Normal crossings divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBP","source_file":"etale.tex","source_line":2477,"source_end_line":2487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2477-L2487","statement_sha256":"055ab10118f440794fe53f8398db8607d8476f84bbdcdd58e12073c7e7c234ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":8111,"rank":8111,"depth":37,"x":1546.458,"y":1163.28,"cluster":"tale-geometry"},{"id":"stacks:0BSF","tag":"0BSF","title":"Normal crossings divisors · Definition 0BSF","summary":"Let X be a locally Noetherian scheme. A normal crossings divisor on X is an effective Cartier divisor D ⊂ X such that for every p ∈ D there exists an étale morphism U → X with p in the image and D ×_X U a strict normal crossings divisor on U.","statement_latex":"Let $X$ be a locally Noetherian scheme. A {\\it normal crossings divisor}\non $X$ is an effective Cartier divisor $D \\subset X$ such that for\nevery $p \\in D$ there exists an \\'etale morphism $U \\to X$ with\n$p$ in the image and $D \\times_X U$ a\nstrict normal crossings divisor on $U$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Normal crossings divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSF","source_file":"etale.tex","source_line":2521,"source_end_line":2528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2521-L2528","statement_sha256":"8f8e4c9dff8a726971fa1bdc1e953feaeecc0982d725fe88e2284a05987b46fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8112,"rank":8112,"depth":0,"x":1586.065,"y":1103.483,"cluster":"tale-geometry"},{"id":"stacks:0CBQ","tag":"0CBQ","title":"Normal crossings divisors · Lemma 0CBQ","summary":"Pullback of a normal crossings divisor by a smooth morphism is a normal crossings divisor. Let X be a locally Noetherian scheme. Let D ⊂ X be a normal crossings divisor. If f : Y → X is a smooth morphism of schemes, then the pullback f^*D is a normal crossings divisor on Y.","statement_latex":"\\begin{slogan}\nPullback of a normal crossings divisor by a smooth\nmorphism is a normal crossings divisor.\n\\end{slogan}\nLet $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a\nnormal crossings divisor. If $f : Y \\to X$ is a smooth\nmorphism of schemes, then the pullback $f^*D$ is a\nnormal crossings divisor on $Y$.","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Normal crossings divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBQ","source_file":"etale.tex","source_line":2536,"source_end_line":2546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2536-L2546","statement_sha256":"2585ebe6684dfc97f17e08694da6d416e5ae0c0ed717d3989b88eb672b00a775","origin":"The Stacks Project","memory_eligible":false,"source_rank":8113,"rank":8113,"depth":40,"x":1605.108,"y":1170.631,"cluster":"tale-geometry"},{"id":"stacks:0CBR","tag":"0CBR","title":"Normal crossings divisors · Lemma 0CBR","summary":"Let X be a locally Noetherian scheme. Let D ⊂ X be a closed subscheme. The following are equivalent • D is a normal crossings divisor in X, • D is reduced, the normalization ν : D^ν → D is unramified, and for any n ≥ 1 the scheme Z_n = D^ν ×_D … ×_D D^ν setminus ((p_1, …, p_n) mid p_i = p_j for some inot = j) is regular, the morphism Z_n → X is a local complete intersection morphism whose conormal sheaf is locally free of rank n.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a closed\nsubscheme. The following are equivalent\n\\begin{enumerate}\n\\item $D$ is a normal crossings divisor in $X$,\n\\item $D$ is reduced, the normalization $\\nu : D^\\nu \\to D$ is unramified,\nand for any $n \\geq 1$ the scheme\n$$\nZ_n = D^\\nu \\times_D \\ldots \\times_D D^\\nu\n\\setminus \\{(p_1, \\ldots, p_n) \\mid p_i = p_j\\text{ for some }i\\not = j\\}\n$$\nis regular, the morphism $Z_n \\to X$ is a local complete intersection\nmorphism whose conormal sheaf is locally free of rank $n$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Normal crossings divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBR","source_file":"etale.tex","source_line":2566,"source_end_line":2581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2566-L2581","statement_sha256":"87ad965581133ef70a2c5ea7c09bdbeed96e31648ada6236999da4c3aff1d3a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8114,"rank":8114,"depth":42,"x":1536.487,"y":1131.59,"cluster":"tale-geometry"},{"id":"stacks:0CBS","tag":"0CBS","title":"Normal crossings divisors · Lemma 0CBS","summary":"Let X be a locally Noetherian scheme. Let D ⊂ X be a closed subscheme. If X is J-2 or Nagata, then the following are equivalent • D is a normal crossings divisor in X, • for every p ∈ D the pullback of D to the spectrum of the strict henselization O_X, p^sh is a strict normal crossings divisor.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $D \\subset X$ be a closed\nsubscheme. If $X$ is J-2 or Nagata, then the following are equivalent\n\\begin{enumerate}\n\\item $D$ is a normal crossings divisor in $X$,\n\\item for every $p \\in D$ the pullback of $D$ to the spectrum of the\nstrict henselization $\\mathcal{O}_{X, p}^{sh}$\nis a strict normal crossings divisor.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Morphisms of Schemes","chapter_id":"etale","section":"Normal crossings divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBS","source_file":"etale.tex","source_line":2678,"source_end_line":2688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale.tex#L2678-L2688","statement_sha256":"34305ce7d246927c71c8df6731fc29adfcc47ea5f24ff7318b623ed8d7868571","origin":"The Stacks Project","memory_eligible":false,"source_rank":8115,"rank":8115,"depth":53,"x":1619.174,"y":1121.357,"cluster":"tale-geometry"},{"id":"stacks:02PG","tag":"02PG","title":"Periodic complexes and Herbrand quotients · Definition 02PG","summary":"Let R be a ring. • A 2-periodic complex over R is given by a quadruple (M, N, φ, ψ) consisting of R-modules M, N and R-module maps φ : M → N, ψ : N → M such that xymatrix … ar[r] & M ar[r]^φ & N ar[r]^ψ & M ar[r]^φ & N ar[r] & … is a complex. In this setting we define the cohomology modules of the complex to be the R-modules H^0(M, N, φ, ψ) = Ker(φ)/Im(ψ) and H^1(M, N, φ, ψ) = Ker(ψ)/Im(φ). We say the 2-periodic complex is exact if the cohomology groups are zero. • A (2,…","statement_latex":"Let $R$ be a ring.\n\\begin{enumerate}\n\\item A {\\it $2$-periodic complex} over $R$ is given\nby a quadruple $(M, N, \\varphi, \\psi)$ consisting of\n$R$-modules $M$, $N$ and $R$-module maps $\\varphi : M \\to N$,\n$\\psi : N \\to M$ such that\n$$\n\\xymatrix{\n\\ldots \\ar[r] &\nM \\ar[r]^\\varphi &\nN \\ar[r]^\\psi &\nM \\ar[r]^\\varphi &\nN \\ar[r] & \\ldots\n}\n$$\nis a complex. In this setting we define the {\\it cohomology modules}\nof the complex to be the $R$-modules\n$$\nH^0(M, N, \\varphi, \\psi) = \\Ker(\\varphi)/\\Im(\\psi)\n\\quad\\text{and}\\quad\nH^1(M, N, \\varphi, \\psi) = \\Ker(\\psi)/\\Im(\\varphi).\n$$\nWe say the $2$-periodic complex is {\\it exact} if the cohomology\ngroups are zero.\n\\item A {\\it $(2, 1)$-periodic complex} over $R$ is given\nby a triple $(M, \\varphi, \\psi)$ consisting of an $R$-module $M$ and\n$R$-module maps $\\varphi : M \\to M$, $\\psi : M \\to M$\nsuch that\n$$\n\\xymatrix{\n\\ldots \\ar[r] &\nM \\ar[r]^\\varphi &\nM \\ar[r]^\\psi &\nM \\ar[r]^\\varphi &\nM \\ar[r] & \\ldots\n}\n$$\nis a complex. Since this is a special case of a $2$-periodic complex\nwe have its {\\it cohomology modules} $H^0(M, \\varphi, \\psi)$,\n$H^1(M, \\varphi, \\psi)$ and a notion of exactness.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Periodic complexes and Herbrand quotients","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PG","source_file":"chow.tex","source_line":109,"source_end_line":152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L109-L152","statement_sha256":"214c35fad04f129692c4e9eafc146d386306367da976ff40ca75ff19036f5745","origin":"The Stacks Project","memory_eligible":false,"source_rank":8116,"rank":8116,"depth":0,"x":2392.516,"y":624.493,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PH","tag":"02PH","title":"Periodic complexes and Herbrand quotients · Definition 02PH","summary":"Let (M, N, φ, ψ) be a 2-periodic complex over a ring R whose cohomology modules have finite length. In this case we define the multiplicity of (M, N, φ, ψ) to be the integer e_R(M, N, φ, ψ) = length_R(H^0(M, N, φ, ψ)) - length_R(H^1(M, N, φ, ψ)) In the case of a (2, 1)-periodic complex (M, φ, ψ), we denote this by e_R(M, φ, ψ) and we will sometimes call this the (additive) Herbrand quotient.","statement_latex":"Let $(M, N, \\varphi, \\psi)$ be a $2$-periodic complex\nover a ring $R$ whose cohomology modules have finite length.\nIn this case we define the {\\it multiplicity} of $(M, N, \\varphi, \\psi)$\nto be the integer\n$$\ne_R(M, N, \\varphi, \\psi) =\n\\text{length}_R(H^0(M, N, \\varphi, \\psi))\n-\n\\text{length}_R(H^1(M, N, \\varphi, \\psi))\n$$\nIn the case of a $(2, 1)$-periodic complex $(M, \\varphi, \\psi)$,\nwe denote this by $e_R(M, \\varphi, \\psi)$ and we will sometimes call this\nthe {\\it (additive) Herbrand quotient}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Periodic complexes and Herbrand quotients","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PH","source_file":"chow.tex","source_line":165,"source_end_line":180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L165-L180","statement_sha256":"1e94e4358fc2f8eaf7dc3bd818e31491c345289ba16716b53974355259a62405","origin":"The Stacks Project","memory_eligible":false,"source_rank":8117,"rank":8117,"depth":0,"x":2589.349,"y":671.274,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EA7","tag":"0EA7","title":"Periodic complexes and Herbrand quotients · Lemma 0EA7","summary":"Let R be a ring. Suppose that we have a short exact sequence of 2-periodic complexes 0 → (M_1, N_1, φ_1, ψ_1) → (M_2, N_2, φ_2, ψ_2) → (M_3, N_3, φ_3, ψ_3) → 0 If two out of three have cohomology modules of finite length so does the third and we have e_R(M_2, N_2, φ_2, ψ_2) = e_R(M_1, N_1, φ_1, ψ_1) + e_R(M_3, N_3, φ_3, ψ_3).","statement_latex":"Let $R$ be a ring. Suppose that we have a short exact sequence of\n$2$-periodic complexes\n$$\n0 \\to (M_1, N_1, \\varphi_1, \\psi_1)\n\\to (M_2, N_2, \\varphi_2, \\psi_2)\n\\to (M_3, N_3, \\varphi_3, \\psi_3)\n\\to 0\n$$\nIf two out of three have cohomology modules of finite length so does\nthe third and we have\n$$\ne_R(M_2, N_2, \\varphi_2, \\psi_2) =\ne_R(M_1, N_1, \\varphi_1, \\psi_1) +\ne_R(M_3, N_3, \\varphi_3, \\psi_3).\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Periodic complexes and Herbrand quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EA7","source_file":"chow.tex","source_line":208,"source_end_line":225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L208-L225","statement_sha256":"acb9afba109141ec96cefd5580d7e31fb365e5baa700828cc70470b14ddab575","origin":"The Stacks Project","memory_eligible":false,"source_rank":8118,"rank":8118,"depth":1,"x":2406.249,"y":748.608,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EA8","tag":"0EA8","title":"Periodic complexes and Herbrand quotients · Lemma 0EA8","summary":"Let R be a ring. If (M, N, φ, ψ) is a 2-periodic complex such that M, N have finite length, then e_R(M, N, φ, ψ) = length_R(M) - length_R(N). In particular, if (M, φ, ψ) is a (2, 1)-periodic complex such that M has finite length, then e_R(M, φ, ψ) = 0.","statement_latex":"Let $R$ be a ring. If $(M, N, \\varphi, \\psi)$ is a $2$-periodic complex\nsuch that $M$, $N$ have finite length, then\n$e_R(M, N, \\varphi, \\psi) = \\text{length}_R(M) - \\text{length}_R(N)$.\nIn particular, if $(M, \\varphi, \\psi)$ is a $(2, 1)$-periodic complex\nsuch that $M$ has finite length, then\n$e_R(M, \\varphi, \\psi) = 0$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Periodic complexes and Herbrand quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EA8","source_file":"chow.tex","source_line":257,"source_end_line":265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L257-L265","statement_sha256":"3250842e7cafa6f2e3f242b5de9db6a9d54d6fc5d570d4ca7cd63cd4eca6d5f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8119,"rank":8119,"depth":2,"x":2479.208,"y":587.392,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EA9","tag":"0EA9","title":"Periodic complexes and Herbrand quotients · Lemma 0EA9","summary":"Let R be a ring. Let f : (M, φ, ψ) → (M', φ', ψ') be a map of (2, 1)-periodic complexes whose cohomology modules have finite length. If Ker(f) and Coker(f) have finite length, then e_R(M, φ, ψ) = e_R(M', φ', ψ').","statement_latex":"Let $R$ be a ring. Let $f : (M, \\varphi, \\psi) \\to (M', \\varphi', \\psi')$\nbe a map of $(2, 1)$-periodic complexes whose cohomology modules\nhave finite length. If $\\Ker(f)$ and $\\Coker(f)$ have finite length,\nthen $e_R(M, \\varphi, \\psi) = e_R(M', \\varphi', \\psi')$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Periodic complexes and Herbrand quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EA9","source_file":"chow.tex","source_line":282,"source_end_line":288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L282-L288","statement_sha256":"a7ff9e518eff5d726bbc8b24e119de95cae8dd6de1badd5e587b466fbcbdc898","origin":"The Stacks Project","memory_eligible":false,"source_rank":8120,"rank":8120,"depth":3,"x":2555.195,"y":747.963,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QF","tag":"02QF","title":"Calculation of some multiplicities · Lemma 02QF","summary":"Let R be a Noetherian local ring. Let M be a finite R-module. Let x ∈ R. Assume that • dim(Supp(M)) ≤ 1, and • dim(Supp(M/xM)) ≤ 0. Write Supp(M) = ( m, q_1, …, q_t). Then e_R(M, 0, x) = ∑_i = 1, …, t ord_R/ q_i(x) length_R_ q_i(M_ q_i).","statement_latex":"Let $R$ be a Noetherian local ring.\nLet $M$ be a finite $R$-module. Let $x \\in R$. Assume that\n\\begin{enumerate}\n\\item $\\dim(\\text{Supp}(M)) \\leq 1$, and\n\\item $\\dim(\\text{Supp}(M/xM)) \\leq 0$.\n\\end{enumerate}\nWrite\n$\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$.\nThen\n$$\ne_R(M, 0, x) =\n\\sum\\nolimits_{i = 1, \\ldots, t}\n\\text{ord}_{R/\\mathfrak q_i}(x)\n\\text{length}_{R_{\\mathfrak q_i}}(M_{\\mathfrak q_i}).\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Calculation of some multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QF","source_file":"chow.tex","source_line":309,"source_end_line":326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L309-L326","statement_sha256":"d31037ac40f6a217401aacbefae3dcbc80e8b7496de369c43e01dac249ba8bff","origin":"The Stacks Project","memory_eligible":false,"source_rank":8121,"rank":8121,"depth":13,"x":2369.698,"y":672.539,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QG","tag":"02QG","title":"Calculation of some multiplicities · Lemma 02QG","summary":"Let R be a Noetherian local ring. Let x ∈ R. If M is a finite Cohen-Macaulay module over R with dim(Supp(M)) = 1 and dim(Supp(M/xM)) = 0, then length_R(M/xM) = ∑_i length_R(R/(x, q_i)) length_R_ q_i(M_ q_i). where q_1, …, q_t are the minimal primes of the support of M. If I ⊂ R is an ideal such that x is a nonzerodivisor on R/I and dim(R/I) = 1, then length_R(R/(x, I)) = ∑_i length_R(R/(x, q_i)) length_R_ q_i((R/I)_ q_i) where q_1, …, q_n are the minimal primes over I.","statement_latex":"Let $R$ be a Noetherian local ring.\nLet $x \\in R$. If $M$ is a finite Cohen-Macaulay module over $R$\nwith $\\dim(\\text{Supp}(M)) = 1$ and $\\dim(\\text{Supp}(M/xM)) = 0$, then\n$$\n\\text{length}_R(M/xM)\n=\n\\sum\\nolimits_i \\text{length}_R(R/(x, \\mathfrak q_i))\n\\text{length}_{R_{\\mathfrak q_i}}(M_{\\mathfrak q_i}).\n$$\nwhere $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ are the\nminimal primes of the support of $M$. If $I \\subset R$ is an ideal\nsuch that $x$ is a nonzerodivisor on $R/I$ and $\\dim(R/I) = 1$, then\n$$\n\\text{length}_R(R/(x, I))\n=\n\\sum\\nolimits_i \\text{length}_R(R/(x, \\mathfrak q_i))\n\\text{length}_{R_{\\mathfrak q_i}}((R/I)_{\\mathfrak q_i})\n$$\nwhere $\\mathfrak q_1, \\ldots, \\mathfrak q_n$ are the minimal\nprimes over $I$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Calculation of some multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QG","source_file":"chow.tex","source_line":372,"source_end_line":394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L372-L394","statement_sha256":"9c2761d5031085ee983fcec86a6bb3f6c96487ce89520595b1fb62cf3e97d01b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8122,"rank":8122,"depth":14,"x":2567.494,"y":622.81,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAB","tag":"0EAB","title":"Calculation of some multiplicities · Lemma 0EAB","summary":"Let R be a ring. Let M be an R-module. Let φ : M → M be an endomorphism and n > 0 such that φ^n = 0 and such that Ker(φ)/Im(φ^n - 1) has finite length as an R-module. Then e_R(M, φ^i, φ^n - i) = 0 for i = 0, …, n.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module.\nLet $\\varphi : M \\to M$ be an endomorphism and $n > 0$\nsuch that $\\varphi^n = 0$ and such that $\\Ker(\\varphi)/\\Im(\\varphi^{n - 1})$\nhas finite length as an $R$-module.\nThen\n$$\ne_R(M, \\varphi^i, \\varphi^{n - i}) = 0\n$$\nfor $i = 0, \\ldots, n$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Calculation of some multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAB","source_file":"chow.tex","source_line":403,"source_end_line":414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L403-L414","statement_sha256":"1f9960c19f2ba65da04b3238a04c21f04013d6b344e4efe374647755a21a9404","origin":"The Stacks Project","memory_eligible":false,"source_rank":8123,"rank":8123,"depth":0,"x":2461.44,"y":771.984,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAC","tag":"0EAC","title":"Calculation of some multiplicities · Lemma 0EAC","summary":"Let (R, m) be a Noetherian local ring. Let (M, φ, ψ) be a (2, 1)-periodic complex over R with M finite and with cohomology groups of finite length over R. Let x ∈ R be such that dim(Supp(M/xM)) ≤ 0. Then e_R(M, xφ, ψ) = e_R(M, φ, ψ) - e_R(Im(φ), 0, x) and e_R(M, φ, xψ) = e_R(M, φ, ψ) + e_R(Im(ψ), 0, x)","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring. Let\n$(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$\nwith $M$ finite and with cohomology groups of finite length over $R$.\nLet $x \\in R$ be such that $\\dim(\\text{Supp}(M/xM)) \\leq 0$. Then\n$$\ne_R(M, x\\varphi, \\psi) = e_R(M, \\varphi, \\psi) - e_R(\\Im(\\varphi), 0, x)\n$$\nand\n$$\ne_R(M, \\varphi, x\\psi) = e_R(M, \\varphi, \\psi) + e_R(\\Im(\\psi), 0, x)\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Calculation of some multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAC","source_file":"chow.tex","source_line":463,"source_end_line":476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L463-L476","statement_sha256":"44e2a082a741f4a5f1fea1830edcb59aea5af92c373d7dd2fc9fba61e2c9045d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8124,"rank":8124,"depth":4,"x":2419.606,"y":601.5,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAE","tag":"0EAE","title":"Preparation for tame symbols · Lemma 0EAE","summary":"Let A be a Noetherian ring. Let m_1, …, m_r be pairwise distinct maximal ideals of A. For i = 1, …, r let φ_i : A_ m_i → B_i be a ring map whose kernel and cokernel are annihilated by a power of m_i. Then there exists a ring map φ : A → B such that • the localization of φ at m_i is isomorphic to φ_i, and • Ker(φ) and Coker(φ) are annihilated by a power of m_1 ∩ … ∩ m_r. Moreover, if each φ_i is finite, injective, or surjective then so is φ.","statement_latex":"Let $A$ be a Noetherian ring. Let $\\mathfrak m_1, \\ldots, \\mathfrak m_r$\nbe pairwise distinct maximal ideals of $A$. For $i = 1, \\ldots, r$ let\n$\\varphi_i : A_{\\mathfrak m_i} \\to B_i$ be a ring map whose\nkernel and cokernel are annihilated by a power\nof $\\mathfrak m_i$. Then there exists a ring map $\\varphi : A \\to B$ such\nthat\n\\begin{enumerate}\n\\item the localization of $\\varphi$ at $\\mathfrak m_i$ is\nisomorphic to $\\varphi_i$, and\n\\item $\\Ker(\\varphi)$ and $\\Coker(\\varphi)$ are annihilated\nby a power of $\\mathfrak m_1 \\cap \\ldots \\cap \\mathfrak m_r$.\n\\end{enumerate}\nMoreover, if each $\\varphi_i$ is finite, injective, or\nsurjective then so is $\\varphi$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for tame symbols","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAE","source_file":"chow.tex","source_line":525,"source_end_line":541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L525-L541","statement_sha256":"b062301b83e702b7290ce39986d05ce0e19b2148424a657d309c6e8ad9d8a481","origin":"The Stacks Project","memory_eligible":false,"source_rank":8125,"rank":8125,"depth":21,"x":2587.855,"y":703.658,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q7","tag":"02Q7","title":"Preparation for tame symbols · Lemma 02Q7","summary":"Let (R, m) be a Noetherian local ring of dimension 1. Let a, b ∈ R be nonzerodivisors. There exists a finite ring extension R ⊂ R' with R'/R annihilated by a power of m and nonzerodivisors t, a', b' ∈ R' such that a = ta' and b = tb' and R' = a'R' + b'R'.","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.\nLet $a, b \\in R$ be nonzerodivisors.\nThere exists a finite ring extension $R \\subset R'$\nwith $R'/R$ annihilated by a power of $\\mathfrak m$\nand nonzerodivisors $t, a', b' \\in R'$ such that\n$a = ta'$ and $b = tb'$ and $R' = a'R' + b'R'$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for tame symbols","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q7","source_file":"chow.tex","source_line":565,"source_end_line":573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L565-L573","statement_sha256":"d16eafc5332e019493b4c5889fee59146f16f0298ce92c3ccee4640bf24f4593","origin":"The Stacks Project","memory_eligible":false,"source_rank":8126,"rank":8126,"depth":42,"x":2381.268,"y":723.833,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAF","tag":"0EAF","title":"Preparation for tame symbols · Lemma 0EAF","summary":"Let (R, m) be a Noetherian local ring of dimension 1. Let a, b ∈ R be nonzerodivisors with a ∈ m. There exists an integer n = n(R, a, b) such that for a finite ring extension R ⊂ R' if b = a^m c for some c ∈ R', then m ≤ n.","statement_latex":"Let $(R, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.\nLet $a, b \\in R$ be nonzerodivisors with $a \\in \\mathfrak m$.\nThere exists an integer $n = n(R, a, b)$ such that for a finite ring\nextension $R \\subset R'$ if $b = a^m c$ for some $c \\in R'$, then $m \\leq n$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for tame symbols","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAF","source_file":"chow.tex","source_line":604,"source_end_line":610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L604-L610","statement_sha256":"dc3d0124cb395236df1c36256c6f98fc5b3c5cd8576cdbd9162f4d998ccff0b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8127,"rank":8127,"depth":11,"x":2517.622,"y":591.498,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAG","tag":"0EAG","title":"Preparation for tame symbols · Lemma 0EAG","summary":"Let (A, m) be a Noetherian local ring of dimension 1. Let r ≥ 2 and let a_1, …, a_r ∈ A be nonzerodivisors not all units. Then there exist • a finite ring extension A ⊂ B with B/A annihilated by a power of m, • for each maximal ideal m_j ⊂ B a nonzerodivisor π_j ∈ B_j = B_ m_j, and • factorizations a_i = u_i, j π_j^e_i, j in B_j with u_i, j ∈ B_j units and e_i, j ≥ 0.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.\nLet $r \\geq 2$ and let $a_1, \\ldots, a_r \\in A$ be nonzerodivisors\nnot all units.\nThen there exist\n\\begin{enumerate}\n\\item a finite ring extension $A \\subset B$ with\n$B/A$ annihilated by a power of $\\mathfrak m$,\n\\item for each maximal ideal $\\mathfrak m_j \\subset B$\na nonzerodivisor $\\pi_j \\in B_j = B_{\\mathfrak m_j}$, and\n\\item factorizations $a_i = u_{i, j} \\pi_j^{e_{i, j}}$ in $B_j$\nwith $u_{i, j} \\in B_j$ units and $e_{i, j} \\geq 0$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for tame symbols","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAG","source_file":"chow.tex","source_line":641,"source_end_line":655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L641-L655","statement_sha256":"33d582a1e3115fb0d4afcbe291432e872f32fd7bfeb8c0a0ba2611b9480872ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":8128,"rank":8128,"depth":43,"x":2523.507,"y":766.762,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAR","tag":"0EAR","title":"Tame symbols · Lemma 0EAR","summary":"The formula ([Tag 0EAQ]) determines a well defined element of kappa( m)^*. In other words, the right hand side does not depend on the choice of the local factorizations or the choice of B.","statement_latex":"The formula (\\ref{equation-tame-symbol}) determines a\nwell defined element of $\\kappa(\\mathfrak m)^*$. In other words, the\nright hand side does not depend on the choice of the\nlocal factorizations or the choice of $B$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Tame symbols","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAR","source_file":"chow.tex","source_line":829,"source_end_line":835,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L829-L835","statement_sha256":"545fd9a1b4a809b1c21ea27cb1c3c4b7b5d8f50fd1d4f88bab5f117a2337f005","origin":"The Stacks Project","memory_eligible":false,"source_rank":8129,"rank":8129,"depth":44,"x":2377.967,"y":640.64,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAS","tag":"0EAS","title":"Tame symbols · Lemma 0EAS","summary":"The tame symbol ([Tag 0EAQ]) satisfies ([Tag 0EAL]), ([Tag 0EAM]), ([Tag 0EAN]), ([Tag 0EAP]) and hence gives a map ∂_A : Q(A)^* × Q(A)^* → kappa( m)^* satisfying ([Tag 0EAI]), ([Tag 0EAJ]), ([Tag 0EAK]).","statement_latex":"The tame symbol (\\ref{equation-tame-symbol}) satisfies\n(\\ref{item-bilinear-better}), (\\ref{item-skew-better}),\n(\\ref{item-normalization}), (\\ref{item-1-x-better}) and hence\ngives a map $\\partial_A : Q(A)^* \\times Q(A)^* \\to \\kappa(\\mathfrak m)^*$\nsatisfying (\\ref{item-bilinear}), (\\ref{item-skew}), (\\ref{item-1-x}).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Tame symbols","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAS","source_file":"chow.tex","source_line":901,"source_end_line":908,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L901-L908","statement_sha256":"657175f882896423bb330e8387ec0cbdb7eb91210ecbb11f8a969dc8ccc5d29e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8130,"rank":8130,"depth":44,"x":2587.078,"y":651.076,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAT","tag":"0EAT","title":"Tame symbols · Lemma 0EAT","summary":"Let (A, m) be a Noetherian local ring of dimension 1. Let A ⊂ B be a finite ring extension with B/A annihilated by a power of m and m not an associated prime of B. For a, b ∈ A nonzerodivisors we have ∂_A(a, b) = ∏ Norm_kappa( m_j)/kappa( m)(∂_B_j(a, b)) where the product is over the maximal ideals m_j of B and B_j = B_ m_j.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $1$.\nLet $A \\subset B$ be a finite ring extension with $B/A$\nannihilated by a power of $\\mathfrak m$ and $\\mathfrak m$ not\nan associated prime of $B$.\nFor $a, b \\in A$ nonzerodivisors we have\n$$\n\\partial_A(a, b) = \\prod\n\\text{Norm}_{\\kappa(\\mathfrak m_j)/\\kappa(\\mathfrak m)}(\\partial_{B_j}(a, b))\n$$\nwhere the product is over the maximal ideals $\\mathfrak m_j$ of $B$\nand $B_j = B_{\\mathfrak m_j}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Tame symbols","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAT","source_file":"chow.tex","source_line":951,"source_end_line":964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L951-L964","statement_sha256":"8993f74ca25fce284aaf277562fafe05368f5593488bd1a21d7368cab7b3b792","origin":"The Stacks Project","memory_eligible":false,"source_rank":8131,"rank":8131,"depth":44,"x":2424.204,"y":762.231,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EPG","tag":"0EPG","title":"Tame symbols · Lemma 0EPG","summary":"Let (A, m, kappa) → (A', m', kappa') be a local homomorphism of Noetherian local rings. Assume A → A' is flat and dim(A) = dim(A') = 1. Set m = length_A'(A'/ mA'). For a_1, a_2 ∈ A nonzerodivisors ∂_A(a_1, a_2)^m maps to ∂_A'(a_1, a_2) via kappa → kappa'.","statement_latex":"Let $(A, \\mathfrak m, \\kappa) \\to (A', \\mathfrak m', \\kappa')$\nbe a local homomorphism of Noetherian local rings. Assume $A \\to A'$\nis flat and $\\dim(A) = \\dim(A') = 1$. Set\n$m = \\text{length}_{A'}(A'/\\mathfrak mA')$.\nFor $a_1, a_2 \\in A$ nonzerodivisors\n$\\partial_A(a_1, a_2)^m$ maps to $\\partial_{A'}(a_1, a_2)$\nvia $\\kappa \\to \\kappa'$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Tame symbols","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPG","source_file":"chow.tex","source_line":999,"source_end_line":1008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L999-L1008","statement_sha256":"c4921819568b4dec503b5b9be77ba55a1adb64a3fc55ddfb545fbf0ab23566bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8132,"rank":8132,"depth":44,"x":2454.972,"y":587.542,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAV","tag":"0EAV","title":"A key lemma · Lemma 0EAV","summary":"Let (A, m) be a 2-dimensional Noetherian local ring. Let t ∈ m be a nonzerodivisor. Say V(t) = ( m, q_1, …, q_r). Let A_ q_i ⊂ B_i be a finite ring extension with B_i/A_ q_i annihilated by a power of t. Then there exists a finite extension A ⊂ B of local rings identifying residue fields with B_i ≅ B_ q_i and B/A annihilated by a power of t.","statement_latex":"Let $(A, \\mathfrak m)$ be a $2$-dimensional Noetherian local ring.\nLet $t \\in \\mathfrak m$ be a nonzerodivisor. Say\n$V(t) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_r\\}$.\nLet $A_{\\mathfrak q_i} \\subset B_i$ be a finite ring\nextension with $B_i/A_{\\mathfrak q_i}$ annihilated by a power of\n$t$. Then there exists a finite extension $A \\subset B$ of\nlocal rings identifying residue fields\nwith $B_i \\cong B_{\\mathfrak q_i}$ and $B/A$ annihilated\nby a power of $t$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAV","source_file":"chow.tex","source_line":1104,"source_end_line":1115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1104-L1115","statement_sha256":"106b659943735a94a425156a4a3207db41e91565d382cbfd5762d05783c8a5de","origin":"The Stacks Project","memory_eligible":false,"source_rank":8133,"rank":8133,"depth":0,"x":2572.97,"y":734.07,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAW","tag":"0EAW","title":"A key lemma · Lemma 0EAW","summary":"Let (A, m) be a 2-dimensional Noetherian local ring. Let a, b ∈ A be nonzerodivisors. Then we have ∑ ord_A/ q(∂_A_ q(a, b)) = 0 where the sum is over the height 1 primes q of A.","statement_latex":"Let $(A, \\mathfrak m)$ be a $2$-dimensional Noetherian local ring.\nLet $a, b \\in A$ be nonzerodivisors.\nThen we have\n$$\n\\sum\n\\text{ord}_{A/\\mathfrak q}(\\partial_{A_{\\mathfrak q}}(a, b))\n=\n0\n$$\nwhere the sum is over the height $1$ primes $\\mathfrak q$ of $A$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAW","source_file":"chow.tex","source_line":1132,"source_end_line":1144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1132-L1144","statement_sha256":"2e338a1660e2fa8a1ccee75be158d66e217880629f1abfe6fccc1c90d7fbc98c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8134,"rank":8134,"depth":45,"x":2367.769,"y":692.906,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EAX","tag":"0EAX","title":"Key Lemma · Lemma 0EAX","summary":"When A is an excellent ring this is [Kato-Milnor-K]. Let A be a 2-dimensional Noetherian local domain with fraction field K. Let f, g ∈ K^*. Let q_1, …, q_t be the height 1 primes q of A such that either f or g is not an element of A^*_ q. Then we have ∑_i = 1, …, t ord_A/ q_i(∂_A_ q_i(f, g)) = 0 We can also write this as ∑_height( q) = 1 ord_A/ q(∂_A_ q(f, g)) = 0 since at any height 1 prime q of A where f, g ∈ A^*_ q we have ∂_A_ q(f, g) = 1.","statement_latex":"\\begin{reference}\nWhen $A$ is an excellent ring this is \\cite[Proposition 1]{Kato-Milnor-K}.\n\\end{reference}\nLet $A$ be a $2$-dimensional Noetherian local domain with fraction field $K$.\nLet $f, g \\in K^*$.\nLet $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the height\n$1$ primes $\\mathfrak q$ of $A$ such that either $f$ or $g$ is not an\nelement of $A^*_{\\mathfrak q}$.\nThen we have\n$$\n\\sum\\nolimits_{i = 1, \\ldots, t}\n\\text{ord}_{A/\\mathfrak q_i}(\\partial_{A_{\\mathfrak q_i}}(f, g))\n=\n0\n$$\nWe can also write this as\n$$\n\\sum\\nolimits_{\\text{height}(\\mathfrak q) = 1}\n\\text{ord}_{A/\\mathfrak q}(\\partial_{A_{\\mathfrak q}}(f, g))\n=\n0\n$$\nsince at any height $1$ prime $\\mathfrak q$\nof $A$ where $f, g \\in A^*_{\\mathfrak q}$\nwe have $\\partial_{A_{\\mathfrak q}}(f, g) = 1$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EAX","source_file":"chow.tex","source_line":1262,"source_end_line":1289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1262-L1289","statement_sha256":"4ffb854ac6e6f368137ee5cb52b4c0aad1708d2be9bcb412707941d20124dfe9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8135,"rank":8135,"depth":46,"x":2552.505,"y":606.674,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QO","tag":"02QO","title":"Setup · Lemma 02QO","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Assume in addition S is a Jacobson scheme, and δ(s) = 0 for every closed point s of S. Let X be locally of finite type over S. Let Z ⊂ X be an integral closed subscheme and let xi ∈ Z be its generic point. The following integers are the same: • δ_X/S(xi), • dim(Z), and • dim(O_Z, z) where z is a closed point of Z.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nAssume in addition $S$ is a Jacobson scheme, and $\\delta(s) = 0$ for every\nclosed point $s$ of $S$. Let $X$ be locally of finite type over $S$.\nLet $Z \\subset X$ be an integral closed subscheme and let\n$\\xi \\in Z$ be its generic point. The following integers are the same:\n\\begin{enumerate}\n\\item $\\delta_{X/S}(\\xi)$,\n\\item $\\dim(Z)$, and\n\\item $\\dim(\\mathcal{O}_{Z, z})$ where $z$ is a closed point of $Z$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Setup","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QO","source_file":"chow.tex","source_line":1457,"source_end_line":1469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1457-L1469","statement_sha256":"bdf2502629c67a7fc69a59e22c1cd34f036903f0b5474c6219a2b4ae805c26f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8136,"rank":8136,"depth":12,"x":2485.512,"y":775.375,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QP","tag":"02QP","title":"Setup · Definition 02QP","summary":"Let (S, δ) as in Situation [Tag 02QL]. For any scheme X locally of finite type over S and any irreducible closed subset Z ⊂ X we define dim_δ(Z) = δ(xi) where xi ∈ Z is the generic point of Z. We will call this the δ-dimension of Z. If Z is a closed subscheme of X, then we define dim_δ(Z) as the supremum of the δ-dimensions of its irreducible components.","statement_latex":"Let $(S, \\delta)$ as in Situation \\ref{situation-setup}.\nFor any scheme $X$ locally of finite type over $S$\nand any irreducible closed subset $Z \\subset X$ we define\n$$\n\\dim_\\delta(Z) = \\delta(\\xi)\n$$\nwhere $\\xi \\in Z$ is the generic point of $Z$.\nWe will call this the {\\it $\\delta$-dimension of $Z$}.\nIf $Z$ is a closed subscheme of $X$, then we define\n$\\dim_\\delta(Z)$ as the supremum of the $\\delta$-dimensions\nof its irreducible components.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Setup","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QP","source_file":"chow.tex","source_line":1503,"source_end_line":1516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1503-L1516","statement_sha256":"c1278438299d9f90b4add58c22029767a23adb2e3dd900e6e1a4444665797c0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8137,"rank":8137,"depth":0,"x":2399.098,"y":612.684,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QR","tag":"02QR","title":"Cycles · Definition 02QR","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let k ∈ Z. • A cycle on X is a formal sum α = ∑ n_Z [Z] where the sum is over integral closed subschemes Z ⊂ X, each n_Z ∈ Z, and the collection (Z; n_Z not = 0) is locally finite (Topology, Definition [Tag 0BDS]). • A k-cycle on X is a cycle α = ∑ n_Z [Z] where n_Z not = 0 ⇒ dim_δ(Z) = k. • The abelian group of all k-cycles on X is denoted Z_k(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $k \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item A {\\it cycle on $X$} is a formal sum\n$$\n\\alpha = \\sum n_Z [Z]\n$$\nwhere the sum is over integral closed subschemes $Z \\subset X$,\neach $n_Z \\in \\mathbf{Z}$, and the collection\n$\\{Z; n_Z \\not = 0\\}$ is locally finite\n(Topology, Definition \\ref{topology-definition-locally-finite}).\n\\item A {\\it $k$-cycle} on $X$ is a cycle\n$$\n\\alpha = \\sum n_Z [Z]\n$$\nwhere $n_Z \\not = 0 \\Rightarrow \\dim_\\delta(Z) = k$.\n\\item The abelian group of all $k$-cycles on $X$ is denoted $Z_k(X)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QR","source_file":"chow.tex","source_line":1534,"source_end_line":1555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1534-L1555","statement_sha256":"354b5660e58a099aa48fc8f541df3456d94a075221c7be3ea74a4a5866ecdf58","origin":"The Stacks Project","memory_eligible":false,"source_rank":8138,"rank":8138,"depth":1,"x":2593.984,"y":683.739,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H46","tag":"0H46","title":"Cycles · Definition 0H46","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. The support of a cycle α = ∑ n_Z [Z] on X is Supp(α) = ⋃_n_Z not = 0 Z ⊂ X","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nThe {\\it support} of a cycle $\\alpha = \\sum n_Z [Z]$ on $X$\nis\n$$\n\\text{Supp}(\\alpha) = \\bigcup\\nolimits_{n_Z \\not = 0} Z \\subset X\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H46","source_file":"chow.tex","source_line":1592,"source_end_line":1601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1592-L1601","statement_sha256":"6aca21081eb2f91bd3e546ce0bc67dc8f4eb1f514d9c2a2a3cc71297177807a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8139,"rank":8139,"depth":0,"x":2392.794,"y":742.027,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H47","tag":"0H47","title":"Cycles · Definition 0H47","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. A cycle α on X is effective if it can be written as α =∑ n_Z [Z] with n_Z ≥ 0 for all Z.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nA cycle $\\alpha$ on $X$ is {\\it effective} if it\ncan be written as $\\alpha =\\sum n_Z [Z]$ with $n_Z \\geq 0$ for all $Z$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H47","source_file":"chow.tex","source_line":1609,"source_end_line":1615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1609-L1615","statement_sha256":"81be2fd1fb48b93e3ff855a5963c87b3b8f470973c1e3680e3b7fc57c5ce4e7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8140,"rank":8140,"depth":0,"x":2494.449,"y":584.618,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QT","tag":"02QT","title":"Cycle associated to a closed subscheme · Lemma 02QT","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let Z ⊂ X be a closed subscheme. • Let Z' ⊂ Z be an irreducible component and let xi ∈ Z' be its generic point. Then length_O_X, xi O_Z, xi < ∞ • If dim_δ(Z) ≤ k and xi ∈ Z with δ(xi) = k, then xi is a generic point of an irreducible component of Z.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $Z \\subset X$ be a closed subscheme.\n\\begin{enumerate}\n\\item Let $Z' \\subset Z$ be an irreducible component and\nlet $\\xi \\in Z'$ be its generic point.\nThen\n$$\n\\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{O}_{Z, \\xi} < \\infty\n$$\n\\item If $\\dim_\\delta(Z) \\leq k$ and $\\xi \\in Z$ with\n$\\delta(\\xi) = k$, then $\\xi$ is a generic point of an\nirreducible component of $Z$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycle associated to a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QT","source_file":"chow.tex","source_line":1628,"source_end_line":1644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1628-L1644","statement_sha256":"5004da1181af7f0fe52445d58108df471341fa219d5db225a5098a9daa0479cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8141,"rank":8141,"depth":9,"x":2546.162,"y":758.665,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QU","tag":"02QU","title":"Cycle associated to a closed subscheme · Definition 02QU","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let Z ⊂ X be a closed subscheme. • For any irreducible component Z' ⊂ Z with generic point xi the integer m_Z', Z = length_O_X, xi O_Z, xi (Lemma [Tag 02QT]) is called the multiplicity of Z' in Z. • Assume dim_δ(Z) ≤ k. The k-cycle associated to Z is [Z]_k = ∑ m_Z', Z[Z'] where the sum is over the irreducible components of Z of δ-dimension k. (This is a k-cycle by Divisors, Lemma [Tag 0BE1].)","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $Z \\subset X$ be a closed subscheme.\n\\begin{enumerate}\n\\item For any irreducible component $Z' \\subset Z$ with generic point $\\xi$\nthe integer\n$m_{Z', Z} = \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{O}_{Z, \\xi}$\n(Lemma \\ref{lemma-multiplicity-finite})\nis called the {\\it multiplicity of $Z'$ in $Z$}.\n\\item Assume $\\dim_\\delta(Z) \\leq k$.\nThe {\\it $k$-cycle associated to $Z$} is\n$$\n[Z]_k\n=\n\\sum m_{Z', Z}[Z']\n$$\nwhere the sum is over the irreducible components of $Z$\nof $\\delta$-dimension $k$. (This is a $k$-cycle by\nDivisors, Lemma \\ref{divisors-lemma-components-locally-finite}.)\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycle associated to a closed subscheme","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QU","source_file":"chow.tex","source_line":1665,"source_end_line":1687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1665-L1687","statement_sha256":"d1c1ff5a9c954bd73df92e7c8df9a37e8f99c88983ede0dc0f04ffd184164490","origin":"The Stacks Project","memory_eligible":false,"source_rank":8142,"rank":8142,"depth":10,"x":2367.763,"y":659.501,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QW","tag":"02QW","title":"Cycle associated to a coherent sheaf · Lemma 02QW","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let F be a coherent O_X-module. • The collection of irreducible components of the support of F is locally finite. • Let Z' ⊂ Supp(F) be an irreducible component and let xi ∈ Z' be its generic point. Then length_O_X, xi F_xi < ∞ • If dim_δ(Supp(F)) ≤ k and xi ∈ Supp(F) with δ(xi) = k, then xi is a generic point of an irreducible component of Supp(F).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item The collection of irreducible components of the support of\n$\\mathcal{F}$ is locally finite.\n\\item Let $Z' \\subset \\text{Supp}(\\mathcal{F})$\nbe an irreducible component and\nlet $\\xi \\in Z'$ be its generic point.\nThen\n$$\n\\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{F}_\\xi < \\infty\n$$\n\\item If $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$\nand $\\xi \\in \\text{Supp}(\\mathcal{F})$ with $\\delta(\\xi) = k$, then $\\xi$ is a\ngeneric point of an irreducible component of $\\text{Supp}(\\mathcal{F})$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycle associated to a coherent sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QW","source_file":"chow.tex","source_line":1701,"source_end_line":1720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1701-L1720","statement_sha256":"b58fbc119e664b430c197b710f68bf0ff6da78eca96064c7fa68d46d64f316d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8143,"rank":8143,"depth":20,"x":2579.414,"y":631.348,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QX","tag":"02QX","title":"Cycle associated to a coherent sheaf · Definition 02QX","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let F be a coherent O_X-module. • For any irreducible component Z' ⊂ Supp(F) with generic point xi the integer m_Z', F = length_O_X, xi F_xi (Lemma [Tag 02QW]) is called the multiplicity of Z' in F. • Assume dim_δ(Supp(F)) ≤ k. The k-cycle associated to F is [F]_k = ∑ m_Z', F[Z'] where the sum is over the irreducible components of Supp(F) of δ-dimension k. (This is a k-cycle by Lemma [Tag…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item For any irreducible component $Z' \\subset \\text{Supp}(\\mathcal{F})$\nwith generic point $\\xi$ the integer\n$m_{Z', \\mathcal{F}} = \\text{length}_{\\mathcal{O}_{X, \\xi}} \\mathcal{F}_\\xi$\n(Lemma \\ref{lemma-length-finite})\nis called the {\\it multiplicity of $Z'$ in $\\mathcal{F}$}.\n\\item Assume $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.\nThe {\\it $k$-cycle associated to $\\mathcal{F}$} is\n$$\n[\\mathcal{F}]_k\n=\n\\sum m_{Z', \\mathcal{F}}[Z']\n$$\nwhere the sum is over the irreducible components of\n$\\text{Supp}(\\mathcal{F})$ of $\\delta$-dimension $k$.\n(This is a $k$-cycle by Lemma \\ref{lemma-length-finite}.)\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycle associated to a coherent sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QX","source_file":"chow.tex","source_line":1743,"source_end_line":1765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1743-L1765","statement_sha256":"779d38183ef392d16d9dcd97567b6267e8dd2196a41b13a6a2ab46c0d48c439a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8144,"rank":8144,"depth":21,"x":2445.761,"y":772.44,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QY","tag":"02QY","title":"Cycle associated to a coherent sheaf · Lemma 02QY","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let Z ⊂ X be a closed subscheme. If dim_δ(Z) ≤ k, then [Z]_k = [ O_Z]_k.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $Z \\subset X$ be a closed subscheme.\nIf $\\dim_\\delta(Z) \\leq k$, then $[Z]_k = [{\\mathcal O}_Z]_k$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycle associated to a coherent sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QY","source_file":"chow.tex","source_line":1775,"source_end_line":1781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1775-L1781","statement_sha256":"c41c19115abd0e5c077c869a3ed0051400ee5607315dc7b84e126194533aa161","origin":"The Stacks Project","memory_eligible":false,"source_rank":8145,"rank":8145,"depth":0,"x":2430.828,"y":592.261,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QZ","tag":"02QZ","title":"Cycle associated to a coherent sheaf · Lemma 02QZ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let 0 → F → G → H → 0 be a short exact sequence of coherent sheaves on X. Assume that the δ-dimension of the supports of F, G, and H is ≤ k. Then [G]_k = [F]_k + [H]_k.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$\nbe a short exact sequence of coherent sheaves on $X$.\nAssume that the $\\delta$-dimension of the supports\nof $\\mathcal{F}$, $\\mathcal{G}$, and $\\mathcal{H}$ is $\\leq k$.\nThen $[\\mathcal{G}]_k = [\\mathcal{F}]_k + [\\mathcal{H}]_k$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycle associated to a coherent sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QZ","source_file":"chow.tex","source_line":1788,"source_end_line":1797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1788-L1797","statement_sha256":"02cb97dcd9b4940cad861d61ea7079d1d0d38df60aca75dda9f23bd38584e462","origin":"The Stacks Project","memory_eligible":false,"source_rank":8146,"rank":8146,"depth":1,"x":2586.994,"y":716.857,"cluster":"divisors-intersection-theory"},{"id":"stacks:02R1","tag":"02R1","title":"Preparation for proper pushforward · Lemma 02R1","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume X, Y integral and dim_δ(X) = dim_δ(Y). Then either f(X) is contained in a proper closed subscheme of Y, or f is dominant and the extension of function fields R(X)/R(Y) is finite.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a morphism.\nAssume $X$, $Y$ integral and $\\dim_\\delta(X) = \\dim_\\delta(Y)$.\nThen either $f(X)$ is contained in a proper closed subscheme\nof $Y$, or $f$ is dominant and the extension of function fields\n$R(X)/R(Y)$ is finite.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02R1","source_file":"chow.tex","source_line":1815,"source_end_line":1824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1815-L1824","statement_sha256":"87d043a8572f8e36e0efa4fe7378d86188412d01089444e00b217dac6e1357be","origin":"The Stacks Project","memory_eligible":false,"source_rank":8147,"rank":8147,"depth":31,"x":2371.284,"y":713.589,"cluster":"divisors-intersection-theory"},{"id":"stacks:02R2","tag":"02R2","title":"Preparation for proper pushforward · Lemma 02R2","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume f is quasi-compact, and (Z_i)_i ∈ I is a locally finite collection of closed subsets of X. Then (overlinef(Z_i))_i ∈ I is a locally finite collection of closed subsets of Y.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a morphism.\nAssume $f$ is quasi-compact, and $\\{Z_i\\}_{i \\in I}$ is a locally\nfinite collection of closed subsets of $X$.\nThen $\\{\\overline{f(Z_i)}\\}_{i \\in I}$ is a locally finite\ncollection of closed subsets of $Y$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02R2","source_file":"chow.tex","source_line":1846,"source_end_line":1855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1846-L1855","statement_sha256":"49d5e8df3e43caa90017dd82ddb17afcc91b3addb7ae3ecb25e6ec1c4007b991","origin":"The Stacks Project","memory_eligible":false,"source_rank":8148,"rank":8148,"depth":0,"x":2533.242,"y":593.4,"cluster":"divisors-intersection-theory"},{"id":"stacks:02R4","tag":"02R4","title":"Proper pushforward · Definition 02R4","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume f is proper. • Let Z ⊂ X be an integral closed subscheme with dim_δ(Z) = k. We define f_*[Z] = ( 0 & if & dim_δ(f(Z))< k, deg(Z/f(Z)) [f(Z)] & if & dim_δ(f(Z)) = k. . Here we think of f(Z) ⊂ Y as an integral closed subscheme. The degree of Z over f(Z) is finite if dim_δ(f(Z)) = dim_δ(Z) by Lemma [Tag 02R1]. • Let α = ∑ n_Z [Z] be a k-cycle on X. The…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a morphism.\nAssume $f$ is proper.\n\\begin{enumerate}\n\\item Let $Z \\subset X$ be an integral closed subscheme\nwith $\\dim_\\delta(Z) = k$. We define\n$$\nf_*[Z] =\n\\left\\{\n\\begin{matrix}\n0 & \\text{if} & \\dim_\\delta(f(Z))< k, \\\\\n\\deg(Z/f(Z)) [f(Z)] & \\text{if} & \\dim_\\delta(f(Z)) = k.\n\\end{matrix}\n\\right.\n$$\nHere we think of $f(Z) \\subset Y$ as an integral closed subscheme.\nThe degree of $Z$ over $f(Z)$ is finite if\n$\\dim_\\delta(f(Z)) = \\dim_\\delta(Z)$\nby Lemma \\ref{lemma-equal-dimension}.\n\\item Let $\\alpha = \\sum n_Z [Z]$ be a $k$-cycle on $X$. The\n{\\it pushforward} of $\\alpha$ as the sum\n$$\nf_* \\alpha = \\sum n_Z f_*[Z]\n$$\nwhere each $f_*[Z]$ is defined as above. The sum is locally finite\nby Lemma \\ref{lemma-quasi-compact-locally-finite} above.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Proper pushforward","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02R4","source_file":"chow.tex","source_line":1882,"source_end_line":1912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1882-L1912","statement_sha256":"49d98e06293764b190953dda2628d79f7370091f57eed2fdb261df72d08a1bb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8149,"rank":8149,"depth":32,"x":2510.431,"y":774.224,"cluster":"divisors-intersection-theory"},{"id":"stacks:02R5","tag":"02R5","title":"Proper pushforward · Lemma 02R5","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y, and Z be locally of finite type over S. Let f : X → Y and g : Y → Z be proper morphisms. Then g_* ∘ f_* = (g ∘ f)_* as maps Z_k(X) → Z_k(Z).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$, and $Z$ be locally of finite type over $S$.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms.\nThen $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02R5","source_file":"chow.tex","source_line":1923,"source_end_line":1929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1923-L1929","statement_sha256":"4bb779d850f4337914407a9c8f163be48690cbfa869a69fdf16b2bee2aebf551","origin":"The Stacks Project","memory_eligible":false,"source_rank":8150,"rank":8150,"depth":32,"x":2381.627,"y":627.702,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F92","tag":"0F92","title":"Proper pushforward · Lemma 0F92","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let X_1, X_2 ⊂ X be closed subschemes such that X = X_1 ∪ X_2 set theoretically. For every k ∈ Z the sequence of abelian groups xymatrix Z_k(X_1 ∩ X_2) ar[r] & Z_k(X_1) ⊕ Z_k(X_2) ar[r] & Z_k(X) ar[r] & 0 is exact. Here X_1 ∩ X_2 is the scheme theoretic intersection and the maps are the pushforward maps with one multiplied by -1.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let $X_1, X_2 \\subset X$\nbe closed subschemes such that $X = X_1 \\cup X_2$ set theoretically.\nFor every $k \\in \\mathbf{Z}$ the sequence of abelian groups\n$$\n\\xymatrix{\nZ_k(X_1 \\cap X_2) \\ar[r] &\nZ_k(X_1) \\oplus Z_k(X_2) \\ar[r] &\nZ_k(X) \\ar[r] &\n0\n}\n$$\nis exact. Here $X_1 \\cap X_2$ is the scheme theoretic intersection and\nthe maps are the pushforward maps with one multiplied by $-1$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F92","source_file":"chow.tex","source_line":1963,"source_end_line":1979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1963-L1979","statement_sha256":"40aba0edd47377347f87df7a2f64feb11c38ec8e81c8ff2c6bbc2a42857ea637","origin":"The Stacks Project","memory_eligible":false,"source_rank":8151,"rank":8151,"depth":0,"x":2594.783,"y":662.716,"cluster":"divisors-intersection-theory"},{"id":"stacks:02R6","tag":"02R6","title":"Proper pushforward · Lemma 02R6","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a proper morphism of schemes which are locally of finite type over S. • Let Z ⊂ X be a closed subscheme with dim_δ(Z) ≤ k. Then f_*[Z]_k = [f_* O_Z]_k. • Let F be a coherent sheaf on X such that dim_δ(Supp(F)) ≤ k. Then f_*[F]_k = [f_* F]_k. Note that the statement makes sense since f_*F and f_*O_Z are coherent O_Y-modules by Cohomology of Schemes, Proposition [Tag 02O5].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a proper morphism of schemes which are\nlocally of finite type over $S$.\n\\begin{enumerate}\n\\item Let $Z \\subset X$ be a closed subscheme with $\\dim_\\delta(Z) \\leq k$.\nThen\n$$\nf_*[Z]_k = [f_*{\\mathcal O}_Z]_k.\n$$\n\\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ such that\n$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. Then\n$$\nf_*[\\mathcal{F}]_k = [f_*{\\mathcal F}]_k.\n$$\n\\end{enumerate}\nNote that the statement makes sense since $f_*\\mathcal{F}$ and\n$f_*\\mathcal{O}_Z$ are coherent $\\mathcal{O}_Y$-modules by\nCohomology of Schemes, Proposition\n\\ref{coherent-proposition-proper-pushforward-coherent}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02R6","source_file":"chow.tex","source_line":1991,"source_end_line":2012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L1991-L2012","statement_sha256":"45e51834a16e10ddd753b8e0ca08966bce994952ea3391a15c2b4189c1451028","origin":"The Stacks Project","memory_eligible":false,"source_rank":8152,"rank":8152,"depth":31,"x":2409.144,"y":758.003,"cluster":"divisors-intersection-theory"},{"id":"stacks:02R8","tag":"02R8","title":"Preparation for flat pullback · Lemma 02R8","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume f is flat of relative dimension r. For any closed subset Z ⊂ Y we have dim_δ(f^-1(Z)) = dim_δ(Z) + r. provided f^-1(Z) is nonempty. If Z is irreducible and Z' ⊂ f^-1(Z) is an irreducible component, then Z' dominates Z and dim_δ(Z') = dim_δ(Z) + r.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a morphism.\nAssume $f$ is flat of relative dimension $r$.\nFor any closed subset $Z \\subset Y$ we have\n$$\n\\dim_\\delta(f^{-1}(Z)) = \\dim_\\delta(Z) + r.\n$$\nprovided $f^{-1}(Z)$ is nonempty.\nIf $Z$ is irreducible and $Z' \\subset f^{-1}(Z)$ is an irreducible\ncomponent, then $Z'$ dominates $Z$ and\n$\\dim_\\delta(Z') = \\dim_\\delta(Z) + r$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02R8","source_file":"chow.tex","source_line":2126,"source_end_line":2140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2126-L2140","statement_sha256":"66d2ad1834a0454ae60fd90f2cc5e8987ca75136a2e976d64dfecdf5c6a42093","origin":"The Stacks Project","memory_eligible":false,"source_rank":8153,"rank":8153,"depth":24,"x":2469.503,"y":582.117,"cluster":"divisors-intersection-theory"},{"id":"stacks:02R9","tag":"02R9","title":"Preparation for flat pullback · Lemma 02R9","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume (Z_i)_i ∈ I is a locally finite collection of closed subsets of Y. Then (f^-1(Z_i))_i ∈ I is a locally finite collection of closed subsets of X.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a morphism.\nAssume $\\{Z_i\\}_{i \\in I}$ is a locally\nfinite collection of closed subsets of $Y$.\nThen $\\{f^{-1}(Z_i)\\}_{i \\in I}$ is a locally finite\ncollection of closed subsets of $X$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02R9","source_file":"chow.tex","source_line":2169,"source_end_line":2178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2169-L2178","statement_sha256":"6b57144f7888070b5355f9acf588d97a6bf686f753683b14c3db0eea8461cb6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8154,"rank":8154,"depth":0,"x":2566.596,"y":746.329,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RB","tag":"02RB","title":"Flat pullback · Definition 02RB","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume f is flat of relative dimension r. • Let Z ⊂ Y be an integral closed subscheme of δ-dimension k. We define f^*[Z] to be the (k+r)-cycle on X to the scheme theoretic inverse image f^*[Z] = [f^-1(Z)]_k+r. This makes sense since dim_δ(f^-1(Z)) = k + r by Lemma [Tag 02R8]. • Let α = ∑ n_i [Z_i] be a k-cycle on Y. The flat pullback of α by f is the sum f^* α…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a morphism.\nAssume $f$ is flat of relative dimension $r$.\n\\begin{enumerate}\n\\item Let $Z \\subset Y$ be an integral closed subscheme of\n$\\delta$-dimension $k$. We define $f^*[Z]$ to be the\n$(k+r)$-cycle on $X$ to the scheme theoretic inverse image\n$$\nf^*[Z] = [f^{-1}(Z)]_{k+r}.\n$$\nThis makes sense since $\\dim_\\delta(f^{-1}(Z)) = k + r$\nby Lemma \\ref{lemma-flat-inverse-image-dimension}.\n\\item Let $\\alpha = \\sum n_i [Z_i]$ be\na $k$-cycle on $Y$. The {\\it flat pullback of $\\alpha$ by $f$}\nis the sum\n$$\nf^* \\alpha = \\sum n_i f^*[Z_i]\n$$\nwhere each $f^*[Z_i]$ is defined as above.\nThe sum is locally finite by Lemma \\ref{lemma-inverse-image-locally-finite}.\n\\item We denote $f^* : Z_k(Y) \\to Z_{k + r}(X)$ the map of abelian\ngroups so obtained.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Flat pullback","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RB","source_file":"chow.tex","source_line":2218,"source_end_line":2244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2218-L2244","statement_sha256":"ab860ca6835e90080e673894126aa1efbe6aa50d22652e7b7918faa4a9eb2ed9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8155,"rank":8155,"depth":25,"x":2362.616,"y":680.227,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RC","tag":"02RC","title":"Flat pullback · Lemma 02RC","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let U ⊂ X be an open subscheme, and denote i : Y = X setminus U → X as a reduced closed subscheme of X. For every k ∈ Z the sequence xymatrix Z_k(Y) ar[r]^i_* & Z_k(X) ar[r]^j^* & Z_k(U) ar[r] & 0 is an exact complex of abelian groups.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $U \\subset X$ be an open subscheme, and denote\n$i : Y = X \\setminus U \\to X$ as a reduced closed subscheme of $X$.\nFor every $k \\in \\mathbf{Z}$ the sequence\n$$\n\\xymatrix{\nZ_k(Y) \\ar[r]^{i_*} & Z_k(X) \\ar[r]^{j^*} & Z_k(U) \\ar[r] & 0\n}\n$$\nis an exact complex of abelian groups.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RC","source_file":"chow.tex","source_line":2262,"source_end_line":2275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2262-L2275","statement_sha256":"961c8e14c78acdc63d6f4017f1750b26fc790548d53511f52bc1bf34afe1e693","origin":"The Stacks Project","memory_eligible":false,"source_rank":8156,"rank":8156,"depth":0,"x":2566.514,"y":613.118,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RD","tag":"02RD","title":"Flat pullback · Lemma 02RD","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y, Z be locally of finite type over S. Let f : X → Y and g : Y → Z be flat morphisms of relative dimensions r and s. Then g ∘ f is flat of relative dimension r + s and f^* ∘ g^* = (g ∘ f)^* as maps Z_k(Z) → Z_k + r + s(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X, Y, Z$ be locally of finite type over $S$.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of relative dimensions\n$r$ and $s$. Then $g \\circ f$ is flat of relative dimension\n$r + s$ and\n$$\nf^* \\circ g^* = (g \\circ f)^*\n$$\nas maps $Z_k(Z) \\to Z_{k + r + s}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RD","source_file":"chow.tex","source_line":2286,"source_end_line":2297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2286-L2297","statement_sha256":"f442d2faaefb3258154a17de0d8ba9d32d49126ea177b6d48c458c05e88f53be","origin":"The Stacks Project","memory_eligible":false,"source_rank":8157,"rank":8157,"depth":26,"x":2469.974,"y":778.566,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RE","tag":"02RE","title":"Flat pullback · Lemma 02RE","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a flat morphism of relative dimension r. • Let Z ⊂ Y be a closed subscheme with dim_δ(Z) ≤ k. Then we have dim_δ(f^-1(Z)) ≤ k + r and [f^-1(Z)]_k + r = f^*[Z]_k in Z_k + r(X). • Let F be a coherent sheaf on Y with dim_δ(Supp(F)) ≤ k. Then we have dim_δ(Supp(f^*F)) ≤ k + r and f^*[ F]_k = [f^* F]_k+r in Z_k + r(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X, Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\n\\begin{enumerate}\n\\item Let $Z \\subset Y$ be a closed subscheme with\n$\\dim_\\delta(Z) \\leq k$. Then we have\n$\\dim_\\delta(f^{-1}(Z)) \\leq k + r$\nand $[f^{-1}(Z)]_{k + r} = f^*[Z]_k$ in $Z_{k + r}(X)$.\n\\item Let $\\mathcal{F}$ be a coherent sheaf on $Y$ with\n$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.\nThen we have $\\dim_\\delta(\\text{Supp}(f^*\\mathcal{F})) \\leq k + r$\nand\n$$\nf^*[{\\mathcal F}]_k = [f^*{\\mathcal F}]_{k+r}\n$$\nin $Z_{k + r}(X)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RE","source_file":"chow.tex","source_line":2330,"source_end_line":2349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2330-L2349","statement_sha256":"0f3f21b621f7a325a8df460dfaaf2167ed3a732ae5ab971c161b4b95a305899f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8158,"rank":8158,"depth":25,"x":2408.014,"y":601.505,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RG","tag":"02RG","title":"Push and pull · Lemma 02RG","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a fibre product diagram of schemes locally of finite type over S. Assume f : X → Y proper and g : Y' → Y flat of relative dimension r. Then also f' is proper and g' is flat of relative dimension r. For any k-cycle α on X we have g^*f_*α = f'_*(g')^*α in Z_k + r(Y').","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a fibre product diagram of schemes locally of finite type over $S$.\nAssume $f : X \\to Y$ proper and $g : Y' \\to Y$ flat of relative dimension $r$.\nThen also $f'$ is proper and $g'$ is flat of relative dimension $r$.\nFor any $k$-cycle $\\alpha$ on $X$ we have\n$$\ng^*f_*\\alpha = f'_*(g')^*\\alpha\n$$\nin $Z_{k + r}(Y')$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RG","source_file":"chow.tex","source_line":2400,"source_end_line":2418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2400-L2418","statement_sha256":"09d8f82756b3ea917b60c84794772368979562b3e3394c6c86a954e3b52937f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8159,"rank":8159,"depth":32,"x":2596.391,"y":697.062,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RH","tag":"02RH","title":"Push and pull · Lemma 02RH","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a finite locally free morphism of degree d (see Morphisms, Definition [Tag 02KA]). Then f is both proper and flat of relative dimension 0, and f_*f^*α = dα for every α ∈ Z_k(Y).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a finite locally free morphism\nof degree $d$ (see\nMorphisms, Definition \\ref{morphisms-definition-finite-locally-free}).\nThen $f$ is both proper and flat of relative dimension $0$, and\n$$\nf_*f^*\\alpha = d\\alpha\n$$\nfor every $\\alpha \\in Z_k(Y)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RH","source_file":"chow.tex","source_line":2438,"source_end_line":2450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2438-L2450","statement_sha256":"142fdfb3969ad223c39cab6fd8d4dd15b2747e5bc7aa074d385f74266e1ab79d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8160,"rank":8160,"depth":32,"x":2380.296,"y":733.547,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RK","tag":"02RK","title":"Preparation for principal divisors · Lemma 02RK","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Assume X is integral. • If Z ⊂ X is an integral closed subscheme, then the following are equivalent: • Z is a prime divisor, • Z has codimension 1 in X, and • dim_δ(Z) = dim_δ(X) - 1. • If Z is an irreducible component of an effective Cartier divisor on X, then dim_δ(Z) = dim_δ(X) - 1.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Assume $X$ is\nintegral.\n\\begin{enumerate}\n\\item If $Z \\subset X$ is an integral closed subscheme, then\nthe following are equivalent:\n\\begin{enumerate}\n\\item $Z$ is a prime divisor,\n\\item $Z$ has codimension $1$ in $X$, and\n\\item $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$.\n\\end{enumerate}\n\\item If $Z$ is an irreducible component of an effective Cartier\ndivisor on $X$, then $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for principal divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RK","source_file":"chow.tex","source_line":2488,"source_end_line":2504,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2488-L2504","statement_sha256":"9377830c49c36c4fce5c6a1f1b50b51b8eacc60986865882dbe5c0eeb2fe2e55","origin":"The Stacks Project","memory_eligible":false,"source_rank":8161,"rank":8161,"depth":11,"x":2510.507,"y":583.788,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RM","tag":"02RM","title":"Preparation for principal divisors · Lemma 02RM","summary":"Let f : X → Y be a morphism of schemes. Let xi ∈ Y be a point. Assume that • X, Y are integral, • Y is locally Noetherian • f is proper, dominant and R(Y) ⊂ R(X) is finite, and • dim(O_Y, xi) = 1. Then there exists an open neighbourhood V ⊂ Y of xi such that f|_f^-1(V) : f^-1(V) → V is finite.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nLet $\\xi \\in Y$ be a point.\nAssume that\n\\begin{enumerate}\n\\item $X$, $Y$ are integral,\n\\item $Y$ is locally Noetherian\n\\item $f$ is proper, dominant and $R(Y) \\subset R(X)$ is finite, and\n\\item $\\dim(\\mathcal{O}_{Y, \\xi}) = 1$.\n\\end{enumerate}\nThen there exists an open neighbourhood $V \\subset Y$ of $\\xi$\nsuch that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for principal divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RM","source_file":"chow.tex","source_line":2523,"source_end_line":2536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2523-L2536","statement_sha256":"80704e484f7e73125a7c38c50f3f83ab17098a33c77538de34049812235dc863","origin":"The Stacks Project","memory_eligible":false,"source_rank":8162,"rank":8162,"depth":42,"x":2534.962,"y":768.395,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RO","tag":"02RO","title":"Principal divisors · Definition 02RO","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Assume X is integral with dim_δ(X) = n. Let f ∈ R(X)^*. The principal divisor associated to f is the (n - 1)-cycle div(f) = div_X(f) = ∑ ord_Z(f) [Z] defined in Divisors, Definition [Tag 0BE3]. This makes sense because prime divisors have δ-dimension n - 1 by Lemma [Tag 02RK].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Assume $X$ is\nintegral with $\\dim_\\delta(X) = n$.\nLet $f \\in R(X)^*$. The {\\it principal divisor\nassociated to $f$} is the $(n - 1)$-cycle\n$$\n\\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z]\n$$\ndefined in Divisors, Definition \\ref{divisors-definition-principal-divisor}.\nThis makes sense because prime divisors have $\\delta$-dimension $n - 1$ by\nLemma \\ref{lemma-divisor-delta-dimension}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Principal divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RO","source_file":"chow.tex","source_line":2565,"source_end_line":2578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2565-L2578","statement_sha256":"ceaff55d28f3cf21c80c769c7d7caf9e5ab1c56bf3369395324522d9c0facd09","origin":"The Stacks Project","memory_eligible":false,"source_rank":8163,"rank":8163,"depth":12,"x":2368.211,"y":645.952,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RR","tag":"02RR","title":"Principal divisors · Lemma 02RR","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Assume X, Y are integral and n = dim_δ(Y). Let f : X → Y be a flat morphism of relative dimension r. Let g ∈ R(Y)^*. Then f^*(div_Y(g)) = div_X(g) in Z_n + r - 1(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$\nare integral and $n = \\dim_\\delta(Y)$.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $g \\in R(Y)^*$. Then\n$$\nf^*(\\text{div}_Y(g)) = \\text{div}_X(g)\n$$\nin $Z_{n + r - 1}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Principal divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RR","source_file":"chow.tex","source_line":2589,"source_end_line":2600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2589-L2600","statement_sha256":"cc4055d5a4e02eedad2ff04a9bbee730158a6cea3b24059e42644ba54994e558","origin":"The Stacks Project","memory_eligible":false,"source_rank":8164,"rank":8164,"depth":5,"x":2589.985,"y":641.615,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RT","tag":"02RT","title":"Principal divisors and pushforward · Lemma 02RT","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Assume X, Y are integral and n = dim_δ(X) = dim_δ(Y). Let p : X → Y be a dominant proper morphism. Let f ∈ R(X)^*. Set g = Nm_R(X)/R(Y)(f). Then we have p_*div(f) = div(g).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$\nare integral and $n = \\dim_\\delta(X) = \\dim_\\delta(Y)$.\nLet $p : X \\to Y$ be a dominant proper morphism.\nLet $f \\in R(X)^*$. Set\n$$\ng = \\text{Nm}_{R(X)/R(Y)}(f).\n$$\nThen we have\n$p_*\\text{div}(f) = \\text{div}(g)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Principal divisors and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RT","source_file":"chow.tex","source_line":2644,"source_end_line":2656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2644-L2656","statement_sha256":"1edac8ab97993edf19ca8482f7bc1cc289d4772b80d9bb12bd63ad7980c3faaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8165,"rank":8165,"depth":43,"x":2429.689,"y":770.855,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RQ","tag":"02RQ","title":"Principal divisors and pushforward · Lemma 02RQ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Assume X is integral and n = dim_δ(X). Let f ∈ R(X)^*. Let U ⊂ X be a nonempty open such that f corresponds to a section f ∈ Γ(U, O_X^*). Let Y ⊂ X ×_S P^1_S be the closure of the graph of f : U → P^1_S. Then • the projection morphism p : Y → X is proper, • p|_p^-1(U) : p^-1(U) → U is an isomorphism, • the pullbacks Y_0 = q^-1D_0 and Y_∞ = q^-1D_∞ via the morphism q : Y → P^1_S are defined…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Assume $X$ is\nintegral and $n = \\dim_\\delta(X)$. Let $f \\in R(X)^*$.\nLet $U \\subset X$ be a nonempty open such that $f$\ncorresponds to a section $f \\in \\Gamma(U, \\mathcal{O}_X^*)$.\nLet $Y \\subset X \\times_S \\mathbf{P}^1_S$ be the\nclosure of the graph of $f : U \\to \\mathbf{P}^1_S$.\nThen\n\\begin{enumerate}\n\\item the projection morphism $p : Y \\to X$ is proper,\n\\item $p|_{p^{-1}(U)} : p^{-1}(U) \\to U$ is an isomorphism,\n\\item the pullbacks $Y_0 = q^{-1}D_0$ and $Y_\\infty = q^{-1}D_\\infty$\nvia the morphism $q : Y \\to \\mathbf{P}^1_S$ are defined\n(Divisors, Definition\n\\ref{divisors-definition-pullback-effective-Cartier-divisor}),\n\\item we have\n$$\n\\text{div}_Y(f) = [Y_0]_{n - 1} - [Y_\\infty]_{n - 1}\n$$\n\\item we have\n$$\n\\text{div}_X(f) = p_*\\text{div}_Y(f)\n$$\n\\item if we view $Y_0$ and $Y_\\infty$ as closed subschemes of $X$\nvia the morphism $p$ then we have\n$$\n\\text{div}_X(f) = [Y_0]_{n - 1} - [Y_\\infty]_{n - 1}\n$$\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Principal divisors and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RQ","source_file":"chow.tex","source_line":2700,"source_end_line":2731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2700-L2731","statement_sha256":"5fa4cf50ebb93a3cab2e6a4747d86101293769c301f8ce7f63cc7d7f8309c237","origin":"The Stacks Project","memory_eligible":false,"source_rank":8166,"rank":8166,"depth":44,"x":2443.98,"y":584.308,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RU","tag":"02RU","title":"Principal divisors and pushforward · Lemma 02RU","summary":"Let K be any field. Let X be a 1-dimensional integral scheme endowed with a proper morphism c : X → Spec(K). Let f ∈ K(X)^* be an invertible rational function. Then ∑_x ∈ X closed [kappa(x) : K] ord_O_X, x(f) = 0 where ord is as in Algebra, Definition [Tag 02MD]. In other words, c_*div(f) = 0.","statement_latex":"Let $K$ be any field. Let $X$ be a $1$-dimensional integral scheme\nendowed with a proper morphism $c : X \\to \\Spec(K)$.\nLet $f \\in K(X)^*$ be an invertible rational function.\nThen\n$$\n\\sum\\nolimits_{x \\in X \\text{ closed}}\n[\\kappa(x) : K] \\text{ord}_{\\mathcal{O}_{X, x}}(f)\n=\n0\n$$\nwhere $\\text{ord}$ is as in\nAlgebra, Definition \\ref{algebra-definition-ord}.\nIn other words, $c_*\\text{div}(f) = 0$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Principal divisors and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RU","source_file":"chow.tex","source_line":2801,"source_end_line":2816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2801-L2816","statement_sha256":"935e4553c64dec13c7b43af09e6f293defa68f332ced025844e2608cb58ca630","origin":"The Stacks Project","memory_eligible":false,"source_rank":8167,"rank":8167,"depth":45,"x":2583.675,"y":730.201,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RW","tag":"02RW","title":"Rational equivalence · Definition 02RW","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let k ∈ Z. • Given any locally finite collection (W_j ⊂ X) of integral closed subschemes with dim_δ(W_j) = k + 1, and any f_j ∈ R(W_j)^* we may consider ∑ (i_j)_*div(f_j) ∈ Z_k(X) where i_j : W_j → X is the inclusion morphism. This makes sense as the morphism coprod i_j : coprod W_j → X is proper. • We say that α ∈ Z_k(X) is rationally equivalent to zero if α is a cycle of the form…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nLet $k \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item Given any locally finite collection $\\{W_j \\subset X\\}$\nof integral closed subschemes with $\\dim_\\delta(W_j) = k + 1$,\nand any $f_j \\in R(W_j)^*$ we may consider\n$$\n\\sum (i_j)_*\\text{div}(f_j) \\in Z_k(X)\n$$\nwhere $i_j : W_j \\to X$ is the inclusion morphism.\nThis makes sense as the morphism\n$\\coprod i_j : \\coprod W_j \\to X$ is proper.\n\\item We say that $\\alpha \\in Z_k(X)$ is {\\it rationally equivalent to zero}\nif $\\alpha$ is a cycle of the form displayed above.\n\\item We say $\\alpha, \\beta \\in Z_k(X)$ are\n{\\it rationally equivalent} and we write $\\alpha \\sim_{rat} \\beta$\nif $\\alpha - \\beta$ is rationally equivalent to zero.\n\\item We define\n$$\n\\CH_k(X) = Z_k(X) / \\sim_{rat}\n$$\nto be the {\\it Chow group of $k$-cycles on $X$}. This is sometimes called\nthe {\\it Chow group of $k$-cycles modulo rational equivalence on $X$}.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational equivalence","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RW","source_file":"chow.tex","source_line":2881,"source_end_line":2908,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2881-L2908","statement_sha256":"84d73660be71c89ec0a608d4bffb3a48d5a4a1dfcee9ea4f7815ae7e925afddb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8168,"rank":8168,"depth":0,"x":2363.0,"y":701.844,"cluster":"divisors-intersection-theory"},{"id":"stacks:02RX","tag":"02RX","title":"Rational equivalence · Lemma 02RX","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let U ⊂ X be an open subscheme, and denote i : Y = X setminus U → X as a reduced closed subscheme of X. Let k ∈ Z. Suppose α, β ∈ Z_k(X). If α|_U sim_rat β|_U then there exist a cycle γ ∈ Z_k(Y) such that α sim_rat β + i_*γ. In other words, the sequence xymatrix CH_k(Y) ar[r]^i_* & CH_k(X) ar[r]^j^* & CH_k(U) ar[r] & 0 is an exact complex of abelian groups.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nLet $U \\subset X$ be an open subscheme, and denote\n$i : Y = X \\setminus U \\to X$ as a reduced closed subscheme of $X$.\nLet $k \\in \\mathbf{Z}$.\nSuppose $\\alpha, \\beta \\in Z_k(X)$.\nIf $\\alpha|_U \\sim_{rat} \\beta|_U$ then there exist a cycle\n$\\gamma \\in Z_k(Y)$ such that\n$$\n\\alpha \\sim_{rat} \\beta + i_*\\gamma.\n$$\nIn other words, the sequence\n$$\n\\xymatrix{\n\\CH_k(Y) \\ar[r]^{i_*} & \\CH_k(X) \\ar[r]^{j^*} & \\CH_k(U) \\ar[r] & 0\n}\n$$\nis an exact complex of abelian groups.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02RX","source_file":"chow.tex","source_line":2948,"source_end_line":2968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2948-L2968","statement_sha256":"3000e02a10eb605a13b8d6f44369abaee6c8096185290dd4ecd1e7233f21f9bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8169,"rank":8169,"depth":1,"x":2548.814,"y":597.376,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F94","tag":"0F94","title":"Rational equivalence · Lemma 0F94","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let X_1, X_2 ⊂ X be closed subschemes such that X = X_1 ∪ X_2 set theoretically. For every k ∈ Z the sequence of abelian groups xymatrix CH_k(X_1 ∩ X_2) ar[r] & CH_k(X_1) ⊕ CH_k(X_2) ar[r] & CH_k(X) ar[r] & 0 is exact. Here X_1 ∩ X_2 is the scheme theoretic intersection and the maps are the pushforward maps with one multiplied by -1.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let $X_1, X_2 \\subset X$\nbe closed subschemes such that $X = X_1 \\cup X_2$ set theoretically.\nFor every $k \\in \\mathbf{Z}$ the sequence of abelian groups\n$$\n\\xymatrix{\n\\CH_k(X_1 \\cap X_2) \\ar[r] &\n\\CH_k(X_1) \\oplus \\CH_k(X_2) \\ar[r] &\n\\CH_k(X) \\ar[r] &\n0\n}\n$$\nis exact. Here $X_1 \\cap X_2$ is the scheme theoretic intersection and the\nmaps are the pushforward maps with one multiplied by $-1$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F94","source_file":"chow.tex","source_line":2990,"source_end_line":3006,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L2990-L3006","statement_sha256":"05e3773e6089ff9ec406b485d0c54779a8092da858f2499ceffdaa0f0f07cc5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8170,"rank":8170,"depth":1,"x":2495.726,"y":780.127,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EPH","tag":"0EPH","title":"Rational equivalence and push and pull · Lemma 0EPH","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be schemes locally of finite type over S. Assume Y integral with dim_δ(Y) = k. Let f : X → Y be a flat morphism of relative dimension r. Then for g ∈ R(Y)^* we have f^*div_Y(g) = ∑ n_j i_j, *div_X_j(g ∘ f|_X_j) as (k + r - 1)-cycles on X where the sum is over the irreducible components X_j of X and n_j is the multiplicity of X_j in X.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be schemes locally of finite type over $S$.\nAssume $Y$ integral with $\\dim_\\delta(Y) = k$.\nLet $f : X \\to Y$ be a flat morphism of\nrelative dimension $r$. Then for $g \\in R(Y)^*$ we have\n$$\nf^*\\text{div}_Y(g) =\n\\sum n_j i_{j, *}\\text{div}_{X_j}(g \\circ f|_{X_j})\n$$\nas $(k + r - 1)$-cycles on $X$ where the sum is over the irreducible\ncomponents $X_j$ of $X$ and $n_j$ is the multiplicity of $X_j$ in $X$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational equivalence and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPH","source_file":"chow.tex","source_line":3097,"source_end_line":3110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3097-L3110","statement_sha256":"5e8f3f7a419cbed763fa964b7fad52569982496c83299dbde06fc0259b3f7a42","origin":"The Stacks Project","memory_eligible":false,"source_rank":8171,"rank":8171,"depth":25,"x":2387.744,"y":614.991,"cluster":"divisors-intersection-theory"},{"id":"stacks:02S1","tag":"02S1","title":"Rational equivalence and push and pull · Lemma 02S1","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be schemes locally of finite type over S. Let f : X → Y be a flat morphism of relative dimension r. Let α sim_rat β be rationally equivalent k-cycles on Y. Then f^*α sim_rat f^*β as (k + r)-cycles on X.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be schemes locally of finite type over $S$.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $\\alpha \\sim_{rat} \\beta$ be rationally equivalent $k$-cycles on $Y$.\nThen $f^*\\alpha \\sim_{rat} f^*\\beta$ as $(k + r)$-cycles on $X$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational equivalence and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02S1","source_file":"chow.tex","source_line":3172,"source_end_line":3179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3172-L3179","statement_sha256":"ec64dd938a81461b17d6ab3a03402ea56d53b1ad0f520e51cfa9ab7bb34c684e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8172,"rank":8172,"depth":33,"x":2600.491,"y":675.581,"cluster":"divisors-intersection-theory"},{"id":"stacks:02S2","tag":"02S2","title":"Rational equivalence and push and pull · Lemma 02S2","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be schemes locally of finite type over S. Let p : X → Y be a proper morphism. Suppose α, β ∈ Z_k(X) are rationally equivalent. Then p_*α is rationally equivalent to p_*β.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be schemes locally of finite type over $S$.\nLet $p : X \\to Y$ be a proper morphism.\nSuppose $\\alpha, \\beta \\in Z_k(X)$ are rationally equivalent.\nThen $p_*\\alpha$ is rationally equivalent to $p_*\\beta$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational equivalence and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02S2","source_file":"chow.tex","source_line":3216,"source_end_line":3223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3216-L3223","statement_sha256":"4f709205ccd63be7340d432884947bd5aba9e4f785ea725b231deb59b470b144","origin":"The Stacks Project","memory_eligible":false,"source_rank":8173,"rank":8173,"depth":46,"x":2394.575,"y":751.737,"cluster":"divisors-intersection-theory"},{"id":"stacks:02S4","tag":"02S4","title":"Rational equivalence and the projective line · Lemma 02S4","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let W ⊂ X ×_S P^1_S be an integral closed subscheme of δ-dimension k + 1. Assume W not = W_0, and W not = W_∞. Then • W_0, W_∞ are effective Cartier divisors of W, • W_0, W_∞ can be viewed as closed subschemes of X and [W_0]_k sim_rat [W_∞]_k, • for any locally finite family of integral closed subschemes W_i ⊂ X ×_S P^1_S of δ-dimension k + 1 with W_i not = (W_i)_0 and W_i not =…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nLet $W \\subset X \\times_S \\mathbf{P}^1_S$ be an integral\nclosed subscheme of $\\delta$-dimension $k + 1$.\nAssume $W \\not = W_0$, and $W \\not = W_\\infty$. Then\n\\begin{enumerate}\n\\item $W_0$, $W_\\infty$ are effective Cartier divisors of $W$,\n\\item $W_0$, $W_\\infty$ can be viewed as closed subschemes\nof $X$ and\n$$\n[W_0]_k \\sim_{rat} [W_\\infty]_k,\n$$\n\\item for any locally finite family of\nintegral closed subschemes\n$W_i \\subset X \\times_S \\mathbf{P}^1_S$\nof $\\delta$-dimension $k + 1$ with $W_i \\not = (W_i)_0$ and\n$W_i \\not = (W_i)_\\infty$ we have\n$\\sum ([(W_i)_0]_k - [(W_i)_\\infty]_k) \\sim_{rat} 0$\non $X$, and\n\\item for any $\\alpha \\in Z_k(X)$ with $\\alpha \\sim_{rat} 0$\nthere exists a locally finite family of\nintegral closed subschemes $W_i \\subset X \\times_S \\mathbf{P}^1_S$\nas above such that $\\alpha = \\sum ([(W_i)_0]_k - [(W_i)_\\infty]_k)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational equivalence and the projective line","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02S4","source_file":"chow.tex","source_line":3334,"source_end_line":3360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3334-L3360","statement_sha256":"514ec6c2d48017da63ce850ee55918cd233f49cbaae5eb63a43aa299432f2fa7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8174,"rank":8174,"depth":47,"x":2485.31,"y":578.475,"cluster":"divisors-intersection-theory"},{"id":"stacks:02S5","tag":"02S5","title":"Rational equivalence and the projective line · Lemma 02S5","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let Z be a closed subscheme of X × P^1. Assume • dim_δ(Z) ≤ k + 1, • dim_δ(Z_0) ≤ k, dim_δ(Z_∞) ≤ k, and • for any embedded point xi (Divisors, Definition [Tag 05AK]) of Z either xi not ∈ Z_0 ∪ Z_∞ or δ(xi) < k. Then [Z_0]_k sim_rat [Z_∞]_k as k-cycles on X.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nLet $Z$ be a closed subscheme of $X \\times \\mathbf{P}^1$.\nAssume\n\\begin{enumerate}\n\\item $\\dim_\\delta(Z) \\leq k + 1$,\n\\item $\\dim_\\delta(Z_0) \\leq k$, $\\dim_\\delta(Z_\\infty) \\leq k$, and\n\\item for any embedded point $\\xi$ (Divisors, Definition\n\\ref{divisors-definition-embedded}) of $Z$ either\n$\\xi \\not \\in Z_0 \\cup Z_\\infty$ or $\\delta(\\xi) < k$.\n\\end{enumerate}\nThen $[Z_0]_k \\sim_{rat} [Z_\\infty]_k$ as $k$-cycles on $X$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational equivalence and the projective line","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02S5","source_file":"chow.tex","source_line":3403,"source_end_line":3417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3403-L3417","statement_sha256":"c7c754b1b406c8229b841e53793ec9eb13bd1cf2ff1945622526a6e11df14542","origin":"The Stacks Project","memory_eligible":false,"source_rank":8175,"rank":8175,"depth":48,"x":2557.846,"y":757.994,"cluster":"divisors-intersection-theory"},{"id":"stacks:02S6","tag":"02S6","title":"Rational equivalence and the projective line · Lemma 02S6","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let F be a coherent sheaf on X × P^1. Let i_0, i_∞ : X → X × P^1 be the closed immersion such that i_t(x) = (x, t). Denote F_0 = i_0^*F and F_∞ = i_∞^*F. Assume • dim_δ(Supp(F)) ≤ k + 1, • dim_δ(Supp(F_0)) ≤ k, dim_δ(Supp(F_∞)) ≤ k, and • for any embedded associated point xi of F either xi not ∈ (X × P^1)_0 ∪ (X × P^1)_∞ or δ(xi) < k. Then [F_0]_k sim_rat [F_∞]_k as k-cycles on X.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nLet $\\mathcal{F}$ be a coherent sheaf on $X \\times \\mathbf{P}^1$.\nLet $i_0, i_\\infty : X \\to X \\times \\mathbf{P}^1$ be the closed immersion\nsuch that $i_t(x) = (x, t)$. Denote $\\mathcal{F}_0 = i_0^*\\mathcal{F}$ and\n$\\mathcal{F}_\\infty = i_\\infty^*\\mathcal{F}$.\nAssume\n\\begin{enumerate}\n\\item $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$,\n\\item $\\dim_\\delta(\\text{Supp}(\\mathcal{F}_0)) \\leq k$,\n$\\dim_\\delta(\\text{Supp}(\\mathcal{F}_\\infty)) \\leq k$, and\n\\item for any embedded associated point $\\xi$ of $\\mathcal{F}$ either\n$\\xi \\not \\in (X \\times \\mathbf{P}^1)_0 \\cup (X \\times \\mathbf{P}^1)_\\infty$\nor $\\delta(\\xi) < k$.\n\\end{enumerate}\nThen $[\\mathcal{F}_0]_k \\sim_{rat} [\\mathcal{F}_\\infty]_k$ as $k$-cycles on $X$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational equivalence and the projective line","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02S6","source_file":"chow.tex","source_line":3469,"source_end_line":3487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3469-L3487","statement_sha256":"407d317050d6cc1aa6202b49ea7333f2324afd82320446bed38ae190777ebb2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8176,"rank":8176,"depth":48,"x":2359.695,"y":666.64,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GU5","tag":"0GU5","title":"Chow groups and envelopes · Definition 0GU5","summary":"[F] Let X be a scheme. An envelope is a proper morphism f : Y → X which is completely decomposed (More on Morphisms, Definition [Tag 0GTI]).","statement_latex":"\\begin{reference}\n\\cite[Definition 18.3]{F}\n\\end{reference}\nLet $X$ be a scheme. An {\\it envelope} is a proper morphism $f : Y \\to X$\nwhich is completely decomposed\n(More on Morphisms, Definition \\ref{more-morphisms-definition-cd-morphism}).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and envelopes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GU5","source_file":"chow.tex","source_line":3552,"source_end_line":3560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3552-L3560","statement_sha256":"4b7b76065507cbe7d26a14248b630ccb7b861c1021bd3dc93b6a4f2b837655fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8177,"rank":8177,"depth":1,"x":2579.605,"y":621.5,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GU6","tag":"0GU6","title":"Chow groups and envelopes · Lemma 0GU6","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. If f : Y → X and g : Z → Y are envelopes, then f ∘ g is an envelope.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nIf $f : Y \\to X$ and $g : Z \\to Y$ are envelopes, then\n$f \\circ g$ is an envelope.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and envelopes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GU6","source_file":"chow.tex","source_line":3566,"source_end_line":3572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3566-L3572","statement_sha256":"bca9fab478468cc5b2bde3cfeef083bd8e5e5bdedaee4048c3cc34ce26cfc673","origin":"The Stacks Project","memory_eligible":false,"source_rank":8178,"rank":8178,"depth":17,"x":2453.558,"y":779.805,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GU7","tag":"0GU7","title":"Chow groups and envelopes · Lemma 0GU7","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X' → X be a morphism of schemes locally of finite type over S. If f : Y → X is an envelope, then the base change f' : Y' → X' of f is an envelope too.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X' \\to X$ be a morphism of schemes locally of finite type over $S$.\nIf $f : Y \\to X$ is an envelope, then the base change $f' : Y' \\to X'$\nof $f$ is an envelope too.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and envelopes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GU7","source_file":"chow.tex","source_line":3579,"source_end_line":3585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3579-L3585","statement_sha256":"c78d4ba8cde1b7486108df5b99eaf88d31b0d42cc6228a99a53e8f9127abfcdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8179,"rank":8179,"depth":17,"x":2419.146,"y":591.268,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GU8","tag":"0GU8","title":"Chow groups and envelopes · Lemma 0GU8","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let f : Y → X be an envelope. Then we have an exact sequence CH_k(Y ×_X Y) xrightarrowp_* - q_* CH_k(Y) xrightarrowf_* CH_k(X) → 0 for all k ∈ Z. Here p, q : Y ×_X Y → Y are the projections.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nLet $f : Y \\to X$ be an envelope. Then\nwe have an exact sequence\n$$\n\\CH_k(Y \\times_X Y) \\xrightarrow{p_* - q_*}\n\\CH_k(Y) \\xrightarrow{f_*}\n\\CH_k(X) \\to 0\n$$\nfor all $k \\in \\mathbf{Z}$. Here $p, q : Y \\times_X Y \\to Y$ are\nthe projections.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and envelopes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GU8","source_file":"chow.tex","source_line":3592,"source_end_line":3605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3592-L3605","statement_sha256":"092ba4e379c2d66a506dedb03cd57eee1ed60681a0b546cb632bf34805699b19","origin":"The Stacks Project","memory_eligible":false,"source_rank":8180,"rank":8180,"depth":33,"x":2596.402,"y":710.946,"cluster":"divisors-intersection-theory"},{"id":"stacks:02S8","tag":"02S8","title":"Chow groups and K-groups · Lemma 02S8","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. The categories Coh_≤ k(X) are Serre subcategories of the abelian category Coh(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nThe categories $\\textit{Coh}_{\\leq k}(X)$ are Serre subcategories\nof the abelian category $\\textit{Coh}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02S8","source_file":"chow.tex","source_line":3742,"source_end_line":3748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3742-L3748","statement_sha256":"6accee9df245e0ff21eccfd9062b17cbeb7dba6116ff9cc677768bcb6204c6b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8181,"rank":8181,"depth":2,"x":2369.116,"y":723.295,"cluster":"divisors-intersection-theory"},{"id":"stacks:02S9","tag":"02S9","title":"Chow groups and K-groups · Lemma 02S9","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. The maps Z_k(X) → K_0(Coh_≤ k(X)/Coh_≤ k - 1(X)), ∑ n_Z[Z] ↦ [bigoplus_n_Z > 0 O_Z^⊕ n_Z] - [bigoplus_n_Z < 0 O_Z^⊕ -n_Z] and K_0(Coh_≤ k(X)/Coh_≤ k - 1(X)) → Z_k(X), F ↦ [F]_k are mutually inverse isomorphisms.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nThe maps\n$$\nZ_k(X)\n\\longrightarrow\nK_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X)),\n\\quad\n\\sum n_Z[Z] \\mapsto\n\\left[\\bigoplus\\nolimits_{n_Z > 0} \\mathcal{O}_Z^{\\oplus n_Z}\\right]\n-\n\\left[\\bigoplus\\nolimits_{n_Z < 0} \\mathcal{O}_Z^{\\oplus -n_Z}\\right]\n$$\nand\n$$\nK_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))\n\\longrightarrow\nZ_k(X),\\quad\n\\mathcal{F} \\longmapsto [\\mathcal{F}]_k\n$$\nare mutually inverse isomorphisms.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02S9","source_file":"chow.tex","source_line":3756,"source_end_line":3779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3756-L3779","statement_sha256":"32298e84f5254dbb8311d5098a4b7e5efcef6f3d3d670bb136acb6b4390285cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8182,"rank":8182,"depth":19,"x":2527.017,"y":585.016,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FDR","tag":"0FDR","title":"Chow groups and K-groups · Lemma 0FDR","summary":"Let π : X → Y be a finite morphism of schemes locally of finite type over (S, δ) as in Situation [Tag 02QL]. Then π_* : Coh(X) → Coh(Y) is an exact functor which sends Coh_≤ k(X) into Coh_≤ k(Y) and induces homomorphisms on K_0 of these categories and their quotients. The maps of Lemma [Tag 02S9] fit into a commutative diagram xymatrix Z_k(X) ar[d]^π_* ar[r] & K_0(Coh_≤ k(X)/Coh_≤ k - 1(X)) ar[d]^π_* ar[r] & Z_k(X) ar[d]^π_* Z_k(Y) ar[r] & K_0(Coh_≤ k(Y)/Coh_≤ k - 1(Y))…","statement_latex":"Let $\\pi : X \\to Y$ be a finite morphism of schemes locally of finite type\nover $(S, \\delta)$ as in Situation \\ref{situation-setup}. Then\n$\\pi_* : \\textit{Coh}(X) \\to \\textit{Coh}(Y)$ is an exact functor\nwhich sends $\\textit{Coh}_{\\leq k}(X)$ into $\\textit{Coh}_{\\leq k}(Y)$\nand induces homomorphisms on $K_0$ of these categories and\ntheir quotients. The maps of Lemma \\ref{lemma-cycles-k-group}\nfit into a commutative diagram\n$$\n\\xymatrix{\nZ_k(X) \\ar[d]^{\\pi_*} \\ar[r] &\nK_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))\n\\ar[d]^{\\pi_*} \\ar[r] &\nZ_k(X) \\ar[d]^{\\pi_*} \\\\\nZ_k(Y) \\ar[r] &\nK_0(\\textit{Coh}_{\\leq k}(Y)/\\textit{Coh}_{\\leq k - 1}(Y)) \\ar[r] &\nZ_k(Y)\n}\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDR","source_file":"chow.tex","source_line":3897,"source_end_line":3917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3897-L3917","statement_sha256":"678040c105da4fb75e19f870211c3bb104fa7060886727b50c3760fc5ba327b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8183,"rank":8183,"depth":32,"x":2521.775,"y":776.862,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FDS","tag":"0FDS","title":"Chow groups and K-groups · Lemma 0FDS","summary":"Let X be a scheme locally of finite type over (S, δ) as in Situation [Tag 02QL]. There is a canonical map CH_k(X) → K_0(Coh_≤ k + 1(X)/Coh_≤ k - 1(X)) induced by the map Z_k(X) → K_0(Coh_≤ k(X)/Coh_≤ k - 1(X)) from Lemma [Tag 02S9].","statement_latex":"Let $X$ be a scheme locally of finite type over $(S, \\delta)$\nas in Situation \\ref{situation-setup}. There is a canonical map\n$$\n\\CH_k(X)\n\\longrightarrow\nK_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))\n$$\ninduced by the map\n$Z_k(X) \\to K_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))$\nfrom Lemma \\ref{lemma-cycles-k-group}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDS","source_file":"chow.tex","source_line":3933,"source_end_line":3945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L3933-L3945","statement_sha256":"18209f95ef3ff1008d86c0b7df079038f74cdf79173572ccbc0e8682682d1b63","origin":"The Stacks Project","memory_eligible":false,"source_rank":8184,"rank":8184,"depth":33,"x":2371.142,"y":632.21,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FDT","tag":"0FDT","title":"Chow groups and K-groups · Lemma 0FDT","summary":"Let X be a locally Noetherian scheme. Let Z ⊂ X be a closed subscheme. Denote Coh_Z(X) ⊂ Coh(X) the Serre subcategory of coherent O_X-modules whose set theoretic support is contained in Z. Then the exact inclusion functor Coh(Z) → Coh_Z(X) induces an isomorphism K'_0(Z) = K_0(Coh(Z)) → K_0(Coh_Z(X))","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $Z \\subset X$ be a closed\nsubscheme. Denote $\\textit{Coh}_Z(X) \\subset \\textit{Coh}(X)$\nthe Serre subcategory of coherent $\\mathcal{O}_X$-modules whose\nset theoretic support is contained in $Z$. Then the exact inclusion\nfunctor $\\textit{Coh}(Z) \\to \\textit{Coh}_Z(X)$ induces\nan isomorphism\n$$\nK'_0(Z) = K_0(\\textit{Coh}(Z)) \\longrightarrow K_0(\\textit{Coh}_Z(X))\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDT","source_file":"chow.tex","source_line":4049,"source_end_line":4060,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4049-L4060","statement_sha256":"e22faa170a8b4173c27e00f3989b91252a4fa2cd11ce835b842cef4c187c806a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8185,"rank":8185,"depth":34,"x":2598.877,"y":653.432,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SJ","tag":"02SJ","title":"The divisor associated to an invertible sheaf · Definition 02SJ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Assume X is integral and n = dim_δ(X). Let L be an invertible O_X-module. • For any nonzero meromorphic section s of L we define the Weil divisor associated to s is the (n - 1)-cycle div_L(s) = ∑ ord_Z, L(s) [Z] defined in Divisors, Definition [Tag 0BE6]. This makes sense because Weil divisors have δ-dimension n - 1 by Lemma [Tag 02RK]. • We define Weil divisor associated to L as c_1(L) ∩…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Assume $X$ is\nintegral and $n = \\dim_\\delta(X)$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$\nwe define the {\\it Weil divisor associated to $s$} is the\n$(n - 1)$-cycle\n$$\n\\text{div}_\\mathcal{L}(s) =\n\\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z]\n$$\ndefined in Divisors, Definition\n\\ref{divisors-definition-divisor-invertible-sheaf}.\nThis makes sense because Weil divisors have $\\delta$-dimension $n - 1$\nby Lemma \\ref{lemma-divisor-delta-dimension}.\n\\item We define {\\it Weil divisor associated to $\\mathcal{L}$} as\n$$\nc_1(\\mathcal{L}) \\cap [X] =\n\\text{class of }\\text{div}_\\mathcal{L}(s) \\in \\CH_{n - 1}(X)\n$$\nwhere $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over\n$X$. This is well defined by\nDivisors, Lemma \\ref{divisors-lemma-divisor-meromorphic-well-defined}.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The divisor associated to an invertible sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SJ","source_file":"chow.tex","source_line":4148,"source_end_line":4175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4148-L4175","statement_sha256":"2a6a834e883e3befecf71005d55be5b52d0a935889e3cb5b638bc6e1aa15e6bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8186,"rank":8186,"depth":12,"x":2413.61,"y":767.172,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SK","tag":"02SK","title":"The divisor associated to an invertible sheaf · Lemma 02SK","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Assume X is integral and n = dim_δ(X). Let L be an invertible O_X-module. Let s ∈ Γ(X, L) be a nonzero global section. Then div_L(s) = [Z(s)]_n - 1 in Z_n - 1(X) and c_1(L) ∩ [X] = [Z(s)]_n - 1 in CH_n - 1(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Assume $X$ is\nintegral and $n = \\dim_\\delta(X)$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$ be a nonzero global section.\nThen\n$$\n\\text{div}_\\mathcal{L}(s) = [Z(s)]_{n - 1}\n$$\nin $Z_{n - 1}(X)$ and\n$$\nc_1(\\mathcal{L}) \\cap [X] = [Z(s)]_{n - 1}\n$$\nin $\\CH_{n - 1}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The divisor associated to an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SK","source_file":"chow.tex","source_line":4197,"source_end_line":4213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4197-L4213","statement_sha256":"170ed351be81e2dd93de3d43830c7760aa2bfeece20b0f4ebd1058a2c4484773","origin":"The Stacks Project","memory_eligible":false,"source_rank":8187,"rank":8187,"depth":1,"x":2458.823,"y":577.9,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SM","tag":"02SM","title":"The divisor associated to an invertible sheaf · Lemma 02SM","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Assume X, Y are integral and n = dim_δ(Y). Let L be an invertible O_Y-module. Let f : X → Y be a flat morphism of relative dimension r. Then f^*(c_1(L) ∩ [Y]) = c_1(f^*L) ∩ [X] in CH_n + r - 1(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$\nare integral and $n = \\dim_\\delta(Y)$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$. Then\n$$\nf^*(c_1(\\mathcal{L}) \\cap [Y]) = c_1(f^*\\mathcal{L}) \\cap [X]\n$$\nin $\\CH_{n + r - 1}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The divisor associated to an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SM","source_file":"chow.tex","source_line":4236,"source_end_line":4247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4236-L4247","statement_sha256":"67ccf77a89d550fd5d6f6acb0977ae510979f68efd0ef5694fd88f50700d8bd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8188,"rank":8188,"depth":6,"x":2577.864,"y":743.363,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SO","tag":"02SO","title":"Intersecting with an invertible sheaf · Definition 02SO","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L be an invertible O_X-module. We define, for every integer k, an operation c_1(L) ∩ - : Z_k + 1(X) → CH_k(X) called intersection with the first Chern class of L. • Given an integral closed subscheme i : W → X with dim_δ(W) = k + 1 we define c_1(L) ∩ [W] = i_*(c_1(i^*L) ∩ [W]) where the right hand side is defined in Definition [Tag 02SJ]. • For a general (k + 1)-cycle α = ∑ n_i [W_i] we…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nWe define, for every integer $k$, an operation\n$$\nc_1(\\mathcal{L}) \\cap - :\nZ_{k + 1}(X) \\to \\CH_k(X)\n$$\ncalled {\\it intersection with the first Chern class of $\\mathcal{L}$}.\n\\begin{enumerate}\n\\item Given an integral closed subscheme $i : W \\to X$ with\n$\\dim_\\delta(W) = k + 1$ we define\n$$\nc_1(\\mathcal{L}) \\cap [W] = i_*(c_1({i^*\\mathcal{L}}) \\cap [W])\n$$\nwhere the right hand side is defined in\nDefinition \\ref{definition-divisor-invertible-sheaf}.\n\\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_i [W_i]$ we set\n$$\nc_1(\\mathcal{L}) \\cap \\alpha = \\sum n_i c_1(\\mathcal{L}) \\cap [W_i]\n$$\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SO","source_file":"chow.tex","source_line":4283,"source_end_line":4307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4283-L4307","statement_sha256":"18b223bef14cdf5bd860e11ce0f3e32a7b591431ae99e5b70b9895d87abc325f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8189,"rank":8189,"depth":13,"x":2356.703,"y":688.821,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SP","tag":"02SP","title":"Intersecting with an invertible sheaf · Lemma 02SP","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L, N be an invertible sheaves on X. Then c_1(L) ∩ α + c_1(N) ∩ α = c_1(L ⊗_O_X N) ∩ α in CH_k(X) for every α ∈ Z_k + 1(X). Moreover, c_1(O_X) ∩ α = 0 for all α.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$, $\\mathcal{N}$ be an invertible sheaves on $X$.\nThen\n$$\nc_1(\\mathcal{L}) \\cap \\alpha  + c_1(\\mathcal{N}) \\cap \\alpha =\nc_1(\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}) \\cap \\alpha\n$$\nin $\\CH_k(X)$ for every $\\alpha \\in Z_{k + 1}(X)$. Moreover,\n$c_1(\\mathcal{O}_X) \\cap \\alpha = 0$ for all $\\alpha$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SP","source_file":"chow.tex","source_line":4329,"source_end_line":4341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4329-L4341","statement_sha256":"29e25ddb19a9b16ac9feabc1883fd6e02eb8179ef43661b1d0c54bae7b708d31","origin":"The Stacks Project","memory_eligible":false,"source_rank":8190,"rank":8190,"depth":1,"x":2563.946,"y":603.423,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EPI","tag":"0EPI","title":"Intersecting with an invertible sheaf · Lemma 0EPI","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let Y be locally of finite type over S. Let L be an invertible O_Y-module. Let s ∈ Γ(Y, L). Assume • dim_δ(Y) ≤ k + 1, • dim_δ(Z(s)) ≤ k, and • for every generic point xi of an irreducible component of Z(s) of δ-dimension k the multiplication by s induces an injection O_Y, xi → L_xi. Write [Y]_k + 1 = ∑ n_i[Y_i] where Y_i ⊂ Y are the irreducible components of Y of δ-dimension k + 1. Set s_i = s|_Y_i ∈ Γ(Y_i, L|_Y_i). Then [Z(s)]_k…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $Y$ be locally of finite type over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module.\nLet $s \\in \\Gamma(Y, \\mathcal{L})$.\nAssume\n\\begin{enumerate}\n\\item $\\dim_\\delta(Y) \\leq k + 1$,\n\\item $\\dim_\\delta(Z(s)) \\leq k$, and\n\\item for every generic point $\\xi$ of an irreducible component of\n$Z(s)$ of $\\delta$-dimension $k$ the multiplication by $s$\ninduces an injection $\\mathcal{O}_{Y, \\xi} \\to \\mathcal{L}_\\xi$.\n\\end{enumerate}\nWrite $[Y]_{k + 1} = \\sum n_i[Y_i]$ where $Y_i \\subset Y$ are the\nirreducible components of $Y$ of $\\delta$-dimension $k + 1$.\nSet $s_i = s|_{Y_i} \\in \\Gamma(Y_i, \\mathcal{L}|_{Y_i})$. Then\n\\begin{equation}\n\n[Z(s)]_k =  \\sum n_i[Z(s_i)]_k\n\\end{equation}\nas $k$-cycles on $Y$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPI","source_file":"chow.tex","source_line":4357,"source_end_line":4379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4357-L4379","statement_sha256":"60ca117522d7d52a8c4b004e75af4ceb5b75b6fece7c43796bf756179b4484fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":8191,"rank":8191,"depth":15,"x":2479.68,"y":784.25,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SQ","tag":"02SQ","title":"Intersecting with an invertible sheaf · Lemma 02SQ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L be an invertible O_X-module. Let Y ⊂ X be a closed subscheme. Let s ∈ Γ(Y, L|_Y). Assume • dim_δ(Y) ≤ k + 1, • dim_δ(Z(s)) ≤ k, and • for every generic point xi of an irreducible component of Z(s) of δ-dimension k the multiplication by s induces an injection O_Y, xi → (L|_Y)_xi|_Y.. Then c_1(L) ∩ [Y]_k + 1 = [Z(s)]_k in CH_k(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $Y \\subset X$ be a closed subscheme.\nLet $s \\in \\Gamma(Y, \\mathcal{L}|_Y)$.\nAssume\n\\begin{enumerate}\n\\item $\\dim_\\delta(Y) \\leq k + 1$,\n\\item $\\dim_\\delta(Z(s)) \\leq k$, and\n\\item for every generic point $\\xi$ of an irreducible component of\n$Z(s)$ of $\\delta$-dimension $k$ the multiplication by $s$\ninduces an injection\n$\\mathcal{O}_{Y, \\xi} \\to (\\mathcal{L}|_Y)_\\xi$\\footnote{For example,\nthis holds if $s$ is a regular section of $\\mathcal{L}|_Y$.}.\n\\end{enumerate}\nThen\n$$\nc_1(\\mathcal{L}) \\cap [Y]_{k + 1} = [Z(s)]_k\n$$\nin $\\CH_k(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SQ","source_file":"chow.tex","source_line":4431,"source_end_line":4453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4431-L4453","statement_sha256":"538433bfcaa0572ac47b21e6596172eb74043e11b40e0f2b33ba39d7a670fba4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8192,"rank":8192,"depth":16,"x":2396.28,"y":602.835,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EPJ","tag":"0EPJ","title":"Intersecting with an invertible sheaf and push and pull · Lemma 0EPJ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a flat morphism of relative dimension r. Let L be an invertible sheaf on Y. Assume Y is integral and n = dim_δ(Y). Let s be a nonzero meromorphic section of L. Then we have f^*div_L(s) = ∑ n_idiv_f^*L|_X_i(s_i) in Z_n + r - 1(X). Here the sum is over the irreducible components X_i ⊂ X of δ-dimension n + r, the section s_i = f|_X_i^*(s) is the pullback of s, and n_i =…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $\\mathcal{L}$ be an invertible sheaf on $Y$.\nAssume $Y$ is integral and $n = \\dim_\\delta(Y)$.\nLet $s$ be a nonzero meromorphic section of $\\mathcal{L}$.\nThen we have\n$$\nf^*\\text{div}_\\mathcal{L}(s) = \\sum n_i\\text{div}_{f^*\\mathcal{L}|_{X_i}}(s_i)\n$$\nin $Z_{n + r - 1}(X)$. Here the sum is over the irreducible\ncomponents $X_i \\subset X$ of $\\delta$-dimension $n + r$,\nthe section $s_i = f|_{X_i}^*(s)$ is the pullback of $s$, and\n$n_i = m_{X_i, X}$ is the multiplicity of $X_i$ in $X$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPJ","source_file":"chow.tex","source_line":4483,"source_end_line":4499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4483-L4499","statement_sha256":"00369cf77084c269280705bd122f2800725c696371cbe1aaa27c7a3b5b2efb45","origin":"The Stacks Project","memory_eligible":false,"source_rank":8193,"rank":8193,"depth":26,"x":2603.966,"y":689.409,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SS","tag":"02SS","title":"Intersecting with an invertible sheaf and push and pull · Lemma 02SS","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a flat morphism of relative dimension r. Let L be an invertible sheaf on Y. Let α be a k-cycle on Y. Then f^*(c_1(L) ∩ α) = c_1(f^*L) ∩ f^*α in CH_k + r - 1(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $\\mathcal{L}$ be an invertible sheaf on $Y$.\nLet $\\alpha$ be a $k$-cycle on $Y$.\nThen\n$$\nf^*(c_1(\\mathcal{L}) \\cap \\alpha) = c_1(f^*\\mathcal{L}) \\cap f^*\\alpha\n$$\nin $\\CH_{k + r - 1}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SS","source_file":"chow.tex","source_line":4532,"source_end_line":4544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4532-L4544","statement_sha256":"ca48705dcc8c5fe852a25ced795dd3416bd344355945c166c7129930a7c4067d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8194,"rank":8194,"depth":27,"x":2380.881,"y":743.494,"cluster":"divisors-intersection-theory"},{"id":"stacks:02ST","tag":"02ST","title":"Intersecting with an invertible sheaf and push and pull · Lemma 02ST","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a proper morphism. Let L be an invertible sheaf on Y. Let s be a nonzero meromorphic section s of L on Y. Assume X, Y integral, f dominant, and dim_δ(X) = dim_δ(Y). Then f_*(div_f^*L(f^*s)) = [R(X) : R(Y)]div_L(s). as cycles on Y. In particular f_*(c_1(f^*L) ∩ [X]) = [R(X) : R(Y)] c_1(L) ∩ [Y] = c_1(L) ∩ f_*[X]","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a proper morphism.\nLet $\\mathcal{L}$ be an invertible sheaf on $Y$.\nLet $s$ be a nonzero meromorphic section $s$ of $\\mathcal{L}$ on $Y$.\nAssume $X$, $Y$ integral, $f$ dominant, and $\\dim_\\delta(X) = \\dim_\\delta(Y)$.\nThen\n$$\nf_*\\left(\\text{div}_{f^*\\mathcal{L}}(f^*s)\\right) =\n[R(X) : R(Y)]\\text{div}_\\mathcal{L}(s).\n$$\nas cycles on $Y$. In particular\n$$\nf_*(c_1(f^*\\mathcal{L}) \\cap [X]) =\n[R(X) : R(Y)] c_1(\\mathcal{L}) \\cap [Y] =\nc_1(\\mathcal{L}) \\cap f_*[X]\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ST","source_file":"chow.tex","source_line":4589,"source_end_line":4608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4589-L4608","statement_sha256":"66d2ba273bef0ec470ff064f3a5f0603a3c55575ba7986ab432c583f1f9514e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8195,"rank":8195,"depth":44,"x":2502.06,"y":576.791,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SU","tag":"02SU","title":"Intersecting with an invertible sheaf and push and pull · Lemma 02SU","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let p : X → Y be a proper morphism. Let α ∈ Z_k + 1(X). Let L be an invertible sheaf on Y. Then p_*(c_1(p^*L) ∩ α) = c_1(L) ∩ p_*α in CH_k(Y).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $p : X \\to Y$ be a proper morphism.\nLet $\\alpha \\in Z_{k + 1}(X)$.\nLet $\\mathcal{L}$ be an invertible sheaf on $Y$.\nThen\n$$\np_*(c_1(p^*\\mathcal{L}) \\cap \\alpha) = c_1(\\mathcal{L}) \\cap p_*\\alpha\n$$\nin $\\CH_k(Y)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SU","source_file":"chow.tex","source_line":4630,"source_end_line":4642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4630-L4642","statement_sha256":"1943894e52ab9692cabe908e61d23835c09b85ea07e8aef338edc256c6993773","origin":"The Stacks Project","memory_eligible":false,"source_rank":8196,"rank":8196,"depth":45,"x":2546.826,"y":768.748,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AYC","tag":"0AYC","title":"Key formula · Lemma 0AYC","summary":"In the situation above the cycle ∑ (Z_i → X)_*( ord_B_i(f_i) div_N|_Z_i(t_i|_Z_i) - ord_B_i(g_i) div_L|_Z_i(s_i|_Z_i) ) is equal to the cycle ∑ (Z_i → X)_*div(∂_B_i(f_i, g_i))","statement_latex":"In the situation above the cycle\n$$\n\\sum\n(Z_i \\to X)_*\\left(\n\\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) -\n\\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \\right)\n$$\nis equal to the cycle\n$$\n\\sum (Z_i \\to X)_*\\text{div}(\\partial_{B_i}(f_i, g_i))\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The key formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYC","source_file":"chow.tex","source_line":4745,"source_end_line":4758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4745-L4758","statement_sha256":"746c89adfa9cbc1576d90c3efa67e11aa1bf5d5faab462a9125f2a32f993af1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8197,"rank":8197,"depth":47,"x":2359.183,"y":652.438,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TH","tag":"02TH","title":"Intersecting with an invertible sheaf and rational equivalence · Lemma 02TH","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Assume X integral and dim_δ(X) = n. Let L, N be invertible on X. Choose a nonzero meromorphic section s of L and a nonzero meromorphic section t of N. Set α = div_L(s) and β = div_N(t). Then c_1(N) ∩ α = c_1(L) ∩ β in CH_n - 2(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nAssume $X$ integral and $\\dim_\\delta(X) = n$.\nLet $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$.\nChoose a nonzero meromorphic section $s$ of $\\mathcal{L}$\nand a nonzero meromorphic section $t$ of $\\mathcal{N}$.\nSet $\\alpha = \\text{div}_\\mathcal{L}(s)$ and\n$\\beta = \\text{div}_\\mathcal{N}(t)$.\nThen\n$$\nc_1(\\mathcal{N}) \\cap \\alpha\n=\nc_1(\\mathcal{L}) \\cap \\beta\n$$\nin $\\CH_{n - 2}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TH","source_file":"chow.tex","source_line":4923,"source_end_line":4940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4923-L4940","statement_sha256":"624258f3f96610a7cc7d7570c0a7571f58ca4ba3874f4e0357b867be5f177fcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8198,"rank":8198,"depth":48,"x":2591.411,"y":631.701,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TI","tag":"02TI","title":"Intersecting with an invertible sheaf and rational equivalence · Lemma 02TI","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L be invertible on X. The operation α ↦ c_1(L) ∩ α factors through rational equivalence to give an operation c_1(L) ∩ - : CH_k + 1(X) → CH_k(X)","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$ be invertible on $X$.\nThe operation $\\alpha \\mapsto c_1(\\mathcal{L}) \\cap \\alpha$\nfactors through rational equivalence to give an operation\n$$\nc_1(\\mathcal{L}) \\cap - : \\CH_{k + 1}(X) \\to \\CH_k(X)\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TI","source_file":"chow.tex","source_line":4947,"source_end_line":4957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L4947-L4957","statement_sha256":"1d49903b0661f31621049dab4f0882ab9b21e33b41c8b0e304e02dc72d0b5421","origin":"The Stacks Project","memory_eligible":false,"source_rank":8199,"rank":8199,"depth":49,"x":2436.627,"y":778.972,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TJ","tag":"02TJ","title":"Intersecting with an invertible sheaf and rational equivalence · Lemma 02TJ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L, N be invertible on X. For any α ∈ CH_k + 2(X) we have c_1(L) ∩ c_1(N) ∩ α = c_1(N) ∩ c_1(L) ∩ α as elements of CH_k(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$.\nFor any $\\alpha \\in \\CH_{k + 2}(X)$ we have\n$$\nc_1(\\mathcal{L}) \\cap c_1(\\mathcal{N}) \\cap \\alpha\n=\nc_1(\\mathcal{N}) \\cap c_1(\\mathcal{L}) \\cap \\alpha\n$$\nas elements of $\\CH_k(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with an invertible sheaf and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TJ","source_file":"chow.tex","source_line":5006,"source_end_line":5018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5006-L5018","statement_sha256":"99862080dc57ea13905839d0e0e3ec2f247c52eda2d06ef22b46e3184676fdc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8200,"rank":8200,"depth":49,"x":2432.325,"y":582.272,"cluster":"divisors-intersection-theory"},{"id":"stacks:02T8","tag":"02T8","title":"Gysin homomorphisms · Definition 02T8","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let (L, s) be a pair consisting of an invertible sheaf and a global section s ∈ Γ(X, L). Let D = Z(s) be the zero scheme of s, and denote i : D → X the closed immersion. We define, for every integer k, a Gysin homomorphism i^* : Z_k + 1(X) → CH_k(D). by the following rules: • Given an integral closed subscheme W ⊂ X with dim_δ(W) = k + 1 we define • if W not ⊂ D, then i^*[W] = [D ∩ W]_k as a…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $(\\mathcal{L}, s)$ be a pair consisting of an invertible\nsheaf and a global section $s \\in \\Gamma(X, \\mathcal{L})$.\nLet $D = Z(s)$ be the zero scheme of $s$, and\ndenote $i : D \\to X$ the closed immersion.\nWe define, for every integer $k$, a {\\it Gysin homomorphism}\n$$\ni^* : Z_{k + 1}(X) \\to \\CH_k(D).\n$$\nby the following rules:\n\\begin{enumerate}\n\\item Given an integral closed subscheme $W \\subset X$ with\n$\\dim_\\delta(W) = k + 1$ we define\n\\begin{enumerate}\n\\item if $W \\not \\subset D$, then $i^*[W] = [D \\cap W]_k$ as a\n$k$-cycle on $D$, and\n\\item if $W \\subset D$, then\n$i^*[W] = i'_*(c_1(\\mathcal{L}|_W) \\cap [W])$,\nwhere $i' : W \\to D$ is the induced closed immersion.\n\\end{enumerate}\n\\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_j[W_j]$\nwe set\n$$\ni^*\\alpha = \\sum n_j i^*[W_j]\n$$\n\\item If $D$ is an effective Cartier divisor, then we denote\n$D \\cdot \\alpha = i_*i^*\\alpha$ the pushforward of the class $i^*\\alpha$\nto a class on $X$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02T8","source_file":"chow.tex","source_line":5068,"source_end_line":5100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5068-L5100","statement_sha256":"63792dfb4a855fb04e27b41c842765eba88e28c9c5f867695e35367b13f1e3e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8201,"rank":8201,"depth":0,"x":2593.905,"y":725.075,"cluster":"divisors-intersection-theory"},{"id":"stacks:02T9","tag":"02T9","title":"Gysin homomorphisms · Lemma 02T9","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let (L, s, i : D → X) be as in Definition [Tag 02T8]. Let α be a (k + 1)-cycle on X. Then i_*i^*α = c_1(L) ∩ α in CH_k(X). In particular, if D is an effective Cartier divisor, then D · α = c_1(O_X(D)) ∩ α.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally\nof finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}. Let $\\alpha$ be a\n$(k + 1)$-cycle on $X$. Then $i_*i^*\\alpha = c_1(\\mathcal{L}) \\cap \\alpha$\nin $\\CH_k(X)$. In particular, if $D$ is an effective Cartier divisor, then\n$D \\cdot \\alpha = c_1(\\mathcal{O}_X(D)) \\cap \\alpha$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02T9","source_file":"chow.tex","source_line":5137,"source_end_line":5145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5137-L5145","statement_sha256":"bc82b2dd8171e0e53aa966b719d54feb75ce2681752889f00a0715ae4056f08f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8202,"rank":8202,"depth":17,"x":2359.592,"y":711.439,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TB","tag":"02TB","title":"Gysin homomorphisms · Lemma 02TB","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let (L, s, i : D → X) be as in Definition [Tag 02T8]. • Let Z ⊂ X be a closed subscheme such that dim_δ(Z) ≤ k + 1 and such that D ∩ Z is an effective Cartier divisor on Z. Then i^*[Z]_k + 1 = [D ∩ Z]_k. • Let F be a coherent sheaf on X such that dim_δ(Supp(F)) ≤ k + 1 and s : F → F ⊗_O_X L is injective. Then i^*[F]_k + 1 = [i^*F]_k in CH_k(D).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}.\n\\begin{enumerate}\n\\item Let $Z \\subset X$ be a closed subscheme such\nthat $\\dim_\\delta(Z) \\leq k + 1$ and such that\n$D \\cap Z$ is an effective Cartier divisor on $Z$. Then\n$i^*[Z]_{k + 1} = [D \\cap Z]_k$.\n\\item Let $\\mathcal{F}$ be a coherent sheaf on $X$\nsuch that $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$ and\n$s : \\mathcal{F} \\to \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}$\nis injective. Then\n$$\ni^*[\\mathcal{F}]_{k + 1} = [i^*\\mathcal{F}]_k\n$$\nin $\\CH_k(D)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TB","source_file":"chow.tex","source_line":5169,"source_end_line":5189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5169-L5189","statement_sha256":"7921b8b655a7abff2225196c13bcb04a518b2b840ccb6a4b1511b33e05e5b26a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8203,"rank":8203,"depth":15,"x":2543.594,"y":588.368,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TA","tag":"02TA","title":"Gysin homomorphisms · Lemma 02TA","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X' → X be a proper morphism of schemes locally of finite type over S. Let (L, s, i : D → X) be as in Definition [Tag 02T8]. Form the diagram xymatrix D' ar[d]_g ar[r]_i' & X' ar[d]^f D ar[r]^i & X as in Remark [Tag 0B6Y]. For any (k + 1)-cycle α' on X' we have i^*f_*α' = g_*(i')^*α' in CH_k(D) (this makes sense as f_* is defined on the level of cycles).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X' \\to X$ be a proper morphism of schemes\nlocally of finite type over $S$.\nLet $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}.\nForm the diagram\n$$\n\\xymatrix{\nD' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\\nD \\ar[r]^i & X\n}\n$$\nas in Remark \\ref{remark-pullback-pairs}.\nFor any $(k + 1)$-cycle $\\alpha'$ on $X'$ we have\n$i^*f_*\\alpha' = g_*(i')^*\\alpha'$ in $\\CH_k(D)$\n(this makes sense as $f_*$ is defined on the level of cycles).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TA","source_file":"chow.tex","source_line":5263,"source_end_line":5281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5263-L5281","statement_sha256":"a06f6068ba8d5070a70f04f900acfd880d9ff7c86257f66812527c39ff20cd0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8204,"rank":8204,"depth":45,"x":2506.833,"y":783.797,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B71","tag":"0B71","title":"Gysin homomorphisms · Lemma 0B71","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X' → X be a flat morphism of relative dimension r of schemes locally of finite type over S. Let (L, s, i : D → X) be as in Definition [Tag 02T8]. Form the diagram xymatrix D' ar[d]_g ar[r]_i' & X' ar[d]^f D ar[r]^i & X as in Remark [Tag 0B6Y]. For any (k + 1)-cycle α on X we have (i')^*f^*α = g^*i^*α in CH_k + r(D') (this makes sense as f^* is defined on the level of cycles).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $f : X' \\to X$\nbe a flat morphism of relative dimension $r$ of schemes locally of finite type\nover $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}. Form the diagram\n$$\n\\xymatrix{\nD' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\\nD \\ar[r]^i & X\n}\n$$\nas in Remark \\ref{remark-pullback-pairs}.\nFor any $(k + 1)$-cycle $\\alpha$ on $X$ we have\n$(i')^*f^*\\alpha = g^*i^*\\alpha$ in $\\CH_{k + r}(D')$\n(this makes sense as $f^*$ is defined on the level of cycles).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B71","source_file":"chow.tex","source_line":5302,"source_end_line":5318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5302-L5318","statement_sha256":"e8a4a938d5c18e8c3fd283bfbccc1feac758159a4518a7f912a1f51fccf96c09","origin":"The Stacks Project","memory_eligible":false,"source_rank":8205,"rank":8205,"depth":28,"x":2376.6,"y":618.602,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TM","tag":"02TM","title":"Gysin homomorphisms and rational equivalence · Lemma 02TM","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let X be integral and n = dim_δ(X). Let i : D → X be an effective Cartier divisor. Let N be an invertible O_X-module and let t be a nonzero meromorphic section of N. Then i^*div_N(t) = c_1(N|_D) ∩ [D]_n - 1 in CH_n - 2(D).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $X$ be integral and $n = \\dim_\\delta(X)$.\nLet $i : D \\to X$ be an effective Cartier divisor.\nLet $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module\nand let $t$ be a nonzero meromorphic section of $\\mathcal{N}$.\nThen $i^*\\text{div}_\\mathcal{N}(t) = c_1(\\mathcal{N}|_D) \\cap [D]_{n - 1}$\nin $\\CH_{n - 2}(D)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TM","source_file":"chow.tex","source_line":5351,"source_end_line":5361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5351-L5361","statement_sha256":"a725156114d5abc10e5ca02b06df8c2ef284458200ad688c94f5ec54f3c1aaf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8206,"rank":8206,"depth":48,"x":2605.794,"y":666.584,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TO","tag":"02TO","title":"Gysin homomorphisms and rational equivalence · Lemma 02TO","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let (L, s, i : D → X) be as in Definition [Tag 02T8]. The Gysin homomorphism factors through rational equivalence to give a map i^* : CH_k + 1(X) → CH_k(D).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}.\nThe Gysin homomorphism factors through rational equivalence to\ngive a map $i^* : \\CH_{k + 1}(X) \\to \\CH_k(D)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TO","source_file":"chow.tex","source_line":5419,"source_end_line":5427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5419-L5427","statement_sha256":"1035ab210d1f0f33e71476a835000b075c6c127e652704aaafe6db4c348dd604","origin":"The Stacks Project","memory_eligible":false,"source_rank":8207,"rank":8207,"depth":50,"x":2397.917,"y":761.384,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F95","tag":"0F95","title":"Gysin homomorphisms and rational equivalence · Lemma 0F95","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let (L, s, i : D → X) be as in Definition [Tag 02T8]. Then i^*i_* : CH_k(D) → CH_k - 1(D) sends α to c_1(L|_D) ∩ α.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally\nof finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}. Then\n$i^*i_* : \\CH_k(D) \\to \\CH_{k - 1}(D)$ sends $\\alpha$ to\n$c_1(\\mathcal{L}|_D) \\cap \\alpha$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F95","source_file":"chow.tex","source_line":5462,"source_end_line":5469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5462-L5469","statement_sha256":"a3157f1bcec2d02a5a2f7e08d8ddb75c5f7644da0f691556466119d3d066fdcf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8208,"rank":8208,"depth":1,"x":2475.073,"y":573.266,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B72","tag":"0B72","title":"Gysin homomorphisms and rational equivalence · Lemma 0B72","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let (L, s, i : D → X) be a triple as in Definition [Tag 02T8]. Let N be an invertible O_X-module. Then i^*(c_1(N) ∩ α) = c_1(i^*N) ∩ i^*α in CH_k - 2(D) for all α ∈ CH_k(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be\nlocally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$\nbe a triple as in Definition \\ref{definition-gysin-homomorphism}.\nLet $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module.\nThen $i^*(c_1(\\mathcal{N}) \\cap \\alpha) = c_1(i^*\\mathcal{N}) \\cap i^*\\alpha$\nin $\\CH_{k - 2}(D)$ for all $\\alpha \\in \\CH_k(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B72","source_file":"chow.tex","source_line":5477,"source_end_line":5485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5477-L5485","statement_sha256":"c00d0d34adf9f9c2c58cf1b62a7bbc82a3bd04c49adb7310d531ca353568c636","origin":"The Stacks Project","memory_eligible":false,"source_rank":8209,"rank":8209,"depth":51,"x":2569.588,"y":756.012,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B73","tag":"0B73","title":"Gysin homomorphisms and rational equivalence · Lemma 0B73","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let (L, s, i : D → X) and (L', s', i' : D' → X) be two triples as in Definition [Tag 02T8]. Then the diagram xymatrix CH_k(X) ar[r]_i^* ar[d]_(i')^* & CH_k - 1(D) ar[d]^j^* CH_k - 1(D') ar[r]^(j')^* & CH_k - 2(D ∩ D') commutes where each of the maps is a Gysin map.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally\nof finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$ and\n$(\\mathcal{L}', s', i' : D' \\to X)$ be two triples as in\nDefinition \\ref{definition-gysin-homomorphism}. Then the diagram\n$$\n\\xymatrix{\n\\CH_k(X) \\ar[r]_{i^*} \\ar[d]_{(i')^*} & \\CH_{k - 1}(D) \\ar[d]^{j^*} \\\\\n\\CH_{k - 1}(D') \\ar[r]^{(j')^*} & \\CH_{k - 2}(D \\cap D')\n}\n$$\ncommutes where each of the maps is a Gysin map.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin homomorphisms and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B73","source_file":"chow.tex","source_line":5495,"source_end_line":5508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5495-L5508","statement_sha256":"f7e07eb033a2cd8d8811746345b6040230e6a260aa168fc8a5b93720532bb34e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8210,"rank":8210,"depth":48,"x":2352.638,"y":674.777,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TR","tag":"02TR","title":"Relative effective Cartier divisors · Lemma 02TR","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let p : X → Y be a flat morphism of relative dimension r. Let i : D → X be a relative effective Cartier divisor (Divisors, Definition [Tag 062T]). Let L = O_X(D). For any α ∈ CH_k + 1(Y) we have i^*p^*α = (p|_D)^*α in CH_k + r(D) and c_1(L) ∩ p^*α = i_* ((p|_D)^*α) in CH_k + r(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $p : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $i : D \\to X$ be a relative effective Cartier divisor\n(Divisors, Definition\n\\ref{divisors-definition-relative-effective-Cartier-divisor}).\nLet $\\mathcal{L} = \\mathcal{O}_X(D)$.\nFor any $\\alpha \\in \\CH_{k + 1}(Y)$ we have\n$$\ni^*p^*\\alpha = (p|_D)^*\\alpha\n$$\nin $\\CH_{k + r}(D)$ and\n$$\nc_1(\\mathcal{L}) \\cap p^*\\alpha = i_* ((p|_D)^*\\alpha)\n$$\nin $\\CH_{k + r}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TR","source_file":"chow.tex","source_line":5597,"source_end_line":5615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5597-L5615","statement_sha256":"e52bccf6be58dc79babf70afda172dd5b61b3c9bd245b1e787b53f1f940ee769","origin":"The Stacks Project","memory_eligible":false,"source_rank":8211,"rank":8211,"depth":18,"x":2578.249,"y":611.49,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TT","tag":"02TT","title":"Affine bundles · Lemma 02TT","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let f : X → Y be a flat morphism of relative dimension r. Assume that for every y ∈ Y, there exists an open neighbourhood U ⊂ Y such that f|_f^-1(U) : f^-1(U) → U is identified with the morphism U × A^r → U. Then f^* : CH_k(Y) → CH_k + r(X) is surjective for all k ∈ Z.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\nAssume that for every $y \\in Y$, there exists an open neighbourhood\n$U \\subset Y$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \\to U$\nis identified with the morphism $U \\times \\mathbf{A}^r \\to U$.\nThen $f^* : \\CH_k(Y) \\to \\CH_{k + r}(X)$ is surjective for all\n$k \\in \\mathbf{Z}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Affine bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TT","source_file":"chow.tex","source_line":5650,"source_end_line":5660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5650-L5660","statement_sha256":"a87cc84fa87ffe99db8248fbbc6c495073816c9f1c01cd455a4cd060f0203946","origin":"The Stacks Project","memory_eligible":false,"source_rank":8212,"rank":8212,"depth":2,"x":2462.623,"y":786.412,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B74","tag":"0B74","title":"Affine bundles · Lemma 0B74","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L be an invertible O_X-module. Let p : L = underlineSpec(Sym^*(L)) → X be the associated vector bundle over X. Then p^* : CH_k(X) → CH_k + 1(L) is an isomorphism for all k.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet\n$$\np :\nL = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{L}))\n\\longrightarrow\nX\n$$\nbe the associated vector bundle over $X$.\nThen $p^* : \\CH_k(X) \\to \\CH_{k + 1}(L)$ is an isomorphism for all $k$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Affine bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B74","source_file":"chow.tex","source_line":5736,"source_end_line":5750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5736-L5750","statement_sha256":"0d95c32ff47cec620f492344d2c3f23d97a8fcbb285986556cf17236b7ff97b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8213,"rank":8213,"depth":19,"x":2407.142,"y":591.554,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F96","tag":"0F96","title":"Affine bundles · Lemma 0F96","summary":"In the situation of Lemma [Tag 0B74] denote o : X → L the zero section (see proof of the lemma). Then we have • o(X) is the zero scheme of a regular global section of p^*L^⊗ -1, • o_* : CH_k(X) → CH_k(L) as o is a closed immersion, • o^* : CH_k + 1(L) → CH_k(X) as o(X) is an effective Cartier divisor, • o^* p^* : CH_k(X) → CH_k(X) is the identity map, • o_*α = - p^*(c_1(L) ∩ α) for any α ∈ CH_k(X), and • o^* o_* : CH_k(X) → CH_k - 1(X) is equal to the map α ↦ - c_1(L) ∩ α.","statement_latex":"In the situation of Lemma \\ref{lemma-linebundle} denote $o : X \\to L$\nthe zero section (see proof of the lemma). Then we have\n\\begin{enumerate}\n\\item $o(X)$ is the zero scheme of a regular global section of\n$p^*\\mathcal{L}^{\\otimes -1}$,\n\\item $o_* : \\CH_k(X) \\to \\CH_k(L)$ as $o$ is a closed immersion,\n\\item $o^* : \\CH_{k + 1}(L) \\to \\CH_k(X)$ as $o(X)$\nis an effective Cartier divisor,\n\\item $o^* p^* : \\CH_k(X) \\to \\CH_k(X)$ is the identity map,\n\\item $o_*\\alpha = - p^*(c_1(\\mathcal{L}) \\cap \\alpha)$ for any\n$\\alpha \\in \\CH_k(X)$, and\n\\item $o^* o_* : \\CH_k(X) \\to \\CH_{k - 1}(X)$ is equal to the map\n$\\alpha \\mapsto - c_1(\\mathcal{L}) \\cap \\alpha$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Affine bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F96","source_file":"chow.tex","source_line":5774,"source_end_line":5790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5774-L5790","statement_sha256":"e2c5f5e9b51a40486c9a2ccc854b42e425489e6abe246415190baef7ecf96d91","origin":"The Stacks Project","memory_eligible":false,"source_rank":8214,"rank":8214,"depth":28,"x":2605.02,"y":703.909,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F97","tag":"0F97","title":"Affine bundles · Lemma 0F97","summary":"Let Y be a scheme. Let L_i, i = 1, 2 be invertible O_Y-modules. Let s be a global section of L_1 ⊗_O_Y L_2. Denote i : D → Y the zero scheme of s. Then there exists a commutative diagram xymatrix D_1 ar[r]_i_1 ar[d]_p_1 & L ar[d]^p & D_2 ar[l]^i_2 ar[d]^p_2 D ar[r]^i & Y & D ar[l]_i and sections s_i of p^*L_i such that the following hold: • p^*s = s_1 ⊗ s_2, • p is of finite type and flat of relative dimension 1, • D_i is the zero scheme of s_i, • D_i ≅…","statement_latex":"Let $Y$ be a scheme. Let $\\mathcal{L}_i$, $i = 1, 2$ be invertible\n$\\mathcal{O}_Y$-modules. Let $s$ be a global section of\n$\\mathcal{L}_1 \\otimes_{\\mathcal{O}_Y} \\mathcal{L}_2$.\nDenote $i : D \\to Y$ the zero scheme of $s$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\nD_1 \\ar[r]_{i_1} \\ar[d]_{p_1} &\nL \\ar[d]^p &\nD_2 \\ar[l]^{i_2} \\ar[d]^{p_2} \\\\\nD \\ar[r]^i &\nY &\nD \\ar[l]_i\n}\n$$\nand sections $s_i$ of $p^*\\mathcal{L}_i$ such that\nthe following hold:\n\\begin{enumerate}\n\\item $p^*s = s_1 \\otimes s_2$,\n\\item $p$ is of finite type and flat of relative dimension $1$,\n\\item $D_i$ is the zero scheme of $s_i$,\n\\item $D_i \\cong\n\\underline{\\Spec}(\\text{Sym}^*(\\mathcal{L}_{3 - i}^{\\otimes -1})|_D))$\nover $D$ for $i = 1, 2$,\n\\item $p^{-1}D = D_1 \\cup D_2$ (scheme theoretic union),\n\\item $D_1 \\cap D_2$ (scheme theoretic intersection) maps\nisomorphically to $D$, and\n\\item $D_1 \\cap D_2 \\to D_i$\nis the zero section of the line bundle $D_i \\to D$ for $i = 1, 2$.\n\\end{enumerate}\nMoreover, the formation of this diagram and the sections $s_i$\ncommutes with arbitrary base change.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Affine bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F97","source_file":"chow.tex","source_line":5821,"source_end_line":5855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5821-L5855","statement_sha256":"8a52334e9ed153a3d74f918f3929e21f47ab17942cfd205a649869bfcd79f597","origin":"The Stacks Project","memory_eligible":false,"source_rank":8215,"rank":8215,"depth":0,"x":2368.435,"y":733.381,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F98","tag":"0F98","title":"Affine bundles · Lemma 0F98","summary":"In the situation of Lemma [Tag 0F97] assume Y is locally of finite type over (S, δ) as in Situation [Tag 02QL]. Then we have i_1^*p^*α = p_1^*i^*α in CH_k(D_1) for all α ∈ CH_k(Y).","statement_latex":"In the situation of Lemma \\ref{lemma-decompose-section}\nassume $Y$ is locally of finite type over $(S, \\delta)$ as in\nSituation \\ref{situation-setup}. Then we have\n$i_1^*p^*\\alpha = p_1^*i^*\\alpha$\nin $\\CH_k(D_1)$ for all $\\alpha \\in \\CH_k(Y)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Affine bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F98","source_file":"chow.tex","source_line":5883,"source_end_line":5890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5883-L5890","statement_sha256":"08d609443b48f4f668b1851fa10245190890c1f4101ceafda9e77971dd03ef0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8216,"rank":8216,"depth":29,"x":2519.392,"y":577.195,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F99","tag":"0F99","title":"Affine bundles · Lemma 0F99","summary":"In Situation [Tag 02QL] let X be a scheme locally of finite type over S. Let (L, s, i : D → X) be a triple as in Definition [Tag 02T8]. There exists a commutative diagram xymatrix D' ar[r]_i' ar[d]_p & X' ar[d]^g D ar[r]^i & X such that • p and g are of finite type and flat of relative dimension 1, • p^* : CH_k(D) → CH_k + 1(D') is injective for all k, • D' ⊂ X' is the zero scheme of a global section s' ∈ Γ(X', O_X'), • p^*i^* = (i')^*g^* as maps CH_k(X) → CH_k(D').…","statement_latex":"In Situation \\ref{situation-setup} let $X$ be a scheme locally\nof finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$\nbe a triple as in Definition \\ref{definition-gysin-homomorphism}.\nThere exists a commutative diagram\n$$\n\\xymatrix{\nD' \\ar[r]_{i'} \\ar[d]_p & X' \\ar[d]^g \\\\\nD \\ar[r]^i & X\n}\n$$\nsuch that\n\\begin{enumerate}\n\\item $p$ and $g$ are of finite type and flat of relative dimension $1$,\n\\item $p^* : \\CH_k(D) \\to \\CH_{k + 1}(D')$ is injective for all $k$,\n\\item $D' \\subset X'$ is the zero scheme of a global section\n$s' \\in \\Gamma(X', \\mathcal{O}_{X'})$,\n\\item $p^*i^* = (i')^*g^*$ as maps $\\CH_k(X) \\to \\CH_k(D')$.\n\\end{enumerate}\nMoreover, these properties remain true after arbitrary base change\nby morphisms $Y \\to X$ which are locally of finite type.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Affine bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F99","source_file":"chow.tex","source_line":5939,"source_end_line":5961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L5939-L5961","statement_sha256":"251b674e1d3e92f280ec99fd6f5ae1b3ef660cd9fa8efb98f8b604d5934ae022","origin":"The Stacks Project","memory_eligible":false,"source_rank":8217,"rank":8217,"depth":30,"x":2533.698,"y":778.291,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B76","tag":"0B76","title":"Bivariant intersection theory · Definition 0B76","summary":"Similar to [F] Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a morphism of schemes locally of finite type over S. Let p ∈ Z. A bivariant class c of degree p for f is given by a rule which assigns to every locally of finite type morphism Y' → Y and every k a map c ∩ - : CH_k(Y') → CH_k - p(X') where X' = Y' ×_Y X, satisfying the following conditions • if Y\" → Y' is a proper, then c ∩ (Y\" → Y')_*α\" = (X\" → X')_*(c ∩ α\") for all α\" on Y\" where X\" = Y\" ×_Y X, •…","statement_latex":"\\begin{reference}\nSimilar to \\cite[Definition 17.1]{F}\n\\end{reference}\nLet $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.\nLet $p \\in \\mathbf{Z}$.\nA {\\it bivariant class $c$ of degree $p$ for $f$} is given by a rule\nwhich assigns to every locally of finite type morphism $Y' \\to Y$\nand every $k$ a map\n$$\nc \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X')\n$$\nwhere $X' = Y' \\times_Y X$, satisfying the following conditions\n\\begin{enumerate}\n\\item if $Y'' \\to Y'$ is a proper, then\n$c \\cap (Y'' \\to Y')_*\\alpha'' = (X'' \\to X')_*(c \\cap \\alpha'')$\nfor all $\\alpha''$ on $Y''$ where $X'' = Y'' \\times_Y X$,\n\\item if $Y'' \\to Y'$ is flat locally of finite type of\nfixed relative dimension, then\n$c \\cap (Y'' \\to Y')^*\\alpha' = (X'' \\to X')^*(c \\cap \\alpha')$\nfor all $\\alpha'$ on $Y'$, and\n\\item if $(\\mathcal{L}', s', i' : D' \\to Y')$ is as in\nDefinition \\ref{definition-gysin-homomorphism}\nwith pullback $(\\mathcal{N}', t', j' : E' \\to X')$ to $X'$,\nthen we have $c \\cap (i')^*\\alpha' = (j')^*(c \\cap \\alpha')$\nfor all $\\alpha'$ on $Y'$.\n\\end{enumerate}\nThe collection of all bivariant classes of degree $p$ for $f$ is\ndenoted $A^p(X \\to Y)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Bivariant intersection theory","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B76","source_file":"chow.tex","source_line":6086,"source_end_line":6117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6086-L6117","statement_sha256":"7f0b95f6e5dd382290ea567ffcdb04de3be98a17cc88d468fdb2f78983d71ba2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8218,"rank":8218,"depth":1,"x":2361.203,"y":637.935,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B78","tag":"0B78","title":"Bivariant intersection theory · Lemma 0B78","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a flat morphism of relative dimension r between schemes locally of finite type over S. Then the rule that to Y' → Y assigns (f')^* : CH_k(Y') → CH_k + r(X') where X' = X ×_Y Y' is a bivariant class of degree -r.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$\nbetween schemes locally of finite type over $S$.\nThen the rule that to $Y' \\to Y$ assigns\n$(f')^* : \\CH_k(Y') \\to \\CH_{k + r}(X')$ where $X' = X \\times_Y Y'$\nis a bivariant class of degree $-r$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Bivariant intersection theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B78","source_file":"chow.tex","source_line":6130,"source_end_line":6138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6130-L6138","statement_sha256":"21469fd024f1a8e9c28954c3b497e41ea4c4c0020c7807f16153fde2b444e439","origin":"The Stacks Project","memory_eligible":false,"source_rank":8219,"rank":8219,"depth":34,"x":2601.589,"y":643.561,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B79","tag":"0B79","title":"Bivariant intersection theory · Lemma 0B79","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let (L, s, i : D → X) be a triple as in Definition [Tag 02T8]. Then the rule that to f : X' → X assigns (i')^* : CH_k(X') → CH_k - 1(D') where D' = D ×_X X' is a bivariant class of degree 1.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in\nDefinition \\ref{definition-gysin-homomorphism}.\nThen the rule that to $f : X' \\to X$ assigns\n$(i')^* : \\CH_k(X') \\to \\CH_{k - 1}(D')$ where $D' = D \\times_X X'$\nis a bivariant class of degree $1$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Bivariant intersection theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B79","source_file":"chow.tex","source_line":6148,"source_end_line":6157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6148-L6157","statement_sha256":"02381dca0492bffe41c3df244cc6fc625504904aa18bd4f4e60c40fbe39555d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8220,"rank":8220,"depth":51,"x":2419.565,"y":775.989,"cluster":"divisors-intersection-theory"},{"id":"stacks:0EPK","tag":"0EPK","title":"Bivariant intersection theory · Lemma 0EPK","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y and g : Y → Z be morphisms of schemes locally of finite type over S. Let c ∈ A^p(X → Z) and assume f is proper. Then the rule that to Z' → Z assigns α ↦ f'_*(c ∩ α) is a bivariant class denoted f_* ∘ c ∈ A^p(Y → Z).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of\nschemes locally of finite type over $S$.\nLet $c \\in A^p(X \\to Z)$ and assume $f$ is proper.\nThen the rule that to $Z' \\to Z$ assigns\n$\\alpha \\longmapsto f'_*(c \\cap \\alpha)$\nis a bivariant class denoted $f_* \\circ c \\in A^p(Y \\to Z)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Bivariant intersection theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPK","source_file":"chow.tex","source_line":6166,"source_end_line":6175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6166-L6175","statement_sha256":"28b3b4582c780a1a48561ca262e56169d6bb53a343495b273284772c5e74f9c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8221,"rank":8221,"depth":46,"x":2447.328,"y":574.787,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B7E","tag":"0B7E","title":"Chow cohomology and the first Chern class · Definition 0B7E","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. The Chow cohomology of X is the graded Z-algebra A^*(X) whose degree p component is A^p(X → X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. The {\\it Chow cohomology}\nof $X$ is the graded $\\mathbf{Z}$-algebra $A^*(X)$ whose degree\n$p$ component is $A^p(X \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow cohomology and the first Chern class","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7E","source_file":"chow.tex","source_line":6275,"source_end_line":6281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6275-L6281","statement_sha256":"8812beb91983448e686b3743507c2dbe4fcf356e61486182b6ee8292cc071510","origin":"The Stacks Project","memory_eligible":false,"source_rank":8222,"rank":8222,"depth":0,"x":2588.845,"y":739.122,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B77","tag":"0B77","title":"Chow cohomology and the first Chern class · Lemma 0B77","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L be an invertible O_X-module. Then the rule that to f : X' → X assigns c_1(f^*L) ∩ - : CH_k(X') → CH_k - 1(X') is a bivariant class of degree 1.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen the rule that to $f : X' \\to X$ assigns\n$c_1(f^*\\mathcal{L}) \\cap - : \\CH_k(X') \\to \\CH_{k - 1}(X')$\nis a bivariant class of degree $1$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow cohomology and the first Chern class","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B77","source_file":"chow.tex","source_line":6297,"source_end_line":6305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6297-L6305","statement_sha256":"a67afe7af03934d029f970c88cdc67ffec60502286c6854b4659baaf43932ea0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8223,"rank":8223,"depth":52,"x":2352.027,"y":698.19,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FDW","tag":"0FDW","title":"Chow cohomology and the first Chern class · Definition 0FDW","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L be an invertible O_X-module. The first Chern class c_1(L) ∈ A^1(X) of L is the bivariant class of Lemma [Tag 0B77].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$\nbe locally of finite type over $S$. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module. The {\\it first Chern class}\n$c_1(\\mathcal{L}) \\in A^1(X)$ of $\\mathcal{L}$\nis the bivariant class of Lemma \\ref{lemma-cap-c1-bivariant}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow cohomology and the first Chern class","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDW","source_file":"chow.tex","source_line":6317,"source_end_line":6324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6317-L6324","statement_sha256":"847a479f509267c4a90f001ff73f6555ec5fc376f56301d2aeecf367d99db54a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8224,"rank":8224,"depth":53,"x":2559.843,"y":593.859,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B7B","tag":"0B7B","title":"Chow cohomology and the first Chern class · Lemma 0B7B","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L be an invertible O_X-module. Then • c_1(L) ∈ A^1(X) is in the center of A^*(X) and • if f : X' → X is locally of finite type and c ∈ A^*(X' → X), then c ∘ c_1(L) = c_1(f^*L) ∘ c.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen\n\\begin{enumerate}\n\\item $c_1(\\mathcal{L}) \\in A^1(X)$ is in the center of $A^*(X)$ and\n\\item if $f : X' \\to X$ is locally of finite type and $c \\in A^*(X' \\to X)$,\nthen $c \\circ c_1(\\mathcal{L}) = c_1(f^*\\mathcal{L}) \\circ c$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow cohomology and the first Chern class","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7B","source_file":"chow.tex","source_line":6331,"source_end_line":6342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6331-L6342","statement_sha256":"e02b646dd3f1b46a85c8b2fde6c8bd14755f4057e5538ae8abbeea44107ea488","origin":"The Stacks Project","memory_eligible":false,"source_rank":8225,"rank":8225,"depth":29,"x":2490.411,"y":788.967,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FDX","tag":"0FDX","title":"Chow cohomology and the first Chern class · Lemma 0FDX","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a finite type scheme over S which has an ample invertible sheaf. Assume d = dim(X) < ∞ (here we really mean dimension and not δ-dimension). Then for any invertible sheaves L_1, …, L_d + 1 on X we have c_1(L_1) ∘ … ∘ c_1(L_d + 1) = 0 in A^d + 1(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a\nfinite type scheme over $S$ which has an ample invertible sheaf.\nAssume $d = \\dim(X) < \\infty$ (here we really mean dimension and\nnot $\\delta$-dimension).\nThen for any invertible sheaves $\\mathcal{L}_1, \\ldots, \\mathcal{L}_{d + 1}$\non $X$ we have\n$c_1(\\mathcal{L}_1) \\circ \\ldots \\circ c_1(\\mathcal{L}_{d + 1}) = 0$\nin $A^{d + 1}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow cohomology and the first Chern class","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDX","source_file":"chow.tex","source_line":6373,"source_end_line":6383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6373-L6383","statement_sha256":"9b64a92142bf06d0beac4618a92066714b862703b3b15f449ed1bf3ab36dcba9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8226,"rank":8226,"depth":52,"x":2384.57,"y":605.46,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B7A","tag":"0B7A","title":"Lemmas on bivariant classes · Lemma 0B7A","summary":"Very weak form of [F] Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a morphism of schemes locally of finite type over S. Let p ∈ Z. Suppose given a rule which assigns to every locally of finite type morphism Y' → Y and every k a map c ∩ - : Z_k(Y') → CH_k - p(X') where Y' = X' ×_X Y, satisfying condition (3) of Definition [Tag 0B76] whenever L'|_D' ≅ O_D'. Then c ∩ - factors through rational equivalence.","statement_latex":"\\begin{reference}\nVery weak form of \\cite[Theorem 17.1]{F}\n\\end{reference}\nLet $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.\nLet $p \\in \\mathbf{Z}$. Suppose given a rule\nwhich assigns to every locally of finite type morphism $Y' \\to Y$\nand every $k$ a map\n$$\nc \\cap - : Z_k(Y') \\longrightarrow \\CH_{k - p}(X')\n$$\nwhere $Y' = X' \\times_X Y$, satisfying condition (3) of\nDefinition \\ref{definition-bivariant-class}\nwhenever $\\mathcal{L}'|_{D'} \\cong \\mathcal{O}_{D'}$. Then\n$c \\cap -$ factors through rational equivalence.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Lemmas on bivariant classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7A","source_file":"chow.tex","source_line":6476,"source_end_line":6493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6476-L6493","statement_sha256":"93090bca2c2cb6438ecd81334446124c3f7dbaf801c3b6004c65a4ed74789abc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8227,"rank":8227,"depth":48,"x":2610.482,"y":680.817,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F9A","tag":"0F9A","title":"Lemmas on bivariant classes · Lemma 0F9A","summary":"Weak form of [F] Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a morphism of schemes locally of finite type over S. Let p ∈ Z. Suppose given a rule which assigns to every locally of finite type morphism Y' → Y and every k a map c ∩ - : CH_k(Y') → CH_k - p(X') where Y' = X' ×_X Y, satisfying conditions (1), (2) of Definition [Tag 0B76] and condition (3) whenever L'|_D' ≅ O_D'. Then c ∩ - is a bivariant class.","statement_latex":"\\begin{reference}\nWeak form of \\cite[Theorem 17.1]{F}\n\\end{reference}\nLet $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.\nLet $p \\in \\mathbf{Z}$. Suppose given a rule\nwhich assigns to every locally of finite type morphism $Y' \\to Y$\nand every $k$ a map\n$$\nc \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X')\n$$\nwhere $Y' = X' \\times_X Y$, satisfying conditions (1), (2) of\nDefinition \\ref{definition-bivariant-class}\nand condition (3) whenever $\\mathcal{L}'|_{D'} \\cong \\mathcal{O}_{D'}$. Then\n$c \\cap -$ is a bivariant class.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Lemmas on bivariant classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9A","source_file":"chow.tex","source_line":6518,"source_end_line":6535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6518-L6535","statement_sha256":"c11e53c415c8433313f812675e68c04a535cb6f2079c5f8ebb25b4cde9652786","origin":"The Stacks Project","memory_eligible":false,"source_rank":8228,"rank":8228,"depth":31,"x":2383.002,"y":753.532,"cluster":"divisors-intersection-theory"},{"id":"stacks:02UC","tag":"02UC","title":"Lemmas on bivariant classes · Lemma 02UC","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a morphism of schemes locally of finite type over S. Let c ∈ A^p(X → Y). For Y\" → Y' → Y set X\" = Y\" ×_Y X and X' = Y' ×_Y X. The following are equivalent • c is zero, • c ∩ [Y'] = 0 in CH_*(X') for every integral scheme Y' locally of finite type over Y, and • for every integral scheme Y' locally of finite type over Y, there exists a proper birational morphism Y\" → Y' such that c ∩ [Y\"] = 0 in CH_*(X\").","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.\nLet $c \\in A^p(X \\to Y)$. For $Y'' \\to Y' \\to Y$ set\n$X'' = Y'' \\times_Y X$ and $X' = Y' \\times_Y X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $c$ is zero,\n\\item $c \\cap [Y'] = 0$ in $\\CH_*(X')$ for every integral scheme $Y'$\nlocally of finite type over $Y$, and\n\\item for every integral scheme $Y'$ locally of finite type over $Y$,\nthere exists a proper birational morphism $Y'' \\to Y'$ such that\n$c \\cap [Y''] = 0$ in $\\CH_*(X'')$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Lemmas on bivariant classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UC","source_file":"chow.tex","source_line":6584,"source_end_line":6599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6584-L6599","statement_sha256":"8090e83f8c1f353559455ac4c112ec5e28173629c43def4be788bc68bb4d96e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8229,"rank":8229,"depth":0,"x":2492.409,"y":570.598,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FDZ","tag":"0FDZ","title":"Lemmas on bivariant classes · Lemma 0FDZ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a morphism of schemes locally of finite type over S. Assume we have disjoint union decompositions X = coprod_i ∈ I X_i and Y = coprod_j ∈ J Y_j by open and closed subschemes and a map a : I → J of sets such that f(X_i) ⊂ Y_a(i). Then A^p(X → Y) = ∏_i ∈ I A^p(X_i → Y_a(i))","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.\nAssume we have disjoint union decompositions\n$X = \\coprod_{i \\in I} X_i$ and $Y = \\coprod_{j \\in J} Y_j$\nby open and closed subschemes\nand a map $a : I \\to J$ of sets such that $f(X_i) \\subset Y_{a(i)}$.\nThen\n$$\nA^p(X \\to Y) = \\prod\\nolimits_{i \\in I} A^p(X_i \\to Y_{a(i)})\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Lemmas on bivariant classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FDZ","source_file":"chow.tex","source_line":6622,"source_end_line":6634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6622-L6634","statement_sha256":"f6ab9e06c5f4432e945c536b18a21b16dfff54ce13ef24f808c3da3f6d70d78c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8230,"rank":8230,"depth":0,"x":2558.93,"y":767.826,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GUC","tag":"0GUC","title":"Lemmas on bivariant classes · Lemma 0GUC","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a morphism of schemes locally of finite type over S. Let g : Y' → Y be an envelope (Definition [Tag 0GU5]) and denote X' = Y' ×_Y X. Let p ∈ Z and let c' ∈ A^p(X' → Y'). If the two restrictions res_1(c') = res_2(c') ∈ A^p(X' ×_X X' → Y' ×_Y Y') are equal (see proof), then there exists a unique c ∈ A^p(X → Y) whose restriction res(c) = c' in A^p(X' → Y').","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a morphism of schemes locally of finite type over $S$.\nLet $g : Y' \\to Y$ be an envelope (Definition \\ref{definition-envelope})\nand denote $X' = Y' \\times_Y X$. Let $p \\in \\mathbf{Z}$ and let\n$c' \\in A^p(X' \\to Y')$. If the two restrictions\n$$\nres_1(c') = res_2(c') \\in A^p(X' \\times_X X' \\to Y' \\times_Y Y')\n$$\nare equal (see proof), then there exists a unique $c \\in A^p(X \\to Y)$\nwhose restriction $res(c) = c'$ in $A^p(X' \\to Y')$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Lemmas on bivariant classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUC","source_file":"chow.tex","source_line":6706,"source_end_line":6718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6706-L6718","statement_sha256":"3b904f6d8e2312ec548295c2f3990c9e564cb92ac0f944a7a55a875e0c276c1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8231,"rank":8231,"depth":34,"x":2351.004,"y":659.999,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TW","tag":"02TW","title":"Projective space bundle formula · Lemma 02TW","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a finite locally free O_X-module E of rank r. Let (π : P → X, O_P(1)) be the projective bundle associated to E. For any α ∈ CH_k(X) the element π_*( c_1(O_P(1))^s ∩ π^*α ) ∈ CH_k + r - 1 - s(X) is 0 if s < r - 1 and is equal to α when s = r - 1.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module\n$\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$\nbe the projective bundle associated to $\\mathcal{E}$.\nFor any $\\alpha \\in \\CH_k(X)$ the element\n$$\n\\pi_*\\left(\nc_1(\\mathcal{O}_P(1))^s \\cap \\pi^*\\alpha\n\\right)\n\\in\n\\CH_{k + r - 1 - s}(X)\n$$\nis $0$ if $s < r - 1$ and is equal to $\\alpha$ when $s = r - 1$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Projective space bundle formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TW","source_file":"chow.tex","source_line":6867,"source_end_line":6883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6867-L6883","statement_sha256":"50c580dc036d44db8c18358621b5377561f630943a4a25a3f9ea33e8997e1c54","origin":"The Stacks Project","memory_eligible":false,"source_rank":8232,"rank":8232,"depth":17,"x":2591.347,"y":621.478,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TX","tag":"02TX","title":"Projective space bundle formula · Lemma 02TX","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a finite locally free O_X-module E of rank r. Let (π : P → X, O_P(1)) be the projective bundle associated to E. The map bigoplus_i = 0^r - 1 CH_k + i(X) → CH_k + r - 1(P), (α_0, …, α_r-1) ↦ π^*α_0 + c_1(O_P(1)) ∩ π^*α_1 + … + c_1(O_P(1))^r - 1 ∩ π^*α_r-1 is an isomorphism.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module\n$\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$\nbe the projective bundle associated to $\\mathcal{E}$.\nThe map\n$$\n\\bigoplus\\nolimits_{i = 0}^{r - 1}\n\\CH_{k + i}(X)\n\\longrightarrow\n\\CH_{k + r - 1}(P),\n$$\n$$\n(\\alpha_0, \\ldots, \\alpha_{r-1})\n\\longmapsto\n\\pi^*\\alpha_0 +\nc_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1\n+ \\ldots +\nc_1(\\mathcal{O}_P(1))^{r - 1} \\cap \\pi^*\\alpha_{r-1}\n$$\nis an isomorphism.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Projective space bundle formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TX","source_file":"chow.tex","source_line":6931,"source_end_line":6954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L6931-L6954","statement_sha256":"7de2dd72e28da674df0bc00d18a8e22c739b0b5dcf75ffdb07a24cc294afa394","origin":"The Stacks Project","memory_eligible":false,"source_rank":8233,"rank":8233,"depth":46,"x":2444.912,"y":786.471,"cluster":"divisors-intersection-theory"},{"id":"stacks:02TY","tag":"02TY","title":"Projective space bundle formula · Lemma 02TY","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. Let p : E = underlineSpec(Sym^*(E)) → X be the associated vector bundle over X. Then p^* : CH_k(X) → CH_k + r(E) is an isomorphism for all k.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.\nLet\n$$\np :\nE = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{E}))\n\\longrightarrow\nX\n$$\nbe the associated vector bundle over $X$.\nThen $p^* : \\CH_k(X) \\to \\CH_{k + r}(E)$ is an isomorphism for all $k$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Projective space bundle formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02TY","source_file":"chow.tex","source_line":7061,"source_end_line":7075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7061-L7075","statement_sha256":"9c1a70f5b187f8a7792c21d4da4c395acc9b0cacdb683067067b6b1fdeca63bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8234,"rank":8234,"depth":47,"x":2420.176,"y":581.454,"cluster":"divisors-intersection-theory"},{"id":"stacks:02U0","tag":"02U0","title":"The Chern classes of a vector bundle · Definition 02U0","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Assume X is integral and n = dim_δ(X). Let E be a finite locally free sheaf of rank r on X. Let (π : P → X, O_P(1)) be the projective space bundle associated to E. • By Lemma [Tag 02TX] there are elements c_i ∈ CH_n - i(X), i = 0, …, r such that c_0 = [X], and ∑_i = 0^r (-1)^i c_1(O_P(1))^i ∩ π^*c_r - i = 0. • With notation as above we set c_i(E) ∩ [X] = c_i as an element of CH_n - i(X). We…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nAssume $X$ is integral and $n = \\dim_\\delta(X)$.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$\non $X$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective space\nbundle associated to $\\mathcal{E}$.\n\\begin{enumerate}\n\\item By Lemma \\ref{lemma-chow-ring-projective-bundle} there are\nelements $c_i \\in \\CH_{n - i}(X)$, $i = 0, \\ldots, r$\nsuch that $c_0 = [X]$, and\n\\begin{equation}\n\n\\sum\\nolimits_{i = 0}^r\n(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap \\pi^*c_{r - i}\n= 0.\n\\end{equation}\n\\item With notation as above we set\n$c_i(\\mathcal{E}) \\cap [X] = c_i$\nas an element of $\\CH_{n - i}(X)$.\nWe call these the {\\it Chern classes of $\\mathcal{E}$ on $X$}.\n\\item The {\\it total Chern class of $\\mathcal{E}$ on $X$}\nis the combination\n$$\nc({\\mathcal E}) \\cap [X] =\nc_0({\\mathcal E}) \\cap [X]\n+ c_1({\\mathcal E}) \\cap [X] + \\ldots\n+ c_r({\\mathcal E}) \\cap [X]\n$$\nwhich is an element of\n$\\CH_*(X) = \\bigoplus_{k \\in \\mathbf{Z}} \\CH_k(X)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The Chern classes of a vector bundle","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02U0","source_file":"chow.tex","source_line":7143,"source_end_line":7176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7143-L7176","statement_sha256":"0955f038c5e5f8c9a227b515ce9952f8530e731cdcc781b4eca6d44811ee61ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":8235,"rank":8235,"depth":47,"x":2603.519,"y":718.77,"cluster":"divisors-intersection-theory"},{"id":"stacks:02U2","tag":"02U2","title":"The Chern classes of a vector bundle · Lemma 02U2","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Assume X is integral and n = dim_δ(X). Let L be an invertible O_X-module. The first Chern class of L on X of Definition [Tag 02U0] is equal to the Weil divisor associated to L by Definition [Tag 02SJ].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nAssume $X$ is integral and $n = \\dim_\\delta(X)$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThe first Chern class of $\\mathcal{L}$ on $X$ of\nDefinition \\ref{definition-chern-classes}\nis equal to the Weil divisor associated to $\\mathcal{L}$\nby Definition \\ref{definition-divisor-invertible-sheaf}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The Chern classes of a vector bundle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02U2","source_file":"chow.tex","source_line":7182,"source_end_line":7192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7182-L7192","statement_sha256":"b9d25261f04e6e83abeabc74011bcd45edfc107f19a666aeb201754003f193f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8236,"rank":8236,"depth":48,"x":2357.585,"y":721.552,"cluster":"divisors-intersection-theory"},{"id":"stacks:02U5","tag":"02U5","title":"Intersecting with Chern classes · Definition 02U5","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. We define, for every integer k and any 0 ≤ j ≤ r, an operation c_j(E) ∩ - : Z_k(X) → CH_k - j(X) called intersection with the jth Chern class of E. • Given an integral closed subscheme i : W → X of δ-dimension k we define c_j(E) ∩ [W] = i_*(c_j(i^*E) ∩ [W]) ∈ CH_k - j(X) where c_j(i^*E) ∩ [W] is as defined in Definition [Tag 02U0]. • For a…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.\nWe define, for every integer $k$ and any $0 \\leq j \\leq r$,\nan operation\n$$\nc_j(\\mathcal{E}) \\cap - : Z_k(X) \\to \\CH_{k - j}(X)\n$$\ncalled {\\it intersection with the $j$th Chern class of $\\mathcal{E}$}.\n\\begin{enumerate}\n\\item Given an integral closed subscheme $i : W \\to X$ of $\\delta$-dimension\n$k$ we define\n$$\nc_j(\\mathcal{E}) \\cap [W] = i_*(c_j({i^*\\mathcal{E}}) \\cap [W])\n\\in\n\\CH_{k - j}(X)\n$$\nwhere $c_j({i^*\\mathcal{E}}) \\cap [W]$ is as defined in\nDefinition \\ref{definition-chern-classes}.\n\\item For a general $k$-cycle $\\alpha = \\sum n_i [W_i]$ we set\n$$\nc_j(\\mathcal{E}) \\cap \\alpha = \\sum n_i c_j(\\mathcal{E}) \\cap [W_i]\n$$\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with Chern classes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02U5","source_file":"chow.tex","source_line":7246,"source_end_line":7272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7246-L7272","statement_sha256":"77f18a87c33b28cc3249e720a26489a170eaa599ce4e4b1e274238e8b206e323","origin":"The Stacks Project","memory_eligible":false,"source_rank":8237,"rank":8237,"depth":48,"x":2536.924,"y":579.772,"cluster":"divisors-intersection-theory"},{"id":"stacks:02U6","tag":"02U6","title":"Intersecting with Chern classes · Lemma 02U6","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. Let (π : P → X, O_P(1)) be the projective bundle associated to E. For α ∈ Z_k(X) the elements c_j(E) ∩ α are the unique elements α_j of CH_k - j(X) such that α_0 = α and ∑_i = 0^r (-1)^i c_1(O_P(1))^i ∩ π^*(α_r - i) = 0 holds in the Chow group of P.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.\nLet $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective bundle\nassociated to $\\mathcal{E}$.\nFor $\\alpha \\in Z_k(X)$ the elements\n$c_j(\\mathcal{E}) \\cap \\alpha$ are the unique elements\n$\\alpha_j$ of $\\CH_{k - j}(X)$\nsuch that $\\alpha_0 = \\alpha$ and\n$$\n\\sum\\nolimits_{i = 0}^r\n(-1)^i c_1(\\mathcal{O}_P(1))^i \\cap\n\\pi^*(\\alpha_{r - i}) = 0\n$$\nholds in the Chow group of $P$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02U6","source_file":"chow.tex","source_line":7279,"source_end_line":7296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7279-L7296","statement_sha256":"474478b8a11b1d45d9d6f17ac5b61548952890ed035166ff1458cfdbfc2273b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8238,"rank":8238,"depth":47,"x":2518.675,"y":786.342,"cluster":"divisors-intersection-theory"},{"id":"stacks:02U7","tag":"02U7","title":"Intersecting with Chern classes · Lemma 02U7","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. If α sim_rat β are rationally equivalent k-cycles on X then c_j(E) ∩ α = c_j(E) ∩ β in CH_k - j(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.\nIf $\\alpha \\sim_{rat} \\beta$ are rationally equivalent $k$-cycles\non $X$ then $c_j(\\mathcal{E}) \\cap \\alpha = c_j(\\mathcal{E}) \\cap \\beta$\nin $\\CH_{k - j}(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02U7","source_file":"chow.tex","source_line":7363,"source_end_line":7371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7363-L7371","statement_sha256":"a8455c4a8d23ed60e2fbdd372d48ab1f34f09c4193e7b0c2da9e86ad97d2fe9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8239,"rank":8239,"depth":48,"x":2365.821,"y":623.459,"cluster":"divisors-intersection-theory"},{"id":"stacks:02U9","tag":"02U9","title":"Intersecting with Chern classes · Lemma 02U9","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. Let p : X → Y be a proper morphism. Let α be a k-cycle on X. Let E be a finite locally free sheaf on Y. Then p_*(c_j(p^*E) ∩ α) = c_j(E) ∩ p_*α","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.\nLet $p : X \\to Y$ be a proper morphism.\nLet $\\alpha$ be a $k$-cycle on $X$.\nLet $\\mathcal{E}$ be a finite locally free sheaf on $Y$.\nThen\n$$\np_*(c_j(p^*\\mathcal{E}) \\cap \\alpha) = c_j(\\mathcal{E}) \\cap p_*\\alpha\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02U9","source_file":"chow.tex","source_line":7393,"source_end_line":7405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7393-L7405","statement_sha256":"582cfcc62790af3236507123040dd6ad053c93047ce4d8edd698707f74658753","origin":"The Stacks Project","memory_eligible":false,"source_rank":8240,"rank":8240,"depth":48,"x":2609.826,"y":656.875,"cluster":"divisors-intersection-theory"},{"id":"stacks:02U8","tag":"02U8","title":"Intersecting with Chern classes · Lemma 02U8","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X, Y be locally of finite type over S. Let E be a finite locally free sheaf of rank r on Y. Let f : X → Y be a flat morphism of relative dimension r. Let α be a k-cycle on Y. Then f^*(c_j(E) ∩ α) = c_j(f^*E) ∩ f^*α","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$, $Y$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $Y$.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $\\alpha$ be a $k$-cycle on $Y$.\nThen\n$$\nf^*(c_j(\\mathcal{E}) \\cap \\alpha) = c_j(f^*\\mathcal{E}) \\cap f^*\\alpha\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02U8","source_file":"chow.tex","source_line":7439,"source_end_line":7450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7439-L7450","statement_sha256":"e2ea448f6413ca4f6866a4101b031be05cbe994e65f85ae75395a3c3f2414194","origin":"The Stacks Project","memory_eligible":false,"source_rank":8241,"rank":8241,"depth":48,"x":2402.767,"y":770.832,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B7G","tag":"0B7G","title":"Intersecting with Chern classes · Lemma 0B7G","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. Let (L, s, i : D → X) be as in Definition [Tag 02T8]. Then c_j(E|_D) ∩ i^*α = i^*(c_j(E) ∩ α) for all α ∈ CH_k(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.\nLet $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}.\nThen $c_j(\\mathcal{E}|_D) \\cap i^*\\alpha = i^*(c_j(\\mathcal{E}) \\cap \\alpha)$\nfor all $\\alpha \\in \\CH_k(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7G","source_file":"chow.tex","source_line":7484,"source_end_line":7493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7484-L7493","statement_sha256":"ef70d49a76474000635ee6ace0d54e6f04015bd366a318413adad0fefb8cf93f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8242,"rank":8242,"depth":52,"x":2463.885,"y":569.059,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B7H","tag":"0B7H","title":"Intersecting with Chern classes · Lemma 0B7H","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a locally free O_X-module of rank r. Let 0 ≤ p ≤ r. Then the rule that to f : X' → X assigns c_p(f^*E) ∩ - : CH_k(X') → CH_k - p(X') is a bivariant class of degree p.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module\nof rank $r$. Let $0 \\leq p \\leq r$.\nThen the rule that to $f : X' \\to X$ assigns\n$c_p(f^*\\mathcal{E}) \\cap - : \\CH_k(X') \\to \\CH_{k - p}(X')$\nis a bivariant class of degree $p$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7H","source_file":"chow.tex","source_line":7532,"source_end_line":7541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7532-L7541","statement_sha256":"abb51cb9a953b518ae41b4906a12ea7d19612e353ba861093c403a3de59ebc31","origin":"The Stacks Project","memory_eligible":false,"source_rank":8243,"rank":8243,"depth":53,"x":2581.225,"y":752.753,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FE1","tag":"0FE1","title":"Intersecting with Chern classes · Definition 0FE1","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a locally free O_X-module of rank r. For i = 0, …, r the ith Chern class of E is the bivariant class c_i(E) ∈ A^i(X) of degree i constructed in Lemma [Tag 0B7H]. The total Chern class of E is the formal sum c(E) = c_0(E) + c_1(E) + … + c_r(E) which is viewed as a nonhomogeneous bivariant class on X.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module\nof rank $r$. For $i = 0, \\ldots, r$ the {\\it $i$th Chern class}\nof $\\mathcal{E}$ is the bivariant class\n$c_i(\\mathcal{E}) \\in A^i(X)$ of degree $i$\nconstructed in Lemma \\ref{lemma-cap-cp-bivariant}. The\n{\\it total Chern class} of $\\mathcal{E}$ is the formal sum\n$$\nc(\\mathcal{E}) = \nc_0(\\mathcal{E}) + c_1(\\mathcal{E}) + \\ldots + c_r(\\mathcal{E})\n$$\nwhich is viewed as a nonhomogeneous bivariant class on $X$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with Chern classes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FE1","source_file":"chow.tex","source_line":7556,"source_end_line":7571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7556-L7571","statement_sha256":"f2e46f0a1a431ed54f2a57ff287305b187e85b96f7e3060cab8974e3fd8e9bfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8244,"rank":8244,"depth":54,"x":2346.685,"y":683.795,"cluster":"divisors-intersection-theory"},{"id":"stacks:02UA","tag":"02UA","title":"Intersecting with Chern classes · Lemma 02UA","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a locally free O_X-module of rank r. Then • c_j(E) ∈ A^j(X) is in the center of A^*(X) and • if f : X' → X is locally of finite type and c ∈ A^*(X' → X), then c ∘ c_j(E) = c_j(f^*E) ∘ c. In particular, if F is a second locally free O_X-module on X of rank s, then c_i(E) ∩ c_j(F) ∩ α = c_j(F) ∩ c_i(E) ∩ α as elements of CH_k - i - j(X) for all α ∈ CH_k(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $r$.\nThen\n\\begin{enumerate}\n\\item $c_j(\\mathcal{E}) \\in A^j(X)$ is in the center of $A^*(X)$ and\n\\item if $f : X' \\to X$ is locally of finite type and $c \\in A^*(X' \\to X)$,\nthen $c \\circ c_j(\\mathcal{E}) = c_j(f^*\\mathcal{E}) \\circ c$.\n\\end{enumerate}\nIn particular, if $\\mathcal{F}$ is a second locally free\n$\\mathcal{O}_X$-module on $X$ of rank $s$, then\n$$\nc_i(\\mathcal{E}) \\cap c_j(\\mathcal{F}) \\cap \\alpha\n=\nc_j(\\mathcal{F}) \\cap c_i(\\mathcal{E}) \\cap \\alpha\n$$\nas elements of $\\CH_{k - i - j}(X)$ for all $\\alpha \\in \\CH_k(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersecting with Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UA","source_file":"chow.tex","source_line":7580,"source_end_line":7599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7580-L7599","statement_sha256":"0014546ef56b051f0b489d82ee63d1c45f2d115578aef52c25ebf42ebae646e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8245,"rank":8245,"depth":48,"x":2575.371,"y":601.458,"cluster":"divisors-intersection-theory"},{"id":"stacks:02UD","tag":"02UD","title":"Polynomial relations among Chern classes · Lemma 02UD","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. Let L be an invertible sheaf on X. Then we have c_i( E ⊗ L) = ∑_j = 0^i binomr - i + jj c_i - j( E) c_1( L)^j in A^*(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free sheaf of\nrank $r$ on $X$. Let $\\mathcal{L}$ be an invertible\nsheaf on $X$. Then we have\n\\begin{equation}\n\nc_i({\\mathcal E} \\otimes {\\mathcal L})\n=\n\\sum\\nolimits_{j = 0}^i\n\\binom{r - i + j}{j} c_{i - j}({\\mathcal E}) c_1({\\mathcal L})^j\n\\end{equation}\nin $A^*(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Polynomial relations among Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UD","source_file":"chow.tex","source_line":7754,"source_end_line":7769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7754-L7769","statement_sha256":"237fc9644ebcc047ca10986ad5616a96edddab25afa59cb49d3578ea565a4f4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8246,"rank":8246,"depth":48,"x":2472.827,"y":792.17,"cluster":"divisors-intersection-theory"},{"id":"stacks:02UG","tag":"02UG","title":"Additivity of Chern classes · Lemma 02UG","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E, F be finite locally free sheaves on X of ranks r, r - 1 which fit into a short exact sequence 0 → O_X → E → F → 0 Then we have c_r(E) = 0, c_j(E) = c_j(F), j = 0, …, r - 1 in A^*(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves\non $X$ of ranks $r$, $r - 1$ which fit into a short\nexact sequence\n$$\n0 \\to \\mathcal{O}_X \\to \\mathcal{E} \\to \\mathcal{F} \\to 0\n$$\nThen we have\n$$\nc_r(\\mathcal{E}) = 0, \\quad\nc_j(\\mathcal{E}) = c_j(\\mathcal{F}), \\quad j = 0, \\ldots, r - 1\n$$\nin $A^*(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Additivity of Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UG","source_file":"chow.tex","source_line":7862,"source_end_line":7878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7862-L7878","statement_sha256":"c260543ca0db6d0b890d653f453bdd392e1dfe564e69a16bcca88e5f9c31c485","origin":"The Stacks Project","memory_eligible":false,"source_rank":8247,"rank":8247,"depth":48,"x":2394.979,"y":593.11,"cluster":"divisors-intersection-theory"},{"id":"stacks:02UH","tag":"02UH","title":"Additivity of Chern classes · Lemma 02UH","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E, F be finite locally free sheaves on X of ranks r, r - 1 which fit into a short exact sequence 0 → L → E → F → 0 where L is an invertible sheaf. Then c(E) = c(L) c(F) in A^*(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves\non $X$ of ranks $r$, $r - 1$ which fit into a short\nexact sequence\n$$\n0 \\to \\mathcal{L} \\to \\mathcal{E} \\to \\mathcal{F} \\to 0\n$$\nwhere $\\mathcal{L}$ is an invertible sheaf.\nThen\n$$\nc(\\mathcal{E}) = c(\\mathcal{L}) c(\\mathcal{F})\n$$\nin $A^*(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Additivity of Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UH","source_file":"chow.tex","source_line":7924,"source_end_line":7940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7924-L7940","statement_sha256":"3f9f34297132a3dac5b0f87c124c79615698ecc37d1fa406b6ec5d5cc2a0164d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8248,"rank":8248,"depth":49,"x":2612.731,"y":695.848,"cluster":"divisors-intersection-theory"},{"id":"stacks:02UI","tag":"02UI","title":"Additivity of Chern classes · Lemma 02UI","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Suppose that E sits in an exact sequence 0 → E_1 → E → E_2 → 0 of finite locally free sheaves E_i of rank r_i. The total Chern classes satisfy c( E) = c( E_1) c( E_2) in A^*(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nSuppose that ${\\mathcal E}$ sits in an\nexact sequence\n$$\n0\n\\to\n{\\mathcal E}_1\n\\to\n{\\mathcal E}\n\\to\n{\\mathcal E}_2\n\\to\n0\n$$\nof finite locally free sheaves $\\mathcal{E}_i$ of rank $r_i$.\nThe total Chern classes satisfy\n$$\nc({\\mathcal E}) = c({\\mathcal E}_1) c({\\mathcal E}_2)\n$$\nin $A^*(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Additivity of Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UI","source_file":"chow.tex","source_line":7964,"source_end_line":7987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L7964-L7987","statement_sha256":"4366880fc930231a5b72ab8dea74d87b0d1f2d1577a3a4af98f57569f4333efd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8249,"rank":8249,"depth":50,"x":2369.245,"y":743.707,"cluster":"divisors-intersection-theory"},{"id":"stacks:02UJ","tag":"02UJ","title":"Additivity of Chern classes · Lemma 02UJ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L_i, i = 1, …, r be invertible O_X-modules on X. Let E be a locally free rank O_X-module endowed with a filtration 0 = E_0 ⊂ E_1 ⊂ E_2 ⊂ … ⊂ E_r = E such that E_i/E_i - 1 ≅ L_i. Set c_1( L_i) = x_i. Then c(E) = ∏_i = 1^r (1 + x_i) in A^*(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet ${\\mathcal L}_i$, $i = 1, \\ldots, r$ be invertible\n$\\mathcal{O}_X$-modules on $X$.\nLet $\\mathcal{E}$ be a locally free rank\n$\\mathcal{O}_X$-module endowed with a filtration\n$$\n0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2\n\\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E}\n$$\nsuch that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$.\nSet $c_1({\\mathcal L}_i) = x_i$. Then\n$$\nc(\\mathcal{E})\n=\n\\prod\\nolimits_{i = 1}^r (1 + x_i)\n$$\nin $A^*(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Additivity of Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UJ","source_file":"chow.tex","source_line":8021,"source_end_line":8041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8021-L8041","statement_sha256":"d0b99c959c4b869e05d5e184b49fa47aeffc28a712a5ac8afb7ea6b43785489a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8250,"rank":8250,"depth":50,"x":2510.475,"y":570.043,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZ1","tag":"0AZ1","title":"Degrees of zero cycles · Definition 0AZ1","summary":"Let k be a field (Example [Tag 02QM]). Let p : X → Spec(k) be proper. The degree of a zero cycle on X is given by proper pushforward p_* : CH_0(X) → CH_0(Spec(k)) (Lemma [Tag 02S2]) combined with the natural isomorphism CH_0(Spec(k)) = Z which maps [Spec(k)] to 1. Notation: deg(α).","statement_latex":"Let $k$ be a field (Example \\ref{example-field}). Let $p : X \\to \\Spec(k)$\nbe proper. The {\\it degree of a zero cycle} on $X$ is given by proper\npushforward\n$$\np_* : \\CH_0(X) \\to \\CH_0(\\Spec(k))\n$$\n(Lemma \\ref{lemma-proper-pushforward-rational-equivalence})\ncombined with the natural isomorphism $\\CH_0(\\Spec(k)) = \\mathbf{Z}$\nwhich maps $[\\Spec(k)]$ to $1$. Notation: $\\deg(\\alpha)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Degrees of zero cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZ1","source_file":"chow.tex","source_line":8061,"source_end_line":8072,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8061-L8072","statement_sha256":"35cdcb47545c71e611c278f82dc564f9ea7272dd4178c881e9a92b02958ef2d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8251,"rank":8251,"depth":47,"x":2546.033,"y":778.493,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZ2","tag":"0AZ2","title":"Degrees of zero cycles · Lemma 0AZ2","summary":"Let k be a field. Let X be proper over k. Let α = ∑ n_i[Z_i] be in Z_0(X). Then deg(α) = ∑ n_ideg(Z_i) where deg(Z_i) is the degree of Z_i → Spec(k), i.e., deg(Z_i) = dim_k Γ(Z_i, O_Z_i).","statement_latex":"Let $k$ be a field. Let $X$ be proper over $k$. Let $\\alpha = \\sum n_i[Z_i]$\nbe in $Z_0(X)$. Then\n$$\n\\deg(\\alpha) = \\sum n_i\\deg(Z_i)\n$$\nwhere $\\deg(Z_i)$ is the degree of $Z_i \\to \\Spec(k)$, i.e.,\n$\\deg(Z_i) = \\dim_k \\Gamma(Z_i, \\mathcal{O}_{Z_i})$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Degrees of zero cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZ2","source_file":"chow.tex","source_line":8077,"source_end_line":8086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8077-L8086","statement_sha256":"c03fa4ac367918df591d32e5694c3c70a16365c9092e0c88bc38f825a8a61085","origin":"The Stacks Project","memory_eligible":false,"source_rank":8252,"rank":8252,"depth":33,"x":2351.946,"y":644.798,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZ3","tag":"0AZ3","title":"Degrees of zero cycles · Lemma 0AZ3","summary":"Let k be a field. Let X be a proper scheme over k of dimension ≤ 1. Let E be a finite locally free O_X-module of constant rank. Then deg(E) = deg(c_1(E) ∩ [X]_1) where the left hand side is defined in Varieties, Definition [Tag 0AYR].","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $\\leq 1$.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module of constant\nrank. Then\n$$\n\\deg(\\mathcal{E}) = \\deg(c_1(\\mathcal{E}) \\cap [X]_1)\n$$\nwhere the left hand side is defined in\nVarieties, Definition \\ref{varieties-definition-degree-invertible-sheaf}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Degrees of zero cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZ3","source_file":"chow.tex","source_line":8098,"source_end_line":8108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8098-L8108","statement_sha256":"ef843afa41c11c588f68207484f09f5c6c521cfc6a3ffdad20cb7f0ca397f001","origin":"The Stacks Project","memory_eligible":false,"source_rank":8253,"rank":8253,"depth":51,"x":2602.883,"y":633.239,"cluster":"divisors-intersection-theory"},{"id":"stacks:0BFI","tag":"0BFI","title":"Degrees of zero cycles · Lemma 0BFI","summary":"Let k be a field. Let X be a proper scheme over k. Let Z ⊂ X be a closed subscheme of dimension d. Let L_1, …, L_d be invertible O_X-modules. Then (L_1 … L_d · Z) = deg( c_1(L_1) ∩ … ∩ c_1(L_d) ∩ [Z]_d) where the left hand side is defined in Varieties, Definition [Tag 0BEP]. In particular, deg_L(Z) = deg(c_1(L)^d ∩ [Z]_d) if L is an ample invertible O_X-module.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\nLet $Z \\subset X$ be a closed subscheme of dimension $d$.\nLet $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible\n$\\mathcal{O}_X$-modules. Then\n$$\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =\n\\deg(\nc_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_d) \\cap [Z]_d)\n$$\nwhere the left hand side is defined in\nVarieties, Definition \\ref{varieties-definition-intersection-number}.\nIn particular,\n$$\n\\deg_\\mathcal{L}(Z) = \\deg(c_1(\\mathcal{L})^d \\cap [Z]_d)\n$$\nif $\\mathcal{L}$ is an ample invertible $\\mathcal{O}_X$-module.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Degrees of zero cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFI","source_file":"chow.tex","source_line":8180,"source_end_line":8198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8180-L8198","statement_sha256":"9200e56230ba0a7737ed4a7db7c1c984586b2c2040ce84e5c3c634817f0f7099","origin":"The Stacks Project","memory_eligible":false,"source_rank":8254,"rank":8254,"depth":46,"x":2426.927,"y":784.334,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FE3","tag":"0FE3","title":"Cycles of given codimension · Lemma 0FE3","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Write δ = δ_X/S as in Section [Tag 02QK]. The following are equivalent • There exists a decomposition X = coprod_n ∈ Z X_n into open and closed subschemes such that δ(xi) = n whenever xi ∈ X_n is a generic point of an irreducible component of X_n. • For all x ∈ X there exists an open neighbourhood U ⊂ X of x and an integer n such that δ(xi) = n whenever xi ∈ U is a generic point of an…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Write\n$\\delta = \\delta_{X/S}$ as in Section \\ref{section-setup}.\nThe following are equivalent\n\\begin{enumerate}\n\\item There exists a decomposition $X = \\coprod_{n \\in \\mathbf{Z}} X_n$\ninto open and closed subschemes such that $\\delta(\\xi) = n$ whenever\n$\\xi \\in X_n$ is a generic point of an irreducible component of $X_n$.\n\\item For all $x \\in X$ there exists an open neighbourhood $U \\subset X$\nof $x$ and an integer $n$ such that $\\delta(\\xi) = n$ whenever\n$\\xi \\in U$ is a generic point of an irreducible component of $U$.\n\\item For all $x \\in X$ there exists an integer $n_x$ such that\n$\\delta(\\xi) = n_x$ for any generic point $\\xi$ of an irreducible\ncomponent of $X$ containing $x$.\n\\end{enumerate}\nThe conditions are satisfied if $X$ is either\nnormal or Cohen-Macaulay\\footnote{In fact, it suffices if\n$X$ is $(S_2)$. Compare with Local Cohomology, Lemma\n\\ref{local-cohomology-lemma-catenary-S2-equidimensional}.}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Cycles of given codimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FE3","source_file":"chow.tex","source_line":8276,"source_end_line":8297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8276-L8297","statement_sha256":"b5ceba9bec7b4bd2b7a5c96f337ceb2f3e544e7a72b3e7e57dfd841dd265a884","origin":"The Stacks Project","memory_eligible":false,"source_rank":8255,"rank":8255,"depth":10,"x":2435.178,"y":572.821,"cluster":"divisors-intersection-theory"},{"id":"stacks:02UL","tag":"02UL","title":"The splitting principle · Lemma 02UL","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E_i be a finite collection of locally free O_X-modules of rank r_i. There exists a projective flat morphism π : P → X of relative dimension d such that • for any morphism f : Y → X the map π_Y^* : CH_*(Y) → CH_* + d(Y ×_X P) is injective, and • each π^*E_i has a filtration whose successive quotients L_i, 1, …, L_i, r_i are invertible O_P-modules. Moreover, when (1) holds the restriction…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally\nof finite type over $S$. Let $\\mathcal{E}_i$ be a finite collection of\nlocally free $\\mathcal{O}_X$-modules of rank $r_i$. There exists a projective\nflat morphism $\\pi : P \\to X$ of relative dimension $d$ such that\n\\begin{enumerate}\n\\item for any morphism $f : Y \\to X$ the map\n$\\pi_Y^* : \\CH_*(Y) \\to \\CH_{* + d}(Y \\times_X P)$ is injective, and\n\\item each $\\pi^*\\mathcal{E}_i$ has a filtration\nwhose successive quotients $\\mathcal{L}_{i, 1}, \\ldots, \\mathcal{L}_{i, r_i}$\nare invertible ${\\mathcal O}_P$-modules.\n\\end{enumerate}\nMoreover, when (1) holds the restriction map $A^*(X) \\to A^*(P)$\n(Remark \\ref{remark-pullback-cohomology}) is injective.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The splitting principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02UL","source_file":"chow.tex","source_line":8428,"source_end_line":8443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8428-L8443","statement_sha256":"ab40c0e79c733342eae9cff7d938f1a509577a43b480945d37b9374ed2d29bf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8256,"rank":8256,"depth":47,"x":2599.386,"y":733.664,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FA5","tag":"0FA5","title":"The splitting principle · Lemma 0FA5","summary":"In Situation [Tag 02QL] let X be locally of finite type over S. Let E be a finite locally free O_X-module with dual E^vee. Then c_i(E^vee) = (-1)^i c_i(E) in A^i(X).","statement_latex":"In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module\nwith dual $\\mathcal{E}^\\vee$. Then\n$$\nc_i(\\mathcal{E}^\\vee) = (-1)^i c_i(\\mathcal{E})\n$$\nin $A^i(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The splitting principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FA5","source_file":"chow.tex","source_line":8519,"source_end_line":8528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8519-L8528","statement_sha256":"7c0b71edbc9382ed2a749cedf711f2374b577951cf1547ddff4606110ab42ca5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8257,"rank":8257,"depth":51,"x":2348.653,"y":708.205,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FA6","tag":"0FA6","title":"The splitting principle · Lemma 0FA6","summary":"In Situation [Tag 02QL] let X be locally of finite type over S. Let E and F be a finite locally free O_X-modules of ranks r and s. Then we have c_1(E ⊗ F) = r c_1(F) + s c_1(E) c_2(E ⊗ F) = r c_2(F) + s c_2(E) + r choose 2 c_1(F)^2 + (rs - 1) c_1(F)c_1(E) + s choose 2 c_1(E)^2 and so on in A^*(X).","statement_latex":"In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ and $\\mathcal{F}$ be a finite locally free\n$\\mathcal{O}_X$-modules of ranks $r$ and $s$. Then we have\n$$\nc_1(\\mathcal{E} \\otimes \\mathcal{F})\n=\nr c_1(\\mathcal{F}) + s c_1(\\mathcal{E})\n$$\n$$\nc_2(\\mathcal{E} \\otimes \\mathcal{F})\n=\nr c_2(\\mathcal{F}) + s c_2(\\mathcal{E}) +\n{r \\choose 2} c_1(\\mathcal{F})^2 +\n(rs - 1) c_1(\\mathcal{F})c_1(\\mathcal{E}) +\n{s \\choose 2} c_1(\\mathcal{E})^2\n$$\nand so on in $A^*(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The splitting principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FA6","source_file":"chow.tex","source_line":8566,"source_end_line":8585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8566-L8585","statement_sha256":"8e3ccf692e18d40cc705e0e998232fcca89b39eae026dafeab6d2522f2bedaa9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8258,"rank":8258,"depth":52,"x":2554.261,"y":584.558,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FA9","tag":"0FA9","title":"Chern classes and sections · Lemma 0FA9","summary":"In the situation described just above assume dim_δ(X') = n, that f^*E has constant rank r, that dim_δ(Z(s)) ≤ n - r, and that for every generic point xi ∈ Z(s) with δ(xi) = n - r the ideal of Z(s) in O_X', xi is generated by a regular sequence of length r. Then c_r(E) ∩ [X']_n = [Z(s)]_n - r in CH_*(X').","statement_latex":"In the situation described just above assume $\\dim_\\delta(X') = n$,\nthat $f^*\\mathcal{E}$ has constant rank $r$, that\n$\\dim_\\delta(Z(s)) \\leq n - r$, and that for every generic point\n$\\xi \\in Z(s)$ with $\\delta(\\xi) = n - r$ the ideal of $Z(s)$\nin $\\mathcal{O}_{X', \\xi}$ is generated by a regular sequence\nof length $r$. Then\n$$\nc_r(\\mathcal{E}) \\cap [X']_n = [Z(s)]_{n - r}\n$$\nin $\\CH_*(X')$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FA9","source_file":"chow.tex","source_line":8749,"source_end_line":8761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8749-L8761","statement_sha256":"7478923f0ed5f61d34cebe427c25b4e87ed2aa72879906db5281c2c05f3c90ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":8259,"rank":8259,"depth":54,"x":2502.02,"y":792.65,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAA","tag":"0FAA","title":"Chern classes and sections · Lemma 0FAA","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let 0 → N' → N → E → 0 be a short exact sequence of finite locally free O_X-modules. Consider the closed embedding i : N' = underlineSpec_X(Sym((N')^vee)) → N = underlineSpec_X(Sym(N^vee)) For α ∈ CH_k(X) we have i_*(p')^*α = p^*(c_top(E) ∩ α) where p' : N' → X and p : N → X are the structure morphisms.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$\nbe a scheme locally of finite type over $S$. Let\n$$\n0 \\to \\mathcal{N}' \\to \\mathcal{N} \\to \\mathcal{E} \\to 0\n$$\nbe a short exact sequence of finite locally free $\\mathcal{O}_X$-modules.\nConsider the closed embedding\n$$\ni :\nN' = \\underline{\\Spec}_X(\\text{Sym}((\\mathcal{N}')^\\vee))\n\\longrightarrow\nN = \\underline{\\Spec}_X(\\text{Sym}(\\mathcal{N}^\\vee))\n$$\nFor $\\alpha \\in \\CH_k(X)$ we have\n$$\ni_*(p')^*\\alpha = p^*(c_{top}(\\mathcal{E}) \\cap \\alpha)\n$$\nwhere $p' : N' \\to X$ and $p : N \\to X$ are the structure morphisms.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAA","source_file":"chow.tex","source_line":8838,"source_end_line":8858,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8838-L8858","statement_sha256":"5fa07281c7d074e836eb109a5a076039244c2812fb9ba303de0505230d2f0c83","origin":"The Stacks Project","memory_eligible":false,"source_rank":8260,"rank":8260,"depth":55,"x":2373.045,"y":609.343,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F9C","tag":"0F9C","title":"The Chern character and tensor products · Lemma 0F9C","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let 0 → E_1 → E → E_2 → 0 be a short exact sequence of finite locally free O_X-modules. Then we have the equality ch(E) = ch(E_1) + ch(E_2) More precisely, we have P_p(E) = P_p(E_1) + P_p(E_2) in A^p(X) where P_p is as in Example [Tag 0F9B].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally\nof finite type over $S$. Let\n$\n0 \\to \\mathcal{E}_1 \\to \\mathcal{E} \\to \\mathcal{E}_2 \\to 0\n$\nbe a short exact sequence of finite locally free $\\mathcal{O}_X$-modules.\nThen we have the equality\n$$\nch(\\mathcal{E}) = ch(\\mathcal{E}_1) + ch(\\mathcal{E}_2)\n$$\nMore precisely, we have\n$P_p(\\mathcal{E}) = P_p(\\mathcal{E}_1) + P_p(\\mathcal{E}_2)$\nin $A^p(X)$ where $P_p$ is as in Example \\ref{example-power-sum}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The Chern character and tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9C","source_file":"chow.tex","source_line":8957,"source_end_line":8972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8957-L8972","statement_sha256":"dfc2569e44ee0d2515b5966c4f705ebe1f66d5cd5f30c11187898910d5497c46","origin":"The Stacks Project","memory_eligible":false,"source_rank":8261,"rank":8261,"depth":0,"x":2615.85,"y":671.402,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F9D","tag":"0F9D","title":"The Chern character and tensor products · Lemma 0F9D","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E_1 and E_2 be finite locally free O_X-modules. Then we have the equality ch(E_1 ⊗_O_X E_2) = ch(E_1) ch(E_2) More precisely, we have P_p(E_1 ⊗_O_X E_2) = ∑_p_1 + p_2 = p p choose p_1 P_p_1(E_1) P_p_2(E_2) in A^p(X) where P_p is as in Example [Tag 0F9B].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be locally\nof finite type over $S$. Let $\\mathcal{E}_1$ and $\\mathcal{E}_2$\nbe finite locally free $\\mathcal{O}_X$-modules.\nThen we have the equality\n$$\nch(\\mathcal{E}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{E}_2) =\nch(\\mathcal{E}_1) ch(\\mathcal{E}_2)\n$$\nMore precisely, we have\n$$\nP_p(\\mathcal{E}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{E}_2) =\n\\sum\\nolimits_{p_1 + p_2 = p}\n{p \\choose p_1} P_{p_1}(\\mathcal{E}_1) P_{p_2}(\\mathcal{E}_2)\n$$\nin $A^p(X)$ where $P_p$ is as in Example \\ref{example-power-sum}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The Chern character and tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9D","source_file":"chow.tex","source_line":8985,"source_end_line":9002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L8985-L9002","statement_sha256":"e63eb4712db05c873e64852ceca20879f6822fdcd21c89ede6120a4358e2056e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8262,"rank":8262,"depth":0,"x":2386.63,"y":763.524,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAB","tag":"0FAB","title":"The Chern character and tensor products · Lemma 0FAB","summary":"In Situation [Tag 02QL] let X be locally of finite type over S. Let E be a finite locally free O_X-module with dual E^vee. Then ch_i(E^vee) = (-1)^i ch_i(E) in A^i(X) ⊗ Q.","statement_latex":"In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module\nwith dual $\\mathcal{E}^\\vee$. Then\n$ch_i(\\mathcal{E}^\\vee) = (-1)^i ch_i(\\mathcal{E})$ in\n$A^i(X) \\otimes \\mathbf{Q}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"The Chern character and tensor products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAB","source_file":"chow.tex","source_line":9017,"source_end_line":9024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9017-L9024","statement_sha256":"85d86da0f037c87eb4c520862f00a6df66a85a007c655fe9f92f304fde11cac4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8263,"rank":8263,"depth":52,"x":2481.684,"y":565.294,"cluster":"divisors-intersection-theory"},{"id":"stacks:0ESZ","tag":"0ESZ","title":"Chern classes and the derived category · Lemma 0ESZ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E ∈ D(O_X) be an object such that there exists a locally bounded complex E^bullet of finite locally free O_X-modules representing E. Then a slight generalization of the above constructions c(E^bullet) ∈ ∏_p ≥ 0 A^p(X), ch(E^bullet) ∈ ∏_p ≥ 0 A^p(X) ⊗ Q, P_p(E^bullet) ∈ A^p(X) are independent of the choice of the complex E^bullet.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$\nbe an object such that there exists a locally bounded complex\n$\\mathcal{E}^\\bullet$ of finite locally free $\\mathcal{O}_X$-modules\nrepresenting $E$. Then a slight generalization of the above constructions\n$$\nc(\\mathcal{E}^\\bullet) \\in \\prod\\nolimits_{p \\geq 0} A^p(X),\\quad\nch(\\mathcal{E}^\\bullet) \\in\n\\prod\\nolimits_{p \\geq 0} A^p(X) \\otimes \\mathbf{Q},\\quad\nP_p(\\mathcal{E}^\\bullet) \\in A^p(X)\n$$\nare independent of the choice of the complex $\\mathcal{E}^\\bullet$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESZ","source_file":"chow.tex","source_line":9090,"source_end_line":9104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9090-L9104","statement_sha256":"b7e3542573690cda8984ea4e8f29f2b09e80ae62dfcc6a7bb0f4a527d269f321","origin":"The Stacks Project","memory_eligible":false,"source_rank":8264,"rank":8264,"depth":51,"x":2571.11,"y":765.64,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GUD","tag":"0GUD","title":"Chern classes and the derived category · Lemma 0GUD","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E ∈ D(O_X) be a perfect object. Assume there exists an envelope f : Y → X (Definition [Tag 0GU5]) such that Lf^*E is isomorphic in D(O_Y) to a locally bounded complex E^bullet of finite locally free O_Y-modules. Then there exists unique bivariant classes c(E) ∈ ∏_p ≥ 0 A^p(X), ch(E) ∈ ∏_p ≥ 0 A^p(X) ⊗ Q, and P_p(E) ∈ A^p(X), independent of the choice of f : Y → X and E^bullet, such that…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$\nbe a perfect object. Assume there exists an envelope\n$f : Y \\to X$ (Definition \\ref{definition-envelope})\nsuch that $Lf^*E$ is isomorphic in $D(\\mathcal{O}_Y)$\nto a locally bounded complex $\\mathcal{E}^\\bullet$ of finite locally free\n$\\mathcal{O}_Y$-modules. Then there exists unique bivariant classes\n$c(E) \\in \\prod_{p \\geq 0} A^p(X)$,\n$ch(E) \\in \\prod_{p \\geq 0} A^p(X) \\otimes \\mathbf{Q}$, and\n$P_p(E) \\in A^p(X)$, independent of the choice of $f : Y \\to X$\nand $\\mathcal{E}^\\bullet$, such that the restriction of these classes\nto $Y$ are equal to $c(\\mathcal{E}^\\bullet)$,\n$ch(\\mathcal{E}^\\bullet)$, and $P_p(\\mathcal{E}^\\bullet)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUD","source_file":"chow.tex","source_line":9212,"source_end_line":9227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9212-L9227","statement_sha256":"5d9d3e8b59a3ca0ccbd236c3c30d383ced729734d25c72dc97da75549aa1cfef","origin":"The Stacks Project","memory_eligible":false,"source_rank":8265,"rank":8265,"depth":54,"x":2343.787,"y":668.535,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F9E","tag":"0F9E","title":"Chern classes and the derived category · Definition 0F9E","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E ∈ D(O_X) be a perfect object. • We say the Chern classes of E are defined if there exists an envelope f : Y → X such that Lf^*E is isomorphic in D(O_Y) to a locally bounded complex of finite locally free O_Y-modules. • If the Chern classes of E are defined, then we define c(E) ∈ ∏_p ≥ 0 A^p(X), ch(E) ∈ ∏_p ≥ 0 A^p(X) ⊗ Q, P_p(E) ∈ A^p(X) by an application of Lemma [Tag 0GUD].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$\nbe a perfect object.\n\\begin{enumerate}\n\\item We say the {\\it Chern classes of $E$ are defined}\\footnote{See\nLemma \\ref{lemma-chern-classes-defined} for some criteria.} if there exists\nan envelope $f : Y \\to X$ such that $Lf^*E$ is isomorphic in\n$D(\\mathcal{O}_Y)$ to a locally bounded complex of finite locally free\n$\\mathcal{O}_Y$-modules.\n\\item If the Chern classes of $E$ are defined, then we define\n$$\nc(E) \\in \\prod\\nolimits_{p \\geq 0} A^p(X),\\quad\nch(E) \\in\n\\prod\\nolimits_{p \\geq 0} A^p(X) \\otimes \\mathbf{Q},\\quad\nP_p(E) \\in A^p(X)\n$$\nby an application of Lemma \\ref{lemma-defined-by-envelope}.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9E","source_file":"chow.tex","source_line":9271,"source_end_line":9291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9271-L9291","statement_sha256":"93c800163369885f507ce7cd2f3ee7a67591af058ebc31b3496604e3b015fabe","origin":"The Stacks Project","memory_eligible":false,"source_rank":8266,"rank":8266,"depth":55,"x":2589.791,"y":611.082,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GUE","tag":"0GUE","title":"Chern classes and the derived category · Lemma 0GUE","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E ∈ D(O_X) be a perfect object. If one of the following conditions hold, then the Chern classes of E are defined: • there exists an envelope f : Y → X such that Lf^*E is isomorphic in D(O_Y) to a locally bounded complex of finite locally free O_Y-modules, • E can be represented by a bounded complex of finite locally free O_X-modules, • the irreducible components of X are quasi-compact, •…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$\nbe a perfect object. If one of the following conditions hold, then\nthe Chern classes of $E$ are defined:\n\\begin{enumerate}\n\\item there exists an envelope $f : Y \\to X$ such that $Lf^*E$\nis isomorphic in $D(\\mathcal{O}_Y)$ to a locally bounded complex of finite\nlocally free $\\mathcal{O}_Y$-modules,\n\\item $E$ can be represented by a bounded complex of finite locally\nfree $\\mathcal{O}_X$-modules,\n\\item the irreducible components of $X$ are quasi-compact,\n\\item $X$ is quasi-compact,\n\\item there exists a morphism $X \\to X'$ of schemes locally of finite type\nover $S$ such that $E$ is the pullback of a perfect object $E'$ on $X'$\nwhose chern classes are defined, or\n\\item add more here.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUE","source_file":"chow.tex","source_line":9299,"source_end_line":9318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9299-L9318","statement_sha256":"1290310180448b400d7ee2f8b7f634e9409668ad0779a13b694a0b6ac66f4dd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8267,"rank":8267,"depth":56,"x":2454.434,"y":793.25,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GUF","tag":"0GUF","title":"Chern classes and the derived category · Lemma 0GUF","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E ∈ D(O_X) be a perfect object. Assume the Chern classes of E are defined. For g : W → X locally of finite type with W integral, there exists a commutative diagram xymatrix W' ar[rd]_g' ar[rr]_b & & W ar[ld]^g & X with W' integral and b : W' → W proper birational such that L(g')^*E is represented by a bounded complex E^bullet of locally free O_W'-modules of constant rank and we have…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let $E \\in D(\\mathcal{O}_X)$\nbe a perfect object. Assume the Chern classes of $E$ are defined.\nFor $g : W \\to X$ locally of finite type with $W$ integral, there exists\na commutative diagram\n$$\n\\xymatrix{\nW' \\ar[rd]_{g'} \\ar[rr]_b & & W \\ar[ld]^g \\\\\n& X\n}\n$$\nwith $W'$ integral and $b : W' \\to W$ proper birational such that $L(g')^*E$\nis represented by a bounded complex $\\mathcal{E}^\\bullet$ of locally free\n$\\mathcal{O}_{W'}$-modules of constant rank and we have\n$res(c_p(E)) = c_p(\\mathcal{E}^\\bullet)$ in $A^p(W')$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUF","source_file":"chow.tex","source_line":9355,"source_end_line":9372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9355-L9372","statement_sha256":"fde194736a91bad66214fbfd11d75f63bb0210591fcb3222af0979c9007e095b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8268,"rank":8268,"depth":18,"x":2407.695,"y":581.869,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAC","tag":"0FAC","title":"Chern classes and the derived category · Lemma 0FAC","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E ∈ D(O_X) be perfect. If the Chern classes of E are defined then • c_p(E) is in the center of the algebra A^*(X), and • if g : X' → X is locally of finite type and c ∈ A^*(X' → X), then c ∘ c_p(E) = c_p(Lg^*E) ∘ c.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $E \\in D(\\mathcal{O}_X)$ be perfect. If the Chern classes\nof $E$ are defined then\n\\begin{enumerate}\n\\item $c_p(E)$ is in the center of the algebra $A^*(X)$, and\n\\item if $g : X' \\to X$ is locally of finite type and $c \\in A^*(X' \\to X)$,\nthen $c \\circ c_p(E) = c_p(Lg^*E) \\circ c$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAC","source_file":"chow.tex","source_line":9393,"source_end_line":9404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9393-L9404","statement_sha256":"d3440361de8821282550dc7084ebd592894711629d1bf6e7325dacc5753a5f54","origin":"The Stacks Project","memory_eligible":false,"source_rank":8269,"rank":8269,"depth":49,"x":2612.386,"y":711.369,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F9F","tag":"0F9F","title":"Chern classes and the derived category · Lemma 0F9F","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let E_1 → E_2 → E_3 → E_1[1] be a distinguished triangle of perfect objects in D(O_X). If one of the following conditions holds • there exists an envelope f : Y → X such that Lf^*E_1 → Lf^*E_2 can be represented by a map of locally bounded complexes of finite locally free O_Y-modules, • E_1 → E_2 can be represented be a map of locally bounded complexes of finite locally free O_X-modules, •…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let\n$$\nE_1 \\to E_2 \\to E_3 \\to E_1[1]\n$$\nbe a distinguished triangle of perfect objects in $D(\\mathcal{O}_X)$.\nIf one of the following conditions holds\n\\begin{enumerate}\n\\item there exists an envelope $f : Y \\to X$ such that\n$Lf^*E_1 \\to Lf^*E_2$ can be represented by a map of locally\nbounded complexes of finite locally free $\\mathcal{O}_Y$-modules,\n\\item $E_1 \\to E_2$ can be represented be a map of locally bounded complexes\nof finite locally free $\\mathcal{O}_X$-modules,\n\\item the irreducible components of $X$ are quasi-compact,\n\\item $X$ is quasi-compact, or\n\\item add more here,\n\\end{enumerate}\nthen the Chern classes of $E_1$, $E_2$, $E_3$ are defined and we have\n$c(E_2) = c(E_1) c(E_3)$, $ch(E_2) = ch(E_1) + ch(E_3)$, and\n$P_p(E_2) = P_p(E_1) + P_p(E_3)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9F","source_file":"chow.tex","source_line":9426,"source_end_line":9448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9426-L9448","statement_sha256":"96b7675c495d62ef07fd4423ab3902648aa4eaa107fda873c6c4a29f8ea17289","origin":"The Stacks Project","memory_eligible":false,"source_rank":8270,"rank":8270,"depth":57,"x":2357.01,"y":732.049,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAE","tag":"0FAE","title":"Chern classes and the derived category · Lemma 0FAE","summary":"In Situation [Tag 02QL] let X be locally of finite type over S. Let E ∈ D(O_X) be a perfect object whose Chern classes are defined. Then c_i(E^vee) = (-1)^i c_i(E), P_i(E^vee) = (-1)^iP_i(E), and ch_i(E^vee) = (-1)^ich_i(E) in A^i(X).","statement_latex":"In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.\nLet $E \\in D(\\mathcal{O}_X)$ be a perfect object whose Chern classes are\ndefined. Then $c_i(E^\\vee) = (-1)^i c_i(E)$, $P_i(E^\\vee) = (-1)^iP_i(E)$,\nand $ch_i(E^\\vee) = (-1)^ich_i(E)$ in $A^i(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAE","source_file":"chow.tex","source_line":9544,"source_end_line":9550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9544-L9550","statement_sha256":"6dc2b0b9f20b4599c967ad353bc722c4c400d2d1750ea14d7325c138e6ee52ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":8271,"rank":8271,"depth":52,"x":2528.892,"y":571.707,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAF","tag":"0FAF","title":"Chern classes and the derived category · Lemma 0FAF","summary":"In Situation [Tag 02QL] let X be locally of finite type over S. Let E be a perfect object of D(O_X) whose Chern classes are defined. Let L be an invertible O_X-module. Then c_i(E ⊗ L) = ∑_j = 0^i binomr - i + jj c_i - j(E) c_1(L)^j provided E has constant rank r ∈ Z.","statement_latex":"In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.\nLet $E$ be a perfect object of $D(\\mathcal{O}_X)$ whose Chern classes\nare defined.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then\n$$\nc_i(E \\otimes \\mathcal{L}) =\n\\sum\\nolimits_{j = 0}^i\n\\binom{r - i + j}{j} c_{i - j}(E) c_1(\\mathcal{L})^j\n$$\nprovided $E$ has constant rank $r \\in \\mathbf{Z}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAF","source_file":"chow.tex","source_line":9564,"source_end_line":9576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9564-L9576","statement_sha256":"563afe3ea8ca6a70a80926b7520dd9dda31ea6323a37a14331b98816680c7928","origin":"The Stacks Project","memory_eligible":false,"source_rank":8272,"rank":8272,"depth":49,"x":2531.094,"y":787.721,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAG","tag":"0FAG","title":"Chern classes and the derived category · Lemma 0FAG","summary":"In Situation [Tag 02QL] let X be locally of finite type over S. Let E and F be perfect objects of D(O_X) whose Chern classes are defined. Then we have c_1(E ⊗_O_X^L F) = r(E) c_1(F) + r(F) c_1(E) and for c_2(E ⊗_O_X^L F) we have the expression r(E) c_2(F) + r(F) c_2(E) + r(E) choose 2 c_1(F)^2 + (r(E)r(F) - 1) c_1(F)c_1(E) + r(F) choose 2 c_1(E)^2 and so on for higher Chern classes in A^*(X). Similarly, we have ch(E ⊗_O_X^L F) = ch(E) ch(F) in A^*(X) ⊗ Q. More precisely,…","statement_latex":"In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.\nLet $E$ and $F$ be perfect objects of $D(\\mathcal{O}_X)$ whose Chern classes\nare defined. Then we have\n$$\nc_1(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F) =\nr(E) c_1(\\mathcal{F}) + r(F) c_1(\\mathcal{E})\n$$\nand for $c_2(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F)$ we have the expression\n$$\nr(E) c_2(F) + r(F) c_2(E) + {r(E) \\choose 2} c_1(F)^2 +\n(r(E)r(F) - 1) c_1(F)c_1(E) + {r(F) \\choose 2} c_1(E)^2\n$$\nand so on for higher Chern classes in $A^*(X)$. Similarly, we have\n$ch(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F) = ch(E) ch(F)$\nin $A^*(X) \\otimes \\mathbf{Q}$. More precisely, we have\n$$\nP_p(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F) = \\sum\\nolimits_{p_1 + p_2 = p}\n{p \\choose p_1} P_{p_1}(E) P_{p_2}(F)\n$$\nin $A^p(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chern classes and the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAG","source_file":"chow.tex","source_line":9600,"source_end_line":9622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9600-L9622","statement_sha256":"7ef9e56a364603531896fbde866cd4473bbeac341fb83b8e195265d257f12608","origin":"The Stacks Project","memory_eligible":false,"source_rank":8273,"rank":8273,"depth":53,"x":2355.553,"y":629.501,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F9H","tag":"0F9H","title":"A baby case of localized Chern classes · Lemma 0F9H","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let i_j : X_j → X, j = 1, 2 be closed immersions such that X = X_1 ∪ X_2 set theoretically. Let E_2 ∈ D(O_X_2) be a perfect object. Assume • Chern classes of E_2 are defined, • the restriction E_2|_X_1 ∩ X_2 is zero, resp. isomorphic to a finite locally free O_X_1 ∩ X_2-module of rank < p sitting in cohomological degree 0. Then there is a canonical bivariant class P'_p(E_2), resp. c'_p(E_2)…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be\nlocally of finite type over $S$. Let $i_j : X_j \\to X$, $j = 1, 2$\nbe closed immersions such that $X = X_1 \\cup X_2$ set theoretically. Let\n$E_2 \\in D(\\mathcal{O}_{X_2})$ be a perfect object. Assume\n\\begin{enumerate}\n\\item Chern classes of $E_2$ are defined,\n\\item the restriction $E_2|_{X_1 \\cap X_2}$ is zero,\nresp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X_1 \\cap X_2}$-module\nof rank $< p$ sitting in cohomological degree $0$.\n\\end{enumerate}\nThen there is a canonical bivariant class\n$$\nP'_p(E_2),\\text{ resp. }c'_p(E_2) \\in A^p(X_2 \\to X)\n$$\ncharacterized by the property\n$$\nP'_p(E_2) \\cap i_{2, *} \\alpha_2 = P_p(E_2) \\cap \\alpha_2\n\\quad\\text{and}\\quad\nP'_p(E_2) \\cap i_{1, *} \\alpha_1 = 0,\n$$\nrespectively\n$$\nc'_p(E_2) \\cap i_{2, *} \\alpha_2 = c_p(E_2) \\cap \\alpha_2\n\\quad\\text{and}\\quad\nc'_p(E_2) \\cap i_{1, *} \\alpha_1 = 0\n$$\nfor $\\alpha_i \\in \\CH_k(X_i)$ and similarly after any base change\n$X' \\to X$ locally of finite type.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9H","source_file":"chow.tex","source_line":9694,"source_end_line":9724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9694-L9724","statement_sha256":"b4247e0d7fbc020a8344842af2bcdd366d8198683a9f88d555a4854e06a21c14","origin":"The Stacks Project","memory_eligible":false,"source_rank":8274,"rank":8274,"depth":2,"x":2612.528,"y":646.584,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAH","tag":"0FAH","title":"A baby case of localized Chern classes · Lemma 0FAH","summary":"In Lemma [Tag 0F9H] the bivariant class P'_p(E_2), resp. c'_p(E_2) in A^p(X_2 → X) does not depend on the choice of X_1.","statement_latex":"In Lemma \\ref{lemma-silly} the bivariant class\n$P'_p(E_2)$, resp.\\ $c'_p(E_2)$ in $A^p(X_2 \\to X)$\ndoes not depend on the choice of $X_1$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAH","source_file":"chow.tex","source_line":9751,"source_end_line":9756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9751-L9756","statement_sha256":"bf0b571b1ab6944b4ec48d586cc386ad2d0bf9cbb2acdf1d9a8987e017bc0c11","origin":"The Stacks Project","memory_eligible":false,"source_rank":8275,"rank":8275,"depth":3,"x":2409.066,"y":779.955,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GUG","tag":"0GUG","title":"A baby case of localized Chern classes · Lemma 0GUG","summary":"In Lemma [Tag 0F9H] let X' → X be a morphism which is locally of finite type. Denote X' = X'_1 ∪ X'_2 and E'_2 ∈ D(O_X'_2) the pullbacks to X'. Then the class P'_p(E_2'), resp. c'_p(E_2') in A^p(X_2' → X') constructed in Lemma [Tag 0F9H] using X' = X'_1 ∪ X'_2 and E_2' is the restriction (Remark [Tag 0F9Z]) of the class P'_p(E_2), resp. c'_p(E_2) in A^p(X_2 → X).","statement_latex":"In Lemma \\ref{lemma-silly} let $X' \\to X$ be a morphism\nwhich is locally of finite type. Denote $X' = X'_1 \\cup X'_2$\nand $E'_2 \\in D(\\mathcal{O}_{X'_2})$ the pullbacks to $X'$.\nThen the class $P'_p(E_2')$, resp.\\ $c'_p(E_2')$ in\n$A^p(X_2' \\to X')$ constructed in Lemma \\ref{lemma-silly} using\n$X' = X'_1 \\cup X'_2$ and $E_2'$ is the restriction\n(Remark \\ref{remark-restriction-bivariant})\nof the class $P'_p(E_2)$, resp.\\ $c'_p(E_2)$ in $A^p(X_2 \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUG","source_file":"chow.tex","source_line":9774,"source_end_line":9784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9774-L9784","statement_sha256":"ff92b48f44a9a51110f9367b680ec7f2528aba1eb19761f4818214868a2fdf5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8276,"rank":8276,"depth":3,"x":2451.892,"y":565.913,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F9I","tag":"0F9I","title":"A baby case of localized Chern classes · Lemma 0F9I","summary":"In Lemma [Tag 0F9H] say E_2 is the restriction of a perfect E ∈ D(O_X) such that E|_X_1 is zero, resp. isomorphic to a finite locally free O_X_1-module of rank < p sitting in cohomological degree 0. If Chern classes of E are defined, then i_2, * ∘ P'_p(E_2) = P_p(E), resp. i_2, * ∘ c'_p(E_2) = c_p(E) (with ∘ as in Lemma [Tag 0EPK]).","statement_latex":"In Lemma \\ref{lemma-silly} say $E_2$ is the restriction of a\nperfect $E \\in D(\\mathcal{O}_X)$ such that $E|_{X_1}$ is zero,\nresp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X_1}$-module\nof rank $< p$ sitting in cohomological degree $0$.\nIf Chern classes of $E$ are defined, then\n$i_{2, *} \\circ P'_p(E_2) = P_p(E)$,\nresp.\\ $i_{2, *} \\circ c'_p(E_2) = c_p(E)$\n(with $\\circ$ as in Lemma \\ref{lemma-push-proper-bivariant}).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9I","source_file":"chow.tex","source_line":9791,"source_end_line":9801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9791-L9801","statement_sha256":"d10363be2733906e3b4628fb6a711e58f51550e690a6fb4ef67c7e00ec31a9ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":8277,"rank":8277,"depth":47,"x":2592.601,"y":748.255,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAI","tag":"0FAI","title":"A baby case of localized Chern classes · Lemma 0FAI","summary":"In Lemma [Tag 0F9H] suppose we have closed subschemes X'_2 ⊂ X_2 and X_1 ⊂ X'_1 ⊂ X such that X = X'_1 ∪ X'_2 set theoretically. Assume E_2|_X'_1 ∩ X_2 is zero, resp. isomorphic to a finite locally free module of rank < p placed in degree 0. Then we have (X'_2 → X_2)_* ∘ P'_p(E_2|_X'_2) = P'_p(E_2), resp. (X'_2 → X_2)_* ∘ c'_p(E_2|_X'_2) = c_p(E_2) (with ∘ as in Lemma [Tag 0EPK]).","statement_latex":"In Lemma \\ref{lemma-silly} suppose we have closed subschemes\n$X'_2 \\subset X_2$ and $X_1 \\subset X'_1 \\subset X$ such that\n$X = X'_1 \\cup X'_2$ set theoretically. Assume $E_2|_{X'_1 \\cap X_2}$\nis zero, resp.\\ isomorphic to a finite locally free module\nof rank $< p$ placed in degree $0$. Then we have\n$(X'_2 \\to X_2)_* \\circ P'_p(E_2|_{X'_2}) = P'_p(E_2)$,\nresp.\\  $(X'_2 \\to X_2)_* \\circ c'_p(E_2|_{X'_2}) = c_p(E_2)$\n(with $\\circ$ as in Lemma \\ref{lemma-push-proper-bivariant}).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAI","source_file":"chow.tex","source_line":9823,"source_end_line":9833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9823-L9833","statement_sha256":"d3bc96b38ef39a3c769030adb3135ae1e443a98818ee5f91fa59964c87f33be4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8278,"rank":8278,"depth":47,"x":2341.923,"y":693.578,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAJ","tag":"0FAJ","title":"A baby case of localized Chern classes · Lemma 0FAJ","summary":"In Lemma [Tag 0F9H] let f : Y → X be locally of finite type and say c ∈ A^*(Y → X). Then c ∘ P'_p(E_2) = P'_p(Lf_2^*E_2) ∘ c resp. c ∘ c'_p(E_2) = c'_p(Lf_2^*E_2) ∘ c in A^*(Y_2 → Y) where f_2 : Y_2 → X_2 is the base change of f.","statement_latex":"In Lemma \\ref{lemma-silly} let $f : Y \\to X$ be locally of finite type\nand say $c \\in A^*(Y \\to X)$. Then\n$$\nc \\circ P'_p(E_2) = P'_p(Lf_2^*E_2) \\circ c\n\\quad\\text{resp.}\\quad\nc \\circ c'_p(E_2) = c'_p(Lf_2^*E_2) \\circ c\n$$\nin $A^*(Y_2 \\to Y)$ where $f_2 : Y_2 \\to X_2$ is the base change of $f$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAJ","source_file":"chow.tex","source_line":9840,"source_end_line":9850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9840-L9850","statement_sha256":"33b90da83e7fe8c06b29d23bdfa01be38ac30dcab903144d37f5d652911252db","origin":"The Stacks Project","memory_eligible":false,"source_rank":8279,"rank":8279,"depth":50,"x":2571.001,"y":591.538,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAK","tag":"0FAK","title":"A baby case of localized Chern classes · Lemma 0FAK","summary":"In Lemma [Tag 0F9H] assume E_2|_X_1 ∩ X_2 is zero. Then P'_1(E_2) & = c'_1(E_2), P'_2(E_2) & = c'_1(E_2)^2 - 2c'_2(E_2), P'_3(E_2) & = c'_1(E_2)^3 - 3c'_1(E_2)c'_2(E_2) + 3c'_3(E_2), P'_4(E_2) & = c'_1(E_2)^4 - 4c'_1(E_2)^2c'_2(E_2) + 4c'_1(E_2)c'_3(E_2) + 2c'_2(E_2)^2 - 4c'_4(E_2), and so on with multiplication as in Remark [Tag 0FA0].","statement_latex":"In Lemma \\ref{lemma-silly} assume $E_2|_{X_1 \\cap X_2}$ is zero. Then\n\\begin{align*}\nP'_1(E_2) & = c'_1(E_2), \\\\\nP'_2(E_2) & = c'_1(E_2)^2 - 2c'_2(E_2), \\\\\nP'_3(E_2) & = c'_1(E_2)^3 - 3c'_1(E_2)c'_2(E_2) + 3c'_3(E_2), \\\\\nP'_4(E_2) & = c'_1(E_2)^4 - 4c'_1(E_2)^2c'_2(E_2) +\n4c'_1(E_2)c'_3(E_2) + 2c'_2(E_2)^2 - 4c'_4(E_2),\n\\end{align*}\nand so on with multiplication as in Remark \\ref{remark-ring-loc-classes}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAK","source_file":"chow.tex","source_line":9866,"source_end_line":9877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9866-L9877","statement_sha256":"93a6d960fd0d44f59f57e39283a8a94023af6041dcf03cd65c3f115abaed1054","origin":"The Stacks Project","memory_eligible":false,"source_rank":8280,"rank":8280,"depth":3,"x":2484.04,"y":797.0,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAL","tag":"0FAL","title":"A baby case of localized Chern classes · Lemma 0FAL","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let i_j : X_j → X, j = 1, 2 be closed immersions such that X = X_1 ∪ X_2 set theoretically. Let E, F ∈ D(O_X) be perfect objects. Assume • Chern classes of E and F are defined, • the restrictions E|_X_1 ∩ X_2 and F|_X_1 ∩ X_2 are isomorphic to a finite locally free O_X_1-modules of rank < p and < q sitting in cohomological degree 0. With notation as in Remark [Tag 0FA0] set c^(p)(E) = 1 +…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be\nlocally of finite type over $S$. Let $i_j : X_j \\to X$, $j = 1, 2$\nbe closed immersions such that $X = X_1 \\cup X_2$ set theoretically. Let\n$E, F \\in D(\\mathcal{O}_X)$ be perfect objects. Assume\n\\begin{enumerate}\n\\item Chern classes of $E$ and $F$ are defined,\n\\item the restrictions $E|_{X_1 \\cap X_2}$ and $F|_{X_1 \\cap X_2}$\nare isomorphic to a finite locally free $\\mathcal{O}_{X_1}$-modules\nof rank $< p$ and $< q$ sitting in cohomological degree $0$.\n\\end{enumerate}\nWith notation as in Remark \\ref{remark-ring-loc-classes} set\n$$\nc^{(p)}(E) = 1 + c_1(E) + \\ldots + c_{p - 1}(E) +\nc'_p(E|_{X_2}) + c'_{p + 1}(E|_{X_2}) + \\ldots \\in A^{(p)}(X_2 \\to X)\n$$\nwith $c'_p(E|_{X_2})$ as in Lemma \\ref{lemma-silly}. Similarly\nfor $c^{(q)}(F)$ and $c^{(p + q)}(E \\oplus F)$.\nThen $c^{(p + q)}(E \\oplus F) = c^{(p)}(E)c^{(q)}(F)$\nin $A^{(p + q)}(X_2 \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAL","source_file":"chow.tex","source_line":9888,"source_end_line":9909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9888-L9909","statement_sha256":"483354a987ce533b83493dd6f759549e9a56ce3546b01f47f1ddc4f88b6fe173","origin":"The Stacks Project","memory_eligible":false,"source_rank":8281,"rank":8281,"depth":58,"x":2382.822,"y":595.923,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAM","tag":"0FAM","title":"A baby case of localized Chern classes · Lemma 0FAM","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let i_j : X_j → X, j = 1, 2 be closed immersions such that X = X_1 ∪ X_2 set theoretically. Let E, F ∈ D(O_X_2) be perfect objects. Assume • Chern classes of E and F are defined, • the restrictions E|_X_1 ∩ X_2 and F|_X_1 ∩ X_2 are zero, Denote P'_p(E), P'_p(F), P'_p(E ⊕ F) ∈ A^p(X_2 → X) for p ≥ 0 the classes constructed in Lemma [Tag 0F9H]. Then P'_p(E ⊕ F) = P'_p(E) + P'_p(F).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be\nlocally of finite type over $S$. Let $i_j : X_j \\to X$, $j = 1, 2$\nbe closed immersions such that $X = X_1 \\cup X_2$ set theoretically. Let\n$E, F \\in D(\\mathcal{O}_{X_2})$ be perfect objects. Assume\n\\begin{enumerate}\n\\item Chern classes of $E$ and $F$ are defined,\n\\item the restrictions $E|_{X_1 \\cap X_2}$ and $F|_{X_1 \\cap X_2}$ are zero,\n\\end{enumerate}\nDenote $P'_p(E), P'_p(F), P'_p(E \\oplus F) \\in A^p(X_2 \\to X)$ for $p \\geq 0$\nthe classes constructed in Lemma \\ref{lemma-silly}. Then\n$P'_p(E \\oplus F) = P'_p(E) + P'_p(F)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAM","source_file":"chow.tex","source_line":9917,"source_end_line":9930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9917-L9930","statement_sha256":"3ff733650f72dfc143a7dd20abc8fff52fe40745072ab0fdac11a6de158979fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":8282,"rank":8282,"depth":58,"x":2619.429,"y":686.864,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAN","tag":"0FAN","title":"A baby case of localized Chern classes · Lemma 0FAN","summary":"In Lemma [Tag 0F9H] assume E_2 has constant rank 0. Let L be an invertible O_X-module. Then c'_i(E_2 ⊗ L) = ∑_j = 0^i binom- i + jj c'_i - j(E_2) c_1(L)^j","statement_latex":"In Lemma \\ref{lemma-silly} assume $E_2$ has constant rank $0$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module. Then\n$$\nc'_i(E_2 \\otimes \\mathcal{L}) =\n\\sum\\nolimits_{j = 0}^i\n\\binom{- i + j}{j} c'_{i - j}(E_2) c_1(\\mathcal{L})^j\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAN","source_file":"chow.tex","source_line":9938,"source_end_line":9947,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9938-L9947","statement_sha256":"94d80fd46ed0f6fb0b5d7ecc4c6b306a06cbfefe29e9791024d61531696156af","origin":"The Stacks Project","memory_eligible":false,"source_rank":8283,"rank":8283,"depth":50,"x":2371.545,"y":754.138,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FE5","tag":"0FE5","title":"A baby case of localized Chern classes · Lemma 0FE5","summary":"In Situation [Tag 02QL] let X be locally of finite type over S. Let X = X_1 ∪ X_2 = X'_1 ∪ X'_2 be two ways of writing X as a set theoretic union of closed subschemes. Let E, E' be perfect objects of D(O_X) whose Chern classes are defined. Assume that E|_X_1 and E'|_X'_1 are zero for i = 1, 2. Denote • r = P'_0(E) ∈ A^0(X_2 → X) and r' = P'_0(E') ∈ A^0(X'_2 → X), • γ_p = c'_p(E|_X_2) ∈ A^p(X_2 → X) and γ'_p = c'_p(E'|_X'_2) ∈ A^p(X'_2 → X), • chi_p = P'_p(E|_X_2) ∈…","statement_latex":"In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.\nLet\n$$\nX = X_1 \\cup X_2 = X'_1 \\cup X'_2\n$$\nbe two ways of writing $X$ as a set theoretic union of closed subschemes.\nLet $E$, $E'$ be perfect objects of $D(\\mathcal{O}_X)$\nwhose Chern classes are defined.\nAssume that $E|_{X_1}$ and $E'|_{X'_1}$ are zero\\footnote{Presumably there\nis a variant of this lemma where we only assume these restrictions are\nisomorphic to a finite locally free modules\nof rank $< p$ and $< p'$.} for $i = 1, 2$. Denote\n\\begin{enumerate}\n\\item $r = P'_0(E) \\in A^0(X_2 \\to X)$ and\n$r' = P'_0(E') \\in A^0(X'_2 \\to X)$,\n\\item $\\gamma_p = c'_p(E|_{X_2}) \\in A^p(X_2 \\to X)$ and\n$\\gamma'_p = c'_p(E'|_{X'_2}) \\in A^p(X'_2 \\to X)$,\n\\item $\\chi_p = P'_p(E|_{X_2}) \\in A^p(X_2 \\to X)$ and\n$\\chi'_p = P'_p(E'|_{X'_2}) \\in A^p(X'_2 \\to X)$\n\\end{enumerate}\nthe classes constructed in Lemma \\ref{lemma-silly}. Then we have\n$$\nc'_1((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E')|_{X_2 \\cap X'_2}) =\nr \\gamma'_1 + r' \\gamma_1\n$$\nin $A^1(X_2 \\cap X'_2 \\to X)$ and\n$$\nc'_2((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E')|_{X_2 \\cap X'_2}) =\nr \\gamma'_2 + r' \\gamma_2 + {r \\choose 2} (\\gamma'_1)^2 +\n(rr' - 1) \\gamma'_1\\gamma_1 + {r' \\choose 2} \\gamma_1^2\n$$\nin $A^2(X_2 \\cap X'_2 \\to X)$ and so on for higher Chern classes.\nSimilarly, we have\n$$\nP'_p((E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E')|_{X_2 \\cap X'_2}) =\n\\sum\\nolimits_{p_1 + p_2 = p}\n{p \\choose p_1} \\chi_{p_1} \\chi'_{p_2}\n$$\nin $A^p(X_2 \\cap X'_2 \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"A baby case of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FE5","source_file":"chow.tex","source_line":9957,"source_end_line":9998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L9957-L9998","statement_sha256":"a6277f2cf2312846b41bf211b9a47608a074237300fb879a7a7d018db6725e2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8284,"rank":8284,"depth":54,"x":2500.377,"y":563.661,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F9J","tag":"0F9J","title":"Gysin at infinity · Lemma 0F9J","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let b : W → P^1_X be a proper morphism of schemes which is an isomorphism over A^1_X. Denote i_∞ : W_∞ → W the inverse image of the divisor D_∞ ⊂ P^1_X with complement A^1_X. Then there is a canonical bivariant class C ∈ A^0(W_∞ → X) with the property that i_∞, *(C ∩ α) = i_0, *α for α ∈ CH_k(X) and similarly after any base change by X' → X locally of finite type.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let\n$b : W \\to \\mathbf{P}^1_X$ be a proper morphism of schemes\nwhich is an isomorphism over $\\mathbf{A}^1_X$.\nDenote $i_\\infty : W_\\infty \\to W$ the inverse image of the divisor\n$D_\\infty \\subset \\mathbf{P}^1_X$ with complement $\\mathbf{A}^1_X$.\nThen there is a canonical bivariant class\n$$\nC \\in A^0(W_\\infty \\to X)\n$$\nwith the property that\n$i_{\\infty, *}(C \\cap \\alpha) = i_{0, *}\\alpha$\nfor $\\alpha \\in \\CH_k(X)$ and similarly after any base change by\n$X' \\to X$ locally of finite type.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin at infinity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9J","source_file":"chow.tex","source_line":10025,"source_end_line":10041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10025-L10041","statement_sha256":"290650491974525fbabf3abec12dfe91dfc26476c74cf8c95a2176e0a0e61ecf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8285,"rank":8285,"depth":52,"x":2558.62,"y":777.459,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GUH","tag":"0GUH","title":"Gysin at infinity · Lemma 0GUH","summary":"In Lemma [Tag 0F9J] let X' → X be a morphism which is locally of finite type. Denote b' : W' → P^1_X' and i'_∞ : W'_∞ → W' the base changes of b and i_∞. Then the class C' ∈ A^0(W'_∞ → X') constructed as in Lemma [Tag 0F9J] using b' is the restriction (Remark [Tag 0F9Z]) of C.","statement_latex":"In Lemma \\ref{lemma-gysin-at-infty} let $X' \\to X$ be a morphism\nwhich is locally of finite type. Denote $b' : W' \\to \\mathbf{P}^1_{X'}$\nand $i'_\\infty : W'_\\infty \\to W'$ the base changes of $b$ and $i_\\infty$.\nThen the class $C' \\in A^0(W'_\\infty \\to X')$ constructed as in\nLemma \\ref{lemma-gysin-at-infty} using $b'$ is the restriction\n(Remark \\ref{remark-restriction-bivariant}) of $C$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin at infinity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUH","source_file":"chow.tex","source_line":10092,"source_end_line":10100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10092-L10100","statement_sha256":"3160b1d15873d0d39358de17d298ae31ee59af2062092a545394283f321f6ae2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8286,"rank":8286,"depth":53,"x":2343.499,"y":652.715,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAQ","tag":"0FAQ","title":"Gysin at infinity · Lemma 0FAQ","summary":"In Lemma [Tag 0F9J] let g : W' → W be a proper morphism which is an isomorphism over A^1_X. Let C' ∈ A^0(W'_∞ → X) and C ∈ A^0(W_∞ → X) be the classes constructed in Lemma [Tag 0F9J]. Then g_∞, * ∘ C' = C in A^0(W_∞ → X).","statement_latex":"In Lemma \\ref{lemma-gysin-at-infty} let $g : W' \\to W$ be a proper morphism\nwhich is an isomorphism over $\\mathbf{A}^1_X$. Let\n$C' \\in A^0(W'_\\infty \\to X)$ and $C \\in A^0(W_\\infty \\to X)$\nbe the classes constructed in Lemma \\ref{lemma-gysin-at-infty}.\nThen $g_{\\infty, *} \\circ C' = C$ in $A^0(W_\\infty \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin at infinity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAQ","source_file":"chow.tex","source_line":10107,"source_end_line":10114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10107-L10114","statement_sha256":"305b7325b3513e9e71e38fd4dca68f2ec6529de7f5e127c6e3ba48824614180d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8287,"rank":8287,"depth":53,"x":2602.733,"y":622.601,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAR","tag":"0FAR","title":"Gysin at infinity · Lemma 0FAR","summary":"In Lemma [Tag 0F9J] we have C ∘ (W_∞ → X)_* ∘ i_∞^* = i_∞^*.","statement_latex":"In Lemma \\ref{lemma-gysin-at-infty} we have\n$C \\circ (W_\\infty \\to X)_* \\circ i_\\infty^* = i_\\infty^*$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin at infinity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAR","source_file":"chow.tex","source_line":10131,"source_end_line":10135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10131-L10135","statement_sha256":"5497141e7419e772fd5080d633a2a41ed996621f797293cb75670c2f777e3cb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8288,"rank":8288,"depth":53,"x":2435.608,"y":792.092,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAS","tag":"0FAS","title":"Gysin at infinity · Lemma 0FAS","summary":"In Lemma [Tag 0F9J] let f : Y → X be a morphism locally of finite type and c ∈ A^*(Y → X). Then C ∘ c = c ∘ C in A^*(W_∞ ×_X Y → X).","statement_latex":"In Lemma \\ref{lemma-gysin-at-infty} let $f : Y \\to X$ be a morphism\nlocally of finite type and $c \\in A^*(Y \\to X)$. Then $C \\circ c = c \\circ C$\nin $A^*(W_\\infty \\times_X Y \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin at infinity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAS","source_file":"chow.tex","source_line":10155,"source_end_line":10160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10155-L10160","statement_sha256":"afc5856348248a4eb6258a3a3da2202a4311d13fc470a1f75e707ad0d71a9216","origin":"The Stacks Project","memory_eligible":false,"source_rank":8289,"rank":8289,"depth":53,"x":2422.529,"y":572.035,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F9K","tag":"0F9K","title":"Preparation for localized Chern classes · Lemma 0F9K","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let Z ⊂ X be a closed subscheme. Let b : W → P^1_X be a proper morphism of schemes. Let Q ∈ D(O_W) be a perfect object. Denote W_∞ ⊂ W the inverse image of the divisor D_∞ ⊂ P^1_X with complement A^1_X. We assume • [(A0)] Chern classes of Q are defined (Section [Tag 0ESY]), • [(A1)] b is an isomorphism over A^1_X, • [(A2)] there exists a closed subscheme T ⊂ W_∞ containing all points of W_∞…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be\nlocally of finite type over $S$. Let $Z \\subset X$ be a closed subscheme.\nLet\n$$\nb : W \\longrightarrow \\mathbf{P}^1_X\n$$\nbe a proper morphism of schemes. Let $Q \\in D(\\mathcal{O}_W)$ be a\nperfect object. Denote $W_\\infty \\subset W$ the inverse image of the divisor\n$D_\\infty \\subset \\mathbf{P}^1_X$ with complement $\\mathbf{A}^1_X$.\nWe assume\n\\begin{enumerate}\n\\item[(A0)] Chern classes of $Q$ are defined\n(Section \\ref{section-pre-derived}),\n\\item[(A1)] $b$ is an isomorphism over $\\mathbf{A}^1_X$,\n\\item[(A2)] there exists a closed subscheme $T \\subset W_\\infty$\ncontaining all points of $W_\\infty$ lying over $X \\setminus Z$ such that\n$Q|_T$ is zero, resp.\\ isomorphic to a finite locally free\n$\\mathcal{O}_T$-module of rank $< p$ sitting in cohomological degree $0$.\n\\end{enumerate}\nThen there exists a canonical bivariant class\n$$\nP'_p(Q),\\text{ resp. }c'_p(Q) \\in A^p(Z \\to X)\n$$\nwith\n$(Z \\to X)_* \\circ P'_p(Q) = P_p(Q|_{X \\times \\{0\\}})$,\nresp.\\ $(Z \\to X)_* \\circ c'_p(Q) = c_p(Q|_{X \\times \\{0\\}})$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9K","source_file":"chow.tex","source_line":10203,"source_end_line":10231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10203-L10231","statement_sha256":"33f968108a9580acbaf3107281635223b4955d16a9320939508196688a2e46d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8290,"rank":8290,"depth":53,"x":2609.345,"y":727.053,"cluster":"divisors-intersection-theory"},{"id":"stacks:0GUI","tag":"0GUI","title":"Preparation for localized Chern classes · Lemma 0GUI","summary":"In Lemma [Tag 0F9K] let X' → X be a morphism which is locally of finite type. Denote Z', b' : W' → P^1_X', and T' ⊂ W'_∞ the base changes of Z, b : W → P^1_X, and T ⊂ W_∞. Set Q' = (W' → W)^*Q. Then the class P'_p(Q'), resp. c'_p(Q') in A^p(Z' → X') constructed as in Lemma [Tag 0F9K] using b', Q', and T' is the restriction (Remark [Tag 0F9Z]) of the class P'_p(Q), resp. c'_p(Q) in A^p(Z → X).","statement_latex":"In Lemma \\ref{lemma-localized-chern-pre} let $X' \\to X$ be a morphism\nwhich is locally of finite type. Denote\n$Z'$, $b' : W' \\to \\mathbf{P}^1_{X'}$, and $T' \\subset W'_\\infty$\nthe base changes of $Z$, $b : W \\to \\mathbf{P}^1_X$, and $T \\subset W_\\infty$.\nSet $Q' = (W' \\to W)^*Q$. Then the class\n$P'_p(Q')$, resp.\\ $c'_p(Q')$ in $A^p(Z' \\to X')$ constructed as in\nLemma \\ref{lemma-localized-chern-pre} using $b'$, $Q'$, and $T'$\nis the restriction (Remark \\ref{remark-restriction-bivariant})\nof the class $P'_p(Q)$, resp.\\ $c'_p(Q)$ in $A^p(Z \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUI","source_file":"chow.tex","source_line":10275,"source_end_line":10286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10275-L10286","statement_sha256":"44a043e7b37c433ca9e558f3756f19cafdcffb6972f993d6791e3871a0a54bfe","origin":"The Stacks Project","memory_eligible":false,"source_rank":8291,"rank":8291,"depth":54,"x":2346.636,"y":718.74,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAU","tag":"0FAU","title":"Preparation for localized Chern classes · Lemma 0FAU","summary":"In Lemma [Tag 0F9K] the bivariant class P'_p(Q), resp. c'_p(Q) is independent of the choice of the closed subscheme T. Moreover, given a proper morphism g : W' → W which is an isomorphism over A^1_X, then setting Q' = g^*Q we have P'_p(Q) = P'_p(Q'), resp. c'_p(Q) = c'_p(Q').","statement_latex":"In Lemma \\ref{lemma-localized-chern-pre} the bivariant class\n$P'_p(Q)$, resp.\\ $c'_p(Q)$\nis independent of the choice of the closed subscheme $T$.\nMoreover, given a proper morphism $g : W' \\to W$ which is an\nisomorphism over $\\mathbf{A}^1_X$, then setting $Q' = g^*Q$\nwe have $P'_p(Q) = P'_p(Q')$, resp.\\ $c'_p(Q) = c'_p(Q')$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAU","source_file":"chow.tex","source_line":10300,"source_end_line":10308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10300-L10308","statement_sha256":"68991211091f07a43d55d695ac6d6a6179002a27b0f0a642e40b6c7e6160faa5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8292,"rank":8292,"depth":54,"x":2547.262,"y":575.644,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAV","tag":"0FAV","title":"Preparation for localized Chern classes · Lemma 0FAV","summary":"In Lemma [Tag 0F9K] assume Q|_T is isomorphic to a finite locally free O_T-module of rank < p. Denote C ∈ A^0(W_∞ → X) the class of Lemma [Tag 0F9J]. Then C ∘ c_p(Q|_X × (0)) = C ∘ (Z → X)_* ∘ c'_p(Q) = c_p(Q|_W_∞) ∘ C","statement_latex":"In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic\nto a finite locally free $\\mathcal{O}_T$-module of rank $< p$.\nDenote $C \\in A^0(W_\\infty \\to X)$ the class of\nLemma \\ref{lemma-gysin-at-infty}. Then\n$$\nC \\circ c_p(Q|_{X \\times \\{0\\}}) =\nC \\circ (Z \\to X)_* \\circ c'_p(Q) = c_p(Q|_{W_\\infty}) \\circ C\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAV","source_file":"chow.tex","source_line":10352,"source_end_line":10362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10352-L10362","statement_sha256":"5877e08f875ad36233b3d37098573722f9d8065f63a9fb7483dc15678f7685d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8293,"rank":8293,"depth":54,"x":2514.361,"y":795.244,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAW","tag":"0FAW","title":"Preparation for localized Chern classes · Lemma 0FAW","summary":"In Lemma [Tag 0F9K] let Y → X be a morphism locally of finite type and let c ∈ A^*(Y → X) be a bivariant class. Then P'_p(Q) ∘ c = c ∘ P'_p(Q) resp. c'_p(Q) ∘ c = c ∘ c'_p(Q) in A^*(Y ×_X Z → X).","statement_latex":"In Lemma \\ref{lemma-localized-chern-pre} let $Y \\to X$ be a morphism\nlocally of finite type and let $c \\in A^*(Y \\to X)$ be a bivariant class.\nThen\n$$\nP'_p(Q) \\circ c = c \\circ P'_p(Q)\n\\quad\\text{resp.}\\quad\nc'_p(Q) \\circ c = c \\circ c'_p(Q)\n$$\nin $A^*(Y \\times_X Z \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAW","source_file":"chow.tex","source_line":10407,"source_end_line":10418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10407-L10418","statement_sha256":"083292417c5a3a9a209334c8b6a7f861d067374106626fa602697202507f9f2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8294,"rank":8294,"depth":54,"x":2361.856,"y":614.445,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAX","tag":"0FAX","title":"Preparation for localized Chern classes · Lemma 0FAX","summary":"In Lemma [Tag 0F9K] assume Q|_T is zero. In A^*(Z → X) we have P'_1(Q) & = c'_1(Q), P'_2(Q) & = c'_1(Q)^2 - 2c'_2(Q), P'_3(Q) & = c'_1(Q)^3 - 3c'_1(Q)c'_2(Q) + 3c'_3(Q), P'_4(Q) & = c'_1(Q)^4 - 4c'_1(Q)^2c'_2(Q) + 4c'_1(Q)c'_3(Q) + 2c'_2(Q)^2 - 4c'_4(Q), and so on with multiplication as in Remark [Tag 0FA0].","statement_latex":"In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is zero. In\n$A^*(Z \\to X)$ we have\n\\begin{align*}\nP'_1(Q) & = c'_1(Q), \\\\\nP'_2(Q) & = c'_1(Q)^2 - 2c'_2(Q), \\\\\nP'_3(Q) & = c'_1(Q)^3 - 3c'_1(Q)c'_2(Q) + 3c'_3(Q), \\\\\nP'_4(Q) & = c'_1(Q)^4 - 4c'_1(Q)^2c'_2(Q) +\n4c'_1(Q)c'_3(Q) + 2c'_2(Q)^2 - 4c'_4(Q),\n\\end{align*}\nand so on with multiplication as in Remark \\ref{remark-ring-loc-classes}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAX","source_file":"chow.tex","source_line":10435,"source_end_line":10447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10435-L10447","statement_sha256":"acb053c46784aa3ec0f3a9172cc175ad8c2d45f8ec77fbe621719b17655eb423","origin":"The Stacks Project","memory_eligible":false,"source_rank":8295,"rank":8295,"depth":55,"x":2619.989,"y":661.281,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAY","tag":"0FAY","title":"Preparation for localized Chern classes · Lemma 0FAY","summary":"In Lemma [Tag 0F9K] assume Q|_T is isomorphic to a finite locally free O_T-module of rank < p. Assume we have another perfect object Q' ∈ D(O_W) whose Chern classes are defined with Q'|_T isomorphic to a finite locally free O_T-module of rank < p' placed in cohomological degree 0. With notation as in Remark [Tag 0FA0] set c^(p)(Q) = 1 + c_1(Q|_X × (0)) + … + c_p - 1(Q|_X × (0)) + c'_p(Q) + c'_p + 1(Q) + … in A^(p)(Z → X) with c'_i(Q) for i ≥ p as in Lemma [Tag 0F9K].…","statement_latex":"In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is isomorphic\nto a finite locally free $\\mathcal{O}_T$-module of rank $< p$.\nAssume we have another perfect object $Q' \\in D(\\mathcal{O}_W)$\nwhose Chern classes are defined with $Q'|_T$ isomorphic to a\nfinite locally free $\\mathcal{O}_T$-module of rank $< p'$ placed\nin cohomological degree $0$. With notation as in\nRemark \\ref{remark-ring-loc-classes} set\n$$\nc^{(p)}(Q) = 1 + c_1(Q|_{X \\times \\{0\\}}) + \\ldots +\nc_{p - 1}(Q|_{X \\times \\{0\\}}) +\nc'_{p}(Q) + c'_{p + 1}(Q) + \\ldots\n$$\nin $A^{(p)}(Z \\to X)$ with $c'_i(Q)$ for $i \\geq p$ as in\nLemma \\ref{lemma-localized-chern-pre}. Similarly for $c^{(p')}(Q')$ and\n$c^{(p + p')}(Q \\oplus Q')$.\nThen $c^{(p + p')}(Q \\oplus Q') = c^{(p)}(Q)c^{(p')}(Q')$\nin $A^{(p + p')}(Z \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAY","source_file":"chow.tex","source_line":10498,"source_end_line":10517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10498-L10517","statement_sha256":"744316fd8f9ffdd98702bdf9c607008784e337382c8c303f4997b47a6e63fa93","origin":"The Stacks Project","memory_eligible":false,"source_rank":8296,"rank":8296,"depth":59,"x":2391.731,"y":773.339,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FAZ","tag":"0FAZ","title":"Preparation for localized Chern classes · Lemma 0FAZ","summary":"In Lemma [Tag 0F9K] assume Q|_T is zero. Assume we have another perfect object Q' ∈ D(O_W) whose Chern classes are defined such that the restriction Q'|_T is zero. In this case the classes P'_p(Q), P'_p(Q'), P'_p(Q ⊕ Q') ∈ A^p(Z → X) constructed in Lemma [Tag 0F9K] satisfy P'_p(Q ⊕ Q') = P'_p(Q) + P'_p(Q').","statement_latex":"In Lemma \\ref{lemma-localized-chern-pre} assume $Q|_T$ is zero.\nAssume we have another perfect object $Q' \\in D(\\mathcal{O}_W)$\nwhose Chern classes are defined such that the restriction $Q'|_T$ is zero.\nIn this case the classes\n$P'_p(Q), P'_p(Q'), P'_p(Q \\oplus Q') \\in A^p(Z \\to X)$\nconstructed in Lemma \\ref{lemma-localized-chern-pre}\nsatisfy $P'_p(Q \\oplus Q') = P'_p(Q) + P'_p(Q')$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Preparation for localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FAZ","source_file":"chow.tex","source_line":10602,"source_end_line":10611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10602-L10611","statement_sha256":"f839b64193c8e787be0554949c07b25f16a3436d146782877125b344151995d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8297,"rank":8297,"depth":59,"x":2470.017,"y":560.957,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FB2","tag":"0FB2","title":"Localized Chern classes · Lemma 0FB2","summary":"In Situation [Tag 0GUJ] there exists a canonical bivariant class P_p(Z → X, E) ∈ A^p(Z → X), resp. c_p(Z → X, E) ∈ A^p(Z → X) with the property that i_* ∘ P_p(Z → X, E) = P_p(E), resp. i_* ∘ c_p(Z → X, E) = c_p(E) as bivariant classes on X (with ∘ as in Lemma [Tag 0EPK]).","statement_latex":"In Situation \\ref{situation-loc-chern} there exists a canonical bivariant class\n$$\nP_p(Z \\to X, E) \\in A^p(Z \\to X),\n\\quad\\text{resp.}\\quad\nc_p(Z \\to X, E) \\in A^p(Z \\to X)\n$$\nwith the property that\n\\begin{equation}\n\ni_* \\circ P_p(Z \\to X, E) = P_p(E),\n\\quad\\text{resp.}\\quad\ni_* \\circ c_p(Z \\to X, E) = c_p(E)\n\\end{equation}\nas bivariant classes on $X$ (with $\\circ$ as in\nLemma \\ref{lemma-push-proper-bivariant}).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FB2","source_file":"chow.tex","source_line":10700,"source_end_line":10717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10700-L10717","statement_sha256":"74cfd69cb124df8ad1f2d0894ff9b3ad31610d6366a48c1b9df7f9c6cf9e09c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8298,"rank":8298,"depth":57,"x":2583.205,"y":762.206,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FB5","tag":"0FB5","title":"Localized Chern classes · Definition 0FB5","summary":"With (S, δ), X, E ∈ D(O_X), and i : Z → X as in Situation [Tag 0GUJ]. • If the restriction E|_X setminus Z is zero, then for all p ≥ 0 we define P_p(Z → X, E) ∈ A^p(Z → X) by the construction in Lemma [Tag 0FB2] and we define the localized Chern character by the formula ch(Z → X, E) = ∑_p = 0, 1, 2, … fracP_p(Z → X, E)p! in ∏_p ≥ 0 A^p(Z → X) ⊗ Q • If the restriction E|_X setminus Z is isomorphic to a finite locally free O_X setminus Z-module of rank < p sitting in…","statement_latex":"With $(S, \\delta)$, $X$, $E \\in D(\\mathcal{O}_X)$, and $i : Z \\to X$ as in\nSituation \\ref{situation-loc-chern}.\n\\begin{enumerate}\n\\item If the restriction $E|_{X \\setminus Z}$ is zero, then for all\n$p \\geq 0$ we define\n$$\nP_p(Z \\to X, E) \\in A^p(Z \\to X)\n$$\nby the construction in Lemma \\ref{lemma-independent-loc-chern}\nand we define the {\\it localized Chern character} by the formula\n$$\nch(Z \\to X, E) =\n\\sum\\nolimits_{p = 0, 1, 2, \\ldots} \\frac{P_p(Z \\to X, E)}{p!}\n\\quad\\text{in}\\quad \\prod\\nolimits_{p \\geq 0} A^p(Z \\to X) \\otimes \\mathbf{Q}\n$$\n\\item If the restriction $E|_{X \\setminus Z}$ is isomorphic to a\nfinite locally free $\\mathcal{O}_{X \\setminus Z}$-module of rank $< p$\nsitting in cohomological degree $0$, then we define the\n{\\it localized $p$th Chern class} $c_p(Z \\to X, E)$ by the construction\nin Lemma \\ref{lemma-independent-loc-chern}.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Localized Chern classes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FB5","source_file":"chow.tex","source_line":10786,"source_end_line":10809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10786-L10809","statement_sha256":"bfb4e229fb6cc514493dbae07381d774fa3bdc90df0e5ba850b2c4fb286ea417","origin":"The Stacks Project","memory_eligible":false,"source_rank":8299,"rank":8299,"depth":58,"x":2337.635,"y":677.942,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FB3","tag":"0FB3","title":"Localized Chern classes · Lemma 0FB3","summary":"In Situation [Tag 0GUJ] let f : X' → X be a morphism of schemes which is locally of finite type. Denote E' = f^*E and Z' = f^-1(Z). Then the bivariant class of Definition [Tag 0FB5] P_p(Z' → X', E') ∈ A^p(Z' → X'), resp. c_p(Z' → X', E') ∈ A^p(Z' → X') constructed as in Lemma [Tag 0FB2] using X', Z', E' is the restriction (Remark [Tag 0F9Z]) of the bivariant class P_p(Z → X, E) ∈ A^p(Z → X), resp. c_p(Z → X, E) ∈ A^p(Z → X).","statement_latex":"In Situation \\ref{situation-loc-chern}\nlet $f : X' \\to X$ be a morphism of schemes which is locally of finite type.\nDenote $E' = f^*E$ and $Z' = f^{-1}(Z)$. Then the bivariant class\nof Definition \\ref{definition-localized-chern}\n$$\nP_p(Z' \\to X', E') \\in A^p(Z' \\to X'),\n\\quad\\text{resp.}\\quad\nc_p(Z' \\to X', E') \\in A^p(Z' \\to X')\n$$\nconstructed as in Lemma \\ref{lemma-independent-loc-chern}\nusing $X', Z', E'$ is the restriction\n(Remark \\ref{remark-restriction-bivariant}) of the\nbivariant class $P_p(Z \\to X, E) \\in A^p(Z \\to X)$,\nresp.\\ $c_p(Z \\to X, E) \\in A^p(Z \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FB3","source_file":"chow.tex","source_line":10823,"source_end_line":10839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10823-L10839","statement_sha256":"46814f932ad1e02447df12922fe30181a3c858a257b26a76a05c9858ccdbb981","origin":"The Stacks Project","memory_eligible":false,"source_rank":8300,"rank":8300,"depth":59,"x":2586.749,"y":600.65,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FB4","tag":"0FB4","title":"Localized Chern classes · Lemma 0FB4","summary":"In Situation [Tag 0GUJ] we have P_p(Z → X, E) ∩ i_*α = P_p(E|_Z) ∩ α, resp. c_p(Z → X, E) ∩ i_*α = c_p(E|_Z) ∩ α in CH_*(Z) for any α ∈ CH_*(Z).","statement_latex":"In Situation \\ref{situation-loc-chern} we have\n$$\nP_p(Z \\to X, E) \\cap i_*\\alpha = P_p(E|_Z) \\cap \\alpha,\n\\quad\\text{resp.}\\quad\nc_p(Z \\to X, E) \\cap i_*\\alpha = c_p(E|_Z) \\cap \\alpha\n$$\nin $\\CH_*(Z)$ for any $\\alpha \\in \\CH_*(Z)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FB4","source_file":"chow.tex","source_line":10917,"source_end_line":10926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10917-L10926","statement_sha256":"89f4647c6ad63d57872d2932620ad594eb554012419e626e40180432242d31ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":8301,"rank":8301,"depth":60,"x":2465.08,"y":799.213,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FB6","tag":"0FB6","title":"Localized Chern classes · Lemma 0FB6","summary":"In Situation [Tag 0GUJ] if α ∈ CH_k(X) has support disjoint from Z, then P_p(Z → X, E) ∩ α = 0, resp. c_p(Z → X, E) ∩ α = 0.","statement_latex":"In Situation \\ref{situation-loc-chern}\nif $\\alpha \\in \\CH_k(X)$ has support disjoint from $Z$, then\n$P_p(Z \\to X, E) \\cap \\alpha = 0$, resp.\\ $c_p(Z \\to X, E) \\cap \\alpha = 0$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FB6","source_file":"chow.tex","source_line":10939,"source_end_line":10944,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10939-L10944","statement_sha256":"dc827aff40f64165b7aa18f010dfb976557fad3ae3587da2b9dfd38c3579d601","origin":"The Stacks Project","memory_eligible":false,"source_rank":8302,"rank":8302,"depth":60,"x":2395.042,"y":583.523,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FB7","tag":"0FB7","title":"Localized Chern classes · Lemma 0FB7","summary":"In Situation [Tag 0GUJ] assume Z ⊂ Z' ⊂ X where Z' is a closed subscheme of X. Then P_p(Z' → X, E) = (Z → Z')_* ∘ P_p(Z → X, E), resp. c_p(Z' → X, E) = (Z → Z')_* ∘ c_p(Z → X, E) (with ∘ as in Lemma [Tag 0EPK]).","statement_latex":"In Situation \\ref{situation-loc-chern}\nassume $Z \\subset Z' \\subset X$ where $Z'$ is a closed subscheme of $X$.\nThen\n$P_p(Z' \\to X, E) = (Z \\to Z')_* \\circ P_p(Z \\to X, E)$,\nresp.\\ $c_p(Z' \\to X, E) = (Z \\to Z')_* \\circ c_p(Z \\to X, E)$\n(with $\\circ$ as in Lemma \\ref{lemma-push-proper-bivariant}).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FB7","source_file":"chow.tex","source_line":10955,"source_end_line":10963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10955-L10963","statement_sha256":"504c03049d0dca3e85dea1a20ce7af2f7099c7957617574550924b300b884455","origin":"The Stacks Project","memory_eligible":false,"source_rank":8303,"rank":8303,"depth":58,"x":2620.383,"y":702.958,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FB8","tag":"0FB8","title":"Localized Chern classes · Lemma 0FB8","summary":"In Lemma [Tag 0F9H] say E_2 is the restriction of a perfect E ∈ D(O_X) whose restriction to X_1 is zero, resp. isomorphic to a finite locally free O_X_1-module of rank < p sitting in cohomological degree 0. Then the class P'_p(E_2), resp. c'_p(E_2) of Lemma [Tag 0F9H] agrees with P_p(X_2 → X, E), resp. c_p(X_2 → X, E) of Definition [Tag 0FB5] provided E satisfies assumption (3) of Situation [Tag 0GUJ].","statement_latex":"In Lemma \\ref{lemma-silly} say $E_2$ is the restriction of a perfect\n$E \\in D(\\mathcal{O}_X)$ whose restriction to $X_1$ is zero,\nresp.\\ isomorphic to a finite locally free $\\mathcal{O}_{X_1}$-module\nof rank $< p$ sitting in cohomological degree $0$. Then the class\n$P'_p(E_2)$, resp.\\ $c'_p(E_2)$ of Lemma \\ref{lemma-silly} agrees with\n$P_p(X_2 \\to X, E)$, resp.\\ $c_p(X_2 \\to X, E)$ of\nDefinition \\ref{definition-localized-chern} provided $E$ satisfies\nassumption (3) of Situation \\ref{situation-loc-chern}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FB8","source_file":"chow.tex","source_line":10974,"source_end_line":10984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L10974-L10984","statement_sha256":"5eb084a62776e0b020b6f66e2d4dc274f178f801d66d977ef3f7d452520770c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8304,"rank":8304,"depth":61,"x":2357.89,"y":742.795,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FE7","tag":"0FE7","title":"Two technical lemmas · Lemma 0FE7","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let b : W → P^1_X be a proper morphism of schemes. Let n ≥ 1. For i = 1, …, n let Z_i ⊂ X be a closed subscheme, let Q_i ∈ D(O_W) be a perfect object, let p_i ≥ 0 be an integer, and let T_i ⊂ W_∞, i = 1, …, n be closed. Denote W_i = b^-1(P^1_Z_i). Assume • for i = 1, …, n the assumption of Lemma [Tag 0F9K] hold for b, Z_i, Q_i, T_i, p_i, • Q_i|_W setminus W_i is zero, resp. isomorphic to a…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be\nlocally of finite type over $S$. Let $b : W \\longrightarrow \\mathbf{P}^1_X$\nbe a proper morphism of schemes. Let $n \\geq 1$. For $i = 1, \\ldots, n$\nlet $Z_i \\subset X$ be a closed subscheme, let $Q_i \\in D(\\mathcal{O}_W)$\nbe a perfect object, let $p_i \\geq 0$ be an integer, and let\n$T_i \\subset W_\\infty$, $i = 1, \\ldots, n$ be closed.\nDenote $W_i = b^{-1}(\\mathbf{P}^1_{Z_i})$. Assume\n\\begin{enumerate}\n\\item for $i = 1, \\ldots, n$ the assumption of\nLemma \\ref{lemma-localized-chern-pre} hold for\n$b, Z_i, Q_i, T_i, p_i$,\n\\item $Q_i|_{W \\setminus W_i}$ is zero, resp.\\ isomorphic to a finite\nlocally free module of rank $< p_i$ placed in cohomological degree $0$,\n\\item $Q_i$ on $W$ satisfies\nassumption (3) of Situation \\ref{situation-loc-chern}.\n\\end{enumerate}\nThen $P'_{p_n}(Q_n) \\circ \\ldots \\circ P'_{p_1}(Q_1)$ is equal to\n$$\n(W_{n, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to\nZ_n \\cap \\ldots \\cap Z_1)_* \\circ\nP'_{p_n}(Q_n|_{W_{n, \\infty}}) \\circ \\ldots \\circ P'_{p_1}(Q_1|_{W_{1, \\infty}})\n\\circ C\n$$\nin $A^{p_n + \\ldots + p_1}(Z_n \\cap \\ldots \\cap Z_1 \\to X)$,\nresp.\\ $c'_{p_n}(Q_n) \\circ \\ldots \\circ c'_{p_1}(Q_1)$ is equal to\n$$\n(W_{n, \\infty} \\cap \\ldots \\cap W_{1, \\infty} \\to\nZ_n \\cap \\ldots \\cap Z_1)_* \\circ\nc'_{p_n}(Q_n|_{W_{n, \\infty}}) \\circ \\ldots \\circ c'_{p_1}(Q_1|_{W_{1, \\infty}})\n\\circ C\n$$\nin $A^{p_n + \\ldots + p_1}(Z_n \\cap \\ldots \\cap Z_1 \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Two technical lemmas","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FE7","source_file":"chow.tex","source_line":11025,"source_end_line":11059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11025-L11059","statement_sha256":"c4717aa2e601fe80c87170866d54d06204d0d5b49750c81d4c2baeb27fa2725e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8305,"rank":8305,"depth":62,"x":2519.587,"y":564.283,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FE8","tag":"0FE8","title":"Two technical lemmas · Lemma 0FE8","summary":"Assume (S, δ), X, Z, b : W → P^1_X, Q, T, p satisfy the assumptions of Lemma [Tag 0F9K]. Let F ∈ D(O_X) be a perfect object such that • the restriction of Q to b^-1(A^1_X) is isomorphic to the pullback of F, • F|_X setminus Z is zero, resp. isomorphic to a finite locally free O_X setminus Z-module of rank < p sitting in cohomological degree 0, and • Q on W and F on X satisfy assumption (3) of Situation [Tag 0GUJ]. Then the class P'_p(Q), resp. c'_p(Q) in A^p(Z → X)…","statement_latex":"Assume $(S, \\delta), X, Z, b : W \\to \\mathbf{P}^1_X, Q, T, p$\nsatisfy the assumptions of Lemma \\ref{lemma-localized-chern-pre}.\nLet $F \\in D(\\mathcal{O}_X)$ be a perfect object such that\n\\begin{enumerate}\n\\item the restriction of $Q$ to $b^{-1}(\\mathbf{A}^1_X)$ is\nisomorphic to the pullback of $F$,\n\\item $F|_{X \\setminus Z}$ is zero, resp.\\ isomorphic to a finite\nlocally free $\\mathcal{O}_{X \\setminus Z}$-module of rank $< p$\nsitting in cohomological degree $0$, and\n\\item $Q$ on $W$ and $F$ on $X$ satisfy assumption (3) of\nSituation \\ref{situation-loc-chern}.\n\\end{enumerate}\nThen the class $P'_p(Q)$, resp.\\ $c'_p(Q)$ in $A^p(Z \\to X)$ constructed\nin Lemma \\ref{lemma-localized-chern-pre}\nis equal to $P_p(Z \\to X, F)$, resp.\\ $c_p(Z \\to X, F)$\nfrom Definition \\ref{definition-localized-chern}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Two technical lemmas","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FE8","source_file":"chow.tex","source_line":11164,"source_end_line":11182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11164-L11182","statement_sha256":"5f4b26585682ab3f7203763aa44f737b1098eb14008ed7048de3095838b39d63","origin":"The Stacks Project","memory_eligible":false,"source_rank":8306,"rank":8306,"depth":62,"x":2543.935,"y":787.906,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBA","tag":"0FBA","title":"Properties of localized Chern classes · Lemma 0FBA","summary":"In Situation [Tag 0GUJ] assume E|_X setminus Z is zero. Then P_1(Z → X, E) & = c_1(Z → X, E), P_2(Z → X, E) & = c_1(Z → X, E)^2 - 2c_2(Z → X, E), P_3(Z → X, E) & = c_1(Z → X, E)^3 - 3c_1(Z → X, E)c_2(Z → X, E) + 3c_3(Z → X, E), and so on where the products are taken in the algebra A^(1)(Z → X) of Remark [Tag 0FA0].","statement_latex":"In Situation \\ref{situation-loc-chern} assume $E|_{X \\setminus Z}$ is zero.\nThen\n\\begin{align*}\nP_1(Z \\to X, E) & = c_1(Z \\to X, E), \\\\\nP_2(Z \\to X, E) & = c_1(Z \\to X, E)^2 - 2c_2(Z \\to X, E), \\\\\nP_3(Z \\to X, E) & = c_1(Z \\to X, E)^3 - 3c_1(Z \\to X, E)c_2(Z \\to X, E)\n+ 3c_3(Z \\to X, E),\n\\end{align*}\nand so on where the products are taken in the algebra $A^{(1)}(Z \\to X)$\nof Remark \\ref{remark-ring-loc-classes}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Properties of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBA","source_file":"chow.tex","source_line":11259,"source_end_line":11271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11259-L11271","statement_sha256":"3b2062b54ded935b4c78022d619104e42d41b6403b7a294ca8a7422bf0389b5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8307,"rank":8307,"depth":56,"x":2345.937,"y":636.663,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBB","tag":"0FBB","title":"Properties of localized Chern classes · Lemma 0FBB","summary":"In Situation [Tag 0GUJ] let Y → X be locally of finite type and c ∈ A^*(Y → X). Then P_p(Z → X, E) ∘ c = c ∘ P_p(Z → X, E), respectively c_p(Z → X, E) ∘ c = c ∘ c_p(Z → X, E) in A^*(Y ×_X Z → X).","statement_latex":"In Situation \\ref{situation-loc-chern}\nlet $Y \\to X$ be locally of finite type and $c \\in A^*(Y \\to X)$.\nThen\n$$\nP_p(Z \\to X, E) \\circ c = c \\circ P_p(Z \\to X, E),\n$$\nrespectively\n$$\nc_p(Z \\to X, E) \\circ c = c \\circ c_p(Z \\to X, E)\n$$\nin $A^*(Y \\times_X Z \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Properties of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBB","source_file":"chow.tex","source_line":11283,"source_end_line":11296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11283-L11296","statement_sha256":"711e144b748a7e6c96b508ea0804457a295d2698cb79a76c4037597f681b9fc6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8308,"rank":8308,"depth":58,"x":2613.85,"y":635.838,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBD","tag":"0FBD","title":"Properties of localized Chern classes · Lemma 0FBD","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let Z → X be a closed immersion. Let E_1 → E_2 → E_3 → E_1[1] be a distinguished triangle of perfect objects in D(O_X). Assume • the restrictions E_1|_X setminus Z and E_3|_X setminus Z are isomorphic to finite locally free O_X setminus Z-modules of rank < p_1 and < p_3 placed in degree 0, and • at least one of the following is true: (a) X is quasi-compact, (b) X has quasi-compact…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let $Z \\to X$ be\na closed immersion. Let\n$$\nE_1 \\to E_2 \\to E_3 \\to E_1[1]\n$$\nbe a distinguished triangle of perfect objects in $D(\\mathcal{O}_X)$.\nAssume\n\\begin{enumerate}\n\\item the restrictions $E_1|_{X \\setminus Z}$ and $E_3|_{X \\setminus Z}$\nare isomorphic to finite locally free $\\mathcal{O}_{X \\setminus Z}$-modules\nof rank $< p_1$ and $< p_3$ placed in degree $0$, and\n\\item at least one of the following is true:\n(a) $X$ is quasi-compact,\n(b) $X$ has quasi-compact irreducible components,\n(c) $E_3 \\to E_1[1]$ can be represented by a map of locally\nbounded complexes of finite locally free $\\mathcal{O}_X$-modules, or\n(d) there exists an envelope $f : Y \\to X$ such that $Lf^*E_3 \\to Lf^*E_1[1]$\ncan be represented by a map of locally bounded complexes of\nfinite locally free $\\mathcal{O}_Y$-modules.\n\\end{enumerate}\nWith notation as in Remark \\ref{remark-loc-chern-classes} we have\n$$\nc^{(p_1 + p_3)}(Z \\to X, E_2) = c^{(p_1)}(Z \\to X, E_1)c^{(p_3)}(Z \\to X, E_3)\n$$\nin $A^{(p_1 + p_3)}(Z \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Properties of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBD","source_file":"chow.tex","source_line":11329,"source_end_line":11357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11329-L11357","statement_sha256":"e16223e95e72eee4e07ff60f7ef485eb4c4aa8fda407658e11cddffa40ff7326","origin":"The Stacks Project","memory_eligible":false,"source_rank":8309,"rank":8309,"depth":63,"x":2416.748,"y":788.63,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBE","tag":"0FBE","title":"Properties of localized Chern classes · Lemma 0FBE","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let Z → X be a closed immersion. Let E_1 → E_2 → E_3 → E_1[1] be a distinguished triangle of perfect objects in D(O_X). Assume • the restrictions E_1|_X setminus Z and E_3|_X setminus Z are zero, and • at least one of the following is true: (a) X is quasi-compact, (b) X has quasi-compact irreducible components, (c) E_3 → E_1[1] can be represented by a map of locally bounded complexes of…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. Let $Z \\to X$ be\na closed immersion. Let\n$$\nE_1 \\to E_2 \\to E_3 \\to E_1[1]\n$$\nbe a distinguished triangle of perfect objects in $D(\\mathcal{O}_X)$.\nAssume\n\\begin{enumerate}\n\\item the restrictions $E_1|_{X \\setminus Z}$ and $E_3|_{X \\setminus Z}$\nare zero, and\n\\item at least one of the following is true:\n(a) $X$ is quasi-compact,\n(b) $X$ has quasi-compact irreducible components,\n(c) $E_3 \\to E_1[1]$ can be represented by a map of locally\nbounded complexes of finite locally free $\\mathcal{O}_X$-modules, or\n(d) there exists an envelope $f : Y \\to X$ such that $Lf^*E_3 \\to Lf^*E_1[1]$\ncan be represented by a map of locally bounded complexes of\nfinite locally free $\\mathcal{O}_Y$-modules.\n\\end{enumerate}\nThen we have\n$$\nP_p(Z \\to X, E_2) = P_p(Z \\to X, E_1) + P_p(Z \\to X, E_3)\n$$\nfor all $p \\in \\mathbf{Z}$ and consequently\n$ch(Z \\to X, E_2) = ch(Z \\to X, E_1) + ch(Z \\to X, E_3)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Properties of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBE","source_file":"chow.tex","source_line":11481,"source_end_line":11509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11481-L11509","statement_sha256":"cbf86a0695fb27cfc6927f8d83760bf494c5a81803fb15f05c894aa6b8432c74","origin":"The Stacks Project","memory_eligible":false,"source_rank":8310,"rank":8310,"depth":64,"x":2439.241,"y":563.883,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBF","tag":"0FBF","title":"Properties of localized Chern classes · Lemma 0FBF","summary":"In Situation [Tag 02QL] let X be locally of finite type over S. Let Z_i ⊂ X, i = 1, 2 be closed subschemes. Let F_i, i = 1, 2 be perfect objects of D(O_X). Assume for i = 1, 2 that F_i|_X setminus Z_i is zero is isomorphic to a finite locally free O_X setminus Z_i-module of rank < p_i. and that F_i on X satisfies assumption (3) of Situation [Tag 0GUJ]. Denote r_i = P_0(Z_i → X, F_i) ∈ A^0(Z_i → X). Then we have c_1(Z_1 ∩ Z_2 → X, F_1 ⊗_O_X^L F_2) = r_1 c_1(Z_2 → X, F_2) +…","statement_latex":"In Situation \\ref{situation-setup} let $X$ be locally of finite type over $S$.\nLet $Z_i \\subset X$, $i = 1, 2$ be closed subschemes. Let $F_i$, $i = 1, 2$\nbe perfect objects of $D(\\mathcal{O}_X)$. Assume for $i = 1, 2$ that\n$F_i|_{X \\setminus Z_i}$ is zero\\footnote{Presumably there\nis a variant of this lemma where we only assume $F_i|_{X \\setminus Z_i}$\nis isomorphic to a finite locally free $\\mathcal{O}_{X \\setminus Z_i}$-module\nof rank $< p_i$.} and that $F_i$ on $X$ satisfies assumption\n(3) of Situation \\ref{situation-loc-chern}. Denote\n$r_i = P_0(Z_i \\to X, F_i) \\in A^0(Z_i \\to X)$.\nThen we have\n$$\nc_1(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) =\nr_1 c_1(Z_2 \\to X, F_2) + r_2 c_1(Z_1 \\to X, F_1)\n$$\nin $A^1(Z_1 \\cap Z_2 \\to X)$ and\n\\begin{align*}\nc_2(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2)\n& =\nr_1 c_2(Z_2 \\to X, F_2) +\nr_2 c_2(Z_1 \\to X, F_1) + \\\\\n& {r_1 \\choose 2} c_1(Z_2 \\to X, F_2)^2 + \\\\\n& (r_1r_2 - 1) c_1(Z_2 \\to X, F_2)c_1(Z_1 \\to X, F_1) + \\\\\n& {r_2 \\choose 2} c_1(Z_1 \\to X, F_1)^2\n\\end{align*}\nin $A^2(Z_1 \\cap Z_2 \\to X)$ and so on for higher Chern classes.\nSimilarly, we have\n$$\nch(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) =\nch(Z_1 \\to X, F_1) ch(Z_2 \\to X, F_2)\n$$\nin $\\prod_{p \\geq 0} A^p(Z_1 \\cap Z_2 \\to X) \\otimes \\mathbf{Q}$.\nMore precisely, we have\n$$\nP_p(Z_1 \\cap Z_2 \\to X, F_1 \\otimes_{\\mathcal{O}_X}^\\mathbf{L} F_2) =\n\\sum\\nolimits_{p_1 + p_2 = p}\n{p \\choose p_1} P_{p_1}(Z_1 \\to X, F_1) P_{p_2}(Z_2 \\to X, F_2)\n$$\nin $A^p(Z_1 \\cap Z_2 \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Properties of localized Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBF","source_file":"chow.tex","source_line":11524,"source_end_line":11564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11524-L11564","statement_sha256":"6828f25bd791dcd9b3e7578c01d51f0ddf718fed4dd8478d197f2e31b22338bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8311,"rank":8311,"depth":63,"x":2603.564,"y":742.561,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBJ","tag":"0FBJ","title":"Higher codimension gysin homomorphisms · Lemma 0FBJ","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let xymatrix Z' ar[r] ar[d]_g & X' ar[d]^f Z ar[r] & X be a cartesian diagram of schemes locally of finite type over S whose horizontal arrows are closed immersions. If N is a virtual normal sheaf for Z in X, then N' = g^*N is a virtual normal sheaf for Z' in X'.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let\n$$\n\\xymatrix{\nZ' \\ar[r] \\ar[d]_g & X' \\ar[d]^f \\\\\nZ \\ar[r] & X\n}\n$$\nbe a cartesian diagram of schemes locally of finite type over $S$\nwhose horizontal arrows are closed immersions.\nIf $\\mathcal{N}$ is a virtual normal sheaf for $Z$ in $X$, then\n$\\mathcal{N}' = g^*\\mathcal{N}$ is a virtual normal sheaf for\n$Z'$ in $X'$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBJ","source_file":"chow.tex","source_line":11850,"source_end_line":11864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11850-L11864","statement_sha256":"65d5063d889fc253fdfd19f723e65d0c882a32a29e450a242d9e85d10f7f64b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8312,"rank":8312,"depth":10,"x":2338.425,"y":704.007,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBK","tag":"0FBK","title":"Higher codimension gysin homomorphisms · Lemma 0FBK","summary":"The construction above defines a bivariant class c(Z → X, N) ∈ A^*(Z → X)^wedge and moreover the construction is compatible with base change as in Lemma [Tag 0FBJ]. If N has constant rank r, then c(Z → X, N) ∈ A^r(Z → X).","statement_latex":"The construction above defines a bivariant class\\footnote{The\nnotation $A^*(Z \\to X)^\\wedge$ is discussed in\nRemark \\ref{remark-completion-bivariant}.\nIf $X$ is quasi-compact, then $A^*(Z \\to X)^\\wedge = A^*(Z \\to X)$.}\n$$\nc(Z \\to X, \\mathcal{N}) \\in A^*(Z \\to X)^\\wedge\n$$\nand moreover the construction is compatible with base change\nas in Lemma \\ref{lemma-pullback-virtual-normal-sheaf}.\nIf $\\mathcal{N}$ has constant rank $r$, then\n$c(Z \\to X, \\mathcal{N}) \\in A^r(Z \\to X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBK","source_file":"chow.tex","source_line":11911,"source_end_line":11924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11911-L11924","statement_sha256":"7bf85de39efeae997c6def827515c4d15b178ff12265ee00d7be65b3fef62eb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8313,"rank":8313,"depth":54,"x":2565.179,"y":581.861,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBL","tag":"0FBL","title":"Higher codimension gysin homomorphisms · Lemma 0FBL","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let N be a virtual normal sheaf for a closed subscheme Z of X. Suppose that we have a short exact sequence 0 → N' → N → E → 0 of finite locally free O_Z-modules such that the given surjection σ : N^vee → C_Z/X factors through a map σ' : (N')^vee → C_Z/X. Then c(Z → X, N) = c_top(E) ∘ c(Z → X, N') as bivariant classes.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme\nlocally of finite type over $S$. Let $\\mathcal{N}$ be a virtual normal\nsheaf for a closed subscheme $Z$ of $X$. Suppose that we have a short\nexact sequence $0 \\to \\mathcal{N}' \\to \\mathcal{N} \\to \\mathcal{E} \\to 0$\nof finite locally free $\\mathcal{O}_Z$-modules such that the given surjection\n$\\sigma : \\mathcal{N}^\\vee \\to \\mathcal{C}_{Z/X}$ factors through a map\n$\\sigma' : (\\mathcal{N}')^\\vee \\to \\mathcal{C}_{Z/X}$.\nThen\n$$\nc(Z \\to X, \\mathcal{N}) = c_{top}(\\mathcal{E}) \\circ c(Z \\to X, \\mathcal{N}')\n$$\nas bivariant classes.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBL","source_file":"chow.tex","source_line":11993,"source_end_line":12007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L11993-L12007","statement_sha256":"f7d61214af1d7ffe43370ba41ea44f59f5e2b6b5d1711365b6f6ddbf9b43ccb2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8314,"rank":8314,"depth":56,"x":2496.128,"y":800.827,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FV7","tag":"0FV7","title":"Higher codimension gysin homomorphisms · Lemma 0FV7","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Consider a cartesian diagram xymatrix Z' ar[r] ar[d]_g & X' ar[d]^f Z ar[r] & X of schemes locally of finite type over S whose horizontal arrows are closed immersions. Let N, resp. N' be a virtual normal sheaf for Z ⊂ X, resp. Z' → X'. Assume given a short exact sequence 0 → N' → g^*N → E → 0 of finite locally free modules on Z' such that the diagram xymatrix g^*N^vee ar[r] ar[d] & (N')^vee ar[d] g^*C_Z/X ar[r] & C_Z'/X' commutes.…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Consider\na cartesian diagram\n$$\n\\xymatrix{\nZ' \\ar[r] \\ar[d]_g & X' \\ar[d]^f \\\\\nZ \\ar[r] & X\n}\n$$\nof schemes locally of finite type over $S$ whose horizontal arrows\nare closed immersions. Let $\\mathcal{N}$, resp.\\ $\\mathcal{N}'$\nbe a virtual normal sheaf for $Z \\subset X$, resp.\\ $Z' \\to X'$.\nAssume given a short exact sequence\n$0 \\to \\mathcal{N}' \\to g^*\\mathcal{N} \\to \\mathcal{E} \\to 0$\nof finite locally free modules on $Z'$ such that the diagram\n$$\n\\xymatrix{\ng^*\\mathcal{N}^\\vee \\ar[r] \\ar[d] &\n(\\mathcal{N}')^\\vee \\ar[d] \\\\\ng^*\\mathcal{C}_{Z/X} \\ar[r] &\n\\mathcal{C}_{Z'/X'}\n}\n$$\ncommutes. Then we have\n$$\nres(c(Z \\to X, \\mathcal{N})) =\nc_{top}(\\mathcal{E}) \\circ c(Z' \\to X', \\mathcal{N}')\n$$\nin $A^*(Z' \\to X')^\\wedge$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV7","source_file":"chow.tex","source_line":12020,"source_end_line":12050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12020-L12050","statement_sha256":"36997f20b9557f8fecdbf610664530e910b45061ee9cfcaa479b7df88e626bd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8315,"rank":8315,"depth":57,"x":2370.827,"y":599.971,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBM","tag":"0FBM","title":"Higher codimension gysin homomorphisms · Lemma 0FBM","summary":"In the situation described just above assume dim_δ(Y) = n and that C_Y ×_X Z/Z has constant rank r. Then c(Z → X, N) ∩ [Y]_n = c_top(E) ∩ [Z ×_X Y]_n - r in CH_*(Z ×_X Y).","statement_latex":"In the situation described just above assume $\\dim_\\delta(Y) = n$\nand that $\\mathcal{C}_{Y \\times_X Z/Z}$ has constant rank $r$.\nThen\n$$\nc(Z \\to X, \\mathcal{N}) \\cap [Y]_n =\nc_{top}(\\mathcal{E}) \\cap [Z \\times_X Y]_{n - r}\n$$\nin $\\CH_*(Z \\times_X Y)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBM","source_file":"chow.tex","source_line":12075,"source_end_line":12085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12075-L12085","statement_sha256":"5ec0bb4773f6349cf44f395f7db3d5cf049366569a32d86f6af6479c638fe255","origin":"The Stacks Project","memory_eligible":false,"source_rank":8316,"rank":8316,"depth":57,"x":2625.012,"y":677.061,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEA","tag":"0FEA","title":"Higher codimension gysin homomorphisms · Lemma 0FEA","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let N be a virtual normal sheaf for a closed subscheme Z of X. Let Y → X be a morphism which is locally of finite type. Given integers r, n assume • N is locally free of rank r, • every irreducible component of Y has δ-dimension n, • dim_δ(Z ×_X Y) ≤ n - r, and • for xi ∈ Z ×_X Y with δ(xi) = n - r the local ring O_Y, xi is Cohen-Macaulay. Then c(Z → X, N) ∩ [Y]_n = [Z ×_X Y]_n - r…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme\nlocally of finite type over $S$. Let $\\mathcal{N}$ be a virtual normal\nsheaf for a closed subscheme $Z$ of $X$. Let $Y \\to X$ be a morphism\nwhich is locally of finite type. Given integers $r$, $n$ assume\n\\begin{enumerate}\n\\item $\\mathcal{N}$ is locally free of rank $r$,\n\\item every irreducible component of $Y$ has $\\delta$-dimension $n$,\n\\item $\\dim_\\delta(Z \\times_X Y) \\leq n - r$, and\n\\item for $\\xi \\in Z \\times_X Y$ with $\\delta(\\xi) = n - r$\nthe local ring $\\mathcal{O}_{Y, \\xi}$ is Cohen-Macaulay.\n\\end{enumerate}\nThen $c(Z \\to X, \\mathcal{N}) \\cap [Y]_n = [Z \\times_X Y]_{n - r}$\nin $\\CH_{n - r}(Z \\times_X Y)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEA","source_file":"chow.tex","source_line":12127,"source_end_line":12142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12127-L12142","statement_sha256":"a5a7e58641df083fe239e22d846e80b46c30b05552f427de5d482876b277446b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8317,"rank":8317,"depth":58,"x":2375.324,"y":764.539,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBN","tag":"0FBN","title":"Higher codimension gysin homomorphisms · Lemma 0FBN","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let (L, s, i : D → X) be a triple as in Definition [Tag 02T8]. The Gysin homomorphism i^* viewed as an element of A^1(D → X) (see Lemma [Tag 0B79]) is the same as the bivariant class c(D → X, N) ∈ A^1(D → X) constructed using N = i^*L viewed as a virtual normal sheaf for D in X.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme\nlocally of finite type over $S$. Let $(\\mathcal{L}, s, i : D \\to X)$\nbe a triple as in Definition \\ref{definition-gysin-homomorphism}.\nThe Gysin homomorphism $i^*$ viewed as an element of $A^1(D \\to X)$\n(see Lemma \\ref{lemma-gysin-bivariant}) is the same as the bivariant class\n$c(D \\to X, \\mathcal{N}) \\in A^1(D \\to X)$\nconstructed using $\\mathcal{N} = i^*\\mathcal{L}$\nviewed as a virtual normal sheaf for $D$ in $X$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBN","source_file":"chow.tex","source_line":12173,"source_end_line":12183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12173-L12183","statement_sha256":"c0c36c48f09c96f2765ebca0e09eec38e7c222ee84a5bcc7b5fab917f5b110a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8318,"rank":8318,"depth":58,"x":2489.213,"y":558.139,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBP","tag":"0FBP","title":"Higher codimension gysin homomorphisms · Lemma 0FBP","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let Z ⊂ X be a closed subscheme with virtual normal sheaf N. Let Y → X be locally of finite type and c ∈ A^*(Y → X). Then c and c(Z → X, N) commute (Remark [Tag 0FA2]).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$ be a scheme\nlocally of finite type over $S$. Let $Z \\subset X$ be a closed subscheme\nwith virtual normal sheaf $\\mathcal{N}$. Let $Y \\to X$ be locally of\nfinite type and $c \\in A^*(Y \\to X)$. Then $c$ and $c(Z \\to X, \\mathcal{N})$\ncommute (Remark \\ref{remark-bivariant-commute}).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBP","source_file":"chow.tex","source_line":12205,"source_end_line":12212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12205-L12212","statement_sha256":"45890e85cd9e25a5c1e9e4fdb5ec2af954034a368d0aaa38f413c8ecf648699d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8319,"rank":8319,"depth":59,"x":2571.299,"y":775.185,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEB","tag":"0FEB","title":"Higher codimension gysin homomorphisms · Lemma 0FEB","summary":"With notation as above we have o^*[C_ZX]_n = [C_Z Y]_n - 1 in CH_n - 1(Y ×_o, C_Y X C_ZX).","statement_latex":"With notation as above we have\n$$\no^*[C_ZX]_n = [C_Z Y]_{n - 1}\n$$\nin $\\CH_{n - 1}(Y \\times_{o, C_Y X} C_ZX)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEB","source_file":"chow.tex","source_line":12259,"source_end_line":12266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12259-L12266","statement_sha256":"0f9b64b3984bfb160335ae1dfdc63e206bd77550b43a5da62865cf8a9fc15bfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8320,"rank":8320,"depth":52,"x":2335.982,"y":661.598,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEC","tag":"0FEC","title":"Higher codimension gysin homomorphisms · Lemma 0FEC","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let Z ⊂ Y ⊂ X be closed subschemes of a scheme locally of finite type over S. Let N be a virtual normal sheaf for Z ⊂ X. Let N' be a virtual normal sheaf for Z ⊂ Y. Let N\" be a virtual normal sheaf for Y ⊂ X. Assume there is a commutative diagram xymatrix (N\")^vee|_Z ar[r] ar[d] & N^vee ar[r] ar[d] & (N')^vee ar[d] C_Y/X|_Z ar[r] & C_Z/X ar[r] & C_Z/Y where the sequence at the bottom is from More on Morphisms, Lemma [Tag 06AE] and…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $Z \\subset Y \\subset X$ be closed subschemes of a scheme locally\nof finite type over $S$.\nLet $\\mathcal{N}$ be a virtual normal sheaf for $Z \\subset X$.\nLet $\\mathcal{N}'$ be a virtual normal sheaf for $Z \\subset Y$.\nLet $\\mathcal{N}''$ be a virtual normal sheaf for $Y \\subset X$.\nAssume there is a commutative diagram\n$$\n\\xymatrix{\n(\\mathcal{N}'')^\\vee|_Z \\ar[r] \\ar[d] &\n\\mathcal{N}^\\vee \\ar[r] \\ar[d] &\n(\\mathcal{N}')^\\vee \\ar[d] \\\\\n\\mathcal{C}_{Y/X}|_Z \\ar[r] &\n\\mathcal{C}_{Z/X} \\ar[r] &\n\\mathcal{C}_{Z/Y}\n}\n$$\nwhere the sequence at the bottom is from More on Morphisms, Lemma\n\\ref{more-morphisms-lemma-transitivity-conormal} and the top\nsequence is a short exact sequence. Then\n$$\nc(Z \\to X, \\mathcal{N}) =\nc(Z \\to Y, \\mathcal{N}') \\circ c(Y \\to X, \\mathcal{N}'')\n$$\nin $A^*(Z \\to X)^\\wedge$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Higher codimension gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEC","source_file":"chow.tex","source_line":12324,"source_end_line":12351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12324-L12351","statement_sha256":"36729487cba8728886ba89be2a47e0afe6694180618db787e0bb994383bbaf3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8321,"rank":8321,"depth":58,"x":2601.121,"y":611.779,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEE","tag":"0FEE","title":"Calculating some classes · Lemma 0FEE","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let E be a locally free O_X-module of rank r. Then ∏_n = 0, …, r c(wedge^n E)^(-1)^n = 1 - (r - 1)! c_r(E) + …","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$\nbe a scheme locally of finite type over $S$. Let $\\mathcal{E}$\nbe a locally free $\\mathcal{O}_X$-module of rank $r$.\nThen\n$$\n\\prod\\nolimits_{n = 0, \\ldots, r} c(\\wedge^n \\mathcal{E})^{(-1)^n} =\n1 - (r - 1)! c_r(\\mathcal{E}) + \\ldots\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Calculating some classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEE","source_file":"chow.tex","source_line":12499,"source_end_line":12509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12499-L12509","statement_sha256":"4b2eb3e602ef79bb3df7e9253a84b1832c221f8531acdb109fe80f580ee32160","origin":"The Stacks Project","memory_eligible":false,"source_rank":8322,"rank":8322,"depth":0,"x":2445.513,"y":799.155,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEF","tag":"0FEF","title":"Calculating some classes · Lemma 0FEF","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let C be a locally free O_X-module of rank r. Consider the morphisms X = underlineProj_X(O_X[T]) xrightarrowi E = underlineProj_X(Sym^*(C)[T]) xrightarrowπ X Then c_t(i_*O_X) = 0 for t = 1, …, r - 1 and in A^0(C → E) we have p^* ∘ π_* ∘ c_r(i_*O_X) = (-1)^r - 1(r - 1)! j^* where j : C → E and p : C → X are the inclusion and structure morphism of the vector bundle C =…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $X$\nbe a scheme locally of finite type over $S$. Let $\\mathcal{C}$\nbe a locally free $\\mathcal{O}_X$-module of rank $r$. Consider the\nmorphisms\n$$\nX = \\underline{\\text{Proj}}_X(\\mathcal{O}_X[T])\n\\xrightarrow{i}\nE = \\underline{\\text{Proj}}_X(\\text{Sym}^*(\\mathcal{C})[T])\n\\xrightarrow{\\pi}\nX\n$$\nThen $c_t(i_*\\mathcal{O}_X) = 0$ for $t = 1, \\ldots, r - 1$ and in\n$A^0(C \\to E)$ we have\n$$\np^* \\circ \\pi_* \\circ c_r(i_*\\mathcal{O}_X) = (-1)^{r - 1}(r - 1)! j^*\n$$\nwhere\n$j : C \\to E$ and $p : C \\to X$ are the inclusion and structure\nmorphism of the vector bundle\n$C = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{C}))$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Calculating some classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEF","source_file":"chow.tex","source_line":12541,"source_end_line":12563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12541-L12563","statement_sha256":"bace9718f09b38829f3f617bc4e92dff82eeeb520b3574bfc4e9c01ca4450a2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8323,"rank":8323,"depth":55,"x":2409.536,"y":572.457,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEG","tag":"0FEG","title":"Calculating some classes · Lemma 0FEG","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let i : Z → X be a regular closed immersion of codimension r between schemes locally of finite type over S. Let N = C_Z/X^vee be the normal sheaf. If X is quasi-compact (or has quasi-compact irreducible components), then c_t(Z → X, i_*O_Z) = 0 for t = 1, …, r - 1 and c_r(Z → X, i_*O_Z) = (-1)^r - 1 (r - 1)! c(Z → X, N) in A^r(Z → X) where c_t(Z → X, i_*O_Z) is the localized Chern class of Definition [Tag 0FB5].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Let $i : Z \\to X$\nbe a regular closed immersion of codimension $r$\nbetween schemes locally of finite type over $S$.\nLet $\\mathcal{N} = \\mathcal{C}_{Z/X}^\\vee$ be the normal sheaf. If $X$\nis quasi-compact (or has quasi-compact irreducible components), then\n$c_t(Z \\to X, i_*\\mathcal{O}_Z) = 0$ for $t = 1, \\ldots, r - 1$ and\n$$\nc_r(Z \\to X, i_*\\mathcal{O}_Z) = (-1)^{r - 1} (r - 1)! c(Z \\to X, \\mathcal{N})\n\\quad\\text{in}\\quad\nA^r(Z \\to X)\n$$\nwhere $c_t(Z \\to X, i_*\\mathcal{O}_Z)$\nis the localized Chern class\nof Definition \\ref{definition-localized-chern}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Calculating some classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEG","source_file":"chow.tex","source_line":12612,"source_end_line":12628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12612-L12628","statement_sha256":"7414b740fdcf33533d97a4ad38fe2ed41bbd961126d6359aa0a699795b179da3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8324,"rank":8324,"depth":63,"x":2618.586,"y":719.358,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEH","tag":"0FEH","title":"Calculating some classes · Lemma 0FEH","summary":"In the situation of Lemma [Tag 0FEG] say dim_δ(X) = n. Then we have • c_t(Z → X, i_*O_Z) ∩ [X]_n = 0 for t = 1, …, r - 1, • c_r(Z → X, i_*O_Z) ∩ [X]_n = (-1)^r - 1(r - 1)![Z]_n - r, • ch_t(Z → X, i_*O_Z) ∩ [X]_n = 0 for t = 0, …, r - 1, and • ch_r(Z → X, i_*O_Z) ∩ [X]_n = [Z]_n - r.","statement_latex":"In the situation of Lemma \\ref{lemma-agreement-with-loc-chern}\nsay $\\dim_\\delta(X) = n$. Then we have\n\\begin{enumerate}\n\\item $c_t(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = 0$ for\n$t = 1, \\ldots, r - 1$,\n\\item $c_r(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n =\n(-1)^{r - 1}(r - 1)![Z]_{n - r}$,\n\\item $ch_t(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = 0$ for\n$t = 0, \\ldots, r - 1$, and\n\\item $ch_r(Z \\to X, i_*\\mathcal{O}_Z) \\cap [X]_n = [Z]_{n - r}$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Calculating some classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEH","source_file":"chow.tex","source_line":12707,"source_end_line":12720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12707-L12720","statement_sha256":"fb2fed6140b7e68f6e959c51cbde0d559d616f64ea91f12056fa638536d592c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8325,"rank":8325,"depth":64,"x":2346.02,"y":729.666,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEJ","tag":"0FEJ","title":"An Adams operator · Lemma 0FEJ","summary":"Let X be a scheme. There is a ring map ψ^2 : K_0(Vect(X)) → K_0(Vect(X)) which sends [L] to [L^⊗ 2] when L is invertible and is compatible with pullbacks.","statement_latex":"Let $X$ be a scheme. There is a ring map\n$$\n\\psi^2 :\nK_0(\\textit{Vect}(X))\n\\longrightarrow\nK_0(\\textit{Vect}(X))\n$$\nwhich sends $[\\mathcal{L}]$ to $[\\mathcal{L}^{\\otimes 2}]$\nwhen $\\mathcal{L}$ is invertible and is compatible with pullbacks.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"An Adams operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEJ","source_file":"chow.tex","source_line":12752,"source_end_line":12763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12752-L12763","statement_sha256":"6fe314489a6cc4c0c978eadb30c4883b1bdb6ee66f96ce535c8624120778fdb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8326,"rank":8326,"depth":0,"x":2538.916,"y":567.238,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FV8","tag":"0FV8","title":"An Adams operator · Lemma 0FV8","summary":"Let X be a scheme. There is a ring map ψ^-1 : K_0(Vect(X)) → K_0(Vect(X)) which sends [E] to [E^vee] when E is finite locally free and is compatible with pullbacks.","statement_latex":"Let $X$ be a scheme. There is a ring map\n$\\psi^{-1} : K_0(\\textit{Vect}(X)) \\to K_0(\\textit{Vect}(X))$\nwhich sends $[\\mathcal{E}]$ to $[\\mathcal{E}^\\vee]$\nwhen $\\mathcal{E}$ is finite locally free\nand is compatible with pullbacks.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"An Adams operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV8","source_file":"chow.tex","source_line":12856,"source_end_line":12863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12856-L12863","statement_sha256":"7e6bb119a3b6b0b98531d9d88a0d6cbbfa7d05f8abde548eaf1d47a37063860a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8327,"rank":8327,"depth":0,"x":2527.285,"y":796.699,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEN","tag":"0FEN","title":"An Adams operator · Lemma 0FEN","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. If ψ^2 is as in Lemma [Tag 0FEJ] and c and ch are as in Remarks [Tag 0FEL] and [Tag 0FEM] then we have c_i(ψ^2(α)) = 2^i c_i(α) and ch_i(ψ^2(α)) = 2^i ch_i(α) for all α ∈ K_0(Vect(X)).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$. If $\\psi^2$ is\nas in Lemma \\ref{lemma-second-adams-operator} and $c$ and $ch$ are as in\nRemarks \\ref{remark-chern-classes-K} and \\ref{remark-chern-character-K}\nthen we have $c_i(\\psi^2(\\alpha)) = 2^i c_i(\\alpha)$ and\n$ch_i(\\psi^2(\\alpha)) = 2^i ch_i(\\alpha)$\nfor all $\\alpha \\in K_0(\\textit{Vect}(X))$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"An Adams operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEN","source_file":"chow.tex","source_line":12901,"source_end_line":12910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L12901-L12910","statement_sha256":"d2678a6583908b6ec88d4fb99d9202061f57128ef915a3333a7e90f9177936b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8328,"rank":8328,"depth":48,"x":2351.154,"y":620.718,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FER","tag":"0FER","title":"An Adams operator · Lemma 0FER","summary":"Let X be a Noetherian regular scheme. Let Z ⊂ X be a closed subscheme. The maps constructed in Remarks [Tag 0FEP] and [Tag 0FEQ] are mutually inverse and we get K'_0(Z) = K_0(D_Z, perf(O_X)).","statement_latex":"Let $X$ be a Noetherian regular scheme.\nLet $Z \\subset X$ be a closed subscheme. The maps constructed\nin Remarks \\ref{remark-perf-Z-cohomology-K} and\n\\ref{remark-perf-Z-regular} are mutually inverse and we get\n$K'_0(Z) = K_0(D_{Z, perf}(\\mathcal{O}_X))$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"An Adams operator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FER","source_file":"chow.tex","source_line":13002,"source_end_line":13009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13002-L13009","statement_sha256":"ddc475a8cbff2b6b5beb71e651eb1623a69abd79788841c6e2c7d733529bf799","origin":"The Stacks Project","memory_eligible":false,"source_rank":8329,"rank":8329,"depth":30,"x":2622.828,"y":650.573,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEW","tag":"0FEW","title":"Chow groups and K-groups revisited · Proposition 0FEW","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Assume given a closed immersion X → Y of schemes locally of finite type over S with Y regular and quasi-compact. Then the composition K'_0(X) → K_0(D_X, perf(O_Y)) → A^*(X → Y) ⊗ Q → CH_*(X) ⊗ Q of the map F ↦ F[0] from Remark [Tag 0FEQ], the map ch(X → Y, -) from Remark [Tag 0FET], and the map c ↦ c ∩ [Y] induces an isomorphism K'_0(X) ⊗ Q → CH_*(X) ⊗ Q which depends on the choice of Y. Moreover, the canonical map CH_k(X) ⊗ Q →…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}. Assume given a\nclosed immersion $X \\to Y$ of schemes locally of finite type over $S$\nwith $Y$ regular and quasi-compact. Then the composition\n$$\nK'_0(X) \\to\nK_0(D_{X, perf}(\\mathcal{O}_Y)) \\to\nA^*(X \\to Y) \\otimes \\mathbf{Q} \\to\n\\CH_*(X) \\otimes \\mathbf{Q}\n$$\nof the map $\\mathcal{F} \\mapsto \\mathcal{F}[0]$ from\nRemark \\ref{remark-perf-Z-regular}, the map $ch(X \\to Y, -)$ from\nRemark \\ref{remark-localized-chern-character-K}, and\nthe map $c \\mapsto c \\cap [Y]$ induces an isomorphism\n$$\nK'_0(X) \\otimes \\mathbf{Q}\n\\longrightarrow\n\\CH_*(X) \\otimes \\mathbf{Q}\n$$\nwhich depends on the choice of $Y$. Moreover, the canonical map\n$$\n\\CH_k(X) \\otimes \\mathbf{Q}\n\\longrightarrow\n\\text{gr}_k K'_0(X) \\otimes \\mathbf{Q}\n$$\n(see above) is an isomorphism of $\\mathbf{Q}$-vector spaces for all\n$k \\in \\mathbf{Z}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Chow groups and K-groups revisited","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEW","source_file":"chow.tex","source_line":13190,"source_end_line":13218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13190-L13218","statement_sha256":"99eb51509e09b07a6d1018088c97f40cbb76b8a3d80c24dfc11fe54d8faf9c82","origin":"The Stacks Project","memory_eligible":false,"source_rank":8330,"rank":8330,"depth":65,"x":2398.262,"y":782.847,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FEY","tag":"0FEY","title":"Rational intersection products on regular schemes · Lemma 0FEY","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a quasi-compact regular scheme of finite type over S with affine diagonal and δ_X/S : X → Z bounded. Then the composition K_0(Vect(X)) ⊗ Q → A^*(X) ⊗ Q → CH_*(X) ⊗ Q of the map ch from Remark [Tag 0FEM] and the map c ↦ c ∩ [X] is an isomorphism.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a quasi-compact regular scheme of finite type over $S$ with\naffine diagonal and $\\delta_{X/S} : X \\to \\mathbf{Z}$ bounded.\nThen the composition\n$$\nK_0(\\textit{Vect}(X)) \\otimes \\mathbf{Q}\n\\longrightarrow\nA^*(X) \\otimes \\mathbf{Q}\n\\longrightarrow\n\\CH_*(X) \\otimes \\mathbf{Q}\n$$\nof the map $ch$ from Remark \\ref{remark-chern-character-K} and\nthe map $c \\mapsto c \\cap [X]$ is an isomorphism.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Rational intersection products on regular schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FEY","source_file":"chow.tex","source_line":13298,"source_end_line":13313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13298-L13313","statement_sha256":"89982ca3715aa4473c21f5d0667d3916f82bf42b62dea6f626881d2052325974","origin":"The Stacks Project","memory_eligible":false,"source_rank":8331,"rank":8331,"depth":66,"x":2457.543,"y":557.657,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FF0","tag":"0FF0","title":"Gysin maps for local complete intersection morphisms · Lemma 0FF0","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let i : X → Y and j : Y → Z be regular immersions of schemes locally of finite type over S. Then j ∘ i is a regular immersion and (j ∘ i)^! = i^! ∘ j^!.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $i : X \\to Y$ and $j : Y \\to Z$ be regular immersions\nof schemes locally of finite type over $S$. Then\n$j \\circ i$ is a regular immersion and\n$(j \\circ i)^! = i^! \\circ j^!$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FF0","source_file":"chow.tex","source_line":13411,"source_end_line":13418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13411-L13418","statement_sha256":"2019ff6f7442b921f35a8c8a47422acd887ed5ba628bcd008ecdbcce1089266e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8332,"rank":8332,"depth":59,"x":2595.061,"y":757.55,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FF1","tag":"0FF1","title":"Gysin maps for local complete intersection morphisms · Lemma 0FF1","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let p : P → X be a smooth morphism of schemes locally of finite type over S and let s : X → P be a section. Then s is a regular immersion and 1 = s^! ∘ p^* in A^*(X)^wedge where p^* ∈ A^*(P → X)^wedge is the bivariant class of Lemma [Tag 0B78].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $p : P \\to X$ be a smooth morphism of schemes locally of finite type\nover $S$ and let $s : X \\to P$ be a section. Then $s$ is a\nregular immersion and $1 = s^! \\circ p^*$ in $A^*(X)^\\wedge$\nwhere $p^* \\in A^*(P \\to X)^\\wedge$ is the bivariant class\nof Lemma \\ref{lemma-flat-pullback-bivariant}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FF1","source_file":"chow.tex","source_line":13434,"source_end_line":13442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13434-L13442","statement_sha256":"77f3539f22edbcf1feb927ef503d17c0958ec818baf9aa48060b7bc2f272806d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8333,"rank":8333,"depth":58,"x":2332.643,"y":688.113,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FF2","tag":"0FF2","title":"Gysin maps for local complete intersection morphisms · Lemma 0FF2","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a local complete intersection morphism of schemes locally of finite type over S. The bivariant class f^! is independent of the choice of the factorization f = g ∘ i with g smooth (provided one exists).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a local complete intersection morphism\nof schemes locally of finite type over $S$.\nThe bivariant class $f^!$ is independent of the choice of\nthe factorization $f = g \\circ i$ with $g$ smooth (provided\none exists).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FF2","source_file":"chow.tex","source_line":13484,"source_end_line":13492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13484-L13492","statement_sha256":"4989a3ad94cd0cc7a9eab987a817b5001e5d3e89c31c829750e565e5392dbaa6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8334,"rank":8334,"depth":60,"x":2582.238,"y":590.313,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FF3","tag":"0FF3","title":"Gysin maps for local complete intersection morphisms · Definition 0FF3","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a local complete intersection morphism of schemes locally of finite type over S. We say the Gysin map for f exists if we can write f = g ∘ i with g smooth and i an immersion. In this case we define the Gysin map f^! = i^! ∘ g^* ∈ A^*(X → Y) as above.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a local complete intersection morphism\nof schemes locally of finite type over $S$. We say\n{\\it the Gysin map for $f$ exists} if we can write\n$f = g \\circ i$ with $g$ smooth and $i$ an immersion.\nIn this case we define the\n{\\it Gysin map} $f^! = i^! \\circ g^* \\in A^*(X \\to Y)$ as above.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FF3","source_file":"chow.tex","source_line":13529,"source_end_line":13538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13529-L13538","statement_sha256":"fa2db3edebcf08ca6e46fa75de4756adbe512473dc89a2900076b042aaafce34","origin":"The Stacks Project","memory_eligible":false,"source_rank":8335,"rank":8335,"depth":0,"x":2476.732,"y":804.271,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FF4","tag":"0FF4","title":"Gysin maps for local complete intersection morphisms · Lemma 0FF4","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a local complete intersection morphism of schemes locally of finite type over S. If the Gysin map exists for f and f is flat, then f^! is equal to the bivariant class of Lemma [Tag 0B78].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a local complete intersection morphism\nof schemes locally of finite type over $S$. If the Gysin map\nexists for $f$ and $f$ is flat, then $f^!$ is equal to the\nbivariant class of Lemma \\ref{lemma-flat-pullback-bivariant}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FF4","source_file":"chow.tex","source_line":13546,"source_end_line":13553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13546-L13553","statement_sha256":"c4110d14234dbc2b405bd90377dc4305f5a9262d44afdf663a45ab51be22aecd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8336,"rank":8336,"depth":58,"x":2382.376,"y":586.415,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FF5","tag":"0FF5","title":"Gysin maps for local complete intersection morphisms · Lemma 0FF5","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y and g : Y → Z be local complete intersection morphisms of schemes locally of finite type over S. Assume the Gysin map exists for g ∘ f and g. Then the Gysin map exists for f and (g ∘ f)^! = f^! ∘ g^!.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be local complete intersection morphisms\nof schemes locally of finite type over $S$. Assume the Gysin\nmap exists for $g \\circ f$ and $g$. Then the Gysin map exists for $f$\nand $(g \\circ f)^! = f^! \\circ g^!$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FF5","source_file":"chow.tex","source_line":13574,"source_end_line":13581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13574-L13581","statement_sha256":"aab6e32e94f2b9fd5e6893ea931a3c21b699dd42992ba0149bbe583ee4a522b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8337,"rank":8337,"depth":60,"x":2627.393,"y":693.627,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FF6","tag":"0FF6","title":"Gysin maps for local complete intersection morphisms · Lemma 0FF6","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Consider a commutative diagram xymatrix X\" ar[d] ar[r] & X' ar[d] ar[r] & X ar[d]^f Y\" ar[r] & Y' ar[r] & Y of schemes locally of finite type over S with both square cartesian. Assume f : X → Y is a local complete intersection morphism such that the Gysin map exists for f. Let c ∈ A^*(Y\" → Y'). Denote res(f^!) ∈ A^*(X' → Y') the restriction of f^! to Y' (Remark [Tag 0F9Z]). Then c and res(f^!) commute (Remark [Tag 0FA2]).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nConsider a commutative diagram\n$$\n\\xymatrix{\nX'' \\ar[d] \\ar[r] &\nX' \\ar[d] \\ar[r] &\nX \\ar[d]^f \\\\\nY'' \\ar[r] &\nY' \\ar[r] &\nY\n}\n$$\nof schemes locally of finite type over $S$ with both square cartesian.\nAssume $f : X \\to Y$ is a local complete intersection morphism\nsuch that the Gysin map exists for $f$. Let $c \\in A^*(Y'' \\to Y')$. Denote\n$res(f^!) \\in A^*(X' \\to Y')$ the restriction of $f^!$ to $Y'$\n(Remark \\ref{remark-restriction-bivariant}). Then $c$ and $res(f^!)$ commute\n(Remark \\ref{remark-bivariant-commute}).","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FF6","source_file":"chow.tex","source_line":13612,"source_end_line":13632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13612-L13632","statement_sha256":"68825a9b35f1127efc2c5c6caa9d6c25b21f435a46aa7f9e05262ff88bbd63c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8338,"rank":8338,"depth":60,"x":2360.236,"y":753.659,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FF7","tag":"0FF7","title":"Gysin maps for local complete intersection morphisms · Lemma 0FF7","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Consider a cartesian diagram xymatrix X' ar[d]_f' ar[r] & X ar[d]^f Y' ar[r] & Y of schemes locally of finite type over S. Assume • f is a local complete intersection morphism and the Gysin map exists for f, • X, X', Y, Y' satisfy the equivalent conditions of Lemma [Tag 0FE3], • for x' ∈ X' with images x, y', and y in X, Y', and Y we have n_x' - n_y' = n_x - n_y where n_x', n_x, n_y', and n_y are as in the lemma, and • for every…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nConsider a cartesian diagram\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r] &\nX \\ar[d]^f \\\\\nY' \\ar[r] &\nY\n}\n$$\nof schemes locally of finite type over $S$. Assume\n\\begin{enumerate}\n\\item $f$ is a local complete intersection morphism and\nthe Gysin map exists for $f$,\n\\item $X$, $X'$, $Y$, $Y'$ satisfy the equivalent conditions of\nLemma \\ref{lemma-locally-equidimensional},\n\\item for $x' \\in X'$ with images $x$, $y'$, and $y$\nin $X$, $Y'$, and $Y$ we have $n_{x'} - n_{y'} = n_x - n_y$\nwhere $n_{x'}$, $n_x$, $n_{y'}$, and $n_y$ are as in the lemma, and\n\\item for every generic point $\\xi \\in X'$ the local ring\n$\\mathcal{O}_{Y', f'(\\xi)}$ is Cohen-Macaulay.\n\\end{enumerate}\nThen $f^![Y'] = [X']$ where $[Y']$ and $[X']$ are as in\nRemark \\ref{remark-fundamental-class}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FF7","source_file":"chow.tex","source_line":13642,"source_end_line":13668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13642-L13668","statement_sha256":"1571cff5336f517562fc6b1fe7d067a89d3c975937e4ba492117ab430e583468","origin":"The Stacks Project","memory_eligible":false,"source_rank":8339,"rank":8339,"depth":59,"x":2509.106,"y":557.605,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FV9","tag":"0FV9","title":"Gysin maps for local complete intersection morphisms · Lemma 0FV9","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Consider a cartesian square xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f Y' ar[r]^g & Y of schemes locally of finite type over S. Assume • both f and f' are local complete intersection morphisms, and • the Gysin map exists for f Then C = Ker(H^-1((g')^*NL_X/Y) → H^-1(NL_X'/Y')) is a finite locally free O_X'-module, the Gysin map exists for f', and we have res(f^!) = c_top(C^vee) ∘ (f')^! in A^*(X' → Y').","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nConsider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} &\nX \\ar[d]^f \\\\\nY' \\ar[r]^g &\nY\n}\n$$\nof schemes locally of finite type over $S$. Assume\n\\begin{enumerate}\n\\item both $f$ and $f'$ are local complete intersection morphisms, and\n\\item the Gysin map exists for $f$\n\\end{enumerate}\nThen $\\mathcal{C} = \\Ker(H^{-1}((g')^*\\NL_{X/Y}) \\to H^{-1}(\\NL_{X'/Y'}))$\nis a finite locally free $\\mathcal{O}_{X'}$-module, the Gysin map\nexists for $f'$, and we have\n$$\nres(f^!) = c_{top}(\\mathcal{C}^\\vee) \\circ (f')^!\n$$\nin $A^*(X' \\to Y')$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FV9","source_file":"chow.tex","source_line":13737,"source_end_line":13761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13737-L13761","statement_sha256":"5eb843618722da9d245a907e5e8086d7381c47a36ee9ee17d8db9316c833467f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8340,"rank":8340,"depth":58,"x":2557.041,"y":786.874,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVA","tag":"0FVA","title":"Blow up formula · Lemma 0FVA","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let i : Z → X be a regular closed immersion of schemes locally of finite type over S. Let b : X' → X be the blowing up with center Z. Picture xymatrix E ar[r]_j ar[d]_π & X' ar[d]^b Z ar[r]^i & X Assume that the Gysin map exists for b. Then we have res(b^!) = c_top(F^vee) ∘ π^* in A^*(E → Z) where F is the kernel of the canonical map π^*C_Z/X → C_E/X'.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $i : Z \\to X$ be a regular closed immersion of schemes\nlocally of finite type over $S$. Let $b : X' \\to X$ be the\nblowing up with center $Z$. Picture\n$$\n\\xymatrix{\nE \\ar[r]_j \\ar[d]_\\pi & X' \\ar[d]^b \\\\\nZ \\ar[r]^i & X\n}\n$$\nAssume that the Gysin map exists for $b$. Then we have\n$$\nres(b^!) = c_{top}(\\mathcal{F}^\\vee) \\circ \\pi^*\n$$\nin $A^*(E \\to Z)$ where $\\mathcal{F}$ is the kernel of the canonical map\n$\\pi^*\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{E/X'}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVA","source_file":"chow.tex","source_line":13807,"source_end_line":13825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13807-L13825","statement_sha256":"625b3c624052bcd2037877ee62b6561017ab6d45bbf620eaa0255dde0bc779cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8341,"rank":8341,"depth":59,"x":2337.104,"y":644.872,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FF9","tag":"0FF9","title":"Gysin maps for local complete intersection morphisms · Lemma 0FF9","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a morphism of schemes locally of finite type over S such that both X and Y are quasi-compact, regular, have affine diagonal, and finite dimension. Then f is a local complete intersection morphism. Assume moreover the Gysin map exists for f. Then f^!(α · β) = f^!α · f^!β in CH^*(X) ⊗ Q where the intersection product is as in Section [Tag 0FEX].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a morphism of schemes locally of finite\ntype over $S$ such that both $X$ and $Y$ are quasi-compact,\nregular, have affine diagonal, and finite dimension.\nThen $f$ is a local complete intersection morphism.\nAssume moreover the Gysin map exists for $f$. Then\n$$\nf^!(\\alpha \\cdot \\beta) = f^!\\alpha \\cdot f^!\\beta\n$$\nin $\\CH^*(X) \\otimes \\mathbf{Q}$ where the intersection product\nis as in Section \\ref{section-intersection-regular}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FF9","source_file":"chow.tex","source_line":13867,"source_end_line":13880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13867-L13880","statement_sha256":"865cf4f9df9bdb152f278386954795ce78306f543eb23321ce4b86ddf2916088","origin":"The Stacks Project","memory_eligible":false,"source_rank":8342,"rank":8342,"depth":61,"x":2613.751,"y":624.765,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FFA","tag":"0FFA","title":"Gysin maps for local complete intersection morphisms · Lemma 0FFA","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y be a morphism of schemes locally of finite type over S such that both X and Y are quasi-compact, regular, have affine diagonal, and finite dimension. Then f is a local complete intersection morphism. Assume moreover the Gysin map exists for f and that f is proper. Then f_*(α · f^!β) = f_*α · β in CH^*(Y) ⊗ Q where the intersection product is as in Section [Tag 0FEX].","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ be a morphism of schemes locally of finite\ntype over $S$ such that both $X$ and $Y$ are quasi-compact,\nregular, have affine diagonal, and finite dimension.\nThen $f$ is a local complete intersection morphism.\nAssume moreover the Gysin map exists for $f$\nand that $f$ is proper. Then\n$$\nf_*(\\alpha \\cdot f^!\\beta) = f_*\\alpha \\cdot \\beta\n$$\nin $\\CH^*(Y) \\otimes \\mathbf{Q}$ where the intersection product\nis as in Section \\ref{section-intersection-regular}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFA","source_file":"chow.tex","source_line":13912,"source_end_line":13926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13912-L13926","statement_sha256":"75ee2af988507f89d8409f67851ef8fd3c2d5f19bdcfeca7c93c9a7c1208f981","origin":"The Stacks Project","memory_eligible":false,"source_rank":8343,"rank":8343,"depth":61,"x":2425.739,"y":796.739,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBS","tag":"0FBS","title":"Gysin maps for diagonals · Lemma 0FBS","summary":"In the situation above we have Δ^! ∘ pr_i^! = 1 in A^0(X).","statement_latex":"In the situation above we have $\\Delta^! \\circ \\text{pr}_i^! = 1$ in $A^0(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBS","source_file":"chow.tex","source_line":13990,"source_end_line":13993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L13990-L13993","statement_sha256":"00a7fb59e66c3e6260263c9aa1bcfae7beb374c5de9d090916db479b57ff3916","origin":"The Stacks Project","memory_eligible":false,"source_rank":8344,"rank":8344,"depth":61,"x":2426.08,"y":563.013,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBT","tag":"0FBT","title":"Gysin maps for diagonals · Proposition 0FBT","summary":"[F] Let (S, δ) be as in Situation [Tag 02QL]. Let f : X → Y and g : Y → Z be morphisms of schemes locally of finite type over S. If g is smooth of relative dimension d, then A^p(X → Y) = A^p - d(X → Z).","statement_latex":"\\begin{reference}\n\\cite[Proposition 17.4.2]{F}\n\\end{reference}\nLet $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of schemes locally\nof finite type over $S$. If $g$ is smooth of relative dimension $d$, then\n$A^p(X \\to Y) = A^{p - d}(X \\to Z)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Gysin maps for diagonals","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBT","source_file":"chow.tex","source_line":14002,"source_end_line":14011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14002-L14011","statement_sha256":"13a813a8c684a25d2c439a9a8f08161f62f92d6c4bcb44146d73c3d5611390d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8345,"rank":8345,"depth":62,"x":2613.97,"y":735.724,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBV","tag":"0FBV","title":"Exterior product · Lemma 0FBV","summary":"The map × : CH_n(X) ⊗_Z CH_m(Y) → CH_n + m(X ×_k Y) is well defined.","statement_latex":"The map\n$\\times : \\CH_n(X) \\otimes_{\\mathbf{Z}} \\CH_m(Y) \\to \\CH_{n + m}(X \\times_k Y)$\nis well defined.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Exterior product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBV","source_file":"chow.tex","source_line":14136,"source_end_line":14141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14136-L14141","statement_sha256":"ccb999c221039a8e6a6c7dd097190325834daa374ea8759465a3d279156fc040","origin":"The Stacks Project","memory_eligible":false,"source_rank":8346,"rank":8346,"depth":47,"x":2336.26,"y":714.962,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBW","tag":"0FBW","title":"Exterior product · Lemma 0FBW","summary":"Let k be a field. Let X be a scheme locally of finite type over k. Then we have a canonical identification A^p(X → Spec(k)) = CH_-p(X) for all p ∈ Z.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$.\nThen we have a canonical identification\n$$\nA^p(X \\to \\Spec(k)) = \\CH_{-p}(X)\n$$\nfor all $p \\in \\mathbf{Z}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Exterior product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBW","source_file":"chow.tex","source_line":14182,"source_end_line":14190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14182-L14190","statement_sha256":"2bf0d090bf221ffe40129cf422562a07aefab3860fca1c35ca040d211adc5aa2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8347,"rank":8347,"depth":47,"x":2557.953,"y":572.552,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBX","tag":"0FBX","title":"Exterior product · Lemma 0FBX","summary":"Let k be a field. Let X be a scheme locally of finite type over k. Let c ∈ A^p(X → Spec(k)). Let Y → Z be a morphism of schemes locally of finite type over k. Let c' ∈ A^q(Y → Z). Then c ∘ c' = c' ∘ c in A^p + q(X ×_k Y → Z).","statement_latex":"Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$.\nLet $c \\in A^p(X \\to \\Spec(k))$. Let $Y \\to Z$ be a morphism of schemes\nlocally of finite type over $k$. Let $c' \\in A^q(Y \\to Z)$. Then\n$c \\circ c' = c' \\circ c$ in $A^{p + q}(X \\times_k Y \\to Z)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Exterior product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBX","source_file":"chow.tex","source_line":14243,"source_end_line":14249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14243-L14249","statement_sha256":"a6974fe2080cd8292cb5c218109d95d7fee9807d06eef5a85b9975c21ae8ed91","origin":"The Stacks Project","memory_eligible":false,"source_rank":8348,"rank":8348,"depth":48,"x":2508.954,"y":803.584,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FBZ","tag":"0FBZ","title":"Exterior product · Lemma 0FBZ","summary":"Exterior product is associative. More precisely, let k be a field, let X, Y, Z be schemes locally of finite type over k, let α ∈ CH_*(X), β ∈ CH_*(Y), γ ∈ CH_*(Z). Then (α × β) × γ = α × (β × γ) in CH_*(X ×_k Y ×_k Z).","statement_latex":"Exterior product is associative. More precisely, let $k$ be a field,\nlet $X, Y, Z$ be schemes locally of finite type over $k$, let\n$\\alpha \\in \\CH_*(X)$, $\\beta \\in \\CH_*(Y)$, $\\gamma \\in \\CH_*(Z)$.\nThen $(\\alpha \\times \\beta) \\times \\gamma =\n\\alpha \\times (\\beta \\times \\gamma)$ in $\\CH_*(X \\times_k Y \\times_k Z)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Exterior product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FBZ","source_file":"chow.tex","source_line":14276,"source_end_line":14283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14276-L14283","statement_sha256":"91e806125f9c52b0c35bf0ff0ce14e02d58391418446f61915ac0aaf9c8149f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8349,"rank":8349,"depth":0,"x":2359.148,"y":605.227,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FC1","tag":"0FC1","title":"Intersection products · Lemma 0FC1","summary":"The product defined above is associative. More precisely, let k be a field, let X be smooth over k, let Y, Z, W be schemes locally of finite type over X, let α ∈ CH_*(Y), β ∈ CH_*(Z), γ ∈ CH_*(W). Then (α · β) · γ = α · (β · γ) in CH_*(Y ×_X Z ×_X W).","statement_latex":"The product defined above is associative. More precisely, let $k$ be a field,\nlet $X$ be smooth over $k$,\nlet $Y, Z, W$ be schemes locally of finite type over $X$, let\n$\\alpha \\in \\CH_*(Y)$, $\\beta \\in \\CH_*(Z)$, $\\gamma \\in \\CH_*(W)$.\nThen $(\\alpha \\cdot \\beta) \\cdot \\gamma =\n\\alpha \\cdot (\\beta \\cdot \\gamma)$ in $\\CH_*(Y \\times_X Z \\times_X W)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersection products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FC1","source_file":"chow.tex","source_line":14341,"source_end_line":14349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14341-L14349","statement_sha256":"01a71fdaef7bb088f8a56131a4fc9c3bb7ed55abb44bbb6ead3e4be770416f9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8350,"rank":8350,"depth":60,"x":2629.391,"y":666.549,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FC2","tag":"0FC2","title":"Intersection products · Lemma 0FC2","summary":"Let k be a field. Let X be a smooth scheme over k, equidimensional of dimension d. The map A^p(X) → CH_d - p(X), c ↦ c ∩ [X]_d is an isomorphism. Via this isomorphism composition of bivariant classes turns into the intersection product defined above.","statement_latex":"Let $k$ be a field. Let $X$ be a smooth scheme over $k$, equidimensional\nof dimension $d$. The map\n$$\nA^p(X) \\longrightarrow \\CH_{d - p}(X),\\quad\nc \\longmapsto c \\cap [X]_d\n$$\nis an isomorphism. Via this isomorphism composition of bivariant\nclasses turns into the intersection product defined above.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersection products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FC2","source_file":"chow.tex","source_line":14402,"source_end_line":14412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14402-L14412","statement_sha256":"c84144e82251f880e7fa660ae09826484e65ae6f8f73690be39aaebdcd9a54d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8351,"rank":8351,"depth":63,"x":2380.561,"y":774.779,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FFB","tag":"0FFB","title":"Intersection products · Lemma 0FFB","summary":"Let k be a field. Let f : X → Y be a morphism of schemes smooth over k. Then the Gysin map exists for f and f^!(α · β) = f^!α · f^!β.","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a morphism of schemes smooth\nover $k$. Then the Gysin map exists for $f$ and\n$f^!(\\alpha \\cdot \\beta) = f^!\\alpha \\cdot f^!\\beta$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersection products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFB","source_file":"chow.tex","source_line":14436,"source_end_line":14441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14436-L14441","statement_sha256":"b3600ebe84049afcfce38c081e6560b141d06a3b47f5f96ce2f412a6303039ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":8352,"rank":8352,"depth":64,"x":2477.103,"y":553.564,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FFC","tag":"0FFC","title":"Intersection products · Lemma 0FFC","summary":"Let k be a field. Let f : X → Y be a proper morphism of schemes smooth over k. Then the Gysin map exists for f and f_*(α · f^!β) = f_*α · β.","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a proper morphism of schemes smooth\nover $k$. Then the Gysin map exists for $f$ and\n$f_*(\\alpha \\cdot f^!\\beta) = f_*\\alpha \\cdot \\beta$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersection products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFC","source_file":"chow.tex","source_line":14481,"source_end_line":14486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14481-L14486","statement_sha256":"6924c8d2975f8aba52613cd06b465ec6f15959ffc3ee9abc09ca9aa3283898af","origin":"The Stacks Project","memory_eligible":false,"source_rank":8353,"rank":8353,"depth":64,"x":2583.913,"y":771.677,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FFD","tag":"0FFD","title":"Intersection products · Lemma 0FFD","summary":"Let k be a field. Let X be an integral scheme smooth over k. Let Y, Z ⊂ X be integral closed subschemes. Set d = dim(Y) + dim(Z) - dim(X). Assume • dim(Y ∩ Z) ≤ d, and • O_Y, xi and O_Z, xi are Cohen-Macaulay for every xi ∈ Y ∩ Z with δ(xi) = d. Then [Y] · [Z] = [Y ∩ Z]_d in CH_d(X).","statement_latex":"Let $k$ be a field. Let $X$ be an integral scheme smooth over $k$.\nLet $Y, Z \\subset X$ be integral closed subschemes. Set\n$d = \\dim(Y) + \\dim(Z) - \\dim(X)$. Assume\n\\begin{enumerate}\n\\item $\\dim(Y \\cap Z) \\leq d$, and\n\\item $\\mathcal{O}_{Y, \\xi}$ and $\\mathcal{O}_{Z, \\xi}$\nare Cohen-Macaulay for every $\\xi \\in Y \\cap Z$ with\n$\\delta(\\xi) = d$.\n\\end{enumerate}\nThen $[Y] \\cdot [Z] = [Y \\cap Z]_d$ in $\\CH_d(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersection products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFD","source_file":"chow.tex","source_line":14522,"source_end_line":14534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14522-L14534","statement_sha256":"4d2d5e6f40bdbad5847578504031a383127c51df1f246e1a0d1821ecd62091c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8354,"rank":8354,"depth":59,"x":2329.506,"y":671.353,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FFE","tag":"0FFE","title":"Intersection products · Lemma 0FFE","summary":"Let k be a field. Let X be a scheme smooth over k. Let i : Y → X be a regular closed immersion. Let α ∈ CH_*(X). If Y is equidimensional of dimension e, then α · [Y]_e = i_*(i^!(α)) in CH_*(X).","statement_latex":"Let $k$ be a field. Let $X$ be a scheme smooth over $k$. Let $i : Y \\to X$ be\na regular closed immersion. Let $\\alpha \\in \\CH_*(X)$. If $Y$ is\nequidimensional of dimension $e$, then\n$\\alpha \\cdot [Y]_e = i_*(i^!(\\alpha))$ in $\\CH_*(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersection products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFE","source_file":"chow.tex","source_line":14562,"source_end_line":14568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14562-L14568","statement_sha256":"1593427a5c0f1067156f518bd914e601ff49b5ba4135c829f2e82d5bf1663ec2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8355,"rank":8355,"depth":64,"x":2598.04,"y":600.906,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FFF","tag":"0FFF","title":"Intersection products · Lemma 0FFF","summary":"Let k be a field. Let X be a smooth scheme over k which is quasi-compact and has affine diagonal. Then the intersection product on CH^*(X) constructed in this section agrees after tensoring with Q with the intersection product constructed in Section [Tag 0FEX].","statement_latex":"Let $k$ be a field. Let $X$ be a smooth scheme over $k$ which is\nquasi-compact and has affine diagonal. Then the intersection\nproduct on $\\CH^*(X)$ constructed in this section agrees\nafter tensoring with $\\mathbf{Q}$ with the intersection product\nconstructed in Section \\ref{section-intersection-regular}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersection products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FFF","source_file":"chow.tex","source_line":14586,"source_end_line":14593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14586-L14593","statement_sha256":"0398b25e5d44adb16dc8561e7a7532603c516f96e79cb5821278f8ed0a0990df","origin":"The Stacks Project","memory_eligible":false,"source_rank":8356,"rank":8356,"depth":64,"x":2456.542,"y":805.423,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FC4","tag":"0FC4","title":"Exterior product over Dedekind domains · Lemma 0FC4","summary":"The map × : CH_n(X) ⊗_Z CH_m(Y) → CH_n + m - 1(X ×_S Y) is well defined.","statement_latex":"The map $\\times : \\CH_n(X) \\otimes_{\\mathbf{Z}} \\CH_m(Y) \\to\n\\CH_{n + m - 1}(X \\times_S Y)$ is well defined.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Exterior product over Dedekind domains","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FC4","source_file":"chow.tex","source_line":14657,"source_end_line":14661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14657-L14661","statement_sha256":"62e1ccb4ce83e02ca04cf4b8cd807afd6e21ab65125d8a5254aec97494fc6eb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8357,"rank":8357,"depth":52,"x":2396.355,"y":574.102,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FC5","tag":"0FC5","title":"Exterior product over Dedekind domains · Lemma 0FC5","summary":"Let (S, δ) be as above. Let X be a scheme locally of finite type over S. Then we have a canonical identification A^p(X → S) = CH_1 - p(X) for all p ∈ Z.","statement_latex":"Let $(S, \\delta)$ be as above. Let $X$ be a scheme locally of finite type\nover $S$. Then we have a canonical identification\n$$\nA^p(X \\to S) = \\CH_{1 - p}(X)\n$$\nfor all $p \\in \\mathbf{Z}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Exterior product over Dedekind domains","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FC5","source_file":"chow.tex","source_line":14733,"source_end_line":14741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14733-L14741","statement_sha256":"583c4ca4df8c6db7ed2a1b54d6145c28d93af5d089aae72a434761106baf5fa7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8358,"rank":8358,"depth":53,"x":2626.98,"y":710.655,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FC6","tag":"0FC6","title":"Exterior product over Dedekind domains · Lemma 0FC6","summary":"Let (S, δ) be as above. Let X be a scheme locally of finite type over S. Let c ∈ A^p(X → S). Let Y → Z be a morphism of schemes locally of finite type over S. Let c' ∈ A^q(Y → Z). Then c ∘ c' = c' ∘ c in A^p + q(X ×_S Y → X ×_S Z).","statement_latex":"Let $(S, \\delta)$ be as above. Let $X$ be a scheme locally of finite type\nover $S$. Let $c \\in A^p(X \\to S)$. Let $Y \\to Z$ be a morphism of schemes\nlocally of finite type over $S$. Let $c' \\in A^q(Y \\to Z)$. Then\n$c \\circ c' = c' \\circ c$ in $A^{p + q}(X \\times_S Y \\to X \\times_S Z)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Exterior product over Dedekind domains","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FC6","source_file":"chow.tex","source_line":14806,"source_end_line":14812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14806-L14812","statement_sha256":"a98058cf6857eef91486db7aa202b5dd094b5ab2dcb3d909678c7f94bb43b4fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8359,"rank":8359,"depth":54,"x":2346.839,"y":740.854,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FC8","tag":"0FC8","title":"Exterior product over Dedekind domains · Lemma 0FC8","summary":"Exterior product is associative. More precisely, let (S, δ) be as above, let X, Y, Z be schemes locally of finite type over S, let α ∈ CH_*(X), β ∈ CH_*(Y), γ ∈ CH_*(Z). Then (α × β) × γ = α × (β × γ) in CH_*(X ×_S Y ×_S Z).","statement_latex":"Exterior product is associative. More precisely, let $(S, \\delta)$ be\nas above, let $X, Y, Z$ be schemes locally of finite type over $S$, let\n$\\alpha \\in \\CH_*(X)$, $\\beta \\in \\CH_*(Y)$, $\\gamma \\in \\CH_*(Z)$.\nThen $(\\alpha \\times \\beta) \\times \\gamma =\n\\alpha \\times (\\beta \\times \\gamma)$ in $\\CH_*(X \\times_S Y \\times_S Z)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Exterior product over Dedekind domains","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FC8","source_file":"chow.tex","source_line":14840,"source_end_line":14847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14840-L14847","statement_sha256":"5d96cad761b61f79543bf8e8f427768f5dc602329882586719ec7fae9f104eda","origin":"The Stacks Project","memory_eligible":false,"source_rank":8360,"rank":8360,"depth":0,"x":2529.301,"y":559.453,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FCA","tag":"0FCA","title":"Intersection products over Dedekind domains · Lemma 0FCA","summary":"The product defined above is associative. More precisely, with (S, δ) as above, let X be smooth over S, let Y, Z, W be schemes locally of finite type over X, let α ∈ CH_*(Y), β ∈ CH_*(Z), γ ∈ CH_*(W). Then (α · β) · γ = α · (β · γ) in CH_*(Y ×_X Z ×_X W).","statement_latex":"The product defined above is associative. More precisely, with\n$(S, \\delta)$ as above, let $X$ be smooth over $S$,\nlet $Y, Z, W$ be schemes locally of finite type over $X$, let\n$\\alpha \\in \\CH_*(Y)$, $\\beta \\in \\CH_*(Z)$, $\\gamma \\in \\CH_*(W)$.\nThen $(\\alpha \\cdot \\beta) \\cdot \\gamma =\n\\alpha \\cdot (\\beta \\cdot \\gamma)$ in $\\CH_*(Y \\times_X Z \\times_X W)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersection products over Dedekind domains","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCA","source_file":"chow.tex","source_line":14917,"source_end_line":14925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14917-L14925","statement_sha256":"bd3dca9d4fc32b218dfbdb9ac239e9cedd9d0c5dedaefc8753b88b19c30c891d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8361,"rank":8361,"depth":60,"x":2540.645,"y":796.976,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FCB","tag":"0FCB","title":"Intersection products over Dedekind domains · Lemma 0FCB","summary":"Let (S, δ) be as above. Let X be a smooth scheme over S, equidimensional of dimension d. The map A^p(X) → CH_d - p(X), c ↦ c ∩ [X]_d is an isomorphism. Via this isomorphism composition of bivariant classes turns into the intersection product defined above.","statement_latex":"Let $(S, \\delta)$ be as above. Let $X$ be a smooth scheme over $S$,\nequidimensional of dimension $d$. The map\n$$\nA^p(X) \\longrightarrow \\CH_{d - p}(X),\\quad\nc \\longmapsto c \\cap [X]_d\n$$\nis an isomorphism. Via this isomorphism composition of bivariant\nclasses turns into the intersection product defined above.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Intersection products over Dedekind domains","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FCB","source_file":"chow.tex","source_line":14978,"source_end_line":14988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L14978-L14988","statement_sha256":"7198b786cb17b0411c590af1af7fa4734f471fbf849fc2f63914a102ea98a254","origin":"The Stacks Project","memory_eligible":false,"source_rank":8362,"rank":8362,"depth":63,"x":2341.08,"y":628.106,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVH","tag":"0FVH","title":"Change of base scheme · Lemma 0FVH","summary":"In Situation [Tag 0FVG] let X → S be locally of finite type. Denote X' → S' the base change by S' → S. If X is integral with dim_δ(X) = k, then every irreducible component Z' of X' has dim_δ'(Z') = k + c,","statement_latex":"In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally\nof finite type. Denote $X' \\to S'$ the base change by $S' \\to S$.\nIf $X$ is integral with $\\dim_\\delta(X) = k$, then\nevery irreducible component $Z'$ of $X'$ has $\\dim_{\\delta'}(Z') = k + c$,","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVH","source_file":"chow.tex","source_line":15138,"source_end_line":15144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15138-L15144","statement_sha256":"f7bf80c3e80244bc7f12987ee1d8400f877dd644afc5dac769499cbb5f0a94e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8363,"rank":8363,"depth":27,"x":2624.307,"y":639.401,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVI","tag":"0FVI","title":"Change of base scheme · Lemma 0FVI","summary":"In Situation [Tag 0FVG] let X → S locally of finite type and let X' → S be the base change by S' → S. • Let Z ⊂ X be a closed subscheme with dim_δ(Z) ≤ k and base change Z' ⊂ X'. Then we have dim_δ'(Z')) ≤ k + c and [Z']_k + c = g^*[Z]_k in Z_k + c(X'). • Let F be a coherent sheaf on X with dim_δ(Supp(F)) ≤ k and base change F' on X'. Then we have dim_δ(Supp(F')) ≤ k + c and g^*[F]_k = [F']_k + c in Z_k + c(X').","statement_latex":"In Situation \\ref{situation-setup-base-change} let $X \\to S$\nlocally of finite type and let $X' \\to S$ be the base change by $S' \\to S$.\n\\begin{enumerate}\n\\item Let $Z \\subset X$ be a closed subscheme with\n$\\dim_\\delta(Z) \\leq k$ and base change $Z' \\subset X'$. Then we have\n$\\dim_{\\delta'}(Z')) \\leq k + c$\nand $[Z']_{k + c} = g^*[Z]_k$ in $Z_{k + c}(X')$.\n\\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ with\n$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$ and base\nchange $\\mathcal{F}'$ on $X'$.\nThen we have $\\dim_\\delta(\\text{Supp}(\\mathcal{F}')) \\leq k + c$\nand $g^*[\\mathcal{F}]_k = [\\mathcal{F}']_{k + c}$\nin $Z_{k + c}(X')$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVI","source_file":"chow.tex","source_line":15192,"source_end_line":15208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15192-L15208","statement_sha256":"17244fae1ae916f52ec3e56aebec7806d5628436518459ee5ae395ac3e59679a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8364,"rank":8364,"depth":28,"x":2406.169,"y":791.927,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVJ","tag":"0FVJ","title":"Change of base scheme · Lemma 0FVJ","summary":"In Situation [Tag 0FVG] let X → S be locally of finite type and let X' → S' be the base change by S' → S. The map g^* : Z_k(X) → Z_k + c(X') above factors through rational equivalence to give a map g^* : CH_k(X) → CH_k + c(X') of chow groups.","statement_latex":"In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally\nof finite type and let $X' \\to S'$ be the base change by $S' \\to S$.\nThe map $g^* : Z_k(X) \\to Z_{k + c}(X')$ above factors through rational\nequivalence to give a map\n$$\ng^* : \\CH_k(X) \\longrightarrow \\CH_{k + c}(X')\n$$\nof chow groups.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVJ","source_file":"chow.tex","source_line":15254,"source_end_line":15264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15254-L15264","statement_sha256":"b4dabc4f84cd2e560f57860130a26c6b7202351644b520a146b9e6a49bcee372","origin":"The Stacks Project","memory_eligible":false,"source_rank":8365,"rank":8365,"depth":49,"x":2444.4,"y":555.455,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVK","tag":"0FVK","title":"Change of base scheme · Lemma 0FVK","summary":"In Situation [Tag 0FVG] let Y → X → S be locally of finite type and let Y' → X' → S' be the base change by S' → S. Assume f : Y → X is flat of relative dimension r. Then f' : Y' → X' is flat of relative dimension r and the diagrams vcenter xymatrix Z_k + r(Y) ar[r]_g^* & Z_k + c + r(Y') Z_k(X) ar[r]^g^* ar[u]^f^* & Z_k + c(X') ar[u]_(f')^* and vcenter xymatrix CH_k + r(Y) ar[r]_g^* & CH_k + c + r(Y') CH_k(X) ar[r]^g^* ar[u]^f^* & CH_k + c(X') ar[u]_(f')^* of cycle and…","statement_latex":"In Situation \\ref{situation-setup-base-change} let $Y \\to X \\to S$ be locally\nof finite type and let $Y' \\to X' \\to S'$ be the base change by $S' \\to S$.\nAssume $f : Y \\to X$ is flat of relative dimension $r$. Then $f' : Y' \\to X'$\nis flat of relative dimension $r$ and the diagrams\n$$\n\\vcenter{\n\\xymatrix{\nZ_{k + r}(Y) \\ar[r]_{g^*} & Z_{k + c + r}(Y') \\\\\nZ_k(X) \\ar[r]^{g^*} \\ar[u]^{f^*} & Z_{k + c}(X') \\ar[u]_{(f')^*}\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\n\\CH_{k + r}(Y) \\ar[r]_{g^*} & \\CH_{k + c + r}(Y') \\\\\n\\CH_k(X) \\ar[r]^{g^*} \\ar[u]^{f^*} & \\CH_{k + c}(X') \\ar[u]_{(f')^*}\n}\n}\n$$\nof cycle and chow groups commutes.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVK","source_file":"chow.tex","source_line":15301,"source_end_line":15323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15301-L15323","statement_sha256":"77b1013f3f271f64677c44b90991baf73678237b6aeb82cb934a5f2e0cd98d3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8366,"rank":8366,"depth":29,"x":2606.526,"y":751.705,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVL","tag":"0FVL","title":"Change of base scheme · Lemma 0FVL","summary":"In Situation [Tag 0FVG] let Y → X → S be locally of finite type and let Y' → X' → S' be the base change by S' → S. Assume f : Y → X is proper. Then f' : Y' → X' is proper and the diagram vcenter xymatrix Z_k(Y) ar[r]_g^* ar[d]_f_* & Z_k + c(Y') ar[d]^f'_* Z_k(X) ar[r]^g^* & Z_k + c(X') and vcenter xymatrix CH_k(Y) ar[r]_g^* ar[d]_f_* & CH_k + c(Y') ar[d]^f'_* CH_k(X) ar[r]^g^* & CH_k + c(X') of cycle and chow groups commutes.","statement_latex":"In Situation \\ref{situation-setup-base-change} let $Y \\to X \\to S$ be locally\nof finite type and let $Y' \\to X' \\to S'$ be the base change by $S' \\to S$.\nAssume $f : Y \\to X$ is proper. Then $f' : Y' \\to X'$ is proper and the diagram\n$$\n\\vcenter{\n\\xymatrix{\nZ_k(Y) \\ar[r]_{g^*} \\ar[d]_{f_*} & Z_{k + c}(Y') \\ar[d]^{f'_*} \\\\\nZ_k(X) \\ar[r]^{g^*} & Z_{k + c}(X')\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\n\\CH_k(Y) \\ar[r]_{g^*} \\ar[d]_{f_*} & \\CH_{k + c}(Y') \\ar[d]^{f'_*} \\\\\n\\CH_k(X) \\ar[r]^{g^*} & \\CH_{k + c}(X')\n}\n}\n$$\nof cycle and chow groups commutes.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVL","source_file":"chow.tex","source_line":15339,"source_end_line":15360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15339-L15360","statement_sha256":"6d83a1191219b7a9f942e887e4bdf6cd2061f32e0c9ee5d0f466de92cab87a8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8367,"rank":8367,"depth":32,"x":2328.895,"y":698.938,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVM","tag":"0FVM","title":"Change of base scheme · Lemma 0FVM","summary":"In Situation [Tag 0FVG] let X → S be locally of finite type and let X' → S' be the base change by S' → S. Let L be an invertible O_X-module with base change L' on X'. Then the diagram xymatrix CH_k(X) ar[r]_g^* ar[d]_c_1(L) ∩ - & CH_k + c(X') ar[d]^c_1(L') ∩ - CH_k - 1(X) ar[r]^g^* & CH_k + c - 1(X') of chow groups commutes.","statement_latex":"In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally\nof finite type and let $X' \\to S'$ be the base change by $S' \\to S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module with\nbase change $\\mathcal{L}'$ on $X'$. Then the\ndiagram\n$$\n\\xymatrix{\n\\CH_k(X) \\ar[r]_{g^*} \\ar[d]_{c_1(\\mathcal{L}) \\cap -} &\n\\CH_{k + c}(X') \\ar[d]^{c_1(\\mathcal{L}') \\cap -} \\\\\n\\CH_{k - 1}(X) \\ar[r]^{g^*} & \\CH_{k + c - 1}(X')\n}\n$$\nof chow groups commutes.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVM","source_file":"chow.tex","source_line":15381,"source_end_line":15396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15381-L15396","statement_sha256":"9127921621cc4a88ec53a5fcbfcef6344612d8058d4bfd4ffb687bc9bddecbb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8368,"rank":8368,"depth":33,"x":2576.284,"y":580.201,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVN","tag":"0FVN","title":"Change of base scheme · Lemma 0FVN","summary":"In Situation [Tag 0FVG] let X → S be locally of finite type and let X' → S' be the base change by S' → S. Let E be a finite locally free O_X-module of rank r with base change E' on X'. Then the diagram xymatrix CH_k(X) ar[r]_g^* ar[d]_c_i(E) ∩ - & CH_k + c(X') ar[d]^c_i(E') ∩ - CH_k - i(X) ar[r]^g^* & CH_k + c - i(X') of chow groups commutes for all i.","statement_latex":"In Situation \\ref{situation-setup-base-change} let $X \\to S$ be locally\nof finite type and let $X' \\to S'$ be the base change by $S' \\to S$.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module of\nrank $r$ with base change $\\mathcal{E}'$ on $X'$. Then the\ndiagram\n$$\n\\xymatrix{\n\\CH_k(X) \\ar[r]_{g^*} \\ar[d]_{c_i(\\mathcal{E}) \\cap -} &\n\\CH_{k + c}(X') \\ar[d]^{c_i(\\mathcal{E}') \\cap -} \\\\\n\\CH_{k - i}(X) \\ar[r]^{g^*} & \\CH_{k + c - i}(X')\n}\n$$\nof chow groups commutes for all $i$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVN","source_file":"chow.tex","source_line":15410,"source_end_line":15425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15410-L15425","statement_sha256":"294e4184cdf1f4c33c2448092828cc51a9204a30eb16bd15f7053b9d3d945bf3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8369,"rank":8369,"depth":48,"x":2489.266,"y":808.345,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVP","tag":"0FVP","title":"Change of base scheme · Lemma 0FVP","summary":"Let (S, δ), (S', δ'), (S\", δ\") be as in Situation [Tag 02QL]. Let g : S' → S and g' : S\" → S' be flat morphisms of schemes and let c, c' ∈ Z be integers such that S, δ, S', δ', g, c and S', δ', S\", g', c' are as in Situation [Tag 0FVG]. Let X → S be locally of finite type and denote X' → S' and X\" → S\" the base changes by S' → S and S\" → S. Then • S, δ, S\", δ\", g ∘ g', c + c' is as in Situation [Tag 0FVG], • the maps g^* : Z_k(X) → Z_k + c(X') and (g')^* : Z_k + c(X') →…","statement_latex":"Let $(S, \\delta)$, $(S', \\delta')$, $(S'', \\delta'')$ be as in\nSituation \\ref{situation-setup}. Let $g : S' \\to S$ and $g' : S'' \\to S'$\nbe flat morphisms of schemes and let $c, c' \\in \\mathbf{Z}$\nbe integers such that $S, \\delta, S', \\delta', g, c$ and\n$S', \\delta', S'', g', c'$ are as in\nSituation \\ref{situation-setup-base-change}.\nLet $X \\to S$ be locally of finite type and denote $X' \\to S'$\nand $X'' \\to S''$ the base changes by $S' \\to S$ and $S'' \\to S$.\nThen\n\\begin{enumerate}\n\\item $S, \\delta, S'', \\delta'', g \\circ g', c + c'$ is as in\nSituation \\ref{situation-setup-base-change},\n\\item the maps $g^* : Z_k(X) \\to Z_{k + c}(X')$ and\n$(g')^* : Z_{k + c}(X') \\to Z_{k + c + c'}(X'')$ of\ncompose to give the map $(g \\circ g')^* : Z_k(X) \\to Z_{k + c + c'}(X'')$, and\n\\item the maps $g^* : \\CH_k(X) \\to \\CH_{k + c}(X')$ and\n$(g')^* : \\CH_{k + c}(X') \\to \\CH_{k + c + c'}(X'')$ of\nLemma \\ref{lemma-pullback-base-change}\ncompose to give the map $(g \\circ g')^* : \\CH_k(X) \\to \\CH_{k + c + c'}(X'')$\nof Lemma \\ref{lemma-pullback-base-change}.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVP","source_file":"chow.tex","source_line":15437,"source_end_line":15460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15437-L15460","statement_sha256":"67d4872c536da0b38ce6cd6427ffe5cf9d3dc6f2fc2b729e42614af9a906f60a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8370,"rank":8370,"depth":50,"x":2369.853,"y":590.534,"cluster":"divisors-intersection-theory"},{"id":"stacks:0FVQ","tag":"0FVQ","title":"Change of base scheme · Lemma 0FVQ","summary":"In Situation [Tag 0FVG] assume c = 0 and assume that S' = lim_i ∈ I S_i is a filtered limit of schemes S_i affine over S such that • with δ_i equal to S_i → S xrightarrowδ Z the pair (S_i, δ_i) is as in Situation [Tag 02QL], • S_i, δ_i, S, δ, S → S_i, c = 0 is as in Situation [Tag 0FVG], • S_i, δ_i, S_i', δ_i', S_i → S_i', c = 0 for i ≥ i' is as in Situation [Tag 0FVG]. Then for a quasi-compact scheme X of finite type over S with base change X' and X_i by S' → S and S_i →…","statement_latex":"In Situation \\ref{situation-setup-base-change} assume $c = 0$\nand assume that $S' = \\lim_{i \\in I} S_i$ is a filtered limit\nof schemes $S_i$ affine over $S$ such that\n\\begin{enumerate}\n\\item with $\\delta_i$ equal to $S_i \\to S \\xrightarrow{\\delta} \\mathbf{Z}$\nthe pair $(S_i, \\delta_i)$ is as in Situation \\ref{situation-setup},\n\\item $S_i, \\delta_i, S, \\delta, S \\to S_i, c = 0$ is as in\nSituation \\ref{situation-setup-base-change},\n\\item $S_i, \\delta_i, S_{i'}, \\delta_{i'}, S_i \\to S_{i'}, c = 0$ \nfor $i \\geq i'$ is as in Situation \\ref{situation-setup-base-change}.\n\\end{enumerate}\nThen for a quasi-compact scheme $X$ of finite type over $S$\nwith base change $X'$ and $X_i$ by $S' \\to S$ and $S_i \\to S$ we have\n$Z_k(X') = \\colim Z_k(X_i)$ and $\\CH_k(X') = \\colim \\CH_k(X_i)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVQ","source_file":"chow.tex","source_line":15482,"source_end_line":15498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15482-L15498","statement_sha256":"3abf8248c9a35286800b264f48b59c8ef13e2424655b98b2aee35d04a1856055","origin":"The Stacks Project","memory_eligible":false,"source_rank":8371,"rank":8371,"depth":51,"x":2633.311,"y":683.474,"cluster":"divisors-intersection-theory"},{"id":"stacks:02P6","tag":"02P6","title":"Appendix A: Alternative approach to key lemma · Definition 02P6","summary":"Let R be a local ring with maximal ideal m and residue field kappa. Let M be a finite length R-module. Say l = length_R(M). • Given elements x_1, …, x_r ∈ M we denote langle x_1, …, x_r rangle = Rx_1 + … + Rx_r the R-submodule of M generated by x_1, …, x_r. • We will say an l-tuple of elements (e_1, …, e_l) of M is admissible if m e_i ⊂ langle e_1, …, e_i - 1 rangle for i = 1, …, l. • A symbol [e_1, …, e_l] will mean (e_1, …, e_l) is an admissible l-tuple. • An admissible…","statement_latex":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and\nresidue field $\\kappa$. Let $M$ be a finite length $R$-module.\nSay $l = \\text{length}_R(M)$.\n\\begin{enumerate}\n\\item Given elements $x_1, \\ldots, x_r \\in M$ we denote\n$\\langle x_1, \\ldots, x_r \\rangle = Rx_1 + \\ldots + Rx_r$ the\n$R$-submodule of $M$ generated by $x_1, \\ldots, x_r$.\n\\item We will say an $l$-tuple of elements\n$(e_1, \\ldots, e_l)$ of $M$ is {\\it admissible} if\n$\\mathfrak m e_i \\subset \\langle e_1, \\ldots, e_{i - 1} \\rangle$\nfor $i = 1, \\ldots, l$.\n\\item A {\\it symbol} $[e_1, \\ldots, e_l]$ will mean\n$(e_1, \\ldots, e_l)$ is an admissible $l$-tuple.\n\\item An {\\it admissible relation} between symbols is one of the following:\n\\begin{enumerate}\n\\item if $(e_1, \\ldots, e_l)$ is an admissible sequence and\nfor some $1 \\leq a \\leq l$ we have\n$e_a \\in \\langle e_1, \\ldots, e_{a - 1}\\rangle$, then\n$[e_1, \\ldots, e_l] = 0$,\n\\item if $(e_1, \\ldots, e_l)$ is an admissible sequence and\nfor some $1 \\leq a \\leq l$ we have $e_a = \\lambda e'_a + x$\nwith $\\lambda \\in R^*$, and\n$x \\in \\langle e_1, \\ldots, e_{a - 1}\\rangle$, then\n$$\n[e_1, \\ldots, e_l] =\n\\overline{\\lambda} [e_1, \\ldots, e_{a - 1}, e'_a, e_{a + 1}, \\ldots, e_l]\n$$\nwhere $\\overline{\\lambda} \\in \\kappa^*$ is the image of $\\lambda$ in\nthe residue field, and\n\\item if $(e_1, \\ldots, e_l)$ is an admissible sequence and\n$\\mathfrak m e_a \\subset \\langle e_1, \\ldots, e_{a - 2}\\rangle$ then\n$$\n[e_1, \\ldots, e_l] =\n- [e_1, \\ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \\ldots, e_l].\n$$\n\\end{enumerate}\n\\item\nWe define the {\\it determinant of the finite length $R$-module $M$} to be\n$$\n\\det\\nolimits_\\kappa(M) =\n\\left\\{\n\\frac{\\kappa\\text{-vector space generated by symbols}}\n{\\kappa\\text{-linear combinations of admissible relations}}\n\\right\\}\n$$\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02P6","source_file":"chow.tex","source_line":15703,"source_end_line":15751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15703-L15751","statement_sha256":"e70ea86ee13914e617b21830c61cf8ee8fc792d8a3cb0e0ca8b487f9aa4412ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":8372,"rank":8372,"depth":0,"x":2364.049,"y":764.51,"cluster":"divisors-intersection-theory"},{"id":"stacks:02P7","tag":"02P7","title":"Appendix A: Alternative approach to key lemma · Lemma 02P7","summary":"With notations as above we have dim_kappa(det_kappa(M)) ≤ 1.","statement_latex":"With notations as above we have $\\dim_\\kappa(\\det_\\kappa(M)) \\leq 1$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02P7","source_file":"chow.tex","source_line":15760,"source_end_line":15763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15760-L15763","statement_sha256":"544e118488d73ad017a4a1aa1b1f74a2747a57ded03a879abe2c28a3fecd4cf9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8373,"rank":8373,"depth":0,"x":2497.556,"y":551.771,"cluster":"divisors-intersection-theory"},{"id":"stacks:02P8","tag":"02P8","title":"Appendix A: Alternative approach to key lemma · Lemma 02P8","summary":"Let R be a local ring with maximal ideal m and residue field kappa. Let M be a finite length R-module which is annihilated by m. Let l = dim_kappa(M). Then the map det_kappa(M) → wedge^l_kappa(M), [e_1, …, e_l] ↦ e_1 wedge … wedge e_l is an isomorphism.","statement_latex":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and\nresidue field $\\kappa$. Let $M$ be a finite length $R$-module\nwhich is annihilated by $\\mathfrak m$. Let $l = \\dim_\\kappa(M)$.\nThen the map\n$$\n\\det\\nolimits_\\kappa(M) \\longrightarrow \\wedge^l_\\kappa(M),\n\\quad\n[e_1, \\ldots, e_l] \\longmapsto e_1 \\wedge \\ldots \\wedge e_l\n$$\nis an isomorphism.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02P8","source_file":"chow.tex","source_line":15826,"source_end_line":15838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15826-L15838","statement_sha256":"9fc09a66b342825b83ca44c29d55f6ee51ca88754e5bbcbc3110131fc642528c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8374,"rank":8374,"depth":1,"x":2570.257,"y":784.614,"cluster":"divisors-intersection-theory"},{"id":"stacks:02P9","tag":"02P9","title":"Appendix A: Alternative approach to key lemma · Lemma 02P9","summary":"Let R be a local ring with maximal ideal m and residue field kappa. Let M be a finite length R-module. The determinant det_kappa(M) defined above is a kappa-vector space of dimension 1. It is generated by the symbol [f_1, …, f_l] for any admissible sequence such that langle f_1, … f_l rangle = M.","statement_latex":"Let $R$ be a local ring with maximal ideal $\\mathfrak m$ and\nresidue field $\\kappa$. Let $M$ be a finite length $R$-module.\nThe determinant $\\det_\\kappa(M)$ defined above is a $\\kappa$-vector\nspace of dimension $1$. It is generated by the symbol\n$[f_1, \\ldots, f_l]$ for any admissible sequence such\nthat $\\langle f_1, \\ldots f_l \\rangle = M$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02P9","source_file":"chow.tex","source_line":15849,"source_end_line":15857,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L15849-L15857","statement_sha256":"651e33bf6748c725ab73e5eab394f0a5133465385aa32fc32f1d2714d13bc7b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8375,"rank":8375,"depth":2,"x":2329.178,"y":654.049,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PA","tag":"02PA","title":"Appendix A: Alternative approach to key lemma · Lemma 02PA","summary":"Let (R, m, kappa) be a local ring. For every short exact sequence 0 → K → L → M → 0 of finite length R-modules there exists a canonical isomorphism γ_K → L → M : det_kappa(K) ⊗_kappa det_kappa(M) → det_kappa(L) defined by the rule on nonzero symbols [e_1, …, e_k] ⊗ [overlinef_1, …, overlinef_m] → [e_1, …, e_k, f_1, …, f_m] with the following properties: • For every isomorphism of short exact sequences, i.e., for every commutative diagram xymatrix 0 ar[r] & K ar[r] ar[d]^u…","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring.\nFor every short exact sequence\n$$\n0 \\to K \\to L \\to M \\to 0\n$$\nof finite length $R$-modules there exists a canonical isomorphism\n$$\n\\gamma_{K \\to L \\to M} :\n\\det\\nolimits_\\kappa(K) \\otimes_\\kappa \\det\\nolimits_\\kappa(M)\n\\longrightarrow\n\\det\\nolimits_\\kappa(L)\n$$\ndefined by the rule on nonzero symbols\n$$\n[e_1, \\ldots, e_k]\n\\otimes\n[\\overline{f}_1, \\ldots, \\overline{f}_m]\n\\longrightarrow\n[e_1, \\ldots, e_k, f_1, \\ldots, f_m]\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item For every isomorphism of short exact sequences, i.e., for\nevery commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\nK \\ar[r] \\ar[d]^u &\nL \\ar[r] \\ar[d]^v &\nM \\ar[r] \\ar[d]^w &\n0 \\\\\n0 \\ar[r] &\nK' \\ar[r] &\nL' \\ar[r] &\nM' \\ar[r] &\n0\n}\n$$\nwith short exact rows and isomorphisms $u, v, w$ we have\n$$\n\\gamma_{K' \\to L' \\to M'} \\circ\n(\\det\\nolimits_\\kappa(u) \\otimes \\det\\nolimits_\\kappa(w))\n=\n\\det\\nolimits_\\kappa(v) \\circ\n\\gamma_{K \\to L \\to M},\n$$\n\\item for every commutative square of finite length $R$-modules\nwith exact rows and columns\n$$\n\\xymatrix{\n& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d] & \\\\\n0 \\ar[r] & A \\ar[r] \\ar[d] & B \\ar[r] \\ar[d] & C \\ar[r] \\ar[d] & 0 \\\\\n0 \\ar[r] & D \\ar[r] \\ar[d] & E \\ar[r] \\ar[d] & F \\ar[r] \\ar[d] & 0 \\\\\n0 \\ar[r] & G \\ar[r] \\ar[d] & H \\ar[r] \\ar[d] & I \\ar[r] \\ar[d] & 0 \\\\\n& 0  & 0  & 0  &\n}\n$$\nthe following diagram is commutative\n$$\n\\xymatrix{\n\\det\\nolimits_\\kappa(A) \\otimes\n\\det\\nolimits_\\kappa(C) \\otimes\n\\det\\nolimits_\\kappa(G) \\otimes\n\\det\\nolimits_\\kappa(I)\n\\ar[dd]_{\\epsilon}\n\\ar[rrr]_-{\\gamma_{A \\to B \\to C} \\otimes \\gamma_{G \\to H \\to I}}\n& & &\n\\det\\nolimits_\\kappa(B) \\otimes\n\\det\\nolimits_\\kappa(H)\n\\ar[d]^{\\gamma_{B \\to E \\to H}}\n\\\\\n& & & \\det\\nolimits_\\kappa(E)\n\\\\\n\\det\\nolimits_\\kappa(A) \\otimes\n\\det\\nolimits_\\kappa(G) \\otimes\n\\det\\nolimits_\\kappa(C) \\otimes\n\\det\\nolimits_\\kappa(I)\n\\ar[rrr]^-{\\gamma_{A \\to D \\to G} \\otimes \\gamma_{C \\to F \\to I}}\n& & &\n\\det\\nolimits_\\kappa(D) \\otimes\n\\det\\nolimits_\\kappa(F)\n\\ar[u]_{\\gamma_{D \\to E \\to F}}\n}\n$$\nwhere $\\epsilon$ is the switch of the factors in the tensor product\ntimes $(-1)^{cg}$ with $c = \\text{length}_R(C)$ and $g = \\text{length}_R(G)$,\nand\n\\item the map $\\gamma_{K \\to L \\to M}$ agrees with the usual isomorphism\nif $0 \\to K \\to L \\to M \\to 0$ is actually a short exact sequence\nof $\\kappa$-vector spaces.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PA","source_file":"chow.tex","source_line":16202,"source_end_line":16295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16202-L16295","statement_sha256":"3cf850709085d8d227a209881fd7269cf3ad87ad9cb9fbba7c99f5791f0592e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8376,"rank":8376,"depth":3,"x":2612.205,"y":613.494,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PB","tag":"02PB","title":"Appendix A: Alternative approach to key lemma · Lemma 02PB","summary":"Let (R, m, kappa) be any local ring. The functor det_kappa : ( finite length R-modules with isomorphisms ) → ( 1-dimensional kappa-vector spaces with isomorphisms ) endowed with the maps γ_K → L → M is characterized by the following properties • its restriction to the subcategory of modules annihilated by m is isomorphic to the usual determinant functor (see Lemma [Tag 02P8]), and • (1), (2) and (3) of Lemma [Tag 02PA] hold.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be any local ring.\nThe functor\n$$\n\\det\\nolimits_\\kappa :\n\\left\\{\n\\begin{matrix}\n\\text{finite length }R\\text{-modules} \\\\\n\\text{with isomorphisms}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\n1\\text{-dimensional }\\kappa\\text{-vector spaces} \\\\\n\\text{with isomorphisms}\n\\end{matrix}\n\\right\\}\n$$\nendowed with the maps $\\gamma_{K \\to L \\to M}$ is characterized by\nthe following properties\n\\begin{enumerate}\n\\item its restriction to the subcategory of modules annihilated\nby $\\mathfrak m$ is isomorphic to the usual determinant functor\n(see Lemma \\ref{lemma-compare-det}), and\n\\item (1), (2) and (3) of Lemma \\ref{lemma-det-exact-sequences}\nhold.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PB","source_file":"chow.tex","source_line":16398,"source_end_line":16427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16398-L16427","statement_sha256":"0e9a70b3760b8037dcd5921a5bb365b22db74cd50e88f5f36d8af2dc40df43c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8377,"rank":8377,"depth":4,"x":2435.955,"y":804.172,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PC","tag":"02PC","title":"Appendix A: Alternative approach to key lemma · Lemma 02PC","summary":"Let (R', m') → (R, m) be a local ring homomorphism which induces an isomorphism on residue fields kappa. Then for every finite length R-module the restriction M_R' is a finite length R'-module and there is a canonical isomorphism det_R, kappa(M) → det_R', kappa(M_R') This isomorphism is functorial in M and compatible with the isomorphisms γ_K → L → M of Lemma [Tag 02PA] defined for det_R, kappa and det_R', kappa.","statement_latex":"Let $(R', \\mathfrak m') \\to (R, \\mathfrak m)$ be a local ring\nhomomorphism which induces an isomorphism on residue fields $\\kappa$.\nThen for every finite length $R$-module the restriction $M_{R'}$\nis a finite length $R'$-module and there is a canonical isomorphism\n$$\n\\det\\nolimits_{R, \\kappa}(M)\n\\longrightarrow\n\\det\\nolimits_{R', \\kappa}(M_{R'})\n$$\nThis isomorphism is functorial in $M$ and compatible with the\nisomorphisms $\\gamma_{K \\to L \\to M}$ of Lemma \\ref{lemma-det-exact-sequences}\ndefined for $\\det_{R, \\kappa}$ and $\\det_{R', \\kappa}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PC","source_file":"chow.tex","source_line":16433,"source_end_line":16447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16433-L16447","statement_sha256":"ac71fc9125c1a0d4c2bfaa2cc0c2eaba871c7c882137741a4d477210caf8107e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8378,"rank":8378,"depth":5,"x":2412.56,"y":563.337,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PD","tag":"02PD","title":"Appendix A: Alternative approach to key lemma · Lemma 02PD","summary":"Let R be a local ring with residue field kappa. Let u ∈ R^* be a unit. Let M be a module of finite length over R. Denote u_M : M → M the map multiplication by u. Then det_kappa(u_M) : det_kappa(M) → det_kappa(M) is multiplication by overlineu^l where l = length_R(M) and overlineu ∈ kappa^* is the image of u.","statement_latex":"Let $R$ be a local ring with residue field $\\kappa$.\nLet $u \\in R^*$ be a unit.\nLet $M$ be a module of finite length over $R$.\nDenote $u_M : M \\to M$ the map multiplication by $u$.\nThen\n$$\n\\det\\nolimits_\\kappa(u_M) :\n\\det\\nolimits_\\kappa(M)\n\\longrightarrow\n\\det\\nolimits_\\kappa(M)\n$$\nis multiplication by $\\overline{u}^l$ where $l = \\text{length}_R(M)$\nand $\\overline{u} \\in \\kappa^*$ is the image of $u$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PD","source_file":"chow.tex","source_line":16504,"source_end_line":16519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16504-L16519","statement_sha256":"9aa2b8f9a5bd63bd033dfbc1e90e8deeca445a811564a5fdb3ca880c9e3e4b67","origin":"The Stacks Project","memory_eligible":false,"source_rank":8379,"rank":8379,"depth":4,"x":2623.679,"y":727.802,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PJ","tag":"02PJ","title":"Appendix A: Alternative approach to key lemma · Definition 02PJ","summary":"Let R be a local ring with residue field kappa. Let (M, φ, ψ) be a (2, 1)-periodic complex over R. Assume that M has finite length and that (M, φ, ψ) is exact. The determinant of (M, φ, ψ) is the element det_kappa(M, φ, ψ) ∈ kappa^* such that the composition det_kappa(M) xrightarrowγ_ψ ∘ σ ∘ γ_φ^-1 det_kappa(M) is multiplication by (-1)^length_R(I_φ)length_R(I_ψ) det_kappa(M, φ, ψ).","statement_latex":"Let $R$ be a local ring with residue field $\\kappa$.\nLet $(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$.\nAssume that $M$ has finite length and that $(M, \\varphi, \\psi)$ is\nexact. The {\\it determinant of $(M, \\varphi, \\psi)$} is\nthe element\n$$\n\\det\\nolimits_\\kappa(M, \\varphi, \\psi) \\in \\kappa^*\n$$\nsuch that the composition\n$$\n\\det\\nolimits_\\kappa(M)\n\\xrightarrow{\\gamma_\\psi \\circ \\sigma \\circ \\gamma_\\varphi^{-1}}\n\\det\\nolimits_\\kappa(M)\n$$\nis multiplication by\n$(-1)^{\\text{length}_R(I_\\varphi)\\text{length}_R(I_\\psi)}\n\\det\\nolimits_\\kappa(M, \\varphi, \\psi)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PJ","source_file":"chow.tex","source_line":16610,"source_end_line":16629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16610-L16629","statement_sha256":"20014b254b74cd6ce9289131409aa5974d4031f930c7b474fb1638f22015c4f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8380,"rank":8380,"depth":0,"x":2335.479,"y":726.321,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PL","tag":"02PL","title":"Appendix A: Alternative approach to key lemma · Lemma 02PL","summary":"Let R be a local ring with residue field kappa. Let (M, φ, ψ) be a (2, 1)-periodic complex over R. Assume that M has finite length and that (M, φ, ψ) is exact. Then det_kappa(M, φ, ψ) det_kappa(M, ψ, φ) = 1.","statement_latex":"Let $R$ be a local ring with residue field $\\kappa$.\nLet $(M, \\varphi, \\psi)$ be a $(2, 1)$-periodic complex over $R$.\nAssume that $M$ has finite length and that $(M, \\varphi, \\psi)$ is\nexact. Then\n$$\n\\det\\nolimits_\\kappa(M, \\varphi, \\psi)\n\\det\\nolimits_\\kappa(M, \\psi, \\varphi)\n= 1.\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PL","source_file":"chow.tex","source_line":16658,"source_end_line":16669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16658-L16669","statement_sha256":"46cf92e5893348cf5082b0f6b84a2d44141e5e5b15308661bf90c14b2ceb7eb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8381,"rank":8381,"depth":0,"x":2549.381,"y":563.732,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PM","tag":"02PM","title":"Appendix A: Alternative approach to key lemma · Lemma 02PM","summary":"Let R be a local ring with residue field kappa. Let (M, φ, φ) be a (2, 1)-periodic complex over R. Assume that M has finite length and that (M, φ, φ) is exact. Then length_R(M) = 2 length_R(Im(φ)) and det_kappa(M, φ, φ) = (-1)^length_R(Im(φ)) = (-1)^frac12length_R(M)","statement_latex":"Let $R$ be a local ring with residue field $\\kappa$.\nLet $(M, \\varphi, \\varphi)$ be a $(2, 1)$-periodic complex over $R$.\nAssume that $M$ has finite length and that $(M, \\varphi, \\varphi)$ is\nexact. Then $\\text{length}_R(M) = 2 \\text{length}_R(\\Im(\\varphi))$\nand\n$$\n\\det\\nolimits_\\kappa(M, \\varphi, \\varphi)\n=\n(-1)^{\\text{length}_R(\\Im(\\varphi))}\n=\n(-1)^{\\frac{1}{2}\\text{length}_R(M)}\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PM","source_file":"chow.tex","source_line":16675,"source_end_line":16689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16675-L16689","statement_sha256":"86230164def87bf8240fd72e9c095fe25a210f3cadebcf789d9ab478c16b06b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8382,"rank":8382,"depth":0,"x":2522.378,"y":805.218,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PN","tag":"02PN","title":"Appendix A: Alternative approach to key lemma · Lemma 02PN","summary":"Let R be a local ring with residue field kappa. Let M be a finite length R-module. • if φ : M → M is an isomorphism then det_kappa(M, φ, 0) = det_kappa(φ). • if ψ : M → M is an isomorphism then det_kappa(M, 0, ψ) = det_kappa(ψ)^-1.","statement_latex":"Let $R$ be a local ring with residue field $\\kappa$.\nLet $M$ be a finite length $R$-module.\n\\begin{enumerate}\n\\item if $\\varphi : M \\to M$ is an isomorphism then\n$\\det_\\kappa(M, \\varphi, 0) = \\det_\\kappa(\\varphi)$.\n\\item if $\\psi : M \\to M$ is an isomorphism then\n$\\det_\\kappa(M, 0, \\psi) = \\det_\\kappa(\\psi)^{-1}$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PN","source_file":"chow.tex","source_line":16695,"source_end_line":16705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16695-L16705","statement_sha256":"b53af671f714bd202099b5ff160ed3183d11f1872c731227f5329982cf8ccf52","origin":"The Stacks Project","memory_eligible":false,"source_rank":8383,"rank":8383,"depth":1,"x":2347.934,"y":611.65,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PO","tag":"02PO","title":"Appendix A: Alternative approach to key lemma · Lemma 02PO","summary":"Let R be a local ring with residue field kappa. Suppose that we have a short exact sequence of (2, 1)-periodic complexes 0 → (M_1, φ_1, ψ_1) → (M_2, φ_2, ψ_2) → (M_3, φ_3, ψ_3) → 0 with all M_i of finite length, and each (M_1, φ_1, ψ_1) exact. Then det_kappa(M_2, φ_2, ψ_2) = det_kappa(M_1, φ_1, ψ_1) det_kappa(M_3, φ_3, ψ_3). in kappa^*.","statement_latex":"Let $R$ be a local ring with residue field $\\kappa$.\nSuppose that we have a short exact sequence of\n$(2, 1)$-periodic complexes\n$$\n0 \\to (M_1, \\varphi_1, \\psi_1)\n\\to (M_2, \\varphi_2, \\psi_2)\n\\to (M_3, \\varphi_3, \\psi_3)\n\\to 0\n$$\nwith all $M_i$ of finite length, and each $(M_1, \\varphi_1, \\psi_1)$ exact.\nThen\n$$\n\\det\\nolimits_\\kappa(M_2, \\varphi_2, \\psi_2) =\n\\det\\nolimits_\\kappa(M_1, \\varphi_1, \\psi_1)\n\\det\\nolimits_\\kappa(M_3, \\varphi_3, \\psi_3).\n$$\nin $\\kappa^*$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PO","source_file":"chow.tex","source_line":16729,"source_end_line":16748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16729-L16748","statement_sha256":"b51c75e969f176cd0bf3fd1cfad81fc81f5f40a0f357bed4d539f71686210f8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8384,"rank":8384,"depth":4,"x":2632.489,"y":655.44,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PU","tag":"02PU","title":"Appendix A: Alternative approach to key lemma · Lemma 02PU","summary":"Let R be a local ring with residue field kappa. Let M be a finite length R-module. Let α, β, γ be endomorphisms of M. Assume that • I_α = K_βγ, and similarly for any permutation of α, β, γ, • K_α = I_βγ, and similarly for any permutation of α, β, γ. Then • The triple (M, α, βγ) is an exact (2, 1)-periodic complex. • The triple (I_γ, α, β) is an exact (2, 1)-periodic complex. • The triple (M/K_β, α, γ) is an exact (2, 1)-periodic complex. • We have det_kappa(M, α, βγ) =…","statement_latex":"Let $R$ be a local ring with residue field $\\kappa$.\nLet $M$ be a finite length $R$-module.\nLet $\\alpha, \\beta, \\gamma$ be endomorphisms of $M$.\nAssume that\n\\begin{enumerate}\n\\item $I_\\alpha = K_{\\beta\\gamma}$, and similarly for any permutation\nof $\\alpha, \\beta, \\gamma$,\n\\item $K_\\alpha = I_{\\beta\\gamma}$, and similarly for any permutation\nof $\\alpha, \\beta, \\gamma$.\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item The triple $(M, \\alpha, \\beta\\gamma)$\nis an exact $(2, 1)$-periodic complex.\n\\item The triple $(I_\\gamma, \\alpha, \\beta)$\nis an exact $(2, 1)$-periodic complex.\n\\item The triple $(M/K_\\beta, \\alpha, \\gamma)$\nis an exact $(2, 1)$-periodic complex.\n\\item We have\n$$\n\\det\\nolimits_\\kappa(M, \\alpha, \\beta\\gamma)\n=\n\\det\\nolimits_\\kappa(I_\\gamma, \\alpha, \\beta)\n\\det\\nolimits_\\kappa(M/K_\\beta, \\alpha, \\gamma).\n$$\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PU","source_file":"chow.tex","source_line":16982,"source_end_line":17010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L16982-L17010","statement_sha256":"3b62b12ea090668d34aa818cc3a0a8966d4d943ad7bfb43269ce4f5de62c2767","origin":"The Stacks Project","memory_eligible":false,"source_rank":8385,"rank":8385,"depth":0,"x":2387.223,"y":784.731,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PV","tag":"02PV","title":"Appendix A: Alternative approach to key lemma · Lemma 02PV","summary":"Let R be a local ring with residue field kappa. Let α : (M, φ, ψ) → (M', φ', ψ') be a morphism of (2, 1)-periodic complexes over R. Assume • M, M' have finite length, • (M, φ, ψ), (M', φ', ψ') are exact, • the maps φ, ψ induce the zero map on K = Ker(α), and • the maps φ, ψ induce the zero map on Q = Coker(α). Denote N = α(M) ⊂ M'. We obtain two short exact sequences of (2, 1)-periodic complexes 0 → (N, φ', ψ') → (M', φ', ψ') → (Q, 0, 0) → 0 0 → (K, 0, 0) → (M, φ, ψ) →…","statement_latex":"Let $R$ be a local ring with residue field $\\kappa$.\nLet $\\alpha : (M, \\varphi, \\psi) \\to (M', \\varphi', \\psi')$\nbe a morphism of $(2, 1)$-periodic complexes over $R$.\nAssume\n\\begin{enumerate}\n\\item $M$, $M'$ have finite length,\n\\item $(M, \\varphi, \\psi)$, $(M', \\varphi', \\psi')$ are exact,\n\\item the maps $\\varphi$, $\\psi$ induce the zero map on\n$K = \\Ker(\\alpha)$, and\n\\item the maps $\\varphi$, $\\psi$ induce the zero map on\n$Q = \\Coker(\\alpha)$.\n\\end{enumerate}\nDenote $N = \\alpha(M) \\subset M'$. We obtain two short exact sequences\nof $(2, 1)$-periodic complexes\n$$\n\\begin{matrix}\n0 \\to (N, \\varphi', \\psi') \\to (M', \\varphi', \\psi') \\to (Q, 0, 0) \\to 0 \\\\\n0 \\to (K, 0, 0) \\to (M, \\varphi, \\psi) \\to (N, \\varphi', \\psi') \\to 0\n\\end{matrix}\n$$\nwhich induce two isomorphisms $\\alpha_i : Q \\to K$, $i = 0, 1$. Then\n$$\n\\det\\nolimits_\\kappa(M, \\varphi, \\psi)\n=\n\\det\\nolimits_\\kappa(\\alpha_0^{-1} \\circ \\alpha_1)\n\\det\\nolimits_\\kappa(M', \\varphi', \\psi')\n$$\nIn particular, if $\\alpha_0 = \\alpha_1$, then\n$\\det\\nolimits_\\kappa(M, \\varphi, \\psi) =\n\\det\\nolimits_\\kappa(M', \\varphi', \\psi')$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PV","source_file":"chow.tex","source_line":17107,"source_end_line":17139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17107-L17139","statement_sha256":"d9ef3816ae1fd71da5a061bab388f4fe533a5ffd0a445c5eb4f9bb4da96b827b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8386,"rank":8386,"depth":0,"x":2464.176,"y":550.01,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PX","tag":"02PX","title":"Appendix A: Alternative approach to key lemma · Lemma 02PX","summary":"Let A be a Noetherian local ring. Let M be a finite A-module of dimension 1. Assume φ, ψ : M → M are two injective A-module maps, and assume φ(ψ(M)) = ψ(φ(M)), for example if φ and ψ commute. Then length_A(M/φψ M) < ∞ and (M/φψ M, φ, ψ) is an exact (2, 1)-periodic complex.","statement_latex":"Let $A$ be a Noetherian local ring.\nLet $M$ be a finite $A$-module of dimension $1$.\nAssume $\\varphi, \\psi : M \\to M$ are two injective\n$A$-module maps, and assume $\\varphi(\\psi(M)) = \\psi(\\varphi(M))$,\nfor example if $\\varphi$ and $\\psi$ commute.\nThen $\\text{length}_A(M/\\varphi\\psi M) < \\infty$\nand $(M/\\varphi\\psi M, \\varphi, \\psi)$ is an exact\n$(2, 1)$-periodic complex.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PX","source_file":"chow.tex","source_line":17370,"source_end_line":17380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17370-L17380","statement_sha256":"e54f2dbf7c63c525e07d5eef230a58b083497eb639af65d117748ad4aa6cae86","origin":"The Stacks Project","memory_eligible":false,"source_rank":8387,"rank":8387,"depth":6,"x":2596.308,"y":766.953,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PY","tag":"02PY","title":"Appendix A: Alternative approach to key lemma · Lemma 02PY","summary":"Let A be a Noetherian local ring. Let a, b ∈ A. • If M is a finite A-module of dimension 1 such that a, b are nonzerodivisors on M, then length_A(M/abM) < ∞ and (M/abM, a, b) is a (2, 1)-periodic exact complex. • If a, b are nonzerodivisors and dim(A) = 1 then length_A(A/(ab)) < ∞ and (A/(ab), a, b) is a (2, 1)-periodic exact complex. In particular, in these cases det_kappa(M/abM, a, b) ∈ kappa^*, resp. det_kappa(A/(ab), a, b) ∈ kappa^* are defined.","statement_latex":"Let $A$ be a Noetherian local ring. Let $a, b \\in A$.\n\\begin{enumerate}\n\\item If $M$ is a finite $A$-module of dimension $1$\nsuch that $a, b$ are nonzerodivisors on $M$, then\n$\\text{length}_A(M/abM) < \\infty$ and\n$(M/abM, a, b)$ is a $(2, 1)$-periodic exact complex.\n\\item If $a, b$ are nonzerodivisors and $\\dim(A) = 1$\nthen $\\text{length}_A(A/(ab)) < \\infty$ and\n$(A/(ab), a, b)$ is a $(2, 1)$-periodic exact complex.\n\\end{enumerate}\nIn particular, in these cases\n$\\det_\\kappa(M/abM, a, b) \\in \\kappa^*$,\nresp.\\ $\\det_\\kappa(A/(ab), a, b) \\in \\kappa^*$\nare defined.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PY","source_file":"chow.tex","source_line":17397,"source_end_line":17413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17397-L17413","statement_sha256":"4952cdf55b55fdc804eac9518bfb9433a0cbfbcad4b54a413997d9015e935fe8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8388,"rank":8388,"depth":7,"x":2324.17,"y":681.88,"cluster":"divisors-intersection-theory"},{"id":"stacks:02PZ","tag":"02PZ","title":"Appendix A: Alternative approach to key lemma · Definition 02PZ","summary":"Let A be a Noetherian local ring with residue field kappa. Let a, b ∈ A. Let M be a finite A-module of dimension 1 such that a, b are nonzerodivisors on M. We define the symbol associated to M, a, b to be the element d_M(a, b) = det_kappa(M/abM, a, b) ∈ kappa^*","statement_latex":"Let $A$ be a Noetherian local ring with residue field $\\kappa$.\nLet $a, b \\in A$.\nLet $M$ be a finite $A$-module of dimension $1$\nsuch that $a, b$ are nonzerodivisors on $M$.\nWe define the {\\it symbol associated to $M, a, b$}\nto be the element\n$$\nd_M(a, b) =\n\\det\\nolimits_\\kappa(M/abM, a, b) \\in \\kappa^*\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02PZ","source_file":"chow.tex","source_line":17419,"source_end_line":17431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17419-L17431","statement_sha256":"020f9d418ec26dd01d0ffdae1f0a90e50b0170f8cdbf33199af874ad5f279f09","origin":"The Stacks Project","memory_eligible":false,"source_rank":8389,"rank":8389,"depth":0,"x":2593.497,"y":590.11,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q0","tag":"02Q0","title":"Appendix A: Alternative approach to key lemma · Lemma 02Q0","summary":"Let A be a Noetherian local ring. Let a, b, c ∈ A. Let M be a finite A-module with dim(Supp(M)) = 1. Assume a, b, c are nonzerodivisors on M. Then d_M(a, bc) = d_M(a, b) d_M(a, c) and d_M(a, b)d_M(b, a) = 1.","statement_latex":"Let $A$ be a Noetherian local ring.\nLet $a, b, c \\in A$. Let $M$ be a finite $A$-module\nwith $\\dim(\\text{Supp}(M)) = 1$. Assume $a, b, c$ are nonzerodivisors on $M$.\nThen\n$$\nd_M(a, bc) = d_M(a, b) d_M(a, c)\n$$\nand $d_M(a, b)d_M(b, a) = 1$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q0","source_file":"chow.tex","source_line":17433,"source_end_line":17443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17433-L17443","statement_sha256":"6faddc1613b1ee512c94085b3211b40feaaef03ab4f949a2e764d3c869c95fd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8390,"rank":8390,"depth":1,"x":2468.585,"y":810.803,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q1","tag":"02Q1","title":"Appendix A: Alternative approach to key lemma · Definition 02Q1","summary":"Let A be a Noetherian local domain of dimension 1 with residue field kappa. Let K be the fraction field of A. We define the tame symbol of A to be the map K^* × K^* → kappa^*, (x, y) ↦ d_A(x, y) where d_A(x, y) is extended to K^* × K^* by the multiplicativity of Lemma [Tag 02Q0].","statement_latex":"Let $A$ be a Noetherian local domain of dimension $1$\nwith residue field $\\kappa$.\nLet $K$ be the fraction field of $A$.\nWe define the {\\it tame symbol} of $A$ to be the map\n$$\nK^* \\times K^* \\longrightarrow \\kappa^*,\n\\quad\n(x, y) \\longmapsto d_A(x, y)\n$$\nwhere $d_A(x, y)$ is extended to $K^* \\times K^*$ by the multiplicativity of\nLemma \\ref{lemma-multiplicativity-symbol}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q1","source_file":"chow.tex","source_line":17452,"source_end_line":17465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17452-L17465","statement_sha256":"dd3171e1474b09217008cc51680b3fb365da18d6a009b250f7a315e6bb6d534a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8391,"rank":8391,"depth":2,"x":2383.142,"y":576.978,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AY9","tag":"0AY9","title":"Appendix A: Alternative approach to key lemma · Lemma 0AY9","summary":"Let A be a Noetherian local ring and M a finite A-module of dimension 1. Let a ∈ A be a nonzerodivisor on M. Then d_M(a, a) = (-1)^length_A(M/aM).","statement_latex":"Let $A$ be a Noetherian local ring and $M$ a finite $A$-module of\ndimension $1$. Let $a \\in A$ be a nonzerodivisor on $M$.\nThen $d_M(a, a) = (-1)^{\\text{length}_A(M/aM)}$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AY9","source_file":"chow.tex","source_line":17471,"source_end_line":17476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17471-L17476","statement_sha256":"2eb0eeac288d15264ede6a6fbf68c03f9f3ad8bf7f93998025eab52368cad198","origin":"The Stacks Project","memory_eligible":false,"source_rank":8392,"rank":8392,"depth":1,"x":2634.408,"y":701.026,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q2","tag":"02Q2","title":"Appendix A: Alternative approach to key lemma · Lemma 02Q2","summary":"Let A be a Noetherian local ring. Let M be a finite A-module of dimension 1. Let b ∈ A be a nonzerodivisor on M, and let u ∈ A^*. Then d_M(u, b) = u^length_A(M/bM) bmod m_A. In particular, if M = A, then d_A(u, b) = u^ord_A(b) bmod m_A.","statement_latex":"Let $A$ be a Noetherian local ring.\nLet $M$ be a finite $A$-module of dimension $1$.\nLet $b \\in A$ be a nonzerodivisor on $M$, and let $u \\in A^*$.\nThen\n$$\nd_M(u, b) = u^{\\text{length}_A(M/bM)} \\bmod \\mathfrak m_A.\n$$\nIn particular, if $M = A$, then\n$d_A(u, b) = u^{\\text{ord}_A(b)} \\bmod \\mathfrak m_A$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q2","source_file":"chow.tex","source_line":17482,"source_end_line":17493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17482-L17493","statement_sha256":"18a944e511d691cc2928fb81e3af85944b593e321dd421eae00d576331301ec9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8393,"rank":8393,"depth":5,"x":2349.116,"y":752.175,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q3","tag":"02Q3","title":"Appendix A: Alternative approach to key lemma · Lemma 02Q3","summary":"Let A be a Noetherian local ring. Let a, b ∈ A. Let 0 → M → M' → M\" → 0 be a short exact sequence of A-modules of dimension 1 such that a, b are nonzerodivisors on all three A-modules. Then d_M'(a, b) = d_M(a, b) d_M\"(a, b) in kappa^*.","statement_latex":"Let $A$ be a Noetherian local ring.\nLet $a, b \\in A$.\nLet\n$$\n0 \\to M \\to M' \\to M'' \\to 0\n$$\nbe a short exact sequence of $A$-modules of dimension $1$\nsuch that $a, b$ are nonzerodivisors on\nall three $A$-modules.\nThen\n$$\nd_{M'}(a, b) = d_M(a, b) d_{M''}(a, b)\n$$\nin $\\kappa^*$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q3","source_file":"chow.tex","source_line":17502,"source_end_line":17518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17502-L17518","statement_sha256":"fa693c24f7ecf2ae7b24b993520986956b053fcd6b3d4dba28c575c8bbc831ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":8394,"rank":8394,"depth":5,"x":2518.505,"y":552.397,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q4","tag":"02Q4","title":"Appendix A: Alternative approach to key lemma · Lemma 02Q4","summary":"Let A be a Noetherian local ring. Let α : M → M' be a homomorphism of finite A-modules of dimension 1. Let a, b ∈ A. Assume • a, b are nonzerodivisors on both M and M', and • dim(Ker(α)), dim(Coker(α)) ≤ 0. Then d_M(a, b) = d_M'(a, b).","statement_latex":"Let $A$ be a Noetherian local ring.\nLet $\\alpha : M \\to M'$ be a homomorphism of\nfinite $A$-modules of dimension $1$.\nLet $a, b \\in A$. Assume\n\\begin{enumerate}\n\\item $a$, $b$ are nonzerodivisors on both $M$ and $M'$, and\n\\item $\\dim(\\Ker(\\alpha)), \\dim(\\Coker(\\alpha)) \\leq 0$.\n\\end{enumerate}\nThen $d_M(a, b) = d_{M'}(a, b)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q4","source_file":"chow.tex","source_line":17532,"source_end_line":17543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17532-L17543","statement_sha256":"0c29c3e3f9003b9f87b858f482b5a1e4ab2b2631a87405413f0f0c5e9edf0c8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8395,"rank":8395,"depth":6,"x":2554.288,"y":796.045,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q5","tag":"02Q5","title":"Appendix A: Alternative approach to key lemma · Lemma 02Q5","summary":"Let A be a Noetherian local ring. Let M be a finite A-module with dim(Supp(M)) = 1. Let a, b ∈ A nonzerodivisors on M. Let q_1, …, q_t be the minimal primes in the support of M. Then d_M(a, b) = ∏_i = 1, …, t d_A/ q_i(a, b)^ length_A_ q_i(M_ q_i) as elements of kappa^*.","statement_latex":"Let $A$ be a Noetherian local ring.\nLet $M$ be a finite $A$-module with $\\dim(\\text{Supp}(M)) = 1$.\nLet $a, b \\in A$ nonzerodivisors on $M$.\nLet $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the minimal\nprimes in the support of $M$. Then\n$$\nd_M(a, b)\n=\n\\prod\\nolimits_{i = 1, \\ldots, t}\nd_{A/\\mathfrak q_i}(a, b)^{\n\\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i})}\n$$\nas elements of $\\kappa^*$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q5","source_file":"chow.tex","source_line":17576,"source_end_line":17591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17576-L17591","statement_sha256":"6274e12ea919b055f7d9d20a3d5bdf3fe1dcc5e866f29dd8a919a5735cc3a856","origin":"The Stacks Project","memory_eligible":false,"source_rank":8396,"rank":8396,"depth":7,"x":2331.769,"y":636.544,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q6","tag":"02Q6","title":"Appendix A: Alternative approach to key lemma · Lemma 02Q6","summary":"Let A be a discrete valuation ring with fraction field K. For nonzero x, y ∈ K we have d_A(x, y) = (-1)^ord_A(x)ord_A(y) fracx^ord_A(y)y^ord_A(x) bmod m_A, in other words the symbol is equal to the usual tame symbol.","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nFor nonzero $x, y \\in K$ we have\n$$\nd_A(x, y)\n=\n(-1)^{\\text{ord}_A(x)\\text{ord}_A(y)}\n\\frac{x^{\\text{ord}_A(y)}}{y^{\\text{ord}_A(x)}} \\bmod \\mathfrak m_A,\n$$\nin other words the symbol is equal to the usual tame symbol.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q6","source_file":"chow.tex","source_line":17623,"source_end_line":17634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17623-L17634","statement_sha256":"8d9430b4966d46dc5e126fdd3f97222bbfabd576d2c4bf2780a455c2bfd9c211","origin":"The Stacks Project","memory_eligible":false,"source_rank":8397,"rank":8397,"depth":0,"x":2624.378,"y":627.887,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q8","tag":"02Q8","title":"Appendix A: Alternative approach to key lemma · Lemma 02Q8","summary":"Let A be a Noetherian local ring. Let a, b ∈ A. Let M be a finite A-module of dimension 1 on which each of a, b, b - a are nonzerodivisors. Then d_M(a, b - a)d_M(b, b) = d_M(b, b - a)d_M(a, b) in kappa^*.","statement_latex":"Let $A$ be a Noetherian local ring.\nLet $a, b \\in A$.\nLet $M$ be a finite $A$-module of dimension $1$ on\nwhich each of $a$, $b$, $b - a$ are nonzerodivisors.\nThen\n$$\nd_M(a, b - a)d_M(b, b) = d_M(b, b - a)d_M(a, b)\n$$\nin $\\kappa^*$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q8","source_file":"chow.tex","source_line":17659,"source_end_line":17670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17659-L17670","statement_sha256":"727b20c0b088384dbbdb051973f4203c48dd6d39a99e9bc59e3ef41adb22d842","origin":"The Stacks Project","memory_eligible":false,"source_rank":8398,"rank":8398,"depth":43,"x":2415.388,"y":800.458,"cluster":"divisors-intersection-theory"},{"id":"stacks:02Q9","tag":"02Q9","title":"Appendix A: Alternative approach to key lemma · Lemma 02Q9","summary":"Let A be a Noetherian local domain of dimension 1 with fraction field K. For x ∈ K setminus (0, 1) we have d_A(x, 1 -x) = 1","statement_latex":"Let $A$ be a Noetherian local domain of dimension $1$\nwith fraction field $K$. For $x \\in K \\setminus \\{0, 1\\}$\nwe have\n$$\nd_A(x, 1 -x) = 1\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Q9","source_file":"chow.tex","source_line":17763,"source_end_line":17771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17763-L17771","statement_sha256":"6320e282a7fb865a5616573563ab8a8b269a05c08fe94d0149e2f9ab0bb06288","origin":"The Stacks Project","memory_eligible":false,"source_rank":8399,"rank":8399,"depth":44,"x":2430.731,"y":554.402,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QB","tag":"02QB","title":"Appendix A: Alternative approach to key lemma · Lemma 02QB","summary":"Let R be a Noetherian local ring. Let q ⊂ R be a prime with dim(R/ q) = 1. Let φ : M → N be a homomorphism of finite R-modules. Assume there exist x_1, …, x_l ∈ M and y_1, …, y_l ∈ M with the following properties • M = langle x_1, …, x_lrangle, • langle x_1, …, x_irangle / langle x_1, …, x_i - 1rangle ≅ R/ q for i = 1, …, l, • N = langle y_1, …, y_lrangle, and • langle y_1, …, y_irangle / langle y_1, …, y_i - 1rangle ≅ R/ q for i = 1, …, l. Then φ is injective if and only…","statement_latex":"Let $R$ be a Noetherian local ring.\nLet $\\mathfrak q \\subset R$ be a prime with $\\dim(R/\\mathfrak q) = 1$.\nLet $\\varphi : M \\to N$ be a homomorphism of finite $R$-modules.\nAssume there exist $x_1, \\ldots, x_l \\in M$ and $y_1, \\ldots, y_l \\in M$\nwith the following properties\n\\begin{enumerate}\n\\item $M = \\langle x_1, \\ldots, x_l\\rangle$,\n\\item $\\langle x_1, \\ldots, x_i\\rangle / \\langle x_1, \\ldots, x_{i - 1}\\rangle\n\\cong R/\\mathfrak q$ for $i = 1, \\ldots, l$,\n\\item $N = \\langle y_1, \\ldots, y_l\\rangle$, and\n\\item $\\langle y_1, \\ldots, y_i\\rangle / \\langle y_1, \\ldots, y_{i - 1}\\rangle\n\\cong R/\\mathfrak q$ for $i = 1, \\ldots, l$.\n\\end{enumerate}\nThen $\\varphi$ is injective if and only if $\\varphi_{\\mathfrak q}$ is an\nisomorphism, and in this case we have\n$$\n\\text{length}_R(\\Coker(\\varphi)) = \\text{ord}_{R/\\mathfrak q}(f)\n$$\nwhere $f \\in \\kappa(\\mathfrak q)$ is the element such that\n$$\n[\\varphi(x_1), \\ldots, \\varphi(x_l)] = f [y_1, \\ldots, y_l]\n$$\nin $\\det_{\\kappa(\\mathfrak q)}(N_{\\mathfrak q})$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QB","source_file":"chow.tex","source_line":17803,"source_end_line":17828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17803-L17828","statement_sha256":"add5c8c7a7cf6e4ba4a5b4b13bb7ae53f4a857bc2311bf52b84dc5cd51ad1fd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8400,"rank":8400,"depth":1,"x":2617.455,"y":744.715,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QC","tag":"02QC","title":"Appendix A: Alternative approach to key lemma · Lemma 02QC","summary":"Let R be a local Noetherian ring. Let q ⊂ R be a prime ideal. Let M be a finite R-module such that q is one of the minimal primes of the support of M. Then there exist x_1, …, x_l ∈ M such that • the support of M / langle x_1, …, x_lrangle does not contain q, and • langle x_1, …, x_irangle / langle x_1, …, x_i - 1rangle ≅ R/ q for i = 1, …, l. Moreover, in this case l = length_R_ q(M_ q).","statement_latex":"Let $R$ be a local Noetherian ring.\nLet $\\mathfrak q \\subset R$ be a prime ideal.\nLet $M$ be a finite $R$-module such that\n$\\mathfrak q$ is one of the minimal primes of the support of $M$.\nThen there exist $x_1, \\ldots, x_l \\in M$ such that\n\\begin{enumerate}\n\\item the support of $M / \\langle x_1, \\ldots, x_l\\rangle$ does not contain\n$\\mathfrak q$, and\n\\item $\\langle x_1, \\ldots, x_i\\rangle / \\langle x_1, \\ldots, x_{i - 1}\\rangle\n\\cong R/\\mathfrak q$ for $i = 1, \\ldots, l$.\n\\end{enumerate}\nMoreover, in this case $l = \\text{length}_{R_\\mathfrak q}(M_\\mathfrak q)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QC","source_file":"chow.tex","source_line":17963,"source_end_line":17977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L17963-L17977","statement_sha256":"d8af4e9ef9e9c1020577c05ba625126a368bb702659adc9a6d4c9e3880be0171","origin":"The Stacks Project","memory_eligible":false,"source_rank":8401,"rank":8401,"depth":6,"x":2326.465,"y":710.302,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QD","tag":"02QD","title":"Appendix A: Alternative approach to key lemma · Proposition 02QD","summary":"Let R be a local Noetherian ring with residue field kappa. Suppose that (M, φ, ψ) is a (2, 1)-periodic complex over R. Assume • M is a finite R-module, • the cohomology modules of (M, φ, ψ) are of finite length, and • dim(Supp(M)) = 1. Let q_i, i = 1, …, t be the minimal primes of the support of M. Then we have - e_R(M, φ, ψ) = ∑_i = 1, …, t ord_R/ q_i( det_kappa( q_i) (M_ q_i, φ_ q_i, ψ_ q_i) )","statement_latex":"Let $R$ be a local Noetherian ring with residue field $\\kappa$.\nSuppose that $(M, \\varphi, \\psi)$ is a $(2, 1)$-periodic\ncomplex over $R$. Assume\n\\begin{enumerate}\n\\item $M$ is a finite $R$-module,\n\\item the cohomology modules of $(M, \\varphi, \\psi)$ are of finite length, and\n\\item $\\dim(\\text{Supp}(M)) = 1$.\n\\end{enumerate}\nLet $\\mathfrak q_i$, $i = 1, \\ldots, t$ be the minimal\nprimes of the support of $M$. Then we have\\footnote{\nObviously we could get rid of the minus sign by redefining\n$\\det_\\kappa(M, \\varphi, \\psi)$ as the inverse of its\ncurrent value, see Definition \\ref{definition-periodic-determinant}.}\n$$\n- e_R(M, \\varphi, \\psi) =\n\\sum\\nolimits_{i = 1, \\ldots, t}\n\\text{ord}_{R/\\mathfrak q_i}\\left(\n\\det\\nolimits_{\\kappa(\\mathfrak q_i)}\n(M_{\\mathfrak q_i}, \\varphi_{\\mathfrak q_i}, \\psi_{\\mathfrak q_i})\n\\right)\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QD","source_file":"chow.tex","source_line":18023,"source_end_line":18046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L18023-L18046","statement_sha256":"67e9103f2e6c8c2091bc859054fb0da5295aa1a41eb50af6ea7f79c78c0e9964","origin":"The Stacks Project","memory_eligible":false,"source_rank":8402,"rank":8402,"depth":7,"x":2568.925,"y":570.44,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QE","tag":"02QE","title":"Appendix A: Alternative approach to key lemma · Lemma 02QE","summary":"Let R be a Noetherian local ring with maximal ideal m. Let M be a finite R-module, and let ψ : M → M be an R-module map. Assume that • Ker(ψ) and Coker(ψ) have finite length, and • dim(Supp(M)) ≤ 1. Write Supp(M) = ( m, q_1, …, q_t) and denote f_i ∈ kappa( q_i)^* the element such that det_kappa( q_i)(ψ_ q_i) : det_kappa( q_i)(M_ q_i) → det_kappa( q_i)(M_ q_i) is multiplication by f_i. Then we have length_R(Coker(ψ)) - length_R(Ker(ψ)) = ∑_i = 1, …, t ord_R/ q_i(f_i).","statement_latex":"Let $R$ be a Noetherian local ring with maximal ideal $\\mathfrak m$.\nLet $M$ be a finite $R$-module, and let $\\psi : M \\to M$ be an\n$R$-module map. Assume that\n\\begin{enumerate}\n\\item $\\Ker(\\psi)$ and $\\Coker(\\psi)$ have finite length, and\n\\item $\\dim(\\text{Supp}(M)) \\leq 1$.\n\\end{enumerate}\nWrite\n$\\text{Supp}(M) = \\{\\mathfrak m, \\mathfrak q_1, \\ldots, \\mathfrak q_t\\}$\nand denote $f_i \\in \\kappa(\\mathfrak q_i)^*$ the element such that\n$\\det_{\\kappa(\\mathfrak q_i)}(\\psi_{\\mathfrak q_i}) :\n\\det_{\\kappa(\\mathfrak q_i)}(M_{\\mathfrak q_i})\n\\to \\det_{\\kappa(\\mathfrak q_i)}(M_{\\mathfrak q_i})$\nis multiplication by $f_i$. Then\nwe have\n$$\n\\text{length}_R(\\Coker(\\psi))\n-\n\\text{length}_R(\\Ker(\\psi))\n=\n\\sum\\nolimits_{i = 1, \\ldots, t}\n\\text{ord}_{R/\\mathfrak q_i}(f_i).\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QE","source_file":"chow.tex","source_line":18277,"source_end_line":18302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L18277-L18302","statement_sha256":"77e1de75aa4f5e079ce753fa90291dfcc57b67cc1eedbc80c2396d26f9912768","origin":"The Stacks Project","memory_eligible":false,"source_rank":8403,"rank":8403,"depth":8,"x":2502.553,"y":811.362,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QJ","tag":"02QJ","title":"Key Lemma · Lemma 02QJ","summary":"When A is an excellent ring this is [Kato-Milnor-K]. Let A be a 2-dimensional Noetherian local domain with fraction field K. Let f, g ∈ K^*. Let q_1, …, q_t be the height 1 primes q of A such that either f or g is not an element of A^*_ q. Then we have ∑_i = 1, …, t ord_A/ q_i(d_A_ q_i(f, g)) = 0 We can also write this as ∑_height( q) = 1 ord_A/ q(d_A_ q(f, g)) = 0 since at any height one prime q of A where f, g ∈ A^*_ q we have d_A_ q(f, g) = 1 by Lemma [Tag 02Q2].","statement_latex":"\\begin{reference}\nWhen $A$ is an excellent ring this is \\cite[Proposition 1]{Kato-Milnor-K}.\n\\end{reference}\nLet $A$ be a $2$-dimensional Noetherian local domain with fraction field $K$.\nLet $f, g \\in K^*$.\nLet $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the height\n$1$ primes $\\mathfrak q$ of $A$ such that either $f$ or $g$ is not an\nelement of $A^*_{\\mathfrak q}$.\nThen we have\n$$\n\\sum\\nolimits_{i = 1, \\ldots, t}\n\\text{ord}_{A/\\mathfrak q_i}(d_{A_{\\mathfrak q_i}}(f, g))\n=\n0\n$$\nWe can also write this as\n$$\n\\sum\\nolimits_{\\text{height}(\\mathfrak q) = 1}\n\\text{ord}_{A/\\mathfrak q}(d_{A_{\\mathfrak q}}(f, g))\n=\n0\n$$\nsince at any height one prime $\\mathfrak q$\nof $A$ where $f, g \\in A^*_{\\mathfrak q}$\nwe have $d_{A_{\\mathfrak q}}(f, g) = 1$ by\nLemma \\ref{lemma-symbol-when-one-is-a-unit}.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix A: Alternative approach to key lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QJ","source_file":"chow.tex","source_line":18339,"source_end_line":18367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L18339-L18367","statement_sha256":"ab73c020f16f99ad4728e5f710decb02fd49ec38425e125e0603504fd74463d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8404,"rank":8404,"depth":11,"x":2357.624,"y":595.859,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SB","tag":"02SB","title":"Appendix B: Alternative approaches · Lemma 02SB","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. Let F be a coherent sheaf on X. Let xymatrix … ar[r] & F ar[r]^φ & F ar[r]^ψ & F ar[r]^φ & F ar[r] & … be a complex as in Homology, Equation ([Tag 02MW]). Assume that • dim_δ(Supp(F)) ≤ k + 1. • dim_δ(Supp(H^i(F, φ, ψ))) ≤ k for i = 0, 1. Then we have [H^0(F, φ, ψ)]_k sim_rat [H^1(F, φ, ψ)]_k as k-cycles on X.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$.\nLet\n$$\n\\xymatrix{\n\\ldots \\ar[r] &\n\\mathcal{F} \\ar[r]^\\varphi &\n\\mathcal{F} \\ar[r]^\\psi &\n\\mathcal{F} \\ar[r]^\\varphi &\n\\mathcal{F} \\ar[r] & \\ldots\n}\n$$\nbe a complex as in Homology, Equation (\\ref{homology-equation-cyclic-complex}).\nAssume that\n\\begin{enumerate}\n\\item $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$.\n\\item $\\dim_\\delta(\\text{Supp}(H^i(\\mathcal{F}, \\varphi, \\psi))) \\leq k$\nfor $i = 0, 1$.\n\\end{enumerate}\nThen we have\n$$\n[H^0(\\mathcal{F}, \\varphi, \\psi)]_k\n\\sim_{rat}\n[H^1(\\mathcal{F}, \\varphi, \\psi)]_k\n$$\nas $k$-cycles on $X$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix B: Alternative approaches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SB","source_file":"chow.tex","source_line":18416,"source_end_line":18445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L18416-L18445","statement_sha256":"e3b88edfe7dac9c1cf7bc4773d606bce0af3168cbe65d58c060cabb59787de57","origin":"The Stacks Project","memory_eligible":false,"source_rank":8405,"rank":8405,"depth":21,"x":2638.042,"y":672.599,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SC","tag":"02SC","title":"Appendix B: Alternative approaches · Lemma 02SC","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be a scheme locally of finite type over S. The map CH_k(X) → K_0(Coh_≤ k + 1(X)/Coh_≤ k - 1(X)) from Lemma [Tag 0FDS] induces a bijection from CH_k(X) onto the image B_k(X) of the map K_0(Coh_≤ k(X)/Coh_≤ k - 1(X)) → K_0(Coh_≤ k + 1(X)/Coh_≤ k - 1(X)).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be a scheme locally of finite type over $S$.\nThe map\n$$\n\\CH_k(X) \\longrightarrow\nK_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))\n$$\nfrom Lemma \\ref{lemma-from-chow-to-K} induces a bijection from\n$\\CH_k(X)$ onto the image $B_k(X)$ of the map\n$$\nK_0(\\textit{Coh}_{\\leq k}(X)/\\textit{Coh}_{\\leq k - 1}(X))\n\\longrightarrow\nK_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X)).\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix B: Alternative approaches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SC","source_file":"chow.tex","source_line":18508,"source_end_line":18524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L18508-L18524","statement_sha256":"65b6b05b7f90affc7f7e5520e0406608815bff22337c991541eabbd2ef93cb7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8406,"rank":8406,"depth":34,"x":2369.317,"y":775.218,"cluster":"divisors-intersection-theory"},{"id":"stacks:02QH","tag":"02QH","title":"Appendix B: Alternative approaches · Lemma 02QH","summary":"Let A be a Noetherian local ring. Let M be a finite A-module. Let a, b ∈ A. Assume • dim(A) = 1, • both a and b are nonzerodivisors in A, • A has no embedded primes, • M has no embedded associated primes, • Supp(M) = Spec(A). Let I = (x ∈ A mid x(a/b) ∈ A). Let q_1, …, q_t be the minimal primes of A. Then (a/b)IM ⊂ M and length_A(M/(a/b)IM) - length_A(M/IM) = ∑_i length_A_ q_i(M_ q_i) ord_A/ q_i(a/b)","statement_latex":"Let $A$ be a Noetherian local ring.\nLet $M$ be a finite $A$-module.\nLet $a, b \\in A$.\nAssume\n\\begin{enumerate}\n\\item $\\dim(A) = 1$,\n\\item both $a$ and $b$ are nonzerodivisors in $A$,\n\\item $A$ has no embedded primes,\n\\item $M$ has no embedded associated primes,\n\\item $\\text{Supp}(M) = \\Spec(A)$.\n\\end{enumerate}\nLet $I = \\{x \\in A \\mid x(a/b) \\in A\\}$.\nLet $\\mathfrak q_1, \\ldots, \\mathfrak q_t$ be the minimal\nprimes of $A$. Then $(a/b)IM \\subset M$ and\n$$\n\\text{length}_A(M/(a/b)IM)\n-\n\\text{length}_A(M/IM)\n=\n\\sum\\nolimits_i\n\\text{length}_{A_{\\mathfrak q_i}}(M_{\\mathfrak q_i})\n\\text{ord}_{A/\\mathfrak q_i}(a/b)\n$$","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix B: Alternative approaches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02QH","source_file":"chow.tex","source_line":18593,"source_end_line":18618,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L18593-L18618","statement_sha256":"e63c0336f8bbca85552227af8a1b2ea00ef2c125569768d5cd99de7ca20f649a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8407,"rank":8407,"depth":15,"x":2485.049,"y":546.867,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SW","tag":"02SW","title":"Appendix B: Alternative approaches · Lemma 02SW","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L be an invertible O_X-module. Let F be a coherent O_X-module. Let s ∈ Γ(X, K_X(L)) be a meromorphic section of L. Assume • dim_δ(X) ≤ k + 1, • X has no embedded points, • F has no embedded associated points, • the support of F is X, and • the section s is regular meromorphic. In this situation let I ⊂ O_X be the ideal of denominators of s, see Divisors, Definition [Tag 02P1]. Then we…","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{K}_X(\\mathcal{L}))$ be a\nmeromorphic section of $\\mathcal{L}$.\nAssume\n\\begin{enumerate}\n\\item $\\dim_\\delta(X) \\leq k + 1$,\n\\item $X$ has no embedded points,\n\\item $\\mathcal{F}$ has no embedded associated points,\n\\item the support of $\\mathcal{F}$ is $X$, and\n\\item the section $s$ is regular meromorphic.\n\\end{enumerate}\nIn this situation let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe the ideal of denominators of $s$, see\nDivisors,\nDefinition \\ref{divisors-definition-regular-meromorphic-ideal-denominators}.\nThen we have the following:\n\\begin{enumerate}\n\\item there are short exact sequences\n$$\n\\begin{matrix}\n0 &\n\\to &\n\\mathcal{I}\\mathcal{F} &\n\\xrightarrow{1} &\n\\mathcal{F} &\n\\to &\n\\mathcal{Q}_1 &\n\\to &\n0 \\\\\n0 &\n\\to &\n\\mathcal{I}\\mathcal{F} &\n\\xrightarrow{s} &\n\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L} &\n\\to &\n\\mathcal{Q}_2 &\n\\to &\n0\n\\end{matrix}\n$$\n\\item the coherent sheaves $\\mathcal{Q}_1$, $\\mathcal{Q}_2$\nare supported in $\\delta$-dimension $\\leq k$,\n\\item the section $s$ restricts to a regular meromorphic\nsection $s_i$ on every irreducible component $X_i$ of\n$X$ of $\\delta$-dimension $k + 1$, and\n\\item writing $[\\mathcal{F}]_{k + 1} = \\sum m_i[X_i]$ we have\n$$\n[\\mathcal{Q}_2]_k - [\\mathcal{Q}_1]_k\n=\n\\sum m_i(X_i \\to X)_*\\text{div}_{\\mathcal{L}|_{X_i}}(s_i)\n$$\nin $Z_k(X)$, in particular\n$$\n[\\mathcal{Q}_2]_k - [\\mathcal{Q}_1]_k\n=\nc_1(\\mathcal{L}) \\cap [\\mathcal{F}]_{k + 1}\n$$\nin $\\CH_k(X)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix B: Alternative approaches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SW","source_file":"chow.tex","source_line":18648,"source_end_line":18712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L18648-L18712","statement_sha256":"b2d5937af7d93c0ce2cf6a6c986e2aac0ec2436e9b878a65250ea7c42b9b0fd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8408,"rank":8408,"depth":16,"x":2583.429,"y":781.123,"cluster":"divisors-intersection-theory"},{"id":"stacks:02SX","tag":"02SX","title":"Appendix B: Alternative approaches · Lemma 02SX","summary":"Let (S, δ) be as in Situation [Tag 02QL]. Let X be locally of finite type over S. Let L be an invertible O_X-module. Let F be a coherent O_X-module. Assume dim_δ(Supp(F)) ≤ k + 1. Then the element [F ⊗_O_X L] - [F] ∈ K_0(Coh_≤ k + 1(X)/Coh_≤ k - 1(X)) lies in the subgroup B_k(X) of Lemma [Tag 02SC] and maps to the element c_1(L) ∩ [F]_k + 1 via the map B_k(X) → CH_k(X).","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nAssume $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k + 1$.\nThen the element\n$$\n[\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}]\n-\n[\\mathcal{F}]\n\\in\nK_0(\\textit{Coh}_{\\leq k + 1}(X)/\\textit{Coh}_{\\leq k - 1}(X))\n$$\nlies in the subgroup $B_k(X)$ of\nLemma \\ref{lemma-cycles-rational-equivalence-K-group} and maps to\nthe element $c_1(\\mathcal{L}) \\cap [\\mathcal{F}]_{k + 1}$ via\nthe map $B_k(X) \\to \\CH_k(X)$.","area":"Divisors & Intersection Theory","chapter":"Chow Homology","chapter_id":"chow","section":"Appendix B: Alternative approaches","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02SX","source_file":"chow.tex","source_line":18755,"source_end_line":18774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/chow.tex#L18755-L18774","statement_sha256":"6e6aff6674ecf645f460041b58e1bd2175c4cf4f36934782c71c1bbae38d38cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8409,"rank":8409,"depth":46,"x":2322.275,"y":664.108,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZD","tag":"0AZD","title":"Proper pushforward · Lemma 0AZD","summary":"See [Serre_algebre_locale]. Suppose that f : X → Y is a proper morphism of varieties. Let F be a coherent sheaf with dim(Supp(F)) ≤ k, then f_*[F]_k = [f_*F]_k. In particular, if Z ⊂ X is a closed subscheme of dimension ≤ k, then f_*[Z]_k = [f_*O_Z]_k.","statement_latex":"\\begin{reference}\nSee \\cite[Chapter V]{Serre_algebre_locale}.\n\\end{reference}\nSuppose that $f : X \\to Y$ is a proper morphism of varieties.\nLet $\\mathcal{F}$ be a coherent sheaf with\n$\\dim(\\text{Supp}(\\mathcal{F})) \\leq k$, then\n$f_*[\\mathcal{F}]_k = [f_*\\mathcal{F}]_k$. In particular, if\n$Z \\subset X$ is a closed subscheme of dimension $\\leq k$, then\n$f_*[Z]_k = [f_*\\mathcal{O}_Z]_k$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZD","source_file":"intersection.tex","source_line":224,"source_end_line":235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L224-L235","statement_sha256":"159f3487f3c2e54d5299e154f6c2314273c115db8975d91e898962e4c8327b69","origin":"The Stacks Project","memory_eligible":false,"source_rank":8410,"rank":8410,"depth":32,"x":2609.197,"y":602.154,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0N","tag":"0B0N","title":"Proper pushforward · Lemma 0B0N","summary":"Let f : X → Y and g : Y → Z be proper morphisms of varieties. Then g_* ∘ f_* = (g ∘ f)_* as maps Z_k(X) → Z_k(Z).","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of\nvarieties. Then $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0N","source_file":"intersection.tex","source_line":241,"source_end_line":245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L241-L245","statement_sha256":"2344121ae8ff9665f630189c6b08ba459b6d43e92d901aea06f710bfa651f6af","origin":"The Stacks Project","memory_eligible":false,"source_rank":8411,"rank":8411,"depth":33,"x":2447.305,"y":810.826,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZF","tag":"0AZF","title":"Flat pullback · Lemma 0AZF","summary":"Let f : X → Y be a flat morphism of varieties. Set r = dim(X) - dim(Y). Then f^*[F]_k = [f^*F]_k + r if F is a coherent sheaf on Y and the dimension of the support of F is at most k.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of varieties. Set $r = \\dim(X) - \\dim(Y)$.\nThen $f^*[\\mathcal{F}]_k = [f^*\\mathcal{F}]_{k + r}$\nif $\\mathcal{F}$ is a coherent sheaf on $Y$ and the dimension of the\nsupport of $\\mathcal{F}$ is at most $k$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZF","source_file":"intersection.tex","source_line":278,"source_end_line":284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L278-L284","statement_sha256":"06b73d3f393842cc199d9480713b7a0df2bbe8c349f065ad773c23f61a052ca9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8412,"rank":8412,"depth":26,"x":2398.832,"y":564.88,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0P","tag":"0B0P","title":"Flat pullback · Lemma 0B0P","summary":"Let f : X → Y and g : Y → Z be flat morphisms of varieties. Then g ∘ f is flat and f^* ∘ g^* = (g ∘ f)^* as maps Z_k(Z) → Z_k + dim(X) - dim(Z)(X).","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of\nvarieties. Then $g \\circ f$ is flat and $f^* \\circ g^* = (g \\circ f)^*$\nas maps $Z_k(Z) \\to Z_{k + \\dim(X) - \\dim(Z)}(X)$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0P","source_file":"intersection.tex","source_line":290,"source_end_line":295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L290-L295","statement_sha256":"401174772a48f5539a654093f44e6402d2ca7ea99e480a39f4362d510d1172c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8413,"rank":8413,"depth":27,"x":2632.561,"y":718.865,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZI","tag":"0AZI","title":"Rational equivalence and rational functions · Lemma 0AZI","summary":"Let X be a variety. Let W ⊂ X be a subvariety of dimension k + 1. Let f ∈ C(W)^* be a nonzero rational function on W. Then div_W(f) is rationally equivalent to zero on X. Conversely, these principal divisors generate the abelian group of cycles rationally equivalent to zero on X.","statement_latex":"Let $X$ be a variety. Let $W \\subset X$ be a subvariety\nof dimension $k + 1$. Let $f \\in \\mathbf{C}(W)^*$ be a nonzero rational\nfunction on $W$. Then $\\text{div}_W(f)$ is rationally equivalent to zero on\n$X$. Conversely, these principal divisors generate the abelian group of\ncycles rationally equivalent to zero on $X$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Rational equivalence and rational functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZI","source_file":"intersection.tex","source_line":397,"source_end_line":404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L397-L404","statement_sha256":"e6fc374c19235e3c4cab5c2ef7847615e66b7e0680410288c80ae727d51a8da8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8414,"rank":8414,"depth":45,"x":2336.125,"y":737.959,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZM","tag":"0AZM","title":"Proper intersections · Lemma 0AZM","summary":"Let X and Y be varieties. Then X × Y is a variety and dim(X × Y) = dim(X) + dim(Y).","statement_latex":"Let $X$ and $Y$ be varieties. Then $X \\times Y$ is a variety and\n$\\dim(X \\times Y) = \\dim(X) + \\dim(Y)$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Proper intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZM","source_file":"intersection.tex","source_line":538,"source_end_line":542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L538-L542","statement_sha256":"d57f4250704fb17354307ff1e6e998771c62be5930e9882aeb79ea2a53b321c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8415,"rank":8415,"depth":28,"x":2539.534,"y":555.517,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZN","tag":"0AZN","title":"Proper intersections · Lemma 0AZN","summary":"Let f : X → Y be a morphism of varieties. • If Z ⊂ Y is a subvariety of dimension d and f is a regular immersion of codimension c, then every irreducible component of f^-1(Z) has dimension ≥ d - c. • If Z ⊂ Y is a subvariety of dimension d and f is a local complete intersection morphism of relative dimension r, then every irreducible component of f^-1(Z) has dimension ≥ d + r.","statement_latex":"Let $f : X \\to Y$ be a morphism of varieties.\n\\begin{enumerate}\n\\item If $Z \\subset Y$ is a subvariety of dimension $d$ and $f$ is a regular\nimmersion of codimension $c$, then every irreducible component\nof $f^{-1}(Z)$ has dimension $\\geq d - c$.\n\\item If $Z \\subset Y$ is a subvariety of dimension $d$ and\n$f$ is a local complete intersection morphism of relative dimension $r$,\nthen every irreducible component of $f^{-1}(Z)$ has dimension $\\geq d + r$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Proper intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZN","source_file":"intersection.tex","source_line":583,"source_end_line":594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L583-L594","statement_sha256":"8590569ee20410ddc430ab289a97e88e5ae2e27a7de51e58e2027c89268700d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8416,"rank":8416,"depth":21,"x":2536.255,"y":805.681,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0Q","tag":"0B0Q","title":"Proper intersections · Lemma 0B0Q","summary":"Let X be a nonsingular variety. Then the diagonal Δ : X → X × X is a regular immersion of codimension dim(X).","statement_latex":"Let $X$ be a nonsingular variety. Then the diagonal\n$\\Delta : X \\to X \\times X$ is a regular immersion of codimension $\\dim(X)$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Proper intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0Q","source_file":"intersection.tex","source_line":621,"source_end_line":625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L621-L625","statement_sha256":"a86530e4ed8381e1524faf7290d8dbbe502c81705f80779fdaa6fc887d125ae3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8417,"rank":8417,"depth":40,"x":2337.326,"y":619.193,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZP","tag":"0AZP","title":"Proper intersections · Lemma 0AZP","summary":"Let X be a nonsingular variety and let W,V ⊂ X be closed subvarieties with dim(W) = s and dim(V) = r. Then every irreducible component Z of V ∩ W has dimension ≥ r + s - dim(X).","statement_latex":"Let $X$ be a nonsingular variety and let $W,V \\subset X$\nbe closed subvarieties with $\\dim(W) = s$ and $\\dim(V) = r$. Then every\nirreducible component $Z$ of $V \\cap W$ has dimension $\\geq r + s - \\dim(X)$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Proper intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZP","source_file":"intersection.tex","source_line":636,"source_end_line":641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L636-L641","statement_sha256":"eb38abb77d0437048c2d0fba16e64cbba7c4d399118c34adb98f466844b4f4f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8418,"rank":8418,"depth":41,"x":2634.237,"y":643.851,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZQ","tag":"0AZQ","title":"Proper intersections · Definition 0AZQ","summary":"Let X be a nonsingular variety. • Let W,V ⊂ X be closed subvarieties with dim(W) = s and dim(V) = r. We say that W and V intersect properly if dim(V ∩ W) ≤ r + s - dim(X). • Let α = ∑ n_i [W_i] be an s-cycle, and β = ∑_j m_j [V_j] be an r-cycle on X. We say that α and β intersect properly if W_i and V_j intersect properly for all i and j.","statement_latex":"Let $X$ be a nonsingular variety.\n\\begin{enumerate}\n\\item Let $W,V \\subset X$ be closed subvarieties with\n$\\dim(W) = s$ and $\\dim(V) = r$. We say that $W$ and $V$\n{\\it intersect properly} if $\\dim(V \\cap W) \\leq r + s - \\dim(X)$.\n\\item Let $\\alpha = \\sum n_i [W_i]$ be an $s$-cycle,\nand $\\beta = \\sum_j m_j [V_j]$ be an $r$-cycle on $X$. We say\nthat $\\alpha$ and $\\beta$ {\\it intersect properly} if\n$W_i$ and $V_j$ intersect properly for all $i$ and $j$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Proper intersections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZQ","source_file":"intersection.tex","source_line":652,"source_end_line":664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L652-L664","statement_sha256":"6ace1abb9daf62edc06f47194ee25f5803862a31ddc75281579f7642a6e38471","origin":"The Stacks Project","memory_eligible":false,"source_rank":8419,"rank":8419,"depth":0,"x":2395.265,"y":794.272,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZS","tag":"0AZS","title":"Intersection multiplicities using Tor formula · Lemma 0AZS","summary":"Let X be a locally Noetherian scheme. • If F and G are coherent O_X-modules, then Tor_p^O_X(F, G) is too. • If L and K are in D^-_Coh(O_X), then so is L ⊗_O_X^L K.","statement_latex":"Let $X$ be a locally Noetherian scheme.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are coherent $\\mathcal{O}_X$-modules,\nthen $\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ is too.\n\\item If $L$ and $K$ are in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then\nso is $L \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Intersection multiplicities using Tor formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZS","source_file":"intersection.tex","source_line":690,"source_end_line":699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L690-L699","statement_sha256":"4be1838e71c2b76f1f5b2edfe9dd8f46a49daffe85da34c2955d0364bda43520","origin":"The Stacks Project","memory_eligible":false,"source_rank":8420,"rank":8420,"depth":28,"x":2450.563,"y":547.544,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZT","tag":"0AZT","title":"Intersection multiplicities using Tor formula · Lemma 0AZT","summary":"Let X be a nonsingular variety. Let F, G be coherent O_X-modules. The O_X-module Tor_p^O_X(F, G) is coherent, has stalk at x equal to Tor_p^O_X, x(F_x, G_x), is supported on Supp(F) ∩ Supp(G), and is nonzero only for p ∈ (0, …, dim(X)).","statement_latex":"Let $X$ be a nonsingular variety.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.\nThe $\\mathcal{O}_X$-module\n$\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$\nis coherent, has stalk at $x$ equal to\n$\\text{Tor}_p^{\\mathcal{O}_{X, x}}(\\mathcal{F}_x, \\mathcal{G}_x)$,\nis supported on\n$\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G})$, and\nis nonzero only for $p \\in \\{0, \\ldots, \\dim(X)\\}$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Intersection multiplicities using Tor formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZT","source_file":"intersection.tex","source_line":728,"source_end_line":739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L728-L739","statement_sha256":"f9e5f1ef42c47df860d75d25dd99b1cefb46cf174af0a23db22c896ffb092ee2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8421,"rank":8421,"depth":29,"x":2608.334,"y":761.037,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1I","tag":"0B1I","title":"Intersection multiplicities using Tor formula · Lemma 0B1I","summary":"Let X be a nonsingular variety. Let V, W ⊂ X be closed subvarieties which intersect properly. Let Z be an irreducible component of V ∩ W and assume that the multiplicity (in the sense of Section [Tag 0AZA]) of Z in the closed subscheme V ∩ W is 1. Then e(X, V · W, Z) = 1 and V and W are smooth in a general point of Z.","statement_latex":"Let $X$ be a nonsingular variety. Let $V, W \\subset X$ be\nclosed subvarieties which intersect properly. Let $Z$ be an irreducible\ncomponent of $V \\cap W$ and assume that the multiplicity\n(in the sense of Section \\ref{section-cycle-of-closed}) of $Z$\nin the closed subscheme $V \\cap W$ is $1$.\nThen $e(X, V \\cdot W, Z) = 1$ and $V$ and $W$ are smooth\nin a general point of $Z$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Intersection multiplicities using Tor formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1I","source_file":"intersection.tex","source_line":795,"source_end_line":804,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L795-L804","statement_sha256":"e3f23e1dec714fb90cbb80b8e525c3e126930466dfb6ebf6219d7c8fb5dfedf3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8422,"rank":8422,"depth":24,"x":2320.065,"y":693.076,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZV","tag":"0AZV","title":"Algebraic multiplicities · Definition 0AZV","summary":"In the situation above, assume d ≥ dim(Supp(M)). In this case, if d > dim(Supp(M)), then we set e_I(M, d) = 0 and if d = dim(Supp(M)), then we set e_I(M, d) equal to d! times the leading coefficient of the numerical polynomial chi_I, M. Thus in both cases we have chi_I, M(n) sim e_I(M, d) fracn^dd! + lower order terms The multiplicity of M for the ideal of definition I is e_I(M) = e_I(M, dim(Supp(M))).","statement_latex":"In the situation above, assume $d \\geq \\dim(\\text{Supp}(M))$.\nIn this case, if  $d > \\dim(\\text{Supp}(M))$, then we set $e_I(M, d) = 0$\nand if $d = \\dim(\\text{Supp}(M))$, then we set $e_I(M, d)$ equal to $d!$\ntimes the leading coefficient of the numerical polynomial $\\chi_{I, M}$.\nThus in both cases we have\n$$\n\\chi_{I, M}(n) \\sim e_I(M, d) \\frac{n^d}{d!} + \\text{lower order terms}\n$$\nThe {\\it multiplicity of $M$ for the ideal of definition $I$}\nis $e_I(M) = e_I(M, \\dim(\\text{Supp}(M)))$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Algebraic multiplicities","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZV","source_file":"intersection.tex","source_line":904,"source_end_line":916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L904-L916","statement_sha256":"02c25e3b7aaed11be9547a0117fd316014a1e56ad82f97140ed896d8a70121e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8423,"rank":8423,"depth":0,"x":2587.51,"y":579.522,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZW","tag":"0AZW","title":"Algebraic multiplicities · Lemma 0AZW","summary":"Let A be a Noetherian local ring. Let I ⊂ A be an ideal of definition. Let 0 → M' → M → M\" → 0 be a short exact sequence of finite A-modules. Let d ≥ dim(Supp(M)). Then e_I(M, d) = e_I(M', d) + e_I(M\", d)","statement_latex":"Let $A$ be a Noetherian local ring. Let $I \\subset A$ be an ideal of\ndefinition. Let $0 \\to M' \\to M \\to M'' \\to 0$ be a short exact sequence\nof finite $A$-modules. Let $d \\geq \\dim(\\text{Supp}(M))$. Then\n$$\ne_I(M, d) = e_I(M', d) + e_I(M'', d)\n$$","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Algebraic multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZW","source_file":"intersection.tex","source_line":921,"source_end_line":929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L921-L929","statement_sha256":"9a87d3c45c9a502a06e2cdb40d5f066cfb6b9ec09a5b3f38533e976abf3cf00d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8424,"rank":8424,"depth":4,"x":2481.527,"y":815.21,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZX","tag":"0AZX","title":"Algebraic multiplicities · Lemma 0AZX","summary":"Let A be a Noetherian local ring. Let I ⊂ A be an ideal of definition. Let M be a finite A-module. Let d ≥ dim(Supp(M)). Then e_I(M, d) = ∑ length_A_ p(M_ p) e_I(A/ p, d) where the sum is over primes p ⊂ A with dim(A/ p) = d.","statement_latex":"Let $A$ be a Noetherian local ring. Let $I \\subset A$ be an ideal of\ndefinition. Let $M$ be a finite $A$-module. Let $d \\geq \\dim(\\text{Supp}(M))$.\nThen\n$$\ne_I(M, d) =\n\\sum \\text{length}_{A_\\mathfrak p}(M_\\mathfrak p) e_I(A/\\mathfrak p, d)\n$$\nwhere the sum is over primes $\\mathfrak p \\subset A$ with\n$\\dim(A/\\mathfrak p) = d$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Algebraic multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZX","source_file":"intersection.tex","source_line":936,"source_end_line":947,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L936-L947","statement_sha256":"7cfe59458e2eb6a079f404992fde2c1b8c9f262d74226c023e9f74998d50df8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8425,"rank":8425,"depth":5,"x":2370.049,"y":581.081,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZY","tag":"0AZY","title":"Algebraic multiplicities · Lemma 0AZY","summary":"Let P be a polynomial of degree r with leading coefficient a. Then r! a = ∑_i = 0, …, r (-1)^ir choose i P(t - i) for any t.","statement_latex":"Let $P$ be a polynomial of degree $r$ with leading coefficient $a$.\nThen\n$$\nr! a = \\sum\\nolimits_{i = 0, \\ldots, r} (-1)^i{r \\choose i} P(t - i)\n$$\nfor any $t$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Algebraic multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZY","source_file":"intersection.tex","source_line":960,"source_end_line":968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L960-L968","statement_sha256":"34e5cb8c007ddbeebbd2919b6451d674faf34aef70ffd118a1c0fbb3be5ed884","origin":"The Stacks Project","memory_eligible":false,"source_rank":8426,"rank":8426,"depth":0,"x":2640.76,"y":690.562,"cluster":"divisors-intersection-theory"},{"id":"stacks:0AZZ","tag":"0AZZ","title":"Algebraic multiplicities · Theorem 0AZZ","summary":"[Serre_algebre_locale] Let A be a Noetherian local ring. Let I = (f_1, …, f_r) ⊂ A be an ideal of definition. Let M be a finite A-module. Then e_I(M, r) = ∑ (-1)^ilength_A H_i(K_bullet(f_1, …, f_r) ⊗_A M)","statement_latex":"\\begin{reference}\n\\cite[Theorem 1 in part B of Chapter IV]{Serre_algebre_locale}\n\\end{reference}\nLet $A$ be a Noetherian local ring. Let $I = (f_1, \\ldots, f_r) \\subset A$\nbe an ideal of definition. Let $M$ be a finite $A$-module. Then\n$$\ne_I(M, r) = \\sum\n(-1)^i\\text{length}_A H_i(K_\\bullet(f_1, \\ldots, f_r) \\otimes_A M)\n$$","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Algebraic multiplicities","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AZZ","source_file":"intersection.tex","source_line":1001,"source_end_line":1012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1001-L1012","statement_sha256":"2ce1837ccacd0430f13b1acf40fbb173162a91fcb24a9768f7c21adce640b311","origin":"The Stacks Project","memory_eligible":false,"source_rank":8427,"rank":8427,"depth":3,"x":2352.857,"y":763.502,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B02","tag":"0B02","title":"Computing intersection multiplicities · Lemma 0B02","summary":"Let X be a nonsingular variety and W, V ⊂ X closed subvarieties which intersect properly. Let Z be an irreducible component of V ∩ W with generic point xi. Assume that O_W, xi and O_V, xi are Cohen-Macaulay. Then e(X, V · W, Z) = length_O_X, xi(O_V ∩ W, xi) where V ∩ W is the scheme theoretic intersection. In particular, if both V and W are Cohen-Macaulay, then V · W = [V ∩ W]_dim(V) + dim(W) - dim(X).","statement_latex":"Let $X$ be a nonsingular variety and $W, V \\subset X$ closed\nsubvarieties which intersect properly. Let $Z$ be an irreducible component\nof $V \\cap W$ with generic point $\\xi$. Assume that $\\mathcal{O}_{W, \\xi}$\nand $\\mathcal{O}_{V, \\xi}$ are Cohen-Macaulay. Then\n$$\ne(X, V \\cdot W, Z) =\n\\text{length}_{\\mathcal{O}_{X, \\xi}}(\\mathcal{O}_{V \\cap W, \\xi})\n$$\nwhere $V \\cap W$ is the scheme theoretic intersection.\nIn particular, if both $V$ and $W$ are Cohen-Macaulay, then\n$V \\cdot W = [V \\cap W]_{\\dim(V) + \\dim(W) - \\dim(X)}$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Computing intersection multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B02","source_file":"intersection.tex","source_line":1138,"source_end_line":1151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1138-L1151","statement_sha256":"b95d04bc8191d1b188e35dbff94e4f07bb62dcee08bd3ac0fb4fb503d58b2f05","origin":"The Stacks Project","memory_eligible":false,"source_rank":8428,"rank":8428,"depth":16,"x":2506.626,"y":546.17,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B03","tag":"0B03","title":"Computing intersection multiplicities · Lemma 0B03","summary":"Let A be a Noetherian local ring. Let I = (f_1, …, f_r) be an ideal generated by a regular sequence. Let M be a finite A-module. Assume that dim(Supp(M/IM)) = 0. Then e_I(M, r) = ∑ (-1)^ilength_A(Tor_i^A(A/I, M)) Here e_I(M, r) is as in Remark [Tag 0B00].","statement_latex":"Let $A$ be a Noetherian local ring. Let $I = (f_1, \\ldots, f_r)$ be an ideal\ngenerated by a regular sequence. Let $M$ be a finite $A$-module. Assume that\n$\\dim(\\text{Supp}(M/IM)) = 0$. Then\n$$\ne_I(M, r) = \\sum (-1)^i\\text{length}_A(\\text{Tor}_i^A(A/I, M))\n$$\nHere $e_I(M, r)$ is as in Remark \\ref{remark-trivial-generalization}.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Computing intersection multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B03","source_file":"intersection.tex","source_line":1173,"source_end_line":1182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1173-L1182","statement_sha256":"b4bf686137f32df26bfcdf1708932598a8e782af1587e1ed7479767fac71aa11","origin":"The Stacks Project","memory_eligible":false,"source_rank":8429,"rank":8429,"depth":6,"x":2568.063,"y":793.888,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B04","tag":"0B04","title":"Computing intersection multiplicities · Lemma 0B04","summary":"Let X be a nonsingular variety. Let W,V ⊂ X be closed subvarieties which intersect properly. Let Z be an irreducible component of V ∩ W with generic point xi. Suppose the ideal of V in O_X, xi is cut out by a regular sequence f_1, …, f_c ∈ O_X, xi. Then e(X, V· W, Z) is equal to c! times the leading coefficient in the Hilbert polynomial t ↦ length_O_X, xi O_W, xi/(f_1, …, f_c)^t, t gg 0. In particular, this coefficient is > 0.","statement_latex":"Let $X$ be a nonsingular variety. Let $W,V \\subset X$ be\nclosed subvarieties which intersect properly. Let $Z$ be an irreducible\ncomponent of $V \\cap W$ with generic point $\\xi$.\nSuppose the ideal of $V$ in $\\mathcal{O}_{X, \\xi}$ is cut out by\na regular sequence $f_1, \\ldots, f_c \\in \\mathcal{O}_{X, \\xi}$.\nThen $e(X, V\\cdot W, Z)$ is equal to $c!$ times the leading coefficient in\nthe Hilbert polynomial\n$$\nt \\mapsto \\text{length}_{\\mathcal{O}_{X, \\xi}}\n\\mathcal{O}_{W, \\xi}/(f_1, \\ldots, f_c)^t,\\quad t \\gg 0.\n$$\nIn particular, this coefficient is $> 0$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Computing intersection multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B04","source_file":"intersection.tex","source_line":1208,"source_end_line":1222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1208-L1222","statement_sha256":"273d7ac4702193b9984f34c33b26fc2686af54318698c6318bd05d3a097acaab","origin":"The Stacks Project","memory_eligible":false,"source_rank":8430,"rank":8430,"depth":23,"x":2323.347,"y":645.961,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B05","tag":"0B05","title":"Computing intersection multiplicities · Lemma 0B05","summary":"In Lemma [Tag 0B04] assume that c = 1, i.e., V is an effective Cartier divisor. Then e(X, V · W, Z) = length_O_X, xi (O_W, xi/f_1O_W, xi).","statement_latex":"In Lemma \\ref{lemma-multiplicity-with-lci} assume that  $c = 1$, i.e., $V$\nis an effective Cartier divisor. Then\n$$\ne(X, V \\cdot W, Z) =\n\\text{length}_{\\mathcal{O}_{X, \\xi}}\n(\\mathcal{O}_{W, \\xi}/f_1\\mathcal{O}_{W, \\xi}).\n$$","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Computing intersection multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B05","source_file":"intersection.tex","source_line":1254,"source_end_line":1263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1254-L1263","statement_sha256":"f44592197b2a2e390c083a9af25e7c559c06ad9b431ce3b2f7b8ac19afc07dc6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8431,"rank":8431,"depth":24,"x":2623.007,"y":616.157,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B06","tag":"0B06","title":"Computing intersection multiplicities · Lemma 0B06","summary":"In Lemma [Tag 0B04] assume that the local ring O_W, xi is Cohen-Macaulay. Then we have e(X, V · W, Z) = length_O_X, xi (O_W, xi/ f_1O_W, xi + … + f_cO_W, xi).","statement_latex":"In Lemma \\ref{lemma-multiplicity-with-lci} assume that\nthe local ring $\\mathcal{O}_{W, \\xi}$ is Cohen-Macaulay. Then we\nhave\n$$\ne(X, V \\cdot W, Z) =\n\\text{length}_{\\mathcal{O}_{X, \\xi}} (\\mathcal{O}_{W, \\xi}/\nf_1\\mathcal{O}_{W, \\xi} + \\ldots + f_c\\mathcal{O}_{W, \\xi}).\n$$","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Computing intersection multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B06","source_file":"intersection.tex","source_line":1280,"source_end_line":1290,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1280-L1290","statement_sha256":"6da1149e9a37bbc1f5b046860346c0ade74a5cd5bd7c8f728735f1f99b8cc458","origin":"The Stacks Project","memory_eligible":false,"source_rank":8432,"rank":8432,"depth":24,"x":2425.844,"y":808.33,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B07","tag":"0B07","title":"Intersection product using Tor formula · Lemma 0B07","summary":"Let X be a nonsingular variety. Let a, b ∈ P^1 be distinct closed points. Let k ≥ 0. • If W ⊂ X × P^1 is a closed subvariety of dimension k + 1 which intersects X × a properly, then • [W_a]_k = W · X × a as cycles on X × P^1, and • [W_a]_k = pr_X, *(W · X × a) as cycles on X. • Let α be a (k + 1)-cycle on X × P^1 which intersects X × a and X × b properly. Then pr_X,*( α · X × a - α · X × b) is rationally equivalent to zero. • Conversely, any k-cycle which is rationally…","statement_latex":"Let $X$ be a nonsingular variety. Let $a, b \\in \\mathbf{P}^1$\nbe distinct closed points. Let $k \\geq 0$.\n\\begin{enumerate}\n\\item If $W \\subset X \\times \\mathbf{P}^1$ is a closed subvariety\nof dimension $k + 1$ which intersects $X \\times a$ properly, then\n\\begin{enumerate}\n\\item $[W_a]_k = W \\cdot X \\times a$ as cycles on $X \\times \\mathbf{P}^1$, and\n\\item $[W_a]_k = \\text{pr}_{X, *}(W \\cdot X \\times a)$ as cycles on $X$.\n\\end{enumerate}\n\\item Let $\\alpha$ be a $(k + 1)$-cycle on $X \\times \\mathbf{P}^1$\nwhich intersects $X \\times a$ and $X \\times b$ properly. Then\n$pr_{X,*}( \\alpha \\cdot X \\times a - \\alpha \\cdot X \\times b)$\nis rationally equivalent to zero.\n\\item Conversely, any $k$-cycle which is\nrationally equivalent to $0$ is of this form.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Intersection product using Tor formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B07","source_file":"intersection.tex","source_line":1328,"source_end_line":1346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1328-L1346","statement_sha256":"b8f6a0e3e42aebe10406aa53f0edd8b684ba298469b3d0969f57e65124d50dcc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8433,"rank":8433,"depth":25,"x":2416.682,"y":554.537,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1J","tag":"0B1J","title":"Intersection product using Tor formula · Lemma 0B1J","summary":"Let X be a nonsingular variety. Let r, s ≥ 0 and let Y, Z ⊂ X be closed subschemes with dim(Y) ≤ r and dim(Z) ≤ s. Assume [Y]_r = ∑ n_i[Y_i] and [Z]_s = ∑ m_j[Z_j] intersect properly. Let T be an irreducible component of Y_i_0 ∩ Z_j_0 for some i_0 and j_0 and assume that the multiplicity (in the sense of Section [Tag 0AZA]) of T in the closed subscheme Y ∩ Z is 1. Then • the coefficient of T in [Y]_r · [Z]_s is 1, • Y and Z are nonsingular at the generic point of T, •…","statement_latex":"Let $X$ be a nonsingular variety. Let $r, s \\geq 0$ and let\n$Y, Z \\subset X$ be closed subschemes with $\\dim(Y) \\leq r$ and\n$\\dim(Z) \\leq s$. Assume $[Y]_r = \\sum n_i[Y_i]$ and\n$[Z]_s = \\sum m_j[Z_j]$ intersect properly.\nLet $T$ be an irreducible component of $Y_{i_0} \\cap Z_{j_0}$\nfor some $i_0$ and $j_0$ and assume that the multiplicity\n(in the sense of Section \\ref{section-cycle-of-closed}) of $T$\nin the closed subscheme $Y \\cap Z$ is $1$.\nThen\n\\begin{enumerate}\n\\item the coefficient of $T$ in $[Y]_r \\cdot [Z]_s$ is $1$,\n\\item $Y$ and $Z$ are nonsingular at the generic point of $T$,\n\\item $n_{i_0} = 1$, $m_{j_0} = 1$, and\n\\item $T$ is not contained in $Y_i$ or $Z_j$ for $i \\not = i_0$ and\n$j \\not = j_0$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Intersection product using Tor formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1J","source_file":"intersection.tex","source_line":1365,"source_end_line":1383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1365-L1383","statement_sha256":"cb07d9c1d285bd6dbb064310ad741e8f982b2ea8876235cd904aa6584f63e591","origin":"The Stacks Project","memory_eligible":false,"source_rank":8434,"rank":8434,"depth":25,"x":2627.707,"y":736.633,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0S","tag":"0B0S","title":"Exterior product · Lemma 0B0S","summary":"Let X and Y be varieties. Let α ∈ Z_r(X) and β ∈ Z_s(Y). If α sim_rat 0 or β sim_rat 0, then α × β sim_rat 0.","statement_latex":"Let $X$ and $Y$ be varieties.\nLet $\\alpha \\in Z_r(X)$ and $\\beta \\in Z_s(Y)$.\nIf $\\alpha \\sim_{rat} 0$ or $\\beta \\sim_{rat} 0$, then\n$\\alpha \\times \\beta \\sim_{rat} 0$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Exterior product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0S","source_file":"intersection.tex","source_line":1450,"source_end_line":1456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1450-L1456","statement_sha256":"e61dc2d77bd4a89308762d1f8f0c6c5f855f71eb2c1aa3dc6266381455c4d698","origin":"The Stacks Project","memory_eligible":false,"source_rank":8435,"rank":8435,"depth":29,"x":2325.412,"y":722.087,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0R","tag":"0B0R","title":"Exterior product · Lemma 0B0R","summary":"Let X and Y be nonsingular varieties. Let α ∈ Z_r(X) and β ∈ Z_s(Y). Then • pr_Y^*(β) = [X] × β and pr_X^*(α) = α × [Y], • α × [Y] and [X]× β intersect properly on X× Y, and • we have α × β = (α × [Y])· ([X]×β) = pr_Y^*(α) · pr_X^*(β) in Z_r + s(X × Y).","statement_latex":"Let $X$ and $Y$ be nonsingular varieties.\nLet $\\alpha \\in Z_r(X)$ and $\\beta \\in Z_s(Y)$.\nThen\n\\begin{enumerate}\n\\item $\\text{pr}_Y^*(\\beta) = [X] \\times \\beta$ and\n$\\text{pr}_X^*(\\alpha) = \\alpha \\times [Y]$,\n\\item $\\alpha \\times [Y]$ and $[X]\\times \\beta$\nintersect properly on $X\\times Y$, and\n\\item we have\n$\\alpha \\times \\beta =\n(\\alpha \\times [Y])\\cdot ([X]\\times\\beta) =\npr_Y^*(\\alpha) \\cdot pr_X^*(\\beta)$\nin $Z_{r + s}(X \\times Y)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Exterior product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0R","source_file":"intersection.tex","source_line":1518,"source_end_line":1534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1518-L1534","statement_sha256":"8f479513c2a51301395296d93c5a951d58b91fcaf79349c6fc737991ce2dfbe7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8436,"rank":8436,"depth":29,"x":2560.212,"y":561.15,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0T","tag":"0B0T","title":"Reduction to the diagonal · Lemma 0B0T","summary":"Let X be a nonsingular variety. • If F and G are coherent O_X-modules, then there are canonical isomorphisms Tor_i^O_X × X(O_Δ, pr_1^*F ⊗_O_X × X pr_2^*G) = Δ_*Tor_i^O_X(F, G) • If K and M are in D_QCoh(O_X), then there is a canonical isomorphism LΔ^* ( Lpr_1^*K ⊗_O_X × X^L Lpr_2^*M ) = K ⊗_O_X^L M in D_QCoh(O_X) and a canonical isomorphism O_Δ ⊗_O_X × X^L Lpr_1^*K ⊗_O_X × X^L Lpr_2^*M = Δ_*(K ⊗_O_X^L M) in D_QCoh(X × X).","statement_latex":"Let $X$ be a nonsingular variety.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are coherent $\\mathcal{O}_X$-modules,\nthen there are canonical isomorphisms\n$$\n\\text{Tor}_i^{\\mathcal{O}_{X \\times X}}(\\mathcal{O}_\\Delta,\n\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times X}}\n\\text{pr}_2^*\\mathcal{G})\n=\n\\Delta_*\\text{Tor}_i^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})\n$$\n\\item If $K$ and $M$ are in $D_\\QCoh(\\mathcal{O}_X)$, then\nthere is a canonical isomorphism\n$$\nL\\Delta^* \\left(\nL\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} L\\text{pr}_2^*M\n\\right)\n= K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M\n$$\nin $D_\\QCoh(\\mathcal{O}_X)$ and a canonical isomorphism\n$$\n\\mathcal{O}_\\Delta \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L}\nL\\text{pr}_1^*K \\otimes_{\\mathcal{O}_{X \\times X}}^\\mathbf{L} L\\text{pr}_2^*M\n= \\Delta_*(K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)\n$$\nin $D_\\QCoh(X \\times X)$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Reduction to the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0T","source_file":"intersection.tex","source_line":1569,"source_end_line":1598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1569-L1598","statement_sha256":"7eb0e99d2b88724851243d03978251972e5b155c48c0be1742cecdccbeda7b27","origin":"The Stacks Project","memory_eligible":false,"source_rank":8437,"rank":8437,"depth":32,"x":2516.459,"y":813.263,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0U","tag":"0B0U","title":"Reduction to the diagonal · Lemma 0B0U","summary":"Let X be a nonsingular variety. Let α, resp. β be an r-cycle, resp. s-cycle on X. Assume α and β intersect properly. Then • α × β and [Δ] intersect properly • we have Δ_*(α · β) = [Δ] · α×β as cycles on X × X, • if X is proper, then pr_1, *([Δ] · α×β) = α·β, where pr_1 : X× X → X is the projection.","statement_latex":"Let $X$ be a nonsingular variety. Let $\\alpha$, resp.\\ $\\beta$\nbe an $r$-cycle, resp.\\ $s$-cycle on $X$. Assume $\\alpha$ and $\\beta$\nintersect properly. Then\n\\begin{enumerate}\n\\item $\\alpha \\times \\beta$ and $[\\Delta]$ intersect properly\n\\item we have $\\Delta_*(\\alpha \\cdot \\beta) = [\\Delta] \\cdot \\alpha\\times\\beta$\nas cycles on $X \\times X$,\n\\item if $X$ is proper, then\n$\\text{pr}_{1, *}([\\Delta] \\cdot \\alpha\\times\\beta) = \\alpha\\cdot\\beta$,\nwhere $pr_1 : X\\times X \\to X$ is the projection.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Reduction to the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0U","source_file":"intersection.tex","source_line":1677,"source_end_line":1690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1677-L1690","statement_sha256":"34b187d4c54384ec9cff3883e9e394fb1bbbb1eb7223b8efd6c57207e02952e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8438,"rank":8438,"depth":34,"x":2345.838,"y":602.357,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0V","tag":"0B0V","title":"Reduction to the diagonal · Proposition 0B0V","summary":"This is one of the main results of [Serre_algebre_locale]. Let X be a nonsingular variety. Let V ⊂ X and W ⊂ Y be closed subvarieties which intersect properly. Let Z ⊂ V ∩ W be an irreducible component. Then e(X, V · W, Z) > 0.","statement_latex":"\\begin{reference}\nThis is one of the main results of \\cite{Serre_algebre_locale}.\n\\end{reference}\nLet $X$ be a nonsingular variety. Let $V \\subset X$ and\n$W \\subset Y$ be closed subvarieties which intersect properly.\nLet $Z \\subset V \\cap W$ be an irreducible component.\nThen $e(X, V \\cdot W, Z) > 0$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Reduction to the diagonal","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0V","source_file":"intersection.tex","source_line":1735,"source_end_line":1744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1735-L1744","statement_sha256":"bd69e2955839412d40fc7b08614060a55e31719f8377909bb826a5e113c94ee9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8439,"rank":8439,"depth":41,"x":2641.501,"y":661.11,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0W","tag":"0B0W","title":"Reduction to the diagonal · Lemma 0B0W","summary":"[Serre_algebre_locale] Let X be a nonsingular variety. Let F and G be coherent sheaves on X with dim(Supp(F)) ≤ r, dim(Supp(G)) ≤ s, and dim(Supp(F) ∩ Supp(G) ) ≤ r + s - dim X. In this case [F]_r and [G]_s intersect properly and [F]_r · [G]_s = ∑ (-1)^p [Tor_p^O_X(F, G)]_r + s - dim(X).","statement_latex":"\\begin{reference}\n\\cite[Chapter V]{Serre_algebre_locale}\n\\end{reference}\nLet $X$ be a nonsingular variety. Let $\\mathcal{F}$ and\n$\\mathcal{G}$ be coherent sheaves on $X$ with\n$\\dim(\\text{Supp}(\\mathcal{F})) \\leq r$,\n$\\dim(\\text{Supp}(\\mathcal{G})) \\leq s$, and\n$\\dim(\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G}) )\n\\leq r + s - \\dim X$. In this case $[\\mathcal{F}]_r$ and $[\\mathcal{G}]_s$\nintersect properly and\n$$\n[\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s =\n\\sum (-1)^p\n[\\text{Tor}_p^{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})]_{r + s - \\dim(X)}.\n$$","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Reduction to the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0W","source_file":"intersection.tex","source_line":1762,"source_end_line":1779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1762-L1779","statement_sha256":"9dcfbdddbfa834241ba9854d8904a94d33c840f029e5fdb8f00a47cfcc67ef48","origin":"The Stacks Project","memory_eligible":false,"source_rank":8440,"rank":8440,"depth":41,"x":2376.016,"y":785.656,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1L","tag":"0B1L","title":"Associativity of intersections · Lemma 0B1L","summary":"Let X be a nonsingular variety. Let U, V, W be closed subvarieties. Assume that U, V, W intersect properly pairwise and that dim(U ∩ V ∩ W) ≤ dim(U) + dim(V) + dim(W) - 2dim(X). Then U · (V · W) = (U · V) · W as cycles on X.","statement_latex":"Let $X$ be a nonsingular variety. Let $U, V, W$ be closed\nsubvarieties. Assume that $U, V, W$ intersect properly pairwise\nand that $\\dim(U \\cap V \\cap W) \\leq \\dim(U) + \\dim(V) + \\dim(W) - 2\\dim(X)$.\nThen\n$$\nU \\cdot (V \\cdot W) = (U \\cdot V) \\cdot W\n$$\nas cycles on $X$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Associativity of intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1L","source_file":"intersection.tex","source_line":1905,"source_end_line":1915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1905-L1915","statement_sha256":"b3c4bd07a2c9a9d6380a29983c25bb6bc24c4a4a0ec8181f684759060c7a7a4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8441,"rank":8441,"depth":42,"x":2471.704,"y":542.975,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0Y","tag":"0B0Y","title":"Flat pullback and intersection products · Lemma 0B0Y","summary":"Let f : X → Y be a flat morphism of nonsingular varieties. Set e = dim(X) - dim(Y). Let F and G be coherent sheaves on Y with dim(Supp(F)) ≤ r, dim(Supp(G)) ≤ s, and dim(Supp(F) ∩ Supp(G) ) ≤ r + s - dim(Y). In this case the cycles [f^*F]_r + e and [f^*G]_s + e intersect properly and f^*([F]_r · [G]_s) = [f^*F]_r + e · [f^*G]_s + e","statement_latex":"Let $f : X \\to Y$ be a flat morphism of nonsingular varieties. Set\n$e = \\dim(X) - \\dim(Y)$. Let $\\mathcal{F}$ and $\\mathcal{G}$ be coherent\nsheaves on $Y$ with $\\dim(\\text{Supp}(\\mathcal{F})) \\leq r$,\n$\\dim(\\text{Supp}(\\mathcal{G})) \\leq s$, and\n$\\dim(\\text{Supp}(\\mathcal{F}) \\cap \\text{Supp}(\\mathcal{G}) )\n\\leq r + s - \\dim(Y)$. In this case the cycles\n$[f^*\\mathcal{F}]_{r + e}$ and $[f^*\\mathcal{G}]_{s + e}$\nintersect properly and\n$$\nf^*([\\mathcal{F}]_r \\cdot [\\mathcal{G}]_s) =\n[f^*\\mathcal{F}]_{r + e} \\cdot [f^*\\mathcal{G}]_{s + e}\n$$","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Flat pullback and intersection products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0Y","source_file":"intersection.tex","source_line":1984,"source_end_line":1998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L1984-L1998","statement_sha256":"a004bf814cfb5ca9f09a0e3c6c05c1e7cf3323ec08d8744fd4b555ee1a265492","origin":"The Stacks Project","memory_eligible":false,"source_rank":8442,"rank":8442,"depth":42,"x":2596.405,"y":776.412,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0Z","tag":"0B0Z","title":"Flat pullback and intersection products · Lemma 0B0Z","summary":"Let f : X → Y be a flat morphism of nonsingular varieties. Let α be a r-cycle on Y and β an s-cycle on Y. Assume that α and β intersect properly. Then f^*α and f^*β intersect properly and f^*( α · β ) = f^*α · f^*β.","statement_latex":"Let $f : X \\to Y$ be a flat morphism of nonsingular varieties.\nLet $\\alpha$ be a $r$-cycle on $Y$ and $\\beta$ an $s$-cycle on $Y$.\nAssume that $\\alpha$ and $\\beta$ intersect properly. Then $f^*\\alpha$\nand $f^*\\beta$ intersect properly and\n$f^*( \\alpha \\cdot \\beta ) = f^*\\alpha \\cdot f^*\\beta$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Flat pullback and intersection products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0Z","source_file":"intersection.tex","source_line":2016,"source_end_line":2023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2016-L2023","statement_sha256":"51598bb1788312f75c74346c1dfd6c5b41f4281b0ec4cb2c9c1e62f25c863b41","origin":"The Stacks Project","memory_eligible":false,"source_rank":8443,"rank":8443,"depth":43,"x":2316.499,"y":674.954,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B10","tag":"0B10","title":"Projection formula for flat proper morphisms · Lemma 0B10","summary":"See [Serre_algebre_locale] for a more general formula. Let f : X → Y be a flat proper morphism of nonsingular varieties. Set e = dim(X) - dim(Y). Let α be an r-cycle on X and let β be a s-cycle on Y. Assume that α and f^*(β) intersect properly. Then f_*(α) and β intersect properly and f_*(α) · β = f_*( α · f^*β)","statement_latex":"\\begin{reference}\nSee \\cite[Chapter V, C), Section 7, formula (10)]{Serre_algebre_locale}\nfor a more general formula.\n\\end{reference}\nLet $f : X \\to Y$ be a flat proper morphism of nonsingular varieties.\nSet $e = \\dim(X) - \\dim(Y)$. Let $\\alpha$ be an $r$-cycle on $X$ and let\n$\\beta$ be a $s$-cycle on $Y$. Assume that $\\alpha$ and $f^*(\\beta)$ intersect\nproperly. Then $f_*(\\alpha)$ and $\\beta$ intersect properly and\n$$\nf_*(\\alpha) \\cdot \\beta = f_*( \\alpha \\cdot f^*\\beta)\n$$","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projection formula for flat proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B10","source_file":"intersection.tex","source_line":2042,"source_end_line":2055,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2042-L2055","statement_sha256":"55c132683f137e97caaa89d497e9edcd24adfdef6a1a800a7014fa68afc355fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8444,"rank":8444,"depth":42,"x":2604.723,"y":590.873,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1M","tag":"0B1M","title":"Projection formula for flat proper morphisms · Lemma 0B1M","summary":"Let X → P be a closed immersion of nonsingular varieties. Let C' ⊂ P × P^1 be a closed subvariety of dimension r + 1. Assume • the fibre C = C'_0 has dimension r, i.e., C' → P^1 is dominant, • C' intersects X × P^1 properly, • [C]_r intersects X properly. Then setting α = [C]_r · X viewed as cycle on X and β = C' · X × P^1 viewed as cycle on X × P^1, we have α = pr_X, *(β · X × 0) as cycles on X where pr_X : X × P^1 → X is the projection.","statement_latex":"Let $X \\to P$ be a closed immersion of nonsingular varieties.\nLet $C' \\subset P \\times \\mathbf{P}^1$ be a closed subvariety of dimension\n$r + 1$. Assume\n\\begin{enumerate}\n\\item the fibre $C = C'_0$ has dimension $r$, i.e., $C' \\to \\mathbf{P}^1$\nis dominant,\n\\item $C'$ intersects $X \\times \\mathbf{P}^1$ properly,\n\\item $[C]_r$ intersects $X$ properly.\n\\end{enumerate}\nThen setting $\\alpha = [C]_r \\cdot X$ viewed as cycle on $X$ and\n$\\beta = C' \\cdot X \\times \\mathbf{P}^1$ viewed as cycle on\n$X \\times \\mathbf{P}^1$, we have\n$$\n\\alpha = \\text{pr}_{X, *}(\\beta \\cdot X \\times 0)\n$$\nas cycles on $X$ where $\\text{pr}_X : X \\times \\mathbf{P}^1 \\to X$ is the\nprojection.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projection formula for flat proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1M","source_file":"intersection.tex","source_line":2145,"source_end_line":2164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2145-L2164","statement_sha256":"51b744eff7d27b5b34a730e1e5033cb1a94bb43feffba212061abcd2257cc1a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8445,"rank":8445,"depth":43,"x":2459.688,"y":816.603,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1P","tag":"0B1P","title":"Projections · Lemma 0B1P","summary":"Let V be a vector space of dimension n + 1. Let X ⊂ P(V) be a closed subscheme. If X not = P(V), then there is a nonempty Zariski open U ⊂ P(V) such that for all closed points p ∈ U the restriction of the projection r_p defines a finite morphism r_p|_X : X → P(W_p).","statement_latex":"Let $V$ be a vector space of dimension $n + 1$.\nLet $X \\subset \\mathbf{P}(V)$ be a closed subscheme.\nIf $X \\not = \\mathbf{P}(V)$, then there is a nonempty Zariski open\n$U \\subset \\mathbf{P}(V)$\nsuch that for all closed points $p \\in U$ the restriction\nof the projection $r_p$ defines a finite morphism\n$r_p|_X : X \\to \\mathbf{P}(W_p)$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1P","source_file":"intersection.tex","source_line":2233,"source_end_line":2242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2233-L2242","statement_sha256":"6da2ee47a06873cf224262fdd020247806b555308d1b7cabd942b68dee84b1d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8446,"rank":8446,"depth":40,"x":2385.048,"y":567.653,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1Q","tag":"0B1Q","title":"Projections · Lemma 0B1Q","summary":"Let V be a vector space of dimension n + 1. Let X ⊂ P(V) be a closed subvariety. Let x ∈ X be a nonsingular point. • If dim(X) < n - 1, then there is a nonempty Zariski open U ⊂ P(V) such that for all closed points p ∈ U the morphism r_p|_X : X → r_p(X) is an isomorphism over an open neighbourhood of r_p(x). • If dim(X) = n - 1, then there is a nonempty Zariski open U ⊂ P(V) such that for all closed points p ∈ U the morphism r_p|_X : X → P(W_p) is étale at x.","statement_latex":"Let $V$ be a vector space of dimension $n + 1$.\nLet $X \\subset \\mathbf{P}(V)$ be a closed subvariety.\nLet $x \\in X$ be a nonsingular point.\n\\begin{enumerate}\n\\item If $\\dim(X) < n - 1$, then there is a nonempty Zariski open\n$U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the\nmorphism $r_p|_X : X \\to r_p(X)$ is an\nisomorphism over an open neighbourhood of $r_p(x)$.\n\\item If $\\dim(X) = n - 1$, then there is a nonempty Zariski open\n$U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the\nmorphism $r_p|_X : X \\to \\mathbf{P}(W_p)$ is \\'etale at $x$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1Q","source_file":"intersection.tex","source_line":2257,"source_end_line":2271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2257-L2271","statement_sha256":"dc97a953a1e36eff12a6ffc7f99bb2782a76104ae41a4d5ec2a634d95680bcb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8447,"rank":8447,"depth":44,"x":2640.494,"y":708.988,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H8A","tag":"0H8A","title":"Projections · Lemma 0H8A","summary":"Let V be a vector space. Let Z ⊂ X ⊂ P(V) be closed subvarieties with Z not = X not = P(V). Let x ∈ Z be a point which is nonsingular on X. Then there is a nonempty Zariski open U ⊂ P(V) such that for all closed points p ∈ U the morphism r_p|_Z : Z → r_p(Z) is an isomorphism over an open neighbourhood of r_p(x).","statement_latex":"Let $V$ be a vector space. Let $Z \\subset X \\subset \\mathbf{P}(V)$ be closed\nsubvarieties with $Z \\not = X \\not = \\mathbf{P}(V)$. Let $x \\in Z$ be a point\nwhich is nonsingular on $X$. Then there is a nonempty Zariski open\n$U \\subset \\mathbf{P}(V)$ such that for all closed points $p \\in U$ the\nmorphism $r_p|_Z : Z \\to r_p(Z)$ is an isomorphism over an open neighbourhood\nof $r_p(x)$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8A","source_file":"intersection.tex","source_line":2318,"source_end_line":2326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2318-L2326","statement_sha256":"2e5c7c58c09a1b465c65505efc5ee938151b15a55c26f852912e6cfe8929caeb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8448,"rank":8448,"depth":45,"x":2338.228,"y":749.749,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1R","tag":"0B1R","title":"Projections · Lemma 0B1R","summary":"Let V be a vector space of dimension n + 1. Let Y, Z ⊂ P(V) be closed subvarieties. There is a nonempty Zariski open U ⊂ P(V) such that for all closed points p ∈ U we have Y ∩ r_p^-1(r_p(Z)) = (Y ∩ Z) ∪ E with E ⊂ Y closed and dim(E) ≤ dim(Y) + dim(Z) + 1 - n.","statement_latex":"Let $V$ be a vector space of dimension $n + 1$.\nLet $Y, Z \\subset \\mathbf{P}(V)$ be closed subvarieties.\nThere is a nonempty Zariski open $U \\subset \\mathbf{P}(V)$\nsuch that for all closed points $p \\in U$ we have\n$$\nY \\cap r_p^{-1}(r_p(Z)) = (Y \\cap Z) \\cup E\n$$\nwith $E \\subset Y$ closed and\n$\\dim(E) \\leq \\dim(Y) + \\dim(Z) + 1 - n$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1R","source_file":"intersection.tex","source_line":2340,"source_end_line":2351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2340-L2351","statement_sha256":"1e1b81f127c126130b755efbcdb3f4c45fc0d180ae66a3577f5c8fc358a6f655","origin":"The Stacks Project","memory_eligible":false,"source_rank":8449,"rank":8449,"depth":35,"x":2528.49,"y":548.016,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B2T","tag":"0B2T","title":"Projections · Lemma 0B2T","summary":"Let V be a vector space. Let B ⊂ P(V) be a closed subvariety of codimension ≥ 2. Let p ∈ P(V) be a closed point, p not ∈ B. Then there exists a line ℓ ⊂ P(V) with p ∈ ℓ and ℓ ∩ B = ∅. Moreover, these lines sweep out an open subset of P(V).","statement_latex":"Let $V$ be a vector space. Let $B \\subset \\mathbf{P}(V)$\nbe a closed subvariety of codimension $\\geq 2$.\nLet $p \\in \\mathbf{P}(V)$ be a closed point, $p \\not \\in B$.\nThen there exists a line $\\ell \\subset \\mathbf{P}(V)$\nwith $p \\in \\ell$ and $\\ell \\cap B = \\emptyset$. Moreover, these lines\nsweep out an open subset of $\\mathbf{P}(V)$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2T","source_file":"intersection.tex","source_line":2379,"source_end_line":2387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2379-L2387","statement_sha256":"4579145cca535a1680a9892004ac42672360f6d5485cb0b2374b78384a00dfbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":8450,"rank":8450,"depth":0,"x":2550.44,"y":804.939,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B2U","tag":"0B2U","title":"Projections · Lemma 0B2U","summary":"Let V be a vector space. Let G = PGL(V). Then G × P(V) → P(V) is doubly transitive.","statement_latex":"Let $V$ be a vector space. Let $G = \\text{PGL}(V)$.\nThen $G \\times \\mathbf{P}(V) \\to \\mathbf{P}(V)$ is\ndoubly transitive.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2U","source_file":"intersection.tex","source_line":2398,"source_end_line":2403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2398-L2403","statement_sha256":"b9b90a55e6b2413fff88fa0f4c66a3a069a56daba7834cef483848aaa6c7cb13","origin":"The Stacks Project","memory_eligible":false,"source_rank":8451,"rank":8451,"depth":0,"x":2327.463,"y":627.799,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B2V","tag":"0B2V","title":"Projections · Lemma 0B2V","summary":"Let k be a field. Let n ≥ 1 be an integer and let x_ij, 1 ≤ i, j ≤ n be variables. Then det ( x_11 & x_12 & … & x_1n x_21 & … & … & … … & … & … & … x_n1 & … & … & x_nn ) is an irreducible element of the polynomial ring k[x_ij].","statement_latex":"Let $k$ be a field. Let $n \\geq 1$ be an integer and let\n$x_{ij}, 1 \\leq i, j \\leq n$ be variables. Then\n$$\n\\det\n\\left(\n\\begin{matrix}\nx_{11} & x_{12} & \\ldots & x_{1n} \\\\\nx_{21} & \\ldots & \\ldots & \\ldots \\\\\n\\ldots & \\ldots & \\ldots & \\ldots \\\\\nx_{n1} & \\ldots & \\ldots & x_{nn}\n\\end{matrix}\n\\right)\n$$\nis an irreducible element of the polynomial ring $k[x_{ij}]$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2V","source_file":"intersection.tex","source_line":2410,"source_end_line":2426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2410-L2426","statement_sha256":"14ea4bd8f3baf1dae8d78260397e0795aff68dc37cc638b5b7e23e34a16b4436","origin":"The Stacks Project","memory_eligible":false,"source_rank":8452,"rank":8452,"depth":0,"x":2634.582,"y":631.9,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1S","tag":"0B1S","title":"Projections · Lemma 0B1S","summary":"With notation as above. Let X, Y be closed subvarieties of P(V) which intersect properly such that X not = P(V) and X ∩ Y not = ∅. For a general line ℓ ⊂ P with [id_V] ∈ ℓ we have • X ⊂ U_g for all [g] ∈ ℓ, • g(X) intersects Y properly for all [g] ∈ ℓ.","statement_latex":"With notation as above. Let $X, Y$ be closed subvarieties of $\\mathbf{P}(V)$\nwhich intersect properly such that $X \\not = \\mathbf{P}(V)$ and\n$X \\cap Y \\not = \\emptyset$. For a general line $\\ell \\subset \\mathbf{P}$\nwith $[\\text{id}_V] \\in \\ell$ we have\n\\begin{enumerate}\n\\item $X \\subset U_g$ for all $[g] \\in \\ell$,\n\\item $g(X)$ intersects $Y$ properly for all $[g] \\in \\ell$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Projections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1S","source_file":"intersection.tex","source_line":2505,"source_end_line":2515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2505-L2515","statement_sha256":"78690716cb06f996cdef0aea447c08f10d55528a62facd0fc6d318e9d1140088","origin":"The Stacks Project","memory_eligible":false,"source_rank":8453,"rank":8453,"depth":36,"x":2404.633,"y":803.281,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B0E","tag":"0B0E","title":"Moving Lemma · Lemma 0B0E","summary":"See [Roberts]. Let X ⊂ P^N be a nonsingular closed subvariety. Let n = dim(X) and 0 ≤ d, d' < n. Let Z ⊂ X be a closed subvariety of dimension d and T_i ⊂ X, i ∈ I be a finite collection of closed subvarieties of dimension d'. Then there exists a subvariety C ⊂ P^N such that C intersects X properly and such that C · X = Z + ∑_j ∈ J m_j Z_j where Z_j ⊂ X are irreducible of dimension d, distinct from Z, and dim(Z_j ∩ T_i) ≤ dim(Z ∩ T_i) with strict inequality if Z does not…","statement_latex":"\\begin{reference}\nSee \\cite{Roberts}.\n\\end{reference}\nLet $X \\subset \\mathbf{P}^N$ be a nonsingular closed subvariety.\nLet $n = \\dim(X)$ and $0 \\leq d, d' < n$. Let $Z \\subset X$ be a closed\nsubvariety of dimension $d$ and $T_i \\subset X$, $i \\in I$ be a finite\ncollection of closed subvarieties of dimension $d'$. Then there exists\na subvariety $C \\subset \\mathbf{P}^N$ such that $C$ intersects $X$\nproperly and such that\n$$\nC \\cdot X = Z + \\sum\\nolimits_{j \\in J} m_j Z_j\n$$\nwhere $Z_j \\subset X$ are irreducible of dimension $d$, distinct from $Z$, and\n$$\n\\dim(Z_j \\cap T_i) \\leq \\dim(Z \\cap T_i)\n$$\nwith strict inequality if $Z$ does not intersect $T_i$ properly in $X$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Moving Lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0E","source_file":"intersection.tex","source_line":2591,"source_end_line":2610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2591-L2610","statement_sha256":"ae1f680350acb437b7091afd586ad44340cbf6ee6264bb2edf14a5ff35edbe9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8454,"rank":8454,"depth":46,"x":2436.4,"y":546.221,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1T","tag":"0B1T","title":"Moving Lemma · Lemma 0B1T","summary":"Let C ⊂ P^N be a closed subvariety. Let X ⊂ P^N be subvariety and let T_i ⊂ X be a finite collection of closed subvarieties. Assume that C and X intersect properly. Then there exists a closed subvariety C' ⊂ P^N × P^1 such that • C' → P^1 is dominant, • C'_0 = C scheme theoretically, • C' and X × P^1 intersect properly, • C'_∞ properly intersects each of the given T_i.","statement_latex":"Let $C \\subset \\mathbf{P}^N$ be a closed subvariety.\nLet $X \\subset \\mathbf{P}^N$ be subvariety and let $T_i \\subset X$\nbe a finite collection of closed subvarieties.\nAssume that $C$ and $X$ intersect properly.\nThen there exists a closed subvariety\n$C' \\subset \\mathbf{P}^N \\times \\mathbf{P}^1$ such that\n\\begin{enumerate}\n\\item $C' \\to \\mathbf{P}^1$ is dominant,\n\\item $C'_0 = C$ scheme theoretically,\n\\item $C'$ and $X \\times \\mathbf{P}^1$ intersect properly,\n\\item $C'_\\infty$ properly intersects each of the given $T_i$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Moving Lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1T","source_file":"intersection.tex","source_line":2699,"source_end_line":2713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2699-L2713","statement_sha256":"7f418bd792753fbe838ce3d23ad5e0d4fe31b397b21ce73b57dac0e10512186e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8455,"rank":8455,"depth":37,"x":2619.844,"y":753.966,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1U","tag":"0B1U","title":"Moving Lemma · Lemma 0B1U","summary":"Let X be a nonsingular projective variety. Let α be an r-cycle and β be an s-cycle on X. Then there exists an r-cycle α' such that α' sim_rat α and such that α' and β intersect properly.","statement_latex":"Let $X$ be a nonsingular projective variety. Let $\\alpha$ be an\n$r$-cycle and $\\beta$ be an $s$-cycle on $X$. Then there exists\nan $r$-cycle $\\alpha'$ such that $\\alpha' \\sim_{rat} \\alpha$ and\nsuch that $\\alpha'$ and $\\beta$ intersect properly.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Moving Lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1U","source_file":"intersection.tex","source_line":2743,"source_end_line":2749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2743-L2749","statement_sha256":"f85f8c5cccd1e5d855830cffed31da708c6c6b39131de38cdea126e4cb505eca","origin":"The Stacks Project","memory_eligible":false,"source_rank":8456,"rank":8456,"depth":47,"x":2317.269,"y":704.829,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B60","tag":"0B60","title":"Intersection products and rational equivalence · Lemma 0B60","summary":"Let X be a nonsingular variety. Let W ⊂ X × P^1 be an (s + 1)-dimensional subvariety dominating P^1. Let W_a, resp. W_b be the fibre of W → P^1 over a, resp. b. Let V be a r-dimensional subvariety of X such that V intersects both W_a and W_b properly. Then [V] · [W_a]_r sim_rat [V] · [W_b]_r.","statement_latex":"Let $X$ be a nonsingular variety. Let\n$W \\subset X \\times \\mathbf{P}^1$ be an $(s + 1)$-dimensional subvariety\ndominating $\\mathbf{P}^1$. Let $W_a$, resp.\\ $W_b$ be the fibre of\n$W \\to \\mathbf{P}^1$ over $a$, resp.\\ $b$. Let $V$ be a $r$-dimensional\nsubvariety of $X$ such that $V$ intersects both $W_a$ and\n$W_b$ properly. Then $[V] \\cdot [W_a]_r \\sim_{rat} [V] \\cdot [W_b]_r$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Intersection products and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B60","source_file":"intersection.tex","source_line":2793,"source_end_line":2801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2793-L2801","statement_sha256":"e2683a354fad0ae556ff2a6b805232632dae70af4b2825dba1d73f688f7d341f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8457,"rank":8457,"depth":43,"x":2580.109,"y":569.265,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B1V","tag":"0B1V","title":"Intersection products and rational equivalence · Theorem 0B1V","summary":"Let X be a nonsingular projective variety. Let α, resp. β be an r, resp. s cycle on X. Assume that α and β intersect properly so that α · β is defined. Finally, assume that α sim_rat 0. Then α · β sim_rat 0.","statement_latex":"Let $X$ be a nonsingular projective variety. Let $\\alpha$, resp.\\ $\\beta$\nbe an $r$, resp.\\ $s$ cycle on $X$. Assume that $\\alpha$ and $\\beta$\nintersect properly so that $\\alpha \\cdot \\beta$ is defined. Finally,\nassume that $\\alpha \\sim_{rat} 0$. Then $\\alpha \\cdot \\beta \\sim_{rat} 0$.","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Intersection products and rational equivalence","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B1V","source_file":"intersection.tex","source_line":2915,"source_end_line":2921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L2915-L2921","statement_sha256":"05a3b483371fabcc18dbb62afc890cb0dea4f070c2e283880719d341b9fc6517","origin":"The Stacks Project","memory_eligible":false,"source_rank":8458,"rank":8458,"depth":47,"x":2495.244,"y":818.569,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B2X","tag":"0B2X","title":"Pullback for a general morphism · Lemma 0B2X","summary":"Let f : X → Y be a morphism of nonsingular projective varieties. The pullback map on chow groups satisfies: • f^* : CH^*(Y) → CH^*(X) is a ring map, • (g ∘ f)^* = f^* ∘ g^* for a composable pair f, g, • the projection formula holds: f_*(α) · β = f_*( α · f^*β), and • if f is flat then it agrees with the previous definition.","statement_latex":"Let $f : X \\to Y$ be a morphism of nonsingular projective varieties.\nThe pullback map on chow groups satisfies:\n\\begin{enumerate}\n\\item $f^* : \\CH^*(Y) \\to \\CH^*(X)$ is a ring map,\n\\item $(g \\circ f)^* = f^* \\circ g^*$ for a composable pair $f, g$,\n\\item the projection formula holds: $f_*(\\alpha) \\cdot \\beta =\nf_*( \\alpha \\cdot f^*\\beta)$, and\n\\item if $f$ is flat then it agrees with the previous definition.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Intersection Theory","chapter_id":"intersection","section":"Pullback for a general morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2X","source_file":"intersection.tex","source_line":3100,"source_end_line":3111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/intersection.tex#L3100-L3111","statement_sha256":"0fde97dbab62404efddfbc0643993d829d33a1cdc56a39e8849c2ad11c66bfe5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8459,"rank":8459,"depth":43,"x":2357.228,"y":586.396,"cluster":"divisors-intersection-theory"},{"id":"stacks:0B95","tag":"0B95","title":"Hilbert scheme of points · Lemma 0B95","summary":"Let X → S be a morphism of schemes. The functor Hilbfunctor^d_X/S satisfies the sheaf property for the fpqc topology (Topologies, Definition [Tag 022G]).","statement_latex":"Let $X \\to S$ be a morphism of schemes. The functor $\\Hilbfunctor^d_{X/S}$\nsatisfies the sheaf property for the fpqc topology\n(Topologies, Definition \\ref{topologies-definition-sheaf-property-fpqc}).","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Hilbert scheme of points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B95","source_file":"pic.tex","source_line":64,"source_end_line":69,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L64-L69","statement_sha256":"ca526033f33a77ce260e7b58a2986b255aa176e6fd9ac0efd4465dc79687ebc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8460,"rank":8460,"depth":47,"x":2487.689,"y":1600.0,"cluster":"moduli-theory"},{"id":"stacks:0B96","tag":"0B96","title":"Hilbert scheme of points · Lemma 0B96","summary":"Let X → S be a morphism of schemes. If X → S is of finite presentation, then the functor Hilbfunctor^d_X/S is limit preserving (Limits, Remark [Tag 05LX]).","statement_latex":"Let $X \\to S$ be a morphism of schemes. If $X \\to S$ is\nof finite presentation, then the functor $\\Hilbfunctor^d_{X/S}$\nis limit preserving (Limits, Remark \\ref{limits-remark-limit-preserving}).","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Hilbert scheme of points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B96","source_file":"pic.tex","source_line":88,"source_end_line":93,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L88-L93","statement_sha256":"8ac200ed6df335a3ac1aad81ad86a1d5ebf907a0cbe4fe788a176e1591e82f60","origin":"The Stacks Project","memory_eligible":false,"source_rank":8461,"rank":8461,"depth":37,"x":2470.18,"y":1607.557,"cluster":"moduli-theory"},{"id":"stacks:0B97","tag":"0B97","title":"Hilbert scheme of points · Lemma 0B97","summary":"Let S be a scheme. Let i : X → Y be a closed immersion of schemes. If Hilbfunctor^d_Y/S is representable by a scheme, so is Hilbfunctor^d_X/S and the corresponding morphism of schemes underlineHilbfunctor^d_X/S → underlineHilbfunctor^d_Y/S is a closed immersion.","statement_latex":"Let $S$ be a scheme. Let $i : X \\to Y$ be a closed immersion of schemes.\nIf $\\Hilbfunctor^d_{Y/S}$ is representable by a scheme, so is\n$\\Hilbfunctor^d_{X/S}$ and the corresponding morphism of schemes\n$\\underline{\\Hilbfunctor}^d_{X/S} \\to \\underline{\\Hilbfunctor}^d_{Y/S}$\nis a closed immersion.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Hilbert scheme of points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B97","source_file":"pic.tex","source_line":123,"source_end_line":130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L123-L130","statement_sha256":"f8ab3ae4c9cd68732a3e0046c6cd6f86a3678f6c682c5f41b9103fc400e2fae4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8462,"rank":8462,"depth":59,"x":2481.503,"y":1585.613,"cluster":"moduli-theory"},{"id":"stacks:0B98","tag":"0B98","title":"Hilbert scheme of points · Lemma 0B98","summary":"Let X → S be a morphism of schemes. If X → S is separated and Hilbfunctor^d_X/S is representable, then underlineHilbfunctor^d_X/S → S is separated.","statement_latex":"Let $X \\to S$ be a morphism of schemes. If $X \\to S$ is separated and\n$\\Hilbfunctor^d_{X/S}$ is representable,\nthen $\\underline{\\Hilbfunctor}^d_{X/S} \\to S$ is separated.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Hilbert scheme of points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B98","source_file":"pic.tex","source_line":163,"source_end_line":168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L163-L168","statement_sha256":"8e8bc052351afe5cdf2a3273eed008ab3697836d9823ee7dc6fb5504837ef066","origin":"The Stacks Project","memory_eligible":false,"source_rank":8463,"rank":8463,"depth":60,"x":2492.378,"y":1613.561,"cluster":"moduli-theory"},{"id":"stacks:0B99","tag":"0B99","title":"Hilbert scheme of points · Lemma 0B99","summary":"Let X → S be a morphism of affine schemes. Let d ≥ 0. Then Hilbfunctor^d_X/S is representable.","statement_latex":"Let $X \\to S$ be a morphism of affine schemes. Let $d \\geq 0$. Then\n$\\Hilbfunctor^d_{X/S}$ is representable.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Hilbert scheme of points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B99","source_file":"pic.tex","source_line":193,"source_end_line":197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L193-L197","statement_sha256":"945939442b8ac3db2cf7a9fb351a96c6716dc411b8a8f9f088b6b79992794c60","origin":"The Stacks Project","memory_eligible":false,"source_rank":8464,"rank":8464,"depth":60,"x":2457.285,"y":1596.625,"cluster":"moduli-theory"},{"id":"stacks:0B9A","tag":"0B9A","title":"Hilbert scheme of points · Proposition 0B9A","summary":"Let X → S be a morphism of schemes. Let d ≥ 0. Assume for all (s, x_1, …, x_d) where s ∈ S and x_1, …, x_d ∈ X_s there exists an affine open U ⊂ X with x_1, …, x_d ∈ U. Then Hilbfunctor^d_X/S is representable by a scheme.","statement_latex":"Let $X \\to S$ be a morphism of schemes. Let $d \\geq 0$. Assume\nfor all $(s, x_1, \\ldots, x_d)$ where $s \\in S$ and\n$x_1, \\ldots, x_d \\in X_s$ there exists an affine open $U \\subset X$\nwith $x_1, \\ldots, x_d \\in U$. Then $\\Hilbfunctor^d_{X/S}$ is\nrepresentable by a scheme.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Hilbert scheme of points","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9A","source_file":"pic.tex","source_line":298,"source_end_line":305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L298-L305","statement_sha256":"65f0df08c461c4e42bc3c39b22c34e5a07cb089999650302c102bed6a1630ec6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8465,"rank":8465,"depth":61,"x":2501.517,"y":1588.503,"cluster":"moduli-theory"},{"id":"stacks:0B9D","tag":"0B9D","title":"Moduli of divisors on smooth curves · Lemma 0B9D","summary":"Let X → S be a smooth morphism of schemes of relative dimension 1. Let D ⊂ X be a closed subscheme. Consider the following conditions • D → S is finite locally free, • D is a relative effective Cartier divisor on X/S, • D → S is locally quasi-finite, flat, and locally of finite presentation, and • D → S is locally quasi-finite and flat. We always have the implications (1) ⇒ (2) ⇔ (3) ⇒ (4) If S is locally Noetherian, then the last arrow is an if and only if. If X → S is…","statement_latex":"Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$.\nLet $D \\subset X$ be a closed subscheme. Consider the following conditions\n\\begin{enumerate}\n\\item $D \\to S$ is finite locally free,\n\\item $D$ is a relative effective Cartier divisor on $X/S$,\n\\item $D \\to S$ is locally quasi-finite, flat, and\nlocally of finite presentation, and\n\\item $D \\to S$ is locally quasi-finite and flat.\n\\end{enumerate}\nWe always have the implications\n$$\n(1) \\Rightarrow (2) \\Leftrightarrow (3) \\Rightarrow (4)\n$$\nIf $S$ is locally Noetherian, then the last arrow is an if and only if.\nIf $X \\to S$ is proper (and $S$ arbitrary), then the first arrow is\nan if and only if.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Moduli of divisors on smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9D","source_file":"pic.tex","source_line":407,"source_end_line":425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L407-L425","statement_sha256":"13c8329ce7e6486d656723c1aeffe486ce37e69aaad2b56b1f7e8d740f32f136","origin":"The Stacks Project","memory_eligible":false,"source_rank":8466,"rank":8466,"depth":45,"x":2472.803,"y":1622.489,"cluster":"moduli-theory"},{"id":"stacks:0B9E","tag":"0B9E","title":"Moduli of divisors on smooth curves · Lemma 0B9E","summary":"Let X → S be a smooth morphism of schemes of relative dimension 1. Let D_1, D_2 ⊂ X be closed subschemes finite locally free of degrees d_1, d_2 over S. Then D_1 + D_2 is finite locally free of degree d_1 + d_2 over S.","statement_latex":"Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$.\nLet $D_1, D_2 \\subset X$ be closed subschemes finite locally free of\ndegrees $d_1$, $d_2$ over $S$. Then $D_1 + D_2$ is finite locally free\nof degree $d_1 + d_2$ over $S$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Moduli of divisors on smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9E","source_file":"pic.tex","source_line":460,"source_end_line":466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L460-L466","statement_sha256":"ec55b8e8f43b94531a4368cb10c31edbca050d7739480be6bb66c5603345fde3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8467,"rank":8467,"depth":46,"x":2466.274,"y":1577.801,"cluster":"moduli-theory"},{"id":"stacks:0B9F","tag":"0B9F","title":"Moduli of divisors on smooth curves · Lemma 0B9F","summary":"Let X → S be a smooth morphism of schemes of relative dimension 1. Let D_1, D_2 ⊂ X be closed subschemes finite locally free of degrees d_1, d_2 over S. If D_1 ⊂ D_2 (as closed subschemes) then there is a closed subscheme D ⊂ X finite locally free of degree d_2 - d_1 over S such that D_2 = D_1 + D.","statement_latex":"Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$.\nLet $D_1, D_2 \\subset X$ be closed subschemes finite locally free of\ndegrees $d_1$, $d_2$ over $S$. If $D_1 \\subset D_2$ (as closed subschemes)\nthen there is a closed subscheme $D \\subset X$ finite locally free of\ndegree $d_2 - d_1$ over $S$ such that $D_2 = D_1 + D$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Moduli of divisors on smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9F","source_file":"pic.tex","source_line":508,"source_end_line":515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L508-L515","statement_sha256":"08d3a0563ea29887bff75527ae50d653936dcc0d41b88f6aeb4b6e49e45e35d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8468,"rank":8468,"depth":47,"x":2509.779,"y":1609.135,"cluster":"moduli-theory"},{"id":"stacks:0B9G","tag":"0B9G","title":"Moduli of divisors on smooth curves · Lemma 0B9G","summary":"Let X → S be a smooth morphism of schemes of relative dimension 1 such that the functors Hilbfunctor^d_X/S are representable. The morphism underlineHilbfunctor^d_X/S ×_S X → underlineHilbfunctor^d + 1_X/S is finite locally free of degree d + 1.","statement_latex":"Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$\nsuch that the functors $\\Hilbfunctor^d_{X/S}$ are representable. The morphism\n$\\underline{\\Hilbfunctor}^d_{X/S} \\times_S X \\to\n\\underline{\\Hilbfunctor}^{d + 1}_{X/S}$\nis finite locally free of degree $d + 1$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Moduli of divisors on smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9G","source_file":"pic.tex","source_line":568,"source_end_line":575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L568-L575","statement_sha256":"7a17e26010ef9a347aa005e696e0fda79c3cd87c998b292d64a419381b0c109d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8469,"rank":8469,"depth":48,"x":2449.02,"y":1610.742,"cluster":"moduli-theory"},{"id":"stacks:0B9H","tag":"0B9H","title":"Moduli of divisors on smooth curves · Lemma 0B9H","summary":"Let X → S be a smooth morphism of schemes of relative dimension 1 such that the functors Hilbfunctor^d_X/S are representable. The schemes underlineHilbfunctor^d_X/S are smooth over S of relative dimension d.","statement_latex":"Let $X \\to S$ be a smooth morphism of schemes of relative dimension $1$\nsuch that the functors $\\Hilbfunctor^d_{X/S}$ are representable. The\nschemes $\\underline{\\Hilbfunctor}^d_{X/S}$ are smooth over $S$ of\nrelative dimension $d$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Moduli of divisors on smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9H","source_file":"pic.tex","source_line":601,"source_end_line":607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L601-L607","statement_sha256":"fdaa6ba0f81c95a6d4a84aa549f15069a1305a0c5aa796c3ae53f4cba1202cd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8470,"rank":8470,"depth":49,"x":2494.935,"y":1573.192,"cluster":"moduli-theory"},{"id":"stacks:0B9I","tag":"0B9I","title":"Moduli of divisors on smooth curves · Proposition 0B9I","summary":"Let X be a geometrically irreducible smooth proper curve over a field k. • The functors Hilbfunctor^d_X/k are representable by smooth proper varieties underlineHilbfunctor^d_X/k of dimension d over k. • For a field extension k'/k the k'-rational points of underlineHilbfunctor^d_X/k are in 1-to-1 bijection with effective Cartier divisors of degree d on X_k'. • For d_1, d_2 ≥ 0 there is a morphism underlineHilbfunctor^d_1_X/k ×_k underlineHilbfunctor^d_2_X/k →…","statement_latex":"Let $X$ be a geometrically irreducible smooth proper curve over a field $k$.\n\\begin{enumerate}\n\\item The functors $\\Hilbfunctor^d_{X/k}$ are representable by smooth\nproper varieties $\\underline{\\Hilbfunctor}^d_{X/k}$ of dimension\n$d$ over $k$.\n\\item For a field extension $k'/k$ the $k'$-rational points\nof $\\underline{\\Hilbfunctor}^d_{X/k}$ are in $1$-to-$1$ bijection\nwith effective Cartier divisors of degree $d$ on $X_{k'}$.\n\\item For $d_1, d_2 \\geq 0$ there is a morphism\n$$\n\\underline{\\Hilbfunctor}^{d_1}_{X/k}\n\\times_k\n\\underline{\\Hilbfunctor}^{d_2}_{X/k}\n\\longrightarrow\n\\underline{\\Hilbfunctor}^{d_1 + d_2}_{X/k}\n$$\nwhich is finite locally free of degree ${d_1 + d_2 \\choose d_1}$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Moduli of divisors on smooth curves","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9I","source_file":"pic.tex","source_line":631,"source_end_line":651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L631-L651","statement_sha256":"5c746a166839ec17cf932537ec0809c77765fa69147f34c79497d0039c46b48b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8471,"rank":8471,"depth":62,"x":2491.036,"y":1629.556,"cluster":"moduli-theory"},{"id":"stacks:0B9L","tag":"0B9L","title":"The Picard functor · Definition 0B9L","summary":"Let Sch_fppf be a big site as in Topologies, Definition [Tag 021S]. Let f : X → S be a morphism of this site. The Picard functor Picardfunctor_X/S is the fppf sheafification of the functor (Sch/S)_fppf → Sets, T ↦ Pic(X_T) If this functor is representable, then we denote underlinePicardfunctor_X/S a scheme representing it.","statement_latex":"Let $\\Sch_{fppf}$ be a big site as in\nTopologies, Definition \\ref{topologies-definition-big-small-fppf}.\nLet $f : X \\to S$ be a morphism of this site. The {\\it Picard functor}\n$\\Picardfunctor_{X/S}$ is the fppf sheafification of the functor\n$$\n(\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad\nT \\longmapsto \\Pic(X_T)\n$$\nIf this functor is representable, then we denote\n$\\underline{\\Picardfunctor}_{X/S}$ a scheme representing it.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard functor","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9L","source_file":"pic.tex","source_line":739,"source_end_line":751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L739-L751","statement_sha256":"0cfe67bae44e8a34676e89d692cc936932355af2ae36763fbe35d3db7f0ba4aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8472,"rank":8472,"depth":1,"x":2446.737,"y":1583.808,"cluster":"moduli-theory"},{"id":"stacks:0B9M","tag":"0B9M","title":"The Picard functor · Lemma 0B9M","summary":"Let f : X → S be as in Definition [Tag 0B9L]. If O_T → f_T, *O_X_T is an isomorphism for all T ∈ Ob((Sch/S)_fppf), then 0 → Pic(T) → Pic(X_T) → Picardfunctor_X/S(T) is an exact sequence for all T.","statement_latex":"Let $f : X \\to S$ be as in Definition \\ref{definition-picard-functor}.\nIf $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism\nfor all $T \\in \\Ob((\\Sch/S)_{fppf})$, then\n$$\n0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/S}(T)\n$$\nis an exact sequence for all $T$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9M","source_file":"pic.tex","source_line":761,"source_end_line":770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L761-L770","statement_sha256":"174491aa65786c18ca23286c61b4f19df8d73f5ea8e4ff38ff7b3a0f58196b86","origin":"The Stacks Project","memory_eligible":false,"source_rank":8473,"rank":8473,"depth":21,"x":2519.022,"y":1592.794,"cluster":"moduli-theory"},{"id":"stacks:0B9N","tag":"0B9N","title":"The Picard functor · Lemma 0B9N","summary":"Let f : X → S be as in Definition [Tag 0B9L]. Assume f has a section σ and that O_T → f_T, *O_X_T is an isomorphism for all T ∈ Ob((Sch/S)_fppf). Then 0 → Pic(T) → Pic(X_T) → Picardfunctor_X/S(T) → 0 is a split exact sequence with splitting given by σ_T^* : Pic(X_T) → Pic(T).","statement_latex":"Let $f : X \\to S$ be as in Definition \\ref{definition-picard-functor}.\nAssume $f$ has a section $\\sigma$ and that\n$\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism\nfor all $T \\in \\Ob((\\Sch/S)_{fppf})$. Then\n$$\n0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/S}(T) \\to 0\n$$\nis a split exact sequence with splitting given by\n$\\sigma_T^* : \\Pic(X_T) \\to \\Pic(T)$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9N","source_file":"pic.tex","source_line":810,"source_end_line":821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L810-L821","statement_sha256":"e7a61bb5b55cc35fbcd796acc905c23ca16d3203a26909bb463c8e0ef71b5a11","origin":"The Stacks Project","memory_eligible":false,"source_rank":8474,"rank":8474,"depth":22,"x":2456.186,"y":1628.454,"cluster":"moduli-theory"},{"id":"stacks:0B9Q","tag":"0B9Q","title":"A representability criterion · Lemma 0B9Q","summary":"Let k be a field. Let G : (Sch/k)^opp → Groups be a functor. With terminology as in Schemes, Definition [Tag 01JI], assume that • G satisfies the sheaf property for the Zariski topology, • there exists a subfunctor F ⊂ G such that • F is representable, • F ⊂ G is representable by open immersion, • for every field extension K of k and g ∈ G(K) there exists a g' ∈ G(k) such that g'g ∈ F(K). Then G is representable by a group scheme over k.","statement_latex":"Let $k$ be a field. Let $G : (\\Sch/k)^{opp} \\to \\textit{Groups}$ be a\nfunctor. With terminology as in\nSchemes, Definition \\ref{schemes-definition-representable-by-open-immersions},\nassume that\n\\begin{enumerate}\n\\item $G$ satisfies the sheaf property for the Zariski topology,\n\\item there exists a subfunctor $F \\subset G$ such that\n\\begin{enumerate}\n\\item $F$ is representable,\n\\item $F \\subset G$ is representable by open immersion,\n\\item for every field extension $K$ of $k$ and $g \\in G(K)$\nthere exists a $g' \\in G(k)$ such that $g'g \\in F(K)$.\n\\end{enumerate}\n\\end{enumerate}\nThen $G$ is representable by a group scheme over $k$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"A representability criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9Q","source_file":"pic.tex","source_line":887,"source_end_line":904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L887-L904","statement_sha256":"61184df2988bdac0e9357453e97b9de9098bb09b8ef6a5207a6dfb6154778e78","origin":"The Stacks Project","memory_eligible":false,"source_rank":8475,"rank":8475,"depth":3,"x":2474.498,"y":1564.337,"cluster":"moduli-theory"},{"id":"stacks:0B9U","tag":"0B9U","title":"The Picard scheme of a curve · Lemma 0B9U","summary":"Let k be a field. Let X be a smooth projective curve over k which has a k-rational point. Then the hypotheses of Lemma [Tag 0B9N] are satisfied.","statement_latex":"Let $k$ be a field. Let $X$ be a smooth projective curve over $k$\nwhich has a $k$-rational point. Then the hypotheses of\nLemma \\ref{lemma-flat-geometrically-connected-fibres-with-section}\nare satisfied.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard scheme of a curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9U","source_file":"pic.tex","source_line":929,"source_end_line":935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L929-L935","statement_sha256":"0daf6afa186fb5a62e3325561740e4bcf3432185093440f29bc260c4bbb7d5f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8476,"rank":8476,"depth":34,"x":2513.775,"y":1623.911,"cluster":"moduli-theory"},{"id":"stacks:0B9V","tag":"0B9V","title":"The Picard scheme of a curve · Lemma 0B9V","summary":"Let k be a field. Let X be a smooth projective curve over k with a k-rational point σ. For a scheme T over k, consider the subset F(T) ⊂ Picardfunctor_X/k, σ(T) consisting of L such that Rf_T, *L is isomorphic to an invertible O_T-module placed in degree 0. Then F ⊂ Picardfunctor_X/k, σ is a subfunctor and the inclusion is representable by open immersions.","statement_latex":"Let $k$ be a field. Let $X$ be a smooth projective curve over $k$\nwith a $k$-rational point $\\sigma$. For a scheme $T$ over $k$,\nconsider the subset $F(T) \\subset \\Picardfunctor_{X/k, \\sigma}(T)$ consisting of\n$\\mathcal{L}$ such that $Rf_{T, *}\\mathcal{L}$ is isomorphic to an invertible\n$\\mathcal{O}_T$-module placed in degree $0$. Then\n$F \\subset \\Picardfunctor_{X/k, \\sigma}$ is a subfunctor and the inclusion is\nrepresentable by open immersions.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard scheme of a curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9V","source_file":"pic.tex","source_line":969,"source_end_line":978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L969-L978","statement_sha256":"a6a2a5aa53e9620cfff7387be0cfd1c8df2dc64b997c8c2f4e82fba003800b60","origin":"The Stacks Project","memory_eligible":false,"source_rank":8477,"rank":8477,"depth":40,"x":2434.55,"y":1601.579,"cluster":"moduli-theory"},{"id":"stacks:0B9W","tag":"0B9W","title":"The Picard scheme of a curve · Definition 0B9W","summary":"Let k be a field. Let X be a smooth projective geometrically irreducible curve over k. The genus of X is g = dim_k H^1(X, O_X).","statement_latex":"Let $k$ be a field. Let $X$ be a smooth projective geometrically irreducible\ncurve over $k$. The {\\it genus} of $X$ is $g = \\dim_k H^1(X, \\mathcal{O}_X)$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard scheme of a curve","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9W","source_file":"pic.tex","source_line":990,"source_end_line":994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L990-L994","statement_sha256":"3818343840896c6ac3e5d2ef5f14c2d11cb92430968bf2452b55af91bd612889","origin":"The Stacks Project","memory_eligible":false,"source_rank":8478,"rank":8478,"depth":0,"x":2513.152,"y":1572.287,"cluster":"moduli-theory"},{"id":"stacks:0B9X","tag":"0B9X","title":"The Picard scheme of a curve · Lemma 0B9X","summary":"Let k be a field. Let X be a smooth projective curve of genus g over k with a k-rational point σ. The open subfunctor F defined in Lemma [Tag 0B9V] is representable by an open subscheme of underlineHilbfunctor^g_X/k.","statement_latex":"Let $k$ be a field. Let $X$ be a smooth projective curve of genus $g$\nover $k$ with a $k$-rational point $\\sigma$. The open subfunctor $F$ defined\nin Lemma \\ref{lemma-define-open} is representable by an open subscheme of\n$\\underline{\\Hilbfunctor}^g_{X/k}$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard scheme of a curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9X","source_file":"pic.tex","source_line":996,"source_end_line":1002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L996-L1002","statement_sha256":"ac2af74590fd52993be4048e3864cb89d6c921666430c447cbfb128a08c238c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8479,"rank":8479,"depth":63,"x":2477.782,"y":1640.292,"cluster":"moduli-theory"},{"id":"stacks:0B9Y","tag":"0B9Y","title":"The Picard scheme of a curve · Lemma 0B9Y","summary":"Let k be a separably closed field. Let X be a smooth projective curve of genus g over k. Let K/k be a field extension and let L be an invertible sheaf on X_K. Then there exists an invertible sheaf L_0 on X such that dim_K H^0(X_K, L ⊗_O_X_K L_0|_X_K) = 1 and dim_K H^1(X_K, L ⊗_O_X_K L_0|_X_K) = 0.","statement_latex":"Let $k$ be a separably closed field. Let $X$ be a smooth projective\ncurve of genus $g$ over $k$. Let $K/k$ be a field extension and let\n$\\mathcal{L}$ be an invertible sheaf on $X_K$. Then there exists an\ninvertible sheaf $\\mathcal{L}_0$ on $X$ such that\n$\\dim_K H^0(X_K,\n\\mathcal{L} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L}_0|_{X_K}) = 1$ and\n$\\dim_K H^1(X_K,\n\\mathcal{L} \\otimes_{\\mathcal{O}_{X_K}} \\mathcal{L}_0|_{X_K}) = 0$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard scheme of a curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9Y","source_file":"pic.tex","source_line":1099,"source_end_line":1109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L1099-L1109","statement_sha256":"079adedcb644a47bb0f9c5edb340ee46c6b5fb35ccca5ece177565b547409caa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8480,"rank":8480,"depth":48,"x":2448.455,"y":1568.247,"cluster":"moduli-theory"},{"id":"stacks:0B9Z","tag":"0B9Z","title":"The Picard scheme of a curve · Proposition 0B9Z","summary":"Let k be a separably closed field. Let X be a smooth projective curve over k. The Picard functor Picardfunctor_X/k is representable.","statement_latex":"Let $k$ be a separably closed field. Let $X$ be a smooth projective\ncurve over $k$. The Picard functor $\\Picardfunctor_{X/k}$ is representable.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard scheme of a curve","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B9Z","source_file":"pic.tex","source_line":1159,"source_end_line":1163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L1159-L1163","statement_sha256":"d98281f33bc332f04674ba117c42886d96eff4ce6ec290924a9fd704fc63e7ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":8481,"rank":8481,"depth":64,"x":2529.97,"y":1605.648,"cluster":"moduli-theory"},{"id":"stacks:0BA0","tag":"0BA0","title":"The Picard scheme of a curve · Lemma 0BA0","summary":"Let k be a separably closed field. Let X be a smooth projective curve of genus g over k. • underlinePicardfunctor_X/k is a disjoint union of g-dimensional smooth proper varieties underlinePicardfunctor^d_X/k, • k-points of underlinePicardfunctor^d_X/k correspond to invertible O_X-modules of degree d, • underlinePicardfunctor^0_X/k is an open and closed subgroup scheme, • for d ≥ 0 there is a canonical morphism γ_d : underlineHilbfunctor^d_X/k →…","statement_latex":"Let $k$ be a separably closed field. Let $X$ be a smooth projective\ncurve of genus $g$ over $k$.\n\\begin{enumerate}\n\\item $\\underline{\\Picardfunctor}_{X/k}$ is a disjoint union of\n$g$-dimensional smooth proper varieties $\\underline{\\Picardfunctor}^d_{X/k}$,\n\\item $k$-points of $\\underline{\\Picardfunctor}^d_{X/k}$\ncorrespond to invertible $\\mathcal{O}_X$-modules of degree $d$,\n\\item $\\underline{\\Picardfunctor}^0_{X/k}$\nis an open and closed subgroup scheme,\n\\item for $d \\geq 0$ there is a canonical morphism\n$\\gamma_d :\n\\underline{\\Hilbfunctor}^d_{X/k} \\to \\underline{\\Picardfunctor}^d_{X/k}$\n\\item the morphisms $\\gamma_d$\nare surjective for $d \\geq g$ and smooth for $d \\geq 2g - 1$,\n\\item the morphism\n$\\underline{\\Hilbfunctor}^g_{X/k} \\to \\underline{\\Picardfunctor}^g_{X/k}$\nis birational.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"The Picard scheme of a curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BA0","source_file":"pic.tex","source_line":1195,"source_end_line":1215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L1195-L1215","statement_sha256":"4f926945855372e7e8fb5e7f65dd3bb5204c1747dd42d6ce5c567804b2654088","origin":"The Stacks Project","memory_eligible":false,"source_rank":8482,"rank":8482,"depth":65,"x":2437.66,"y":1624.745,"cluster":"moduli-theory"},{"id":"stacks:0CDT","tag":"0CDT","title":"Some remarks on Picard groups · Lemma 0CDT","summary":"Let k be a field. Let X be a quasi-compact and quasi-separated scheme over k with H^0(X, O_X) = k. If X has a k-rational point, then for any Galois extension k'/k we have Pic(X) = Pic(X_k')^Gal(k'/k) Moreover the action of Gal(k'/k) on Pic(X_k') is continuous.","statement_latex":"Let $k$ be a field. Let $X$ be a quasi-compact and quasi-separated scheme\nover $k$ with $H^0(X, \\mathcal{O}_X) = k$. If $X$ has a $k$-rational point,\nthen for any Galois extension $k'/k$ we have\n$$\n\\Pic(X) = \\Pic(X_{k'})^{\\text{Gal}(k'/k)}\n$$\nMoreover the action of $\\text{Gal}(k'/k)$ on $\\Pic(X_{k'})$\nis continuous.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Some remarks on Picard groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDT","source_file":"pic.tex","source_line":1341,"source_end_line":1351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L1341-L1351","statement_sha256":"f54ffe611e4303b445c2c90b7319a7fd75d156f426cc76920450bca935afb60d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8483,"rank":8483,"depth":36,"x":2491.57,"y":1556.8,"cluster":"moduli-theory"},{"id":"stacks:0CD5","tag":"0CD5","title":"Some remarks on Picard groups · Lemma 0CD5","summary":"Let k be a field of characteristic p > 0. Let X be a quasi-compact and quasi-separated scheme over k with H^0(X, O_X) = k. Let n be an integer prime to p. Then the map Pic(X)[n] → Pic(X_k')[n] is bijective for any purely inseparable extension k'/k.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $X$ be a\nquasi-compact and quasi-separated scheme over $k$ with\n$H^0(X, \\mathcal{O}_X) = k$. Let $n$ be an integer prime to $p$.\nThen the map\n$$\n\\Pic(X)[n] \\longrightarrow \\Pic(X_{k'})[n]\n$$\nis bijective for any purely inseparable extension $k'/k$.","area":"Moduli Theory","chapter":"Picard Schemes of Curves","chapter_id":"pic","section":"Some remarks on Picard groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CD5","source_file":"pic.tex","source_line":1408,"source_end_line":1418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pic.tex#L1408-L1418","statement_sha256":"dc7ff51f4cba663cd1bbae117bb739910ded82362ed953c9033ec9c91289f302","origin":"The Stacks Project","memory_eligible":false,"source_rank":8484,"rank":8484,"depth":33,"x":2506.76,"y":1639.228,"cluster":"moduli-theory"},{"id":"stacks:0FG0","tag":"0FG0","title":"Correspondences · Lemma 0FG0","summary":"We have the following for correspondences: • composition of correspondences is Q-bilinear and associative, • there is a canonical isomorphism CH_-r(X) ⊗ Q = Corr^r(X, Spec(k)) such that pullback by correspondences corresponds to composition, • there is a canonical isomorphism CH^r(X) ⊗ Q = Corr^r(Spec(k), X) such that pushforward by correspondences corresponds to composition, • composition of correspondences is compatible with pushforward and pullback of cycles.","statement_latex":"We have the following for correspondences:\n\\begin{enumerate}\n\\item composition of correspondences is $\\mathbf{Q}$-bilinear\nand associative,\n\\item there is a canonical isomorphism\n$$\n\\CH_{-r}(X) \\otimes \\mathbf{Q} = \\text{Corr}^r(X, \\Spec(k))\n$$\nsuch that pullback by correspondences corresponds to composition,\n\\item there is a canonical isomorphism\n$$\n\\CH^r(X) \\otimes \\mathbf{Q} = \\text{Corr}^r(\\Spec(k), X)\n$$\nsuch that pushforward by correspondences corresponds to composition,\n\\item composition of correspondences is compatible with pushforward and\npullback of cycles.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Correspondences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FG0","source_file":"weil.tex","source_line":223,"source_end_line":242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L223-L242","statement_sha256":"0bd52e67a03c56e4109cd5313966864e41cc7beaf057fdb3591c8ad09366042c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8485,"rank":8485,"depth":65,"x":1982.213,"y":1080.468,"cluster":"duality-cohomology"},{"id":"stacks:0FG2","tag":"0FG2","title":"Correspondences · Lemma 0FG2","summary":"Smooth projective schemes over k with correspondences and composition of correspondences as defined above form a graded category over Q (Differential Graded Algebra, Definition [Tag 09L2]).","statement_latex":"Smooth projective schemes over $k$ with correspondences and composition\nof correspondences as defined above form a graded category over\n$\\mathbf{Q}$\n(Differential Graded Algebra, Definition \\ref{dga-definition-graded-category}).","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Correspondences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FG2","source_file":"weil.tex","source_line":332,"source_end_line":338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L332-L338","statement_sha256":"4d2d33c290819943b27b59cc692d8329412bae53628ac5159fae1f6a93336cbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8486,"rank":8486,"depth":66,"x":2113.427,"y":1156.847,"cluster":"duality-cohomology"},{"id":"stacks:0FG3","tag":"0FG3","title":"Correspondences · Lemma 0FG3","summary":"There is a contravariant functor from the category of smooth projective schemes over k to the category of correspondences which is the identity on objects and sends f : Y → X to the element [Γ_f] ∈ Corr^0(X, Y).","statement_latex":"There is a contravariant functor from the category of smooth\nprojective schemes over $k$ to the category of correspondences\nwhich is the identity on objects and sends $f : Y \\to X$ to\nthe element $[\\Gamma_f] \\in \\text{Corr}^0(X, Y)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Correspondences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FG3","source_file":"weil.tex","source_line":392,"source_end_line":398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L392-L398","statement_sha256":"5f461bd91cd066366abaf704a3ca5bcc14456114c72ffe17d5b8bb43d373fe0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8487,"rank":8487,"depth":67,"x":1954.651,"y":1175.048,"cluster":"duality-cohomology"},{"id":"stacks:0FG5","tag":"0FG5","title":"Correspondences · Lemma 0FG5","summary":"Let f : Y → X be a morphism of smooth projective schemes over k. Let [Γ_f] ∈ Corr^0(X, Y) be as in Example [Tag 0FG1]. Then • pushforward of cycles by the correspondence [Γ_f] agrees with the Gysin map f^! : CH^*(X) → CH^*(Y), • pullback of cycles by the correspondence [Γ_f] agrees with the pushforward map f_* : CH_*(Y) → CH_*(X), • if X and Y are equidimensional of dimensions d and e, then • pushforward of cycles by the correspondence [Γ_f^t] of Remark [Tag 0FG4]…","statement_latex":"Let $f : Y \\to X$ be a morphism of smooth projective schemes over $k$.\nLet $[\\Gamma_f] \\in \\text{Corr}^0(X, Y)$ be as in\nExample \\ref{example-graph-correspondence}. Then\n\\begin{enumerate}\n\\item pushforward of cycles by the correspondence $[\\Gamma_f]$\nagrees with the Gysin map $f^! : \\CH^*(X) \\to \\CH^*(Y)$,\n\\item pullback of cycles by the correspondence $[\\Gamma_f]$\nagrees with the pushforward map $f_* : \\CH_*(Y) \\to \\CH_*(X)$,\n\\item if $X$ and $Y$ are equidimensional of dimensions $d$ and $e$,\nthen\n\\begin{enumerate}\n\\item pushforward of cycles by the correspondence\n$[\\Gamma_f^t]$ of Remark \\ref{remark-transpose}\ncorresponds to pushforward of cycles by $f$, and\n\\item pullback of cycles by the correspondence\n$[\\Gamma_f^t]$ of Remark \\ref{remark-transpose}\ncorresponds to the Gysin map $f^!$.\n\\end{enumerate}\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Correspondences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FG5","source_file":"weil.tex","source_line":443,"source_end_line":464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L443-L464","statement_sha256":"a21e0daba559e22438b20b0efbc42335040bf98799fb422e9f574de17ab0666e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8488,"rank":8488,"depth":65,"x":2057.479,"y":1071.145,"cluster":"duality-cohomology"},{"id":"stacks:0FG7","tag":"0FG7","title":"Correspondences · Lemma 0FG7","summary":"The tensor product of correspondences defined above turns the category of correspondences into a symmetric monoidal category with unit Spec(k).","statement_latex":"The tensor product of correspondences defined above turns the category of\ncorrespondences into a symmetric monoidal category with unit $\\Spec(k)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Correspondences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FG7","source_file":"weil.tex","source_line":549,"source_end_line":553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L549-L553","statement_sha256":"859b4614c1e140086f3eacd0f129250461f4edeb52fcc6bd695ab46508f8d6d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8489,"rank":8489,"depth":0,"x":2065.24,"y":1206.612,"cluster":"duality-cohomology"},{"id":"stacks:0FG8","tag":"0FG8","title":"Correspondences · Lemma 0FG8","summary":"Let f : Y → X be a morphism of smooth projective schemes over k. Assume X and Y equidimensional of dimensions d and e. Denote a = [Γ_f] ∈ Corr^0(X, Y) and a^t = [Γ_f^t] ∈ Corr^d - e(Y, X). Set eta_X = [Γ_X → X × X] ∈ Corr^0(X × X, X), eta_Y = [Γ_Y → Y × Y] ∈ Corr^0(Y × Y, Y), [X] ∈ Corr^-d(X, Spec(k)), and [Y] ∈ Corr^-e(Y, Spec(k)). The diagram xymatrix X ⊗ Y ar[r]_a ⊗ id ar[d]_id ⊗ a^t & Y ⊗ Y ar[r]_eta_Y & Y ar[d]^[Y] X ⊗ X ar[r]^eta_X & X ar[r]^[X] & Spec(k) is…","statement_latex":"Let $f : Y \\to X$ be a morphism of smooth projective schemes over $k$.\nAssume $X$ and $Y$ equidimensional of dimensions $d$ and $e$.\nDenote $a = [\\Gamma_f] \\in \\text{Corr}^0(X, Y)$ and\n$a^t = [\\Gamma_f^t] \\in \\text{Corr}^{d - e}(Y, X)$.\nSet\n$\\eta_X = [\\Gamma_{X \\to X \\times X}] \\in \\text{Corr}^0(X \\times X, X)$,\n$\\eta_Y = [\\Gamma_{Y \\to Y \\times Y}] \\in \\text{Corr}^0(Y \\times Y, Y)$,\n$[X] \\in \\text{Corr}^{-d}(X, \\Spec(k))$, and\n$[Y] \\in \\text{Corr}^{-e}(Y, \\Spec(k))$. The diagram\n$$\n\\xymatrix{\nX \\otimes Y \\ar[r]_{a \\otimes \\text{id}} \\ar[d]_{\\text{id} \\otimes a^t} &\nY \\otimes Y \\ar[r]_{\\eta_Y} &\nY \\ar[d]^{[Y]} \\\\\nX \\otimes X \\ar[r]^{\\eta_X} &\nX \\ar[r]^{[X]} &\n\\Spec(k)\n}\n$$\nis commutative in the category of correspondences.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Correspondences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FG8","source_file":"weil.tex","source_line":559,"source_end_line":581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L559-L581","statement_sha256":"2437afcbbbc2ed50f64f570b444814fda2b44dee05304b8f47eb5f47f6de41e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8490,"rank":8490,"depth":66,"x":1950.154,"y":1110.77,"cluster":"duality-cohomology"},{"id":"stacks:0FGA","tag":"0FGA","title":"Chow motives · Lemma 0FGA","summary":"The category M_k whose objects are motives over k and morphisms are morphisms of motives over k is a Q-linear category. There is a contravariant functor h : (smooth projective schemes over k) → M_k defined by h(X) = (X, 1, 0) and h(f) = [Γ_f].","statement_latex":"The category $M_k$ whose objects are motives over $k$ and morphisms\nare morphisms of motives over $k$ is a $\\mathbf{Q}$-linear category.\nThere is a contravariant functor\n$$\nh : \\{\\text{smooth projective schemes over }k\\} \\longrightarrow M_k\n$$\ndefined by $h(X) = (X, 1, 0)$ and $h(f) = [\\Gamma_f]$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGA","source_file":"weil.tex","source_line":656,"source_end_line":665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L656-L665","statement_sha256":"a0114b182d0f20d8e20fe9f824f6803b1b6a0e13f3f5afdb28da64b7228c579b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8491,"rank":8491,"depth":68,"x":2112.685,"y":1116.16,"cluster":"duality-cohomology"},{"id":"stacks:0FGB","tag":"0FGB","title":"Chow motives · Lemma 0FGB","summary":"The category M_k is Karoubian.","statement_latex":"The category $M_k$ is Karoubian.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGB","source_file":"weil.tex","source_line":671,"source_end_line":674,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L671-L674","statement_sha256":"215aaf5c0f0bfad8f68d7c89c097452ad0791aaa371ae535f72233f6ca402e1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8492,"rank":8492,"depth":1,"x":1988.049,"y":1204.731,"cluster":"duality-cohomology"},{"id":"stacks:0FGC","tag":"0FGC","title":"Chow motives · Lemma 0FGC","summary":"The category M_k with tensor product defined as above is symmetric monoidal with the obvious associativity and commutativity constraints and with unit 1 = (Spec(k), 1, 0).","statement_latex":"The category $M_k$ with tensor product defined as above\nis symmetric monoidal with the obvious associativity and commutativity\nconstraints and with unit $\\mathbf{1} = (\\Spec(k), 1, 0)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGC","source_file":"weil.tex","source_line":725,"source_end_line":730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L725-L730","statement_sha256":"51c42f87bfa51713b0daac65fd960232883698ede51df12a8f9b0bbf625b24ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":8493,"rank":8493,"depth":1,"x":2008.803,"y":1068.206,"cluster":"duality-cohomology"},{"id":"stacks:0FGD","tag":"0FGD","title":"Chow motives · Lemma 0FGD","summary":"With notation as in Example [Tag 0FG6] • the motive (X, c_0, 0) is isomorphic to the motive 1 = (Spec(k), 1, 0). • the motive (X, c_2, 0) is isomorphic to the motive 1(-1) = (Spec(k), 1, -1).","statement_latex":"With notation as in Example \\ref{example-decompose-P1}\n\\begin{enumerate}\n\\item\nthe motive $(X, c_0, 0)$ is isomorphic to the motive\n$\\mathbf{1} = (\\Spec(k), 1, 0)$.\n\\item\nthe motive $(X, c_2, 0)$ is isomorphic to the motive\n$\\mathbf{1}(-1) = (\\Spec(k), 1, -1)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGD","source_file":"weil.tex","source_line":748,"source_end_line":759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L748-L759","statement_sha256":"63418d0553438788c173a2395f01f51ca5d3bdf62a930bc8adbc6e124990b236","origin":"The Stacks Project","memory_eligible":false,"source_rank":8494,"rank":8494,"depth":68,"x":2103.628,"y":1181.059,"cluster":"duality-cohomology"},{"id":"stacks:0FGF","tag":"0FGF","title":"Chow motives · Lemma 0FGF","summary":"The category M_k is additive.","statement_latex":"The category $M_k$ is additive.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGF","source_file":"weil.tex","source_line":829,"source_end_line":832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L829-L832","statement_sha256":"f67910db117cc3c8808e093853b9ed3f9400d4f132523090e7c326578ee1813b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8495,"rank":8495,"depth":69,"x":1942.379,"y":1151.543,"cluster":"duality-cohomology"},{"id":"stacks:0FGG","tag":"0FGG","title":"Chow motives · Lemma 0FGG","summary":"In M_k we have h(P^1_k) ≅ 1 ⊕ 1(-1).","statement_latex":"In $M_k$ we have $h(\\mathbf{P}^1_k) \\cong \\mathbf{1} \\oplus \\mathbf{1}(-1)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGG","source_file":"weil.tex","source_line":856,"source_end_line":859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L856-L859","statement_sha256":"d93832704115e5b7fdfa6a5e74b59fc0e8c61278113c3712ee693683ea0b50df","origin":"The Stacks Project","memory_eligible":false,"source_rank":8496,"rank":8496,"depth":69,"x":2085.523,"y":1081.565,"cluster":"duality-cohomology"},{"id":"stacks:0FGH","tag":"0FGH","title":"Chow motives · Lemma 0FGH","summary":"Let X, c_2 be as in Example [Tag 0FG6]. Let C be a Q-linear Karoubian symmetric monoidal category. Any Q-linear functor F : ( smooth projective schemes over k morphisms are correspondences of degree 0 ) → C of symmetric monoidal categories such that the image of F(c_2) on F(X) is an invertible object, factors uniquely through a functor F : M_k → C of symmetric monoidal categories.","statement_latex":"Let $X$, $c_2$ be as in Example \\ref{example-decompose-P1}.\nLet $\\mathcal{C}$ be a $\\mathbf{Q}$-linear Karoubian symmetric\nmonoidal category. Any $\\mathbf{Q}$-linear functor\n$$\nF :\n\\left\\{\n\\begin{matrix}\n\\text{smooth projective schemes over }k\\\\\n\\text{morphisms are correspondences of degree }0\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\mathcal{C}\n$$\nof symmetric monoidal categories such that the image of $F(c_2)$ on\n$F(X)$ is an invertible object, factors uniquely through a functor\n$F : M_k \\to \\mathcal{C}$ of symmetric monoidal categories.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGH","source_file":"weil.tex","source_line":866,"source_end_line":885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L866-L885","statement_sha256":"e88e9eab42a24256b6069fd04992cc1a5c879a00182305358dd4d1e88c748226","origin":"The Stacks Project","memory_eligible":false,"source_rank":8497,"rank":8497,"depth":70,"x":2036.071,"y":1214.856,"cluster":"duality-cohomology"},{"id":"stacks:0FGI","tag":"0FGI","title":"Chow motives · Lemma 0FGI","summary":"Let X be a smooth projective scheme over k which is equidimensional of dimension d. Then h(X)(d) is a left dual to h(X) in M_k.","statement_latex":"Let $X$ be a smooth projective scheme over $k$ which is equidimensional\nof dimension $d$. Then $h(X)(d)$ is a left dual to $h(X)$ in $M_k$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGI","source_file":"weil.tex","source_line":968,"source_end_line":972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L968-L972","statement_sha256":"d706986ec6081a04e07650fe80e60e1334c8d72a30c9d1733dbc3d57d09fc957","origin":"The Stacks Project","memory_eligible":false,"source_rank":8498,"rank":8498,"depth":66,"x":1965.101,"y":1088.067,"cluster":"duality-cohomology"},{"id":"stacks:0FGJ","tag":"0FGJ","title":"Chow motives · Lemma 0FGJ","summary":"Every object of M_k has a left dual.","statement_latex":"Every object of $M_k$ has a left dual.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGJ","source_file":"weil.tex","source_line":1020,"source_end_line":1023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1020-L1023","statement_sha256":"80630491bd93aeeb78ccea5e78e183c16c861d91e9737e707c38fd8cbdd842fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8499,"rank":8499,"depth":67,"x":2119.928,"y":1141.474,"cluster":"duality-cohomology"},{"id":"stacks:0FGL","tag":"0FGL","title":"Chow groups of motives · Definition 0FGL","summary":"Let k be a base field. Let M = (X, p, m) be a Chow motive over k. For i ∈ Z we define the ith Chow group of M by the formula CH^i(M) = p(CH^i + m(X) ⊗ Q)","statement_latex":"Let $k$ be a base field. Let $M = (X, p, m)$ be a Chow motive over $k$.\nFor $i \\in \\mathbf{Z}$ we define the {\\it $i$th Chow group of $M$}\nby the formula\n$$\n\\CH^i(M) = p\\left(\\CH^{i + m}(X) \\otimes \\mathbf{Q}\\right)\n$$","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow groups of motives","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGL","source_file":"weil.tex","source_line":1052,"source_end_line":1060,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1052-L1060","statement_sha256":"b388c423882598d00d99a8dd93c11cfdd4275dc5990f351834aa8853a51f61cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8500,"rank":8500,"depth":0,"x":1962.271,"y":1190.112,"cluster":"duality-cohomology"},{"id":"stacks:0FGM","tag":"0FGM","title":"Chow groups of motives · Lemma 0FGM","summary":"Let k be a base field. The functor CH^i(-) on the category of motives M_k is representable by 1(-i), i.e., we have CH^i(M) = Hom_M_k(1(-i), M) functorially in M in M_k.","statement_latex":"Let $k$ be a base field. The functor $\\CH^i(-)$ on the category\nof motives $M_k$ is representable by $\\mathbf{1}(-i)$, i.e., we\nhave\n$$\n\\CH^i(M) = \\Hom_{M_k}(\\mathbf{1}(-i), M)\n$$\nfunctorially in $M$ in $M_k$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow groups of motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGM","source_file":"weil.tex","source_line":1088,"source_end_line":1097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1088-L1097","statement_sha256":"e32475053107f7160ac4977b6c2db675f205b39459a772e1438dae60d6ce0701","origin":"The Stacks Project","memory_eligible":false,"source_rank":8501,"rank":8501,"depth":66,"x":2039.677,"y":1064.36,"cluster":"duality-cohomology"},{"id":"stacks:0FGN","tag":"0FGN","title":"Manin · Lemma 0FGN","summary":"Let k be a base field. Let c : M → N be a morphism of motives. If for every smooth projective scheme X over k the map c ⊗ 1 : M ⊗ h(X) → N ⊗ h(X) induces an isomorphism on Chow groups, then c is an isomorphism.","statement_latex":"Let $k$ be a base field. Let $c : M \\to N$ be a morphism of motives.\nIf for every smooth projective scheme $X$ over $k$ the map\n$c \\otimes 1 : M \\otimes h(X) \\to N \\otimes h(X)$ induces an isomorphism on\nChow groups, then $c$ is an isomorphism.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chow groups of motives","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGN","source_file":"weil.tex","source_line":1109,"source_end_line":1115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1109-L1115","statement_sha256":"f22a591ef58912d5f291ff6fc86fbe0c09c06fd1c6cdda85f91c237f32a5a38d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8502,"rank":8502,"depth":68,"x":2083.874,"y":1201.474,"cluster":"duality-cohomology"},{"id":"stacks:0FGQ","tag":"0FGQ","title":"Projective space bundle formula · Lemma 0FGQ","summary":"In the situation above, the map ∑_i = 0, …, r - 1 c_i : bigoplus_i = 0, …, r - 1 h(X)(-i) → h(P) is an isomorphism in the category of motives.","statement_latex":"In the situation above, the map\n$$\n\\sum\\nolimits_{i = 0, \\ldots, r - 1} c_i :\n\\bigoplus\\nolimits_{i = 0, \\ldots, r - 1} h(X)(-i)\n\\longrightarrow\nh(P)\n$$\nis an isomorphism in the category of motives.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Projective space bundle formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGQ","source_file":"weil.tex","source_line":1173,"source_end_line":1183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1173-L1183","statement_sha256":"bffe82deb13c578a3f915c872f43dd3ba0ca65a05c871bbd42b5a6b97c04abd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8503,"rank":8503,"depth":69,"x":1940.539,"y":1125.19,"cluster":"duality-cohomology"},{"id":"stacks:0FGR","tag":"0FGR","title":"Projective space bundle formula · Lemma 0FGR","summary":"Let p : P → X be as in Lemma [Tag 0FGQ]. The class [Δ_P] of the diagonal of P in CH^*(P × P) can be written as [Δ_P] = (∑_i = 0, …, r - 1 r - 1 choose i c_r - 1 - i(pr_1^*S^vee) ∩ c_1(pr_2^*O_P(1))^i) ∩ (p × p)^*[Δ_X] where S is the kernel of the canonical surjection p^*E → O_P(1).","statement_latex":"Let $p : P \\to X$ be as in Lemma \\ref{lemma-projective-space-bundle-formula}.\nThe class $[\\Delta_P]$ of the diagonal of $P$ in $\\CH^*(P \\times P)$\ncan be written\nas\n$$\n[\\Delta_P] =\n\\left(\\sum\\nolimits_{i = 0, \\ldots, r - 1}\n{r - 1 \\choose i} c_{r - 1 - i}(\\text{pr}_1^*\\mathcal{S}^\\vee) \\cap\nc_1(\\text{pr}_2^*\\mathcal{O}_P(1))^i\\right)\n\\cap\n(p \\times p)^*[\\Delta_X]\n$$\nwhere $\\mathcal{S}$ is the kernel of the canonical surjection\n$p^*\\mathcal{E} \\to \\mathcal{O}_P(1)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Projective space bundle formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGR","source_file":"weil.tex","source_line":1248,"source_end_line":1264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1248-L1264","statement_sha256":"88864b4ecda21526cda79860eb9f01de1fc5d44ad1d48e689428698982253805","origin":"The Stacks Project","memory_eligible":false,"source_rank":8504,"rank":8504,"depth":70,"x":2108.137,"y":1100.025,"cluster":"duality-cohomology"},{"id":"stacks:0FGT","tag":"0FGT","title":"Classical Weil cohomology theories · Lemma 0FGT","summary":"Assume given (D1) and (D3) satisfying (A). For f : X → Y a morphism of smooth projective varieties we have f_*(f^*b ∪ a) = b ∪ f_*a. If g : Y → Z is a second morphism of smooth projective varieties, then g_* ∘ f_* = (g ∘ f)_*.","statement_latex":"Assume given (D1) and (D3) satisfying (A). For $f : X \\to Y$\na morphism of smooth projective varieties we have\n$f_*(f^*b \\cup a) = b \\cup f_*a$. If $g : Y \\to Z$ is a second morphism\nof smooth projective varieties, then $g_* \\circ f_* = (g \\circ f)_*$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGT","source_file":"weil.tex","source_line":1386,"source_end_line":1392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1386-L1392","statement_sha256":"5723f268fde3624a7562d19b4663d924890fc9e0fbe607b8ba7881209c56aab8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8505,"rank":8505,"depth":0,"x":2004.445,"y":1214.058,"cluster":"duality-cohomology"},{"id":"stacks:0FGV","tag":"0FGV","title":"Classical Weil cohomology theories · Definition 0FGV","summary":"Let k be an algebraically closed field. Let F be a field of characteristic 0. A classical Weil cohomology theory over k with coefficients in F is given by data (D1), (D2), and (D3) satisfying Poincaré duality, the K\\\"unneth formula, and compatibility with cycle classes, more precisely, satisfying (A), (B), and (C).","statement_latex":"Let $k$ be an algebraically closed field.\nLet $F$ be a field of characteristic $0$.\nA {\\it classical Weil cohomology theory} over $k$ with coefficients in $F$\nis given by data (D1), (D2), and (D3) satisfying\nPoincar\\'e duality, the K\\\"unneth formula, and compatibility\nwith cycle classes, more precisely, satisfying (A), (B), and (C).","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGV","source_file":"weil.tex","source_line":1464,"source_end_line":1472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1464-L1472","statement_sha256":"8c41cad51da9419346b0090ad83cefa18c04ba71c406b3f8dbb40846e9888d37","origin":"The Stacks Project","memory_eligible":false,"source_rank":8506,"rank":8506,"depth":0,"x":1989.154,"y":1070.662,"cluster":"duality-cohomology"},{"id":"stacks:0FGW","tag":"0FGW","title":"Classical Weil cohomology theories · Lemma 0FGW","summary":"Let H^* be a classical Weil cohomology theory (Definition [Tag 0FGV]). Let X be a smooth projective variety of dimension d. The diagram xymatrix CH^d(X) ar[r]_-γ ar@=[d] & H^2d(X) ar[d]^int_X CH_0(X) ar[r]^deg & F commutes where deg : CH_0(X) → Z is the degree of zero cycles discussed in Chow Homology, Section [Tag 0AZ0].","statement_latex":"Let $H^*$ be a classical Weil cohomology theory\n(Definition \\ref{definition-weil-cohomology-theory-classical}).\nLet $X$ be a smooth projective variety of dimension $d$. The diagram\n$$\n\\xymatrix{\n\\CH^d(X) \\ar[r]_-\\gamma \\ar@{=}[d] &\nH^{2d}(X) \\ar[d]^{\\int_X} \\\\\n\\CH_0(X) \\ar[r]^\\deg & F\n}\n$$\ncommutes where $\\deg : \\CH_0(X) \\to \\mathbf{Z}$ is the degree of\nzero cycles discussed in Chow Homology, Section\n\\ref{chow-section-degree-zero-cycles}.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGW","source_file":"weil.tex","source_line":1477,"source_end_line":1492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1477-L1492","statement_sha256":"04f0039124935c652432238b31312df1e2e9844ad6fd5e9fe8f0736cf1a22a0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8507,"rank":8507,"depth":1,"x":2116.16,"y":1168.043,"cluster":"duality-cohomology"},{"id":"stacks:0FGX","tag":"0FGX","title":"Classical Weil cohomology theories · Lemma 0FGX","summary":"Let H^* be a classical Weil cohomology theory (Definition [Tag 0FGV]). Let X and Y be smooth projective varieties. Then int_X × Y = int_X ⊗ int_Y.","statement_latex":"Let $H^*$ be a classical Weil cohomology theory\n(Definition \\ref{definition-weil-cohomology-theory-classical}).\nLet $X$ and $Y$ be smooth projective varieties.\nThen $\\int_{X \\times Y} = \\int_X \\otimes \\int_Y$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGX","source_file":"weil.tex","source_line":1501,"source_end_line":1507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1501-L1507","statement_sha256":"81b12651dd7b4e4595ffc994e4faa50e2f2ae41918a3cdd25015c990f0da909c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8508,"rank":8508,"depth":2,"x":1943.636,"y":1168.302,"cluster":"duality-cohomology"},{"id":"stacks:0FGY","tag":"0FGY","title":"Classical Weil cohomology theories · Lemma 0FGY","summary":"Let H^* be a classical Weil cohomology theory (Definition [Tag 0FGV]). Let X and Y be smooth projective varieties. Then pr_2, * : H^*(X × Y) → H^*(Y) sends a ⊗ b to (int_X a) b.","statement_latex":"Let $H^*$ be a classical Weil cohomology theory\n(Definition \\ref{definition-weil-cohomology-theory-classical}).\nLet $X$ and $Y$ be smooth projective varieties.\nThen $\\text{pr}_{2, *} : H^*(X \\times Y) \\to H^*(Y)$\nsends $a \\otimes b$ to $(\\int_X a) b$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGY","source_file":"weil.tex","source_line":1522,"source_end_line":1529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1522-L1529","statement_sha256":"37123b6dad7a5e05c425aeea1cc1b4b60cfb2799292291452bac415291d7cd91","origin":"The Stacks Project","memory_eligible":false,"source_rank":8509,"rank":8509,"depth":3,"x":2071.056,"y":1069.899,"cluster":"duality-cohomology"},{"id":"stacks:0FGZ","tag":"0FGZ","title":"Classical Weil cohomology theories · Lemma 0FGZ","summary":"Let H^* be a classical Weil cohomology theory (Definition [Tag 0FGV]). Let X be a smooth projective variety of dimension d. Choose a basis e_i, j, j = 1, …, β_i of H^i(X) over F. Using K\\\"unneth write γ([Δ]) = ∑_i = 0, …, 2d ∑_j e_i, j ⊗ e'_2d - i , j in bigoplus_i H^i(X) ⊗_F H^2d - i(X) with e'_2d - i, j ∈ H^2d - i(X). Then int_X e_i, j ∪ e'_2d - i, j' = (-1)^iδ_jj'.","statement_latex":"Let $H^*$ be a classical Weil cohomology theory\n(Definition \\ref{definition-weil-cohomology-theory-classical}).\nLet $X$ be a smooth projective variety of dimension $d$.\nChoose a basis $e_{i, j}, j = 1, \\ldots, \\beta_i$ of $H^i(X)$ over $F$.\nUsing K\\\"unneth write\n$$\n\\gamma([\\Delta]) =\n\\sum\\nolimits_{i = 0, \\ldots, 2d}\n\\sum\\nolimits_j e_{i, j} \\otimes e'_{2d - i , j}\n\\quad\\text{in}\\quad\n\\bigoplus\\nolimits_i H^i(X) \\otimes_F H^{2d - i}(X)\n$$\nwith $e'_{2d - i, j} \\in H^{2d - i}(X)$.\nThen $\\int_X e_{i, j} \\cup e'_{2d - i, j'} = (-1)^i\\delta_{jj'}$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FGZ","source_file":"weil.tex","source_line":1535,"source_end_line":1551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1535-L1551","statement_sha256":"58f260448ecdc8c604f303482a46330f422a6bfe0477fa835ace97b633293fc9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8510,"rank":8510,"depth":3,"x":2056.183,"y":1215.228,"cluster":"duality-cohomology"},{"id":"stacks:0FH0","tag":"0FH0","title":"Classical Weil cohomology theories · Lemma 0FH0","summary":"Let H^* be a classical Weil cohomology theory (Definition [Tag 0FGV]). Let X be a smooth projective variety. We have ∑_i = 0, …, 2dim(X) (-1)^idim_F H^i(X) = deg([Δ] · [Δ]) = deg(c_d(T_X) ∩ [X])","statement_latex":"Let $H^*$ be a classical Weil cohomology theory\n(Definition \\ref{definition-weil-cohomology-theory-classical}).\nLet $X$ be a smooth projective variety. We have\n$$\n\\sum\\nolimits_{i = 0, \\ldots, 2\\dim(X)} (-1)^i\\dim_F H^i(X) =\n\\deg([\\Delta] \\cdot [\\Delta]) = \\deg(c_d(\\mathcal{T}_X) \\cap [X])\n$$","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FH0","source_file":"weil.tex","source_line":1584,"source_end_line":1593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1584-L1593","statement_sha256":"36771f83f3a528fee6fc879e1d94323103924175cbf48fe09226cc151b2cf8d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8511,"rank":8511,"depth":65,"x":1949.943,"y":1099.255,"cluster":"duality-cohomology"},{"id":"stacks:0FH1","tag":"0FH1","title":"Classical Weil cohomology theories · Lemma 0FH1","summary":"Let k be an algebraically closed field. Let F be a field of characteristic 0. Consider a Q-linear functor G : M_k → graded F-vector spaces of symmetric monoidal categories such that G(1(1)) is nonzero only in degree -2. Then we obtain data (D1), (D2), (D3) satisfying all of (A), (B), (C) except for possibly (A)(c) and (A)(d).","statement_latex":"Let $k$ be an algebraically closed field. Let $F$ be a field of\ncharacteristic $0$. Consider a $\\mathbf{Q}$-linear functor\n$$\nG : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}\n$$\nof symmetric monoidal categories such that $G(\\mathbf{1}(1))$\nis nonzero only in degree $-2$. Then we obtain data (D1), (D2), (D3)\nsatisfying all of (A), (B), (C) except for possibly (A)(c) and (A)(d).","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FH1","source_file":"weil.tex","source_line":1640,"source_end_line":1650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1640-L1650","statement_sha256":"02fc8a58e3909b245da0c0d045a058f9233a82dbc2b52cdbc0edd2040c7ea9a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8512,"rank":8512,"depth":67,"x":2122.09,"y":1124.569,"cluster":"duality-cohomology"},{"id":"stacks:0FH2","tag":"0FH2","title":"Classical Weil cohomology theories · Lemma 0FH2","summary":"Let k be an algebraically closed field. Let F be a field of characteristic 0. Let H^* be a classical Weil cohomology theory. Then we can construct a Q-linear functor G : M_k → graded F-vector spaces of symmetric monoidal categories such that H^*(X) = G(h(X)).","statement_latex":"Let $k$ be an algebraically closed field. Let $F$ be a field of\ncharacteristic $0$. Let $H^*$ be a classical Weil cohomology theory.\nThen we can construct a $\\mathbf{Q}$-linear functor\n$$\nG : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}\n$$\nof symmetric monoidal categories such that $H^*(X) = G(h(X))$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FH2","source_file":"weil.tex","source_line":1812,"source_end_line":1821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1812-L1821","statement_sha256":"389c2707707e495ec334b866d137020dd18b16b8bc3fabdab6714e84c5f5c442","origin":"The Stacks Project","memory_eligible":false,"source_rank":8513,"rank":8513,"depth":71,"x":1974.328,"y":1203.833,"cluster":"duality-cohomology"},{"id":"stacks:0FH3","tag":"0FH3","title":"Classical Weil cohomology theories · Proposition 0FH3","summary":"Let k be an algebraically closed field. Let F be a field of characteristic 0. A classical Weil cohomology theory is the same thing as a Q-linear functor G : M_k → graded F-vector spaces of symmetric monoidal categories together with an isomorphism F[2] → G(1(1)) of graded F-vector spaces such that in addition • G(h(X)) lives in nonnegative degrees, and • dim_F G^0(h(X)) = 1 for any smooth projective variety X.","statement_latex":"Let $k$ be an algebraically closed field. Let $F$ be a field of\ncharacteristic $0$. A classical Weil cohomology theory is the same thing\nas a $\\mathbf{Q}$-linear functor\n$$\nG : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}\n$$\nof symmetric monoidal categories together with an isomorphism\n$F[2] \\to G(\\mathbf{1}(1))$ of graded $F$-vector spaces such that\nin addition\n\\begin{enumerate}\n\\item $G(h(X))$ lives in nonnegative degrees, and\n\\item $\\dim_F G^0(h(X)) = 1$\n\\end{enumerate}\nfor any smooth projective variety $X$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Classical Weil cohomology theories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FH3","source_file":"weil.tex","source_line":1957,"source_end_line":1973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L1957-L1973","statement_sha256":"10b7e13aed3f5ed2482d1ec40c39cc35d8f921f244839e5eb2f85bb12c362307","origin":"The Stacks Project","memory_eligible":false,"source_rank":8514,"rank":8514,"depth":72,"x":2019.687,"y":1061.095,"cluster":"duality-cohomology"},{"id":"stacks:0FH5","tag":"0FH5","title":"Cycles over non-closed fields · Lemma 0FH5","summary":"Let k be a field. Let X be a smooth projective scheme over k. Then CH_0(X) is generated by classes of closed points whose residue fields are separable over k.","statement_latex":"Let $k$ be a field. Let $X$ be a smooth projective scheme over $k$.\nThen $\\CH_0(X)$ is generated by classes of closed points whose residue\nfields are separable over $k$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Cycles over non-closed fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FH5","source_file":"weil.tex","source_line":2015,"source_end_line":2020,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2015-L2020","statement_sha256":"1b1e9823f37bba1b713aa6f68f043d2ff84f35bcf834fd47dabc0cb1066748ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":8515,"rank":8515,"depth":60,"x":2101.279,"y":1192.494,"cluster":"duality-cohomology"},{"id":"stacks:0FH6","tag":"0FH6","title":"Cycles over non-closed fields · Lemma 0FH6","summary":"Let K/k be an algebraic field extension. Let X be a finite type scheme over k. Then CH_i(X_K) = colim CH_i(X_k') where the colimit is over the subextensions K/k'/k with k'/k finite.","statement_latex":"Let $K/k$ be an algebraic field extension. Let $X$ be a finite type\nscheme over $k$. Then $\\CH_i(X_K) = \\colim \\CH_i(X_{k'})$ where the\ncolimit is over the subextensions $K/k'/k$ with $k'/k$ finite.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Cycles over non-closed fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FH6","source_file":"weil.tex","source_line":2095,"source_end_line":2100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2095-L2100","statement_sha256":"1bf19c109bc8191e072657020c5afbcd443c48520a4587c662ef0cd004f211cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8516,"rank":8516,"depth":52,"x":1934.934,"y":1141.742,"cluster":"duality-cohomology"},{"id":"stacks:0FH7","tag":"0FH7","title":"Cycles over non-closed fields · Lemma 0FH7","summary":"Let k be a field. Let X be a geometrically irreducible smooth projective scheme over k. Let x, x' ∈ X be k-rational points. Let n be an integer invertible in k. Then there exists a finite separable extension k'/k such that the pullback of [x] - [x'] to X_k' is divisible by n in CH_0(X_k').","statement_latex":"Let $k$ be a field. Let $X$ be a geometrically irreducible\nsmooth projective scheme over $k$. Let $x, x' \\in X$ be $k$-rational points.\nLet $n$ be an integer invertible in $k$.\nThen there exists a finite separable extension $k'/k$ such that\nthe pullback of $[x] - [x']$ to $X_{k'}$\nis divisible by $n$ in $\\CH_0(X_{k'})$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Cycles over non-closed fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FH7","source_file":"weil.tex","source_line":2107,"source_end_line":2115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2107-L2115","statement_sha256":"a19ac6d6c564a03e01bbdabf3d81878b8fc9e97349db2f1d65b33d89f94aa9f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8517,"rank":8517,"depth":66,"x":2098.91,"y":1084.603,"cluster":"duality-cohomology"},{"id":"stacks:0FH8","tag":"0FH8","title":"Cycles over non-closed fields · Lemma 0FH8","summary":"Let K/k be an algebraic extension of fields. Let X be a finite type scheme over k. The kernel of the map CH_i(X) → CH_i(X_K) constructed in Lemma [Tag 0FH6] is torsion.","statement_latex":"Let $K/k$ be an algebraic extension of fields.\nLet $X$ be a finite type scheme over $k$.\nThe kernel of the map $\\CH_i(X) \\to \\CH_i(X_K)$\nconstructed in Lemma \\ref{lemma-chow-limit}\nis torsion.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Cycles over non-closed fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FH8","source_file":"weil.tex","source_line":2167,"source_end_line":2174,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2167-L2174","statement_sha256":"f666dcae484e530929b1cbed562f677e82f4b290cb3577fa73bcdbcd139284b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8518,"rank":8518,"depth":53,"x":2023.717,"y":1220.194,"cluster":"duality-cohomology"},{"id":"stacks:0FH9","tag":"0FH9","title":"Voevodsky · Lemma 0FH9","summary":"[nilpotence] Let k be a field. Let X be a geometrically irreducible smooth projective scheme over k. Let x, x' ∈ X be k-rational points. For n large enough the class of the zero cycle ([x] - [x']) × … × ([x] - [x']) ∈ CH_0(X^n) is torsion.","statement_latex":"\\begin{reference}\n\\cite{nilpotence}\n\\end{reference}\nLet $k$ be a field. Let $X$ be a geometrically irreducible\nsmooth projective scheme over $k$. Let $x, x' \\in X$ be $k$-rational points.\nFor $n$ large enough the class of the zero cycle\n$$\n([x] - [x']) \\times \\ldots \\times ([x] - [x']) \\in\n\\CH_0(X^n)\n$$\nis torsion.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Cycles over non-closed fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FH9","source_file":"weil.tex","source_line":2185,"source_end_line":2198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2185-L2198","statement_sha256":"c0f7c35cd6c7e0bfd84050e72508bb623fc622dfee2eab57db5c9977f1918533","origin":"The Stacks Project","memory_eligible":false,"source_rank":8519,"rank":8519,"depth":67,"x":1969.962,"y":1077.11,"cluster":"duality-cohomology"},{"id":"stacks:0FHB","tag":"0FHB","title":"Weil cohomology theories, I · Lemma 0FHB","summary":"Assume given (D0), (D1), and (D3) satisfying (A). For f : X → Y a morphism of nonempty equidimensional smooth projective schemes over k we have f_*(f^*b ∪ a) = b ∪ f_*a. If g : Y → Z is a second morphism with Z nonempty smooth projective and equidimensional, then g_* ∘ f_* = (g ∘ f)_*.","statement_latex":"Assume given (D0), (D1), and (D3) satisfying (A). For $f : X \\to Y$\na morphism of nonempty equidimensional smooth projective schemes over $k$\nwe have $f_*(f^*b \\cup a) = b \\cup f_*a$. If $g : Y \\to Z$ is a second morphism\nwith $Z$ nonempty smooth projective and equidimensional, then\n$g_* \\circ f_* = (g \\circ f)_*$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHB","source_file":"weil.tex","source_line":2442,"source_end_line":2449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2442-L2449","statement_sha256":"3a0a423e6ac27caf217c5e9c4a56fc2736afb2e401a6af72a41965943438940a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8520,"rank":8520,"depth":0,"x":2125.131,"y":1152.344,"cluster":"duality-cohomology"},{"id":"stacks:0FHC","tag":"0FHC","title":"Weil cohomology theories, I · Lemma 0FHC","summary":"Assume given (D0), (D1), and (D3) satisfying (A) and (B). Let X and Y be nonempty smooth projective schemes over k equidimensional of dimensions d and e. Then pr_2, * : H^*(X × Y)(d + e) → H^*(Y)(e) sends a ⊗ b to (int_X a) b.","statement_latex":"Assume given (D0), (D1), and (D3) satisfying (A) and (B).\nLet $X$ and $Y$ be nonempty smooth projective schemes over $k$\nequidimensional of dimensions $d$ and $e$. Then\n$\\text{pr}_{2, *} : H^*(X \\times Y)(d + e) \\to H^*(Y)(e)$ sends\n$a \\otimes b$ to $(\\int_X a) b$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHC","source_file":"weil.tex","source_line":2481,"source_end_line":2488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2481-L2488","statement_sha256":"8e269c26d6d783be783cb85229412ad8f1d5103d8bb8981728ab0f4133be69ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":8521,"rank":8521,"depth":0,"x":1949.685,"y":1185.012,"cluster":"duality-cohomology"},{"id":"stacks:0FHE","tag":"0FHE","title":"Weil cohomology theories, I · Lemma 0FHE","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Then H^i(Spec(k)) = 0 for i not = 0 and there is a unique F-algebra isomorphism F = H^0(Spec(k)). We have γ([Spec(k)]) = 1 and int_Spec(k) 1 = 1.","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nThen $H^i(\\Spec(k)) = 0$ for $i \\not = 0$ and there is a\nunique $F$-algebra isomorphism $F = H^0(\\Spec(k))$.\nWe have $\\gamma([\\Spec(k)]) = 1$ and $\\int_{\\Spec(k)} 1 = 1$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHE","source_file":"weil.tex","source_line":2553,"source_end_line":2559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2553-L2559","statement_sha256":"d7d90c76ee69b8b8193a86ae21be559273c0d8e5d46ad6be8b50b6efa2d0db8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8522,"rank":8522,"depth":0,"x":2053.094,"y":1061.002,"cluster":"duality-cohomology"},{"id":"stacks:0FHF","tag":"0FHF","title":"Weil cohomology theories, I · Lemma 0FHF","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let X be a smooth projective scheme over k. If X = ∅, then H^*(X) = 0. If X is nonempty, then γ([X]) = 1 and 1 not = 0 in H^0(X).","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $X$ be a smooth projective scheme over $k$.\nIf $X = \\emptyset$, then $H^*(X) = 0$.\nIf $X$ is nonempty, then $\\gamma([X]) = 1$ and $1 \\not = 0$ in $H^0(X)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHF","source_file":"weil.tex","source_line":2582,"source_end_line":2588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2582-L2588","statement_sha256":"9a171a5a0fabf3e6fbafd5f62b8075580995d576d9cf46876954e92da64e2117","origin":"The Stacks Project","memory_eligible":false,"source_rank":8523,"rank":8523,"depth":46,"x":2076.637,"y":1211.567,"cluster":"duality-cohomology"},{"id":"stacks:0FHG","tag":"0FHG","title":"Weil cohomology theories, I · Lemma 0FHG","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let i : X → Y be a closed immersion of nonempty smooth projective equidimensional schemes over k. Then γ([X]) = i_*1 in H^2c(Y)(c) where c = dim(Y) - dim(X).","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $i : X \\to Y$ be a closed immersion of nonempty smooth projective\nequidimensional schemes over $k$. Then\n$\\gamma([X]) = i_*1$ in $H^{2c}(Y)(c)$ where $c = \\dim(Y) - \\dim(X)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHG","source_file":"weil.tex","source_line":2630,"source_end_line":2636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2630-L2636","statement_sha256":"31b0ce1f85418b1f753a22bb48e629c1feb3b6403c39cf561bfdd373a3e72ebf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8524,"rank":8524,"depth":47,"x":1937.788,"y":1113.613,"cluster":"duality-cohomology"},{"id":"stacks:0FHH","tag":"0FHH","title":"Weil cohomology theories, I · Lemma 0FHH","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let X be a nonempty smooth projective scheme over k equidimensional of dimension d. Choose a basis e_i, j, j = 1, …, β_i of H^i(X) over F. Using K\\\"unneth write γ([Δ]) = ∑_i ∑_j e_i, j ⊗ e'_2d - i , j in bigoplus_i H^i(X) ⊗_F H^2d - i(X)(d) with e'_2d - i, j ∈ H^2d - i(X)(d). Then int_X e_i, j ∪ e'_2d - i, j' = (-1)^iδ_jj'.","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $X$ be a nonempty smooth projective scheme over $k$ equidimensional\nof dimension $d$. Choose a basis $e_{i, j}, j = 1, \\ldots, \\beta_i$ of\n$H^i(X)$ over $F$. Using K\\\"unneth write\n$$\n\\gamma([\\Delta]) =\n\\sum\\nolimits_i\n\\sum\\nolimits_j e_{i, j} \\otimes e'_{2d - i , j}\n\\quad\\text{in}\\quad\n\\bigoplus\\nolimits_i H^i(X) \\otimes_F H^{2d - i}(X)(d)\n$$\nwith $e'_{2d - i, j} \\in H^{2d - i}(X)(d)$.\nThen $\\int_X e_{i, j} \\cup e'_{2d - i, j'} = (-1)^i\\delta_{jj'}$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHH","source_file":"weil.tex","source_line":2643,"source_end_line":2658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2643-L2658","statement_sha256":"a3346ffb83fd4930ff7b9f63162c476472e1254a63764fcd538aa390a5018094","origin":"The Stacks Project","memory_eligible":false,"source_rank":8525,"rank":8525,"depth":48,"x":2119.476,"y":1107.038,"cluster":"duality-cohomology"},{"id":"stacks:0FHI","tag":"0FHI","title":"Weil cohomology theories, I · Lemma 0FHI","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Then H^*(P^1_k) is 1-dimensional in dimensions 0 and 2 and zero in other degrees.","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nThen $H^*(\\mathbf{P}^1_k)$ is $1$-dimensional in dimensions $0$ and $2$\nand zero in other degrees.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHI","source_file":"weil.tex","source_line":2684,"source_end_line":2689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2684-L2689","statement_sha256":"2c4979f817c5746d9bea06d0ee2a2d2f9d986a096f6a4efc2c59a9c754d6daa4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8526,"rank":8526,"depth":49,"x":1990.414,"y":1215.295,"cluster":"duality-cohomology"},{"id":"stacks:0FHJ","tag":"0FHJ","title":"Weil cohomology theories, I · Lemma 0FHJ","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). If X and Y are smooth projective schemes over k, then H^*(X amalg Y) → H^*(X) × H^*(Y), a ↦ (i^*a, j^*a) is an isomorphism where i, j are the coprojections.","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nIf $X$ and $Y$ are smooth projective schemes over $k$, then\n$H^*(X \\amalg Y) \\to H^*(X) \\times H^*(Y)$,\n$a \\mapsto (i^*a, j^*a)$ is an isomorphism where $i$, $j$\nare the coprojections.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHJ","source_file":"weil.tex","source_line":2714,"source_end_line":2721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2714-L2721","statement_sha256":"1baea206e753c2c1d4f09e3f2edd4d124ca3fd667794d0b51b9dd4afdb449911","origin":"The Stacks Project","memory_eligible":false,"source_rank":8527,"rank":8527,"depth":48,"x":1998.55,"y":1061.791,"cluster":"duality-cohomology"},{"id":"stacks:0FHK","tag":"0FHK","title":"Weil cohomology theories, I · Lemma 0FHK","summary":"Let k be a field. Let F be a field of characteristic 0. Assume given a Q-linear functor G : M_k → graded F-vector spaces of symmetric monoidal categories such that G(1(1)) is nonzero only in degree -2. Then we obtain data (D0), (D1), (D2), and (D3) satisfying all of (A), (B), and (C) above.","statement_latex":"Let $k$ be a field. Let $F$ be a field of characteristic $0$.\nAssume given a $\\mathbf{Q}$-linear functor\n$$\nG : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}\n$$\nof symmetric monoidal categories such that $G(\\mathbf{1}(1))$\nis nonzero only in degree $-2$. Then we obtain data (D0), (D1), (D2), and (D3)\nsatisfying all of (A), (B), and (C) above.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHK","source_file":"weil.tex","source_line":2792,"source_end_line":2802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2792-L2802","statement_sha256":"9f4308ca2b8f7c833e68c90d77eae091e3184b82721fd5082f113fefc77ba7b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8528,"rank":8528,"depth":68,"x":2116.332,"y":1179.939,"cluster":"duality-cohomology"},{"id":"stacks:0FHL","tag":"0FHL","title":"Weil cohomology theories, I · Lemma 0FHL","summary":"Let k be a field. Let F be a field of characteristic 0. Given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C) we can construct a Q-linear functor G : M_k → graded F-vector spaces of symmetric monoidal categories such that H^*(X) = G(h(X)).","statement_latex":"Let $k$ be a field. Let $F$ be a field of characteristic $0$. Given\n(D0), (D1), (D2), and (D3) satisfying (A), (B), and (C)\nwe can construct a $\\mathbf{Q}$-linear functor\n$$\nG : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}\n$$\nof symmetric monoidal categories such that $H^*(X) = G(h(X))$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHL","source_file":"weil.tex","source_line":2979,"source_end_line":2988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L2979-L2988","statement_sha256":"c540532385c1568ee2e1fbc23ef30bdf71b492f4309c4aa66da96fe7b27330dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8529,"rank":8529,"depth":72,"x":1933.949,"y":1159.592,"cluster":"duality-cohomology"},{"id":"stacks:0FHM","tag":"0FHM","title":"Weil cohomology theories, I · Proposition 0FHM","summary":"Let k be a field. Let F be a field of characteristic 0. There is a 1-to-1 correspondence between the following • data (D0), (D1), (D2), and (D3) satisfying (A), (B), and(C), and • Q-linear symmetric monoidal functors G : M_k → graded F-vector spaces such that G(1(1)) is nonzero only in degree -2.","statement_latex":"Let $k$ be a field. Let $F$ be a field of characteristic $0$. There is a\n$1$-to-$1$ correspondence between the following\n\\begin{enumerate}\n\\item data (D0), (D1), (D2), and (D3) satisfying (A), (B), and(C), and\n\\item $\\mathbf{Q}$-linear symmetric monoidal functors\n$$\nG : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}\n$$\nsuch that $G(\\mathbf{1}(1))$ is nonzero only in degree $-2$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, I","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHM","source_file":"weil.tex","source_line":3143,"source_end_line":3155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3143-L3155","statement_sha256":"0298956ac6fd783dab8cfdc84dd48cdbf184eaba527906526dcf08d97b5f06a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8530,"rank":8530,"depth":73,"x":2085.227,"y":1070.856,"cluster":"duality-cohomology"},{"id":"stacks:0FHP","tag":"0FHP","title":"Further properties · Lemma 0FHP","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let X, Y be nonempty smooth projective schemes both equidimensional of dimension d over k. Then int_X amalg Y = int_X + int_Y.","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $X, Y$ be nonempty smooth projective schemes both equidimensional\nof dimension $d$ over $k$. Then $\\int_{X \\amalg Y} = \\int_X + \\int_Y$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Further properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHP","source_file":"weil.tex","source_line":3190,"source_end_line":3195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3190-L3195","statement_sha256":"fd2837329192adcefd800583472bf6bb2e1351f3221cc69f9d1f516e6fb0cbe9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8531,"rank":8531,"depth":49,"x":2044.922,"y":1222.556,"cluster":"duality-cohomology"},{"id":"stacks:0FHQ","tag":"0FHQ","title":"Further properties · Lemma 0FHQ","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let X be a smooth projective scheme of dimension zero over k. Then • H^i(X) = 0 for i not = 0, • H^0(X) is a finite separable algebra over F, • dim_F H^0(X) = deg(X → Spec(F)), • int_X : H^0(X) → F is the trace map, • γ([X]) = 1, and • int_X γ([X]) = deg(X → Spec(k)).","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $X$ be a smooth projective scheme of dimension zero over $k$.\nThen\n\\begin{enumerate}\n\\item $H^i(X) = 0$ for $i \\not = 0$,\n\\item $H^0(X)$ is a finite separable algebra over $F$,\n\\item $\\dim_F H^0(X) = \\deg(X \\to \\Spec(F))$,\n\\item $\\int_X : H^0(X) \\to F$ is the trace map,\n\\item $\\gamma([X]) = 1$, and\n\\item $\\int_X \\gamma([X]) = \\deg(X \\to \\Spec(k))$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Further properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHQ","source_file":"weil.tex","source_line":3214,"source_end_line":3227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3214-L3227","statement_sha256":"a2a17c77a137923c00e28bb40195cf3d993b351dfa3f67c0917ca22f3f774236","origin":"The Stacks Project","memory_eligible":false,"source_rank":8532,"rank":8532,"depth":50,"x":1952.39,"y":1087.445,"cluster":"duality-cohomology"},{"id":"stacks:0FHR","tag":"0FHR","title":"Further properties · Lemma 0FHR","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let X be a nonempty smooth projective scheme equidimensional of dimension d over k. The diagram xymatrix CH^d(X) ar[r]_-γ ar@=[d] & H^2d(X)(d) ar[d]^int_X CH_0(X) ar[r]^deg & F commutes where deg : CH_0(X) → Z is the degree of zero cycles discussed in Chow Homology, Section [Tag 0AZ0].","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $X$ be a nonempty smooth projective scheme\nequidimensional of dimension $d$ over $k$. The diagram\n$$\n\\xymatrix{\n\\CH^d(X) \\ar[r]_-\\gamma \\ar@{=}[d] &\nH^{2d}(X)(d) \\ar[d]^{\\int_X} \\\\\n\\CH_0(X) \\ar[r]^\\deg & F\n}\n$$\ncommutes where $\\deg : \\CH_0(X) \\to \\mathbf{Z}$ is the degree of\nzero cycles discussed in Chow Homology, Section\n\\ref{chow-section-degree-zero-cycles}.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Further properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHR","source_file":"weil.tex","source_line":3287,"source_end_line":3302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3287-L3302","statement_sha256":"caa028e3c767f91652eac6ce302344419694edeb8d12c16a1f37e2c43df325f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8533,"rank":8533,"depth":61,"x":2129.769,"y":1134.701,"cluster":"duality-cohomology"},{"id":"stacks:0FHS","tag":"0FHS","title":"Further properties · Lemma 0FHS","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let X be a nonempty smooth projective scheme over k which is equidimensional of dimension d. We have ∑_i (-1)^idim_F H^i(X) = deg(Δ · Δ) = deg(c_d(T_X/k))","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $X$ be a nonempty smooth projective scheme over $k$ which is\nequidimensional of dimension $d$. We have\n$$\n\\sum\\nolimits_i (-1)^i\\dim_F H^i(X) =\n\\deg(\\Delta \\cdot \\Delta) = \\deg(c_d(\\mathcal{T}_{X/k}))\n$$","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Further properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHS","source_file":"weil.tex","source_line":3320,"source_end_line":3329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3320-L3329","statement_sha256":"964264b67f713562daa3b092728bc2d548f6f991abaa490a490c4d1f2fcad581","origin":"The Stacks Project","memory_eligible":false,"source_rank":8534,"rank":8534,"depth":65,"x":1960.501,"y":1200.687,"cluster":"duality-cohomology"},{"id":"stacks:0FHT","tag":"0FHT","title":"Further properties · Lemma 0FHT","summary":"Let F be a field of characteristic 0. Let F' and F_i, i = 1, …, r be finite separable F-algebras. Let A be a finite F-algebra. Let σ, σ' : A → F' and σ_i : A → F_i be F-algebra maps. Assume σ and σ' surjective. If there is a relation Tr_F'/F ∘ σ - Tr_F'/F ∘ σ' = n(∑ m_i Tr_F_i/F ∘ σ_i) where n > 1 and m_i are integers, then σ = σ'.","statement_latex":"Let $F$ be a field of characteristic $0$.\nLet $F'$ and $F_i$, $i = 1, \\ldots, r$\nbe finite separable $F$-algebras. Let $A$ be a finite $F$-algebra.\nLet $\\sigma, \\sigma' : A \\to F'$ and $\\sigma_i : A \\to F_i$\nbe $F$-algebra maps. Assume $\\sigma$ and $\\sigma'$ surjective.\nIf there is a relation\n$$\n\\text{Tr}_{F'/F} \\circ \\sigma - \\text{Tr}_{F'/F} \\circ \\sigma' =\nn(\\sum m_i \\text{Tr}_{F_i/F} \\circ \\sigma_i)\n$$\nwhere $n > 1$ and $m_i$ are integers, then $\\sigma = \\sigma'$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Further properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHT","source_file":"weil.tex","source_line":3373,"source_end_line":3386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3373-L3386","statement_sha256":"f25f53fb1da042346d706c0b06ba0e01d0b4aa57a6975c6f2ba89d3399f58349","origin":"The Stacks Project","memory_eligible":false,"source_rank":8535,"rank":8535,"depth":9,"x":2032.452,"y":1055.582,"cluster":"duality-cohomology"},{"id":"stacks:0FHU","tag":"0FHU","title":"Further properties · Lemma 0FHU","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let k'/k be a finite separable extension. Let X be a smooth projective scheme over k'. Let x, x' ∈ X be k'-rational points. If γ(x) not = γ(x'), then [x] - [x'] is not divisible by any integer n > 1 in CH_0(X).","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $k'/k$ be a finite separable extension.\nLet $X$ be a smooth projective scheme over $k'$.\nLet $x, x' \\in X$ be $k'$-rational points.\nIf $\\gamma(x) \\not = \\gamma(x')$, then\n$[x] - [x']$ is not divisible by any integer $n > 1$ in $\\CH_0(X)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Further properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHU","source_file":"weil.tex","source_line":3438,"source_end_line":3446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3438-L3446","statement_sha256":"db789e9fe58ddb05b11f274a1702ea22973132091445e5f7fae85ca8e3ea128f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8536,"rank":8536,"depth":61,"x":2096.26,"y":1203.814,"cluster":"duality-cohomology"},{"id":"stacks:0FHV","tag":"0FHV","title":"Further properties · Lemma 0FHV","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let k'/k be a finite separable extension. Let X be a geometrically irreducible smooth projective scheme over k' of dimension d. Then γ : CH_0(X) → H^2d(X)(d) factors through deg : CH_0(X) → Z.","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $k'/k$ be a finite separable extension. Let $X$ be a geometrically\nirreducible smooth projective scheme over $k'$ of dimension $d$.\nThen $\\gamma : \\CH_0(X) \\to H^{2d}(X)(d)$ factors through\n$\\deg : \\CH_0(X) \\to \\mathbf{Z}$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Further properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHV","source_file":"weil.tex","source_line":3479,"source_end_line":3486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3479-L3486","statement_sha256":"70e5163ec79eacdec89c279d3a48f20ffb32cb231abb46134891d907f9bf1c76","origin":"The Stacks Project","memory_eligible":false,"source_rank":8537,"rank":8537,"depth":67,"x":1929.55,"y":1130.516,"cluster":"duality-cohomology"},{"id":"stacks:0FHW","tag":"0FHW","title":"Further properties · Lemma 0FHW","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let f : X → Y be a dominant morphism of irreducible smooth projective schemes over k. Then H^*(Y) → H^*(X) is injective.","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let\n$f : X \\to Y$ be a dominant morphism of irreducible smooth projective schemes\nover $k$. Then $H^*(Y) \\to H^*(X)$ is injective.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Further properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHW","source_file":"weil.tex","source_line":3519,"source_end_line":3524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3519-L3524","statement_sha256":"737e8a0c00ac9fc3d2a9d3132d59d91c960bdfc9caf40b9056b35af4a22ab1f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8538,"rank":8538,"depth":47,"x":2111.919,"y":1089.86,"cluster":"duality-cohomology"},{"id":"stacks:0FHX","tag":"0FHX","title":"Further properties · Lemma 0FHX","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let k\"/k'/k be finite separable algebras and let X be a smooth projective scheme over k'. Then H^*(X) ⊗_H^0(Spec(k')) H^0(Spec(k\")) = H^*(X ×_Spec(k') Spec(k\"))","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let\n$k''/k'/k$ be finite separable algebras and let $X$ be a\nsmooth projective scheme over $k'$. Then\n$$\nH^*(X) \\otimes_{H^0(\\Spec(k'))} H^0(\\Spec(k'')) =\nH^*(X \\times_{\\Spec(k')} \\Spec(k''))\n$$","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Further properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FHX","source_file":"weil.tex","source_line":3535,"source_end_line":3544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3535-L3544","statement_sha256":"2ca1bd64f3b3bd0b52606d965e0f02f85b1b11db9ced979226d9010a2b14de17","origin":"The Stacks Project","memory_eligible":false,"source_rank":8539,"rank":8539,"depth":51,"x":2009.861,"y":1223.681,"cluster":"duality-cohomology"},{"id":"stacks:0FI0","tag":"0FI0","title":"Weil cohomology theories, II · Lemma 0FI0","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). Let X be a smooth projective scheme over k. Set k' = Γ(X, O_X). The following are equivalent • there exist finitely many closed points x_1, …, x_r ∈ X whose residue fields are separable over k such that H^0(X) → H^0(x_1) ⊕ … ⊕ H^0(x_r) is injective, • the map H^0(Spec(k')) → H^0(X) is an isomorphism. If this is true, then H^0(X) is a finite separable algebra over F. If X is equidimensional of dimension…","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nLet $X$ be a smooth projective scheme over $k$.\nSet $k' = \\Gamma(X, \\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item there exist finitely many closed points $x_1, \\ldots, x_r \\in X$\nwhose residue fields are separable over $k$ such that\n$H^0(X) \\to H^0(x_1) \\oplus \\ldots \\oplus H^0(x_r)$ is injective,\n\\item the map $H^0(\\Spec(k')) \\to H^0(X)$ is an isomorphism.\n\\end{enumerate}\nIf this is true, then $H^0(X)$ is a finite separable algebra over $F$.\nIf $X$ is equidimensional of dimension $d$, then (1) and (2)\nare also equivalent to\n\\begin{enumerate}\n\\item[(3)] the classes of closed points generate $H^{2d}(X)(d)$\nas a module over $H^0(X)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FI0","source_file":"weil.tex","source_line":3642,"source_end_line":3660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3642-L3660","statement_sha256":"62d84fea0538ead9f92feb245568736d392ba03e5396e7f0669809a8a98b181c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8540,"rank":8540,"depth":68,"x":1977.416,"y":1066.672,"cluster":"duality-cohomology"},{"id":"stacks:0FI1","tag":"0FI1","title":"Weil cohomology theories, II · Lemma 0FI1","summary":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). If there exists a smooth projective scheme Y over k such that H^i(Y) is nonzero for some i < 0, then there exists an equidimensional smooth projective scheme X over k such that the equivalent conditions of Lemma [Tag 0FI0] fail for X.","statement_latex":"Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C).\nIf there exists a smooth projective scheme $Y$ over $k$ such that\n$H^i(Y)$ is nonzero for some $i < 0$, then there exists an equidimensional\nsmooth projective scheme $X$ over $k$ such that the equivalent conditions\nof Lemma \\ref{lemma-H-0-separable} fail for $X$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FI1","source_file":"weil.tex","source_line":3711,"source_end_line":3718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3711-L3718","statement_sha256":"80f97027a2daad9688274e54069a8405a063f5aa52012042f78f980eee8e17e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8541,"rank":8541,"depth":69,"x":2128.005,"y":1164.298,"cluster":"duality-cohomology"},{"id":"stacks:0FI2","tag":"0FI2","title":"Weil cohomology theories, II · Definition 0FI2","summary":"Let k be a field. Let F be a field of characteristic 0. A Weil cohomology theory over k with coefficients in F is given by data (D0), (D1), (D2), and (D3) satisfying Poincaré duality, the K\\\"unneth formula, and compatibility with cycle classes, more precisely, satisfying axioms (A), (B), and (C) of Section [Tag 0FHA] and in addition such that the equivalent conditions (1) and (2) of Lemma [Tag 0FI0] hold for every smooth projective X over k.","statement_latex":"Let $k$ be a field. Let $F$ be a field of characteristic $0$.\nA {\\it Weil cohomology theory} over $k$ with coefficients in $F$\nis given by data (D0), (D1), (D2), and (D3) satisfying\nPoincar\\'e duality, the K\\\"unneth formula, and compatibility\nwith cycle classes, more precisely, satisfying axioms (A), (B), and (C)\nof Section \\ref{section-axioms}\nand in addition such that the equivalent conditions (1) and (2) of\nLemma \\ref{lemma-H-0-separable} hold for every smooth projective $X$ over $k$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, II","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FI2","source_file":"weil.tex","source_line":3739,"source_end_line":3749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3739-L3749","statement_sha256":"e736a517ad00e77859c79ec06a602ca8a08baf23fc1504da905856d300391a5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8542,"rank":8542,"depth":69,"x":1937.949,"y":1177.793,"cluster":"duality-cohomology"},{"id":"stacks:0GIJ","tag":"0GIJ","title":"Weil cohomology theories, II · Proposition 0GIJ","summary":"Let k be a field. Let F be a field of characteristic 0. A Weil cohomology theory is the same thing as a Q-linear symmetric monoidal functor G : M_k → graded F-vector spaces such that • G(1(1)) is nonzero only in degree -2, and • for every smooth projective scheme X over k with k' = Γ(X, O_X) the homomorphism G(h(Spec(k'))) → G(h(X)) of graded F-vector spaces is an isomorphism in degree 0.","statement_latex":"Let $k$ be a field. Let $F$ be a field of characteristic $0$.\nA Weil cohomology theory is the same thing as a $\\mathbf{Q}$-linear\nsymmetric monoidal functor\n$$\nG : M_k \\longrightarrow \\text{graded }F\\text{-vector spaces}\n$$\nsuch that\n\\begin{enumerate}\n\\item $G(\\mathbf{1}(1))$ is nonzero only in degree $-2$, and\n\\item for every smooth projective scheme $X$ over $k$ with\n$k' = \\Gamma(X, \\mathcal{O}_X)$ the homomorphism\n$G(h(\\Spec(k'))) \\to G(h(X))$ of graded $F$-vector spaces\nis an isomorphism in degree $0$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIJ","source_file":"weil.tex","source_line":3794,"source_end_line":3810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3794-L3810","statement_sha256":"2871c9a7e834d1aedf3505eb7af53cbf1b11bdd19138f1154ad6ed44b3a7f585","origin":"The Stacks Project","memory_eligible":false,"source_rank":8543,"rank":8543,"depth":74,"x":2067.584,"y":1059.689,"cluster":"duality-cohomology"},{"id":"stacks:0FI5","tag":"0FI5","title":"Chern classes · Proposition 0FI5","summary":"In the situation above there is a unique rule which assigns to every X ∈ Ob(C) a \"total Chern class\" c^A : K_0(Vect(X)) → ∏_i ≥ 0 A^i(X) with the following properties • For X ∈ Ob(C) we have c^A(α + β) = c^A(α) c^A(β) and c^A(0) = 1. • If f : X' → X is a morphism of C, then f^* ∘ c^A = c^A ∘ f^*. • Given X ∈ Ob(C) and L ∈ Pic(X) we have c^A([L]) = 1 + c_1^A(L).","statement_latex":"In the situation above there is a unique rule which assigns to\nevery $X \\in \\Ob(\\mathcal{C})$ a ``total Chern class''\n$$\nc^A : K_0(\\textit{Vect}(X)) \\longrightarrow  \\prod\\nolimits_{i \\geq 0} A^i(X)\n$$\nwith the following properties\n\\begin{enumerate}\n\\item For $X \\in \\Ob(\\mathcal{C})$ we have\n$c^A(\\alpha + \\beta) = c^A(\\alpha) c^A(\\beta)$\nand $c^A(0) = 1$.\n\\item If $f : X' \\to X$ is a morphism of $\\mathcal{C}$, then\n$f^* \\circ c^A =  c^A \\circ f^*$.\n\\item Given $X \\in \\Ob(\\mathcal{C})$ and $\\mathcal{L} \\in \\Pic(X)$\nwe have $c^A([\\mathcal{L}]) = 1 + c_1^A(\\mathcal{L})$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chern classes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FI5","source_file":"weil.tex","source_line":3907,"source_end_line":3924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L3907-L3924","statement_sha256":"a2d4ae92e1e9299148931afba1811dcef101cdc61613d6fc8cd32b5a92522e7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8544,"rank":8544,"depth":0,"x":2066.968,"y":1220.758,"cluster":"duality-cohomology"},{"id":"stacks:0FI7","tag":"0FI7","title":"Chern classes · Lemma 0FI7","summary":"In the situation above. Let X ∈ Ob(C). Let E_i be a finite collection of locally free O_X-modules of rank r_i. There exists a morphism p : P → X in C such that • p^* : A(X) → A(P) is injective, • each p^*E_i has a filtration whose successive quotients L_i, 1, …, L_i, r_i are invertible O_P-modules.","statement_latex":"In the situation above. Let $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{E}_i$\nbe a finite collection of locally free $\\mathcal{O}_X$-modules of rank $r_i$.\nThere exists a morphism $p : P \\to X$ in $\\mathcal{C}$ such that\n\\begin{enumerate}\n\\item $p^* : A(X) \\to A(P)$ is injective,\n\\item each $p^*\\mathcal{E}_i$ has a filtration whose successive quotients\n$\\mathcal{L}_{i, 1}, \\ldots, \\mathcal{L}_{i, r_i}$\nare invertible $\\mathcal{O}_P$-modules.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FI7","source_file":"weil.tex","source_line":4068,"source_end_line":4079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4068-L4079","statement_sha256":"4f7280121aa56474ee7a6a3c46dfd4041510a61e72492fd22a324cedc14f2c59","origin":"The Stacks Project","memory_eligible":false,"source_rank":8545,"rank":8545,"depth":0,"x":1937.556,"y":1101.326,"cluster":"duality-cohomology"},{"id":"stacks:0FI8","tag":"0FI8","title":"Chern classes · Lemma 0FI8","summary":"Let X ∈ Ob(C). Let E be a finite locally free O_X-module. Let L be an invertible O_X-module. Then c^A_i( E ⊗ L) = ∑_j = 0^i binomr - i + jj c^A_i - j( E) ∪ c^A_1( L)^j","statement_latex":"Let $X \\in \\Ob(\\mathcal{C})$. Let $\\mathcal{E}$ be a finite locally free\n$\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module. Then\n$$\nc^A_i({\\mathcal E} \\otimes {\\mathcal L})\n=\n\\sum\\nolimits_{j = 0}^i\n\\binom{r - i + j}{j} c^A_{i - j}({\\mathcal E}) \\cup c^A_1({\\mathcal L})^j\n$$","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FI8","source_file":"weil.tex","source_line":4100,"source_end_line":4111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4100-L4111","statement_sha256":"96f66e455c6d756889e336c216b7f63eec9fc0497daea7b112b24b937ca0004b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8546,"rank":8546,"depth":1,"x":2129.525,"y":1116.001,"cluster":"duality-cohomology"},{"id":"stacks:0FI9","tag":"0FI9","title":"Chern classes · Proposition 0FI9","summary":"In the situation above assume A(X) is a Q-algebra for all X ∈ Ob(C). Then there is a unique rule which assigns to every X ∈ Ob(C) a \"chern character\" ch^A : K_0(Vect(X)) → ∏_i ≥ 0 A^i(X) with the following properties • ch^A is a ring map for all X ∈ Ob(C). • If f : X' → X is a morphism of C, then f^* ∘ ch^A = ch^A ∘ f^*. • Given X ∈ Ob(C) and L ∈ Pic(X) we have ch^A([L]) = exp(c_1^A(L)).","statement_latex":"In the situation above assume $A(X)$ is a $\\mathbf{Q}$-algebra for all\n$X \\in \\Ob(\\mathcal{C})$. Then there is a unique rule which assigns to\nevery $X \\in \\Ob(\\mathcal{C})$ a ``chern character''\n$$\nch^A : K_0(\\textit{Vect}(X)) \\longrightarrow\n\\prod\\nolimits_{i \\geq 0} A^i(X)\n$$\nwith the following properties\n\\begin{enumerate}\n\\item $ch^A$ is a ring map for all $X \\in \\Ob(\\mathcal{C})$.\n\\item If $f : X' \\to X$ is a morphism of $\\mathcal{C}$, then\n$f^* \\circ ch^A =  ch^A \\circ f^*$.\n\\item Given $X \\in \\Ob(\\mathcal{C})$ and $\\mathcal{L} \\in \\Pic(X)$\nwe have $ch^A([\\mathcal{L}]) = \\exp(c_1^A(\\mathcal{L}))$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chern classes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FI9","source_file":"weil.tex","source_line":4130,"source_end_line":4147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4130-L4147","statement_sha256":"1d34e4ed450c09dcb8931697942f43cd7cd32b02562314d890524a435e8312c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8547,"rank":8547,"depth":1,"x":1975.771,"y":1214.361,"cluster":"duality-cohomology"},{"id":"stacks:0FIA","tag":"0FIA","title":"Chern classes · Lemma 0FIA","summary":"In the situation above let X ∈ Ob(C). If ψ^2 is as in Chow Homology, Lemma [Tag 0FEJ] and c^A and ch^A are as in Propositions [Tag 0FI5] and [Tag 0FI9] then we have c^A_i(ψ^2(α)) = 2^i c^A_i(α) and ch^A_i(ψ^2(α)) = 2^i ch^A_i(α) for all α ∈ K_0(Vect(X)).","statement_latex":"In the situation above let $X \\in \\Ob(\\mathcal{C})$.\nIf $\\psi^2$ is as in\nChow Homology, Lemma \\ref{chow-lemma-second-adams-operator}\nand $c^A$ and $ch^A$ are as in\nPropositions \\ref{proposition-chern-class} and\n\\ref{proposition-chern-character}\nthen we have $c^A_i(\\psi^2(\\alpha)) = 2^i c^A_i(\\alpha)$ and\n$ch^A_i(\\psi^2(\\alpha)) = 2^i ch^A_i(\\alpha)$\nfor all $\\alpha \\in K_0(\\textit{Vect}(X))$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIA","source_file":"weil.tex","source_line":4243,"source_end_line":4254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4243-L4254","statement_sha256":"2ede26617134c5432e3f0d66fb198529e3fdb51b87420c402b1503fcd85db70c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8548,"rank":8548,"depth":2,"x":2010.137,"y":1054.176,"cluster":"duality-cohomology"},{"id":"stacks:0FIC","tag":"0FIC","title":"Exterior powers and K-groups · Lemma 0FIC","summary":"Let X be a scheme. There are maps λ^r : K_0(Vect(X)) → K_0(Vect(X)) which sends [E] to [wedge^r(E)] when E is a finite locally free O_X-module and which are compatible with pullbacks.","statement_latex":"Let $X$ be a scheme. There are maps\n$$\n\\lambda^r : K_0(\\textit{Vect}(X)) \\longrightarrow K_0(\\textit{Vect}(X))\n$$\nwhich sends $[\\mathcal{E}]$ to $[\\wedge^r(\\mathcal{E})]$\nwhen $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module\nand which are compatible with pullbacks.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Exterior powers and K-groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIC","source_file":"weil.tex","source_line":4306,"source_end_line":4315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4306-L4315","statement_sha256":"561b1107d1c1da2b8d0918a3529a6a0303ed1e0e271cbef857561103a64d7b42","origin":"The Stacks Project","memory_eligible":false,"source_rank":8549,"rank":8549,"depth":0,"x":2113.878,"y":1192.148,"cluster":"duality-cohomology"},{"id":"stacks:0FIE","tag":"0FIE","title":"Weil cohomology theories, III · Lemma 0FIE","summary":"Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4). There is a unique rule which assigns to every smooth projective X over k a ring homomorphism ch^H : K_0(Vect(X)) → ∏_i ≥ 0 H^2i(X)(i) compatible with pullbacks such that ch^H(L) = exp(c_1^H(L)) for any invertible O_X-module L.","statement_latex":"Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4).\nThere is a unique rule which assigns to every smooth projective $X$ over $k$\na ring homomorphism\n$$\nch^H :\nK_0(\\textit{Vect}(X))\n\\longrightarrow\n\\prod\\nolimits_{i \\geq 0} H^{2i}(X)(i)\n$$\ncompatible with pullbacks such that\n$ch^H(\\mathcal{L}) = \\exp(c_1^H(\\mathcal{L}))$\nfor any invertible $\\mathcal{O}_X$-module $\\mathcal{L}$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIE","source_file":"weil.tex","source_line":4472,"source_end_line":4486,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4472-L4486","statement_sha256":"63d290c4613330e72f512dc759781dc3bb8f94e98a0f7bdbbe1bff7ebe7d3d28","origin":"The Stacks Project","memory_eligible":false,"source_rank":8550,"rank":8550,"depth":2,"x":1925.95,"y":1149.165,"cluster":"duality-cohomology"},{"id":"stacks:0FIF","tag":"0FIF","title":"Weil cohomology theories, III · Lemma 0FIF","summary":"Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4). There is a unique rule which assigns to every smooth projective X over k a graded ring homomorphism γ : CH^*(X) → bigoplus_i ≥ 0 H^2i(X)(i) compatible with pullbacks such that ch^H(α) = γ(ch(α)) for α in K_0(Vect(X)).","statement_latex":"Assume given (D0), (D1), and (D2') satisfying axioms (A1), (A2), (A3), and (A4).\nThere is a unique rule which assigns to every smooth projective $X$ over $k$\na graded ring homomorphism\n$$\n\\gamma : \\CH^*(X) \\longrightarrow \\bigoplus\\nolimits_{i \\geq 0} H^{2i}(X)(i)\n$$\ncompatible with pullbacks such that $ch^H(\\alpha) = \\gamma(ch(\\alpha))$\nfor $\\alpha$ in $K_0(\\textit{Vect}(X))$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIF","source_file":"weil.tex","source_line":4495,"source_end_line":4505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4495-L4505","statement_sha256":"0040064aa880d58838fda9c8571884d8ab00bec271ba1609910d5b4a426327cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8551,"rank":8551,"depth":67,"x":2099.531,"y":1074.035,"cluster":"duality-cohomology"},{"id":"stacks:0FIH","tag":"0FIH","title":"Weil cohomology theories, III · Lemma 0FIH","summary":"Let b : X' → X be the blowing up of a smooth projective scheme over k in a smooth closed subscheme Z ⊂ X. Picture xymatrix E ar[r]_j ar[d]_π & X' ar[d]^b Z ar[r]^i & X Assume there exists an element of K_0(X) whose restriction to Z is equal to the class of C_Z/X in K_0(Z). Assume every irreducible component of Z has codimension r in X. Then there exists a cycle theta ∈ CH^r - 1(X') such that b^![Z] = [E] · theta in CH^r(X') and π_*j^!(theta) = [Z] in CH^r(Z).","statement_latex":"Let $b : X' \\to X$ be the blowing up of a smooth projective\nscheme over $k$ in a smooth closed subscheme $Z \\subset X$.\nPicture\n$$\n\\xymatrix{\nE \\ar[r]_j \\ar[d]_\\pi & X' \\ar[d]^b \\\\\nZ \\ar[r]^i & X\n}\n$$\nAssume there exists an element of $K_0(X)$ whose restriction to\n$Z$ is equal to the class of $\\mathcal{C}_{Z/X}$ in $K_0(Z)$.\nAssume every irreducible component of $Z$ has codimension $r$ in $X$.\nThen there exists a cycle $\\theta \\in \\CH^{r - 1}(X')$\nsuch that $b^![Z] = [E] \\cdot \\theta$ in $\\CH^r(X')$ and\n$\\pi_*j^!(\\theta) = [Z]$ in $\\CH^r(Z)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIH","source_file":"weil.tex","source_line":4543,"source_end_line":4560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4543-L4560","statement_sha256":"102b6b34159980ef91a06959c44841ddb9a2e1596af22f22e8b534bf1877b345","origin":"The Stacks Project","memory_eligible":false,"source_rank":8552,"rank":8552,"depth":64,"x":2031.781,"y":1228.317,"cluster":"duality-cohomology"},{"id":"stacks:0FII","tag":"0FII","title":"Weil cohomology theories, III · Lemma 0FII","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A4) and (A7). Let X be a smooth projective scheme over k. Let Z ⊂ X be a smooth closed subscheme such that every irreducible component of Z has codimension r in X. Assume the class of C_Z/X in K_0(Z) is the restriction of an element of K_0(X). If a ∈ H^*(X) and a|_Z = 0 in H^*(Z), then γ([Z]) ∪ a = 0.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A4)\nand (A7). Let $X$ be a smooth projective scheme over $k$. Let $Z \\subset X$\nbe a smooth closed subscheme such that every irreducible component of $Z$\nhas codimension $r$ in $X$. Assume the class of\n$\\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the restriction of an element of $K_0(X)$.\nIf $a \\in H^*(X)$ and $a|_Z = 0$ in $H^*(Z)$, then\n$\\gamma([Z]) \\cup a = 0$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FII","source_file":"weil.tex","source_line":4608,"source_end_line":4617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4608-L4617","statement_sha256":"50d2891693b29a6e4029246f47ae1864ce015b266713d316eb891cab3635a5b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8553,"rank":8553,"depth":65,"x":1957.483,"y":1075.726,"cluster":"duality-cohomology"},{"id":"stacks:0FIJ","tag":"0FIJ","title":"Weil cohomology theories, III · Lemma 0FIJ","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Then axiom (A) of Section [Tag 0FHA] holds with int_X = λ as in axiom (A6).","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).\nThen axiom (A) of Section \\ref{section-axioms} holds with\n$\\int_X = \\lambda$ as in axiom (A6).","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIJ","source_file":"weil.tex","source_line":4635,"source_end_line":4640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4635-L4640","statement_sha256":"276a812d6bb9600b2f54d0ee08302741f2e523954e6319d82c9ae0b169311555","origin":"The Stacks Project","memory_eligible":false,"source_rank":8554,"rank":8554,"depth":66,"x":2135.422,"y":1146.262,"cluster":"duality-cohomology"},{"id":"stacks:0FIL","tag":"0FIL","title":"Weil cohomology theories, III · Lemma 0FIL","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Then axiom (B) of Section [Tag 0FHA] holds.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).\nThen axiom (B) of Section \\ref{section-axioms} holds.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIL","source_file":"weil.tex","source_line":4729,"source_end_line":4733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4729-L4733","statement_sha256":"4f1814e88791dcb81d799d5df8ff0f53287f46a825613200a8cd91c661a39cf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8555,"rank":8555,"depth":0,"x":1947.022,"y":1195.338,"cluster":"duality-cohomology"},{"id":"stacks:0FIM","tag":"0FIM","title":"Weil cohomology theories, III · Lemma 0FIM","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Then axiom (C)(d) of Section [Tag 0FHA] holds.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).\nThen axiom (C)(d) of Section \\ref{section-axioms} holds.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIM","source_file":"weil.tex","source_line":4776,"source_end_line":4780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4776-L4780","statement_sha256":"30ade15365f4c5a2a01c4b592e4c5f0160db1ba4b46ebd8648637207453aee3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8556,"rank":8556,"depth":1,"x":2046.726,"y":1051.893,"cluster":"duality-cohomology"},{"id":"stacks:0FIN","tag":"0FIN","title":"Weil cohomology theories, III · Lemma 0FIN","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Let p : P → X be as in axiom (A3) with X nonempty equidimensional. Then γ commutes with pushforward along p.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).\nLet $p : P \\to X$ be as in axiom (A3) with $X$ nonempty equidimensional.\nThen $\\gamma$ commutes with pushforward along $p$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIN","source_file":"weil.tex","source_line":4808,"source_end_line":4813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4808-L4813","statement_sha256":"d9a6d2bdcc11f5ac16a873798b0310e3120c49862aed95d947f794cf7b80f076","origin":"The Stacks Project","memory_eligible":false,"source_rank":8557,"rank":8557,"depth":71,"x":2088.663,"y":1214.642,"cluster":"duality-cohomology"},{"id":"stacks:0FVR","tag":"0FVR","title":"Weil cohomology theories, III · Lemma 0FVR","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). If k'/k is a Galois extension, then we have int_Spec(k') 1 = [k' : k].","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).\nIf $k'/k$ is a Galois extension, then we have\n$\\int_{\\Spec(k')} 1 = [k' : k]$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVR","source_file":"weil.tex","source_line":4846,"source_end_line":4851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4846-L4851","statement_sha256":"1f2c3f352f0e3316b2b370e891dcdfe61085ba33d3e7f3fdae6141d0971a42f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8558,"rank":8558,"depth":0,"x":1926.465,"y":1118.193,"cluster":"duality-cohomology"},{"id":"stacks:0FIP","tag":"0FIP","title":"Weil cohomology theories, III · Lemma 0FIP","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). In order to show that γ commutes with pushforward it suffices to show that i_*(1) = γ([Z]) if i : Z → X is a closed immersion of nonempty smooth projective equidimensional schemes over k.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).\nIn order to show that $\\gamma$ commutes with pushforward it suffices\nto show that $i_*(1) = \\gamma([Z])$ if $i : Z \\to X$ is a closed\nimmersion of nonempty smooth projective equidimensional schemes over $k$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIP","source_file":"weil.tex","source_line":4885,"source_end_line":4891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L4885-L4891","statement_sha256":"338d7bd7c1a5352806d9b721070556b3ad679ad3905285b4a98adce488da7c4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8559,"rank":8559,"depth":72,"x":2124.11,"y":1097.228,"cluster":"duality-cohomology"},{"id":"stacks:0FIQ","tag":"0FIQ","title":"Weil cohomology theories, III · Lemma 0FIQ","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). Given integers 0 < l < n and a nonempty equidimensional smooth projective scheme X over k consider the projection morphism p : X × G(l, n) → X. Then γ commutes with pushforward along p.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).\nGiven integers $0 < l < n$ and a nonempty equidimensional\nsmooth projective scheme $X$ over $k$ consider the projection morphism\n$p : X \\times \\mathbf{G}(l, n) \\to X$.\nThen $\\gamma$ commutes with pushforward along $p$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIQ","source_file":"weil.tex","source_line":5010,"source_end_line":5017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L5010-L5017","statement_sha256":"5502354cc44d6a8e23e26fc184cc6ab0c1b04329fdfda95d154e5e06210ca75a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8560,"rank":8560,"depth":72,"x":1994.916,"y":1225.145,"cluster":"duality-cohomology"},{"id":"stacks:0FIR","tag":"0FIR","title":"Weil cohomology theories, III · Lemma 0FIR","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). In order to show that γ commutes with pushforward it suffices to show that i_*(1) = γ([Z]) if i : Z → X is a closed immersion of nonempty smooth projective equidimensional schemes over k such that the class of C_Z/X in K_0(Z) is the pullback of a class in K_0(X).","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).\nIn order to show that $\\gamma$ commutes with pushforward it suffices\nto show that $i_*(1) = \\gamma([Z])$ if $i : Z \\to X$ is a closed\nimmersion of nonempty smooth projective equidimensional schemes over $k$\nsuch that the class of $\\mathcal{C}_{Z/X}$ in $K_0(Z)$ is the\npullback of a class in $K_0(X)$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIR","source_file":"weil.tex","source_line":5040,"source_end_line":5048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L5040-L5048","statement_sha256":"389c77797f4c272d3f5e6955c88c1f25c6e67d82b08de47918dc688f4e8a2e44","origin":"The Stacks Project","memory_eligible":false,"source_rank":8561,"rank":8561,"depth":73,"x":1987.296,"y":1057.109,"cluster":"duality-cohomology"},{"id":"stacks:0FVS","tag":"0FVS","title":"Weil cohomology theories, III · Lemma 0FVS","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7). If k\"/k'/k are finite separable field extensions, then H^0(Spec(k')) → H^0(Spec(k\")) is injective.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A7).\nIf $k''/k'/k$ are finite separable field extensions, then\n$H^0(\\Spec(k')) \\to H^0(\\Spec(k''))$ is injective.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVS","source_file":"weil.tex","source_line":5085,"source_end_line":5090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L5085-L5090","statement_sha256":"a0090e81142e82bf12d2793bef3af286d2f76aa24ae472613928cd21a79f7afa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8562,"rank":8562,"depth":7,"x":2128.383,"y":1176.98,"cluster":"duality-cohomology"},{"id":"stacks:0FIS","tag":"0FIS","title":"Weil cohomology theories, III · Lemma 0FIS","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8). Let b : X' → X be a blowing up of a smooth projective scheme X over k which is nonempty equidimensional of dimension d in a nonwhere dense smooth center Z. Then b_*(1) = 1.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8).\nLet $b : X' \\to X$ be a blowing up of a smooth projective scheme $X$\nover $k$ which is nonempty equidimensional of dimension $d$\nin a nonwhere dense smooth center $Z$. Then $b_*(1) = 1$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIS","source_file":"weil.tex","source_line":5118,"source_end_line":5124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L5118-L5124","statement_sha256":"64257a5ae2ff1c3072152ea6f2e70810c1f815941c1416175d69390f6282c11d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8563,"rank":8563,"depth":46,"x":1927.472,"y":1168.625,"cluster":"duality-cohomology"},{"id":"stacks:0FIT","tag":"0FIT","title":"Weil cohomology theories, III · Lemma 0FIT","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8). Then the cycle class map γ commutes with pushforward.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8).\nThen the cycle class map $\\gamma$ commutes with pushforward.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIT","source_file":"weil.tex","source_line":5167,"source_end_line":5171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L5167-L5171","statement_sha256":"e6912a9f982bdb612085a13abda2cc6b4212f3b20a0991d35231b108cfcc1388","origin":"The Stacks Project","memory_eligible":false,"source_rank":8564,"rank":8564,"depth":74,"x":2082.709,"y":1060.527,"cluster":"duality-cohomology"},{"id":"stacks:0FIU","tag":"0FIU","title":"Weil cohomology theories, III · Proposition 0FIU","summary":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8). Then we have a Weil cohomology theory.","statement_latex":"Assume given data (D0), (D1), and (D2') satisfying axioms (A1) -- (A8).\nThen we have a Weil cohomology theory.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIU","source_file":"weil.tex","source_line":5204,"source_end_line":5208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L5204-L5208","statement_sha256":"029fbf8c89d8adce54804ca0817f670fb1fbbbbf431e188bef86821cd121b8c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8565,"rank":8565,"depth":75,"x":2055.102,"y":1228.719,"cluster":"duality-cohomology"},{"id":"stacks:0FVT","tag":"0FVT","title":"Weil cohomology theories, III · Lemma 0FVT","summary":"Let k'/k be an extension of fields. Let F'/F be an extension of fields of characteristic 0. Assume given • data (D0), (D1), (D2') for k and F denoted F(1), H^*, c_1^H, • data (D0), (D1), (D2') for k' and F' denoted F'(1), (H')^*, c_1^H', and • an isomorphism F(1) ⊗_F F' → F'(1), functorial isomorphisms H^*(X) ⊗_F F' → (H')^*(X_k') on the category of smooth projective schemes X over k such that the diagrams xymatrix Pic(X) ar[r]_c_1^H ar[d] & H^2(X)(1) ar[d] Pic(X_k')…","statement_latex":"Let $k'/k$ be an extension of fields. Let $F'/F$ be an extension\nof fields of characteristic $0$. Assume given\n\\begin{enumerate}\n\\item data (D0), (D1), (D2') for $k$ and $F$ denoted\n$F(1), H^*, c_1^H$,\n\\item data (D0), (D1), (D2') for $k'$ and $F'$ denoted\n$F'(1), (H')^*, c_1^{H'}$, and\n\\item an isomorphism $F(1) \\otimes_F F' \\to F'(1)$, functorial isomorphisms\n$H^*(X) \\otimes_F F' \\to (H')^*(X_{k'})$ on the category of smooth projective\nschemes $X$ over $k$ such that the diagrams\n$$\n\\xymatrix{\n\\Pic(X) \\ar[r]_{c_1^H} \\ar[d] & H^2(X)(1) \\ar[d] \\\\\n\\Pic(X_{k'}) \\ar[r]^{c_1^{H'}} & (H')^2(X_{k'})(1)\n}\n$$\ncommute.\n\\end{enumerate}\nIn this case, if $F'(1), (H')^*, c_1^{H'}$ satisfy axioms (A1) -- (A9),\nthen the same is true for $F(1), H^*, c_1^H$.","area":"Duality & Cohomology","chapter":"Weil Cohomology Theories","chapter_id":"weil","section":"Weil cohomology theories, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVT","source_file":"weil.tex","source_line":5227,"source_end_line":5249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/weil.tex#L5227-L5249","statement_sha256":"a119511e8f899dbc55f8af283b697d3f5d1514c59e05d5329f55248468ed31af","origin":"The Stacks Project","memory_eligible":false,"source_rank":8566,"rank":8566,"depth":68,"x":1939.933,"y":1088.703,"cluster":"duality-cohomology"},{"id":"stacks:06Z4","tag":"06Z4","title":"Adequate functors · Definition 06Z4","summary":"Let A be a ring. A module-valued functor is a functor F : Alg_A → Ab such that • for every object B of Alg_A the group F(B) is endowed with the structure of a B-module, and • for any morphism B → B' of Alg_A the map F(B) → F(B') is B-linear. A morphism of module-valued functors is a transformation of functors φ : F → G such that F(B) → G(B) is B-linear for all B ∈ Ob(Alg_A).","statement_latex":"Let $A$ be a ring. A {\\it module-valued functor} is a functor\n$F : \\textit{Alg}_A \\to \\textit{Ab}$ such that\n\\begin{enumerate}\n\\item for every object $B$ of $\\textit{Alg}_A$ the group\n$F(B)$ is endowed with the structure of a $B$-module, and\n\\item for any morphism $B \\to B'$ of $\\textit{Alg}_A$ the map\n$F(B) \\to F(B')$ is $B$-linear.\n\\end{enumerate}\nA {\\it morphism of module-valued functors} is a transformation of\nfunctors $\\varphi : F \\to G$ such that $F(B) \\to G(B)$ is $B$-linear\nfor all $B \\in \\Ob(\\textit{Alg}_A)$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Z4","source_file":"adequate.tex","source_line":111,"source_end_line":124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L111-L124","statement_sha256":"0e7b31d4a2bf66506241defe4955c2ea0e980c39e12b55d063756581d84e88e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8567,"rank":8567,"depth":0,"x":2486.54,"y":1140.0,"cluster":"local-crystalline-methods"},{"id":"stacks:06UT","tag":"06UT","title":"Adequate functors · Definition 06UT","summary":"Let A be a ring. A module-valued functor F on Alg_A is called • adequate if there exists a map of A-modules M → N such that F is isomorphic to Ker(underlineM → underlineN). • linearly adequate if F is isomorphic to the kernel of a map underlineA^⊕ n → underlineA^⊕ m.","statement_latex":"Let $A$ be a ring. A module-valued functor $F$ on $\\textit{Alg}_A$ is\ncalled\n\\begin{enumerate}\n\\item {\\it adequate} if there exists a\nmap of $A$-modules $M \\to N$ such that $F$ is isomorphic to\n$\\Ker(\\underline{M} \\to \\underline{N})$.\n\\item {\\it linearly adequate} if $F$ is isomorphic to the\nkernel of a map $\\underline{A^{\\oplus n}} \\to \\underline{A^{\\oplus m}}$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UT","source_file":"adequate.tex","source_line":161,"source_end_line":172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L161-L172","statement_sha256":"c5d884dd0deda69831f85e67fee2f974f04890273e5c1ce8d9943659a1d6eb0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8568,"rank":8568,"depth":0,"x":2471.647,"y":1146.427,"cluster":"local-crystalline-methods"},{"id":"stacks:06UU","tag":"06UU","title":"Adequate functors · Lemma 06UU","summary":"Let A be a ring. Let F be an adequate functor on Alg_A. If B = colim B_i is a filtered colimit of A-algebras, then F(B) = colim F(B_i).","statement_latex":"Let $A$ be a ring.\nLet $F$ be an adequate functor on $\\textit{Alg}_A$.\nIf $B = \\colim B_i$ is a filtered\ncolimit of $A$-algebras, then $F(B) = \\colim F(B_i)$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UU","source_file":"adequate.tex","source_line":192,"source_end_line":198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L192-L198","statement_sha256":"f5787e97e7fc60510b7430a1c064dcb5970d7969e116d1b0570a583dbad76b0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8569,"rank":8569,"depth":4,"x":2481.279,"y":1127.763,"cluster":"local-crystalline-methods"},{"id":"stacks:06UW","tag":"06UW","title":"Adequate functors · Lemma 06UW","summary":"Let A be a ring. Let F be an adequate functor on Alg_A. If B → B' is flat, then F(B) ⊗_B B' → F(B') is an isomorphism.","statement_latex":"Let $A$ be a ring.\nLet $F$ be an adequate functor on $\\textit{Alg}_A$.\nIf $B \\to B'$ is flat, then $F(B) \\otimes_B B' \\to F(B')$\nis an isomorphism.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UW","source_file":"adequate.tex","source_line":224,"source_end_line":230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L224-L230","statement_sha256":"281716ba65a5221c2b811e5616c6cb754e26cd451d4004791b62c4c5aa4b8b49","origin":"The Stacks Project","memory_eligible":false,"source_rank":8570,"rank":8570,"depth":0,"x":2490.528,"y":1151.535,"cluster":"local-crystalline-methods"},{"id":"stacks:06UX","tag":"06UX","title":"Adequate functors · Lemma 06UX","summary":"Let A be a ring. Let F be an adequate functor on Alg_A. Then there exists a surjection L → F with L a direct sum of linearly adequate functors.","statement_latex":"Let $A$ be a ring.\nLet $F$ be an adequate functor on $\\textit{Alg}_A$. Then there exists a\nsurjection $L \\to F$ with $L$ a direct sum of linearly adequate functors.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UX","source_file":"adequate.tex","source_line":250,"source_end_line":255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L250-L255","statement_sha256":"1f719cc7fb3f1e72e8cdd7b30f7f31776f4891c8d509ed3e0c845a66346eca8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8571,"rank":8571,"depth":5,"x":2460.68,"y":1137.129,"cluster":"local-crystalline-methods"},{"id":"stacks:06UZ","tag":"06UZ","title":"Adequate functors · Lemma 06UZ","summary":"Let A be a ring. Let F be a module-valued functor on Alg_A. Assume that for B → B' flat the map F(B) ⊗_B B' → F(B') is an isomorphism. Let B be a graded A-algebra. Then • F(B) = bigoplus_k ∈ Z F(B)^(k), and • the map B → B_0 → B induces map F(B) → F(B) whose image is contained in F(B)^(0).","statement_latex":"Let $A$ be a ring.\nLet $F$ be a module-valued functor on $\\textit{Alg}_A$.\nAssume that for $B \\to B'$ flat the map\n$F(B) \\otimes_B B' \\to F(B')$ is an isomorphism.\nLet $B$ be a graded $A$-algebra. Then\n\\begin{enumerate}\n\\item $F(B) = \\bigoplus_{k \\in \\mathbf{Z}} F(B)^{(k)}$, and\n\\item the map $B \\to B_0 \\to B$ induces map $F(B) \\to F(B)$\nwhose image is contained in $F(B)^{(0)}$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UZ","source_file":"adequate.tex","source_line":315,"source_end_line":327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L315-L327","statement_sha256":"35f029b86be38e4a8ee5df3ba37fe1d38d20eabd867181a3316819a97e156923","origin":"The Stacks Project","memory_eligible":false,"source_rank":8572,"rank":8572,"depth":0,"x":2498.302,"y":1130.221,"cluster":"local-crystalline-methods"},{"id":"stacks:06V0","tag":"06V0","title":"Adequate functors · Lemma 06V0","summary":"Let A be a ring. Given a solid diagram xymatrix 0 ar[r] & L ar[d]_φ ar[r] & underlineA^⊕ n ar[r] ar@..>[ld] & underlineA^⊕ m & underlineM of module-valued functors on Alg_A with exact row there exists a dotted arrow making the diagram commute.","statement_latex":"Let $A$ be a ring. Given a solid diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\nL \\ar[d]_\\varphi \\ar[r] &\n\\underline{A^{\\oplus n}} \\ar[r] \\ar@{..>}[ld] &\n\\underline{A^{\\oplus m}} \\\\\n& \\underline{M}\n}\n$$\nof module-valued functors on $\\textit{Alg}_A$\nwith exact row there exists a dotted arrow making the diagram commute.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06V0","source_file":"adequate.tex","source_line":387,"source_end_line":401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L387-L401","statement_sha256":"b520c4f954df3e0a04f4b8749b6dd6b58b34ff1e4ae23dc438a9e6b1107fcda8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8573,"rank":8573,"depth":1,"x":2473.878,"y":1159.129,"cluster":"local-crystalline-methods"},{"id":"stacks:06V1","tag":"06V1","title":"Adequate functors · Lemma 06V1","summary":"Let A be a ring. Let φ : F → underlineM be a map of module-valued functors on Alg_A with F adequate. Then Coker(φ) is adequate.","statement_latex":"Let $A$ be a ring.\nLet $\\varphi : F \\to \\underline{M}$ be a map of module-valued functors\non $\\textit{Alg}_A$ with $F$ adequate.\nThen $\\Coker(\\varphi)$ is adequate.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06V1","source_file":"adequate.tex","source_line":429,"source_end_line":435,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L429-L435","statement_sha256":"b2b6b3d3871db77ecf3db891d0cde64004f50dfc2e67b0951242c7890e9a1773","origin":"The Stacks Project","memory_eligible":false,"source_rank":8574,"rank":8574,"depth":6,"x":2468.325,"y":1121.118,"cluster":"local-crystalline-methods"},{"id":"stacks:06V2","tag":"06V2","title":"Adequate functors · Lemma 06V2","summary":"The cokernel of a map of adequate functors on the category of algebras over a ring is adequate. Let A be a ring. Let φ : F → G be a map of adequate functors on Alg_A. Then Coker(φ) is adequate.","statement_latex":"\\begin{slogan}\nThe cokernel of a map of adequate functors on the category of algebras\nover a ring is adequate.\n\\end{slogan}\nLet $A$ be a ring.\nLet $\\varphi : F \\to G$ be a map of adequate functors on $\\textit{Alg}_A$.\nThen $\\Coker(\\varphi)$ is adequate.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06V2","source_file":"adequate.tex","source_line":466,"source_end_line":475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L466-L475","statement_sha256":"30f53af6b405bbbb41f4b1fc7c139187a64f50069f6edbaea38db67b2c9589e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8575,"rank":8575,"depth":7,"x":2505.329,"y":1147.77,"cluster":"local-crystalline-methods"},{"id":"stacks:06V3","tag":"06V3","title":"Adequate functors · Lemma 06V3","summary":"Let A be a ring. Let φ : F → G be a map of adequate functors on Alg_A. Then Ker(φ) is adequate.","statement_latex":"Let $A$ be a ring.\nLet $\\varphi : F \\to G$ be a map of adequate functors on $\\textit{Alg}_A$.\nThen $\\Ker(\\varphi)$ is adequate.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06V3","source_file":"adequate.tex","source_line":491,"source_end_line":496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L491-L496","statement_sha256":"14b82dc1132a4fc147dd88b36fb0732f4473f573be9b4ba0f44c2def2877a661","origin":"The Stacks Project","memory_eligible":false,"source_rank":8576,"rank":8576,"depth":8,"x":2453.649,"y":1149.137,"cluster":"local-crystalline-methods"},{"id":"stacks:06V4","tag":"06V4","title":"Adequate functors · Lemma 06V4","summary":"Let A be a ring. An arbitrary direct sum of adequate functors on Alg_A is adequate. A colimit of adequate functors is adequate.","statement_latex":"Let $A$ be a ring.\nAn arbitrary direct sum of adequate functors on $\\textit{Alg}_A$\nis adequate. A colimit of adequate functors is adequate.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06V4","source_file":"adequate.tex","source_line":516,"source_end_line":521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L516-L521","statement_sha256":"76f23868d8f5015c1574be207114459d9a8dccd9c739ec79645bddd6d92911b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8577,"rank":8577,"depth":9,"x":2492.703,"y":1117.198,"cluster":"local-crystalline-methods"},{"id":"stacks:06V5","tag":"06V5","title":"Adequate functors · Lemma 06V5","summary":"Let A be a ring. Let F, G be module-valued functors on Alg_A. Let φ : F → G be a transformation of functors. Assume • φ is additive, • for every A-algebra B and xi ∈ F(B) and unit u ∈ B^* we have φ(uxi) = uφ(xi) in G(B), and • for any flat ring map B → B' we have G(B) ⊗_B B' = G(B'). Then φ is a morphism of module-valued functors.","statement_latex":"Let $A$ be a ring.\nLet $F, G$ be module-valued functors on $\\textit{Alg}_A$.\nLet $\\varphi : F \\to G$ be a transformation of functors. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is additive,\n\\item for every $A$-algebra $B$ and $\\xi \\in F(B)$ and unit\n$u \\in B^*$ we have $\\varphi(u\\xi) = u\\varphi(\\xi)$ in $G(B)$, and\n\\item for any flat ring map $B \\to B'$ we have\n$G(B) \\otimes_B B' = G(B')$.\n\\end{enumerate}\nThen $\\varphi$ is a morphism of module-valued functors.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06V5","source_file":"adequate.tex","source_line":532,"source_end_line":545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L532-L545","statement_sha256":"012ea29b82783af5f9b27a7c47a7fb6e4b4cb17b8d46448cf472e31b455e30bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8578,"rank":8578,"depth":0,"x":2489.387,"y":1165.139,"cluster":"local-crystalline-methods"},{"id":"stacks:06V6","tag":"06V6","title":"Adequate functors · Lemma 06V6","summary":"Let A be a ring. Let 0 → underlineM → G → L → 0 be a short exact sequence of module-valued functors on Alg_A with L linearly adequate. Then G is adequate.","statement_latex":"Let $A$ be a ring.\nLet $0 \\to \\underline{M} \\to G \\to L \\to 0$ be a short exact sequence\nof module-valued functors on $\\textit{Alg}_A$ with $L$ linearly adequate.\nThen $G$ is adequate.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06V6","source_file":"adequate.tex","source_line":563,"source_end_line":569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L563-L569","statement_sha256":"1a5edecb24672bbaf44fd15f15016c99bb875d73ee714e4f514a68733efdaef4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8579,"rank":8579,"depth":1,"x":2451.707,"y":1126.227,"cluster":"local-crystalline-methods"},{"id":"stacks:06V8","tag":"06V8","title":"Adequate functors · Lemma 06V8","summary":"Let A be a ring. Let 0 → F → G → H → 0 be a short exact sequence of module-valued functors on Alg_A. If F and H are adequate, so is G.","statement_latex":"Let $A$ be a ring.\nLet $0 \\to F \\to G \\to H \\to 0$ be a short exact sequence of\nmodule-valued functors on $\\textit{Alg}_A$.\nIf $F$ and $H$ are adequate, so is $G$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06V8","source_file":"adequate.tex","source_line":692,"source_end_line":698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L692-L698","statement_sha256":"2840b641f85fe159aeb64d2b772d65d2115e1c7e067b5a7d43cfcd3e1968565c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8580,"rank":8580,"depth":10,"x":2513.191,"y":1133.871,"cluster":"local-crystalline-methods"},{"id":"stacks:06V9","tag":"06V9","title":"Adequate functors · Lemma 06V9","summary":"Let A → A' be a ring map. If F is an adequate functor on Alg_A, then its restriction F' to Alg_A' is adequate too.","statement_latex":"Let $A \\to A'$ be a ring map. If $F$ is an adequate functor on\n$\\textit{Alg}_A$, then its restriction $F'$ to\n$\\textit{Alg}_{A'}$ is adequate too.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06V9","source_file":"adequate.tex","source_line":735,"source_end_line":740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L735-L740","statement_sha256":"aaaf0c522d15e7461b94baac4404f1f8ad5d4b5d257cddad188736d0d943f795","origin":"The Stacks Project","memory_eligible":false,"source_rank":8581,"rank":8581,"depth":0,"x":2459.744,"y":1164.202,"cluster":"local-crystalline-methods"},{"id":"stacks:06VA","tag":"06VA","title":"Adequate functors · Lemma 06VA","summary":"Let A → A' be a ring map. If F' is an adequate functor on Alg_A', then the module-valued functor F : B ↦ F'(A' ⊗_A B) on Alg_A is adequate too.","statement_latex":"Let $A \\to A'$ be a ring map. If $F'$ is an adequate functor on\n$\\textit{Alg}_{A'}$, then the module-valued functor\n$F : B \\mapsto F'(A' \\otimes_A B)$ on $\\textit{Alg}_A$ is adequate too.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VA","source_file":"adequate.tex","source_line":750,"source_end_line":755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L750-L755","statement_sha256":"3a09dbd17d76f7e6d36b34ce9b5f586eba600b8615353bdb457c9b3bd3ca4ef4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8582,"rank":8582,"depth":0,"x":2475.32,"y":1109.666,"cluster":"local-crystalline-methods"},{"id":"stacks:06VB","tag":"06VB","title":"Adequate functors · Lemma 06VB","summary":"Let A = A_1 × … × A_n be a product of rings. An adequate functor over A is the same thing as a sequence F_1, …, F_n of adequate functors F_i over A_i.","statement_latex":"Let $A = A_1 \\times \\ldots \\times A_n$ be a product of rings.\nAn adequate functor over $A$ is the same thing as a sequence\n$F_1, \\ldots, F_n$ of adequate functors $F_i$ over $A_i$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VB","source_file":"adequate.tex","source_line":771,"source_end_line":776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L771-L776","statement_sha256":"80b55a9d06895e734c5d206086c324c777eb4c9a818e86be88749c0c4d8f9496","origin":"The Stacks Project","memory_eligible":false,"source_rank":8583,"rank":8583,"depth":0,"x":2508.728,"y":1160.338,"cluster":"local-crystalline-methods"},{"id":"stacks:06VH","tag":"06VH","title":"Adequate functors · Lemma 06VH","summary":"Let A → A' be a ring map and let F be a module-valued functor on Alg_A such that • the restriction F' of F to the category of A'-algebras is adequate, and • for any A-algebra B the sequence 0 → F(B) → F(B ⊗_A A') → F(B ⊗_A A' ⊗_A A') is exact. Then F is adequate.","statement_latex":"Let $A \\to A'$ be a ring map and let $F$ be a module-valued functor on\n$\\textit{Alg}_A$ such that\n\\begin{enumerate}\n\\item the restriction $F'$ of $F$ to the category of $A'$-algebras is\nadequate, and\n\\item for any $A$-algebra $B$ the sequence\n$$\n0 \\to F(B) \\to F(B \\otimes_A A') \\to F(B \\otimes_A A' \\otimes_A A')\n$$\nis exact.\n\\end{enumerate}\nThen $F$ is adequate.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VH","source_file":"adequate.tex","source_line":785,"source_end_line":799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L785-L799","statement_sha256":"637c147f76a1968750cc3579818a9e85b36275d7e394721b920444907e7a6ae9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8584,"rank":8584,"depth":9,"x":2441.342,"y":1141.343,"cluster":"local-crystalline-methods"},{"id":"stacks:06Z6","tag":"06Z6","title":"Higher exts of adequate functors · Lemma 06Z6","summary":"Let A be a ring. For every module-valued functor F on Alg_A there exists a morphism Q(F) → F of module-valued functors on Alg_A such that (1) Q(F) is adequate and (2) for every adequate functor G the map Hom(G, Q(F)) → Hom(G, F) is a bijection.","statement_latex":"Let $A$ be a ring.\nFor every module-valued functor $F$ on $\\textit{Alg}_A$\nthere exists a morphism $Q(F) \\to F$ of module-valued functors on\n$\\textit{Alg}_A$ such that (1) $Q(F)$ is adequate and (2) for every\nadequate functor $G$ the map $\\Hom(G, Q(F)) \\to \\Hom(G, F)$\nis a bijection.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Z6","source_file":"adequate.tex","source_line":825,"source_end_line":833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L825-L833","statement_sha256":"46757631c17ca8165c4a0f0327dfcfd79bd286b01bcfe9b4b0498e5076deab90","origin":"The Stacks Project","memory_eligible":false,"source_rank":8585,"rank":8585,"depth":10,"x":2508.198,"y":1116.429,"cluster":"local-crystalline-methods"},{"id":"stacks:06Z7","tag":"06Z7","title":"Higher exts of adequate functors · Lemma 06Z7","summary":"Let A be a ring. Denote P the category of module-valued functors on Alg_A and A the category of adequate functors on Alg_A. Denote i : A → P the inclusion functor. Denote Q : P → A the construction of Lemma [Tag 06Z6]. Then • i is fully faithful, exact, and its image is a weak Serre subcategory, • P has enough injectives, • the functor Q is a right adjoint to i hence left exact, • Q transforms injectives into injectives, • A has enough injectives.","statement_latex":"Let $A$ be a ring. Denote $\\mathcal{P}$ the category of module-valued\nfunctors on $\\textit{Alg}_A$ and $\\mathcal{A}$ the category of adequate\nfunctors on $\\textit{Alg}_A$. Denote $i : \\mathcal{A} \\to \\mathcal{P}$\nthe inclusion functor. Denote $Q : \\mathcal{P} \\to \\mathcal{A}$\nthe construction of Lemma \\ref{lemma-adjoint}.\nThen\n\\begin{enumerate}\n\\item $i$ is fully faithful, exact, and its image is a weak Serre subcategory,\n\\item $\\mathcal{P}$ has enough injectives,\n\\item the functor $Q$ is a right adjoint to $i$ hence left exact,\n\\item $Q$ transforms injectives into injectives,\n\\item $\\mathcal{A}$ has enough injectives.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Z7","source_file":"adequate.tex","source_line":868,"source_end_line":883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L868-L883","statement_sha256":"777b28da75f9805ec3b207deb98ece3388b6db1522851984d3e9a68698c273ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":8586,"rank":8586,"depth":11,"x":2478.113,"y":1174.271,"cluster":"local-crystalline-methods"},{"id":"stacks:06Z8","tag":"06Z8","title":"Higher exts of adequate functors · Lemma 06Z8","summary":"Let A be a ring. Let F be a module valued functor. For every B ∈ Ob(Alg_A) and B-module N there is a canonical decomposition F(B[N]) = F(B) ⊕ TF(B, N) characterized by the following properties • TF(B, N) = Ker(F(B[N]) → F(B)), • there is a B-module structure TF(B, N) compatible with B[N]-module structure on F(B[N]), • TF is a functor from the category of pairs (B, N), • there are canonical maps N ⊗_B F(B) → TF(B, N) inducing a transformation between functors defined on…","statement_latex":"Let $A$ be a ring. Let $F$ be a module valued functor.\nFor every $B \\in \\Ob(\\textit{Alg}_A)$ and $B$-module $N$\nthere is a canonical decomposition\n$$\nF(B[N]) = F(B) \\oplus TF(B, N)\n$$\ncharacterized by the following properties\n\\begin{enumerate}\n\\item $TF(B, N) = \\Ker(F(B[N]) \\to F(B))$,\n\\item there is a $B$-module structure $TF(B, N)$\ncompatible with $B[N]$-module structure on $F(B[N])$,\n\\item $TF$ is a functor from the category of pairs $(B, N)$,\n\\item\n\nthere are canonical maps $N \\otimes_B F(B) \\to TF(B, N)$\ninducing a transformation between functors defined on the category\nof pairs $(B, N)$,\n\\item $TF(B, 0) = 0$ and the map $TF(B, N) \\to TF(B, N')$ is\nzero when $N \\to N'$ is the zero map.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Z8","source_file":"adequate.tex","source_line":920,"source_end_line":942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L920-L942","statement_sha256":"a2fec85983b39c8bb31421960517fe899455a16af7def13f073acfd82e71c031","origin":"The Stacks Project","memory_eligible":false,"source_rank":8587,"rank":8587,"depth":0,"x":2453.169,"y":1112.992,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZA","tag":"06ZA","title":"Higher exts of adequate functors · Lemma 06ZA","summary":"Let A be a ring. Let I be an injective object of the category of module-valued functors. Then for any B ∈ Ob(Alg_A) and short exact sequence 0 → N_1 → N → N_2 → 0 of B-modules the sequence TI(B, N_1) → TI(B, N) → TI(B, N_2) → 0 is exact.","statement_latex":"Let $A$ be a ring. Let $I$ be an injective object of the category\nof module-valued functors. Then for any $B \\in \\Ob(\\textit{Alg}_A)$\nand short exact sequence\n$0 \\to N_1 \\to N \\to N_2 \\to 0$\nof $B$-modules the sequence\n$$\nTI(B, N_1) \\to TI(B, N) \\to TI(B, N_2) \\to 0\n$$\nis exact.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZA","source_file":"adequate.tex","source_line":977,"source_end_line":988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L977-L988","statement_sha256":"4ddfeb79e4500e5b664fae3c88388853ec8e86496797f65ddfc4e0fb02fd8d39","origin":"The Stacks Project","memory_eligible":false,"source_rank":8588,"rank":8588,"depth":1,"x":2522.503,"y":1144.804,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZB","tag":"06ZB","title":"Higher exts of adequate functors · Lemma 06ZB","summary":"Let A be a ring. Let F be a module-valued functor such that for any B ∈ Ob(Alg_A) the functor TF(B, -) on B-modules transforms a short exact sequence of B-modules into a right exact sequence. Then • TF(B, N_1 ⊕ N_2) = TF(B, N_1) ⊕ TF(B, N_2), • there is a second functorial B-module structure on TF(B, N) defined by setting x · b = TF(B, b· 1_N)(x) for x ∈ TF(B, N) and b ∈ B, • the canonical map N ⊗_B F(B) → TF(B, N) of Lemma [Tag 06Z8] is B-linear also with respect to the…","statement_latex":"Let $A$ be a ring. Let $F$ be a module-valued functor\nsuch that for any $B \\in \\Ob(\\textit{Alg}_A)$ the\nfunctor $TF(B, -)$ on $B$-modules transforms a short exact sequence\nof $B$-modules into a right exact sequence. Then\n\\begin{enumerate}\n\\item $TF(B, N_1 \\oplus N_2) = TF(B, N_1) \\oplus TF(B, N_2)$,\n\\item there is a second functorial $B$-module structure on $TF(B, N)$\ndefined by setting $x \\cdot b = TF(B, b\\cdot 1_N)(x)$ for $x \\in TF(B, N)$\nand $b \\in B$,\n\\item\n\nthe canonical map $N \\otimes_B F(B) \\to TF(B, N)$ of\nLemma \\ref{lemma-tangent-functor}\nis $B$-linear also with respect to the second $B$-module structure,\n\\item\n\ngiven a finitely presented $B$-module $N$ there is a canonical\nisomorphism $TF(B, B) \\otimes_B N \\to TF(B, N)$ where the tensor\nproduct uses the second $B$-module structure on $TF(B, B)$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZB","source_file":"adequate.tex","source_line":1053,"source_end_line":1075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1053-L1075","statement_sha256":"9ad141649a88550e4719fc9108ab0239c9d45ad56cfa9d558f895c1fdaa09f45","origin":"The Stacks Project","memory_eligible":false,"source_rank":8589,"rank":8589,"depth":1,"x":2443.987,"y":1161.048,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZF","tag":"06ZF","title":"Higher exts of adequate functors · Lemma 06ZF","summary":"Let A be a ring. For F a module-valued functor on Alg_A say (*) holds if for all B ∈ Ob(Alg_A) the functor TF(B, -) on B-modules transforms a short exact sequence of B-modules into a right exact sequence. Let 0 → F → G → H → 0 be a short exact sequence of module-valued functors on Alg_A. • If (*) holds for F, G then (*) holds for H. • If (*) holds for F, H then (*) holds for G. • If H' → H is morphism of module-valued functors on Alg_A and (*) holds for F, G, H, and H',…","statement_latex":"Let $A$ be a ring. For $F$ a module-valued functor on $\\textit{Alg}_A$\nsay $(*)$ holds if for all $B \\in \\Ob(\\textit{Alg}_A)$ the\nfunctor $TF(B, -)$ on $B$-modules transforms a short exact sequence\nof $B$-modules into a right exact sequence. Let\n$0 \\to F \\to G \\to H \\to 0$ be a short exact sequence of\nmodule-valued functors on $\\textit{Alg}_A$.\n\\begin{enumerate}\n\\item If $(*)$ holds for $F, G$ then $(*)$ holds for $H$.\n\\item If $(*)$ holds for $F, H$ then $(*)$ holds for $G$.\n\\item If $H' \\to H$ is morphism of module-valued functors on $\\textit{Alg}_A$\nand $(*)$ holds for $F$, $G$, $H$, and $H'$, then $(*)$ holds for\n$G \\times_H H'$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZF","source_file":"adequate.tex","source_line":1141,"source_end_line":1156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1141-L1156","statement_sha256":"2ff5d238a6390feefd271f00a9e72826234316a78454f5fdb12514f0c8d43638","origin":"The Stacks Project","memory_eligible":false,"source_rank":8590,"rank":8590,"depth":0,"x":2489.841,"y":1103.256,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZG","tag":"06ZG","title":"Higher exts of adequate functors · Lemma 06ZG","summary":"Let A be a ring. Let M, P be A-modules with P of finite presentation. Then Ext^i_P(underlineP, underlineM) = 0 for i > 0 where P is the category of module-valued functors on Alg_A.","statement_latex":"Let $A$ be a ring. Let $M$, $P$ be $A$-modules with $P$ of finite\npresentation. Then\n$\\Ext^i_\\mathcal{P}(\\underline{P}, \\underline{M}) = 0$\nfor $i > 0$ where $\\mathcal{P}$ is the category of module-valued\nfunctors on $\\textit{Alg}_A$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZG","source_file":"adequate.tex","source_line":1207,"source_end_line":1214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1207-L1214","statement_sha256":"a9f34a976a01bbf56ef4d7257f1798c4b461e51cfb7cdeed2f2e1cff99b59379","origin":"The Stacks Project","memory_eligible":false,"source_rank":8591,"rank":8591,"depth":12,"x":2502.761,"y":1173.366,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZH","tag":"06ZH","title":"Higher exts of adequate functors · Lemma 06ZH","summary":"Let A be a ring. Let M be an A-module. Let L be a linearly adequate functor on Alg_A. Then Ext^i_P(L, underlineM) = 0 for i > 0 where P is the category of module-valued functors on Alg_A.","statement_latex":"Let $A$ be a ring. Let $M$ be an $A$-module. Let $L$ be a linearly\nadequate functor on $\\textit{Alg}_A$. Then\n$\\Ext^i_\\mathcal{P}(L, \\underline{M}) = 0$\nfor $i > 0$ where $\\mathcal{P}$ is the category of module-valued\nfunctors on $\\textit{Alg}_A$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZH","source_file":"adequate.tex","source_line":1339,"source_end_line":1346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1339-L1346","statement_sha256":"d3b641e69ece4d853d18bebae643164be5f05c68731ec8144ef9acca69cd1f7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8592,"rank":8592,"depth":13,"x":2435.504,"y":1128.076,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZI","tag":"06ZI","title":"Higher exts of adequate functors · Lemma 06ZI","summary":"With notation as in Lemma [Tag 06Z7] we have R^pQ(F) = 0 for all p > 0 and any adequate functor F.","statement_latex":"With notation as in\nLemma \\ref{lemma-enough-injectives}\nwe have $R^pQ(F) = 0$ for all $p > 0$ and any adequate functor $F$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of adequate functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZI","source_file":"adequate.tex","source_line":1373,"source_end_line":1378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1373-L1378","statement_sha256":"47db5f07aea8ce418430db091e43ac3cd5882b5fa34a1e0129da5b1ba46d97fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8593,"rank":8593,"depth":14,"x":2523.222,"y":1123.225,"cluster":"local-crystalline-methods"},{"id":"stacks:06VG","tag":"06VG","title":"Adequate modules · Definition 06VG","summary":"A sheaf of O-modules F on (Sch/S)_τ is adequate if there exists a τ-covering (Spec(A_i) → S)_i ∈ I such that F_F, A_i is adequate for all i ∈ I.","statement_latex":"A sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on $(\\Sch/S)_\\tau$ is\n{\\it adequate} if there exists a $\\tau$-covering\n$\\{\\Spec(A_i) \\to S\\}_{i \\in I}$ such that $F_{\\mathcal{F}, A_i}$ is\nadequate for all $i \\in I$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VG","source_file":"adequate.tex","source_line":1477,"source_end_line":1483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1477-L1483","statement_sha256":"58bc2e64ca44cd0cd70732bc4e0d178ab2597b5bc5997a22688b8a03d26f12d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8594,"rank":8594,"depth":0,"x":2461.275,"y":1177.583,"cluster":"local-crystalline-methods"},{"id":"stacks:06VI","tag":"06VI","title":"Adequate modules · Lemma 06VI","summary":"Let S be a scheme. Let F be an adequate O-module on (Sch/S)_τ. For any affine scheme Spec(A) over S the functor F_F, A is adequate.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on\n$(\\Sch/S)_\\tau$. For any affine scheme $\\Spec(A)$ over $S$\nthe functor $F_{\\mathcal{F}, A}$ is adequate.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VI","source_file":"adequate.tex","source_line":1489,"source_end_line":1494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1489-L1494","statement_sha256":"e1a627077e7938e6b8397f6d7bdd57ead7f313295bf3737e4178937caf9b26c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8595,"rank":8595,"depth":10,"x":2463.288,"y":1100.972,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZJ","tag":"06ZJ","title":"Adequate modules · Lemma 06ZJ","summary":"Let S = Spec(A) be an affine scheme. The category of adequate O-modules on (Sch/S)_τ is equivalent to the category of adequate module-valued functors on Alg_A.","statement_latex":"Let $S = \\Spec(A)$ be an affine scheme. The category of adequate\n$\\mathcal{O}$-modules on $(\\Sch/S)_\\tau$ is equivalent to the\ncategory of adequate module-valued functors on $\\textit{Alg}_A$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZJ","source_file":"adequate.tex","source_line":1520,"source_end_line":1525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1520-L1525","statement_sha256":"696b4138e4ff494e06fde4e8d94bb94b37d1574b3546d2613faa7a354d4c61bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8596,"rank":8596,"depth":11,"x":2524.468,"y":1159.632,"cluster":"local-crystalline-methods"},{"id":"stacks:06VJ","tag":"06VJ","title":"Adequate modules · Lemma 06VJ","summary":"Let f : T → S be a morphism of schemes. The pullback f^*F of an adequate O-module F on (Sch/S)_τ is an adequate O-module on (Sch/T)_τ.","statement_latex":"Let $f : T \\to S$ be a morphism of schemes.\nThe pullback $f^*\\mathcal{F}$ of an adequate $\\mathcal{O}$-module\n$\\mathcal{F}$ on $(\\Sch/S)_\\tau$ is an adequate\n$\\mathcal{O}$-module on $(\\Sch/T)_\\tau$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VJ","source_file":"adequate.tex","source_line":1541,"source_end_line":1547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1541-L1547","statement_sha256":"dbe44e439dba2a393a35343c6e89c822ef36a7df1621bb3817721b3e08c93d13","origin":"The Stacks Project","memory_eligible":false,"source_rank":8597,"rank":8597,"depth":11,"x":2430.608,"y":1150.937,"cluster":"local-crystalline-methods"},{"id":"stacks:06VK","tag":"06VK","title":"Adequate modules · Lemma 06VK","summary":"Let S be a scheme. Let F be an O-module on (Sch/S)_τ. The following are equivalent • F is adequate, • there exists an affine open covering S = ⋃ S_i and maps of quasi-coherent O_S_i-modules G_i → H_i such that F|_(Sch/S_i)_τ is the kernel of G_i^a → H_i^a • there exists a τ-covering (S_i → S)_i ∈ I and maps of O_S_i-quasi-coherent modules G_i → H_i such that F|_(Sch/S_i)_τ is the kernel of G_i^a → H_i^a, • there exists a τ-covering (f_i : S_i → S)_i ∈ I such that each…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}$-module on\n$(\\Sch/S)_\\tau$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is adequate,\n\\item there exists an affine open covering $S = \\bigcup S_i$ and\nmaps of quasi-coherent $\\mathcal{O}_{S_i}$-modules\n$\\mathcal{G}_i \\to \\mathcal{H}_i$\nsuch that $\\mathcal{F}|_{(\\Sch/S_i)_\\tau}$ is the\nkernel of $\\mathcal{G}_i^a \\to \\mathcal{H}_i^a$\n\\item there exists a $\\tau$-covering $\\{S_i \\to S\\}_{i \\in I}$ and\nmaps of $\\mathcal{O}_{S_i}$-quasi-coherent modules\n$\\mathcal{G}_i \\to \\mathcal{H}_i$\nsuch that $\\mathcal{F}|_{(\\Sch/S_i)_\\tau}$ is the\nkernel of $\\mathcal{G}_i^a \\to \\mathcal{H}_i^a$,\n\\item there exists a $\\tau$-covering $\\{f_i : S_i \\to S\\}_{i \\in I}$\nsuch that each $f_i^*\\mathcal{F}$ is adequate,\n\\item for any affine scheme $U$ over $S$ the restriction\n$\\mathcal{F}|_{(\\Sch/U)_\\tau}$ is the kernel\nof a map $\\mathcal{G}^a \\to \\mathcal{H}^a$ of quasi-coherent\n$\\mathcal{O}_U$-modules.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VK","source_file":"adequate.tex","source_line":1569,"source_end_line":1592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1569-L1592","statement_sha256":"a57f43d861e07821250abfb5320d8de6fae0c754b37d7107ed53249d557f8d88","origin":"The Stacks Project","memory_eligible":false,"source_rank":8598,"rank":8598,"depth":12,"x":2508.075,"y":1103.323,"cluster":"local-crystalline-methods"},{"id":"stacks:06VL","tag":"06VL","title":"Adequate modules · Lemma 06VL","summary":"Let F be an adequate O-module on (Sch/S)_τ. For any surjective flat morphism Spec(B) → Spec(A) of affines over S the extended v Cech complex 0 → F(Spec(A)) → F(Spec(B)) → F(Spec(B ⊗_A B)) → … is exact. In particular F satisfies the sheaf condition for fpqc coverings, and is a sheaf of O-modules on (Sch/S)_fppf.","statement_latex":"Let $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on\n$(\\Sch/S)_\\tau$. For any surjective flat morphism\n$\\Spec(B) \\to \\Spec(A)$ of affines over $S$\nthe extended {\\v C}ech complex\n$$\n0 \\to \\mathcal{F}(\\Spec(A)) \\to\n\\mathcal{F}(\\Spec(B)) \\to\n\\mathcal{F}(\\Spec(B \\otimes_A B)) \\to \\ldots\n$$\nis exact. In particular $\\mathcal{F}$ satisfies the sheaf condition\nfor fpqc coverings, and is a sheaf of $\\mathcal{O}$-modules\non $(\\Sch/S)_{fppf}$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VL","source_file":"adequate.tex","source_line":1623,"source_end_line":1637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1623-L1637","statement_sha256":"8e6f50de2eb816d4e330b12f3353dddf04abce1a1a0ced38efccfe772d4abd7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8599,"rank":8599,"depth":21,"x":2488.931,"y":1183.651,"cluster":"local-crystalline-methods"},{"id":"stacks:07AH","tag":"07AH","title":"Adequate modules · Definition 07AH","summary":"Let S be a scheme. The category of adequate O-modules on (Sch/S)_τ is denoted Adeq(O) or Adeq((Sch/S)_τ, O). If we want to think just about the abelian category of adequate modules without choosing a topology we simply write Adeq(S).","statement_latex":"Let $S$ be a scheme. The category of adequate $\\mathcal{O}$-modules on\n$(\\Sch/S)_\\tau$ is denoted {\\it $\\textit{Adeq}(\\mathcal{O})$} or\n{\\it $\\textit{Adeq}((\\Sch/S)_\\tau, \\mathcal{O})$}. If we want to think just\nabout the abelian category of adequate modules without choosing a\ntopology we simply write {\\it $\\textit{Adeq}(S)$}.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AH","source_file":"adequate.tex","source_line":1669,"source_end_line":1676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1669-L1676","statement_sha256":"0e9beaab4e35b2aba482f1272978206caf53682c239d34987e907064db3b0078","origin":"The Stacks Project","memory_eligible":false,"source_rank":8600,"rank":8600,"depth":0,"x":2437.676,"y":1112.466,"cluster":"local-crystalline-methods"},{"id":"stacks:06VM","tag":"06VM","title":"Adequate modules · Lemma 06VM","summary":"Let S be a scheme. Let F be an adequate O-module on (Sch/S)_τ. • The restriction F|_S_Zar is a quasi-coherent O_S-module on the scheme S. • The restriction F|_S_etale is the quasi-coherent module associated to F|_S_Zar. • For any affine scheme U over S we have H^q(U, F) = 0 for all q > 0. • There is a canonical isomorphism H^q(S, F|_S_Zar) = H^q((Sch/S)_τ, F).","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be an adequate\n$\\mathcal{O}$-module on $(\\Sch/S)_\\tau$.\n\\begin{enumerate}\n\\item The restriction $\\mathcal{F}|_{S_{Zar}}$ is a quasi-coherent\n$\\mathcal{O}_S$-module on the scheme $S$.\n\\item The restriction $\\mathcal{F}|_{S_\\etale}$ is the\nquasi-coherent module associated to $\\mathcal{F}|_{S_{Zar}}$.\n\\item For any affine scheme $U$ over $S$ we have $H^q(U, \\mathcal{F}) = 0$\nfor all $q > 0$.\n\\item There is a canonical isomorphism\n$$\nH^q(S, \\mathcal{F}|_{S_{Zar}}) =\nH^q((\\Sch/S)_\\tau, \\mathcal{F}).\n$$\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VM","source_file":"adequate.tex","source_line":1678,"source_end_line":1695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1678-L1695","statement_sha256":"d5afa85a69f748499310d6551e8b8c7d25685b318b4f9aa299ec746633bb126c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8601,"rank":8601,"depth":26,"x":2534.14,"y":1136.232,"cluster":"local-crystalline-methods"},{"id":"stacks:06VP","tag":"06VP","title":"Adequate modules · Lemma 06VP","summary":"Let S be a scheme. Let F be a presheaf of O-modules on (Sch/S)_τ. If for every affine scheme Spec(A) over S the functor F_F, A is adequate, then the sheafification of F is an adequate O-module.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a presheaf of $\\mathcal{O}$-modules\non $(\\Sch/S)_\\tau$. If for every affine scheme\n$\\Spec(A)$ over $S$ the functor $F_{\\mathcal{F}, A}$ is\nadequate, then the sheafification of $\\mathcal{F}$ is an adequate\n$\\mathcal{O}$-module.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VP","source_file":"adequate.tex","source_line":1767,"source_end_line":1774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1767-L1774","statement_sha256":"9ed737bdb23f1db7549f32282bfb70a6cdd90d45505d73a68f46ee2e233b8ec6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8602,"rank":8602,"depth":8,"x":2442.578,"y":1173.98,"cluster":"local-crystalline-methods"},{"id":"stacks:06VQ","tag":"06VQ","title":"Adequate modules · Lemma 06VQ","summary":"Let S be a scheme. • The category Adeq(O) is abelian. • The functor Adeq(O) → Mod((Sch/S)_τ, O) is exact. • If 0 → F_1 → F_2 → F_3 → 0 is a short exact sequence of O-modules and F_1 and F_3 are adequate, then F_2 is adequate. • The category Adeq(O) has colimits and Adeq(O) → Mod((Sch/S)_τ, O) commutes with them.","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item The category $\\textit{Adeq}(\\mathcal{O})$ is abelian.\n\\item The functor\n$\\textit{Adeq}(\\mathcal{O}) \\to\n\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$\nis exact.\n\\item If $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nis a short exact sequence of $\\mathcal{O}$-modules and\n$\\mathcal{F}_1$ and $\\mathcal{F}_3$ are adequate, then\n$\\mathcal{F}_2$ is adequate.\n\\item The category $\\textit{Adeq}(\\mathcal{O})$ has colimits and\n$\\textit{Adeq}(\\mathcal{O}) \\to\n\\textit{Mod}((\\Sch/S)_\\tau, \\mathcal{O})$\ncommutes with them.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VQ","source_file":"adequate.tex","source_line":1808,"source_end_line":1826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1808-L1826","statement_sha256":"2c3736dc19bd49c070eca1832bc255983790554050bb9591579e99c8bc5aa991","origin":"The Stacks Project","memory_eligible":false,"source_rank":8603,"rank":8603,"depth":27,"x":2480.273,"y":1093.063,"cluster":"local-crystalline-methods"},{"id":"stacks:06VR","tag":"06VR","title":"Adequate modules · Lemma 06VR","summary":"Let f : T → S be a quasi-compact and quasi-separated morphism of schemes. For any adequate O_T-module on (Sch/T)_τ the pushforward f_*F and the higher direct images R^if_*F are adequate O_S-modules on (Sch/S)_τ.","statement_latex":"Let $f : T \\to S$ be a quasi-compact and quasi-separated morphism\nof schemes. For any adequate $\\mathcal{O}_T$-module on\n$(\\Sch/T)_\\tau$ the pushforward\n$f_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$\nare adequate $\\mathcal{O}_S$-modules on $(\\Sch/S)_\\tau$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VR","source_file":"adequate.tex","source_line":1890,"source_end_line":1897,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L1890-L1897","statement_sha256":"92b840d450e2472f2a10d1eca9d8a8e4758d393a830564025569344ddd53316b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8604,"rank":8604,"depth":27,"x":2518.055,"y":1175.238,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZM","tag":"06ZM","title":"Parasitic adequate modules · Lemma 06ZM","summary":"Let S be a scheme. Let F be an adequate O-module on (Sch/S)_τ. The following are equivalent: • vF = 0, • F is parasitic, • F is parasitic for the τ-topology, • F(U) = 0 for all U ⊂ S open, and • there exists an affine open covering S = ⋃ U_i such that F(U_i) = 0 for all i.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{F}$ be an adequate $\\mathcal{O}$-module on\n$(\\Sch/S)_\\tau$. The following are equivalent:\n\\begin{enumerate}\n\\item $v\\mathcal{F} = 0$,\n\\item $\\mathcal{F}$ is parasitic,\n\\item $\\mathcal{F}$ is parasitic for the $\\tau$-topology,\n\\item $\\mathcal{F}(U) = 0$ for all $U \\subset S$ open, and\n\\item there exists an affine open covering $S = \\bigcup U_i$\nsuch that $\\mathcal{F}(U_i) = 0$ for all $i$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Parasitic adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZM","source_file":"adequate.tex","source_line":2039,"source_end_line":2052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2039-L2052","statement_sha256":"2b632cd882556dff4d0c3720bb5d8db1100f3facff7479200ca6ad4533da9c06","origin":"The Stacks Project","memory_eligible":false,"source_rank":8605,"rank":8605,"depth":11,"x":2422.856,"y":1135.551,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZN","tag":"06ZN","title":"Parasitic adequate modules · Lemma 06ZN","summary":"Let S be a scheme. The subcategory C ⊂ Adeq(O) of parasitic adequate modules is a Serre subcategory. Moreover, the functor v induces an equivalence of categories Adeq(O) / C = QCoh(O_S).","statement_latex":"Let $S$ be a scheme. The subcategory\n$\\mathcal{C} \\subset \\textit{Adeq}(\\mathcal{O})$ of parasitic adequate\nmodules is a Serre subcategory. Moreover, the functor $v$ induces\nan equivalence of categories\n$$\n\\textit{Adeq}(\\mathcal{O}) / \\mathcal{C} = \\QCoh(\\mathcal{O}_S).\n$$","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Parasitic adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZN","source_file":"adequate.tex","source_line":2084,"source_end_line":2093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2084-L2093","statement_sha256":"4bdf88dd1e0fbfda6153061a4eb1b502a250879c491060aca20352efa7411af0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8606,"rank":8606,"depth":12,"x":2526.303,"y":1110.48,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZP","tag":"06ZP","title":"Parasitic adequate modules · Lemma 06ZP","summary":"Let f : T → S be a quasi-compact and quasi-separated morphism of schemes. For any parasitic adequate O_T-module on (Sch/T)_τ the pushforward f_*F and the higher direct images R^if_*F are parasitic adequate O_S-modules on (Sch/S)_τ.","statement_latex":"Let $f : T \\to S$ be a quasi-compact and quasi-separated morphism\nof schemes. For any parasitic adequate $\\mathcal{O}_T$-module on\n$(\\Sch/T)_\\tau$ the pushforward\n$f_*\\mathcal{F}$ and the higher direct images $R^if_*\\mathcal{F}$\nare parasitic adequate $\\mathcal{O}_S$-modules on $(\\Sch/S)_\\tau$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Parasitic adequate modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZP","source_file":"adequate.tex","source_line":2116,"source_end_line":2123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2116-L2123","statement_sha256":"d11fec584f371eb996a3b5d25e838372984281b256396bd4befeac1c27e51d3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8607,"rank":8607,"depth":28,"x":2469.465,"y":1188.644,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZQ","tag":"06ZQ","title":"Derived categories of adequate modules, I · Lemma 06ZQ","summary":"Let S be a scheme. Let C ⊂ Adeq(O) denote the full subcategory consisting of parasitic adequate modules. Then D(Adeq(O))/D_C(Adeq(O)) = D(QCoh(O_S)) and similarly for the bounded versions.","statement_latex":"Let $S$ be a scheme. Let\n$\\mathcal{C} \\subset \\textit{Adeq}(\\mathcal{O})$ denote the\nfull subcategory consisting of parasitic adequate modules.\nThen\n$$\nD(\\textit{Adeq}(\\mathcal{O}))/D_\\mathcal{C}(\\textit{Adeq}(\\mathcal{O}))\n= D(\\QCoh(\\mathcal{O}_S))\n$$\nand similarly for the bounded versions.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Derived categories of adequate modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZQ","source_file":"adequate.tex","source_line":2164,"source_end_line":2175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2164-L2175","statement_sha256":"453ab27727e91f1ac1a3af702b8bd558d8062e3615bd509559daf93e009fe5a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8608,"rank":8608,"depth":10,"x":2448.266,"y":1097.64,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZR","tag":"06ZR","title":"Derived categories of adequate modules, I · Lemma 06ZR","summary":"Let U = Spec(A) be an affine scheme. The bounded below derived category D^+(Adeq(O)) is the localization of K^+(QCoh(O_U)) at the multiplicative subset of universal quasi-isomorphisms.","statement_latex":"Let $U = \\Spec(A)$ be an affine scheme.\nThe bounded below derived category\n$D^+(\\textit{Adeq}(\\mathcal{O}))$ is the localization\nof $K^+(\\QCoh(\\mathcal{O}_U))$ at the multiplicative subset of\nuniversal quasi-isomorphisms.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Derived categories of adequate modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZR","source_file":"adequate.tex","source_line":2202,"source_end_line":2209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2202-L2209","statement_sha256":"f46a7d186c86e1014d353bef70ec9bbaf54f1b3a22eae37ce985475ac3ea1981","origin":"The Stacks Project","memory_eligible":false,"source_rank":8609,"rank":8609,"depth":13,"x":2538.152,"y":1153.387,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZS","tag":"06ZS","title":"Derived categories of adequate modules, I · Lemma 06ZS","summary":"Let U = Spec(A) be an affine scheme. The inclusion functor Adeq(O) → Mod((Sch/U)_τ, O) has a right adjoint A. Moreover, the adjunction mapping A(F) → F is an isomorphism for every adequate module F.","statement_latex":"Let $U = \\Spec(A)$ be an affine scheme.\nThe inclusion functor\n$$\n\\textit{Adeq}(\\mathcal{O}) \\to\n\\textit{Mod}((\\Sch/U)_\\tau, \\mathcal{O})\n$$\nhas a right adjoint $A$\\footnote{This is the ``adequator''.}.\nMoreover, the adjunction mapping\n$A(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism for every\nadequate module $\\mathcal{F}$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Derived categories of adequate modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZS","source_file":"adequate.tex","source_line":2256,"source_end_line":2268,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2256-L2268","statement_sha256":"828fd947e2dbac148bdbe1b09ef54142e4ff83190f9b8eb0d07a66fea511b596","origin":"The Stacks Project","memory_eligible":false,"source_rank":8610,"rank":8610,"depth":20,"x":2425.728,"y":1163.395,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZU","tag":"06ZU","title":"Derived categories of adequate modules, I · Lemma 06ZU","summary":"Let U = Spec(A) be an affine scheme. For any object F of Adeq(O) we have R^pA(F) = 0 for all p > 0 where A is as in Lemma [Tag 06ZS].","statement_latex":"Let $U = \\Spec(A)$ be an affine scheme.\nFor any object $\\mathcal{F}$ of $\\textit{Adeq}(\\mathcal{O})$\nwe have $R^pA(\\mathcal{F}) = 0$ for all $p > 0$ where $A$ is\nas in\nLemma \\ref{lemma-right-adjoint-adequate}.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Derived categories of adequate modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZU","source_file":"adequate.tex","source_line":2316,"source_end_line":2323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2316-L2323","statement_sha256":"47c93fba90e5500e381999fd13977c2481094e80c3013c0c27d52da839601d47","origin":"The Stacks Project","memory_eligible":false,"source_rank":8611,"rank":8611,"depth":27,"x":2501.448,"y":1091.405,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZV","tag":"06ZV","title":"Derived categories of adequate modules, I · Lemma 06ZV","summary":"If U = Spec(A) is an affine scheme, then the bounded below version D^+(Adeq(O)) → D^+_Adeq(O) of the functor above is an equivalence.","statement_latex":"If $U = \\Spec(A)$ is an affine scheme, then the bounded\nbelow version\n\\begin{equation}\n\nD^+(\\textit{Adeq}(\\mathcal{O}))\n\\longrightarrow\nD^+_{\\textit{Adeq}}(\\mathcal{O})\n\\end{equation}\nof the functor above is an equivalence.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Derived categories of adequate modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZV","source_file":"adequate.tex","source_line":2368,"source_end_line":2379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2368-L2379","statement_sha256":"7a216f0a6eb709ee8140172127a9397b9a8339fe93efcaa0da9d619628be410c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8612,"rank":8612,"depth":28,"x":2503.523,"y":1188.538,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZW","tag":"06ZW","title":"Derived categories of adequate modules, I · Lemma 06ZW","summary":"Let U = Spec(A) be an affine scheme. Let F and G be adequate O-modules. For any i ≥ 0 the natural map Ext^i_Adeq(O)(F, G) → Ext^i_Mod(O)(F, G) is an isomorphism.","statement_latex":"Let $U = \\Spec(A)$ be an affine scheme.\nLet $\\mathcal{F}$ and $\\mathcal{G}$ be adequate $\\mathcal{O}$-modules.\nFor any $i \\geq 0$ the natural map\n$$\n\\Ext^i_{\\textit{Adeq}(\\mathcal{O})}(\\mathcal{F}, \\mathcal{G})\n\\longrightarrow\n\\Ext^i_{\\textit{Mod}(\\mathcal{O})}(\\mathcal{F}, \\mathcal{G})\n$$\nis an isomorphism.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Derived categories of adequate modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZW","source_file":"adequate.tex","source_line":2410,"source_end_line":2421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2410-L2421","statement_sha256":"fdf3d3c0a36c4748e88f1fea458deebb1dbd3718ff8a438f2e0fad97ff41145a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8613,"rank":8613,"depth":29,"x":2423.006,"y":1117.311,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZY","tag":"06ZY","title":"Pure extensions · Definition 06ZY","summary":"Let A be a ring. • An A-module P is said to be pure projective if for every universally exact sequence 0 → K → M → N → 0 of A-module the sequence 0 → Hom_A(P, K) → Hom_A(P, M) → Hom_A(P, N) → 0 is exact. • An A-module I is said to be pure injective if for every universally exact sequence 0 → K → M → N → 0 of A-module the sequence 0 → Hom_A(N, I) → Hom_A(M, I) → Hom_A(K, I) → 0 is exact.","statement_latex":"Let $A$ be a ring.\n\\begin{enumerate}\n\\item An $A$-module $P$ is said to be {\\it pure projective}\nif for every universally exact sequence\n$0 \\to K \\to M \\to N \\to 0$ of $A$-module the sequence\n$0 \\to \\Hom_A(P, K) \\to \\Hom_A(P, M) \\to \\Hom_A(P, N) \\to 0$\nis exact.\n\\item An $A$-module $I$ is said to be {\\it pure injective}\nif for every universally exact sequence\n$0 \\to K \\to M \\to N \\to 0$ of $A$-module the sequence\n$0 \\to \\Hom_A(N, I) \\to \\Hom_A(M, I) \\to \\Hom_A(K, I) \\to 0$\nis exact.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Pure extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZY","source_file":"adequate.tex","source_line":2445,"source_end_line":2460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2445-L2460","statement_sha256":"5726da1fd0d00a680209fc8faa09aad2f83d87b21619c2c5a5ef9a496295c73e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8614,"rank":8614,"depth":0,"x":2540.915,"y":1124.224,"cluster":"local-crystalline-methods"},{"id":"stacks:06ZZ","tag":"06ZZ","title":"Pure extensions · Lemma 06ZZ","summary":"Let A be a ring. • A module is pure projective if and only if it is a direct summand of a direct sum of finitely presented A-modules. • For any module M there exists a universally exact sequence 0 → N → P → M → 0 with P pure projective.","statement_latex":"Let $A$ be a ring.\n\\begin{enumerate}\n\\item A module is pure projective if and only if\nit is a direct summand of a direct sum of finitely presented $A$-modules.\n\\item For any module $M$ there exists a universally exact sequence\n$0 \\to N \\to P \\to M \\to 0$ with $P$ pure projective.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Pure extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ZZ","source_file":"adequate.tex","source_line":2465,"source_end_line":2474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2465-L2474","statement_sha256":"b1fce13155de2aa4658258f71f89eb5448f49f052e52813896175239eae64ebc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8615,"rank":8615,"depth":2,"x":2447.432,"y":1186.68,"cluster":"local-crystalline-methods"},{"id":"stacks:0700","tag":"0700","title":"Pure extensions · Lemma 0700","summary":"Let A be a ring. For any A-module M set M^vee = Hom_Z(M, Q/Z). • For any A-module M the A-module M^vee is pure injective. • An A-module I is pure injective if and only if the map I → (I^vee)^vee splits. • For any module M there exists a universally exact sequence 0 → M → I → N → 0 with I pure injective.","statement_latex":"Let $A$ be a ring. For any $A$-module $M$ set\n$M^\\vee = \\Hom_\\mathbf{Z}(M, \\mathbf{Q}/\\mathbf{Z})$.\n\\begin{enumerate}\n\\item For any $A$-module $M$ the $A$-module $M^\\vee$ is pure injective.\n\\item An $A$-module $I$ is pure injective if and only if the map\n$I \\to (I^\\vee)^\\vee$ splits.\n\\item For any module $M$ there exists a universally exact sequence\n$0 \\to M \\to I \\to N \\to 0$ with $I$ pure injective.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Pure extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0700","source_file":"adequate.tex","source_line":2499,"source_end_line":2510,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2499-L2510","statement_sha256":"c491a964faf9b6cc4a693448cfbee5970239b5ef8785db70009d8e9e5da7f8cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8616,"rank":8616,"depth":0,"x":2466.338,"y":1086.557,"cluster":"local-crystalline-methods"},{"id":"stacks:0701","tag":"0701","title":"Pure extensions · Lemma 0701","summary":"Let A be a ring. • Let L → M → N be a universally exact sequence of A-modules. Let K = Im(M → N). Then K → N is universally injective. • Any universally exact complex can be split into universally exact short exact sequences.","statement_latex":"Let $A$ be a ring.\n\\begin{enumerate}\n\\item Let $L \\to M \\to N$ be a universally exact sequence\nof $A$-modules. Let $K = \\Im(M \\to N)$.\nThen $K \\to N$ is universally injective.\n\\item Any universally exact complex\ncan be split into universally exact short exact sequences.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Pure extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0701","source_file":"adequate.tex","source_line":2542,"source_end_line":2552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2542-L2552","statement_sha256":"2f90058c52b14d67dc1e671948713adcbaecbace993e7e5038e80e5fa824b070","origin":"The Stacks Project","memory_eligible":false,"source_rank":8617,"rank":8617,"depth":0,"x":2533.584,"y":1171.973,"cluster":"local-crystalline-methods"},{"id":"stacks:0702","tag":"0702","title":"Pure extensions · Definition 0702","summary":"Let A be a ring. Let M be an A-module. • A pure projective resolution P_bullet → M is a universally exact sequence … → P_1 → P_0 → M → 0 with each P_i pure projective. • A pure injective resolution M → I^bullet is a universally exact sequence 0 → M → I^0 → I^1 → … with each I^i pure injective.","statement_latex":"Let $A$ be a ring. Let $M$ be an $A$-module.\n\\begin{enumerate}\n\\item A {\\it pure projective resolution} $P_\\bullet \\to M$\nis a universally exact sequence\n$$\n\\ldots \\to P_1 \\to P_0 \\to M \\to 0\n$$\nwith each $P_i$ pure projective.\n\\item A {\\it pure injective resolution} $M \\to I^\\bullet$ is a universally\nexact sequence\n$$\n0 \\to M \\to I^0 \\to I^1 \\to \\ldots\n$$\nwith each $I^i$ pure injective.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Pure extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0702","source_file":"adequate.tex","source_line":2568,"source_end_line":2585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2568-L2585","statement_sha256":"945158ab3363a564d6bb91967a5e9a2fd36ae46fc1b115a07675e571fa18d191","origin":"The Stacks Project","memory_eligible":false,"source_rank":8618,"rank":8618,"depth":0,"x":2414.135,"y":1146.895,"cluster":"local-crystalline-methods"},{"id":"stacks:0703","tag":"0703","title":"Pure extensions · Lemma 0703","summary":"Let A be a ring. • Any A-module has a pure projective resolution. Let M → N be a map of A-modules. Let P_bullet → M be a pure projective resolution and let N_bullet → N be a universally exact resolution. • [(2)] There exists a map of complexes P_bullet → N_bullet inducing the given map M = Coker(P_1 → P_0) → Coker(N_1 → N_0) = N • [(3)] two maps α, β : P_bullet → N_bullet inducing the same map M → N are homotopic.","statement_latex":"Let $A$ be a ring.\n\\begin{enumerate}\n\\item Any $A$-module has a pure projective resolution.\n\\end{enumerate}\nLet $M \\to N$ be a map of $A$-modules.\nLet $P_\\bullet \\to M$ be a pure projective resolution and\nlet $N_\\bullet \\to N$ be a universally exact resolution.\n\\begin{enumerate}\n\\item[(2)] There exists a map of complexes $P_\\bullet \\to N_\\bullet$\ninducing the given map\n$$\nM = \\Coker(P_1 \\to P_0) \\to \\Coker(N_1 \\to N_0) = N\n$$\n\\item[(3)] two maps $\\alpha, \\beta : P_\\bullet \\to N_\\bullet$\ninducing the same map $M \\to N$ are homotopic.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Pure extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0703","source_file":"adequate.tex","source_line":2591,"source_end_line":2609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2591-L2609","statement_sha256":"d9d0307069d702482fdf750e443ad05d17a64d02e6a81a56c42b4b97be7efbd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8619,"rank":8619,"depth":3,"x":2523.437,"y":1097.133,"cluster":"local-crystalline-methods"},{"id":"stacks:0704","tag":"0704","title":"Pure extensions · Lemma 0704","summary":"Let A be a ring. • Any A-module has a pure injective resolution. Let M → N be a map of A-modules. Let M → M^bullet be a universally exact resolution and let N → I^bullet be a pure injective resolution. • [(2)] There exists a map of complexes M^bullet → I^bullet inducing the given map M = Ker(M^0 → M^1) → Ker(I^0 → I^1) = N • [(3)] two maps α, β : M^bullet → I^bullet inducing the same map M → N are homotopic.","statement_latex":"Let $A$ be a ring.\n\\begin{enumerate}\n\\item Any $A$-module has a pure injective resolution.\n\\end{enumerate}\nLet $M \\to N$ be a map of $A$-modules.\nLet $M \\to M^\\bullet$ be a universally exact resolution and\nlet $N \\to I^\\bullet$ be a pure injective resolution.\n\\begin{enumerate}\n\\item[(2)] There exists a map of complexes $M^\\bullet \\to I^\\bullet$\ninducing the given map\n$$\nM = \\Ker(M^0 \\to M^1) \\to \\Ker(I^0 \\to I^1) = N\n$$\n\\item[(3)] two maps $\\alpha, \\beta : M^\\bullet \\to I^\\bullet$\ninducing the same map $M \\to N$ are homotopic.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Pure extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0704","source_file":"adequate.tex","source_line":2643,"source_end_line":2661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2643-L2661","statement_sha256":"f0d90f0f33e122a281b0e0ba87acd0a72e521c4d5fcec21c1b373fe57f381d6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8620,"rank":8620,"depth":4,"x":2482.465,"y":1196.789,"cluster":"local-crystalline-methods"},{"id":"stacks:0705","tag":"0705","title":"Pure extensions · Definition 0705","summary":"Let A be a ring and let M, N be A-modules. The ith pure extension module Pext^i_A(M, N) is the ith cohomology module of the complex Hom_A(M, I^bullet) where I^bullet is a pure injective resolution of N.","statement_latex":"Let $A$ be a ring and let $M$, $N$ be $A$-modules.\nThe $i$th {\\it pure extension module} $\\text{Pext}^i_A(M, N)$\nis the $i$th cohomology module of the complex\n$\\Hom_A(M, I^\\bullet)$ where $I^\\bullet$ is a pure injective\nresolution of $N$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Pure extensions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0705","source_file":"adequate.tex","source_line":2681,"source_end_line":2688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2681-L2688","statement_sha256":"4d6197e2cfb6908cacdaad8891cf4d78311985ccfec0c4582d02aebd0babaa2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8621,"rank":8621,"depth":0,"x":2432.073,"y":1099.148,"cluster":"local-crystalline-methods"},{"id":"stacks:0706","tag":"0706","title":"Pure extensions · Lemma 0706","summary":"Let A be a ring. • Pext^i_A(M, N) = 0 for i > 0 whenever N is pure injective, • Pext^i_A(M, N) = 0 for i > 0 whenever M is pure projective, in particular if M is an A-module of finite presentation, • Pext^i_A(M, N) is also the ith cohomology module of the complex Hom_A(P_bullet, N) where P_bullet is a pure projective resolution of M.","statement_latex":"Let $A$ be a ring.\n\\begin{enumerate}\n\\item $\\text{Pext}^i_A(M, N) = 0$ for $i > 0$ whenever $N$ is pure injective,\n\\item $\\text{Pext}^i_A(M, N) = 0$ for $i > 0$ whenever $M$ is pure projective,\nin particular if $M$ is an $A$-module of finite presentation,\n\\item $\\text{Pext}^i_A(M, N)$ is also the $i$th cohomology module\nof the complex $\\Hom_A(P_\\bullet, N)$ where $P_\\bullet$\nis a pure projective resolution of $M$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Pure extensions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0706","source_file":"adequate.tex","source_line":2696,"source_end_line":2707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2696-L2707","statement_sha256":"5d2c69b03acda121b4ad7ecad471a3ffc9d5f33db8b26d09f5cbefa6b75ca7fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8622,"rank":8622,"depth":0,"x":2548.814,"y":1142.956,"cluster":"local-crystalline-methods"},{"id":"stacks:0708","tag":"0708","title":"Higher exts of quasi-coherent sheaves on the big site · Lemma 0708","summary":"Let A be a ring. Let A be the category of adequate functors on Alg_A. The injective objects of A are exactly the functors underlineI where I is a pure injective A-module.","statement_latex":"Let $A$ be a ring.\nLet $\\mathcal{A}$ be the category of adequate functors on $\\textit{Alg}_A$.\nThe injective objects of $\\mathcal{A}$ are exactly the functors\n$\\underline{I}$ where $I$ is a pure injective $A$-module.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of quasi-coherent sheaves on the big site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0708","source_file":"adequate.tex","source_line":2737,"source_end_line":2743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2737-L2743","statement_sha256":"120c7a52b1804632ab2489c3d600df6755980e4193b48a91ae5f2cdbcd5c3fe2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8623,"rank":8623,"depth":12,"x":2426.406,"y":1177.197,"cluster":"local-crystalline-methods"},{"id":"stacks:0709","tag":"0709","title":"Higher exts of quasi-coherent sheaves on the big site · Lemma 0709","summary":"Let U = Spec(A) be an affine scheme. Let M, N be A-modules. For all i we have a canonical isomorphism Ext^i_Mod(O)(M^a, N^a) = Pext^i_A(M, N) functorial in M and N.","statement_latex":"Let $U = \\Spec(A)$ be an affine scheme. Let $M$, $N$ be $A$-modules.\nFor all $i$ we have a canonical isomorphism\n$$\n\\Ext^i_{\\textit{Mod}(\\mathcal{O})}(M^a, N^a) = \\text{Pext}^i_A(M, N)\n$$\nfunctorial in $M$ and $N$.","area":"Local & Crystalline Methods","chapter":"Adequate Modules","chapter_id":"adequate","section":"Higher exts of quasi-coherent sheaves on the big site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0709","source_file":"adequate.tex","source_line":2778,"source_end_line":2786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/adequate.tex#L2778-L2786","statement_sha256":"cee4a5a0c54151c5f523d8d96aaea132c7aa6634019baa86455163bb2fc8b680","origin":"The Stacks Project","memory_eligible":false,"source_rank":8624,"rank":8624,"depth":30,"x":2489.691,"y":1081.652,"cluster":"local-crystalline-methods"},{"id":"stacks:08XJ","tag":"08XJ","title":"Essential surjections and injections · Definition 08XJ","summary":"Let A be an abelian category. • An injection A ⊂ B of A is essential, or we say that B is an essential extension of A, if every nonzero subobject B' ⊂ B has nonzero intersection with A. • A surjection f : A → B of A is essential if for every proper subobject A' ⊂ A we have f(A') not = B.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item An injection $A \\subset B$ of $\\mathcal{A}$ is {\\it essential},\nor we say that $B$ is an {\\it essential extension of} $A$,\nif every nonzero subobject $B' \\subset B$ has nonzero intersection with $A$.\n\\item A surjection $f : A \\to B$ of $\\mathcal{A}$ is {\\it essential}\nif for every proper subobject $A' \\subset A$ we have $f(A') \\not = B$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Essential surjections and injections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XJ","source_file":"dualizing.tex","source_line":105,"source_end_line":115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L105-L115","statement_sha256":"254efd9fbec368d311ba7324cd2179067da277d01b9b147719ab0c520e2cbb0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8625,"rank":8625,"depth":0,"x":2137.917,"y":1126.688,"cluster":"duality-cohomology"},{"id":"stacks:08XK","tag":"08XK","title":"Essential surjections and injections · Lemma 08XK","summary":"Let A be an abelian category. • If A ⊂ B and B ⊂ C are essential extensions, then A ⊂ C is an essential extension. • If A ⊂ B is an essential extension and C ⊂ B is a subobject, then A ∩ C ⊂ C is an essential extension. • If A → B and B → C are essential surjections, then A → C is an essential surjection. • Given an essential surjection f : A → B and a surjection A → C with kernel K, the morphism C → B/f(K) is an essential surjection.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item If $A \\subset B$ and $B \\subset C$ are essential extensions, then\n$A \\subset C$ is an essential extension.\n\\item If $A \\subset B$ is an essential extension and $C \\subset B$\nis a subobject, then $A \\cap C \\subset C$ is an essential extension.\n\\item If $A \\to B$ and $B \\to C$ are essential surjections, then\n$A \\to C$ is an essential surjection.\n\\item Given an essential surjection $f : A \\to B$ and a surjection\n$A \\to C$ with kernel $K$, the morphism $C \\to B/f(K)$ is an essential\nsurjection.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Essential surjections and injections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XK","source_file":"dualizing.tex","source_line":120,"source_end_line":134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L120-L134","statement_sha256":"869f40df124105676f6a0ba94f0d5e6c098abe34a3eb3815979ff0585998c115","origin":"The Stacks Project","memory_eligible":false,"source_rank":8626,"rank":8626,"depth":0,"x":1960.968,"y":1211.216,"cluster":"duality-cohomology"},{"id":"stacks:08XL","tag":"08XL","title":"Essential surjections and injections · Lemma 08XL","summary":"Let R be a ring. Let M be an R-module. Let E = colim E_i be a filtered colimit of R-modules. Suppose given a compatible system of essential injections M → E_i of R-modules. Then M → E is an essential injection.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module. Let $E = \\colim E_i$\nbe a filtered colimit of $R$-modules. Suppose given a compatible\nsystem of essential injections $M \\to E_i$ of $R$-modules.\nThen $M \\to E$ is an essential injection.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Essential surjections and injections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XL","source_file":"dualizing.tex","source_line":140,"source_end_line":146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L140-L146","statement_sha256":"c967e837f29d0763d2a16f37194263e8413b072f4e800785e7d6f3f34dd257ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":8627,"rank":8627,"depth":2,"x":2023.618,"y":1048.103,"cluster":"duality-cohomology"},{"id":"stacks:08XM","tag":"08XM","title":"Essential surjections and injections · Lemma 08XM","summary":"Let R be a ring. Let M ⊂ N be R-modules. The following are equivalent • M ⊂ N is an essential extension, • for all x ∈ N nonzero there exists an f ∈ R such that fx ∈ M and fx not = 0.","statement_latex":"Let $R$ be a ring. Let $M \\subset N$ be $R$-modules. The following\nare equivalent\n\\begin{enumerate}\n\\item $M \\subset N$ is an essential extension,\n\\item for all $x \\in N$ nonzero there exists an $f \\in R$ such that $fx \\in M$\nand $fx \\not = 0$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Essential surjections and injections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XM","source_file":"dualizing.tex","source_line":153,"source_end_line":162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L153-L162","statement_sha256":"2493a90ad468959fc44800ef0b3a31102571bf32387ad6c41188a0031a522849","origin":"The Stacks Project","memory_eligible":false,"source_rank":8628,"rank":8628,"depth":0,"x":2108.788,"y":1204.29,"cluster":"duality-cohomology"},{"id":"stacks:08XP","tag":"08XP","title":"Injective modules · Lemma 08XP","summary":"Let R be a ring. Any product of injective R-modules is injective.","statement_latex":"Let $R$ be a ring. Any product of injective $R$-modules is injective.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XP","source_file":"dualizing.tex","source_line":183,"source_end_line":186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L183-L186","statement_sha256":"e7e65d82247d3987d6cdf91165281981ab872d47ef7b077201d86be0eb010a12","origin":"The Stacks Project","memory_eligible":false,"source_rank":8629,"rank":8629,"depth":1,"x":1919.951,"y":1137.293,"cluster":"duality-cohomology"},{"id":"stacks:08XQ","tag":"08XQ","title":"Injective modules · Lemma 08XQ","summary":"Let R → S be a flat ring map. If E is an injective S-module, then E is injective as an R-module.","statement_latex":"Let $R \\to S$ be a flat ring map. If $E$ is an injective $S$-module,\nthen $E$ is injective as an $R$-module.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XQ","source_file":"dualizing.tex","source_line":192,"source_end_line":196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L192-L196","statement_sha256":"3150d578027572a76a328711a1d19313a3c9739bb8b00e295b3a0b7d53f21397","origin":"The Stacks Project","memory_eligible":false,"source_rank":8630,"rank":8630,"depth":1,"x":2113.516,"y":1079.413,"cluster":"duality-cohomology"},{"id":"stacks:08YV","tag":"08YV","title":"Injective modules · Lemma 08YV","summary":"Let R → S be an epimorphism of rings. Let E be an S-module. If E is injective as an R-module, then E is an injective S-module.","statement_latex":"Let $R \\to S$ be an epimorphism of rings. Let $E$ be an $S$-module.\nIf $E$ is injective as an $R$-module, then $E$ is an injective $S$-module.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YV","source_file":"dualizing.tex","source_line":204,"source_end_line":208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L204-L208","statement_sha256":"6c1a35b45deae72ba2b1c8756c043e499693992b3b22c3e138e5bb78b1aef6fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8631,"rank":8631,"depth":2,"x":2017.11,"y":1232.274,"cluster":"duality-cohomology"},{"id":"stacks:08XR","tag":"08XR","title":"Injective modules · Lemma 08XR","summary":"Let R → S be a ring map. If E is an injective R-module, then Hom_R(S, E) is an injective S-module.","statement_latex":"Let $R \\to S$ be a ring map. If $E$ is an injective $R$-module,\nthen $\\Hom_R(S, E)$ is an injective $S$-module.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XR","source_file":"dualizing.tex","source_line":215,"source_end_line":219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L215-L219","statement_sha256":"ad1fde9fe454d9aeca2c09cc9dec8065805014cdeea04f1692d073c111dbad91","origin":"The Stacks Project","memory_eligible":false,"source_rank":8632,"rank":8632,"depth":1,"x":1965.154,"y":1064.474,"cluster":"duality-cohomology"},{"id":"stacks:08XS","tag":"08XS","title":"Injective modules · Lemma 08XS","summary":"Let R be a ring. Let I be an injective R-module. Let E ⊂ I be a submodule. The following are equivalent • E is injective, and • for all E ⊂ E' ⊂ I with E ⊂ E' essential we have E = E'. In particular, an R-module is injective if and only if every essential extension is trivial.","statement_latex":"Let $R$ be a ring. Let $I$ be an injective $R$-module. Let $E \\subset I$\nbe a submodule. The following are equivalent\n\\begin{enumerate}\n\\item $E$ is injective, and\n\\item for all $E \\subset E' \\subset I$ with $E \\subset E'$ essential\nwe have $E = E'$.\n\\end{enumerate}\nIn particular, an $R$-module is injective if and only if every essential\nextension is trivial.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XS","source_file":"dualizing.tex","source_line":226,"source_end_line":237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L226-L237","statement_sha256":"542f55529e8e69c807c3824566c711c2888fb004d2f373583f982261bd202082","origin":"The Stacks Project","memory_eligible":false,"source_rank":8633,"rank":8633,"depth":0,"x":2138.797,"y":1158.939,"cluster":"duality-cohomology"},{"id":"stacks:08XV","tag":"08XV","title":"Injective modules · Lemma 08XV","summary":"Let R be a Noetherian ring. A direct sum of injective modules is injective.","statement_latex":"Let $R$ be a Noetherian ring. A direct sum of injective modules\nis injective.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XV","source_file":"dualizing.tex","source_line":291,"source_end_line":295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L291-L295","statement_sha256":"2179e3696e5ef1e0743828945c90023a1b02c9a356f50b84c1a6c7854a03186f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8634,"rank":8634,"depth":3,"x":1934.33,"y":1187.876,"cluster":"duality-cohomology"},{"id":"stacks:0A6I","tag":"0A6I","title":"Injective modules · Lemma 0A6I","summary":"Let R be a Noetherian ring. Let S ⊂ R be a multiplicative subset. If E is an injective R-module, then S^-1E is an injective S^-1R-module.","statement_latex":"Let $R$ be a Noetherian ring. Let $S \\subset R$ be a multiplicative\nsubset. If $E$ is an injective $R$-module, then $S^{-1}E$ is an\ninjective $S^{-1}R$-module.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6I","source_file":"dualizing.tex","source_line":309,"source_end_line":314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L309-L314","statement_sha256":"945f0aad2e002fcead0ef0b881317ce992686898ea9b71737bf65b932f5ecfd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8635,"rank":8635,"depth":3,"x":2062.117,"y":1050.212,"cluster":"duality-cohomology"},{"id":"stacks:08XW","tag":"08XW","title":"Injective modules · Lemma 08XW","summary":"Let R be a Noetherian ring. Let I be an injective R-module. • Let f ∈ R. Then E = ⋃ I[f^n] = I[f^∞] is an injective submodule of I. • Let J ⊂ R be an ideal. Then the J-power torsion submodule I[J^∞] is an injective submodule of I.","statement_latex":"Let $R$ be a Noetherian ring. Let $I$ be an injective $R$-module.\n\\begin{enumerate}\n\\item Let $f \\in R$. Then $E = \\bigcup I[f^n] = I[f^\\infty]$\nis an injective submodule of $I$.\n\\item Let $J \\subset R$ be an ideal. Then the $J$-power torsion\nsubmodule $I[J^\\infty]$ is an injective submodule of $I$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XW","source_file":"dualizing.tex","source_line":335,"source_end_line":344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L335-L344","statement_sha256":"9927b940220c3beb28c170c8155db970269bcaa9a3b3992ff8a99ccd5f6d4e36","origin":"The Stacks Project","memory_eligible":false,"source_rank":8636,"rank":8636,"depth":1,"x":2078.631,"y":1224.62,"cluster":"duality-cohomology"},{"id":"stacks:0A6J","tag":"0A6J","title":"Injective modules · Lemma 0A6J","summary":"Let A be a Noetherian ring. Let E be an injective A-module. Then E ⊗_A A[x] has injective-amplitude [0, 1] as an object of D(A[x]). In particular, E ⊗_A A[x] has finite injective dimension as an A[x]-module.","statement_latex":"Let $A$ be a Noetherian ring. Let $E$ be an injective $A$-module.\nThen $E \\otimes_A A[x]$ has injective-amplitude $[0, 1]$\nas an object of $D(A[x])$. In particular, $E \\otimes_A A[x]$\nhas finite injective dimension as an $A[x]$-module.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6J","source_file":"dualizing.tex","source_line":373,"source_end_line":379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L373-L379","statement_sha256":"1d127ad90dc4853818be0e713b25fa85ad8f763643488a348c0f982e2a0eb839","origin":"The Stacks Project","memory_eligible":false,"source_rank":8637,"rank":8637,"depth":4,"x":1925.861,"y":1105.12,"cluster":"duality-cohomology"},{"id":"stacks:08XY","tag":"08XY","title":"Projective covers · Definition 08XY","summary":"Let R be a ring. A surjection P → M of R-modules is said to be a projective cover, or sometimes a projective envelope, if P is a projective R-module and P → M is an essential surjection.","statement_latex":"Let $R$ be a ring. A surjection $P \\to M$ of $R$-modules is said\nto be a {\\it projective cover}, or sometimes a {\\it projective envelope},\nif $P$ is a projective $R$-module and $P \\to M$ is an essential\nsurjection.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Projective covers","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XY","source_file":"dualizing.tex","source_line":406,"source_end_line":412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L406-L412","statement_sha256":"4fc94e092fa8654cbc68ecc97edb4eb318bc43e662f38ae640755466c3dda107","origin":"The Stacks Project","memory_eligible":false,"source_rank":8638,"rank":8638,"depth":0,"x":2135.071,"y":1106.556,"cluster":"duality-cohomology"},{"id":"stacks:08XZ","tag":"08XZ","title":"Projective covers · Lemma 08XZ","summary":"Let R be a ring and let M be an R-module. If a projective cover of M exists, then it is unique up to isomorphism.","statement_latex":"Let $R$ be a ring and let $M$ be an $R$-module. If a projective cover\nof $M$ exists, then it is unique up to isomorphism.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Projective covers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08XZ","source_file":"dualizing.tex","source_line":421,"source_end_line":425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L421-L425","statement_sha256":"5b4aac3335d63899b6c66b0d249cf04ee9d66aab0387c8004ce9c5058dca57ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":8639,"rank":8639,"depth":0,"x":1979.306,"y":1224.466,"cluster":"duality-cohomology"},{"id":"stacks:08Y0","tag":"08Y0","title":"Projective covers · Lemma 08Y0","summary":"Let (R, m, kappa) be a local ring. Any finite R-module has a projective cover.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring. Any finite $R$-module has\na projective cover.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Projective covers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Y0","source_file":"dualizing.tex","source_line":441,"source_end_line":445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L441-L445","statement_sha256":"d6669bf20845e05c9f1b696aa904d3eafa59ebdd16c7b25355eb1db766f30ab2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8640,"rank":8640,"depth":3,"x":1999.389,"y":1048.752,"cluster":"duality-cohomology"},{"id":"stacks:08Y2","tag":"08Y2","title":"Injective hulls · Definition 08Y2","summary":"Let R be a ring. An injection M → I of R-modules is said to be an injective hull if I is an injective R-module and M → I is an essential injection.","statement_latex":"Let $R$ be a ring. An injection $M \\to I$ of $R$-modules is said\nto be an {\\it injective hull} if $I$ is an injective $R$-module and\n$M \\to I$ is an essential injection.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hulls","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Y2","source_file":"dualizing.tex","source_line":470,"source_end_line":475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L470-L475","statement_sha256":"9ec158b7706435e189b8a6fa6c3e782ade32677ab9db8dd70f77ac921f481250","origin":"The Stacks Project","memory_eligible":false,"source_rank":8641,"rank":8641,"depth":0,"x":2126.159,"y":1190.024,"cluster":"duality-cohomology"},{"id":"stacks:08Y3","tag":"08Y3","title":"Injective hulls · Lemma 08Y3","summary":"Let R be a ring. Any R-module has an injective hull.","statement_latex":"Let $R$ be a ring. Any $R$-module has an injective hull.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hulls","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Y3","source_file":"dualizing.tex","source_line":480,"source_end_line":483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L480-L483","statement_sha256":"44f445ea156eb35eed87b05025cff7162b8c23adc327187450e499931e53f187","origin":"The Stacks Project","memory_eligible":false,"source_rank":8642,"rank":8642,"depth":3,"x":1918.625,"y":1157.715,"cluster":"duality-cohomology"},{"id":"stacks:08Y4","tag":"08Y4","title":"Injective hulls · Lemma 08Y4","summary":"Let R be a ring. Let M, N be R-modules and let M → E and N → E' be injective hulls. Then • for any R-module map φ : M → N there exists an R-module map ψ : E → E' such that xymatrix M ar[r] ar[d]_φ & E ar[d]^ψ N ar[r] & E' commutes, • if φ is injective, then ψ is injective, • if φ is an essential injection, then ψ is an isomorphism, • if φ is an isomorphism, then ψ is an isomorphism, • if M → I is an embedding of M into an injective R-module, then there is an isomorphism I…","statement_latex":"Let $R$ be a ring. Let $M$, $N$ be $R$-modules and let $M \\to E$\nand $N \\to E'$ be injective hulls. Then\n\\begin{enumerate}\n\\item for any $R$-module map $\\varphi : M \\to N$ there exists an\n$R$-module map $\\psi : E \\to E'$ such that\n$$\n\\xymatrix{\nM \\ar[r] \\ar[d]_\\varphi & E \\ar[d]^\\psi \\\\\nN \\ar[r] & E'\n}\n$$\ncommutes,\n\\item if $\\varphi$ is injective, then $\\psi$ is injective,\n\\item if $\\varphi$ is an essential injection, then $\\psi$ is an isomorphism,\n\\item if $\\varphi$ is an isomorphism, then $\\psi$ is an isomorphism,\n\\item if $M \\to I$ is an embedding of $M$ into an injective $R$-module,\nthen there is an isomorphism $I \\cong E \\oplus I'$ compatible with\nthe embeddings of $M$,\n\\end{enumerate}\nIn particular, the injective hull $E$ of $M$ is unique up to isomorphism.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hulls","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Y4","source_file":"dualizing.tex","source_line":501,"source_end_line":523,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L501-L523","statement_sha256":"0f5dd2e543d317ddcd03828a9b4ba320b4257e7a4a8483eaa31b7db89a2e9a33","origin":"The Stacks Project","memory_eligible":false,"source_rank":8643,"rank":8643,"depth":1,"x":2098.027,"y":1063.577,"cluster":"duality-cohomology"},{"id":"stacks:08Y6","tag":"08Y6","title":"Injective hulls · Definition 08Y6","summary":"An object X of an additive category is called indecomposable if it is nonzero and if X = Y ⊕ Z, then either Y = 0 or Z = 0.","statement_latex":"An object $X$ of an additive category is called {\\it indecomposable}\nif it is nonzero and if $X = Y \\oplus Z$, then either $Y = 0$ or $Z = 0$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hulls","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Y6","source_file":"dualizing.tex","source_line":551,"source_end_line":555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L551-L555","statement_sha256":"ff989baa5183c1ce7c8d67dee5e47784a3fc1bbe86c66ea3ac3f2119403e2c01","origin":"The Stacks Project","memory_eligible":false,"source_rank":8644,"rank":8644,"depth":0,"x":2041.32,"y":1235.157,"cluster":"duality-cohomology"},{"id":"stacks:08Y7","tag":"08Y7","title":"Injective hulls · Lemma 08Y7","summary":"Let R be a ring. Let E be an indecomposable injective R-module. Then • E is the injective hull of any nonzero submodule of E, • the intersection of any two nonzero submodules of E is nonzero, • End_R(E) is a noncommutative local ring with maximal ideal those φ : E → E whose kernel is nonzero, and • the set of zerodivisors on E is a prime ideal p of R and E is an injective R_ p-module.","statement_latex":"Let $R$ be a ring. Let $E$ be an indecomposable injective $R$-module.\nThen\n\\begin{enumerate}\n\\item $E$ is the injective hull of any nonzero submodule of $E$,\n\\item the intersection of any two nonzero submodules of $E$ is nonzero,\n\\item $\\text{End}_R(E)$ is a noncommutative local ring with maximal\nideal those $\\varphi : E \\to E$ whose kernel is nonzero, and\n\\item the set of zerodivisors on $E$ is a prime ideal $\\mathfrak p$ of $R$\nand $E$ is an injective $R_\\mathfrak p$-module.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hulls","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Y7","source_file":"dualizing.tex","source_line":557,"source_end_line":569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L557-L569","statement_sha256":"e09a47e9c9a1defc074b494b28edd2f2038174927b6957be2ed69f3fc7367074","origin":"The Stacks Project","memory_eligible":false,"source_rank":8645,"rank":8645,"depth":3,"x":1944.949,"y":1076.12,"cluster":"duality-cohomology"},{"id":"stacks:08Y8","tag":"08Y8","title":"Injective hulls · Lemma 08Y8","summary":"Let p ⊂ R be a prime of a ring R. Let E be the injective hull of R/ p. Then • E is indecomposable, • E is the injective hull of kappa( p), • E is the injective hull of kappa( p) over the ring R_ p.","statement_latex":"Let $\\mathfrak p \\subset R$ be a prime of a ring $R$.\nLet $E$ be the injective hull of $R/\\mathfrak p$. Then\n\\begin{enumerate}\n\\item $E$ is indecomposable,\n\\item $E$ is the injective hull of $\\kappa(\\mathfrak p)$,\n\\item $E$ is the injective hull of $\\kappa(\\mathfrak p)$\nover the ring $R_\\mathfrak p$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hulls","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Y8","source_file":"dualizing.tex","source_line":596,"source_end_line":606,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L596-L606","statement_sha256":"3e340d840c41438f7cecae1dc2c620ae42f39e5a4c6279a6489f841a7101d539","origin":"The Stacks Project","memory_eligible":false,"source_rank":8646,"rank":8646,"depth":3,"x":2144.329,"y":1138.842,"cluster":"duality-cohomology"},{"id":"stacks:08Y9","tag":"08Y9","title":"Injective hulls · Lemma 08Y9","summary":"Let R be a Noetherian ring. Let E be an indecomposable injective R-module. Then there exists a prime ideal p of R such that E is the injective hull of kappa( p).","statement_latex":"Let $R$ be a Noetherian ring. Let $E$ be an indecomposable injective\n$R$-module. Then there exists a prime ideal $\\mathfrak p$ of $R$ such that\n$E$ is the injective hull of $\\kappa(\\mathfrak p)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hulls","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Y9","source_file":"dualizing.tex","source_line":628,"source_end_line":633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L628-L633","statement_sha256":"0958db145c6584224f51d4da753d4360df3d4006db64e9452b1370a14d1b2edb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8647,"rank":8647,"depth":4,"x":1946.45,"y":1205.866,"cluster":"duality-cohomology"},{"id":"stacks:08YA","tag":"08YA","title":"Structure of injective modules over Noetherian rings · Proposition 08YA","summary":"Let R be a Noetherian ring. Every injective module is a direct sum of indecomposable injective modules. Every indecomposable injective module is the injective hull of the residue field at a prime.","statement_latex":"Let $R$ be a Noetherian ring.\nEvery injective module is a direct sum of indecomposable injective modules.\nEvery indecomposable injective module is the injective hull of\nthe residue field at a prime.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hulls","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YA","source_file":"dualizing.tex","source_line":645,"source_end_line":651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L645-L651","statement_sha256":"c18da79022881f6835e757475bd2136720f6c98c4a9180754df9e9a645429d02","origin":"The Stacks Project","memory_eligible":false,"source_rank":8648,"rank":8648,"depth":9,"x":2038.659,"y":1043.821,"cluster":"duality-cohomology"},{"id":"stacks:08YX","tag":"08YX","title":"Duality over Artinian local rings · Lemma 08YX","summary":"Let (R, m, kappa) be an artinian local ring. Let E be an injective hull of kappa. For every finite R-module M we have length_R(M) = length_R(Hom_R(M, E)) In particular, the injective hull E of kappa is a finite R-module.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be an artinian local ring.\nLet $E$ be an injective hull of $\\kappa$. For every finite\n$R$-module $M$ we have\n$$\n\\text{length}_R(M) = \\text{length}_R(\\Hom_R(M, E))\n$$\nIn particular, the injective hull $E$ of $\\kappa$ is a finite $R$-module.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Duality over Artinian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YX","source_file":"dualizing.tex","source_line":694,"source_end_line":703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L694-L703","statement_sha256":"9173b1c88077db6efcbbf69797f781ac1e77b66de311bfc66d7d011fd1ac2630","origin":"The Stacks Project","memory_eligible":false,"source_rank":8649,"rank":8649,"depth":0,"x":2101.109,"y":1215.993,"cluster":"duality-cohomology"},{"id":"stacks:08YY","tag":"08YY","title":"Duality over Artinian local rings · Lemma 08YY","summary":"Let (R, m, kappa) be an artinian local ring. Let E be an injective hull of kappa. For any finite R-module M the evaluation map M → Hom_R(Hom_R(M, E), E) is an isomorphism. In particular R = Hom_R(E, E).","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be an artinian local ring.\nLet $E$ be an injective hull of $\\kappa$.\nFor any finite $R$-module $M$ the evaluation map\n$$\nM \\longrightarrow \\Hom_R(\\Hom_R(M, E), E)\n$$\nis an isomorphism. In particular $R = \\Hom_R(E, E)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Duality over Artinian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YY","source_file":"dualizing.tex","source_line":724,"source_end_line":733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L724-L733","statement_sha256":"cbdca6a886c3cd3e0a0378d997598107ced101536fc21ed378a1dab8dfa59f12","origin":"The Stacks Project","memory_eligible":false,"source_rank":8650,"rank":8650,"depth":1,"x":1916.216,"y":1124.28,"cluster":"duality-cohomology"},{"id":"stacks:08YZ","tag":"08YZ","title":"Duality over Artinian local rings · Lemma 08YZ","summary":"Let (R, m, kappa) be an artinian local ring. Let E be an injective hull of kappa. The functor D(-) = Hom_R(-, E) induces an exact anti-equivalence Mod^fg_R → Mod^fg_R and D ∘ D ≅ id.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be an artinian local ring.\nLet $E$ be an injective hull of $\\kappa$.\nThe functor $D(-) = \\Hom_R(-, E)$ induces an exact anti-equivalence\n$\\text{Mod}^{fg}_R \\to \\text{Mod}^{fg}_R$ and\n$D \\circ D \\cong \\text{id}$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Duality over Artinian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08YZ","source_file":"dualizing.tex","source_line":749,"source_end_line":756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L749-L756","statement_sha256":"14ff7d74bc64e583ed39f55f0daf83d0c01399fb5f69be6e2a10daf21b15f6a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8651,"rank":8651,"depth":2,"x":2126.745,"y":1086.918,"cluster":"duality-cohomology"},{"id":"stacks:08Z0","tag":"08Z0","title":"Duality over Artinian local rings · Lemma 08Z0","summary":"Assumptions and notation as in Lemma [Tag 08YZ]. Let I ⊂ R be an ideal and M a finite R-module. Then D(M[I]) = D(M)/ID(M) and D(M/IM) = D(M)[I]","statement_latex":"Assumptions and notation as in Lemma \\ref{lemma-duality}.\nLet $I \\subset R$ be an ideal and $M$ a finite $R$-module.\nThen\n$$\nD(M[I]) = D(M)/ID(M) \\quad\\text{and}\\quad D(M/IM) = D(M)[I]\n$$","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Duality over Artinian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Z0","source_file":"dualizing.tex","source_line":764,"source_end_line":772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L764-L772","statement_sha256":"1657c5d857d0d667194b40c2da221e8eccb31fa184f3c0e88165b7b626c34306","origin":"The Stacks Project","memory_eligible":false,"source_rank":8652,"rank":8652,"depth":3,"x":2001.289,"y":1234.232,"cluster":"duality-cohomology"},{"id":"stacks:08Z2","tag":"08Z2","title":"Injective hull of the residue field · Lemma 08Z2","summary":"Let R → S be a surjective map of local rings with kernel I. Let E be the injective hull of the residue field of R over R. Then E[I] is the injective hull of the residue field of S over S.","statement_latex":"Let $R \\to S$ be a surjective map of local rings with kernel $I$.\nLet $E$ be the injective hull of the residue field of $R$ over $R$.\nThen $E[I]$ is the injective hull of the residue field of $S$ over $S$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hull of the residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Z2","source_file":"dualizing.tex","source_line":791,"source_end_line":796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L791-L796","statement_sha256":"139f75f1d6d7276407aefb71ad3868785b7a7aa87e1455a2f8ab02361dc71525","origin":"The Stacks Project","memory_eligible":false,"source_rank":8653,"rank":8653,"depth":2,"x":1975.279,"y":1054.046,"cluster":"duality-cohomology"},{"id":"stacks:08Z3","tag":"08Z3","title":"Injective hull of the residue field · Lemma 08Z3","summary":"Let (R, m, kappa) be a local ring. Let E be the injective hull of kappa. Let M be a m-power torsion R-module with n = dim_kappa(M[ m]) < ∞. Then M is isomorphic to a submodule of E^⊕ n.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring.\nLet $E$ be the injective hull of $\\kappa$.\nLet $M$ be a $\\mathfrak m$-power torsion $R$-module\nwith $n = \\dim_\\kappa(M[\\mathfrak m]) < \\infty$.\nThen $M$ is isomorphic to a submodule of $E^{\\oplus n}$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hull of the residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Z3","source_file":"dualizing.tex","source_line":805,"source_end_line":812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L805-L812","statement_sha256":"3b0a907898f21e66fffff3330c20048360a70a92404191367edbaa1f79f6e8ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":8654,"rank":8654,"depth":1,"x":2139.697,"y":1172.397,"cluster":"duality-cohomology"},{"id":"stacks:08Z4","tag":"08Z4","title":"Injective hull of the residue field · Lemma 08Z4","summary":"Let (R, m, kappa) be a Noetherian local ring. Let E be an injective hull of kappa over R. Let E_n be an injective hull of kappa over R/ m^n. Then E = ⋃ E_n and E_n = E[ m^n].","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $E$ be an injective hull of $\\kappa$ over $R$.\nLet $E_n$ be an injective hull of $\\kappa$ over $R/\\mathfrak m^n$.\nThen $E = \\bigcup E_n$ and $E_n = E[\\mathfrak m^n]$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hull of the residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Z4","source_file":"dualizing.tex","source_line":824,"source_end_line":830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L824-L830","statement_sha256":"cda470a09768fa83ad4ac3d1810a8bd7f4e8cc8a73de161537a4937a9479785d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8655,"rank":8655,"depth":3,"x":1922.839,"y":1178.434,"cluster":"duality-cohomology"},{"id":"stacks:08Z5","tag":"08Z5","title":"Injective hull of the residue field · Lemma 08Z5","summary":"Let R → S be a flat local homomorphism of local Noetherian rings such that R/ m_R ≅ S/ m_R S. Then the injective hull of the residue field of R is the injective hull of the residue field of S.","statement_latex":"Let $R \\to S$ be a flat local homomorphism of local Noetherian rings\nsuch that $R/\\mathfrak m_R \\cong S/\\mathfrak m_R S$.\nThen the injective hull of the residue field\nof $R$ is the injective hull of the residue field of $S$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hull of the residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Z5","source_file":"dualizing.tex","source_line":843,"source_end_line":849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L843-L849","statement_sha256":"82adeaa4e166a64862912ffb1d097cbf27bd8b4fa4794659c8d9df22128a286c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8656,"rank":8656,"depth":4,"x":2078.209,"y":1050.672,"cluster":"duality-cohomology"},{"id":"stacks:08Z6","tag":"08Z6","title":"Injective hull of the residue field · Lemma 08Z6","summary":"Let (R, m, kappa) be a Noetherian local ring. Let E be an injective hull of kappa over R. Then Hom_R(E, E) is canonically isomorphic to the completion of R.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $E$ be an injective hull of $\\kappa$ over $R$. Then\n$\\Hom_R(E, E)$ is canonically isomorphic to the completion of $R$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hull of the residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Z6","source_file":"dualizing.tex","source_line":865,"source_end_line":870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L865-L870","statement_sha256":"5a8b80aced4cf232f24de5d90fd826d2f7962ce2599efc80a55da31a4bc225a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8657,"rank":8657,"depth":4,"x":2066.36,"y":1233.413,"cluster":"duality-cohomology"},{"id":"stacks:08Z7","tag":"08Z7","title":"Injective hull of the residue field · Lemma 08Z7","summary":"Let (R, m, kappa) be a Noetherian local ring. Let E be an injective hull of kappa over R. Then E satisfies the descending chain condition.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $E$ be an injective hull of $\\kappa$ over $R$. Then\n$E$ satisfies the descending chain condition.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hull of the residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Z7","source_file":"dualizing.tex","source_line":884,"source_end_line":889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L884-L889","statement_sha256":"edabb9d218a73974c2b9c8db548100afd8975652bfe0da6ddd2424c34b2fe029","origin":"The Stacks Project","memory_eligible":false,"source_rank":8658,"rank":8658,"depth":7,"x":1927.862,"y":1091.652,"cluster":"duality-cohomology"},{"id":"stacks:08Z8","tag":"08Z8","title":"Injective hull of the residue field · Lemma 08Z8","summary":"Let (R, m, kappa) be a Noetherian local ring. Let E be an injective hull of kappa. • For an R-module M the following are equivalent: • M satisfies the ascending chain condition, • M is a finite R-module, and • there exist n, m and an exact sequence R^⊕ m → R^⊕ n → M → 0. • For an R-module M the following are equivalent: • M satisfies the descending chain condition, • M is m-power torsion and dim_kappa(M[ m]) < ∞, and • there exist n, m and an exact sequence 0 → M → E^⊕ n…","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $E$ be an injective hull of $\\kappa$.\n\\begin{enumerate}\n\\item For an $R$-module $M$ the following are equivalent:\n\\begin{enumerate}\n\\item $M$ satisfies the ascending chain condition,\n\\item $M$ is a finite $R$-module, and\n\\item there exist $n, m$ and an exact sequence\n$R^{\\oplus m} \\to R^{\\oplus n} \\to M \\to 0$.\n\\end{enumerate}\n\\item For an $R$-module $M$ the following are equivalent:\n\\begin{enumerate}\n\\item $M$ satisfies the descending chain condition,\n\\item $M$ is $\\mathfrak m$-power torsion and\n$\\dim_\\kappa(M[\\mathfrak m]) < \\infty$, and\n\\item there exist $n, m$ and an exact sequence\n$0 \\to M \\to E^{\\oplus n} \\to E^{\\oplus m}$.\n\\end{enumerate}\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hull of the residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Z8","source_file":"dualizing.tex","source_line":909,"source_end_line":930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L909-L930","statement_sha256":"d9bb2bc2b5239860f25de59d5d78ff2590ed48847349961e9ce332c78874a034","origin":"The Stacks Project","memory_eligible":false,"source_rank":8659,"rank":8659,"depth":8,"x":2144.424,"y":1117.651,"cluster":"duality-cohomology"},{"id":"stacks:08Z9","tag":"08Z9","title":"Matlis duality · Proposition 08Z9","summary":"Let (R, m, kappa) be a complete local Noetherian ring. Let E be an injective hull of kappa over R. The functor D(-) = Hom_R(-, E) induces an anti-equivalence ( R-modules with the descending chain condition ) longleftrightarrow ( R-modules with the ascending chain condition ) and we have D ∘ D = id on either side of the equivalence.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a complete local Noetherian ring.\nLet $E$ be an injective hull of $\\kappa$ over $R$. The functor\n$D(-) = \\Hom_R(-, E)$ induces an anti-equivalence\n$$\n\\left\\{\n\\begin{matrix}\nR\\text{-modules with the} \\\\\n\\text{descending chain condition}\n\\end{matrix}\n\\right\\}\n\\longleftrightarrow\n\\left\\{\n\\begin{matrix}\nR\\text{-modules with the} \\\\\n\\text{ascending chain condition}\n\\end{matrix}\n\\right\\}\n$$\nand we have $D \\circ D = \\text{id}$ on either side of the equivalence.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Injective hull of the residue field","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08Z9","source_file":"dualizing.tex","source_line":967,"source_end_line":988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L967-L988","statement_sha256":"cfc754f09b2e877e27a6306ea2182ddb499455e31030d75eaf35e2af200342bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8660,"rank":8660,"depth":9,"x":1963.465,"y":1221.57,"cluster":"duality-cohomology"},{"id":"stacks:0A6L","tag":"0A6L","title":"Deriving torsion · Lemma 0A6L","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. The functor RΓ_I is right adjoint to the functor D(I^∞-torsion) → D(A).","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nThe functor $R\\Gamma_I$ is right adjoint to the functor\n$D(I^\\infty\\text{-torsion}) \\to D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Deriving torsion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6L","source_file":"dualizing.tex","source_line":1076,"source_end_line":1081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1076-L1081","statement_sha256":"3a86fd8756c44a5c8a3531dcafcd19afa20075f5b4a3c23e8bd11a0748f13a4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8661,"rank":8661,"depth":5,"x":2013.432,"y":1041.902,"cluster":"duality-cohomology"},{"id":"stacks:0954","tag":"0954","title":"Deriving torsion · Lemma 0954","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. For any object K of D(A) we have RΓ_I(K) = hocolim RHom_A(A/I^n, K) in D(A) and R^qΓ_I(K) = colim_n Ext_A^q(A/I^n, K) as modules for all q ∈ Z.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nFor any object $K$ of $D(A)$ we have\n$$\nR\\Gamma_I(K) = \\text{hocolim}\\ R\\Hom_A(A/I^n, K)\n$$\nin $D(A)$ and\n$$\nR^q\\Gamma_I(K) = \\colim_n \\Ext_A^q(A/I^n, K)\n$$\nas modules for all $q \\in \\mathbf{Z}$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Deriving torsion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0954","source_file":"dualizing.tex","source_line":1090,"source_end_line":1102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1090-L1102","statement_sha256":"660758870c1da4344db16967d9cf0d2402e2eb37e87feb6c18a34752a7a67fa4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8662,"rank":8662,"depth":3,"x":2121.286,"y":1203.06,"cluster":"duality-cohomology"},{"id":"stacks:0A6M","tag":"0A6M","title":"Deriving torsion · Lemma 0A6M","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. Let K^bullet be a complex of A-modules such that f : K^bullet → K^bullet is an isomorphism for some f ∈ I, i.e., K^bullet is a complex of A_f-modules. Then RΓ_I(K^bullet) = 0.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nLet $K^\\bullet$ be a complex of $A$-modules such that\n$f : K^\\bullet \\to K^\\bullet$ is an isomorphism for some\n$f \\in I$, i.e., $K^\\bullet$ is a complex of $A_f$-modules. Then\n$R\\Gamma_I(K^\\bullet) = 0$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Deriving torsion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6M","source_file":"dualizing.tex","source_line":1119,"source_end_line":1126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1119-L1126","statement_sha256":"70afccba71ae0015537572d4960240370158679686578133bc04bc7732559ec2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8663,"rank":8663,"depth":0,"x":1911.742,"y":1145.309,"cluster":"duality-cohomology"},{"id":"stacks:0A6P","tag":"0A6P","title":"Deriving torsion · Lemma 0A6P","summary":"Let A be a ring and let I be a finitely generated ideal. Let M and N be I-power torsion modules. • Hom_D(A)(M, N) = Hom_D(I^∞-torsion)(M, N), • Ext^1_D(A)(M, N) = Ext^1_D(I^∞-torsion)(M, N), • Ext^2_D(I^∞-torsion)(M, N) → Ext^2_D(A)(M, N) is not surjective in general, • ([Tag 0A6N]) is not an equivalence in general.","statement_latex":"Let $A$ be a ring and let $I$ be a finitely generated ideal.\nLet $M$ and $N$ be $I$-power torsion modules.\n\\begin{enumerate}\n\\item $\\Hom_{D(A)}(M, N) = \\Hom_{D({I^\\infty\\text{-torsion}})}(M, N)$,\n\\item $\\Ext^1_{D(A)}(M, N) =\n\\Ext^1_{D({I^\\infty\\text{-torsion}})}(M, N)$,\n\\item $\\Ext^2_{D({I^\\infty\\text{-torsion}})}(M, N) \\to\n\\Ext^2_{D(A)}(M, N)$ is not surjective in general,\n\\item (\\ref{equation-compare-torsion}) is not an equivalence in general.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Deriving torsion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6P","source_file":"dualizing.tex","source_line":1148,"source_end_line":1160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1148-L1160","statement_sha256":"ef534a2e97bb3a1205f49b039bd5c24b0bb01cdcb4a7522509a3eebea76c2701","origin":"The Stacks Project","memory_eligible":false,"source_rank":8664,"rank":8664,"depth":0,"x":2113.096,"y":1068.843,"cluster":"duality-cohomology"},{"id":"stacks:0A6R","tag":"0A6R","title":"Local cohomology · Lemma 0A6R","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. There exists a right adjoint RΓ_Z ([Tag 0A6Q]) to the inclusion functor D_I^∞-torsion(A) → D(A). In fact, if I is generated by f_1, …, f_r ∈ A, then we have RΓ_Z(K) = (A → ∏_i_0 A_f_i_0 → ∏_i_0 < i_1 A_f_i_0f_i_1 → … → A_f_1… f_r) ⊗_A^L K functorially in K ∈ D(A).","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nThere exists a right adjoint $R\\Gamma_Z$ (\\ref{equation-local-cohomology})\nto the inclusion functor $D_{I^\\infty\\text{-torsion}}(A) \\to D(A)$.\nIn fact, if $I$ is generated by $f_1, \\ldots, f_r \\in A$, then we have\n$$\nR\\Gamma_Z(K) =\n(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to\n\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}}\n\\to \\ldots \\to A_{f_1\\ldots f_r}) \\otimes_A^\\mathbf{L} K\n$$\nfunctorially in $K \\in D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6R","source_file":"dualizing.tex","source_line":1248,"source_end_line":1261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1248-L1261","statement_sha256":"e3781f351bf0765b3e05812b0648c558b64226421dd51e0038ce668e47b7ac8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8665,"rank":8665,"depth":2,"x":2025.941,"y":1239.816,"cluster":"duality-cohomology"},{"id":"stacks:0BJB","tag":"0BJB","title":"Local cohomology · Lemma 0BJB","summary":"Let A → B be a ring homomorphism and let I ⊂ A be a finitely generated ideal. Set J = IB. Set Z = V(I) and Y = V(J). Then RΓ_Z(M_A) = RΓ_Y(M)_A functorially in M ∈ D(B). Here (-)_A denotes the restriction functors D(B) → D(A) and D_J^∞-torsion(B) → D_I^∞-torsion(A).","statement_latex":"Let $A \\to B$ be a ring homomorphism and let $I \\subset A$\nbe a finitely generated ideal. Set $J = IB$. Set $Z = V(I)$\nand $Y = V(J)$. Then\n$$\nR\\Gamma_Z(M_A) = R\\Gamma_Y(M)_A\n$$\nfunctorially in $M \\in D(B)$. Here $(-)_A$ denotes the restriction\nfunctors $D(B) \\to D(A)$ and\n$D_{J^\\infty\\text{-torsion}}(B) \\to D_{I^\\infty\\text{-torsion}}(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJB","source_file":"dualizing.tex","source_line":1324,"source_end_line":1335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1324-L1335","statement_sha256":"e16fbb05edc69b195b9ca3dd981d5c41c09f4ac431ae05cc7d21e010b8dfef32","origin":"The Stacks Project","memory_eligible":false,"source_rank":8666,"rank":8666,"depth":3,"x":1952.573,"y":1063.945,"cluster":"duality-cohomology"},{"id":"stacks:0ALZ","tag":"0ALZ","title":"Local cohomology · Lemma 0ALZ","summary":"Let A → B be a ring homomorphism and let I ⊂ A be a finitely generated ideal. Set J = IB. Let Z = V(I) and Y = V(J). Then RΓ_Z(K) ⊗_A^L B = RΓ_Y(K ⊗_A^L B) functorially in K ∈ D(A).","statement_latex":"Let $A \\to B$ be a ring homomorphism and let $I \\subset A$\nbe a finitely generated ideal. Set $J = IB$. Let $Z = V(I)$ and $Y = V(J)$.\nThen\n$$\nR\\Gamma_Z(K) \\otimes_A^\\mathbf{L} B = R\\Gamma_Y(K \\otimes_A^\\mathbf{L} B)\n$$\nfunctorially in $K \\in D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALZ","source_file":"dualizing.tex","source_line":1361,"source_end_line":1370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1361-L1370","statement_sha256":"50fbc8a7d1c2650251fb4dd62418522a3ef5effa4b9c2d0cdee9d5fee6a1ce28","origin":"The Stacks Project","memory_eligible":false,"source_rank":8667,"rank":8667,"depth":3,"x":2148.485,"y":1152.171,"cluster":"duality-cohomology"},{"id":"stacks:0A6S","tag":"0A6S","title":"Local cohomology · Lemma 0A6S","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. Let K^bullet be a complex of A-modules such that f : K^bullet → K^bullet is an isomorphism for some f ∈ I, i.e., K^bullet is a complex of A_f-modules. Then RΓ_Z(K^bullet) = 0.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nLet $K^\\bullet$ be a complex of $A$-modules such that\n$f : K^\\bullet \\to K^\\bullet$ is an isomorphism for some\n$f \\in I$, i.e., $K^\\bullet$ is a complex of $A_f$-modules. Then\n$R\\Gamma_Z(K^\\bullet) = 0$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6S","source_file":"dualizing.tex","source_line":1394,"source_end_line":1401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1394-L1401","statement_sha256":"f12951de352d5552e5afb1b3afb2127c827996502495abee58c2cc59d492a9e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8668,"rank":8668,"depth":0,"x":1932.654,"y":1198.37,"cluster":"duality-cohomology"},{"id":"stacks:0ALY","tag":"0ALY","title":"Local cohomology · Lemma 0ALY","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. For K, L ∈ D(A) we have RΓ_Z(K ⊗_A^L L) = K ⊗_A^L RΓ_Z(L) = RΓ_Z(K) ⊗_A^L L = RΓ_Z(K) ⊗_A^L RΓ_Z(L) If K or L is in D_I^∞-torsion(A) then so is K ⊗_A^L L.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nFor $K, L \\in D(A)$ we have\n$$\nR\\Gamma_Z(K \\otimes_A^\\mathbf{L} L) =\nK \\otimes_A^\\mathbf{L} R\\Gamma_Z(L) =\nR\\Gamma_Z(K) \\otimes_A^\\mathbf{L} L =\nR\\Gamma_Z(K) \\otimes_A^\\mathbf{L} R\\Gamma_Z(L)\n$$\nIf $K$ or $L$ is in $D_{I^\\infty\\text{-torsion}}(A)$ then so is\n$K \\otimes_A^\\mathbf{L} L$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALY","source_file":"dualizing.tex","source_line":1409,"source_end_line":1421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1409-L1421","statement_sha256":"912e63c475028e79a6169c9667f9341a8bac66c3e3871d975d994428878dcea9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8669,"rank":8669,"depth":3,"x":2054.89,"y":1041.533,"cluster":"duality-cohomology"},{"id":"stacks:0BJC","tag":"0BJC","title":"Local cohomology · Lemma 0BJC","summary":"Let A be a ring and let I, J ⊂ A be finitely generated ideals. Set Z = V(I) and Y = V(J). Then Z ∩ Y = V(I + J) and RΓ_Y ∘ RΓ_Z = RΓ_Y ∩ Z as functors D(A) → D_(I + J)^∞-torsion(A). For K ∈ D^+(A) there is a spectral sequence E_2^p, q = H^p_Y(H^q_Z(K)) ⇒ H^p + q_Y ∩ Z(K) as in Derived Categories, Lemma [Tag 015N].","statement_latex":"Let $A$ be a ring and let $I, J \\subset A$ be finitely generated\nideals. Set $Z = V(I)$ and $Y = V(J)$. Then $Z \\cap Y = V(I + J)$\nand $R\\Gamma_Y \\circ R\\Gamma_Z = R\\Gamma_{Y \\cap Z}$ as functors\n$D(A) \\to D_{(I + J)^\\infty\\text{-torsion}}(A)$. For $K \\in D^+(A)$\nthere is a spectral sequence\n$$\nE_2^{p, q} = H^p_Y(H^q_Z(K)) \\Rightarrow H^{p + q}_{Y \\cap Z}(K)\n$$\nas in Derived Categories, Lemma\n\\ref{derived-lemma-grothendieck-spectral-sequence}.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJC","source_file":"dualizing.tex","source_line":1436,"source_end_line":1448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1436-L1448","statement_sha256":"6ff62ca7d1d9071379270106c69d88c09390268823d1818899c89d422653826b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8670,"rank":8670,"depth":13,"x":2090.948,"y":1226.898,"cluster":"duality-cohomology"},{"id":"stacks:0AM0","tag":"0AM0","title":"Local cohomology · Lemma 0AM0","summary":"Let A → B be a flat ring map and let I ⊂ A be a finitely generated ideal such that A/I = B/IB. Then base change and restriction induce quasi-inverse equivalences D_I^∞-torsion(A) = D_(IB)^∞-torsion(B).","statement_latex":"Let $A \\to B$ be a flat ring map and let $I \\subset A$ be a finitely\ngenerated ideal such that $A/I = B/IB$. Then base change and\nrestriction induce quasi-inverse equivalences\n$D_{I^\\infty\\text{-torsion}}(A) = D_{(IB)^\\infty\\text{-torsion}}(B)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AM0","source_file":"dualizing.tex","source_line":1479,"source_end_line":1485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1479-L1485","statement_sha256":"bb24c95dc3964097174baeb83ce29c1e5e0484660edd468544e1a0539fedd3fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8671,"rank":8671,"depth":5,"x":1914.956,"y":1110.449,"cluster":"duality-cohomology"},{"id":"stacks:05EH","tag":"05EH","title":"Local cohomology · Lemma 05EH","summary":"Let A → B be a flat ring map and let I ⊂ A be a finitely generated ideal such that A/I → B/IB is an isomorphism. For K ∈ D_I^∞-torsion(A) and L ∈ D(A) the map RHom_A(K, L) → RHom_B(K ⊗_A B, L ⊗_A B) is a quasi-isomorphism. In particular, if M, N are A-modules and M is I-power torsion, then the canonical map Ext^i_A(M, N) → Ext^i_B(M ⊗_A B, N ⊗_A B) is an isomorphism for all i.","statement_latex":"Let $A \\to B$ be a flat ring map and let $I \\subset A$ be a\nfinitely generated ideal such that $A/I \\to B/IB$ is an isomorphism.\nFor $K \\in D_{I^\\infty\\text{-torsion}}(A)$ and $L \\in D(A)$\nthe map\n$$\nR\\Hom_A(K, L) \\longrightarrow R\\Hom_B(K \\otimes_A B, L \\otimes_A B)\n$$\nis a quasi-isomorphism. In particular, if $M$, $N$ are $A$-modules and\n$M$ is $I$-power torsion, then the canonical map\n$$\n\\Ext^i_A(M, N)\n\\longrightarrow\n\\Ext^i_B(M \\otimes_A B, N \\otimes_A B)\n$$\nis an isomorphism for all $i$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05EH","source_file":"dualizing.tex","source_line":1501,"source_end_line":1518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1501-L1518","statement_sha256":"577a3e3818dd0932390906902ffec8f5e05ce2b79590a89d89b4c5bf72342916","origin":"The Stacks Project","memory_eligible":false,"source_rank":8672,"rank":8672,"depth":6,"x":2138.803,"y":1096.428,"cluster":"duality-cohomology"},{"id":"stacks:0955","tag":"0955","title":"Local cohomology for Noetherian rings · Lemma 0955","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. • the adjunction RΓ_I(K) → K is an isomorphism for K ∈ D_I^∞-torsion(A), • the functor ([Tag 0A6N]) D(I^∞-torsion) → D_I^∞-torsion(A) is an equivalence, • the transformation of functors ([Tag 0A6U]) is an isomorphism, in other words RΓ_I(K) = RΓ_Z(K) for K ∈ D(A).","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\n\\begin{enumerate}\n\\item the adjunction $R\\Gamma_I(K) \\to K$ is an isomorphism\nfor $K \\in D_{I^\\infty\\text{-torsion}}(A)$,\n\\item the functor\n(\\ref{equation-compare-torsion})\n$D(I^\\infty\\text{-torsion}) \\to D_{I^\\infty\\text{-torsion}}(A)$\nis an equivalence,\n\\item the transformation of functors\n(\\ref{equation-compare-torsion-functors}) is an isomorphism,\nin other words $R\\Gamma_I(K) = R\\Gamma_Z(K)$ for $K \\in D(A)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0955","source_file":"dualizing.tex","source_line":1571,"source_end_line":1585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1571-L1585","statement_sha256":"8645a219057eb03a4ed1dce9069bc59e56b974333dce87d28d1779b80ee9232c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8673,"rank":8673,"depth":8,"x":1984.723,"y":1234.045,"cluster":"duality-cohomology"},{"id":"stacks:0956","tag":"0956","title":"Local cohomology for Noetherian rings · Lemma 0956","summary":"Let A be a Noetherian ring and let I = (f_1, …, f_r) be an ideal of A. Set Z = V(I) ⊂ Spec(A). There are canonical isomorphisms RΓ_I(A) → (A → ∏_i_0 A_f_i_0 → ∏_i_0 < i_1 A_f_i_0f_i_1 → … → A_f_1… f_r) → RΓ_Z(A) in D(A). If M is an A-module, then we have similarly RΓ_I(M) ≅ (M → ∏_i_0 M_f_i_0 → ∏_i_0 < i_1 M_f_i_0f_i_1 → … → M_f_1… f_r) ≅ RΓ_Z(M) in D(A).","statement_latex":"Let $A$ be a Noetherian ring and let $I = (f_1, \\ldots, f_r)$ be an ideal\nof $A$. Set $Z = V(I) \\subset \\Spec(A)$. There are canonical isomorphisms\n$$\nR\\Gamma_I(A) \\to\n(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to\n\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to\n\\ldots \\to A_{f_1\\ldots f_r}) \\to R\\Gamma_Z(A)\n$$\nin $D(A)$. If $M$ is an $A$-module, then we have similarly\n$$\nR\\Gamma_I(M) \\cong\n(M \\to \\prod\\nolimits_{i_0} M_{f_{i_0}} \\to\n\\prod\\nolimits_{i_0 < i_1} M_{f_{i_0}f_{i_1}} \\to\n\\ldots \\to M_{f_1\\ldots f_r}) \\cong R\\Gamma_Z(M)\n$$\nin $D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0956","source_file":"dualizing.tex","source_line":1648,"source_end_line":1666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1648-L1666","statement_sha256":"8a4257308442a2e10c0e1bd9ee781cfdf43f1961eb6b976ae70180ad16ab4f39","origin":"The Stacks Project","memory_eligible":false,"source_rank":8674,"rank":8674,"depth":9,"x":1987.678,"y":1044.781,"cluster":"duality-cohomology"},{"id":"stacks:0957","tag":"0957","title":"Local cohomology for Noetherian rings · Lemma 0957","summary":"If A → B is a homomorphism of Noetherian rings and I ⊂ A is an ideal, then in D(B) we have RΓ_I(A) ⊗_A^L B = RΓ_Z(A) ⊗_A^L B = RΓ_Y(B) = RΓ_IB(B) where Y = V(IB) ⊂ Spec(B).","statement_latex":"If $A \\to B$ is a homomorphism of Noetherian rings and $I \\subset A$\nis an ideal, then in $D(B)$ we have\n$$\nR\\Gamma_I(A) \\otimes_A^\\mathbf{L} B =\nR\\Gamma_Z(A) \\otimes_A^\\mathbf{L} B =\nR\\Gamma_Y(B) = R\\Gamma_{IB}(B)\n$$\nwhere $Y = V(IB) \\subset \\Spec(B)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Local cohomology for Noetherian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0957","source_file":"dualizing.tex","source_line":1674,"source_end_line":1684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1674-L1684","statement_sha256":"c7bbfca788690b310239e04b16aeb1ce230ec94fcad044b430b7109e01dd7461","origin":"The Stacks Project","memory_eligible":false,"source_rank":8675,"rank":8675,"depth":10,"x":2137.984,"y":1186.285,"cluster":"duality-cohomology"},{"id":"stacks:0AVZ","tag":"0AVZ","title":"Depth · Lemma 0AVZ","summary":"Let A be a Noetherian ring, let I ⊂ A be an ideal, and let M be a finite A-module such that IM not = M. Then the following integers are equal: • depth_I(M), • the smallest integer i such that Ext_A^i(A/I, M) is nonzero, and • the smallest integer i such that H^i_I(M) is nonzero. Moreover, we have Ext^i_A(N, M) = 0 for i < depth_I(M) for any finite A-module N annihilated by a power of I.","statement_latex":"Let $A$ be a Noetherian ring, let $I \\subset A$ be an ideal, and\nlet $M$ be a finite $A$-module such that $IM \\not = M$. Then\nthe following integers are equal:\n\\begin{enumerate}\n\\item $\\text{depth}_I(M)$,\n\\item the smallest integer $i$ such that $\\Ext_A^i(A/I, M)$\nis nonzero, and\n\\item the smallest integer $i$ such that $H^i_I(M)$ is nonzero.\n\\end{enumerate}\nMoreover, we have $\\Ext^i_A(N, M) = 0$ for $i < \\text{depth}_I(M)$\nfor any finite $A$-module $N$ annihilated by a power of $I$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AVZ","source_file":"dualizing.tex","source_line":1703,"source_end_line":1716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1703-L1716","statement_sha256":"a58902d1ed5ea1dce7811f0a676e49fcf2348a71b3e6cb47ec272bc00e055762","origin":"The Stacks Project","memory_eligible":false,"source_rank":8676,"rank":8676,"depth":14,"x":1912.933,"y":1167.193,"cluster":"duality-cohomology"},{"id":"stacks:0BUV","tag":"0BUV","title":"Depth · Lemma 0BUV","summary":"Let A be a Noetherian ring. Let 0 → N' → N → N\" → 0 be a short exact sequence of finite A-modules. Let I ⊂ A be an ideal. • depth_I(N) ≥ min(depth_I(N'), depth_I(N\")) • depth_I(N\") ≥ min(depth_I(N), depth_I(N') - 1) • depth_I(N') ≥ min(depth_I(N), depth_I(N\") + 1)","statement_latex":"Let $A$ be a Noetherian ring. Let $0 \\to N' \\to N \\to N'' \\to 0$\nbe a short exact sequence of finite $A$-modules.\nLet $I \\subset A$ be an ideal.\n\\begin{enumerate}\n\\item\n$\\text{depth}_I(N) \\geq \\min\\{\\text{depth}_I(N'), \\text{depth}_I(N'')\\}$\n\\item\n$\\text{depth}_I(N'') \\geq \\min\\{\\text{depth}_I(N), \\text{depth}_I(N') - 1\\}$\n\\item\n$\\text{depth}_I(N') \\geq \\min\\{\\text{depth}_I(N), \\text{depth}_I(N'') + 1\\}$\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUV","source_file":"dualizing.tex","source_line":1781,"source_end_line":1794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1781-L1794","statement_sha256":"0c3ea9658b8764781e0976dabf5bb6fc87f250a43c5bbf01460578d3fdc9753a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8677,"rank":8677,"depth":15,"x":2094.576,"y":1053.359,"cluster":"duality-cohomology"},{"id":"stacks:0BUW","tag":"0BUW","title":"Depth · Lemma 0BUW","summary":"Let A be a Noetherian ring, let I ⊂ A be an ideal, and let M a finite A-module with IM not = M. • If x ∈ I is a nonzerodivisor on M, then depth_I(M/xM) = depth_I(M) - 1. • Any M-regular sequence x_1, …, x_r in I can be extended to an M-regular sequence in I of length depth_I(M).","statement_latex":"Let $A$ be a Noetherian ring, let $I \\subset A$ be an ideal, and\nlet $M$ a finite $A$-module with $IM \\not = M$.\n\\begin{enumerate}\n\\item If $x \\in I$ is a nonzerodivisor on $M$, then\n$\\text{depth}_I(M/xM) = \\text{depth}_I(M) - 1$.\n\\item Any $M$-regular sequence $x_1, \\ldots, x_r$ in $I$ can be extended to an\n$M$-regular sequence in $I$ of length $\\text{depth}_I(M)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUW","source_file":"dualizing.tex","source_line":1821,"source_end_line":1831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1821-L1831","statement_sha256":"69a93244e68341e672c54411744a0f365c3c5f4c60653d896c8ef7f4e084414d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8678,"rank":8678,"depth":16,"x":2052.098,"y":1240.717,"cluster":"duality-cohomology"},{"id":"stacks:0BUX","tag":"0BUX","title":"Depth · Lemma 0BUX","summary":"Let R be a Noetherian local ring. If M is a finite Cohen-Macaulay R-module and I ⊂ R a nontrivial ideal. Then depth_I(M) = dim(Supp(M)) - dim(Supp(M/IM)).","statement_latex":"Let $R$ be a Noetherian local ring. If $M$ is a finite Cohen-Macaulay\n$R$-module and $I \\subset R$ a nontrivial ideal. Then\n$$\n\\text{depth}_I(M) = \\dim(\\text{Supp}(M)) - \\dim(\\text{Supp}(M/IM)).\n$$","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUX","source_file":"dualizing.tex","source_line":1843,"source_end_line":1850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1843-L1850","statement_sha256":"c4edc67fb1bf59f17b4deee7532d6e7ccd6c6d868f7fbbf18a004e422eb418a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8679,"rank":8679,"depth":17,"x":1932.531,"y":1078.157,"cluster":"duality-cohomology"},{"id":"stacks:0BUY","tag":"0BUY","title":"Depth · Lemma 0BUY","summary":"Let R → S be a flat local ring homomorphism of Noetherian local rings. Denote m ⊂ R the maximal ideal. Let I ⊂ S be an ideal. If S/ mS is Cohen-Macaulay, then depth_I(S) ≥ dim(S/ mS) - dim(S/ mS + I)","statement_latex":"Let $R \\to S$ be a flat local ring homomorphism of Noetherian local\nrings. Denote $\\mathfrak m \\subset R$ the maximal ideal.\nLet $I \\subset S$ be an ideal.\nIf $S/\\mathfrak mS$ is Cohen-Macaulay, then\n$$\n\\text{depth}_I(S) \\geq \\dim(S/\\mathfrak mS) - \\dim(S/\\mathfrak mS + I)\n$$","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUY","source_file":"dualizing.tex","source_line":1876,"source_end_line":1885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1876-L1885","statement_sha256":"d9a16bfab4e5476d8abfb9701e497ff60d6e1d802510b55b937049a79f83593d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8680,"rank":8680,"depth":18,"x":2151.83,"y":1130.278,"cluster":"duality-cohomology"},{"id":"stacks:0AW0","tag":"0AW0","title":"Depth · Lemma 0AW0","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. Let M be an A-module. Let Z = V(I). Then H^0_I(M) = H^0_Z(M). Let N be the common value and set M' = M/N. Then • H^0_I(M') = 0 and H^p_I(M) = H^p_I(M') and H^p_I(N) = 0 for all p > 0, • H^0_Z(M') = 0 and H^p_Z(M) = H^p_Z(M') and H^p_Z(N) = 0 for all p > 0.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nLet $M$ be an $A$-module. Let $Z = V(I)$.\nThen $H^0_I(M) = H^0_Z(M)$. Let $N$ be the common value and\nset $M' = M/N$. Then\n\\begin{enumerate}\n\\item $H^0_I(M') = 0$ and $H^p_I(M) = H^p_I(M')$ and $H^p_I(N) = 0$\nfor all $p > 0$,\n\\item $H^0_Z(M') = 0$ and $H^p_Z(M) = H^p_Z(M')$ and $H^p_Z(N) = 0$\nfor all $p > 0$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Depth","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AW0","source_file":"dualizing.tex","source_line":1894,"source_end_line":1906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1894-L1906","statement_sha256":"40b3bb80e97973c520c993ca8d2688925cf9b92e146b3c95d5c4ef58490d70a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8681,"rank":8681,"depth":6,"x":1947.831,"y":1216.439,"cluster":"duality-cohomology"},{"id":"stacks:0A6W","tag":"0A6W","title":"Torsion versus complete modules · Lemma 0A6W","summary":"Results of this nature are sometimes referred to as Greenlees-May duality. Let A be a ring and let I be a finitely generated ideal. Let RΓ_Z be as in Lemma [Tag 0A6R]. Let ^wedge denote derived completion as in More on Algebra, Lemma [Tag 091V]. For an object K in D(A) we have RΓ_Z(K^wedge) = RΓ_Z(K) and (RΓ_Z(K))^wedge = K^wedge in D(A).","statement_latex":"\\begin{slogan}\nResults of this nature are sometimes referred to as Greenlees-May duality.\n\\end{slogan}\nLet $A$ be a ring and let $I$ be a finitely generated ideal.\nLet $R\\Gamma_Z$ be as in Lemma \\ref{lemma-local-cohomology-adjoint}.\nLet ${\\ }^\\wedge$ denote derived completion as in\nMore on Algebra, Lemma \\ref{more-algebra-lemma-derived-completion}.\nFor an object $K$ in $D(A)$ we have\n$$\nR\\Gamma_Z(K^\\wedge) = R\\Gamma_Z(K)\n\\quad\\text{and}\\quad\n(R\\Gamma_Z(K))^\\wedge = K^\\wedge\n$$\nin $D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Torsion versus complete modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6W","source_file":"dualizing.tex","source_line":1952,"source_end_line":1968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L1952-L1968","statement_sha256":"b0088778c81957571ce3f10e007de3171b901dfcddabda70c34ac2a97efc2216","origin":"The Stacks Project","memory_eligible":false,"source_rank":8682,"rank":8682,"depth":6,"x":2029.118,"y":1036.821,"cluster":"duality-cohomology"},{"id":"stacks:0A6X","tag":"0A6X","title":"Torsion versus complete modules · Proposition 0A6X","summary":"This is a special case of [Porta-Liran-Yekutieli]. Let A be a ring and let I ⊂ A be a finitely generated ideal. The functors RΓ_Z and ^wedge define quasi-inverse equivalences of categories D_I^∞-torsion(A) ↔ D_comp(A, I)","statement_latex":"\\begin{reference}\nThis is a special case of \\cite[Theorem 1.1]{Porta-Liran-Yekutieli}.\n\\end{reference}\nLet $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nThe functors $R\\Gamma_Z$ and ${\\ }^\\wedge$\ndefine quasi-inverse equivalences of categories\n$$\nD_{I^\\infty\\text{-torsion}}(A) \\leftrightarrow D_{comp}(A, I)\n$$","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Torsion versus complete modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6X","source_file":"dualizing.tex","source_line":2018,"source_end_line":2029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2018-L2029","statement_sha256":"77204dd64f3429961f61c403c635b04298a06a97b99ef701f639f834b3fa8696","origin":"The Stacks Project","memory_eligible":false,"source_rank":8683,"rank":8683,"depth":7,"x":2113.777,"y":1215.721,"cluster":"duality-cohomology"},{"id":"stacks:0A6Y","tag":"0A6Y","title":"Torsion versus complete modules · Lemma 0A6Y","summary":"With notation as in Lemma [Tag 0A6W]. For objects K, L in D(A) there is a canonical isomorphism RHom_A(K^wedge, L^wedge) → RHom_A(RΓ_Z(K), RΓ_Z(L)) in D(A).","statement_latex":"With notation as in Lemma \\ref{lemma-complete-and-local}.\nFor objects $K, L$ in $D(A)$ there is a canonical isomorphism\n$$\nR\\Hom_A(K^\\wedge, L^\\wedge) \\longrightarrow R\\Hom_A(R\\Gamma_Z(K), R\\Gamma_Z(L))\n$$\nin $D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Torsion versus complete modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6Y","source_file":"dualizing.tex","source_line":2039,"source_end_line":2047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2039-L2047","statement_sha256":"8ffbd5b7298eb4f9d8a04415163906140e3136666f101623e962d01f752adaf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8684,"rank":8684,"depth":8,"x":1907.107,"y":1131.687,"cluster":"duality-cohomology"},{"id":"stacks:0EEW","tag":"0EEW","title":"Torsion versus complete modules · Lemma 0EEW","summary":"Let I and J be ideals in a Noetherian ring A. Let M be a finite A-module. Set Z =V(J). Consider the derived I-adic completion RΓ_Z(M)^wedge of local cohomology. Then • we have RΓ_Z(M)^wedge = Rlim RΓ_Z(M/I^nM), and • there are short exact sequences 0 → R^1lim H^i - 1_Z(M/I^nM) → H^i(RΓ_Z(M)^wedge) → lim H^i_Z(M/I^nM) → 0 In particular RΓ_Z(M)^wedge has vanishing cohomology in negative degrees.","statement_latex":"Let $I$ and $J$ be ideals in a Noetherian ring $A$. Let $M$ be a finite\n$A$-module. Set $Z =V(J)$. Consider the derived $I$-adic completion\n$R\\Gamma_Z(M)^\\wedge$ of local cohomology. Then\n\\begin{enumerate}\n\\item we have $R\\Gamma_Z(M)^\\wedge = R\\lim R\\Gamma_Z(M/I^nM)$, and\n\\item there are short exact sequences\n$$\n0 \\to R^1\\lim H^{i - 1}_Z(M/I^nM) \\to H^i(R\\Gamma_Z(M)^\\wedge) \\to\n\\lim H^i_Z(M/I^nM) \\to 0\n$$\n\\end{enumerate}\nIn particular $R\\Gamma_Z(M)^\\wedge$ has vanishing cohomology\nin negative degrees.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Torsion versus complete modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEW","source_file":"dualizing.tex","source_line":2109,"source_end_line":2124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2109-L2124","statement_sha256":"2eac6e4f247dc2a6294338814d01f106244cdfb8af2505b3be5d0874d509ff6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8685,"rank":8685,"depth":24,"x":2127.481,"y":1076.282,"cluster":"duality-cohomology"},{"id":"stacks:0EEX","tag":"0EEX","title":"Torsion versus complete modules · Lemma 0EEX","summary":"With notation and hypotheses as in Lemma [Tag 0EEW] assume A is I-adically complete. Then H^0(RΓ_Z(M)^wedge) = colim H^0_V(J')(M) where the filtered colimit is over J' ⊂ J such that V(J') ∩ V(I) = V(J) ∩ V(I).","statement_latex":"With notation and hypotheses as in Lemma \\ref{lemma-completion-local}\nassume $A$ is $I$-adically complete. Then\n$$\nH^0(R\\Gamma_Z(M)^\\wedge) = \\colim H^0_{V(J')}(M)\n$$\nwhere the filtered colimit is over $J' \\subset J$ such that\n$V(J') \\cap V(I) = V(J) \\cap V(I)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Torsion versus complete modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEX","source_file":"dualizing.tex","source_line":2152,"source_end_line":2161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2152-L2161","statement_sha256":"1f12a9470109199675b9f9b1533d637f94c27a5180d2c8dca2e5af230452d3c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8686,"rank":8686,"depth":25,"x":2009.322,"y":1242.483,"cluster":"duality-cohomology"},{"id":"stacks:0A70","tag":"0A70","title":"Trivial duality for a ring map · Lemma 0A70","summary":"Let A → B be a ring homomorphism. The functor RHom(B, -) constructed above is right adjoint to the restriction functor D(B) → D(A).","statement_latex":"Let $A \\to B$ be a ring homomorphism. The functor $R\\Hom(B, -)$\nconstructed above is right adjoint to the restriction functor\n$D(B) \\to D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Trivial duality for a ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A70","source_file":"dualizing.tex","source_line":2210,"source_end_line":2215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2210-L2215","statement_sha256":"d5412627ea5a84c7bfd6e21366dcb921b3ed1dceaec326f40a9d75255d77ad75","origin":"The Stacks Project","memory_eligible":false,"source_rank":8687,"rank":8687,"depth":5,"x":1962.713,"y":1052.54,"cluster":"duality-cohomology"},{"id":"stacks:0C0F","tag":"0C0F","title":"Trivial duality for a ring map · Lemma 0C0F","summary":"Let A → B → C be ring maps. Then RHom(C, -) ∘ RHom(B, -) : D(A) → D(C) is the functor RHom(C, -) : D(A) → D(C).","statement_latex":"Let $A \\to B \\to C$ be ring maps. Then\n$R\\Hom(C, -) \\circ R\\Hom(B, -) : D(A) \\to D(C)$\nis the functor $R\\Hom(C, -) : D(A) \\to D(C)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Trivial duality for a ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0F","source_file":"dualizing.tex","source_line":2223,"source_end_line":2228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2223-L2228","statement_sha256":"cfa928c8f26a6e307084736c3ebe85159fa5457cfcd8fddcb0ba2bf19982e76c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8688,"rank":8688,"depth":6,"x":2150.166,"y":1166.359,"cluster":"duality-cohomology"},{"id":"stacks:0A71","tag":"0A71","title":"Trivial duality for a ring map · Lemma 0A71","summary":"Let φ : A → B be a ring homomorphism. For K in D(A) we have φ_*RHom(B, K) = RHom_A(B, K) where φ_* : D(B) → D(A) is restriction. In particular R^qHom(B, K) = Ext_A^q(B, K).","statement_latex":"Let $\\varphi : A \\to B$ be a ring homomorphism. For $K$ in $D(A)$ we have\n$$\n\\varphi_*R\\Hom(B, K) = R\\Hom_A(B, K)\n$$\nwhere $\\varphi_* : D(B) \\to D(A)$ is restriction. In particular\n$R^q\\Hom(B, K) = \\Ext_A^q(B, K)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Trivial duality for a ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A71","source_file":"dualizing.tex","source_line":2234,"source_end_line":2242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2234-L2242","statement_sha256":"7b0cacab535d2997f48e08fe71d2adb2a7574dc31715e4ef661bcaf3d8cd3abd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8689,"rank":8689,"depth":0,"x":1919.998,"y":1188.836,"cluster":"duality-cohomology"},{"id":"stacks:0A72","tag":"0A72","title":"Trivial duality for a ring map · Lemma 0A72","summary":"With notation as above, assume A → B is a finite ring map of Noetherian rings. Then RHom(B, -) maps D^+_Coh(A) into D^+_Coh(B).","statement_latex":"With notation as above, assume $A \\to B$ is a finite ring map of\nNoetherian rings. Then $R\\Hom(B, -)$ maps\n$D^+_{\\textit{Coh}}(A)$ into $D^+_{\\textit{Coh}}(B)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Trivial duality for a ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A72","source_file":"dualizing.tex","source_line":2262,"source_end_line":2267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2262-L2267","statement_sha256":"4d2511b97a19ebe5e5d08ab713e19e8205e15f6c7f967b1064cb429350a532db","origin":"The Stacks Project","memory_eligible":false,"source_rank":8690,"rank":8690,"depth":2,"x":2071.916,"y":1041.396,"cluster":"duality-cohomology"},{"id":"stacks:0BZC","tag":"0BZC","title":"Trivial duality for a ring map · Lemma 0BZC","summary":"In Situation [Tag 0BZB] the functor RHom(A, -) is equal to the composition of RHom(E, -) : D(R) → D(E, d) and the equivalence - ⊗^L_E A : D(E, d) → D(A).","statement_latex":"In Situation \\ref{situation-resolution} the functor $R\\Hom(A, -)$\nis equal to the composition of\n$R\\Hom(E, -) : D(R) \\to D(E, \\text{d})$\nand the equivalence $- \\otimes^\\mathbf{L}_E A : D(E, \\text{d}) \\to D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Trivial duality for a ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZC","source_file":"dualizing.tex","source_line":2330,"source_end_line":2336,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2330-L2336","statement_sha256":"a757c627928d3c06153cd06c0cd25e1f46889af04e66b22b4a0449e30b202e85","origin":"The Stacks Project","memory_eligible":false,"source_rank":8691,"rank":8691,"depth":16,"x":2078.472,"y":1236.665,"cluster":"duality-cohomology"},{"id":"stacks:0BZD","tag":"0BZD","title":"Trivial duality for a ring map · Lemma 0BZD","summary":"In Situation [Tag 0BZB] assume that • E viewed as an object of D(R) is compact, and • N = Hom^bullet_R(E^bullet, R) computes RHom(E, R). Then RHom(E, -) : D(R) → D(E) is isomorphic to K ↦ K ⊗_R^L N.","statement_latex":"In Situation \\ref{situation-resolution} assume that\n\\begin{enumerate}\n\\item $E$ viewed as an object of $D(R)$ is compact, and\n\\item $N = \\Hom^\\bullet_R(E^\\bullet, R)$ computes $R\\Hom(E, R)$.\n\\end{enumerate}\nThen $R\\Hom(E, -) : D(R) \\to D(E)$ is isomorphic to\n$K \\mapsto K \\otimes_R^\\mathbf{L} N$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Trivial duality for a ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZD","source_file":"dualizing.tex","source_line":2348,"source_end_line":2357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2348-L2357","statement_sha256":"2ec589154a96d9f95e1ffad13e9d85f72a6381bd52efd61e5f5e16ff610b27d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8692,"rank":8692,"depth":2,"x":1916.321,"y":1096.147,"cluster":"duality-cohomology"},{"id":"stacks:0BZE","tag":"0BZE","title":"Trivial duality for a ring map · Lemma 0BZE","summary":"In Situation [Tag 0BZB] assume A is a perfect R-module. Then RHom(A, -) : D(R) → D(A) is given by K ↦ K ⊗_R^L M where M = RHom(A, R) ∈ D(A).","statement_latex":"In Situation \\ref{situation-resolution} assume $A$ is a perfect $R$-module.\nThen\n$$\nR\\Hom(A, -) : D(R) \\to D(A)\n$$\nis given by $K \\mapsto K \\otimes_R^\\mathbf{L} M$\nwhere $M = R\\Hom(A, R) \\in D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Trivial duality for a ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZE","source_file":"dualizing.tex","source_line":2364,"source_end_line":2373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2364-L2373","statement_sha256":"b2e37568c9dab75bcd5145daf86343eac89be2676ca84c0b36a74749db2d6b69","origin":"The Stacks Project","memory_eligible":false,"source_rank":8693,"rank":8693,"depth":17,"x":2149.301,"y":1107.775,"cluster":"duality-cohomology"},{"id":"stacks:0BZH","tag":"0BZH","title":"Trivial duality for a ring map · Lemma 0BZH","summary":"Let R → A be a surjective ring map whose kernel I is an invertible R-module. The functor RHom(A, -) : D(R) → D(A) is isomorphic to K ↦ K ⊗_R^L N[-1] where N is inverse of the invertible A-module I ⊗_R A.","statement_latex":"Let $R \\to A$ be a surjective ring map whose kernel $I$\nis an invertible $R$-module. The functor\n$R\\Hom(A, -) : D(R) \\to D(A)$\nis isomorphic to $K \\mapsto K \\otimes_R^\\mathbf{L} N[-1]$\nwhere $N$ is inverse of the invertible $A$-module $I \\otimes_R A$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Trivial duality for a ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZH","source_file":"dualizing.tex","source_line":2403,"source_end_line":2410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2403-L2410","statement_sha256":"b1eb9f1108168205981ab697aac3b249dde011c0465096f5c2f6d2f638308756","origin":"The Stacks Project","memory_eligible":false,"source_rank":8694,"rank":8694,"depth":18,"x":1967.835,"y":1231.617,"cluster":"duality-cohomology"},{"id":"stacks:0E2A","tag":"0E2A","title":"Base change for trivial duality · Lemma 0E2A","summary":"In the situation above, the map ([Tag 0E29]) is an isomorphism if and only if the map RHom_R(A, K) ⊗_R^L R' → RHom_R(A, K ⊗_R^L R') of More on Algebra, Lemma [Tag 0BYN] is an isomorphism.","statement_latex":"In the situation above, the map (\\ref{equation-base-change})\nis an isomorphism if and only if the map\n$$\nR\\Hom_R(A, K) \\otimes_R^\\mathbf{L} R'\n\\longrightarrow\nR\\Hom_R(A, K \\otimes_R^\\mathbf{L} R')\n$$\nof More on Algebra, Lemma\n\\ref{more-algebra-lemma-internal-hom-diagonal-better} is an isomorphism.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Base change for trivial duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2A","source_file":"dualizing.tex","source_line":2478,"source_end_line":2489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2478-L2489","statement_sha256":"7971c157474de2d78c6f5882a79a909b3dc78ba0dc627f4168de81db2b554cba","origin":"The Stacks Project","memory_eligible":false,"source_rank":8695,"rank":8695,"depth":20,"x":2002.115,"y":1036.988,"cluster":"duality-cohomology"},{"id":"stacks:0BZM","tag":"0BZM","title":"Base change for trivial duality · Lemma 0BZM","summary":"Let R → A and R → R' be ring maps and A' = A ⊗_R R'. Assume • A is pseudo-coherent as an R-module, • R' has finite tor dimension as an R-module (for example R → R' is flat), • A and R' are tor independent over R. Then ([Tag 0E29]) is an isomorphism for K ∈ D^+(R).","statement_latex":"Let $R \\to A$ and $R \\to R'$ be ring maps and $A' = A \\otimes_R R'$.\nAssume\n\\begin{enumerate}\n\\item $A$ is pseudo-coherent as an $R$-module,\n\\item $R'$ has finite tor dimension as an $R$-module (for example\n$R \\to R'$ is flat),\n\\item $A$ and $R'$ are tor independent over $R$.\n\\end{enumerate}\nThen (\\ref{equation-base-change}) is an isomorphism for $K \\in D^+(R)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Base change for trivial duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZM","source_file":"dualizing.tex","source_line":2508,"source_end_line":2519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2508-L2519","statement_sha256":"503bc4dd97352b0b8feeee9d2b86bceda3d68aebedd18ace78ac1f687bc9aaa6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8696,"rank":8696,"depth":21,"x":2133.582,"y":1200.241,"cluster":"duality-cohomology"},{"id":"stacks:0BZP","tag":"0BZP","title":"Base change for trivial duality · Lemma 0BZP","summary":"Let R → A and R → R' be ring maps and A' = A ⊗_R R'. Assume • A is perfect as an R-module, • A and R' are tor independent over R. Then ([Tag 0E29]) is an isomorphism for all K ∈ D(R).","statement_latex":"Let $R \\to A$ and $R \\to R'$ be ring maps and $A' = A \\otimes_R R'$.\nAssume\n\\begin{enumerate}\n\\item $A$ is perfect as an $R$-module,\n\\item $A$ and $R'$ are tor independent over $R$.\n\\end{enumerate}\nThen (\\ref{equation-base-change}) is an isomorphism for all $K \\in D(R)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Base change for trivial duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZP","source_file":"dualizing.tex","source_line":2527,"source_end_line":2536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2527-L2536","statement_sha256":"601ea76f7fc4445bce49de5a8826e8b852ad50322dcdb4617316daf996938105","origin":"The Stacks Project","memory_eligible":false,"source_rank":8697,"rank":8697,"depth":21,"x":1904.959,"y":1154.379,"cluster":"duality-cohomology"},{"id":"stacks:0A7B","tag":"0A7B","title":"Dualizing complexes · Definition 0A7B","summary":"Let A be a Noetherian ring. A dualizing complex is a complex of A-modules ω_A^bullet such that • ω_A^bullet has finite injective dimension, • H^i(ω_A^bullet) is a finite A-module for all i, and • A → RHom_A(ω_A^bullet, ω_A^bullet) is a quasi-isomorphism.","statement_latex":"Let $A$ be a Noetherian ring. A {\\it dualizing complex} is a\ncomplex of $A$-modules $\\omega_A^\\bullet$ such that\n\\begin{enumerate}\n\\item $\\omega_A^\\bullet$ has finite injective dimension,\n\\item $H^i(\\omega_A^\\bullet)$ is a finite $A$-module for all $i$, and\n\\item $A \\to R\\Hom_A(\\omega_A^\\bullet, \\omega_A^\\bullet)$\nis a quasi-isomorphism.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7B","source_file":"dualizing.tex","source_line":2556,"source_end_line":2566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2556-L2566","statement_sha256":"9a07e2ebce1fa4e981083e4618697de0c2319e83ffcbcdbf7babc35efbfb9d1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8698,"rank":8698,"depth":0,"x":2110.781,"y":1058.304,"cluster":"duality-cohomology"},{"id":"stacks:0G4H","tag":"0G4H","title":"Dualizing complexes · Lemma 0G4H","summary":"Let A be a Noetherian ring. Let K, L ∈ D_Coh(A) and assume L has finite injective dimension. Then RHom_A(K, L) is in D_Coh(A).","statement_latex":"Let $A$ be a Noetherian ring. Let $K, L \\in D_{\\textit{Coh}}(A)$\nand assume $L$ has finite injective dimension. Then\n$R\\Hom_A(K, L)$ is in $D_{\\textit{Coh}}(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4H","source_file":"dualizing.tex","source_line":2573,"source_end_line":2578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2573-L2578","statement_sha256":"c8ef31a669938c23047d737d2131a68d546aa481944c88dd5e8009462d150ecc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8699,"rank":8699,"depth":31,"x":2036.142,"y":1246.261,"cluster":"duality-cohomology"},{"id":"stacks:0A7C","tag":"0A7C","title":"Dualizing complexes · Lemma 0A7C","summary":"Let A be a Noetherian ring. If ω_A^bullet is a dualizing complex, then the functor D : K ↦ RHom_A(K, ω_A^bullet) is an anti-equivalence D_Coh(A) → D_Coh(A) which exchanges D^+_Coh(A) and D^-_Coh(A) and induces an anti-equivalence D^b_Coh(A) → D^b_Coh(A). Moreover D ∘ D is isomorphic to the identity functor.","statement_latex":"Let $A$ be a Noetherian ring. If $\\omega_A^\\bullet$ is a dualizing\ncomplex, then the functor\n$$\nD : K \\longmapsto R\\Hom_A(K, \\omega_A^\\bullet)\n$$\nis an anti-equivalence $D_{\\textit{Coh}}(A) \\to D_{\\textit{Coh}}(A)$\nwhich exchanges $D^+_{\\textit{Coh}}(A)$ and $D^-_{\\textit{Coh}}(A)$\nand induces an anti-equivalence\n$D^b_{\\textit{Coh}}(A) \\to D^b_{\\textit{Coh}}(A)$.\nMoreover $D \\circ D$ is isomorphic to the identity functor.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7C","source_file":"dualizing.tex","source_line":2600,"source_end_line":2612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2600-L2612","statement_sha256":"8ce69f50e53df397b455fcf53d93fbcaf0d32ef0246662e859118e20898df0be","origin":"The Stacks Project","memory_eligible":false,"source_rank":8700,"rank":8700,"depth":32,"x":1939.864,"y":1065.001,"cluster":"duality-cohomology"},{"id":"stacks:0A7E","tag":"0A7E","title":"Dualizing complexes · Lemma 0A7E","summary":"Let A be a Noetherian ring. Let F : D^b_Coh(A) → D^b_Coh(A) be an A-linear equivalence of categories. Then F(A) is an invertible object of D(A).","statement_latex":"Let $A$ be a Noetherian ring. Let\n$F : D^b_{\\textit{Coh}}(A) \\to D^b_{\\textit{Coh}}(A)$ be an $A$-linear\nequivalence of categories. Then $F(A)$ is an invertible object of $D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7E","source_file":"dualizing.tex","source_line":2641,"source_end_line":2646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2641-L2646","statement_sha256":"a679a1bfe2c66fefbaf8b1446fb4b261e84a19999735df380920cefef6d140d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8701,"rank":8701,"depth":14,"x":2156.995,"y":1144.165,"cluster":"duality-cohomology"},{"id":"stacks:0A7F","tag":"0A7F","title":"Dualizing complexes · Lemma 0A7F","summary":"Let A be a Noetherian ring. If ω_A^bullet and (ω'_A)^bullet are dualizing complexes, then (ω'_A)^bullet is quasi-isomorphic to ω_A^bullet ⊗_A^L L for some invertible object L of D(A).","statement_latex":"Let $A$ be a Noetherian ring. If $\\omega_A^\\bullet$ and\n$(\\omega'_A)^\\bullet$ are dualizing complexes, then\n$(\\omega'_A)^\\bullet$ is quasi-isomorphic to\n$\\omega_A^\\bullet \\otimes_A^\\mathbf{L} L$\nfor some invertible object $L$ of $D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7F","source_file":"dualizing.tex","source_line":2701,"source_end_line":2708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2701-L2708","statement_sha256":"d0b46437073fab61c3bbba21ea5726e9b92c9cac13f17059a2109f816d55b721","origin":"The Stacks Project","memory_eligible":false,"source_rank":8702,"rank":8702,"depth":33,"x":1932.84,"y":1209.107,"cluster":"duality-cohomology"},{"id":"stacks:0A7G","tag":"0A7G","title":"Dualizing complexes · Lemma 0A7G","summary":"Let A be a Noetherian ring. Let B = S^-1A be a localization. If ω_A^bullet is a dualizing complex, then ω_A^bullet ⊗_A B is a dualizing complex for B.","statement_latex":"Let $A$ be a Noetherian ring. Let $B = S^{-1}A$ be a localization.\nIf $\\omega_A^\\bullet$ is a dualizing\ncomplex, then $\\omega_A^\\bullet \\otimes_A B$ is a dualizing\ncomplex for $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7G","source_file":"dualizing.tex","source_line":2732,"source_end_line":2738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2732-L2738","statement_sha256":"90d8f3893542dbb7df98420f743a025bf63c58f633b3d14afa15b51e9aafec68","origin":"The Stacks Project","memory_eligible":false,"source_rank":8703,"rank":8703,"depth":20,"x":2046.098,"y":1033.73,"cluster":"duality-cohomology"},{"id":"stacks:0A7H","tag":"0A7H","title":"Dualizing complexes · Lemma 0A7H","summary":"Let A be a Noetherian ring. Let f_1, …, f_n ∈ A generate the unit ideal. If ω_A^bullet is a complex of A-modules such that (ω_A^bullet)_f_i is a dualizing complex for A_f_i for all i, then ω_A^bullet is a dualizing complex for A.","statement_latex":"Let $A$ be a Noetherian ring. Let $f_1, \\ldots, f_n \\in A$\ngenerate the unit ideal. If $\\omega_A^\\bullet$ is a complex\nof $A$-modules such that $(\\omega_A^\\bullet)_{f_i}$ is a dualizing\ncomplex for $A_{f_i}$ for all $i$, then $\\omega_A^\\bullet$ is a dualizing\ncomplex for $A$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7H","source_file":"dualizing.tex","source_line":2759,"source_end_line":2766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2759-L2766","statement_sha256":"e81474548b76c5a5759e56c2d82a86631ef801ce06297c815b827fbb80d63bd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8704,"rank":8704,"depth":25,"x":2103.714,"y":1227.644,"cluster":"duality-cohomology"},{"id":"stacks:0AX0","tag":"0AX0","title":"Dualizing complexes · Lemma 0AX0","summary":"Let A → B be a finite ring map of Noetherian rings. Let ω_A^bullet be a dualizing complex. Then RHom(B, ω_A^bullet) is a dualizing complex for B.","statement_latex":"Let $A \\to B$ be a finite ring map of Noetherian rings.\nLet $\\omega_A^\\bullet$ be a dualizing complex.\nThen $R\\Hom(B, \\omega_A^\\bullet)$ is a dualizing complex for $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AX0","source_file":"dualizing.tex","source_line":2807,"source_end_line":2812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2807-L2812","statement_sha256":"1c8e436f7919ca893e17af7cce3ef52e5e3c320be066a6a0b2fcee8be6a57f66","origin":"The Stacks Project","memory_eligible":false,"source_rank":8705,"rank":8705,"depth":33,"x":1904.952,"y":1117.161,"cluster":"duality-cohomology"},{"id":"stacks:0A7I","tag":"0A7I","title":"Dualizing complexes · Lemma 0A7I","summary":"Let A → B be a surjective homomorphism of Noetherian rings. Let ω_A^bullet be a dualizing complex. Then RHom(B, ω_A^bullet) is a dualizing complex for B.","statement_latex":"Let $A \\to B$ be a surjective homomorphism of Noetherian rings.\nLet $\\omega_A^\\bullet$ be a dualizing complex.\nThen $R\\Hom(B, \\omega_A^\\bullet)$ is a dualizing complex for $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7I","source_file":"dualizing.tex","source_line":2838,"source_end_line":2843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2838-L2843","statement_sha256":"8a6fc660fb91a2d49d7f7db555c49423d2d5c1ca834ae9f5914c032708e6010f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8706,"rank":8706,"depth":34,"x":2140.762,"y":1085.795,"cluster":"duality-cohomology"},{"id":"stacks:0A7J","tag":"0A7J","title":"Dualizing complexes · Lemma 0A7J","summary":"Let A be a Noetherian ring. If ω_A^bullet is a dualizing complex, then ω_A^bullet ⊗_A A[x] is a dualizing complex for A[x].","statement_latex":"Let $A$ be a Noetherian ring. If $\\omega_A^\\bullet$ is a dualizing\ncomplex, then $\\omega_A^\\bullet \\otimes_A A[x]$ is a dualizing\ncomplex for $A[x]$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7J","source_file":"dualizing.tex","source_line":2849,"source_end_line":2854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2849-L2854","statement_sha256":"13da015563c67689e030d40de2c7830ac0e6fdbd96b40c04607fdbc738b2f138","origin":"The Stacks Project","memory_eligible":false,"source_rank":8707,"rank":8707,"depth":20,"x":1991.852,"y":1242.992,"cluster":"duality-cohomology"},{"id":"stacks:0A7K","tag":"0A7K","title":"Dualizing complexes · Proposition 0A7K","summary":"Let A be a Noetherian ring which has a dualizing complex. Then any A-algebra essentially of finite type over A has a dualizing complex.","statement_latex":"Let $A$ be a Noetherian ring which has a dualizing complex.\nThen any $A$-algebra essentially of finite type over $A$\nhas a dualizing complex.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7K","source_file":"dualizing.tex","source_line":2872,"source_end_line":2877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2872-L2877","statement_sha256":"f65e4ac9d817af85058a7fa6ea6e548376f89ddb141690832f2be26b4d86da7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8708,"rank":8708,"depth":35,"x":1975.215,"y":1042.246,"cluster":"duality-cohomology"},{"id":"stacks:0A7L","tag":"0A7L","title":"Dualizing complexes · Lemma 0A7L","summary":"Let A be a Noetherian ring. Let ω_A^bullet be a dualizing complex. Let m ⊂ A be a maximal ideal and set kappa = A/ m. Then RHom_A(kappa, ω_A^bullet) ≅ kappa[n] for some n ∈ Z.","statement_latex":"Let $A$ be a Noetherian ring. Let $\\omega_A^\\bullet$ be a dualizing\ncomplex. Let $\\mathfrak m \\subset A$ be a maximal ideal and set\n$\\kappa = A/\\mathfrak m$. Then\n$R\\Hom_A(\\kappa, \\omega_A^\\bullet) \\cong \\kappa[n]$ for some\n$n \\in \\mathbf{Z}$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7L","source_file":"dualizing.tex","source_line":2885,"source_end_line":2892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2885-L2892","statement_sha256":"c5772c3bbb775396658afa42c134e9e730787d66e1345ee728232aba5758db26","origin":"The Stacks Project","memory_eligible":false,"source_rank":8709,"rank":8709,"depth":35,"x":2149.207,"y":1181.064,"cluster":"duality-cohomology"},{"id":"stacks:0AX1","tag":"0AX1","title":"Dualizing complexes over local rings · Lemma 0AX1","summary":"Let (A, m, kappa) → (B, m', kappa') be a finite local map of Noetherian local rings. Let ω_A^bullet be a normalized dualizing complex. Then ω_B^bullet = RHom(B, ω_A^bullet) is a normalized dualizing complex for B.","statement_latex":"Let $(A, \\mathfrak m, \\kappa) \\to (B, \\mathfrak m', \\kappa')$\nbe a finite local map of Noetherian local rings. Let $\\omega_A^\\bullet$\nbe a normalized dualizing complex. Then\n$\\omega_B^\\bullet = R\\Hom(B, \\omega_A^\\bullet)$ is a\nnormalized dualizing complex for $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AX1","source_file":"dualizing.tex","source_line":2918,"source_end_line":2925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2918-L2925","statement_sha256":"e8a02ec2cfabf68186384c2d6978aa9ae331a306b5720010e6862d34b3e87424","origin":"The Stacks Project","memory_eligible":false,"source_rank":8710,"rank":8710,"depth":34,"x":1908.875,"y":1177.423,"cluster":"duality-cohomology"},{"id":"stacks:0A7N","tag":"0A7N","title":"Dualizing complexes over local rings · Lemma 0A7N","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Let A → B be surjective. Then ω_B^bullet = RHom_A(B, ω_A^bullet) is a normalized dualizing complex for B.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local\nring with normalized dualizing complex $\\omega_A^\\bullet$.\nLet $A \\to B$ be surjective. Then\n$\\omega_B^\\bullet = R\\Hom_A(B, \\omega_A^\\bullet)$ is a\nnormalized dualizing complex for $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7N","source_file":"dualizing.tex","source_line":2942,"source_end_line":2949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2942-L2949","statement_sha256":"34dfd499dce21aaa291259d0ed3f00402e14482b700257671ac78caac54665e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8711,"rank":8711,"depth":35,"x":2089.32,"y":1043.516,"cluster":"duality-cohomology"},{"id":"stacks:0A7P","tag":"0A7P","title":"Dualizing complexes over local rings · Lemma 0A7P","summary":"Let (A, m, kappa) be a Noetherian local ring. Let F be an A-linear self-equivalence of the category of finite length A-modules. Then F is isomorphic to the identity functor.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local\nring. Let $F$ be an $A$-linear self-equivalence of the category of\nfinite length $A$-modules. Then $F$ is isomorphic to the identity functor.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7P","source_file":"dualizing.tex","source_line":2955,"source_end_line":2960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2955-L2960","statement_sha256":"75f6b211cfee5f86c0c44faa04c4a4102cc4ca87007322006c68469611698501","origin":"The Stacks Project","memory_eligible":false,"source_rank":8712,"rank":8712,"depth":0,"x":2063.905,"y":1244.979,"cluster":"duality-cohomology"},{"id":"stacks:0A7Q","tag":"0A7Q","title":"Dualizing complexes over local rings · Lemma 0A7Q","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Let E be an injective hull of kappa. Then there exists a functorial isomorphism RHom_A(N, ω_A^bullet) = Hom_A(N, E)[0] for N running through the finite length A-modules.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local\nring with normalized dualizing complex $\\omega_A^\\bullet$.\nLet $E$ be an injective hull of $\\kappa$. Then there exists\na functorial isomorphism\n$$\nR\\Hom_A(N, \\omega_A^\\bullet) = \\Hom_A(N, E)[0]\n$$\nfor $N$ running through the finite length $A$-modules.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7Q","source_file":"dualizing.tex","source_line":2986,"source_end_line":2996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L2986-L2996","statement_sha256":"3bfae7cb1a36052301e999339d8e6b94bead70af714b50336711abede8a975b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8713,"rank":8713,"depth":10,"x":1920.398,"y":1081.733,"cluster":"duality-cohomology"},{"id":"stacks:0A7U","tag":"0A7U","title":"Dualizing complexes over local rings · Lemma 0A7U","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Let M be a finite A-module and let d = dim(Supp(M)). Then • if Ext^i_A(M, ω_A^bullet) is nonzero, then i ∈ (-d, …, 0), • the dimension of the support of Ext^i_A(M, ω_A^bullet) is at most -i, • depth(M) is the smallest integer δ ≥ 0 such that Ext^-δ_A(M, ω_A^bullet) not = 0.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with\nnormalized dualizing complex $\\omega_A^\\bullet$. Let $M$ be a finite\n$A$-module and let $d = \\dim(\\text{Supp}(M))$. Then\n\\begin{enumerate}\n\\item if $\\Ext^i_A(M, \\omega_A^\\bullet)$ is nonzero, then\n$i \\in \\{-d, \\ldots, 0\\}$,\n\\item the dimension of the support of $\\Ext^i_A(M, \\omega_A^\\bullet)$\nis at most $-i$,\n\\item $\\text{depth}(M)$ is the smallest integer $\\delta \\geq 0$ such that\n$\\Ext^{-\\delta}_A(M, \\omega_A^\\bullet) \\not = 0$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7U","source_file":"dualizing.tex","source_line":3014,"source_end_line":3027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3014-L3027","statement_sha256":"7ead519dfe33e0d1cf5983bb5b388522fa79250470b67775d119c08fa109c71c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8714,"rank":8714,"depth":14,"x":2157.885,"y":1120.743,"cluster":"duality-cohomology"},{"id":"stacks:0B5A","tag":"0B5A","title":"Dualizing complexes over local rings · Lemma 0B5A","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Let M be a finite A-module. The following are equivalent • M is Cohen-Macaulay, • Ext^i_A(M, ω_A^bullet) is nonzero for at most one i, • Ext^-i_A(M, ω_A^bullet) is zero for i not = dim(Supp(M)). Denote CM_d the category of finite Cohen-Macaulay A-modules of depth d. Then M ↦ Ext^-d_A(M, ω_A^bullet) defines an anti-auto-equivalence of CM_d.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring\nwith normalized dualizing complex $\\omega_A^\\bullet$. Let $M$\nbe a finite $A$-module. The following are equivalent\n\\begin{enumerate}\n\\item $M$ is Cohen-Macaulay,\n\\item $\\Ext^i_A(M, \\omega_A^\\bullet)$ is nonzero for at most one $i$,\n\\item $\\Ext^{-i}_A(M, \\omega_A^\\bullet)$ is zero for\n$i \\not = \\dim(\\text{Supp}(M))$.\n\\end{enumerate}\nDenote $CM_d$ the category of finite Cohen-Macaulay $A$-modules\nof depth $d$. Then $M \\mapsto \\Ext^{-d}_A(M, \\omega_A^\\bullet)$\ndefines an anti-auto-equivalence of $CM_d$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5A","source_file":"dualizing.tex","source_line":3089,"source_end_line":3103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3089-L3103","statement_sha256":"72a555fa29092cbe33d5cc584bfc67c4d4bd496af71b2e350e963e4ab443e15a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8715,"rank":8715,"depth":33,"x":1951.057,"y":1226.907,"cluster":"duality-cohomology"},{"id":"stacks:0A7R","tag":"0A7R","title":"Dualizing complexes over local rings · Lemma 0A7R","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. If dim(A) = 0, then ω_A^bullet ≅ E[0] where E is an injective hull of the residue field.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local\nring with normalized dualizing complex $\\omega_A^\\bullet$.\nIf $\\dim(A) = 0$, then $\\omega_A^\\bullet \\cong E[0]$\nwhere $E$ is an injective hull of the residue field.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7R","source_file":"dualizing.tex","source_line":3133,"source_end_line":3139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3133-L3139","statement_sha256":"e047409560c68aeca296d2ca134c24aa794418535f5b2fe5bc4d3c178296db44","origin":"The Stacks Project","memory_eligible":false,"source_rank":8716,"rank":8716,"depth":11,"x":2018.305,"y":1030.944,"cluster":"duality-cohomology"},{"id":"stacks:0A7S","tag":"0A7S","title":"Dualizing complexes over local rings · Lemma 0A7S","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex. Let I ⊂ m be an ideal of finite length. Set B = A/I. Then there is a distinguished triangle ω_B^bullet → ω_A^bullet → Hom_A(I, E)[0] → ω_B^bullet[1] in D(A) where E is an injective hull of kappa and ω_B^bullet is a normalized dualizing complex for B.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local\nring with normalized dualizing complex. Let $I \\subset \\mathfrak m$ be an\nideal of finite length. Set $B = A/I$. Then there is a distinguished\ntriangle\n$$\n\\omega_B^\\bullet \\to \\omega_A^\\bullet \\to \\Hom_A(I, E)[0] \\to\n\\omega_B^\\bullet[1]\n$$\nin $D(A)$ where $E$ is an injective hull of $\\kappa$ and\n$\\omega_B^\\bullet$ is a normalized dualizing complex for $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7S","source_file":"dualizing.tex","source_line":3145,"source_end_line":3157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3145-L3157","statement_sha256":"f78d205d83a3ebd12459d013667254c6f531a28a6af7ce8c2c081ba23407903e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8717,"rank":8717,"depth":36,"x":2126.48,"y":1213.9,"cluster":"duality-cohomology"},{"id":"stacks:0A7T","tag":"0A7T","title":"Dualizing complexes over local rings · Lemma 0A7T","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Let f ∈ m be a nonzerodivisor. Set B = A/(f). Then there is a distinguished triangle ω_B^bullet → ω_A^bullet → ω_A^bullet → ω_B^bullet[1] in D(A) where ω_B^bullet is a normalized dualizing complex for B.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local\nring with normalized dualizing complex $\\omega_A^\\bullet$.\nLet $f \\in \\mathfrak m$ be a\nnonzerodivisor. Set $B = A/(f)$. Then there is a distinguished\ntriangle\n$$\n\\omega_B^\\bullet \\to \\omega_A^\\bullet \\to \\omega_A^\\bullet \\to\n\\omega_B^\\bullet[1]\n$$\nin $D(A)$ where $\\omega_B^\\bullet$ is a normalized dualizing complex\nfor $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7T","source_file":"dualizing.tex","source_line":3165,"source_end_line":3178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3165-L3178","statement_sha256":"7df36ccbc5e0ac3b5dab729f265c37f571570a4857f1a3a5586b0777a0954609","origin":"The Stacks Project","memory_eligible":false,"source_rank":8718,"rank":8718,"depth":36,"x":1899.218,"y":1140.253,"cluster":"duality-cohomology"},{"id":"stacks:0A7V","tag":"0A7V","title":"Dualizing complexes over local rings · Lemma 0A7V","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Let p be a minimal prime of A with dim(A/ p) = e. Then H^i(ω_A^bullet)_ p is nonzero if and only if i = -e.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with\nnormalized dualizing complex $\\omega_A^\\bullet$.\nLet $\\mathfrak p$ be a minimal prime of $A$ with\n$\\dim(A/\\mathfrak p) = e$. Then\n$H^i(\\omega_A^\\bullet)_\\mathfrak p$ is nonzero\nif and only if $i = -e$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7V","source_file":"dualizing.tex","source_line":3185,"source_end_line":3193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3185-L3193","statement_sha256":"d690b224a7dcfd6e6c3050beaa8ff095dab05c789c9bb7f94a5708662500bc93","origin":"The Stacks Project","memory_eligible":false,"source_rank":8719,"rank":8719,"depth":36,"x":2126.389,"y":1065.484,"cluster":"duality-cohomology"},{"id":"stacks:0A7X","tag":"0A7X","title":"Dualizing complexes and dimension functions · Lemma 0A7X","summary":"Let A be a Noetherian ring. Let p be a minimal prime of A. Then H^i(ω_A^bullet)_ p is nonzero for exactly one i.","statement_latex":"Let $A$ be a Noetherian ring. Let $\\mathfrak p$ be a minimal prime\nof $A$. Then $H^i(\\omega_A^\\bullet)_\\mathfrak p$ is nonzero\nfor exactly one $i$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes and dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7X","source_file":"dualizing.tex","source_line":3231,"source_end_line":3236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3231-L3236","statement_sha256":"fc393757b2b5829cdce0d4ffffb52eed7c35434015e15d927889a70dfcec8d45","origin":"The Stacks Project","memory_eligible":false,"source_rank":8720,"rank":8720,"depth":21,"x":2018.83,"y":1249.816,"cluster":"duality-cohomology"},{"id":"stacks:0A7Y","tag":"0A7Y","title":"Dualizing complexes and dimension functions · Lemma 0A7Y","summary":"Let A be a Noetherian ring and let ω_A^bullet be a dualizing complex. Let A → B be a surjective ring map and let ω_B^bullet = RHom(B, ω_A^bullet) be the dualizing complex for B of Lemma [Tag 0A7I]. Then we have δ_ω_B^bullet = δ_ω_A^bullet|_Spec(B)","statement_latex":"Let $A$ be a Noetherian ring and let $\\omega_A^\\bullet$ be a dualizing\ncomplex. Let $A \\to B$ be a surjective ring map and let\n$\\omega_B^\\bullet = R\\Hom(B, \\omega_A^\\bullet)$ be the dualizing\ncomplex for $B$ of Lemma \\ref{lemma-dualizing-quotient}. Then we have\n$$\n\\delta_{\\omega_B^\\bullet} = \\delta_{\\omega_A^\\bullet}|_{\\Spec(B)}\n$$","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes and dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7Y","source_file":"dualizing.tex","source_line":3264,"source_end_line":3273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3264-L3273","statement_sha256":"9ba11e99d24c5d15f1b1ff93b0bc10fba3fdee58f91135fba25e65d277b2d9bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8721,"rank":8721,"depth":36,"x":1949.797,"y":1052.545,"cluster":"duality-cohomology"},{"id":"stacks:0A7Z","tag":"0A7Z","title":"Dualizing complexes and dimension functions · Lemma 0A7Z","summary":"Let A be a Noetherian ring and let ω_A^bullet be a dualizing complex. The function δ = δ_ω_A^bullet defined above is a dimension function (Topology, Definition [Tag 02I9]).","statement_latex":"Let $A$ be a Noetherian ring and let $\\omega_A^\\bullet$ be a dualizing\ncomplex. The function $\\delta = \\delta_{\\omega_A^\\bullet}$\ndefined above is a dimension function\n(Topology, Definition \\ref{topology-definition-dimension-function}).","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes and dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A7Z","source_file":"dualizing.tex","source_line":3280,"source_end_line":3286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3280-L3286","statement_sha256":"0b6ba55070f9ef0b1248ff8d38665e884aecdcc204f70f6f55759274b4d1d3da","origin":"The Stacks Project","memory_eligible":false,"source_rank":8722,"rank":8722,"depth":37,"x":2159.676,"y":1159.009,"cluster":"duality-cohomology"},{"id":"stacks:0A80","tag":"0A80","title":"Dualizing complexes and dimension functions · Lemma 0A80","summary":"Let A be a Noetherian ring which has a dualizing complex. Then A is universally catenary of finite dimension.","statement_latex":"Let $A$ be a Noetherian ring which has a dualizing\ncomplex. Then $A$ is universally catenary of finite dimension.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes and dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A80","source_file":"dualizing.tex","source_line":3323,"source_end_line":3327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3323-L3327","statement_sha256":"569ab5350ab1024f51975e415ecf72315ba839b00959735cf877d7480af6f31c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8723,"rank":8723,"depth":38,"x":1918.915,"y":1199.659,"cluster":"duality-cohomology"},{"id":"stacks:0AWE","tag":"0AWE","title":"Dualizing complexes and dimension functions · Lemma 0AWE","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Let d = dim(A) and ω_A = H^-d(ω_A^bullet). Then • the support of ω_A is the union of the irreducible components of Spec(A) of dimension d, • ω_A satisfies (S_2), see Algebra, Definition [Tag 031P].","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with\nnormalized dualizing complex $\\omega_A^\\bullet$. Let $d = \\dim(A)$\nand $\\omega_A = H^{-d}(\\omega_A^\\bullet)$. Then\n\\begin{enumerate}\n\\item the support of $\\omega_A$ is the union of the irreducible components\nof $\\Spec(A)$ of dimension $d$,\n\\item $\\omega_A$ satisfies $(S_2)$, see\nAlgebra, Definition \\ref{algebra-definition-conditions}.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Dualizing complexes and dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWE","source_file":"dualizing.tex","source_line":3358,"source_end_line":3369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3358-L3369","statement_sha256":"0d4d3b37b1025da147e85dd3e46bff366daadef5c93deafd5bfb294dfe03f088","origin":"The Stacks Project","memory_eligible":false,"source_rank":8724,"rank":8724,"depth":39,"x":2063.99,"y":1032.806,"cluster":"duality-cohomology"},{"id":"stacks:0A82","tag":"0A82","title":"The local duality theorem · Lemma 0A82","summary":"Let (A, m, kappa) be a Noetherian local ring. Let ω_A^bullet be a normalized dualizing complex. Let Z = V( m) ⊂ Spec(A). Then E = R^0Γ_Z(ω_A^bullet) is an injective hull of kappa and RΓ_Z(ω_A^bullet) = E[0].","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $\\omega_A^\\bullet$ be a normalized dualizing complex.\nLet $Z = V(\\mathfrak m) \\subset \\Spec(A)$.\nThen $E = R^0\\Gamma_Z(\\omega_A^\\bullet)$ is an injective hull of\n$\\kappa$ and $R\\Gamma_Z(\\omega_A^\\bullet) = E[0]$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"The local duality theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A82","source_file":"dualizing.tex","source_line":3424,"source_end_line":3431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3424-L3431","statement_sha256":"55eabd49988044d47924951ed230f67a9c7106e20666170474ac2e68ff356380","origin":"The Stacks Project","memory_eligible":false,"source_rank":8725,"rank":8725,"depth":11,"x":2091.236,"y":1238.485,"cluster":"duality-cohomology"},{"id":"stacks:0A84","tag":"0A84","title":"The local duality theorem · Theorem 0A84","summary":"Let (A, m, kappa) be a Noetherian local ring. Let ω_A^bullet be a normalized dualizing complex. Let E be an injective hull of the residue field. Let Z = V( m) ⊂ Spec(A). Denote ^wedge derived completion with respect to m. Then RHom_A(K, ω_A^bullet)^wedge ≅ RHom_A(RΓ_Z(K), E[0]) for K in D(A).","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $\\omega_A^\\bullet$ be a normalized dualizing complex.\nLet $E$ be an injective hull of the residue field.\nLet $Z = V(\\mathfrak m) \\subset \\Spec(A)$.\nDenote ${}^\\wedge$ derived completion with respect to $\\mathfrak m$.\nThen\n$$\nR\\Hom_A(K, \\omega_A^\\bullet)^\\wedge \\cong R\\Hom_A(R\\Gamma_Z(K), E[0])\n$$\nfor $K$ in $D(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"The local duality theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A84","source_file":"dualizing.tex","source_line":3475,"source_end_line":3487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3475-L3487","statement_sha256":"00180ae7e53f3af92dbcf94536a033cf1742629da33ac568592f53523f6c5355","origin":"The Stacks Project","memory_eligible":false,"source_rank":8726,"rank":8726,"depth":12,"x":1905.451,"y":1102.065,"cluster":"duality-cohomology"},{"id":"stacks:0AAK","tag":"0AAK","title":"The local duality theorem · Lemma 0AAK","summary":"Let (A, m, kappa) be a Noetherian local ring. Let ω_A^bullet be a normalized dualizing complex. Let E be an injective hull of the residue field. Let K ∈ D_Coh(A). Then Ext^-i_A(K, ω_A^bullet)^wedge = Hom_A(H^i_ m(K), E) where ^wedge denotes m-adic completion.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $\\omega_A^\\bullet$ be a normalized dualizing complex.\nLet $E$ be an injective hull of the residue field.\nLet $K \\in D_{\\textit{Coh}}(A)$. Then\n$$\n\\Ext^{-i}_A(K, \\omega_A^\\bullet)^\\wedge =\n\\Hom_A(H^i_{\\mathfrak m}(K), E)\n$$\nwhere ${}^\\wedge$ denotes $\\mathfrak m$-adic completion.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"The local duality theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAK","source_file":"dualizing.tex","source_line":3509,"source_end_line":3520,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3509-L3520","statement_sha256":"0d3aa0514049424e9c2e25ad7677e26514d2231c4da8e9916f7b0b5c85dcb78a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8727,"rank":8727,"depth":33,"x":2152.539,"y":1097.233,"cluster":"duality-cohomology"},{"id":"stacks:0AWR","tag":"0AWR","title":"Cohen-Macaulay rings · Lemma 0AWR","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Then depth(A) is equal to the smallest integer δ ≥ 0 such that H^-δ(ω_A^bullet) not = 0.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring with\nnormalized dualizing complex $\\omega_A^\\bullet$.\nThen $\\text{depth}(A)$ is equal to the smallest integer $\\delta \\geq 0$\nsuch that $H^{-\\delta}(\\omega_A^\\bullet) \\not = 0$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWR","source_file":"dualizing.tex","source_line":3591,"source_end_line":3597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3591-L3597","statement_sha256":"f47f4f258cc73724cd5faabb5d4218a7dc810e4e26fa786f1280817d6b1904a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":8728,"rank":8728,"depth":34,"x":1973.946,"y":1241.226,"cluster":"duality-cohomology"},{"id":"stacks:0AWS","tag":"0AWS","title":"Cohen-Macaulay rings · Lemma 0AWS","summary":"Let (A, m, kappa) be a Noetherian local ring with normalized dualizing complex ω_A^bullet and dualizing module ω_A = H^-dim(A)(ω_A^bullet). The following are equivalent • A is Cohen-Macaulay, • ω_A^bullet is concentrated in a single degree, and • ω_A^bullet = ω_A[dim(A)]. In this case ω_A is a maximal Cohen-Macaulay module.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring\nwith normalized dualizing complex $\\omega_A^\\bullet$\nand dualizing module $\\omega_A = H^{-\\dim(A)}(\\omega_A^\\bullet)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A$ is Cohen-Macaulay,\n\\item $\\omega_A^\\bullet$ is concentrated in a single degree, and\n\\item $\\omega_A^\\bullet = \\omega_A[\\dim(A)]$.\n\\end{enumerate}\nIn this case $\\omega_A$ is a maximal Cohen-Macaulay module.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWS","source_file":"dualizing.tex","source_line":3624,"source_end_line":3636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3624-L3636","statement_sha256":"a3ea9e7cc586337511464e8ff1a83bbad77e8427cf73ec798fd40162bd04a8ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":8729,"rank":8729,"depth":34,"x":1989.868,"y":1033.385,"cluster":"duality-cohomology"},{"id":"stacks:0DW5","tag":"0DW5","title":"Cohen-Macaulay rings · Lemma 0DW5","summary":"Let A be a Noetherian ring. If there exists a finite A-module ω_A such that ω_A[0] is a dualizing complex, then A is Cohen-Macaulay.","statement_latex":"Let $A$ be a Noetherian ring. If there exists a finite $A$-module\n$\\omega_A$ such that $\\omega_A[0]$ is a dualizing complex, then\n$A$ is Cohen-Macaulay.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DW5","source_file":"dualizing.tex","source_line":3642,"source_end_line":3647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3642-L3647","statement_sha256":"8a32a916ad6cafd64e9fe633223b9310961580ace26b0d0582c5d86961ddaf17","origin":"The Stacks Project","memory_eligible":false,"source_rank":8730,"rank":8730,"depth":35,"x":2145.509,"y":1195.931,"cluster":"duality-cohomology"},{"id":"stacks:0EHS","tag":"0EHS","title":"Cohen-Macaulay rings · Lemma 0EHS","summary":"Let A be a Noetherian ring with dualizing complex ω_A^bullet. Let M be a finite A-module. Then U = ( p ∈ Spec(A) mid M_ p is Cohen-Macaulay) is an open subset of Spec(A) whose intersection with Supp(M) is dense.","statement_latex":"Let $A$ be a Noetherian ring with dualizing complex $\\omega_A^\\bullet$.\nLet $M$ be a finite $A$-module. Then\n$$\nU = \\{\\mathfrak p \\in \\Spec(A) \\mid M_\\mathfrak p\\text{ is Cohen-Macaulay}\\}\n$$\nis an open subset of $\\Spec(A)$ whose intersection with\n$\\text{Supp}(M)$ is dense.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHS","source_file":"dualizing.tex","source_line":3657,"source_end_line":3666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3657-L3666","statement_sha256":"907a57460a38931c13ac4b2abeddf7cb40f9f7a1db440e1c40e450f1af7459ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":8731,"rank":8731,"depth":34,"x":1899.645,"y":1164.337,"cluster":"duality-cohomology"},{"id":"stacks:0EHT","tag":"0EHT","title":"Cohen-Macaulay rings · Lemma 0EHT","summary":"Let A be a Noetherian ring. If A has a dualizing complex ω_A^bullet, then ( p ∈ Spec(A) mid A_ p is Cohen-Macaulay) is a dense open subset of Spec(A).","statement_latex":"Let $A$ be a Noetherian ring. If $A$ has a dualizing complex\n$\\omega_A^\\bullet$, then\n$\\{\\mathfrak p \\in \\Spec(A) \\mid A_\\mathfrak p\\text{ is Cohen-Macaulay}\\}$\nis a dense open subset of $\\Spec(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Cohen-Macaulay rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHT","source_file":"dualizing.tex","source_line":3689,"source_end_line":3695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3689-L3695","statement_sha256":"d9af11234c4c888bd6aea5be4ba4763308ddffac766ecf4b2238b1b0de5fc273","origin":"The Stacks Project","memory_eligible":false,"source_rank":8732,"rank":8732,"depth":35,"x":2106.669,"y":1047.945,"cluster":"duality-cohomology"},{"id":"stacks:0DW7","tag":"0DW7","title":"Gorenstein rings · Definition 0DW7","summary":"Gorenstein rings. • Let A be a Noetherian local ring. We say A is Gorenstein if A[0] is a dualizing complex for A. • Let A be a Noetherian ring. We say A is Gorenstein if A_ p is Gorenstein for every prime p of A.","statement_latex":"Gorenstein rings.\n\\begin{enumerate}\n\\item Let $A$ be a Noetherian local ring. We say $A$ is {\\it Gorenstein}\nif $A[0]$ is a dualizing complex for $A$.\n\\item Let $A$ be a Noetherian ring. We say $A$ is {\\it Gorenstein}\nif $A_\\mathfrak p$ is Gorenstein for every prime $\\mathfrak p$ of $A$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DW7","source_file":"dualizing.tex","source_line":3731,"source_end_line":3740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3731-L3740","statement_sha256":"b8e191bedd9132c17f4bb97784d37c3cf2436116397cb85e0ef1984a4200aa37","origin":"The Stacks Project","memory_eligible":false,"source_rank":8733,"rank":8733,"depth":0,"x":2047.521,"y":1251.555,"cluster":"duality-cohomology"},{"id":"stacks:0DW8","tag":"0DW8","title":"Gorenstein rings · Lemma 0DW8","summary":"A Gorenstein ring is Cohen-Macaulay.","statement_latex":"A Gorenstein ring is Cohen-Macaulay.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DW8","source_file":"dualizing.tex","source_line":3750,"source_end_line":3753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3750-L3753","statement_sha256":"e11b272b4705c1d8c73c4e5b4056231cad7f98e54b01da68b785d9b377a8cc83","origin":"The Stacks Project","memory_eligible":false,"source_rank":8734,"rank":8734,"depth":35,"x":1927.213,"y":1067.571,"cluster":"duality-cohomology"},{"id":"stacks:0AWX","tag":"0AWX","title":"Gorenstein rings · Lemma 0AWX","summary":"A regular local ring is Gorenstein. A regular ring is Gorenstein.","statement_latex":"A regular local ring is Gorenstein.\nA regular ring is Gorenstein.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWX","source_file":"dualizing.tex","source_line":3762,"source_end_line":3766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3762-L3766","statement_sha256":"655e43c0ab2d1db16c1897a4f7a82b9e518327860c0f0e39d98435b2f31d37a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8735,"rank":8735,"depth":18,"x":2164.244,"y":1135.077,"cluster":"duality-cohomology"},{"id":"stacks:0DW9","tag":"0DW9","title":"Gorenstein rings · Lemma 0DW9","summary":"Let A be a Noetherian ring. • If A has a dualizing complex ω_A^bullet, then • A is Gorenstein ⇔ ω_A^bullet is an invertible object of D(A), • A_ p is Gorenstein ⇔ (ω_A^bullet)_ p is an invertible object of D(A_ p), • ( p ∈ Spec(A) mid A_ p is Gorenstein) is an open subset. • If A is Gorenstein, then A has a dualizing complex if and only if A[0] is a dualizing complex.","statement_latex":"Let $A$ be a Noetherian ring.\n\\begin{enumerate}\n\\item If $A$ has a dualizing complex $\\omega_A^\\bullet$, then\n\\begin{enumerate}\n\\item $A$ is Gorenstein $\\Leftrightarrow$ $\\omega_A^\\bullet$ is an invertible\nobject of $D(A)$,\n\\item $A_\\mathfrak p$ is Gorenstein $\\Leftrightarrow$\n$(\\omega_A^\\bullet)_\\mathfrak p$ is an invertible object of\n$D(A_\\mathfrak p)$,\n\\item $\\{\\mathfrak p \\in \\Spec(A) \\mid A_\\mathfrak p\\text{ is Gorenstein}\\}$\nis an open subset.\n\\end{enumerate}\n\\item If $A$ is Gorenstein, then $A$ has a dualizing complex if and\nonly if $A[0]$ is a dualizing complex.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DW9","source_file":"dualizing.tex","source_line":3780,"source_end_line":3797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3780-L3797","statement_sha256":"1545bd3127f0f643c1ced25ff062c24a3297eb0b6601bc15c74c0fab75cbf887","origin":"The Stacks Project","memory_eligible":false,"source_rank":8736,"rank":8736,"depth":34,"x":1934.824,"y":1219.926,"cluster":"duality-cohomology"},{"id":"stacks:0BJI","tag":"0BJI","title":"Gorenstein rings · Lemma 0BJI","summary":"Let (A, m, kappa) be a Noetherian local ring. Then A is Gorenstein if and only if Ext^i_A(kappa, A) is zero for i gg 0.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nThen $A$ is Gorenstein if and only if $\\Ext^i_A(\\kappa, A)$\nis zero for $i \\gg 0$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJI","source_file":"dualizing.tex","source_line":3845,"source_end_line":3850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3845-L3850","statement_sha256":"da87c2a88c8835222704b9533f0456e8ed5fdf1477d249da6d458cdc656f7f61","origin":"The Stacks Project","memory_eligible":false,"source_rank":8737,"rank":8737,"depth":5,"x":2035.916,"y":1026.887,"cluster":"duality-cohomology"},{"id":"stacks:0HB2","tag":"0HB2","title":"Gorenstein rings · Lemma 0HB2","summary":"Let A be a Noetherian ring. The following are equivalent • A has finite injective dimension as a module over itself, • A is Gorenstein and finite dimensional. In this case A[0] is a dualizing complex.","statement_latex":"Let $A$ be a Noetherian ring. The following are equivalent\n\\begin{enumerate}\n\\item $A$ has finite injective dimension as a module over itself,\n\\item $A$ is Gorenstein and finite dimensional.\n\\end{enumerate}\nIn this case $A[0]$ is a dualizing complex.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HB2","source_file":"dualizing.tex","source_line":3860,"source_end_line":3868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3860-L3868","statement_sha256":"347c9e7be10a08e79ed65c8ef1afdf708ebd61f72fef760fdb766e944d326b1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8738,"rank":8738,"depth":39,"x":2116.731,"y":1226.896,"cluster":"duality-cohomology"},{"id":"stacks:0BJJ","tag":"0BJJ","title":"Gorenstein rings · Lemma 0BJJ","summary":"Let (A, m, kappa) be a Noetherian local ring. Let f ∈ m be a nonzerodivisor. Set B = A/(f). Then A is Gorenstein if and only if B is Gorenstein.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$\nbe a Noetherian local ring. Let $f \\in \\mathfrak m$ be a\nnonzerodivisor. Set $B = A/(f)$. Then $A$ is Gorenstein if and\nonly if $B$ is Gorenstein.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJJ","source_file":"dualizing.tex","source_line":3891,"source_end_line":3897,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3891-L3897","statement_sha256":"04c8ec823987a2be400731ccbcb73c69597e70568e50afac983af179e3946046","origin":"The Stacks Project","memory_eligible":false,"source_rank":8739,"rank":8739,"depth":37,"x":1895.963,"y":1125.115,"cluster":"duality-cohomology"},{"id":"stacks:0DWA","tag":"0DWA","title":"Gorenstein rings · Lemma 0DWA","summary":"If A → B is a local complete intersection homomorphism of rings and A is a Noetherian Gorenstein ring, then B is a Gorenstein ring.","statement_latex":"If $A \\to B$ is a local complete intersection homomorphism of rings and\n$A$ is a Noetherian Gorenstein ring, then $B$ is a Gorenstein ring.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWA","source_file":"dualizing.tex","source_line":3918,"source_end_line":3922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3918-L3922","statement_sha256":"772afc2083f9ef904acfa411dff11ef4e79362f3312fd5bf873d956078af28bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8740,"rank":8740,"depth":38,"x":2140.974,"y":1074.822,"cluster":"duality-cohomology"},{"id":"stacks:0BJL","tag":"0BJL","title":"Gorenstein rings · Lemma 0BJL","summary":"Let A → B be a flat local homomorphism of Noetherian local rings. The following are equivalent • B is Gorenstein, and • A and B/ m_A B are Gorenstein.","statement_latex":"Let $A \\to B$ be a flat local homomorphism of Noetherian local rings.\nThe following are equivalent\n\\begin{enumerate}\n\\item $B$ is Gorenstein, and\n\\item $A$ and $B/\\mathfrak m_A B$ are Gorenstein.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJL","source_file":"dualizing.tex","source_line":3946,"source_end_line":3954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L3946-L3954","statement_sha256":"2f93fa8c84cc385612de9f60fe4f1bd6ffdd8d986fd776571c1fd6389d4ffe70","origin":"The Stacks Project","memory_eligible":false,"source_rank":8741,"rank":8741,"depth":36,"x":2000.54,"y":1251.197,"cluster":"duality-cohomology"},{"id":"stacks:0EBT","tag":"0EBT","title":"Gorenstein rings · Lemma 0EBT","summary":"Let (A, m, kappa) be a Noetherian local Gorenstein ring of dimension d. Let E be the injective hull of kappa. Then Tor_i^A(E, kappa) is zero for i not = d and Tor_d^A(E, kappa) = kappa.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local Gorenstein ring\nof dimension $d$. Let $E$ be the injective hull of $\\kappa$. Then\n$\\text{Tor}_i^A(E, \\kappa)$ is zero for $i \\not = d$\nand $\\text{Tor}_d^A(E, \\kappa) = \\kappa$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Gorenstein rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBT","source_file":"dualizing.tex","source_line":4004,"source_end_line":4010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4004-L4010","statement_sha256":"d345f19029aa3148eee59fbaae74879c0bebdcd1a310431384b82298d03abb6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8742,"rank":8742,"depth":12,"x":1962.2,"y":1041.14,"cluster":"duality-cohomology"},{"id":"stacks:0AWD","tag":"0AWD","title":"The ubiquity of dualizing complexes · Lemma 0AWD","summary":"Let A → B be a local homomorphism of Noetherian local rings. Let ω_A^bullet be a normalized dualizing complex. If A → B is flat and m_A B = m_B, then ω_A^bullet ⊗_A B is a normalized dualizing complex for B.","statement_latex":"Let $A \\to B$ be a local homomorphism of Noetherian local rings.\nLet $\\omega_A^\\bullet$ be a normalized dualizing complex.\nIf $A \\to B$ is flat and $\\mathfrak m_A B = \\mathfrak m_B$,\nthen $\\omega_A^\\bullet \\otimes_A B$ is a normalized dualizing\ncomplex for $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"The ubiquity of dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWD","source_file":"dualizing.tex","source_line":4037,"source_end_line":4044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4037-L4044","statement_sha256":"0854e9b95bdf226b4a6a8f1da1f8db524a2285bb46749bc732bc826b83273e12","origin":"The Stacks Project","memory_eligible":false,"source_rank":8743,"rank":8743,"depth":20,"x":2159.688,"y":1174.478,"cluster":"duality-cohomology"},{"id":"stacks:0DWC","tag":"0DWC","title":"The ubiquity of dualizing complexes · Lemma 0DWC","summary":"Let A → B be a flat map of Noetherian rings. Let I ⊂ A be an ideal such that A/I = B/IB and such that IB is contained in the Jacobson radical of B. Let ω_A^bullet be a dualizing complex. Then ω_A^bullet ⊗_A B is a dualizing complex for B.","statement_latex":"Let $A \\to B$ be a flat map of Noetherian rings. Let\n$I \\subset A$ be an ideal such that $A/I = B/IB$ and\nsuch that $IB$ is contained in the Jacobson radical of $B$.\nLet $\\omega_A^\\bullet$ be a dualizing complex.\nThen $\\omega_A^\\bullet \\otimes_A B$ is a dualizing\ncomplex for $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"The ubiquity of dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWC","source_file":"dualizing.tex","source_line":4067,"source_end_line":4075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4067-L4075","statement_sha256":"6364de9477baf55d37e838db09b621eda6e4cd4199f5dff32909d18c40d07bef","origin":"The Stacks Project","memory_eligible":false,"source_rank":8744,"rank":8744,"depth":20,"x":1906.46,"y":1188.236,"cluster":"duality-cohomology"},{"id":"stacks:0DWD","tag":"0DWD","title":"The ubiquity of dualizing complexes · Lemma 0DWD","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let ω_A^bullet be a dualizing complex. • ω_A^bullet ⊗_A A^h is a dualizing complex on the henselization (A^h, I^h) of the pair (A, I), • ω_A^bullet ⊗_A A^wedge is a dualizing complex on the I-adic completion A^wedge, and • if A is local, then ω_A^bullet ⊗_A A^h, resp. ω_A^bullet ⊗_A A^sh is a dualzing complex on the henselization, resp. strict henselization of A.","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nLet $\\omega_A^\\bullet$ be a dualizing complex.\n\\begin{enumerate}\n\\item $\\omega_A^\\bullet \\otimes_A A^h$ is a dualizing complex on the\nhenselization $(A^h, I^h)$ of the pair $(A, I)$,\n\\item $\\omega_A^\\bullet \\otimes_A A^\\wedge$ is a dualizing complex on\nthe $I$-adic completion $A^\\wedge$, and\n\\item if $A$ is local, then $\\omega_A^\\bullet \\otimes_A A^h$,\nresp.\\ $\\omega_A^\\bullet \\otimes_A A^{sh}$ is a dualzing complex\non the henselization, resp.\\ strict henselization of $A$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"The ubiquity of dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWD","source_file":"dualizing.tex","source_line":4099,"source_end_line":4112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4099-L4112","statement_sha256":"a0755ed51f590b3b4b231eebe02b9b8962a3a60726c3d8cde85b02152666188b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8745,"rank":8745,"depth":21,"x":2082.384,"y":1034.174,"cluster":"duality-cohomology"},{"id":"stacks:0BFR","tag":"0BFR","title":"The ubiquity of dualizing complexes · Lemma 0BFR","summary":"The following types of rings have a dualizing complex: • fields, • Noetherian complete local rings, • Z, • Dedekind domains, • any ring which is obtained from one of the rings above by taking an algebra essentially of finite type, or by taking an ideal-adic completion, or by taking a henselization, or by taking a strict henselization.","statement_latex":"The following types of rings have a dualizing complex:\n\\begin{enumerate}\n\\item fields,\n\\item Noetherian complete local rings,\n\\item $\\mathbf{Z}$,\n\\item Dedekind domains,\n\\item any ring which is obtained from one of the rings above by\ntaking an algebra essentially of finite type, or by taking an\nideal-adic completion, or by taking a henselization, \nor by taking a strict henselization.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"The ubiquity of dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFR","source_file":"dualizing.tex","source_line":4125,"source_end_line":4138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4125-L4138","statement_sha256":"e8cb83ed1527068d00a90335449fe92268170fe77487ada9f2e26c8670a29fd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8746,"rank":8746,"depth":36,"x":2076.544,"y":1247.918,"cluster":"duality-cohomology"},{"id":"stacks:0BJN","tag":"0BJN","title":"Formal fibres · Lemma 0BJN","summary":"Properties (A), (B), (C), (D), and (E) of More on Algebra, Section [Tag 0BIR] hold for P(k → R) =\"R is a Gorenstein ring\".","statement_latex":"Properties (A), (B), (C), (D), and (E) of\nMore on Algebra, Section \\ref{more-algebra-section-properties-formal-fibres}\nhold for $P(k \\to R) =$``$R$ is a Gorenstein ring''.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJN","source_file":"dualizing.tex","source_line":4190,"source_end_line":4195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4190-L4195","statement_sha256":"ca4185cc796b0a7f5351a944be2152f318e41728ab4f6ae6618a85b1a33e218c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8747,"rank":8747,"depth":41,"x":1908.716,"y":1086.755,"cluster":"duality-cohomology"},{"id":"stacks:0AWY","tag":"0AWY","title":"Formal fibres · Lemma 0AWY","summary":"Let A be a Noetherian local ring. If A has a dualizing complex, then the formal fibres of A are Gorenstein.","statement_latex":"Let $A$ be a Noetherian local ring. If $A$ has a dualizing complex,\nthen the formal fibres of $A$ are Gorenstein.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWY","source_file":"dualizing.tex","source_line":4234,"source_end_line":4238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4234-L4238","statement_sha256":"23ca7aa5247a1c266054291c2665f04d8fdf18da4dc7a5169f1fbcd4b9b0df13","origin":"The Stacks Project","memory_eligible":false,"source_rank":8748,"rank":8748,"depth":35,"x":2162.446,"y":1110.399,"cluster":"duality-cohomology"},{"id":"stacks:0BJP","tag":"0BJP","title":"Formal fibres · Lemma 0BJP","summary":"Properties (A), (B), (C), (D), and (E) of More on Algebra, Section [Tag 0BIR] hold for P(k → R) =\"R is a local complete intersection\". See Divided Power Algebra, Definition [Tag 09Q3].","statement_latex":"Properties (A), (B), (C), (D), and (E) of\nMore on Algebra, Section \\ref{more-algebra-section-properties-formal-fibres}\nhold for $P(k \\to R) =$``$R$ is a local complete intersection''.\nSee Divided Power Algebra, Definition \\ref{dpa-definition-lci}.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Formal fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJP","source_file":"dualizing.tex","source_line":4270,"source_end_line":4276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4270-L4276","statement_sha256":"8fa89bd18b397f66bffe96f5817c6c121b43e8f881299ba45536b5b4bf42ed87","origin":"The Stacks Project","memory_eligible":false,"source_rank":8749,"rank":8749,"depth":41,"x":1956.032,"y":1237.122,"cluster":"duality-cohomology"},{"id":"stacks:0BZJ","tag":"0BZJ","title":"Upper shriek algebraically · Lemma 0BZJ","summary":"Let φ : R → A be a finite type homomorphism of Noetherian rings. The functor φ^! is well defined up to isomorphism.","statement_latex":"Let $\\varphi : R \\to A$ be a finite type homomorphism of\nNoetherian rings. The functor $\\varphi^!$ is well defined\nup to isomorphism.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZJ","source_file":"dualizing.tex","source_line":4364,"source_end_line":4369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4364-L4369","statement_sha256":"91b92519d7b79b2c524b7f534f5b4f4e19cc016f6fb66ef4b224bef250baae61","origin":"The Stacks Project","memory_eligible":false,"source_rank":8750,"rank":8750,"depth":19,"x":2006.405,"y":1026.246,"cluster":"duality-cohomology"},{"id":"stacks:0BZK","tag":"0BZK","title":"Upper shriek algebraically · Lemma 0BZK","summary":"Let φ : R → A be a finite type homomorphism of Noetherian rings. • φ^! maps D^+(R) into D^+(A) and D^+_Coh(R) into D^+_Coh(A). • if φ is perfect, then φ^! maps D^-(R) into D^-(A), D^-_Coh(R) into D^-_Coh(A), and D^b_Coh(R) into D^b_Coh(A).","statement_latex":"Let $\\varphi : R \\to A$ be a finite type homomorphism of Noetherian rings.\n\\begin{enumerate}\n\\item $\\varphi^!$ maps $D^+(R)$ into $D^+(A)$ and\n$D^+_{\\textit{Coh}}(R)$ into $D^+_{\\textit{Coh}}(A)$.\n\\item if $\\varphi$ is perfect, then $\\varphi^!$ maps\n$D^-(R)$ into $D^-(A)$,\n$D^-_{\\textit{Coh}}(R)$ into $D^-_{\\textit{Coh}}(A)$, and\n$D^b_{\\textit{Coh}}(R)$ into $D^b_{\\textit{Coh}}(A)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZK","source_file":"dualizing.tex","source_line":4411,"source_end_line":4422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4411-L4422","statement_sha256":"a542883d4beec83cd2a7e81383892e747a0a57ff5360dc775e36efd74a804be6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8751,"rank":8751,"depth":17,"x":2139.035,"y":1210.595,"cluster":"duality-cohomology"},{"id":"stacks:0BZL","tag":"0BZL","title":"Upper shriek algebraically · Lemma 0BZL","summary":"Let φ be a finite type homomorphism of Noetherian rings. If ω_R^bullet is a dualizing complex for R, then φ^!(ω_R^bullet) is a dualizing complex for A.","statement_latex":"Let $\\varphi$ be a finite type homomorphism of Noetherian rings.\nIf $\\omega_R^\\bullet$ is a dualizing complex for $R$, then\n$\\varphi^!(\\omega_R^\\bullet)$ is a dualizing complex for $A$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZL","source_file":"dualizing.tex","source_line":4447,"source_end_line":4452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4447-L4452","statement_sha256":"d41fac639528ee479aeae0be093b7d6c82c466ac425b4aa0176679961ce90b46","origin":"The Stacks Project","memory_eligible":false,"source_rank":8752,"rank":8752,"depth":35,"x":1892.629,"y":1149.828,"cluster":"duality-cohomology"},{"id":"stacks:0BZN","tag":"0BZN","title":"Upper shriek algebraically · Lemma 0BZN","summary":"Let R → R' be a flat homomorphism of Noetherian rings. Let φ : R → A be a finite type ring map. Let φ' : R' → A' = A ⊗_R R' be the map induced by φ. Then we have a functorial maps φ^!(K) ⊗_A^L A' → (φ')^!(K ⊗_R^L R') for K in D(R) which are isomorphisms for K ∈ D^+(R).","statement_latex":"Let $R \\to R'$ be a flat homomorphism of Noetherian rings.\nLet $\\varphi : R \\to A$ be a finite type ring map.\nLet $\\varphi' : R' \\to A' = A \\otimes_R R'$ be the map induced by $\\varphi$.\nThen we have a functorial maps\n$$\n\\varphi^!(K) \\otimes_A^\\mathbf{L} A' \\longrightarrow\n(\\varphi')^!(K \\otimes_R^\\mathbf{L} R')\n$$\nfor $K$ in $D(R)$ which are isomorphisms for $K \\in D^+(R)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZN","source_file":"dualizing.tex","source_line":4460,"source_end_line":4471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4460-L4471","statement_sha256":"7a8f844ee53f37a194ee8ecdda3c8e1071fce1cca592eb42bb6d824a868458ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":8753,"rank":8753,"depth":22,"x":2123.528,"y":1054.682,"cluster":"duality-cohomology"},{"id":"stacks:0BZR","tag":"0BZR","title":"Upper shriek algebraically · Lemma 0BZR","summary":"Let R → R' be a homomorphism of Noetherian rings. Let φ : R → A be a perfect ring map (More on Algebra, Definition [Tag 067H]) such that R' and A are tor independent over R. Let φ' : R' → A' = A ⊗_R R' be the map induced by φ. Then we have a functorial isomorphism φ^!(K) ⊗_A^L A' = (φ')^!(K ⊗_R^L R') for K in D(R).","statement_latex":"Let $R \\to R'$ be a homomorphism of Noetherian rings.\nLet $\\varphi : R \\to A$ be a perfect ring map\n(More on Algebra, Definition\n\\ref{more-algebra-definition-pseudo-coherent-perfect})\nsuch that $R'$ and $A$ are tor independent over $R$.\nLet $\\varphi' : R' \\to A' = A \\otimes_R R'$ be the map induced by $\\varphi$.\nThen we have a functorial isomorphism\n$$\n\\varphi^!(K) \\otimes_A^\\mathbf{L} A' =\n(\\varphi')^!(K \\otimes_R^\\mathbf{L} R')\n$$\nfor $K$ in $D(R)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZR","source_file":"dualizing.tex","source_line":4492,"source_end_line":4506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4492-L4506","statement_sha256":"e594853db0ca86d09bf560302f82333a9c3c5f2079af932d502578c79c4e88dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8754,"rank":8754,"depth":22,"x":2029.644,"y":1256.15,"cluster":"duality-cohomology"},{"id":"stacks:0BZS","tag":"0BZS","title":"Upper shriek algebraically · Lemma 0BZS","summary":"Let R → R' be a homomorphism of Noetherian rings. Let φ : R → A be flat of finite type. Let φ' : R' → A' = A ⊗_R R' be the map induced by φ. Then we have a functorial isomorphism φ^!(K) ⊗_A^L A' = (φ')^!(K ⊗_R^L R') for K in D(R).","statement_latex":"Let $R \\to R'$ be a homomorphism of Noetherian rings.\nLet $\\varphi : R \\to A$ be flat of finite type.\nLet $\\varphi' : R' \\to A' = A \\otimes_R R'$ be the map induced by $\\varphi$.\nThen we have a functorial isomorphism\n$$\n\\varphi^!(K) \\otimes_A^\\mathbf{L} A' =\n(\\varphi')^!(K \\otimes_R^\\mathbf{L} R')\n$$\nfor $K$ in $D(R)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZS","source_file":"dualizing.tex","source_line":4538,"source_end_line":4549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4538-L4549","statement_sha256":"e7752898cb0bfea15ccb9001ba5bbef7fb9a0162d506089fa41ad3cd54c7b32c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8755,"rank":8755,"depth":36,"x":1936.724,"y":1054.027,"cluster":"duality-cohomology"},{"id":"stacks:0BZT","tag":"0BZT","title":"Upper shriek algebraically · Lemma 0BZT","summary":"Let A xrightarrowa B xrightarrowb C be finite type homomorphisms of Noetherian rings. Then there is a transformation of functors b^! ∘ a^! → (b ∘ a)^! which is an isomorphism on D^+(A).","statement_latex":"Let $A \\xrightarrow{a} B \\xrightarrow{b} C$ be finite type homomorphisms of\nNoetherian rings. Then there is a transformation of functors\n$b^! \\circ a^! \\to (b \\circ a)^!$ which is an isomorphism on $D^+(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZT","source_file":"dualizing.tex","source_line":4557,"source_end_line":4562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4557-L4562","statement_sha256":"3aa5a99395e5462a36d03136811503db3da1fefe9e20b1482cf731c2f28f28cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8756,"rank":8756,"depth":22,"x":2168.115,"y":1150.483,"cluster":"duality-cohomology"},{"id":"stacks:0C0G","tag":"0C0G","title":"Upper shriek algebraically · Lemma 0C0G","summary":"Let φ : R → A be a finite map of Noetherian rings. Then φ^! is isomorphic to the functor RHom(A, -) : D(R) → D(A) from Section [Tag 0A6Z].","statement_latex":"Let $\\varphi : R \\to A$ be a finite map of Noetherian rings.\nThen $\\varphi^!$ is isomorphic to the functor\n$R\\Hom(A, -) : D(R) \\to D(A)$ from\nSection \\ref{section-trivial}.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0G","source_file":"dualizing.tex","source_line":4598,"source_end_line":4604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4598-L4604","statement_sha256":"45dca7744613b53434c3f1e24cc51221c1d6f2ea9747f8fb06012ebbdb468849","origin":"The Stacks Project","memory_eligible":false,"source_rank":8757,"rank":8757,"depth":23,"x":1919.568,"y":1210.741,"cluster":"duality-cohomology"},{"id":"stacks:0C0H","tag":"0C0H","title":"Upper shriek algebraically · Lemma 0C0H","summary":"Let R be a Noetherian ring and let f ∈ R. If φ denotes the map R → R_f, then φ^! is isomorphic to - ⊗_R^L R_f. More generally, if φ : R → R' is a map such that Spec(R') → Spec(R) is an open immersion, then φ^! is isomorphic to - ⊗_R^L R'.","statement_latex":"Let $R$ be a Noetherian ring and let $f \\in R$.\nIf $\\varphi$ denotes the map $R \\to R_f$, then $\\varphi^!$\nis isomorphic to $- \\otimes_R^\\mathbf{L} R_f$.\nMore generally, if $\\varphi : R \\to R'$ is a map such that\n$\\Spec(R') \\to \\Spec(R)$ is an open immersion, then\n$\\varphi^!$ is isomorphic to $- \\otimes_R^\\mathbf{L} R'$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0H","source_file":"dualizing.tex","source_line":4647,"source_end_line":4655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4647-L4655","statement_sha256":"d0cb1f90662d93a9829df860b55dc3f7af1b8a7c8945a9163cc34c656f60023e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8758,"rank":8758,"depth":23,"x":2054.578,"y":1025.01,"cluster":"duality-cohomology"},{"id":"stacks:0BZU","tag":"0BZU","title":"Upper shriek algebraically · Lemma 0BZU","summary":"Let φ : R → A be a perfect homomorphism of Noetherian rings (for example φ is flat of finite type). Then φ^!(K) = K ⊗_R^L φ^!(R) for K ∈ D(R).","statement_latex":"Let $\\varphi : R \\to A$ be a perfect homomorphism of Noetherian rings\n(for example $\\varphi$ is flat of finite type).\nThen $\\varphi^!(K) = K \\otimes_R^\\mathbf{L} \\varphi^!(R)$\nfor $K \\in D(R)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZU","source_file":"dualizing.tex","source_line":4672,"source_end_line":4678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4672-L4678","statement_sha256":"fdba2107b1817a07d0905036400be347c44d9dd9cb309e4319d2a16bd3e8e8ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":8759,"rank":8759,"depth":36,"x":2104.454,"y":1238.878,"cluster":"duality-cohomology"},{"id":"stacks:0E9L","tag":"0E9L","title":"Upper shriek algebraically · Lemma 0E9L","summary":"Let φ : A → B be a finite type homomorphism of Noetherian rings. Let ω_A^bullet be a dualizing complex for A. Set ω_B^bullet = φ^!(ω_A^bullet). Denote D_A(K) = RHom_A(K, ω_A^bullet) for K ∈ D_Coh(A) and D_B(L) = RHom_B(L, ω_B^bullet) for L ∈ D_Coh(B). Then there is a functorial isomorphism φ^!(K) = D_B(D_A(K) ⊗_A^L B) for K ∈ D_Coh(A).","statement_latex":"Let $\\varphi : A \\to B$ be a finite type homomorphism of Noetherian rings.\nLet $\\omega_A^\\bullet$ be a dualizing complex for $A$. Set\n$\\omega_B^\\bullet = \\varphi^!(\\omega_A^\\bullet)$. Denote\n$D_A(K) = R\\Hom_A(K, \\omega_A^\\bullet)$ for $K \\in D_{\\textit{Coh}}(A)$\nand\n$D_B(L) = R\\Hom_B(L, \\omega_B^\\bullet)$ for $L \\in D_{\\textit{Coh}}(B)$.\nThen there is a functorial isomorphism\n$$\n\\varphi^!(K) = D_B(D_A(K) \\otimes_A^\\mathbf{L} B)\n$$\nfor $K \\in D_{\\textit{Coh}}(A)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Upper shriek algebraically","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9L","source_file":"dualizing.tex","source_line":4693,"source_end_line":4706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4693-L4706","statement_sha256":"7a344bbaaf800b7954d16209627e8b6c5c79af3e334638e00fa700b2c8322300","origin":"The Stacks Project","memory_eligible":false,"source_rank":8760,"rank":8760,"depth":36,"x":1895.393,"y":1109.292,"cluster":"duality-cohomology"},{"id":"stacks:0BZV","tag":"0BZV","title":"Relative dualizing complexes in the Noetherian case · Lemma 0BZV","summary":"Let R → R' be a homomorphism of Noetherian rings. Let R → A be of finite type. Set A' = A ⊗_R R'. If • R → R' is flat, or • R → A is flat, or • R → A is perfect and R' and A are tor independent over R, then there is an isomorphism ω_A/R^bullet ⊗_A^L A' → ω^bullet_A'/R' in D(A').","statement_latex":"Let $R \\to R'$ be a homomorphism of Noetherian rings.\nLet $R \\to A$ be of finite type. Set $A' = A \\otimes_R R'$. If\n\\begin{enumerate}\n\\item $R \\to R'$ is flat, or\n\\item $R \\to A$ is flat, or\n\\item $R \\to A$ is perfect\nand $R'$ and $A$ are tor independent over $R$,\n\\end{enumerate}\nthen there is an isomorphism\n$\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L} A' \\to \\omega^\\bullet_{A'/R'}$\nin $D(A')$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZV","source_file":"dualizing.tex","source_line":4805,"source_end_line":4818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4805-L4818","statement_sha256":"ba802f3d589ed44ca1d1c04be7d7c6ff5913badcdca6723f62b4d1ee98a06da7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8761,"rank":8761,"depth":37,"x":2154.127,"y":1086.188,"cluster":"duality-cohomology"},{"id":"stacks:0BZW","tag":"0BZW","title":"Relative dualizing complexes in the Noetherian case · Lemma 0BZW","summary":"Let φ : R → A be a finite type map of Noetherian rings. • If φ is perfect, then • ω_A/R^bullet is in D^b_Coh(A), • ω_A/R^bullet has finite tor dimension over R, and • A → RHom_A(ω_A/R^bullet, ω_A/R^bullet) is an isomorphism. • If φ is flat, then (1)(a), (1)(b), and (1)(c) hold and • ω_A/R^bullet is R-perfect (More on Algebra, Definition [Tag 0DHS]), and • for every map R → k to a field the base change ω_A/R^bullet ⊗_A^L (A ⊗_R k) is a dualizing complex for A ⊗_R k.","statement_latex":"Let $\\varphi : R \\to A$ be a finite type map of Noetherian rings.\n\\begin{enumerate}\n\\item If $\\varphi$ is perfect, then\n\\begin{enumerate}\n\\item $\\omega_{A/R}^\\bullet$ is in $D^b_{\\textit{Coh}}(A)$,\n\\item $\\omega_{A/R}^\\bullet$ has finite tor dimension over $R$, and\n\\item $A \\to R\\Hom_A(\\omega_{A/R}^\\bullet, \\omega_{A/R}^\\bullet)$\nis an isomorphism.\n\\end{enumerate}\n\\item If $\\varphi$ is flat, then (1)(a), (1)(b), and (1)(c) hold and\n\\begin{enumerate}\n\\item $\\omega_{A/R}^\\bullet$ is $R$-perfect\n(More on Algebra,\nDefinition \\ref{more-algebra-definition-relatively-perfect}), and\n\\item for every map $R \\to k$ to a field the base change\n$\\omega_{A/R}^\\bullet \\otimes_A^\\mathbf{L} (A \\otimes_R k)$\nis a dualizing complex for $A \\otimes_R k$.\n\\end{enumerate}\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZW","source_file":"dualizing.tex","source_line":4825,"source_end_line":4846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4825-L4846","statement_sha256":"944cad9ef4f943c8c1a8154fe9801bdb788bde6f91e1291a83c47876d44d41ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":8762,"rank":8762,"depth":38,"x":1981.676,"y":1250.268,"cluster":"duality-cohomology"},{"id":"stacks:0E0P","tag":"0E0P","title":"Relative dualizing complexes in the Noetherian case · Lemma 0E0P","summary":"Let K/k be an extension of fields. Let A be a finite type k-algebra. Let A_K = A ⊗_k K. If ω_A^bullet is a dualizing complex for A, then ω_A^bullet ⊗_A A_K is a dualizing complex for A_K.","statement_latex":"Let $K/k$ be an extension of fields. Let $A$ be a finite type\n$k$-algebra. Let $A_K = A \\otimes_k K$. If\n$\\omega_A^\\bullet$ is a dualizing complex for $A$, then\n$\\omega_A^\\bullet \\otimes_A A_K$ is a dualizing complex for $A_K$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0P","source_file":"dualizing.tex","source_line":4892,"source_end_line":4898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4892-L4898","statement_sha256":"6c3587f2b5542e12a5c46b984eeefc4b04f4a4a52bc688724de600d25a6fc336","origin":"The Stacks Project","memory_eligible":false,"source_rank":8763,"rank":8763,"depth":39,"x":1976.884,"y":1031.117,"cluster":"duality-cohomology"},{"id":"stacks:0E4B","tag":"0E4B","title":"Relative dualizing complexes in the Noetherian case · Lemma 0E4B","summary":"Let φ : R → A be a local complete intersection homomorphism of Noetherian rings. Then ω_A/R^bullet is an invertible object of D(A) and φ^!(K) = K ⊗_R^L ω_A/R^bullet for all K ∈ D(R).","statement_latex":"Let $\\varphi : R \\to A$ be a local complete intersection homomorphism of\nNoetherian rings. Then $\\omega_{A/R}^\\bullet$ is an invertible object of\n$D(A)$ and $\\varphi^!(K) = K \\otimes_R^\\mathbf{L} \\omega_{A/R}^\\bullet$\nfor all $K \\in D(R)$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4B","source_file":"dualizing.tex","source_line":4910,"source_end_line":4916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4910-L4916","statement_sha256":"dfaf408818d687d159362661ffd377f3ced81e013b6822e706e57bb1e958b493","origin":"The Stacks Project","memory_eligible":false,"source_rank":8764,"rank":8764,"depth":37,"x":2156.906,"y":1190.22,"cluster":"duality-cohomology"},{"id":"stacks:0E4C","tag":"0E4C","title":"Relative dualizing complexes in the Noetherian case · Lemma 0E4C","summary":"Let φ : R → A be a flat finite type homomorphism of Noetherian rings. The following are equivalent • the fibres A ⊗_R kappa( p) are Gorenstein for all primes p ⊂ R, and • ω_A/R^bullet is an invertible object of D(A), see More on Algebra, Lemma [Tag 0FNT].","statement_latex":"Let $\\varphi : R \\to A$ be a flat finite type homomorphism of Noetherian rings.\nThe following are equivalent\n\\begin{enumerate}\n\\item the fibres $A \\otimes_R \\kappa(\\mathfrak p)$ are Gorenstein\nfor all primes $\\mathfrak p \\subset R$, and\n\\item $\\omega_{A/R}^\\bullet$ is an invertible object of $D(A)$, see\nMore on Algebra, Lemma \\ref{more-algebra-lemma-invertible-derived}.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4C","source_file":"dualizing.tex","source_line":4955,"source_end_line":4965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L4955-L4965","statement_sha256":"6e0af88f28930161cc1b73c08a53ca0e5a5844974f3aa56de66075f6300a7d87","origin":"The Stacks Project","memory_eligible":false,"source_rank":8765,"rank":8765,"depth":39,"x":1895.848,"y":1175.027,"cluster":"duality-cohomology"},{"id":"stacks:0E9N","tag":"0E9N","title":"Relative dualizing complexes in the Noetherian case · Lemma 0E9N","summary":"Let φ : R → A be a finite type homomorphism of Noetherian rings. Assume R local and let m ⊂ A be a maximal ideal lying over the maximal ideal of R. If ω_R^bullet is a normalized dualizing complex for R, then φ^!(ω_R^bullet)_ m is a normalized dualizing complex for A_ m.","statement_latex":"Let $\\varphi : R \\to A$ be a finite type homomorphism of Noetherian rings.\nAssume $R$ local and let $\\mathfrak m \\subset A$ be a maximal\nideal lying over the maximal ideal of $R$. If $\\omega_R^\\bullet$\nis a normalized dualizing complex for $R$, then\n$\\varphi^!(\\omega_R^\\bullet)_\\mathfrak m$ is a normalized\ndualizing complex for $A_\\mathfrak m$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9N","source_file":"dualizing.tex","source_line":5003,"source_end_line":5011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5003-L5011","statement_sha256":"e4cf62809f03c165a55d59185d58ff3c76b2057bce7a51f4421f0a653ac4e2a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8766,"rank":8766,"depth":37,"x":2100.852,"y":1037.908,"cluster":"duality-cohomology"},{"id":"stacks:0E9P","tag":"0E9P","title":"Relative dualizing complexes in the Noetherian case · Lemma 0E9P","summary":"Let R → A be a finite type homomorphism of Noetherian rings. Let q ⊂ A be a prime ideal lying over p ⊂ R. Then H^i(ω_A/R^bullet)_ q not = 0 ⇒ - d ≤ i where d is the dimension of the fibre of Spec(A) → Spec(R) over p at the point q.","statement_latex":"Let $R \\to A$ be a finite type homomorphism of Noetherian rings.\nLet $\\mathfrak q \\subset A$ be a prime ideal lying over\n$\\mathfrak p \\subset R$. Then\n$$\nH^i(\\omega_{A/R}^\\bullet)_\\mathfrak q \\not = 0\n\\Rightarrow - d \\leq i\n$$\nwhere $d$ is the dimension of the fibre of $\\Spec(A) \\to \\Spec(R)$\nover $\\mathfrak p$ at the point $\\mathfrak q$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9P","source_file":"dualizing.tex","source_line":5039,"source_end_line":5050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5039-L5050","statement_sha256":"811b632ae7741bffdd1af1a8706922b0dc6e930ecc4ea22c62c05d4f8135c677","origin":"The Stacks Project","memory_eligible":false,"source_rank":8767,"rank":8767,"depth":24,"x":2059.895,"y":1255.645,"cluster":"duality-cohomology"},{"id":"stacks:0E9Q","tag":"0E9Q","title":"Relative dualizing complexes in the Noetherian case · Lemma 0E9Q","summary":"Let R → A be a flat finite type homomorphism of Noetherian rings. Let q ⊂ A be a prime ideal lying over p ⊂ R. Then H^i(ω_A/R^bullet)_ q not = 0 ⇒ - d ≤ i ≤ 0 where d is the dimension of the fibre of Spec(A) → Spec(R) over p at the point q. If all fibres of Spec(A) → Spec(R) have dimension ≤ d, then ω_A/R^bullet has tor amplitude in [-d, 0] as a complex of R-modules.","statement_latex":"Let $R \\to A$ be a flat finite type homomorphism of Noetherian rings.\nLet $\\mathfrak q \\subset A$ be a prime ideal lying over\n$\\mathfrak p \\subset R$. Then\n$$\nH^i(\\omega_{A/R}^\\bullet)_\\mathfrak q \\not = 0\n\\Rightarrow - d \\leq i \\leq 0\n$$\nwhere $d$ is the dimension of the fibre of $\\Spec(A) \\to \\Spec(R)$\nover $\\mathfrak p$ at the point $\\mathfrak q$. If all fibres of\n$\\Spec(A) \\to \\Spec(R)$ have dimension $\\leq d$, then\n$\\omega_{A/R}^\\bullet$ has tor amplitude in $[-d, 0]$\nas a complex of $R$-modules.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9Q","source_file":"dualizing.tex","source_line":5081,"source_end_line":5095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5081-L5095","statement_sha256":"2a29b21415cf2ae9906ad48a9b6e52caa08533c2d1e5d1b1ed0680d93547f89e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8768,"rank":8768,"depth":38,"x":1914.797,"y":1071.594,"cluster":"duality-cohomology"},{"id":"stacks:0E9R","tag":"0E9R","title":"Relative dualizing complexes in the Noetherian case · Lemma 0E9R","summary":"Let R → A be a finite type homomorphism of Noetherian rings. Let p ⊂ R be a prime ideal. Assume • R_ p is Cohen-Macaulay, and • for any minimal prime q ⊂ A we have trdeg_kappa(R ∩ q) kappa( q) ≤ r. Then H^i(ω_A/R^bullet)_ p not = 0 ⇒ - r ≤ i and H^-r(ω_A/R^bullet)_ p is (S_2) as an A_ p-module.","statement_latex":"Let $R \\to A$ be a finite type homomorphism of Noetherian rings.\nLet $\\mathfrak p \\subset R$ be a prime ideal. Assume\n\\begin{enumerate}\n\\item $R_\\mathfrak p$ is Cohen-Macaulay, and\n\\item for any minimal prime $\\mathfrak q \\subset A$ we have\n$\\text{trdeg}_{\\kappa(R \\cap \\mathfrak q)} \\kappa(\\mathfrak q) \\leq r$.\n\\end{enumerate}\nThen\n$$\nH^i(\\omega_{A/R}^\\bullet)_\\mathfrak p \\not = 0 \\Rightarrow - r \\leq i\n$$\nand $H^{-r}(\\omega_{A/R}^\\bullet)_\\mathfrak p$ is $(S_2)$\nas an $A_\\mathfrak p$-module.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes in the Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9R","source_file":"dualizing.tex","source_line":5131,"source_end_line":5146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5131-L5146","statement_sha256":"7a7b0eec517e013932106ca3ca36e96e650c593c0218b4867855a83161e83b31","origin":"The Stacks Project","memory_eligible":false,"source_rank":8769,"rank":8769,"depth":40,"x":2170.153,"y":1125.052,"cluster":"duality-cohomology"},{"id":"stacks:0E4A","tag":"0E4A","title":"More on dualizing complexes · Lemma 0E4A","summary":"Let A → B be a faithfully flat map of Noetherian rings. If K ∈ D(A) and K ⊗_A^L B is a dualizing complex for B, then K is a dualizing complex for A.","statement_latex":"Let $A \\to B$ be a faithfully flat map of Noetherian rings.\nIf $K \\in D(A)$ and $K \\otimes_A^\\mathbf{L} B$\nis a dualizing complex for $B$, then $K$ is a dualizing complex\nfor $A$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"More on dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4A","source_file":"dualizing.tex","source_line":5223,"source_end_line":5229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5223-L5229","statement_sha256":"1850a0634f82eafcb607d170edd827d4895699931ba20261f276c8484b318795","origin":"The Stacks Project","memory_eligible":false,"source_rank":8770,"rank":8770,"depth":20,"x":1938.548,"y":1230.673,"cluster":"duality-cohomology"},{"id":"stacks:0E4D","tag":"0E4D","title":"More on dualizing complexes · Lemma 0E4D","summary":"Let φ : A → B be a homomorphism of Noetherian rings. Assume • A → B is syntomic and induces a surjective map on spectra, or • A → B is a faithfully flat local complete intersection, or • A → B is faithfully flat of finite type with Gorenstein fibres. Then K ∈ D(A) is a dualizing complex for A if and only if K ⊗_A^L B is a dualizing complex for B.","statement_latex":"Let $\\varphi : A \\to B$ be a homomorphism of Noetherian rings. Assume\n\\begin{enumerate}\n\\item $A \\to B$ is syntomic and induces a surjective map on spectra, or\n\\item $A \\to B$ is a faithfully flat local complete intersection, or\n\\item $A \\to B$ is faithfully flat of finite type with Gorenstein fibres.\n\\end{enumerate}\nThen $K \\in D(A)$ is a dualizing complex for $A$ if and only if\n$K \\otimes_A^\\mathbf{L} B$ is a dualizing complex for $B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"More on dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4D","source_file":"dualizing.tex","source_line":5259,"source_end_line":5269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5259-L5269","statement_sha256":"5a01fbaeab2a32b99d81274e463e18b43fd7158c5079a4b0267742dfe5209f5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8771,"rank":8771,"depth":40,"x":2024.511,"y":1021.084,"cluster":"duality-cohomology"},{"id":"stacks:0E4E","tag":"0E4E","title":"More on dualizing complexes · Lemma 0E4E","summary":"Let (A, m, kappa) → (B, n, l) be a flat local homorphism of Noetherian rings such that n = m B. If E is the injective hull of kappa, then E ⊗_A B is the injective hull of l.","statement_latex":"Let $(A, \\mathfrak m, \\kappa) \\to (B, \\mathfrak n, l)$\nbe a flat local homorphism of Noetherian rings such that\n$\\mathfrak n = \\mathfrak m B$. If $E$ is the injective\nhull of $\\kappa$, then $E \\otimes_A B$ is the injective\nhull of $l$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"More on dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4E","source_file":"dualizing.tex","source_line":5294,"source_end_line":5301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5294-L5301","statement_sha256":"19b240314afbcd39716067315b1cd23e14424ca0c08d8c2f8d3bd1c6dd1d990c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8772,"rank":8772,"depth":5,"x":2129.814,"y":1224.689,"cluster":"duality-cohomology"},{"id":"stacks:0E4F","tag":"0E4F","title":"More on dualizing complexes · Lemma 0E4F","summary":"Let φ : A → B be a flat homorphism of Noetherian rings such that for all primes q ⊂ B we have p B_ q = qB_ q where p = φ^-1( q), for example if φ is étale. If I is an injective A-module, then I ⊗_A B is an injective B-module.","statement_latex":"Let $\\varphi : A \\to B$ be a flat homorphism of Noetherian rings such\nthat for all primes $\\mathfrak q \\subset B$ we have\n$\\mathfrak p B_\\mathfrak q = \\mathfrak qB_\\mathfrak q$\nwhere $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$, for example\nif $\\varphi$ is \\'etale.\nIf $I$ is an injective $A$-module, then $I \\otimes_A B$ is\nan injective $B$-module.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"More on dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4F","source_file":"dualizing.tex","source_line":5328,"source_end_line":5337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5328-L5337","statement_sha256":"aee26f5fa00646256fcee6f58d218180e0d19a19851f9b39e10130bc98203fa7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8773,"rank":8773,"depth":41,"x":1888.101,"y":1134.181,"cluster":"duality-cohomology"},{"id":"stacks:0E2C","tag":"0E2C","title":"Relative dualizing complexes · Definition 0E2C","summary":"Let R → A be a flat ring map of finite presentation. A relative dualizing complex is an object K ∈ D(A) such that • K is R-perfect (More on Algebra, Definition [Tag 0DHS]), and • RHom_A ⊗_R A(A, K ⊗_A^L (A ⊗_R A)) is isomorphic to A.","statement_latex":"Let $R \\to A$ be a flat ring map of finite presentation.\nA {\\it relative dualizing complex} is an object $K \\in D(A)$ such that\n\\begin{enumerate}\n\\item $K$ is $R$-perfect (More on Algebra, Definition\n\\ref{more-algebra-definition-relatively-perfect}), and\n\\item $R\\Hom_{A \\otimes_R A}(A, K \\otimes_A^\\mathbf{L} (A \\otimes_R A))$\nis isomorphic to $A$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2C","source_file":"dualizing.tex","source_line":5395,"source_end_line":5405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5395-L5405","statement_sha256":"3699482c29094dffa4a44b6cdcb56b29032d8da783e82ed17e0b2fa3f585484f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8774,"rank":8774,"depth":2,"x":2139.463,"y":1063.67,"cluster":"duality-cohomology"},{"id":"stacks:0E2D","tag":"0E2D","title":"Relative dualizing complexes · Lemma 0E2D","summary":"Let R → A be a flat ring map of finite presentation. Any two relative dualizing complexes for R → A are isomorphic.","statement_latex":"Let $R \\to A$ be a flat ring map of finite presentation.\nAny two relative dualizing complexes for $R \\to A$ are isomorphic.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2D","source_file":"dualizing.tex","source_line":5414,"source_end_line":5418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5414-L5418","statement_sha256":"eeab515174dfe2bfc97c3b819855f51ee96dc24c320c63e0bea496016e35954e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8775,"rank":8775,"depth":17,"x":2010.64,"y":1258.558,"cluster":"duality-cohomology"},{"id":"stacks:0E2E","tag":"0E2E","title":"Relative dualizing complexes · Lemma 0E2E","summary":"Let φ : R → A be a flat finite type ring map of Noetherian rings. Then the relative dualizing complex ω_A/R^bullet = φ^!(R) of Section [Tag 0E9M] is a relative dualizing complex in the sense of Definition [Tag 0E2C].","statement_latex":"Let $\\varphi : R \\to A$ be a flat finite type ring map of Noetherian rings.\nThen the relative dualizing complex $\\omega_{A/R}^\\bullet = \\varphi^!(R)$\nof Section \\ref{section-relative-dualizing-complexes-Noetherian}\nis a relative dualizing complex in the sense of\nDefinition \\ref{definition-relative-dualizing-complex}.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2E","source_file":"dualizing.tex","source_line":5439,"source_end_line":5446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5439-L5446","statement_sha256":"a26c3604c34ab186709e8ea84b5d40cc78f52f59f94529e88bad4690b6cfe1d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8776,"rank":8776,"depth":39,"x":1948.825,"y":1041.458,"cluster":"duality-cohomology"},{"id":"stacks:0E2F","tag":"0E2F","title":"Relative dualizing complexes · Lemma 0E2F","summary":"Let R → A be a flat ring map of finite presentation. Then • there exists a relative dualizing complex K in D(A), and • for any ring map R → R' setting A' = A ⊗_R R' and K' = K ⊗_A^L A', then K' is a relative dualizing complex for R' → A'. Moreover, if xi : A → K ⊗_A^L (A ⊗_R A) is a generator for the cyclic module Hom_D(A ⊗_R A)(A, K ⊗_A^L (A ⊗_R A)) then in (2) the derived base change of xi by A ⊗_R A → A' ⊗_R' A' is a generator for the cyclic module Hom_D(A' ⊗_R'…","statement_latex":"Let $R \\to A$ be a flat ring map of finite presentation. Then\n\\begin{enumerate}\n\\item there exists a relative dualizing complex $K$ in $D(A)$, and\n\\item for any ring map $R \\to R'$ setting $A' = A \\otimes_R R'$\nand $K' = K \\otimes_A^\\mathbf{L} A'$, then $K'$ is a\nrelative dualizing complex for $R' \\to A'$.\n\\end{enumerate}\nMoreover, if\n$$\n\\xi : A \\longrightarrow K \\otimes_A^\\mathbf{L} (A \\otimes_R A)\n$$\nis a generator for the cyclic module\n$\\Hom_{D(A \\otimes_R A)}(A, K \\otimes_A^\\mathbf{L} (A \\otimes_R A))$\nthen in (2) the derived base change of $\\xi$ by\n$A \\otimes_R A \\to A' \\otimes_{R'} A'$ is a generator for\nthe cyclic module\n$\\Hom_{D(A' \\otimes_{R'} A')}(A',\nK' \\otimes_{A'}^\\mathbf{L} (A' \\otimes_{R'} A'))$","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2F","source_file":"dualizing.tex","source_line":5473,"source_end_line":5493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5473-L5493","statement_sha256":"a6468a160848d3db736f18fa3e797881b953a477c84a38810c48c119ba9120e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8777,"rank":8777,"depth":40,"x":2169.29,"y":1166.639,"cluster":"duality-cohomology"},{"id":"stacks:0E2G","tag":"0E2G","title":"Relative dualizing complexes · Lemma 0E2G","summary":"Let R → A be a flat ring map of finite presentation. Let K be a relative dualizing complex. Then A → RHom_A(K, K) is an isomorphism.","statement_latex":"Let $R \\to A$ be a flat ring map of finite presentation.\nLet $K$ be a relative dualizing complex.\nThen $A \\to R\\Hom_A(K, K)$ is an isomorphism.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2G","source_file":"dualizing.tex","source_line":5580,"source_end_line":5585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5580-L5585","statement_sha256":"0ded4751404ab8921caf1c1e22c71b534e7031d2862468b2476f639e7c2fbe25","origin":"The Stacks Project","memory_eligible":false,"source_rank":8778,"rank":8778,"depth":41,"x":1905.7,"y":1199.474,"cluster":"duality-cohomology"},{"id":"stacks:0E2H","tag":"0E2H","title":"Relative dualizing complexes · Lemma 0E2H","summary":"Let R → A → B be a ring maps which are flat and of finite presentation. Let K_A/R and K_B/A be relative dualizing complexes for R → A and A → B. Then K = K_A/R ⊗_A^L K_B/A is a relative dualizing complex for R → B.","statement_latex":"Let $R \\to A \\to B$ be a ring maps which are flat and of finite presentation.\nLet $K_{A/R}$ and $K_{B/A}$ be relative dualizing complexes for $R \\to A$\nand $A \\to B$. Then $K = K_{A/R} \\otimes_A^\\mathbf{L} K_{B/A}$\nis a relative dualizing complex for $R \\to B$.","area":"Duality & Cohomology","chapter":"Dualizing Complexes","chapter_id":"dualizing","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2H","source_file":"dualizing.tex","source_line":5611,"source_end_line":5617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/dualizing.tex#L5611-L5617","statement_sha256":"09e097ea46f5a8da403c88b6edb025ec210c93888122255c51f9d5b9e10d1d76","origin":"The Stacks Project","memory_eligible":false,"source_rank":8779,"rank":8779,"depth":41,"x":2073.889,"y":1025.461,"cluster":"duality-cohomology"},{"id":"stacks:0A86","tag":"0A86","title":"Dualizing complexes on schemes · Lemma 0A86","summary":"Let X be a locally Noetherian scheme. Let K be an object of D(O_X). The following are equivalent • For every affine open U = Spec(A) ⊂ X there exists a dualizing complex ω_A^bullet for A such that K|_U is isomorphic to the image of ω_A^bullet by the functor widetilde : D(A) → D(O_U). • There is an affine open covering X = ⋃ U_i, U_i = Spec(A_i) such that for each i there exists a dualizing complex ω_i^bullet for A_i such that K|_U_i is isomorphic to the image of…","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $K$ be an object of\n$D(\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item For every affine open $U = \\Spec(A) \\subset X$ there exists\na dualizing complex $\\omega_A^\\bullet$ for $A$ such that\n$K|_U$ is isomorphic to the image of $\\omega_A^\\bullet$ by\nthe functor $\\widetilde{} : D(A) \\to D(\\mathcal{O}_U)$.\n\\item There is an affine open covering $X = \\bigcup U_i$, $U_i = \\Spec(A_i)$\nsuch that for each $i$ there exists a dualizing complex $\\omega_i^\\bullet$ for\n$A_i$ such that $K|_{U_i}$ is isomorphic to the image of $\\omega_i^\\bullet$ by\nthe functor $\\widetilde{} : D(A_i) \\to D(\\mathcal{O}_{U_i})$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing complexes on schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A86","source_file":"duality.tex","source_line":90,"source_end_line":104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L90-L104","statement_sha256":"e97061fa0f4984d899ad1b746dcf1683f2d310d69bfd3c1d9577228523d53a69","origin":"The Stacks Project","memory_eligible":false,"source_rank":8780,"rank":8780,"depth":27,"x":2089.827,"y":1249.51,"cluster":"duality-cohomology"},{"id":"stacks:0A87","tag":"0A87","title":"Dualizing complexes on schemes · Definition 0A87","summary":"Let X be a locally Noetherian scheme. An object K of D(O_X) is called a dualizing complex if K satisfies the equivalent conditions of Lemma [Tag 0A86].","statement_latex":"Let $X$ be a locally Noetherian scheme. An object $K$ of\n$D(\\mathcal{O}_X)$ is called a {\\it dualizing complex} if\n$K$ satisfies the equivalent conditions of\nLemma \\ref{lemma-equivalent-definitions}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing complexes on schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A87","source_file":"duality.tex","source_line":137,"source_end_line":143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L137-L143","statement_sha256":"3243d35cb98add6f8817551d1ae18cd97ae165442480db7bd385bfb136e43520","origin":"The Stacks Project","memory_eligible":false,"source_rank":8781,"rank":8781,"depth":28,"x":1897.643,"y":1093.134,"cluster":"duality-cohomology"},{"id":"stacks:0A88","tag":"0A88","title":"Dualizing complexes on schemes · Lemma 0A88","summary":"Let A be a Noetherian ring and let X = Spec(A). Let K, L be objects of D(A). If K ∈ D_Coh(A) and L has finite injective dimension, then RSheafHom_O_X(widetildeK, widetildeL) = widetildeRHom_A(K, L) in D(O_X).","statement_latex":"Let $A$ be a Noetherian ring and let $X = \\Spec(A)$. Let $K, L$ be objects\nof $D(A)$. If $K \\in D_{\\textit{Coh}}(A)$ and $L$ has finite injective\ndimension, then\n$$\nR\\SheafHom_{\\mathcal{O}_X}(\\widetilde{K}, \\widetilde{L})\n=\n\\widetilde{R\\Hom_A(K, L)}\n$$\nin $D(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing complexes on schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A88","source_file":"duality.tex","source_line":148,"source_end_line":159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L148-L159","statement_sha256":"94df05a78ecf0a49e943ead6d01bf30f04cb80be6bc77df6bca4902f735c908d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8782,"rank":8782,"depth":30,"x":2165.467,"y":1099.402,"cluster":"duality-cohomology"},{"id":"stacks:0G4I","tag":"0G4I","title":"Dualizing complexes on schemes · Lemma 0G4I","summary":"Let X be a Noetherian scheme. Let K, L, M ∈ D_QCoh(O_X). Then the map RSheafHom(L, M) ⊗_O_X^L K → RSheafHom(RSheafHom(K, L), M) of Cohomology, Lemma [Tag 0A8U] is an isomorphism in the following two cases • K ∈ D^-_Coh(O_X), L ∈ D^+_Coh(O_X), and M affine locally has finite injective dimension (see proof), or • K and L are in D_Coh(O_X), the object RSheafHom(L, M) has finite tor dimension, and L and M affine locally have finite injective dimension (in particular L and M…","statement_latex":"Let $X$ be a Noetherian scheme. Let $K, L, M \\in D_\\QCoh(\\mathcal{O}_X)$. \nThen the map\n$$\nR\\SheafHom(L, M) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K\n\\longrightarrow\nR\\SheafHom(R\\SheafHom(K, L), M)\n$$\nof Cohomology, Lemma \\ref{cohomology-lemma-internal-hom-evaluate}\nis an isomorphism in the following two cases\n\\begin{enumerate}\n\\item $K \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$,\n$L \\in D^+_{\\textit{Coh}}(\\mathcal{O}_X)$, and $M$ affine locally has\nfinite injective dimension (see proof), or\n\\item $K$ and $L$ are in $D_{\\textit{Coh}}(\\mathcal{O}_X)$,\nthe object $R\\SheafHom(L, M)$ has finite tor dimension, and\n$L$ and $M$ affine locally have finite injective dimension\n(in particular $L$ and $M$ are bounded).\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing complexes on schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4I","source_file":"duality.tex","source_line":197,"source_end_line":217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L197-L217","statement_sha256":"4cec4cdc361e05cf4a19e1ab03e1facc5efb99e693ffb5ff6f78c958688731c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8783,"rank":8783,"depth":32,"x":1962.667,"y":1246.946,"cluster":"duality-cohomology"},{"id":"stacks:0A89","tag":"0A89","title":"Dualizing complexes on schemes · Lemma 0A89","summary":"Let K be a dualizing complex on a locally Noetherian scheme X. Then K is an object of D_Coh(O_X) and D = RSheafHom_O_X(-, K) induces an anti-equivalence D : D_Coh(O_X) → D_Coh(O_X) which comes equipped with a canonical isomorphism id → D ∘ D. If X is quasi-compact, then D exchanges D^+_Coh(O_X) and D^-_Coh(O_X) and induces an anti-equivalence D^b_Coh(O_X) → D^b_Coh(O_X).","statement_latex":"Let $K$ be a dualizing complex on a locally Noetherian scheme $X$.\nThen $K$ is an object of $D_{\\textit{Coh}}(\\mathcal{O}_X)$\nand $D = R\\SheafHom_{\\mathcal{O}_X}(-, K)$ induces an anti-equivalence\n$$\nD :\nD_{\\textit{Coh}}(\\mathcal{O}_X)\n\\longrightarrow\nD_{\\textit{Coh}}(\\mathcal{O}_X)\n$$\nwhich comes equipped with a canonical isomorphism\n$\\text{id} \\to D \\circ D$. If $X$ is quasi-compact, then\n$D$ exchanges $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and\n$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and induces an anti-equivalence\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing complexes on schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A89","source_file":"duality.tex","source_line":254,"source_end_line":270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L254-L270","statement_sha256":"9f3f75e183fcc5b04cab76e32de8fbfe1802645a59a957571c0f97d48ece0663","origin":"The Stacks Project","memory_eligible":false,"source_rank":8784,"rank":8784,"depth":33,"x":1993.599,"y":1022.778,"cluster":"duality-cohomology"},{"id":"stacks:0ATP","tag":"0ATP","title":"Dualizing complexes on schemes · Lemma 0ATP","summary":"Let X be a locally Noetherian scheme. If K and K' are dualizing complexes on X, then K' is isomorphic to K ⊗_O_X^L L for some invertible object L of D(O_X).","statement_latex":"Let $X$ be a locally Noetherian scheme. If $K$ and $K'$ are dualizing\ncomplexes on $X$, then $K'$ is isomorphic to\n$K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$\nfor some invertible object $L$ of $D(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing complexes on schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATP","source_file":"duality.tex","source_line":329,"source_end_line":335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L329-L335","statement_sha256":"bd357394efbc4d335c3b36eca2eaabb7901f227ff80279f102dd4f7612b82bcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8785,"rank":8785,"depth":34,"x":2151.268,"y":1205.87,"cluster":"duality-cohomology"},{"id":"stacks:0AWF","tag":"0AWF","title":"Dualizing complexes on schemes · Lemma 0AWF","summary":"Let X be a locally Noetherian scheme. Let ω_X^bullet be a dualizing complex on X. Then X is universally catenary and the function X → Z defined by x ↦ δ(x) such that ω_X, x^bullet[-δ(x)] is a normalized dualizing complex over O_X, x is a dimension function.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\omega_X^\\bullet$\nbe a dualizing complex on $X$. Then $X$ is universally catenary\nand the function\n$X \\to \\mathbf{Z}$ defined by\n$$\nx \\longmapsto \\delta(x)\\text{ such that }\n\\omega_{X, x}^\\bullet[-\\delta(x)]\n\\text{ is a normalized dualizing complex over }\n\\mathcal{O}_{X, x}\n$$\nis a dimension function.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing complexes on schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWF","source_file":"duality.tex","source_line":349,"source_end_line":362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L349-L362","statement_sha256":"39c35e76aaf580fb6e806438a50dba4fc4fc5c17ff9d19472b3746c962291a09","origin":"The Stacks Project","memory_eligible":false,"source_rank":8786,"rank":8786,"depth":38,"x":1887.42,"y":1160.266,"cluster":"duality-cohomology"},{"id":"stacks:0ECM","tag":"0ECM","title":"Dualizing complexes on schemes · Lemma 0ECM","summary":"Let X be a locally Noetherian scheme. Let ω_X^bullet be a dualizing complex on X with associated dimension function δ. Let F be a coherent O_X-module. Set E^i = SheafExt^-i_O_X(F, ω_X^bullet). Then E^i is a coherent O_X-module and for x ∈ X we have • E^i_x is nonzero only for δ(x) ≤ i ≤ δ(x) + dim(Supp(F_x)), • dim(Supp(E^i + δ(x)_x)) ≤ i, • depth(F_x) is the smallest integer i ≥ 0 such that E_x^i + δ(x) not = 0, and • we have x ∈ Supp(bigoplus_j ≤ i E^j) ⇔ depth_O_X,…","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\omega_X^\\bullet$\nbe a dualizing complex on $X$ with associated dimension function $\\delta$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Set\n$\\mathcal{E}^i = \\SheafExt^{-i}_{\\mathcal{O}_X}(\\mathcal{F}, \\omega_X^\\bullet)$.\nThen $\\mathcal{E}^i$ is a coherent $\\mathcal{O}_X$-module and\nfor $x \\in X$ we have\n\\begin{enumerate}\n\\item $\\mathcal{E}^i_x$ is nonzero only for\n$\\delta(x) \\leq i \\leq \\delta(x) + \\dim(\\text{Supp}(\\mathcal{F}_x))$,\n\\item $\\dim(\\text{Supp}(\\mathcal{E}^{i + \\delta(x)}_x)) \\leq i$,\n\\item $\\text{depth}(\\mathcal{F}_x)$ is the smallest integer\n$i \\geq 0$ such that $\\mathcal{E}_x^{i + \\delta(x)} \\not = 0$, and\n\\item we have\n$x \\in \\text{Supp}(\\bigoplus_{j \\leq i} \\mathcal{E}^j)\n\\Leftrightarrow\n\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) + \\delta(x) \\leq i$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing complexes on schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECM","source_file":"duality.tex","source_line":370,"source_end_line":389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L370-L389","statement_sha256":"3108c661767ef15aaea2d618216c0577272e54eefe0a159da366ee65a1a17d83","origin":"The Stacks Project","memory_eligible":false,"source_rank":8787,"rank":8787,"depth":39,"x":2118.956,"y":1044.027,"cluster":"duality-cohomology"},{"id":"stacks:0A9E","tag":"0A9E","title":"Right adjoint of pushforward · Lemma 0A9E","summary":"This is almost the same as [Neeman-Grothendieck]. Let f : X → Y be a morphism between quasi-separated and quasi-compact schemes. The functor Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) has a right adjoint.","statement_latex":"\\begin{reference}\nThis is almost the same as \\cite[Example 4.2]{Neeman-Grothendieck}.\n\\end{reference}\nLet $f : X \\to Y$ be a morphism between quasi-separated and quasi-compact\nschemes. The functor $Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$\nhas a right adjoint.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9E","source_file":"duality.tex","source_line":435,"source_end_line":443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L435-L443","statement_sha256":"9f28b746cdb282aa485af98dab247ffa5f7193f30571f01bd207fdc48bb9d1e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8788,"rank":8788,"depth":38,"x":2041.599,"y":1261.404,"cluster":"duality-cohomology"},{"id":"stacks:0A9I","tag":"0A9I","title":"Right adjoint of pushforward · Lemma 0A9I","summary":"Let f : X → Y be a morphism of quasi-compact and quasi-separated schemes. Let a : D_QCoh(O_Y) → D_QCoh(O_X) be the right adjoint to Rf_* of Lemma [Tag 0A9E]. Then a maps D^+_QCoh(O_Y) into D^+_QCoh(O_X). In fact, there exists an integer N such that H^i(K) = 0 for i ≤ c implies H^i(a(K)) = 0 for i ≤ c - N.","statement_latex":"Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated\nschemes. Let $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$\nbe the right adjoint to $Rf_*$ of Lemma \\ref{lemma-twisted-inverse-image}.\nThen $a$ maps $D^+_\\QCoh(\\mathcal{O}_Y)$ into $D^+_\\QCoh(\\mathcal{O}_X)$.\nIn fact, there exists an integer $N$ such that\n$H^i(K) = 0$ for $i \\leq c$ implies $H^i(a(K)) = 0$ for $i \\leq c - N$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9I","source_file":"duality.tex","source_line":508,"source_end_line":516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L508-L516","statement_sha256":"1a9dbe5e1405f99414404d8c3fc466bd32fb90ebe22fc6eade0645dc95529838","origin":"The Stacks Project","memory_eligible":false,"source_rank":8789,"rank":8789,"depth":39,"x":1923.678,"y":1056.951,"cluster":"duality-cohomology"},{"id":"stacks:0A9Q","tag":"0A9Q","title":"Right adjoint of pushforward · Lemma 0A9Q","summary":"Let f : X → Y be a morphism of quasi-compact and quasi-separated schemes. Let a be the right adjoint to Rf_* : D_QCoh(O_X) → D_QCoh(O_Y). Let L ∈ D_QCoh(O_X) and K ∈ D_QCoh(O_Y). Then the map ([Tag 0B6H]) Rf_*RSheafHom_O_X(L, a(K)) → RSheafHom_O_Y(Rf_*L, K) becomes an isomorphism after applying the functor DQ_Y : D(O_Y) → D_QCoh(O_Y) discussed in Derived Categories of Schemes, Section [Tag 0CQZ].","statement_latex":"Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes.\nLet $a$ be the right adjoint to\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$.\nLet $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$.\nThen the map (\\ref{equation-sheafy-trace})\n$$\nRf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K))\n\\longrightarrow\nR\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K)\n$$\nbecomes an isomorphism after applying the functor\n$DQ_Y : D(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_Y)$\ndiscussed in Derived Categories of Schemes, Section\n\\ref{perfect-section-better-coherator}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9Q","source_file":"duality.tex","source_line":557,"source_end_line":573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L557-L573","statement_sha256":"fc14e9c598254f737959c3a97df7e9691a5a0d88884db90d07012fce4b72dc78","origin":"The Stacks Project","memory_eligible":false,"source_rank":8790,"rank":8790,"depth":32,"x":2175.375,"y":1140.91,"cluster":"duality-cohomology"},{"id":"stacks:0B6I","tag":"0B6I","title":"Right adjoint of pushforward · Lemma 0B6I","summary":"Let f : X → Y be a morphism of quasi-separated and quasi-compact schemes. For all L ∈ D_QCoh(O_X) and K ∈ D_QCoh(O_Y) ([Tag 0B6H]) induces an isomorphism RHom_X(L, a(K)) → RHom_Y(Rf_*L, K) of global derived homs.","statement_latex":"Let $f : X \\to Y$ be a morphism of quasi-separated and quasi-compact\nschemes.\nFor all $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$\n(\\ref{equation-sheafy-trace}) induces an isomorphism\n$R\\Hom_X(L, a(K)) \\to R\\Hom_Y(Rf_*L, K)$ of global derived homs.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6I","source_file":"duality.tex","source_line":660,"source_end_line":667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L660-L667","statement_sha256":"0215ce2633390b83cdd1a81a298379fbc0e52642af8bedf8a3e5ae0e1048e808","origin":"The Stacks Project","memory_eligible":false,"source_rank":8791,"rank":8791,"depth":33,"x":1921.93,"y":1221.925,"cluster":"duality-cohomology"},{"id":"stacks:0A9K","tag":"0A9K","title":"Right adjoint of pushforward and restriction to opens · Lemma 0A9K","summary":"In diagram ([Tag 0A9J]) assume that g is flat or more generally that f and g are Tor independent. Then a ∘ Rg_* ← Rg'_* ∘ a' is an isomorphism.","statement_latex":"In diagram (\\ref{equation-base-change}) assume that $g$ is flat or\nmore generally that $f$ and $g$ are Tor independent. Then\n$a \\circ Rg_* \\leftarrow Rg'_* \\circ a'$ is an isomorphism.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and restriction to opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9K","source_file":"duality.tex","source_line":739,"source_end_line":744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L739-L744","statement_sha256":"f779a0efae676967c200331020f4c023bbf84eaf2bcf46219d77eb7c100f3b53","origin":"The Stacks Project","memory_eligible":false,"source_rank":8792,"rank":8792,"depth":33,"x":2043.825,"y":1018.11,"cluster":"duality-cohomology"},{"id":"stacks:0A9N","tag":"0A9N","title":"Right adjoint of pushforward and restriction to opens · Lemma 0A9N","summary":"Let f : X → Y be a morphism of quasi-compact and quasi-separated schemes. Let a be the right adjoint to Rf_* : D_QCoh(O_X) → D_QCoh(O_Y). Let V ⊂ Y be quasi-compact open with inverse image U ⊂ X. • For every Q ∈ D_QCoh^+(O_Y) supported on Y setminus V the image a(Q) is supported on X setminus U if and only if ([Tag 0A9L]) is an isomorphism on all K in D_QCoh^+(O_Y). • For every Q ∈ D_QCoh(O_Y) supported on Y setminus V the image a(Q) is supported on X setminus U if and…","statement_latex":"Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated\nschemes. Let $a$ be the right adjoint to\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$.\nLet $V \\subset Y$ be quasi-compact open with inverse image $U \\subset X$.\n\\begin{enumerate}\n\\item For every $Q \\in D_\\QCoh^+(\\mathcal{O}_Y)$\nsupported on $Y \\setminus V$ the image $a(Q)$ is supported on\n$X \\setminus U$ if and only if (\\ref{equation-sheafy})\nis an isomorphism on all $K$ in $D_\\QCoh^+(\\mathcal{O}_Y)$.\n\\item For every $Q \\in D_\\QCoh(\\mathcal{O}_Y)$\nsupported on $Y \\setminus V$ the image $a(Q)$ is supported on\n$X \\setminus U$ if and only if (\\ref{equation-sheafy})\nis an isomorphism on all $K$ in $D_\\QCoh(\\mathcal{O}_Y)$.\n\\item If $a$ commutes with direct sums, then the equivalent conditions of\n(1) imply the equivalent conditions of (2).\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and restriction to opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9N","source_file":"duality.tex","source_line":817,"source_end_line":835,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L817-L835","statement_sha256":"5ebf86f0e447489c8c81d80a1759210907215617cb9b49fbe1b8103782a7c830","origin":"The Stacks Project","memory_eligible":false,"source_rank":8793,"rank":8793,"depth":37,"x":2117.939,"y":1237.851,"cluster":"duality-cohomology"},{"id":"stacks:0A9P","tag":"0A9P","title":"Right adjoint of pushforward and restriction to opens · Lemma 0A9P","summary":"Let Y be a quasi-compact and quasi-separated scheme. Let f : X → Y be a proper morphism. If • f is flat and of finite presentation, or • Y is Noetherian then the equivalent conditions of Lemma [Tag 0A9N] part (1) hold for all quasi-compact opens V of Y.","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to Y$ be a proper morphism. If\\footnote{This proof works for those\nmorphisms of quasi-compact and quasi-separated schemes such that\n$Rf_*P$ is pseudo-coherent for all $P$ perfect on $X$. It follows\neasily from a theorem of Kiehl \\cite{Kiehl} that this holds if\n$f$ is proper and pseudo-coherent. This is the correct generality\nfor this lemma and some of the other results in this chapter.}\n\\begin{enumerate}\n\\item $f$ is flat and of finite presentation, or\n\\item $Y$ is Noetherian\n\\end{enumerate}\nthen the equivalent conditions of Lemma \\ref{lemma-when-sheafy} part (1)\nhold for all quasi-compact opens $V$ of $Y$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and restriction to opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9P","source_file":"duality.tex","source_line":888,"source_end_line":903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L888-L903","statement_sha256":"1a99db021ae92c2b31045c6fa0e75041b0c2a157c2f85ca67786ae1b68747c52","origin":"The Stacks Project","memory_eligible":false,"source_rank":8794,"rank":8794,"depth":38,"x":1886.28,"y":1117.716,"cluster":"duality-cohomology"},{"id":"stacks:0ATQ","tag":"0ATQ","title":"Right adjoint of pushforward and base change, I · Lemma 0ATQ","summary":"Consider a commutative diagram xymatrix X' ar[r]_k ar[d]_f' & X ar[d]^f Y' ar[r]^l ar[d]_g' & Y ar[d]^g Z' ar[r]^m & Z of quasi-compact and quasi-separated schemes where both diagrams are cartesian and where f and l as well as g and m are Tor independent. Then the maps ([Tag 0AA6]) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\\nZ' \\ar[r]^m & Z\n}\n$$\nof quasi-compact and quasi-separated schemes where\nboth diagrams are cartesian and where $f$ and $l$\nas well as $g$ and $m$ are Tor independent.\nThen the maps (\\ref{equation-base-change-map})\nfor the two squares compose to give the base\nchange map for the outer rectangle (see proof for a precise statement).","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATQ","source_file":"duality.tex","source_line":1020,"source_end_line":1036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L1020-L1036","statement_sha256":"28047d583d40210b924472c1f3c70b5779ab6cae7ab227053c6551e62fcf97ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":8795,"rank":8795,"depth":39,"x":2154.056,"y":1074.798,"cluster":"duality-cohomology"},{"id":"stacks:0ATR","tag":"0ATR","title":"Right adjoint of pushforward and base change, I · Lemma 0ATR","summary":"Consider a commutative diagram xymatrix X\" ar[r]_g' ar[d]_f\" & X' ar[r]_g ar[d]_f' & X ar[d]^f Y\" ar[r]^h' & Y' ar[r]^h & Y of quasi-compact and quasi-separated schemes where both diagrams are cartesian and where f and h as well as f' and h' are Tor independent. Then the maps ([Tag 0AA6]) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX'' \\ar[r]_{g'} \\ar[d]_{f''} & X' \\ar[r]_g \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY'' \\ar[r]^{h'} & Y' \\ar[r]^h & Y\n}\n$$\nof quasi-compact and quasi-separated schemes where\nboth diagrams are cartesian and where $f$ and $h$\nas well as $f'$ and $h'$ are Tor independent.\nThen the maps (\\ref{equation-base-change-map})\nfor the two squares compose to give the base\nchange map for the outer rectangle (see proof for a precise statement).","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATR","source_file":"duality.tex","source_line":1157,"source_end_line":1172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L1157-L1172","statement_sha256":"961cd865883b08a66b06a192a8db4c7657128f67e738e5e33c7a9febef103bae","origin":"The Stacks Project","memory_eligible":false,"source_rank":8796,"rank":8796,"depth":39,"x":1990.907,"y":1258.624,"cluster":"duality-cohomology"},{"id":"stacks:0AA8","tag":"0AA8","title":"Right adjoint of pushforward and base change, II · Lemma 0AA8","summary":"In diagram ([Tag 0A9J]) assume • g : Y' → Y is a morphism of affine schemes, • f : X → Y is proper, and • f and g are Tor independent. Then the base change map ([Tag 0AA6]) induces an isomorphism L(g')^*a(K) → a'(Lg^*K) in the following cases • for all K ∈ D_QCoh(O_Y) if f is flat of finite presentation, • for all K ∈ D_QCoh(O_Y) if f is perfect and Y Noetherian, • for K ∈ D_QCoh^+(O_Y) if g has finite Tor dimension and Y Noetherian.","statement_latex":"In diagram (\\ref{equation-base-change}) assume\n\\begin{enumerate}\n\\item $g : Y' \\to Y$ is a morphism of affine schemes,\n\\item $f : X \\to Y$ is proper, and\n\\item $f$ and $g$ are Tor independent.\n\\end{enumerate}\nThen the base change map (\\ref{equation-base-change-map}) induces an\nisomorphism\n$$\nL(g')^*a(K) \\longrightarrow a'(Lg^*K)\n$$\nin the following cases\n\\begin{enumerate}\n\\item for all $K \\in D_\\QCoh(\\mathcal{O}_Y)$ if $f$\nis flat of finite presentation,\n\\item for all $K \\in D_\\QCoh(\\mathcal{O}_Y)$ if $f$\nis perfect and $Y$ Noetherian,\n\\item for $K \\in D_\\QCoh^+(\\mathcal{O}_Y)$ if $g$ has finite Tor dimension\nand $Y$ Noetherian.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and base change, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AA8","source_file":"duality.tex","source_line":1418,"source_end_line":1440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L1418-L1440","statement_sha256":"b2b5e8a46b7a6013d05b14735215bc4171a46ee7f9eb82a3b27109634471f97a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8797,"rank":8797,"depth":39,"x":1963.348,"y":1030.205,"cluster":"duality-cohomology"},{"id":"stacks:0B6J","tag":"0B6J","title":"Trace map and base change · Lemma 0B6J","summary":"Suppose we have a diagram ([Tag 0A9J]) where f and g are tor independent. Then the maps 1 star Tr_f : Lg^* ∘ Rf_* ∘ a → Lg^* and Tr_f' star 1 : Rf'_* ∘ a' ∘ Lg^* → Lg^* agree via the base change maps β : Lg^* ∘ Rf_* → Rf'_* ∘ L(g')^* (Cohomology, Remark [Tag 08HY]) and α : L(g')^* ∘ a → a' ∘ Lg^* ([Tag 0AA6]). More precisely, the diagram xymatrix Lg^* ∘ Rf_* ∘ a ar[d]_β star 1 ar[r]_-1 star Tr_f & Lg^* Rf'_* ∘ L(g')^* ∘ a ar[r]^1 star α & Rf'_* ∘ a' ∘ Lg^* ar[u]_Tr_f'…","statement_latex":"Suppose we have a diagram (\\ref{equation-base-change}) where $f$ and $g$\nare tor independent. Then the maps\n$1 \\star \\text{Tr}_f : Lg^* \\circ Rf_* \\circ a \\to Lg^*$ and\n$\\text{Tr}_{f'} \\star 1 : Rf'_* \\circ a' \\circ Lg^* \\to Lg^*$\nagree via the base change maps\n$\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$\n(Cohomology, Remark \\ref{cohomology-remark-base-change})\nand $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$\n(\\ref{equation-base-change-map}).\nMore precisely, the diagram\n$$\n\\xymatrix{\nLg^* \\circ Rf_* \\circ a\n\\ar[d]_{\\beta \\star 1} \\ar[r]_-{1 \\star \\text{Tr}_f} &\nLg^* \\\\\nRf'_* \\circ L(g')^* \\circ a \\ar[r]^{1 \\star \\alpha} &\nRf'_* \\circ a' \\circ Lg^* \\ar[u]_{\\text{Tr}_{f'} \\star 1}\n}\n$$\nof transformations of functors commutes.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and trace maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6J","source_file":"duality.tex","source_line":1649,"source_end_line":1671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L1649-L1671","statement_sha256":"be1fdb4276d583b772aa5a42afa3849f410127ec1b5f87118a85fd56a2f228c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8798,"rank":8798,"depth":34,"x":2167.618,"y":1183.196,"cluster":"duality-cohomology"},{"id":"stacks:0B6K","tag":"0B6K","title":"Right adjoint of pushforward and trace maps · Lemma 0B6K","summary":"Suppose we have a diagram ([Tag 0A9J]) where f and g are tor independent. Then the maps 1 star eta_f : L(g')^* → L(g')^* ∘ a ∘ Rf_* and eta_f' star 1 : L(g')^* → a' ∘ Rf'_* ∘ L(g')^* agree via the base change maps β : Lg^* ∘ Rf_* → Rf'_* ∘ L(g')^* (Cohomology, Remark [Tag 08HY]) and α : L(g')^* ∘ a → a' ∘ Lg^* ([Tag 0AA6]). More precisely, the diagram xymatrix L(g')^* ar[r]_-1 star eta_f ar[d]_eta_f' star 1 & L(g')^* ∘ a ∘ Rf_* ar[d]^α a' ∘ Rf'_* ∘ L(g')^* & a' ∘ Lg^* ∘…","statement_latex":"Suppose we have a diagram (\\ref{equation-base-change}) where $f$ and $g$\nare tor independent. Then the maps\n$1 \\star \\eta_f : L(g')^* \\to L(g')^* \\circ a \\circ Rf_*$ and\n$\\eta_{f'} \\star 1 : L(g')^* \\to a' \\circ Rf'_* \\circ L(g')^*$\nagree via the base change maps\n$\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$\n(Cohomology, Remark \\ref{cohomology-remark-base-change})\nand $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$\n(\\ref{equation-base-change-map}).\nMore precisely, the diagram\n$$\n\\xymatrix{\nL(g')^* \\ar[r]_-{1 \\star \\eta_f} \\ar[d]_{\\eta_{f'} \\star 1} &\nL(g')^* \\circ a \\circ Rf_* \\ar[d]^\\alpha \\\\\na' \\circ Rf'_* \\circ L(g')^* &\na' \\circ Lg^* \\circ Rf_* \\ar[l]_-\\beta\n}\n$$\nof transformations of functors commutes.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and trace maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6K","source_file":"duality.tex","source_line":1738,"source_end_line":1759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L1738-L1759","statement_sha256":"f10f360f97a3f1f33afac00be087ec6251322c27bde3db40328a352deb608dac","origin":"The Stacks Project","memory_eligible":false,"source_rank":8799,"rank":8799,"depth":35,"x":1893.612,"y":1186.295,"cluster":"duality-cohomology"},{"id":"stacks:0A9T","tag":"0A9T","title":"Right adjoint of pushforward and pullback · Lemma 0A9T","summary":"Let f : X → Y be a morphism of quasi-compact and quasi-separated schemes. The map Lf^*K ⊗^L_O_X a(L) → a(K ⊗_O_Y^L L) defined above for K, L ∈ D_QCoh(O_Y) is an isomorphism if K is perfect. In particular, ([Tag 0A9S]) is an isomorphism if K is perfect.","statement_latex":"Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated\nschemes. The map\n$Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(L) \\to\na(K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} L)$\ndefined above for $K, L \\in D_\\QCoh(\\mathcal{O}_Y)$\nis an isomorphism if $K$ is perfect. In particular,\n(\\ref{equation-compare-with-pullback}) is an isomorphism if $K$ is perfect.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9T","source_file":"duality.tex","source_line":1867,"source_end_line":1876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L1867-L1876","statement_sha256":"bffbd3618ada101e07ef0b5d058dc16bc4bb8c5c1b43b1f97527b6643d5dd6e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8800,"rank":8800,"depth":32,"x":2093.422,"y":1028.331,"cluster":"duality-cohomology"},{"id":"stacks:0B6P","tag":"0B6P","title":"Right adjoint of pushforward and pullback · Lemma 0B6P","summary":"Suppose we have a diagram ([Tag 0A9J]) where f and g are tor independent. Let K ∈ D_QCoh(O_Y). The diagram xymatrix L(g')^*(Lf^*K ⊗^L_O_X a(O_Y)) ar[r] ar[d] & L(g')^*a(K) ar[d] L(f')^*Lg^*K ⊗_O_X'^L a'(O_Y') ar[r] & a'(Lg^*K) commutes where the horizontal arrows are the maps ([Tag 0A9S]) for K and Lg^*K and the vertical maps are constructed using Cohomology, Remark [Tag 08HY] and ([Tag 0AA6]).","statement_latex":"Suppose we have a diagram (\\ref{equation-base-change}) where $f$ and $g$\nare tor independent. Let $K \\in D_\\QCoh(\\mathcal{O}_Y)$. The diagram\n$$\n\\xymatrix{\nL(g')^*(Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(\\mathcal{O}_Y))\n\\ar[r] \\ar[d] & L(g')^*a(K) \\ar[d] \\\\\nL(f')^*Lg^*K \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} a'(\\mathcal{O}_{Y'})\n\\ar[r] & a'(Lg^*K)\n}\n$$\ncommutes where the horizontal arrows are the maps\n(\\ref{equation-compare-with-pullback}) for $K$ and $Lg^*K$\nand the vertical maps are constructed using\nCohomology, Remark \\ref{cohomology-remark-base-change} and\n(\\ref{equation-base-change-map}).","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6P","source_file":"duality.tex","source_line":1901,"source_end_line":1918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L1901-L1918","statement_sha256":"c1b9bd4aa0d4714cc0ed40e2b6af2289974d3354a5ff8b28c54f1b2b18c4f3c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8801,"rank":8801,"depth":36,"x":2073.089,"y":1258.48,"cluster":"duality-cohomology"},{"id":"stacks:0B6Q","tag":"0B6Q","title":"Right adjoint of pushforward and pullback · Lemma 0B6Q","summary":"Let f : X → Y be a proper morphism of Noetherian schemes. Let V ⊂ Y be an open such that f^-1(V) → V is an isomorphism. Then for K ∈ D_QCoh^+(O_Y) the map ([Tag 0A9S]) restricts to an isomorphism over f^-1(V).","statement_latex":"Let $f : X \\to Y$ be a proper morphism of Noetherian schemes. Let $V \\subset Y$\nbe an open such that $f^{-1}(V) \\to V$ is an isomorphism. Then for\n$K \\in D_\\QCoh^+(\\mathcal{O}_Y)$ the map (\\ref{equation-compare-with-pullback})\nrestricts to an isomorphism over $f^{-1}(V)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6Q","source_file":"duality.tex","source_line":1953,"source_end_line":1959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L1953-L1959","statement_sha256":"4a5cc45b17eb0e5fa99bcdcd6194b40a0d351700f33adc997212f7f993c1b2bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8802,"rank":8802,"depth":39,"x":1902.788,"y":1077.005,"cluster":"duality-cohomology"},{"id":"stacks:0B6R","tag":"0B6R","title":"Right adjoint of pushforward and pullback · Lemma 0B6R","summary":"Let f : X → Y and g : Y → Z be composable morphisms of quasi-compact and quasi-separated schemes and set h = g ∘ f. Let a, b, c be the adjoints of Lemma [Tag 0A9E] for f, g, h. For any K ∈ D_QCoh(O_Z) the diagram xymatrix Lf^*(Lg^*K ⊗_O_Y^L b(O_Z)) ⊗_O_X^L a(O_Y) ar@=[d] ar[r] & a(Lg^*K ⊗_O_Y^L b(O_Z)) ar[r] & a(b(K)) ar@=[d] Lh^*K ⊗_O_X^L Lf^*b(O_Z) ⊗_O_X^L a(O_Y) ar[r] & Lh^*K ⊗_O_X^L c(O_Z) ar[r] & c(K) is commutative where the arrows are ([Tag 0A9S]) and we have used…","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be composable morphisms of quasi-compact\nand quasi-separated schemes and set $h = g \\circ f$. Let $a, b, c$ be the\nadjoints of Lemma \\ref{lemma-twisted-inverse-image} for $f, g, h$.\nFor any $K \\in D_\\QCoh(\\mathcal{O}_Z)$ the diagram\n$$\n\\xymatrix{\nLf^*(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L}\nb(\\mathcal{O}_Z)) \\otimes_{\\mathcal{O}_X}^\\mathbf{L} a(\\mathcal{O}_Y)\n\\ar@{=}[d] \\ar[r] &\na(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} b(\\mathcal{O}_Z)) \\ar[r] &\na(b(K)) \\ar@{=}[d] \\\\\nLh^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*b(\\mathcal{O}_Z)\n\\otimes_{\\mathcal{O}_X}^\\mathbf{L} a(\\mathcal{O}_Y) \\ar[r] &\nLh^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} c(\\mathcal{O}_Z) \\ar[r] &\nc(K)\n}\n$$\nis commutative where the arrows are (\\ref{equation-compare-with-pullback})\nand we have used $Lh^* = Lf^* \\circ Lg^*$ and $c = a \\circ b$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward and pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6R","source_file":"duality.tex","source_line":1971,"source_end_line":1992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L1971-L1992","statement_sha256":"3276d2474d014bf5a47338a38bd5c15882b61d284ff3c17f108fa2f161d8b52b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8803,"rank":8803,"depth":39,"x":2174.645,"y":1114.236,"cluster":"duality-cohomology"},{"id":"stacks:0A75","tag":"0A75","title":"Right adjoint of pushforward for closed immersions · Lemma 0A75","summary":"With notation as above. The functor SheafHom(O_Z, -) is a right adjoint to the functor i_* : Mod(O_Z) → Mod(O_X). For V ⊂ Z open we have Γ(V, SheafHom(O_Z, F)) = (s ∈ Γ(U, F) mid Is = 0) where U ⊂ X is an open whose intersection with Z is V.","statement_latex":"With notation as above. The functor $\\SheafHom(\\mathcal{O}_Z, -)$ is a\nright adjoint to the functor\n$i_* : \\textit{Mod}(\\mathcal{O}_Z) \\to \\textit{Mod}(\\mathcal{O}_X)$.\nFor $V \\subset Z$ open we have\n$$\n\\Gamma(V, \\SheafHom(\\mathcal{O}_Z, \\mathcal{F})) =\n\\{s \\in \\Gamma(U, \\mathcal{F}) \\mid \\mathcal{I}s = 0\\}\n$$\nwhere $U \\subset X$ is an open whose intersection with $Z$ is $V$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A75","source_file":"duality.tex","source_line":2065,"source_end_line":2076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2065-L2076","statement_sha256":"ac61b37832161a2a3acd4ae77a670389c7947df42ce436d4335e880fa3600d7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8804,"rank":8804,"depth":4,"x":1943.951,"y":1241.2,"cluster":"duality-cohomology"},{"id":"stacks:0A76","tag":"0A76","title":"Right adjoint of pushforward for closed immersions · Lemma 0A76","summary":"With notation as above. The functor RSheafHom(O_Z, -) is the right adjoint of the functor Ri_* : D(O_Z) → D(O_X).","statement_latex":"With notation as above. The functor $R\\SheafHom(\\mathcal{O}_Z, -)$\nis the right adjoint of the functor\n$Ri_* : D(\\mathcal{O}_Z) \\to D(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A76","source_file":"duality.tex","source_line":2107,"source_end_line":2112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2107-L2112","statement_sha256":"5fa28f4c06205c4c1c2831ad5d24b72449332c486d69d4103f99bf9f0da39880","origin":"The Stacks Project","memory_eligible":false,"source_rank":8805,"rank":8805,"depth":5,"x":2012.046,"y":1016.395,"cluster":"duality-cohomology"},{"id":"stacks:0A77","tag":"0A77","title":"Right adjoint of pushforward for closed immersions · Lemma 0A77","summary":"With notation as above. We have Ri_*RSheafHom(O_Z, K) = RSheafHom_O_X(i_*O_Z, K) in D(O_X) for all K in D(O_X).","statement_latex":"With notation as above. We have\n$$\nRi_*R\\SheafHom(\\mathcal{O}_Z, K) =\nR\\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Z, K)\n$$\nin $D(\\mathcal{O}_X)$ for all $K$ in $D(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A77","source_file":"duality.tex","source_line":2121,"source_end_line":2129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2121-L2129","statement_sha256":"cdd03b50420b893aca6ee751676a2288668d16f982e7dc658bae598eddd63ad1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8806,"rank":8806,"depth":0,"x":2142.779,"y":1221.058,"cluster":"duality-cohomology"},{"id":"stacks:0E2I","tag":"0E2I","title":"Right adjoint of pushforward for closed immersions · Lemma 0E2I","summary":"With notation as above. For M ∈ D(O_Z) we have RSheafHom_O_X(Ri_*M, K) = Ri_*RSheafHom_O_Z(M, RSheafHom(O_Z, K)) in D(O_Z) for all K in D(O_X).","statement_latex":"With notation as above. For $M \\in D(\\mathcal{O}_Z)$ we have\n$$\nR\\SheafHom_{\\mathcal{O}_X}(Ri_*M, K) =\nRi_*R\\SheafHom_{\\mathcal{O}_Z}(M, R\\SheafHom(\\mathcal{O}_Z, K))\n$$\nin $D(\\mathcal{O}_Z)$ for all $K$ in $D(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2I","source_file":"duality.tex","source_line":2136,"source_end_line":2144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2136-L2144","statement_sha256":"51220b3c77dc9d814023613dc822c7a26857ec7b1b6f4cbc848aab7195ce013f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8807,"rank":8807,"depth":1,"x":1881.468,"y":1144.228,"cluster":"duality-cohomology"},{"id":"stacks:0A78","tag":"0A78","title":"Right adjoint of pushforward for closed immersions · Lemma 0A78","summary":"Let i : Z → X be a pseudo-coherent closed immersion of schemes (any closed immersion if X is locally Noetherian). Then • RSheafHom(O_Z, -) maps D^+_QCoh(O_X) into D^+_QCoh(O_Z), and • if X = Spec(A) and Z = Spec(B), then the diagram xymatrix D^+(B) ar[r] & D_QCoh^+(O_Z) D^+(A) ar[r] ar[u]^RHom(B, -) & D_QCoh^+(O_X) ar[u]_RSheafHom(O_Z, -) is commutative.","statement_latex":"Let $i : Z \\to X$ be a pseudo-coherent closed immersion of schemes\n(any closed immersion if $X$ is locally Noetherian).\nThen\n\\begin{enumerate}\n\\item $R\\SheafHom(\\mathcal{O}_Z, -)$ maps $D^+_\\QCoh(\\mathcal{O}_X)$\ninto $D^+_\\QCoh(\\mathcal{O}_Z)$, and\n\\item if $X = \\Spec(A)$ and $Z = \\Spec(B)$, then the diagram\n$$\n\\xymatrix{\nD^+(B) \\ar[r] & D_\\QCoh^+(\\mathcal{O}_Z) \\\\\nD^+(A) \\ar[r] \\ar[u]^{R\\Hom(B, -)} &\nD_\\QCoh^+(\\mathcal{O}_X) \\ar[u]_{R\\SheafHom(\\mathcal{O}_Z, -)}\n}\n$$\nis commutative.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A78","source_file":"duality.tex","source_line":2155,"source_end_line":2173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2155-L2173","statement_sha256":"0ce54657ee7bfd43f636eda5365069c85d030cb25f6eb3c714132b5bb2dc6be8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8808,"rank":8808,"depth":30,"x":2136.257,"y":1052.493,"cluster":"duality-cohomology"},{"id":"stacks:0A79","tag":"0A79","title":"Right adjoint of pushforward for closed immersions · Lemma 0A79","summary":"Let i : Z → X be a closed immersion of schemes. Assume X is a locally Noetherian. Then RSheafHom(O_Z, -) maps D^+_Coh(O_X) into D^+_Coh(O_Z).","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nAssume $X$ is a locally Noetherian.\nThen $R\\SheafHom(\\mathcal{O}_Z, -)$ maps $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$\ninto $D^+_{\\textit{Coh}}(\\mathcal{O}_Z)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A79","source_file":"duality.tex","source_line":2231,"source_end_line":2237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2231-L2237","statement_sha256":"39237261c40ca440aa87663e24fa6c3f2517614a518477b123f0ed148bf1dfae","origin":"The Stacks Project","memory_eligible":false,"source_rank":8809,"rank":8809,"depth":31,"x":2022.009,"y":1264.974,"cluster":"duality-cohomology"},{"id":"stacks:0A9X","tag":"0A9X","title":"Right adjoint of pushforward for closed immersions · Lemma 0A9X","summary":"Let X be a quasi-compact and quasi-separated scheme. Let i : Z → X be a pseudo-coherent closed immersion (if X is Noetherian, then any closed immersion is pseudo-coherent). Let a : D_QCoh(O_X) → D_QCoh(O_Z) be the right adjoint to Ri_*. Then there is a functorial isomorphism a(K) = RSheafHom(O_Z, K) for K ∈ D_QCoh^+(O_X).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $i : Z \\to X$ be a pseudo-coherent closed immersion\n(if $X$ is Noetherian, then any closed immersion is pseudo-coherent).\nLet $a : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Z)$ be the\nright adjoint to $Ri_*$. Then there is a functorial isomorphism\n$$\na(K) = R\\SheafHom(\\mathcal{O}_Z, K)\n$$\nfor $K \\in D_\\QCoh^+(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9X","source_file":"duality.tex","source_line":2249,"source_end_line":2260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2249-L2260","statement_sha256":"1a39b30762cf14ec19dca95f7fd73ba9cb790f136cc245e7c06cce59b269be64","origin":"The Stacks Project","memory_eligible":false,"source_rank":8810,"rank":8810,"depth":40,"x":1935.275,"y":1043.192,"cluster":"duality-cohomology"},{"id":"stacks:0E2L","tag":"0E2L","title":"Right adjoint of pushforward for closed immersions and base change · Lemma 0E2L","summary":"In the situation above, the map ([Tag 0E2K]) is an isomorphism if and only if the base change map Lf^*RSheafHom_O_X(O_Z, K) → RSheafHom_O_X'(O_Z', Lf^*K) of Cohomology, Remark [Tag 08I3] is an isomorphism.","statement_latex":"In the situation above, the map (\\ref{equation-base-change-exact-support})\nis an isomorphism if and only if the base change map\n$$\nLf^*R\\SheafHom_{\\mathcal{O}_X}(\\mathcal{O}_Z, K)\n\\longrightarrow\nR\\SheafHom_{\\mathcal{O}_{X'}}(\\mathcal{O}_{Z'}, Lf^*K)\n$$\nof Cohomology, Remark \\ref{cohomology-remark-prepare-fancy-base-change}\nis an isomorphism.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions and base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2L","source_file":"duality.tex","source_line":2343,"source_end_line":2354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2343-L2354","statement_sha256":"c6613e8c836af1e5e1ec617b707ca65cfa90287bdafc6ae466cb7065df0f271a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8811,"rank":8811,"depth":2,"x":2177.882,"y":1157.657,"cluster":"duality-cohomology"},{"id":"stacks:0E2M","tag":"0E2M","title":"Right adjoint of pushforward for closed immersions and base change · Lemma 0E2M","summary":"In the situation above, assume f is flat and i pseudo-coherent. Then ([Tag 0E2K]) is an isomorphism for K in D^+_QCoh(O_X).","statement_latex":"In the situation above, assume $f$ is flat and $i$ pseudo-coherent.\nThen (\\ref{equation-base-change-exact-support}) is an isomorphism\nfor $K$ in $D^+_\\QCoh(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions and base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2M","source_file":"duality.tex","source_line":2377,"source_end_line":2382,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2377-L2382","statement_sha256":"5bc47ea4f9cc6806113ff06290dbd103e1ccc7d9dad35d97e624c9e174e34f4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8812,"rank":8812,"depth":30,"x":1906.603,"y":1210.979,"cluster":"duality-cohomology"},{"id":"stacks:0E2N","tag":"0E2N","title":"Right adjoint of pushforward for closed immersions and base change · Lemma 0E2N","summary":"Let i : Z → X be a pseudo-coherent closed immersion of schemes. Let M ∈ D_QCoh(O_X) locally have tor-amplitude in [a, ∞). Let K ∈ D_QCoh^+(O_X). Then there is a canonical isomorphism RSheafHom(O_Z, K) ⊗_O_Z^L Li^*M = RSheafHom(O_Z, K ⊗_O_X^L M) in D(O_Z).","statement_latex":"Let $i : Z \\to X$ be a pseudo-coherent closed immersion of schemes.\nLet $M \\in D_\\QCoh(\\mathcal{O}_X)$ locally have tor-amplitude in $[a, \\infty)$.\nLet $K \\in D_\\QCoh^+(\\mathcal{O}_X)$. Then there is a canonical isomorphism\n$$\nR\\SheafHom(\\mathcal{O}_Z, K) \\otimes_{\\mathcal{O}_Z}^\\mathbf{L} Li^*M =\nR\\SheafHom(\\mathcal{O}_Z, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M)\n$$\nin $D(\\mathcal{O}_Z)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for closed immersions and base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2N","source_file":"duality.tex","source_line":2424,"source_end_line":2434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2424-L2434","statement_sha256":"5fc149ff23dfcd76b6943a2e993490c25cd6702d1f1bb49799b8ca2368ab1af1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8813,"rank":8813,"depth":31,"x":2063.954,"y":1017.492,"cluster":"duality-cohomology"},{"id":"stacks:0BUZ","tag":"0BUZ","title":"Right adjoint of pushforward for finite morphisms · Lemma 0BUZ","summary":"With notation as above. The functor SheafHom(f_*O_Y, -) is a right adjoint to the restriction functor Mod(f_*O_Y) → Mod(O_X). For an affine open U ⊂ X we have Γ(U, SheafHom(f_*O_Y, F)) = Hom_A(B, F(U)) where A = O_X(U) and B = O_Y(f^-1(U)).","statement_latex":"With notation as above. The functor $\\SheafHom(f_*\\mathcal{O}_Y, -)$ is a\nright adjoint to the restriction functor\n$\\textit{Mod}(f_*\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$.\nFor an affine open $U \\subset X$ we have\n$$\n\\Gamma(U, \\SheafHom(f_*\\mathcal{O}_Y, \\mathcal{F})) =\n\\Hom_A(B, \\mathcal{F}(U))\n$$\nwhere $A = \\mathcal{O}_X(U)$ and $B = \\mathcal{O}_Y(f^{-1}(U))$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUZ","source_file":"duality.tex","source_line":2487,"source_end_line":2498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2487-L2498","statement_sha256":"2a30d82e907ba0a40ae69f832828fd9789bff20e72bb93331891c61c4d7cc4c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8814,"rank":8814,"depth":9,"x":2103.57,"y":1249.736,"cluster":"duality-cohomology"},{"id":"stacks:0BV0","tag":"0BV0","title":"Right adjoint of pushforward for finite morphisms · Lemma 0BV0","summary":"With notation as above. The functor RSheafHom(f_*O_Y, -) is the right adjoint of the functor D(f_*O_Y) → D(O_X).","statement_latex":"With notation as above. The functor $R\\SheafHom(f_*\\mathcal{O}_Y, -)$\nis the right adjoint of the functor $D(f_*\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BV0","source_file":"duality.tex","source_line":2519,"source_end_line":2523,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2519-L2523","statement_sha256":"2952c9fa965750c122c6b5284af1bb44c596c8643bc7cb4264e7cd1d4c539dc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8815,"rank":8815,"depth":10,"x":1887.329,"y":1100.78,"cluster":"duality-cohomology"},{"id":"stacks:0BV1","tag":"0BV1","title":"Right adjoint of pushforward for finite morphisms · Lemma 0BV1","summary":"With notation as above. The composition D(O_X) xrightarrowRSheafHom(f_*O_Y, -) D(f_*O_Y) → D(O_X) is the functor K ↦ RSheafHom_O_X(f_*O_Y, K).","statement_latex":"With notation as above. The composition\n$$\nD(\\mathcal{O}_X) \\xrightarrow{R\\SheafHom(f_*\\mathcal{O}_Y, -)}\nD(f_*\\mathcal{O}_Y) \\to D(\\mathcal{O}_X)\n$$\nis the functor $K \\mapsto R\\SheafHom_{\\mathcal{O}_X}(f_*\\mathcal{O}_Y, K)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BV1","source_file":"duality.tex","source_line":2531,"source_end_line":2539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2531-L2539","statement_sha256":"d228cb9777a6889270562d3847000337613a5eb2adff63746ad2a260b15f2f2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8816,"rank":8816,"depth":0,"x":2166.909,"y":1087.902,"cluster":"duality-cohomology"},{"id":"stacks:0AX2","tag":"0AX2","title":"Right adjoint of pushforward for finite morphisms · Lemma 0AX2","summary":"Let f : Y → X be a finite pseudo-coherent morphism of schemes (a finite morphism of Noetherian schemes is pseudo-coherent). The functor RSheafHom(f_*O_Y, -) maps D_QCoh^+(O_X) into D_QCoh^+(f_*O_Y). If X is quasi-compact and quasi-separated, then the diagram xymatrix D_QCoh^+(O_X) ar[rr]_a ar[rd]_RSheafHom(f_*O_Y, -) & & D_QCoh^+(O_Y) ar[ld]^Φ & D_QCoh^+(f_*O_Y) is commutative, where a is the right adjoint of Lemma [Tag 0A9E] for f and Φ is the equivalence of Derived…","statement_latex":"Let $f : Y \\to X$ be a finite pseudo-coherent morphism of schemes\n(a finite morphism of Noetherian schemes is pseudo-coherent).\nThe functor $R\\SheafHom(f_*\\mathcal{O}_Y, -)$ maps\n$D_\\QCoh^+(\\mathcal{O}_X)$ into $D_\\QCoh^+(f_*\\mathcal{O}_Y)$.\nIf $X$ is quasi-compact and quasi-separated, then the diagram\n$$\n\\xymatrix{\nD_\\QCoh^+(\\mathcal{O}_X) \\ar[rr]_a \\ar[rd]_{R\\SheafHom(f_*\\mathcal{O}_Y, -)}\n& & D_\\QCoh^+(\\mathcal{O}_Y) \\ar[ld]^\\Phi \\\\\n& D_\\QCoh^+(f_*\\mathcal{O}_Y)\n}\n$$\nis commutative, where $a$ is the right adjoint of\nLemma \\ref{lemma-twisted-inverse-image} for $f$ and $\\Phi$ is the equivalence\nof Derived Categories of Schemes, Lemma\n\\ref{perfect-lemma-affine-morphism-equivalence}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AX2","source_file":"duality.tex","source_line":2545,"source_end_line":2563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2545-L2563","statement_sha256":"78eb27434eabb6eec1d341670e208d77e11b428c5fc544d916bcd8bc5bd35fe6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8817,"rank":8817,"depth":40,"x":1970.869,"y":1256.243,"cluster":"duality-cohomology"},{"id":"stacks:0E4I","tag":"0E4I","title":"Right adjoint of pushforward for proper flat morphisms · Lemma 0E4I","summary":"Let Y be a quasi-compact and quasi-separated scheme. Let f : X → Y be a morphism of schemes which is proper, flat, and of finite presentation. Let a be the right adjoint for Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) of Lemma [Tag 0A9E]. Then a commutes with direct sums.","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to Y$ be a morphism of schemes which is proper, flat, and\nof finite presentation.\nLet $a$ be the right adjoint for\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of\nLemma \\ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4I","source_file":"duality.tex","source_line":2613,"source_end_line":2621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2613-L2621","statement_sha256":"3a3ba41e736f609f3e467ce29055507b048dfb63914a4b151ee79bf7cf6ead3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8818,"rank":8818,"depth":39,"x":1980.062,"y":1020.587,"cluster":"duality-cohomology"},{"id":"stacks:0E4J","tag":"0E4J","title":"Right adjoint of pushforward for proper flat morphisms · Lemma 0E4J","summary":"Let Y be a quasi-compact and quasi-separated scheme. Let f : X → Y be a morphism of schemes which is proper, flat, and of finite presentation. Let a be the right adjoint for Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) of Lemma [Tag 0A9E]. Then • for every closed T ⊂ Y if Q ∈ D_QCoh(Y) is supported on T, then a(Q) is supported on f^-1(T), • for every quasi-compact open V ⊂ Y and any K ∈ D_QCoh(O_Y) the map ([Tag 0A9L]) is an isomorphism.","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to Y$ be a morphism of schemes which is proper, flat, and\nof finite presentation.\nLet $a$ be the right adjoint for\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of\nLemma \\ref{lemma-twisted-inverse-image}. Then\n\\begin{enumerate}\n\\item for every closed $T \\subset Y$ if $Q \\in D_\\QCoh(Y)$ is supported on $T$,\nthen $a(Q)$ is supported on $f^{-1}(T)$,\n\\item for every quasi-compact open $V \\subset Y$ and any\n$K \\in D_\\QCoh(\\mathcal{O}_Y)$ the map (\\ref{equation-sheafy})\nis an isomorphism.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4J","source_file":"duality.tex","source_line":2648,"source_end_line":2663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2648-L2663","statement_sha256":"b06b3d69b68328f9e8a520ef86393fee7ce542982aad38b588b9811d2245c1cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8819,"rank":8819,"depth":40,"x":2163.013,"y":1199.79,"cluster":"duality-cohomology"},{"id":"stacks:0E4K","tag":"0E4K","title":"Right adjoint of pushforward for proper flat morphisms · Lemma 0E4K","summary":"Let Y be a quasi-compact and quasi-separated scheme. Let f : X → Y be a morphism of schemes which is proper, flat, and of finite presentation. The map ([Tag 0A9S]) is an isomorphism for every object K of D_QCoh(O_Y).","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to Y$ be a morphism of schemes which is proper, flat, and\nof finite presentation.\nThe map (\\ref{equation-compare-with-pullback}) is an isomorphism\nfor every object $K$ of $D_\\QCoh(\\mathcal{O}_Y)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4K","source_file":"duality.tex","source_line":2674,"source_end_line":2681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2674-L2681","statement_sha256":"ffb98309469a6bbe4862268145a3c42d3f4f818e487b7e94d72f092b08daf2c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8820,"rank":8820,"depth":40,"x":1883.661,"y":1171.425,"cluster":"duality-cohomology"},{"id":"stacks:0AAB","tag":"0AAB","title":"Right adjoint of pushforward for proper flat morphisms · Lemma 0AAB","summary":"Let g : Y' → Y be a morphism of quasi-compact and quasi-separated schemes. Let f : X → Y be a proper, flat morphism of finite presentation. Then the base change map ([Tag 0AA6]) is an isomorphism for all K ∈ D_QCoh(O_Y).","statement_latex":"Let $g : Y' \\to Y$ be a morphism of quasi-compact and quasi-separated schemes.\nLet $f : X \\to Y$ be a proper, flat morphism of finite presentation.\nThen the base change map (\\ref{equation-base-change-map}) is an isomorphism\nfor all $K \\in D_\\QCoh(\\mathcal{O}_Y)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAB","source_file":"duality.tex","source_line":2711,"source_end_line":2717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2711-L2717","statement_sha256":"6644c4723f9109044342d0354655cb16b354872d7b10e5c44bd3750276c4fe56","origin":"The Stacks Project","memory_eligible":false,"source_rank":8821,"rank":8821,"depth":41,"x":2112.737,"y":1033.663,"cluster":"duality-cohomology"},{"id":"stacks:0E4L","tag":"0E4L","title":"Right adjoint of pushforward for proper flat morphisms · Lemma 0E4L","summary":"Let Y be a quasi-compact and quasi-separated scheme. Let f : X → Y be a morphism of schemes which is proper, flat, and of finite presentation with relative dualizing complex ω_X/Y^bullet (Remark [Tag 0B6S]). Then • ω_X/Y^bullet is a Y-perfect object of D(O_X), • Rf_*ω_X/Y^bullet has vanishing cohomology sheaves in positive degrees, • O_X → RSheafHom_O_X(ω_X/Y^bullet, ω_X/Y^bullet) is an isomorphism.","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to Y$ be a morphism of schemes which is\nproper, flat, and of finite presentation with\nrelative dualizing complex $\\omega_{X/Y}^\\bullet$\n(Remark \\ref{remark-relative-dualizing-complex}).\nThen\n\\begin{enumerate}\n\\item $\\omega_{X/Y}^\\bullet$ is a $Y$-perfect object of $D(\\mathcal{O}_X)$,\n\\item $Rf_*\\omega_{X/Y}^\\bullet$ has vanishing cohomology sheaves\nin positive degrees,\n\\item $\\mathcal{O}_X \\to\nR\\SheafHom_{\\mathcal{O}_X}(\\omega_{X/Y}^\\bullet, \\omega_{X/Y}^\\bullet)$\nis an isomorphism.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4L","source_file":"duality.tex","source_line":2810,"source_end_line":2826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2810-L2826","statement_sha256":"11f2f51b366ad24a2465e06a3b64f9e252dc339feddf9babb7450a30147cd3be","origin":"The Stacks Project","memory_eligible":false,"source_rank":8822,"rank":8822,"depth":41,"x":2054.533,"y":1265.509,"cluster":"duality-cohomology"},{"id":"stacks:0E2P","tag":"0E2P","title":"Rigidity · Lemma 0E2P","summary":"Let Y be a quasi-compact and quasi-separated scheme. Let f : X → Y be a proper, flat morphism of finite presentation with relative dualizing complex ω_X/Y^bullet (Remark [Tag 0B6S]). There is a canonical isomorphism O_X = c(Lpr_1^*ω_X/Y^bullet) = c(Lpr_2^*ω_X/Y^bullet) and a canonical isomorphism ω_X/Y^bullet = c(Lpr_1^*ω_X/Y^bullet ⊗_O_X ×_Y X^L Lpr_2^*ω_X/Y^bullet) where c is the right adjoint of Lemma [Tag 0A9E] for the diagonal Δ : X → X ×_Y X.","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to Y$ be a proper, flat morphism of finite presentation\nwith relative dualizing complex $\\omega_{X/Y}^\\bullet$\n(Remark \\ref{remark-relative-dualizing-complex}).\nThere is a canonical isomorphism\n\\begin{equation}\n\n\\mathcal{O}_X =\nc(L\\text{pr}_1^*\\omega_{X/Y}^\\bullet) =\nc(L\\text{pr}_2^*\\omega_{X/Y}^\\bullet)\n\\end{equation}\nand a canonical isomorphism\n\\begin{equation}\n\n\\omega_{X/Y}^\\bullet =\nc\\left(L\\text{pr}_1^*\\omega_{X/Y}^\\bullet\n\\otimes_{\\mathcal{O}_{X \\times_Y X}}^\\mathbf{L}\nL\\text{pr}_2^*\\omega_{X/Y}^\\bullet\\right)\n\\end{equation}\nwhere $c$ is the right adjoint of\nLemma \\ref{lemma-twisted-inverse-image}\nfor the diagonal $\\Delta : X \\to X \\times_Y X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2P","source_file":"duality.tex","source_line":2941,"source_end_line":2965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L2941-L2965","statement_sha256":"0bc8ddc1c9e91fafca5bdd977b65166a570430b8057ba4fba6974be5eb1aeb44","origin":"The Stacks Project","memory_eligible":false,"source_rank":8823,"rank":8823,"depth":42,"x":1910.836,"y":1061.278,"cluster":"duality-cohomology"},{"id":"stacks:0A9R","tag":"0A9R","title":"Right adjoint of pushforward for perfect proper morphisms · Lemma 0A9R","summary":"Let f : X → Y be a perfect proper morphism of Noetherian schemes. Let a be the right adjoint for Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) of Lemma [Tag 0A9E]. Then a commutes with direct sums.","statement_latex":"Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes.\nLet $a$ be the right adjoint for\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of\nLemma \\ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for perfect proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9R","source_file":"duality.tex","source_line":3035,"source_end_line":3041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3035-L3041","statement_sha256":"c4774149bc2d4cd9555d09c61883754bf4c5713f61f6c666def0d2b7e1b49e09","origin":"The Stacks Project","memory_eligible":false,"source_rank":8824,"rank":8824,"depth":39,"x":2181.356,"y":1130.42,"cluster":"duality-cohomology"},{"id":"stacks:0AAA","tag":"0AAA","title":"Right adjoint of pushforward for perfect proper morphisms · Lemma 0AAA","summary":"Let f : X → Y be a perfect proper morphism of Noetherian schemes. Let a be the right adjoint for Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) of Lemma [Tag 0A9E]. Then • for every closed T ⊂ Y if Q ∈ D_QCoh(Y) is supported on T, then a(Q) is supported on f^-1(T), • for every open V ⊂ Y and any K ∈ D_QCoh(O_Y) the map ([Tag 0A9L]) is an isomorphism, and","statement_latex":"Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes.\nLet $a$ be the right adjoint for\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of\nLemma \\ref{lemma-twisted-inverse-image}. Then\n\\begin{enumerate}\n\\item for every closed $T \\subset Y$ if $Q \\in D_\\QCoh(Y)$ is supported on $T$,\nthen $a(Q)$ is supported on $f^{-1}(T)$,\n\\item for every open $V \\subset Y$ and any $K \\in D_\\QCoh(\\mathcal{O}_Y)$\nthe map (\\ref{equation-sheafy}) is an isomorphism, and\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for perfect proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AAA","source_file":"duality.tex","source_line":3068,"source_end_line":3080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3068-L3080","statement_sha256":"af2e1c6304ff5b377bc87c0f8acf1503876f5c3c8c789ab3c9626d7c0b119e25","origin":"The Stacks Project","memory_eligible":false,"source_rank":8825,"rank":8825,"depth":40,"x":1925.972,"y":1233.058,"cluster":"duality-cohomology"},{"id":"stacks:0A9U","tag":"0A9U","title":"Right adjoint of pushforward for perfect proper morphisms · Lemma 0A9U","summary":"Let f : X → Y be a perfect proper morphism of Noetherian schemes. The map ([Tag 0A9S]) is an isomorphism for every object K of D_QCoh(O_Y).","statement_latex":"Let $f : X \\to Y$ be a perfect proper morphism of Noetherian\nschemes. The map (\\ref{equation-compare-with-pullback}) is an isomorphism\nfor every object $K$ of $D_\\QCoh(\\mathcal{O}_Y)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for perfect proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9U","source_file":"duality.tex","source_line":3088,"source_end_line":3093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3088-L3093","statement_sha256":"f660b56e8334f3a295ed7d811e9a9ea3c9ffa2a1bce44fa457c7b9e1f467d5cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":8826,"rank":8826,"depth":40,"x":2031.876,"y":1012.201,"cluster":"duality-cohomology"},{"id":"stacks:0BZG","tag":"0BZG","title":"Right adjoint of pushforward for perfect proper morphisms · Lemma 0BZG","summary":"Let f : X → Y be a perfect proper morphism of Noetherian schemes. Let g : Y' → Y be a morphism with Y' Noetherian. If X and Y' are tor independent over Y, then the base change map ([Tag 0AA6]) is an isomorphism for all K ∈ D_QCoh(O_Y).","statement_latex":"Let $f : X \\to Y$ be a perfect proper morphism of Noetherian schemes.\nLet $g : Y' \\to Y$ be a morphism with $Y'$ Noetherian. If $X$ and\n$Y'$ are tor independent over $Y$, then the base\nchange map (\\ref{equation-base-change-map}) is an isomorphism\nfor all $K \\in D_\\QCoh(\\mathcal{O}_Y)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for perfect proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZG","source_file":"duality.tex","source_line":3114,"source_end_line":3121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3114-L3121","statement_sha256":"59a309158cbe7bec209c5d5f63396f94fa2faa92293572a28f83e8a3d4d3c8ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":8827,"rank":8827,"depth":41,"x":2131.509,"y":1235.415,"cluster":"duality-cohomology"},{"id":"stacks:0B4B","tag":"0B4B","title":"Right adjoint of pushforward for effective Cartier divisors · Lemma 0B4B","summary":"As above, let X be a scheme and let D ⊂ X be an effective Cartier divisor. There is a canonical isomorphism RSheafHom(O_D, O_X) = N[-1] in D(O_D).","statement_latex":"As above, let $X$ be a scheme and let $D \\subset X$ be an\neffective Cartier divisor. There is a canonical isomorphism\n$R\\SheafHom(\\mathcal{O}_D, \\mathcal{O}_X) = \\mathcal{N}[-1]$\nin $D(\\mathcal{O}_D)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4B","source_file":"duality.tex","source_line":3150,"source_end_line":3156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3150-L3156","statement_sha256":"e119a764ebf9b2f6ccf09d656016c6c35eee5efbcd4329b5a8f3431c81c21d7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8828,"rank":8828,"depth":1,"x":1878.239,"y":1127.226,"cluster":"duality-cohomology"},{"id":"stacks:0AA4","tag":"0AA4","title":"Right adjoint of pushforward for effective Cartier divisors · Lemma 0AA4","summary":"As above, let X be a scheme and let D ⊂ X be an effective Cartier divisor. Then ([Tag 0B4C]) combined with Lemma [Tag 0B4B] defines an isomorphism Li^*K ⊗_O_D^L N[-1] → RSheafHom(O_D, K) functorial in K in D(O_X).","statement_latex":"As above, let $X$ be a scheme and let $D \\subset X$ be an\neffective Cartier divisor. Then (\\ref{equation-map-effective-Cartier})\ncombined with Lemma \\ref{lemma-compute-for-effective-Cartier}\ndefines an isomorphism\n$$\nLi^*K \\otimes_{\\mathcal{O}_D}^\\mathbf{L} \\mathcal{N}[-1]\n\\longrightarrow\nR\\SheafHom(\\mathcal{O}_D, K)\n$$\nfunctorial in $K$ in $D(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward for effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AA4","source_file":"duality.tex","source_line":3227,"source_end_line":3239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3227-L3239","statement_sha256":"50687adcefe7ccfb9e72ac1debbccbfac9fa7e85cec24a7c7240d705620fa8d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8829,"rank":8829,"depth":11,"x":2152.323,"y":1063.216,"cluster":"duality-cohomology"},{"id":"stacks:0A9W","tag":"0A9W","title":"Right adjoint of pushforward in examples · Lemma 0A9W","summary":"Let Y be a quasi-compact and quasi-separated scheme. Let E be a finite locally free O_Y-module of rank n + 1 with determinant L = wedge^n + 1(E). Let f : X = P(E) → Y be the projection. Let a be the right adjoint for Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) of Lemma [Tag 0A9E]. Then there is an isomorphism c : f^*L(-n - 1)[n] → a(O_Y) In particular, if E = O_Y^⊕ n + 1, then X = P^n_Y and we obtain a(O_Y) = O_X(-n - 1)[n].","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated scheme.\nLet $\\mathcal{E}$ be a finite locally\nfree $\\mathcal{O}_Y$-module of rank $n + 1$ with determinant\n$\\mathcal{L} = \\wedge^{n + 1}(\\mathcal{E})$.\nLet $f : X = \\mathbf{P}(\\mathcal{E}) \\to Y$ be the projection.\nLet $a$ be the right adjoint for\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of\nLemma \\ref{lemma-twisted-inverse-image}.\nThen there is an isomorphism\n$$\nc : f^*\\mathcal{L}(-n - 1)[n] \\longrightarrow a(\\mathcal{O}_Y)\n$$\nIn particular, if $\\mathcal{E} = \\mathcal{O}_Y^{\\oplus n + 1}$, then\n$X = \\mathbf{P}^n_Y$ and we obtain\n$a(\\mathcal{O}_Y) = \\mathcal{O}_X(-n - 1)[n]$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward in examples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A9W","source_file":"duality.tex","source_line":3294,"source_end_line":3311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3294-L3311","statement_sha256":"5c8f971876d86be0faf5eabaac09334e0a37d5d27aa4c722486a3c7ace5747f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8830,"rank":8830,"depth":39,"x":2001.515,"y":1266.177,"cluster":"duality-cohomology"},{"id":"stacks:0BQW","tag":"0BQW","title":"Right adjoint of pushforward in examples · Lemma 0BQW","summary":"Let Y be a ringed space. Let I ⊂ O_Y be a sheaf of ideals. Set O_X = O_Y/I and N = SheafHom_O_Y(I/I^2, O_X). There is a canonical isomorphism c : N → SheafExt^1_O_Y(O_X, O_X) .","statement_latex":"Let $Y$ be a ringed space. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$\nbe a sheaf of ideals. Set $\\mathcal{O}_X = \\mathcal{O}_Y/\\mathcal{I}$ and\n$\\mathcal{N} =\n\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{I}/\\mathcal{I}^2, \\mathcal{O}_X)$.\nThere is a canonical isomorphism\n$c : \\mathcal{N} \\to\n\\SheafExt^1_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)\n$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward in examples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQW","source_file":"duality.tex","source_line":3402,"source_end_line":3412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3402-L3412","statement_sha256":"c9435dce58e54239bb0a4dee009dcc3320666aaafd6fd11dc94e3b83632f1d5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8831,"rank":8831,"depth":6,"x":1949.442,"y":1030.668,"cluster":"duality-cohomology"},{"id":"stacks:0BQY","tag":"0BQY","title":"Right adjoint of pushforward in examples · Lemma 0BQY","summary":"Let Y be a ringed space. Let I ⊂ O_Y be a sheaf of ideals. Set O_X = O_Y/I. If I is Koszul-regular (Divisors, Definition [Tag 063D]) then composition on RSheafHom_O_Y(O_X, O_X) defines isomorphisms wedge^i(SheafExt^1_O_Y(O_X, O_X)) → SheafExt^i_O_Y(O_X, O_X) for all i.","statement_latex":"Let $Y$ be a ringed space. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$\nbe a sheaf of ideals. Set $\\mathcal{O}_X = \\mathcal{O}_Y/\\mathcal{I}$.\nIf $\\mathcal{I}$ is Koszul-regular\n(Divisors, Definition \\ref{divisors-definition-regular-ideal-sheaf})\nthen composition on $R\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)$\ndefines isomorphisms\n$$\n\\wedge^i(\\SheafExt^1_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X))\n\\longrightarrow\n\\SheafExt^i_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_X)\n$$\nfor all $i$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward in examples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQY","source_file":"duality.tex","source_line":3443,"source_end_line":3457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3443-L3457","statement_sha256":"d2aa44fd8dd37f40c2d6aeab9155da3874d6f0a91dc35fdd2076ddec6b4c7a4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8832,"rank":8832,"depth":5,"x":2177.497,"y":1174.95,"cluster":"duality-cohomology"},{"id":"stacks:0BQZ","tag":"0BQZ","title":"Right adjoint of pushforward in examples · Lemma 0BQZ","summary":"Let Y be a ringed space. Let I ⊂ O_Y be a sheaf of ideals. Set O_X = O_Y/I and N = SheafHom_O_Y(I/I^2, O_X). If I is Koszul-regular (Divisors, Definition [Tag 063D]) then RSheafHom_O_Y(O_X, O_Y) = wedge^r N[-r] where r : Y → (1, 2, 3, … ) sends y to the minimal number of generators of I needed in a neighbourhood of y.","statement_latex":"Let $Y$ be a ringed space. Let $\\mathcal{I} \\subset \\mathcal{O}_Y$\nbe a sheaf of ideals. Set $\\mathcal{O}_X = \\mathcal{O}_Y/\\mathcal{I}$ and\n$\\mathcal{N} =\n\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{I}/\\mathcal{I}^2, \\mathcal{O}_X)$.\nIf $\\mathcal{I}$ is Koszul-regular\n(Divisors, Definition \\ref{divisors-definition-regular-ideal-sheaf}) then\n$$\nR\\SheafHom_{\\mathcal{O}_Y}(\\mathcal{O}_X, \\mathcal{O}_Y) =\n\\wedge^r \\mathcal{N}[-r]\n$$\nwhere $r : Y \\to \\{1, 2, 3, \\ldots \\}$ sends $y$ to\nthe minimal number of generators of $\\mathcal{I}$ needed in a neighbourhood\nof $y$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward in examples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQZ","source_file":"duality.tex","source_line":3543,"source_end_line":3558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3543-L3558","statement_sha256":"1241e03408f55702ed2bc0ff55dd0870666df865b1de692ef432853e34e91573","origin":"The Stacks Project","memory_eligible":false,"source_rank":8833,"rank":8833,"depth":7,"x":1892.972,"y":1197.99,"cluster":"duality-cohomology"},{"id":"stacks:0BR0","tag":"0BR0","title":"Right adjoint of pushforward in examples · Lemma 0BR0","summary":"Let Y be a quasi-compact and quasi-separated scheme. Let i : X → Y be a Koszul-regular closed immersion. Let a be the right adjoint of Ri_* : D_QCoh(O_X) → D_QCoh(O_Y) of Lemma [Tag 0A9E]. Then there is an isomorphism wedge^rN[-r] → a(O_Y) where N = SheafHom_O_X(C_X/Y, O_X) is the normal sheaf of i (Morphisms, Section [Tag 01R1]) and r is its rank viewed as a locally constant function on X.","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated scheme.\nLet $i : X \\to Y$ be a Koszul-regular closed immersion.\nLet $a$ be the right adjoint of\n$Ri_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of\nLemma \\ref{lemma-twisted-inverse-image}. Then there is an isomorphism\n$$\n\\wedge^r\\mathcal{N}[-r] \\longrightarrow a(\\mathcal{O}_Y)\n$$\nwhere\n$\\mathcal{N} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{C}_{X/Y}, \\mathcal{O}_X)$\nis the normal sheaf of $i$\n(Morphisms, Section \\ref{morphisms-section-conormal-sheaf})\nand $r$ is its rank viewed as a locally constant\nfunction on $X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward in examples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BR0","source_file":"duality.tex","source_line":3634,"source_end_line":3650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3634-L3650","statement_sha256":"1fba94ef4e3cf4e4a04361cc025d2dd0910506f8a0312d724cd0776e4382dca3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8834,"rank":8834,"depth":41,"x":2084.472,"y":1019.346,"cluster":"duality-cohomology"},{"id":"stacks:0BRT","tag":"0BRT","title":"Right adjoint of pushforward in examples · Lemma 0BRT","summary":"Let S be a Noetherian scheme. Let f : X → S be a smooth proper morphism of relative dimension d. Let a be the right adjoint of Rf_* : D_QCoh(O_X) → D_QCoh(O_S) as in Lemma [Tag 0A9E]. Then there is an isomorphism wedge^d Ω_X/S[d] → a(O_S) in D(O_X).","statement_latex":"Let $S$ be a Noetherian scheme.\nLet $f : X \\to S$ be a smooth proper morphism of relative dimension $d$.\nLet $a$ be the right adjoint of\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_S)$ as in\nLemma \\ref{lemma-twisted-inverse-image}. Then there is an isomorphism\n$$\n\\wedge^d \\Omega_{X/S}[d] \\longrightarrow a(\\mathcal{O}_S)\n$$\nin $D(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Right adjoint of pushforward in examples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BRT","source_file":"duality.tex","source_line":3661,"source_end_line":3672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3661-L3672","statement_sha256":"1799191e570d98e03fc103b89e6f3f9910bc942ecee30c87d446284da91d0e20","origin":"The Stacks Project","memory_eligible":false,"source_rank":8835,"rank":8835,"depth":43,"x":2086.926,"y":1260.017,"cluster":"duality-cohomology"},{"id":"stacks:0AA0","tag":"0AA0","title":"Upper shriek functors · Lemma 0AA0","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. The functor f^! is, up to canonical isomorphism, independent of the choice of the compactification.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism\nof $\\textit{FTS}_S$. The functor $f^!$ is, up to canonical isomorphism,\nindependent of the choice of the compactification.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AA0","source_file":"duality.tex","source_line":3768,"source_end_line":3773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3768-L3773","statement_sha256":"69a4a32c1b7e82cd38f0e3c6a54f529b77b4bc1dda042629bddefffce213ad5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8836,"rank":8836,"depth":61,"x":1891.348,"y":1083.737,"cluster":"duality-cohomology"},{"id":"stacks:0ATX","tag":"0ATX","title":"Upper shriek functors · Lemma 0ATX","summary":"In Situation [Tag 0F42] let f : X → Y and g : Y → Z be composable morphisms of FTS_S. Then there is a canonical isomorphism (g ∘ f)^! → f^! ∘ g^!.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ and $g : Y \\to Z$\nbe composable morphisms of $\\textit{FTS}_S$. Then there is a canonical\nisomorphism $(g \\circ f)^! \\to f^! \\circ g^!$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATX","source_file":"duality.tex","source_line":3943,"source_end_line":3948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L3943-L3948","statement_sha256":"774e26c24cba0cd4c32b6f3dd94a992e4b72d6d91f34bbd9fd100447fc85ad4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8837,"rank":8837,"depth":62,"x":2177.656,"y":1102.77,"cluster":"duality-cohomology"},{"id":"stacks:0ATY","tag":"0ATY","title":"Upper shriek functors · Lemma 0ATY","summary":"In Situation [Tag 0F42] the constructions of Lemmas [Tag 0AA0] and [Tag 0ATX] define a pseudo functor from the category FTS_S into the 2-category of categories (see Categories, Definition [Tag 003N]).","statement_latex":"In Situation \\ref{situation-shriek} the constructions of\nLemmas \\ref{lemma-shriek-well-defined} and \\ref{lemma-upper-shriek-composition}\ndefine a pseudo functor from the category $\\textit{FTS}_S$\ninto the $2$-category of categories (see Categories, Definition\n\\ref{categories-definition-functor-into-2-category}).","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATY","source_file":"duality.tex","source_line":4152,"source_end_line":4159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4152-L4159","statement_sha256":"6dd0f9b70354ea258fa323a2eadf31d4a0d32eec2dc9d7ce5aa96b348ed302dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8838,"rank":8838,"depth":63,"x":1950.969,"y":1251.365,"cluster":"duality-cohomology"},{"id":"stacks:0B6T","tag":"0B6T","title":"Upper shriek functors · Lemma 0B6T","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. There are canonical maps μ_f, K : Lf^*K ⊗_O_X^L f^!O_Y → f^!K functorial in K in D^+_QCoh(O_Y). If g : Y → Z is another morphism of FTS_S, then the diagram xymatrix Lf^*(Lg^*K ⊗_O_Y^L g^!O_Z) ⊗_O_X^L f^!O_Y ar@=[d] ar[r]_-μ_f & f^!(Lg^*K ⊗_O_Y^L g^!O_Z) ar[r]_-f^!μ_g & f^!g^!K ar@=[d] Lf^*Lg^*K ⊗_O_X^L Lf^* g^!O_Z ⊗_O_X^L f^!O_Y ar[r]^-μ_f & Lf^*Lg^*K ⊗_O_X^L f^!g^!O_Z ar[r]^-μ_g ∘ f & f^!g^!K commutes for all…","statement_latex":"In Situation \\ref{situation-shriek} let\n$f : X \\to Y$ be a morphism of $\\textit{FTS}_S$. There are canonical maps\n$$\n\\mu_{f, K} :\nLf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y\n\\longrightarrow\nf^!K\n$$\nfunctorial in $K$ in $D^+_\\QCoh(\\mathcal{O}_Y)$.\nIf $g : Y \\to Z$ is another morphism of $\\textit{FTS}_S$, then\nthe diagram\n$$\n\\xymatrix{\nLf^*(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} g^!\\mathcal{O}_Z)\n\\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y\n\\ar@{=}[d] \\ar[r]_-{\\mu_f} &\nf^!(Lg^*K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} g^!\\mathcal{O}_Z)\n\\ar[r]_-{f^!\\mu_g} &\nf^!g^!K \\ar@{=}[d] \\\\\nLf^*Lg^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^* g^!\\mathcal{O}_Z\n\\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y \\ar[r]^-{\\mu_f} &\nLf^*Lg^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!g^!\\mathcal{O}_Z\n\\ar[r]^-{\\mu_{g \\circ f}} & f^!g^!K\n}\n$$\ncommutes for all $K \\in D^+_\\QCoh(\\mathcal{O}_Z)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6T","source_file":"duality.tex","source_line":4209,"source_end_line":4237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4209-L4237","statement_sha256":"f0fb888181d9691f0ad969091edbf39306d4db0a7020f3e3c17d5ed6ee5d2c0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8839,"rank":8839,"depth":63,"x":1998.684,"y":1012.891,"cluster":"duality-cohomology"},{"id":"stacks:0AU0","tag":"0AU0","title":"Properties of upper shriek functors · Lemma 0AU0","summary":"In Situation [Tag 0F42] let Y be an object of FTS_S and let j : X → Y be an open immersion. Then there is a canonical isomorphism j^! = j^* of functors.","statement_latex":"In Situation \\ref{situation-shriek} let $Y$ be an object\nof $\\textit{FTS}_S$ and let $j : X \\to Y$ be an open immersion.\nThen there is a canonical isomorphism $j^! = j^*$ of functors.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AU0","source_file":"duality.tex","source_line":4350,"source_end_line":4355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4350-L4355","statement_sha256":"ead594656da6aad1c3f917f48d038dd72b04c369dc4e215924b43c4175f3a6fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8840,"rank":8840,"depth":0,"x":2155.454,"y":1216.046,"cluster":"duality-cohomology"},{"id":"stacks:0G4J","tag":"0G4J","title":"Properties of upper shriek functors · Lemma 0G4J","summary":"In Situation [Tag 0F42] let xymatrix U ar[r]_j ar[d]_g & X ar[d]^f V ar[r]^j' & Y be a commutative diagram of FTS_S where j and j' are open immersions. Then j^* ∘ f^! = g^! ∘ (j')^* as functors D^+_QCoh(O_Y) → D^+(O_U).","statement_latex":"In Situation \\ref{situation-shriek} let\n$$\n\\xymatrix{\nU \\ar[r]_j \\ar[d]_g & X \\ar[d]^f \\\\\nV \\ar[r]^{j'} & Y\n}\n$$\nbe a commutative diagram of $\\textit{FTS}_S$ where $j$ and $j'$ are\nopen immersions. Then $j^* \\circ f^! = g^! \\circ (j')^*$ as functors\n$D^+_\\QCoh(\\mathcal{O}_Y) \\to D^+(\\mathcal{O}_U)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4J","source_file":"duality.tex","source_line":4369,"source_end_line":4381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4369-L4381","statement_sha256":"1e7df2cd9e489500cbfd3b58569981f2aaa1721cefc68de9952cecc88fa2c3a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8841,"rank":8841,"depth":63,"x":1876.162,"y":1155.128,"cluster":"duality-cohomology"},{"id":"stacks:0AA1","tag":"0AA1","title":"Properties of upper shriek functors · Lemma 0AA1","summary":"In Situation [Tag 0F42] let Y be an object of FTS_S and let f : X = A^1_Y → Y be the projection. Then there is a (noncanonical) isomorphism f^!(-) ≅ Lf^*(-) [1] of functors.","statement_latex":"In Situation \\ref{situation-shriek} let $Y$ be an object of $\\textit{FTS}_S$\nand let $f : X = \\mathbf{A}^1_Y \\to Y$ be\nthe projection. Then there is a (noncanonical) isomorphism\n$f^!(-) \\cong Lf^*(-) [1]$ of functors.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AA1","source_file":"duality.tex","source_line":4391,"source_end_line":4397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4391-L4397","statement_sha256":"e871f12240da36d796800b4d27c41f604146bde819d038db10374bdb7ec2a7a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8842,"rank":8842,"depth":41,"x":2131.389,"y":1041.441,"cluster":"duality-cohomology"},{"id":"stacks:0AA2","tag":"0AA2","title":"Properties of upper shriek functors · Lemma 0AA2","summary":"In Situation [Tag 0F42] let Y be an object of FTS_S and let i : X → Y be a closed immersion. Then there is a canonical isomorphism i^!(-) = RSheafHom(O_X, -) of functors.","statement_latex":"In Situation \\ref{situation-shriek} let $Y$ be an object of\n$\\textit{FTS}_S$ and let $i : X \\to Y$ be a closed immersion.\nThen there is a canonical isomorphism\n$i^!(-) = R\\SheafHom(\\mathcal{O}_X, -)$ of functors.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AA2","source_file":"duality.tex","source_line":4406,"source_end_line":4412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4406-L4412","statement_sha256":"605352d7d2ac8e386d076e3ca92b9896291429dde09052521a75cdaa38c6fc6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8843,"rank":8843,"depth":41,"x":2034.501,"y":1270.355,"cluster":"duality-cohomology"},{"id":"stacks:0AU1","tag":"0AU1","title":"Properties of upper shriek functors · Lemma 0AU1","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Then f^! maps D_Coh^+(O_Y) into D_Coh^+(O_X).","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Then $f^!$ maps\n$D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$ into $D_{\\textit{Coh}}^+(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AU1","source_file":"duality.tex","source_line":4468,"source_end_line":4473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4468-L4473","statement_sha256":"188ba48dd6161f4c801d15c5e9a0f62603955ef4bc76ab94c2d524fe694ee6d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":8844,"rank":8844,"depth":42,"x":1921.73,"y":1046.326,"cluster":"duality-cohomology"},{"id":"stacks:0AA3","tag":"0AA3","title":"Properties of upper shriek functors · Lemma 0AA3","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. If K is a dualizing complex for Y, then f^!K is a dualizing complex for X.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. If $K$ is a dualizing complex\nfor $Y$, then $f^!K$ is a dualizing complex for $X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AA3","source_file":"duality.tex","source_line":4490,"source_end_line":4495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4490-L4495","statement_sha256":"043912f9055c9398b0fd39e88038804f97ecdbbb71bf2032d4135ea97302c031","origin":"The Stacks Project","memory_eligible":false,"source_rank":8845,"rank":8845,"depth":42,"x":2185.343,"y":1147.647,"cluster":"duality-cohomology"},{"id":"stacks:0AU2","tag":"0AU2","title":"Properties of upper shriek functors · Lemma 0AU2","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Let K be a dualizing complex on Y. Set D_Y(M) = RSheafHom_O_Y(M, K) for M ∈ D_Coh(O_Y) and D_X(E) = RSheafHom_O_X(E, f^!K) for E ∈ D_Coh(O_X). Then there is a canonical isomorphism f^!M → D_X(Lf^*D_Y(M)) for M ∈ D_Coh^+(O_Y).","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Let $K$ be a dualizing complex\non $Y$. Set $D_Y(M) = R\\SheafHom_{\\mathcal{O}_Y}(M, K)$ for\n$M \\in D_{\\textit{Coh}}(\\mathcal{O}_Y)$ and\n$D_X(E) = R\\SheafHom_{\\mathcal{O}_X}(E, f^!K)$ for\n$E \\in D_{\\textit{Coh}}(\\mathcal{O}_X)$. Then there is a canonical\nisomorphism\n$$\nf^!M \\longrightarrow D_X(Lf^*D_Y(M))\n$$\nfor $M \\in D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AU2","source_file":"duality.tex","source_line":4516,"source_end_line":4529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4516-L4529","statement_sha256":"550e57ba96399323da7fd6c55d564154153a3a08a625770b5d45d5391e56287a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8846,"rank":8846,"depth":61,"x":1909.165,"y":1222.6,"cluster":"duality-cohomology"},{"id":"stacks:0B6U","tag":"0B6U","title":"Properties of upper shriek functors · Lemma 0B6U","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Assume f is perfect (e.g., flat). Then • [(a)] f^!O_Y is in D_Coh^b(O_X), • [(b)] f^!O_Y has finite tor dimension in D(f^-1O_Y), • [(c)] O_X → RSheafHom_O_X(f^!O_Y, f^!O_Y) is an isomorphism, • [(d)] f^! maps D_Coh^b(O_Y) into D_Coh^b(O_X), • [(e)] the map μ_f, K : Lf^*K ⊗_O_X^L f^!O_Y → f^!K of Lemma [Tag 0B6T] is an isomorphism for all K ∈ D_QCoh^+(O_Y).","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Assume $f$ is perfect (e.g., flat). Then\n\\begin{enumerate}\n\\item[(a)] $f^!\\mathcal{O}_Y$ is in $D_{\\textit{Coh}}^b(\\mathcal{O}_X)$,\n\\item[(b)] $f^!\\mathcal{O}_Y$ has finite tor dimension in\n$D(f^{-1}\\mathcal{O}_Y)$,\n\\item[(c)] $\\mathcal{O}_X \\to\nR\\SheafHom_{\\mathcal{O}_X}(f^!\\mathcal{O}_Y, f^!\\mathcal{O}_Y)$\nis an isomorphism,\n\\item[(d)] $f^!$ maps $D_{\\textit{Coh}}^b(\\mathcal{O}_Y)$ into\n$D_{\\textit{Coh}}^b(\\mathcal{O}_X)$,\n\\item[(e)] the map\n$\\mu_{f,  K} :\nLf^*K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} f^!\\mathcal{O}_Y\n\\to\nf^!K$\nof Lemma \\ref{lemma-map-pullback-to-shriek-well-defined}\nis an isomorphism for all $K \\in D_\\QCoh^+(\\mathcal{O}_Y)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6U","source_file":"duality.tex","source_line":4580,"source_end_line":4601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4580-L4601","statement_sha256":"4c0c81ad2d3eb0dee7f4d919c8a23ec158b10941fec8400de6ddc9988b12b34d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8847,"rank":8847,"depth":64,"x":2052.703,"y":1010.381,"cluster":"duality-cohomology"},{"id":"stacks:0E9T","tag":"0E9T","title":"Properties of upper shriek functors · Lemma 0E9T","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. If f is flat, then f^!O_Y is a Y-perfect object of D(O_X) and O_X → RSheafHom_O_X(f^!O_Y, f^!O_Y) is an isomorphism.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. If $f$ is flat, then\n$f^!\\mathcal{O}_Y$ is a $Y$-perfect object of $D(\\mathcal{O}_X)$ and\n$\\mathcal{O}_X \\to\nR\\SheafHom_{\\mathcal{O}_X}(f^!\\mathcal{O}_Y, f^!\\mathcal{O}_Y)$\nis an isomorphism.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9T","source_file":"duality.tex","source_line":4672,"source_end_line":4680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4672-L4680","statement_sha256":"316f0499bf4c89e04e8237319a3b59ba0f7e887d81374cf2e6ae7aa2dc66b493","origin":"The Stacks Project","memory_eligible":false,"source_rank":8848,"rank":8848,"depth":39,"x":2117.594,"y":1248.583,"cluster":"duality-cohomology"},{"id":"stacks:0B6V","tag":"0B6V","title":"Properties of upper shriek functors · Lemma 0B6V","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Assume f : X → Y is a local complete intersection morphism. Then • f^!O_Y is an invertible object of D(O_X), and • f^! maps perfect complexes to perfect complexes.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Assume $f : X \\to Y$ is a local complete\nintersection morphism. Then\n\\begin{enumerate}\n\\item $f^!\\mathcal{O}_Y$ is an invertible object of $D(\\mathcal{O}_X)$, and\n\\item $f^!$ maps perfect complexes to perfect complexes.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Properties of upper shriek functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B6V","source_file":"duality.tex","source_line":4691,"source_end_line":4700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4691-L4700","statement_sha256":"af64bb8d044e4f6687724b67d922dfe7b1f01556913d043a920a65f86efabfa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8849,"rank":8849,"depth":65,"x":1877.919,"y":1109.601,"cluster":"duality-cohomology"},{"id":"stacks:0E9U","tag":"0E9U","title":"Base change for upper shriek · Lemma 0E9U","summary":"In Situation [Tag 0F42] let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian diagram of FTS_S with g flat. Then there is an isomorphism L(g')^* ∘ f^! → (f')^! ∘ Lg^* on D_QCoh^+(O_Y).","statement_latex":"In Situation \\ref{situation-shriek} let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian diagram of $\\textit{FTS}_S$ with $g$ flat.\nThen there is an isomorphism\n$L(g')^* \\circ f^! \\to (f')^! \\circ Lg^*$ on\n$D_\\QCoh^+(\\mathcal{O}_Y)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Base change for upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9U","source_file":"duality.tex","source_line":4779,"source_end_line":4792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4779-L4792","statement_sha256":"e1236b747b5a8e7b7b21959933ecd8bf8d81ef5e313756fa94997cecf6165e7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8850,"rank":8850,"depth":40,"x":2166.742,"y":1076.049,"cluster":"duality-cohomology"},{"id":"stacks:0FWI","tag":"0FWI","title":"Base change for upper shriek · Lemma 0FWI","summary":"In Situation [Tag 0F42] let f : X → Y be an étale morphism of FTS_S. Then f^! ≅ f^* as functors on D^+_QCoh(O_Y).","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be an \\'etale\nmorphism of $\\textit{FTS}_S$. Then $f^! \\cong f^*$ as functors on\n$D^+_\\QCoh(\\mathcal{O}_Y)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Base change for upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWI","source_file":"duality.tex","source_line":4826,"source_end_line":4831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4826-L4831","statement_sha256":"a1ef4dc18ec7bb459fb35cb49d6d9938f43ccb443e90382df021f81c88dce7e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8851,"rank":8851,"depth":66,"x":1980.541,"y":1264.887,"cluster":"duality-cohomology"},{"id":"stacks:0BZY","tag":"0BZY","title":"Makeshift base change · Lemma 0BZY","summary":"In Situation [Tag 0F42] let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian diagram of FTS_S. Let E ∈ D^+_QCoh(O_Y) be an object such that Lg^*E is in D^+(O_Y). If f is flat, then L(g')^*f^!E and (f')^!Lg^*E restrict to isomorphic objects of D(O_U') for U' ⊂ X' affine open mapping into affine opens of Y, Y', and X.","statement_latex":"In Situation \\ref{situation-shriek} let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian diagram of $\\textit{FTS}_S$.\nLet $E \\in D^+_\\QCoh(\\mathcal{O}_Y)$ be an object\nsuch that $Lg^*E$ is in $D^+(\\mathcal{O}_Y)$.\nIf $f$ is flat, then $L(g')^*f^!E$ and $(f')^!Lg^*E$\nrestrict to isomorphic objects of $D(\\mathcal{O}_{U'})$\nfor $U' \\subset X'$ affine open mapping into affine opens of $Y$, $Y'$, and $X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Base change for upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZY","source_file":"duality.tex","source_line":4876,"source_end_line":4891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4876-L4891","statement_sha256":"99fa4d7be56c4946840e4805972c380f1f1a1a10cfcd3ee081ba32ad597d02d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8852,"rank":8852,"depth":37,"x":1965.969,"y":1019.712,"cluster":"duality-cohomology"},{"id":"stacks:0BZZ","tag":"0BZZ","title":"Base change for upper shriek · Lemma 0BZZ","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Assume f is flat. Set ω_X/Y^bullet = f^!O_Y in D^b_Coh(X). Let y ∈ Y and h : X_y → X the projection. Then Lh^*ω_X/Y^bullet is a dualizing complex on X_y.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Assume $f$ is flat. Set\n$\\omega_{X/Y}^\\bullet = f^!\\mathcal{O}_Y$ in $D^b_{\\textit{Coh}}(X)$.\nLet $y \\in Y$ and $h : X_y \\to X$ the projection.\nThen $Lh^*\\omega_{X/Y}^\\bullet$ is a dualizing complex\non $X_y$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Base change for upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BZZ","source_file":"duality.tex","source_line":4912,"source_end_line":4920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L4912-L4920","statement_sha256":"3fe31a719588718fbe9cbd2b8d593655acd0589f372340b1e4eb3bbe3fd70dda","origin":"The Stacks Project","memory_eligible":false,"source_rank":8853,"rank":8853,"depth":65,"x":2174.109,"y":1192.424,"cluster":"duality-cohomology"},{"id":"stacks:0AU7","tag":"0AU7","title":"Glueing dualizing complexes · Lemma 0AU7","summary":"In Situation [Tag 0AU4] let X be a scheme of finite type over S and let U be a finite open covering of X by schemes separated over S. If there exists a dualizing complex normalized relative to ω_S^bullet and U, then it is unique up to unique isomorphism.","statement_latex":"In Situation \\ref{situation-dualizing} let $X$ be a scheme of finite type\nover $S$ and let $\\mathcal{U}$ be a finite open covering of $X$\nby schemes separated over $S$. If there exists a dualizing complex\nnormalized relative to $\\omega_S^\\bullet$ and $\\mathcal{U}$, then it is unique\nup to unique isomorphism.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Glueing dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AU7","source_file":"duality.tex","source_line":5099,"source_end_line":5106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5099-L5106","statement_sha256":"32290050a9d39dd729e9c39b24a3ad2c076dc1a22c7b6a06f831b242419d8cc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8854,"rank":8854,"depth":35,"x":1881.413,"y":1183.162,"cluster":"duality-cohomology"},{"id":"stacks:0AU8","tag":"0AU8","title":"Glueing dualizing complexes · Lemma 0AU8","summary":"In Situation [Tag 0AU4] let X be a scheme of finite type over S and let U, V be two finite open coverings of X by schemes separated over S. If there exists a dualizing complex normalized relative to ω_S^bullet and U, then there exists a dualizing complex normalized relative to ω_S^bullet and V and these complexes are canonically isomorphic.","statement_latex":"In Situation \\ref{situation-dualizing} let $X$ be a scheme of finite type\nover $S$ and let $\\mathcal{U}$, $\\mathcal{V}$ be two finite open coverings\nof $X$ by schemes separated over $S$.\nIf there exists a dualizing complex normalized\nrelative to $\\omega_S^\\bullet$ and $\\mathcal{U}$, then\nthere exists a dualizing complex normalized relative to\n$\\omega_S^\\bullet$ and $\\mathcal{V}$ and these complexes are\ncanonically isomorphic.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Glueing dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AU8","source_file":"duality.tex","source_line":5130,"source_end_line":5140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5130-L5140","statement_sha256":"3e75547d38f9be47b900f45cf8369c4d04f7ff7ecac64db5d2ce53a7d536024c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8855,"rank":8855,"depth":36,"x":2104.939,"y":1023.732,"cluster":"duality-cohomology"},{"id":"stacks:0AU9","tag":"0AU9","title":"Glueing dualizing complexes · Lemma 0AU9","summary":"In Situation [Tag 0AU4] let X be a scheme of finite type over S and let U be a finite open covering of X by schemes separated over S. Then there exists a dualizing complex normalized relative to ω_S^bullet and U.","statement_latex":"In Situation \\ref{situation-dualizing} let $X$ be a scheme of finite type\nover $S$ and let $\\mathcal{U}$ be a finite open covering\nof $X$ by schemes separated over $S$. Then there exists\na dualizing complex normalized relative to $\\omega_S^\\bullet$ and\n$\\mathcal{U}$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Glueing dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AU9","source_file":"duality.tex","source_line":5174,"source_end_line":5181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5174-L5181","statement_sha256":"a5996eba51241ac91d20bebc06c30212e1e65ae280c1c68dc259d2d9120836cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8856,"rank":8856,"depth":36,"x":2068.283,"y":1268.399,"cluster":"duality-cohomology"},{"id":"stacks:0AUA","tag":"0AUA","title":"Glueing dualizing complexes · Definition 0AUA","summary":"Let S be a Noetherian scheme and let ω_S^bullet be a dualizing complex on S. Let X be a scheme of finite type over S. The complex K constructed above is called the dualizing complex normalized relative to ω_S^bullet and is denoted ω_X^bullet.","statement_latex":"Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a dualizing\ncomplex on $S$. Let $X$ be a scheme of finite type over $S$.\nThe complex $K$ constructed above is called the\n{\\it dualizing complex normalized relative to $\\omega_S^\\bullet$}\nand is denoted $\\omega_X^\\bullet$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Glueing dualizing complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUA","source_file":"duality.tex","source_line":5251,"source_end_line":5258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5251-L5258","statement_sha256":"de2cbd7fa0fe50e27a269f6888efcc5c8f7c5d322419d9233bd204b69e872779","origin":"The Stacks Project","memory_eligible":false,"source_rank":8857,"rank":8857,"depth":0,"x":1898.371,"y":1066.963,"cluster":"duality-cohomology"},{"id":"stacks:0AUB","tag":"0AUB","title":"Glueing dualizing complexes · Lemma 0AUB","summary":"Let (S, ω_S^bullet) be as in Situation [Tag 0AU4]. Let f : X → Y be a morphism of finite type schemes over S. Let ω_X^bullet and ω_Y^bullet be dualizing complexes normalized relative to ω_S^bullet. Then ω_X^bullet is a dualizing complex normalized relative to ω_Y^bullet.","statement_latex":"Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}.\nLet $f : X \\to Y$ be a morphism of finite type schemes over $S$.\nLet $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes\nnormalized relative to $\\omega_S^\\bullet$. Then $\\omega_X^\\bullet$\nis a dualizing complex normalized relative to $\\omega_Y^\\bullet$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Glueing dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUB","source_file":"duality.tex","source_line":5269,"source_end_line":5276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5269-L5276","statement_sha256":"7f4eb7d82300887094763a27b467249af55a853358f4975c73538e7b92dde9e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8858,"rank":8858,"depth":0,"x":2185.967,"y":1119.144,"cluster":"duality-cohomology"},{"id":"stacks:0AUC","tag":"0AUC","title":"Glueing dualizing complexes · Lemma 0AUC","summary":"Let (S, ω_S^bullet) be as in Situation [Tag 0AU4]. Let j : X → Y be an open immersion of schemes of finite type over S. Let ω_X^bullet and ω_Y^bullet be dualizing complexes normalized relative to ω_S^bullet. Then there is a canonical isomorphism ω_X^bullet = ω_Y^bullet|_X.","statement_latex":"Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}.\nLet $j : X \\to Y$ be an open immersion of schemes of finite type over $S$.\nLet $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes\nnormalized relative to $\\omega_S^\\bullet$. Then there is a canonical\nisomorphism $\\omega_X^\\bullet = \\omega_Y^\\bullet|_X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Glueing dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUC","source_file":"duality.tex","source_line":5308,"source_end_line":5315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5308-L5315","statement_sha256":"43f408a16976c92834862e1bd3382555fe01813a036c4aedb992c223b39826b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8859,"rank":8859,"depth":1,"x":1931.656,"y":1243.993,"cluster":"duality-cohomology"},{"id":"stacks:0AUD","tag":"0AUD","title":"Glueing dualizing complexes · Lemma 0AUD","summary":"Let (S, ω_S^bullet) be as in Situation [Tag 0AU4]. Let f : X → Y be a proper morphism of schemes of finite type over S. Let ω_X^bullet and ω_Y^bullet be dualizing complexes normalized relative to ω_S^bullet. Let a be the right adjoint of Lemma [Tag 0A9E] for f. Then there is a canonical isomorphism a(ω_Y^bullet) = ω_X^bullet.","statement_latex":"Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}.\nLet $f : X \\to Y$ be a proper morphism of schemes of finite type over $S$.\nLet $\\omega_X^\\bullet$ and $\\omega_Y^\\bullet$ be dualizing complexes\nnormalized relative to $\\omega_S^\\bullet$. Let $a$ be the\nright adjoint of Lemma \\ref{lemma-twisted-inverse-image} for\n$f$. Then there is a canonical isomorphism\n$a(\\omega_Y^\\bullet) = \\omega_X^\\bullet$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Glueing dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUD","source_file":"duality.tex","source_line":5323,"source_end_line":5332,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5323-L5332","statement_sha256":"53649e3f270c9d41bee7101a941b5d534f906dff73a298f2ce639f63bac51e6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8860,"rank":8860,"depth":63,"x":2018.877,"y":1007.369,"cluster":"duality-cohomology"},{"id":"stacks:0AUE","tag":"0AUE","title":"Glueing dualizing complexes · Lemma 0AUE","summary":"Let (S, ω_S^bullet) be as in Situation [Tag 0AU4]. With f^!_new and ω_X^bullet defined for all (morphisms of) schemes of finite type over S as above: • the functors f^!_new and the arrows (g ∘ f)^!_new → f^!_new ∘ g^!_new turn D_Coh^+ into a pseudo functor from the category of schemes of finite type over S into the 2-category of categories, • ω_X^bullet = (X → S)^!_new ω_S^bullet, • the functor D_X defines an involution of D_Coh(O_X) switching D_Coh^+(O_X) and…","statement_latex":"Let $(S, \\omega_S^\\bullet)$ be as in Situation \\ref{situation-dualizing}.\nWith $f^!_{new}$ and $\\omega_X^\\bullet$ defined for all (morphisms of)\nschemes of finite type over $S$ as above:\n\\begin{enumerate}\n\\item the functors $f^!_{new}$ and the arrows\n$(g \\circ f)^!_{new} \\to f^!_{new} \\circ g^!_{new}$\nturn $D_{\\textit{Coh}}^+$ into a pseudo functor from the category of\nschemes of finite type over $S$ into the $2$-category of categories,\n\\item $\\omega_X^\\bullet = (X \\to S)^!_{new} \\omega_S^\\bullet$,\n\\item the functor $D_X$\ndefines an involution of $D_{\\textit{Coh}}(\\mathcal{O}_X)$\nswitching $D_{\\textit{Coh}}^+(\\mathcal{O}_X)$ and\n$D_{\\textit{Coh}}^-(\\mathcal{O}_X)$ and fixing\n$D_{\\textit{Coh}}^b(\\mathcal{O}_X)$,\n\\item $\\omega_X^\\bullet = f^!_{new}\\omega_Y^\\bullet$ for\n$f : X \\to Y$ a morphism of finite type schemes over $S$,\n\\item $f^!_{new}M = D_X(Lf^*D_Y(M))$ for\n$M \\in D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$, and\n\\item if in addition $f$ is proper, then $f^!_{new}$ is isomorphic\nto the restriction of the right adjoint of\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ to\n$D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$ and there is a canonical isomorphism\n$$\nRf_*R\\SheafHom_{\\mathcal{O}_X}(K, f^!_{new}M)\n\\to\nR\\SheafHom_{\\mathcal{O}_Y}(Rf_*K, M)\n$$\nfor $K \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and\n$M \\in D_{\\textit{Coh}}^+(\\mathcal{O}_Y)$, and\n$$\nRf_*R\\SheafHom_{\\mathcal{O}_X}(K, \\omega_X^\\bullet) =\nR\\SheafHom_{\\mathcal{O}_Y}(Rf_*K, \\omega_Y^\\bullet)\n$$\nfor $K \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and\n\\end{enumerate}\nIf $X$ is separated over $S$, then\n$\\omega_X^\\bullet$ is canonically isomorphic to\n$(X \\to S)^!\\omega_S^\\bullet$ and\nif $f$ is a morphism between schemes separated\nover $S$, then there is a canonical isomorphism\\footnote{We haven't\nchecked that these are compatible with the isomorphisms\n$(g \\circ f)^! \\to f^! \\circ g^!$ and\n$(g \\circ f)^!_{new} \\to f^!_{new} \\circ g^!_{new}$. We will do this\nhere if we need this later.}\n$f_{new}^!K = f^!K$ for $K$ in $D_{\\textit{Coh}}^+$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Glueing dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AUE","source_file":"duality.tex","source_line":5370,"source_end_line":5417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5370-L5417","statement_sha256":"b07745204499ac4f57194fb70447e94351088d767d3e75bcd8e32fea783733d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8861,"rank":8861,"depth":64,"x":2144.986,"y":1231.589,"cluster":"duality-cohomology"},{"id":"stacks:0AWL","tag":"0AWL","title":"Dimension functions · Lemma 0AWL","summary":"Let S be a Noetherian scheme and let ω_S^bullet be a dualizing complex. Let X be a scheme of finite type over S and let ω_X^bullet be the dualizing complex normalized relative to ω_S^bullet. If x ∈ X is a closed point lying over a closed point s of S, then ω_X, x^bullet is a normalized dualizing complex over O_X, x provided that ω_S, s^bullet is a normalized dualizing complex over O_S, s.","statement_latex":"Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a\ndualizing complex. Let $X$ be a scheme of finite type over $S$ and let\n$\\omega_X^\\bullet$ be the dualizing complex normalized relative\nto $\\omega_S^\\bullet$. If $x \\in X$ is a closed point lying over\na closed point $s$ of $S$, then $\\omega_{X, x}^\\bullet$\nis a normalized dualizing complex over $\\mathcal{O}_{X, x}$\nprovided that $\\omega_{S, s}^\\bullet$ is a normalized dualizing\ncomplex over $\\mathcal{O}_{S, s}$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWL","source_file":"duality.tex","source_line":5587,"source_end_line":5597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5587-L5597","statement_sha256":"a0761895b35197bb6fe5de0c742b6d3c78587938d68f909708a7e37b87abf922","origin":"The Stacks Project","memory_eligible":false,"source_rank":8862,"rank":8862,"depth":42,"x":1871.385,"y":1137.707,"cluster":"duality-cohomology"},{"id":"stacks:0AWM","tag":"0AWM","title":"Dimension functions · Lemma 0AWM","summary":"Let S be a Noetherian scheme and let ω_S^bullet be a dualizing complex. Let f : X → S be of finite type and let ω_X^bullet be the dualizing complex normalized relative to ω_S^bullet. For all x ∈ X we have δ_X(x) - δ_S(f(x)) = trdeg_kappa(f(x))(kappa(x)) where δ_S, resp. δ_X is the dimension function of ω_S^bullet, resp. ω_X^bullet, see Lemma [Tag 0AWF].","statement_latex":"Let $S$ be a Noetherian scheme and let $\\omega_S^\\bullet$ be a\ndualizing complex. Let $f : X \\to S$ be of finite type\nand let $\\omega_X^\\bullet$ be the dualizing complex\nnormalized relative to $\\omega_S^\\bullet$. For all $x \\in X$ we have\n$$\n\\delta_X(x) - \\delta_S(f(x)) = \\text{trdeg}_{\\kappa(f(x))}(\\kappa(x))\n$$\nwhere $\\delta_S$, resp.\\ $\\delta_X$\nis the dimension function of\n$\\omega_S^\\bullet$, resp.\\ $\\omega_X^\\bullet$, see\nLemma \\ref{lemma-dimension-function-scheme}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWM","source_file":"duality.tex","source_line":5638,"source_end_line":5651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5638-L5651","statement_sha256":"1f8147a8d027d2fdb54594df0968e3042be48aad7122803341ebce25fc638e7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8863,"rank":8863,"depth":43,"x":2148.934,"y":1051.593,"cluster":"duality-cohomology"},{"id":"stacks:0BV5","tag":"0BV5","title":"Dimension functions · Lemma 0BV5","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Let x ∈ X with image y ∈ Y. Then H^i(f^!O_Y)_x not = 0 ⇒ - dim_x(X_y) ≤ i.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Let $x \\in X$ with image $y \\in Y$. Then\n$$\nH^i(f^!\\mathcal{O}_Y)_x \\not = 0\n\\Rightarrow - \\dim_x(X_y) \\leq i.\n$$","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BV5","source_file":"duality.tex","source_line":5674,"source_end_line":5682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5674-L5682","statement_sha256":"86b455a94303c2277ee6d49c4ae6552822a4ad6b5a12fed7a22ac8e057b4af0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8864,"rank":8864,"depth":25,"x":2013.377,"y":1272.821,"cluster":"duality-cohomology"},{"id":"stacks:0BV6","tag":"0BV6","title":"Dimension functions · Lemma 0BV6","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Let x ∈ X with image y ∈ Y. If f is flat, then H^i(f^!O_Y)_x not = 0 ⇒ - dim_x(X_y) ≤ i ≤ 0. In fact, if all fibres of f have dimension ≤ d, then f^!O_Y has tor-amplitude in [-d, 0] as an object of D(X, f^-1O_Y).","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Let $x \\in X$ with image $y \\in Y$.\nIf $f$ is flat, then\n$$\nH^i(f^!\\mathcal{O}_Y)_x \\not = 0\n\\Rightarrow - \\dim_x(X_y) \\leq i \\leq 0.\n$$\nIn fact, if all fibres of $f$ have dimension $\\leq d$, then\n$f^!\\mathcal{O}_Y$ has tor-amplitude in $[-d, 0]$ as an object\nof $D(X, f^{-1}\\mathcal{O}_Y)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BV6","source_file":"duality.tex","source_line":5697,"source_end_line":5709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5697-L5709","statement_sha256":"cd55752651f3608c24d24b5672c246c534916935b8c99df394c9aca246779aef","origin":"The Stacks Project","memory_eligible":false,"source_rank":8865,"rank":8865,"depth":39,"x":1935.345,"y":1032.511,"cluster":"duality-cohomology"},{"id":"stacks:0E9V","tag":"0E9V","title":"Dimension functions · Lemma 0E9V","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Let x ∈ X with image y ∈ Y. Assume • O_Y, y is Cohen-Macaulay, and • trdeg_kappa(f(xi))(kappa(xi)) ≤ r for any generic point xi of an irreducible component of X containing x. Then H^i(f^!O_Y)_x not = 0 ⇒ - r ≤ i and the stalk H^-r(f^!O_Y)_x is (S_2) as an O_X, x-module.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Let $x \\in X$ with image $y \\in Y$. Assume\n\\begin{enumerate}\n\\item $\\mathcal{O}_{Y, y}$ is Cohen-Macaulay, and\n\\item $\\text{trdeg}_{\\kappa(f(\\xi))}(\\kappa(\\xi)) \\leq r$\nfor any generic point $\\xi$ of an irreducible component\nof $X$ containing $x$.\n\\end{enumerate}\nThen\n$$\nH^i(f^!\\mathcal{O}_Y)_x \\not = 0\n\\Rightarrow - r \\leq i\n$$\nand the stalk $H^{-r}(f^!\\mathcal{O}_Y)_x$ is $(S_2)$ as an\n$\\mathcal{O}_{X, x}$-module.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9V","source_file":"duality.tex","source_line":5717,"source_end_line":5734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5717-L5734","statement_sha256":"c0591c14f4c71f3f5a33170b5dd087b14371b1663e30e137497380177d9c5ec9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8866,"rank":8866,"depth":41,"x":2186.406,"y":1165.579,"cluster":"duality-cohomology"},{"id":"stacks:0BV7","tag":"0BV7","title":"Dimension functions · Lemma 0BV7","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. If f is flat and quasi-finite, then f^!O_Y = ω_X/Y[0] for some coherent O_X-module ω_X/Y flat over Y.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. If $f$ is flat and quasi-finite, then\n$$\nf^!\\mathcal{O}_Y = \\omega_{X/Y}[0]\n$$\nfor some coherent $\\mathcal{O}_X$-module $\\omega_{X/Y}$ flat over $Y$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BV7","source_file":"duality.tex","source_line":5744,"source_end_line":5752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5744-L5752","statement_sha256":"561c4dea8a3f386dc548cf4d8a687d3239e0fa968603c00ae36cfe298a294937","origin":"The Stacks Project","memory_eligible":false,"source_rank":8867,"rank":8867,"depth":43,"x":1893.952,"y":1209.963,"cluster":"duality-cohomology"},{"id":"stacks:0BV8","tag":"0BV8","title":"Dimension functions · Lemma 0BV8","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. If f is Cohen-Macaulay (More on Morphisms, Definition [Tag 045R]), then f^!O_Y = ω_X/Y[d] for some coherent O_X-module ω_X/Y flat over Y where d is the locally constant function on X which gives the relative dimension of X over Y.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. If $f$ is Cohen-Macaulay (More on Morphisms, Definition\n\\ref{more-morphisms-definition-CM}), then\n$$\nf^!\\mathcal{O}_Y = \\omega_{X/Y}[d]\n$$\nfor some coherent $\\mathcal{O}_X$-module $\\omega_{X/Y}$ flat over $Y$\nwhere $d$ is the locally constant\nfunction on $X$ which gives the relative dimension of $X$ over $Y$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dimension functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BV8","source_file":"duality.tex","source_line":5760,"source_end_line":5771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5760-L5771","statement_sha256":"92413ddf3a3da24f7f6021dffeed2875ac111d1e6299b92dcce2a6f665fb305d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8868,"rank":8868,"depth":44,"x":2074.105,"y":1011.075,"cluster":"duality-cohomology"},{"id":"stacks:0AWK","tag":"0AWK","title":"Dualizing modules · Lemma 0AWK","summary":"Let X be a connected Noetherian scheme and let ω_X be a dualizing module on X. The support of ω_X is the union of the irreducible components of maximal dimension with respect to any dimension function and ω_X is a coherent O_X-module having property (S_2).","statement_latex":"Let $X$ be a connected Noetherian scheme and let $\\omega_X$ be a dualizing\nmodule on $X$. The support of $\\omega_X$ is the union of the irreducible\ncomponents of maximal dimension with respect to any dimension function\nand $\\omega_X$ is a coherent $\\mathcal{O}_X$-module having property $(S_2)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWK","source_file":"duality.tex","source_line":5883,"source_end_line":5889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5883-L5889","statement_sha256":"9294663693cebd78ee29e059255cd2cc0bfefee526ba8eb95a570b5abb45ce22","origin":"The Stacks Project","memory_eligible":false,"source_rank":8869,"rank":8869,"depth":40,"x":2101.232,"y":1260.223,"cluster":"duality-cohomology"},{"id":"stacks:0AWN","tag":"0AWN","title":"Dualizing modules · Lemma 0AWN","summary":"Let X/A with ω_X^bullet and ω_X be as in Example [Tag 0AWI]. Then • H^i(ω_X^bullet) not = 0 ⇒ i ∈ (-dim(X), …, 0), • the dimension of the support of H^i(ω_X^bullet) is at most -i, • Supp(ω_X) is the union of the components of dimension dim(X), and • ω_X has property (S_2).","statement_latex":"Let $X/A$ with $\\omega_X^\\bullet$ and $\\omega_X$ be as in\nExample \\ref{example-proper-over-local}. Then\n\\begin{enumerate}\n\\item $H^i(\\omega_X^\\bullet) \\not = 0 \\Rightarrow\ni \\in \\{-\\dim(X), \\ldots, 0\\}$,\n\\item the dimension of the support of $H^i(\\omega_X^\\bullet)$ is at most $-i$,\n\\item $\\text{Supp}(\\omega_X)$ is the union of\nthe components of dimension $\\dim(X)$, and\n\\item $\\omega_X$ has property $(S_2)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWN","source_file":"duality.tex","source_line":5907,"source_end_line":5919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5907-L5919","statement_sha256":"1f1c0a84b6c0acc036e90b12c1dcc1186f61082f41b36bf42954f58ff5ecc779","origin":"The Stacks Project","memory_eligible":false,"source_rank":8870,"rank":8870,"depth":44,"x":1880.634,"y":1091.717,"cluster":"duality-cohomology"},{"id":"stacks:0AWP","tag":"0AWP","title":"Dualizing modules · Lemma 0AWP","summary":"Let X/A with dualizing module ω_X be as in Example [Tag 0AWI]. Let d = dim(X_s) be the dimension of the closed fibre. If dim(X) = d + dim(A), then the dualizing module ω_X represents the functor F ↦ Hom_A(H^d(X, F), ω_A) on the category of coherent O_X-modules.","statement_latex":"Let $X/A$ with dualizing module $\\omega_X$ be as in\nExample \\ref{example-proper-over-local}.\nLet $d = \\dim(X_s)$ be the dimension\nof the closed fibre. If $\\dim(X) = d + \\dim(A)$, then\nthe dualizing module $\\omega_X$ represents the functor\n$$\n\\mathcal{F} \\longmapsto \\Hom_A(H^d(X, \\mathcal{F}), \\omega_A)\n$$\non the category of coherent $\\mathcal{O}_X$-modules.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Dualizing modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWP","source_file":"duality.tex","source_line":5939,"source_end_line":5950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5939-L5950","statement_sha256":"645b6ffe51adc4c9d1074eb287f09ebbca9846215d78fddda71c042d924f012b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8871,"rank":8871,"depth":45,"x":2179.128,"y":1090.797,"cluster":"duality-cohomology"},{"id":"stacks:0AWT","tag":"0AWT","title":"Cohen-Macaulay schemes · Lemma 0AWT","summary":"Let X be a locally Noetherian scheme with dualizing complex ω_X^bullet. • X is Cohen-Macaulay ⇔ ω_X^bullet locally has a unique nonzero cohomology sheaf, • O_X, x is Cohen-Macaulay ⇔ ω_X, x^bullet has a unique nonzero cohomology, • U = (x ∈ X mid O_X, x is Cohen-Macaulay) is open and Cohen-Macaulay. If X is connected and Cohen-Macaulay, then there is an integer n and a coherent Cohen-Macaulay O_X-module ω_X such that ω_X^bullet = ω_X[-n].","statement_latex":"Let $X$ be a locally Noetherian scheme with dualizing complex\n$\\omega_X^\\bullet$.\n\\begin{enumerate}\n\\item $X$ is Cohen-Macaulay $\\Leftrightarrow$ $\\omega_X^\\bullet$\nlocally has a unique nonzero cohomology sheaf,\n\\item $\\mathcal{O}_{X, x}$ is Cohen-Macaulay $\\Leftrightarrow$\n$\\omega_{X, x}^\\bullet$ has a unique nonzero cohomology,\n\\item $U = \\{x \\in X \\mid \\mathcal{O}_{X, x}\\text{ is Cohen-Macaulay}\\}$\nis open and Cohen-Macaulay.\n\\end{enumerate}\nIf $X$ is connected and Cohen-Macaulay, then there is an integer $n$\nand a coherent Cohen-Macaulay $\\mathcal{O}_X$-module $\\omega_X$\nsuch that $\\omega_X^\\bullet = \\omega_X[-n]$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Cohen-Macaulay schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWT","source_file":"duality.tex","source_line":5996,"source_end_line":6011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L5996-L6011","statement_sha256":"87b1bfa589f30f12a7602b6caeeff6e1f87937672825e910a3c7fa65520d7176","origin":"The Stacks Project","memory_eligible":false,"source_rank":8872,"rank":8872,"depth":35,"x":1959.528,"y":1261.029,"cluster":"duality-cohomology"},{"id":"stacks:0AWU","tag":"0AWU","title":"Cohen-Macaulay schemes · Lemma 0AWU","summary":"Let X be a locally Noetherian scheme. If there exists a coherent sheaf ω_X such that ω_X[0] is a dualizing complex on X, then X is a Cohen-Macaulay scheme.","statement_latex":"Let $X$ be a locally Noetherian scheme. If there exists a coherent sheaf\n$\\omega_X$ such that $\\omega_X[0]$ is a dualizing complex on $X$, then\n$X$ is a Cohen-Macaulay scheme.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Cohen-Macaulay schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWU","source_file":"duality.tex","source_line":6053,"source_end_line":6058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6053-L6058","statement_sha256":"220aa476ab0ba0383ba57cdb31de122c66befbea78fb4e05c75cdc894f5c3bc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8873,"rank":8873,"depth":36,"x":1984.588,"y":1010.629,"cluster":"duality-cohomology"},{"id":"stacks:0C0Z","tag":"0C0Z","title":"Cohen-Macaulay schemes · Lemma 0C0Z","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Let x ∈ X. If f is flat, then the following are equivalent • f is Cohen-Macaulay at x, • f^!O_Y has a unique nonzero cohomology sheaf in a neighbourhood of x.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism\nof $\\textit{FTS}_S$. Let $x \\in X$. If $f$ is flat, then\nthe following are equivalent\n\\begin{enumerate}\n\\item $f$ is Cohen-Macaulay at $x$,\n\\item $f^!\\mathcal{O}_Y$ has a unique nonzero cohomology sheaf\nin a neighbourhood of $x$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Cohen-Macaulay schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0Z","source_file":"duality.tex","source_line":6066,"source_end_line":6076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6066-L6076","statement_sha256":"982912000a6c21ab08d91b4f9c08faba6bb8d42cef2ddac0d63acd280b43bc3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8874,"rank":8874,"depth":66,"x":2167.668,"y":1209.702,"cluster":"duality-cohomology"},{"id":"stacks:0AWW","tag":"0AWW","title":"Gorenstein schemes · Definition 0AWW","summary":"Let X be a scheme. We say X is Gorenstein if X is locally Noetherian and O_X, x is Gorenstein for all x ∈ X.","statement_latex":"Let $X$ be a scheme. We say $X$ is {\\it Gorenstein} if $X$ is\nlocally Noetherian and $\\mathcal{O}_{X, x}$ is Gorenstein for all $x \\in X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWW","source_file":"duality.tex","source_line":6149,"source_end_line":6153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6149-L6153","statement_sha256":"03606c31a9f64f0dece9aeb08e3da7230c3eb182e303a62c0bc5ee0cd2bea890","origin":"The Stacks Project","memory_eligible":false,"source_rank":8875,"rank":8875,"depth":0,"x":1872.266,"y":1166.747,"cluster":"duality-cohomology"},{"id":"stacks:0C00","tag":"0C00","title":"Gorenstein schemes · Lemma 0C00","summary":"A Gorenstein scheme is Cohen-Macaulay.","statement_latex":"A Gorenstein scheme is Cohen-Macaulay.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C00","source_file":"duality.tex","source_line":6160,"source_end_line":6163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6160-L6163","statement_sha256":"e2d6654a31b4223c661e44eeacd7afcbdbe979099877eb1aad81e90adbf332fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":8876,"rank":8876,"depth":36,"x":2124.902,"y":1030.659,"cluster":"duality-cohomology"},{"id":"stacks:0DWG","tag":"0DWG","title":"Gorenstein schemes · Lemma 0DWG","summary":"A regular scheme is Gorenstein.","statement_latex":"A regular scheme is Gorenstein.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWG","source_file":"duality.tex","source_line":6171,"source_end_line":6174,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6171-L6174","statement_sha256":"38eaa7c7a1a108156417385759b957b1585d2b05f055e5fab1395ac2633b2b16","origin":"The Stacks Project","memory_eligible":false,"source_rank":8877,"rank":8877,"depth":19,"x":2047.969,"y":1274.618,"cluster":"duality-cohomology"},{"id":"stacks:0BFQ","tag":"0BFQ","title":"Gorenstein schemes · Lemma 0BFQ","summary":"Let X be a locally Noetherian scheme. • If X has a dualizing complex ω_X^bullet, then • X is Gorenstein ⇔ ω_X^bullet is an invertible object of D(O_X), • O_X, x is Gorenstein ⇔ ω_X, x^bullet is an invertible object of D(O_X, x), • U = (x ∈ X mid O_X, x is Gorenstein) is an open Gorenstein subscheme. • If X is Gorenstein, then X has a dualizing complex if and only if O_X[0] is a dualizing complex.","statement_latex":"Let $X$ be a locally Noetherian scheme.\n\\begin{enumerate}\n\\item If $X$ has a dualizing complex $\\omega_X^\\bullet$, then\n\\begin{enumerate}\n\\item $X$ is Gorenstein $\\Leftrightarrow$ $\\omega_X^\\bullet$ is an invertible\nobject of $D(\\mathcal{O}_X)$,\n\\item $\\mathcal{O}_{X, x}$ is Gorenstein $\\Leftrightarrow$\n$\\omega_{X, x}^\\bullet$ is an invertible object of $D(\\mathcal{O}_{X, x})$,\n\\item $U = \\{x \\in X \\mid \\mathcal{O}_{X, x}\\text{ is Gorenstein}\\}$\nis an open Gorenstein subscheme.\n\\end{enumerate}\n\\item If $X$ is Gorenstein, then $X$ has a dualizing complex if and\nonly if $\\mathcal{O}_X[0]$ is a dualizing complex.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFQ","source_file":"duality.tex","source_line":6182,"source_end_line":6198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6182-L6198","statement_sha256":"26744668f38bc5e497cd921e3f548b7a00e11818dbcbf9eedbab8c5f53626e46","origin":"The Stacks Project","memory_eligible":false,"source_rank":8878,"rank":8878,"depth":35,"x":1908.366,"y":1050.837,"cluster":"duality-cohomology"},{"id":"stacks:0BVA","tag":"0BVA","title":"Gorenstein schemes · Lemma 0BVA","summary":"If f : Y → X is a local complete intersection morphism with X a Gorenstein scheme, then Y is Gorenstein.","statement_latex":"If $f : Y \\to X$ is a local complete intersection morphism\nwith $X$ a Gorenstein scheme, then $Y$ is Gorenstein.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVA","source_file":"duality.tex","source_line":6206,"source_end_line":6210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6206-L6210","statement_sha256":"95b09f1291f626441abc174573cd768cee7de010cba814df6e82f60e380862f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8879,"rank":8879,"depth":39,"x":2191.564,"y":1136.726,"cluster":"duality-cohomology"},{"id":"stacks:0C01","tag":"0C01","title":"Gorenstein schemes · Lemma 0C01","summary":"The property P(S) =\"S is Gorenstein\" is local in the syntomic topology.","statement_latex":"The property $\\mathcal{P}(S) =$``$S$ is Gorenstein''\nis local in the syntomic topology.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C01","source_file":"duality.tex","source_line":6218,"source_end_line":6222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6218-L6222","statement_sha256":"31a1225c0ff717e0db697966ac9226cabdb24f47c87c26ceeccc403887a5fede","origin":"The Stacks Project","memory_eligible":false,"source_rank":8880,"rank":8880,"depth":40,"x":1913.376,"y":1234.188,"cluster":"duality-cohomology"},{"id":"stacks:0C03","tag":"0C03","title":"Gorenstein morphisms · Lemma 0C03","summary":"Let X be a locally Noetherian scheme over the field k. Let k'/k be a finitely generated field extension. Let x ∈ X be a point, and let x' ∈ X_k' be a point lying over x. Then we have O_X, x is Gorenstein ⇔ O_X_k', x' is Gorenstein If X is locally of finite type over k, the same holds for any field extension k'/k.","statement_latex":"Let $X$ be a locally Noetherian scheme over the field $k$.\nLet $k'/k$ be a finitely generated field extension.\nLet $x \\in X$ be a point, and let $x' \\in X_{k'}$ be a point lying\nover $x$. Then we have\n$$\n\\mathcal{O}_{X, x}\\text{ is Gorenstein}\n\\Leftrightarrow\n\\mathcal{O}_{X_{k'}, x'}\\text{ is Gorenstein}\n$$\nIf $X$ is locally of finite type over $k$, the same holds for any\nfield extension $k'/k$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C03","source_file":"duality.tex","source_line":6262,"source_end_line":6275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6262-L6275","statement_sha256":"05a3fc4a8cc145fd70007bf78956fed7d862a2e40cf8f19aedbeda7031961300","origin":"The Stacks Project","memory_eligible":false,"source_rank":8881,"rank":8881,"depth":42,"x":2040.264,"y":1004.23,"cluster":"duality-cohomology"},{"id":"stacks:0C04","tag":"0C04","title":"Gorenstein morphisms · Definition 0C04","summary":"Let f : X → Y be a morphism of schemes. Assume that all the fibres X_y are locally Noetherian schemes. • Let x ∈ X, and y = f(x). We say that f is Gorenstein at x if f is flat at x, and the local ring of the scheme X_y at x is Gorenstein. • We say f is a Gorenstein morphism if f is Gorenstein at every point of X.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume that all the fibres $X_y$ are locally Noetherian schemes.\n\\begin{enumerate}\n\\item Let $x \\in X$, and $y = f(x)$. We say that $f$ is\n{\\it Gorenstein at $x$} if $f$ is flat at $x$, and the\nlocal ring of the scheme $X_y$ at $x$ is Gorenstein.\n\\item We say $f$ is a {\\it Gorenstein morphism} if $f$ is\nGorenstein at every point of $X$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C04","source_file":"duality.tex","source_line":6301,"source_end_line":6312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6301-L6312","statement_sha256":"5c21e61d937a5ba7fd375308af447e7bbca1164a2ef54d4f99ee041824cb70c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8882,"rank":8882,"depth":0,"x":2131.72,"y":1246.05,"cluster":"duality-cohomology"},{"id":"stacks:0C05","tag":"0C05","title":"Gorenstein morphisms · Lemma 0C05","summary":"Let f : X → Y be a morphism of schemes. Assume all fibres of f are locally Noetherian. The following are equivalent • f is Gorenstein, and • f is flat and its fibres are Gorenstein schemes.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume all fibres of $f$ are locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is Gorenstein, and\n\\item $f$ is flat and its fibres are Gorenstein schemes.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C05","source_file":"duality.tex","source_line":6317,"source_end_line":6326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6317-L6326","statement_sha256":"b5c2487e0ef12345154130baa8fb155c2114eeb8d746006e1635d1fd6c26e661","origin":"The Stacks Project","memory_eligible":false,"source_rank":8883,"rank":8883,"depth":0,"x":1869.544,"y":1119.498,"cluster":"duality-cohomology"},{"id":"stacks:0C06","tag":"0C06","title":"Gorenstein morphisms · Lemma 0C06","summary":"A Gorenstein morphism is Cohen-Macaulay.","statement_latex":"A Gorenstein morphism is Cohen-Macaulay.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C06","source_file":"duality.tex","source_line":6332,"source_end_line":6335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6332-L6335","statement_sha256":"2ec39aea3d7b574daabed16b19e0fb28ed66c810da86e6c777c8148a56ece6c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8884,"rank":8884,"depth":37,"x":2164.946,"y":1063.992,"cluster":"duality-cohomology"},{"id":"stacks:0C15","tag":"0C15","title":"Gorenstein morphisms · Lemma 0C15","summary":"A syntomic morphism is Gorenstein. Equivalently a flat local complete intersection morphism is Gorenstein.","statement_latex":"A syntomic morphism is Gorenstein. Equivalently a flat\nlocal complete intersection morphism is Gorenstein.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C15","source_file":"duality.tex","source_line":6341,"source_end_line":6345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6341-L6345","statement_sha256":"93fc6321ac8bbc15a66eb9d498feafc102f208af829b318eea5db7b449054f5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8885,"rank":8885,"depth":40,"x":1991.577,"y":1272.756,"cluster":"duality-cohomology"},{"id":"stacks:0C11","tag":"0C11","title":"Gorenstein morphisms · Lemma 0C11","summary":"Let f : X → Y and g : Y → Z be morphisms. Assume that the fibres X_y, Y_z and X_z of f, g, and g ∘ f are locally Noetherian. • If f is Gorenstein at x and g is Gorenstein at f(x), then g ∘ f is Gorenstein at x. • If f and g are Gorenstein, then g ∘ f is Gorenstein. • If g ∘ f is Gorenstein at x and f is flat at x, then f is Gorenstein at x and g is Gorenstein at f(x). • If g ∘ f is Gorenstein and f is flat, then f is Gorenstein and g is Gorenstein at every point in the…","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms. Assume that the\nfibres $X_y$, $Y_z$ and $X_z$ of $f$, $g$, and $g \\circ f$ are\nlocally Noetherian.\n\\begin{enumerate}\n\\item If $f$ is Gorenstein at $x$ and $g$ is Gorenstein\nat $f(x)$, then $g \\circ f$ is Gorenstein at $x$.\n\\item If $f$ and $g$ are Gorenstein, then $g \\circ f$ is Gorenstein.\n\\item If $g \\circ f$ is Gorenstein at $x$ and $f$ is flat at $x$,\nthen $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$.\n\\item If $g \\circ f$ is Gorenstein and $f$ is flat, then\n$f$ is Gorenstein and $g$ is Gorenstein at every point in\nthe image of $f$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C11","source_file":"duality.tex","source_line":6358,"source_end_line":6373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6358-L6373","statement_sha256":"8e79a7bab2697ea2c939f6d126b4aee13cf6933398a9ae2bfe13a34618df499b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8886,"rank":8886,"depth":37,"x":1951.493,"y":1020.181,"cluster":"duality-cohomology"},{"id":"stacks:0C12","tag":"0C12","title":"Gorenstein morphisms · Lemma 0C12","summary":"Gorensteinnes of the total space of a flat fibration implies same for base and fibres Let f : X → Y be a flat morphism of locally Noetherian schemes. If X is Gorenstein, then f is Gorenstein and O_Y, f(x) is Gorenstein for all x ∈ X.","statement_latex":"\\begin{slogan}\nGorensteinnes of the total space of a flat fibration implies\nsame for base and fibres\n\\end{slogan}\nLet $f : X \\to Y$ be a flat morphism of locally Noetherian schemes.\nIf $X$ is Gorenstein, then $f$ is Gorenstein and $\\mathcal{O}_{Y, f(x)}$\nis Gorenstein for all $x \\in X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C12","source_file":"duality.tex","source_line":6380,"source_end_line":6389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6380-L6389","statement_sha256":"43bb4b9665bd32917ebfb8954bbf3e2a68f9d2967b155d2dec112968c739afca","origin":"The Stacks Project","memory_eligible":false,"source_rank":8887,"rank":8887,"depth":37,"x":2184.404,"y":1183.851,"cluster":"duality-cohomology"},{"id":"stacks:0C07","tag":"0C07","title":"Gorenstein morphisms · Lemma 0C07","summary":"Let f : X → Y be a morphism of schemes. Assume that all the fibres X_y are locally Noetherian schemes. Let Y' → Y be locally of finite type. Let f' : X' → Y' be the base change of f. Let x' ∈ X' be a point with image x ∈ X. • If f is Gorenstein at x, then f' : X' → Y' is Gorenstein at x'. • If f is flat at x and f' is Gorenstein at x', then f is Gorenstein at x. • If Y' → Y is flat at f'(x') and f' is Gorenstein at x', then f is Gorenstein at x.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume that all the fibres $X_y$ are locally Noetherian schemes.\nLet $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$\nbe the base change of $f$.\nLet $x' \\in X'$ be a point with image $x \\in X$.\n\\begin{enumerate}\n\\item If $f$ is Gorenstein at $x$, then\n$f' : X' \\to Y'$ is Gorenstein at $x'$.\n\\item If $f$ is flat at $x$ and $f'$ is Gorenstein at $x'$, then $f$\nis Gorenstein at $x$.\n\\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Gorenstein at\n$x'$, then $f$ is Gorenstein at $x$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C07","source_file":"duality.tex","source_line":6396,"source_end_line":6411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6396-L6411","statement_sha256":"85f373aa0b41421f3d758de443ea0f2c59c442cbc2a07d0556a1c2402ed32e52","origin":"The Stacks Project","memory_eligible":false,"source_rank":8888,"rank":8888,"depth":43,"x":1880.727,"y":1195.335,"cluster":"duality-cohomology"},{"id":"stacks:0E0Q","tag":"0E0Q","title":"Gorenstein morphisms · Lemma 0E0Q","summary":"Let f : X → Y be a morphism of schemes which is flat and locally of finite type. Then formation of the set (x ∈ X mid f is Gorenstein at x) commutes with arbitrary base change.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is flat and\nlocally of finite type. Then formation of the set\n$\\{x \\in X \\mid f\\text{ is Gorenstein at }x\\}$\ncommutes with arbitrary base change.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0Q","source_file":"duality.tex","source_line":6465,"source_end_line":6471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6465-L6471","statement_sha256":"153cf9c14f17068b8bbc45f614c49b4f77ef9c901478a88b1ee22aadb0ee4521","origin":"The Stacks Project","memory_eligible":false,"source_rank":8889,"rank":8889,"depth":40,"x":2095.64,"y":1014.367,"cluster":"duality-cohomology"},{"id":"stacks:0C08","tag":"0C08","title":"Gorenstein morphisms · Lemma 0C08","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Let x ∈ X. If f is flat, then the following are equivalent • f is Gorenstein at x, • f^!O_Y is isomorphic to an invertible object in a neighbourhood of x. In particular, the set of points where f is Gorenstein is open in X.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism\nof $\\textit{FTS}_S$. Let $x \\in X$. If $f$ is flat, then\nthe following are equivalent\n\\begin{enumerate}\n\\item $f$ is Gorenstein at $x$,\n\\item $f^!\\mathcal{O}_Y$ is isomorphic to an invertible object\nin a neighbourhood of $x$.\n\\end{enumerate}\nIn particular, the set of points where $f$ is Gorenstein is\nopen in $X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C08","source_file":"duality.tex","source_line":6504,"source_end_line":6516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6504-L6516","statement_sha256":"fa54c6a00167db1fc2f260e17cfaf19804269c6256af27640603bbf23d1b7678","origin":"The Stacks Project","memory_eligible":false,"source_rank":8890,"rank":8890,"depth":66,"x":2082.683,"y":1270.019,"cluster":"duality-cohomology"},{"id":"stacks:0C09","tag":"0C09","title":"Gorenstein morphisms · Lemma 0C09","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. Let x ∈ X with image s ∈ S. Set d = dim_x(X_s). The following are equivalent • f is Gorenstein at x, • there exists an open neighbourhood U ⊂ X of x and a locally quasi-finite morphism U → A^d_S over S which is Gorenstein at x, • there exists an open neighbourhood U ⊂ X of x and a locally quasi-finite Gorenstein morphism U → A^d_S over S, • for any S-morphism g : U → A^d_S of an open…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and locally\nof finite presentation. Let $x \\in X$ with image $s \\in S$.\nSet $d = \\dim_x(X_s)$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is Gorenstein at $x$,\n\\item there exists an open neighbourhood $U \\subset X$ of $x$\nand a locally quasi-finite morphism $U \\to \\mathbf{A}^d_S$ over $S$\nwhich is Gorenstein at $x$,\n\\item there exists an open neighbourhood $U \\subset X$ of $x$\nand a locally quasi-finite Gorenstein morphism $U \\to \\mathbf{A}^d_S$ over $S$,\n\\item for any $S$-morphism $g : U \\to \\mathbf{A}^d_S$\nof an open neighbourhood $U \\subset X$ of $x$ we have:\n$g$ is quasi-finite at $x$ $\\Rightarrow$ $g$ is Gorenstein at $x$.\n\\end{enumerate}\nIn particular, the set of points where $f$ is Gorenstein is open in $X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C09","source_file":"duality.tex","source_line":6550,"source_end_line":6567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6550-L6567","statement_sha256":"c69f1a3cd48f2830a214bd3ff078e026552f29166e8e7854a0ca77a416a2e508","origin":"The Stacks Project","memory_eligible":false,"source_rank":8891,"rank":8891,"depth":67,"x":1886.447,"y":1073.951,"cluster":"duality-cohomology"},{"id":"stacks:0C0A","tag":"0C0A","title":"Gorenstein morphisms · Lemma 0C0A","summary":"The property P(f)=\"the fibres of f are locally Noetherian and f is Gorenstein\" is local in the fppf topology on the target and local in the syntomic topology on the source.","statement_latex":"The property\n$\\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is\nGorenstein'' is local in the fppf topology on the target and\nlocal in the syntomic topology on the source.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0A","source_file":"duality.tex","source_line":6626,"source_end_line":6632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6626-L6632","statement_sha256":"7ba9a1dedd1aa7c0ca7396b730b0cedf88a16d25b2de22cbb30b76a98692578c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8892,"rank":8892,"depth":43,"x":2189.13,"y":1107.215,"cluster":"duality-cohomology"},{"id":"stacks:0E4N","tag":"0E4N","title":"More on dualizing complexes · Lemma 0E4N","summary":"Let f : X → Y be a morphism of locally Noetherian schemes. Assume • f is syntomic and surjective, or • f is a surjective flat local complete intersection morphism, or • f is a surjective Gorenstein morphism of finite type. Then K ∈ D_QCoh(O_Y) is a dualizing complex on Y if and only if Lf^*K is a dualizing complex on X.","statement_latex":"Let $f : X \\to Y$ be a morphism of locally Noetherian schemes. Assume\n\\begin{enumerate}\n\\item $f$ is syntomic and surjective, or\n\\item $f$ is a surjective flat local complete intersection morphism, or\n\\item $f$ is a surjective Gorenstein morphism of finite type.\n\\end{enumerate}\nThen $K \\in D_\\QCoh(\\mathcal{O}_Y)$ is a dualizing complex on $Y$ if and only\nif $Lf^*K$ is a dualizing complex on $X$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"More on dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4N","source_file":"duality.tex","source_line":6683,"source_end_line":6693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6683-L6693","statement_sha256":"870b898e2c4135a7159046b0c7ef4333c4be7fa4bca6722033494aea07dcc3df","origin":"The Stacks Project","memory_eligible":false,"source_rank":8893,"rank":8893,"depth":41,"x":1938.932,"y":1254.587,"cluster":"duality-cohomology"},{"id":"stacks:0FVV","tag":"0FVV","title":"Duality for proper schemes over fields · Lemma 0FVV","summary":"Let X be a proper scheme over a field k. There exists a dualizing complex ω_X^bullet with the following properties • H^i(ω_X^bullet) is nonzero only for i ∈ [-dim(X), 0], • ω_X = H^-dim(X)(ω_X^bullet) is a coherent (S_2)-module whose support is the irreducible components of dimension dim(X), • the dimension of the support of H^i(ω_X^bullet) is at most -i, • for x ∈ X closed the module H^i(ω_X, x^bullet) ⊕ … ⊕ H^0(ω_X, x^bullet) is nonzero if and only if depth(O_X, x) ≤…","statement_latex":"Let $X$ be a proper scheme over a field $k$. There exists a dualizing complex\n$\\omega_X^\\bullet$ with the following properties\n\\begin{enumerate}\n\\item $H^i(\\omega_X^\\bullet)$ is nonzero only for $i \\in [-\\dim(X), 0]$,\n\\item $\\omega_X = H^{-\\dim(X)}(\\omega_X^\\bullet)$ is a coherent\n$(S_2)$-module whose support is the irreducible components\nof dimension $\\dim(X)$,\n\\item the dimension of the support of $H^i(\\omega_X^\\bullet)$ is at most $-i$,\n\\item for $x \\in X$ closed the module\n$H^i(\\omega_{X, x}^\\bullet) \\oplus \\ldots \\oplus H^0(\\omega_{X, x}^\\bullet)$\nis nonzero if and only if $\\text{depth}(\\mathcal{O}_{X, x}) \\leq -i$,\n\\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ there are functorial\nisomorphisms\\footnote{This property\ncharacterizes $\\omega_X^\\bullet$ in $D_\\QCoh(\\mathcal{O}_X)$\nup to unique isomorphism by the Yoneda lemma. Since $\\omega_X^\\bullet$\nis in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ in fact it suffices to consider\n$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.}\n$$\n\\Ext^i_X(K, \\omega_X^\\bullet) = \\Hom_k(H^{-i}(X, K), k)\n$$\ncompatible with shifts and distinguished triangles,\n\\item there are functorial isomorphisms\n$\\Hom(\\mathcal{F}, \\omega_X) = \\Hom_k(H^{\\dim(X)}(X, \\mathcal{F}), k)$\nfor $\\mathcal{F}$ quasi-coherent on $X$, and\n\\item if $X \\to \\Spec(k)$ is smooth of relative dimension $d$,\nthen $\\omega_X^\\bullet \\cong \\wedge^d\\Omega_{X/k}[d]$ and\n$\\omega_X \\cong \\wedge^d\\Omega_{X/k}$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Duality for proper schemes over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVV","source_file":"duality.tex","source_line":6716,"source_end_line":6746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6716-L6746","statement_sha256":"1496dc681e4f342cfc0dc033747e641fea86d63f72919b395829bb0c86ba5c2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8894,"rank":8894,"depth":46,"x":2004.979,"y":1003.693,"cluster":"duality-cohomology"},{"id":"stacks:0FVY","tag":"0FVY","title":"Duality for proper schemes over fields · Lemma 0FVY","summary":"Let k, X, and ω_X^bullet be as in Lemma [Tag 0FVV]. Let t : H^0(X, ω_X^bullet) → k be as in Remark [Tag 0FVW]. Let E ∈ D(O_X) be perfect. Then the pairings H^i(X, ω_X^bullet ⊗_O_X^L E^vee) × H^-i(X, E) → k, (xi, eta) ↦ t((1_ω_X^bullet ⊗ ε)(xi ∪ eta)) are perfect for all i. Here ∪ denotes the cup product of Cohomology, Section [Tag 0FKU] and ε : E^vee ⊗_O_X^L E → O_X is as in Cohomology, Example [Tag 0FPC].","statement_latex":"Let $k$, $X$, and $\\omega_X^\\bullet$\nbe as in Lemma \\ref{lemma-duality-proper-over-field}.\nLet $t : H^0(X, \\omega_X^\\bullet) \\to k$ be as in\nRemark \\ref{remark-duality-proper-over-field}.\nLet $E \\in D(\\mathcal{O}_X)$ be perfect. Then the pairings\n$$\nH^i(X, \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E^\\vee)\n\\times\nH^{-i}(X, E) \\longrightarrow k, \\quad\n(\\xi, \\eta) \\longmapsto\nt((1_{\\omega_X^\\bullet} \\otimes \\epsilon)(\\xi \\cup \\eta))\n$$\nare perfect for all $i$. Here $\\cup$ denotes the cup product\nof Cohomology, Section \\ref{cohomology-section-cup-product} and\n$\\epsilon : E^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E \\to \\mathcal{O}_X$\nis as in Cohomology, Example \\ref{cohomology-example-dual-derived}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Duality for proper schemes over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVY","source_file":"duality.tex","source_line":6860,"source_end_line":6878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6860-L6878","statement_sha256":"d5cd15c1ae7fdff4c05859cbd028b4a251585dfb14be932d83ac944d6dcdae77","origin":"The Stacks Project","memory_eligible":false,"source_rank":8895,"rank":8895,"depth":47,"x":2158.195,"y":1226.401,"cluster":"duality-cohomology"},{"id":"stacks:0FVZ","tag":"0FVZ","title":"Duality for proper schemes over fields · Lemma 0FVZ","summary":"Let X be a proper scheme over a field k which is Cohen-Macaulay and equidimensional of dimension d. The module ω_X of Lemma [Tag 0FVV] has the following properties • ω_X is a dualizing module on X (Section [Tag 0AWH]), • ω_X is a coherent Cohen-Macaulay module whose support is X, • there are functorial isomorphisms Ext^i_X(K, ω_X[d]) = Hom_k(H^-i(X, K), k) compatible with shifts and distinguished triangles for K ∈ D_QCoh(X), • there are functorial isomorphisms Ext^d -…","statement_latex":"Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay\nand equidimensional of dimension $d$. The module $\\omega_X$\nof Lemma \\ref{lemma-duality-proper-over-field} has the following properties\n\\begin{enumerate}\n\\item $\\omega_X$ is a dualizing module on $X$\n(Section \\ref{section-dualizing-module}),\n\\item $\\omega_X$ is a coherent Cohen-Macaulay module whose support is $X$,\n\\item there are functorial isomorphisms\n$\\Ext^i_X(K, \\omega_X[d]) = \\Hom_k(H^{-i}(X, K), k)$\ncompatible with shifts and distinguished triangles for $K \\in D_\\QCoh(X)$,\n\\item there are functorial isomorphisms\n$\\Ext^{d - i}(\\mathcal{F}, \\omega_X) = \\Hom_k(H^i(X, \\mathcal{F}), k)$\nfor $\\mathcal{F}$ quasi-coherent on $X$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Duality for proper schemes over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FVZ","source_file":"duality.tex","source_line":6888,"source_end_line":6904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6888-L6904","statement_sha256":"b6ff5cfd23f66e9f6de544b80ca1aed12e95fd236489fa5eaaddac4e12aa048b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8896,"rank":8896,"depth":47,"x":1865.823,"y":1149.039,"cluster":"duality-cohomology"},{"id":"stacks:0FW1","tag":"0FW1","title":"Duality for proper schemes over fields · Lemma 0FW1","summary":"Let X be a proper scheme over a field k. Let ω_X^bullet and ω_X be as in Lemma [Tag 0FVV]. • If X → Spec(k) factors as X → Spec(k') → Spec(k) for some field k', then ω_X^bullet and ω_X are as in Lemma [Tag 0FVV] for the morphism X → Spec(k'). • If K/k is a field extension, then the pullback of ω_X^bullet and ω_X to the base change X_K are as in Lemma [Tag 0FVV] for the morphism X_K → Spec(K).","statement_latex":"Let $X$ be a proper scheme over a field $k$. Let $\\omega_X^\\bullet$ and\n$\\omega_X$ be as in Lemma \\ref{lemma-duality-proper-over-field}.\n\\begin{enumerate}\n\\item If $X \\to \\Spec(k)$ factors as $X \\to \\Spec(k') \\to \\Spec(k)$\nfor some field $k'$, then $\\omega_X^\\bullet$ and $\\omega_X$\nare as in Lemma \\ref{lemma-duality-proper-over-field} for the morphism\n$X \\to \\Spec(k')$.\n\\item If $K/k$ is a field extension, then the pullback of\n$\\omega_X^\\bullet$ and $\\omega_X$ to the base change $X_K$\nare as in  Lemma \\ref{lemma-duality-proper-over-field} for the morphism\n$X_K \\to \\Spec(K)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Duality for proper schemes over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FW1","source_file":"duality.tex","source_line":6945,"source_end_line":6959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L6945-L6959","statement_sha256":"02213a46d0926d216f8ad697300fb4f156f0233888fd396b0f589765e573fb8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8897,"rank":8897,"depth":47,"x":2143.908,"y":1040.076,"cluster":"duality-cohomology"},{"id":"stacks:0E2T","tag":"0E2T","title":"Relative dualizing complexes · Definition 0E2T","summary":"Let X → S be a morphism of schemes which is flat and locally of finite presentation. Let W ⊂ X ×_S X be any open such that the diagonal Δ_X/S : X → X ×_S X factors through a closed immersion Δ : X → W. A relative dualizing complex is a pair (K, xi) consisting of an object K ∈ D(O_X) and a map xi : Δ_*O_X → Lpr_1^*K|_W in D(O_W) such that • K is S-perfect (Derived Categories of Schemes, Definition [Tag 0DI0]), and • xi defines an isomorphism of Δ_*O_X with RSheafHom_O_W(…","statement_latex":"Let $X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation. Let $W \\subset X \\times_S X$\nbe any open such that the diagonal $\\Delta_{X/S} : X \\to X \\times_S X$\nfactors through a closed immersion $\\Delta : X \\to W$.\nA {\\it relative dualizing complex} is a\npair $(K, \\xi)$ consisting of an object $K \\in D(\\mathcal{O}_X)$\nand a map\n$$\n\\xi : \\Delta_*\\mathcal{O}_X \\longrightarrow L\\text{pr}_1^*K|_W\n$$\nin $D(\\mathcal{O}_W)$ such that\n\\begin{enumerate}\n\\item $K$ is $S$-perfect (Derived Categories of Schemes, Definition\n\\ref{perfect-definition-relatively-perfect}), and\n\\item $\\xi$ defines an isomorphism of $\\Delta_*\\mathcal{O}_X$\nwith\n$R\\SheafHom_{\\mathcal{O}_W}(\n\\Delta_*\\mathcal{O}_X, L\\text{pr}_1^*K|_W)$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Relative dualizing complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2T","source_file":"duality.tex","source_line":7016,"source_end_line":7037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7016-L7037","statement_sha256":"7ead86e04e708602bc2df9e0221896f039c62d2dfac6985eb67fe090f8915939","origin":"The Stacks Project","memory_eligible":false,"source_rank":8898,"rank":8898,"depth":2,"x":2026.359,"y":1278.456,"cluster":"duality-cohomology"},{"id":"stacks:0E2U","tag":"0E2U","title":"Relative dualizing complexes · Lemma 0E2U","summary":"Let X → S be a morphism of schemes which is flat and locally of finite presentation. Let (K, xi) be a relative dualizing complex. Then for any commutative diagram xymatrix Spec(A) ar[d] ar[r] & X ar[d] Spec(R) ar[r] & S whose horizontal arrows are open immersions, the restriction of K to Spec(A) corresponds via Derived Categories of Schemes, Lemma [Tag 06Z0] to a relative dualizing complex for R → A in the sense of Dualizing Complexes, Definition [Tag 0E2C].","statement_latex":"Let $X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation. Let $(K, \\xi)$ be a relative\ndualizing complex. Then for any commutative diagram\n$$\n\\xymatrix{\n\\Spec(A) \\ar[d] \\ar[r] & X \\ar[d] \\\\\n\\Spec(R) \\ar[r] & S\n}\n$$\nwhose horizontal arrows are open immersions, the\nrestriction of $K$ to $\\Spec(A)$ corresponds via\nDerived Categories of Schemes, Lemma\n\\ref{perfect-lemma-affine-compare-bounded}\nto a relative dualizing complex for $R \\to A$\nin the sense of Dualizing Complexes, Definition\n\\ref{dualizing-definition-relative-dualizing-complex}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2U","source_file":"duality.tex","source_line":7053,"source_end_line":7071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7053-L7071","statement_sha256":"14c68a26aba3ca9d67a602ac6288a86ed20c77e50ad86fda9e64f9b5a5192bc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8899,"rank":8899,"depth":37,"x":1921.233,"y":1035.733,"cluster":"duality-cohomology"},{"id":"stacks:0E2V","tag":"0E2V","title":"Relative dualizing complexes · Lemma 0E2V","summary":"Let X → S be a morphism of schemes which is flat and locally of finite presentation. Let (K, xi) be a relative dualizing complex. Then O_X → RSheafHom_O_X(K, K) is an isomorphism.","statement_latex":"Let $X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation. Let $(K, \\xi)$ be a relative\ndualizing complex. Then\n$\\mathcal{O}_X \\to R\\SheafHom_{\\mathcal{O}_X}(K, K)$\nis an isomorphism.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2V","source_file":"duality.tex","source_line":7100,"source_end_line":7107,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7100-L7107","statement_sha256":"791d788802a4cfde7e0b643c75a5a3fa350b4ad6ce3d278262e1240660d0337a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8900,"rank":8900,"depth":42,"x":2194.217,"y":1155.183,"cluster":"duality-cohomology"},{"id":"stacks:0E2W","tag":"0E2W","title":"Relative dualizing complexes · Lemma 0E2W","summary":"Let X → S be a morphism of schemes which is flat and locally of finite presentation. If (K, xi) and (L, eta) are two relative dualizing complexes on X/S, then there is a unique isomorphism K → L sending xi to eta.","statement_latex":"Let $X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation. If $(K, \\xi)$ and $(L, \\eta)$\nare two relative dualizing complexes on $X/S$, then there is a unique\nisomorphism $K \\to L$ sending $\\xi$ to $\\eta$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2W","source_file":"duality.tex","source_line":7116,"source_end_line":7122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7116-L7122","statement_sha256":"6d075ecd3bdfaf4ec954f1bc71cd3afd79af8c7c550d4fe7d5ba6e4105f196fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":8901,"rank":8901,"depth":43,"x":1896.566,"y":1222.067,"cluster":"duality-cohomology"},{"id":"stacks:0E2X","tag":"0E2X","title":"Relative dualizing complexes · Lemma 0E2X","summary":"Let X → S be a morphism of schemes which is flat and locally of finite presentation. There exists a relative dualizing complex (K, xi).","statement_latex":"Let $X \\to S$ be a morphism of schemes which is\nflat and locally of finite presentation.\nThere exists a relative dualizing complex $(K, \\xi)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2X","source_file":"duality.tex","source_line":7161,"source_end_line":7166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7161-L7166","statement_sha256":"f5f6f6cd12e5cc3b430e6d5ba5b1ffd7bd556e4a3d793e9cba5e0cf777211758","origin":"The Stacks Project","memory_eligible":false,"source_rank":8902,"rank":8902,"depth":44,"x":2062.429,"y":1003.635,"cluster":"duality-cohomology"},{"id":"stacks:0E2Y","tag":"0E2Y","title":"Relative dualizing complexes · Lemma 0E2Y","summary":"Consider a cartesian square xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f S' ar[r]^g & S of schemes. Assume X → S is flat and locally of finite presentation. Let (K, xi) be a relative dualizing complex for f. Set K' = L(g')^*K. Let xi' be the derived base change of xi (see proof). Then (K', xi') is a relative dualizing complex for f'.","statement_latex":"Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nof schemes. Assume $X \\to S$ is flat and locally of finite presentation.\nLet $(K, \\xi)$ be a relative dualizing complex for $f$.\nSet $K' = L(g')^*K$. Let $\\xi'$ be the derived base change of $\\xi$\n(see proof). Then $(K', \\xi')$ is a relative dualizing complex for $f'$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2Y","source_file":"duality.tex","source_line":7253,"source_end_line":7266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7253-L7266","statement_sha256":"3014b64051590301c1f46828e3ff923c8ba115a38d0814e1779c09fe809867d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8903,"rank":8903,"depth":41,"x":2115.834,"y":1259.073,"cluster":"duality-cohomology"},{"id":"stacks:0E2Z","tag":"0E2Z","title":"Relative dualizing complexes · Lemma 0E2Z","summary":"Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a proper, flat morphism of finite presentation. The relative dualizing complex ω_X/S^bullet of Remark [Tag 0B6S] together with ([Tag 0E2Q]) is a relative dualizing complex in the sense of Definition [Tag 0E2T].","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to S$ be a proper, flat morphism of finite presentation.\nThe relative dualizing complex $\\omega_{X/S}^\\bullet$ of\nRemark \\ref{remark-relative-dualizing-complex}\ntogether with (\\ref{equation-pre-rigid}) is a relative\ndualizing complex in the sense of\nDefinition \\ref{definition-relative-dualizing-complex}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E2Z","source_file":"duality.tex","source_line":7312,"source_end_line":7321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7312-L7321","statement_sha256":"e437e743d8c564bd5a6cde085f3f2ab3c9de0151e718358fb4de3f1f5b736c19","origin":"The Stacks Project","memory_eligible":false,"source_rank":8904,"rank":8904,"depth":43,"x":1870.793,"y":1100.863,"cluster":"duality-cohomology"},{"id":"stacks:0E9W","tag":"0E9W","title":"Relative dualizing complexes · Lemma 0E9W","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. If f is flat, then f^!O_Y is (the first component of) a relative dualizing complex for X over Y in the sense of Definition [Tag 0E2T].","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$\nbe a morphism of $\\textit{FTS}_S$. If $f$ is flat, then\n$f^!\\mathcal{O}_Y$ is (the first component of)\na relative dualizing complex for $X$ over $Y$ in the sense of\nDefinition \\ref{definition-relative-dualizing-complex}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9W","source_file":"duality.tex","source_line":7371,"source_end_line":7378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7371-L7378","statement_sha256":"abcd7ee0784303f378a728b6ccff20410929dfcf3e9e8f0d07a33e50bc2ff16c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8905,"rank":8905,"depth":41,"x":2179.018,"y":1078.46,"cluster":"duality-cohomology"},{"id":"stacks:0E30","tag":"0E30","title":"Relative dualizing complexes · Lemma 0E30","summary":"Let f : Y → X and X → S be morphisms of schemes which are flat and of finite presentation. Let (K, xi) and (M, eta) be a relative dualizing complex for X → S and Y → X. Set E = M ⊗_O_Y^L Lf^*K. Then (E, zeta) is a relative dualizing complex for Y → S for a suitable zeta.","statement_latex":"Let $f : Y \\to X$ and $X \\to S$ be morphisms of schemes\nwhich are flat and of finite presentation.\nLet $(K, \\xi)$ and $(M, \\eta)$\nbe a relative dualizing complex for $X \\to S$ and $Y \\to X$.\nSet $E = M \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Lf^*K$.\nThen $(E, \\zeta)$ is a relative dualizing complex for $Y \\to S$ for\na suitable $\\zeta$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Relative dualizing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E30","source_file":"duality.tex","source_line":7405,"source_end_line":7414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7405-L7414","statement_sha256":"e431c7bcfced014383cc61279f175e2380664e239a97b356ce5268bd5c0ed40a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8906,"rank":8906,"depth":42,"x":1969.545,"y":1270.064,"cluster":"duality-cohomology"},{"id":"stacks:0E9Y","tag":"0E9Y","title":"The fundamental class of an lci morphism · Lemma 0E9Y","summary":"Let X be a locally ringed space. Let E_1 xrightarrowα E_0 → F → 0 be a short exact sequence of O_X-modules. Assume E_1 and E_0 are locally free of ranks r_1, r_0. Then there is a canonical map wedge^r_0 - r_1F → wedge^r_1(E_1^vee) ⊗ wedge^r_0E_0 which is an isomorphism on the stalk at x ∈ X if and only if F is locally free of rank r_0 - r_1 in an open neighbourhood of x.","statement_latex":"Let $X$ be a locally ringed space. Let\n$$\n\\mathcal{E}_1 \\xrightarrow{\\alpha} \\mathcal{E}_0 \\to \\mathcal{F} \\to 0\n$$\nbe a short exact sequence of $\\mathcal{O}_X$-modules.\nAssume $\\mathcal{E}_1$ and $\\mathcal{E}_0$ are locally\nfree of ranks $r_1, r_0$. Then there is a canonical map\n$$\n\\wedge^{r_0 - r_1}\\mathcal{F}\n\\longrightarrow\n\\wedge^{r_1}(\\mathcal{E}_1^\\vee) \\otimes \\wedge^{r_0}\\mathcal{E}_0\n$$\nwhich is an isomorphism on the stalk at $x \\in X$\nif and only if $\\mathcal{F}$ is locally free of rank $r_0 - r_1$\nin an open neighbourhood of $x$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"The fundamental class of an lci morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9Y","source_file":"duality.tex","source_line":7509,"source_end_line":7526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7509-L7526","statement_sha256":"ef6b35041f0b484a1b48e14099249752468801cc74ae7857c978b613bb612728","origin":"The Stacks Project","memory_eligible":false,"source_rank":8907,"rank":8907,"depth":0,"x":1969.925,"y":1009.66,"cluster":"duality-cohomology"},{"id":"stacks:0E9Z","tag":"0E9Z","title":"The fundamental class of an lci morphism · Lemma 0E9Z","summary":"Let Y be a Noetherian scheme. Let f : X → Y be a local complete intersection morphism which factors as an immersion X → P followed by a proper smooth morphism P → Y. Let r be the locally constant function on X such that ω_X/Y = H^-r(f^!O_Y) is the unique nonzero cohomology sheaf of f^!O_Y, see Lemma [Tag 0B6V]. Then there is a map wedge^rΩ_X/Y → ω_X/Y which is an isomorphism on the stalk at a point x if and only if f is smooth at x.","statement_latex":"Let $Y$ be a Noetherian scheme. Let $f : X \\to Y$ be a\nlocal complete intersection morphism which factors\nas an immersion $X \\to P$ followed by a proper smooth morphism $P \\to Y$.\nLet $r$ be the locally constant function on\n$X$ such that $\\omega_{X/Y} = H^{-r}(f^!\\mathcal{O}_Y)$\nis the unique nonzero cohomology sheaf of $f^!\\mathcal{O}_Y$, see\nLemma \\ref{lemma-lci-shriek}.\nThen there is a map\n$$\n\\wedge^r\\Omega_{X/Y} \\longrightarrow \\omega_{X/Y}\n$$\nwhich is an isomorphism on the stalk at a point $x$ if and only\nif $f$ is smooth at $x$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"The fundamental class of an lci morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9Z","source_file":"duality.tex","source_line":7556,"source_end_line":7571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7556-L7571","statement_sha256":"1ef7397ba6c255fff03ff3aaeded303ed06beea49e298ece6d32b42433e58a12","origin":"The Stacks Project","memory_eligible":false,"source_rank":8908,"rank":8908,"depth":66,"x":2179.262,"y":1202.084,"cluster":"duality-cohomology"},{"id":"stacks:0EA0","tag":"0EA0","title":"The fundamental class of an lci morphism · Lemma 0EA0","summary":"Let f : X → Y be a morphism of schemes. Let r ≥ 0. Assume • Y is Cohen-Macaulay (Properties, Definition [Tag 02IO]), • f factors as X → P → Y where the first morphism is an immersion and the second is smooth and proper, • if x ∈ X and dim(O_X, x) ≤ 1, then f is Koszul at x (More on Morphisms, Definition [Tag 069F]), and • if xi is a generic point of an irreducible component of X, then we have trdeg_kappa(f(xi)) kappa(xi) = r. Then with ω_X/Y = H^-r(f^!O_Y) there is a map…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $r \\geq 0$. Assume\n\\begin{enumerate}\n\\item $Y$ is Cohen-Macaulay (Properties, Definition\n\\ref{properties-definition-Cohen-Macaulay}),\n\\item $f$ factors as $X \\to P \\to Y$ where the first morphism is\nan immersion and the second is smooth and proper,\n\\item if $x \\in X$ and $\\dim(\\mathcal{O}_{X, x}) \\leq 1$,\nthen $f$ is Koszul at $x$ (More on Morphisms, Definition\n\\ref{more-morphisms-definition-lci}), and\n\\item if $\\xi$ is a generic point of an irreducible component of $X$, then\nwe have\n$\\text{trdeg}_{\\kappa(f(\\xi))} \\kappa(\\xi) = r$.\n\\end{enumerate}\nThen with $\\omega_{X/Y} = H^{-r}(f^!\\mathcal{O}_Y)$ there is a map\n$$\n\\wedge^r\\Omega_{X/Y} \\longrightarrow \\omega_{X/Y}\n$$\nwhich is an isomorphism on the locus where $f$ is smooth.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"The fundamental class of an lci morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EA0","source_file":"duality.tex","source_line":7627,"source_end_line":7647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7627-L7647","statement_sha256":"111a2ffd99fe3453bb61b64b055e1fb001dcb7ef3388ae438fb6d13255ab6301","origin":"The Stacks Project","memory_eligible":false,"source_rank":8909,"rank":8909,"depth":67,"x":1869.853,"y":1178.953,"cluster":"duality-cohomology"},{"id":"stacks:0G2H","tag":"0G2H","title":"Extension by zero for coherent modules · Lemma 0G2H","summary":"Let j : U → X be an open immersion of Noetherian schemes. Let (K_n) be a Deligne system and denote K ∈ D^b_Coh(O_U) the value of the constant system (K_n|_U). Let L be an object of D^b_Coh(O_X). Then colim Hom_X(K_n, L) = Hom_U(K, L|_U).","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes.\nLet $(K_n)$ be a Deligne system and denote\n$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ the value\nof the constant system $(K_n|_U)$. Let $L$ be an object of\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.\nThen $\\colim \\Hom_X(K_n, L) = \\Hom_U(K, L|_U)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Extension by zero for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2H","source_file":"duality.tex","source_line":7702,"source_end_line":7710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7702-L7710","statement_sha256":"6a2a5e3b2cb6cb14c5586414f48326c75ea4581622cc7fd17859bfe2aa897b27","origin":"The Stacks Project","memory_eligible":false,"source_rank":8910,"rank":8910,"depth":34,"x":2116.851,"y":1020.287,"cluster":"duality-cohomology"},{"id":"stacks:0G4K","tag":"0G4K","title":"Extension by zero for coherent modules · Lemma 0G4K","summary":"The result of Lemma [Tag 0G2H] holds even for L ∈ D^+_Coh(O_X).","statement_latex":"The result of Lemma \\ref{lemma-lift-map} holds even for\n$L \\in D^+_{\\textit{Coh}}(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Extension by zero for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4K","source_file":"duality.tex","source_line":7871,"source_end_line":7875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7871-L7875","statement_sha256":"afcb95a497a92a56bbff32c2db09c7b2a52faee3975ab30c44d3ffd7180e70cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8911,"rank":8911,"depth":35,"x":2062.259,"y":1277.691,"cluster":"duality-cohomology"},{"id":"stacks:0G4L","tag":"0G4L","title":"Extension by zero for coherent modules · Lemma 0G4L","summary":"Let j : U → X be an open immersion of Noetherian schemes. • Let (K_n) and (L_n) be Deligne systems. Let K and L be the values of the constant systems (K_n|_U) and (L_n|_U). Given a morphism α : K → L of D(O_U) there is a unique morphism of pro-systems (K_n) → (L_n) of D^b_Coh(O_X) whose restriction to U is α. • Given K ∈ D^b_Coh(O_U) there exists a Deligne system (K_n) such that (K_n|_U) is constant with value K. • The pro-object (K_n) of D^b_Coh(O_X) of (2) is unique up…","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes.\n\\begin{enumerate}\n\\item Let $(K_n)$ and $(L_n)$ be Deligne systems.\nLet $K$ and $L$ be the values of the constant systems\n$(K_n|_U)$ and $(L_n|_U)$. Given a morphism $\\alpha : K \\to L$\nof $D(\\mathcal{O}_U)$\nthere is a unique morphism of pro-systems $(K_n) \\to (L_n)$\nof $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ whose restriction to $U$ is $\\alpha$.\n\\item Given $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_U)$ there exists a\nDeligne system $(K_n)$ such that $(K_n|_U)$ is constant\nwith value $K$.\n\\item The pro-object $(K_n)$ of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$\nof (2) is unique up to unique isomorphism (as a pro-object).\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Extension by zero for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4L","source_file":"duality.tex","source_line":7885,"source_end_line":7901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7885-L7901","statement_sha256":"253b6650cfdcec0774dac5f0f6953867be62ac14d557bdd3c50c862dcf1741cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8912,"rank":8912,"depth":35,"x":1895.354,"y":1056.692,"cluster":"duality-cohomology"},{"id":"stacks:0G4M","tag":"0G4M","title":"Extension by zero for coherent modules · Lemma 0G4M","summary":"Let j : U → X be an open immersion of Noetherian schemes. Let K → L → M → K[1] be a distinguished triangle of D^b_Coh(O_U). Then there exists an inverse system of distinguished triangles K_n → L_n → M_n → K_n[1] in D^b_Coh(O_X) such that (K_n), (L_n), (M_n) are Deligne systems and such that the restriction of these distinguished triangles to U is isomorphic to the distinguished triangle we started out with.","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let\n$$\nK \\to L \\to M \\to K[1]\n$$\nbe a distinguished triangle of $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$.\nThen there exists an inverse system of distinguished triangles\n$$\nK_n \\to L_n \\to M_n \\to K_n[1]\n$$\nin $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that $(K_n)$, $(L_n)$, $(M_n)$\nare Deligne systems and such that the restriction of these\ndistinguished triangles to $U$ is isomorphic to the distinguished triangle\nwe started out with.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Extension by zero for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4M","source_file":"duality.tex","source_line":7927,"source_end_line":7942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L7927-L7942","statement_sha256":"ee01fd230fc740d01ac26bcd3c38cc52a5bbfe193d1f399363ecee317be731e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8913,"rank":8913,"depth":36,"x":2196.443,"y":1125.014,"cluster":"duality-cohomology"},{"id":"stacks:0G4Q","tag":"0G4Q","title":"Extension by zero for coherent modules · Lemma 0G4Q","summary":"Let j : U → X be an open immersion of Noetherian schemes. Let K_n → L_n → M_n → K_n[1] be an inverse system of distinguished triangles in D^b_Coh(O_X). If (K_n) and (M_n) are pro-isomorphic to Deligne systems, then so is (L_n).","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes.\nLet\n$$\nK_n \\to L_n \\to M_n \\to K_n[1]\n$$\nbe an inverse system of distinguished triangles in\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. If $(K_n)$ and $(M_n)$\nare pro-isomorphic to Deligne systems, then so is $(L_n)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Extension by zero for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4Q","source_file":"duality.tex","source_line":8018,"source_end_line":8028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8018-L8028","statement_sha256":"51ede7e042b8b8e9e9a2d78cb645f57054de0055dbed5b93a3e4dae965100b32","origin":"The Stacks Project","memory_eligible":false,"source_rank":8914,"rank":8914,"depth":36,"x":1919.211,"y":1245.597,"cluster":"duality-cohomology"},{"id":"stacks:0G4R","tag":"0G4R","title":"Extension by zero for coherent modules · Lemma 0G4R","summary":"Let X be a Noetherian scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let F^bullet be a complex of coherent O_X-modules. Let p ∈ Z. Set H = H^p(F^bullet) and H_n = H^p(I^nF^bullet). Then there are canonical O_X-module maps … → H_3 → H_2 → H_1 → H There exists a c > 0 such that for n ≥ c the image of H_n → H is contained in I^n - cH and there is a canonical O_X-module map I^nH → H_n - c such that the compositions I^n H → H_n - c → I^n - 2cH and H_n → I^n - cH →…","statement_latex":"Let $X$ be a Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. Let $\\mathcal{F}^\\bullet$ be a\ncomplex of coherent $\\mathcal{O}_X$-modules. Let $p \\in \\mathbf{Z}$.\nSet $\\mathcal{H} = H^p(\\mathcal{F}^\\bullet)$ and\n$\\mathcal{H}_n = H^p(\\mathcal{I}^n\\mathcal{F}^\\bullet)$.\nThen there are canonical $\\mathcal{O}_X$-module maps\n$$\n\\ldots \\to \\mathcal{H}_3 \\to \\mathcal{H}_2 \\to \\mathcal{H}_1 \\to \\mathcal{H}\n$$\nThere exists a $c > 0$ such that for $n \\geq c$ the image of\n$\\mathcal{H}_n \\to \\mathcal{H}$ is contained in\n$\\mathcal{I}^{n - c}\\mathcal{H}$ and there is a canonical\n$\\mathcal{O}_X$-module map\n$\\mathcal{I}^n\\mathcal{H} \\to \\mathcal{H}_{n - c}$ such that the compositions\n$$\n\\mathcal{I}^n \\mathcal{H} \\to \\mathcal{H}_{n - c} \\to\n\\mathcal{I}^{n - 2c}\\mathcal{H}\n\\quad\\text{and}\\quad\n\\mathcal{H}_n \\to \\mathcal{I}^{n - c}\\mathcal{H} \\to \\mathcal{H}_{n - 2c}\n$$\nare the canonical ones. In particular, the inverse systems\n$(\\mathcal{H}_n)$ and $(\\mathcal{I}^n\\mathcal{H})$\nare isomorphic as pro-objects of $\\textit{Mod}(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Extension by zero for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4R","source_file":"duality.tex","source_line":8066,"source_end_line":8091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8066-L8091","statement_sha256":"c03e530e25f3d45b7ba879a687b30f1792c792fb9d5f30ea3bd97816a551bde4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8915,"rank":8915,"depth":4,"x":2026.773,"y":999.132,"cluster":"duality-cohomology"},{"id":"stacks:0G4S","tag":"0G4S","title":"Extension by zero for coherent modules · Lemma 0G4S","summary":"Let j : U → X be an open immersion of Noetherian schemes. Let a ≤ b be integers. Let (K_n) be an inverse system of D^b_Coh(O_X) such that H^i(K_n) = 0 for i not ∈ [a, b]. The following are equivalent • (K_n) is pro-isomorphic to a Deligne system, • for every p ∈ Z there exists a coherent O_X-module F such that the pro-systems (H^p(K_n)) and (I^nF) are pro-isomorphic.","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes.\nLet $a \\leq b$ be integers. Let $(K_n)$ be an inverse system of\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$\nsuch that $H^i(K_n) = 0$ for $i \\not \\in [a, b]$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $(K_n)$ is pro-isomorphic to a Deligne system,\n\\item for every $p \\in \\mathbf{Z}$ there exists a coherent\n$\\mathcal{O}_X$-module $\\mathcal{F}$ such that the pro-systems\n$(H^p(K_n))$ and $(\\mathcal{I}^n\\mathcal{F})$ are pro-isomorphic.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Extension by zero for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4S","source_file":"duality.tex","source_line":8102,"source_end_line":8115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8102-L8115","statement_sha256":"32e1d4c3e287ff45afe42e88e08284a65b0638e17172990dcaf1027521ac0ac8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8916,"rank":8916,"depth":37,"x":2145.774,"y":1242.142,"cluster":"duality-cohomology"},{"id":"stacks:0G4T","tag":"0G4T","title":"Extension by zero for coherent modules · Lemma 0G4T","summary":"Let j : U → X be an open immersion of Noetherian schemes. Let (K_n) be an inverse system in D^b_Coh(O_X). Let X = W_1 ∪ … ∪ W_r be an open covering. The following are equivalent • (K_n) is pro-isomorphic to a Deligne system, • for each i the restriction (K_n|_W_i) is pro-isomorphic to a Deligne system with respect to the open immersion U ∩ W_i → W_i.","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let\n$(K_n)$ be an inverse system in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.\nLet $X = W_1 \\cup \\ldots \\cup W_r$ be an open covering.\nThe following are equivalent\n\\begin{enumerate}\n\\item $(K_n)$ is pro-isomorphic to a Deligne system,\n\\item for each $i$ the restriction $(K_n|_{W_i})$\nis pro-isomorphic to a Deligne system with respect to\nthe open immersion $U \\cap W_i \\to W_i$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Extension by zero for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4T","source_file":"duality.tex","source_line":8140,"source_end_line":8152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8140-L8152","statement_sha256":"fe6153ac0ce44199027305d6d7a427374b86203e02469695aed3d36548232b52","origin":"The Stacks Project","memory_eligible":false,"source_rank":8917,"rank":8917,"depth":36,"x":1862.328,"y":1130.366,"cluster":"duality-cohomology"},{"id":"stacks:0G4U","tag":"0G4U","title":"Extension by zero for coherent modules · Lemma 0G4U","summary":"Let j : U → X be an open immersion of Noetherian schemes. Let I ⊂ O_X be a quasi-coherent sheaf of ideals with V(I) = X setminus U. Let K be in D^b_Coh(O_X). Then K ⊗_O_X^L I^n is pro-isomorphic to a Deligne system with constant value K|_U over U.","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes. Let\n$\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of\nideals with $V(\\mathcal{I}) = X \\setminus U$.\nLet $K$ be in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.\nThen\n$$\nK \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{I}^n\n$$\nis pro-isomorphic to a Deligne system with constant value $K|_U$ over $U$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Extension by zero for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4U","source_file":"duality.tex","source_line":8210,"source_end_line":8221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8210-L8221","statement_sha256":"2d0fb551e189134dcd3c808a40034964060aeb539fa0367e35f3748189408f0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":8918,"rank":8918,"depth":37,"x":2161.514,"y":1051.877,"cluster":"duality-cohomology"},{"id":"stacks:0G4W","tag":"0G4W","title":"Preliminaries to compactly supported cohomology · Lemma 0G4W","summary":"Let f : X → Y be a proper morphism of Noetherian schemes. Let V ⊂ Y be an open subscheme and set U = f^-1(V). Picture xymatrix U ar[r]_j ar[d]_g & X ar[d]^f V ar[r]^j' & Y Then we have a canonical isomorphism Rj'_! ∘ Rg_* → Rf_* ∘ Rj_! of functors D^b_Coh(O_U) → Pro-D^b_Coh(O_Y) where Rj_! and Rj'_! are as in Remark [Tag 0G4N].","statement_latex":"Let $f : X \\to Y$ be a proper morphism of Noetherian schemes.\nLet $V \\subset Y$ be an open subscheme and set $U = f^{-1}(V)$.\nPicture\n$$\n\\xymatrix{\nU \\ar[r]_j \\ar[d]_g & X \\ar[d]^f \\\\\nV \\ar[r]^{j'} & Y\n}\n$$\nThen we have a canonical isomorphism $Rj'_! \\circ Rg_* \\to Rf_* \\circ Rj_!$\nof functors $D^b_{\\textit{Coh}}(\\mathcal{O}_U) \\to\n\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ where\n$Rj_!$ and $Rj'_!$ are as in Remark \\ref{remark-extension-by-zero}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Preliminaries to compactly supported cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4W","source_file":"duality.tex","source_line":8254,"source_end_line":8269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8254-L8269","statement_sha256":"39f2c9fcaf9c4affdce450e2ec629e10f4d4bd9afeab4bc331d2000aa6f12c66","origin":"The Stacks Project","memory_eligible":false,"source_rank":8919,"rank":8919,"depth":38,"x":2003.864,"y":1279.739,"cluster":"duality-cohomology"},{"id":"stacks:0G4X","tag":"0G4X","title":"Preliminaries to compactly supported cohomology · Lemma 0G4X","summary":"Let j : U → X be an open immersion of Noetherian schemes. Let j' : U → X' be a compactification of U over X (see proof) and denote f : X' → X the structure morphism. Then we have a canonical isomorphism Rj_! → Rf_* ∘ R(j')_! of functors D^b_Coh(O_U) → Pro-D^b_Coh(O_X) where Rj_! and Rj'_! are as in Remark [Tag 0G4N].","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes.\nLet $j' : U \\to X'$ be a compactification of $U$ over $X$ (see proof)\nand denote $f : X' \\to X$ the structure morphism.\nThen we have a canonical isomorphism $Rj_! \\to Rf_* \\circ R(j')_!$\nof functors $D^b_{\\textit{Coh}}(\\mathcal{O}_U) \\to\n\\text{Pro-}D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ where\n$Rj_!$ and $Rj'_!$ are as in Remark \\ref{remark-extension-by-zero}.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Preliminaries to compactly supported cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G4X","source_file":"duality.tex","source_line":8359,"source_end_line":8368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8359-L8368","statement_sha256":"1a1590e247afc90d9b71038825e434c2ad82e375745f952899dfb6386c02b028","origin":"The Stacks Project","memory_eligible":false,"source_rank":8920,"rank":8920,"depth":39,"x":1936.808,"y":1022.014,"cluster":"duality-cohomology"},{"id":"stacks:0G50","tag":"0G50","title":"Compactly supported cohomology for coherent modules · Lemma 0G50","summary":"The functor Rf_! is, up to isomorphism, independent of the choice of the compactification.","statement_latex":"The functor $Rf_!$ is, up to isomorphism, independent\nof the choice of the compactification.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Compactly supported cohomology for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G50","source_file":"duality.tex","source_line":8470,"source_end_line":8474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8470-L8474","statement_sha256":"f665eca61be7f48eb73ed2930bb5403c803dc1fc76917e7cb272bc1d6826479c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8921,"rank":8921,"depth":0,"x":2193.757,"y":1174.154,"cluster":"duality-cohomology"},{"id":"stacks:0G51","tag":"0G51","title":"Compactly supported cohomology for coherent modules · Proposition 0G51","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Then the functors Rf_! and f^! are adjoint in the following sense: for all K ∈ D^b_Coh(O_X) and L ∈ D^+_Coh(O_Y) we have Hom_X(K, f^!L) = Hom_Pro-D^+_Coh(O_Y)(Rf_!K, L) bifunctorially in K and L.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Then the functors $Rf_!$ and $f^!$ are adjoint in\nthe following sense: for all $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$\nand $L \\in D^+_{\\textit{Coh}}(\\mathcal{O}_Y)$ we have\n$$\n\\Hom_X(K, f^!L) =\n\\Hom_{\\text{Pro-}D^+_{\\textit{Coh}}(\\mathcal{O}_Y)}(Rf_!K, L)\n$$\nbifunctorially in $K$ and $L$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Compactly supported cohomology for coherent modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G51","source_file":"duality.tex","source_line":8503,"source_end_line":8514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8503-L8514","statement_sha256":"fe6b5df1908b99ad2bbf753a0d26effde35b0ecf8e0fb8aa19383dd923f2fb64","origin":"The Stacks Project","memory_eligible":false,"source_rank":8922,"rank":8922,"depth":43,"x":1881.64,"y":1207.8,"cluster":"duality-cohomology"},{"id":"stacks:0G52","tag":"0G52","title":"Compactly supported cohomology for coherent modules · Lemma 0G52","summary":"In Situation [Tag 0F42] let f : X → Y be a morphism of FTS_S. Let K → L → M → K[1] be a distinguished triangle of D^b_Coh(O_X). Then there exists an inverse system of distinguished triangles K_n → L_n → M_n → K_n[1] in D^b_Coh(O_Y) such that the pro-systems (K_n), (L_n), and (M_n) give Rf_!K, Rf_!L, and Rf_!M.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ be a morphism of\n$\\textit{FTS}_S$. Let\n$$\nK \\to L \\to M \\to K[1]\n$$\nbe a distinguished triangle of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.\nThen there exists an inverse system of distinguished triangles\n$$\nK_n \\to L_n \\to M_n \\to K_n[1]\n$$\nin $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$ such that the pro-systems\n$(K_n)$, $(L_n)$, and $(M_n)$ give $Rf_!K$, $Rf_!L$, and $Rf_!M$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Compactly supported cohomology for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G52","source_file":"duality.tex","source_line":8543,"source_end_line":8557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8543-L8557","statement_sha256":"4d1243f45d9dbfbf57feb733e3c5a8cda81fa392362f0fdba2fbe6ffb2acccd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8923,"rank":8923,"depth":37,"x":2084.928,"y":1005.694,"cluster":"duality-cohomology"},{"id":"stacks:0G55","tag":"0G55","title":"Compactly supported cohomology for coherent modules · Lemma 0G55","summary":"In Situation [Tag 0F42] let f : X → Y and g : Y → Z be composable morphisms of FTS_S. With notation as in Remark [Tag 0G54] we have Rg_! ∘ Rf_! = R(g ∘ f)_!.","statement_latex":"In Situation \\ref{situation-shriek} let $f : X \\to Y$ and $g : Y \\to Z$\nbe composable morphisms of $\\textit{FTS}_S$. With notation as in\nRemark \\ref{remark-composition-lower-shriek} we have\n$Rg_! \\circ Rf_! = R(g \\circ f)_!$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Compactly supported cohomology for coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G55","source_file":"duality.tex","source_line":8664,"source_end_line":8670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8664-L8670","statement_sha256":"feec3df583e93d56e7c246e3a19782dba079817629fd9bd1d429698ad18f8f05","origin":"The Stacks Project","memory_eligible":false,"source_rank":8924,"rank":8924,"depth":63,"x":2097.567,"y":1270.328,"cluster":"duality-cohomology"},{"id":"stacks:0G5A","tag":"0G5A","title":"Duality for compactly supported cohomology · Lemma 0G5A","summary":"Let p : U → Spec(k) be separated of finite type where k is a field. Let ω_U/k^bullet = p^!O_Spec(k). There are canonical isomorphisms Hom_k(H^i(U, K), k) = H^-i_c(U, RSheafHom_O_U(K, ω_U/k^bullet)) of topological k-vector spaces functorial for K in D^b_Coh(O_U).","statement_latex":"Let $p : U \\to \\Spec(k)$ be separated of finite type where $k$ is a field.\nLet $\\omega_{U/k}^\\bullet = p^!\\mathcal{O}_{\\Spec(k)}$.\nThere are canonical isomorphisms\n$$\n\\Hom_k(H^i(U, K), k) =\nH^{-i}_c(U, R\\SheafHom_{\\mathcal{O}_U}(K, \\omega_{U/k}^\\bullet))\n$$\nof topological $k$-vector spaces\nfunctorial for $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_U)$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Duality for compactly supported cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5A","source_file":"duality.tex","source_line":8898,"source_end_line":8909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8898-L8909","statement_sha256":"d936f5f7b088e70bdf74875063a98a35b6015dfa87948c161458eb41f6b62b1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":8925,"rank":8925,"depth":47,"x":1875.223,"y":1082.182,"cluster":"duality-cohomology"},{"id":"stacks:0G5B","tag":"0G5B","title":"Duality for compactly supported cohomology · Lemma 0G5B","summary":"With notation as in Lemma [Tag 0G5A] suppose U' ⊂ U is an open subscheme. Then the diagram xymatrix Hom_k(H^i(U, K), k) ar[rr] & & H^-i_c(U, RSheafHom_O_U(K, ω_U/k^bullet)) Hom_k(H^i(U', K|_U'), k) ar[rr] ar[u] & & H^-i_c(U', RSheafHom_O_U'(K, ω_U'/k^bullet)) ar[u] is commutative. Here the horizontal arrows are the isomorphisms of Lemma [Tag 0G5A], the vertical arrow on the left is the contragredient to the restriction map H^i(U, K) → H^i(U', K|_U'), and the right…","statement_latex":"With notation as in Lemma \\ref{lemma-duality-compact-support}\nsuppose $U' \\subset U$ is an open subscheme. Then the diagram\n$$\n\\xymatrix{\n\\Hom_k(H^i(U, K), k) \\ar[rr] & &\nH^{-i}_c(U, R\\SheafHom_{\\mathcal{O}_U}(K, \\omega_{U/k}^\\bullet)) \\\\\n\\Hom_k(H^i(U', K|_{U'}), k) \\ar[rr] \\ar[u] & &\nH^{-i}_c(U', R\\SheafHom_{\\mathcal{O}_{U'}}(K, \\omega_{U'/k}^\\bullet)) \\ar[u]\n}\n$$\nis commutative. Here the horizontal arrows are the isomorphisms of\nLemma \\ref{lemma-duality-compact-support}, the vertical arrow on the\nleft is the contragredient to the restriction map\n$H^i(U, K) \\to H^i(U', K|_{U'})$, and the right vertical arrow\nis Remark \\ref{remark-covariance-open-lower-shriek} (see discussion\nbefore the lemma).","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Duality for compactly supported cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5B","source_file":"duality.tex","source_line":8972,"source_end_line":8990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L8972-L8990","statement_sha256":"16b54c7b3c95bfc1eb5ad23bfc0eceb0d938f3ef9f2ac6acfbc4a8f7fc262da0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8926,"rank":8926,"depth":48,"x":2190.778,"y":1094.767,"cluster":"duality-cohomology"},{"id":"stacks:0G5C","tag":"0G5C","title":"Duality for compactly supported cohomology · Lemma 0G5C","summary":"Let X be a proper scheme over a field k. Let K ∈ D^b_Coh(O_X) with H^i(K) = 0 for i < 0. Set F = H^0(K). Let Z ⊂ X be closed with complement U = X setminus U. Then H^0_c(U, K|_U) ⊂ H^0(X, F) is given by those global sections of F which vanish in an open neighbourhood of Z.","statement_latex":"Let $X$ be a proper scheme over a field $k$. Let\n$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ with $H^i(K) = 0$\nfor $i < 0$. Set $\\mathcal{F} = H^0(K)$.\nLet $Z \\subset X$ be closed with complement $U = X \\setminus U$.\nThen\n$$\nH^0_c(U, K|_U) \\subset H^0(X, \\mathcal{F})\n$$\nis given by those global sections of $\\mathcal{F}$ which\nvanish in an open neighbourhood of $Z$.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Duality for compactly supported cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5C","source_file":"duality.tex","source_line":9058,"source_end_line":9070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L9058-L9070","statement_sha256":"ba918634c416db47ff5d10fbcd57516b411f12068b33901844c66643d928e260","origin":"The Stacks Project","memory_eligible":false,"source_rank":8927,"rank":8927,"depth":4,"x":1947.741,"y":1264.702,"cluster":"duality-cohomology"},{"id":"stacks:0G5E","tag":"0G5E","title":"Lichtenbaum's theorem · Lemma 0G5E","summary":"Let U be a variety. Let F be a coherent O_U-module. If H^d(U, F) is nonzero, then dim(U) ≥ d and if equality holds, then U is proper.","statement_latex":"Let $U$ be a variety. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.\nIf $H^d(U, \\mathcal{F})$ is nonzero, then $\\dim(U) \\geq d$ and if\nequality holds, then $U$ is proper.","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Lichtenbaum's theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5E","source_file":"duality.tex","source_line":9111,"source_end_line":9116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L9111-L9116","statement_sha256":"c0f6cf69b6ec68caf0618cb957c1222fbfb35e9482284f603a4178dc25b0b4d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8928,"rank":8928,"depth":61,"x":1990.336,"y":1001.239,"cluster":"duality-cohomology"},{"id":"stacks:0G5F","tag":"0G5F","title":"Lichtenbaum's theorem · Theorem 0G5F","summary":"Let X be a nonempty separated scheme of finite type over a field k. Let d = dim(X). The following are equivalent • H^d(X, F) = 0 for all coherent O_X-modules F on X, • H^d(X, F) = 0 for all quasi-coherent O_X-modules F on X, and • no irreducible component X' ⊂ X of dimension d is proper over k.","statement_latex":"Let $X$ be a nonempty separated scheme of finite type over a field $k$.\nLet $d = \\dim(X)$. The following are equivalent\n\\begin{enumerate}\n\\item $H^d(X, \\mathcal{F}) = 0$ for all coherent $\\mathcal{O}_X$-modules\n$\\mathcal{F}$ on $X$,\n\\item $H^d(X, \\mathcal{F}) = 0$ for all quasi-coherent $\\mathcal{O}_X$-modules\n$\\mathcal{F}$ on $X$, and\n\\item no irreducible component $X' \\subset X$ of dimension $d$\nis proper over $k$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Schemes","chapter_id":"duality","section":"Lichtenbaum's theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5F","source_file":"duality.tex","source_line":9163,"source_end_line":9175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/duality.tex#L9163-L9175","statement_sha256":"c34b401d6262c4a32a23efd195b41445afd856d98b1fd75f826ef4ee899e1035","origin":"The Stacks Project","memory_eligible":false,"source_rank":8929,"rank":8929,"depth":62,"x":2170.968,"y":1219.89,"cluster":"duality-cohomology"},{"id":"stacks:0BT0","tag":"0BT0","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BT0","summary":"Let A → B be a quasi-finite ring map. Given two factorizations A → B' → B and A → B\" → B with A → B' and A → B\" finite and Spec(B) → Spec(B') and Spec(B) → Spec(B\") open immersions, there exists an A-subalgebra B\"' ⊂ B finite over A such that Spec(B) → Spec(B\"') an open immersion and B' → B and B\" → B factor through B\"'.","statement_latex":"Let $A \\to B$ be a quasi-finite ring map. Given two factorizations\n$A \\to B' \\to B$ and $A \\to B'' \\to B$ with\n$A \\to B'$ and $A \\to B''$ finite and $\\Spec(B) \\to \\Spec(B')$\nand $\\Spec(B) \\to \\Spec(B'')$ open immersions, there exists\nan $A$-subalgebra $B''' \\subset B$ finite over $A$ such that\n$\\Spec(B) \\to \\Spec(B''')$ an open immersion and $B' \\to B$ and\n$B'' \\to B$ factor through $B'''$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT0","source_file":"discriminant.tex","source_line":103,"source_end_line":112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L103-L112","statement_sha256":"097a3b3461b7aeee60147c3a64f2fe999e02b7c6a6de0c9a5d1ea1b7c1f8ec18","origin":"The Stacks Project","memory_eligible":false,"source_rank":8930,"rank":8930,"depth":6,"x":2520.119,"y":1188.943,"cluster":"local-crystalline-methods"},{"id":"stacks:0BT1","tag":"0BT1","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BT1","summary":"The module ([Tag 0BSZ]) is well defined, i.e., independent of the choice of the factorization.","statement_latex":"The module (\\ref{equation-dualizing}) is well defined, i.e.,\nindependent of the choice of the factorization.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT1","source_file":"discriminant.tex","source_line":139,"source_end_line":143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L139-L143","statement_sha256":"be1226421838bc9fa714956ca98ab670f98aac537a73597202a949c09c4c2deb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8931,"rank":8931,"depth":7,"x":2410.473,"y":1126.562,"cluster":"local-crystalline-methods"},{"id":"stacks:0BT2","tag":"0BT2","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BT2","summary":"Let A → B be a quasi-finite map of Noetherian rings. • If A → B factors as A → A_f → B for some f ∈ A, then ω_B/A = ω_B/A_f. • If g ∈ B, then (ω_B/A)_g = ω_B_g/A. • If f ∈ A, then ω_B_f/A_f = (ω_B/A)_f.","statement_latex":"Let $A \\to B$ be a quasi-finite map of Noetherian rings.\n\\begin{enumerate}\n\\item If $A \\to B$ factors as $A \\to A_f \\to B$ for some $f \\in A$,\nthen $\\omega_{B/A} = \\omega_{B/A_f}$.\n\\item If $g \\in B$, then $(\\omega_{B/A})_g = \\omega_{B_g/A}$.\n\\item If $f \\in A$, then $\\omega_{B_f/A_f} = (\\omega_{B/A})_f$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT2","source_file":"discriminant.tex","source_line":162,"source_end_line":171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L162-L171","statement_sha256":"8a7874c02eaee8960c14d73485e3b3ed97e1dd1df4560670f3df60c533f37126","origin":"The Stacks Project","memory_eligible":false,"source_rank":8932,"rank":8932,"depth":2,"x":2542.593,"y":1110.211,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVC","tag":"0BVC","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BVC","summary":"The base change map ([Tag 0BVB]) is independent of the choice of the factorization A → B' → B. Given ring maps A → A_1 → A_2 the composition of the base change maps for A → A_1 and A_1 → A_2 is the base change map for A → A_2.","statement_latex":"The base change map (\\ref{equation-bc-dualizing})\nis independent of the choice of the\nfactorization $A \\to B' \\to B$. Given ring maps $A \\to A_1 \\to A_2$\nthe composition of the base change maps for $A \\to A_1$ and $A_1 \\to A_2$\nis the base change map for $A \\to A_2$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVC","source_file":"discriminant.tex","source_line":217,"source_end_line":224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L217-L224","statement_sha256":"aecca57f22f8a6fade7d36945d2ca8977e9d4dd8dffe3fc84507287896f057c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8933,"rank":8933,"depth":8,"x":2457.618,"y":1197.954,"cluster":"local-crystalline-methods"},{"id":"stacks:0BT3","tag":"0BT3","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BT3","summary":"If A → A_1 is flat, then the base change map ([Tag 0BVB]) induces an isomorphism ω_B/A ⊗_B B_1 → ω_B_1/A_1.","statement_latex":"If $A \\to A_1$ is flat, then\nthe base change map (\\ref{equation-bc-dualizing}) induces an isomorphism\n$\\omega_{B/A} \\otimes_B B_1 \\to \\omega_{B_1/A_1}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT3","source_file":"discriminant.tex","source_line":232,"source_end_line":237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L232-L237","statement_sha256":"85895da9c4184509a0e04f2762f3145abdeef6e046e6f2087c4cb6de037300f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":8934,"rank":8934,"depth":14,"x":2449.656,"y":1084.118,"cluster":"local-crystalline-methods"},{"id":"stacks:0BT4","tag":"0BT4","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BT4","summary":"Let A → B → C be quasi-finite homomorphisms of Noetherian rings. There is a canonical map ω_B/A ⊗_B ω_C/B → ω_C/A.","statement_latex":"Let $A \\to B \\to C$ be quasi-finite homomorphisms of Noetherian rings.\nThere is a canonical map\n$\\omega_{B/A} \\otimes_B \\omega_{C/B} \\to \\omega_{C/A}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT4","source_file":"discriminant.tex","source_line":247,"source_end_line":252,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L247-L252","statement_sha256":"467e8d721480982013000bc84cc7ed3174f4590a778a413646ca45ecdb49e5af","origin":"The Stacks Project","memory_eligible":false,"source_rank":8935,"rank":8935,"depth":0,"x":2547.849,"y":1164.179,"cluster":"local-crystalline-methods"},{"id":"stacks:0BT5","tag":"0BT5","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BT5","summary":"Let A → B and A → C be quasi-finite maps of Noetherian rings. Then ω_B × C/A = ω_B/A × ω_C/A as modules over B × C.","statement_latex":"Let $A \\to B$ and $A \\to C$ be quasi-finite maps of Noetherian rings.\nThen $\\omega_{B \\times C/A} = \\omega_{B/A} \\times \\omega_{C/A}$\nas modules over $B \\times C$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT5","source_file":"discriminant.tex","source_line":275,"source_end_line":280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L275-L280","statement_sha256":"ef68b645e41cc68e7a6d53227152b4382b5a8a14e43ab73a78e6a022d4a28d61","origin":"The Stacks Project","memory_eligible":false,"source_rank":8936,"rank":8936,"depth":0,"x":2409.981,"y":1160.831,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVD","tag":"0BVD","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BVD","summary":"Let A → B be a quasi-finite homomorphism of Noetherian rings. Then Ass_B(ω_B/A) is the set of primes of B lying over associated primes of A.","statement_latex":"Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings.\nThen $\\text{Ass}_B(\\omega_{B/A})$ is the set of primes of $B$\nlying over associated primes of $A$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVD","source_file":"discriminant.tex","source_line":291,"source_end_line":296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L291-L296","statement_sha256":"3c693bb89e076fd84c66f82b520ca2954feb1620b6971546582cef2f13dbeb95","origin":"The Stacks Project","memory_eligible":false,"source_rank":8937,"rank":8937,"depth":15,"x":2515.147,"y":1084.485,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVE","tag":"0BVE","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BVE","summary":"Let A → B be a flat quasi-finite homomorphism of Noetherian rings. Then ω_B/A is a flat A-module.","statement_latex":"Let $A \\to B$ be a flat quasi-finite homomorphism of Noetherian rings.\nThen $\\omega_{B/A}$ is a flat $A$-module.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVE","source_file":"discriminant.tex","source_line":337,"source_end_line":341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L337-L341","statement_sha256":"078217ee921930ac14adecad1ac60eb42b11be3e1f4d319abd0c0d799d563df8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8938,"rank":8938,"depth":29,"x":2498.869,"y":1201.341,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVF","tag":"0BVF","title":"Dualizing modules for quasi-finite ring maps · Lemma 0BVF","summary":"If A → B is flat, then the base change map ([Tag 0BVB]) induces an isomorphism ω_B/A ⊗_B B_1 → ω_B_1/A_1.","statement_latex":"If $A \\to B$ is flat, then the base change map (\\ref{equation-bc-dualizing})\ninduces an isomorphism $\\omega_{B/A} \\otimes_B B_1 \\to \\omega_{B_1/A_1}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVF","source_file":"discriminant.tex","source_line":372,"source_end_line":376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L372-L376","statement_sha256":"bc5692d2b03ac879692fbc1d6328dbf67962d07f5977a4d48292ce15c090415a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8939,"rank":8939,"depth":29,"x":2416.285,"y":1105.217,"cluster":"local-crystalline-methods"},{"id":"stacks:0C0I","tag":"0C0I","title":"Dualizing modules for quasi-finite ring maps · Lemma 0C0I","summary":"Let A → B be a quasi-finite homomorphism of Noetherian rings. Let ω_B/A^bullet ∈ D(B) be the algebraic relative dualizing complex discussed in Dualizing Complexes, Section [Tag 0E9M]. Then there is a (nonunique) isomorphism ω_B/A = H^0(ω_B/A^bullet).","statement_latex":"Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings.\nLet $\\omega_{B/A}^\\bullet \\in D(B)$ be the algebraic relative dualizing\ncomplex discussed in Dualizing Complexes, Section\n\\ref{dualizing-section-relative-dualizing-complexes-Noetherian}.\nThen there is a (nonunique) isomorphism\n$\\omega_{B/A} = H^0(\\omega_{B/A}^\\bullet)$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Dualizing modules for quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0I","source_file":"discriminant.tex","source_line":445,"source_end_line":453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L445-L453","statement_sha256":"ebc8bfd2a4b485e49277f883f039ac6023bfcdcf971f2ac19b86df3cf014c294","origin":"The Stacks Project","memory_eligible":false,"source_rank":8940,"rank":8940,"depth":24,"x":2555.506,"y":1129.418,"cluster":"local-crystalline-methods"},{"id":"stacks:0BJF","tag":"0BJF","title":"Discriminant of a finite locally free morphism · Lemma 0BJF","summary":"Let π : X → Y be a morphism of schemes which is finite locally free. Then π is étale if and only if its discriminant is empty.","statement_latex":"Let $\\pi : X \\to Y$ be a morphism of schemes which is finite locally\nfree. Then $\\pi$ is \\'etale if and only if its discriminant is empty.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Discriminant of a finite locally free morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJF","source_file":"discriminant.tex","source_line":538,"source_end_line":542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L538-L542","statement_sha256":"192f0524d526c360da2314ff13e3247395ad932fa82f8291cbc2e74c9b7d47cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":8941,"rank":8941,"depth":46,"x":2432.491,"y":1191.014,"cluster":"local-crystalline-methods"},{"id":"stacks:0BT6","tag":"0BT6","title":"Traces for flat quasi-finite ring maps · Definition 0BT6","summary":"Let A → B be a flat quasi-finite map of Noetherian rings. The trace element is the unique element τ_B/A ∈ ω_B/A with the following property: for any Noetherian A-algebra A_1 such that B_1 = B ⊗_A A_1 comes with a product decomposition B_1 = C × D with A_1 → C finite the image of τ_B/A in ω_C/A_1 is Trace_C/A_1. Here we use the base change map ([Tag 0BVB]) and Lemma [Tag 0BT5] to get ω_B/A → ω_B_1/A_1 → ω_C/A_1.","statement_latex":"Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.\nThe {\\it trace element} is the unique\\footnote{Uniqueness\nand existence will be justified in\nLemmas \\ref{lemma-trace-unique} and \\ref{lemma-dualizing-tau}.}\nelement\n$\\tau_{B/A} \\in \\omega_{B/A}$\nwith the following property: for any Noetherian $A$-algebra $A_1$\nsuch that $B_1 = B \\otimes_A A_1$ comes with a\nproduct decomposition $B_1 = C \\times D$ with $A_1 \\to C$ finite\nthe image of $\\tau_{B/A}$ in $\\omega_{C/A_1}$\nis $\\text{Trace}_{C/A_1}$.\nHere we use the base change map (\\ref{equation-bc-dualizing}) and\nLemma \\ref{lemma-dualizing-product} to get\n$\\omega_{B/A} \\to \\omega_{B_1/A_1} \\to \\omega_{C/A_1}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Traces for flat quasi-finite ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT6","source_file":"discriminant.tex","source_line":603,"source_end_line":619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L603-L619","statement_sha256":"7e438c813018ea6450c50d04e94bfc012570e97f7be664182eb2e2e27aefeb44","origin":"The Stacks Project","memory_eligible":false,"source_rank":8942,"rank":8942,"depth":1,"x":2473.966,"y":1074.964,"cluster":"local-crystalline-methods"},{"id":"stacks:0BT7","tag":"0BT7","title":"Traces for flat quasi-finite ring maps · Lemma 0BT7","summary":"Let A → B be a flat quasi-finite map of Noetherian rings. Then there is at most one trace element in ω_B/A.","statement_latex":"Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.\nThen there is at most one trace element in $\\omega_{B/A}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Traces for flat quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT7","source_file":"discriminant.tex","source_line":625,"source_end_line":629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L625-L629","statement_sha256":"626d064c722766300d67aef78bf24d7f01614ee996e85304e5d59b2df538d814","origin":"The Stacks Project","memory_eligible":false,"source_rank":8943,"rank":8943,"depth":29,"x":2537.15,"y":1184.846,"cluster":"local-crystalline-methods"},{"id":"stacks:0BT8","tag":"0BT8","title":"Traces for flat quasi-finite ring maps · Lemma 0BT8","summary":"Let A → B be a finite flat map of Noetherian rings. Then Trace_B/A ∈ ω_B/A is the trace element.","statement_latex":"Let $A \\to B$ be a finite flat map of Noetherian rings.\nThen $\\text{Trace}_{B/A} \\in \\omega_{B/A}$ is the trace element.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Traces for flat quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT8","source_file":"discriminant.tex","source_line":656,"source_end_line":660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L656-L660","statement_sha256":"08e1bbc4c1a5222f2939088dd3243225277d6569c637aa2ea92222844946e577","origin":"The Stacks Project","memory_eligible":false,"source_rank":8944,"rank":8944,"depth":1,"x":2401.251,"y":1139.355,"cluster":"local-crystalline-methods"},{"id":"stacks:0BT9","tag":"0BT9","title":"Traces for flat quasi-finite ring maps · Lemma 0BT9","summary":"Let A → B be a flat quasi-finite map of Noetherian rings. Let τ ∈ ω_B/A be a trace element. • If A → A_1 is a map with A_1 Noetherian, then with B_1 = A_1 ⊗_A B the image of τ in ω_B_1/A_1 is a trace element. • If A = R_f for some ring R and f ∈ R, then τ is a trace element in ω_B/R. • If g ∈ B, then the image of τ in ω_B_g/A is a trace element. • If B = B_1 × B_2, then τ maps to a trace element in both ω_B_1/A and ω_B_2/A.","statement_latex":"Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.\nLet $\\tau \\in \\omega_{B/A}$ be a trace element.\n\\begin{enumerate}\n\\item If $A \\to A_1$ is a map with $A_1$ Noetherian, then with\n$B_1 = A_1 \\otimes_A B$ the image of $\\tau$ in $\\omega_{B_1/A_1}$ is a\ntrace element.\n\\item If $A = R_f$ for some ring $R$ and $f \\in R$, then\n$\\tau$ is a trace element in $\\omega_{B/R}$.\n\\item If $g \\in B$, then the image of $\\tau$ in $\\omega_{B_g/A}$\nis a trace element.\n\\item If $B = B_1 \\times B_2$, then $\\tau$ maps to a trace element\nin both $\\omega_{B_1/A}$ and $\\omega_{B_2/A}$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Traces for flat quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BT9","source_file":"discriminant.tex","source_line":676,"source_end_line":691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L676-L691","statement_sha256":"529c45da87214f2aa0f9eadf1a743e041fa455790aedb7df00ddc6ea79b06f29","origin":"The Stacks Project","memory_eligible":false,"source_rank":8945,"rank":8945,"depth":3,"x":2538.989,"y":1095.489,"cluster":"local-crystalline-methods"},{"id":"stacks:0BTA","tag":"0BTA","title":"Traces for flat quasi-finite ring maps · Lemma 0BTA","summary":"Let A → B be a flat quasi-finite map of Noetherian rings. Let g_1, …, g_m ∈ B be elements generating the unit ideal. Let τ ∈ ω_B/A be an element whose image in ω_B_g_i/A is a trace element for A → B_g_i. Then τ is a trace element.","statement_latex":"Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.\nLet $g_1, \\ldots, g_m \\in B$ be elements generating the unit ideal.\nLet $\\tau \\in \\omega_{B/A}$ be an element whose image in\n$\\omega_{B_{g_i}/A}$ is a trace element for $A \\to B_{g_i}$.\nThen $\\tau$ is a trace element.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Traces for flat quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTA","source_file":"discriminant.tex","source_line":737,"source_end_line":744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L737-L744","statement_sha256":"3a5b2b06d1f7c3e645fd9736d96bf0936c4943a96c16acb4da9deb16c7ffe2cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8946,"rank":8946,"depth":30,"x":2472.245,"y":1206.741,"cluster":"local-crystalline-methods"},{"id":"stacks:0BTB","tag":"0BTB","title":"Traces for flat quasi-finite ring maps · Lemma 0BTB","summary":"Let A → B be a flat quasi-finite map of Noetherian rings. There exists a trace element τ ∈ ω_B/A.","statement_latex":"Let $A \\to B$ be a flat quasi-finite map of Noetherian rings.\nThere exists a trace element $\\tau \\in \\omega_{B/A}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Traces for flat quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTB","source_file":"discriminant.tex","source_line":780,"source_end_line":784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L780-L784","statement_sha256":"c8cf1f4d5d26c573b6c750b8fffd26d498150bb33a967a1cfdd10551f9ec81ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":8947,"rank":8947,"depth":31,"x":2431.727,"y":1086.028,"cluster":"local-crystalline-methods"},{"id":"stacks:0C13","tag":"0C13","title":"Traces for flat quasi-finite ring maps · Lemma 0C13","summary":"Let k be a field and let A be a finite k-algebra. Assume A is local with residue field k'. The following are equivalent • Trace_A/k is nonzero, • τ_A/k ∈ ω_A/k is nonzero, and • k'/k is separable and length_A(A) is prime to the characteristic of k.","statement_latex":"Let $k$ be a field and let $A$ be a finite $k$-algebra. Assume $A$\nis local with residue field $k'$. The following are equivalent\n\\begin{enumerate}\n\\item $\\text{Trace}_{A/k}$ is nonzero,\n\\item $\\tau_{A/k} \\in \\omega_{A/k}$ is nonzero, and\n\\item $k'/k$ is separable and $\\text{length}_A(A)$ is prime\nto the characteristic of $k$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Traces for flat quasi-finite ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C13","source_file":"discriminant.tex","source_line":909,"source_end_line":919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L909-L919","statement_sha256":"74702f946a4408e773832d7677976da8324a06155b1c1d69e785f35e88c1a9fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":8948,"rank":8948,"depth":6,"x":2559.518,"y":1152.488,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVL","tag":"0BVL","title":"The Noether different · Lemma 0BVL","summary":"Let A → B_i, i = 1, 2 be ring maps. Set B = B_1 × B_2. • The annihilator J of Ker(B ⊗_A B → B) is J_1 × J_2 where J_i is the annihilator of Ker(B_i ⊗_A B_i → B_i). • The Noether different D of B over A is D_1 × D_2, where D_i is the Noether different of B_i over A.","statement_latex":"Let $A \\to B_i$, $i = 1, 2$ be ring maps. Set $B = B_1 \\times B_2$.\n\\begin{enumerate}\n\\item The annihilator $J$ of $\\Ker(B \\otimes_A B \\to B)$ is $J_1 \\times J_2$\nwhere $J_i$ is the annihilator of $\\Ker(B_i \\otimes_A B_i \\to B_i)$.\n\\item The Noether different $\\mathfrak{D}$ of $B$ over $A$ is\n$\\mathfrak{D}_1 \\times \\mathfrak{D}_2$, where $\\mathfrak{D}_i$ is\nthe Noether different of $B_i$ over $A$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Noether different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVL","source_file":"discriminant.tex","source_line":1077,"source_end_line":1087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1077-L1087","statement_sha256":"090480d56118a091ecfa94cf77de77c24669d18f8cc2159c1e765280e8e89c13","origin":"The Stacks Project","memory_eligible":false,"source_rank":8949,"rank":8949,"depth":0,"x":2410.876,"y":1176.145,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVM","tag":"0BVM","title":"The Noether different · Lemma 0BVM","summary":"Let A → B be a finite type ring map. Let A → A' be a flat ring map. Set B' = B ⊗_A A'. • The annihilator J' of Ker(B' ⊗_A' B' → B') is J ⊗_A A' where J is the annihilator of Ker(B ⊗_A B → B). • The Noether different D' of B' over A' is DB', where D is the Noether different of B over A.","statement_latex":"Let $A \\to B$ be a finite type ring map. Let $A \\to A'$ be a flat ring map.\nSet $B' = B \\otimes_A A'$.\n\\begin{enumerate}\n\\item The annihilator $J'$ of $\\Ker(B' \\otimes_{A'} B' \\to B')$ is\n$J \\otimes_A A'$ where $J$ is the annihilator of $\\Ker(B \\otimes_A B \\to B)$.\n\\item The Noether different $\\mathfrak{D}'$ of $B'$ over $A'$ is\n$\\mathfrak{D}B'$, where $\\mathfrak{D}$ is\nthe Noether different of $B$ over $A$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Noether different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVM","source_file":"discriminant.tex","source_line":1093,"source_end_line":1104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1093-L1104","statement_sha256":"61529a3cdb0ad897b118724aefa03cb0ca8a08dc70972107ed7105599ad28c9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8950,"rank":8950,"depth":0,"x":2502.045,"y":1073.702,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVN","tag":"0BVN","title":"The Noether different · Lemma 0BVN","summary":"Let A → B' → B be ring maps with A → B' of finite type and B' → B inducing an open immersion of spectra. • The annihilator J of Ker(B ⊗_A B → B) is J' ⊗_B' B where J' is the annihilator of Ker(B' ⊗_A B' → B'). • The Noether different D of B over A is D'B, where D' is the Noether different of B' over A.","statement_latex":"Let $A \\to B' \\to B$ be ring maps with $A \\to B'$\nof finite type and $B' \\to B$ inducing an open immersion of spectra.\n\\begin{enumerate}\n\\item The annihilator $J$ of $\\Ker(B \\otimes_A B \\to B)$ is\n$J' \\otimes_{B'} B$ where $J'$ is the annihilator of\n$\\Ker(B' \\otimes_A B' \\to B')$.\n\\item The Noether different $\\mathfrak{D}$ of $B$ over $A$ is\n$\\mathfrak{D}'B$, where $\\mathfrak{D}'$ is\nthe Noether different of $B'$ over $A$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Noether different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVN","source_file":"discriminant.tex","source_line":1117,"source_end_line":1129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1117-L1129","statement_sha256":"13b16b58aa278361ef7df484170bf92125d1111720c3abec305a894ad38772d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":8951,"rank":8951,"depth":2,"x":2517.294,"y":1201.784,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVR","tag":"0BVR","title":"The Noether different · Lemma 0BVR","summary":"Let A → B be a quasi-finite homomorphism of Noetherian rings. • If A → A' is a flat map of Noetherian rings, then xymatrix ω_B/A × J ar[r] ar[d] & B ar[d] ω_B'/A' × J' ar[r] & B' is commutative where notation as in Lemma [Tag 0BVM] and horizontal arrows are given by ([Tag 0BVQ]). • If B = B_1 × B_2, then xymatrix ω_B/A × J ar[r] ar[d] & B ar[d] ω_B_i/A × J_i ar[r] & B_i is commutative for i = 1, 2 where notation as in Lemma [Tag 0BVL] and horizontal arrows are given by…","statement_latex":"Let $A \\to B$ be a quasi-finite homomorphism of Noetherian rings.\n\\begin{enumerate}\n\\item If $A \\to A'$ is a flat map of Noetherian rings, then\n$$\n\\xymatrix{\n\\omega_{B/A} \\times J \\ar[r] \\ar[d] & B \\ar[d]  \\\\\n\\omega_{B'/A'} \\times J' \\ar[r] & B'\n}\n$$\nis commutative where notation as in\nLemma \\ref{lemma-noether-different-base-change}\nand horizontal arrows are given by\n(\\ref{equation-pairing-noether}).\n\\item If $B = B_1 \\times B_2$, then\n$$\n\\xymatrix{\n\\omega_{B/A} \\times J \\ar[r] \\ar[d] & B \\ar[d]  \\\\\n\\omega_{B_i/A} \\times J_i \\ar[r] & B_i\n}\n$$\nis commutative for $i = 1, 2$ where notation as in\nLemma \\ref{lemma-noether-different-product}\nand horizontal arrows are given by\n(\\ref{equation-pairing-noether}).\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Noether different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVR","source_file":"discriminant.tex","source_line":1168,"source_end_line":1195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1168-L1195","statement_sha256":"4af443f10fa42b66287a19467fd1d76acab3667978c4ed492c4829fb78fdd5fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":8952,"rank":8952,"depth":1,"x":2402.333,"y":1115.45,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVS","tag":"0BVS","title":"The Noether different · Lemma 0BVS","summary":"Let A → B be a flat quasi-finite homomorphism of Noetherian rings. The pairing of Remark [Tag 0BVP] induces an isomorphism J → Hom_B(ω_B/A, B).","statement_latex":"Let $A \\to B$ be a flat quasi-finite homomorphism of Noetherian rings.\nThe pairing of Remark \\ref{remark-construction-pairing} induces an isomorphism\n$J \\to \\Hom_B(\\omega_{B/A}, B)$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Noether different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVS","source_file":"discriminant.tex","source_line":1210,"source_end_line":1215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1210-L1215","statement_sha256":"913adc3e655009b7ede1caffb1aee556199f8a536453bb948e56c3f17bf4741c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8953,"rank":8953,"depth":29,"x":2557.488,"y":1113.872,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVT","tag":"0BVT","title":"The Noether different · Lemma 0BVT","summary":"Let A → B be a flat quasi-finite homomorphism of Noetherian rings. The diagram xymatrix J ar[rr] ar[rd]_μ & & Hom_B(ω_B/A, B) ar[ld]^φ ↦ φ(τ_B/A) & B commutes where the horizontal arrow is the isomorphism of Lemma [Tag 0BVS]. Hence the Noether different of B over A is the image of the map Hom_B(ω_B/A, B) → B.","statement_latex":"Let $A \\to B$ be a flat quasi-finite homomorphism of Noetherian rings.\nThe diagram\n$$\n\\xymatrix{\nJ \\ar[rr] \\ar[rd]_\\mu & &\n\\Hom_B(\\omega_{B/A}, B) \\ar[ld]^{\\varphi \\mapsto \\varphi(\\tau_{B/A})} \\\\\n& B\n}\n$$\ncommutes where the horizontal arrow is the isomorphism of\nLemma \\ref{lemma-noether-pairing-flat-quasi-finite}.\nHence the Noether different of $B$ over $A$\nis the image of the map $\\Hom_B(\\omega_{B/A}, B) \\to B$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Noether different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVT","source_file":"discriminant.tex","source_line":1267,"source_end_line":1282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1267-L1282","statement_sha256":"d963848396d15e3817c476d3d9695581979c14244febc38a39dbdc6d5f4fb4f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":8954,"rank":8954,"depth":30,"x":2443.653,"y":1203.62,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVU","tag":"0BVU","title":"The Noether different · Lemma 0BVU","summary":"Let A → B be a finite type ring map. Let D ⊂ B be the Noether different. Then V(D) is the set of primes q ⊂ B such that A → B is not unramified at q.","statement_latex":"Let $A \\to B$ be a finite type ring map. Let $\\mathfrak{D} \\subset B$\nbe the Noether different. Then $V(\\mathfrak{D})$ is the set of primes\n$\\mathfrak q \\subset B$ such that $A \\to B$ is not unramified at $\\mathfrak q$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Noether different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVU","source_file":"discriminant.tex","source_line":1307,"source_end_line":1312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1307-L1312","statement_sha256":"ad994276035841307035f89a4ec058ea02defdad6648f8604fdb6828ac6a3e06","origin":"The Stacks Project","memory_eligible":false,"source_rank":8955,"rank":8955,"depth":4,"x":2455.494,"y":1072.056,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVW","tag":"0BVW","title":"The K\\\"ahler different · Definition 0BVW","summary":"Let f : Y → X be a morphism of schemes which is locally of finite type. The K\\\"ahler different is the 0th fitting ideal of Ω_Y/X.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes which is locally of finite type.\nThe {\\it K\\\"ahler different} is the $0$th fitting ideal of $\\Omega_{Y/X}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The K\\\"ahler different","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVW","source_file":"discriminant.tex","source_line":1352,"source_end_line":1356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1352-L1356","statement_sha256":"95c4e834e2015790e4f3518b230aa9b530e02645386a23daa2c27bc747a39b73","origin":"The Stacks Project","memory_eligible":false,"source_rank":8956,"rank":8956,"depth":0,"x":2553.141,"y":1176.41,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVX","tag":"0BVX","title":"The K\\\"ahler different · Lemma 0BVX","summary":"Consider a cartesian diagram of schemes xymatrix Y' ar[d]_f' ar[r] & Y ar[d]^f X' ar[r]^g & X with f locally of finite type. Let R ⊂ Y, resp. R' ⊂ Y' be the closed subscheme cut out by the K\\\"ahler different of f, resp. f'. Then Y' → Y induces an isomorphism R' → R ×_Y Y'.","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nY' \\ar[d]_{f'} \\ar[r] & Y \\ar[d]^f \\\\\nX' \\ar[r]^g & X\n}\n$$\nwith $f$ locally of finite type. Let $R \\subset Y$, resp.\\ $R' \\subset Y'$\nbe the closed subscheme cut out by the K\\\"ahler different of $f$, resp.\\ $f'$.\nThen $Y' \\to Y$ induces an isomorphism $R' \\to R \\times_Y Y'$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The K\\\"ahler different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVX","source_file":"discriminant.tex","source_line":1361,"source_end_line":1373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1361-L1373","statement_sha256":"a3ba84c8713b1b6851750ccaa1069f068584c4197193b7c5420a0a63d386344c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8957,"rank":8957,"depth":17,"x":2396.297,"y":1154.74,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVY","tag":"0BVY","title":"The K\\\"ahler different · Lemma 0BVY","summary":"Let f : Y → X be a morphism of schemes which is locally of finite type. Let R ⊂ Y be the closed subscheme defined by the K\\\"ahler different. Then R ⊂ Y is exactly the set of points where f is not unramified.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes which is locally of finite type.\nLet $R \\subset Y$ be the closed subscheme defined by\nthe K\\\"ahler different. Then $R \\subset Y$ is exactly\nthe set of points where $f$ is not unramified.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The K\\\"ahler different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVY","source_file":"discriminant.tex","source_line":1382,"source_end_line":1388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1382-L1388","statement_sha256":"0aaffa579dc08d7329ee9ee3b9ef995a267606c4e22e189f8897ac872eccf516","origin":"The Stacks Project","memory_eligible":false,"source_rank":8958,"rank":8958,"depth":21,"x":2530.157,"y":1081.297,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVZ","tag":"0BVZ","title":"The K\\\"ahler different · Lemma 0BVZ","summary":"Let A be a ring. Let n ≥ 1 and f_1, …, f_n ∈ A[x_1, …, x_n]. Set B = A[x_1, …, x_n]/(f_1, …, f_n). The K\\\"ahler different of B over A is the ideal of B generated by det(∂ f_i/∂ x_j).","statement_latex":"Let $A$ be a ring. Let $n \\geq 1$ and\n$f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$.\nSet $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$.\nThe K\\\"ahler different of $B$ over $A$ is the ideal\nof $B$ generated by $\\det(\\partial f_i/\\partial x_j)$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The K\\\"ahler different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVZ","source_file":"discriminant.tex","source_line":1395,"source_end_line":1402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1395-L1402","statement_sha256":"cf92c7869aaa08c815c24dbe0d16881649c73613c2fab0a74c7ff72007f14f3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8959,"rank":8959,"depth":2,"x":2490.281,"y":1212.159,"cluster":"local-crystalline-methods"},{"id":"stacks:0BW1","tag":"0BW1","title":"The Dedekind different · Lemma 0BW1","summary":"Assume the Dedekind different of A → B is defined. Consider the statements • A → B is flat, • A is a normal ring, • Trace_L/K(B) ⊂ A, • 1 ∈ L_B/A, and • the Dedekind different D_B/A is an ideal of B. Then we have (1) ⇒ (3), (2) ⇒ (3), (3) ⇔ (4), and (4) ⇒ (5).","statement_latex":"Assume the Dedekind different of $A \\to B$ is defined. Consider the statements\n\\begin{enumerate}\n\\item $A \\to B$ is flat,\n\\item $A$ is a normal ring,\n\\item $\\text{Trace}_{L/K}(B) \\subset A$,\n\\item $1 \\in \\mathcal{L}_{B/A}$, and\n\\item the Dedekind different $\\mathfrak{D}_{B/A}$ is an ideal of $B$.\n\\end{enumerate}\nThen we have (1) $\\Rightarrow$ (3), (2) $\\Rightarrow$ (3),\n(3) $\\Leftrightarrow$ (4), and (4) $\\Rightarrow$ (5).","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Dedekind different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BW1","source_file":"discriminant.tex","source_line":1439,"source_end_line":1451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1439-L1451","statement_sha256":"6fcc1ca7be6a8b38a3977f179a1ad67a5433e83baff912a109e33ffa1fda32d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8960,"rank":8960,"depth":8,"x":2414.018,"y":1092.355,"cluster":"local-crystalline-methods"},{"id":"stacks:0BW2","tag":"0BW2","title":"The Dedekind different · Lemma 0BW2","summary":"If the Dedekind different of A → B is defined, then there is a canonical isomorphism L_B/A → ω_B/A.","statement_latex":"If the Dedekind different of $A \\to B$ is defined, then\nthere is a canonical isomorphism\n$\\mathcal{L}_{B/A} \\to \\omega_{B/A}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Dedekind different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BW2","source_file":"discriminant.tex","source_line":1477,"source_end_line":1482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1477-L1482","statement_sha256":"94affcecf91b7201a875a3ae899f1f03a8d241c43d55b78110ac66cde9bd8a05","origin":"The Stacks Project","memory_eligible":false,"source_rank":8961,"rank":8961,"depth":47,"x":2567.456,"y":1137.68,"cluster":"local-crystalline-methods"},{"id":"stacks:0BW3","tag":"0BW3","title":"The Dedekind different · Lemma 0BW3","summary":"If the Dedekind different of A → B is defined and A → B is flat, then • the canonical isomorphism L_B/A → ω_B/A sends 1 ∈ L_B/A to the trace element τ_B/A ∈ ω_B/A, and • the Dedekind different is D_B/A = (b ∈ B mid bω_B/A ⊂ Bτ_B/A).","statement_latex":"If the Dedekind different of $A \\to B$ is defined and $A \\to B$ is flat, then\n\\begin{enumerate}\n\\item the canonical isomorphism $\\mathcal{L}_{B/A} \\to \\omega_{B/A}$\nsends $1 \\in \\mathcal{L}_{B/A}$ to the trace element\n$\\tau_{B/A} \\in \\omega_{B/A}$, and\n\\item the Dedekind different is\n$\\mathfrak{D}_{B/A} = \\{b \\in B \\mid b\\omega_{B/A} \\subset B\\tau_{B/A}\\}$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Dedekind different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BW3","source_file":"discriminant.tex","source_line":1496,"source_end_line":1506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1496-L1506","statement_sha256":"edf3084e41f5d6d9afeffcc410671e8f804459e3bf780960c0e6c91dfb0a816a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8962,"rank":8962,"depth":9,"x":2417.029,"y":1191.621,"cluster":"local-crystalline-methods"},{"id":"stacks:0BW4","tag":"0BW4","title":"The different · Definition 0BW4","summary":"Let f : Y → X be a flat locally quasi-finite morphism of locally Noetherian schemes. Let ω_Y/X be the relative dualizing module and let τ_Y/X ∈ Γ(Y, ω_Y/X) be the trace element (Remarks [Tag 0BVG] and [Tag 0BVJ]). The annihilator of Coker(O_Y xrightarrowτ_Y/X ω_Y/X) is the different of Y/X. It is a coherent ideal D_f ⊂ O_Y.","statement_latex":"Let $f : Y \\to X$ be a flat locally quasi-finite morphism of\nlocally Noetherian schemes.\nLet $\\omega_{Y/X}$ be the relative dualizing module and let\n$\\tau_{Y/X} \\in \\Gamma(Y, \\omega_{Y/X})$ be the trace element\n(Remarks \\ref{remark-relative-dualizing-for-quasi-finite} and\n\\ref{remark-relative-dualizing-for-flat-quasi-finite}).\nThe annihilator of\n$$\n\\Coker(\\mathcal{O}_Y \\xrightarrow{\\tau_{Y/X}} \\omega_{Y/X})\n$$\nis the {\\it different} of $Y/X$. It is a coherent ideal\n$\\mathfrak{D}_f \\subset \\mathcal{O}_Y$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The different","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BW4","source_file":"discriminant.tex","source_line":1525,"source_end_line":1539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1525-L1539","statement_sha256":"37924f80fdd2e96e99618df1b7c5e62609ff30c370d349ab7cf3522210cb7f46","origin":"The Stacks Project","memory_eligible":false,"source_rank":8963,"rank":8963,"depth":0,"x":2484.949,"y":1065.8,"cluster":"local-crystalline-methods"},{"id":"stacks:0BW5","tag":"0BW5","title":"The different · Lemma 0BW5","summary":"Let f : Y → X be a flat quasi-finite morphism of Noetherian schemes. Let V = Spec(B) ⊂ Y, U = Spec(A) ⊂ X be affine open subschemes with f(V) ⊂ U. If the Dedekind different of A → B is defined, then D_f|_V = widetildeD_B/A as coherent ideal sheaves on V.","statement_latex":"Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes.\nLet $V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$\nbe affine open subschemes with $f(V) \\subset U$.\nIf the Dedekind different of $A \\to B$ is defined, then\n$$\n\\mathfrak{D}_f|_V = \\widetilde{\\mathfrak{D}_{B/A}}\n$$\nas coherent ideal sheaves on $V$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BW5","source_file":"discriminant.tex","source_line":1547,"source_end_line":1557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1547-L1557","statement_sha256":"b97082384fe31065448f99ce01573a1c1bf87bf22786a5db8120ca24f78db116","origin":"The Stacks Project","memory_eligible":false,"source_rank":8964,"rank":8964,"depth":10,"x":2536.325,"y":1197.834,"cluster":"local-crystalline-methods"},{"id":"stacks:0BW6","tag":"0BW6","title":"The different · Lemma 0BW6","summary":"Let f : Y → X be a flat quasi-finite morphism of Noetherian schemes. Let V = Spec(B) ⊂ Y, U = Spec(A) ⊂ X be affine open subschemes with f(V) ⊂ U. If ω_Y/X|_V is invertible, i.e., if ω_B/A is an invertible B-module, then D_f|_V = widetildeD as coherent ideal sheaves on V where D ⊂ B is the Noether different of B over A.","statement_latex":"Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes.\nLet $V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$\nbe affine open subschemes with $f(V) \\subset U$.\nIf $\\omega_{Y/X}|_V$ is invertible, i.e., if $\\omega_{B/A}$\nis an invertible $B$-module, then\n$$\n\\mathfrak{D}_f|_V = \\widetilde{\\mathfrak{D}}\n$$\nas coherent ideal sheaves on $V$ where\n$\\mathfrak{D} \\subset B$ is the Noether different of $B$ over $A$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BW6","source_file":"discriminant.tex","source_line":1564,"source_end_line":1576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1564-L1576","statement_sha256":"58ef48bc68f9005f116ff2f240b95ff91ae969765b3f6232d91f8a52e649e0c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8965,"rank":8965,"depth":31,"x":2391.485,"y":1129.256,"cluster":"local-crystalline-methods"},{"id":"stacks:0BW7","tag":"0BW7","title":"The different · Lemma 0BW7","summary":"Consider a cartesian diagram of Noetherian schemes xymatrix Y' ar[d]_f' ar[r] & Y ar[d]^f X' ar[r]^g & X with f flat and quasi-finite. Let R ⊂ Y, resp. R' ⊂ Y' be the closed subscheme cut out by the different D_f, resp. D_f'. Then Y' → Y induces a bijective closed immersion R' → R ×_Y Y'. If g is flat or if ω_Y/X is invertible, then R' = R ×_Y Y'.","statement_latex":"Consider a cartesian diagram of Noetherian schemes\n$$\n\\xymatrix{\nY' \\ar[d]_{f'} \\ar[r] & Y \\ar[d]^f \\\\\nX' \\ar[r]^g & X\n}\n$$\nwith $f$ flat and quasi-finite. Let $R \\subset Y$, resp.\\ $R' \\subset Y'$\nbe the closed subscheme cut out by the different\n$\\mathfrak{D}_f$, resp.\\ $\\mathfrak{D}_{f'}$.\nThen $Y' \\to Y$ induces a bijective closed immersion $R' \\to R \\times_Y Y'$.\nIf $g$ is flat or if $\\omega_{Y/X}$ is invertible, then\n$R' = R \\times_Y Y'$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BW7","source_file":"discriminant.tex","source_line":1595,"source_end_line":1610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1595-L1610","statement_sha256":"ff0a758949af13029af8240798437148e5ace5e8d98a487c4b74cddfb0f27dd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8966,"rank":8966,"depth":30,"x":2554.302,"y":1097.472,"cluster":"local-crystalline-methods"},{"id":"stacks:0BW8","tag":"0BW8","title":"The different · Lemma 0BW8","summary":"Let f : Y → X be a finite flat morphism of Noetherian schemes. Then Norm_f : f_*O_Y → O_X maps f_*D_f into the ideal sheaf of the discriminant D_f.","statement_latex":"Let $f : Y \\to X$ be a finite flat morphism of Noetherian schemes.\nThen $\\text{Norm}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$ maps\n$f_*\\mathfrak{D}_f$ into the ideal sheaf of the discriminant $D_f$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BW8","source_file":"discriminant.tex","source_line":1645,"source_end_line":1650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1645-L1650","statement_sha256":"e967a1b3c2162cea5b6be0c2e45634adb36bc4cadfa972cb1629aa472db3aea7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8967,"rank":8967,"depth":1,"x":2459.303,"y":1213.906,"cluster":"local-crystalline-methods"},{"id":"stacks:0BW9","tag":"0BW9","title":"The different · Lemma 0BW9","summary":"Let f : Y → X be a flat quasi-finite morphism of Noetherian schemes. The closed subscheme R ⊂ Y defined by the different D_f is exactly the set of points where f is not étale (equivalently not unramified).","statement_latex":"Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes.\nThe closed subscheme $R \\subset Y$ defined by the different $\\mathfrak{D}_f$\nis exactly the set of points where $f$ is not \\'etale\n(equivalently not unramified).","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BW9","source_file":"discriminant.tex","source_line":1689,"source_end_line":1695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1689-L1695","statement_sha256":"b210a484f81bdfa1f379c805738e9f14548bb68155e5158c713a4695da097b88","origin":"The Stacks Project","memory_eligible":false,"source_rank":8968,"rank":8968,"depth":47,"x":2435.599,"y":1073.414,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWA","tag":"0BWA","title":"The different · Lemma 0BWA","summary":"Let f : Y → X be a flat quasi-finite morphism of Noetherian schemes. Let R ⊂ Y be the closed subscheme defined by D_f. • If ω_Y/X is invertible, then R is a locally principal closed subscheme of Y. • If ω_Y/X is invertible and f is finite, then the norm of R is the discriminant D_f of f. • If ω_Y/X is invertible and f is étale at the associated points of Y, then R is an effective Cartier divisor and there is an isomorphism O_Y(R) = ω_Y/X.","statement_latex":"Let $f : Y \\to X$ be a flat quasi-finite morphism of Noetherian schemes.\nLet $R \\subset Y$ be the closed subscheme defined by $\\mathfrak{D}_f$.\n\\begin{enumerate}\n\\item If $\\omega_{Y/X}$ is invertible,\nthen $R$ is a locally principal closed subscheme of $Y$.\n\\item If $\\omega_{Y/X}$ is invertible and $f$ is finite, then\nthe norm of $R$ is the discriminant $D_f$ of $f$.\n\\item If $\\omega_{Y/X}$ is invertible and $f$\nis \\'etale at the associated points of $Y$, then $R$\nis an effective Cartier divisor and there is an\nisomorphism $\\mathcal{O}_Y(R) = \\omega_{Y/X}$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWA","source_file":"discriminant.tex","source_line":1724,"source_end_line":1738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1724-L1738","statement_sha256":"b8c810ba014acb9c7e997c46501d7cec4a41bcfb63d14b914149a7c2363d17a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":8969,"rank":8969,"depth":48,"x":2566.732,"y":1164.028,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWE","tag":"0BWE","title":"Quasi-finite syntomic morphisms · Lemma 0BWE","summary":"Let f : Y → X be a morphism of schemes. The following are equivalent • f is locally quasi-finite and syntomic, • f is locally quasi-finite, flat, and a local complete intersection morphism, • f is locally quasi-finite, flat, locally of finite presentation, and the fibres of f are local complete intersections, • f is locally quasi-finite and for every y ∈ Y there are affine opens y ∈ V = Spec(B) ⊂ Y, U = Spec(A) ⊂ X with f(V) ⊂ U an integer n and h, f_1, …, f_n ∈ A[x_1, …,…","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite and syntomic,\n\\item $f$ is locally quasi-finite, flat, and a local complete intersection\nmorphism,\n\\item $f$ is locally quasi-finite, flat, locally of finite presentation,\nand the fibres of $f$ are local complete intersections,\n\\item $f$ is locally quasi-finite and for every $y \\in Y$ there are\naffine opens $y \\in V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$\nwith $f(V) \\subset U$ an integer $n$ and\n$h, f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$ such that\n$B = A[x_1, \\ldots, x_n, 1/h]/(f_1, \\ldots, f_n)$,\n\\item for every $y \\in Y$ there are affine opens\n$y \\in V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$\nwith $f(V) \\subset U$ such that $A \\to B$ is a relative global complete\nintersection of the form $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$,\n\\item $f$ is locally quasi-finite, flat, locally of finite presentation,\nand $\\NL_{Y/X}$ has tor-amplitude in $[-1, 0]$, and\n\\item $f$ is flat, locally of finite presentation,\n$\\NL_{Y/X}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$,\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Quasi-finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWE","source_file":"discriminant.tex","source_line":1787,"source_end_line":1810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1787-L1810","statement_sha256":"6a581cbba1d6b9b66fb202fe21b01ca6342734b3482caee6987a76e3d045e412","origin":"The Stacks Project","memory_eligible":false,"source_rank":8970,"rank":8970,"depth":39,"x":2396.298,"y":1171.656,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWK","tag":"0DWK","title":"Quasi-finite syntomic morphisms · Lemma 0DWK","summary":"Invertibility of the relative dualizing module. • If A → B is a quasi-finite flat homomorphism of Noetherian rings, then ω_B/A is an invertible B-module if and only if ω_B ⊗_A kappa( p)/kappa( p) is an invertible B ⊗_A kappa( p)-module for all primes p ⊂ A. • If Y → X is a quasi-finite flat morphism of Noetherian schemes, then ω_Y/X is invertible if and only if ω_Y_x/x is invertible for all x ∈ X.","statement_latex":"Invertibility of the relative dualizing module.\n\\begin{enumerate}\n\\item If $A \\to B$ is a quasi-finite flat homomorphism of Noetherian rings,\nthen $\\omega_{B/A}$ is an invertible $B$-module if and only if\n$\\omega_{B \\otimes_A \\kappa(\\mathfrak p)/\\kappa(\\mathfrak p)}$\nis an invertible $B \\otimes_A \\kappa(\\mathfrak p)$-module\nfor all primes $\\mathfrak p \\subset A$.\n\\item If $Y \\to X$ is a quasi-finite flat morphism of\nNoetherian schemes, then $\\omega_{Y/X}$ is invertible\nif and only if $\\omega_{Y_x/x}$ is invertible for all $x \\in X$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Quasi-finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWK","source_file":"discriminant.tex","source_line":1874,"source_end_line":1887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1874-L1887","statement_sha256":"6184b7d36a2ed01e8774985d9ad9eab0323acf35c422773ba840b0bb76dc049f","origin":"The Stacks Project","memory_eligible":false,"source_rank":8971,"rank":8971,"depth":37,"x":2516.446,"y":1068.806,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWL","tag":"0DWL","title":"Quasi-finite syntomic morphisms · Lemma 0DWL","summary":"Let k be a field. Let B = k[x_1, …, x_n]/(f_1, …, f_n) be a global complete intersection over k of dimension 0. Then ω_B/k is invertible.","statement_latex":"Let $k$ be a field. Let $B = k[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$\nbe a global complete intersection over $k$ of dimension $0$.\nThen $\\omega_{B/k}$ is invertible.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Quasi-finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWL","source_file":"discriminant.tex","source_line":1911,"source_end_line":1916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1911-L1916","statement_sha256":"5746c47cb8a6fe3f694eb207c3a9c924b082415715ca8f14e16f68fd71aec736","origin":"The Stacks Project","memory_eligible":false,"source_rank":8972,"rank":8972,"depth":39,"x":2510.529,"y":1213.543,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWF","tag":"0BWF","title":"Quasi-finite syntomic morphisms · Lemma 0BWF","summary":"Let f : Y → X be a morphism of locally Noetherian schemes. If f satisfies the equivalent conditions of Lemma [Tag 0BWE] then ω_Y/X is an invertible O_Y-module.","statement_latex":"Let $f : Y \\to X$ be a morphism of locally Noetherian schemes. If $f$\nsatisfies the equivalent conditions of Lemma \\ref{lemma-syntomic-quasi-finite}\nthen $\\omega_{Y/X}$ is an invertible $\\mathcal{O}_Y$-module.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Quasi-finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWF","source_file":"discriminant.tex","source_line":1935,"source_end_line":1940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1935-L1940","statement_sha256":"1757dd7d58890507d99b580b2894fd9e55c2d4f2d5e45df6ca60139b8fa536b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8973,"rank":8973,"depth":40,"x":2397.944,"y":1102.911,"cluster":"local-crystalline-methods"},{"id":"stacks:0FK9","tag":"0FK9","title":"Quasi-finite syntomic morphisms · Lemma 0FK9","summary":"With notation as in Example [Tag 0FK8] the schemes X_n, d and Y_n, d are regular and irreducible, the morphism Y_n, d → X_n, d is locally quasi-finite and syntomic, and there is a dense open subscheme V ⊂ Y_n, d such that Y_n, d → X_n, d restricts to an étale morphism V → X_n, d.","statement_latex":"With notation as in Example \\ref{example-universal-quasi-finite-syntomic}\nthe schemes $X_{n, d}$ and $Y_{n, d}$ are regular and irreducible,\nthe morphism $Y_{n, d} \\to X_{n, d}$ is locally quasi-finite and\nsyntomic, and there is a dense open subscheme $V \\subset Y_{n, d}$\nsuch that $Y_{n, d} \\to X_{n, d}$ restricts to an \\'etale morphism\n$V \\to X_{n, d}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Quasi-finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FK9","source_file":"discriminant.tex","source_line":1970,"source_end_line":1978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L1970-L1978","statement_sha256":"d1c2343f51cea762c2b778626fe9b0b879faecfbcdd50910022c0d719351d7eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8974,"rank":8974,"depth":45,"x":2570.775,"y":1120.694,"cluster":"local-crystalline-methods"},{"id":"stacks:0FKA","tag":"0FKA","title":"Quasi-finite syntomic morphisms · Lemma 0FKA","summary":"Let f : Y → X be a morphism of schemes. If f satisfies the equivalent conditions of Lemma [Tag 0BWE] then for every y ∈ Y there exist n, d and a commutative diagram xymatrix Y ar[d] & V ar[d] ar[l] ar[r] & Y_n, d ar[d] X & U ar[l] ar[r] & X_n, d where U ⊂ X and V ⊂ Y are open with y ∈ V, where Y_n, d → X_n, d is as in Example [Tag 0FK8], and where the square on the right hand side is cartesian.","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes. If $f$ satisfies the equivalent\nconditions of Lemma \\ref{lemma-syntomic-quasi-finite} then for every\n$y \\in Y$ there exist $n, d$ and a commutative diagram\n$$\n\\xymatrix{\nY \\ar[d] &\nV \\ar[d] \\ar[l] \\ar[r] &\nY_{n, d} \\ar[d] \\\\\nX & U \\ar[l] \\ar[r] &\nX_{n, d}\n}\n$$\nwhere $U \\subset X$ and $V \\subset Y$ are open with $y \\in V$,\nwhere $Y_{n, d} \\to X_{n, d}$\nis as in Example \\ref{example-universal-quasi-finite-syntomic}, and\nwhere the square on the right hand side is cartesian.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Quasi-finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKA","source_file":"discriminant.tex","source_line":2007,"source_end_line":2025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2007-L2025","statement_sha256":"7e88141776fa6b4340de5527d0e74923e0693fca01b41bf84906e3f1ec2b2ab1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8975,"rank":8975,"depth":40,"x":2428.34,"y":1206.062,"cluster":"local-crystalline-methods"},{"id":"stacks:0FKY","tag":"0FKY","title":"Finite syntomic morphisms · Lemma 0FKY","summary":"Let f : Y → X be a morphism of schemes. The following are equivalent • f is finite and syntomic, • f is finite, flat, and a local complete intersection morphism, • f is finite, flat, locally of finite presentation, and the fibres of f are local complete intersections, • f is finite and for every x ∈ X there is an affine open x ∈ U = Spec(A) ⊂ X an integer n and f_1, …, f_n ∈ A[x_1, …, x_n] such that f^-1(U) is isomorphic to the spectrum of A[x_1, …, x_n]/(f_1, …, f_n), •…","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is finite and syntomic,\n\\item $f$ is finite, flat, and a local complete intersection morphism,\n\\item $f$ is finite, flat, locally of finite presentation,\nand the fibres of $f$ are local complete intersections,\n\\item $f$ is finite and for every $x \\in X$ there is an\naffine open $x \\in U = \\Spec(A) \\subset X$ an integer $n$\nand $f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$ such that\n$f^{-1}(U)$ is isomorphic to the spectrum of\n$A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$,\n\\item $f$ is finite, flat, locally of finite presentation,\nand $\\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$, and\n\\item $f$ is finite, flat, locally of finite presentation, and\n$\\NL_{X/Y}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$,\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKY","source_file":"discriminant.tex","source_line":2064,"source_end_line":2082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2064-L2082","statement_sha256":"ec3cedb39089596819a0e7441beb64b988605c96311f616cfbf4d4126991a99c","origin":"The Stacks Project","memory_eligible":false,"source_rank":8976,"rank":8976,"depth":40,"x":2464.896,"y":1061.599,"cluster":"local-crystalline-methods"},{"id":"stacks:0FL0","tag":"0FL0","title":"Finite syntomic morphisms · Lemma 0FL0","summary":"With notation as in Example [Tag 0FKZ] there is an open subscheme U_d ⊂ X_d with the following property: a morphism of schemes X → X_d factors through U_d if and only if Y_d ×_X_d X → X is syntomic.","statement_latex":"With notation as in Example \\ref{example-universal-finite-syntomic}\nthere is an open subscheme $U_d \\subset X_d$ with the following property:\na morphism of schemes $X \\to X_d$ factors through $U_d$ if and only\nif $Y_d \\times_{X_d} X \\to X$ is syntomic.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FL0","source_file":"discriminant.tex","source_line":2145,"source_end_line":2151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2145-L2151","statement_sha256":"ac9f9b23a66456f5ee97a7c7b1e71d771e139910d8557e930515638df9261be1","origin":"The Stacks Project","memory_eligible":false,"source_rank":8977,"rank":8977,"depth":39,"x":2554.538,"y":1189.475,"cluster":"local-crystalline-methods"},{"id":"stacks:0FL1","tag":"0FL1","title":"Finite syntomic morphisms · Lemma 0FL1","summary":"With notation as in Example [Tag 0FKZ] and U_d as in Lemma [Tag 0FL0] then U_d is smooth over Spec(Z).","statement_latex":"With notation as in Example \\ref{example-universal-finite-syntomic}\nand $U_d$ as in Lemma \\ref{lemma-universal-finite-syntomic}\nthen $U_d$ is smooth over $\\Spec(\\mathbf{Z})$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FL1","source_file":"discriminant.tex","source_line":2166,"source_end_line":2171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2166-L2171","statement_sha256":"16872c73a7ea7d266226d0b82505a893aea2970feef939a29ec18d5a3eebaf48","origin":"The Stacks Project","memory_eligible":false,"source_rank":8978,"rank":8978,"depth":47,"x":2384.805,"y":1145.84,"cluster":"local-crystalline-methods"},{"id":"stacks:0FL2","tag":"0FL2","title":"Finite syntomic morphisms · Lemma 0FL2","summary":"With notation as in Example [Tag 0FKZ] consider the open subscheme U'_d ⊂ X_d over which π_d is étale. Then U'_d is a dense subset of the open U_d of Lemma [Tag 0FL0].","statement_latex":"With notation as in Example \\ref{example-universal-finite-syntomic}\nconsider the open subscheme $U'_d \\subset X_d$ over which\n$\\pi_d$ is \\'etale. Then $U'_d$ is a dense subset of the\nopen $U_d$ of Lemma \\ref{lemma-universal-finite-syntomic}.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FL2","source_file":"discriminant.tex","source_line":2200,"source_end_line":2206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2200-L2206","statement_sha256":"8817c23281a9a1d04960730016449a0224261dc1391a8467a3ea729f984aa934","origin":"The Stacks Project","memory_eligible":false,"source_rank":8979,"rank":8979,"depth":49,"x":2545.805,"y":1081.406,"cluster":"local-crystalline-methods"},{"id":"stacks:0FL4","tag":"0FL4","title":"Finite syntomic morphisms · Lemma 0FL4","summary":"Let f : Y → X be a morphism of schemes. If f satisfies the equivalent conditions of Lemma [Tag 0FKY] then for every x ∈ X there exist a d and a commutative diagram xymatrix Y ar[d] & V ar[d] ar[l] ar[r] & V_d ar[d] ar[r] & Y_d ar[d]^π_d X & U ar[l] ar[r] & U_d ar[r] & X_d with the following properties • U ⊂ X is open, x ∈ U, and V = f^-1(U), • π_d : Y_d → X_d is as in Example [Tag 0FKZ], • U_d ⊂ X_d is as in Lemma [Tag 0FL0] and V_d = π_d^-1(U_d) ⊂ Y_d, • where the middle…","statement_latex":"Let $f : Y \\to X$ be a morphism of schemes. If $f$ satisfies the equivalent\nconditions of Lemma \\ref{lemma-syntomic-finite} then for every\n$x \\in X$ there exist a $d$ and a commutative diagram\n$$\n\\xymatrix{\nY \\ar[d] &\nV \\ar[d] \\ar[l] \\ar[r] &\nV_d \\ar[d] \\ar[r] &\nY_d \\ar[d]^{\\pi_d}\\\\\nX &\nU \\ar[l] \\ar[r] &\nU_d \\ar[r] &\nX_d\n}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item $U \\subset X$ is open, $x \\in U$, and $V = f^{-1}(U)$,\n\\item $\\pi_d : Y_d \\to X_d$ is as in\nExample \\ref{example-universal-finite-syntomic},\n\\item $U_d \\subset X_d$ is as in Lemma \\ref{lemma-universal-finite-syntomic}\nand $V_d = \\pi_d^{-1}(U_d) \\subset Y_d$,\n\\item where the middle square is cartesian.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Finite syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FL4","source_file":"discriminant.tex","source_line":2291,"source_end_line":2317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2291-L2317","statement_sha256":"1daa1a3567bdda6c43ff143f058530ddbe1ae64351156a36eb0d31b6e7e82a74","origin":"The Stacks Project","memory_eligible":false,"source_rank":8980,"rank":8980,"depth":41,"x":2478.59,"y":1220.918,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWC","tag":"0BWC","title":"A formula for the different · Lemma 0BWC","summary":"[Mazur-Roberts] Let A → P be a ring map. Let f_1, …, f_n ∈ P be a Koszul regular sequence. Assume B = P/(f_1, …, f_n) is flat over A. Let g_1, …, g_n ∈ P ⊗_A B be a Koszul regular sequence generating the kernel of the multiplication map P ⊗_A B → B. Write f_i ⊗ 1 = ∑ g_ij g_j. Then the annihilator of Ker(B ⊗_A B → B) is a principal ideal generated by the image of det(g_ij).","statement_latex":"\\begin{reference}\n\\cite[Appendix]{Mazur-Roberts}\n\\end{reference}\nLet $A \\to P$ be a ring map. Let $f_1, \\ldots, f_n \\in P$ be a\nKoszul regular sequence. Assume $B = P/(f_1, \\ldots, f_n)$\nis flat over $A$. Let $g_1, \\ldots, g_n \\in P \\otimes_A B$\nbe a Koszul regular sequence generating the kernel of the multiplication\nmap $P \\otimes_A B \\to B$. Write $f_i \\otimes 1 = \\sum g_{ij} g_j$.\nThen the annihilator of $\\Ker(B \\otimes_A B \\to B)$ is a principal\nideal generated by the image of $\\det(g_{ij})$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"A formula for the different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWC","source_file":"discriminant.tex","source_line":2355,"source_end_line":2367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2355-L2367","statement_sha256":"e6ac7f12d792e207dd64ddf0b428d918f1830c79f1eefd7babba56a4bf7a7e02","origin":"The Stacks Project","memory_eligible":false,"source_rank":8981,"rank":8981,"depth":1,"x":2415.675,"y":1079.256,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWD","tag":"0BWD","title":"A formula for the different · Lemma 0BWD","summary":"Let A be a ring. Let n ≥ 1 and h, f_1, …, f_n ∈ A[x_1, …, x_n]. Set B = A[x_1, …, x_n, 1/h]/(f_1, …, f_n). Assume that B is quasi-finite over A. Then • B is flat over A and A → B is a relative local complete intersection, • the annihilator J of I = Ker(B ⊗_A B → B) is free of rank 1 over B, • the Noether different of B over A is generated by det(∂ f_i/∂ x_j) in B.","statement_latex":"Let $A$ be a ring. Let $n \\geq 1$ and\n$h, f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$.\nSet $B = A[x_1, \\ldots, x_n, 1/h]/(f_1, \\ldots, f_n)$.\nAssume that $B$ is quasi-finite over $A$.\nThen\n\\begin{enumerate}\n\\item $B$ is flat over $A$ and $A \\to B$ is a relative local complete\nintersection,\n\\item the annihilator $J$ of $I = \\Ker(B \\otimes_A B \\to B)$\nis free of rank $1$ over $B$,\n\\item the Noether different of $B$ over $A$ is generated\nby $\\det(\\partial f_i/\\partial x_j)$ in $B$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"A formula for the different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWD","source_file":"discriminant.tex","source_line":2394,"source_end_line":2409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2394-L2409","statement_sha256":"3db96e9aa7fbdec61d351d92fef7c0a7870d66f0000b49ef4e7274674c8d2e2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":8982,"rank":8982,"depth":34,"x":2576.718,"y":1148.33,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWG","tag":"0BWG","title":"A formula for the different · Lemma 0BWG","summary":"Let f : Y → X be a morphism of Noetherian schemes. If f satisfies the equivalent conditions of Lemma [Tag 0BWE] then the different D_f of f is the K\\\"ahler different of f.","statement_latex":"Let $f : Y \\to X$ be a morphism of Noetherian schemes. If $f$\nsatisfies the equivalent conditions of Lemma \\ref{lemma-syntomic-quasi-finite}\nthen the different $\\mathfrak{D}_f$ of $f$ is the K\\\"ahler different\nof $f$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"A formula for the different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWG","source_file":"discriminant.tex","source_line":2490,"source_end_line":2496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2490-L2496","statement_sha256":"3fd14931dac2255382f899db7ae5871f27c493f4d1903d561ad1a03975e14538","origin":"The Stacks Project","memory_eligible":false,"source_rank":8983,"rank":8983,"depth":41,"x":2401.633,"y":1188.957,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWH","tag":"0BWH","title":"A formula for the different · Lemma 0BWH","summary":"Let A be a ring. Let n ≥ 1 and h, f_1, …, f_n ∈ A[x_1, …, x_n]. Set B = A[x_1, …, x_n, 1/h]/(f_1, …, f_n). Assume that B is quasi-finite over A. Then there is an isomorphism B → ω_B/A mapping det(∂ f_i/∂ x_j) to τ_B/A.","statement_latex":"Let $A$ be a ring. Let $n \\geq 1$ and\n$h, f_1, \\ldots, f_n \\in A[x_1, \\ldots, x_n]$.\nSet $B = A[x_1, \\ldots, x_n, 1/h]/(f_1, \\ldots, f_n)$.\nAssume that $B$ is quasi-finite over $A$.\nThen there is an isomorphism $B \\to \\omega_{B/A}$\nmapping $\\det(\\partial f_i/\\partial x_j)$ to $\\tau_{B/A}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"A formula for the different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWH","source_file":"discriminant.tex","source_line":2509,"source_end_line":2517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2509-L2517","statement_sha256":"a24f9295dc67bf7826710ada4e3a1ecc437d6cafe6c69c74cfac79a1bb5ed484","origin":"The Stacks Project","memory_eligible":false,"source_rank":8984,"rank":8984,"depth":41,"x":2498.499,"y":1059.074,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWJ","tag":"0BWJ","title":"A formula for the different · Lemma 0BWJ","summary":"Let S be a Noetherian scheme. Let X, Y be smooth schemes of relative dimension n over S. Let f : Y → X be a locally quasi-finite morphism over S. Then f is flat and the closed subscheme R ⊂ Y cut out by the different of f is the locally principal closed subscheme cut out by wedge^n(df) ∈ Γ(Y, (f^*Ω^n_X/S)^⊗ -1 ⊗_O_Y Ω^n_Y/S) If f is étale at the associated points of Y, then R is an effective Cartier divisor and f^*Ω^n_X/S ⊗_O_Y O(R) = Ω^n_Y/S as invertible sheaves on Y.","statement_latex":"Let $S$ be a Noetherian scheme. Let $X$, $Y$ be smooth schemes\nof relative dimension $n$ over $S$. Let $f : Y \\to X$ be a\nlocally quasi-finite morphism over $S$.\nThen $f$ is flat and the closed subscheme $R \\subset Y$\ncut out by the different of $f$ is the locally principal\nclosed subscheme cut out by\n$$\n\\wedge^n(\\text{d}f) \\in\n\\Gamma(Y,\n(f^*\\Omega^n_{X/S})^{\\otimes -1} \\otimes_{\\mathcal{O}_Y} \\Omega^n_{Y/S})\n$$\nIf $f$ is \\'etale at the associated points of $Y$, then $R$ is an\neffective Cartier divisor and\n$$\nf^*\\Omega^n_{X/S} \\otimes_{\\mathcal{O}_Y} \\mathcal{O}(R) =\n\\Omega^n_{Y/S}\n$$\nas invertible sheaves on $Y$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"A formula for the different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWJ","source_file":"discriminant.tex","source_line":2548,"source_end_line":2568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2548-L2568","statement_sha256":"9486718ed9870d26310a1ecae51daaf32b76704ba0fd6ee0e115218c4fc922b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":8985,"rank":8985,"depth":42,"x":2531.665,"y":1210.48,"cluster":"local-crystalline-methods"},{"id":"stacks:0FKD","tag":"0FKD","title":"The Tate map · Proposition 0FKD","summary":"There exists a unique rule that to every locally quasi-finite syntomic morphism of locally Noetherian schemes Y → X assigns an isomorphism c_Y/X : det(NL_Y/X) → ω_Y/X satisfying the following two properties • the section δ(NL_Y/X) is mapped to τ_Y/X, and • the rule is compatible with restriction to opens and with base change.","statement_latex":"There exists a unique rule that to every locally quasi-finite syntomic\nmorphism of locally Noetherian schemes $Y \\to X$ assigns an isomorphism\n$$\nc_{Y/X} : \\det(\\NL_{Y/X}) \\longrightarrow \\omega_{Y/X}\n$$\nsatisfying the following two properties\n\\begin{enumerate}\n\\item the section $\\delta(\\NL_{Y/X})$ is mapped to $\\tau_{Y/X}$, and\n\\item the rule is compatible with restriction to opens and with\nbase change.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"The Tate map","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FKD","source_file":"discriminant.tex","source_line":2716,"source_end_line":2729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2716-L2729","statement_sha256":"5316b216fc037c552468a8cf039ee7070c676f5903fb51cce4f2eef5d4c14279","origin":"The Stacks Project","memory_eligible":false,"source_rank":8986,"rank":8986,"depth":48,"x":2384.81,"y":1117.245,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWL","tag":"0BWL","title":"A generalization of the different · Lemma 0BWL","summary":"Let A → B be a map of Noetherian rings. Consider the conditions • nonzerodivisors of A map to nonzerodivisors of B, • (1) holds and Q(A) → Q(A) ⊗_A B is flat, • A → B_ q is flat for every q ∈ Ass(B), • (3) holds and A → B_ q is flat for every q lying over an element in Ass(A). Then we have the following implications xymatrix (1) & (2) ar@=>[l] ar@=>[d] (3) ar@=>[u] & (4) ar@=>[l] If going up holds for A → B then (2) and (4) are equivalent.","statement_latex":"Let $A \\to B$ be a map of Noetherian rings. Consider the conditions\n\\begin{enumerate}\n\\item nonzerodivisors of $A$ map to nonzerodivisors of $B$,\n\\item (1) holds and $Q(A) \\to Q(A) \\otimes_A B$ is flat,\n\\item $A \\to B_\\mathfrak q$ is flat for every\n$\\mathfrak q \\in \\text{Ass}(B)$,\n\\item (3) holds and $A \\to B_\\mathfrak q$ is flat for every $\\mathfrak q$\nlying over an element in $\\text{Ass}(A)$.\n\\end{enumerate}\nThen we have the following implications\n$$\n\\xymatrix{\n(1) & (2) \\ar@{=>}[l] \\ar@{=>}[d] \\\\\n(3) \\ar@{=>}[u] & (4) \\ar@{=>}[l]\n}\n$$\nIf going up holds for $A \\to B$ then (2) and (4) are equivalent.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"A generalization of the different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWL","source_file":"discriminant.tex","source_line":2970,"source_end_line":2989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L2970-L2989","statement_sha256":"af83ace512c2858676f6040756dfc34e5dbb60fe0b9896a22b6758640446dabb","origin":"The Stacks Project","memory_eligible":false,"source_rank":8987,"rank":8987,"depth":12,"x":2568.874,"y":1102.605,"cluster":"local-crystalline-methods"},{"id":"stacks:0BWN","tag":"0BWN","title":"A generalization of the different · Lemma 0BWN","summary":"Assume the Dedekind different is defined for A → B. Set X = Spec(A) and Y = Spec(B). The generalization of Remark [Tag 0BWM] applies to the morphism f : Y → X if and only if 1 ∈ L_B/A (e.g., if A is normal, see Lemma [Tag 0BW1]). In this case D_B/A is an ideal of B and we have D_f = widetildeD_B/A as coherent ideal sheaves on Y.","statement_latex":"Assume the Dedekind different is defined for $A \\to B$.\nSet $X = \\Spec(A)$ and $Y = \\Spec(B)$. The generalization of\nRemark \\ref{remark-different-generalization}\napplies to the morphism $f : Y \\to X$ if and only if\n$1 \\in \\mathcal{L}_{B/A}$ (e.g., if $A$ is normal, see\nLemma \\ref{lemma-dedekind-different-ideal}).\nIn this case $\\mathfrak{D}_{B/A}$ is an ideal of $B$ and we have\n$$\n\\mathfrak{D}_f = \\widetilde{\\mathfrak{D}_{B/A}}\n$$\nas coherent ideal sheaves on $Y$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"A generalization of the different","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BWN","source_file":"discriminant.tex","source_line":3051,"source_end_line":3064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L3051-L3064","statement_sha256":"efaa159523a55b10af6a3563f24ed901034eccc848cd452cdba46e828880e955","origin":"The Stacks Project","memory_eligible":false,"source_rank":8988,"rank":8988,"depth":48,"x":2444.386,"y":1218.339,"cluster":"local-crystalline-methods"},{"id":"stacks:0BUL","tag":"0BUL","title":"Comparison with duality theory · Lemma 0BUL","summary":"Let f : Y → X be a quasi-finite separated morphism of Noetherian schemes. For every pair of affine opens Spec(B) = V ⊂ Y, Spec(A) = U ⊂ X with f(V) ⊂ U there is an isomorphism H^0(V, f^!O_X) = ω_B/A where f^! is as in Duality for Schemes, Section [Tag 0A9Y]. These isomorphisms are compatible with restriction maps and define a canonical isomorphism H^0(f^!O_X) = ω_Y/X with ω_Y/X as in Remark [Tag 0BVG]. Similarly, if f : Y → X is a quasi-finite morphism of schemes of…","statement_latex":"Let $f : Y \\to X$ be a quasi-finite separated morphism of Noetherian schemes.\nFor every pair of affine opens $\\Spec(B) = V \\subset Y$,\n$\\Spec(A) = U \\subset X$ with $f(V) \\subset U$ there is an isomorphism\n$$\nH^0(V, f^!\\mathcal{O}_X) = \\omega_{B/A}\n$$\nwhere $f^!$ is as in\nDuality for Schemes, Section \\ref{duality-section-upper-shriek}.\nThese isomorphisms are compatible with restriction maps and define a canonical\nisomorphism $H^0(f^!\\mathcal{O}_X) = \\omega_{Y/X}$ with\n$\\omega_{Y/X}$ as in Remark \\ref{remark-relative-dualizing-for-quasi-finite}.\nSimilarly, if $f : Y \\to X$ is a quasi-finite morphism of schemes of\nfinite type over a Noetherian base $S$ endowed with a dualizing complex\n$\\omega_S^\\bullet$, then $H^0(f_{new}^!\\mathcal{O}_X) = \\omega_{Y/X}$.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Comparison with duality theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUL","source_file":"discriminant.tex","source_line":3142,"source_end_line":3158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L3142-L3158","statement_sha256":"f843b5e8f33f6214008d84a7ffb426722dcc806747bdd1cf9293ed977eb3ec30","origin":"The Stacks Project","memory_eligible":false,"source_rank":8989,"rank":8989,"depth":65,"x":2443.107,"y":1061.693,"cluster":"local-crystalline-methods"},{"id":"stacks:0BVI","tag":"0BVI","title":"Comparison with duality theory · Lemma 0BVI","summary":"Let f : Y → X be a finite flat morphism of Noetherian schemes. The map Trace_f : f_*O_Y → O_X of Section [Tag 0BVH] corresponds to a map O_Y → f^!O_X (see proof). Denote τ_Y/X ∈ H^0(Y, f^!O_X) the image of 1. Via the isomorphism H^0(f^!O_X) = ω_X/Y of Lemma [Tag 0BUL] this agrees with the construction in Remark [Tag 0BVJ].","statement_latex":"Let $f : Y \\to X$ be a finite flat morphism of Noetherian schemes.\nThe map\n$$\n\\text{Trace}_f : f_*\\mathcal{O}_Y \\longrightarrow \\mathcal{O}_X\n$$\nof Section \\ref{section-discriminant}\ncorresponds to a map $\\mathcal{O}_Y \\to f^!\\mathcal{O}_X$ (see proof).\nDenote $\\tau_{Y/X} \\in H^0(Y, f^!\\mathcal{O}_X)$ the image of $1$.\nVia the isomorphism $H^0(f^!\\mathcal{O}_X) = \\omega_{X/Y}$ of\nLemma \\ref{lemma-compare-dualizing}\nthis agrees with the construction in\nRemark \\ref{remark-relative-dualizing-for-flat-quasi-finite}.","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Comparison with duality theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BVI","source_file":"discriminant.tex","source_line":3216,"source_end_line":3230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L3216-L3230","statement_sha256":"1dbc92e2d2019afb275d4dcb7256ee4da1a1ac81a63c0531438de264df623a19","origin":"The Stacks Project","memory_eligible":false,"source_rank":8990,"rank":8990,"depth":66,"x":2570.558,"y":1176.963,"cluster":"local-crystalline-methods"},{"id":"stacks:0C16","tag":"0C16","title":"Quasi-finite Gorenstein morphisms · Lemma 0C16","summary":"Let f : Y → X be a quasi-finite morphism of Noetherian schemes. The following are equivalent • f is Gorenstein, • f is flat and the fibres of f are Gorenstein, • f is flat and ω_Y/X is invertible (Remark [Tag 0BVG]), • for every y ∈ Y there are affine opens y ∈ V = Spec(B) ⊂ Y, U = Spec(A) ⊂ X with f(V) ⊂ U such that A → B is flat and ω_B/A is an invertible B-module.","statement_latex":"Let $f : Y \\to X$ be a quasi-finite morphism of Noetherian schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is Gorenstein,\n\\item $f$ is flat and the fibres of $f$ are Gorenstein,\n\\item $f$ is flat and $\\omega_{Y/X}$ is invertible\n(Remark \\ref{remark-relative-dualizing-for-quasi-finite}),\n\\item for every $y \\in Y$ there are affine opens\n$y \\in V = \\Spec(B) \\subset Y$, $U = \\Spec(A) \\subset X$\nwith $f(V) \\subset U$ such that $A \\to B$ is flat\nand $\\omega_{B/A}$ is an invertible $B$-module.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Discriminants and Differents","chapter_id":"discriminant","section":"Quasi-finite Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C16","source_file":"discriminant.tex","source_line":3290,"source_end_line":3304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/discriminant.tex#L3290-L3304","statement_sha256":"351de7b3c694bb3d6be4db54534eb797737a2f60e18f436147aeac3641114496","origin":"The Stacks Project","memory_eligible":false,"source_rank":8991,"rank":8991,"depth":67,"x":2383.095,"y":1164.229,"cluster":"local-crystalline-methods"},{"id":"stacks:0FL5","tag":"0FL5","title":"The de Rham complex · Lemma 0FL5","summary":"Let xymatrix X' ar[r]_f ar[d] & X ar[d] S' ar[r] & S be a cartesian diagram of schemes. Then the maps discussed above induce isomorphisms f^*Ω^p_X/S → Ω^p_X'/S'.","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\\nS' \\ar[r] & S\n}\n$$\nbe a cartesian diagram of schemes. Then the maps discussed\nabove induce isomorphisms\n$f^*\\Omega^p_{X/S} \\to \\Omega^p_{X'/S'}$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"The de Rham complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FL5","source_file":"derham.tex","source_line":73,"source_end_line":85,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L73-L85","statement_sha256":"a10b9e0325e78adc57273156160deb120429d936a9e6ec757e59bc7cbd4146d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8992,"rank":8992,"depth":17,"x":2532.187,"y":1066.845,"cluster":"local-crystalline-methods"},{"id":"stacks:0FLV","tag":"0FLV","title":"The de Rham complex · Lemma 0FLV","summary":"Consider a commutative diagram of schemes xymatrix X' ar[r]_f ar[d] & X ar[d] S' ar[r] & S If X' → X and S' → S are étale, then the maps discussed above induce isomorphisms f^*Ω^p_X/S → Ω^p_X'/S'.","statement_latex":"Consider a commutative diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\\nS' \\ar[r] & S\n}\n$$\nIf $X' \\to X$ and $S' \\to S$ are \\'etale, then the maps discussed\nabove induce isomorphisms\n$f^*\\Omega^p_{X/S} \\to \\Omega^p_{X'/S'}$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"The de Rham complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLV","source_file":"derham.tex","source_line":92,"source_end_line":104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L92-L104","statement_sha256":"ac17a0581c60997edf67605d7aae9cd69aeedfdc9c044923120a49acd49c038a","origin":"The Stacks Project","memory_eligible":false,"source_rank":8993,"rank":8993,"depth":45,"x":2500.431,"y":1223.9,"cluster":"local-crystalline-methods"},{"id":"stacks:0FLW","tag":"0FLW","title":"de Rham cohomology · Lemma 0FLW","summary":"Let X → S be a morphism of affine schemes given by the ring map R → A. Then RΓ(X, Ω^bullet_X/S) = Ω^bullet_A/R in D(R) and H^i_dR(X/S) = H^i(Ω^bullet_A/R).","statement_latex":"Let $X \\to S$ be a morphism of affine schemes given by the ring map\n$R \\to A$. Then $R\\Gamma(X, \\Omega^\\bullet_{X/S}) = \\Omega^\\bullet_{A/R}$\nin $D(R)$ and $H^i_{dR}(X/S) = H^i(\\Omega^\\bullet_{A/R})$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLW","source_file":"derham.tex","source_line":158,"source_end_line":163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L158-L163","statement_sha256":"347935e0677620f735ea2906e3cdb5091dc42d05be37bfa2d9ab9e910198a172","origin":"The Stacks Project","memory_eligible":false,"source_rank":8994,"rank":8994,"depth":24,"x":2397.127,"y":1089.521,"cluster":"local-crystalline-methods"},{"id":"stacks:0FLX","tag":"0FLX","title":"de Rham cohomology · Lemma 0FLX","summary":"Let p : X → S be a morphism of schemes. If p is quasi-compact and quasi-separated, then Rp_*Ω^bullet_X/S is an object of D_QCoh(O_S).","statement_latex":"Let $p : X \\to S$ be a morphism of schemes. If $p$ is quasi-compact\nand quasi-separated, then $Rp_*\\Omega^\\bullet_{X/S}$ is an object\nof $D_\\QCoh(\\mathcal{O}_S)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLX","source_file":"derham.tex","source_line":172,"source_end_line":177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L172-L177","statement_sha256":"5f99e8b408a03d0885f1a235277cc5db69120ca6d9eb6481949bf8a1a7b45604","origin":"The Stacks Project","memory_eligible":false,"source_rank":8995,"rank":8995,"depth":28,"x":2582.115,"y":1130.159,"cluster":"local-crystalline-methods"},{"id":"stacks:0FLY","tag":"0FLY","title":"de Rham cohomology · Lemma 0FLY","summary":"Let p : X → S be a proper morphism of schemes with S locally Noetherian. Then Rp_*Ω^bullet_X/S is an object of D_Coh(O_S).","statement_latex":"Let $p : X \\to S$ be a proper morphism of schemes with $S$ locally\nNoetherian. Then $Rp_*\\Omega^\\bullet_{X/S}$ is an object\nof $D_{\\textit{Coh}}(\\mathcal{O}_S)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLY","source_file":"derham.tex","source_line":190,"source_end_line":195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L190-L195","statement_sha256":"539c65094ded14ab7cd18ea237e054420ce35162aa235fd8173eb7c699fa2801","origin":"The Stacks Project","memory_eligible":false,"source_rank":8996,"rank":8996,"depth":31,"x":2412.345,"y":1205.461,"cluster":"local-crystalline-methods"},{"id":"stacks:0FLZ","tag":"0FLZ","title":"de Rham cohomology · Lemma 0FLZ","summary":"Let A be a Noetherian ring. Let X be a proper scheme over S = Spec(A). Then H^i_dR(X/S) is a finite A-module for all i.","statement_latex":"Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $S = \\Spec(A)$.\nThen $H^i_{dR}(X/S)$ is a finite $A$-module for all $i$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLZ","source_file":"derham.tex","source_line":205,"source_end_line":209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L205-L209","statement_sha256":"6e8635995b2f5dd44c63f357e57c87105ff0b301dfd6ca6562ba99d43f9c0d36","origin":"The Stacks Project","memory_eligible":false,"source_rank":8997,"rank":8997,"depth":32,"x":2477.234,"y":1052.995,"cluster":"local-crystalline-methods"},{"id":"stacks:0FM0","tag":"0FM0","title":"de Rham cohomology · Lemma 0FM0","summary":"Let f : X → S be a proper smooth morphism of schemes. Then Rf_*Ω^p_X/S, p ≥ 0 and Rf_*Ω^bullet_X/S are perfect objects of D(O_S) whose formation commutes with arbitrary change of base.","statement_latex":"Let $f : X \\to S$ be a proper smooth morphism of schemes. Then\n$Rf_*\\Omega^p_{X/S}$, $p \\geq 0$ and $Rf_*\\Omega^\\bullet_{X/S}$ are\nperfect objects of $D(\\mathcal{O}_S)$ whose formation commutes\nwith arbitrary change of base.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FM0","source_file":"derham.tex","source_line":215,"source_end_line":221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L215-L221","statement_sha256":"d67d10b686ed82c4c8c0a05569acb471b59162522ea4a2dda390bb79fd551bd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":8998,"rank":8998,"depth":39,"x":2552.291,"y":1202.834,"cluster":"local-crystalline-methods"},{"id":"stacks:0FM3","tag":"0FM3","title":"Cup product · Lemma 0FM3","summary":"Let p : X → S be a morphism of schemes. The cup product on H^*_dR(X/S) is associative and graded commutative.","statement_latex":"Let $p : X \\to S$ be a morphism of schemes.\nThe cup product on $H^*_{dR}(X/S)$ is associative and\ngraded commutative.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Cup product","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FM3","source_file":"derham.tex","source_line":315,"source_end_line":320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L315-L320","statement_sha256":"37f08dd8cacc79b760842041babe89d9fbe44e621d661dfa4b1585a5237d65b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":8999,"rank":8999,"depth":1,"x":2375.757,"y":1134.666,"cluster":"local-crystalline-methods"},{"id":"stacks:0FM5","tag":"0FM5","title":"Hodge cohomology · Lemma 0FM5","summary":"Let p : X → S be a morphism of schemes. The cup product on H^*_Hodge(X/S) is associative and graded commutative.","statement_latex":"Let $p : X \\to S$ be a morphism of schemes.\nThe cup product on $H^*_{Hodge}(X/S)$ is associative and graded commutative.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Hodge cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FM5","source_file":"derham.tex","source_line":424,"source_end_line":428,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L424-L428","statement_sha256":"7b35bee0fe7f60d4ad20759af22407ded86e7b0455027954d3b00018b36a03b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9000,"rank":9000,"depth":2,"x":2561.473,"y":1084.568,"cluster":"local-crystalline-methods"},{"id":"stacks:0FM8","tag":"0FM8","title":"The Hodge filtration · Definition 0FM8","summary":"Let X → S be a morphism of schemes. The Hodge filtration on H^n_dR(X/S) is the filtration with terms F^pH^n_dR(X/S) = Im(H^n(X, σ_≥ pΩ^bullet_X/S) → H^n_dR(X/S)) where σ_≥ pΩ^bullet_X/S is as in Homology, Section [Tag 0118].","statement_latex":"Let $X \\to S$ be a morphism of schemes. The {\\it Hodge filtration}\non $H^n_{dR}(X/S)$ is the filtration with terms\n$$\nF^pH^n_{dR}(X/S) = \\Im\\left(H^n(X, \\sigma_{\\geq p}\\Omega^\\bullet_{X/S})\n\\longrightarrow H^n_{dR}(X/S)\\right)\n$$\nwhere $\\sigma_{\\geq p}\\Omega^\\bullet_{X/S}$ is as in\nHomology, Section \\ref{homology-section-truncations}.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"The Hodge filtration","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FM8","source_file":"derham.tex","source_line":556,"source_end_line":566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L556-L566","statement_sha256":"6bc10e5125fc32f49f5b99568f4925795eb267a8b30132d560cc61a85238d37a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9001,"rank":9001,"depth":0,"x":2464.44,"y":1227.44,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMA","tag":"0FMA","title":"K\\\"unneth formula · Lemma 0FMA","summary":"In the situation above there is a canonical isomorphism Tot(Ω^bullet_X/S boxtimes Ω^bullet_Y/S) → Ω^bullet_X ×_S Y/S of complexes of f^-1O_S-modules.","statement_latex":"In the situation above there is a canonical isomorphism\n$$\n\\text{Tot}(\\Omega^\\bullet_{X/S} \\boxtimes \\Omega^\\bullet_{Y/S})\n\\longrightarrow\n\\Omega^\\bullet_{X \\times_S Y/S}\n$$\nof complexes of $f^{-1}\\mathcal{O}_S$-modules.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"K\\\"unneth formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMA","source_file":"derham.tex","source_line":653,"source_end_line":662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L653-L662","statement_sha256":"ef5e40aca3993581aa553b97514e00eb008445259f52e19cab6f332e8631dfee","origin":"The Stacks Project","memory_eligible":false,"source_rank":9002,"rank":9002,"depth":18,"x":2420.931,"y":1066.413,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMB","tag":"0FMB","title":"K\\\"unneth formula · Lemma 0FMB","summary":"Assume X and Y are smooth, quasi-compact, with affine diagonal over S = Spec(A). Then the map RΓ(X, Ω^bullet_X/S) ⊗_A^L RΓ(Y, Ω^bullet_Y/S) → RΓ(X ×_S Y, Ω^bullet_X ×_S Y/S) is an isomorphism in D(A).","statement_latex":"Assume $X$ and $Y$ are smooth, quasi-compact, with affine diagonal over\n$S = \\Spec(A)$. Then the map\n$$\nR\\Gamma(X, \\Omega^\\bullet_{X/S})\n\\otimes_A^\\mathbf{L}\nR\\Gamma(Y, \\Omega^\\bullet_{Y/S})\n\\longrightarrow\nR\\Gamma(X \\times_S Y, \\Omega^\\bullet_{X \\times_S Y/S})\n$$\nis an isomorphism in $D(A)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"K\\\"unneth formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMB","source_file":"derham.tex","source_line":709,"source_end_line":721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L709-L721","statement_sha256":"bba1b38739b146a9a03cc2570def1ea6080bd32cb1712d0199e8d3ea12d5e533","origin":"The Stacks Project","memory_eligible":false,"source_rank":9003,"rank":9003,"depth":37,"x":2583.124,"y":1160.824,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMC","tag":"0FMC","title":"K\\\"unneth formula · Lemma 0FMC","summary":"Assume X → S and Y → S are smooth and quasi-compact and the morphisms X → X ×_S X and Y → Y ×_S Y are affine. Then the relative cup product Ra_*Ω^bullet_X/S ⊗_O_S^L Rb_*Ω^bullet_Y/S → Rf_*Ω^bullet_X ×_S Y/S is an isomorphism in D(O_S).","statement_latex":"Assume $X \\to S$ and $Y \\to S$ are smooth and quasi-compact\nand the morphisms $X \\to X \\times_S X$ and $Y \\to Y \\times_S Y$ are affine.\nThen the relative cup product\n$$\nRa_*\\Omega^\\bullet_{X/S}\n\\otimes_{\\mathcal{O}_S}^\\mathbf{L}\nRb_*\\Omega^\\bullet_{Y/S}\n\\longrightarrow\nRf_*\\Omega^\\bullet_{X \\times_S Y/S}\n$$\nis an isomorphism in $D(\\mathcal{O}_S)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"K\\\"unneth formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMC","source_file":"derham.tex","source_line":792,"source_end_line":805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L792-L805","statement_sha256":"8a142e0e763625239e04b2858dc6b37efc13dedec3de48c85c50ad0354b85352","origin":"The Stacks Project","memory_eligible":false,"source_rank":9004,"rank":9004,"depth":38,"x":2386.862,"y":1183.323,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMD","tag":"0FMD","title":"First Chern class in de Rham cohomology · Lemma 0FMD","summary":"Given a commutative diagram xymatrix X' ar[r]_f ar[d] & X ar[d] S' ar[r] & S of schemes the diagrams xymatrix Pic(X') ar[d]_c_1^dR & Pic(X) ar[d]^c_1^dR ar[l]^f^* H^2_dR(X'/S') & H^2_dR(X/S) ar[l]_f^* xymatrix Pic(X') ar[d]_c_1^Hodge & Pic(X) ar[d]^c_1^Hodge ar[l]^f^* H^1(X', Ω^1_X'/S') & H^1(X, Ω^1_X/S) ar[l]_f^* commute.","statement_latex":"Given a commutative diagram\n$$\n\\xymatrix{\nX' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\\nS' \\ar[r] & S\n}\n$$\nof schemes the diagrams\n$$\n\\xymatrix{\n\\Pic(X') \\ar[d]_{c_1^{dR}} &\n\\Pic(X) \\ar[d]^{c_1^{dR}} \\ar[l]^{f^*} \\\\\nH^2_{dR}(X'/S') &\nH^2_{dR}(X/S) \\ar[l]_{f^*}\n}\n\\quad\n\\xymatrix{\n\\Pic(X') \\ar[d]_{c_1^{Hodge}} &\n\\Pic(X) \\ar[d]^{c_1^{Hodge}} \\ar[l]^{f^*} \\\\\nH^1(X', \\Omega^1_{X'/S'}) &\nH^1(X, \\Omega^1_{X/S}) \\ar[l]_{f^*}\n}\n$$\ncommute.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"First Chern class in de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMD","source_file":"derham.tex","source_line":846,"source_end_line":872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L846-L872","statement_sha256":"5167cac75357dac1789432b4eee0925c95b95d639b09e1d4f321dcb87bd2514d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9005,"rank":9005,"depth":0,"x":2513.967,"y":1054.888,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUF","tag":"0FUF","title":"de Rham cohomology of a line bundle · Lemma 0FUF","summary":"With notation as above, there is a short exact sequence of complexes 0 → Ω^bullet_X/S → Ω^bullet_L^star/S, 0 → Ω^bullet_X/S[-1] → 0","statement_latex":"With notation as above, there is a short exact sequence of complexes\n$$\n0 \\to \\Omega^\\bullet_{X/S} \\to\n\\Omega^\\bullet_{L^\\star/S, 0} \\to\n\\Omega^\\bullet_{X/S}[-1] \\to 0\n$$","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology of a line bundle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUF","source_file":"derham.tex","source_line":1130,"source_end_line":1138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1130-L1138","statement_sha256":"4fa29a44f609c8208162e6631edb144458eb95803689b0c6aa915801035e3baf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9006,"rank":9006,"depth":1,"x":2523.559,"y":1222.339,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUG","tag":"0FUG","title":"de Rham cohomology of a line bundle · Lemma 0FUG","summary":"The \"boundary\" map δ : Ω^bullet_X/S → Ω^bullet_X/S[2] in D(X, f^-1O_S) coming from the short exact sequence in Lemma [Tag 0FUF] is the map of Remark [Tag 0FU7] for xi = c_1^dR(L).","statement_latex":"The ``boundary'' map\n$\\delta : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[2]$\nin $D(X, f^{-1}\\mathcal{O}_S)$ coming from\nthe short exact sequence in Lemma \\ref{lemma-the-complex-for-L-star}\nis the map of Remark \\ref{remark-cup-product-as-a-map}\nfor $\\xi = c_1^{dR}(\\mathcal{L})$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology of a line bundle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUG","source_file":"derham.tex","source_line":1204,"source_end_line":1212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1204-L1212","statement_sha256":"53de2626be3db6a0559a982119d5567ec8e6f35f79c17f802460eef25646fc06","origin":"The Stacks Project","memory_eligible":false,"source_rank":9007,"rank":9007,"depth":2,"x":2381.303,"y":1103.869,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUH","tag":"0FUH","title":"de Rham cohomology of a line bundle · Lemma 0FUH","summary":"With notation as above we have • Ω^p_L^star/S, n = Ω^p_L^star/S, 0 ⊗_O_X L^⊗ n for all n ∈ Z as quasi-coherent O_X-modules, • Ω^bullet_X/S = Ω^bullet_L/X, 0 as complexes, and • for n > 0 and p ≥ 0 we have Ω^p_L/X, n = Ω^p_L^star/S, n.","statement_latex":"With notation as above we have\n\\begin{enumerate}\n\\item $\\Omega^p_{L^\\star/S, n} =\n\\Omega^p_{L^\\star/S, 0} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}$\nfor all $n \\in \\mathbf{Z}$ as quasi-coherent $\\mathcal{O}_X$-modules,\n\\item $\\Omega^\\bullet_{X/S} = \\Omega^\\bullet_{L/X, 0}$\nas complexes, and\n\\item for $n > 0$ and $p \\geq 0$ we have\n$\\Omega^p_{L/X, n} = \\Omega^p_{L^\\star/S, n}$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology of a line bundle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUH","source_file":"derham.tex","source_line":1291,"source_end_line":1303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1291-L1303","statement_sha256":"f8ade7f6f7198171608cbe7bdd331feae6d4306feb954a0088e28d245029ba2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9008,"rank":9008,"depth":0,"x":2582.207,"y":1110.532,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUI","tag":"0FUI","title":"de Rham cohomology of a line bundle · Lemma 0FUI","summary":"In the situation above, assume there is a morphism S → Spec(Q). Then Ω^bullet_X/S → π_*Ω^bullet_L/S is a quasi-isomorphism and H_dR^*(X/S) = H_dR^*(L/S).","statement_latex":"In the situation above, assume there is a morphism $S \\to \\Spec(\\mathbf{Q})$.\nThen $\\Omega^\\bullet_{X/S} \\to \\pi_*\\Omega^\\bullet_{L/S}$ is a\nquasi-isomorphism and $H_{dR}^*(X/S) = H_{dR}^*(L/S)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology of a line bundle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUI","source_file":"derham.tex","source_line":1310,"source_end_line":1315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1310-L1315","statement_sha256":"ea1cfa52bace8a53cbf0b539ed39c87c41f8f9b200f496bbb304f80d55b2f4a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9009,"rank":9009,"depth":0,"x":2428.144,"y":1220.014,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMH","tag":"0FMH","title":"de Rham cohomology of projective space · Lemma 0FMH","summary":"There exists a short exact sequence 0 → Ω → O(-1)^⊕ n + 1 → O → 0","statement_latex":"There exists a short exact sequence\n$$\n0 \\to \\Omega \\to \\mathcal{O}(-1)^{\\oplus n + 1} \\to \\mathcal{O} \\to 0\n$$","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology of projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMH","source_file":"derham.tex","source_line":1376,"source_end_line":1382,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1376-L1382","statement_sha256":"e704b18b28b7103cf4ce0f927703f73f4d08f5ee5a45185f46261e390b64ed1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9010,"rank":9010,"depth":0,"x":2453.799,"y":1051.256,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUK","tag":"0FUK","title":"de Rham cohomology of projective space · Lemma 0FUK","summary":"In the situation above we have the following cohomology groups • H^q(P^n_A, Ω^p) = 0 unless 0 ≤ p = q ≤ n, • for 0 ≤ p ≤ n the A-module H^p(P^n_A, Ω^p) free of rank 1. • for q > 0, k > 0, and p arbitrary we have H^q(P^n_A, Ω^p(k)) = 0, and • add more here.","statement_latex":"In the situation above we have the following cohomology groups\n\\begin{enumerate}\n\\item $H^q(\\mathbf{P}^n_A, \\Omega^p) = 0$\nunless $0 \\leq p = q \\leq n$,\n\\item for $0 \\leq p \\leq n$ the $A$-module\n$H^p(\\mathbf{P}^n_A, \\Omega^p)$ free of rank $1$.\n\\item for $q > 0$, $k > 0$, and $p$ arbitrary we have\n$H^q(\\mathbf{P}^n_A, \\Omega^p(k)) = 0$, and\n\\item add more here.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology of projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUK","source_file":"derham.tex","source_line":1426,"source_end_line":1438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1426-L1438","statement_sha256":"1670d7a11c6a1fb70392ea3a50ce20ba7d185bf4665113da9e357e3376296cf3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9011,"rank":9011,"depth":26,"x":2571.011,"y":1190.753,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMI","tag":"0FMI","title":"de Rham cohomology of projective space · Lemma 0FMI","summary":"We have H^q(P^n_A, Ω^p) = 0 unless 0 ≤ p = q ≤ n. For 0 ≤ p ≤ n the A-module H^p(P^n_A, Ω^p) free of rank 1 with basis element c_1^Hodge(O(1))^p.","statement_latex":"We have $H^q(\\mathbf{P}^n_A, \\Omega^p) = 0$\nunless $0 \\leq p = q \\leq n$. For $0 \\leq p \\leq n$ the $A$-module\n$H^p(\\mathbf{P}^n_A, \\Omega^p)$ free of rank $1$ with basis element\n$c_1^{Hodge}(\\mathcal{O}(1))^p$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology of projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMI","source_file":"derham.tex","source_line":1487,"source_end_line":1493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1487-L1493","statement_sha256":"2ec66dece821e2ce636bc20de355ad4a470e33a3ff951039b1fdd47799f7389c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9012,"rank":9012,"depth":27,"x":2371.693,"y":1154.268,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMJ","tag":"0FMJ","title":"de Rham cohomology of projective space · Lemma 0FMJ","summary":"For 0 ≤ i ≤ n the de Rham cohomology H^2i_dR(P^n_A/A) is a free A-module of rank 1 with basis element c_1^dR(O(1))^i. In all other degrees the de Rham cohomology of P^n_A over A is zero.","statement_latex":"For $0 \\leq i \\leq n$ the de Rham cohomology\n$H^{2i}_{dR}(\\mathbf{P}^n_A/A)$ is a free $A$-module of rank $1$\nwith basis element $c_1^{dR}(\\mathcal{O}(1))^i$.\nIn all other degrees the de Rham cohomology of $\\mathbf{P}^n_A$\nover $A$ is zero.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"de Rham cohomology of projective space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMJ","source_file":"derham.tex","source_line":1584,"source_end_line":1591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1584-L1591","statement_sha256":"8b939e362856cdd4136c8b928e4961d3817ad2144e0dbec2ea01ebd25a1a162a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9013,"rank":9013,"depth":28,"x":2548.632,"y":1067.769,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMM","tag":"0FMM","title":"The spectral sequence for a smooth morphism · Lemma 0FMM","summary":"Let f : X → Y be a quasi-compact, quasi-separated, and smooth morphism of schemes over a base scheme S. There is a bounded spectral sequence with first page E_1^p, q = H^q(Ω^p_Y/S ⊗_O_Y^L Rf_*Ω^bullet_X/Y) converging to R^p + qf_*Ω^bullet_X/S.","statement_latex":"Let $f : X \\to Y$ be a quasi-compact, quasi-separated, and smooth\nmorphism of schemes over a base scheme $S$. There is a bounded spectral\nsequence with first page\n$$\nE_1^{p, q} =\nH^q(\\Omega^p_{Y/S} \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} Rf_*\\Omega^\\bullet_{X/Y})\n$$\nconverging to $R^{p + q}f_*\\Omega^\\bullet_{X/S}$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"The spectral sequence for a smooth morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMM","source_file":"derham.tex","source_line":1693,"source_end_line":1703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1693-L1703","statement_sha256":"f38e4770c6e15223eb9411819b99c845695a644dd4b247242a89321ac885a46f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9014,"rank":9014,"depth":33,"x":2487.505,"y":1232.529,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMP","tag":"0FMP","title":"Leray-Hirsch type theorems · Lemma 0FMP","summary":"Let f : X → Y be a smooth proper morphism of schemes. Let N and n_1, …, n_N ≥ 0 be integers and let xi_i ∈ H^n_i_dR(X/Y), 1 ≤ i ≤ N. Assume for all points y ∈ Y the images of xi_1, …, xi_N in H^*_dR(X_y/y) form a basis over kappa(y). Then the map bigoplus_i = 1^N O_Y[-n_i] → Rf_*Ω^bullet_X/Y associated to xi_1, …, xi_N is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a smooth proper morphism of schemes.\nLet $N$ and $n_1, \\ldots, n_N \\geq 0$ be integers and let\n$\\xi_i \\in H^{n_i}_{dR}(X/Y)$, $1 \\leq i \\leq N$.\nAssume for all points $y \\in Y$ the images of $\\xi_1, \\ldots, \\xi_N$\nin $H^*_{dR}(X_y/y)$ form a basis over $\\kappa(y)$. Then the map\n$$\n\\bigoplus\\nolimits_{i = 1}^N \\mathcal{O}_Y[-n_i]\n\\longrightarrow\nRf_*\\Omega^\\bullet_{X/Y}\n$$\nassociated to $\\xi_1, \\ldots, \\xi_N$ is an isomorphism.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Leray-Hirsch type theorems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMP","source_file":"derham.tex","source_line":1769,"source_end_line":1782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1769-L1782","statement_sha256":"dbed9e86c9427cf56c048720d5257472b7fe241ce1271ac13a03b2589a9b50b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9015,"rank":9015,"depth":40,"x":2399.779,"y":1075.806,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUM","tag":"0FUM","title":"Leray-Hirsch type theorems · Lemma 0FUM","summary":"Let f : X → Y be a smooth proper morphism of schemes over a base S. Assume • Y and S are affine, and • there exist integers N and n_1, …, n_N ≥ 0 and xi_i ∈ H^n_i_dR(X/S), 1 ≤ i ≤ N such that for all points y ∈ Y the images of xi_1, …, xi_N in H^*_dR(X_y/y) form a basis over kappa(y). Then the map bigoplus_i = 1^N H^*_dR(Y/S) → H^*_dR(X/S), (a_1, …, a_N) ↦ ∑ xi_i ∪ f^*a_i is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a smooth proper morphism of schemes over a base $S$.\nAssume\n\\begin{enumerate}\n\\item $Y$ and $S$ are affine, and\n\\item there exist integers $N$ and $n_1, \\ldots, n_N \\geq 0$ and\n$\\xi_i \\in H^{n_i}_{dR}(X/S)$, $1 \\leq i \\leq N$ such that\nfor all points $y \\in Y$ the images of $\\xi_1, \\ldots, \\xi_N$\nin $H^*_{dR}(X_y/y)$ form a basis over $\\kappa(y)$.\n\\end{enumerate}\nThen the map\n$$\n\\bigoplus\\nolimits_{i = 1}^N H^*_{dR}(Y/S) \\longrightarrow\nH^*_{dR}(X/S), \\quad\n(a_1, \\ldots, a_N) \\longmapsto  \\sum \\xi_i \\cup f^*a_i\n$$\nis an isomorphism.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Leray-Hirsch type theorems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUM","source_file":"derham.tex","source_line":1806,"source_end_line":1824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1806-L1824","statement_sha256":"301a7bdfa9fe3cee705324f5f1550a7b1a89ae00a30d4fe3b3226d355b739b26","origin":"The Stacks Project","memory_eligible":false,"source_rank":9016,"rank":9016,"depth":41,"x":2591.16,"y":1141.822,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMR","tag":"0FMR","title":"Leray-Hirsch type theorems · Proposition 0FMR","summary":"Let f : X → Y be a smooth proper morphism of schemes over a base S. Let N and n_1, …, n_N ≥ 0 be integers and let xi_i ∈ H^n_i_dR(X/S), 1 ≤ i ≤ N. Assume for all points y ∈ Y the images of xi_1, …, xi_N in H^*_dR(X_y/y) form a basis over kappa(y). The map tilde xi = bigoplus tilde xi_i[-n_i] : bigoplus Ω^bullet_Y/S[-n_i] → Rf_*Ω^bullet_X/S (see proof) is an isomorphism in D(Y, (Y → S)^-1O_S) and correspondingly the map bigoplus_i = 1^N H^*_dR(Y/S) → H^*_dR(X/S), (a_1, …,…","statement_latex":"Let $f : X \\to Y$ be a smooth proper morphism of schemes over a base $S$.\nLet $N$ and $n_1, \\ldots, n_N \\geq 0$ be integers and let\n$\\xi_i \\in H^{n_i}_{dR}(X/S)$, $1 \\leq i \\leq N$.\nAssume for all points $y \\in Y$ the images of $\\xi_1, \\ldots, \\xi_N$\nin $H^*_{dR}(X_y/y)$ form a basis over $\\kappa(y)$. The map\n$$\n\\tilde \\xi = \\bigoplus \\tilde \\xi_i[-n_i] :\n\\bigoplus \\Omega^\\bullet_{Y/S}[-n_i]\n\\longrightarrow\nRf_*\\Omega^\\bullet_{X/S}\n$$\n(see proof) is an isomorphism in $D(Y, (Y \\to S)^{-1}\\mathcal{O}_S)$ and\ncorrespondingly the map\n$$\n\\bigoplus\\nolimits_{i = 1}^N H^*_{dR}(Y/S) \\longrightarrow\nH^*_{dR}(X/S), \\quad\n(a_1, \\ldots, a_N) \\longmapsto  \\sum \\xi_i \\cup f^*a_i\n$$\nis an isomorphism.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Leray-Hirsch type theorems","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMR","source_file":"derham.tex","source_line":1940,"source_end_line":1961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L1940-L1961","statement_sha256":"bb2f6420027914b100d17dcb77815ef46660c72e1aa104363dd4ad0eae44f74d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9017,"rank":9017,"depth":42,"x":2396.281,"y":1201.943,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMT","tag":"0FMT","title":"Projective space bundle formula · Proposition 0FMT","summary":"Let X → S be a morphism of schemes. Let E be a locally free O_X-module of constant rank r. Consider the morphism p : P = P(E) → X. Then the map bigoplus_i = 0, …, r - 1 H^*_dR(X/S) → H^*_dR(P/S) given by the rule (a_0, …, a_r - 1) ↦ ∑_i = 0, …, r - 1 c_1^dR(O_P(1))^i ∪ p^*(a_i) is an isomorphism.","statement_latex":"Let $X \\to S$ be a morphism of schemes. Let $\\mathcal{E}$ be a locally\nfree $\\mathcal{O}_X$-module of constant rank $r$. Consider the morphism\n$p : P = \\mathbf{P}(\\mathcal{E}) \\to X$.\nThen the map\n$$\n\\bigoplus\\nolimits_{i = 0, \\ldots, r - 1} H^*_{dR}(X/S)\n\\longrightarrow\nH^*_{dR}(P/S)\n$$\ngiven by the rule\n$$\n(a_0, \\ldots, a_{r - 1}) \\longmapsto\n\\sum\\nolimits_{i = 0, \\ldots, r - 1} c_1^{dR}(\\mathcal{O}_P(1))^i \\cup p^*(a_i)\n$$\nis an isomorphism.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Projective space bundle formula","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMT","source_file":"derham.tex","source_line":2028,"source_end_line":2045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2028-L2045","statement_sha256":"6c13479fc5c511ace86887f7a7adffcc2e1116ed2fb8c9e1b1e13597f77622c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9018,"rank":9018,"depth":43,"x":2491.961,"y":1046.502,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMV","tag":"0FMV","title":"Log poles along a divisor · Definition 0FMV","summary":"Let X → S be a morphism of schemes. Let Y ⊂ X be an effective Cartier divisor. We say the de Rham complex of log poles is defined for Y ⊂ X over S if for all y ∈ Y and local equation f ∈ O_X, y of Y we have • O_X, y → Ω_X/S, y, g ↦ g df is a split injection, and • Ω^p_X/S, y is f-torsion free for all p.","statement_latex":"Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an\neffective Cartier divisor. We say the\n{\\it de Rham complex of log poles is defined for $Y \\subset X$ over $S$}\nif for all $y \\in Y$ and local equation $f \\in \\mathcal{O}_{X, y}$\nof $Y$ we have\n\\begin{enumerate}\n\\item $\\mathcal{O}_{X, y} \\to \\Omega_{X/S, y}$, $g \\mapsto g \\text{d}f$\nis a split injection, and\n\\item $\\Omega^p_{X/S, y}$ is $f$-torsion free for all $p$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Log poles along a divisor","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMV","source_file":"derham.tex","source_line":2155,"source_end_line":2167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2155-L2167","statement_sha256":"169e688994a63df9c5d5a0035e0afdeb6514c1070d28368dccffd2f28dc4438e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9019,"rank":9019,"depth":0,"x":2546.593,"y":1215.987,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMW","tag":"0FMW","title":"Log poles along a divisor · Lemma 0FMW","summary":"Let X → S be a morphism of schemes. Let Y ⊂ X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y ⊂ X over S. There is a canonical short exact sequence of complexes 0 → Ω^bullet_X/S → Ω^bullet_X/S(log Y) → Ω^bullet_Y/S[-1] → 0","statement_latex":"Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an\neffective Cartier divisor.\nAssume the de Rham complex of log poles is defined for $Y \\subset X$ over $S$.\nThere is a canonical short exact sequence\nof complexes\n$$\n0 \\to \\Omega^\\bullet_{X/S} \\to\n\\Omega^\\bullet_{X/S}(\\log Y) \\to\n\\Omega^\\bullet_{Y/S}[-1] \\to 0\n$$","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Log poles along a divisor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMW","source_file":"derham.tex","source_line":2173,"source_end_line":2185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2173-L2185","statement_sha256":"f9198533c91e4810705a34551aeade4d9933efeb0bd96bfd7b8f16c6060d8d3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9020,"rank":9020,"depth":1,"x":2369.418,"y":1121.693,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUA","tag":"0FUA","title":"Log poles along a divisor · Definition 0FUA","summary":"Let X → S be a morphism of schemes. Let Y ⊂ X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y ⊂ X over S. Then the complex Ω^bullet_X/S(log Y) constructed in Lemma [Tag 0FMW] is the de Rham complex of log poles for Y ⊂ X over S.","statement_latex":"Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an\neffective Cartier divisor. Assume the de Rham complex of log poles\nis defined for $Y \\subset X$ over $S$. Then the complex\n$$\n\\Omega^\\bullet_{X/S}(\\log Y)\n$$\nconstructed in Lemma \\ref{lemma-log-complex} is the\n{\\it de Rham complex of log poles for $Y \\subset X$ over $S$}.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Log poles along a divisor","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUA","source_file":"derham.tex","source_line":2254,"source_end_line":2264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2254-L2264","statement_sha256":"2baa611b55a35575dcdc7f75f9c940e1fb5425fe487ee98da1e2c6600a008f14","origin":"The Stacks Project","memory_eligible":false,"source_rank":9021,"rank":9021,"depth":2,"x":2576.585,"y":1090.588,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUP","tag":"0FUP","title":"Log poles along a divisor · Lemma 0FUP","summary":"Let p : X → S be a morphism of schemes. Let Y ⊂ X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y ⊂ X over S. • The maps wedge : Ω^p_X/S × Ω^q_X/S → Ω^p + q_X/S extend uniquely to O_X-bilinear maps wedge : Ω^p_X/S(log Y) × Ω^q_X/S(log Y) → Ω^p + q_X/S(log Y) satisfying the Leibniz rule d(ω wedge eta) = d(ω) wedge eta + (-1)^deg(ω) ω wedge d(eta), • with multiplication as in (1) the map Ω^bullet_X/S → Ω^bullet_X/S(log(Y) is a…","statement_latex":"Let $p : X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an\neffective Cartier divisor. Assume the de Rham complex of log poles\nis defined for $Y \\subset X$ over $S$.\n\\begin{enumerate}\n\\item The maps\n$\\wedge : \\Omega^p_{X/S} \\times \\Omega^q_{X/S} \\to \\Omega^{p + q}_{X/S}$\nextend uniquely to $\\mathcal{O}_X$-bilinear maps\n$$\n\\wedge : \\Omega^p_{X/S}(\\log Y) \\times \\Omega^q_{X/S}(\\log Y)\n\\to \\Omega^{p + q}_{X/S}(\\log Y)\n$$\nsatisfying the Leibniz rule\n$\n\\text{d}(\\omega \\wedge \\eta) = \\text{d}(\\omega) \\wedge \\eta +\n(-1)^{\\deg(\\omega)} \\omega \\wedge \\text{d}(\\eta)$,\n\\item with multiplication as in (1) the map\n$\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}(\\log(Y)$\nis a homomorphism of differential graded $\\mathcal{O}_S$-algebras,\n\\item via the maps in (1) we have $\\Omega^p_{X/S}(\\log Y) =\n\\wedge^p(\\Omega^1_{X/S}(\\log Y))$, and\n\\item the map\n$\\text{Res} : \\Omega^\\bullet_{X/S}(\\log Y) \\to \\Omega^\\bullet_{Y/S}[-1]$\nsatisfies\n$$\n\\text{Res}(\\omega \\wedge \\eta) = \\text{Res}(\\omega) \\wedge \\eta|_Y\n$$\nfor $\\omega$ a local section of $\\Omega^p_{X/S}(\\log Y)$ and $\\eta$\na local section of $\\Omega^q_{X/S}$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Log poles along a divisor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUP","source_file":"derham.tex","source_line":2269,"source_end_line":2300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2269-L2300","statement_sha256":"ad3edb58cbb23af2d38485b2a657d5aa43542efea214531a51de2791116c9e86","origin":"The Stacks Project","memory_eligible":false,"source_rank":9022,"rank":9022,"depth":2,"x":2448.411,"y":1231.543,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUQ","tag":"0FUQ","title":"Log poles along a divisor · Lemma 0FUQ","summary":"Let f : X → S be a morphism of schemes. Let Y ⊂ X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y ⊂ X over S. Denote δ : Ω^bullet_Y/S → Ω^bullet_X/S[2] in D(X, f^-1O_S) the \"boundary\" map coming from the short exact sequence in Lemma [Tag 0FMW]. Denote xi' : Ω^bullet_X/S → Ω^bullet_X/S[2] in D(X, f^-1O_S) the map of Remark [Tag 0FU7] corresponding to xi = c_1^dR(O_X(-Y)). Denote zeta' : Ω^bullet_Y/S → Ω^bullet_Y/S[2] in D(Y,…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an effective\nCartier divisor. Assume the de Rham complex of log poles is defined for\n$Y \\subset X$ over $S$. Denote\n$$\n\\delta : \\Omega^\\bullet_{Y/S} \\to \\Omega^\\bullet_{X/S}[2]\n$$\nin $D(X, f^{-1}\\mathcal{O}_S)$ the ``boundary'' map coming from the\nshort exact sequence in Lemma \\ref{lemma-log-complex}. Denote\n$$\n\\xi' : \\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X/S}[2]\n$$\nin $D(X, f^{-1}\\mathcal{O}_S)$ the map of\nRemark \\ref{remark-cup-product-as-a-map}\ncorresponding to $\\xi = c_1^{dR}(\\mathcal{O}_X(-Y))$. Denote\n$$\n\\zeta' : \\Omega^\\bullet_{Y/S} \\to \\Omega^\\bullet_{Y/S}[2]\n$$\nin $D(Y, f|_Y^{-1}\\mathcal{O}_S)$ the map of\nRemark \\ref{remark-cup-product-as-a-map} corresponding to\n$\\zeta = c_1^{dR}(\\mathcal{O}_X(-Y)|_Y)$. Then the diagram\n$$\n\\xymatrix{\n\\Omega^\\bullet_{X/S} \\ar[d]_{\\xi'} \\ar[r] &\n\\Omega^\\bullet_{Y/S} \\ar[d]^{\\zeta'} \\ar[ld]_\\delta \\\\\n\\Omega^\\bullet_{X/S}[2] \\ar[r] &\n\\Omega^\\bullet_{Y/S}[2]\n}\n$$\nis commutative in $D(X, f^{-1}\\mathcal{O}_S)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Log poles along a divisor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUQ","source_file":"derham.tex","source_line":2342,"source_end_line":2373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2342-L2373","statement_sha256":"bffe22fbe9db8feb7ad39bd668ec04ec48c50f923eedb3a2e7354a3f3dda34f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9023,"rank":9023,"depth":3,"x":2429.511,"y":1054.292,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMX","tag":"0FMX","title":"Log poles along a divisor · Lemma 0FMX","summary":"Let X → S be a morphism of schemes. Let Y ⊂ X be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for Y ⊂ X over S. Let b ∈ H^m_dR(X/S) be a de Rham cohomology class whose restriction to Y is zero. Then c_1^dR(O_X(Y)) ∪ b = 0 in H^m + 2_dR(X/S).","statement_latex":"Let $X \\to S$ be a morphism of schemes. Let $Y \\subset X$ be an effective\nCartier divisor. Assume the de Rham complex of log poles is defined for\n$Y \\subset X$ over $S$. Let $b \\in H^m_{dR}(X/S)$ be a de Rham cohomology\nclass whose restriction to $Y$ is zero. Then\n$c_1^{dR}(\\mathcal{O}_X(Y)) \\cup b = 0$ in $H^{m + 2}_{dR}(X/S)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Log poles along a divisor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMX","source_file":"derham.tex","source_line":2459,"source_end_line":2466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2459-L2466","statement_sha256":"c382657b85d1a9ee5e54db2141a1278e15fcaddbcda3ca09a2e87c05e00af629","origin":"The Stacks Project","memory_eligible":false,"source_rank":9024,"rank":9024,"depth":4,"x":2586.502,"y":1174.664,"cluster":"local-crystalline-methods"},{"id":"stacks:0FMY","tag":"0FMY","title":"Log poles along a divisor · Lemma 0FMY","summary":"Let X → T → S be morphisms of schemes. Let Y ⊂ X be an effective Cartier divisor. If both X → T and Y → T are smooth, then the de Rham complex of log poles is defined for Y ⊂ X over S.","statement_latex":"Let $X \\to T \\to S$ be morphisms of schemes. Let $Y \\subset X$ be an effective\nCartier divisor. If both $X \\to T$ and $Y \\to T$ are smooth, then\nthe de Rham complex of log poles is defined for $Y \\subset X$ over $S$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Log poles along a divisor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FMY","source_file":"derham.tex","source_line":2479,"source_end_line":2484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2479-L2484","statement_sha256":"7a7cd45cc47dbba8766f8d79c9efec0f9dda6008012517a726c2923964632054","origin":"The Stacks Project","memory_eligible":false,"source_rank":9025,"rank":9025,"depth":46,"x":2373.245,"y":1174.984,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUR","tag":"0FUR","title":"Calculations · Lemma 0FUR","summary":"For a ≥ 0 we have • the map Ω^a_X/S → b_*Ω^a_L/S is an isomorphism, • the map Ω^a_Z/S → p_*Ω^a_P/S is an isomorphism, and • the map Rb_*Ω^a_L/S → i_*Rp_*Ω^a_P/S is an isomorphism on cohomology sheaves in degree ≥ 1.","statement_latex":"For $a \\geq 0$ we have\n\\begin{enumerate}\n\\item the map\n$\\Omega^a_{X/S} \\to b_*\\Omega^a_{L/S}$ is an isomorphism,\n\\item the map $\\Omega^a_{Z/S} \\to p_*\\Omega^a_{P/S}$ is an isomorphism,\nand\n\\item the map $Rb_*\\Omega^a_{L/S} \\to i_*Rp_*\\Omega^a_{P/S}$ is an isomorphism\non cohomology sheaves in degree $\\geq 1$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Calculations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUR","source_file":"derham.tex","source_line":2637,"source_end_line":2648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2637-L2648","statement_sha256":"42185faba29bd997d179fd89951c2eeeea320238375c39c459f424d7ba4730ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":9026,"rank":9026,"depth":28,"x":2530.749,"y":1053.349,"cluster":"local-crystalline-methods"},{"id":"stacks:0G5G","tag":"0G5G","title":"Calculations · Lemma 0G5G","summary":"For a ≥ 0 there are canonical maps b^*Ω^a_X/S → Ω^a_L/S → b^*Ω^a_X/S ⊗_O_L O_L((n - 1)E) whose composition is induced by the inclusion O_L ⊂ O_L((n - 1)E).","statement_latex":"For $a \\geq 0$ there are canonical maps\n$$\nb^*\\Omega^a_{X/S} \\longrightarrow\n\\Omega^a_{L/S} \\longrightarrow\nb^*\\Omega^a_{X/S} \\otimes_{\\mathcal{O}_L} \\mathcal{O}_L((n - 1)E)\n$$\nwhose composition is induced by the inclusion\n$\\mathcal{O}_L \\subset \\mathcal{O}_L((n - 1)E)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Calculations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5G","source_file":"derham.tex","source_line":2733,"source_end_line":2743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2733-L2743","statement_sha256":"d0bb0d878cef598ac154906ee33293dfd16ca55e6a335c4e57036f02de7c3b9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9027,"rank":9027,"depth":4,"x":2512.364,"y":1232.989,"cluster":"local-crystalline-methods"},{"id":"stacks:0G5H","tag":"0G5H","title":"Calculations · Lemma 0G5H","summary":"Let E = 0(P) be the exceptional divisor of the blowing up b. For any locally free O_X-module E and 0 ≤ i ≤ n - 1 the map E → Rb_*(b^*E ⊗_O_L O_L(iE)) is an isomorphism in D(O_X).","statement_latex":"Let $E = 0(P)$ be the exceptional divisor of the blowing up $b$.\nFor any locally free $\\mathcal{O}_X$-module $\\mathcal{E}$ and\n$0 \\leq i \\leq n - 1$ the map\n$$\n\\mathcal{E}\n\\longrightarrow\nRb_*(b^*\\mathcal{E} \\otimes_{\\mathcal{O}_L} \\mathcal{O}_L(iE))\n$$\nis an isomorphism in $D(\\mathcal{O}_X)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Calculations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5H","source_file":"derham.tex","source_line":2782,"source_end_line":2793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2782-L2793","statement_sha256":"7d7b58ecf120af6cd1f361f19c458a842bdc2d81c0ef31c622c19c0d4f7350f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9028,"rank":9028,"depth":29,"x":2381.041,"y":1089.635,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUU","tag":"0FUU","title":"Blowing up and de Rham cohomology · Lemma 0FUU","summary":"With notation as in More on Morphisms, Lemma [Tag 0FUT] for a ≥ 0 we have • the map Ω^a_X/S → b_*Ω^a_X'/S is an isomorphism, • the map Ω^a_Z/S → p_*Ω^a_E/S is an isomorphism, • the map Rb_*Ω^a_X'/S → i_*Rp_*Ω^a_E/S is an isomorphism on cohomology sheaves in degree ≥ 1.","statement_latex":"With notation as in More on Morphisms, Lemma \\ref{more-morphisms-lemma-blowup}\nfor $a \\geq 0$ we have\n\\begin{enumerate}\n\\item the map\n$\\Omega^a_{X/S} \\to b_*\\Omega^a_{X'/S}$ is an isomorphism,\n\\item the map $\\Omega^a_{Z/S} \\to p_*\\Omega^a_{E/S}$ is an isomorphism,\n\\item the map $Rb_*\\Omega^a_{X'/S} \\to i_*Rp_*\\Omega^a_{E/S}$ is an isomorphism\non cohomology sheaves in degree $\\geq 1$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Blowing up and de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUU","source_file":"derham.tex","source_line":2881,"source_end_line":2892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2881-L2892","statement_sha256":"9fa5f23015534acc6f08b00afe9eee59bd8a6cfaec4ed55123c726a63d37dd70","origin":"The Stacks Project","memory_eligible":false,"source_rank":9029,"rank":9029,"depth":47,"x":2593.832,"y":1120.926,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUV","tag":"0FUV","title":"Blowing up and de Rham cohomology · Lemma 0FUV","summary":"With notation as in More on Morphisms, Lemma [Tag 0FUT] and denoting f : X → S the structure morphism there is a canonical distinguished triangle Ω^bullet_X/S → Rb_*(Ω^bullet_X'/S) ⊕ i_*Ω^bullet_Z/S → i_*Rp_*(Ω^bullet_E/S) → Ω^bullet_X/S[1] in D(X, f^-1O_S) where the four maps Ω^bullet_X/S & → & Rb_*(Ω^bullet_X'/S), Ω^bullet_X/S & → & i_*Ω^bullet_Z/S, Rb_*(Ω^bullet_X'/S) & → & i_*Rp_*(Ω^bullet_E/S), i_*Ω^bullet_Z/S & → & i_*Rp_*(Ω^bullet_E/S) are the canonical ones…","statement_latex":"With notation as in More on Morphisms, Lemma \\ref{more-morphisms-lemma-blowup}\nand denoting $f : X \\to S$ the structure morphism there is a canonical\ndistinguished triangle\n$$\n\\Omega^\\bullet_{X/S} \\to\nRb_*(\\Omega^\\bullet_{X'/S}) \\oplus i_*\\Omega^\\bullet_{Z/S} \\to\ni_*Rp_*(\\Omega^\\bullet_{E/S}) \\to\n\\Omega^\\bullet_{X/S}[1]\n$$\nin $D(X, f^{-1}\\mathcal{O}_S)$ where the four maps\n$$\n\\begin{matrix}\n\\Omega^\\bullet_{X/S} & \\to & Rb_*(\\Omega^\\bullet_{X'/S}), \\\\\n\\Omega^\\bullet_{X/S} & \\to & i_*\\Omega^\\bullet_{Z/S}, \\\\\nRb_*(\\Omega^\\bullet_{X'/S}) & \\to & i_*Rp_*(\\Omega^\\bullet_{E/S}), \\\\\ni_*\\Omega^\\bullet_{Z/S} & \\to & i_*Rp_*(\\Omega^\\bullet_{E/S})\n\\end{matrix}\n$$\nare the canonical ones (Section \\ref{section-de-rham-complex}),\nexcept with sign reversed for one of them.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Blowing up and de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUV","source_file":"derham.tex","source_line":2914,"source_end_line":2936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2914-L2936","statement_sha256":"6fc586ea6cf404d64830b0ddf9e99bb18063fdca624d88bd21c7ce2b55f35d75","origin":"The Stacks Project","memory_eligible":false,"source_rank":9030,"rank":9030,"depth":48,"x":2411.185,"y":1218.903,"cluster":"local-crystalline-methods"},{"id":"stacks:0FUW","tag":"0FUW","title":"Blowing up and de Rham cohomology · Proposition 0FUW","summary":"With notation as in More on Morphisms, Lemma [Tag 0FUT] the map Ω^bullet_X/S → Rb_*Ω^bullet_X'/S has a splitting in D(X, (X → S)^-1O_S).","statement_latex":"With notation as in More on Morphisms, Lemma \\ref{more-morphisms-lemma-blowup}\nthe map $\\Omega^\\bullet_{X/S} \\to Rb_*\\Omega^\\bullet_{X'/S}$\nhas a splitting in $D(X, (X \\to S)^{-1}\\mathcal{O}_S)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Blowing up and de Rham cohomology","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FUW","source_file":"derham.tex","source_line":2970,"source_end_line":2975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L2970-L2975","statement_sha256":"d65c976ae678f8ecd2e3bbd5135d1fbb84eab9c08b60f1eae92a68affc83e8f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9031,"rank":9031,"depth":49,"x":2467.252,"y":1042.466,"cluster":"local-crystalline-methods"},{"id":"stacks:0G5I","tag":"0G5I","title":"Blowing up and de Rham cohomology · Lemma 0G5I","summary":"Let i : Z → X be a closed immersion of schemes which is regular of codimension c. Then Ext^q_O_X(i_*F, E) = 0 for q < c for E locally free on X and F any O_Z-module.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes which is regular of\ncodimension $c$. Then $\\Ext^q_{\\mathcal{O}_X}(i_*\\mathcal{F}, \\mathcal{E}) = 0$\nfor $q < c$ for $\\mathcal{E}$ locally free on $X$ and $\\mathcal{F}$\nany $\\mathcal{O}_Z$-module.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Blowing up and de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5I","source_file":"derham.tex","source_line":3103,"source_end_line":3109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L3103-L3109","statement_sha256":"e037b898771436340e2d3d3d821f5c4830eb0860f7261c77529c04307b5b16dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9032,"rank":9032,"depth":0,"x":2568.107,"y":1204.887,"cluster":"local-crystalline-methods"},{"id":"stacks:0G5J","tag":"0G5J","title":"Blowing up and de Rham cohomology · Lemma 0G5J","summary":"With notation as in More on Morphisms, Lemma [Tag 0FUT] for a ≥ 0 there is a unique arrow Rb_*Ω^a_X'/S → Ω^a_X/S in D(O_X) whose composition with Ω^a_X/S → Rb_*Ω^a_X'/S is the identity on Ω^a_X/S.","statement_latex":"With notation as in More on Morphisms, Lemma \\ref{more-morphisms-lemma-blowup}\nfor $a \\geq 0$ there is a unique arrow\n$Rb_*\\Omega^a_{X'/S} \\to \\Omega^a_{X/S}$ in $D(\\mathcal{O}_X)$\nwhose composition with $\\Omega^a_{X/S} \\to Rb_*\\Omega^a_{X'/S}$\nis the identity on $\\Omega^a_{X/S}$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Blowing up and de Rham cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5J","source_file":"derham.tex","source_line":3142,"source_end_line":3149,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L3142-L3149","statement_sha256":"2233bb81f37bfd1d0396a0ddd365af31343e2ed18dfb944fcd48992bb01a8410","origin":"The Stacks Project","memory_eligible":false,"source_rank":9033,"rank":9033,"depth":48,"x":2362.489,"y":1142.154,"cluster":"local-crystalline-methods"},{"id":"stacks:0FL8","tag":"0FL8","title":"Comparing sheaves of differential forms · Lemma 0FL8","summary":"Let R be a ring and consider a commutative diagram xymatrix 0 ar[r] & K^0 ar[r] & L^0 ar[r] & M^0 ar[r] & 0 & & L^-1 ar[u]_∂ ar@=[r] & M^-1 ar[u] of R-modules with exact top row and M^0 and M^-1 finite free of the same rank. For p ≥ 0 there are canonical maps c^p : wedge^p(H^0(L^bullet)) → wedge^p(K^0) ⊗_R det(M^bullet) with the following properties: • c^0(1) = δ(M^bullet), and • c^q + p(ω wedge eta) = ω wedge c^p(eta) for ω ∈ wedge^q(K^0) and eta ∈…","statement_latex":"Let $R$ be a ring and consider a commutative diagram\n$$\n\\xymatrix{\n0 \\ar[r] &\nK^0 \\ar[r] &\nL^0 \\ar[r] &\nM^0 \\ar[r] & 0 \\\\\n& & L^{-1} \\ar[u]_\\partial \\ar@{=}[r] &\nM^{-1} \\ar[u]\n}\n$$\nof $R$-modules with exact top row and $M^0$ and $M^{-1}$\nfinite free of the same rank. For $p \\geq 0$ there are canonical maps\n$$\nc^p : \n\\wedge^p(H^0(L^\\bullet))\n\\longrightarrow\n\\wedge^p(K^0) \\otimes_R \\det(M^\\bullet)\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item $c^0(1) = \\delta(M^\\bullet)$, and\n\\item $c^{q + p}(\\omega \\wedge \\eta) = \\omega \\wedge c^p(\\eta)$\nfor $\\omega \\in \\wedge^q(K^0)$ and $\\eta \\in \\wedge^p(H^0(L^\\bullet))$.\n\\end{enumerate}\nIn particular, the composition of $c^p$ with the map\n$\\wedge^p(K^0) \\to \\wedge^p(H^0(L^\\bullet))$ is multiplication\nwith $\\delta(M^\\bullet)$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Comparing sheaves of differential forms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FL8","source_file":"derham.tex","source_line":3254,"source_end_line":3284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L3254-L3284","statement_sha256":"df057f56d39ac015409bf4a3cb47e309a26a817c07196a56ba63d54b15a8686c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9034,"rank":9034,"depth":0,"x":2565.18,"y":1071.524,"cluster":"local-crystalline-methods"},{"id":"stacks:0H9F","tag":"0H9F","title":"Comparing sheaves of differential forms · Lemma 0H9F","summary":"Let R_1 → R_2 be a ring homomorphism. For i = 1, 2 consider commutative diagrams xymatrix 0 ar[r] & K_i^0 ar[r] & L_i^0 ar[r] & M_i^0 ar[r] & 0 & & L_i^-1 ar[u]_∂_i ar@=[r] & M_i^-1 ar[u] of R_i-modules as in Lemma [Tag 0FL8]. Assume we have maps K_1^0 → K_2^0, L_1^j → L_2^j, M_1^j → M_2^j compatible with the given ring map R_1 → R_2 and compatible with the maps in the displayed diagrams. If the maps M_1^j ⊗_R_1 R_2 → M_2^j are isomorphisms, then the diagrams xymatrix…","statement_latex":"Let $R_1 \\to R_2$ be a ring homomorphism. For $i = 1, 2$ consider\ncommutative diagrams\n$$\n\\xymatrix{\n0 \\ar[r] &\nK_i^0 \\ar[r] &\nL_i^0 \\ar[r] &\nM_i^0 \\ar[r] & 0 \\\\\n& & L_i^{-1} \\ar[u]_{\\partial_i} \\ar@{=}[r] &\nM_i^{-1} \\ar[u]\n}\n$$\nof $R_i$-modules as in Lemma \\ref{lemma-funny-map}. Assume we have\nmaps $K_1^0 \\to K_2^0$, $L_1^j \\to L_2^j$, $M_1^j \\to M_2^j$\ncompatible with the given ring map $R_1 \\to R_2$ and\ncompatible with the maps in the displayed diagrams. If the maps\n$M_1^j \\otimes_{R_1} R_2 \\to M_2^j$ are isomorphisms, then\nthe diagrams\n$$\n\\xymatrix{ \n\\wedge^p(H^0(L_1^\\bullet)) \\ar[d]\n\\ar[r]_-{c^p} &\n\\wedge^p(K_1^0) \\otimes_{R_1} \\det(M_1^\\bullet) \\ar[d] \\\\\n\\wedge^p(H^0(L_2^\\bullet))\n\\ar[r]^-{c^p} &\n\\wedge^p(K_2^0) \\otimes_{R_2} \\det(M_2^\\bullet)\n}\n$$\ncommute.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Comparing sheaves of differential forms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9F","source_file":"derham.tex","source_line":3325,"source_end_line":3356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L3325-L3356","statement_sha256":"080f067d232ba1dbca36f2850e048a87ed243e54af4de99b4a65975bc0017998","origin":"The Stacks Project","memory_eligible":false,"source_rank":9035,"rank":9035,"depth":1,"x":2472.233,"y":1239.128,"cluster":"local-crystalline-methods"},{"id":"stacks:0H9G","tag":"0H9G","title":"Comparing sheaves of differential forms · Lemma 0H9G","summary":"Consider a commutative diagram xymatrix Y' ar[d]_f' ar[r]_b & Y ar[d]^f X' ar[r]^a & X of schemes which induces an isomorphism of Y' with an open subscheme of X' ×_X Y. Assume f is locally quasi-finite and syntomic and assume given maps c^p_Y/X : Ω^p_Y/Z → f^*Ω^p_X/Z ⊗_O_Y det(NL_Y/X) for p ≥ 0 satisfying ([Tag 0H9C]) and ([Tag 0H9D]). Then there is at most one collection of maps c^p_Y'/X' : Ω^p_Y'/Z → (f')^*Ω^p_X'/Z ⊗_O_Y' det(NL_Y'/X') for p ≥ 0 satisfying ([Tag 0H9C])…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nY' \\ar[d]_{f'} \\ar[r]_b & Y \\ar[d]^f \\\\\nX' \\ar[r]^a & X\n}\n$$\nof schemes which induces an isomorphism of $Y'$ with an open subscheme of\n$X' \\times_X Y$. Assume $f$ is locally quasi-finite and syntomic and\nassume given maps $c^p_{Y/X} : \\Omega^p_{Y/\\mathbf{Z}} \\to\nf^*\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_Y} \\det(\\NL_{Y/X})$\nfor $p \\geq 0$ satisfying\n(\\ref{item-degree-zero}) and (\\ref{item-multiplicative}).\nThen there is at most one collection of maps\n$c^p_{Y'/X'} : \\Omega^p_{Y'/\\mathbf{Z}} \\to\n(f')^*\\Omega^p_{X'/\\mathbf{Z}} \\otimes_{\\mathcal{O}_{Y'}} \\det(\\NL_{Y'/X'})$\nfor $p \\geq 0$\nsatisfying (\\ref{item-degree-zero}) and (\\ref{item-multiplicative})\nsuch that the diagrams\n$$\n\\xymatrix{\nb^*\\Omega^p_{Y/\\mathbf{Z}} \\ar[rr]_-{b^*c^p_{Y/X}} \\ar[d] & &\nb^*(f^*\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_Y} \\det(\\NL_{Y/X}))\n\\ar@{=}[r] &\n(f')^*a^*\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_Y} b^*\\det(\\NL_{Y/X}))\n\\ar[d] \\\\\n\\Omega^p_{Y'/\\mathbf{Z}} \\ar[rrr]^-{c^p_{Y'/X'}} & & &\n(f')^*\\Omega^p_{X'/\\mathbf{Z}} \\otimes_{\\mathcal{O}_{Y'}} \\det(\\NL_{Y'/X'})\n}\n$$\ncommute for all $p \\geq 0$. Here the vertical arrows use the maps\n$b^*\\Omega^p_{Y/\\mathbf{Z}} \\to \\Omega^p_{Y'/\\mathbf{Z}}$ and\n$a^*\\Omega^p_{X/\\mathbf{Z}} \\to \\Omega^p_{X'/\\mathbf{Z}}$\nof Section \\ref{section-de-rham-complex}\nas well as the identification $b^*\\det(\\NL_{Y/X}) = \\det(\\NL_{Y'/X'})$ of\nDiscriminants, Section \\ref{discriminant-section-tate-map}.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Comparing sheaves of differential forms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9G","source_file":"derham.tex","source_line":3405,"source_end_line":3443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L3405-L3443","statement_sha256":"84209e951a31851db42cff45d23e651357813db333e9b04adcd96121426edbed","origin":"The Stacks Project","memory_eligible":false,"source_rank":9036,"rank":9036,"depth":0,"x":2405.787,"y":1062.262,"cluster":"local-crystalline-methods"},{"id":"stacks:0H9H","tag":"0H9H","title":"Comparing sheaves of differential forms · Lemma 0H9H","summary":"Consider a commutative diagram xymatrix Y' ar[d]_f' ar[r]_b & Y ar[d]^f X' ar[r]^a & X of schemes which induces an isomorphism of Y' with an open subscheme of X' ×_X Y. Assume f is locally quasi-finite and syntomic. Then for every y' ∈ Y' we can find opens V' ⊂ Y', V ⊂ Y, U' ⊂ X, U ⊂ X with y' ∈ V', with f'(V') ⊂ U', b(V') ⊂ V, a(U') ⊂ U, f(V) ⊂ U, and such that for p ≥ 0 there are maps c^p_V/U and c^p_V'/U' satisfying ([Tag 0H9C]) and ([Tag 0H9D]) which are compatible…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nY' \\ar[d]_{f'} \\ar[r]_b & Y \\ar[d]^f \\\\\nX' \\ar[r]^a & X\n}\n$$\nof schemes which induces an isomorphism of $Y'$ with an open subscheme of\n$X' \\times_X Y$. Assume $f$ is locally quasi-finite and syntomic. Then for\nevery $y' \\in Y'$ we can find opens $V' \\subset Y'$,\n$V \\subset Y$, $U' \\subset X$, $U \\subset X$ with $y' \\in V'$, with\n$f'(V') \\subset U'$, $b(V') \\subset V$, $a(U') \\subset U$,\n$f(V) \\subset U$, and such that for $p \\geq 0$ there are maps\n$c^p_{V/U}$ and $c^p_{V'/U'}$ satisfying (\\ref{item-degree-zero})\nand (\\ref{item-multiplicative}) which are compatible with the diagram\n$$\n\\xymatrix{\nV' \\ar[d] \\ar[r]_b & V \\ar[d] \\\\\nU' \\ar[r] & U\n}\n$$\nin the sense explained in Lemma \\ref{lemma-base-change-Garel-upstairs}.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Comparing sheaves of differential forms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H9H","source_file":"derham.tex","source_line":3469,"source_end_line":3493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L3469-L3493","statement_sha256":"70b7ef513e1784a413fe9b409b5e0df361b37b966fa4fbb8604bc3aef0a2fe6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9037,"rank":9037,"depth":40,"x":2597.591,"y":1155.258,"cluster":"local-crystalline-methods"},{"id":"stacks:0FLA","tag":"0FLA","title":"Comparing sheaves of differential forms · Lemma 0FLA","summary":"There exists a unique rule that to every locally quasi-finite syntomic morphism of schemes f : Y → X assigns O_Y-module maps c^p_Y/X : Ω^p_Y/Z → f^*Ω^p_X/Z ⊗_O_Y det(NL_Y/X) for p ≥ 0 satisfying ([Tag 0H9C]), ([Tag 0H9D]), and ([Tag 0H9E]). In particular, the composition of c^p_Y/X with f^*Ω^p_X/Z → Ω^p_Y/Z is multiplication by δ(NL_Y/X).","statement_latex":"There exists a unique rule that to every locally quasi-finite syntomic\nmorphism of schemes $f : Y \\to X$ assigns $\\mathcal{O}_Y$-module maps\n$$\nc^p_{Y/X} :\n\\Omega^p_{Y/\\mathbf{Z}}\n\\longrightarrow\nf^*\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_Y} \\det(\\NL_{Y/X})\n$$\nfor $p \\geq 0$ satisfying (\\ref{item-degree-zero}), (\\ref{item-multiplicative}),\nand (\\ref{item-base-change}). In particular, the composition of\n$c^p_{Y/X}$ with $f^*\\Omega^p_{X/\\mathbf{Z}} \\to \\Omega^p_{Y/\\mathbf{Z}}$\nis multiplication by $\\delta(\\NL_{Y/X})$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Comparing sheaves of differential forms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLA","source_file":"derham.tex","source_line":3520,"source_end_line":3534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L3520-L3534","statement_sha256":"78b57bb775b95032f01c884bde3081f7e78ce124587a26b62de9b9bf714f782d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9038,"rank":9038,"depth":49,"x":2380.723,"y":1195.639,"cluster":"local-crystalline-methods"},{"id":"stacks:0FLB","tag":"0FLB","title":"Trace maps on de Rham complexes · Lemma 0FLB","summary":"There exists a unique rule that to every finite syntomic morphism of schemes f : Y → X assigns O_X-module maps Theta^p_Y/X : f_*Ω^p_Y/Z → Ω^p_X/Z satisfying the following properties • the composition with Ω^p_X/Z ⊗_O_X f_*O_Y → f_*Ω^p_Y/Z is equal to id ⊗ Trace_f where Trace_f : f_*O_Y → O_X is the map from Discriminants, Section [Tag 0BVH], • the rule is compatible with base change.","statement_latex":"There exists a unique rule that to every finite syntomic\nmorphism of schemes $f : Y \\to X$ assigns $\\mathcal{O}_X$-module maps\n$$\n\\Theta^p_{Y/X} :\nf_*\\Omega^p_{Y/\\mathbf{Z}}\n\\longrightarrow\n\\Omega^p_{X/\\mathbf{Z}}\n$$\nsatisfying the following properties\n\\begin{enumerate}\n\\item the composition with\n$\\Omega^p_{X/\\mathbf{Z}} \\otimes_{\\mathcal{O}_X} f_*\\mathcal{O}_Y\n\\to f_*\\Omega^p_{Y/\\mathbf{Z}}$ is equal to\n$\\text{id} \\otimes \\text{Trace}_f$\nwhere $\\text{Trace}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$\nis the map from\nDiscriminants, Section \\ref{discriminant-section-discriminant},\n\\item the rule is compatible with base change.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Trace maps on de Rham complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLB","source_file":"derham.tex","source_line":3820,"source_end_line":3841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L3820-L3841","statement_sha256":"becc5bf97b031cac889eecc3077c1d18e453ac893528d6920fb5cd7bf3624640","origin":"The Stacks Project","memory_eligible":false,"source_rank":9039,"rank":9039,"depth":50,"x":2508.546,"y":1042.351,"cluster":"local-crystalline-methods"},{"id":"stacks:0FLC","tag":"0FLC","title":"Trace maps on de Rham complexes · Proposition 0FLC","summary":"[Garel] Let f : Y → X be a finite syntomic morphism of schemes. The maps Theta^p_Y/X of Lemma [Tag 0FLB] define a map of complexes Theta_Y/X : f_*Ω^bullet_Y/Z → Ω^bullet_X/Z with the following properties • in degree 0 we get Trace_f : f_*O_Y → O_X, see Discriminants, Section [Tag 0BVH], • we have Theta_Y/X(ω wedge eta) = ω wedge Theta_Y/X(eta) for ω in Ω^bullet_X/Z and eta in f_*Ω^bullet_Y/Z, • if f is a morphism over a base scheme S, then Theta_Y/X induces a map of…","statement_latex":"\\begin{reference}\n\\cite{Garel}\n\\end{reference}\nLet $f : Y \\to X$ be a finite syntomic morphism of schemes.\nThe maps $\\Theta^p_{Y/X}$ of Lemma \\ref{lemma-Garel} define a map of complexes\n$$\n\\Theta_{Y/X} :\nf_*\\Omega^\\bullet_{Y/\\mathbf{Z}}\n\\longrightarrow\n\\Omega^\\bullet_{X/\\mathbf{Z}}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item in degree $0$ we get\n$\\text{Trace}_f : f_*\\mathcal{O}_Y \\to \\mathcal{O}_X$, see\nDiscriminants, Section \\ref{discriminant-section-discriminant},\n\\item we have\n$\\Theta_{Y/X}(\\omega \\wedge \\eta) = \\omega \\wedge \\Theta_{Y/X}(\\eta)$\nfor $\\omega$ in $\\Omega^\\bullet_{X/\\mathbf{Z}}$ and $\\eta$\nin $f_*\\Omega^\\bullet_{Y/\\mathbf{Z}}$,\n\\item if $f$ is a morphism over a base scheme $S$, then\n$\\Theta_{Y/X}$ induces a map of complexes\n$f_*\\Omega^\\bullet_{Y/S} \\to \\Omega^\\bullet_{X/S}$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Trace maps on de Rham complexes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FLC","source_file":"derham.tex","source_line":3970,"source_end_line":3996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L3970-L3996","statement_sha256":"f7007725d25950e97913ba911dd2a44b4c30792713437bd60222e43cb4c3d645","origin":"The Stacks Project","memory_eligible":false,"source_rank":9040,"rank":9040,"depth":51,"x":2537.647,"y":1228.464,"cluster":"local-crystalline-methods"},{"id":"stacks:0FW2","tag":"0FW2","title":"Trace maps on de Rham complexes · Lemma 0FW2","summary":"Let p be a prime number. Let X → S be a smooth morphism of relative dimension d of schemes in characteristic p. The relative Frobenius F_X/S : X → X^(p) of X/S (Varieties, Definition [Tag 0CC9]) is finite syntomic and the corresponding map Theta_X/X^(p) : F_X/S, *Ω^bullet_X/S → Ω^bullet_X^(p)/S is zero in all degrees except in degree d where it defines a surjection.","statement_latex":"Let $p$ be a prime number. Let $X \\to S$ be a smooth morphism\nof relative dimension $d$ of schemes in characteristic $p$.\nThe relative Frobenius $F_{X/S} : X \\to X^{(p)}$ of $X/S$\n(Varieties, Definition \\ref{varieties-definition-relative-frobenius})\nis finite syntomic and the corresponding map\n$$\n\\Theta_{X/X^{(p)}} :\nF_{X/S, *}\\Omega^\\bullet_{X/S} \\to \\Omega^\\bullet_{X^{(p)}/S}\n$$\nis zero in all degrees except in degree $d$ where it defines a\nsurjection.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Trace maps on de Rham complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FW2","source_file":"derham.tex","source_line":4150,"source_end_line":4163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L4150-L4163","statement_sha256":"731b2244ab75b82990bcdca345429a4f4747b04a809b271bbabd702023f8b5df","origin":"The Stacks Project","memory_eligible":false,"source_rank":9041,"rank":9041,"depth":46,"x":2366.017,"y":1107.383,"cluster":"local-crystalline-methods"},{"id":"stacks:0FW4","tag":"0FW4","title":"Poincaré duality · Lemma 0FW4","summary":"Let k be a field. Let X be a nonempty smooth proper scheme over k equidimensional of dimension d. There exists a k-linear map t : H^d(X, Ω^d_X/k) → k unique up to precomposing by multiplication by a unit of H^0(X, O_X) with the following property: for all p, q the pairing H^q(X, Ω^p_X/k) × H^d - q(X, Ω^d - p_X/k) → k, (xi, xi') ↦ t(xi ∪ xi') is perfect.","statement_latex":"Let $k$ be a field. Let $X$ be a nonempty smooth proper scheme over $k$\nequidimensional of dimension $d$. There exists a $k$-linear map\n$$\nt : H^d(X, \\Omega^d_{X/k}) \\longrightarrow k\n$$\nunique up to precomposing by multiplication by a unit of\n$H^0(X, \\mathcal{O}_X)$ with the following property: for all $p, q$ the pairing\n$$\nH^q(X, \\Omega^p_{X/k}) \\times H^{d - q}(X, \\Omega^{d - p}_{X/k})\n\\longrightarrow\nk, \\quad\n(\\xi, \\xi') \\longmapsto t(\\xi \\cup \\xi')\n$$\nis perfect.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Poincaré duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FW4","source_file":"derham.tex","source_line":4278,"source_end_line":4294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L4278-L4294","statement_sha256":"deaa149c643049a853bf794ac752323698c41350bfeb00ee3b1a0b0400ae986d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9042,"rank":9042,"depth":48,"x":2590.601,"y":1099.256,"cluster":"local-crystalline-methods"},{"id":"stacks:0FW5","tag":"0FW5","title":"Poincaré duality · Lemma 0FW5","summary":"Let k be a field. Let X be a smooth proper scheme over k. The map d : H^0(X, O_X) → H^0(X, Ω^1_X/k) is zero.","statement_latex":"Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. The map\n$$\n\\text{d} : H^0(X, \\mathcal{O}_X) \\to H^0(X, \\Omega^1_{X/k})\n$$\nis zero.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Poincaré duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FW5","source_file":"derham.tex","source_line":4330,"source_end_line":4337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L4330-L4337","statement_sha256":"319539d14b08649b7fcf856f3edbbc0cc99c09d75e2c4e634d8e45f00ab874f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9043,"rank":9043,"depth":44,"x":2431.068,"y":1233.072,"cluster":"local-crystalline-methods"},{"id":"stacks:0FW6","tag":"0FW6","title":"Poincaré duality · Lemma 0FW6","summary":"Let k be a field. Let X be a smooth proper scheme over k equidimensional of dimension d. The map d : H^d(X, Ω^d - 1_X/k) → H^d(X, Ω^d_X/k) is zero.","statement_latex":"Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$\nequidimensional of dimension $d$. The map\n$$\n\\text{d} : H^d(X, \\Omega^{d - 1}_{X/k}) \\to H^d(X, \\Omega^d_{X/k})\n$$\nis zero.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Poincaré duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FW6","source_file":"derham.tex","source_line":4359,"source_end_line":4367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L4359-L4367","statement_sha256":"272cdae1d484c3434b60312039d3192cbf8dd88ee487a293321169928b036b10","origin":"The Stacks Project","memory_eligible":false,"source_rank":9044,"rank":9044,"depth":49,"x":2441.124,"y":1043.326,"cluster":"local-crystalline-methods"},{"id":"stacks:0FW7","tag":"0FW7","title":"Poincaré duality · Proposition 0FW7","summary":"Let k be a field. Let X be a nonempty smooth proper scheme over k equidimensional of dimension d. There exists a k-linear map t : H^2d_dR(X/k) → k unique up to precomposing by multiplication by a unit of H^0(X, O_X) with the following property: for all i the pairing H^i_dR(X/k) × H_dR^2d - i(X/k) → k, (xi, xi') ↦ t(xi ∪ xi') is perfect.","statement_latex":"Let $k$ be a field. Let $X$ be a nonempty smooth proper scheme over $k$\nequidimensional of dimension $d$. There exists a $k$-linear map\n$$\nt : H^{2d}_{dR}(X/k) \\longrightarrow k\n$$\nunique up to precomposing by multiplication by a unit of\n$H^0(X, \\mathcal{O}_X)$ with the following property: for all $i$ the pairing\n$$\nH^i_{dR}(X/k) \\times H_{dR}^{2d - i}(X/k)\n\\longrightarrow\nk, \\quad\n(\\xi, \\xi') \\longmapsto t(\\xi \\cup \\xi')\n$$\nis perfect.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Poincaré duality","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FW7","source_file":"derham.tex","source_line":4472,"source_end_line":4488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L4472-L4488","statement_sha256":"63c9d8aa50a30039ac659b6201657ce84312224e7bce826619c540419d982e5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9045,"rank":9045,"depth":50,"x":2586.714,"y":1189.368,"cluster":"local-crystalline-methods"},{"id":"stacks:0FW9","tag":"0FW9","title":"Chern classes · Lemma 0FW9","summary":"There is a unique rule which assigns to every quasi-compact and quasi-separated scheme X a total Chern class c^dR : K_0(Vect(X)) → ∏_i ≥ 0 H^2i_dR(X/Z) with the following properties • we have c^dR(α + β) = c^dR(α) c^dR(β) for α, β ∈ K_0(Vect(X)), • if f : X → X' is a morphism of quasi-compact and quasi-separated schemes, then c^dR(f^*α) = f^*c^dR(α), • given L ∈ Pic(X) we have c^dR([L]) = 1 + c_1^dR(L)","statement_latex":"There is a unique rule which assigns to every quasi-compact and\nquasi-separated scheme $X$ a total Chern class\n$$\nc^{dR} :\nK_0(\\textit{Vect}(X))\n\\longrightarrow\n\\prod\\nolimits_{i \\geq 0} H^{2i}_{dR}(X/\\mathbf{Z})\n$$\nwith the following properties\n\\begin{enumerate}\n\\item we have $c^{dR}(\\alpha + \\beta) = c^{dR}(\\alpha) c^{dR}(\\beta)$\nfor $\\alpha, \\beta \\in K_0(\\textit{Vect}(X))$,\n\\item if $f : X \\to X'$ is a morphism of quasi-compact and\nquasi-separated schemes, then $c^{dR}(f^*\\alpha) = f^*c^{dR}(\\alpha)$,\n\\item given $\\mathcal{L} \\in \\Pic(X)$ we have\n$c^{dR}([\\mathcal{L}]) = 1 + c_1^{dR}(\\mathcal{L})$\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FW9","source_file":"derham.tex","source_line":4678,"source_end_line":4697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L4678-L4697","statement_sha256":"1b37efd7a6d1ca34266a6b3083088dcb5c175764e42d43682919392b0753e1ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":9046,"rank":9046,"depth":0,"x":2361.272,"y":1164.218,"cluster":"local-crystalline-methods"},{"id":"stacks:0FWB","tag":"0FWB","title":"Chern classes · Lemma 0FWB","summary":"There is a unique rule which assigns to every quasi-compact and quasi-separated scheme X over Q a \"chern character\" ch^dR : K_0(Vect(X)) → ∏_i ≥ 0 H_dR^2i(X/Q) with the following properties • ch^dR is a ring map for all X, • if f : X' → X is a morphism of quasi-compact and quasi-separated schemes over Q, then f^* ∘ ch^dR = ch^dR ∘ f^*, and • given L ∈ Pic(X) we have ch^dR([L]) = exp(c_1^dR(L)).","statement_latex":"There is a unique rule which assigns to every quasi-compact and quasi-separated\nscheme $X$ over $\\mathbf{Q}$ a ``chern character''\n$$\nch^{dR} : K_0(\\textit{Vect}(X)) \\longrightarrow\n\\prod\\nolimits_{i \\geq 0} H_{dR}^{2i}(X/\\mathbf{Q})\n$$\nwith the following properties\n\\begin{enumerate}\n\\item $ch^{dR}$ is a ring map for all $X$,\n\\item if $f : X' \\to X$ is a morphism of quasi-compact and quasi-separated\nschemes over $\\mathbf{Q}$, then $f^* \\circ ch^{dR} =  ch^{dR} \\circ f^*$, and\n\\item given $\\mathcal{L} \\in \\Pic(X)$\nwe have $ch^{dR}([\\mathcal{L}]) = \\exp(c_1^{dR}(\\mathcal{L}))$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWB","source_file":"derham.tex","source_line":4776,"source_end_line":4792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L4776-L4792","statement_sha256":"b4cc358b2b9d8ac7888673871525ddc1e804dcadfc01d2d80b7edf9f7107cfce","origin":"The Stacks Project","memory_eligible":false,"source_rank":9047,"rank":9047,"depth":0,"x":2548.266,"y":1054.531,"cluster":"local-crystalline-methods"},{"id":"stacks:0FWD","tag":"0FWD","title":"A Weil cohomology theory · Proposition 0FWD","summary":"Let k be a field of characteristic zero. The functor that sends a smooth projective scheme X over k to H_dR^*(X/k) is a Weil cohomology theory in the sense of Weil Cohomology Theories, Definition [Tag 0FI2].","statement_latex":"Let $k$ be a field of characteristic zero. The functor that\nsends a smooth projective scheme $X$ over $k$ to $H_{dR}^*(X/k)$\nis a Weil cohomology theory in the sense of\nWeil Cohomology Theories, Definition\n\\ref{weil-definition-weil-cohomology-theory}.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"A Weil cohomology theory","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FWD","source_file":"derham.tex","source_line":5106,"source_end_line":5113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5106-L5113","statement_sha256":"43de78d76904de937c0724ab72f02cb42086761f1151616d675ecf9c9ba4c986","origin":"The Stacks Project","memory_eligible":false,"source_rank":9048,"rank":9048,"depth":76,"x":2498.445,"y":1242.047,"cluster":"local-crystalline-methods"},{"id":"stacks:0G85","tag":"0G85","title":"Gysin maps for closed immersions · Lemma 0G85","summary":"The Gysin map ([Tag 0G84]) is compatible with the de Rham differentials on Ω^bullet_X/S and Ω^bullet_Z/S.","statement_latex":"The Gysin map (\\ref{equation-gysin}) is compatible with the de Rham\ndifferentials on $\\Omega^\\bullet_{X/S}$ and $\\Omega^\\bullet_{Z/S}$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Gysin maps for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G85","source_file":"derham.tex","source_line":5214,"source_end_line":5218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5214-L5218","statement_sha256":"c9067c7c4309fa7b6847fd6ce483ca4276b3e685543dde4631d4b9d284b08f9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9049,"rank":9049,"depth":24,"x":2384.067,"y":1075.037,"cluster":"local-crystalline-methods"},{"id":"stacks:0G86","tag":"0G86","title":"Gysin maps for closed immersions · Lemma 0G86","summary":"Let X → S be a morphism of schemes. Let Z → X be a closed immersion of finite presentation whose conormal sheaf C_Z/X is locally free of rank c. Then there is a canonical map γ^p : Ω^p_Z/S → H^c_Z(Ω^p + c_X/S) which is locally given by the maps γ^p_f_1, …, f_c of Remark [Tag 0G83].","statement_latex":"Let $X \\to S$ be a morphism of schemes. Let $Z \\to X$ be a closed immersion\nof finite presentation whose conormal sheaf $\\mathcal{C}_{Z/X}$ is\nlocally free of rank $c$. Then there is a canonical map\n$$\n\\gamma^p : \\Omega^p_{Z/S} \\to \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S})\n$$\nwhich is locally given by the maps $\\gamma^p_{f_1, \\ldots, f_c}$\nof Remark \\ref{remark-gysin-equations}.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Gysin maps for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G86","source_file":"derham.tex","source_line":5255,"source_end_line":5265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5255-L5265","statement_sha256":"a49e0c6b0dfa164fdc748e96abf239adb71e4efb53123cceb32c80b3e4375055","origin":"The Stacks Project","memory_eligible":false,"source_rank":9050,"rank":9050,"depth":27,"x":2603.324,"y":1133.45,"cluster":"local-crystalline-methods"},{"id":"stacks:0G87","tag":"0G87","title":"Gysin maps for closed immersions · Lemma 0G87","summary":"Let X → S and i : Z → X be as in Lemma [Tag 0G86]. The Gysin map γ^p is compatible with the de Rham differentials on Ω^bullet_X/S and Ω^bullet_Z/S.","statement_latex":"Let $X \\to S$ and $i : Z \\to X$ be as in Lemma \\ref{lemma-gysin-global}.\nThe Gysin map $\\gamma^p$ is compatible with the de Rham\ndifferentials on $\\Omega^\\bullet_{X/S}$ and $\\Omega^\\bullet_{Z/S}$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Gysin maps for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G87","source_file":"derham.tex","source_line":5290,"source_end_line":5295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5290-L5295","statement_sha256":"3575e3fa9aa5a062ce176da1f39065e888b1fa7e2f46eaeb9466d6b5ac1f6257","origin":"The Stacks Project","memory_eligible":false,"source_rank":9051,"rank":9051,"depth":28,"x":2394.092,"y":1215.015,"cluster":"local-crystalline-methods"},{"id":"stacks:0G88","tag":"0G88","title":"Gysin maps for closed immersions · Lemma 0G88","summary":"Let X → S and i : Z → X be as in Lemma [Tag 0G86]. Given α ∈ H^q(X, Ω^p_X/S) we have γ^p(α|_Z) = i^-1α wedge γ^0(1) in H^q(Z, H^c_Z(Ω^p + c_X/S)). Please see proof for notation.","statement_latex":"Let $X \\to S$ and $i : Z \\to X$ be as in Lemma \\ref{lemma-gysin-global}.\nGiven $\\alpha \\in H^q(X, \\Omega^p_{X/S})$ we have\n$\\gamma^p(\\alpha|_Z) = i^{-1}\\alpha \\wedge \\gamma^0(1)$ in\n$H^q(Z, \\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S}))$.\nPlease see proof for notation.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Gysin maps for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G88","source_file":"derham.tex","source_line":5302,"source_end_line":5309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5302-L5309","statement_sha256":"7100c08280eebd98f5469af6f92234e2e921478525a5cf5a072c6c87a35d067d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9052,"rank":9052,"depth":28,"x":2483.031,"y":1035.652,"cluster":"local-crystalline-methods"},{"id":"stacks:0G89","tag":"0G89","title":"Gysin maps for closed immersions · Lemma 0G89","summary":"Let c ≥ 0 be an integer. Let xymatrix Z' ar[d]_h ar[r] & X' ar[d]_g ar[r] & S' ar[d] Z ar[r] & X ar[r] & S be a commutative diagram of schemes. Assume • Z → X and Z' → X' satisfy the assumptions of Lemma [Tag 0G86], • the left square in the diagram is cartesian, and • h^*C_Z/X → C_Z'/X' (Morphisms, Lemma [Tag 01R4]) is an isomorphism. Then the diagram xymatrix h^*Ω^p_Z/S ar[rr]_-h^-1γ^p ar[d] & & O_X'|_Z' ⊗_h^-1O_X|_Z h^-1H^c_Z(Ω^p + c_X/S) ar[d] Ω^p_Z'/S' ar[rr]^γ^p & &…","statement_latex":"Let $c \\geq 0$ be an integer. Let\n$$\n\\xymatrix{\nZ' \\ar[d]_h \\ar[r] & X' \\ar[d]_g \\ar[r] & S' \\ar[d] \\\\\nZ \\ar[r] & X \\ar[r] & S\n}\n$$\nbe a commutative diagram of schemes.\nAssume\n\\begin{enumerate}\n\\item $Z \\to X$ and $Z' \\to X'$\nsatisfy the assumptions of Lemma \\ref{lemma-gysin-global},\n\\item the left square in the diagram is cartesian, and\n\\item $h^*\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z'/X'}$\n(Morphisms, Lemma \\ref{morphisms-lemma-conormal-functorial})\nis an isomorphism.\n\\end{enumerate}\nThen the diagram\n$$\n\\xymatrix{\nh^*\\Omega^p_{Z/S} \\ar[rr]_-{h^{-1}\\gamma^p} \\ar[d] & &\n\\mathcal{O}_{X'}|_{Z'} \\otimes_{h^{-1}\\mathcal{O}_X|_Z}\nh^{-1}\\mathcal{H}^c_Z(\\Omega^{p + c}_{X/S}) \\ar[d] \\\\\n\\Omega^p_{Z'/S'} \\ar[rr]^{\\gamma^p} & &\n\\mathcal{H}^c_{Z'}(\\Omega^{p + c}_{X'/S'})\n}\n$$\nis commutative. The left vertical arrow is functoriality of modules of\ndifferentials and the right vertical arrow uses\nCohomology, Remark \\ref{cohomology-remark-support-functorial}.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Gysin maps for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G89","source_file":"derham.tex","source_line":5379,"source_end_line":5411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5379-L5411","statement_sha256":"8c90cef976b98c2e4eb07fb4cfe5d51812307c921de98c2196a11f11ed4947eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9053,"rank":9053,"depth":28,"x":2561.903,"y":1218.88,"cluster":"local-crystalline-methods"},{"id":"stacks:0G8B","tag":"0G8B","title":"Gysin maps for closed immersions · Lemma 0G8B","summary":"Let X → S and i : Z → X be as in Lemma [Tag 0G86]. Assume X → S is smooth and Z → X Koszul regular. The Gysin maps γ^p, q are compatible with the de Rham differentials on Ω^bullet_X/S and Ω^bullet_Z/S.","statement_latex":"Let $X \\to S$ and $i : Z \\to X$ be as in Lemma \\ref{lemma-gysin-global}.\nAssume $X \\to S$ is smooth and $Z \\to X$ Koszul regular.\nThe Gysin maps $\\gamma^{p, q}$ are compatible with the de Rham\ndifferentials on $\\Omega^\\bullet_{X/S}$ and $\\Omega^\\bullet_{Z/S}$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Gysin maps for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8B","source_file":"derham.tex","source_line":5488,"source_end_line":5494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5488-L5494","statement_sha256":"5810f7b4f51fc040fae96ab3fe615bfce13e8ed15a2dba584e4f258a2aad0029","origin":"The Stacks Project","memory_eligible":false,"source_rank":9054,"rank":9054,"depth":29,"x":2355.834,"y":1128.277,"cluster":"local-crystalline-methods"},{"id":"stacks:0G8C","tag":"0G8C","title":"Gysin maps for closed immersions · Lemma 0G8C","summary":"Let X → S, i : Z → X, and c ≥ 0 be as in Lemma [Tag 0G86]. Assume X → S smooth and Z → X Koszul regular. Given α ∈ H^q(X, Ω^p_X/S) we have γ^p, q(α|_Z) = α ∪ γ^0, 0(1) in H^q + c(X, Ω^p + c_X/S) with γ^a, b as in Remark [Tag 0G8A].","statement_latex":"Let $X \\to S$, $i : Z \\to X$, and $c \\geq 0$ be as in\nLemma \\ref{lemma-gysin-global}. Assume $X \\to S$ smooth and\n$Z \\to X$ Koszul regular. Given $\\alpha \\in H^q(X, \\Omega^p_{X/S})$ we have\n$\\gamma^{p, q}(\\alpha|_Z) = \\alpha \\cup \\gamma^{0, 0}(1)$ in\n$H^{q + c}(X, \\Omega^{p + c}_{X/S})$ with $\\gamma^{a, b}$ as in\nRemark \\ref{remark-how-to-use}.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Gysin maps for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8C","source_file":"derham.tex","source_line":5501,"source_end_line":5509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5501-L5509","statement_sha256":"b821974ad0f66f0f9cb922827ad3dc8b4e1553d0e16852df26e8f26b68fa01f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9055,"rank":9055,"depth":29,"x":2581.259,"y":1078.022,"cluster":"local-crystalline-methods"},{"id":"stacks:0G8D","tag":"0G8D","title":"Gysin maps for closed immersions · Lemma 0G8D","summary":"Let c ≥ 0 and xymatrix Z' ar[d]_h ar[r] & X' ar[d]_g ar[r] & S' ar[d] Z ar[r] & X ar[r] & S satisfy the assumptions of Lemma [Tag 0G89] and assume in addition that X → S and X' → S' are smooth and that Z → X and Z' → X' are Koszul regular immersions. Then the diagram xymatrix H^q(Z, Ω^p_Z/S) ar[rr]_-γ^p, q ar[d] & & H^q + c(X, Ω^p + c_X/S) ar[d] H^q(Z', Ω^p_Z'/S') ar[rr]^γ^p, q & & H^q + c(X', Ω^p + c_X'/S') is commutative where γ^p, q is as in Remark [Tag 0G8A].","statement_latex":"Let $c \\geq 0$ and\n$$\n\\xymatrix{\nZ' \\ar[d]_h \\ar[r] & X' \\ar[d]_g \\ar[r] & S' \\ar[d] \\\\\nZ \\ar[r] & X \\ar[r] & S\n}\n$$\nsatisfy the assumptions of Lemma \\ref{lemma-gysin-transverse} and assume\nin addition that $X \\to S$ and $X' \\to S'$ are smooth and that\n$Z \\to X$ and $Z' \\to X'$ are Koszul regular immersions.\nThen the diagram\n$$\n\\xymatrix{\nH^q(Z, \\Omega^p_{Z/S}) \\ar[rr]_-{\\gamma^{p, q}} \\ar[d] & &\nH^{q + c}(X, \\Omega^{p + c}_{X/S}) \\ar[d] \\\\\nH^q(Z', \\Omega^p_{Z'/S'}) \\ar[rr]^{\\gamma^{p, q}} & &\nH^{q + c}(X', \\Omega^{p + c}_{X'/S'})\n}\n$$\nis commutative where $\\gamma^{p, q}$ is as in Remark \\ref{remark-how-to-use}.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Gysin maps for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8D","source_file":"derham.tex","source_line":5545,"source_end_line":5567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5545-L5567","statement_sha256":"58d627da83ae1df7e4ce33a719d1e573ed34c65fc2cb7ff6242deb89578da3a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9056,"rank":9056,"depth":29,"x":2455.107,"y":1243.437,"cluster":"local-crystalline-methods"},{"id":"stacks:0G8E","tag":"0G8E","title":"Gysin maps for closed immersions · Lemma 0G8E","summary":"Let k be a field. Let X be an irreducible smooth proper scheme over k of dimension d. Let Z ⊂ X be the reduced closed subscheme consisting of a single k-rational point x. Then the image of 1 ∈ k = H^0(Z, O_Z) = H^0(Z, Ω^0_Z/k) by the map H^0(Z, Ω^0_Z/k) → H^d(X, Ω^d_X/k) of Remark [Tag 0G8A] is nonzero.","statement_latex":"Let $k$ be a field. Let $X$ be an irreducible smooth proper scheme over $k$\nof dimension $d$. Let $Z \\subset X$ be the reduced closed subscheme consisting\nof a single $k$-rational point $x$. Then the image of\n$1 \\in k = H^0(Z, \\mathcal{O}_Z) = H^0(Z, \\Omega^0_{Z/k})$\nby the map $H^0(Z, \\Omega^0_{Z/k}) \\to H^d(X, \\Omega^d_{X/k})$\nof Remark \\ref{remark-how-to-use} is nonzero.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Gysin maps for closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8E","source_file":"derham.tex","source_line":5574,"source_end_line":5582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5574-L5582","statement_sha256":"7d98734f518adf728c7c6c439232cc5a6283a3c549de879fcdad197b9f88fae8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9057,"rank":9057,"depth":47,"x":2415.001,"y":1049.359,"cluster":"local-crystalline-methods"},{"id":"stacks:0G8H","tag":"0G8H","title":"Relative Poincaré duality · Lemma 0G8H","summary":"In Situation [Tag 0G8G] the pushforward f_*O_X is a finite étale O_S-algebra and locally on S we have Rf_*O_X = f_*O_X ⊕ P in D(O_S) with P perfect of tor amplitude in [1, ∞). The map d : f_*O_X → f_*Ω_X/S is zero.","statement_latex":"In Situation \\ref{situation-relative-duality} the pushforward\n$f_*\\mathcal{O}_X$ is a finite \\'etale $\\mathcal{O}_S$-algebra\nand locally on $S$ we have $Rf_*\\mathcal{O}_X = f_*\\mathcal{O}_X \\oplus P$\nin $D(\\mathcal{O}_S)$ with $P$ perfect of tor amplitude in $[1, \\infty)$.\nThe map $\\text{d} : f_*\\mathcal{O}_X \\to f_*\\Omega_{X/S}$ is zero.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Relative Poincaré duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8H","source_file":"derham.tex","source_line":5637,"source_end_line":5644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5637-L5644","statement_sha256":"dd07b5f073b62b93fa1e3da10e04938e2dddba4ed78d905faff268970af5a6fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9058,"rank":9058,"depth":48,"x":2601.143,"y":1170.035,"cluster":"local-crystalline-methods"},{"id":"stacks:0G8I","tag":"0G8I","title":"Relative Poincaré duality · Lemma 0G8I","summary":"In Situation [Tag 0G8G] there exists an O_S-module map t : Rf_*Ω^n_X/S[n] → O_S unique up to precomposing by multiplication by a unit of H^0(X, O_X) with the following property: for all p the pairing Rf_*Ω^p_X/S ⊗_O_S^L Rf_*Ω^n - p_X/S[n] → O_S given by the relative cup product composed with t is a perfect pairing of perfect complexes on S.","statement_latex":"In Situation \\ref{situation-relative-duality} there exists an\n$\\mathcal{O}_S$-module map\n$$\nt : Rf_*\\Omega^n_{X/S}[n] \\longrightarrow \\mathcal{O}_S\n$$\nunique up to precomposing by multiplication by a unit of\n$H^0(X, \\mathcal{O}_X)$ with the following property: for all $p$ the pairing\n$$\nRf_*\\Omega^p_{X/S}\n\\otimes_{\\mathcal{O}_S}^\\mathbf{L}\nRf_*\\Omega^{n - p}_{X/S}[n]\n\\longrightarrow\n\\mathcal{O}_S\n$$\ngiven by the relative cup product composed with $t$\nis a perfect pairing of perfect complexes on $S$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Relative Poincaré duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8I","source_file":"derham.tex","source_line":5659,"source_end_line":5677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5659-L5677","statement_sha256":"ed8367c3b1fbd7ec03922c6e092c3cdabde96888908dba3c04d8864a332140a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9059,"rank":9059,"depth":49,"x":2366.217,"y":1186.716,"cluster":"local-crystalline-methods"},{"id":"stacks:0G8J","tag":"0G8J","title":"Relative Poincaré duality · Lemma 0G8J","summary":"In Situation [Tag 0G8G] the map d : R^nf_*Ω^n - 1_X/S → R^nf_*Ω^n_X/S is zero.","statement_latex":"In Situation \\ref{situation-relative-duality} the map\n$\\text{d} : R^nf_*\\Omega^{n - 1}_{X/S} \\to R^nf_*\\Omega^n_{X/S}$\nis zero.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Relative Poincaré duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8J","source_file":"derham.tex","source_line":5761,"source_end_line":5766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L5761-L5766","statement_sha256":"6b9de81c7d7f90967a615ac938dc032a2ed0c9942f9588ee561efd9ff1f876fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9060,"rank":9060,"depth":0,"x":2526.457,"y":1040.727,"cluster":"local-crystalline-methods"},{"id":"stacks:0G8K","tag":"0G8K","title":"Relative Poincaré duality · Proposition 0G8K","summary":"Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a proper smooth morphism of schemes all of whose fibres are nonempty and equidimensional of dimension n. There exists an O_S-module map t : R^2nf_*Ω^bullet_X/S → O_S unique up to precomposing by multiplication by a unit of H^0(X, O_X) with the following property: the pairing Rf_*Ω^bullet_X/S ⊗_O_S^L Rf_*Ω^bullet_X/S[2n] → O_S, (xi, xi') ↦ t(xi ∪ xi') is a perfect pairing of perfect complexes on S.","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \\to S$\nbe a proper smooth morphism of schemes all of whose fibres are nonempty\nand equidimensional of dimension $n$. There exists an\n$\\mathcal{O}_S$-module map\n$$\nt : R^{2n}f_*\\Omega^\\bullet_{X/S} \\longrightarrow \\mathcal{O}_S\n$$\nunique up to precomposing by multiplication by a unit of\n$H^0(X, \\mathcal{O}_X)$ with the following property: the pairing\n$$\nRf_*\\Omega^\\bullet_{X/S}\n\\otimes_{\\mathcal{O}_S}^\\mathbf{L}\nRf_*\\Omega^\\bullet_{X/S}[2n]\n\\longrightarrow\n\\mathcal{O}_S, \\quad\n(\\xi, \\xi') \\longmapsto t(\\xi \\cup \\xi')\n$$\nis a perfect pairing of perfect complexes on $S$.","area":"Local & Crystalline Methods","chapter":"de Rham Cohomology","chapter_id":"derham","section":"Relative Poincaré duality","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G8K","source_file":"derham.tex","source_line":6100,"source_end_line":6120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/derham.tex#L6100-L6120","statement_sha256":"3d2eaae88e56e174bd51b69e98580d5dede0713b0d82193a48ca1266ef9ee139","origin":"The Stacks Project","memory_eligible":false,"source_rank":9061,"rank":9061,"depth":51,"x":2525.695,"y":1239.825,"cluster":"local-crystalline-methods"},{"id":"stacks:0A6T","tag":"0A6T","title":"Generalities · Lemma 0A6T","summary":"Let A be a ring and let I be a finitely generated ideal. Set Z = V(I) ⊂ X = Spec(A). For K ∈ D(A) corresponding to widetildeK ∈ D_QCoh(O_X) via Derived Categories of Schemes, Lemma [Tag 06Z0] there is a functorial isomorphism RΓ_Z(K) = RΓ_Z(X, widetildeK) where on the left we have Dualizing Complexes, Equation ([Tag 0A6Q]) and on the right we have the functor of Cohomology, Section [Tag 0G6Y].","statement_latex":"Let $A$ be a ring and let $I$ be a finitely generated ideal.\nSet $Z = V(I) \\subset X = \\Spec(A)$. For $K \\in D(A)$ corresponding\nto $\\widetilde{K} \\in D_\\QCoh(\\mathcal{O}_X)$ via\nDerived Categories of Schemes, Lemma \\ref{perfect-lemma-affine-compare-bounded}\nthere is a functorial isomorphism\n$$\nR\\Gamma_Z(K) = R\\Gamma_Z(X, \\widetilde{K})\n$$\nwhere on the left we have\nDualizing Complexes, Equation (\\ref{dualizing-equation-local-cohomology})\nand on the right we have the functor of\nCohomology, Section \\ref{cohomology-section-cohomology-support-bis}.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Generalities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A6T","source_file":"local-cohomology.tex","source_line":59,"source_end_line":73,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L59-L73","statement_sha256":"476420b39fc409a380cdcd06adeea84c51dd8923288d5386ff6f6759503a7d40","origin":"The Stacks Project","memory_eligible":false,"source_rank":9062,"rank":9062,"depth":27,"x":2365.731,"y":1092.195,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWR","tag":"0DWR","title":"Generalities · Lemma 0DWR","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. Set X = Spec(A), Z = V(I), U = X setminus Z, and j : U → X the inclusion morphism. Let F be a quasi-coherent O_U-module. Then • there exists an A-module M such that F is the restriction of widetildeM to U, • given M there is an exact sequence 0 → H^0_Z(M) → M → H^0(U, F) → H^1_Z(M) → 0 and isomorphisms H^p(U, F) = H^p + 1_Z(M) for p ≥ 1, • we may take M = H^0(U, F) in which case we have H^0_Z(M) = H^1_Z(M) = 0.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nSet $X = \\Spec(A)$, $Z = V(I)$, $U = X \\setminus Z$, and $j : U \\to X$\nthe inclusion morphism. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_U$-module. Then\n\\begin{enumerate}\n\\item there exists an $A$-module $M$ such that $\\mathcal{F}$ is the\nrestriction of $\\widetilde{M}$ to $U$,\n\\item given $M$ there is an exact sequence\n$$\n0 \\to H^0_Z(M) \\to M \\to H^0(U, \\mathcal{F}) \\to H^1_Z(M) \\to 0\n$$\nand isomorphisms $H^p(U, \\mathcal{F}) = H^{p + 1}_Z(M)$ for $p \\geq 1$,\n\\item we may take $M = H^0(U, \\mathcal{F})$ in which case\nwe have $H^0_Z(M) = H^1_Z(M) = 0$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Generalities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWR","source_file":"local-cohomology.tex","source_line":148,"source_end_line":165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L148-L165","statement_sha256":"e3369ef256a61c1f3d8f7295eb67de5310bb48fdc9e50b56b748805dc3e8780f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9063,"rank":9063,"depth":28,"x":2603.023,"y":1110.335,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWS","tag":"0DWS","title":"Generalities · Lemma 0DWS","summary":"Let I, J ⊂ A be finitely generated ideals of a ring A. If M is an I-power torsion module, then the canonical map H^i_V(I) ∩ V(J)(M) → H^i_V(J)(M) is an isomorphism for all i.","statement_latex":"Let $I, J \\subset A$ be finitely generated ideals of a ring $A$.\nIf $M$ is an $I$-power torsion module, then the\ncanonical map\n$$\nH^i_{V(I) \\cap V(J)}(M) \\to H^i_{V(J)}(M)\n$$\nis an isomorphism for all $i$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Generalities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWS","source_file":"local-cohomology.tex","source_line":198,"source_end_line":207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L198-L207","statement_sha256":"a29c6650c953718f72ceba7f5bb8a07ec3a3f893ecd2a4b9a1e21050bddf86f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9064,"rank":9064,"depth":14,"x":2412.968,"y":1231.918,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWT","tag":"0DWT","title":"Generalities · Lemma 0DWT","summary":"Let S ⊂ A be a multiplicative set of a ring A. Let M be an A-module with S^-1M = 0. Then colim_f ∈ S H^0_V(f)(M) = M and colim_f ∈ S H^1_V(f)(M) = 0.","statement_latex":"Let $S \\subset A$ be a multiplicative set of a ring $A$.\nLet $M$ be an $A$-module with $S^{-1}M = 0$. Then\n$\\colim_{f \\in S} H^0_{V(f)}(M) = M$ and\n$\\colim_{f \\in S} H^1_{V(f)}(M) = 0$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Generalities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWT","source_file":"local-cohomology.tex","source_line":217,"source_end_line":223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L217-L223","statement_sha256":"f2a74a7eaee4f18760fa24e0da3307b31057c725bd21e9a4144776ca23629d80","origin":"The Stacks Project","memory_eligible":false,"source_rank":9065,"rank":9065,"depth":15,"x":2455.447,"y":1033.913,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWU","tag":"0DWU","title":"Generalities · Lemma 0DWU","summary":"Let I ⊂ A be a finitely generated ideal of a ring A. Let p be a prime ideal. Let M be an A-module. Let i ≥ 0 be an integer and consider the map Ψ : colim_f ∈ A, f not ∈ p H^i_V((I, f))(M) → H^i_V(I)(M) Then • Im(Ψ) is the set of elements which map to zero in H^i_V(I)(M)_ p, • if H^i - 1_V(I)(M)_ p = 0, then Ψ is injective, • if H^i - 1_V(I)(M)_ p = H^i_V(I)(M)_ p = 0, then Ψ is an isomorphism.","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring $A$.\nLet $\\mathfrak p$ be a prime ideal. Let $M$ be an $A$-module.\nLet $i \\geq 0$ be an integer and consider the map\n$$\n\\Psi :\n\\colim_{f \\in A, f \\not \\in \\mathfrak p} H^i_{V((I, f))}(M)\n\\longrightarrow\nH^i_{V(I)}(M)\n$$\nThen\n\\begin{enumerate}\n\\item $\\Im(\\Psi)$ is the set of elements which map to zero in\n$H^i_{V(I)}(M)_\\mathfrak p$,\n\\item if $H^{i - 1}_{V(I)}(M)_\\mathfrak p = 0$, then $\\Psi$ is injective,\n\\item if $H^{i - 1}_{V(I)}(M)_\\mathfrak p = H^i_{V(I)}(M)_\\mathfrak p = 0$,\nthen $\\Psi$ is an isomorphism.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Generalities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWU","source_file":"local-cohomology.tex","source_line":234,"source_end_line":253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L234-L253","statement_sha256":"a659e019bbe59d5474d14ed3c3594d03dbf6e913696399d53277111df756dc84","origin":"The Stacks Project","memory_eligible":false,"source_rank":9066,"rank":9066,"depth":16,"x":2583.681,"y":1204.46,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWV","tag":"0DWV","title":"Generalities · Lemma 0DWV","summary":"Let I ⊂ I' ⊂ A be finitely generated ideals of a Noetherian ring A. Let M be an A-module. Let i ≥ 0 be an integer. Consider the map Ψ : H^i_V(I')(M) → H^i_V(I)(M) The following are true: • if H^i_ pA_ p(M_ p) = 0 for all p ∈ V(I) setminus V(I'), then Ψ is surjective, • if H^i - 1_ pA_ p(M_ p) = 0 for all p ∈ V(I) setminus V(I'), then Ψ is injective, • if H^i_ pA_ p(M_ p) = H^i - 1_ pA_ p(M_ p) = 0 for all p ∈ V(I) setminus V(I'), then Ψ is an isomorphism.","statement_latex":"Let $I \\subset I' \\subset A$ be finitely generated ideals of a\nNoetherian ring $A$. Let $M$ be an $A$-module. Let $i \\geq 0$ be an integer.\nConsider the map\n$$\n\\Psi : H^i_{V(I')}(M) \\to H^i_{V(I)}(M)\n$$\nThe following are true:\n\\begin{enumerate}\n\\item if $H^i_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$\nfor all $\\mathfrak p \\in V(I) \\setminus V(I')$, then\n$\\Psi$ is surjective,\n\\item if $H^{i - 1}_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$\nfor all $\\mathfrak p \\in V(I) \\setminus V(I')$, then\n$\\Psi$ is injective,\n\\item if $H^i_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) =\nH^{i - 1}_{\\mathfrak pA_\\mathfrak p}(M_\\mathfrak p) = 0$\nfor all $\\mathfrak p \\in V(I) \\setminus V(I')$, then\n$\\Psi$ is an isomorphism.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Generalities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWV","source_file":"local-cohomology.tex","source_line":268,"source_end_line":289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L268-L289","statement_sha256":"435d444ce3a81359b8324c0633d7148e579e0fbe5d1dcc40fc9d23abbf9ae170","origin":"The Stacks Project","memory_eligible":false,"source_rank":9067,"rank":9067,"depth":17,"x":2351.384,"y":1151.329,"cluster":"local-crystalline-methods"},{"id":"stacks:0BLR","tag":"0BLR","title":"Hartshorne's connectedness lemma · Lemma 0BLR","summary":"[Hartshorne-connectedness] Hartshorne's connectedness Let A be a Noetherian local ring of depth ≥ 2. Then the punctured spectra of A, A^h, and A^sh are connected.","statement_latex":"\\begin{reference}\n\\cite[Proposition 2.1]{Hartshorne-connectedness}\n\\end{reference}\n\\begin{slogan}\nHartshorne's connectedness\n\\end{slogan}\nLet $A$ be a Noetherian local ring of depth $\\geq 2$.\nThen the punctured spectra of $A$, $A^h$, and $A^{sh}$ are connected.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Hartshorne's connectedness lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLR","source_file":"local-cohomology.tex","source_line":327,"source_end_line":337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L327-L337","statement_sha256":"b872fd65a9d4090b8becd3b4edd823aed2d2015a77dce0de781ee1cac16b7d41","origin":"The Stacks Project","memory_eligible":false,"source_rank":9068,"rank":9068,"depth":49,"x":2565.947,"y":1058.461,"cluster":"local-crystalline-methods"},{"id":"stacks:0FIW","tag":"0FIW","title":"Hartshorne's connectedness lemma · Lemma 0FIW","summary":"[EGA] Let A be a Noetherian local ring which is catenary and (S_2). Then Spec(A) is equidimensional.","statement_latex":"\\begin{reference}\n\\cite[IV Corollary 5.10.9]{EGA}\n\\end{reference}\nLet $A$ be a Noetherian local ring which is catenary and $(S_2)$.\nThen $\\Spec(A)$ is equidimensional.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Hartshorne's connectedness lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FIW","source_file":"local-cohomology.tex","source_line":353,"source_end_line":360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L353-L360","statement_sha256":"c48eb29a51631dc6d70b5d528b69871b3bcbd8c4a7d0c5059be88ded69d5f205","origin":"The Stacks Project","memory_eligible":false,"source_rank":9069,"rank":9069,"depth":50,"x":2482.202,"y":1249.168,"cluster":"local-crystalline-methods"},{"id":"stacks:0DX7","tag":"0DX7","title":"Cohomological dimension · Lemma 0DX7","summary":"Let I ⊂ A be a finitely generated ideal of a ring A. Set Y = V(I) ⊂ X = Spec(A). Let d ≥ -1 be an integer. The following are equivalent • H^i_Y(A) = 0 for i > d, • H^i_Y(M) = 0 for i > d for every A-module M, and • if d = -1, then Y = ∅, if d = 0, then Y is open and closed in X, and if d > 0 then H^i(X setminus Y, F) = 0 for i ≥ d for every quasi-coherent O_X setminus Y-module F.","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring $A$.\nSet $Y = V(I) \\subset X = \\Spec(A)$. Let $d \\geq -1$ be an integer.\nThe following are equivalent\n\\begin{enumerate}\n\\item $H^i_Y(A) = 0$ for $i > d$,\n\\item $H^i_Y(M) = 0$ for $i > d$ for every $A$-module $M$, and\n\\item if $d = -1$, then $Y = \\emptyset$, if $d = 0$, then\n$Y$ is open and closed in $X$, and if $d > 0$ then\n$H^i(X \\setminus Y, \\mathcal{F}) = 0$ for $i \\geq d$\nfor every quasi-coherent $\\mathcal{O}_{X \\setminus Y}$-module $\\mathcal{F}$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DX7","source_file":"local-cohomology.tex","source_line":394,"source_end_line":407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L394-L407","statement_sha256":"85940abd5ed1f2580173058eeb8f031de8dd67d51bb268bfd32065fb29b3b37b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9070,"rank":9070,"depth":29,"x":2390.362,"y":1060.551,"cluster":"local-crystalline-methods"},{"id":"stacks:0DX8","tag":"0DX8","title":"Cohomological dimension · Definition 0DX8","summary":"Let I ⊂ A be a finitely generated ideal of a ring A. The smallest integer d ≥ -1 satisfying the equivalent conditions of Lemma [Tag 0DX7] is called the cohomological dimension of I in A and is denoted cd(A, I).","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring $A$.\nThe smallest integer $d \\geq -1$ satisfying the equivalent conditions\nof Lemma \\ref{lemma-cd} is called the\n{\\it cohomological dimension of $I$ in $A$} and is\ndenoted $\\text{cd}(A, I)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DX8","source_file":"local-cohomology.tex","source_line":458,"source_end_line":465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L458-L465","statement_sha256":"644bff8e47db07560e5895ea5f10d014545adf0131a3bbe951b7cdeef74050ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":9071,"rank":9071,"depth":30,"x":2610.312,"y":1147.741,"cluster":"local-crystalline-methods"},{"id":"stacks:0DX9","tag":"0DX9","title":"Cohomological dimension · Lemma 0DX9","summary":"Let I ⊂ A be a finitely generated ideal of a ring A. Then • cd(A, I) is at most equal to the number of generators of I, • cd(A, I) ≤ r if there exist f_1, …, f_r ∈ A such that V(f_1, …, f_r) = V(I), • cd(A, I) ≤ c if Spec(A) setminus V(I) can be covered by c affine opens.","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring $A$.\nThen\n\\begin{enumerate}\n\\item $\\text{cd}(A, I)$ is at most equal to the number of\ngenerators of $I$,\n\\item $\\text{cd}(A, I) \\leq r$ if there exist $f_1, \\ldots, f_r \\in A$\nsuch that $V(f_1, \\ldots, f_r) = V(I)$,\n\\item $\\text{cd}(A, I) \\leq c$ if $\\Spec(A) \\setminus V(I)$\ncan be covered by $c$ affine opens.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DX9","source_file":"local-cohomology.tex","source_line":473,"source_end_line":485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L473-L485","statement_sha256":"29286af202925b28d0d556f85a60ce314a3d5832fb487bb13e3d3fc430e14fa5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9072,"rank":9072,"depth":30,"x":2377.431,"y":1208.404,"cluster":"local-crystalline-methods"},{"id":"stacks:0ECP","tag":"0ECP","title":"Cohomological dimension · Lemma 0ECP","summary":"Let I, J ⊂ A be finitely generated ideals of a ring A. Then cd(A, I + J) ≤ cd(A, I) + cd(A, J).","statement_latex":"Let $I, J \\subset A$ be finitely generated ideals of a ring $A$.\nThen $\\text{cd}(A, I + J) \\leq \\text{cd}(A, I) + \\text{cd}(A, J)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECP","source_file":"local-cohomology.tex","source_line":504,"source_end_line":508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L504-L508","statement_sha256":"c8a6f724d2ea659a65de719e59f4a685af7bddc60e9b2e188460141d26e80bba","origin":"The Stacks Project","memory_eligible":false,"source_rank":9073,"rank":9073,"depth":14,"x":2500.675,"y":1031.092,"cluster":"local-crystalline-methods"},{"id":"stacks:0DXA","tag":"0DXA","title":"Cohomological dimension · Lemma 0DXA","summary":"Let A → B be a ring map. Let I ⊂ A be a finitely generated ideal. Then cd(B, IB) ≤ cd(A, I). If A → B is faithfully flat, then equality holds.","statement_latex":"Let $A \\to B$ be a ring map. Let $I \\subset A$ be a finitely generated ideal.\nThen $\\text{cd}(B, IB) \\leq \\text{cd}(A, I)$. If $A \\to B$ is faithfully\nflat, then equality holds.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXA","source_file":"local-cohomology.tex","source_line":515,"source_end_line":520,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L515-L520","statement_sha256":"3053a1155d8f05f4f9094fa41aaf5b6b9063d80330dbe866bc521e129bcd015b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9074,"rank":9074,"depth":4,"x":2552.513,"y":1232.265,"cluster":"local-crystalline-methods"},{"id":"stacks:0DXB","tag":"0DXB","title":"Cohomological dimension · Lemma 0DXB","summary":"Let I ⊂ A be a finitely generated ideal of a ring A. Then cd(A, I) = max cd(A_ p, I_ p).","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring $A$.\nThen $\\text{cd}(A, I) = \\max \\text{cd}(A_\\mathfrak p, I_\\mathfrak p)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXB","source_file":"local-cohomology.tex","source_line":527,"source_end_line":531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L527-L531","statement_sha256":"1accc23106decb560d939d79bc0dd870dd4101a3b5d72dc1dc0b3403c38068f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9075,"rank":9075,"depth":4,"x":2352.021,"y":1113.045,"cluster":"local-crystalline-methods"},{"id":"stacks:0DXC","tag":"0DXC","title":"Cohomological dimension · Lemma 0DXC","summary":"Let I ⊂ A be a finitely generated ideal of a ring A. If M is a finite A-module, then H^i_V(I)(M) = 0 for i > dim(Supp(M)). In particular, we have cd(A, I) ≤ dim(A).","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring $A$.\nIf $M$ is a finite $A$-module, then\n$H^i_{V(I)}(M) = 0$ for $i > \\dim(\\text{Supp}(M))$.\nIn particular, we have $\\text{cd}(A, I) \\leq \\dim(A)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXC","source_file":"local-cohomology.tex","source_line":541,"source_end_line":547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L541-L547","statement_sha256":"4c1c2e070c1d15d93105eae0eb47e298864776bac144f84ccbb6b6b21346ddc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9076,"rank":9076,"depth":30,"x":2596.329,"y":1087.13,"cluster":"local-crystalline-methods"},{"id":"stacks:0DXD","tag":"0DXD","title":"Cohomological dimension · Lemma 0DXD","summary":"Let I ⊂ A be a finitely generated ideal of a ring A. If cd(A, I) = 1 then Spec(A) setminus V(I) is nonempty affine.","statement_latex":"Let $I \\subset A$ be a finitely generated ideal of a ring $A$. If\n$\\text{cd}(A, I) = 1$ then $\\Spec(A) \\setminus V(I)$ is nonempty affine.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXD","source_file":"local-cohomology.tex","source_line":582,"source_end_line":586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L582-L586","statement_sha256":"2173bd0ca57009c8672583d1e402b57d029eb7d30cc352dd09a2bb622c39dbf8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9077,"rank":9077,"depth":30,"x":2436.633,"y":1245.248,"cluster":"local-crystalline-methods"},{"id":"stacks:0DXE","tag":"0DXE","title":"Cohomological dimension · Lemma 0DXE","summary":"Let (A, m) be a Noetherian local ring of dimension d. Then H^d_ m(A) is nonzero and cd(A, m) = d.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring of dimension $d$.\nThen $H^d_\\mathfrak m(A)$ is nonzero and $\\text{cd}(A, \\mathfrak m) = d$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXE","source_file":"local-cohomology.tex","source_line":594,"source_end_line":598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L594-L598","statement_sha256":"709943e77b21a86a4ae03b7af14af78a766d60c7511e9fbce057e80109c94597","origin":"The Stacks Project","memory_eligible":false,"source_rank":9078,"rank":9078,"depth":37,"x":2427.214,"y":1037.539,"cluster":"local-crystalline-methods"},{"id":"stacks:0DXF","tag":"0DXF","title":"Cohomological dimension · Lemma 0DXF","summary":"Let (A, m) be a Noetherian local ring. Let I ⊂ A be a proper ideal. Let p ⊂ A be a prime ideal such that V( p) ∩ V(I) = ( m). Then dim(A/ p) ≤ cd(A, I).","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $I \\subset A$ be a proper ideal.\nLet $\\mathfrak p \\subset A$ be a prime ideal\nsuch that $V(\\mathfrak p) \\cap V(I) = \\{\\mathfrak m\\}$.\nThen $\\dim(A/\\mathfrak p) \\leq \\text{cd}(A, I)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXF","source_file":"local-cohomology.tex","source_line":635,"source_end_line":642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L635-L642","statement_sha256":"450adb93bf5d096e56ea5d44751e352e68c79a45ee3e8a58a8c478fbfadad54b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9079,"rank":9079,"depth":38,"x":2601.611,"y":1185.711,"cluster":"local-crystalline-methods"},{"id":"stacks:0EHU","tag":"0EHU","title":"Cohomological dimension · Lemma 0EHU","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let b : X' → X = Spec(A) be the blowing up of I. If the fibres of b have dimension ≤ d - 1, then cd(A, I) ≤ d.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nLet $b : X' \\to X = \\Spec(A)$ be the blowing up of $I$.\nIf the fibres of $b$ have dimension $\\leq d - 1$, then\n$\\text{cd}(A, I) \\leq d$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHU","source_file":"local-cohomology.tex","source_line":653,"source_end_line":659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L653-L659","statement_sha256":"e605a5e8550240f8f0edd91b3d38c9d1d60dfcb0e1cb838193aa299a2bfba480","origin":"The Stacks Project","memory_eligible":false,"source_rank":9080,"rank":9080,"depth":40,"x":2353.265,"y":1175.383,"cluster":"local-crystalline-methods"},{"id":"stacks:0EEZ","tag":"0EEZ","title":"More general supports · Lemma 0EEZ","summary":"Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. For an A-module M the following are equivalent • H^0_T(M) = M, and • Supp(M) ⊂ T. The category of such A-modules is a Serre subcategory of the category A-modules closed under direct sums.","statement_latex":"Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable\nunder specialization. For an $A$-module $M$ the following are equivalent\n\\begin{enumerate}\n\\item $H^0_T(M) = M$, and\n\\item $\\text{Supp}(M) \\subset T$.\n\\end{enumerate}\nThe category of such $A$-modules is a Serre subcategory\nof the category $A$-modules closed under direct sums.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"More general supports","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEZ","source_file":"local-cohomology.tex","source_line":711,"source_end_line":721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L711-L721","statement_sha256":"c60609f2d43cb2a89a0a25852c3b4d07f5be8447d7d3351112093e96c251fb19","origin":"The Stacks Project","memory_eligible":false,"source_rank":9081,"rank":9081,"depth":4,"x":2545.153,"y":1041.764,"cluster":"local-crystalline-methods"},{"id":"stacks:0EF0","tag":"0EF0","title":"More general supports · Lemma 0EF0","summary":"Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. The functor RH^0_T is the right adjoint to the functor D(Mod_A, T) → D(A).","statement_latex":"Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$\nbe a subset stable under specialization. The functor\n$RH^0_T$ is the right adjoint to the functor\n$D(\\text{Mod}_{A, T}) \\to D(A)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"More general supports","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EF0","source_file":"local-cohomology.tex","source_line":764,"source_end_line":770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L764-L770","statement_sha256":"550b8823a7ecaba7e57aa5872854fde58b71c9ab64bb459053169afca7867be3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9082,"rank":9082,"depth":5,"x":2511.029,"y":1249.665,"cluster":"local-crystalline-methods"},{"id":"stacks:0EF1","tag":"0EF1","title":"More general supports · Lemma 0EF1","summary":"Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. For any object K of D(A) we have H^i(RH^0_T(K)) = colim_Z ⊂ T closed H^i_Z(K)","statement_latex":"Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$\nbe a subset stable under specialization.\nFor any object $K$ of $D(A)$ we have\n$$\nH^i(RH^0_T(K)) = \\colim_{Z \\subset T\\text{ closed}} H^i_Z(K)\n$$","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"More general supports","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EF1","source_file":"local-cohomology.tex","source_line":779,"source_end_line":787,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L779-L787","statement_sha256":"08d32111e632604a240e3f0a1aa74e9b0711e913d846a4c47295b6ba8c0e3f1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9083,"rank":9083,"depth":0,"x":2368.669,"y":1076.592,"cluster":"local-crystalline-methods"},{"id":"stacks:0EF2","tag":"0EF2","title":"More general supports · Lemma 0EF2","summary":"Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. The functor D^+(Mod_A, T) → D^+_T(A) is an equivalence.","statement_latex":"Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable\nunder specialization. The functor $D^+(\\text{Mod}_{A, T}) \\to D^+_T(A)$\nis an equivalence.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"More general supports","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EF2","source_file":"local-cohomology.tex","source_line":802,"source_end_line":807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L802-L807","statement_sha256":"f442793ae0c1da2ea600cbc70d8f7f26d4f62649ece4c404796494fea00b3a28","origin":"The Stacks Project","memory_eligible":false,"source_rank":9084,"rank":9084,"depth":10,"x":2613.396,"y":1123.545,"cluster":"local-crystalline-methods"},{"id":"stacks:0EF3","tag":"0EF3","title":"More general supports · Lemma 0EF3","summary":"Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. If dim(A) < ∞, then functor D(Mod_A, T) → D_T(A) is an equivalence.","statement_latex":"Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable\nunder specialization. If $\\dim(A) < \\infty$, then functor\n$D(\\text{Mod}_{A, T}) \\to D_T(A)$ is an equivalence.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"More general supports","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EF3","source_file":"local-cohomology.tex","source_line":837,"source_end_line":842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L837-L842","statement_sha256":"ab0027af6f3feb534df8bd329161575bc4d951d980374d5cc26e3ef7e79f4eeb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9085,"rank":9085,"depth":31,"x":2394.67,"y":1228.03,"cluster":"local-crystalline-methods"},{"id":"stacks:0EF5","tag":"0EF5","title":"More general supports · Lemma 0EF5","summary":"Let A → B be a flat homomorphism of Noetherian rings. Let T ⊂ Spec(A) be a subset stable under specialization. Let T' ⊂ Spec(B) be the inverse image of T. Then the canonical map RΓ_T(K) ⊗_A^L B → RΓ_T'(K ⊗_A^L B) is an isomorphism for K ∈ D^+(A). If A and B have finite dimension, then this is true for K ∈ D(A).","statement_latex":"Let $A \\to B$ be a flat homomorphism of Noetherian rings.\nLet $T \\subset \\Spec(A)$ be a subset stable under specialization.\nLet $T' \\subset \\Spec(B)$ be the inverse image of $T$.\nThen the canonical map\n$$\nR\\Gamma_T(K) \\otimes_A^\\mathbf{L} B\n\\longrightarrow\nR\\Gamma_{T'}(K \\otimes_A^\\mathbf{L} B)\n$$\nis an isomorphism for $K \\in D^+(A)$. If $A$ and $B$ have finite\ndimension, then this is true for $K \\in D(A)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"More general supports","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EF5","source_file":"local-cohomology.tex","source_line":881,"source_end_line":894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L881-L894","statement_sha256":"18bad20dc85315d6c754a0c07526db06961a0c5bf9c14baa7c8e6666d2c146cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9086,"rank":9086,"depth":4,"x":2472.112,"y":1026.407,"cluster":"local-crystalline-methods"},{"id":"stacks:0EF6","tag":"0EF6","title":"More general supports · Lemma 0EF6","summary":"Let A be a ring and let T, T' ⊂ Spec(A) subsets stable under specialization. For K ∈ D^+(A) there is a spectral sequence E_2^p, q = H^p_T(H^p_T'(K)) ⇒ H^p + q_T ∩ T'(K) as in Derived Categories, Lemma [Tag 015N].","statement_latex":"Let $A$ be a ring and let $T, T' \\subset \\Spec(A)$ subsets\nstable under specialization. For $K \\in D^+(A)$\nthere is a spectral sequence\n$$\nE_2^{p, q} = H^p_T(H^p_{T'}(K)) \\Rightarrow H^{p + q}_{T \\cap T'}(K)\n$$\nas in Derived Categories, Lemma\n\\ref{derived-lemma-grothendieck-spectral-sequence}.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"More general supports","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EF6","source_file":"local-cohomology.tex","source_line":913,"source_end_line":923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L913-L923","statement_sha256":"4082545192bcf9afdea5914e151ad930c234f1120bb642bf3516f4a7f76ff614","origin":"The Stacks Project","memory_eligible":false,"source_rank":9087,"rank":9087,"depth":13,"x":2577.389,"y":1219.469,"cluster":"local-crystalline-methods"},{"id":"stacks:0EF7","tag":"0EF7","title":"More general supports · Lemma 0EF7","summary":"Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. Assume A has finite dimension. Then RΓ_T(K) = RΓ_T(A) ⊗_A^L K for K ∈ D(A). For K, L ∈ D(A) we have RΓ_T(K ⊗_A^L L) = K ⊗_A^L RΓ_T(L) = RΓ_T(K) ⊗_A^L L = RΓ_T(K) ⊗_A^L RΓ_T(L) If K or L is in D_T(A) then so is K ⊗_A^L L.","statement_latex":"Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset\nstable under specialization. Assume $A$ has finite dimension. Then\n$$\nR\\Gamma_T(K) = R\\Gamma_T(A) \\otimes_A^\\mathbf{L} K\n$$\nfor $K \\in D(A)$. For $K, L \\in D(A)$ we have\n$$\nR\\Gamma_T(K \\otimes_A^\\mathbf{L} L) =\nK \\otimes_A^\\mathbf{L} R\\Gamma_T(L) =\nR\\Gamma_T(K) \\otimes_A^\\mathbf{L} L =\nR\\Gamma_T(K) \\otimes_A^\\mathbf{L} R\\Gamma_T(L)\n$$\nIf $K$ or $L$ is in $D_T(A)$ then so is $K \\otimes_A^\\mathbf{L} L$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"More general supports","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EF7","source_file":"local-cohomology.tex","source_line":945,"source_end_line":960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L945-L960","statement_sha256":"3c2c438c72afb7a23a40a688b488ea26bf74637cdde14010d8ad0cda53c61101","origin":"The Stacks Project","memory_eligible":false,"source_rank":9088,"rank":9088,"depth":3,"x":2343.969,"y":1136.654,"cluster":"local-crystalline-methods"},{"id":"stacks:0EF8","tag":"0EF8","title":"Filtrations on local cohomology · Lemma 0EF8","summary":"Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. Let T' ⊂ T be the set of nonminimal primes in T. Then T' is a subset of Spec(A) stable under specialization and for every A-module M there is an exact sequence 0 → colim_Z, f H^1_f(H^i - 1_Z(M)) → H^i_T'(M) → H^i_T(M) → bigoplus_ p ∈ T setminus T' H^i_ p A_ p(M_ p) where the colimit is over closed subsets Z ⊂ T and f ∈ A with V(f) ∩ Z ⊂ T'.","statement_latex":"Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$\nbe a subset stable under specialization. Let $T' \\subset T$ be\nthe set of nonminimal primes in $T$. Then $T'$\nis a subset of $\\Spec(A)$ stable under specialization\nand for every $A$-module $M$ there is an exact sequence\n$$\n0 \\to\n\\colim_{Z, f} H^1_f(H^{i - 1}_Z(M)) \\to\nH^i_{T'}(M) \\to H^i_T(M) \\to\n\\bigoplus\\nolimits_{\\mathfrak p \\in T \\setminus T'}\nH^i_{\\mathfrak p A_\\mathfrak p}(M_\\mathfrak p)\n$$\nwhere the colimit is over closed subsets $Z \\subset T$\nand $f \\in A$ with $V(f) \\cap Z \\subset T'$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Filtrations on local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EF8","source_file":"local-cohomology.tex","source_line":999,"source_end_line":1015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L999-L1015","statement_sha256":"222dc44c7a460d47f6e010c5aab8399eb5473e46127eade9317aee79853ed938","origin":"The Stacks Project","memory_eligible":false,"source_rank":9089,"rank":9089,"depth":14,"x":2583.233,"y":1065.109,"cluster":"local-crystalline-methods"},{"id":"stacks:0EF9","tag":"0EF9","title":"Filtrations on local cohomology · Lemma 0EF9","summary":"Let A be a Noetherian ring of finite dimension. Let T ⊂ Spec(A) be a subset stable under specialization. Let (M_n)_n ≥ 0 be an inverse system of A-modules. Let i ≥ 0 be an integer. Assume that for every m there exists an integer m'(m) ≥ m such that for all p ∈ T the induced map H^i_ p A_ p(M_k, p) → H^i_ p A_ p(M_m, p) is zero for k ≥ m'(m). Let m\" : N → N be the 2^dim(T)-fold self-composition of m'. Then the map H^i_T(M_k) → H^i_T(M_m) is zero for all k ≥ m\"(m).","statement_latex":"Let $A$ be a Noetherian ring of finite dimension.\nLet $T \\subset \\Spec(A)$ be a subset stable under specialization.\nLet $\\{M_n\\}_{n \\geq 0}$ be an inverse system of $A$-modules.\nLet $i \\geq 0$ be an integer. Assume that for every $m$ there\nexists an integer $m'(m) \\geq m$ such  that for all\n$\\mathfrak p \\in T$ the induced map\n$$\nH^i_{\\mathfrak p A_\\mathfrak p}(M_{k, \\mathfrak p})\n\\longrightarrow\nH^i_{\\mathfrak p A_\\mathfrak p}(M_{m, \\mathfrak p})\n$$\nis zero for $k \\geq m'(m)$. Let $m'' : \\mathbf{N} \\to \\mathbf{N}$\nbe the $2^{\\dim(T)}$-fold self-composition of $m'$. Then the map\n$H^i_T(M_k) \\to H^i_T(M_m)$ is zero for all $k \\geq m''(m)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Filtrations on local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EF9","source_file":"local-cohomology.tex","source_line":1048,"source_end_line":1064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1048-L1064","statement_sha256":"ea89741b7e5e258d0bd9191b7b8de6afc7478234811f05295a511a006f466608","origin":"The Stacks Project","memory_eligible":false,"source_rank":9090,"rank":9090,"depth":15,"x":2464.067,"y":1254.06,"cluster":"local-crystalline-methods"},{"id":"stacks:0EFA","tag":"0EFA","title":"Filtrations on local cohomology · Lemma 0EFA","summary":"Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. Let (M_n)_n ≥ 0 be an inverse system of A-modules. Let i ≥ 0 be an integer. Assume the dimension of A is finite and that for every m there exists an integer m'(m) ≥ m such that for all p ∈ T we have • H^i - 1_ p A_ p(M_k, p) → H^i - 1_ p A_ p(M_m, p) is zero for k ≥ m'(m), and • H^i_ p A_ p(M_k, p) → H^i_ p A_ p(M_m, p) has image G( p, m) independent of k ≥ m'(m) and moreover G( p, m)…","statement_latex":"Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset\nstable under specialization. Let $\\{M_n\\}_{n \\geq 0}$ be an inverse system\nof $A$-modules. Let $i \\geq 0$ be an integer. Assume the dimension of $A$\nis finite and that for every $m$ there exists an integer $m'(m) \\geq m$\nsuch that for all $\\mathfrak p \\in T$ we have\n\\begin{enumerate}\n\\item $H^{i - 1}_{\\mathfrak p A_\\mathfrak p}(M_{k, \\mathfrak p})\n\\to H^{i - 1}_{\\mathfrak p A_\\mathfrak p}(M_{m, \\mathfrak p})$\nis zero for $k \\geq m'(m)$, and\n\\item $ H^i_{\\mathfrak p A_\\mathfrak p}(M_{k, \\mathfrak p}) \\to\nH^i_{\\mathfrak p A_\\mathfrak p}(M_{m, \\mathfrak p})$\nhas image $G(\\mathfrak p, m)$ independent of $k \\geq m'(m)$ and moreover\n$G(\\mathfrak p, m)$ maps injectively into\n$H^i_{\\mathfrak p A_\\mathfrak p}(M_{0, \\mathfrak p})$.\n\\end{enumerate}\nThen there exists an integer $m_0$ such that for every $m \\geq m_0$\nthere exists an integer $m''(m) \\geq m$ such that\nfor $k \\geq m''(m)$ the image of $H^i_T(M_k) \\to H^i_T(M_m)$\nmaps injectively into $H^i_T(M_{m_0})$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Filtrations on local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFA","source_file":"local-cohomology.tex","source_line":1099,"source_end_line":1120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1099-L1120","statement_sha256":"e274911abcf9b43f2e7076ace078b3f2f19839d0d7ca455afa7e31a9df1c2ee0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9091,"rank":9091,"depth":16,"x":2399.844,"y":1046.642,"cluster":"local-crystalline-methods"},{"id":"stacks:0AW8","tag":"0AW8","title":"Finiteness of local cohomology, I · Lemma 0AW8","summary":"[Faltings-annulators] Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. Let M be a finite A-module. Let n ≥ 0. The following are equivalent • H^i_T(M) is finite for i ≤ n, • there exists an ideal J ⊂ A with V(J) ⊂ T such that J annihilates H^i_T(M) for i ≤ n. If T = V(I) = Z for an ideal I ⊂ A, then these are also equivalent to • [(3)] there exists an e ≥ 0 such that I^e annihilates H^i_Z(M) for i ≤ n.","statement_latex":"\\begin{reference}\n\\cite[Lemma 3]{Faltings-annulators}\n\\end{reference}\nLet $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable\nunder specialization. Let $M$ be a finite $A$-module. Let $n \\geq 0$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $H^i_T(M)$ is finite for $i \\leq n$,\n\\item there exists an ideal $J \\subset A$ with $V(J) \\subset T$\nsuch that $J$ annihilates $H^i_T(M)$ for $i \\leq n$.\n\\end{enumerate}\nIf $T = V(I) = Z$ for an ideal $I \\subset A$, then these are also\nequivalent to\n\\begin{enumerate}\n\\item[(3)] there exists an $e \\geq 0$ such that $I^e$ annihilates\n$H^i_Z(M)$ for $i \\leq n$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AW8","source_file":"local-cohomology.tex","source_line":1202,"source_end_line":1221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1202-L1221","statement_sha256":"ead736e261baf2fd4f005d7115f5b599b220479ae199dcbb88275d409c7b34f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9092,"rank":9092,"depth":9,"x":2614.484,"y":1163.411,"cluster":"local-crystalline-methods"},{"id":"stacks:0AW9","tag":"0AW9","title":"Finiteness of local cohomology, I · Lemma 0AW9","summary":"This is a special case of [Faltings-finiteness]. Let A be a Noetherian ring, I ⊂ A an ideal, M a finite A-module, and n ≥ 0 an integer. Let Z = V(I). The following are equivalent • the modules H^i_Z(M) are finite for i ≤ n, and • for all p ∈ Spec(A) the modules H^i_Z(M)_ p, i ≤ n are finite A_ p-modules.","statement_latex":"\\begin{reference}\nThis is a special case of \\cite[Satz 1]{Faltings-finiteness}.\n\\end{reference}\nLet $A$ be a Noetherian ring, $I \\subset A$ an ideal, $M$ a finite\n$A$-module, and $n \\geq 0$ an integer. Let $Z = V(I)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item the modules $H^i_Z(M)$ are finite for $i \\leq n$, and\n\\item for all $\\mathfrak p \\in \\Spec(A)$ the modules\n$H^i_Z(M)_\\mathfrak p$, $i \\leq n$ are finite $A_\\mathfrak p$-modules.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AW9","source_file":"local-cohomology.tex","source_line":1274,"source_end_line":1287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1274-L1287","statement_sha256":"52dcb217d3b8f4c70e9f0e78bf7d61a8a8659adbbdc1656481a963f94c502ee8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9093,"rank":9093,"depth":15,"x":2361.743,"y":1199.179,"cluster":"local-crystalline-methods"},{"id":"stacks:0BPX","tag":"0BPX","title":"Finiteness of local cohomology, I · Lemma 0BPX","summary":"Let A be a ring and let J ⊂ I ⊂ A be finitely generated ideals. Let i ≥ 0 be an integer. Set Z = V(I). If H^i_Z(A) is annihilated by J^n for some n, then H^i_Z(M) annihilated by J^m for some m = m(M) for every finitely presented A-module M such that M_f is a finite locally free A_f-module for all f ∈ I.","statement_latex":"Let $A$ be a ring and let $J \\subset I \\subset A$ be finitely generated ideals.\nLet $i \\geq 0$ be an integer. Set $Z = V(I)$. If\n$H^i_Z(A)$ is annihilated by $J^n$ for some $n$, then\n$H^i_Z(M)$ annihilated by $J^m$ for some $m = m(M)$\nfor every finitely presented $A$-module $M$ such that\n$M_f$ is a finite locally free $A_f$-module for all $f \\in I$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPX","source_file":"local-cohomology.tex","source_line":1344,"source_end_line":1352,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1344-L1352","statement_sha256":"f55f5a67c731201ad77c90cb88303991c5b9cc4b57a953a07aedccb0d6d643ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":9094,"rank":9094,"depth":0,"x":2519.699,"y":1029.013,"cluster":"local-crystalline-methods"},{"id":"stacks:0BPY","tag":"0BPY","title":"Finiteness of local cohomology, I · Lemma 0BPY","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Set Z = V(I). Let n ≥ 0 be an integer. If H^i_Z(A) is finite for 0 ≤ i ≤ n, then the same is true for H^i_Z(M), 0 ≤ i ≤ n for any finite A-module M such that M_f is a finite locally free A_f-module for all f ∈ I.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal. Set $Z = V(I)$.\nLet $n \\geq 0$ be an integer. If $H^i_Z(A)$ is finite for $0 \\leq i \\leq n$,\nthen the same is true for $H^i_Z(M)$, $0 \\leq i \\leq n$ for\nany finite $A$-module $M$ such that $M_f$ is a finite locally free\n$A_f$-module for all $f \\in I$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPY","source_file":"local-cohomology.tex","source_line":1381,"source_end_line":1388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1381-L1388","statement_sha256":"01e122f83858252876c2f81a0a761ec5758c8cdfbdc623740a042b96d4bc3ec8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9095,"rank":9095,"depth":10,"x":2540.112,"y":1244.599,"cluster":"local-crystalline-methods"},{"id":"stacks:0BJZ","tag":"0BJZ","title":"Finiteness of pushforwards, I · Lemma 0BJZ","summary":"Let X be a locally Noetherian scheme. Let j : U → X be the inclusion of an open subscheme with complement Z. For x ∈ U let i_x : W_x → U be the integral closed subscheme with generic point x. Let F be a coherent O_U-module. The following are equivalent • for all x ∈ Ass(F) the O_X-module j_*i_x, *O_W_x is coherent, • j_*F is coherent.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion\nof an open subscheme with complement $Z$. For $x \\in U$ let\n$i_x : W_x \\to U$ be the integral closed subscheme with generic point $x$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item for all $x \\in \\text{Ass}(\\mathcal{F})$ the\n$\\mathcal{O}_X$-module $j_*i_{x, *}\\mathcal{O}_{W_x}$ is coherent,\n\\item $j_*\\mathcal{F}$ is coherent.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJZ","source_file":"local-cohomology.tex","source_line":1425,"source_end_line":1437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1425-L1437","statement_sha256":"0f7dce5a46116fac889969288ed90e947af3f04ca2c11b69a2e605ce55ec6070","origin":"The Stacks Project","memory_eligible":false,"source_rank":9096,"rank":9096,"depth":18,"x":2351.275,"y":1096.885,"cluster":"local-crystalline-methods"},{"id":"stacks:0BK0","tag":"0BK0","title":"Finiteness of pushforwards, I · Lemma 0BK0","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Set X = Spec(A), Z = V(I), U = X setminus Z, and j : U → X the inclusion morphism. Let F be a coherent O_U-module. Then • there exists a finite A-module M such that F is the restriction of widetildeM to U, • given M there is an exact sequence 0 → H^0_Z(M) → M → H^0(U, F) → H^1_Z(M) → 0 and isomorphisms H^p(U, F) = H^p + 1_Z(M) for p ≥ 1, • given M and p ≥ 0 the following are equivalent • R^pj_*F is coherent, • H^p(U,…","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nSet $X = \\Spec(A)$, $Z = V(I)$, $U = X \\setminus Z$, and $j : U \\to X$\nthe inclusion morphism. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.\nThen\n\\begin{enumerate}\n\\item there exists a finite $A$-module $M$ such that $\\mathcal{F}$ is the\nrestriction of $\\widetilde{M}$ to $U$,\n\\item given $M$ there is an exact sequence\n$$\n0 \\to H^0_Z(M) \\to M \\to H^0(U, \\mathcal{F}) \\to H^1_Z(M) \\to 0\n$$\nand isomorphisms $H^p(U, \\mathcal{F}) = H^{p + 1}_Z(M)$ for $p \\geq 1$,\n\\item given $M$ and $p \\geq 0$ the following are equivalent\n\\begin{enumerate}\n\\item $R^pj_*\\mathcal{F}$ is coherent,\n\\item $H^p(U, \\mathcal{F})$ is a finite $A$-module,\n\\item $H^{p + 1}_Z(M)$ is a finite $A$-module,\n\\end{enumerate}\n\\item if the equivalent conditions in (3) hold for $p = 0$, we may take\n$M = \\Gamma(U, \\mathcal{F})$ in which case we have $H^0_Z(M) = H^1_Z(M) = 0$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BK0","source_file":"local-cohomology.tex","source_line":1506,"source_end_line":1529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1506-L1529","statement_sha256":"8f8423e7fafde50edadb7ce08298e2665b40c4a51d2947bed4bfea068317f0df","origin":"The Stacks Project","memory_eligible":false,"source_rank":9097,"rank":9097,"depth":29,"x":2609.878,"y":1098.66,"cluster":"local-crystalline-methods"},{"id":"stacks:0AWA","tag":"0AWA","title":"Finiteness of pushforwards, I · Lemma 0AWA","summary":"Let X be a locally Noetherian scheme. Let j : U → X be the inclusion of an open subscheme with complement Z. Let F be a coherent O_U-module. Assume • X is Nagata, • X is universally catenary, and • for x ∈ Ass(F) and z ∈ Z ∩ overline(x) we have dim(O_overline(x), z) ≥ 2. Then j_*F is coherent.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $j : U \\to X$ be the inclusion of an\nopen subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent\n$\\mathcal{O}_U$-module. Assume\n\\begin{enumerate}\n\\item $X$ is Nagata,\n\\item $X$ is universally catenary, and\n\\item for $x \\in \\text{Ass}(\\mathcal{F})$ and\n$z \\in Z \\cap \\overline{\\{x\\}}$ we have\n$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) \\geq 2$.\n\\end{enumerate}\nThen $j_*\\mathcal{F}$ is coherent.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWA","source_file":"local-cohomology.tex","source_line":1558,"source_end_line":1572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1558-L1572","statement_sha256":"925f91e4cec5916e970969ca97ae5560722686062e227788f296000d5fb7b792","origin":"The Stacks Project","memory_eligible":false,"source_rank":9098,"rank":9098,"depth":32,"x":2417.337,"y":1244.408,"cluster":"local-crystalline-methods"},{"id":"stacks:0BK1","tag":"0BK1","title":"Finiteness of pushforwards, I · Lemma 0BK1","summary":"Let X be an integral locally Noetherian scheme. Let j : U → X be the inclusion of a nonempty open subscheme with complement Z. Assume that for all z ∈ Z and any associated prime p of the completion O_X, z^wedge we have dim(O_X, z^wedge/ p) ≥ 2. Then j_*O_U is coherent.","statement_latex":"Let $X$ be an integral locally Noetherian scheme. Let $j : U \\to X$\nbe the inclusion of a nonempty open subscheme with complement $Z$. Assume\nthat for all $z \\in Z$ and any associated prime $\\mathfrak p$ of\nthe completion $\\mathcal{O}_{X, z}^\\wedge$\nwe have $\\dim(\\mathcal{O}_{X, z}^\\wedge/\\mathfrak p) \\geq 2$.\nThen $j_*\\mathcal{O}_U$ is coherent.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BK1","source_file":"local-cohomology.tex","source_line":1618,"source_end_line":1626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1618-L1626","statement_sha256":"4071ff1f46da62ba6cff898504e747a92ca4ed963287210f2ce54002eb40a569","origin":"The Stacks Project","memory_eligible":false,"source_rank":9099,"rank":9099,"depth":33,"x":2442.162,"y":1027.209,"cluster":"local-crystalline-methods"},{"id":"stacks:0BK3","tag":"0BK3","title":"Koll\\'ar · Proposition 0BK3","summary":"See [k-coherent] and see [EGA] for a special case. Weak analogue of Hartogs' Theorem: On Noetherian schemes, the restriction of a coherent sheaf to an open set with complement of codimension 2 in the sheaf's support, is coherent. Let j : U → X be an open immersion of locally Noetherian schemes with complement Z. Let F be a coherent O_U-module. The following are equivalent • j_*F is coherent, • for x ∈ Ass(F) and z ∈ Z ∩ overline(x) and any associated prime p of the…","statement_latex":"\\begin{reference}\nSee \\cite{k-coherent} and see \\cite[IV, Proposition 7.2.2]{EGA}\nfor a special case.\n\\end{reference}\n\\begin{slogan}\nWeak analogue of Hartogs' Theorem: On Noetherian schemes, the\nrestriction of a coherent sheaf to an open set with complement\nof codimension 2 in the sheaf's support, is coherent.\n\\end{slogan}\nLet $j : U \\to X$ be an open immersion of locally Noetherian schemes\nwith complement $Z$. Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $j_*\\mathcal{F}$ is coherent,\n\\item for $x \\in \\text{Ass}(\\mathcal{F})$ and\n$z \\in Z \\cap \\overline{\\{x\\}}$ and any associated prime\n$\\mathfrak p$ of the completion $\\mathcal{O}_{\\overline{\\{x\\}}, z}^\\wedge$\nwe have $\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}^\\wedge/\\mathfrak p) \\geq 2$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, I","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BK3","source_file":"local-cohomology.tex","source_line":1727,"source_end_line":1748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1727-L1748","statement_sha256":"fc733a9926df3495f2411c5d727c07d2cb7da4fa37d250c8cd7bb5328fcb40e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9100,"rank":9100,"depth":34,"x":2598.862,"y":1201.834,"cluster":"local-crystalline-methods"},{"id":"stacks:0BL9","tag":"0BL9","title":"Finiteness of pushforwards, I · Lemma 0BL9","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Set Z = V(I). Let M be a finite A-module. The following are equivalent • H^1_Z(M) is a finite A-module, and • for all p ∈ Ass(M), p not ∈ Z and all q ∈ V( p + I) the completion of (A/ p)_ q does not have associated primes of dimension 1.","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nSet $Z = V(I)$. Let $M$ be a finite $A$-module. The following\nare equivalent\n\\begin{enumerate}\n\\item $H^1_Z(M)$ is a finite $A$-module, and\n\\item for all $\\mathfrak p \\in \\text{Ass}(M)$, $\\mathfrak p \\not \\in Z$\nand all $\\mathfrak q \\in V(\\mathfrak p + I)$ the completion of\n$(A/\\mathfrak p)_\\mathfrak q$ does not have associated primes\nof dimension $1$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BL9","source_file":"local-cohomology.tex","source_line":1763,"source_end_line":1775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1763-L1775","statement_sha256":"a2fb44de97bbf0264743ab087cff7ed1f0b1186048f301f828f541fb863b4154","origin":"The Stacks Project","memory_eligible":false,"source_rank":9101,"rank":9101,"depth":35,"x":2342.33,"y":1161.897,"cluster":"local-crystalline-methods"},{"id":"stacks:0AWB","tag":"0AWB","title":"Finiteness of pushforwards, I · Lemma 0AWB","summary":"Let X be a locally Noetherian scheme. Let j : U → X be the inclusion of an open subscheme with complement Z. Let F be a coherent O_U-module. Assume • X is universally catenary, • for every z ∈ Z the formal fibres of O_X, z are (S_1). In this situation the following are equivalent • [(a)] for x ∈ Ass(F) and z ∈ Z ∩ overline(x) we have dim(O_overline(x), z) ≥ 2, and • [(b)] j_*F is coherent.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $j : U \\to X$ be the inclusion of an\nopen subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent\n$\\mathcal{O}_U$-module. Assume\n\\begin{enumerate}\n\\item $X$ is universally catenary,\n\\item for every $z \\in Z$ the formal fibres of $\\mathcal{O}_{X, z}$\nare $(S_1)$.\n\\end{enumerate}\nIn this situation the following are equivalent\n\\begin{enumerate}\n\\item[(a)] for $x \\in \\text{Ass}(\\mathcal{F})$ and\n$z \\in Z \\cap \\overline{\\{x\\}}$ we have\n$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) \\geq 2$, and\n\\item[(b)] $j_*\\mathcal{F}$ is coherent.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AWB","source_file":"local-cohomology.tex","source_line":1788,"source_end_line":1806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1788-L1806","statement_sha256":"6e4fb9d083d79dbf8b908c41493b67edb1e865eff783313a1ba05ca0ca249a82","origin":"The Stacks Project","memory_eligible":false,"source_rank":9102,"rank":9102,"depth":50,"x":2564.087,"y":1045.533,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWX","tag":"0DWX","title":"Depth and dimension · Lemma 0DWX","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let M be a finite A-module. Let p ∈ V(I) be a prime ideal. Assume e = depth_IA_ p(M_ p) < ∞. Then there exists a nonempty open U ⊂ V( p) such that depth_IA_ q(M_ q) ≥ e for all q ∈ U.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nLet $M$ be a finite $A$-module. Let $\\mathfrak p \\in V(I)$\nbe a prime ideal. Assume\n$e = \\text{depth}_{IA_\\mathfrak p}(M_\\mathfrak p) < \\infty$.\nThen there exists a nonempty open $U \\subset V(\\mathfrak p)$\nsuch that $\\text{depth}_{IA_\\mathfrak q}(M_\\mathfrak q) \\geq e$\nfor all $\\mathfrak q \\in U$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Depth and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWX","source_file":"local-cohomology.tex","source_line":1842,"source_end_line":1851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1842-L1851","statement_sha256":"414d705ecbe4ea78421123dadbc93af74fc327636863702a499edd947e99130f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9103,"rank":9103,"depth":1,"x":2493.993,"y":1257.623,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWY","tag":"0DWY","title":"Depth and dimension · Lemma 0DWY","summary":"Let A be a Noetherian ring. Let M be a finite A-module. Let p be a prime ideal. Assume e = depth_A_ p(M_ p) < ∞. Then there exists a nonempty open U ⊂ V( p) such that depth_A_ q(M_ q) ≥ e for all q ∈ U and for all but finitely many q ∈ U we have depth_A_ q(M_ q) > e.","statement_latex":"Let $A$ be a Noetherian ring. Let $M$ be a finite $A$-module.\nLet $\\mathfrak p$ be a prime ideal. Assume\n$e = \\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) < \\infty$.\nThen there exists a nonempty open $U \\subset V(\\mathfrak p)$\nsuch that $\\text{depth}_{A_\\mathfrak q}(M_\\mathfrak q) \\geq e$\nfor all $\\mathfrak q \\in U$ and\nfor all but finitely many $\\mathfrak q \\in U$ we have\n$\\text{depth}_{A_\\mathfrak q}(M_\\mathfrak q) > e$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Depth and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWY","source_file":"local-cohomology.tex","source_line":1870,"source_end_line":1880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1870-L1880","statement_sha256":"41dbd0ee02ff220df0ec9f29b552d6274ce734f1d62d83966dc4231cb07b959c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9104,"rank":9104,"depth":3,"x":2374.868,"y":1061.038,"cluster":"local-crystalline-methods"},{"id":"stacks:0ECN","tag":"0ECN","title":"Depth and dimension · Lemma 0ECN","summary":"Let X be a Noetherian scheme with dualizing complex ω_X^bullet. Let F be a coherent O_X-module. Let k ≥ 0 be an integer. Assume F is (S_k). Then there is a finite number of points x ∈ X such that depth(F_x) = k and dim(Supp(F_x)) > k","statement_latex":"Let $X$ be a Noetherian scheme with dualizing complex $\\omega_X^\\bullet$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $k \\geq 0$\nbe an integer. Assume $\\mathcal{F}$ is $(S_k)$.\nThen there is a finite number of points $x \\in X$ such that\n$$\n\\text{depth}(\\mathcal{F}_x) = k\n\\quad\\text{and}\\quad\n\\dim(\\text{Supp}(\\mathcal{F}_x)) > k\n$$","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Depth and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECN","source_file":"local-cohomology.tex","source_line":1904,"source_end_line":1915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1904-L1915","statement_sha256":"c476b33cfb4138087fe8805d2723cb41cf2a3bc7cd6f7f096862655e6676e411","origin":"The Stacks Project","memory_eligible":false,"source_rank":9105,"rank":9105,"depth":35,"x":2621.322,"y":1138.568,"cluster":"local-crystalline-methods"},{"id":"stacks:0DWZ","tag":"0DWZ","title":"Depth and dimension · Lemma 0DWZ","summary":"Let (A, m) be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Let M be a finite A-module. Set E^i = Ext_A^-i(M, ω_A^bullet). Then • E^i is a finite A-module nonzero only for 0 ≤ i ≤ dim(Supp(M)), • dim(Supp(E^i)) ≤ i, • depth(M) is the smallest integer δ ≥ 0 such that E^δ not = 0, • p ∈ Supp(E^0 ⊕ … ⊕ E^i) ⇔ depth_A_ p(M_ p) + dim(A/ p) ≤ i, • the annihilator of E^i is equal to the annihilator of H^i_ m(M).","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring with\nnormalized dualizing complex $\\omega_A^\\bullet$.\nLet $M$ be a finite $A$-module.\nSet $E^i = \\text{Ext}_A^{-i}(M, \\omega_A^\\bullet)$.\nThen\n\\begin{enumerate}\n\\item $E^i$ is a finite $A$-module nonzero only for\n$0 \\leq i \\leq \\dim(\\text{Supp}(M))$,\n\\item $\\dim(\\text{Supp}(E^i)) \\leq i$,\n\\item $\\text{depth}(M)$ is the smallest integer $\\delta \\geq 0$ such that\n$E^\\delta \\not = 0$,\n\\item $\\mathfrak p \\in \\text{Supp}(E^0 \\oplus \\ldots \\oplus E^i)\n\\Leftrightarrow\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) \\leq i$,\n\\item the annihilator of $E^i$ is equal to the annihilator\nof $H^i_\\mathfrak m(M)$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Depth and dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DWZ","source_file":"local-cohomology.tex","source_line":1967,"source_end_line":1986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L1967-L1986","statement_sha256":"9b0642b3514280f59e075da6e86221d9f131f2921710185bd3bf47ef12d34dd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9106,"rank":9106,"depth":38,"x":2376.725,"y":1221.417,"cluster":"local-crystalline-methods"},{"id":"stacks:0EFC","tag":"0EFC","title":"Annihilators of local cohomology, I · Proposition 0EFC","summary":"[Faltings-annulators]. Let A be a Noetherian ring which has a dualizing complex. Let T ⊂ T' ⊂ Spec(A) be subsets stable under specialization. Let s ≥ 0 an integer. Let M be a finite A-module. The following are equivalent • there exists an ideal J ⊂ A with V(J) ⊂ T' such that J annihilates H^i_T(M) for i ≤ s, and • for all p not ∈ T', q ∈ T with p ⊂ q we have depth_A_ p(M_ p) + dim((A/ p)_ q) > s","statement_latex":"\\begin{reference}\n\\cite{Faltings-annulators}.\n\\end{reference}\nLet $A$ be a Noetherian ring which has a dualizing complex.\nLet $T \\subset T' \\subset \\Spec(A)$ be subsets stable under\nspecialization. Let $s \\geq 0$ an integer. Let $M$ be a finite $A$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists an ideal $J \\subset A$ with $V(J) \\subset T'$\nsuch that $J$ annihilates $H^i_T(M)$ for $i \\leq s$, and\n\\item for all $\\mathfrak p \\not \\in T'$,\n$\\mathfrak q \\in T$ with $\\mathfrak p \\subset \\mathfrak q$\nwe have\n$$\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q) > s\n$$\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Annihilators of local cohomology, I","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFC","source_file":"local-cohomology.tex","source_line":2034,"source_end_line":2054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2034-L2054","statement_sha256":"fd4c655b9264ca4e0b05e6b39e8b9830eff59538cbb04a512eb2388c7d976375","origin":"The Stacks Project","memory_eligible":false,"source_rank":9107,"rank":9107,"depth":34,"x":2490.703,"y":1021.113,"cluster":"local-crystalline-methods"},{"id":"stacks:0EFE","tag":"0EFE","title":"Annihilators of local cohomology, I · Lemma 0EFE","summary":"Let I be an ideal of a Noetherian ring A. Let M be a finite A-module, let p ⊂ A be a prime ideal, and let s ≥ 0 be an integer. Assume • A has a dualizing complex, • p not ∈ V(I), and • for all primes p' ⊂ p and q ∈ V(I) with p' ⊂ q we have depth_A_ p'(M_ p') + dim((A/ p')_ q) > s Then there exists an f ∈ A, f not ∈ p which annihilates H^i_V(I)(M) for i ≤ s.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$.\nLet $M$ be a finite $A$-module, let $\\mathfrak p \\subset A$ be a prime\nideal, and let $s \\geq 0$ be an integer. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex,\n\\item $\\mathfrak p \\not \\in V(I)$, and\n\\item for all primes $\\mathfrak p' \\subset \\mathfrak p$\nand $\\mathfrak q \\in V(I)$ with $\\mathfrak p' \\subset \\mathfrak q$ we have\n$$\n\\text{depth}_{A_{\\mathfrak p'}}(M_{\\mathfrak p'}) +\n\\dim((A/\\mathfrak p')_\\mathfrak q) > s\n$$\n\\end{enumerate}\nThen there exists an $f \\in A$, $f \\not \\in \\mathfrak p$ which annihilates\n$H^i_{V(I)}(M)$ for $i \\leq s$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Annihilators of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFE","source_file":"local-cohomology.tex","source_line":2208,"source_end_line":2225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2208-L2225","statement_sha256":"898540b830a9201688c24f785b91a206b3dba627d2818937a0b6e66e9ccc0e69","origin":"The Stacks Project","memory_eligible":false,"source_rank":9108,"rank":9108,"depth":35,"x":2567.897,"y":1233.935,"cluster":"local-crystalline-methods"},{"id":"stacks:0EFD","tag":"0EFD","title":"Finiteness of local cohomology, II · Proposition 0EFD","summary":"[Faltings-annulators]. Let A be a Noetherian ring which has a dualizing complex. Let T ⊂ Spec(A) be a subset stable under specialization. Let s ≥ 0 an integer. Let M be a finite A-module. The following are equivalent • H^i_T(M) is a finite A-module for i ≤ s, and • for all p not ∈ T, q ∈ T with p ⊂ q we have depth_A_ p(M_ p) + dim((A/ p)_ q) > s","statement_latex":"\\begin{reference}\n\\cite{Faltings-annulators}.\n\\end{reference}\nLet $A$ be a Noetherian ring which has a dualizing complex.\nLet $T \\subset \\Spec(A)$ be a subset stable under specialization.\nLet $s \\geq 0$ an integer. Let $M$ be a finite $A$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $H^i_T(M)$ is a finite $A$-module for $i \\leq s$, and\n\\item for all $\\mathfrak p \\not \\in T$, $\\mathfrak q \\in T$ with\n$\\mathfrak p \\subset \\mathfrak q$ we have\n$$\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q) > s\n$$\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFD","source_file":"local-cohomology.tex","source_line":2257,"source_end_line":2275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2257-L2275","statement_sha256":"34aa65e65f7395d8164f5d37cd855699c7fbc68e5ba4a33e9d2cd36a397a0a4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9109,"rank":9109,"depth":35,"x":2339.352,"y":1120.568,"cluster":"local-crystalline-methods"},{"id":"stacks:0BJS","tag":"0BJS","title":"Finiteness of local cohomology, II · Lemma 0BJS","summary":"Let A → B be a finite homomorphism of Noetherian rings. Let I ⊂ A be an ideal and set J = IB. Let M be a finite B-module. If A is universally catenary, then s_B, J(M) = s_A, I(M).","statement_latex":"Let $A \\to B$ be a finite homomorphism of Noetherian rings.\nLet $I \\subset A$ be an ideal and set $J = IB$. Let $M$ be\na finite $B$-module. If $A$ is universally catenary, then\n$s_{B, J}(M) = s_{A, I}(M)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJS","source_file":"local-cohomology.tex","source_line":2309,"source_end_line":2315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2309-L2315","statement_sha256":"3d49226b75b83d359c35b48f8c14ff63576e4feca5265a48fea20444759c9d68","origin":"The Stacks Project","memory_eligible":false,"source_rank":9110,"rank":9110,"depth":15,"x":2599.586,"y":1074.386,"cluster":"local-crystalline-methods"},{"id":"stacks:0EHW","tag":"0EHW","title":"Finiteness of local cohomology, II · Lemma 0EHW","summary":"Let A be a Noetherian ring which has a dualizing complex. Let I ⊂ A be an ideal. Let M be a finite A-module. Let A', M' be the I-adic completions of A, M. Let p' ⊂ q' be prime ideals of A' with q' ∈ V(IA') lying over p ⊂ q in A. Then depth_A_ p'(M'_ p') ≥ depth_A_ p(M_ p) and depth_A_ p'(M'_ p') + dim((A'/ p')_ q') = depth_A_ p(M_ p) + dim((A/ p)_ q)","statement_latex":"Let $A$ be a Noetherian ring which has a dualizing complex.\nLet $I \\subset A$ be an ideal.\nLet $M$ be a finite $A$-module. Let $A', M'$ be the $I$-adic\ncompletions of $A, M$. Let $\\mathfrak p' \\subset \\mathfrak q'$\nbe prime ideals of $A'$ with $\\mathfrak q' \\in V(IA')$\nlying over $\\mathfrak p \\subset \\mathfrak q$ in $A$. Then\n$$\n\\text{depth}_{A_{\\mathfrak p'}}(M'_{\\mathfrak p'})\n\\geq\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p)\n$$\nand\n$$\n\\text{depth}_{A_{\\mathfrak p'}}(M'_{\\mathfrak p'}) +\n\\dim((A'/\\mathfrak p')_{\\mathfrak q'}) =\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q)\n$$","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHW","source_file":"local-cohomology.tex","source_line":2345,"source_end_line":2365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2345-L2365","statement_sha256":"23463be9c629097ba56154021a41fa2b52e3d7ee6139fdc63155f1b7124116cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9111,"rank":9111,"depth":36,"x":2444.511,"y":1256.48,"cluster":"local-crystalline-methods"},{"id":"stacks:0BJT","tag":"0BJT","title":"Finiteness of local cohomology, II · Lemma 0BJT","summary":"Let A be a universally catenary Noetherian local ring. Let I ⊂ A be an ideal. Let M be a finite A-module. Then s_A, I(M) ≥ s_A^wedge, I^wedge(M^wedge) If the formal fibres of A are (S_n), then min(n + 1, s_A, I(M)) ≤ s_A^wedge, I^wedge(M^wedge).","statement_latex":"Let $A$ be a universally catenary Noetherian local ring.\nLet $I \\subset A$ be an ideal. Let $M$ be\na finite $A$-module. Then\n$$\ns_{A, I}(M) \\geq s_{A^\\wedge, I^\\wedge}(M^\\wedge)\n$$\nIf the formal fibres of $A$ are $(S_n)$, then\n$\\min(n + 1, s_{A, I}(M)) \\leq s_{A^\\wedge, I^\\wedge}(M^\\wedge)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJT","source_file":"local-cohomology.tex","source_line":2410,"source_end_line":2420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2410-L2420","statement_sha256":"15ce31e59f85661d43d1b4458a7f0e3b00b23eaf37268764511da0d57bd92d62","origin":"The Stacks Project","memory_eligible":false,"source_rank":9112,"rank":9112,"depth":50,"x":2412.36,"y":1033.753,"cluster":"local-crystalline-methods"},{"id":"stacks:0BJU","tag":"0BJU","title":"Finiteness of local cohomology, II · Lemma 0BJU","summary":"This is a special case of [Faltings-annulators]. Let A be a Gorenstein Noetherian local ring. Let I ⊂ A be an ideal and set Z = V(I) ⊂ Spec(A). Let M be a finite A-module. Let s = s_A, I(M) as in ([Tag 0BJR]). Then H^i_Z(M) is finite for i < s, but H^s_Z(M) is not finite.","statement_latex":"\\begin{reference}\nThis is a special case of\n\\cite[Satz 1]{Faltings-annulators}.\n\\end{reference}\nLet $A$ be a Gorenstein Noetherian local ring. Let $I \\subset A$\nbe an ideal and set $Z = V(I) \\subset \\Spec(A)$.\nLet $M$ be a finite $A$-module. Let $s = s_{A, I}(M)$ as in\n(\\ref{equation-cutoff}). Then $H^i_Z(M)$ is finite for $i < s$,\nbut $H^s_Z(M)$ is not finite.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJU","source_file":"local-cohomology.tex","source_line":2533,"source_end_line":2544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2533-L2544","statement_sha256":"bf9994278e81fe9ab312ee728bc926c5bb0e90abb674c300e3f828e48ba436a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9113,"rank":9113,"depth":36,"x":2615.596,"y":1180.046,"cluster":"local-crystalline-methods"},{"id":"stacks:0BJV","tag":"0BJV","title":"Finiteness of local cohomology, II · Theorem 0BJV","summary":"This is a special case of [Faltings-finiteness]. Let A be a Noetherian ring and let I ⊂ A be an ideal. Set Z = V(I) ⊂ Spec(A). Let M be a finite A-module. Set s = s_A, I(M) as in ([Tag 0BJR]). Assume that • A is universally catenary, • the formal fibres of the local rings of A are Cohen-Macaulay. Then H^i_Z(M) is finite for 0 ≤ i < s and H^s_Z(M) is not finite.","statement_latex":"\\begin{reference}\nThis is a special case of \\cite[Satz 2]{Faltings-finiteness}.\n\\end{reference}\nLet $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nSet $Z = V(I) \\subset \\Spec(A)$. Let $M$ be a finite $A$-module.\nSet $s = s_{A, I}(M)$ as in (\\ref{equation-cutoff}).\nAssume that\n\\begin{enumerate}\n\\item $A$ is universally catenary,\n\\item the formal fibres of the local rings of $A$ are Cohen-Macaulay.\n\\end{enumerate}\nThen $H^i_Z(M)$ is finite for $0 \\leq i < s$ and\n$H^s_Z(M)$ is not finite.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, II","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJV","source_file":"local-cohomology.tex","source_line":2562,"source_end_line":2577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2562-L2577","statement_sha256":"bc2b2af3886d92ecfa7c5aa54a77f49ef0ad0e2706b54efd7172a1e11c4610f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9114,"rank":9114,"depth":51,"x":2347.539,"y":1187.508,"cluster":"local-crystalline-methods"},{"id":"stacks:0BJY","tag":"0BJY","title":"Finiteness of pushforwards, II · Lemma 0BJY","summary":"Let X be a locally Noetherian scheme. Let j : U → X be the inclusion of an open subscheme with complement Z. Let F be a coherent O_U-module. Let n ≥ 0 be an integer. Assume • X is universally catenary, • for every z ∈ Z the formal fibres of O_X, z are (S_n). In this situation the following are equivalent • [(a)] for x ∈ Supp(F) and z ∈ Z ∩ overline(x) we have depth_O_X, x(F_x) + dim(O_overline(x), z) > n, • [(b)] R^pj_*F is coherent for 0 ≤ p < n.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion\nof an open subscheme with complement $Z$. Let $\\mathcal{F}$ be a coherent\n$\\mathcal{O}_U$-module. Let $n \\geq 0$ be an integer. Assume\n\\begin{enumerate}\n\\item $X$ is universally catenary,\n\\item for every $z \\in Z$ the formal fibres of\n$\\mathcal{O}_{X, z}$ are $(S_n)$.\n\\end{enumerate}\nIn this situation the following are equivalent\n\\begin{enumerate}\n\\item[(a)] for $x \\in \\text{Supp}(\\mathcal{F})$ and\n$z \\in Z \\cap \\overline{\\{x\\}}$ we have\n$\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) +\n\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > n$,\n\\item[(b)] $R^pj_*\\mathcal{F}$ is coherent for $0 \\leq p < n$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJY","source_file":"local-cohomology.tex","source_line":2645,"source_end_line":2663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2645-L2663","statement_sha256":"3f2260fb8ae029c7d86d207c551b412420a42e469975282a2c792bd68c4be1c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9115,"rank":9115,"depth":52,"x":2539.591,"y":1029.582,"cluster":"local-crystalline-methods"},{"id":"stacks:0BLT","tag":"0BLT","title":"Finiteness of pushforwards, II · Lemma 0BLT","summary":"Let X be a locally Noetherian scheme. Let j : U → X be the inclusion of an open subscheme with complement Z. Let n ≥ 0 be an integer. If R^pj_*O_U is coherent for 0 ≤ p < n, then the same is true for R^pj_*F, 0 ≤ p < n for any finite locally free O_U-module F.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $j : U \\to X$ be the inclusion\nof an open subscheme with complement $Z$. Let $n \\geq 0$ be an integer.\nIf $R^pj_*\\mathcal{O}_U$ is coherent for $0 \\leq p < n$, then\nthe same is true for $R^pj_*\\mathcal{F}$, $0 \\leq p < n$\nfor any finite locally free $\\mathcal{O}_U$-module $\\mathcal{F}$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLT","source_file":"local-cohomology.tex","source_line":2682,"source_end_line":2689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2682-L2689","statement_sha256":"7f5ac6b66e0238146a6b1806bcc9087503d26a8e1b75181c374e9d487a8e6a31","origin":"The Stacks Project","memory_eligible":false,"source_rank":9116,"rank":9116,"depth":30,"x":2524.944,"y":1255.468,"cluster":"local-crystalline-methods"},{"id":"stacks:0BM5","tag":"0BM5","title":"Finiteness of pushforwards, II · Lemma 0BM5","summary":"[Bhatt-local] Let A be a ring and let J ⊂ I ⊂ A be finitely generated ideals. Let p ≥ 0 be an integer. Set U = Spec(A) setminus V(I). If H^p(U, O_U) is annihilated by J^n for some n, then H^p(U, F) annihilated by J^m for some m = m(F) for every finite locally free O_U-module F.","statement_latex":"\\begin{reference}\n\\cite[Lemma 1.9]{Bhatt-local}\n\\end{reference}\nLet $A$ be a ring and let $J \\subset I \\subset A$ be finitely generated ideals.\nLet $p \\geq 0$ be an integer. Set $U = \\Spec(A) \\setminus V(I)$. If\n$H^p(U, \\mathcal{O}_U)$ is annihilated by $J^n$ for some $n$, then\n$H^p(U, \\mathcal{F})$ annihilated by $J^m$ for some $m = m(\\mathcal{F})$\nfor every finite locally free $\\mathcal{O}_U$-module $\\mathcal{F}$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of pushforwards, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BM5","source_file":"local-cohomology.tex","source_line":2699,"source_end_line":2709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2699-L2709","statement_sha256":"63726b8aa68b74523028b54f36b7cfa983e591c6f6b4e98ad7cdcea8037262a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9117,"rank":9117,"depth":19,"x":2353.748,"y":1080.238,"cluster":"local-crystalline-methods"},{"id":"stacks:0EHY","tag":"0EHY","title":"Annihilators of local cohomology, II · Definition 0EHY","summary":"Let I be an ideal of a Noetherian ring A. Let K ∈ D^+_Coh(A). We define the I-depth of K, denoted depth_I(K), to be the maximal m ∈ Z ∪ (∞) such that H^i_I(K) = 0 for all i < m. If A is local with maximal ideal m then we call depth_ m(K) simply the depth of K.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let\n$K \\in D^+_{\\textit{Coh}}(A)$. We define the {\\it $I$-depth} of $K$,\ndenoted $\\text{depth}_I(K)$, to be the maximal\n$m \\in \\mathbf{Z} \\cup \\{\\infty\\}$ such that $H^i_I(K) = 0$ for all $i < m$.\nIf $A$ is local with maximal ideal $\\mathfrak m$\nthen we call $\\text{depth}_\\mathfrak m(K)$ simply the {\\it depth} of $K$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Annihilators of local cohomology, II","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHY","source_file":"local-cohomology.tex","source_line":2761,"source_end_line":2769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2761-L2769","statement_sha256":"9a9decca99fa0a26f9623364e899545361456f38207fdf987d5b330aadf35e6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9118,"rank":9118,"depth":0,"x":2621.44,"y":1112.374,"cluster":"local-crystalline-methods"},{"id":"stacks:0EHZ","tag":"0EHZ","title":"Annihilators of local cohomology, II · Proposition 0EHZ","summary":"Let A be a Noetherian ring which has a dualizing complex. Let T ⊂ T' ⊂ Spec(A) be subsets stable under specialization. Let s ∈ Z. Let K be an object of D_Coh^+(A). The following are equivalent • there exists an ideal J ⊂ A with V(J) ⊂ T' such that J annihilates H^i_T(K) for i ≤ s, and • for all p not ∈ T', q ∈ T with p ⊂ q we have depth_A_ p(K_ p) + dim((A/ p)_ q) > s","statement_latex":"Let $A$ be a Noetherian ring which has a dualizing complex.\nLet $T \\subset T' \\subset \\Spec(A)$ be subsets stable under\nspecialization. Let $s \\in \\mathbf{Z}$. Let $K$ be an object of\n$D_{\\textit{Coh}}^+(A)$. The following are equivalent\n\\begin{enumerate}\n\\item there exists an ideal $J \\subset A$ with $V(J) \\subset T'$\nsuch that $J$ annihilates $H^i_T(K)$ for $i \\leq s$, and\n\\item for all $\\mathfrak p \\not \\in T'$,\n$\\mathfrak q \\in T$ with $\\mathfrak p \\subset \\mathfrak q$\nwe have\n$$\n\\text{depth}_{A_\\mathfrak p}(K_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q) > s\n$$\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Annihilators of local cohomology, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHZ","source_file":"local-cohomology.tex","source_line":2776,"source_end_line":2793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2776-L2793","statement_sha256":"bbbb4f16b0ba9796c591b46ed1f3f70c65e6eaf10f7578b4339a6d5d90d48779","origin":"The Stacks Project","memory_eligible":false,"source_rank":9119,"rank":9119,"depth":35,"x":2397.756,"y":1240.829,"cluster":"local-crystalline-methods"},{"id":"stacks:0EI1","tag":"0EI1","title":"Finiteness of local cohomology, III · Lemma 0EI1","summary":"Let A be a Noetherian ring. Let T ⊂ Spec(A) be a subset stable under specialization. Let K be an object of D_Coh^+(A). Let n ∈ Z. The following are equivalent • H^i_T(K) is finite for i ≤ n, • there exists an ideal J ⊂ A with V(J) ⊂ T such that J annihilates H^i_T(K) for i ≤ n. If T = V(I) = Z for an ideal I ⊂ A, then these are also equivalent to • [(3)] there exists an e ≥ 0 such that I^e annihilates H^i_Z(K) for i ≤ n.","statement_latex":"Let $A$ be a Noetherian ring. Let $T \\subset \\Spec(A)$ be a subset stable\nunder specialization. Let $K$ be an object of $D_{\\textit{Coh}}^+(A)$.\nLet $n \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $H^i_T(K)$ is finite for $i \\leq n$,\n\\item there exists an ideal $J \\subset A$ with $V(J) \\subset T$\nsuch that $J$ annihilates $H^i_T(K)$ for $i \\leq n$.\n\\end{enumerate}\nIf $T = V(I) = Z$ for an ideal $I \\subset A$, then these are also\nequivalent to\n\\begin{enumerate}\n\\item[(3)] there exists an $e \\geq 0$ such that $I^e$ annihilates\n$H^i_Z(K)$ for $i \\leq n$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EI1","source_file":"local-cohomology.tex","source_line":2963,"source_end_line":2979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L2963-L2979","statement_sha256":"f486090038f049da63919d83bbd30ac6dc22fb1a4152321a202987c649bd4728","origin":"The Stacks Project","memory_eligible":false,"source_rank":9120,"rank":9120,"depth":10,"x":2459.521,"y":1018.742,"cluster":"local-crystalline-methods"},{"id":"stacks:0EI2","tag":"0EI2","title":"Finiteness of local cohomology, III · Proposition 0EI2","summary":"Let A be a Noetherian ring which has a dualizing complex. Let T ⊂ Spec(A) be a subset stable under specialization. Let s ∈ Z. Let K ∈ D_Coh^+(A). The following are equivalent • H^i_T(K) is a finite A-module for i ≤ s, and • for all p not ∈ T, q ∈ T with p ⊂ q we have depth_A_ p(K_ p) + dim((A/ p)_ q) > s","statement_latex":"Let $A$ be a Noetherian ring which has a dualizing complex.\nLet $T \\subset \\Spec(A)$ be a subset stable under specialization.\nLet $s \\in \\mathbf{Z}$. Let $K \\in D_{\\textit{Coh}}^+(A)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $H^i_T(K)$ is a finite $A$-module for $i \\leq s$, and\n\\item for all $\\mathfrak p \\not \\in T$, $\\mathfrak q \\in T$ with\n$\\mathfrak p \\subset \\mathfrak q$ we have\n$$\n\\text{depth}_{A_\\mathfrak p}(K_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q) > s\n$$\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Finiteness of local cohomology, III","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EI2","source_file":"local-cohomology.tex","source_line":3042,"source_end_line":3057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3042-L3057","statement_sha256":"9ec52c811f6e69598e253fa73f7ad6b29a815fe2310a73bf255513bb14a5c56d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9121,"rank":9121,"depth":36,"x":2592.838,"y":1217.948,"cluster":"local-crystalline-methods"},{"id":"stacks:0DX3","tag":"0DX3","title":"Improving coherent modules · Lemma 0DX3","summary":"Let X be a Noetherian scheme. Let T ⊂ X be a subset stable under specialization. Let F be a coherent O_X-module. Then there is a unique map F → F' of coherent O_X-modules such that • F → F' is surjective, • F_x → F'_x is an isomorphism for x not ∈ T, • depth_O_X, x(F'_x) ≥ 1 for x ∈ T. If f : Y → X is a flat morphism with Y Noetherian, then f^*F → f^*F' is the corresponding quotient for f^-1(T) ⊂ Y and f^*F.","statement_latex":"Let $X$ be a Noetherian scheme. Let $T \\subset X$ be a subset\nstable under specialization. Let $\\mathcal{F}$ be a coherent\n$\\mathcal{O}_X$-module. Then there is a unique map\n$\\mathcal{F} \\to \\mathcal{F}'$ of coherent $\\mathcal{O}_X$-modules\nsuch that\n\\begin{enumerate}\n\\item $\\mathcal{F} \\to \\mathcal{F}'$ is surjective,\n\\item $\\mathcal{F}_x \\to \\mathcal{F}'_x$ is an isomorphism for $x \\not \\in T$,\n\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}'_x) \\geq 1$ for $x \\in T$.\n\\end{enumerate}\nIf $f : Y \\to X$ is a flat morphism with $Y$ Noetherian, then\n$f^*\\mathcal{F} \\to f^*\\mathcal{F}'$ is the corresponding\nquotient for $f^{-1}(T) \\subset Y$ and $f^*\\mathcal{F}$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Improving coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DX3","source_file":"local-cohomology.tex","source_line":3077,"source_end_line":3092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3077-L3092","statement_sha256":"f50d7f5eca7e0b152ee923d321382ef686cd25b507fcbe5fbe95a8e208fd30cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9122,"rank":9122,"depth":14,"x":2333.822,"y":1146.562,"cluster":"local-crystalline-methods"},{"id":"stacks:0DX4","tag":"0DX4","title":"Improving coherent modules · Lemma 0DX4","summary":"Let j : U → X be an open immersion of Noetherian schemes. Let F be a coherent O_X-module. Assume F' = j_*(F|_U) is coherent. Then F → F' is the unique map of coherent O_X-modules such that • F|_U → F'|_U is an isomorphism, • depth_O_X, x(F'_x) ≥ 2 for x ∈ X, x not ∈ U. If f : Y → X is a flat morphism with Y Noetherian, then f^*F → f^*F' is the corresponding map for f^-1(U) ⊂ Y.","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nAssume $\\mathcal{F}' = j_*(\\mathcal{F}|_U)$ is coherent.\nThen $\\mathcal{F} \\to \\mathcal{F}'$ is the unique map\nof coherent $\\mathcal{O}_X$-modules such that\n\\begin{enumerate}\n\\item $\\mathcal{F}|_U \\to \\mathcal{F}'|_U$\nis an isomorphism,\n\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}'_x) \\geq 2$\nfor $x \\in X$, $x \\not \\in U$.\n\\end{enumerate}\nIf $f : Y \\to X$ is a flat morphism with $Y$ Noetherian, then\n$f^*\\mathcal{F} \\to f^*\\mathcal{F}'$ is the corresponding\nmap for $f^{-1}(U) \\subset Y$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Improving coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DX4","source_file":"local-cohomology.tex","source_line":3119,"source_end_line":3135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3119-L3135","statement_sha256":"c99a225266c322d9f195f42b113f6c16eba6bddb5c95167ba40a43cea0e3d473","origin":"The Stacks Project","memory_eligible":false,"source_rank":9123,"rank":9123,"depth":32,"x":2582.715,"y":1052.043,"cluster":"local-crystalline-methods"},{"id":"stacks:0DX5","tag":"0DX5","title":"Improving coherent modules · Lemma 0DX5","summary":"Let X be a Noetherian scheme. Let Z ⊂ X be a closed subscheme. Let F be a coherent O_X-module. Assume X is universally catenary and the formal fibres of local rings have (S_1). Then there exists a unique map F → F\" of coherent O_X-modules such that • F_x → F\"_x is an isomorphism for x ∈ X setminus Z, • F_x → F\"_x is surjective and depth_O_X, x(F\"_x) = 1 for x ∈ Z such that there exists an immediate specialization x' leadsto x with x' not ∈ Z and x' ∈ Ass(F), • depth_O_X,…","statement_latex":"Let $X$ be a Noetherian scheme. Let $Z \\subset X$ be a closed subscheme.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Assume\n$X$ is universally catenary and the formal fibres of local rings have $(S_1)$.\nThen there exists a unique map $\\mathcal{F} \\to \\mathcal{F}''$\nof coherent $\\mathcal{O}_X$-modules such that\n\\begin{enumerate}\n\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$\nis an isomorphism for $x \\in X \\setminus Z$,\n\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is surjective and\n$\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) = 1$\nfor $x \\in Z$ such that there exists an immediate specialization\n$x' \\leadsto x$ with $x' \\not \\in Z$ and $x' \\in \\text{Ass}(\\mathcal{F})$,\n\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 2$\nfor the remaining $x \\in Z$.\n\\end{enumerate}\nIf $f : Y \\to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian,\nthen $f^*\\mathcal{F} \\to f^*\\mathcal{F}''$ satisfies the same properties\nwith respect to $f^{-1}(Z) \\subset Y$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Improving coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DX5","source_file":"local-cohomology.tex","source_line":3147,"source_end_line":3167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3147-L3167","statement_sha256":"3598ce97458304d79f5b7756d1ef678ae5d99374d87ab53aadbfcc2926029b10","origin":"The Stacks Project","memory_eligible":false,"source_rank":9124,"rank":9124,"depth":51,"x":2474.982,"y":1263.382,"cluster":"local-crystalline-methods"},{"id":"stacks:0EI3","tag":"0EI3","title":"Improving coherent modules · Lemma 0EI3","summary":"Let X be a Noetherian scheme which locally has a dualizing complex. Let T' ⊂ X be a subset stable under specialization. Let F be a coherent O_X-module. Assume that if x leadsto x' is an immediate specialization of points in X with x' ∈ T' and x not ∈ T', then depth(F_x) ≥ 1. Then there exists a unique map F → F\" of coherent O_X-modules such that • F_x → F\"_x is an isomorphism for x not ∈ T', • depth_O_X, x(F\"_x) ≥ 2 for x ∈ T'. If f : Y → X is a Cohen-Macaulay morphism…","statement_latex":"Let $X$ be a Noetherian scheme which locally has a dualizing complex.\nLet $T' \\subset X$ be a subset stable under specialization.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nAssume that if $x \\leadsto x'$ is an immediate specialization\nof points in $X$ with $x' \\in T'$ and $x \\not \\in T'$, then\n$\\text{depth}(\\mathcal{F}_x) \\geq 1$.\nThen there exists a unique map $\\mathcal{F} \\to \\mathcal{F}''$\nof coherent $\\mathcal{O}_X$-modules such that\n\\begin{enumerate}\n\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is an isomorphism\nfor $x \\not \\in T'$,\n\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 2$\nfor $x \\in T'$.\n\\end{enumerate}\nIf $f : Y \\to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian,\nthen $f^*\\mathcal{F} \\to f^*\\mathcal{F}''$ satisfies the same properties\nwith respect to $f^{-1}(T') \\subset Y$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Improving coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EI3","source_file":"local-cohomology.tex","source_line":3230,"source_end_line":3249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3230-L3249","statement_sha256":"8f5171e034d61602e8cbc4120395876ef824c4fa21cef30cbbbd05ecafe7473a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9125,"rank":9125,"depth":36,"x":2384.292,"y":1045.989,"cluster":"local-crystalline-methods"},{"id":"stacks:0EI4","tag":"0EI4","title":"Improving coherent modules · Lemma 0EI4","summary":"Let X be a Noetherian scheme which locally has a dualizing complex. Let T' ⊂ T ⊂ X be subsets stable under specialization such that if x leadsto x' is an immediate specialization of points in X and x' ∈ T', then x ∈ T. Let F be a coherent O_X-module. Then there exists a unique map F → F\" of coherent O_X-modules such that • F_x → F\"_x is an isomorphism for x not ∈ T, • F_x → F\"_x is surjective and depth_O_X, x(F\"_x) ≥ 1 for x ∈ T, x not ∈ T', and • depth_O_X, x(F\"_x) ≥ 2…","statement_latex":"Let $X$ be a Noetherian scheme which locally has a dualizing complex.\nLet $T' \\subset T \\subset X$ be subsets stable under specialization\nsuch that if $x \\leadsto x'$ is an immediate specialization\nof points in $X$ and $x' \\in T'$, then $x \\in T$. Let $\\mathcal{F}$\nbe a coherent $\\mathcal{O}_X$-module.\nThen there exists a unique map $\\mathcal{F} \\to \\mathcal{F}''$\nof coherent $\\mathcal{O}_X$-modules such that\n\\begin{enumerate}\n\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is an isomorphism\nfor $x \\not \\in T$,\n\\item $\\mathcal{F}_x \\to \\mathcal{F}''_x$ is surjective\nand $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 1$\nfor $x \\in T$, $x \\not \\in T'$, and\n\\item $\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}''_x) \\geq 2$\nfor $x \\in T'$.\n\\end{enumerate}\nIf $f : Y \\to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian,\nthen $f^*\\mathcal{F} \\to f^*\\mathcal{F}''$ satisfies the same properties\nwith respect to $f^{-1}(T') \\subset f^{-1}(T) \\subset Y$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Improving coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EI4","source_file":"local-cohomology.tex","source_line":3308,"source_end_line":3329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3308-L3329","statement_sha256":"537bdcaf11f26414995b7f8ad2129dea2c2530e861cd596cb115bdedeba5e027","origin":"The Stacks Project","memory_eligible":false,"source_rank":9126,"rank":9126,"depth":37,"x":2626.459,"y":1155.045,"cluster":"local-crystalline-methods"},{"id":"stacks:0EB1","tag":"0EB1","title":"Hartshorne-Lichtenbaum vanishing · Lemma 0EB1","summary":"Let A be a Noetherian ring of dimension d. Let I ⊂ I' ⊂ A be ideals. If I' is contained in the Jacobson radical of A and cd(A, I') < d, then cd(A, I) < d.","statement_latex":"Let $A$ be a Noetherian ring of dimension $d$. Let $I \\subset I' \\subset A$\nbe ideals. If $I'$ is contained in the Jacobson radical\nof $A$ and $\\text{cd}(A, I') < d$, then $\\text{cd}(A, I) < d$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Hartshorne-Lichtenbaum vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EB1","source_file":"local-cohomology.tex","source_line":3358,"source_end_line":3363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3358-L3363","statement_sha256":"4b81e2436db0bef3c37d1d2e54cf1b44ac040dab77b0fac59899f66227720859","origin":"The Stacks Project","memory_eligible":false,"source_rank":9127,"rank":9127,"depth":31,"x":2359.671,"y":1212.151,"cluster":"local-crystalline-methods"},{"id":"stacks:0EB2","tag":"0EB2","title":"Hartshorne-Lichtenbaum vanishing · Lemma 0EB2","summary":"Let A be a Noetherian ring of dimension d. Let I ⊂ A be an ideal. If H^d_V(I)(M) = 0 for some finite A-module whose support contains all the irreducible components of dimension d, then cd(A, I) < d.","statement_latex":"Let $A$ be a Noetherian ring of dimension $d$. Let $I \\subset A$\nbe an ideal. If $H^d_{V(I)}(M) = 0$ for some finite $A$-module\nwhose support contains all the irreducible components of\ndimension $d$, then $\\text{cd}(A, I) < d$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Hartshorne-Lichtenbaum vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EB2","source_file":"local-cohomology.tex","source_line":3379,"source_end_line":3385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3379-L3385","statement_sha256":"375ea60aa03ff53c7d87c37d4e35900fb1b03d36a35586f7641319e7c233c827","origin":"The Stacks Project","memory_eligible":false,"source_rank":9128,"rank":9128,"depth":31,"x":2510.766,"y":1018.285,"cluster":"local-crystalline-methods"},{"id":"stacks:0EB3","tag":"0EB3","title":"Hartshorne-Lichtenbaum vanishing · Lemma 0EB3","summary":"Let A be a Noetherian local ring of dimension d. Let f ∈ A be an element which is not contained in any minimal prime of dimension d. Then f : H^d_V(I)(M) → H^d_V(I)(M) is surjective for any finite A-module M and any ideal I ⊂ A.","statement_latex":"Let $A$ be a Noetherian local ring of dimension $d$. Let $f \\in A$\nbe an element which is not contained in any minimal prime of\ndimension $d$. Then $f : H^d_{V(I)}(M) \\to H^d_{V(I)}(M)$\nis surjective for any finite $A$-module $M$ and any ideal $I \\subset A$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Hartshorne-Lichtenbaum vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EB3","source_file":"local-cohomology.tex","source_line":3414,"source_end_line":3420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3414-L3420","statement_sha256":"9d8022a547a4255e5ea4cfcdbd37b8f12a6c142ea8f049c473655ca5be5a5407","origin":"The Stacks Project","memory_eligible":false,"source_rank":9129,"rank":9129,"depth":31,"x":2555.338,"y":1247.415,"cluster":"local-crystalline-methods"},{"id":"stacks:0EB4","tag":"0EB4","title":"Hartshorne-Lichtenbaum vanishing · Lemma 0EB4","summary":"Let A be a Noetherian local ring with normalized dualizing complex ω_A^bullet. Let I ⊂ A be an ideal. If H^0_V(I)(ω_A^bullet) = 0, then cd(A, I) < dim(A).","statement_latex":"Let $A$ be a Noetherian local ring with\nnormalized dualizing complex $\\omega_A^\\bullet$.\nLet $I \\subset A$ be an ideal.\nIf $H^0_{V(I)}(\\omega_A^\\bullet) = 0$, then $\\text{cd}(A, I) < \\dim(A)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Hartshorne-Lichtenbaum vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EB4","source_file":"local-cohomology.tex","source_line":3431,"source_end_line":3437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3431-L3437","statement_sha256":"439d79318968a1ac2709556a45629f37fbb7277f69d6fcd94efde64d061a4250","origin":"The Stacks Project","memory_eligible":false,"source_rank":9130,"rank":9130,"depth":39,"x":2337.795,"y":1103.473,"cluster":"local-crystalline-methods"},{"id":"stacks:0EB5","tag":"0EB5","title":"Hartshorne-Lichtenbaum vanishing · Lemma 0EB5","summary":"Let (A, m) be a complete Noetherian local domain. Let p ⊂ A be a prime ideal of dimension 1. For every n ≥ 1 there is an m ≥ n such that p^(m) ⊂ p^n.","statement_latex":"Let $(A, \\mathfrak m)$ be a complete Noetherian local domain. Let\n$\\mathfrak p \\subset A$ be a prime ideal of dimension $1$.\nFor every $n \\geq 1$ there is an $m \\geq n$ such that\n$\\mathfrak p^{(m)} \\subset \\mathfrak p^n$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Hartshorne-Lichtenbaum vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EB5","source_file":"local-cohomology.tex","source_line":3476,"source_end_line":3482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3476-L3482","statement_sha256":"3ce2072b5f586ee91ef636bcabc5140fd0c4018432491ff2c3d5227ecd9ebbd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9131,"rank":9131,"depth":0,"x":2614.491,"y":1086.141,"cluster":"local-crystalline-methods"},{"id":"stacks:0EB6","tag":"0EB6","title":"Hartshorne-Lichtenbaum vanishing · Proposition 0EB6","summary":"[CD] Let A be a Noetherian local ring with completion A^wedge. Let I ⊂ A be an ideal such that dim V(IA^wedge + p) ≥ 1 for every minimal prime p ⊂ A^wedge of dimension dim(A). Then cd(A, I) < dim(A).","statement_latex":"\\begin{reference}\n\\cite[Theorem 3.1]{CD}\n\\end{reference}\nLet $A$ be a Noetherian local ring with completion $A^\\wedge$.\nLet $I \\subset A$ be an ideal such that\n$$\n\\dim V(IA^\\wedge + \\mathfrak p) \\geq 1\n$$\nfor every minimal prime $\\mathfrak p \\subset A^\\wedge$ of dimension $\\dim(A)$.\nThen $\\text{cd}(A, I) < \\dim(A)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Hartshorne-Lichtenbaum vanishing","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EB6","source_file":"local-cohomology.tex","source_line":3504,"source_end_line":3516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3504-L3516","statement_sha256":"0c62275f46ba61e0a2dff6d7a13afef26c403b4ff6d9297cbd49dc9b4ef92d66","origin":"The Stacks Project","memory_eligible":false,"source_rank":9132,"rank":9132,"depth":40,"x":2424.034,"y":1256.249,"cluster":"local-crystalline-methods"},{"id":"stacks:0EB7","tag":"0EB7","title":"Hartshorne-Lichtenbaum vanishing · Lemma 0EB7","summary":"Let (A, m) be a Noetherian local ring. Let I ⊂ A be an ideal. Assume A is excellent, normal, and dim V(I) ≥ 1. Then cd(A, I) < dim(A). In particular, if dim(A) = 2, then Spec(A) setminus V(I) is affine.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $I \\subset A$ be an ideal. Assume $A$ is excellent,\nnormal, and $\\dim V(I) \\geq 1$. Then $\\text{cd}(A, I) < \\dim(A)$.\nIn particular, if $\\dim(A) = 2$, then $\\Spec(A) \\setminus V(I)$ is affine.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Hartshorne-Lichtenbaum vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EB7","source_file":"local-cohomology.tex","source_line":3558,"source_end_line":3564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3558-L3564","statement_sha256":"7b5632eab2173a464588250f1073371e8f41b63b98b316e34e32e2189c6f6f53","origin":"The Stacks Project","memory_eligible":false,"source_rank":9133,"rank":9133,"depth":41,"x":2427.686,"y":1022.301,"cluster":"local-crystalline-methods"},{"id":"stacks:0EBV","tag":"0EBV","title":"Frobenius action · Lemma 0EBV","summary":"Let p be a prime number. Let (A, m, kappa) be a Noetherian local ring with p = 0 in A. Let M be a finite A-module such that M ⊗_A, F A ≅ M. Then M is finite free.","statement_latex":"Let $p$ be a prime number. Let $(A, \\mathfrak m, \\kappa)$\nbe a Noetherian local ring\nwith $p = 0$ in $A$. Let $M$ be a finite $A$-module\nsuch that $M \\otimes_{A, F} A \\cong M$. Then $M$ is finite free.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Frobenius action","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBV","source_file":"local-cohomology.tex","source_line":3605,"source_end_line":3611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3605-L3611","statement_sha256":"d501d0ebfddadce467ea72835696fc61c6c338593865312adce5f7cb81055fcf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9134,"rank":9134,"depth":6,"x":2613.478,"y":1197.213,"cluster":"local-crystalline-methods"},{"id":"stacks:0EBW","tag":"0EBW","title":"Frobenius action · Lemma 0EBW","summary":"See [Lech-inequalities] and [MatCA]. Let A be a ring. If f_1, …, f_r - 1, f_rg_r are independent, then f_1, …, f_r are independent.","statement_latex":"\\begin{reference}\nSee \\cite{Lech-inequalities} and \\cite[Lemma 1 page 299]{MatCA}.\n\\end{reference}\nLet $A$ be a ring. If $f_1, \\ldots, f_{r - 1}, f_rg_r$\nare independent, then $f_1, \\ldots, f_r$ are independent.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Frobenius action","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBW","source_file":"local-cohomology.tex","source_line":3641,"source_end_line":3648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3641-L3648","statement_sha256":"ddc172078b253d5c770c91f6b50f91f314bf464aebbc263fa818c27d33854242","origin":"The Stacks Project","memory_eligible":false,"source_rank":9135,"rank":9135,"depth":0,"x":2335.294,"y":1173.614,"cluster":"local-crystalline-methods"},{"id":"stacks:0EBX","tag":"0EBX","title":"Frobenius action · Lemma 0EBX","summary":"See [Lech-inequalities] and [MatCA]. Let A be a ring. If f_1, …, f_r - 1, f_rg_r are independent and if the A-module A/(f_1, …, f_r - 1, f_rg_r) has finite length, then & length_A(A/(f_1, …, f_r - 1, f_rg_r)) & = length_A(A/(f_1, …, f_r - 1, f_r)) + length_A(A/(f_1, …, f_r - 1, g_r))","statement_latex":"\\begin{reference}\nSee \\cite{Lech-inequalities} and \\cite[Lemma 2 page 300]{MatCA}.\n\\end{reference}\nLet $A$ be a ring. If $f_1, \\ldots, f_{r - 1}, f_rg_r$\nare independent and if the $A$-module\n$A/(f_1, \\ldots, f_{r - 1}, f_rg_r)$ has finite length, then\n\\begin{align*}\n& \\text{length}_A(A/(f_1, \\ldots, f_{r - 1}, f_rg_r)) \\\\\n& =\n\\text{length}_A(A/(f_1, \\ldots, f_{r - 1}, f_r)) +\n\\text{length}_A(A/(f_1, \\ldots, f_{r - 1}, g_r))\n\\end{align*}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Frobenius action","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBX","source_file":"local-cohomology.tex","source_line":3659,"source_end_line":3673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3659-L3673","statement_sha256":"cca28c5acd79422eace4920529236672e254eeac2a31906cfe73499e5d9d51b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9136,"rank":9136,"depth":1,"x":2559.822,"y":1032.904,"cluster":"local-crystalline-methods"},{"id":"stacks:0EBY","tag":"0EBY","title":"Frobenius action · Lemma 0EBY","summary":"See [Lech-inequalities] and [MatCA]. Let (A, m) be a local ring. If m = (x_1, …, x_r) and x_1^e_1, …, x_r^e_r are independent for some e_i > 0, then length_A(A/(x_1^e_1, …, x_r^e_r)) = e_1… e_r.","statement_latex":"\\begin{reference}\nSee \\cite{Lech-inequalities} and \\cite[Lemma 3 page 300]{MatCA}.\n\\end{reference}\nLet $(A, \\mathfrak m)$ be a local ring. If $\\mathfrak m = (x_1, \\ldots, x_r)$\nand $x_1^{e_1}, \\ldots, x_r^{e_r}$ are independent for some $e_i > 0$,\nthen $\\text{length}_A(A/(x_1^{e_1}, \\ldots, x_r^{e_r})) = e_1\\ldots e_r$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Frobenius action","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBY","source_file":"local-cohomology.tex","source_line":3694,"source_end_line":3702,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3694-L3702","statement_sha256":"cfb2654e165ecf1b35dea4aa24cf1d24057270259e1839e74e5c968444093bb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9137,"rank":9137,"depth":2,"x":2507.315,"y":1264.496,"cluster":"local-crystalline-methods"},{"id":"stacks:0EBZ","tag":"0EBZ","title":"Frobenius action · Lemma 0EBZ","summary":"Let φ : A → B be a flat ring map. If f_1, …, f_r ∈ A are independent, then φ(f_1), …, φ(f_r) ∈ B are independent.","statement_latex":"Let $\\varphi : A \\to B$ be a flat ring map.\nIf $f_1, \\ldots, f_r \\in A$ are independent, then\n$\\varphi(f_1), \\ldots, \\varphi(f_r) \\in B$ are independent.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Frobenius action","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EBZ","source_file":"local-cohomology.tex","source_line":3708,"source_end_line":3713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3708-L3713","statement_sha256":"3ed7a80bdb049b9b24666612ba40438fa583edc6b6584c480fea4e940b34c793","origin":"The Stacks Project","memory_eligible":false,"source_rank":9138,"rank":9138,"depth":0,"x":2359.518,"y":1063.556,"cluster":"local-crystalline-methods"},{"id":"stacks:0EC0","tag":"0EC0","title":"Kunz · Lemma 0EC0","summary":"[Kunz-flat] Let p be a prime number. Let A be a Noetherian ring with p = 0. The following are equivalent • A is regular, and • F : A → A, a ↦ a^p is flat.","statement_latex":"\\begin{reference}\n\\cite{Kunz-flat}\n\\end{reference}\nLet $p$ be a prime number.\nLet $A$ be a Noetherian ring with $p = 0$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A$ is regular, and\n\\item $F : A \\to A$, $a \\mapsto a^p$ is flat.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Frobenius action","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EC0","source_file":"local-cohomology.tex","source_line":3721,"source_end_line":3733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3721-L3733","statement_sha256":"cf543da5a8806ba7fa3c8f2388749bf8c1b6837d35e1ad6a1554665455e71562","origin":"The Stacks Project","memory_eligible":false,"source_rank":9139,"rank":9139,"depth":19,"x":2630.594,"y":1127.982,"cluster":"local-crystalline-methods"},{"id":"stacks:0EC2","tag":"0EC2","title":"Structure of certain modules · Lemma 0EC2","summary":"Special case of [Lyubeznik] Let k be a field of characteristic 0. Let d ≥ 1. Let A = k[[x_1, …, x_d]] with maximal ideal m. Let M be an m-power torsion A-module endowed with additive operators D_1, …, D_d satisfying the leibniz rule D_i(fz) = ∂_i(f) z + f D_i(z) for f ∈ A and z ∈ M. Here ∂_i is differentiation with respect to x_i. Then M is isomorphic to a direct sum of copies of the injective hull E of k.","statement_latex":"\\begin{reference}\nSpecial case of \\cite[Theorem 2.4]{Lyubeznik}\n\\end{reference}\nLet $k$ be a field of characteristic $0$. Let $d \\geq 1$.\nLet $A = k[[x_1, \\ldots, x_d]]$ with maximal ideal $\\mathfrak m$.\nLet $M$ be an $\\mathfrak m$-power torsion $A$-module endowed with\nadditive operators $D_1, \\ldots, D_d$ satisfying the leibniz rule\n$$\nD_i(fz) = \\partial_i(f) z + f D_i(z)\n$$\nfor $f \\in A$ and $z \\in M$. Here $\\partial_i$ is\ndifferentiation with respect to $x_i$.\nThen $M$ is isomorphic to a direct sum\nof copies of the injective hull $E$ of $k$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Structure of certain modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EC2","source_file":"local-cohomology.tex","source_line":3790,"source_end_line":3806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3790-L3806","statement_sha256":"876a162cd6dc5475d4ff23c4ea1cf0ace4f2ee6593ec4235b3b20a9f265d4db5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9140,"rank":9140,"depth":4,"x":2378.432,"y":1234.486,"cluster":"local-crystalline-methods"},{"id":"stacks:0EC3","tag":"0EC3","title":"Structure of certain modules · Lemma 0EC3","summary":"Follows from [Huneke-Sharp] with a little bit of work. Also follows directly from [Lyubeznik2]. Let p be a prime number. Let (A, m, k) be a regular local ring with p = 0. Denote F : A → A, a ↦ a^p be the Frobenius endomorphism. Let M be a m-power torsion module such that M ⊗_A, F A ≅ M. Then M is isomorphic to a direct sum of copies of the injective hull E of k.","statement_latex":"\\begin{reference}\nFollows from \\cite[Corollary 3.6]{Huneke-Sharp} with a\nlittle bit of work. Also follows directly from\n\\cite[Theorem 1.4]{Lyubeznik2}.\n\\end{reference}\nLet $p$ be a prime number. Let $(A, \\mathfrak m, k)$\nbe a regular local ring with $p = 0$. Denote $F : A \\to A$, $a \\mapsto a^p$\nbe the Frobenius endomorphism. Let $M$ be a $\\mathfrak m$-power torsion module\nsuch that $M \\otimes_{A, F} A \\cong M$. Then $M$ is isomorphic to a direct sum\nof copies of the injective hull $E$ of $k$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Structure of certain modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EC3","source_file":"local-cohomology.tex","source_line":3857,"source_end_line":3869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3857-L3869","statement_sha256":"f9b2c6e431c9bd4f78baf06a053ccbad066d196f9e6dd4b0a2d981983fb90648","origin":"The Stacks Project","memory_eligible":false,"source_rank":9141,"rank":9141,"depth":20,"x":2478.91,"y":1012.461,"cluster":"local-crystalline-methods"},{"id":"stacks:0EC5","tag":"0EC5","title":"Additional structure on local cohomology · Lemma 0EC5","summary":"Let A be a ring. Let I ⊂ A be a finitely generated ideal. Set Z = V(I). For each derivation theta : A → A there exists a canonical additive operator D on the local cohomology modules H^i_Z(A) satisfying the Leibniz rule with respect to theta.","statement_latex":"Let $A$ be a ring. Let $I \\subset A$ be a finitely generated ideal.\nSet $Z = V(I)$.\nFor each derivation $\\theta : A \\to A$ there exists a canonical\nadditive operator $D$ on the local cohomology modules\n$H^i_Z(A)$ satisfying the Leibniz rule with respect to $\\theta$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Additional structure on local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EC5","source_file":"local-cohomology.tex","source_line":3963,"source_end_line":3970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3963-L3970","statement_sha256":"08d0e64f6e0ac8bd21624afffa1f439ad26f95ded3fdb6c2049c4bc6795ccdd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9142,"rank":9142,"depth":3,"x":2583.557,"y":1233.598,"cluster":"local-crystalline-methods"},{"id":"stacks:0EC6","tag":"0EC6","title":"Additional structure on local cohomology · Lemma 0EC6","summary":"Let p be a prime number. Let A be a ring with p = 0. Denote F : A → A, a ↦ a^p the Frobenius endomorphism. Let I ⊂ A be a finitely generated ideal. Set Z = V(I). There exists an isomorphism RΓ_Z(A) ⊗_A, F^L A ≅ RΓ_Z(A).","statement_latex":"Let $p$ be a prime number. Let $A$ be a ring with $p = 0$.\nDenote $F : A \\to A$, $a \\mapsto a^p$ the Frobenius endomorphism.\nLet $I \\subset A$ be a finitely generated ideal. Set $Z = V(I)$.\nThere exists an isomorphism\n$R\\Gamma_Z(A) \\otimes_{A, F}^\\mathbf{L} A \\cong R\\Gamma_Z(A)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Additional structure on local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EC6","source_file":"local-cohomology.tex","source_line":3986,"source_end_line":3993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L3986-L3993","statement_sha256":"d5fdc6e703f2f808933792b57c385eed6270f2abf6edf490bbe8e0f27a22c137","origin":"The Stacks Project","memory_eligible":false,"source_rank":9143,"rank":9143,"depth":4,"x":2328.093,"y":1129.724,"cluster":"local-crystalline-methods"},{"id":"stacks:0EC7","tag":"0EC7","title":"Additional structure on local cohomology · Lemma 0EC7","summary":"Let A be a ring. Let V → Spec(A) be quasi-compact, quasi-separated, and étale. For each derivation theta : A → A there exists a canonical additive operator D on H^i(V, O_V) satisfying the Leibniz rule with respect to theta.","statement_latex":"Let $A$ be a ring. Let $V \\to \\Spec(A)$ be quasi-compact, quasi-separated,\nand \\'etale. For each derivation $\\theta : A \\to A$ there exists a canonical\nadditive operator $D$ on $H^i(V, \\mathcal{O}_V)$\nsatisfying the Leibniz rule with respect to $\\theta$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Additional structure on local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EC7","source_file":"local-cohomology.tex","source_line":4002,"source_end_line":4008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4002-L4008","statement_sha256":"0640ae8358a405189f94052f14ae63c355ad86d67629cb84f9dbabf5193cae11","origin":"The Stacks Project","memory_eligible":false,"source_rank":9144,"rank":9144,"depth":28,"x":2600.496,"y":1061.238,"cluster":"local-crystalline-methods"},{"id":"stacks:0EC9","tag":"0EC9","title":"Additional structure on local cohomology · Lemma 0EC9","summary":"Let p be a prime number. Let A be a ring with p = 0. Denote F : A → A, a ↦ a^p the Frobenius endomorphism. If V → Spec(A) is quasi-compact, quasi-separated, and étale, then there exists an isomorphism RΓ(V, O_V) ⊗_A, F^L A ≅ RΓ(V, O_V).","statement_latex":"Let $p$ be a prime number. Let $A$ be a ring with $p = 0$.\nDenote $F : A \\to A$, $a \\mapsto a^p$ the Frobenius endomorphism.\nIf $V \\to \\Spec(A)$ is quasi-compact, quasi-separated,\nand \\'etale, then there exists an isomorphism\n$R\\Gamma(V, \\mathcal{O}_V) \\otimes_{A, F}^\\mathbf{L} A \\cong\nR\\Gamma(V, \\mathcal{O}_V)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"Additional structure on local cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EC9","source_file":"local-cohomology.tex","source_line":4047,"source_end_line":4055,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4047-L4055","statement_sha256":"e084a7402b2e27af9ee00776a2971630d5c1a70da58c00315681ae329a69e3ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":9145,"rank":9145,"depth":45,"x":2454.44,"y":1266.679,"cluster":"local-crystalline-methods"},{"id":"stacks:0G9T","tag":"0G9T","title":"A bit of uniformity, I · Lemma 0G9T","summary":"Let R be a ring. Let M → M' be a map of R-modules with M of finite presentation such that Tor_1^R(M, N) → Tor_1^R(M', N) is zero for all R-modules N. Then M → M' factors through a free R-module.","statement_latex":"Let $R$ be a ring. Let $M \\to M'$ be a map of $R$-modules with\n$M$ of finite presentation\nsuch that $\\text{Tor}_1^R(M, N) \\to \\text{Tor}_1^R(M', N)$ is zero for all\n$R$-modules $N$. Then $M \\to M'$ factors through a free $R$-module.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9T","source_file":"local-cohomology.tex","source_line":4081,"source_end_line":4087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4081-L4087","statement_sha256":"8c6fe13502e096d69c4eb2f0a63b5ad2a5a873c28e9b004c4ae4673b22b9c660","origin":"The Stacks Project","memory_eligible":false,"source_rank":9146,"rank":9146,"depth":3,"x":2396.826,"y":1031.891,"cluster":"local-crystalline-methods"},{"id":"stacks:0G9V","tag":"0G9V","title":"A bit of uniformity, I · Lemma 0G9V","summary":"Let R be a ring. Let α : M → M' be a map of R-modules. Let P_bullet → M and P'_bullet → M' be resolutions by projective R-modules. Let e ≥ 0 be an integer. Consider the following conditions • We can find a map of complexes a_bullet : P_bullet → P'_bullet inducing α on cohomology with a_i = 0 for i > e. • We can find a map of complexes a_bullet : P_bullet → P'_bullet inducing α on cohomology with a_e + 1 = 0. • The map Ext^i_R(M', N) → Ext^i_R(M, N) is zero for all…","statement_latex":"Let $R$ be a ring. Let $\\alpha : M \\to M'$ be a map of $R$-modules.\nLet $P_\\bullet \\to M$ and $P'_\\bullet \\to M'$ be resolutions by\nprojective $R$-modules. Let $e \\geq 0$ be an integer.\nConsider the following conditions\n\\begin{enumerate}\n\\item We can find a map of complexes $a_\\bullet : P_\\bullet \\to P'_\\bullet$\ninducing $\\alpha$ on cohomology with $a_i = 0$ for $i > e$.\n\\item We can find a map of complexes $a_\\bullet : P_\\bullet \\to P'_\\bullet$\ninducing $\\alpha$ on cohomology with $a_{e + 1} = 0$.\n\\item The map $\\Ext^i_R(M', N) \\to \\Ext^i_R(M, N)$ is zero\nfor all $R$-modules $N$ and $i > e$.\n\\item The map $\\Ext^{e + 1}_R(M', N) \\to \\Ext^{e + 1}_R(M, N)$ is zero\nfor all $R$-modules $N$.\n\\item Let $N = \\Im(P'_{e + 1} \\to P'_e)$ and denote\n$\\xi \\in \\Ext^{e + 1}_R(M', N)$ the canonical element (see proof).\nThen $\\xi$ maps to zero in $\\Ext^{e + 1}_R(M, N)$.\n\\item The map $\\text{Tor}_i^R(M, N) \\to \\text{Tor}_i^R(M', N)$\nis zero for all $R$-modules $N$ and $i > e$.\n\\item The map $\\text{Tor}_{e + 1}^R(M, N) \\to \\text{Tor}_{e + 1}^R(M', N)$\nis zero for all $R$-modules $N$.\n\\end{enumerate}\nThen we always have the implications\n$$\n(1) \\Leftrightarrow (2) \\Leftrightarrow (3) \\Leftrightarrow (4)\n\\Leftrightarrow (5) \\Rightarrow (6) \\Leftrightarrow (7)\n$$\nIf $M$ is $(-e - 1)$-pseudo-coherent (for example if $R$ is Noetherian\nand $M$ is a finite $R$-module), then all conditions\nare equivalent.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9V","source_file":"local-cohomology.tex","source_line":4122,"source_end_line":4153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4122-L4153","statement_sha256":"e136e7cce48cfcc63c9f3cfb1ffc4ae7d1631d8c73d0e025706f70daefb5b08e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9147,"rank":9147,"depth":13,"x":2628.536,"y":1172.582,"cluster":"local-crystalline-methods"},{"id":"stacks:0EH1","tag":"0EH1","title":"A bit of uniformity, I · Lemma 0EH1","summary":"Let I be an ideal of a Noetherian ring A. For all n ≥ 1 there exists an m > n such that the map A/I^m → A/I^n satisfies the equivalent conditions of Lemma [Tag 0G9V] with e = cd(A, I).","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$.\nFor all $n \\geq 1$ there exists an $m > n$ such that the map\n$A/I^m \\to A/I^n$ satisfies the equivalent conditions of\nLemma \\ref{lemma-characterize-vanishing-tor-ext-above-e} with\n$e = \\text{cd}(A, I)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EH1","source_file":"local-cohomology.tex","source_line":4243,"source_end_line":4250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4243-L4250","statement_sha256":"02edb729bb48d7cd83c8bbf66ab3454a08654228b8730cea9f31bec18d12605c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9148,"rank":9148,"depth":14,"x":2344.027,"y":1200.366,"cluster":"local-crystalline-methods"},{"id":"stacks:0G9X","tag":"0G9X","title":"A bit of uniformity, II · Lemma 0G9X","summary":"Let I be an ideal of a Noetherian ring A. For every m ≥ 0 and i > 0 there exist a c = c(A, I, m, i) ≥ 0 such that for every A-module M annihilated by I^m the map Tor^A_i(M, A/I^n) → Tor^A_i(M, A/I^n - c) is zero for all n ≥ c.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. For every $m \\geq 0$\nand $i > 0$ there exist a $c = c(A, I, m, i) \\geq 0$ such that\nfor every $A$-module $M$ annihilated by $I^m$ the map\n$$\n\\text{Tor}^A_i(M, A/I^n) \\to \\text{Tor}^A_i(M, A/I^{n - c})\n$$\nis zero for all $n \\geq c$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9X","source_file":"local-cohomology.tex","source_line":4293,"source_end_line":4302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4293-L4302","statement_sha256":"0248e544c3b526f4bc6c88bd70bff3fa56cfe8249667a7d75814a8daeffd5198","origin":"The Stacks Project","memory_eligible":false,"source_rank":9149,"rank":9149,"depth":3,"x":2531.812,"y":1018.116,"cluster":"local-crystalline-methods"},{"id":"stacks:0G9Y","tag":"0G9Y","title":"A bit of uniformity, II · Lemma 0G9Y","summary":"Let I = (a_1, …, a_t) be an ideal of a Noetherian ring A. Set a = a_1 and denote B = A[fracIa] the affine blowup algebra. There exists a c > 0 such that Tor_i^A(B, M) is annihilated by I^c for all A-modules M and i ≥ t.","statement_latex":"Let $I = (a_1, \\ldots, a_t)$ be an ideal of a Noetherian ring $A$.\nSet $a = a_1$ and denote $B = A[\\frac{I}{a}]$ the affine blowup algebra.\nThere exists a $c > 0$ such that $\\text{Tor}_i^A(B, M)$ is annihilated\nby $I^c$ for all $A$-modules $M$ and $i \\geq t$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9Y","source_file":"local-cohomology.tex","source_line":4337,"source_end_line":4343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4337-L4343","statement_sha256":"0f4232123b19cade7d6c0374aed410bf6cc3a9c94dcf6aa5c43db9311f3e9790","origin":"The Stacks Project","memory_eligible":false,"source_rank":9150,"rank":9150,"depth":3,"x":2539.917,"y":1259.487,"cluster":"local-crystalline-methods"},{"id":"stacks:0G9Z","tag":"0G9Z","title":"A bit of uniformity, II · Lemma 0G9Z","summary":"In the situation above, for q ≥ q(A, I) and any A-module M we have RΓ(X, Lp^*widetildeM(q)) ≅ M ⊗_A^L I^q in D(A).","statement_latex":"In the situation above, for $q \\geq q(A, I)$ and any $A$-module $M$ we have\n$$\nR\\Gamma(X, Lp^*\\widetilde{M}(q)) \\cong M \\otimes_A^\\mathbf{L} I^q\n$$\nin $D(A)$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G9Z","source_file":"local-cohomology.tex","source_line":4401,"source_end_line":4408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4401-L4408","statement_sha256":"2e14bc1ff6af751bed8ccaabce99037ad5126bc1878d88e68af45fd86b8bbf9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9151,"rank":9151,"depth":32,"x":2339.481,"y":1085.794,"cluster":"local-crystalline-methods"},{"id":"stacks:0GA0","tag":"0GA0","title":"A bit of uniformity, II · Lemma 0GA0","summary":"In the situation above, let t be an upper bound on the number of generators for I. There exists an integer c = c(A, I) ≥ 0 such that for any A-module M the cohomology sheaves H^j(Lp^*widetildeM) are annihilated by I^c for j ≤ -t.","statement_latex":"In the situation above, let $t$ be an upper bound on the number of\ngenerators for $I$. There exists an integer $c = c(A, I) \\geq 0$\nsuch that for any $A$-module $M$ the cohomology sheaves\n$H^j(Lp^*\\widetilde{M})$ are annihilated by $I^c$ for $j \\leq -t$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GA0","source_file":"local-cohomology.tex","source_line":4425,"source_end_line":4431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4425-L4431","statement_sha256":"04ec7dbd8d068f884fa35dba58ef3041f510ccbbd86fc9524b6e129179c0fd18","origin":"The Stacks Project","memory_eligible":false,"source_rank":9152,"rank":9152,"depth":28,"x":2627.467,"y":1100.167,"cluster":"local-crystalline-methods"},{"id":"stacks:0GA1","tag":"0GA1","title":"A bit of uniformity, II · Lemma 0GA1","summary":"In the situation above, let t be an upper bound on the number of generators for I. There exists an integer c = c(A, I) ≥ 0 such that for any A-module M the tor modules Tor_i^A(M, A/I^q) are annihilated by I^c for i > t and all q ≥ 0.","statement_latex":"In the situation above, let $t$ be an upper bound on the number of\ngenerators for $I$. There exists an integer $c = c(A, I) \\geq 0$\nsuch that for any $A$-module $M$ the tor modules\n$\\text{Tor}_i^A(M, A/I^q)$ are annihilated by $I^c$ for $i > t$\nand all $q \\geq 0$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GA1","source_file":"local-cohomology.tex","source_line":4451,"source_end_line":4458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4451-L4458","statement_sha256":"bbc3712ce34f6a1932ec181c1a5c824cc38d8b464eca6388f464e334af13091f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9153,"rank":9153,"depth":33,"x":2403.158,"y":1253.248,"cluster":"local-crystalline-methods"},{"id":"stacks:0GA2","tag":"0GA2","title":"A bit of uniformity, II · Lemma 0GA2","summary":"Let I be an ideal of a Noetherian ring A. Let t ≥ 0 be an upper bound on the number of generators of I. There exist N, c ≥ 0 such that the maps Tor_t + 1^A(M, A/I^n) → Tor_t + 1^A(M, A/I^n - c) are zero for any A-module M and all n ≥ N.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let $t \\geq 0$\nbe an upper bound on the number of generators of $I$.\nThere exist $N, c \\geq 0$ such that the maps\n$$\n\\text{Tor}_{t + 1}^A(M, A/I^n) \\to \\text{Tor}_{t + 1}^A(M, A/I^{n - c})\n$$\nare zero for any $A$-module $M$ and all $n \\geq N$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GA2","source_file":"local-cohomology.tex","source_line":4496,"source_end_line":4505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4496-L4505","statement_sha256":"4e719f0be74e11a43c2f24b94d23637b9b7616351d81507f83e73b39b9823500","origin":"The Stacks Project","memory_eligible":false,"source_rank":9154,"rank":9154,"depth":34,"x":2445.531,"y":1012.668,"cluster":"local-crystalline-methods"},{"id":"stacks:0GA3","tag":"0GA3","title":"A bit of uniformity, II · Proposition 0GA3","summary":"Let I be an ideal of a Noetherian ring A. Let t ≥ 0 be an upper bound on the number of generators of I. There exist N, c ≥ 0 such that for n ≥ N the maps A/I^n → A/I^n - c satisfy the equivalent conditions of Lemma [Tag 0G9V] with e = t.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let $t \\geq 0$\nbe an upper bound on the number of generators of $I$.\nThere exist $N, c \\geq 0$ such that for $n \\geq N$ the maps\n$$\nA/I^n \\to A/I^{n - c}\n$$\nsatisfy the equivalent conditions of\nLemma \\ref{lemma-characterize-vanishing-tor-ext-above-e} with $e = t$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GA3","source_file":"local-cohomology.tex","source_line":4570,"source_end_line":4580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4570-L4580","statement_sha256":"2e206f2fd917a1c2b6738baaa1e0c7764f40f8603025ca6bbe29fc5c867a191b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9155,"rank":9155,"depth":35,"x":2608.037,"y":1214.464,"cluster":"local-crystalline-methods"},{"id":"stacks:0GA7","tag":"0GA7","title":"A bit of uniformity, III · Lemma 0GA7","summary":"With q_0 = q(S) and d = d(S) as above, we have • for n ≥ 1, q ≥ q_0, and i > 0 we have H^i(X, O_Y_n(q)) = 0, • for n ≥ 1 and q ≥ q_0 we have H^0(X, O_Y_n(q)) = I^q/I^q + n, • for q ≥ q_0 and i > 0 we have H^i(X, O_X(q)) = 0, • for q ≥ q_0 we have H^0(X, O_X(q)) = I^q.","statement_latex":"With $q_0 = q(S)$ and $d = d(S)$ as above, we have\n\\begin{enumerate}\n\\item for $n \\geq 1$, $q \\geq q_0$, and $i > 0$ we have\n$H^i(X, \\mathcal{O}_{Y_n}(q)) = 0$,\n\\item for $n \\geq 1$ and $q \\geq q_0$ we have\n$H^0(X, \\mathcal{O}_{Y_n}(q)) = I^q/I^{q + n}$,\n\\item for $q \\geq q_0$ and $i > 0$ we have\n$H^i(X, \\mathcal{O}_X(q)) = 0$,\n\\item for $q \\geq q_0$ we have\n$H^0(X, \\mathcal{O}_X(q)) = I^q$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GA7","source_file":"local-cohomology.tex","source_line":4796,"source_end_line":4809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4796-L4809","statement_sha256":"51b11ff6593a6df3dba1d655fbcb16e7a6ccf5deecdbaeaa50e29010fcda6875","origin":"The Stacks Project","memory_eligible":false,"source_rank":9156,"rank":9156,"depth":0,"x":2325.437,"y":1157.773,"cluster":"local-crystalline-methods"},{"id":"stacks:0GA8","tag":"0GA8","title":"A bit of uniformity, III · Lemma 0GA8","summary":"Let 0 → K → L → M → 0 be a short exact sequence of A-modules such that K and L are annihilated by I^n and M is an (A, n, c)-module. Then the kernel of p^*K → p^*L is scheme theoretically supported on Y_c.","statement_latex":"Let $0 \\to K \\to L \\to M \\to 0$ be a short exact sequence of $A$-modules\nsuch that $K$ and $L$ are annihilated by $I^n$ and $M$ is an\n$(A, n, c)$-module. Then the kernel of $p^*K \\to p^*L$\nis scheme theoretically supported on $Y_c$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GA8","source_file":"local-cohomology.tex","source_line":4877,"source_end_line":4883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4877-L4883","statement_sha256":"490f7bae3c87f9e980422a2c4ada048a47630822856df865008d720ed7af262c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9157,"rank":9157,"depth":0,"x":2579.853,"y":1039.016,"cluster":"local-crystalline-methods"},{"id":"stacks:0GA9","tag":"0GA9","title":"A bit of uniformity, III · Lemma 0GA9","summary":"Let F be a coherent O_X-module. Then F is scheme theoretically supported on Y_c if and only if the canonical map F → F(c) is zero.","statement_latex":"Let $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThen $\\mathcal{F}$ is scheme theoretically\nsupported on $Y_c$ if and only if the canonical map\n$\\mathcal{F} \\to \\mathcal{F}(c)$ is zero.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GA9","source_file":"local-cohomology.tex","source_line":4914,"source_end_line":4920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4914-L4920","statement_sha256":"7e3861af3a3d39f9c740ba4d460ca9fa7b5179da105aa285c4ce2a43adf1f372","origin":"The Stacks Project","memory_eligible":false,"source_rank":9158,"rank":9158,"depth":0,"x":2487.592,"y":1271.349,"cluster":"local-crystalline-methods"},{"id":"stacks:0GAA","tag":"0GAA","title":"A bit of uniformity, III · Lemma 0GAA","summary":"With q_0 = q(S) and d = d(S) as above, suppose we have integers n ≥ c ≥ 0, an (A, n, c)-module M, an index i ∈ (0, 1, …, d), and an integer q. Then we distinguish the following cases • In the case i = d ≥ 1 and q ≥ q_0 we have H^d(X, p^*M(q)) = 0. • In the case i = d - 1 ≥ 1 and q ≥ q_0 we have H^d - 1(X, p^*M(q)) = 0. • In the case d - 1 > i > 0 and q ≥ q_0 + (d - 1 - i)c the map H^i(X, p^*M(q)) → H^i(X, p^*M(q - (d - 1 - i)c)) is zero. • In the case i = 0, d ∈ (0, 1),…","statement_latex":"With $q_0 = q(S)$ and $d = d(S)$ as above, suppose we have\nintegers $n \\geq c \\geq 0$, an $(A, n, c)$-module $M$,\nan index $i \\in \\{0, 1, \\ldots, d\\}$, and an integer $q$.\nThen we distinguish the following cases\n\\begin{enumerate}\n\\item In the case $i = d \\geq 1$ and $q \\geq q_0$ we have\n$H^d(X, p^*M(q)) = 0$.\n\\item In the case $i = d - 1 \\geq 1$ and $q \\geq q_0$ we have\n$H^{d - 1}(X, p^*M(q)) = 0$.\n\\item In the case $d - 1 > i > 0$ and $q \\geq q_0 + (d - 1 - i)c$\nthe map\n$H^i(X, p^*M(q)) \\to H^i(X, p^*M(q - (d - 1 - i)c))$\nis zero.\n\\item In the case $i = 0$, $d \\in \\{0, 1\\}$, and $q \\geq q_0$, there\nis a surjection\n$$\nI^qM \\longrightarrow H^0(X, p^*M(q))\n$$\n\\item In the case $i = 0$, $d > 1$, and $q \\geq q_0 + (d - 1)c$ the map\n$$\nH^0(X, p^*M(q)) \\to H^0(X, p^*M(q - (d - 1)c))\n$$\nhas image contained in the image of the canonical map\n$I^{q - (d - 1)c}M \\to H^0(X, p^*M(q - (d - 1)c))$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAA","source_file":"local-cohomology.tex","source_line":4927,"source_end_line":4954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L4927-L4954","statement_sha256":"dad0a40e470444fc08627a9daa8c250376c182e15d6e892ee9c8f6875ab0118a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9159,"rank":9159,"depth":1,"x":2368.583,"y":1047.293,"cluster":"local-crystalline-methods"},{"id":"stacks:0GAC","tag":"0GAC","title":"A bit of uniformity, III · Lemma 0GAC","summary":"With q_0 = q(S) and d = d(S) as above, let M be an (A, n, c)-module and let φ : M → I^n/I^2n be an A-linear map. Assume n ≥ max(q_0 + (1 + d)c, (2 + d)c) and if d = 0 assume n ≥ q_0 + 2c. Then the composition M xrightarrowφ I^n/I^2n → I^n - (1 + d)c/I^2n - (1 + d)c is of the form ∑ a_i ψ_i with a_i ∈ I^c and ψ_i : M → I^n - (2 + d)c/I^2n - (2 + d)c.","statement_latex":"With $q_0 = q(S)$ and $d = d(S)$ as above, let $M$ be an $(A, n, c)$-module\nand let $\\varphi : M \\to I^n/I^{2n}$ be an $A$-linear map. Assume\n$n \\geq \\max(q_0 + (1 + d)c, (2 + d)c)$ and if $d = 0$ assume\n$n \\geq q_0 + 2c$. Then the composition\n$$\nM \\xrightarrow{\\varphi} I^n/I^{2n} \\to\nI^{n - (1 + d)c}/I^{2n - (1 + d)c}\n$$\nis of the form $\\sum a_i \\psi_i$ with $a_i \\in I^c$ and\n$\\psi_i : M \\to I^{n - (2 + d)c}/I^{2n - (2 + d)c}$.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAC","source_file":"local-cohomology.tex","source_line":5113,"source_end_line":5125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L5113-L5125","statement_sha256":"e11233585b044f994aeec67438370d98d82fa3cd95a5c2ef866b3c94ba3c4973","origin":"The Stacks Project","memory_eligible":false,"source_rank":9160,"rank":9160,"depth":2,"x":2636.978,"y":1145.149,"cluster":"local-crystalline-methods"},{"id":"stacks:0GAD","tag":"0GAD","title":"A bit of uniformity, III · Lemma 0GAD","summary":"With d = d(S) and q_0 = q(S) as above. Then • for integers n ≥ c ≥ 0 with n ≥ max(q_0 + (1 + d)c, (2 + d)c), • for K of D(A/I^n) with H^i(K) = 0 for i not = -1, 0 and H^i(K) finite for i = -1, 0 such that Ext^1_A/I^c(K, N) is annihilated by I^c for all finite A/I^n-modules N the map Ext^1_A/I^n(K, I^n/I^2n) → Ext^1_A/I^n(K, I^n - (1 + d)c/I^2n - 2(1 + d)c) is zero.","statement_latex":"With $d = d(S)$ and $q_0 = q(S)$ as above. Then\n\\begin{enumerate}\n\\item for integers $n \\geq c \\geq 0$ with\n$n \\geq \\max(q_0 + (1 + d)c, (2 + d)c)$,\n\\item for $K$ of $D(A/I^n)$ with $H^i(K) = 0$ for $i \\not = -1, 0$\nand $H^i(K)$ finite for $i = -1, 0$ such that $\\Ext^1_{A/I^c}(K, N)$\nis annihilated by $I^c$ for all finite $A/I^n$-modules $N$\n\\end{enumerate}\nthe map\n$$\n\\Ext^1_{A/I^n}(K, I^n/I^{2n})\n\\longrightarrow\n\\Ext^1_{A/I^n}(K, I^{n - (1 + d)c}/I^{2n - 2(1 + d)c})\n$$\nis zero.","area":"Local & Crystalline Methods","chapter":"Local Cohomology","chapter_id":"local-cohomology","section":"A bit of uniformity, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAD","source_file":"local-cohomology.tex","source_line":5233,"source_end_line":5250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/local-cohomology.tex#L5233-L5250","statement_sha256":"aba29fea1238e8837c667e81b9a5b9f86eea47325c38e481bd4914834486f619","origin":"The Stacks Project","memory_eligible":false,"source_rank":9161,"rank":9161,"depth":12,"x":2359.901,"y":1225.422,"cluster":"local-crystalline-methods"},{"id":"stacks:0EI8","tag":"0EI8","title":"Formal sections, I · Lemma 0EI8","summary":"Let X be a scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let … → F_3 → F_2 → F_1 be an inverse system of quasi-coherent O_X-modules such that F_n = F_n + 1/I^nF_n + 1. Set F = lim F_n. Then • F = Rlim F_n, • for any affine open U ⊂ X we have H^p(U, F) = 0 for p > 0, and • for each p there is a short exact sequence 0 → R^1lim H^p - 1(X, F_n) → H^p(X, F) → lim H^p(X, F_n) → 0. If moreover I is of finite type, then • [(4)] F_n = F/I^nF, and • [(5)] I^n F = lim_m ≥…","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of quasi-coherent $\\mathcal{O}_X$-modules\nsuch that\n$\\mathcal{F}_n = \\mathcal{F}_{n + 1}/\\mathcal{I}^n\\mathcal{F}_{n + 1}$.\nSet $\\mathcal{F} = \\lim \\mathcal{F}_n$. Then\n\\begin{enumerate}\n\\item $\\mathcal{F} = R\\lim \\mathcal{F}_n$,\n\\item for any affine open $U \\subset X$ we have\n$H^p(U, \\mathcal{F}) = 0$ for $p > 0$, and\n\\item for each $p$ there is a short exact sequence\n$0 \\to R^1\\lim H^{p - 1}(X, \\mathcal{F}_n) \\to\nH^p(X, \\mathcal{F}) \\to \\lim H^p(X, \\mathcal{F}_n) \\to 0$.\n\\end{enumerate}\nIf moreover $\\mathcal{I}$ is of finite type, then\n\\begin{enumerate}\n\\item[(4)]\n$\\mathcal{F}_n = \\mathcal{F}/\\mathcal{I}^n\\mathcal{F}$, and\n\\item[(5)]\n$\\mathcal{I}^n \\mathcal{F} = \\lim_{m \\geq n} \\mathcal{I}^n\\mathcal{F}_m$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Formal sections, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EI8","source_file":"algebraization.tex","source_line":153,"source_end_line":179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L153-L179","statement_sha256":"7ba8b28bf2f1a2f0c1e6c98a91b41eb50be940582f998f59dd3103d6d584d443","origin":"The Stacks Project","memory_eligible":false,"source_rank":9162,"rank":9162,"depth":25,"x":1135.696,"y":1600.0,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EH9","tag":"0EH9","title":"Formal sections, II · Lemma 0EH9","summary":"Let X be a scheme. Let f ∈ Γ(X, O_X). Let … → F_3 → F_2 → F_1 be an inverse system of quasi-coherent O_X-modules. The following are equivalent • for all n ≥ 1 the map f : F_n + 1 → F_n + 1 factors through F_n + 1 → F_n to give a short exact sequence 0 → F_n → F_n + 1 → F_1 → 0, • for all n ≥ 1 the map f^n : F_n + 1 → F_n + 1 factors through F_n + 1 → F_1 to give a short exact sequence 0 → F_1 → F_n + 1 → F_n → 0 • there exists an O_X-module G which is f-divisible such…","statement_latex":"Let $X$ be a scheme. Let $f \\in \\Gamma(X, \\mathcal{O}_X)$. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of quasi-coherent $\\mathcal{O}_X$-modules.\nThe following are equivalent\n\\begin{enumerate}\n\\item for all $n \\geq 1$ the map\n$f : \\mathcal{F}_{n + 1} \\to \\mathcal{F}_{n + 1}$ factors\nthrough $\\mathcal{F}_{n + 1} \\to \\mathcal{F}_n$ to give a\nshort exact sequence\n$0 \\to \\mathcal{F}_n \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_1 \\to 0$,\n\\item for all $n \\geq 1$ the map\n$f^n : \\mathcal{F}_{n + 1} \\to \\mathcal{F}_{n + 1}$\nfactors through $\\mathcal{F}_{n + 1} \\to \\mathcal{F}_1$\nto give a short exact sequence\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_{n + 1} \\to \\mathcal{F}_n \\to 0$\n\\item there exists an $\\mathcal{O}_X$-module $\\mathcal{G}$\nwhich is $f$-divisible such that $\\mathcal{F}_n = \\mathcal{G}[f^n]$.\n\\item there exists an $\\mathcal{O}_X$-module $\\mathcal{F}$ which is\n$f$-torsion free such that $\\mathcal{F}_n = \\mathcal{F}/f^n\\mathcal{F}$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Formal sections, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EH9","source_file":"algebraization.tex","source_line":219,"source_end_line":243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L219-L243","statement_sha256":"eaf4cb0120f72c8ad018189481f0efe26554d0d8c9ee9f8076d38de4a49b49b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9163,"rank":9163,"depth":26,"x":1122.726,"y":1605.598,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0BLD","tag":"0BLD","title":"Formal sections, II · Lemma 0BLD","summary":"Slightly improved version of [Bhatt-local] Let A be a ring and f ∈ A. Let X be a scheme over A. Let F be a quasi-coherent O_X-module. Assume that F[f^n] = Ker(f^n : F → F) stabilizes. Then RΓ(X, lim F/f^nF) = RΓ(X, F)^wedge where the right hand side indicates the derived completion with respect to the ideal (f) ⊂ A. Consequently, for p ∈ Z we obtain a commutative diagram xymatrix & 0 & 0 0 ar[r] & widehatH^p(X, F) ar[r] ar[u] & lim H^p(X, F/f^nF) ar[r] ar[u] & T_f(H^p +…","statement_latex":"\\begin{reference}\nSlightly improved version of \\cite[Lemma 1.6]{Bhatt-local}\n\\end{reference}\nLet $A$ be a ring and $f \\in A$. Let $X$ be a scheme over $A$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume that $\\mathcal{F}[f^n] = \\Ker(f^n : \\mathcal{F} \\to \\mathcal{F})$\nstabilizes. Then\n$$\nR\\Gamma(X, \\lim \\mathcal{F}/f^n\\mathcal{F}) =\nR\\Gamma(X, \\mathcal{F})^\\wedge\n$$\nwhere the right hand side indicates the derived completion\nwith respect to the ideal $(f) \\subset A$. Consequently, for\n$p \\in \\mathbf{Z}$ we obtain a commutative diagram\n$$\n\\xymatrix{\n& 0 & 0 \\\\\n0 \\ar[r] &\n\\widehat{H^p(X, \\mathcal{F})} \\ar[r] \\ar[u] &\n\\lim H^p(X, \\mathcal{F}/f^n\\mathcal{F}) \\ar[r] \\ar[u] &\nT_f(H^{p + 1}(X, \\mathcal{F})) \\ar[r] &\n0 \\\\\n0 \\ar[r] &\nH^0(H^p(X, \\mathcal{F})^\\wedge) \\ar[r] \\ar[u] &\nH^p(X, \\lim \\mathcal{F}/f^n\\mathcal{F}) \\ar[r] \\ar[u] &\nT_f(H^{p + 1}(X, \\mathcal{F})) \\ar[r] \\ar@{=}[u] &\n0 \\\\\n&\nR^1\\lim H^p(X, \\mathcal{F})[f^n] \\ar[u] \\ar[r]^\\cong &\nR^1\\lim H^{p - 1}(X, \\mathcal{F}/f^n\\mathcal{F}) \\ar[u] \\\\\n& 0 \\ar[u] & 0 \\ar[u]\n}\n$$\nwith exact rows and columns where\n$\\widehat{H^p(X, \\mathcal{F})} =\n\\lim H^p(X, \\mathcal{F})/f^n H^p(X, \\mathcal{F})$\nis the usual $f$-adic completion\nand $T_f(-)$ denotes the $f$-adic Tate module as in\nMore on Algebra, Example\n\\ref{more-algebra-example-spectral-sequence-principal}.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Formal sections, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLD","source_file":"algebraization.tex","source_line":259,"source_end_line":301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L259-L301","statement_sha256":"6162e8c5203354214d4971606df63d080a0ead59c45c006338724cb4678754fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9164,"rank":9164,"depth":26,"x":1131.113,"y":1589.343,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIA","tag":"0EIA","title":"Formal sections, III · Lemma 0EIA","summary":"Let I = (f_1, …, f_r) be an ideal of a Noetherian ring A. If cd(A, I) = 1, then there exist c ≥ 1 and maps φ_j : I^c → A such that ∑ f_j φ_j : I^c → I is the inclusion map.","statement_latex":"Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$.\nIf $\\text{cd}(A, I) = 1$, then there exist $c \\geq 1$ and maps\n$\\varphi_j : I^c \\to A$ such that $\\sum f_j \\varphi_j : I^c \\to I$\nis the inclusion map.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Formal sections, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIA","source_file":"algebraization.tex","source_line":328,"source_end_line":334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L328-L334","statement_sha256":"340bb30f32fb30bacd903a31525927b4daf56aac649847770652f2983dddd841","origin":"The Stacks Project","memory_eligible":false,"source_rank":9165,"rank":9165,"depth":31,"x":1139.169,"y":1610.046,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIB","tag":"0EIB","title":"Formal sections, III · Lemma 0EIB","summary":"Let I = (f_1, …, f_r) be an ideal of a Noetherian ring A with cd(A, I) = 1. Let c ≥ 1 and φ_j : I^c → A, j = 1, …, r be as in Lemma [Tag 0EIA]. Then there is a unique graded A-algebra map Φ : bigoplus_n ≥ 0 I^nc → A[T_1, …, T_r] with Φ(g) = ∑ φ_j(g) T_j for g ∈ I^c. Moreover, the composition of Φ with the map A[T_1, …, T_r] → bigoplus_n ≥ 0 I^n, T_j ↦ f_j is the inclusion map bigoplus_n ≥ 0 I^nc → bigoplus_n ≥ 0 I^n.","statement_latex":"Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$\nwith $\\text{cd}(A, I) = 1$. Let $c \\geq 1$ and $\\varphi_j : I^c \\to A$,\n$j = 1, \\ldots, r$ be as in Lemma \\ref{lemma-cd-one}.\nThen there is a unique graded $A$-algebra map\n$$\n\\Phi : \\bigoplus\\nolimits_{n \\geq 0} I^{nc} \\to A[T_1, \\ldots, T_r]\n$$\nwith $\\Phi(g) = \\sum \\varphi_j(g) T_j$ for $g \\in I^c$.\nMoreover, the composition of $\\Phi$ with the map\n$A[T_1, \\ldots, T_r] \\to \\bigoplus_{n \\geq 0} I^n$,\n$T_j \\mapsto f_j$ is the inclusion map\n$\\bigoplus_{n \\geq 0} I^{nc} \\to \\bigoplus_{n \\geq 0} I^n$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Formal sections, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIB","source_file":"algebraization.tex","source_line":349,"source_end_line":363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L349-L363","statement_sha256":"6fc9f8f22de042c823588f1bd1a06780d8fcedb676ec54f8a1304693299497f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9166,"rank":9166,"depth":32,"x":1113.174,"y":1597.5,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIC","tag":"0EIC","title":"Formal sections, III · Lemma 0EIC","summary":"Let I = (f_1, …, f_r) be an ideal of a Noetherian ring A with cd(A, I) = 1. Let c ≥ 1 and φ_j : I^c → A, j = 1, …, r be as in Lemma [Tag 0EIA]. Let A → B be a ring map with B Noetherian and let N be a finite B-module. Then, after possibly increasing c and adjusting φ_j accordingly, there is a unique unique graded B-module map Φ_N : bigoplus_n ≥ 0 I^ncN → N[T_1, …, T_r] with Φ_N(g x) = Φ(g) x for g ∈ I^nc and x ∈ N where Φ is as in Lemma [Tag 0EIB]. The composition of Φ_N…","statement_latex":"Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$\nwith $\\text{cd}(A, I) = 1$. Let $c \\geq 1$ and $\\varphi_j : I^c \\to A$,\n$j = 1, \\ldots, r$ be as in Lemma \\ref{lemma-cd-one}.\nLet $A \\to B$ be a ring map with $B$ Noetherian and let $N$ be\na finite $B$-module. Then, after possibly increasing $c$\nand adjusting $\\varphi_j$ accordingly, there is a unique\nunique graded $B$-module map\n$$\n\\Phi_N : \\bigoplus\\nolimits_{n \\geq 0} I^{nc}N \\to N[T_1, \\ldots, T_r]\n$$\nwith $\\Phi_N(g x) = \\Phi(g) x$ for $g \\in I^{nc}$ and $x \\in N$\nwhere $\\Phi$ is as in Lemma \\ref{lemma-cd-one-extend}.\nThe composition of $\\Phi_N$ with the map\n$N[T_1, \\ldots, T_r] \\to \\bigoplus_{n \\geq 0} I^nN$,\n$T_j \\mapsto f_j$ is the inclusion map\n$\\bigoplus_{n \\geq 0} I^{nc}N \\to \\bigoplus_{n \\geq 0} I^nN$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Formal sections, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIC","source_file":"algebraization.tex","source_line":393,"source_end_line":411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L393-L411","statement_sha256":"dd41117a7e81d2009e2714cc97da22095986cb0e1274405b9e35aced242a03e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9167,"rank":9167,"depth":33,"x":1145.939,"y":1591.483,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EH6","tag":"0EH6","title":"Formal sections, III · Lemma 0EH6","summary":"Let I = (f_1, …, f_r) be an ideal of a Noetherian ring A with cd(A, I) = 1. Let c ≥ 1 and φ_j : I^c → A, j = 1, …, r be as in Lemma [Tag 0EIA]. Let X be a Noetherian scheme over Spec(A). Let … → F_3 → F_2 → F_1 be an inverse system of coherent O_X-modules such that F_n = F_n + 1/I^nF_n + 1. Set F = lim F_n. Then, after possibly increasing c and adjusting φ_j accordingly, there exists a unique graded O_X-module map Φ_F : bigoplus_n ≥ 0 I^ncF → F[T_1, …, T_r] with Φ_F(g s)…","statement_latex":"Let $I = (f_1, \\ldots, f_r)$ be an ideal of a Noetherian ring $A$ with\n$\\text{cd}(A, I) = 1$. Let $c \\geq 1$ and $\\varphi_j : I^c \\to A$,\n$j = 1, \\ldots, r$ be as in Lemma \\ref{lemma-cd-one}.\nLet $X$ be a Noetherian scheme over $\\Spec(A)$. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of coherent $\\mathcal{O}_X$-modules\nsuch that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nSet $\\mathcal{F} = \\lim \\mathcal{F}_n$.\nThen, after possibly increasing $c$ and adjusting $\\varphi_j$ accordingly,\nthere exists a unique graded $\\mathcal{O}_X$-module map\n$$\n\\Phi_\\mathcal{F} :\n\\bigoplus\\nolimits_{n \\geq 0} I^{nc}\\mathcal{F}\n\\longrightarrow\n\\mathcal{F}[T_1, \\ldots, T_r]\n$$\nwith $\\Phi_\\mathcal{F}(g s) = \\Phi(g) s$ for $g \\in I^{nc}$ and\n$s$ a local section of $\\mathcal{F}$ where $\\Phi$ is as in\nLemma \\ref{lemma-cd-one-extend}. The composition of $\\Phi_\\mathcal{F}$\nwith the map\n$\\mathcal{F}[T_1, \\ldots, T_r] \\to \\bigoplus_{n \\geq 0} I^n\\mathcal{F}$,\n$T_j \\mapsto f_j$\nis the canonical inclusion\n$\\bigoplus_{n \\geq 0} I^{nc}\\mathcal{F} \\to\n\\bigoplus_{n \\geq 0} I^n\\mathcal{F}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Formal sections, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EH6","source_file":"algebraization.tex","source_line":440,"source_end_line":469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L440-L469","statement_sha256":"395583c82836e36f3ce8485bc59853f446f52d54de674e089a40d532fc148172","origin":"The Stacks Project","memory_eligible":false,"source_rank":9168,"rank":9168,"depth":34,"x":1124.669,"y":1616.659,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EH7","tag":"0EH7","title":"Formal sections, III · Lemma 0EH7","summary":"Let I be an ideal of a Noetherian ring A. Let X be a Noetherian scheme over Spec(A). Let … → F_3 → F_2 → F_1 be an inverse system of coherent O_X-modules such that F_n = F_n + 1/I^nF_n + 1. If cd(A, I) = 1, then for all p ∈ Z the limit topology on lim H^p(X, F_n) is I-adic.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let $X$ be a Noetherian scheme\nover $\\Spec(A)$. Let\n$$\n\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1\n$$\nbe an inverse system of coherent $\\mathcal{O}_X$-modules\nsuch that $\\mathcal{F}_n = \\mathcal{F}_{n + 1}/I^n\\mathcal{F}_{n + 1}$.\nIf $\\text{cd}(A, I) = 1$, then for all $p \\in \\mathbf{Z}$ the limit topology on\n$\\lim H^p(X, \\mathcal{F}_n)$ is $I$-adic.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Formal sections, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EH7","source_file":"algebraization.tex","source_line":510,"source_end_line":521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L510-L521","statement_sha256":"b29d5b3ed5702f891ad1b62d34e0406628926e22a2cbf7482229df0a6b6d181a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9169,"rank":9169,"depth":35,"x":1119.833,"y":1583.556,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DX0","tag":"0DX0","title":"Mittag-Leffler conditions · Lemma 0DX0","summary":"Let (A, m) be a Noetherian local ring. • Let M be a finite A-module. Then the A-module H^i_ m(M) satisfies the descending chain condition for any i. • Let U = Spec(A) setminus ( m) be the punctured spectrum of A. Let F be a coherent O_U-module. Then the A-module H^i(U, F) satisfies the descending chain condition for i > 0.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\n\\begin{enumerate}\n\\item Let $M$ be a finite $A$-module. Then the $A$-module\n$H^i_\\mathfrak m(M)$ satisfies the descending chain condition\nfor any $i$.\n\\item Let $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$ be the\npunctured spectrum of $A$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.\nThen the $A$-module $H^i(U, \\mathcal{F})$\nsatisfies the descending chain condition for $i > 0$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Mittag-Leffler conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DX0","source_file":"algebraization.tex","source_line":601,"source_end_line":614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L601-L614","statement_sha256":"42451e0f010bf6fb68bec29287a11305058eb81d712f97d463bec05fd32ae704","origin":"The Stacks Project","memory_eligible":false,"source_rank":9170,"rank":9170,"depth":30,"x":1152.059,"y":1606.767,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DX1","tag":"0DX1","title":"Mittag-Leffler conditions · Lemma 0DX1","summary":"Let (A, m) be a Noetherian local ring. • Let (M_n) be an inverse system of finite A-modules. Then the inverse system H^i_ m(M_n) satisfies the Mittag-Leffler condition for any i. • Let U = Spec(A) setminus ( m) be the punctured spectrum of A. Let F_n be an inverse system of coherent O_U-modules. Then the inverse system H^i(U, F_n) satisfies the Mittag-Leffler condition for i > 0.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\n\\begin{enumerate}\n\\item Let $(M_n)$ be an inverse system of finite $A$-modules. Then the\ninverse system $H^i_\\mathfrak m(M_n)$ satisfies the Mittag-Leffler\ncondition for any $i$.\n\\item Let $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$ be the\npunctured spectrum of $A$.\nLet $\\mathcal{F}_n$ be an inverse system of\ncoherent $\\mathcal{O}_U$-modules.\nThen the inverse system $H^i(U, \\mathcal{F}_n)$\nsatisfies the Mittag-Leffler condition for $i > 0$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Mittag-Leffler conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DX1","source_file":"algebraization.tex","source_line":663,"source_end_line":677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L663-L677","statement_sha256":"9807be841c060c55ba79c15049f221544fb4e2dd834c68ef1e135c67e14c55bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9171,"rank":9171,"depth":31,"x":1107.051,"y":1607.957,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EHB","tag":"0EHB","title":"Mittag-Leffler conditions · Lemma 0EHB","summary":"Let (A, m) be a Noetherian local ring. Let (M_n) be an inverse system of finite A-modules. Let M → lim M_n be a map where M is a finite A-module such that for some i the map H^i_ m(M) → lim H^i_ m(M_n) is an isomorphism. Then the inverse system H^i_ m(M_n) is essentially constant with value H^i_ m(M).","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $(M_n)$ be an inverse system of finite $A$-modules.\nLet $M \\to \\lim M_n$ be a map where $M$ is a finite $A$-module\nsuch that for some $i$ the map\n$H^i_\\mathfrak m(M) \\to \\lim H^i_\\mathfrak m(M_n)$\nis an isomorphism.\nThen the inverse system $H^i_\\mathfrak m(M_n)$\nis essentially constant with value $H^i_\\mathfrak m(M)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Mittag-Leffler conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHB","source_file":"algebraization.tex","source_line":683,"source_end_line":693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L683-L693","statement_sha256":"0f129dc3236413878bb79bf28176350b66ad94dcb1f8f72a96aad36762b98e73","origin":"The Stacks Project","memory_eligible":false,"source_rank":9172,"rank":9172,"depth":32,"x":1141.063,"y":1580.142,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXJ","tag":"0DXJ","title":"Mittag-Leffler conditions · Lemma 0DXJ","summary":"Let (A, m) be a Noetherian local ring. Let I ⊂ A be an ideal. Let M be a finite A-module. Then H^i(RΓ_ m(M)^wedge) = lim H^i_ m(M/I^nM) for all i where RΓ_ m(M)^wedge denotes the derived I-adic completion.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module.\nThen\n$$\nH^i(R\\Gamma_\\mathfrak m(M)^\\wedge) = \\lim H^i_\\mathfrak m(M/I^nM)\n$$\nfor all $i$ where $R\\Gamma_\\mathfrak m(M)^\\wedge$ denotes\nthe derived $I$-adic completion.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Mittag-Leffler conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXJ","source_file":"algebraization.tex","source_line":709,"source_end_line":719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L709-L719","statement_sha256":"194d95cb5890dff2467f0a1387db907ba5854f39b1603c31ea2efaf543b3a9a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9173,"rank":9173,"depth":32,"x":1138.175,"y":1621.894,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0996","tag":"0996","title":"Derived completion on a ringed site · Lemma 0996","summary":"Let (C, O) be a ringed site. Let f be a global section of O. • For L, N ∈ D(O_f) we have RSheafHom_O(L, N) = RSheafHom_O_f(L, N). In particular the two O_f-structures on RSheafHom_O(L, N) agree. • For K ∈ D(O) and L ∈ D(O_f) we have RSheafHom_O(L, K) = RSheafHom_O_f(L, RSheafHom_O(O_f, K)) In particular RSheafHom_O(O_f, RSheafHom_O(O_f, K)) = RSheafHom_O(O_f, K). • If g is a second global section of O, then RSheafHom_O(O_f, RSheafHom_O(O_g, K)) = RSheafHom_O(O_gf, K).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $f$ be a global\nsection of $\\mathcal{O}$.\n\\begin{enumerate}\n\\item For $L, N \\in D(\\mathcal{O}_f)$ we have\n$R\\SheafHom_\\mathcal{O}(L, N) = R\\SheafHom_{\\mathcal{O}_f}(L, N)$.\nIn particular the two $\\mathcal{O}_f$-structures on\n$R\\SheafHom_\\mathcal{O}(L, N)$ agree.\n\\item For $K \\in D(\\mathcal{O})$ and\n$L \\in D(\\mathcal{O}_f)$ we have\n$$\nR\\SheafHom_\\mathcal{O}(L, K) =\nR\\SheafHom_{\\mathcal{O}_f}(L, R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K))\n$$\nIn particular\n$R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f,\nR\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)) =\nR\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K)$.\n\\item If $g$ is a second global\nsection of $\\mathcal{O}$, then\n$$\nR\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, R\\SheafHom_\\mathcal{O}(\\mathcal{O}_g, K))\n= R\\SheafHom_\\mathcal{O}(\\mathcal{O}_{gf}, K).\n$$\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0996","source_file":"algebraization.tex","source_line":747,"source_end_line":773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L747-L773","statement_sha256":"5fc2d319d11f7624c3b7f813a787caf98e512e8464eca559105211c4b0ef0ac7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9174,"rank":9174,"depth":5,"x":1105.36,"y":1588.005,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0997","tag":"0997","title":"Derived completion on a ringed site · Lemma 0997","summary":"Let (C, O) be a ringed site. Let f be a global section of O. Let K ∈ D(O). The following are equivalent • RSheafHom_O(O_f, K) = 0, • RSheafHom_O(L, K) = 0 for all L in D(O_f), • T(K, f) = 0.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $f$ be a global\nsection of $\\mathcal{O}$. Let $K \\in D(\\mathcal{O})$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $R\\SheafHom_\\mathcal{O}(\\mathcal{O}_f, K) = 0$,\n\\item $R\\SheafHom_\\mathcal{O}(L, K) = 0$ for all $L$ in $D(\\mathcal{O}_f)$,\n\\item $T(K, f) = 0$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0997","source_file":"algebraization.tex","source_line":832,"source_end_line":842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L832-L842","statement_sha256":"a2e47ffc3994bd623226faf96182ba17590316565ad7ebdbd7e7f4b89049a764","origin":"The Stacks Project","memory_eligible":false,"source_rank":9175,"rank":9175,"depth":8,"x":1158.906,"y":1594.662,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0998","tag":"0998","title":"Derived completion on a ringed site · Lemma 0998","summary":"Let (C, O) be a ringed site. Let K ∈ D(O). The rule which associates to U the set I(U) of sections f ∈ O(U) such that T(K|_U, f) = 0 is a sheaf of ideals in O.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let $K \\in D(\\mathcal{O})$.\nThe rule which associates to $U$ the set $\\mathcal{I}(U)$\nof sections $f \\in \\mathcal{O}(U)$ such that $T(K|_U, f) = 0$\nis a sheaf of ideals in $\\mathcal{O}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0998","source_file":"algebraization.tex","source_line":871,"source_end_line":877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L871-L877","statement_sha256":"0a12e43ca3e886763a9c69841b38b1eabc88bfd909ba46cf3aa40d79d5a83370","origin":"The Stacks Project","memory_eligible":false,"source_rank":9176,"rank":9176,"depth":9,"x":1112.359,"y":1621.077,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0999","tag":"0999","title":"Derived completion on a ringed site · Definition 0999","summary":"Let (C, O) be a ringed site. Let I ⊂ O be a sheaf of ideals. Let K ∈ D(O). We say that K is derived complete with respect to I if for every object U of C and f ∈ I(U) the object T(K|_U, f) of D(O_U) is zero.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals.\nLet $K \\in D(\\mathcal{O})$. We say that $K$ is\n{\\it derived complete with respect to $\\mathcal{I}$}\nif for every object $U$ of $\\mathcal{C}$ and $f \\in \\mathcal{I}(U)$\nthe object $T(K|_U, f)$ of $D(\\mathcal{O}_U)$ is zero.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0999","source_file":"algebraization.tex","source_line":908,"source_end_line":916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L908-L916","statement_sha256":"bd30b62c4ecf405483ecc3c45f00921bf650ec478336a9ef9285b7f8696be58b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9177,"rank":9177,"depth":0,"x":1125.925,"y":1573.582,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099A","tag":"099A","title":"Derived completion on a ringed site · Lemma 099A","summary":"Let (C, O) be a ringed site. Let I ⊂ O be a sheaf of ideals. If K ∈ D(O) and L ∈ D_comp(O), then RSheafHom_O(K, L) ∈ D_comp(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{I} \\subset \\mathcal{O}$ be a sheaf of ideals.\nIf $K \\in D(\\mathcal{O})$ and $L \\in D_{comp}(\\mathcal{O})$, then\n$R\\SheafHom_\\mathcal{O}(K, L) \\in D_{comp}(\\mathcal{O})$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099A","source_file":"algebraization.tex","source_line":929,"source_end_line":935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L929-L935","statement_sha256":"401ac3171f4db27b39447c6b5b82eaad08a7d1a677d9e7c6483074bbca0482c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9178,"rank":9178,"depth":9,"x":1155.019,"y":1617.712,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099C","tag":"099C","title":"Derived completion on a ringed site · Lemma 099C","summary":"Let C be a site. Let O → O' be a homomorphism of sheaves of rings. Let I ⊂ O be a sheaf of ideals. The inverse image of D_comp(O, I) under the restriction functor D(O') → D(O) is D_comp(O', IO').","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$\nbe a homomorphism of sheaves of rings. Let $\\mathcal{I} \\subset \\mathcal{O}$\nbe a sheaf of ideals. The inverse image of $D_{comp}(\\mathcal{O}, \\mathcal{I})$\nunder the restriction functor $D(\\mathcal{O}') \\to D(\\mathcal{O})$ is\n$D_{comp}(\\mathcal{O}', \\mathcal{I}\\mathcal{O}')$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099C","source_file":"algebraization.tex","source_line":959,"source_end_line":966,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L959-L966","statement_sha256":"5f86260046ed4e06b1441b2a0343ac17a1a309c9614de62f4d84efc7ef5704bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9179,"rank":9179,"depth":10,"x":1096.332,"y":1601.17,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099J","tag":"099J","title":"Derived completion on a ringed site · Lemma 099J","summary":"Let f : (Sh(D), O') → (Sh(C), O) be a morphism of ringed topoi. Let I ⊂ O and I' ⊂ O' be sheaves of ideals such that f^sharp sends f^-1I into I'. Then Rf_* sends D_comp(O', I') into D_comp(O, I).","statement_latex":"Let $f : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nbe a morphism of ringed topoi. Let $\\mathcal{I} \\subset \\mathcal{O}$\nand $\\mathcal{I}' \\subset \\mathcal{O}'$ be sheaves of ideals such\nthat $f^\\sharp$ sends $f^{-1}\\mathcal{I}$ into $\\mathcal{I}'$.\nThen $Rf_*$ sends $D_{comp}(\\mathcal{O}', \\mathcal{I}')$\ninto $D_{comp}(\\mathcal{O}, \\mathcal{I})$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099J","source_file":"algebraization.tex","source_line":981,"source_end_line":989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L981-L989","statement_sha256":"b3c81db20ec433ffa25b1f4eab18b4d01d4f4df3349d8cf31f7f1867ffd80ab9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9180,"rank":9180,"depth":19,"x":1154.558,"y":1579.472,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099B","tag":"099B","title":"Derived completion on a ringed site · Lemma 099B","summary":"Let (C, O) be a ringed on a site. Let f_1, …, f_r be global sections of O. Let I ⊂ O be the ideal sheaf generated by f_1, …, f_r. Then the inclusion functor D_comp(O) → D(O) has a left adjoint, i.e., given any object K of D(O) there exists a map K → K^wedge with K^wedge in D_comp(O) such that the map Hom_D(O)(K^wedge, E) → Hom_D(O)(K, E) is bijective whenever E is in D_comp(O). In fact we have K^wedge = RSheafHom_O (O → ∏_i_0 O_f_i_0 → ∏_i_0 < i_1 O_f_i_0f_i_1 → … →…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed on a site. Let $f_1, \\ldots, f_r$\nbe global sections of $\\mathcal{O}$. Let $\\mathcal{I} \\subset \\mathcal{O}$ be\nthe ideal sheaf generated by $f_1, \\ldots, f_r$.\nThen the inclusion functor $D_{comp}(\\mathcal{O}) \\to D(\\mathcal{O})$\nhas a left adjoint, i.e., given any object $K$ of $D(\\mathcal{O})$\nthere exists a map $K \\to K^\\wedge$ with $K^\\wedge$ in $D_{comp}(\\mathcal{O})$\nsuch that the map\n$$\n\\Hom_{D(\\mathcal{O})}(K^\\wedge, E) \\longrightarrow \\Hom_{D(\\mathcal{O})}(K, E)\n$$\nis bijective whenever $E$ is in $D_{comp}(\\mathcal{O})$. In fact\nwe have\n$$\nK^\\wedge =\nR\\SheafHom_\\mathcal{O}\n(\\mathcal{O} \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{f_{i_0}} \\to\n\\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{f_{i_0}f_{i_1}} \\to\n\\ldots \\to \\mathcal{O}_{f_1\\ldots f_r}, K)\n$$\nfunctorially in $K$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099B","source_file":"algebraization.tex","source_line":1023,"source_end_line":1045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1023-L1045","statement_sha256":"642d4677ff4ca247d21443934a7b62c1bf4480d0a700c8f0bf5917dcfbed1118","origin":"The Stacks Project","memory_eligible":false,"source_rank":9181,"rank":9181,"depth":6,"x":1128.357,"y":1629.847,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0A0E","tag":"0A0E","title":"Derived completion on a ringed site · Lemma 0A0E","summary":"Let (C, O) be a ringed on a site. Let f_1, …, f_r be global sections of O. Let I ⊂ O be the ideal sheaf generated by f_1, …, f_r. Let K ∈ D(O). The derived completion K^wedge of Lemma [Tag 099B] is given by the formula K^wedge = Rlim K ⊗^L_O K_n where K_n = K(O, f_1^n, …, f_r^n) is the Koszul complex on f_1^n, …, f_r^n over O.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed on a site. Let $f_1, \\ldots, f_r$\nbe global sections of $\\mathcal{O}$. Let $\\mathcal{I} \\subset \\mathcal{O}$ be\nthe ideal sheaf generated by $f_1, \\ldots, f_r$. Let $K \\in D(\\mathcal{O})$.\nThe derived completion $K^\\wedge$ of Lemma \\ref{lemma-derived-completion}\nis given by the formula\n$$\nK^\\wedge = R\\lim K \\otimes^\\mathbf{L}_\\mathcal{O} K_n\n$$\nwhere $K_n = K(\\mathcal{O}, f_1^n, \\ldots, f_r^n)$\nis the Koszul complex on $f_1^n, \\ldots, f_r^n$ over $\\mathcal{O}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0E","source_file":"algebraization.tex","source_line":1090,"source_end_line":1102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1090-L1102","statement_sha256":"e62a1f015285d5286a30dfcbee1cd4c79cd74b1af93594740d436e1b93bc6735","origin":"The Stacks Project","memory_eligible":false,"source_rank":9182,"rank":9182,"depth":7,"x":1106.633,"y":1576.479,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099D","tag":"099D","title":"Derived completion on a ringed site · Lemma 099D","summary":"There exist a way to construct • for every pair (A, I) consisting of a ring A and a finitely generated ideal I ⊂ A a complex K(A, I) of A-modules, • a map K(A, I) → A of complexes of A-modules, • for every ring map A → B and finitely generated ideal I ⊂ A a map of complexes K(A, I) → K(B, IB), such that • [(a)] for A → B and I ⊂ A finitely generated the diagram xymatrix K(A, I) ar[r] ar[d] & A ar[d] K(B, IB) ar[r] & B commutes, • [(b)] for A → B → C and I ⊂ A finitely…","statement_latex":"There exist a way to construct\n\\begin{enumerate}\n\\item for every pair $(A, I)$ consisting of a ring $A$ and a finitely\ngenerated ideal $I \\subset A$ a complex $K(A, I)$ of $A$-modules,\n\\item a map $K(A, I) \\to A$ of complexes of $A$-modules,\n\\item for every ring map $A \\to B$ and finitely generated ideal $I \\subset A$\na map of complexes $K(A, I) \\to K(B, IB)$,\n\\end{enumerate}\nsuch that\n\\begin{enumerate}\n\\item[(a)] for $A \\to B$ and $I \\subset A$ finitely generated the diagram\n$$\n\\xymatrix{\nK(A, I) \\ar[r] \\ar[d] & A \\ar[d] \\\\\nK(B, IB) \\ar[r] & B\n}\n$$\ncommutes,\n\\item[(b)] for $A \\to B \\to C$ and $I \\subset A$ finitely generated\nthe composition of the maps\n$K(A, I) \\to K(B, IB) \\to K(C, IC)$ is the map $K(A, I) \\to K(C, IC)$.\n\\item[(c)] for $A \\to B$ and a finitely generated ideal $I \\subset A$\nthe induced map $K(A, I) \\otimes_A^\\mathbf{L} B \\to K(B, IB)$\nis an isomorphism in $D(B)$, and\n\\item[(d)] if $I = (f_1, \\ldots, f_r) \\subset A$ then there is a commutative\ndiagram\n$$\n\\xymatrix{\n(A \\to \\prod\\nolimits_{i_0} A_{f_{i_0}} \\to\n\\prod\\nolimits_{i_0 < i_1} A_{f_{i_0}f_{i_1}} \\to\n\\ldots \\to A_{f_1\\ldots f_r}) \\ar[r] \\ar[d] &  K(A, I) \\ar[d] \\\\\nA \\ar[r]^1 & A\n}\n$$\nin $D(A)$ whose horizontal arrows are isomorphisms.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099D","source_file":"algebraization.tex","source_line":1130,"source_end_line":1168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1130-L1168","statement_sha256":"94cd0c8c461e6e9ebfe195c757cf01daf875601d8fe9e626a02ac90fc226cfa8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9183,"rank":9183,"depth":11,"x":1167.016,"y":1604.184,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099E","tag":"099E","title":"Derived completion on a ringed site · Lemma 099E","summary":"Let (C, O) be a ringed site. Let I ⊂ O be a finite type sheaf of ideals. There exists a map K → O in D(O) such that for every U ∈ Ob(C) such that I|_U is generated by f_1, …, f_r ∈ I(U) there is an isomorphism (O_U → ∏_i_0 O_U, f_i_0 → ∏_i_0 < i_1 O_U, f_i_0f_i_1 → … → O_U, f_1… f_r) → K|_U compatible with maps to O_U.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site. Let\n$\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of ideals.\nThere exists a map $K \\to \\mathcal{O}$ in $D(\\mathcal{O})$\nsuch that for every $U \\in \\Ob(\\mathcal{C})$ such that\n$\\mathcal{I}|_U$ is generated by $f_1, \\ldots, f_r \\in \\mathcal{I}(U)$\nthere is an isomorphism\n$$\n(\\mathcal{O}_U \\to \\prod\\nolimits_{i_0} \\mathcal{O}_{U, f_{i_0}} \\to\n\\prod\\nolimits_{i_0 < i_1} \\mathcal{O}_{U, f_{i_0}f_{i_1}} \\to\n\\ldots \\to \\mathcal{O}_{U, f_1\\ldots f_r}) \\longrightarrow K|_U\n$$\ncompatible with maps to $\\mathcal{O}_U$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099E","source_file":"algebraization.tex","source_line":1234,"source_end_line":1248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1234-L1248","statement_sha256":"c793b3159a4a819b027c494b834f2f0ba1b8f9bed474d9077c2c7f6e16454e2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9184,"rank":9184,"depth":12,"x":1098.636,"y":1618.33,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099F","tag":"099F","title":"Derived completion on a ringed site · Proposition 099F","summary":"Let (C, O) be a ringed site. Let I ⊂ O be a finite type sheaf of ideals. There exists a left adjoint to the inclusion functor D_comp(O) → D(O).","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\mathcal{I} \\subset \\mathcal{O}$ be a finite type sheaf of\nideals. There exists a left adjoint to the inclusion\nfunctor $D_{comp}(\\mathcal{O}) \\to D(\\mathcal{O})$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099F","source_file":"algebraization.tex","source_line":1297,"source_end_line":1303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1297-L1303","statement_sha256":"57220f10a2cdcc5fe7b20c321543854dae323a4501f176844726e222ceaf603c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9185,"rank":9185,"depth":13,"x":1138.57,"y":1567.999,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099H","tag":"099H","title":"Derived completion on a ringed site · Lemma 099H","summary":"Let C be a site. Assume φ : O → O' is a flat homomorphism of sheaves of rings. Let f_1, …, f_r be global sections of O such that O/(f_1, …, f_r) ≅ O'/(f_1, …, f_r)O'. Then the map of extended alternating v Cech complexes xymatrix O → ∏_i_0 O_f_i_0 → ∏_i_0 < i_1 O_f_i_0f_i_1 → … → O_f_1… f_r ar[d] O' → ∏_i_0 O'_f_i_0 → ∏_i_0 < i_1 O'_f_i_0f_i_1 → … → O'_f_1… f_r is a quasi-isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site.\nAssume $\\varphi : \\mathcal{O} \\to \\mathcal{O}'$ is a flat homomorphism\nof sheaves of rings. Let $f_1, \\ldots, f_r$ be global sections\nof $\\mathcal{O}$ such that $\\mathcal{O}/(f_1, \\ldots, f_r) \\cong\n\\mathcal{O}'/(f_1, \\ldots, f_r)\\mathcal{O}'$.\nThen the map of extended alternating {\\v C}ech complexes\n$$\n\\xymatrix{\n\\mathcal{O} \\to\n\\prod_{i_0} \\mathcal{O}_{f_{i_0}} \\to\n\\prod_{i_0 < i_1} \\mathcal{O}_{f_{i_0}f_{i_1}} \\to \\ldots \\to\n\\mathcal{O}_{f_1\\ldots f_r} \\ar[d] \\\\\n\\mathcal{O}' \\to\n\\prod_{i_0} \\mathcal{O}'_{f_{i_0}} \\to\n\\prod_{i_0 < i_1} \\mathcal{O}'_{f_{i_0}f_{i_1}} \\to \\ldots \\to\n\\mathcal{O}'_{f_1\\ldots f_r}\n}\n$$\nis a quasi-isomorphism.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099H","source_file":"algebraization.tex","source_line":1384,"source_end_line":1405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1384-L1405","statement_sha256":"b5e3a075b420d12978cc36888d3b0acb313d95d5e1fb56a6293cb6dba3b12217","origin":"The Stacks Project","memory_eligible":false,"source_rank":9186,"rank":9186,"depth":6,"x":1149.823,"y":1629.058,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099I","tag":"099I","title":"Derived completion on a ringed site · Lemma 099I","summary":"Let C be a site. Let O → O' be a homomorphism of sheaves of rings. Let I ⊂ O be a finite type sheaf of ideals. If O → O' is flat and O/I ≅ O'/IO', then the restriction functor D(O') → D(O) induces an equivalence D_comp(O', IO') → D_comp(O, I).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}'$ be a\nhomomorphism of sheaves of rings. Let $\\mathcal{I} \\subset \\mathcal{O}$\nbe a finite type sheaf of ideals.\nIf $\\mathcal{O} \\to \\mathcal{O}'$ is flat and\n$\\mathcal{O}/\\mathcal{I} \\cong \\mathcal{O}'/\\mathcal{I}\\mathcal{O}'$,\nthen the restriction functor $D(\\mathcal{O}') \\to D(\\mathcal{O})$\ninduces an equivalence\n$D_{comp}(\\mathcal{O}', \\mathcal{I}\\mathcal{O}') \\to\nD_{comp}(\\mathcal{O}, \\mathcal{I})$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099I","source_file":"algebraization.tex","source_line":1426,"source_end_line":1437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1426-L1437","statement_sha256":"2112a744f025e710ff13fc99a7855b542ba788379731777fb3062f5b73ab262b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9187,"rank":9187,"depth":20,"x":1091.248,"y":1589.615,"cluster":"algebraic-formal-geometry"},{"id":"stacks:099K","tag":"099K","title":"Derived completion on a ringed site · Lemma 099K","summary":"Let f : (Sh(D), O') → (Sh(C), O) be a morphism of ringed topoi. Let I ⊂ O and I' ⊂ O' be finite type sheaves of ideals such that f^sharp sends f^-1I into I'. Then Rf_* sends D_comp(O', I') into D_comp(O, I) and has a left adjoint Lf_comp^* which is Lf^* followed by derived completion.","statement_latex":"Let $f : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nbe a morphism of ringed topoi. Let $\\mathcal{I} \\subset \\mathcal{O}$\nand $\\mathcal{I}' \\subset \\mathcal{O}'$ \nbe finite type sheaves of ideals such that $f^\\sharp$ sends\n$f^{-1}\\mathcal{I}$ into $\\mathcal{I}'$.\nThen $Rf_*$ sends $D_{comp}(\\mathcal{O}', \\mathcal{I}')$\ninto $D_{comp}(\\mathcal{O}, \\mathcal{I})$ and has a left adjoint\n$Lf_{comp}^*$ which is $Lf^*$ followed by derived completion.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099K","source_file":"algebraization.tex","source_line":1480,"source_end_line":1490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1480-L1490","statement_sha256":"9572907a9ce782bf7b00bb84152fbccce3ceabaec868618304be05d89920cd4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9188,"rank":9188,"depth":20,"x":1167.642,"y":1585.391,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0A0G","tag":"0A0G","title":"Derived completion on a ringed site · Lemma 0A0G","summary":"Generalization of [BS]. Compare with [HL-P] in the setting of quasi-coherent modules and morphisms of (derived) algebraic stacks. Let f : (Sh(D), O') → (Sh(C), O) be a morphism of ringed topoi. Let I ⊂ O be a finite type sheaf of ideals. Let I' ⊂ O' be the ideal generated by f^sharp(f^-1I). Then Rf_* commutes with derived completion, i.e., Rf_*(K^wedge) = (Rf_*K)^wedge.","statement_latex":"\\begin{reference}\nGeneralization of \\cite[Lemma 6.5.9 (2)]{BS}. Compare with\n\\cite[Theorem 6.5]{HL-P} in the setting of quasi-coherent modules\nand morphisms of (derived) algebraic stacks.\n\\end{reference}\nLet $f : (\\Sh(\\mathcal{D}), \\mathcal{O}') \\to (\\Sh(\\mathcal{C}), \\mathcal{O})$\nbe a morphism of ringed topoi. Let $\\mathcal{I} \\subset \\mathcal{O}$\nbe a finite type sheaf of ideals. Let $\\mathcal{I}' \\subset \\mathcal{O}'$\nbe the ideal generated by $f^\\sharp(f^{-1}\\mathcal{I})$.\nThen $Rf_*$ commutes with derived completion, i.e.,\n$Rf_*(K^\\wedge) = (Rf_*K)^\\wedge$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0G","source_file":"algebraization.tex","source_line":1506,"source_end_line":1519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1506-L1519","statement_sha256":"9b8fa6b6bd79f6888fe13ff06c107127f9cbe476f2bf56be7468d77662cf64a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9189,"rank":9189,"depth":20,"x":1113.692,"y":1632.732,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0BLX","tag":"0BLX","title":"Derived completion on a ringed site · Lemma 0BLX","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. Let C be a site and let O be a sheaf of A-algebras. Let F be a sheaf of O-modules. Then we have RΓ(C, F)^wedge = RΓ(C, F^wedge) in D(A) where F^wedge is the derived completion of F with respect to IO and on the left hand wide we have the derived completion with respect to I. This produces two spectral sequences E_2^i, j = H^i(H^j(C, F)^wedge) and E_2^p, q = H^p(C, H^q(F^wedge)) both converging to H^*(RΓ(C,…","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nLet $\\mathcal{C}$ be a site and let $\\mathcal{O}$ be a sheaf\nof $A$-algebras. Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nThen we have\n$$\nR\\Gamma(\\mathcal{C}, \\mathcal{F})^\\wedge =\nR\\Gamma(\\mathcal{C}, \\mathcal{F}^\\wedge)\n$$\nin $D(A)$ where $\\mathcal{F}^\\wedge$ is the derived\ncompletion of $\\mathcal{F}$ with respect to $I\\mathcal{O}$ and on the\nleft hand wide we have the derived completion with respect to $I$.\nThis produces two spectral sequences\n$$\nE_2^{i, j} = H^i(H^j(\\mathcal{C}, \\mathcal{F})^\\wedge)\n\\quad\\text{and}\\quad\nE_2^{p, q} = H^p(\\mathcal{C}, H^q(\\mathcal{F}^\\wedge))\n$$\nboth converging to\n$H^*(R\\Gamma(\\mathcal{C}, \\mathcal{F})^\\wedge) =\nH^*(\\mathcal{C}, \\mathcal{F}^\\wedge)$","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Derived completion on a ringed site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLX","source_file":"algebraization.tex","source_line":1559,"source_end_line":1581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1559-L1581","statement_sha256":"3b078741d446eb54a89327517aba1bd431fde0cc97cd077ac961f8d207e75df3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9190,"rank":9190,"depth":21,"x":1115.446,"y":1566.01,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0A0L","tag":"0A0L","title":"The theorem on formal functions · Lemma 0A0L","summary":"Let X be a locally Noetherian scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let K be a pseudo-coherent object of D(O_X) with derived completion K^wedge. Then H^p(U, K^wedge) = lim H^p(U, K)/I^nH^p(U, K) = H^p(U, K)^wedge for any affine open U ⊂ X where I = I(U) and where on the right we have the derived completion with respect to I.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. Let $K$ be a\npseudo-coherent object of $D(\\mathcal{O}_X)$ with derived completion\n$K^\\wedge$. Then\n$$\nH^p(U, K^\\wedge) = \\lim H^p(U, K)/I^nH^p(U, K) =\nH^p(U, K)^\\wedge\n$$\nfor any affine open $U \\subset X$\nwhere $I = \\mathcal{I}(U)$ and where on the right we have the derived\ncompletion with respect to $I$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0L","source_file":"algebraization.tex","source_line":1651,"source_end_line":1664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1651-L1664","statement_sha256":"cf7468d13ef6172fd4467bcd8d64557beda5c82ebd948fed254186a9afe9e08e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9191,"rank":9191,"depth":23,"x":1168.727,"y":1617.097,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0A0K","tag":"0A0K","title":"The theorem on formal functions · Lemma 0A0K","summary":"Let X be a locally Noetherian scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let K be an object of D(O_X). Then • the derived completion K^wedge is equal to Rlim (K ⊗_O_X^L O_X/I^n). Let K is a pseudo-coherent object of D(O_X). Then • [(2)] the cohomology sheaf H^q(K^wedge) is equal to lim H^q(K)/I^nH^q(K). Let F be a coherent O_X-module. Then • [(3)] the derived completion F^wedge is equal to lim F/I^nF, • [(4)] lim F/I^n F = Rlim F/I^n F, • [(5)] H^p(U,…","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals.\nLet $K$ be an object of $D(\\mathcal{O}_X)$. Then\n\\begin{enumerate}\n\\item the derived completion $K^\\wedge$ is equal to\n$R\\lim (K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_X/\\mathcal{I}^n)$.\n\\end{enumerate}\nLet $K$ is a pseudo-coherent object of $D(\\mathcal{O}_X)$. Then\n\\begin{enumerate}\n\\item[(2)] the cohomology sheaf $H^q(K^\\wedge)$ is equal to\n$\\lim H^q(K)/\\mathcal{I}^nH^q(K)$.\n\\end{enumerate}\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module\\footnote{For example\n$H^q(K)$ for $K$ pseudo-coherent on our locally Noetherian $X$.}. Then\n\\begin{enumerate}\n\\item[(3)] the derived completion $\\mathcal{F}^\\wedge$ is equal to\n$\\lim \\mathcal{F}/\\mathcal{I}^n\\mathcal{F}$,\n\\item[(4)]\n$\\lim \\mathcal{F}/\\mathcal{I}^n \\mathcal{F} =\nR\\lim \\mathcal{F}/\\mathcal{I}^n \\mathcal{F}$,\n\\item[(5)] $H^p(U, \\mathcal{F}^\\wedge) = 0$ for $p \\not = 0$ for all\naffine opens $U \\subset X$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0K","source_file":"algebraization.tex","source_line":1684,"source_end_line":1709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1684-L1709","statement_sha256":"477eef915c5b4681eefd01e2b194be05d39b726e0c0e4067377655b4f7e06702","origin":"The Stacks Project","memory_eligible":false,"source_rank":9192,"rank":9192,"depth":27,"x":1086.984,"y":1609.525,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0A0M","tag":"0A0M","title":"The theorem on formal functions · Lemma 0A0M","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let X be a Noetherian scheme over A. Let F be a coherent O_X-module. Assume that H^p(X, F) is a finite A-module for all p. Then there are short exact sequences 0 → R^1lim H^p - 1(X, F/I^nF) → H^p(X, F)^wedge → lim H^p(X, F/I^nF) → 0 of A-modules where H^p(X, F)^wedge is the usual I-adic completion. If f is proper, then the R^1lim term is zero.","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Let $X$ be a\nNoetherian scheme over $A$. Let $\\mathcal{F}$ be a coherent\n$\\mathcal{O}_X$-module. Assume that $H^p(X, \\mathcal{F})$ is\na finite $A$-module for all $p$. Then there are short exact sequences\n$$\n0 \\to R^1\\lim H^{p - 1}(X, \\mathcal{F}/I^n\\mathcal{F}) \\to\nH^p(X, \\mathcal{F})^\\wedge \\to \\lim H^p(X, \\mathcal{F}/I^n\\mathcal{F}) \\to 0\n$$\nof $A$-modules where $H^p(X, \\mathcal{F})^\\wedge$ is the usual $I$-adic\ncompletion. If $f$ is proper, then the $R^1\\lim$ term is zero.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0M","source_file":"algebraization.tex","source_line":1758,"source_end_line":1770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1758-L1770","statement_sha256":"f89d79954bc9a88fa015b8b95096ae4ee89177cda877bda78b8b0a64802f3b3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9193,"rank":9193,"depth":36,"x":1154.451,"y":1568.058,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFH","tag":"0EFH","title":"Algebraization of local cohomology, I · Lemma 0EFH","summary":"Let I, J be ideals of a Noetherian ring A. Let M be a finite A-module. Let p ⊂ A be a prime. Let s and d be integers. Assume • A has a dualizing complex, • p not ∈ V(J) ∩ V(I), • cd(A, I) ≤ d, and • for all primes p' ⊂ p we have depth_A_ p'(M_ p') + dim((A/ p')_ q) > d + s for all q ∈ V( p') ∩ V(J) ∩ V(I). Then there exists an f ∈ A, f not ∈ p which annihilates H^i(RΓ_J(M)^wedge) for i ≤ s where ^wedge indicates I-adic completion.","statement_latex":"Let $I, J$ be ideals of a Noetherian ring $A$.\nLet $M$ be a finite $A$-module. Let $\\mathfrak p \\subset A$ be a prime.\nLet $s$ and $d$ be integers. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex,\n\\item $\\mathfrak p \\not \\in V(J) \\cap V(I)$,\n\\item $\\text{cd}(A, I) \\leq d$, and\n\\item for all primes $\\mathfrak p' \\subset \\mathfrak p$\nwe have\n$\\text{depth}_{A_{\\mathfrak p'}}(M_{\\mathfrak p'}) +\n\\dim((A/\\mathfrak p')_\\mathfrak q) > d + s$\nfor all $\\mathfrak q \\in V(\\mathfrak p') \\cap V(J) \\cap V(I)$.\n\\end{enumerate}\nThen there exists an $f \\in A$, $f \\not \\in \\mathfrak p$ which annihilates\n$H^i(R\\Gamma_J(M)^\\wedge)$ for $i \\leq s$ where ${}^\\wedge$\nindicates $I$-adic completion.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFH","source_file":"algebraization.tex","source_line":1867,"source_end_line":1885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1867-L1885","statement_sha256":"ff73cc0a4dc42da3f5986c2ef84ace3b76dbb91e9c8f8aa3b46c0e63dfec43f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9194,"rank":9194,"depth":36,"x":1137.778,"y":1638.016,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFI","tag":"0EFI","title":"Algebraization of local cohomology, I · Lemma 0EFI","summary":"Let I, J be ideals of a Noetherian ring. Let M be a finite A-module. Let s and d be integers. With T as in ([Tag 0EFG]) assume • A has a dualizing complex, • if p ∈ V(I), then no condition, • if p not ∈ V(I), p ∈ T, then dim((A/ p)_ q) ≤ d for some q ∈ V( p) ∩ V(J) ∩ V(I), • if p not ∈ V(I), p not ∈ T, then depth_A_ p(M_ p) ≥ s or depth_A_ p(M_ p) + dim((A/ p)_ q) > d + s for all q ∈ V( p) ∩ V(J) ∩ V(I). Then there exists an ideal J_0 ⊂ J with V(J_0) ∩ V(I) = V(J) ∩ V(I)…","statement_latex":"Let $I, J$ be ideals of a Noetherian ring. Let $M$ be a finite $A$-module.\nLet $s$ and $d$ be integers. With $T$ as in\n(\\ref{equation-associated-subset}) assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex,\n\\item if $\\mathfrak p \\in V(I)$, then no condition,\n\\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\in T$, then\n$\\dim((A/\\mathfrak p)_\\mathfrak q) \\leq d$ for some\n$\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$,\n\\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\not \\in T$, then\n$$\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s\n\\quad\\text{or}\\quad\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s\n$$\nfor all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$.\n\\end{enumerate}\nThen there exists an ideal $J_0 \\subset J$ with\n$V(J_0) \\cap V(I) = V(J) \\cap V(I)$ such that for any $J' \\subset J_0$ with\n$V(J') \\cap V(I) = V(J) \\cap V(I)$ the map\n$$\nR\\Gamma_{J'}(M) \\longrightarrow R\\Gamma_{J_0}(M)\n$$\ninduces an isomorphism in cohomology in degrees $\\leq s$\nand moreover these modules are annihilated by a power of $J_0I$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFI","source_file":"algebraization.tex","source_line":1924,"source_end_line":1952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L1924-L1952","statement_sha256":"339e117356f1d7e0ad4f8b398db7946ba5a18155e739fa225b9529de8ceac40f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9195,"rank":9195,"depth":40,"x":1093.14,"y":1576.021,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFJ","tag":"0EFJ","title":"Algebraization of local cohomology, I · Lemma 0EFJ","summary":"In Lemma [Tag 0EFI] if instead of the empty condition (2) we assume • [(2')] if p ∈ V(I), p not ∈ V(J) ∩ V(I), then depth_A_ p(M_ p) + dim((A/ p)_ q) > s for all q ∈ V( p) ∩ V(J) ∩ V(I), then the conditions also imply that H^i_J_0(M) is a finite A-module for i ≤ s.","statement_latex":"In Lemma \\ref{lemma-kill-colimit-weak-general} if instead of the empty\ncondition (2) we assume\n\\begin{enumerate}\n\\item[(2')] if $\\mathfrak p \\in V(I)$, $\\mathfrak p \\not \\in V(J) \\cap V(I)$,\nthen\n$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q) > s$\nfor all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$,\n\\end{enumerate}\nthen the conditions also imply that $H^i_{J_0}(M)$ is a finite\n$A$-module for $i \\leq s$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFJ","source_file":"algebraization.tex","source_line":2063,"source_end_line":2076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2063-L2076","statement_sha256":"ecf67706ed4253558dc625ffd3202b1cb2d598a22d7d8959fd6e55057e47aebe","origin":"The Stacks Project","memory_eligible":false,"source_rank":9196,"rank":9196,"depth":41,"x":1177.151,"y":1596.719,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFK","tag":"0EFK","title":"Algebraization of local cohomology, I · Lemma 0EFK","summary":"If in Lemma [Tag 0EFI] we additionally assume • [(6)] if p not ∈ V(I), p ∈ T, then depth_A_ p(M_ p) > s, then H^i_J_0(M) = H^i_J(M) = H^i_J + I(M) for i ≤ s and these modules are annihilated by a power of I.","statement_latex":"If in Lemma \\ref{lemma-kill-colimit-weak-general} we additionally assume\n\\begin{enumerate}\n\\item[(6)] if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\in T$, then\n$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$,\n\\end{enumerate}\nthen $H^i_{J_0}(M) = H^i_J(M) = H^i_{J + I}(M)$ for $i \\leq s$ and these\nmodules are annihilated by a power of $I$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFK","source_file":"algebraization.tex","source_line":2093,"source_end_line":2102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2093-L2102","statement_sha256":"4c6534c944252910b0768a77f15d09580f06211eb32d464cb74ef7eee76dda9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9197,"rank":9197,"depth":41,"x":1097.409,"y":1629.593,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFL","tag":"0EFL","title":"Algebraization of local cohomology, I · Lemma 0EFL","summary":"Let I, J be ideals of a Noetherian ring A. Let M be a finite A-module. Let s and d be integers. With T as in ([Tag 0EFG]) assume • A is I-adically complete and has a dualizing complex, • if p ∈ V(I) no condition, • cd(A, I) ≤ d, • if p not ∈ V(I), p not ∈ T then depth_A_ p(M_ p) ≥ s or depth_A_ p(M_ p) + dim((A/ p)_ q) > d + s for all q ∈ V( p) ∩ V(J) ∩ V(I), • if p not ∈ V(I), p not ∈ T, V( p) ∩ V(J) ∩ V(I) not = ∅, and depth(M_ p) < s, then one of the following holds: •…","statement_latex":"Let $I, J$ be ideals of a Noetherian ring $A$.\nLet $M$ be a finite $A$-module.\nLet $s$ and $d$ be integers. With $T$ as in\n(\\ref{equation-associated-subset}) assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item if $\\mathfrak p \\in V(I)$ no condition,\n\\item $\\text{cd}(A, I) \\leq d$,\n\\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\not \\in T$ then\n$$\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s\n\\quad\\text{or}\\quad\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s\n$$\nfor all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$,\n\\item if $\\mathfrak p \\not \\in V(I)$, $\\mathfrak p \\not \\in T$,\n$V(\\mathfrak p) \\cap V(J) \\cap V(I) \\not = \\emptyset$, and\n$\\text{depth}(M_\\mathfrak p) < s$, then one\nof the following holds\\footnote{Our method\nforces this additional condition. We will return to this\n(insert future reference).}:\n\\begin{enumerate}\n\\item $\\dim(\\text{Supp}(M_\\mathfrak p)) < s + 2$\\footnote{For example\nif $M$ satisfies Serre's condition $(S_s)$\non the complement of $V(I) \\cup T$.}, or\n\\item  $\\delta(\\mathfrak p) > d + \\delta_{max} - 1$\nwhere $\\delta$ is a dimension function and $\\delta_{max}$\nis the maximum of $\\delta$ on $V(J) \\cap V(I)$, or\n\\item $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s + \\delta_{max} - \\delta_{min} - 2$\nfor all $\\mathfrak q \\in V(\\mathfrak p) \\cap V(J) \\cap V(I)$.\n\\end{enumerate}\n\\end{enumerate}\nThen there exists an ideal $J_0 \\subset J$ with\n$V(J_0) \\cap V(I) = V(J) \\cap V(I)$\nsuch that for any $J' \\subset J_0$ with\n$V(J') \\cap V(I) = V(J) \\cap V(I)$ the map\n$$\nR\\Gamma_{J'}(M) \\longrightarrow R\\Gamma_J(M)^\\wedge\n$$\ninduces an isomorphism on cohomology in degrees $\\leq s$.\nHere ${}^\\wedge$ denotes derived $I$-adic completion.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFL","source_file":"algebraization.tex","source_line":2116,"source_end_line":2161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2116-L2161","statement_sha256":"b729db263fd5875b6bebf1e1f45ce50efeafc0b4f6c380a246d1e7eec7357996","origin":"The Stacks Project","memory_eligible":false,"source_rank":9198,"rank":9198,"depth":0,"x":1130.237,"y":1559.122,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXK","tag":"0DXK","title":"Algebraization of local cohomology, II · Lemma 0DXK","summary":"Let (A, m) be a Noetherian local ring. Let I ⊂ A be an ideal. Let M be a finite A-module and let p ⊂ A be a prime. Let s and d be integers. Assume • A has a dualizing complex, • cd(A, I) ≤ d, and • depth_A_ p(M_ p) + dim(A/ p) > d + s. Then there exists an f ∈ A setminus p which annihilates H^i(RΓ_ m(M)^wedge) for i ≤ s where ^wedge indicates I-adic completion.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module and\nlet $\\mathfrak p \\subset A$ be a prime. Let $s$ and $d$ be integers. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex,\n\\item $\\text{cd}(A, I) \\leq d$, and\n\\item\n$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s$.\n\\end{enumerate}\nThen there exists an $f \\in A \\setminus \\mathfrak p$ which annihilates\n$H^i(R\\Gamma_\\mathfrak m(M)^\\wedge)$ for $i \\leq s$ where ${}^\\wedge$\nindicates $I$-adic completion.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXK","source_file":"algebraization.tex","source_line":2398,"source_end_line":2412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2398-L2412","statement_sha256":"3ff226189d344e6d5729116b913a41943b57ca538d397ccd67b0ffd31f1c137b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9199,"rank":9199,"depth":39,"x":1163.142,"y":1630.689,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXM","tag":"0DXM","title":"Algebraization of local cohomology, II · Lemma 0DXM","summary":"Let (A, m) be a Noetherian local ring. Let I ⊂ A be an ideal. Let M be a finite A-module. Let s and d be integers. Assume • A has a dualizing complex, • if p ∈ V(I), then no condition, • if p not ∈ V(I) and V( p) ∩ V(I) = ( m), then dim(A/ p) ≤ d, • if p not ∈ V(I) and V( p) ∩ V(I) not = ( m), then depth_A_ p(M_ p) ≥ s or depth_A_ p(M_ p) + dim(A/ p) > d + s Then there exists an ideal J_0 ⊂ A with V(J_0) ∩ V(I) = ( m) such that for any J ⊂ J_0 with V(J) ∩ V(I) = ( m) the…","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module.\nLet $s$ and $d$ be integers. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex,\n\\item if $\\mathfrak p \\in V(I)$, then no condition,\n\\item if $\\mathfrak p \\not \\in V(I)$ and\n$V(\\mathfrak p) \\cap V(I) = \\{\\mathfrak m\\}$, then\n$\\dim(A/\\mathfrak p) \\leq d$,\n\\item if $\\mathfrak p \\not \\in V(I)$ and\n$V(\\mathfrak p) \\cap V(I) \\not = \\{\\mathfrak m\\}$, then\n$$\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s\n\\quad\\text{or}\\quad\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s\n$$\n\\end{enumerate}\nThen there exists an ideal $J_0 \\subset A$ with\n$V(J_0) \\cap V(I) = \\{\\mathfrak m\\}$ such that for any $J \\subset J_0$ with\n$V(J) \\cap V(I) = \\{\\mathfrak m\\}$ the map\n$$\nR\\Gamma_J(M) \\longrightarrow R\\Gamma_{J_0}(M)\n$$\ninduces an isomorphism in cohomology in degrees $\\leq s$\nand moreover these modules are annihilated by a power of $J_0I$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXM","source_file":"algebraization.tex","source_line":2437,"source_end_line":2464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2437-L2464","statement_sha256":"c903efa3d1e1006497c1740a740d9af34cc212bbf257dbb1211994177bac8c30","origin":"The Stacks Project","memory_eligible":false,"source_rank":9200,"rank":9200,"depth":41,"x":1080.233,"y":1596.126,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXN","tag":"0DXN","title":"Algebraization of local cohomology, II · Lemma 0DXN","summary":"In Lemma [Tag 0DXM] if instead of the empty condition (2) we assume • [(2')] if p ∈ V(I) and p not = m, then depth_A_ p(M_ p) + dim(A/ p) > s, then the conditions also imply that H^i_J_0(M) is a finite A-module for i ≤ s.","statement_latex":"In Lemma \\ref{lemma-kill-colimit-weak} if instead of the empty\ncondition (2) we assume\n\\begin{enumerate}\n\\item[(2')] if $\\mathfrak p \\in V(I)$ and $\\mathfrak p \\not = \\mathfrak m$,\nthen $\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > s$,\n\\end{enumerate}\nthen the conditions also imply that $H^i_{J_0}(M)$ is a finite\n$A$-module for $i \\leq s$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXN","source_file":"algebraization.tex","source_line":2470,"source_end_line":2480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2470-L2480","statement_sha256":"d553be8466afe31feaa6fafaae4d1fb2fd920fca08fe3f47639ebefcb9e6e1d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9201,"rank":9201,"depth":42,"x":1170.326,"y":1574.291,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFQ","tag":"0EFQ","title":"Algebraization of local cohomology, II · Lemma 0EFQ","summary":"If in Lemma [Tag 0DXM] we additionally assume • [(6)] if p not ∈ V(I) and V( p) ∩ V(I) = ( m), then depth_A_ p(M_ p) > s, then H^i_J_0(M) = H^i_J(M) = H^i_ m(M) for i ≤ s and these modules are annihilated by a power of I.","statement_latex":"If in Lemma \\ref{lemma-kill-colimit-weak} we additionally assume\n\\begin{enumerate}\n\\item[(6)] if $\\mathfrak p \\not \\in V(I)$ and\n$V(\\mathfrak p) \\cap V(I) = \\{\\mathfrak m\\}$, then\n$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$,\n\\end{enumerate}\nthen $H^i_{J_0}(M) = H^i_J(M) = H^i_\\mathfrak m(M)$ for $i \\leq s$\nand these modules are annihilated by a power of $I$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFQ","source_file":"algebraization.tex","source_line":2486,"source_end_line":2496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2486-L2496","statement_sha256":"e97e7fd9f16fbb438787e57dbf3fe3efb15d7afc407706140b0737959d7b4e90","origin":"The Stacks Project","memory_eligible":false,"source_rank":9202,"rank":9202,"depth":42,"x":1120.825,"y":1642.365,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXP","tag":"0DXP","title":"Algebraization of local cohomology, II · Lemma 0DXP","summary":"Let (A, m) be a Noetherian local ring. Let I ⊂ A be an ideal. Let M be a finite A-module. Let s and d be integers. Assume • A is I-adically complete and has a dualizing complex, • if p ∈ V(I), no condition, • cd(A, I) ≤ d, • if p not ∈ V(I) and V( p) ∩ V(I) not = ( m) then depth_A_ p(M_ p) ≥ s or depth_A_ p(M_ p) + dim(A/ p) > d + s Then there exists an ideal J_0 ⊂ A with V(J_0) ∩ V(I) = ( m) such that for any J ⊂ J_0 with V(J) ∩ V(I) = ( m) the map RΓ_J(M) →…","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $I \\subset A$ be an ideal. Let $M$ be a finite $A$-module.\nLet $s$ and $d$ be integers. Assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item if $\\mathfrak p \\in V(I)$, no condition,\n\\item $\\text{cd}(A, I) \\leq d$,\n\\item if $\\mathfrak p \\not \\in V(I)$ and\n$V(\\mathfrak p) \\cap V(I) \\not = \\{\\mathfrak m\\}$ then\n$$\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) \\geq s\n\\quad\\text{or}\\quad\n\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) + \\dim(A/\\mathfrak p) > d + s\n$$\n\\end{enumerate}\nThen there exists an ideal $J_0 \\subset A$ with\n$V(J_0) \\cap V(I) = \\{\\mathfrak m\\}$ such that for any $J \\subset J_0$ with\n$V(J) \\cap V(I) = \\{\\mathfrak m\\}$ the map\n$$\nR\\Gamma_J(M) \\longrightarrow\nR\\Gamma_J(M)^\\wedge = R\\Gamma_\\mathfrak m(M)^\\wedge\n$$\ninduces an isomorphism in cohomology in degrees $\\leq s$.\nHere ${}^\\wedge$ denotes derived $I$-adic completion.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXP","source_file":"algebraization.tex","source_line":2502,"source_end_line":2528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2502-L2528","statement_sha256":"85012801b5b526ef594475a6e3235493932717c640ffb63f3c2ea82b1bef52a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9203,"rank":9203,"depth":42,"x":1102.363,"y":1563.108,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EID","tag":"0EID","title":"Algebraization of local cohomology, III · Lemma 0EID","summary":"In Situation [Tag 0EFU] let p ⊂ q be primes of A with p not ∈ V(I) and q ∈ T. If there does not exist an r ∈ V(I) setminus T with p ⊂ r ⊂ q then depth(M_ p) > s.","statement_latex":"In Situation \\ref{situation-bootstrap} let $\\mathfrak p \\subset \\mathfrak q$\nbe primes of $A$ with $\\mathfrak p \\not \\in V(I)$ and\n$\\mathfrak q \\in T$. If there does not exist an\n$\\mathfrak r \\in V(I) \\setminus T$ with\n$\\mathfrak p \\subset \\mathfrak r \\subset \\mathfrak q$\nthen $\\text{depth}(M_\\mathfrak p) > s$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EID","source_file":"algebraization.tex","source_line":2681,"source_end_line":2689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2681-L2689","statement_sha256":"857c21963c94409ecfbff5a73dd2fa493f1a2b36a027acda5787e9255669490d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9204,"rank":9204,"depth":15,"x":1180.645,"y":1611.659,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFV","tag":"0EFV","title":"Algebraization of local cohomology, III · Lemma 0EFV","summary":"In Situation [Tag 0EFU] we have • [(E)] if T' ⊂ T is a smaller specialization stable subset, then A, I, T', M satisfies the assumptions of Situation [Tag 0EFU], • [(F)] if S ⊂ A is a multiplicative subset, then S^-1A, S^-1I, T', S^-1M satisfies the assumptions of Situation [Tag 0EFU] where T' ⊂ V(S^-1I) is the inverse image of T, • [(G)] the quadruple A', I', T', M' satisfies the assumptions of Situation [Tag 0EFU] where A', I', M' are the usual I-adic completions of A,…","statement_latex":"In Situation \\ref{situation-bootstrap} we have\n\\begin{enumerate}\n\\item[(E)] if $T' \\subset T$ is a smaller specialization stable subset, then\n$A, I, T', M$ satisfies the assumptions of Situation \\ref{situation-bootstrap},\n\\item[(F)] if $S \\subset A$ is a multiplicative subset, then\n$S^{-1}A, S^{-1}I, T', S^{-1}M$\nsatisfies the assumptions of Situation \\ref{situation-bootstrap}\nwhere $T' \\subset V(S^{-1}I)$ is the inverse image of $T$,\n\\item[(G)] the quadruple $A', I', T', M'$\nsatisfies the assumptions of Situation \\ref{situation-bootstrap}\nwhere $A', I', M'$ are the usual $I$-adic completions of $A, I, M$\nand $T' \\subset V(I')$ is the inverse image of $T$.\n\\end{enumerate}\nLet $I \\subset \\mathfrak a \\subset A$ be an ideal such that\n$V(\\mathfrak a) \\subset T$. Then\n\\begin{enumerate}\n\\item[(A)] if $I$ is contained in the Jacobson radical of $A$,\nthen all hypotheses of\nLemmas \\ref{lemma-kill-colimit-weak-general} and\n\\ref{lemma-kill-colimit-support-general} are satisfied\nfor $A, I, \\mathfrak a, M$,\n\\item[(B)] if $A$ is complete with respect to $I$, then\nall hypotheses except for possibly (5) of\nLemma \\ref{lemma-algebraize-local-cohomology-general}\nare satisfied for $A, I, \\mathfrak a, M$,\n\\item[(C)] if $A$ is local with maximal ideal $\\mathfrak m = \\mathfrak a$,\nthen all hypotheses of\nLemmas \\ref{lemma-kill-colimit-weak} and \\ref{lemma-kill-colimit-support}\nhold for $A, \\mathfrak m, I, M$,\n\\item[(D)] if $A$ is local with maximal ideal $\\mathfrak m = \\mathfrak a$\nand $I$-adically complete, then all hypotheses of\nLemma \\ref{lemma-algebraize-local-cohomology}\nhold for $A, \\mathfrak m, I, M$,\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFV","source_file":"algebraization.tex","source_line":2726,"source_end_line":2762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2726-L2762","statement_sha256":"18a328a06e1ff982605a24ce7c528d5a347fb1b83698f868ae10ad1f67c0423c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9205,"rank":9205,"depth":43,"x":1082.734,"y":1620.375,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFR","tag":"0EFR","title":"Algebraization of local cohomology, III · Lemma 0EFR","summary":"In Situation [Tag 0EFU] assume A is local with maximal ideal m and T = ( m). Then H^i_ m(M) → lim H^i_ m(M/I^nM) is an isomorphism for i ≤ s and these modules are annihilated by a power of I.","statement_latex":"In Situation \\ref{situation-bootstrap} assume $A$ is local with\nmaximal ideal $\\mathfrak m$ and $T = \\{\\mathfrak m\\}$. Then\n$H^i_\\mathfrak m(M) \\to \\lim H^i_\\mathfrak m(M/I^nM)$\nis an isomorphism for $i \\leq s$ and these modules are\nannihilated by a power of $I$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFR","source_file":"algebraization.tex","source_line":2921,"source_end_line":2928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2921-L2928","statement_sha256":"26afad2b521904aadb78c1330bfbad2e631eaa3d75ead887adeb897be2fefeec","origin":"The Stacks Project","memory_eligible":false,"source_rank":9206,"rank":9206,"depth":44,"x":1148.679,"y":1557.679,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFW","tag":"0EFW","title":"Algebraization of local cohomology, III · Lemma 0EFW","summary":"Let I ⊂ a be ideals of a Noetherian ring A. Let M be a finite A-module. Let s and d be integers. If we assume • [(a)] A has a dualizing complex, • [(b)] cd(A, I) ≤ d, • [(c)] if p not ∈ V(I) and q ∈ V( p) ∩ V( a) then depth_A_ p(M_ p) > s or depth_A_ p(M_ p) + dim((A/ p)_ q) > d + s. Then A, I, V( a), M, s, d are as in Situation [Tag 0EFU].","statement_latex":"Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.\nLet $M$ be a finite $A$-module. Let $s$ and $d$ be integers.\nIf we assume\n\\begin{enumerate}\n\\item[(a)] $A$ has a dualizing complex,\n\\item[(b)] $\\text{cd}(A, I) \\leq d$,\n\\item[(c)] if $\\mathfrak p \\not \\in V(I)$ and\n$\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ then\n$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) > s$ or\n$\\text{depth}_{A_\\mathfrak p}(M_\\mathfrak p) +\n\\dim((A/\\mathfrak p)_\\mathfrak q) > d + s$.\n\\end{enumerate}\nThen $A, I, V(\\mathfrak a), M, s, d$ are as in\nSituation \\ref{situation-bootstrap}.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFW","source_file":"algebraization.tex","source_line":2966,"source_end_line":2982,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L2966-L2982","statement_sha256":"6ceae50ddf779cd2a6f4926558d8f0a974566dc2161b6e59e99e51aded014985","origin":"The Stacks Project","memory_eligible":false,"source_rank":9207,"rank":9207,"depth":39,"x":1150.486,"y":1642.272,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFX","tag":"0EFX","title":"Algebraization of local cohomology, III · Lemma 0EFX","summary":"In Situation [Tag 0EFU] the inverse systems (H^i_T(I^nM))_n ≥ 0 are pro-zero for i ≤ s. Moreover, there exists an integer m_0 such that for all m ≥ m_0 there exists an integer m'(m) ≥ m such that for k ≥ m'(m) the image of H^s + 1_T(I^kM) → H^s + 1_T(I^mM) maps injectively to H^s + 1_T(I^m_0M).","statement_latex":"In Situation \\ref{situation-bootstrap} the inverse systems\n$\\{H^i_T(I^nM)\\}_{n \\geq 0}$ are pro-zero for $i \\leq s$.\nMoreover, there exists an integer $m_0$ such that for all\n$m \\geq m_0$ there exists an integer $m'(m) \\geq m$ such that for\n$k \\geq m'(m)$ the image of\n$H^{s + 1}_T(I^kM) \\to H^{s + 1}_T(I^mM)$\nmaps injectively to $H^{s + 1}_T(I^{m_0}M)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFX","source_file":"algebraization.tex","source_line":3018,"source_end_line":3027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3018-L3027","statement_sha256":"e75ddaa21aecc814cfa30d24593adc95381d2ba962b43e1f66a8cf680d20d4ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":9208,"rank":9208,"depth":45,"x":1080.364,"y":1580.24,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EFY","tag":"0EFY","title":"Algebraization of local cohomology, III · Lemma 0EFY","summary":"In Situation [Tag 0EFU] there exists an integer m_0 ≥ 0 such that • H^i_T(M) → H^i_T(M/I^nM) is injective for all i ≤ s and n ≥ m_0, • (H^i_T(M/I^nM))_n ≥ 0 satisfies the Mittag-Leffler condition for i < s. • (H^i_T(I^m_0M/I^nM))_n ≥ m_0 satisfies the Mittag-Leffler condition for i ≤ s, • H^i_T(M) → lim H^i_T(M/I^nM) is an isomorphism for i < s, • H^s_T(I^m_0M) → lim H^s_T(I^m_0M/I^nM) is an isomorphism for i ≤ s, • H^s_T(M) → lim H^s_T(M/I^nM) is injective with cokernel…","statement_latex":"In Situation \\ref{situation-bootstrap} there exists an integer $m_0 \\geq 0$\nsuch that\n\\begin{enumerate}\n\\item $H^i_T(M) \\to H^i_T(M/I^nM)$ is injective for all $i \\leq s$\nand $n \\geq m_0$,\n\\item $\\{H^i_T(M/I^nM)\\}_{n \\geq 0}$\nsatisfies the Mittag-Leffler condition for $i < s$.\n\\item $\\{H^i_T(I^{m_0}M/I^nM)\\}_{n \\geq m_0}$\nsatisfies the Mittag-Leffler condition for $i \\leq s$,\n\\item $H^i_T(M) \\to \\lim H^i_T(M/I^nM)$\nis an isomorphism for $i < s$,\n\\item $H^s_T(I^{m_0}M) \\to \\lim H^s_T(I^{m_0}M/I^nM)$\nis an isomorphism for $i \\leq s$,\n\\item $H^s_T(M) \\to \\lim H^s_T(M/I^nM)$ is\ninjective with cokernel killed by $I^{m_0}$, and\n\\item $R^1\\lim H^s_T(M/I^nM)$ is killed by $I^{m_0}$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EFY","source_file":"algebraization.tex","source_line":3170,"source_end_line":3189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3170-L3189","statement_sha256":"eb7d807805ebdfc6d7805f6e13995ac77a3aa5131c7f655f920127d81494c397","origin":"The Stacks Project","memory_eligible":false,"source_rank":9209,"rank":9209,"depth":46,"x":1183.051,"y":1586.261,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIE","tag":"0EIE","title":"Algebraization of local cohomology, III · Theorem 0EIE","summary":"In Situation [Tag 0EFU] for i ≤ s the inverse system (H^i_T(M/I^nM))_n ≥ 0 is essentially constant with value H^i_T(M). In particular, (H^i_T(M/I^nM))_n ≥ 0 is Mittag-Leffler, H^i_T(M) is annihilated by a power of I, and H^i_T(M) = lim H^i_T(M/I^nM).","statement_latex":"In Situation \\ref{situation-bootstrap} for $i \\leq s$ the inverse system\n$\\{H^i_T(M/I^nM)\\}_{n \\geq 0}$ is essentially constant with value $H^i_T(M)$.\nIn particular, $\\{H^i_T(M/I^nM)\\}_{n \\geq 0}$ is Mittag-Leffler,\n$H^i_T(M)$ is annihilated by a power of $I$, and\n$H^i_T(M) = \\lim H^i_T(M/I^nM)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, III","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIE","source_file":"algebraization.tex","source_line":3264,"source_end_line":3271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3264-L3271","statement_sha256":"8a2160b352be80c5bf80c5285be2cecedda486f8452ca7f5cf81c4d5d907d0dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9210,"rank":9210,"depth":47,"x":1101.636,"y":1640.654,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EG0","tag":"0EG0","title":"Algebraization of local cohomology, III · Lemma 0EG0","summary":"Let I ⊂ a ⊂ A be ideals of a Noetherian ring A and let M be a finite A-module. Let s and d be integers. Suppose that • A, I, V( a), M satisfy the assumptions of Situation [Tag 0EFU] for s and d, and • A, I, a, M satisfy the conditions of Lemma [Tag 0EFL] for s + 1 and d with J = a. Then there exists an ideal J_0 ⊂ a with V(J_0) ∩ V(I) = V( a) such that for any J ⊂ J_0 with V(J) ∩ V(I) = V( a) the map H^s + 1_J(M) → lim H^s + 1_ a(M/I^nM) is an isomorphism.","statement_latex":"Let $I \\subset \\mathfrak a \\subset A$ be ideals of a Noetherian ring $A$\nand let $M$ be a finite $A$-module. Let $s$ and $d$ be integers.\nSuppose that\n\\begin{enumerate}\n\\item $A, I, V(\\mathfrak a), M$ satisfy the assumptions of\nSituation \\ref{situation-bootstrap} for $s$ and $d$, and\n\\item $A, I, \\mathfrak a, M$ satisfy the conditions of\nLemma \\ref{lemma-algebraize-local-cohomology-general}\nfor $s + 1$ and $d$ with $J = \\mathfrak a$.\n\\end{enumerate}\nThen there exists an ideal\n$J_0 \\subset \\mathfrak a$ with $V(J_0) \\cap V(I) = V(\\mathfrak a)$\nsuch that for any $J \\subset J_0$ with $V(J) \\cap V(I) = V(\\mathfrak a)$\nthe map\n$$\nH^{s + 1}_J(M) \\longrightarrow \\lim H^{s + 1}_\\mathfrak a(M/I^nM)\n$$\nis an isomorphism.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of local cohomology, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EG0","source_file":"algebraization.tex","source_line":3450,"source_end_line":3470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3450-L3470","statement_sha256":"01715185c608d623a3e860d0aa46127df791a076eddf3e0cb804069456bae6d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9211,"rank":9211,"depth":48,"x":1118.102,"y":1553.456,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXI","tag":"0DXI","title":"Algebraization of formal sections, I · Lemma 0DXI","summary":"Let U be the punctured spectrum of a Noetherian local ring A. Let F be a coherent O_U-module. Let I ⊂ A be an ideal. Then H^i(RΓ(U, F)^wedge) = lim H^i(U, F/I^nF) for all i where RΓ(U, F)^wedge denotes the derived I-adic completion.","statement_latex":"Let $U$ be the punctured spectrum of a Noetherian local ring $A$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_U$-module.\nLet $I \\subset A$ be an ideal. Then\n$$\nH^i(R\\Gamma(U, \\mathcal{F})^\\wedge) =\n\\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F})\n$$\nfor all $i$ where $R\\Gamma(U, \\mathcal{F})^\\wedge$ denotes\nthe derived $I$-adic completion.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXI","source_file":"algebraization.tex","source_line":3518,"source_end_line":3529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3518-L3529","statement_sha256":"0f3d3bd9f0d388f60fce966bd7981dd0ea24f6e93cfffbaf3fa65a51acabd541","origin":"The Stacks Project","memory_eligible":false,"source_rank":9212,"rank":9212,"depth":32,"x":1176.666,"y":1627.846,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXQ","tag":"0DXQ","title":"Algebraization of formal sections, I · Theorem 0DXQ","summary":"The method of proof follows roughly the method of proof of [Faltings-algebraisation] and [Faltings-uber]. The result is almost the same as [MRaynaud-paper] (affine complement case) and [MRaynaud-book] (complement is union of few affines). Let (A, m) be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal I. Set X = Spec(A), Y = V(I), and U = X setminus ( m). Let F be a coherent sheaf on U. Assume • cd(A, I) ≤ d, i.e., H^i(X…","statement_latex":"\\begin{reference}\nThe method of proof follows roughly the method of\nproof of \\cite[Theorem 1]{Faltings-algebraisation}\nand \\cite[Satz 2]{Faltings-uber}.\nThe result is almost the same as\n\\cite[Theorem 1.1]{MRaynaud-paper} (affine complement case) and\n\\cite[Theorem 3.9]{MRaynaud-book} (complement is union of few affines).\n\\end{reference}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring which has a\ndualizing complex and is complete with respect to an ideal $I$.\nSet $X = \\Spec(A)$, $Y = V(I)$, and $U = X \\setminus \\{\\mathfrak m\\}$.\nLet $\\mathcal{F}$ be a coherent sheaf on $U$.\nAssume\n\\begin{enumerate}\n\\item $\\text{cd}(A, I) \\leq d$, i.e.,\n$H^i(X \\setminus Y, \\mathcal{G}) = 0$ for $i \\geq d$ and\nquasi-coherent $\\mathcal{G}$ on $X$,\n\\item for any $x \\in X \\setminus Y$ whose closure $\\overline{\\{x\\}}$\nin $X$ meets $U \\cap Y$ we have\n$$\n\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) \\geq s\n\\quad\\text{or}\\quad\n\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x)\n+ \\dim(\\overline{\\{x\\}}) > d + s\n$$\n\\end{enumerate}\nThen there exists an open $V_0 \\subset U$ containing $U \\cap Y$\nsuch that for any open $V \\subset V_0$ containing $U \\cap Y$\nthe map\n$$\nH^i(V, \\mathcal{F}) \\to \\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F})\n$$\nis an isomorphism for $i < s$. If in addition\n$\n\\text{depth}_{\\mathcal{O}_{X, x}}(\\mathcal{F}_x) +\n\\dim(\\overline{\\{x\\}}) > s\n$\nfor all $x \\in U \\cap Y$, then these cohomology groups are finite $A$-modules.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, I","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXQ","source_file":"algebraization.tex","source_line":3551,"source_end_line":3591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3551-L3591","statement_sha256":"313303138da08f471a0ebcb3a241875cc87bbf98dcd1312366b117dedd8c375c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9213,"rank":9213,"depth":43,"x":1072.638,"y":1606.005,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXR","tag":"0DXR","title":"Algebraization of formal sections, I · Lemma 0DXR","summary":"Let (A, m) be a Noetherian local ring which has a dualizing complex and is complete with respect to an ideal I. Set X = Spec(A), Y = V(I), and U = X setminus ( m). Let F be a coherent sheaf on U. Assume for any associated point x ∈ U of F we have dim(overline(x)) > cd(A, I) + 1 where overline(x) is the closure in X. Then the map colim H^0(V, F) → lim H^0(U, F/I^nF) is an isomorphism of finite A-modules where the colimit is over opens V ⊂ U containing U ∩ Y.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring which has a\ndualizing complex and is complete with respect to an ideal $I$.\nSet $X = \\Spec(A)$, $Y = V(I)$, and $U = X \\setminus \\{\\mathfrak m\\}$.\nLet $\\mathcal{F}$ be a coherent sheaf on $U$.\nAssume for any associated point $x \\in U$ of $\\mathcal{F}$\nwe have $\\dim(\\overline{\\{x\\}}) > \\text{cd}(A, I) + 1$\nwhere $\\overline{\\{x\\}}$ is the closure in $X$.\nThen the map\n$$\n\\colim H^0(V, \\mathcal{F})\n\\longrightarrow\n\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})\n$$\nis an isomorphism of finite $A$-modules\nwhere the colimit is over opens $V \\subset U$\ncontaining $U \\cap Y$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXR","source_file":"algebraization.tex","source_line":3631,"source_end_line":3649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3631-L3649","statement_sha256":"fe9d547658baaee95d332ca422143b9891f5d446bca329442fe22fcf178adb19","origin":"The Stacks Project","memory_eligible":false,"source_rank":9214,"rank":9214,"depth":44,"x":1167.83,"y":1562.667,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0H48","tag":"0H48","title":"Algebraization of formal sections, II · Lemma 0H48","summary":"Let I ⊂ a be ideals of a Noetherian ring A. Let 0 → F' → F → F\" → 0 be a short exact sequence of coherent modules on U = Spec(A) setminus V( a). Let V be the set of open subschemes V ⊂ U containing U ∩ V(I) ordered by reverse inclusion. Consider the commutative diagram xymatrix colim_V H^0(V, F') ar[d] ar[r] & colim_V H^0(V, F) ar[d] ar[r] & colim_V H^0(V, F\") ar[d] lim H^0(U, F'/I^nF') ar[r] & lim H^0(U, F'/I^nF) ar[r] & lim H^0(U, F'/I^nF\") If the left and right…","statement_latex":"Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.\nLet $0 \\to \\mathcal{F}' \\to \\mathcal{F} \\to \\mathcal{F}'' \\to 0$\nbe a short exact sequence of coherent modules on\n$U = \\Spec(A) \\setminus V(\\mathfrak a)$. Let $\\mathcal{V}$\nbe the set of open subschemes $V \\subset U$ containing\n$U \\cap V(I)$ ordered by reverse inclusion.\nConsider the commutative diagram\n$$\n\\xymatrix{\n\\colim_\\mathcal{V} H^0(V, \\mathcal{F}') \\ar[d] \\ar[r] &\n\\colim_\\mathcal{V} H^0(V, \\mathcal{F}) \\ar[d] \\ar[r] &\n\\colim_\\mathcal{V} H^0(V, \\mathcal{F}'') \\ar[d] \\\\\n\\lim H^0(U, \\mathcal{F}'/I^n\\mathcal{F}') \\ar[r] &\n\\lim H^0(U, \\mathcal{F}'/I^n\\mathcal{F}) \\ar[r] &\n\\lim H^0(U, \\mathcal{F}'/I^n\\mathcal{F}'')\n}\n$$\nIf the left and right downarrows are isomorphisms so is the middle.\nIf the middle and left downarrows are isomorphisms, so is the left.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H48","source_file":"algebraization.tex","source_line":3673,"source_end_line":3694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3673-L3694","statement_sha256":"ef3aaeef3ed1ee3876d21ceba6b905ebb717ea31b4db11c256abb37279f17ae9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9215,"rank":9215,"depth":30,"x":1132.147,"y":1649.458,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIF","tag":"0EIF","title":"Algebraization of formal sections, II · Lemma 0EIF","summary":"Let I ⊂ a be ideals of a Noetherian ring A. Let F be a coherent module on U = Spec(A) setminus V( a). Assume • A is I-adically complete and has a dualizing complex, • if x ∈ Ass(F), x not ∈ V(I), overline(x) ∩ V(I) not ⊂ V( a), and z ∈ overline(x) ∩ V( a), then dim(O_overline(x), z) > cd(A, I) + 1, • one of the following holds: • the restriction of F to U setminus V(I) is (S_1) • the dimension of V( a) is at most 2. Then we obtain an isomorphism colim H^0(V, F) → lim…","statement_latex":"Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.\nLet $\\mathcal{F}$ be a coherent module on\n$U = \\Spec(A) \\setminus V(\\mathfrak a)$.\nAssume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$,\n$\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$,\nand $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then\n$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$,\n\\item one of the following holds:\n\\begin{enumerate}\n\\item the restriction of $\\mathcal{F}$ to $U \\setminus V(I)$ is $(S_1)$\n\\item the dimension of $V(\\mathfrak a)$ is at most $2$\\footnote{In\nthe sense that the difference of the maximal and minimal values\non $V(\\mathfrak a)$ of a dimension function on $\\Spec(A)$ is at most $2$.}.\n\\end{enumerate}\n\\end{enumerate}\nThen we obtain an isomorphism\n$$\n\\colim H^0(V, \\mathcal{F})\n\\longrightarrow\n\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})\n$$\nwhere the colimit is over opens $V \\subset U$ containing $U \\cap V(I)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIF","source_file":"algebraization.tex","source_line":3768,"source_end_line":3795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3768-L3795","statement_sha256":"da0f78a8d8eeaec4ec1b00e857fe0d7d405a91583326e34046b56efa12f0ad7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9216,"rank":9216,"depth":30,"x":1088.26,"y":1564.422,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EG2","tag":"0EG2","title":"Algebraization of formal sections, II · Proposition 0EG2","summary":"Let I ⊂ a be ideals of a Noetherian ring A. Let F be a coherent module on U = Spec(A) setminus V( a). Assume • A is I-adically complete and has a dualizing complex, • if x ∈ Ass(F), x not ∈ V(I), overline(x) ∩ V(I) not ⊂ V( a), and z ∈ overline(x) ∩ V( a), then dim(O_overline(x), z) > cd(A, I) + 1. Then we obtain an isomorphism colim H^0(V, F) → lim H^0(U, F/I^nF) where the colimit is over opens V ⊂ U containing U ∩ V(I).","statement_latex":"Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.\nLet $\\mathcal{F}$ be a coherent module on\n$U = \\Spec(A) \\setminus V(\\mathfrak a)$.\nAssume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$,\n$\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$,\nand $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then\n$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$.\n\\end{enumerate}\nThen we obtain an isomorphism\n$$\n\\colim H^0(V, \\mathcal{F})\n\\longrightarrow\n\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})\n$$\nwhere the colimit is over opens $V \\subset U$ containing $U \\cap V(I)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EG2","source_file":"algebraization.tex","source_line":3883,"source_end_line":3903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3883-L3903","statement_sha256":"58497f60b5d9ebf20fb0438a74141572b79ca9158234786fa55c7c96c9357b6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9217,"rank":9217,"depth":31,"x":1189.93,"y":1602.574,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIG","tag":"0EIG","title":"Algebraization of formal sections, II · Lemma 0EIG","summary":"Let I ⊂ a be ideals of a Noetherian ring A. Let F be a coherent module on U = Spec(A) setminus V( a). Assume • A is I-adically complete and has a dualizing complex, • if x ∈ Ass(F), x not ∈ V(I), overline(x) ∩ V(I) not ⊂ V( a), and z ∈ V( a) ∩ overline(x), then dim(O_overline(x), z) > cd(A, I) + 1, • for x ∈ U with overline(x) ∩ V(I) ⊂ V( a) we have depth(F_x) ≥ 2, Then we obtain an isomorphism H^0(U, F) → lim H^0(U, F/I^nF)","statement_latex":"Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.\nLet $\\mathcal{F}$ be a coherent module on\n$U = \\Spec(A) \\setminus V(\\mathfrak a)$.\nAssume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$,\n$\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$, and\n$z \\in V(\\mathfrak a) \\cap \\overline{\\{x\\}}$, then\n$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$,\n\\item for $x \\in U$ with $\\overline{\\{x\\}} \\cap V(I) \\subset V(\\mathfrak a)$\nwe have $\\text{depth}(\\mathcal{F}_x) \\geq 2$,\n\\end{enumerate}\nThen we obtain an isomorphism\n$$\nH^0(U, \\mathcal{F})\n\\longrightarrow\n\\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})\n$$","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIG","source_file":"algebraization.tex","source_line":3974,"source_end_line":3995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L3974-L3995","statement_sha256":"b0f3d92128a7418871a9db65419f5ad0eb7ee037570fd02162a5141faf3c3ca2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9218,"rank":9218,"depth":32,"x":1083.324,"y":1632.395,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIH","tag":"0EIH","title":"Algebraization of formal sections, II · Lemma 0EIH","summary":"Let A be a Noetherian ring. Let f ∈ a ⊂ A be an element of an ideal of A. Let M be a finite A-module. Assume • A is f-adically complete, • f is a nonzerodivisor on M, • H^1_ a(M/fM) is a finite A-module. Then with U = Spec(A) setminus V( a) the map colim_V Γ(V, widetildeM) → lim Γ(U, widetildeM/f^nM) is an isomorphism where the colimit is over opens V ⊂ U containing U ∩ V(f).","statement_latex":"Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$\nbe an element of an ideal of $A$. Let $M$ be a finite $A$-module.\nAssume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor on $M$,\n\\item $H^1_\\mathfrak a(M/fM)$ is a finite $A$-module.\n\\end{enumerate}\nThen with $U = \\Spec(A) \\setminus V(\\mathfrak a)$ the map\n$$\n\\colim_V \\Gamma(V, \\widetilde{M})\n\\longrightarrow\n\\lim \\Gamma(U, \\widetilde{M/f^nM})\n$$\nis an isomorphism where the colimit is over opens $V \\subset U$\ncontaining $U \\cap V(f)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIH","source_file":"algebraization.tex","source_line":4009,"source_end_line":4027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4009-L4027","statement_sha256":"1c77049e7e7299eb1d0d497d7841ab037210daf918aab70f3ca568c1c9b0c3fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9219,"rank":9219,"depth":30,"x":1138.44,"y":1549.185,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0H49","tag":"0H49","title":"Algebraization of formal sections, II · Proposition 0H49","summary":"Let A be a Noetherian ring. Let f ∈ a ⊂ A be an element of an ideal of A. Let F be a coherent module on U = Spec(A) setminus V( a). Assume • A is f-adically complete and has a dualizing complex, • if x ∈ Ass(F), x not ∈ V(f), overline(x) ∩ V(f) not ⊂ V( a), and z ∈ overline(x) ∩ V( a), then dim(O_overline(x), z) > 2. Then the map colim_V Γ(V, F) → lim Γ(U, F/f^nF) is an isomorphism where the colimit is over opens V ⊂ U containing U ∩ V(f).","statement_latex":"Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$\nbe an element of an ideal of $A$. Let $\\mathcal{F}$ be a coherent\nmodule on $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete and has a dualizing complex,\n\\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(f)$,\n$\\overline{\\{x\\}} \\cap V(f) \\not \\subset V(\\mathfrak a)$,\nand $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then\n$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > 2$.\n\\end{enumerate}\nThen the map\n$$\n\\colim_V \\Gamma(V, \\mathcal{F})\n\\longrightarrow\n\\lim \\Gamma(U, \\mathcal{F}/f^n\\mathcal{F})\n$$\nis an isomorphism where the colimit is over opens $V \\subset U$\ncontaining $U \\cap V(f)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H49","source_file":"algebraization.tex","source_line":4060,"source_end_line":4080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4060-L4080","statement_sha256":"07b741cf7eacf6dead63ed238b873a9508b4365f832879e5188da524ff3e89a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9220,"rank":9220,"depth":53,"x":1164.94,"y":1642.624,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EII","tag":"0EII","title":"Algebraization of formal sections, II · Lemma 0EII","summary":"Let A be a Noetherian ring. Let f ∈ a ⊂ A be an element of an ideal of A. Let M be a finite A-module. Assume • A is f-adically complete, • H^1_ a(M) and H^2_ a(M) are annihilated by a power of f. Then with U = Spec(A) setminus V( a) the map Γ(U, widetildeM) → lim Γ(U, widetildeM/f^nM) is an isomorphism.","statement_latex":"Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$\nbe an element of an ideal of $A$. Let $M$ be a finite $A$-module.\nAssume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $H^1_\\mathfrak a(M)$ and $H^2_\\mathfrak a(M)$ are\nannihilated by a power of $f$.\n\\end{enumerate}\nThen with $U = \\Spec(A) \\setminus V(\\mathfrak a)$ the map\n$$\n\\Gamma(U, \\widetilde{M})\n\\longrightarrow\n\\lim \\Gamma(U, \\widetilde{M/f^nM})\n$$\nis an isomorphism.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EII","source_file":"algebraization.tex","source_line":4143,"source_end_line":4160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4143-L4160","statement_sha256":"21816f9b126ff58faa1bfb979e8f272a2aafc66dc084aa2d1dcf8fee40d69639","origin":"The Stacks Project","memory_eligible":false,"source_rank":9221,"rank":9221,"depth":30,"x":1069.449,"y":1588.297,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EG4","tag":"0EG4","title":"Algebraization of formal sections, III · Proposition 0EG4","summary":"Let I ⊂ a be ideals of a Noetherian ring A. Let F be a coherent module on U = Spec(A) setminus V( a). Let s ≥ 0. Assume • A is I-adically complete and has a dualizing complex, • if x ∈ U setminus V(I) then depth(F_x) > s or depth(F_x) + dim(O_overline(x), z) > cd(A, I) + s + 1 for all z ∈ V( a) ∩ overline(x), • one of the following conditions holds: • the restriction of F to U setminus V(I) is (S_s + 1), or • the dimension of V( a) is at most 2. Then the maps H^i(U, F) →…","statement_latex":"Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.\nLet $\\mathcal{F}$ be a coherent module on\n$U = \\Spec(A) \\setminus V(\\mathfrak a)$.\nLet $s \\geq 0$.\nAssume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item if $x \\in U \\setminus V(I)$ then\n$\\text{depth}(\\mathcal{F}_x) > s$ or\n$$\n\\text{depth}(\\mathcal{F}_x) +\n\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + s + 1\n$$\nfor all $z \\in V(\\mathfrak a) \\cap \\overline{\\{x\\}}$,\n\\item one of the following conditions holds:\n\\begin{enumerate}\n\\item the restriction of $\\mathcal{F}$ to $U \\setminus V(I)$\nis $(S_{s + 1})$, or\n\\item the dimension of $V(\\mathfrak a)$ is at most $2$\\footnote{In\nthe sense that the difference of the maximal and minimal values\non $V(\\mathfrak a)$ of a dimension function on $\\Spec(A)$ is at most $2$.}.\n\\end{enumerate}\n\\end{enumerate}\nThen the maps\n$$\nH^i(U, \\mathcal{F})\n\\longrightarrow\n\\lim H^i(U, \\mathcal{F}/I^n\\mathcal{F})\n$$\nare isomorphisms for $i < s$. Moreover we have an isomorphism\n$$\n\\colim H^s(V, \\mathcal{F})\n\\longrightarrow\n\\lim H^s(U, \\mathcal{F}/I^n\\mathcal{F})\n$$\nwhere the colimit is over opens $V \\subset U$ containing $U \\cap V(I)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, III","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EG4","source_file":"algebraization.tex","source_line":4202,"source_end_line":4240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4202-L4240","statement_sha256":"623f4ec29938724edacff3227d39ba09291dfea1901cc5b54a26e817327fc3ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":9222,"rank":9222,"depth":49,"x":1184.512,"y":1574.057,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKM","tag":"0EKM","title":"Algebraization of formal sections, III · Lemma 0EKM","summary":"Let A be a Noetherian ring. Let f ∈ a ⊂ A be an element of an ideal of A. Let M be a finite A-module. Let s ≥ 0. Assume • A is f-adically complete, • H^i_ a(M) is annihilated by a power of f for i ≤ s + 1. Then with U = Spec(A) setminus V( a) the map H^i(U, widetildeM) → lim H^i(U, widetildeM/f^nM) is an isomorphism for i < s.","statement_latex":"Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$\nbe an element of an ideal of $A$. Let $M$ be a finite $A$-module.\nLet $s \\geq 0$. Assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $H^i_\\mathfrak a(M)$ is annihilated by a power of $f$\nfor $i \\leq s + 1$.\n\\end{enumerate}\nThen with $U = \\Spec(A) \\setminus V(\\mathfrak a)$ the map\n$$\nH^i(U, \\widetilde{M})\n\\longrightarrow\n\\lim H^i(U, \\widetilde{M/f^nM})\n$$\nis an isomorphism for $i < s$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of formal sections, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKM","source_file":"algebraization.tex","source_line":4311,"source_end_line":4328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4311-L4328","statement_sha256":"f4dc56be9b34a7c8dc3725625db7552e7c80c4a1abc9034bd70f84ba5e65be3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9223,"rank":9223,"depth":31,"x":1110.507,"y":1650.473,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ECR","tag":"0ECR","title":"Application to connectedness · Lemma 0ECR","summary":"[Varbaro] Let (A, m) be a Noetherian complete local ring. Let I be a proper ideal of A. Set X = Spec(A) and Y = V(I). Denote • d the minimal dimension of an irreducible component of X, and • c the minimal dimension of a closed subset Z ⊂ X such that X setminus Z is disconnected. Then for Z ⊂ Y closed we have Y setminus Z is connected if dim(Z) < min(c, d - 1) - cd(A, I). In particular, the punctured spectrum of A/I is connected if cd(A, I) < min(c, d - 1).","statement_latex":"\\begin{reference}\n\\cite[Theorem 1.6]{Varbaro}\n\\end{reference}\nLet $(A, \\mathfrak m)$ be a Noetherian complete local ring.\nLet $I$ be a proper ideal of $A$.\nSet $X = \\Spec(A)$ and $Y = V(I)$.\nDenote\n\\begin{enumerate}\n\\item $d$ the minimal dimension of an irreducible component of $X$, and\n\\item $c$ the minimal dimension of a closed subset $Z \\subset X$\nsuch that $X \\setminus Z$ is disconnected.\n\\end{enumerate}\nThen for $Z \\subset Y$ closed we have $Y \\setminus Z$ is connected if\n$\\dim(Z) < \\min(c, d - 1) - \\text{cd}(A, I)$. In particular, the punctured\nspectrum of $A/I$ is connected if $\\text{cd}(A, I) < \\min(c, d - 1)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECR","source_file":"algebraization.tex","source_line":4368,"source_end_line":4385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4368-L4385","statement_sha256":"f82b736474edf07c5b3981623dafb880ab4d62f341f943ef281a93abc865def9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9224,"rank":9224,"depth":45,"x":1103.573,"y":1551.332,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EG5","tag":"0EG5","title":"Application to connectedness · Lemma 0EG5","summary":"Let I ⊂ a be ideals of a Noetherian ring A. Assume • A is I-adically complete and has a dualizing complex, • if p ⊂ A is a minimal prime not contained in V(I) and q ∈ V( p) ∩ V( a), then dim((A/ p)_ q) > cd(A, I) + 1, • any nonempty open V ⊂ Spec(A) which contains V(I) setminus V( a) is connected. Then V(I) setminus V( a) is either empty or connected.","statement_latex":"Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.\nAssume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item if $\\mathfrak p \\subset A$ is a minimal prime not contained\nin $V(I)$ and $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$, then\n$\\dim((A/\\mathfrak p)_\\mathfrak q) > \\text{cd}(A, I) + 1$,\n\\item any nonempty open $V \\subset \\Spec(A)$ which contains\n$V(I) \\setminus V(\\mathfrak a)$ is connected\\footnote{For example\nif $A$ is a domain.}.\n\\end{enumerate}\nThen $V(I) \\setminus V(\\mathfrak a)$ is either empty or connected.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EG5","source_file":"algebraization.tex","source_line":4447,"source_end_line":4461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4447-L4461","statement_sha256":"d326c04eee4b50470220fecd70b497adcced12947404a4929749a40a1cead656","origin":"The Stacks Project","memory_eligible":false,"source_rank":9225,"rank":9225,"depth":32,"x":1189.09,"y":1621.058,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EG3","tag":"0EG3","title":"Application to connectedness · Lemma 0EG3","summary":"Let A be a Noetherian domain which has a dualizing complex and which is complete with respect to a nonzero f ∈ A. Let f ∈ a ⊂ A be an ideal. Assume every irreducible component of Z = V( a) has codimension > 2 in X = Spec(A), i.e., assume every irreducible component of Z has codimension > 1 in Y = V(f). Then Y setminus Z is connected.","statement_latex":"Let $A$ be a Noetherian domain which has a dualizing complex\nand which is complete with respect to a nonzero $f \\in A$.\nLet $f \\in \\mathfrak a \\subset A$ be an ideal.\nAssume every irreducible component of $Z = V(\\mathfrak a)$\nhas codimension $> 2$ in $X = \\Spec(A)$, i.e., assume every\nirreducible component of $Z$ has codimension $> 1$ in $Y = V(f)$.\nThen $Y \\setminus Z$ is connected.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EG3","source_file":"algebraization.tex","source_line":4478,"source_end_line":4487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4478-L4487","statement_sha256":"64e2207244db9cddbe6b29c4f5066849d287c756147f6382a99cc1527c4f10fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":9226,"rank":9226,"depth":54,"x":1069.02,"y":1618.142,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKP","tag":"0EKP","title":"The completion functor · Lemma 0EKP","summary":"Let X be a Noetherian scheme and let Y ⊂ X be a closed subscheme. Let Y_n ⊂ X be the nth infinitesimal neighbourhood of Y in X. Consider the following conditions • X is quasi-affine and Γ(X, O_X) → lim Γ(Y_n, O_Y_n) is an isomorphism, • X has an ample invertible module L and Γ(X, L^⊗ m) → lim Γ(Y_n, L^⊗ m|_Y_n) is an isomorphism for all m gg 0, • for every finite locally free O_X-module E the map Γ(X, E) → lim Γ(Y_n, E|_Y_n) is an isomorphism, and • the completion functor…","statement_latex":"Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme.\nLet $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$.\nConsider the following conditions\n\\begin{enumerate}\n\\item $X$ is quasi-affine and\n$\\Gamma(X, \\mathcal{O}_X) \\to \\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})$\nis an isomorphism,\n\\item $X$ has an ample invertible module $\\mathcal{L}$ and\n$\\Gamma(X, \\mathcal{L}^{\\otimes m}) \\to\n\\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})$\nis an isomorphism for all $m \\gg 0$,\n\\item for every finite locally free $\\mathcal{O}_X$-module\n$\\mathcal{E}$ the map\n$\\Gamma(X, \\mathcal{E}) \\to \\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})$\nis an isomorphism, and\n\\item the completion functor\n$\\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(X, \\mathcal{I})$\nis fully faithful on the full subcategory of finite locally free\nobjects.\n\\end{enumerate}\nThen (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4)\nand (4) $\\Rightarrow$ (3).","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKP","source_file":"algebraization.tex","source_line":4541,"source_end_line":4565,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4541-L4565","statement_sha256":"f998934fec226dd9e41f9eccb1a7079e322ca23195925f5979c8cf9920de8ed3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9227,"rank":9227,"depth":21,"x":1160.61,"y":1551.651,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EK2","tag":"0EK2","title":"The completion functor · Lemma 0EK2","summary":"Let X be a Noetherian scheme and let Y ⊂ X be a closed subscheme with ideal sheaf I ⊂ O_X. Let Y_n ⊂ X be the nth infinitesimal neighbourhood of Y in X. Let V be the set of open subschemes V ⊂ X containing Y ordered by reverse inclusion. • X is quasi-affine and colim_V Γ(V, O_V) → lim Γ(Y_n, O_Y_n) is an isomorphism, • X has an ample invertible module L and colim_V Γ(V, L^⊗ m) → lim Γ(Y_n, L^⊗ m|_Y_n) is an isomorphism for all m gg 0, • for every V ∈ V and every finite…","statement_latex":"Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme\nwith ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.\nLet $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$.\nLet $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$\nordered by reverse inclusion.\n\\begin{enumerate}\n\\item $X$ is quasi-affine and\n$$\n\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{O}_V)\n\\longrightarrow\n\\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})\n$$\nis an isomorphism,\n\\item $X$ has an ample invertible module $\\mathcal{L}$ and\n$$\n\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{L}^{\\otimes m})\n\\longrightarrow\n\\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})\n$$\nis an isomorphism for all $m \\gg 0$,\n\\item for every $V \\in \\mathcal{V}$ and every finite locally free\n$\\mathcal{O}_V$-module $\\mathcal{E}$ the map\n$$\n\\colim_{V' \\geq V} \\Gamma(V', \\mathcal{E}|_{V'})\n\\longrightarrow\n\\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})\n$$\nis an isomorphism, and\n\\item the completion functor\n$$\n\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)\n\\longrightarrow\n\\textit{Coh}(X, \\mathcal{I}),\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\nis fully faithful on the full subcategory of\nfinite locally free objects (see explanation above).\n\\end{enumerate}\nThen (1) $\\Rightarrow$ (2) $\\Rightarrow$ (3) $\\Rightarrow$ (4)\nand (4) $\\Rightarrow$ (3).","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EK2","source_file":"algebraization.tex","source_line":4623,"source_end_line":4666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4623-L4666","statement_sha256":"8a1a10b974f1f562040052ca458a9101ff0653699d089918eb8e39cf81bae9b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9228,"rank":9228,"depth":22,"x":1146.433,"y":1653.422,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIQ","tag":"0EIQ","title":"The completion functor · Lemma 0EIQ","summary":"Let X be a Noetherian scheme. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. The functor Coh(X, I) → Pro-QCoh(O_X) is fully faithful, see Categories, Remark [Tag 05PX].","statement_latex":"Let $X$ be a Noetherian scheme. Let $\\mathcal{I} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. The functor\n$$\n\\textit{Coh}(X, \\mathcal{I}) \\longrightarrow \\text{Pro-}\\QCoh(\\mathcal{O}_X)\n$$\nis fully faithful, see Categories, Remark \\ref{categories-remark-pro-category}.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIQ","source_file":"algebraization.tex","source_line":4720,"source_end_line":4728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4720-L4728","statement_sha256":"51bcba3a6c94967c5ab674f7eb1fc8b8b317d8151eac670d8a73773bbefd7c63","origin":"The Stacks Project","memory_eligible":false,"source_rank":9229,"rank":9229,"depth":0,"x":1074.51,"y":1569.707,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKQ","tag":"0EKQ","title":"The completion functor · Lemma 0EKQ","summary":"Let I ⊂ a be ideals of a Noetherian ring A. Let U = Spec(A) setminus V( a). Assume • A is I-adically complete and has a dualizing complex, • for any associated prime p ⊂ A with p not ∈ V(I) and V( p) ∩ V(I) not ⊂ V( a) and q ∈ V( p) ∩ V( a) we have dim((A/ p)_ q) > cd(A, I) + 1, • for p ⊂ A with p not ∈ V(I) and V( p) ∩ V(I) ⊂ V( a) we have depth(A_ p) ≥ 2. Then the completion functor Coh(O_U) → Coh(U, IO_U), F ↦ F^wedge is fully faithful on the full subcategory of finite…","statement_latex":"Let $I \\subset \\mathfrak a$ be ideals of a Noetherian ring $A$.\nLet $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item for any associated prime $\\mathfrak p \\subset A$ with\n$\\mathfrak p \\not \\in V(I)$ and\n$V(\\mathfrak p) \\cap V(I) \\not \\subset V(\\mathfrak a)$ and\n$\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have\n$\\dim((A/\\mathfrak p)_\\mathfrak q) > \\text{cd}(A, I) + 1$,\n\\item for $\\mathfrak p \\subset A$ with $\\mathfrak p \\not \\in V(I)$\nand $V(\\mathfrak p) \\cap V(I) \\subset V(\\mathfrak a)$\nwe have $\\text{depth}(A_\\mathfrak p) \\geq 2$.\n\\end{enumerate}\nThen the completion functor\n$$\n\\textit{Coh}(\\mathcal{O}_U)\n\\longrightarrow\n\\textit{Coh}(U, I\\mathcal{O}_U),\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\nis fully faithful on the full subcategory of\nfinite locally free objects.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKQ","source_file":"algebraization.tex","source_line":4747,"source_end_line":4772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4747-L4772","statement_sha256":"fc5cf540f666fb7d60159781b5eb1c0b1afd5ed9702c136ceb41274e895ff53c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9230,"rank":9230,"depth":33,"x":1195.758,"y":1590.784,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKS","tag":"0EKS","title":"The completion functor · Lemma 0EKS","summary":"Let A be a Noetherian ring. Let f ∈ a ⊂ A be an element of an ideal of A. Let U = Spec(A) setminus V( a). Assume • A is f-adically complete, • H^1_ a(A) and H^2_ a(A) are annihilated by a power of f. Then the completion functor Coh(O_U) → Coh(U, IO_U), F ↦ F^wedge is fully faithful on the full subcategory of finite locally free objects.","statement_latex":"Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$\nbe an element of an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$.\nAssume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $H^1_\\mathfrak a(A)$ and $H^2_\\mathfrak a(A)$ are\nannihilated by a power of $f$.\n\\end{enumerate}\nThen the completion functor\n$$\n\\textit{Coh}(\\mathcal{O}_U)\n\\longrightarrow\n\\textit{Coh}(U, I\\mathcal{O}_U),\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\nis fully faithful on the full subcategory of\nfinite locally free objects.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKS","source_file":"algebraization.tex","source_line":4785,"source_end_line":4805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4785-L4805","statement_sha256":"843625715097f1da933bdd0d5af37797efc9a191110e71db73bde34920d7ff39","origin":"The Stacks Project","memory_eligible":false,"source_rank":9231,"rank":9231,"depth":31,"x":1088.624,"y":1644.428,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKT","tag":"0EKT","title":"The completion functor · Lemma 0EKT","summary":"Let A be a Noetherian ring. Let f ∈ a be an element of an ideal of A. Let U = Spec(A) setminus V( a). Assume • A has a dualizing complex and is complete with respect to f, • for every prime p ⊂ A, f not ∈ p and q ∈ V( p) ∩ V( a) we have depth(A_ p) + dim((A/ p)_ q) > 2. Then the completion functor Coh(O_U) → Coh(U, IO_U), F ↦ F^wedge is fully faithful on the full subcategory of finite locally free objects.","statement_latex":"Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a$ be an element of\nan ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and is complete with respect to $f$,\n\\item for every prime $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$\nand $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have\n$\\text{depth}(A_\\mathfrak p) + \\dim((A/\\mathfrak p)_\\mathfrak q) > 2$.\n\\end{enumerate}\nThen the completion functor\n$$\n\\textit{Coh}(\\mathcal{O}_U)\n\\longrightarrow\n\\textit{Coh}(U, I\\mathcal{O}_U),\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\nis fully faithful on the full subcategory of finite locally free objects.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKT","source_file":"algebraization.tex","source_line":4818,"source_end_line":4837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4818-L4837","statement_sha256":"a541ed29fe3869b11d942eff00ccecce16efdce6b1f043b7dfc4c1e66f7134f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9232,"rank":9232,"depth":35,"x":1124.745,"y":1543.36,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKU","tag":"0EKU","title":"The completion functor · Lemma 0EKU","summary":"Let I ⊂ a ⊂ A be ideals of a Noetherian ring A. Let U = Spec(A) setminus V( a). Let V be the set of open subschemes of U containing U ∩ V(I) ordered by reverse inclusion. Assume • A is I-adically complete and has a dualizing complex, • for any associated prime p ⊂ A with I not ⊂ p and V( p) ∩ V(I) not ⊂ V( a) and q ∈ V( p) ∩ V( a) we have dim((A/ p)_ q) > cd(A, I) + 1. Then the completion functor colim_V Coh(O_V) → Coh(U, IO_U), F ↦ F^wedge is fully faithful on the full…","statement_latex":"Let $I \\subset \\mathfrak a \\subset A$ be ideals of a Noetherian ring $A$.\nLet $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Let $\\mathcal{V}$ be\nthe set of open subschemes of $U$ containing $U \\cap V(I)$\nordered by reverse inclusion. Assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item for any associated prime\n$\\mathfrak p \\subset A$ with\n$I \\not \\subset \\mathfrak p$ and\n$V(\\mathfrak p) \\cap V(I) \\not \\subset V(\\mathfrak a)$\nand $\\mathfrak q \\in V(\\mathfrak p) \\cap V(\\mathfrak a)$ we have\n$\\dim((A/\\mathfrak p)_\\mathfrak q) > \\text{cd}(A, I) + 1$.\n\\end{enumerate}\nThen the completion functor\n$$\n\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)\n\\longrightarrow\n\\textit{Coh}(U, I\\mathcal{O}_U),\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\nis fully faithful on the full subcategory of\nfinite locally free objects.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKU","source_file":"algebraization.tex","source_line":4844,"source_end_line":4869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4844-L4869","statement_sha256":"f9054a9a345d2f0d26d5cd403bda5dcd5b17fc8faa14752a725e6a594a44f1a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9233,"rank":9233,"depth":32,"x":1179.772,"y":1639.057,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKV","tag":"0EKV","title":"The completion functor · Lemma 0EKV","summary":"Let A be a Noetherian ring. Let f ∈ a ⊂ A be an element of an ideal of A. Let U = Spec(A) setminus V( a). Let V be the set of open subschemes of U containing U ∩ V(f) ordered by reverse inclusion. Assume • A is f-adically complete, • f is a nonzerodivisor, • H^1_ a(A/fA) is a finite A-module. Then the completion functor colim_V Coh(O_V) → Coh(U, fO_U), F ↦ F^wedge is fully faithful on the full subcategory of finite locally free objects.","statement_latex":"Let $A$ be a Noetherian ring. Let $f \\in \\mathfrak a \\subset A$\nbe an element of an ideal of $A$. Let $U = \\Spec(A) \\setminus V(\\mathfrak a)$.\nLet $\\mathcal{V}$ be the set of open subschemes of $U$ containing $U \\cap V(f)$\nordered by reverse inclusion. Assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak a(A/fA)$ is a finite $A$-module.\n\\end{enumerate}\nThen the completion functor\n$$\n\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)\n\\longrightarrow\n\\textit{Coh}(U, f\\mathcal{O}_U),\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\nis fully faithful on the full subcategory of finite locally free objects.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKV","source_file":"algebraization.tex","source_line":4881,"source_end_line":4901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4881-L4901","statement_sha256":"a19f8e1402f85f38276e076644b4b9de2d0629acf868512197c46244955ef232","origin":"The Stacks Project","memory_eligible":false,"source_rank":9234,"rank":9234,"depth":31,"x":1061.417,"y":1599.438,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIV","tag":"0EIV","title":"The completion functor · Lemma 0EIV","summary":"Let I ⊂ a ⊂ A be ideals of a Noetherian ring A. Let U = Spec(A) setminus V( a). Let V be the set of open subschemes of U containing U ∩ V(I) ordered by reverse inclusion. Let F and G be coherent O_V-modules for some V ∈ V. The map colim_V' ≥ V Hom_V(G|_V', F|_V') → Hom_Coh(U, IO_U)(G^wedge, F^wedge) is bijective if the following assumptions hold: • A is I-adically complete and has a dualizing complex, • if x ∈ Ass(F), x not ∈ V(I), overline(x) ∩ V(I) not ⊂ V( a) and z ∈…","statement_latex":"Let $I \\subset \\mathfrak a \\subset A$ be ideals of a Noetherian ring $A$.\nLet $U = \\Spec(A) \\setminus V(\\mathfrak a)$. Let $\\mathcal{V}$ be the set\nof open subschemes of $U$ containing $U \\cap V(I)$ ordered by reverse\ninclusion. Let $\\mathcal{F}$ and\n$\\mathcal{G}$ be coherent $\\mathcal{O}_V$-modules for some\n$V \\in \\mathcal{V}$. The map\n$$\n\\colim_{V' \\geq V} \\Hom_V(\\mathcal{G}|_{V'}, \\mathcal{F}|_{V'})\n\\longrightarrow\n\\Hom_{\\textit{Coh}(U, I\\mathcal{O}_U)}(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge)\n$$\nis bijective if the following assumptions hold:\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete and has a dualizing complex,\n\\item if $x \\in \\text{Ass}(\\mathcal{F})$, $x \\not \\in V(I)$,\n$\\overline{\\{x\\}} \\cap V(I) \\not \\subset V(\\mathfrak a)$\nand $z \\in \\overline{\\{x\\}} \\cap V(\\mathfrak a)$, then\n$\\dim(\\mathcal{O}_{\\overline{\\{x\\}}, z}) > \\text{cd}(A, I) + 1$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIV","source_file":"algebraization.tex","source_line":4913,"source_end_line":4934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4913-L4934","statement_sha256":"52d4af129558881f16676f6e7429a525628f20a86c8ac247c388940064aac551","origin":"The Stacks Project","memory_eligible":false,"source_rank":9235,"rank":9235,"depth":32,"x":1181.374,"y":1561.235,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXT","tag":"0DXT","title":"Algebraization of coherent formal modules, I · Lemma 0DXT","summary":"In Situation [Tag 0EHC]. Consider an inverse system (M_n) of A-modules such that • M_n is a finite A-module, • M_n is annihilated by I^n, • the kernel and cokernel of M_n + 1/I^nM_n + 1 → M_n are a-power torsion. Then (widetildeM_n|_U) is in Coh(U, IO_U). Conversely, every object of Coh(U, IO_U) arises in this manner.","statement_latex":"In Situation \\ref{situation-algebraize}.\nConsider an inverse system $(M_n)$ of $A$-modules such\nthat\n\\begin{enumerate}\n\\item $M_n$ is a finite $A$-module,\n\\item $M_n$ is annihilated by $I^n$,\n\\item the kernel and cokernel of $M_{n + 1}/I^nM_{n + 1} \\to M_n$\nare $\\mathfrak a$-power torsion.\n\\end{enumerate}\nThen $(\\widetilde{M}_n|_U)$ is in $\\textit{Coh}(U, I\\mathcal{O}_U)$.\nConversely, every object of $\\textit{Coh}(U, I\\mathcal{O}_U)$\narises in this manner.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXT","source_file":"algebraization.tex","source_line":4996,"source_end_line":5010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L4996-L5010","statement_sha256":"f3e9b644630e920b838c69716e7a68ae73163fd25951dcb1cbf5c86812c0dc32","origin":"The Stacks Project","memory_eligible":false,"source_rank":9236,"rank":9236,"depth":30,"x":1123.246,"y":1658.125,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIL","tag":"0EIL","title":"Algebraization of coherent formal modules, I · Lemma 0EIL","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Consider the following conditions: • (F_n) is in the essential image of the functor ([Tag 0EIK]), • (F_n) is the completion of a coherent O_U-module, • (F_n) is the completion of a coherent O_V-module for U ∩ Y ⊂ V ⊂ U open, • (F_n) is the completion of the restriction to U of a coherent O_X-module, • (F_n) is the restriction to U of the completion of a coherent O_X-module, • there exists an object (G_n) of…","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$\nbe an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Consider the\nfollowing conditions:\n\\begin{enumerate}\n\\item $(\\mathcal{F}_n)$ is in the essential image\nof the functor (\\ref{equation-completion}),\n\\item $(\\mathcal{F}_n)$ is the completion of a\ncoherent $\\mathcal{O}_U$-module,\n\\item $(\\mathcal{F}_n)$ is the completion of a coherent\n$\\mathcal{O}_V$-module for $U \\cap Y \\subset V \\subset U$ open,\n\\item $(\\mathcal{F}_n)$ is the completion of\nthe restriction to $U$ of a coherent $\\mathcal{O}_X$-module,\n\\item $(\\mathcal{F}_n)$ is the restriction to $U$ of\nthe completion of a coherent $\\mathcal{O}_X$-module,\n\\item there exists an object $(\\mathcal{G}_n)$ of\n$\\textit{Coh}(X, I\\mathcal{O}_X)$ whose restriction\nto $U$ is $(\\mathcal{F}_n)$.\n\\end{enumerate}\nThen conditions (1), (2), (3), (4), and (5) are equivalent and imply (6).\nIf $A$ is $I$-adically complete then condition (6) implies the others.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIL","source_file":"algebraization.tex","source_line":5053,"source_end_line":5075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5053-L5075","statement_sha256":"89b4cb2d4a3e57a42778548cadff2353298b11c35979c1d84047cd45d0fa263f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9237,"rank":9237,"depth":18,"x":1087.959,"y":1552.996,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIM","tag":"0EIM","title":"Algebraization of coherent formal modules, I · Definition 0EIM","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). We say (F_n) extends to X if there exists an object (G_n) of Coh(X, IO_X) whose restriction to U is isomorphic to (F_n).","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an\nobject of $\\textit{Coh}(U, I\\mathcal{O}_U)$. We say\n{\\it $(\\mathcal{F}_n)$ extends to $X$} if there exists an object\n$(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I\\mathcal{O}_X)$ whose restriction\nto $U$ is isomorphic to $(\\mathcal{F}_n)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, I","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIM","source_file":"algebraization.tex","source_line":5158,"source_end_line":5165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5158-L5165","statement_sha256":"37875b38c97b49f905714cdc2acf807095de312df2f6fdfb285d2a188b478d08","origin":"The Stacks Project","memory_eligible":false,"source_rank":9238,"rank":9238,"depth":0,"x":1199.253,"y":1610.876,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIN","tag":"0EIN","title":"Algebraization of coherent formal modules, I · Lemma 0EIN","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Let A', I', a' be the I-adic completions of A, I, a. Set X' = Spec(A') and U' = X' setminus V( a'). The following are equivalent • (F_n) extends to X, and • the pullback of (F_n) to U' is the completion of a coherent O_U'-module.","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an\nobject of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $A', I', \\mathfrak a'$\nbe the $I$-adic completions of $A, I, \\mathfrak a$. Set $X' = \\Spec(A')$\nand $U' = X' \\setminus V(\\mathfrak a')$. The following are equivalent\n\\begin{enumerate}\n\\item $(\\mathcal{F}_n)$ extends to $X$, and\n\\item the pullback of $(\\mathcal{F}_n)$ to $U'$ is the completion\nof a coherent $\\mathcal{O}_{U'}$-module.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIN","source_file":"algebraization.tex","source_line":5170,"source_end_line":5181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5170-L5181","statement_sha256":"4ec2b162a8ac1fc8b5b4237c5d13489cfda3c9e896c2c5cb01c99cf1cc778007","origin":"The Stacks Project","memory_eligible":false,"source_rank":9239,"rank":9239,"depth":19,"x":1069.799,"y":1631.479,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIP","tag":"0EIP","title":"Algebraization of coherent formal modules, I · Definition 0EIP","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). We say (F_n) canonically extends to X if the the inverse system (widetildeH^0(U, F_n))_n ≥ 1 in QCoh(O_X) is pro-isomorphic to an object (G_n) of Coh(X, IO_X).","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an\nobject of $\\textit{Coh}(U, I\\mathcal{O}_U)$. We say\n{\\it $(\\mathcal{F}_n)$ canonically extends to $X$} if the the\ninverse system\n$$\n\\{\\widetilde{H^0(U, \\mathcal{F}_n)}\\}_{n \\geq 1}\n$$\nin $\\QCoh(\\mathcal{O}_X)$ is pro-isomorphic to an object\n$(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I\\mathcal{O}_X)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, I","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIP","source_file":"algebraization.tex","source_line":5230,"source_end_line":5241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5230-L5241","statement_sha256":"07b0b1b0c159cf2112511cb02e875ece4ae6a51884717b5abadc848acc745878","origin":"The Stacks Project","memory_eligible":false,"source_rank":9240,"rank":9240,"depth":0,"x":1149.199,"y":1542.261,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIR","tag":"0EIR","title":"Algebraization of coherent formal modules, I · Lemma 0EIR","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). If (F_n) canonically extends to X, then • (widetildeH^0(U, F_n)) is pro-isomorphic to an object (G_n) of Coh(X, I O_X) unique up to unique isomorphism, • the restriction of (G_n) to U is isomorphic to (F_n), i.e., (F_n) extends to X, • the inverse system (H^0(U, F_n)) satisfies the Mittag-Leffler condition, and • the module M in ([Tag 0EHD]) is finite over the I-adic completion of A and the limit topology on…","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an\nobject of $\\textit{Coh}(U, I\\mathcal{O}_U)$. If $(\\mathcal{F}_n)$\ncanonically extends to $X$, then\n\\begin{enumerate}\n\\item $(\\widetilde{H^0(U, \\mathcal{F}_n)})$ is pro-isomorphic to an\nobject $(\\mathcal{G}_n)$ of $\\textit{Coh}(X, I \\mathcal{O}_X)$\nunique up to unique isomorphism,\n\\item the restriction of $(\\mathcal{G}_n)$ to $U$ is isomorphic\nto $(\\mathcal{F}_n)$, i.e., $(\\mathcal{F}_n)$ extends to $X$,\n\\item the inverse system $\\{H^0(U, \\mathcal{F}_n)\\}$\nsatisfies the Mittag-Leffler condition, and\n\\item the module $M$ in (\\ref{equation-guess}) is finite over the\n$I$-adic completion of $A$ and the limit topology on\n$M$ is the $I$-adic topology.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIR","source_file":"algebraization.tex","source_line":5248,"source_end_line":5265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5248-L5265","statement_sha256":"3bacc22d7c775278055381314959c6f57e4e5d39d5994a72d86aedddaf880897","origin":"The Stacks Project","memory_eligible":false,"source_rank":9241,"rank":9241,"depth":15,"x":1162.479,"y":1653.808,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIS","tag":"0EIS","title":"Algebraization of coherent formal modules, I · Lemma 0EIS","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Let A → A' be a flat ring map. Set X' = Spec(A'), let U' ⊂ X' be the inverse image of U, and denote g : U' → U the induced morphism. Set (F'_n) = (g^*F_n), see Cohomology of Schemes, Lemma [Tag 0887]. If (F_n) canonically extends to X, then (F'_n) canonically extends to X'. Moreover, the extension found in Lemma [Tag 0EIR] for (F_n) pulls back to the extension for (F'_n).","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an\nobject of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $A \\to A'$ be a flat ring\nmap. Set $X' = \\Spec(A')$, let $U' \\subset X'$ be the inverse image of $U$,\nand denote $g : U' \\to U$ the induced morphism. Set\n$(\\mathcal{F}'_n) = (g^*\\mathcal{F}_n)$, see\nCohomology of Schemes, Lemma \\ref{coherent-lemma-inverse-systems-pullback}.\nIf $(\\mathcal{F}_n)$ canonically extends to $X$, then\n$(\\mathcal{F}'_n)$ canonically extends to $X'$.\nMoreover, the extension found in Lemma \\ref{lemma-canonically-algebraizable}\nfor $(\\mathcal{F}_n)$ pulls back to the extension for\n$(\\mathcal{F}'_n)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIS","source_file":"algebraization.tex","source_line":5296,"source_end_line":5309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5296-L5309","statement_sha256":"b7b5a03a559da53447ddd81ebac9e729c3ae25d079bfc10d703720f33e1616ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":9242,"rank":9242,"depth":30,"x":1062.359,"y":1578.62,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EHH","tag":"0EHH","title":"Algebraization of coherent formal modules, I · Lemma 0EHH","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Let M be as in ([Tag 0EHD]). Assume • [(a)] the inverse system H^0(U, F_n) has Mittag-Leffler, • [(b)] the limit topology on M agrees with the I-adic topology, and • [(c)] the image of M → H^0(U, F_n) is a finite A-module for all n. Then (F_n) extends canonically to X. In particular, if A is I-adically complete, then (F_n) is the completion of a coherent O_U-module.","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $M$ be as in (\\ref{equation-guess}).\nAssume\n\\begin{enumerate}\n\\item[(a)] the inverse system $H^0(U, \\mathcal{F}_n)$ has Mittag-Leffler,\n\\item[(b)] the limit topology on $M$ agrees with the $I$-adic topology, and\n\\item[(c)] the image of $M \\to H^0(U, \\mathcal{F}_n)$ is a finite $A$-module\nfor all $n$.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends canonically to $X$.\nIn particular, if $A$ is $I$-adically complete, then\n$(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHH","source_file":"algebraization.tex","source_line":5327,"source_end_line":5341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5327-L5341","statement_sha256":"e012fef6e5f398523b14764fd29b618c8b23adf23cb5ee57c37256de7552b4c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9243,"rank":9243,"depth":6,"x":1197.485,"y":1577.245,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXW","tag":"0DXW","title":"Algebraization of coherent formal modules, I · Lemma 0DXW","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • I = (f) is a principal ideal for a nonzerodivisor f ∈ a, • F_n is a finite locally free O_U/f^nO_U-module, • H^1_ a(A/fA) and H^2_ a(A/fA) are finite A-modules. Then (F_n) extends canonically to X. In particular, if A is complete, then (F_n) is the completion of a coherent O_U-module.","statement_latex":"In Situation \\ref{situation-algebraize} let\n$(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$.\nAssume\n\\begin{enumerate}\n\\item $I = (f)$ is a principal ideal for a nonzerodivisor $f \\in \\mathfrak a$,\n\\item $\\mathcal{F}_n$ is a finite locally free\n$\\mathcal{O}_U/f^n\\mathcal{O}_U$-module,\n\\item $H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$\nare finite $A$-modules.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$\nis complete, then $(\\mathcal{F}_n)$ is the completion of a coherent\n$\\mathcal{O}_U$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXW","source_file":"algebraization.tex","source_line":5365,"source_end_line":5380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5365-L5380","statement_sha256":"77f24efbab6da37a8bde48e317dc8292cae5b90bd0b43fec0403ae95b95bd3fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9244,"rank":9244,"depth":31,"x":1098.345,"y":1655.407,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIU","tag":"0EIU","title":"Algebraization of coherent formal modules, II · Lemma 0EIU","summary":"In Situation [Tag 0EHC]. Let (F_n) → (F'_n) be a morphism of Coh(U, IO_U) whose kernel and cokernel are annihilated by a power of I. Then • (F_n) extends to X if and only if (F'_n) extends to X, and • (F_n) is the completion of a coherent O_U-module if and only if (F'_n) is.","statement_latex":"In Situation \\ref{situation-algebraize}. Let\n$(\\mathcal{F}_n) \\to (\\mathcal{F}'_n)$ be a morphism of\n$\\textit{Coh}(U, I\\mathcal{O}_U)$\nwhose kernel and cokernel are annihilated by a power of $I$. Then\n\\begin{enumerate}\n\\item $(\\mathcal{F}_n)$ extends to $X$ if and only if\n$(\\mathcal{F}'_n)$ extends to $X$, and\n\\item $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module\nif and only if $(\\mathcal{F}'_n)$ is.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIU","source_file":"algebraization.tex","source_line":5457,"source_end_line":5469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5457-L5469","statement_sha256":"0cba61d86ea02b2d574be7986f4f0fc451cd573b73f9c0aaa0efd0a999548051","origin":"The Stacks Project","memory_eligible":false,"source_rank":9245,"rank":9245,"depth":22,"x":1108.657,"y":1540.828,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EHF","tag":"0EHF","title":"Algebraization of coherent formal modules, II · Lemma 0EHF","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). If the inverse system H^0(U, F_n) has Mittag-Leffler, then the canonical maps widetildeM/I^nM|_U → F_n are surjective for all n where M is as in ([Tag 0EHD]).","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. If the inverse system\n$H^0(U, \\mathcal{F}_n)$ has Mittag-Leffler, then the canonical maps\n$$\n\\widetilde{M/I^nM}|_U \\to \\mathcal{F}_n\n$$\nare surjective for all $n$ where $M$ is as in (\\ref{equation-guess}).","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHF","source_file":"algebraization.tex","source_line":5485,"source_end_line":5494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5485-L5494","statement_sha256":"c7f4d3ac97ee2cfdc718ef7d508bae681b1ff94b12c0ca0c16c23c2a2a09f68d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9246,"rank":9246,"depth":17,"x":1193.699,"y":1631.71,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EHG","tag":"0EHG","title":"Algebraization of coherent formal modules, II · Lemma 0EHG","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Let M be as in ([Tag 0EHD]). Set G_n = widetildeM/I^nM. If the limit topology on M agrees with the I-adic topology, then G_n|_U is a coherent O_U-module and the map of inverse systems (G_n|_U) → (F_n) is injective in the abelian category Coh(U, IO_U).","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $M$ be as in (\\ref{equation-guess}).\nSet\n$$\n\\mathcal{G}_n = \\widetilde{M/I^nM}.\n$$\nIf the limit topology on $M$ agrees with the $I$-adic topology, then\n$\\mathcal{G}_n|_U$ is a coherent\n$\\mathcal{O}_U$-module and the map of inverse systems\n$$\n(\\mathcal{G}_n|_U) \\longrightarrow (\\mathcal{F}_n)\n$$\nis injective in the abelian category $\\textit{Coh}(U, I\\mathcal{O}_U)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHG","source_file":"algebraization.tex","source_line":5522,"source_end_line":5537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5522-L5537","statement_sha256":"9ff443a04ed5bdb5cab8b48df782b0bc8695e54e1a5853561d2d01e69f2e1740","origin":"The Stacks Project","memory_eligible":false,"source_rank":9247,"rank":9247,"depth":17,"x":1057.103,"y":1612.837,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIY","tag":"0EIY","title":"A distance function · Lemma 0EIY","summary":"Let Y be a Noetherian scheme and let Z ⊂ Y be a closed subset. • For y ∈ Y we have δ_Z(y) = 0 ⇔ y ∈ Z. • The subsets (y ∈ Y mid δ_Z(y) ≤ k) are stable under specialization. • For y ∈ Y and z ∈ overline(y) ∩ Z we have dim(O_overline(y), z) ≥ δ_Z(y). • If δ is a dimension function on Y, then δ_Z(y) ≥ δ(y) - δ_max where δ_max is the maximum value of δ on Z. • If Y = Spec(A) is the spectrum of a catenary Noetherian local ring with maximal ideal m and Z = ( m), then δ_Z(y) =…","statement_latex":"Let $Y$ be a Noetherian scheme and let $Z \\subset Y$ be a closed subset.\n\\begin{enumerate}\n\\item For $y \\in Y$ we have $\\delta_Z(y) = 0 \\Leftrightarrow y \\in Z$.\n\\item The subsets $\\{y \\in Y \\mid \\delta_Z(y) \\leq k\\}$ are\nstable under specialization.\n\\item For $y \\in Y$ and $z \\in \\overline{\\{y\\}} \\cap Z$ we have\n$\\dim(\\mathcal{O}_{\\overline{\\{y\\}}, z}) \\geq \\delta_Z(y)$.\n\\item If $\\delta$ is a dimension function on $Y$, then\n$\\delta_Z(y) \\geq \\delta(y) - \\delta_{max}$ where $\\delta_{max}$\nis the maximum value of $\\delta$ on $Z$.\n\\item If $Y = \\Spec(A)$ is the spectrum of a catenary Noetherian local ring\nwith maximal ideal $\\mathfrak m$ and $Z = \\{\\mathfrak m\\}$, then\n$\\delta_Z(y) = \\dim(\\overline{\\{y\\}})$.\n\\item If $Y' \\subset Y$ is an open subscheme, then\n$\\delta^{Y'}_{Y' \\cap Z}(y') \\geq \\delta^Y_Z(y')$ for $y' \\in Y'$.\n\\end{enumerate}\nAssume $Y$ is catenary. Then\n\\begin{enumerate}\n\\setcounter{enumi}{6}\n\\item Let $y' \\leadsto y$ be an immediate specialization of points of $Y$.\nIf $Y$ is catenary, then $\\delta_Z(y') \\leq \\delta_Z(y) + 1$.\n\\item Given a pattern of specializations\n$$\n\\xymatrix{\n& y'_0 \\ar@{~>}[ld] \\ar@{~>}[rd] &\n& y'_1 \\ar@{~>}[ld] & \\ldots\n& y'_{k - 1} \\ar@{~>}[rd] &\n\\\\\ny_0 & &\ny_1 & &\n\\ldots & &\ny_k = y\n}\n$$\nbetween points of $Y$ with $y_0 \\in Z$ and $y_i' \\leadsto y_i$\nan immediate specialization, then $\\delta_Z(y_k) \\leq k$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"A distance function","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIY","source_file":"algebraization.tex","source_line":5621,"source_end_line":5660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5621-L5660","statement_sha256":"9e16cad64e6b65e314df830d66dd19193ea7681d0ccc18fd365f8e73ac191f34","origin":"The Stacks Project","memory_eligible":false,"source_rank":9248,"rank":9248,"depth":4,"x":1173.682,"y":1548.875,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EIZ","tag":"0EIZ","title":"A distance function · Lemma 0EIZ","summary":"Let Y be a universally catenary Noetherian scheme. Let Z ⊂ Y be a closed subscheme. Let f : Y' → Y be a finite type morphism all of whose fibres have dimension ≤ e. Set Z' = f^-1(Z). Then δ_Z(y) ≤ δ_Z'(y') + e - trdeg_kappa(y)(kappa(y')) for y' ∈ Y' with image y ∈ Y.","statement_latex":"Let $Y$ be a universally catenary Noetherian scheme. Let $Z \\subset Y$\nbe a closed subscheme. Let $f : Y' \\to Y$ be a finite type\nmorphism all of whose fibres have dimension $\\leq e$. Set $Z' = f^{-1}(Z)$.\nThen\n$$\n\\delta_Z(y) \\leq \\delta_{Z'}(y') + e - \\text{trdeg}_{\\kappa(y)}(\\kappa(y'))\n$$\nfor $y' \\in Y'$ with image $y \\in Y$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"A distance function","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EIZ","source_file":"algebraization.tex","source_line":5759,"source_end_line":5769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5759-L5769","statement_sha256":"7b28e690bfcb05da6f37c98d1fc038acdc6f54a4934d5198760be15a98e481db","origin":"The Stacks Project","memory_eligible":false,"source_rank":9249,"rank":9249,"depth":14,"x":1138.954,"y":1662.844,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJ3","tag":"0EJ3","title":"Algebraization of coherent formal modules, III · Definition 0EJ3","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Let a, b be integers. Let δ^Y_Z be as in ([Tag 0EIX]). We say (F_n) satisfies the (a, b)-inequalities if for y ∈ U ∩ Y and a prime p ⊂ O_X, y^wedge with p not ∈ V(IO_X, y^wedge) • if V( p) ∩ V(IO_X, y^wedge) not = ( m_y^wedge), then depth((F^wedge_y)_ p) + δ^Y_Z(y) ≥ a or depth((F^wedge_y)_ p) + dim(O_X, y^wedge/ p) + δ^Y_Z(y) > b • if V( p) ∩ V(IO_X, y^wedge) = ( m_y^wedge), then depth((F^wedge_y)_ p) +…","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $a, b$ be integers.\nLet $\\delta^Y_Z$ be as in (\\ref{equation-delta-Z}).\nWe say\n{\\it $(\\mathcal{F}_n)$ satisfies the $(a, b)$-inequalities} if for\n$y \\in U \\cap Y$ and a prime $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$\nwith $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$\n\\begin{enumerate}\n\\item if $V(\\mathfrak p) \\cap V(I\\mathcal{O}_{X, y}^\\wedge) \\not =\n\\{\\mathfrak m_y^\\wedge\\}$, then\n$$\n\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\delta^Y_Z(y) \\geq a\n\\quad\\text{or}\\quad\n\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +\n\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > b\n$$\n\\item if $V(\\mathfrak p) \\cap V(I\\mathcal{O}_{X, y}^\\wedge) =\n\\{\\mathfrak m_y^\\wedge\\}$, then\n$$\n\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\delta^Y_Z(y) > a\n$$\n\\end{enumerate}\nWe say {\\it $(\\mathcal{F}_n)$ satisfies the strict $(a, b)$-inequalities}\nif for $y \\in U \\cap Y$ and a prime\n$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with\n$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$\nwe have\n$$\n\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) + \\delta^Y_Z(y) > a\n\\quad\\text{or}\\quad\n\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +\n\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > b\n$$","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, III","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJ3","source_file":"algebraization.tex","source_line":5927,"source_end_line":5962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5927-L5962","statement_sha256":"3b01a09ca4dcfa6f60c2877da9267d5a43ffc4c7157dbcba6c674e1458c8ba14","origin":"The Stacks Project","memory_eligible":false,"source_rank":9250,"rank":9250,"depth":0,"x":1072.536,"y":1558.506,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJ4","tag":"0EJ4","title":"Algebraization of coherent formal modules, III · Lemma 0EJ4","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Let a, b be integers. • If (F_n) is annihilated by a power of I, then (F_n) satisfies the (a, b)-inequalities for any a, b. • If (F_n) satisfies the (a + 1, b)-inequalities, then (F_n) satisfies the strict (a, b)-inequalities. If cd(A, I) ≤ d and A has a dualizing complex, then • [(3)] (F_n) satisfies the (s, s + d)-inequalities if and only if for all y ∈ U ∩ Y the tuple O_X, y^wedge, IO_X, y^wedge, (…","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $a, b$ be integers.\n\\begin{enumerate}\n\\item If $(\\mathcal{F}_n)$ is annihilated by a power of $I$, then\n$(\\mathcal{F}_n)$ satisfies the $(a, b)$-inequalities for any $a, b$.\n\\item If $(\\mathcal{F}_n)$ satisfies the $(a + 1, b)$-inequalities, then\n$(\\mathcal{F}_n)$ satisfies the strict $(a, b)$-inequalities.\n\\end{enumerate}\nIf $\\text{cd}(A, I) \\leq d$ and $A$ has a dualizing complex, then\n\\begin{enumerate}\n\\item[(3)] $(\\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities\nif and only if for all $y \\in U \\cap Y$ the tuple\n$\\mathcal{O}_{X, y}^\\wedge, I\\mathcal{O}_{X, y}^\\wedge,\n\\{\\mathfrak m_y^\\wedge\\}, \\mathcal{F}_y^\\wedge, s - \\delta^Y_Z(y), d$\nis as in Situation \\ref{situation-bootstrap}.\n\\item[(4)]\nIf $(\\mathcal{F}_n)$ satisfies the strict $(s, s + d)$-inequalities, then\n$(\\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJ4","source_file":"algebraization.tex","source_line":5967,"source_end_line":5988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5967-L5988","statement_sha256":"333add8fdd7ac276f8de9b17b5525dd8684da7b45bea00d30a0dccd19306dc5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9251,"rank":9251,"depth":40,"x":1206.166,"y":1597.979,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKW","tag":"0EKW","title":"Algebraization of coherent formal modules, III · Lemma 0EKW","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). If cd(A, I) = 1, then F satisfies the (2, 3)-inequalities if and only if depth((F^wedge_y)_ p) + dim(O_X, y^wedge/ p) + δ^Y_Z(y) > 3 for all y ∈ U ∩ Y and p ⊂ O_X, y^wedge with p not ∈ V(IO_X, y^wedge).","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. If $\\text{cd}(A, I) = 1$, then\n$\\mathcal{F}$ satisfies the $(2, 3)$-inequalities if and only if\n$$\n\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +\n\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > 3\n$$\nfor all $y \\in U \\cap Y$ and $\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$\nwith $\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKW","source_file":"algebraization.tex","source_line":5995,"source_end_line":6006,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L5995-L6006","statement_sha256":"7738e9e5ceebe1445e2c9d0d87ca684046d34d48838bb0af28ae7f60bce37756","origin":"The Stacks Project","memory_eligible":false,"source_rank":9252,"rank":9252,"depth":39,"x":1075.159,"y":1644.957,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJ5","tag":"0EJ5","title":"Algebraization of coherent formal modules, III · Lemma 0EJ5","summary":"In Situation [Tag 0EHC] let F be a coherent O_U-module and d ≥ 1. Assume • A is I-adically complete, has a dualizing complex, and cd(A, I) ≤ d, • the completion F^wedge of F satisfies the strict (1, 1 + d)-inequalities. Let x ∈ X be a point. Let W = overline(x). If W ∩ Y has an irreducible component contained in Z and one which is not, then depth(F_x) ≥ 1.","statement_latex":"In Situation \\ref{situation-algebraize} let $\\mathcal{F}$ be a\ncoherent $\\mathcal{O}_U$-module and $d \\geq 1$. Assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete, has a dualizing complex, and\n$\\text{cd}(A, I) \\leq d$,\n\\item the completion $\\mathcal{F}^\\wedge$ of $\\mathcal{F}$\nsatisfies the strict $(1, 1 + d)$-inequalities.\n\\end{enumerate}\nLet $x \\in X$ be a point. Let $W = \\overline{\\{x\\}}$.\nIf $W \\cap Y$ has an irreducible component contained in $Z$\nand one which is not, then $\\text{depth}(\\mathcal{F}_x) \\geq 1$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJ5","source_file":"algebraization.tex","source_line":6041,"source_end_line":6054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6041-L6054","statement_sha256":"d1dca4ff1a1c5123a488d53ceb0335787606a032e2825089916608fe5f83ec48","origin":"The Stacks Project","memory_eligible":false,"source_rank":9253,"rank":9253,"depth":37,"x":1134.31,"y":1535.379,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJ6","tag":"0EJ6","title":"Algebraization of coherent formal modules, III · Lemma 0EJ6","summary":"In Situation [Tag 0EHC] let F be a coherent O_U-module and d ≥ 1. Assume • A is I-adically complete, has a dualizing complex, and cd(A, I) ≤ d, • the completion F^wedge of F satisfies the strict (1, 1+ d)-inequalities, and • for x ∈ U with overline(x) ∩ Y ⊂ Z we have depth(F_x) ≥ 2. Then H^0(U, F) → lim H^0(U, F/I^nF) is an isomorphism.","statement_latex":"In Situation \\ref{situation-algebraize} let $\\mathcal{F}$ be a\ncoherent $\\mathcal{O}_U$-module and $d \\geq 1$. Assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete, has a dualizing complex, and\n$\\text{cd}(A, I) \\leq d$,\n\\item the completion $\\mathcal{F}^\\wedge$ of $\\mathcal{F}$\nsatisfies the strict $(1, 1+ d)$-inequalities, and\n\\item for $x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$\nwe have $\\text{depth}(\\mathcal{F}_x) \\geq 2$.\n\\end{enumerate}\nThen $H^0(U, \\mathcal{F}) \\to \\lim H^0(U, \\mathcal{F}/I^n\\mathcal{F})$\nis an isomorphism.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJ6","source_file":"algebraization.tex","source_line":6086,"source_end_line":6100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6086-L6100","statement_sha256":"1692e170dd8433483d60a41215f35fa4eb43224030235ecb38cd108692fa0242","origin":"The Stacks Project","memory_eligible":false,"source_rank":9254,"rank":9254,"depth":38,"x":1179.054,"y":1650.368,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJ7","tag":"0EJ7","title":"Algebraization of coherent formal modules, III · Lemma 0EJ7","summary":"In Situation [Tag 0EHC] let F be a coherent O_U-module and d ≥ 1. Assume • A is I-adically complete, has a dualizing complex, and cd(A, I) ≤ d, • the completion F^wedge of F satisfies the strict (1, 1 + d)-inequalities, and • for x ∈ U with overline(x) ∩ Y ⊂ Z we have depth(F_x) ≥ 2. Then the map Hom_U(G, F) → Hom_Coh(U, IO_U)(G^wedge, F^wedge) is bijective for every coherent O_U-module G.","statement_latex":"In Situation \\ref{situation-algebraize} let $\\mathcal{F}$ be a\ncoherent $\\mathcal{O}_U$-module and $d \\geq 1$. Assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete, has a dualizing complex, and\n$\\text{cd}(A, I) \\leq d$,\n\\item the completion $\\mathcal{F}^\\wedge$ of $\\mathcal{F}$\nsatisfies the strict $(1, 1 + d)$-inequalities, and\n\\item for $x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$\nwe have $\\text{depth}(\\mathcal{F}_x) \\geq 2$.\n\\end{enumerate}\nThen the map\n$$\n\\Hom_U(\\mathcal{G}, \\mathcal{F})\n\\longrightarrow\n\\Hom_{\\textit{Coh}(U, I\\mathcal{O}_U)}(\\mathcal{G}^\\wedge, \\mathcal{F}^\\wedge)\n$$\nis bijective for every coherent $\\mathcal{O}_U$-module $\\mathcal{G}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJ7","source_file":"algebraization.tex","source_line":6118,"source_end_line":6137,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6118-L6137","statement_sha256":"bfe554f998f09005af5c66792edd00b1a845209c24e9793662d59add30e8c68c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9255,"rank":9255,"depth":39,"x":1052.912,"y":1590.643,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJ8","tag":"0EJ8","title":"Algebraization of coherent formal modules, III · Lemma 0EJ8","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U) and d ≥ 1. Assume • A is I-adically complete, has a dualizing complex, and cd(A, I) ≤ d, • (F_n) is the completion of a coherent O_U-module, • (F_n) satisfies the strict (1, 1 + d)-inequalities. Then there exists a unique coherent O_U-module F whose completion is (F_n) such that for x ∈ U with overline(x) ∩ Y ⊂ Z we have depth(F_x) ≥ 2.","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an\nobject of $\\textit{Coh}(U, I\\mathcal{O}_U)$ and $d \\geq 1$. Assume\n\\begin{enumerate}\n\\item $A$ is $I$-adically complete, has a dualizing complex, and\n$\\text{cd}(A, I) \\leq d$,\n\\item $(\\mathcal{F}_n)$ is the completion of a coherent $\\mathcal{O}_U$-module,\n\\item $(\\mathcal{F}_n)$ satisfies the strict $(1, 1 + d)$-inequalities.\n\\end{enumerate}\nThen there exists a unique coherent $\\mathcal{O}_U$-module $\\mathcal{F}$\nwhose completion is $(\\mathcal{F}_n)$ such that for\n$x \\in U$ with $\\overline{\\{x\\}} \\cap Y \\subset Z$\nwe have $\\text{depth}(\\mathcal{F}_x) \\geq 2$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJ8","source_file":"algebraization.tex","source_line":6160,"source_end_line":6174,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6160-L6174","statement_sha256":"1ebf62553e12ea2eb96b42be8a10b31761986db1cc4aac17b7db73d8f28d238e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9256,"rank":9256,"depth":40,"x":1194.71,"y":1562.962,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DXU","tag":"0DXU","title":"Algebraization of coherent formal modules, IV · Lemma 0DXU","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A is local and a = m is the maximal ideal, • A has a dualizing complex, • I = (f) is a principal ideal for a nonzerodivisor f ∈ m, • F_n is a finite locally free O_U/f^nO_U-module, • if p ∈ V(f) setminus ( m), then depth((A/f)_ p) + dim(A/ p) > 1, and • if p not ∈ V(f) and V( p) ∩ V(f) not = ( m), then depth(A_ p) + dim(A/ p) > 3. Then (F_n) extends canonically to X. In particular, if A is complete,…","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ is local and $\\mathfrak a = \\mathfrak m$ is the maximal ideal,\n\\item $A$ has a dualizing complex,\n\\item $I = (f)$ is a principal ideal for a nonzerodivisor $f \\in \\mathfrak m$,\n\\item $\\mathcal{F}_n$ is a finite locally free\n$\\mathcal{O}_U/f^n\\mathcal{O}_U$-module,\n\\item if $\\mathfrak p \\in V(f) \\setminus \\{\\mathfrak m\\}$, then\n$\\text{depth}((A/f)_\\mathfrak p) + \\dim(A/\\mathfrak p) > 1$, and\n\\item if $\\mathfrak p \\not \\in V(f)$ and\n$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$\nis complete, then $(\\mathcal{F}_n)$ is the completion of a coherent\n$\\mathcal{O}_U$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXU","source_file":"algebraization.tex","source_line":6236,"source_end_line":6255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6236-L6255","statement_sha256":"034aaa40f93566b0741c4a280c06e265480af7ceead5ad6a2c180f353a3231b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9257,"rank":9257,"depth":39,"x":1111.975,"y":1664.365,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EHK","tag":"0EHK","title":"Algebraization of coherent formal modules, IV · Lemma 0EHK","summary":"In Situation [Tag 0EHC] let (M_n) be an inverse system of A-modules as in Lemma [Tag 0DXT] and let (F_n) be the corresponding object of Coh(U, IO_U). Let d ≥ cd(A, I) and s ≥ 0 be integers. With notation as above assume • A is local with maximal ideal m = a, • A has a dualizing complex, and • (F_n) satisfies the (s, s + d)-inequalities (Definition [Tag 0EJ3]). Let E be an injective hull of the residue field of A. Then for i ≤ s there exists a finite A-module N annihilated…","statement_latex":"In Situation \\ref{situation-algebraize} let $(M_n)$ be an inverse system of\n$A$-modules as in Lemma \\ref{lemma-system-of-modules} and let\n$(\\mathcal{F}_n)$ be the corresponding object of\n$\\textit{Coh}(U, I\\mathcal{O}_U)$. Let $d \\geq \\text{cd}(A, I)$\nand $s \\geq 0$ be integers.\nWith notation as above assume\n\\begin{enumerate}\n\\item $A$ is local with maximal ideal $\\mathfrak m = \\mathfrak a$,\n\\item $A$ has a dualizing complex, and\n\\item $(\\mathcal{F}_n)$ satisfies the $(s, s + d)$-inequalities\n(Definition \\ref{definition-s-d-inequalities}).\n\\end{enumerate}\nLet $E$ be an injective hull of the residue field of $A$. Then for $i \\leq s$\nthere exists a finite $A$-module $N$ annihilated by a power\nof $I$ and for $n \\gg 0$ compatible maps\n$$\nH^i_\\mathfrak m(M_n) \\to \\Hom_A(N, E)\n$$\nwhose cokernels are finite length $A$-modules and whose kernels $K_n$\nform an inverse system such that $\\Im(K_{n''} \\to K_{n'})$ has finite\nlength for $n'' \\gg n' \\gg 0$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EHK","source_file":"algebraization.tex","source_line":6380,"source_end_line":6403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6380-L6403","statement_sha256":"6d3de9b097b0e79540106d72adb2f40762de8521bbabb06b7e692249bc2b5ae4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9258,"rank":9258,"depth":46,"x":1091.331,"y":1542.01,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJ9","tag":"0EJ9","title":"Algebraization of coherent formal modules, IV · Lemma 0EJ9","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A is local and a = m is the maximal ideal, • A has a dualizing complex, • I = (f) is a principal ideal, • (F_n) satisfies the (2, 3)-inequalities. Then (F_n) extends to X. In particular, if A is I-adically complete, then (F_n) is the completion of a coherent O_U-module.","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ is local and $\\mathfrak a = \\mathfrak m$ is the maximal ideal,\n\\item $A$ has a dualizing complex,\n\\item $I = (f)$ is a principal ideal,\n\\item $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is\n$I$-adically complete, then $(\\mathcal{F}_n)$ is the completion\nof a coherent $\\mathcal{O}_U$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJ9","source_file":"algebraization.tex","source_line":6518,"source_end_line":6531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6518-L6531","statement_sha256":"7ddb8123e94c27a136547127e1df747ed1a6f1b63925d4d7f6bd1f8fa1b4e21f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9259,"rank":9259,"depth":47,"x":1205.535,"y":1620.926,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJA","tag":"0EJA","title":"Algebraization of coherent formal modules, IV · Lemma 0EJA","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A is local with maximal ideal a = m, • cd(A, I) = 1. Then (F_n) satisfies the (2, 3)-inequalities if and only if for all y ∈ U ∩ Y with dim((y)) = 1 and every prime p ⊂ O_X, y^wedge, p not ∈ V(IO_X, y^wedge) we have depth((F_y^wedge)_ p) + dim(O_X, y^wedge/ p) > 2","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ is local with maximal ideal $\\mathfrak a = \\mathfrak m$,\n\\item $\\text{cd}(A, I) = 1$.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities if and only\nif for all $y \\in U \\cap Y$ with $\\dim(\\{y\\}) = 1$ and every prime\n$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$,\n$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$ we have\n$$\n\\text{depth}((\\mathcal{F}_y^\\wedge)_\\mathfrak p) +\n\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) > 2\n$$","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJA","source_file":"algebraization.tex","source_line":6660,"source_end_line":6676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6660-L6676","statement_sha256":"4cfda3b23d55c82bb0ba3d501f756ca451f6482bac6080ced2c9e4e50596b2fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9260,"rank":9260,"depth":40,"x":1057.104,"y":1627.569,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJB","tag":"0EJB","title":"Algebraization of coherent formal modules, IV · Lemma 0EJB","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A is local with maximal ideal a = m, • A has a dualizing complex, • cd(A, I) = 1, • for y ∈ U ∩ Y the module F_y^wedge is finite locally free outside V(IO_X, y^wedge), for example if F_n is a finite locally free O_U/I^nO_U-module, and • one of the following is true • A_f is (S_2) and every irreducible component of X not contained in Y has dimension ≥ 4, or • if p not ∈ V(f) and V( p) ∩ V(f) not = (…","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an object\nof $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ is local with maximal ideal $\\mathfrak a = \\mathfrak m$,\n\\item $A$ has a dualizing complex,\n\\item $\\text{cd}(A, I) = 1$,\n\\item for $y \\in U \\cap Y$ the module $\\mathcal{F}_y^\\wedge$\nis finite locally free outside $V(I\\mathcal{O}_{X, y}^\\wedge)$,\nfor example if $\\mathcal{F}_n$ is a finite locally free\n$\\mathcal{O}_U/I^n\\mathcal{O}_U$-module, and\n\\item one of the following is true\n\\begin{enumerate}\n\\item $A_f$ is $(S_2)$ and every irreducible component of $X$\nnot contained in $Y$ has dimension $\\geq 4$, or\n\\item if $\\mathfrak p \\not \\in V(f)$ and\n$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.\n\\end{enumerate}\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJB","source_file":"algebraization.tex","source_line":6721,"source_end_line":6743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6721-L6743","statement_sha256":"f6fa7471dc1093c606836ab7966fbbdacdff190d41de3ef91a337b4280a0350b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9261,"rank":9261,"depth":41,"x":1161.741,"y":1537.997,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJE","tag":"0EJE","title":"Improving coherent formal modules · Lemma 0EJE","summary":"In the situation above assume X locally has a dualizing complex. Let T ⊂ Y be a subset stable under specialization. Assume for y ∈ T and for a nonmaximal prime p ⊂ O_X, y^wedge with V( p) ∩ V(I^wedge_y) = ( m_y^wedge) we have depth_(O_X, y)_ p ((F^wedge_y)_ p) > 0 Then there exists a canonical map (F_n) → (F_n') of inverse systems of coherent O_X-modules with the following properties • for y ∈ T we have depth(F'_n, y) ≥ 1, • (F'_n) is isomorphic as a pro-system to an…","statement_latex":"In the situation above assume $X$ locally has a dualizing complex.\nLet $T \\subset Y$ be a subset stable under specialization.\nAssume for $y \\in T$ and for a nonmaximal prime\n$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with\n$V(\\mathfrak p) \\cap V(\\mathcal{I}^\\wedge_y) = \\{\\mathfrak m_y^\\wedge\\}$\nwe have\n$$\n\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}\n((\\mathcal{F}^\\wedge_y)_\\mathfrak p) > 0\n$$\nThen there exists a canonical map\n$(\\mathcal{F}_n) \\to (\\mathcal{F}_n')$\nof inverse systems of coherent $\\mathcal{O}_X$-modules\nwith the following properties\n\\begin{enumerate}\n\\item for $y \\in T$ we have $\\text{depth}(\\mathcal{F}'_{n, y}) \\geq 1$,\n\\item $(\\mathcal{F}'_n)$ is isomorphic as a pro-system to an object\n$(\\mathcal{G}_n)$ of $\\textit{Coh}(X, \\mathcal{I})$,\n\\item the induced morphism\n$(\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ of\n$\\textit{Coh}(X, \\mathcal{I})$ is surjective with kernel\nannihilated by a power of $\\mathcal{I}$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Improving coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJE","source_file":"algebraization.tex","source_line":6869,"source_end_line":6894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6869-L6894","statement_sha256":"289e77fbcc297a17e0d0415c4d7b1d5aec49d6bff915ec7ae9fc996db18e7c5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9262,"rank":9262,"depth":48,"x":1156.588,"y":1664.049,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJF","tag":"0EJF","title":"Improving coherent formal modules · Lemma 0EJF","summary":"In the situation above assume X locally has a dualizing complex. Let T' ⊂ T ⊂ Y be subsets stable under specialization. Let d ≥ 0 be an integer. Assume • [(a)] affine locally we have X = Spec(A_0) and Y = V(I_0) and cd(A_0, I_0) ≤ d, • [(b)] for y ∈ T and a nonmaximal prime p ⊂ O_X, y^wedge with V( p) ∩ V(I_y^wedge) = ( m_y^wedge) we have depth_(O_X, y)_ p ((F^wedge_y)_ p) > 0 • [(c)] for y ∈ T' and for a prime p ⊂ O_X, y^wedge with p not ∈ V(I_y^wedge) and V( p) ∩…","statement_latex":"In the situation above assume $X$ locally has a dualizing complex.\nLet $T' \\subset T \\subset Y$ be subsets stable under specialization.\nLet $d \\geq 0$ be an integer. Assume\n\\begin{enumerate}\n\\item[(a)] affine locally we have $X = \\Spec(A_0)$ and $Y = V(I_0)$\nand $\\text{cd}(A_0, I_0) \\leq d$,\n\\item[(b)] for $y \\in T$ and a nonmaximal prime\n$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with\n$V(\\mathfrak p) \\cap V(\\mathcal{I}_y^\\wedge) = \\{\\mathfrak m_y^\\wedge\\}$\nwe have\n$$\n\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}\n((\\mathcal{F}^\\wedge_y)_\\mathfrak p) > 0\n$$\n\\item[(c)] for $y \\in T'$ and for a prime\n$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with\n$\\mathfrak p \\not \\in V(\\mathcal{I}_y^\\wedge)$\nand $V(\\mathfrak p) \\cap V(\\mathcal{I}_y^\\wedge) \\not =\n\\{\\mathfrak m_y^\\wedge\\}$ we have\n$$\n\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}\n((\\mathcal{F}^\\wedge_y)_\\mathfrak p) \\geq 1\n\\quad\\text{or}\\quad\n\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}\n((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +\n\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) > 1 + d\n$$\n\\item[(d)] for $y \\in T'$ and a nonmaximal prime\n$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with\n$V(\\mathfrak p) \\cap V(\\mathcal{I}_y^\\wedge) = \\{\\mathfrak m_y^\\wedge\\}$\nwe have\n$$\n\\text{depth}_{(\\mathcal{O}_{X, y})_\\mathfrak p}\n((\\mathcal{F}^\\wedge_y)_\\mathfrak p) > 1\n$$\n\\item[(e)] if $y \\leadsto y'$ is an immediate specialization and\n$y' \\in T'$, then $y \\in T$.\n\\end{enumerate}\nThen there exists a canonical map $(\\mathcal{F}_n) \\to (\\mathcal{F}_n'')$\nof inverse systems of coherent $\\mathcal{O}_X$-modules\nwith the following properties\n\\begin{enumerate}\n\\item for $y \\in T$ we have $\\text{depth}(\\mathcal{F}''_{n, y}) \\geq 1$,\n\\item for $y' \\in T'$ we have $\\text{depth}(\\mathcal{F}''_{n, y'}) \\geq 2$,\n\\item $(\\mathcal{F}''_n)$ is isomorphic as a pro-system to an object\n$(\\mathcal{H}_n)$ of $\\textit{Coh}(X, \\mathcal{I})$,\n\\item the induced morphism $(\\mathcal{F}_n) \\to (\\mathcal{H}_n)$ of\n$\\textit{Coh}(X, \\mathcal{I})$ has kernel and cokernel\nannihilated by a power of $\\mathcal{I}$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Improving coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJF","source_file":"algebraization.tex","source_line":6951,"source_end_line":7003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L6951-L7003","statement_sha256":"27d2d95f5e32bcb24f6758f2d1ac6567f0eea41b445065ca2002175f37c0bf42","origin":"The Stacks Project","memory_eligible":false,"source_rank":9263,"rank":9263,"depth":49,"x":1058.537,"y":1567.699,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJG","tag":"0EJG","title":"Improving coherent formal modules · Lemma 0EJG","summary":"In Situation [Tag 0EHC] assume that A has a dualizing complex. Let d ≥ cd(A, I). Let (F_n) be an object of Coh(U, IO_U). Assume (F_n) satisfies the (2, 2 + d)-inequalities, see Definition [Tag 0EJ3]. Then there exists a canonical map (F_n) → (F_n\") of inverse systems of coherent O_U-modules with the following properties • depth(F\"_n, y) + δ^Y_Z(y) ≥ 3 for all y ∈ U ∩ Y, • (F\"_n) is isomorphic as a pro-system to an object (H_n) of Coh(U, IO_U), • the induced morphism (F_n)…","statement_latex":"In Situation \\ref{situation-algebraize} assume that $A$ has\na dualizing complex. Let $d \\geq \\text{cd}(A, I)$. Let $(\\mathcal{F}_n)$\nbe an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n$(\\mathcal{F}_n)$ satisfies the $(2, 2 + d)$-inequalities, see\nDefinition \\ref{definition-s-d-inequalities}.\nThen there exists a canonical map $(\\mathcal{F}_n) \\to (\\mathcal{F}_n'')$\nof inverse systems of coherent $\\mathcal{O}_U$-modules\nwith the following properties\n\\begin{enumerate}\n\\item $\\text{depth}(\\mathcal{F}''_{n, y}) + \\delta^Y_Z(y) \\geq 3$\nfor all $y \\in U \\cap Y$,\n\\item $(\\mathcal{F}''_n)$ is isomorphic as a pro-system to an object\n$(\\mathcal{H}_n)$ of $\\textit{Coh}(U, I\\mathcal{O}_U)$,\n\\item the induced morphism $(\\mathcal{F}_n) \\to (\\mathcal{H}_n)$ of\n$\\textit{Coh}(U, I\\mathcal{O}_U)$ has kernel and cokernel\nannihilated by a power of $I$,\n\\item the modules $H^0(U, \\mathcal{F}''_n)$ and $H^1(U, \\mathcal{F}''_n)$\nare finite $A$-modules for all $n$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Improving coherent formal modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJG","source_file":"algebraization.tex","source_line":7074,"source_end_line":7095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7074-L7095","statement_sha256":"6e05d4938372b7f282e895508749e7c97125d80382825c6f7d013b19f83b4a9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9264,"rank":9264,"depth":53,"x":1209.056,"y":1583.186,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJI","tag":"0EJI","title":"Algebraization of coherent formal modules, V · Lemma 0EJI","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A has a dualizing complex and cd(A, I) = 1, • (F_n) is pro-isomorphic to an inverse system (F_n\") of coherent O_U-modules such that depth(F\"_n, y) + δ^Y_Z(y) ≥ 3 for all y ∈ U ∩ Y. Then (F_n) extends canonically to X, see Definition [Tag 0EIP].","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$\nbe an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and $\\text{cd}(A, I) = 1$,\n\\item $(\\mathcal{F}_n)$ is pro-isomorphic to an inverse system\n$(\\mathcal{F}_n'')$ of coherent $\\mathcal{O}_U$-modules such that\n$\\text{depth}(\\mathcal{F}''_{n, y}) + \\delta^Y_Z(y) \\geq 3$\nfor all $y \\in U \\cap Y$.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends canonically to $X$, see\nDefinition \\ref{definition-canonically-algebraizable}.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJI","source_file":"algebraization.tex","source_line":7128,"source_end_line":7141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7128-L7141","statement_sha256":"4d6b228f657f32415ae0050a930106f42a792749bc5281b24df617f178d5b1e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9265,"rank":9265,"depth":53,"x":1085.009,"y":1657.534,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJJ","tag":"0EJJ","title":"Algebraization in cohomological dimension 1 · Proposition 0EJJ","summary":"The local case of this result is [MRaynaud-book]. In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A has a dualizing complex and cd(A, I) = 1, • (F_n) satisfies the (2, 3)-inequalities, see Definition [Tag 0EJ3]. Then (F_n) extends to X. In particular, if A is I-adically complete, then (F_n) is the completion of a coherent O_U-module.","statement_latex":"\\begin{reference}\nThe local case of this result is \\cite[IV Corollaire 2.9]{MRaynaud-book}.\n\\end{reference}\nIn Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$\nbe an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and $\\text{cd}(A, I) = 1$,\n\\item $(\\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities, see\nDefinition \\ref{definition-s-d-inequalities}.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is\n$I$-adically complete, then $(\\mathcal{F}_n)$ is the completion\nof a coherent $\\mathcal{O}_U$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, V","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJJ","source_file":"algebraization.tex","source_line":7262,"source_end_line":7277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7262-L7277","statement_sha256":"d79b5e6d4f81e194bca1dc7617adbd50747195dded41bda93fed8537bc19c77b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9266,"rank":9266,"depth":54,"x":1116.846,"y":1531.72,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJK","tag":"0EJK","title":"Algebraization of coherent formal modules, V · Lemma 0EJK","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A has a dualizing complex, • all fibres of the blowing up b : X' → X of I have dimension ≤ d - 1, • one of the following is true • (F_n) satisfies the (d + 1, d + 2)-inequalities (Definition [Tag 0EJ3]), or • for y ∈ U ∩ Y and a prime p ⊂ O_X, y^wedge with p not ∈ V(IO_X, y^wedge) we have depth((F^wedge_y)_ p) + dim(O_X, y^wedge/ p) + δ^Y_Z(y) > d + 2 Then (F_n) extends to X.","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$\nbe an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex,\n\\item all fibres of the blowing up $b : X' \\to X$ of $I$\nhave dimension $\\leq d - 1$,\n\\item one of the following is true\n\\begin{enumerate}\n\\item $(\\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities\n(Definition \\ref{definition-s-d-inequalities}), or\n\\item for $y \\in U \\cap Y$ and a prime\n$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with\n$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$\nwe have\n$$\n\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +\n\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > d + 2\n$$\n\\end{enumerate}\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends to $X$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJK","source_file":"algebraization.tex","source_line":7292,"source_end_line":7315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7292-L7315","statement_sha256":"b83840de3c530b03e4452f2c650ef979fd2a1635b050c3d4286bf994e85c6ff4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9267,"rank":9267,"depth":54,"x":1194.915,"y":1643.088,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJN","tag":"0EJN","title":"Algebraization for ideals with few generators · Proposition 0EJN","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A has a dualizing complex, • V(I) = V(f_1, …, f_d) for some d ≥ 1 and f_1, …, f_d ∈ A, • one of the following is true • (F_n) satisfies the (d + 1, d + 2)-inequalities (Definition [Tag 0EJ3]), or • for y ∈ U ∩ Y and a prime p ⊂ O_X, y^wedge with p not ∈ V(IO_X, y^wedge) we have depth((F^wedge_y)_ p) + dim(O_X, y^wedge/ p) + δ^Y_Z(y) > d + 2 Then (F_n) extends to X. In particular, if A is I-adically…","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$\nbe an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex,\n\\item $V(I) = V(f_1, \\ldots, f_d)$ for some $d \\geq 1$ and\n$f_1, \\ldots, f_d \\in A$,\n\\item one of the following is true\n\\begin{enumerate}\n\\item $(\\mathcal{F}_n)$ satisfies the $(d + 1, d + 2)$-inequalities\n(Definition \\ref{definition-s-d-inequalities}), or\n\\item for $y \\in U \\cap Y$ and a prime\n$\\mathfrak p \\subset \\mathcal{O}_{X, y}^\\wedge$ with\n$\\mathfrak p \\not \\in V(I\\mathcal{O}_{X, y}^\\wedge)$\nwe have\n$$\n\\text{depth}((\\mathcal{F}^\\wedge_y)_\\mathfrak p) +\n\\dim(\\mathcal{O}_{X, y}^\\wedge/\\mathfrak p) + \\delta^Y_Z(y) > d + 2\n$$\n\\end{enumerate}\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends to $X$. In particular, if $A$ is\n$I$-adically complete, then $(\\mathcal{F}_n)$ is the completion\nof a coherent $\\mathcal{O}_U$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, V","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJN","source_file":"algebraization.tex","source_line":7583,"source_end_line":7608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7583-L7608","statement_sha256":"f0595cdfafe4c0c61162163aa6d5001955542ca03a73056b09bf0a04bb9dd436","origin":"The Stacks Project","memory_eligible":false,"source_rank":9268,"rank":9268,"depth":55,"x":1047.094,"y":1605.086,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJP","tag":"0EJP","title":"Algebraization of coherent formal modules, V · Lemma 0EJP","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A is a local ring which has a dualizing complex, • all irreducible components of X have the same dimension, • the scheme X setminus Y is Cohen-Macaulay, • I is generated by d elements, • dim(X) - dim(Z) > d + 2, and • for y ∈ U ∩ Y the module F_y^wedge is finite locally free outside V(IO_X, y^wedge), for example if F_n is a finite locally free O_U/I^nO_U-module. Then (F_n) extends to X. In particular…","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$\nbe an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ is a local ring which has a dualizing complex,\n\\item all irreducible components of $X$ have the same dimension,\n\\item the scheme $X \\setminus Y$ is Cohen-Macaulay,\n\\item $I$ is generated by $d$ elements,\n\\item $\\dim(X) - \\dim(Z) > d + 2$, and\n\\item for $y \\in U \\cap Y$ the module $\\mathcal{F}_y^\\wedge$\nis finite locally free outside $V(I\\mathcal{O}_{X, y}^\\wedge)$,\nfor example if $\\mathcal{F}_n$ is a finite locally free\n$\\mathcal{O}_U/I^n\\mathcal{O}_U$-module.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends to $X$. In particular if $A$ is $I$-adically\ncomplete, then $(\\mathcal{F}_n)$ is the completion of a coherent\n$\\mathcal{O}_U$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJP","source_file":"algebraization.tex","source_line":7627,"source_end_line":7645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7627-L7645","statement_sha256":"ce5e488ffc83f49fd50f9ec7dc4fcc12a246bd2ee55abe654025bcf968678bf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9269,"rank":9269,"depth":56,"x":1187.31,"y":1548.97,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJS","tag":"0EJS","title":"Algebraization of coherent formal modules, VI · Proposition 0EJS","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • there exist f_1, …, f_d ∈ I such that for y ∈ U ∩ Y the ideal IO_X, y is generated by f_1, …, f_d and f_1, …, f_d form a F_y^wedge-regular sequence, • H^0(U, F_1) and H^1(U, F_1) are finite A-modules. Then (F_n) extends canonically to X. In particular, if A is complete, then (F_n) is the completion of a coherent O_U-module.","statement_latex":"In Situation \\ref{situation-algebraize} let\n$(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$.\nAssume\n\\begin{enumerate}\n\\item there exist $f_1, \\ldots, f_d \\in I$ such that\nfor $y \\in U \\cap Y$ the ideal $I\\mathcal{O}_{X, y}$\nis generated by $f_1, \\ldots, f_d$ and\n$f_1, \\ldots, f_d$ form a $\\mathcal{F}_y^\\wedge$-regular sequence,\n\\item $H^0(U, \\mathcal{F}_1)$ and $H^1(U, \\mathcal{F}_1)$\nare finite $A$-modules.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$\nis complete, then $(\\mathcal{F}_n)$ is the completion of a coherent\n$\\mathcal{O}_U$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, VI","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJS","source_file":"algebraization.tex","source_line":7708,"source_end_line":7724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7708-L7724","statement_sha256":"ba48b26fccb40754862a201f5bbad3761a7fc8a9a21bc31848ebe9ee9d718df2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9270,"rank":9270,"depth":7,"x":1128.772,"y":1670.472,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EJU","tag":"0EJU","title":"Algebraization of coherent formal modules, VI · Proposition 0EJU","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume there is Noetherian local ring (R, m) and a ring map R → A such that • I = m A, • for y ∈ U ∩ Y the stalk F_y^wedge is R-flat, • H^0(U, F_1) and H^1(U, F_1) are finite A-modules. Then (F_n) extends canonically to X. In particular, if A is complete, then (F_n) is the completion of a coherent O_U-module.","statement_latex":"In Situation \\ref{situation-algebraize} let\n$(\\mathcal{F}_n)$ be an object of $\\textit{Coh}(U, I\\mathcal{O}_U)$.\nAssume there is Noetherian local ring $(R, \\mathfrak m)$ and a ring\nmap $R \\to A$ such that\n\\begin{enumerate}\n\\item $I = \\mathfrak m A$,\n\\item for $y \\in U \\cap Y$ the stalk $\\mathcal{F}_y^\\wedge$ is $R$-flat,\n\\item $H^0(U, \\mathcal{F}_1)$ and $H^1(U, \\mathcal{F}_1)$ are finite\n$A$-modules.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ extends canonically to $X$. In particular, if $A$\nis complete, then $(\\mathcal{F}_n)$ is the completion of a coherent\n$\\mathcal{O}_U$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Algebraization of coherent formal modules, VI","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJU","source_file":"algebraization.tex","source_line":7780,"source_end_line":7795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7780-L7795","statement_sha256":"3df72dbc8e1c605980e357f7793ad856cf2b84d9265eb5574344a86af0405edc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9271,"rank":9271,"depth":8,"x":1073.979,"y":1547.098,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKY","tag":"0EKY","title":"Application to the completion functor · Lemma 0EKY","summary":"In Situation [Tag 0EHC] assume • A has a dualizing complex and is I-adically complete, • I = (f) generated by a single element, • A is local with maximal ideal a = m, • one of the following is true • A_f is (S_2) and for p ⊂ A, f not ∈ p minimal we have dim(A/ p) ≥ 4, or • if p not ∈ V(f) and V( p) ∩ V(f) not = ( m), then depth(A_ p) + dim(A/ p) > 3. Then with U_0 = U ∩ V(f) the completion functor colim_U_0 ⊂ U' ⊂ U open Coh(O_U') → Coh(U, fO_U) is an equivalence on the…","statement_latex":"In Situation \\ref{situation-algebraize} assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and is $I$-adically complete,\n\\item $I = (f)$ generated by a single element,\n\\item $A$ is local with maximal ideal $\\mathfrak a = \\mathfrak m$,\n\\item one of the following is true\n\\begin{enumerate}\n\\item $A_f$ is $(S_2)$ and for $\\mathfrak p \\subset A$,\n$f \\not \\in \\mathfrak p$ minimal we have $\\dim(A/\\mathfrak p) \\geq 4$, or\n\\item if $\\mathfrak p \\not \\in V(f)$ and\n$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.\n\\end{enumerate}\n\\end{enumerate}\nThen with $U_0 = U \\cap V(f)$ the completion functor\n$$\n\\colim_{U_0 \\subset U' \\subset U\\text{ open}}\n\\textit{Coh}(\\mathcal{O}_{U'})\n\\longrightarrow\n\\textit{Coh}(U, f\\mathcal{O}_U)\n$$\nis an equivalence on the full subcategories of finite locally free objects.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to the completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKY","source_file":"algebraization.tex","source_line":7872,"source_end_line":7896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7872-L7896","statement_sha256":"a021f1679fece080d2d34bb0f58bec579f830e7834ed85c1c618eb7470fe6df3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9272,"rank":9272,"depth":55,"x":1214.232,"y":1607.254,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EKZ","tag":"0EKZ","title":"Application to the completion functor · Lemma 0EKZ","summary":"In Situation [Tag 0EHC] assume • I = (f) is principal, • A is f-adically complete, • f is a nonzerodivisor, • H^1_ a(A/fA) and H^2_ a(A/fA) are finite A-modules. Then with U_0 = U ∩ V(f) the completion functor colim_U_0 ⊂ U' ⊂ U open Coh(O_U') → Coh(U, fO_U) is an equivalence on the full subcategories of finite locally free objects.","statement_latex":"In Situation \\ref{situation-algebraize} assume\n\\begin{enumerate}\n\\item $I = (f)$ is principal,\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak a(A/fA)$ and $H^2_\\mathfrak a(A/fA)$\nare finite $A$-modules.\n\\end{enumerate}\nThen with $U_0 = U \\cap V(f)$ the completion functor\n$$\n\\colim_{U_0 \\subset U' \\subset U\\text{ open}}\n\\textit{Coh}(\\mathcal{O}_{U'})\n\\longrightarrow\n\\textit{Coh}(U, f\\mathcal{O}_U)\n$$\nis an equivalence on the full subcategories of finite locally free objects.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to the completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKZ","source_file":"algebraization.tex","source_line":7916,"source_end_line":7934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7916-L7934","statement_sha256":"d7a829a169314b61322709bf3df064870efaea87ecc22b9ac48cf3879c71559e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9273,"rank":9273,"depth":32,"x":1061.749,"y":1642.637,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F23","tag":"0F23","title":"Coherent triples · Lemma 0F23","summary":"For any coherent triple (F, F_0, α) there exists a coherent O_X-module F' such that f : F' → F' is injective, an isomorphism α' : F'|_U → F, and a map α'_0 : F'/fF' → F_0 such that α ∘ (α' bmod f) = α'_0|_U_0.","statement_latex":"For any coherent triple $(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$\nthere exists a coherent $\\mathcal{O}_X$-module $\\mathcal{F}'$\nsuch that $f : \\mathcal{F}' \\to \\mathcal{F}'$ is injective,\nan isomorphism $\\alpha' : \\mathcal{F}'|_U \\to \\mathcal{F}$, and a map\n$\\alpha'_0 : \\mathcal{F}'/f\\mathcal{F}' \\to \\mathcal{F}_0$\nsuch that $\\alpha \\circ (\\alpha' \\bmod f) = \\alpha'_0|_{U_0}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Coherent triples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F23","source_file":"algebraization.tex","source_line":7995,"source_end_line":8003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L7995-L8003","statement_sha256":"de3308ddd600413a9a7b3dc1382d9c2ecac303a329b5ca8f5e83985f8aa890e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9274,"rank":9274,"depth":30,"x":1146.111,"y":1529.521,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F25","tag":"0F25","title":"Coherent triples · Lemma 0F25","summary":"The quantity chi(F, F_0, α) in ([Tag 0F24]) does not depend on the choice of F', α', α'_0 as in Lemma [Tag 0F23].","statement_latex":"The quantity $\\chi(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$ in\n(\\ref{equation-chi-triple}) does not depend on the choice of\n$\\mathcal{F}', \\alpha', \\alpha'_0$ as in Lemma \\ref{lemma-prepare-chi-triple}.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Coherent triples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F25","source_file":"algebraization.tex","source_line":8040,"source_end_line":8045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8040-L8045","statement_sha256":"aeab8424d36d117f6c8ca8bbe3ffbd238fa3707c363d3fea9ffa0d8e29291d53","origin":"The Stacks Project","memory_eligible":false,"source_rank":9275,"rank":9275,"depth":31,"x":1174.995,"y":1661.382,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F26","tag":"0F26","title":"Coherent triples · Lemma 0F26","summary":"We have chi(G, G_0, β) = chi(F, F_0, α) + chi(H, H_0, γ) if 0 → (F, F_0, α) → (G, G_0, β) → (H, H_0, γ) → 0 is a short exact sequence of coherent triples.","statement_latex":"We have\n$\\chi(\\mathcal{G}, \\mathcal{G}_0, \\beta) =\n\\chi(\\mathcal{F}, \\mathcal{F}_0, \\alpha) +\n\\chi(\\mathcal{H}, \\mathcal{H}_0, \\gamma)$ if\n$$\n0 \\to\n(\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\to\n(\\mathcal{G}, \\mathcal{G}_0, \\beta) \\to\n(\\mathcal{H}, \\mathcal{H}_0, \\gamma)\n\\to 0\n$$\nis a short exact sequence of coherent triples.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Coherent triples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F26","source_file":"algebraization.tex","source_line":8132,"source_end_line":8146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8132-L8146","statement_sha256":"95776428307f10dcfbc396a8ced3d3ce8fa565bc6f643d8d222260226ac38613","origin":"The Stacks Project","memory_eligible":false,"source_rank":9276,"rank":9276,"depth":31,"x":1047.099,"y":1580.183,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F27","tag":"0F27","title":"Coherent triples · Proposition 0F27","summary":"Let (F, F_0, α) be a coherent triple. Let (L, L_0, λ) be an invertible coherent triple. Then the function Z → Z, n ↦ chi((F, F_0, α) ⊗ (L, L_0, λ)^⊗ n) is a polynomial of degree ≤ dim(Supp(F)).","statement_latex":"Let $(\\mathcal{F}, \\mathcal{F}_0, \\alpha)$ be a coherent triple.\nLet $(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$ be an invertible coherent\ntriple. Then the function\n$$\n\\mathbf{Z} \\longrightarrow \\mathbf{Z},\\quad\nn \\longmapsto\n\\chi((\\mathcal{F}, \\mathcal{F}_0, \\alpha) \\otimes\n(\\mathcal{L}, \\mathcal{L}_0, \\lambda)^{\\otimes n})\n$$\nis a polynomial of degree $\\leq \\dim(\\text{Supp}(\\mathcal{F}))$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Coherent triples","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F27","source_file":"algebraization.tex","source_line":8285,"source_end_line":8297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8285-L8297","statement_sha256":"a366b53da90013502ffebbb0a3746a02d1bf549e6e435c91ad06ca5d4c9645d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9277,"rank":9277,"depth":0,"x":1207.4,"y":1567.432,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F28","tag":"0F28","title":"Coherent triples · Lemma 0F28","summary":"Assume depth(A) ≥ 3 or equivalently depth(A/fA) ≥ 2. Let (L, L_0, λ) be an invertible coherent triple. Then chi(L, L_0, λ) = length_A Coker(Γ(U, L) → Γ(U_0, L_0)) and in particular this is ≥ 0. Moreover, chi(L, L_0, λ) = 0 if and only if L ≅ O_U.","statement_latex":"Assume $\\text{depth}(A) \\geq 3$ or equivalently\n$\\text{depth}(A/fA) \\geq 2$. Let $(\\mathcal{L}, \\mathcal{L}_0, \\lambda)$\nbe an invertible coherent triple. Then\n$$\n\\chi(\\mathcal{L}, \\mathcal{L}_0, \\lambda) =\n\\text{length}_A \\Coker(\\Gamma(U, \\mathcal{L}) \\to \\Gamma(U_0, \\mathcal{L}_0))\n$$\nand in particular this is $\\geq 0$. Moreover, \n$\\chi(\\mathcal{L}, \\mathcal{L}_0, \\lambda) = 0$ if and only if\n$\\mathcal{L} \\cong \\mathcal{O}_U$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Coherent triples","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F28","source_file":"algebraization.tex","source_line":8394,"source_end_line":8406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8394-L8406","statement_sha256":"350e8dedf7586ae4b002fbb8af68e89f4029c57d2604475e2d9f4381d21898c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9278,"rank":9278,"depth":32,"x":1098.984,"y":1668.226,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F2A","tag":"0F2A","title":"Invertible modules on punctured spectra, I · Lemma 0F2A","summary":"Let (A, m) be a Noetherian local ring. Let f ∈ m be a nonzerodivisor and assume that depth(A/fA) ≥ 2, or equivalently depth(A) ≥ 3. Let U, resp. U_0 be the punctured spectrum of A, resp. A/fA. The map Pic(U) → Pic(U_0) is injective on torsion.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$\nbe a nonzerodivisor and assume that $\\text{depth}(A/fA) \\geq 2$, or equivalently\n$\\text{depth}(A) \\geq 3$. Let $U$, resp.\\ $U_0$ be the punctured\nspectrum of $A$, resp.\\ $A/fA$. The map\n$$\n\\Pic(U) \\to \\Pic(U_0)\n$$\nis injective on torsion.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Invertible modules on punctured spectra, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2A","source_file":"algebraization.tex","source_line":8481,"source_end_line":8491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8481-L8491","statement_sha256":"a79adda80d8fd4a5bffb6f76983ced4996cc4f8a5c420086641be0ee9d9159b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9279,"rank":9279,"depth":33,"x":1097.869,"y":1531.802,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F2B","tag":"0F2B","title":"Koll\\'ar · Proposition 0F2B","summary":"[Kollar-map-pic] Let (A, m) be a Noetherian local ring. Let f ∈ m. Assume • A has a dualizing complex, • f is a nonzerodivisor, • depth(A/fA) ≥ 2, or equivalently depth(A) ≥ 3, • if f ∈ p ⊂ A is a prime ideal with dim(A/ p) = 2, then depth(A_ p) ≥ 2. Let U, resp. U_0 be the punctured spectrum of A, resp. A/fA. The map Pic(U) → Pic(U_0) is injective. Finally, if (1), (2), (3), A is (S_2), and dim(A) ≥ 4, then (4) holds.","statement_latex":"\\begin{reference}\n\\cite[Theorem 1.9]{Kollar-map-pic}\n\\end{reference}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring. Let $f \\in \\mathfrak m$.\nAssume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex,\n\\item $f$ is a nonzerodivisor,\n\\item $\\text{depth}(A/fA) \\geq 2$, or equivalently $\\text{depth}(A) \\geq 3$,\n\\item if $f \\in \\mathfrak p \\subset A$ is a prime ideal with\n$\\dim(A/\\mathfrak p) = 2$, then $\\text{depth}(A_\\mathfrak p) \\geq 2$.\n\\end{enumerate}\nLet $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$. The map\n$$\n\\Pic(U) \\to \\Pic(U_0)\n$$\nis injective. Finally, if (1), (2), (3), $A$ is $(S_2)$, and\n$\\dim(A) \\geq 4$, then (4) holds.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Invertible modules on punctured spectra, I","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2B","source_file":"algebraization.tex","source_line":8512,"source_end_line":8532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8512-L8532","statement_sha256":"97e093a2ca5aedb8ddbdb7c9c5a16543f60e8669b83178872494c779363c3ec5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9280,"rank":9280,"depth":51,"x":1208.867,"y":1632.192,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F2D","tag":"0F2D","title":"Invertible modules on punctured spectra, II · Lemma 0F2D","summary":"Let (A, m) be a Noetherian local ring and f ∈ m. Assume • A is f-adically complete, • f is a nonzerodivisor, • H^1_ m(A/fA) and H^2_ m(A/fA) are finite A-modules, and • H^3_ m(A/fA) = 0(A/fA) ≥ 4, or equivalently depth(A) ≥ 5.. Let U, resp. U_0 be the punctured spectrum of A, resp. A/fA. Then colim_U_0 ⊂ U' ⊂ U open Pic(U') → Pic(U_0) is surjective.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring and $f \\in \\mathfrak m$.\nAssume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$\nare finite $A$-modules, and\n\\item $H^3_\\mathfrak m(A/fA) = 0$\\footnote{Observe that (3) and (4) hold\nif $\\text{depth}(A/fA) \\geq 4$, or equivalently $\\text{depth}(A) \\geq 5$.}.\n\\end{enumerate}\nLet $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$.\nThen\n$$\n\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\Pic(U')\n\\longrightarrow\n\\Pic(U_0)\n$$\nis surjective.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Invertible modules on punctured spectra, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2D","source_file":"algebraization.tex","source_line":8734,"source_end_line":8754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8734-L8754","statement_sha256":"8cdf518c66fed2505a14d2958b7c919e787ba19f1dcd12332eed56289a6addba","origin":"The Stacks Project","memory_eligible":false,"source_rank":9281,"rank":9281,"depth":33,"x":1045.605,"y":1621.102,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F2F","tag":"0F2F","title":"Invertible modules on punctured spectra, II · Lemma 0F2F","summary":"Let (A, m) be a Noetherian local ring and f ∈ m. Assume • the conditions of Lemma [Tag 0F2D] hold, and • for every maximal ideal p ⊂ A_f the punctured spectrum of (A_f)_ p has trivial Picard group. Let U, resp. U_0 be the punctured spectrum of A, resp. A/fA. Then Pic(U) → Pic(U_0) is surjective.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring and $f \\in \\mathfrak m$.\nAssume\n\\begin{enumerate}\n\\item the conditions of Lemma \\ref{lemma-surjective-Pic-first} hold, and\n\\item for every maximal ideal $\\mathfrak p \\subset A_f$\nthe punctured spectrum of $(A_f)_\\mathfrak p$ has trivial Picard group.\n\\end{enumerate}\nLet $U$, resp.\\ $U_0$ be the punctured spectrum of $A$, resp.\\ $A/fA$.\nThen\n$$\n\\Pic(U) \\longrightarrow \\Pic(U_0)\n$$\nis surjective.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Invertible modules on punctured spectra, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2F","source_file":"algebraization.tex","source_line":8804,"source_end_line":8819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8804-L8819","statement_sha256":"178e81dea2fab04ae4e555a6126d7d6018efaa25c7fd786cd035bd2cd6d027df","origin":"The Stacks Project","memory_eligible":false,"source_rank":9282,"rank":9282,"depth":34,"x":1175.45,"y":1536.289,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F2G","tag":"0F2G","title":"Invertible modules on punctured spectra, II · Lemma 0F2G","summary":"Let (A, m) be a Noetherian local ring of depth ≥ 2. Let A^wedge be its completion. Let U, resp. U^wedge be the punctured spectrum of A, resp. A^wedge. Then Pic(U) → Pic(U^wedge) is injective.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring of depth $\\geq 2$.\nLet $A^\\wedge$ be its completion. Let $U$, resp.\\ $U^\\wedge$\nbe the punctured spectrum of $A$, resp.\\ $A^\\wedge$. Then\n$\\Pic(U) \\to \\Pic(U^\\wedge)$ is injective.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Invertible modules on punctured spectra, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2G","source_file":"algebraization.tex","source_line":8840,"source_end_line":8846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8840-L8846","statement_sha256":"23d22dd7a445c6999781ae280f1b2c9d79ce22af4318c4dfa652d9b33914fb6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9283,"rank":9283,"depth":30,"x":1147.793,"y":1673.069,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F2H","tag":"0F2H","title":"Invertible modules on punctured spectra, II · Lemma 0F2H","summary":"Let (A, m) be a regular local ring. Then the Picard group of the punctured spectrum of A is trivial.","statement_latex":"Let $(A, \\mathfrak m)$ be a regular local ring. Then the Picard\ngroup of the punctured spectrum of $A$ is trivial.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Invertible modules on punctured spectra, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2H","source_file":"algebraization.tex","source_line":8865,"source_end_line":8869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8865-L8869","statement_sha256":"680427b63fd18c4f1a9c827a712f71556c02fd5ad729c9755d19193ec71e9163","origin":"The Stacks Project","memory_eligible":false,"source_rank":9284,"rank":9284,"depth":26,"x":1057.826,"y":1556.038,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F2I","tag":"0F2I","title":"Grothendieck · Proposition 0F2I","summary":"Let (A, m) be a Noetherian local ring. If A is a complete intersection of dimension ≥ 4, then the Picard group of the punctured spectrum of A is trivial.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is a\ncomplete intersection of dimension $\\geq 4$, then the Picard\ngroup of the punctured spectrum of $A$ is trivial.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Invertible modules on punctured spectra, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F2I","source_file":"algebraization.tex","source_line":8885,"source_end_line":8890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L8885-L8890","statement_sha256":"a7c99e51ae5ffb1aa30f16fba3d812ff0c8b07648aa6e882553218111dd3c314","origin":"The Stacks Project","memory_eligible":false,"source_rank":9285,"rank":9285,"depth":35,"x":1218.933,"y":1591.429,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EL1","tag":"0EL1","title":"Application to Lefschetz theorems · Proposition 0EL1","summary":"In the situation above assume there exists an integer σ such that for all points p ∈ P setminus Q we have depth(F_p) + dim(overline(p)) > σ Then the map H^i(P, F) → lim H^i(Q_n, F_n) is an isomorphism for 0 ≤ i < σ.","statement_latex":"In the situation above assume there exists an integer $\\sigma$ such that\nfor all points $p \\in P \\setminus Q$ we have\n$$\n\\text{depth}(\\mathcal{F}_p) + \\dim(\\overline{\\{p\\}}) > \\sigma\n$$\nThen the map\n$$\nH^i(P, \\mathcal{F}) \\longrightarrow \\lim H^i(Q_n, \\mathcal{F}_n)\n$$\nis an isomorphism for $0 \\leq i < \\sigma$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to Lefschetz theorems","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EL1","source_file":"algebraization.tex","source_line":9020,"source_end_line":9032,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L9020-L9032","statement_sha256":"cb89e92a81cdc1dd82e899576810c2b83e906dfc76be461264c9ebb40fb0083f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9286,"rank":9286,"depth":35,"x":1071.079,"y":1657.011,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EL2","tag":"0EL2","title":"Application to Lefschetz theorems · Lemma 0EL2","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module. Let s ∈ Γ(X, L). Let Y = Z(s) be the zero scheme of s with nth infinitesimal neighbourhood Y_n = Z(s^n). Let F be a coherent O_X-module. Assume that for all x ∈ X setminus Y we have depth(F_x) + dim(overline(x)) > 1 Then Γ(V, F) → lim Γ(Y_n, F|_Y_n) is an isomorphism for any open subscheme V ⊂ X containing Y.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\nLet $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the\nzero scheme of $s$ with $n$th infinitesimal neighbourhood $Y_n = Z(s^n)$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nAssume that for all $x \\in X \\setminus Y$ we have\n$$\n\\text{depth}(\\mathcal{F}_x) + \\dim(\\overline{\\{x\\}}) > 1\n$$\nThen $\\Gamma(V, \\mathcal{F}) \\to \\lim \\Gamma(Y_n, \\mathcal{F}|_{Y_n})$\nis an isomorphism for any open subscheme $V \\subset X$ containing $Y$.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to Lefschetz theorems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EL2","source_file":"algebraization.tex","source_line":9154,"source_end_line":9167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L9154-L9167","statement_sha256":"c1ef47d31f99af523c614f9c744f43c60524bb1e4e9c31850b6217ebb2a23f4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9287,"rank":9287,"depth":36,"x":1127.591,"y":1524.227,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EL4","tag":"0EL4","title":"Application to Lefschetz theorems · Lemma 0EL4","summary":"In Situation [Tag 0EHC] let (F_n) be an object of Coh(U, IO_U). Assume • A is a graded ring, a = A_+, and I is a homogeneous ideal, • (F_n) = (widetildeM_n|_U) where (M_n) is an inverse system of graded A-modules, and • (F_n) extends canonically to X. Then there is a finite graded A-module N such that • [(a)] the inverse systems (N/I^nN) and (M_n) are pro-isomorphic in the category of graded A-modules modulo A_+-power torsion modules, and • [(b)] (F_n) is the completion…","statement_latex":"In Situation \\ref{situation-algebraize} let $(\\mathcal{F}_n)$ be an\nobject of $\\textit{Coh}(U, I\\mathcal{O}_U)$. Assume\n\\begin{enumerate}\n\\item $A$ is a graded ring, $\\mathfrak a = A_+$, and\n$I$ is a homogeneous ideal,\n\\item $(\\mathcal{F}_n) = (\\widetilde{M_n}|_U)$ where $(M_n)$\nis an inverse system of graded $A$-modules, and\n\\item $(\\mathcal{F}_n)$ extends canonically to $X$.\n\\end{enumerate}\nThen there is a finite graded $A$-module $N$ such that\n\\begin{enumerate}\n\\item[(a)] the inverse systems $(N/I^nN)$ and $(M_n)$ are pro-isomorphic\nin the category of graded $A$-modules modulo $A_+$-power torsion\nmodules, and\n\\item[(b)] $(\\mathcal{F}_n)$ is the completion of of the coherent\nmodule associated to $N$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to Lefschetz theorems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EL4","source_file":"algebraization.tex","source_line":9220,"source_end_line":9239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L9220-L9239","statement_sha256":"cb21e14064fe5c86643cae546a231898046b1496da4150d9c052e09c9989b402","origin":"The Stacks Project","memory_eligible":false,"source_rank":9288,"rank":9288,"depth":31,"x":1192.958,"y":1654.723,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EL5","tag":"0EL5","title":"Application to Lefschetz theorems · Proposition 0EL5","summary":"In the situation above let (F_n) be an object of Coh(P, I). Assume for all q ∈ Q and for all primes p ∈ O_P, q^wedge, p not ∈ V(I_q^wedge) we have depth((F_q^wedge)_ p) + dim(O_P, q^wedge/ p) + dim(overline(q)) > 2 Then (F_n) is the completion of a coherent O_P-module.","statement_latex":"In the situation above let $(\\mathcal{F}_n)$ be an object of\n$\\textit{Coh}(P, \\mathcal{I})$. Assume for all $q \\in Q$ and for\nall primes $\\mathfrak p \\in \\mathcal{O}_{P, q}^\\wedge$,\n$\\mathfrak p \\not \\in V(\\mathcal{I}_q^\\wedge)$ we have\n$$\n\\text{depth}((\\mathcal{F}_q^\\wedge)_\\mathfrak p) +\n\\dim(\\mathcal{O}_{P, q}^\\wedge/\\mathfrak p) +\n\\dim(\\overline{\\{q\\}}) > 2\n$$\nThen $(\\mathcal{F}_n)$ is the completion of a coherent\n$\\mathcal{O}_P$-module.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to Lefschetz theorems","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EL5","source_file":"algebraization.tex","source_line":9321,"source_end_line":9334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L9321-L9334","statement_sha256":"37eb080561546dd4a14c3e1a5084f3314b1d22f1a833231cbaec94fca393444d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9289,"rank":9289,"depth":54,"x":1039.214,"y":1595.354,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0EL7","tag":"0EL7","title":"Application to Lefschetz theorems · Proposition 0EL7","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module and let s ∈ Γ(X, L). Let Y = Z(s) be the zero scheme of s and denote I ⊂ O_X the corresponding sheaf of ideals. Let V be the set of open subschemes of X containing Y ordered by reverse inclusion. Assume that for all x ∈ X setminus Y we have depth(O_X, x) + dim(overline(x)) > 2 Then the completion functor colim_V Coh(O_V) → Coh(X, I) is an equivalence on the full subcategories of…","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\nLet $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module\nand let $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$\nbe the zero scheme of $s$ and denote $\\mathcal{I} \\subset \\mathcal{O}_X$\nthe corresponding sheaf of ideals.\nLet $\\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$\nordered by reverse inclusion.\nAssume that for all $x \\in X \\setminus Y$ we have\n$$\n\\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2\n$$\nThen the completion functor\n$$\n\\colim_\\mathcal{V}\n\\textit{Coh}(\\mathcal{O}_V)\n\\longrightarrow\n\\textit{Coh}(X, \\mathcal{I})\n$$\nis an equivalence on the full subcategories of finite locally free objects.","area":"Algebraic & Formal Geometry","chapter":"Algebraic and Formal Geometry","chapter_id":"algebraization","section":"Application to Lefschetz theorems","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EL7","source_file":"algebraization.tex","source_line":9447,"source_end_line":9468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraization.tex#L9447-L9468","statement_sha256":"125cc236d2311547d23407322db7c1383d11a17722c1a7057984e2a7a0c8cdbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":9290,"rank":9290,"depth":55,"x":1200.955,"y":1551.724,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0BXY","tag":"0BXY","title":"Curves and function fields · Lemma 0BXY","summary":"Let k be a field. Let X be a curve and Y a proper variety. Let U ⊂ X be a nonempty open and let f : U → Y be a morphism. If x ∈ X is a closed point such that O_X, x is a discrete valuation ring, then there exist an open U ⊂ U' ⊂ X containing x and a morphism of varieties f' : U' → Y extending f.","statement_latex":"Let $k$ be a field. Let $X$ be a curve and $Y$ a proper variety.\nLet $U \\subset X$ be a nonempty open and let $f : U \\to Y$ be a morphism.\nIf $x \\in X$ is a closed point such that $\\mathcal{O}_{X, x}$\nis a discrete valuation ring, then there exist an open\n$U \\subset U' \\subset X$ containing $x$ and a morphism of\nvarieties $f' : U' \\to Y$ extending $f$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves and function fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXY","source_file":"curves.tex","source_line":54,"source_end_line":62,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L54-L62","statement_sha256":"f7cd26c0b8026489c06d27dd198d5ad2d81d91d7ec12a1e651a657d820ff47b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9291,"rank":9291,"depth":18,"x":109.507,"y":1131.849,"cluster":"varieties-curves"},{"id":"stacks:0BXZ","tag":"0BXZ","title":"Curves and function fields · Lemma 0BXZ","summary":"Let k be a field. Let X be a normal curve and Y a proper variety. The set of rational maps from X to Y is the same as the set of morphisms X → Y.","statement_latex":"Let $k$ be a field. Let $X$ be a normal curve and $Y$ a proper variety.\nThe set of rational maps from $X$ to $Y$ is the same as the set\nof morphisms $X \\to Y$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves and function fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BXZ","source_file":"curves.tex","source_line":69,"source_end_line":74,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L69-L74","statement_sha256":"9fb0755ec1e01edce1f9a8021f14021cc4061fa8b057ffc8b1551fc195a81366","origin":"The Stacks Project","memory_eligible":false,"source_rank":9292,"rank":9292,"depth":47,"x":325.578,"y":1077.526,"cluster":"varieties-curves"},{"id":"stacks:0CCK","tag":"0CCK","title":"Curves and function fields · Lemma 0CCK","summary":"Let k be a field. Let f : X → Y be a nonconstant morphism of curves over k. If Y is normal, then f is flat.","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a nonconstant morphism\nof curves over $k$. If $Y$ is normal, then $f$ is flat.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves and function fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCK","source_file":"curves.tex","source_line":85,"source_end_line":89,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L85-L89","statement_sha256":"f2cbe94280cb761d3be5e0ed70bef13c1d6c458a8d401c0f1d5b62f75469d7e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9293,"rank":9293,"depth":47,"x":209.725,"y":1240.482,"cluster":"varieties-curves"},{"id":"stacks:0CCL","tag":"0CCL","title":"Curves and function fields · Lemma 0CCL","summary":"Let k be a field. Let f : X → Y be a morphism of schemes over k. Assume • Y is separated over k, • X is proper of dimension ≤ 1 over k, • f(Z) has at least two points for every irreducible component Z ⊂ X of dimension 1. Then f is finite.","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a morphism of\nschemes over $k$. Assume\n\\begin{enumerate}\n\\item $Y$ is separated over $k$,\n\\item $X$ is proper of dimension $\\leq 1$ over $k$,\n\\item $f(Z)$ has at least two points for every irreducible\ncomponent $Z \\subset X$ of dimension $1$.\n\\end{enumerate}\nThen $f$ is finite.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves and function fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCL","source_file":"curves.tex","source_line":105,"source_end_line":116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L105-L116","statement_sha256":"ad0f430f9a3de27e24e67323dccef7b92dfcacf6ce58999e84baa623548b59bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9294,"rank":9294,"depth":46,"x":164.026,"y":1054.247,"cluster":"varieties-curves"},{"id":"stacks:0BY0","tag":"0BY0","title":"Curves and function fields · Lemma 0BY0","summary":"Let k be a field. Let X → Y be a morphism of varieties with Y proper and X a curve. There exists a factorization X → overlineX → Y where X → overlineX is an open immersion and overlineX is a projective curve.","statement_latex":"Let $k$ be a field. Let $X \\to Y$ be a morphism of varieties\nwith $Y$ proper and $X$ a curve.\nThere exists a factorization $X \\to \\overline{X} \\to Y$\nwhere $X \\to \\overline{X}$ is an open immersion\nand $\\overline{X}$ is a projective curve.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves and function fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BY0","source_file":"curves.tex","source_line":137,"source_end_line":144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L137-L144","statement_sha256":"35c5829d251102ccdc06ecaab8c6958c5da942c28ccd1af04b5a99c034d8008c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9295,"rank":9295,"depth":37,"x":347.82,"y":1165.844,"cluster":"varieties-curves"},{"id":"stacks:0BY1","tag":"0BY1","title":"Curves and function fields · Theorem 0BY1","summary":"Let k be a field. The following categories are canonically equivalent • The category of finitely generated field extensions K/k of transcendence degree 1. • The category of curves and dominant rational maps. • The category of normal projective curves and nonconstant morphisms. • The category of nonsingular projective curves and nonconstant morphisms. • The category of regular projective curves and nonconstant morphisms. • The category of normal proper curves and…","statement_latex":"Let $k$ be a field. The following categories are canonically equivalent\n\\begin{enumerate}\n\\item The category of finitely generated field extensions $K/k$ of\ntranscendence degree $1$.\n\\item The category of curves and dominant rational maps.\n\\item The category of normal projective curves and nonconstant morphisms.\n\\item The category of nonsingular projective curves and nonconstant morphisms.\n\\item The category of regular projective curves and nonconstant morphisms.\n\\item The category of normal proper curves and nonconstant morphisms.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves and function fields","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BY1","source_file":"curves.tex","source_line":158,"source_end_line":170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L158-L170","statement_sha256":"525a3178c05e5d84ce810b5c8f12c53b7ada383cc9dc3017b4de970d2dd8035e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9296,"rank":9296,"depth":48,"x":122.145,"y":1187.882,"cluster":"varieties-curves"},{"id":"stacks:0BY2","tag":"0BY2","title":"Curves and function fields · Definition 0BY2","summary":"Let k be a field. Let X be a curve. A nonsingular projective model of X is a pair (Y, φ) where Y is a nonsingular projective curve and φ : k(X) → k(Y) is an isomorphism of function fields.","statement_latex":"Let $k$ be a field. Let $X$ be a curve.\nA {\\it nonsingular projective model of $X$}\nis a pair $(Y, \\varphi)$ where $Y$ is a nonsingular projective\ncurve and $\\varphi : k(X) \\to k(Y)$ is an isomorphism\nof function fields.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves and function fields","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BY2","source_file":"curves.tex","source_line":230,"source_end_line":237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L230-L237","statement_sha256":"4287b374770a35e3a91f43f927f1454f08c3fe0594f67b80b04614af98996dfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9297,"rank":9297,"depth":0,"x":271.098,"y":1043.321,"cluster":"varieties-curves"},{"id":"stacks:0BY3","tag":"0BY3","title":"Curves and function fields · Lemma 0BY3","summary":"Let k be a field. Let X be a curve and let Y be the nonsingular projective model of X. If k is perfect, then Y is a smooth projective curve.","statement_latex":"Let $k$ be a field. Let $X$ be a curve and let $Y$ be the nonsingular\nprojective model of $X$. If $k$ is perfect, then $Y$ is a smooth\nprojective curve.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves and function fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BY3","source_file":"curves.tex","source_line":248,"source_end_line":253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L248-L253","statement_sha256":"83c509664552c7778d6cd1d89f887ec0149ee104514e680c0942275aa9b7a095","origin":"The Stacks Project","memory_eligible":false,"source_rank":9298,"rank":9298,"depth":47,"x":277.526,"y":1234.778,"cluster":"varieties-curves"},{"id":"stacks:0BY4","tag":"0BY4","title":"Curves and function fields · Lemma 0BY4","summary":"Let k be a field. Let X be a geometrically irreducible curve over k. For a field extension K/k denote Y_K a nonsingular projective model of (X_K)_red. • If X is proper, then Y_K is the normalization of X_K. • There exists K/k finite purely inseparable such that Y_K is smooth. • Whenever Y_K is smooth we have H^0(Y_K, O_Y_K) = K. • Given a commutative diagram xymatrix Ω & K' ar[l] K ar[u] & k ar[l] ar[u] of fields such that Y_K and Y_K' are smooth, then Y_Ω = (Y_K)_Ω =…","statement_latex":"Let $k$ be a field. Let $X$ be a geometrically irreducible curve over $k$.\nFor a field extension $K/k$ denote $Y_K$ a nonsingular projective model\nof $(X_K)_{red}$.\n\\begin{enumerate}\n\\item If $X$ is proper, then $Y_K$ is the normalization of $X_K$.\n\\item There exists $K/k$ finite purely inseparable such that $Y_K$ is smooth.\n\\item Whenever $Y_K$ is smooth\\footnote{Or even geometrically reduced.}\nwe have $H^0(Y_K, \\mathcal{O}_{Y_K}) = K$.\n\\item Given a commutative diagram\n$$\n\\xymatrix{\n\\Omega & K' \\ar[l] \\\\\nK \\ar[u] & k \\ar[l] \\ar[u]\n}\n$$\nof fields such that $Y_K$ and $Y_{K'}$ are smooth, then\n$Y_\\Omega = (Y_K)_\\Omega = (Y_{K'})_\\Omega$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves and function fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BY4","source_file":"curves.tex","source_line":260,"source_end_line":280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L260-L280","statement_sha256":"19ca0c9e24237fe7aab4457f42d6091a4c51102e3da1032941cda46107f07894","origin":"The Stacks Project","memory_eligible":false,"source_rank":9299,"rank":9299,"depth":44,"x":118.54,"y":1097.004,"cluster":"varieties-curves"},{"id":"stacks:0CCN","tag":"0CCN","title":"Linear series · Definition 0CCN","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. Let d ≥ 0 and r ≥ 0. A linear series of degree d and dimension r is a pair (L, V) where L is an invertible O_X-module of degree d (Varieties, Definition [Tag 0AYR]) and V ⊂ H^0(X, L) is a k-subvector space of dimension r + 1. We will abbreviate this by saying (L, V) is a g^r_d on X.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.\nLet $d \\geq 0$ and $r \\geq 0$.\nA {\\it linear series of degree $d$ and dimension $r$}\nis a pair $(\\mathcal{L}, V)$ where $\\mathcal{L}$ is an\ninvertible $\\mathcal{O}_X$-module of degree $d$\n(Varieties, Definition \\ref{varieties-definition-degree-invertible-sheaf})\nand $V \\subset H^0(X, \\mathcal{L})$ is a $k$-subvector space\nof dimension $r + 1$. We will abbreviate this by saying\n$(\\mathcal{L}, V)$ is a {\\it $\\mathfrak g^r_d$} on $X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Linear series","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCN","source_file":"curves.tex","source_line":338,"source_end_line":349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L338-L349","statement_sha256":"73ef89997266dca3298b03bc50c1e5f9acc937e11fff09be46832364a8a9e928","origin":"The Stacks Project","memory_eligible":false,"source_rank":9300,"rank":9300,"depth":1,"x":346.972,"y":1108.404,"cluster":"varieties-curves"},{"id":"stacks:0CCP","tag":"0CCP","title":"Linear series · Lemma 0CCP","summary":"Let k be a field. Let X be a nonsingular proper curve over k. Let (L, V) be a g^r_d on X. Then there exists a morphism φ : X → P^r_k = Proj(k[T_0, …, T_r]) of varieties over k and a map α : φ^*O_P^r_k(1) → L such that φ^*T_0, …, φ^*T_r are sent to a basis of V by α.","statement_latex":"Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$.\nLet $(\\mathcal{L}, V)$ be a $\\mathfrak g^r_d$ on $X$. Then\nthere exists a morphism\n$$\n\\varphi : X \\longrightarrow \\mathbf{P}^r_k = \\text{Proj}(k[T_0, \\ldots, T_r])\n$$\nof varieties over $k$ and a map\n$\\alpha : \\varphi^*\\mathcal{O}_{\\mathbf{P}^r_k}(1) \\to \\mathcal{L}$\nsuch that $\\varphi^*T_0, \\ldots, \\varphi^*T_r$\nare sent to a basis of $V$ by $\\alpha$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Linear series","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCP","source_file":"curves.tex","source_line":365,"source_end_line":377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L365-L377","statement_sha256":"fdb5a803b3a4dc320fdcd3a16373dc356665d79c8b956696b7666cb140138b6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9301,"rank":9301,"depth":19,"x":169.048,"y":1229.829,"cluster":"varieties-curves"},{"id":"stacks:0CCQ","tag":"0CCQ","title":"Linear series · Lemma 0CCQ","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. If X has a g^r_d, then X has a g^s_d for all 0 ≤ s ≤ r.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.\nIf $X$ has a $\\mathfrak g^r_d$, then $X$ has a $\\mathfrak g^s_d$ for\nall $0 \\leq s \\leq r$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Linear series","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCQ","source_file":"curves.tex","source_line":395,"source_end_line":400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L395-L400","statement_sha256":"8c2aa426493de899d751d9c59bed61716b1e4450884c8847497a1bd17b5e7e79","origin":"The Stacks Project","memory_eligible":false,"source_rank":9302,"rank":9302,"depth":0,"x":202.659,"y":1039.0,"cluster":"varieties-curves"},{"id":"stacks:0CCR","tag":"0CCR","title":"Linear series · Lemma 0CCR","summary":"Let k be a field. Let X be a nonsingular proper curve over k. Let (L, V) be a g^1_d on X. Then the morphism φ : X → P^1_k of Lemma [Tag 0CCP] either • is nonconstant and has degree ≤ d, or • factors through a closed point of P^1_k and in this case H^0(X, O_X) not = k.","statement_latex":"Let $k$ be a field. Let $X$ be a nonsingular proper curve over $k$.\nLet $(\\mathcal{L}, V)$ be a $\\mathfrak g^1_d$ on $X$. Then the morphism\n$\\varphi : X \\to \\mathbf{P}^1_k$ of Lemma \\ref{lemma-linear-series}\neither\n\\begin{enumerate}\n\\item is nonconstant and has degree $\\leq d$, or\n\\item factors through a closed point of $\\mathbf{P}^1_k$ and in this\ncase $H^0(X, \\mathcal{O}_X) \\not = k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Linear series","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCR","source_file":"curves.tex","source_line":408,"source_end_line":419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L408-L419","statement_sha256":"2758342686b5070569fc1892df4878fa65ecf75d11a7ed4b8abf57c639248e81","origin":"The Stacks Project","memory_eligible":false,"source_rank":9303,"rank":9303,"depth":45,"x":331.56,"y":1199.065,"cluster":"varieties-curves"},{"id":"stacks:0CCS","tag":"0CCS","title":"Linear series · Lemma 0CCS","summary":"Let k be a field. Let X be a proper curve over k with H^0(X, O_X) = k. If X has a g^r_d, then r ≤ d. If equality holds, then H^1(X, O_X) = 0, i.e., the genus of X (Definition [Tag 0BY7]) is 0.","statement_latex":"Let $k$ be a field. Let $X$ be a proper curve over $k$ with\n$H^0(X, \\mathcal{O}_X) = k$. If $X$ has a $\\mathfrak g^r_d$, then\n$r \\leq d$. If equality holds, then $H^1(X, \\mathcal{O}_X) = 0$, i.e.,\nthe genus of $X$ (Definition \\ref{definition-genus}) is $0$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Linear series","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCS","source_file":"curves.tex","source_line":442,"source_end_line":448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L442-L448","statement_sha256":"7392167629a82374576e983f3c7598d8b9fe267f56ef81ae96cf6b4716f7299b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9304,"rank":9304,"depth":46,"x":107.4,"y":1154.098,"cluster":"varieties-curves"},{"id":"stacks:0BS2","tag":"0BS2","title":"Duality · Lemma 0BS2","summary":"Let X be a proper scheme of dimension ≤ 1 over a field k. There exists a dualizing complex ω_X^bullet with the following properties • H^i(ω_X^bullet) is nonzero only for i = -1, 0, • ω_X = H^-1(ω_X^bullet) is a coherent Cohen-Macaulay module whose support is the irreducible components of dimension 1, • for x ∈ X closed, the module H^0(ω_X, x^bullet) is nonzero if and only if either • dim(O_X, x) = 0 or • dim(O_X, x) = 1 and O_X, x is not Cohen-Macaulay, • for K ∈…","statement_latex":"Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$.\nThere exists a dualizing complex $\\omega_X^\\bullet$ with the\nfollowing properties\n\\begin{enumerate}\n\\item $H^i(\\omega_X^\\bullet)$ is nonzero only for $i = -1, 0$,\n\\item $\\omega_X = H^{-1}(\\omega_X^\\bullet)$\nis a coherent Cohen-Macaulay module whose support is the\nirreducible components of dimension $1$,\n\\item for $x \\in X$ closed, the module $H^0(\\omega_{X, x}^\\bullet)$\nis nonzero if and only if either\n\\begin{enumerate}\n\\item $\\dim(\\mathcal{O}_{X, x}) = 0$ or\n\\item $\\dim(\\mathcal{O}_{X, x}) = 1$\nand $\\mathcal{O}_{X, x}$ is not Cohen-Macaulay,\n\\end{enumerate}\n\\item for $K \\in D_\\QCoh(\\mathcal{O}_X)$ there are functorial\nisomorphisms\\footnote{This property\ncharacterizes $\\omega_X^\\bullet$ in $D_\\QCoh(\\mathcal{O}_X)$\nup to unique isomorphism by the Yoneda lemma. Since $\\omega_X^\\bullet$\nis in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ in fact it suffices to consider\n$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.}\n$$\n\\Ext^i_X(K, \\omega_X^\\bullet) = \\Hom_k(H^{-i}(X, K), k)\n$$\ncompatible with shifts and distinguished triangles,\n\\item there are functorial isomorphisms\n$\\Hom(\\mathcal{F}, \\omega_X) = \\Hom_k(H^1(X, \\mathcal{F}), k)$\nfor $\\mathcal{F}$ quasi-coherent on $X$,\n\\item if $X \\to \\Spec(k)$ is smooth of relative dimension $1$,\nthen $\\omega_X \\cong \\Omega_{X/k}$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BS2","source_file":"curves.tex","source_line":578,"source_end_line":611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L578-L611","statement_sha256":"87b7aa74a4d54040b2d4868d8814896ae0c7cd19328c5e958589c6feb82d1ae4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9305,"rank":9305,"depth":46,"x":309.204,"y":1059.899,"cluster":"varieties-curves"},{"id":"stacks:0BS3","tag":"0BS3","title":"Duality · Lemma 0BS3","summary":"Let X be a proper scheme over a field k which is Cohen-Macaulay and equidimensional of dimension 1. The module ω_X of Lemma [Tag 0BS2] has the following properties • ω_X is a dualizing module on X (Duality for Schemes, Section [Tag 0AWH]), • ω_X is a coherent Cohen-Macaulay module whose support is X, • there are functorial isomorphisms Ext^i_X(K, ω_X[1]) = Hom_k(H^-i(X, K), k) compatible with shifts for K ∈ D_QCoh(X), • there are functorial isomorphisms Ext^1 + i(F, ω_X)…","statement_latex":"Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay\nand equidimensional of dimension $1$. The module $\\omega_X$\nof Lemma \\ref{lemma-duality-dim-1} has the following properties\n\\begin{enumerate}\n\\item $\\omega_X$ is a dualizing module on $X$\n(Duality for Schemes, Section \\ref{duality-section-dualizing-module}),\n\\item $\\omega_X$ is a coherent Cohen-Macaulay module whose support is $X$,\n\\item there are functorial isomorphisms\n$\\Ext^i_X(K, \\omega_X[1]) = \\Hom_k(H^{-i}(X, K), k)$\ncompatible with shifts for $K \\in D_\\QCoh(X)$,\n\\item there are functorial isomorphisms\n$\\Ext^{1 + i}(\\mathcal{F}, \\omega_X) = \\Hom_k(H^{-i}(X, \\mathcal{F}), k)$\nfor $\\mathcal{F}$ quasi-coherent on $X$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BS3","source_file":"curves.tex","source_line":646,"source_end_line":662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L646-L662","statement_sha256":"5bed9d3cc1d37e64b87bd3f6108fa4caa4160855f795a5869ccca06fc4a85889","origin":"The Stacks Project","memory_eligible":false,"source_rank":9306,"rank":9306,"depth":47,"x":236.022,"y":1244.187,"cluster":"varieties-curves"},{"id":"stacks:0E32","tag":"0E32","title":"Duality · Lemma 0E32","summary":"Let X be a proper scheme of dimension ≤ 1 over a field k. Let ω_X^bullet and ω_X be as in Lemma [Tag 0BS2]. • If X → Spec(k) factors as X → Spec(k') → Spec(k) for some field k', then ω_X^bullet and ω_X satisfy properties (4), (5), (6) with k replaced with k'. • If K/k is a field extension, then the pullback of ω_X^bullet and ω_X to the base change X_K are as in Lemma [Tag 0BS2] for the morphism X_K → Spec(K).","statement_latex":"Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$.\nLet $\\omega_X^\\bullet$ and $\\omega_X$ be as in Lemma \\ref{lemma-duality-dim-1}.\n\\begin{enumerate}\n\\item If $X \\to \\Spec(k)$ factors as $X \\to \\Spec(k') \\to \\Spec(k)$\nfor some field $k'$, then $\\omega_X^\\bullet$ and $\\omega_X$\nsatisfy properties (4), (5), (6) with $k$ replaced with $k'$.\n\\item If $K/k$ is a field extension, then the pullback of\n$\\omega_X^\\bullet$ and $\\omega_X$ to the base change $X_K$\nare as in  Lemma \\ref{lemma-duality-dim-1} for the morphism\n$X_K \\to \\Spec(K)$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E32","source_file":"curves.tex","source_line":709,"source_end_line":722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L709-L722","statement_sha256":"2cc19081770b644dff22608d6c53452ce5c3edb60119d8df0a360470427b9998","origin":"The Stacks Project","memory_eligible":false,"source_rank":9307,"rank":9307,"depth":47,"x":141.624,"y":1066.465,"cluster":"varieties-curves"},{"id":"stacks:0E33","tag":"0E33","title":"Duality · Lemma 0E33","summary":"Let X be a proper scheme of dimension ≤ 1 over a field k. Let i : Y → X be a closed immersion. Let ω_X^bullet, ω_X, ω_Y^bullet, ω_Y be as in Lemma [Tag 0BS2]. Then • ω_Y^bullet = RSheafHom(O_Y, ω_X^bullet), • ω_Y = SheafHom(O_Y, ω_X) and i_*ω_Y = SheafHom_O_X(i_*O_Y, ω_X).","statement_latex":"Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$.\nLet $i : Y \\to X$ be a closed immersion.\nLet $\\omega_X^\\bullet$, $\\omega_X$, $\\omega_Y^\\bullet$, $\\omega_Y$\nbe as in Lemma \\ref{lemma-duality-dim-1}. Then\n\\begin{enumerate}\n\\item $\\omega_Y^\\bullet = R\\SheafHom(\\mathcal{O}_Y, \\omega_X^\\bullet)$,\n\\item $\\omega_Y = \\SheafHom(\\mathcal{O}_Y, \\omega_X)$ and\n$i_*\\omega_Y = \\SheafHom_{\\mathcal{O}_X}(i_*\\mathcal{O}_Y, \\omega_X)$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E33","source_file":"curves.tex","source_line":758,"source_end_line":769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L758-L769","statement_sha256":"8ff2bfbffe493d8bc149373751e8ff3a5d227158cd70ef0d19d2471c0d74fc70","origin":"The Stacks Project","memory_eligible":false,"source_rank":9308,"rank":9308,"depth":47,"x":354.516,"y":1144.084,"cluster":"varieties-curves"},{"id":"stacks:0E34","tag":"0E34","title":"Duality · Lemma 0E34","summary":"Let X be a proper scheme over a field k which is Gorenstein, reduced, and equidimensional of dimension 1. Let i : Y → X be a reduced closed subscheme equidimensional of dimension 1. Let j : Z → X be the scheme theoretic closure of X setminus Y. Then • Y and Z are Cohen-Macaulay, • if I ⊂ O_X, resp. J ⊂ O_X is the ideal sheaf of Y, resp. Z in X, then I = i_*I' and J = j_*J' where I' ⊂ O_Z, resp. J' ⊂ O_Y is the ideal sheaf of Y ∩ Z in Z, resp. Y, • ω_Y = J'(i^*ω_X) and…","statement_latex":"Let $X$ be a proper scheme over a field $k$ which is\nGorenstein, reduced, and equidimensional of dimension $1$.\nLet $i : Y \\to X$ be a reduced closed subscheme equidimensional\nof dimension $1$. Let $j : Z \\to X$ be the scheme theoretic\nclosure of $X \\setminus Y$. Then\n\\begin{enumerate}\n\\item $Y$ and $Z$ are Cohen-Macaulay,\n\\item if $\\mathcal{I} \\subset \\mathcal{O}_X$,\nresp.\\ $\\mathcal{J} \\subset \\mathcal{O}_X$ is the ideal sheaf of\n$Y$, resp.\\ $Z$ in $X$, then\n$$\n\\mathcal{I} = i_*\\mathcal{I}'\n\\quad\\text{and}\\quad\n\\mathcal{J} = j_*\\mathcal{J}'\n$$\nwhere $\\mathcal{I}' \\subset \\mathcal{O}_Z$,\nresp.\\ $\\mathcal{J}' \\subset \\mathcal{O}_Y$ is the ideal sheaf\nof $Y \\cap Z$ in $Z$, resp.\\ $Y$,\n\\item $\\omega_Y = \\mathcal{J}'(i^*\\omega_X)$ and\n$i_*(\\omega_Y) = \\mathcal{J}\\omega_X$,\n\\item $\\omega_Z = \\mathcal{I}'(i^*\\omega_X)$ and\n$i_*(\\omega_Z) = \\mathcal{I}\\omega_X$,\n\\item we have the following short exact sequences\n\\begin{align*}\n0 \\to \\omega_X \\to i_*i^*\\omega_X \\oplus j_*j^*\\omega_X \\to\n\\mathcal{O}_{Y \\cap Z} \\to 0 \\\\\n0 \\to i_*\\omega_Y \\to \\omega_X \\to j_*j^*\\omega_X \\to 0 \\\\\n0 \\to j_*\\omega_Z \\to \\omega_X \\to i_*i^*\\omega_X \\to 0 \\\\\n0 \\to i_*\\omega_Y \\oplus j_*\\omega_Z \\to \\omega_X \\to\n\\mathcal{O}_{Y \\cap Z} \\to 0 \\\\\n0 \\to \\omega_Y \\to i^*\\omega_X \\to \\mathcal{O}_{Y \\cap Z} \\to 0 \\\\\n0 \\to \\omega_Z \\to j^*\\omega_X \\to \\mathcal{O}_{Y \\cap Z} \\to 0\n\\end{align*}\n\\end{enumerate}\nHere $\\omega_X$, $\\omega_Y$, $\\omega_Z$ are as in\nLemma \\ref{lemma-duality-dim-1}.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Duality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E34","source_file":"curves.tex","source_line":794,"source_end_line":832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L794-L832","statement_sha256":"dcaf10f648ed93d531d57da180f7bfc09d9cbbe37a5a9c7f2ae11724f70077fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9309,"rank":9309,"depth":48,"x":134.737,"y":1207.757,"cluster":"varieties-curves"},{"id":"stacks:0BS5","tag":"0BS5","title":"Riemann-Roch · Lemma 0BS5","summary":"Let X be a proper scheme of dimension ≤ 1 over a field k. With ω_X^bullet and ω_X as in Lemma [Tag 0BS2] we have chi(X, O_X) = chi(X, ω_X^bullet) If X is Cohen-Macaulay and equidimensional of dimension 1, then chi(X, O_X) = - chi(X, ω_X)","statement_latex":"Let $X$ be a proper scheme of dimension $\\leq 1$ over a field $k$.\nWith $\\omega_X^\\bullet$ and $\\omega_X$ as in Lemma \\ref{lemma-duality-dim-1}\nwe have\n$$\n\\chi(X, \\mathcal{O}_X) = \\chi(X, \\omega_X^\\bullet)\n$$\nIf $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then\n$$\n\\chi(X, \\mathcal{O}_X) = - \\chi(X, \\omega_X)\n$$","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Riemann-Roch","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BS5","source_file":"curves.tex","source_line":956,"source_end_line":968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L956-L968","statement_sha256":"b71fa34b067063e47e637c7aa011fffe227fdf293edd134fc23214b3264b3601","origin":"The Stacks Project","memory_eligible":false,"source_rank":9310,"rank":9310,"depth":48,"x":245.784,"y":1035.805,"cluster":"varieties-curves"},{"id":"stacks:0BS6","tag":"0BS6","title":"Riemann-Roch · Lemma 0BS6","summary":"Let X be a proper scheme over a field k which is Gorenstein and equidimensional of dimension 1. Let ω_X be as in Lemma [Tag 0BS2]. Then • ω_X is an invertible O_X-module, • deg(ω_X) = -2chi(X, O_X), • for a locally free O_X-module E of constant rank we have chi(X, E) = deg(E) - textstylefrac12 rank(E) deg(ω_X) and dim_k(H^i(X, E)) = dim_k(H^1 - i(X, E^vee ⊗_O_X ω_X)) for all i ∈ Z.","statement_latex":"Let $X$ be a proper scheme over a field $k$ which is Gorenstein and\nequidimensional of dimension $1$. Let $\\omega_X$ be as in\nLemma \\ref{lemma-duality-dim-1}. Then\n\\begin{enumerate}\n\\item $\\omega_X$ is an invertible $\\mathcal{O}_X$-module,\n\\item $\\deg(\\omega_X) = -2\\chi(X, \\mathcal{O}_X)$,\n\\item for a locally free $\\mathcal{O}_X$-module $\\mathcal{E}$\nof constant rank we have\n$$\n\\chi(X, \\mathcal{E}) = \\deg(\\mathcal{E}) -\n\\textstyle{\\frac{1}{2}} \\text{rank}(\\mathcal{E}) \\deg(\\omega_X)\n$$\nand $\\dim_k(H^i(X, \\mathcal{E})) =\n\\dim_k(H^{1 - i}(X, \\mathcal{E}^\\vee \\otimes_{\\mathcal{O}_X} \\omega_X))$\nfor all $i \\in \\mathbf{Z}$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Riemann-Roch","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BS6","source_file":"curves.tex","source_line":1000,"source_end_line":1018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1000-L1018","statement_sha256":"990f9366290380640d9f022753d69eb5c2ee318059f3e51ab069d03b7624208f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9311,"rank":9311,"depth":47,"x":302.275,"y":1225.933,"cluster":"varieties-curves"},{"id":"stacks:0BY5","tag":"0BY5","title":"Some vanishing results · Lemma 0BY5","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Then X is connected, Cohen-Macaulay, and equidimensional of dimension 1.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Then $X$ is connected, Cohen-Macaulay,\nand equidimensional of dimension $1$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Some vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BY5","source_file":"curves.tex","source_line":1059,"source_end_line":1064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1059-L1064","statement_sha256":"9b2b4409600af0f6ffd842bca85ddce3252612dcdae366a4f8e4563bd5cd6ecb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9312,"rank":9312,"depth":27,"x":107.393,"y":1117.607,"cluster":"varieties-curves"},{"id":"stacks:0B5E","tag":"0B5E","title":"Some vanishing results · Lemma 0B5E","summary":"In Situation [Tag 0B5D]. Given an exact sequence ω_X → F → Q → 0 of coherent O_X-modules with H^1(X, Q) = 0 (for example if dim(Supp(Q)) = 0), then either H^1(X, F) = 0 or F = ω_X ⊕ Q.","statement_latex":"In Situation \\ref{situation-Cohen-Macaulay-curve}. Given an exact sequence\n$$\n\\omega_X \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0\n$$\nof coherent $\\mathcal{O}_X$-modules with $H^1(X, \\mathcal{Q}) = 0$\n(for example if $\\dim(\\text{Supp}(\\mathcal{Q})) = 0$), then\neither $H^1(X, \\mathcal{F}) = 0$ or\n$\\mathcal{F} = \\omega_X \\oplus \\mathcal{Q}$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Some vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5E","source_file":"curves.tex","source_line":1103,"source_end_line":1113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1103-L1113","statement_sha256":"e562d3c819c8e90b859b167df07a39f142f13762686f19156a816c8cc6502b5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9313,"rank":9313,"depth":46,"x":338.599,"y":1086.853,"cluster":"varieties-curves"},{"id":"stacks:0B62","tag":"0B62","title":"Some vanishing results · Lemma 0B62","summary":"In Situation [Tag 0B5D]. Let L be an invertible O_X-module which is globally generated and not isomorphic to O_X. Then H^1(X, ω_X ⊗ L) = 0.","statement_latex":"In Situation \\ref{situation-Cohen-Macaulay-curve}. Let\n$\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module which is\nglobally generated and not isomorphic to $\\mathcal{O}_X$. Then\n$H^1(X, \\omega_X \\otimes \\mathcal{L}) = 0$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Some vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B62","source_file":"curves.tex","source_line":1136,"source_end_line":1142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1136-L1142","statement_sha256":"947095e35559a6e82f8a45e1285f8ba2b4432beb06e1502d39d05ab382e12ea7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9314,"rank":9314,"depth":0,"x":192.597,"y":1240.981,"cluster":"varieties-curves"},{"id":"stacks:0B5F","tag":"0B5F","title":"Some vanishing results · Lemma 0B5F","summary":"In Situation [Tag 0B5D]. Given an exact sequence ω_X → F → Q → 0 of coherent O_X-modules with dim(Supp(Q)) = 0 and dim_k H^0(X, Q) ≥ 2 and such that there is no nonzero submodule Q' ⊂ F such that Q' → Q is injective. Then the submodule of F generated by global sections surjects onto Q.","statement_latex":"In Situation \\ref{situation-Cohen-Macaulay-curve}. Given an exact sequence\n$$\n\\omega_X \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0\n$$\nof coherent $\\mathcal{O}_X$-modules with $\\dim(\\text{Supp}(\\mathcal{Q})) = 0$\nand $\\dim_k H^0(X, \\mathcal{Q}) \\geq 2$ and such that there is no nonzero\nsubmodule $\\mathcal{Q}' \\subset \\mathcal{F}$ such that\n$\\mathcal{Q}' \\to \\mathcal{Q}$ is injective.\nThen the submodule of $\\mathcal{F}$ generated by global\nsections surjects onto $\\mathcal{Q}$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Some vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5F","source_file":"curves.tex","source_line":1154,"source_end_line":1166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1154-L1166","statement_sha256":"002dbe2097eede2e930beda9bc791e957b90811287e7916751b0787d1e07f126","origin":"The Stacks Project","memory_eligible":false,"source_rank":9315,"rank":9315,"depth":47,"x":176.284,"y":1044.155,"cluster":"varieties-curves"},{"id":"stacks:0B5G","tag":"0B5G","title":"Some vanishing results · Lemma 0B5G","summary":"In Situation [Tag 0B5D] assume that X is integral. Let 0 → ω_X → F → Q → 0 be a short exact sequence of coherent O_X-modules with F torsion free, dim(Supp(Q)) = 0, and dim_k H^0(X, Q) ≥ 2. Then F is globally generated.","statement_latex":"In Situation \\ref{situation-Cohen-Macaulay-curve} assume that\n$X$ is integral. Let $0 \\to \\omega_X \\to \\mathcal{F} \\to \\mathcal{Q} \\to 0$\nbe a short exact sequence of coherent $\\mathcal{O}_X$-modules with\n$\\mathcal{F}$ torsion free, $\\dim(\\text{Supp}(\\mathcal{Q})) = 0$,\nand $\\dim_k H^0(X, \\mathcal{Q}) \\geq 2$. Then $\\mathcal{F}$\nis globally generated.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Some vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5G","source_file":"curves.tex","source_line":1189,"source_end_line":1197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1189-L1197","statement_sha256":"2a0f1eb6df49f38d23895170270a899c9729923af6bed10f040f1d280e5d1a43","origin":"The Stacks Project","memory_eligible":false,"source_rank":9316,"rank":9316,"depth":48,"x":346.88,"y":1180.263,"cluster":"varieties-curves"},{"id":"stacks:0B5H","tag":"0B5H","title":"Some vanishing results · Lemma 0B5H","summary":"In Situation [Tag 0B5D]. Let L be a very ample invertible O_X-module with deg(L) ≥ 2. Then ω_X ⊗_O_X L is globally generated.","statement_latex":"In Situation \\ref{situation-Cohen-Macaulay-curve}. Let $\\mathcal{L}$\nbe a very ample invertible $\\mathcal{O}_X$-module with\n$\\deg(\\mathcal{L}) \\geq 2$. Then\n$\\omega_X \\otimes_{\\mathcal{O}_X} \\mathcal{L}$ is globally generated.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Some vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B5H","source_file":"curves.tex","source_line":1219,"source_end_line":1225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1219-L1225","statement_sha256":"6048b9a72674b5564a47e8eba9c67dc6f73fb6d14c3c28b4368ecfb93f48070b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9317,"rank":9317,"depth":48,"x":111.239,"y":1176.692,"cluster":"varieties-curves"},{"id":"stacks:0E8V","tag":"0E8V","title":"Very ample invertible sheaves · Lemma 0E8V","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Let L be an invertible O_X-module. Assume • L has a regular global section, • H^1(X, L) = 0, and • L is ample. Then L^⊗ 6 is very ample on X over k.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module. Assume\n\\begin{enumerate}\n\\item $\\mathcal{L}$ has a regular global section,\n\\item $H^1(X, \\mathcal{L}) = 0$, and\n\\item $\\mathcal{L}$ is ample.\n\\end{enumerate}\nThen $\\mathcal{L}^{\\otimes 6}$ is very ample on $X$ over $k$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Very ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8V","source_file":"curves.tex","source_line":1288,"source_end_line":1299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1288-L1299","statement_sha256":"1696847fec987e209283ef18010f8d178b9ebed879ff263c60691f40dcbcb0dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9318,"rank":9318,"depth":27,"x":288.162,"y":1045.399,"cluster":"varieties-curves"},{"id":"stacks:0E8W","tag":"0E8W","title":"Very ample invertible sheaves · Lemma 0E8W","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Let L be an invertible O_X-module. Assume • L is globally generated, • H^1(X, L) = 0, and • L is ample. Then L^⊗ 2 is very ample on X over k.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module. Assume\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is globally generated,\n\\item $H^1(X, \\mathcal{L}) = 0$, and\n\\item $\\mathcal{L}$ is ample.\n\\end{enumerate}\nThen $\\mathcal{L}^{\\otimes 2}$ is very ample on $X$ over $k$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Very ample invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8W","source_file":"curves.tex","source_line":1376,"source_end_line":1387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1376-L1387","statement_sha256":"f3f0cdd68551ff7bf411ee35b46ac907bb9cf1257285d353a86b3a2062835f38","origin":"The Stacks Project","memory_eligible":false,"source_rank":9319,"rank":9319,"depth":43,"x":263.243,"y":1242.929,"cluster":"varieties-curves"},{"id":"stacks:0BY7","tag":"0BY7","title":"The genus of a curve · Definition 0BY7","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Then the genus of X is g = dim_k H^1(X, O_X).","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having\ndimension $1$ and $H^0(X, \\mathcal{O}_X) = k$.\nThen the {\\it genus} of $X$ is $g = \\dim_k H^1(X, \\mathcal{O}_X)$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"The genus of a curve","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BY7","source_file":"curves.tex","source_line":1459,"source_end_line":1464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1459-L1464","statement_sha256":"c43aea7d0dff213ac3e1bea3cabfd9e8721acea49d483f8e3f1eb1d211c862e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9320,"rank":9320,"depth":0,"x":122.538,"y":1082.87,"cluster":"varieties-curves"},{"id":"stacks:0BY9","tag":"0BY9","title":"The genus of a curve · Lemma 0BY9","summary":"Let k'/k be a field extension. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Then X_k' is a proper scheme over k' having dimension 1 and H^0(X_k', O_X_k') = k'. Moreover the genus of X_k' is equal to the genus of X.","statement_latex":"Let $k'/k$ be a field extension. Let $X$ be a proper scheme over $k$ having\ndimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. Then $X_{k'}$ is a\nproper scheme over $k'$\nhaving dimension $1$ and $H^0(X_{k'}, \\mathcal{O}_{X_{k'}}) = k'$.\nMoreover the genus of $X_{k'}$ is equal to the genus of $X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"The genus of a curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BY9","source_file":"curves.tex","source_line":1506,"source_end_line":1513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1506-L1513","statement_sha256":"b2262e5bdf6f912f5857d4542993539ad707e18f5535b5a29ef87d06078cbad9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9321,"rank":9321,"depth":30,"x":355.388,"y":1121.119,"cluster":"varieties-curves"},{"id":"stacks:0C19","tag":"0C19","title":"The genus of a curve · Lemma 0C19","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. If X is Gorenstein, then deg(ω_X) = 2g - 2 where g is the genus of X and ω_X is as in Lemma [Tag 0BS2].","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having\ndimension $1$ and $H^0(X, \\mathcal{O}_X) = k$. If $X$ is Gorenstein,\nthen\n$$\n\\deg(\\omega_X) = 2g - 2\n$$\nwhere $g$ is the genus of $X$ and $\\omega_X$ is as in\nLemma \\ref{lemma-duality-dim-1}.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"The genus of a curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C19","source_file":"curves.tex","source_line":1526,"source_end_line":1536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1526-L1536","statement_sha256":"32d6124b944fd3fdd8229a0157e68e3a8bb8b04db4c6cfdb32c6ea1898dde7b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9322,"rank":9322,"depth":48,"x":152.598,"y":1225.21,"cluster":"varieties-curves"},{"id":"stacks:0C1A","tag":"0C1A","title":"The genus of a curve · Lemma 0C1A","summary":"Let X be a smooth proper curve over a field k with H^0(X, O_X) = k. Then dim_k H^0(X, Ω_X/k) = g and deg(Ω_X/k) = 2g - 2 where g is the genus of X.","statement_latex":"Let $X$ be a smooth proper curve over a field $k$\nwith $H^0(X, \\mathcal{O}_X) = k$. Then\n$$\n\\dim_k H^0(X, \\Omega_{X/k}) = g\n\\quad\\text{and}\\quad\n\\deg(\\Omega_{X/k}) = 2g - 2\n$$\nwhere $g$ is the genus of $X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"The genus of a curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1A","source_file":"curves.tex","source_line":1542,"source_end_line":1552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1542-L1552","statement_sha256":"d94c1c7d74b47b45dac32637e921191b4d04a356f1b5518372d76f5c68af3f4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9323,"rank":9323,"depth":49,"x":218.533,"y":1033.073,"cluster":"varieties-curves"},{"id":"stacks:0BYB","tag":"0BYB","title":"Plane curves · Lemma 0BYB","summary":"Let Z ⊂ P^2_k be a closed subscheme which is equidimensional of dimension 1 and has no embedded points (equivalently Z is Cohen-Macaulay). Then the ideal I(Z) ⊂ k[T_0, T_1, T_2] corresponding to Z is principal.","statement_latex":"Let $Z \\subset \\mathbf{P}^2_k$ be a closed subscheme which\nis equidimensional of dimension $1$ and has no embedded points\n(equivalently $Z$ is Cohen-Macaulay).\nThen the ideal $I(Z) \\subset k[T_0, T_1, T_2]$ corresponding\nto $Z$ is principal.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Plane curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYB","source_file":"curves.tex","source_line":1641,"source_end_line":1648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1641-L1648","statement_sha256":"7b29669b42f0d855da54ade9e33b346024a8522fbe538558d7c82f163979150e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9324,"rank":9324,"depth":30,"x":324.596,"y":1212.457,"cluster":"varieties-curves"},{"id":"stacks:0BYC","tag":"0BYC","title":"Plane curves · Lemma 0BYC","summary":"Let Z ⊂ P^2_k be as in Lemma [Tag 0BYB] and let I(Z) = (F) for some F ∈ k[T_0, T_1, T_2]. Then Z is a curve if and only if F is irreducible.","statement_latex":"Let $Z \\subset \\mathbf{P}^2_k$ be as in Lemma \\ref{lemma-equation-plane-curve}\nand let $I(Z) = (F)$ for some $F \\in k[T_0, T_1, T_2]$.\nThen $Z$ is a curve if and only if $F$ is irreducible.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Plane curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYC","source_file":"curves.tex","source_line":1661,"source_end_line":1666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1661-L1666","statement_sha256":"129a73da29d1ceecd5a95f662fcbc9c22cf7886e1a12579db98db9f5ffa5e9a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9325,"rank":9325,"depth":31,"x":101.771,"y":1140.248,"cluster":"varieties-curves"},{"id":"stacks:0BYD","tag":"0BYD","title":"Plane curves · Lemma 0BYD","summary":"Let Z ⊂ P^2_k be as in Lemma [Tag 0BYB] and let I(Z) = (F) for some F ∈ k[T_0, T_1, T_2]. Then H^0(Z, O_Z) = k and the genus of Z is (d - 1)(d - 2)/2 where d = deg(F).","statement_latex":"Let $Z \\subset \\mathbf{P}^2_k$ be as in Lemma \\ref{lemma-equation-plane-curve}\nand let $I(Z) = (F)$ for some $F \\in k[T_0, T_1, T_2]$.\nThen $H^0(Z, \\mathcal{O}_Z) = k$ and the genus of $Z$ is\n$(d - 1)(d - 2)/2$ where $d = \\deg(F)$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Plane curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYD","source_file":"curves.tex","source_line":1686,"source_end_line":1692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1686-L1692","statement_sha256":"5e78bc7638c705166ca81714724290a06ed57bd96a7a887a38506033a1bedec7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9326,"rank":9326,"depth":31,"x":324.507,"y":1066.939,"cluster":"varieties-curves"},{"id":"stacks:0CCU","tag":"0CCU","title":"Plane curves · Lemma 0CCU","summary":"Let Z ⊂ P^2_k be as in Lemma [Tag 0BYB] and let I(Z) = (F) for some F ∈ k[T_0, T_1, T_2]. If Z → Spec(k) is smooth in at least one point and k is infinite, then there exists a closed point z ∈ Z contained in the smooth locus such that kappa(z)/k is finite separable of degree at most d.","statement_latex":"Let $Z \\subset \\mathbf{P}^2_k$ be as in Lemma \\ref{lemma-equation-plane-curve}\nand let $I(Z) = (F)$ for some $F \\in k[T_0, T_1, T_2]$.\nIf $Z \\to \\Spec(k)$ is smooth in at least one point and $k$ is infinite, then\nthere exists a closed point $z \\in Z$ contained in the smooth\nlocus such that $\\kappa(z)/k$ is finite separable of degree\nat most $d$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Plane curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCU","source_file":"curves.tex","source_line":1732,"source_end_line":1740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1732-L1740","statement_sha256":"e2f25c1086ad5acd9765ff10cf542bbc82b641136a496c65c6dc9c780dec2433","origin":"The Stacks Project","memory_eligible":false,"source_rank":9327,"rank":9327,"depth":31,"x":219.048,"y":1247.672,"cluster":"varieties-curves"},{"id":"stacks:0C6M","tag":"0C6M","title":"Curves of genus zero · Lemma 0C6M","summary":"Let X be a proper curve over a field k with H^0(X, O_X) = k. If X has genus 0, then every invertible O_X-module L of degree 0 is trivial.","statement_latex":"Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$.\nIf $X$ has genus $0$, then every invertible $\\mathcal{O}_X$-module\n$\\mathcal{L}$ of degree $0$ is trivial.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves of genus zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6M","source_file":"curves.tex","source_line":1787,"source_end_line":1792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1787-L1792","statement_sha256":"d68ce75aebb87ba10e4f60cbf28418a558ac4c7b9f5d8d074801f292401d6d0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9328,"rank":9328,"depth":48,"x":151.363,"y":1054.252,"cluster":"varieties-curves"},{"id":"stacks:0C6T","tag":"0C6T","title":"Curves of genus zero · Lemma 0C6T","summary":"Let X be a proper curve over a field k with H^0(X, O_X) = k. Assume X has genus 0. Let L be an invertible O_X-module of degree d > 0. Then we have • dim_k H^0(X, L) = d + 1 and dim_k H^1(X, L) = 0, • L is very ample and defines a closed immersion into P^d_k.","statement_latex":"Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$.\nAssume $X$ has genus $0$. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module of degree $d > 0$. Then we have\n\\begin{enumerate}\n\\item $\\dim_k H^0(X, \\mathcal{L}) = d + 1$ and $\\dim_k H^1(X, \\mathcal{L}) = 0$,\n\\item $\\mathcal{L}$ is very ample and defines a closed immersion into\n$\\mathbf{P}^d_k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves of genus zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6T","source_file":"curves.tex","source_line":1801,"source_end_line":1811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1801-L1811","statement_sha256":"8208f3e59f03111401f0fd61f5ebe9b29e608aaa9a387eb1dfaff599cfa0d8e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9329,"rank":9329,"depth":46,"x":357.145,"y":1158.638,"cluster":"varieties-curves"},{"id":"stacks:0C6N","tag":"0C6N","title":"Curves of genus zero · Lemma 0C6N","summary":"Let X be a proper curve over a field k with H^0(X, O_X) = k. If X is Gorenstein and has genus 0, then X is isomorphic to a plane curve of degree 2.","statement_latex":"Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$.\nIf $X$ is Gorenstein and has genus $0$, then $X$\nis isomorphic to a plane curve of degree $2$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves of genus zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6N","source_file":"curves.tex","source_line":1890,"source_end_line":1895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1890-L1895","statement_sha256":"7dbcc38e6734c5803ce8043491e992275006cb6c9d5b7b85fcb119011f2770f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9330,"rank":9330,"depth":49,"x":121.084,"y":1198.494,"cluster":"varieties-curves"},{"id":"stacks:0C6U","tag":"0C6U","title":"Characterization of the projective line · Proposition 0C6U","summary":"Let k be a field. Let X be a proper curve over k. The following are equivalent • X ≅ P^1_k, • X is smooth and geometrically irreducible over k, X has genus 0, and X has an invertible module of odd degree, • X is geometrically integral over k, X has genus 0, X is Gorenstein, and X has an invertible sheaf of odd degree, • H^0(X, O_X) = k, X has genus 0, X is Gorenstein, and X has an invertible sheaf of odd degree, • X is geometrically integral over k, X has genus 0, and X…","statement_latex":"Let $k$ be a field. Let $X$ be a proper curve over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X \\cong \\mathbf{P}^1_k$,\n\\item $X$ is smooth and geometrically irreducible over $k$,\n$X$ has genus $0$, and $X$ has an invertible module of odd degree,\n\\item $X$ is geometrically integral over $k$, $X$ has genus $0$,\n$X$ is Gorenstein, and $X$ has an invertible sheaf of odd degree,\n\\item $H^0(X, \\mathcal{O}_X) = k$, $X$ has genus $0$, $X$ is Gorenstein,\nand $X$ has an invertible sheaf of odd degree,\n\\item $X$ is geometrically integral over $k$, $X$ has genus $0$,\nand $X$ has an invertible $\\mathcal{O}_X$-module of degree $1$,\n\\item $H^0(X, \\mathcal{O}_X) = k$, $X$ has genus $0$,\nand $X$ has an invertible $\\mathcal{O}_X$-module of degree $1$,\n\\item $H^1(X, \\mathcal{O}_X) = 0$ and $X$ has an invertible\n$\\mathcal{O}_X$-module of degree $1$,\n\\item $H^1(X, \\mathcal{O}_X) = 0$ and $X$\nhas closed points $x_1, \\ldots, x_n$ such that\n$\\mathcal{O}_{X, x_i}$ is normal and $\\gcd([\\kappa(x_i) : k]) = 1$, and\n\\item add more here.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves of genus zero","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6U","source_file":"curves.tex","source_line":1920,"source_end_line":1943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L1920-L1943","statement_sha256":"ac01c1f981c55c2eac917f9da4e4905874e68f014a208b0b27060a881a7d8a3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9331,"rank":9331,"depth":49,"x":263.326,"y":1034.899,"cluster":"varieties-curves"},{"id":"stacks:0DJB","tag":"0DJB","title":"Curves of genus zero · Lemma 0DJB","summary":"Let X be a proper curve over a field k with H^0(X, O_X) = k. Assume X is singular and has genus 0. Then there exists a diagram xymatrix x' ar[d] ar[r] & X' ar[d]^ν ar[r] & Spec(k') ar[d] x ar[r] & X ar[r] & Spec(k) where • k'/k is a nontrivial finite extension, • X' ≅ P^1_k', • x' is a k'-rational point of X', • x is a k-rational point of X, • X' setminus (x') → X setminus (x) is an isomorphism, • 0 → O_X → ν_*O_X' → k'/k → 0 is a short exact sequence where k'/k =…","statement_latex":"Let $X$ be a proper curve over a field $k$ with $H^0(X, \\mathcal{O}_X) = k$.\nAssume $X$ is singular and has genus $0$. Then there exists a diagram\n$$\n\\xymatrix{\nx' \\ar[d] \\ar[r] & X' \\ar[d]^\\nu \\ar[r] & \\Spec(k') \\ar[d] \\\\\nx \\ar[r] & X \\ar[r] & \\Spec(k)\n}\n$$\nwhere\n\\begin{enumerate}\n\\item $k'/k$ is a nontrivial finite extension,\n\\item $X' \\cong \\mathbf{P}^1_{k'}$,\n\\item $x'$ is a $k'$-rational point of $X'$,\n\\item $x$ is a $k$-rational point of $X$,\n\\item $X' \\setminus \\{x'\\} \\to X \\setminus \\{x\\}$ is an isomorphism,\n\\item $0 \\to \\mathcal{O}_X \\to \\nu_*\\mathcal{O}_{X'} \\to k'/k \\to 0$\nis a short exact sequence\nwhere $k'/k = \\kappa(x')/\\kappa(x)$ indicates the skyscraper sheaf\non the point $x$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Curves of genus zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DJB","source_file":"curves.tex","source_line":2004,"source_end_line":2026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2004-L2026","statement_sha256":"6ea0db48013285d353d71098a23efcae23c64ef8b1877ccd8a86c070e989cb76","origin":"The Stacks Project","memory_eligible":false,"source_rank":9332,"rank":9332,"depth":50,"x":290.04,"y":1236.562,"cluster":"varieties-curves"},{"id":"stacks:0BYF","tag":"0BYF","title":"Geometric genus · Definition 0BYF","summary":"Let k be a field. Let X be a geometrically irreducible curve over k. The geometric genus of X is the genus of a smooth projective model of X possibly defined over an extension field of k as in Lemma [Tag 0BY4].","statement_latex":"Let $k$ be a field. Let $X$ be a geometrically irreducible\ncurve over $k$. The {\\it geometric genus} of $X$ is the genus\nof a smooth projective model of $X$ possibly defined over\nan extension field of $k$ as in\nLemma \\ref{lemma-smooth-models}.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Geometric genus","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BYF","source_file":"curves.tex","source_line":2154,"source_end_line":2161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2154-L2161","statement_sha256":"29032a19e863b9541b6cdf843b7d86b8d02267ed1cc059b0d382865423b7e778","origin":"The Stacks Project","memory_eligible":false,"source_rank":9333,"rank":9333,"depth":45,"x":107.882,"y":1102.806,"cluster":"varieties-curves"},{"id":"stacks:0C1C","tag":"0C1C","title":"Riemann-Hurwitz · Lemma 0C1C","summary":"A morphism of smooth curves is separable iff it is etale almost everywhere Let k be a field. Let f : X → Y be a morphism of smooth curves over k. The following are equivalent • df : f^*Ω_Y/k → Ω_X/k is nonzero, • Ω_X/Y is supported on a proper closed subset of X, • there exists a nonempty open U ⊂ X such that f|_U : U → Y is unramified, • there exists a nonempty open U ⊂ X such that f|_U : U → Y is étale, • the extension k(X)/k(Y) of function fields is finite separable.","statement_latex":"\\begin{slogan}\nA morphism of smooth curves is separable iff it is etale almost everywhere\n\\end{slogan}\nLet $k$ be a field. Let $f : X \\to Y$ be a morphism of smooth curves over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\text{d}f : f^*\\Omega_{Y/k} \\to \\Omega_{X/k}$ is nonzero,\n\\item $\\Omega_{X/Y}$ is supported on a proper closed subset of $X$,\n\\item there exists a nonempty open $U \\subset X$ such that\n$f|_U : U \\to Y$ is unramified,\n\\item there exists a nonempty open $U \\subset X$ such that\n$f|_U : U \\to Y$ is \\'etale,\n\\item the extension $k(X)/k(Y)$ of function fields is\nfinite separable.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Riemann-Hurwitz","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1C","source_file":"curves.tex","source_line":2346,"source_end_line":2363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2346-L2363","statement_sha256":"e911ba77f793c04bb0bec246b795de30bd152fd84c5e6e2dbde5eab66ddb5889","origin":"The Stacks Project","memory_eligible":false,"source_rank":9334,"rank":9334,"depth":48,"x":350.147,"y":1098.068,"cluster":"varieties-curves"},{"id":"stacks:0C1D","tag":"0C1D","title":"Riemann-Hurwitz · Lemma 0C1D","summary":"Let f : X → Y be a morphism of smooth proper curves over a field k which satisfies the equivalent conditions of Lemma [Tag 0C1C]. If k = H^0(Y, O_Y) = H^0(X, O_X) and X and Y have genus g_X and g_Y, then 2g_X - 2 = (2g_Y - 2) deg(f) + deg(R) where R ⊂ X is the effective Cartier divisor cut out by the different of f.","statement_latex":"Let $f : X \\to Y$ be a morphism of smooth proper curves\nover a field $k$ which satisfies the equivalent conditions of\nLemma \\ref{lemma-generically-etale}. If\n$k = H^0(Y, \\mathcal{O}_Y) = H^0(X, \\mathcal{O}_X)$\nand $X$ and $Y$ have genus $g_X$ and $g_Y$, then\n$$\n2g_X - 2 = (2g_Y - 2) \\deg(f) + \\deg(R)\n$$\nwhere $R \\subset X$ is the effective Cartier divisor cut out by\nthe different of $f$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Riemann-Hurwitz","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1D","source_file":"curves.tex","source_line":2388,"source_end_line":2400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2388-L2400","statement_sha256":"3971da929588e082c3b3432a7a9e24971a992cf3300bbd79cfd7e497f80681fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9335,"rank":9335,"depth":50,"x":175.04,"y":1239.25,"cluster":"varieties-curves"},{"id":"stacks:0C1E","tag":"0C1E","title":"Riemann-Hurwitz · Lemma 0C1E","summary":"Let X → Spec(k) be smooth of relative dimension 1 at a closed point x ∈ X. If kappa(x) is separable over k, then for any uniformizer s in the discrete valuation ring O_X, x the element ds freely generates Ω_X/k, x over O_X, x.","statement_latex":"Let $X \\to \\Spec(k)$ be smooth of relative dimension $1$ at a closed\npoint $x \\in X$. If $\\kappa(x)$ is separable over $k$, then for\nany uniformizer $s$ in the discrete valuation ring $\\mathcal{O}_{X, x}$\nthe element $\\text{d}s$ freely generates $\\Omega_{X/k, x}$\nover $\\mathcal{O}_{X, x}$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Riemann-Hurwitz","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1E","source_file":"curves.tex","source_line":2411,"source_end_line":2418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2411-L2418","statement_sha256":"ae405b98e0b8970fa9a1f1d647d4c280ba1343add2f40dcdc4e781a379fa8808","origin":"The Stacks Project","memory_eligible":false,"source_rank":9336,"rank":9336,"depth":37,"x":190.651,"y":1035.466,"cluster":"varieties-curves"},{"id":"stacks:0C1F","tag":"0C1F","title":"Riemann-Hurwitz · Lemma 0C1F","summary":"Notation and assumptions as in Lemma [Tag 0C1D]. For a closed point x ∈ X let d_x be the multiplicity of x in R. Then 2g_X - 2 = (2g_Y - 2) deg(f) + ∑ d_x [kappa(x) : k] Moreover, we have the following results • d_x = length_O_X, x(Ω_X/Y, x), • d_x ≥ e_x - 1 where e_x is the ramification index of O_X, x over O_Y, y, • d_x = e_x - 1 if and only if O_X, x is tamely ramified over O_Y, y.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-rh}. For a closed point\n$x \\in X$ let $d_x$ be the multiplicity of $x$ in $R$. Then\n$$\n2g_X - 2 = (2g_Y - 2) \\deg(f) + \\sum\\nolimits d_x [\\kappa(x) : k]\n$$\nMoreover, we have the following results\n\\begin{enumerate}\n\\item $d_x = \\text{length}_{\\mathcal{O}_{X, x}}(\\Omega_{X/Y, x})$,\n\\item $d_x \\geq e_x - 1$ where $e_x$ is the ramification index\nof $\\mathcal{O}_{X, x}$ over $\\mathcal{O}_{Y, y}$,\n\\item $d_x = e_x - 1$ if and only if $\\mathcal{O}_{X, x}$ is tamely\nramified over $\\mathcal{O}_{Y, y}$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Riemann-Hurwitz","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1F","source_file":"curves.tex","source_line":2432,"source_end_line":2447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2432-L2447","statement_sha256":"2746146b64a494d0e7fe9ecd75cc3d4600283888f0fcdc69a6474bee977e37af","origin":"The Stacks Project","memory_eligible":false,"source_rank":9337,"rank":9337,"depth":68,"x":343.254,"y":1194.839,"cluster":"varieties-curves"},{"id":"stacks:0CCW","tag":"0CCW","title":"Inseparable maps · Lemma 0CCW","summary":"Let k be a field. Let f : X → Y be a surjective morphism of curves over k. If X is smooth over k and Y is normal, then Y is smooth over k.","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a surjective morphism\nof curves over $k$. If $X$ is smooth over $k$ and\n$Y$ is normal, then $Y$ is smooth over $k$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCW","source_file":"curves.tex","source_line":2536,"source_end_line":2541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2536-L2541","statement_sha256":"7463c7be043f825385abe8f624a818911e00dac44300ab74d24b29fba7e8d8a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9338,"rank":9338,"depth":48,"x":102.19,"y":1163.862,"cluster":"varieties-curves"},{"id":"stacks:0CCX","tag":"0CCX","title":"Inseparable maps · Lemma 0CCX","summary":"Let k be a field of characteristic p > 0. Let f : X → Y be a nonconstant morphism of proper nonsingular curves over k. If the extension k(X)/k(Y) of function fields is purely inseparable, then there exists a factorization X = X_0 → X_1 → … → X_n = Y such that each X_i is a proper nonsingular curve and X_i → X_i + 1 is a degree p morphism with k(X_i + 1) ⊂ k(X_i) inseparable.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a\nnonconstant morphism of proper nonsingular curves over $k$.\nIf the extension $k(X)/k(Y)$ of function fields\nis purely inseparable, then there exists a factorization\n$$\nX = X_0 \\to X_1 \\to \\ldots \\to X_n = Y\n$$\nsuch that each $X_i$ is a proper nonsingular curve\nand $X_i \\to X_{i + 1}$ is a degree $p$\nmorphism with $k(X_{i + 1}) \\subset k(X_i)$\ninseparable.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCX","source_file":"curves.tex","source_line":2550,"source_end_line":2563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2550-L2563","statement_sha256":"290da768f15c781acf6202098a38cd4c9d2adf165a119aad007e00ec287c61b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9339,"rank":9339,"depth":49,"x":305.172,"y":1049.742,"cluster":"varieties-curves"},{"id":"stacks:0CCY","tag":"0CCY","title":"Inseparable maps · Lemma 0CCY","summary":"Let k be a field of characteristic p > 0. Let f : X → Y be a nonconstant morphism of proper nonsingular curves over k. If X is smooth and k(Y) ⊂ k(X) is inseparable of degree p, then there is a unique isomorphism Y = X^(p) such that f is F_X/k.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a\nnonconstant morphism of proper nonsingular curves over $k$.\nIf $X$ is smooth and $k(Y) \\subset k(X)$ is inseparable of degree $p$,\nthen there is a unique isomorphism $Y = X^{(p)}$ such that\n$f$ is $F_{X/k}$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCY","source_file":"curves.tex","source_line":2572,"source_end_line":2579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2572-L2579","statement_sha256":"449c1a1f8d6cc343c9edaf8811d943a9aeed61f6d90329e45042c6d5c12a808a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9340,"rank":9340,"depth":49,"x":247.179,"y":1249.378,"cluster":"varieties-curves"},{"id":"stacks:0CCZ","tag":"0CCZ","title":"Inseparable maps · Lemma 0CCZ","summary":"Let k be a field of characteristic p > 0. Let f : X → Y be a nonconstant morphism of proper nonsingular curves over k. If X is smooth and k(Y) ⊂ k(X) is purely inseparable, then there is a unique n ≥ 0 and a unique isomorphism Y = X^(p^n) such that f is the n-fold relative Frobenius of X/k.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a\nnonconstant morphism of proper nonsingular curves over $k$.\nIf $X$ is smooth and $k(Y) \\subset k(X)$ is purely inseparable,\nthen there is a unique $n \\geq 0$ and a unique isomorphism $Y = X^{(p^n)}$\nsuch that $f$ is the $n$-fold relative Frobenius of $X/k$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CCZ","source_file":"curves.tex","source_line":2607,"source_end_line":2614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2607-L2614","statement_sha256":"9ea4008a80c8c9dfcb5c2a42a0baff56faf47c49b225b7c0eac9cd3feefb0a6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9341,"rank":9341,"depth":50,"x":129.22,"y":1068.985,"cluster":"varieties-curves"},{"id":"stacks:0CD0","tag":"0CD0","title":"Inseparable maps · Lemma 0CD0","summary":"Let k be a field of characteristic p > 0. Let f : X → Y be a nonconstant morphism of proper nonsingular curves over k. Assume • X is smooth, • H^0(X, O_X) = k, • k(X)/k(Y) is purely inseparable. Then Y is smooth, H^0(Y, O_Y) = k, and the genus of Y is equal to the genus of X.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a\nnonconstant morphism of proper nonsingular curves over $k$.\nAssume\n\\begin{enumerate}\n\\item $X$ is smooth,\n\\item $H^0(X, \\mathcal{O}_X) = k$,\n\\item $k(X)/k(Y)$ is purely inseparable.\n\\end{enumerate}\nThen $Y$ is smooth, $H^0(Y, \\mathcal{O}_Y) = k$, and the genus of $Y$\nis equal to the genus of $X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CD0","source_file":"curves.tex","source_line":2623,"source_end_line":2635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2623-L2635","statement_sha256":"01a5e6fd3b7e01382362fca35b5db08166ac15927855950e5226459ed9341679","origin":"The Stacks Project","memory_eligible":false,"source_rank":9342,"rank":9342,"depth":51,"x":361.623,"y":1135.173,"cluster":"varieties-curves"},{"id":"stacks:0CD2","tag":"0CD2","title":"Inseparable maps · Proposition 0CD2","summary":"Let k be a field of characteristic p > 0. Let f : X → Y be a nonconstant morphism of proper smooth curves over k. Then we can factor f as X → X^(p^n) → Y where X^(p^n) → Y is a nonconstant morphism of proper smooth curves inducing a separable field extension k(X^(p^n))/k(Y), we have X^(p^n) = X ×_Spec(k), F_Spec(k)^n Spec(k), and X → X^(p^n) is the n-fold relative frobenius of X.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $f : X \\to Y$ be a\nnonconstant morphism of proper smooth curves over $k$.\nThen we can factor $f$ as\n$$\nX \\longrightarrow X^{(p^n)} \\longrightarrow Y\n$$\nwhere $X^{(p^n)} \\to Y$ is a nonconstant morphism of proper smooth curves\ninducing a separable field extension $k(X^{(p^n)})/k(Y)$, we have\n$$\nX^{(p^n)} = X \\times_{\\Spec(k), F_{\\Spec(k)}^n} \\Spec(k),\n$$\nand $X \\to X^{(p^n)}$ is the $n$-fold relative frobenius of $X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Inseparable maps","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CD2","source_file":"curves.tex","source_line":2670,"source_end_line":2684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2670-L2684","statement_sha256":"2752b41ea3c8b8e91f8cb9568c9b517f416131403be4fc17b3f8c2d661b81a00","origin":"The Stacks Project","memory_eligible":false,"source_rank":9343,"rank":9343,"depth":51,"x":136.683,"y":1218.365,"cluster":"varieties-curves"},{"id":"stacks:0CD3","tag":"0CD3","title":"Inseparable maps · Lemma 0CD3","summary":"Let k be a field of characteristic p > 0. Let X be a smooth proper curve over k. Let (L, V) be a g^r_d with r ≥ 1. Then one of the following two is true • there exists a g^1_d whose corresponding morphism X → P^1_k (Lemma [Tag 0CCP]) is generically étale (i.e., is as in Lemma [Tag 0C1C]), or • there exists a g^r_d' on X^(p) where d' ≤ d/p.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $X$ be a smooth proper\ncurve over $k$. Let $(\\mathcal{L}, V)$ be a $\\mathfrak g^r_d$ with $r \\geq 1$.\nThen one of the following two is true\n\\begin{enumerate}\n\\item there exists a $\\mathfrak g^1_d$ whose corresponding morphism\n$X \\to \\mathbf{P}^1_k$ (Lemma \\ref{lemma-linear-series})\nis generically \\'etale (i.e., is as in Lemma \\ref{lemma-generically-etale}), or\n\\item there exists a $\\mathfrak g^r_{d'}$ on $X^{(p)}$ where\n$d' \\leq d/p$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CD3","source_file":"curves.tex","source_line":2697,"source_end_line":2709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2697-L2709","statement_sha256":"ac82693d40bb9c62e20099debc669a5277db3d569064b9703d608ae303de378e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9344,"rank":9344,"depth":52,"x":235.8,"y":1029.095,"cluster":"varieties-curves"},{"id":"stacks:0CD4","tag":"0CD4","title":"Inseparable maps · Lemma 0CD4","summary":"Let k be a field. Let X be a smooth proper curve over k with H^0(X, O_X) = k and genus g ≥ 2. Then there exists a closed point x ∈ X with kappa(x)/k separable of degree ≤ 2g - 2.","statement_latex":"Let $k$ be a field. Let $X$ be a smooth proper curve over $k$\nwith $H^0(X, \\mathcal{O}_X) = k$ and genus $g \\geq 2$.\nThen there exists a closed point $x \\in X$ with\n$\\kappa(x)/k$ separable of degree $\\leq 2g - 2$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CD4","source_file":"curves.tex","source_line":2753,"source_end_line":2759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2753-L2759","statement_sha256":"67d29f08ec25e89b02865989c978b0f361a1c92bf9c3697bde24ebd7ec7bd828","origin":"The Stacks Project","memory_eligible":false,"source_rank":9345,"rank":9345,"depth":53,"x":315.038,"y":1225.2,"cluster":"varieties-curves"},{"id":"stacks:0C1G","tag":"0C1G","title":"Inseparable maps · Lemma 0C1G","summary":"Let X be a smooth curve over a field k. Let overlinex ∈ X_overlinek be a closed point with image x ∈ X. The ramification index of O_X, x ⊂ O_X_overlinek, overlinex is the inseparable degree of kappa(x)/k.","statement_latex":"Let $X$ be a smooth curve over a field $k$. Let\n$\\overline{x} \\in X_{\\overline{k}}$ be a closed\npoint with image $x \\in X$. The ramification index of\n$\\mathcal{O}_{X, x} \\subset \\mathcal{O}_{X_{\\overline{k}}, \\overline{x}}$\nis the inseparable degree of $\\kappa(x)/k$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1G","source_file":"curves.tex","source_line":2817,"source_end_line":2824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2817-L2824","statement_sha256":"6e62243b3196a2039e8a9fb94bbeda426994c8ef72955d445b1097f56599f106","origin":"The Stacks Project","memory_eligible":false,"source_rank":9346,"rank":9346,"depth":37,"x":98.58,"y":1125.406,"cluster":"varieties-curves"},{"id":"stacks:0E36","tag":"0E36","title":"Pushouts · Lemma 0E36","summary":"In the situation above, let Z = Spec(k') where k' is a field and Z' = Spec(k'_1 × … × k'_n) with k'_i/k' finite extensions of fields. Let x ∈ X be the image of Z → X and x'_i ∈ X' the image of Spec(k'_i) → X'. Then we have a fibre product diagram xymatrix ∏_i = 1, …, n k'_i & ∏_i = 1, …, n O_X', x'_i^wedge ar[l] k' ar[u] & O_X, x^wedge ar[u] ar[l] where the horizontal arrows are given by the maps to the residue fields.","statement_latex":"In the situation above, let $Z = \\Spec(k')$ where $k'$ is a field and\n$Z' = \\Spec(k'_1 \\times \\ldots \\times k'_n)$ with $k'_i/k'$\nfinite extensions of fields. Let $x \\in X$ be the image of $Z \\to X$\nand $x'_i \\in X'$ the image of $\\Spec(k'_i) \\to X'$.\nThen we have a fibre product diagram\n$$\n\\xymatrix{\n\\prod\\nolimits_{i = 1, \\ldots, n} k'_i &\n\\prod\\nolimits_{i = 1, \\ldots, n} \\mathcal{O}_{X', x'_i}^\\wedge \\ar[l] \\\\\nk' \\ar[u] &\n\\mathcal{O}_{X, x}^\\wedge \\ar[u] \\ar[l]\n}\n$$\nwhere the horizontal arrows are given by the maps to the residue fields.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Pushouts","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E36","source_file":"curves.tex","source_line":2915,"source_end_line":2931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2915-L2931","statement_sha256":"bea10ba7db563849b301bf48ab99e775070a1ef3222f06bf7c04a7f738d2b7ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":9347,"rank":9347,"depth":10,"x":338.807,"y":1076.095,"cluster":"varieties-curves"},{"id":"stacks:0C1I","tag":"0C1I","title":"Glueing and squishing · Lemma 0C1I","summary":"Let k be an algebraically closed field. Let k ⊂ A be a ring extension such that A has exactly two k-sub algebras, then either A = k × k or A = k[ε].","statement_latex":"Let $k$ be an algebraically closed field. Let $k \\subset A$ be a ring\nextension such that $A$ has exactly two $k$-sub algebras, then\neither $A = k \\times k$ or $A = k[\\epsilon]$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Glueing and squishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1I","source_file":"curves.tex","source_line":2965,"source_end_line":2970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L2965-L2970","statement_sha256":"8af9424824a7d6ac90a6dcee482057576b9db224b985f8603b47d7ab32902a2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9348,"rank":9348,"depth":0,"x":201.115,"y":1249.026,"cluster":"varieties-curves"},{"id":"stacks:0C1L","tag":"0C1L","title":"Glueing and squishing · Lemma 0C1L","summary":"Let k be an algebraically closed field. Let f : X' → X be a finite morphism of algebraic k-schemes such that O_X ⊂ f_*O_X' and such that f is an isomorphism away from a finite set of points. Then there is a factorization X' = X_n → X_n - 1 → … → X_1 → X_0 = X such that each X_i → X_i - 1 is either the glueing of two points or the squishing of a tangent vector (see Examples [Tag 0C1J] and [Tag 0C1K]).","statement_latex":"Let $k$ be an algebraically closed field. Let $f : X' \\to X$ be a\nfinite morphism of algebraic $k$-schemes such that\n$\\mathcal{O}_X \\subset f_*\\mathcal{O}_{X'}$ and such that $f$ is an\nisomorphism away from a finite set of points. Then there is a factorization\n$$\nX' = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X\n$$\nsuch that each $X_i \\to X_{i - 1}$ is either the glueing of\ntwo points or the squishing of a tangent vector\n(see Examples \\ref{example-glue-points} and\n\\ref{example-squish-tangent-vector}).","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Glueing and squishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1L","source_file":"curves.tex","source_line":3039,"source_end_line":3052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3039-L3052","statement_sha256":"fc69fb52cfbcca090967ba8026ba23ae6193b77475618ba6b318b24cc5541b24","origin":"The Stacks Project","memory_eligible":false,"source_rank":9349,"rank":9349,"depth":1,"x":163.524,"y":1043.07,"cluster":"varieties-curves"},{"id":"stacks:0C1M","tag":"0C1M","title":"Glueing and squishing · Lemma 0C1M","summary":"Let k be an algebraically closed field. If f : X' → X is the glueing of two points a, b as in Example [Tag 0C1J], then there is an exact sequence k^* → Pic(X) → Pic(X') → 0 The first map is zero if a and b are on different connected components of X' and injective if X' is proper and a and b are on the same connected component of X'.","statement_latex":"Let $k$ be an algebraically closed field. If $f : X' \\to X$ is the\nglueing of two points $a, b$ as in Example \\ref{example-glue-points}, then\nthere is an exact sequence\n$$\nk^* \\to \\Pic(X) \\to \\Pic(X') \\to 0\n$$\nThe first map is zero if $a$ and $b$ are on different\nconnected components of $X'$ and injective\nif $X'$ is proper and $a$ and $b$ are on the same connected component of $X'$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Glueing and squishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1M","source_file":"curves.tex","source_line":3105,"source_end_line":3116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3105-L3116","statement_sha256":"b86a9a62ab1c2ee8ab724f15dc2605f8eda819a4e817e55a870893de6c0aae7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9350,"rank":9350,"depth":33,"x":357.156,"y":1173.805,"cluster":"varieties-curves"},{"id":"stacks:0C1N","tag":"0C1N","title":"Glueing and squishing · Lemma 0C1N","summary":"Let k be an algebraically closed field. If f : X' → X is the squishing of a tangent vector vartheta as in Example [Tag 0C1K], then there is an exact sequence (k, +) → Pic(X) → Pic(X') → 0 and the first map is injective if X' is proper and reduced.","statement_latex":"Let $k$ be an algebraically closed field. If $f : X' \\to X$ is the\nsquishing of a tangent vector $\\vartheta$ as in\nExample \\ref{example-squish-tangent-vector}, then\nthere is an exact sequence\n$$\n(k, +) \\to \\Pic(X) \\to \\Pic(X') \\to 0\n$$\nand the first map is injective if $X'$ is proper and reduced.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Glueing and squishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1N","source_file":"curves.tex","source_line":3146,"source_end_line":3156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3146-L3156","statement_sha256":"8d34078cae3d3d0114c78bd230e71c06db8ede53e3566b3ef8e5e112954f901b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9351,"rank":9351,"depth":33,"x":108.872,"y":1187.295,"cluster":"varieties-curves"},{"id":"stacks:0C1V","tag":"0C1V","title":"Multicross and nodal singularities · Lemma 0C1V","summary":"Let k be a separably closed field. Let A be a 1-dimensional reduced Nagata local k-algebra with residue field k. Then δ-invariant A ≥ number of branches of A - 1 If equality holds, then A^wedge is as in ([Tag 0C1U]).","statement_latex":"Let $k$ be a separably closed field. Let $A$ be a $1$-dimensional\nreduced Nagata local $k$-algebra with residue field $k$. Then\n$$\n\\delta\\text{-invariant }A \\geq \\text{number of branches of }A - 1\n$$\nIf equality holds, then $A^\\wedge$ is as in (\\ref{equation-multicross}).","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Multicross and nodal singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1V","source_file":"curves.tex","source_line":3213,"source_end_line":3221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3213-L3221","statement_sha256":"1f49a925c1ed3c98f145fe74359b8073b28d6177b50fa61c9d870642638a0d3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9352,"rank":9352,"depth":56,"x":281.361,"y":1036.24,"cluster":"varieties-curves"},{"id":"stacks:0C1W","tag":"0C1W","title":"Multicross and nodal singularities · Definition 0C1W","summary":"Let k be an algebraically closed field. Let X be an algebraic 1-dimensional k-scheme. Let x ∈ X be a closed point. We say x defines a multicross singularity if the completion O_X, x^wedge is isomorphic to ([Tag 0C1U]) for some n ≥ 2. We say x is a node, or an ordinary double point, or defines a nodal singularity if n = 2.","statement_latex":"Let $k$ be an algebraically closed field. Let $X$ be an algebraic\n$1$-dimensional $k$-scheme. Let $x \\in X$ be a closed point.\nWe say $x$ defines a {\\it multicross singularity} if the completion\n$\\mathcal{O}_{X, x}^\\wedge$\nis isomorphic to (\\ref{equation-multicross}) for some $n \\geq 2$.\nWe say $x$ is a {\\it node}, or an {\\it ordinary double point}, or\n{\\it defines a nodal singularity} if $n = 2$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Multicross and nodal singularities","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1W","source_file":"curves.tex","source_line":3245,"source_end_line":3254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3245-L3254","statement_sha256":"b3ba91c64ecd575d46407e0047976dbca9be2a9f1a9afa7448e3b7726140a15a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9353,"rank":9353,"depth":0,"x":275.635,"y":1245.811,"cluster":"varieties-curves"},{"id":"stacks:0C1X","tag":"0C1X","title":"Multicross and nodal singularities · Lemma 0C1X","summary":"Let k be an algebraically closed field. Let X be a reduced algebraic 1-dimensional k-scheme. Let x ∈ X. The following are equivalent • x defines a multicross singularity, • the δ-invariant of X at x is the number of branches of X at x minus 1, • there is a sequence of morphisms U_n → U_n - 1 → … → U_0 = U ⊂ X where U is an open neighbourhood of x, where U_n is nonsingular, and where each U_i → U_i - 1 is the glueing of two points as in Example [Tag 0C1J].","statement_latex":"Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic\n$1$-dimensional $k$-scheme. Let $x \\in X$. The following are equivalent\n\\begin{enumerate}\n\\item $x$ defines a multicross singularity,\n\\item the $\\delta$-invariant of $X$ at $x$ is the\nnumber of branches of $X$ at $x$ minus $1$,\n\\item there is a sequence of morphisms\n$U_n \\to U_{n - 1} \\to \\ldots \\to U_0 = U \\subset X$\nwhere $U$ is an open neighbourhood of $x$, where\n$U_n$ is nonsingular, and where each $U_i \\to U_{i - 1}$\nis the glueing of two points as in Example \\ref{example-glue-points}.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Multicross and nodal singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1X","source_file":"curves.tex","source_line":3260,"source_end_line":3274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3260-L3274","statement_sha256":"fe5e01a655757fe7f0842aaa4bed4acabb8c079c5378f32aadfada7b3fda3082","origin":"The Stacks Project","memory_eligible":false,"source_rank":9354,"rank":9354,"depth":57,"x":111.084,"y":1087.794,"cluster":"varieties-curves"},{"id":"stacks:0CDZ","tag":"0CDZ","title":"Multicross and nodal singularities · Lemma 0CDZ","summary":"Let k be an algebraically closed field. Let X be a reduced algebraic 1-dimensional k-scheme. Let x ∈ X be a multicross singularity (Definition [Tag 0C1W]). If X is Gorenstein, then x is a node.","statement_latex":"Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic\n$1$-dimensional $k$-scheme. Let $x \\in X$ be a multicross singularity\n(Definition \\ref{definition-multicross}).\nIf $X$ is Gorenstein, then $x$ is a node.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Multicross and nodal singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDZ","source_file":"curves.tex","source_line":3317,"source_end_line":3323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3317-L3323","statement_sha256":"37807f3c504ab94c677b44dacedfdf42a73a3bf728d323717ae8bf0ff28e82e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9355,"rank":9355,"depth":40,"x":359.86,"y":1110.977,"cluster":"varieties-curves"},{"id":"stacks:0C1Z","tag":"0C1Z","title":"Torsion in the Picard group · Lemma 0C1Z","summary":"Let k be an algebraically closed field. Let X be a smooth projective curve of genus g over k. • If n ≥ 1 is invertible in k, then Pic(X)[n] ≅ (Z/nZ)^⊕ 2g. • If the characteristic of k is p > 0, then there exists an integer 0 ≤ f ≤ g such that Pic(X)[p^m] ≅ (Z/p^mZ)^⊕ f for all m ≥ 1.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $X$ be a smooth projective curve of genus $g$ over $k$.\n\\begin{enumerate}\n\\item If $n \\geq 1$ is invertible in $k$, then\n$\\Pic(X)[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2g}$.\n\\item If the characteristic of $k$ is $p > 0$, then there exists\nan integer $0 \\leq f \\leq g$ such that\n$\\Pic(X)[p^m] \\cong (\\mathbf{Z}/p^m\\mathbf{Z})^{\\oplus f}$ for\nall $m \\geq 1$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Torsion in the Picard group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C1Z","source_file":"curves.tex","source_line":3393,"source_end_line":3405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3393-L3405","statement_sha256":"365c7c75f8692ae9171524dedfddf2c7dff8a61d77822ab7a268f5a403982536","origin":"The Stacks Project","memory_eligible":false,"source_rank":9356,"rank":9356,"depth":66,"x":157.476,"y":1235.226,"cluster":"varieties-curves"},{"id":"stacks:0CDU","tag":"0CDU","title":"Torsion in the Picard group · Lemma 0CDU","summary":"Let k be a field. Let n be prime to the characteristic of k. Let X be a smooth proper curve over k with H^0(X, O_X) = k and of genus g. • If g = 1 then there exists a finite separable extension k'/k such that X_k' has a k'-rational point and Pic(X_k')[n] ≅ (Z/nZ)^⊕ 2. • If g ≥ 2 then there exists a finite separable extension k'/k with [k' : k] ≤ (2g - 2)(n^2g)! such that X_k' has a k'-rational point and Pic(X_k')[n] ≅ (Z/nZ)^⊕ 2g.","statement_latex":"Let $k$ be a field. Let $n$ be prime to the characteristic of $k$.\nLet $X$ be a smooth proper curve over $k$ with $H^0(X, \\mathcal{O}_X) = k$\nand of genus $g$.\n\\begin{enumerate}\n\\item If $g = 1$ then there exists a finite separable extension\n$k'/k$ such that $X_{k'}$ has a $k'$-rational point and\n$\\Pic(X_{k'})[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2}$.\n\\item If $g \\geq 2$ then there exists a finite separable extension\n$k'/k$ with $[k' : k] \\leq (2g - 2)(n^{2g})!$\nsuch that $X_{k'}$ has a $k'$-rational point and\n$\\Pic(X_{k'})[n] \\cong (\\mathbf{Z}/n\\mathbf{Z})^{\\oplus 2g}$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Torsion in the Picard group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDU","source_file":"curves.tex","source_line":3424,"source_end_line":3438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3424-L3438","statement_sha256":"0da7d25d7adef946788b774925f407129f2b9450196b1a8dd5f7a8a81495cd5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9357,"rank":9357,"depth":67,"x":206.866,"y":1028.467,"cluster":"varieties-curves"},{"id":"stacks:0C20","tag":"0C20","title":"Torsion in the Picard group · Proposition 0C20","summary":"Let k be an algebraically closed field. Let X be a proper scheme over k which is reduced, connected, and has dimension 1. Let g be the genus of X and let g_geom be the sum of the geometric genera of the irreducible components of X. For any prime ℓ different from the characteristic of k we have dim_F_ℓ Pic(X)[ℓ] ≤ g + g_geom and equality holds if and only if all the singularities of X are multicross.","statement_latex":"Let $k$ be an algebraically closed field. Let $X$ be a proper scheme over $k$\nwhich is reduced, connected, and has dimension $1$. Let $g$ be the genus\nof $X$ and let $g_{geom}$ be the sum of the geometric genera of the\nirreducible components of $X$. For any prime $\\ell$ different from\nthe characteristic of $k$ we have\n$$\n\\dim_{\\mathbf{F}_\\ell} \\Pic(X)[\\ell]\n\\leq g + g_{geom}\n$$\nand equality holds if and only if all the singularities of $X$\nare multicross.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Torsion in the Picard group","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C20","source_file":"curves.tex","source_line":3483,"source_end_line":3496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3483-L3496","statement_sha256":"10e298f35f61a739f07ac3dec0dbbfbeaf7996eb5e02f5e95a48eb1a9aa76623","origin":"The Stacks Project","memory_eligible":false,"source_rank":9358,"rank":9358,"depth":67,"x":336.906,"y":1209.217,"cluster":"varieties-curves"},{"id":"stacks:0CE1","tag":"0CE1","title":"Genus versus geometric genus · Lemma 0CE1","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. Then g_geom(X/k) = ∑_C ⊂ X g_geom(C/k) where the sum is over irreducible components C ⊂ X of dimension 1.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.\nThen\n$$\ng_{geom}(X/k) = \\sum\\nolimits_{C \\subset X} g_{geom}(C/k)\n$$\nwhere the sum is over irreducible components $C \\subset X$ of dimension $1$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Genus versus geometric genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CE1","source_file":"curves.tex","source_line":3581,"source_end_line":3589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3581-L3589","statement_sha256":"212496e5468d6f1d0751f6ae744e3390cfdb3ef8ff9aa899a8ffa0ec9a1922c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9359,"rank":9359,"depth":27,"x":95.311,"y":1149.636,"cluster":"varieties-curves"},{"id":"stacks:0CE2","tag":"0CE2","title":"Genus versus geometric genus · Lemma 0CE2","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. Then • We have g_geom(X/k) = g_geom(X_red/k). • If X' → X is a birational proper morphism, then g_geom(X'/k) = g_geom(X/k). • If X^ν → X is the normalization morphism, then g_geom(X^ν/k) = g_geom(X/k).","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.\nThen\n\\begin{enumerate}\n\\item We have $g_{geom}(X/k) = g_{geom}(X_{red}/k)$.\n\\item If $X' \\to X$ is a birational proper morphism, then\n$g_{geom}(X'/k) = g_{geom}(X/k)$.\n\\item If $X^\\nu \\to X$ is the normalization morphism, then\n$g_{geom}(X^\\nu/k) = g_{geom}(X/k)$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Genus versus geometric genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CE2","source_file":"curves.tex","source_line":3601,"source_end_line":3612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3601-L3612","statement_sha256":"15fd553683cb2ad028e5c2276a592fc847f2f6416acf03dd0281472ffbba6ab6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9360,"rank":9360,"depth":43,"x":321.702,"y":1056.348,"cluster":"varieties-curves"},{"id":"stacks:0CE3","tag":"0CE3","title":"Genus versus geometric genus · Lemma 0CE3","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. Let f : Y → X be a finite morphism such that there exists a dense open U ⊂ X over which f is a closed immersion. Then dim_k H^1(X, O_X) ≥ dim_k H^1(Y, O_Y)","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension\n$\\leq 1$ over $k$. Let $f : Y \\to X$ be a finite morphism\nsuch that there exists a dense open $U \\subset X$ over\nwhich $f$ is a closed immersion. Then\n$$\n\\dim_k H^1(X, \\mathcal{O}_X) \\geq \\dim_k H^1(Y, \\mathcal{O}_Y)\n$$","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Genus versus geometric genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CE3","source_file":"curves.tex","source_line":3632,"source_end_line":3641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3632-L3641","statement_sha256":"32c8d72133c23d1e533e583385676b5578ab3d04c5c000fb304b6580b486f32e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9361,"rank":9361,"depth":27,"x":229.651,"y":1253.882,"cluster":"varieties-curves"},{"id":"stacks:0CE4","tag":"0CE4","title":"Genus versus geometric genus · Lemma 0CE4","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. If X' → X is a birational proper morphism, then dim_k H^1(X, O_X) ≥ dim_k H^1(X', O_X') If X is reduced, H^0(X, O_X) → H^0(X', O_X') is surjective, and equality holds, then X' = X.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.\nIf $X' \\to X$ is a birational proper morphism, then\n$$\n\\dim_k H^1(X, \\mathcal{O}_X) \\geq \\dim_k H^1(X', \\mathcal{O}_{X'})\n$$\nIf $X$ is reduced, $H^0(X, \\mathcal{O}_X) \\to H^0(X', \\mathcal{O}_{X'})$\nis surjective, and equality holds, then $X' = X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Genus versus geometric genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CE4","source_file":"curves.tex","source_line":3661,"source_end_line":3670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3661-L3670","statement_sha256":"5be1c1a2e9816c954b78031f33a75c604d2bca3047b942a1eda5321dcbe117e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9362,"rank":9362,"depth":43,"x":138.545,"y":1055.706,"cluster":"varieties-curves"},{"id":"stacks:0CE5","tag":"0CE5","title":"Genus versus geometric genus · Lemma 0CE5","summary":"Let k be a field. Let C be a proper curve over k. Set kappa = H^0(C, O_C). Then [kappa : k]_s dim_kappa H^1(C, O_C) ≥ g_geom(C/k)","statement_latex":"Let $k$ be a field. Let $C$ be a proper curve over $k$.\nSet $\\kappa = H^0(C, \\mathcal{O}_C)$. Then\n$$\n[\\kappa : k]_s \\dim_\\kappa H^1(C, \\mathcal{O}_C) \\geq g_{geom}(C/k)\n$$","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Genus versus geometric genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CE5","source_file":"curves.tex","source_line":3692,"source_end_line":3699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3692-L3699","statement_sha256":"b9f52302718777923ca4fed6c6a5c8f411e5b7ed96fd98765387af1fd3b4dd30","origin":"The Stacks Project","memory_eligible":false,"source_rank":9363,"rank":9363,"depth":34,"x":365.419,"y":1150.278,"cluster":"varieties-curves"},{"id":"stacks:0CE6","tag":"0CE6","title":"Genus versus geometric genus · Lemma 0CE6","summary":"Let k be a field. Let X be a proper scheme of dimension ≤ 1 over k. Let ℓ be a prime number invertible in k. Then dim_F_ℓ Pic(X)[ℓ] ≤ dim_k H^1(X, O_X) + g_geom(X/k) where g_geom(X/k) is as defined above.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme of dimension $\\leq 1$ over $k$.\nLet $\\ell$ be a prime number invertible in $k$. Then\n$$\n\\dim_{\\mathbf{F}_\\ell} \\Pic(X)[\\ell] \\leq\n\\dim_k H^1(X, \\mathcal{O}_X) + g_{geom}(X/k)\n$$\nwhere $g_{geom}(X/k)$ is as defined above.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Genus versus geometric genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CE6","source_file":"curves.tex","source_line":3741,"source_end_line":3750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3741-L3750","statement_sha256":"129aa80e780e80e8d17cbf95183ce8dfa37b43676711ba99edc29b4555e3e3fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9364,"rank":9364,"depth":68,"x":121.724,"y":1209.36,"cluster":"varieties-curves"},{"id":"stacks:0C47","tag":"0C47","title":"Nodal curves · Definition 0C47","summary":"Let k be a field. Let X be a 1-dimensional locally algebraic k-scheme. • We say a closed point x ∈ X is a node, or an ordinary double point, or defines a nodal singularity if there exists an ordinary double point overlinex ∈ X_overlinek mapping to x. • We say the singularities of X are at-worst-nodal if all closed points of X are either in the smooth locus of the structure morphism X → Spec(k) or are ordinary double points.","statement_latex":"Let $k$ be a field. Let $X$ be a $1$-dimensional locally algebraic $k$-scheme.\n\\begin{enumerate}\n\\item We say a closed point $x \\in X$ is a {\\it node}, or an\n{\\it ordinary double point}, or {\\it defines a nodal singularity}\nif there exists an ordinary double point $\\overline{x} \\in X_{\\overline{k}}$\nmapping to $x$.\n\\item We say the {\\it singularities of $X$ are at-worst-nodal} if\nall closed points of $X$ are either in the smooth locus of\nthe structure morphism $X \\to \\Spec(k)$ or are ordinary double points.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C47","source_file":"curves.tex","source_line":3795,"source_end_line":3807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3795-L3807","statement_sha256":"a2e6146281a1a6afd3a63fac0124475f2334495fde20695fbf85cd2f37ade348","origin":"The Stacks Project","memory_eligible":false,"source_rank":9365,"rank":9365,"depth":0,"x":254.098,"y":1027.255,"cluster":"varieties-curves"},{"id":"stacks:0C48","tag":"0C48","title":"Nodal curves · Lemma 0C48","summary":"Let (A, m) be a regular local ring of dimension 2. Let I ⊂ m be an ideal. • If A/I is reduced, then I = (0), I = m, or I = (f) for some nonzero f ∈ m. • If A/I has depth 1, then I = (f) for some nonzero f ∈ m.","statement_latex":"Let $(A, \\mathfrak m)$ be a regular local ring of dimension $2$.\nLet $I \\subset \\mathfrak m$ be an ideal.\n\\begin{enumerate}\n\\item If $A/I$ is reduced, then $I = (0)$, $I = \\mathfrak m$, or\n$I = (f)$ for some nonzero $f \\in \\mathfrak m$.\n\\item If $A/I$ has depth $1$, then $I = (f)$ for some nonzero\n$f \\in \\mathfrak m$. \n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C48","source_file":"curves.tex","source_line":3816,"source_end_line":3826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3816-L3826","statement_sha256":"42216870c037d95db9ad0db59f9a254831016e50f4031165a810f0a78bf5ce9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9366,"rank":9366,"depth":19,"x":303.0,"y":1236.948,"cluster":"varieties-curves"},{"id":"stacks:0C49","tag":"0C49","title":"Nodal curves · Lemma 0C49","summary":"Let k be a field. Let (A, m, kappa) be a Noetherian local k-algebra. The following are equivalent • kappa/k is separable, A is reduced, dim_kappa( m/ m^2) = 2, and there exists a nondegenerate q ∈ Sym^2_kappa( m/ m^2) which maps to zero in m^2/ m^3, • kappa/k is separable, depth(A) = 1, dim_kappa( m/ m^2) = 2, and there exists a nondegenerate q ∈ Sym^2_kappa( m/ m^2) which maps to zero in m^2/ m^3, • kappa/k is separable, A^wedge ≅ kappa[[x, y]]/(ax^2 + bxy + cy^2) as a…","statement_latex":"Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian\nlocal $k$-algebra. The following are equivalent\n\\begin{enumerate}\n\\item $\\kappa/k$ is separable, $A$ is reduced,\n$\\dim_\\kappa(\\mathfrak m/\\mathfrak m^2) = 2$, and there exists a nondegenerate\n$q \\in \\text{Sym}^2_\\kappa(\\mathfrak m/\\mathfrak m^2)$\nwhich maps to zero in $\\mathfrak m^2/\\mathfrak m^3$,\n\\item $\\kappa/k$ is separable, $\\text{depth}(A) = 1$,\n$\\dim_\\kappa(\\mathfrak m/\\mathfrak m^2) = 2$, and there exists a nondegenerate\n$q \\in \\text{Sym}^2_\\kappa(\\mathfrak m/\\mathfrak m^2)$\nwhich maps to zero in $\\mathfrak m^2/\\mathfrak m^3$,\n\\item $\\kappa/k$ is separable,\n$A^\\wedge \\cong \\kappa[[x, y]]/(ax^2 + bxy + cy^2)$\nas a $k$-algebra where $ax^2 + bxy + cy^2$ is a nondegenerate quadratic form\nover $\\kappa$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C49","source_file":"curves.tex","source_line":3872,"source_end_line":3890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3872-L3890","statement_sha256":"0b155a1f7cff6a0d5485cf47ed8cd380405b2be0aa0e6178bc3bdda5984b117b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9367,"rank":9367,"depth":20,"x":98.021,"y":1109.891,"cluster":"varieties-curves"},{"id":"stacks:0C4A","tag":"0C4A","title":"Nodal curves · Lemma 0C4A","summary":"Let k be a field. Let (A, m, kappa) be a Nagata local k-algebra. The following are equivalent • k → A is as in Lemma [Tag 0C49], • kappa/k is separable, A is reduced of dimension 1, the δ-invariant of A is 1, and A has 2 geometric branches. If this holds, then the integral closure A' of A in its total ring of fractions has either 1 or 2 maximal ideals m' and the extensions kappa( m')/k are separable.","statement_latex":"Let $k$ be a field. Let $(A, \\mathfrak m, \\kappa)$ be a\nNagata local $k$-algebra. The following are equivalent\n\\begin{enumerate}\n\\item $k \\to A$ is as in Lemma \\ref{lemma-nodal-algebraic},\n\\item $\\kappa/k$ is separable, $A$ is reduced of dimension $1$,\nthe $\\delta$-invariant of $A$ is $1$, and $A$ has $2$ geometric branches.\n\\end{enumerate}\nIf this holds, then the integral closure $A'$ of $A$\nin its total ring of fractions has either $1$ or $2$\nmaximal ideals $\\mathfrak m'$ and the extensions\n$\\kappa(\\mathfrak m')/k$ are separable.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4A","source_file":"curves.tex","source_line":3963,"source_end_line":3976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L3963-L3976","statement_sha256":"a79ad4dd4aaa54198ae6e638a4549abef9d53e02bc132bfe5c53b3c03310f24d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9368,"rank":9368,"depth":55,"x":351.704,"y":1087.239,"cluster":"varieties-curves"},{"id":"stacks:0C4B","tag":"0C4B","title":"Nodal curves · Lemma 0C4B","summary":"Let k be a field. Let A = k[[x_1, …, x_n]]. Let I = (f_1, …, f_m) ⊂ A be an ideal. For any r ≥ 0 the ideal in A/I generated by the r × r-minors of the matrix (∂ f_j/∂ x_i) is independent of the choice of the generators of I or the regular system of parameters x_1, …, x_n of A.","statement_latex":"Let $k$ be a field. Let $A = k[[x_1, \\ldots, x_n]]$. Let\n$I = (f_1, \\ldots, f_m) \\subset A$ be an ideal. For any\n$r \\geq 0$ the ideal in $A/I$ generated by the $r \\times r$-minors\nof the matrix $(\\partial f_j/\\partial x_i)$ is independent\nof the choice of the generators of $I$ or the\nregular system of parameters $x_1, \\ldots, x_n$ of $A$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4B","source_file":"curves.tex","source_line":4064,"source_end_line":4072,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4064-L4072","statement_sha256":"d13df3bbbbfc49ebf920083617430e243e4ecf9fbe008cbb85e3063b6658c6ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":9369,"rank":9369,"depth":0,"x":182.62,"y":1248.116,"cluster":"varieties-curves"},{"id":"stacks:0C4C","tag":"0C4C","title":"Nodal curves · Lemma 0C4C","summary":"Let k be a field. Let A = k[[x_1, …, x_n]]. Let I = (f_1, …, f_m) ⊂ m_A be an ideal. The following are equivalent • k → A/I is as in Lemma [Tag 0C49], • A/I is reduced and the (n - 1) × (n - 1) minors of the matrix (∂ f_j/∂ x_i) generate I + m_A, • depth(A/I) = 1 and the (n - 1) × (n - 1) minors of the matrix (∂ f_j/∂ x_i) generate I + m_A.","statement_latex":"Let $k$ be a field. Let $A = k[[x_1, \\ldots, x_n]]$. Let\n$I = (f_1, \\ldots, f_m) \\subset \\mathfrak m_A$ be an ideal. The following\nare equivalent\n\\begin{enumerate}\n\\item $k \\to A/I$ is as in Lemma \\ref{lemma-nodal-algebraic},\n\\item $A/I$ is reduced and the\n$(n - 1) \\times (n - 1)$ minors of the matrix\n$(\\partial f_j/\\partial x_i)$ generate $I + \\mathfrak m_A$,\n\\item $\\text{depth}(A/I) = 1$ and the\n$(n - 1) \\times (n - 1)$ minors of the matrix\n$(\\partial f_j/\\partial x_i)$ generate $I + \\mathfrak m_A$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4C","source_file":"curves.tex","source_line":4094,"source_end_line":4108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4094-L4108","statement_sha256":"6b4f10674f9ea1ef9fe610b2b0438f0f379750bd3a49c53e8170d9bf36b6d014","origin":"The Stacks Project","memory_eligible":false,"source_rank":9370,"rank":9370,"depth":21,"x":177.921,"y":1033.242,"cluster":"varieties-curves"},{"id":"stacks:0C4D","tag":"0C4D","title":"Nodal curves · Lemma 0C4D","summary":"Let k be a field. Let X be a 1-dimensional algebraic k-scheme. Let x ∈ X be a closed point. The following are equivalent • x is a node, • k → O_X, x is as in Lemma [Tag 0C49], • any overlinex ∈ X_overlinek mapping to x defines a nodal singularity, • kappa(x)/k is separable, O_X, x is reduced, and the first Fitting ideal of Ω_X/k generates m_x in O_X, x, • kappa(x)/k is separable, depth(O_X, x) = 1, and the first Fitting ideal of Ω_X/k generates m_x in O_X, x, • kappa(x)/k…","statement_latex":"Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme.\nLet $x \\in X$ be a closed point. The following are equivalent\n\\begin{enumerate}\n\\item $x$ is a node,\n\\item $k \\to \\mathcal{O}_{X, x}$ is as in Lemma \\ref{lemma-nodal-algebraic},\n\\item any $\\overline{x} \\in X_{\\overline{k}}$ mapping to $x$ defines\na nodal singularity,\n\\item $\\kappa(x)/k$ is separable, $\\mathcal{O}_{X, x}$ is reduced, and\nthe first Fitting ideal of $\\Omega_{X/k}$ generates $\\mathfrak m_x$\nin $\\mathcal{O}_{X, x}$,\n\\item $\\kappa(x)/k$ is separable, $\\text{depth}(\\mathcal{O}_{X, x}) = 1$, and\nthe first Fitting ideal of $\\Omega_{X/k}$ generates $\\mathfrak m_x$\nin $\\mathcal{O}_{X, x}$,\n\\item $\\kappa(x)/k$ is separable and $\\mathcal{O}_{X, x}$ is reduced, has\n$\\delta$-invariant $1$, and has $2$ geometric branches.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4D","source_file":"curves.tex","source_line":4173,"source_end_line":4191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4173-L4191","statement_sha256":"92d61ac3f32ee28e95beca1ecd8b4c47bfbe6ab19fc5ba09289d26b6dfa18945","origin":"The Stacks Project","memory_eligible":false,"source_rank":9371,"rank":9371,"depth":57,"x":354.429,"y":1189.24,"cluster":"varieties-curves"},{"id":"stacks:0CBV","tag":"0CBV","title":"Nodal curves · Definition 0CBV","summary":"Let k be a field. Let X be a 1-dimensional algebraic k-scheme. Let x ∈ X be a closed point. We say x is a split node if x is a node, kappa(x) = k, and the equivalent assertions of Remark [Tag 0CBU] hold for A = O_X, x.","statement_latex":"Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme.\nLet $x \\in X$ be a closed point. We say $x$ is a {\\it split node}\nif $x$ is a node, $\\kappa(x) = k$, and the equivalent assertions of\nRemark \\ref{remark-trivial-quadratic-extension}\nhold for $A = \\mathcal{O}_{X, x}$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBV","source_file":"curves.tex","source_line":4354,"source_end_line":4361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4354-L4361","statement_sha256":"899ac70ebaeb47d5480f7537a74434114943c6470d4f5e003b78a2fb8cb506a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9372,"rank":9372,"depth":0,"x":98.467,"y":1174.343,"cluster":"varieties-curves"},{"id":"stacks:0CBW","tag":"0CBW","title":"Nodal curves · Lemma 0CBW","summary":"Let k be a field. Let X be a 1-dimensional algebraic k-scheme. Let x ∈ X be a closed point. The following are equivalent • x is a split node, • x is a node and there are exactly two points x_1, x_2 of the normalization X^ν lying over x with k = kappa(x_1) = kappa(x_2), • O_X, x^wedge ≅ k[[x, y]]/(xy) as a k-algebra, and • add more here.","statement_latex":"Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme.\nLet $x \\in X$ be a closed point. The following are equivalent\n\\begin{enumerate}\n\\item $x$ is a split node,\n\\item $x$ is a node and there are exactly two points $x_1, x_2$\nof the normalization $X^\\nu$ lying over $x$ with\n$k = \\kappa(x_1) = \\kappa(x_2)$,\n\\item $\\mathcal{O}_{X, x}^\\wedge \\cong k[[x, y]]/(xy)$ as a $k$-algebra, and\n\\item add more here.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBW","source_file":"curves.tex","source_line":4367,"source_end_line":4379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4367-L4379","statement_sha256":"e2a1346869dbe89a9f184296db279cb9727354bd502c54a44ffd74afd25c1331","origin":"The Stacks Project","memory_eligible":false,"source_rank":9373,"rank":9373,"depth":58,"x":299.469,"y":1039.901,"cluster":"varieties-curves"},{"id":"stacks:0C56","tag":"0C56","title":"Nodal curves · Lemma 0C56","summary":"Let K/k be an extension of fields. Let X be a locally algebraic k-scheme of dimension 1. Let y ∈ X_K be a point with image x ∈ X. The following are equivalent • x is a closed point of X and a node, and • y is a closed point of Y and a node.","statement_latex":"Let $K/k$ be an extension of fields. Let $X$ be a locally algebraic\n$k$-scheme of dimension $1$. Let $y \\in X_K$ be a point with image\n$x \\in X$. The following are equivalent\n\\begin{enumerate}\n\\item $x$ is a closed point of $X$ and a node, and\n\\item $y$ is a closed point of $Y$ and a node.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C56","source_file":"curves.tex","source_line":4387,"source_end_line":4396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4387-L4396","statement_sha256":"05d469b872c72c256c82ea82759f6a3478784d1d29f0dbea8a8d521c07d8c68b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9374,"rank":9374,"depth":58,"x":259.312,"y":1253.387,"cluster":"varieties-curves"},{"id":"stacks:0C57","tag":"0C57","title":"Nodal curves · Lemma 0C57","summary":"Let k be a field. Let X be a locally algebraic k-scheme of dimension 1. Let Y → X be an étale morphism. Let y ∈ Y be a point with image x ∈ X. The following are equivalent • x is a closed point of X and a node, and • y is a closed point of Y and a node.","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic\n$k$-scheme of dimension $1$. Let $Y \\to X$ be an \\'etale morphism.\nLet $y \\in Y$ be a point with image $x \\in X$. The following are equivalent\n\\begin{enumerate}\n\\item $x$ is a closed point of $X$ and a node, and\n\\item $y$ is a closed point of $Y$ and a node.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C57","source_file":"curves.tex","source_line":4444,"source_end_line":4453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4444-L4453","statement_sha256":"c0f9bfa4095227e1b2979b3fe70a54713e4a6dc7dbe893920d3f0a2ec0937bba","origin":"The Stacks Project","memory_eligible":false,"source_rank":9375,"rank":9375,"depth":59,"x":117.046,"y":1072.929,"cluster":"varieties-curves"},{"id":"stacks:0CD6","tag":"0CD6","title":"Nodal curves · Lemma 0CD6","summary":"Let k'/k be a finite separable field extension. Let X be a locally algebraic k'-scheme of dimension 1. Let x ∈ X be a closed point. The following are equivalent • x is a node, and • x is a node when X viewed as a locally algebraic k-scheme.","statement_latex":"Let $k'/k$ be a finite separable field extension.\nLet $X$ be a locally algebraic $k'$-scheme of dimension $1$.\nLet $x \\in X$ be a closed point. The following are equivalent\n\\begin{enumerate}\n\\item $x$ is a node, and\n\\item $x$ is a node when $X$ viewed as a locally algebraic $k$-scheme.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CD6","source_file":"curves.tex","source_line":4466,"source_end_line":4475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4466-L4475","statement_sha256":"588d85b3d32e78d3d0bddeaf0ded83dacfdf4394091561e0e4722d5d71dd120f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9376,"rank":9376,"depth":58,"x":367.417,"y":1125.344,"cluster":"varieties-curves"},{"id":"stacks:0C4E","tag":"0C4E","title":"Nodal curves · Lemma 0C4E","summary":"Let k be a field. Let X be a locally algebraic k-scheme equidimensional of dimension 1. The following are equivalent • the singularities of X are at-worst-nodal, and • X is a local complete intersection over k and the closed subscheme Z ⊂ X cut out by the first fitting ideal of Ω_X/k is unramified over k.","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme\nequidimensional of dimension $1$.\nThe following are equivalent\n\\begin{enumerate}\n\\item the singularities of $X$ are at-worst-nodal, and\n\\item $X$ is a local complete intersection over $k$\nand the closed subscheme $Z \\subset X$ cut out by the\nfirst fitting ideal of $\\Omega_{X/k}$ is unramified over $k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C4E","source_file":"curves.tex","source_line":4482,"source_end_line":4493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4482-L4493","statement_sha256":"a20c57fe46151a907456ba1d00142209b388c508ee243bfd2c246861721d8064","origin":"The Stacks Project","memory_eligible":false,"source_rank":9377,"rank":9377,"depth":58,"x":140.333,"y":1228.903,"cluster":"varieties-curves"},{"id":"stacks:0E37","tag":"0E37","title":"Nodal curves · Lemma 0E37","summary":"Let k be a field. Let X be a locally algebraic k-scheme equidimensional of dimension 1 whose singularities are at-worst-nodal. Then X is Gorenstein and geometrically reduced.","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme\nequidimensional of dimension $1$ whose singularities are at-worst-nodal.\nThen $X$ is Gorenstein and geometrically reduced.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E37","source_file":"curves.tex","source_line":4545,"source_end_line":4550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4545-L4550","statement_sha256":"31ac92a6b382afe573257a5b6c69f103eed00dfd0d551a609e6645194083e243","origin":"The Stacks Project","memory_eligible":false,"source_rank":9378,"rank":9378,"depth":59,"x":224.618,"y":1023.405,"cluster":"varieties-curves"},{"id":"stacks:0E38","tag":"0E38","title":"Nodal curves · Lemma 0E38","summary":"Let k be a field. Let X be a locally algebraic k-scheme equidimensional of dimension 1 whose singularities are at-worst-nodal. If Y ⊂ X is a reduced closed subscheme equidimensional of dimension 1, then • the singularities of Y are at-worst-nodal, and • if Z ⊂ X is the scheme theoretic closure of X setminus Y, then • the scheme theoretic intersection Y ∩ Z is the disjoint union of spectra of finite separable extensions of k, • each point of Y ∩ Z is a node of X, and • Y →…","statement_latex":"Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme\nequidimensional of dimension $1$ whose singularities are at-worst-nodal.\nIf $Y \\subset X$ is a reduced closed subscheme\nequidimensional of dimension $1$, then\n\\begin{enumerate}\n\\item the singularities of $Y$ are at-worst-nodal, and\n\\item if $Z \\subset X$ is the scheme theoretic closure of\n$X \\setminus Y$, then\n\\begin{enumerate}\n\\item the scheme theoretic intersection $Y \\cap Z$ is\nthe disjoint union of spectra of finite separable extensions of $k$,\n\\item each point of $Y \\cap Z$ is a node of $X$, and\n\\item $Y \\to \\Spec(k)$ is smooth at every point of $Y \\cap Z$.\n\\end{enumerate}\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E38","source_file":"curves.tex","source_line":4576,"source_end_line":4593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4576-L4593","statement_sha256":"ab30ea5b307bf5b4fd93a234ebc424e91f239344cece096d98f7782b3b09140a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9379,"rank":9379,"depth":58,"x":327.865,"y":1223.035,"cluster":"varieties-curves"},{"id":"stacks:0C59","tag":"0C59","title":"Families of nodal curves · Lemma 0C59","summary":"Let f : X → S be a morphism of schemes. The following are equivalent • f is flat, locally of finite presentation, every nonempty fibre X_s is equidimensional of dimension 1, and X_s has at-worst-nodal singularities, and • f is syntomic of relative dimension 1 and the closed subscheme Sing(f) ⊂ X defined by the first Fitting ideal of Ω_X/S is unramified over S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is flat, locally of finite presentation, every nonempty fibre\n$X_s$ is equidimensional of dimension $1$, and $X_s$ has\nat-worst-nodal singularities, and\n\\item $f$ is syntomic of relative dimension $1$ and the closed subscheme\n$\\text{Sing}(f) \\subset X$ defined by the first Fitting ideal of\n$\\Omega_{X/S}$ is unramified over $S$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C59","source_file":"curves.tex","source_line":4661,"source_end_line":4672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4661-L4672","statement_sha256":"685f160ca328a0a6fc884a9229f3ec72d61b8de3c47ff6d31350cc42e5c2f8f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9380,"rank":9380,"depth":59,"x":90.871,"y":1134.294,"cluster":"varieties-curves"},{"id":"stacks:0C5A","tag":"0C5A","title":"Families of nodal curves · Definition 0C5A","summary":"Let f : X → S be a morphism of schemes. We say f is at-worst-nodal of relative dimension 1 if f satisfies the equivalent conditions of Lemma [Tag 0C59].","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. We say $f$ is\n{\\it at-worst-nodal of relative dimension $1$} if $f$ satisfies\nthe equivalent conditions of Lemma \\ref{lemma-nodal-family}.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5A","source_file":"curves.tex","source_line":4686,"source_end_line":4691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4686-L4691","statement_sha256":"210a5140eb592b41961625d992f4174ed77c40340f9cb70e968065eeb78021dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9381,"rank":9381,"depth":60,"x":337.326,"y":1065.16,"cluster":"varieties-curves"},{"id":"stacks:0CD7","tag":"0CD7","title":"Families of nodal curves · Lemma 0CD7","summary":"A smooth morphism of relative dimension 1 is at-worst-nodal of relative dimension 1.","statement_latex":"A smooth morphism of relative dimension $1$ is\nat-worst-nodal of relative dimension $1$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CD7","source_file":"curves.tex","source_line":4704,"source_end_line":4708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4704-L4708","statement_sha256":"c32f986172dbdb37d5e871039b963512158bd47e8f097092fa5387ce6d737961","origin":"The Stacks Project","memory_eligible":false,"source_rank":9382,"rank":9382,"depth":0,"x":211.018,"y":1256.243,"cluster":"varieties-curves"},{"id":"stacks:0C5B","tag":"0C5B","title":"Families of nodal curves · Lemma 0C5B","summary":"Let f : X → S be at-worst-nodal of relative dimension 1. Then the same is true for any base change of f.","statement_latex":"Let $f : X \\to S$ be at-worst-nodal of relative dimension $1$.\nThen the same is true for any base change of $f$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5B","source_file":"curves.tex","source_line":4714,"source_end_line":4718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4714-L4718","statement_sha256":"cf9d83039ae4424fd392c9794c45aba5eca31ce10306a1f58089cd4b31e19b44","origin":"The Stacks Project","memory_eligible":false,"source_rank":9383,"rank":9383,"depth":38,"x":150.41,"y":1043.382,"cluster":"varieties-curves"},{"id":"stacks:0DSC","tag":"0DSC","title":"Families of nodal curves · Lemma 0DSC","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation. Then there is a maximal open subscheme U ⊂ X such that f|_U : U → S is at-worst-nodal of relative dimension 1. Moreover, formation of U commutes with arbitrary base change.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation. Then there is a maximal open\nsubscheme $U \\subset X$ such that $f|_U : U \\to S$ is\nat-worst-nodal of relative dimension $1$. Moreover, formation\nof $U$ commutes with arbitrary base change.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSC","source_file":"curves.tex","source_line":4737,"source_end_line":4744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4737-L4744","statement_sha256":"2f88417c89b6761ed24eec4152cce770ec76a918cf7f012d58e7c9be899fd246","origin":"The Stacks Project","memory_eligible":false,"source_rank":9384,"rank":9384,"depth":44,"x":366.57,"y":1166.119,"cluster":"varieties-curves"},{"id":"stacks:0C5C","tag":"0C5C","title":"Families of nodal curves · Lemma 0C5C","summary":"Let f : X → S be at-worst-nodal of relative dimension 1. If Y → X is an étale morphism, then the composition g : Y → S is at-worst-nodal of relative dimension 1.","statement_latex":"Let $f : X \\to S$ be at-worst-nodal of relative dimension $1$.\nIf $Y \\to X$ is an \\'etale morphism, then the composition $g : Y \\to S$\nis at-worst-nodal of relative dimension $1$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5C","source_file":"curves.tex","source_line":4777,"source_end_line":4782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4777-L4782","statement_sha256":"795f3e0588c9540484925d8185abef3df98fb8f330b836e0e512efd24d340680","origin":"The Stacks Project","memory_eligible":false,"source_rank":9385,"rank":9385,"depth":60,"x":108.127,"y":1198.313,"cluster":"varieties-curves"},{"id":"stacks:0CD8","tag":"0CD8","title":"Families of nodal curves · Lemma 0CD8","summary":"Let S' → S be an étale morphism of schemes. Let f : X → S' be at-worst-nodal of relative dimension 1. Then the composition g : X → S is at-worst-nodal of relative dimension 1.","statement_latex":"Let $S' \\to S$ be an \\'etale morphism of schemes.\nLet $f : X \\to S'$ be at-worst-nodal of relative dimension $1$.\nThen the composition $g : X \\to S$\nis at-worst-nodal of relative dimension $1$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CD8","source_file":"curves.tex","source_line":4800,"source_end_line":4806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4800-L4806","statement_sha256":"a4d7bac388faa0822b709dc662c882a06e36ab861ba51132cd4691064b80fe59","origin":"The Stacks Project","memory_eligible":false,"source_rank":9386,"rank":9386,"depth":59,"x":273.032,"y":1027.697,"cluster":"varieties-curves"},{"id":"stacks:0C5D","tag":"0C5D","title":"Families of nodal curves · Lemma 0C5D","summary":"Let f : X → S be a morphism of schemes. Let (U_i → X) be an étale covering. The following are equivalent • f is at-worst-nodal of relative dimension 1, • each U_i → S is at-worst-nodal of relative dimension 1. In other words, being at-worst-nodal of relative dimension 1 is étale local on the source.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $\\{U_i \\to X\\}$\nbe an \\'etale covering. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is at-worst-nodal of relative dimension $1$,\n\\item each $U_i \\to S$ is at-worst-nodal of relative dimension $1$.\n\\end{enumerate}\nIn other words, being at-worst-nodal of relative dimension $1$\nis \\'etale local on the source.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5D","source_file":"curves.tex","source_line":4822,"source_end_line":4832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4822-L4832","statement_sha256":"f7916dbeded67279a3587811e48ab78990c436d24fc26bdffa43d188bbe29c2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9387,"rank":9387,"depth":61,"x":288.659,"y":1247.372,"cluster":"varieties-curves"},{"id":"stacks:0C5E","tag":"0C5E","title":"Families of nodal curves · Lemma 0C5E","summary":"Let f : X → S be a morphism of schemes. Let (U_i → S) be an fpqc covering. The following are equivalent • f is at-worst-nodal of relative dimension 1, • each X ×_S U_i → U_i is at-worst-nodal of relative dimension 1. In other words, being at-worst-nodal of relative dimension 1 is fpqc local on the target.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $\\{U_i \\to S\\}$\nbe an fpqc covering. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is at-worst-nodal of relative dimension $1$,\n\\item each $X \\times_S U_i \\to U_i$ is at-worst-nodal of relative\ndimension $1$.\n\\end{enumerate}\nIn other words, being at-worst-nodal of relative dimension $1$\nis fpqc local on the target.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5E","source_file":"curves.tex","source_line":4849,"source_end_line":4860,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4849-L4860","statement_sha256":"8bb4623948eeece0d2572b2667b7bafe58ab3707c9b217e30af22ef9a871a6c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9388,"rank":9388,"depth":59,"x":100.227,"y":1094.049,"cluster":"varieties-curves"},{"id":"stacks:0C5F","tag":"0C5F","title":"Families of nodal curves · Lemma 0C5F","summary":"Let S = lim S_i be a limit of a directed system of schemes with affine transition morphisms. Let 0 ∈ I and let f_0 : X_0 → Y_0 be a morphism of schemes over S_0. Assume S_0, X_0, Y_0 are quasi-compact and quasi-separated. Let f_i : X_i → Y_i be the base change of f_0 to S_i and let f : X → Y be the base change of f_0 to S. If • f is at-worst-nodal of relative dimension 1, and • f_0 is locally of finite presentation, then there exists an i ≥ 0 such that f_i is…","statement_latex":"Let $S = \\lim S_i$ be a limit of a directed system of schemes\nwith affine transition morphisms.\nLet $0 \\in I$ and let $f_0 : X_0 \\to Y_0$ be a morphism of schemes over $S_0$.\nAssume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated.\nLet $f_i : X_i \\to Y_i$ be the base change of $f_0$ to $S_i$ and\nlet $f : X \\to Y$ be the base change of $f_0$ to $S$.\nIf\n\\begin{enumerate}\n\\item $f$ is at-worst-nodal of relative dimension $1$, and\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen there exists an $i \\geq 0$ such that $f_i$ is at-worst-nodal\nof relative dimension $1$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5F","source_file":"curves.tex","source_line":4878,"source_end_line":4893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4878-L4893","statement_sha256":"6cd3cc90716f4dffd1758ba8d65fe4c1d3e41cfbb164688992e6a4d4e44c2720","origin":"The Stacks Project","memory_eligible":false,"source_rank":9389,"rank":9389,"depth":37,"x":362.823,"y":1100.194,"cluster":"varieties-curves"},{"id":"stacks:0CBX","tag":"0CBX","title":"Families of nodal curves · Lemma 0CBX","summary":"Let f : T → S be a morphism of schemes. Let t ∈ T with image s ∈ S. Assume • f is flat at t, • O_S, s is Noetherian, • f is locally of finite type, • t is a split node of the fibre T_s. Then there exists an h ∈ m_s^wedge and an isomorphism O_T, t^wedge ≅ O_S, s^wedge[[x, y]]/(xy - h) of O_S, s^wedge-algebras.","statement_latex":"Let $f : T \\to S$ be a morphism of schemes. Let $t \\in T$\nwith image $s \\in S$. Assume\n\\begin{enumerate}\n\\item $f$ is flat at $t$,\n\\item $\\mathcal{O}_{S, s}$ is Noetherian,\n\\item $f$ is locally of finite type,\n\\item $t$ is a split node of the fibre $T_s$.\n\\end{enumerate}\nThen there exists an $h \\in \\mathfrak m_s^\\wedge$ and an isomorphism\n$$\n\\mathcal{O}_{T, t}^\\wedge \\cong\n\\mathcal{O}_{S, s}^\\wedge[[x, y]]/(xy - h)\n$$\nof $\\mathcal{O}_{S, s}^\\wedge$-algebras.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBX","source_file":"curves.tex","source_line":4919,"source_end_line":4935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L4919-L4935","statement_sha256":"a9ad7f9c328962ec26906870b73029a1a23252e5a115f85e7f402492375244e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9390,"rank":9390,"depth":59,"x":163.981,"y":1244.858,"cluster":"varieties-curves"},{"id":"stacks:0GKA","tag":"0GKA","title":"Families of nodal curves · Lemma 0GKA","summary":"Let f : X → S be a morphism of schemes. Assume • f is proper, • f is at-worst-nodal of relative dimension 1, and • the geometric fibres of f are connected. Then (a) f_*O_X = O_S and this holds after any base change, (b) R^1f_*O_X is a finite locally free O_S-module whose formation commutes with any base change, and (c) R^qf_*O_X = 0 for q ≥ 2.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item $f$ is at-worst-nodal of relative dimension $1$, and\n\\item the geometric fibres of $f$ are connected.\n\\end{enumerate}\nThen (a) $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds after\nany base change, (b) $R^1f_*\\mathcal{O}_X$ is a finite locally free\n$\\mathcal{O}_S$-module whose formation commutes with any base change,\nand (c) $R^qf_*\\mathcal{O}_X = 0$ for $q \\geq 2$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKA","source_file":"curves.tex","source_line":5017,"source_end_line":5029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5017-L5029","statement_sha256":"c61cab29bf8e105d047c85744df37605e9be13290322d2625ed3c5201a5af090","origin":"The Stacks Project","memory_eligible":false,"source_rank":9391,"rank":9391,"depth":41,"x":194.31,"y":1025.066,"cluster":"varieties-curves"},{"id":"stacks:0CBY","tag":"0CBY","title":"Étale local structure of nodal families · Lemma 0CBY","summary":"Let f : X → S be a morphism of schemes. Assume that f is at-worst-nodal of relative dimension 1. Let x ∈ X be a point which is a singular point of the fibre X_s. Then there exists a commutative diagram of schemes xymatrix X ar[d] & U ar[rr] ar[l] ar[rd] & & W ar[r] ar[ld] & Spec(Z[u, v, a]/(uv - a)) ar[d] S & & V ar[ll] ar[rr] & & Spec(Z[a]) with X ← U, S ← V, and U → W étale morphisms, and with the right hand square cartesian, such that there exists a point u ∈ U mapping…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume that\n$f$ is at-worst-nodal of relative dimension $1$. Let\n$x \\in X$ be a point which is a singular point of the\nfibre $X_s$. Then there exists a commutative diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d] &\nU \\ar[rr] \\ar[l] \\ar[rd] & &\nW \\ar[r] \\ar[ld] &\n\\Spec(\\mathbf{Z}[u, v, a]/(uv - a)) \\ar[d] \\\\\nS & &\nV \\ar[ll] \\ar[rr] & & \\Spec(\\mathbf{Z}[a])\n}\n$$\nwith $X \\leftarrow U$, $S \\leftarrow V$, and $U \\to W$ \\'etale morphisms,\nand with the right hand square cartesian, such that there exists\na point $u \\in U$ mapping to $x$ in $X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Étale local structure of nodal families","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBY","source_file":"curves.tex","source_line":5076,"source_end_line":5095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5076-L5095","statement_sha256":"fe75615aa60a486b79c247b6573fc673b4750babcc87d92f90019a58042269db","origin":"The Stacks Project","memory_eligible":false,"source_rank":9392,"rank":9392,"depth":60,"x":348.901,"y":1204.584,"cluster":"varieties-curves"},{"id":"stacks:0E3A","tag":"0E3A","title":"More vanishing results · Lemma 0E3A","summary":"In Situation [Tag 0B5D] assume X is integral and has genus g. Let L be an invertible O_X-module. Let Z ⊂ X be a 0-dimensional closed subscheme with ideal sheaf I ⊂ O_X. If H^1(X, IL) is nonzero, then deg(L) ≤ 2g - 2 + deg(Z) with strict inequality unless IL ≅ ω_X.","statement_latex":"In Situation \\ref{situation-Cohen-Macaulay-curve} assume $X$ is integral and\nhas genus $g$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $Z \\subset X$ be a $0$-dimensional closed subscheme with ideal\nsheaf $\\mathcal{I} \\subset \\mathcal{O}_X$. If $H^1(X, \\mathcal{I}\\mathcal{L})$\nis nonzero, then\n$$\n\\deg(\\mathcal{L}) \\leq 2g - 2 + \\deg(Z)\n$$\nwith strict inequality unless $\\mathcal{I}\\mathcal{L} \\cong \\omega_X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"More vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3A","source_file":"curves.tex","source_line":5290,"source_end_line":5301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5290-L5301","statement_sha256":"8d6b5305fd9eba3c70e9395a9af4a935ff6ed1f74125ed7017214510b2bf053e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9393,"rank":9393,"depth":49,"x":90.203,"y":1159.87,"cluster":"varieties-curves"},{"id":"stacks:0E3B","tag":"0E3B","title":"More vanishing results · Lemma 0E3B","summary":"[Jongmin] In Situation [Tag 0B5D] assume X is integral and has genus g. Let L be an invertible O_X-module. Let Z ⊂ X be a 0-dimensional closed subscheme with ideal sheaf I ⊂ O_X. If deg(L) > 2g - 2 + deg(Z), then H^1(X, IL) = 0 and one of the following possibilities occurs • H^0(X, IL) not = 0, or • g = 0 and deg(L) = deg(Z) - 1. In case (2) if Z = ∅, then X ≅ P^1_k and L corresponds to O_P^1(-1).","statement_latex":"\\begin{reference}\n\\cite[Lemma 2]{Jongmin}\n\\end{reference}\nIn Situation \\ref{situation-Cohen-Macaulay-curve}\nassume $X$ is integral and has genus $g$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $Z \\subset X$ be a $0$-dimensional closed subscheme with ideal\nsheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.\nIf $\\deg(\\mathcal{L}) > 2g - 2 + \\deg(Z)$, then\n$H^1(X, \\mathcal{I}\\mathcal{L}) = 0$ and one of the following possibilities\noccurs\n\\begin{enumerate}\n\\item $H^0(X, \\mathcal{I}\\mathcal{L}) \\not = 0$, or\n\\item $g = 0$ and $\\deg(\\mathcal{L}) = \\deg(Z) - 1$.\n\\end{enumerate}\nIn case (2) if $Z = \\emptyset$, then $X \\cong \\mathbf{P}^1_k$ and $\\mathcal{L}$\ncorresponds to $\\mathcal{O}_{\\mathbf{P}^1}(-1)$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"More vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3B","source_file":"curves.tex","source_line":5339,"source_end_line":5358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5339-L5358","statement_sha256":"2463774b404113973babe2156bb49b2e31f39d32f30bf5f489afec2d8c639cd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9394,"rank":9394,"depth":50,"x":317.22,"y":1045.9,"cluster":"varieties-curves"},{"id":"stacks:0E3C","tag":"0E3C","title":"More vanishing results · Lemma 0E3C","summary":"[Jongmin] In Situation [Tag 0B5D] assume X is integral and has genus g. Let L be an invertible O_X-module. If deg(L) ≥ 2g, then L is globally generated.","statement_latex":"\\begin{reference}\n\\cite[Lemma 3]{Jongmin}\n\\end{reference}\nIn Situation \\ref{situation-Cohen-Macaulay-curve}\nassume $X$ is integral and has genus $g$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nIf $\\deg(\\mathcal{L}) \\geq 2g$, then $\\mathcal{L}$\nis globally generated.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"More vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3C","source_file":"curves.tex","source_line":5375,"source_end_line":5385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5375-L5385","statement_sha256":"477693089c6b1da5e2468f6121bd58e5cc1c10ea2bc460740199980e559f7367","origin":"The Stacks Project","memory_eligible":false,"source_rank":9395,"rank":9395,"depth":51,"x":241.373,"y":1259.034,"cluster":"varieties-curves"},{"id":"stacks:0E3D","tag":"0E3D","title":"More vanishing results · Lemma 0E3D","summary":"In Situation [Tag 0B5D] assume X is integral and has genus g. Let L be an invertible O_X-module. Let Z ⊂ X be a nonempty 0-dimensional closed subscheme. If deg(L) ≥ 2g - 1 + deg(Z), then L is globally generated and H^0(X, L) → H^0(X, L|_Z) is surjective.","statement_latex":"In Situation \\ref{situation-Cohen-Macaulay-curve}\nassume $X$ is integral and has genus $g$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $Z \\subset X$ be a nonempty $0$-dimensional closed subscheme.\nIf $\\deg(\\mathcal{L}) \\geq 2g - 1 + \\deg(Z)$, then $\\mathcal{L}$\nis globally generated and $H^0(X, \\mathcal{L}) \\to H^0(X, \\mathcal{L}|_Z)$\nis surjective.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"More vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3D","source_file":"curves.tex","source_line":5416,"source_end_line":5425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5416-L5425","statement_sha256":"bc6e764424cb4c215e6286cc909b34e714e086ae7cc9ca6dea1c9833620a58af","origin":"The Stacks Project","memory_eligible":false,"source_rank":9396,"rank":9396,"depth":52,"x":125.753,"y":1058.573,"cluster":"varieties-curves"},{"id":"stacks:0H2V","tag":"0H2V","title":"More vanishing results · Lemma 0H2V","summary":"In Situation [Tag 0B5D], assume X is geometrically integral over k and has genus g. Let L be an invertible O_X-module. If deg(L) ≥ 2g + 1, then L is very ample.","statement_latex":"In Situation \\ref{situation-Cohen-Macaulay-curve},\nassume $X$ is geometrically integral over $k$ and has genus $g$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nIf $\\deg(\\mathcal{L}) \\geq 2g + 1$, then $\\mathcal{L}$\nis very ample.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"More vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2V","source_file":"curves.tex","source_line":5434,"source_end_line":5441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5434-L5441","statement_sha256":"b105fd227920240b1d99bf6012dda41932ee7b9cd911478bcb77a5c1e62a6c91","origin":"The Stacks Project","memory_eligible":false,"source_rank":9397,"rank":9397,"depth":53,"x":372.537,"y":1140.892,"cluster":"varieties-curves"},{"id":"stacks:0E3E","tag":"0E3E","title":"More vanishing results · Lemma 0E3E","summary":"Weak version of [Jongmin] Let k be a field. Let X be a proper scheme over k which is reduced, connected, and of dimension 1. Let L be an invertible O_X-module. Let Z ⊂ X be a 0-dimensional closed subscheme with ideal sheaf I ⊂ O_X. If H^1(X, IL) not = 0, then there exists a reduced connected closed subscheme Y ⊂ X of dimension 1 such that deg(L|_Y) ≤ -2chi(Y, O_Y) + deg(Z ∩ Y) where Z ∩ Y is the scheme theoretic intersection.","statement_latex":"\\begin{reference}\nWeak version of \\cite[Lemma 4]{Jongmin}\n\\end{reference}\nLet $k$ be a field. Let $X$ be a proper scheme over $k$\nwhich is reduced, connected, and of dimension $1$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $Z \\subset X$ be a $0$-dimensional closed subscheme with ideal\nsheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.\nIf $H^1(X, \\mathcal{I}\\mathcal{L}) \\not = 0$, then there exists\na reduced connected closed subscheme $Y \\subset X$\nof dimension $1$ such that\n$$\n\\deg(\\mathcal{L}|_Y) \\leq -2\\chi(Y, \\mathcal{O}_Y) + \\deg(Z \\cap Y)\n$$\nwhere $Z \\cap Y$ is the scheme theoretic intersection.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"More vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3E","source_file":"curves.tex","source_line":5462,"source_end_line":5479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5462-L5479","statement_sha256":"de8df76562bfbad05a048e8ba8bd9e39d96b4867a22b3b3f71c0eb0d88f53c6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9398,"rank":9398,"depth":49,"x":124.04,"y":1220.325,"cluster":"varieties-curves"},{"id":"stacks:0E3F","tag":"0E3F","title":"More vanishing results · Lemma 0E3F","summary":"Let k be a field. Let X be a proper scheme over k which is reduced, connected, and of dimension 1. Let L be an invertible O_X-module. Assume that for every reduced connected closed subscheme Y ⊂ X of dimension 1 we have deg(L|_Y) ≥ 2dim_k H^1(Y, O_Y) Then L is globally generated.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$\nwhich is reduced, connected, and of dimension $1$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume that for every reduced connected closed subscheme\n$Y \\subset X$ of dimension $1$ we have\n$$\n\\deg(\\mathcal{L}|_Y) \\geq 2\\dim_k H^1(Y, \\mathcal{O}_Y)\n$$\nThen $\\mathcal{L}$ is globally generated.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"More vanishing results","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3F","source_file":"curves.tex","source_line":5537,"source_end_line":5548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5537-L5548","statement_sha256":"db7d9e4368a0caf9adb97581722634bd8200f19732c5d7f89e44e68411907eab","origin":"The Stacks Project","memory_eligible":false,"source_rank":9399,"rank":9399,"depth":52,"x":243.556,"y":1020.49,"cluster":"varieties-curves"},{"id":"stacks:0E63","tag":"0E63","title":"Contracting rational tails · Lemma 0E63","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Assume the singularities of X are at-worst-nodal. Let C ⊂ X be a rational tail (Example [Tag 0E3H]). Then deg(ω_X|_C) < 0.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are\nat-worst-nodal. Let $C \\subset X$ be a rational tail\n(Example \\ref{example-rational-tail}). Then $\\deg(\\omega_X|_C) < 0$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational tails","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E63","source_file":"curves.tex","source_line":5769,"source_end_line":5775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5769-L5775","statement_sha256":"3a7cbeb93f50b539448b5e116a053ca44a908fef87e1d6a8b8b9370963d7d7b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9400,"rank":9400,"depth":60,"x":316.223,"y":1235.941,"cluster":"varieties-curves"},{"id":"stacks:0E3I","tag":"0E3I","title":"Contracting rational tails · Lemma 0E3I","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Assume the singularities of X are at-worst-nodal. Let C ⊂ X be a rational tail (Example [Tag 0E3H]). For any field extension K/k the base change C_K ⊂ X_K is a finite disjoint union of rational tails.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are\nat-worst-nodal. Let $C \\subset X$ be a rational tail\n(Example \\ref{example-rational-tail}).\nFor any field extension $K/k$ the base change $C_K \\subset X_K$\nis a finite disjoint union of rational tails.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational tails","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3I","source_file":"curves.tex","source_line":5792,"source_end_line":5800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5792-L5800","statement_sha256":"8f9a9dc3aa11a2fc486316f94d508f6c06a788ae048650235e01794079ed3cf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9401,"rank":9401,"depth":50,"x":89.086,"y":1118.151,"cluster":"varieties-curves"},{"id":"stacks:0E3J","tag":"0E3J","title":"Contracting rational tails · Lemma 0E3J","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Assume the singularities of X are at-worst-nodal. If X does not have a rational tail (Example [Tag 0E3H]), then for every reduced connected closed subscheme Y ⊂ X, Y not = X of dimension 1 we have deg(ω_X|_Y) ≥ dim_k H^1(Y, O_Y).","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are\nat-worst-nodal. If $X$ does not have a rational tail\n(Example \\ref{example-rational-tail}),\nthen for every reduced connected closed subscheme\n$Y \\subset X$, $Y \\not = X$ of dimension $1$ we have\n$\\deg(\\omega_X|_Y) \\geq \\dim_k H^1(Y, \\mathcal{O}_Y)$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational tails","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3J","source_file":"curves.tex","source_line":5814,"source_end_line":5823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5814-L5823","statement_sha256":"b52fec5d4d5287f77ffcf117b502344d2b9a0e3dbc2e54aecd613a4a78140be0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9402,"rank":9402,"depth":60,"x":351.634,"y":1076.071,"cluster":"varieties-curves"},{"id":"stacks:0E3K","tag":"0E3K","title":"Contracting rational tails · Lemma 0E3K","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Assume the singularities of X are at-worst-nodal. Assume X does not have a rational tail (Example [Tag 0E3H]). If • the genus of X is 0, then X is isomorphic to an irreducible plane conic and ω_X^⊗ -1 is very ample, • the genus of X is 1, then ω_X ≅ O_X, • the genus of X is ≥ 2, then ω_X^⊗ m is globally generated for m ≥ 2.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are\nat-worst-nodal. Assume $X$ does not have a rational tail\n(Example \\ref{example-rational-tail}). If\n\\begin{enumerate}\n\\item the genus of $X$ is $0$, then $X$ is isomorphic to an\nirreducible plane conic and $\\omega_X^{\\otimes -1}$ is very ample,\n\\item the genus of $X$ is $1$, then $\\omega_X \\cong \\mathcal{O}_X$,\n\\item the genus of $X$ is $\\geq 2$, then\n$\\omega_X^{\\otimes m}$ is globally generated for $m \\geq 2$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational tails","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3K","source_file":"curves.tex","source_line":5862,"source_end_line":5875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5862-L5875","statement_sha256":"514481a529a8ca7636d4d40dc67389c7c529201020323e32d4346bdf7ccb7b25","origin":"The Stacks Project","memory_eligible":false,"source_rank":9403,"rank":9403,"depth":61,"x":191.67,"y":1256.308,"cluster":"varieties-curves"},{"id":"stacks:0E3L","tag":"0E3L","title":"Contracting rational tails · Lemma 0E3L","summary":"Let k be a field. Let X be a proper scheme over k of dimension 1 with H^0(X, O_X) = k. Assume the singularities of X are at-worst-nodal. Consider a sequence X = X_0 → X_1 → … → X_n = X' of contractions of rational tails (Example [Tag 0E3H]) until none are left. Then • if the genus of X is 0, then X' is an irreducible plane conic, • if the genus of X is 1, then ω_X' ≅ O_X, • if the genus of X is > 1, then ω_X'^⊗ m is globally generated for m ≥ 2. If the genus of X is ≥ 1,…","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$\nwith $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are\nat-worst-nodal. Consider a sequence\n$$\nX = X_0 \\to X_1 \\to \\ldots \\to X_n = X'\n$$\nof contractions of rational tails (Example \\ref{example-rational-tail})\nuntil none are left. Then\n\\begin{enumerate}\n\\item if the genus of $X$ is $0$, then $X'$ is an irreducible\nplane conic,\n\\item if the genus of $X$ is $1$, then $\\omega_{X'} \\cong \\mathcal{O}_X$,\n\\item if the genus of $X$ is $> 1$, then\n$\\omega_{X'}^{\\otimes m}$ is globally generated for $m \\geq 2$.\n\\end{enumerate}\nIf the genus of $X$ is $\\geq 1$, then the morphism $X \\to X'$\nis independent of choices and formation of this morphism\ncommutes with base field extensions.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational tails","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3L","source_file":"curves.tex","source_line":5917,"source_end_line":5937,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L5917-L5937","statement_sha256":"2530a63e3939128bb75b9e3f794e5ebece854018aba86a2afabfcfdb00c88884","origin":"The Stacks Project","memory_eligible":false,"source_rank":9404,"rank":9404,"depth":62,"x":164.649,"y":1032.349,"cluster":"varieties-curves"},{"id":"stacks:0E64","tag":"0E64","title":"Contracting rational bridges · Lemma 0E64","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Assume the singularities of X are at-worst-nodal. Let C ⊂ X be a rational bridge (Example [Tag 0E3M]). Then deg(ω_X|_C) = 0.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are\nat-worst-nodal. Let $C \\subset X$ be a rational bridge\n(Example \\ref{example-rational-bridge}). Then $\\deg(\\omega_X|_C) = 0$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational bridges","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E64","source_file":"curves.tex","source_line":6099,"source_end_line":6105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6099-L6105","statement_sha256":"cba9d994305ebbba45f59e709a03ee16a0574c5b979e59dd48195d2812a71bcc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9405,"rank":9405,"depth":60,"x":364.931,"y":1182.352,"cluster":"varieties-curves"},{"id":"stacks:0E65","tag":"0E65","title":"Contracting rational bridges · Lemma 0E65","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Assume the singularities of X are at-worst-nodal. Let C ⊂ X be a rational bridge (Example [Tag 0E3M]). For any field extension K/k the base change C_K ⊂ X_K is a finite disjoint union of rational bridges.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Assume the singularities of $X$ are\nat-worst-nodal. Let $C \\subset X$ be a rational bridge\n(Example \\ref{example-rational-bridge}).\nFor any field extension $K/k$ the base change $C_K \\subset X_K$\nis a finite disjoint union of rational bridges.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational bridges","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E65","source_file":"curves.tex","source_line":6123,"source_end_line":6131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6123-L6131","statement_sha256":"5d8bc7c9f8f73b488d8dc0bb6bd6c90e269182e7e1f0148d8a87354e1e822665","origin":"The Stacks Project","memory_eligible":false,"source_rank":9406,"rank":9406,"depth":0,"x":96.275,"y":1185.391,"cluster":"varieties-curves"},{"id":"stacks:0E3N","tag":"0E3N","title":"Contracting rational bridges · Lemma 0E3N","summary":"Let c : X → Y be the contraction of a rational bridge (Example [Tag 0E3M]). Then c^*ω_Y ≅ ω_X.","statement_latex":"Let $c : X \\to Y$ be the contraction of a rational bridge\n(Example \\ref{example-rational-bridge}).\nThen $c^*\\omega_Y \\cong \\omega_X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational bridges","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3N","source_file":"curves.tex","source_line":6149,"source_end_line":6154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6149-L6154","statement_sha256":"c53bb57e39a98f45e6550ff79abb588c6ad17b96edfe1f28cd6999d137cc3420","origin":"The Stacks Project","memory_eligible":false,"source_rank":9407,"rank":9407,"depth":49,"x":292.184,"y":1030.512,"cluster":"varieties-curves"},{"id":"stacks:0E3P","tag":"0E3P","title":"Contracting rational bridges · Lemma 0E3P","summary":"Let k be a field. Let X be a proper scheme over k having dimension 1 and H^0(X, O_X) = k. Assume • the singularities of X are at-worst-nodal, • X does not have a rational tail (Example [Tag 0E3H]), • X does not have a rational bridge (Example [Tag 0E3M]), • the genus g of X is ≥ 2. Then ω_X is ample.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$\nand $H^0(X, \\mathcal{O}_X) = k$. Assume\n\\begin{enumerate}\n\\item the singularities of $X$ are at-worst-nodal,\n\\item $X$ does not have a rational tail\n(Example \\ref{example-rational-tail}),\n\\item $X$ does not have a rational bridge\n(Example \\ref{example-rational-bridge}),\n\\item the genus $g$ of $X$ is $\\geq 2$.\n\\end{enumerate}\nThen $\\omega_X$ is ample.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational bridges","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3P","source_file":"curves.tex","source_line":6215,"source_end_line":6228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6215-L6228","statement_sha256":"31332a7d64d07b4e8f783baacbe5e668a2a0e286b3cb6ab312644d96f8aa62be","origin":"The Stacks Project","memory_eligible":false,"source_rank":9408,"rank":9408,"depth":60,"x":272.248,"y":1256.167,"cluster":"varieties-curves"},{"id":"stacks:0E3Q","tag":"0E3Q","title":"Contracting rational bridges · Lemma 0E3Q","summary":"Let k be a field. Let X be a proper scheme over k of dimension 1 with H^0(X, O_X) = k having genus g ≥ 2. Assume the singularities of X are at-worst-nodal and that X has no rational tails. Consider a sequence X = X_0 → X_1 → … → X_n = X' of contractions of rational bridges (Example [Tag 0E3M]) until none are left. Then ω_X' ample. The morphism X → X' is independent of choices and formation of this morphism commutes with base field extensions.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$\nwith $H^0(X, \\mathcal{O}_X) = k$ having genus $g \\geq 2$.\nAssume the singularities of $X$ are at-worst-nodal and that\n$X$ has no rational tails. Consider a sequence\n$$\nX = X_0 \\to X_1 \\to \\ldots \\to X_n = X'\n$$\nof contractions of rational bridges\n(Example \\ref{example-rational-bridge}) until none are left.\nThen $\\omega_{X'}$ ample.\nThe morphism $X \\to X'$ is independent of choices and\nformation of this morphism commutes with base field extensions.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting rational bridges","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3Q","source_file":"curves.tex","source_line":6273,"source_end_line":6287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6273-L6287","statement_sha256":"152412636fde63478c7f632ab38ae2afc36117119db2aee6011090778c75d2fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9409,"rank":9409,"depth":61,"x":105.271,"y":1078.235,"cluster":"varieties-curves"},{"id":"stacks:0E7P","tag":"0E7P","title":"Contracting to a stable curve · Lemma 0E7P","summary":"Let k be a field. Let c : X → Y be a morphism of proper schemes over k. Assume • O_Y = c_*O_X and R^1c_*O_X = 0, • X and Y are reduced, Gorenstein, and have dimension 1, • ∃ m ∈ Z with H^1(X, ω_X^⊗ m) = 0 and ω_X^⊗ m generated by global sections. Then c^*ω_Y ≅ ω_X.","statement_latex":"Let $k$ be a field. Let $c : X \\to Y$ be a morphism of proper schemes over $k$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$,\n\\item $X$ and $Y$ are reduced, Gorenstein, and have dimension $1$,\n\\item $\\exists\\ m \\in \\mathbf{Z}$ with\n$H^1(X, \\omega_X^{\\otimes m}) = 0$ and $\\omega_X^{\\otimes m}$\ngenerated by global sections.\n\\end{enumerate}\nThen $c^*\\omega_Y \\cong \\omega_X$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting to a stable curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7P","source_file":"curves.tex","source_line":6354,"source_end_line":6366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6354-L6366","statement_sha256":"7209350996f7ff9a0c074613b47a776e8bca58cd12cd23d064b66124ff1ff83c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9410,"rank":9410,"depth":48,"x":371.821,"y":1114.739,"cluster":"varieties-curves"},{"id":"stacks:0E7Q","tag":"0E7Q","title":"Contracting to a stable curve · Lemma 0E7Q","summary":"Let k be a field. Let X be a proper scheme over k of dimension 1 with H^0(X, O_X) = k having genus g ≥ 2. Assume the singularities of X are at-worst-nodal. There is a unique morphism (up to unique isomorphism) c : X → Y of schemes over k having the following properties: • Y is proper over k, dim(Y) = 1, the singularities of Y are at-worst-nodal, • O_Y = c_*O_X and R^1c_*O_X = 0, and • ω_Y is ample on Y.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension\n$1$ with $H^0(X, \\mathcal{O}_X) = k$ having genus $g \\geq 2$.\nAssume the singularities of $X$ are at-worst-nodal.\nThere is a unique morphism (up to unique isomorphism)\n$$\nc : X \\longrightarrow Y\n$$\nof schemes over $k$ having the following properties:\n\\begin{enumerate}\n\\item $Y$ is proper over $k$, $\\dim(Y) = 1$, the singularities of $Y$\nare at-worst-nodal,\n\\item $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$, and\n\\item $\\omega_Y$ is ample on $Y$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting to a stable curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7Q","source_file":"curves.tex","source_line":6433,"source_end_line":6449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6433-L6449","statement_sha256":"c5c795cc2ac37261731fb488563e8a09ade5e10ace69d06ebedb48c75c859dc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9411,"rank":9411,"depth":63,"x":145.631,"y":1239.225,"cluster":"varieties-curves"},{"id":"stacks:0E8X","tag":"0E8X","title":"Contracting to a stable curve · Lemma 0E8X","summary":"Let k be a field. Let X be a proper scheme over k of dimension 1 with H^0(X, O_X) = k having genus g ≥ 2. Assume the singularities of X are at-worst-nodal and ω_X is ample. Then ω_X^⊗ 3 is very ample and H^1(X, ω_X^⊗ 3) = 0.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with\n$H^0(X, \\mathcal{O}_X) = k$ having genus $g \\geq 2$. Assume the singularities\nof $X$ are at-worst-nodal and $\\omega_X$ is ample. Then\n$\\omega_X^{\\otimes 3}$ is very ample and $H^1(X, \\omega_X^{\\otimes 3}) = 0$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Contracting to a stable curve","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8X","source_file":"curves.tex","source_line":6573,"source_end_line":6579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6573-L6579","statement_sha256":"9d8766a0d1e843354d139aebbfa50f84e44bdba000cd3093ffed5325f2586b47","origin":"The Stacks Project","memory_eligible":false,"source_rank":9412,"rank":9412,"depth":63,"x":212.396,"y":1018.809,"cluster":"varieties-curves"},{"id":"stacks:0E67","tag":"0E67","title":"Vector fields · Lemma 0E67","summary":"Let k be an algebraically closed field. Let X be a smooth, proper, connected curve over k. Let g be the genus of X. • If g ≥ 2, then Der_k(O_X, O_X) is zero, • if g = 1 and D ∈ Der_k(O_X, O_X) is nonzero, then D does not fix any closed point of X, and • if g = 0 and D ∈ Der_k(O_X, O_X) is nonzero, then D fixes at most 2 closed points of X.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $X$ be a smooth, proper, connected curve over $k$.\nLet $g$ be the genus of $X$.\n\\begin{enumerate}\n\\item If $g \\geq 2$, then $\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$\nis zero,\n\\item if $g = 1$ and $D \\in \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$\nis nonzero, then $D$ does not fix any closed point of $X$, and\n\\item if $g = 0$ and $D \\in \\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)$\nis nonzero, then $D$ fixes at most $2$ closed points of $X$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Vector fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E67","source_file":"curves.tex","source_line":6660,"source_end_line":6673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6660-L6673","statement_sha256":"a38dad0ae974ad901843e1e4f4e195ba6ac94a004efd31b74c9172741e332308","origin":"The Stacks Project","memory_eligible":false,"source_rank":9413,"rank":9413,"depth":48,"x":340.578,"y":1219.475,"cluster":"varieties-curves"},{"id":"stacks:0E68","tag":"0E68","title":"Vector fields · Lemma 0E68","summary":"Let k be an algebraically closed field. Let X be an at-worst-nodal, proper, connected 1-dimensional scheme over k. Let ν : X^ν → X be the normalization. Let S ⊂ X^ν be the set of points where ν is not an isomorphism. Then Der_k(O_X, O_X) = (D' ∈ Der_k(O_X^ν, O_X^ν) mid D' fixes every x^ν ∈ S)","statement_latex":"Let $k$ be an algebraically closed field.\nLet $X$ be an at-worst-nodal, proper, connected\n$1$-dimensional scheme over $k$. Let $\\nu : X^\\nu \\to X$ be the normalization.\nLet $S \\subset X^\\nu$ be the set of points where $\\nu$ is not an\nisomorphism. Then\n$$\n\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) =\n\\{D' \\in \\text{Der}_k(\\mathcal{O}_{X^\\nu}, \\mathcal{O}_{X^\\nu}) \\mid\nD' \\text{ fixes every }x^\\nu \\in S\\}\n$$","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Vector fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E68","source_file":"curves.tex","source_line":6698,"source_end_line":6710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6698-L6710","statement_sha256":"f54d1a713625e3e3f1a0eec8f1283792bf08326e02bc1f7d7f24994ec00066bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9414,"rank":9414,"depth":59,"x":84.368,"y":1144.146,"cluster":"varieties-curves"},{"id":"stacks:0E69","tag":"0E69","title":"Vector fields · Lemma 0E69","summary":"Let k be an algebraically closed field. Let X be an at-worst-nodal, proper, connected 1-dimensional scheme over k. Assume the genus of X is at least 2 and that X has no rational tails or bridges. Then Der_k(O_X, O_X) = 0.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $X$ be an at-worst-nodal, proper, connected\n$1$-dimensional scheme over $k$. Assume the genus of $X$ is at least\n$2$ and that $X$ has no rational tails\nor bridges. Then\n$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$.","area":"Varieties & Curves","chapter":"Algebraic Curves","chapter_id":"curves","section":"Vector fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E69","source_file":"curves.tex","source_line":6741,"source_end_line":6749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/curves.tex#L6741-L6749","statement_sha256":"51b879a9a965743d09914a1dcb260a56032efadf03da23c1991caa8f28d97710","origin":"The Stacks Project","memory_eligible":false,"source_rank":9415,"rank":9415,"depth":60,"x":334.182,"y":1054.201,"cluster":"varieties-curves"},{"id":"stacks:0ADZ","tag":"0ADZ","title":"A trace map in positive characteristic · Lemma 0ADZ","summary":"Let φ : R[x]/(x^p - a) → R[y]/(y^p - b) be an R-algebra homomorphism. Then Tr_x = Tr_y ∘ φ.","statement_latex":"Let $\\varphi : R[x]/(x^p - a) \\to R[y]/(y^p - b)$ be an $R$-algebra\nhomomorphism. Then $\\text{Tr}_x = \\text{Tr}_y \\circ \\varphi$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"A trace map in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADZ","source_file":"resolve.tex","source_line":182,"source_end_line":186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L182-L186","statement_sha256":"3a5248da8d5408dd6308a12a3df12ecb34879ff22a994095ecb82e45baafce96","origin":"The Stacks Project","memory_eligible":false,"source_rank":9416,"rank":9416,"depth":0,"x":222.165,"y":1262.534,"cluster":"varieties-curves"},{"id":"stacks:0AX5","tag":"0AX5","title":"A trace map in positive characteristic · Lemma 0AX5","summary":"Let F_p ⊂ Lambda ⊂ R ⊂ S be ring extensions and assume that S is isomorphic to R[x]/(x^p - a) for some a ∈ R. Then there are canonical R-linear maps Tr : Ω^t + 1_S/Lambda → Ω_R/Lambda^t + 1 for t ≥ 0 such that eta_1 wedge … wedge eta_t wedge x^idx ↦ ( 0 & if & 0 ≤ i ≤ p - 2, eta_1 wedge … wedge eta_t wedge da & if & i = p - 1 . for eta_i ∈ Ω_R/Lambda and such that Tr annihilates the image of S ⊗_R Ω_R/Lambda^t + 1 → Ω_S/Lambda^t + 1.","statement_latex":"Let $\\mathbf{F}_p \\subset \\Lambda \\subset R \\subset S$ be ring extensions\nand assume that $S$ is isomorphic to $R[x]/(x^p - a)$ for some $a \\in R$.\nThen there are canonical $R$-linear maps\n$$\n\\text{Tr} :\n\\Omega^{t + 1}_{S/\\Lambda}\n\\longrightarrow\n\\Omega_{R/\\Lambda}^{t + 1}\n$$\nfor $t \\geq 0$ such that\n$$\n\\eta_1 \\wedge \\ldots \\wedge \\eta_t \\wedge x^i\\text{d}x\n\\longmapsto\n\\left\\{\n\\begin{matrix}\n0 & \\text{if} & 0 \\leq i \\leq p - 2, \\\\\n\\eta_1 \\wedge \\ldots \\wedge \\eta_t \\wedge \\text{d}a & \\text{if} & i = p - 1\n\\end{matrix}\n\\right.\n$$\nfor $\\eta_i \\in \\Omega_{R/\\Lambda}$ and such that $\\text{Tr}$ annihilates the\nimage of\n$S \\otimes_R \\Omega_{R/\\Lambda}^{t + 1} \\to \\Omega_{S/\\Lambda}^{t + 1}$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"A trace map in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AX5","source_file":"resolve.tex","source_line":283,"source_end_line":308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L283-L308","statement_sha256":"14fb0fa64fd1a2a6aa0e781d79d98dd119ab26cadd0da8094b9e6a33b53dd36b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9417,"rank":9417,"depth":3,"x":137.124,"y":1045.082,"cluster":"varieties-curves"},{"id":"stacks:0AX6","tag":"0AX6","title":"A trace map in positive characteristic · Lemma 0AX6","summary":"Let S be a scheme over F_p. Let f : Y → X be a finite morphism of Noetherian normal integral schemes over S. Assume • the extension of function fields is purely inseparable of degree p, and • Ω_X/S is a coherent O_X-module (for example if X is of finite type over S). For i ≥ 1 there is a canonical map Tr : f_*Ω^i_Y/S → (Ω_X/S^i)^** whose stalk in the generic point of X recovers the trace map of Lemma [Tag 0AX5].","statement_latex":"Let $S$ be a scheme over $\\mathbf{F}_p$. Let $f : Y \\to X$ be a finite morphism\nof Noetherian normal integral schemes over $S$. Assume\n\\begin{enumerate}\n\\item the extension of function fields is purely inseparable of degree $p$, and\n\\item $\\Omega_{X/S}$ is a coherent $\\mathcal{O}_X$-module (for example\nif $X$ is of finite type over $S$).\n\\end{enumerate}\nFor $i \\geq 1$ there is a canonical map\n$$\n\\text{Tr} : f_*\\Omega^i_{Y/S} \\longrightarrow (\\Omega_{X/S}^i)^{**}\n$$\nwhose stalk in the generic point of $X$ recovers the trace map of\nLemma \\ref{lemma-trace-higher}.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"A trace map in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AX6","source_file":"resolve.tex","source_line":373,"source_end_line":388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L373-L388","statement_sha256":"2a44cb81d24f76f5ca5dd530084abc470fd3e6142ca1ff0ced1920083beb07f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9418,"rank":9418,"depth":21,"x":374.995,"y":1157.313,"cluster":"varieties-curves"},{"id":"stacks:0AGQ","tag":"0AGQ","title":"Quadratic transformations · Lemma 0AGQ","summary":"Let (A, m, kappa) be a regular local ring of dimension 2. Let f : X → S = Spec(A) be the blowing up of A in m wotj exceptional divisor E. There is a closed immersion r : X → P^1_S over S such that • r|_E : E → P^1_kappa is an isomorphism, • O_X(E) = O_X(-1) = r^*O_P^1(-1), and • C_E/X = (r|_E)^*O_P^1(1) and N_E/X = (r|_E)^*O_P^1(-1).","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.\nLet $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$\nwotj exceptional divisor $E$. There is a closed immersion\n$$\nr : X \\longrightarrow \\mathbf{P}^1_S\n$$\nover $S$ such that\n\\begin{enumerate}\n\\item $r|_E : E \\to \\mathbf{P}^1_\\kappa$ is an isomorphism,\n\\item $\\mathcal{O}_X(E) = \\mathcal{O}_X(-1) =\nr^*\\mathcal{O}_{\\mathbf{P}^1}(-1)$, and\n\\item $\\mathcal{C}_{E/X} = (r|_E)^*\\mathcal{O}_{\\mathbf{P}^1}(1)$ and\n$\\mathcal{N}_{E/X} = (r|_E)^*\\mathcal{O}_{\\mathbf{P}^1}(-1)$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGQ","source_file":"resolve.tex","source_line":448,"source_end_line":464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L448-L464","statement_sha256":"13f4b42a809b7afef50547252437c6f57ded9b22a06d28daa1d084709136fea2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9419,"rank":9419,"depth":20,"x":109.012,"y":1209.593,"cluster":"varieties-curves"},{"id":"stacks:0AGR","tag":"0AGR","title":"Quadratic transformations · Lemma 0AGR","summary":"Let (A, m, kappa) be a regular local ring of dimension 2. Let f : X → S = Spec(A) be the blowing up of A in m. Then X is an irreducible regular scheme.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.\nLet $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.\nThen $X$ is an irreducible regular scheme.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGR","source_file":"resolve.tex","source_line":509,"source_end_line":514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L509-L514","statement_sha256":"7e63ea0669b24862f10914bf6fd51079067b3d0a8b42fbff1150a4a349d824dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9420,"rank":9420,"depth":15,"x":263.291,"y":1019.884,"cluster":"varieties-curves"},{"id":"stacks:0C5G","tag":"0C5G","title":"Quadratic transformations · Lemma 0C5G","summary":"Let (A, m, kappa) be a regular local ring of dimension 2. Let f : X → S = Spec(A) be the blowing up of A in m. Then Pic(X) = Z generated by O_X(E).","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.\nLet $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.\nThen $\\Pic(X) = \\mathbf{Z}$ generated by $\\mathcal{O}_X(E)$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5G","source_file":"resolve.tex","source_line":538,"source_end_line":543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L538-L543","statement_sha256":"29e8dd6010a5c0f882e96020ce853ce59a02ddf2ce6cf66067fa985323c4b410","origin":"The Stacks Project","memory_eligible":false,"source_rank":9421,"rank":9421,"depth":28,"x":302.134,"y":1247.593,"cluster":"varieties-curves"},{"id":"stacks:0AGS","tag":"0AGS","title":"Quadratic transformations · Lemma 0AGS","summary":"Let (A, m, kappa) be a regular local ring of dimension 2. Let f : X → S = Spec(A) be the blowing up of A in m. Let F be a quasi-coherent O_X-module. • H^p(X, F) = 0 for p not ∈ (0, 1), • H^1(X, O_X(n)) = 0 for n ≥ -1, • H^1(X, F) = 0 if F or F(1) is globally generated, • H^0(X, O_X(n)) = m^max(0, n), • length_A H^1(X, O_X(n)) = -n(-n - 1)/2 if n < 0.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.\nLet $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item $H^p(X, \\mathcal{F}) = 0$ for $p \\not \\in \\{0, 1\\}$,\n\\item $H^1(X, \\mathcal{O}_X(n)) = 0$ for $n \\geq -1$,\n\\item $H^1(X, \\mathcal{F}) = 0$ if $\\mathcal{F}$ or $\\mathcal{F}(1)$\nis globally generated,\n\\item $H^0(X, \\mathcal{O}_X(n)) = \\mathfrak m^{\\max(0, n)}$,\n\\item $\\text{length}_A H^1(X, \\mathcal{O}_X(n)) = -n(-n - 1)/2$\nif $n < 0$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGS","source_file":"resolve.tex","source_line":568,"source_end_line":582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L568-L582","statement_sha256":"b0dfb3012f83ea3632bf4239c4a71eec76cbd0b08a280f5f154322d4c4f6664a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9422,"rank":9422,"depth":30,"x":90.115,"y":1101.546,"cluster":"varieties-curves"},{"id":"stacks:0AGT","tag":"0AGT","title":"Quadratic transformations · Lemma 0AGT","summary":"Let (A, m) be a regular local ring of dimension 2. Let f : X → S = Spec(A) be the blowing up of A in m. Let m^n ⊂ I ⊂ m be an ideal. Let d ≥ 0 be the largest integer such that I O_X ⊂ O_X(-dE) where E is the exceptional divisor. Set I' = IO_X(dE) ⊂ O_X. Then d > 0, the sheaf O_X/I' is supported in finitely many closed points x_1, …, x_r of X, and length_A(A/I) & > length_A Γ(X, O_X/I') & ≥ ∑_i = 1, …, r length_O_X, x_i (O_X, x_i/I'_x_i)","statement_latex":"Let $(A, \\mathfrak m)$ be a regular local ring of dimension $2$.\nLet $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.\nLet $\\mathfrak m^n \\subset I \\subset \\mathfrak m$ be an ideal.\nLet $d \\geq 0$ be the largest integer such that\n$$\nI \\mathcal{O}_X \\subset \\mathcal{O}_X(-dE)\n$$\nwhere $E$ is the exceptional divisor. Set\n$\\mathcal{I}' = I\\mathcal{O}_X(dE) \\subset \\mathcal{O}_X$.\nThen $d > 0$, the sheaf\n$\\mathcal{O}_X/\\mathcal{I}'$ is supported in finitely many\nclosed points $x_1, \\ldots, x_r$ of $X$, and\n\\begin{align*}\n\\text{length}_A(A/I)\n& >\n\\text{length}_A \\Gamma(X, \\mathcal{O}_X/\\mathcal{I}') \\\\\n& \\geq\n\\sum\\nolimits_{i = 1, \\ldots, r}\n\\text{length}_{\\mathcal{O}_{X, x_i}}\n(\\mathcal{O}_{X, x_i}/\\mathcal{I}'_{x_i})\n\\end{align*}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGT","source_file":"resolve.tex","source_line":700,"source_end_line":723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L700-L723","statement_sha256":"fa1152297f705e24be70ea705d0c75511b33a6f682cedf1f03026d2143888500","origin":"The Stacks Project","memory_eligible":false,"source_rank":9423,"rank":9423,"depth":31,"x":364.236,"y":1088.919,"cluster":"varieties-curves"},{"id":"stacks:0B4L","tag":"0B4L","title":"Quadratic transformations · Lemma 0B4L","summary":"Let (A, m, kappa) be a regular local ring of dimension 2. Let f : X → S = Spec(A) be the blowing up of A in m. Then Ω_X/S = i_*Ω_E/kappa, where i : E → X is the immersion of the exceptional divisor.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of dimension $2$.\nLet $f : X \\to S = \\Spec(A)$ be the blowing up of $A$ in $\\mathfrak m$.\nThen $\\Omega_{X/S} = i_*\\Omega_{E/\\kappa}$, where $i : E \\to X$\nis the immersion of the exceptional divisor.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4L","source_file":"resolve.tex","source_line":794,"source_end_line":800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L794-L800","statement_sha256":"135f8153197bcbb33d0d23a60c26d9a2a546978df3b2e87a8b02234aa858b685","origin":"The Stacks Project","memory_eligible":false,"source_rank":9424,"rank":9424,"depth":21,"x":172.024,"y":1253.974,"cluster":"varieties-curves"},{"id":"stacks:0AHH","tag":"0AHH","title":"Dominating by quadratic transformations · Lemma 0AHH","summary":"Let X be a Noetherian scheme. Let T ⊂ X be a finite set of closed points x such that O_X, x is regular of dimension 2 for x ∈ T. Let I ⊂ O_X be a quasi-coherent sheaf of ideals such that O_X/I is supported on T. Then there exists a sequence X_n → X_n - 1 → … → X_1 → X_0 = X where X_i + 1 → X_i is the blowing up of X_i at a closed point lying above a point of T such that IO_X_n is an invertible ideal sheaf.","statement_latex":"Let $X$ be a Noetherian scheme. Let $T \\subset X$ be a finite set of\nclosed points $x$ such that $\\mathcal{O}_{X, x}$ is\nregular of dimension $2$ for $x \\in T$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent\nsheaf of ideals such that $\\mathcal{O}_X/\\mathcal{I}$ is supported\non $T$.\nThen there exists a sequence\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X\n$$\nwhere $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed\npoint lying above a point of $T$ such that\n$\\mathcal{I}\\mathcal{O}_{X_n}$ is an invertible ideal sheaf.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Dominating by quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHH","source_file":"resolve.tex","source_line":844,"source_end_line":859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L844-L859","statement_sha256":"3c3c39f5074623b59d3933efbd160a7edb07a88a0e242f82f2cd6dc8a4d89e66","origin":"The Stacks Project","memory_eligible":false,"source_rank":9425,"rank":9425,"depth":32,"x":181.037,"y":1022.918,"cluster":"varieties-curves"},{"id":"stacks:0AHI","tag":"0AHI","title":"Dominating by quadratic transformations · Lemma 0AHI","summary":"Let X be a Noetherian scheme. Let T ⊂ X be a finite set of closed points x such that O_X, x is a regular local ring of dimension 2. Let f : Y → X be a proper morphism of schemes which is an isomorphism over U = X setminus T. Then there exists a sequence X_n → X_n - 1 → … → X_1 → X_0 = X where X_i + 1 → X_i is the blowing up of X_i at a closed point x_i lying above a point of T and a factorization X_n → Y → X of the composition.","statement_latex":"Let $X$ be a Noetherian scheme. Let $T \\subset X$ be a finite set of\nclosed points $x$ such that $\\mathcal{O}_{X, x}$ is a regular local\nring of dimension $2$. Let $f : Y \\to X$ be a proper morphism of\nschemes which is an isomorphism over $U = X \\setminus T$.\nThen there exists a sequence\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X\n$$\nwhere $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed\npoint $x_i$ lying above a point of $T$ and a factorization $X_n \\to Y \\to X$\nof the composition.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Dominating by quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHI","source_file":"resolve.tex","source_line":928,"source_end_line":941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L928-L941","statement_sha256":"d2d722f4804694dcd35269756bee3fcf917f00112df70d232df05954bcac3d11","origin":"The Stacks Project","memory_eligible":false,"source_rank":9426,"rank":9426,"depth":59,"x":360.416,"y":1198.622,"cluster":"varieties-curves"},{"id":"stacks:0C5H","tag":"0C5H","title":"Dominating by quadratic transformations · Lemma 0C5H","summary":"Let S be a scheme. Let X be a scheme over S which is regular and has dimension 2. Let Y be a proper scheme over S. Given an S-rational map f : U → Y from X to Y there exists a sequence X_n → X_n - 1 → … → X_1 → X_0 = X and an S-morphism f_n : X_n → Y such that X_i + 1 → X_i is the blowing up of X_i at a closed point not lying over U and f_n and f agree.","statement_latex":"Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is\nregular and has dimension $2$. Let $Y$ be a proper\nscheme over $S$. Given an $S$-rational map $f : U \\to Y$ from\n$X$ to $Y$ there exists a sequence\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X\n$$\nand an $S$-morphism $f_n : X_n \\to Y$ such that $X_{i + 1} \\to X_i$\nis the blowing up of $X_i$ at a closed point not lying over $U$\nand $f_n$ and $f$ agree.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Dominating by quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5H","source_file":"resolve.tex","source_line":964,"source_end_line":976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L964-L976","statement_sha256":"de703a2023a0ba886885155ebfbf10680254027f131d148f1af3c7e0598852ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":9427,"rank":9427,"depth":60,"x":86.517,"y":1170.811,"cluster":"varieties-curves"},{"id":"stacks:0BBS","tag":"0BBS","title":"Dominating by normalized blowups · Definition 0BBS","summary":"Let X be a scheme such that every quasi-compact open has finitely many irreducible components. Let x ∈ X be a closed point. The normalized blowup of X at x is the composition X\" → X' → X where X' → X is the blowup of X in x and X\" → X' is the normalization of X'.","statement_latex":"Let $X$ be a scheme such that every quasi-compact open has finitely\nmany irreducible components. Let $x \\in X$ be a closed point.\nThe {\\it normalized blowup of $X$ at $x$} is the composition\n$X'' \\to X' \\to X$ where $X' \\to X$ is the blowup\nof $X$ in $x$ and $X'' \\to X'$ is the normalization of $X'$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Dominating by normalized blowups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBS","source_file":"resolve.tex","source_line":1004,"source_end_line":1011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1004-L1011","statement_sha256":"7dd72539cc4a87b43b831531fb0ca2dfac1cc035b0c0f878cd154b15bb55f75c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9428,"rank":9428,"depth":0,"x":311.122,"y":1035.739,"cluster":"varieties-curves"},{"id":"stacks:0BFT","tag":"0BFT","title":"Dominating by normalized blowups · Lemma 0BFT","summary":"In Definition [Tag 0BBS] if X is Nagata, then the normalized blowing up of X at x is normal, Nagata, and proper over X.","statement_latex":"In Definition \\ref{definition-normalized-blowup} if $X$ is Nagata,\nthen the normalized blowing up of $X$ at $x$ is\nnormal, Nagata, and proper over $X$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Dominating by normalized blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFT","source_file":"resolve.tex","source_line":1027,"source_end_line":1032,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1027-L1032","statement_sha256":"143e4df1f1424e15dc94577165f6aaaf78185f1f98350a282e9a2f6c8f91e84e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9429,"rank":9429,"depth":34,"x":254.054,"y":1263.058,"cluster":"varieties-curves"},{"id":"stacks:0BBT","tag":"0BBT","title":"Dominating by normalized blowups · Lemma 0BBT","summary":"Let X be a scheme which is Noetherian, Nagata, and has dimension 2. Let f : Y → X be a proper birational morphism. Then there exists a commutative diagram xymatrix X_n ar[r] ar[d] & X_n - 1 ar[r] & … ar[r] & X_1 ar[r] & X_0 ar[d] Y ar[rrrr] & & & & X where X_0 → X is the normalization and where X_i + 1 → X_i is the normalized blowing up of X_i at a closed point.","statement_latex":"Let $X$ be a scheme which is Noetherian, Nagata, and has dimension $2$.\nLet $f : Y \\to X$ be a proper birational morphism.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\nX_n \\ar[r] \\ar[d] &\nX_{n - 1} \\ar[r] &\n\\ldots \\ar[r] &\nX_1 \\ar[r] &\nX_0 \\ar[d] \\\\\nY \\ar[rrrr]  & & & & X\n}\n$$\nwhere $X_0 \\to X$ is the normalization and\nwhere $X_{i + 1} \\to X_i$ is the normalized blowing up of $X_i$ at a closed\npoint.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Dominating by normalized blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBT","source_file":"resolve.tex","source_line":1059,"source_end_line":1077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1059-L1077","statement_sha256":"f0dc5002bb5be216977e8ce8f81dc8f48360fa0d11b31e9f4866cd18d676c6ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":9430,"rank":9430,"depth":43,"x":113.163,"y":1062.815,"cluster":"varieties-curves"},{"id":"stacks:0C5I","tag":"0C5I","title":"Dominating by normalized blowups · Lemma 0C5I","summary":"Let S be a scheme. Let X be a scheme over S which is Noetherian, Nagata, and has dimension 2. Let Y be a proper scheme over S. Given an S-rational map f : U → Y from X to Y there exists a sequence X_n → X_n - 1 → … → X_1 → X_0 → X and an S-morphism f_n : X_n → Y such that X_0 → X is the normalization, X_i + 1 → X_i is the normalized blowing up of X_i at a closed point, and f_n and f agree.","statement_latex":"Let $S$ be a scheme. Let $X$ be a scheme over $S$ which is\nNoetherian, Nagata, and has dimension $2$. Let $Y$ be a proper\nscheme over $S$. Given an $S$-rational map $f : U \\to Y$ from\n$X$ to $Y$ there exists a sequence\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 \\to X\n$$\nand an $S$-morphism $f_n : X_n \\to Y$ such that $X_0 \\to X$ is the\nnormalization, $X_{i + 1} \\to X_i$ is the normalized blowing up of\n$X_i$ at a closed point, and $f_n$ and $f$ agree.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Dominating by normalized blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5I","source_file":"resolve.tex","source_line":1172,"source_end_line":1184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1172-L1184","statement_sha256":"e7770451f95202348889e1d18e7fcda09eab56b2d0a265e8adf6376caadd212a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9431,"rank":9431,"depth":44,"x":378.401,"y":1130.607,"cluster":"varieties-curves"},{"id":"stacks:0BFV","tag":"0BFV","title":"Modifying over local rings · Lemma 0BFV","summary":"The functor F ([Tag 0BFU]) is an equivalence.","statement_latex":"The functor $F$ (\\ref{equation-equivalence}) is an equivalence.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Modifying over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFV","source_file":"resolve.tex","source_line":1230,"source_end_line":1233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1230-L1233","statement_sha256":"26ad1c9a1dca18d034b2442fb2c5705c17c7795e80ace0c822336457f5908b10","origin":"The Stacks Project","memory_eligible":false,"source_rank":9432,"rank":9432,"depth":28,"x":128.003,"y":1231.241,"cluster":"varieties-curves"},{"id":"stacks:0BFW","tag":"0BFW","title":"Modifying over local rings · Lemma 0BFW","summary":"Let S, s_i, S_i be as in ([Tag 0BFU]). If f : X → S corresponds to g_i : Y_i → S_i under F, then f is separated, proper, finite, if and only if g_i is so for i = 1, …, n.","statement_latex":"Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence}).\nIf $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$,\nthen $f$ is separated, proper, finite, if and only if $g_i$ is so\nfor $i = 1, \\ldots, n$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Modifying over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFW","source_file":"resolve.tex","source_line":1253,"source_end_line":1259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1253-L1259","statement_sha256":"19654d3500710c94790a71ab10d0a2eee1f5e8e4411c3c8929598ed3f0a30851","origin":"The Stacks Project","memory_eligible":false,"source_rank":9433,"rank":9433,"depth":42,"x":231.84,"y":1014.696,"cluster":"varieties-curves"},{"id":"stacks:0BFX","tag":"0BFX","title":"Modifying over local rings · Lemma 0BFX","summary":"Let S, s_i, S_i be as in ([Tag 0BFU]). If f : X → S corresponds to g_i : Y_i → S_i under F, then X_s_i ≅ (Y_i)_s_i as schemes over kappa(s_i).","statement_latex":"Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence}).\nIf $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$,\nthen $X_{s_i} \\cong (Y_i)_{s_i}$ as schemes over $\\kappa(s_i)$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Modifying over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFX","source_file":"resolve.tex","source_line":1266,"source_end_line":1271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1266-L1271","statement_sha256":"40b3d9e7cb9495ec436d00c1db7b762e235d6f9ca53adc253af3bc170cea9a15","origin":"The Stacks Project","memory_eligible":false,"source_rank":9434,"rank":9434,"depth":0,"x":329.528,"y":1233.552,"cluster":"varieties-curves"},{"id":"stacks:0BFY","tag":"0BFY","title":"Modifying over local rings · Lemma 0BFY","summary":"Let S, s_i, S_i be as in ([Tag 0BFU]) and assume f : X → S corresponds to g_i : Y_i → S_i under F. Then there exists a factorization X = Z_m → Z_m - 1 → … → Z_1 → Z_0 = S of f where Z_j + 1 → Z_j is the blowing up of Z_j at a closed point z_j lying over (s_1, …, s_n) if and only if for each i there exists a factorization Y_i = Z_i, m_i → Z_i, m_i - 1 → … → Z_i, 1 → Z_i, 0 = S_i of g_i where Z_i, j + 1 → Z_i, j is the blowing up of Z_i, j at a closed point z_i, j lying…","statement_latex":"Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence})\nand assume $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$.\nThen there exists a factorization\n$$\nX = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = S\n$$\nof $f$ where $Z_{j + 1} \\to Z_j$ is the blowing up of $Z_j$ at a closed\npoint $z_j$ lying over $\\{s_1, \\ldots, s_n\\}$ if and only if for each\n$i$ there exists a factorization\n$$\nY_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = S_i\n$$\nof $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the blowing up of $Z_{i, j}$\nat a closed point $z_{i, j}$ lying over $s_i$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Modifying over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFY","source_file":"resolve.tex","source_line":1277,"source_end_line":1293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1277-L1293","statement_sha256":"ec74c9c2d068e2721758d0e98365b44ceacbb1a905224bea8a22e42e08fbc541","origin":"The Stacks Project","memory_eligible":false,"source_rank":9435,"rank":9435,"depth":1,"x":81.202,"y":1127.476,"cluster":"varieties-curves"},{"id":"stacks:0BFZ","tag":"0BFZ","title":"Modifying over local rings · Lemma 0BFZ","summary":"Let S, s_i, S_i be as in ([Tag 0BFU]) and assume f : X → S corresponds to g_i : Y_i → S_i under F. Assume every quasi-compact open of S has finitely many irreducible components. Then there exists a factorization X = Z_m → Z_m - 1 → … → Z_1 → Z_0 = S of f where Z_j + 1 → Z_j is the normalized blowing up of Z_j at a closed point z_j lying over (x_1, …, x_n) if and only if for each i there exists a factorization Y_i = Z_i, m_i → Z_i, m_i - 1 → … → Z_i, 1 → Z_i, 0 = S_i of…","statement_latex":"Let $S, s_i, S_i$ be as in (\\ref{equation-equivalence})\nand assume $f : X \\to S$ corresponds to $g_i : Y_i \\to S_i$ under $F$.\nAssume every quasi-compact open of $S$ has finitely many irreducible\ncomponents.  Then there exists a factorization\n$$\nX = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = S\n$$\nof $f$ where $Z_{j + 1} \\to Z_j$ is the normalized blowing up of $Z_j$\nat a closed point $z_j$ lying over $\\{x_1, \\ldots, x_n\\}$ if and only if\nfor each $i$ there exists a factorization\n$$\nY_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = S_i\n$$\nof $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the normalized blowing up of\n$Z_{i, j}$ at a closed point $z_{i, j}$ lying over $s_i$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Modifying over local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BFZ","source_file":"resolve.tex","source_line":1323,"source_end_line":1340,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1323-L1340","statement_sha256":"74c9dbdb0f60161af438a9db2d377754481222f63ea60279e80ed8f9e06aaf65","origin":"The Stacks Project","memory_eligible":false,"source_rank":9436,"rank":9436,"depth":2,"x":349.935,"y":1064.715,"cluster":"varieties-curves"},{"id":"stacks:0B4M","tag":"0B4M","title":"Vanishing · Lemma 0B4M","summary":"In Situation [Tag 0AX8] there exists a U-admissible blowup X' → S which dominates X.","statement_latex":"In Situation \\ref{situation-vanishing} there exists a $U$-admissible\nblowup $X' \\to S$ which dominates $X$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4M","source_file":"resolve.tex","source_line":1394,"source_end_line":1398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1394-L1398","statement_sha256":"81cc9b35c180ee4726b1f256952d8aed686ecbd4f20418c8c92c33540997b1a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9437,"rank":9437,"depth":59,"x":202.071,"y":1263.714,"cluster":"varieties-curves"},{"id":"stacks:0AX9","tag":"0AX9","title":"Vanishing · Lemma 0AX9","summary":"In Situation [Tag 0AX8] there exists a nonzero f ∈ m such that for every i = 1, …, r there exist • a closed point x_i ∈ C_i with x_i not ∈ C_j for j not = i, • a factorization f = g_i f_i of f in O_X, x_i such that g_i ∈ m_x_i maps to a nonzero element of O_C_i, x_i.","statement_latex":"In Situation \\ref{situation-vanishing} there exists a nonzero\n$f \\in \\mathfrak m$ such that for every $i = 1, \\ldots, r$ there exist\n\\begin{enumerate}\n\\item a closed point $x_i \\in C_i$ with $x_i \\not \\in C_j$ for $j \\not = i$,\n\\item a factorization $f = g_i f_i$ of $f$ in $\\mathcal{O}_{X, x_i}$\nsuch that $g_i \\in \\mathfrak m_{x_i}$ maps to a nonzero element\nof $\\mathcal{O}_{C_i, x_i}$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AX9","source_file":"resolve.tex","source_line":1405,"source_end_line":1415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1405-L1415","statement_sha256":"2392b97fa518479d07d92a5afda95b489ea3f27a2bc1e87d1043ab355a0d3241","origin":"The Stacks Project","memory_eligible":false,"source_rank":9438,"rank":9438,"depth":0,"x":151.014,"y":1032.802,"cluster":"varieties-curves"},{"id":"stacks:0AXA","tag":"0AXA","title":"Vanishing · Lemma 0AXA","summary":"In Situation [Tag 0AX8] assume X is normal. Let Z ⊂ X be a nonempty effective Cartier divisor such that Z ⊂ X_s set theoretically. Then the conormal sheaf of Z is not trivial. More precisely, there exists an i such that C_i ⊂ Z and deg(C_Z/X|_C_i) > 0.","statement_latex":"In Situation \\ref{situation-vanishing} assume $X$ is normal.\nLet $Z \\subset X$ be a nonempty effective Cartier divisor such that\n$Z \\subset X_s$ set theoretically.\nThen the conormal sheaf of $Z$ is not trivial.\nMore precisely, there exists an $i$ such that $C_i \\subset Z$\nand $\\deg(\\mathcal{C}_{Z/X}|_{C_i}) > 0$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXA","source_file":"resolve.tex","source_line":1426,"source_end_line":1434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1426-L1434","statement_sha256":"6874268e6740206cc085dc5c8a94f4a1e5e076b7601e4e0e49afdf407c49d29b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9439,"rank":9439,"depth":45,"x":374.618,"y":1174.268,"cluster":"varieties-curves"},{"id":"stacks:0AXB","tag":"0AXB","title":"Vanishing · Lemma 0AXB","summary":"In Situation [Tag 0AX8] assume X is normal and A Nagata. The map H^1(X, O_X) → H^1(f^-1(U), O_X) is injective.","statement_latex":"In Situation \\ref{situation-vanishing} assume $X$ is normal\nand $A$ Nagata. The map\n$$\nH^1(X, \\mathcal{O}_X) \\longrightarrow H^1(f^{-1}(U), \\mathcal{O}_X)\n$$\nis injective.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXB","source_file":"resolve.tex","source_line":1474,"source_end_line":1482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1474-L1482","statement_sha256":"c2bf5e5b28ff3d10d445be6328d22dc3d7f2a46c5ee5ad77a2f3657964a86d8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9440,"rank":9440,"depth":46,"x":95.647,"y":1196.857,"cluster":"varieties-curves"},{"id":"stacks:0AXC","tag":"0AXC","title":"Vanishing · Lemma 0AXC","summary":"In Situation [Tag 0AX8] assume X is normal and A Nagata. Then Hom_D(A)(kappa[-1], Rf_*O_X) is zero. This uses D(A) = D_QCoh(O_S) to think of Rf_*O_X as an object of D(A).","statement_latex":"In Situation \\ref{situation-vanishing} assume $X$ is normal and $A$ Nagata.\nThen\n$$\n\\Hom_{D(A)}(\\kappa[-1], Rf_*\\mathcal{O}_X)\n$$\nis zero. This uses $D(A) = D_\\QCoh(\\mathcal{O}_S)$ to think of\n$Rf_*\\mathcal{O}_X$ as an object of $D(A)$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXC","source_file":"resolve.tex","source_line":1539,"source_end_line":1548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1539-L1548","statement_sha256":"69e061e4d0ff4bc3e57d4ede1455e4823ab06cb4dd59e9f90332c36e83d79c67","origin":"The Stacks Project","memory_eligible":false,"source_rank":9441,"rank":9441,"depth":47,"x":283.409,"y":1021.701,"cluster":"varieties-curves"},{"id":"stacks:0AXD","tag":"0AXD","title":"Grauert-Riemenschneider · Proposition 0AXD","summary":"In Situation [Tag 0AX8] assume • X is a normal scheme, • A is Nagata and has a dualizing complex ω_A^bullet. Let ω_X be the dualizing module of X (Remark [Tag 0B4R]). Then R^1f_*ω_X = 0.","statement_latex":"In Situation \\ref{situation-vanishing} assume\n\\begin{enumerate}\n\\item $X$ is a normal scheme,\n\\item $A$ is Nagata and has a dualizing complex $\\omega_A^\\bullet$.\n\\end{enumerate}\nLet $\\omega_X$ be the dualizing module of $X$\n(Remark \\ref{remark-dualizing-setup}). Then $R^1f_*\\omega_X = 0$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Vanishing","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXD","source_file":"resolve.tex","source_line":1620,"source_end_line":1629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1620-L1629","statement_sha256":"e96da69f5d3e548568a04f1f7165de345468c92a71cdeb89008062c300ab4788","origin":"The Stacks Project","memory_eligible":false,"source_rank":9442,"rank":9442,"depth":48,"x":285.815,"y":1257.674,"cluster":"varieties-curves"},{"id":"stacks:0AXF","tag":"0AXF","title":"Boundedness · Lemma 0AXF","summary":"Let (A, m, kappa) be a Noetherian normal local domain of dimension 2. Consider a commutative diagram xymatrix X' ar[rd]_f' ar[rr]_g & & X ar[ld]^f & Spec(A) where f and f' are modifications as in Situation [Tag 0AX8] and X normal. Then we have a short exact sequence 0 → H^1(X, O_X) → H^1(X', O_X') → H^0(X, R^1g_*O_X') → 0 Also dim(Supp(R^1g_*O_X')) = 0 and R^1g_*O_X' is generated by global sections.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian normal local domain\nof dimension $2$. Consider a commutative diagram\n$$\n\\xymatrix{\nX' \\ar[rd]_{f'} \\ar[rr]_g & & X \\ar[ld]^f \\\\\n& \\Spec(A)\n}\n$$\nwhere $f$ and $f'$ are modifications as in Situation \\ref{situation-vanishing}\nand $X$ normal. Then we have a short exact sequence\n$$\n0 \\to H^1(X, \\mathcal{O}_X) \\to H^1(X', \\mathcal{O}_{X'}) \\to\nH^0(X, R^1g_*\\mathcal{O}_{X'}) \\to 0\n$$\nAlso $\\dim(\\text{Supp}(R^1g_*\\mathcal{O}_{X'})) = 0$\nand $R^1g_*\\mathcal{O}_{X'}$ is generated by global sections.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXF","source_file":"resolve.tex","source_line":1683,"source_end_line":1701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1683-L1701","statement_sha256":"1aaa6245cdb87f9273852007743cd22d8b4f26c9d28d7cb15b0610e5bee61c32","origin":"The Stacks Project","memory_eligible":false,"source_rank":9443,"rank":9443,"depth":41,"x":94.055,"y":1084.835,"cluster":"varieties-curves"},{"id":"stacks:0AXJ","tag":"0AXJ","title":"Boundedness · Lemma 0AXJ","summary":"Let (A, m, kappa) be a local normal Nagata domain of dimension 2. Let a ∈ A be nonzero. There exists an integer N such that for every modification f : X → Spec(A) with X normal the A-module M_X, a = Coker(A → H^0(Z, O_Z)) where Z ⊂ X is cut out by a has length bounded by N.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain\nof dimension $2$. Let $a \\in A$ be nonzero. There exists an integer $N$ such\nthat for every modification $f : X \\to \\Spec(A)$ with $X$ normal the\n$A$-module\n$$\nM_{X, a} = \\Coker(A \\longrightarrow H^0(Z, \\mathcal{O}_Z))\n$$\nwhere $Z \\subset X$ is cut out by $a$ has length bounded by $N$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXJ","source_file":"resolve.tex","source_line":1729,"source_end_line":1739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1729-L1739","statement_sha256":"c99efe6e6057f10d2356034a7463ab41dc737888b0f45f54e9e06b224ac08941","origin":"The Stacks Project","memory_eligible":false,"source_rank":9444,"rank":9444,"depth":47,"x":374.773,"y":1103.497,"cluster":"varieties-curves"},{"id":"stacks:0B4N","tag":"0B4N","title":"Boundedness · Definition 0B4N","summary":"Let (A, m, kappa) be a local normal Nagata domain of dimension 2. • We say A defines a rational singularity if for every normal modification X → Spec(A) we have H^1(X, O_X) = 0. • We say that reduction to rational singularities is possible for A if the length of the A-modules H^1(X, O_X) is bounded for all modifications X → Spec(A) with X normal.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain\nof dimension $2$.\n\\begin{enumerate}\n\\item We say $A$ {\\it defines a rational singularity} if for every\nnormal modification $X \\to \\Spec(A)$ we have $H^1(X, \\mathcal{O}_X) = 0$.\n\\item We say that {\\it reduction to rational singularities\nis possible for $A$} if the length of the $A$-modules\n$$\nH^1(X, \\mathcal{O}_X)\n$$\nis bounded for all modifications $X \\to \\Spec(A)$ with $X$ normal.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4N","source_file":"resolve.tex","source_line":1794,"source_end_line":1808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1794-L1808","statement_sha256":"df78568d3fe6406a24b66358978049caf001ec8e0dabba2531ffceb0b6f50dd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9445,"rank":9445,"depth":0,"x":152.512,"y":1249.191,"cluster":"varieties-curves"},{"id":"stacks:0BG0","tag":"0BG0","title":"Boundedness · Lemma 0BG0","summary":"Let (A, m, kappa) be a local normal Nagata domain of dimension 2 which defines a rational singularity. Let A ⊂ B be a local extension of domains with the same fraction field which is essentially of finite type such that dim(B) = 2 and B normal. Then B defines a rational singularity.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of\ndimension $2$ which defines a rational singularity. Let $A \\subset B$\nbe a local extension of domains with the same fraction field\nwhich is essentially of finite type such\nthat $\\dim(B) = 2$ and $B$ normal. Then $B$ defines a rational singularity.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BG0","source_file":"resolve.tex","source_line":1817,"source_end_line":1824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1817-L1824","statement_sha256":"2fd4d0ab73c715c92efc2553047b2bcab3d745c588f0290ea2609d410bbafa2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9446,"rank":9446,"depth":42,"x":199.295,"y":1015.372,"cluster":"varieties-curves"},{"id":"stacks:0B4P","tag":"0B4P","title":"Boundedness · Lemma 0B4P","summary":"Let (A, m, kappa) be a local normal Nagata domain of dimension 2. If reduction to rational singularities is possible for A, then there exists a finite sequence of normalized blowups X = X_n → X_n - 1 → … → X_1 → X_0 = Spec(A) in closed points such that for any closed point x ∈ X the local ring O_X, x defines a rational singularity. In particular X → Spec(A) is a modification and X is a normal scheme projective over A.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain\nof dimension $2$. If reduction to rational singularities is possible\nfor $A$, then there exists a finite sequence of normalized blowups\n$$\nX = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A)\n$$\nin closed points such that for any closed point $x \\in X$\nthe local ring $\\mathcal{O}_{X, x}$ defines a rational singularity.\nIn particular $X \\to \\Spec(A)$ is a modification and $X$\nis a normal scheme projective over $A$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4P","source_file":"resolve.tex","source_line":1869,"source_end_line":1881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1869-L1881","statement_sha256":"674792a6089436a48d7446bf1016f00d91e598676ad945add7ce3b66ef8bb480","origin":"The Stacks Project","memory_eligible":false,"source_rank":9447,"rank":9447,"depth":44,"x":353.005,"y":1214.561,"cluster":"varieties-curves"},{"id":"stacks:0AXL","tag":"0AXL","title":"Boundedness · Lemma 0AXL","summary":"Let A → B be a finite injective local ring map of local normal Nagata domains of dimension 2. Assume that the induced extension of fraction fields is separable. If reduction to rational singularities is possible for A then it is possible for B.","statement_latex":"Let $A \\to B$ be a finite injective local ring map of local normal\nNagata domains of dimension $2$. Assume that the induced extension of\nfraction fields is separable. If reduction to rational singularities\nis possible for $A$ then it is possible for $B$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AXL","source_file":"resolve.tex","source_line":1921,"source_end_line":1927,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1921-L1927","statement_sha256":"a4b81d0e839aebd142fea1047eaa869a63f0bc7341f18171875deac0f3059e00","origin":"The Stacks Project","memory_eligible":false,"source_rank":9448,"rank":9448,"depth":58,"x":79.165,"y":1154.832,"cluster":"varieties-curves"},{"id":"stacks:0B4Q","tag":"0B4Q","title":"Boundedness · Lemma 0B4Q","summary":"Let A be a Nagata regular local ring of dimension 2. Then A defines a rational singularity.","statement_latex":"Let $A$ be a Nagata regular local ring of dimension $2$. Then $A$ defines\na rational singularity.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4Q","source_file":"resolve.tex","source_line":1987,"source_end_line":1991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L1987-L1991","statement_sha256":"f53bce3e24de05e6d2178140395747084d1ab5963e03d62a58b31f7442d78463","origin":"The Stacks Project","memory_eligible":false,"source_rank":9449,"rank":9449,"depth":60,"x":329.409,"y":1043.365,"cluster":"varieties-curves"},{"id":"stacks:0B4S","tag":"0B4S","title":"Boundedness · Lemma 0B4S","summary":"Let A be a local normal Nagata domain of dimension 2 which has a dualizing complex ω_A^bullet. If there exists a nonzero d ∈ A such that for all normal modifications X → Spec(A) the cokernel of the trace map Γ(X, ω_X) → ω_A is annihilated by d, then reduction to rational singularities is possible for A.","statement_latex":"Let $A$ be a local normal Nagata domain of dimension $2$ which has a\ndualizing complex $\\omega_A^\\bullet$. If there exists a nonzero $d \\in A$\nsuch that for all normal modifications $X \\to \\Spec(A)$ the cokernel of the\ntrace map\n$$\n\\Gamma(X, \\omega_X) \\to \\omega_A\n$$\nis annihilated by $d$, then reduction to rational singularities\nis possible for $A$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4S","source_file":"resolve.tex","source_line":2015,"source_end_line":2026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2015-L2026","statement_sha256":"087f994d26989ab2a3ce55be5bf474ee45edd14681dc86f310e63db10280204b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9450,"rank":9450,"depth":65,"x":234.413,"y":1267.81,"cluster":"varieties-curves"},{"id":"stacks:0B4T","tag":"0B4T","title":"Boundedness · Lemma 0B4T","summary":"Let p be a prime number. Let A be a regular local ring of dimension 2 and characteristic p. Let A_0 ⊂ A be a subring such that Ω_A/A_0 is free of rank r < ∞. Set ω_A = Ω^r_A/A_0. If X → Spec(A) is the result of a sequence of blowups in closed points, then there exists a map φ_X : (Ω^r_X/Spec(A_0))^** → ω_X extending the given identification in the generic point.","statement_latex":"Let $p$ be a prime number.\nLet $A$ be a regular local ring of dimension $2$ and characteristic $p$.\nLet $A_0 \\subset A$ be a subring such that $\\Omega_{A/A_0}$ is free\nof rank $r < \\infty$. Set $\\omega_A = \\Omega^r_{A/A_0}$. If $X \\to \\Spec(A)$\nis the result of a sequence of blowups in closed points, then\nthere exists a map\n$$\n\\varphi_X : (\\Omega^r_{X/\\Spec(A_0)})^{**} \\longrightarrow \\omega_X\n$$\nextending the given identification in the generic point.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4T","source_file":"resolve.tex","source_line":2073,"source_end_line":2085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2073-L2085","statement_sha256":"5ef265caa373296323646461ce702cd5b95fd9f4ccd6fe856a82501e118722a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9451,"rank":9451,"depth":31,"x":123.843,"y":1048.155,"cluster":"varieties-curves"},{"id":"stacks:0B4U","tag":"0B4U","title":"Boundedness · Lemma 0B4U","summary":"Let p be a prime number. Let A be a complete regular local ring of dimension 2 and characteristic p. Let L/K be a degree p inseparable extension of the fraction field K of A. Let B ⊂ L be the integral closure of A. Then reduction to rational singularities is possible for B.","statement_latex":"Let $p$ be a prime number. Let $A$ be a complete regular local ring of\ndimension $2$ and characteristic $p$. Let $L/K$ be a degree $p$ inseparable\nextension of the fraction field $K$ of $A$. Let $B \\subset L$ be the integral\nclosure of $A$. Then reduction to rational singularities is possible for $B$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4U","source_file":"resolve.tex","source_line":2131,"source_end_line":2137,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2131-L2137","statement_sha256":"1a343a9f55074876efeec341e1ba530557bd8fe69b66fff478803477213f96b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9452,"rank":9452,"depth":66,"x":382.311,"y":1147.498,"cluster":"varieties-curves"},{"id":"stacks:0B4X","tag":"0B4X","title":"Rational singularities · Lemma 0B4X","summary":"In Situation [Tag 0B4W]. Let F be a quasi-coherent O_X-module. Then • H^p(X, F) = 0 for p not ∈ (0, 1), and • H^1(X, F) = 0 if F is globally generated.","statement_latex":"In Situation \\ref{situation-rational}.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then\n\\begin{enumerate}\n\\item $H^p(X, \\mathcal{F}) = 0$ for $p \\not \\in \\{0, 1\\}$, and\n\\item $H^1(X, \\mathcal{F}) = 0$ if $\\mathcal{F}$ is globally generated.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Rational singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4X","source_file":"resolve.tex","source_line":2247,"source_end_line":2255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2247-L2255","statement_sha256":"156ae975915ec018a9016518f41297be2394a6beb7ef0f286142842247ce49f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9453,"rank":9453,"depth":39,"x":111.524,"y":1220.987,"cluster":"varieties-curves"},{"id":"stacks:0B4Y","tag":"0B4Y","title":"Rational singularities · Lemma 0B4Y","summary":"In Situation [Tag 0B4W] assume E = X_s is an effective Cartier divisor. Let I be the ideal sheaf of E. Then H^0(X, I^n) = m^n and H^1(X, I^n) = 0.","statement_latex":"In Situation \\ref{situation-rational} assume\n$E = X_s$ is an effective Cartier divisor.\nLet $\\mathcal{I}$ be the ideal sheaf of $E$. Then\n$H^0(X, \\mathcal{I}^n) = \\mathfrak m^n$ and\n$H^1(X, \\mathcal{I}^n) = 0$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Rational singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4Y","source_file":"resolve.tex","source_line":2270,"source_end_line":2277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2270-L2277","statement_sha256":"e9e128affedce5f862cc7b346824c3aabd6ed7e9be1988ac002c6ef7c41b884c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9454,"rank":9454,"depth":40,"x":252.259,"y":1012.912,"cluster":"varieties-curves"},{"id":"stacks:0B4Z","tag":"0B4Z","title":"Rational singularities · Lemma 0B4Z","summary":"In Situation [Tag 0B4W] the blowup of Spec(A) in m is normal.","statement_latex":"In Situation \\ref{situation-rational}\nthe blowup of $\\Spec(A)$ in $\\mathfrak m$ is normal.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Rational singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4Z","source_file":"resolve.tex","source_line":2316,"source_end_line":2320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2316-L2320","statement_sha256":"e6755463132073940e187c0f47f064772d5dd00f31493c56ba9279b4b7bed805","origin":"The Stacks Project","memory_eligible":false,"source_rank":9455,"rank":9455,"depth":41,"x":315.884,"y":1246.464,"cluster":"varieties-curves"},{"id":"stacks:0B63","tag":"0B63","title":"Rational singularities · Lemma 0B63","summary":"In Situation [Tag 0B4W]. Let X be the blowup of Spec(A) in m. Let E ⊂ X be the exceptional divisor. With O_X(1) = I as usual and O_E(1) = O_X(1)|_E we have • E is a proper Cohen-Macaulay curve over kappa. • O_E(1) is very ample • deg(O_E(1)) ≥ 1 and equality holds only if A is a regular local ring, • H^1(E, O_E(n)) = 0 for n ≥ 0, and • H^0(E, O_E(n)) = m^n/ m^n + 1 for n ≥ 0.","statement_latex":"In Situation \\ref{situation-rational}.\nLet $X$ be the blowup of $\\Spec(A)$ in $\\mathfrak m$. Let $E \\subset X$\nbe the exceptional divisor. With $\\mathcal{O}_X(1) = \\mathcal{I}$ as\nusual and $\\mathcal{O}_E(1) = \\mathcal{O}_X(1)|_E$ we have\n\\begin{enumerate}\n\\item $E$ is a proper Cohen-Macaulay curve over $\\kappa$.\n\\item $\\mathcal{O}_E(1)$ is very ample\n\\item $\\deg(\\mathcal{O}_E(1)) \\geq 1$ and equality holds only if\n$A$ is a regular local ring,\n\\item $H^1(E, \\mathcal{O}_E(n)) = 0$ for $n \\geq 0$, and\n\\item $H^0(E, \\mathcal{O}_E(n)) = \\mathfrak m^n/\\mathfrak m^{n + 1}$\nfor $n \\geq 0$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Rational singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B63","source_file":"resolve.tex","source_line":2363,"source_end_line":2378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2363-L2378","statement_sha256":"7709e41b9f1d3a22126816047667debc3061d76d303d21fcb7f02e6fe032867f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9456,"rank":9456,"depth":42,"x":80.888,"y":1110.195,"cluster":"varieties-curves"},{"id":"stacks:0BBU","tag":"0BBU","title":"Rational singularities · Lemma 0BBU","summary":"In Situation [Tag 0B4W] assume A has a dualizing complex ω_A^bullet. With ω_X the dualizing module of X, the trace map H^0(X, ω_X) → ω_A is an isomorphism and consequently there is a canonical map f^*ω_A → ω_X.","statement_latex":"In Situation \\ref{situation-rational} assume $A$ has a\ndualizing complex $\\omega_A^\\bullet$. With $\\omega_X$ the dualizing\nmodule of $X$, the trace map $H^0(X, \\omega_X) \\to \\omega_A$ is an\nisomorphism and consequently there is a canonical map\n$f^*\\omega_A \\to \\omega_X$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Rational singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBU","source_file":"resolve.tex","source_line":2400,"source_end_line":2407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2400-L2407","statement_sha256":"1070d2eb4c3416e9066705dddf7493dd1f7e3f413e5ff78d050cc7908f492604","origin":"The Stacks Project","memory_eligible":false,"source_rank":9457,"rank":9457,"depth":66,"x":364.072,"y":1077.298,"cluster":"varieties-curves"},{"id":"stacks:0B64","tag":"0B64","title":"Rational singularities · Lemma 0B64","summary":"In Situation [Tag 0B4W] assume A has a dualizing complex ω_A^bullet and is not regular. Let X be the blowup of Spec(A) in m with exceptional divisor E ⊂ X. Let ω_X be the dualizing module of X. Then • ω_E = ω_X|_E ⊗ O_E(-1), • H^1(X, ω_X(n)) = 0 for n ≥ 0, • the map f^*ω_A → ω_X of Lemma [Tag 0BBU] is surjective.","statement_latex":"In Situation \\ref{situation-rational} assume $A$ has a\ndualizing complex $\\omega_A^\\bullet$ and is not regular.\nLet $X$ be the blowup of $\\Spec(A)$ in $\\mathfrak m$ with\nexceptional divisor $E \\subset X$. Let $\\omega_X$\nbe the dualizing module of $X$. Then\n\\begin{enumerate}\n\\item $\\omega_E = \\omega_X|_E \\otimes \\mathcal{O}_E(-1)$,\n\\item $H^1(X, \\omega_X(n)) = 0$ for $n \\geq 0$,\n\\item the map $f^*\\omega_A \\to \\omega_X$ of\nLemma \\ref{lemma-dualizing-rational} is surjective.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Rational singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B64","source_file":"resolve.tex","source_line":2423,"source_end_line":2436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2423-L2436","statement_sha256":"4037d778b3876eb7af7291b06a3c0d2fbbd372c4bd2a04b5efca6bf636cf5d7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9458,"rank":9458,"depth":67,"x":181.506,"y":1262.449,"cluster":"varieties-curves"},{"id":"stacks:0BBV","tag":"0BBV","title":"Rational singularities · Lemma 0BBV","summary":"Let (A, m, kappa) be a local normal Nagata domain of dimension 2 which defines a rational singularity. Assume A has a dualizing complex. Then there exists a finite sequence of blowups in singular closed points X = X_n → X_n - 1 → … → X_1 → X_0 = Spec(A) such that X_i is normal for each i and such that the dualizing sheaf ω_X of X is an invertible O_X-module.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata domain of\ndimension $2$ which defines a rational singularity. Assume $A$ has\na dualizing complex. Then there exists a finite sequence of blowups in\nsingular closed points\n$$\nX = X_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A)\n$$\nsuch that $X_i$ is normal for each $i$ and such that\nthe dualizing sheaf $\\omega_X$ of $X$ is an invertible\n$\\mathcal{O}_X$-module.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Rational singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBV","source_file":"resolve.tex","source_line":2470,"source_end_line":2482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2470-L2482","statement_sha256":"c387f298c848865f5570d60f4459dd40f89b1854d5bc8e9b6f1515c9b2a87879","origin":"The Stacks Project","memory_eligible":false,"source_rank":9459,"rank":9459,"depth":68,"x":167.219,"y":1022.06,"cluster":"varieties-curves"},{"id":"stacks:0BG2","tag":"0BG2","title":"Formal arcs · Lemma 0BG2","summary":"Let X be a locally Noetherian scheme. Let (X, p) = (X_0, p_0) ← (X_1, p_1) ← (X_2, p_2) ← (X_3, p_3) ← … be a sequence of blowups such that • p_i is closed, maps to p_i - 1, and kappa(p_i) = kappa(p_i - 1), • there exists an x_1 ∈ m_p whose image in m_p_i, i > 0 defines the exceptional divisor E_i ⊂ X_i. Then the sequence is obtained from a nonsingular arc a : T → X as above.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let\n$$\n(X, p) = (X_0, p_0) \\leftarrow (X_1, p_1) \\leftarrow (X_2, p_2) \\leftarrow\n(X_3, p_3) \\leftarrow \\ldots\n$$\nbe a sequence of blowups such that\n\\begin{enumerate}\n\\item $p_i$ is closed, maps to $p_{i - 1}$, and\n$\\kappa(p_i) = \\kappa(p_{i - 1})$,\n\\item there exists an $x_1 \\in \\mathfrak m_p$ whose image\nin $\\mathfrak m_{p_i}$, $i > 0$ defines the exceptional divisor\n$E_i \\subset X_i$.\n\\end{enumerate}\nThen the sequence is obtained from a nonsingular arc $a : T \\to X$\nas above.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Formal arcs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BG2","source_file":"resolve.tex","source_line":2585,"source_end_line":2602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2585-L2602","statement_sha256":"8e2cbfdfcfad2225773594fd565a9c8c51c2b84c0ba216e7e68d6a6e99f4409e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9460,"rank":9460,"depth":22,"x":371.296,"y":1191.401,"cluster":"varieties-curves"},{"id":"stacks:0BG3","tag":"0BG3","title":"Formal arcs · Lemma 0BG3","summary":"Let (A, m, kappa) be a Noetherian local domain of dimension 2. Let A → R be a surjection onto a complete discrete valuation ring. This defines a nonsingular arc a : T = Spec(R) → Spec(A). Let Spec(A) = X_0 ← X_1 ← X_2 ← X_3 ← … be the sequence of blowing ups constructed from a. If A_ p is a regular local ring where p = Ker(A → R), then for some i the scheme X_i is regular at x_i.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local domain of\ndimension $2$. Let $A \\to R$ be a surjection onto a\ncomplete discrete valuation ring.\nThis defines a nonsingular arc $a : T = \\Spec(R) \\to \\Spec(A)$. Let\n$$\n\\Spec(A) = X_0 \\leftarrow X_1 \\leftarrow X_2 \\leftarrow X_3 \\leftarrow \\ldots\n$$\nbe the sequence of blowing ups constructed from $a$.\nIf $A_\\mathfrak p$ is a regular local ring where\n$\\mathfrak p = \\Ker(A \\to R)$, then\nfor some $i$ the scheme $X_i$ is regular at $x_i$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Formal arcs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BG3","source_file":"resolve.tex","source_line":2679,"source_end_line":2692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2679-L2692","statement_sha256":"9b7e2e15319c099852ee9072539b4b9747b53f1144b97d43ba3f489c2953c9ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":9461,"rank":9461,"depth":0,"x":84.314,"y":1182.319,"cluster":"varieties-curves"},{"id":"stacks:0BG5","tag":"0BG5","title":"Base change to the completion · Lemma 0BG5","summary":"Let (A, m, kappa) be a local ring with finitely generated maximal ideal m. Let X be a scheme over A. Let Y = X ×_Spec(A) Spec(A^wedge) where A^wedge is the m-adic completion of A. For a point q ∈ Y with image p ∈ X lying over the closed point of Spec(A) the local ring map O_X, p → O_Y, q induces an isomorphism on completions.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local ring with finitely generated\nmaximal ideal $\\mathfrak m$. Let $X$ be a scheme over $A$.\nLet $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$ where\n$A^\\wedge$ is the $\\mathfrak m$-adic completion of $A$.\nFor a point $q \\in Y$ with image $p \\in X$ lying\nover the closed point of $\\Spec(A)$ the\nlocal ring map $\\mathcal{O}_{X, p} \\to \\mathcal{O}_{Y, q}$\ninduces an isomorphism on completions.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BG5","source_file":"resolve.tex","source_line":2748,"source_end_line":2758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2748-L2758","statement_sha256":"9e14ad82135193d82b1362b38291a395687174c03c364460db2a1605fabb3d05","origin":"The Stacks Project","memory_eligible":false,"source_rank":9462,"rank":9462,"depth":5,"x":303.476,"y":1026.002,"cluster":"varieties-curves"},{"id":"stacks:0BG6","tag":"0BG6","title":"Base change to the completion · Lemma 0BG6","summary":"Let (A, m, kappa) be a Noetherian local ring. Let X → Spec(A) be a morphism which is locally of finite type. Set Y = X ×_Spec(A) Spec(A^wedge). Let y ∈ Y with image x ∈ X. Then • if O_Y, y is regular, then O_X, x is regular, • if y is in the closed fibre, then O_Y, y is regular ⇔ O_X, x is regular, and • If X is proper over A, then X is regular if and only if Y is regular.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $X \\to \\Spec(A)$ be a morphism which is locally of finite type.\nSet $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$. Let $y \\in Y$ with\nimage $x \\in X$. Then\n\\begin{enumerate}\n\\item if $\\mathcal{O}_{Y, y}$ is regular, then $\\mathcal{O}_{X, x}$\nis regular,\n\\item if $y$ is in the closed fibre, then $\\mathcal{O}_{Y, y}$ is regular\n$\\Leftrightarrow \\mathcal{O}_{X, x}$ is regular, and\n\\item If $X$ is proper over $A$, then $X$ is regular\nif and only if $Y$ is regular.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BG6","source_file":"resolve.tex","source_line":2779,"source_end_line":2793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2779-L2793","statement_sha256":"7837eff610abb7d027be10fc20963363ad3843cbb2cab0f9fbf57164fcdb937d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9463,"rank":9463,"depth":19,"x":267.535,"y":1265.892,"cluster":"varieties-curves"},{"id":"stacks:0AFK","tag":"0AFK","title":"Base change to the completion · Lemma 0AFK","summary":"Let (A, m) be a Noetherian local ring with completion A^wedge. Let U ⊂ Spec(A) and U^wedge ⊂ Spec(A^wedge) be the punctured spectra. If Y → Spec(A^wedge) is a U^wedge-admissible blowup, then there exists a U-admissible blowup X → Spec(A) such that Y = X ×_Spec(A) Spec(A^wedge).","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring with completion $A^\\wedge$.\nLet $U \\subset \\Spec(A)$ and $U^\\wedge \\subset \\Spec(A^\\wedge)$ be the\npunctured spectra. If $Y \\to \\Spec(A^\\wedge)$ is a $U^\\wedge$-admissible\nblowup, then there exists a $U$-admissible blowup $X \\to \\Spec(A)$\nsuch that $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFK","source_file":"resolve.tex","source_line":2811,"source_end_line":2818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2811-L2818","statement_sha256":"4a529cfc4e0a6b1485aad541e535902abe039a525bca95ade8c16f8ad4cc9547","origin":"The Stacks Project","memory_eligible":false,"source_rank":9464,"rank":9464,"depth":15,"x":100.941,"y":1068.389,"cluster":"varieties-curves"},{"id":"stacks:0BG7","tag":"0BG7","title":"Base change to the completion · Lemma 0BG7","summary":"Let (A, m, kappa) be a Nagata local normal domain of dimension 2. Assume A defines a rational singularity and that the completion A^wedge of A is normal. Then • A^wedge defines a rational singularity, and • if X → Spec(A) is the blowing up in m, then for a closed point x ∈ X the completion O_X, x is normal.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Nagata local normal domain of\ndimension $2$. Assume $A$ defines a rational singularity and that\nthe completion $A^\\wedge$ of $A$ is normal. Then\n\\begin{enumerate}\n\\item $A^\\wedge$ defines a rational singularity, and\n\\item if $X \\to \\Spec(A)$ is the blowing up in $\\mathfrak m$, then\nfor a closed point $x \\in X$ the completion $\\mathcal{O}_{X, x}$ is normal.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BG7","source_file":"resolve.tex","source_line":2835,"source_end_line":2845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2835-L2845","statement_sha256":"9bfc9bd0c5f7206c04f64ce97f74a55b3c86b00f00e9dcfffe2fac6b99091e5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9465,"rank":9465,"depth":60,"x":382.923,"y":1119.551,"cluster":"varieties-curves"},{"id":"stacks:0BG8","tag":"0BG8","title":"Base change to the completion · Lemma 0BG8","summary":"Let (A, m) be a local Noetherian ring. Let X be a scheme over A. Assume • A is analytically unramified (Algebra, Definition [Tag 032X]), • X is locally of finite type over A, and • X → Spec(A) is étale at the generic points of irreducible components of X. Then the normalization of X is finite over X.","statement_latex":"Let $(A, \\mathfrak m)$ be a local Noetherian ring. Let\n$X$ be a scheme over $A$. Assume\n\\begin{enumerate}\n\\item $A$ is analytically unramified\n(Algebra, Definition \\ref{algebra-definition-analytically-unramified}),\n\\item $X$ is locally of finite type over $A$, and\n\\item $X \\to \\Spec(A)$ is \\'etale at the generic points of irreducible\ncomponents of $X$.\n\\end{enumerate}\nThen the normalization of $X$ is finite over $X$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BG8","source_file":"resolve.tex","source_line":2895,"source_end_line":2907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2895-L2907","statement_sha256":"c7b2b4e7c19ac1ce4f2049881f1217d0887b17c6d978162bead1addb3395fe78","origin":"The Stacks Project","memory_eligible":false,"source_rank":9466,"rank":9466,"depth":38,"x":133.576,"y":1241.963,"cluster":"varieties-curves"},{"id":"stacks:0BG9","tag":"0BG9","title":"Base change to the completion · Lemma 0BG9","summary":"Let (A, m, kappa) be a Noetherian local ring. Let X → Spec(A) be a morphism which is locally of finite type. Set Y = X ×_Spec(A) Spec(A^wedge). If the complement of the special fibre in Y is normal, then the normalization X^ν → X is finite and the base change of X^ν to Spec(A^wedge) recovers the normalization of Y.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $X \\to \\Spec(A)$ be a morphism which is locally of finite type.\nSet $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$.\nIf the complement of the special fibre in $Y$ is normal, then\nthe normalization $X^\\nu \\to X$ is finite and the base change\nof $X^\\nu$ to $\\Spec(A^\\wedge)$ recovers the normalization of $Y$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BG9","source_file":"resolve.tex","source_line":2972,"source_end_line":2980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L2972-L2980","statement_sha256":"565b43673459d8ea1ae7b57d8a532f9e7fa3b16181a73827648dc6c4db6e63ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":9467,"rank":9467,"depth":31,"x":219.094,"y":1009.958,"cluster":"varieties-curves"},{"id":"stacks:0BGA","tag":"0BGA","title":"Base change to the completion · Lemma 0BGA","summary":"Let (A, m, kappa) be a Noetherian local domain whose completion A^wedge is normal. Then given any sequence Y_n → Y_n - 1 → … → Y_1 → Spec(A^wedge) of normalized blowups, there exists a sequence of (proper) normalized blowups X_n → X_n - 1 → … → X_1 → Spec(A) whose base change to A^wedge recovers the given sequence.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local domain whose completion\n$A^\\wedge$ is normal. Then given any sequence\n$$\nY_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to \\Spec(A^\\wedge)\n$$\nof normalized blowups, there exists a sequence of (proper) normalized blowups\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to \\Spec(A)\n$$\nwhose base change to $A^\\wedge$ recovers the given sequence.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGA","source_file":"resolve.tex","source_line":3018,"source_end_line":3030,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3018-L3030","statement_sha256":"38e03189599720690ea82089204491ddb6451af62c17723f9d49c9bad768c020","origin":"The Stacks Project","memory_eligible":false,"source_rank":9468,"rank":9468,"depth":32,"x":342.741,"y":1229.801,"cluster":"varieties-curves"},{"id":"stacks:0BGD","tag":"0BGD","title":"Rational double points · Lemma 0BGD","summary":"Let kappa be a field. Let I ⊂ kappa[x, y] be an ideal. Let a + b x + c y + d x^2 + exy + f y^2 ∈ I^2 for some a, b, c, d, e, f ∈ k not all zero. If the colength of I in kappa[x, y] is > 1, then a + b x + c y + d x^2 + exy + f y^2 = j(g + hx + iy)^2 for some j, g, h, i ∈ kappa.","statement_latex":"Let $\\kappa$ be a field. Let $I \\subset \\kappa[x, y]$ be an ideal. Let\n$$\na + b x + c y + d x^2 + exy + f y^2 \\in I^2\n$$\nfor some $a, b, c, d, e, f \\in k$ not all zero. If the colength\nof $I$ in $\\kappa[x, y]$ is $> 1$, then\n$a + b x + c y + d x^2 + exy + f y^2 = j(g + hx + iy)^2$\nfor some $j, g, h, i \\in \\kappa$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Rational double points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGD","source_file":"resolve.tex","source_line":3098,"source_end_line":3108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3098-L3108","statement_sha256":"f2714feeb7c2b51aff0bf7e12d765507d335b32b091ec5cba93575b771ff30e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9469,"rank":9469,"depth":3,"x":74.481,"y":1137.752,"cluster":"varieties-curves"},{"id":"stacks:0BGE","tag":"0BGE","title":"Rational double points · Lemma 0BGE","summary":"Let (A, m, kappa) be a local normal Nagata domain of dimension 2 which defines a rational singularity, whose completion is normal, and which is Gorenstein. Then there exists a finite sequence of blowups in singular closed points X_n → X_n - 1 → … → X_1 → X_0 = Spec(A) such that X_n is regular and such that each intervening schemes X_i is normal with finitely many singular points of the same type.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local normal Nagata\ndomain of dimension $2$ which defines a rational singularity,\nwhose completion is normal, and which is Gorenstein.\nThen there exists a finite sequence of blowups in\nsingular closed points\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = \\Spec(A)\n$$\nsuch that $X_n$ is regular and such that each intervening\nschemes $X_i$ is normal with finitely many singular points\nof the same type.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Rational double points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGE","source_file":"resolve.tex","source_line":3498,"source_end_line":3511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3498-L3511","statement_sha256":"15ffb31d7d77062d15a890412b5a0fa7b70e4cb87e6722aa55b546295e981794","origin":"The Stacks Project","memory_eligible":false,"source_rank":9470,"rank":9470,"depth":0,"x":346.612,"y":1053.319,"cluster":"varieties-curves"},{"id":"stacks:0BGG","tag":"0BGG","title":"Implied properties · Lemma 0BGG","summary":"Let Y be a Noetherian integral scheme. Assume there exists an alteration f : X → Y with X regular. Then the normalization Y^ν → Y is finite and Y has a dense open which is regular.","statement_latex":"Let $Y$ be a Noetherian integral scheme. Assume there exists an alteration\n$f : X \\to Y$ with $X$ regular. Then the normalization $Y^\\nu \\to Y$\nis finite and $Y$ has a dense open which is regular.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Implied properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGG","source_file":"resolve.tex","source_line":3529,"source_end_line":3534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3529-L3534","statement_sha256":"62e986b90d136f6a0ab3caf25d8b42ff6944226033c9f7c237a582b8c4a9e542","origin":"The Stacks Project","memory_eligible":false,"source_rank":9471,"rank":9471,"depth":32,"x":213.701,"y":1270.228,"cluster":"varieties-curves"},{"id":"stacks:0BGH","tag":"0BGH","title":"Implied properties · Lemma 0BGH","summary":"Let (A, m) be a local Noetherian ring. Let B ⊂ C be finite A-algebras. Assume that (a) B is a normal ring, and (b) the m-adic completion C^wedge is a normal ring. Then B^wedge is a normal ring.","statement_latex":"Let $(A, \\mathfrak m)$ be a local Noetherian ring. Let $B \\subset C$\nbe finite $A$-algebras. Assume that (a) $B$ is a normal ring, and\n(b) the $\\mathfrak m$-adic completion $C^\\wedge$ is a normal ring.\nThen $B^\\wedge$ is a normal ring.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Implied properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGH","source_file":"resolve.tex","source_line":3558,"source_end_line":3564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3558-L3564","statement_sha256":"4613dcc67ca7fae430a17f3519e6c2aae250310aa7b7f97b85898d327a48ae3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9472,"rank":9472,"depth":19,"x":137.193,"y":1034.609,"cluster":"varieties-curves"},{"id":"stacks:0BGI","tag":"0BGI","title":"Implied properties · Lemma 0BGI","summary":"Let (A, m, kappa) be a local Noetherian domain. Assume there exists an alteration f : X → Spec(A) with X regular. Then • there exists a nonzero f ∈ A such that A_f is regular, • the integral closure B of A in its fraction field is finite over A, • the m-adic completion of B is a normal ring, i.e., the completions of B at its maximal ideals are normal domains, and • the generic formal fibre of A is regular.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local Noetherian domain.\nAssume there exists an alteration $f : X \\to \\Spec(A)$\nwith $X$ regular. Then\n\\begin{enumerate}\n\\item there exists a nonzero $f \\in A$ such that $A_f$ is regular,\n\\item the integral closure $B$ of $A$ in its fraction field is finite over $A$,\n\\item the $\\mathfrak m$-adic completion of $B$ is a normal ring, i.e., the\ncompletions of $B$ at its maximal ideals are normal domains, and\n\\item the generic formal fibre of $A$ is regular.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Implied properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGI","source_file":"resolve.tex","source_line":3609,"source_end_line":3621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3609-L3621","statement_sha256":"acc412734b740ec7d51049faa54a32c34e32578eff1ab2cd99ab2fc9af0eabd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9473,"rank":9473,"depth":33,"x":383.353,"y":1165.079,"cluster":"varieties-curves"},{"id":"stacks:0BGK","tag":"0BGK","title":"Resolution · Definition 0BGK","summary":"Let Y be a Noetherian integral scheme. A resolution of singularities of Y is a modification f : X → Y such that X is regular.","statement_latex":"Let $Y$ be a Noetherian integral scheme. A {\\it resolution of singularities}\nof $Y$ is a modification $f : X \\to Y$ such that $X$ is regular.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Resolution","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGK","source_file":"resolve.tex","source_line":3701,"source_end_line":3705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3701-L3705","statement_sha256":"268c8efd4833ea37e19a6c942dc6da2f5763f7a0da565221126b20ee67145d6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9474,"rank":9474,"depth":0,"x":96.608,"y":1208.597,"cluster":"varieties-curves"},{"id":"stacks:0BGL","tag":"0BGL","title":"Resolution · Definition 0BGL","summary":"Let Y be a 2-dimensional Noetherian integral scheme. We say Y has a resolution of singularities by normalized blowups if there exists a sequence Y_n → Y_n - 1 → … → Y_1 → Y_0 → Y where • Y_i is proper over Y for i = 0, …, n, • Y_0 → Y is the normalization, • Y_i → Y_i - 1 is a normalized blowup for i = 1, …, n, and • Y_n is regular.","statement_latex":"Let $Y$ be a $2$-dimensional Noetherian integral scheme.\nWe say $Y$ has a {\\it resolution of singularities by normalized blowups}\nif there exists a sequence\n$$\nY_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to Y_0 \\to Y\n$$\nwhere\n\\begin{enumerate}\n\\item $Y_i$ is proper over $Y$ for $i = 0, \\ldots, n$,\n\\item $Y_0 \\to Y$ is the normalization,\n\\item $Y_i \\to Y_{i - 1}$ is a normalized blowup for $i = 1, \\ldots, n$, and\n\\item $Y_n$ is regular.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Resolution","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGL","source_file":"resolve.tex","source_line":3710,"source_end_line":3725,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3710-L3725","statement_sha256":"bcf9946ede290a114b24466dbc01ec30311742d8478060418ff5a5680ed78d01","origin":"The Stacks Project","memory_eligible":false,"source_rank":9475,"rank":9475,"depth":0,"x":273.244,"y":1013.592,"cluster":"varieties-curves"},{"id":"stacks:0BGM","tag":"0BGM","title":"Resolution · Lemma 0BGM","summary":"Let (A, m, kappa) be a Noetherian local ring. Assume A is normal and has dimension 2. If Spec(A) has a resolution of singularities, then Spec(A) has a resolution by normalized blowups.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nAssume $A$ is normal and has dimension $2$.\nIf $\\Spec(A)$ has a resolution of singularities,\nthen $\\Spec(A)$ has a resolution by normalized blowups.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGM","source_file":"resolve.tex","source_line":3732,"source_end_line":3738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3732-L3738","statement_sha256":"475437f455698fe7eff63ae9166161ab61ce6c06cf575830e0b34f632d063f4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9476,"rank":9476,"depth":69,"x":299.842,"y":1257.876,"cluster":"varieties-curves"},{"id":"stacks:0BGN","tag":"0BGN","title":"Resolution · Lemma 0BGN","summary":"Let (A, m, kappa) be a Noetherian complete local ring. Assume A is a normal domain of dimension 2. Then Spec(A) has a resolution of singularities.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian complete local ring.\nAssume $A$ is a normal domain of dimension $2$. Then $\\Spec(A)$ has a\nresolution of singularities.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGN","source_file":"resolve.tex","source_line":3833,"source_end_line":3838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3833-L3838","statement_sha256":"b025c786c16aea56f5a130253e03533dd98a608e9c3df5881355ccea7545fd80","origin":"The Stacks Project","memory_eligible":false,"source_rank":9477,"rank":9477,"depth":70,"x":83.55,"y":1092.659,"cluster":"varieties-curves"},{"id":"stacks:0BGP","tag":"0BGP","title":"Lipman · Theorem 0BGP","summary":"[Lipman] Let Y be a two dimensional integral Noetherian scheme. The following are equivalent • there exists an alteration X → Y with X regular, • there exists a resolution of singularities of Y, • Y has a resolution of singularities by normalized blowups, • the normalization Y^ν → Y is finite, Y^ν has finitely many singular points y_1, …, y_m, and for each y_i the completion of O_Y^ν, y_i is normal.","statement_latex":"\\begin{reference}\n\\cite[Theorem on page 151]{Lipman}\n\\end{reference}\nLet $Y$ be a two dimensional integral Noetherian scheme. The following are\nequivalent\n\\begin{enumerate}\n\\item there exists an alteration $X \\to Y$ with $X$ regular,\n\\item there exists a resolution of singularities of $Y$,\n\\item $Y$ has a resolution of singularities by normalized blowups,\n\\item the normalization $Y^\\nu \\to Y$ is finite, $Y^\\nu$ has\nfinitely many singular points $y_1, \\ldots, y_m$, and for each\n$y_i$ the completion of $\\mathcal{O}_{Y^\\nu, y_i}$ is normal.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Resolution","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGP","source_file":"resolve.tex","source_line":3944,"source_end_line":3959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L3944-L3959","statement_sha256":"833a3500456703ccda1dcbef583977c0762be69dcc2087209cee47100f1da531","origin":"The Stacks Project","memory_eligible":false,"source_rank":9478,"rank":9478,"depth":71,"x":376.216,"y":1091.758,"cluster":"varieties-curves"},{"id":"stacks:0BI4","tag":"0BI4","title":"Embedded resolution · Lemma 0BI4","summary":"Let Y be a one dimensional integral Noetherian scheme. The following are equivalent • there exists an alteration X → Y with X regular, • there exists a resolution of singularities of Y, • there exists a finite sequence Y_n → Y_n - 1 → … → Y_1 → Y of blowups in closed points with Y_n regular, and • the normalization Y^ν → Y is finite.","statement_latex":"Let $Y$ be a one dimensional integral Noetherian scheme.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists an alteration $X \\to Y$ with $X$ regular,\n\\item there exists a resolution of singularities of $Y$,\n\\item there exists a finite sequence\n$Y_n \\to Y_{n - 1} \\to \\ldots \\to Y_1 \\to Y$ of blowups\nin closed points with $Y_n$ regular, and\n\\item the normalization $Y^\\nu \\to Y$ is finite.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Embedded resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BI4","source_file":"resolve.tex","source_line":4030,"source_end_line":4042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4030-L4042","statement_sha256":"dadd887b66120d7c4e4af483974cfdb1fa925d0f6051111d8e0a35f46b135752","origin":"The Stacks Project","memory_eligible":false,"source_rank":9479,"rank":9479,"depth":42,"x":160.904,"y":1258.667,"cluster":"varieties-curves"},{"id":"stacks:0BI5","tag":"0BI5","title":"Embedded resolution · Lemma 0BI5","summary":"Let X be a Noetherian scheme. Let Y ⊂ X be an integral closed subscheme of dimension 1 satisfying the equivalent conditions of Lemma [Tag 0BI4]. Then there exists a finite sequence X_n → X_n - 1 → … → X_1 → X of blowups in closed points such that the strict transform of Y in X_n is a regular curve.","statement_latex":"Let $X$ be a Noetherian scheme. Let $Y \\subset X$ be an integral closed\nsubscheme of dimension $1$ satisfying the equivalent conditions of\nLemma \\ref{lemma-resolve-curve}. Then there exists a finite sequence\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X\n$$\nof blowups in closed points such that the strict transform of $Y$\nin $X_n$ is a regular curve.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Embedded resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BI5","source_file":"resolve.tex","source_line":4089,"source_end_line":4099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4089-L4099","statement_sha256":"c5f0e9c7f0c399c0e7410ad33de005df27ab65729f2f0da315238844fe9d2c97","origin":"The Stacks Project","memory_eligible":false,"source_rank":9480,"rank":9480,"depth":43,"x":185.475,"y":1013.155,"cluster":"varieties-curves"},{"id":"stacks:0BI7","tag":"0BI7","title":"Embedded resolution · Lemma 0BI7","summary":"In the situation above let X' → X be the blowing up of X in p. Let Y', Z' ⊂ X' be the strict transforms of Y, Z. If O_Y, p is regular, then • Y' → Y is an isomorphism, • Y' meets the exceptional fibre E ⊂ X' in one point q and m_q(Y ∩ E) = 1, • if q ∈ Z' too, then m_q(Y ∩ Z') < m_p(Y ∩ Z).","statement_latex":"In the situation above let $X' \\to X$ be the blowing up of $X$ in $p$.\nLet $Y', Z' \\subset X'$ be the strict transforms of $Y, Z$.\nIf $\\mathcal{O}_{Y, p}$ is regular, then\n\\begin{enumerate}\n\\item $Y' \\to Y$ is an isomorphism,\n\\item $Y'$ meets the exceptional fibre $E \\subset X'$ in one point\n$q$ and $m_q(Y \\cap E) = 1$,\n\\item if $q \\in Z'$ too, then $m_q(Y \\cap Z') < m_p(Y \\cap Z)$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Embedded resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BI7","source_file":"resolve.tex","source_line":4128,"source_end_line":4139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4128-L4139","statement_sha256":"93bc08d7bbedd4328fd195d9689b9b70f102e2a6a6fc4d02650b563853ef2ab1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9481,"rank":9481,"depth":22,"x":364.981,"y":1208.341,"cluster":"varieties-curves"},{"id":"stacks:0BI8","tag":"0BI8","title":"Embedded resolution · Lemma 0BI8","summary":"Let X be a Noetherian scheme. Let Y_i ⊂ X, i = 1, …, n be an integral closed subschemes of dimension 1 each satisfying the equivalent conditions of Lemma [Tag 0BI4]. Then there exists a finite sequence X_n → X_n - 1 → … → X_1 → X of blowups in closed points such that the strict transform Y'_i ⊂ X_n of Y_i in X_n are pairwise disjoint regular curves.","statement_latex":"Let $X$ be a Noetherian scheme. Let $Y_i \\subset X$, $i = 1, \\ldots, n$\nbe an integral closed subschemes of dimension $1$ each satisfying the\nequivalent conditions of Lemma \\ref{lemma-resolve-curve}. Then there\nexists a finite sequence\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X\n$$\nof blowups in closed points such that the strict transform $Y'_i \\subset X_n$\nof $Y_i$ in $X_n$ are pairwise disjoint regular curves.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Embedded resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BI8","source_file":"resolve.tex","source_line":4177,"source_end_line":4188,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4177-L4188","statement_sha256":"177433fdf122e59493c7eb278d5d3a3519556bd612395d99a077b6472f36ee4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9482,"rank":9482,"depth":44,"x":75.345,"y":1166.225,"cluster":"varieties-curves"},{"id":"stacks:0BIB","tag":"0BIB","title":"Embedded resolution · Lemma 0BIB","summary":"Let X be a regular scheme of dimension 2. Let Z ⊂ X be a proper closed subscheme. There exists a sequence X_n → … → X_1 → X of blowing ups in closed points such that the inverse image Z_n of Z in X_n is an effective Cartier divisor.","statement_latex":"Let $X$ be a regular scheme of dimension $2$. Let $Z \\subset X$\nbe a proper closed subscheme. There exists a sequence\n$$\nX_n \\to \\ldots \\to X_1 \\to X\n$$\nof blowing ups in closed points such that the inverse image $Z_n$ of $Z$\nin $X_n$ is an effective Cartier divisor.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Embedded resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIB","source_file":"resolve.tex","source_line":4206,"source_end_line":4215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4206-L4215","statement_sha256":"61f06286b1177d340899beeee1ad2cc785e98ae7864a47453f70759774f7ec40","origin":"The Stacks Project","memory_eligible":false,"source_rank":9483,"rank":9483,"depth":33,"x":323.049,"y":1032.794,"cluster":"varieties-curves"},{"id":"stacks:0BIC","tag":"0BIC","title":"Embedded resolution · Lemma 0BIC","summary":"Let X be a regular scheme of dimension 2. Let Z ⊂ X be a proper closed subscheme such that every irreducible component Y ⊂ Z of dimension 1 satisfies the equivalent conditions of Lemma [Tag 0BI4]. Then there exists a sequence X_n → … → X_1 → X of blowups in closed points such that the inverse image Z_n of Z in X_n is an effective Cartier divisor supported on a strict normal crossings divisor.","statement_latex":"Let $X$ be a regular scheme of dimension $2$. Let $Z \\subset X$\nbe a proper closed subscheme such that every irreducible component\n$Y \\subset Z$ of dimension $1$ satisfies the equivalent conditions of\nLemma \\ref{lemma-resolve-curve}. Then there exists a sequence\n$$\nX_n \\to \\ldots \\to X_1 \\to X\n$$\nof blowups in closed points such that the inverse image $Z_n$ of $Z$\nin $X_n$ is an effective Cartier divisor supported on a strict normal crossings\ndivisor.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Embedded resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BIC","source_file":"resolve.tex","source_line":4230,"source_end_line":4242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4230-L4242","statement_sha256":"217188fb9e96462d4d472c64b6d18daa5df1ad29570c1d1d494dd15a320c4afa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9484,"rank":9484,"depth":45,"x":247.618,"y":1271.99,"cluster":"varieties-curves"},{"id":"stacks:0C5J","tag":"0C5J","title":"Contracting exceptional curves · Lemma 0C5J","summary":"Let X be a Noetherian scheme. Let E ⊂ X be an exceptional curve of the first kind. If a contraction X → X' of E exists, then it has the following universal property: for every morphism φ : X → Y such that φ(E) is a point, there is a unique factorization X → X' → Y of φ.","statement_latex":"Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an\nexceptional curve of the first kind. If a contraction $X \\to X'$\nof $E$ exists, then it has the following universal property:\nfor every morphism $\\varphi : X \\to Y$ such that $\\varphi(E)$\nis a point, there is a unique factorization\n$X \\to X' \\to Y$ of $\\varphi$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5J","source_file":"resolve.tex","source_line":4321,"source_end_line":4329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4321-L4329","statement_sha256":"16845e7ed385cad96de09a7aaf928b1254142ea115a6dc15c873ab5dac4f27e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9485,"rank":9485,"depth":31,"x":110.74,"y":1052.577,"cluster":"varieties-curves"},{"id":"stacks:0C5K","tag":"0C5K","title":"Contracting exceptional curves · Lemma 0C5K","summary":"Let X be a Noetherian scheme. Let E ⊂ X be an exceptional curve of the first kind. If there exists a contraction of E, then it is unique up to unique isomorphism.","statement_latex":"Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an\nexceptional curve of the first kind. If there exists a contraction\nof $E$, then it is unique up to unique isomorphism.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5K","source_file":"resolve.tex","source_line":4374,"source_end_line":4379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4374-L4379","statement_sha256":"ac709191022ab59795a05d537a645ef0ebbd794bd72835edcd44cf2419904a2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9486,"rank":9486,"depth":32,"x":388.41,"y":1136.79,"cluster":"varieties-curves"},{"id":"stacks:0C2K","tag":"0C2K","title":"Contracting exceptional curves · Lemma 0C2K","summary":"Let X be a Noetherian scheme. Let E ⊂ X be an exceptional curve of the first kind. Let E_n = nE and denote O_n its structure sheaf. Then A = lim H^0(E_n, O_n) is a complete local Noetherian regular local ring of dimension 2 and Ker(A → H^0(E_n, O_n)) is the nth power of its maximal ideal.","statement_latex":"Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an\nexceptional curve of the first kind. Let $E_n = nE$ and\ndenote $\\mathcal{O}_n$ its structure sheaf. Then\n$$\nA = \\lim H^0(E_n, \\mathcal{O}_n)\n$$\nis a complete local Noetherian regular local ring of dimension $2$\nand $\\Ker(A \\to H^0(E_n, \\mathcal{O}_n))$ is the $n$th power of\nits maximal ideal.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2K","source_file":"resolve.tex","source_line":4386,"source_end_line":4397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4386-L4397","statement_sha256":"7a03998d7271154daa6883176e4119faef9aa2bff056b050b54097ee6b057518","origin":"The Stacks Project","memory_eligible":false,"source_rank":9487,"rank":9487,"depth":10,"x":115.653,"y":1232.349,"cluster":"varieties-curves"},{"id":"stacks:0C2L","tag":"0C2L","title":"Contracting exceptional curves · Lemma 0C2L","summary":"Let X be a Noetherian scheme. Let E ⊂ X be an exceptional curve of the first kind. If there exists a morphism f : X → Y such that • Y is Noetherian, • f is proper, • f maps E to a point y of Y, • f is quasi-finite at every point not in E, Then there exists a contraction of E and it is the Stein factorization of f.","statement_latex":"Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an\nexceptional curve of the first kind. If there exists a morphism\n$f : X \\to Y$ such that\n\\begin{enumerate}\n\\item $Y$ is Noetherian,\n\\item $f$ is proper,\n\\item $f$ maps $E$ to a point $y$ of $Y$,\n\\item $f$ is quasi-finite at every point not in $E$,\n\\end{enumerate}\nThen there exists a contraction of $E$ and it is the Stein\nfactorization of $f$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2L","source_file":"resolve.tex","source_line":4460,"source_end_line":4473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4460-L4473","statement_sha256":"1258addadc1a3fd800ed038dfab865248524603df738f5fe89a4224f0f89aef6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9488,"rank":9488,"depth":45,"x":240.064,"y":1006.88,"cluster":"varieties-curves"},{"id":"stacks:0C5L","tag":"0C5L","title":"Contracting exceptional curves · Lemma 0C5L","summary":"Let b : X → X' be the contraction of an exceptional curve of the first kind E ⊂ X. Then there is a short exact sequence 0 → Pic(X') → Pic(X) → Z → 0 where the first map is pullback by b and the second map sends L to the degree of L on the exceptional curve E. The sequence is split by the map n ↦ O_X(-nE).","statement_latex":"Let $b : X \\to X'$ be the contraction of an\nexceptional curve of the first kind $E \\subset X$.\nThen there is a short exact sequence\n$$\n0 \\to \\Pic(X') \\to \\Pic(X) \\to \\mathbf{Z} \\to 0\n$$\nwhere the first map is pullback by $b$ and the second map sends\n$\\mathcal{L}$ to the degree of $\\mathcal{L}$ on the exceptional\ncurve $E$. The sequence is split by the map\n$n \\mapsto \\mathcal{O}_X(-nE)$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5L","source_file":"resolve.tex","source_line":4559,"source_end_line":4571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4559-L4571","statement_sha256":"d83b3837f4a51fdfec92ffe75c26e4a40405699af0ef6d32a77d5ba2e2efc0cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9489,"rank":9489,"depth":31,"x":329.734,"y":1243.979,"cluster":"varieties-curves"},{"id":"stacks:0C2J","tag":"0C2J","title":"Contracting exceptional curves · Lemma 0C2J","summary":"Let X be a Noetherian scheme. Let E ⊂ X be an exceptional curve of the first kind. Let L be an invertible O_X-module. Let n be the integer such that L|_E has degree n viewed as an invertible module on P^1. Then • If H^1(X, L) = 0 and n ≥ 0, then H^1(X, L(iE)) = 0 for 0 ≤ i ≤ n + 1. • If n ≤ 0, then H^1(X, L) ⊂ H^1(X, L(E)).","statement_latex":"Let $X$ be a Noetherian scheme. Let $E \\subset X$ be an\nexceptional curve of the first kind. Let $\\mathcal{L}$ be\nan invertible $\\mathcal{O}_X$-module.\nLet $n$ be the integer such that $\\mathcal{L}|_E$ has degree $n$\nviewed as an invertible module on $\\mathbf{P}^1$. Then\n\\begin{enumerate}\n\\item If $H^1(X, \\mathcal{L}) = 0$ and $n \\geq 0$, then\n$H^1(X, \\mathcal{L}(iE)) = 0$ for $0 \\leq i \\leq n + 1$.\n\\item If $n \\leq 0$, then\n$H^1(X, \\mathcal{L}) \\subset H^1(X, \\mathcal{L}(E))$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2J","source_file":"resolve.tex","source_line":4637,"source_end_line":4650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4637-L4650","statement_sha256":"adce45fd11eb099463b3018e937c06a10836c71f99b106bad2d1cb742cb0d915","origin":"The Stacks Project","memory_eligible":false,"source_rank":9490,"rank":9490,"depth":28,"x":72.676,"y":1119.898,"cluster":"varieties-curves"},{"id":"stacks:0C2M","tag":"0C2M","title":"Contracting exceptional curves · Lemma 0C2M","summary":"Let S = Spec(R) be an affine Noetherian scheme. Let X → S be a proper morphism. Let L be an ample invertible sheaf on X. Let E ⊂ X be an exceptional curve of the first kind. Then • there exists a contraction b : X → X' of E, • X' is proper over S, and • the invertible O_X'-module L' is ample with L' as in Remark [Tag 0C5M].","statement_latex":"Let $S = \\Spec(R)$ be an affine Noetherian scheme.\nLet $X \\to S$ be a proper morphism. Let $\\mathcal{L}$ be an\nample invertible sheaf on $X$. Let $E \\subset X$ be an\nexceptional curve of the first kind. Then\n\\begin{enumerate}\n\\item there exists a contraction $b : X \\to X'$ of $E$,\n\\item $X'$ is proper over $S$, and\n\\item the invertible $\\mathcal{O}_{X'}$-module $\\mathcal{L}'$\nis ample with $\\mathcal{L}'$ as in Remark \\ref{remark-pic-blowup}.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2M","source_file":"resolve.tex","source_line":4665,"source_end_line":4677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4665-L4677","statement_sha256":"f13188a1e4bfe3221f204a90169a8540ec3238c3358aac2a1f7879f19717b481","origin":"The Stacks Project","memory_eligible":false,"source_rank":9491,"rank":9491,"depth":46,"x":362.311,"y":1065.476,"cluster":"varieties-curves"},{"id":"stacks:0C2N","tag":"0C2N","title":"Contracting exceptional curves · Lemma 0C2N","summary":"Let S be a Noetherian scheme. Let f : X → S be a morphism of finite type. Let E ⊂ X be an exceptional curve of the first kind which is in a fibre of f. • If X is projective over S, then there exists a contraction X → X' of E and X' is projective over S. • If X is quasi-projective over S, then there exists a contraction X → X' of E and X' is quasi-projective over S.","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a morphism of finite type.\nLet $E \\subset X$ be an exceptional curve of the first kind which is in a\nfibre of $f$.\n\\begin{enumerate}\n\\item If $X$ is projective over $S$, then there exists a contraction\n$X \\to X'$ of $E$ and $X'$ is projective over $S$.\n\\item If $X$ is quasi-projective over $S$, then there exists a contraction\n$X \\to X'$ of $E$ and $X'$ is quasi-projective over $S$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2N","source_file":"resolve.tex","source_line":4768,"source_end_line":4779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4768-L4779","statement_sha256":"ffd071e4e2416eac5050ed88574ded5b6e61214a9834eae41a13e74479358caf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9492,"rank":9492,"depth":47,"x":192.327,"y":1270.164,"cluster":"varieties-curves"},{"id":"stacks:0C5N","tag":"0C5N","title":"Contracting exceptional curves · Lemma 0C5N","summary":"Let S be a Noetherian scheme. Let f : X → S be a separated morphism of finite type with X regular of dimension 2. Then X is quasi-projective over S.","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a\nseparated morphism of finite type with $X$ regular of dimension $2$.\nThen $X$ is quasi-projective over $S$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5N","source_file":"resolve.tex","source_line":4832,"source_end_line":4837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4832-L4837","statement_sha256":"fad1fd4a6fd0bf09b5d14d1d2b945c1c2f318fc2e74b7dbe66dbf99d3870d307","origin":"The Stacks Project","memory_eligible":false,"source_rank":9493,"rank":9493,"depth":61,"x":153.025,"y":1022.52,"cluster":"varieties-curves"},{"id":"stacks:0C5P","tag":"0C5P","title":"Contracting exceptional curves · Lemma 0C5P","summary":"Let S be a Noetherian scheme. Let f : X → S be a proper morphism with X regular of dimension 2. Then X is projective over S.","statement_latex":"Let $S$ be a Noetherian scheme. Let $f : X \\to S$ be a\nproper morphism with $X$ regular of dimension $2$.\nThen $X$ is projective over $S$.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Contracting exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5P","source_file":"resolve.tex","source_line":4865,"source_end_line":4870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4865-L4870","statement_sha256":"5488f3c0ddc9a1a3c8896d92fc84137669668ec9002b009af4a9086f10aaea5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9494,"rank":9494,"depth":62,"x":381.39,"y":1182.995,"cluster":"varieties-curves"},{"id":"stacks:0C5R","tag":"0C5R","title":"Factorization birational maps · Lemma 0C5R","summary":"Let f : X → Y be a proper birational morphism between integral Noetherian schemes regular of dimension 2. Then f is a sequence of blowups in closed points.","statement_latex":"Let $f : X \\to Y$ be a proper birational morphism between\nintegral Noetherian schemes regular of dimension $2$.\nThen $f$ is a sequence of blowups in closed points.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Factorization birational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5R","source_file":"resolve.tex","source_line":4890,"source_end_line":4895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4890-L4895","statement_sha256":"2ed02f0552d20e3765ff98dfe69f6a0e5152f35c86b4981c7371d50b670f21fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9495,"rank":9495,"depth":61,"x":83.641,"y":1194.255,"cluster":"varieties-curves"},{"id":"stacks:0C5S","tag":"0C5S","title":"Factorization birational maps · Lemma 0C5S","summary":"Let S be a Noetherian scheme. Let X and Y be proper integral schemes over S which are regular of dimension 2. Then X and Y are S-birational if and only if there exists a diagram of S-morphisms X = X_0 ← X_1 ← … ← X_n = Y_m → … → Y_1 → Y_0 = Y where each morphism is a blowup in a closed point.","statement_latex":"Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be proper\nintegral schemes over $S$ which are regular of dimension $2$.\nThen $X$ and $Y$ are $S$-birational if and only if there\nexists a diagram of $S$-morphisms\n$$\nX = X_0 \\leftarrow X_1 \\leftarrow \\ldots \\leftarrow X_n = Y_m\n\\to \\ldots \\to Y_1 \\to Y_0 = Y\n$$\nwhere each morphism is a blowup in a closed point.","area":"Varieties & Curves","chapter":"Resolution of Surfaces","chapter_id":"resolve","section":"Factorization birational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5S","source_file":"resolve.tex","source_line":4974,"source_end_line":4985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/resolve.tex#L4974-L4985","statement_sha256":"22d87a54ef77cce54c8b4e28c979a6808fa84e3ba3e2f17d0159337f1fa2a91c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9496,"rank":9496,"depth":62,"x":294.359,"y":1016.819,"cluster":"varieties-curves"},{"id":"stacks:0C5U","tag":"0C5U","title":"Linear algebra · Lemma 0C5U","summary":"[Taussky] Let A = (a_ij) be a complex n × n matrix. • If |a_ii| > ∑_j not = i |a_ij| for each i, then det(A) is nonzero. • If there exists a real vector m = (m_1, …, m_n) with m_i > 0 such that |a_ii m_i| > ∑_j not = i |a_ijm_j| for each i, then det(A) is nonzero.","statement_latex":"\\begin{reference}\n\\cite[Theorem I]{Taussky}\n\\end{reference}\nLet $A = (a_{ij})$ be a complex $n \\times n$ matrix.\n\\begin{enumerate}\n\\item If $|a_{ii}| > \\sum_{j \\not = i} |a_{ij}|$ for each $i$, then\n$\\det(A)$ is nonzero.\n\\item If there exists a real vector $m = (m_1, \\ldots, m_n)$\nwith $m_i > 0$ such that $|a_{ii} m_i| > \\sum_{j \\not = i} |a_{ij}m_j|$\nfor each $i$, then $\\det(A)$ is nonzero.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Linear algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5U","source_file":"models.tex","source_line":86,"source_end_line":99,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L86-L99","statement_sha256":"3a56e16b1da0e6baa7bea5a0e05771433863f5d416f98a360168ac4b08073b60","origin":"The Stacks Project","memory_eligible":false,"source_rank":9497,"rank":9497,"depth":0,"x":281.654,"y":1267.481,"cluster":"varieties-curves"},{"id":"stacks:0C5V","tag":"0C5V","title":"Linear algebra · Lemma 0C5V","summary":"Let A = (a_ij) be a real n × n matrix with a_ij ≥ 0 for i not = j. Let m = (m_1, …, m_n) be a real vector with m_i > 0. For I ⊂ (1, …, n) let x_I ∈ R^n be the vector whose ith coordinate is m_i if i ∈ I and 0 otherwise. If -a_iim_i ≥ ∑_j not = i a_ijm_j for each i, then Ker(A) is the vector space spanned by the vectors x_I such that • a_ij = 0 for i ∈ I, j not ∈ I, and • equality holds in ([Tag 0C5W]) for i ∈ I.","statement_latex":"Let $A = (a_{ij})$ be a real $n \\times n$ matrix with\n$a_{ij} \\geq 0$ for $i \\not = j$. Let $m = (m_1, \\ldots, m_n)$ be a real\nvector with $m_i > 0$. For $I \\subset \\{1, \\ldots, n\\}$ let\n$x_I \\in \\mathbf{R}^n$\nbe the vector whose $i$th coordinate is $m_i$ if $i \\in I$\nand $0$ otherwise. If\n\\begin{equation}\n\n-a_{ii}m_i \\geq \\sum\\nolimits_{j \\not = i} a_{ij}m_j\n\\end{equation}\nfor each $i$, then $\\Ker(A)$ is the vector space\nspanned by the vectors $x_I$ such that\n\\begin{enumerate}\n\\item $a_{ij} = 0$ for $i \\in I$, $j \\not \\in I$, and\n\\item equality holds in (\\ref{equation-ineq}) for $i \\in I$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Linear algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5V","source_file":"models.tex","source_line":113,"source_end_line":131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L113-L131","statement_sha256":"b6cedb91e5517b905dca6581c8e96f10f7264e24b0e20a98103cf09d57691fd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9498,"rank":9498,"depth":0,"x":89.25,"y":1075.241,"cluster":"varieties-curves"},{"id":"stacks:0C5X","tag":"0C5X","title":"Linear algebra · Lemma 0C5X","summary":"Let A = (a_ij) be a symmetric real n × n matrix with a_ij ≥ 0 for i not = j. Let m = (m_1, …, m_n) be a real vector with m_i > 0. Assume • Am = 0, • there is no proper nonempty subset I ⊂ (1, …, n) such that a_ij = 0 for i ∈ I and j not ∈ I. Then x^t A x ≤ 0 with equality if and only if x = qm for some q ∈ R.","statement_latex":"Let $A = (a_{ij})$ be a symmetric real $n \\times n$ matrix with\n$a_{ij} \\geq 0$ for $i \\not = j$.\nLet $m = (m_1, \\ldots, m_n)$ be a real vector with $m_i > 0$.\nAssume\n\\begin{enumerate}\n\\item $Am = 0$,\n\\item there is no proper nonempty subset $I \\subset \\{1, \\ldots, n\\}$\nsuch that $a_{ij} = 0$ for $i \\in I$ and $j \\not \\in I$.\n\\end{enumerate}\nThen $x^t A x \\leq 0$ with equality if and only if $x = qm$\nfor some $q \\in \\mathbf{R}$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Linear algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5X","source_file":"models.tex","source_line":155,"source_end_line":168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L155-L168","statement_sha256":"0d36e89ebe20bd25e81db5277e5f41c4f53f0d934d45da680c8b2b30166d1d6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9499,"rank":9499,"depth":0,"x":386.024,"y":1107.855,"cluster":"varieties-curves"},{"id":"stacks:0C6V","tag":"0C6V","title":"Linear algebra · Lemma 0C6V","summary":"Let L be a finite free Z-module endowed with an integral symmetric bilinear positive definite form langle , rangle : L × L → Z. Let A ⊂ L be a submodule with L/A torsion free. Set B = (b ∈ L mid langle a, brangle = 0, ∀ a ∈ A). Then we have injective maps A^\\#/A ← L/(A ⊕ B) → B^\\#/B whose cokernels are quotients of L^\\#/L. Here A^\\# = (a' ∈ A ⊗ Q mid langle a, a'rangle ∈ Z, ∀ a ∈ A) and similarly for B and L.","statement_latex":"Let $L$ be a finite free $\\mathbf{Z}$-module endowed\nwith an integral symmetric bilinear positive definite\nform $\\langle\\ ,\\ \\rangle : L \\times L \\to \\mathbf{Z}$.\nLet $A \\subset L$ be a submodule with $L/A$ torsion free. Set\n$B = \\{b \\in L \\mid \\langle a, b\\rangle = 0,\\ \\forall a \\in A\\}$.\nThen we have injective maps\n$$\nA^\\#/A \\leftarrow L/(A \\oplus B) \\rightarrow B^\\#/B\n$$\nwhose cokernels are quotients of $L^\\#/L$. Here\n$A^\\# = \\{a' \\in A \\otimes \\mathbf{Q} \\mid\n\\langle a, a'\\rangle \\in \\mathbf{Z},\\ \\forall a \\in A\\}$\nand similarly for $B$ and $L$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Linear algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6V","source_file":"models.tex","source_line":206,"source_end_line":221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L206-L221","statement_sha256":"219cf0ee54f6811bb0c19eb672b5751ae8c9b09cf6a37b10878bde6e2685f5e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9500,"rank":9500,"depth":0,"x":140.71,"y":1252.35,"cluster":"varieties-curves"},{"id":"stacks:0C6W","tag":"0C6W","title":"Linear algebra · Lemma 0C6W","summary":"Let L_0, L_1 be a finite free Z-modules endowed with integral symmetric bilinear positive definite forms langle , rangle : L_i × L_i → Z. Let d : L_0 → L_1 and d^* : L_1 → L_0 be adjoint. If langle , rangle on L_0 is unimodular, then there is an isomorphism Φ : Coker(d^*d)_torsion → Im(d)^\\#/Im(d) with notation as in Lemma [Tag 0C6V].","statement_latex":"Let $L_0$, $L_1$ be a finite free $\\mathbf{Z}$-modules endowed\nwith integral symmetric bilinear positive definite\nforms $\\langle\\ ,\\ \\rangle : L_i \\times L_i \\to \\mathbf{Z}$.\nLet $\\text{d} : L_0 \\to L_1$ and $\\text{d}^* : L_1 \\to L_0$\nbe adjoint. If $\\langle\\ ,\\ \\rangle$ on $L_0$ is unimodular, then\nthere is an isomorphism\n$$\n\\Phi :\n\\Coker(\\text{d}^*\\text{d})_{torsion}\n\\longrightarrow\n\\Im(\\text{d})^\\#/\\Im(\\text{d})\n$$\nwith notation as in Lemma \\ref{lemma-orthogonal-direct-sum}.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Linear algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6W","source_file":"models.tex","source_line":244,"source_end_line":259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L244-L259","statement_sha256":"5a820aed9762196324983a11e5ae00d775c3a0c0008deba775b18dd2aeb8f527","origin":"The Stacks Project","memory_eligible":false,"source_rank":9501,"rank":9501,"depth":1,"x":205.468,"y":1006.354,"cluster":"varieties-curves"},{"id":"stacks:0C6X","tag":"0C6X","title":"Linear algebra · Lemma 0C6X","summary":"Let A = (a_ij) be a symmetric n × n integer matrix with a_ij ≥ 0 for i not = j. Let m = (m_1, …, m_n) be an integer vector with m_i > 0. Assume • Am = 0, • there is no proper nonempty subset I ⊂ (1, …, n) such that a_ij = 0 for i ∈ I and j not ∈ I. Let e be the number of pairs (i, j) with i < j and a_ij > 0. Then for ℓ a prime number coprime with all a_ij and m_i we have dim_F_ℓ(Coker(A)[ℓ]) ≤ 1 - n + e","statement_latex":"Let $A = (a_{ij})$ be a symmetric $n \\times n$ integer matrix with\n$a_{ij} \\geq 0$ for $i \\not = j$. Let $m = (m_1, \\ldots, m_n)$ be an\ninteger vector with $m_i > 0$. Assume\n\\begin{enumerate}\n\\item $Am = 0$,\n\\item there is no proper nonempty subset $I \\subset \\{1, \\ldots, n\\}$\nsuch that $a_{ij} = 0$ for $i \\in I$ and $j \\not \\in I$.\n\\end{enumerate}\nLet $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$.\nThen for $\\ell$ a prime number coprime with all $a_{ij}$ and $m_i$\nwe have\n$$\n\\dim_{\\mathbf{F}_\\ell}(\\Coker(A)[\\ell]) \\leq 1 - n + e\n$$","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Linear algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6X","source_file":"models.tex","source_line":300,"source_end_line":316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L300-L316","statement_sha256":"265df673b931015fb5c79bf8da1ce137bd239e86875b050cdfb1ce9337c93f1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9502,"rank":9502,"depth":2,"x":355.692,"y":1224.715,"cluster":"varieties-curves"},{"id":"stacks:0C6Z","tag":"0C6Z","title":"Numerical types · Definition 0C6Z","summary":"A numerical type T is given by n, m_i, a_ij, w_i, g_i where n ≥ 1 is an integer and m_i, a_ij, w_i, g_i are integers for 1 ≤ i, j ≤ n subject to the following conditions • m_i > 0, w_i > 0, g_i ≥ 0, • the matrix A = (a_ij) is symmetric and a_ij ≥ 0 for i not = j, • there is no proper nonempty subset I ⊂ (1, …, n) such that a_ij = 0 for i ∈ I, j not ∈ I, • for each i we have ∑_j a_ijm_j = 0, and • w_i | a_ij.","statement_latex":"A {\\it numerical type} $T$ is given by\n$$\nn, m_i, a_{ij}, w_i, g_i\n$$\nwhere $n \\geq 1$ is an integer and $m_i$, $a_{ij}$, $w_i$, $g_i$\nare integers for $1 \\leq i, j \\leq n$ subject to the following conditions\n\\begin{enumerate}\n\\item $m_i > 0$, $w_i > 0$, $g_i \\geq 0$,\n\\item the matrix $A = (a_{ij})$ is symmetric and $a_{ij} \\geq 0$\nfor $i \\not = j$,\n\\item there is no proper nonempty subset $I \\subset \\{1, \\ldots, n\\}$\nsuch that $a_{ij} = 0$ for $i \\in I$, $j \\not \\in I$,\n\\item for each $i$ we have $\\sum_j a_{ij}m_j = 0$, and\n\\item $w_i | a_{ij}$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6Z","source_file":"models.tex","source_line":428,"source_end_line":445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L428-L445","statement_sha256":"de5982f887d3103eb20dbd02f480574d36edea95e98ca1a37446fdba085f15da","origin":"The Stacks Project","memory_eligible":false,"source_rank":9503,"rank":9503,"depth":0,"x":69.028,"y":1148.862,"cluster":"varieties-curves"},{"id":"stacks:0C70","tag":"0C70","title":"Numerical types · Definition 0C70","summary":"We say two numerical types n, m_i, a_ij, w_i, g_i and n', m'_i, a'_ij, w'_i, g'_i are equivalent types if there exists a permutation σ of (1, …, n) such that m_i = m'_σ(i), a_ij = a'_σ(i)σ(j), w_i = w'_σ(i), and g_i = g'_σ(i).","statement_latex":"We say two numerical types $n, m_i, a_{ij}, w_i, g_i$ and\n$n', m'_i, a'_{ij}, w'_i, g'_i$ are {\\it equivalent types} if\nthere exists a permutation $\\sigma$ of $\\{1, \\ldots, n\\}$\nsuch that $m_i = m'_{\\sigma(i)}$, $a_{ij} = a'_{\\sigma(i)\\sigma(j)}$,\n$w_i = w'_{\\sigma(i)}$, and $g_i = g'_{\\sigma(i)}$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C70","source_file":"models.tex","source_line":453,"source_end_line":460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L453-L460","statement_sha256":"dac942558d162af698a61881a2888a620347c29b6662a27c3dcc91121f4bff0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9504,"rank":9504,"depth":0,"x":341.684,"y":1042.027,"cluster":"varieties-curves"},{"id":"stacks:0C71","tag":"0C71","title":"Numerical types · Lemma 0C71","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type. Then the expression g = 1 + ∑ m_i(w_i(g_i - 1) - frac12 a_ii) is an integer.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type. Then the expression\n$$\ng = 1 + \\sum m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})\n$$\nis an integer.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C71","source_file":"models.tex","source_line":465,"source_end_line":472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L465-L472","statement_sha256":"5ce78b26b02f6adb1f6f8ce111f60648cd2060a760bd9dbab2a5c9791ef60e5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9505,"rank":9505,"depth":0,"x":226.43,"y":1275.753,"cluster":"varieties-curves"},{"id":"stacks:0C72","tag":"0C72","title":"Numerical types · Definition 0C72","summary":"We say n, m_i, a_ij, w_i, g_i is a numerical type of genus g if g = 1 + ∑ m_i(w_i(g_i - 1) - frac12 a_ii) is the integer from Lemma [Tag 0C71].","statement_latex":"We say $n, m_i, a_{ij}, w_i, g_i$ is a {\\it numerical type of genus $g$}\nif $g = 1 + \\sum m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$ is the integer\nfrom Lemma \\ref{lemma-genus}.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C72","source_file":"models.tex","source_line":495,"source_end_line":500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L495-L500","statement_sha256":"e81c6d70806527ecccde9903f3654ec112c49eb385fe289ccc337aa74eeaec98","origin":"The Stacks Project","memory_eligible":false,"source_rank":9506,"rank":9506,"depth":1,"x":123.356,"y":1037.769,"cluster":"varieties-curves"},{"id":"stacks:0C73","tag":"0C73","title":"Numerical types · Lemma 0C73","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type of genus g. If n = 1, then a_11 = 0 and g = 1 + m_1w_1(g_1 - 1). Moreover, we can classify all such numerical types as follows • If g < 0, then g_1 = 0 and there are finitely many possible numerical types of genus g with n = 1 corresponding to factorizations m_1w_1 = 1 - g. • If g = 0, then m_1 = 1, w_1 = 1, g_1 = 0 as in Lemma [Tag 0C8S]. • If g = 1, then we conclude g_1 = 1 but m_1, w_1 can be arbitrary positive integers;…","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.\nIf $n = 1$, then $a_{11} = 0$ and $g = 1 + m_1w_1(g_1 - 1)$.\nMoreover, we can classify all such numerical types as follows\n\\begin{enumerate}\n\\item If $g < 0$, then $g_1 = 0$ and there are finitely many possible\nnumerical types of genus $g$ with $n = 1$ corresponding to factorizations\n$m_1w_1 = 1 - g$.\n\\item If $g = 0$, then $m_1 = 1$, $w_1 = 1$, $g_1 = 0$\nas in Lemma \\ref{lemma-genus-zero}.\n\\item If $g = 1$, then we conclude $g_1 = 1$ but $m_1, w_1$ can be arbitrary\npositive integers; this is case\n(\\ref{item-one}) of Lemma \\ref{lemma-genus-one}.\n\\item If $g > 1$, then $g_1 > 1$ and there are finitely many possible\nnumerical types of genus $g$ with $n = 1$ corresponding to\nfactorizations $m_1w_1(g_1 - 1) = g - 1$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C73","source_file":"models.tex","source_line":507,"source_end_line":525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L507-L525","statement_sha256":"36195bd0600b380f25ad8f37ddbc35ec50c390af52c666dc4616346e2219e9fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9507,"rank":9507,"depth":0,"x":391.011,"y":1154.886,"cluster":"varieties-curves"},{"id":"stacks:0C74","tag":"0C74","title":"Numerical types · Lemma 0C74","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type of genus g. If n > 1, then a_ii < 0 for all i.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.\nIf $n > 1$, then $a_{ii} < 0$ for all $i$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C74","source_file":"models.tex","source_line":531,"source_end_line":535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L531-L535","statement_sha256":"f7c21c188faff6693ed98758478443d6b561cb34724ad2b4cb94f182c9c2d37e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9508,"rank":9508,"depth":1,"x":99.171,"y":1220.465,"cluster":"varieties-curves"},{"id":"stacks:0C75","tag":"0C75","title":"Numerical types · Lemma 0C75","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type of genus g. Assume n > 1. If i is such that the contribution m_i(w_i(g_i - 1) - frac12 a_ii) to the genus g is < 0, then g_i = 0 and a_ii = -w_i.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.\nAssume $n > 1$. If $i$ is such that the contribution\n$m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$\nto the genus $g$ is $< 0$, then $g_i = 0$ and $a_{ii} = -w_i$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C75","source_file":"models.tex","source_line":541,"source_end_line":547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L541-L547","statement_sha256":"bbb5a9f32a9c7930b5601c5c782714f47fb638d5793fe081d93619bde63680ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":9509,"rank":9509,"depth":2,"x":261.796,"y":1006.297,"cluster":"varieties-curves"},{"id":"stacks:0C76","tag":"0C76","title":"Numerical types · Definition 0C76","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type. We say i is a (-1)-index if g_i = 0 and a_ii = -w_i.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type.\nWe say $i$ is a {\\it $(-1)$-index} if $g_i = 0$ and $a_{ii} = -w_i$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C76","source_file":"models.tex","source_line":554,"source_end_line":558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L554-L558","statement_sha256":"3fa02e50b75a009bc78b612bb9b76e49319b5f6cc6d52a093d854f1cd8934b96","origin":"The Stacks Project","memory_eligible":false,"source_rank":9510,"rank":9510,"depth":0,"x":314.159,"y":1256.749,"cluster":"varieties-curves"},{"id":"stacks:0C77","tag":"0C77","title":"Numerical types · Lemma 0C77","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type T. Assume n is a (-1)-index. Then there is a numerical type T' given by n', m'_i, a'_ij, w'_i, g'_i with • n' = n - 1, • m'_i = m_i, • a'_ij = a_ij - a_ina_jn/a_nn, • w'_i = w_i/2 if a_in/w_n even and a_in/w_i odd and w'_i = w_i else, • g'_i = fracw_iw'_i(g_i - 1) + 1 + fraca_in^2 - w_na_in2w'_iw_n. Moreover, we have g = g'.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.\nAssume $n$ is a $(-1)$-index. Then there is a numerical\ntype $T'$ given by $n', m'_i, a'_{ij}, w'_i, g'_i$ with\n\\begin{enumerate}\n\\item $n' = n - 1$,\n\\item $m'_i = m_i$,\n\\item $a'_{ij} = a_{ij} - a_{in}a_{jn}/a_{nn}$,\n\\item $w'_i = w_i/2$ if $a_{in}/w_n$ even and $a_{in}/w_i$ odd\nand $w'_i = w_i$ else,\n\\item $g'_i =\n\\frac{w_i}{w'_i}(g_i - 1) + 1 + \\frac{a_{in}^2 - w_na_{in}}{2w'_iw_n}$.\n\\end{enumerate}\nMoreover, we have $g = g'$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C77","source_file":"models.tex","source_line":563,"source_end_line":578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L563-L578","statement_sha256":"d5dbff77e88b4a4f7d7ffb1984e63badad6780bda1e15767fa6bbecbf1cf7d5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9511,"rank":9511,"depth":2,"x":73.901,"y":1101.627,"cluster":"varieties-curves"},{"id":"stacks:0C78","tag":"0C78","title":"Numerical types · Lemma 0C78","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type. Let e be the number of pairs (i, j) with i < j and a_ij > 0. Then the expression g_top = 1 - n + e is ≥ 0.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type.\nLet $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$.\nThen the expression $g_{top} = 1 - n + e$ is $\\geq 0$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C78","source_file":"models.tex","source_line":644,"source_end_line":649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L644-L649","statement_sha256":"e6ea90fb3f60b1f204b826a20a5c56d86e7fd6f47a8d7941890b4f8d29bf2df3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9512,"rank":9512,"depth":1,"x":376.109,"y":1079.662,"cluster":"varieties-curves"},{"id":"stacks:0C79","tag":"0C79","title":"Numerical types · Definition 0C79","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type T. The topological genus of T is the nonnegative integer g_top = 1 - n + e from Lemma [Tag 0C78].","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The\n{\\it topological genus of $T$} is the nonnegative integer\n$g_{top} = 1 - n + e$ from Lemma \\ref{lemma-top-genus}.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C79","source_file":"models.tex","source_line":657,"source_end_line":662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L657-L662","statement_sha256":"be503a09b92eeeab6638fb7384a719bf947f2a71a25c958d4d6ec91b8657a13e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9513,"rank":9513,"depth":2,"x":170.725,"y":1267.525,"cluster":"varieties-curves"},{"id":"stacks:0C7A","tag":"0C7A","title":"Numerical types · Definition 0C7A","summary":"We say the numerical type n, m_i, a_ij, w_i, g_i of genus g is minimal if there does not exist an i with g_i = 0 and a_ii = -w_i, in other words, if there does not exist a (-1)-index.","statement_latex":"We say the numerical type $n, m_i, a_{ij}, w_i, g_i$ of genus $g$\nis {\\it minimal} if there does not exist an $i$\nwith $g_i = 0$ and $a_{ii} = -w_i$, in other words, if there\ndoes not exist a $(-1)$-index.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7A","source_file":"models.tex","source_line":670,"source_end_line":676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L670-L676","statement_sha256":"031ab063fdc72b0bc210b319cae4e1702ab0c5fe7cf045224c70c20848fdf2aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9514,"rank":9514,"depth":0,"x":171.098,"y":1012.205,"cluster":"varieties-curves"},{"id":"stacks:0C7B","tag":"0C7B","title":"Numerical types · Lemma 0C7B","summary":"If n, m_i, a_ij, w_i, g_i is a minimal numerical type with n > 1, then g ≥ 1.","statement_latex":"If $n, m_i, a_{ij}, w_i, g_i$ is a minimal numerical type\nwith $n > 1$, then $g \\geq 1$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7B","source_file":"models.tex","source_line":682,"source_end_line":686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L682-L686","statement_sha256":"de6eb316933fdad5e98b0e7a595d7f3e870f4754aebb6fa465dbe3de25df3507","origin":"The Stacks Project","memory_eligible":false,"source_rank":9515,"rank":9515,"depth":3,"x":376.348,"y":1200.872,"cluster":"varieties-curves"},{"id":"stacks:0C7C","tag":"0C7C","title":"Numerical types · Lemma 0C7C","summary":"If n, m_i, a_ij, w_i, g_i is a minimal numerical type with n > 1, then g ≥ g_top.","statement_latex":"If $n, m_i, a_{ij}, w_i, g_i$ is a minimal numerical type\nwith $n > 1$, then $g \\geq g_{top}$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7C","source_file":"models.tex","source_line":694,"source_end_line":698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L694-L698","statement_sha256":"2ec80b34d0a14b4712ba9572cc64364a17b998b8ca8ebcaff6a2fd2186c0597a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9516,"rank":9516,"depth":4,"x":72.979,"y":1178.193,"cluster":"varieties-curves"},{"id":"stacks:0C7D","tag":"0C7D","title":"Numerical types · Lemma 0C7D","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type of genus g. Assume n > 1. If i is such that the contribution m_i(w_i(g_i - 1) - frac12 a_ii) to the genus g is 0, then g_i = 0 and a_ii = -2w_i.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.\nAssume $n > 1$. If $i$ is such that the contribution\n$m_i(w_i(g_i - 1) - \\frac{1}{2} a_{ii})$\nto the genus $g$ is $0$, then $g_i = 0$ and $a_{ii} = -2w_i$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7D","source_file":"models.tex","source_line":845,"source_end_line":851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L845-L851","statement_sha256":"cbaf01feb4a8ef068eb81ae764d387032193770552d564ab9230ab7f79bb3b87","origin":"The Stacks Project","memory_eligible":false,"source_rank":9517,"rank":9517,"depth":2,"x":315.155,"y":1022.624,"cluster":"varieties-curves"},{"id":"stacks:0C7E","tag":"0C7E","title":"Numerical types · Definition 0C7E","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type of genus g. We say i is a (-2)-index if g_i = 0 and a_ii = -2w_i.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.\nWe say $i$ is a {\\it $(-2)$-index} if $g_i = 0$ and $a_{ii} = -2w_i$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Numerical types","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7E","source_file":"models.tex","source_line":863,"source_end_line":867,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L863-L867","statement_sha256":"0d8aca9a3ddd18c2508d10e93d8fd465c03c54629a116594a976bc22fb735112","origin":"The Stacks Project","memory_eligible":false,"source_rank":9518,"rank":9518,"depth":0,"x":261.629,"y":1275.003,"cluster":"varieties-curves"},{"id":"stacks:0C7H","tag":"0C7H","title":"The Picard group of a numerical type · Definition 0C7H","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type T. The Picard group of T is the cokernel of the matrix (a_ij/w_i), more precisely Pic(T) = Coker( Z^⊕ n → Z^⊕ n, e_i ↦ ∑ fraca_ijw_je_j ) where e_i denotes the ith standard basis vector for Z^⊕ n.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$. The\n{\\it Picard group of $T$} is the cokernel of the matrix\n$(a_{ij}/w_i)$, more precisely\n$$\n\\Pic(T) =\n\\Coker\\left(\n\\mathbf{Z}^{\\oplus n} \\to \\mathbf{Z}^{\\oplus n},\\quad\ne_i\n\\mapsto\n\\sum \\frac{a_{ij}}{w_j}e_j\n\\right)\n$$\nwhere $e_i$ denotes the $i$th standard basis vector for $\\mathbf{Z}^{\\oplus n}$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The Picard group of a numerical type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7H","source_file":"models.tex","source_line":894,"source_end_line":909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L894-L909","statement_sha256":"13db9aa40ea8436622a253535c84167c309f613b19ab45da675b27ff132a5a64","origin":"The Stacks Project","memory_eligible":false,"source_rank":9519,"rank":9519,"depth":0,"x":97.982,"y":1058.318,"cluster":"varieties-curves"},{"id":"stacks:0C7I","tag":"0C7I","title":"The Picard group of a numerical type · Lemma 0C7I","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type T. The Picard group of T is a finitely generated abelian group of rank 1.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.\nThe Picard group of $T$ is a finitely generated abelian group of rank $1$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The Picard group of a numerical type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7I","source_file":"models.tex","source_line":911,"source_end_line":915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L911-L915","statement_sha256":"a2fe3838f3779de70984d2d82c6c54afb77ea2c9a614d5392168f51025c7942c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9520,"rank":9520,"depth":1,"x":393.194,"y":1125.307,"cluster":"varieties-curves"},{"id":"stacks:0CE7","tag":"0CE7","title":"The Picard group of a numerical type · Lemma 0CE7","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type T. Then Pic(T) ⊂ Coker(A) where A = (a_ij).","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.\nThen $\\Pic(T) \\subset \\Coker(A)$ where $A = (a_{ij})$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The Picard group of a numerical type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CE7","source_file":"models.tex","source_line":926,"source_end_line":930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L926-L930","statement_sha256":"2a56f3f9ec06c0373778c88919e87b80fdfc5277fa49cdbffd3803c66e7df46d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9521,"rank":9521,"depth":0,"x":121.373,"y":1243.536,"cluster":"varieties-curves"},{"id":"stacks:0C7J","tag":"0C7J","title":"The Picard group of a numerical type · Lemma 0C7J","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type T. Assume n is a (-1)-index. Let T' be the numerical type constructed in Lemma [Tag 0C77]. There exists an injective map Pic(T) → Pic(T') whose cokernel is an elementary abelian 2-group.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.\nAssume $n$ is a $(-1)$-index. Let $T'$ be the numerical\ntype constructed in Lemma \\ref{lemma-contract}. There exists an\ninjective map\n$$\n\\Pic(T) \\to \\Pic(T')\n$$\nwhose cokernel is an elementary abelian $2$-group.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The Picard group of a numerical type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7J","source_file":"models.tex","source_line":951,"source_end_line":961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L951-L961","statement_sha256":"4680d348c06e628c848703ba35f64b0d22a8405d0320976b967f165335a65cf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9522,"rank":9522,"depth":3,"x":226.836,"y":1001.882,"cluster":"varieties-curves"},{"id":"stacks:0C7K","tag":"0C7K","title":"The Picard group of a numerical type · Lemma 0C7K","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type T. If the genus g of T is ≤ 0, then Pic(T) = Z.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type $T$.\nIf the genus $g$ of $T$ is $\\leq 0$, then $\\Pic(T) = \\mathbf{Z}$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The Picard group of a numerical type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7K","source_file":"models.tex","source_line":1004,"source_end_line":1008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L1004-L1008","statement_sha256":"fa89d767b3689f90b3cf418e808a8f2e8fa1b79fbcfa01e7d6ef0b0ba5b48216","origin":"The Stacks Project","memory_eligible":false,"source_rank":9523,"rank":9523,"depth":4,"x":343.514,"y":1240.148,"cluster":"varieties-curves"},{"id":"stacks:0C7M","tag":"0C7M","title":"Classification of proper subgraphs · Lemma 0C7M","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet If n > 2, then given a pair i, j of (-2)-indices with a_ij > 0, then up to ordering we have the m's, a's, w's • are given by ( m_1 m_2 ), ( -2w & w w & -2w ), ( w w ) with w arbitrary and 2m_1 ≥ m_2 and 2m_2 ≥ m_1, or • are given by ( m_1 m_2 ), ( -2w & 2w 2w & -4w ), ( w 2w ) with w arbitrary and m_1 ≥ m_2 and 2m_2 ≥ m_1, or • are given by ( m_1 m_2 ), ( -2w & 3w 3w & -6w ), ( w 3w ) with w…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet\n}\n$$\nIf $n > 2$, then given a pair $i, j$ of $(-2)$-indices with $a_{ij} > 0$,\nthen up to ordering we have the $m$'s, $a$'s, $w$'s\n\\begin{enumerate}\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w \\\\\nw & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $w$ arbitrary and $2m_1 \\geq m_2$ and $2m_2 \\geq m_1$, or\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 2w \\\\\n2w & -4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n2w\n\\end{matrix}\n\\right)\n$$\nwith $w$ arbitrary and $m_1 \\geq m_2$ and $2m_2 \\geq m_1$, or\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 3w \\\\\n3w & -6w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n3w\n\\end{matrix}\n\\right)\n$$\nwith $w$ arbitrary and $2m_1 \\geq 3m_2$ and $2m_2 \\geq m_1$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7M","source_file":"models.tex","source_line":1124,"source_end_line":1214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L1124-L1214","statement_sha256":"40625ce13ef4c2ff6dd63a03d9e510b6dc48f4e82d456409edb1c32b8c7cf2b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9524,"rank":9524,"depth":0,"x":65.601,"y":1130.554,"cluster":"varieties-curves"},{"id":"stacks:0C7R","tag":"0C7R","title":"Classification of proper subgraphs · Lemma 0C7R","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet If n > 3, then given a triple i, j, k of (-2)-indices with at least two a_ij, a_ik, a_jk nonzero, then up to ordering we have the m's, a's, w's • are given by ( m_1 m_2 m_3 ), ( -2w & w & 0 w & -2w & w 0 & w & -2w ), ( w w w ) with 2m_1 ≥ m_2, 2m_2 ≥ m_1 + m_3, 2m_3 ≥ m_2, or • are given by ( m_1 m_2 m_3 ), ( -2w & w & 0 w & -2w & 2w 0 & 2w & -4w ), ( w w 2w ) with 2m_1 ≥ m_2,…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet\n}\n$$\nIf $n > 3$, then given a triple $i, j, k$ of $(-2)$-indices\nwith at least two $a_{ij}, a_{ik}, a_{jk}$ nonzero, then up\nto ordering we have the $m$'s, $a$'s, $w$'s\n\\begin{enumerate}\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 \\\\\nw & -2w & w \\\\\n0 & w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2$, or\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 \\\\\nw & -2w & 2w \\\\\n0 & 2w & -4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\n2w\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + 2m_3$, $2m_3 \\geq m_2$, or\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-4w & 2w & 0 \\\\\n2w & -4w & 2w \\\\\n0 & 2w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n2w \\\\\n2w \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $m_3 \\geq m_2$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7R","source_file":"models.tex","source_line":1290,"source_end_line":1392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L1290-L1392","statement_sha256":"beac4c12cf97aadd4e36d458adb051d50662b5a8b7b5f8a57f46605e1dfb8c32","origin":"The Stacks Project","memory_eligible":false,"source_rank":9525,"rank":9525,"depth":0,"x":358.946,"y":1053.598,"cluster":"varieties-curves"},{"id":"stacks:0C7V","tag":"0C7V","title":"Classification of proper subgraphs · Lemma 0C7V","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet If n > 4, then given four (-2)-indices i, j, k, l with a_ij, a_jk, a_kl nonzero, then up to ordering we have the m's, a's, w's • are given by ( m_1 m_2 m_3 m_4 ), ( -2w & w & 0 & 0 w & -2w & w & 0 0 & w & -2w & w 0 & 0 & w & -2w ), ( w w w w ) with 2m_1 ≥ m_2, 2m_2 ≥ m_1 + m_3, 2m_3 ≥ m_2 + m_4, and 2m_4 ≥ m_3, or • are given by ( m_1 m_2 m_3 m_4 ), ( -2w & w…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet\n}\n$$\nIf $n > 4$, then given four $(-2)$-indices $i, j, k, l$\nwith $a_{ij}, a_{jk}, a_{kl}$ nonzero, then up\nto ordering we have the $m$'s, $a$'s, $w$'s\n\\begin{enumerate}\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 \\\\\nw & -2w & w & 0 \\\\\n0 & w & -2w & w \\\\\n0 & 0 & w & -2w \n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,\nand $2m_4 \\geq m_3$, or\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 \\\\\nw & -2w & w & 0 \\\\\n0 & w & -2w & 2w \\\\\n0 & 0 & 2w & -4w \n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\n2w\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + 2m_4$,\nand $2m_4 \\geq m_3$, or\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-4w & 2w & 0 & 0 \\\\\n2w & -4w & 2w & 0 \\\\\n0 & 2w & -4w & 2w \\\\\n0 & 0 & 2w & -2w \n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n2w \\\\\n2w \\\\\n2w \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,\nand $m_4 \\geq m_3$, or\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 \\\\\nw & -2w & 2w & 0 \\\\\n0 & 2w & -4w & 2w \\\\\n0 & 0 & 2w & -4w \n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\n2w \\\\\n2w\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + 2m_3$, $2m_3 \\geq m_2 + m_4$,\nand $2m_4 \\geq m_3$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C7V","source_file":"models.tex","source_line":1471,"source_end_line":1619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L1471-L1619","statement_sha256":"143522730aceabce210ea56171e13baa2688c5a94471ad5375d4edae045e64b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9526,"rank":9526,"depth":0,"x":204.374,"y":1277.011,"cluster":"varieties-curves"},{"id":"stacks:0C80","tag":"0C80","title":"Classification of proper subgraphs · Lemma 0C80","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] ar@-[d] & bullet & bullet If n > 4, then given four (-2)-indices i, j, k, l with a_ij, a_ik, a_il nonzero, then up to ordering we have the m's, a's, w's • are given by ( m_1 m_2 m_3 m_4 ), ( -2w & w & w & w w & -2w & 0 & 0 w & 0 & -2w & 0 w & 0 & 0 & -2w ), ( w w w w ) with 2m_1 ≥ m_2 + m_3 + m_4, 2m_2 ≥ m_1, 2m_3 ≥ m_1, 2m_4 ≥ m_1. Observe that this implies m_1 ≥ max(m_2, m_3, m_4).","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\\n& \\bullet\n}\n$$\nIf $n > 4$, then given four $(-2)$-indices $i, j, k, l$\nwith $a_{ij}, a_{ik}, a_{il}$ nonzero, then up\nto ordering we have the $m$'s, $a$'s, $w$'s\n\\begin{enumerate}\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & w & w \\\\\nw & -2w & 0 & 0 \\\\\nw & 0 & -2w & 0 \\\\\nw & 0 & 0 & -2w \n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2 + m_3 + m_4$, $2m_2 \\geq m_1$, $2m_3 \\geq m_1$,\n$2m_4 \\geq m_1$. Observe that this implies $m_1 \\geq \\max(m_2, m_3, m_4)$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C80","source_file":"models.tex","source_line":1667,"source_end_line":1714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L1667-L1714","statement_sha256":"79d98114cf6d4c23cd6451dae9dd7f970714a61b14cc1d65fa752228a4a1368c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9527,"rank":9527,"depth":0,"x":138.627,"y":1024.318,"cluster":"varieties-curves"},{"id":"stacks:0C82","tag":"0C82","title":"Classification of proper subgraphs · Lemma 0C82","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet If n > 5, then given five (-2)-indices h, i, j, k, l with a_hi, a_ij, a_jk, a_kl nonzero, then up to ordering we have the m's, a's, w's • are given by ( m_1 m_2 m_3 m_4 m_5 ), ( -2w & w & 0 & 0 & 0 w & -2w & w & 0 & 0 0 & w & -2w & w & 0 0 & 0 & w & -2w & w 0 & 0 & 0 & w & -2w ), ( w w w w w ) with 2m_1 ≥ m_2, 2m_2 ≥ m_1 + m_3, 2m_3 ≥ m_2 +…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet\n}\n$$\nIf $n > 5$, then given five $(-2)$-indices $h, i, j, k, l$\nwith $a_{hi}, a_{ij}, a_{jk}, a_{kl}$ nonzero, then up\nto ordering we have the $m$'s, $a$'s, $w$'s\n\\begin{enumerate}\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4 \\\\\nm_5\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 \\\\\n0 & w & -2w & w & 0 \\\\\n0 & 0 & w & -2w & w \\\\\n0 & 0 & 0 & w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,\n$2m_4 \\geq m_3 + m_5$, and $2m_5 \\geq m_4$, or\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4 \\\\\nm_5\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 \\\\\n0 & w & -2w & w & 0 \\\\\n0 & 0 & w & -2w & 2w \\\\\n0 & 0 & 0 & 2w & -4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\n2w\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + 2m_4$,\n$2m_4 \\geq m_3 + m_5$, and $2m_5 \\geq m_4$, or\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4 \\\\\nm_5\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-4w & 2w & 0 & 0 & 0 \\\\\n2w & -4w & 2w & 0 & 0 \\\\\n0 & 2w & -4w & 2w & 0 \\\\\n0 & 0 & 2w & -4w & 2w \\\\\n0 & 0 & 0 & 2w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n2w \\\\\n2w \\\\\n2w \\\\\n2w \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,\n$2m_4 \\geq m_3 + m_5$, and $m_4 \\geq m_3$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C82","source_file":"models.tex","source_line":1829,"source_end_line":1954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L1829-L1954","statement_sha256":"118a477bb40dd0dbc83c50f27b506090a4356a554abbeaca5a3fe332e5bb56ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":9528,"rank":9528,"depth":0,"x":390.56,"y":1173.487,"cluster":"varieties-curves"},{"id":"stacks:0C86","tag":"0C86","title":"Classification of proper subgraphs · Lemma 0C86","summary":"Nonexistence of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[ld] ar@-[r] ar@-[d] & bullet bullet & bullet If n > 5, there do not exist five (-2)-indices h, i, j, k with a_hi > 0, a_hj > 0, a_hk > 0, and a_hl > 0.","statement_latex":"Nonexistence of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[ld] \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\\n\\bullet & \\bullet\n}\n$$\nIf $n > 5$, there do {\\bf not} exist five $(-2)$-indices\n$h$, $i$, $j$, $k$ with $a_{hi} > 0$, $a_{hj} > 0$, $a_{hk} > 0$, and\n$a_{hl} > 0$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C86","source_file":"models.tex","source_line":1985,"source_end_line":1997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L1985-L1997","statement_sha256":"591ae0cfe77d82f0e779f7661dc802270eb2f0073a2a0a867948106bc23e8078","origin":"The Stacks Project","memory_eligible":false,"source_rank":9529,"rank":9529,"depth":0,"x":84.536,"y":1206.476,"cluster":"varieties-curves"},{"id":"stacks:0C87","tag":"0C87","title":"Classification of proper subgraphs · Lemma 0C87","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] ar@-[d] & bullet & & bullet If n > 5, then given five (-2)-indices h, i, j, k, l with a_hi, a_ij, a_jk, a_jl nonzero, then up to ordering we have the m's, a's, w's • are given by ( m_1 m_2 m_3 m_4 m_5 ), ( -2w & w & 0 & 0 & 0 w & -2w & w & 0 & 0 0 & w & -2w & w & w 0 & 0 & w & -2w & 0 0 & 0 & w & 0 & -2w ), ( w w w w w ) with 2m_1 ≥ m_2, 2m_2 ≥ m_1 + m_3, 2m_3 ≥ m_2 +…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\\n& & \\bullet\n}\n$$\nIf $n > 5$, then given five $(-2)$-indices $h, i, j, k, l$\nwith $a_{hi}, a_{ij}, a_{jk}, a_{jl}$ nonzero, then up\nto ordering we have the $m$'s, $a$'s, $w$'s\n\\begin{enumerate}\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4 \\\\\nm_5\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 \\\\\n0 & w & -2w & w & w \\\\\n0 & 0 & w & -2w & 0 \\\\\n0 & 0 & w & 0 & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4 + m_5$,\n$2m_4 \\geq m_3$, and $2m_5 \\geq m_3$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C87","source_file":"models.tex","source_line":2034,"source_end_line":2085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2034-L2085","statement_sha256":"7c6e57aab9d8c21a73bed7c574a46453d80e7d70187c1904e8fe4b4b8e909f9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9530,"rank":9530,"depth":0,"x":283.856,"y":1008.316,"cluster":"varieties-curves"},{"id":"stacks:0C89","tag":"0C89","title":"Classification of proper subgraphs · Lemma 0C89","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@..[r] & bullet ar@-[r] & bullet ar@-[r] & bullet Let t > 5 and n > t. Then given t distinct (-2)-indices i_1, …, i_t such that a_i_ji_j + 1 is nonzero for j = 1, …, t - 1, then up to reversing the order of these indices we have the a's and w's • are given by w_i_1 = w_i_2 = … = w_i_t = w, a_i_ji_j + 1 = w, and a_i_ji_k = 0 if k > j + 1, or • are given by w_i_1 = w_i_2 = … =…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{..}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet\n}\n$$\nLet $t > 5$ and $n > t$. Then given $t$ distinct $(-2)$-indices\n$i_1, \\ldots, i_t$ such that $a_{i_ji_{j + 1}}$ is nonzero for\n$j = 1, \\ldots, t - 1$, then up to reversing the order of these indices\nwe have the $a$'s and $w$'s\n\\begin{enumerate}\n\\item\n\nare given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_t} = w$,\n$a_{i_ji_{j + 1}} = w$, and $a_{i_ji_k} = 0$ if $k > j + 1$, or\n\\item\n\nare given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_{t - 1}} = w$,\n$w_{j_t} = 2w$, $a_{i_ji_{j + 1}} = w$ for $j < t - 1$,\n$a_{i_{t - 1}i_t} = 2w$, and $a_{i_ji_k} = 0$ if $k > j + 1$, or\n\\item\n\nare given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_{t - 1}} = 2w$,\n$w_{j_t} = w$, $a_{i_ji_{j + 1}} = 2w$, and\n$a_{i_{t - 1}i_t} = 2w$, and $a_{i_ji_k} = 0$ if $k > j + 1$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C89","source_file":"models.tex","source_line":2135,"source_end_line":2168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2135-L2168","statement_sha256":"85b54c7384fd8daba9c9b258da6f3833366c08985b3738cdcdfedaa2358293aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9531,"rank":9531,"depth":0,"x":296.248,"y":1267.783,"cluster":"varieties-curves"},{"id":"stacks:0C8D","tag":"0C8D","title":"Classification of proper subgraphs · Lemma 0C8D","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@..[r] & bullet ar@-[r] & bullet ar@-[r] ar@-[d] & bullet & & & bullet Let t > 4 and n > t + 1. Then given t + 1 distinct (-2)-indices i_1, …, i_t + 1 such that a_i_ji_j + 1 is nonzero for j = 1, …, t - 1 and a_i_t - 1i_t + 1 is nonzero, then we have the a's and w's • are given by w_i_1 = w_i_2 = … = w_i_t + 1 = w, a_i_ji_j + 1 = w for j = 1, …, t - 1, a_i_t - 1i_t + 1 = w and a_i_ji_k = 0…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{..}[r] & \\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\\\\n& & & \\bullet\n}\n$$\nLet $t > 4$ and $n > t + 1$. Then given $t + 1$ distinct\n$(-2)$-indices $i_1, \\ldots, i_{t + 1}$ such that $a_{i_ji_{j + 1}}$\nis nonzero for $j = 1, \\ldots, t - 1$ and $a_{i_{t - 1}i_{t + 1}}$\nis nonzero, then we have the $a$'s and $w$'s\n\\begin{enumerate}\n\\item\n\nare given by $w_{i_1} = w_{i_2} = \\ldots = w_{i_{t + 1}} = w$,\n$a_{i_ji_{j + 1}} = w$ for $j = 1, \\ldots, t - 1$,\n$a_{i_{t - 1}i_{t + 1}} = w$ and $a_{i_ji_k} = 0$ for other\npairs $(j, k)$ with $j > k$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8D","source_file":"models.tex","source_line":2206,"source_end_line":2228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2206-L2228","statement_sha256":"a6606b1e4152156be0fde598507fe54a4afb0348254e6bd8927118526ab560d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9532,"rank":9532,"depth":0,"x":78.245,"y":1083.311,"cluster":"varieties-curves"},{"id":"stacks:0C8F","tag":"0C8F","title":"Classification of proper subgraphs · Lemma 0C8F","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] ar@-[d] & bullet ar@-[r] & bullet & & bullet Let n > 6. Then given 6 distinct (-2)-indices i_1, …, i_6 such that a_12, a_23, a_34, a_45, a_36 are nonzero, then we have the m's, a's, and w's • are given by ( m_1 m_2 m_3 m_4 m_5 m_6 ), ( -2w & w & 0 & 0 & 0 & 0 w & -2w & w & 0 & 0 & 0 0 & w & -2w & w & 0 & w 0 & 0 & w & -2w & w & 0 0 & 0 & 0 & w & -2w & 0 0 & 0 & w & 0 &…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &\n\\bullet \\ar@{-}[r] & \\bullet \\\\\n& & \\bullet\n}\n$$\nLet $n > 6$. Then given $6$ distinct $(-2)$-indices $i_1, \\ldots, i_6$\nsuch that $a_{12}, a_{23}, a_{34}, a_{45}, a_{36}$ are nonzero, then\nwe have the $m$'s, $a$'s, and $w$'s\n\\begin{enumerate}\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4 \\\\\nm_5 \\\\\nm_6\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 & 0 \\\\\n0 & w & -2w & w & 0 & w \\\\\n0 & 0 & w & -2w & w & 0 \\\\\n0 & 0 & 0 & w & -2w & 0 \\\\\n0 & 0 & w & 0 & 0 & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4 + m_6$,\n$2m_4 \\geq m_3 + m_5$, $2m_5 \\geq m_3$, and $2m_6 \\geq m_3$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8F","source_file":"models.tex","source_line":2242,"source_end_line":2296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2242-L2296","statement_sha256":"dd3034fe1a7cb542089355a241a3541d030dc7fab98a0f7b7cb524e43f974a69","origin":"The Stacks Project","memory_eligible":false,"source_rank":9533,"rank":9533,"depth":0,"x":387.64,"y":1095.65,"cluster":"varieties-curves"},{"id":"stacks:0C8H","tag":"0C8H","title":"Classification of proper subgraphs · Lemma 0C8H","summary":"Nonexistence of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@..[r] ar@-[d] & bullet ar@-[d] ar@-[r] & bullet & bullet & bullet Assume t ≥ 4 and n > t + 2. There do not exist t + 2 distinct (-2)-indices i_0, …, i_t + 1 such that a_i_ji_j + 1 > 0 for j = 1, …, t - 1 and a_i_0i_2 > 0 and a_i_t - 1i_t + 1 > 0.","statement_latex":"Nonexistence of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{..}[r] \\ar@{-}[d] &\n\\bullet \\ar@{-}[d] \\ar@{-}[r] & \\bullet \\\\\n& \\bullet & \\bullet\n}\n$$\nAssume $t \\geq 4$ and $n > t + 2$.\nThere do {\\bf not} exist $t + 2$ distinct\n$(-2)$-indices $i_0, \\ldots, i_{t + 1}$ such that\n$a_{i_ji_{j + 1}} > 0$ for $j = 1, \\ldots, t - 1$\nand $a_{i_0i_2} > 0$ and $a_{i_{t - 1}i_{t + 1}} > 0$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8H","source_file":"models.tex","source_line":2318,"source_end_line":2333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2318-L2333","statement_sha256":"eae7650098e4433a492583a8d8ec1d90c246138eaa62e7fbc1d027f1de60a14c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9534,"rank":9534,"depth":0,"x":149.347,"y":1262.268,"cluster":"varieties-curves"},{"id":"stacks:0C8I","tag":"0C8I","title":"Classification of proper subgraphs · Lemma 0C8I","summary":"Nonexistence of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] ar@-[d] & bullet ar@-[r] & bullet & & bullet ar@-[d] & & bullet Assume n > 7. There do not exist 7 distinct (-2)-indices f, g, h, i, j, k, l such that a_fg, a_gh, a_ij, a_jh, a_kl, a_lh are nonzero.","statement_latex":"Nonexistence of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &\n\\bullet \\ar@{-}[r] & \\bullet \\\\\n& & \\bullet \\ar@{-}[d] \\\\\n& & \\bullet\n}\n$$\nAssume $n > 7$. There do {\\bf not} exist $7$ distinct\n$(-2)$-indices $f, g, h, i, j, k, l$\nsuch that $a_{fg}, a_{gh}, a_{ij}, a_{jh}, a_{kl}, a_{lh}$ are nonzero.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8I","source_file":"models.tex","source_line":2362,"source_end_line":2376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2362-L2376","statement_sha256":"7713b55919cf2a96024db78a9e2876091bacd7ef913766166044ab84ed5ade75","origin":"The Stacks Project","memory_eligible":false,"source_rank":9535,"rank":9535,"depth":0,"x":191.111,"y":1003.948,"cluster":"varieties-curves"},{"id":"stacks:0C8J","tag":"0C8J","title":"Classification of proper subgraphs · Lemma 0C8J","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] ar@-[d] & bullet ar@-[r] & bullet & & & bullet Let n > 7. Then given 7 distinct (-2)-indices i_1, …, i_7 such that a_12, a_23, a_34, a_45, a_56, a_47 are nonzero, then we have the m's, a's, and w's • are given by ( m_1 m_2 m_3 m_4 m_5 m_6 m_7 ), ( -2w & w & 0 & 0 & 0 & 0 & 0 w & -2w & w & 0 & 0 & 0 & 0 0 & w & -2w & w & 0 & 0 & 0 0 & 0 & w & -2w & w &…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] \\ar@{-}[d] & \\bullet \\ar@{-}[r] & \\bullet \\\\\n& & & \\bullet\n}\n$$\nLet $n > 7$. Then given $7$ distinct $(-2)$-indices $i_1, \\ldots, i_7$\nsuch that $a_{12}, a_{23}, a_{34}, a_{45}, a_{56}, a_{47}$ are nonzero,\nthen we have the $m$'s, $a$'s, and $w$'s\n\\begin{enumerate}\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4 \\\\\nm_5 \\\\\nm_6 \\\\\nm_7\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 & 0 & 0 \\\\\n0 & w & -2w & w & 0 & 0 & 0 \\\\\n0 & 0 & w & -2w & w & 0 & w \\\\\n0 & 0 & 0 & w & -2w & w & 0 \\\\\n0 & 0 & 0 & 0 & w & -2w & 0 \\\\\n0 & 0 & 0 & w & 0 & 0 & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,\n$2m_4 \\geq m_3 + m_5 + m_7$, $2m_5 \\geq m_4 + m_6$, $2m_6 \\geq m_5$,\nand $2m_7 \\geq m_4$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8J","source_file":"models.tex","source_line":2392,"source_end_line":2450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2392-L2450","statement_sha256":"c477931a32ab495e5aed08db89674b0b86fd0e953635cb371d6f0628e4fbb49c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9536,"rank":9536,"depth":0,"x":368.216,"y":1218.33,"cluster":"varieties-curves"},{"id":"stacks:0C8L","tag":"0C8L","title":"Classification of proper subgraphs · Lemma 0C8L","summary":"Classification of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] ar@-[d] & bullet ar@-[r] & bullet & & & & bullet Let n > 8. Then given 8 distinct (-2)-indices i_1, …, i_8 such that a_12, a_23, a_34, a_45, a_56, a_65, a_57 are nonzero, then we have the m's, a's, and w's • are given by ( m_1 m_2 m_3 m_4 m_5 m_6 m_7 m_8 ), ( -2w & w & 0 & 0 & 0 & 0 & 0 & 0 w & -2w & w & 0 & 0 & 0 & 0 & 0 0 & w & -2w &…","statement_latex":"Classification of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &\n\\bullet \\ar@{-}[r] & \\bullet \\\\\n& & & & \\bullet\n}\n$$\nLet $n > 8$. Then given $8$ distinct $(-2)$-indices $i_1, \\ldots, i_8$\nsuch that $a_{12}, a_{23}, a_{34}, a_{45}, a_{56}, a_{65}, a_{57}$\nare nonzero, then we have the $m$'s, $a$'s, and $w$'s\n\\begin{enumerate}\n\\item\n\nare given by\n$$\n\\left(\n\\begin{matrix}\nm_1 \\\\\nm_2 \\\\\nm_3 \\\\\nm_4 \\\\\nm_5 \\\\\nm_6 \\\\\nm_7 \\\\\nm_8\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 & 0 & 0 & 0 \\\\\n0 & w & -2w & w & 0 & 0 & 0 & 0 \\\\\n0 & 0 & w & -2w & w & 0 & 0 & 0 \\\\\n0 & 0 & 0 & w & -2w & w & 0 & w \\\\\n0 & 0 & 0 & 0 & w & -2w & w & 0 \\\\\n0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\\\\n0 & 0 & 0 & 0 & w & 0 & 0 & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right)\n$$\nwith $2m_1 \\geq m_2$, $2m_2 \\geq m_1 + m_3$, $2m_3 \\geq m_2 + m_4$,\n$2m_4 \\geq m_3 + m_5$, $2m_5 \\geq m_4 + m_6 + m_8$, $2m_6 \\geq m_5 + m_7$,\n$2m_7 \\geq m_6$, and $2m_8 \\geq m_5$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8L","source_file":"models.tex","source_line":2485,"source_end_line":2547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2485-L2547","statement_sha256":"cf1611c64a0e5946f038c2d68bf70369bb8a0f8996625e58ade4712aad2312d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9537,"rank":9537,"depth":0,"x":64.935,"y":1160.687,"cluster":"varieties-curves"},{"id":"stacks:0C8N","tag":"0C8N","title":"Classification of proper subgraphs · Lemma 0C8N","summary":"Nonexistence of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] ar@-[d] & bullet ar@-[r] & bullet ar@-[r] & bullet & & & bullet Assume n > 8. There do not exist 8 distinct (-2)-indices e, f, g, h, i, j, k, l such that a_ef, a_fg, a_gh, a_hi, a_ij, a_jk, a_lh are nonzero.","statement_latex":"Nonexistence of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] & \\bullet \\\\\n& & & \\bullet\n}\n$$\nAssume $n > 8$. There do {\\bf not} exist $8$ distinct\n$(-2)$-indices $e, f, g, h, i, j, k, l$\nsuch that $a_{ef}, a_{fg}, a_{gh}, a_{hi}, a_{ij}, a_{jk}, a_{lh}$\nare nonzero.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8N","source_file":"models.tex","source_line":2553,"source_end_line":2569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2553-L2569","statement_sha256":"2bca5b9e1667ef2c3d2edbcda0be271c3ffa2cfd45c7fc57439a413d18b5094c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9538,"rank":9538,"depth":0,"x":335.18,"y":1030.98,"cluster":"varieties-curves"},{"id":"stacks:0C8P","tag":"0C8P","title":"Classification of proper subgraphs · Lemma 0C8P","summary":"Nonexistence of proper subgraphs of the form xymatrix bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] & bullet ar@-[r] ar@-[d] & bullet ar@-[r] & bullet & & & & & bullet Assume n > 9. There do not exist 9 distinct (-2)-indices d, e, f, g, h, i, j, k, l such that a_de, a_ef, a_fg, a_gh, a_hi, a_ij, a_jk, a_lh are nonzero.","statement_latex":"Nonexistence of proper subgraphs of the form\n$$\n\\xymatrix{\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] &\n\\bullet \\ar@{-}[r] & \\bullet \\ar@{-}[r] \\ar@{-}[d] &\n\\bullet \\ar@{-}[r] & \\bullet \\\\\n& & & & & \\bullet\n}\n$$\nAssume $n > 9$. There do {\\bf not} exist $9$ distinct\n$(-2)$-indices $d, e, f, g, h, i, j, k, l$\nsuch that $a_{de}, a_{ef}, a_{fg}, a_{gh}, a_{hi}, a_{ij}, a_{jk}, a_{lh}$\nare nonzero.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8P","source_file":"models.tex","source_line":2592,"source_end_line":2608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2592-L2608","statement_sha256":"544ec3ab51f5818db816ce54292e53f3e34e87173b3bfb3227f9b1f4188b3630","origin":"The Stacks Project","memory_eligible":false,"source_rank":9539,"rank":9539,"depth":0,"x":240.122,"y":1280.203,"cluster":"varieties-curves"},{"id":"stacks:0C8Q","tag":"0C8Q","title":"Classification of proper subgraphs · Proposition 0C8Q","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type of genus g. Let I ⊂ (1, …, n) be a proper subset of cardinality ≥ 2 consisting of (-2)-indices such that there does not exist a nonempty proper subset I' ⊂ I with a_i'i = 0 for i' ∈ I, i ∈ I setminus I'. Then up to reordering the m_i's, a_ij's, w_i's for i, j ∈ I are as listed in Lemmas [Tag 0C7M], [Tag 0C7R], [Tag 0C7V], [Tag 0C80], [Tag 0C82], [Tag 0C87], [Tag 0C89], [Tag 0C8D], [Tag 0C8F], [Tag 0C8J], or [Tag 0C8L].","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.\nLet $I \\subset \\{1, \\ldots, n\\}$ be a proper subset of cardinality $\\geq 2$\nconsisting of $(-2)$-indices such that there\ndoes not exist a nonempty proper subset $I' \\subset I$\nwith $a_{i'i} = 0$ for $i' \\in I$, $i \\in I \\setminus I'$.\nThen up to reordering the $m_i$'s, $a_{ij}$'s, $w_i$'s\nfor $i, j \\in I$ are as listed in\nLemmas \\ref{lemma-two-by-two},\n\\ref{lemma-three-by-three},\n\\ref{lemma-four-by-four},\n\\ref{lemma-D4},\n\\ref{lemma-five-by-five},\n\\ref{lemma-D5},\n\\ref{lemma-long},\n\\ref{lemma-Dn},\n\\ref{lemma-E6},\n\\ref{lemma-E7}, or\n\\ref{lemma-E8}.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of proper subgraphs","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8Q","source_file":"models.tex","source_line":2617,"source_end_line":2637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2617-L2637","statement_sha256":"27f8b14a81c523ed8595e9573b808eeecfdf6d5b38cc06ef9eff9907a7dc0b1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9540,"rank":9540,"depth":1,"x":109.676,"y":1042.268,"cluster":"varieties-curves"},{"id":"stacks:0C8S","tag":"0C8S","title":"Genus zero · Lemma 0C8S","summary":"The only minimal numerical type of genus zero is n = 1, m_1 = 1, a_11 = 0, w_1 = 1, g_1 = 0.","statement_latex":"The only minimal numerical type of genus zero is\n$n = 1$, $m_1 = 1$, $a_{11} = 0$, $w_1 = 1$, $g_1 = 0$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of minimal type for genus zero and one","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8S","source_file":"models.tex","source_line":2662,"source_end_line":2666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2662-L2666","statement_sha256":"7f977adad755766714a59c45ed777e828750f4f646255ee88c0226b4db11d653","origin":"The Stacks Project","memory_eligible":false,"source_rank":9541,"rank":9541,"depth":4,"x":397.475,"y":1143.795,"cluster":"varieties-curves"},{"id":"stacks:0C8T","tag":"0C8T","title":"Genus one · Lemma 0C8T","summary":"The minimal numerical types of genus one are up to equivalence • n = 1, a_11 = 0, g_1 = 1, m_1, w_1 ≥ 1 arbitrary, • n = 2, and m_i, a_ij, w_i, g_i given by ( m m ), ( -2w & 2w 2w & -2w ), ( w w ), ( 0 0 ) with w and m arbitrary, • n = 2, and m_i, a_ij, w_i, g_i given by ( 2m m ), ( -2w & 4w 4w & -8w ), ( w 4w ), ( 0 0 ) with w and m arbitrary, • n = 3, and m_i, a_ij, w_i, g_i given by ( m m m ), ( -2w & w & w w & -2w & w w & w & -2w ), ( w w w ), ( 0 0 0 ) with w and m…","statement_latex":"The minimal numerical types of genus one are up to equivalence\n\\begin{enumerate}\n\\item\n\n$n = 1$, $a_{11} = 0$, $g_1 = 1$, $m_1, w_1 \\geq 1$ arbitrary,\n\\item\n\n$n = 2$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 2w \\\\\n2w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 2$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\n2m \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 4w \\\\\n4w & -8w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & w \\\\\nw & -2w & w \\\\\nw & w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 \\\\\nw & -2w & 3w \\\\\n0 & 3w & -6w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\n3w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\n3m\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-6w & 3w & 0 \\\\\n3w & -6w & 3w \\\\\n0 & 3w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n3w \\\\\n3w \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\n2m \\\\\n2m \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 2w & 0 \\\\\n2w & -4w & 4w \\\\\n0 & 4w & -8w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n2w \\\\\n4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 2w & 0 \\\\\n2w & -4w & 2w \\\\\n0 & 2w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n2w \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 3$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-4w & 2w & 0 \\\\\n2w & -2w & 2w \\\\\n0 & 2w & -4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n2w \\\\\nw \\\\\n2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\nm \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & w \\\\\nw & -2w & w & 0 \\\\\n0 & w & -2w & w \\\\\nw & 0 & w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\n2m \\\\\n2m \\\\\n2m \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 2w & 0 & 0 \\\\\n2w & -4w & 2w & 0 \\\\\n0 & 2w & -4w & 4w \\\\\n0 & 0 & 4w & -8w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n2w \\\\\n2w \\\\\n4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\nm \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 2w & 0 & 0 \\\\\n2w & -4w & 2w & 0 \\\\\n0 & 2w & -4w & 2w \\\\\n0 & 0 & 2w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n2w \\\\\n2w \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\n2m \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-4w & 2w & 0 & 0 \\\\\n2w & -2w & w & 0 \\\\\n0 & w & -2w & 2w \\\\\n0 & 0 & 2w & -4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n2w \\\\\nw \\\\\nw \\\\\n2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\n2m \\\\\nm \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & w & 2w \\\\\nw & -2w & 0 & 0 \\\\\nw & 0 & -2w & 0 \\\\\n2w & 0 & 0 & -4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\n2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 4$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\n2m \\\\\nm \\\\\nm \\\\\n2m\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-4w & 2w & 2w & 2w \\\\\n2w & -4w & 0 & 0 \\\\\n2w & 0 & -4w & 0 \\\\\n2w & 0 & 0 & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n2w \\\\\n2w \\\\\n2w \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\nm \\\\\nm \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & w \\\\\nw & -2w & w & 0 & 0 \\\\\n0 & w & -2w & w & 0 \\\\\n0 & 0 & w & -2w & w \\\\\nw & 0 & 0 & w & -2w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\n3m \\\\\n2m \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 \\\\\n0 & w & -2w & 2w & 0 \\\\\n0 & 0 & 2w & -4w & 2w \\\\\n0 & 0 & 0 & 2w & -4w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\n2w \\\\\n2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\n3m \\\\\n4m \\\\\n2m\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-4w & 2w & 0 & 0 & 0 \\\\\n2w & -4w & 2w & 0 & 0 \\\\\n0 & 2w & -4w & 2w & 0 \\\\\n0 & 0 & 2w & -2w & w \\\\\n0 & 0 & 0 & w & -2w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n2w \\\\\n2w \\\\\n2w \\\\\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\n2m \\\\\n2m \\\\\n2m \\\\\n2m \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 2w & 0 & 0 & 0 \\\\\n2w & -4w & 2w & 0 & 0 \\\\\n0 & 2w & -4w & 2w & 0 \\\\\n0 & 0 & 2w & -4w & 4w \\\\\n0 & 0 & 0 & 4w & -8w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n2w \\\\\n2w \\\\\n2w \\\\\n4w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\nm \\\\\nm \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 2w & 0 & 0 & 0 \\\\\n2w & -4w & 2w & 0 & 0 \\\\\n0 & 2w & -4w & 2w & 0 \\\\\n0 & 0 & 2w & -4w & 2w \\\\\n0 & 0 & 0 & 2w & -2w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n2w \\\\\n2w \\\\\n2w \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\n2m \\\\\n2m \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-4w & 2w & 0 & 0 & 0 \\\\\n2w & -2w & w & 0 & 0 \\\\\n0 & w & -2w & w & 0 \\\\\n0 & 0 & w & -2w & 2w \\\\\n0 & 0 & 0 & 2w & -4w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n2w \\\\\nw \\\\\nw \\\\\nw \\\\\n2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\n2m \\\\\nm \\\\\nm \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & w & w & w \\\\\nw & -2w & 0 & 0 & 0 \\\\\nw & 0 & -2w & 0 & 0 \\\\\nw & 0 & 0 & -2w & 0 \\\\\nw & 0 & 0 & 0 & -2w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\n2m \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-4w & 2w & 0 & 0 & 0 \\\\\n2w & -2w & w & 0 & 0 \\\\\n0 & w & -2w & w & w \\\\\n0 & 0 & w & -2w & 0 \\\\\n0 & 0 & w & 0 & -2w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n2w \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 5$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\n2m \\\\\n2m \\\\\n2m \\\\\nm \\\\\nm\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & 2w & 0 & 0 & 0 \\\\\n2w & -4w & 2w & 0 & 0 \\\\\n0 & 2w & -4w & 2w & 2w \\\\\n0 & 0 & 2w & -4w & 0 \\\\\n0 & 0 & 2w & 0 & -4w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\n2w \\\\\n2w \\\\\n2w \\\\\n2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n \\geq 6$ and we have an $n$-cycle generalizing (\\ref{item-five-cycle}):\n\\begin{enumerate}\n\\item $m_1 = \\ldots = m_n = m$,\n\\item $a_{12} = \\ldots = a_{(n - 1) n} = w$, $a_{1n} = w$,\nand for other $i < j$ we have $a_{ij} = 0$,\n\\item $w_1 = \\ldots = w_n = w$\n\\end{enumerate}\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n \\geq 6$ and we have a chain generalizing (\\ref{item-up-equal-equal-up}):\n\\begin{enumerate}\n\\item $m_1 = \\ldots = m_{n - 1} = 2m$, $m_n = m$,\n\\item $a_{12} = \\ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 1) n} = 4w$,\nand for other $i < j$ we have $a_{ij} = 0$,\n\\item $w_1 = w$, $w_2 = \\ldots = w_{n - 1} = 2w$, $w_n = 4w$\n\\end{enumerate}\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n \\geq 6$ and we have a chain generalizing (\\ref{item-up-equal-equal-down}):\n\\begin{enumerate}\n\\item $m_1 = \\ldots = m_n = m$,\n\\item $a_{12} = \\ldots = a_{(n - 1) n} = w$,\nand for other $i < j$ we have $a_{ij} = 0$,\n\\item $w_1 = w$, $w_2 = \\ldots = w_{n - 1} = 2w$, $w_n = w$\n\\end{enumerate}\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n \\geq 6$ and we have a chain generalizing (\\ref{item-down-equal-equal-up}):\n\\begin{enumerate}\n\\item $m_1 = w$, $w_2 = \\ldots = m_{n - 1} = 2m$, $m_n = m$,\n\\item $a_{12} = 2w$, $a_{23} = \\ldots = a_{(n - 2) (n - 1)} = w$,\n$a_{(n - 1) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$,\n\\item $w_1 = 2w$, $w_2 = \\ldots = w_{n - 1} = w$, $w_n = 2w$\n\\end{enumerate}\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n \\geq 6$ and we have a type generalizing (\\ref{item-triple-extended-up}):\n\\begin{enumerate}\n\\item $m_1 = m$, $m_2 = \\ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_n = m$,\n\\item $a_{12} = 2w$, $a_{23} = \\ldots = a_{(n - 2) (n - 1)} = w$,\n$a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,\n\\item $w_1 = 2w$, $w_2 = \\ldots = w_n = w$\n\\end{enumerate}\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n \\geq 6$ and we have a type generalizing (\\ref{item-triple-extended-down}):\n\\begin{enumerate}\n\\item $m_1 = \\ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_n = m$,\n\\item $a_{12} = \\ldots = a_{(n - 2) (n - 1)} = 2w$,\n$a_{(n - 2) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$,\n\\item $w_1 = w$, $w_2 = \\ldots = w_n = 2w$\n\\end{enumerate}\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n \\geq 6$ and we have a type generalizing (\\ref{item-quadruple}):\n\\begin{enumerate}\n\\item $m_1 = m_2 = m$, $m_3 = \\ldots = m_{n - 2} = 2m$, $m_{n - 1} = m_n = m$,\n\\item $a_{13} = w$, $a_{23} = \\ldots = a_{(n - 2) (n - 1)} = w$,\n$a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,\n\\item $w_1 = \\ldots = w_n = w$,\n\\end{enumerate}\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 7$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\n3m \\\\\nm \\\\\n2m \\\\\nm \\\\\n2m\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 & 0 & 0 \\\\\n0 & w & -2w & 0 & w & 0 & w \\\\\n0 & 0 & 0 & -2w & w & 0 & 0 \\\\\n0 & 0 & w & w & -2w & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & -2w & w \\\\\n0 & 0 & w & 0 & 0 & w & -2w\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 8$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\n3m \\\\\n4m \\\\\n3m \\\\\n2m \\\\\nm \\\\\n2m\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 & 0 & 0 & 0 \\\\\n0 & w & -2w & w & 0 & 0 & 0 & 0 \\\\\n0 & 0 & w & -2w & w & 0 & 0 & w \\\\\n0 & 0 & 0 & w & -2w & w & 0 & 0 \\\\\n0 & 0 & 0 & 0 & w & -2w & w & 0 \\\\\n0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\\\\n0 & 0 & 0 & w & 0 & 0 & 0 & -2w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary,\n\\item\n\n$n = 9$, and $m_i, a_{ij}, w_i, g_i$ given by\n$$\n\\left(\n\\begin{matrix}\nm \\\\\n2m \\\\\n3m \\\\\n4m \\\\\n5m \\\\\n6m \\\\\n4m \\\\\n2m \\\\\n3m\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n-2w & w & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nw & -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & w & -2w & w & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & w & -2w & w & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & w & -2w & w & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & w & -2w & w & 0 & w \\\\\n0 & 0 & 0 & 0 & 0 & w & -2w & w & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\\\\n0 & 0 & 0 & 0 & 0 & w & 0 & 0 & -2w \\\\\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw \\\\\nw\n\\end{matrix}\n\\right),\n\\quad\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0 \\\\\n0\n\\end{matrix}\n\\right)\n$$\nwith $w$ and $m$ arbitrary.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Classification of minimal type for genus zero and one","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C8T","source_file":"models.tex","source_line":2673,"source_end_line":3861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L2673-L3861","statement_sha256":"4582dadae6b60c52fe3c6fb88baba5ebe136763853fb3dd2447085ac4971182f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9542,"rank":9542,"depth":0,"x":103.336,"y":1232.318,"cluster":"varieties-curves"},{"id":"stacks:0C9U","tag":"0C9U","title":"Bounding invariants of numerical types · Lemma 0C9U","summary":"Let n, m_i, a_ij, w_i, g_i be a numerical type of genus g. Given i, j with a_ij > 0 we have m_ia_ij ≤ m_j|a_jj| and m_iw_i ≤ m_j|a_jj|.","statement_latex":"Let $n, m_i, a_{ij}, w_i, g_i$ be a numerical type of genus $g$.\nGiven $i, j$ with $a_{ij} > 0$ we have\n$m_ia_{ij} \\leq m_j|a_{jj}|$ and $m_iw_i \\leq m_j|a_{jj}|$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Bounding invariants of numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C9U","source_file":"models.tex","source_line":3882,"source_end_line":3887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L3882-L3887","statement_sha256":"cf9a518e7fc04c18b63411c8589902ed72d1a4098b5850bf9ff4e9613ed4e731","origin":"The Stacks Project","memory_eligible":false,"source_rank":9543,"rank":9543,"depth":0,"x":249.178,"y":999.923,"cluster":"varieties-curves"},{"id":"stacks:0C9V","tag":"0C9V","title":"Bounding invariants of numerical types · Lemma 0C9V","summary":"Fix g ≥ 2. For every minimal numerical type n, m_i, a_ij, w_i, g_i of genus g with n > 1 we have • the set J ⊂ (1, …, n) of non-(-2)-indices has at most 2g - 2 elements, • for j ∈ J we have g_j < g, • for j ∈ J we have m_j|a_jj| ≤ 6g - 6, and • for j ∈ J and i ∈ (1, …, n) we have m_ia_ij ≤ 6g - 6.","statement_latex":"Fix $g \\geq 2$. For every minimal numerical type $n, m_i, a_{ij}, w_i, g_i$\nof genus $g$ with $n > 1$ we have\n\\begin{enumerate}\n\\item the set $J \\subset \\{1, \\ldots, n\\}$ of non-$(-2)$-indices\nhas at most $2g - 2$ elements,\n\\item for $j \\in J$ we have $g_j < g$,\n\\item for $j \\in J$ we have $m_j|a_{jj}| \\leq 6g - 6$, and\n\\item for $j \\in J$ and $i \\in \\{1, \\ldots, n\\}$\nwe have $m_ia_{ij} \\leq 6g - 6$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Bounding invariants of numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C9V","source_file":"models.tex","source_line":3897,"source_end_line":3909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L3897-L3909","statement_sha256":"41a39a23496c2ab3213429d149bd62c249a547260ce64c85673b809568da0eaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9544,"rank":9544,"depth":3,"x":328.596,"y":1254.279,"cluster":"varieties-curves"},{"id":"stacks:0C9W","tag":"0C9W","title":"Bounding invariants of numerical types · Lemma 0C9W","summary":"Fix g ≥ 2. For every minimal numerical type n, m_i, a_ij, w_i, g_i of genus g we have m_i|a_ij| ≤ 768g.","statement_latex":"Fix $g \\geq 2$. For every minimal numerical type $n, m_i, a_{ij}, w_i, g_i$\nof genus $g$ we have $m_i|a_{ij}| \\leq 768g$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Bounding invariants of numerical types","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C9W","source_file":"models.tex","source_line":3954,"source_end_line":3958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L3954-L3958","statement_sha256":"f4bc84b509b80ad853e9e31c01174560f2b836b60a862caf31c4571ebf7b50c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9545,"rank":9545,"depth":4,"x":65.243,"y":1111.652,"cluster":"varieties-curves"},{"id":"stacks:0C9X","tag":"0C9X","title":"Bounding invariants of numerical types · Proposition 0C9X","summary":"Let g ≥ 2. For every numerical type T of genus g and prime number ℓ > 768g we have dim_F_ℓ Pic(T)[ℓ] ≤ g where Pic(T) is as in Definition [Tag 0C7H]. If T is minimal, then we even have dim_F_ℓ Pic(T)[ℓ] ≤ g_top ≤ g where g_top as in Definition [Tag 0C79].","statement_latex":"Let $g \\geq 2$. For every numerical type $T$ of genus $g$\nand prime number $\\ell > 768g$ we have\n$$\n\\dim_{\\mathbf{F}_\\ell} \\Pic(T)[\\ell] \\leq g\n$$\nwhere $\\Pic(T)$ is as in Definition \\ref{definition-picard-group}.\nIf $T$ is minimal, then we even have\n$$\n\\dim_{\\mathbf{F}_\\ell} \\Pic(T)[\\ell] \\leq g_{top} \\leq g\n$$\nwhere $g_{top}$ as in Definition \\ref{definition-top-genus}.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Bounding invariants of numerical types","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C9X","source_file":"models.tex","source_line":4068,"source_end_line":4081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4068-L4081","statement_sha256":"da39b95bf289c069cdfa64d6e91b60d5a473b618b46b42e2efa5cfccf82b7dfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9546,"rank":9546,"depth":5,"x":374.42,"y":1067.349,"cluster":"varieties-curves"},{"id":"stacks:0C2S","tag":"0C2S","title":"Models · Lemma 0C2S","summary":"Let V_1 → V_2 be a closed immersion of algebraic schemes over K. If X_2 is a model for V_2, then the scheme theoretic image of V_1 → X_2 is a model for V_1.","statement_latex":"Let $V_1 \\to V_2$ be a closed immersion of algebraic schemes over $K$.\nIf $X_2$ is a model for $V_2$, then the scheme theoretic image\nof $V_1 \\to X_2$ is a model for $V_1$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Models","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2S","source_file":"models.tex","source_line":4157,"source_end_line":4162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4157-L4162","statement_sha256":"b5bdb22e9430ee204f6a1450ad9bf0288d36bba6523df364bf9a7a24e3265655","origin":"The Stacks Project","memory_eligible":false,"source_rank":9547,"rank":9547,"depth":19,"x":181.886,"y":1275.645,"cluster":"varieties-curves"},{"id":"stacks:0C2T","tag":"0C2T","title":"Models · Lemma 0C2T","summary":"Let X be a model of a geometrically normal variety V over K. Then the normalization ν : X^ν → X is finite and the base change of X^ν to the completion R^wedge is the normalization of the base change of X. Moreover, for each x ∈ X^ν the completion of O_X^ν, x is normal.","statement_latex":"Let $X$ be a model of a geometrically normal variety $V$ over $K$.\nThen the normalization $\\nu : X^\\nu \\to X$ is finite and\nthe base change of $X^\\nu$ to the completion $R^\\wedge$\nis the normalization of the base change of $X$. Moreover, for\neach $x \\in X^\\nu$ the completion of $\\mathcal{O}_{X^\\nu, x}$\nis normal.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Models","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2T","source_file":"models.tex","source_line":4178,"source_end_line":4186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4178-L4186","statement_sha256":"c4833b661395489bf2f7d166fc8344fedaf42e6833ed3e87a1738a1d07a2d2a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9548,"rank":9548,"depth":51,"x":156.329,"y":1012.559,"cluster":"varieties-curves"},{"id":"stacks:0C2U","tag":"0C2U","title":"Models · Lemma 0C2U","summary":"Let X be a model of a smooth curve C over K. Then there exists a resolution of singularities of X and any resolution is a model of C.","statement_latex":"Let $X$ be a model of a smooth curve $C$ over $K$. Then\nthere exists a resolution of singularities of $X$\nand any resolution is a model of $C$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Models","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2U","source_file":"models.tex","source_line":4221,"source_end_line":4226,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4221-L4226","statement_sha256":"ac054118cac68184887c2070ed64d4c5f04c787d44d7f064e2ef3b33d4a02884","origin":"The Stacks Project","memory_eligible":false,"source_rank":9549,"rank":9549,"depth":72,"x":386.954,"y":1192.219,"cluster":"varieties-curves"},{"id":"stacks:0C2V","tag":"0C2V","title":"Models · Definition 0C2V","summary":"Let C be a smooth projective curve over K with H^0(C, O_C) = K. A minimal model will be a regular, proper model X for C such that X does not contain an exceptional curve of the first kind (Resolution of Surfaces, Section [Tag 0C2I]).","statement_latex":"Let $C$ be a smooth projective curve over $K$ with\n$H^0(C, \\mathcal{O}_C) = K$. A {\\it minimal model}\nwill be a regular, proper model $X$ for $C$ such that\n$X$ does not contain an exceptional curve of the first kind\n(Resolution of Surfaces, Section \\ref{resolve-section-minus-one}).","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Models","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2V","source_file":"models.tex","source_line":4247,"source_end_line":4254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4247-L4254","statement_sha256":"704c9a1a08d161f2d838a9248fe105dc0e5d97b9faa7a211626b44eaccb948cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9550,"rank":9550,"depth":0,"x":72.126,"y":1190.601,"cluster":"varieties-curves"},{"id":"stacks:0CD9","tag":"0CD9","title":"Models · Lemma 0CD9","summary":"A regular proper model of a curve is obtained by successive blowups from a minimal model Let C be a smooth projective curve over K with H^0(C, O_C) = K. If X is a regular proper model for C, then there exists a sequence of morphisms X = X_m → X_m - 1 → … → X_1 → X_0 of proper regular models of C, such that each morphism is a contraction of an exceptional curve of the first kind, and such that X_0 is a minimal model.","statement_latex":"\\begin{slogan}\nA regular proper model of a curve is obtained by successive blowups\nfrom a minimal model\n\\end{slogan}\nLet $C$ be a smooth projective curve over $K$ with\n$H^0(C, \\mathcal{O}_C) = K$. If $X$ is a regular proper\nmodel for $C$, then there exists a sequence of morphisms\n$$\nX = X_m \\to X_{m - 1} \\to \\ldots \\to X_1 \\to X_0\n$$\nof proper regular models of $C$, such that each morphism is a\ncontraction of an exceptional curve of the first kind, and such\nthat $X_0$ is a minimal model.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Models","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CD9","source_file":"models.tex","source_line":4267,"source_end_line":4282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4267-L4282","statement_sha256":"8ab5c558656602592d613199c5d95d0eb61dd9b4831a7026a4c4ac0bd62b9c0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9551,"rank":9551,"depth":63,"x":305.792,"y":1012.99,"cluster":"varieties-curves"},{"id":"stacks:0C2W","tag":"0C2W","title":"Models · Proposition 0C2W","summary":"Let C be a smooth projective curve over K with H^0(C, O_C) = K. A minimal model exists.","statement_latex":"Let $C$ be a smooth projective curve over $K$ with\n$H^0(C, \\mathcal{O}_C) = K$. A minimal model exists.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Models","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C2W","source_file":"models.tex","source_line":4301,"source_end_line":4305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4301-L4305","statement_sha256":"bc91cf7830723ccfe7248869511af958b00634a9b56d642229fa43239dd04171","origin":"The Stacks Project","memory_eligible":false,"source_rank":9552,"rank":9552,"depth":73,"x":276.293,"y":1276.787,"cluster":"varieties-curves"},{"id":"stacks:0C5Z","tag":"0C5Z","title":"The geometry of a regular model · Lemma 0C5Z","summary":"Let X be a regular model of a smooth curve C over K. • the special fibre X_k is an effective Cartier divisor on X, • each irreducible component C_i of X_k is an effective Cartier divisor on X, • X_k = ∑ m_i C_i (sum of effective Cartier divisors) where m_i is the multiplicity of C_i in X_k, • O_X(X_k) ≅ O_X.","statement_latex":"Let $X$ be a regular model of a smooth curve $C$ over $K$.\n\\begin{enumerate}\n\\item the special fibre $X_k$ is an effective Cartier divisor on $X$,\n\\item each irreducible component $C_i$ of $X_k$ is an effective\nCartier divisor on $X$,\n\\item $X_k = \\sum m_i C_i$ (sum of effective Cartier divisors)\nwhere $m_i$ is the multiplicity of $C_i$ in $X_k$,\n\\item $\\mathcal{O}_X(X_k) \\cong \\mathcal{O}_X$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C5Z","source_file":"models.tex","source_line":4330,"source_end_line":4341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4330-L4341","statement_sha256":"04adb3dcb1af612adf8592dc42ac185ae2cc1678d740a4fa1acb4903e81a68fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9553,"rank":9553,"depth":21,"x":85.732,"y":1065.335,"cluster":"varieties-curves"},{"id":"stacks:0C60","tag":"0C60","title":"The geometry of a regular model · Lemma 0C60","summary":"Let X be a regular model of a smooth curve C over K. Then • X → Spec(R) is a Gorenstein morphism of relative dimension 1, • each of the irreducible components C_i of X_k is Gorenstein.","statement_latex":"Let $X$ be a regular model of a smooth curve $C$ over $K$. Then\n\\begin{enumerate}\n\\item $X \\to \\Spec(R)$ is a Gorenstein morphism of relative dimension $1$,\n\\item each of the irreducible components $C_i$ of $X_k$ is Gorenstein.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C60","source_file":"models.tex","source_line":4389,"source_end_line":4396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4389-L4396","statement_sha256":"a3d5e5c02662e49a94caf23546adda99cab3db3961e1b88860a03c3f5b23544a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9554,"rank":9554,"depth":38,"x":396.577,"y":1113.171,"cluster":"varieties-curves"},{"id":"stacks:0C62","tag":"0C62","title":"The geometry of a regular model · Lemma 0C62","summary":"In Situation [Tag 0C61] the special fibre X_k is connected.","statement_latex":"In Situation \\ref{situation-regular-model} the special fibre $X_k$ is connected.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C62","source_file":"models.tex","source_line":4422,"source_end_line":4425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4422-L4425","statement_sha256":"f9e112176d83549f8c2fb8736fa94c409381d46c1280749981012ffa050f9264","origin":"The Stacks Project","memory_eligible":false,"source_rank":9555,"rank":9555,"depth":41,"x":128.651,"y":1254.408,"cluster":"varieties-curves"},{"id":"stacks:0C63","tag":"0C63","title":"The geometry of a regular model · Lemma 0C63","summary":"In Situation [Tag 0C61] there is an exact sequence 0 → Z → Z^⊕ n → Pic(X) → Pic(C) → 0 where the first map sends 1 to (m_1, …, m_n) and the second maps sends the ith basis vector to O_X(C_i).","statement_latex":"In Situation \\ref{situation-regular-model} there is an exact sequence\n$$\n0 \\to \\mathbf{Z} \\to \\mathbf{Z}^{\\oplus n} \\to\n\\Pic(X) \\to \\Pic(C) \\to 0\n$$\nwhere the first map sends $1$ to $(m_1, \\ldots, m_n)$ and the second\nmaps sends the $i$th basis vector to $\\mathcal{O}_X(C_i)$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C63","source_file":"models.tex","source_line":4432,"source_end_line":4441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4432-L4441","statement_sha256":"13e87f4a3091550568f0e2c4bda0f31bb7a63140f4b086eeb49346745e3b4e1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9556,"rank":9556,"depth":26,"x":212.714,"y":998.0,"cluster":"varieties-curves"},{"id":"stacks:0C64","tag":"0C64","title":"The geometry of a regular model · Lemma 0C64","summary":"In Situation [Tag 0C61] given L an invertible O_X-module and a = (a_1, …, a_n) ∈ Z^⊕ n we define langle a, L rangle = ∑ a_ideg(L|_C_i) Then langle , rangle is bilinear and for b = (b_1, …, b_n) ∈ Z^⊕ n we have langle a, O_X(∑ b_i C_i) rangle = langle b, O_X(∑ a_i C_i) rangle","statement_latex":"In Situation \\ref{situation-regular-model} given $\\mathcal{L}$ an invertible\n$\\mathcal{O}_X$-module and\n$a = (a_1, \\ldots, a_n) \\in \\mathbf{Z}^{\\oplus n}$ we define\n$$\n\\langle a, \\mathcal{L} \\rangle = \\sum a_i\\deg(\\mathcal{L}|_{C_i})\n$$\nThen $\\langle , \\rangle$ is bilinear and for\n$b = (b_1, \\ldots, b_n) \\in \\mathbf{Z}^{\\oplus n}$ we have\n$$\n\\left\\langle a, \\mathcal{O}_X(\\sum b_i C_i) \\right\\rangle =\n\\left\\langle b, \\mathcal{O}_X(\\sum a_i C_i) \\right\\rangle\n$$","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C64","source_file":"models.tex","source_line":4480,"source_end_line":4494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4480-L4494","statement_sha256":"a729fa8e51ce3e1847733ecc448c88b90e2de768ad96be6ff876c718be4117bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9557,"rank":9557,"depth":44,"x":357.054,"y":1234.986,"cluster":"varieties-curves"},{"id":"stacks:0C66","tag":"0C66","title":"The geometry of a regular model · Lemma 0C66","summary":"In Situation [Tag 0C61] the symmetric bilinear form ([Tag 0C65]) has the following properties • (C_i · C_j) ≥ 0 if i not = j with equality if and only if C_i ∩ C_j = ∅, • (∑ m_i C_i · C_j) = 0, • there is no nonempty proper subset I ⊂ (1, …, n) such that (C_i · C_j) = 0 for i ∈ I, j not ∈ I. • (∑ a_i C_i · ∑ a_i C_i) ≤ 0 with equality if and only if there exists a q ∈ Q such that a_i = qm_i for i = 1, …, n,","statement_latex":"In Situation \\ref{situation-regular-model} the symmetric bilinear form\n(\\ref{equation-form}) has the following properties\n\\begin{enumerate}\n\\item $(C_i \\cdot C_j) \\geq 0$ if $i \\not = j$ with equality if and only\nif $C_i \\cap C_j = \\emptyset$,\n\\item $(\\sum m_i C_i \\cdot C_j) = 0$,\n\\item there is no nonempty proper subset $I \\subset \\{1, \\ldots, n\\}$\nsuch that $(C_i \\cdot C_j) = 0$ for $i \\in I$, $j \\not \\in I$.\n\\item $(\\sum a_i C_i \\cdot \\sum a_i C_i) \\leq 0$ with equality if and\nonly if there exists a $q \\in \\mathbf{Q}$ such that $a_i = qm_i$\nfor $i = 1, \\ldots, n$,\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C66","source_file":"models.tex","source_line":4538,"source_end_line":4552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4538-L4552","statement_sha256":"989a3221a945274f50c648c6f6113b1bc9313ec901bf1f9a05057b3bdfaab188","origin":"The Stacks Project","memory_eligible":false,"source_rank":9558,"rank":9558,"depth":45,"x":59.772,"y":1142.054,"cluster":"varieties-curves"},{"id":"stacks:0C67","tag":"0C67","title":"The geometry of a regular model · Lemma 0C67","summary":"In Situation [Tag 0C61] set d = gcd(m_1, …, m_n) and let D = ∑ (m_i/d)C_i as an effective Cartier divisor. Then O_X(D) has order dividing d in Pic(X) and C_D/X an invertible O_D-module of order dividing d in Pic(D).","statement_latex":"In Situation \\ref{situation-regular-model} set $d = \\gcd(m_1, \\ldots, m_n)$\nand let $D = \\sum (m_i/d)C_i$ as an effective Cartier divisor.\nThen $\\mathcal{O}_X(D)$ has order dividing $d$ in $\\Pic(X)$\nand $\\mathcal{C}_{D/X}$ an invertible $\\mathcal{O}_D$-module\nof order dividing $d$ in $\\Pic(D)$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C67","source_file":"models.tex","source_line":4575,"source_end_line":4582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4575-L4582","statement_sha256":"fc726e6c55e831e0155ff1e87d96130620834490b8b8e1f42614f444c44521cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9559,"rank":9559,"depth":22,"x":353.984,"y":1041.805,"cluster":"varieties-curves"},{"id":"stacks:0C68","tag":"0C68","title":"The geometry of a regular model · Lemma 0C68","summary":"[Artin-Winters] In Situation [Tag 0C61] let d = gcd(m_1, …, m_n). Let D = ∑ (m_i/d) C_i as an effective Cartier divisor. Then there exists a sequence of effective Cartier divisors (X_k)_red = Z_0 ⊂ Z_1 ⊂ … ⊂ Z_m = D such that Z_j = Z_j - 1 + C_i_j for some i_j ∈ (1, …, n) for j = 1, …, m and such that H^0(Z_j, O_Z_j) is a field finite over k for j = 0, … m.","statement_latex":"\\begin{reference}\n\\cite[Lemma 2.6]{Artin-Winters}\n\\end{reference}\nIn Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$.\nLet $D = \\sum (m_i/d) C_i$ as an effective Cartier divisor. Then there exists\na sequence of effective Cartier divisors\n$$\n(X_k)_{red} = Z_0 \\subset Z_1 \\subset \\ldots \\subset Z_m = D\n$$\nsuch that $Z_j = Z_{j - 1} + C_{i_j}$ for some $i_j \\in \\{1, \\ldots, n\\}$\nfor $j = 1, \\ldots, m$ and such that $H^0(Z_j, \\mathcal{O}_{Z_j})$\nis a field finite over $k$ for $j = 0, \\ldots m$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C68","source_file":"models.tex","source_line":4595,"source_end_line":4609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4595-L4609","statement_sha256":"021243d06c0557ed0a586bfe34b86e4f96c3e2ee98a250656fde5abf4425da33","origin":"The Stacks Project","memory_eligible":false,"source_rank":9560,"rank":9560,"depth":46,"x":217.53,"y":1282.888,"cluster":"varieties-curves"},{"id":"stacks:0C69","tag":"0C69","title":"The geometry of a regular model · Lemma 0C69","summary":"[Artin-Winters] In Situation [Tag 0C61] let d = gcd(m_1, …, m_n). Let D = ∑ (m_i/d) C_i as an effective Cartier divisor on X. Then 1 - g_C = d [kappa : k] (1 - g_D) where g_C is the genus of C, g_D is the genus of D, and kappa = H^0(D, O_D).","statement_latex":"\\begin{reference}\n\\cite[Lemma 2.6]{Artin-Winters}\n\\end{reference}\nIn Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$.\nLet $D = \\sum (m_i/d) C_i$ as an effective Cartier divisor on $X$. Then\n$$\n1 - g_C = d [\\kappa : k] (1 - g_D)\n$$\nwhere $g_C$ is the genus of $C$, $g_D$ is the genus of $D$, and\n$\\kappa = H^0(D, \\mathcal{O}_D)$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C69","source_file":"models.tex","source_line":4650,"source_end_line":4662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4650-L4662","statement_sha256":"b970b7c71d7e4587cb31fe7eeee73088c0233ee2846c3000b5a4b5751a7cd9fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9561,"rank":9561,"depth":47,"x":124.193,"y":1027.46,"cluster":"varieties-curves"},{"id":"stacks:0C6A","tag":"0C6A","title":"The geometry of a regular model · Lemma 0C6A","summary":"In Situation [Tag 0C61] given a pair of indices i, j such that C_i and C_j are exceptional curves of the first kind and C_i ∩ C_j not = ∅, then n = 2, m_1 = m_2 = 1, C_1 ≅ P^1_k, C_2 ≅ P^1_k, C_1 and C_2 meet in a k-rational point, and C has genus 0.","statement_latex":"In Situation \\ref{situation-regular-model} given a pair of indices $i, j$\nsuch that $C_i$ and $C_j$ are exceptional curves of the first kind\nand $C_i \\cap C_j \\not = \\emptyset$, then\n$n = 2$, $m_1 = m_2 = 1$, $C_1 \\cong \\mathbf{P}^1_k$,\n$C_2 \\cong \\mathbf{P}^1_k$, $C_1$ and $C_2$ meet in a $k$-rational point,\nand $C$ has genus $0$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"The geometry of a regular model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6A","source_file":"models.tex","source_line":4710,"source_end_line":4718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4710-L4718","statement_sha256":"b5cd35f130c0198d9f2c36c3d1b7ef81922895804086810da55b6b013a2d1994","origin":"The Stacks Project","memory_eligible":false,"source_rank":9562,"rank":9562,"depth":48,"x":398.674,"y":1162.969,"cluster":"varieties-curves"},{"id":"stacks:0C6B","tag":"0C6B","title":"Uniqueness of the minimal model · Lemma 0C6B","summary":"Let C be a smooth projective curve over K with H^0(C, O_C) = K and genus > 0. There is a unique minimal model for C.","statement_latex":"Let $C$ be a smooth projective curve over $K$ with\n$H^0(C, \\mathcal{O}_C) = K$ and genus $> 0$.\nThere is a unique minimal model for $C$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Uniqueness of the minimal model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6B","source_file":"models.tex","source_line":4778,"source_end_line":4783,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4778-L4783","statement_sha256":"3fa207872858e7631e94cb08159cef792f0385b84d5dcf9add4b2d8ab5cc55ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":9563,"rank":9563,"depth":74,"x":87.023,"y":1218.844,"cluster":"varieties-curves"},{"id":"stacks:0C9Z","tag":"0C9Z","title":"Uniqueness of the minimal model · Lemma 0C9Z","summary":"Let C be a smooth projective curve over K with H^0(C, O_C) = K and genus > 0. Let X be the minimal model for C (Lemma [Tag 0C6B]). Let Y be a regular proper model for C. Then there is a unique morphism of models Y → X which is a sequence of contractions of exceptional curves of the first kind.","statement_latex":"Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$\nand genus $> 0$. Let $X$ be the minimal model for $C$\n(Lemma \\ref{lemma-minimal-model-unique}).\nLet $Y$ be a regular proper model for $C$. Then there is a unique\nmorphism of models $Y \\to X$ which is a sequence of contractions of\nexceptional curves of the first kind.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Uniqueness of the minimal model","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C9Z","source_file":"models.tex","source_line":4833,"source_end_line":4841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4833-L4841","statement_sha256":"8e30a43a1ba66dc1a014c5a7b69e6941f1adfd468d50d9187a8715319f7865f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9564,"rank":9564,"depth":75,"x":272.063,"y":1000.608,"cluster":"varieties-curves"},{"id":"stacks:0CA2","tag":"0CA2","title":"A formula for the genus · Lemma 0CA2","summary":"In Situation [Tag 0C61] suppose we have an effective Cartier divisors D, D' ⊂ X such that D' = D + C_i for some i ∈ (1, …, n) and D' ⊂ X_k. Then chi(X_k, O_D') - chi(X_k, O_D) = chi(X_k, O_X(-D)|_C_i) = -(D · C_i) + chi(C_i, O_C_i)","statement_latex":"In Situation \\ref{situation-regular-model} suppose we have an\neffective Cartier divisors $D, D' \\subset X$ such that\n$D' = D + C_i$ for some $i \\in \\{1, \\ldots, n\\}$ and $D' \\subset X_k$.\nThen\n$$\n\\chi(X_k, \\mathcal{O}_{D'}) - \\chi(X_k, \\mathcal{O}_D) =\n\\chi(X_k, \\mathcal{O}_X(-D)|_{C_i}) =\n-(D \\cdot C_i) + \\chi(C_i, \\mathcal{O}_{C_i})\n$$","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"A formula for the genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CA2","source_file":"models.tex","source_line":4893,"source_end_line":4904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4893-L4904","statement_sha256":"1a482e69f6afe103a4ae6ae00c134b6c1690e005a4d54d0b3a5710dd6fc83c79","origin":"The Stacks Project","memory_eligible":false,"source_rank":9565,"rank":9565,"depth":45,"x":311.151,"y":1266.767,"cluster":"varieties-curves"},{"id":"stacks:0CA3","tag":"0CA3","title":"A formula for the genus · Lemma 0CA3","summary":"In Situation [Tag 0C61] we have g_C = 1 + ∑_i = 1, …, n m_i([kappa_i : k] (g_i - 1) - frac12(C_i · C_i)) where kappa_i = H^0(C_i, O_C_i), g_i is the genus of C_i, and g_C is the genus of C.","statement_latex":"In Situation \\ref{situation-regular-model} we have\n$$\ng_C = 1 + \\sum\\nolimits_{i = 1, \\ldots, n}\nm_i\\left([\\kappa_i : k] (g_i - 1) - \\frac{1}{2}(C_i \\cdot C_i)\\right)\n$$\nwhere $\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$,\n$g_i$ is the genus of $C_i$, and $g_C$ is the genus of $C$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"A formula for the genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CA3","source_file":"models.tex","source_line":4939,"source_end_line":4948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L4939-L4948","statement_sha256":"a2d914388112e65f76c658f7b092d890cf9b7fd14d25e89a6dabf9c473d7a559","origin":"The Stacks Project","memory_eligible":false,"source_rank":9566,"rank":9566,"depth":46,"x":68.073,"y":1092.529,"cluster":"varieties-curves"},{"id":"stacks:0CA4","tag":"0CA4","title":"A formula for the genus · Lemma 0CA4","summary":"In Situation [Tag 0C61] with kappa_i = H^0(C_i, O_C_i) and g_i the genus of C_i the data n, m_i, (C_i · C_j), [kappa_i : k], g_i is a numerical type of genus equal to the genus of C.","statement_latex":"In Situation \\ref{situation-regular-model} with\n$\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$ and $g_i$ the genus of $C_i$\nthe data\n$$\nn, m_i, (C_i \\cdot C_j), [\\kappa_i : k], g_i\n$$\nis a numerical type of genus equal to the genus of $C$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"A formula for the genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CA4","source_file":"models.tex","source_line":5004,"source_end_line":5013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5004-L5013","statement_sha256":"b6fddf8a664b72c0a11d48afc2ada5be8c90457195530499d08f784ae5c0f31c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9567,"rank":9567,"depth":47,"x":387.718,"y":1083.072,"cluster":"varieties-curves"},{"id":"stacks:0CA5","tag":"0CA5","title":"A formula for the genus · Definition 0CA5","summary":"In Situation [Tag 0C61] the numerical type associated to X is the numerical type described in Lemma [Tag 0CA4].","statement_latex":"In Situation \\ref{situation-regular-model} the\n{\\it numerical type associated to $X$} is the numerical\ntype described in Lemma \\ref{lemma-numerical-type-of-model}.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"A formula for the genus","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CA5","source_file":"models.tex","source_line":5049,"source_end_line":5054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5049-L5054","statement_sha256":"c2e90fe05346dcd533df83db314e9d9f80cb62735c57f49b047640685a7252cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9568,"rank":9568,"depth":48,"x":159.418,"y":1271.588,"cluster":"varieties-curves"},{"id":"stacks:0CA6","tag":"0CA6","title":"A formula for the genus · Lemma 0CA6","summary":"In Situation [Tag 0C61]. The following are equivalent • X is a minimal model, and • the numerical type associated to X is minimal.","statement_latex":"In Situation \\ref{situation-regular-model}. The following\nare equivalent\n\\begin{enumerate}\n\\item $X$ is a minimal model, and\n\\item the numerical type associated to $X$ is minimal.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"A formula for the genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CA6","source_file":"models.tex","source_line":5059,"source_end_line":5067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5059-L5067","statement_sha256":"efc9dfcea87576f5d012865174e1a7174eb51c7a34e7658603a55dba67e3cf5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9569,"rank":9569,"depth":50,"x":176.179,"y":1002.797,"cluster":"varieties-curves"},{"id":"stacks:0CE8","tag":"0CE8","title":"A formula for the genus · Lemma 0CE8","summary":"In Situation [Tag 0C61] assume C has a K-rational point. Then • X_k has a k-rational point x which is a smooth point of X_k over k, • if x ∈ C_i, then H^0(C_i, O_C_i) = k and m_i = 1, and • H^0(X_k, O_X_k) = k and X_k has genus equal to the genus of C.","statement_latex":"In Situation \\ref{situation-regular-model} assume $C$ has a $K$-rational point.\nThen\n\\begin{enumerate}\n\\item $X_k$ has a $k$-rational point $x$ which is a smooth point of $X_k$\nover $k$,\n\\item if $x \\in C_i$, then $H^0(C_i, \\mathcal{O}_{C_i}) = k$ and\n$m_i = 1$, and\n\\item $H^0(X_k, \\mathcal{O}_{X_k}) = k$ and $X_k$ has genus equal to\nthe genus of $C$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"A formula for the genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CE8","source_file":"models.tex","source_line":5115,"source_end_line":5127,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5115-L5127","statement_sha256":"b87abbf29ee2cfc8daf68240e1dbfaa6ab78e2394c4dbc1b69285d6631ef94b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9570,"rank":9570,"depth":48,"x":380.154,"y":1210.695,"cluster":"varieties-curves"},{"id":"stacks:0CE9","tag":"0CE9","title":"A formula for the genus · Lemma 0CE9","summary":"In Situation [Tag 0C61] assume X is a minimal model, gcd(m_1, …, m_n) = 1, and H^0((X_k)_red, O) = k. Then the map H^1(X_k, O_X_k) → H^1((X_k)_red, O_(X_k)_red) is surjective and has a nontrivial kernel as soon as (X_k)_red not = X_k.","statement_latex":"In Situation \\ref{situation-regular-model} assume $X$ is a minimal model,\n$\\gcd(m_1, \\ldots, m_n) = 1$, and $H^0((X_k)_{red}, \\mathcal{O}) = k$. Then\nthe map\n$$\nH^1(X_k, \\mathcal{O}_{X_k}) \\to H^1((X_k)_{red}, \\mathcal{O}_{(X_k)_{red}})\n$$\nis surjective and has a nontrivial kernel as soon as $(X_k)_{red} \\not = X_k$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"A formula for the genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CE9","source_file":"models.tex","source_line":5157,"source_end_line":5166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5157-L5166","statement_sha256":"986b716055c3c285572788c95901503f79df9329a8388dc6c76c62517f265c78","origin":"The Stacks Project","memory_eligible":false,"source_rank":9571,"rank":9571,"depth":51,"x":62.279,"y":1173.101,"cluster":"varieties-curves"},{"id":"stacks:0CEA","tag":"0CEA","title":"A formula for the genus · Lemma 0CEA","summary":"In Situation [Tag 0C61] assume X_k has a k-rational point x which is a smooth point of X_k → Spec(k). Then dim_k H^1((X_k)_red, O_(X_k)_red) ≥ g_top + g_geom(X_k/k) where g_geom is as in Algebraic Curves, Section [Tag 0CE0] and g_top is the topological genus (Definition [Tag 0C79]) of the numerical type associated to X_k (Definition [Tag 0CA5]).","statement_latex":"In Situation \\ref{situation-regular-model} assume $X_k$ has a $k$-rational\npoint $x$ which is a smooth point of $X_k \\to \\Spec(k)$. Then\n$$\n\\dim_k H^1((X_k)_{red}, \\mathcal{O}_{(X_k)_{red}}) \\geq\ng_{top} + g_{geom}(X_k/k)\n$$\nwhere $g_{geom}$ is as in\nAlgebraic Curves, Section \\ref{curves-section-genus-geometric-genus}\nand $g_{top}$ is the topological genus\n(Definition \\ref{definition-top-genus})\nof the numerical type associated to $X_k$\n(Definition \\ref{definition-numerical-type-model}).","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"A formula for the genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEA","source_file":"models.tex","source_line":5214,"source_end_line":5228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5214-L5228","statement_sha256":"2633864adbb7259906b603ddc7e425e46fcb9e2804e6f4735402c55ff4458393","origin":"The Stacks Project","memory_eligible":false,"source_rank":9572,"rank":9572,"depth":49,"x":327.141,"y":1020.317,"cluster":"varieties-curves"},{"id":"stacks:0CEE","tag":"0CEE","title":"A formula for the genus · Lemma 0CEE","summary":"If equality holds in Lemma [Tag 0CEA] then • the unique irreducible component of X_k containing x is a smooth projective geometrically irreducible curve over k, • if C ⊂ X_k is another irreducible component, then kappa = H^0(C, O_C) is a finite separable extension of k, C has a kappa-rational point, and C is smooth over kappa","statement_latex":"If equality holds in Lemma \\ref{lemma-genus-reduction-bigger-than}\nthen\n\\begin{enumerate}\n\\item the unique irreducible component of $X_k$ containing\n$x$ is a smooth projective geometrically irreducible curve\nover $k$,\n\\item if $C \\subset X_k$ is another irreducible component, then\n$\\kappa = H^0(C, \\mathcal{O}_C)$ is a finite separable extension\nof $k$, $C$ has a $\\kappa$-rational point, and $C$ is smooth over $\\kappa$\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"A formula for the genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CEE","source_file":"models.tex","source_line":5323,"source_end_line":5335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5323-L5335","statement_sha256":"d21dc449601ebdf9d9d34c947ced39d2256a6a1bf813f7a9edf46f4fb20f0687","origin":"The Stacks Project","memory_eligible":false,"source_rank":9573,"rank":9573,"depth":50,"x":254.637,"y":1283.499,"cluster":"varieties-curves"},{"id":"stacks:0C6C","tag":"0C6C","title":"Blowing down exceptional curves · Lemma 0C6C","summary":"In Situation [Tag 0C61] assume that C_n is an exceptional curve of the first kind. Let f : X → X' be the contraction of C_n. Let C'_i = f(C_i). Write X'_k = ∑ m'_i C'_i. Then X', C'_i, i = 1, …, n' = n - 1, and m'_i = m_i is as in Situation [Tag 0C61] and we have • for i, j < n we have (C'_i · C'_j) = (C_i · C_j) - (C_i · C_n) (C_j · C_n) /(C_n · C_n), • for i < n if C_i ∩ C_n not = ∅, then there are maps kappa_i ← kappa'_i → kappa_n. Here kappa_i = H^0(C_i, O_C_i) and…","statement_latex":"In Situation \\ref{situation-regular-model} assume that $C_n$ is\nan exceptional curve of the first kind. Let $f : X \\to X'$ be the\ncontraction of $C_n$. Let $C'_i = f(C_i)$. Write $X'_k = \\sum m'_i C'_i$.\nThen $X'$, $C'_i$, $i = 1, \\ldots, n' = n - 1$, and $m'_i = m_i$\nis as in Situation \\ref{situation-regular-model} and we have\n\\begin{enumerate}\n\\item for $i, j < n$ we have\n$(C'_i \\cdot C'_j) =\n(C_i \\cdot C_j) - (C_i \\cdot C_n) (C_j \\cdot C_n) /(C_n \\cdot C_n)$,\n\\item for $i < n$ if $C_i \\cap C_n \\not = \\emptyset$, then there are\nmaps $\\kappa_i \\leftarrow \\kappa'_i \\rightarrow \\kappa_n$.\n\\end{enumerate}\nHere $\\kappa_i = H^0(C_i, \\mathcal{O}_{C_i})$ and\n$\\kappa'_i = H^0(C'_i, \\mathcal{O}_{C'_i})$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Blowing down exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C6C","source_file":"models.tex","source_line":5406,"source_end_line":5422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5406-L5422","statement_sha256":"75daf5e0de876a81251d860270231670ea054370c21164651f434fb9ed2132d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9574,"rank":9574,"depth":47,"x":96.317,"y":1048.084,"cluster":"varieties-curves"},{"id":"stacks:0CDA","tag":"0CDA","title":"Blowing down exceptional curves · Lemma 0CDA","summary":"Let C be a smooth projective curve over K with H^0(C, O_C) = K and genus 0. If there is more than one minimal model for C, then the special fibre of every minimal model is isomorphic to P^1_k.","statement_latex":"Let $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$\nand genus $0$. If there is more than one minimal model for $C$, then\nthe special fibre of every minimal model is isomorphic to $\\mathbf{P}^1_k$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Blowing down exceptional curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDA","source_file":"models.tex","source_line":5543,"source_end_line":5548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5543-L5548","statement_sha256":"2dc300e567c5d8f3a3268340b8b04c976f72f1a00b339decf2c4215d0cb78228","origin":"The Stacks Project","memory_eligible":false,"source_rank":9575,"rank":9575,"depth":0,"x":402.643,"y":1131.915,"cluster":"varieties-curves"},{"id":"stacks:0CAB","tag":"0CAB","title":"Picard groups of models · Lemma 0CAB","summary":"In Situation [Tag 0C61] let d = gcd(m_1, …, m_n). If L is an invertible O_X-module which • restricts to the trivial invertible module on C, and • has degree 0 on each C_i, then L^⊗ d ≅ O_X.","statement_latex":"In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$.\nIf $\\mathcal{L}$ is an invertible $\\mathcal{O}_X$-module which\n\\begin{enumerate}\n\\item restricts to the trivial invertible module on $C$, and\n\\item has degree $0$ on each $C_i$,\n\\end{enumerate}\nthen $\\mathcal{L}^{\\otimes d} \\cong \\mathcal{O}_X$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Picard groups of models","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAB","source_file":"models.tex","source_line":5669,"source_end_line":5678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5669-L5678","statement_sha256":"8be4ed7a9904f4649afba03c8de5412f9d1fa4a71d28a11f78c853a7e713a773","origin":"The Stacks Project","memory_eligible":false,"source_rank":9576,"rank":9576,"depth":46,"x":109.091,"y":1244.015,"cluster":"varieties-curves"},{"id":"stacks:0CAC","tag":"0CAC","title":"Picard groups of models · Lemma 0CAC","summary":"In Situation [Tag 0C61] let T be the numerical type associated to X. There exists a canonical map Pic(C) → Pic(T) whose kernel is exactly those invertible modules on C which are the restriction of invertible modules L on X with deg_C_i(L|_C_i) = 0 for i = 1, …, n.","statement_latex":"In Situation \\ref{situation-regular-model}\nlet $T$ be the numerical type associated to $X$.\nThere exists a canonical map\n$$\n\\Pic(C) \\to \\Pic(T)\n$$\nwhose kernel is exactly those invertible modules on $C$\nwhich are the restriction of invertible modules $\\mathcal{L}$\non $X$ with $\\deg_{C_i}(\\mathcal{L}|_{C_i}) = 0$ for\n$i = 1, \\ldots, n$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Picard groups of models","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAC","source_file":"models.tex","source_line":5694,"source_end_line":5706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5694-L5706","statement_sha256":"6b4fade17c760efb6b735c90fcb3e0d8051fca010cd417b749ed0d3812339076","origin":"The Stacks Project","memory_eligible":false,"source_rank":9577,"rank":9577,"depth":34,"x":235.515,"y":994.567,"cluster":"varieties-curves"},{"id":"stacks:0CAD","tag":"0CAD","title":"Picard groups of models · Lemma 0CAD","summary":"In Situation [Tag 0C61] let d = gcd(m_1, …, m_n) and let T be the numerical type associated to X. Let h ≥ 1 be an integer prime to d. There exists an exact sequence 0 → Pic(X)[h] → Pic(C)[h] → Pic(T)[h]","statement_latex":"In Situation \\ref{situation-regular-model} let $d = \\gcd(m_1, \\ldots, m_n)$\nand let $T$ be the numerical type associated to $X$.\nLet $h \\geq 1$ be an integer prime to $d$. There exists an exact sequence\n$$\n0 \\to \\Pic(X)[h] \\to \\Pic(C)[h] \\to \\Pic(T)[h]\n$$","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Picard groups of models","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAD","source_file":"models.tex","source_line":5745,"source_end_line":5753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5745-L5753","statement_sha256":"141ae5a47912cfb78155f43f996f6a9c5ff20fc202f954e88cccdb42a0727e10","origin":"The Stacks Project","memory_eligible":false,"source_rank":9578,"rank":9578,"depth":47,"x":342.985,"y":1250.466,"cluster":"varieties-curves"},{"id":"stacks:0CAE","tag":"0CAE","title":"Picard groups of models · Lemma 0CAE","summary":"In Situation [Tag 0C61] let h be an integer prime to the characteristic of k. Then the map Pic(X)[h] → Pic((X_k)_red)[h] is injective.","statement_latex":"In Situation \\ref{situation-regular-model} let $h$ be an integer\nprime to the characteristic of $k$. Then the map\n$$\n\\Pic(X)[h] \\longrightarrow \\Pic((X_k)_{red})[h]\n$$\nis injective.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Picard groups of models","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAE","source_file":"models.tex","source_line":5777,"source_end_line":5785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5777-L5785","statement_sha256":"ef39f161e7480d5716ffc428013785d4829c8780c66c6a70f50845f270110fb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9579,"rank":9579,"depth":37,"x":57.702,"y":1122.639,"cluster":"varieties-curves"},{"id":"stacks:0CDD","tag":"0CDD","title":"Semistable reduction · Lemma 0CDD","summary":"Let R be a discrete valuation ring. Let X be a scheme which is at-worst-nodal of relative dimension 1 over R. Let x ∈ X be a point of the special fibre of X over R. Then there exists a commutative diagram xymatrix X ar[d] & U ar[r] ar[d] ar[l] & Spec(A) ar[dl] Spec(R) & Spec(R') ar[l] where R ⊂ R' is an étale extension of discrete valuation rings, the morphism U → X is étale, the morphism U → Spec(A) is étale, there is a point x' ∈ U mapping to x, and A = R'[u, v]/(uv) or…","statement_latex":"Let $R$ be a discrete valuation ring. Let $X$ be a scheme which is\nat-worst-nodal of relative dimension $1$ over $R$.\nLet $x \\in X$ be a point of the special fibre\nof $X$ over $R$. Then there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] &\nU \\ar[r] \\ar[d] \\ar[l] &\n\\Spec(A) \\ar[dl] \\\\\n\\Spec(R) &\n\\Spec(R') \\ar[l]\n}\n$$\nwhere $R \\subset R'$ is an \\'etale extension of discrete valuation rings,\nthe morphism $U \\to X$ is \\'etale, the morphism $U \\to \\Spec(A)$ is \\'etale,\nthere is a point $x' \\in U$ mapping to $x$, and\n$$\nA = R'[u, v]/(uv)\n\\quad\\text{or}\\quad\nA = R'[u, v]/(uv - \\pi^n)\n$$\nwhere $n \\geq 0$ and $\\pi \\in R'$ is a uniformizer.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Semistable reduction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDD","source_file":"models.tex","source_line":5889,"source_end_line":5913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5889-L5913","statement_sha256":"61bee01f5fa920cc26494331e5121925ec03c6ce5c6e2d27c2460070bea0f08b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9580,"rank":9580,"depth":61,"x":371.134,"y":1054.962,"cluster":"varieties-curves"},{"id":"stacks:0CDE","tag":"0CDE","title":"Semistable reduction · Lemma 0CDE","summary":"Let R be a discrete valuation ring. Let X be a quasi-compact scheme which is at-worst-nodal of relative dimension 1 with smooth generic fibre over R. Then there exists m ≥ 0 and a sequence X_m → … → X_1 → X_0 = X such that • X_i + 1 → X_i is the blowing up of a closed point x_i where X_i is singular, • X_i → Spec(R) is at-worst-nodal of relative dimension 1, • X_m is regular.","statement_latex":"Let $R$ be a discrete valuation ring. Let $X$ be a quasi-compact scheme which\nis at-worst-nodal of relative dimension $1$ with smooth generic fibre over $R$.\nThen there exists $m \\geq 0$ and a sequence\n$$\nX_m \\to \\ldots \\to X_1 \\to X_0 = X\n$$\nsuch that\n\\begin{enumerate}\n\\item $X_{i + 1} \\to X_i$ is the blowing up of a closed point\n$x_i$ where $X_i$ is singular,\n\\item $X_i \\to \\Spec(R)$ is at-worst-nodal of relative dimension $1$,\n\\item $X_m$ is regular.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Semistable reduction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDE","source_file":"models.tex","source_line":5973,"source_end_line":5988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L5973-L5988","statement_sha256":"51e29e6c859df692b3c450abadb173346b803042e9123db3d920762afc64b56e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9581,"rank":9581,"depth":0,"x":194.285,"y":1282.913,"cluster":"varieties-curves"},{"id":"stacks:0CDF","tag":"0CDF","title":"Semistable reduction · Lemma 0CDF","summary":"Let R be a discrete valuation ring with fraction field K and residue field k. Assume X → Spec(R) is at-worst-nodal of relative dimension 1 over R. Let X → X' be the contraction of an exceptional curve E ⊂ X of the first kind. Then X' is at-worst-nodal of relative dimension 1 over R.","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$\nand residue field $k$. Assume $X \\to \\Spec(R)$ is\nat-worst-nodal of relative dimension $1$ over $R$.\nLet $X \\to X'$ be the contraction of an\nexceptional curve $E \\subset X$ of the first kind.\nThen $X'$ is at-worst-nodal of relative dimension $1$ over $R$.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Semistable reduction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDF","source_file":"models.tex","source_line":6094,"source_end_line":6102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L6094-L6102","statement_sha256":"abed5896bdef0d76fd7c064f186dd624881dce03506d3b89b9064fc44de65aed","origin":"The Stacks Project","memory_eligible":false,"source_rank":9582,"rank":9582,"depth":58,"x":141.332,"y":1014.243,"cluster":"varieties-curves"},{"id":"stacks:0CDG","tag":"0CDG","title":"Semistable reduction · Lemma 0CDG","summary":"Let R be a discrete valuation ring with fraction field K. Let C be a smooth projective curve over K with H^0(C, O_C) = K. The following are equivalent • there exists a proper model of C which is at-worst-nodal of relative dimension 1 over R, • there exists a minimal model of C which is at-worst-nodal of relative dimension 1 over R, and • any minimal model of C is at-worst-nodal of relative dimension 1 over R.","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$.\nLet $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists a proper model of $C$ which is\nat-worst-nodal of relative dimension $1$ over $R$,\n\\item there exists a minimal model of $C$ which is at-worst-nodal\nof relative dimension $1$ over $R$, and\n\\item any minimal model of $C$ is at-worst-nodal\nof relative dimension $1$ over $R$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Semistable reduction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDG","source_file":"models.tex","source_line":6140,"source_end_line":6153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L6140-L6153","statement_sha256":"455676e9dba636a843d2626a769fe299a4623b13a14fab020ee714efd43e690e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9583,"rank":9583,"depth":75,"x":396.657,"y":1182.456,"cluster":"varieties-curves"},{"id":"stacks:0CDH","tag":"0CDH","title":"Semistable reduction · Definition 0CDH","summary":"Let R be a discrete valuation ring with fraction field K. Let C be a smooth projective curve over K with H^0(C, O_C) = K. We say that C has semistable reduction if the equivalent conditions of Lemma [Tag 0CDG] are satisfied.","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$.\nLet $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.\nWe say that $C$ has {\\it semistable reduction} if the equivalent\nconditions of Lemma \\ref{lemma-semistable} are satisfied.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Semistable reduction","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDH","source_file":"models.tex","source_line":6190,"source_end_line":6196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L6190-L6196","statement_sha256":"4baafd542b61049431c9522d873204cb094529c1744272a2b0b093962cd1e60a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9584,"rank":9584,"depth":76,"x":72.832,"y":1203.314,"cluster":"varieties-curves"},{"id":"stacks:0CDI","tag":"0CDI","title":"Semistable reduction · Lemma 0CDI","summary":"Let R be a discrete valuation ring with fraction field K. Let C be a smooth projective curve over K with H^0(C, O_C) = K. The following are equivalent • there exists a proper smooth model for C, • there exists a minimal model for C which is smooth over R, • any minimal model is smooth over R.","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$.\nLet $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists a proper smooth model for $C$,\n\\item there exists a minimal model for $C$ which is smooth over $R$,\n\\item any minimal model is smooth over $R$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Semistable reduction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDI","source_file":"models.tex","source_line":6198,"source_end_line":6208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L6198-L6208","statement_sha256":"0aef344e99fce67e3b261c1c03968f0d71b144351b3e3c45b8da23969e97c809","origin":"The Stacks Project","memory_eligible":false,"source_rank":9585,"rank":9585,"depth":75,"x":295.034,"y":1004.016,"cluster":"varieties-curves"},{"id":"stacks:0CDJ","tag":"0CDJ","title":"Semistable reduction · Definition 0CDJ","summary":"Let R be a discrete valuation ring with fraction field K. Let C be a smooth projective curve over K with H^0(C, O_C) = K. We say that C has good reduction if the equivalent conditions of Lemma [Tag 0CDI] are satisfied.","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$.\nLet $C$ be a smooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.\nWe say that $C$ has {\\it good reduction} if the equivalent\nconditions of Lemma \\ref{lemma-good} are satisfied.","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Semistable reduction","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDJ","source_file":"models.tex","source_line":6224,"source_end_line":6230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L6224-L6230","statement_sha256":"d03bc59e255ecd40d21572ade2123ee6745242499aa9fd635296bb41740ab982","origin":"The Stacks Project","memory_eligible":false,"source_rank":9586,"rank":9586,"depth":76,"x":291.453,"y":1277.293,"cluster":"varieties-curves"},{"id":"stacks:0CDN","tag":"0CDN","title":"Semistable reduction for curves · Theorem 0CDN","summary":"[DM] Let R be a discrete valuation ring with fraction field K. Let C be a smooth projective curve over K with H^0(C, O_C) = K. Then there exists an extension of discrete valuation rings R ⊂ R' which induces a finite separable extension of fraction fields K'/K such that C_K' has semistable reduction. More precisely, we have the following • If the genus of C is zero, then there exists a degree 2 separable extension K'/K such that C_K' ≅ P^1_K' and hence C_K' is isomorphic…","statement_latex":"\\begin{reference}\n\\cite[Corollary 2.7]{DM}\n\\end{reference}\nLet $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a\nsmooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$.\nThen there exists an extension of discrete valuation rings\n$R \\subset R'$ which induces a finite separable extension of\nfraction fields $K'/K$ such that $C_{K'}$ has semistable reduction.\nMore precisely, we have the following\n\\begin{enumerate}\n\\item If the genus of $C$ is zero, then there exists a degree $2$\nseparable extension $K'/K$ such that $C_{K'} \\cong \\mathbf{P}^1_{K'}$\nand hence $C_{K'}$ is isomorphic to the generic fibre of the\nsmooth projective scheme $\\mathbf{P}^1_{R'}$ over the integral closure\n$R'$ of $R$ in $K'$.\n\\item If the genus of $C$ is one, then there exists a finite separable\nextension $K'/K$ such that $C_{K'}$ has semistable reduction\nover $R'_\\mathfrak m$ for every maximal ideal $\\mathfrak m$\nof the integral closure $R'$ of $R$ in $K'$. Moreover, the special\nfibre of the (unique) minimal model of $C_{K'}$ over $R'_\\mathfrak m$\nis either a smooth genus one curve or a cycle of rational curves.\n\\item If the genus $g$ of $C$ is greater than one, then there exists a\nfinite separable extension $K'/K$ of degree at most\n$B_g$ (\\ref{equation-bound}) such that $C_{K'}$ has semistable reduction\nover $R'_\\mathfrak m$ for every maximal ideal $\\mathfrak m$\nof the integral closure $R'$ of $R$ in $K'$.\n\\end{enumerate}","area":"Varieties & Curves","chapter":"Semistable Reduction","chapter_id":"models","section":"Semistable reduction for curves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CDN","source_file":"models.tex","source_line":6639,"source_end_line":6668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/models.tex#L6639-L6668","statement_sha256":"9f36bd4365637a2f4069461c322c34eade4ad3f304bd3dba0235a4a19a3b7a17","origin":"The Stacks Project","memory_eligible":false,"source_rank":9587,"rank":9587,"depth":68,"x":74.144,"y":1073.575,"cluster":"varieties-curves"},{"id":"stacks:0GNJ","tag":"0GNJ","title":"Functors on module categories · Lemma 0GNJ","summary":"Let A be a ring. Let B be a category having filtered colimits. Let F : Mod^fp_A → B be a functor. Then F extends uniquely to a functor F' : Mod_A → B which commutes with filtered colimits.","statement_latex":"Let $A$ be a ring. Let $\\mathcal{B}$ be a category having filtered\ncolimits. Let $F : \\text{Mod}^{fp}_A \\to \\mathcal{B}$ be a functor. Then $F$\nextends uniquely to a functor $F' : \\text{Mod}_A \\to \\mathcal{B}$\nwhich commutes with filtered colimits.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors on module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNJ","source_file":"functors.tex","source_line":40,"source_end_line":46,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L40-L46","statement_sha256":"1b21cb20e966abd4f1ecfacb4d93ba98b0f6781a23a37efb6c9a87a4685fd1d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9588,"rank":9588,"depth":3,"x":2427.687,"y":1585.981,"cluster":"moduli-theory"},{"id":"stacks:0GNK","tag":"0GNK","title":"Functors on module categories · Lemma 0GNK","summary":"Let A, B, F be as in Lemma [Tag 0GNJ]. Assume B is additive and F is additive. Then F' is additive and commutes with arbitrary direct sums.","statement_latex":"Let $A$, $\\mathcal{B}$, $F$ be as in Lemma \\ref{lemma-functor-on-fp-modules}.\nAssume $\\mathcal{B}$ is additive and $F$ is additive. Then\n$F'$ is additive and commutes with arbitrary direct sums.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors on module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNK","source_file":"functors.tex","source_line":65,"source_end_line":70,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L65-L70","statement_sha256":"b1c3217791092b8720022e81115a465849d6ebbdefceb0431ae02060d2b66d88","origin":"The Stacks Project","memory_eligible":false,"source_rank":9589,"rank":9589,"depth":4,"x":2530.816,"y":1580.278,"cluster":"moduli-theory"},{"id":"stacks:0GNL","tag":"0GNL","title":"Functors on module categories · Lemma 0GNL","summary":"Let A, B, F be as in Lemma [Tag 0GNJ]. Assume B is additive, has cokernels, and F is right exact. Then F' is additive, right exact, and commutes with arbitrary direct sums.","statement_latex":"Let $A$, $\\mathcal{B}$, $F$ be as in Lemma \\ref{lemma-functor-on-fp-modules}.\nAssume $\\mathcal{B}$ is additive, has cokernels, and $F$ is right exact. Then\n$F'$ is additive, right exact, and commutes with arbitrary direct sums.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors on module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNL","source_file":"functors.tex","source_line":115,"source_end_line":120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L115-L120","statement_sha256":"00c92b145fc662fafc638ad82ea84a4259a3a0de8dceead2b6825906866b0ffa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9590,"rank":9590,"depth":5,"x":2457.985,"y":1644.186,"cluster":"moduli-theory"},{"id":"stacks:0GNM","tag":"0GNM","title":"Functors on module categories · Lemma 0GNM","summary":"Let A, B, F be as in Lemma [Tag 0GNJ]. Assume A is a coherent ring (Algebra, Definition [Tag 05CV]), B is additive, has kernels, filtered colimits commute with taking kernels, and F is left exact. Then F' is additive, left exact, and commutes with arbitrary direct sums.","statement_latex":"Let $A$, $\\mathcal{B}$, $F$ be as in Lemma \\ref{lemma-functor-on-fp-modules}.\nAssume $A$ is a coherent ring (Algebra, Definition\n\\ref{algebra-definition-coherent}), $\\mathcal{B}$ is additive, has kernels,\nfiltered colimits commute with taking kernels, and $F$ is left exact. Then\n$F'$ is additive, left exact, and commutes with arbitrary direct sums.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors on module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNM","source_file":"functors.tex","source_line":186,"source_end_line":193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L186-L193","statement_sha256":"446e6517e677055a82d50007a3eb631da192ec7a000899c4b478dca4a6c5ea0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9591,"rank":9591,"depth":5,"x":2460.352,"y":1554.115,"cluster":"moduli-theory"},{"id":"stacks:0GNN","tag":"0GNN","title":"Functors on module categories · Lemma 0GNN","summary":"Let A be a ring. Let B be an additive category with cokernels. There is an equivalence of categories between • the category of functors F : Mod^fp_A → B which are right exact, and • the category of pairs (K, kappa) where K ∈ Ob(B) and kappa : A → End_B(K) is a ring homomorphism given by the rule sending F to F(A) with its natural A-action.","statement_latex":"Let $A$ be a ring. Let $\\mathcal{B}$ be an additive category\nwith cokernels. There is an equivalence of categories between\n\\begin{enumerate}\n\\item the category of functors $F : \\text{Mod}^{fp}_A \\to \\mathcal{B}$\nwhich are right exact, and\n\\item the category of pairs $(K, \\kappa)$ where $K \\in \\Ob(\\mathcal{B})$\nand $\\kappa : A \\to \\text{End}_\\mathcal{B}(K)$ is a ring homomorphism\n\\end{enumerate}\ngiven by the rule sending $F$ to $F(A)$ with its natural $A$-action.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors on module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNN","source_file":"functors.tex","source_line":257,"source_end_line":268,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L257-L268","statement_sha256":"8f1367898a63c281c84e3717b9230931f716ffbc910a0cdfa193393df5c021e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9592,"rank":9592,"depth":4,"x":2532.28,"y":1623.081,"cluster":"moduli-theory"},{"id":"stacks:0GNQ","tag":"0GNQ","title":"Functors on module categories · Lemma 0GNQ","summary":"Let A be a ring. Let B be an additive category with arbitrary direct sums and cokernels. There is an equivalence of categories between • the category of functors F : Mod_A → B which are right exact and commute with arbitrary direct sums, and • the category of pairs (K, kappa) where K ∈ Ob(B) and kappa : A → End_B(K) is a ring homomorphism given by the rule sending F to F(A) with its natural A-action.","statement_latex":"Let $A$ be a ring. Let $\\mathcal{B}$ be an additive category\nwith arbitrary direct sums and cokernels. There is an equivalence\nof categories between\n\\begin{enumerate}\n\\item the category of functors $F : \\text{Mod}_A \\to \\mathcal{B}$\nwhich are right exact and commute with arbitrary direct sums, and\n\\item the category of pairs $(K, \\kappa)$ where $K \\in \\Ob(\\mathcal{B})$\nand $\\kappa : A \\to \\text{End}_\\mathcal{B}(K)$ is a ring homomorphism\n\\end{enumerate}\ngiven by the rule sending $F$ to $F(A)$ with its natural $A$-action.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors on module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNQ","source_file":"functors.tex","source_line":415,"source_end_line":427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L415-L427","statement_sha256":"b496fdfbe59f887cdf976f871bd173c2639f1ef8909ebc3f19ea969ae3296860","origin":"The Stacks Project","memory_eligible":false,"source_rank":9593,"rank":9593,"depth":6,"x":2421.93,"y":1612.858,"cluster":"moduli-theory"},{"id":"stacks:0GNS","tag":"0GNS","title":"Functors between categories of modules · Lemma 0GNS","summary":"Let A and B be rings. Let F : Mod_A → Mod_B be a functor. The following are equivalent • F is isomorphic to the functor M ↦ M ⊗_A K for some A ⊗_Z B-module K, • F is right exact and commutes with all direct sums, • F commutes with all colimits, • F has a right adjoint G.","statement_latex":"Let $A$ and $B$ be rings. Let $F : \\text{Mod}_A \\to \\text{Mod}_B$\nbe a functor. The following are equivalent\n\\begin{enumerate}\n\\item $F$ is isomorphic to the functor $M \\mapsto M \\otimes_A K$\nfor some $A \\otimes_\\mathbf{Z} B$-module $K$,\n\\item $F$ is right exact and commutes with all direct sums,\n\\item $F$ commutes with all colimits,\n\\item $F$ has a right adjoint $G$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNS","source_file":"functors.tex","source_line":445,"source_end_line":456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L445-L456","statement_sha256":"a472bfcf15831a563598d9c5c13c7f38877612ac11ff02e4b909d1939643b59e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9594,"rank":9594,"depth":4,"x":2513.007,"y":1556.879,"cluster":"moduli-theory"},{"id":"stacks:0GNV","tag":"0GNV","title":"Functors between categories of modules · Lemma 0GNV","summary":"Let R be a ring. Let A and B be R-algebras. There is an equivalence of categories between • the category of R-linear functors F : Mod_A → Mod_B which are right exact and commute with arbitrary direct sums, and • the category Mod_A ⊗_R B. given by sending K to the functor F in ([Tag 0GNU]).","statement_latex":"Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. There is an\nequivalence of categories between\n\\begin{enumerate}\n\\item the category of $R$-linear functors\n$F : \\text{Mod}_A \\to \\text{Mod}_B$ which\nare right exact and commute with arbitrary direct sums, and\n\\item the category $\\text{Mod}_{A \\otimes_R B}$.\n\\end{enumerate}\ngiven by sending $K$ to the functor $F$ in (\\ref{equation-FM-modules}).","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GNV","source_file":"functors.tex","source_line":521,"source_end_line":532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L521-L532","statement_sha256":"241b273438c54169650a14013a0334f78e6e24e55f134e34762c1e151a5242eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9595,"rank":9595,"depth":5,"x":2490.5,"y":1651.32,"cluster":"moduli-theory"},{"id":"stacks:0GP0","tag":"0GP0","title":"Functors between categories of modules · Lemma 0GP0","summary":"Let A and B be rings. If F : Mod_A → Mod_B is an equivalence of categories, then there exists an isomorphism A → B of rings and an invertible B-module L such that F is isomorphic to the functor M ↦ (M ⊗_A B) ⊗_B L.","statement_latex":"Let $A$ and $B$ be rings. If\n$$\nF : \\text{Mod}_A \\longrightarrow \\text{Mod}_B\n$$\nis an equivalence of categories, then there exists an isomorphism\n$A \\to B$ of rings and an invertible $B$-module $L$ such that\n$F$ is isomorphic to the functor $M \\mapsto (M \\otimes_A B) \\otimes_B L$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GP0","source_file":"functors.tex","source_line":591,"source_end_line":600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L591-L600","statement_sha256":"b37a475bf7d7f1ff27467b2400eb7e85db7a0b48b2d23364c88cb8426f95cd11","origin":"The Stacks Project","memory_eligible":false,"source_rank":9596,"rank":9596,"depth":6,"x":2430.24,"y":1567.629,"cluster":"moduli-theory"},{"id":"stacks:0GP1","tag":"0GP1","title":"Functors between categories of modules · Lemma 0GP1","summary":"Let R be a ring. Let A and B be R-algebras. If F : Mod_A → Mod_B is an R-linear equivalence of categories, then there exists an isomorphism A → B of R-algebras and an invertible B-module L such that F is isomorphic to the functor M ↦ (M ⊗_A B) ⊗_B L.","statement_latex":"Let $R$ be a ring. Let $A$ and $B$ be $R$-algebras. If\n$$\nF : \\text{Mod}_A \\longrightarrow \\text{Mod}_B\n$$\nis an $R$-linear equivalence of categories, then there exists an isomorphism\n$A \\to B$ of $R$-algebras and an invertible $B$-module $L$ such that\n$F$ is isomorphic to the functor $M \\mapsto (M \\otimes_A B) \\otimes_B L$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GP1","source_file":"functors.tex","source_line":662,"source_end_line":671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L662-L671","statement_sha256":"3070b4d8507a619396057fd60628284a96044692a4bfd89fde77057c3b372350","origin":"The Stacks Project","memory_eligible":false,"source_rank":9597,"rank":9597,"depth":7,"x":2543.652,"y":1595.57,"cluster":"moduli-theory"},{"id":"stacks:0GP4","tag":"0GP4","title":"Extending functors on categories of modules · Lemma 0GP4","summary":"Let A and B be rings. Let F : Mod^fp_A → Mod^fp_B be a functor. Then F extends uniquely to a functor F' : Mod_A → Mod_B which commutes with filtered colimits.","statement_latex":"Let $A$ and $B$ be rings. Let\n$F : \\text{Mod}^{fp}_A \\to \\text{Mod}^{fp}_B$ be a functor.\nThen $F$ extends uniquely to a functor\n$F' : \\text{Mod}_A \\to \\text{Mod}_B$\nwhich commutes with filtered colimits.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Extending functors on categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GP4","source_file":"functors.tex","source_line":713,"source_end_line":720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L713-L720","statement_sha256":"750e87948a35aa2adc0cbda2fa33eed183ec3c19a8c26a82aea56b52ba6b74f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9598,"rank":9598,"depth":4,"x":2436.003,"y":1639.95,"cluster":"moduli-theory"},{"id":"stacks:0GP6","tag":"0GP6","title":"Extending functors on categories of modules · Lemma 0GP6","summary":"With A, B, F, and F' as in Lemma [Tag 0GP4]. • If F is additive, then F' is additive and commutes with arbitrary direct sums, and • if F is right exact, then F' is right exact.","statement_latex":"With $A$, $B$, $F$, and $F'$ as in Lemma \\ref{lemma-functor-fp-modules}.\n\\begin{enumerate}\n\\item If $F$ is additive, then $F'$ is additive and\ncommutes with arbitrary direct sums, and\n\\item if $F$ is right exact, then $F'$ is right exact.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Extending functors on categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GP6","source_file":"functors.tex","source_line":739,"source_end_line":747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L739-L747","statement_sha256":"1dcd86bedbbf6e8408b1bac2a211b798d6b0b278f05bf9c478d5a77e31f9b9fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9599,"rank":9599,"depth":6,"x":2480.32,"y":1544.817,"cluster":"moduli-theory"},{"id":"stacks:0GP8","tag":"0GP8","title":"Extending functors on categories of modules · Lemma 0GP8","summary":"With A, B, F, and F' as in Lemma [Tag 0GP4]. Assume A is a coherent ring (Algebra, Definition [Tag 05CV]). If F is left exact, then F' is left exact.","statement_latex":"With $A$, $B$, $F$, and $F'$ as in Lemma \\ref{lemma-functor-fp-modules}.\nAssume $A$ is a coherent ring\n(Algebra, Definition \\ref{algebra-definition-coherent}).\nIf $F$ is left exact, then $F'$ is left exact.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Extending functors on categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GP8","source_file":"functors.tex","source_line":764,"source_end_line":770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L764-L770","statement_sha256":"7460a038d206e162dcf9b2963d24193d3573623df8eadeffc5e42507806ce922","origin":"The Stacks Project","memory_eligible":false,"source_rank":9600,"rank":9600,"depth":6,"x":2524.74,"y":1641.429,"cluster":"moduli-theory"},{"id":"stacks:0GP9","tag":"0GP9","title":"Extending functors on categories of modules · Lemma 0GP9","summary":"Let A and B be Noetherian rings. Let F : Mod^fg_A → Mod^fg_B be a functor. Then F extends uniquely to a functor F' : Mod_A → Mod_B which commutes with filtered colimits. If F is additive, then F' is additive and commutes with arbitrary direct sums. If F is exact, left exact, or right exact, so is F'.","statement_latex":"Let $A$ and $B$ be Noetherian rings. Let\n$F : \\text{Mod}^{fg}_A \\to \\text{Mod}^{fg}_B$ be a functor.\nThen $F$ extends uniquely to a functor $F' : \\text{Mod}_A \\to \\text{Mod}_B$\nwhich commutes with filtered colimits. If $F$ is additive, then\n$F'$ is additive and commutes with arbitrary direct sums.\nIf $F$ is exact, left exact, or right exact, so is $F'$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Extending functors on categories of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GP9","source_file":"functors.tex","source_line":780,"source_end_line":788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L780-L788","statement_sha256":"de3b979bc8dd235cd49ef660c082a6be9938ba3e9ed64da2c1005a106071a3f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9601,"rank":9601,"depth":7,"x":2412.817,"y":1594.77,"cluster":"moduli-theory"},{"id":"stacks:0FZD","tag":"0FZD","title":"Functors between categories of quasi-coherent modules · Lemma 0FZD","summary":"Let R be a ring. Let X and Y be schemes over R with X affine. There is an equivalence of categories between • the category of R-linear functors F : QCoh(O_X) → QCoh(O_Y) which are right exact and commute with arbitrary direct sums, and • the category QCoh(O_X ×_R Y) given by sending K to the functor F in ([Tag 0FZC]).","statement_latex":"Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$ with $X$ affine.\nThere is an equivalence of categories between\n\\begin{enumerate}\n\\item the category of $R$-linear functors\n$F : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$\nwhich are right exact and commute with arbitrary direct sums, and\n\\item the category $\\QCoh(\\mathcal{O}_{X \\times_R Y})$\n\\end{enumerate}\ngiven by sending $\\mathcal{K}$ to the functor $F$ in (\\ref{equation-FM-QCoh}).","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZD","source_file":"functors.tex","source_line":841,"source_end_line":852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L841-L852","statement_sha256":"c7755a7259b019872a06828c0e210d3535904a594eec6914ff5011168c2df2d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9602,"rank":9602,"depth":25,"x":2534.438,"y":1565.294,"cluster":"moduli-theory"},{"id":"stacks:0FZF","tag":"0FZF","title":"Functors between categories of quasi-coherent modules · Lemma 0FZF","summary":"In Lemma [Tag 0FZD] let F correspond to K in QCoh(O_X ×_R Y). We have • If f : X' → X is an affine morphism, then F ∘ f_* corresponds to (f × id_Y)^*K. • If g : Y' → Y is a flat morphism, then g^* ∘ F corresponds to (id_X × g)^*K. • If j : V → Y is an open immersion, then j^* ∘ F corresponds to K|_X ×_R V.","statement_latex":"In Lemma \\ref{lemma-functor-quasi-coherent-from-affine} let $F$\ncorrespond to $\\mathcal{K}$ in $\\QCoh(\\mathcal{O}_{X \\times_R Y})$.\nWe have\n\\begin{enumerate}\n\\item If $f : X' \\to X$ is an affine morphism, then $F \\circ f_*$\ncorresponds to $(f \\times \\text{id}_Y)^*\\mathcal{K}$.\n\\item If $g : Y' \\to Y$ is a flat morphism, then $g^* \\circ F$ corresponds to\n$(\\text{id}_X \\times g)^*\\mathcal{K}$.\n\\item If $j : V \\to Y$ is an open immersion, then $j^* \\circ F$\ncorresponds to $\\mathcal{K}|_{X \\times_R V}$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZF","source_file":"functors.tex","source_line":902,"source_end_line":915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L902-L915","statement_sha256":"811ceddca60a9e13a4211431be63b26cf64953fed4d1cffc937c163731eee008","origin":"The Stacks Project","memory_eligible":false,"source_rank":9603,"rank":9603,"depth":30,"x":2467.614,"y":1657.191,"cluster":"moduli-theory"},{"id":"stacks:0GPA","tag":"0GPA","title":"Functors between categories of quasi-coherent modules · Lemma 0GPA","summary":"Let R be a ring. Let X and Y be schemes over R. Assume X is quasi-compact with affine diagonal. Let F : QCoh(O_X) → QCoh(O_Y) be an R-linear, right exact functor which commutes with arbitrary direct sums. Then we can construct • a quasi-coherent module K on X ×_R Y, and • a natural transformation t : F → F_K where F_K denotes the functor ([Tag 0FZC]) such that t : F ∘ f_* → F_K ∘ f_* is an isomorphism for every morphism f : X' → X whose source is an affine scheme.","statement_latex":"Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$. Assume $X$\nis quasi-compact with affine diagonal. Let\n$F : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$\nbe an $R$-linear, right exact functor which commutes\nwith arbitrary direct sums. Then we can construct\n\\begin{enumerate}\n\\item a quasi-coherent module $\\mathcal{K}$ on $X \\times_R Y$, and\n\\item a natural transformation $t : F \\to F_\\mathcal{K}$\nwhere $F_\\mathcal{K}$ denotes the functor (\\ref{equation-FM-QCoh})\n\\end{enumerate}\nsuch that $t : F \\circ f_* \\to F_\\mathcal{K} \\circ f_*$ is an isomorphism\nfor every morphism $f : X' \\to X$ whose source is an affine scheme.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPA","source_file":"functors.tex","source_line":983,"source_end_line":997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L983-L997","statement_sha256":"b4d4ffcb2dd22b57f8fa163937f8f2a089add7aa5c33a49c69d330b9cb768e6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9604,"rank":9604,"depth":31,"x":2442.691,"y":1550.198,"cluster":"moduli-theory"},{"id":"stacks:0FZG","tag":"0FZG","title":"Functors between categories of quasi-coherent modules · Lemma 0FZG","summary":"In Lemma [Tag 0FZD] or in Lemma [Tag 0GPA] if F is an exact functor, then the corresponding object K of QCoh(O_X ×_R Y) is flat over X.","statement_latex":"In Lemma \\ref{lemma-functor-quasi-coherent-from-affine}\nor in Lemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal-pre}\nif $F$ is an exact functor, then the corresponding object\n$\\mathcal{K}$ of $\\QCoh(\\mathcal{O}_{X \\times_R Y})$ is flat over $X$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZG","source_file":"functors.tex","source_line":1129,"source_end_line":1135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1129-L1135","statement_sha256":"0b666c659a6d4478dbbd981d07d33cdb701e82bb24e20b5ee4d40d038a804e3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9605,"rank":9605,"depth":32,"x":2548.369,"y":1615.739,"cluster":"moduli-theory"},{"id":"stacks:0FZH","tag":"0FZH","title":"Functors between categories of quasi-coherent modules · Lemma 0FZH","summary":"Let R be a ring. Let X and Y be schemes over R. Assume X is quasi-compact with affine diagonal. There is an equivalence of categories between • the category of R-linear exact functors F : QCoh(O_X) → QCoh(O_Y) which commute with arbitrary direct sums, and • the full subcategory of QCoh(O_X ×_R Y) consisting of K such that • K is flat over X, • for F ∈ QCoh(O_X) we have R^qpr_2, *(pr_1^*F ⊗_O_X ×_R Y K) = 0 for q > 0. given by sending K to the functor F in ([Tag 0FZC]).","statement_latex":"Let $R$ be a ring. Let $X$ and $Y$ be schemes over $R$. Assume $X$ is\nquasi-compact with affine diagonal.\nThere is an equivalence of categories between\n\\begin{enumerate}\n\\item the category of $R$-linear exact functors\n$F : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$\nwhich commute with arbitrary direct sums, and\n\\item the full subcategory of $\\QCoh(\\mathcal{O}_{X \\times_R Y})$ consisting\nof $\\mathcal{K}$ such that\n\\begin{enumerate}\n\\item $\\mathcal{K}$ is flat over $X$,\n\\item for $\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have\n$R^q\\text{pr}_{2, *}(\\text{pr}_1^*\\mathcal{F}\n\\otimes_{\\mathcal{O}_{X \\times_R Y}} \\mathcal{K}) = 0$ for $q > 0$.\n\\end{enumerate}\n\\end{enumerate}\ngiven by sending $\\mathcal{K}$ to the functor $F$ in (\\ref{equation-FM-QCoh}).","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZH","source_file":"functors.tex","source_line":1145,"source_end_line":1164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1145-L1164","statement_sha256":"18bc5d7c46f90cc1ded054c985deb2d60511de6195567627e1d3bd4c95d67a2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9606,"rank":9606,"depth":33,"x":2416.193,"y":1627.506,"cluster":"moduli-theory"},{"id":"stacks:0GPC","tag":"0GPC","title":"Functors between categories of quasi-coherent modules · Lemma 0GPC","summary":"Let R be a ring. Let X, Y, Z be schemes over R. Assume X and Y are quasi-compact and have affine diagonal. Let F : QCoh(O_X) → QCoh(O_Y) and G : QCoh(O_Y) → QCoh(O_Z) be R-linear exact functors which commute with arbitrary direct sums. Let K in QCoh(O_X ×_R Y) and L in QCoh(O_Y ×_R Z) be the corresponding \"kernels\", see Lemma [Tag 0FZH]. Then G ∘ F corresponds to pr_13, *(pr_12^*K ⊗_O_X ×_R Y ×_R Z pr_23^*L) in QCoh(O_X ×_R Z).","statement_latex":"Let $R$ be a ring. Let $X$, $Y$, $Z$ be schemes over $R$. Assume\n$X$ and $Y$ are quasi-compact and have affine diagonal. Let\n$$\nF : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)\n\\quad\\text{and}\\quad\nG : \\QCoh(\\mathcal{O}_Y) \\to \\QCoh(\\mathcal{O}_Z)\n$$\nbe $R$-linear exact functors which commute with arbitrary direct sums.\nLet $\\mathcal{K}$ in $\\QCoh(\\mathcal{O}_{X \\times_R Y})$\nand $\\mathcal{L}$ in $\\QCoh(\\mathcal{O}_{Y \\times_R Z})$\nbe the corresponding ``kernels'', see\nLemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal}.\nThen $G \\circ F$ corresponds to\n$\\text{pr}_{13, *}(\\text{pr}_{12}^*\\mathcal{K}\n\\otimes_{\\mathcal{O}_{X \\times_R Y \\times_R Z}}\n\\text{pr}_{23}^*\\mathcal{L})$ in $\\QCoh(\\mathcal{O}_{X \\times_R Z})$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPC","source_file":"functors.tex","source_line":1257,"source_end_line":1275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1257-L1275","statement_sha256":"1d4df07916030750c6fe80dcb054d3185662cd85ee13115169cb81075935242c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9607,"rank":9607,"depth":34,"x":2505.216,"y":1542.868,"cluster":"moduli-theory"},{"id":"stacks:0FZI","tag":"0FZI","title":"Functors between categories of quasi-coherent modules · Lemma 0FZI","summary":"Let R, X, Y, and K be as in Lemma [Tag 0FZH] part (2). Then for any scheme T over R we have R^qpr_13, *(pr_12^*F ⊗_O_T ×_R X ×_R Y pr_23^*K) = 0 for F quasi-coherent on T ×_R X and q > 0.","statement_latex":"Let $R$, $X$, $Y$, and $\\mathcal{K}$ be as in\nLemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal} part (2).\nThen for any scheme $T$ over $R$ we have\n$$\nR^q\\text{pr}_{13, *}(\\text{pr}_{12}^*\\mathcal{F}\n\\otimes_{\\mathcal{O}_{T \\times_R X \\times_R Y}}\n\\text{pr}_{23}^*\\mathcal{K}) = 0\n$$\nfor $\\mathcal{F}$ quasi-coherent on $T \\times_R X$ and $q > 0$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZI","source_file":"functors.tex","source_line":1384,"source_end_line":1395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1384-L1395","statement_sha256":"1092b1f8a3e8bfe4f719512e7c185aaddc69c8ef487f849567094575d7d4aa6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9608,"rank":9608,"depth":34,"x":2507.656,"y":1657.066,"cluster":"moduli-theory"},{"id":"stacks:0FZJ","tag":"0FZJ","title":"Functors between categories of quasi-coherent modules · Lemma 0FZJ","summary":"In Lemma [Tag 0FZH] let F and K correspond. If X is separated and flat over R, then there is a surjection O_X boxtimes F(O_X) → K.","statement_latex":"In Lemma \\ref{lemma-functor-quasi-coherent-from-affine-diagonal}\nlet $F$ and $\\mathcal{K}$ correspond. If $X$ is separated and\nflat over $R$, then there is a surjection\n$\\mathcal{O}_X \\boxtimes F(\\mathcal{O}_X) \\to \\mathcal{K}$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZJ","source_file":"functors.tex","source_line":1431,"source_end_line":1437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1431-L1437","statement_sha256":"a9f865d5a8a5e44928e50a9ea8abf84fc12f08fe0693280430da24e0170e363b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9609,"rank":9609,"depth":35,"x":2412.993,"y":1573.325,"cluster":"moduli-theory"},{"id":"stacks:0GPE","tag":"0GPE","title":"Gabriel-Rosenberg reconstruction · Lemma 0GPE","summary":"Let X be a quasi-compact and quasi-separated scheme. Let F be a quasi-coherent O_X-module. Then F is a categorically compact object of QCoh(O_X) if and only if F is of finite presentation.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $\\mathcal{F}$ is a categorically compact object of\n$\\QCoh(\\mathcal{O}_X)$ if and only if $\\mathcal{F}$ is of\nfinite presentation.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Gabriel-Rosenberg reconstruction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPE","source_file":"functors.tex","source_line":1511,"source_end_line":1518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1511-L1518","statement_sha256":"79bfe90daefec70648d81258f4d07f09f505db48e7df8092afd1f8c59af5659d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9610,"rank":9610,"depth":19,"x":2551.617,"y":1581.453,"cluster":"moduli-theory"},{"id":"stacks:0GPF","tag":"0GPF","title":"Gabriel-Rosenberg reconstruction · Lemma 0GPF","summary":"Let X be an affine scheme. Let F be a finitely presented O_X-module. Let E be a nonzero quasi-coherent O_X-module. If Supp(E) ⊂ Supp(F), then there exists a nonzero map F → E.","statement_latex":"Let $X$ be an affine scheme. Let $\\mathcal{F}$ be a finitely presented\n$\\mathcal{O}_X$-module. Let $\\mathcal{E}$ be a nonzero quasi-coherent\n$\\mathcal{O}_X$-module. If\n$\\text{Supp}(\\mathcal{E}) \\subset \\text{Supp}(\\mathcal{F})$,\nthen there exists a nonzero map $\\mathcal{F} \\to \\mathcal{E}$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Gabriel-Rosenberg reconstruction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPF","source_file":"functors.tex","source_line":1537,"source_end_line":1544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1537-L1544","statement_sha256":"af497b190430d39ae52a422cfa097928c5985f036d2a623a4f482c9324cddfe6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9611,"rank":9611,"depth":16,"x":2441.71,"y":1654.881,"cluster":"moduli-theory"},{"id":"stacks:0GPG","tag":"0GPG","title":"Gabriel-Rosenberg reconstruction · Lemma 0GPG","summary":"Let X be a quasi-compact and quasi-separated scheme. Let F be a finitely presented O_X-module. The following two subcategories of QCoh(O_X) are equal • the full subcategory A ⊂ QCoh(O_X) whose objects are the quasi-coherent modules whose support is (set theoretically) contained in Supp(F), • the smallest Serre subcategory B ⊂ QCoh(O_X) containing F closed under extensions and arbitrary direct sums.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module.\nThe following two subcategories of $\\QCoh(\\mathcal{O}_X)$ are equal\n\\begin{enumerate}\n\\item the full subcategory $\\mathcal{A} \\subset \\QCoh(\\mathcal{O}_X)$\nwhose objects are the quasi-coherent modules\nwhose support is (set theoretically) contained in $\\text{Supp}(\\mathcal{F})$,\n\\item the smallest Serre subcategory $\\mathcal{B} \\subset \\QCoh(\\mathcal{O}_X)$\ncontaining\n$\\mathcal{F}$ closed under extensions and arbitrary direct sums.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Gabriel-Rosenberg reconstruction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPG","source_file":"functors.tex","source_line":1575,"source_end_line":1588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1575-L1588","statement_sha256":"0d51a21d132d06e1605a5182938c3ce950bf0e987e21686c99ed523da01a796f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9612,"rank":9612,"depth":17,"x":2463.938,"y":1537.168,"cluster":"moduli-theory"},{"id":"stacks:0GPH","tag":"0GPH","title":"Gabriel-Rosenberg reconstruction · Lemma 0GPH","summary":"Let X be a quasi-compact and quasi-separated scheme. Let Z ⊂ X be a closed subset such that U = X setminus Z is quasi-compact. Let A ⊂ QCoh(O_X) be the full subcategory whose objects are the quasi-coherent modules supported on Z. Then the restriction functor QCoh(O_X) → QCoh(O_U) induces an equivalence QCoh(O_X)/A ≅ QCoh(O_U).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $Z \\subset X$ be a closed subset such that $U = X \\setminus Z$\nis quasi-compact. Let $\\mathcal{A} \\subset \\QCoh(\\mathcal{O}_X)$\nbe the full subcategory whose objects are the quasi-coherent modules\nsupported on $Z$. Then the restriction functor\n$\\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_U)$ induces\nan equivalence $\\QCoh(\\mathcal{O}_X)/\\mathcal{A} \\cong \\QCoh(\\mathcal{O}_U)$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Gabriel-Rosenberg reconstruction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPH","source_file":"functors.tex","source_line":1644,"source_end_line":1653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1644-L1653","statement_sha256":"a0849127d4970725449c734d9233fcdd55e2417ccf311404d0d857747d5814ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":9613,"rank":9613,"depth":8,"x":2542.997,"y":1637.59,"cluster":"moduli-theory"},{"id":"stacks:0GPI","tag":"0GPI","title":"Gabriel-Rosenberg reconstruction · Lemma 0GPI","summary":"Let X be a quasi-compact and quasi-separated scheme. If QCoh(O_X) is equivalent to the category of modules over a ring, then X is affine.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nIf $\\QCoh(\\mathcal{O}_X)$ is equivalent to the category\nof modules over a ring, then $X$ is affine.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Gabriel-Rosenberg reconstruction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPI","source_file":"functors.tex","source_line":1666,"source_end_line":1671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1666-L1671","statement_sha256":"252796c9a502731bb2e779681e1d9e4582b6e6821fbd6ffadfcc98df1ad6ecb2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9614,"rank":9614,"depth":23,"x":2402.564,"y":1608.107,"cluster":"moduli-theory"},{"id":"stacks:0GPJ","tag":"0GPJ","title":"Gabriel-Rosenberg reconstruction · Proposition 0GPJ","summary":"Special case of [Brandenburg] Let X and Y be quasi-compact and quasi-separated schemes. If F : QCoh(O_X) → QCoh(O_Y) is an equivalence, then there exists an isomorphism f : Y → X of schemes and an invertible O_Y-module L such that F(F) = f^*F ⊗ L.","statement_latex":"\\begin{reference}\nSpecial case of \\cite[Theorem 1.2]{Brandenburg}\n\\end{reference}\nLet $X$ and $Y$ be quasi-compact and quasi-separated schemes.\nIf $F : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$\nis an equivalence, then there exists an isomorphism\n$f : Y \\to X$ of schemes and an invertible $\\mathcal{O}_Y$-module\n$\\mathcal{L}$ such that $F(\\mathcal{F}) = f^*\\mathcal{F} \\otimes \\mathcal{L}$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Gabriel-Rosenberg reconstruction","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPJ","source_file":"functors.tex","source_line":1747,"source_end_line":1757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1747-L1757","statement_sha256":"b6506e1ba6178275db5a603e7a716dc1ab8573409587e0b2960a1976a83b9ea7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9615,"rank":9615,"depth":24,"x":2531.069,"y":1549.602,"cluster":"moduli-theory"},{"id":"stacks:0FZL","tag":"0FZL","title":"Functors between categories of coherent modules · Lemma 0FZL","summary":"Let X and Y be Noetherian schemes. Let F : Coh(O_X) → Coh(O_Y) be a functor. Then F extends uniquely to a functor QCoh(O_X) → QCoh(O_Y) which commutes with filtered colimits. If F is additive, then its extension commutes with arbitrary direct sums. If F is exact, left exact, or right exact, so is its extension.","statement_latex":"Let $X$ and $Y$ be Noetherian schemes. Let\n$F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$\nbe a functor. Then $F$ extends uniquely to a functor\n$\\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$\nwhich commutes with filtered colimits.\nIf $F$ is additive, then its extension commutes with arbitrary direct sums.\nIf $F$ is exact, left exact, or right exact, so is its extension.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZL","source_file":"functors.tex","source_line":1825,"source_end_line":1834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1825-L1834","statement_sha256":"16e8e2453a35895091b7fc14d6ed74b576881fbce6fd9a5d1dd471b3d9d2540d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9616,"rank":9616,"depth":17,"x":2482.899,"y":1666.766,"cluster":"moduli-theory"},{"id":"stacks:0GPK","tag":"0GPK","title":"Functors between categories of coherent modules · Lemma 0GPK","summary":"Let X and Y be Noetherian schemes. Let F : Coh(O_X) → Coh(O_Y) be an equivalence of categories. Then there is an isomorphism f : Y → X and an invertible O_Y-module L such that F(F) = f^*F ⊗ L.","statement_latex":"Let $X$ and $Y$ be Noetherian schemes. Let\n$F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$\nbe an equivalence of categories. Then there is an isomorphism $f : Y \\to X$\nand an invertible $\\mathcal{O}_Y$-module $\\mathcal{L}$\nsuch that $F(\\mathcal{F}) = f^*\\mathcal{F} \\otimes \\mathcal{L}$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPK","source_file":"functors.tex","source_line":1892,"source_end_line":1899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1892-L1899","statement_sha256":"0641c24efe7c3d3010539f9a6d311414c95383b3cf2c77d04fdfa0dee3736705","origin":"The Stacks Project","memory_eligible":false,"source_rank":9617,"rank":9617,"depth":25,"x":2423.653,"y":1551.971,"cluster":"moduli-theory"},{"id":"stacks:0FZM","tag":"0FZM","title":"Functors between categories of coherent modules · Lemma 0FZM","summary":"Let f : V → X be a quasi-finite separated morphism of Noetherian schemes. If there exists a coherent O_V-module K whose support is V such that f_*K is coherent and R^qf_*K = 0, then f is finite.","statement_latex":"Let $f : V \\to X$ be a quasi-finite separated morphism of Noetherian\nschemes. If there exists a coherent $\\mathcal{O}_V$-module $\\mathcal{K}$\nwhose support is $V$ such that $f_*\\mathcal{K}$ is coherent and\n$R^qf_*\\mathcal{K} = 0$, then $f$ is finite.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZM","source_file":"functors.tex","source_line":1920,"source_end_line":1926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1920-L1926","statement_sha256":"e0ee46e886b44a46858cee6de8163e5e587c3a4e17812801b6f54796d49288f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9618,"rank":9618,"depth":50,"x":2560.904,"y":1603.475,"cluster":"moduli-theory"},{"id":"stacks:0FZN","tag":"0FZN","title":"Functors between categories of coherent modules · Lemma 0FZN","summary":"Let k be a field. Let X, Y be finite type schemes over k with X separated. There is an equivalence of categories between • the category of k-linear exact functors F : Coh(O_X) → Coh(O_Y), and • the category of coherent O_X × Y-modules K which are flat over X and have support finite over Y given by sending K to the restriction of the functor ([Tag 0FZC]) to Coh(O_X).","statement_latex":"Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with\n$X$ separated. There is an equivalence of categories between\n\\begin{enumerate}\n\\item the category of $k$-linear exact functors\n$F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$, and\n\\item the category of coherent $\\mathcal{O}_{X \\times Y}$-modules\n$\\mathcal{K}$ which are flat over $X$ and have support finite over $Y$\n\\end{enumerate}\ngiven by sending $\\mathcal{K}$ to the restriction of the functor\n(\\ref{equation-FM-QCoh}) to $\\textit{Coh}(\\mathcal{O}_X)$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZN","source_file":"functors.tex","source_line":1963,"source_end_line":1975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L1963-L1975","statement_sha256":"3fa9a12b18b20de4ebe7350b0a65dc87b5068c65f585d6efeb62fe2e02dac0f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9619,"rank":9619,"depth":51,"x":2416.99,"y":1643.732,"cluster":"moduli-theory"},{"id":"stacks:0FZP","tag":"0FZP","title":"Functors between categories of coherent modules · Lemma 0FZP","summary":"Let f : X → Y be a finite type separated morphism of schemes. Let F be a finite type quasi-coherent module on X with support finite over Y and with L = f_*F an invertible O_X-module. Then there exists a section s : Y → X such that F ≅ s_*L.","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of schemes. Let\n$\\mathcal{F}$ be a finite type quasi-coherent module on $X$\nwith support finite over $Y$\nand with $\\mathcal{L} = f_*\\mathcal{F}$ an invertible $\\mathcal{O}_X$-module.\nThen there exists a section $s : Y \\to X$ such that\n$\\mathcal{F} \\cong s_*\\mathcal{L}$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZP","source_file":"functors.tex","source_line":2038,"source_end_line":2046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L2038-L2046","statement_sha256":"56b6ee631ff8f657fb555b9e9b967d55b4021437e201bf9e219f4956fd64d54e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9620,"rank":9620,"depth":0,"x":2491.394,"y":1531.401,"cluster":"moduli-theory"},{"id":"stacks:0FZQ","tag":"0FZQ","title":"Functors between categories of coherent modules · Lemma 0FZQ","summary":"Let f : X → Y be a finite type separated morphism of schemes with a section s : Y → X. Let F be a finite type quasi-coherent module on X, set theoretically supported on s(Y) with L = f_*F an invertible O_X-module. If Y is reduced, then F ≅ s_*L.","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of schemes with a section\n$s : Y \\to X$. Let $\\mathcal{F}$ be a finite type quasi-coherent module\non $X$, set theoretically supported on $s(Y)$ with\n$\\mathcal{L} = f_*\\mathcal{F}$\nan invertible $\\mathcal{O}_X$-module. If $Y$ is reduced, then\n$\\mathcal{F} \\cong s_*\\mathcal{L}$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZQ","source_file":"functors.tex","source_line":2055,"source_end_line":2063,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L2055-L2063","statement_sha256":"d513e9c73ef8feb5c326da6784ab909293668c3079853b8ba909ad06151efe61","origin":"The Stacks Project","memory_eligible":false,"source_rank":9621,"rank":9621,"depth":1,"x":2527.167,"y":1657.541,"cluster":"moduli-theory"},{"id":"stacks:0FZR","tag":"0FZR","title":"Functors between categories of coherent modules · Lemma 0FZR","summary":"Weak version of the result in [Gabriel] stating that the category of quasi-coherent modules determines the isomorphism class of a scheme. Let k be a field. Let X, Y be finite type schemes over k with X separated and Y reduced. If there is a k-linear equivalence F : Coh(O_X) → Coh(O_Y) of categories, then there is an isomorphism f : Y → X over k and an invertible O_Y-module L such that F(F) = f^*F ⊗ L.","statement_latex":"\\begin{reference}\nWeak version of the result in \\cite{Gabriel}\nstating that the category of quasi-coherent modules\ndetermines the isomorphism class of a scheme.\n\\end{reference}\nLet $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with\n$X$ separated and $Y$ reduced. If there is a $k$-linear equivalence\n$F : \\textit{Coh}(\\mathcal{O}_X) \\to \\textit{Coh}(\\mathcal{O}_Y)$\nof categories, then there is an isomorphism $f : Y \\to X$\nover $k$ and an invertible $\\mathcal{O}_Y$-module $\\mathcal{L}$\nsuch that $F(\\mathcal{F}) = f^*\\mathcal{F} \\otimes \\mathcal{L}$.","area":"Moduli Theory","chapter":"Functors and Morphisms","chapter_id":"functors","section":"Functors between categories of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZR","source_file":"functors.tex","source_line":2074,"source_end_line":2087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/functors.tex#L2074-L2087","statement_sha256":"45d76103bc4839896730d9abe277fdf867c80d3bdcba72e8e3a033ab8d16f08f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9622,"rank":9622,"depth":26,"x":2398.258,"y":1584.201,"cluster":"moduli-theory"},{"id":"stacks:0FY4","tag":"0FY4","title":"Serre functors · Lemma 0FY4","summary":"Let k be a field. Let T be a k-linear triangulated category such that dim_k Hom_T(X, Y) < ∞ for all X, Y ∈ Ob(T). The following are equivalent • there exists a k-linear equivalence S : T → T and k-linear isomorphisms c_X, Y : Hom_T(X, Y) → Hom_T(Y, S(X))^vee functorial in X, Y ∈ Ob(T), • for every X ∈ Ob(T) the functor Y ↦ Hom_T(X, Y)^vee is representable and the functor Y ↦ Hom_T(Y, X)^vee is corepresentable.","statement_latex":"Let $k$ be a field. Let $\\mathcal{T}$ be a $k$-linear\ntriangulated category such that $\\dim_k \\Hom_\\mathcal{T}(X, Y) < \\infty$\nfor all $X, Y \\in \\Ob(\\mathcal{T})$. The following are equivalent\n\\begin{enumerate}\n\\item there exists a $k$-linear equivalence\n$S : \\mathcal{T} \\to \\mathcal{T}$ and $k$-linear isomorphisms\n$c_{X, Y} : \\Hom_\\mathcal{T}(X, Y) \\to \\Hom_\\mathcal{T}(Y, S(X))^\\vee$\nfunctorial in $X, Y \\in \\Ob(\\mathcal{T})$,\n\\item for every $X \\in \\Ob(\\mathcal{T})$\nthe functor $Y \\mapsto \\Hom_\\mathcal{T}(X, Y)^\\vee$\nis representable and the functor $Y \\mapsto \\Hom_\\mathcal{T}(Y, X)^\\vee$\nis corepresentable.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Serre functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FY4","source_file":"equiv.tex","source_line":110,"source_end_line":125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L110-L125","statement_sha256":"f14a29cd97c5d2440e505de6a9a7dbb70445e251ab023e68d22716b47c3d86ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":9623,"rank":9623,"depth":1,"x":215.08,"y":799.213,"cluster":"derived-categories"},{"id":"stacks:0FY5","tag":"0FY5","title":"Serre functors · Definition 0FY5","summary":"Let k be a field. Let T be a k-linear triangulated category such that dim_k Hom_T(X, Y) < ∞ for all X, Y ∈ Ob(T). We say a Serre functor exists if the equivalent conditions of Lemma [Tag 0FY4] are satisfied. In this case a Serre functor is a k-linear equivalence S : T → T endowed with k-linear isomorphisms c_X, Y : Hom_T(X, Y) → Hom_T(Y, S(X))^vee functorial in X, Y ∈ Ob(T).","statement_latex":"Let $k$ be a field. Let $\\mathcal{T}$ be a $k$-linear\ntriangulated category such that $\\dim_k \\Hom_\\mathcal{T}(X, Y) < \\infty$\nfor all $X, Y \\in \\Ob(\\mathcal{T})$. We say {\\it a Serre functor\nexists} if the equivalent conditions of Lemma \\ref{lemma-Serre-functor-exists}\nare satisfied. In this case a {\\it Serre functor} is a $k$-linear equivalence\n$S : \\mathcal{T} \\to \\mathcal{T}$ endowed with $k$-linear isomorphisms\n$c_{X, Y} : \\Hom_\\mathcal{T}(X, Y) \\to \\Hom_\\mathcal{T}(Y, S(X))^\\vee$\nfunctorial in $X, Y \\in \\Ob(\\mathcal{T})$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Serre functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FY5","source_file":"equiv.tex","source_line":160,"source_end_line":170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L160-L170","statement_sha256":"0108de5a46abb670cd0e7e4be9049a532848d4c6547a53cb0bd1e21bc8ab4f87","origin":"The Stacks Project","memory_eligible":false,"source_rank":9624,"rank":9624,"depth":2,"x":145.042,"y":583.523,"cluster":"derived-categories"},{"id":"stacks:0FY6","tag":"0FY6","title":"Serre functors · Lemma 0FY6","summary":"In the situation of Definition [Tag 0FY5]. If a Serre functor exists, then it is unique up to unique isomorphism and it is an exact functor of triangulated categories.","statement_latex":"In the situation of Definition \\ref{definition-Serre-functor}.\nIf a Serre functor exists, then it is unique up to unique isomorphism and\nit is an exact functor of triangulated categories.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Serre functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FY6","source_file":"equiv.tex","source_line":244,"source_end_line":249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L244-L249","statement_sha256":"cb6b827bac6b819507dac233332998802b7a9333f13ca3dbdf70f421d921ed46","origin":"The Stacks Project","memory_eligible":false,"source_rank":9625,"rank":9625,"depth":8,"x":370.383,"y":702.958,"cluster":"derived-categories"},{"id":"stacks:0FY8","tag":"0FY8","title":"Examples of Serre functors · Lemma 0FY8","summary":"Let k be a field. Let X be a proper scheme over k which is Gorenstein. Consider the complex ω_X^bullet of Duality for Schemes, Lemmas [Tag 0FVV]. Then the functor S : D_perf(O_X) → D_perf(O_X), K ↦ S(K) = ω_X^bullet ⊗_O_X^L K is a Serre functor.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is Gorenstein.\nConsider the complex $\\omega_X^\\bullet$ of\nDuality for Schemes, Lemmas \\ref{duality-lemma-duality-proper-over-field}.\nThen the functor\n$$\nS : D_{perf}(\\mathcal{O}_X) \\longrightarrow D_{perf}(\\mathcal{O}_X),\\quad\nK \\longmapsto S(K) = \\omega_X^\\bullet \\otimes_{\\mathcal{O}_X}^\\mathbf{L} K\n$$\nis a Serre functor.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Examples of Serre functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FY8","source_file":"equiv.tex","source_line":372,"source_end_line":383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L372-L383","statement_sha256":"683c0328351039b983a45940b13c7c1a35a6599650c87bc1dd2c810a8de93570","origin":"The Stacks Project","memory_eligible":false,"source_rank":9626,"rank":9626,"depth":48,"x":107.89,"y":742.795,"cluster":"derived-categories"},{"id":"stacks:0FYA","tag":"0FYA","title":"Characterizing coherent modules · Lemma 0FYA","summary":"With k, n, and R as above, for an object K of D(R) the following are equivalent • ∑_i ∈ Z dim_k H^i(K) < ∞, and • K is a compact object.","statement_latex":"With $k$, $n$, and $R$ as above, for an object $K$ of $D(R)$\nthe following are equivalent\n\\begin{enumerate}\n\\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k H^i(K) < \\infty$, and\n\\item $K$ is a compact object.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Characterizing coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYA","source_file":"equiv.tex","source_line":458,"source_end_line":466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L458-L466","statement_sha256":"334289ccb3ff6553f4aa2173f817fe3673908805082eaf83a70dff7e9f11a52a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9627,"rank":9627,"depth":18,"x":269.587,"y":564.283,"cluster":"derived-categories"},{"id":"stacks:0FYB","tag":"0FYB","title":"Characterizing coherent modules · Lemma 0FYB","summary":"Let k be a field. Let n ≥ 0. Let K ∈ D_QCoh(O_P^n_k). The following are equivalent • K is in D^b_Coh(O_P^n_k), • ∑_i ∈ Z dim_k H^i(P^n_k, E ⊗^L K) < ∞ for each perfect object E of D(O_P^n_k), • ∑_i ∈ Z dim_k Ext^i_P^n_k(E, K) < ∞ for each perfect object E of D(O_P^n_k), • ∑_i ∈ Z dim_k H^i(P^n_k, K ⊗^L O_P^n_k(d)) < ∞ for d = 0, 1, …, n.","statement_latex":"Let $k$ be a field. Let $n \\geq 0$. Let\n$K \\in D_\\QCoh(\\mathcal{O}_{\\mathbf{P}^n_k})$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ is in $D^b_{\\textit{Coh}}(\\mathcal{O}_{\\mathbf{P}^n_k})$,\n\\item $\\sum_{i \\in \\mathbf{Z}}\n\\dim_k H^i(\\mathbf{P}^n_k, E \\otimes^\\mathbf{L} K) < \\infty$\nfor each perfect object $E$ of\n$D(\\mathcal{O}_{\\mathbf{P}^n_k})$,\n\\item $\\sum_{i \\in \\mathbf{Z}}\n\\dim_k \\Ext^i_{\\mathbf{P}^n_k}(E, K) < \\infty$\nfor each perfect object $E$ of $D(\\mathcal{O}_{\\mathbf{P}^n_k})$,\n\\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k H^i(\\mathbf{P}^n_k,\nK \\otimes^\\mathbf{L} \\mathcal{O}_{\\mathbf{P}^n_k}(d)) < \\infty$\nfor $d = 0, 1, \\ldots, n$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Characterizing coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYB","source_file":"equiv.tex","source_line":549,"source_end_line":567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L549-L567","statement_sha256":"87e9e793976e5c7bb8aba5915e36427f7faa0e69fbc422a25bbc3ca7181550dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9628,"rank":9628,"depth":33,"x":293.935,"y":787.906,"cluster":"derived-categories"},{"id":"stacks:0FYC","tag":"0FYC","title":"Characterizing coherent modules · Lemma 0FYC","summary":"Let X be a scheme proper over a field k. Let K ∈ D^b_Coh(O_X) and let E in D(O_X) be perfect. Then ∑_i ∈ Z dim_k Ext^i_X(E, K) < ∞.","statement_latex":"Let $X$ be a scheme proper over a field $k$. Let\n$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and let $E$ in $D(\\mathcal{O}_X)$\nbe perfect. Then\n$\\sum_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_X(E, K) < \\infty$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Characterizing coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYC","source_file":"equiv.tex","source_line":620,"source_end_line":626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L620-L626","statement_sha256":"236f9a76ea47bee447a1098fa9133ca136ea2f0b905aebcd81787da638b0e133","origin":"The Stacks Project","memory_eligible":false,"source_rank":9629,"rank":9629,"depth":35,"x":95.937,"y":636.663,"cluster":"derived-categories"},{"id":"stacks:0FYD","tag":"0FYD","title":"Characterizing coherent modules · Lemma 0FYD","summary":"In the projective case this is [Rouquier-dimensions] and implicit in [BvdB] Let X be a proper scheme over a field k. Let K ∈ Ob(D_QCoh(O_X)). The following are equivalent • K ∈ D^b_Coh(O_X), and • ∑_i ∈ Z dim_k Ext^i_X(E, K) < ∞ for all perfect E in D(O_X).","statement_latex":"\\begin{reference}\nIn the projective case this is \\cite[Lemma 7.46]{Rouquier-dimensions}\nand implicit in \\cite[Theorem A.1]{BvdB}\n\\end{reference}\nLet $X$ be a proper scheme over a field $k$. Let\n$K \\in \\Ob(D_\\QCoh(\\mathcal{O}_X))$. The following are equivalent\n\\begin{enumerate}\n\\item $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$, and\n\\item $\\sum_{i \\in \\mathbf{Z}} \\dim_k \\Ext^i_X(E, K) < \\infty$\nfor all perfect $E$ in $D(\\mathcal{O}_X)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Characterizing coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYD","source_file":"equiv.tex","source_line":638,"source_end_line":651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L638-L651","statement_sha256":"3334accd4eab032e34696293f0358986d659212978a225cb5b8428908e33d95f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9630,"rank":9630,"depth":41,"x":363.85,"y":635.838,"cluster":"derived-categories"},{"id":"stacks:0FYF","tag":"0FYF","title":"A representability theorem · Lemma 0FYF","summary":"Let D be a triangulated category. Let D' ⊂ D be a full triangulated subcategory. Let X ∈ Ob(D). The category of arrows E → X with E ∈ Ob(D') is filtered.","statement_latex":"Let $\\mathcal{D}$ be a triangulated category. Let\n$\\mathcal{D}' \\subset \\mathcal{D}$ be a full triangulated subcategory. Let\n$X \\in \\Ob(\\mathcal{D})$. The category of arrows $E \\to X$ with\n$E \\in \\Ob(\\mathcal{D}')$ is filtered.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"A representability theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYF","source_file":"equiv.tex","source_line":704,"source_end_line":710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L704-L710","statement_sha256":"abb01bcae0c2260a3d5aaf1e99552a91646040dd0a7adedc1b61188e614523e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9631,"rank":9631,"depth":1,"x":166.748,"y":788.63,"cluster":"derived-categories"},{"id":"stacks:0FYG","tag":"0FYG","title":"A representability theorem · Lemma 0FYG","summary":"[CKN] Let k be a field. Let D be a k-linear triangulated category which has direct sums and is compactly generated. Denote D_c the full subcategory of compact objects. Let H : D_c^opp → Vect_k be a k-linear cohomological functor such that dim_k H(X) < ∞ for all X ∈ Ob(D_c). Then H is isomorphic to the functor X ↦ Hom(X, Y) for some Y ∈ Ob(D).","statement_latex":"\\begin{reference}\n\\cite[Lemma 2.14]{CKN}\n\\end{reference}\nLet $k$ be a field. Let $\\mathcal{D}$ be a $k$-linear triangulated category\nwhich has direct sums and is compactly generated.\nDenote $\\mathcal{D}_c$ the full\nsubcategory of compact objects. Let $H : \\mathcal{D}_c^{opp} \\to \\text{Vect}_k$\nbe a $k$-linear cohomological functor such that\n$\\dim_k H(X) < \\infty$ for all $X \\in \\Ob(\\mathcal{D}_c)$.\nThen $H$ is isomorphic to the functor $X \\mapsto \\Hom(X, Y)$\nfor some $Y \\in \\Ob(\\mathcal{D})$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"A representability theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYG","source_file":"equiv.tex","source_line":734,"source_end_line":747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L734-L747","statement_sha256":"7d01e23d30a5dd370b1afa5b6b10742335849f92d9239fdfc20cbb36769c3814","origin":"The Stacks Project","memory_eligible":false,"source_rank":9632,"rank":9632,"depth":9,"x":189.241,"y":563.883,"cluster":"derived-categories"},{"id":"stacks:0FYH","tag":"0FYH","title":"A representability theorem · Theorem 0FYH","summary":"In the projective case this is [BvdB] Let X be a proper scheme over a field k. Let F : D_perf(O_X)^opp → Vect_k be a k-linear cohomological functor such that ∑_n ∈ Z dim_k F(E[n]) < ∞ for all E ∈ D_perf(O_X). Then F is isomorphic to a functor of the form E ↦ Hom_X(E, K) for some K ∈ D^b_Coh(O_X).","statement_latex":"\\begin{reference}\nIn the projective case this is \\cite[Theorem A.1]{BvdB}\n\\end{reference}\nLet $X$ be a proper scheme over a field $k$.\nLet $F : D_{perf}(\\mathcal{O}_X)^{opp} \\to \\text{Vect}_k$\nbe a $k$-linear cohomological functor such that\n$$\n\\sum\\nolimits_{n \\in \\mathbf{Z}} \\dim_k F(E[n]) < \\infty\n$$\nfor all $E \\in D_{perf}(\\mathcal{O}_X)$. Then $F$ is isomorphic to a functor\nof the form $E \\mapsto \\Hom_X(E, K)$ for some\n$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"A representability theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYH","source_file":"equiv.tex","source_line":814,"source_end_line":828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L814-L828","statement_sha256":"b99366f3a6d63784edb6c673ef3679fdec709bdbd42f6bf8dfc44219112f4ea7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9633,"rank":9633,"depth":42,"x":353.564,"y":742.561,"cluster":"derived-categories"},{"id":"stacks:0H4A","tag":"0H4A","title":"A representability theorem · Lemma 0H4A","summary":"Let X be a proper scheme over a field k which is regular. Let G : D_perf(O_X) → Vect_k be a k-linear homological functor such that ∑_n ∈ Z dim_k G(E[n]) < ∞ for all E ∈ D_perf(O_X). Then G is isomorphic to a functor of the form E ↦ Hom_X(K, E) for some K ∈ D_perf(O_X).","statement_latex":"Let $X$ be a proper scheme over a field $k$ which is regular.\nLet $G : D_{perf}(\\mathcal{O}_X) \\to \\text{Vect}_k$\nbe a $k$-linear homological functor such that\n$$\n\\sum\\nolimits_{n \\in \\mathbf{Z}} \\dim_k G(E[n]) < \\infty\n$$\nfor all $E \\in D_{perf}(\\mathcal{O}_X)$. Then $G$ is isomorphic to a functor\nof the form $E \\mapsto \\Hom_X(K, E)$ for some $K \\in D_{perf}(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"A representability theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4A","source_file":"equiv.tex","source_line":844,"source_end_line":854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L844-L854","statement_sha256":"f76e706217e82520b5d5442cfb54ab044a2413c41e41f7cb36470279b6decc8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9634,"rank":9634,"depth":43,"x":88.425,"y":704.007,"cluster":"derived-categories"},{"id":"stacks:0FYN","tag":"0FYN","title":"Existence of adjoints · Lemma 0FYN","summary":"Let k be a field. Let X and Y be proper schemes over k. If X is regular, then any k-linear exact functor F : D_perf(O_X) → D_perf(O_Y) has an exact right adjoint and an exact left adjoint.","statement_latex":"Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$.\nIf $X$ is regular, then any $k$-linear exact functor\n$F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nhas an exact right adjoint and an exact left adjoint.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Existence of adjoints","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYN","source_file":"equiv.tex","source_line":880,"source_end_line":886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L880-L886","statement_sha256":"f89bebaf742f61cfd6a36cb9a9bac938469358acec8c4bbdcf07aa7fba4ad712","origin":"The Stacks Project","memory_eligible":false,"source_rank":9635,"rank":9635,"depth":44,"x":315.179,"y":581.861,"cluster":"derived-categories"},{"id":"stacks:0FYQ","tag":"0FYQ","title":"Fourier-Mukai functors · Definition 0FYQ","summary":"Let S be a scheme. Let X and Y be schemes over S. Let K ∈ D(O_X ×_S Y). The exact functor Φ_K : D(O_X) → D(O_Y), M ↦ Rpr_2, *( Lpr_1^*M ⊗_O_X ×_S Y^L K) of triangulated categories is called a Fourier-Mukai functor and K is called a Fourier-Mukai kernel for this functor. Moreover, • if Φ_K sends D_QCoh(O_X) into D_QCoh(O_Y) then the resulting exact functor Φ_K : D_QCoh(O_X) → D_QCoh(O_Y) is called a Fourier-Mukai functor, • if Φ_K sends D_perf(O_X) into D_perf(O_Y) then…","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$.\nLet $K \\in D(\\mathcal{O}_{X \\times_S Y})$. The exact functor\n$$\n\\Phi_K : D(\\mathcal{O}_X) \\longrightarrow D(\\mathcal{O}_Y),\\quad\nM \\longmapsto R\\text{pr}_{2, *}(\nL\\text{pr}_1^*M \\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K)\n$$\nof triangulated categories is called a {\\it Fourier-Mukai functor}\nand $K$ is called a {\\it Fourier-Mukai kernel} for this functor.\nMoreover,\n\\begin{enumerate}\n\\item if $\\Phi_K$ sends $D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$\nthen the resulting exact functor\n$\\Phi_K : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$\nis called a Fourier-Mukai functor,\n\\item if $\\Phi_K$ sends $D_{perf}(\\mathcal{O}_X)$ into\n$D_{perf}(\\mathcal{O}_Y)$ then the resulting exact functor\n$\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nis called a Fourier-Mukai functor, and\n\\item if $X$ and $Y$ are Noetherian and $\\Phi_K$ sends\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ into $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$\nthen the resulting exact functor\n$\\Phi_K : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to\nD^b_{\\textit{Coh}}(\\mathcal{O}_Y)$\nis called a Fourier-Mukai functor.\nSimilarly for $D_{\\textit{Coh}}$, $D^+_{\\textit{Coh}}$, $D^-_{\\textit{Coh}}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fourier-Mukai functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYQ","source_file":"equiv.tex","source_line":922,"source_end_line":951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L922-L951","statement_sha256":"c4b3495813b79be861cac799666acce24ac74f0811bc40a4c90c5e4fb85c6d29","origin":"The Stacks Project","memory_eligible":false,"source_rank":9636,"rank":9636,"depth":0,"x":246.128,"y":800.827,"cluster":"derived-categories"},{"id":"stacks:0FYR","tag":"0FYR","title":"Fourier-Mukai functors · Lemma 0FYR","summary":"Let S be a scheme. Let X and Y be schemes over S. Let K ∈ D(O_X ×_S Y). The corresponding Fourier-Mukai functor Φ_K sends D_QCoh(O_X) into D_QCoh(O_Y) if K is in D_QCoh(O_X ×_S Y) and X → S is quasi-compact and quasi-separated.","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$.\nLet $K \\in D(\\mathcal{O}_{X \\times_S Y})$.\nThe corresponding Fourier-Mukai functor $\\Phi_K$ sends\n$D_\\QCoh(\\mathcal{O}_X)$ into $D_\\QCoh(\\mathcal{O}_Y)$\nif $K$ is in $D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$ and $X \\to S$ is\nquasi-compact and quasi-separated.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fourier-Mukai functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYR","source_file":"equiv.tex","source_line":953,"source_end_line":961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L953-L961","statement_sha256":"49f26c1ed438bf88590d25060a3ec78843e7d4773c163a41e3c296c664a2571a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9637,"rank":9637,"depth":30,"x":120.827,"y":599.971,"cluster":"derived-categories"},{"id":"stacks:0FYS","tag":"0FYS","title":"Fourier-Mukai functors · Lemma 0FYS","summary":"Let S be a scheme. Let X, Y, Z be schemes over S. Assume X → S, Y → S, and Z → S are quasi-compact and quasi-separated. Let K ∈ D_QCoh(O_X ×_S Y). Let K' ∈ D_QCoh(O_Y ×_S Z). Consider the Fourier-Mukai functors Φ_K : D_QCoh(O_X) → D_QCoh(O_Y) and Φ_K' : D_QCoh(O_Y) → D_QCoh(O_Z). If X and Z are tor independent over S and Y → S is flat, then Φ_K' ∘ Φ_K = Φ_K\" : D_QCoh(O_X) → D_QCoh(O_Z) where K\" = Rpr_13, *( Lpr_12^*K ⊗_O_X ×_S Y ×_S Z^L Lpr_23^*K') in D_QCoh(O_X ×_S Z).","statement_latex":"Let $S$ be a scheme. Let $X, Y, Z$ be schemes over $S$. Assume\n$X \\to S$, $Y \\to S$, and $Z \\to S$ are quasi-compact and quasi-separated.\nLet $K \\in D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$.\nLet $K' \\in D_\\QCoh(\\mathcal{O}_{Y \\times_S Z})$.\nConsider the Fourier-Mukai functors\n$\\Phi_K : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$\nand $\\Phi_{K'} : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_Z)$.\nIf $X$ and $Z$ are tor independent over $S$ and $Y \\to S$ is flat,\nthen\n$$\n\\Phi_{K'} \\circ \\Phi_K = \\Phi_{K''} :\nD_\\QCoh(\\mathcal{O}_X)\n\\longrightarrow\nD_\\QCoh(\\mathcal{O}_Z)\n$$\nwhere\n$$\nK'' = R\\text{pr}_{13, *}(\nL\\text{pr}_{12}^*K\n\\otimes_{\\mathcal{O}_{X \\times_S Y \\times_S Z}}^\\mathbf{L}\nL\\text{pr}_{23}^*K')\n$$\nin $D_\\QCoh(\\mathcal{O}_{X \\times_S Z})$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fourier-Mukai functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYS","source_file":"equiv.tex","source_line":982,"source_end_line":1007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L982-L1007","statement_sha256":"7706c9a0255d14acb1861653f1208b63ac1a209959c5f1925fc5dbfbdd2b8f76","origin":"The Stacks Project","memory_eligible":false,"source_rank":9638,"rank":9638,"depth":33,"x":375.012,"y":677.061,"cluster":"derived-categories"},{"id":"stacks:0FYT","tag":"0FYT","title":"Fourier-Mukai functors · Lemma 0FYT","summary":"Let S be a scheme. Let X and Y be schemes over S. Let K ∈ D(O_X ×_S Y). The corresponding Fourier-Mukai functor Φ_K sends D_perf(O_X) into D_perf(O_Y) if at least one of the following conditions is satisfied: • S is Noetherian, X → S and Y → S are of finite type, K ∈ D^b_Coh(O_X ×_S Y), the support of H^i(K) is proper over Y for all i, and K has finite tor dimension as an object of D(pr_2^-1O_Y), • X → S is of finite presentation and K can be represented by a bounded…","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$.\nLet $K \\in D(\\mathcal{O}_{X \\times_S Y})$.\nThe corresponding Fourier-Mukai functor $\\Phi_K$ sends\n$D_{perf}(\\mathcal{O}_X)$ into $D_{perf}(\\mathcal{O}_Y)$ if at least\none of the following conditions is satisfied:\n\\begin{enumerate}\n\\item $S$ is Noetherian, $X \\to S$ and $Y \\to S$ are of finite type,\n$K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_{X \\times_S Y})$, the support of $H^i(K)$\nis proper over $Y$ for all $i$, and $K$ has finite tor dimension\nas an object of $D(\\text{pr}_2^{-1}\\mathcal{O}_Y)$,\n\\item $X \\to S$ is of finite presentation and $K$ can be represented\nby a bounded complex $\\mathcal{K}^\\bullet$ of finitely presented\n$\\mathcal{O}_{X \\times_S Y}$-modules, flat over $Y$, with support\nproper over $Y$,\n\\item $X \\to S$ is a proper flat morphism of finite presentation\nand $K$ is perfect,\n\\item $S$ is Noetherian, $X \\to S$ is flat and proper, and $K$ is perfect\n\\item $X \\to S$ is a proper flat morphism of finite presentation\nand $K$ is $Y$-perfect,\n\\item $S$ is Noetherian, $X \\to S$ is flat and proper, and $K$ is\n$Y$-perfect.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fourier-Mukai functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYT","source_file":"equiv.tex","source_line":1064,"source_end_line":1088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1064-L1088","statement_sha256":"bc337ad617f29fb0eebdc35ab9ee0f8d3a2afea05a483e2d3553011b940dda06","origin":"The Stacks Project","memory_eligible":false,"source_rank":9639,"rank":9639,"depth":40,"x":125.324,"y":764.539,"cluster":"derived-categories"},{"id":"stacks:0FYU","tag":"0FYU","title":"Fourier-Mukai functors · Lemma 0FYU","summary":"Let S be a Noetherian scheme. Let X and Y be schemes of finite type over S. Let K ∈ D^b_Coh(O_X ×_S Y). The corresponding Fourier-Mukai functor Φ_K sends D^b_Coh(O_X) into D^b_Coh(O_Y) if at least one of the following conditions is satisfied: • the support of H^i(K) is proper over Y for all i, and K has finite tor dimension as an object of D(pr_1^-1O_X), • K can be represented by a bounded complex K^bullet of coherent O_X ×_S Y-modules, flat over X, with support proper…","statement_latex":"Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be schemes of finite type\nover $S$. Let $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_{X \\times_S Y})$.\nThe corresponding Fourier-Mukai functor $\\Phi_K$ sends\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ into $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$\nif at least one of the following conditions is satisfied:\n\\begin{enumerate}\n\\item the support of $H^i(K)$ is proper over $Y$ for all $i$, and $K$\nhas finite tor dimension as an object of $D(\\text{pr}_1^{-1}\\mathcal{O}_X)$,\n\\item $K$ can be represented by a bounded complex $\\mathcal{K}^\\bullet$\nof coherent $\\mathcal{O}_{X \\times_S Y}$-modules, flat over $X$, with support\nproper over $Y$,\n\\item the support of $H^i(K)$ is proper over $Y$ for all $i$\nand $X$ is a regular scheme,\n\\item $K$ is perfect, the support of $H^i(K)$ is proper over $Y$ for all $i$,\nand $Y \\to S$ is flat.\n\\end{enumerate}\nFurthermore in each case the support condition is automatic\nif $X \\to S$ is proper.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fourier-Mukai functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYU","source_file":"equiv.tex","source_line":1137,"source_end_line":1157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1137-L1157","statement_sha256":"6a8917f8253b29cc71ef7e7f9544f9226bde1051f434bde385542808e814a020","origin":"The Stacks Project","memory_eligible":false,"source_rank":9640,"rank":9640,"depth":33,"x":239.213,"y":558.139,"cluster":"derived-categories"},{"id":"stacks:0FYW","tag":"0FYW","title":"Fourier-Mukai functors · Lemma 0FYW","summary":"Compare with discussion in [Rizzardo]. Let X → S and Y → S be morphisms of quasi-compact and quasi-separated schemes. Let Φ : D_QCoh(O_X) → D_QCoh(O_Y) be a Fourier-Mukai functor with pseudo-coherent kernel K ∈ D_QCoh(O_X ×_S Y). Let a : D_QCoh(O_Y) → D_QCoh(O_X ×_S Y) be the right adjoint to Rpr_2, *, see Duality for Schemes, Lemma [Tag 0A9E]. Denote K' = (Y ×_S X → X ×_S Y)^* RSheafHom_O_X ×_S Y(K, a(O_Y)) ∈ D_QCoh(O_Y ×_S X) and denote Φ' : D_QCoh(O_Y) → D_QCoh(O_X)…","statement_latex":"\\begin{reference}\nCompare with discussion in \\cite{Rizzardo}.\n\\end{reference}\nLet $X \\to S$ and $Y \\to S$ be morphisms of quasi-compact and quasi-separated\nschemes. Let $\\Phi : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$\nbe a Fourier-Mukai functor with pseudo-coherent kernel\n$K \\in D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$.\nLet $a : D_\\QCoh(\\mathcal{O}_Y) \\to  D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$\nbe the right adjoint to $R\\text{pr}_{2, *}$, see\nDuality for Schemes, Lemma \\ref{duality-lemma-twisted-inverse-image}.\nDenote\n$$\nK' = (Y \\times_S X \\to X \\times_S Y)^*\nR\\SheafHom_{\\mathcal{O}_{X \\times_S Y}}(K, a(\\mathcal{O}_Y)) \\in\nD_\\QCoh(\\mathcal{O}_{Y \\times_S X})\n$$\nand denote $\\Phi' : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$\nthe corresponding Fourier-Mukai transform. There is a canonical map\n$$\n\\Hom_X(M, \\Phi'(N)) \\longrightarrow \\Hom_Y(\\Phi(M), N)\n$$\nfunctorial in $M$ in $D_\\QCoh(\\mathcal{O}_X)$ and $N$ in\n$D_\\QCoh(\\mathcal{O}_Y)$ which is an isomorphism if\n\\begin{enumerate}\n\\item $N$ is perfect, or\n\\item $K$ is perfect and $X \\to S$ is proper flat and of finite presentation.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fourier-Mukai functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYW","source_file":"equiv.tex","source_line":1269,"source_end_line":1298,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1269-L1298","statement_sha256":"d48497efa320e8dccd21c7402d471e3677f50bc88dda34244c79ce44e6f48eb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9641,"rank":9641,"depth":41,"x":321.299,"y":775.185,"cluster":"derived-categories"},{"id":"stacks:0FYX","tag":"0FYX","title":"Fourier-Mukai functors · Lemma 0FYX","summary":"Compare with discussion in [Rizzardo]. Let S be a Noetherian scheme. Let Y → S be a flat proper Gorenstein morphism and let X → S be a finite type morphism. Denote ω^bullet_Y/S the relative dualizing complex of Y over S. Let Φ : D_QCoh(O_X) → D_QCoh(O_Y) be a Fourier-Mukai functor with perfect kernel K ∈ D_QCoh(O_X ×_S Y). Denote K' = (Y ×_S X → X ×_S Y)^*(K^vee ⊗_O_X ×_S Y^L Lpr_2^*ω^bullet_Y/S) ∈ D_QCoh(O_Y ×_S X) and denote Φ' : D_QCoh(O_Y) → D_QCoh(O_X) the…","statement_latex":"\\begin{reference}\nCompare with discussion in \\cite{Rizzardo}.\n\\end{reference}\nLet $S$ be a Noetherian scheme. Let $Y \\to S$ be a flat proper\nGorenstein morphism and let $X \\to S$ be a finite type morphism.\nDenote $\\omega^\\bullet_{Y/S}$ the relative dualizing complex of\n$Y$ over $S$. Let $\\Phi : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$\nbe a Fourier-Mukai functor with perfect kernel\n$K \\in D_\\QCoh(\\mathcal{O}_{X \\times_S Y})$. Denote\n$$\nK' = (Y \\times_S X \\to X \\times_S Y)^*(K^\\vee\n\\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L}\nL\\text{pr}_2^*\\omega^\\bullet_{Y/S})\n\\in\nD_\\QCoh(\\mathcal{O}_{Y \\times_S X})\n$$\nand denote $\\Phi' : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$\nthe corresponding Fourier-Mukai transform. There is a canonical\nisomorphism\n$$\n\\Hom_Y(N, \\Phi(M)) \\longrightarrow \\Hom_X(\\Phi'(N), M)\n$$\nfunctorial in $M$ in $D_\\QCoh(\\mathcal{O}_X)$ and $N$ in\n$D_\\QCoh(\\mathcal{O}_Y)$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fourier-Mukai functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYX","source_file":"equiv.tex","source_line":1359,"source_end_line":1385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1359-L1385","statement_sha256":"f1903345ff8a57b86fa11d5df33fb0224f2fc9fe8c93a6d2df69af6d7382a437","origin":"The Stacks Project","memory_eligible":false,"source_rank":9642,"rank":9642,"depth":67,"x":85.982,"y":661.598,"cluster":"derived-categories"},{"id":"stacks:0FYY","tag":"0FYY","title":"Fourier-Mukai functors · Lemma 0FYY","summary":"Let S be a Noetherian scheme. • For X, Y proper and flat over S and K in D_perf(O_X ×_S Y) we obtain a Fourier-Mukai functor Φ_K : D_perf(O_X) → D_perf(O_Y). • For X, Y, Z proper and flat over S, K ∈ D_perf(O_X ×_S Y), K' ∈ D_perf(O_Y ×_S Z) the composition Φ_K' ∘ Φ_K : D_perf(O_X) → D_perf(O_Z) is equal to Φ_K\" with K\" ∈ D_perf(O_X ×_S Z) computed as in Lemma [Tag 0FYS], • For X, Y, K, Φ_K as in (1) if X → S is Gorenstein, then Φ_K' : D_perf(O_Y) → D_perf(O_X) is a right…","statement_latex":"Let $S$ be a Noetherian scheme.\n\\begin{enumerate}\n\\item For $X$, $Y$ proper and flat over $S$ and $K$ in\n$D_{perf}(\\mathcal{O}_{X \\times_S Y})$ we obtain a Fourier-Mukai functor\n$\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$.\n\\item For $X$, $Y$, $Z$ proper and flat over $S$, $K \\in\nD_{perf}(\\mathcal{O}_{X \\times_S Y})$, $K' \\in\nD_{perf}(\\mathcal{O}_{Y \\times_S Z})$ the composition\n$\\Phi_{K'} \\circ \\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Z)$\nis equal to $\\Phi_{K''}$ with $K'' \\in D_{perf}(\\mathcal{O}_{X \\times_S Z})$\ncomputed as in Lemma \\ref{lemma-compose-fourier-mukai},\n\\item For $X$, $Y$, $K$, $\\Phi_K$ as in (1) if $X \\to S$ is Gorenstein, then\n$\\Phi_{K'} : D_{perf}(\\mathcal{O}_Y) \\to D_{perf}(\\mathcal{O}_X)$ is a right\nadjoint to $\\Phi_K$ where $K' \\in D_{perf}(\\mathcal{O}_{Y \\times_S X})$\nis the pullback of $L\\text{pr}_1^*\\omega_{X/S}^\\bullet\n\\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K^\\vee$ by\n$Y \\times_S X \\to X \\times_S Y$.\n\\item For $X$, $Y$, $K$, $\\Phi_K$ as in (1) if $Y \\to S$ is Gorenstein, then\n$\\Phi_{K''} : D_{perf}(\\mathcal{O}_Y) \\to D_{perf}(\\mathcal{O}_X)$ is a left\nadjoint to $\\Phi_K$ where $K'' \\in D_{perf}(\\mathcal{O}_{Y \\times_S X})$\nis the pullback of $L\\text{pr}_2^*\\omega_{Y/S}^\\bullet\n\\otimes_{\\mathcal{O}_{X \\times_S Y}}^\\mathbf{L} K^\\vee$ by\n$Y \\times_S X \\to X \\times_S Y$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fourier-Mukai functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FYY","source_file":"equiv.tex","source_line":1427,"source_end_line":1453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1427-L1453","statement_sha256":"1ab58e34c21b493b116f3f4f59f052bca88260ae970469dd06302d424d52f137","origin":"The Stacks Project","memory_eligible":false,"source_rank":9643,"rank":9643,"depth":68,"x":351.121,"y":611.779,"cluster":"derived-categories"},{"id":"stacks:0FZ0","tag":"0FZ0","title":"Resolutions and bounds · Lemma 0FZ0","summary":"Let R be a Noetherian ring. Let X, Y be finite type schemes over R having the resolution property. For any coherent O_X ×_R Y-module F there exist a surjection E boxtimes G → F where E is a finite locally free O_X-module and G is a finite locally free O_Y-module.","statement_latex":"Let $R$ be a Noetherian ring. Let $X$, $Y$ be finite type schemes over $R$\nhaving the resolution property. For any coherent\n$\\mathcal{O}_{X \\times_R Y}$-module $\\mathcal{F}$ there exist\na surjection $\\mathcal{E} \\boxtimes \\mathcal{G} \\to \\mathcal{F}$\nwhere $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module\nand $\\mathcal{G}$ is a finite locally free $\\mathcal{O}_Y$-module.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ0","source_file":"equiv.tex","source_line":1532,"source_end_line":1540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1532-L1540","statement_sha256":"771fe7a53dc2bd0f19974c0460a9652bb52fe6fec340027020777b8c89219aa4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9644,"rank":9644,"depth":18,"x":195.513,"y":799.155,"cluster":"derived-categories"},{"id":"stacks:0FZ1","tag":"0FZ1","title":"Resolutions and bounds · Lemma 0FZ1","summary":"Let R be a ring. Let X, Y be quasi-compact and quasi-separated schemes over R having the resolution property. For any finite type quasi-coherent O_X ×_R Y-module F there exist a surjection E boxtimes G → F where E is a finite locally free O_X-module and G is a finite locally free O_Y-module.","statement_latex":"Let $R$ be a ring. Let $X$, $Y$ be quasi-compact and quasi-separated\nschemes over $R$ having the resolution property. For any finite\ntype quasi-coherent $\\mathcal{O}_{X \\times_R Y}$-module $\\mathcal{F}$\nthere exist a surjection $\\mathcal{E} \\boxtimes \\mathcal{G} \\to \\mathcal{F}$\nwhere $\\mathcal{E}$ is a finite locally free $\\mathcal{O}_X$-module\nand $\\mathcal{G}$ is a finite locally free $\\mathcal{O}_Y$-module.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ1","source_file":"equiv.tex","source_line":1586,"source_end_line":1594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1586-L1594","statement_sha256":"a769173b6cdfaa57f5832b0c4f834178b5d2cafb92aed0e585b05d54100500cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9645,"rank":9645,"depth":38,"x":159.536,"y":572.457,"cluster":"derived-categories"},{"id":"stacks:0FZ2","tag":"0FZ2","title":"Resolutions and bounds · Lemma 0FZ2","summary":"Let R be a Noetherian ring. Let X be a separated finite type scheme over R which has the resolution property. Set O_Δ = Δ_*(O_X) where Δ : X → X ×_R X is the diagonal of X/k. There exists a resolution … → E_2 boxtimes G_2 → E_1 boxtimes G_1 → E_0 boxtimes G_0 → O_Δ → 0 where each E_i and G_i is a finite locally free O_X-module.","statement_latex":"Let $R$ be a Noetherian ring. Let $X$ be a separated finite type scheme\nover $R$ which has the resolution property. Set\n$\\mathcal{O}_\\Delta = \\Delta_*(\\mathcal{O}_X)$ where\n$\\Delta : X \\to X \\times_R X$ is the diagonal of $X/k$.\nThere exists a resolution\n$$\n\\ldots \\to\n\\mathcal{E}_2 \\boxtimes \\mathcal{G}_2 \\to\n\\mathcal{E}_1 \\boxtimes \\mathcal{G}_1 \\to\n\\mathcal{E}_0 \\boxtimes \\mathcal{G}_0 \\to\n\\mathcal{O}_\\Delta \\to 0\n$$\nwhere each $\\mathcal{E}_i$ and $\\mathcal{G}_i$ is a finite locally\nfree $\\mathcal{O}_X$-module.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ2","source_file":"equiv.tex","source_line":1618,"source_end_line":1634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1618-L1634","statement_sha256":"4097349bda8bb84465526837b12c997be0a6ffbdacf1dac12f563545d66a3996","origin":"The Stacks Project","memory_eligible":false,"source_rank":9646,"rank":9646,"depth":19,"x":368.586,"y":719.358,"cluster":"derived-categories"},{"id":"stacks:0FZ3","tag":"0FZ3","title":"Resolutions and bounds · Lemma 0FZ3","summary":"Let X be a regular Noetherian scheme of dimension d < ∞. Then • for F, G coherent O_X-modules we have Ext^n_X(F, G) = 0 for n > d, and • for K, L ∈ D^b_Coh(O_X) and a ∈ Z if H^i(K) = 0 for i < a + d and H^i(L) = 0 for i ≥ a then Hom_X(K, L) = 0.","statement_latex":"Let $X$ be a regular Noetherian scheme of dimension $d < \\infty$. Then\n\\begin{enumerate}\n\\item for $\\mathcal{F}$, $\\mathcal{G}$ coherent $\\mathcal{O}_X$-modules\nwe have $\\Ext^n_X(\\mathcal{F}, \\mathcal{G}) = 0$ for $n > d$, and\n\\item for $K, L \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$\nif $H^i(K) = 0$ for $i < a + d$ and $H^i(L) = 0$ for $i \\geq a$ then\n$\\Hom_X(K, L) = 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ3","source_file":"equiv.tex","source_line":1644,"source_end_line":1654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1644-L1654","statement_sha256":"7a571f85a4852c7899512d2286c0c3e873b7c966b4b68750c45db29d02946091","origin":"The Stacks Project","memory_eligible":false,"source_rank":9647,"rank":9647,"depth":28,"x":96.02,"y":729.666,"cluster":"derived-categories"},{"id":"stacks:0FZ4","tag":"0FZ4","title":"Resolutions and bounds · Lemma 0FZ4","summary":"Let X be a regular Noetherian scheme of dimension d < ∞. Let K ∈ D^b_Coh(O_X) and a ∈ Z. If H^i(K) = 0 for a < i < a + d, then K = τ_≤ aK ⊕ τ_≥ a + dK.","statement_latex":"Let $X$ be a regular Noetherian scheme of dimension $d < \\infty$.\nLet $K \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ and $a \\in \\mathbf{Z}$.\nIf $H^i(K) = 0$ for $a < i < a + d$, then\n$K = \\tau_{\\leq a}K \\oplus \\tau_{\\geq a + d}K$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ4","source_file":"equiv.tex","source_line":1702,"source_end_line":1708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1702-L1708","statement_sha256":"8409af108bfd91e0b458bbb68382f3edafb56bd18c4634143ce0a2862dfc4273","origin":"The Stacks Project","memory_eligible":false,"source_rank":9648,"rank":9648,"depth":29,"x":288.916,"y":567.238,"cluster":"derived-categories"},{"id":"stacks:0FZ5","tag":"0FZ5","title":"Resolutions and bounds · Lemma 0FZ5","summary":"Let k be a field. Let X be a quasi-compact separated smooth scheme over k. There exist finite locally free O_X-modules E and G such that O_Δ ∈ langle E boxtimes G rangle in D(O_X × X) where the notation is as in Derived Categories, Section [Tag 09SI].","statement_latex":"Let $k$ be a field. Let $X$ be a quasi-compact separated smooth scheme over $k$.\nThere exist finite locally free $\\mathcal{O}_X$-modules\n$\\mathcal{E}$ and $\\mathcal{G}$ such that\n$$\n\\mathcal{O}_\\Delta \\in \\langle \\mathcal{E} \\boxtimes \\mathcal{G} \\rangle\n$$\nin $D(\\mathcal{O}_{X \\times X})$ where the notation is as in\nDerived Categories, Section \\ref{derived-section-generators}.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ5","source_file":"equiv.tex","source_line":1724,"source_end_line":1734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1724-L1734","statement_sha256":"0e2911839e04e6cf8c06348a0d308be0175974fb7e88c16ca0021ecfc075e02d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9649,"rank":9649,"depth":40,"x":277.285,"y":796.699,"cluster":"derived-categories"},{"id":"stacks:0FZ6","tag":"0FZ6","title":"Resolutions and bounds · Lemma 0FZ6","summary":"Let k be a field. Let X be a scheme proper and smooth over k. Then D_perf(O_X) has a strong generator.","statement_latex":"Let $k$ be a field. Let $X$ be a scheme proper and smooth over $k$.\nThen $D_{perf}(\\mathcal{O}_X)$\nhas a strong generator.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ6","source_file":"equiv.tex","source_line":1777,"source_end_line":1782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1777-L1782","statement_sha256":"2bddaa1547abb011c34dd1a894f8d380cdb02f2b6f8b49957acaa4a1505448f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9650,"rank":9650,"depth":41,"x":101.154,"y":620.718,"cluster":"derived-categories"},{"id":"stacks:0FZ7","tag":"0FZ7","title":"Resolutions and bounds · Lemma 0FZ7","summary":"Let k be a field. Let X be a proper smooth scheme over k. There exists integers m, n ≥ 1 and a finite locally free O_X-module G such that every coherent O_X-module is contained in smd(add(G[-m, m])^star n) with notation as in Derived Categories, Section [Tag 0FX0].","statement_latex":"Let $k$ be a field. Let $X$ be a proper smooth scheme over $k$.\nThere exists integers $m, n \\geq 1$ and a finite locally free\n$\\mathcal{O}_X$-module $\\mathcal{G}$ such that every coherent\n$\\mathcal{O}_X$-module is contained in $smd(add(\\mathcal{G}[-m, m])^{\\star n})$\nwith notation as in Derived Categories, Section\n\\ref{derived-section-operate-on-full}.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ7","source_file":"equiv.tex","source_line":1845,"source_end_line":1853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1845-L1853","statement_sha256":"abefcaa16cc7afd52a5b73ffdd0018a3bb15fa0f233b2d3f37b2cbc420b60c48","origin":"The Stacks Project","memory_eligible":false,"source_rank":9651,"rank":9651,"depth":42,"x":372.828,"y":650.573,"cluster":"derived-categories"},{"id":"stacks:0FZ8","tag":"0FZ8","title":"Resolutions and bounds · Lemma 0FZ8","summary":"Let k be a field. Let X be a smooth proper scheme over k. Let A be an abelian category. Let H : D_perf(O_X) → A be a homological functor (Derived Categories, Definition [Tag 0147]) such that for all K in D_perf(O_X) the object H^i(K) is nonzero for only a finite number of i ∈ Z. Then there exists an integer m ≥ 1 such that H^i(F) = 0 for any coherent O_X-module F and i not ∈ [-m, m]. Similarly for cohomological functors.","statement_latex":"Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$.\nLet $\\mathcal{A}$ be an abelian category. Let\n$H : D_{perf}(\\mathcal{O}_X) \\to \\mathcal{A}$ be a homological\nfunctor (Derived Categories, Definition \\ref{derived-definition-homological})\nsuch that for all $K$ in $D_{perf}(\\mathcal{O}_X)$ the object\n$H^i(K)$ is nonzero for only a finite number of $i \\in \\mathbf{Z}$.\nThen there exists an integer $m \\geq 1$ such that\n$H^i(\\mathcal{F}) = 0$ for any coherent $\\mathcal{O}_X$-module\n$\\mathcal{F}$ and $i \\not \\in [-m, m]$.\nSimilarly for cohomological functors.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ8","source_file":"equiv.tex","source_line":1871,"source_end_line":1883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1871-L1883","statement_sha256":"09af9eac5fe4f666a4bcc8d1a7d5dbe5dd275489d11e2a8ae7a5a123c19a092a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9652,"rank":9652,"depth":43,"x":148.262,"y":782.847,"cluster":"derived-categories"},{"id":"stacks:0FZ9","tag":"0FZ9","title":"Resolutions and bounds · Lemma 0FZ9","summary":"Let k be a field. Let X, Y be finite type schemes over k. Let K_0 → K_1 → K_2 → … be a system of objects of D_perf(O_X × Y) and m ≥ 0 an integer such that • H^q(K_i) is nonzero only for q ≤ m, • for every coherent O_X-module F with dim(Supp(F)) = 0 the object Rpr_2, *( pr_1^*F ⊗_O_X × Y^L K_n) has vanishing cohomology sheaves in degrees outside [-m, m] ∪ [-m - n, m - n] and for n > 2m the transition maps induce isomorphisms on cohomology sheaves in degrees in [-m, m].…","statement_latex":"Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$.\nLet $K_0 \\to K_1 \\to K_2 \\to \\ldots$ be a system of objects\nof $D_{perf}(\\mathcal{O}_{X \\times Y})$ and $m \\geq 0$ an integer such that\n\\begin{enumerate}\n\\item $H^q(K_i)$ is nonzero only for $q \\leq m$,\n\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ with\n$\\dim(\\text{Supp}(\\mathcal{F})) = 0$ the object\n$$\nR\\text{pr}_{2, *}(\n\\text{pr}_1^*\\mathcal{F} \\otimes_{\\mathcal{O}_{X \\times Y}}^\\mathbf{L}\nK_n)\n$$\nhas vanishing cohomology sheaves in degrees outside\n$[-m, m] \\cup [-m - n, m - n]$ and for $n > 2m$ the transition maps\ninduce isomorphisms on cohomology sheaves in degrees in $[-m, m]$.\n\\end{enumerate}\nThen $K_n$ has vanishing cohomology sheaves in degrees outside\n$[-m, m] \\cup [-m - n, m - n]$ and for $n > 2m$ the\ntransition maps induce isomorphisms on cohomology sheaves in degrees in\n$[-m, m]$. Moreover, if $X$ and $Y$ are smooth over $k$, then for $n$\nlarge enough we find $K_n = K \\oplus C_n$ in\n$D_{perf}(\\mathcal{O}_{X \\times Y})$\nwhere $K$ has cohomology only indegrees $[-m, m]$ and $C_n$ only in\ndegrees $[-m - n, m - n]$ and the transition maps\ndefine isomorphisms between various copies of $K$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Resolutions and bounds","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZ9","source_file":"equiv.tex","source_line":1890,"source_end_line":1917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1890-L1917","statement_sha256":"77a35fcdf13b715ecc502a95cce6cb524e7b14a266fdc2f69524740c49edb216","origin":"The Stacks Project","memory_eligible":false,"source_rank":9653,"rank":9653,"depth":40,"x":207.543,"y":557.657,"cluster":"derived-categories"},{"id":"stacks:0FZT","tag":"0FZT","title":"Sibling functors · Definition 0FZT","summary":"Let A be an abelian category. Let D be a triangulated category. We say two exact functors of triangulated categories F, F' : D^b(A) → D are siblings, or we say F' is a sibling of F, if the following two conditions are satisfied • the functors F ∘ i and F' ∘ i are isomorphic where i : A → D^b(A) is the inclusion functor, and • F(K) ≅ F'(K) for any K in D^b(A).","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D}$ be a\ntriangulated category. We say two exact functors of triangulated categories\n$$\nF, F' : D^b(\\mathcal{A}) \\longrightarrow \\mathcal{D}\n$$\nare {\\it siblings}, or we say $F'$ is a {\\it sibling} of $F$,\nif the following two conditions are satisfied\n\\begin{enumerate}\n\\item the functors $F \\circ i$ and $F' \\circ i$ are isomorphic\nwhere $i : \\mathcal{A} \\to D^b(\\mathcal{A})$ is the inclusion functor, and\n\\item $F(K) \\cong F'(K)$ for any $K$ in $D^b(\\mathcal{A})$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Sibling functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZT","source_file":"equiv.tex","source_line":1971,"source_end_line":1985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1971-L1985","statement_sha256":"b4e891cba5c0fef41230976320c344579daba29dbc3cc656c736db1e6e0bdb88","origin":"The Stacks Project","memory_eligible":false,"source_rank":9654,"rank":9654,"depth":0,"x":345.061,"y":757.55,"cluster":"derived-categories"},{"id":"stacks:0FZU","tag":"0FZU","title":"Sibling functors · Lemma 0FZU","summary":"Let A be an abelian category. Let D be a triangulated category. Let F, F' : D^b(A) → D be exact functors of triangulated categories. Assume • the functors F ∘ i and F' ∘ i are isomorphic where i : A → D^b(A) is the inclusion functor, and • for all X, Y ∈ Ob(A) we have Ext^q_D(F(X), F(Y)) = 0 for q < 0 (for example if F is fully faithful). Then F and F' are siblings.","statement_latex":"Let $\\mathcal{A}$ be an abelian category. Let $\\mathcal{D}$ be a\ntriangulated category. Let\n$F, F' : D^b(\\mathcal{A}) \\longrightarrow \\mathcal{D}$\nbe exact functors of triangulated categories. Assume\n\\begin{enumerate}\n\\item the functors $F \\circ i$ and $F' \\circ i$ are isomorphic\nwhere $i : \\mathcal{A} \\to D^b(\\mathcal{A})$ is the inclusion functor, and\n\\item for all $X, Y \\in \\Ob(\\mathcal{A})$ we have\n$\\Ext^q_\\mathcal{D}(F(X), F(Y)) = 0$ for $q < 0$ (for example\nif $F$ is fully faithful).\n\\end{enumerate}\nThen $F$ and $F'$ are siblings.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Sibling functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZU","source_file":"equiv.tex","source_line":1990,"source_end_line":2004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L1990-L2004","statement_sha256":"eb52aaf9cee9c9dcd424b10fc284a66cc53bc1fada9b1c9fd29727662a2097d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9655,"rank":9655,"depth":10,"x":82.643,"y":688.113,"cluster":"derived-categories"},{"id":"stacks:0FZV","tag":"0FZV","title":"Sibling functors · Lemma 0FZV","summary":"Let F and F' be siblings as in Definition [Tag 0FZT]. Then • if F is essentially surjective, then F' is essentially surjective, • if F is fully faithful, then F' is fully faithful.","statement_latex":"Let $F$ and $F'$ be siblings as in Definition \\ref{definition-siblings}.\nThen\n\\begin{enumerate}\n\\item if $F$ is essentially surjective, then $F'$ is essentially\nsurjective,\n\\item if $F$ is fully faithful, then $F'$ is fully faithful.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Sibling functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZV","source_file":"equiv.tex","source_line":2035,"source_end_line":2044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2035-L2044","statement_sha256":"97d86f7d0c799529d83c61a4e2ac085eba8cb46f640f27adb74a1f1c6dfdbaac","origin":"The Stacks Project","memory_eligible":false,"source_rank":9656,"rank":9656,"depth":8,"x":332.238,"y":590.313,"cluster":"derived-categories"},{"id":"stacks:0GWF","tag":"0GWF","title":"Sibling functors · Lemma 0GWF","summary":"Let A be an abelian category with enough negative objects. Let X ∈ D^b(A). Let b ∈ Z with H^i(X) = 0 for i > b. Then there exists a map N[-b] → X such that the induced map N → H^b(X) is surjective and Hom(H^b(X), N) = 0.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough negative objects.\nLet $X \\in D^b(\\mathcal{A})$. Let $b \\in \\mathbf{Z}$ with\n$H^i(X) = 0$ for $i > b$. Then\nthere exists a map $N[-b] \\to X$ such that the induced map\n$N \\to H^b(X)$ is surjective and $\\Hom(H^b(X), N) = 0$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Sibling functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWF","source_file":"equiv.tex","source_line":2207,"source_end_line":2214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2207-L2214","statement_sha256":"e6d8977f4256330fc0b8113286cc02aca6f227a6426226cd5af273e2ff52ecfe","origin":"The Stacks Project","memory_eligible":false,"source_rank":9657,"rank":9657,"depth":0,"x":226.732,"y":804.271,"cluster":"derived-categories"},{"id":"stacks:0GWG","tag":"0GWG","title":"Sibling functors · Lemma 0GWG","summary":"Let A be an abelian category with enough negative objects. Let f : X → X' be a morphism of D^b(A). Let b ∈ Z such that H^i(X) = 0 for i > b and H^i(X') = 0 for i ≥ b. Then there exists a map N[-b] → X such that the induced map N → H^b(X) is surjective, such that Hom(H^b(X), N) = 0, and such that the composition N[-b] → X → X' is zero.","statement_latex":"Let $\\mathcal{A}$ be an abelian category with enough negative objects.\nLet $f : X \\to X'$ be a morphism of $D^b(\\mathcal{A})$. Let $b \\in \\mathbf{Z}$\nsuch that $H^i(X) = 0$ for $i > b$ and $H^i(X') = 0$ for $i \\geq b$.\nThen there exists a map $N[-b] \\to X$ such that the induced map\n$N \\to H^b(X)$ is surjective, such that $\\Hom(H^b(X), N) = 0$, and\nsuch that the composition $N[-b] \\to X \\to X'$ is zero.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Sibling functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWG","source_file":"equiv.tex","source_line":2224,"source_end_line":2232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2224-L2232","statement_sha256":"ace3b4daa25055f1a7f40776b25255a131ccb8dee2e5c037c116db2b14a331d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9658,"rank":9658,"depth":16,"x":132.376,"y":586.415,"cluster":"derived-categories"},{"id":"stacks:0FZW","tag":"0FZW","title":"Sibling functors · Proposition 0FZW","summary":"[Orlov-K3]; the fact that we do not need to assume vanishing of Ext^q(N, X) for q > 0 in the definition of negative objects above is due to [Canonaco-Stellari]. Let F and F' be siblings as in Definition [Tag 0FZT]. Assume that F is fully faithful and that A has enough negative objects (see above). Then F and F' are isomorphic functors.","statement_latex":"\\begin{reference}\n\\cite[Proposition 2.16]{Orlov-K3}; the fact that we do not need\nto assume vanishing of $\\Ext^q(N, X)$ for $q > 0$ in the definition\nof negative objects above is due to \\cite{Canonaco-Stellari}.\n\\end{reference}\nLet $F$ and $F'$ be siblings as in Definition \\ref{definition-siblings}.\nAssume that $F$ is fully faithful and that $\\mathcal{A}$ has enough\nnegative objects (see above). Then $F$ and $F'$ are isomorphic functors.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Sibling functors","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZW","source_file":"equiv.tex","source_line":2257,"source_end_line":2267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2257-L2267","statement_sha256":"b1c42a912493662bce72066e82b001370efd8401b771b44b0f3083464c2780a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9659,"rank":9659,"depth":17,"x":377.393,"y":693.627,"cluster":"derived-categories"},{"id":"stacks:0G24","tag":"0G24","title":"Deducing fully faithfulness · Lemma 0G24","summary":"Variant of [Orlov-K3] Let F : D → D' be an exact functor of triangulated categories. Let S ⊂ Ob(D) be a set of objects. Assume • F has both right and left adjoints, • for K ∈ D if Hom(E, K[i]) = 0 for all E ∈ S and i ∈ Z then K = 0, • for K ∈ D if Hom(K, E[i]) = 0 for all E ∈ S and i ∈ Z then K = 0, • the map Hom(E, E'[i]) → Hom(F(E), F(E')[i]) induced by F is bijective for all E, E' ∈ S and i ∈ Z. Then F is fully faithful.","statement_latex":"\\begin{reference}\nVariant of \\cite[Lemma 2.15]{Orlov-K3}\n\\end{reference}\nLet $F : \\mathcal{D} \\to \\mathcal{D}'$ be an exact functor of\ntriangulated categories. Let $S \\subset \\Ob(\\mathcal{D})$ be\na set of objects. Assume\n\\begin{enumerate}\n\\item $F$ has both right and left adjoints,\n\\item for $K \\in \\mathcal{D}$ if $\\Hom(E, K[i]) = 0$ for all\n$E \\in S$ and $i \\in \\mathbf{Z}$ then $K = 0$,\n\\item for $K \\in \\mathcal{D}$ if $\\Hom(K, E[i]) = 0$ for all\n$E \\in S$ and $i \\in \\mathbf{Z}$ then $K = 0$,\n\\item the map $\\Hom(E, E'[i]) \\to \\Hom(F(E), F(E')[i])$ induced by $F$\nis bijective for all $E, E' \\in S$ and $i \\in \\mathbf{Z}$.\n\\end{enumerate}\nThen $F$ is fully faithful.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Deducing fully faithfulness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G24","source_file":"equiv.tex","source_line":2475,"source_end_line":2493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2475-L2493","statement_sha256":"f58c792826a4bac0721037f69dfb7e104dff467d79875924a05f2629acbe41aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9660,"rank":9660,"depth":3,"x":110.236,"y":753.659,"cluster":"derived-categories"},{"id":"stacks:0G02","tag":"0G02","title":"Deducing fully faithfulness · Lemma 0G02","summary":"Let k be a field. Let X be a scheme of finite type over k which is regular. Let x ∈ X be a closed point. For a coherent O_X-module F supported at x choose a coherent O_X-module F' supported at x such that F_x and F'_x are Matlis dual. Then there is an isomorphism Hom_X(F, M) = H^0(X, M ⊗_O_X^L F'[-d_x]) where d_x = dim(O_X, x) functorial in M in D_perf(O_X).","statement_latex":"Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which\nis regular. Let $x \\in X$ be a closed point. For a coherent\n$\\mathcal{O}_X$-module $\\mathcal{F}$ supported at $x$ choose\na coherent $\\mathcal{O}_X$-module $\\mathcal{F}'$ supported at $x$\nsuch that $\\mathcal{F}_x$ and $\\mathcal{F}'_x$ are Matlis dual.\nThen there is an isomorphism\n$$\n\\Hom_X(\\mathcal{F}, M) =\nH^0(X, M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}'[-d_x])\n$$\nwhere $d_x = \\dim(\\mathcal{O}_{X, x})$\nfunctorial in $M$ in $D_{perf}(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Deducing fully faithfulness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G02","source_file":"equiv.tex","source_line":2530,"source_end_line":2544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2530-L2544","statement_sha256":"64ff2b41bea7ae5a17a8a46fa5a91b98a3833a324fbdc660c08231e71e6a7870","origin":"The Stacks Project","memory_eligible":false,"source_rank":9661,"rank":9661,"depth":19,"x":259.106,"y":557.605,"cluster":"derived-categories"},{"id":"stacks:0G03","tag":"0G03","title":"Deducing fully faithfulness · Lemma 0G03","summary":"Let k be a field. Let X be a scheme of finite type over k which is regular. Let x ∈ X be a closed point and denote O_x the skyscraper sheaf at x with value kappa(x). Let K in D_perf(O_X). • If Ext^i_X(O_x, K) = 0 then there exists an open neighbourhood U of x such that H^i - d_x(K)|_U = 0 where d_x = dim(O_X, x). • If Hom_X(O_x, K[i]) = 0 for all i ∈ Z, then K is zero in an open neighbourhood of x. • If Ext^i_X(K, O_x) = 0 then there exists an open neighbourhood U of x…","statement_latex":"Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which\nis regular. Let $x \\in X$ be a closed point and denote $\\mathcal{O}_x$\nthe skyscraper sheaf at $x$ with value $\\kappa(x)$. Let $K$ in\n$D_{perf}(\\mathcal{O}_X)$.\n\\begin{enumerate}\n\\item If $\\Ext^i_X(\\mathcal{O}_x, K) = 0$ then there exists an open\nneighbourhood $U$ of $x$ such that $H^{i - d_x}(K)|_U = 0$ where\n$d_x = \\dim(\\mathcal{O}_{X, x})$.\n\\item If $\\Hom_X(\\mathcal{O}_x, K[i]) = 0$ for all\n$i \\in \\mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$.\n\\item If $\\Ext^i_X(K, \\mathcal{O}_x) = 0$ then there exists an open\nneighbourhood $U$ of $x$ such that $H^i(K^\\vee)|_U = 0$.\n\\item If $\\Hom_X(K, \\mathcal{O}_x[i]) = 0$ for all\n$i \\in \\mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$.\n\\item If $H^i(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_x) = 0$\nthen there exists an open neighbourhood $U$ of $x$ such that\n$H^i(K)|_U = 0$.\n\\item If $H^i(X, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_x) = 0$\nfor $i \\in \\mathbf{Z}$ then $K$ is zero in an\nopen neighbourhood of $x$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Deducing fully faithfulness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G03","source_file":"equiv.tex","source_line":2577,"source_end_line":2600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2577-L2600","statement_sha256":"46962e313fbebb7951a186ef3140110bfc928d459998dcd3120d7325409dd9c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9662,"rank":9662,"depth":20,"x":307.041,"y":786.874,"cluster":"derived-categories"},{"id":"stacks:0GWZ","tag":"0GWZ","title":"Deducing fully faithfulness · Lemma 0GWZ","summary":"Let X be a Noetherian scheme. Let x ∈ X be a closed point and denote O_x the skyscraper sheaf at x with value kappa(x). Let K in D^b_Coh(O_X). Let b ∈ Z. The following are equivalent • H^i(K)_x = 0 for all i > b and • Hom_X(K, O_x[-i]) = 0 for all i > b.","statement_latex":"Let $X$ be a Noetherian scheme. Let $x \\in X$ be a closed point and\ndenote $\\mathcal{O}_x$ the skyscraper sheaf at $x$ with value $\\kappa(x)$.\nLet $K$ in $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$. Let $b \\in \\mathbf{Z}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $H^i(K)_x = 0$ for all $i > b$ and\n\\item $\\Hom_X(K, \\mathcal{O}_x[-i]) = 0$ for all $i > b$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Deducing fully faithfulness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWZ","source_file":"equiv.tex","source_line":2619,"source_end_line":2629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2619-L2629","statement_sha256":"d641560f7062f3cfbcd9c90f0129aeaf3e064cd21cf777a969c87832f24715fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9663,"rank":9663,"depth":12,"x":87.104,"y":644.872,"cluster":"derived-categories"},{"id":"stacks:0G25","tag":"0G25","title":"Deducing fully faithfulness · Lemma 0G25","summary":"Let k be a field. Let X and Y be proper schemes over k. Assume X is regular. Then a k-linear exact functor F : D_perf(O_X) → D_perf(O_Y) is fully faithful if and only if for any closed points x, x' ∈ X the maps F : Ext^i_X(O_x, O_x') → Ext^i_Y(F(O_x), F(O_x')) are isomorphisms for all i ∈ Z. Here O_x is the skyscraper sheaf at x with value kappa(x).","statement_latex":"Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$.\nAssume $X$ is regular. Then a $k$-linear exact functor\n$F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nis fully faithful if and only if\nfor any closed points $x, x' \\in X$ the maps\n$$\nF : \\Ext^i_X(\\mathcal{O}_x, \\mathcal{O}_{x'})\n\\longrightarrow\n\\Ext^i_Y(F(\\mathcal{O}_x), F(\\mathcal{O}_{x'}))\n$$\nare isomorphisms for all $i \\in \\mathbf{Z}$.\nHere $\\mathcal{O}_x$ is the skyscraper sheaf at $x$ with value $\\kappa(x)$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Deducing fully faithfulness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G25","source_file":"equiv.tex","source_line":2650,"source_end_line":2664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2650-L2664","statement_sha256":"4a8184c194dc8303c9fdaf7117935afaf25efbb79324b7325c504c0c7b867bb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9664,"rank":9664,"depth":45,"x":363.751,"y":624.765,"cluster":"derived-categories"},{"id":"stacks:0G26","tag":"0G26","title":"Deducing fully faithfulness · Lemma 0G26","summary":"Email from Noah Olander of Jun 9, 2020 Let k be a field. Let X be a proper scheme over k which is regular. Let F : D_perf(O_X) → D_perf(O_X) be a k-linear exact functor. Assume for every coherent O_X-module F with dim(Supp(F)) = 0 there is an isomorphism F ≅ F(F). Then F is fully faithful.","statement_latex":"\\begin{reference}\nEmail from Noah Olander of Jun 9, 2020\n\\end{reference}\nLet $k$ be a field. Let $X$ be a proper scheme over $k$ which is regular.\nLet $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$\nbe a $k$-linear exact functor. Assume for every coherent\n$\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$\nthere is an isomorphism $\\mathcal{F} \\cong F(\\mathcal{F})$.\nThen $F$ is fully faithful.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Deducing fully faithfulness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G26","source_file":"equiv.tex","source_line":2674,"source_end_line":2685,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2674-L2685","statement_sha256":"4ba0ac9300f52902bddec88b3fb38265337ef3cfa7ef8a968a9cf4b8b041b6f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9665,"rank":9665,"depth":46,"x":175.739,"y":796.739,"cluster":"derived-categories"},{"id":"stacks:0FZZ","tag":"0FZZ","title":"Special functors · Definition 0FZZ","summary":"Let k be a field. Let X, Y be finite type schemes over k. Recall that D^b_Coh(O_X) = D^b(Coh(O_X)) by Derived Categories of Schemes, Proposition [Tag 0FDB]. We say two k-linear exact functors F, F' : D^b_Coh(O_X) = D^b(Coh(O_X)) → D^b_Coh(O_Y) are siblings, or we say F' is a sibling of F if F and F' are siblings in the sense of Definition [Tag 0FZT] with abelian category being Coh(O_X). If X is regular then D_perf(O_X) = D^b_Coh(O_X) by Derived Categories of Schemes,…","statement_latex":"Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$.\nRecall that\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X) = D^b(\\textit{Coh}(\\mathcal{O}_X))$\nby Derived Categories of Schemes, Proposition \\ref{perfect-proposition-DCoh}.\nWe say two $k$-linear exact functors\n$$\nF, F' :\nD^b_{\\textit{Coh}}(\\mathcal{O}_X) = D^b(\\textit{Coh}(\\mathcal{O}_X))\n\\longrightarrow\nD^b_{\\textit{Coh}}(\\mathcal{O}_Y)\n$$\nare {\\it siblings}, or we say $F'$ is a {\\it sibling} of $F$ if $F$ and $F'$\nare siblings in the sense of Definition \\ref{definition-siblings}\nwith abelian category being $\\textit{Coh}(\\mathcal{O}_X)$.\nIf $X$ is regular then\n$D_{perf}(\\mathcal{O}_X) = D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ by\nDerived Categories of Schemes, Lemma \\ref{perfect-lemma-perfect-on-noetherian}\nand we use the same terminology for $k$-linear exact functors\n$F, F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Special functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0FZZ","source_file":"equiv.tex","source_line":2748,"source_end_line":2769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2748-L2769","statement_sha256":"510d0ebb6b69fdb0c402edba2b74b75392354e403d0bc9ee42f31b5528231dd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9666,"rank":9666,"depth":34,"x":176.08,"y":563.013,"cluster":"derived-categories"},{"id":"stacks:0G00","tag":"0G00","title":"Special functors · Lemma 0G00","summary":"Let k be a field. Let X, Y be finite type schemes over k with X separated. Let F : D^b_Coh(O_X) → D^b_Coh(O_Y) be a k-linear exact functor sending Coh(O_X) ⊂ D^b_Coh(O_X) into Coh(O_Y) ⊂ D^b_Coh(O_Y). Then there exists a Fourier-Mukai functor F' : D^b_Coh(O_X) → D^b_Coh(O_Y) whose kernel is a coherent O_X × Y-module K flat over X and with support finite over Y which is a sibling of F.","statement_latex":"Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with\n$X$ separated. Let\n$F : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$\nbe a $k$-linear exact functor sending\n$\\textit{Coh}(\\mathcal{O}_X) \\subset D^b_{\\textit{Coh}}(\\mathcal{O}_X)$\ninto\n$\\textit{Coh}(\\mathcal{O}_Y) \\subset D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$.\nThen there exists a Fourier-Mukai functor\n$F' : D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$\nwhose kernel is a coherent $\\mathcal{O}_{X \\times Y}$-module $\\mathcal{K}$\nflat over $X$ and with support finite over $Y$ which is a sibling of $F$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Special functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G00","source_file":"equiv.tex","source_line":2771,"source_end_line":2784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2771-L2784","statement_sha256":"018edab2bb24d074860c991a1d8b76c285bec6fbfde09b4550b345d1afb34116","origin":"The Stacks Project","memory_eligible":false,"source_rank":9667,"rank":9667,"depth":52,"x":363.97,"y":735.724,"cluster":"derived-categories"},{"id":"stacks:0GX0","tag":"0GX0","title":"Special functors · Lemma 0GX0","summary":"Let k be a field. Let X, Y be proper schemes over k. Assume X is regular. Let F, G : D_perf(O_X) → D_perf(O_Y) be k-linear exact functors such that • F(F) ≅ G(F) for any coherent O_X-module F with dim(Supp(F)) = 0, • F is fully faithful. Then the essential image of G is contained in the essential image of F.","statement_latex":"Let $k$ be a field. Let $X$, $Y$ be proper schemes over $k$. Assume\n$X$ is regular. Let\n$F, G : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nbe $k$-linear exact functors such that\n\\begin{enumerate}\n\\item $F(\\mathcal{F}) \\cong G(\\mathcal{F})$ for any coherent\n$\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$,\n\\item $F$ is fully faithful.\n\\end{enumerate}\nThen the essential image of $G$ is contained in the essential\nimage of $F$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Special functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GX0","source_file":"equiv.tex","source_line":2821,"source_end_line":2834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2821-L2834","statement_sha256":"57e244e44aa26a1bef99ff430b26d9ef38a3e12fa504b7fbc9ffccc6f1eae8c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9668,"rank":9668,"depth":45,"x":86.26,"y":714.962,"cluster":"derived-categories"},{"id":"stacks:0G27","tag":"0G27","title":"Special functors · Lemma 0G27","summary":"Email from Noah Olander of Jun 8, 2020 Let k be a field. Let X be a proper scheme over k which is regular. Let F : D_perf(O_X) → D_perf(O_X) be a k-linear exact functor. Assume for every coherent O_X-module F with dim(Supp(F)) = 0 there is an isomorphism F ≅ F(F). Then there exists an automorphism f : X → X over k which induces the identity on the underlying topological space and an invertible O_X-module L such that F and F'(M) = f^*M ⊗_O_X^L L are siblings.","statement_latex":"\\begin{reference}\nEmail from Noah Olander of Jun 8, 2020\n\\end{reference}\nLet $k$ be a field. Let $X$ be a proper scheme over $k$ which is regular.\nLet $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$\nbe a $k$-linear exact functor. Assume for every coherent\n$\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$\nthere is an isomorphism $\\mathcal{F} \\cong F(\\mathcal{F})$.\nThen there exists an automorphism $f : X \\to X$ over $k$\nwhich induces the identity on the\nunderlying topological space\\footnote{This often forces $f$\nto be the identity, see Varieties, Lemma \\ref{varieties-lemma-automorphism}.}\nand an invertible $\\mathcal{O}_X$-module $\\mathcal{L}$\nsuch that $F$ and $F'(M) = f^*M \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{L}$\nare siblings.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Special functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G27","source_file":"equiv.tex","source_line":2866,"source_end_line":2883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2866-L2883","statement_sha256":"c97a8b4e1af666ef6e6a618b73e035004a2afee15b548b3928e40719a6f02da4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9669,"rank":9669,"depth":47,"x":307.953,"y":572.552,"cluster":"derived-categories"},{"id":"stacks:0G06","tag":"0G06","title":"Special functors · Lemma 0G06","summary":"Let k be a field. Let X, Y be proper schemes over k. Assume X regular. Let F, G : D_perf(O_X) → D_perf(O_Y) be k-linear exact functors such that • F(F) ≅ G(F) for any coherent O_X-module F with dim(Supp(F)) = 0, • F is fully faithful, and • G is a Fourier-Mukai functor whose kernel is in D_perf(O_X × Y). Then there exists a Fourier-Mukai functor F' : D_perf(O_X) → D_perf(O_Y) whose kernel is in D_perf(O_X × Y) such that F and F' are siblings.","statement_latex":"Let $k$ be a field. Let $X$, $Y$ be proper schemes over $k$.\nAssume $X$ regular.\nLet $F, G : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nbe $k$-linear exact functors such that\n\\begin{enumerate}\n\\item $F(\\mathcal{F}) \\cong G(\\mathcal{F})$ for any coherent\n$\\mathcal{O}_X$-module $\\mathcal{F}$ with $\\dim(\\text{Supp}(\\mathcal{F})) = 0$,\n\\item $F$ is fully faithful, and\n\\item $G$ is a Fourier-Mukai functor whose kernel is in\n$D_{perf}(\\mathcal{O}_{X \\times Y})$.\n\\end{enumerate}\nThen there exists a Fourier-Mukai functor\n$F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nwhose kernel is in $D_{perf}(\\mathcal{O}_{X \\times Y})$\nsuch that $F$ and $F'$ are siblings.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Special functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G06","source_file":"equiv.tex","source_line":2946,"source_end_line":2963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L2946-L2963","statement_sha256":"51103283cdd2a76baef62d61d1b3f8231fbe5fa069f1a86bdb00673a9435a880","origin":"The Stacks Project","memory_eligible":false,"source_rank":9670,"rank":9670,"depth":48,"x":258.954,"y":803.584,"cluster":"derived-categories"},{"id":"stacks:0G0B","tag":"0G0B","title":"Fully faithful functors · Lemma 0G0B","summary":"Let k be a field. Let X and Y be smooth proper schemes over k. Given a k-linear, exact, fully faithful functor F : D_perf(O_X) → D_perf(O_Y) there exists a Fourier-Mukai functor F' : D_perf(O_X) → D_perf(O_Y) whose kernel is in D_perf(O_X × Y) which is a sibling to F.","statement_latex":"Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$.\nGiven a $k$-linear, exact, fully faithful functor\n$F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nthere exists a Fourier-Mukai functor\n$F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ whose kernel\nis in $D_{perf}(\\mathcal{O}_{X \\times Y})$ which is a sibling to $F$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fully faithful functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0B","source_file":"equiv.tex","source_line":3246,"source_end_line":3254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3246-L3254","statement_sha256":"23982d7911628ff751aa7622d2928515236ea4bf9f932b5ae00abc04be1fe888","origin":"The Stacks Project","memory_eligible":false,"source_rank":9671,"rank":9671,"depth":49,"x":109.148,"y":605.227,"cluster":"derived-categories"},{"id":"stacks:0G0C","tag":"0G0C","title":"Orlov · Theorem 0G0C","summary":"[Orlov-K3]; this is shown in [Noah] without the assumption that X be projective Let k be a field. Let X and Y be smooth proper schemes over k with X projective over k. Any k-linear fully faithful exact functor F : D_perf(O_X) → D_perf(O_Y) is a Fourier-Mukai functor for some kernel in D_perf(O_X × Y).","statement_latex":"\\begin{reference}\n\\cite[Theorem 2.2]{Orlov-K3}; this is shown in \\cite{Noah}\nwithout the assumption that $X$ be projective\n\\end{reference}\nLet $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$\nwith $X$ projective over $k$. Any $k$-linear fully faithful exact \nfunctor $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nis a Fourier-Mukai functor for some kernel in\n$D_{perf}(\\mathcal{O}_{X \\times Y})$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fully faithful functors","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0C","source_file":"equiv.tex","source_line":3265,"source_end_line":3276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3265-L3276","statement_sha256":"d95a710d69e2e9ba6df91c20696b8e92961e84559133649838abbbd7b19a74cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9672,"rank":9672,"depth":50,"x":379.391,"y":666.549,"cluster":"derived-categories"},{"id":"stacks:0G0D","tag":"0G0D","title":"Fully faithful functors · Proposition 0G0D","summary":"Let k be a field. Let X and Y be smooth proper schemes over k. If F : D_perf(O_X) → D_perf(O_Y) is a k-linear exact equivalence of triangulated categories then there exists a Fourier-Mukai functor F' : D_perf(O_X) → D_perf(O_Y) whose kernel is in D_perf(O_X × Y) which is an equivalence and a sibling of F.","statement_latex":"Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$.\nIf $F : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nis a $k$-linear exact equivalence of triangulated categories then\nthere exists a Fourier-Mukai functor\n$F' : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$ whose\nkernel is in $D_{perf}(\\mathcal{O}_{X \\times Y})$\nwhich is an equivalence and a sibling of $F$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fully faithful functors","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0D","source_file":"equiv.tex","source_line":3335,"source_end_line":3344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3335-L3344","statement_sha256":"804d1eb06f8d6f2838f2fef45563ddac58394db8c418118b65dce9417f8d2e6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9673,"rank":9673,"depth":50,"x":130.561,"y":774.779,"cluster":"derived-categories"},{"id":"stacks:0G0E","tag":"0G0E","title":"Fully faithful functors · Lemma 0G0E","summary":"Let k be a field. Let X be a smooth proper scheme over k. Let K ∈ D_perf(O_X × X). If the Fourier-Mukai functor Φ_K : D_perf(O_X) → D_perf(O_X) is isomorphic to the identity functor, then K ≅ Δ_*O_X in _perf(O_X × X).","statement_latex":"Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$.\nLet $K \\in D_{perf}(\\mathcal{O}_{X \\times X})$. If the Fourier-Mukai\nfunctor $\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_X)$\nis isomorphic to the identity functor, then\n$K \\cong \\Delta_*\\mathcal{O}_X$ in $_{perf}(\\mathcal{O}_{X \\times X})$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Fully faithful functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0E","source_file":"equiv.tex","source_line":3351,"source_end_line":3358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3351-L3358","statement_sha256":"abfdd4d287e3cce1382a07baa77eacd541ef3f81856091ed9efcadd9ecc43608","origin":"The Stacks Project","memory_eligible":false,"source_rank":9674,"rank":9674,"depth":19,"x":227.103,"y":553.564,"cluster":"derived-categories"},{"id":"stacks:0G0G","tag":"0G0G","title":"A category of Fourier-Mukai kernels · Lemma 0G0G","summary":"Let S' → S be a morphism of schemes. The rule which sends • a smooth proper scheme X over S to X' = S' ×_S X, and • the isomorphism class of an object K of D_perf(O_X ×_S Y) to the isomorphism class of L(X' ×_S' Y' → X ×_S Y)^*K in D_perf(O_X' ×_S' Y') is a functor from the category defined for S to the category defined for S'.","statement_latex":"Let $S' \\to S$ be a morphism of schemes.\nThe rule which sends\n\\begin{enumerate}\n\\item a smooth proper scheme $X$ over $S$ to $X' =  S' \\times_S X$, and\n\\item the isomorphism class of an object $K$\nof $D_{perf}(\\mathcal{O}_{X \\times_S Y})$ to the isomorphism class of\n$L(X' \\times_{S'} Y' \\to X \\times_S Y)^*K$\nin $D_{perf}(\\mathcal{O}_{X' \\times_{S'} Y'})$\n\\end{enumerate}\nis a functor from the category defined for $S$ to the category\ndefined for $S'$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"A category of Fourier-Mukai kernels","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0G","source_file":"equiv.tex","source_line":3511,"source_end_line":3524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3511-L3524","statement_sha256":"be9b30f84570101a9fdcc9f1991f4c65b004134402e81ea25c2f67b702a647b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9675,"rank":9675,"depth":39,"x":333.913,"y":771.677,"cluster":"derived-categories"},{"id":"stacks:0G0I","tag":"0G0I","title":"Relative equivalences · Definition 0G0I","summary":"Let S be a scheme. Let X → S and Y → S be smooth proper morphisms. An object K ∈ D_perf(O_X ×_S Y) is said to be the Fourier-Mukai kernel of a relative equivalence from X to Y over S if there exist an object K' ∈ D_perf(O_X ×_S Y) such that Δ_X/S, *O_X ≅ Rpr_13, *(Lpr_12^*K ⊗_O_X ×_S Y ×_S X^L Lpr_23^*K') in D(O_X ×_S X) and Δ_Y/S, *O_Y ≅ Rpr_13, *(Lpr_12^*K' ⊗_O_Y ×_S X ×_S Y^L Lpr_23^*K) in D(O_Y ×_S Y). In other words, the isomorphism class of K defines an invertible…","statement_latex":"Let $S$ be a scheme. Let $X \\to S$ and $Y \\to S$ be smooth proper morphisms.\nAn object $K \\in D_{perf}(\\mathcal{O}_{X \\times_S Y})$\nis said to be {\\it the Fourier-Mukai kernel of a relative equivalence\nfrom $X$ to $Y$ over $S$}\nif there exist an object $K' \\in D_{perf}(\\mathcal{O}_{X \\times_S Y})$\nsuch that\n$$\n\\Delta_{X/S, *}\\mathcal{O}_X \\cong\nR\\text{pr}_{13, *}(L\\text{pr}_{12}^*K\n\\otimes_{\\mathcal{O}_{X \\times_S Y \\times_S X}}^\\mathbf{L}\nL\\text{pr}_{23}^*K')\n$$\nin $D(\\mathcal{O}_{X \\times_S X})$ and\n$$\n\\Delta_{Y/S, *}\\mathcal{O}_Y \\cong\nR\\text{pr}_{13, *}(L\\text{pr}_{12}^*K'\n\\otimes_{\\mathcal{O}_{Y \\times_S X \\times_S Y}}^\\mathbf{L}\nL\\text{pr}_{23}^*K)\n$$\nin $D(\\mathcal{O}_{Y \\times_S Y})$. In other words, the isomorphism class\nof $K$ defines an invertible arrow in the category defined in\nSection \\ref{section-category-Fourier-Mukai-kernels}.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Relative equivalences","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0I","source_file":"equiv.tex","source_line":3574,"source_end_line":3598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3574-L3598","statement_sha256":"9918f22065d0cc0f3b264828a0ab097687d11b0179b407d8122dc3fd1e22c053","origin":"The Stacks Project","memory_eligible":false,"source_rank":9676,"rank":9676,"depth":0,"x":79.506,"y":671.353,"cluster":"derived-categories"},{"id":"stacks:0G0J","tag":"0G0J","title":"Relative equivalences · Lemma 0G0J","summary":"With notation as in Definition [Tag 0G0I] let K be the Fourier-Mukai kernel of a relative equivalence from X to Y over S. Then the corresponding Fourier-Mukai functors Φ_K : D_QCoh(O_X) → D_QCoh(O_Y) (Lemma [Tag 0FYR]) and Φ_K : D_perf(O_X) → D_perf(O_Y) (Lemma [Tag 0FYT]) are equivalences.","statement_latex":"With notation as in Definition \\ref{definition-relative-equivalence-kernel}\nlet $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$\nto $Y$ over $S$. Then the corresponding Fourier-Mukai functors\n$\\Phi_K : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$\n(Lemma \\ref{lemma-fourier-Mukai-QCoh})\nand $\\Phi_K : D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\n(Lemma \\ref{lemma-fourier-mukai})\nare equivalences.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Relative equivalences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0J","source_file":"equiv.tex","source_line":3603,"source_end_line":3613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3603-L3613","statement_sha256":"21cab9ed2c158601d25f9d3b7669c4c688e563c2224917e76771549ab660eee7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9677,"rank":9677,"depth":41,"x":348.04,"y":600.906,"cluster":"derived-categories"},{"id":"stacks:0G0K","tag":"0G0K","title":"Relative equivalences · Lemma 0G0K","summary":"With notation as in Definition [Tag 0G0I] let K be the Fourier-Mukai kernel of a relative equivalence from X to Y over S. Let S_1 → S be a morphism of schemes. Let X_1 = S_1 ×_S X and Y_1 = S_1 ×_S Y. Then the pullback K_1 = L(X_1 ×_S_1 Y_1 → X ×_S Y)^*K is the Fourier-Mukai kernel of a relative equivalence from X_1 to Y_1 over S_1.","statement_latex":"With notation as in Definition \\ref{definition-relative-equivalence-kernel}\nlet $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$\nto $Y$ over $S$. Let $S_1 \\to S$ be a morphism of schemes. Let\n$X_1 = S_1 \\times_S X$ and $Y_1 = S_1 \\times_S Y$. Then the pullback\n$K_1 = L(X_1 \\times_{S_1} Y_1 \\to X \\times_S Y)^*K$ is\nthe Fourier-Mukai kernel of a relative equivalence from $X_1$\nto $Y_1$ over $S_1$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Relative equivalences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0K","source_file":"equiv.tex","source_line":3620,"source_end_line":3629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3620-L3629","statement_sha256":"d6716ea930d59b0405fb839f47f10faa9ea5e86811904d687cd004e1f8cc6285","origin":"The Stacks Project","memory_eligible":false,"source_rank":9678,"rank":9678,"depth":39,"x":206.542,"y":805.423,"cluster":"derived-categories"},{"id":"stacks:0G0L","tag":"0G0L","title":"Relative equivalences · Lemma 0G0L","summary":"Let S = lim_i ∈ I S_i be a limit of a directed system of schemes with affine transition morphisms g_i'i : S_i' → S_i. We assume that S_i is quasi-compact and quasi-separated for all i ∈ I. Let 0 ∈ I. Let X_0 → S_0 and Y_0 → S_0 be smooth proper morphisms. We set X_i = S_i ×_S_0 X_0 for i ≥ 0 and X = S ×_S_0 X_0 and similarly for Y_0. If K is the Fourier-Mukai kernel of a relative equivalence from X to Y over S then for some i ≥ 0 there exists a Fourier-Mukai kernel of a…","statement_latex":"Let $S = \\lim_{i \\in I} S_i$ be a limit of a directed system of schemes\nwith affine transition morphisms $g_{i'i} : S_{i'} \\to S_i$.\nWe assume that $S_i$ is quasi-compact and quasi-separated for all $i \\in I$.\nLet $0 \\in I$. Let $X_0 \\to S_0$ and $Y_0 \\to S_0$ be smooth proper morphisms.\nWe set $X_i = S_i \\times_{S_0} X_0$ for $i \\geq 0$\nand $X = S \\times_{S_0} X_0$ and similarly for $Y_0$. If $K$ is the\nFourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$\nthen for some $i \\geq 0$ there exists a\nFourier-Mukai kernel of a relative equivalence from $X_i$ to $Y_i$ over $S_i$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Relative equivalences","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0L","source_file":"equiv.tex","source_line":3670,"source_end_line":3681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3670-L3681","statement_sha256":"92fdc9026fdc7225ef25c396f8911d9c93488463f7b6e918ad5b1e3d7030dff5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9679,"rank":9679,"depth":40,"x":146.355,"y":574.102,"cluster":"derived-categories"},{"id":"stacks:0G0N","tag":"0G0N","title":"No deformations · Lemma 0G0N","summary":"Let (R, m, kappa) → (A, n, λ) be a flat local ring homorphism of local rings which is essentially of finite presentation. Let overlinef_1, …, overlinef_r ∈ n/ m A ⊂ A/ m A be a regular sequence. Let K ∈ D(A). Assume • K is perfect, • K ⊗_A^L A/ m A is isomorphic in D(A/ m A) to the Koszul complex on overlinef_1, …, overlinef_r. Then K is isomorphic in D(A) to a Koszul complex on a regular sequence f_1, …, f_r ∈ A lifting the given elements overlinef_1, …, overlinef_r.…","statement_latex":"Let $(R, \\mathfrak m, \\kappa) \\to (A, \\mathfrak n, \\lambda)$\nbe a flat local ring homorphism of local rings\nwhich is essentially of finite presentation.\nLet $\\overline{f}_1, \\ldots, \\overline{f}_r \\in \\mathfrak n/\\mathfrak m A\n\\subset A/\\mathfrak m A$ be a regular sequence. Let $K \\in D(A)$. Assume\n\\begin{enumerate}\n\\item $K$ is perfect,\n\\item $K \\otimes_A^\\mathbf{L} A/\\mathfrak m A$ is isomorphic in\n$D(A/\\mathfrak m A)$ to the\nKoszul complex on $\\overline{f}_1, \\ldots, \\overline{f}_r$.\n\\end{enumerate}\nThen $K$ is isomorphic in $D(A)$ to a Koszul complex on a regular sequence\n$f_1, \\ldots, f_r \\in A$ lifting the given elements\n$\\overline{f}_1, \\ldots, \\overline{f}_r$. Moreover, $A/(f_1, \\ldots, f_r)$\nis flat over $R$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"No deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0N","source_file":"equiv.tex","source_line":3744,"source_end_line":3761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3744-L3761","statement_sha256":"661653a4986bb2be0cc82edc83b955977f7cca260afaf2997bb55bc998c444b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9680,"rank":9680,"depth":14,"x":376.98,"y":710.655,"cluster":"derived-categories"},{"id":"stacks:0G0P","tag":"0G0P","title":"No deformations · Lemma 0G0P","summary":"Let R → S be a finite type flat ring map of Noetherian rings. Let q ⊂ S be a prime ideal lying over p ⊂ R. Let K ∈ D(S) be perfect. Let f_1, …, f_r ∈ q S_ q be a regular sequence such that S_ q/(f_1, …, f_r) is flat over R and such that K ⊗_S^L S_ q is isomorphic to the Koszul complex on f_1, …, f_r. Then there exists a g ∈ S, g not ∈ q such that • f_1, …, f_r are the images of f'_1, …, f'_r ∈ S_g, • f'_1, …, f'_r form a regular sequence in S_g, • S_g/(f'_1, …, f'_r) is…","statement_latex":"Let $R \\to S$ be a finite type flat ring map of Noetherian rings.\nLet $\\mathfrak q \\subset S$ be a prime ideal lying over\n$\\mathfrak p \\subset R$. Let $K \\in D(S)$ be perfect.\nLet $f_1, \\ldots, f_r \\in \\mathfrak q S_\\mathfrak q$\nbe a regular sequence such that $S_\\mathfrak q/(f_1, \\ldots, f_r)$\nis flat over $R$ and such that\n$K \\otimes_S^\\mathbf{L} S_\\mathfrak q$ is isomorphic to the\nKoszul complex on $f_1, \\ldots, f_r$. Then there exists a\n$g \\in S$, $g \\not \\in \\mathfrak q$ such that\n\\begin{enumerate}\n\\item $f_1, \\ldots, f_r$ are the images of\n$f'_1, \\ldots, f'_r \\in S_g$,\n\\item $f'_1, \\ldots, f'_r$ form a regular sequence in $S_g$,\n\\item $S_g/(f'_1, \\ldots, f'_r)$ is flat over $R$,\n\\item $K \\otimes_S^\\mathbf{L} S_g$ is isomorphic to the\nKoszul complex on $f_1, \\ldots, f_r$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"No deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0P","source_file":"equiv.tex","source_line":3821,"source_end_line":3840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3821-L3840","statement_sha256":"265350e3b3b4b00eb5e2efae98721a47fa2bbff15b0978c25ef648c22679c2a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9681,"rank":9681,"depth":34,"x":96.839,"y":740.854,"cluster":"derived-categories"},{"id":"stacks:0G0Q","tag":"0G0Q","title":"No deformations · Lemma 0G0Q","summary":"Let S be a Noetherian scheme. Let s ∈ S. Let p : X → Y be a morphism of schemes over S. Assume • Y → S and X → S proper, • X is flat over S, • X_s → Y_s an isomorphism. Then there exists an open neighbourhood U ⊂ S of s such that the base change X_U → Y_U is an isomorphism.","statement_latex":"Let $S$ be a Noetherian scheme. Let $s \\in S$.\nLet $p : X \\to Y$ be a morphism of schemes over $S$.\nAssume\n\\begin{enumerate}\n\\item $Y \\to S$ and $X \\to S$ proper,\n\\item $X$ is flat over $S$,\n\\item $X_s \\to Y_s$ an isomorphism.\n\\end{enumerate}\nThen there exists an open neighbourhood $U \\subset S$ of $s$\nsuch that the base change $X_U \\to Y_U$ is an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"No deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0Q","source_file":"equiv.tex","source_line":3862,"source_end_line":3874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3862-L3874","statement_sha256":"826813dda7149e0047f7842f13dfead94e916ea0a45de9f80afe3ca1d3fe06ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":9682,"rank":9682,"depth":41,"x":279.301,"y":559.453,"cluster":"derived-categories"},{"id":"stacks:0G0R","tag":"0G0R","title":"No deformations · Lemma 0G0R","summary":"Let k be a field. Let S be a finite type scheme over k with k-rational point s. Let Y → S be a smooth proper morphism. Let X = Y_s × S → S be the constant family with fibre Y_s. Let K be the Fourier-Mukai kernel of a relative equivalence from X to Y over S. Assume the restriction L(Y_s ×_S Y_s → X ×_S Y)^*K ≅ Δ_Y_s/k, * O_Y_s in D(O_Y_s × Y_s). Then there is an open neighbourhood s ∈ U ⊂ S such that Y|_U is isomorphic to Y_s × U over U.","statement_latex":"Let $k$ be a field. Let $S$ be a finite type scheme over $k$\nwith $k$-rational point $s$. Let $Y \\to S$ be a smooth proper morphism.\nLet $X = Y_s \\times S \\to S$ be the constant family with fibre\n$Y_s$. Let $K$ be the Fourier-Mukai kernel of a relative equivalence\nfrom $X$ to $Y$ over $S$. Assume the restriction\n$$\nL(Y_s \\times_S Y_s \\to X \\times_S Y)^*K \\cong \n\\Delta_{Y_s/k, *} \\mathcal{O}_{Y_s}\n$$\nin $D(\\mathcal{O}_{Y_s \\times Y_s})$. Then there is an open neighbourhood\n$s \\in U \\subset S$ such that $Y|_U$ is isomorphic to $Y_s \\times U$ over $U$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"No deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0R","source_file":"equiv.tex","source_line":3913,"source_end_line":3926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L3913-L3926","statement_sha256":"637ff8d522441d3cc0a437672876af5729e3c91c9403935e05965d4139cf7447","origin":"The Stacks Project","memory_eligible":false,"source_rank":9683,"rank":9683,"depth":42,"x":290.645,"y":796.976,"cluster":"derived-categories"},{"id":"stacks:0G0S","tag":"0G0S","title":"No deformations · Lemma 0G0S","summary":"Let k be an algebraically closed field. Let X be a smooth proper scheme over k. Let f : Y → S be a smooth proper morphism with S of finite type over k. Let K be the Fourier-Mukai kernel of a relative equivalence from X × S to Y over S. Then S can be covered by open subschemes U such that there is a U-isomorphism f^-1(U) ≅ Y_0 × U for some Y_0 proper and smooth over k.","statement_latex":"Let $k$ be an algebraically closed field. Let $X$\nbe a smooth proper scheme over $k$.\nLet $f : Y \\to S$ be a smooth proper morphism with $S$ of finite type over $k$.\nLet $K$ be the Fourier-Mukai kernel of a relative equivalence\nfrom $X \\times S$ to $Y$ over $S$. Then $S$ can be covered by\nopen subschemes $U$ such that there is a $U$-isomorphism\n$f^{-1}(U) \\cong Y_0 \\times U$ for some $Y_0$ proper and smooth over $k$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"No deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0S","source_file":"equiv.tex","source_line":4009,"source_end_line":4018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L4009-L4018","statement_sha256":"f81f5e72c970ba42b3e0a5733f6c02e2f713626524d65aef89115c56392594ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":9684,"rank":9684,"depth":43,"x":91.08,"y":628.106,"cluster":"derived-categories"},{"id":"stacks:0G0U","tag":"0G0U","title":"Countability · Lemma 0G0U","summary":"Let R be a countable Noetherian ring. Then the category of schemes of finite type over R is countable.","statement_latex":"Let $R$ be a countable Noetherian ring. Then the category of schemes of finite\ntype over $R$ is countable.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Countability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0U","source_file":"equiv.tex","source_line":4091,"source_end_line":4095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L4091-L4095","statement_sha256":"8c13405dd84960145d46500fa6986901b2933a225fc6211e22285ee0f36a90d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9685,"rank":9685,"depth":0,"x":374.307,"y":639.401,"cluster":"derived-categories"},{"id":"stacks:0G0V","tag":"0G0V","title":"Countability · Lemma 0G0V","summary":"Let A be a countable abelian category. Then D^b(A) is countable.","statement_latex":"Let $\\mathcal{A}$ be a countable abelian category.\nThen $D^b(\\mathcal{A})$ is countable.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Countability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0V","source_file":"equiv.tex","source_line":4101,"source_end_line":4105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L4101-L4105","statement_sha256":"ed2eca3fc877b1c1dd01bac8adf6d26b98762677948be5d8f92ea7200c312b86","origin":"The Stacks Project","memory_eligible":false,"source_rank":9686,"rank":9686,"depth":16,"x":156.169,"y":791.927,"cluster":"derived-categories"},{"id":"stacks:0G0W","tag":"0G0W","title":"Countability · Lemma 0G0W","summary":"Let X be a scheme of finite type over a countable Noetherian ring. Then the categories D_perf(O_X) and D^b_Coh(O_X) are countable.","statement_latex":"Let $X$ be a scheme of finite type over a countable Noetherian ring.\nThen the categories $D_{perf}(\\mathcal{O}_X)$ and\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ are countable.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Countability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0W","source_file":"equiv.tex","source_line":4125,"source_end_line":4130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L4125-L4130","statement_sha256":"bb2e299a9fe5216839328051247c02a869d1f8ddd24e632cc24713e6d5bd8fd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9687,"rank":9687,"depth":34,"x":194.4,"y":555.455,"cluster":"derived-categories"},{"id":"stacks:0G0X","tag":"0G0X","title":"Countability · Lemma 0G0X","summary":"Let K be an algebraically closed field. Let S be a finite type scheme over K. Let X → S and Y → S be finite type morphisms. There exists a countable set I and for i ∈ I a pair (S_i → S, h_i) with the following properties • S_i → S is a morphism of finite type, set X_i = X ×_S S_i and Y_i = Y ×_S S_i, • h_i : X_i → Y_i is an isomorphism over S_i, and • for any closed point s ∈ S(K) if X_s ≅ Y_s over K = kappa(s) then s is in the image of S_i → S for some i.","statement_latex":"Let $K$ be an algebraically closed field.\nLet $S$ be a finite type scheme over $K$.\nLet $X \\to S$ and $Y \\to S$ be finite type morphisms.\nThere exists a countable set $I$ and for $i \\in I$ a pair\n$(S_i \\to S, h_i)$ with the following properties\n\\begin{enumerate}\n\\item $S_i \\to S$ is a morphism of finite type, set\n$X_i = X \\times_S S_i$ and $Y_i = Y \\times_S S_i$,\n\\item $h_i : X_i \\to Y_i$ is an isomorphism over $S_i$, and\n\\item for any closed point $s \\in S(K)$ if $X_s \\cong Y_s$\nover $K = \\kappa(s)$ then $s$ is in the image of $S_i \\to S$\nfor some $i$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Countability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0X","source_file":"equiv.tex","source_line":4149,"source_end_line":4164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L4149-L4164","statement_sha256":"16a5f6d683436b7b7a230c6e94a90408c2d755d3a8085472e3161b4e41d316b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9688,"rank":9688,"depth":26,"x":356.526,"y":751.705,"cluster":"derived-categories"},{"id":"stacks:0G0Y","tag":"0G0Y","title":"Countability · Lemma 0G0Y","summary":"Let K be an algebraically closed field. There exists a countable set I and for i ∈ I a pair (S_i/K, X_i → S_i, Y_i → S_i, M_i) with the following properties • S_i is a scheme of finite type over K, • X_i → S_i and Y_i → S_i are proper smooth morphisms of schemes, • M_i ∈ D_perf(O_X_i ×_S_i Y_i) is the Fourier-Mukai kernel of a relative equivalence from X_i to Y_i over S_i, and • for any smooth proper schemes X and Y over K such that there is a K-linear exact equivalence…","statement_latex":"Let $K$ be an algebraically closed field. There exists a countable set $I$\nand for $i \\in I$ a pair $(S_i/K, X_i \\to S_i, Y_i \\to S_i, M_i)$\nwith the following properties\n\\begin{enumerate}\n\\item $S_i$ is a scheme of finite type over $K$,\n\\item $X_i \\to S_i$ and $Y_i \\to S_i$ are proper smooth\nmorphisms of schemes,\n\\item $M_i \\in D_{perf}(\\mathcal{O}_{X_i \\times_{S_i} Y_i})$\nis the Fourier-Mukai kernel of a relative equivalence from\n$X_i$ to $Y_i$ over $S_i$, and\n\\item for any smooth proper schemes $X$ and $Y$ over $K$\nsuch that there is a $K$-linear exact equivalence\n$D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$\nthere exists an $i \\in I$ and a $s \\in S_i(K)$\nsuch that $X \\cong (X_i)_s$ and $Y \\cong (Y_i)_s$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Countability","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G0Y","source_file":"equiv.tex","source_line":4222,"source_end_line":4240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L4222-L4240","statement_sha256":"a3ea7c07000323c65f49539133e4b672b36873c68cbc5b15d5d56aff21e971e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9689,"rank":9689,"depth":69,"x":78.895,"y":698.938,"cluster":"derived-categories"},{"id":"stacks:0G10","tag":"0G10","title":"Countability of derived equivalent varieties · Definition 0G10","summary":"Let k be a field. Let X and Y be smooth projective schemes over k. We say X and Y are derived equivalent if there exists a k-linear exact equivalence D_perf(O_X) → D_perf(O_Y).","statement_latex":"Let $k$ be a field. Let $X$ and $Y$ be smooth projective schemes over $k$.\nWe say $X$ and $Y$ are {\\it derived equivalent} if there exists a $k$-linear\nexact equivalence\n$D_{perf}(\\mathcal{O}_X) \\to D_{perf}(\\mathcal{O}_Y)$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Countability of derived equivalent varieties","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G10","source_file":"equiv.tex","source_line":4335,"source_end_line":4341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L4335-L4341","statement_sha256":"2acacea4156d4fe69bfb58ab619949db38544c7f3489fac0aa17f21750158849","origin":"The Stacks Project","memory_eligible":false,"source_rank":9690,"rank":9690,"depth":0,"x":326.284,"y":580.201,"cluster":"derived-categories"},{"id":"stacks:0G11","tag":"0G11","title":"Countability of derived equivalent varieties · Theorem 0G11","summary":"Slight improvement of [AT] Let K be an algebraically closed field. Let X be a smooth proper scheme over K. There are at most countably many isomorphism classes of smooth proper schemes Y over K which are derived equivalent to X.","statement_latex":"\\begin{reference}\nSlight improvement of \\cite{AT}\n\\end{reference}\nLet $K$ be an algebraically closed field. Let $\\mathbf{X}$ be a smooth proper\nscheme over $K$. There are at most countably many isomorphism classes\nof smooth proper schemes $\\mathbf{Y}$ over $K$ which are derived\nequivalent to $\\mathbf{X}$.","area":"Derived Categories","chapter":"Derived Categories of Varieties","chapter_id":"equiv","section":"Countability of derived equivalent varieties","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G11","source_file":"equiv.tex","source_line":4346,"source_end_line":4355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/equiv.tex#L4346-L4355","statement_sha256":"0bf38d44d52f90b25a323b17e7c04894ceae11eb75b493954318836c7049bb8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9691,"rank":9691,"depth":70,"x":239.266,"y":808.345,"cluster":"derived-categories"},{"id":"stacks:04JJ","tag":"04JJ","title":"Schemes étale over a point · Definition 04JJ","summary":"Let G be a topological group. A G-set, sometimes called a discrete G-set, is a set X endowed with a left action a : G × X → X such that a is continuous when X is given the discrete topology and G × X the product topology. A morphism of G-sets f : X → Y is simply any G-equivariant map from X to Y. The category of G-sets is denoted G-Sets.","statement_latex":"Let $G$ be a topological group.\nA {\\it $G$-set}, sometimes called a {\\it discrete $G$-set},\nis a set $X$ endowed with a left action $a : G \\times X \\to X$\nsuch that $a$ is continuous when $X$ is given the discrete topology and\n$G \\times X$ the product topology.\nA {\\it morphism of $G$-sets} $f : X \\to Y$ is simply any $G$-equivariant\nmap from $X$ to $Y$.\nThe category of $G$-sets is denoted {\\it $G\\textit{-Sets}$}.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Schemes étale over a point","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JJ","source_file":"pione.tex","source_line":43,"source_end_line":53,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L43-L53","statement_sha256":"444caf186e067707f8fe357bfd07c58d97943f0a6e65dc8d0943558fffb7d8c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9692,"rank":9692,"depth":0,"x":1565.992,"y":1176.27,"cluster":"tale-geometry"},{"id":"stacks:03QR","tag":"03QR","title":"Schemes étale over a point · Lemma 03QR","summary":"Let K be a field. Let K^sep be a separable closure of K. Consider the profinite group G = Gal(K^sep/K). The functor schemes étale over K & → & G-Sets X/K & ↦ & Mor_Spec(K)(Spec(K^sep), X) is an equivalence of categories.","statement_latex":"Let $K$ be a field. Let $K^{sep}$ be a separable closure of $K$.\nConsider the profinite group $G = \\text{Gal}(K^{sep}/K)$.\nThe functor\n$$\n\\begin{matrix}\n\\text{schemes \\'etale over }K &\n\\longrightarrow &\nG\\textit{-Sets} \\\\\nX/K & \\longmapsto &\n\\Mor_{\\Spec(K)}(\\Spec(K^{sep}), X)\n\\end{matrix}\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Schemes étale over a point","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QR","source_file":"pione.tex","source_line":68,"source_end_line":83,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L68-L83","statement_sha256":"2daf5ffe01611aab2507b397b1794225db1bb37a27ef107bf067942158802874","origin":"The Stacks Project","memory_eligible":false,"source_rank":9693,"rank":9693,"depth":45,"x":1561.009,"y":1105.026,"cluster":"tale-geometry"},{"id":"stacks:0BMR","tag":"0BMR","title":"Galois categories · Lemma 0BMR","summary":"Let C be a category and let F : C → Sets be a functor. The map ([Tag 0BS7]) identifies Aut(F) with a closed subgroup of ∏_X ∈ Ob(C) Aut(F(X)). In particular, if F(X) is finite for all X, then Aut(F) is a profinite group.","statement_latex":"Let $\\mathcal{C}$ be a category and let $F : \\mathcal{C} \\to \\textit{Sets}$\nbe a functor. The map (\\ref{equation-embedding-product}) identifies\n$\\text{Aut}(F)$ with a closed subgroup of\n$\\prod_{X \\in \\Ob(\\mathcal{C})} \\text{Aut}(F(X))$.\nIn particular, if $F(X)$ is finite for all $X$, then\n$\\text{Aut}(F)$ is a profinite group.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Galois categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMR","source_file":"pione.tex","source_line":165,"source_end_line":173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L165-L173","statement_sha256":"fae7f6a0c3e29dc200f14f0ab2d7dbe2d69f5efa66a2fed7a8a83edf18f15d84","origin":"The Stacks Project","memory_eligible":false,"source_rank":9694,"rank":9694,"depth":0,"x":1622.463,"y":1155.133,"cluster":"tale-geometry"},{"id":"stacks:0BMU","tag":"0BMU","title":"Galois categories · Lemma 0BMU","summary":"Let G be a topological group. The automorphism group of the functor ([Tag 0BMT]) endowed with its profinite topology from Lemma [Tag 0BMR] is the profinite completion of G.","statement_latex":"Let $G$ be a topological group. The automorphism group of the functor\n(\\ref{equation-forgetful}) endowed with its profinite topology from\nLemma \\ref{lemma-aut-inverse-limit} is the profinite completion of $G$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Galois categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMU","source_file":"pione.tex","source_line":219,"source_end_line":224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L219-L224","statement_sha256":"8b8a4fdc7e26c65d98a1aa478866753ebde77b8753d1ff0158966f8b5174a726","origin":"The Stacks Project","memory_eligible":false,"source_rank":9695,"rank":9695,"depth":2,"x":1536.179,"y":1153.037,"cluster":"tale-geometry"},{"id":"stacks:0BMV","tag":"0BMV","title":"Galois categories · Lemma 0BMV","summary":"Let G be a topological group. Let F : Finite-G-Sets → Sets be an exact functor with F(X) finite for all X. Then F is isomorphic to the functor ([Tag 0BMT]).","statement_latex":"Let $G$ be a topological group. Let\n$F : \\textit{Finite-}G\\textit{-Sets} \\to \\textit{Sets}$\nbe an exact functor with $F(X)$ finite for all $X$.\nThen $F$ is isomorphic to the functor (\\ref{equation-forgetful}).","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Galois categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMV","source_file":"pione.tex","source_line":279,"source_end_line":285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L279-L285","statement_sha256":"bf6ff315be90228a6e2f67287689a78551c6ef459dbe54af30ea680d30ad9caf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9696,"rank":9696,"depth":3,"x":1601.997,"y":1105.256,"cluster":"tale-geometry"},{"id":"stacks:0BMY","tag":"0BMY","title":"Galois categories · Definition 0BMY","summary":"Different from the definition in [SGA1]. Compare with [BS]. Let C be a category and let F : C → Sets be a functor. The pair (C, F) is a Galois category if • C has finite limits and finite colimits, • every object of C is a finite (possibly empty) coproduct of connected objects, • F(X) is finite for all X ∈ Ob(C), and • F reflects isomorphisms if F(f) is an isomorphism, then f is an isomorphism. and is exact. Here we say X ∈ Ob(C) is connected if it is not initial and for…","statement_latex":"\\begin{reference}\nDifferent from the definition in \\cite[Expos\\'e V, Definition 5.1]{SGA1}.\nCompare with \\cite[Definition 7.2.1]{BS}.\n\\end{reference}\nLet $\\mathcal{C}$ be a category and let $F : \\mathcal{C} \\to \\textit{Sets}$\nbe a functor. The pair $(\\mathcal{C}, F)$ is a {\\it Galois category} if\n\\begin{enumerate}\n\\item $\\mathcal{C}$ has finite limits and finite colimits,\n\\item\n\nevery object of $\\mathcal{C}$ is a finite (possibly empty)\ncoproduct of connected objects,\n\\item $F(X)$ is finite for all $X \\in \\Ob(\\mathcal{C})$, and\n\\item $F$ reflects isomorphisms\\footnote{Namely, given a morphism\n$f$ of $\\mathcal{C}$ if $F(f)$ is an isomorphism, then\n$f$ is an isomorphism.} and is exact\\footnote{This means that\n$F$ commutes with finite limits and colimits, see\nCategories, Section \\ref{categories-section-exact-functor}.}.\n\\end{enumerate}\nHere we say $X \\in \\Ob(\\mathcal{C})$ is connected if\nit is not initial and for any monomorphism $Y \\to X$\neither $Y$ is initial or $Y \\to X$ is an isomorphism.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Galois categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMY","source_file":"pione.tex","source_line":368,"source_end_line":392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L368-L392","statement_sha256":"15c4ff48f3b7efe091c171605afe18ac2c27927d057017e98f51a63565e20289","origin":"The Stacks Project","memory_eligible":false,"source_rank":9697,"rank":9697,"depth":0,"x":1591.809,"y":1178.39,"cluster":"tale-geometry"},{"id":"stacks:0BN0","tag":"0BN0","title":"Galois categories · Lemma 0BN0","summary":"Let (C, F) be a Galois category. Let X → Y ∈ Arrows(C). Then • F is faithful, • X → Y is a monomorphism ⇔ F(X) → F(Y) is injective, • X → Y is an epimorphism ⇔ F(X) → F(Y) is surjective, • an object A of C is initial if and only if F(A) = ∅, • an object Z of C is final if and only if F(Z) is a singleton, • if X and Y are connected, then X → Y is an epimorphism, • if X is connected and a, b : X → Y are two morphisms then a = b as soon as F(a) and F(b) agree on one element…","statement_latex":"Let $(\\mathcal{C}, F)$ be a Galois category. Let\n$X \\to Y \\in \\text{Arrows}(\\mathcal{C})$. Then\n\\begin{enumerate}\n\\item $F$ is faithful,\n\\item $X \\to Y$ is a monomorphism\n$\\Leftrightarrow F(X) \\to F(Y)$ is injective,\n\\item $X \\to Y$ is an epimorphism\n$\\Leftrightarrow F(X) \\to F(Y)$ is surjective,\n\\item an object $A$ of $\\mathcal{C}$ is initial if and only if\n$F(A) = \\emptyset$,\n\\item an object $Z$ of $\\mathcal{C}$ is final if and only if\n$F(Z)$ is a singleton,\n\\item if $X$ and $Y$ are connected, then $X \\to Y$ is an epimorphism,\n\\item\n\nif $X$ is connected and $a, b : X \\to Y$ are two morphisms\nthen $a = b$ as soon as $F(a)$ and $F(b)$ agree on one element of $F(X)$,\n\\item if $X = \\coprod_{i = 1, \\ldots, n} X_i$ and\n$Y = \\coprod_{j = 1, \\ldots, m} Y_j$ where $X_i$, $Y_j$ are connected,\nthen there is map $\\alpha : \\{1, \\ldots, n\\} \\to \\{1, \\ldots, m\\}$\nsuch that $X \\to Y$ comes from a collection of morphisms\n$X_i \\to Y_{\\alpha(i)}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Galois categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BN0","source_file":"pione.tex","source_line":406,"source_end_line":431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L406-L431","statement_sha256":"42bb122bde0aaaf14736114b0eb9a3158a06fa4e2347805692c8c9d2c07b6757","origin":"The Stacks Project","memory_eligible":false,"source_rank":9698,"rank":9698,"depth":0,"x":1540.124,"y":1118.231,"cluster":"tale-geometry"},{"id":"stacks:0BN2","tag":"0BN2","title":"Galois categories · Lemma 0BN2","summary":"Let (C, F) be a Galois category. For any connected object X of C there exists a Galois object Y and a morphism Y → X.","statement_latex":"Let $(\\mathcal{C}, F)$ be a Galois category. For any connected object $X$\nof $\\mathcal{C}$ there exists a Galois object $Y$ and a morphism $Y \\to X$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Galois categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BN2","source_file":"pione.tex","source_line":487,"source_end_line":491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L487-L491","statement_sha256":"4c4a90d77f642ed8d8b9ec098b08aa9d6a55bee9b7c27d6deff16485c5f845b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9699,"rank":9699,"depth":1,"x":1627.255,"y":1133.377,"cluster":"tale-geometry"},{"id":"stacks:0BN3","tag":"0BN3","title":"Galois categories · Lemma 0BN3","summary":"Compare with [BS]. Let (C, F) be a Galois category. Let G = Aut(F) be as in Example [Tag 0BMW]. For any connected X in C the action of G on F(X) is transitive.","statement_latex":"\\begin{reference}\nCompare with \\cite[Definition 7.2.4]{BS}.\n\\end{reference}\nLet $(\\mathcal{C}, F)$ be a Galois category. Let $G = \\text{Aut}(F)$\nbe as in Example \\ref{example-from-C-F-to-G-sets}. For any connected\n$X$ in $\\mathcal{C}$ the action of $G$ on $F(X)$ is transitive.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Galois categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BN3","source_file":"pione.tex","source_line":530,"source_end_line":538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L530-L538","statement_sha256":"cb5c55958a0fbf2afb8111d6b65c526d5b400c8be7f1e236e6d14575ef4bedf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9700,"rank":9700,"depth":3,"x":1550.267,"y":1171.927,"cluster":"tale-geometry"},{"id":"stacks:0BN4","tag":"0BN4","title":"Galois categories · Proposition 0BN4","summary":"This is a weak version of [SGA1]. The proof is borrowed from [BS]. Let (C, F) be a Galois category. Let G = Aut(F) be as in Example [Tag 0BMW]. The functor F : C → Finite-G-Sets ([Tag 0BMX]) an equivalence.","statement_latex":"\\begin{reference}\nThis is a weak version of \\cite[Expos\\'e V]{SGA1}.\nThe proof is borrowed from \\cite[Theorem 7.2.5]{BS}.\n\\end{reference}\nLet $(\\mathcal{C}, F)$ be a Galois category. Let $G = \\text{Aut}(F)$\nbe as in Example \\ref{example-from-C-F-to-G-sets}. The functor\n$F : \\mathcal{C} \\to \\textit{Finite-}G\\textit{-Sets}$\n(\\ref{equation-remember}) an equivalence.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Galois categories","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BN4","source_file":"pione.tex","source_line":611,"source_end_line":621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L611-L621","statement_sha256":"f8ae2b8b37315c6fef2fa6de2c026fc4221e159f86446ea4a38e9179f4382104","origin":"The Stacks Project","memory_eligible":false,"source_rank":9701,"rank":9701,"depth":4,"x":1576.223,"y":1099.297,"cluster":"tale-geometry"},{"id":"stacks:0BN5","tag":"0BN5","title":"Galois categories · Lemma 0BN5","summary":"Let (C, F) and (C', F') be Galois categories. Let H : C → C' be an exact functor. There exists an isomorphism t : F' ∘ H → F. The choice of t determines a continuous homomorphism h : G' = Aut(F') → Aut(F) = G and a 2-commutative diagram xymatrix C ar[r]_H ar[d] & C' ar[d] Finite-G-Sets ar[r]^h & Finite-G'-Sets The map h is independent of t up to an inner automorphism of G. Conversely, given a continuous homomorphism h : G' → G there is an exact functor H : C → C' and an…","statement_latex":"Let $(\\mathcal{C}, F)$ and $(\\mathcal{C}', F')$ be Galois categories.\nLet $H : \\mathcal{C} \\to \\mathcal{C}'$ be an exact functor.\nThere exists an isomorphism $t : F' \\circ H \\to F$.\nThe choice of $t$ determines a continuous homomorphism\n$h : G' = \\text{Aut}(F') \\to \\text{Aut}(F) = G$ and\na $2$-commutative diagram\n$$\n\\xymatrix{\n\\mathcal{C} \\ar[r]_H \\ar[d] & \\mathcal{C}' \\ar[d] \\\\\n\\textit{Finite-}G\\textit{-Sets} \\ar[r]^h &\n\\textit{Finite-}G'\\textit{-Sets}\n}\n$$\nThe map $h$ is independent of $t$ up\nto an inner automorphism of $G$.\nConversely, given a continuous homomorphism $h : G' \\to G$ there\nis an exact functor $H : \\mathcal{C} \\to \\mathcal{C}'$ and an\nisomorphism $t$ recovering $h$ as above.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Galois categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BN5","source_file":"pione.tex","source_line":671,"source_end_line":691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L671-L691","statement_sha256":"7388d98627f1253745abcd775f1fa144235629170266cc88fa92e3a786ca3773","origin":"The Stacks Project","memory_eligible":false,"source_rank":9702,"rank":9702,"depth":5,"x":1615.767,"y":1168.067,"cluster":"tale-geometry"},{"id":"stacks:0BN6","tag":"0BN6","title":"Functors and homomorphisms · Lemma 0BN6","summary":"In diagram ([Tag 0BTR]) the following are equivalent • h : G' → G is surjective, • H : C → C' is fully faithful, • if X ∈ Ob(C) is connected, then H(X) is connected, • if X ∈ Ob(C) is connected and there is a morphism *' → H(X) in C', then there is a morphism * → X, and • for any object X of C the map Mor_C(*, X) → Mor_C'(*', H(X)) is bijective. Here * and *' are final objects of C and C'.","statement_latex":"In diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item $h : G' \\to G$ is surjective,\n\\item $H : \\mathcal{C} \\to \\mathcal{C}'$ is fully faithful,\n\\item if $X \\in \\Ob(\\mathcal{C})$ is connected, then $H(X)$ is connected,\n\\item if $X \\in \\Ob(\\mathcal{C})$ is connected and there is\na morphism $*' \\to H(X)$ in $\\mathcal{C}'$, then\nthere is a morphism $* \\to X$, and\n\\item for any object $X$ of $\\mathcal{C}$ the map\n$\\Mor_\\mathcal{C}(*, X) \\to \\Mor_{\\mathcal{C}'}(*', H(X))$\nis bijective.\n\\end{enumerate}\nHere $*$ and $*'$ are final objects of $\\mathcal{C}$ and $\\mathcal{C}'$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Functors and homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BN6","source_file":"pione.tex","source_line":736,"source_end_line":751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L736-L751","statement_sha256":"8239416b6c2f82a7ce621e60065296e1cb4dd6f4054c245831d540e3e5c0c551","origin":"The Stacks Project","memory_eligible":false,"source_rank":9703,"rank":9703,"depth":0,"x":1530.715,"y":1139.596,"cluster":"tale-geometry"},{"id":"stacks:0BS8","tag":"0BS8","title":"Functors and homomorphisms · Lemma 0BS8","summary":"In diagram ([Tag 0BTR]) the following are equivalent • h ∘ h' is trivial, and • the image of H' ∘ H consists of objects isomorphic to finite coproducts of final objects.","statement_latex":"In diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item $h \\circ h'$ is trivial, and\n\\item the image of $H' \\circ H$ consists of objects isomorphic to finite\ncoproducts of final objects.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Functors and homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BS8","source_file":"pione.tex","source_line":786,"source_end_line":794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L786-L794","statement_sha256":"7e96a8926db33545ac5c508690702578d4afdbe5d790ef9c7c8e2066cdac31e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9704,"rank":9704,"depth":0,"x":1616.918,"y":1112.143,"cluster":"tale-geometry"},{"id":"stacks:0BS9","tag":"0BS9","title":"Functors and homomorphisms · Lemma 0BS9","summary":"In diagram ([Tag 0BTR]) the following are equivalent • the sequence G\" xrightarrowh' G' xrightarrowh G → 1 is exact in the following sense: h is surjective, h ∘ h' is trivial, and Ker(h) is the smallest closed normal subgroup containing Im(h'), • H is fully faithful and an object X' of C' is in the essential image of H if and only if H'(X') is isomorphic to a finite coproduct of final objects, and • H is fully faithful, H ∘ H' sends every object to a finite coproduct of…","statement_latex":"In diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item the sequence $G'' \\xrightarrow{h'} G' \\xrightarrow{h} G \\to 1$\nis exact in the following sense: $h$ is surjective, $h \\circ h'$ is trivial,\nand $\\Ker(h)$ is the smallest closed normal subgroup containing $\\Im(h')$,\n\\item $H$ is fully faithful and an object $X'$ of $\\mathcal{C}'$ is in\nthe essential image of $H$ if and only if $H'(X')$ is isomorphic to a\nfinite coproduct of final objects, and\n\\item $H$ is fully faithful, $H \\circ H'$ sends every object to a finite\ncoproduct of final objects, and for an object $X'$ of $\\mathcal{C}'$\nsuch that $H'(X')$ is a finite coproduct of final objects there exists\nan object $X$ of $\\mathcal{C}$ and an epimorphism $H(X) \\to X'$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Functors and homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BS9","source_file":"pione.tex","source_line":804,"source_end_line":819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L804-L819","statement_sha256":"9fa00e011960f7f083b8f1fbc2203a2dc6e322fc8c1b2817eec906c8b2e23ade","origin":"The Stacks Project","memory_eligible":false,"source_rank":9705,"rank":9705,"depth":1,"x":1575.147,"y":1181.77,"cluster":"tale-geometry"},{"id":"stacks:0BN7","tag":"0BN7","title":"Functors and homomorphisms · Lemma 0BN7","summary":"In diagram ([Tag 0BTR]) the following are equivalent • h' is injective, and • for every connected object X\" of C\" there exists an object X' of C' and a diagram X\" ← Y\" → H(X') in C\" where Y\" → X\" is an epimorphism and Y\" → H(X') is a monomorphism.","statement_latex":"In diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item $h'$ is injective, and\n\\item for every connected object $X''$ of $\\mathcal{C}''$\nthere exists an object $X'$ of $\\mathcal{C}'$ and a diagram\n$$\nX'' \\leftarrow Y'' \\rightarrow H(X')\n$$\nin $\\mathcal{C}''$ where $Y'' \\to X''$ is an epimorphism and\n$Y'' \\to H(X')$ is a monomorphism.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Functors and homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BN7","source_file":"pione.tex","source_line":857,"source_end_line":870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L857-L870","statement_sha256":"265dd61983a76d146e8ab179a1be62e5b961e63d0ea67992c04bf9d5de42a0de","origin":"The Stacks Project","memory_eligible":false,"source_rank":9706,"rank":9706,"depth":0,"x":1549.788,"y":1106.222,"cluster":"tale-geometry"},{"id":"stacks:0BTS","tag":"0BTS","title":"Functors and homomorphisms · Lemma 0BTS","summary":"In diagram ([Tag 0BTR]) the following are equivalent • the image of h' is normal, and • for every connected object X' of C' such that there is a morphism from the final object of C\" to H'(X') we have that H'(X') is isomorphic to a finite coproduct of final objects.","statement_latex":"In diagram (\\ref{equation-translation}) the following are equivalent\n\\begin{enumerate}\n\\item the image of $h'$ is normal, and\n\\item for every connected object $X'$ of $\\mathcal{C}'$ such that\nthere is a morphism from the final object of $\\mathcal{C}''$\nto $H'(X')$ we have that $H'(X')$ is isomorphic to a finite coproduct\nof final objects.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Functors and homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTS","source_file":"pione.tex","source_line":899,"source_end_line":909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L899-L909","statement_sha256":"76898ac2d07f9ddbd11e511fd00a10caf98b02f19c7ab00acdce453f6bbb1498","origin":"The Stacks Project","memory_eligible":false,"source_rank":9707,"rank":9707,"depth":0,"x":1629.766,"y":1147.816,"cluster":"tale-geometry"},{"id":"stacks:0BN9","tag":"0BN9","title":"Finite étale morphisms · Lemma 0BN9","summary":"Let X be a scheme. The category FÉt_X has finite limits and finite colimits and for any morphism X' → X the base change functor FÉt_X → FÉt_X' is exact.","statement_latex":"Let $X$ be a scheme. The category $\\textit{F\\'Et}_X$ has finite limits and\nfinite colimits and for any morphism $X' \\to X$ the base change functor\n$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X'}$ is exact.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BN9","source_file":"pione.tex","source_line":966,"source_end_line":971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L966-L971","statement_sha256":"a0620085a3364460819be6822178cf175e8129b3e85443ad16ccb737275bea36","origin":"The Stacks Project","memory_eligible":false,"source_rank":9708,"rank":9708,"depth":47,"x":1536.739,"y":1162.621,"cluster":"tale-geometry"},{"id":"stacks:0BL7","tag":"0BL7","title":"Finite étale morphisms · Lemma 0BL7","summary":"Let X be a scheme. Given U, V finite étale over X there exists a scheme W finite étale over X such that Mor_X(X, W) = Mor_X(U, V) and such that the same remains true after any base change.","statement_latex":"Let $X$ be a scheme. Given $U, V$ finite \\'etale over $X$ there\nexists a scheme $W$ finite \\'etale over $X$ such that\n$$\n\\Mor_X(X, W) = \\Mor_X(U, V)\n$$\nand such that the same remains true after any base change.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BL7","source_file":"pione.tex","source_line":1067,"source_end_line":1075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1067-L1075","statement_sha256":"6165bc8f6557b65d61237a471c5ce7eb419a5c25ab86faa8841a1dd9e2d8c8f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9709,"rank":9709,"depth":51,"x":1593.797,"y":1098.508,"cluster":"tale-geometry"},{"id":"stacks:0BNB","tag":"0BNB","title":"Finite étale morphisms · Lemma 0BNB","summary":"Let X be a connected scheme. Let overlinex be a geometric point. The functor F_overlinex : FÉt_X → Sets, Y ↦ |Y_overlinex| defines a Galois category (Definition [Tag 0BMY]).","statement_latex":"Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point.\nThe functor\n$$\nF_{\\overline{x}} : \\textit{F\\'Et}_X \\longrightarrow \\textit{Sets},\\quad\nY \\longmapsto |Y_{\\overline{x}}|\n$$\ndefines a Galois category (Definition \\ref{definition-galois-category}).","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNB","source_file":"pione.tex","source_line":1135,"source_end_line":1144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1135-L1144","statement_sha256":"4d281a74052451cfebd0678de9fcce660ef01461469dc61f888c4a9185bf2c7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9710,"rank":9710,"depth":48,"x":1603.34,"y":1178.667,"cluster":"tale-geometry"},{"id":"stacks:0BNC","tag":"0BNC","title":"Fundamental groups · Definition 0BNC","summary":"Let X be a connected scheme. Let overlinex be a geometric point of X. The fundamental group of X with base point overlinex is the group π_1(X, overlinex) = Aut(F_overlinex) of automorphisms of the fibre functor F_overlinex : FÉt_X → Sets endowed with its canonical profinite topology from Lemma [Tag 0BMR].","statement_latex":"Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point\nof $X$. The {\\it fundamental group} of $X$ with\n{\\it base point} $\\overline{x}$ is the group\n$$\n\\pi_1(X, \\overline{x}) = \\text{Aut}(F_{\\overline{x}})\n$$\nof automorphisms of the fibre functor\n$F_{\\overline{x}} : \\textit{F\\'Et}_X \\to \\textit{Sets}$\nendowed with its canonical profinite topology from\nLemma \\ref{lemma-aut-inverse-limit}.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Fundamental groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNC","source_file":"pione.tex","source_line":1198,"source_end_line":1210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1198-L1210","statement_sha256":"1779aed921fb0a31012d83d211874f16936df531f046cc03c4ac847b9f7a6166","origin":"The Stacks Project","memory_eligible":false,"source_rank":9711,"rank":9711,"depth":1,"x":1531.392,"y":1124.636,"cluster":"tale-geometry"},{"id":"stacks:0BND","tag":"0BND","title":"Fundamental groups · Theorem 0BND","summary":"Let X be a connected scheme. Let overlinex be a geometric point of X. • The fibre functor F_overlinex defines an equivalence of categories FÉt_X → Finite-π_1(X, overlinex)-Sets • Given a second geometric point overlinex' of X there exists an isomorphism t : F_overlinex → F_overlinex'. This gives an isomorphism π_1(X, overlinex) → π_1(X, overlinex') compatible with the equivalences in (1). This isomorphism is independent of t up to inner conjugation. • Given a morphism f :…","statement_latex":"Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point\nof $X$.\n\\begin{enumerate}\n\\item The fibre functor $F_{\\overline{x}}$ defines an equivalence of\ncategories\n$$\n\\textit{F\\'Et}_X \\longrightarrow\n\\textit{Finite-}\\pi_1(X, \\overline{x})\\textit{-Sets}\n$$\n\\item Given a second geometric point $\\overline{x}'$ of $X$ there\nexists an isomorphism $t : F_{\\overline{x}} \\to F_{\\overline{x}'}$.\nThis gives an isomorphism $\\pi_1(X, \\overline{x}) \\to \\pi_1(X, \\overline{x}')$\ncompatible with the equivalences in (1). This isomorphism is\nindependent of $t$ up to inner conjugation.\n\\item Given a morphism $f : X \\to Y$ of connected schemes denote\n$\\overline{y} = f \\circ \\overline{x}$. There is a canonical\ncontinuous homomorphism\n$$\nf_* : \\pi_1(X, \\overline{x}) \\to \\pi_1(Y, \\overline{y})\n$$\nsuch that the diagram\n$$\n\\xymatrix{\n\\textit{F\\'Et}_Y \\ar[r]_{\\text{base change}} \\ar[d]_{F_{\\overline{y}}} &\n\\textit{F\\'Et}_X \\ar[d]^{F_{\\overline{x}}} \\\\\n\\textit{Finite-}\\pi_1(Y, \\overline{y})\\textit{-Sets} \\ar[r]^{f_*} &\n\\textit{Finite-}\\pi_1(X, \\overline{x})\\textit{-Sets}\n}\n$$\nis commutative.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Fundamental groups","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BND","source_file":"pione.tex","source_line":1216,"source_end_line":1249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1216-L1249","statement_sha256":"c136a16c5fdbb7f2989722f0df17ea095246e0ba536a35d6625fb867a05c873b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9712,"rank":9712,"depth":49,"x":1628.496,"y":1123.648,"cluster":"tale-geometry"},{"id":"stacks:0BNE","tag":"0BNE","title":"Fundamental groups · Lemma 0BNE","summary":"Let K be a field and set X = Spec(K). Let overlineK be an algebraic closure and denote overlinex : Spec(overlineK) → X the corresponding geometric point. Let K^sep ⊂ overlineK be the separable algebraic closure. • The functor of Lemma [Tag 03QR] induces an equivalence FÉt_X → Finite-Gal(K^sep/K)-Sets. compatible with F_overlinex and the functor Finite-Gal(K^sep/K)-Sets → Sets. • This induces a canonical isomorphism Gal(K^sep/K) → π_1(X, overlinex) of profinite topological…","statement_latex":"Let $K$ be a field and set $X = \\Spec(K)$. Let $\\overline{K}$ be an\nalgebraic closure and denote $\\overline{x} : \\Spec(\\overline{K}) \\to X$\nthe corresponding geometric point. Let $K^{sep} \\subset \\overline{K}$\nbe the separable algebraic closure.\n\\begin{enumerate}\n\\item The functor of Lemma \\ref{lemma-sheaves-point} induces an equivalence\n$$\n\\textit{F\\'Et}_X \\longrightarrow\n\\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-Sets}.\n$$\ncompatible with $F_{\\overline{x}}$ and the functor\n$\\textit{Finite-}\\text{Gal}(K^{sep}/K)\\textit{-Sets} \\to \\textit{Sets}$.\n\\item This induces a canonical isomorphism\n$$\n\\text{Gal}(K^{sep}/K) \\longrightarrow \\pi_1(X, \\overline{x})\n$$\nof profinite topological groups.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Fundamental groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BNE","source_file":"pione.tex","source_line":1269,"source_end_line":1289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1269-L1289","statement_sha256":"f78abebbce10a000d9644f511515d75dc65ffd94e5dad2f025921c7e878767d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9713,"rank":9713,"depth":46,"x":1557.252,"y":1179.817,"cluster":"tale-geometry"},{"id":"stacks:0BQA","tag":"0BQA","title":"Topological invariance of the fundamental group · Lemma 0BQA","summary":"Let f : X → Y be a morphism of quasi-compact and quasi-separated schemes such that the base change functor FÉt_Y → FÉt_X is an equivalence of categories. In this case • f induces a homeomorphism π_0(X) → π_0(Y), • if X or equivalently Y is connected, then π_1(X, overlinex) = π_1(Y, overliney).","statement_latex":"Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated schemes\nsuch that the base change functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$\nis an equivalence of categories. In this case\n\\begin{enumerate}\n\\item $f$ induces a homeomorphism $\\pi_0(X) \\to \\pi_0(Y)$,\n\\item if $X$ or equivalently $Y$ is connected, then\n$\\pi_1(X, \\overline{x}) = \\pi_1(Y, \\overline{y})$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Topological invariance of the fundamental group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQA","source_file":"pione.tex","source_line":1419,"source_end_line":1429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1419-L1429","statement_sha256":"43ef17daebc13de1f0dc3cc07052db84b1ef954802307607673fef13c75887fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9714,"rank":9714,"depth":5,"x":1564.663,"y":1097.478,"cluster":"tale-geometry"},{"id":"stacks:09ZS","tag":"09ZS","title":"Topological invariance of the fundamental group · Lemma 09ZS","summary":"Let (A, I) be a henselian pair. Set X = Spec(A) and Z = Spec(A/I). The functor FÉt_X → FÉt_Z, U ↦ U ×_X Z is an equivalence of categories.","statement_latex":"Let $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and $Z = \\Spec(A/I)$.\nThe functor\n$$\n\\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_Z,\\quad\nU \\longmapsto U \\times_X Z\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Topological invariance of the fundamental group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZS","source_file":"pione.tex","source_line":1470,"source_end_line":1479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1470-L1479","statement_sha256":"65ac0c48d4d634642611be0fc6398be5a119ac44cb680b14755e3cf77c141294","origin":"The Stacks Project","memory_eligible":false,"source_rank":9715,"rank":9715,"depth":48,"x":1625.775,"y":1162.787,"cluster":"tale-geometry"},{"id":"stacks:0BQB","tag":"0BQB","title":"Topological invariance of the fundamental group · Lemma 0BQB","summary":"Let X ⊂ X' be a thickening of schemes. The functor FÉt_X' → FÉt_X, U' ↦ U' ×_X' X is an equivalence of categories.","statement_latex":"Let $X \\subset X'$ be a thickening of schemes. The functor\n$$\n\\textit{F\\'Et}_{X'} \\longrightarrow \\textit{F\\'Et}_X,\\quad\nU' \\longmapsto U' \\times_{X'} X\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Topological invariance of the fundamental group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQB","source_file":"pione.tex","source_line":1492,"source_end_line":1500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1492-L1500","statement_sha256":"b6b86760d6497104ac9ec6ecf39b8e293562fc2609f86019b14e72505ca7805a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9716,"rank":9716,"depth":49,"x":1527.615,"y":1149.225,"cluster":"tale-geometry"},{"id":"stacks:0BQN","tag":"0BQN","title":"Topological invariance of the fundamental group · Proposition 0BQN","summary":"Let f : X → Y be a universal homeomorphism of schemes. Then FÉt_Y → FÉt_X, V ↦ V ×_Y X is an equivalence. Thus if X and Y are connected, then f induces an isomorphism π_1(X, overlinex) → π_1(Y, overliney) of fundamental groups.","statement_latex":"Let $f : X \\to Y$ be a universal homeomorphism of schemes. Then\n$$\n\\textit{F\\'Et}_Y \\longrightarrow \\textit{F\\'Et}_X,\\quad\nV \\longmapsto V \\times_Y X\n$$\nis an equivalence. Thus if $X$ and $Y$ are connected, then\n$f$ induces an isomorphism $\\pi_1(X, \\overline{x}) \\to \\pi_1(Y, \\overline{y})$\nof fundamental groups.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Topological invariance of the fundamental group","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQN","source_file":"pione.tex","source_line":1546,"source_end_line":1556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1546-L1556","statement_sha256":"833e844f7ff31b7c01d0b7fb069e7c69672a734ae59d6b19d69f5ffd404585ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":9717,"rank":9717,"depth":52,"x":1611.391,"y":1103.261,"cluster":"tale-geometry"},{"id":"stacks:0A48","tag":"0A48","title":"Finite étale covers of proper schemes · Lemma 0A48","summary":"Let A be a henselian local ring. Let X be a proper scheme over A with closed fibre X_0. Then the functor FÉt_X → FÉt_X_0, U ↦ U_0 = U ×_X X_0 is an equivalence of categories.","statement_latex":"Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$\nwith closed fibre $X_0$. Then the functor\n$$\n\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X_0},\\quad\nU \\longmapsto U_0 = U \\times_X X_0\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A48","source_file":"pione.tex","source_line":1621,"source_end_line":1630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1621-L1630","statement_sha256":"2f03b0b826b5f03ce8f9e2c520d0d214a320e99996c2b92d09ff8703b0a1d755","origin":"The Stacks Project","memory_eligible":false,"source_rank":9718,"rank":9718,"depth":53,"x":1586.434,"y":1185.161,"cluster":"tale-geometry"},{"id":"stacks:0GS2","tag":"0GS2","title":"Finite étale covers of proper schemes · Lemma 0GS2","summary":"Let (A, I) be a henselian pair. Let X be a proper scheme over A. Set X_0 = X ×_Spec(A) Spec(A/I). Then the functor FÉt_X → FÉt_X_0, U ↦ U_0 = U ×_X X_0 is an equivalence of categories.","statement_latex":"Let $(A, I)$ be a henselian pair. Let $X$ be a proper scheme over $A$.\nSet $X_0 = X \\times_{\\Spec(A)} \\Spec(A/I)$. Then the functor\n$$\n\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_{X_0},\\quad\nU \\longmapsto U_0 = U \\times_X X_0\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GS2","source_file":"pione.tex","source_line":1748,"source_end_line":1757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1748-L1757","statement_sha256":"e4472315885ef1370d63f6b801a22171c8c0cac4922d59d2d2447eae315c71fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9719,"rank":9719,"depth":54,"x":1538.706,"y":1110.182,"cluster":"tale-geometry"},{"id":"stacks:0A49","tag":"0A49","title":"Finite étale covers of proper schemes · Lemma 0A49","summary":"Let k'/k be an extension of algebraically closed fields. Let X be a proper scheme over k. Then the functor U ↦ U_k' is an equivalence of categories between schemes finite étale over X and schemes finite étale over X_k'.","statement_latex":"Let $k'/k$ be an extension of algebraically closed fields.\nLet $X$ be a proper scheme over $k$. Then the functor\n$$\nU \\longmapsto U_{k'}\n$$\nis an equivalence of categories between schemes finite \\'etale over\n$X$ and schemes finite \\'etale over $X_{k'}$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A49","source_file":"pione.tex","source_line":1882,"source_end_line":1891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1882-L1891","statement_sha256":"cf0a0ffdb029a64b0611c5b9569c3f59b4b66ee8403b95ba8585804478d5e3bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9720,"rank":9720,"depth":54,"x":1634.734,"y":1138.548,"cluster":"tale-geometry"},{"id":"stacks:0BQE","tag":"0BQE","title":"Local connectedness · Lemma 0BQE","summary":"Let f : X → Y be a morphism of schemes. If f(X) is dense in Y then the base change functor FÉt_Y → FÉt_X is faithful.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. If $f(X)$ is dense in $Y$\nthen the base change functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_X$\nis faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Local connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQE","source_file":"pione.tex","source_line":1936,"source_end_line":1941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1936-L1941","statement_sha256":"76f497f532d73b67ff6c8819ae0a170ee0d303e512ea836b46ece6290ca91975","origin":"The Stacks Project","memory_eligible":false,"source_rank":9721,"rank":9721,"depth":52,"x":1540.59,"y":1172.307,"cluster":"tale-geometry"},{"id":"stacks:0BLQ","tag":"0BLQ","title":"Local connectedness · Lemma 0BLQ","summary":"Let (A, m) be a local ring. Set X = Spec(A) and let U = X setminus ( m). If the punctured spectrum of the strict henselization of A is connected, then FÉt_X → FÉt_U, Y ↦ Y ×_X U is a fully faithful functor.","statement_latex":"Let $(A, \\mathfrak m)$ be a local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$. If the punctured spectrum\nof the strict henselization of $A$ is connected, then\n$$\n\\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_U,\\quad\nY \\longmapsto Y \\times_X U\n$$\nis a fully faithful functor.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Local connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLQ","source_file":"pione.tex","source_line":1968,"source_end_line":1978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L1968-L1978","statement_sha256":"c3739e3a0d0492235dcd1b44ec7cb3028c2077d4bbada9afe6d770c217a0a84e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9722,"rank":9722,"depth":53,"x":1583.097,"y":1093.562,"cluster":"tale-geometry"},{"id":"stacks:0BQF","tag":"0BQF","title":"Local connectedness · Lemma 0BQF","summary":"Let X be a scheme. Let U ⊂ X be a dense open. Assume • the underlying topological space of X is Noetherian, and • for every x ∈ X setminus U the punctured spectrum of the strict henselization of O_X, x is connected. Then FÉt_X → Fét_U is fully faithful.","statement_latex":"Let $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume\n\\begin{enumerate}\n\\item the underlying topological space of $X$ is Noetherian, and\n\\item for every $x \\in X \\setminus U$ the punctured spectrum of the\nstrict henselization of $\\mathcal{O}_{X, x}$ is connected.\n\\end{enumerate}\nThen $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Local connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQF","source_file":"pione.tex","source_line":2018,"source_end_line":2027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2018-L2027","statement_sha256":"dcc3f567ff250c492f3ad90591fab295bc90456f30f9df5da81cbd0f9bb49574","origin":"The Stacks Project","memory_eligible":false,"source_rank":9723,"rank":9723,"depth":54,"x":1615.251,"y":1176.196,"cluster":"tale-geometry"},{"id":"stacks:0BSA","tag":"0BSA","title":"Local connectedness · Lemma 0BSA","summary":"Let X be a scheme. Let U ⊂ X be a dense open. Assume • U → X is quasi-compact, • every point of X setminus U is closed, and • for every x ∈ X setminus U the punctured spectrum of the strict henselization of O_X, x is connected. Then FÉt_X → Fét_U is fully faithful.","statement_latex":"Let $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume\n\\begin{enumerate}\n\\item $U \\to X$ is quasi-compact,\n\\item every point of $X \\setminus U$ is closed, and\n\\item for every $x \\in X \\setminus U$ the punctured spectrum of the\nstrict henselization of $\\mathcal{O}_{X, x}$ is connected.\n\\end{enumerate}\nThen $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Local connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSA","source_file":"pione.tex","source_line":2052,"source_end_line":2062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2052-L2062","statement_sha256":"4768cf18bb4b81d7c064b1e73c5dd3e3f11f9d7bbcd6790dba4d7eb965331b38","origin":"The Stacks Project","memory_eligible":false,"source_rank":9724,"rank":9724,"depth":54,"x":1524.603,"y":1133.276,"cluster":"tale-geometry"},{"id":"stacks:0BQG","tag":"0BQG","title":"Local connectedness · Lemma 0BQG","summary":"Let X be a scheme. Let U ⊂ X be a dense open. Assume • every quasi-compact open of X has finitely many irreducible components, • for every x ∈ X setminus U the punctured spectrum of the strict henselization of O_X, x is connected. Then FÉt_X → Fét_U is fully faithful.","statement_latex":"Let $X$ be a scheme. Let $U \\subset X$ be a dense open. Assume\n\\begin{enumerate}\n\\item every quasi-compact open of $X$ has finitely many\nirreducible components,\n\\item for every $x \\in X \\setminus U$ the punctured spectrum of the\nstrict henselization of $\\mathcal{O}_{X, x}$ is connected.\n\\end{enumerate}\nThen $\\textit{F\\'Et}_X \\to \\textit{F\\'et}_U$ is fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Local connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQG","source_file":"pione.tex","source_line":2089,"source_end_line":2099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2089-L2099","statement_sha256":"c5648e943edaa22950b332b01348eac6ffc129dd6cb37e24253d19d9f149124c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9725,"rank":9725,"depth":54,"x":1626.501,"y":1113.384,"cluster":"tale-geometry"},{"id":"stacks:0BSB","tag":"0BSB","title":"Local connectedness · Lemma 0BSB","summary":"Let (A, m) be a local ring. Set X = Spec(A) and U = X setminus ( m). Let U^sh be the punctured spectrum of the strict henselization A^sh of A. Assume U is quasi-compact and U^sh is connected. Then the sequence π_1(U^sh, overlineu) → π_1(U, overlineu) → π_1(X, overlineu) → 1 is exact in the sense of Lemma [Tag 0BS9] part (1).","statement_latex":"Let $(A, \\mathfrak m)$ be a local ring. Set $X = \\Spec(A)$ and\n$U = X \\setminus \\{\\mathfrak m\\}$. Let $U^{sh}$ be the punctured spectrum\nof the strict henselization $A^{sh}$ of $A$.\nAssume $U$ is quasi-compact and $U^{sh}$ is connected. Then the sequence\n$$\n\\pi_1(U^{sh}, \\overline{u}) \\to \\pi_1(U, \\overline{u}) \\to\n\\pi_1(X, \\overline{u}) \\to 1\n$$\nis exact in the sense of Lemma \\ref{lemma-functoriality-galois-ses} part (1).","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Local connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSB","source_file":"pione.tex","source_line":2148,"source_end_line":2159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2148-L2159","statement_sha256":"f62f55a60e0a619ff2444b234b543426d6d0c4357902d36c5f3b48078525df36","origin":"The Stacks Project","memory_eligible":false,"source_rank":9726,"rank":9726,"depth":55,"x":1567.047,"y":1186.254,"cluster":"tale-geometry"},{"id":"stacks:0BQI","tag":"0BQI","title":"Local connectedness · Lemma 0BQI","summary":"Let X be an irreducible, geometrically unibranch scheme. For any nonempty open U ⊂ X the canonical map π_1(U, overlineu) → π_1(X, overlineu) is surjective. The map ([Tag 0BQH]) π_1(eta, overlineeta) → π_1(X, overlineeta) is surjective as well.","statement_latex":"Let $X$ be an irreducible, geometrically unibranch scheme.\nFor any nonempty open $U \\subset X$ the canonical map\n$$\n\\pi_1(U, \\overline{u}) \\longrightarrow \\pi_1(X, \\overline{u})\n$$\nis surjective. The map (\\ref{equation-inclusion-generic-point})\n$\\pi_1(\\eta, \\overline{\\eta}) \\to \\pi_1(X, \\overline{\\eta})$\nis surjective as well.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Local connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQI","source_file":"pione.tex","source_line":2247,"source_end_line":2257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2247-L2257","statement_sha256":"3a6c25ecbafa6301f8018cfec2476ab5a9cb090065e2a3539f4884eaaa6c8118","origin":"The Stacks Project","memory_eligible":false,"source_rank":9727,"rank":9727,"depth":55,"x":1552.212,"y":1098.327,"cluster":"tale-geometry"},{"id":"stacks:0BSC","tag":"0BSC","title":"Local connectedness · Lemma 0BSC","summary":"Let X be a scheme. Let x_1, …, x_n ∈ X be a finite number of closed points such that • U = X setminus (x_1, …, x_n) is connected and is a retrocompact open of X, and • for each i the punctured spectrum U_i^sh of the strict henselization of O_X, x_i is connected. Then the map π_1(U) → π_1(X) is surjective and the kernel is the smallest closed normal subgroup of π_1(U) containing the image of π_1(U_i^sh) → π_1(U) for i = 1, …, n.","statement_latex":"Let $X$ be a scheme. Let $x_1, \\ldots, x_n \\in X$ be a finite\nnumber of closed points such that\n\\begin{enumerate}\n\\item $U = X \\setminus \\{x_1, \\ldots, x_n\\}$ is connected and is\na retrocompact open of $X$, and\n\\item for each $i$ the punctured spectrum $U_i^{sh}$ of the\nstrict henselization of $\\mathcal{O}_{X, x_i}$ is connected.\n\\end{enumerate}\nThen the map $\\pi_1(U) \\to \\pi_1(X)$ is surjective and the kernel\nis the smallest closed normal subgroup of $\\pi_1(U)$ containing\nthe image of $\\pi_1(U_i^{sh}) \\to \\pi_1(U)$ for $i = 1, \\ldots, n$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Local connectedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSC","source_file":"pione.tex","source_line":2288,"source_end_line":2301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2288-L2301","statement_sha256":"05c25b840d22be50c20bdc6ac97eee6628dfa1ee48e8c1f0fc56e0b936614b60","origin":"The Stacks Project","memory_eligible":false,"source_rank":9728,"rank":9728,"depth":56,"x":1634.281,"y":1155.038,"cluster":"tale-geometry"},{"id":"stacks:0BQK","tag":"0BQK","title":"Fundamental groups of normal schemes · Lemma 0BQK","summary":"In the situation above the following are equivalent • X is unramified in L, • Y → X is étale, and • Y → X is finite étale.","statement_latex":"In the situation above the following are equivalent\n\\begin{enumerate}\n\\item $X$ is unramified in $L$,\n\\item $Y \\to X$ is \\'etale, and\n\\item $Y \\to X$ is finite \\'etale.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Fundamental groups of normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQK","source_file":"pione.tex","source_line":2371,"source_end_line":2379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2371-L2379","statement_sha256":"bd5c365b07254d1488e9cc5814703422f28f14ba233c70827475d0c70eb5e841","origin":"The Stacks Project","memory_eligible":false,"source_rank":9729,"rank":9729,"depth":57,"x":1527.616,"y":1159.812,"cluster":"tale-geometry"},{"id":"stacks:0BQL","tag":"0BQL","title":"Fundamental groups of normal schemes · Lemma 0BQL","summary":"Let X be a normal integral scheme with function field K. Let Y → X be a finite étale morphism. If Y is connected, then Y is an integral normal scheme and Y is the normalization of X in the function field of Y.","statement_latex":"Let $X$ be a normal integral scheme with function field $K$.\nLet $Y \\to X$ be a finite \\'etale morphism. If $Y$ is connected,\nthen $Y$ is an integral normal scheme and $Y$ is the normalization\nof $X$ in the function field of $Y$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Fundamental groups of normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQL","source_file":"pione.tex","source_line":2429,"source_end_line":2435,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2429-L2435","statement_sha256":"fdc3adcc4ae4dc38ff5d5f6266dc9fa06e7378dd3bc5b83700c77e521519822f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9730,"rank":9730,"depth":38,"x":1602.81,"y":1095.443,"cluster":"tale-geometry"},{"id":"stacks:0BQM","tag":"0BQM","title":"Fundamental groups of normal schemes · Proposition 0BQM","summary":"Let X be a normal integral scheme with function field K. Then the canonical map ([Tag 0BQH]) Gal(K^sep/K) = π_1(eta, overlineeta) → π_1(X, overlineeta) is identified with the quotient map Gal(K^sep/K) → Gal(M/K) where M ⊂ K^sep is the union of the finite subextensions L such that X is unramified in L.","statement_latex":"Let $X$ be a normal integral scheme with function field $K$.\nThen the canonical map (\\ref{equation-inclusion-generic-point})\n$$\n\\text{Gal}(K^{sep}/K) = \\pi_1(\\eta, \\overline{\\eta})\n\\longrightarrow \\pi_1(X, \\overline{\\eta})\n$$\nis identified with the quotient map\n$\\text{Gal}(K^{sep}/K) \\to \\text{Gal}(M/K)$ where $M \\subset K^{sep}$\nis the union of the finite subextensions $L$\nsuch that $X$ is unramified in $L$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Fundamental groups of normal schemes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BQM","source_file":"pione.tex","source_line":2459,"source_end_line":2471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2459-L2471","statement_sha256":"733fc6ef90f0a42cbbced8058772b7bbf5978fb95fbc4d3d9f07f61d3f310e9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9731,"rank":9731,"depth":58,"x":1599.106,"y":1186.027,"cluster":"tale-geometry"},{"id":"stacks:0BSM","tag":"0BSM","title":"Fundamental groups of normal schemes · Lemma 0BSM","summary":"Let (A, m) be a normal local ring. Set X = Spec(A). Let A^sh be the strict henselization of A. Let K and K^sh be the fraction fields of A and A^sh. Then the sequence π_1(Spec(K^sh)) → π_1(Spec(K)) → π_1(X) → 1 is exact in the sense of Lemma [Tag 0BS9] part (1).","statement_latex":"Let $(A, \\mathfrak m)$ be a normal local ring.\nSet $X = \\Spec(A)$. Let $A^{sh}$ be the strict henselization of $A$.\nLet $K$ and $K^{sh}$ be the fraction fields of $A$ and $A^{sh}$.\nThen the sequence\n$$\n\\pi_1(\\Spec(K^{sh})) \\to \\pi_1(\\Spec(K)) \\to \\pi_1(X) \\to 1\n$$\nis exact in the sense of Lemma \\ref{lemma-functoriality-galois-ses} part (1).","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Fundamental groups of normal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSM","source_file":"pione.tex","source_line":2521,"source_end_line":2531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2521-L2531","statement_sha256":"2254c3585f6f8d4cd3bbdc6519dd3211e41d0d4b14b027a839177e64cc74d663","origin":"The Stacks Project","memory_eligible":false,"source_rank":9732,"rank":9732,"depth":59,"x":1528.645,"y":1116.788,"cluster":"tale-geometry"},{"id":"stacks:0BSP","tag":"0BSP","title":"Group actions and integral closure · Lemma 0BSP","summary":"Let A be a normal domain whose fraction field K is separably algebraically closed. Let p ⊂ A be a nonzero prime ideal. Then the residue field kappa( p) is algebraically closed.","statement_latex":"Let $A$ be a normal domain whose fraction field $K$ is separably algebraically\nclosed. Let $\\mathfrak p \\subset A$ be a nonzero prime ideal.\nThen the residue field $\\kappa(\\mathfrak p)$ is algebraically closed.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSP","source_file":"pione.tex","source_line":2587,"source_end_line":2592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2587-L2592","statement_sha256":"d0393592fb62a636d3f115a4a4507396cb050a94ef6f509bb1c55a6f279d6dae","origin":"The Stacks Project","memory_eligible":false,"source_rank":9733,"rank":9733,"depth":7,"x":1636.811,"y":1127.917,"cluster":"tale-geometry"},{"id":"stacks:0BSQ","tag":"0BSQ","title":"Group actions and integral closure · Lemma 0BSQ","summary":"A normal local ring with separably closed fraction field is strictly henselian.","statement_latex":"A normal local ring with separably closed fraction field is\nstrictly henselian.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSQ","source_file":"pione.tex","source_line":2611,"source_end_line":2615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2611-L2615","statement_sha256":"f7d26c8f4d7ceaef82a741e6ebab21201aac4af05fdea48bc61047195e0f1873","origin":"The Stacks Project","memory_eligible":false,"source_rank":9734,"rank":9734,"depth":8,"x":1547.669,"y":1181.345,"cluster":"tale-geometry"},{"id":"stacks:0BSS","tag":"0BSS","title":"Group actions and integral closure · Lemma 0BSS","summary":"Let G be a finite group acting on a ring R. Let R^G → A be a ring map. Let q' ⊂ A ⊗_R^G R be a prime lying over the prime q ⊂ R. Then I_ q = (σ ∈ G mid σ( q) = q and σ bmod q = id_kappa( q)) is equal to I_ q' = (σ ∈ G mid σ( q') = q' and σ bmod q' = id_kappa( q'))","statement_latex":"Let $G$ be a finite group acting on a ring $R$. Let $R^G \\to A$ be a ring\nmap. Let $\\mathfrak q' \\subset A \\otimes_{R^G} R$ be a prime lying\nover the prime $\\mathfrak q \\subset R$. Then\n$$\nI_\\mathfrak q = \\{\\sigma \\in G \\mid\n\\sigma(\\mathfrak q) = \\mathfrak q\\text{ and }\n\\sigma \\bmod \\mathfrak q = \\text{id}_{\\kappa(\\mathfrak q)}\\}\n$$\nis equal to\n$$\nI_{\\mathfrak q'} = \\{\\sigma \\in G \\mid\n\\sigma(\\mathfrak q') = \\mathfrak q'\\text{ and }\n\\sigma \\bmod \\mathfrak q' = \\text{id}_{\\kappa(\\mathfrak q')}\\}\n$$","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSS","source_file":"pione.tex","source_line":2636,"source_end_line":2652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2636-L2652","statement_sha256":"fbf97f7300d1cdeef81bc5094dd40e45fa219e6263e5ff944058b8d88d95180e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9735,"rank":9735,"depth":0,"x":1570.547,"y":1090.933,"cluster":"tale-geometry"},{"id":"stacks:0BST","tag":"0BST","title":"Group actions and integral closure · Lemma 0BST","summary":"Let G be a finite group acting on a ring R. Let q ⊂ R be a prime. Set I = (σ ∈ G mid σ( q) = q and σ bmod q = id_ q) Then R^G → R^I is étale at R^I ∩ q.","statement_latex":"Let $G$ be a finite group acting on a ring $R$. Let $\\mathfrak q \\subset R$\nbe a prime. Set\n$$\nI = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak q) = \\mathfrak q\n\\text{ and } \\sigma \\bmod \\mathfrak q = \\text{id}_\\mathfrak q\\}\n$$\nThen $R^G \\to R^I$ is \\'etale at $R^I \\cap \\mathfrak q$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BST","source_file":"pione.tex","source_line":2664,"source_end_line":2673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2664-L2673","statement_sha256":"75a50d08d6cd920fb17956b032032d09056f4f18e2360ebcc13a7670e489df08","origin":"The Stacks Project","memory_eligible":false,"source_rank":9736,"rank":9736,"depth":46,"x":1626.649,"y":1170.964,"cluster":"tale-geometry"},{"id":"stacks:0BSU","tag":"0BSU","title":"Group actions and integral closure · Lemma 0BSU","summary":"Let A be a normal domain with fraction field K. Let L/K be a (possibly infinite) Galois extension. Let G = Gal(L/K) and let B be the integral closure of A in L. Let q ⊂ B. Set I = (σ ∈ G mid σ( q) = q and σ bmod q = id_kappa( q)) Then (B^I)_B^I ∩ q is a filtered colimit of étale A-algebras.","statement_latex":"Let $A$ be a normal domain with fraction field $K$.\nLet $L/K$ be a (possibly infinite) Galois extension.\nLet $G = \\text{Gal}(L/K)$ and let\n$B$ be the integral closure of $A$ in $L$.\nLet $\\mathfrak q \\subset B$. Set\n$$\nI = \\{\\sigma \\in G \\mid\n\\sigma(\\mathfrak q) = \\mathfrak q \\text{ and }\n\\sigma \\bmod \\mathfrak q = \\text{id}_{\\kappa(\\mathfrak q)}\\}\n$$\nThen $(B^I)_{B^I \\cap \\mathfrak q}$ is a filtered colimit\nof \\'etale $A$-algebras.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Group actions and integral closure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSU","source_file":"pione.tex","source_line":2784,"source_end_line":2798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2784-L2798","statement_sha256":"77ecd3c6f39504ff061b70c0390f098fc44a7492dfc2aa56bf4e5225cbcb276a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9737,"rank":9737,"depth":47,"x":1520.422,"y":1143.655,"cluster":"tale-geometry"},{"id":"stacks:0BSW","tag":"0BSW","title":"Ramification theory · Lemma 0BSW","summary":"In the situation described above, via the isomorphism π_1(U) = Gal(K^sep/K) the diagram ([Tag 0BSV]) translates into the diagram xymatrix I ar[r] ar[rd]_1 & D ar[d] ar[r] & Gal(K^sep/K) ar[d] & Gal(kappa( m^sh)/kappa) ar[r] & Gal(M/K) where K^sep/M/K is the maximal subextension unramified with respect to A. Moreover, the vertical arrows are surjective, the kernel of the left vertical arrow is I and the kernel of the right vertical arrow is the smallest closed normal…","statement_latex":"In the situation described above, via the isomorphism\n$\\pi_1(U) = \\text{Gal}(K^{sep}/K)$ the diagram\n(\\ref{equation-inertia-diagram-pione})\ntranslates into the diagram\n$$\n\\xymatrix{\nI \\ar[r] \\ar[rd]_1 & D \\ar[d] \\ar[r] & \\text{Gal}(K^{sep}/K) \\ar[d] \\\\\n& \\text{Gal}(\\kappa(\\mathfrak m^{sh})/\\kappa) \\ar[r] & \\text{Gal}(M/K)\n}\n$$\nwhere $K^{sep}/M/K$ is the maximal subextension unramified\nwith respect to $A$. Moreover, the vertical arrows are surjective,\nthe kernel of the left vertical arrow is $I$ and the kernel of the\nright vertical arrow is\nthe smallest closed normal subgroup of $\\text{Gal}(K^{sep}/K)$\ncontaining $I$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Ramification theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BSW","source_file":"pione.tex","source_line":2882,"source_end_line":2900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2882-L2900","statement_sha256":"a3997bf0e2cc92809482e3a48e6bd703ff3bf6f50502b3d59b87dc89a2a356fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":9738,"rank":9738,"depth":60,"x":1621.184,"y":1103.329,"cluster":"tale-geometry"},{"id":"stacks:0BTD","tag":"0BTD","title":"Ramification theory · Lemma 0BTD","summary":"Let X be a normal integral scheme with function field K. With notation as above, the following three subgroups of Gal(K^sep/K) = π_1(Spec(K)) are equal • the kernel of the surjection Gal(K^sep/K) → π_1(X), • the smallest normal closed subgroup containing I_y for all y ∈ X^sep, and • the smallest normal closed subgroup containing Gal(K^sep/K_x^sh) for all x ∈ X.","statement_latex":"Let $X$ be a normal integral scheme with function field $K$.\nWith notation as above, the following three subgroups of\n$\\text{Gal}(K^{sep}/K) = \\pi_1(\\Spec(K))$\nare equal\n\\begin{enumerate}\n\\item the kernel of the surjection\n$\\text{Gal}(K^{sep}/K) \\longrightarrow \\pi_1(X)$,\n\\item the smallest normal closed subgroup containing $I_y$\nfor all $y \\in X^{sep}$, and\n\\item the smallest normal closed subgroup containing\n$\\text{Gal}(K^{sep}/K_x^{sh})$ for all $x \\in  X$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Ramification theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTD","source_file":"pione.tex","source_line":2983,"source_end_line":2997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L2983-L2997","statement_sha256":"8589589de18e7196f843f901b5c58af953a718aafba702fa326bd8106c6d9169","origin":"The Stacks Project","memory_eligible":false,"source_rank":9739,"rank":9739,"depth":61,"x":1579.118,"y":1190.642,"cluster":"tale-geometry"},{"id":"stacks:0BTF","tag":"0BTF","title":"Ramification theory · Lemma 0BTF","summary":"Let X be an integral normal scheme with function field K. Let L/K be a finite extension. Let Y → X be the normalization of X in L. The following are equivalent • X is unramified in L as defined in Section [Tag 0BQJ], • Y → X is an unramified morphism of schemes, • Y → X is an étale morphism of schemes, • Y → X is a finite étale morphism of schemes, • for x ∈ X the projection Y ×_X Spec(O_X, x) → Spec(O_X, x) is unramified, • same as in (5) but with O_X, x^h, • same as in…","statement_latex":"Let $X$ be an integral normal scheme with function field $K$.\nLet $L/K$ be a finite extension. Let $Y \\to X$ be the normalization\nof $X$ in $L$. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is unramified in $L$ as defined in Section \\ref{section-normal},\n\\item $Y \\to X$ is an unramified morphism of schemes,\n\\item $Y \\to X$ is an \\'etale morphism of schemes,\n\\item $Y \\to X$ is a finite \\'etale morphism of schemes,\n\\item for $x \\in X$ the projection\n$Y \\times_X \\Spec(\\mathcal{O}_{X, x}) \\to \\Spec(\\mathcal{O}_{X, x})$\nis unramified,\n\\item same as in (5) but with $\\mathcal{O}_{X, x}^h$,\n\\item same as in (5) but with $\\mathcal{O}_{X, x}^{sh}$,\n\\item for $x \\in X$ the scheme theoretic fibre $Y_x$\nis \\'etale over $x$ of degree $\\geq [L : K]$.\n\\end{enumerate}\nIf $L/K$ is Galois with Galois group $G$, then these are also\nequivalent to\n\\begin{enumerate}\n\\item[(9)] for $y \\in Y$ the group\n$I_y = \\{g \\in G \\mid g(y) = y\\text{ and }\ng \\bmod \\mathfrak m_y = \\text{id}_{\\kappa(y)}\\}$ is trivial.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Ramification theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTF","source_file":"pione.tex","source_line":3057,"source_end_line":3082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3057-L3082","statement_sha256":"95427e81dd450d5a192977f79fc117a22790209e025da55e249779bf1da5163c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9740,"rank":9740,"depth":58,"x":1539.742,"y":1101.983,"cluster":"tale-geometry"},{"id":"stacks:0BUA","tag":"0BUA","title":"Ramification theory · Lemma 0BUA","summary":"Let A be a discrete valuation ring with fraction field K. Let L/K be a (possibly infinite) Galois extension. Let B be the integral closure of A in L. Let m be a maximal ideal of B. Let G = Gal(L/K), D = (σ ∈ G mid σ( m) = m), and I = (σ ∈ D mid σ bmod m = id_kappa( m)). The decomposition group D fits into a canonical exact sequence 1 → I → D → Aut(kappa( m)/kappa_A) → 1 The inertia group I fits into a canonical exact sequence 1 → P → I → I_t → 1 such that • P is a normal…","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $L/K$ be a (possibly infinite) Galois extension.\nLet $B$ be the integral closure of $A$ in $L$.\nLet $\\mathfrak m$ be a maximal ideal of $B$.\nLet $G = \\text{Gal}(L/K)$,\n$D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m) = \\mathfrak m\\}$, and\n$I = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m =\n\\text{id}_{\\kappa(\\mathfrak m)}\\}$.\nThe decomposition group $D$ fits into a canonical exact sequence\n$$\n1 \\to I \\to D \\to \\text{Aut}(\\kappa(\\mathfrak m)/\\kappa_A) \\to 1\n$$\nThe inertia group $I$ fits into a canonical exact sequence\n$$\n1 \\to P \\to I \\to I_t \\to 1\n$$\nsuch that\n\\begin{enumerate}\n\\item $P$ is a normal subgroup of $D$,\n\\item $P$ is a pro-$p$-group if the characteristic of\n$\\kappa_A$ is $p > 1$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$\nis zero,\n\\item there is a multiplicatively directed $S \\subset \\mathbf{N}$\nsuch that $\\kappa(\\mathfrak m)$ contains a primitive $n$th root of unity\nfor each $n \\in S$ (elements of $S$ are prime to $p$),\n\\item there exists a canonical surjective map\n$$\n\\theta_{can} : I \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m))\n$$\nwhose kernel is $P$, which satisfies\n$\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$\nfor $\\tau \\in D$, $\\sigma \\in I$, and which induces an isomorphism\n$I_t \\to \\lim_{n \\in S} \\mu_n(\\kappa(\\mathfrak m))$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Ramification theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUA","source_file":"pione.tex","source_line":3174,"source_end_line":3210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3174-L3210","statement_sha256":"279a97bf6bba62c212242be4003e89c6bb4f716a6f2fab169b41db1fff73b8a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9741,"rank":9741,"depth":43,"x":1640.531,"y":1145.213,"cluster":"tale-geometry"},{"id":"stacks:0BUB","tag":"0BUB","title":"Ramification theory · Lemma 0BUB","summary":"Let A be a discrete valuation ring with fraction field K. Let K^sep be a separable closure of K. Let A^sep be the integral closure of A in K^sep. Let m^sep be a maximal ideal of A^sep. Let m = m^sep ∩ A, let kappa = A/ m, and let overlinekappa = A^sep/ m^sep. Then overlinekappa is an algebraic closure of kappa. Let G = Gal(K^sep/K), D = (σ ∈ G mid σ( m^sep) = m^sep), and I = (σ ∈ D mid σ bmod m^sep = id_kappa( m^sep)). The decomposition group D fits into a canonical exact…","statement_latex":"Let $A$ be a discrete valuation ring with fraction field $K$.\nLet $K^{sep}$ be a separable closure of $K$.\nLet $A^{sep}$ be the integral closure of $A$ in $K^{sep}$.\nLet $\\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$.\nLet $\\mathfrak m = \\mathfrak m^{sep} \\cap A$, let\n$\\kappa = A/\\mathfrak m$, and let\n$\\overline{\\kappa} = A^{sep}/\\mathfrak m^{sep}$.\nThen $\\overline{\\kappa}$ is an algebraic closure of $\\kappa$.\nLet $G = \\text{Gal}(K^{sep}/K)$,\n$D = \\{\\sigma \\in G \\mid \\sigma(\\mathfrak m^{sep}) = \\mathfrak m^{sep}\\}$, and\n$I = \\{\\sigma \\in D \\mid \\sigma \\bmod \\mathfrak m^{sep} =\n\\text{id}_{\\kappa(\\mathfrak m^{sep})}\\}$.\nThe decomposition group $D$ fits into a canonical exact sequence\n$$\n1 \\to I \\to D \\to \\text{Gal}(\\kappa^{sep}/\\kappa) \\to 1\n$$\nwhere $\\kappa^{sep} \\subset \\overline{\\kappa}$ is the separable\nclosure of $\\kappa$.\nThe inertia group $I$ fits into a canonical exact sequence\n$$\n1 \\to P \\to I \\to I_t \\to 1\n$$\nsuch that\n\\begin{enumerate}\n\\item $P$ is a normal subgroup of $D$,\n\\item $P$ is a pro-$p$-group if the characteristic of\n$\\kappa_A$ is $p > 1$ and $P = \\{1\\}$ if the characteristic of $\\kappa_A$\nis zero,\n\\item there exists a canonical surjective map\n$$\n\\theta_{can} : I \\to \\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep})\n$$\nwhose kernel is $P$, which satisfies\n$\\theta_{can}(\\tau \\sigma \\tau^{-1}) = \\tau(\\theta_{can}(\\sigma))$\nfor $\\tau \\in D$, $\\sigma \\in I$, and which induces an isomorphism\n$I_t \\to \\lim_{n\\text{ prime to }p} \\mu_n(\\kappa^{sep})$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Ramification theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUB","source_file":"pione.tex","source_line":3344,"source_end_line":3383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3344-L3383","statement_sha256":"4a6784f63577b880202e942d90029d41279b46b09046935b49c028682724f6be","origin":"The Stacks Project","memory_eligible":false,"source_rank":9742,"rank":9742,"depth":44,"x":1530.954,"y":1170.639,"cluster":"tale-geometry"},{"id":"stacks:0BTV","tag":"0BTV","title":"Geometric and arithmetic fundamental groups · Lemma 0BTV","summary":"Let I be a directed set. Let X_i be an inverse system of quasi-compact and quasi-separated schemes over I with affine transition morphisms. Let X = lim X_i as in Limits, Section [Tag 01YV]. Then there is an equivalence of categories colim FÉt_X_i = FÉt_X If X_i is connected for all sufficiently large i and overlinex is a geometric point of X, then π_1(X, overlinex) = lim π_1(X_i, overlinex)","statement_latex":"Let $I$ be a directed set. Let $X_i$ be an\ninverse system of quasi-compact and quasi-separated schemes\nover $I$ with affine transition morphisms.\nLet $X = \\lim X_i$ as in Limits, Section \\ref{limits-section-limits}.\nThen there is an equivalence of categories\n$$\n\\colim \\textit{F\\'Et}_{X_i} = \\textit{F\\'Et}_X\n$$\nIf $X_i$ is connected for all sufficiently large $i$ and $\\overline{x}$\nis a geometric point of $X$, then\n$$\n\\pi_1(X, \\overline{x}) = \\lim \\pi_1(X_i, \\overline{x})\n$$","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Geometric and arithmetic fundamental groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTV","source_file":"pione.tex","source_line":3427,"source_end_line":3442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3427-L3442","statement_sha256":"5322224bd07be44cdf7c21abb85c86ea41b4131129d47918fb789b0874720209","origin":"The Stacks Project","memory_eligible":false,"source_rank":9743,"rank":9743,"depth":40,"x":1591.578,"y":1089.352,"cluster":"tale-geometry"},{"id":"stacks:0BTW","tag":"0BTW","title":"Geometric and arithmetic fundamental groups · Lemma 0BTW","summary":"Let k be a field with perfection k^perf. Let X be a connected scheme over k. Then X_k^perf is connected and π_1(X_k^perf) → π_1(X) is an isomorphism.","statement_latex":"Let $k$ be a field with perfection $k^{perf}$. Let $X$ be a connected scheme\nover $k$. Then $X_{k^{perf}}$ is connected and\n$\\pi_1(X_{k^{perf}}) \\to \\pi_1(X)$ is an isomorphism.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Geometric and arithmetic fundamental groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTW","source_file":"pione.tex","source_line":3453,"source_end_line":3458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3453-L3458","statement_sha256":"58ccc746c9a80ee30b0af871b23dda9597398c8f2ad5ee2ea7f9ae17f7846183","origin":"The Stacks Project","memory_eligible":false,"source_rank":9744,"rank":9744,"depth":53,"x":1612.334,"y":1184.11,"cluster":"tale-geometry"},{"id":"stacks:0BTX","tag":"0BTX","title":"Geometric and arithmetic fundamental groups · Lemma 0BTX","summary":"Let k be a field with algebraic closure overlinek. Let X be a quasi-compact and quasi-separated scheme over k. If the base change X_overlinek is connected, then there is a short exact sequence 1 → π_1(X_overlinek) → π_1(X) → π_1(Spec(k)) → 1 of profinite topological groups.","statement_latex":"Let $k$ be a field with algebraic closure $\\overline{k}$.\nLet $X$ be a quasi-compact and quasi-separated scheme over $k$.\nIf the base change $X_{\\overline{k}}$ is connected, then\nthere is a short exact sequence\n$$\n1 \\to \\pi_1(X_{\\overline{k}}) \\to \\pi_1(X) \\to \\pi_1(\\Spec(k)) \\to 1\n$$\nof profinite topological groups.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Geometric and arithmetic fundamental groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTX","source_file":"pione.tex","source_line":3468,"source_end_line":3478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3468-L3478","statement_sha256":"14de3cb35d009a0fe06fbbda15248c872829366ced28779bbd1aa55185766a73","origin":"The Stacks Project","memory_eligible":false,"source_rank":9745,"rank":9745,"depth":54,"x":1520.426,"y":1125.759,"cluster":"tale-geometry"},{"id":"stacks:0BUN","tag":"0BUN","title":"Homotopy exact sequence · Lemma 0BUN","summary":"[SGA1]. Let f : X → S be a proper morphism of schemes. Let X → S' → S be the Stein factorization of f, see More on Morphisms, Theorem [Tag 03H2]. If f is of finite presentation, flat, with geometrically reduced fibres, then S' → S is finite étale.","statement_latex":"\\begin{reference}\n\\cite[Expose X, Proposition 1.2, p. 262]{SGA1}.\n\\end{reference}\nLet $f : X \\to S$ be a proper morphism of schemes.\nLet $X \\to S' \\to S$ be the Stein factorization of $f$, see\nMore on Morphisms, Theorem\n\\ref{more-morphisms-theorem-stein-factorization-general}.\nIf $f$ is of finite presentation, flat, with geometrically\nreduced fibres, then $S' \\to S$ is finite \\'etale.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Homotopy exact sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BUN","source_file":"pione.tex","source_line":3580,"source_end_line":3591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3580-L3591","statement_sha256":"41f1cc9003d5a101b89581ef3ad4333966fe6a3f328338f84fad49bed579f7c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9746,"rank":9746,"depth":48,"x":1635.621,"y":1116.596,"cluster":"tale-geometry"},{"id":"stacks:0C0J","tag":"0C0J","title":"Homotopy exact sequence · Proposition 0C0J","summary":"Let f : X → S be a flat proper morphism of finite presentation whose geometric fibres are connected and reduced. Assume S is connected and let overlines be a geometric point of S. Then there is an exact sequence π_1(X_overlines) → π_1(X) → π_1(S) → 1 of fundamental groups.","statement_latex":"Let $f : X \\to S$ be a flat proper morphism of finite presentation whose\ngeometric fibres are connected and reduced. Assume $S$ is connected and\nlet $\\overline{s}$ be a geometric point of $S$. Then there is an exact\nsequence\n$$\n\\pi_1(X_{\\overline{s}}) \\to \\pi_1(X) \\to \\pi_1(S) \\to 1\n$$\nof fundamental groups.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Homotopy exact sequence","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0J","source_file":"pione.tex","source_line":3601,"source_end_line":3611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3601-L3611","statement_sha256":"09fa41aed0cc61abedfbc56574e74562b87fa3be93fac82cc8d3dda855218ee8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9747,"rank":9747,"depth":49,"x":1557.711,"y":1189.028,"cluster":"tale-geometry"},{"id":"stacks:0C0K","tag":"0C0K","title":"Specialization maps · Lemma 0C0K","summary":"Consider a commutative diagram xymatrix Y ar[d]_g ar[r] & X ar[d]^f T ar[r] & S of schemes where f and g are proper with geometrically connected fibres. Let t' leadsto t be a specialization of points in T and consider a specialization map sp : π_1(Y_overlinet') → π_1(Y_overlinet) as above. Then there is a commutative diagram xymatrix π_1(Y_overlinet') ar[r]_sp ar[d] & π_1(Y_overlinet) ar[d] π_1(X_overlines') ar[r]^sp & π_1(X_overlines) of specialization maps where…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nY \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\\nT \\ar[r] & S\n}\n$$\nof schemes where $f$ and $g$ are proper with geometrically connected\nfibres. Let $t' \\leadsto t$ be a specialization of points in $T$\nand consider a specialization map\n$sp : \\pi_1(Y_{\\overline{t}'}) \\to \\pi_1(Y_{\\overline{t}})$ as above.\nThen there is a commutative diagram\n$$\n\\xymatrix{\n\\pi_1(Y_{\\overline{t}'}) \\ar[r]_{sp} \\ar[d] & \\pi_1(Y_{\\overline{t}}) \\ar[d] \\\\\n\\pi_1(X_{\\overline{s}'}) \\ar[r]^{sp} & \\pi_1(X_{\\overline{s}})\n}\n$$\nof specialization maps where $\\overline{s}$ and $\\overline{s}'$\nare the images of $\\overline{t}$ and $\\overline{t}'$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Specialization maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0K","source_file":"pione.tex","source_line":3744,"source_end_line":3766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3744-L3766","statement_sha256":"3cb11f2c7edda1c7a3d3cc3178d894ceedc3b64b77720088defc73d38912ab6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9748,"rank":9748,"depth":47,"x":1556.91,"y":1090.992,"cluster":"tale-geometry"},{"id":"stacks:0C0L","tag":"0C0L","title":"Specialization maps · Lemma 0C0L","summary":"Let f : X → S be a proper morphism with geometrically connected fibres. Let s\" leadsto s' leadsto s be specializations of points of S. A composition of specialization maps π_1(X_overlines\") → π_1(X_overlines') → π_1(X_overlines) is a specialization map π_1(X_overlines\") → π_1(X_overlines).","statement_latex":"Let $f : X \\to S$ be a proper morphism with geometrically connected fibres.\nLet $s'' \\leadsto s' \\leadsto s$ be specializations of points of $S$.\nA composition of specialization maps\n$\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}'}) \\to\n\\pi_1(X_{\\overline{s}})$ is a specialization map\n$\\pi_1(X_{\\overline{s}''}) \\to \\pi_1(X_{\\overline{s}})$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Specialization maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0L","source_file":"pione.tex","source_line":3800,"source_end_line":3808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3800-L3808","statement_sha256":"fb7baeb39daf59146bf6fb5d703d9ac13609656ed76e296274e48a8bcb25289d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9749,"rank":9749,"depth":46,"x":1636.675,"y":1163.133,"cluster":"tale-geometry"},{"id":"stacks:0C0M","tag":"0C0M","title":"Specialization maps · Lemma 0C0M","summary":"Let f : X → S be a proper morphism with geometrically connected fibres. Let s' leadsto s be a specialization of points of S and let sp : π_1(X_overlines') → π_1(X_overlines) be a specialization map. Then there exists a strictly henselian valuation ring R over S with algebraically closed fraction field such that sp is isomorphic to sp_R defined above.","statement_latex":"Let $f : X \\to S$ be a proper morphism with geometrically connected fibres.\nLet $s' \\leadsto s$ be a specialization of points of $S$ and let\n$sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$\nbe a specialization map. Then there exists a strictly henselian\nvaluation ring $R$ over $S$ with algebraically closed fraction field\nsuch that $sp$ is isomorphic to $sp_R$ defined above.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Specialization maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0M","source_file":"pione.tex","source_line":3841,"source_end_line":3849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3841-L3849","statement_sha256":"f8159d709227375529c08131b31eaa843374a785f9e4ab7f59da467ebdfdebb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9750,"rank":9750,"depth":9,"x":1519.352,"y":1155.164,"cluster":"tale-geometry"},{"id":"stacks:0C0N","tag":"0C0N","title":"Specialization maps · Lemma 0C0N","summary":"Let f : X → S be a proper morphism with geometrically connected fibres. Let s' leadsto s be a specialization of points of S and let sp : π_1(X_overlines') → π_1(X_overlines) be a specialization map. If S is Noetherian, then there exists a strictly henselian discrete valuation ring R over S such that sp is isomorphic to sp_R defined above.","statement_latex":"Let $f : X \\to S$ be a proper morphism with geometrically connected fibres.\nLet $s' \\leadsto s$ be a specialization of points of $S$ and let\n$sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$\nbe a specialization map. If $S$ is Noetherian, then\nthere exists a strictly henselian\ndiscrete valuation ring $R$ over $S$ such that $sp$ is isomorphic to $sp_R$\ndefined above.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Specialization maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0N","source_file":"pione.tex","source_line":3875,"source_end_line":3884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3875-L3884","statement_sha256":"f0e9f2047751f8d7c7a9da3ce2e8c7a5513e57caf3a751494ec9992cd9d79bb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9751,"rank":9751,"depth":55,"x":1612.661,"y":1094.216,"cluster":"tale-geometry"},{"id":"stacks:0EL8","tag":"0EL8","title":"Restriction to a closed subscheme · Lemma 0EL8","summary":"Let X be a Noetherian scheme and let Y ⊂ X be a closed subscheme with ideal sheaf I ⊂ O_X. Assume the completion functor Coh(O_X) → Coh(X, I), F ↦ F^wedge is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor FÉt_X → FÉt_Y is fully faithful.","statement_latex":"Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme\nwith ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.\nAssume the completion functor\n$$\n\\textit{Coh}(\\mathcal{O}_X)\n\\longrightarrow \n\\textit{Coh}(X, \\mathcal{I}),\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\nis fully faithful on the full subcategory of finite locally free objects\n(see above).\nThen the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Restriction to a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EL8","source_file":"pione.tex","source_line":3957,"source_end_line":3972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L3957-L3972","statement_sha256":"a5977358832d239fb1bf106d283dfb51d910004578d69eee9c8bb58d911d9d8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9752,"rank":9752,"depth":52,"x":1592.787,"y":1192.509,"cluster":"tale-geometry"},{"id":"stacks:0EL9","tag":"0EL9","title":"Restriction to a closed subscheme · Lemma 0EL9","summary":"Let X be a Noetherian scheme and let Y ⊂ X be a closed subscheme with ideal sheaf I ⊂ O_X. Assume the completion functor Coh(O_X) → Coh(X, I), F ↦ F^wedge is an equivalence on full subcategories of finite locally free objects (see above). Then the restriction functor FÉt_X → FÉt_Y is an equivalence.","statement_latex":"Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme\nwith ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.\nAssume the completion functor\n$$\n\\textit{Coh}(\\mathcal{O}_X)\n\\longrightarrow \n\\textit{Coh}(X, \\mathcal{I}),\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\nis an equivalence on full subcategories of finite locally free objects\n(see above).\nThen the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$\nis an equivalence.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Restriction to a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EL9","source_file":"pione.tex","source_line":4018,"source_end_line":4033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4018-L4033","statement_sha256":"0bfa7df35ee58b1f3911cda69d90133ebbc77e90667341103486576186e3b7e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":9753,"rank":9753,"depth":53,"x":1528.134,"y":1108.408,"cluster":"tale-geometry"},{"id":"stacks:0ELA","tag":"0ELA","title":"Restriction to a closed subscheme · Lemma 0ELA","summary":"Let X be a Noetherian scheme and let Y ⊂ X be a closed subscheme with ideal sheaf I ⊂ O_X. Let V be the set of open subschemes V ⊂ X containing Y ordered by reverse inclusion. Assume the completion functor colim_V Coh(O_V) → Coh(X, I), F ↦ F^wedge defines is fully faithful on the full subcategory of finite locally free objects (see above). Then the restriction functor colim_V FÉt_V → FÉt_Y is fully faithful.","statement_latex":"Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme\nwith ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.\nLet $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$\nordered by reverse inclusion. Assume the completion functor\n$$\n\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)\n\\longrightarrow\n\\textit{Coh}(X, \\mathcal{I}),\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\ndefines is fully faithful on the full subcategory of\nfinite locally free objects (see above).\nThen the restriction functor\n$\\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Restriction to a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ELA","source_file":"pione.tex","source_line":4103,"source_end_line":4121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4103-L4121","statement_sha256":"50d7552d68b7e2f839f4e3e2923296975816af3e877dc6faca4718c5b3e72d75","origin":"The Stacks Project","memory_eligible":false,"source_rank":9754,"rank":9754,"depth":53,"x":1643.909,"y":1133.841,"cluster":"tale-geometry"},{"id":"stacks:0EK1","tag":"0EK1","title":"Restriction to a closed subscheme · Lemma 0EK1","summary":"Let X be a Noetherian scheme and let Y ⊂ X be a closed subscheme with ideal sheaf I ⊂ O_X. Let V be the set of open subschemes V ⊂ X containing Y ordered by reverse inclusion. Assume the completion functor colim_V Coh(O_V) → Coh(X, I), F ↦ F^wedge defines an equivalence of the full subcategories of finite locally free objects (see explanation above). Then the restriction functor colim_V FÉt_V → FÉt_Y is an equivalence.","statement_latex":"Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme\nwith ideal sheaf $\\mathcal{I} \\subset \\mathcal{O}_X$.\nLet $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$\nordered by reverse inclusion. Assume the completion functor\n$$\n\\colim_\\mathcal{V} \\textit{Coh}(\\mathcal{O}_V)\n\\longrightarrow\n\\textit{Coh}(X, \\mathcal{I}),\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}^\\wedge\n$$\ndefines an equivalence of the full subcategories of\nfinite locally free objects (see explanation above).\nThen the restriction functor\n$$\n\\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y\n$$\nis an equivalence.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Restriction to a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EK1","source_file":"pione.tex","source_line":4178,"source_end_line":4198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4178-L4198","statement_sha256":"33b3d4897dcf239b07868e8cf1edc3ffde6faaae134109cf842fdbab472dd994","origin":"The Stacks Project","memory_eligible":false,"source_rank":9755,"rank":9755,"depth":54,"x":1537.658,"y":1180.969,"cluster":"tale-geometry"},{"id":"stacks:0EJX","tag":"0EJX","title":"Restriction to a closed subscheme · Lemma 0EJX","summary":"Let X be a scheme and let Y ⊂ X be a closed subscheme. If every connected component of X meets Y, then the restriction functor FÉt_X → FÉt_Y is faithful.","statement_latex":"Let $X$ be a scheme and let $Y \\subset X$ be a closed subscheme.\nIf every connected component of $X$ meets $Y$, then\nthe restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$\nis faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Restriction to a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJX","source_file":"pione.tex","source_line":4268,"source_end_line":4274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4268-L4274","statement_sha256":"238287d29f136333f9e2fe57c2c2dec70ebc7e3d3f052db48cc234d5366fb1c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9756,"rank":9756,"depth":19,"x":1578.269,"y":1085.548,"cluster":"tale-geometry"},{"id":"stacks:0EJZ","tag":"0EJZ","title":"Restriction to a closed subscheme · Lemma 0EJZ","summary":"Let X be a Noetherian scheme and let Y ⊂ X be a closed subscheme. Let Y_n ⊂ X be the nth infinitesimal neighbourhood of Y in X. Assume one of the following holds • X is quasi-affine and Γ(X, O_X) → lim Γ(Y_n, O_Y_n) is an isomorphism, or • X has an ample invertible module L and Γ(X, L^⊗ m) → lim Γ(Y_n, L^⊗ m|_Y_n) is an isomorphism for all m gg 0, or • for every finite locally free O_X-module E the map Γ(X, E) → lim Γ(Y_n, E|_Y_n) is an isomorphism. Then the restriction…","statement_latex":"Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme.\nLet $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$.\nAssume one of the following holds\n\\begin{enumerate}\n\\item $X$ is quasi-affine and\n$\\Gamma(X, \\mathcal{O}_X) \\to \\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})$\nis an isomorphism, or\n\\item $X$ has an ample invertible module $\\mathcal{L}$ and\n$\\Gamma(X, \\mathcal{L}^{\\otimes m}) \\to\n\\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})$\nis an isomorphism for all $m \\gg 0$, or\n\\item for every finite locally free $\\mathcal{O}_X$-module\n$\\mathcal{E}$ the map\n$\\Gamma(X, \\mathcal{E}) \\to \\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})$\nis an isomorphism.\n\\end{enumerate}\nThen the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Restriction to a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EJZ","source_file":"pione.tex","source_line":4290,"source_end_line":4310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4290-L4310","statement_sha256":"dcf26fff62eca156425a6d21743fc6c4875cc2ee248a5eb0cf8277c92cf88e63","origin":"The Stacks Project","memory_eligible":false,"source_rank":9757,"rank":9757,"depth":53,"x":1625.243,"y":1179.325,"cluster":"tale-geometry"},{"id":"stacks:0EK0","tag":"0EK0","title":"Restriction to a closed subscheme · Lemma 0EK0","summary":"Let X be a Noetherian scheme and let Y ⊂ X be a closed subscheme. Let Y_n ⊂ X be the nth infinitesimal neighbourhood of Y in X. Let V be the set of open subschemes V ⊂ X containing Y ordered by reverse inclusion. Assume one of the following holds • X is quasi-affine and colim_V Γ(V, O_V) → lim Γ(Y_n, O_Y_n) is an isomorphism, or • X has an ample invertible module L and colim_V Γ(V, L^⊗ m) → lim Γ(Y_n, L^⊗ m|_Y_n) is an isomorphism for all m gg 0, or • for every V ∈ V and…","statement_latex":"Let $X$ be a Noetherian scheme and let $Y \\subset X$ be a closed subscheme.\nLet $Y_n \\subset X$ be the $n$th infinitesimal neighbourhood of $Y$ in $X$.\nLet $\\mathcal{V}$ be the set of open subschemes $V \\subset X$ containing $Y$\nordered by reverse inclusion. Assume one of the following holds\n\\begin{enumerate}\n\\item $X$ is quasi-affine and\n$$\n\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{O}_V)\n\\longrightarrow\n\\lim \\Gamma(Y_n, \\mathcal{O}_{Y_n})\n$$\nis an isomorphism, or\n\\item $X$ has an ample invertible module $\\mathcal{L}$ and\n$$\n\\colim_\\mathcal{V} \\Gamma(V, \\mathcal{L}^{\\otimes m})\n\\longrightarrow\n\\lim \\Gamma(Y_n, \\mathcal{L}^{\\otimes m}|_{Y_n})\n$$\nis an isomorphism for all $m \\gg 0$, or\n\\item for every $V \\in \\mathcal{V}$ and every finite locally free\n$\\mathcal{O}_V$-module $\\mathcal{E}$ the map\n$$\n\\colim_{V' \\geq V} \\Gamma(V', \\mathcal{E}|_{V'})\n\\longrightarrow\n\\lim \\Gamma(Y_n, \\mathcal{E}|_{Y_n})\n$$\nis an isomorphism.\n\\end{enumerate}\nThen the functor\n$$\n\\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y\n$$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Restriction to a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EK0","source_file":"pione.tex","source_line":4319,"source_end_line":4354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4319-L4354","statement_sha256":"82a34fc019d3fb2801375f5ba818d847514a34da5c2bfca59a41b720f15530bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9758,"rank":9758,"depth":54,"x":1514.76,"y":1136.662,"cluster":"tale-geometry"},{"id":"stacks:0EK4","tag":"0EK4","title":"Pushouts and fundamental groups · Lemma 0EK4","summary":"In More on Morphisms, Situation [Tag 0ECI], for example if Z → Y and Z → X are closed immersions of schemes, there is an equivalence of categories FÉt_Y amalg_Z X → FÉt_Y ×_FÉt_Z FÉt_X","statement_latex":"In More on Morphisms, Situation\n\\ref{more-morphisms-situation-pushout-along-closed-immersion-and-integral},\nfor example if $Z \\to Y$ and $Z \\to X$ are closed immersions of schemes,\nthere is an equivalence of categories\n$$\n\\textit{F\\'Et}_{Y \\amalg_Z X}\n\\longrightarrow\n\\textit{F\\'Et}_Y\n\\times_{\\textit{F\\'Et}_Z}\n\\textit{F\\'Et}_X\n$$","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Pushouts and fundamental groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EK4","source_file":"pione.tex","source_line":4374,"source_end_line":4387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4374-L4387","statement_sha256":"16de82cf59875288222a654ddb5ab18023813bdb61d74d87d477fb04408b0bbf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9759,"rank":9759,"depth":52,"x":1630.99,"y":1105.308,"cluster":"tale-geometry"},{"id":"stacks:0BLG","tag":"0BLG","title":"Finite étale covers of punctured spectra, I · Lemma 0BLG","summary":"In Situation [Tag 0BLF]. Assume one of the following holds • dim(A/ p) ≥ 2 for every minimal prime p ⊂ A with f not ∈ p, or • every connected component of U meets U_0. Then FÉt_U → FÉt_U_0, V ↦ V_0 = V ×_U U_0 is a faithful functor.","statement_latex":"In Situation \\ref{situation-local-lefschetz}.\nAssume one of the following holds\n\\begin{enumerate}\n\\item $\\dim(A/\\mathfrak p) \\geq 2$ for every minimal prime\n$\\mathfrak p \\subset A$ with $f \\not \\in \\mathfrak p$, or\n\\item every connected component of $U$ meets $U_0$.\n\\end{enumerate}\nThen\n$$\n\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0},\\quad\nV \\longmapsto V_0 = V \\times_U U_0\n$$\nis a faithful functor.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLG","source_file":"pione.tex","source_line":4456,"source_end_line":4471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4456-L4471","statement_sha256":"902476274f5f13fd7f3579718f70e7b81392a1adbf2923d8a5840ca2a49dc03b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9760,"rank":9760,"depth":20,"x":1570.262,"y":1194.724,"cluster":"tale-geometry"},{"id":"stacks:0BLH","tag":"0BLH","title":"Finite étale covers of punctured spectra, I · Lemma 0BLH","summary":"In Situation [Tag 0BLF]. Let V → U be a finite morphism. Let A^wedge be the m-adic completion of A, let X' = Spec(A^wedge) and let U' and V' be the base changes of U and V to X'. If Y' → X' is a finite morphism such that V' = Y' ×_X' U', then there exists a finite morphism Y → X such that V = Y ×_X U and Y' = Y ×_X X'.","statement_latex":"In Situation \\ref{situation-local-lefschetz}. Let $V \\to U$ be a finite\nmorphism.  Let $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$,\nlet $X' = \\Spec(A^\\wedge)$ and let $U'$ and $V'$ be the base changes of\n$U$ and $V$ to $X'$. If $Y' \\to X'$ is a finite morphism such that\n$V' = Y' \\times_{X'} U'$, then there exists a finite morphism $Y \\to X$\nsuch that $V = Y \\times_X U$ and $Y' = Y \\times_X X'$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLH","source_file":"pione.tex","source_line":4483,"source_end_line":4491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4483-L4491","statement_sha256":"f11eec440aea84fea97ef0bbe84d7e304e4b551b3befc75a2b3d4095cbe5f4bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9761,"rank":9761,"depth":20,"x":1543.032,"y":1093.946,"cluster":"tale-geometry"},{"id":"stacks:0BLI","tag":"0BLI","title":"Finite étale covers of punctured spectra, I · Lemma 0BLI","summary":"In Situation [Tag 0BLF] assume A is henselian or more generally that (A, (f)) is a henselian pair. Let A^wedge be the m-adic completion of A, let X' = Spec(A^wedge) and let U' and U'_0 be the base changes of U and U_0 to X'. If FÉt_U' → FÉt_U'_0 is fully faithful, then FÉt_U → FÉt_U_0 is fully faithful.","statement_latex":"In Situation \\ref{situation-local-lefschetz} assume $A$ is henselian\nor more generally that $(A, (f))$ is a henselian pair.\nLet $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$,\nlet $X' = \\Spec(A^\\wedge)$ and let $U'$ and $U'_0$ be the base changes of\n$U$ and $U_0$ to $X'$. If $\\textit{F\\'Et}_{U'} \\to \\textit{F\\'Et}_{U'_0}$\nis fully faithful, then $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLI","source_file":"pione.tex","source_line":4515,"source_end_line":4524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4515-L4524","statement_sha256":"7468a172bb340a333999205829a7847f344fa3a97c9346ef2e46443d382d3780","origin":"The Stacks Project","memory_eligible":false,"source_rank":9762,"rank":9762,"depth":52,"x":1644.54,"y":1153.033,"cluster":"tale-geometry"},{"id":"stacks:0EK5","tag":"0EK5","title":"Finite étale covers of punctured spectra, I · Lemma 0EK5","summary":"In Situation [Tag 0BLF]. Assume • [(a)] A has a dualizing complex, • [(b)] the pair (A, (f)) is henselian, • [(c)] one of the following is true • [(i)] A_f is (S_2) and every irreducible component of X not contained in X_0 has dimension ≥ 3, or • [(ii)] for every prime p ⊂ A, f not ∈ p we have depth(A_ p) + dim(A/ p) > 2. Then the restriction functor FÉt_U → FÉt_U_0 is fully faithful.","statement_latex":"In Situation \\ref{situation-local-lefschetz}. Assume\n\\begin{enumerate}\n\\item[(a)] $A$ has a dualizing complex,\n\\item[(b)] the pair $(A, (f))$ is henselian,\n\\item[(c)] one of the following is true\n\\begin{enumerate}\n\\item[(i)] $A_f$ is $(S_2)$ and every irreducible component of $X$\nnot contained in $X_0$ has dimension $\\geq 3$, or\n\\item[(ii)] for every prime\n$\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$ we have\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 2$.\n\\end{enumerate}\n\\end{enumerate}\nThen the restriction functor\n$\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0}$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EK5","source_file":"pione.tex","source_line":4599,"source_end_line":4617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4599-L4617","statement_sha256":"9e3e945024aa65390739860f719d76c6bad8763bf2dcc3bc1b22bb196978c964","origin":"The Stacks Project","memory_eligible":false,"source_rank":9763,"rank":9763,"depth":53,"x":1521.71,"y":1167.113,"cluster":"tale-geometry"},{"id":"stacks:0BM6","tag":"0BM6","title":"Finite étale covers of punctured spectra, I · Lemma 0BM6","summary":"[Bhatt-local] In Situation [Tag 0BLF]. Assume • H^1_ m(A) and H^2_ m(A) are annihilated by a power of f, and • A is henselian or more generally (A, (f)) is a henselian pair. Then the restriction functor FÉt_U → FÉt_U_0 is fully faithful.","statement_latex":"\\begin{reference}\n\\cite[Corollary 1.11]{Bhatt-local}\n\\end{reference}\nIn Situation \\ref{situation-local-lefschetz}. Assume\n\\begin{enumerate}\n\\item $H^1_\\mathfrak m(A)$ and $H^2_\\mathfrak m(A)$ are\nannihilated by a power of $f$, and\n\\item $A$ is henselian or more generally $(A, (f))$ is a henselian pair.\n\\end{enumerate}\nThen the restriction functor\n$\\textit{F\\'Et}_U \\longrightarrow \\textit{F\\'Et}_{U_0}$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BM6","source_file":"pione.tex","source_line":4645,"source_end_line":4659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4645-L4659","statement_sha256":"9f6d025e28a0db5ce1dff90731b1b878db4978d1774924f8528192d098bbd1c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9764,"rank":9764,"depth":53,"x":1601.258,"y":1086.733,"cluster":"tale-geometry"},{"id":"stacks:0BLJ","tag":"0BLJ","title":"Finite étale covers of punctured spectra, I · Lemma 0BLJ","summary":"In Situation [Tag 0BLF] assume A has depth ≥ 3 and A is henselian or more generally (A, (f)) is a henselian pair. Then the restriction functor FÉt_U → FÉt_U_0 is fully faithful.","statement_latex":"In Situation \\ref{situation-local-lefschetz} assume $A$ has depth $\\geq 3$\nand $A$ is henselian or more generally $(A, (f))$ is a henselian pair. Then\nthe restriction functor\n$\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLJ","source_file":"pione.tex","source_line":4672,"source_end_line":4679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4672-L4679","statement_sha256":"73e91652616f55ae6741773c570ad455917f568b1fdcc65f25eca86999a98812","origin":"The Stacks Project","memory_eligible":false,"source_rank":9765,"rank":9765,"depth":54,"x":1607.262,"y":1191.531,"cluster":"tale-geometry"},{"id":"stacks:0BM8","tag":"0BM8","title":"Purity in local case, I · Lemma 0BM8","summary":"Let (A, m) be a Noetherian local ring. Set X = Spec(A) and let U = X setminus ( m). Let π : Y → X be a finite morphism such that depth(O_Y, y) ≥ 2 for all closed points y ∈ Y. Then Y is the spectrum of B = O_Y(π^-1(U)).","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$.\nLet $\\pi : Y \\to X$ be a finite morphism such that\n$\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$ for all closed points\n$y \\in Y$.\nThen $Y$ is the spectrum of $B = \\mathcal{O}_Y(\\pi^{-1}(U))$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BM8","source_file":"pione.tex","source_line":4734,"source_end_line":4742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4734-L4742","statement_sha256":"b8c80dcf02dd634af49252b04a78f9ce5401093d6c364085afffaf60c5b6e759","origin":"The Stacks Project","memory_eligible":false,"source_rank":9766,"rank":9766,"depth":32,"x":1518.231,"y":1117.387,"cluster":"tale-geometry"},{"id":"stacks:0BLK","tag":"0BLK","title":"Purity in local case, I · Lemma 0BLK","summary":"Let (A, m) be a Noetherian local ring. Set X = Spec(A) and let U = X setminus ( m). Let V be finite étale over U. Assume A has depth ≥ 2. The following are equivalent • V = Y ×_X U for some Y → X finite étale, • B = Γ(V, O_V) is finite étale over A.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$.\nLet $V$ be finite \\'etale\nover $U$. Assume $A$ has depth $\\geq 2$. The following are equivalent\n\\begin{enumerate}\n\\item $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale,\n\\item $B = \\Gamma(V, \\mathcal{O}_V)$ is finite \\'etale over $A$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLK","source_file":"pione.tex","source_line":4763,"source_end_line":4773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4763-L4773","statement_sha256":"56eecea32ff3d2b550efef67e0c0c58e050eff10ae9ec3d62c91a43d052b8597","origin":"The Stacks Project","memory_eligible":false,"source_rank":9767,"rank":9767,"depth":33,"x":1643.966,"y":1121.558,"cluster":"tale-geometry"},{"id":"stacks:0BM9","tag":"0BM9","title":"Purity in local case, I · Lemma 0BM9","summary":"Let (A, m) be a Noetherian local ring. Set X = Spec(A) and let U = X setminus ( m). Assume A is normal of dimension ≥ 2. The functor FÉt_U → ( finite normal A-algebras B such that Spec(B) → X is étale over U ), V ↦ Γ(V, O_V) is an equivalence. Moreover, V = Y ×_X U for some Y → X finite étale if and only if B = Γ(V, O_V) is finite étale over A.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$. Assume $A$ is normal\nof dimension $\\geq 2$. The functor\n$$\n\\textit{F\\'Et}_U \\longrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{finite normal }A\\text{-algebras }B\\text{ such} \\\\\n\\text{that }\\Spec(B) \\to X\\text{ is \\'etale over }U\n\\end{matrix}\n\\right\\},\n\\quad\nV \\longmapsto \\Gamma(V, \\mathcal{O}_V)\n$$\nis an equivalence. Moreover, $V = Y \\times_X U$ for some $Y \\to X$\nfinite \\'etale if and only if $B = \\Gamma(V, \\mathcal{O}_V)$\nis finite \\'etale over $A$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BM9","source_file":"pione.tex","source_line":4792,"source_end_line":4811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4792-L4811","statement_sha256":"6f4f42501ca49c69396b7c4b688e953feb1576ddc604dedb50f438616f6f7e65","origin":"The Stacks Project","memory_eligible":false,"source_rank":9768,"rank":9768,"depth":45,"x":1547.546,"y":1190.076,"cluster":"tale-geometry"},{"id":"stacks:0BLL","tag":"0BLL","title":"Purity in local case, I · Lemma 0BLL","summary":"Let (A, m) be a Noetherian local ring. Set X = Spec(A) and let U = X setminus ( m). Let V be finite étale over U. Let A^wedge be the m-adic completion of A, let X' = Spec(A^wedge) and let U' and V' be the base changes of U and V to X'. The following are equivalent • V = Y ×_X U for some Y → X finite étale, and • V' = Y' ×_X' U' for some Y' → X' finite étale.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. Set $X = \\Spec(A)$\nand let $U = X \\setminus \\{\\mathfrak m\\}$.\nLet $V$ be finite \\'etale over $U$.\nLet $A^\\wedge$ be the $\\mathfrak m$-adic completion of $A$,\nlet $X' = \\Spec(A^\\wedge)$ and let $U'$ and $V'$ be the base changes of\n$U$ and $V$ to $X'$. The following are equivalent\n\\begin{enumerate}\n\\item $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale, and\n\\item $V' = Y' \\times_{X'} U'$ for some $Y' \\to X'$ finite \\'etale.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLL","source_file":"pione.tex","source_line":4859,"source_end_line":4871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4859-L4871","statement_sha256":"e38e318bfca6e9a146d1a82b5af55dd30ab9f5c2d13449e760a8e31805873a72","origin":"The Stacks Project","memory_eligible":false,"source_rank":9769,"rank":9769,"depth":38,"x":1563.602,"y":1084.46,"cluster":"tale-geometry"},{"id":"stacks:0EK6","tag":"0EK6","title":"Purity in local case, I · Lemma 0EK6","summary":"In Situation [Tag 0BLF]. Let V be finite étale over U. Assume • [(a)] A has a dualizing complex, • [(b)] the pair (A, (f)) is henselian, • [(c)] one of the following is true • [(i)] A_f is (S_2) and every irreducible component of X not contained in X_0 has dimension ≥ 3, or • [(ii)] for every prime p ⊂ A, f not ∈ p we have depth(A_ p) + dim(A/ p) > 2. • [(d)] V_0 = V ×_U U_0 is equal to Y_0 ×_X_0 U_0 for some Y_0 → X_0 finite étale. Then V = Y ×_X U for some Y → X finite…","statement_latex":"In Situation \\ref{situation-local-lefschetz}. Let $V$ be finite\n\\'etale over $U$. Assume\n\\begin{enumerate}\n\\item[(a)] $A$ has a dualizing complex,\n\\item[(b)] the pair $(A, (f))$ is henselian,\n\\item[(c)] one of the following is true\n\\begin{enumerate}\n\\item[(i)] $A_f$ is $(S_2)$ and every irreducible component of $X$\nnot contained in $X_0$ has dimension $\\geq 3$, or\n\\item[(ii)] for every prime $\\mathfrak p \\subset A$, $f \\not \\in \\mathfrak p$\nwe have $\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 2$.\n\\end{enumerate}\n\\item[(d)] $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$\nfor some $Y_0 \\to X_0$ finite \\'etale.\n\\end{enumerate}\nThen $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EK6","source_file":"pione.tex","source_line":4889,"source_end_line":4907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4889-L4907","statement_sha256":"f2b197b67b4d692e2c27e319114ca916ff53dae213c736a725150d2486a0a1a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9770,"rank":9770,"depth":54,"x":1636.959,"y":1171.763,"cluster":"tale-geometry"},{"id":"stacks:0BLS","tag":"0BLS","title":"Purity in local case, I · Lemma 0BLS","summary":"In Situation [Tag 0BLF]. Let V be finite étale over U. Assume • H^1_ m(A) and H^2_ m(A) are annihilated by a power of f, • V_0 = V ×_U U_0 is equal to Y_0 ×_X_0 U_0 for some Y_0 → X_0 finite étale. Then V = Y ×_X U for some Y → X finite étale.","statement_latex":"In Situation \\ref{situation-local-lefschetz}.\nLet $V$ be finite \\'etale over $U$. Assume\n\\begin{enumerate}\n\\item $H^1_\\mathfrak m(A)$ and $H^2_\\mathfrak m(A)$\nare annihilated by a power of $f$,\n\\item $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$\nfor some $Y_0 \\to X_0$ finite \\'etale.\n\\end{enumerate}\nThen $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLS","source_file":"pione.tex","source_line":4925,"source_end_line":4936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4925-L4936","statement_sha256":"ed015dfddc0d271cbc6a53e421fe2703c04b7f20a412d71d013aefd45d37ad86","origin":"The Stacks Project","memory_eligible":false,"source_rank":9771,"rank":9771,"depth":54,"x":1512.216,"y":1148.93,"cluster":"tale-geometry"},{"id":"stacks:0BLM","tag":"0BLM","title":"Purity in local case, I · Lemma 0BLM","summary":"In Situation [Tag 0BLF]. Let V be finite étale over U. Assume • A has depth ≥ 3, • V_0 = V ×_U U_0 is equal to Y_0 ×_X_0 U_0 for some Y_0 → X_0 finite étale. Then V = Y ×_X U for some Y → X finite étale.","statement_latex":"In Situation \\ref{situation-local-lefschetz}.\nLet $V$ be finite \\'etale over $U$. Assume\n\\begin{enumerate}\n\\item $A$ has depth $\\geq 3$,\n\\item $V_0 = V \\times_U U_0$ is equal to $Y_0 \\times_{X_0} U_0$\nfor some $Y_0 \\to X_0$ finite \\'etale.\n\\end{enumerate}\nThen $V = Y \\times_X U$ for some $Y \\to X$ finite \\'etale.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLM","source_file":"pione.tex","source_line":4954,"source_end_line":4964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4954-L4964","statement_sha256":"f2382d595d1cf0e540032f6a5ca630c8f0e912a1cad0f35cb124dcf5a1f7f37b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9772,"rank":9772,"depth":55,"x":1622.953,"y":1094.794,"cluster":"tale-geometry"},{"id":"stacks:0BJG","tag":"0BJG","title":"Purity of branch locus · Lemma 0BJG","summary":"Let (A, m) be a Noetherian local ring with dim(A) ≥ 1. Let f ∈ m. Then there exist a p ∈ V(f) with dim(A_ p) = 1.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring with $\\dim(A) \\geq 1$.\nLet $f \\in \\mathfrak m$. Then there exist a $\\mathfrak p \\in V(f)$ with\n$\\dim(A_\\mathfrak p) = 1$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity of branch locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJG","source_file":"pione.tex","source_line":4989,"source_end_line":4994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L4989-L4994","statement_sha256":"d2a70c76e13432f9f76b0f20eceb0efff7a08d9cc6435c11ae3155a77b3b4471","origin":"The Stacks Project","memory_eligible":false,"source_rank":9773,"rank":9773,"depth":16,"x":1584.697,"y":1197.909,"cluster":"tale-geometry"},{"id":"stacks:0BJH","tag":"0BJH","title":"Purity of branch locus · Lemma 0BJH","summary":"Let f : X → Y be a morphism of locally Noetherian schemes. Let x ∈ X. Assume • f is flat, • f is quasi-finite at x, • x is not a generic point of an irreducible component of X, • for specializations x' leadsto x with dim(O_X, x') = 1 our f is unramified at x'. Then f is étale at x.","statement_latex":"Let $f : X \\to Y$ be a morphism of locally Noetherian schemes.\nLet $x \\in X$. Assume\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item $f$ is quasi-finite at $x$,\n\\item $x$ is not a generic point of an irreducible component of $X$,\n\\item for specializations $x' \\leadsto x$ with\n$\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$.\n\\end{enumerate}\nThen $f$ is \\'etale at $x$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity of branch locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BJH","source_file":"pione.tex","source_line":5013,"source_end_line":5025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5013-L5025","statement_sha256":"f33338c6ccda59aed5446d5d9603adfd1fb60a4fb60ac370b705682ec175c2ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":9774,"rank":9774,"depth":47,"x":1529.794,"y":1099.824,"cluster":"tale-geometry"},{"id":"stacks:0BMA","tag":"0BMA","title":"Purity of branch locus · Lemma 0BMA","summary":"Let (A, m) be a regular local ring of dimension d ≥ 2. Set X = Spec(A) and U = X setminus ( m). Then • the functor FÉt_X → FÉt_U is essentially surjective, i.e., purity holds for A, • any finite A → B with B normal which induces a finite étale morphism on punctured spectra is étale.","statement_latex":"Let $(A, \\mathfrak m)$ be a regular local ring of dimension $d \\geq 2$.\nSet $X = \\Spec(A)$ and $U = X \\setminus \\{\\mathfrak m\\}$. Then\n\\begin{enumerate}\n\\item the functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$\nis essentially surjective, i.e., purity holds for $A$,\n\\item any finite $A \\to B$ with $B$ normal which\ninduces a finite \\'etale morphism on punctured spectra is \\'etale.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity of branch locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMA","source_file":"pione.tex","source_line":5066,"source_end_line":5076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5066-L5076","statement_sha256":"fc823df7dbbeb97264995604c126cc41c0d12fdecab5a62bbb4a98d42c78218c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9775,"rank":9775,"depth":56,"x":1649.569,"y":1141.14,"cluster":"tale-geometry"},{"id":"stacks:0BMB","tag":"0BMB","title":"Purity of branch locus · Lemma 0BMB","summary":"[Nagata-Purity] and [SGA1] This result was first stated and proved by Zariski in geometric form in [Zariski-Purity]. The generalization to nonperfect ground fields by Nagata was published as the next article in the same volume of the Proceedings of the National Academy of Sciences of the United States of America in [Nagata-Remarks-Purity]. In the following year Nagata proved the result for Noetherian local rings in [Nagata-Purity]. His proof uses a result of Chow which is…","statement_latex":"\\begin{reference}\n\\cite{Nagata-Purity} and \\cite[Exp. X, Thm. 3.1]{SGA1}\n\\end{reference}\n\\begin{history}\nThis result was first stated and proved by Zariski in\ngeometric form in \\cite{Zariski-Purity}.\nThe generalization to nonperfect ground fields by Nagata\nwas published as the next article in the same volume of the\nProceedings of the National Academy of Sciences of the United States of America\nin \\cite{Nagata-Remarks-Purity}. In the following year Nagata\nproved the result for Noetherian local rings in \\cite{Nagata-Purity}.\nHis proof uses a result of Chow which is a Bertini theorem for\ncomplete local domains, see \\cite{Chow-Bertini};\nthe history of Bertini's theorems is discussed in\nKleiman's historical article \\cite{Kleiman-Bertini}.\nA few years later a completely different proof was found by\nAuslander, see \\cite{Auslander-Purity}.\n\\end{history}\nLet $f : X \\to Y$ be a morphism of locally Noetherian schemes.\nLet $x \\in X$ and set $y = f(x)$. Assume\n\\begin{enumerate}\n\\item $\\mathcal{O}_{X, x}$ is normal,\n\\item $\\mathcal{O}_{Y, y}$ is regular,\n\\item $f$ is quasi-finite at $x$,\n\\item $\\dim(\\mathcal{O}_{X, x}) = \\dim(\\mathcal{O}_{Y, y}) \\geq 1$\n\\item for specializations $x' \\leadsto x$ with\n$\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is unramified at $x'$.\n\\end{enumerate}\nThen $f$ is \\'etale at $x$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity of branch locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BMB","source_file":"pione.tex","source_line":5117,"source_end_line":5148,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5117-L5148","statement_sha256":"4cb5887ba4bbbb3f2637cddb8c59d9fdd8515127606ca98219e013f8454f3d8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9776,"rank":9776,"depth":57,"x":1527.604,"y":1178.767,"cluster":"tale-geometry"},{"id":"stacks:0EY6","tag":"0EY6","title":"Purity of branch locus · Lemma 0EY6","summary":"Let j : U → X be an open immersion of locally Noetherian schemes such that depth(O_X, x) ≥ 2 for x not ∈ U. Let π : V → U be finite étale. Then • B = j_*π_*O_V is a reflexive coherent O_X-algebra, set Y = underlineSpec_X(B), • Y → X is the unique finite morphism such that V = Y ×_X U and depth(O_Y, y) ≥ 2 for y ∈ Y setminus V, • Y → X is étale at y if and only if Y → X is flat at y, and • Y → X is étale if and only if B is finite locally free as an O_X-module. Moreover,…","statement_latex":"Let $j : U \\to X$ be an open immersion of locally Noetherian schemes\nsuch that $\\text{depth}(\\mathcal{O}_{X, x}) \\geq 2$ for $x \\not \\in U$.\nLet $\\pi : V \\to U$ be finite \\'etale. Then\n\\begin{enumerate}\n\\item $\\mathcal{B} = j_*\\pi_*\\mathcal{O}_V$ is a reflexive coherent\n$\\mathcal{O}_X$-algebra, set $Y = \\underline{\\Spec}_X(\\mathcal{B})$,\n\\item $Y \\to X$ is the unique finite morphism such that\n$V = Y \\times_X U$ and $\\text{depth}(\\mathcal{O}_{Y, y}) \\geq 2$\nfor $y \\in Y \\setminus V$,\n\\item $Y \\to X$ is \\'etale at $y$ if and only if $Y \\to X$ is flat at $y$, and\n\\item $Y \\to X$ is \\'etale if and only if $\\mathcal{B}$\nis finite locally free as an $\\mathcal{O}_X$-module.\n\\end{enumerate}\nMoreover, (a) the construction of $\\mathcal{B}$ and $Y \\to X$ commutes\nwith base change by flat morphisms $X' \\to X$ of locally Noetherian\nschemes, and (b) if $V' \\to U'$ is a finite \\'etale morphism with\n$U \\subset U' \\subset X$ open which restricts to $V \\to U$ over $U$,\nthen there is a unique isomorphism $Y' \\times_X U' = V'$ over $U'$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity of branch locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EY6","source_file":"pione.tex","source_line":5219,"source_end_line":5239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5219-L5239","statement_sha256":"1c487e2787d570dc3ecd14caaea3444bf451010ed421aa3b3016a694a0a7e102","origin":"The Stacks Project","memory_eligible":false,"source_rank":9777,"rank":9777,"depth":48,"x":1587.486,"y":1081.485,"cluster":"tale-geometry"},{"id":"stacks:0EY7","tag":"0EY7","title":"Purity of branch locus · Lemma 0EY7","summary":"Let j : U → X be an open immersion of Noetherian schemes such that purity holds for O_X, x for all x not ∈ U. Then FÉt_X → FÉt_U is essentially surjective.","statement_latex":"Let $j : U \\to X$ be an open immersion of Noetherian schemes\nsuch that purity holds for $\\mathcal{O}_{X, x}$ for all $x \\not \\in U$.\nThen\n$$\n\\textit{F\\'Et}_X \\longrightarrow \\textit{F\\'Et}_U\n$$\nis essentially surjective.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity of branch locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EY7","source_file":"pione.tex","source_line":5303,"source_end_line":5312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5303-L5312","statement_sha256":"9036d61290f1e97aaa66cfadae9a544736bac04011ebf14137530614f2d1c940","origin":"The Stacks Project","memory_eligible":false,"source_rank":9778,"rank":9778,"depth":41,"x":1621.677,"y":1187.556,"cluster":"tale-geometry"},{"id":"stacks:0BLN","tag":"0BLN","title":"Finite étale covers of punctured spectra, II · Lemma 0BLN","summary":"In Situation [Tag 0BLF]. Let U' ⊂ U be open and contain U_0. Assume for p ⊂ A minimal with p ∈ U', p not ∈ U_0 we have dim(A/ p) ≥ 2. Then FÉt_U' → FÉt_U_0, V' ↦ V_0 = V' ×_U' U_0 is a faithful functor. Moreover, there exists a U' satisfying the assumption and any smaller open U\" ⊂ U' containing U_0 also satisfies this assumption. In particular, the restriction functor colim_U_0 ⊂ U' ⊂ U open FÉt_U' → FÉt_U_0 is faithful.","statement_latex":"In Situation \\ref{situation-local-lefschetz}. Let $U' \\subset U$\nbe open and contain $U_0$. Assume for $\\mathfrak p \\subset A$ minimal\nwith $\\mathfrak p \\in U'$, $\\mathfrak p \\not \\in U_0$ we have\n$\\dim(A/\\mathfrak p) \\geq 2$. Then\n$$\n\\textit{F\\'Et}_{U'} \\longrightarrow \\textit{F\\'Et}_{U_0},\\quad\nV' \\longmapsto V_0 = V' \\times_{U'} U_0\n$$\nis a faithful functor. Moreover, there exists a $U'$ satisfying\nthe assumption and any smaller open $U'' \\subset U'$ containing\n$U_0$ also satisfies this assumption. In particular, the restriction\nfunctor\n$$\n\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}\n\\longrightarrow\n\\textit{F\\'Et}_{U_0}\n$$\nis faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLN","source_file":"pione.tex","source_line":5372,"source_end_line":5392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5372-L5392","statement_sha256":"c09d9e8cf209dfa25d85fd234165f5225d3140a4995b7fc1aedd57a1c4adbc30","origin":"The Stacks Project","memory_eligible":false,"source_rank":9779,"rank":9779,"depth":20,"x":1510.792,"y":1128.543,"cluster":"tale-geometry"},{"id":"stacks:0DXX","tag":"0DXX","title":"Finite étale covers of punctured spectra, II · Lemma 0DXX","summary":"In Situation [Tag 0BLF] assume • A has a dualizing complex and is f-adically complete, • every irreducible component of X not contained in X_0 has dimension ≥ 3. Then the restriction functor colim_U_0 ⊂ U' ⊂ U open FÉt_U' → FÉt_U_0 is fully faithful.","statement_latex":"In Situation \\ref{situation-local-lefschetz} assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and is $f$-adically complete,\n\\item every irreducible component of $X$ not contained in $X_0$\nhas dimension $\\geq 3$.\n\\end{enumerate}\nThen the restriction functor\n$$\n\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}\n\\longrightarrow\n\\textit{F\\'Et}_{U_0}\n$$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXX","source_file":"pione.tex","source_line":5409,"source_end_line":5424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5409-L5424","statement_sha256":"7b04e41355002bc76fc5b9dffb55f52c225064417b9bc7b756ccb80188fbc690","origin":"The Stacks Project","memory_eligible":false,"source_rank":9780,"rank":9780,"depth":54,"x":1640.447,"y":1109.075,"cluster":"tale-geometry"},{"id":"stacks:0BLP","tag":"0BLP","title":"Finite étale covers of punctured spectra, II · Lemma 0BLP","summary":"In Situation [Tag 0BLF] assume • A is f-adically complete, • f is a nonzerodivisor. • H^1_ m(A/fA) is a finite A-module. Then the restriction functor colim_U_0 ⊂ U' ⊂ U open FÉt_U' → FÉt_U_0 is fully faithful.","statement_latex":"In Situation \\ref{situation-local-lefschetz} assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor.\n\\item $H^1_\\mathfrak m(A/fA)$ is a finite $A$-module.\n\\end{enumerate}\nThen the restriction functor\n$$\n\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}\n\\longrightarrow\n\\textit{F\\'Et}_{U_0}\n$$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLP","source_file":"pione.tex","source_line":5435,"source_end_line":5450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5435-L5450","statement_sha256":"c98b7e2ea51e769fe6cd6254af24a1d48cc76fd29b7da2cc93955bbf57bce9f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9781,"rank":9781,"depth":54,"x":1560.23,"y":1197.292,"cluster":"tale-geometry"},{"id":"stacks:0DXY","tag":"0DXY","title":"Finite étale covers of punctured spectra, III · Lemma 0DXY","summary":"In Situation [Tag 0BLF] assume • A has a dualizing complex and is f-adically complete, • one of the following is true • A_f is (S_2) and every irreducible component of X not contained in X_0 has dimension ≥ 4, or • if p not ∈ V(f) and V( p) ∩ V(f) not = ( m), then depth(A_ p) + dim(A/ p) > 3. Then the restriction functor colim_U_0 ⊂ U' ⊂ U open FÉt_U' → FÉt_U_0 is an equivalence.","statement_latex":"In Situation \\ref{situation-local-lefschetz} assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and is $f$-adically complete,\n\\item one of the following is true\n\\begin{enumerate}\n\\item $A_f$ is $(S_2)$ and every irreducible component of $X$\nnot contained in $X_0$ has dimension $\\geq 4$, or\n\\item if $\\mathfrak p \\not \\in V(f)$ and\n$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.\n\\end{enumerate}\n\\end{enumerate}\nThen the restriction functor\n$$\n\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}\n\\longrightarrow\n\\textit{F\\'Et}_{U_0}\n$$\nis an equivalence.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXY","source_file":"pione.tex","source_line":5479,"source_end_line":5500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5479-L5500","statement_sha256":"4dc13ca6b34c10c46f7c8913f2d1b2731d1bcc6e0b604a1a45a06a570873c0a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9782,"rank":9782,"depth":56,"x":1548.401,"y":1086.36,"cluster":"tale-geometry"},{"id":"stacks:0BLV","tag":"0BLV","title":"Finite étale covers of punctured spectra, III · Lemma 0BLV","summary":"In Situation [Tag 0BLF] assume • A is f-adically complete, • f is a nonzerodivisor, • H^1_ m(A/fA) and H^2_ m(A/fA) are finite A-modules. Then the restriction functor colim_U_0 ⊂ U' ⊂ U open FÉt_U' → FÉt_U_0 is an equivalence.","statement_latex":"In Situation \\ref{situation-local-lefschetz} assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$\nare finite $A$-modules.\n\\end{enumerate}\nThen the restriction functor\n$$\n\\colim_{U_0 \\subset U' \\subset U\\text{ open}} \\textit{F\\'Et}_{U'}\n\\longrightarrow\n\\textit{F\\'Et}_{U_0}\n$$\nis an equivalence.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLV","source_file":"pione.tex","source_line":5509,"source_end_line":5525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5509-L5525","statement_sha256":"16ecdf69127deae36c8eee240b1ecec4f219c69a8691cd0f40c3a6943d3ea98c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9783,"rank":9783,"depth":55,"x":1646.654,"y":1161.694,"cluster":"tale-geometry"},{"id":"stacks:0EK9","tag":"0EK9","title":"Finite étale covers of punctured spectra, IV · Lemma 0EK9","summary":"In Situation [Tag 0BLF] assume • A has a dualizing complex and is f-adically complete, • one of the following is true • A_f is (S_2) and every irreducible component of X not contained in X_0 has dimension ≥ 4, or • if p not ∈ V(f) and V( p) ∩ V(f) not = ( m), then depth(A_ p) + dim(A/ p) > 3. • for every maximal ideal p ⊂ A_f purity holds for (A_f)_ p. Then the restriction functor FÉt_U → FÉt_U_0 is essentially surjective.","statement_latex":"In Situation \\ref{situation-local-lefschetz} assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and is $f$-adically complete,\n\\item one of the following is true\n\\begin{enumerate}\n\\item $A_f$ is $(S_2)$ and every irreducible component of $X$\nnot contained in $X_0$ has dimension $\\geq 4$, or\n\\item if $\\mathfrak p \\not \\in V(f)$ and\n$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.\n\\end{enumerate}\n\\item for every maximal ideal $\\mathfrak p \\subset A_f$\npurity holds for $(A_f)_\\mathfrak p$.\n\\end{enumerate}\nThen the restriction functor $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$\nis essentially surjective.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EK9","source_file":"pione.tex","source_line":5572,"source_end_line":5590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5572-L5590","statement_sha256":"b398cfe1af007988c06ffc72815a30447c8c5fa33a7c4d0ea8ea296c60f13e3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9784,"rank":9784,"depth":57,"x":1513.188,"y":1161.895,"cluster":"tale-geometry"},{"id":"stacks:0EKA","tag":"0EKA","title":"Finite étale covers of punctured spectra, IV · Lemma 0EKA","summary":"Let (A, m) be a Noetherian local ring. Let f ∈ m. Assume • A is f-adically complete, • f is a nonzerodivisor, • H^1_ m(A/fA) and H^2_ m(A/fA) are finite A-modules, • for every maximal ideal p ⊂ A_f purity holds for (A_f)_ p. Then the restriction functor FÉt_U → FÉt_U_0 is essentially surjective.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $f \\in \\mathfrak m$. Assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$ are finite\n$A$-modules,\n\\item for every maximal ideal $\\mathfrak p \\subset A_f$\npurity holds for $(A_f)_\\mathfrak p$.\n\\end{enumerate}\nThen the restriction functor $\\textit{F\\'Et}_U \\to \\textit{F\\'Et}_{U_0}$\nis essentially surjective.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Finite étale covers of punctured spectra, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EKA","source_file":"pione.tex","source_line":5608,"source_end_line":5622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5608-L5622","statement_sha256":"96238aa591d61cc775a85097bbe89b662e290ace2106313d3703735b2c2c4eaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9785,"rank":9785,"depth":58,"x":1611.761,"y":1085.77,"cluster":"tale-geometry"},{"id":"stacks:0DXZ","tag":"0DXZ","title":"Purity in local case, II · Lemma 0DXZ","summary":"Let (A, m) be a Noetherian local ring. Let f ∈ m. Assume • A has a dualizing complex and is f-adically complete, • one of the following is true • A_f is (S_2) and every irreducible component of X not contained in X_0 has dimension ≥ 4, or • if p not ∈ V(f) and V( p) ∩ V(f) not = ( m), then depth(A_ p) + dim(A/ p) > 3. • for every maximal ideal p ⊂ A_f purity holds for (A_f)_ p, and • purity holds for A. Then purity holds for A/fA.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $f \\in \\mathfrak m$. Assume\n\\begin{enumerate}\n\\item $A$ has a dualizing complex and is $f$-adically complete,\n\\item one of the following is true\n\\begin{enumerate}\n\\item $A_f$ is $(S_2)$ and every irreducible component of $X$\nnot contained in $X_0$ has dimension $\\geq 4$, or\n\\item if $\\mathfrak p \\not \\in V(f)$ and\n$V(\\mathfrak p) \\cap V(f) \\not = \\{\\mathfrak m\\}$, then\n$\\text{depth}(A_\\mathfrak p) + \\dim(A/\\mathfrak p) > 3$.\n\\end{enumerate}\n\\item for every maximal ideal $\\mathfrak p \\subset A_f$\npurity holds for $(A_f)_\\mathfrak p$, and\n\\item purity holds for $A$.\n\\end{enumerate}\nThen purity holds for $A/fA$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DXZ","source_file":"pione.tex","source_line":5645,"source_end_line":5664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5645-L5664","statement_sha256":"2a2072f2256c3bda15a5267f6c9fa5f34fb86f05b7ceb687194c8579d97fd4be","origin":"The Stacks Project","memory_eligible":false,"source_rank":9786,"rank":9786,"depth":58,"x":1600.255,"y":1198.197,"cluster":"tale-geometry"},{"id":"stacks:0BPC","tag":"0BPC","title":"Purity in local case, II · Lemma 0BPC","summary":"Let (A, m) be a Noetherian local ring. Let f ∈ m. Assume • A is f-adically complete, • f is a nonzerodivisor, • H^1_ m(A/fA) and H^2_ m(A/fA) are finite A-modules, • for every maximal ideal p ⊂ A_f purity holds for (A_f)_ p, • purity holds for A. Then purity holds for A/fA.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring.\nLet $f \\in \\mathfrak m$. Assume\n\\begin{enumerate}\n\\item $A$ is $f$-adically complete,\n\\item $f$ is a nonzerodivisor,\n\\item $H^1_\\mathfrak m(A/fA)$ and $H^2_\\mathfrak m(A/fA)$ are finite\n$A$-modules,\n\\item for every maximal ideal $\\mathfrak p \\subset A_f$\npurity holds for $(A_f)_\\mathfrak p$,\n\\item purity holds for $A$.\n\\end{enumerate}\nThen purity holds for $A/fA$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPC","source_file":"pione.tex","source_line":5680,"source_end_line":5694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5680-L5694","statement_sha256":"63c1fe879a384f49157da38e27448c3f489549a832e47be31bd8e4e479e1b273","origin":"The Stacks Project","memory_eligible":false,"source_rank":9787,"rank":9787,"depth":59,"x":1518.067,"y":1108.479,"cluster":"tale-geometry"},{"id":"stacks:0BPD","tag":"0BPD","title":"Purity in local case, II · Proposition 0BPD","summary":"Let (A, m) be a Noetherian local ring. If A is a complete intersection of dimension ≥ 3, then purity holds for A in the sense that any finite étale cover of the punctured spectrum extends.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring. If $A$ is a\ncomplete intersection of dimension $\\geq 3$, then purity\nholds for $A$ in the sense that any finite \\'etale cover of\nthe punctured spectrum extends.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, II","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPD","source_file":"pione.tex","source_line":5713,"source_end_line":5719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5713-L5719","statement_sha256":"9bb7eb4f9eca58b7b71060a8038d4aebf019584bdf599dcad7f26429c2a76164","origin":"The Stacks Project","memory_eligible":false,"source_rank":9788,"rank":9788,"depth":60,"x":1651.241,"y":1128.063,"cluster":"tale-geometry"},{"id":"stacks:0EY9","tag":"0EY9","title":"Purity in local case, III · Lemma 0EY9","summary":"Let (A, m) be a Noetherian local ring of depth ≥ 2. Let B = A[[x_1, …, x_d]] with d ≥ 1. Set Y = Spec(B) and Y_0 = V(x_1, …, x_d). For any open subscheme V ⊂ Y with V_0 = V ∩ Y_0 equal to Y_0 setminus ( m_B) the restriction functor FÉt_V → FÉt_V_0 is fully faithful.","statement_latex":"Let $(A, \\mathfrak m)$ be a Noetherian local ring of depth $\\geq 2$.\nLet $B = A[[x_1, \\ldots, x_d]]$ with $d \\geq 1$.\nSet $Y = \\Spec(B)$ and $Y_0 = V(x_1, \\ldots, x_d)$.\nFor any open subscheme $V \\subset Y$ with\n$V_0 = V \\cap Y_0$ equal to $Y_0 \\setminus \\{\\mathfrak m_B\\}$\nthe restriction functor\n$$\n\\textit{F\\'Et}_V \\longrightarrow \\textit{F\\'Et}_{V_0}\n$$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EY9","source_file":"pione.tex","source_line":5758,"source_end_line":5770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5758-L5770","statement_sha256":"f14fc41bf145fee37d52e3ff00ba56bd2dfcf53767277bff8b325b3839244fd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9789,"rank":9789,"depth":53,"x":1536.932,"y":1189.381,"cluster":"tale-geometry"},{"id":"stacks:0EYA","tag":"0EYA","title":"Purity in local case, III · Lemma 0EYA","summary":"Ramanujam-Samuel for finite étale covers Let (A, m) be a Noetherian local ring of depth ≥ 2. Let B = A[[x_1, …, x_d]] with d ≥ 1. For any open V ⊂ Y = Spec(B) which contains • any prime q ⊂ B such that q ∩ A not = m, • the prime m B the functor FÉt_Y → FÉt_V is an equivalence. In particular purity holds for B.","statement_latex":"\\begin{slogan}\nRamanujam-Samuel for finite \\'etale covers\n\\end{slogan}\nLet $(A, \\mathfrak m)$ be a Noetherian local ring of depth $\\geq 2$. Let\n$B = A[[x_1, \\ldots, x_d]]$ with $d \\geq 1$. For any open\n$V \\subset Y = \\Spec(B)$ which contains\n\\begin{enumerate}\n\\item any prime $\\mathfrak q \\subset B$ such that\n$\\mathfrak q \\cap A \\not = \\mathfrak m$,\n\\item the prime $\\mathfrak m B$\n\\end{enumerate}\nthe functor\n$\n\\textit{F\\'Et}_Y\n\\to\n\\textit{F\\'Et}_V\n$\nis an equivalence. In particular purity holds for $B$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYA","source_file":"pione.tex","source_line":5806,"source_end_line":5826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5806-L5826","statement_sha256":"ab6be2311809da9cb343aad719e067792346e8cbf6a9466ce0b64e4608b12716","origin":"The Stacks Project","memory_eligible":false,"source_rank":9790,"rank":9790,"depth":55,"x":1572.022,"y":1078.959,"cluster":"tale-geometry"},{"id":"stacks:0EYB","tag":"0EYB","title":"Purity in local case, III · Lemma 0EYB","summary":"Let f : X → S be a morphism of schemes. Let U ⊂ X be an open subscheme. Assume • f is smooth, • S is Noetherian, • for s ∈ S with depth(O_S, s) ≤ 1 we have X_s = U_s, • U_s ⊂ X_s is dense for all s ∈ S. Then FÉt_X → FÉt_U is an equivalence.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $U \\subset X$\nbe an open subscheme. Assume\n\\begin{enumerate}\n\\item $f$ is smooth,\n\\item $S$ is Noetherian,\n\\item for $s \\in S$ with $\\text{depth}(\\mathcal{O}_{S, s}) \\leq 1$\nwe have $X_s = U_s$,\n\\item $U_s \\subset X_s$ is dense for all $s \\in S$.\n\\end{enumerate}\nThen $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_U$ is an equivalence.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, III","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYB","source_file":"pione.tex","source_line":5883,"source_end_line":5895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5883-L5895","statement_sha256":"67fb6efba4e429d110258aceeca66b02d1e745739f56c64e3b479f330fc22507","origin":"The Stacks Project","memory_eligible":false,"source_rank":9791,"rank":9791,"depth":56,"x":1635.142,"y":1180.61,"cluster":"tale-geometry"},{"id":"stacks:0EYC","tag":"0EYC","title":"Purity in local case, III · Proposition 0EYC","summary":"Let A → B be a local homomorphism of local Noetherian rings. Assume A has depth ≥ 2, A → B is formally smooth for the m_B-adic topology, and dim(B) > dim(A). For any open V ⊂ Y = Spec(B) which contains • any prime q ⊂ B such that q ∩ A not = m_A, • the prime m_A B the functor FÉt_Y → FÉt_V is an equivalence. In particular purity holds for B.","statement_latex":"Let $A \\to B$ be a local homomorphism of local Noetherian rings.\nAssume $A$ has depth $\\geq 2$, $A \\to B$ is formally smooth for the\n$\\mathfrak m_B$-adic topology, and $\\dim(B) > \\dim(A)$. For any open\n$V \\subset Y = \\Spec(B)$ which contains\n\\begin{enumerate}\n\\item any prime $\\mathfrak q \\subset B$ such that\n$\\mathfrak q \\cap A \\not = \\mathfrak m_A$,\n\\item the prime $\\mathfrak m_A B$\n\\end{enumerate}\nthe functor $\\textit{F\\'Et}_Y \\to \\textit{F\\'Et}_V$\nis an equivalence. In particular purity holds for $B$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity in local case, III","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYC","source_file":"pione.tex","source_line":5959,"source_end_line":5972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L5959-L5972","statement_sha256":"af45507f25c74fe8126c7070c6604544ec462afdbb94f7edc43811c3d897d2e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9792,"rank":9792,"depth":57,"x":1506.456,"y":1141.348,"cluster":"tale-geometry"},{"id":"stacks:0ELC","tag":"0ELC","title":"Lefschetz for the fundamental group · Proposition 0ELC","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module. Let s ∈ Γ(X, L). Let Y = Z(s) be the zero scheme of s. Assume that for all x ∈ X setminus Y we have depth(O_X, x) + dim(overline(x)) > 1 Then the restriction functor FÉt_X → FÉt_Y is fully faithful. In fact, for any open subscheme V ⊂ X containing Y the restriction functor FÉt_V → FÉt_Y is fully faithful.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\nLet $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the\nzero scheme of $s$. Assume that for all $x \\in X \\setminus Y$\nwe have\n$$\n\\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 1\n$$\nThen the restriction functor $\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$\nis fully faithful. In fact, for any open subscheme $V \\subset X$\ncontaining $Y$ the restriction functor\n$\\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y$\nis fully faithful.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Lefschetz for the fundamental group","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ELC","source_file":"pione.tex","source_line":6047,"source_end_line":6062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6047-L6062","statement_sha256":"d3b318d0e974d8c0e034e586e230b202229f3e24953e9fefaab2910b51c916dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9793,"rank":9793,"depth":54,"x":1633.309,"y":1097.144,"cluster":"tale-geometry"},{"id":"stacks:0ELD","tag":"0ELD","title":"Lefschetz for the fundamental group · Proposition 0ELD","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module. Let s ∈ Γ(X, L). Let Y = Z(s) be the zero scheme of s. Let V be the set of open subschemes of X containing Y ordered by reverse inclusion. Assume that for all x ∈ X setminus Y we have depth(O_X, x) + dim(overline(x)) > 2 Then the restriction functor colim_V FÉt_V → FÉt_Y is an equivalence.","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\nLet $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the\nzero scheme of $s$. Let $\\mathcal{V}$ be the set of open\nsubschemes of $X$ containing $Y$ ordered by reverse inclusion.\nAssume that for all $x \\in X \\setminus Y$ we have\n$$\n\\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2\n$$\nThen the restriction functor\n$$\n\\colim_\\mathcal{V} \\textit{F\\'Et}_V \\to \\textit{F\\'Et}_Y\n$$\nis an equivalence.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Lefschetz for the fundamental group","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ELD","source_file":"pione.tex","source_line":6077,"source_end_line":6093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6077-L6093","statement_sha256":"8f41cb40a141f7b354231e2a1386c92f87f4003136cd3db1658874be67a33a2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9794,"rank":9794,"depth":56,"x":1575.139,"y":1202.039,"cluster":"tale-geometry"},{"id":"stacks:0ELE","tag":"0ELE","title":"Lefschetz for the fundamental group · Proposition 0ELE","summary":"Let k be a field. Let X be a proper scheme over k. Let L be an ample invertible O_X-module. Let s ∈ Γ(X, L). Let Y = Z(s) be the zero scheme of s. Assume that for all x ∈ X setminus Y we have depth(O_X, x) + dim(overline(x)) > 2 and that for x ∈ X setminus Y closed purity holds for O_X, x. Then the restriction functor FÉt_X → FÉt_Y is an equivalence. If X or equivalently Y is connected, then π_1(Y, overliney) → π_1(X, overliney) is an isomorphism for any geometric point…","statement_latex":"Let $k$ be a field. Let $X$ be a proper scheme over $k$.\nLet $\\mathcal{L}$ be an ample invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$. Let $Y = Z(s)$ be the\nzero scheme of $s$.\nAssume that for all $x \\in X \\setminus Y$ we have\n$$\n\\text{depth}(\\mathcal{O}_{X, x}) + \\dim(\\overline{\\{x\\}}) > 2\n$$\nand that for $x \\in X \\setminus Y$ closed purity holds for\n$\\mathcal{O}_{X, x}$. Then the restriction functor\n$\\textit{F\\'Et}_X \\to \\textit{F\\'Et}_Y$\nis an equivalence. If $X$ or equivalently $Y$ is connected, then\n$$\n\\pi_1(Y, \\overline{y}) \\to \\pi_1(X, \\overline{y})\n$$\nis an isomorphism for any geometric point $\\overline{y}$ of $Y$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Lefschetz for the fundamental group","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ELE","source_file":"pione.tex","source_line":6102,"source_end_line":6120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6102-L6120","statement_sha256":"fd58d9b20f1cf8c11154476ff04c0a02489a1f39c85b3670d940c6e06b50447f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9795,"rank":9795,"depth":57,"x":1533.554,"y":1091.348,"cluster":"tale-geometry"},{"id":"stacks:0EA2","tag":"0EA2","title":"Purity of ramification locus · Lemma 0EA2","summary":"Let A be a Noetherian normal local domain of dimension 2. Assume A is Nagata, has a dualizing module ω_A, and has a resolution of singularities f : X → Spec(A). Let ω_X be as in Resolution of Surfaces, Remark [Tag 0B4R]. If ω_X ≅ O_X(E) for some effective Cartier divisor E ⊂ X supported on the exceptional fibre, then A defines a rational singularity. If f is a minimal resolution, then E = 0.","statement_latex":"Let $A$ be a Noetherian normal local domain of dimension $2$.\nAssume $A$ is Nagata, has a dualizing module $\\omega_A$, and has a\nresolution of singularities $f : X \\to \\Spec(A)$.\nLet $\\omega_X$ be as in Resolution of Surfaces,\nRemark \\ref{resolve-remark-dualizing-setup}.\nIf $\\omega_X \\cong \\mathcal{O}_X(E)$ for some effective\nCartier divisor $E \\subset X$ supported on the exceptional\nfibre, then $A$ defines a rational singularity.\nIf $f$ is a minimal resolution, then $E = 0$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity of ramification locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EA2","source_file":"pione.tex","source_line":6167,"source_end_line":6178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6167-L6178","statement_sha256":"2d0fdfb0e72f181e417c05ab19361dcaaaebfae16aa36c66d1105f9e92257822","origin":"The Stacks Project","memory_eligible":false,"source_rank":9796,"rank":9796,"depth":68,"x":1653.594,"y":1149.549,"cluster":"tale-geometry"},{"id":"stacks:0EA3","tag":"0EA3","title":"Purity of ramification locus · Lemma 0EA3","summary":"Let f : X → Spec(A) be a finite type morphism. Let x ∈ X be a point. Assume • A is an excellent regular local ring, • O_X, x is normal of dimension 2, • f is étale outside of overline(x). Then f is étale at x.","statement_latex":"Let $f : X \\to \\Spec(A)$ be a finite type morphism.\nLet $x \\in X$ be a point. Assume\n\\begin{enumerate}\n\\item $A$ is an excellent regular local ring,\n\\item $\\mathcal{O}_{X, x}$ is normal of dimension $2$,\n\\item $f$ is \\'etale outside of $\\overline{\\{x\\}}$.\n\\end{enumerate}\nThen $f$ is \\'etale at $x$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity of ramification locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EA3","source_file":"pione.tex","source_line":6219,"source_end_line":6229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6219-L6229","statement_sha256":"33f098df1ebbde1f23813be15bc16499f1870b314ad4697466ed03b5251fec24","origin":"The Stacks Project","memory_eligible":false,"source_rank":9797,"rank":9797,"depth":72,"x":1517.868,"y":1174.822,"cluster":"tale-geometry"},{"id":"stacks:0EA4","tag":"0EA4","title":"Purity of ramification locus · Lemma 0EA4","summary":"This result for complex spaces can be found on page 170 of [Fischer]. In general this is [Zong] attributed to Gabber. Let f : X → Y be a morphism of locally Noetherian schemes. Let x ∈ X and set y = f(x). Assume • O_X, x is normal of dimension ≥ 1, • O_Y, y is regular, • f is locally of finite type, and • for specializations x' leadsto x with dim(O_X, x') = 1 our f is étale at x'. Then f is étale at x.","statement_latex":"\\begin{reference}\nThis result for complex spaces can be found on page 170 of \\cite{Fischer}.\nIn general this is \\cite[Theorem 2.4]{Zong} attributed to Gabber.\n\\end{reference}\nLet $f : X \\to Y$ be a morphism of locally Noetherian schemes.\nLet $x \\in X$ and set $y = f(x)$. Assume\n\\begin{enumerate}\n\\item $\\mathcal{O}_{X, x}$ is normal of dimension $\\geq 1$,\n\\item $\\mathcal{O}_{Y, y}$ is regular,\n\\item $f$ is locally of finite type, and\n\\item for specializations $x' \\leadsto x$ with\n$\\dim(\\mathcal{O}_{X, x'}) = 1$ our $f$ is \\'etale at $x'$.\n\\end{enumerate}\nThen $f$ is \\'etale at $x$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Purity of ramification locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EA4","source_file":"pione.tex","source_line":6307,"source_end_line":6323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6307-L6323","statement_sha256":"dc175c99d208b804c3560eb8362d1bea6f65ccdfc03cc8a1e3532d0ed808a04f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9798,"rank":9798,"depth":73,"x":1597.866,"y":1078.886,"cluster":"tale-geometry"},{"id":"stacks:0ECB","tag":"0ECB","title":"Affineness of complement of ramification locus · Lemma 0ECB","summary":"Let (A, m) be a regular local ring which contains a field. Let f : V → Spec(A) be étale and quasi-compact. Assume that m not ∈ f(V) and assume that g : V → Spec(A) setminus ( m) is affine. Then H^i(V, O_V), i > 0 is isomorphic to a direct sum of copies of the injective hull of the residue field of A.","statement_latex":"Let $(A, \\mathfrak m)$ be a regular local ring which contains a field.\nLet $f : V \\to \\Spec(A)$ be \\'etale and quasi-compact.\nAssume that $\\mathfrak m \\not \\in f(V)$ and assume that\n$g : V \\to \\Spec(A) \\setminus \\{\\mathfrak m\\}$ is affine.\nThen $H^i(V, \\mathcal{O}_V)$, $i > 0$ is isomorphic to a direct\nsum of copies of the injective hull of the residue field of $A$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Affineness of complement of ramification locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECB","source_file":"pione.tex","source_line":6408,"source_end_line":6416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6408-L6416","statement_sha256":"233070e418d94906630bcff9f961df6ecf7a12ea13d7d5f24c6709bbde9da2fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9799,"rank":9799,"depth":45,"x":1616.078,"y":1195.365,"cluster":"tale-geometry"},{"id":"stacks:0ECC","tag":"0ECC","title":"Affineness of complement of ramification locus · Lemma 0ECC","summary":"In the situation of Lemma [Tag 0ECB] assume that H^i(V, O_V) = 0 for i ≥ dim(A) - 1. Then V is affine.","statement_latex":"In the situation of Lemma \\ref{lemma-structure-cohomology}\nassume that $H^i(V, \\mathcal{O}_V) = 0$ for $i \\geq \\dim(A) - 1$.\nThen $V$ is affine.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Affineness of complement of ramification locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECC","source_file":"pione.tex","source_line":6474,"source_end_line":6479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6474-L6479","statement_sha256":"0549da715621ae08cc337bd358ced8393aae485474e4acb12f4ad570d072a2b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9800,"rank":9800,"depth":46,"x":1508.664,"y":1119.587,"cluster":"tale-geometry"},{"id":"stacks:0ECD","tag":"0ECD","title":"Affineness of complement of ramification locus · Theorem 0ECD","summary":"Let Y be an excellent regular scheme over a field. Let f : X → Y be a finite type morphism of schemes with X normal. Let V ⊂ X be the maximal open subscheme where f is étale. Then the inclusion morphism V → X is affine.","statement_latex":"Let $Y$ be an excellent regular scheme over a field. Let $f : X \\to Y$\nbe a finite type morphism of schemes with $X$ normal. Let $V \\subset X$\nbe the maximal open subscheme where $f$ is \\'etale. Then the inclusion\nmorphism $V \\to X$ is affine.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Affineness of complement of ramification locus","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECD","source_file":"pione.tex","source_line":6512,"source_end_line":6518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6512-L6518","statement_sha256":"1d14f2abfef04dbdb72aa535b20d6b8592a3085c565b34d5f170468a805861e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9801,"rank":9801,"depth":74,"x":1649.22,"y":1114.5,"cluster":"tale-geometry"},{"id":"stacks:0C0P","tag":"0C0P","title":"Specialization maps in the smooth proper case · Lemma 0C0P","summary":"Let f : X → S be a flat proper morphism with geometrically connected fibres. Let s' leadsto s be a specialization. If X_s is geometrically reduced, then the specialization map sp : π_1(X_overlines') → π_1(X_overlines) is surjective.","statement_latex":"Let $f : X \\to S$ be a flat proper morphism with geometrically\nconnected fibres. Let $s' \\leadsto s$ be a specialization.\nIf $X_s$ is geometrically reduced, then the specialization\nmap $sp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})$\nis surjective.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Specialization maps in the smooth proper case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0P","source_file":"pione.tex","source_line":6597,"source_end_line":6604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6597-L6604","statement_sha256":"b27b89b1b4e54461268a43bb64995f1de5388506a922ae4c36032c45bd6f1aa9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9802,"rank":9802,"depth":50,"x":1549.376,"y":1198.249,"cluster":"tale-geometry"},{"id":"stacks:0C0Q","tag":"0C0Q","title":"Specialization maps in the smooth proper case · Proposition 0C0Q","summary":"Let f : X → S be a smooth proper morphism with geometrically connected fibres. Let s' leadsto s be a specialization. If the characteristic to kappa(s) is zero, then the specialization map sp : π_1(X_overlines') → π_1(X_overlines) is an isomorphism.","statement_latex":"Let $f : X \\to S$ be a smooth proper morphism with geometrically\nconnected fibres. Let $s' \\leadsto s$ be a specialization.\nIf the characteristic to $\\kappa(s)$ is zero, then the specialization\nmap\n$$\nsp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})\n$$\nis an isomorphism.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Specialization maps in the smooth proper case","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0Q","source_file":"pione.tex","source_line":6622,"source_end_line":6632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6622-L6632","statement_sha256":"d7d9c2ac1df7e7c79b36b623abb9ad0ef8aba4a0993df14131980070b0e92f0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9803,"rank":9803,"depth":58,"x":1555.67,"y":1079.497,"cluster":"tale-geometry"},{"id":"stacks:0C0R","tag":"0C0R","title":"Specialization maps in the smooth proper case · Theorem 0C0R","summary":"Let f : X → S be a smooth proper morphism with geometrically connected fibres. Let s' leadsto s be a specialization. If the characteristic of kappa(s) is p, then the specialization map sp : π_1(X_overlines') → π_1(X_overlines) is surjective and induces an isomorphism π'_1(X_overlines') ≅ π'_1(X_overlines) of the maximal prime-to-p quotients","statement_latex":"Let $f : X \\to S$ be a smooth proper morphism with geometrically\nconnected fibres. Let $s' \\leadsto s$ be a specialization.\nIf the characteristic of $\\kappa(s)$ is $p$, then the specialization\nmap\n$$\nsp : \\pi_1(X_{\\overline{s}'}) \\to \\pi_1(X_{\\overline{s}})\n$$\nis surjective and induces an isomorphism\n$$\n\\pi'_1(X_{\\overline{s}'}) \\cong \\pi'_1(X_{\\overline{s}})\n$$\nof the maximal prime-to-p quotients","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Specialization maps in the smooth proper case","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0C0R","source_file":"pione.tex","source_line":6742,"source_end_line":6756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6742-L6756","statement_sha256":"46088414b98905979e214a56b22afc05c07559ec3a60f198d26c8f6d66ac25d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9804,"rank":9804,"depth":59,"x":1646.787,"y":1170.897,"cluster":"tale-geometry"},{"id":"stacks:0EYD","tag":"0EYD","title":"Tame ramification · Lemma 0EYD","summary":"Let X' → X be a morphism of locally Noetherian schemes. Let U ⊂ X be a dense open. Assume • U' = f^-1(U) is dense open in X', • for every prime divisor Z ⊂ X with Z ∩ U = ∅ the local ring O_X, xi of X at the generic point xi of Z is a discrete valuation ring, • for every prime divisor Z' ⊂ X' with Z' ∩ U' = ∅ the local ring O_X', xi' of X' at the generic point xi' of Z' is a discrete valuation ring, • if xi' ∈ X' is as in (3), then xi = f(xi') is as in (2). Then if f : Y…","statement_latex":"Let $X' \\to X$ be a morphism of locally Noetherian schemes.\nLet $U \\subset X$ be a dense open. Assume\n\\begin{enumerate}\n\\item $U' = f^{-1}(U)$ is dense open in $X'$,\n\\item for every prime divisor $Z \\subset X$ with $Z \\cap U = \\emptyset$\nthe local ring $\\mathcal{O}_{X, \\xi}$ of $X$ at the generic point $\\xi$\nof $Z$ is a discrete valuation ring,\n\\item for every prime divisor $Z' \\subset X'$\nwith $Z' \\cap U' = \\emptyset$ the local ring $\\mathcal{O}_{X', \\xi'}$\nof $X'$ at the generic point $\\xi'$ of $Z'$ is a discrete valuation ring,\n\\item if $\\xi' \\in X'$ is as in (3), then $\\xi = f(\\xi')$ is as in (2).\n\\end{enumerate}\nThen if $f : Y \\to U$ is finite \\'etale and\n$Y$ is unramified, resp.\\ tamely ramified over $X$\nin codimension $1$, then $Y' = Y \\times_X X' \\to U'$ is finite \\'etale\nand $Y'$ is unramified, resp.\\ tamely ramified over $X'$ in codimension $1$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYD","source_file":"pione.tex","source_line":6840,"source_end_line":6858,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6840-L6858","statement_sha256":"62dcf4e060507e6e4572e9d278b9b66c6fba08974ca91c717bf343dfcabbe167","origin":"The Stacks Project","memory_eligible":false,"source_rank":9805,"rank":9805,"depth":51,"x":1505.694,"y":1155.157,"cluster":"tale-geometry"},{"id":"stacks:0H2W","tag":"0H2W","title":"Tame ramification · Lemma 0H2W","summary":"Let X be a locally Noetherian scheme. Let U ⊂ X be open and dense. Let Y → U be a finite étale morphism. Assume • Y is unramified over X in codimension 1, and • O_X, x is regular for all x ∈ X setminus U. Then there exists a finite étale morphism Y' → X whose restriction to U is Y.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $U \\subset X$ be open and dense.\nLet $Y \\to U$ be a finite \\'etale morphism. Assume\n\\begin{enumerate}\n\\item $Y$ is unramified over $X$ in codimension $1$, and\n\\item $\\mathcal{O}_{X, x}$ is regular for all $x \\in X \\setminus U$.\n\\end{enumerate}\nThen there exists a finite \\'etale morphism $Y' \\to X$\nwhose restriction to $U$ is $Y$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2W","source_file":"pione.tex","source_line":6869,"source_end_line":6879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6869-L6879","statement_sha256":"7cae07b4d6672fca572916ea853e0fb343811aea6f4be8ae05de1e70004e033d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9806,"rank":9806,"depth":57,"x":1622.724,"y":1086.509,"cluster":"tale-geometry"},{"id":"stacks:0EYE","tag":"0EYE","title":"Tame ramification · Lemma 0EYE","summary":"Let X be a locally Noetherian scheme. Let D ⊂ X be an effective Cartier divisor such that D is a regular scheme. Let Y → X setminus D be a finite étale morphism. If Y is unramified over X in codimension 1, then there exists a finite étale morphism Y' → X whose restriction to X setminus D is Y.","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $D \\subset X$\nbe an effective Cartier divisor such that $D$ is a regular scheme.\nLet $Y \\to X \\setminus D$ be a finite \\'etale morphism.\nIf $Y$ is unramified over $X$ in codimension $1$, then\nthere exists a finite \\'etale morphism $Y' \\to X$\nwhose restriction to $X \\setminus D$ is $Y$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYE","source_file":"pione.tex","source_line":6916,"source_end_line":6924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6916-L6924","statement_sha256":"43f0c30fca31e3ade3c4f7a2b2639a303cb7fa6ea570e7ac5591f8a239133ef6","origin":"The Stacks Project","memory_eligible":false,"source_rank":9807,"rank":9807,"depth":58,"x":1591.544,"y":1203.866,"cluster":"tale-geometry"},{"id":"stacks:0EYG","tag":"0EYG","title":"Abhyankar's lemma for regular divisor · Lemma 0EYG","summary":"Let X be a locally Noetherian scheme. Let D ⊂ X be an effective Cartier divisor such that D is a regular scheme. Let Y → X setminus D be a finite étale morphism. If Y is tamely ramified over X in codimension 1, then étale locally on X the morphism Y → X is as given as a finite disjoint union of standard tamely ramified morphisms as described in Example [Tag 0EYF].","statement_latex":"Let $X$ be a locally Noetherian scheme. Let $D \\subset X$\nbe an effective Cartier divisor such that $D$ is a regular scheme.\nLet $Y \\to X \\setminus D$ be a finite \\'etale morphism.\nIf $Y$ is tamely ramified over $X$ in codimension $1$, then\n\\'etale locally on $X$ the morphism $Y \\to X$ is as given\nas a finite disjoint union of standard tamely ramified\nmorphisms as described in Example \\ref{example-tamely-ramified}.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYG","source_file":"pione.tex","source_line":6960,"source_end_line":6969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L6960-L6969","statement_sha256":"a7d9351376694a92b5c627587f7538c617e5e8b4a6cb89330f74dd5a06382300","origin":"The Stacks Project","memory_eligible":false,"source_rank":9808,"rank":9808,"depth":59,"x":1519.961,"y":1099.343,"cluster":"tale-geometry"},{"id":"stacks:0EYH","tag":"0EYH","title":"Tame ramification · Lemma 0EYH","summary":"In the situation of Lemma [Tag 0EYG] the normalization of X in Y is a finite locally free morphism π : Y' → X such that • the restriction of Y' to X setminus D is isomorphic to Y, • D' = π^-1(D)_red is an effective Cartier divisor on Y', and • D' is a regular scheme. Moreover, étale locally on X the morphism Y' → X is a finite disjoint union of morphisms Spec(A[x]/(x^e - f)) → Spec(A) where A is a Noetherian ring, f ∈ A is a nonzerodivisor with A/fA regular, and e ≥ 1 is…","statement_latex":"In the situation of Lemma \\ref{lemma-abhyankar-one-divisor}\nthe normalization of $X$ in $Y$ is a finite locally free morphism\n$\\pi : Y' \\to X$ such that\n\\begin{enumerate}\n\\item the restriction of $Y'$ to $X \\setminus D$ is isomorphic to $Y$,\n\\item $D' = \\pi^{-1}(D)_{red}$ is an effective Cartier divisor on $Y'$, and\n\\item $D'$ is a regular scheme.\n\\end{enumerate}\nMoreover, \\'etale locally on $X$ the morphism $Y' \\to X$ is a finite disjoint\nunion of morphisms\n$$\n\\Spec(A[x]/(x^e - f)) \\to \\Spec(A)\n$$\nwhere $A$ is a Noetherian ring, $f \\in A$ is a nonzerodivisor with\n$A/fA$ regular, and $e \\geq 1$ is invertible in $A$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYH","source_file":"pione.tex","source_line":7086,"source_end_line":7103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L7086-L7103","statement_sha256":"8edfae66609b637ec78bcce748ef29ccb62199c2c376912ce2204997ec4a9946","origin":"The Stacks Project","memory_eligible":false,"source_rank":9809,"rank":9809,"depth":60,"x":1657.182,"y":1135.901,"cluster":"tale-geometry"},{"id":"stacks:0EYI","tag":"0EYI","title":"Tame ramification · Lemma 0EYI","summary":"In the situation of Lemma [Tag 0EYG] let Y' → X be as in Lemma [Tag 0EYH]. Let R be a discrete valuation ring with fraction field K. Let t : Spec(R) → X be a morphism such that the scheme theoretic inverse image t^-1D is the reduced closed point of Spec(R). • If t|_Spec(K) lifts to a point of Y, then we get a lift t' : Spec(R) → Y' such that Y' → X is étale along t'(Spec(R)). • If Spec(K) ×_X Y is isomorphic to a disjoint union of copies of Spec(K), then Y' → X is finite…","statement_latex":"In the situation of Lemma \\ref{lemma-abhyankar-one-divisor}\nlet $Y' \\to X$ be as in Lemma \\ref{lemma-extend-tame-covering-normal}.\nLet $R$ be a discrete valuation ring with fraction field $K$.\nLet\n$$\nt : \\Spec(R) \\to X\n$$\nbe a morphism such that the scheme theoretic inverse image\n$t^{-1}D$ is the reduced closed point of $\\Spec(R)$.\n\\begin{enumerate}\n\\item If $t|_{\\Spec(K)}$ lifts to a point of $Y$, then\nwe get a lift $t' : \\Spec(R) \\to Y'$ such that $Y' \\to X$\nis \\'etale along $t'(\\Spec(R))$.\n\\item If $\\Spec(K) \\times_X Y$ is isomorphic to a disjoint union\nof copies of $\\Spec(K)$, then $Y' \\to X$ is finite \\'etale\nover an open neighbourhood of $t(\\Spec(R))$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYI","source_file":"pione.tex","source_line":7126,"source_end_line":7145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L7126-L7145","statement_sha256":"63d537f0537202c253e9be21f5e6d3c03faa4c356698df98f08ddcf5e3467201","origin":"The Stacks Project","memory_eligible":false,"source_rank":9810,"rank":9810,"depth":61,"x":1526.235,"y":1186.948,"cluster":"tale-geometry"},{"id":"stacks:0EYJ","tag":"0EYJ","title":"Tame ramification · Lemma 0EYJ","summary":"Let S be an integral normal Noetherian scheme with generic point eta. Let f : X → S be a smooth morphism with geometrically connected fibres. Let σ : S → X be a section of f. Let Z → X_eta be a finite étale Galois cover (Section [Tag 03SF]) with group G of order invertible on S such that Z has a kappa(eta)-rational point mapping to σ(eta). Then there exists a finite étale Galois cover Y → X with group G whose restriction to X_eta is Z.","statement_latex":"Let $S$ be an integral normal Noetherian scheme with generic point $\\eta$.\nLet $f : X \\to S$ be a smooth morphism with geometrically connected fibres.\nLet $\\sigma : S \\to X$ be a section of $f$. Let $Z \\to X_\\eta$ be a\nfinite \\'etale Galois cover (Section \\ref{section-finite-etale-under-galois})\nwith group $G$ of order invertible on $S$ such that\n$Z$ has a $\\kappa(\\eta)$-rational point mapping to $\\sigma(\\eta)$.\nThen there exists a finite \\'etale Galois cover $Y \\to X$ with group $G$\nwhose restriction to $X_\\eta$ is $Z$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYJ","source_file":"pione.tex","source_line":7197,"source_end_line":7207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L7197-L7207","statement_sha256":"aff26f51cf0367efd89cee0c04653a9a767e17fb179af19745c40dfa0a5891ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":9811,"rank":9811,"depth":62,"x":1581.897,"y":1074.694,"cluster":"tale-geometry"},{"id":"stacks:0EZJ","tag":"0EZJ","title":"Tame ramification · Lemma 0EZJ","summary":"Let S be a quasi-compact and quasi-separated integral normal scheme with generic point eta. Let f : X → S be a quasi-compact and quasi-separated smooth morphism with geometrically connected fibres. Let σ : S → X be a section of f. Let Z → X_eta be a finite étale Galois cover (Section [Tag 03SF]) with group G of order invertible on S such that Z has a kappa(eta)-rational point mapping to σ(eta). Then there exists a finite étale Galois cover Y → X with group G whose…","statement_latex":"Let $S$ be a quasi-compact and quasi-separated integral normal scheme\nwith generic point $\\eta$. Let $f : X \\to S$ be a quasi-compact and\nquasi-separated smooth morphism with geometrically connected fibres.\nLet $\\sigma : S \\to X$ be a section of $f$. Let $Z \\to X_\\eta$ be a\nfinite \\'etale Galois cover (Section \\ref{section-finite-etale-under-galois})\nwith group $G$ of order invertible on $S$ such that\n$Z$ has a $\\kappa(\\eta)$-rational point mapping to $\\sigma(\\eta)$.\nThen there exists a finite \\'etale Galois cover $Y \\to X$ with group $G$\nwhose restriction to $X_\\eta$ is $Z$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tame ramification","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZJ","source_file":"pione.tex","source_line":7333,"source_end_line":7344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L7333-L7344","statement_sha256":"1555404314e5273151a50a347af6b540e83a9071fa614116483cd5c922032a35","origin":"The Stacks Project","memory_eligible":false,"source_rank":9812,"rank":9812,"depth":63,"x":1631.259,"y":1189.367,"cluster":"tale-geometry"},{"id":"stacks:0G1F","tag":"0G1F","title":"Tricks in positive characteristic · Lemma 0G1F","summary":"In the situation above, if NF(C, φ, τ) > 0, then there exist an étale k-algebra map φ' and a surjective k-algebra map τ' fitting into the commutative diagram xymatrix & B C ar[r] & C/φ'(I)C ar[u]_τ' k[x_1, …, x_n] ar[u]^φ' ar[r] & A ar[u] ar@/_3em/[uu]_π with NF(C, φ', τ') < NF(C, φ, τ).","statement_latex":"In the situation above, if $NF(C, \\varphi, \\tau) > 0$, then there exist\nan \\'etale $k$-algebra map $\\varphi'$ and a surjective $k$-algebra map\n$\\tau'$ fitting into the commutative diagram\n$$\n\\xymatrix{\n& B \\\\\nC \\ar[r] & C/\\varphi'(I)C \\ar[u]_{\\tau'} \\\\\nk[x_1, \\ldots, x_n] \\ar[u]^{\\varphi'} \\ar[r] &\nA \\ar[u] \\ar@/_3em/[uu]_\\pi\n}\n$$\nwith $NF(C, \\varphi', \\tau') < NF(C, \\varphi, \\tau)$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tricks in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1F","source_file":"pione.tex","source_line":7473,"source_end_line":7487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L7473-L7487","statement_sha256":"3cae6905a7163a79a71d067632e1a609298719627ce4644cd39ce6c935f1acca","origin":"The Stacks Project","memory_eligible":false,"source_rank":9813,"rank":9813,"depth":43,"x":1502.291,"y":1132.663,"cluster":"tale-geometry"},{"id":"stacks:0G1G","tag":"0G1G","title":"Tricks in positive characteristic · Lemma 0G1G","summary":"Let k be a field of characteristic p > 0. Let X → A^n_k be an étale morphism with X affine. Then there exists a finite étale morphism X → A^n_k.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $X \\to \\mathbf{A}^n_k$ be an\n\\'etale morphism with $X$ affine. Then there exists a finite \\'etale\nmorphism $X \\to \\mathbf{A}^n_k$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tricks in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1G","source_file":"pione.tex","source_line":7567,"source_end_line":7572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L7567-L7572","statement_sha256":"bb43084c28db0d3fd695254f6ee6adcf406479b8c56950f596c096c9c625f93b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9814,"rank":9814,"depth":44,"x":1643.373,"y":1101.211,"cluster":"tale-geometry"},{"id":"stacks:0G1H","tag":"0G1H","title":"Tricks in positive characteristic · Lemma 0G1H","summary":"Let k be a field of characteristic p > 0. Let Z ⊂ A^n_k be a closed subscheme. Let Y → Z be finite étale. There exists a finite étale morphism f : U → A^n_k such that there is an open and closed immersion Y → f^-1(Z) over Z.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $Z \\subset \\mathbf{A}^n_k$\nbe a closed subscheme. Let $Y \\to Z$ be finite \\'etale. There exists a\nfinite \\'etale morphism $f : U \\to \\mathbf{A}^n_k$ such that\nthere is an open and closed immersion $Y \\to f^{-1}(Z)$ over $Z$.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tricks in positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1H","source_file":"pione.tex","source_line":7583,"source_end_line":7589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L7583-L7589","statement_sha256":"a894c2dec839a2e02fe0b55761a8a7a306f80993c854a153fef8333b92cf2a33","origin":"The Stacks Project","memory_eligible":false,"source_rank":9815,"rank":9815,"depth":44,"x":1564.421,"y":1204.736,"cluster":"tale-geometry"},{"id":"stacks:0G1I","tag":"0G1I","title":"Tricks in positive characteristic · Proposition 0G1I","summary":"Let p be a prime number. Let i : Z → X be a closed immersion of connected affine schemes over F_p. For any geometric point overlinez of Z the map π_1(Z, overlinez) → π_1(X, overlinez) is injective.","statement_latex":"Let $p$ be a prime number. Let $i : Z \\to X$ be a closed immersion\nof connected affine schemes over $\\mathbf{F}_p$. For any geometric\npoint $\\overline{z}$ of $Z$ the map\n$$\n\\pi_1(Z, \\overline{z}) \\to \\pi_1(X, \\overline{z})\n$$\nis injective.","area":"Étale Geometry","chapter":"Fundamental Groups of Schemes","chapter_id":"pione","section":"Tricks in positive characteristic","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G1I","source_file":"pione.tex","source_line":7623,"source_end_line":7632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/pione.tex#L7623-L7632","statement_sha256":"75069c1b698dda7f8b49ea2b287fac753560f545d91f6f5860cff9e2cbd4b81d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9816,"rank":9816,"depth":45,"x":1539.32,"y":1083.273,"cluster":"tale-geometry"},{"id":"stacks:03N5","tag":"03N5","title":"The étale topology · Definition 03N5","summary":"A family of morphisms ( φ_i : U_i → X)_i ∈ I is called an étale covering if each φ_i is an étale morphism and their images cover X, i.e., X = ⋃_i ∈ I φ_i(U_i).","statement_latex":"A family of morphisms $\\{ \\varphi_i : U_i \\to X\\}_{i \\in I}$ is\ncalled an {\\it \\'etale covering} if each $\\varphi_i$ is an \\'etale morphism\nand their images cover $X$, i.e.,\n$X = \\bigcup_{i \\in I} \\varphi_i(U_i)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03N5","source_file":"etale-cohomology.tex","source_line":149,"source_end_line":155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L149-L155","statement_sha256":"8c9e64617794b89f0f28b78b26379f00f58c28692e7e7cabf7596f106b504cde","origin":"The Stacks Project","memory_eligible":false,"source_rank":9817,"rank":9817,"depth":0,"x":1655.817,"y":1158.797,"cluster":"tale-geometry"},{"id":"stacks:03NC","tag":"03NC","title":"Presheaves · Definition 03NC","summary":"Let C be a category. A presheaf of sets (respectively, an abelian presheaf) on C is a functor C^opp → Sets (resp. Ab).","statement_latex":"Let $\\mathcal{C}$ be a category. A {\\it presheaf of sets} (respectively, an\n{\\it abelian presheaf}) on $\\mathcal{C}$ is a functor $\\mathcal{C}^{opp} \\to\n\\textit{Sets}$ (resp.\\ $\\textit{Ab}$).","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NC","source_file":"etale-cohomology.tex","source_line":421,"source_end_line":426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L421-L426","statement_sha256":"324ac6a3e08c99203b40cf575b75f4914f85e4ff8ba1699d6ddfc9780827788a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9818,"rank":9818,"depth":0,"x":1508.789,"y":1169.237,"cluster":"tale-geometry"},{"id":"stacks:03NE","tag":"03NE","title":"Yoneda · Lemma 03NE","summary":"Morphisms between objects are in bijection with natural transformations between the functors they represent. Let C be a category, and X, Y ∈ Ob(C). There is a natural bijection Mor_C(X, Y) & → & Mor_PSh(C) (h_X, h_Y) ψ & ↦ & h_ψ = ψ ∘ - : h_X → h_Y.","statement_latex":"\\begin{slogan}\nMorphisms between objects are in bijection with natural transformations\nbetween the functors they represent.\n\\end{slogan}\nLet $\\mathcal{C}$ be a category, and $X, Y \\in\n\\Ob(\\mathcal{C})$. There is a natural bijection\n$$\n\\begin{matrix}\n\\Mor_\\mathcal{C}(X, Y) &\n\\longrightarrow &\n\\Mor_{\\textit{PSh}(\\mathcal{C})} (h_X, h_Y) \\\\\n\\psi &\n\\longmapsto &\nh_\\psi = \\psi \\circ - : h_X \\to h_Y.\n\\end{matrix}\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NE","source_file":"etale-cohomology.tex","source_line":475,"source_end_line":493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L475-L493","statement_sha256":"f74c1b5171f3bce839279794619d473d18ab323e95a07b1ee7db41258f5ffca9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9819,"rank":9819,"depth":1,"x":1609.075,"y":1077.87,"cluster":"tale-geometry"},{"id":"stacks:03NG","tag":"03NG","title":"Sites · Definition 03NG","summary":"Let C be a category. A family of morphisms with fixed target U = (φ_i : U_i → U)_i∈ I is the data of • an object U ∈ C, • a set I (possibly empty), and • for all i∈ I, a morphism φ_i : U_i → U of C with target U.","statement_latex":"Let $\\mathcal{C}$ be a category. A {\\it family of morphisms with fixed target}\n$\\mathcal{U} = \\{\\varphi_i : U_i \\to U\\}_{i\\in I}$ is the data of\n\\begin{enumerate}\n\\item an object $U \\in \\mathcal{C}$,\n\\item a set $I$ (possibly empty), and\n\\item for all $i\\in I$, a morphism $\\varphi_i : U_i \\to U$ of $\\mathcal{C}$\nwith target $U$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Sites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NG","source_file":"etale-cohomology.tex","source_line":507,"source_end_line":517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L507-L517","statement_sha256":"adde55e7408e1b0ddfe8b48f5d92d22a265d9813953c8ec68e9d0014eae3cec2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9820,"rank":9820,"depth":0,"x":1608.598,"y":1202.475,"cluster":"tale-geometry"},{"id":"stacks:03NH","tag":"03NH","title":"Sites · Definition 03NH","summary":"A site consists of a category C and a set Cov(C) consisting of families of morphisms with fixed target called coverings, such that • (isomorphism) if φ : V → U is an isomorphism in C, then (φ : V → U) is a covering, • (locality) if (φ_i : U_i → U)_i∈ I is a covering and for all i ∈ I we are given a covering (ψ_ij : U_ij → U_i )_j∈ I_i, then ( φ_i ∘ ψ_ij : U_ij → U )_(i, j)∈ ∏_i∈ I (i) × I_i is also a covering, and • (base change) if (U_i → U)_i∈ I is a covering and V → U…","statement_latex":"A {\\it site}\\footnote{What we call a site is a called a category endowed with\na pretopology in \\cite[Expos\\'e II, D\\'efinition 1.3]{SGA4}.\nIn \\cite{ArtinTopologies} it is called a category with a Grothendieck\ntopology.} consists of a category $\\mathcal{C}$ and a set\n$\\text{Cov}(\\mathcal{C})$ consisting of families of morphisms with fixed target\ncalled {\\it coverings}, such that\n\\begin{enumerate}\n\\item (isomorphism) if $\\varphi : V \\to U$ is an isomorphism in $\\mathcal{C}$,\nthen $\\{\\varphi : V \\to U\\}$ is a covering,\n\\item (locality) if $\\{\\varphi_i : U_i \\to U\\}_{i\\in I}$ is a covering and\nfor all $i \\in I$ we are given a covering\n$\\{\\psi_{ij} : U_{ij} \\to U_i \\}_{j\\in I_i}$, then\n$$\n\\{\n\\varphi_i \\circ \\psi_{ij} : U_{ij} \\to U\n\\}_{(i, j)\\in \\prod_{i\\in I} \\{i\\} \\times I_i}\n$$\nis also a covering, and\n\\item (base change) if $\\{U_i \\to U\\}_{i\\in I}$\nis a covering and $V \\to U$ is a morphism in $\\mathcal{C}$, then\n\\begin{enumerate}\n\\item for all $i \\in I$ the fibre product\n$U_i \\times_U V$ exists in $\\mathcal{C}$, and\n\\item $\\{U_i \\times_U V \\to V\\}_{i\\in I}$ is a covering.\n\\end{enumerate}\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Sites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NH","source_file":"etale-cohomology.tex","source_line":525,"source_end_line":553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L525-L553","statement_sha256":"b8e3140ccb7c342d765f46da7f18a6e1359bb27264de2caee4987ba94466c957","origin":"The Stacks Project","memory_eligible":false,"source_rank":9821,"rank":9821,"depth":0,"x":1508.485,"y":1110.081,"cluster":"tale-geometry"},{"id":"stacks:03NK","tag":"03NK","title":"Sheaves · Definition 03NK","summary":"A presheaf F of sets (resp. abelian presheaf) on a site C is said to be a separated presheaf if for all coverings (φ_i : U_i → U)_i∈ I ∈ Cov (C) the map F(U) → ∏_i∈ I F(U_i) is injective. Here the map is s ↦ (s|_U_i)_i∈ I. The presheaf F is a sheaf if for all coverings (φ_i : U_i → U)_i∈ I ∈ Cov (C), the diagram xymatrix F(U) ar[r] & ∏_i∈ I F(U_i) ar@<1ex>[r] ar@<-1ex>[r] & ∏_i, j ∈ I F(U_i ×_U U_j), where the first map is s ↦ (s|_U_i)_i∈ I and the two maps on the right…","statement_latex":"A presheaf $\\mathcal{F}$ of sets (resp. abelian presheaf) on a site\n$\\mathcal{C}$ is said to be a {\\it separated presheaf} if for all coverings\n$\\{\\varphi_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov} (\\mathcal{C})$\nthe map\n$$\n\\mathcal{F}(U) \\longrightarrow \\prod\\nolimits_{i\\in I} \\mathcal{F}(U_i)\n$$\nis injective. Here the map is $s \\mapsto (s|_{U_i})_{i\\in I}$.\nThe presheaf $\\mathcal{F}$ is a {\\it sheaf} if for all coverings\n$\\{\\varphi_i : U_i \\to U\\}_{i\\in I} \\in \\text{Cov} (\\mathcal{C})$, the\ndiagram\n\\begin{equation}\n\n\\xymatrix{\n\\mathcal{F}(U) \\ar[r] &\n\\prod_{i\\in I} \\mathcal{F}(U_i) \\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\prod_{i, j \\in I} \\mathcal{F}(U_i \\times_U U_j),\n}\n\\end{equation}\nwhere the first map is $s \\mapsto (s|_{U_i})_{i\\in I}$ and the two\nmaps on the right are\n$(s_i)_{i\\in I} \\mapsto (s_i |_{U_i \\times_U U_j})$ and\n$(s_i)_{i\\in I} \\mapsto (s_j |_{U_i \\times_U U_j})$,\nis an equalizer diagram in the category of sets (resp.\\ abelian groups).","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NK","source_file":"etale-cohomology.tex","source_line":580,"source_end_line":606,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L580-L606","statement_sha256":"f565f5fc133791b7b7f0afd03ad51bdf9774e6fd911680fbcdffac4616f0270a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9822,"rank":9822,"depth":0,"x":1656.993,"y":1121.434,"cluster":"tale-geometry"},{"id":"stacks:03NO","tag":"03NO","title":"Sheaves · Definition 03NO","summary":"We denote Sh(C) (resp. Ab(C)) the full subcategory of PSh(C) (resp. PAb(C)) whose objects are sheaves. This is the category of sheaves of sets (resp. abelian sheaves) on C.","statement_latex":"We denote $\\Sh(\\mathcal{C})$ (resp.\\ $\\textit{Ab}(\\mathcal{C})$)\nthe full subcategory of $\\textit{PSh}(\\mathcal{C})$\n(resp.\\ $\\textit{PAb}(\\mathcal{C})$) whose objects are sheaves. This is the\n{\\it category of sheaves of sets} (resp.\\ {\\it abelian sheaves}) on\n$\\mathcal{C}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NO","source_file":"etale-cohomology.tex","source_line":623,"source_end_line":630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L623-L630","statement_sha256":"db034ac13a79d278e26f1dc4011820368658c73a1189fc598dce1aa60d77a867","origin":"The Stacks Project","memory_eligible":false,"source_rank":9823,"rank":9823,"depth":0,"x":1538.048,"y":1197.527,"cluster":"tale-geometry"},{"id":"stacks:03NR","tag":"03NR","title":"Sheafification · Definition 03NR","summary":"Let F be a presheaf on the site C and U = (U_i → U) ∈ Cov (C). We define the zeroth v Cech cohomology group of F with respect to U by check H^0 (U, F) = ( (s_i)_i∈ I ∈ ∏_i∈ I F(U_i) such that s_i|_U_i ×_U U_j = s_j |_U_i ×_U U_j ).","statement_latex":"Let $\\mathcal{F}$ be a presheaf on the site $\\mathcal{C}$ and\n$\\mathcal{U} = \\{U_i \\to U\\} \\in \\text{Cov} (\\mathcal{C})$.\nWe define the {\\it zeroth {\\v C}ech cohomology group} of\n$\\mathcal{F}$ with respect to $\\mathcal{U}$ by\n$$\n\\check H^0 (\\mathcal{U}, \\mathcal{F}) =\n\\left\\{\n(s_i)_{i\\in I} \\in \\prod\\nolimits_{i\\in I }\\mathcal{F}(U_i)\n\\text{ such that }\ns_i|_{U_i \\times_U U_j} = s_j |_{U_i \\times_U U_j}\n\\right\\}.\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Sheafification","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NR","source_file":"etale-cohomology.tex","source_line":727,"source_end_line":741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L727-L741","statement_sha256":"61199a738f58673d98e51e63305c09fe9dbd81d4248d51011a266fb35fc49930","origin":"The Stacks Project","memory_eligible":false,"source_rank":9824,"rank":9824,"depth":0,"x":1564.634,"y":1073.606,"cluster":"tale-geometry"},{"id":"stacks:03NS","tag":"03NS","title":"Sheafification · Theorem 03NS","summary":"Let C be a site and F a presheaf on C. • The rule U ↦ F^+(U) := colim_U covering of U check H^0(U, F) is a presheaf. And the colimit is a directed one. • There is a canonical map of presheaves F → F^+. • If F is a separated presheaf then F^+ is a sheaf and the map in (2) is injective. • F^+ is a separated presheaf. • F^\\# = (F^+)^+ is a sheaf, and the canonical map induces a functorial isomorphism Hom_PSh(C)(F, G) = Hom_Sh(C)(F^\\#, G) for any G ∈ Sh(C).","statement_latex":"Let $\\mathcal{C}$ be a site and $\\mathcal{F}$ a presheaf on $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The rule\n$$\nU \\mapsto \\mathcal{F}^+(U) :=\n\\colim_{\\mathcal{U} \\text{ covering of }U}\n\\check H^0(\\mathcal{U}, \\mathcal{F})\n$$\nis a presheaf. And the colimit is a directed one.\n\\item There is a canonical map of presheaves $\\mathcal{F} \\to \\mathcal{F}^+$.\n\\item If $\\mathcal{F}$ is a separated presheaf then $\\mathcal{F}^+$ is a sheaf\nand the map in (2) is injective.\n\\item $\\mathcal{F}^+$ is a separated presheaf.\n\\item $\\mathcal{F}^\\# = (\\mathcal{F}^+)^+$ is a sheaf, and the canonical\nmap induces a functorial isomorphism\n$$\n\\Hom_{\\textit{PSh}(\\mathcal{C})}(\\mathcal{F}, \\mathcal{G}) =\n\\Hom_{\\Sh(\\mathcal{C})}(\\mathcal{F}^\\#, \\mathcal{G})\n$$\nfor any $\\mathcal{G} \\in \\Sh(\\mathcal{C})$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Sheafification","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NS","source_file":"etale-cohomology.tex","source_line":798,"source_end_line":821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L798-L821","statement_sha256":"a37af3c30e4f224f2abdca802d7ca77f1d5589e693844e8ea020ce5e323380b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":9825,"rank":9825,"depth":3,"x":1644.889,"y":1180.342,"cluster":"tale-geometry"},{"id":"stacks:03NU","tag":"03NU","title":"Cohomology · Theorem 03NU","summary":"The category of abelian sheaves on a site is an abelian category which has enough injectives.","statement_latex":"The category of abelian sheaves on a site is an abelian category\nwhich has enough injectives.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NU","source_file":"etale-cohomology.tex","source_line":842,"source_end_line":846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L842-L846","statement_sha256":"520215d277789309c51c57ac3446849f0455ba857d0bfb8db27f8bd0cf682ab3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9826,"rank":9826,"depth":6,"x":1499.506,"y":1147.09,"cluster":"tale-geometry"},{"id":"stacks:03NW","tag":"03NW","title":"The fpqc topology · Definition 03NW","summary":"Let T be a scheme. An fpqc covering of T is a family ( φ_i : T_i → T)_i ∈ I such that • each φ_i is a flat morphism and ⋃_i∈ I φ_i(T_i) = T, and • for each affine open U ⊂ T there exists a finite set K, a map i : K → I and affine opens U_i(k) ⊂ T_i(k) such that U = ⋃_k ∈ K φ_i(k)(U_i(k)).","statement_latex":"Let $T$ be a scheme. An {\\it fpqc covering} of $T$ is a family\n$\\{ \\varphi_i : T_i \\to T\\}_{i \\in I}$ such that\n\\begin{enumerate}\n\\item each $\\varphi_i$ is a flat morphism and\n$\\bigcup_{i\\in I} \\varphi_i(T_i) = T$, and\n\\item for each affine open $U \\subset T$ there exists a finite\nset $K$, a map $\\mathbf{i} : K \\to I$ and affine opens\n$U_{\\mathbf{i}(k)} \\subset T_{\\mathbf{i}(k)}$ such that\n$U = \\bigcup_{k \\in K} \\varphi_{\\mathbf{i}(k)}(U_{\\mathbf{i}(k)})$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The fpqc topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NW","source_file":"etale-cohomology.tex","source_line":882,"source_end_line":894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L882-L894","statement_sha256":"e04133af9a6a3891247a5f0ff71dde769fdd22a7faf8f814d16c60f7e55eb516","origin":"The Stacks Project","memory_eligible":false,"source_rank":9827,"rank":9827,"depth":0,"x":1633.789,"y":1088.969,"cluster":"tale-geometry"},{"id":"stacks:03NZ","tag":"03NZ","title":"The fpqc topology · Lemma 03NZ","summary":"The collection of fpqc coverings on the category of schemes satisfies the axioms of site.","statement_latex":"The collection of fpqc coverings on the category of schemes\nsatisfies the axioms of site.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03NZ","source_file":"etale-cohomology.tex","source_line":930,"source_end_line":934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L930-L934","statement_sha256":"e1d70ed30e2facd8566ce932b30afc8d78ddb88abf7b918657e6d962b3769055","origin":"The Stacks Project","memory_eligible":false,"source_rank":9828,"rank":9828,"depth":20,"x":1581.378,"y":1208.323,"cluster":"tale-geometry"},{"id":"stacks:03X6","tag":"03X6","title":"The fpqc topology · Definition 03X6","summary":"Let S be a scheme. The category of schemes over S is denoted Sch/S. Consider a functor F : (Sch/S)^opp → Sets, in other words a presheaf of sets. We say F satisfies the sheaf property for the fpqc topology if for every fpqc covering (U_i → U)_i ∈ I of schemes over S the diagram ([Tag 03NL]) is an equalizer diagram.","statement_latex":"Let $S$ be a scheme. The category of schemes over $S$ is denoted\n$\\Sch/S$. Consider a functor\n$\\mathcal{F} : (\\Sch/S)^{opp} \\to \\textit{Sets}$, in other words\na presheaf of sets. We say $\\mathcal{F}$\n{\\it satisfies the sheaf property for the fpqc topology}\nif for every fpqc covering $\\{U_i \\to U\\}_{i \\in I}$ of schemes over $S$\nthe diagram (\\ref{equation-sheaf-axiom}) is an equalizer diagram.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The fpqc topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03X6","source_file":"etale-cohomology.tex","source_line":953,"source_end_line":962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L953-L962","statement_sha256":"ea7934f10d2209ab61f2a1f2921b6d47ec8eeadd7c0677059b2618d6efc39a40","origin":"The Stacks Project","memory_eligible":false,"source_rank":9829,"rank":9829,"depth":1,"x":1523.9,"y":1090.277,"cluster":"tale-geometry"},{"id":"stacks:03O1","tag":"03O1","title":"The fpqc topology · Lemma 03O1","summary":"Let F be a presheaf on Sch/S. Then F satisfies the sheaf property for the fpqc topology if and only if • F satisfies the sheaf property with respect to the Zariski topology, and • for every faithfully flat morphism Spec(B) → Spec(A) of affine schemes over S, the sheaf axiom holds for the covering (Spec(B) → Spec(A)). Namely, this means that xymatrix F(Spec(A)) ar[r] & F(Spec(B)) ar@<1ex>[r] ar@<-1ex>[r] & F(Spec(B ⊗_A B)) is an equalizer diagram.","statement_latex":"Let $\\mathcal{F}$ be a presheaf on $\\Sch/S$. Then\n$\\mathcal{F}$ satisfies the sheaf property for the fpqc topology\nif and only if\n\\begin{enumerate}\n\\item $\\mathcal{F}$ satisfies the sheaf property with respect to the\nZariski topology, and\n\\item for every faithfully flat morphism $\\Spec(B) \\to \\Spec(A)$\nof affine schemes over $S$, the sheaf axiom holds for the covering\n$\\{\\Spec(B) \\to \\Spec(A)\\}$. Namely, this means that\n$$\n\\xymatrix{\n\\mathcal{F}(\\Spec(A)) \\ar[r] &\n\\mathcal{F}(\\Spec(B)) \\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\mathcal{F}(\\Spec(B \\otimes_A B))\n}\n$$\nis an equalizer diagram.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03O1","source_file":"etale-cohomology.tex","source_line":973,"source_end_line":993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L973-L993","statement_sha256":"ab9c4bcb5c4e9a160ecd187a727a535be160781f20b2a3a34d00d9b3f0449a79","origin":"The Stacks Project","memory_eligible":false,"source_rank":9830,"rank":9830,"depth":2,"x":1661.555,"y":1144.845,"cluster":"tale-geometry"},{"id":"stacks:03O3","tag":"03O3","title":"The fpqc topology · Lemma 03O3","summary":"Any representable presheaf on Sch/S satisfies the sheaf condition for the fpqc topology.","statement_latex":"Any representable presheaf on $\\Sch/S$ satisfies the\nsheaf condition for the fpqc topology.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03O3","source_file":"etale-cohomology.tex","source_line":1066,"source_end_line":1070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1066-L1070","statement_sha256":"2845ca22da23f3d8bf7d8bf7682a85e2c36957d11367ba899a1b58cd299e16d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9831,"rank":9831,"depth":41,"x":1515.807,"y":1182.81,"cluster":"tale-geometry"},{"id":"stacks:03O7","tag":"03O7","title":"Faithfully flat descent · Definition 03O7","summary":"Let U = ( t_i : T_i → T)_i ∈ I be a family of morphisms of schemes with fixed target. A descent datum for quasi-coherent sheaves with respect to U is a collection ((F_i)_i ∈ I, (φ_ij)_i, j ∈ I) where • F_i is a quasi-coherent sheaf on T_i, and • φ_ij : pr_0^* F_i → pr_1^* F_j is an isomorphism of modules on T_i ×_T T_j, such that the cocycle condition holds: the diagrams xymatrix pr_0^*F_i ar[dr]_pr_02^*φ_ik ar[rr]^pr_01^*φ_ij & & pr_1^*F_j ar[dl]^pr_12^*φ_jk & pr_2^*F_k…","statement_latex":"Let $\\mathcal{U} = \\{ t_i : T_i \\to T\\}_{i \\in I}$ be a family of\nmorphisms of schemes with fixed target. A {\\it descent datum} for\nquasi-coherent sheaves with respect to $\\mathcal{U}$ is a collection\n$((\\mathcal{F}_i)_{i \\in I}, (\\varphi_{ij})_{i, j \\in I})$ where\n\\begin{enumerate}\n\\item $\\mathcal{F}_i$ is a quasi-coherent sheaf on $T_i$, and\n\\item $\\varphi_{ij} : \\text{pr}_0^* \\mathcal{F}_i \\to\n\\text{pr}_1^* \\mathcal{F}_j$ is an isomorphism of modules\non $T_i \\times_T T_j$,\n\\end{enumerate}\nsuch that the {\\it cocycle condition} holds: the diagrams\n$$\n\\xymatrix{\n\\text{pr}_0^*\\mathcal{F}_i \\ar[dr]_{\\text{pr}_{02}^*\\varphi_{ik}}\n\\ar[rr]^{\\text{pr}_{01}^*\\varphi_{ij}} & &\n\\text{pr}_1^*\\mathcal{F}_j \\ar[dl]^{\\text{pr}_{12}^*\\varphi_{jk}} \\\\\n& \\text{pr}_2^*\\mathcal{F}_k\n}\n$$\ncommute on $T_i \\times_T T_j \\times_T T_k$.\nThis descent datum is called {\\it effective} if there exist a quasi-coherent\nsheaf $\\mathcal{F}$ over $T$ and $\\mathcal{O}_{T_i}$-module isomorphisms\n$\\varphi_i : t_i^* \\mathcal{F} \\cong \\mathcal{F}_i$ compatible with\nthe maps $\\varphi_{ij}$, namely\n$$\n\\varphi_{ij} = \\text{pr}_1^* (\\varphi_j) \\circ \\text{pr}_0^* (\\varphi_i)^{-1}.\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Faithfully flat descent","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03O7","source_file":"etale-cohomology.tex","source_line":1140,"source_end_line":1169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1140-L1169","statement_sha256":"58a595874d8e0e714442fee4ba134968e47cb0a1d8a384025ecd417f1267cb14","origin":"The Stacks Project","memory_eligible":false,"source_rank":9832,"rank":9832,"depth":0,"x":1592.94,"y":1071.84,"cluster":"tale-geometry"},{"id":"stacks:03O8","tag":"03O8","title":"Faithfully flat descent · Theorem 03O8","summary":"If V = (T_i → T)_i∈ I is an fpqc covering, then all descent data for quasi-coherent sheaves with respect to V are effective.","statement_latex":"If $\\mathcal{V} = \\{T_i \\to T\\}_{i\\in I}$ is an fpqc covering, then all\ndescent data for quasi-coherent sheaves with respect to $\\mathcal{V}$\nare effective.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Faithfully flat descent","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03O8","source_file":"etale-cohomology.tex","source_line":1175,"source_end_line":1180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1175-L1180","statement_sha256":"29868c136e7020291726e80623288430629f15464170d4276a77702ff04fdf6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9833,"rank":9833,"depth":21,"x":1625.382,"y":1197.744,"cluster":"tale-geometry"},{"id":"stacks:03O9","tag":"03O9","title":"Faithfully flat descent · Lemma 03O9","summary":"If A → B is faithfully flat, then the complex (B/A)_bullet is exact in positive degrees, and H^0((B/A)_bullet) = A.","statement_latex":"If $A \\to B$ is faithfully flat, then the complex $(B/A)_\\bullet$ is exact in\npositive degrees, and $H^0((B/A)_\\bullet) = A$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Faithfully flat descent","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03O9","source_file":"etale-cohomology.tex","source_line":1214,"source_end_line":1218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1214-L1218","statement_sha256":"70afd0b7dc7980e81e7b6e9163501c7f811431c010e7848bd07a4ecb301c9c8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9834,"rank":9834,"depth":8,"x":1499.905,"y":1123.13,"cluster":"tale-geometry"},{"id":"stacks:03OA","tag":"03OA","title":"Faithfully flat descent · Lemma 03OA","summary":"If A → B is faithfully flat and M is an A-module, then the complex (B/A)_bullet ⊗_A M is exact in positive degrees, and H^0((B/A)_bullet ⊗_A M) = M.","statement_latex":"If $A \\to B$ is faithfully flat and $M$ is an $A$-module, then the\ncomplex $(B/A)_\\bullet \\otimes_A M$ is exact in positive degrees, and\n$H^0((B/A)_\\bullet \\otimes_A M) = M$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Faithfully flat descent","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OA","source_file":"etale-cohomology.tex","source_line":1236,"source_end_line":1241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1236-L1241","statement_sha256":"f7adcdffdb3585249785d52874e6df158348c6dd25268b555a2821b4ef209c1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9835,"rank":9835,"depth":8,"x":1652.804,"y":1106.911,"cluster":"tale-geometry"},{"id":"stacks:03OB","tag":"03OB","title":"Faithfully flat descent · Definition 03OB","summary":"Let A → B be a ring map and N a B-module. A descent datum for N with respect to A → B is an isomorphism φ : N ⊗_A B ≅ B ⊗_A N of B ⊗_A B-modules such that the diagram of B ⊗_A B ⊗_A B-modules xymatrix N ⊗_A B ⊗_A B ar[dr]_φ_02 ar[rr]^φ_01 & & B ⊗_A N ⊗_A B ar[dl]^φ_12 & B ⊗_A B ⊗_A N commutes where φ_01 = φ ⊗ id_B and similarly for φ_12 and φ_02.","statement_latex":"Let $A \\to B$ be a ring map and $N$ a $B$-module. A {\\it descent datum} for\n$N$ with respect to $A \\to B$ is an isomorphism\n$\\varphi : N \\otimes_A B \\cong B \\otimes_A N$ of $B \\otimes_A B$-modules such\nthat the diagram of $B \\otimes_A B \\otimes_A B$-modules\n$$\n\\xymatrix{\n{N \\otimes_A B \\otimes_A B} \\ar[dr]_{\\varphi_{02}} \\ar[rr]^{\\varphi_{01}} & &\n{B \\otimes_A N \\otimes_A B} \\ar[dl]^{\\varphi_{12}} \\\\\n& {B \\otimes_A B \\otimes_A N}\n}\n$$\ncommutes where $\\varphi_{01} = \\varphi \\otimes \\text{id}_B$ and similarly\nfor $\\varphi_{12}$ and $\\varphi_{02}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Faithfully flat descent","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OB","source_file":"etale-cohomology.tex","source_line":1247,"source_end_line":1262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1247-L1262","statement_sha256":"1d007f8883f5e5b5230581e8d5424e46f98a9c6093383d535dd0abf25b19b2d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":9836,"rank":9836,"depth":0,"x":1552.859,"y":1205.869,"cluster":"tale-geometry"},{"id":"stacks:03OC","tag":"03OC","title":"Faithfully flat descent · Definition 03OC","summary":"A descent datum (N, φ) is called effective if there exists an A-module M such that (N, φ) ≅ (B ⊗_A M, φ_can), with the obvious notion of isomorphism of descent data.","statement_latex":"A descent datum $(N, \\varphi)$ is called {\\it effective} if there exists an\n$A$-module $M$ such that $(N, \\varphi) \\cong (B \\otimes_A M,\n\\varphi_\\text{can})$, with the obvious notion of isomorphism of descent data.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Faithfully flat descent","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OC","source_file":"etale-cohomology.tex","source_line":1274,"source_end_line":1279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1274-L1279","statement_sha256":"727680d17eae55deab0436bfaa8d4e290847fc3673784dd6e16787f2c89139ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":9837,"rank":9837,"depth":0,"x":1546.964,"y":1075.875,"cluster":"tale-geometry"},{"id":"stacks:03OD","tag":"03OD","title":"Faithfully flat descent · Theorem 03OD","summary":"If A → B is faithfully flat then descent data with respect to A→ B are effective.","statement_latex":"If $A \\to B$ is faithfully flat then descent data with respect to $A\\to B$\nare effective.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Faithfully flat descent","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OD","source_file":"etale-cohomology.tex","source_line":1285,"source_end_line":1289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1285-L1289","statement_sha256":"701dd77f28ea5e2690279d784c2aadf9cac8416615adafd31a9cf21f1d1fb140","origin":"The Stacks Project","memory_eligible":false,"source_rank":9838,"rank":9838,"depth":11,"x":1656.11,"y":1168.608,"cluster":"tale-geometry"},{"id":"stacks:03OG","tag":"03OG","title":"Quasi-coherent sheaves · Proposition 03OG","summary":"For any quasi-coherent sheaf F on S the presheaf F^a : & Sch/S & → & Ab & (f: T → S) & ↦ & Γ(T, f^*F) is an O-module which satisfies the sheaf condition for the fpqc topology.","statement_latex":"For any quasi-coherent sheaf $\\mathcal{F}$ on $S$ the presheaf\n$$\n\\begin{matrix}\n\\mathcal{F}^a : & \\Sch/S & \\to & \\textit{Ab}\\\\\n& (f: T \\to S) & \\mapsto & \\Gamma(T, f^*\\mathcal{F})\n\\end{matrix}\n$$\nis an $\\mathcal{O}$-module which satisfies the sheaf condition for the\nfpqc topology.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Quasi-coherent sheaves","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OG","source_file":"etale-cohomology.tex","source_line":1331,"source_end_line":1342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1331-L1342","statement_sha256":"cd34a1c87d6dfffa5d618068a68bda0e7ec71c81e4c715ea4aa8d8b3dcb4559e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9839,"rank":9839,"depth":40,"x":1500.683,"y":1162.144,"cluster":"tale-geometry"},{"id":"stacks:03OH","tag":"03OH","title":"Quasi-coherent sheaves · Definition 03OH","summary":"Let C be a ringed site, i.e., a site endowed with a sheaf of rings O. A sheaf of O-modules F on C is called quasi-coherent if for all U ∈ Ob(C) there exists a covering (U_i → U)_i∈ I of C such that the restriction F|_C/U_i is isomorphic to the cokernel of an O-linear map of free O-modules bigoplus_k ∈ K O|_C/U_i → bigoplus_l ∈ L O|_C/U_i. The direct sum over K is the sheaf associated to the presheaf V ↦ bigoplus_k ∈ K O(V) and similarly for the other.","statement_latex":"Let $\\mathcal{C}$ be a {\\it ringed site}, i.e., a site endowed with a\nsheaf of rings $\\mathcal{O}$. A sheaf of $\\mathcal{O}$-modules $\\mathcal{F}$ on\n$\\mathcal{C}$ is called {\\it quasi-coherent} if for all\n$U \\in \\Ob(\\mathcal{C})$ there exists a covering\n$\\{U_i \\to U\\}_{i\\in I}$ of $\\mathcal{C}$ such that the restriction\n$\\mathcal{F}|_{\\mathcal{C}/U_i}$ is isomorphic to the cokernel of\nan $\\mathcal{O}$-linear map of free $\\mathcal{O}$-modules\n$$\n\\bigoplus\\nolimits_{k \\in K} \\mathcal{O}|_{\\mathcal{C}/U_i}\n\\longrightarrow\n\\bigoplus\\nolimits_{l \\in L} \\mathcal{O}|_{\\mathcal{C}/U_i}.\n$$\nThe direct sum over $K$ is the sheaf associated to the presheaf\n$V \\mapsto \\bigoplus_{k \\in K} \\mathcal{O}(V)$ and similarly for the other.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Quasi-coherent sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OH","source_file":"etale-cohomology.tex","source_line":1385,"source_end_line":1401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1385-L1401","statement_sha256":"af685b79df5663d1bb03f56196d9325a19a0413d717be8548f9e0210f771fecd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9840,"rank":9840,"depth":0,"x":1620.776,"y":1078.519,"cluster":"tale-geometry"},{"id":"stacks:03OJ","tag":"03OJ","title":"Meta theorem on quasi-coherent sheaves · Theorem 03OJ","summary":"Let S be a scheme. Let C be a site. Assume that • the underlying category C is a full subcategory of Sch/S, • any Zariski covering of T ∈ Ob(C) can be refined by a covering of C, • S/S is an object of C, • every covering of C is an fpqc covering of schemes. Then the presheaf O is a sheaf on C and any quasi-coherent O-module on (C, O) is of the form F^a for some quasi-coherent sheaf F on S.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{C}$ be a site. Assume that\n\\begin{enumerate}\n\\item the underlying category $\\mathcal{C}$ is a\nfull subcategory of $\\Sch/S$,\n\\item any Zariski covering of $T \\in \\Ob(\\mathcal{C})$\ncan be refined by a covering of $\\mathcal{C}$,\n\\item $S/S$ is an object of $\\mathcal{C}$,\n\\item every covering of $\\mathcal{C}$ is an fpqc covering of schemes.\n\\end{enumerate}\nThen the presheaf $\\mathcal{O}$ is a sheaf on $\\mathcal{C}$ and\nany quasi-coherent $\\mathcal{O}$-module on $(\\mathcal{C}, \\mathcal{O})$\nis of the form $\\mathcal{F}^a$ for some quasi-coherent sheaf\n$\\mathcal{F}$ on $S$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Quasi-coherent sheaves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OJ","source_file":"etale-cohomology.tex","source_line":1415,"source_end_line":1431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1415-L1431","statement_sha256":"ca63a790565da0bf34ec80ae92d095a6ecf797b834c276ee563ab384ee121466","origin":"The Stacks Project","memory_eligible":false,"source_rank":9841,"rank":9841,"depth":22,"x":1599.419,"y":1208.634,"cluster":"tale-geometry"},{"id":"stacks:03OL","tag":"03OL","title":"v Cech cohomology · Definition 03OL","summary":"Let C be a category. Let U = (U_i → U)_i ∈ I be a family of morphisms of C with fixed target. Assume that all the fibre products U_i_0 ×_U … ×_U U_i_p exists in C. Let F ∈ PAb(C) be an abelian presheaf. We define the v Cech complex checkC^bullet(U, F) by ∏_i_0 ∈ I F(U_i_0) → ∏_i_0, i_1 ∈ I F(U_i_0 ×_U U_i_1) → ∏_i_0, i_1, i_2 ∈ I F(U_i_0 ×_U U_i_1 ×_U U_i_2) → … where the first term is in degree 0 and the maps are the usual ones. The v Cech cohomology groups are defined…","statement_latex":"Let $\\mathcal{C}$ be a category. Let\n$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ be a family of morphisms of\n$\\mathcal{C}$ with fixed target. Assume that all the fibre products\n$U_{i_0} \\times_U \\ldots \\times_U U_{i_p}$ exists in $\\mathcal{C}$.\nLet $\\mathcal{F} \\in \\textit{PAb}(\\mathcal{C})$ be an abelian\npresheaf. We define the {\\it {\\v C}ech complex}\n$\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$ by\n$$\n\\prod_{i_0 \\in I} \\mathcal{F}(U_{i_0}) \\to\n\\prod_{i_0, i_1 \\in I} \\mathcal{F}(U_{i_0} \\times_U U_{i_1}) \\to\n\\prod_{i_0, i_1, i_2 \\in I}\n\\mathcal{F}(U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2}) \\to \\ldots\n$$\nwhere the first term is in degree 0 and the maps are the usual ones.\nThe {\\it {\\v C}ech cohomology groups} are defined by\n$$\n\\check{H}^p(\\mathcal{U}, \\mathcal{F}) =\nH^p(\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})).\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"v Cech cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OL","source_file":"etale-cohomology.tex","source_line":1476,"source_end_line":1497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1476-L1497","statement_sha256":"f7f48c6315f8186c543da97b9b57b4b141a06457762f3561a6dd15bd18d1fc60","origin":"The Stacks Project","memory_eligible":false,"source_rank":9842,"rank":9842,"depth":0,"x":1510.324,"y":1100.316,"cluster":"tale-geometry"},{"id":"stacks:03OM","tag":"03OM","title":"v Cech cohomology · Lemma 03OM","summary":"Notation and assumptions as in Definition [Tag 03OL]. The functor checkC^bullet(U, -) is exact on the category PAb(C).","statement_latex":"Notation and assumptions as in Definition \\ref{definition-cech-complex}.\nThe functor $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, -)$\nis exact on the category $\\textit{PAb}(\\mathcal{C})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"v Cech cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OM","source_file":"etale-cohomology.tex","source_line":1503,"source_end_line":1508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1503-L1508","statement_sha256":"fe7b4b8c5284684af7659d9eb0249ea4b80756cd205588f318027216cf912be1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9843,"rank":9843,"depth":1,"x":1663.486,"y":1129.702,"cluster":"tale-geometry"},{"id":"stacks:03ON","tag":"03ON","title":"Yoneda Lemma · Lemma 03ON","summary":"For any presheaf F on a category C and U ∈ Ob(C) there is a functorial isomorphism Hom_PSh(C)(h_U, F) = F(U).","statement_latex":"For any presheaf $\\mathcal{F}$ on a category $\\mathcal{C}$\nand $U \\in \\Ob(\\mathcal{C})$ there is a functorial isomorphism\n$$\n\\Hom_{\\textit{PSh}(\\mathcal{C})}(h_U, \\mathcal{F}) =\n\\mathcal{F}(U).\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"v Cech cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ON","source_file":"etale-cohomology.tex","source_line":1539,"source_end_line":1547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1539-L1547","statement_sha256":"feb4093f5421db51b5bb26e203ae27cdd80e1f16d88279d0661784255dbacd20","origin":"The Stacks Project","memory_eligible":false,"source_rank":9844,"rank":9844,"depth":1,"x":1526.596,"y":1195.093,"cluster":"tale-geometry"},{"id":"stacks:03OO","tag":"03OO","title":"v Cech cohomology · Definition 03OO","summary":"Let C be a category. Given a presheaf of sets G, we define the free abelian presheaf on G, denoted Z_G, by the rule Z_G(U) = Z[G(U)] for U ∈ Ob(C) with restriction maps induced by the restriction maps of G. In the special case G = h_U we write simply Z_U = Z_h_U.","statement_latex":"Let $\\mathcal{C}$ be a category.\nGiven a presheaf of sets $\\mathcal{G}$, we define the\n{\\it free abelian presheaf on $\\mathcal{G}$},\ndenoted $\\mathbf{Z}_\\mathcal{G}$, by the rule\n$$\n\\mathbf{Z}_\\mathcal{G}(U)\n=\n\\mathbf{Z}[\\mathcal{G}(U)]\n$$\nfor $U \\in \\Ob(\\mathcal{C})$\nwith restriction maps induced by the restriction maps of $\\mathcal{G}$.\nIn the special case $\\mathcal{G} = h_U$ we write simply\n$\\mathbf{Z}_U = \\mathbf{Z}_{h_U}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"v Cech cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OO","source_file":"etale-cohomology.tex","source_line":1561,"source_end_line":1576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1561-L1576","statement_sha256":"0f6af5d53d6f18e09436c1e0203027190b1e19a1da4719a31ca40d4851ca65a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9845,"rank":9845,"depth":0,"x":1575.063,"y":1068.908,"cluster":"tale-geometry"},{"id":"stacks:03OP","tag":"03OP","title":"v Cech cohomology · Lemma 03OP","summary":"Notation and assumptions as in Definition [Tag 03OL]. The v Cech complex checkC^bullet(U, F) can be described explicitly as follows checkC^bullet(U, F) & = & ( ∏_i_0 ∈ I Hom_PAb(C)(Z_U_i_0, F) → ∏_i_0, i_1 ∈ I Hom_PAb(C)( Z_U_i_0 ×_U U_i_1, F) → … ) & = & Hom_PAb(C)( ( bigoplus_i_0 ∈ I Z_U_i_0 ← bigoplus_i_0, i_1 ∈ I Z_U_i_0 ×_U U_i_1 ← … ), F)","statement_latex":"Notation and assumptions as in Definition \\ref{definition-cech-complex}.\nThe {\\v C}ech complex $\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})$\ncan be described explicitly as follows\n\\begin{eqnarray*}\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U}, \\mathcal{F})\n& = &\n\\left(\n\\prod_{i_0 \\in I}\n\\Hom_{\\textit{PAb}(\\mathcal{C})}(\\mathbf{Z}_{U_{i_0}}, \\mathcal{F}) \\to\n\\prod_{i_0, i_1 \\in I}\n\\Hom_{\\textit{PAb}(\\mathcal{C})}(\n\\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}}, \\mathcal{F}) \\to \\ldots\n\\right) \\\\\n& = &\n\\Hom_{\\textit{PAb}(\\mathcal{C})}\\left(\n\\left(\n\\bigoplus_{i_0 \\in I} \\mathbf{Z}_{U_{i_0}} \\leftarrow\n\\bigoplus_{i_0, i_1 \\in I} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}} \\leftarrow\n\\ldots\n\\right), \\mathcal{F}\\right)\n\\end{eqnarray*}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"v Cech cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OP","source_file":"etale-cohomology.tex","source_line":1592,"source_end_line":1615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1592-L1615","statement_sha256":"3eee6250226b7596d4d57caeb63204ed2fbf55fc0746e1d3f5d7ea60d4f99465","origin":"The Stacks Project","memory_eligible":false,"source_rank":9846,"rank":9846,"depth":2,"x":1640.951,"y":1189.735,"cluster":"tale-geometry"},{"id":"stacks:03OQ","tag":"03OQ","title":"v Cech cohomology · Lemma 03OQ","summary":"Notation and assumptions as in Definition [Tag 03OL]. The complex of abelian presheaves Z_U^bullet : bigoplus_i_0 ∈ I Z_U_i_0 ← bigoplus_i_0, i_1 ∈ I Z_U_i_0 ×_U U_i_1 ← bigoplus_i_0, i_1, i_2 ∈ I Z_U_i_0 ×_U U_i_1 ×_U U_i_2 ← … is exact in all degrees except 0 in PAb(C).","statement_latex":"Notation and assumptions as in Definition \\ref{definition-cech-complex}.\nThe complex of abelian presheaves\n\\begin{align*}\n\\mathbf{Z}_\\mathcal{U}^\\bullet \\quad : \\quad\n\\bigoplus_{i_0 \\in I} \\mathbf{Z}_{U_{i_0}} \\leftarrow\n\\bigoplus_{i_0, i_1 \\in I} \\mathbf{Z}_{U_{i_0} \\times_U U_{i_1}} \\leftarrow\n\\bigoplus_{i_0, i_1, i_2 \\in I}\n\\mathbf{Z}_{U_{i_0} \\times_U U_{i_1} \\times_U U_{i_2}} \\leftarrow\n\\ldots\n\\end{align*}\nis exact in all degrees except $0$ in $\\textit{PAb}(\\mathcal{C})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"v Cech cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OQ","source_file":"etale-cohomology.tex","source_line":1626,"source_end_line":1639,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1626-L1639","statement_sha256":"30875b25b2175875a1474c6b2bcdf4e2340c0bcbe2d2861b041bbc42be749d68","origin":"The Stacks Project","memory_eligible":false,"source_rank":9847,"rank":9847,"depth":3,"x":1494.865,"y":1137.906,"cluster":"tale-geometry"},{"id":"stacks:03OR","tag":"03OR","title":"v Cech cohomology · Lemma 03OR","summary":"Notation and assumptions as in Definition [Tag 03OL]. If I is an injective object of PAb(C), then check H^p(U, I) = 0 for all p > 0.","statement_latex":"Notation and assumptions as in Definition \\ref{definition-cech-complex}.\nIf $\\mathcal{I}$ is an injective object of $\\textit{PAb}(\\mathcal{C})$,\nthen $\\check H^p(\\mathcal{U}, \\mathcal{I}) = 0$ for all $p > 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"v Cech cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OR","source_file":"etale-cohomology.tex","source_line":1701,"source_end_line":1706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1701-L1706","statement_sha256":"3dbfd2113722860068c183786eb7674238ef3cee6f80536da1052551dcfc00ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":9848,"rank":9848,"depth":1,"x":1644.608,"y":1093.129,"cluster":"tale-geometry"},{"id":"stacks:03OS","tag":"03OS","title":"v Cech cohomology · Theorem 03OS","summary":"Notation and assumptions as in Definition [Tag 03OL]. On PAb(C) the functors checkH^p(U, -) are the right derived functors of checkH^0(U, -).","statement_latex":"Notation and assumptions as in Definition \\ref{definition-cech-complex}.\nOn $\\textit{PAb}(\\mathcal{C})$ the functors $\\check{H}^p(\\mathcal{U}, -)$ are\nthe right derived functors of $\\check{H}^0(\\mathcal{U}, -)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"v Cech cohomology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OS","source_file":"etale-cohomology.tex","source_line":1723,"source_end_line":1728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1723-L1728","statement_sha256":"586d141141f92384c2895db38167ed8048ba6cdc64b0eb43996622eb30e94344","origin":"The Stacks Project","memory_eligible":false,"source_rank":9849,"rank":9849,"depth":20,"x":1570.028,"y":1211.384,"cluster":"tale-geometry"},{"id":"stacks:03OV","tag":"03OV","title":"The v Cech-to-cohomology spectral sequence · Lemma 03OV","summary":"The forgetful functor Ab(C)→ PAb(C) transforms injectives into injectives.","statement_latex":"The forgetful functor $\\textit{Ab}(\\mathcal{C})\\to \\textit{PAb}(\\mathcal{C})$\ntransforms injectives into injectives.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The v Cech-to-cohomology spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OV","source_file":"etale-cohomology.tex","source_line":1757,"source_end_line":1761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1757-L1761","statement_sha256":"c49fbfd906baba20c5b184647a1ab7dd26434ad5d6b910c5548e2b1d622f48f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9850,"rank":9850,"depth":9,"x":1529.835,"y":1081.572,"cluster":"tale-geometry"},{"id":"stacks:03OW","tag":"03OW","title":"The v Cech-to-cohomology spectral sequence · Theorem 03OW","summary":"Let C be a site. For any covering U = (U_i → U)_i ∈ I of U ∈ Ob(C) and any abelian sheaf F on C there is a spectral sequence E_2^p, q = check H^p(U, underlineH^q(F)) ⇒ H^p+q(U, F), where underlineH^q(F) is the abelian presheaf V ↦ H^q(V, F).","statement_latex":"Let $\\mathcal{C}$ be a site. For any covering\n$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ of $U \\in \\Ob(\\mathcal{C})$\nand any abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}$\nthere is a spectral sequence\n$$\nE_2^{p, q}\n=\n\\check H^p(\\mathcal{U}, \\underline{H}^q(\\mathcal{F}))\n\\Rightarrow\nH^{p+q}(U, \\mathcal{F}),\n$$\nwhere $\\underline{H}^q(\\mathcal{F})$ is the abelian presheaf\n$V \\mapsto H^q(V, \\mathcal{F})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The v Cech-to-cohomology spectral sequence","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OW","source_file":"etale-cohomology.tex","source_line":1770,"source_end_line":1785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1770-L1785","statement_sha256":"6abb1bc70d1983f99139bab075f49310f4134108ff75ee1670f7af32b286f3e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9851,"rank":9851,"depth":22,"x":1664.166,"y":1154.652,"cluster":"tale-geometry"},{"id":"stacks:03X8","tag":"03X8","title":"Big and small sites of schemes · Definition 03X8","summary":"(See Topologies, Definitions [Tag 021M], [Tag 0225], [Tag 021Z], [Tag 0215], and [Tag 020O].) Let τ ∈ (fppf, syntomic, smooth, etale, Zariski). A family of morphisms of schemes (f_i : T_i → T)_i ∈ I with fixed target is called a τ-covering if and only if each f_i is flat of finite presentation, syntomic, smooth, étale, resp. an open immersion, and we have ⋃ f_i(T_i) = T.","statement_latex":"(See\nTopologies, Definitions\n\\ref{topologies-definition-fppf-covering},\n\\ref{topologies-definition-syntomic-covering},\n\\ref{topologies-definition-smooth-covering},\n\\ref{topologies-definition-etale-covering}, and\n\\ref{topologies-definition-zariski-covering}.)\nLet $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.\nA family of morphisms of schemes $\\{f_i : T_i \\to T\\}_{i \\in I}$ with fixed\ntarget is called a {\\it $\\tau$-covering} if and only if\neach $f_i$ is flat of finite presentation, syntomic, smooth, \\'etale,\nresp.\\ an open immersion, and we have $\\bigcup f_i(T_i) = T$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big and small sites of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03X8","source_file":"etale-cohomology.tex","source_line":1865,"source_end_line":1879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1865-L1879","statement_sha256":"70f8596b2240748fac4795c9903cc34a91791e4f0fa92d69ccfec0a34dcce83c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9852,"rank":9852,"depth":1,"x":1505.989,"y":1177.037,"cluster":"tale-geometry"},{"id":"stacks:03XB","tag":"03XB","title":"Big and small sites of schemes · Definition 03XB","summary":"Let S be a scheme. Let τ ∈ (fppf, syntomic, smooth, etale, linebreak[0] Zariski). • A big τ-site of S is any of the sites (Sch/S)_τ constructed as explained above and in more detail in Topologies, Definitions [Tag 021S], [Tag 03X3], [Tag 03X0], [Tag 021B], and [Tag 020T]. • If τ ∈ (etale, Zariski), then the small τ-site of S is the full subcategory S_τ of (Sch/S)_τ whose objects are schemes T over S whose structure morphism T → S is étale, resp. an open immersion. A…","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{fppf, syntomic, smooth, \\etale, \\linebreak[0] Zariski\\}$.\n\\begin{enumerate}\n\\item A {\\it big $\\tau$-site of $S$} is any of the sites\n$(\\Sch/S)_\\tau$ constructed as explained above and in more detail in\nTopologies, Definitions\n\\ref{topologies-definition-big-small-fppf},\n\\ref{topologies-definition-big-small-syntomic},\n\\ref{topologies-definition-big-small-smooth},\n\\ref{topologies-definition-big-small-etale}, and\n\\ref{topologies-definition-big-small-Zariski}.\n\\item If $\\tau \\in \\{\\etale, Zariski\\}$, then the\n{\\it small $\\tau$-site of $S$}\nis the full subcategory $S_\\tau$ of $(\\Sch/S)_\\tau$ whose objects\nare schemes $T$ over $S$ whose structure morphism $T \\to S$ is \\'etale,\nresp.\\ an open immersion. A covering in $S_\\tau$ is a covering\n$\\{U_i \\to U\\}$ in $(\\Sch/S)_\\tau$\nsuch that $U$ is an object of $S_\\tau$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big and small sites of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XB","source_file":"etale-cohomology.tex","source_line":1908,"source_end_line":1929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1908-L1929","statement_sha256":"7cebd31c3c4e151d9b9f283a2dcbd1b0662ee39a8cf2b4a8458be079f7110a98","origin":"The Stacks Project","memory_eligible":false,"source_rank":9853,"rank":9853,"depth":1,"x":1604.846,"y":1070.539,"cluster":"tale-geometry"},{"id":"stacks:03YX","tag":"03YX","title":"Big and small sites of schemes · Lemma 03YX","summary":"Let τ ∈ (etale, Zariski). If F is an abelian sheaf defined on (Sch/S)_τ, then the cohomology groups of F over S agree with the cohomology groups of F|_S_τ over S.","statement_latex":"Let $\\tau \\in \\{\\etale, Zariski\\}$.\nIf $\\mathcal{F}$ is an abelian sheaf defined on\n$(\\Sch/S)_\\tau$, then\nthe cohomology groups of $\\mathcal{F}$ over $S$ agree with the cohomology\ngroups of $\\mathcal{F}|_{S_\\tau}$ over $S$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big and small sites of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YX","source_file":"etale-cohomology.tex","source_line":1976,"source_end_line":1983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L1976-L1983","statement_sha256":"cec58e7c550aefdfcd6b43b65292a23023e60f051c89a740c38ab2614f0c674b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9854,"rank":9854,"depth":9,"x":1617.621,"y":1205.463,"cluster":"tale-geometry"},{"id":"stacks:03X9","tag":"03X9","title":"Big and small sites of schemes · Definition 03X9","summary":"(See Topologies, Definitions [Tag 021Q], [Tag 0229], [Tag 0223], [Tag 0219], and [Tag 020R].) Let τ ∈ (fppf, syntomic, smooth, etale, Zariski). Let T be an affine scheme. A standard τ-covering of T is a family (f_j : U_j → T)_j = 1, …, m with each U_j is affine, and each f_j flat and of finite presentation, standard syntomic, standard smooth, étale, resp. the immersion of a standard principal open in T and T = ⋃ f_j(U_j).","statement_latex":"(See\nTopologies, Definitions\n\\ref{topologies-definition-standard-fppf},\n\\ref{topologies-definition-standard-syntomic},\n\\ref{topologies-definition-standard-smooth},\n\\ref{topologies-definition-standard-etale}, and\n\\ref{topologies-definition-standard-Zariski}.)\nLet $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.\nLet $T$ be an affine scheme.\nA {\\it standard $\\tau$-covering} of $T$ is a family\n$\\{f_j : U_j \\to T\\}_{j = 1, \\ldots, m}$ with each $U_j$ is affine,\nand each $f_j$ flat and of finite presentation,\nstandard syntomic, standard smooth, \\'etale, resp.\\ the immersion of a\nstandard principal open in $T$ and $T = \\bigcup f_j(U_j)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big and small sites of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03X9","source_file":"etale-cohomology.tex","source_line":2008,"source_end_line":2024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2008-L2024","statement_sha256":"930ea3f8e6ba96613322be4cf5e1f78d2e362efd9ee2c8c4e32d8c3100de9447","origin":"The Stacks Project","memory_eligible":false,"source_rank":9855,"rank":9855,"depth":2,"x":1499.438,"y":1113.016,"cluster":"tale-geometry"},{"id":"stacks:03XA","tag":"03XA","title":"Big and small sites of schemes · Lemma 03XA","summary":"Let τ ∈ (fppf, syntomic, smooth, etale, Zariski). Any τ-covering of an affine scheme can be refined by a standard τ-covering.","statement_latex":"Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.\nAny $\\tau$-covering of an affine scheme can be refined by a\nstandard $\\tau$-covering.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big and small sites of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XA","source_file":"etale-cohomology.tex","source_line":2026,"source_end_line":2031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2026-L2031","statement_sha256":"472d9dd0458cf3a58821d49978be8c2a01831f77acf4b70d418d13fd72c94cef","origin":"The Stacks Project","memory_eligible":false,"source_rank":9856,"rank":9856,"depth":36,"x":1661.284,"y":1114.127,"cluster":"tale-geometry"},{"id":"stacks:03YY","tag":"03YY","title":"Big and small sites of schemes · Lemma 03YY","summary":"Let τ ∈ (fppf, syntomic, smooth, etale, Zariski). Let S be a scheme. Let (Sch/S)_τ and (Sch'/S)_τ be two big τ-sites of S, and assume that the first is contained in the second. In this case • for any abelian sheaf F' defined on (Sch'/S)_τ and any object U of (Sch/S)_τ we have H^p_τ(U, F'|_(Sch/S)_τ) = H^p_τ(U, F') In words: the cohomology of F' over U computed in the bigger site agrees with the cohomology of F' restricted to the smaller site over U. • for any abelian…","statement_latex":"Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.\nLet $S$ be a scheme.\nLet $(\\Sch/S)_\\tau$ and $(\\Sch'/S)_\\tau$ be two\nbig $\\tau$-sites of $S$, and assume that the first is contained in the second.\nIn this case\n\\begin{enumerate}\n\\item for any abelian sheaf $\\mathcal{F}'$ defined on $(\\Sch'/S)_\\tau$ and\nany object $U$ of $(\\Sch/S)_\\tau$ we have\n$$\nH^p_\\tau(U, \\mathcal{F}'|_{(\\Sch/S)_\\tau}) =\nH^p_\\tau(U, \\mathcal{F}')\n$$\nIn words: the cohomology of $\\mathcal{F}'$ over $U$ computed in the bigger site\nagrees with the cohomology of $\\mathcal{F}'$ restricted to the smaller site\nover $U$.\n\\item for any abelian sheaf $\\mathcal{F}$ on $(\\Sch/S)_\\tau$ there is an\nabelian sheaf $\\mathcal{F}'$ on $(\\Sch/S)_\\tau'$ whose restriction to\n$(\\Sch/S)_\\tau$ is isomorphic to $\\mathcal{F}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big and small sites of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YY","source_file":"etale-cohomology.tex","source_line":2051,"source_end_line":2072,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2051-L2072","statement_sha256":"7e8af71ab475b981d3f40f9b30d5e62a06e5f2acc8000c55c444d8721a53e7b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9857,"rank":9857,"depth":20,"x":1540.783,"y":1205.343,"cluster":"tale-geometry"},{"id":"stacks:04HQ","tag":"04HQ","title":"The étale topos · Definition 04HQ","summary":"Let S be a scheme. • The étale topos, or the small étale topos of S is the category Sh(S_etale) of sheaves of sets on the small étale site of S. • The Zariski topos, or the small Zariski topos of S is the category Sh(S_Zar) of sheaves of sets on the small Zariski site of S. • For τ ∈ (fppf, syntomic, smooth, etale, Zariski) a big τ-topos is the category of sheaves of set on a big τ-topos of S.","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item The {\\it \\'etale topos}, or the {\\it small \\'etale topos}\nof $S$ is the category $\\Sh(S_\\etale)$ of sheaves of sets on\nthe small \\'etale site of $S$.\n\\item The {\\it Zariski topos}, or the {\\it small Zariski topos}\nof $S$ is the category $\\Sh(S_{Zar})$ of sheaves of sets on the\nsmall Zariski site of $S$.\n\\item For $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$ a\n{\\it big $\\tau$-topos} is the category of sheaves of set on a\nbig $\\tau$-topos of $S$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The étale topos","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HQ","source_file":"etale-cohomology.tex","source_line":2095,"source_end_line":2109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2095-L2109","statement_sha256":"b61de982beb9dfe14727a2d7c7109392c794b01382fb258f08e307d0e05d43fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9858,"rank":9858,"depth":0,"x":1556.319,"y":1069.41,"cluster":"tale-geometry"},{"id":"stacks:0958","tag":"0958","title":"The étale topos · Lemma 0958","summary":"Let S be a scheme. The étale topos of S is independent (up to canonical equivalence) of the construction of the small étale site in Definition [Tag 03XB].","statement_latex":"Let $S$ be a scheme. The \\'etale topos of $S$ is independent\n(up to canonical equivalence) of the construction of the small\n\\'etale site in Definition \\ref{definition-tau-site}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The étale topos","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0958","source_file":"etale-cohomology.tex","source_line":2127,"source_end_line":2132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2127-L2132","statement_sha256":"8fcb3a9b3377407853936e65f97ae4c915f9037a0319422d30d1beed2147b7cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9859,"rank":9859,"depth":10,"x":1654.39,"y":1178.699,"cluster":"tale-geometry"},{"id":"stacks:03OZ","tag":"03OZ","title":"Cohomology of quasi-coherent sheaves · Lemma 03OZ","summary":"Let τ ∈ (fppf, syntomic, smooth, etale, Zariski). Let S be a scheme. Let F be an abelian sheaf on (Sch/S)_τ, or on S_τ in case τ = etale, and let U = (U_i → U)_i ∈ I be a standard τ-covering of this site. Let V = coprod_i ∈ I U_i. Then • V is an affine scheme, • V = (V → U) is an fpqc covering and also a τ-covering unless τ = Zariski, • the v Cech complexes checkC^bullet (U, F) and checkC^bullet (V, F) agree.","statement_latex":"Let $\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$.\nLet $S$ be a scheme.\nLet $\\mathcal{F}$ be an abelian sheaf on $(\\Sch/S)_\\tau$, or on\n$S_\\tau$ in case $\\tau = \\etale$, and let\n$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$\nbe a standard $\\tau$-covering of this site.\nLet $V = \\coprod_{i \\in I} U_i$. Then\n\\begin{enumerate}\n\\item $V$ is an affine scheme,\n\\item $\\mathcal{V} = \\{V \\to U\\}$ is an fpqc covering\nand also a $\\tau$-covering unless $\\tau = Zariski$,\n\\item the {\\v C}ech complexes\n$\\check{\\mathcal{C}}^\\bullet (\\mathcal{U}, \\mathcal{F})$ and\n$\\check{\\mathcal{C}}^\\bullet (\\mathcal{V}, \\mathcal{F})$ agree.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03OZ","source_file":"etale-cohomology.tex","source_line":2166,"source_end_line":2183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2166-L2183","statement_sha256":"5640856529c64927be97cfc2cc0d2d7edf1868942d9088b01a6d6b2542b5e34f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9860,"rank":9860,"depth":2,"x":1493.84,"y":1153.704,"cluster":"tale-geometry"},{"id":"stacks:03P1","tag":"03P1","title":"Locality of cohomology · Lemma 03P1","summary":"Let C be a site, F an abelian sheaf on C, U an object of C, p > 0 an integer and xi ∈ H^p(U, F). Then there exists a covering U = (U_i → U)_i ∈ I of U in C such that xi |_U_i = 0 for all i ∈ I.","statement_latex":"Let $\\mathcal{C}$ be a site, $\\mathcal{F}$ an abelian sheaf on $\\mathcal{C}$,\n$U$ an object of $\\mathcal{C}$, $p > 0$ an integer and $\\xi \\in\nH^p(U, \\mathcal{F})$. Then there exists a covering\n$\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ of $U$ in $\\mathcal{C}$\nsuch that $\\xi |_{U_i} = 0$ for all $i \\in I$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03P1","source_file":"etale-cohomology.tex","source_line":2254,"source_end_line":2261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2254-L2261","statement_sha256":"96da7f66abe9ed0496ef752b5118f842cc04c128a978c678b898a039b40cdc92","origin":"The Stacks Project","memory_eligible":false,"source_rank":9861,"rank":9861,"depth":1,"x":1632.626,"y":1080.878,"cluster":"tale-geometry"},{"id":"stacks:03P2","tag":"03P2","title":"Cohomology of quasi-coherent sheaves · Theorem 03P2","summary":"Let S be a scheme and F a quasi-coherent O_S-module. Let C be either (Sch/S)_τ for τ ∈ (fppf, syntomic, smooth, etale, Zariski) or S_etale. Then H^p(S, F) = H^p_τ(S, F^a) for all p ≥ 0 where • the left hand side indicates the usual cohomology of the sheaf F on the underlying topological space of the scheme S, and • the right hand side indicates cohomology of the abelian sheaf F^a (see Proposition [Tag 03OG]) on the site C.","statement_latex":"Let $S$ be a scheme and $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_S$-module.\nLet $\\mathcal{C}$ be either $(\\Sch/S)_\\tau$ for\n$\\tau \\in \\{fppf, syntomic, smooth, \\etale, Zariski\\}$ or\n$S_\\etale$. Then\n$$\nH^p(S, \\mathcal{F}) = H^p_\\tau(S, \\mathcal{F}^a)\n$$\nfor all $p \\geq 0$ where\n\\begin{enumerate}\n\\item the left hand side indicates the usual cohomology of the sheaf\n$\\mathcal{F}$ on the underlying topological space of the scheme $S$, and\n\\item the right hand side indicates cohomology\nof the abelian sheaf $\\mathcal{F}^a$ (see\nProposition \\ref{proposition-quasi-coherent-sheaf-fpqc})\non the site $\\mathcal{C}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of quasi-coherent sheaves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03P2","source_file":"etale-cohomology.tex","source_line":2279,"source_end_line":2297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2279-L2297","statement_sha256":"cfcb112e48dcab71e64236e5c899041afcced0335db0c91b102fd00f9d6fee82","origin":"The Stacks Project","memory_eligible":false,"source_rank":9862,"rank":9862,"depth":41,"x":1588.757,"y":1213.614,"cluster":"tale-geometry"},{"id":"stacks:03P4","tag":"03P4","title":"Examples of sheaves · Definition 03P4","summary":"On any of the sites (Sch/S)_τ or S_τ of Section [Tag 03X7]. • The sheaf T ↦ Γ(T, O_T) is denoted O_S, or G_a, or G_a, S if we want to indicate the base scheme. • Similarly, the sheaf T ↦ Γ(T, O^*_T) is denoted O_S^*, or G_m, or G_m, S if we want to indicate the base scheme. • The constant sheaf underlineZ/nZ on any site is the sheafification of the constant presheaf U ↦ Z/nZ.","statement_latex":"On any of the sites $(\\Sch/S)_\\tau$ or $S_\\tau$ of\nSection \\ref{section-big-small}.\n\\begin{enumerate}\n\\item The sheaf $T \\mapsto \\Gamma(T, \\mathcal{O}_T)$ is denoted\n$\\mathcal{O}_S$, or $\\mathbf{G}_a$, or $\\mathbf{G}_{a, S}$ if we\nwant to indicate the base scheme.\n\\item Similarly, the sheaf\n$T \\mapsto \\Gamma(T, \\mathcal{O}^*_T)$ is denoted $\\mathcal{O}_S^*$, or\n$\\mathbf{G}_m$, or $\\mathbf{G}_{m, S}$ if we want\nto indicate the base scheme.\n\\item The {\\it constant sheaf} $\\underline{\\mathbf{Z}/n\\mathbf{Z}}$ on any\nsite is the sheafification of the constant presheaf\n$U \\mapsto \\mathbf{Z}/n\\mathbf{Z}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Examples of sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03P4","source_file":"etale-cohomology.tex","source_line":2403,"source_end_line":2419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2403-L2419","statement_sha256":"67497526af5c961de57a73d50790ecd42512c2a4555ef257fd6d955a8aacc6eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9863,"rank":9863,"depth":0,"x":1514.204,"y":1090.582,"cluster":"tale-geometry"},{"id":"stacks:04HS","tag":"04HS","title":"Examples of sheaves · Definition 04HS","summary":"Let S be a scheme. The structure sheaf of S is the sheaf of rings O_S on any of the sites S_Zar, S_etale, or (Sch/S)_τ discussed above.","statement_latex":"Let $S$ be a scheme. The {\\it structure sheaf} of $S$ is the sheaf of rings\n$\\mathcal{O}_S$\non any of the sites $S_{Zar}$, $S_\\etale$, or $(\\Sch/S)_\\tau$\ndiscussed above.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Examples of sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HS","source_file":"etale-cohomology.tex","source_line":2458,"source_end_line":2464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2458-L2464","statement_sha256":"21a4dfdc5b64851999d5b1249f473d2197ce3d530db96d3a937cdf0bf659c330","origin":"The Stacks Project","memory_eligible":false,"source_rank":9864,"rank":9864,"depth":0,"x":1668.446,"y":1139.104,"cluster":"tale-geometry"},{"id":"stacks:03P8","tag":"03P8","title":"Picard groups · Theorem 03P8","summary":"For any scheme X we have canonical identifications H_fppf^1(X, G_m) & = H^1_syntomic(X, G_m) & = H^1_smooth(X, G_m) & = H_etale^1(X, G_m) & = H^1_Zar(X, G_m) & = Pic(X) & = H^1(X, O_X^*)","statement_latex":"For any scheme $X$ we have canonical identifications\n\\begin{align*}\nH_{fppf}^1(X, \\mathbf{G}_m) & = H^1_{syntomic}(X, \\mathbf{G}_m) \\\\\n& = H^1_{smooth}(X, \\mathbf{G}_m) \\\\\n& = H_\\etale^1(X, \\mathbf{G}_m) \\\\\n& = H^1_{Zar}(X, \\mathbf{G}_m) \\\\\n& = \\Pic(X) \\\\\n& = H^1(X, \\mathcal{O}_X^*)\n\\end{align*}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Picard groups","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03P8","source_file":"etale-cohomology.tex","source_line":2497,"source_end_line":2508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2497-L2508","statement_sha256":"f4c11a30a36b6c3a041a6a682c24e0791ea91eecd56ae27755bd26aaec34aac8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9865,"rank":9865,"depth":23,"x":1515.365,"y":1190.955,"cluster":"tale-geometry"},{"id":"stacks:03PB","tag":"03PB","title":"Étale morphisms · Definition 03PB","summary":"A morphism of schemes is étale if it is smooth of relative dimension 0.","statement_latex":"A morphism of schemes is {\\it \\'etale} if it is smooth of relative dimension 0.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PB","source_file":"etale-cohomology.tex","source_line":2582,"source_end_line":2585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2582-L2585","statement_sha256":"4326372bb95cad14786ee22fada33487e1aedc7a2862dce1d55550b5c51f18c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9866,"rank":9866,"depth":0,"x":1586.698,"y":1065.595,"cluster":"tale-geometry"},{"id":"stacks:03PC","tag":"03PC","title":"Étale morphisms · Proposition 03PC","summary":"Facts on étale morphisms. • Let k be a field. A morphism of schemes U → Spec(k) is étale if and only if U ≅ coprod_i ∈ I Spec(k_i) such that for each i ∈ I the ring k_i is a field which is a finite separable extension of k. • Let φ : U → S be a morphism of schemes. The following conditions are equivalent: • φ is étale, • φ is locally finitely presented, flat, and all its fibres are étale, • φ is flat, unramified and locally of finite presentation. • A ring map A → B is…","statement_latex":"Facts on \\'etale morphisms.\n\\begin{enumerate}\n\\item Let $k$ be a field. A morphism of schemes $U \\to \\Spec(k)$ is\n\\'etale if and only if $U \\cong \\coprod_{i \\in I} \\Spec(k_i)$\nsuch that for each $i \\in I$\nthe ring $k_i$ is a field which is a finite separable extension of $k$.\n\\item Let $\\varphi : U \\to S$ be a morphism of schemes. The following\nconditions are equivalent:\n\\begin{enumerate}\n\\item $\\varphi$ is \\'etale,\n\\item $\\varphi$ is locally finitely presented, flat, and all its fibres are\n\\'etale,\n\\item $\\varphi$ is flat, unramified and locally of finite presentation.\n\\end{enumerate}\n\\item A ring map $A \\to B$ is \\'etale if and only if\n$B \\cong A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_n)$\nsuch that $\\Delta = \\det \\left( \\frac{\\partial f_i}{\\partial x_j} \\right)$\nis invertible in $B$.\n\\item The base change of an \\'etale morphism is \\'etale.\n\\item Compositions of \\'etale morphisms are \\'etale.\n\\item Fibre products and products of \\'etale morphisms are \\'etale.\n\\item An \\'etale morphism has relative dimension 0.\n\\item Let $Y \\to X$ be an \\'etale morphism.\nIf $X$ is reduced (respectively regular) then so is $Y$.\n\\item \\'Etale morphisms are open.\n\\item If $X \\to S$ and $Y \\to S$ are \\'etale, then any\n$S$-morphism $X \\to Y$ is also \\'etale.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale morphisms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PC","source_file":"etale-cohomology.tex","source_line":2591,"source_end_line":2621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2591-L2621","statement_sha256":"5f753c9592e0d2375315919d8da0da544a83b186b871f136606d06b50b7b9534","origin":"The Stacks Project","memory_eligible":false,"source_rank":9867,"rank":9867,"depth":46,"x":1635.01,"y":1198.789,"cluster":"tale-geometry"},{"id":"stacks:03PD","tag":"03PD","title":"Étale morphisms · Definition 03PD","summary":"A ring map A → B is called standard étale if B ≅ (A[t]/(f))_g with f, g ∈ A[t], with f monic, and df/dt invertible in B.","statement_latex":"A ring map $A \\to B$ is called {\\it standard \\'etale} if\n$B \\cong \\left(A[t]/(f)\\right)_g$ with $f, g \\in A[t]$, with $f$ monic,\nand $\\text{d}f/\\text{d}t$ invertible in $B$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PD","source_file":"etale-cohomology.tex","source_line":2643,"source_end_line":2648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2643-L2648","statement_sha256":"178c4be8db4c1e04e2c1a33602b386024f8376c13753480e7f26c521c915b4cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9868,"rank":9868,"depth":0,"x":1491.976,"y":1127.839,"cluster":"tale-geometry"},{"id":"stacks:03PE","tag":"03PE","title":"Étale morphisms · Theorem 03PE","summary":"A ring map A → B is étale at a prime q if and only if there exists g ∈ B, g not ∈ q such that B_g is standard étale over A.","statement_latex":"A ring map $A \\to B$ is \\'etale at a prime $\\mathfrak q$ if and only if there\nexists $g \\in B$, $g \\not \\in \\mathfrak q$ such that $B_g$ is standard\n\\'etale over $A$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PE","source_file":"etale-cohomology.tex","source_line":2700,"source_end_line":2705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2700-L2705","statement_sha256":"6ed8ac0537b37d162ed5e0d725dfd44709de6642897c2c80eae4c5503f9c6aef","origin":"The Stacks Project","memory_eligible":false,"source_rank":9869,"rank":9869,"depth":43,"x":1654.843,"y":1098.935,"cluster":"tale-geometry"},{"id":"stacks:03PG","tag":"03PG","title":"Étale coverings · Definition 03PG","summary":"An étale covering of a scheme U is a family of morphisms of schemes (φ_i : U_i → U)_i ∈ I such that • each φ_i is an étale morphism, • the U_i cover U, i.e., U = ⋃_i∈ Iφ_i(U_i).","statement_latex":"An {\\it \\'etale covering} of a scheme $U$ is a family of morphisms\nof schemes\n$\\{\\varphi_i : U_i \\to U\\}_{i \\in I}$ such that\n\\begin{enumerate}\n\\item each $\\varphi_i$ is an \\'etale morphism,\n\\item the $U_i$ cover $U$, i.e., $U = \\bigcup_{i\\in I}\\varphi_i(U_i)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale coverings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PG","source_file":"etale-cohomology.tex","source_line":2722,"source_end_line":2731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2722-L2731","statement_sha256":"b84f3a39c773a632aa4cc77c49551c0320b652fd781d72c472aed0af21bb9f13","origin":"The Stacks Project","memory_eligible":false,"source_rank":9870,"rank":9870,"depth":0,"x":1557.789,"y":1212.899,"cluster":"tale-geometry"},{"id":"stacks:03PH","tag":"03PH","title":"Étale coverings · Lemma 03PH","summary":"Any étale covering is an fpqc covering.","statement_latex":"Any \\'etale covering is an fpqc covering.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PH","source_file":"etale-cohomology.tex","source_line":2733,"source_end_line":2736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2733-L2736","statement_sha256":"22365628a5a5ac3edf84a4625e599ea0fabe7c7e72ae22d69edf5cfd5d36f414","origin":"The Stacks Project","memory_eligible":false,"source_rank":9871,"rank":9871,"depth":39,"x":1537.667,"y":1073.506,"cluster":"tale-geometry"},{"id":"stacks:03PI","tag":"03PI","title":"Étale coverings · Definition 03PI","summary":"(For more details see Section [Tag 03X7], or Topologies, Section [Tag 0214].) Let S be a scheme. The big étale site over S is the site (Sch/S)_etale, see Definition [Tag 03XB]. The small étale site over S is the site S_etale, see Definition [Tag 03XB]. We define similarly the big and small Zariski sites on S, denoted (Sch/S)_Zar and S_Zar.","statement_latex":"(For more details see Section \\ref{section-big-small}, or\nTopologies, Section \\ref{topologies-section-etale}.)\nLet $S$ be a scheme.\nThe {\\it big \\'etale site over $S$} is the site\n$(\\Sch/S)_\\etale$, see\nDefinition \\ref{definition-tau-site}.\nThe {\\it small \\'etale site over $S$} is the site $S_\\etale$, see\nDefinition \\ref{definition-tau-site}.\nWe define similarly the {\\it big} and {\\it small Zariski sites} on $S$,\ndenoted $(\\Sch/S)_{Zar}$ and $S_{Zar}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale coverings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PI","source_file":"etale-cohomology.tex","source_line":2760,"source_end_line":2772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2760-L2772","statement_sha256":"2b864df4bfc18ca19c09fff4a3ab6abfb5269ec2bd224dc926b681691cffeeaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9872,"rank":9872,"depth":2,"x":1664.862,"y":1165.063,"cluster":"tale-geometry"},{"id":"stacks:03PJ","tag":"03PJ","title":"Étale coverings · Proposition 03PJ","summary":"Let S be a scheme and F an abelian sheaf on (Sch/S)_etale. Then F|_S_etale is a sheaf on S_etale and H^p_etale(S, F|_S_etale) = H^p_etale(S, F) for all p ≥ 0.","statement_latex":"Let $S$ be a scheme and $\\mathcal{F}$ an abelian sheaf on\n$(\\Sch/S)_\\etale$.\nThen $\\mathcal{F}|_{S_\\etale}$ is a sheaf on $S_\\etale$ and\n$$\nH^p_\\etale(S, \\mathcal{F}|_{S_\\etale}) =\nH^p_\\etale(S, \\mathcal{F})\n$$\nfor all $p \\geq 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale coverings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PJ","source_file":"etale-cohomology.tex","source_line":2792,"source_end_line":2802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2792-L2802","statement_sha256":"46b77e7dbb8ade8aae01af44a26d6a0a676ed158ac08fee9077e3c7346d75c56","origin":"The Stacks Project","memory_eligible":false,"source_rank":9873,"rank":9873,"depth":10,"x":1497.1,"y":1169.733,"cluster":"tale-geometry"},{"id":"stacks:03PL","tag":"03PL","title":"Kummer theory · Lemma 03PL","summary":"If n∈ O_S^* then 0 → μ_n, S → G_m, S xrightarrow(·)^n G_m, S → 0 is a short exact sequence of sheaves on both the small and big étale site of S.","statement_latex":"If $n\\in \\mathcal{O}_S^*$ then\n$$\n0 \\to\n\\mu_{n, S} \\to\n\\mathbf{G}_{m, S} \\xrightarrow{(\\cdot)^n}\n\\mathbf{G}_{m, S} \\to 0\n$$\nis a short exact sequence of sheaves on both the small and\nbig \\'etale site of $S$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Kummer theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PL","source_file":"etale-cohomology.tex","source_line":2841,"source_end_line":2852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2841-L2852","statement_sha256":"545840ef727b11e79f213db8f567c0292c67debfdae5561925492956c3adcf66","origin":"The Stacks Project","memory_eligible":false,"source_rank":9874,"rank":9874,"depth":0,"x":1617.295,"y":1070.895,"cluster":"tale-geometry"},{"id":"stacks:040N","tag":"040N","title":"Kummer theory · Lemma 040N","summary":"For any n ∈ N the sequence 0 → μ_n, S → G_m, S xrightarrow(·)^n G_m, S → 0 is a short exact sequence of sheaves on the site (Sch/S)_fppf and (Sch/S)_syntomic.","statement_latex":"For any $n \\in \\mathbf{N}$ the sequence\n$$\n0 \\to\n\\mu_{n, S} \\to\n\\mathbf{G}_{m, S} \\xrightarrow{(\\cdot)^n}\n\\mathbf{G}_{m, S} \\to 0\n$$\nis a short exact sequence of sheaves on the site\n$(\\Sch/S)_{fppf}$ and $(\\Sch/S)_{syntomic}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Kummer theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040N","source_file":"etale-cohomology.tex","source_line":2920,"source_end_line":2931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L2920-L2931","statement_sha256":"046a336ed8ee563ace41a8ddbdb16a4da9c89d2e7d80045b0a548380ff1aa882","origin":"The Stacks Project","memory_eligible":false,"source_rank":9875,"rank":9875,"depth":38,"x":1608.128,"y":1212.265,"cluster":"tale-geometry"},{"id":"stacks:040Q","tag":"040Q","title":"Kummer theory · Lemma 040Q","summary":"Let S be a scheme. There is a canonical identification H_etale^1(S, μ_n) = group of pairs (L, α) up to isomorphism as above if n is invertible on S. In general we have H_fppf^1(S, μ_n) = group of pairs (L, α) up to isomorphism as above. The same result holds with fppf replaced by syntomic.","statement_latex":"Let $S$ be a scheme. There is a canonical identification\n$$\nH_\\etale^1(S, \\mu_n) =\n\\text{group of pairs }(\\mathcal{L}, \\alpha)\\text{ up to isomorphism as above}\n$$\nif $n$ is invertible on $S$. In general we have\n$$\nH_{fppf}^1(S, \\mu_n) =\n\\text{group of pairs }(\\mathcal{L}, \\alpha)\\text{ up to isomorphism as above}.\n$$\nThe same result holds with fppf replaced by syntomic.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Kummer theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040Q","source_file":"etale-cohomology.tex","source_line":3056,"source_end_line":3069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3056-L3069","statement_sha256":"b7a73a1c1e0aff304f18e2cd13bd9539e5325363693cede0295186062dd1043b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9876,"rank":9876,"depth":24,"x":1500.986,"y":1102.598,"cluster":"tale-geometry"},{"id":"stacks:03PO","tag":"03PO","title":"Neighborhoods, stalks and points · Definition 03PO","summary":"Let S be a scheme. • A geometric point of S is a morphism Spec(k) → S where k is algebraically closed. Such a point is usually denoted overlines, i.e., by an overlined small case letter. We often use overlines to denote the scheme Spec(k) as well as the morphism, and we use kappa(overlines) to denote k. • We say overlines lies over s to indicate that s ∈ S is the image of overlines. • An étale neighborhood of a geometric point overlines of S is a commutative diagram…","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item A {\\it geometric point} of $S$ is a morphism\n$\\Spec(k) \\to S$ where $k$ is algebraically closed.\nSuch a point is usually denoted $\\overline{s}$, i.e., by an overlined\nsmall case letter. We often use $\\overline{s}$ to denote the scheme\n$\\Spec(k)$ as well as the morphism, and we use $\\kappa(\\overline{s})$\nto denote $k$.\n\\item We say $\\overline{s}$ {\\it lies over} $s$\nto indicate that $s \\in S$ is the image of $\\overline{s}$.\n\\item An {\\it \\'etale neighborhood} of a geometric point $\\overline{s}$\nof $S$ is a commutative diagram\n$$\n\\xymatrix{\n& U \\ar[d]^\\varphi \\\\\n{\\overline{s}} \\ar[r]^{\\overline{s}} \\ar[ur]^{\\bar u} & S\n}\n$$\nwhere $\\varphi$ is an \\'etale morphism of schemes.\nWe write $(U, \\overline{u}) \\to (S, \\overline{s})$.\n\\item A {\\it morphism of \\'etale neighborhoods}\n$(U, \\overline{u}) \\to (U', \\overline{u}')$\nis an $S$-morphism $h: U \\to U'$\nsuch that $\\overline{u}' = h \\circ \\overline{u}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Neighborhoods, stalks and points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PO","source_file":"etale-cohomology.tex","source_line":3200,"source_end_line":3227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3200-L3227","statement_sha256":"c9b395a268f1171adfc1c47da89eb258953aae639c4ee147a97f2e8c23f9acc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":9877,"rank":9877,"depth":0,"x":1668.52,"y":1122.71,"cluster":"tale-geometry"},{"id":"stacks:03PQ","tag":"03PQ","title":"Neighborhoods, stalks and points · Lemma 03PQ","summary":"Let S be a scheme, and let overlines be a geometric point of S. The category of étale neighborhoods is cofiltered. More precisely: • Let (U_i, overlineu_i)_i = 1, 2 be two étale neighborhoods of overlines in S. Then there exists a third étale neighborhood (U, overlineu) and morphisms (U, overlineu) → (U_i, overlineu_i), i = 1, 2. • Let h_1, h_2: (U, overlineu) → (U', overlineu') be two morphisms between étale neighborhoods of overlines. Then there exist an étale…","statement_latex":"Let $S$ be a scheme, and let $\\overline{s}$ be a geometric point of $S$.\nThe category of \\'etale neighborhoods is cofiltered. More precisely:\n\\begin{enumerate}\n\\item Let $(U_i, \\overline{u}_i)_{i = 1, 2}$ be two \\'etale neighborhoods of\n$\\overline{s}$ in $S$. Then there exists a third \\'etale neighborhood\n$(U, \\overline{u})$ and morphisms\n$(U, \\overline{u}) \\to (U_i, \\overline{u}_i)$, $i = 1, 2$.\n\\item Let $h_1, h_2: (U, \\overline{u}) \\to (U', \\overline{u}')$ be two\nmorphisms between \\'etale neighborhoods of $\\overline{s}$. Then there exist an\n\\'etale neighborhood $(U'', \\overline{u}'')$ and a morphism\n$h : (U'', \\overline{u}'') \\to (U, \\overline{u})$\nwhich equalizes $h_1$ and $h_2$, i.e., such that\n$h_1 \\circ h = h_2 \\circ h$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Neighborhoods, stalks and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PQ","source_file":"etale-cohomology.tex","source_line":3259,"source_end_line":3275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3259-L3275","statement_sha256":"96950df17cc1428db2de3fb1fb15f8730ffa132b2469317c972880d85fb4306d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9878,"rank":9878,"depth":47,"x":1528.528,"y":1203.104,"cluster":"tale-geometry"},{"id":"stacks:03PR","tag":"03PR","title":"Neighborhoods, stalks and points · Lemma 03PR","summary":"Let S be a scheme. Let overlines be a geometric point of S. Let (U, overlineu) be an étale neighborhood of overlines. Let U = (φ_i : U_i → U )_i∈ I be an étale covering. Then there exist i ∈ I and overlineu_i : overlines → U_i such that φ_i : (U_i, overlineu_i) → (U, overlineu) is a morphism of étale neighborhoods.","statement_latex":"Let $S$ be a scheme.\nLet $\\overline{s}$ be a geometric point of $S$.\nLet $(U, \\overline{u})$ be an \\'etale neighborhood of $\\overline{s}$.\nLet $\\mathcal{U} = \\{\\varphi_i : U_i \\to U \\}_{i\\in I}$ be an \\'etale covering.\nThen there exist $i \\in I$ and $\\overline{u}_i : \\overline{s} \\to U_i$\nsuch that $\\varphi_i : (U_i, \\overline{u}_i) \\to (U, \\overline{u})$\nis a morphism of \\'etale neighborhoods.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Neighborhoods, stalks and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PR","source_file":"etale-cohomology.tex","source_line":3304,"source_end_line":3313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3304-L3313","statement_sha256":"7f4f355a4ec63f5500a43ea3f5d071bfb1731f1d61881fde6cca7c63a487b463","origin":"The Stacks Project","memory_eligible":false,"source_rank":9879,"rank":9879,"depth":47,"x":1567.183,"y":1064.111,"cluster":"tale-geometry"},{"id":"stacks:040R","tag":"040R","title":"Neighborhoods, stalks and points · Definition 040R","summary":"Let S be a scheme. Let F be a presheaf on S_etale. Let overlines be a geometric point of S. The stalk of F at overlines is F_overlines = colim_(U, overlineu) F(U) where (U, overlineu) runs over all étale neighborhoods of overlines in S.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{F}$ be a presheaf on $S_\\etale$.\nLet $\\overline{s}$ be a geometric point of $S$.\nThe {\\it stalk} of $\\mathcal{F}$ at $\\overline{s}$ is\n$$\n\\mathcal{F}_{\\overline{s}}\n=\n\\colim_{(U, \\overline{u})} \\mathcal{F}(U)\n$$\nwhere $(U, \\overline{u})$ runs over all \\'etale\nneighborhoods of $\\overline{s}$ in $S$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Neighborhoods, stalks and points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040R","source_file":"etale-cohomology.tex","source_line":3340,"source_end_line":3353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3340-L3353","statement_sha256":"f5d3f80aa702def5ff41ba16abf06fd6160d39a45583b35e273387d2246a2836","origin":"The Stacks Project","memory_eligible":false,"source_rank":9880,"rank":9880,"depth":0,"x":1650.619,"y":1188.784,"cluster":"tale-geometry"},{"id":"stacks:04FM","tag":"04FM","title":"Neighborhoods, stalks and points · Lemma 04FM","summary":"Let S be a scheme. Let overlines be a geometric point of S. Consider the functor u : S_etale & → Sets, U & ↦ |U_overlines| = (overlineu such that (U, overlineu) is an étale neighbourhood of overlines). Here |U_overlines| denotes the underlying set of the geometric fibre. Then u defines a point p of the site S_etale (Sites, Definition [Tag 00Y5]) and its associated stalk functor F ↦ F_p (Sites, Equation [Tag 04EH]) is the functor F ↦ F_overlines defined above.","statement_latex":"Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$.\nConsider the functor\n\\begin{align*}\nu : S_\\etale & \\longrightarrow \\textit{Sets}, \\\\\nU & \\longmapsto\n|U_{\\overline{s}}|\n=\n\\{\\overline{u} \\text{ such that }(U, \\overline{u})\n\\text{ is an \\'etale neighbourhood of }\\overline{s}\\}.\n\\end{align*}\nHere $|U_{\\overline{s}}|$ denotes the underlying set of the geometric fibre.\nThen $u$ defines a point $p$ of the site $S_\\etale$\n(Sites, Definition \\ref{sites-definition-point})\nand its associated stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$\n(Sites, Equation \\ref{sites-equation-stalk})\nis the functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$\ndefined above.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Neighborhoods, stalks and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FM","source_file":"etale-cohomology.tex","source_line":3369,"source_end_line":3388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3369-L3388","statement_sha256":"60c379914d001527d22c14998d07cc1633649efca3b09eaa636001a7402695bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9881,"rank":9881,"depth":48,"x":1488.515,"y":1144.107,"cluster":"tale-geometry"},{"id":"stacks:03PT","tag":"03PT","title":"Neighborhoods, stalks and points · Lemma 03PT","summary":"Let S be a scheme. Let overlines be a geometric point of S. • The stalk functor PAb(S_etale) → Ab, F ↦ F_overlines is exact. • We have (F^\\#)_overlines = F_overlines for any presheaf of sets F on S_etale. • The functor Ab(S_etale) → Ab, F ↦ F_overlines is exact. • Similarly the functors PSh(S_etale) → Sets and Sh(S_etale) → Sets given by the stalk functor F ↦ F_overlines are exact (see Categories, Definition [Tag 0034]) and commute with arbitrary colimits.","statement_latex":"Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$.\n\\begin{enumerate}\n\\item The stalk functor\n$\\textit{PAb}(S_\\etale) \\to \\textit{Ab}$,\n$\\mathcal{F}  \\mapsto  \\mathcal{F}_{\\overline{s}}$\nis exact.\n\\item We have $(\\mathcal{F}^\\#)_{\\overline{s}} = \\mathcal{F}_{\\overline{s}}$\nfor any presheaf of sets $\\mathcal{F}$ on $S_\\etale$.\n\\item The functor\n$\\textit{Ab}(S_\\etale) \\to \\textit{Ab}$,\n$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$ is exact.\n\\item Similarly the functors\n$\\textit{PSh}(S_\\etale) \\to \\textit{Sets}$ and\n$\\Sh(S_\\etale) \\to \\textit{Sets}$ given by the stalk functor\n$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{s}}$ are exact (see\nCategories, Definition \\ref{categories-definition-exact})\nand commute with arbitrary colimits.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Neighborhoods, stalks and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PT","source_file":"etale-cohomology.tex","source_line":3455,"source_end_line":3475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3455-L3475","statement_sha256":"97ee5a94b77b47f07a48d3b1bd2f436d12711c31f6a1907030344ba7a84b9868","origin":"The Stacks Project","memory_eligible":false,"source_rank":9882,"rank":9882,"depth":49,"x":1644.284,"y":1084.953,"cluster":"tale-geometry"},{"id":"stacks:03PU","tag":"03PU","title":"Neighborhoods, stalks and points · Theorem 03PU","summary":"Let S be a scheme. A map a : F → G of sheaves of sets is injective (resp. surjective) if and only if the map on stalks a_overlines : F_overlines → G_overlines is injective (resp. surjective) for all geometric points of S. A sequence of abelian sheaves on S_etale is exact if and only if it is exact on all stalks at geometric points of S.","statement_latex":"Let $S$ be a scheme.\nA map $a : \\mathcal{F} \\to \\mathcal{G}$ of sheaves of sets is injective\n(resp.\\ surjective) if and only if the map on stalks\n$a_{\\overline{s}} : \\mathcal{F}_{\\overline{s}} \\to \\mathcal{G}_{\\overline{s}}$\nis injective (resp.\\ surjective) for all geometric points of $S$.\nA sequence of abelian sheaves on $S_\\etale$ is exact\nif and only if it is exact on all stalks at geometric points of $S$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Neighborhoods, stalks and points","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PU","source_file":"etale-cohomology.tex","source_line":3545,"source_end_line":3554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3545-L3554","statement_sha256":"74beefe70e318ff38017ca19c18fbaaea997d05da09b33f17a711bc003533335","origin":"The Stacks Project","memory_eligible":false,"source_rank":9883,"rank":9883,"depth":50,"x":1576.86,"y":1217.219,"cluster":"tale-geometry"},{"id":"stacks:04HU","tag":"04HU","title":"Neighborhoods, stalks and points · Lemma 04HU","summary":"Let S be a scheme. • Let p be a point of the small étale site S_etale of S given by a functor u : S_etale → Sets. Then there exists a geometric point overlines of S such that p is isomorphic to the point of S_etale associated to overlines in Lemma [Tag 04FM]. • Let p : Sh(pt) → Sh(S_etale) be a point of the small étale topos of S. Then p comes from a geometric point of S, i.e., the stalk functor F ↦ F_p is isomorphic to a stalk functor as defined in Definition [Tag 040R].","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item Let $p$ be a point of the small \\'etale site\n$S_\\etale$ of $S$ given by a functor\n$u : S_\\etale \\to \\textit{Sets}$.\nThen there exists a geometric point $\\overline{s}$ of $S$ such that\n$p$ is isomorphic to the point of $S_\\etale$ associated to\n$\\overline{s}$ in\nLemma \\ref{lemma-stalk-gives-point}.\n\\item Let $p : \\Sh(pt) \\to \\Sh(S_\\etale)$ be a point\nof the small \\'etale topos of $S$. Then $p$ comes from a geometric point\nof $S$, i.e., the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$\nis isomorphic to a stalk functor as defined in\nDefinition \\ref{definition-stalk}.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Neighborhoods, stalks and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HU","source_file":"etale-cohomology.tex","source_line":3609,"source_end_line":3626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3609-L3626","statement_sha256":"e31f46f29ca622f4ee13919d6f78fe572947cc69ac24594b94bf19c855faf1ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":9884,"rank":9884,"depth":49,"x":1520.102,"y":1081.163,"cluster":"tale-geometry"},{"id":"stacks:06VX","tag":"06VX","title":"Points in other topologies · Lemma 06VX","summary":"Let S be a scheme. All of the following sites have enough points S_affine, Zar, S_Zar, S_affine, etale, S_etale, (Sch/S)_Zar, (Aff/S)_Zar, (Sch/S)_etale, (Aff/S)_etale, (Sch/S)_smooth, (Aff/S)_smooth, (Sch/S)_syntomic, (Aff/S)_syntomic, (Sch/S)_fppf, and (Aff/S)_fppf.","statement_latex":"Let $S$ be a scheme. All of the following sites have enough points\n$S_{affine, Zar}$, $S_{Zar}$, $S_{affine, \\etale}$, $S_\\etale$,\n$(\\Sch/S)_{Zar}$, $(\\textit{Aff}/S)_{Zar}$,\n$(\\Sch/S)_\\etale$, $(\\textit{Aff}/S)_\\etale$,\n$(\\Sch/S)_{smooth}$, $(\\textit{Aff}/S)_{smooth}$,\n$(\\Sch/S)_{syntomic}$, $(\\textit{Aff}/S)_{syntomic}$,\n$(\\Sch/S)_{fppf}$, and $(\\textit{Aff}/S)_{fppf}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Points in other topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VX","source_file":"etale-cohomology.tex","source_line":3810,"source_end_line":3819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3810-L3819","statement_sha256":"b21908bf648a2f1d03ccf2d40bc267cbeb0bf42a7f872b580007d086399cfd57","origin":"The Stacks Project","memory_eligible":false,"source_rank":9885,"rank":9885,"depth":51,"x":1671.661,"y":1149.416,"cluster":"tale-geometry"},{"id":"stacks:04HV","tag":"04HV","title":"Supports of abelian sheaves · Lemma 04HV","summary":"Let S be a scheme. Let F be a subsheaf of the final object of the étale topos of S (see Sites, Example [Tag 00W3]). Then there exists a unique open W ⊂ S such that F = h_W.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a subsheaf of the final\nobject of the \\'etale topos of $S$ (see\nSites, Example \\ref{sites-example-singleton-sheaf}).\nThen there exists a unique open\n$W \\subset S$ such that $\\mathcal{F} = h_W$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Supports of abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HV","source_file":"etale-cohomology.tex","source_line":3922,"source_end_line":3929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3922-L3929","statement_sha256":"1b891c08c3169b8b8c6ac8ba5d563fefed23207f42b5f19be3450231df09ad6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9886,"rank":9886,"depth":0,"x":1504.693,"y":1185.155,"cluster":"tale-geometry"},{"id":"stacks:04FR","tag":"04FR","title":"Supports of abelian sheaves · Lemma 04FR","summary":"Let S be a scheme. Let F be an abelian sheaf on S_etale. Let σ ∈ F(U) be a local section. There exists an open subset W ⊂ U such that • W ⊂ U is the largest Zariski open subset of U such that σ|_W = 0, • for every φ : V → U in S_etale we have σ|_V = 0 ⇔ φ(V) ⊂ W, • for every geometric point overlineu of U we have (U, overlineu, σ) = 0 in F_overlines ⇔ overlineu ∈ W where overlines = (U → S) ∘ overlineu.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{F}$ be an abelian sheaf on $S_\\etale$.\nLet $\\sigma \\in \\mathcal{F}(U)$ be a local section.\nThere exists an open subset $W \\subset U$ such that\n\\begin{enumerate}\n\\item $W \\subset U$ is the largest Zariski open subset of $U$ such\nthat $\\sigma|_W = 0$,\n\\item for every $\\varphi : V \\to U$ in $S_\\etale$ we have\n$$\n\\sigma|_V = 0 \\Leftrightarrow \\varphi(V) \\subset W,\n$$\n\\item for every geometric point $\\overline{u}$ of $U$ we have\n$$\n(U, \\overline{u}, \\sigma) = 0\\text{ in }\\mathcal{F}_{\\overline{s}}\n\\Leftrightarrow\n\\overline{u} \\in W\n$$\nwhere $\\overline{s} = (U \\to S) \\circ \\overline{u}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Supports of abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FR","source_file":"etale-cohomology.tex","source_line":3942,"source_end_line":3963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3942-L3963","statement_sha256":"c1ed6fea10ec75d060e4661c0ef79710b5f59660cc8213d00817a46accd5e852","origin":"The Stacks Project","memory_eligible":false,"source_rank":9887,"rank":9887,"depth":1,"x":1599.255,"y":1063.825,"cluster":"tale-geometry"},{"id":"stacks:04FS","tag":"04FS","title":"Supports of abelian sheaves · Definition 04FS","summary":"Let S be a scheme. Let F be an abelian sheaf on S_etale. • The support of F is the set of points s ∈ S such that F_overlines not = 0 for any (some) geometric point overlines lying over s. • Let σ ∈ F(U) be a section. The support of σ is the closed subset U setminus W, where W ⊂ U is the largest open subset of U on which σ restricts to zero (see Lemma [Tag 04FR]).","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{F}$ be an abelian sheaf on $S_\\etale$.\n\\begin{enumerate}\n\\item The {\\it support of $\\mathcal{F}$} is the set of\npoints $s \\in S$ such that $\\mathcal{F}_{\\overline{s}} \\not = 0$\nfor any (some) geometric point $\\overline{s}$ lying over $s$.\n\\item Let $\\sigma \\in \\mathcal{F}(U)$ be a section.\nThe {\\it support of $\\sigma$} is the closed subset $U \\setminus W$, where\n$W \\subset U$ is the largest open subset of $U$ on which $\\sigma$\nrestricts to zero (see\nLemma \\ref{lemma-zero-over-image}).\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Supports of abelian sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FS","source_file":"etale-cohomology.tex","source_line":3991,"source_end_line":4005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L3991-L4005","statement_sha256":"6655861724a4e4a4c2b56b46b8e64a9f6a8f8ea01ff7de94fd56ecb3ff32b44a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9888,"rank":9888,"depth":2,"x":1627.15,"y":1207.225,"cluster":"tale-geometry"},{"id":"stacks:04FT","tag":"04FT","title":"Supports of abelian sheaves · Lemma 04FT","summary":"Let S be a scheme. Let F be an abelian sheaf on S_etale. Let U ∈ Ob(S_etale) and σ ∈ F(U). • The support of σ is closed in U. • The support of σ + σ' is contained in the union of the supports of σ, σ' ∈ F(U). • If φ : F → G is a map of abelian sheaves on S_etale, then the support of φ(σ) is contained in the support of σ ∈ F(U). • The support of F is the union of the images of the supports of all local sections of F. • If F → G is surjective then the support of G is a…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{F}$ be an abelian sheaf on $S_\\etale$.\nLet $U \\in \\Ob(S_\\etale)$ and $\\sigma \\in \\mathcal{F}(U)$.\n\\begin{enumerate}\n\\item The support of $\\sigma$ is closed in $U$.\n\\item The support of $\\sigma + \\sigma'$ is contained in the union of\nthe supports of $\\sigma, \\sigma' \\in \\mathcal{F}(U)$.\n\\item If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a map of\nabelian sheaves on $S_\\etale$, then the support of $\\varphi(\\sigma)$\nis contained in the support of $\\sigma \\in \\mathcal{F}(U)$.\n\\item The support of $\\mathcal{F}$ is the union of the images of the\nsupports of all local sections of $\\mathcal{F}$.\n\\item If $\\mathcal{F} \\to \\mathcal{G}$ is surjective then the support\nof $\\mathcal{G}$ is a subset of the support of $\\mathcal{F}$.\n\\item If $\\mathcal{F} \\to \\mathcal{G}$ is injective then the support\nof $\\mathcal{F}$ is a subset of the support of $\\mathcal{G}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Supports of abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FT","source_file":"etale-cohomology.tex","source_line":4025,"source_end_line":4044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4025-L4044","statement_sha256":"d8534c9c304922997188cc183ca41c0d6ba131ad1739b2aeb1e480a341dc0496","origin":"The Stacks Project","memory_eligible":false,"source_rank":9889,"rank":9889,"depth":2,"x":1491.001,"y":1117.139,"cluster":"tale-geometry"},{"id":"stacks:04FU","tag":"04FU","title":"Supports of abelian sheaves · Lemma 04FU","summary":"The support of a sheaf of rings on S_etale is closed.","statement_latex":"The support of a sheaf of rings on $S_\\etale$ is closed.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Supports of abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FU","source_file":"etale-cohomology.tex","source_line":4056,"source_end_line":4059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4056-L4059","statement_sha256":"bd54bfc7b61e21a060c12a35e38002af53f16dd944ba0504ec494a615448b96a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9890,"rank":9890,"depth":0,"x":1664.171,"y":1106.293,"cluster":"tale-geometry"},{"id":"stacks:03QE","tag":"03QE","title":"Henselian rings · Theorem 03QE","summary":"Let A→ B be finite type ring map and p ⊂ A a prime ideal. Then there exist an étale ring map A → A' and a prime p' ⊂ A' lying over p such that • kappa( p) = kappa( p'), • B ⊗_A A' = B_1× … × B_r × C, • A'→ B_i is finite and there exists a unique prime q_i⊂ B_i lying over p', and • all irreducible components of the fibre Spec(C ⊗_A' kappa( p')) of C over p' have dimension at least 1.","statement_latex":"Let $A\\to B$ be finite type ring map and $\\mathfrak p \\subset A$ a prime\nideal. Then there exist an \\'etale ring map $A \\to A'$ and a prime\n$\\mathfrak p' \\subset A'$ lying over $\\mathfrak p$ such that\n\\begin{enumerate}\n\\item\n$\\kappa(\\mathfrak p) = \\kappa(\\mathfrak p')$,\n\\item\n$ B \\otimes_A A' = B_1\\times \\ldots \\times B_r \\times C$,\n\\item\n$ A'\\to B_i$ is finite and there exists a unique prime $q_i\\subset B_i$ lying\nover $\\mathfrak p'$, and\n\\item all irreducible components of the fibre\n$\\Spec(C \\otimes_{A'} \\kappa(\\mathfrak p'))$ of $C$ over $\\mathfrak p'$\nhave dimension at least 1.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Henselian rings","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QE","source_file":"etale-cohomology.tex","source_line":4079,"source_end_line":4096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4079-L4096","statement_sha256":"41a89ff44bcd8eaa25f62559892d28773a021abb8b3ddfd69de70edfb08280d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9891,"rank":9891,"depth":30,"x":1544.974,"y":1212.754,"cluster":"tale-geometry"},{"id":"stacks:03QF","tag":"03QF","title":"Henselian rings · Definition 03QF","summary":"(See Algebra, Definition [Tag 04GF].) A local ring (R, m, kappa) is called henselian if for all f ∈ R[T] monic, for all a_0 ∈ kappa such that bar f(a_0) = 0 and bar f'(a_0) ≠ 0, there exists an a ∈ R such that f(a) = 0 and a bmod m = a_0.","statement_latex":"(See Algebra, Definition \\ref{algebra-definition-henselian}.)\nA local ring $(R, \\mathfrak m, \\kappa)$ is called\n{\\it henselian} if for all\n$f \\in R[T]$ monic, for all $a_0 \\in \\kappa$ such that\n$\\bar f(a_0) = 0$ and $\\bar f'(a_0) \\neq 0$, there exists\nan $a \\in R$ such that $f(a) = 0$ and $a \\bmod \\mathfrak m = a_0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Henselian rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QF","source_file":"etale-cohomology.tex","source_line":4127,"source_end_line":4135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4127-L4135","statement_sha256":"8b50be484e63d27423cc1bc418b0e210e0d7bd941bbb013b1a3f74643e6185f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":9892,"rank":9892,"depth":1,"x":1547.259,"y":1066.338,"cluster":"tale-geometry"},{"id":"stacks:03QG","tag":"03QG","title":"Henselian rings · Theorem 03QG","summary":"Complete local rings are henselian.","statement_latex":"Complete local rings are henselian.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Henselian rings","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QG","source_file":"etale-cohomology.tex","source_line":4146,"source_end_line":4149,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4146-L4149","statement_sha256":"a57968d00cab918f0e7e0d7ad6801d5ef17c862343cfc3d11defb6409e4fdb56","origin":"The Stacks Project","memory_eligible":false,"source_rank":9893,"rank":9893,"depth":1,"x":1663.537,"y":1175.807,"cluster":"tale-geometry"},{"id":"stacks:03QH","tag":"03QH","title":"Henselian rings · Theorem 03QH","summary":"Let (R, m, kappa) be a local ring. The following are equivalent: • R is henselian, • for any f∈ R[T] and any factorization bar f = g_0 h_0 in kappa[T] with gcd(g_0, h_0)=1, there exists a factorization f = gh in R[T] with bar g = g_0 and bar h = h_0, • any finite R-algebra S is isomorphic to a finite product of local rings finite over R, • any finite type R-algebra A is isomorphic to a product A ≅ A' × C where A' ≅ A_1 × … × A_r is a product of finite local R-algebras and…","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring. The following are equivalent:\n\\begin{enumerate}\n\\item $R$ is henselian,\n\\item for any $f\\in R[T]$ and any factorization $\\bar f = g_0 h_0$ in\n$\\kappa[T]$ with $\\gcd(g_0, h_0)=1$, there exists a factorization $f = gh$ in\n$R[T]$ with $\\bar g = g_0$ and $\\bar h = h_0$,\n\\item any finite $R$-algebra $S$ is isomorphic to a finite product of\nlocal rings finite over $R$,\n\\item any finite type $R$-algebra $A$ is isomorphic to a product\n$A \\cong A' \\times C$ where $A' \\cong A_1 \\times \\ldots \\times A_r$\nis a product of finite local $R$-algebras and all the irreducible\ncomponents of $C \\otimes_R \\kappa$ have dimension at least 1,\n\\item if $A$ is an \\'etale $R$-algebra and $\\mathfrak n$ is a maximal ideal of\n$A$ lying over $\\mathfrak m$ such that $\\kappa \\cong A/\\mathfrak n$, then there\nexists an isomorphism $\\varphi : A \\cong R \\times A'$ such that\n$\\varphi(\\mathfrak n) = \\mathfrak m \\times A' \\subset R \\times A'$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Henselian rings","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QH","source_file":"etale-cohomology.tex","source_line":4156,"source_end_line":4175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4156-L4175","statement_sha256":"aebda27306179ab1458c629a36364a7da24fed927d2603c9ae99ca30a20f9536","origin":"The Stacks Project","memory_eligible":false,"source_rank":9894,"rank":9894,"depth":44,"x":1489.436,"y":1161.037,"cluster":"tale-geometry"},{"id":"stacks:03QJ","tag":"03QJ","title":"Henselian rings · Lemma 03QJ","summary":"If R is henselian and A is a finite R-algebra, then A is a finite product of henselian local rings.","statement_latex":"If $R$ is henselian and $A$ is a finite $R$-algebra, then $A$ is a finite\nproduct of henselian local rings.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Henselian rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QJ","source_file":"etale-cohomology.tex","source_line":4185,"source_end_line":4189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4185-L4189","statement_sha256":"c2d53e1b971dc80bdc3ebe2ace18cbf5201dcf5e83edc10a0bc4f3168a4bcc8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9895,"rank":9895,"depth":45,"x":1629.956,"y":1072.974,"cluster":"tale-geometry"},{"id":"stacks:03QK","tag":"03QK","title":"Henselian rings · Definition 03QK","summary":"A local ring R is called strictly henselian if it is henselian and its residue field is separably closed.","statement_latex":"A local ring $R$ is called {\\it strictly henselian} if it is henselian and its\nresidue field is separably closed.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Henselian rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QK","source_file":"etale-cohomology.tex","source_line":4196,"source_end_line":4200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4196-L4200","statement_sha256":"df3bdf0fa8a062b69b0df04b622f29b8ca59f4c66c7b02d882c5084c7236105f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9896,"rank":9896,"depth":0,"x":1597.095,"y":1217.915,"cluster":"tale-geometry"},{"id":"stacks:03QL","tag":"03QL","title":"Henselian rings · Theorem 03QL","summary":"Let (R, m, kappa) be a local ring and kappa⊂kappa^sep a separable algebraic closure. There exist canonical flat local ring maps R → R^h → R^sh where • R^h, R^sh are filtered colimits of étale R-algebras, • R^h is henselian, R^sh is strictly henselian, • m R^h (resp. m R^sh) is the maximal ideal of R^h (resp. R^sh), and • kappa = R^h/ m R^h, and kappa^sep = R^sh/ m R^sh as extensions of kappa.","statement_latex":"Let $(R, \\mathfrak m, \\kappa)$ be a local ring and\n$\\kappa\\subset\\kappa^{sep}$ a separable algebraic closure.\nThere exist canonical flat local ring maps $R \\to R^h \\to R^{sh}$ where\n\\begin{enumerate}\n\\item $R^h$, $R^{sh}$ are filtered colimits of \\'etale $R$-algebras,\n\\item $R^h$ is henselian, $R^{sh}$ is strictly henselian,\n\\item $\\mathfrak m R^h$ (resp.\\ $\\mathfrak m R^{sh}$) is the\nmaximal ideal of $R^h$ (resp.\\ $R^{sh}$), and\n\\item $\\kappa = R^h/\\mathfrak m R^h$, and\n$\\kappa^{sep} = R^{sh}/\\mathfrak m R^{sh}$ as extensions of $\\kappa$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Henselian rings","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QL","source_file":"etale-cohomology.tex","source_line":4215,"source_end_line":4228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4215-L4228","statement_sha256":"386123511ecd70093a092fc31bea078c1f483a892ec6c1dbb254964b9fb4d1be","origin":"The Stacks Project","memory_eligible":false,"source_rank":9897,"rank":9897,"depth":44,"x":1504.597,"y":1092.158,"cluster":"tale-geometry"},{"id":"stacks:04HX","tag":"04HX","title":"Stalks of the structure sheaf · Lemma 04HX","summary":"The stalk of the structure sheaf of a scheme in the etale topology is the strict henselization. Let S be a scheme. Let overlines be a geometric point of S lying over s ∈ S. Let kappa = kappa(s) and let kappa ⊂ kappa^sep ⊂ kappa(overlines) denote the separable algebraic closure of kappa in kappa(overlines). Then there is a canonical identification (O_S, s)^sh ≅ (O_S)_overlines where the left hand side is the strict henselization of the local ring O_S, s as described in…","statement_latex":"\\begin{slogan}\nThe stalk of the structure sheaf of a scheme\nin the etale topology is the strict henselization.\n\\end{slogan}\nLet $S$ be a scheme.\nLet $\\overline{s}$ be a geometric point of $S$ lying over $s \\in S$.\nLet $\\kappa = \\kappa(s)$ and let\n$\\kappa \\subset \\kappa^{sep} \\subset \\kappa(\\overline{s})$ denote\nthe separable algebraic closure of $\\kappa$ in $\\kappa(\\overline{s})$.\nThen there is a canonical identification\n$$\n(\\mathcal{O}_{S, s})^{sh}\n\\cong\n(\\mathcal{O}_S)_{\\overline{s}}\n$$\nwhere the left hand side is the strict henselization of the local ring\n$\\mathcal{O}_{S, s}$ as described in\nTheorem \\ref{theorem-henselization}\nand right hand side is the stalk of the structure sheaf\n$\\mathcal{O}_S$ on $S_\\etale$ at\nthe geometric point $\\overline{s}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Stalks of the structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HX","source_file":"etale-cohomology.tex","source_line":4256,"source_end_line":4279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4256-L4279","statement_sha256":"bb4e1f54e568e4e8d897a8449a09f10887ded34419c2d07c20b97753bf0b38fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9898,"rank":9898,"depth":46,"x":1674.249,"y":1132.479,"cluster":"tale-geometry"},{"id":"stacks:03PS","tag":"03PS","title":"Stalks of the structure sheaf · Definition 03PS","summary":"Let S be a scheme. Let overlines be a geometric point of S lying over the point s ∈ S. • The étale local ring of S at overlines is the stalk of the structure sheaf O_S on S_etale at overlines. We sometimes call this the strict henselization of O_S, s relative to the geometric point overlines. Notation used: O_S, overlines^sh. • The henselization of O_S, s is the henselization of the local ring of S at s. See Algebra, Definition [Tag 04GQ], and Theorem [Tag 03QL].…","statement_latex":"Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$\nlying over the point $s \\in S$.\n\\begin{enumerate}\n\\item The {\\it \\'etale local ring of $S$ at $\\overline{s}$}\nis the stalk of the structure sheaf $\\mathcal{O}_S$ on $S_\\etale$\nat $\\overline{s}$. We sometimes call this the\n{\\it strict henselization of $\\mathcal{O}_{S, s}$} relative\nto the geometric point $\\overline{s}$.\nNotation used: $\\mathcal{O}_{S, \\overline{s}}^{sh}$.\n\\item The {\\it henselization of $\\mathcal{O}_{S, s}$} is the\nhenselization of the local ring of $S$ at $s$. See\nAlgebra, Definition \\ref{algebra-definition-henselization},\nand\nTheorem \\ref{theorem-henselization}.\nNotation: $\\mathcal{O}_{S, s}^h$.\n\\item The {\\it strict henselization of $S$ at $\\overline{s}$}\nis the scheme $\\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh})$.\n\\item The {\\it henselization of $S$ at $s$} is the scheme\n$\\Spec(\\mathcal{O}_{S, s}^h)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Stalks of the structure sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PS","source_file":"etale-cohomology.tex","source_line":4311,"source_end_line":4333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4311-L4333","statement_sha256":"e62708ab106bdd21d2d0d67747748f8df67f2c5d3dc9df7c80bbd148e88ed5bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9899,"rank":9899,"depth":45,"x":1516.434,"y":1199.134,"cluster":"tale-geometry"},{"id":"stacks:04HY","tag":"04HY","title":"Stalks of the structure sheaf · Lemma 04HY","summary":"Let S be a scheme. Let s ∈ S. Then we have O_S, s^h = colim_(U, u) O(U) where the colimit is over the filtered category of étale neighbourhoods (U, u) of (S, s) such that kappa(s) = kappa(u).","statement_latex":"Let $S$ be a scheme. Let $s \\in S$. Then we have\n$$\n\\mathcal{O}_{S, s}^h =\n\\colim_{(U, u)} \\mathcal{O}(U)\n$$\nwhere the colimit is over the filtered category of\n\\'etale neighbourhoods $(U, u)$ of $(S, s)$ such that\n$\\kappa(s) = \\kappa(u)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Stalks of the structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HY","source_file":"etale-cohomology.tex","source_line":4356,"source_end_line":4366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4356-L4366","statement_sha256":"8fb492533017d3a0873c11a1fe5f9a63c15895ceb23a048263bf771d3c84540b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9900,"rank":9900,"depth":45,"x":1579.318,"y":1060.18,"cluster":"tale-geometry"},{"id":"stacks:04HZ","tag":"04HZ","title":"Stalks of the structure sheaf · Lemma 04HZ","summary":"Let S be a scheme. The small étale site S_etale endowed with its structure sheaf O_S is a locally ringed site, see Modules on Sites, Definition [Tag 04EU].","statement_latex":"Let $S$ be a scheme. The small \\'etale site $S_\\etale$ endowed with\nits structure sheaf $\\mathcal{O}_S$ is a locally ringed site, see\nModules on Sites, Definition \\ref{sites-modules-definition-locally-ringed}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Stalks of the structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04HZ","source_file":"etale-cohomology.tex","source_line":4392,"source_end_line":4397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4392-L4397","statement_sha256":"515028b18c60182c1be1d07c8e9b964ca79e6c4663808221d02cd65b45c83e91","origin":"The Stacks Project","memory_eligible":false,"source_rank":9901,"rank":9901,"depth":51,"x":1644.811,"y":1198.578,"cluster":"tale-geometry"},{"id":"stacks:03PW","tag":"03PW","title":"Direct images · Definition 03PW","summary":"Let f: X→ Y be a morphism of schemes. Let F a presheaf of sets on X_etale. The direct image, or pushforward of F (under f) is f_*F : Y_etale^opp → Sets, (V/Y) ↦ F(X ×_Y V/X). We sometimes write f_* = f_small, * to distinguish from other direct image functors (such as usual Zariski pushforward or f_big, *).","statement_latex":"Let $f: X\\to Y$ be a morphism of schemes.\nLet $\\mathcal{F} $ a presheaf of sets on $X_\\etale$.\nThe {\\it direct image}, or {\\it pushforward} of $\\mathcal{F}$\n(under $f$) is\n$$\nf_*\\mathcal{F} : Y_\\etale^{opp} \\longrightarrow \\textit{Sets}, \\quad\n(V/Y) \\longmapsto \\mathcal{F}(X \\times_Y V/X).\n$$\nWe sometimes write $f_* = f_{small, *}$ to distinguish from other\ndirect image functors (such as usual Zariski pushforward or $f_{big, *}$).","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Direct images","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PW","source_file":"etale-cohomology.tex","source_line":4468,"source_end_line":4480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4468-L4480","statement_sha256":"f110b6b00e2c2b938705b33c448373e789f2794cada82953880a6508a49e4b56","origin":"The Stacks Project","memory_eligible":false,"source_rank":9902,"rank":9902,"depth":0,"x":1484.929,"y":1133.569,"cluster":"tale-geometry"},{"id":"stacks:03PY","tag":"03PY","title":"Direct images · Definition 03PY","summary":"Let f: X→ Y be a morphism of schemes. Let F a sheaf of sets on X_etale. The direct image, or pushforward of F (under f) is f_*F : Y_etale^opp → Sets, (V/Y) ↦ F(X ×_Y V/X) which is a sheaf by Remark [Tag 03PX]. We sometimes write f_* = f_small, * to distinguish from other direct image functors (such as usual Zariski pushforward or f_big, *).","statement_latex":"Let $f: X\\to Y$ be a morphism of schemes.\nLet $\\mathcal{F} $ a sheaf of sets on $X_\\etale$.\nThe {\\it direct image}, or {\\it pushforward} of $\\mathcal{F}$\n(under $f$) is\n$$\nf_*\\mathcal{F} : Y_\\etale^{opp} \\longrightarrow \\textit{Sets}, \\quad\n(V/Y) \\longmapsto \\mathcal{F}(X \\times_Y V/X)\n$$\nwhich is a sheaf by\nRemark \\ref{remark-direct-image-sheaf}.\nWe sometimes write $f_* = f_{small, *}$ to distinguish from other\ndirect image functors (such as usual Zariski pushforward or $f_{big, *}$).","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Direct images","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03PY","source_file":"etale-cohomology.tex","source_line":4517,"source_end_line":4531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4517-L4531","statement_sha256":"0a462b58736a72512920c68066196bb16784cb89227b6bc1d5525de650857d42","origin":"The Stacks Project","memory_eligible":false,"source_rank":9903,"rank":9903,"depth":0,"x":1655.412,"y":1090.707,"cluster":"tale-geometry"},{"id":"stacks:04I2","tag":"04I2","title":"Direct images · Definition 04I2","summary":"Let f: X → Y be a morphism of schemes. The right derived functors (R^pf_*)_p ≥ 1 of f_* : Ab(X_etale) → Ab(Y_etale) are called higher direct images.","statement_latex":"Let $f: X \\to Y$ be a morphism of schemes.\nThe right derived functors $\\{R^pf_*\\}_{p \\geq 1}$ of\n$f_* : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$\nare called {\\it higher direct images}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Direct images","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04I2","source_file":"etale-cohomology.tex","source_line":4565,"source_end_line":4571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4565-L4571","statement_sha256":"b4b0829a1fd48c035d70a93d9370f4cff4ad728b295d9a40a61bd45a8c508793","origin":"The Stacks Project","memory_eligible":false,"source_rank":9904,"rank":9904,"depth":0,"x":1564.003,"y":1219.282,"cluster":"tale-geometry"},{"id":"stacks:03Q0","tag":"03Q0","title":"Inverse image · Definition 03Q0","summary":"Let f: X→ Y be a morphism of schemes. The inverse image, or pullback for pullbacks of sheaves of sets or sheaves of abelian groups, and we reserve f^* for pullbacks of sheaves of modules via a morphism of ringed sites/topoi. functors are the functors f^-1 = f_small^-1 : Sh(Y_etale) → Sh(X_etale) and f^-1 = f_small^-1 : Ab(Y_etale) → Ab(X_etale) which are left adjoint to f_* = f_small, *. Thus f^-1 is characterized by the fact that Hom_Sh(X_etale) (f^-1G, F) =…","statement_latex":"Let $f: X\\to Y$ be a morphism of schemes. The {\\it inverse image}, or\n{\\it pullback}\\footnote{We use the notation $f^{-1}$ for pullbacks of\nsheaves of sets or sheaves of abelian groups, and we reserve $f^*$ for\npullbacks of sheaves of modules via a morphism of ringed sites/topoi.}\nfunctors are the functors\n$$\nf^{-1} = f_{small}^{-1} :\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\Sh(X_\\etale)\n$$\nand\n$$\nf^{-1} = f_{small}^{-1} :\n\\textit{Ab}(Y_\\etale)\n\\longrightarrow\n\\textit{Ab}(X_\\etale)\n$$\nwhich are left adjoint to $f_* = f_{small, *}$. Thus\n$f^{-1}$ is characterized by the fact that\n$$\n\\Hom_{{\\Sh(X_\\etale)}} (f^{-1}\\mathcal{G}, \\mathcal{F})\n=\n\\Hom_{\\Sh(Y_\\etale)} (\\mathcal{G}, f_*\\mathcal{F})\n$$\nfunctorially, for any $\\mathcal{F} \\in \\Sh(X_\\etale)$ and\n$\\mathcal{G} \\in \\Sh(Y_\\etale)$. We similarly have\n$$\n\\Hom_{{\\textit{Ab}(X_\\etale)}} (f^{-1}\\mathcal{G}, \\mathcal{F})\n=\n\\Hom_{\\textit{Ab}(Y_\\etale)} (\\mathcal{G}, f_*\\mathcal{F})\n$$\nfor $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ and\n$\\mathcal{G} \\in \\textit{Ab}(Y_\\etale)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Inverse image","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Q0","source_file":"etale-cohomology.tex","source_line":4587,"source_end_line":4623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4587-L4623","statement_sha256":"6d580db3cb0d831df72b7a9ce24c5223a5a42e468f20cae76e043a96d0118d05","origin":"The Stacks Project","memory_eligible":false,"source_rank":9905,"rank":9905,"depth":0,"x":1527.946,"y":1072.34,"cluster":"tale-geometry"},{"id":"stacks:03Q1","tag":"03Q1","title":"Inverse image · Lemma 03Q1","summary":"Let f : X → Y be a morphism of schemes. • The functor f^-1 : Ab(Y_etale) → Ab(X_etale) is exact. • The functor f^-1 : Sh(Y_etale) → Sh(X_etale) is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition [Tag 0034]. • Let overlinex → X be a geometric point. Let G be a sheaf on Y_etale. Then there is a canonical identification (f^-1G)_overlinex = G_overliney. where overliney = f ∘ overlinex. • For any V → Y étale we have f^-1h_V = h_X ×_Y V.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\n\\begin{enumerate}\n\\item The functor\n$f^{-1} : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$\nis exact.\n\\item The functor\n$f^{-1} : \\Sh(Y_\\etale) \\to \\Sh(X_\\etale)$\nis exact, i.e., it commutes with finite limits and colimits, see\nCategories, Definition \\ref{categories-definition-exact}.\n\\item Let $\\overline{x} \\to X$ be a geometric point.\nLet $\\mathcal{G}$ be a sheaf on $Y_\\etale$.\nThen there is a canonical identification\n$$\n(f^{-1}\\mathcal{G})_{\\overline{x}} = \\mathcal{G}_{\\overline{y}}.\n$$\nwhere $\\overline{y} = f \\circ \\overline{x}$.\n\\item For any $V \\to Y$ \\'etale we have $f^{-1}h_V = h_{X \\times_Y V}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Inverse image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Q1","source_file":"etale-cohomology.tex","source_line":4648,"source_end_line":4668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4648-L4668","statement_sha256":"3c40909342f177413f59a37e47fc74c7b80f1c0fcd8550eaf24e25ad01d98421","origin":"The Stacks Project","memory_eligible":false,"source_rank":9906,"rank":9906,"depth":51,"x":1672.961,"y":1160.391,"cluster":"tale-geometry"},{"id":"stacks:09XM","tag":"09XM","title":"Comparing topologies · Lemma 09XM","summary":"Let S be a scheme. Let F be a sheaf of sets on S_etale. Let s, t ∈ F(S). Then there exists an open W ⊂ S characterized by the following property: A morphism f : T → S factors through W if and only if s|_T = t|_T (restriction is pullback by f_small).","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a sheaf of sets on $S_\\etale$.\nLet $s, t \\in \\mathcal{F}(S)$. Then there exists an open $W \\subset S$\ncharacterized by the following property: A morphism $f : T \\to S$\nfactors through $W$ if and only if $s|_T = t|_T$ (restriction is\npullback by $f_{small}$).","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XM","source_file":"etale-cohomology.tex","source_line":4937,"source_end_line":4944,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4937-L4944","statement_sha256":"b11f6e3433f9d7580f31318b4920f23414dcec19a8ac0adce9c190e7e27d796b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9907,"rank":9907,"depth":52,"x":1494.902,"y":1177.78,"cluster":"tale-geometry"},{"id":"stacks:09XN","tag":"09XN","title":"Comparing topologies · Lemma 09XN","summary":"Let S be a scheme. Let τ ∈ (Zariski, etale). Consider the morphism π_S : (Sch/S)_τ → S_τ of Topologies, Lemma [Tag 020Z] or [Tag 021G]. Let F be a sheaf on S_τ. Then π_S^-1F is given by the rule (π_S^-1F)(T) = Γ(T_τ, f_small^-1F) where f : T → S. Moreover, π_S^-1F satisfies the sheaf condition with respect to fpqc coverings.","statement_latex":"Let $S$ be a scheme. Let $\\tau \\in \\{Zariski, \\etale\\}$. Consider the morphism\n$$\n\\pi_S : (\\Sch/S)_\\tau \\longrightarrow S_\\tau\n$$\nof Topologies, Lemma \\ref{topologies-lemma-at-the-bottom} or\n\\ref{topologies-lemma-at-the-bottom-etale}. Let $\\mathcal{F}$ be a sheaf on\n$S_\\tau$. Then $\\pi_S^{-1}\\mathcal{F}$ is given by the rule\n$$\n(\\pi_S^{-1}\\mathcal{F})(T) = \\Gamma(T_\\tau, f_{small}^{-1}\\mathcal{F})\n$$\nwhere $f : T \\to S$. Moreover, $\\pi_S^{-1}\\mathcal{F}$ satisfies the\nsheaf condition with respect to fpqc coverings.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XN","source_file":"etale-cohomology.tex","source_line":4959,"source_end_line":4973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L4959-L4973","statement_sha256":"eded84137220b5db5737f5693f6fae321e13e9e4a8a3c55acbbdb02391fb5ae4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9908,"rank":9908,"depth":53,"x":1612.427,"y":1063.719,"cluster":"tale-geometry"},{"id":"stacks:0A3H","tag":"0A3H","title":"Comparing topologies · Lemma 0A3H","summary":"Let S be a scheme. Let f : T → S be a morphism such that • f is flat and quasi-compact, and • the geometric fibres of f are connected. Let F be a sheaf on S_etale. Then Γ(S, F) = Γ(T, f^-1_smallF).","statement_latex":"Let $S$ be a scheme. Let $f : T \\to S$ be a morphism such that\n\\begin{enumerate}\n\\item $f$ is flat and quasi-compact, and\n\\item the geometric fibres of $f$ are connected.\n\\end{enumerate}\nLet $\\mathcal{F}$ be a sheaf on $S_\\etale$.\nThen $\\Gamma(S, \\mathcal{F}) = \\Gamma(T, f^{-1}_{small}\\mathcal{F})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3H","source_file":"etale-cohomology.tex","source_line":5029,"source_end_line":5038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5029-L5038","statement_sha256":"d883028e7b4b21dc7f1a97ca5a7a0df6434666ccff01f0c5365826851020d55a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9909,"rank":9909,"depth":54,"x":1617.499,"y":1214.781,"cluster":"tale-geometry"},{"id":"stacks:0EZK","tag":"0EZK","title":"Comparing topologies · Lemma 0EZK","summary":"Let S be a scheme. Let f : X → S be a morphism such that • f is submersive, and • the geometric fibres of f are connected. Let F be a sheaf on S_etale. Then Γ(S, F) = Γ(X, f^-1_smallF).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to S$ be a morphism such that\n\\begin{enumerate}\n\\item $f$ is submersive, and\n\\item the geometric fibres of $f$ are connected.\n\\end{enumerate}\nLet $\\mathcal{F}$ be a sheaf on $S_\\etale$.\nThen $\\Gamma(S, \\mathcal{F}) = \\Gamma(X, f^{-1}_{small}\\mathcal{F})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZK","source_file":"etale-cohomology.tex","source_line":5076,"source_end_line":5085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5076-L5085","statement_sha256":"16c54a7881109cb114652760a96c422ee8f19915f20f93ff22667059454d527d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9910,"rank":9910,"depth":53,"x":1492.057,"y":1106.075,"cluster":"tale-geometry"},{"id":"stacks:0A3I","tag":"0A3I","title":"Comparing topologies · Lemma 0A3I","summary":"Let K/k be an extension of fields with k separably algebraically closed. Let S be a scheme over k. Denote p : S_K = S ×_Spec(k) Spec(K) → S the projection. Let F be a sheaf on S_etale. Then Γ(S, F) = Γ(S_K, p^-1_smallF).","statement_latex":"Let $K/k$ be an extension of fields with $k$ separably\nalgebraically closed. Let $S$ be a scheme over $k$. Denote\n$p : S_K = S \\times_{\\Spec(k)} \\Spec(K) \\to S$ the projection.\nLet $\\mathcal{F}$ be a sheaf on $S_\\etale$.\nThen $\\Gamma(S, \\mathcal{F}) = \\Gamma(S_K, p^{-1}_{small}\\mathcal{F})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3I","source_file":"etale-cohomology.tex","source_line":5142,"source_end_line":5149,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5142-L5149","statement_sha256":"8196b838ff1591c064c2b21303b814adad9a87c29b3e62038b58b57da98f3867","origin":"The Stacks Project","memory_eligible":false,"source_rank":9911,"rank":9911,"depth":55,"x":1672.292,"y":1115.07,"cluster":"tale-geometry"},{"id":"stacks:04I5","tag":"04I5","title":"Recovering morphisms · Lemma 04I5","summary":"Let f : X → Y be a morphism of schemes. The morphism of ringed sites (f_small, f_small^sharp) associated to f is a morphism of locally ringed sites, see Modules on Sites, Definition [Tag 04HA].","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nThe morphism of ringed sites $(f_{small}, f_{small}^\\sharp)$\nassociated to $f$ is a morphism of locally ringed sites, see\nModules on Sites,\nDefinition \\ref{sites-modules-definition-morphism-locally-ringed-topoi}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Recovering morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04I5","source_file":"etale-cohomology.tex","source_line":5177,"source_end_line":5184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5177-L5184","statement_sha256":"bb258c02e0b356d9d7540ad3cd4209ff7548df73fc11213c8075a5c08bc44523","origin":"The Stacks Project","memory_eligible":false,"source_rank":9912,"rank":9912,"depth":52,"x":1531.909,"y":1210.876,"cluster":"tale-geometry"},{"id":"stacks:04IJ","tag":"04IJ","title":"Recovering morphisms · Lemma 04IJ","summary":"Let X, Y be schemes. Let f : X → Y be a morphism of schemes. Let t be a 2-morphism from (f_small, f_small^sharp) to itself, see Modules on Sites, Definition [Tag 04IC]. Then t = id.","statement_latex":"Let $X$, $Y$ be schemes. Let $f : X \\to Y$ be a morphism of schemes.\nLet $t$ be a $2$-morphism from $(f_{small}, f_{small}^\\sharp)$ to itself, see\nModules on Sites,\nDefinition \\ref{sites-modules-definition-2-morphism-ringed-topoi}.\nThen $t = \\text{id}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Recovering morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04IJ","source_file":"etale-cohomology.tex","source_line":5226,"source_end_line":5233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5226-L5233","statement_sha256":"0c8a06ccf12dc06e970a5971193703054f21155bdea35bbec40ccab7448f82bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9913,"rank":9913,"depth":52,"x":1558.428,"y":1060.31,"cluster":"tale-geometry"},{"id":"stacks:04LW","tag":"04LW","title":"Recovering morphisms · Lemma 04LW","summary":"Let X, Y be schemes. Any two morphisms a, b : X → Y of schemes for which there exists a 2-isomorphism (a_small, a_small^sharp) ≅ (b_small, b_small^sharp) in the 2-category of ringed topoi are equal.","statement_latex":"Let $X$, $Y$ be schemes.\nAny two morphisms $a, b : X \\to Y$ of schemes\nfor which there exists a $2$-isomorphism\n$(a_{small}, a_{small}^\\sharp) \\cong (b_{small}, b_{small}^\\sharp)$\nin the $2$-category of ringed topoi are equal.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Recovering morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04LW","source_file":"etale-cohomology.tex","source_line":5297,"source_end_line":5304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5297-L5304","statement_sha256":"b5f1f22a5804057b312e2869a9c106ec757ed49ecfcb60c27c4438c5aa4aa499","origin":"The Stacks Project","memory_eligible":false,"source_rank":9914,"rank":9914,"depth":0,"x":1660.132,"y":1186.603,"cluster":"tale-geometry"},{"id":"stacks:04I6","tag":"04I6","title":"Recovering morphisms · Lemma 04I6","summary":"Let X, Y be affine schemes. Let (g, g^\\#) : (Sh(X_etale), O_X) → (Sh(Y_etale), O_Y) be a morphism of locally ringed topoi. Then there exists a unique morphism of schemes f : X → Y such that (g, g^\\#) is 2-isomorphic to (f_small, f_small^sharp), see Modules on Sites, Definition [Tag 04IC].","statement_latex":"Let $X$, $Y$ be affine schemes.\nLet\n$$\n(g, g^\\#) :\n(\\Sh(X_\\etale), \\mathcal{O}_X)\n\\longrightarrow\n(\\Sh(Y_\\etale), \\mathcal{O}_Y)\n$$\nbe a morphism of locally ringed topoi. Then there exists a\nunique morphism of schemes $f : X \\to Y$ such that\n$(g, g^\\#)$ is $2$-isomorphic to $(f_{small}, f_{small}^\\sharp)$,\nsee\nModules on Sites,\nDefinition \\ref{sites-modules-definition-2-morphism-ringed-topoi}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Recovering morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04I6","source_file":"etale-cohomology.tex","source_line":5346,"source_end_line":5362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5346-L5362","statement_sha256":"3ecdf65e24a4006e22c7323f2ce5a35e19dc1bf617fbbce5394ed284e6865daf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9915,"rank":9915,"depth":51,"x":1483.267,"y":1151.123,"cluster":"tale-geometry"},{"id":"stacks:04I7","tag":"04I7","title":"Recovering morphisms · Theorem 04I7","summary":"Let X, Y be schemes. Let (g, g^\\#) : (Sh(X_etale), O_X) → (Sh(Y_etale), O_Y) be a morphism of locally ringed topoi. Then there exists a unique morphism of schemes f : X → Y such that (g, g^\\#) is isomorphic to (f_small, f_small^sharp). In other words, the construction Sch → Locally ringed topoi, X → (X_etale, O_X) is fully faithful (morphisms up to 2-isomorphisms on the right hand side).","statement_latex":"Let $X$, $Y$ be schemes. Let\n$$\n(g, g^\\#) :\n(\\Sh(X_\\etale), \\mathcal{O}_X)\n\\longrightarrow\n(\\Sh(Y_\\etale), \\mathcal{O}_Y)\n$$\nbe a morphism of locally ringed topoi. Then there exists a\nunique morphism of schemes $f : X \\to Y$ such that\n$(g, g^\\#)$ is isomorphic to $(f_{small}, f_{small}^\\sharp)$.\nIn other words, the construction\n$$\n\\Sch \\longrightarrow \\textit{Locally ringed topoi},\n\\quad\nX \\longrightarrow (X_\\etale, \\mathcal{O}_X)\n$$\nis fully faithful (morphisms up to $2$-isomorphisms on the right hand side).","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Recovering morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04I7","source_file":"etale-cohomology.tex","source_line":5552,"source_end_line":5571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5552-L5571","statement_sha256":"a3a3beed5db5c926aa3d7158acd0107586e84dba2e2070d26a679603ba9ef1a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9916,"rank":9916,"depth":53,"x":1642.492,"y":1076.8,"cluster":"tale-geometry"},{"id":"stacks:04DK","tag":"04DK","title":"Property (A) · Lemma 04DK","summary":"Let f : X → Y be a morphism of schemes. Assume (A). • f_small, * : Ab(X_etale) → Ab(Y_etale) reflects injections and surjections, • f_small^-1f_small, *F → F is surjective for any abelian sheaf F on X_etale, • f_small, * : Ab(X_etale) → Ab(Y_etale) is faithful.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume (A).\n\\begin{enumerate}\n\\item\n$f_{small, *} :\n\\textit{Ab}(X_\\etale)\n\\to\n\\textit{Ab}(Y_\\etale)$\nreflects injections and surjections,\n\\item $f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$\nis surjective for any abelian sheaf $\\mathcal{F}$ on $X_\\etale$,\n\\item\n$f_{small, *} :\n\\textit{Ab}(X_\\etale)\n\\to\n\\textit{Ab}(Y_\\etale)$\nis faithful.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (A)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DK","source_file":"etale-cohomology.tex","source_line":5737,"source_end_line":5757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5737-L5757","statement_sha256":"0af230dbebf952140401f411c516c7826ad350623dfb755b1642392e997ed4e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9917,"rank":9917,"depth":1,"x":1584.751,"y":1222.204,"cluster":"tale-geometry"},{"id":"stacks:04DL","tag":"04DL","title":"Property (A) · Lemma 04DL","summary":"Let f : X → Y be a separated locally quasi-finite morphism of schemes. Then property (A) above holds.","statement_latex":"Let $f : X \\to Y$ be a separated locally quasi-finite morphism of schemes.\nThen property (A) above holds.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (A)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DL","source_file":"etale-cohomology.tex","source_line":5794,"source_end_line":5798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5794-L5798","statement_sha256":"620657d4e0cdca2e48eddf73d4b5de246e1e1dfad64ae855cfa79d349a8d0aa8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9918,"rank":9918,"depth":45,"x":1510.27,"y":1081.98,"cluster":"tale-geometry"},{"id":"stacks:04DM","tag":"04DM","title":"Property (A) · Lemma 04DM","summary":"Let f : X → Y be an integral morphism of schemes. Then property (A) holds.","statement_latex":"Let $f : X \\to Y$ be an integral morphism of schemes.\nThen property (A) holds.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (A)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DM","source_file":"etale-cohomology.tex","source_line":5851,"source_end_line":5855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5851-L5855","statement_sha256":"a7999df9b57a83b5426e49ba095557e064db7b402204fc9c75df27eb685bc48b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9919,"rank":9919,"depth":46,"x":1678.244,"y":1143.222,"cluster":"tale-geometry"},{"id":"stacks:04C9","tag":"04C9","title":"Property (A) · Lemma 04C9","summary":"Let f : X → Y be a morphism of schemes. Denote f_small : Sh(X_etale) → Sh(Y_etale) the associated morphism of small étale topoi. Assume at least one of the following • f is integral, or • f is separated and locally quasi-finite. Then the functor f_small, * : Ab(X_etale) → Ab(Y_etale) has the following properties • the map f_small^-1f_small, *F → F is always surjective, • f_small, * is faithful, and • f_small, * reflects injections and surjections.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Denote\n$f_{small} :\n\\Sh(X_\\etale)\n\\to\n\\Sh(Y_\\etale)$\nthe associated morphism of small \\'etale topoi. Assume at least one\nof the following\n\\begin{enumerate}\n\\item $f$ is integral, or\n\\item $f$ is separated and locally quasi-finite.\n\\end{enumerate}\nThen the functor\n$f_{small, *} :\n\\textit{Ab}(X_\\etale)\n\\to\n\\textit{Ab}(Y_\\etale)$\nhas the following properties\n\\begin{enumerate}\n\\item the map\n$f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$\nis always surjective,\n\\item $f_{small, *}$ is faithful, and\n\\item $f_{small, *}$ reflects injections and surjections.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (A)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04C9","source_file":"etale-cohomology.tex","source_line":5875,"source_end_line":5901,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5875-L5901","statement_sha256":"69366b41a6b95d24ab16396dd625c997e6d4c042e250ff0979a4d250487f5ac8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9920,"rank":9920,"depth":47,"x":1504.837,"y":1193.461,"cluster":"tale-geometry"},{"id":"stacks:04DO","tag":"04DO","title":"Property (B) · Lemma 04DO","summary":"Let f : X → Y be a morphism of schemes. Assume (B) holds. Then the functor f_small, * : Sh(X_etale) → Sh(Y_etale) transforms surjections into surjections.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume (B) holds.\nThen the functor\n$f_{small, *} :\n\\Sh(X_\\etale)\n\\to\n\\Sh(Y_\\etale)$\ntransforms surjections into surjections.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (B)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DO","source_file":"etale-cohomology.tex","source_line":5919,"source_end_line":5928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5919-L5928","statement_sha256":"e01de72809da1ed931a89c2bd4da35688513598a1679390e65df539d86717938","origin":"The Stacks Project","memory_eligible":false,"source_rank":9921,"rank":9921,"depth":1,"x":1592.453,"y":1057.789,"cluster":"tale-geometry"},{"id":"stacks:04DP","tag":"04DP","title":"Property (B) · Lemma 04DP","summary":"Let f : X → Y be a morphism of schemes. Suppose • V → Y is an étale morphism of schemes, • (U_i → X ×_Y V) is an étale covering, and • v ∈ V is a point. Assume that for any such data there exists an étale neighbourhood (V', v') → (V, v), a disjoint union decomposition X ×_Y V' = coprod W'_i, and morphisms W'_i → U_i over X ×_Y V. Then property (B) holds.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Suppose\n\\begin{enumerate}\n\\item $V \\to Y$ is an \\'etale morphism of schemes,\n\\item $\\{U_i \\to X \\times_Y V\\}$ is an \\'etale covering, and\n\\item $v \\in V$ is a point.\n\\end{enumerate}\nAssume that for any such data there exists an \\'etale neighbourhood\n$(V', v') \\to (V, v)$, a disjoint union decomposition\n$X \\times_Y V' = \\coprod W'_i$, and morphisms $W'_i \\to U_i$\nover $X \\times_Y V$. Then property (B) holds.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (B)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DP","source_file":"etale-cohomology.tex","source_line":5935,"source_end_line":5947,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5935-L5947","statement_sha256":"21f1c57758c5933b8fafc5e61edabcee5d0e43d3bd87091b90c85bc7bfe4f13d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9922,"rank":9922,"depth":0,"x":1637.026,"y":1207.802,"cluster":"tale-geometry"},{"id":"stacks:04DQ","tag":"04DQ","title":"Property (B) · Lemma 04DQ","summary":"Let f : X → Y be a finite morphism of schemes. Then property (B) holds.","statement_latex":"Let $f : X \\to Y$ be a finite morphism of schemes.\nThen property (B) holds.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (B)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DQ","source_file":"etale-cohomology.tex","source_line":5953,"source_end_line":5957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L5953-L5957","statement_sha256":"67c1dcb8d236d17e45eee8be5f722f5ae2311cdb2943b7d6d7b9f93d7eacad97","origin":"The Stacks Project","memory_eligible":false,"source_rank":9923,"rank":9923,"depth":47,"x":1483.262,"y":1122.331,"cluster":"tale-geometry"},{"id":"stacks:04DR","tag":"04DR","title":"Property (B) · Lemma 04DR","summary":"Let f : X → Y be an integral morphism of schemes. Then property (B) holds.","statement_latex":"Let $f : X \\to Y$ be an integral morphism of schemes.\nThen property (B) holds.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (B)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DR","source_file":"etale-cohomology.tex","source_line":6025,"source_end_line":6029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6025-L6029","statement_sha256":"f177bd6a1c7714b25d8bec9ffadfb444b8d2df378b2b10c08495b05b27590deb","origin":"The Stacks Project","memory_eligible":false,"source_rank":9924,"rank":9924,"depth":48,"x":1665.686,"y":1098.067,"cluster":"tale-geometry"},{"id":"stacks:04C2","tag":"04C2","title":"Property (B) · Lemma 04C2","summary":"Let f : X → Y be a morphism of schemes. Assume f is integral (for example finite). Then • f_small, * transforms surjections into surjections (on sheaves of sets and on abelian sheaves), • f_small^-1f_small, *F → F is surjective for any abelian sheaf F on X_etale, • f_small, * : Ab(X_etale) → Ab(Y_etale) is faithful and reflects injections and surjections, and • f_small, * : Ab(X_etale) → Ab(Y_etale) is exact.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes.\nAssume $f$ is integral (for example finite).\nThen\n\\begin{enumerate}\n\\item $f_{small, *}$ transforms surjections into surjections (on sheaves\nof sets and on abelian sheaves),\n\\item $f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$\nis surjective for any abelian sheaf $\\mathcal{F}$ on $X_\\etale$,\n\\item\n$f_{small, *} :\n\\textit{Ab}(X_\\etale)\n\\to\n\\textit{Ab}(Y_\\etale)$\nis faithful and reflects injections and surjections, and\n\\item\n$f_{small, *} :\n\\textit{Ab}(X_\\etale)\n\\to\n\\textit{Ab}(Y_\\etale)$\nis exact.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (B)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04C2","source_file":"etale-cohomology.tex","source_line":6055,"source_end_line":6078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6055-L6078","statement_sha256":"159c794ed2ddfd8cbc924d8b6df468caa968a8016248f4f3c52cf6bfb1acbdbf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9925,"rank":9925,"depth":49,"x":1550.489,"y":1219.675,"cluster":"tale-geometry"},{"id":"stacks:04DT","tag":"04DT","title":"Property (C) · Lemma 04DT","summary":"Let f : X → Y be a morphism of schemes. Assume (C) holds. Then the functor f_small, * : Sh(X_etale) → Sh(Y_etale) reflects injections and surjections.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume (C) holds. Then the functor\n$f_{small, *} :\n\\Sh(X_\\etale)\n\\to\n\\Sh(Y_\\etale)$\nreflects injections and surjections.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (C)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DT","source_file":"etale-cohomology.tex","source_line":6103,"source_end_line":6111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6103-L6111","statement_sha256":"71fa35f57ee984a30a65b3a06758167534cc6b9dbc54be48666aa29653fce512","origin":"The Stacks Project","memory_eligible":false,"source_rank":9926,"rank":9926,"depth":1,"x":1537.618,"y":1064.377,"cluster":"tale-geometry"},{"id":"stacks:04DV","tag":"04DV","title":"Property (C) · Lemma 04DV","summary":"Let f : X → Y be a morphism of schemes. Assume that for any V → Y étale we have that • X ×_Y V → V has property (C), and • X ×_Y V → V is closed. Then the functor Y_etale → X_etale, V ↦ X ×_Y V is almost cocontinuous, see Sites, Definition [Tag 04B7].","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume that\nfor any $V \\to Y$ \\'etale we have that\n\\begin{enumerate}\n\\item $X \\times_Y V \\to V$ has property (C), and\n\\item $X \\times_Y V \\to V$ is closed.\n\\end{enumerate}\nThen the functor\n$Y_\\etale \\to X_\\etale$, $V \\mapsto X \\times_Y V$\nis almost cocontinuous, see\nSites, Definition \\ref{sites-definition-almost-cocontinuous}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (C)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DV","source_file":"etale-cohomology.tex","source_line":6131,"source_end_line":6143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6131-L6143","statement_sha256":"8a580ec53a6eb398ac30a1578471c0b9664bff7687031fa3757d6b2f241dae43","origin":"The Stacks Project","memory_eligible":false,"source_rank":9927,"rank":9927,"depth":1,"x":1672.22,"y":1171.768,"cluster":"tale-geometry"},{"id":"stacks:04DW","tag":"04DW","title":"Property (C) · Lemma 04DW","summary":"Let f : X → Y be an integral morphism of schemes which defines a homeomorphism of X with a closed subset of Y. Then property (C) holds.","statement_latex":"Let $f : X \\to Y$ be an integral morphism of schemes which defines\na homeomorphism of $X$ with a closed subset of $Y$.\nThen property (C) holds.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (C)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DW","source_file":"etale-cohomology.tex","source_line":6161,"source_end_line":6166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6161-L6166","statement_sha256":"16b14662c3ba5b134010e01b3be99a1da2f08a1a4af2c678c7a3e5341b18d04f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9928,"rank":9928,"depth":49,"x":1486.296,"y":1168.951,"cluster":"tale-geometry"},{"id":"stacks:04DX","tag":"04DX","title":"Property (C) · Lemma 04DX","summary":"Let f : X → Y be a morphism of schemes. Assume that f is universally injective and integral (for example a closed immersion). Then • f_small, * : Sh(X_etale) → Sh(Y_etale) reflects injections and surjections, • f_small, * : Sh(X_etale) → Sh(Y_etale) commutes with pushouts and coequalizers (and more generally finite connected colimits), • f_small, * transforms surjections into surjections (on sheaves of sets and on abelian sheaves), • the map f_small^-1f_small, *F → F is…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Assume that $f$ is\nuniversally injective and integral (for example a closed immersion).\nThen\n\\begin{enumerate}\n\\item\n$f_{small, *} :\n\\Sh(X_\\etale)\n\\to\n\\Sh(Y_\\etale)$\nreflects injections and surjections,\n\\item\n$f_{small, *} :\n\\Sh(X_\\etale)\n\\to\n\\Sh(Y_\\etale)$\ncommutes with pushouts and coequalizers (and more generally\nfinite connected colimits),\n\\item $f_{small, *}$ transforms surjections into surjections (on sheaves\nof sets and on abelian sheaves),\n\\item the map\n$f_{small}^{-1}f_{small, *}\\mathcal{F} \\to \\mathcal{F}$\nis surjective for any sheaf (of sets or of abelian groups)\n$\\mathcal{F}$ on $X_\\etale$,\n\\item the functor $f_{small, *}$ is faithful (on sheaves of sets and\non abelian sheaves),\n\\item\n$f_{small, *} :\n\\textit{Ab}(X_\\etale)\n\\to\n\\textit{Ab}(Y_\\etale)$\nis exact, and\n\\item the functor\n$Y_\\etale \\to X_\\etale$, $V \\mapsto X \\times_Y V$ is\nalmost cocontinuous.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Property (C)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DX","source_file":"etale-cohomology.tex","source_line":6204,"source_end_line":6241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6204-L6241","statement_sha256":"200df9c2541541c109cc1108d1ffeb77c39b2c9b84150c38279383f92016b350","origin":"The Stacks Project","memory_eligible":false,"source_rank":9929,"rank":9929,"depth":50,"x":1625.89,"y":1065.359,"cluster":"tale-geometry"},{"id":"stacks:0BTY","tag":"0BTY","title":"Topological invariance of the small étale site · Theorem 0BTY","summary":"Let X and Y be two schemes over a base scheme S. Let S' → S be a universal homeomorphism. Denote X' (resp. Y') the base change to S'. If X is étale over S, then the map Mor_S(Y, X) → Mor_S'(Y', X') is bijective.","statement_latex":"Let $X$ and $Y$ be two schemes over a base scheme $S$. Let\n$S' \\to S$ be a universal homeomorphism.\nDenote $X'$ (resp.\\ $Y'$) the base change to $S'$.\nIf $X$ is \\'etale over $S$, then the map\n$$\n\\Mor_S(Y, X) \\longrightarrow \\Mor_{S'}(Y', X')\n$$\nis bijective.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Topological invariance of the small étale site","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BTY","source_file":"etale-cohomology.tex","source_line":6283,"source_end_line":6293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6283-L6293","statement_sha256":"8d81d4277bd6d2fce5965178100fe1570b9773d26b78bb4252d58e0213dc89af","origin":"The Stacks Project","memory_eligible":false,"source_rank":9930,"rank":9930,"depth":46,"x":1606.229,"y":1221.212,"cluster":"tale-geometry"},{"id":"stacks:04DZ","tag":"04DZ","title":"Topological invariance of the small étale site · Theorem 04DZ","summary":"[EGA] Let f : X → Y be a morphism of schemes. Assume f is integral, universally injective and surjective (i.e., f is a universal homeomorphism, see Morphisms, Lemma [Tag 04DF]). The functor V ↦ V_X = X ×_Y V defines an equivalence of categories ( schemes V étale over Y ) ↔ ( schemes U étale over X )","statement_latex":"\\begin{reference}\n\\cite[IV Theorem 18.1.2]{EGA}\n\\end{reference}\nLet $f : X \\to Y$ be a morphism of schemes.\nAssume $f$ is integral, universally injective and surjective\n(i.e., $f$ is a universal homeomorphism, see\nMorphisms, Lemma \\ref{morphisms-lemma-universal-homeomorphism}).\nThe functor\n$$\nV \\longmapsto V_X = X \\times_Y V\n$$\ndefines an equivalence of categories\n$$\n\\{\n\\text{schemes }V\\text{ \\'etale over }Y\n\\}\n\\leftrightarrow\n\\{\n\\text{schemes }U\\text{ \\'etale over }X\n\\}\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Topological invariance of the small étale site","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04DZ","source_file":"etale-cohomology.tex","source_line":6353,"source_end_line":6376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6353-L6376","statement_sha256":"375e7d8914c3cb579196e22eef0956eab9b3a6ba3acc2fcd2a7931793487df6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9931,"rank":9931,"depth":21,"x":1495.211,"y":1094.924,"cluster":"tale-geometry"},{"id":"stacks:03SI","tag":"03SI","title":"Topological invariance of étale cohomology · Proposition 03SI","summary":"Let X_0 → X be a universal homeomorphism of schemes (for example the closed immersion defined by a nilpotent sheaf of ideals). Then • the étale sites X_etale and (X_0)_etale are isomorphic, • the étale topoi Sh(X_etale) and Sh((X_0)_etale) are equivalent, and • H^q_etale(X, F) = H^q_etale(X_0, F|_X_0) for all q and for any abelian sheaf F on X_etale.","statement_latex":"Let $X_0 \\to X$ be a universal homeomorphism of schemes\n(for example the closed immersion defined by a nilpotent sheaf of ideals).\nThen\n\\begin{enumerate}\n\\item the \\'etale sites $X_\\etale$ and $(X_0)_\\etale$ are isomorphic,\n\\item the \\'etale topoi $\\Sh(X_\\etale)$ and $\\Sh((X_0)_\\etale)$\nare equivalent, and\n\\item $H^q_\\etale(X, \\mathcal{F}) = H^q_\\etale(X_0, \\mathcal{F}|_{X_0})$\nfor all $q$ and\nfor any abelian sheaf $\\mathcal{F}$ on $X_\\etale$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Topological invariance of the small étale site","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SI","source_file":"etale-cohomology.tex","source_line":6547,"source_end_line":6560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6547-L6560","statement_sha256":"6cc5dcbf6e76ee31fd1ad4256b16c7cbdd83536e815129022f299650c927725a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9932,"rank":9932,"depth":22,"x":1678.933,"y":1125.103,"cluster":"tale-geometry"},{"id":"stacks:04FV","tag":"04FV","title":"Closed immersions and pushforward · Lemma 04FV","summary":"Let i : Z → X be a closed immersion of schemes. Let U, U' be schemes étale over X. Let h : U_Z → U'_Z be a morphism over Z. Then there exists a diagram xymatrix U & W ar[l]_a ar[r]^b & U' in X_etale such that a_Z : W_Z → U_Z is an isomorphism and h = b_Z ∘ (a_Z)^-1.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nLet $U, U'$ be schemes \\'etale over $X$. Let $h : U_Z \\to U'_Z$\nbe a morphism over $Z$. Then there exists a diagram\n$$\n\\xymatrix{\nU & W \\ar[l]_a \\ar[r]^b & U'\n}\n$$\nin $X_\\etale$ such that $a_Z : W_Z \\to U_Z$\nis an isomorphism and $h = b_Z \\circ (a_Z)^{-1}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Closed immersions and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FV","source_file":"etale-cohomology.tex","source_line":6596,"source_end_line":6608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6596-L6608","statement_sha256":"8c79bcd521b45a7a68f4f835405c21a0d3851d4fcad8f1ca35b8bca899a63068","origin":"The Stacks Project","memory_eligible":false,"source_rank":9933,"rank":9933,"depth":17,"x":1518.929,"y":1207.232,"cluster":"tale-geometry"},{"id":"stacks:04FW","tag":"04FW","title":"Closed immersions and pushforward · Lemma 04FW","summary":"Let i : Z → X be a closed immersion of schemes. Let V → Z be an étale morphism of schemes. There exist étale morphisms U_i → X and morphisms U_i, Z → V such that (U_i, Z → V) is a Zariski covering of V.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nLet $V \\to Z$ be an \\'etale morphism of schemes.\nThere exist \\'etale morphisms $U_i \\to X$ and morphisms\n$U_{i, Z} \\to V$ such that $\\{U_{i, Z} \\to V\\}$\nis a Zariski covering of $V$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Closed immersions and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FW","source_file":"etale-cohomology.tex","source_line":6620,"source_end_line":6627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6620-L6627","statement_sha256":"beaafc20de646b1ec048c566b6b87326e15405cf8bb9f3f7621e3806768fb687","origin":"The Stacks Project","memory_eligible":false,"source_rank":9934,"rank":9934,"depth":7,"x":1570.952,"y":1055.634,"cluster":"tale-geometry"},{"id":"stacks:04FX","tag":"04FX","title":"Closed immersions and pushforward · Lemma 04FX","summary":"Let i : Z → X be a closed immersion of schemes. Let G be a sheaf of sets on Z_etale. Let overlinex be a geometric point of X. Then (i_small, *G)_overlinex = ( * & if & overlinex not ∈ Z G_overlinex & if & overlinex ∈ Z . where * denotes a singleton set.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nLet $\\mathcal{G}$ be a sheaf of sets on $Z_\\etale$.\nLet $\\overline{x}$ be a geometric point of $X$.\nThen\n$$\n(i_{small, *}\\mathcal{G})_{\\overline{x}} =\n\\left\\{\n\\begin{matrix}\n* & \\text{if} & \\overline{x} \\not \\in Z \\\\\n\\mathcal{G}_{\\overline{x}} & \\text{if} & \\overline{x} \\in Z\n\\end{matrix}\n\\right.\n$$\nwhere $*$ denotes a singleton set.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Closed immersions and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FX","source_file":"etale-cohomology.tex","source_line":6643,"source_end_line":6659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6643-L6659","statement_sha256":"721aa521dfce4d9f6f3e608d8d6cfd2b77f1b70eea9cc87496f759d4c521fdaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9935,"rank":9935,"depth":18,"x":1654.638,"y":1197.169,"cluster":"tale-geometry"},{"id":"stacks:04CA","tag":"04CA","title":"Closed immersions and pushforward · Proposition 04CA","summary":"Let i : Z → X be a closed immersion of schemes. • The functor i_small, * : Sh(Z_etale) → Sh(X_etale) is fully faithful and its essential image is those sheaves of sets F on X_etale whose restriction to X setminus Z is isomorphic to *, and • the functor i_small, * : Ab(Z_etale) → Ab(X_etale) is fully faithful and its essential image is those abelian sheaves on X_etale whose support is contained in Z. In both cases i_small^-1 is a left inverse to the functor i_small, *.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\n\\begin{enumerate}\n\\item The functor\n$$\ni_{small, *} :\n\\Sh(Z_\\etale)\n\\longrightarrow\n\\Sh(X_\\etale)\n$$\nis fully faithful and its essential image is those sheaves of sets\n$\\mathcal{F}$ on $X_\\etale$ whose restriction to $X \\setminus Z$ is\nisomorphic to $*$, and\n\\item the functor\n$$\ni_{small, *} :\n\\textit{Ab}(Z_\\etale)\n\\longrightarrow\n\\textit{Ab}(X_\\etale)\n$$\nis fully faithful and its essential image is those abelian sheaves on\n$X_\\etale$ whose support is contained in $Z$.\n\\end{enumerate}\nIn both cases $i_{small}^{-1}$ is a left inverse to the functor\n$i_{small, *}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Closed immersions and pushforward","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CA","source_file":"etale-cohomology.tex","source_line":6701,"source_end_line":6727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6701-L6727","statement_sha256":"80e19899e34a370a12a9d86ce6b654e11a914efa2e7676e07c3b06df1da0050c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9936,"rank":9936,"depth":52,"x":1478.826,"y":1140.196,"cluster":"tale-geometry"},{"id":"stacks:04FZ","tag":"04FZ","title":"Integral universally injective morphisms · Proposition 04FZ","summary":"Let f : X → Y be a morphism of schemes which is integral and universally injective. • The functor f_small, * : Sh(X_etale) → Sh(Y_etale) is fully faithful and its essential image is those sheaves of sets F on Y_etale whose restriction to Y setminus f(X) is isomorphic to *, and • the functor f_small, * : Ab(X_etale) → Ab(Y_etale) is fully faithful and its essential image is those abelian sheaves on Y_etale whose support is contained in f(X). In both cases f_small^-1 is a…","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is integral\nand universally injective.\n\\begin{enumerate}\n\\item The functor\n$$\nf_{small, *} :\n\\Sh(X_\\etale)\n\\longrightarrow\n\\Sh(Y_\\etale)\n$$\nis fully faithful and its essential image is those sheaves of sets\n$\\mathcal{F}$ on $Y_\\etale$ whose restriction to $Y \\setminus f(X)$ is\nisomorphic to $*$, and\n\\item the functor\n$$\nf_{small, *} :\n\\textit{Ab}(X_\\etale)\n\\longrightarrow\n\\textit{Ab}(Y_\\etale)\n$$\nis fully faithful and its essential image is those abelian sheaves on\n$Y_\\etale$ whose support is contained in $f(X)$.\n\\end{enumerate}\nIn both cases $f_{small}^{-1}$ is a left inverse to the functor\n$f_{small, *}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Integral universally injective morphisms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04FZ","source_file":"etale-cohomology.tex","source_line":6764,"source_end_line":6791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6764-L6791","statement_sha256":"d56c0b15256e643aa7d61c8c9e7a6f1c5a409b35dc0aded3348521361205e3db","origin":"The Stacks Project","memory_eligible":false,"source_rank":9937,"rank":9937,"depth":53,"x":1654.567,"y":1082.354,"cluster":"tale-geometry"},{"id":"stacks:04C7","tag":"04C7","title":"Big sites and pushforward · Lemma 04C7","summary":"Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Let f : X → Y be a monomorphism of schemes. Then the canonical map f_big^-1f_big, *F → F is an isomorphism for any sheaf F on (Sch/X)_τ.","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nLet $f : X \\to Y$ be a monomorphism of schemes.\nThen the canonical map\n$f_{big}^{-1}f_{big, *}\\mathcal{F} \\to \\mathcal{F}$\nis an isomorphism for any sheaf $\\mathcal{F}$ on\n$(\\Sch/X)_\\tau$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big sites and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04C7","source_file":"etale-cohomology.tex","source_line":6824,"source_end_line":6832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6824-L6832","statement_sha256":"c2e026ae3ac44079e65c298664479a4a03ea1e88ba7266d0c5b5dfca97e12624","origin":"The Stacks Project","memory_eligible":false,"source_rank":9938,"rank":9938,"depth":7,"x":1571.359,"y":1224.955,"cluster":"tale-geometry"},{"id":"stacks:04E3","tag":"04E3","title":"Big sites and pushforward · Lemma 04E3","summary":"Let f : X → Y be a closed immersion of schemes. Let U → X be a syntomic (resp. smooth, resp. étale) morphism. Then there exist syntomic (resp. smooth, resp. étale) morphisms V_i → Y and morphisms V_i ×_Y X → U such that (V_i ×_Y X → U) is a Zariski covering of U.","statement_latex":"Let $f : X \\to Y$ be a closed immersion of schemes.\nLet $U \\to X$ be a syntomic (resp.\\ smooth, resp.\\ \\'etale) morphism.\nThen there exist syntomic (resp.\\ smooth, resp.\\ \\'etale) morphisms\n$V_i \\to Y$ and morphisms $V_i \\times_Y X \\to U$ such that\n$\\{V_i \\times_Y X \\to U\\}$ is a Zariski covering of $U$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big sites and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04E3","source_file":"etale-cohomology.tex","source_line":6856,"source_end_line":6863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6856-L6863","statement_sha256":"789136e8edc9bd3bf7a4cb08cdb91cbfd7bb0fd292a1fca64402fa7916b9594e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9939,"rank":9939,"depth":36,"x":1517.954,"y":1072.344,"cluster":"tale-geometry"},{"id":"stacks:04E4","tag":"04E4","title":"Big sites and pushforward · Lemma 04E4","summary":"Let f : X → Y be a closed immersion of schemes. Let (U_i → X) be a syntomic (resp. smooth, resp. étale) covering. There exists a syntomic (resp. smooth, resp. étale) covering (V_j → Y) such that for each j, either V_j ×_Y X = ∅, or the morphism V_j ×_Y X → X factors through U_i for some i.","statement_latex":"Let $f : X \\to Y$ be a closed immersion of schemes.\nLet $\\{U_i \\to X\\}$ be a syntomic (resp.\\ smooth, resp.\\ \\'etale) covering.\nThere exists a syntomic (resp.\\ smooth, resp.\\ \\'etale) covering $\\{V_j \\to Y\\}$\nsuch that for each $j$, either $V_j \\times_Y X = \\emptyset$, or the\nmorphism $V_j \\times_Y X \\to X$ factors through $U_i$ for some $i$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big sites and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04E4","source_file":"etale-cohomology.tex","source_line":6885,"source_end_line":6892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6885-L6892","statement_sha256":"05ca446dd7f25805cd12de02076e1aa26b3567f2ab4fb37ea3fa9f1797a4e49b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9940,"rank":9940,"depth":37,"x":1680.318,"y":1154.706,"cluster":"tale-geometry"},{"id":"stacks:04C3","tag":"04C3","title":"Big sites and pushforward · Lemma 04C3","summary":"Let f : X → Y be a closed immersion of schemes. Let τ ∈ (syntomic, smooth, etale). The functor V ↦ X ×_Y V defines an almost cocontinuous functor (see Sites, Definition [Tag 04B7]) (Sch/Y)_τ → (Sch/X)_τ between big τ sites.","statement_latex":"Let $f : X \\to Y$ be a closed immersion of schemes.\nLet $\\tau \\in \\{syntomic, smooth, \\etale\\}$.\nThe functor $V \\mapsto X \\times_Y V$ defines an almost\ncocontinuous functor (see\nSites, Definition \\ref{sites-definition-almost-cocontinuous})\n$(\\Sch/Y)_\\tau \\to (\\Sch/X)_\\tau$ between\nbig $\\tau$ sites.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big sites and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04C3","source_file":"etale-cohomology.tex","source_line":6906,"source_end_line":6915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6906-L6915","statement_sha256":"054165f1d2c6da5c4aabae3aab347a2abedf3b409001950184479783f748d872","origin":"The Stacks Project","memory_eligible":false,"source_rank":9941,"rank":9941,"depth":38,"x":1494.064,"y":1186.153,"cluster":"tale-geometry"},{"id":"stacks:04C4","tag":"04C4","title":"Big sites and pushforward · Lemma 04C4","summary":"Let f : X → Y be a closed immersion of schemes. Let τ ∈ (syntomic, smooth, etale). • The pushforward f_big, * : Sh((Sch/X)_τ) → Sh((Sch/Y)_τ) commutes with coequalizers and pushouts. • The pushforward f_big, * : Ab((Sch/X)_τ) → Ab((Sch/Y)_τ) is exact.","statement_latex":"Let $f : X \\to Y$ be a closed immersion of schemes.\nLet $\\tau \\in \\{syntomic, smooth, \\etale\\}$.\n\\begin{enumerate}\n\\item The pushforward\n$f_{big, *} :\n\\Sh((\\Sch/X)_\\tau)\n\\to\n\\Sh((\\Sch/Y)_\\tau)$\ncommutes with coequalizers and pushouts.\n\\item The pushforward\n$f_{big, *} :\n\\textit{Ab}((\\Sch/X)_\\tau)\n\\to\n\\textit{Ab}((\\Sch/Y)_\\tau)$\nis exact.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Big sites and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04C4","source_file":"etale-cohomology.tex","source_line":6929,"source_end_line":6947,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6929-L6947","statement_sha256":"bddf015446dc7e9f00e4ad2e45b036b83e4c84f38645443c7c6d6b09697b2303","origin":"The Stacks Project","memory_eligible":false,"source_rank":9942,"rank":9942,"depth":39,"x":1606.295,"y":1057.074,"cluster":"tale-geometry"},{"id":"stacks:04CC","tag":"04CC","title":"Exactness of big lower shriek · Lemma 04CC","summary":"Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Let f : X → Y be a morphism of schemes. Let f_big : Sh((Sch/X)_τ) → Sh((Sch/Y)_τ) be the corresponding morphism of topoi as in Topologies, Lemma [Tag 0210], [Tag 021H], [Tag 04HC], [Tag 04HD], or [Tag 021W]. • The functor f_big^-1 : Ab((Sch/Y)_τ) → Ab((Sch/X)_τ) has a left adjoint f_big! : Ab((Sch/X)_τ) → Ab((Sch/Y)_τ) which is exact. • The functor f_big^* : Mod((Sch/Y)_τ, O) → Mod((Sch/X)_τ, O) has a left adjoint f_big! :…","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nLet $f : X \\to Y$ be a morphism of schemes. Let\n$$\nf_{big} :\n\\Sh((\\Sch/X)_\\tau)\n\\longrightarrow\n\\Sh((\\Sch/Y)_\\tau)\n$$\nbe the corresponding morphism of topoi as in\nTopologies, Lemma\n\\ref{topologies-lemma-morphism-big},\n\\ref{topologies-lemma-morphism-big-etale},\n\\ref{topologies-lemma-morphism-big-smooth},\n\\ref{topologies-lemma-morphism-big-syntomic}, or\n\\ref{topologies-lemma-morphism-big-fppf}.\n\\begin{enumerate}\n\\item The functor\n$f_{big}^{-1} : \\textit{Ab}((\\Sch/Y)_\\tau) \\to \\textit{Ab}((\\Sch/X)_\\tau)$\nhas a left adjoint\n$$\nf_{big!} : \\textit{Ab}((\\Sch/X)_\\tau) \\to \\textit{Ab}((\\Sch/Y)_\\tau)\n$$\nwhich is exact.\n\\item The functor\n$f_{big}^* :\n\\textit{Mod}((\\Sch/Y)_\\tau, \\mathcal{O})\n\\to\n\\textit{Mod}((\\Sch/X)_\\tau, \\mathcal{O})$\nhas a left adjoint\n$$\nf_{big!} :\n\\textit{Mod}((\\Sch/X)_\\tau, \\mathcal{O})\n\\to\n\\textit{Mod}((\\Sch/Y)_\\tau, \\mathcal{O})\n$$\nwhich is exact.\n\\end{enumerate}\nMoreover, the two functors $f_{big!}$ agree on underlying sheaves\nof abelian groups.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Exactness of big lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CC","source_file":"etale-cohomology.tex","source_line":6996,"source_end_line":7037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L6996-L7037","statement_sha256":"a1acff02170a1adcc3c62f2c5836cd0d406c5ce96e083541d4d454b997852581","origin":"The Stacks Project","memory_eligible":false,"source_rank":9943,"rank":9943,"depth":9,"x":1627.372,"y":1216.189,"cluster":"tale-geometry"},{"id":"stacks:07AJ","tag":"07AJ","title":"Exactness of big lower shriek · Lemma 07AJ","summary":"Let X be a scheme. Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Let C_1 ⊂ C_2 ⊂ (Sch/X)_τ be full subcategories with the following properties: • For an object U/X of C_t, • if (U_i → U) is a covering of (Sch/X)_τ, then U_i/X is an object of C_t, • U × A^1/X is an object of C_t. • X/X is an object of C_t. We endow C_t with the structure of a site whose coverings are exactly those coverings (U_i → U) of (Sch/X)_τ with U ∈ Ob(C_t). Then • [(a)] The functor C_1 → C_2 is…","statement_latex":"Let $X$ be a scheme. Let\n$\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nLet $\\mathcal{C}_1 \\subset \\mathcal{C}_2 \\subset (\\Sch/X)_\\tau$ be full\nsubcategories with the following properties:\n\\begin{enumerate}\n\\item For an object $U/X$ of $\\mathcal{C}_t$,\n\\begin{enumerate}\n\\item if $\\{U_i \\to U\\}$ is a covering of $(\\Sch/X)_\\tau$, then\n$U_i/X$ is an object of $\\mathcal{C}_t$,\n\\item $U \\times \\mathbf{A}^1/X$ is an object of $\\mathcal{C}_t$.\n\\end{enumerate}\n\\item $X/X$ is an object of $\\mathcal{C}_t$.\n\\end{enumerate}\nWe endow $\\mathcal{C}_t$ with the structure of a site whose coverings are\nexactly those coverings $\\{U_i \\to U\\}$ of $(\\Sch/X)_\\tau$ with\n$U \\in \\Ob(\\mathcal{C}_t)$. Then\n\\begin{enumerate}\n\\item[(a)] The functor $\\mathcal{C}_1 \\to \\mathcal{C}_2$\nis fully faithful, continuous, and cocontinuous.\n\\end{enumerate}\nDenote $g : \\Sh(\\mathcal{C}_1) \\to \\Sh(\\mathcal{C}_2)$ the corresponding\nmorphism of topoi. Denote $\\mathcal{O}_t$ the restriction of $\\mathcal{O}$\nto $\\mathcal{C}_t$. Denote $g_!$ the functor of\nModules on Sites, Definition \\ref{sites-modules-definition-g-shriek}.\n\\begin{enumerate}\n\\item[(b)] The canonical map $g_!\\mathcal{O}_1 \\to \\mathcal{O}_2$\nis an isomorphism.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Exactness of big lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AJ","source_file":"etale-cohomology.tex","source_line":7069,"source_end_line":7099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7069-L7099","statement_sha256":"637748921626661fe7159558e3f7ce333ad076542d2413633cde6684ea47f128","origin":"The Stacks Project","memory_eligible":false,"source_rank":9944,"rank":9944,"depth":1,"x":1483.648,"y":1110.653,"cluster":"tale-geometry"},{"id":"stacks:0A50","tag":"0A50","title":"Mayer-Vietoris for étale cohomology · Lemma 0A50","summary":"Let X be a scheme. Suppose that X = U ∪ V is a union of two opens. For any abelian sheaf F on X_etale there exists a long exact cohomology sequence 0 → H^0_etale(X, F) → H^0_etale(U, F) ⊕ H^0_etale(V, F) → H^0_etale(U ∩ V, F) phantom→ … phantom0 → H^1_etale(X, F) → H^1_etale(U, F) ⊕ H^1_etale(V, F) → H^1_etale(U ∩ V, F) → … This long exact sequence is functorial in F.","statement_latex":"Let $X$ be a scheme. Suppose that $X = U \\cup V$ is a\nunion of two opens. For any abelian sheaf $\\mathcal{F}$ on $X_\\etale$\nthere exists a long exact cohomology sequence\n$$\n\\begin{matrix}\n0 \\to\nH^0_\\etale(X, \\mathcal{F}) \\to\nH^0_\\etale(U, \\mathcal{F}) \\oplus H^0_\\etale(V, \\mathcal{F}) \\to\nH^0_\\etale(U \\cap V, \\mathcal{F}) \\phantom{\\to \\ldots} \\\\\n\\phantom{0} \\to H^1_\\etale(X, \\mathcal{F}) \\to\nH^1_\\etale(U, \\mathcal{F}) \\oplus H^1_\\etale(V, \\mathcal{F}) \\to\nH^1_\\etale(U \\cap V, \\mathcal{F}) \\to \\ldots\n\\end{matrix}\n$$\nThis long exact sequence is functorial in $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A50","source_file":"etale-cohomology.tex","source_line":7175,"source_end_line":7192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7175-L7192","statement_sha256":"aa92cb293d374e838f9f9f099f5e9a6f39f0dcbea5a1f4239029e4e460f58187","origin":"The Stacks Project","memory_eligible":false,"source_rank":9945,"rank":9945,"depth":10,"x":1674.797,"y":1106.915,"cluster":"tale-geometry"},{"id":"stacks:0EYK","tag":"0EYK","title":"Relative Mayer-Vietoris · Lemma 0EYK","summary":"Let f : X → Y be a morphism of schemes. Suppose that X = U ∪ V is a union of two open subschemes. Denote a = f|_U : U → Y, b = f|_V : V → Y, and c = f|_U ∩ V : U ∩ V → Y. For every abelian sheaf F on X_etale there exists a long exact sequence 0 → f_*F → a_*(F|_U) ⊕ b_*(F|_V) → c_*(F|_U ∩ V) → R^1f_*F → … on Y_etale. This long exact sequence is functorial in F.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Suppose that $X = U \\cup V$\nis a union of two open subschemes. Denote\n$a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and\n$c = f|_{U \\cap V} : U \\cap V \\to Y$.\nFor every abelian sheaf $\\mathcal{F}$ on $X_\\etale$\nthere exists a long exact sequence\n$$\n0 \\to\nf_*\\mathcal{F} \\to\na_*(\\mathcal{F}|_U) \\oplus b_*(\\mathcal{F}|_V) \\to\nc_*(\\mathcal{F}|_{U \\cap V}) \\to\nR^1f_*\\mathcal{F} \\to \\ldots\n$$\non $Y_\\etale$.\nThis long exact sequence is functorial in $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYK","source_file":"etale-cohomology.tex","source_line":7222,"source_end_line":7239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7222-L7239","statement_sha256":"59928240cc56295015136a4706847b54dda1994c7fccc0f1a1e1c924d241fc25","origin":"The Stacks Project","memory_eligible":false,"source_rank":9946,"rank":9946,"depth":11,"x":1536.636,"y":1218.309,"cluster":"tale-geometry"},{"id":"stacks:0EZL","tag":"0EZL","title":"Colimits · Definition 0EZL","summary":"Let I be a preordered set. Let (X_i, f_i'i) be an inverse system of schemes over I. A system (F_i, φ_i'i) of sheaves on (X_i, f_i'i) is given by • a sheaf F_i on (X_i)_etale for all i ∈ I, • for i' ≥ i a map φ_i'i : f_i'i^-1F_i → F_i' of sheaves on (X_i')_etale such that φ_i\"i = φ_i\"i' ∘ f_i\" i'^-1φ_i'i whenever i\" ≥ i' ≥ i.","statement_latex":"Let $I$ be a preordered set. Let $(X_i, f_{i'i})$ be an inverse\nsystem of schemes over $I$.\nA {\\it system $(\\mathcal{F}_i, \\varphi_{i'i})$ of sheaves\non $(X_i, f_{i'i})$} is given by\n\\begin{enumerate}\n\\item a sheaf $\\mathcal{F}_i$ on $(X_i)_\\etale$ for all $i \\in I$,\n\\item for $i' \\geq i$ a map\n$\\varphi_{i'i} : f_{i'i}^{-1}\\mathcal{F}_i \\to \\mathcal{F}_{i'}$\nof sheaves on $(X_{i'})_\\etale$\n\\end{enumerate}\nsuch that $\\varphi_{i''i} = \\varphi_{i''i'} \\circ f_{i'' i'}^{-1}\\varphi_{i'i}$\nwhenever $i'' \\geq i' \\geq i$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZL","source_file":"etale-cohomology.tex","source_line":7298,"source_end_line":7312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7298-L7312","statement_sha256":"15a590471cd6e096dbc52ba2083038bc36cfba408f11efae22be51e86e464563","origin":"The Stacks Project","memory_eligible":false,"source_rank":9947,"rank":9947,"depth":0,"x":1548.954,"y":1057.522,"cluster":"tale-geometry"},{"id":"stacks:0EYL","tag":"0EYL","title":"Colimits · Lemma 0EYL","summary":"Let I be a directed set. Let (X_i, f_i'i) be an inverse system of schemes over I with affine transition morphisms. Let X = lim_i ∈ I X_i. With notation as in Topologies, Lemma [Tag 04HR] we have X_affine, etale = colim (X_i)_affine, etale as sites in the sense of Sites, Lemma [Tag 09YL].","statement_latex":"Let $I$ be a directed set. Let $(X_i, f_{i'i})$ be an inverse\nsystem of schemes over $I$ with affine transition morphisms.\nLet $X = \\lim_{i \\in I} X_i$. With\nnotation as in Topologies, Lemma \\ref{topologies-lemma-alternative} we have\n$$\nX_{affine, \\etale} = \\colim (X_i)_{affine, \\etale}\n$$\nas sites in the sense of\nSites, Lemma \\ref{sites-lemma-colimit-sites}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYL","source_file":"etale-cohomology.tex","source_line":7335,"source_end_line":7346,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7335-L7346","statement_sha256":"0fcc5269cb86c7c742959412485b10283e788a4df6d89c7422912f624e967ffa","origin":"The Stacks Project","memory_eligible":false,"source_rank":9948,"rank":9948,"depth":40,"x":1669.359,"y":1183.269,"cluster":"tale-geometry"},{"id":"stacks:09YQ","tag":"09YQ","title":"Colimits · Theorem 09YQ","summary":"Let X = lim_i ∈ I X_i be a limit of a directed system of schemes with affine transition morphisms f_i'i : X_i' → X_i. We assume that X_i is quasi-compact and quasi-separated for all i ∈ I. Let (F_i, φ_i'i) be a system of abelian sheaves on (X_i, f_i'i). Denote f_i : X → X_i the projection and set F = colim f_i^-1F_i. Then colim_i∈ I H_etale^p(X_i, F_i) = H_etale^p(X, F). for all p ≥ 0.","statement_latex":"Let $X = \\lim_{i \\in I} X_i$ be a limit of a directed system of schemes\nwith affine transition morphisms $f_{i'i} : X_{i'} \\to X_i$. We assume\nthat $X_i$ is quasi-compact and quasi-separated for all $i \\in I$.\nLet $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of abelian sheaves\non $(X_i, f_{i'i})$. Denote $f_i : X \\to X_i$ the projection and set\n$\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Then\n$$\n\\colim_{i\\in I} H_\\etale^p(X_i, \\mathcal{F}_i) = H_\\etale^p(X, \\mathcal{F}).\n$$\nfor all $p \\geq 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YQ","source_file":"etale-cohomology.tex","source_line":7399,"source_end_line":7411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7399-L7411","statement_sha256":"5c6abe34b76f2747e159bb325262f8f724f2a89e2d4951240987a8406f9ed8fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9949,"rank":9949,"depth":41,"x":1479.156,"y":1158.828,"cluster":"tale-geometry"},{"id":"stacks:03Q5","tag":"03Q5","title":"Colimits · Lemma 03Q5","summary":"Let X be a quasi-compact and quasi-separated scheme. Let I be a directed set. Let (F_i, φ_ij) be a system of abelian sheaves on X_etale over I. Then colim_i∈ I H_etale^p(X, F_i) = H_etale^p(X, colim_i∈ I F_i).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let $I$\nbe a directed set. Let $(\\mathcal{F}_i, \\varphi_{ij})$ be a system\nof abelian sheaves on $X_\\etale$ over $I$. Then\n$$\n\\colim_{i\\in I} H_\\etale^p(X, \\mathcal{F}_i) = H_\\etale^p(X,\n\\colim_{i\\in I} \\mathcal{F}_i).\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Q5","source_file":"etale-cohomology.tex","source_line":7426,"source_end_line":7435,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7426-L7435","statement_sha256":"c88b0e04026caf8033296a2ce37639a4ab44c159419c730aef85cc4440d06093","origin":"The Stacks Project","memory_eligible":false,"source_rank":9950,"rank":9950,"depth":42,"x":1639.312,"y":1068.786,"cluster":"tale-geometry"},{"id":"stacks:03Q6","tag":"03Q6","title":"Colimits · Lemma 03Q6","summary":"Let A be a ring, (I, ≤) a directed set and (B_i, φ_ij) a system of A-algebras. Set B = colim_i∈ I B_i. Let X → Spec(A) be a quasi-compact and quasi-separated morphism of schemes. Let F an abelian sheaf on X_etale. Denote Y_i = X ×_Spec(A) Spec(B_i), Y = X ×_Spec(A) Spec(B), G_i = (Y_i → X)^-1F and G = (Y → X)^-1F. Then H_etale^p(Y, G) = colim_i∈ I H_etale^p (Y_i, G_i).","statement_latex":"Let $A$ be a ring, $(I, \\leq)$ a directed set and $(B_i, \\varphi_{ij})$ a\nsystem of $A$-algebras. Set $B = \\colim_{i\\in I} B_i$. Let $X \\to \\Spec(A)$\nbe a quasi-compact and quasi-separated morphism of schemes. Let\n$\\mathcal{F}$ an abelian sheaf on $X_\\etale$.\nDenote $Y_i = X \\times_{\\Spec(A)} \\Spec(B_i)$,\n$Y = X \\times_{\\Spec(A)} \\Spec(B)$,\n$\\mathcal{G}_i = (Y_i \\to X)^{-1}\\mathcal{F}$ and\n$\\mathcal{G} = (Y \\to X)^{-1}\\mathcal{F}$. Then\n$$\nH_\\etale^p(Y, \\mathcal{G}) =\n\\colim_{i\\in I} H_\\etale^p (Y_i, \\mathcal{G}_i).\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Q6","source_file":"etale-cohomology.tex","source_line":7463,"source_end_line":7477,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7463-L7477","statement_sha256":"17cf39107fe5456da39bf22a8c25885ddd98d97b90cda9c881632c50910418bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":9951,"rank":9951,"depth":42,"x":1593.554,"y":1226.3,"cluster":"tale-geometry"},{"id":"stacks:03Q8","tag":"03Q8","title":"Colimits · Lemma 03Q8","summary":"Let f: X→ Y be a morphism of schemes and F∈ Ab(X_etale). Then R^pf_*F is the sheaf associated to the presheaf (V → Y) ↦ H_etale^p(X ×_Y V, F|_X ×_Y V). More generally, for K ∈ D(X_etale) we have that R^pf_*K is the sheaf associated to the presheaf (V → Y) ↦ H_etale^p(X ×_Y V, K|_X ×_Y V).","statement_latex":"Let $f: X\\to Y$ be a morphism of schemes and $\\mathcal{F}\\in\n\\textit{Ab}(X_\\etale)$. Then $R^pf_*\\mathcal{F}$ is the sheaf\nassociated to the presheaf\n$$\n(V \\to Y) \\longmapsto H_\\etale^p(X \\times_Y V, \\mathcal{F}|_{X \\times_Y V}).\n$$\nMore generally, for $K \\in D(X_\\etale)$ we have that $R^pf_*K$ is the\nsheaf associated to the presheaf\n$$\n(V \\to Y) \\longmapsto H_\\etale^p(X \\times_Y V, K|_{X \\times_Y V}).\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Q8","source_file":"etale-cohomology.tex","source_line":7509,"source_end_line":7522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7509-L7522","statement_sha256":"c671a65282cfc05ff9f0d6cf092bc0d0b990896abbed7555b6de2371cb36237a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9952,"rank":9952,"depth":4,"x":1500.483,"y":1083.968,"cluster":"tale-geometry"},{"id":"stacks:09Z1","tag":"09Z1","title":"Colimits · Lemma 09Z1","summary":"Let S be a scheme. Let X = lim_i ∈ I X_i be a limit of a directed system of schemes over S with affine transition morphisms f_i'i : X_i' → X_i. We assume the structure morphisms g_i : X_i → S and g : X → S are quasi-compact and quasi-separated. Let (F_i, φ_i'i) be a system of abelian sheaves on (X_i, f_i'i). Denote f_i : X → X_i the projection and set F = colim f_i^-1F_i. Then colim_i∈ I R^p g_i, * F_i = R^p g_* F for all p ≥ 0.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be a limit of a\ndirected system of schemes over $S$ with affine transition morphisms\n$f_{i'i} : X_{i'} \\to X_i$. We assume the structure morphisms\n$g_i : X_i \\to S$ and $g : X \\to S$ are quasi-compact and quasi-separated.\nLet $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of abelian sheaves\non $(X_i, f_{i'i})$. Denote $f_i : X \\to X_i$ the projection and set\n$\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Then\n$$\n\\colim_{i\\in I} R^p g_{i, *} \\mathcal{F}_i = R^p g_* \\mathcal{F}\n$$\nfor all $p \\geq 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Z1","source_file":"etale-cohomology.tex","source_line":7534,"source_end_line":7547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7534-L7547","statement_sha256":"e84cec94d2be1c9c90d4d4eb39965f9a628825f28e5f7607a40c61f59534c2bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9953,"rank":9953,"depth":42,"x":1683.852,"y":1136.191,"cluster":"tale-geometry"},{"id":"stacks:0EYM","tag":"0EYM","title":"Colimits · Lemma 0EYM","summary":"Let I be a directed set. Let g_i : X_i → S_i be an inverse system of morphisms of schemes over I. Assume g_i is quasi-compact and quasi-separated and for i' ≥ i the transition morphisms f_i'i : X_i' → X_i and h_i'i : S_i' → S_i are affine. Let g : X → S be the limit of the morphisms g_i, see Limits, Section [Tag 01YV]. Denote f_i : X → X_i and h_i : S → S_i the projections. Let (F_i, φ_i'i) be a system of sheaves on (X_i, f_i'i). Set F = colim f_i^-1F_i. Then R^p g_* F =…","statement_latex":"Let $I$ be a directed set. Let $g_i : X_i \\to S_i$ be an inverse system of\nmorphisms of schemes over $I$. Assume $g_i$ is quasi-compact and\nquasi-separated and for $i' \\geq i$ the transition morphisms\n$f_{i'i} : X_{i'} \\to X_i$ and $h_{i'i} : S_{i'} \\to S_i$ are affine.\nLet $g : X \\to S$ be the limit of the morphisms $g_i$, see\nLimits, Section \\ref{limits-section-limits}.\nDenote $f_i : X \\to X_i$ and $h_i : S \\to S_i$ the projections.\nLet $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of sheaves\non $(X_i, f_{i'i})$. Set $\\mathcal{F} = \\colim f_i^{-1}\\mathcal{F}_i$. Then\n$$\nR^p g_* \\mathcal{F} =\n\\colim_{i \\in I} h_i^{-1}R^p g_{i, *} \\mathcal{F}_i\n$$\nfor all $p \\geq 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYM","source_file":"etale-cohomology.tex","source_line":7566,"source_end_line":7582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7566-L7582","statement_sha256":"dbef3411553c94e24fbba6137c5341950f24ee80abc95d621e8082e92b58c50f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9954,"rank":9954,"depth":42,"x":1506.371,"y":1201.831,"cluster":"tale-geometry"},{"id":"stacks:0EYN","tag":"0EYN","title":"Colimits · Lemma 0EYN","summary":"Let X = lim_i ∈ I X_i be a directed limit of schemes with affine transition morphisms f_i'i and projection morphisms f_i : X → X_i. Let F be a sheaf on X_etale. Then • there are canonical maps φ_i'i : f_i'i^-1f_i, *F → f_i', *F such that (f_i, *F, φ_i'i) is a system of sheaves on (X_i, f_i'i) as in Definition [Tag 0EZL], and • F = colim f_i^-1f_i, *F.","statement_latex":"Let $X = \\lim_{i \\in I} X_i$ be a directed limit of schemes\nwith affine transition morphisms $f_{i'i}$ and projection morphisms\n$f_i : X \\to X_i$. Let $\\mathcal{F}$ be a sheaf on $X_\\etale$. Then\n\\begin{enumerate}\n\\item there are canonical maps\n$\\varphi_{i'i} : f_{i'i}^{-1}f_{i, *}\\mathcal{F} \\to f_{i', *}\\mathcal{F}$\nsuch that $(f_{i, *}\\mathcal{F}, \\varphi_{i'i})$ is a system of\nsheaves on $(X_i, f_{i'i})$ as in\nDefinition \\ref{definition-inverse-system-sheaves}, and\n\\item $\\mathcal{F} = \\colim f_i^{-1}f_{i, *}\\mathcal{F}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYN","source_file":"etale-cohomology.tex","source_line":7721,"source_end_line":7734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7721-L7734","statement_sha256":"5d51641afa4486a5286698ca0e8c429806873a3615c2556d217609b166de652b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9955,"rank":9955,"depth":41,"x":1584.577,"y":1052.495,"cluster":"tale-geometry"},{"id":"stacks:0DV2","tag":"0DV2","title":"Colimits · Lemma 0DV2","summary":"Let I be a directed set. Let g_i : X_i → S_i be an inverse system of morphisms of schemes over I. Assume g_i is quasi-compact and quasi-separated and for i' ≥ i the transition morphisms X_i' → X_i and S_i' → S_i are affine. Let g : X → S be the limit of the morphisms g_i, see Limits, Section [Tag 01YV]. Denote f_i : X → X_i and h_i : S → S_i the projections. Let F be an abelian sheaf on X. Then we have R^pg_*F = colim_i ∈ I h_i^-1R^pg_i, *(f_i, *F)","statement_latex":"Let $I$ be a directed set. Let $g_i : X_i \\to S_i$ be an inverse system of\nmorphisms of schemes over $I$. Assume $g_i$ is quasi-compact and\nquasi-separated and for $i' \\geq i$ the transition morphisms\n$X_{i'} \\to X_i$ and $S_{i'} \\to S_i$ are affine.\nLet $g : X \\to S$ be the limit of the morphisms $g_i$, see\nLimits, Section \\ref{limits-section-limits}.\nDenote $f_i : X \\to X_i$ and $h_i : S \\to S_i$ the projections.\nLet $\\mathcal{F}$ be an  abelian sheaf on $X$. Then we have\n$$\nR^pg_*\\mathcal{F} = \\colim_{i \\in I} h_i^{-1}R^pg_{i, *}(f_{i, *}\\mathcal{F})\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DV2","source_file":"etale-cohomology.tex","source_line":7742,"source_end_line":7755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7742-L7755","statement_sha256":"3a81f53e6c682f27c8246b2954ec5f35a93f129462245f471d4449e9988d6e45","origin":"The Stacks Project","memory_eligible":false,"source_rank":9956,"rank":9956,"depth":43,"x":1647.096,"y":1207.224,"cluster":"tale-geometry"},{"id":"stacks:0EZM","tag":"0EZM","title":"Colimits and complexes · Lemma 0EZM","summary":"Let X = lim_i ∈ I X_i be a limit of a directed system of schemes with affine transition morphisms f_i'i : X_i' → X_i. We assume that X_i is quasi-compact and quasi-separated for all i ∈ I. Let F_i^bullet be a complex of abelian sheaves on X_i, etale. Let φ_i'i : f_i'i^-1F_i^bullet → F_i'^bullet be a map of complexes on X_i, etale such that φ_i\"i = φ_i\"i' ∘ f_i\" i'^-1φ_i'i whenever i\" ≥ i' ≥ i. Assume there is an integer a such that F_i^n = 0 for n < a and all i ∈ I. Then…","statement_latex":"Let $X = \\lim_{i \\in I} X_i$ be a limit of a directed system of schemes\nwith affine transition morphisms $f_{i'i} : X_{i'} \\to X_i$. We assume\nthat $X_i$ is quasi-compact and quasi-separated for all $i \\in I$.\nLet $\\mathcal{F}_i^\\bullet$ be a complex of abelian sheaves on\n$X_{i, \\etale}$. Let $\\varphi_{i'i} : f_{i'i}^{-1}\\mathcal{F}_i^\\bullet \\to\n\\mathcal{F}_{i'}^\\bullet$ be a map of complexes on $X_{i, \\etale}$\nsuch that $\\varphi_{i''i} = \\varphi_{i''i'} \\circ f_{i'' i'}^{-1}\\varphi_{i'i}$\nwhenever $i'' \\geq i' \\geq i$. Assume there is an integer $a$ such that\n$\\mathcal{F}_i^n = 0$ for $n < a$ and all $i \\in I$.\nThen we have\n$$\nH^p_\\etale(X, \\colim f_i^{-1}\\mathcal{F}_i^\\bullet) =\n\\colim H^p_\\etale(X_i, \\mathcal{F}^\\bullet_i)\n$$\nwhere $f_i : X \\to X_i$ is the projection.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits and complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZM","source_file":"etale-cohomology.tex","source_line":7782,"source_end_line":7799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7782-L7799","statement_sha256":"275e2794ac63ea47f3b2f6372a04a3631e37793b56d4afb609d86d0f16c26acd","origin":"The Stacks Project","memory_eligible":false,"source_rank":9957,"rank":9957,"depth":42,"x":1476.308,"y":1128.485,"cluster":"tale-geometry"},{"id":"stacks:0GIS","tag":"0GIS","title":"Colimits and complexes · Lemma 0GIS","summary":"Let X be a quasi-compact and quasi-sepated scheme. Let K_i ∈ D(X_etale), i ∈ I be a family of objects. Assume given a ∈ Z such that H^n(K_i) = 0 for n < a and i ∈ I. Then RΓ(X, bigoplus_i K_i) = bigoplus_i RΓ(X, K_i).","statement_latex":"Let $X$ be a quasi-compact and quasi-sepated scheme. Let\n$K_i \\in D(X_\\etale)$, $i \\in I$ be a family of objects.\nAssume given $a \\in \\mathbf{Z}$ such that $H^n(K_i) = 0$ for $n < a$\nand $i \\in I$. Then $R\\Gamma(X, \\bigoplus_i K_i) =\n\\bigoplus_i R\\Gamma(X, K_i)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits and complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIS","source_file":"etale-cohomology.tex","source_line":7830,"source_end_line":7837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7830-L7837","statement_sha256":"3c3ba11bf137a598c896f676e7d7b96ac9f238e36699d76696739a45a784ab48","origin":"The Stacks Project","memory_eligible":false,"source_rank":9958,"rank":9958,"depth":43,"x":1665.85,"y":1089.578,"cluster":"tale-geometry"},{"id":"stacks:0GIT","tag":"0GIT","title":"Colimits and complexes · Lemma 0GIT","summary":"Let S be a scheme. Let X = lim_i ∈ I X_i be a limit of a directed system of schemes over S with affine transition morphisms f_i'i : X_i' → X_i. We assume that X_i is quasi-compact and quasi-separated for all i ∈ I. Let K ∈ D^+(S_etale). Then colim_i ∈ I H_etale^p(X_i, K|_X_i) = H_etale^p(X, K|_X). for all p ∈ Z where K|_X_i and K|_X are the pullbacks of K to X_i and X.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be a limit of a\ndirected system of schemes over $S$ with affine transition morphisms\n$f_{i'i} : X_{i'} \\to X_i$. We assume that $X_i$ is quasi-compact and\nquasi-separated for all $i \\in I$. Let $K \\in D^+(S_\\etale)$. Then\n$$\n\\colim_{i \\in I} H_\\etale^p(X_i, K|_{X_i}) = H_\\etale^p(X, K|_X).\n$$\nfor all $p \\in \\mathbf{Z}$ where $K|_{X_i}$ and $K|_X$\nare the pullbacks of $K$ to $X_i$ and $X$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits and complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIT","source_file":"etale-cohomology.tex","source_line":7852,"source_end_line":7863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7852-L7863","statement_sha256":"4db587237b93708a1edeb9e6d0b7c2d06d4e8e7eae883b74ffdf80686b857d18","origin":"The Stacks Project","memory_eligible":false,"source_rank":9959,"rank":9959,"depth":43,"x":1557.209,"y":1226.023,"cluster":"tale-geometry"},{"id":"stacks:0GIU","tag":"0GIU","title":"Colimits and complexes · Lemma 0GIU","summary":"Let I, g_i : X_i → S_i, g : X → S, f_i, g_i, h_i be as in Lemma [Tag 0EYM]. Let 0 ∈ I and K_0 ∈ D^+(X_0, etale). For i ≥ 0 denote K_i the pullback of K_0 to X_i. Denote K the pullback of K_0 to X. Then R^pg_*K = colim_i ≥ 0 h_i^-1R^pg_i, *K_i for all p ∈ Z.","statement_latex":"Let $I$, $g_i : X_i \\to S_i$, $g : X \\to S$, $f_i$, $g_i$, $h_i$ be as in\nLemma \\ref{lemma-relative-colimit-general}.\nLet $0 \\in I$ and $K_0 \\in D^+(X_{0, \\etale})$.\nFor $i \\geq 0$ denote $K_i$ the pullback of $K_0$ to $X_i$.\nDenote $K$ the pullback of $K_0$ to $X$. Then\n$$\nR^pg_*K = \\colim_{i \\geq 0} h_i^{-1}R^pg_{i, *}K_i\n$$\nfor all $p \\in \\mathbf{Z}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits and complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIU","source_file":"etale-cohomology.tex","source_line":7876,"source_end_line":7887,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7876-L7887","statement_sha256":"329bcd725febe5a068797a233ada35a436c8b5c7594d28a5ab277c0131631eed","origin":"The Stacks Project","memory_eligible":false,"source_rank":9960,"rank":9960,"depth":43,"x":1527.55,"y":1063.521,"cluster":"tale-geometry"},{"id":"stacks:0GIV","tag":"0GIV","title":"Colimits and complexes · Lemma 0GIV","summary":"Let I, g_i : X_i → S_i, g : X → S, f_ii', f_i, g_i, h_i be as in Lemma [Tag 0EYM]. Let F_i^bullet be a complex of abelian sheaves on X_i, etale. Let φ_i'i : f_i'i^-1F_i^bullet → F_i'^bullet be a map of complexes on X_i, etale such that φ_i\"i = φ_i\"i' ∘ f_i\" i'^-1φ_i'i whenever i\" ≥ i' ≥ i. Assume there is an integer a such that F_i^n = 0 for n < a and all i ∈ I. Then R^pg_*(colim f_i^-1F_i^bullet) = colim_i ≥ 0 h_i^-1R^pg_i, *F_i^bullet for all p ∈ Z.","statement_latex":"Let $I$, $g_i : X_i \\to S_i$, $g : X \\to S$, $f_{ii'}$, $f_i$, $g_i$, $h_i$\nbe as in Lemma \\ref{lemma-relative-colimit-general}.\nLet $\\mathcal{F}_i^\\bullet$ be a complex of abelian sheaves on\n$X_{i, \\etale}$. Let $\\varphi_{i'i} : f_{i'i}^{-1}\\mathcal{F}_i^\\bullet \\to\n\\mathcal{F}_{i'}^\\bullet$ be a map of complexes on $X_{i, \\etale}$\nsuch that $\\varphi_{i''i} = \\varphi_{i''i'} \\circ f_{i'' i'}^{-1}\\varphi_{i'i}$\nwhenever $i'' \\geq i' \\geq i$. Assume there is an integer $a$ such that\n$\\mathcal{F}_i^n = 0$ for $n < a$ and all $i \\in I$. Then\n$$\nR^pg_*(\\colim f_i^{-1}\\mathcal{F}_i^\\bullet) =\n\\colim_{i \\geq 0} h_i^{-1}R^pg_{i, *}\\mathcal{F}_i^\\bullet\n$$\nfor all $p \\in \\mathbf{Z}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits and complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIV","source_file":"etale-cohomology.tex","source_line":7930,"source_end_line":7945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7930-L7945","statement_sha256":"9260978956e0a3ba3cf762f1e9107fa9066432b23c597808cdde5377c2a0a1d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9961,"rank":9961,"depth":43,"x":1680.328,"y":1166.672,"cluster":"tale-geometry"},{"id":"stacks:0GIW","tag":"0GIW","title":"Colimits and complexes · Lemma 0GIW","summary":"Let f : X → Y be a quasi-compact and quasi-sepated morphism of schemes. Let K_i ∈ D(X_etale), i ∈ I be a family of objects. Assume given a ∈ Z such that H^n(K_i) = 0 for n < a and i ∈ I. Then Rf_*(bigoplus_i K_i) = bigoplus_i Rf_*K_i.","statement_latex":"Let $f : X \\to Y$ be a quasi-compact and quasi-sepated morphism of\nschemes. Let $K_i \\in D(X_\\etale)$, $i \\in I$ be a family of objects.\nAssume given $a \\in \\mathbf{Z}$ such that $H^n(K_i) = 0$ for $n < a$\nand $i \\in I$. Then $Rf_*(\\bigoplus_i K_i) = \\bigoplus_i Rf_*K_i$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Colimits and complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIW","source_file":"etale-cohomology.tex","source_line":7974,"source_end_line":7980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L7974-L7980","statement_sha256":"9dc57a041f75b5ee98bd579ee889aaa1c830b8e4b98783791854e564f2069b8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9962,"rank":9962,"depth":44,"x":1484.428,"y":1177.316,"cluster":"tale-geometry"},{"id":"stacks:03Q9","tag":"03Q9","title":"Stalks of higher direct images · Theorem 03Q9","summary":"Let f: X → S be a quasi-compact and quasi-separated morphism of schemes, F an abelian sheaf on X_etale, and overlines a geometric point of S lying over s ∈ S. Then (R^nf_* F)_overlines = H_etale^n( X ×_S Spec(O_S, overlines^sh), p^-1F) where p : X ×_S Spec(O_S, overlines^sh) → X is the projection. For K ∈ D^+(X_etale) and n ∈ Z we have (R^nf_*K)_overlines = H_etale^n(X ×_S Spec(O_S, overlines^sh), p^-1K) In fact, we have (Rf_*K)_overlines = RΓ_etale(X ×_S Spec(O_S,…","statement_latex":"Let $f: X \\to S$ be a quasi-compact and quasi-separated morphism of schemes,\n$\\mathcal{F}$ an abelian sheaf on $X_\\etale$, and $\\overline{s}$ a\ngeometric point of $S$ lying over $s \\in S$. Then\n$$\n\\left(R^nf_* \\mathcal{F}\\right)_{\\overline{s}} =\nH_\\etale^n( X \\times_S \\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh}),\np^{-1}\\mathcal{F})\n$$\nwhere $p : X \\times_S \\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh}) \\to X$\nis the projection. For $K \\in D^+(X_\\etale)$ and $n \\in \\mathbf{Z}$\nwe have\n$$\n\\left(R^nf_*K\\right)_{\\overline{s}} =\nH_\\etale^n(X \\times_S \\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh}), p^{-1}K)\n$$\nIn fact, we have\n$$\n\\left(Rf_*K\\right)_{\\overline{s}}\n=\nR\\Gamma_\\etale(X \\times_S \\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh}), p^{-1}K)\n$$\nin $D^+(\\textit{Ab})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Stalks of higher direct images","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Q9","source_file":"etale-cohomology.tex","source_line":8014,"source_end_line":8038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8014-L8038","statement_sha256":"7b7661d987ce5e0772c2dcc80407ca7ff52b7f35e667559a14fcced8a39b4c78","origin":"The Stacks Project","memory_eligible":false,"source_rank":9963,"rank":9963,"depth":47,"x":1620.525,"y":1058.133,"cluster":"tale-geometry"},{"id":"stacks:03QB","tag":"03QB","title":"The Leray spectral sequence · Lemma 03QB","summary":"Let f: X → Y be a morphism and I an injective object of Ab(X_etale). Let V ∈ Ob(Y_etale). Then • for any covering V = (V_j→ V)_j ∈ J we have check H^p(V, f_*I) = 0 for all p > 0, • f_*I is acyclic for the functor Γ(V, -), and • if g : Y → Z, then f_*I is acyclic for g_*.","statement_latex":"Let $f: X \\to Y$ be a morphism and $\\mathcal{I}$ an injective object of\n$\\textit{Ab}(X_\\etale)$. Let $V \\in \\Ob(Y_\\etale)$. Then\n\\begin{enumerate}\n\\item for any covering $\\mathcal{V} = \\{V_j\\to V\\}_{j \\in J}$ we have\n$\\check H^p(\\mathcal{V}, f_*\\mathcal{I}) = 0$ for all $p > 0$,\n\\item $f_*\\mathcal{I}$ is acyclic for the functor $\\Gamma(V, -)$, and\n\\item if $g : Y \\to Z$, then $f_*\\mathcal{I}$ is acyclic for $g_*$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Leray spectral sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QB","source_file":"etale-cohomology.tex","source_line":8145,"source_end_line":8155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8145-L8155","statement_sha256":"958e3d0ce30e97af633b97a853d0145e9c7b0906f08df2bfe729d881b968ae1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9964,"rank":9964,"depth":23,"x":1616.007,"y":1223.486,"cluster":"tale-geometry"},{"id":"stacks:03QC","tag":"03QC","title":"Leray spectral sequence · Proposition 03QC","summary":"Let f: X → Y be a morphism of schemes and F an étale sheaf on X. Then there is a spectral sequence E_2^p, q = H_etale^p(Y, R^qf_*F) ⇒ H_etale^p+q(X, F).","statement_latex":"Let $f: X \\to Y$ be a morphism of schemes and $\\mathcal{F}$ an \\'etale sheaf on\n$X$. Then there is a spectral sequence\n$$\nE_2^{p, q} = H_\\etale^p(Y, R^qf_*\\mathcal{F}) \\Rightarrow\nH_\\etale^{p+q}(X, \\mathcal{F}).\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Leray spectral sequence","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QC","source_file":"etale-cohomology.tex","source_line":8175,"source_end_line":8183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8175-L8183","statement_sha256":"7ef81559ab532ac1065db0c31d0bd280f3a49da8256c8946cb88252e4db9ff71","origin":"The Stacks Project","memory_eligible":false,"source_rank":9965,"rank":9965,"depth":24,"x":1486.174,"y":1098.809,"cluster":"tale-geometry"},{"id":"stacks:03QO","tag":"03QO","title":"Vanishing of finite higher direct images · Lemma 03QO","summary":"Let R be a strictly henselian local ring. Set S = Spec(R) and let overlines be its closed point. Then the global sections functor Γ(S, -) : Ab(S_etale) → Ab is exact. In fact we have Γ(S, F) = F_overlines for any sheaf of sets F. In particular ∀ p≥ 1, H_etale^p(S, F)=0 for all F∈ Ab(S_etale).","statement_latex":"Let $R$ be a strictly henselian local ring. Set $S = \\Spec(R)$ and let\n$\\overline{s}$ be its closed point. Then the global\nsections functor\n$\\Gamma(S, -) : \\textit{Ab}(S_\\etale) \\to \\textit{Ab}$\nis exact. In fact we have $\\Gamma(S, \\mathcal{F}) = \\mathcal{F}_{\\overline{s}}$\nfor any sheaf of sets $\\mathcal{F}$. In particular\n$$\n\\forall p\\geq 1, \\quad H_\\etale^p(S, \\mathcal{F})=0\n$$\nfor all $\\mathcal{F}\\in \\textit{Ab}(S_\\etale)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Vanishing of finite higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QO","source_file":"etale-cohomology.tex","source_line":8206,"source_end_line":8218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8206-L8218","statement_sha256":"52f7599b1464a5ea7dae062c578ff2f2c34ec0b97df7a594d90ec060284c48a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9966,"rank":9966,"depth":45,"x":1682.461,"y":1117.101,"cluster":"tale-geometry"},{"id":"stacks:03QP","tag":"03QP","title":"Vanishing of finite higher direct images · Proposition 03QP","summary":"Let f : X → Y be a finite morphism of schemes. • For any geometric point overliney : Spec(k) → Y we have (f_*F)_overliney = ∏_overlinex : Spec(k) → X, f(overlinex) = overliney F_overlinex. for F in Sh(X_etale) and (f_*F)_overliney = bigoplus_overlinex : Spec(k) → X, f(overlinex) = overliney F_overlinex. for F in Ab(X_etale). • For any q ≥ 1 we have R^q f_*F = 0 for F in Ab(X_etale).","statement_latex":"Let $f : X \\to Y$ be a finite morphism of schemes.\n\\begin{enumerate}\n\\item For any geometric point $\\overline{y} : \\Spec(k) \\to Y$ we have\n$$\n(f_*\\mathcal{F})_{\\overline{y}} =\n\\prod\\nolimits_{\\overline{x} : \\Spec(k) \\to X,\\ f(\\overline{x}) =\n\\overline{y}} \\mathcal{F}_{\\overline{x}}.\n$$\nfor $\\mathcal{F}$ in $\\Sh(X_\\etale)$ and\n$$\n(f_*\\mathcal{F})_{\\overline{y}} =\n\\bigoplus\\nolimits_{\\overline{x} : \\Spec(k) \\to X,\\ f(\\overline{x}) =\n\\overline{y}} \\mathcal{F}_{\\overline{x}}.\n$$\nfor $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$.\n\\item For any $q \\geq 1$ we have $R^q f_*\\mathcal{F} = 0$\nfor $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Vanishing of finite higher direct images","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QP","source_file":"etale-cohomology.tex","source_line":8244,"source_end_line":8264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8244-L8264","statement_sha256":"db49ee571f549c364a6f4609057af9545b9bc8435d80aaf18b60422080260238","origin":"The Stacks Project","memory_eligible":false,"source_rank":9967,"rank":9967,"depth":51,"x":1522.778,"y":1215.135,"cluster":"tale-geometry"},{"id":"stacks:0959","tag":"0959","title":"Vanishing of finite higher direct images · Lemma 0959","summary":"Consider a cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y of schemes with f a finite morphism. For any sheaf of sets F on X_etale we have f'_*(g')^-1F = g^-1f_*F.","statement_latex":"Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nof schemes with $f$ a finite morphism.\nFor any sheaf of sets $\\mathcal{F}$ on $X_\\etale$ we have\n$f'_*(g')^{-1}\\mathcal{F} = g^{-1}f_*\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Vanishing of finite higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0959","source_file":"etale-cohomology.tex","source_line":8291,"source_end_line":8303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8291-L8303","statement_sha256":"75e27cf9d5e9c6466b96469d6ea86e61bb6f254567e1b20ac0c23273de396186","origin":"The Stacks Project","memory_eligible":false,"source_rank":9968,"rank":9968,"depth":52,"x":1561.747,"y":1051.999,"cluster":"tale-geometry"},{"id":"stacks:0EYP","tag":"0EYP","title":"Vanishing of finite higher direct images · Lemma 0EYP","summary":"Consider a cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y of schemes with f an integral morphism. For any sheaf of sets F on X_etale we have f'_*(g')^-1F = g^-1f_*F.","statement_latex":"Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nof schemes with $f$ an integral morphism.\nFor any sheaf of sets $\\mathcal{F}$ on $X_\\etale$ we have\n$f'_*(g')^{-1}\\mathcal{F} = g^{-1}f_*\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Vanishing of finite higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYP","source_file":"etale-cohomology.tex","source_line":8349,"source_end_line":8361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8349-L8361","statement_sha256":"a0f1a9293b9c3464fd112ffd6a2dae689b5cfb6a564d5b7fba9b63df040a718c","origin":"The Stacks Project","memory_eligible":false,"source_rank":9969,"rank":9969,"depth":53,"x":1664.35,"y":1194.613,"cluster":"tale-geometry"},{"id":"stacks:09Z2","tag":"09Z2","title":"Vanishing of finite higher direct images · Lemma 09Z2","summary":"Let f : X → Y be a surjective finite morphism of schemes. Set f_n : X_n → Y equal to the (n + 1)-fold fibre product of X over Y. For F ∈ Ab(Y_etale) set F_n = f_n, *f_n^-1F. There is an exact sequence 0 → F → F_0 → F_1 → F_2 → … on Y_etale. Moreover, there is a spectral sequence E_1^p, q = H^q_etale(X_p, f_p^-1F) converging to H^p + q(Y_etale, F). This spectral sequence is functorial in F.","statement_latex":"Let $f : X \\to Y$ be a surjective finite morphism of schemes.\nSet $f_n : X_n \\to Y$ equal to the $(n + 1)$-fold fibre product\nof $X$ over $Y$. For $\\mathcal{F} \\in \\textit{Ab}(Y_\\etale)$ set\n$\\mathcal{F}_n = f_{n, *}f_n^{-1}\\mathcal{F}$. There is an exact\nsequence\n$$\n0 \\to \\mathcal{F} \\to \\mathcal{F}_0 \\to \\mathcal{F}_1 \\to\n\\mathcal{F}_2 \\to \\ldots\n$$\non $Y_\\etale$. Moreover, there is a spectral sequence\n$$\nE_1^{p, q} = H^q_\\etale(X_p, f_p^{-1}\\mathcal{F})\n$$\nconverging to $H^{p + q}(Y_\\etale, \\mathcal{F})$.\nThis spectral sequence is functorial in $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Vanishing of finite higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Z2","source_file":"etale-cohomology.tex","source_line":8411,"source_end_line":8428,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8411-L8428","statement_sha256":"c05b2a1e1c67842d5c6036816a80b116b70a76070e4b5189ed3606f90ffdbb8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9970,"rank":9970,"depth":52,"x":1473.729,"y":1147.603,"cluster":"tale-geometry"},{"id":"stacks:03QX","tag":"03QX","title":"Galois action on stalks · Definition 03QX","summary":"Let S be a scheme. Let overlines be a geometric point lying over the point s of S. Let kappa(s) ⊂ kappa(s)^sep ⊂ kappa(overlines) denote the separable algebraic closure of kappa(s) in the algebraically closed field kappa(overlines). • In this situation the absolute Galois group of kappa(s) is Gal(kappa(s)^sep/kappa(s)). It is sometimes denoted Gal_kappa(s). • The geometric point overlines is called algebraic if kappa(s) ⊂ kappa(overlines) is an algebraic closure of kappa(s).","statement_latex":"Let $S$ be a scheme.\nLet $\\overline{s}$ be a geometric point lying over the point $s$ of $S$.\nLet $\\kappa(s) \\subset \\kappa(s)^{sep} \\subset \\kappa(\\overline{s})$\ndenote the separable algebraic closure of $\\kappa(s)$ in the algebraically\nclosed field $\\kappa(\\overline{s})$.\n\\begin{enumerate}\n\\item In this situation the {\\it absolute Galois group} of $\\kappa(s)$\nis $\\text{Gal}(\\kappa(s)^{sep}/\\kappa(s))$. It is sometimes denoted\n$\\text{Gal}_{\\kappa(s)}$.\n\\item The geometric point $\\overline{s}$ is called\n{\\it algebraic} if $\\kappa(s) \\subset \\kappa(\\overline{s})$ is\nan algebraic closure of $\\kappa(s)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois action on stalks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QX","source_file":"etale-cohomology.tex","source_line":8562,"source_end_line":8577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8562-L8577","statement_sha256":"283dd4c721eeef7de37400e9e49e1e7a1927d4a6957d0499187b310104f41acf","origin":"The Stacks Project","memory_eligible":false,"source_rank":9971,"rank":9971,"depth":0,"x":1652.354,"y":1073.997,"cluster":"tale-geometry"},{"id":"stacks:03QT","tag":"03QT","title":"Galois action on stalks · Theorem 03QT","summary":"Let S = Spec(K) with K a field. Let overlines be a geometric point of S. Let G = Gal_kappa(s) denote the absolute Galois group. Taking stalks induces an equivalence of categories Sh(S_etale) → G-Sets, F ↦ F_overlines.","statement_latex":"Let $S = \\Spec(K)$ with $K$ a field.\nLet $\\overline{s}$ be a geometric point of $S$.\nLet $G = \\text{Gal}_{\\kappa(s)}$ denote the absolute Galois group.\nTaking stalks induces an equivalence of categories\n$$\n\\Sh(S_\\etale) \\longrightarrow G\\textit{-Sets},\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{s}}.\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois action on stalks","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QT","source_file":"etale-cohomology.tex","source_line":8625,"source_end_line":8636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8625-L8636","statement_sha256":"20eac12c9eef15006996294ddb6de5dfa1fc24a4d409e4fc24691e23e3a229b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9972,"rank":9972,"depth":46,"x":1579.725,"y":1229.854,"cluster":"tale-geometry"},{"id":"stacks:04JM","tag":"04JM","title":"Galois action on stalks · Lemma 04JM","summary":"Assumptions and notations as in Theorem [Tag 03QT]. There is a functorial bijection Γ(S, F) = (F_overlines)^G","statement_latex":"Assumptions and notations as in\nTheorem \\ref{theorem-equivalence-sheaves-point}.\nThere is a functorial bijection\n$$\n\\Gamma(S, \\mathcal{F}) = (\\mathcal{F}_{\\overline{s}})^G\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois action on stalks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JM","source_file":"etale-cohomology.tex","source_line":8667,"source_end_line":8675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8667-L8675","statement_sha256":"0789738ee37cce78233bb6a82477f789b8242398d027d68ccc7baa88d0520fd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9973,"rank":9973,"depth":47,"x":1507.841,"y":1073.491,"cluster":"tale-geometry"},{"id":"stacks:04JP","tag":"04JP","title":"Group cohomology · Definition 04JP","summary":"Let G be a topological group. • A G-module, sometimes called a discrete G-module, is an abelian group M endowed with a left action a : G × M → M by group homomorphisms such that a is continuous when M is given the discrete topology. • A morphism of G-modules f : M → N is a G-equivariant homomorphism from M to N. • The category of G-modules is denoted Mod_G. Let R be a ring. • An R-G-module is an R-module M endowed with a left action a : G × M → M by R-linear maps such…","statement_latex":"Let $G$ be a topological group.\n\\begin{enumerate}\n\\item A {\\it $G$-module}, sometimes called a {\\it discrete $G$-module},\nis an abelian group $M$ endowed with a left action $a : G \\times M \\to M$\nby group homomorphisms such that $a$ is continuous when $M$ is given the\ndiscrete topology.\n\\item A {\\it morphism of $G$-modules} $f : M \\to N$ is a\n$G$-equivariant homomorphism from $M$ to $N$.\n\\item The category of $G$-modules is denoted $\\text{Mod}_G$.\n\\end{enumerate}\nLet $R$ be a ring.\n\\begin{enumerate}\n\\item An {\\it $R\\text{-}G$-module} is an $R$-module $M$ endowed with\na left action $a : G \\times M \\to M$ by $R$-linear maps such that $a$\nis continuous when $M$ is given the discrete topology.\n\\item A {\\it morphism of $R\\text{-}G$-modules} $f : M \\to N$ is a\n$G$-equivariant $R$-module map from $M$ to $N$.\n\\item The category of $R\\text{-}G$-modules is denoted $\\text{Mod}_{R, G}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Group cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JP","source_file":"etale-cohomology.tex","source_line":8785,"source_end_line":8806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8785-L8806","statement_sha256":"34460ea24c6bb6e452614cedcec6f93f07b51f2e40131d728ae05753a15fcaba","origin":"The Stacks Project","memory_eligible":false,"source_rank":9974,"rank":9974,"depth":0,"x":1686.847,"y":1148.11,"cluster":"tale-geometry"},{"id":"stacks:04JR","tag":"04JR","title":"Group cohomology · Definition 04JR","summary":"Let G be a topological group. Let M be a discrete G-module with continuous G-action. In other words, M is an object of the category Mod_G introduced in Definition [Tag 04JP]. • The right derived functors H^i(G, M) of H^0(G, M) on the category Mod_G are called the continuous group cohomology groups of M. • If G is an abstract group endowed with the discrete topology then the H^i(G, M) are called the group cohomology groups of M. • If G is a Galois group, then the groups…","statement_latex":"Let $G$ be a topological group. Let $M$ be a discrete $G$-module\nwith continuous $G$-action. In other words, $M$ is an object\nof the category $\\text{Mod}_G$ introduced in\nDefinition \\ref{definition-G-module-continuous}.\n\\begin{enumerate}\n\\item The right derived functors $H^i(G, M)$ of $H^0(G, M)$ on the\ncategory $\\text{Mod}_G$ are called the\n{\\it continuous group cohomology groups} of $M$.\n\\item If $G$ is an abstract group endowed with the discrete topology\nthen the $H^i(G, M)$ are called the {\\it group cohomology groups} of $M$.\n\\item If $G$ is a Galois group, then the groups $H^i(G, M)$ are called\nthe {\\it Galois cohomology groups} of $M$.\n\\item If $G$ is the absolute Galois group of a field $K$, then the groups\n$H^i(G, M)$ are sometimes called the {\\it Galois cohomology groups of $K$\nwith coefficients in $M$}. In this case we sometimes write\n$H^i(K, M)$ instead of $H^i(G, M)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Group cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JR","source_file":"etale-cohomology.tex","source_line":8836,"source_end_line":8855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8836-L8855","statement_sha256":"bc1416f390d65d62c65fd609befef2c79349795ef4ba4cced041f2adcea4f119","origin":"The Stacks Project","memory_eligible":false,"source_rank":9975,"rank":9975,"depth":1,"x":1494.569,"y":1194.726,"cluster":"tale-geometry"},{"id":"stacks:0DVD","tag":"0DVD","title":"Group cohomology · Lemma 0DVD","summary":"Let G be a topological group. Let R be a ring. For every i ≥ 0 the diagram xymatrix Mod_R, G ar[rr]_H^i(G, -) ar[d] & & Mod_R ar[d] Mod_G ar[rr]^H^i(G, -) & & Ab whose vertical arrows are the forgetful functors is commutative.","statement_latex":"Let $G$ be a topological group. Let $R$ be a ring.\nFor every $i \\geq 0$ the diagram\n$$\n\\xymatrix{\n\\text{Mod}_{R, G} \\ar[rr]_{H^i(G, -)} \\ar[d] & &\n\\text{Mod}_R \\ar[d] \\\\\n\\text{Mod}_G \\ar[rr]^{H^i(G, -)} & &\n\\textit{Ab}\n}\n$$\nwhose vertical arrows are the forgetful functors is commutative.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Group cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVD","source_file":"etale-cohomology.tex","source_line":8857,"source_end_line":8870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8857-L8870","statement_sha256":"17991d27555758a43b40a01dcaa89f58fd1e526bb09f64acec5809c9e7e98867","origin":"The Stacks Project","memory_eligible":false,"source_rank":9976,"rank":9976,"depth":15,"x":1599.014,"y":1051.043,"cluster":"tale-geometry"},{"id":"stacks:0DVE","tag":"0DVE","title":"Group cohomology · Lemma 0DVE","summary":"Let G be a topological group. Let R be a ring. Let M, N be R-G-modules. If M is finite projective as an R-module, then Ext^i(M, N) = H^i(G, M^vee ⊗_R N) (for notation see proof).","statement_latex":"Let $G$ be a topological group. Let $R$ be a ring.\nLet $M$, $N$ be $R\\text{-}G$-modules. If $M$ is finite projective\nas an $R$-module, then\n$\\text{Ext}^i(M, N) = H^i(G, M^\\vee \\otimes_R N)$ (for notation\nsee proof).","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Group cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVE","source_file":"etale-cohomology.tex","source_line":8905,"source_end_line":8912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8905-L8912","statement_sha256":"84e75ba987a9065fffd32dcc766bfa9ae6ad5527e975df2871bb7fb1d92c55ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":9977,"rank":9977,"depth":6,"x":1637.598,"y":1216.493,"cluster":"tale-geometry"},{"id":"stacks:0DVF","tag":"0DVF","title":"Group cohomology · Lemma 0DVF","summary":"Let G be a topological group. Let k be a field. Let V be a k-G-module. If G is topologically finitely generated and dim_k(V) < ∞, then dim_k H^1(G, V) < ∞.","statement_latex":"Let $G$ be a topological group. Let $k$ be a field.\nLet $V$ be a $k\\text{-}G$-module.\nIf $G$ is topologically finitely generated and\n$\\dim_k(V) < \\infty$, then $\\dim_k H^1(G, V) < \\infty$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Group cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVF","source_file":"etale-cohomology.tex","source_line":8940,"source_end_line":8946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8940-L8946","statement_sha256":"02e6dc32edc517d3ac594e047a7233ebf00598e1adc8a5a1938bb6559c5433e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":9978,"rank":9978,"depth":7,"x":1475.867,"y":1116.244,"cluster":"tale-geometry"},{"id":"stacks:0DV3","tag":"0DV3","title":"Group cohomology · Lemma 0DV3","summary":"Let G be a profinite topological group. Then • H^i(G, M) is torsion for i > 0 and any G-module M, and • H^i(G, M) = 0 if M is a Q-vector space.","statement_latex":"Let $G$ be a profinite topological group.\nThen\n\\begin{enumerate}\n\\item $H^i(G, M)$ is torsion for $i > 0$ and any $G$-module $M$, and\n\\item $H^i(G, M) = 0$ if $M$ is a $\\mathbf{Q}$-vector space.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Group cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DV3","source_file":"etale-cohomology.tex","source_line":8978,"source_end_line":8986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L8978-L8986","statement_sha256":"e80a2854213e51236ef579e09b4a6a2c355bbdba71e24def456eab57eae14972","origin":"The Stacks Project","memory_eligible":false,"source_rank":9979,"rank":9979,"depth":0,"x":1676.026,"y":1098.371,"cluster":"tale-geometry"},{"id":"stacks:04JQ","tag":"04JQ","title":"Cohomology of a point · Lemma 04JQ","summary":"Let S = Spec(K) with K a field. Let overlines be a geometric point of S. Let G = Gal_kappa(s) denote the absolute Galois group. The stalk functor induces an equivalence of categories Ab(S_etale) → Mod_G, F ↦ F_overlines.","statement_latex":"Let $S = \\Spec(K)$ with $K$ a field.\nLet $\\overline{s}$ be a geometric point of $S$.\nLet $G = \\text{Gal}_{\\kappa(s)}$ denote the absolute Galois group.\nThe stalk functor induces an equivalence of categories\n$$\n\\textit{Ab}(S_\\etale) \\longrightarrow \\text{Mod}_G,\n\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}_{\\overline{s}}.\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of a point","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JQ","source_file":"etale-cohomology.tex","source_line":9077,"source_end_line":9088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9077-L9088","statement_sha256":"28095c0daf776238e648d7ccca4aef7aa1f1ddbe882b33539efc73ab058a7741","origin":"The Stacks Project","memory_eligible":false,"source_rank":9980,"rank":9980,"depth":47,"x":1542.616,"y":1225.305,"cluster":"tale-geometry"},{"id":"stacks:03QU","tag":"03QU","title":"Cohomology of a point · Lemma 03QU","summary":"Notation and assumptions as in Lemma [Tag 04JQ]. Let F be an abelian sheaf on Spec(K)_etale which corresponds to the G-module M. Then • in D(Ab) we have a canonical isomorphism RΓ(S, F) = RΓ_G(M), • H_etale^0(S, F) = M^G, and • H_etale^q(S, F) = H^q(G, M).","statement_latex":"Notation and assumptions as in\nLemma \\ref{lemma-equivalence-abelian-sheaves-point}.\nLet $\\mathcal{F}$ be an abelian sheaf on $\\Spec(K)_\\etale$\nwhich corresponds to the $G$-module $M$.\nThen\n\\begin{enumerate}\n\\item in $D(\\textit{Ab})$ we have a canonical isomorphism\n$R\\Gamma(S, \\mathcal{F}) = R\\Gamma_G(M)$,\n\\item $H_\\etale^0(S, \\mathcal{F}) = M^G$, and\n\\item $H_\\etale^q(S, \\mathcal{F}) = H^q(G, M)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of a point","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03QU","source_file":"etale-cohomology.tex","source_line":9099,"source_end_line":9112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9099-L9112","statement_sha256":"e53509d3f74e1c5b37f1edd42dae3cb18abb76cc7ace5294e13c65527184392d","origin":"The Stacks Project","memory_eligible":false,"source_rank":9981,"rank":9981,"depth":48,"x":1538.909,"y":1055.767,"cluster":"tale-geometry"},{"id":"stacks:0D1W","tag":"0D1W","title":"Cohomology of a point · Lemma 0D1W","summary":"Let R be a local ring of dimension 0. Let S = Spec(R). Then every O_S-module on S_etale is quasi-coherent.","statement_latex":"Let $R$ be a local ring of dimension $0$. Let $S = \\Spec(R)$.\nThen every $\\mathcal{O}_S$-module on $S_\\etale$ is quasi-coherent.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of a point","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1W","source_file":"etale-cohomology.tex","source_line":9176,"source_end_line":9180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9176-L9180","statement_sha256":"775ddb0d173bc0caa1577ef7fecdffbceb464b9731376be4ab5925bfca9fae95","origin":"The Stacks Project","memory_eligible":false,"source_rank":9982,"rank":9982,"depth":46,"x":1678.176,"y":1178.851,"cluster":"tale-geometry"},{"id":"stacks:03R2","tag":"03R2","title":"Brauer groups · Theorem 03R2","summary":"Let K be a field. For a unital, associative (not necessarily commutative) K-algebra A the following are equivalent • A is finite central simple K-algebra, • A is a finite dimensional K-vector space, K is the center of A, and A has no nontrivial two-sided ideal, • there exists d ≥ 1 such that A ⊗_K bar K ≅ Mat(d × d, bar K), • there exists d ≥ 1 such that A ⊗_K K^sep ≅ Mat(d × d, K^sep), • there exist d ≥ 1 and a finite Galois extension K'/K such that A ⊗_K K' ≅ Mat(d × d,…","statement_latex":"Let $K$ be a field. For a unital, associative (not necessarily commutative)\n$K$-algebra $A$ the following are equivalent\n\\begin{enumerate}\n\\item $A$ is finite central simple $K$-algebra,\n\\item $A$ is a finite dimensional $K$-vector space, $K$ is the center of $A$,\nand $A$ has no nontrivial two-sided ideal,\n\\item there exists $d \\geq 1$ such that\n$A \\otimes_K \\bar K \\cong \\text{Mat}(d \\times d, \\bar K)$,\n\\item there exists $d \\geq 1$ such that\n$A \\otimes_K K^{sep} \\cong \\text{Mat}(d \\times d, K^{sep})$,\n\\item there exist $d \\geq 1$ and a finite Galois extension $K'/K$\nsuch that\n$A \\otimes_K K' \\cong \\text{Mat}(d \\times d, K')$,\n\\item there exist $n \\geq 1$ and a finite central skew field $D$\nover $K$ such that $A \\cong \\text{Mat}(n \\times n, D)$.\n\\end{enumerate}\nThe integer $d$ is called the {\\it degree} of $A$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Brauer groups","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03R2","source_file":"etale-cohomology.tex","source_line":9272,"source_end_line":9291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9272-L9291","statement_sha256":"7bfeecd5af46346f83c3d2e0ca2267c33c6c54bd4ec2ce3e5f5c2a17ccdb20ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":9983,"rank":9983,"depth":10,"x":1476.219,"y":1167.097,"cluster":"tale-geometry"},{"id":"stacks:03R4","tag":"03R4","title":"Brauer groups · Lemma 03R4","summary":"Let A be a finite central simple algebra over K. Then A ⊗_K A^opp & → & End_K(A) a ⊗ a' & ↦ & (x ↦ a x a') is an isomorphism of algebras over K.","statement_latex":"Let $A$ be a finite central simple algebra over $K$. Then\n$$\n\\begin{matrix}\nA \\otimes_K A^{opp} & \\longrightarrow & \\text{End}_K(A) \\\\\n\\ a \\otimes a' & \\longmapsto & (x \\mapsto a x a')\n\\end{matrix}\n$$\nis an isomorphism of algebras over $K$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Brauer groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03R4","source_file":"etale-cohomology.tex","source_line":9298,"source_end_line":9308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9298-L9308","statement_sha256":"5a21e802d8ff00a96736456d63d5abaf3c5d1aaa6c6e5c5078a983161f0c5c91","origin":"The Stacks Project","memory_eligible":false,"source_rank":9984,"rank":9984,"depth":5,"x":1634.812,"y":1061.021,"cluster":"tale-geometry"},{"id":"stacks:03R3","tag":"03R3","title":"Brauer groups · Definition 03R3","summary":"Two finite central simple algebras A_1 and A_2 over K are called similar, or equivalent if there exist m, n ≥ 1 such that Mat(n × n, A_1) ≅ Mat(m × m, A_2). We write A_1 sim A_2.","statement_latex":"Two finite central simple algebras $A_1$ and $A_2$ over $K$ are called\n{\\it similar}, or {\\it equivalent} if there exist $m, n \\geq 1$\nsuch that $\\text{Mat}(n \\times n, A_1)\n\\cong \\text{Mat}(m \\times m, A_2)$. We write $A_1 \\sim A_2$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Brauer groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03R3","source_file":"etale-cohomology.tex","source_line":9315,"source_end_line":9321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9315-L9321","statement_sha256":"9d91f766f805a40a4e406557f609a2856651080cfe2c582299800a19e15b912a","origin":"The Stacks Project","memory_eligible":false,"source_rank":9985,"rank":9985,"depth":0,"x":1603.127,"y":1229.464,"cluster":"tale-geometry"},{"id":"stacks:03R5","tag":"03R5","title":"Brauer groups · Definition 03R5","summary":"Let K be a field. The Brauer group of K is the set Br (K) of similarity classes of finite central simple algebras over K, endowed with the group law induced by tensor product (over K). The class of A in Br(K) is denoted by [A]. The neutral element is [K] = [Mat(d × d, K)] for any d ≥ 1.","statement_latex":"Let $K$ be a field. The {\\it Brauer group} of $K$ is the set $\\text{Br} (K)$\nof similarity classes of finite central simple algebras over $K$, endowed with\nthe group law induced by tensor product (over $K$). The class of $A$ in\n$\\text{Br}(K)$ is denoted by $[A]$. The neutral element is\n$[K] = [\\text{Mat}(d \\times d, K)]$ for any $d \\geq 1$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Brauer groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03R5","source_file":"etale-cohomology.tex","source_line":9327,"source_end_line":9334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9327-L9334","statement_sha256":"b4c4c09ac21762378e379fc6cb752f8754d4b83370490a5b195e82603a30509f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9986,"rank":9986,"depth":0,"x":1490.878,"y":1087.08,"cluster":"tale-geometry"},{"id":"stacks:03R6","tag":"03R6","title":"Brauer groups · Lemma 03R6","summary":"Central simple algebras are classified by Galois cohomology of PGL. Let K be a field and let K^sep be a separable algebraic closure. Then the set of isomorphism classes of central simple algebras of degree d over K is in bijection with the non-abelian cohomology H^1(Gal(K^sep/K), PGL_d(K^sep)).","statement_latex":"\\begin{slogan}\nCentral simple algebras are classified by Galois cohomology of PGL.\n\\end{slogan}\nLet $K$ be a field and let $K^{sep}$ be a separable algebraic closure.\nThen the set of isomorphism classes of central simple algebras of degree\n$d$ over $K$ is in bijection with the non-abelian cohomology\n$H^1(\\text{Gal}(K^{sep}/K), \\text{PGL}_d(K^{sep}))$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Brauer groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03R6","source_file":"etale-cohomology.tex","source_line":9346,"source_end_line":9355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9346-L9355","statement_sha256":"69fc6cf5d4e0fe33d34d3fcc0be5dc013838407f59702038094520a84a5b82e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9987,"rank":9987,"depth":11,"x":1688.423,"y":1128.436,"cluster":"tale-geometry"},{"id":"stacks:03R7","tag":"03R7","title":"Brauer groups · Theorem 03R7","summary":"Let K be a field with separable algebraic closure K^sep. The map δ : Br(K) → H^2(Gal(K^sep/K), (K^sep)^*) defined above is a group isomorphism.","statement_latex":"Let $K$ be a field with separable algebraic closure $K^{sep}$. The map\n$\\delta : \\text{Br}(K) \\to H^2(\\text{Gal}(K^{sep}/K), (K^{sep})^*)$\ndefined above is a group isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Brauer groups","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03R7","source_file":"etale-cohomology.tex","source_line":9414,"source_end_line":9419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9414-L9419","statement_sha256":"6e970998243bd7b563067a23d2f2248f5ec3ecacfff1e0406e56e9eec1159c70","origin":"The Stacks Project","memory_eligible":false,"source_rank":9988,"rank":9988,"depth":0,"x":1509.252,"y":1210.145,"cluster":"tale-geometry"},{"id":"stacks:0A2K","tag":"0A2K","title":"The Brauer group of a scheme · Lemma 0A2K","summary":"Let S be a scheme. Let F and G be finite locally free sheaves of O_S-modules of positive rank. If there exists an isomorphism SheafHom_O_S(F, F) ≅ SheafHom_O_S(G, G) of O_S-algebras, then there exists an invertible sheaf L on S such that F ⊗_O_S L ≅ G and such that this isomorphism induces the given isomorphism of endomorphism algebras.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ and $\\mathcal{G}$ be finite locally\nfree sheaves of $\\mathcal{O}_S$-modules of positive rank. If there\nexists an isomorphism\n$\\SheafHom_{\\mathcal{O}_S}(\\mathcal{F}, \\mathcal{F}) \\cong\n\\SheafHom_{\\mathcal{O}_S}(\\mathcal{G}, \\mathcal{G})$ of\n$\\mathcal{O}_S$-algebras, then there exists an invertible sheaf\n$\\mathcal{L}$ on $S$ such that\n$\\mathcal{F} \\otimes_{\\mathcal{O}_S} \\mathcal{L} \\cong \\mathcal{G}$\nand such that this isomorphism induces the given isomorphism of\nendomorphism algebras.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Brauer group of a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2K","source_file":"etale-cohomology.tex","source_line":9483,"source_end_line":9495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9483-L9495","statement_sha256":"5af1b988e68c9e8cce4908f8a0c9452600b3787f6ef5875d0faa1ad07a7eef9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":9989,"rank":9989,"depth":0,"x":1575.754,"y":1048.005,"cluster":"tale-geometry"},{"id":"stacks:0A2L","tag":"0A2L","title":"The Brauer group of a scheme · Lemma 0A2L","summary":"Argument taken from [Saltman-torsion]. Let S be a scheme. Let A be an Azumaya algebra which is locally free of rank d^2 over S. Then the class of A in the Brauer group of S is annihilated by d.","statement_latex":"\\begin{reference}\nArgument taken from \\cite{Saltman-torsion}.\n\\end{reference}\nLet $S$ be a scheme. Let $\\mathcal{A}$ be an Azumaya algebra which is\nlocally free of rank $d^2$ over $S$. Then the class\nof $\\mathcal{A}$ in the Brauer group of $S$ is annihilated by $d$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Brauer group of a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A2L","source_file":"etale-cohomology.tex","source_line":9518,"source_end_line":9526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9518-L9526","statement_sha256":"29e1e4c306b2b9c58376c932d367272cee5db09a31190886470a81bf58128c4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":9990,"rank":9990,"depth":41,"x":1657.217,"y":1205.516,"cluster":"tale-geometry"},{"id":"stacks:0A3K","tag":"0A3K","title":"The Artin-Schreier sequence · Lemma 0A3K","summary":"Let p be a prime. Let S be a scheme of characteristic p. • If S is affine, then H_etale^q(S, underlineZ/pZ) = 0 for all q ≥ 2. • If S is a quasi-compact and quasi-separated scheme of dimension d, then H_etale^q(S, underlineZ/pZ) = 0 for all q ≥ 2 + d.","statement_latex":"Let $p$ be a prime. Let $S$ be a scheme of characteristic $p$.\n\\begin{enumerate}\n\\item If $S$ is affine, then\n$H_\\etale^q(S, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$ for all\n$q \\geq 2$.\n\\item If $S$ is a quasi-compact and quasi-separated scheme of\ndimension $d$, then $H_\\etale^q(S, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$\nfor all $q \\geq 2 + d$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Artin-Schreier sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3K","source_file":"etale-cohomology.tex","source_line":9611,"source_end_line":9622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9611-L9622","statement_sha256":"2561f51f6771c566daf5d04ce3b1a76aae51663022d09a050cc4c6adac9fe29f","origin":"The Stacks Project","memory_eligible":false,"source_rank":9991,"rank":9991,"depth":42,"x":1470.226,"y":1135.498,"cluster":"tale-geometry"},{"id":"stacks:0A3L","tag":"0A3L","title":"The Artin-Schreier sequence · Lemma 0A3L","summary":"Let k be an algebraically closed field of characteristic p > 0. Let V be a finite dimensional k-vector space. Let F : V → V be a frobenius linear map, i.e., an additive map such that F(λ v) = λ^p F(v) for all λ ∈ k and v ∈ V. Then F - 1 : V → V is surjective with kernel a finite dimensional F_p-vector space of dimension ≤ dim_k(V).","statement_latex":"Let $k$ be an algebraically closed field of characteristic $p > 0$.\nLet $V$ be a finite dimensional $k$-vector space. Let $F : V \\to V$\nbe a frobenius linear map, i.e., an additive map such that\n$F(\\lambda v) = \\lambda^p F(v)$ for all $\\lambda \\in k$ and $v \\in V$.\nThen $F - 1 : V \\to V$ is surjective with kernel a finite dimensional\n$\\mathbf{F}_p$-vector space of dimension $\\leq \\dim_k(V)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Artin-Schreier sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3L","source_file":"etale-cohomology.tex","source_line":9640,"source_end_line":9648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9640-L9648","statement_sha256":"8bc06bcae0fe56a44270f642beb81bb539ad10555cd4c2dc744105f28d0a7ec4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9992,"rank":9992,"depth":1,"x":1664.682,"y":1080.951,"cluster":"tale-geometry"},{"id":"stacks:0HAN","tag":"0HAN","title":"The Artin-Schreier sequence · Lemma 0HAN","summary":"In the situation of Lemma [Tag 0A3L] let k'/k be an extension of algebraically closed fields. Set V' = V ⊗_k k' and F' : V' → V' be the Frobenius linear map induced by F. Then Ker(F - 1) maps isomorphically onto Ker(F' - 1).","statement_latex":"In the situation of Lemma \\ref{lemma-F-1} let $k'/k$ be an extension\nof algebraically closed fields. Set $V' = V \\otimes_k k'$ and\n$F' : V' \\to V'$ be the Frobenius linear map induced by $F$.\nThen $\\Ker(F - 1)$ maps isomorphically onto $\\Ker(F' - 1)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Artin-Schreier sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAN","source_file":"etale-cohomology.tex","source_line":9691,"source_end_line":9697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9691-L9697","statement_sha256":"693ca0a08a2e2aebb2cee6fe8d34e3ab4ff5a2ada2210b84e9cd8c50d5c8c4ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":9993,"rank":9993,"depth":2,"x":1565.023,"y":1231.717,"cluster":"tale-geometry"},{"id":"stacks:0A3M","tag":"0A3M","title":"The Artin-Schreier sequence · Lemma 0A3M","summary":"Let X be a separated scheme of finite type over a field k. Let F be a coherent sheaf of O_X-modules. Then dim_k H^d(X, F) < ∞ where d = dim(X).","statement_latex":"Let $X$ be a separated scheme of finite type over a field $k$.\nLet $\\mathcal{F}$ be a coherent sheaf of $\\mathcal{O}_X$-modules.\nThen $\\dim_k H^d(X, \\mathcal{F}) < \\infty$ where $d = \\dim(X)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Artin-Schreier sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3M","source_file":"etale-cohomology.tex","source_line":9709,"source_end_line":9714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9709-L9714","statement_sha256":"6dd206f3bf96e629b5197caf4b71812fde205e17f22a644d1637e1572fff4fb2","origin":"The Stacks Project","memory_eligible":false,"source_rank":9994,"rank":9994,"depth":30,"x":1517.203,"y":1063.767,"cluster":"tale-geometry"},{"id":"stacks:0A3N","tag":"0A3N","title":"The Artin-Schreier sequence · Lemma 0A3N","summary":"Let X be separated of finite type over an algebraically closed field k of characteristic p > 0. Then H_etale^q(X, underlineZ/pZ) = 0 for q ≥ dim(X) + 1.","statement_latex":"Let $X$ be separated of finite type over an algebraically closed\nfield $k$ of characteristic $p > 0$. Then\n$H_\\etale^q(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) = 0$ for\n$q \\geq dim(X) + 1$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Artin-Schreier sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3N","source_file":"etale-cohomology.tex","source_line":9837,"source_end_line":9843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9837-L9843","statement_sha256":"4820937b8c941072a17e46cfe002c5c4218461fb38a4a8e9fb1725795a12dcc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9995,"rank":9995,"depth":43,"x":1687.756,"y":1160.608,"cluster":"tale-geometry"},{"id":"stacks:0A3P","tag":"0A3P","title":"The Artin-Schreier sequence · Lemma 0A3P","summary":"Let X be a proper scheme over an algebraically closed field k of characteristic p > 0. Then • H_etale^q(X, underlineZ/pZ) is a finite Z/pZ-module for all q, and • H^q_etale(X, underlineZ/pZ) → H^q_etale(X_k', underlineZ/pZ) is an isomorphism if k'/k is an extension of algebraically closed fields.","statement_latex":"Let $X$ be a proper scheme over an algebraically closed\nfield $k$ of characteristic $p > 0$. Then\n\\begin{enumerate}\n\\item $H_\\etale^q(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}})$\nis a finite $\\mathbf{Z}/p\\mathbf{Z}$-module for all $q$, and\n\\item $H^q_\\etale(X, \\underline{\\mathbf{Z}/p\\mathbf{Z}}) \\to\nH^q_\\etale(X_{k'}, \\underline{\\mathbf{Z}/p\\mathbf{Z}})$\nis an isomorphism if $k'/k$ is an extension of algebraically\nclosed fields.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The Artin-Schreier sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3P","source_file":"etale-cohomology.tex","source_line":9862,"source_end_line":9874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9862-L9874","statement_sha256":"b13ce892e9bf175cae7d80133762af03e8209a177faa0fadead42c33f4af50a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":9996,"rank":9996,"depth":42,"x":1483.841,"y":1186.009,"cluster":"tale-geometry"},{"id":"stacks:03RU","tag":"03RU","title":"Locally constant sheaves · Definition 03RU","summary":"Let X be a scheme. Let F be a sheaf of sets on X_etale. • Let E be a set. We say F is the constant sheaf with value E if F is the sheafification of the presheaf U ↦ E. Notation: underlineE_X or underlineE. • We say F is a constant sheaf if it is isomorphic to a sheaf as in (1). • We say F is locally constant if there exists a covering (U_i → X) such that F|_U_i is a constant sheaf. • We say that F is finite locally constant if it is locally constant and the constant…","statement_latex":"Let $X$ be a scheme.\nLet $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$.\n\\begin{enumerate}\n\\item Let $E$ be a set. We say $\\mathcal{F}$ is the\n{\\it constant sheaf with value $E$} if $\\mathcal{F}$ is the\nsheafification of the presheaf $U \\mapsto E$.\nNotation: $\\underline{E}_X$ or $\\underline{E}$.\n\\item We say $\\mathcal{F}$ is a {\\it constant sheaf} if it is\nisomorphic to a sheaf as in (1).\n\\item We say $\\mathcal{F}$ is {\\it locally constant} if there exists a\ncovering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf.\n\\item We say that $\\mathcal{F}$ is {\\it finite locally constant} if it\nis locally constant and the constant values are finite sets.\n\\end{enumerate}\nLet $\\mathcal{F}$ be a sheaf of abelian groups on $X_\\etale$.\n\\begin{enumerate}\n\\item Let $A$ be an abelian group.\nWe say $\\mathcal{F}$ is the {\\it constant sheaf with value $A$} if\n$\\mathcal{F}$ is the sheafification of the presheaf $U \\mapsto A$.\nNotation: $\\underline{A}_X$ or $\\underline{A}$.\n\\item We say $\\mathcal{F}$ is a {\\it constant sheaf} if it is isomorphic\nas an abelian sheaf to a sheaf as in (1).\n\\item We say $\\mathcal{F}$ is {\\it locally constant} if there exists a\ncovering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf.\n\\item We say that $\\mathcal{F}$ is {\\it finite locally constant} if it\nis locally constant and the constant values are finite abelian groups.\n\\end{enumerate}\nLet $\\Lambda$ be a ring. Let $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules\non $X_\\etale$.\n\\begin{enumerate}\n\\item Let $M$ be a $\\Lambda$-module.\nWe say $\\mathcal{F}$ is the {\\it constant sheaf with value $M$} if\n$\\mathcal{F}$ is the sheafification of the presheaf $U \\mapsto M$.\nNotation: $\\underline{M}_X$ or $\\underline{M}$.\n\\item We say $\\mathcal{F}$ is a {\\it constant sheaf} if it is isomorphic\nas a sheaf of $\\Lambda$-modules to a sheaf as in (1).\n\\item We say $\\mathcal{F}$ is {\\it locally constant} if there exists a\ncovering $\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i}$ is a constant sheaf.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RU","source_file":"etale-cohomology.tex","source_line":9916,"source_end_line":9957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9916-L9957","statement_sha256":"c79d80476e2cbd3c90ae7b9ee25fb5f4954e001642b47b026529311f189ed0ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":9997,"rank":9997,"depth":0,"x":1613.953,"y":1051.391,"cluster":"tale-geometry"},{"id":"stacks:095A","tag":"095A","title":"Locally constant sheaves · Lemma 095A","summary":"Let f : X → Y be a morphism of schemes. If G is a locally constant sheaf of sets, abelian groups, or Lambda-modules on Y_etale, the same is true for f^-1G on X_etale.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. If $\\mathcal{G}$ is a\nlocally constant sheaf of sets, abelian groups, or $\\Lambda$-modules\non $Y_\\etale$, the same is true for $f^{-1}\\mathcal{G}$\non $X_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095A","source_file":"etale-cohomology.tex","source_line":9959,"source_end_line":9965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9959-L9965","statement_sha256":"8b537acc917b5cfcba3c46711549086e1f5782721bb4fe95e742bc4f3d863391","origin":"The Stacks Project","memory_eligible":false,"source_rank":9998,"rank":9998,"depth":1,"x":1626.282,"y":1224.718,"cluster":"tale-geometry"},{"id":"stacks:095B","tag":"095B","title":"Locally constant sheaves · Lemma 095B","summary":"Let f : X → Y be a finite étale morphism of schemes. If F is a (finite) locally constant sheaf of sets, (finite) locally constant sheaf of abelian groups, or (finite type) locally constant sheaf of Lambda-modules on X_etale, the same is true for f_*F on Y_etale.","statement_latex":"Let $f : X \\to Y$ be a finite \\'etale morphism of schemes.\nIf $\\mathcal{F}$ is a (finite) locally constant sheaf of sets,\n(finite) locally constant sheaf of abelian groups, or\n(finite type) locally constant sheaf of $\\Lambda$-modules\non $X_\\etale$, the same is true for $f_*\\mathcal{F}$\non $Y_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095B","source_file":"etale-cohomology.tex","source_line":9972,"source_end_line":9980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9972-L9980","statement_sha256":"544de3805691bf834efc29e349d7e632aedfcff3998bc6326f5c02455161e4a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":9999,"rank":9999,"depth":46,"x":1477.608,"y":1103.744,"cluster":"tale-geometry"},{"id":"stacks:03RV","tag":"03RV","title":"Locally constant sheaves · Lemma 03RV","summary":"Let X be a scheme and F a sheaf of sets on X_etale. Then the following are equivalent • F is finite locally constant, and • F = h_U for some finite étale morphism U → X.","statement_latex":"Let $X$ be a scheme and $\\mathcal{F}$ a sheaf of sets on $X_\\etale$.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is finite locally constant, and\n\\item $\\mathcal{F} = h_U$ for some finite \\'etale morphism $U \\to X$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RV","source_file":"etale-cohomology.tex","source_line":9993,"source_end_line":10001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L9993-L10001","statement_sha256":"b40fbb495990691e4ceeeb19a6bf35bed79a849f0e7274e5b554f22ee4fb353d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10000,"rank":10000,"depth":47,"x":1684.798,"y":1108.593,"cluster":"tale-geometry"},{"id":"stacks:095C","tag":"095C","title":"Locally constant sheaves · Lemma 095C","summary":"Let X be a scheme. • Let φ : F → G be a map of locally constant sheaves of sets on X_etale. If F is finite locally constant, there exists an étale covering (U_i → X) such that φ|_U_i is the map of constant sheaves associated to a map of sets. • Let φ : F → G be a map of locally constant sheaves of abelian groups on X_etale. If F is finite locally constant, there exists an étale covering (U_i → X) such that φ|_U_i is the map of constant abelian sheaves associated to a map…","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map\nof locally constant sheaves of sets on $X_\\etale$.\nIf $\\mathcal{F}$ is finite locally constant, there exists an\n\\'etale covering $\\{U_i \\to X\\}$ such that\n$\\varphi|_{U_i}$ is the map of constant sheaves associated to\na map of sets.\n\\item Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map\nof locally constant sheaves of abelian groups on $X_\\etale$.\nIf $\\mathcal{F}$ is finite locally constant, there exists an \\'etale\ncovering $\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of\nconstant abelian sheaves associated to a map of abelian groups.\n\\item Let $\\Lambda$ be a ring.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map\nof locally constant sheaves of $\\Lambda$-modules on $X_\\etale$.\nIf $\\mathcal{F}$ is of finite type, then there exists an \\'etale covering\n$\\{U_i \\to X\\}$ such that $\\varphi|_{U_i}$ is the map of constant\nsheaves of $\\Lambda$-modules associated to a map of $\\Lambda$-modules.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095C","source_file":"etale-cohomology.tex","source_line":10020,"source_end_line":10042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10020-L10042","statement_sha256":"47383502af6102dc3b34fff643202a3ce1ec82ed03a3819cb76be1447e5b9f1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10001,"rank":10001,"depth":1,"x":1527.911,"y":1222.734,"cluster":"tale-geometry"},{"id":"stacks:03RX","tag":"03RX","title":"Locally constant sheaves · Lemma 03RX","summary":"Let X be a scheme. • The category of finite locally constant sheaves of sets is closed under finite limits and colimits inside Sh(X_etale). • The category of finite locally constant abelian sheaves is a weak Serre subcategory of Ab(X_etale). • Let Lambda be a Noetherian ring. The category of finite type, locally constant sheaves of Lambda-modules on X_etale is a weak Serre subcategory of Mod(X_etale, Lambda).","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item The category of finite locally constant sheaves of sets\nis closed under finite limits and colimits inside $\\Sh(X_\\etale)$.\n\\item The category of finite locally constant abelian sheaves is a\nweak Serre subcategory of $\\textit{Ab}(X_\\etale)$.\n\\item Let $\\Lambda$ be a Noetherian ring. The category of\nfinite type, locally constant sheaves of $\\Lambda$-modules on\n$X_\\etale$ is a weak Serre subcategory of\n$\\textit{Mod}(X_\\etale, \\Lambda)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RX","source_file":"etale-cohomology.tex","source_line":10049,"source_end_line":10062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10049-L10062","statement_sha256":"d2d4e0ce53fe0e5fd2018e0eeeea051cb31ad6126380c6bc529aa291e2c6bf7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10002,"rank":10002,"depth":2,"x":1551.84,"y":1049.316,"cluster":"tale-geometry"},{"id":"stacks:095D","tag":"095D","title":"Locally constant sheaves · Lemma 095D","summary":"Let X be a scheme. Let Lambda be a ring. The tensor product of two locally constant sheaves of Lambda-modules on X_etale is a locally constant sheaf of Lambda-modules.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a ring.\nThe tensor product of two locally constant sheaves of $\\Lambda$-modules\non $X_\\etale$ is a locally constant sheaf of $\\Lambda$-modules.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095D","source_file":"etale-cohomology.tex","source_line":10070,"source_end_line":10075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10070-L10075","statement_sha256":"83753d2a92f97e947ed421dee9c358a4f5990f1d80a5e7e9170e7188ea2dc3a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10003,"rank":10003,"depth":1,"x":1673.814,"y":1190.958,"cluster":"tale-geometry"},{"id":"stacks:09BF","tag":"09BF","title":"Locally constant sheaves · Lemma 09BF","summary":"Let X be a connected scheme. Let Lambda be a ring and let F be a locally constant sheaf of Lambda-modules. Then there exists a Lambda-module M and an étale covering (U_i → X) such that F|_U_i ≅ underlineM|_U_i.","statement_latex":"Let $X$ be a connected scheme. Let $\\Lambda$ be a ring and let\n$\\mathcal{F}$ be a locally constant sheaf of $\\Lambda$-modules.\nThen there exists a $\\Lambda$-module $M$ and an \\'etale covering\n$\\{U_i \\to X\\}$ such that $\\mathcal{F}|_{U_i} \\cong \\underline{M}|_{U_i}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BF","source_file":"etale-cohomology.tex","source_line":10083,"source_end_line":10089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10083-L10089","statement_sha256":"30e5002f702d7299dde5f5493de3976deef4a9f45ff32674c371827e24958d87","origin":"The Stacks Project","memory_eligible":false,"source_rank":10004,"rank":10004,"depth":0,"x":1469.699,"y":1155.678,"cluster":"tale-geometry"},{"id":"stacks:0DV5","tag":"0DV5","title":"Locally constant sheaves and the fundamental group · Lemma 0DV5","summary":"Let X be a connected scheme. Let overlinex be a geometric point of X. • There is an equivalence of categories ( finite locally constant sheaves of sets on X_etale ) longleftrightarrow ( finite π_1(X, overlinex)-sets ) • There is an equivalence of categories ( finite locally constant sheaves of abelian groups on X_etale ) longleftrightarrow ( finite π_1(X, overlinex)-modules ) • Let Lambda be a finite ring. There is an equivalence of categories ( finite type, locally…","statement_latex":"Let $X$ be a connected scheme. Let $\\overline{x}$ be a geometric point of $X$.\n\\begin{enumerate}\n\\item There is an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{finite locally constant}\\\\\n\\text{sheaves of sets on }X_\\etale\n\\end{matrix}\n\\right\\}\n\\longleftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{finite }\\pi_1(X, \\overline{x})\\text{-sets}\n\\end{matrix}\n\\right\\}\n$$\n\\item There is an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{finite locally constant}\\\\\n\\text{sheaves of abelian groups on }X_\\etale\n\\end{matrix}\n\\right\\}\n\\longleftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{finite }\\pi_1(X, \\overline{x})\\text{-modules}\n\\end{matrix}\n\\right\\}\n$$\n\\item Let $\\Lambda$ be a finite ring. There is an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{finite type, locally constant}\\\\\n\\text{sheaves of }\\Lambda\\text{-modules on }X_\\etale\n\\end{matrix}\n\\right\\}\n\\longleftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{finite }\\pi_1(X, \\overline{x})\\text{-modules endowed}\\\\\n\\text{with commuting }\\Lambda\\text{-module structure}\n\\end{matrix}\n\\right\\}\n$$\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves and the fundamental group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DV5","source_file":"etale-cohomology.tex","source_line":10117,"source_end_line":10168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10117-L10168","statement_sha256":"80c4e559207c699ae097187ebf85459a61d60d546b32a6782531f9086f3964db","origin":"The Stacks Project","memory_eligible":false,"source_rank":10005,"rank":10005,"depth":50,"x":1648.817,"y":1065.754,"cluster":"tale-geometry"},{"id":"stacks:0GIY","tag":"0GIY","title":"Locally constant sheaves and the fundamental group · Lemma 0GIY","summary":"Let X be an irreducible, geometrically unibranch scheme. Let overlinex be a geometric point of X. Let Lambda be a ring. There is an equivalence of categories ( finite type, locally constant sheaves of Lambda-modules on X_etale ) longleftrightarrow ( finite Lambda-modules M endowed with a continuous π_1(X, overlinex)-action )","statement_latex":"Let $X$ be an irreducible, geometrically unibranch scheme.\nLet $\\overline{x}$ be a geometric point of $X$.\nLet $\\Lambda$ be a ring. There is an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{finite type, locally constant}\\\\\n\\text{sheaves of }\\Lambda\\text{-modules on }X_\\etale\n\\end{matrix}\n\\right\\}\n\\longleftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{finite }\\Lambda\\text{-modules }M\\text{ endowed}\\\\\n\\text{with a continuous }\\pi_1(X, \\overline{x})\\text{-action}\n\\end{matrix}\n\\right\\}\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Locally constant sheaves and the fundamental group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIY","source_file":"etale-cohomology.tex","source_line":10199,"source_end_line":10219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10199-L10219","statement_sha256":"84a58f94c6d0f650ed0ec385f43a50b66c7c8ab823e130b35a06c17c13c04932","origin":"The Stacks Project","memory_eligible":false,"source_rank":10006,"rank":10006,"depth":62,"x":1588.973,"y":1233.919,"cluster":"tale-geometry"},{"id":"stacks:03SE","tag":"03SE","title":"Méthode de la trace · Definition 03SE","summary":"Let f : Y → X be a finite étale morphism of schemes. The map f_* f^-1 → id described above and explicitly below is called the trace.","statement_latex":"Let $f : Y \\to X$ be a finite \\'etale morphism of schemes.\nThe map $f_* f^{-1} \\to \\text{id}$ described above and explicitly below\nis called the {\\it trace}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Méthode de la trace","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SE","source_file":"etale-cohomology.tex","source_line":10352,"source_end_line":10357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10352-L10357","statement_sha256":"a5f4c7ddd89ac356845b9f2baf35ee6766b83551643b13a6412ccf0eb9081087","origin":"The Stacks Project","memory_eligible":false,"source_rank":10007,"rank":10007,"depth":0,"x":1497.748,"y":1075.753,"cluster":"tale-geometry"},{"id":"stacks:0A3R","tag":"0A3R","title":"Méthode de la trace · Lemma 0A3R","summary":"Let S be a connected scheme. Let ℓ be a prime number. Let F be a finite type, locally constant sheaf of F_ℓ-vector spaces on S_etale. Then there exists a finite étale morphism f : T → S of degree prime to ℓ such that f^-1F has a finite filtration whose successive quotients are underlineZ/ℓZ_T.","statement_latex":"Let $S$ be a connected scheme. Let $\\ell$ be a prime number. Let\n$\\mathcal{F}$ be a finite type, locally constant sheaf of\n$\\mathbf{F}_\\ell$-vector spaces on $S_\\etale$.\nThen there exists a finite \\'etale morphism\n$f : T \\to S$ of degree prime to $\\ell$ such that $f^{-1}\\mathcal{F}$\nhas a finite filtration whose successive quotients are\n$\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}_T$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Méthode de la trace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3R","source_file":"etale-cohomology.tex","source_line":10413,"source_end_line":10422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10413-L10422","statement_sha256":"1ddb6fa4c1b91a75ede50a01d322c03052416f2b04f548685c45fa82012e7112","origin":"The Stacks Project","memory_eligible":false,"source_rank":10008,"rank":10008,"depth":51,"x":1692.463,"y":1140.704,"cluster":"tale-geometry"},{"id":"stacks:0GIZ","tag":"0GIZ","title":"Méthode de la trace · Lemma 0GIZ","summary":"Let Lambda be a Noetherian ring. Let ℓ be a prime number and n ≥ 1. Let H be a finite ℓ-group. Let M be a finite Lambda[H]-module annihilated by ℓ^n. Then there is a finite filtration 0 = M_0 ⊂ M_1 ⊂ … ⊂ M_t = M by Lambda[H]-submodules such that H acts trivially on M_i + 1/M_i for all i = 0, …, t - 1.","statement_latex":"Let $\\Lambda$ be a Noetherian ring.\nLet $\\ell$ be a prime number and $n \\geq 1$.\nLet $H$ be a finite $\\ell$-group.\nLet $M$ be a finite $\\Lambda[H]$-module annihilated by $\\ell^n$.\nThen there is a finite filtration\n$0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_t = M$\nby $\\Lambda[H]$-submodules\nsuch that $H$ acts trivially on $M_{i + 1}/M_i$ for all\n$i = 0, \\ldots, t - 1$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Méthode de la trace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIZ","source_file":"etale-cohomology.tex","source_line":10459,"source_end_line":10470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10459-L10470","statement_sha256":"911f59d5bc8a1652fff1ce755a4bc9beed6641f5a464e70134bb2921d5f5235f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10009,"rank":10009,"depth":0,"x":1496.396,"y":1203.378,"cluster":"tale-geometry"},{"id":"stacks:0GJ0","tag":"0GJ0","title":"Méthode de la trace · Lemma 0GJ0","summary":"Let S be an irreducible, geometrically unibranch scheme. Let ℓ be a prime number and n ≥ 1. Let Lambda be a Noetherian ring. Let F be a finite type, locally constant sheaf of Lambda-modules on S_etale which is annihilated by ℓ^n. Then there exists a finite étale morphism f : T → S of degree prime to ℓ such that f^-1F has a finite filtration whose successive quotients are of the form underlineM_T for some finite Lambda-modules M.","statement_latex":"Let $S$ be an irreducible, geometrically unibranch scheme.\nLet $\\ell$ be a prime number and $n \\geq 1$. Let $\\Lambda$\nbe a Noetherian ring. Let $\\mathcal{F}$ be a finite type,\nlocally constant sheaf of $\\Lambda$-modules on $S_\\etale$\nwhich is annihilated by $\\ell^n$. Then there exists a\nfinite \\'etale morphism $f : T \\to S$ of degree prime to $\\ell$\nsuch that $f^{-1}\\mathcal{F}$ has a finite filtration whose\nsuccessive quotients are of the form $\\underline{M}_T$\nfor some finite $\\Lambda$-modules $M$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Méthode de la trace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJ0","source_file":"etale-cohomology.tex","source_line":10478,"source_end_line":10489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10478-L10489","statement_sha256":"682e7e8515945739a132e07322c81f7434bc2b64d11a1c23c5a05e8f30381612","origin":"The Stacks Project","memory_eligible":false,"source_rank":10010,"rank":10010,"depth":63,"x":1590.695,"y":1045.705,"cluster":"tale-geometry"},{"id":"stacks:0DV7","tag":"0DV7","title":"Galois cohomology · Lemma 0DV7","summary":"Let ℓ be a prime number and n an integer > 0. Let S be a quasi-compact and quasi-separated scheme. Let X = lim_i ∈ I X_i be the limit of a directed system of S-schemes each X_i → S being finite étale of constant degree relatively prime to ℓ. The following are equivalent: • there exists an ℓ-power torsion sheaf G on S such that H_etale^n(S, G) ≠ 0 and • there exists an ℓ-power torsion sheaf F on X such that H_etale^n(X, F) ≠ 0. In fact, given G we can take F = g^-1F and…","statement_latex":"Let $\\ell$ be a prime number and $n$ an integer $> 0$.\nLet $S$ be a quasi-compact and quasi-separated scheme.\nLet $X = \\lim_{i \\in I} X_i$ be the limit of a\ndirected system of $S$-schemes each $X_i \\to S$\nbeing finite \\'etale of constant degree relatively prime to $\\ell$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item there exists an $\\ell$-power torsion sheaf\n$\\mathcal{G}$ on $S$ such that $H_\\etale^n(S, \\mathcal{G}) \\neq 0$ and\n\\item there exists an $\\ell$-power torsion sheaf $\\mathcal{F}$ on $X$\nsuch that $H_\\etale^n(X, \\mathcal{F}) \\neq 0$.\n\\end{enumerate}\nIn fact, given\n$\\mathcal{G}$ we can take $\\mathcal{F} = g^{-1}\\mathcal{F}$\nand given\n$\\mathcal{F}$ we can take $\\mathcal{G} = g_*\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DV7","source_file":"etale-cohomology.tex","source_line":10542,"source_end_line":10560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10542-L10560","statement_sha256":"21fb27d65e23890a10e70add6c4efa0967d2cf1b82ddd693972f1460cd717d59","origin":"The Stacks Project","memory_eligible":false,"source_rank":10011,"rank":10011,"depth":52,"x":1648.031,"y":1215.698,"cluster":"tale-geometry"},{"id":"stacks:0DV8","tag":"0DV8","title":"Galois cohomology · Lemma 0DV8","summary":"Let ℓ be a prime number and n an integer > 0. Let K be a field with G = Gal(K^sep/K) and let H ⊂ G be a maximal pro-ℓ subgroup with L/K being the corresponding field extension. Then H^n_etale(Spec(K), F) = 0 for all ℓ-power torsion F if and only if H^n_etale(Spec(L), underlineZ/ℓZ) = 0.","statement_latex":"Let $\\ell$ be a prime number and $n$ an integer $> 0$.\nLet $K$ be a field with $G = Gal(K^{sep}/K)$ and let\n$H \\subset G$ be a maximal pro-$\\ell$ subgroup with $L/K$\nbeing the corresponding field extension. Then\n$H^n_\\etale(\\Spec(K), \\mathcal{F}) = 0$ for all\n$\\ell$-power torsion $\\mathcal{F}$ if and only if\n$H^n_\\etale(\\Spec(L), \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}) = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DV8","source_file":"etale-cohomology.tex","source_line":10598,"source_end_line":10607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10598-L10607","statement_sha256":"b416f161274c051d3d0a77932a532d1f4fb44e2c114c75ca62fbcbb661bc4563","origin":"The Stacks Project","memory_eligible":false,"source_rank":10012,"rank":10012,"depth":53,"x":1468.817,"y":1122.761,"cluster":"tale-geometry"},{"id":"stacks:0DV9","tag":"0DV9","title":"Galois cohomology · Lemma 0DV9","summary":"Let ℓ be a prime number and n an integer > 0. Let K be a field with G = Gal(K^sep/K) and let H ⊂ G be a maximal pro-ℓ subgroup with L/K being the corresponding field extension. Then H^q_etale(Spec(K),F) = 0 for q ≥ n and all ℓ-torsion sheaves F if and only if H^n_etale(Spec(L), underlineZ/ℓZ) = 0.","statement_latex":"Let $\\ell$ be a prime number and $n$ an integer $> 0$.\nLet $K$ be a field with $G = Gal(K^{sep}/K)$ and let\n$H \\subset G$ be a maximal pro-$\\ell$ subgroup \nwith $L/K$ being the corresponding field extension.\nThen $H^q_\\etale(\\Spec(K),\\mathcal{F}) = 0$ for $q \\geq n$ and all\n$\\ell$-torsion sheaves $\\mathcal{F}$ if  and only if\n$H^n_\\etale(\\Spec(L), \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}) = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DV9","source_file":"etale-cohomology.tex","source_line":10657,"source_end_line":10666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10657-L10666","statement_sha256":"6bc14f9162b183a550153b86d2cfbcac5c30920ea7a056b7c28091ed964fbbef","origin":"The Stacks Project","memory_eligible":false,"source_rank":10013,"rank":10013,"depth":54,"x":1675.971,"y":1089.559,"cluster":"tale-geometry"},{"id":"stacks:03R8","tag":"03R8","title":"Galois cohomology · Proposition 03R8","summary":"[SerreGaloisCohomology] Let K be a field with separable algebraic closure K^sep. Assume that for any finite extension K' of K we have Br(K') = 0. Then • H^q(Gal(K^sep/K), (K^sep)^*) = 0 for all q ≥ 1, and • H^q(Gal(K^sep/K), M) = 0 for any torsion Gal(K^sep/K)-module M and any q ≥ 2,","statement_latex":"\\begin{reference}\n\\cite[Chapter II, Section 3, Proposition 5]{SerreGaloisCohomology}\n\\end{reference}\nLet $K$ be a field with separable algebraic closure $K^{sep}$.\nAssume that for any finite extension $K'$ of $K$ we have\n$\\text{Br}(K') = 0$. Then\n\\begin{enumerate}\n\\item $H^q(\\text{Gal}(K^{sep}/K), (K^{sep})^*) = 0$\nfor all $q \\geq 1$, and\n\\item $H^q(\\text{Gal}(K^{sep}/K), M) = 0$\nfor any torsion $\\text{Gal}(K^{sep}/K)$-module $M$ and any $q \\geq 2$,\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03R8","source_file":"etale-cohomology.tex","source_line":10693,"source_end_line":10707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10693-L10707","statement_sha256":"df6a3fa5452d31ef21d764807160954bcac0a2109f5e17cb5470df659e541759","origin":"The Stacks Project","memory_eligible":false,"source_rank":10014,"rank":10014,"depth":55,"x":1549.757,"y":1231.768,"cluster":"tale-geometry"},{"id":"stacks:03R9","tag":"03R9","title":"Galois cohomology · Definition 03R9","summary":"A field K is called C_r if for every 0 < d^r < n and every f ∈ K[T_1, …, T_n] homogeneous of degree d, there exist α = (α_1, …, α_n), α_i ∈ K not all zero, such that f(α) = 0. Such an α is called a nontrivial solution of f.","statement_latex":"A field $K$ is called {\\it $C_r$}\nif for every $0 < d^r < n$ and every $f \\in K[T_1,\n\\ldots, T_n]$ homogeneous of degree $d$, there exist $\\alpha = (\\alpha_1,\n\\ldots, \\alpha_n)$, $\\alpha_i \\in K$ not all zero, such that $f(\\alpha) = 0$.\nSuch an $\\alpha$ is called a {\\it nontrivial solution} of $f$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03R9","source_file":"etale-cohomology.tex","source_line":10798,"source_end_line":10805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10798-L10805","statement_sha256":"f722118ed2b166957fa76d3489a8975893cf5cb00cb27e4ff97aa5067f57e02e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10015,"rank":10015,"depth":0,"x":1528.437,"y":1055.062,"cluster":"tale-geometry"},{"id":"stacks:03RB","tag":"03RB","title":"Galois cohomology · Lemma 03RB","summary":"Let k be an algebraically closed field. Let f_1, …, f_s ∈ k[T_1, …, T_n] be homogeneous polynomials of degree d_1, …, d_s with d_i > 0. If s < n, then f_1 = … = f_s = 0 have a common nontrivial solution.","statement_latex":"Let $k$ be an algebraically closed field. Let\n$f_1, \\ldots, f_s \\in k[T_1, \\ldots, T_n]$\nbe homogeneous polynomials of degree $d_1, \\ldots, d_s$ with $d_i\n> 0$. If $s < n$, then $f_1 = \\ldots = f_s = 0$ have a common nontrivial\nsolution.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RB","source_file":"etale-cohomology.tex","source_line":10815,"source_end_line":10822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10815-L10822","statement_sha256":"d371485aa68e979f63e9c689f74d0b0a9075f9c929e1dc307af32002262f548e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10016,"rank":10016,"depth":29,"x":1686.462,"y":1173.417,"cluster":"tale-geometry"},{"id":"stacks:03RC","tag":"03RC","title":"Galois cohomology · Theorem 03RC","summary":"Let K be a C_1 field. Then Br(K) = 0.","statement_latex":"Let $K$ be a $C_1$ field. Then $\\text{Br}(K) = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RC","source_file":"etale-cohomology.tex","source_line":10833,"source_end_line":10836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10833-L10836","statement_sha256":"931ab5ccfb51536ccebf54f4d187f07cfd69b166e237b6b3600adf08e9d7b3e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10017,"rank":10017,"depth":0,"x":1474.489,"y":1175.814,"cluster":"tale-geometry"},{"id":"stacks:03RE","tag":"03RE","title":"Galois cohomology · Definition 03RE","summary":"Let k be a field. A variety is separated, integral scheme of finite type over k. A curve is a variety of dimension 1.","statement_latex":"Let $k$ be a field. A {\\it variety} is separated, integral scheme of\nfinite type over $k$. A {\\it curve} is a variety of dimension $1$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RE","source_file":"etale-cohomology.tex","source_line":10855,"source_end_line":10859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10855-L10859","statement_sha256":"2c2b5dc6c834ad5db23eb270930b16c15555a2fd5250a9748afb011755852d4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10018,"rank":10018,"depth":0,"x":1629.064,"y":1053.612,"cluster":"tale-geometry"},{"id":"stacks:03RD","tag":"03RD","title":"Tsen's theorem · Theorem 03RD","summary":"The function field of a variety of dimension r over an algebraically closed field k is C_r.","statement_latex":"The function field of a variety of dimension $r$ over an algebraically closed\nfield $k$ is $C_r$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RD","source_file":"etale-cohomology.tex","source_line":10861,"source_end_line":10865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10861-L10865","statement_sha256":"bb72a0b9b298ba1beea4a05f799a0034549402f7d25aaca63c2b3e160b667af6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10019,"rank":10019,"depth":32,"x":1613.334,"y":1231.657,"cluster":"tale-geometry"},{"id":"stacks:03RF","tag":"03RF","title":"Galois cohomology · Lemma 03RF","summary":"Let C be a curve over an algebraically closed field k. Then the Brauer group of the function field of C is zero: Br(k(C)) = 0.","statement_latex":"Let $C$ be a curve over an algebraically closed field $k$. Then\nthe Brauer group of the function field of $C$ is zero:\n$\\text{Br}(k(C)) = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RF","source_file":"etale-cohomology.tex","source_line":10893,"source_end_line":10898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10893-L10898","statement_sha256":"da44c84663d2c66d9885a99a62ed13e7db47de57893c1616ad83a1028a7ee071","origin":"The Stacks Project","memory_eligible":false,"source_rank":10020,"rank":10020,"depth":33,"x":1481.588,"y":1091.267,"cluster":"tale-geometry"},{"id":"stacks:03RG","tag":"03RG","title":"Galois cohomology · Lemma 03RG","summary":"Let k be an algebraically closed field and K/k a field extension of transcendence degree 1. Then for all q ≥ 1, H_etale^q(Spec(K), G_m) = 0.","statement_latex":"Let $k$ be an algebraically closed field and $K/k$ a field extension\nof transcendence degree 1. Then for all $q \\geq 1$,\n$H_\\etale^q(\\Spec(K), \\mathbf{G}_m) = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Galois cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RG","source_file":"etale-cohomology.tex","source_line":10906,"source_end_line":10911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10906-L10911","statement_sha256":"b1e7ad1e30d55e8e2e72a86dcbcf616f776e14a5622c77a7531187799c4da604","origin":"The Stacks Project","memory_eligible":false,"source_rank":10021,"rank":10021,"depth":56,"x":1691.899,"y":1120.069,"cluster":"tale-geometry"},{"id":"stacks:03RI","tag":"03RI","title":"The Fundamental Exact Sequence · Theorem 03RI","summary":"There is a short exact sequence of étale sheaves on X 0 → G_m, X → j_* G_m, eta → bigoplus_x ∈ X_0 i_x_* underlineZ → 0.","statement_latex":"There is a short exact sequence of \\'etale sheaves on $X$\n$$\n0 \\longrightarrow\n\\mathbf{G}_{m, X} \\longrightarrow\nj_* \\mathbf{G}_{m, \\eta} \\longrightarrow\n\\bigoplus\\nolimits_{x \\in X_0} {i_x}_* \\underline{\\mathbf{Z}}\n\\longrightarrow 0.\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Higher vanishing for the multiplicative group","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RI","source_file":"etale-cohomology.tex","source_line":10943,"source_end_line":10953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10943-L10953","statement_sha256":"05a52f7a2671a029d41ab718cf673f9e0934a517b9b7d82f08590a2e7731425d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10022,"rank":10022,"depth":47,"x":1513.432,"y":1218.289,"cluster":"tale-geometry"},{"id":"stacks:03RJ","tag":"03RJ","title":"Higher vanishing for the multiplicative group · Lemma 03RJ","summary":"For any q ≥ 1, R^q j_*G_m, eta = 0.","statement_latex":"For any $q \\geq 1$, $R^q j_*\\mathbf{G}_{m, \\eta} = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Higher vanishing for the multiplicative group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RJ","source_file":"etale-cohomology.tex","source_line":10987,"source_end_line":10990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L10987-L10990","statement_sha256":"c407b8e1cca72f1c2ba28a478c6fdfdbbba6fae1e1cc0e2cfc03499d9f240f63","origin":"The Stacks Project","memory_eligible":false,"source_rank":10023,"rank":10023,"depth":57,"x":1566.111,"y":1044.379,"cluster":"tale-geometry"},{"id":"stacks:03RK","tag":"03RK","title":"Higher vanishing for the multiplicative group · Lemma 03RK","summary":"For all p ≥ 1, H_etale^p(X, j_*G_m, eta) = 0.","statement_latex":"For all $p \\geq 1$, $H_\\etale^p(X, j_*\\mathbf{G}_{m, \\eta}) = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Higher vanishing for the multiplicative group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RK","source_file":"etale-cohomology.tex","source_line":11034,"source_end_line":11037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11034-L11037","statement_sha256":"b88310fd50c5fbbde211b73bb65256bdca5a88d995dd5ced81127381ffc9744a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10024,"rank":10024,"depth":58,"x":1667.247,"y":1202.707,"cluster":"tale-geometry"},{"id":"stacks:03RL","tag":"03RL","title":"Higher vanishing for the multiplicative group · Lemma 03RL","summary":"For all q ≥ 1, H_etale^q(X, bigoplus_x ∈ X_0 i_x_* underlineZ) = 0.","statement_latex":"For all $q \\geq 1$, $H_\\etale^q(X, \\bigoplus_{x \\in X_0} {i_x}_*\n\\underline{\\mathbf{Z}}) = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Higher vanishing for the multiplicative group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RL","source_file":"etale-cohomology.tex","source_line":11051,"source_end_line":11055,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11051-L11055","statement_sha256":"b46a99b89d98c1bca8095f804bc9f6e6f38a8eb35afcb00527ba69298c7f8aa1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10025,"rank":10025,"depth":52,"x":1465.095,"y":1143.271,"cluster":"tale-geometry"},{"id":"stacks:03RM","tag":"03RM","title":"Higher vanishing for the multiplicative group · Theorem 03RM","summary":"Let X be a smooth curve over an algebraically closed field. Then H_etale^q(X, G_m) = 0 for all q ≥ 2.","statement_latex":"Let $X$ be a smooth curve over an algebraically closed field. Then\n$$\nH_\\etale^q(X, \\mathbf{G}_m) = 0 \\ \\ \\text{ for all } q \\geq 2.\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Higher vanishing for the multiplicative group","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RM","source_file":"etale-cohomology.tex","source_line":11073,"source_end_line":11079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11073-L11079","statement_sha256":"7392446fdeb6cb473b4a78ad25f18a0efb264d744e0a6050a1a2b4c97be9f807","origin":"The Stacks Project","memory_eligible":false,"source_rank":10026,"rank":10026,"depth":0,"x":1662.201,"y":1072.304,"cluster":"tale-geometry"},{"id":"stacks:03RQ","tag":"03RQ","title":"Picard groups of curves · Lemma 03RQ","summary":"Let X be a smooth projective curve of genus g over an algebraically closed field k and let n ≥ 1 be invertible in k. Then there are canonical identifications H_etale^q(X, μ_n) = ( μ_n(k) & if q = 0, Pic^0(X)[n] & if q = 1, Z/nZ & if q = 2, 0 & if q ≥ 3. . Since μ_n ≅ underlineZ/nZ, this gives (noncanonical) identifications H_etale^q(X, underlineZ/nZ) ≅ ( Z/nZ & if q = 0, (Z/nZ)^2g & if q = 1, Z/nZ & if q = 2, 0 & if q ≥ 3. .","statement_latex":"Let $X$ be a smooth projective curve of genus $g$ over an\nalgebraically closed field $k$ and let $n \\geq 1$ be invertible in $k$.\nThen there are canonical identifications\n$$\nH_\\etale^q(X, \\mu_n) =\n\\left\\{\n\\begin{matrix}\n\\mu_n(k) & \\text{ if }q = 0, \\\\\n\\Pic^0(X)[n] & \\text{ if }q = 1, \\\\\n\\mathbf{Z}/n\\mathbf{Z} & \\text{ if }q = 2, \\\\\n0 & \\text{ if }q \\geq 3.\n\\end{matrix}\n\\right.\n$$\nSince $\\mu_n \\cong \\underline{\\mathbf{Z}/n\\mathbf{Z}}$, this gives\n(noncanonical) identifications\n$$\nH_\\etale^q(X, \\underline{\\mathbf{Z}/n\\mathbf{Z}}) \\cong\n\\left\\{\n\\begin{matrix}\n\\mathbf{Z}/n\\mathbf{Z} & \\text{ if }q = 0, \\\\\n(\\mathbf{Z}/n\\mathbf{Z})^{2g} & \\text{ if }q = 1, \\\\\n\\mathbf{Z}/n\\mathbf{Z} & \\text{ if }q = 2, \\\\\n0 & \\text{ if }q \\geq 3.\n\\end{matrix}\n\\right.\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Picard groups of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RQ","source_file":"etale-cohomology.tex","source_line":11134,"source_end_line":11163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11134-L11163","statement_sha256":"78873a3001bd7e63dd39c83ba4cc773dbe6228941a4d33b77174b8bc8483fcb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10027,"rank":10027,"depth":66,"x":1573.818,"y":1236.681,"cluster":"tale-geometry"},{"id":"stacks:0AMB","tag":"0AMB","title":"Picard groups of curves · Lemma 0AMB","summary":"Let π : X → Y be a nonconstant morphism of smooth projective curves over an algebraically closed field k and let n ≥ 1 be invertible in k. The map π^* : H^2_etale(Y, μ_n) → H^2_etale(X, μ_n) is given by multiplication by the degree of π.","statement_latex":"Let $\\pi : X \\to Y$ be a nonconstant morphism of smooth projective curves\nover an algebraically closed field $k$ and let $n \\geq 1$ be invertible in $k$.\nThe map\n$$\n\\pi^* : H^2_\\etale(Y, \\mu_n) \\longrightarrow H^2_\\etale(X, \\mu_n)\n$$\nis given by multiplication by the degree of $\\pi$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Picard groups of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMB","source_file":"etale-cohomology.tex","source_line":11219,"source_end_line":11228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11219-L11228","statement_sha256":"07bcebaca2c9e3d3872cc328430c4c5bd7b0746d229ee98b67feecf26c143c49","origin":"The Stacks Project","memory_eligible":false,"source_rank":10028,"rank":10028,"depth":67,"x":1506.72,"y":1065.109,"cluster":"tale-geometry"},{"id":"stacks:03RR","tag":"03RR","title":"Picard groups of curves · Lemma 03RR","summary":"Let X be an affine smooth curve over an algebraically closed field k and n ∈ k^*. Let X ⊂ overlineX be a smooth projective compactification (Varieties, Remark [Tag 0H1F]). Let g be the genus of overlineX and let r be the number of points of overlineX setminus X. Then • H_etale^0(X, μ_n) = μ_n(k); • H_etale^1(X, μ_n) ≅ (Z/nZ)^2g+r-1, and • H_etale^q(X, μ_n) = 0 for all q ≥ 2.","statement_latex":"Let $X$ be an affine smooth curve over an algebraically closed field $k$ and\n$n \\in k^*$. Let $X \\subset \\overline{X}$ be a smooth projective\ncompactification\n(Varieties, Remark \\ref{varieties-remark-smooth-projective-compactification}).\nLet $g$ be the genus of $\\overline{X}$ and let $r$ be the number of\npoints of $\\overline{X} \\setminus X$. Then\n\\begin{enumerate}\n\\item $H_\\etale^0(X, \\mu_n) = \\mu_n(k)$;\n\\item $H_\\etale^1(X, \\mu_n) \\cong (\\mathbf{Z}/n\\mathbf{Z})^{2g+r-1}$, and\n\\item $H_\\etale^q(X, \\mu_n) = 0$ for all $q \\geq 2$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Picard groups of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RR","source_file":"etale-cohomology.tex","source_line":11245,"source_end_line":11258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11245-L11258","statement_sha256":"c69e576f77ae8b2cdec76583bb7b24d5db83d39add3f0c11371a281c6a6b6927","origin":"The Stacks Project","memory_eligible":false,"source_rank":10029,"rank":10029,"depth":67,"x":1694.402,"y":1153.66,"cluster":"tale-geometry"},{"id":"stacks:03S3","tag":"03S3","title":"Extension by zero · Definition 03S3","summary":"Let j : U → X be an étale morphism of schemes. • The restriction functor j^-1 : Sh(X_etale) → Sh(U_etale) has a left adjoint j_!^Sh : Sh(U_etale) → Sh(X_etale). • The restriction functor j^-1 : Ab(X_etale) → Ab(U_etale) has a left adjoint which is denoted j_! : Ab(U_etale) → Ab(X_etale) and called extension by zero. • Let Lambda be a ring. The restriction functor j^-1 : Mod(X_etale, Lambda) → Mod(U_etale, Lambda) has a left adjoint which is denoted j_! : Mod(U_etale,…","statement_latex":"Let $j : U \\to X$ be an \\'etale morphism of schemes.\n\\begin{enumerate}\n\\item The restriction functor\n$j^{-1} : \\Sh(X_\\etale) \\to \\Sh(U_\\etale)$\nhas a left adjoint\n$j_!^{Sh} : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$.\n\\item The restriction functor\n$j^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(U_\\etale)$\nhas a left adjoint which is denoted\n$j_! : \\textit{Ab}(U_\\etale) \\to \\textit{Ab}(X_\\etale)$\nand called {\\it extension by zero}.\n\\item Let $\\Lambda$ be a ring. The restriction functor\n$j^{-1} : \\textit{Mod}(X_\\etale, \\Lambda) \\to\n\\textit{Mod}(U_\\etale, \\Lambda)$\nhas a left adjoint which is denoted\n$j_! : \\textit{Mod}(U_\\etale, \\Lambda) \\to\n\\textit{Mod}(X_\\etale, \\Lambda)$\nand called {\\it extension by zero}.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Extension by zero","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03S3","source_file":"etale-cohomology.tex","source_line":11360,"source_end_line":11381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11360-L11381","statement_sha256":"de7e3b367f8d5dde3329ec86726942144dbc249bbbd39c71920b340ac007ed8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10030,"rank":10030,"depth":0,"x":1484.539,"y":1194.91,"cluster":"tale-geometry"},{"id":"stacks:03S5","tag":"03S5","title":"Extension by zero · Proposition 03S5","summary":"Let j : U → X be an étale morphism of schemes. Let F in Ab(U_etale). If overlinex : Spec(k) → X is a geometric point of X, then (j_!F)_overlinex = bigoplus_overlineu : Spec(k) → U, j(overlineu) = overlinex F_baru. In particular, j_! is an exact functor.","statement_latex":"Let $j : U \\to X$ be an \\'etale morphism of schemes.\nLet $\\mathcal{F}$ in $\\textit{Ab}(U_\\etale)$.\nIf $\\overline{x} : \\Spec(k) \\to X$ is a geometric point of $X$, then \n$$\n(j_!\\mathcal{F})_{\\overline{x}} =\n\\bigoplus\\nolimits_{\\overline{u} : \\Spec(k) \\to U,\\ j(\\overline{u}) =\n\\overline{x}} \\mathcal{F}_{\\bar{u}}.\n$$\nIn particular, $j_!$ is an exact functor.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Extension by zero","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03S5","source_file":"etale-cohomology.tex","source_line":11424,"source_end_line":11435,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11424-L11435","statement_sha256":"dbf99601fab95ac03619ce5eaad9f300e6b75119ceef0a3ac49c3d623bfa1fb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10031,"rank":10031,"depth":50,"x":1606.267,"y":1045.227,"cluster":"tale-geometry"},{"id":"stacks:0F70","tag":"0F70","title":"Extension by zero · Lemma 0F70","summary":"Let j : U → X be an open immersion of schemes. For any abelian sheaf F on U_etale, the adjunction mappings j^-1j_*F → F and F → j^-1j_!F are isomorphisms. In fact, j_!F is the unique abelian sheaf on X_etale whose restriction to U is F and whose stalks at geometric points of X setminus U are zero.","statement_latex":"Let $j : U \\to X$ be an open immersion of schemes. For any\nabelian sheaf $\\mathcal{F}$ on $U_\\etale$, the adjunction mappings\n$j^{-1}j_*\\mathcal{F} \\to \\mathcal{F}$ and\n$\\mathcal{F} \\to j^{-1}j_!\\mathcal{F}$ are isomorphisms.\nIn fact, $j_!\\mathcal{F}$ is the unique abelian sheaf on $X_\\etale$\nwhose restriction to $U$ is $\\mathcal{F}$ and whose stalks at\ngeometric points of $X \\setminus U$ are zero.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F70","source_file":"etale-cohomology.tex","source_line":11467,"source_end_line":11476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11467-L11476","statement_sha256":"56d051eef7de95da51fd3550d3a6564737cf603bb91b137996cdb6d5949c76ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":10032,"rank":10032,"depth":51,"x":1636.915,"y":1224.892,"cluster":"tale-geometry"},{"id":"stacks:03S6","tag":"03S6","title":"Extension by zero commutes with base change · Lemma 03S6","summary":"Let f: Y → X be a morphism of schemes. Let j: V → X be an étale morphism. Consider the fibre product xymatrix V' = Y ×_X V ar[d]_f' ar[r]_-j' & Y ar[d]^f V ar[r]^j & X Then we have j'_! f'^-1 = f^-1 j_! on abelian sheaves and on sheaves of modules.","statement_latex":"Let $f: Y \\to X$ be a morphism of schemes. Let $j: V \\to X$ be an \\'etale\nmorphism. Consider the fibre product\n$$\n\\xymatrix{\nV' = Y \\times_X V \\ar[d]_{f'} \\ar[r]_-{j'} & Y \\ar[d]^f \\\\\nV \\ar[r]^j & X\n}\n$$\nThen we have $j'_! f'^{-1} = f^{-1} j_!$ on abelian sheaves and on\nsheaves of modules.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03S6","source_file":"etale-cohomology.tex","source_line":11492,"source_end_line":11504,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11492-L11504","statement_sha256":"b04d44149b5d3cefbd44db827057c1c80ff22dc936e91fb5e5f08bd372f0500c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10033,"rank":10033,"depth":10,"x":1469.629,"y":1109.659,"cluster":"tale-geometry"},{"id":"stacks:0F4L","tag":"0F4L","title":"Extension by zero · Lemma 0F4L","summary":"Let j : U → X be separated and étale. Then there is a functorial injective map j_!F → j_*F on abelian sheaves and sheaves of Lambda-modules.","statement_latex":"Let $j : U \\to X$ be separated and \\'etale. Then there is a functorial\ninjective map $j_!\\mathcal{F} \\to j_*\\mathcal{F}$\non abelian sheaves and sheaves of $\\Lambda$-modules.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4L","source_file":"etale-cohomology.tex","source_line":11516,"source_end_line":11521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11516-L11521","statement_sha256":"43407eea6b5f282a6c71b5de92a3ee1e586cb437190be335c9eb04402c1c63fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10034,"rank":10034,"depth":17,"x":1685.914,"y":1099.697,"cluster":"tale-geometry"},{"id":"stacks:03S7","tag":"03S7","title":"Extension by zero · Lemma 03S7","summary":"Let j : U → X be finite and étale. Then the map j_! → j_* of Lemma [Tag 0F4L] is an isomorphism on abelian sheaves and sheaves of Lambda-modules.","statement_latex":"Let $j : U \\to X$ be finite and \\'etale. Then the map\n$j_! \\to j_*$ of Lemma \\ref{lemma-shriek-into-star-separated-etale}\nis an isomorphism\non abelian sheaves and sheaves of $\\Lambda$-modules.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03S7","source_file":"etale-cohomology.tex","source_line":11570,"source_end_line":11576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11570-L11576","statement_sha256":"99ff175ac252c3a499740dc374addee1ed92e425fa25c987efddc3b875a273a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10035,"rank":10035,"depth":46,"x":1534.256,"y":1229.927,"cluster":"tale-geometry"},{"id":"stacks:095L","tag":"095L","title":"Extension by zero · Lemma 095L","summary":"Let X be a scheme. Let Z ⊂ X be a closed subscheme and let U ⊂ X be the complement. Denote i : Z → X and j : U → X the inclusion morphisms. For every abelian sheaf F on X_etale there is a canonical short exact sequence 0 → j_!j^-1F → F → i_*i^-1F → 0 on X_etale.","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme and let\n$U \\subset X$ be the complement. Denote $i : Z \\to X$ and $j : U \\to X$\nthe inclusion morphisms. For every abelian sheaf $\\mathcal{F}$ on $X_\\etale$\nthere is a canonical short exact sequence\n$$\n0 \\to j_!j^{-1}\\mathcal{F} \\to \\mathcal{F} \\to i_*i^{-1}\\mathcal{F} \\to 0\n$$\non $X_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095L","source_file":"etale-cohomology.tex","source_line":11587,"source_end_line":11597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11587-L11597","statement_sha256":"29987e43e02aee00ccfbeb0db43ce5d4c2ee5030fb2a4134a237c123471365b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10036,"rank":10036,"depth":51,"x":1541.368,"y":1047.621,"cluster":"tale-geometry"},{"id":"stacks:0GJ1","tag":"0GJ1","title":"Extension by zero · Lemma 0GJ1","summary":"Consider a cartesian diagram of schemes xymatrix U ar[d]_g ar[r]_j' & X ar[d]^f V ar[r]^j & Y where f is finite and j is an open immersion. Then f_* ∘ j'_! = j_! ∘ g_* as functors Ab(U_etale) → Ab(Y_etale).","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nU \\ar[d]_g \\ar[r]_{j'} & X \\ar[d]^f \\\\\nV \\ar[r]^j & Y\n}\n$$\nwhere $f$ is finite and $j$ is an open immersion.\nThen $f_* \\circ j'_! = j_! \\circ g_*$ as functors\n$\\textit{Ab}(U_\\etale) \\to \\textit{Ab}(Y_\\etale)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJ1","source_file":"etale-cohomology.tex","source_line":11611,"source_end_line":11623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11611-L11623","statement_sha256":"e252e39365c6d446ddb77837e2bd0d202a955c53559bcf82b7ce5fe8a14d3c5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10037,"rank":10037,"depth":53,"x":1682.9,"y":1186.254,"cluster":"tale-geometry"},{"id":"stacks:03RW","tag":"03RW","title":"Constructible sheaves · Definition 03RW","summary":"Let X be a scheme. • A sheaf of sets on X_etale is constructible if for every affine open U ⊂ X there exists a finite decomposition of U into constructible locally closed subschemes U = coprod_i U_i such that F|_U_i is finite locally constant for all i. • A sheaf of abelian groups on X_etale is constructible if for every affine open U ⊂ X there exists a finite decomposition of U into constructible locally closed subschemes U = coprod_i U_i such that F|_U_i is finite…","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item A sheaf of sets on $X_\\etale$ is {\\it constructible}\nif for every affine open $U \\subset X$ there exists a finite decomposition\nof $U$ into constructible locally closed subschemes $U = \\coprod_i U_i$\nsuch that $\\mathcal{F}|_{U_i}$ is finite locally constant for all $i$.\n\\item A sheaf of abelian groups on $X_\\etale$ is {\\it constructible}\nif for every affine open $U \\subset X$ there exists a finite decomposition\nof $U$ into constructible locally closed subschemes $U = \\coprod_i U_i$\nsuch that $\\mathcal{F}|_{U_i}$ is finite locally constant for all $i$.\n\\item Let $\\Lambda$ be a Noetherian ring. A sheaf of $\\Lambda$-modules\non $X_\\etale$ is {\\it constructible} if for every affine open\n$U \\subset X$ there exists a finite decomposition\nof $U$ into constructible locally closed subschemes\n$U = \\coprod_i U_i$ such that\n$\\mathcal{F}|_{U_i}$ is of finite type and locally constant for all $i$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RW","source_file":"etale-cohomology.tex","source_line":11671,"source_end_line":11690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11671-L11690","statement_sha256":"86b371ad11b7b99666cfdb5f236e6698a7617cc460fd2c1f325eef6ae9046f17","origin":"The Stacks Project","memory_eligible":false,"source_rank":10038,"rank":10038,"depth":0,"x":1466.791,"y":1164.31,"cluster":"tale-geometry"},{"id":"stacks:095E","tag":"095E","title":"Constructible sheaves · Lemma 095E","summary":"Let X be a quasi-compact and quasi-separated scheme. Let F be a sheaf of sets on X_etale. The following are equivalent • F is constructible, • there exists an open covering X = ⋃ U_i such that F|_U_i is constructible, and • there exists a partition X = ⋃ X_i by constructible locally closed subschemes such that F|_X_i is finite locally constant. A similar statement holds for abelian sheaves and sheaves of Lambda-modules if Lambda is Noetherian.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\mathcal{F}$\nbe a sheaf of sets on $X_\\etale$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is constructible,\n\\item there exists an open covering $X = \\bigcup U_i$ such that\n$\\mathcal{F}|_{U_i}$ is constructible, and\n\\item there exists a partition $X = \\bigcup X_i$ by constructible\nlocally closed subschemes such that $\\mathcal{F}|_{X_i}$ is finite\nlocally constant.\n\\end{enumerate}\nA similar statement holds for abelian sheaves and sheaves of\n$\\Lambda$-modules if $\\Lambda$ is Noetherian.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095E","source_file":"etale-cohomology.tex","source_line":11716,"source_end_line":11730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11716-L11730","statement_sha256":"b14ff4b3ed7f09321308d3f517514307f8b3848c8fb6536f152da3e055237d4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10039,"rank":10039,"depth":4,"x":1644.006,"y":1057.737,"cluster":"tale-geometry"},{"id":"stacks:09YR","tag":"09YR","title":"Constructible sheaves · Lemma 09YR","summary":"Let X be a quasi-compact and quasi-separated scheme. Let F be a sheaf of sets, abelian groups, Lambda-modules (with Lambda Noetherian) on X_etale. If there exist constructible locally closed subschemes T_i ⊂ X such that (a) X = ⋃ T_j and (b) F|_T_j is constructible, then F is constructible.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $\\mathcal{F}$ be a sheaf of sets, abelian groups,\n$\\Lambda$-modules (with $\\Lambda$ Noetherian) on $X_\\etale$.\nIf there exist constructible locally closed subschemes $T_i \\subset X$\nsuch that (a) $X = \\bigcup T_j$ and (b) $\\mathcal{F}|_{T_j}$ is\nconstructible, then $\\mathcal{F}$ is constructible.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YR","source_file":"etale-cohomology.tex","source_line":11758,"source_end_line":11766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11758-L11766","statement_sha256":"0f6fc36102e5b4734b75e708b3b4e2c5cb7c3f5c30ba533e1d3feb84481bf33e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10040,"rank":10040,"depth":5,"x":1598.979,"y":1237.094,"cluster":"tale-geometry"},{"id":"stacks:095F","tag":"095F","title":"Constructible sheaves · Lemma 095F","summary":"Let X be a scheme. Checking constructibility of a sheaf of sets, abelian groups, Lambda-modules (with Lambda Noetherian) can be done Zariski locally on X.","statement_latex":"Let $X$ be a scheme. Checking constructibility of a sheaf\nof sets, abelian groups, $\\Lambda$-modules (with $\\Lambda$ Noetherian)\ncan be done Zariski locally on $X$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095F","source_file":"etale-cohomology.tex","source_line":11788,"source_end_line":11793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11788-L11793","statement_sha256":"c8382da7ddb6119e4c240c9942f729bca73ef7730fc942c2484d75bad27ba88a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10041,"rank":10041,"depth":5,"x":1487.814,"y":1079.1,"cluster":"tale-geometry"},{"id":"stacks:095G","tag":"095G","title":"Constructible sheaves · Lemma 095G","summary":"Let f : X → Y be a morphism of schemes. If F is a constructible sheaf of sets, abelian groups, or Lambda-modules (with Lambda Noetherian) on Y_etale, the same is true for f^-1F on X_etale.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. If $\\mathcal{F}$ is a\nconstructible sheaf of sets, abelian groups, or $\\Lambda$-modules\n(with $\\Lambda$ Noetherian) on $Y_\\etale$, the same\nis true for $f^{-1}\\mathcal{F}$ on $X_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095G","source_file":"etale-cohomology.tex","source_line":11807,"source_end_line":11813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11807-L11813","statement_sha256":"ecf51f291fedd2c2c2262b8f091a2893794e8da6223049d61b46cd1f018ea187","origin":"The Stacks Project","memory_eligible":false,"source_rank":10042,"rank":10042,"depth":6,"x":1697.09,"y":1132.589,"cluster":"tale-geometry"},{"id":"stacks:03RZ","tag":"03RZ","title":"Constructible sheaves · Lemma 03RZ","summary":"Let X be a scheme. • The category of constructible sheaves of sets is closed under finite limits and colimits inside Sh(X_etale). • The category of constructible abelian sheaves is a weak Serre subcategory of Ab(X_etale). • Let Lambda be a Noetherian ring. The category of constructible sheaves of Lambda-modules on X_etale is a weak Serre subcategory of Mod(X_etale, Lambda).","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item The category of constructible sheaves of sets\nis closed under finite limits and colimits inside $\\Sh(X_\\etale)$.\n\\item The category of constructible abelian sheaves is a\nweak Serre subcategory of $\\textit{Ab}(X_\\etale)$.\n\\item Let $\\Lambda$ be a Noetherian ring. The category of\nconstructible sheaves of $\\Lambda$-modules on\n$X_\\etale$ is a weak Serre subcategory of\n$\\textit{Mod}(X_\\etale, \\Lambda)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RZ","source_file":"etale-cohomology.tex","source_line":11827,"source_end_line":11840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11827-L11840","statement_sha256":"00f6142e3d2e13d395554b815c0d84913211f3ce399277487c6fc928518c35d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10043,"rank":10043,"depth":3,"x":1499.523,"y":1211.99,"cluster":"tale-geometry"},{"id":"stacks:09YS","tag":"09YS","title":"Constructible sheaves · Lemma 09YS","summary":"Let X be a quasi-compact and quasi-separated scheme. • Let F → G be a map of constructible sheaves of sets on X_etale. Then the set of points x ∈ X where F_overlinex → G_overlinex is surjective, resp. injective, resp. is isomorphic to a given map of sets, is constructible in X. • Let F be a constructible abelian sheaf on X_etale. The support of F is constructible. • Let Lambda be a Noetherian ring. Let F be a constructible sheaf of Lambda-modules on X_etale. The support…","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\n\\begin{enumerate}\n\\item Let $\\mathcal{F} \\to \\mathcal{G}$ be a map of constructible\nsheaves of sets on $X_\\etale$. Then the set of points $x \\in X$\nwhere $\\mathcal{F}_{\\overline{x}} \\to \\mathcal{G}_{\\overline{x}}$\nis surjective, resp.\\ injective, resp.\\ is isomorphic to a given map\nof sets, is constructible in $X$.\n\\item Let $\\mathcal{F}$ be a constructible abelian sheaf on $X_\\etale$.\nThe support of $\\mathcal{F}$ is constructible.\n\\item Let $\\Lambda$ be a Noetherian ring.\nLet $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$.\nThe support of $\\mathcal{F}$ is constructible.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YS","source_file":"etale-cohomology.tex","source_line":11866,"source_end_line":11881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11866-L11881","statement_sha256":"2079ee8ca8f47054e78f8d0ac0b228d491ed6357e2d9bc2bcebe0a2d2fbcd121","origin":"The Stacks Project","memory_eligible":false,"source_rank":10044,"rank":10044,"depth":5,"x":1581.451,"y":1041.134,"cluster":"tale-geometry"},{"id":"stacks:095P","tag":"095P","title":"Constructible sheaves · Lemma 095P","summary":"Let X be a quasi-compact and quasi-separated scheme. Let F = colim_i ∈ I F_i be a filtered colimit of sheaves of sets, abelian sheaves, or sheaves of modules. • If F and F_i are constructible sheaves of sets, then the ind-object F_i is essentially constant with value F. • If F and F_i are constructible sheaves of abelian groups, then the ind-object F_i is essentially constant with value F. • Let Lambda be a Noetherian ring. If F and F_i are constructible sheaves of…","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let\n$\\mathcal{F} = \\colim_{i \\in I} \\mathcal{F}_i$ be a filtered\ncolimit of sheaves of sets, abelian sheaves, or sheaves of modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ and $\\mathcal{F}_i$ are constructible sheaves of\nsets, then the ind-object $\\mathcal{F}_i$ is essentially constant with\nvalue $\\mathcal{F}$.\n\\item If $\\mathcal{F}$ and $\\mathcal{F}_i$ are constructible sheaves of\nabelian groups, then the ind-object $\\mathcal{F}_i$ is essentially constant\nwith value $\\mathcal{F}$.\n\\item Let $\\Lambda$ be a Noetherian ring.\nIf $\\mathcal{F}$ and $\\mathcal{F}_i$ are constructible sheaves of\n$\\Lambda$-modules, then the ind-object $\\mathcal{F}_i$ is essentially constant\nwith value $\\mathcal{F}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095P","source_file":"etale-cohomology.tex","source_line":11902,"source_end_line":11919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11902-L11919","statement_sha256":"c0d26cc88694fb0243fd3050cae871982fb84f954d9709187758f45db1275221","origin":"The Stacks Project","memory_eligible":false,"source_rank":10045,"rank":10045,"depth":6,"x":1658.528,"y":1213.814,"cluster":"tale-geometry"},{"id":"stacks:095I","tag":"095I","title":"Constructible sheaves · Lemma 095I","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. The tensor product of two constructible sheaves of Lambda-modules on X_etale is a constructible sheaf of Lambda-modules.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nThe tensor product of two constructible sheaves of $\\Lambda$-modules\non $X_\\etale$ is a constructible sheaf of $\\Lambda$-modules.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095I","source_file":"etale-cohomology.tex","source_line":11990,"source_end_line":11995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L11990-L11995","statement_sha256":"c7583e7c57e95db2848c18ff8fc61183a478106b35eb1355befe88c8ec52bfdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10046,"rank":10046,"depth":2,"x":1462.596,"y":1130.118,"cluster":"tale-geometry"},{"id":"stacks:0GKB","tag":"0GKB","title":"Constructible sheaves · Lemma 0GKB","summary":"Let Lambda → Lambda' be a homomorphism of Noetherian rings. Let X be a scheme. Let F be a constructible sheaf of Lambda-modules on X_etale. Then F ⊗_underlineLambda underlineLambda' is a constructible sheaf of Lambda'-modules.","statement_latex":"Let $\\Lambda \\to \\Lambda'$ be a homomorphism of Noetherian rings.\nLet $X$ be a scheme. Let $\\mathcal{F}$ be a constructible\nsheaf of $\\Lambda$-modules on $X_\\etale$. Then\n$\\mathcal{F} \\otimes_{\\underline{\\Lambda}} \\underline{\\Lambda'}$\nis a constructible sheaf of $\\Lambda'$-modules.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKB","source_file":"etale-cohomology.tex","source_line":12005,"source_end_line":12012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12005-L12012","statement_sha256":"34b4e0e87deaecf2aa74eaa669063e04b9bcef3887181b4ac1327ab891c0ab5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10047,"rank":10047,"depth":0,"x":1674.63,"y":1080.599,"cluster":"tale-geometry"},{"id":"stacks:03S0","tag":"03S0","title":"Auxiliary lemmas on morphisms · Lemma 03S0","summary":"Let U → X be an étale morphism of quasi-compact and quasi-separated schemes (for example an étale morphism of Noetherian schemes). Then there exists a partition X = coprod_i X_i by constructible locally closed subschemes such that X_i ×_X U → X_i is finite étale for all i.","statement_latex":"Let $U \\to X$ be an \\'etale morphism of quasi-compact and quasi-separated\nschemes (for example an \\'etale morphism of Noetherian schemes). Then there\nexists a partition $X = \\coprod_i X_i$ by constructible locally closed\nsubschemes such that $X_i \\times_X U \\to X_i$ is finite \\'etale for all $i$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Auxiliary lemmas on morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03S0","source_file":"etale-cohomology.tex","source_line":12034,"source_end_line":12040,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12034-L12040","statement_sha256":"8d53397f77af5c4ef65d5a7edbc075c66e7b9571a6e9b05ced55371e84722deb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10048,"rank":10048,"depth":47,"x":1557.964,"y":1237.611,"cluster":"tale-geometry"},{"id":"stacks:03S1","tag":"03S1","title":"Auxiliary lemmas on morphisms · Lemma 03S1","summary":"Let f: X → Y be a morphism of schemes which is quasi-compact, quasi-separated, and locally of finite type. If eta is a generic point of an irreducible component of Y such that f^-1(eta) is finite, then there exists an open V ⊂ Y containing eta such that f^-1(V) → V is finite.","statement_latex":"Let $f: X \\to Y$ be a morphism of schemes which is quasi-compact,\nquasi-separated, and locally of finite type. If $\\eta$ is a generic point\nof an irreducible component of $Y$ such that $f^{-1}(\\eta)$ is finite, then\nthere exists an open $V \\subset Y$ containing $\\eta$ such that\n$f^{-1}(V) \\to V$ is finite.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Auxiliary lemmas on morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03S1","source_file":"etale-cohomology.tex","source_line":12068,"source_end_line":12075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12068-L12075","statement_sha256":"9dd4d7362295105796a8b644c63b754dd0ae7ca2375e61fa2198502440d7fdf8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10049,"rank":10049,"depth":29,"x":1517.679,"y":1055.42,"cluster":"tale-geometry"},{"id":"stacks:095K","tag":"095K","title":"Auxiliary lemmas on morphisms · Lemma 095K","summary":"Let f : Y → X be a quasi-finite and finitely presented morphism of affine schemes. • There exists a surjective morphism of affine schemes X' → X and a closed subscheme Z' ⊂ Y' = X' ×_X Y such that • Z' ⊂ Y' is a thickening, and • Z' → X' is a finite étale morphism. • There exists a finite partition X = coprod X_i by locally closed, constructible, affine strata, and surjective finite locally free morphisms X'_i → X_i such that the reduction of Y'_i = X'_i ×_X Y → X'_i is…","statement_latex":"Let $f : Y \\to X$ be a quasi-finite and finitely presented\nmorphism of affine schemes.\n\\begin{enumerate}\n\\item There exists a surjective morphism of affine schemes $X' \\to X$ and a\nclosed subscheme $Z' \\subset Y' = X' \\times_X Y$ such that\n\\begin{enumerate}\n\\item $Z' \\subset Y'$ is a thickening, and\n\\item $Z' \\to X'$ is a finite \\'etale morphism.\n\\end{enumerate}\n\\item There exists a finite partition $X = \\coprod X_i$ by\nlocally closed, constructible, affine strata, and surjective finite locally\nfree morphisms $X'_i \\to X_i$ such that the reduction of\n$Y'_i = X'_i \\times_X Y \\to X'_i$ is isomorphic to\n$\\coprod_{j = 1}^{n_i} (X'_i)_{red} \\to (X'_i)_{red}$ for some $n_i$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Auxiliary lemmas on morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095K","source_file":"etale-cohomology.tex","source_line":12084,"source_end_line":12101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12084-L12101","statement_sha256":"d7efb223d16701a3708d28c4ed17d49806c2c6f6b28534e94ae37d1a8efa5471","origin":"The Stacks Project","memory_eligible":false,"source_rank":10050,"rank":10050,"depth":36,"x":1694.105,"y":1167.038,"cluster":"tale-geometry"},{"id":"stacks:03S8","tag":"03S8","title":"More on constructible sheaves · Lemma 03S8","summary":"Let j : U → X be an étale morphism of quasi-compact and quasi-separated schemes. • The sheaf h_U is a constructible sheaf of sets. • The sheaf j_!underlineM is a constructible abelian sheaf for a finite abelian group M. • If Lambda is a Noetherian ring and M is a finite Lambda-module, then j_!underlineM is a constructible sheaf of Lambda-modules on X_etale.","statement_latex":"Let $j : U \\to X$ be an \\'etale morphism of quasi-compact and\nquasi-separated schemes.\n\\begin{enumerate}\n\\item The sheaf $h_U$ is a constructible sheaf of sets.\n\\item The sheaf $j_!\\underline{M}$ is a constructible abelian sheaf\nfor a finite abelian group $M$.\n\\item If $\\Lambda$ is a Noetherian ring and $M$ is a finite $\\Lambda$-module,\nthen $j_!\\underline{M}$ is a constructible sheaf of $\\Lambda$-modules\non $X_\\etale$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03S8","source_file":"etale-cohomology.tex","source_line":12184,"source_end_line":12196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12184-L12196","statement_sha256":"0e6c58af97ba82fc18548aaacaad4552887f6224084442e44bf4f11bf2cb0919","origin":"The Stacks Project","memory_eligible":false,"source_rank":10051,"rank":10051,"depth":48,"x":1473.994,"y":1184.861,"cluster":"tale-geometry"},{"id":"stacks:03SA","tag":"03SA","title":"More on constructible sheaves · Lemma 03SA","summary":"Let X be a quasi-compact and quasi-separated scheme. • Let F be a sheaf of sets on X_etale. Then F is a filtered colimit of constructible sheaves of sets. • Let F be a torsion abelian sheaf on X_etale. Then F is a filtered colimit of constructible abelian sheaves. • Let Lambda be a Noetherian ring and F a sheaf of Lambda-modules on X_etale. Then F is a filtered colimit of constructible sheaves of Lambda-modules.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$.\nThen $\\mathcal{F}$ is a filtered colimit of constructible\nsheaves of sets.\n\\item Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$.\nThen $\\mathcal{F}$ is a filtered colimit of constructible abelian sheaves.\n\\item Let $\\Lambda$ be a Noetherian ring and $\\mathcal{F}$ a sheaf\nof $\\Lambda$-modules on $X_\\etale$. Then\n$\\mathcal{F}$ is a filtered colimit of constructible sheaves of\n$\\Lambda$-modules.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SA","source_file":"etale-cohomology.tex","source_line":12215,"source_end_line":12229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12215-L12229","statement_sha256":"8ce87a63ac7ad021c5e641c6a0702f5e9b4a804ae0d5f0d467f6859e69672ebc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10052,"rank":10052,"depth":49,"x":1622.14,"y":1046.661,"cluster":"tale-geometry"},{"id":"stacks:095Q","tag":"095Q","title":"More on constructible sheaves · Lemma 095Q","summary":"Let f : X → Y be a surjective morphism of quasi-compact and quasi-separated schemes. • Let F be a sheaf of sets on Y_etale. Then F is constructible if and only if f^-1F is constructible. • Let F be an abelian sheaf on Y_etale. Then F is constructible if and only if f^-1F is constructible. • Let Lambda be a Noetherian ring. Let F be sheaf of Lambda-modules on Y_etale. Then F is constructible if and only if f^-1F is constructible.","statement_latex":"Let $f : X \\to Y$ be a surjective morphism of quasi-compact and\nquasi-separated schemes.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a sheaf of sets on $Y_\\etale$. Then $\\mathcal{F}$\nis constructible if and only if $f^{-1}\\mathcal{F}$ is constructible.\n\\item Let $\\mathcal{F}$ be an abelian sheaf on $Y_\\etale$. Then $\\mathcal{F}$\nis constructible if and only if $f^{-1}\\mathcal{F}$ is constructible.\n\\item Let $\\Lambda$ be a Noetherian ring.\nLet $\\mathcal{F}$ be sheaf of $\\Lambda$-modules on $Y_\\etale$.\nThen $\\mathcal{F}$ is constructible if and only if $f^{-1}\\mathcal{F}$\nis constructible.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095Q","source_file":"etale-cohomology.tex","source_line":12275,"source_end_line":12289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12275-L12289","statement_sha256":"fb1f5f881bf53b36da747538caa45e1c9da7542d036a7b018e519e85f55dac4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10053,"rank":10053,"depth":52,"x":1624.038,"y":1232.846,"cluster":"tale-geometry"},{"id":"stacks:095H","tag":"095H","title":"More on constructible sheaves · Lemma 095H","summary":"Let f : X → Y be a finite étale morphism of schemes. Let Lambda be a Noetherian ring. If F is a constructible sheaf of sets, constructible sheaf of abelian groups, or constructible sheaf of Lambda-modules on X_etale, the same is true for f_*F on Y_etale.","statement_latex":"Let $f : X \\to Y$ be a finite \\'etale morphism of schemes. Let $\\Lambda$ be a\nNoetherian ring. If $\\mathcal{F}$ is a constructible sheaf of sets,\nconstructible sheaf of abelian groups, or constructible sheaf of\n$\\Lambda$-modules on $X_\\etale$, the same is true for\n$f_*\\mathcal{F}$ on $Y_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095H","source_file":"etale-cohomology.tex","source_line":12320,"source_end_line":12327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12320-L12327","statement_sha256":"96c5695249aef2e5dcd78b7ef3dfd871fefc3b2a54c78b8f4ad0a5219ad1ddbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10054,"rank":10054,"depth":53,"x":1472.738,"y":1096.475,"cluster":"tale-geometry"},{"id":"stacks:09Y9","tag":"09Y9","title":"More on constructible sheaves · Lemma 09Y9","summary":"Let X be a quasi-compact and quasi-separated scheme. The category of constructible sheaves of sets is the full subcategory of Sh(X_etale) consisting of sheaves F which are coequalizers xymatrix F_1 ar@<1ex>[r] ar@<-1ex>[r] & F_0 ar[r] & F such that F_i, i = 0, 1 is a finite coproduct of sheaves of the form h_U with U a quasi-compact and quasi-separated object of X_etale.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. The category of\nconstructible sheaves of sets is the full subcategory of $\\Sh(X_\\etale)$\nconsisting of sheaves $\\mathcal{F}$ which are coequalizers\n$$\n\\xymatrix{\n\\mathcal{F}_1\n\\ar@<1ex>[r] \\ar@<-1ex>[r]\n&\n\\mathcal{F}_0 \\ar[r]\n&\n\\mathcal{F}}\n$$\nsuch that $\\mathcal{F}_i$, $i = 0, 1$ is a finite coproduct of sheaves of\nthe form $h_U$ with $U$ a quasi-compact and quasi-separated\nobject of $X_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Y9","source_file":"etale-cohomology.tex","source_line":12342,"source_end_line":12359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12342-L12359","statement_sha256":"8363d6f2010f38c95d9bd9f153421e84c343b1008505a8b154167909dabf5c45","origin":"The Stacks Project","memory_eligible":false,"source_rank":10055,"rank":10055,"depth":50,"x":1694.227,"y":1111.199,"cluster":"tale-geometry"},{"id":"stacks:095N","tag":"095N","title":"More on constructible sheaves · Lemma 095N","summary":"Let X be a quasi-compact and quasi-separated scheme. Let Lambda be a Noetherian ring. The category of constructible sheaves of Lambda-modules is exactly the category of modules of the form Coker( bigoplus_j = 1, …, m j_V_j!underlineLambda_V_j → bigoplus_i = 1, …, n j_U_i!underlineLambda_U_i ) with V_j and U_i quasi-compact and quasi-separated objects of X_etale. In fact, we can even assume U_i and V_j affine.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\Lambda$\nbe a Noetherian ring. The category of constructible sheaves of\n$\\Lambda$-modules is exactly the category of modules of the form\n$$\n\\Coker\\left(\n\\bigoplus\\nolimits_{j = 1, \\ldots, m} j_{V_j!}\\underline{\\Lambda}_{V_j}\n\\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} j_{U_i!}\\underline{\\Lambda}_{U_i}\n\\right)\n$$\nwith $V_j$ and $U_i$ quasi-compact and quasi-separated objects of\n$X_\\etale$. In fact, we can even assume $U_i$ and $V_j$ affine.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095N","source_file":"etale-cohomology.tex","source_line":12411,"source_end_line":12425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12411-L12425","statement_sha256":"ff846bc5c9ac06328494c264883c1863a5d61649098af7cc7e326b9152b10949","origin":"The Stacks Project","memory_eligible":false,"source_rank":10056,"rank":10056,"depth":50,"x":1518.861,"y":1226.153,"cluster":"tale-geometry"},{"id":"stacks:09YT","tag":"09YT","title":"More on constructible sheaves · Lemma 09YT","summary":"Let X be a quasi-compact and quasi-separated scheme. The category of constructible abelian sheaves is exactly the category of abelian sheaves of the form Coker( bigoplus_j = 1, …, m j_V_j!underlineZ/m_jZ_V_j → bigoplus_i = 1, …, n j_U_i!underlineZ/n_iZ_U_i ) with V_j and U_i quasi-compact and quasi-separated objects of X_etale and m_j, n_i positive integers. In fact, we can even assume U_i and V_j affine.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. The category of\nconstructible abelian sheaves is exactly the category of abelian\nsheaves of the form\n$$\n\\Coker\\left(\n\\bigoplus\\nolimits_{j = 1, \\ldots, m}\nj_{V_j!}\\underline{\\mathbf{Z}/m_j\\mathbf{Z}}_{V_j}\n\\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n}\nj_{U_i!}\\underline{\\mathbf{Z}/n_i\\mathbf{Z}}_{U_i}\n\\right)\n$$\nwith $V_j$ and $U_i$ quasi-compact and quasi-separated objects of\n$X_\\etale$ and $m_j$, $n_i$ positive integers.\nIn fact, we can even assume $U_i$ and $V_j$ affine.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YT","source_file":"etale-cohomology.tex","source_line":12470,"source_end_line":12487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12470-L12487","statement_sha256":"6e3976da06d9686af06c619a69a8a2d35478d5e2b1d27c3692b1c5540e8af3a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10057,"rank":10057,"depth":51,"x":1555.773,"y":1041.668,"cluster":"tale-geometry"},{"id":"stacks:09Z4","tag":"09Z4","title":"More on constructible sheaves · Lemma 09Z4","summary":"Let X be a quasi-compact and quasi-separated scheme. Let Lambda be a Noetherian ring. Let F be a constructible sheaf of sets, abelian groups, or Lambda-modules on X_etale. Let G = colim G_i be a filtered colimit of sheaves of sets, abelian groups, or Lambda-modules. Then Mor(F, G) = colim Mor(F, G_i) in the category of sheaves of sets, abelian groups, or Lambda-modules on X_etale.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let $\\Lambda$ be a\nNoetherian ring. Let $\\mathcal{F}$ be a constructible sheaf of sets, abelian\ngroups, or $\\Lambda$-modules on $X_\\etale$. Let\n$\\mathcal{G} = \\colim \\mathcal{G}_i$ be a filtered colimit of sheaves of\nsets, abelian groups, or $\\Lambda$-modules. Then\n$$\n\\Mor(\\mathcal{F}, \\mathcal{G}) = \\colim \\Mor(\\mathcal{F}, \\mathcal{G}_i)\n$$\nin the category of sheaves of sets, abelian groups, or $\\Lambda$-modules on\n$X_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Z4","source_file":"etale-cohomology.tex","source_line":12496,"source_end_line":12508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12496-L12508","statement_sha256":"adc3a85be8a753eba0cb2b9b0932181b020626400bdae97c9fa5c152e9310d57","origin":"The Stacks Project","memory_eligible":false,"source_rank":10058,"rank":10058,"depth":52,"x":1677.052,"y":1198.83,"cluster":"tale-geometry"},{"id":"stacks:095R","tag":"095R","title":"More on constructible sheaves · Lemma 095R","summary":"Let f : X → Y be a finite and finitely presented morphism of schemes. Let Lambda be a Noetherian ring. If F is a constructible sheaf of sets, abelian groups, or Lambda-modules on X_etale, then f_*F is too.","statement_latex":"Let $f : X \\to Y$ be a finite and finitely presented morphism of schemes.\nLet $\\Lambda$ be a Noetherian ring. If $\\mathcal{F}$ is a constructible\nsheaf of sets, abelian groups, or $\\Lambda$-modules on $X_\\etale$,\nthen $f_*\\mathcal{F}$ is too.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095R","source_file":"etale-cohomology.tex","source_line":12527,"source_end_line":12533,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12527-L12533","statement_sha256":"6ddc5af3960409447cb8c23e7c74c35deeb0e19c9f3bd019c81212be615bd463","origin":"The Stacks Project","memory_eligible":false,"source_rank":10059,"rank":10059,"depth":54,"x":1460.99,"y":1151.703,"cluster":"tale-geometry"},{"id":"stacks:09YU","tag":"09YU","title":"More on constructible sheaves · Lemma 09YU","summary":"Let X = lim_i ∈ I X_i be a limit of a directed system of schemes with affine transition morphisms. We assume that X_i is quasi-compact and quasi-separated for all i ∈ I. • The category of constructible sheaves of sets on X_etale is the colimit of the categories of constructible sheaves of sets on (X_i)_etale. • The category of constructible abelian sheaves on X_etale is the colimit of the categories of constructible abelian sheaves on (X_i)_etale. • Let Lambda be a…","statement_latex":"Let $X = \\lim_{i \\in I} X_i$ be a limit of a directed\nsystem of schemes with affine transition morphisms.\nWe assume that $X_i$ is quasi-compact and quasi-separated\nfor all $i \\in I$.\n\\begin{enumerate}\n\\item The category of constructible sheaves of sets on $X_\\etale$\nis the colimit of the categories of constructible sheaves of sets\non $(X_i)_\\etale$.\n\\item The category of constructible abelian sheaves on $X_\\etale$\nis the colimit of the categories of constructible abelian sheaves\non $(X_i)_\\etale$.\n\\item Let $\\Lambda$ be a Noetherian ring. The category of constructible\nsheaves of $\\Lambda$-modules on $X_\\etale$ is the colimit of the\ncategories of constructible sheaves of $\\Lambda$-modules on $(X_i)_\\etale$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YU","source_file":"etale-cohomology.tex","source_line":12551,"source_end_line":12568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12551-L12568","statement_sha256":"72a3dce827fe2f05471ec5114ec88f3eed25de37ccfea4895c1d9e66ab7e8795","origin":"The Stacks Project","memory_eligible":false,"source_rank":10060,"rank":10060,"depth":51,"x":1658.435,"y":1063.754,"cluster":"tale-geometry"},{"id":"stacks:0GL2","tag":"0GL2","title":"More on constructible sheaves · Lemma 0GL2","summary":"Let X = lim_i ∈ I X_i be a limit of a directed system of schemes with affine transition morphisms. We assume that X_i is quasi-compact and quasi-separated for all i ∈ I. • The category of finite locally constant sheaves on X_etale is the colimit of the categories of finite locally constant sheaves on (X_i)_etale. • The category of finite locally constant abelian sheaves on X_etale is the colimit of the categories of finite locally constant abelian sheaves on (X_i)_etale.…","statement_latex":"Let $X = \\lim_{i \\in I} X_i$ be a limit of a directed\nsystem of schemes with affine transition morphisms.\nWe assume that $X_i$ is quasi-compact and quasi-separated\nfor all $i \\in I$.\n\\begin{enumerate}\n\\item The category of finite locally constant sheaves on $X_\\etale$\nis the colimit of the categories of finite locally constant sheaves\non $(X_i)_\\etale$.\n\\item The category of finite locally constant abelian sheaves on $X_\\etale$\nis the colimit of the categories of finite locally constant abelian sheaves\non $(X_i)_\\etale$.\n\\item Let $\\Lambda$ be a Noetherian ring. The category of finite type,\nlocally constant sheaves of $\\Lambda$-modules on $X_\\etale$\nis the colimit of the categories of finite type, locally constant\nsheaves of $\\Lambda$-modules on $(X_i)_\\etale$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GL2","source_file":"etale-cohomology.tex","source_line":12631,"source_end_line":12649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12631-L12649","statement_sha256":"82e2285db3eff5475524b624c0b632733059c277b30fd3e37365f1aee1f76658","origin":"The Stacks Project","memory_eligible":false,"source_rank":10061,"rank":10061,"depth":52,"x":1583.482,"y":1240.844,"cluster":"tale-geometry"},{"id":"stacks:09BG","tag":"09BG","title":"More on constructible sheaves · Lemma 09BG","summary":"Let X be an irreducible scheme with generic point eta. • Let S' ⊂ S be an inclusion of sets. If we have underlineS' ⊂ G ⊂ underlineS in Sh(X_etale) and S' = G_overlineeta, then G = underlineS'. • Let A' ⊂ A be an inclusion of abelian groups. If we have underlineA' ⊂ G ⊂ underlineA in Ab(X_etale) and A' = G_overlineeta, then G = underlineA'. • Let M' ⊂ M be an inclusion of modules over a ring Lambda. If we have underlineM' ⊂ G ⊂ underlineM in Mod(X_etale, underlineLambda)…","statement_latex":"Let $X$ be an irreducible scheme with generic point $\\eta$.\n\\begin{enumerate}\n\\item Let $S' \\subset S$ be an inclusion of sets. If we have\n$\\underline{S'} \\subset \\mathcal{G} \\subset \\underline{S}$\nin $\\Sh(X_\\etale)$ and $S' = \\mathcal{G}_{\\overline{\\eta}}$, then\n$\\mathcal{G} = \\underline{S'}$.\n\\item Let $A' \\subset A$ be an inclusion of abelian groups. If we have\n$\\underline{A'} \\subset \\mathcal{G} \\subset \\underline{A}$\nin $\\textit{Ab}(X_\\etale)$ and $A' = \\mathcal{G}_{\\overline{\\eta}}$, then\n$\\mathcal{G} = \\underline{A'}$.\n\\item Let $M' \\subset M$ be an inclusion of modules over a ring $\\Lambda$.\nIf we have $\\underline{M'} \\subset \\mathcal{G} \\subset \\underline{M}$\nin $\\textit{Mod}(X_\\etale, \\underline{\\Lambda})$\nand $M' = \\mathcal{G}_{\\overline{\\eta}}$, then\n$\\mathcal{G} = \\underline{M'}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BG","source_file":"etale-cohomology.tex","source_line":12674,"source_end_line":12692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12674-L12692","statement_sha256":"378cb7ce50f36fd16679de828b04f600aa89cf1c587ca890a6a80dd18b81f289","origin":"The Stacks Project","memory_eligible":false,"source_rank":10062,"rank":10062,"depth":0,"x":1496.241,"y":1067.533,"cluster":"tale-geometry"},{"id":"stacks:09Z5","tag":"09Z5","title":"More on constructible sheaves · Lemma 09Z5","summary":"Let X be an integral normal scheme with function field K. Let E be a set. • Let g : Spec(K) → X be the inclusion of the generic point. Then g_*underlineE = underlineE. • Let j : U → X be the inclusion of a nonempty open. Then j_*underlineE = underlineE.","statement_latex":"Let $X$ be an integral normal scheme with function field $K$.\nLet $E$ be a set.\n\\begin{enumerate}\n\\item Let $g : \\Spec(K) \\to X$ be the inclusion of the generic point.\nThen $g_*\\underline{E} = \\underline{E}$.\n\\item Let $j : U \\to X$ be the inclusion of a nonempty open. Then\n$j_*\\underline{E} = \\underline{E}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Z5","source_file":"etale-cohomology.tex","source_line":12699,"source_end_line":12709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12699-L12709","statement_sha256":"e80d06311bf7865303836dbce0e9e718103d1601355d55852cae2b35889ee259","origin":"The Stacks Project","memory_eligible":false,"source_rank":10063,"rank":10063,"depth":48,"x":1700.175,"y":1145.916,"cluster":"tale-geometry"},{"id":"stacks:0F0M","tag":"0F0M","title":"More on constructible sheaves · Lemma 0F0M","summary":"Let X be a quasi-compact and quasi-separated scheme. Let eta ∈ X be a generic point of an irreducible component of X. • Let F be a torsion abelian sheaf on X_etale whose stalk F_overlineeta is zero. Then F = colim F_i is a filtered colimit of constructible abelian sheaves F_i such that for each i the support of F_i is contained in a closed subscheme not containing eta. • Let Lambda be a Noetherian ring and F a sheaf of Lambda-modules on X_etale whose stalk F_overlineeta…","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme. Let\n$\\eta \\in X$ be a generic point of an irreducible component of $X$.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$\nwhose stalk $\\mathcal{F}_{\\overline{\\eta}}$ is zero.\nThen $\\mathcal{F} = \\colim \\mathcal{F}_i$ is a filtered colimit of\nconstructible abelian sheaves $\\mathcal{F}_i$ such that for each $i$\nthe support of $\\mathcal{F}_i$ is contained\nin a closed subscheme not containing $\\eta$.\n\\item Let $\\Lambda$ be a Noetherian ring and $\\mathcal{F}$ a sheaf\nof $\\Lambda$-modules on $X_\\etale$ whose stalk\n$\\mathcal{F}_{\\overline{\\eta}}$ is zero. Then\n$\\mathcal{F} = \\colim \\mathcal{F}_i$\nis a filtered colimit of constructible sheaves of\n$\\Lambda$-modules $\\mathcal{F}_i$ such that for each $i$\nthe support of $\\mathcal{F}_i$ is contained in a closed subscheme\nnot containing $\\eta$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"More on constructible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0M","source_file":"etale-cohomology.tex","source_line":12730,"source_end_line":12750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12730-L12750","statement_sha256":"d417c48460242287fd9010b892bdf5aee62f9f90ea388214d415f4b13132213c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10064,"rank":10064,"depth":50,"x":1486.521,"y":1203.9,"cluster":"tale-geometry"},{"id":"stacks:09BH","tag":"09BH","title":"Constructible sheaves on Noetherian schemes · Proposition 09BH","summary":"Let X be a Noetherian scheme. Let Lambda be a Noetherian ring. • Any sub or quotient sheaf of a constructible sheaf of sets is constructible. • The category of constructible abelian sheaves on X_etale is a (strong) Serre subcategory of Ab(X_etale). In particular, every sub and quotient sheaf of a constructible abelian sheaf on X_etale is constructible. • The category of constructible sheaves of Lambda-modules on X_etale is a (strong) Serre subcategory of Mod(X_etale,…","statement_latex":"Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring.\n\\begin{enumerate}\n\\item Any sub or quotient sheaf of a constructible sheaf of sets\nis constructible.\n\\item The category of constructible abelian sheaves on $X_\\etale$ is a\n(strong) Serre subcategory of $\\textit{Ab}(X_\\etale)$. In particular,\nevery sub and quotient sheaf of a constructible abelian sheaf\non $X_\\etale$ is constructible.\n\\item The category of constructible sheaves of $\\Lambda$-modules\non $X_\\etale$ is a (strong) Serre subcategory of\n$\\textit{Mod}(X_\\etale, \\Lambda)$. In particular, every submodule\nand quotient module of a constructible sheaf of $\\Lambda$-modules\non $X_\\etale$ is constructible.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves on Noetherian schemes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BH","source_file":"etale-cohomology.tex","source_line":12803,"source_end_line":12819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12803-L12819","statement_sha256":"5e3aca459217b3ee3848a24fa1c16ee817f9edb21b580847338453ea57c6b211","origin":"The Stacks Project","memory_eligible":false,"source_rank":10065,"rank":10065,"depth":4,"x":1597.563,"y":1039.726,"cluster":"tale-geometry"},{"id":"stacks:09YV","tag":"09YV","title":"Constructible sheaves on Noetherian schemes · Lemma 09YV","summary":"Let X be a Noetherian scheme. Let Lambda be a Noetherian ring. Consider inclusions F_1 ⊂ F_2 ⊂ F_3 ⊂ … ⊂ F in the category of sheaves of sets, abelian groups, or Lambda-modules. If F is constructible, then for some n we have F_n = F_n + 1 = F_n + 2 = ….","statement_latex":"Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring.\nConsider inclusions\n$$\n\\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\mathcal{F}_3 \\subset \\ldots\n\\subset \\mathcal{F}\n$$\nin the category of sheaves of sets, abelian groups, or $\\Lambda$-modules.\nIf $\\mathcal{F}$ is constructible, then for some $n$\nwe have $\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\mathcal{F}_{n + 2} = \\ldots$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YV","source_file":"etale-cohomology.tex","source_line":12900,"source_end_line":12911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12900-L12911","statement_sha256":"4c427dfae97f0a34f1ad225571867f28cfe9fc890601921ecbe2344aa267c9eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10066,"rank":10066,"depth":7,"x":1647.763,"y":1224.001,"cluster":"tale-geometry"},{"id":"stacks:09Z6","tag":"09Z6","title":"Constructible sheaves on Noetherian schemes · Lemma 09Z6","summary":"Let X be a Noetherian scheme. • Let F be a constructible sheaf of sets on X_etale. There exist an injective map of sheaves F → ∏_i = 1, …, n f_i, *underlineE_i where f_i : Y_i → X is a finite morphism and E_i is a finite set. • Let F be a constructible abelian sheaf on X_etale. There exist an injective map of abelian sheaves F → bigoplus_i = 1, …, n f_i, *underlineM_i where f_i : Y_i → X is a finite morphism and M_i is a finite abelian group. • Let Lambda be a Noetherian…","statement_latex":"Let $X$ be a Noetherian scheme.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a constructible sheaf of sets on $X_\\etale$.\nThere exist an injective map of sheaves\n$$\n\\mathcal{F} \\longrightarrow\n\\prod\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{E_i}\n$$\nwhere $f_i : Y_i \\to X$ is a finite morphism and $E_i$ is a finite set.\n\\item Let $\\mathcal{F}$ be a constructible abelian sheaf on $X_\\etale$.\nThere exist an injective map of abelian sheaves\n$$\n\\mathcal{F} \\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i}\n$$\nwhere $f_i : Y_i \\to X$ is a finite morphism and\n$M_i$ is a finite abelian group.\n\\item Let $\\Lambda$ be a Noetherian ring.\nLet $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$.\nThere exist an injective map of sheaves of modules\n$$\n\\mathcal{F} \\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i}\n$$\nwhere $f_i : Y_i \\to X$ is a finite morphism and\n$M_i$ is a finite $\\Lambda$-module.\n\\end{enumerate}\nMoreover, we may assume each $Y_i$ is irreducible, reduced, maps onto\nan irreducible and reduced closed subscheme $Z_i \\subset X$ such that\n$Y_i \\to Z_i$ is finite \\'etale over a nonempty open of $Z_i$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Z6","source_file":"etale-cohomology.tex","source_line":12920,"source_end_line":12952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L12920-L12952","statement_sha256":"99a902422e2e527122cda64e1174052532544ea37d9eb0c4464a0b987cb49522","origin":"The Stacks Project","memory_eligible":false,"source_rank":10067,"rank":10067,"depth":53,"x":1462.349,"y":1116.483,"cluster":"tale-geometry"},{"id":"stacks:09Z7","tag":"09Z7","title":"Constructible sheaves on Noetherian schemes · Lemma 09Z7","summary":"[SGA4] Let X be a quasi-compact and quasi-separated scheme. • Let F be a constructible sheaf of sets on X_etale. There exist an injective map of sheaves F → ∏_i = 1, …, n f_i, *underlineE_i where f_i : Y_i → X is a finite and finitely presented morphism and E_i is a finite set. • Let F be a constructible abelian sheaf on X_etale. There exist an injective map of abelian sheaves F → bigoplus_i = 1, …, n f_i, *underlineM_i where f_i : Y_i → X is a finite and finitely…","statement_latex":"\\begin{reference}\n\\cite[Exposee IX, Proposition 2.14]{SGA4}\n\\end{reference}\nLet $X$ be a quasi-compact and quasi-separated scheme.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a constructible sheaf of sets on $X_\\etale$.\nThere exist an injective map of sheaves\n$$\n\\mathcal{F} \\longrightarrow\n\\prod\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{E_i}\n$$\nwhere $f_i : Y_i \\to X$ is a finite and finitely presented morphism and\n$E_i$ is a finite set.\n\\item Let $\\mathcal{F}$ be a constructible abelian sheaf on $X_\\etale$.\nThere exist an injective map of abelian sheaves\n$$\n\\mathcal{F} \\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i}\n$$\nwhere $f_i : Y_i \\to X$ is a finite and finitely presented morphism and\n$M_i$ is a finite abelian group.\n\\item Let $\\Lambda$ be a Noetherian ring.\nLet $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$.\nThere exist an injective map of sheaves of modules\n$$\n\\mathcal{F} \\longrightarrow\n\\bigoplus\\nolimits_{i = 1, \\ldots, n} f_{i, *}\\underline{M_i}\n$$\nwhere $f_i : Y_i \\to X$ is a finite and finitely presented morphism and\n$M_i$ is a finite $\\Lambda$-module.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Z7","source_file":"etale-cohomology.tex","source_line":13027,"source_end_line":13060,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13027-L13060","statement_sha256":"bf54d476573c2a1add20d13cb60729d7bacd56cf9fb92dc352415b0e2b48801d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10068,"rank":10068,"depth":54,"x":1685.785,"y":1090.527,"cluster":"tale-geometry"},{"id":"stacks:0F0N","tag":"0F0N","title":"Constructible sheaves on Noetherian schemes · Lemma 0F0N","summary":"Let X be a Noetherian scheme. Let E ⊂ X be a subset closed under specialization. • Let F be a torsion abelian sheaf on X_etale whose support is contained in E. Then F = colim F_i is a filtered colimit of constructible abelian sheaves F_i such that for each i the support of F_i is contained in a closed subset contained in E. • Let Lambda be a Noetherian ring and F a sheaf of Lambda-modules on X_etale whose support is contained in E. Then F = colim F_i is a filtered colimit…","statement_latex":"Let $X$ be a Noetherian scheme. Let $E \\subset X$ be a subset closed\nunder specialization.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$\nwhose support is contained in $E$. Then $\\mathcal{F} = \\colim \\mathcal{F}_i$\nis a filtered colimit of constructible abelian sheaves $\\mathcal{F}_i$\nsuch that for each $i$ the support of $\\mathcal{F}_i$ is contained in\na closed subset contained in $E$.\n\\item Let $\\Lambda$ be a Noetherian ring and $\\mathcal{F}$ a sheaf\nof $\\Lambda$-modules on $X_\\etale$ whose support is contained in $E$.\nThen $\\mathcal{F} = \\colim \\mathcal{F}_i$\nis a filtered colimit of constructible sheaves of\n$\\Lambda$-modules $\\mathcal{F}_i$ such that for each $i$\nthe support of $\\mathcal{F}_i$ is contained in a closed subset\ncontained in $E$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Constructible sheaves on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0N","source_file":"etale-cohomology.tex","source_line":13090,"source_end_line":13108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13090-L13108","statement_sha256":"297db0be989a9f07753fb05cff8a114d7133d093a7894bea7c4adac7dc130991","origin":"The Stacks Project","memory_eligible":false,"source_rank":10069,"rank":10069,"depth":50,"x":1541.738,"y":1236.613,"cluster":"tale-geometry"},{"id":"stacks:0GJ6","tag":"0GJ6","title":"Specializations and étale sheaves · Lemma 0GJ6","summary":"Let g : S' → S be a morphism of schemes. Let F be a sheaf on S_etale. Let overlines' be a geometric point of S', and let overlinet' be a geometric point of Spec(O^sh_S', overlines'). Denote overlines = g(overlines') and overlinet = h(overlinet') where h : Spec(O^sh_S', overlines') → Spec(O^sh_S, overlines) is the canonical morphism. For any sheaf F on S_etale the specialization map sp : (g^-1F)_overlines' → (g^-1F)_overlinet' is equal to the specialization map sp :…","statement_latex":"Let $g : S' \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a sheaf\non $S_\\etale$. Let $\\overline{s}'$ be a geometric point of $S'$, and let\n$\\overline{t}'$ be a geometric point of\n$\\Spec(\\mathcal{O}^{sh}_{S', \\overline{s}'})$. Denote\n$\\overline{s} = g(\\overline{s}')$ and $\\overline{t} = h(\\overline{t}')$\nwhere $h : \\Spec(\\mathcal{O}^{sh}_{S', \\overline{s}'}) \\to\n\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$ is the canonical morphism.\nFor any sheaf $\\mathcal{F}$ on $S_\\etale$ the specialization map\n$$\nsp :\n(g^{-1}\\mathcal{F})_{\\overline{s}'}\n\\longrightarrow\n(g^{-1}\\mathcal{F})_{\\overline{t}'}\n$$\nis equal to the specialization map\n$sp : \\mathcal{F}_{\\overline{s}} \\to \\mathcal{F}_{\\overline{t}}$\nvia the identifications\n$(g^{-1}\\mathcal{F})_{\\overline{s}'} = \\mathcal{F}_{\\overline{s}}$ and\n$(g^{-1}\\mathcal{F})_{\\overline{t}'} = \\mathcal{F}_{\\overline{t}}$\nof Lemma \\ref{lemma-stalk-pullback}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Specializations and étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJ6","source_file":"etale-cohomology.tex","source_line":13329,"source_end_line":13351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13329-L13351","statement_sha256":"b97c3133562d4205579a1488921c8cc6a523e8e2af5c2f1fe38ffae5b306c723","origin":"The Stacks Project","memory_eligible":false,"source_rank":10070,"rank":10070,"depth":52,"x":1530.465,"y":1046.944,"cluster":"tale-geometry"},{"id":"stacks:0GJ7","tag":"0GJ7","title":"Specializations and étale sheaves · Lemma 0GJ7","summary":"Let S be a scheme such that every quasi-compact open of S has finite number of irreducible components (for example if S has a Noetherian underlying topological space, or if S is locally Noetherian). Let F be a sheaf of sets on S_etale. The following are equivalent • F is finite locally constant, and • all stalks of F are finite sets and all specialization maps sp : F_overlines → F_overlinet are bijective.","statement_latex":"Let $S$ be a scheme such that every quasi-compact open of $S$ has\nfinite number of irreducible components (for example if $S$ has a\nNoetherian underlying topological space, or if $S$ is locally Noetherian).\nLet $\\mathcal{F}$ be a sheaf of sets on $S_\\etale$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is finite locally constant, and\n\\item all stalks of $\\mathcal{F}$ are finite sets and all specialization maps\n$sp : \\mathcal{F}_{\\overline{s}} \\to \\mathcal{F}_{\\overline{t}}$\nare bijective.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Specializations and étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJ7","source_file":"etale-cohomology.tex","source_line":13357,"source_end_line":13370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13357-L13370","statement_sha256":"5506449c4d2b67f7967f7ebf8607829ee9bafb003c3937bd22499de72b1d67ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":10071,"rank":10071,"depth":53,"x":1691.484,"y":1180.556,"cluster":"tale-geometry"},{"id":"stacks:0GKC","tag":"0GKC","title":"Specializations and étale sheaves · Lemma 0GKC","summary":"Let S be a scheme such that every quasi-compact open of S has finite number of irreducible components (for example if S has a Noetherian underlying topological space, or if S is locally Noetherian). Let Lambda be a Noetherian ring. Let F be a sheaf of Lambda-modules on S_etale. The following are equivalent • F is a finite type, locally constant sheaf of Lambda-modules, and • all stalks of F are finite Lambda-modules and all specialization maps sp : F_overlines →…","statement_latex":"Let $S$ be a scheme such that every quasi-compact open of $S$ has\nfinite number of irreducible components (for example if $S$ has a\nNoetherian underlying topological space, or if $S$ is locally Noetherian).\nLet $\\Lambda$ be a Noetherian ring.\nLet $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules on $S_\\etale$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a finite type, locally constant sheaf\nof $\\Lambda$-modules, and\n\\item all stalks of $\\mathcal{F}$ are finite $\\Lambda$-modules and\nall specialization maps\n$sp : \\mathcal{F}_{\\overline{s}} \\to \\mathcal{F}_{\\overline{t}}$\nare bijective.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Specializations and étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKC","source_file":"etale-cohomology.tex","source_line":13418,"source_end_line":13434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13418-L13434","statement_sha256":"a130f2e91c478c0f07da01d7fbdd860aef9ea8ac7e748902ffb2d96c9f7a9f1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10072,"rank":10072,"depth":54,"x":1465.052,"y":1173.39,"cluster":"tale-geometry"},{"id":"stacks:0GJ8","tag":"0GJ8","title":"Specializations and étale sheaves · Lemma 0GJ8","summary":"Let f : X → S be a quasi-compact and quasi-separated morphism of schemes. Let K ∈ D^+(X_etale). Let overlines be a geometric point of S and let overlinet be a geometric point of Spec(O^sh_S, overlines). We have a commutative diagram xymatrix (Rf_*K)_overlines ar[r]_sp ar@=[d] & (Rf_*K)_overlinet ar@=[d] RΓ(X ×_S Spec(O^sh_S, overlines), K) ar[r] & RΓ(X ×_S Spec(O^sh_S, overlinet), K) where the bottom horizontal arrow arises as pullback by the morphism id_X × c where c :…","statement_latex":"Let $f : X \\to S$ be a quasi-compact and quasi-separated morphism of schemes.\nLet $K \\in D^+(X_\\etale)$. Let $\\overline{s}$ be a geometric point of $S$\nand let $\\overline{t}$ be a geometric point of\n$\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$. We have a commutative\ndiagram\n$$\n\\xymatrix{\n(Rf_*K)_{\\overline{s}} \\ar[r]_{sp} \\ar@{=}[d] &\n(Rf_*K)_{\\overline{t}} \\ar@{=}[d] \\\\\nR\\Gamma(X \\times_S \\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}}), K)\n\\ar[r] &\nR\\Gamma(X \\times_S \\Spec(\\mathcal{O}^{sh}_{S, \\overline{t}}), K)\n}\n$$\nwhere the bottom horizontal arrow arises as pullback by the morphism\n$\\text{id}_X \\times c$ where\n$c : \\Spec(\\mathcal{O}^{sh}_{S, \\overline{t}}) \\to\n\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$\nis the morphism introduced in Remark \\ref{remark-another-sp}.\nThe vertical arrows are given by\nTheorem \\ref{theorem-higher-direct-images}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Specializations and étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJ8","source_file":"etale-cohomology.tex","source_line":13490,"source_end_line":13513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13490-L13513","statement_sha256":"be4871f9b54e1e18e08be4d6e5802aee2808ea68d1f1d1eaa75a0fada73f74d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10073,"rank":10073,"depth":48,"x":1637.974,"y":1050.054,"cluster":"tale-geometry"},{"id":"stacks:095W","tag":"095W","title":"Complexes with constructible cohomology · Definition 095W","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. We denote D_c(X_etale, Lambda) the full subcategory of D(X_etale, Lambda) of complexes whose cohomology sheaves are constructible sheaves of Lambda-modules.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nWe denote {\\it $D_c(X_\\etale, \\Lambda)$} the full subcategory\nof $D(X_\\etale, \\Lambda)$ of complexes whose cohomology sheaves\nare constructible sheaves of $\\Lambda$-modules.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Complexes with constructible cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095W","source_file":"etale-cohomology.tex","source_line":13568,"source_end_line":13574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13568-L13574","statement_sha256":"0d8cff97e288204263ae55108bf07dccf549ebd839a2c928152f036065571054","origin":"The Stacks Project","memory_eligible":false,"source_rank":10074,"rank":10074,"depth":0,"x":1609.616,"y":1239.33,"cluster":"tale-geometry"},{"id":"stacks:095X","tag":"095X","title":"Complexes with constructible cohomology · Lemma 095X","summary":"Let Lambda be a Noetherian ring. If j : U → X is an étale morphism of schemes, then • K|_U ∈ D_c(U_etale, Lambda) if K ∈ D_c(X_etale, Lambda), and • j_!M ∈ D_c(X_etale, Lambda) if M ∈ D_c(U_etale, Lambda) and the morphism j is quasi-compact and quasi-separated.","statement_latex":"Let $\\Lambda$ be a Noetherian ring.\nIf $j : U \\to X$ is an \\'etale morphism of schemes, then\n\\begin{enumerate}\n\\item $K|_U \\in D_c(U_\\etale, \\Lambda)$ if $K \\in D_c(X_\\etale, \\Lambda)$, and\n\\item $j_!M \\in D_c(X_\\etale, \\Lambda)$ if $M \\in D_c(U_\\etale, \\Lambda)$ and\nthe morphism $j$ is quasi-compact and quasi-separated.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Complexes with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095X","source_file":"etale-cohomology.tex","source_line":13582,"source_end_line":13591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13582-L13591","statement_sha256":"a1e8e6e573cb327ea7d8dd6d56261b532d3f6197ad93fd315cf1b09edcefb9de","origin":"The Stacks Project","memory_eligible":false,"source_rank":10075,"rank":10075,"depth":49,"x":1478.171,"y":1083.498,"cluster":"tale-geometry"},{"id":"stacks:095Y","tag":"095Y","title":"Complexes with constructible cohomology · Lemma 095Y","summary":"Let Lambda be a Noetherian ring. Let f : X → Y be a morphism of schemes. If K ∈ D_c(Y_etale, Lambda) then Lf^*K ∈ D_c(X_etale, Lambda).","statement_latex":"Let $\\Lambda$ be a Noetherian ring.\nLet $f : X \\to Y$ be a morphism of schemes. If $K \\in D_c(Y_\\etale, \\Lambda)$\nthen $Lf^*K \\in D_c(X_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Complexes with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095Y","source_file":"etale-cohomology.tex","source_line":13598,"source_end_line":13603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13598-L13603","statement_sha256":"bb6ca1bc46b12ce5c985d9fa63465a5d5c13e72888fc5148d992d6f6b9b2c01c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10076,"rank":10076,"depth":7,"x":1700.658,"y":1123.866,"cluster":"tale-geometry"},{"id":"stacks:095Z","tag":"095Z","title":"Complexes with constructible cohomology · Lemma 095Z","summary":"Let X be a quasi-compact and quasi-separated scheme. Let Lambda be a Noetherian ring. Let K ∈ D(X_etale, Lambda) and b ∈ Z such that H^b(K) is constructible. Then there exist a sheaf F which is a finite direct sum of j_U!underlineLambda with U ∈ Ob(X_etale) affine and a map F[-b] → K in D(X_etale, Lambda) inducing a surjection F → H^b(K).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $\\Lambda$ be a Noetherian ring. Let $K \\in D(X_\\etale, \\Lambda)$\nand $b \\in \\mathbf{Z}$ such that $H^b(K)$ is constructible.\nThen there exist a sheaf $\\mathcal{F}$ which is a finite direct sum\nof $j_{U!}\\underline{\\Lambda}$ with $U \\in \\Ob(X_\\etale)$ affine and\na map $\\mathcal{F}[-b] \\to K$ in $D(X_\\etale, \\Lambda)$\ninducing a surjection $\\mathcal{F} \\to H^b(K)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Complexes with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095Z","source_file":"etale-cohomology.tex","source_line":13610,"source_end_line":13619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13610-L13619","statement_sha256":"8b2690ddc748fdc91117a9fae88e2baee68515fb0994f3140526ce740a9cc727","origin":"The Stacks Project","memory_eligible":false,"source_rank":10077,"rank":10077,"depth":7,"x":1503.92,"y":1220.45,"cluster":"tale-geometry"},{"id":"stacks:0960","tag":"0960","title":"Complexes with constructible cohomology · Lemma 0960","summary":"Let X be a quasi-compact and quasi-separated scheme. Let Lambda be a Noetherian ring. Let K ∈ D^-(X_etale, Lambda). Then the following are equivalent • K is in D_c(X_etale, Lambda), • K can be represented by a bounded above complex whose terms are finite direct sums of j_U!underlineLambda with U ∈ Ob(X_etale) affine, • K can be represented by a bounded above complex of flat constructible sheaves of Lambda-modules.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $\\Lambda$ be a Noetherian ring. Let $K \\in D^-(X_\\etale, \\Lambda)$. Then\nthe following are equivalent\n\\begin{enumerate}\n\\item $K$ is in $D_c(X_\\etale, \\Lambda)$,\n\\item $K$ can be represented by a bounded above complex\nwhose terms are finite direct sums of $j_{U!}\\underline{\\Lambda}$\nwith $U \\in \\Ob(X_\\etale)$ affine,\n\\item $K$ can be represented by a bounded above complex\nof flat constructible sheaves of $\\Lambda$-modules.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Complexes with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0960","source_file":"etale-cohomology.tex","source_line":13643,"source_end_line":13656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13643-L13656","statement_sha256":"d159bd4ac6e22e24907aeda94f8f488874bc91fd5c55d3d15e38c1c053739e71","origin":"The Stacks Project","memory_eligible":false,"source_rank":10078,"rank":10078,"depth":8,"x":1571.395,"y":1037.396,"cluster":"tale-geometry"},{"id":"stacks:0961","tag":"0961","title":"Complexes with constructible cohomology · Lemma 0961","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. Let K, L ∈ D_c^-(X_etale, Lambda). Then K ⊗_Lambda^L L is in D_c^-(X_etale, Lambda).","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nLet $K, L \\in D_c^-(X_\\etale, \\Lambda)$. Then\n$K \\otimes_\\Lambda^\\mathbf{L} L$ is in $D_c^-(X_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Complexes with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0961","source_file":"etale-cohomology.tex","source_line":13687,"source_end_line":13692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13687-L13692","statement_sha256":"c310a6b0e7b88aa2c4855df610a8dd883538cb801aaa51da3f7d0da428e4061a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10079,"rank":10079,"depth":9,"x":1668.954,"y":1210.854,"cluster":"tale-geometry"},{"id":"stacks:03TQ","tag":"03TQ","title":"Tor finite with constructible cohomology · Definition 03TQ","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. We denote D_ctf(X_etale, Lambda) the full subcategory of D_c(X_etale, Lambda) consisting of objects having locally finite tor dimension.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring. We denote\n{\\it $D_{ctf}(X_\\etale, \\Lambda)$} the full subcategory of\n$D_c(X_\\etale, \\Lambda)$\nconsisting of objects having locally finite tor dimension.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Tor finite with constructible cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TQ","source_file":"etale-cohomology.tex","source_line":13711,"source_end_line":13717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13711-L13717","statement_sha256":"232f4271bb3fd44fb9a2ff59f9bd66e51b57b2235e06733aa16c0502b189bfed","origin":"The Stacks Project","memory_eligible":false,"source_rank":10080,"rank":10080,"depth":0,"x":1457.294,"y":1138.226,"cluster":"tale-geometry"},{"id":"stacks:03TT","tag":"03TT","title":"Tor finite with constructible cohomology · Lemma 03TT","summary":"Let Lambda be a Noetherian ring. Let X be a quasi-compact and quasi-separated scheme. Let K ∈ D(X_etale, Lambda). The following are equivalent • K ∈ D_ctf(X_etale, Lambda), and • K can be represented by a finite complex of constructible flat sheaves of Lambda-modules. In fact, if K has tor amplitude in [a, b] then we can represent K by a complex F^a → … → F^b with F^p a constructible flat sheaf of Lambda-modules.","statement_latex":"Let $\\Lambda$ be a Noetherian ring. Let $X$ be a quasi-compact\nand quasi-separated scheme. Let $K \\in D(X_\\etale, \\Lambda)$. The following\nare equivalent\n\\begin{enumerate}\n\\item $K \\in D_{ctf}(X_\\etale, \\Lambda)$, and\n\\item $K$ can be represented by a finite complex of constructible\nflat sheaves of $\\Lambda$-modules.\n\\end{enumerate}\nIn fact, if $K$ has tor amplitude in $[a, b]$ then we can represent\n$K$ by a complex $\\mathcal{F}^a \\to \\ldots \\to \\mathcal{F}^b$ with\n$\\mathcal{F}^p$ a constructible flat sheaf of $\\Lambda$-modules.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Tor finite with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TT","source_file":"etale-cohomology.tex","source_line":13743,"source_end_line":13756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13743-L13756","statement_sha256":"89922d9bde37d065c86fc027471a193bcc8f328125518b71b9b03e816c08a0d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10081,"rank":10081,"depth":51,"x":1672.008,"y":1071.607,"cluster":"tale-geometry"},{"id":"stacks:0962","tag":"0962","title":"Tor finite with constructible cohomology · Lemma 0962","summary":"Let Lambda be a Noetherian ring. If j : U → X is an étale morphism of schemes, then • K|_U ∈ D_ctf(U_etale, Lambda) if K ∈ D_ctf(X_etale, Lambda), and • j_!M ∈ D_ctf(X_etale, Lambda) if M ∈ D_ctf(U_etale, Lambda) and the morphism j is quasi-compact and quasi-separated.","statement_latex":"Let $\\Lambda$ be a Noetherian ring.\nIf $j : U \\to X$ is an \\'etale morphism of schemes, then\n\\begin{enumerate}\n\\item $K|_U \\in D_{ctf}(U_\\etale, \\Lambda)$ if\n$K \\in D_{ctf}(X_\\etale, \\Lambda)$, and\n\\item $j_!M \\in D_{ctf}(X_\\etale, \\Lambda)$ if\n$M \\in D_{ctf}(U_\\etale, \\Lambda)$ and\nthe morphism $j$ is quasi-compact and quasi-separated.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Tor finite with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0962","source_file":"etale-cohomology.tex","source_line":13814,"source_end_line":13825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13814-L13825","statement_sha256":"bc3aa57a36b69b5fb4f97f5f60a9dfe8c8cff32467c74e0d987fbe3c16733814","origin":"The Stacks Project","memory_eligible":false,"source_rank":10082,"rank":10082,"depth":52,"x":1567.14,"y":1242.751,"cluster":"tale-geometry"},{"id":"stacks:0963","tag":"0963","title":"Tor finite with constructible cohomology · Lemma 0963","summary":"Let Lambda be a Noetherian ring. Let f : X → Y be a morphism of schemes. If K ∈ D_ctf(Y_etale, Lambda) then Lf^*K ∈ D_ctf(X_etale, Lambda).","statement_latex":"Let $\\Lambda$ be a Noetherian ring. Let $f : X \\to Y$ be a morphism of schemes.\nIf $K \\in D_{ctf}(Y_\\etale, \\Lambda)$ then\n$Lf^*K \\in D_{ctf}(X_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Tor finite with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0963","source_file":"etale-cohomology.tex","source_line":13835,"source_end_line":13840,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13835-L13840","statement_sha256":"bb0342d40bae306a13961c317cf443b13d2cd1a225a8421979bf451ffe87acb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10083,"rank":10083,"depth":52,"x":1506.774,"y":1056.846,"cluster":"tale-geometry"},{"id":"stacks:09BI","tag":"09BI","title":"Tor finite with constructible cohomology · Lemma 09BI","summary":"Let X be a connected scheme. Let Lambda be a Noetherian ring. Let K ∈ D_ctf(X_etale, Lambda) have locally constant cohomology sheaves. Then there exists a finite complex of finite projective Lambda-modules M^bullet and an étale covering (U_i → X) such that K|_U_i ≅ underlineM^bullet|_U_i in D(U_i, etale, Lambda).","statement_latex":"Let $X$ be a connected scheme. Let $\\Lambda$ be a Noetherian ring.\nLet $K \\in D_{ctf}(X_\\etale, \\Lambda)$ have locally constant cohomology sheaves.\nThen there exists a finite complex of finite projective $\\Lambda$-modules\n$M^\\bullet$ and an \\'etale covering $\\{U_i \\to X\\}$ such that\n$K|_{U_i} \\cong \\underline{M^\\bullet}|_{U_i}$ in $D(U_{i, \\etale}, \\Lambda)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Tor finite with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BI","source_file":"etale-cohomology.tex","source_line":13849,"source_end_line":13856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13849-L13856","statement_sha256":"dd50131f9f4c1d0a2916d4972eb6b7573d36a37422c324998417c1a0b50213bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10084,"rank":10084,"depth":11,"x":1700.997,"y":1159.788,"cluster":"tale-geometry"},{"id":"stacks:0DDC","tag":"0DDC","title":"Torsion sheaves · Lemma 0DDC","summary":"Let X be a quasi-compact and quasi-separated scheme. • If F is a torsion abelian sheaf on X_etale, then H^n_etale(X, F) is a torsion abelian group for all n. • If K in D^+(X_etale) has torsion cohomology sheaves, then H^n_etale(X, K) is a torsion abelian group for all n.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is a torsion abelian sheaf on $X_\\etale$, then\n$H^n_\\etale(X, \\mathcal{F})$ is a torsion abelian group for all $n$.\n\\item If $K$ in $D^+(X_\\etale)$ has torsion cohomology sheaves, then\n$H^n_\\etale(X, K)$ is a torsion abelian group for all $n$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Torsion sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDC","source_file":"etale-cohomology.tex","source_line":13898,"source_end_line":13907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13898-L13907","statement_sha256":"6255c148f8dd655d708d3e2b5e7679ee585a22e3b4a9ac819476fb78aca6e605","origin":"The Stacks Project","memory_eligible":false,"source_rank":10085,"rank":10085,"depth":43,"x":1474.752,"y":1194.124,"cluster":"tale-geometry"},{"id":"stacks:0DDD","tag":"0DDD","title":"Torsion sheaves · Lemma 0DDD","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of schemes. • If F is a torsion abelian sheaf on X_etale, then R^nf_*F is a torsion abelian sheaf on Y_etale for all n. • If K in D^+(X_etale) has torsion cohomology sheaves, then Rf_*K is an object of D^+(Y_etale) whose cohomology sheaves are torsion abelian sheaves.","statement_latex":"Let $f : X \\to Y$ be a quasi-compact and quasi-separated\nmorphism of schemes.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is a torsion abelian sheaf on $X_\\etale$, then\n$R^nf_*\\mathcal{F}$ is a torsion abelian sheaf on $Y_\\etale$ for all $n$.\n\\item If $K$ in $D^+(X_\\etale)$ has torsion cohomology sheaves, then\n$Rf_*K$ is an object of $D^+(Y_\\etale)$ whose cohomology sheaves are\ntorsion abelian sheaves.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Torsion sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDD","source_file":"etale-cohomology.tex","source_line":13923,"source_end_line":13934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L13923-L13934","statement_sha256":"7de0dcc5df68db6a389d4415c7aaec1f52aef7fa7b0a667de3cb988aa7c39889","origin":"The Stacks Project","memory_eligible":false,"source_rank":10086,"rank":10086,"depth":44,"x":1614.12,"y":1040.262,"cluster":"tale-geometry"},{"id":"stacks:09XQ","tag":"09XQ","title":"Cohomology with support in a closed subscheme · Lemma 09XQ","summary":"Let i : Z → X be a closed immersion of schemes. Let I be an injective abelian sheaf on X_etale. Then H_Z(I) is an injective abelian sheaf on Z_etale.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nLet $\\mathcal{I}$ be an injective abelian sheaf on $X_\\etale$.\nThen $\\mathcal{H}_Z(\\mathcal{I})$ is an injective abelian sheaf\non $Z_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology with support in a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XQ","source_file":"etale-cohomology.tex","source_line":14022,"source_end_line":14028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14022-L14028","statement_sha256":"bf64fd1fb4b5fe9d5a0fd3cb7c66f56e36dce95faadc1935a0db90086e755536","origin":"The Stacks Project","memory_eligible":false,"source_rank":10087,"rank":10087,"depth":0,"x":1635.106,"y":1233.005,"cluster":"tale-geometry"},{"id":"stacks:09XR","tag":"09XR","title":"Cohomology with support in a closed subscheme · Lemma 09XR","summary":"Let i : Z → X be a closed immersion of schemes. Let G be an injective abelian sheaf on Z_etale. Then H^p_Z(i_*G) = 0 for p > 0.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nLet $\\mathcal{G}$ be an injective abelian sheaf on $Z_\\etale$.\nThen $\\mathcal{H}^p_Z(i_*\\mathcal{G}) = 0$ for $p > 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology with support in a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XR","source_file":"etale-cohomology.tex","source_line":14057,"source_end_line":14062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14057-L14062","statement_sha256":"f7be8a975ab6bd785a332610873d124a0ed5a9eeb94cee03819a5079ddd618df","origin":"The Stacks Project","memory_eligible":false,"source_rank":10088,"rank":10088,"depth":9,"x":1464.449,"y":1102.648,"cluster":"tale-geometry"},{"id":"stacks:0A45","tag":"0A45","title":"Cohomology with support in a closed subscheme · Lemma 0A45","summary":"Let i : Z → X be a closed immersion of schemes. Let j : U → X be the inclusion of the complement of Z. Let F be an abelian sheaf on X_etale. There is a distinguished triangle i_*RH_Z(F) → F → Rj_*(F|_U) → i_*RH_Z(F)[1] in D(X_etale). This produces an exact sequence 0 → i_*H_Z(F) → F → j_*(F|_U) → i_*H^1_Z(F) → 0 and isomorphisms R^pj_*(F|_U) ≅ i_*H^p + 1_Z(F) for p ≥ 1.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nLet $j : U \\to X$ be the inclusion of the complement of $Z$.\nLet $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.\nThere is a distinguished triangle\n$$\ni_*R\\mathcal{H}_Z(\\mathcal{F}) \\to \\mathcal{F} \\to Rj_*(\\mathcal{F}|_U) \\to\ni_*R\\mathcal{H}_Z(\\mathcal{F})[1]\n$$\nin $D(X_\\etale)$. This produces an exact sequence\n$$\n0 \\to i_*\\mathcal{H}_Z(\\mathcal{F}) \\to \\mathcal{F} \\to\nj_*(\\mathcal{F}|_U) \\to i_*\\mathcal{H}^1_Z(\\mathcal{F}) \\to 0\n$$\nand isomorphisms\n$R^pj_*(\\mathcal{F}|_U) \\cong i_*\\mathcal{H}^{p + 1}_Z(\\mathcal{F})$\nfor $p \\geq 1$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology with support in a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A45","source_file":"etale-cohomology.tex","source_line":14071,"source_end_line":14089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14071-L14089","statement_sha256":"9bc34a01009f44144496b421a861ea7abb589ce046303d5cf609663eceb2c883","origin":"The Stacks Project","memory_eligible":false,"source_rank":10089,"rank":10089,"depth":0,"x":1695.366,"y":1101.936,"cluster":"tale-geometry"},{"id":"stacks:0AEG","tag":"0AEG","title":"Cohomology with support in a closed subscheme · Lemma 0AEG","summary":"Let i : Z → X be a closed immersion of schemes. The map Ri_small, * = i_small, * : D(Z_etale) → D(X_etale) induces an equivalence D(Z_etale) → D_Z(X_etale) with quasi-inverse i_small^-1|_D_Z(X_etale) = RH_Z|_D_Z(X_etale)","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nThe map $Ri_{small, *} = i_{small, *} : D(Z_\\etale) \\to D(X_\\etale)$\ninduces an equivalence $D(Z_\\etale) \\to D_Z(X_\\etale)$ with quasi-inverse\n$$\ni_{small}^{-1}|_{D_Z(X_\\etale)} = R\\mathcal{H}_Z|_{D_Z(X_\\etale)}\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology with support in a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEG","source_file":"etale-cohomology.tex","source_line":14111,"source_end_line":14119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14111-L14119","statement_sha256":"61e2bca8d48a0f4fca1c1aabb86452be6dfd359a93d14610e40cc433fafc43ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":10090,"rank":10090,"depth":53,"x":1525.482,"y":1233.629,"cluster":"tale-geometry"},{"id":"stacks:0A46","tag":"0A46","title":"Cohomology with support in a closed subscheme · Lemma 0A46","summary":"Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let F be a quasi-coherent O_X-module and denote F^a the associated quasi-coherent sheaf on the small étale site of X (Proposition [Tag 03OG]). Then • H^q_Z(X, F) agrees with H^q_Z(X_etale, F^a), • if the complement of Z is retrocompact in X, then i_*H^q_Z(F^a) is a quasi-coherent sheaf of O_X-modules equal to (i_*H^q_Z(F))^a.","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nand denote $\\mathcal{F}^a$ the associated quasi-coherent sheaf\non the small \\'etale site of $X$\n(Proposition \\ref{proposition-quasi-coherent-sheaf-fpqc}). Then\n\\begin{enumerate}\n\\item $H^q_Z(X, \\mathcal{F})$ agrees with $H^q_Z(X_\\etale, \\mathcal{F}^a)$,\n\\item if the complement of $Z$ is retrocompact in $X$, then\n$i_*\\mathcal{H}^q_Z(\\mathcal{F}^a)$ is a quasi-coherent sheaf of\n$\\mathcal{O}_X$-modules equal to $(i_*\\mathcal{H}^q_Z(\\mathcal{F}))^a$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology with support in a closed subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A46","source_file":"etale-cohomology.tex","source_line":14155,"source_end_line":14168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14155-L14168","statement_sha256":"a7d6965511872642067052dd8bd561d5d3d4e86baa6777417146cab0b5c20770","origin":"The Stacks Project","memory_eligible":false,"source_rank":10091,"rank":10091,"depth":42,"x":1544.869,"y":1039.918,"cluster":"tale-geometry"},{"id":"stacks:09AX","tag":"09AX","title":"Schemes with strictly henselian local rings · Lemma 09AX","summary":"Let S be a scheme all of whose local rings are strictly henselian. Then for any abelian sheaf F on S_etale we have H^i(S_etale, F) = H^i(S_Zar, F).","statement_latex":"Let $S$ be a scheme all of whose local rings are strictly henselian.\nThen for any abelian sheaf $\\mathcal{F}$ on $S_\\etale$ we have\n$H^i(S_\\etale, \\mathcal{F}) = H^i(S_{Zar}, \\mathcal{F})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Schemes with strictly henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AX","source_file":"etale-cohomology.tex","source_line":14204,"source_end_line":14209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14204-L14209","statement_sha256":"506a80531f0bf5f621d66d70be8993045330e4a1dbb8bbc33081100f37d4d13d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10092,"rank":10092,"depth":46,"x":1686.501,"y":1193.922,"cluster":"tale-geometry"},{"id":"stacks:0GY0","tag":"0GY0","title":"Schemes with strictly henselian local rings · Lemma 0GY0","summary":"Let R be a ring all of whose local rings are strictly henselian. Let F be a sheaf on Spec(R)_etale. Assume that for all f, g ∈ R the kernel of H^1_etale(D(f + g), F) → H^1_etale(D(f(f + g)), F) ⊕ H^1_etale(D(g(f + g)), F) is zero. Then H^q_etale(Spec(R), F) = 0 for q > 0.","statement_latex":"Let $R$ be a ring all of whose local rings are strictly henselian.\nLet $\\mathcal{F}$ be a sheaf on $\\Spec(R)_\\etale$.\nAssume that for all $f, g \\in R$ the kernel of\n$$\nH^1_\\etale(D(f + g), \\mathcal{F})\n\\longrightarrow\nH^1_\\etale(D(f(f + g)), \\mathcal{F}) \\oplus\nH^1_\\etale(D(g(f + g)), \\mathcal{F})\n$$\nis zero. Then $H^q_\\etale(\\Spec(R), \\mathcal{F}) = 0$ for $q > 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Schemes with strictly henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GY0","source_file":"etale-cohomology.tex","source_line":14227,"source_end_line":14239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14227-L14239","statement_sha256":"c6691a4736d962d538f4b8b82da2991f564ad265b6591cd72d868b2050071a9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10093,"rank":10093,"depth":47,"x":1457.975,"y":1160.692,"cluster":"tale-geometry"},{"id":"stacks:09AY","tag":"09AY","title":"Schemes with strictly henselian local rings · Lemma 09AY","summary":"Let S be an affine scheme such that (1) all points are closed, and (2) all residue fields are separably algebraically closed. Then for any abelian sheaf F on S_etale we have H^i(S_etale, F) = 0 for i > 0.","statement_latex":"Let $S$ be an affine scheme such that\n(1) all points are closed, and (2) all residue fields are separably\nalgebraically closed. Then\nfor any abelian sheaf $\\mathcal{F}$ on $S_\\etale$ we have\n$H^i(S_\\etale, \\mathcal{F}) = 0$ for $i > 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Schemes with strictly henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AY","source_file":"etale-cohomology.tex","source_line":14279,"source_end_line":14286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14279-L14286","statement_sha256":"6ee781e724e928e543195108aa6e2d956d698fd31b92a48a90b96773aef4ea24","origin":"The Stacks Project","memory_eligible":false,"source_rank":10094,"rank":10094,"depth":47,"x":1653.417,"y":1055.413,"cluster":"tale-geometry"},{"id":"stacks:09Z9","tag":"09Z9","title":"Schemes with strictly henselian local rings · Lemma 09Z9","summary":"Let X be an integral normal scheme with separably closed function field. • A separated étale morphism U → X is a disjoint union of open immersions. • All local rings of X are strictly henselian.","statement_latex":"Let $X$ be an integral normal scheme with separably closed\nfunction field.\n\\begin{enumerate}\n\\item A separated \\'etale morphism $U \\to X$ is a\ndisjoint union of open immersions.\n\\item All local rings of $X$ are strictly henselian.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Schemes with strictly henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09Z9","source_file":"etale-cohomology.tex","source_line":14311,"source_end_line":14320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14311-L14320","statement_sha256":"a2457d57b784ad62589160c4b6def2b90d05918f153d3d30fbad14a6b06a66c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10095,"rank":10095,"depth":47,"x":1593.901,"y":1244.142,"cluster":"tale-geometry"},{"id":"stacks:0EZP","tag":"0EZP","title":"Schemes with strictly henselian local rings · Lemma 0EZP","summary":"Let f : X → Y be a morphism of schemes where X is an integral normal scheme with separably closed function field. Then R^qf_*underlineM = 0 for q > 0 and any abelian group M.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes where $X$ is an integral\nnormal scheme with separably closed function field. Then\n$R^qf_*\\underline{M} = 0$ for $q > 0$ and any abelian group $M$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Schemes with strictly henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZP","source_file":"etale-cohomology.tex","source_line":14368,"source_end_line":14373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14368-L14373","statement_sha256":"99d290abafba5376c9814d008cd776717d71edc9e6d7243a0b0548ea1a9ea344","origin":"The Stacks Project","memory_eligible":false,"source_rank":10096,"rank":10096,"depth":48,"x":1485.903,"y":1071.023,"cluster":"tale-geometry"},{"id":"stacks:09ZA","tag":"09ZA","title":"Schemes with strictly henselian local rings · Lemma 09ZA","summary":"Let X be an affine integral normal scheme with separably closed function field. Let Z ⊂ X be a closed subscheme. Let V → Z be an étale morphism with V affine. Then V is a finite disjoint union of open subschemes of Z. If V → Z is surjective and finite étale, then V → Z has a section.","statement_latex":"Let $X$ be an affine integral normal scheme with separably closed\nfunction field. Let $Z \\subset X$ be a closed subscheme. Let\n$V \\to Z$ be an \\'etale morphism with $V$ affine. Then $V$ is a finite\ndisjoint union of open subschemes of $Z$. If $V \\to Z$ is\nsurjective and finite \\'etale, then $V \\to Z$ has a section.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Schemes with strictly henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZA","source_file":"etale-cohomology.tex","source_line":14389,"source_end_line":14396,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14389-L14396","statement_sha256":"ac21fcabd1f89b035b995595eb160406b51e63aa1650c903bc52abbcf4bcac2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10097,"rank":10097,"depth":48,"x":1704.987,"y":1137.467,"cluster":"tale-geometry"},{"id":"stacks:09ZB","tag":"09ZB","title":"Schemes with strictly henselian local rings · Lemma 09ZB","summary":"Let X be a normal integral affine scheme with separably closed function field. Let Z ⊂ X be a closed subscheme. For any finite abelian group M we have H^1_etale(Z, underlineM) = 0.","statement_latex":"Let $X$ be a normal integral affine scheme with separably closed\nfunction field. Let $Z \\subset X$ be a closed subscheme.\nFor any finite abelian group $M$ we have $H^1_\\etale(Z, \\underline{M}) = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Schemes with strictly henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZB","source_file":"etale-cohomology.tex","source_line":14424,"source_end_line":14429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14424-L14429","statement_sha256":"82378ac5d840956f2cc29520db99534ceed1752643141e5eb3f43d868e1f026d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10098,"rank":10098,"depth":49,"x":1489.779,"y":1212.865,"cluster":"tale-geometry"},{"id":"stacks:09ZC","tag":"09ZC","title":"Schemes with strictly henselian local rings · Lemma 09ZC","summary":"Let X be a normal integral affine scheme with separably closed function field. Let Z ⊂ X be a closed subscheme. For any finite abelian group M we have H^q_etale(Z, underlineM) = 0 for q ≥ 1.","statement_latex":"Let $X$ be a normal integral affine scheme with separably closed\nfunction field. Let $Z \\subset X$ be a closed subscheme.\nFor any finite abelian group $M$ we have\n$H^q_\\etale(Z, \\underline{M}) = 0$ for $q \\geq 1$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Schemes with strictly henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZC","source_file":"etale-cohomology.tex","source_line":14444,"source_end_line":14450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14444-L14450","statement_sha256":"201df2bc3f9d4de86a0f6b2eabfcc14690e50f794f30b321fe3dee40591d7c15","origin":"The Stacks Project","memory_eligible":false,"source_rank":10099,"rank":10099,"depth":51,"x":1587.94,"y":1034.967,"cluster":"tale-geometry"},{"id":"stacks:09ZD","tag":"09ZD","title":"Schemes with strictly henselian local rings · Lemma 09ZD","summary":"Let X be an affine scheme. • There exists an integral surjective morphism X' → X such that for every closed subscheme Z' ⊂ X', every finite abelian group M, and every q ≥ 1 we have H^q_etale(Z', underlineM) = 0. • For any closed subscheme Z ⊂ X, finite abelian group M, q ≥ 1, and xi ∈ H^q_etale(Z, underlineM) there exists a finite surjective morphism X' → X of finite presentation such that xi pulls back to zero in H^q_etale(X' ×_X Z, underlineM).","statement_latex":"Let $X$ be an affine scheme.\n\\begin{enumerate}\n\\item There exists an integral surjective morphism $X' \\to X$ such that for\nevery closed subscheme $Z' \\subset X'$, every finite abelian group $M$, and\nevery $q \\geq 1$ we have $H^q_\\etale(Z', \\underline{M}) = 0$.\n\\item For any closed subscheme $Z \\subset X$, finite abelian group $M$,\n$q \\geq 1$, and $\\xi \\in H^q_\\etale(Z, \\underline{M})$ there exists a\nfinite surjective morphism $X' \\to X$ of finite presentation such that\n$\\xi$ pulls back to zero in $H^q_\\etale(X' \\times_X Z, \\underline{M})$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Schemes with strictly henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZD","source_file":"etale-cohomology.tex","source_line":14466,"source_end_line":14478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14466-L14478","statement_sha256":"213ab4705f63e6e4e1232806089341297c07c483e0670cb1c6ce47eea9507495","origin":"The Stacks Project","memory_eligible":false,"source_rank":10100,"rank":10100,"depth":52,"x":1658.692,"y":1222.041,"cluster":"tale-geometry"},{"id":"stacks:0GY2","tag":"0GY2","title":"Absolutely integrally closed vanishing · Lemma 0GY2","summary":"Let A be a ring. Let a, b ∈ A such that aA + bA = A and a bmod bA is a root of unity. Then there exists a monogenic extension A ⊂ B and an element y ∈ B such that u = a - by is a unit.","statement_latex":"Let $A$ be a ring. Let $a, b \\in A$ such that\n$aA + bA = A$ and $a \\bmod bA$ is a root of unity.\nThen there exists a monogenic extension $A \\subset B$\nand an element $y \\in B$ such that $u = a - by$ is a unit.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Absolutely integrally closed vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GY2","source_file":"etale-cohomology.tex","source_line":14522,"source_end_line":14528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14522-L14528","statement_sha256":"050e0d5ce7a808bb2fceee28550731fc6335910dcf18d733bae98dfb390dcb0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10101,"rank":10101,"depth":0,"x":1455.87,"y":1124.139,"cluster":"tale-geometry"},{"id":"stacks:0GY3","tag":"0GY3","title":"Absolutely integrally closed vanishing · Lemma 0GY3","summary":"We have A[s, frac1π s + 1] = (A[t, frac1π^ℓ t + 1])[s]/(Φ(s) - t) In particular, the Hopf algebra of G is a monogenic extension of the Hopf algebra of H.","statement_latex":"We have\n$$\nA[s, \\frac{1}{\\pi s + 1}] =\n\\left(A[t, \\frac{1}{\\pi^\\ell t + 1}]\\right)[s]/(\\Phi(s) - t)\n$$\nIn particular, the Hopf algebra of $G$ is a monogenic extension\nof the Hopf algebra of $H$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Absolutely integrally closed vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GY3","source_file":"etale-cohomology.tex","source_line":14631,"source_end_line":14640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14631-L14640","statement_sha256":"3833476b77cc924a84f40551f5011105738c0df3981cf6858ba1c4525d37938d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10102,"rank":10102,"depth":0,"x":1684.395,"y":1081.2,"cluster":"tale-geometry"},{"id":"stacks:0GY5","tag":"0GY5","title":"Absolutely integrally closed vanishing · Lemma 0GY5","summary":"Let R be an A-algebra which is absolutely integrally closed. Then G(R) → H(R) is surjective.","statement_latex":"Let $R$ be an $A$-algebra which is absolutely integrally closed. Then\n$G(R) \\to H(R)$ is surjective.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Absolutely integrally closed vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GY5","source_file":"etale-cohomology.tex","source_line":14682,"source_end_line":14686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14682-L14686","statement_sha256":"eba3b606125d73ac67e469ad1eb35ba1e964941785df55dd737c6a3521c79926","origin":"The Stacks Project","memory_eligible":false,"source_rank":10103,"rank":10103,"depth":1,"x":1550.275,"y":1242.701,"cluster":"tale-geometry"},{"id":"stacks:0GY6","tag":"0GY6","title":"Absolutely integrally closed vanishing · Lemma 0GY6","summary":"Let R be an A-algebra which is absolutely integrally closed. Let I, J ⊂ R be ideals with I + J = R. There exists a g ∈ G(R) such that g bmod I = σ_0 and g bmod J = σ_1.","statement_latex":"Let $R$ be an $A$-algebra which is absolutely integrally closed.\nLet $I, J \\subset R$ be ideals with $I + J = R$.\nThere exists a $g \\in G(R)$ such\nthat $g \\bmod I = \\sigma_0$ and $g \\bmod J = \\sigma_1$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Absolutely integrally closed vanishing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GY6","source_file":"etale-cohomology.tex","source_line":14700,"source_end_line":14706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14700-L14706","statement_sha256":"15970ef0f668ce599bfbd1af8231702fd3aef8728287816fe983676d2908715d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10104,"rank":10104,"depth":1,"x":1519.266,"y":1047.307,"cluster":"tale-geometry"},{"id":"stacks:0GY7","tag":"0GY7","title":"Absolutely integrally closed vanishing · Proposition 0GY7","summary":"Let R be an absolutely integrally closed ring. Let M be a finite abelian group. Then H^i_etale(Spec(R), underlineM) = 0 for i > 0.","statement_latex":"Let $R$ be an absolutely integrally closed ring.\nLet $M$ be a finite abelian group.\nThen $H^i_\\etale(\\Spec(R), \\underline{M}) = 0$ for $i > 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Absolutely integrally closed vanishing","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GY7","source_file":"etale-cohomology.tex","source_line":14730,"source_end_line":14735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14730-L14735","statement_sha256":"0e261a866d336b8a316bb712bb7d90586143b10a7b2f9c9020fd7efa83ae8c1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10105,"rank":10105,"depth":48,"x":1699.448,"y":1173.923,"cluster":"tale-geometry"},{"id":"stacks:09ZE","tag":"09ZE","title":"Affine analog of proper base change · Lemma 09ZE","summary":"Let X be an affine scheme. Let F be a torsion abelian sheaf on X_etale. Let Z ⊂ X be a closed subscheme. Let xi ∈ H^q_etale(Z, F|_Z) for some q > 0. Then there exists an injective map F → F' of torsion abelian sheaves on X_etale such that the image of xi in H^q_etale(Z, F'|_Z) is zero.","statement_latex":"Let $X$ be an affine scheme. Let $\\mathcal{F}$ be a torsion abelian sheaf\non  $X_\\etale$. Let $Z \\subset X$ be a closed subscheme. Let\n$\\xi \\in H^q_\\etale(Z, \\mathcal{F}|_Z)$ for some $q > 0$.\nThen there exists an injective map $\\mathcal{F} \\to \\mathcal{F}'$\nof torsion abelian sheaves on $X_\\etale$ such that\nthe image of $\\xi$ in $H^q_\\etale(Z, \\mathcal{F}'|_Z)$ is zero.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Affine analog of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZE","source_file":"etale-cohomology.tex","source_line":14900,"source_end_line":14908,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14900-L14908","statement_sha256":"a22de77302bbc5011a6f88190f7250c8ff9da4dc66d9e59854d5ac4f2cb49727","origin":"The Stacks Project","memory_eligible":false,"source_rank":10106,"rank":10106,"depth":55,"x":1464.521,"y":1182.807,"cluster":"tale-geometry"},{"id":"stacks:09ZF","tag":"09ZF","title":"Affine analog of proper base change · Lemma 09ZF","summary":"Let X be a quasi-compact and quasi-separated scheme. Let i : Z → X be a closed immersion. Assume that • for any sheaf F on X_Zar the map Γ(X, F) → Γ(Z, i^-1F) is bijective, and • for any finite morphism X' → X assumption (1) holds for Z ×_X X' → X'. Then for any sheaf F on X_etale we have Γ(X, F) = Γ(Z, i^-1_smallF).","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $i : Z \\to X$ be a closed immersion. Assume that\n\\begin{enumerate}\n\\item for any sheaf $\\mathcal{F}$ on $X_{Zar}$ the map\n$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(Z, i^{-1}\\mathcal{F})$\nis bijective, and\n\\item for any finite morphism $X' \\to X$ assumption (1) holds\nfor $Z \\times_X X' \\to X'$.\n\\end{enumerate}\nThen for any sheaf $\\mathcal{F}$ on $X_\\etale$ we have\n$\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, i^{-1}_{small}\\mathcal{F})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Affine analog of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZF","source_file":"etale-cohomology.tex","source_line":14974,"source_end_line":14987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L14974-L14987","statement_sha256":"e14f2526434546e1d9f8ea6c6f04576545b849002ae908e8b052653a4f224113","origin":"The Stacks Project","memory_eligible":false,"source_rank":10107,"rank":10107,"depth":50,"x":1630.78,"y":1042.809,"cluster":"tale-geometry"},{"id":"stacks:0CAM","tag":"0CAM","title":"Affine analog of proper base change · Lemma 0CAM","summary":"Let Z ⊂ X be a closed subset of a topological space X. Assume • X is a spectral space (Topology, Definition [Tag 08YG]), and • for x ∈ X the intersection Z ∩ overline(x) is connected (in particular nonempty). If Z = Z_1 amalg Z_2 with Z_i closed in Z, then there exists a decomposition X = X_1 amalg X_2 with X_i closed in X and Z_i = Z ∩ X_i.","statement_latex":"Let $Z \\subset X$ be a closed subset of a topological space $X$.\nAssume\n\\begin{enumerate}\n\\item $X$ is a spectral space\n(Topology, Definition \\ref{topology-definition-spectral-space}), and\n\\item for $x \\in X$ the intersection $Z \\cap \\overline{\\{x\\}}$\nis connected (in particular nonempty).\n\\end{enumerate}\nIf $Z = Z_1 \\amalg Z_2$ with $Z_i$ closed in $Z$,\nthen there exists a decomposition $X = X_1 \\amalg X_2$ with\n$X_i$ closed in $X$ and $Z_i = Z \\cap X_i$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Affine analog of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CAM","source_file":"etale-cohomology.tex","source_line":15071,"source_end_line":15084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15071-L15084","statement_sha256":"f6a280a7b9edbdc34194bd973fbef2ef101552b5381d24dd378fd5b4c9de5990","origin":"The Stacks Project","memory_eligible":false,"source_rank":10108,"rank":10108,"depth":6,"x":1620.756,"y":1240.584,"cluster":"tale-geometry"},{"id":"stacks:09ZG","tag":"09ZG","title":"Affine analog of proper base change · Lemma 09ZG","summary":"Let Z ⊂ X be a closed subset of a topological space X. Assume • X is a spectral space (Topology, Definition [Tag 08YG]), and • for x ∈ X the intersection Z ∩ overline(x) is connected (in particular nonempty). Then for any sheaf F on X we have Γ(X, F) = Γ(Z, F|_Z).","statement_latex":"Let $Z \\subset X$ be a closed subset of a topological space $X$.\nAssume\n\\begin{enumerate}\n\\item $X$ is a spectral space\n(Topology, Definition \\ref{topology-definition-spectral-space}), and\n\\item for $x \\in X$ the intersection $Z \\cap \\overline{\\{x\\}}$\nis connected (in particular nonempty).\n\\end{enumerate}\nThen for any sheaf $\\mathcal{F}$ on $X$ we have\n$\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, \\mathcal{F}|_Z)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Affine analog of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZG","source_file":"etale-cohomology.tex","source_line":15098,"source_end_line":15110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15098-L15110","statement_sha256":"fef9a409074c424651e57cb5c53baef06eaf67eec9e7332431eea9381d5e410e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10109,"rank":10109,"depth":10,"x":1468.947,"y":1088.904,"cluster":"tale-geometry"},{"id":"stacks:09ZH","tag":"09ZH","title":"Affine analog of proper base change · Lemma 09ZH","summary":"Let (A, I) be a henselian pair. Set X = Spec(A) and Z = Spec(A/I). For any sheaf F on X_etale we have Γ(X, F) = Γ(Z, F|_Z).","statement_latex":"Let $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and\n$Z = \\Spec(A/I)$. For any sheaf $\\mathcal{F}$ on $X_\\etale$\nwe have $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, \\mathcal{F}|_Z)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Affine analog of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZH","source_file":"etale-cohomology.tex","source_line":15179,"source_end_line":15184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15179-L15184","statement_sha256":"4086fba7f94b802dc1d47be794cdede16b9d45b74b570d48a06f7c1cb303bb9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10110,"rank":10110,"depth":51,"x":1703.104,"y":1114.637,"cluster":"tale-geometry"},{"id":"stacks:09ZI","tag":"09ZI","title":"Gabber · Theorem 09ZI","summary":"Let (A, I) be a henselian pair. Set X = Spec(A) and Z = Spec(A/I). For any torsion abelian sheaf F on X_etale we have H^q_etale(X, F) = H^q_etale(Z, F|_Z).","statement_latex":"Let $(A, I)$ be a henselian pair. Set $X = \\Spec(A)$ and\n$Z = \\Spec(A/I)$. For any torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$\nwe have $H^q_\\etale(X, \\mathcal{F}) = H^q_\\etale(Z, \\mathcal{F}|_Z)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Affine analog of proper base change","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZI","source_file":"etale-cohomology.tex","source_line":15204,"source_end_line":15209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15204-L15209","statement_sha256":"82040f52d88ad745239140b6e394f55dac4be2875230d6c50a9bb6999f6f4fae","origin":"The Stacks Project","memory_eligible":false,"source_rank":10111,"rank":10111,"depth":56,"x":1509.549,"y":1228.645,"cluster":"tale-geometry"},{"id":"stacks:0A51","tag":"0A51","title":"Affine analog of proper base change · Lemma 0A51","summary":"Let X be a scheme with affine diagonal which can be covered by n + 1 affine opens. Let Z ⊂ X be a closed subscheme. Let A be a torsion sheaf of rings on X_etale and let I be an injective sheaf of A-modules on X_etale. Then H^q_etale(Z, I|_Z) = 0 for q > n.","statement_latex":"Let $X$ be a scheme with affine diagonal which can be covered by\n$n + 1$ affine opens. Let $Z \\subset X$ be a closed subscheme.\nLet $\\mathcal{A}$ be a torsion sheaf of rings on $X_\\etale$\nand let $\\mathcal{I}$ be an injective sheaf of $\\mathcal{A}$-modules\non $X_\\etale$.\nThen $H^q_\\etale(Z, \\mathcal{I}|_Z) = 0$ for $q > n$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Affine analog of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A51","source_file":"etale-cohomology.tex","source_line":15258,"source_end_line":15266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15258-L15266","statement_sha256":"ce2b43c26e17f25667982e4bae8f8d292b5c0afa5d0e8465bfdf1569bf82eea8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10112,"rank":10112,"depth":57,"x":1560.644,"y":1034.552,"cluster":"tale-geometry"},{"id":"stacks:0A5B","tag":"0A5B","title":"Cohomology of torsion sheaves on curves · Lemma 0A5B","summary":"In Situation [Tag 0A52] assume X is smooth and F = underlineZ/ℓZ for some prime number ℓ. Then statements ([Tag 0A53]) -- ([Tag 0A5A]) hold for F.","statement_latex":"In Situation \\ref{situation-what-to-prove}\nassume $X$ is smooth and $\\mathcal{F} = \\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$\nfor some prime number $\\ell$. Then statements\n(\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold\nfor $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5B","source_file":"etale-cohomology.tex","source_line":15376,"source_end_line":15383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15376-L15383","statement_sha256":"4d82dd21235ad0bd91ed3ba47b00cf871e16e22f336359e13b28c407843e31de","origin":"The Stacks Project","memory_eligible":false,"source_rank":10113,"rank":10113,"depth":68,"x":1679.172,"y":1206.841,"cluster":"tale-geometry"},{"id":"stacks:0A5C","tag":"0A5C","title":"Cohomology of torsion sheaves on curves · Lemma 0A5C","summary":"Let k be an algebraically closed field. Let X be a separated finite type scheme over k of dimension ≤ 1. Let 0 → F_1 → F → F_2 → 0 be a short exact sequence of torsion abelian sheaves on X. If statements ([Tag 0A53]) -- ([Tag 0A5A]) hold for F_1 and F_2, then they hold for F.","statement_latex":"Let $k$ be an algebraically closed field. Let $X$ be a separated finite\ntype scheme over $k$ of dimension $\\leq 1$. Let\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0$\nbe a short exact sequence of torsion abelian sheaves on $X$.\nIf statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold\nfor $\\mathcal{F}_1$ and $\\mathcal{F}_2$, then they hold\nfor $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5C","source_file":"etale-cohomology.tex","source_line":15449,"source_end_line":15458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15449-L15458","statement_sha256":"6d5aa865bbd3e51c788481a1e4ea84044ba5020ec6c4ab67c835441ba6ffa2da","origin":"The Stacks Project","memory_eligible":false,"source_rank":10114,"rank":10114,"depth":5,"x":1452.992,"y":1146.993,"cluster":"tale-geometry"},{"id":"stacks:0A5D","tag":"0A5D","title":"Cohomology of torsion sheaves on curves · Lemma 0A5D","summary":"Let k be an algebraically closed field. Let f : X → Y be a finite morphism of separated finite type schemes over k of dimension ≤ 1. Let F be a torsion abelian sheaf on X. If statements ([Tag 0A53]) -- ([Tag 0A5A]) hold for F, then they hold for f_*F.","statement_latex":"Let $k$ be an algebraically closed field. Let $f : X \\to Y$ be a\nfinite morphism of separated finite type schemes over $k$ of\ndimension $\\leq 1$. Let $\\mathcal{F}$ be a torsion abelian sheaf on $X$.\nIf statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold\nfor $\\mathcal{F}$, then they hold for $f_*\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5D","source_file":"etale-cohomology.tex","source_line":15470,"source_end_line":15477,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15470-L15477","statement_sha256":"597ec26d42183703ef9af9218bbc6ea2767bd05aaa7561be2dc0c66d1161c2f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10115,"rank":10115,"depth":53,"x":1668.12,"y":1062.698,"cluster":"tale-geometry"},{"id":"stacks:0GJA","tag":"0GJA","title":"Cohomology of torsion sheaves on curves · Lemma 0GJA","summary":"In Situation [Tag 0A52] assume F constructible. Let j : X' → X be the inclusion of a dense open subscheme. Then statements ([Tag 0A53]) -- ([Tag 0A5A]) hold for F if and only if they hold for j_!j^-1F.","statement_latex":"In Situation \\ref{situation-what-to-prove} assume $\\mathcal{F}$\nconstructible. Let $j : X' \\to X$ be the inclusion of a dense open subscheme.\nThen statements\n(\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold for $\\mathcal{F}$\nif and only if they hold for $j_!j^{-1}\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJA","source_file":"etale-cohomology.tex","source_line":15490,"source_end_line":15497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15490-L15497","statement_sha256":"9a191be76559e0c5daa5d10bb710d236fc85199e1d04fa5c1ddb03ba982deb32","origin":"The Stacks Project","memory_eligible":false,"source_rank":10116,"rank":10116,"depth":52,"x":1577.183,"y":1247.111,"cluster":"tale-geometry"},{"id":"stacks:03SG","tag":"03SG","title":"Cohomology of torsion sheaves on curves · Lemma 03SG","summary":"In Situation [Tag 0A52] assume X is smooth. Let j : U → X an open immersion. Let ℓ be a prime number. Let F = j_!underlineZ/ℓZ. Then statements ([Tag 0A53]) -- ([Tag 0A5A]) hold for F.","statement_latex":"In Situation \\ref{situation-what-to-prove} assume $X$ is smooth.\nLet $j : U \\to X$ an open immersion. Let $\\ell$ be a prime number.\nLet $\\mathcal{F} = j_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$.\nThen statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold\nfor $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SG","source_file":"etale-cohomology.tex","source_line":15547,"source_end_line":15554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15547-L15554","statement_sha256":"1e57a443964f4c32e7a492fc0028ad4f5425361896c92e332442b9f481516d1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10117,"rank":10117,"depth":69,"x":1495.857,"y":1059.339,"cluster":"tale-geometry"},{"id":"stacks:0A3Q","tag":"0A3Q","title":"Cohomology of torsion sheaves on curves · Lemma 0A3Q","summary":"In Situation [Tag 0A52] assume X reduced. Let j : U → X an open immersion. Let ℓ be a prime number and F = j_!underlineZ/ℓZ. Then statements ([Tag 0A53]) -- ([Tag 0A5A]) hold for F.","statement_latex":"In Situation \\ref{situation-what-to-prove} assume $X$ reduced.\nLet $j : U \\to X$ an open immersion. Let $\\ell$ be a prime number\nand $\\mathcal{F} = j_!\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$.\nThen statements (\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold\nfor $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3Q","source_file":"etale-cohomology.tex","source_line":15565,"source_end_line":15572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15565-L15572","statement_sha256":"33fa276e1cea5dafa79ad32723c9a37ec81dba9cf5d9dc9e4411624dc44b6a99","origin":"The Stacks Project","memory_eligible":false,"source_rank":10118,"rank":10118,"depth":70,"x":1707.039,"y":1151.746,"cluster":"tale-geometry"},{"id":"stacks:03SD","tag":"03SD","title":"Cohomology of torsion sheaves on curves · Lemma 03SD","summary":"In Situation [Tag 0A52] assume X reduced. Let j : U → X an open immersion with U connected. Let ℓ be a prime number. Let G a finite locally constant sheaf of F_ℓ-vector spaces on U. Let F = j_!G. Then statements ([Tag 0A53]) -- ([Tag 0A5A]) hold for F.","statement_latex":"In Situation \\ref{situation-what-to-prove} assume $X$ reduced.\nLet $j : U \\to X$ an open immersion with $U$ connected. Let\n$\\ell$ be a prime number. Let $\\mathcal{G}$ a finite locally\nconstant sheaf of $\\mathbf{F}_\\ell$-vector spaces on $U$. Let\n$\\mathcal{F} = j_!\\mathcal{G}$. Then statements\n(\\ref{item-vanishing}) -- (\\ref{item-surjective}) hold for $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SD","source_file":"etale-cohomology.tex","source_line":15604,"source_end_line":15612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15604-L15612","statement_sha256":"c41afc24302ac66d5695dbad8afe2ed428671cd0ebfb1624bd1e6e9ba7e641f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10119,"rank":10119,"depth":71,"x":1476.774,"y":1203.488,"cluster":"tale-geometry"},{"id":"stacks:03SC","tag":"03SC","title":"Cohomology of torsion sheaves on curves · Theorem 03SC","summary":"If k is an algebraically closed field, X is a separated, finite type scheme of dimension ≤ 1 over k, and F is a torsion abelian sheaf on X_etale, then • H^q_etale(X, F) = 0 for q > 2, • H^q_etale(X, F) = 0 for q > 1 if X is affine, • H^q_etale(X, F) = 0 for q > 1 if p = char(k) > 0 and F is p-power torsion, • H^q_etale(X, F) is finite if F is constructible and torsion prime to char(k), • H^q_etale(X, F) is finite if X is proper and F constructible, • H^q_etale(X, F) →…","statement_latex":"If $k$ is an algebraically closed field, $X$ is a separated, finite type\nscheme of dimension $\\leq 1$ over $k$, and $\\mathcal{F}$ is a torsion\nabelian sheaf on $X_\\etale$, then\n\\begin{enumerate}\n\\item\n$H^q_\\etale(X, \\mathcal{F}) = 0$ for $q > 2$,\n\\item\n$H^q_\\etale(X, \\mathcal{F}) = 0$ for $q > 1$ if $X$ is affine,\n\\item\n$H^q_\\etale(X, \\mathcal{F}) = 0$ for $q > 1$ if $p = \\text{char}(k) > 0$\nand $\\mathcal{F}$ is $p$-power torsion,\n\\item\n$H^q_\\etale(X, \\mathcal{F})$ is finite if $\\mathcal{F}$ is\nconstructible and torsion prime to $\\text{char}(k)$,\n\\item\n$H^q_\\etale(X, \\mathcal{F})$ is finite if $X$ is proper and\n$\\mathcal{F}$ constructible,\n\\item\n$H^q_\\etale(X, \\mathcal{F}) \\to\nH^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism\nfor any extension $k'/k$ of algebraically closed fields\nif $\\mathcal{F}$ is torsion prime to $\\text{char}(k)$,\n\\item\n$H^q_\\etale(X, \\mathcal{F}) \\to\nH^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism\nfor any extension $k'/k$ of algebraically closed fields\nif $X$ is proper,\n\\item\n$H^2_\\etale(X, \\mathcal{F}) \\to H^2_\\etale(U, \\mathcal{F})$\nis surjective for all $U \\subset X$ open.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SC","source_file":"etale-cohomology.tex","source_line":15655,"source_end_line":15688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15655-L15688","statement_sha256":"1fdf92a0b5f44419e30ba395af40347fa58fd412be6de1e8149e01fb850be67d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10120,"rank":10120,"depth":72,"x":1605.087,"y":1034.507,"cluster":"tale-geometry"},{"id":"stacks:03RT","tag":"03RT","title":"Cohomology of torsion sheaves on curves · Theorem 03RT","summary":"Let X be a finite type, dimension 1 scheme over an algebraically closed field k. Let F be a torsion sheaf on X_etale. Then H_etale^q(X, F) = 0, ∀ q ≥ 3. If X affine then also H_etale^2(X, F) = 0.","statement_latex":"Let $X$ be a finite type, dimension $1$ scheme over an\nalgebraically closed field $k$. Let $\\mathcal{F}$ be a torsion sheaf\non $X_\\etale$. Then\n$$\nH_\\etale^q(X, \\mathcal{F}) = 0, \\quad \\forall q \\geq 3.\n$$\nIf $X$ affine then also $H_\\etale^2(X, \\mathcal{F}) = 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03RT","source_file":"etale-cohomology.tex","source_line":15739,"source_end_line":15748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15739-L15748","statement_sha256":"7899db251d5543054a81683900b19e0f85d76649e5e7b6193c2e99504d34de09","origin":"The Stacks Project","memory_eligible":false,"source_rank":10121,"rank":10121,"depth":73,"x":1646.402,"y":1232.116,"cluster":"tale-geometry"},{"id":"stacks:0A5E","tag":"0A5E","title":"Cohomology of torsion sheaves on curves · Lemma 0A5E","summary":"Let k'/k be an extension of separably closed fields. Let X be a proper scheme over k of dimension ≤ 1. Let F be a torsion abelian sheaf on X. Then the map H^q_etale(X, F) → H^q_etale(X_k', F|_X_k') is an isomorphism for q ≥ 0.","statement_latex":"Let $k'/k$ be an extension of separably closed fields.\nLet $X$ be a proper scheme over $k$ of dimension $\\leq 1$.\nLet $\\mathcal{F}$ be a torsion abelian sheaf on $X$.\nThen the map $H^q_\\etale(X, \\mathcal{F}) \\to\nH^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism\nfor $q \\geq 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion sheaves on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5E","source_file":"etale-cohomology.tex","source_line":15768,"source_end_line":15776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15768-L15776","statement_sha256":"49389cfd3c414613d2062db1dd43149b96009886fcd938cbda855d9f270c5fc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10122,"rank":10122,"depth":73,"x":1456.836,"y":1109.723,"cluster":"tale-geometry"},{"id":"stacks:0GJC","tag":"0GJC","title":"Cohomology of torsion modules on curves · Lemma 0GJC","summary":"Let Lambda be a Noetherian ring, let M be a finite Lambda-module which is annihilated by an integer n > 0, let k be an algebraically closed field, and let X be a separated, finite type scheme of dimension ≤ 1 over k. Then • H^q_etale(X, underlineM) is a finite Lambda-module if n is prime to char(k), • H^q_etale(X, underlineM) is a finite Lambda-module if X is proper.","statement_latex":"Let $\\Lambda$ be a Noetherian ring, let $M$ be a finite $\\Lambda$-module which\nis annihilated by an integer $n > 0$, let $k$ be an algebraically closed field,\nand let $X$ be a separated, finite type scheme of dimension $\\leq 1$ over $k$.\nThen\n\\begin{enumerate}\n\\item $H^q_\\etale(X, \\underline{M})$ is a finite $\\Lambda$-module\nif $n$ is prime to $\\text{char}(k)$,\n\\item $H^q_\\etale(X, \\underline{M})$ is a finite $\\Lambda$-module\nif $X$ is proper.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion modules on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJC","source_file":"etale-cohomology.tex","source_line":15817,"source_end_line":15829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15817-L15829","statement_sha256":"8d0adc11bfb875e9733617d90d83bbaf7fedd7ed1a7cd7012111354bedbc1b77","origin":"The Stacks Project","memory_eligible":false,"source_rank":10123,"rank":10123,"depth":73,"x":1695.282,"y":1092.392,"cluster":"tale-geometry"},{"id":"stacks:0GJD","tag":"0GJD","title":"Cohomology of torsion modules on curves · Lemma 0GJD","summary":"Let Lambda be a Noetherian ring, let k be an algebraically closed field, let f : X → Y be a finite morphism of separated finite type schemes over k of dimension ≤ 1, and let F be a sheaf of Lambda-modules on X_etale. If H^q_etale(X, F) is a finite Lambda-module, then so is H^q_etale(Y, f_*F).","statement_latex":"Let $\\Lambda$ be a Noetherian ring, let $k$ be an algebraically closed field,\nlet $f : X \\to Y$ be a finite morphism of separated finite type schemes\nover $k$ of dimension $\\leq 1$, and let $\\mathcal{F}$ be a sheaf\nof $\\Lambda$-modules on $X_\\etale$. If\n$H^q_\\etale(X, \\mathcal{F})$ is a finite $\\Lambda$-module, then\nso is $H^q_\\etale(Y, f_*\\mathcal{F})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion modules on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJD","source_file":"etale-cohomology.tex","source_line":15908,"source_end_line":15916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15908-L15916","statement_sha256":"501f2c57de5978ff2bd67ef20a7a20ef62516fde606a646ac926340390ec8938","origin":"The Stacks Project","memory_eligible":false,"source_rank":10124,"rank":10124,"depth":52,"x":1533.232,"y":1240.619,"cluster":"tale-geometry"},{"id":"stacks:0GJE","tag":"0GJE","title":"Cohomology of torsion modules on curves · Lemma 0GJE","summary":"Let Lambda be a Noetherian ring, let k be an algebraically closed field, let X be a separated finite type scheme over k of dimension ≤ 1, let F be a constructible sheaf of Lambda-modules on X_etale, and let j : X' → X be the inclusion of a dense open subscheme. Then H^q_etale(X, F) is a finite Lambda-module if and only if H^q_etale(X, j_!j^-1F) is a finite Lambda-module.","statement_latex":"Let $\\Lambda$ be a Noetherian ring, let $k$ be an algebraically closed field,\nlet $X$ be a separated finite type scheme over $k$ of dimension $\\leq 1$,\nlet $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on $X_\\etale$,\nand let $j : X' \\to X$ be the inclusion of a dense open subscheme. Then\n$H^q_\\etale(X, \\mathcal{F})$ is a finite $\\Lambda$-module if and only if\n$H^q_\\etale(X, j_!j^{-1}\\mathcal{F})$ is a finite $\\Lambda$-module.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion modules on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJE","source_file":"etale-cohomology.tex","source_line":15926,"source_end_line":15934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15926-L15934","statement_sha256":"ce23edfeba4f743e0c9273295544b0fd8646f737834e55f8568ba26c4f19c910","origin":"The Stacks Project","memory_eligible":false,"source_rank":10125,"rank":10125,"depth":52,"x":1533.526,"y":1039.168,"cluster":"tale-geometry"},{"id":"stacks:0GJF","tag":"0GJF","title":"Cohomology of torsion modules on curves · Lemma 0GJF","summary":"Let Lambda be a Noetherian ring, let M be a finite Lambda-module which is annihilated by an integer n > 0, let k be an algebraically closed field, let X be a separated, finite type scheme of dimension ≤ 1 over k, and let j : U → X be an open immersion. Then • H^q_etale(X, j_!underlineM) is a finite Lambda-module if n is prime to char(k), • H^q_etale(X, j_!underlineM) is a finite Lambda-module if X is proper.","statement_latex":"Let $\\Lambda$ be a Noetherian ring, let $M$ be a finite $\\Lambda$-module which\nis annihilated by an integer $n > 0$, let $k$ be an algebraically closed field,\nlet $X$ be a separated, finite type scheme of dimension $\\leq 1$ over $k$, and\nlet $j : U \\to X$ be an open immersion. Then\n\\begin{enumerate}\n\\item $H^q_\\etale(X, j_!\\underline{M})$ is a finite $\\Lambda$-module\nif $n$ is prime to $\\text{char}(k)$,\n\\item $H^q_\\etale(X, j_!\\underline{M})$ is a finite $\\Lambda$-module\nif $X$ is proper.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion modules on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJF","source_file":"etale-cohomology.tex","source_line":15972,"source_end_line":15984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15972-L15984","statement_sha256":"b38b74415fd193c49f6b4c59f08f902777f4091ea77ee90bcd78e04b3a6398b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10126,"rank":10126,"depth":74,"x":1695.47,"y":1188.029,"cluster":"tale-geometry"},{"id":"stacks:0GJG","tag":"0GJG","title":"Cohomology of torsion modules on curves · Lemma 0GJG","summary":"Let Lambda be a Noetherian ring, let k be an algebraically closed field, let X be a separated finite type scheme over k of dimension ≤ 1, and let 0 → F_1 → F → F_2 → 0 be a short exact sequence of sheaves of Lambda-modules on X_etale. If H^q_etale(X, F_i), i = 1, 2 are finite Lambda-modules then H^q_etale(X, F) is a finite Lambda-module.","statement_latex":"Let $\\Lambda$ be a Noetherian ring, let $k$ be an algebraically closed field,\nlet $X$ be a separated finite type scheme over $k$ of dimension $\\leq 1$, and\nlet $0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0$\nbe a short exact sequence of sheaves of $\\Lambda$-modules on $X_\\etale$. If\n$H^q_\\etale(X, \\mathcal{F}_i)$, $i = 1, 2$ are finite $\\Lambda$-modules\nthen $H^q_\\etale(X, \\mathcal{F})$ is a finite $\\Lambda$-module.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion modules on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJG","source_file":"etale-cohomology.tex","source_line":15996,"source_end_line":16004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L15996-L16004","statement_sha256":"299b921a5eb1ef3609529e2341b6c7061d949b0800c4961c5fc269c0a07b0836","origin":"The Stacks Project","memory_eligible":false,"source_rank":10127,"rank":10127,"depth":0,"x":1456.109,"y":1170.134,"cluster":"tale-geometry"},{"id":"stacks:0GJH","tag":"0GJH","title":"Cohomology of torsion modules on curves · Lemma 0GJH","summary":"Let Lambda be a Noetherian ring, let k be an algebraically closed field, let X be a separated, finite type scheme of dimension ≤ 1 over k, let j : U → X be an open immersion with U connected, let ℓ be a prime number, let n > 0, and let G be a finite type, locally constant sheaf of Lambda-modules on U_etale annihilated by ℓ^n. Then • H^q_etale(X, j_!G) is a finite Lambda-module if ℓ is prime to char(k), • H^q_etale(X, j_!G) is a finite Lambda-module if X is proper.","statement_latex":"Let $\\Lambda$ be a Noetherian ring, let $k$ be an algebraically closed field,\nlet $X$ be a separated, finite type scheme of dimension $\\leq 1$ over $k$,\nlet $j : U \\to X$ be an open immersion with $U$ connected, let\n$\\ell$ be a prime number, let $n > 0$, and let $\\mathcal{G}$ be a finite type,\nlocally constant sheaf of $\\Lambda$-modules on $U_\\etale$ annihilated by\n$\\ell^n$. Then\n\\begin{enumerate}\n\\item $H^q_\\etale(X, j_!\\mathcal{G})$ is a finite $\\Lambda$-module\nif $\\ell$ is prime to $\\text{char}(k)$,\n\\item $H^q_\\etale(X, j_!\\mathcal{G})$ is a finite $\\Lambda$-module\nif $X$ is proper.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion modules on curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJH","source_file":"etale-cohomology.tex","source_line":16010,"source_end_line":16024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16010-L16024","statement_sha256":"fcc0372cac005a61bcbcabd72d33f65ed7a2d6212daa1c369dd8978166a6b68c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10128,"rank":10128,"depth":75,"x":1647.188,"y":1047.389,"cluster":"tale-geometry"},{"id":"stacks:0GJI","tag":"0GJI","title":"Cohomology of torsion modules on curves · Theorem 0GJI","summary":"Let Lambda be a Noetherian ring, let k be an algebraically closed field, let X be a separated, finite type scheme of dimension ≤ 1 over k, and let F be a constructible sheaf of Lambda-modules on X_etale which is torsion. Then • H^q_etale(X, F) is a finite Lambda-module if F is torsion prime to char(k), • H^q_etale(X, F) is a finite Lambda-module if X is proper.","statement_latex":"Let $\\Lambda$ be a Noetherian ring, let $k$ be an algebraically closed field,\nlet $X$ be a separated, finite type scheme of dimension $\\leq 1$ over $k$,\nand let $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules\non $X_\\etale$ which is torsion. Then\n\\begin{enumerate}\n\\item\n\n$H^q_\\etale(X, \\mathcal{F})$ is a finite $\\Lambda$-module\nif $\\mathcal{F}$ is torsion prime to $\\text{char}(k)$,\n\\item\n\n$H^q_\\etale(X, \\mathcal{F})$ is a finite $\\Lambda$-module if $X$ is proper.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of torsion modules on curves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJI","source_file":"etale-cohomology.tex","source_line":16061,"source_end_line":16076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16061-L16076","statement_sha256":"aca020897adc7220d109217e16d9f9ed5f1c81b208b91ad66fa475c710493f74","origin":"The Stacks Project","memory_eligible":false,"source_rank":10129,"rank":10129,"depth":76,"x":1604.956,"y":1246.519,"cluster":"tale-geometry"},{"id":"stacks:0A5G","tag":"0A5G","title":"First cohomology of proper schemes · Lemma 0A5G","summary":"Let A be a henselian local ring. Let X be a proper scheme over A with closed fibre X_0. Let M be a finite abelian group. Then H^1_etale(X, underlineM) = H^1_etale(X_0, underlineM).","statement_latex":"Let $A$ be a henselian local ring. Let $X$ be a proper scheme over $A$\nwith closed fibre $X_0$. Let $M$ be a finite abelian group.\nThen $H^1_\\etale(X, \\underline{M}) = H^1_\\etale(X_0, \\underline{M})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"First cohomology of proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5G","source_file":"etale-cohomology.tex","source_line":16133,"source_end_line":16138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16133-L16138","statement_sha256":"cfab1e2680062aad40d781166d6ee07d1c9ae9b4f04c3d4fe1dfce8d7998f426","origin":"The Stacks Project","memory_eligible":false,"source_rank":10130,"rank":10130,"depth":54,"x":1475.837,"y":1075.552,"cluster":"tale-geometry"},{"id":"stacks:0A5H","tag":"0A5H","title":"First cohomology of proper schemes · Lemma 0A5H","summary":"Let A be a henselian local ring. Let X = P^1_A. Let X_0 ⊂ X be the closed fibre. Let ℓ be a prime number. Let I be an injective sheaf of Z/ℓZ-modules on X_etale. Then H^q_etale(X_0, I|_X_0) = 0 for q > 0.","statement_latex":"Let $A$ be a henselian local ring. Let $X = \\mathbf{P}^1_A$.\nLet $X_0 \\subset X$ be the closed fibre. Let $\\ell$ be a prime\nnumber. Let $\\mathcal{I}$ be an injective sheaf of\n$\\mathbf{Z}/\\ell\\mathbf{Z}$-modules on $X_\\etale$. Then\n$H^q_\\etale(X_0, \\mathcal{I}|_{X_0}) = 0$ for $q > 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"First cohomology of proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A5H","source_file":"etale-cohomology.tex","source_line":16164,"source_end_line":16171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16164-L16171","statement_sha256":"379df3b53383f6153af2cec7f76609ba486aa3877b11f1e0701737c30217aba2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10131,"rank":10131,"depth":58,"x":1708.762,"y":1128.407,"cluster":"tale-geometry"},{"id":"stacks:0EZR","tag":"0EZR","title":"Preliminaries on base change · Lemma 0EZR","summary":"Consider a cartesian diagram of schemes xymatrix X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g Let (U_i → X) be an étale covering such that U_i → S factors as U_i → V_i → S with V_i → S étale and consider the cartesian diagrams xymatrix U_i ar[d]_f_i & U_i ×_X Y ar[l]^h_i ar[d]^e_i V_i & V_i ×_S T ar[l]_g_i Let F be a sheaf on T_etale. Let K in D(T_etale). Set K_i = K|_V_i ×_S T and F_i = F|_V_i ×_S T. • If f_i^-1g_i, *F_i = h_i, *e_i^-1F_i for all i, then f^-1g_*F =…","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nLet $\\{U_i \\to X\\}$ be an \\'etale covering such that $U_i \\to S$\nfactors as $U_i \\to V_i \\to S$ with $V_i \\to S$ \\'etale\nand consider the cartesian diagrams\n$$\n\\xymatrix{\nU_i \\ar[d]_{f_i} & U_i \\times_X Y \\ar[l]^{h_i} \\ar[d]^{e_i} \\\\\nV_i & V_i \\times_S T \\ar[l]_{g_i}\n}\n$$\nLet $\\mathcal{F}$ be a sheaf on $T_\\etale$. Let $K$ in $D(T_\\etale)$.\nSet $K_i = K|_{V_i \\times_S T}$ and\n$\\mathcal{F}_i = \\mathcal{F}|_{V_i \\times_S T}$.\n\\begin{enumerate}\n\\item If $f_i^{-1}g_{i, *}\\mathcal{F}_i = h_{i, *}e_i^{-1}\\mathcal{F}_i$\nfor all $i$, then $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$.\n\\item If $f_i^{-1}Rg_{i, *}K_i = Rh_{i, *}e_i^{-1}K_i$\nfor all $i$, then $f^{-1}Rg_*K = Rh_*e^{-1}K$.\n\\item If $\\mathcal{F}$ is an abelian sheaf and\n$f_i^{-1}R^qg_{i, *}\\mathcal{F}_i = R^qh_{i, *}e_i^{-1}\\mathcal{F}_i$\nfor all $i$, then\n$f^{-1}R^qg_*\\mathcal{F} = R^qh_*e^{-1}\\mathcal{F}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Preliminaries on base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZR","source_file":"etale-cohomology.tex","source_line":16306,"source_end_line":16337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16306-L16337","statement_sha256":"fa50ba4ffb521e3567a62f3a36b7128ee045aa60a97e243e7918edff7fd6e7cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10132,"rank":10132,"depth":11,"x":1494.292,"y":1221.691,"cluster":"tale-geometry"},{"id":"stacks:0EZS","tag":"0EZS","title":"Preliminaries on base change · Lemma 0EZS","summary":"Consider a tower of cartesian diagrams of schemes xymatrix W ar[d]_i & Z ar[l]^j ar[d]^k X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g Let K in D(T_etale). If f^-1Rg_*K → Rh_*e^-1K and i^-1Rh_*e^-1K → Rj_*k^-1e^-1K are isomorphisms, then (f ∘ i)^-1Rg_*K → Rj_*(e ∘ k)^-1K is an isomorphism. Similarly, if F is an abelian sheaf on T_etale and if f^-1R^qg_*F → R^qh_*e^-1F and i^-1R^qh_*e^-1F → R^qj_*k^-1e^-1F are isomorphisms, then (f ∘ i)^-1R^qg_*F → R^qj_*(e ∘ k)^-1F is an…","statement_latex":"Consider a tower of cartesian diagrams of schemes\n$$\n\\xymatrix{\nW \\ar[d]_i & Z \\ar[l]^j \\ar[d]^k \\\\\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nLet $K$ in $D(T_\\etale)$. If\n$$\nf^{-1}Rg_*K \\to Rh_*e^{-1}K\n\\quad\\text{and}\\quad\ni^{-1}Rh_*e^{-1}K \\to Rj_*k^{-1}e^{-1}K\n$$\nare isomorphisms, then\n$(f \\circ i)^{-1}Rg_*K \\to Rj_*(e \\circ k)^{-1}K$\nis an isomorphism.\nSimilarly, if $\\mathcal{F}$ is an abelian sheaf on $T_\\etale$ and if\n$$\nf^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}\n\\quad\\text{and}\\quad\ni^{-1}R^qh_*e^{-1}\\mathcal{F} \\to R^qj_*k^{-1}e^{-1}\\mathcal{F}\n$$\nare isomorphisms, then\n$(f \\circ i)^{-1}R^qg_*\\mathcal{F} \\to R^qj_*(e \\circ k)^{-1}\\mathcal{F}$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Preliminaries on base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZS","source_file":"etale-cohomology.tex","source_line":16372,"source_end_line":16400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16372-L16400","statement_sha256":"656142ca79e9af690243416ac14a637c33bb1855d026e58ddf45e61e26c97914","origin":"The Stacks Project","memory_eligible":false,"source_rank":10133,"rank":10133,"depth":0,"x":1577.503,"y":1031.024,"cluster":"tale-geometry"},{"id":"stacks:0EZT","tag":"0EZT","title":"Preliminaries on base change · Lemma 0EZT","summary":"Let I be a directed set. Consider an inverse system of cartesian diagrams of schemes xymatrix X_i ar[d]_f_i & Y_i ar[l]^h_i ar[d]^e_i S_i & T_i ar[l]_g_i with affine transition morphisms and with g_i quasi-compact and quasi-separated. Set X = lim X_i, S = lim S_i, T = lim T_i and Y = lim Y_i to obtain the cartesian diagram xymatrix X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g Let (F_i, φ_i'i) be a system of sheaves on (T_i) as in Definition [Tag 0EZL]. Set F = colim…","statement_latex":"Let $I$ be a directed set. Consider an inverse system of\ncartesian diagrams of schemes\n$$\n\\xymatrix{\nX_i \\ar[d]_{f_i} & Y_i \\ar[l]^{h_i} \\ar[d]^{e_i} \\\\\nS_i & T_i \\ar[l]_{g_i}\n}\n$$\nwith affine transition morphisms and with $g_i$ quasi-compact and\nquasi-separated. Set $X = \\lim X_i$,\n$S = \\lim S_i$, $T = \\lim T_i$ and $Y = \\lim Y_i$ to\nobtain the cartesian diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nLet $(\\mathcal{F}_i, \\varphi_{i'i})$ be a system of sheaves on\n$(T_i)$ as in Definition \\ref{definition-inverse-system-sheaves}. Set\n$\\mathcal{F} = \\colim p_i^{-1}\\mathcal{F}_i$ on $T$\nwhere $p_i : T \\to T_i$ is the projection.\nThen we have the following\n\\begin{enumerate}\n\\item If $f_i^{-1}g_{i, *}\\mathcal{F}_i = h_{i, *}e_i^{-1}\\mathcal{F}_i$\nfor all $i$, then\n$f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$.\n\\item If $\\mathcal{F}_i$ is an abelian sheaf for all $i$ and\n$f_i^{-1}R^qg_{i, *}\\mathcal{F}_i = R^qh_{i, *}e_i^{-1}\\mathcal{F}_i$\nfor all $i$, then\n$f^{-1}R^qg_*\\mathcal{F} = R^qh_*e^{-1}\\mathcal{F}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Preliminaries on base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZT","source_file":"etale-cohomology.tex","source_line":16409,"source_end_line":16443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16409-L16443","statement_sha256":"5eee98d65a2a66009d369cc3f533e41dc1b198ab72dab075ea1f31b945cbe73a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10134,"rank":10134,"depth":43,"x":1669.564,"y":1219.018,"cluster":"tale-geometry"},{"id":"stacks:0GJL","tag":"0GJL","title":"Preliminaries on base change · Lemma 0GJL","summary":"Let I, X_i, Y_i, S_i, T_i, f_i, h_i, e_i, g_i, X, Y, S, T, f, h, e, g be as in the statement of Lemma [Tag 0EZT]. Let 0 ∈ I and let K_0 ∈ D^+(T_0, etale). For i ∈ I, i ≥ 0 denote K_i the pullback of K_0 to T_i. Denote K the pullback of K_0 to T. If f_i^-1Rg_i, *K_i = Rh_i, *e_i^-1K_i for all i ≥ 0, then f^-1Rg_*K = Rh_*e^-1K.","statement_latex":"Let $I$, $X_i$, $Y_i$, $S_i$, $T_i$, $f_i$, $h_i$, $e_i$, $g_i$,\n$X$, $Y$, $S$, $T$, $f$, $h$, $e$, $g$ be as in the statement\nof Lemma \\ref{lemma-base-change-Rf-star-colim}.\nLet $0 \\in I$ and let $K_0 \\in D^+(T_{0, \\etale})$.\nFor $i \\in I$, $i \\geq 0$ denote $K_i$ the pullback of\n$K_0$ to $T_i$. Denote $K$ the pullback of $K_0$ to $T$.\nIf $f_i^{-1}Rg_{i, *}K_i = Rh_{i, *}e_i^{-1}K_i$\nfor all $i \\geq 0$, then $f^{-1}Rg_*K = Rh_*e^{-1}K$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Preliminaries on base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJL","source_file":"etale-cohomology.tex","source_line":16468,"source_end_line":16478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16468-L16478","statement_sha256":"b8029d6ea4b38f0e7cb68f6beb3eff97e848a38481b77f4443605afa8387e191","origin":"The Stacks Project","memory_eligible":false,"source_rank":10135,"rank":10135,"depth":44,"x":1450.287,"y":1132.547,"cluster":"tale-geometry"},{"id":"stacks:0EZU","tag":"0EZU","title":"Preliminaries on base change · Lemma 0EZU","summary":"Consider a cartesian diagram of schemes xymatrix X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g where g : T → S is quasi-compact and quasi-separated. Let F be an abelian sheaf on T_etale. Let q ≥ 0. The following are equivalent • For every geometric point overlinex of X with image overlines = f(overlinex) we have H^q(Spec(O^sh_X, overlinex) ×_S T, F) = H^q(Spec(O^sh_S, overlines) ×_S T, F) • f^-1R^qg_*F → R^qh_*e^-1F is an isomorphism.","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nwhere $g : T \\to S$ is quasi-compact and quasi-separated.\nLet $\\mathcal{F}$ be an\nabelian sheaf on $T_\\etale$. Let $q \\geq 0$. The following are equivalent\n\\begin{enumerate}\n\\item For every geometric point $\\overline{x}$ of $X$ with image\n$\\overline{s} = f(\\overline{x})$ we have\n$$\nH^q(\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_S T, \\mathcal{F})\n=\nH^q(\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}}) \\times_S T, \\mathcal{F})\n$$\n\\item $f^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$\nis an isomorphism.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Preliminaries on base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZU","source_file":"etale-cohomology.tex","source_line":16496,"source_end_line":16519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16496-L16519","statement_sha256":"345cc6c2493bd2ecf46244f25d03260b4594a1978ccf8b58a313add7bb84e694","origin":"The Stacks Project","memory_eligible":false,"source_rank":10136,"rank":10136,"depth":52,"x":1681.74,"y":1071.828,"cluster":"tale-geometry"},{"id":"stacks:0EZV","tag":"0EZV","title":"Preliminaries on base change · Lemma 0EZV","summary":"Let f : X → S be a morphism of schemes. Let overlinex be a geometric point of X with image overlines in S. Let Spec(K) → Spec(O^sh_S, overlines) be a morphism with K a separably closed field. Let F be an abelian sheaf on Spec(K)_etale. Let q ≥ 0. The following are equivalent • H^q(Spec(O^sh_X, overlinex) ×_S Spec(K), F) = H^q(Spec(O^sh_S, overlines) ×_S Spec(K), F) • H^q(Spec(O^sh_X, overlinex) ×_Spec(O^sh_S, overlines) Spec(K), F) = H^q(Spec(K), F)","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\overline{x}$ be a geometric point of $X$ with image $\\overline{s}$ in $S$.\nLet $\\Spec(K) \\to \\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$\nbe a morphism with $K$ a separably closed field. Let $\\mathcal{F}$ be an\nabelian sheaf on $\\Spec(K)_\\etale$. Let $q \\geq 0$. The following are\nequivalent\n\\begin{enumerate}\n\\item\n$H^q(\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}}) \\times_S \\Spec(K), \\mathcal{F}) =\nH^q(\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}}) \\times_S \\Spec(K), \\mathcal{F})$\n\\item\n$H^q(\\Spec(\\mathcal{O}^{sh}_{X, \\overline{x}})\n\\times_{\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})} \\Spec(K), \\mathcal{F}) =\nH^q(\\Spec(K), \\mathcal{F})$\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Preliminaries on base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZV","source_file":"etale-cohomology.tex","source_line":16531,"source_end_line":16548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16531-L16548","statement_sha256":"d26cb9104cc45d5a89bf9dddcf761c4faabe9dce5e686c7e6b6906b45991684c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10137,"rank":10137,"depth":42,"x":1559.781,"y":1248.103,"cluster":"tale-geometry"},{"id":"stacks:0EZX","tag":"0EZX","title":"Base change for pushforward · Lemma 0EZX","summary":"Consider the cartesian diagram of schemes xymatrix X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g Assume that f is flat and every object U of X_etale has a covering (U_i → U) such that U_i → S factors as U_i → V_i → S with V_i → S étale and U_i → V_i quasi-compact with geometrically connected fibres. Then for any sheaf F of sets on T_etale we have f^-1g_*F = h_*e^-1F.","statement_latex":"Consider the cartesian diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nAssume that $f$ is flat and every object $U$ of $X_\\etale$ has\na covering $\\{U_i \\to U\\}$ such that $U_i \\to S$\nfactors as $U_i \\to V_i \\to S$ with $V_i \\to S$\n\\'etale and $U_i \\to V_i$ quasi-compact with\ngeometrically connected fibres.\nThen for any sheaf $\\mathcal{F}$ of sets on $T_\\etale$ we have\n$f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZX","source_file":"etale-cohomology.tex","source_line":16609,"source_end_line":16625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16609-L16625","statement_sha256":"2ba7aa3a2f779b1f64ea3b4cc78e7b5c38155bf4c318e8b85ea3c441c23fb21f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10138,"rank":10138,"depth":55,"x":1507.906,"y":1048.726,"cluster":"tale-geometry"},{"id":"stacks:0EYS","tag":"0EYS","title":"Base change for pushforward · Lemma 0EYS","summary":"Consider a cartesian diagram of schemes xymatrix X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g where f is flat and locally of finite presentation with geometrically reduced fibres. Then f^-1g_*F = h_*e^-1F for any sheaf F on T_etale.","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nwhere $f$ is flat and locally of finite presentation\nwith geometrically reduced fibres.\nThen $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$\nfor any sheaf $\\mathcal{F}$ on $T_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYS","source_file":"etale-cohomology.tex","source_line":16663,"source_end_line":16676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16663-L16676","statement_sha256":"2276d6792ae865a7e2ff9d610b52f644e2b26b5bf0b4e916bc9569b1d4def05f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10139,"rank":10139,"depth":56,"x":1706.683,"y":1166.422,"cluster":"tale-geometry"},{"id":"stacks:0EZY","tag":"0EZY","title":"Base change for pushforward · Lemma 0EZY","summary":"Consider the cartesian diagrams of schemes xymatrix X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g Assume that S is the spectrum of a separably closed field. Then f^-1g_*F = h_*e^-1F for any sheaf F on T_etale.","statement_latex":"Consider the cartesian diagrams of schemes\n$$\n\\xymatrix{\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nAssume that $S$ is the spectrum of a separably closed field.\nThen $f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$\nfor any sheaf $\\mathcal{F}$ on $T_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZY","source_file":"etale-cohomology.tex","source_line":16684,"source_end_line":16696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16684-L16696","statement_sha256":"c58546b97171bce3a2c6c713482d3c1d7a0a913625fbc4eb1b290ad9fe6ba06d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10140,"rank":10140,"depth":56,"x":1465.228,"y":1192.451,"cluster":"tale-geometry"},{"id":"stacks:0EZZ","tag":"0EZZ","title":"Base change for pushforward · Lemma 0EZZ","summary":"Consider a cartesian diagram of schemes xymatrix X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g Assume that • f is flat and open, • the residue fields of S are separably algebraically closed, • given an étale morphism U → X with U affine we can write U as a finite disjoint union of open subschemes of X (for example if X is a normal integral scheme with separably closed function field), • any nonempty open of a fibre X_s of f is connected (for example if X_s is irreducible or…","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nAssume that\n\\begin{enumerate}\n\\item $f$ is flat and open,\n\\item the residue fields of $S$ are separably algebraically closed,\n\\item given an \\'etale morphism $U \\to X$ with $U$ affine\nwe can write $U$ as a finite disjoint union of open subschemes\nof $X$ (for example if $X$ is a normal integral scheme\nwith separably closed function field),\n\\item any nonempty open of a fibre $X_s$ of $f$ is connected\n(for example if $X_s$ is irreducible or empty).\n\\end{enumerate}\nThen for any sheaf $\\mathcal{F}$ of sets on $T_\\etale$ we have\n$f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EZZ","source_file":"etale-cohomology.tex","source_line":16711,"source_end_line":16733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16711-L16733","statement_sha256":"e92d2af631e18158424e9c65ae671e91196fbd352f843b4c557f6ad765c2e408","origin":"The Stacks Project","memory_eligible":false,"source_rank":10141,"rank":10141,"depth":56,"x":1622.493,"y":1036.1,"cluster":"tale-geometry"},{"id":"stacks:0EYR","tag":"0EYR","title":"Base change for pushforward · Lemma 0EYR","summary":"Let f : X → S be a morphism of schemes which is flat and locally of finite presentation with geometrically reduced fibres. Then f^-1 : Sh(S_etale) → Sh(X_etale) commutes with products.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is flat and\nlocally of finite presentation with geometrically reduced fibres.\nThen $f^{-1} : \\Sh(S_\\etale) \\to \\Sh(X_\\etale)$ commutes\nwith products.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYR","source_file":"etale-cohomology.tex","source_line":16746,"source_end_line":16752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16746-L16752","statement_sha256":"011b235b2a93c41cbd121c3c93ad83edc408dd07fd59e943604a065f2f09dbac","origin":"The Stacks Project","memory_eligible":false,"source_rank":10142,"rank":10142,"depth":55,"x":1632.27,"y":1240.822,"cluster":"tale-geometry"},{"id":"stacks:0F00","tag":"0F00","title":"Base change for pushforward · Lemma 0F00","summary":"Let f : X → S be a flat morphism of schemes such that for every geometric point overlinex of X the map O_S, f(overlinex)^sh → O_X, overlinex^sh has geometrically connected fibres. Then for every cartesian diagram of schemes xymatrix X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g with g quasi-compact and quasi-separated we have f^-1g_*F = h_*e^-1F for any sheaf F of sets on T_etale.","statement_latex":"Let $f : X \\to S$ be a flat morphism of schemes such\nthat for every geometric point $\\overline{x}$ of $X$ the map\n$$\n\\mathcal{O}_{S, f(\\overline{x})}^{sh}\n\\longrightarrow\n\\mathcal{O}_{X, \\overline{x}}^{sh}\n$$\nhas geometrically connected fibres. Then for every\ncartesian diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nwith $g$ quasi-compact and quasi-separated we have\n$f^{-1}g_*\\mathcal{F} = h_*e^{-1}\\mathcal{F}$\nfor any sheaf $\\mathcal{F}$ of sets on $T_\\etale$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F00","source_file":"etale-cohomology.tex","source_line":16784,"source_end_line":16804,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16784-L16804","statement_sha256":"6c1d5ebb6880b9ef1e903051683982f538c19254337aa09c9773ab5485fdb211","origin":"The Stacks Project","memory_eligible":false,"source_rank":10143,"rank":10143,"depth":55,"x":1460.264,"y":1095.272,"cluster":"tale-geometry"},{"id":"stacks:0F03","tag":"0F03","title":"Base change for higher direct images · Lemma 0F03","summary":"With f : X → S and n as in Remark [Tag 0F02] assume for some q ≥ 1 we have BC(f, n, q - 1). Then for every commutative diagram xymatrix X ar[d]_f & X' ar[l] ar[d]_f' & Y ar[l]^h ar[d]^e S & S' ar[l] & T ar[l]_g with X' = X ×_S S' and Y = X' ×_S' T and g quasi-compact and quasi-separated, and every abelian sheaf F on T_etale annihilated by n • the base change map (f')^-1R^qg_*F→ R^qh_*e^-1F is injective, • if F ⊂ G where G on T_etale is annihilated by n, then Coker(…","statement_latex":"With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}\nassume for some $q \\geq 1$ we have $BC(f, n, q - 1)$. Then\nfor every commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & X' \\ar[l] \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\\nS & S' \\ar[l] & T \\ar[l]_g\n}\n$$\nwith $X' = X \\times_S S'$ and $Y = X' \\times_{S'} T$ and\n$g$ quasi-compact and quasi-separated, and every abelian sheaf\n$\\mathcal{F}$ on $T_\\etale$ annihilated by $n$\n\\begin{enumerate}\n\\item the base change map\n$(f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F}$\nis injective,\n\\item if $\\mathcal{F} \\subset \\mathcal{G}$ where $\\mathcal{G}$\non $T_\\etale$ is annihilated by $n$, then\n$$\n\\Coker\\left(\n(f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F}\n\\right)\n\\subset\n\\Coker\\left(\n(f')^{-1}R^qg_*\\mathcal{G}\\to R^qh_*e^{-1}\\mathcal{G}\n\\right)\n$$\n\\item if in (2) the sheaf $\\mathcal{G}$ is an injective sheaf\nof $\\mathbf{Z}/n\\mathbf{Z}$-modules, then\n$$\n\\Coker\\left((f')^{-1}R^qg_*\\mathcal{F}\\to R^qh_*e^{-1}\\mathcal{F} \\right)\n\\subset R^qh_*e^{-1}\\mathcal{G}\n$$\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F03","source_file":"etale-cohomology.tex","source_line":16868,"source_end_line":16904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16868-L16904","statement_sha256":"1e129cedeb690b013703d987d494d42860fe66ccbf0be4f319f3f38957761e41","origin":"The Stacks Project","memory_eligible":false,"source_rank":10144,"rank":10144,"depth":0,"x":1704.379,"y":1105.007,"cluster":"tale-geometry"},{"id":"stacks:0F04","tag":"0F04","title":"Base change for higher direct images · Lemma 0F04","summary":"With f : X → S and n as in Remark [Tag 0F02] assume for some q ≥ 1 we have BC(f, n, q - 1). Consider commutative diagrams vcenter xymatrix X ar[d]_f & X' ar[d]_f' ar[l] & Y ar[l]^h ar[d]^e & Y' ar[l]^π' ar[d]^e' S & S' ar[l] & T ar[l]_g & T' ar[l]_π and vcenter xymatrix X' ar[d]_f' & & Y' ar[ll]^h' = h ∘ π' ar[d]^e' S' & & T' ar[ll]_g' = g ∘ π where all squares are cartesian, g quasi-compact and quasi-separated, and π is integral surjective. Let F be an abelian sheaf on…","statement_latex":"With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}\nassume for some $q \\geq 1$ we have $BC(f, n, q - 1)$. Consider\ncommutative diagrams\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[d]_f &\nX' \\ar[d]_{f'} \\ar[l] &\nY \\ar[l]^h \\ar[d]^e &\nY' \\ar[l]^{\\pi'} \\ar[d]^{e'} \\\\\nS &\nS' \\ar[l] &\nT \\ar[l]_g &\nT' \\ar[l]_\\pi\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nX' \\ar[d]_{f'} & & Y' \\ar[ll]^{h' = h \\circ \\pi'} \\ar[d]^{e'} \\\\\nS' & & T' \\ar[ll]_{g' = g \\circ \\pi}\n}\n}\n$$\nwhere all squares are cartesian, $g$ quasi-compact and quasi-separated, and\n$\\pi$ is integral surjective. Let $\\mathcal{F}$ be an abelian sheaf\non $T_\\etale$ annihilated by $n$ and set $\\mathcal{F}' = \\pi^{-1}\\mathcal{F}$.\nIf the base change map\n$$\n(f')^{-1}R^qg'_*\\mathcal{F}' \\longrightarrow R^qh'_*(e')^{-1}\\mathcal{F}'\n$$\nis an isomorphism, then the base change map\n$(f')^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F04","source_file":"etale-cohomology.tex","source_line":16938,"source_end_line":16974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16938-L16974","statement_sha256":"37dfc56fb3706e1277775c254c37f286d2535c76f6dd7ec9e063cb28e621a6af","origin":"The Stacks Project","memory_eligible":false,"source_rank":10145,"rank":10145,"depth":54,"x":1516.364,"y":1236.471,"cluster":"tale-geometry"},{"id":"stacks:0F05","tag":"0F05","title":"Base change for higher direct images · Lemma 0F05","summary":"With f : X → S and n as in Remark [Tag 0F02] assume for some q ≥ 1 we have BC(f, n, q - 1). Consider commutative diagrams vcenter xymatrix X ar[d]_f & X' ar[d]_f' ar[l] & X\" ar[l]^π' ar[d]_f\" & Y ar[l]^h' ar[d]^e S & S' ar[l] & S\" ar[l]_π & T ar[l]_g' and vcenter xymatrix X' ar[d]_f' & & Y ar[ll]^h = h' ∘ π' ar[d]^e S' & & T ar[ll]_g = g' ∘ π where all squares are cartesian, g' quasi-compact and quasi-separated, and π is integral. Let F be an abelian sheaf on T_etale…","statement_latex":"With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}\nassume for some $q \\geq 1$ we have $BC(f, n, q - 1)$. Consider\ncommutative diagrams\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[d]_f &\nX' \\ar[d]_{f'} \\ar[l] &\nX'' \\ar[l]^{\\pi'} \\ar[d]_{f''} &\nY \\ar[l]^{h'} \\ar[d]^e \\\\\nS &\nS' \\ar[l] &\nS'' \\ar[l]_\\pi &\nT \\ar[l]_{g'}\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nX' \\ar[d]_{f'} & & Y \\ar[ll]^{h = h' \\circ \\pi'} \\ar[d]^e \\\\\nS' & & T \\ar[ll]_{g = g' \\circ \\pi}\n}\n}\n$$\nwhere all squares are cartesian, $g'$ quasi-compact and quasi-separated, and\n$\\pi$ is integral. Let $\\mathcal{F}$ be an abelian sheaf\non $T_\\etale$ annihilated by $n$. If the base change map\n$$\n(f')^{-1}R^qg_*\\mathcal{F} \\longrightarrow R^qh_*e^{-1}\\mathcal{F}\n$$\nis an isomorphism, then the base change map\n$(f'')^{-1}R^qg'_*\\mathcal{F} \\to R^qh'_*e^{-1}\\mathcal{F}$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F05","source_file":"etale-cohomology.tex","source_line":16996,"source_end_line":17031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L16996-L17031","statement_sha256":"53e79c4e531e44bac5dec619935e6601e3788d0715c1000702e20871efefa8ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":10146,"rank":10146,"depth":50,"x":1549.316,"y":1032.654,"cluster":"tale-geometry"},{"id":"stacks:0F06","tag":"0F06","title":"Base change for higher direct images · Lemma 0F06","summary":"Let T be a quasi-compact and quasi-separated scheme. Let P be a property for quasi-compact and quasi-separated schemes over T. Assume • If T\" → T' is a thickening of quasi-compact and quasi-separated schemes over T, then P(T\") if and only if P(T'). • If T' = lim T_i is a limit of an inverse system of quasi-compact and quasi-separated schemes over T with affine transition morphisms and P(T_i) holds for all i, then P(T') holds. • If Z ⊂ T' is a closed subscheme with…","statement_latex":"Let $T$ be a quasi-compact and quasi-separated scheme.\nLet $P$ be a property for quasi-compact and quasi-separated\nschemes over $T$. Assume\n\\begin{enumerate}\n\\item If $T'' \\to T'$ is a thickening of quasi-compact and\nquasi-separated schemes over $T$, then $P(T'')$ if and only if $P(T')$.\n\\item If $T' = \\lim T_i$ is a limit of an inverse system of\nquasi-compact and quasi-separated schemes over $T$ with affine\ntransition morphisms and $P(T_i)$ holds for all $i$, then\n$P(T')$ holds.\n\\item If $Z \\subset T'$ is a closed subscheme with\nquasi-compact complement $V \\subset T'$ and $P(T')$ holds,\nthen either $P(V)$ or $P(Z)$ holds.\n\\end{enumerate}\nThen $P(T)$ implies $P(\\Spec(K))$ for some morphism $\\Spec(K) \\to T$\nwhere $K$ is a field.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F06","source_file":"etale-cohomology.tex","source_line":17045,"source_end_line":17063,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L17045-L17063","statement_sha256":"c9da720f2b2c8543e31613f8f52875a7f014841e089e5b8e607277bfd6c4afb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10147,"rank":10147,"depth":0,"x":1689.054,"y":1201.803,"cluster":"tale-geometry"},{"id":"stacks:0F07","tag":"0F07","title":"Base change for higher direct images · Lemma 0F07","summary":"With f : X → S and n as in Remark [Tag 0F02] assume for some q ≥ 1 we have that BC(f, n, q - 1) is true, but BC(f, n, q) is not. Then there exist a commutative diagram xymatrix X ar[d]_f & X' ar[d]_f' ar[l] & Y ar[l]^h ar[d]^e S & S' ar[l] & Spec(K) ar[l]_g where X' = X ×_S S', Y = X' ×_S' Spec(K), K is a field, and F is an abelian sheaf on Spec(K) annihilated by n such that (f')^-1R^qg_*F → R^qh_*e^-1F is not an isomorphism.","statement_latex":"With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}\nassume for some $q \\geq 1$ we have that $BC(f, n, q - 1)$ is true, but\n$BC(f, n, q)$ is not. Then there exist a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & X' \\ar[d]_{f'} \\ar[l] & Y \\ar[l]^h \\ar[d]^e \\\\\nS & S' \\ar[l] & \\Spec(K) \\ar[l]_g\n}\n$$\nwhere $X' = X \\times_S S'$, $Y = X' \\times_{S'} \\Spec(K)$,\n$K$ is a field, and $\\mathcal{F}$ is an abelian sheaf\non $\\Spec(K)$ annihilated by $n$ such that\n$(f')^{-1}R^qg_*\\mathcal{F} \\to R^qh_*e^{-1}\\mathcal{F}$\nis not an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F07","source_file":"etale-cohomology.tex","source_line":17078,"source_end_line":17094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L17078-L17094","statement_sha256":"0f339145a016c9f5a9d3e2ca3cdbd2db646ee410f96f5dc9e2e3fbbbb57f2bd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10148,"rank":10148,"depth":53,"x":1449.762,"y":1156.323,"cluster":"tale-geometry"},{"id":"stacks:0F08","tag":"0F08","title":"Base change for higher direct images · Lemma 0F08","summary":"With f : X → S and n as in Remark [Tag 0F02] assume for some q ≥ 1 we have that BC(f, n, q - 1) is true, but BC(f, n, q) is not. Then there exist a commutative diagram xymatrix X ar[d]_f & X' ar[d] ar[l] & Y ar[l]^h ar[d] S & S' ar[l] & Spec(K) ar[l] with both squares cartesian, where • S' is affine, integral, and normal with algebraically closed function field, • K is algebraically closed and Spec(K) → S' is dominant (in other words K is an extension of the function…","statement_latex":"With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}\nassume for some $q \\geq 1$ we have that\n$BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not.\nThen there exist a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & X' \\ar[d] \\ar[l] & Y \\ar[l]^h \\ar[d] \\\\\nS & S' \\ar[l] & \\Spec(K) \\ar[l]\n}\n$$\nwith both squares cartesian, where\n\\begin{enumerate}\n\\item $S'$ is affine, integral, and normal with algebraically\nclosed function field,\n\\item $K$ is algebraically closed and $\\Spec(K) \\to S'$\nis dominant (in other words $K$ is an extension of\nthe function field of $S'$)\n\\end{enumerate}\nand there exists an integer $d | n$\nsuch that $R^qh_*(\\mathbf{Z}/d\\mathbf{Z})$ is nonzero.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Base change for higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F08","source_file":"etale-cohomology.tex","source_line":17295,"source_end_line":17317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L17295-L17317","statement_sha256":"e83dec57d8b2a46e46c2008bfb7e6cacd8969efba1fc1a111b33c20b5f2a8b20","origin":"The Stacks Project","memory_eligible":false,"source_rank":10149,"rank":10149,"depth":0,"x":1662.988,"y":1053.982,"cluster":"tale-geometry"},{"id":"stacks:0EYT","tag":"0EYT","title":"Smooth base change · Lemma 0EYT","summary":"Let K/k be an extension of fields. Let X be a smooth affine curve over k with a rational point x ∈ X(k). Let F be an abelian sheaf on Spec(K) annihilated by an integer n invertible in k. Let q > 0 and xi ∈ H^q(X_K, (X_K → Spec(K))^-1F) There exist • finite extensions K'/K and k'/k with k' ⊂ K', • a finite étale Galois cover Z → X_k' with group G such that the order of G divides a power of n, such that Z → X_k' is split over x_k', and such that xi dies in H^q(Z_K', (Z_K' →…","statement_latex":"Let $K/k$ be an extension of fields. Let $X$ be a smooth affine curve\nover $k$ with a rational point $x \\in X(k)$. Let $\\mathcal{F}$ be an abelian\nsheaf on $\\Spec(K)$ annihilated by an integer $n$ invertible in $k$.\nLet $q > 0$ and\n$$\n\\xi \\in H^q(X_K, (X_K \\to \\Spec(K))^{-1}\\mathcal{F})\n$$\nThere exist\n\\begin{enumerate}\n\\item finite extensions $K'/K$ and $k'/k$ with $k' \\subset K'$,\n\\item a finite \\'etale Galois cover $Z \\to X_{k'}$ with group $G$\n\\end{enumerate}\nsuch that the order of $G$ divides a power of $n$, such that\n$Z \\to X_{k'}$ is split over $x_{k'}$, and\nsuch that $\\xi$ dies in $H^q(Z_{K'}, (Z_{K'} \\to \\Spec(K))^{-1}\\mathcal{F})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Smooth base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYT","source_file":"etale-cohomology.tex","source_line":17356,"source_end_line":17373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L17356-L17373","statement_sha256":"882e118d4f4c6c35b0cf4f7e87fb1df7d0d8947b0e01cc5feceb13dd3f73976e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10150,"rank":10150,"depth":73,"x":1587.986,"y":1250.622,"cluster":"tale-geometry"},{"id":"stacks:0EYU","tag":"0EYU","title":"Smooth base change · Theorem 0EYU","summary":"Consider a cartesian diagram of schemes xymatrix X ar[d]_f & Y ar[l]^h ar[d]^e S & T ar[l]_g where f is smooth and g quasi-compact and quasi-separated. Then f^-1R^qg_*F = R^qh_*e^-1F for any q and any abelian sheaf F on T_etale all of whose stalks at geometric points are torsion of orders invertible on S.","statement_latex":"Consider a cartesian diagram of schemes\n$$\n\\xymatrix{\nX \\ar[d]_f & Y \\ar[l]^h \\ar[d]^e \\\\\nS & T \\ar[l]_g\n}\n$$\nwhere $f$ is smooth and $g$ quasi-compact and quasi-separated. Then\n$$\nf^{-1}R^qg_*\\mathcal{F} = R^qh_*e^{-1}\\mathcal{F}\n$$\nfor any $q$ and any abelian sheaf $\\mathcal{F}$\non $T_\\etale$ all of whose stalks at geometric points are torsion of\norders invertible on $S$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Smooth base change","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EYU","source_file":"etale-cohomology.tex","source_line":17445,"source_end_line":17461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L17445-L17461","statement_sha256":"fc9a2daffbb5e97396cb7c170e65a1bc7947a486cd5577c123b747b4ac39c7c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10151,"rank":10151,"depth":74,"x":1485.063,"y":1062.889,"cluster":"tale-geometry"},{"id":"stacks:0F09","tag":"0F09","title":"Smooth base change · Lemma 0F09","summary":"Let S be a scheme. Let S' = lim S_i be a directed inverse limit of schemes S_i smooth over S with affine transition morphisms. Let f : X → S be quasi-compact and quasi-separated and form the fibre square xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f S' ar[r]^g & S Then g^-1Rf_*E = R(f')_*(g')^-1E for any E ∈ D^+(X_etale) whose cohomology sheaves H^q(E) have stalks which are torsion of orders invertible on S.","statement_latex":"Let $S$ be a scheme. Let $S' = \\lim S_i$ be a directed inverse\nlimit of schemes $S_i$ smooth over $S$ with affine transition\nmorphisms. Let $f : X \\to S$ be quasi-compact and quasi-separated\nand form the fibre square\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nThen\n$$\ng^{-1}Rf_*E = R(f')_*(g')^{-1}E\n$$\nfor any $E \\in D^+(X_\\etale)$ whose cohomology sheaves $H^q(E)$\nhave stalks which are torsion of orders invertible on $S$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Smooth base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F09","source_file":"etale-cohomology.tex","source_line":17974,"source_end_line":17992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L17974-L17992","statement_sha256":"4876ac4e576c7f972b42cd662764bc12c00ed70e089fee190e8a5c56ae67de53","origin":"The Stacks Project","memory_eligible":false,"source_rank":10152,"rank":10152,"depth":75,"x":1712.14,"y":1142.994,"cluster":"tale-geometry"},{"id":"stacks:0F1C","tag":"0F1C","title":"Applications of smooth base change · Lemma 0F1C","summary":"Let L/K be an extension of fields. Let g : T → S be a quasi-compact and quasi-separated morphism of schemes over K. Denote g_L : T_L → S_L the base change of g to Spec(L). Let E ∈ D^+(T_etale) have cohomology sheaves whose stalks are torsion of orders invertible in K. Let E_L be the pullback of E to (T_L)_etale. Then Rg_L, *E_L is the pullback of Rg_*E to S_L.","statement_latex":"Let $L/K$ be an extension of fields. Let $g : T \\to S$ be a quasi-compact\nand quasi-separated morphism of schemes over $K$. Denote\n$g_L : T_L \\to S_L$ the base change of $g$ to $\\Spec(L)$.\nLet $E \\in D^+(T_\\etale)$ have cohomology sheaves whose stalks\nare torsion of orders invertible in $K$. Let $E_L$ be the\npullback of $E$ to $(T_L)_\\etale$. Then\n$Rg_{L, *}E_L$ is the pullback of $Rg_*E$ to $S_L$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Applications of smooth base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1C","source_file":"etale-cohomology.tex","source_line":18031,"source_end_line":18040,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18031-L18040","statement_sha256":"16bf0dff25a8bec62150dd65415f362d7cb10d59b2bd22081716b2c686f8fdc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10153,"rank":10153,"depth":76,"x":1480.061,"y":1212.84,"cluster":"tale-geometry"},{"id":"stacks:0F0B","tag":"0F0B","title":"Applications of smooth base change · Lemma 0F0B","summary":"Let K/k be an extension of separably closed fields. Let X be a quasi-compact and quasi-separated scheme over k. Let E ∈ D^+(X_etale) have cohomology sheaves whose stalks are torsion of orders invertible in k. Then • the maps H^q_etale(X, E) → H^q_etale(X_K, E|_X_K) are isomorphisms, and • E → R(X_K → X)_*E|_X_K is an isomorphism.","statement_latex":"Let $K/k$ be an extension of separably closed fields. Let $X$\nbe a quasi-compact and quasi-separated scheme over $k$.\nLet $E \\in D^+(X_\\etale)$ have cohomology sheaves whose stalks\nare torsion of orders invertible in $k$. Then\n\\begin{enumerate}\n\\item the maps $H^q_\\etale(X, E) \\to H^q_\\etale(X_K, E|_{X_K})$\nare isomorphisms, and\n\\item $E \\to R(X_K \\to X)_*E|_{X_K}$ is an isomorphism.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Applications of smooth base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0B","source_file":"etale-cohomology.tex","source_line":18065,"source_end_line":18076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18065-L18076","statement_sha256":"269b3e368ac57e1f5ddbe282951f4e43403d621dda73094913cac34d278cbfa7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10154,"rank":10154,"depth":76,"x":1595.132,"y":1029.478,"cluster":"tale-geometry"},{"id":"stacks:0F1D","tag":"0F1D","title":"Applications of smooth base change · Lemma 0F1D","summary":"With f : X → S and n as in Remark [Tag 0F02] assume n is invertible on S and that for some q ≥ 1 we have that BC(f, n, q - 1) is true, but BC(f, n, q) is not. Then there exist a commutative diagram xymatrix X ar[d]_f & X' ar[d] ar[l] & Y ar[l]^h ar[d] S & S' ar[l] & Spec(K) ar[l] with both squares cartesian, where S' is affine, integral, and normal with algebraically closed function field K and there exists an integer d | n such that R^qh_*(Z/dZ) is nonzero.","statement_latex":"With $f : X \\to S$ and $n$ as in Remark \\ref{remark-base-change-holds}\nassume $n$ is invertible on $S$ and that for some $q \\geq 1$\nwe have that $BC(f, n, q - 1)$ is true, but $BC(f, n, q)$ is not.\nThen there exist a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & X' \\ar[d] \\ar[l] & Y \\ar[l]^h \\ar[d] \\\\\nS & S' \\ar[l] & \\Spec(K) \\ar[l]\n}\n$$\nwith both squares cartesian, where $S'$ is affine, integral, and normal\nwith algebraically closed function field $K$ and there exists an integer\n$d | n$ such that $R^qh_*(\\mathbf{Z}/d\\mathbf{Z})$ is nonzero.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Applications of smooth base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1D","source_file":"etale-cohomology.tex","source_line":18105,"source_end_line":18120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18105-L18120","statement_sha256":"e400b4c69aadd963c0d3a4287f2c60941f9cd5afe9b32126da9df1c3eea63128","origin":"The Stacks Project","memory_eligible":false,"source_rank":10155,"rank":10155,"depth":77,"x":1657.793,"y":1230.168,"cluster":"tale-geometry"},{"id":"stacks:0A0B","tag":"0A0B","title":"The proper base change theorem · Lemma 0A0B","summary":"Let (A, I) be a henselian pair. Let f : X → Spec(A) be a proper morphism of schemes. Let Z = X ×_Spec(A) Spec(A/I). For any sheaf F on the topological space associated to X we have Γ(X, F) = Γ(Z, F|_Z).","statement_latex":"Let $(A, I)$ be a henselian pair. Let $f : X \\to \\Spec(A)$ be a proper morphism\nof schemes. Let $Z = X \\times_{\\Spec(A)} \\Spec(A/I)$. For any\nsheaf $\\mathcal{F}$ on the topological space associated to $X$ we\nhave $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, \\mathcal{F}|_Z)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0B","source_file":"etale-cohomology.tex","source_line":18167,"source_end_line":18173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18167-L18173","statement_sha256":"8f333482d8ed7939bac16db4b514a9cce65fbbf2edcef30a180c2a80097397f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10156,"rank":10156,"depth":49,"x":1450.005,"y":1117.633,"cluster":"tale-geometry"},{"id":"stacks:0A0C","tag":"0A0C","title":"The proper base change theorem · Lemma 0A0C","summary":"Let (A, I) be a henselian pair. Let f : X → Spec(A) be a proper morphism of schemes. Let i : Z → X be the closed immersion of X ×_Spec(A) Spec(A/I) into X. For any sheaf F on X_etale we have Γ(X, F) = Γ(Z, i_small^-1F).","statement_latex":"Let $(A, I)$ be a henselian pair. Let $f : X \\to \\Spec(A)$ be a proper morphism\nof schemes. Let $i : Z \\to X$ be the closed immersion of\n$X \\times_{\\Spec(A)} \\Spec(A/I)$ into $X$. For any\nsheaf $\\mathcal{F}$ on $X_\\etale$ we\nhave $\\Gamma(X, \\mathcal{F}) = \\Gamma(Z, i_{small}^{-1}\\mathcal{F})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0C","source_file":"etale-cohomology.tex","source_line":18199,"source_end_line":18206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18199-L18206","statement_sha256":"cf4aad8f5ae517ada2420d15e2feea2fc3af7a0fc6e30b81dd4ed687ff80736d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10157,"rank":10157,"depth":51,"x":1693.949,"y":1082.678,"cluster":"tale-geometry"},{"id":"stacks:0A3S","tag":"0A3S","title":"The proper base change theorem · Lemma 0A3S","summary":"Let A be a henselian local ring. Let f : X → Spec(A) be a proper morphism of schemes. Let X_0 ⊂ X be the fibre of f over the closed point. For any sheaf F on X_etale we have Γ(X, F) = Γ(X_0, F|_X_0).","statement_latex":"Let $A$ be a henselian local ring. Let $f : X \\to \\Spec(A)$\nbe a proper morphism of schemes. Let $X_0 \\subset X$ be the fibre of\n$f$ over the closed point. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we\nhave $\\Gamma(X, \\mathcal{F}) = \\Gamma(X_0, \\mathcal{F}|_{X_0})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3S","source_file":"etale-cohomology.tex","source_line":18214,"source_end_line":18220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18214-L18220","statement_sha256":"257cc1ae054bea47c747d887a8afbb506894f2422ecf18760a4ae1ccc4191429","origin":"The Stacks Project","memory_eligible":false,"source_rank":10158,"rank":10158,"depth":52,"x":1542.037,"y":1247.025,"cluster":"tale-geometry"},{"id":"stacks:0A3T","tag":"0A3T","title":"The proper base change theorem · Lemma 0A3T","summary":"Let f : X → S be a proper morphism of schemes. Let overlines → S be a geometric point. For any sheaf F on X_etale the canonical map (f_*F)_overlines → Γ(X_overlines, F_overlines) is bijective.","statement_latex":"Let $f : X \\to S$ be a proper morphism of schemes. Let\n$\\overline{s} \\to S$ be a geometric point.\nFor any sheaf $\\mathcal{F}$ on $X_\\etale$\nthe canonical map\n$$\n(f_*\\mathcal{F})_{\\overline{s}} \\longrightarrow\n\\Gamma(X_{\\overline{s}}, \\mathcal{F}_{\\overline{s}})\n$$\nis bijective.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3T","source_file":"etale-cohomology.tex","source_line":18253,"source_end_line":18264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18253-L18264","statement_sha256":"715d0aae50866b23939821fed3f7616230b457be87d26476ddd190f1f0f47bc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10159,"rank":10159,"depth":53,"x":1521.873,"y":1039.447,"cluster":"tale-geometry"},{"id":"stacks:0A3U","tag":"0A3U","title":"The proper base change theorem · Lemma 0A3U","summary":"Let f : X → Y be a proper morphism of schemes. Let g : Y' → Y be a morphism of schemes. Set X' = Y' ×_Y X with projections f' : X' → Y' and g' : X' → X. Let F be any sheaf on X_etale. Then g^-1f_*F = f'_*(g')^-1F.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$\nbe a morphism of schemes. Set $X' = Y' \\times_Y X$ with projections\n$f' : X' \\to Y'$ and $g' : X' \\to X$. Let $\\mathcal{F}$ be any sheaf on\n$X_\\etale$. Then $g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3U","source_file":"etale-cohomology.tex","source_line":18281,"source_end_line":18287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18281-L18287","statement_sha256":"ce72de28d4f6bd0e6e5909a05ff5bc9ddd0cb938013ae5c591305646705a0e6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10160,"rank":10160,"depth":54,"x":1703.838,"y":1181.201,"cluster":"tale-geometry"},{"id":"stacks:0A4B","tag":"0A4B","title":"The proper base change theorem · Lemma 0A4B","summary":"Let f : X → Y be a proper morphism of schemes. The following are equivalent • cohomology commutes with base change for f (see above), • for every prime number ℓ and every injective sheaf of Z/ℓZ-modules I on X_etale and every diagram ([Tag 0A29]) where X' = Y' ×_Y X the sheaves R^qf'_*(g')^-1I are zero for q > 0.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes.\nThe following are equivalent\n\\begin{enumerate}\n\\item cohomology commutes with base change for $f$ (see above),\n\\item for every prime number $\\ell$ and every injective\nsheaf of $\\mathbf{Z}/\\ell\\mathbf{Z}$-modules $\\mathcal{I}$\non $X_\\etale$ and every diagram (\\ref{equation-base-change-diagram})\nwhere $X' = Y' \\times_Y X$ the sheaves\n$R^qf'_*(g')^{-1}\\mathcal{I}$ are zero for $q > 0$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4B","source_file":"etale-cohomology.tex","source_line":18359,"source_end_line":18371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18359-L18371","statement_sha256":"bb7c0b2e07128eb38d0e51327c61d9fca6653827ad9c1ea0564a7b986c9ac89b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10161,"rank":10161,"depth":55,"x":1455.436,"y":1179.925,"cluster":"tale-geometry"},{"id":"stacks:0A4C","tag":"0A4C","title":"The proper base change theorem · Lemma 0A4C","summary":"Let f : X → Y and g : Y → Z be proper morphisms of schemes. Assume • cohomology commutes with base change for f, • cohomology commutes with base change for g ∘ f, and • f is surjective. Then cohomology commutes with base change for g.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of schemes. Assume\n\\begin{enumerate}\n\\item cohomology commutes with base change for $f$,\n\\item cohomology commutes with base change for $g \\circ f$, and\n\\item $f$ is surjective.\n\\end{enumerate}\nThen cohomology commutes with base change for $g$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4C","source_file":"etale-cohomology.tex","source_line":18430,"source_end_line":18439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18430-L18439","statement_sha256":"a55106efb58854fef0105fa8b71b3815b0f907a84c42dc40a1b4a1cf5fcc96ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":10162,"rank":10162,"depth":56,"x":1639.8,"y":1039.787,"cluster":"tale-geometry"},{"id":"stacks:0A4D","tag":"0A4D","title":"The proper base change theorem · Lemma 0A4D","summary":"Let f : X → Y and g : Y → Z be proper morphisms of schemes. Assume • cohomology commutes with base change for f, and • cohomology commutes with base change for g. Then cohomology commutes with base change for g ∘ f.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms of schemes. Assume\n\\begin{enumerate}\n\\item cohomology commutes with base change for $f$, and\n\\item cohomology commutes with base change for $g$.\n\\end{enumerate}\nThen cohomology commutes with base change for $g \\circ f$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4D","source_file":"etale-cohomology.tex","source_line":18476,"source_end_line":18484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18476-L18484","statement_sha256":"215423a00b0f3990be4173a53faf3704c34f51a825ba078252cb7d10ed0f2afb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10163,"rank":10163,"depth":56,"x":1616.526,"y":1247.926,"cluster":"tale-geometry"},{"id":"stacks:0A4E","tag":"0A4E","title":"The proper base change theorem · Lemma 0A4E","summary":"Proper base change for étale cohomology holds for finite morphisms. Let f : X → Y be a finite morphism of schemes. Then cohomology commutes with base change for f.","statement_latex":"\\begin{slogan}\nProper base change for \\'etale cohomology holds for finite morphisms.\n\\end{slogan}\nLet $f : X \\to Y$ be a finite morphism of schemes.\nThen cohomology commutes with base change for $f$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4E","source_file":"etale-cohomology.tex","source_line":18518,"source_end_line":18525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18518-L18525","statement_sha256":"1e94a322f8d911efac0cda76f18b813fd8aea8ebd8fd5fa75ebfe3be2284bcde","origin":"The Stacks Project","memory_eligible":false,"source_rank":10164,"rank":10164,"depth":56,"x":1466.171,"y":1081.088,"cluster":"tale-geometry"},{"id":"stacks:0A4F","tag":"0A4F","title":"The proper base change theorem · Lemma 0A4F","summary":"To prove that cohomology commutes with base change for every proper morphism of schemes it suffices to prove it holds for the morphism P^1_S → S for every scheme S.","statement_latex":"To prove that cohomology commutes with base change for\nevery proper morphism of schemes it suffices to prove it\nholds for the morphism $\\mathbf{P}^1_S \\to S$ for every scheme $S$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4F","source_file":"etale-cohomology.tex","source_line":18538,"source_end_line":18543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18538-L18543","statement_sha256":"b7f16620557c9fdfa9f127c1ed84c6ef3591c23d3375640d1d13dd83ac37f7f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10165,"rank":10165,"depth":57,"x":1711.431,"y":1118.832,"cluster":"tale-geometry"},{"id":"stacks:095T","tag":"095T","title":"Proper Base Change · Theorem 095T","summary":"Let f : X → Y be a proper morphism of schemes. Let g : Y' → Y be a morphism of schemes. Set X' = Y' ×_Y X and consider the cartesian diagram xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y Let F be an abelian torsion sheaf on X_etale. Then the base change map g^-1Rf_*F → Rf'_*(g')^-1F is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be\na morphism of schemes. Set $X' = Y' \\times_Y X$\nand consider the cartesian diagram\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nLet $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$.\nThen the base change map\n$$\ng^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F}\n$$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095T","source_file":"etale-cohomology.tex","source_line":18609,"source_end_line":18626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18609-L18626","statement_sha256":"647756c7bfa532e382bf3de8507417e954e20922f33d94dcd17f81044757073d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10166,"rank":10166,"depth":74,"x":1500.034,"y":1230.269,"cluster":"tale-geometry"},{"id":"stacks:0DDE","tag":"0DDE","title":"The proper base change theorem · Lemma 0DDE","summary":"Let f : X → Y be a proper morphism of schemes. Let g : Y' → Y be a morphism of schemes. Set X' = Y' ×_Y X and denote f' : X' → Y' and g' : X' → X the projections. Let E ∈ D^+(X_etale) have torsion cohomology sheaves. Then the base change map ([Tag 0A2A]) g^-1Rf_*E → Rf'_*(g')^-1E is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be\na morphism of schemes. Set $X' = Y' \\times_Y X$ and denote\n$f' : X' \\to Y'$ and $g' : X' \\to X$ the projections.\nLet $E \\in D^+(X_\\etale)$ have torsion cohomology sheaves.\nThen the base change map (\\ref{equation-base-change})\n$g^{-1}Rf_*E \\to Rf'_*(g')^{-1}E$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDE","source_file":"etale-cohomology.tex","source_line":18707,"source_end_line":18716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18707-L18716","statement_sha256":"ac842d70af4c8a2f945e9a470b4ea35dc40e34cdebad54bc4380e3bdd644bf96","origin":"The Stacks Project","memory_eligible":false,"source_rank":10167,"rank":10167,"depth":75,"x":1566.361,"y":1027.961,"cluster":"tale-geometry"},{"id":"stacks:0DDF","tag":"0DDF","title":"The proper base change theorem · Lemma 0DDF","summary":"Let f : X → Y be a proper morphism of schemes. Let overliney → Y be a geometric point. • For a torsion abelian sheaf F on X_etale we have (R^nf_*F)_overliney = H^n_etale(X_overliney, F_overliney). • For E ∈ D^+(X_etale) with torsion cohomology sheaves we have (R^nf_*E)_overliney = H^n_etale(X_overliney, E|_X_overliney).","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $\\overline{y} \\to Y$\nbe a geometric point.\n\\begin{enumerate}\n\\item For a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have\n$(R^nf_*\\mathcal{F})_{\\overline{y}} =\nH^n_\\etale(X_{\\overline{y}}, \\mathcal{F}_{\\overline{y}})$.\n\\item For $E \\in D^+(X_\\etale)$ with torsion cohomology sheaves we have\n$(R^nf_*E)_{\\overline{y}} =\nH^n_\\etale(X_{\\overline{y}}, E|_{X_{\\overline{y}}})$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The proper base change theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDF","source_file":"etale-cohomology.tex","source_line":18733,"source_end_line":18745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18733-L18745","statement_sha256":"15271e9f1c47c579895bbbc1e4d1f845bbb7f633c370c0879cf1291266975471","origin":"The Stacks Project","memory_eligible":false,"source_rank":10168,"rank":10168,"depth":76,"x":1680.247,"y":1214.945,"cluster":"tale-geometry"},{"id":"stacks:0DDG","tag":"0DDG","title":"Applications of proper base change · Lemma 0DDG","summary":"Let K/k be an extension of separably closed fields. Let X be a proper scheme over k. Let F be a torsion abelian sheaf on X_etale. Then the map H^q_etale(X, F) → H^q_etale(X_K, F|_X_K) is an isomorphism for q ≥ 0.","statement_latex":"Let $K/k$ be an extension of separably closed fields.\nLet $X$ be a proper scheme over $k$.\nLet $\\mathcal{F}$ be a torsion abelian sheaf on $X_\\etale$.\nThen the map $H^q_\\etale(X, \\mathcal{F}) \\to\nH^q_\\etale(X_K, \\mathcal{F}|_{X_K})$ is an isomorphism\nfor $q \\geq 0$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Applications of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDG","source_file":"etale-cohomology.tex","source_line":18769,"source_end_line":18777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18769-L18777","statement_sha256":"cad0a0ac5c69db012d492c5f43f60c1229c4ca6559d9f4e53bcbb6a5b2f5e3ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":10169,"rank":10169,"depth":75,"x":1445.689,"y":1141.621,"cluster":"tale-geometry"},{"id":"stacks:095U","tag":"095U","title":"Applications of proper base change · Lemma 095U","summary":"Let f : X → Y be a proper morphism of schemes all of whose fibres have dimension ≤ n. Then for any abelian torsion sheaf F on X_etale we have R^qf_*F = 0 for q > 2n.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes\nall of whose fibres have dimension $\\leq n$.\nThen for any abelian torsion sheaf $\\mathcal{F}$ on $X_\\etale$\nwe have $R^qf_*\\mathcal{F} = 0$ for $q > 2n$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Applications of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/095U","source_file":"etale-cohomology.tex","source_line":18785,"source_end_line":18791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18785-L18791","statement_sha256":"7667e106303e2d4ab640376054232994fea01fc8d2eb44d1d2c651b897b32dd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10170,"rank":10170,"depth":77,"x":1677.825,"y":1062.523,"cluster":"tale-geometry"},{"id":"stacks:0F0C","tag":"0F0C","title":"Applications of proper base change · Lemma 0F0C","summary":"Let f : X → Y be a proper morphism of schemes. Let g : Y' → Y be a morphism of schemes. Set X' = Y' ×_Y X and denote f' : X' → Y' and g' : X' → X the projections. Let n ≥ 1 be an integer. Let E ∈ D(X_etale, Z/nZ). Then the base change map ([Tag 0A2A]) g^-1Rf_*E → Rf'_*(g')^-1E is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be\na morphism of schemes. Set $X' = Y' \\times_Y X$ and denote\n$f' : X' \\to Y'$ and $g' : X' \\to X$ the projections.\nLet $n \\geq 1$ be an integer.\nLet $E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$.\nThen the base change map (\\ref{equation-base-change})\n$g^{-1}Rf_*E \\to Rf'_*(g')^{-1}E$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Applications of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0C","source_file":"etale-cohomology.tex","source_line":18886,"source_end_line":18896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18886-L18896","statement_sha256":"c4fc9c6249f555f6c646e02c9f2e41b23a1fa84eba7fd66000f4365a9b1a25fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10171,"rank":10171,"depth":78,"x":1570.161,"y":1252.74,"cluster":"tale-geometry"},{"id":"stacks:0F0E","tag":"0F0E","title":"Applications of proper base change · Lemma 0F0E","summary":"Let X be a quasi-compact and quasi-separated scheme. Let E ∈ D^+(X_etale) and K ∈ D^+(Z). Then RΓ(X, E ⊗_Z^L underlineK) = RΓ(X, E) ⊗_Z^L K","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $E \\in D^+(X_\\etale)$ and $K \\in D^+(\\mathbf{Z})$.\nThen\n$$\nR\\Gamma(X, E \\otimes_\\mathbf{Z}^\\mathbf{L} \\underline{K}) =\nR\\Gamma(X, E) \\otimes_\\mathbf{Z}^\\mathbf{L} K\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Applications of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0E","source_file":"etale-cohomology.tex","source_line":18937,"source_end_line":18946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L18937-L18946","statement_sha256":"0a0166757bcd1dd1a85936ad841cfb71707436b84db1f99a488659929e41a511","origin":"The Stacks Project","memory_eligible":false,"source_rank":10172,"rank":10172,"depth":42,"x":1496.517,"y":1051.204,"cluster":"tale-geometry"},{"id":"stacks:0F0F","tag":"0F0F","title":"Applications of proper base change · Lemma 0F0F","summary":"Let f : X → Y be a proper morphism of schemes. Let E ∈ D^+(X_etale) have torsion cohomology sheaves. Let K ∈ D^+(Y_etale). Then Rf_*E ⊗_Z^L K = Rf_*(E ⊗_Z^L f^-1K) in D^+(Y_etale).","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes.\nLet $E \\in D^+(X_\\etale)$ have torsion cohomology sheaves.\nLet $K \\in D^+(Y_\\etale)$. Then\n$$\nRf_*E \\otimes_\\mathbf{Z}^\\mathbf{L} K =\nRf_*(E \\otimes_\\mathbf{Z}^\\mathbf{L} f^{-1}K)\n$$\nin $D^+(Y_\\etale)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Applications of proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0F","source_file":"etale-cohomology.tex","source_line":19000,"source_end_line":19010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19000-L19010","statement_sha256":"c50a6a8486e19f3254df18ab3ffdcbf8eec9b6dcc3e7b690517373bbc714ae11","origin":"The Stacks Project","memory_eligible":false,"source_rank":10173,"rank":10173,"depth":77,"x":1713.086,"y":1158.123,"cluster":"tale-geometry"},{"id":"stacks:0GJP","tag":"0GJP","title":"Local acyclicity · Definition 0GJP","summary":"[SGA4.5] and [SGA4.5] Let f : X → S be a morphism of schemes. Let K be an object of D(X_etale). • Let overlinex be a geometric point of X with image overlines = f(overlinex). We say f is locally acyclic at overlinex relative to K if for every geometric point overlinet of Spec(O^sh_S, overlines) the map ([Tag 0GJN]) is an isomorphism is an algebraic geometric point of Spec(O^sh_S, overlines). Often using Lemma [Tag 0F0B] one may reduce to this case.. • We say f is locally…","statement_latex":"\\begin{reference}\n\\cite[Definition 2.12, page 242]{SGA4.5} and\n\\cite[Definition (1.3), page 54]{SGA4.5}\n\\end{reference}\nLet $f : X \\to S$ be a morphism of schemes.\nLet $K$ be an object of $D(X_\\etale)$.\n\\begin{enumerate}\n\\item\nLet $\\overline{x}$ be a geometric point of $X$ with image\n$\\overline{s} = f(\\overline{x})$.\nWe say $f$ is {\\it locally acyclic at $\\overline{x}$ relative to $K$}\nif for every geometric point $\\overline{t}$ of\n$\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$ the\nmap (\\ref{equation-alpha-K}) is an isomorphism\\footnote{We do not\nassume $\\overline{t}$ is an algebraic geometric point of\n$\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$. Often using\nLemma \\ref{lemma-smooth-base-change-separably-closed}\none may reduce to this case.}.\n\\item We say $f$ is {\\it locally acyclic relative to $K$}\nif $f$ is locally acyclic at $\\overline{x}$ relative to $K$\nfor every geometric point $\\overline{x}$ of $X$.\n\\item We say $f$ is {\\it universally locally acyclic relative to $K$}\nif for any morphism $S' \\to S$ of schemes the base change $f' : X' \\to S'$\nis locally acyclic relative to the pullback of $K$ to $X'$.\n\\item We say $f$ is {\\it locally acyclic} if for all geometric\npoints $\\overline{x}$ of $X$ and any integer $n$ prime to the characteristic\nof $\\kappa(\\overline{x})$, the morphism $f$ is locally acyclic\nat $\\overline{x}$ relative to the constant sheaf with value\n$\\mathbf{Z}/n\\mathbf{Z}$.\n\\item We say $f$ is {\\it universally locally acyclic} if\nfor any morphism $S' \\to S$ of schemes the base change $f' : X' \\to S'$\nis locally acyclic.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Local acyclicity","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJP","source_file":"etale-cohomology.tex","source_line":19109,"source_end_line":19144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19109-L19144","statement_sha256":"15108d6413c9678fdde61e3aa7cd5fab84517fe07bfac05155b06a1c7be01484","origin":"The Stacks Project","memory_eligible":false,"source_rank":10174,"rank":10174,"depth":77,"x":1467.189,"y":1202.209,"cluster":"tale-geometry"},{"id":"stacks:0GJQ","tag":"0GJQ","title":"Local acyclicity · Proposition 0GJQ","summary":"Let f : X → S be a smooth morphism of schemes. Then f is universally locally acyclic.","statement_latex":"Let $f : X \\to S$ be a smooth morphism of schemes.\nThen $f$ is universally locally acyclic.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Local acyclicity","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJQ","source_file":"etale-cohomology.tex","source_line":19166,"source_end_line":19170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19166-L19170","statement_sha256":"2d6c4863b5717b7f3a3566d8111487a25f3e32c1be9f5639cadffb52edece2a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10175,"rank":10175,"depth":78,"x":1613.188,"y":1030.018,"cluster":"tale-geometry"},{"id":"stacks:0GJR","tag":"0GJR","title":"Local acyclicity · Lemma 0GJR","summary":"Let f : X → S be a morphism of schemes. Let F be a locally constant abelian sheaf on X_etale such that for every geometric point overlinex of X the abelian group F_overlinex is a torsion group all of whose elements have order prime to the characteristic of the residue field of overlinex. If f is locally acyclic, then f is locally acyclic relative to F.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$ be a\nlocally constant abelian sheaf on $X_\\etale$ such that for every geometric\npoint $\\overline{x}$ of $X$ the abelian group\n$\\mathcal{F}_{\\overline{x}}$ is a torsion group all of whose elements have\norder prime to the characteristic of the\nresidue field of $\\overline{x}$. If $f$ is locally acyclic, then $f$ is\nlocally acyclic relative to $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Local acyclicity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJR","source_file":"etale-cohomology.tex","source_line":19233,"source_end_line":19242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19233-L19242","statement_sha256":"1f339a90245fdf3574029e80da17cf809c100588f621ddf91fdbcf7271ecb0cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10176,"rank":10176,"depth":78,"x":1644.029,"y":1240.02,"cluster":"tale-geometry"},{"id":"stacks:0GJS","tag":"0GJS","title":"Local acyclicity · Lemma 0GJS","summary":"Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a cartesian diagram of schemes. Let K be an object of D(X_etale). Let overlinex' be a geometric point of X' with image overlinex in X. If • f is locally acyclic at overlinex relative to K and • g is locally quasi-finite, or S' = lim S_i is a directed inverse limit of schemes locally quasi-finite over S with affine transition morphisms, or g : S' → S is integral, then f' locally acyclic at overlinex' relative…","statement_latex":"Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nbe a cartesian diagram of schemes. Let $K$ be an object of $D(X_\\etale)$.\nLet $\\overline{x}'$ be a geometric point of $X'$ with image $\\overline{x}$\nin $X$. If\n\\begin{enumerate}\n\\item $f$ is locally acyclic at $\\overline{x}$ relative to $K$ and\n\\item $g$ is locally quasi-finite, or $S' = \\lim S_i$\nis a directed inverse limit of schemes locally quasi-finite over $S$\nwith affine transition morphisms, or $g : S' \\to S$ is integral,\n\\end{enumerate}\nthen $f'$ locally acyclic at $\\overline{x}'$ relative to $(g')^{-1}K$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Local acyclicity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJS","source_file":"etale-cohomology.tex","source_line":19288,"source_end_line":19307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19288-L19307","statement_sha256":"a1723e57b369a3a633f4705ce91786baef2ff01e3aa832c6d99e8c3071672ef6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10177,"rank":10177,"depth":51,"x":1452.238,"y":1102.545,"cluster":"tale-geometry"},{"id":"stacks:0GJU","tag":"0GJU","title":"The cospecialization map · Lemma 0GJU","summary":"The map i^-1β_K, overlines, overlinet is an isomorphism.","statement_latex":"The map $i^{-1}\\beta_{K, \\overline{s}, \\overline{t}}$ is an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The cospecialization map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJU","source_file":"etale-cohomology.tex","source_line":19419,"source_end_line":19422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19419-L19422","statement_sha256":"66b386d633d92ac58dab075dfadcde02580b7c1b62a5dc15726e03344dcaa70e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10178,"rank":10178,"depth":78,"x":1704.441,"y":1095.084,"cluster":"tale-geometry"},{"id":"stacks:0GJV","tag":"0GJV","title":"The cospecialization map · Lemma 0GJV","summary":"In the situation above, if in addition f is quasi-compact and quasi-separated, then the diagram xymatrix (Rf_*K)_overlines ar[r] ar[d]_sp & RΓ(X_overlines, K) (Rf_*K)_overlinet ar[r] & RΓ(X_overlinet, K) ar[u]_cosp is commutative.","statement_latex":"In the situation above, if in addition\n$f$ is quasi-compact and quasi-separated, then the diagram\n$$\n\\xymatrix{\n(Rf_*K)_{\\overline{s}} \\ar[r] \\ar[d]_{sp} &\nR\\Gamma(X_{\\overline{s}}, K) \\\\\n(Rf_*K)_{\\overline{t}} \\ar[r] &\nR\\Gamma(X_{\\overline{t}}, K) \\ar[u]_{cosp}\n}\n$$\nis commutative.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The cospecialization map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJV","source_file":"etale-cohomology.tex","source_line":19468,"source_end_line":19481,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19468-L19481","statement_sha256":"3a234ab763b2be6feb03212c0d5feeafdbefca369965c45189bb5de2b741d5ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":10179,"rank":10179,"depth":79,"x":1524.31,"y":1243.824,"cluster":"tale-geometry"},{"id":"stacks:0GJW","tag":"0GJW","title":"The cospecialization map · Lemma 0GJW","summary":"Let f : X → S be a morphism of schemes. Let K ∈ D(X_etale). Assume • K is bounded below, i.e., K ∈ D^+(X_etale), • f is locally acyclic relative to K, • f is proper, and • K has torsion cohomology sheaves. Then for every geometric point overlines of S and every geometric point overlinet of Spec(O^sh_S, overlines) both the specialization map sp : (Rf_*K)_overlines → (Rf_*K)_overlinet and the cospecialization map cosp : RΓ(X_overlinet, K) → RΓ(X_overlines, K) are isomorphisms.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $K \\in D(X_\\etale)$.\nAssume\n\\begin{enumerate}\n\\item $K$ is bounded below, i.e., $K \\in D^+(X_\\etale)$,\n\\item $f$ is locally acyclic relative to $K$,\n\\item $f$ is proper, and\n\\item $K$ has torsion cohomology sheaves.\n\\end{enumerate}\nThen for every geometric point $\\overline{s}$ of $S$ and every geometric\npoint $\\overline{t}$ of $\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$\nboth the specialization map\n$sp : (Rf_*K)_{\\overline{s}} \\to (Rf_*K)_{\\overline{t}}$\nand the cospecialization map\n$cosp : R\\Gamma(X_{\\overline{t}}, K) \\to R\\Gamma(X_{\\overline{s}}, K)$\nare isomorphisms.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The cospecialization map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJW","source_file":"etale-cohomology.tex","source_line":19506,"source_end_line":19523,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19506-L19523","statement_sha256":"8ea53ca1f56009578344b2698c0adac989869af5344381bc0e383d0dd8bf70e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10180,"rank":10180,"depth":80,"x":1537.535,"y":1031.746,"cluster":"tale-geometry"},{"id":"stacks:0GKD","tag":"0GKD","title":"The cospecialization map · Lemma 0GKD","summary":"Let f : X → S be a morphism of schemes. Let F be an abelian sheaf on X_etale. Assume • f is smooth and proper • F is locally constant, and • F_overlinex is a torsion group all of whose elements have order prime to the residue characteristic of overlinex for every geometric point overlinex of X. Then for every geometric point overlines of S and every geometric point overlinet of Spec(O^sh_S, overlines) the specialization map sp : (Rf_*F)_overlines → (Rf_*F)_overlinet is an…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $\\mathcal{F}$\nbe an abelian sheaf on $X_\\etale$. Assume\n\\begin{enumerate}\n\\item $f$ is smooth and proper\n\\item $\\mathcal{F}$ is locally constant, and\n\\item $\\mathcal{F}_{\\overline{x}}$ is a torsion group all of\nwhose elements have order prime to the residue characteristic of\n$\\overline{x}$ for every geometric point $\\overline{x}$ of $X$.\n\\end{enumerate}\nThen for every geometric point $\\overline{s}$ of $S$ and every geometric\npoint $\\overline{t}$ of $\\Spec(\\mathcal{O}^{sh}_{S, \\overline{s}})$\nthe specialization map\n$sp : (Rf_*\\mathcal{F})_{\\overline{s}} \\to (Rf_*\\mathcal{F})_{\\overline{t}}$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"The cospecialization map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKD","source_file":"etale-cohomology.tex","source_line":19562,"source_end_line":19578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19562-L19578","statement_sha256":"570dd95e114bf6955519908ec2b3fc172c4c03481a60c280715a821bdf1c7fae","origin":"The Stacks Project","memory_eligible":false,"source_rank":10181,"rank":10181,"depth":81,"x":1698.473,"y":1195.779,"cluster":"tale-geometry"},{"id":"stacks:0F0Q","tag":"0F0Q","title":"Cohomological dimension · Definition 0F0Q","summary":"Let X be a quasi-compact and quasi-separated scheme. The cohomological dimension of X is the smallest element cd(X) ∈ (0, 1, 2, …) ∪ (∞) such that for any abelian torsion sheaf F on X_etale we have H^i_etale(X, F) = 0 for i > cd(X). If X = Spec(A) we sometimes call this the cohomological dimension of A.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nThe {\\it cohomological dimension of $X$} is the smallest\nelement\n$$\n\\text{cd}(X) \\in \\{0, 1, 2, \\ldots\\} \\cup \\{\\infty\\}\n$$\nsuch that for any abelian torsion sheaf $\\mathcal{F}$\non $X_\\etale$ we have $H^i_\\etale(X, \\mathcal{F}) = 0$\nfor $i > \\text{cd}(X)$. If $X = \\Spec(A)$ we sometimes\ncall this the cohomological dimension of $A$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomological dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0Q","source_file":"etale-cohomology.tex","source_line":19612,"source_end_line":19624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19612-L19624","statement_sha256":"3e622f2c9847eedc25fa9b0e5ef088e9477a2cd98f2de144f293a1606a9b8032","origin":"The Stacks Project","memory_eligible":false,"source_rank":10182,"rank":10182,"depth":0,"x":1447.667,"y":1166.117,"cluster":"tale-geometry"},{"id":"stacks:0F0R","tag":"0F0R","title":"Cohomological dimension · Lemma 0F0R","summary":"Let X = lim X_i be a directed limit of a system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then cd(X) ≤ max cd(X_i).","statement_latex":"Let $X = \\lim X_i$ be a directed limit of a system of\nquasi-compact and quasi-separated schemes with affine\ntransition morphisms. Then $\\text{cd}(X) \\leq \\max \\text{cd}(X_i)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0R","source_file":"etale-cohomology.tex","source_line":19632,"source_end_line":19637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19632-L19637","statement_sha256":"18177244745025e0f511fae4eb75d4b31856777cbf08ff1238c96df3efa61321","origin":"The Stacks Project","memory_eligible":false,"source_rank":10183,"rank":10183,"depth":42,"x":1656.646,"y":1045.569,"cluster":"tale-geometry"},{"id":"stacks:0F0S","tag":"0F0S","title":"Cohomological dimension · Lemma 0F0S","summary":"Let K be a field. Let X be a 1-dimensional affine scheme of finite type over K. Then cd(X) ≤ 1 + cd(K).","statement_latex":"Let $K$ be a field. Let $X$ be a $1$-dimensional\naffine scheme of finite type over $K$. Then\n$\\text{cd}(X) \\leq 1 + \\text{cd}(K)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0S","source_file":"etale-cohomology.tex","source_line":19650,"source_end_line":19655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19650-L19655","statement_sha256":"0da61601582d72ff644e98e310c11a1ee1e8dff4864dc5f8f406359005222065","origin":"The Stacks Project","memory_eligible":false,"source_rank":10184,"rank":10184,"depth":73,"x":1599.439,"y":1253.223,"cluster":"tale-geometry"},{"id":"stacks:0F0T","tag":"0F0T","title":"Cohomological dimension · Lemma 0F0T","summary":"Let L/K be a field extension. Then we have cd(L) ≤ cd(K) + trdeg_K(L).","statement_latex":"Let $L/K$ be a field extension. Then we have\n$\\text{cd}(L) \\leq \\text{cd}(K) + \\text{trdeg}_K(L)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0T","source_file":"etale-cohomology.tex","source_line":19672,"source_end_line":19676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19672-L19676","statement_sha256":"f4513d5a6b4c8597767fc2f09f3c0bb4c5b9fcd81fd99bc6f1373225f66fd1c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10185,"rank":10185,"depth":74,"x":1474.523,"y":1067.478,"cluster":"tale-geometry"},{"id":"stacks:0F0U","tag":"0F0U","title":"Cohomological dimension · Lemma 0F0U","summary":"Let K be a field. Let X be a scheme of finite type over K. Let x ∈ X. Set a = trdeg_K(kappa(x)) and d = dim_x(X). Then there is a map K(t_1, …, t_a)^sep → O_X, x^sh such that • the residue field of O_X, x^sh is a purely inseparable extension of K(t_1, …, t_a)^sep, • O_X, x^sh is a filtered colimit of finite type K(t_1, …, t_a)^sep-algebras of dimension ≤ d - a.","statement_latex":"Let $K$ be a field. Let $X$ be a scheme of finite type over $K$.\nLet $x \\in X$. Set $a = \\text{trdeg}_K(\\kappa(x))$\nand $d = \\dim_x(X)$. Then there is a map\n$$\nK(t_1, \\ldots, t_a)^{sep} \\longrightarrow \\mathcal{O}_{X, x}^{sh}\n$$\nsuch that\n\\begin{enumerate}\n\\item the residue field of $\\mathcal{O}_{X, x}^{sh}$ is a purely inseparable\nextension of $K(t_1, \\ldots, t_a)^{sep}$,\n\\item $\\mathcal{O}_{X, x}^{sh}$ is a filtered colimit of finite\ntype $K(t_1, \\ldots, t_a)^{sep}$-algebras of dimension $\\leq d - a$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0U","source_file":"etale-cohomology.tex","source_line":19691,"source_end_line":19706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19691-L19706","statement_sha256":"d776e7642d1e8b5381d256c38255c7c0eb7296007e2a32083f25066ddf088ce3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10186,"rank":10186,"depth":12,"x":1716.218,"y":1133.621,"cluster":"tale-geometry"},{"id":"stacks:0F0V","tag":"0F0V","title":"Cohomological dimension · Proposition 0F0V","summary":"Let K be a field. Let X be an affine scheme of finite type over K. Then we have cd(X) ≤ dim(X) + cd(K).","statement_latex":"Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$.\nThen we have $\\text{cd}(X) \\leq \\dim(X) + \\text{cd}(K)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomological dimension","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0V","source_file":"etale-cohomology.tex","source_line":19762,"source_end_line":19766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19762-L19766","statement_sha256":"c44b44bc00a0886631d3649d6899d561595320e09b0627bb128ef98ccc490d65","origin":"The Stacks Project","memory_eligible":false,"source_rank":10187,"rank":10187,"depth":78,"x":1484.601,"y":1222.069,"cluster":"tale-geometry"},{"id":"stacks:0F0W","tag":"0F0W","title":"Cohomological dimension · Lemma 0F0W","summary":"Let K be a field. Let X be an affine scheme of finite type over K. Let E_a ⊂ X be the set of points x ∈ X with trdeg_K(kappa(x)) ≤ a. Let F be an abelian torsion sheaf on X_etale whose support is contained in E_a. Then H^b_etale(X, F) = 0 for b > a + cd(K).","statement_latex":"Let $K$ be a field. Let $X$ be an affine scheme of finite type over $K$.\nLet $E_a \\subset X$ be the set of points\n$x \\in X$ with $\\text{trdeg}_K(\\kappa(x)) \\leq a$.\nLet $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$\nwhose support is contained in $E_a$. Then\n$H^b_\\etale(X, \\mathcal{F}) = 0$ for $b > a + \\text{cd}(K)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0W","source_file":"etale-cohomology.tex","source_line":19915,"source_end_line":19923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19915-L19923","statement_sha256":"aa6a78ca6b3aa4ceaccc15d59c290e53e7ad95fff6c5466c1886a361c6e32866","origin":"The Stacks Project","memory_eligible":false,"source_rank":10188,"rank":10188,"depth":79,"x":1584.352,"y":1025.252,"cluster":"tale-geometry"},{"id":"stacks:0F0X","tag":"0F0X","title":"Cohomological dimension · Lemma 0F0X","summary":"Let f : X → Y be an affine morphism of schemes of finite type over a field K. Let E_a(X) be the set of points x ∈ X with trdeg_K(kappa(x)) ≤ a. Let F be an abelian torsion sheaf on X_etale whose support is contained in E_a. Then R^qf_*F has support contained in E_a - q(Y).","statement_latex":"Let $f : X \\to Y$ be an affine morphism of schemes of finite\ntype over a field $K$. Let $E_a(X)$ be the set of points $x \\in X$\nwith $\\text{trdeg}_K(\\kappa(x)) \\leq a$.\nLet $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$\nwhose support is contained in $E_a$. Then\n$R^qf_*\\mathcal{F}$ has support contained in\n$E_{a - q}(Y)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0X","source_file":"etale-cohomology.tex","source_line":19940,"source_end_line":19949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L19940-L19949","statement_sha256":"2289b5f2db5cc449163fbbc0c797e630883a4555735752fdb02fdf847f0e3366","origin":"The Stacks Project","memory_eligible":false,"source_rank":10189,"rank":10189,"depth":80,"x":1669.147,"y":1227.159,"cluster":"tale-geometry"},{"id":"stacks:0F0Z","tag":"0F0Z","title":"Finite cohomological dimension · Definition 0F0Z","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of schemes. The cohomological dimension of f is the smallest element cd(f) ∈ (0, 1, 2, …) ∪ (∞) such that for any abelian torsion sheaf F on X_etale we have R^if_*F = 0 for i > cd(f).","statement_latex":"Let $f : X \\to Y$ be a quasi-compact and quasi-separated\nmorphism of schemes.\nThe {\\it cohomological dimension of $f$} is the smallest\nelement\n$$\n\\text{cd}(f) \\in \\{0, 1, 2, \\ldots\\} \\cup \\{\\infty\\}\n$$\nsuch that for any abelian torsion sheaf $\\mathcal{F}$\non $X_\\etale$ we have $R^if_*\\mathcal{F} = 0$\nfor $i > \\text{cd}(f)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Finite cohomological dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0Z","source_file":"etale-cohomology.tex","source_line":20026,"source_end_line":20038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20026-L20038","statement_sha256":"a55af91a8a4924f9df5d929ec3a90fa271bd45646eba4f1a7971e0233b3d0da6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10190,"rank":10190,"depth":0,"x":1444.055,"y":1126.302,"cluster":"tale-geometry"},{"id":"stacks:0F10","tag":"0F10","title":"Finite cohomological dimension · Lemma 0F10","summary":"Let K be a field. • If f : X → Y is a morphism of finite type schemes over K, then cd(f) < ∞. • If cd(K) < ∞, then cd(X) < ∞ for any finite type scheme X over K.","statement_latex":"Let $K$ be a field.\n\\begin{enumerate}\n\\item If $f : X \\to Y$ is a morphism of finite type schemes over $K$,\nthen $\\text{cd}(f) < \\infty$.\n\\item If $\\text{cd}(K) < \\infty$, then $\\text{cd}(X) < \\infty$\nfor any finite type scheme $X$ over $K$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Finite cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F10","source_file":"etale-cohomology.tex","source_line":20040,"source_end_line":20049,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20040-L20049","statement_sha256":"1fe5851e8196fd410cb9ff97e1acaad724ed33ae3a86627c0048ca54fd26568e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10191,"rank":10191,"depth":81,"x":1691.356,"y":1072.904,"cluster":"tale-geometry"},{"id":"stacks:0F11","tag":"0F11","title":"Finite cohomological dimension · Lemma 0F11","summary":"Cohomology and direct sums. Let n ≥ 1 be an integer. • Let f : X → Y be a quasi-compact and quasi-separated morphism of schemes with cd(f) < ∞. Then the functor Rf_* : D(X_etale, Z/nZ) → D(Y_etale, Z/nZ) commutes with direct sums. • Let X be a quasi-compact and quasi-separated scheme with cd(X) < ∞. Then the functor RΓ(X, -) : D(X_etale, Z/nZ) → D(Z/nZ) commutes with direct sums.","statement_latex":"Cohomology and direct sums. Let $n \\geq 1$ be an integer.\n\\begin{enumerate}\n\\item Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism\nof schemes with $\\text{cd}(f) < \\infty$. Then the functor\n$$\nRf_* :\nD(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})\n\\longrightarrow\nD(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})\n$$\ncommutes with direct sums.\n\\item Let $X$ be a quasi-compact and quasi-separated scheme with\n$\\text{cd}(X) < \\infty$. Then the functor\n$$\nR\\Gamma(X, -) :\nD(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})\n\\longrightarrow\nD(\\mathbf{Z}/n\\mathbf{Z})\n$$\ncommutes with direct sums.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Finite cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F11","source_file":"etale-cohomology.tex","source_line":20073,"source_end_line":20096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20073-L20096","statement_sha256":"bab3995d0bbca4454873b347ce31cb975131b06eac49120848596b0b4658b027","origin":"The Stacks Project","memory_eligible":false,"source_rank":10192,"rank":10192,"depth":43,"x":1551.82,"y":1252.76,"cluster":"tale-geometry"},{"id":"stacks:0F0D","tag":"0F0D","title":"Finite cohomological dimension · Lemma 0F0D","summary":"Let f : X → Y be a proper morphism of schemes. Let n ≥ 1 be an integer. Then the functor Rf_* : D(X_etale, Z/nZ) → D(Y_etale, Z/nZ) commutes with direct sums.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $n \\geq 1$\nbe an integer. Then the functor\n$$\nRf_* :\nD(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})\n\\longrightarrow\nD(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})\n$$\ncommutes with direct sums.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Finite cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0D","source_file":"etale-cohomology.tex","source_line":20140,"source_end_line":20151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20140-L20151","statement_sha256":"b6d6dd41dbdba3bc177ab2c95a9a2960b1ed8968f27d6f7511b3ec58b7f485c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10193,"rank":10193,"depth":78,"x":1510.04,"y":1040.777,"cluster":"tale-geometry"},{"id":"stacks:0F12","tag":"0F12","title":"Finite cohomological dimension · Lemma 0F12","summary":"Let X be a quasi-compact and quasi-separated scheme such that cd(X) < ∞. Let Lambda be a torsion ring. Let E ∈ D(X_etale, Lambda) and K ∈ D(Lambda). Then RΓ(X, E ⊗_Lambda^L underlineK) = RΓ(X, E) ⊗_Lambda^L K","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme\nsuch that $\\text{cd}(X) < \\infty$. Let $\\Lambda$ be a torsion ring.\nLet $E \\in D(X_\\etale, \\Lambda)$ and $K \\in D(\\Lambda)$. Then\n$$\nR\\Gamma(X, E \\otimes_\\Lambda^\\mathbf{L} \\underline{K}) =\nR\\Gamma(X, E) \\otimes_\\Lambda^\\mathbf{L} K\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Finite cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F12","source_file":"etale-cohomology.tex","source_line":20163,"source_end_line":20172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20163-L20172","statement_sha256":"dcbb59d68222691d81df42ff8ce6c3fae9fd8d160fbe386e20858b90d7fd2661","origin":"The Stacks Project","memory_eligible":false,"source_rank":10194,"rank":10194,"depth":44,"x":1711.494,"y":1173.498,"cluster":"tale-geometry"},{"id":"stacks:0F0G","tag":"0F0G","title":"Finite cohomological dimension · Lemma 0F0G","summary":"Let f : X → Y be a proper morphism of schemes. Let Lambda be a torsion ring. Let E ∈ D(X_etale, Lambda) and K ∈ D(Y_etale, Lambda). Then Rf_*E ⊗_Lambda^L K = Rf_*(E ⊗_Lambda^L f^-1K) in D(Y_etale, Lambda).","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $\\Lambda$\nbe a torsion ring. Let $E \\in D(X_\\etale, \\Lambda)$ and\n$K \\in D(Y_\\etale, \\Lambda)$. Then\n$$\nRf_*E \\otimes_\\Lambda^\\mathbf{L} K =\nRf_*(E \\otimes_\\Lambda^\\mathbf{L} f^{-1}K)\n$$\nin $D(Y_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Finite cohomological dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0G","source_file":"etale-cohomology.tex","source_line":20192,"source_end_line":20202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20192-L20202","statement_sha256":"036d1d4a86569f08ddac142c74145dd62ef82011a444aa771823375f563c43df","origin":"The Stacks Project","memory_eligible":false,"source_rank":10195,"rank":10195,"depth":79,"x":1455.993,"y":1189.955,"cluster":"tale-geometry"},{"id":"stacks:0F14","tag":"0F14","title":"K\\\"unneth in étale cohomology · Lemma 0F14","summary":"Let k be a separably closed field. Let X be a proper scheme over k. Let Y be a quasi-compact and quasi-separated scheme over k. • If E ∈ D^+(X_etale) has torsion cohomology sheaves and K ∈ D^+(Y_etale), then RΓ(X ×_Spec(k) Y, pr_1^-1E ⊗_Z^L pr_2^-1K ) = RΓ(X, E) ⊗_Z^L RΓ(Y, K) • If n ≥ 1 is an integer, Y is of finite type over k, E ∈ D(X_etale, Z/nZ), and K ∈ D(Y_etale, Z/nZ), then RΓ(X ×_Spec(k) Y, pr_1^-1E ⊗_Z/nZ^L pr_2^-1K ) = RΓ(X, E) ⊗_Z/nZ^L RΓ(Y, K)","statement_latex":"Let $k$ be a separably closed field. Let $X$ be a proper scheme over $k$.\nLet $Y$ be a quasi-compact and quasi-separated scheme over $k$.\n\\begin{enumerate}\n\\item If $E \\in D^+(X_\\etale)$ has torsion cohomology sheaves and\n$K \\in D^+(Y_\\etale)$, then\n$$\nR\\Gamma(X \\times_{\\Spec(k)} Y,\n\\text{pr}_1^{-1}E\n\\otimes_\\mathbf{Z}^\\mathbf{L}\n\\text{pr}_2^{-1}K\n)\n=\nR\\Gamma(X, E)\n\\otimes_\\mathbf{Z}^\\mathbf{L}\nR\\Gamma(Y, K)\n$$\n\\item If $n \\geq 1$ is an integer, $Y$ is of finite type over $k$,\n$E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$, and\n$K \\in D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$, then\n$$\nR\\Gamma(X \\times_{\\Spec(k)} Y,\n\\text{pr}_1^{-1}E\n\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}\n\\text{pr}_2^{-1}K\n)\n=\nR\\Gamma(X, E)\n\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}\nR\\Gamma(Y, K)\n$$\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"K\\\"unneth in étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F14","source_file":"etale-cohomology.tex","source_line":20266,"source_end_line":20299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20266-L20299","statement_sha256":"b91e60a4b79f68951eafc3934272a0ff1e0bfc60d2f625991c84275c703934db","origin":"The Stacks Project","memory_eligible":false,"source_rank":10196,"rank":10196,"depth":82,"x":1631.313,"y":1032.707,"cluster":"tale-geometry"},{"id":"stacks:0F1F","tag":"0F1F","title":"K\\\"unneth in étale cohomology · Lemma 0F1F","summary":"Let K be a separably closed field. Let X be a scheme of finite type over K. Let F be an abelian sheaf on X_etale whose support is contained in the set of closed points of X. Then H^q(X, F) = 0 for q > 0 and F is globally generated.","statement_latex":"Let $K$ be a separably closed field. Let $X$ be a scheme of finite\ntype over $K$. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$\nwhose support is contained in the set of closed points of $X$.\nThen $H^q(X, \\mathcal{F}) = 0$ for $q > 0$ and $\\mathcal{F}$\nis globally generated.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"K\\\"unneth in étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1F","source_file":"etale-cohomology.tex","source_line":20332,"source_end_line":20339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20332-L20339","statement_sha256":"f44267be1e163e41662cfabc1e7cdb08e898bdab5fdeb3af71fc9dbd6c234e8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10197,"rank":10197,"depth":80,"x":1628.487,"y":1248.326,"cluster":"tale-geometry"},{"id":"stacks:0F1G","tag":"0F1G","title":"K\\\"unneth in étale cohomology · Lemma 0F1G","summary":"Let K be a separably closed field. Let X be a scheme of finite type over K. Let Q ∈ D(X_etale). Assume that Q_overlinex is nonzero only if x is a closed point of X. Then Q = 0 ⇔ H^i(X, Q) = 0 for all i","statement_latex":"Let $K$ be a separably closed field. Let $X$ be a scheme of finite\ntype over $K$. Let $Q \\in D(X_\\etale)$. Assume that $Q_{\\overline{x}}$\nis nonzero only if $x$ is a closed point of $X$. Then\n$$\nQ = 0 \\Leftrightarrow H^i(X, Q) = 0 \\text{ for all }i\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"K\\\"unneth in étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1G","source_file":"etale-cohomology.tex","source_line":20369,"source_end_line":20377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20369-L20377","statement_sha256":"17aaa1339b3564176642936a2e3a1ba34d67f8179bc6f875ab837754e86bd76b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10198,"rank":10198,"depth":81,"x":1457.028,"y":1087.59,"cluster":"tale-geometry"},{"id":"stacks:0F1H","tag":"0F1H","title":"K\\\"unneth in étale cohomology · Lemma 0F1H","summary":"Let K be a field. Let j : U → X be an open immersion of schemes of finite type over K. Let Y be a scheme of finite type over K. Consider the diagram xymatrix Y ×_Spec(K) X ar[d]_q & Y ×_Spec(K) U ar[l]^h ar[d]^p X & U ar[l]_j Then the base change map q^-1Rj_*F → Rh_*p^-1F is an isomorphism for F an abelian sheaf on U_etale whose stalks are torsion of orders invertible in K.","statement_latex":"Let $K$ be a field. Let $j : U \\to X$ be an open immersion of\nschemes of finite type over $K$. Let $Y$ be a scheme of finite type\nover $K$. Consider the diagram\n$$\n\\xymatrix{\nY \\times_{\\Spec(K)} X \\ar[d]_q  &\nY \\times_{\\Spec(K)} U \\ar[l]^h \\ar[d]^p \\\\\nX & U \\ar[l]_j\n}\n$$\nThen the base change map $q^{-1}Rj_*\\mathcal{F} \\to Rh_*p^{-1}\\mathcal{F}$\nis an isomorphism for $\\mathcal{F}$ an abelian sheaf on $U_\\etale$\nwhose stalks are torsion of orders invertible in $K$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"K\\\"unneth in étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1H","source_file":"etale-cohomology.tex","source_line":20402,"source_end_line":20417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20402-L20417","statement_sha256":"cf436beeddeaacc23d37ef25dbbc3d94c2562233eda509073eda1c67915e80dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10199,"rank":10199,"depth":83,"x":1712.939,"y":1108.843,"cluster":"tale-geometry"},{"id":"stacks:0F1I","tag":"0F1I","title":"K\\\"unneth in étale cohomology · Lemma 0F1I","summary":"Let K be a field. For any commutative diagram xymatrix X ar[d] & X' ar[l] ar[d]_f' & Y ar[l]^h ar[d]^e Spec(K) & S' ar[l] & T ar[l]_g of schemes over K with X' = X ×_Spec(K) S' and Y = X' ×_S' T and g quasi-compact and quasi-separated, and every abelian sheaf F on T_etale whose stalks are torsion of orders invertible in K the base change map (f')^-1Rg_*F → Rh_*e^-1F is an isomorphism.","statement_latex":"Let $K$ be a field. For any commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l] \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\\n\\Spec(K) & S' \\ar[l] & T \\ar[l]_g\n}\n$$\nof schemes over $K$ with\n$X' = X \\times_{\\Spec(K)} S'$ and $Y = X' \\times_{S'} T$ and\n$g$ quasi-compact and quasi-separated, and every abelian sheaf\n$\\mathcal{F}$ on $T_\\etale$ whose stalks are torsion of orders\ninvertible in $K$ the base change map\n$$\n(f')^{-1}Rg_*\\mathcal{F}\n\\longrightarrow\nRh_*e^{-1}\\mathcal{F}\n$$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"K\\\"unneth in étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1I","source_file":"etale-cohomology.tex","source_line":20596,"source_end_line":20616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20596-L20616","statement_sha256":"b76e2d37dcae7c80881809f025aa1ddda8eb59a71e5edb7745eacd30d0c26a7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10200,"rank":10200,"depth":84,"x":1506.966,"y":1238.492,"cluster":"tale-geometry"},{"id":"stacks:0F1J","tag":"0F1J","title":"K\\\"unneth in étale cohomology · Lemma 0F1J","summary":"Let K be a field. Let n ≥ 1 be invertible in K. Consider a commutative diagram xymatrix X ar[d] & X' ar[l]^p ar[d]_f' & Y ar[l]^h ar[d]^e Spec(K) & S' ar[l] & T ar[l]_g of schemes with X' = X ×_Spec(K) S' and Y = X' ×_S' T and g quasi-compact and quasi-separated. The canonical map p^-1E ⊗_Z/nZ^L (f')^-1Rg_*F → Rh_*(h^-1p^-1E ⊗_Z/nZ^L e^-1F) is an isomorphism if E in D^+(X_etale, Z/nZ) has tor amplitude in [a, ∞] for some a ∈ Z and F in D^+(T_etale, Z/nZ).","statement_latex":"Let $K$ be a field. Let $n \\geq 1$ be invertible in $K$.\nConsider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l]^p \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\\n\\Spec(K) & S' \\ar[l] & T \\ar[l]_g\n}\n$$\nof schemes with\n$X' = X \\times_{\\Spec(K)} S'$ and $Y = X' \\times_{S'} T$ and\n$g$ quasi-compact and quasi-separated. The canonical map\n$$\np^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} (f')^{-1}Rg_*F\n\\longrightarrow\nRh_*(h^{-1}p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} e^{-1}F)\n$$\nis an isomorphism if $E$ in $D^+(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$\nhas tor amplitude in $[a, \\infty]$ for some $a \\in \\mathbf{Z}$ and\n$F$ in $D^+(T_\\etale, \\mathbf{Z}/n\\mathbf{Z})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"K\\\"unneth in étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1J","source_file":"etale-cohomology.tex","source_line":20705,"source_end_line":20726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20705-L20726","statement_sha256":"96963e73e30ef26ef62c55fcf1d620ec92c9727bd690e2136d731315be8cfcc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10201,"rank":10201,"depth":85,"x":1554.628,"y":1025.835,"cluster":"tale-geometry"},{"id":"stacks:0F1N","tag":"0F1N","title":"K\\\"unneth in étale cohomology · Lemma 0F1N","summary":"Let K be a field. Let n ≥ 1 be invertible in K. Consider a commutative diagram xymatrix X ar[d] & X' ar[l]^p ar[d]_f' & Y ar[l]^h ar[d]^e Spec(K) & S' ar[l] & T ar[l]_g of schemes of finite type over K with X' = X ×_Spec(K) S' and Y = X' ×_S' T. The canonical map p^-1E ⊗_Z/nZ^L (f')^-1Rg_*F → Rh_*(h^-1p^-1E ⊗_Z/nZ^L e^-1F) is an isomorphism for E in D(X_etale, Z/nZ) and F in D(T_etale, Z/nZ).","statement_latex":"Let $K$ be a field. Let $n \\geq 1$ be invertible in $K$.\nConsider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l]^p \\ar[d]_{f'} & Y \\ar[l]^h \\ar[d]^e \\\\\n\\Spec(K) & S' \\ar[l] & T \\ar[l]_g\n}\n$$\nof schemes of finite type over $K$ with\n$X' = X \\times_{\\Spec(K)} S'$ and $Y = X' \\times_{S'} T$.\nThe canonical map\n$$\np^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} (f')^{-1}Rg_*F\n\\longrightarrow\nRh_*(h^{-1}p^{-1}E \\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L} e^{-1}F)\n$$\nis an isomorphism for $E$ in $D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$\nand $F$ in $D(T_\\etale, \\mathbf{Z}/n\\mathbf{Z})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"K\\\"unneth in étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1N","source_file":"etale-cohomology.tex","source_line":20878,"source_end_line":20898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20878-L20898","statement_sha256":"20907c2290e5ee080ade5ebb131d771e54c770be7ce136554d225419404c8db0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10202,"rank":10202,"depth":86,"x":1690.612,"y":1209.846,"cluster":"tale-geometry"},{"id":"stacks:0F1P","tag":"0F1P","title":"K\\\"unneth in étale cohomology · Lemma 0F1P","summary":"Let k be a separably closed field. Let X and Y be finite type schemes over k. Let n ≥ 1 be an integer invertible in k. Then for E ∈ D(X_etale, Z/nZ) and K ∈ D(Y_etale, Z/nZ) we have RΓ(X ×_Spec(k) Y, pr_1^-1E ⊗_Z/nZ^L pr_2^-1K ) = RΓ(X, E) ⊗_Z/nZ^L RΓ(Y, K)","statement_latex":"Let $k$ be a separably closed field. Let $X$ and $Y$ be\nfinite type schemes over $k$. Let $n \\geq 1$ be an integer\ninvertible in $k$. Then for\n$E \\in D(X_\\etale, \\mathbf{Z}/n\\mathbf{Z})$ and\n$K \\in D(Y_\\etale, \\mathbf{Z}/n\\mathbf{Z})$\nwe have\n$$\nR\\Gamma(X \\times_{\\Spec(k)} Y,\n\\text{pr}_1^{-1}E\n\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}\n\\text{pr}_2^{-1}K\n)\n=\nR\\Gamma(X, E)\n\\otimes_{\\mathbf{Z}/n\\mathbf{Z}}^\\mathbf{L}\nR\\Gamma(Y, K)\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"K\\\"unneth in étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1P","source_file":"etale-cohomology.tex","source_line":20937,"source_end_line":20956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L20937-L20956","statement_sha256":"e5f4c07866043ef018d5726b11a0123a3a4d98bed16579a26f685d32160661c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10203,"rank":10203,"depth":87,"x":1442.15,"y":1151.27,"cluster":"tale-geometry"},{"id":"stacks:0F1L","tag":"0F1L","title":"Comparing chaotic and Zariski topologies · Lemma 0F1L","summary":"In the situation above let K be an object of D^+(X_affine). Then K is in the essential image of the (fully faithful) functor Rε_* ; D(X_affine, Zar) → D(X_affine) if and only if the following two conditions hold • RΓ(∅, K) is zero in D(Ab), and • if U = V ∪ W with U, V, W ⊂ X affine open and V, W ⊂ U standard open (Algebra, Definition [Tag 00E1]), then the map c^K_U, V, W, V ∩ W of Cohomology on Sites, Lemma [Tag 0F16] is a quasi-isomorphism.","statement_latex":"In the situation above let $K$ be an object of $D^+(X_{affine})$.\nThen $K$ is in the essential image of the (fully faithful) functor\n$R\\epsilon_* ; D(X_{affine, Zar}) \\to D(X_{affine})$ if and only\nif the following two conditions hold\n\\begin{enumerate}\n\\item $R\\Gamma(\\emptyset, K)$ is zero in $D(\\textit{Ab})$, and\n\\item if $U = V \\cup W$ with $U, V, W \\subset X$ affine open and\n$V, W \\subset U$ standard open\n(Algebra, Definition \\ref{algebra-definition-Zariski-topology}), then\nthe map $c^K_{U, V, W, V \\cap W}$ of\nCohomology on Sites, Lemma \\ref{sites-cohomology-lemma-c-square}\nis a quasi-isomorphism.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing chaotic and Zariski topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1L","source_file":"etale-cohomology.tex","source_line":21009,"source_end_line":21024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21009-L21024","statement_sha256":"68ee6890962226cb8eef81a95a5b7c98ffc657b36438dd068701c64189ac0fe4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10204,"rank":10204,"depth":15,"x":1672.664,"y":1053.397,"cluster":"tale-geometry"},{"id":"stacks:0758","tag":"0758","title":"Comparing big and small topoi · Lemma 0758","summary":"Let S be a scheme. Let T be an object of (Sch/S)_etale. • If I is injective in Ab((Sch/S)_etale), then • i_f^-1I is injective in Ab(T_etale), • I|_S_etale is injective in Ab(S_etale), • If I^bullet is a K-injective complex in Ab((Sch/S)_etale), then • i_f^-1I^bullet is a K-injective complex in Ab(T_etale), • I^bullet|_S_etale is a K-injective complex in Ab(S_etale), The corresponding statements for modules do not hold.","statement_latex":"Let $S$ be a scheme. Let $T$ be an object of $(\\Sch/S)_\\etale$.\n\\begin{enumerate}\n\\item If $\\mathcal{I}$ is injective in $\\textit{Ab}((\\Sch/S)_\\etale)$, then\n\\begin{enumerate}\n\\item $i_f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(T_\\etale)$,\n\\item $\\mathcal{I}|_{S_\\etale}$ is injective in $\\textit{Ab}(S_\\etale)$,\n\\end{enumerate}\n\\item If $\\mathcal{I}^\\bullet$ is a K-injective complex\nin $\\textit{Ab}((\\Sch/S)_\\etale)$, then\n\\begin{enumerate}\n\\item $i_f^{-1}\\mathcal{I}^\\bullet$ is a K-injective complex in\n$\\textit{Ab}(T_\\etale)$,\n\\item $\\mathcal{I}^\\bullet|_{S_\\etale}$ is a K-injective complex in\n$\\textit{Ab}(S_\\etale)$,\n\\end{enumerate}\n\\end{enumerate}\nThe corresponding statements for modules do not hold.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0758","source_file":"etale-cohomology.tex","source_line":21101,"source_end_line":21120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21101-L21120","statement_sha256":"1f4fa29776c4bff7736d0810b92fdca19018b9ff60cafb576bcaff71db924443","origin":"The Stacks Project","memory_eligible":false,"source_rank":10205,"rank":10205,"depth":10,"x":1581.316,"y":1256.539,"cluster":"tale-geometry"},{"id":"stacks:075A","tag":"075A","title":"Comparing big and small topoi · Lemma 075A","summary":"Let f : T → S be a morphism of schemes. • For K in D((Sch/T)_etale) we have (Rf_big, *K)|_S_etale = Rf_small, *(K|_T_etale) in D(S_etale). • For K in D((Sch/T)_etale, O) we have (Rf_big, *K)|_S_etale = Rf_small, *(K|_T_etale) in D(Mod(S_etale, O_S)). More generally, let g : S' → S be an object of (Sch/S)_etale. Consider the fibre product xymatrix T' ar[r]_g' ar[d]_f' & T ar[d]^f S' ar[r]^g & S Then • [(3)] For K in D((Sch/T)_etale) we have i_g^-1(Rf_big, *K) = Rf'_small,…","statement_latex":"Let $f : T \\to S$ be a morphism of schemes.\n\\begin{enumerate}\n\\item For $K$ in $D((\\Sch/T)_\\etale)$ we have\n$\n(Rf_{big, *}K)|_{S_\\etale} = Rf_{small, *}(K|_{T_\\etale})\n$\nin $D(S_\\etale)$.\n\\item For $K$ in $D((\\Sch/T)_\\etale, \\mathcal{O})$ we have\n$\n(Rf_{big, *}K)|_{S_\\etale} = Rf_{small, *}(K|_{T_\\etale})\n$\nin $D(\\textit{Mod}(S_\\etale, \\mathcal{O}_S))$.\n\\end{enumerate}\nMore generally, let $g : S' \\to S$ be an object of $(\\Sch/S)_\\etale$.\nConsider the fibre product\n$$\n\\xymatrix{\nT' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nThen\n\\begin{enumerate}\n\\item[(3)] For $K$ in $D((\\Sch/T)_\\etale)$ we have\n$i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$\nin $D(S'_\\etale)$.\n\\item[(4)] For $K$ in $D((\\Sch/T)_\\etale, \\mathcal{O})$ we have\n$i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$\nin $D(\\textit{Mod}(S'_\\etale, \\mathcal{O}_{S'}))$.\n\\item[(5)] For $K$ in $D((\\Sch/T)_\\etale)$ we have\n$g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$\nin $D((\\Sch/S')_\\etale)$.\n\\item[(6)] For $K$ in $D((\\Sch/T)_\\etale, \\mathcal{O})$ we have\n$g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$\nin $D(\\textit{Mod}(S'_\\etale, \\mathcal{O}_{S'}))$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075A","source_file":"etale-cohomology.tex","source_line":21195,"source_end_line":21233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21195-L21233","statement_sha256":"8baa2dd1509ab82ea50e8f37b5065cc55e384a294bb9ee528369df973d613a86","origin":"The Stacks Project","memory_eligible":false,"source_rank":10206,"rank":10206,"depth":24,"x":1485.232,"y":1054.741,"cluster":"tale-geometry"},{"id":"stacks:0DDH","tag":"0DDH","title":"Comparing big and small topoi · Lemma 0DDH","summary":"Let f : T → S be a morphism of schemes. Then • For K in D(S_etale) we have H^n_etale(S, π_S^-1K) = H^n(S_etale, K). • For K in D(S_etale, O_S) we have H^n_etale(S, Lπ_S^*K) = H^n(S_etale, K). • For K in D(S_etale) we have H^n_etale(T, π_S^-1K) = H^n(T_etale, f_small^-1K). • For K in D(S_etale, O_S) we have H^n_etale(T, Lπ_S^*K) = H^n(T_etale, Lf_small^*K). • For M in D((Sch/S)_etale) we have H^n_etale(T, M) = H^n(T_etale, i_f^-1M). • For M in D((Sch/S)_etale, O) we have…","statement_latex":"Let $f : T \\to S$ be a morphism of schemes. Then\n\\begin{enumerate}\n\\item For $K$ in $D(S_\\etale)$ we have\n$H^n_\\etale(S, \\pi_S^{-1}K) = H^n(S_\\etale, K)$.\n\\item For $K$ in $D(S_\\etale, \\mathcal{O}_S)$ we have\n$H^n_\\etale(S, L\\pi_S^*K) = H^n(S_\\etale, K)$.\n\\item For $K$ in $D(S_\\etale)$ we have\n$H^n_\\etale(T, \\pi_S^{-1}K) = H^n(T_\\etale, f_{small}^{-1}K)$.\n\\item For $K$ in $D(S_\\etale, \\mathcal{O}_S)$ we have\n$H^n_\\etale(T, L\\pi_S^*K) = H^n(T_\\etale, Lf_{small}^*K)$.\n\\item For $M$ in $D((\\Sch/S)_\\etale)$ we have\n$H^n_\\etale(T, M) = H^n(T_\\etale, i_f^{-1}M)$.\n\\item For $M$ in $D((\\Sch/S)_\\etale, \\mathcal{O})$ we have\n$H^n_\\etale(T, M) = H^n(T_\\etale, i_f^*M)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDH","source_file":"etale-cohomology.tex","source_line":21274,"source_end_line":21291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21274-L21291","statement_sha256":"32b33d93b5e73defce1b72d098d3891efcc30ab68c771ed9ef798c5e58212071","origin":"The Stacks Project","memory_eligible":false,"source_rank":10207,"rank":10207,"depth":26,"x":1718.561,"y":1149.103,"cluster":"tale-geometry"},{"id":"stacks:0DDI","tag":"0DDI","title":"Comparing big and small topoi · Lemma 0DDI","summary":"Let S be a scheme. For K ∈ D(S_etale) the map K → Rπ_S, *π_S^-1K is an isomorphism.","statement_latex":"Let $S$ be a scheme. For $K \\in D(S_\\etale)$ the map\n$$\nK \\longrightarrow R\\pi_{S, *}\\pi_S^{-1}K\n$$\nis an isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDI","source_file":"etale-cohomology.tex","source_line":21307,"source_end_line":21314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21307-L21314","statement_sha256":"da58c65d3194b93b971cde59cff8fa77e5070f1650ff4b972ae40ce136b2fc05","origin":"The Stacks Project","memory_eligible":false,"source_rank":10208,"rank":10208,"depth":0,"x":1470.414,"y":1211.971,"cluster":"tale-geometry"},{"id":"stacks:0DDJ","tag":"0DDJ","title":"Comparing big and small topoi · Lemma 0DDJ","summary":"Let f : T → S be a proper morphism of schemes. Then we have • π_S^-1 ∘ f_small, * = f_big, * ∘ π_T^-1 as functors Sh(T_etale) → Sh((Sch/S)_etale), • π_S^-1Rf_small, *K = Rf_big, *π_T^-1K for K in D^+(T_etale) whose cohomology sheaves are torsion, • π_S^-1Rf_small, *K = Rf_big, *π_T^-1K for K in D(T_etale, Z/nZ), and • π_S^-1Rf_small, *K = Rf_big, *π_T^-1K for all K in D(T_etale) if f is finite.","statement_latex":"Let $f : T \\to S$ be a proper morphism of schemes. Then we have\n\\begin{enumerate}\n\\item $\\pi_S^{-1} \\circ f_{small, *} = f_{big, *} \\circ \\pi_T^{-1}$\nas functors $\\Sh(T_\\etale) \\to \\Sh((\\Sch/S)_\\etale)$,\n\\item $\\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_T^{-1}K$\nfor $K$ in $D^+(T_\\etale)$ whose cohomology sheaves are torsion,\n\\item $\\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_T^{-1}K$\nfor $K$ in $D(T_\\etale, \\mathbf{Z}/n\\mathbf{Z})$, and\n\\item $\\pi_S^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_T^{-1}K$\nfor all $K$ in $D(T_\\etale)$ if $f$ is finite.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDJ","source_file":"etale-cohomology.tex","source_line":21322,"source_end_line":21335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21322-L21335","statement_sha256":"e79453013df0958a6c1e1e39ca386f304891affefc45c36e0a12206248d16b76","origin":"The Stacks Project","memory_eligible":false,"source_rank":10209,"rank":10209,"depth":79,"x":1602.949,"y":1024.651,"cluster":"tale-geometry"},{"id":"stacks:0DDL","tag":"0DDL","title":"Comparing fppf and étale topologies · Lemma 0DDL","summary":"With notation as above. Let F be a sheaf on S_etale. The rule (Sch/S)_fppf → Sets, (f : X → S) ↦ Γ(X, f_small^-1F) is a sheaf and a fortiori a sheaf on (Sch/S)_etale. In fact this sheaf is equal to π_S^-1F on (Sch/S)_etale and ε_S^-1π_S^-1F on (Sch/S)_fppf.","statement_latex":"With notation as above.\nLet $\\mathcal{F}$ be a sheaf on $S_\\etale$. The rule\n$$\n(\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad\n(f : X \\to S) \\longmapsto \\Gamma(X, f_{small}^{-1}\\mathcal{F})\n$$\nis a sheaf and a fortiori a sheaf on $(\\Sch/S)_\\etale$.\nIn fact this sheaf is equal to\n$\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_\\etale$ and\n$\\epsilon_S^{-1}\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_{fppf}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDL","source_file":"etale-cohomology.tex","source_line":21453,"source_end_line":21465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21453-L21465","statement_sha256":"bb6db4bfe8430d4fa7e88e4c0e278966d423551626148ed8cb4899f214e1f8bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10210,"rank":10210,"depth":54,"x":1655.902,"y":1238.161,"cluster":"tale-geometry"},{"id":"stacks:0DDM","tag":"0DDM","title":"Comparing fppf and étale topologies · Lemma 0DDM","summary":"With notation as above. Let f : X → Y be a morphism of (Sch/S)_fppf. Then there are commutative diagrams of topoi xymatrix Sh((Sch/X)_fppf) ar[rr]_f_big, fppf ar[d]_ε_X & & Sh((Sch/Y)_fppf) ar[d]^ε_Y Sh((Sch/X)_etale) ar[rr]^f_big, etale & & Sh((Sch/Y)_etale) and xymatrix Sh((Sch/X)_fppf) ar[rr]_f_big, fppf ar[d]_a_X & & Sh((Sch/Y)_fppf) ar[d]^a_Y Sh(X_etale) ar[rr]^f_small & & Sh(Y_etale) with a_X = π_X ∘ ε_X and a_Y = π_X ∘ ε_X.","statement_latex":"With notation as above.\nLet $f : X \\to Y$ be a morphism of $(\\Sch/S)_{fppf}$.\nThen there are commutative diagrams of topoi\n$$\n\\xymatrix{\n\\Sh((\\Sch/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{\\epsilon_X} & &\n\\Sh((\\Sch/Y)_{fppf}) \\ar[d]^{\\epsilon_Y} \\\\\n\\Sh((\\Sch/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &\n\\Sh((\\Sch/Y)_\\etale)\n}\n$$\nand\n$$\n\\xymatrix{\n\\Sh((\\Sch/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{a_X} & &\n\\Sh((\\Sch/Y)_{fppf}) \\ar[d]^{a_Y} \\\\\n\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &\n\\Sh(Y_\\etale)\n}\n$$\nwith $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDM","source_file":"etale-cohomology.tex","source_line":21483,"source_end_line":21506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21483-L21506","statement_sha256":"03a6ec4f79961bcc0c2aa358526c3738163620c592333b529a348c5361bca2c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10211,"rank":10211,"depth":0,"x":1444.979,"y":1110.662,"cluster":"tale-geometry"},{"id":"stacks:0DDN","tag":"0DDN","title":"Comparing fppf and étale topologies · Lemma 0DDN","summary":"In Lemma [Tag 0DDM] if f is proper, then we have a_Y^-1 ∘ f_small, * = f_big, fppf, * ∘ a_X^-1.","statement_latex":"In Lemma \\ref{lemma-push-pull-fppf-etale} if $f$ is proper, then we have\n$a_Y^{-1} \\circ f_{small, *} = f_{big, fppf, *} \\circ a_X^{-1}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDN","source_file":"etale-cohomology.tex","source_line":21513,"source_end_line":21517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21513-L21517","statement_sha256":"78289a0f7bcb0ba058a02bafa6aea856f32e06d13b09fba244380843c0aa1fa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10212,"rank":10212,"depth":80,"x":1703.259,"y":1084.973,"cluster":"tale-geometry"},{"id":"stacks:0DEU","tag":"0DEU","title":"Comparing fppf and étale topologies · Lemma 0DEU","summary":"In Lemma [Tag 0DDM] assume f is flat, locally of finite presentation, and surjective. Then the functor Sh(Y_etale) → ( (G, H, α) middle| G ∈ Sh(X_etale), H ∈ Sh((Sch/Y)_fppf), α : a_X^-1G → f_big, fppf^-1H an isomorphism ) sending F to (f_small^-1F, a_Y^-1F, can) is an equivalence.","statement_latex":"In Lemma \\ref{lemma-push-pull-fppf-etale} assume\n$f$ is flat, locally of finite presentation, and surjective.\nThen the functor\n$$\n\\Sh(Y_\\etale) \\longrightarrow\n\\left\\{\n(\\mathcal{G}, \\mathcal{H}, \\alpha)\n\\middle|\n\\begin{matrix}\n\\mathcal{G} \\in \\Sh(X_\\etale),\\ \\mathcal{H} \\in \\Sh((\\Sch/Y)_{fppf}), \\\\\n\\alpha : a_X^{-1}\\mathcal{G} \\to f_{big, fppf}^{-1}\\mathcal{H}\n\\text{ an isomorphism}\n\\end{matrix}\n\\right\\}\n$$\nsending $\\mathcal{F}$ to\n$(f_{small}^{-1}\\mathcal{F}, a_Y^{-1}\\mathcal{F}, can)$ is an equivalence.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEU","source_file":"etale-cohomology.tex","source_line":21545,"source_end_line":21564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21545-L21564","statement_sha256":"f44f90fa8e650da72981ac0aba3804ffdf73f06260e741f1a953300c7fc94cd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10213,"rank":10213,"depth":81,"x":1533.322,"y":1250.609,"cluster":"tale-geometry"},{"id":"stacks:0F0H","tag":"0F0H","title":"Comparing fppf and étale topologies · Lemma 0F0H","summary":"Consider the comparison morphism ε : (Sch/S)_fppf → (Sch/S)_etale. Let P denote the class of finite morphisms of schemes. For X in (Sch/S)_etale denote A'_X ⊂ Ab((Sch/X)_etale) the full subcategory consisting of sheaves of the form π_X^-1F with F in Ab(X_etale). Then Cohomology on Sites, Properties ([Tag 0EZ4]), ([Tag 0EZ5]), ([Tag 0EZ6]), ([Tag 0EZ7]), and ([Tag 0EZ8]) of Cohomology on Sites, Situation [Tag 0EZ3] hold.","statement_latex":"Consider the comparison morphism\n$\\epsilon : (\\Sch/S)_{fppf} \\to (\\Sch/S)_\\etale$.\nLet $\\mathcal{P}$ denote the class of finite morphisms of schemes.\nFor $X$ in $(\\Sch/S)_\\etale$ denote\n$\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$\nthe full subcategory consisting of sheaves of the form\n$\\pi_X^{-1}\\mathcal{F}$ with $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$.\nThen Cohomology on Sites, Properties\n(\\ref{sites-cohomology-item-base-change-P}),\n(\\ref{sites-cohomology-item-restriction-A}),\n(\\ref{sites-cohomology-item-A-sheaf}),\n(\\ref{sites-cohomology-item-A-and-P}), and\n(\\ref{sites-cohomology-item-refine-tau-by-P})\nof Cohomology on Sites, Situation\n\\ref{sites-cohomology-situation-compare} hold.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0H","source_file":"etale-cohomology.tex","source_line":21665,"source_end_line":21682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21665-L21682","statement_sha256":"e928a5a7219ef2aff9a0168fd190ad3033a3bb9dc02c269587b7b0389c369806","origin":"The Stacks Project","memory_eligible":false,"source_rank":10214,"rank":10214,"depth":80,"x":1525.425,"y":1031.863,"cluster":"tale-geometry"},{"id":"stacks:0DDS","tag":"0DDS","title":"Comparing fppf and étale topologies · Lemma 0DDS","summary":"With notation as above. • For X ∈ Ob((Sch/S)_fppf) and an abelian sheaf F on X_etale we have ε_X, *a_X^-1F = π_X^-1F and R^iε_X, *(a_X^-1F) = 0 for i > 0. • For a finite morphism f : X → Y in (Sch/S)_fppf and abelian sheaf F on X we have a_Y^-1(R^if_small, *F) = R^if_big, fppf, *(a_X^-1F) for all i. • For a scheme X and K in D^+(X_etale) the map π_X^-1K → Rε_X, *(a_X^-1K) is an isomorphism. • For a finite morphism f : X → Y of schemes and K in D^+(X_etale) we have…","statement_latex":"With notation as above.\n\\begin{enumerate}\n\\item For $X \\in \\Ob((\\Sch/S)_{fppf})$ and an abelian sheaf $\\mathcal{F}$\non $X_\\etale$ we have\n$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$\nand $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.\n\\item For a finite morphism $f : X \\to Y$ in $(\\Sch/S)_{fppf}$\nand abelian sheaf $\\mathcal{F}$ on $X$ we have\n$a_Y^{-1}(R^if_{small, *}\\mathcal{F}) =\nR^if_{big, fppf, *}(a_X^{-1}\\mathcal{F})$\nfor all $i$.\n\\item For a scheme $X$ and $K$ in $D^+(X_\\etale)$ the map\n$\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism.\n\\item For a finite morphism $f : X \\to Y$ of schemes\nand $K$ in $D^+(X_\\etale)$ we have\n$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$.\n\\item For a proper morphism $f : X \\to Y$ of schemes\nand $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves we have\n$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDS","source_file":"etale-cohomology.tex","source_line":21724,"source_end_line":21746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21724-L21746","statement_sha256":"5f7bed57f5ec24c0962c796544edea04a8cb17708a0a335de41b0e3d83937313","origin":"The Stacks Project","memory_eligible":false,"source_rank":10215,"rank":10215,"depth":81,"x":1707.31,"y":1188.813,"cluster":"tale-geometry"},{"id":"stacks:0DDT","tag":"0DDT","title":"Comparing fppf and étale topologies · Lemma 0DDT","summary":"Let X be a scheme. For K ∈ D^+(X_etale) the map K → Ra_X, *a_X^-1K is an isomorphism with a_X : Sh((Sch/X)_fppf) → Sh(X_etale) as above.","statement_latex":"Let $X$ be a scheme. For $K \\in D^+(X_\\etale)$ the map\n$$\nK \\longrightarrow Ra_{X, *}a_X^{-1}K\n$$\nis an isomorphism with $a_X : \\Sh((\\Sch/X)_{fppf}) \\to \\Sh(X_\\etale)$\nas above.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDT","source_file":"etale-cohomology.tex","source_line":21820,"source_end_line":21828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21820-L21828","statement_sha256":"8f00d6d4efa0965bdbecd441703be839ed5ec96d5605953d66c99d9dc4a627c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10216,"rank":10216,"depth":82,"x":1446.759,"y":1176.275,"cluster":"tale-geometry"},{"id":"stacks:0DDU","tag":"0DDU","title":"Comparing fppf and étale topologies · Lemma 0DDU","summary":"For a scheme X and a_X : Sh((Sch/X)_fppf) → Sh(X_etale) as above: • H^q(X_etale, F) = H^q_fppf(X, a_X^-1F) for an abelian sheaf F on X_etale, • H^q(X_etale, K) = H^q_fppf(X, a_X^-1K) for K ∈ D^+(X_etale). Example: if A is an abelian group, then H^q_etale(X, underlineA) = H^q_fppf(X, underlineA).","statement_latex":"For a scheme $X$ and $a_X : \\Sh((\\Sch/X)_{fppf}) \\to \\Sh(X_\\etale)$\nas above:\n\\begin{enumerate}\n\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\\mathcal{F})$\nfor an abelian sheaf $\\mathcal{F}$ on $X_\\etale$,\n\\item $H^q(X_\\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$.\n\\end{enumerate}\nExample: if $A$ is an abelian group, then\n$H^q_\\etale(X, \\underline{A}) = H^q_{fppf}(X, \\underline{A})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDU","source_file":"etale-cohomology.tex","source_line":21858,"source_end_line":21869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21858-L21869","statement_sha256":"3b190f81477c1d1297323a91a9bb05990c9f1503bb1ada5c5ec39ee9e39d1c93","origin":"The Stacks Project","memory_eligible":false,"source_rank":10217,"rank":10217,"depth":83,"x":1649.136,"y":1037.562,"cluster":"tale-geometry"},{"id":"stacks:0DEW","tag":"0DEW","title":"Comparing fppf and étale topologies: modules · Lemma 0DEW","summary":"Let S be a scheme. Let F be a quasi-coherent O_S-module on S_etale. • The rule F^a : (Sch/S)_etale → Ab, (f : T → S) ↦ Γ(T, f_small^*F) satisfies the sheaf condition for fppf and a fortiori étale coverings, • F^a = π_S^*F on (Sch/S)_etale, • F^a = a_S^*F on (Sch/S)_fppf, • the rule F ↦ F^a defines an equivalence between quasi-coherent O_S-modules and quasi-coherent modules on ((Sch/S)_etale, O), • the rule F ↦ F^a defines an equivalence between quasi-coherent O_S-modules…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_S$-module on $S_\\etale$.\n\\begin{enumerate}\n\\item The rule\n$$\n\\mathcal{F}^a : (\\Sch/S)_\\etale \\longrightarrow \\textit{Ab},\\quad\n(f : T \\to S) \\longmapsto \\Gamma(T, f_{small}^*\\mathcal{F})\n$$\nsatisfies the sheaf condition for fppf and a fortiori \\'etale coverings,\n\\item $\\mathcal{F}^a = \\pi_S^*\\mathcal{F}$ on $(\\Sch/S)_\\etale$,\n\\item $\\mathcal{F}^a = a_S^*\\mathcal{F}$ on $(\\Sch/S)_{fppf}$,\n\\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines\nan equivalence between quasi-coherent $\\mathcal{O}_S$-modules\nand quasi-coherent modules on\n$((\\Sch/S)_\\etale, \\mathcal{O})$,\n\\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines\nan equivalence between quasi-coherent $\\mathcal{O}_S$-modules\nand quasi-coherent modules on\n$((\\Sch/S)_{fppf}, \\mathcal{O})$,\n\\item we have $\\epsilon_{S, *}a_S^*\\mathcal{F} = \\pi_S^*\\mathcal{F}$\nand $a_{S, *}a_S^*\\mathcal{F} = \\mathcal{F}$,\n\\item we have $R^i\\epsilon_{S, *}(a_S^*\\mathcal{F}) = 0$\nand $R^ia_{S, *}(a_S^*\\mathcal{F}) = 0$ for $i > 0$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEW","source_file":"etale-cohomology.tex","source_line":21926,"source_end_line":21952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L21926-L21952","statement_sha256":"21ca864b0e8225fadeb65c951dc632d769b6bb44685de54bf6c2cf30faa0490d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10218,"rank":10218,"depth":42,"x":1611.424,"y":1254.861,"cluster":"tale-geometry"},{"id":"stacks:0DEX","tag":"0DEX","title":"Comparing fppf and étale topologies: modules · Lemma 0DEX","summary":"Let S be a scheme. For F a quasi-coherent O_S-module on S_etale the maps π_S^*F → Rε_S, *(a_S^*F) and F → Ra_S, *(a_S^*F) are isomorphisms with a_S : Sh((Sch/S)_fppf) → Sh(S_etale) as above.","statement_latex":"Let $S$ be a scheme. For $\\mathcal{F}$ a quasi-coherent\n$\\mathcal{O}_S$-module on $S_\\etale$ the maps\n$$\n\\pi_S^*\\mathcal{F} \\longrightarrow R\\epsilon_{S, *}(a_S^*\\mathcal{F})\n\\quad\\text{and}\\quad\n\\mathcal{F} \\longrightarrow Ra_{S, *}(a_S^*\\mathcal{F})\n$$\nare isomorphisms with\n$a_S : \\Sh((\\Sch/S)_{fppf}) \\to \\Sh(S_\\etale)$ as above.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEX","source_file":"etale-cohomology.tex","source_line":22027,"source_end_line":22038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22027-L22038","statement_sha256":"b71ca43b3f86d79da30a7c6c196203dcb78472189c5e46b995288e938b4ec2ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":10219,"rank":10219,"depth":43,"x":1464.364,"y":1073.079,"cluster":"tale-geometry"},{"id":"stacks:0H0U","tag":"0H0U","title":"Comparing fppf and étale topologies: modules · Lemma 0H0U","summary":"Let S = Spec(A) be an affine scheme. Let M^bullet be a complex of A-modules. Consider the complex F^bullet of presheaves of O-modules on (Aff/S)_fppf given by the rule (U/S) = (Spec(B)/Spec(A)) ↦ M^bullet ⊗_A B Then this is a complex of modules and the canonical map M^bullet → RΓ((Aff/S)_fppf, F^bullet) is a quasi-isomorphism.","statement_latex":"Let $S = \\Spec(A)$ be an affine scheme. Let $M^\\bullet$ be a complex\nof $A$-modules. Consider the complex $\\mathcal{F}^\\bullet$ of\npresheaves of $\\mathcal{O}$-modules on\n$(\\textit{Aff}/S)_{fppf}$ given by the rule\n$$\n(U/S) = (\\Spec(B)/\\Spec(A)) \\longmapsto M^\\bullet \\otimes_A B\n$$\nThen this is a complex of modules and the canonical map\n$$\nM^\\bullet \\longrightarrow\nR\\Gamma((\\textit{Aff}/S)_{fppf}, \\mathcal{F}^\\bullet)\n$$\nis a quasi-isomorphism.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing fppf and étale topologies: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0U","source_file":"etale-cohomology.tex","source_line":22046,"source_end_line":22061,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22046-L22061","statement_sha256":"806f3d53ad07e0f719dca81859a58d3c4f9481475c9e41c1e0b17fc896c9f063","origin":"The Stacks Project","memory_eligible":false,"source_rank":10220,"rank":10220,"depth":43,"x":1719.199,"y":1123.719,"cluster":"tale-geometry"},{"id":"stacks:0DDW","tag":"0DDW","title":"Comparing ph and étale topologies · Lemma 0DDW","summary":"With notation as above. Let F be a sheaf on S_etale. The rule (Sch/S)_ph → Sets, (f : X → S) ↦ Γ(X, f_small^-1F) is a sheaf and a fortiori a sheaf on (Sch/S)_etale. In fact this sheaf is equal to π_S^-1F on (Sch/S)_etale and ε_S^-1π_S^-1F on (Sch/S)_ph.","statement_latex":"With notation as above.\nLet $\\mathcal{F}$ be a sheaf on $S_\\etale$. The rule\n$$\n(\\Sch/S)_{ph} \\longrightarrow \\textit{Sets},\\quad\n(f : X \\to S) \\longmapsto \\Gamma(X, f_{small}^{-1}\\mathcal{F})\n$$\nis a sheaf and a fortiori a sheaf on $(\\Sch/S)_\\etale$.\nIn fact this sheaf is equal to\n$\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_\\etale$ and\n$\\epsilon_S^{-1}\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_{ph}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDW","source_file":"etale-cohomology.tex","source_line":22174,"source_end_line":22186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22174-L22186","statement_sha256":"c4ca60da212256e0da35ae54aa0c0bba34bb2094457d026b42d058ceabfae097","origin":"The Stacks Project","memory_eligible":false,"source_rank":10221,"rank":10221,"depth":54,"x":1490.375,"y":1231.066,"cluster":"tale-geometry"},{"id":"stacks:0DDX","tag":"0DDX","title":"Comparing ph and étale topologies · Lemma 0DDX","summary":"With notation as above. Let f : X → Y be a morphism of (Sch/S)_ph. Then there are commutative diagrams of topoi xymatrix Sh((Sch/X)_ph) ar[rr]_f_big, ph ar[d]_ε_X & & Sh((Sch/Y)_ph) ar[d]^ε_Y Sh((Sch/X)_etale) ar[rr]^f_big, etale & & Sh((Sch/Y)_etale) and xymatrix Sh((Sch/X)_ph) ar[rr]_f_big, ph ar[d]_a_X & & Sh((Sch/Y)_ph) ar[d]^a_Y Sh(X_etale) ar[rr]^f_small & & Sh(Y_etale) with a_X = π_X ∘ ε_X and a_Y = π_X ∘ ε_X.","statement_latex":"With notation as above.\nLet $f : X \\to Y$ be a morphism of $(\\Sch/S)_{ph}$.\nThen there are commutative diagrams of topoi\n$$\n\\xymatrix{\n\\Sh((\\Sch/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{\\epsilon_X} & &\n\\Sh((\\Sch/Y)_{ph}) \\ar[d]^{\\epsilon_Y} \\\\\n\\Sh((\\Sch/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &\n\\Sh((\\Sch/Y)_\\etale)\n}\n$$\nand\n$$\n\\xymatrix{\n\\Sh((\\Sch/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{a_X} & &\n\\Sh((\\Sch/Y)_{ph}) \\ar[d]^{a_Y} \\\\\n\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &\n\\Sh(Y_\\etale)\n}\n$$\nwith $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDX","source_file":"etale-cohomology.tex","source_line":22259,"source_end_line":22282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22259-L22282","statement_sha256":"b26ac4e92acdc04509a4079302788c40d91e8c3e014310a7569ab63cc34a6cd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10222,"rank":10222,"depth":0,"x":1572.849,"y":1021.897,"cluster":"tale-geometry"},{"id":"stacks:0DDY","tag":"0DDY","title":"Comparing ph and étale topologies · Lemma 0DDY","summary":"In Lemma [Tag 0DDX] if f is proper, then we have a_Y^-1 ∘ f_small, * = f_big, ph, * ∘ a_X^-1.","statement_latex":"In Lemma \\ref{lemma-push-pull-ph-etale} if $f$ is proper, then we have\n$a_Y^{-1} \\circ f_{small, *} = f_{big, ph, *} \\circ a_X^{-1}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DDY","source_file":"etale-cohomology.tex","source_line":22289,"source_end_line":22293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22289-L22293","statement_sha256":"36ee5ab9b6d9e9db65c04aa81810cc7b0c5441a8bd7ece373ebe37a9cf4545f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10223,"rank":10223,"depth":80,"x":1680.33,"y":1223.098,"cluster":"tale-geometry"},{"id":"stacks:0F0I","tag":"0F0I","title":"Comparing ph and étale topologies · Lemma 0F0I","summary":"Consider the comparison morphism ε : (Sch/S)_ph → (Sch/S)_etale. Let P denote the class of proper morphisms of schemes. For X in (Sch/S)_etale denote A'_X ⊂ Ab((Sch/X)_etale) the full subcategory consisting of sheaves of the form π_X^-1F where F is a torsion abelian sheaf on X_etale Then Cohomology on Sites, Properties ([Tag 0EZ4]), ([Tag 0EZ5]), ([Tag 0EZ6]), ([Tag 0EZ7]), and ([Tag 0EZ8]) of Cohomology on Sites, Situation [Tag 0EZ3] hold.","statement_latex":"Consider the comparison morphism\n$\\epsilon : (\\Sch/S)_{ph} \\to (\\Sch/S)_\\etale$.\nLet $\\mathcal{P}$ denote the class of proper morphisms of schemes.\nFor $X$ in $(\\Sch/S)_\\etale$ denote\n$\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$\nthe full subcategory consisting of sheaves of the form\n$\\pi_X^{-1}\\mathcal{F}$ where $\\mathcal{F}$ is a\ntorsion abelian sheaf on $X_\\etale$\nThen Cohomology on Sites, Properties\n(\\ref{sites-cohomology-item-base-change-P}),\n(\\ref{sites-cohomology-item-restriction-A}),\n(\\ref{sites-cohomology-item-A-sheaf}),\n(\\ref{sites-cohomology-item-A-and-P}), and\n(\\ref{sites-cohomology-item-refine-tau-by-P})\nof Cohomology on Sites, Situation\n\\ref{sites-cohomology-situation-compare} hold.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0I","source_file":"etale-cohomology.tex","source_line":22321,"source_end_line":22339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22321-L22339","statement_sha256":"f763b76c61b3ddb8be450888dfd910387b67e466a5cca271e7528a56461b184b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10224,"rank":10224,"depth":80,"x":1439.077,"y":1135.651,"cluster":"tale-geometry"},{"id":"stacks:0DE4","tag":"0DE4","title":"Comparing ph and étale topologies · Lemma 0DE4","summary":"With notation as above. • For X ∈ Ob((Sch/S)_ph) and an abelian torsion sheaf F on X_etale we have ε_X, *a_X^-1F = π_X^-1F and R^iε_X, *(a_X^-1F) = 0 for i > 0. • For a proper morphism f : X → Y in (Sch/S)_ph and abelian torsion sheaf F on X we have a_Y^-1(R^if_small, *F) = R^if_big, ph, *(a_X^-1F) for all i. • For a scheme X and K in D^+(X_etale) with torsion cohomology sheaves the map π_X^-1K → Rε_X, *(a_X^-1K) is an isomorphism. • For a proper morphism f : X → Y of…","statement_latex":"With notation as above.\n\\begin{enumerate}\n\\item For $X \\in \\Ob((\\Sch/S)_{ph})$ and an abelian torsion sheaf $\\mathcal{F}$\non $X_\\etale$ we have\n$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$\nand $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.\n\\item For a proper morphism $f : X \\to Y$ in $(\\Sch/S)_{ph}$\nand abelian torsion sheaf $\\mathcal{F}$ on $X$ we have\n$a_Y^{-1}(R^if_{small, *}\\mathcal{F}) =\nR^if_{big, ph, *}(a_X^{-1}\\mathcal{F})$\nfor all $i$.\n\\item For a scheme $X$ and $K$ in $D^+(X_\\etale)$ with torsion\ncohomology sheaves the map\n$\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism.\n\\item For a proper morphism $f : X \\to Y$ of schemes\nand $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves we have\n$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DE4","source_file":"etale-cohomology.tex","source_line":22388,"source_end_line":22408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22388-L22408","statement_sha256":"5820a2eb976a0347e034647fc38d789dbf4941a6dde01abce1a4707fab1ab299","origin":"The Stacks Project","memory_eligible":false,"source_rank":10225,"rank":10225,"depth":81,"x":1687.5,"y":1063.181,"cluster":"tale-geometry"},{"id":"stacks:0DE5","tag":"0DE5","title":"Comparing ph and étale topologies · Lemma 0DE5","summary":"Let X be a scheme. For K ∈ D^+(X_etale) with torsion cohomology sheaves the map K → Ra_X, *a_X^-1K is an isomorphism with a_X : Sh((Sch/X)_ph) → Sh(X_etale) as above.","statement_latex":"Let $X$ be a scheme. For $K \\in D^+(X_\\etale)$ with torsion cohomology\nsheaves the map\n$$\nK \\longrightarrow Ra_{X, *}a_X^{-1}K\n$$\nis an isomorphism with $a_X : \\Sh((\\Sch/X)_{ph}) \\to \\Sh(X_\\etale)$ as above.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DE5","source_file":"etale-cohomology.tex","source_line":22440,"source_end_line":22448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22440-L22448","statement_sha256":"3d46ef07675f4c7c2898a9beaa4d81c871faf894ac8a09b56ac3bb19668e75ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":10226,"rank":10226,"depth":82,"x":1562.493,"y":1257.74,"cluster":"tale-geometry"},{"id":"stacks:0DE6","tag":"0DE6","title":"Comparing ph and étale topologies · Lemma 0DE6","summary":"For a scheme X and a_X : Sh((Sch/X)_ph) → Sh(X_etale) as above: • H^q(X_etale, F) = H^q_ph(X, a_X^-1F) for a torsion abelian sheaf F on X_etale, • H^q(X_etale, K) = H^q_ph(X, a_X^-1K) for K ∈ D^+(X_etale) with torsion cohomology sheaves. Example: if A is a torsion abelian group, then H^q_etale(X, underlineA) = H^q_ph(X, underlineA).","statement_latex":"For a scheme $X$ and $a_X : \\Sh((\\Sch/X)_{ph}) \\to \\Sh(X_\\etale)$\nas above:\n\\begin{enumerate}\n\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{ph}(X, a_X^{-1}\\mathcal{F})$\nfor a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$,\n\\item $H^q(X_\\etale, K) = H^q_{ph}(X, a_X^{-1}K)$\nfor $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves.\n\\end{enumerate}\nExample: if $A$ is a torsion abelian group, then\n$H^q_\\etale(X, \\underline{A}) = H^q_{ph}(X, \\underline{A})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DE6","source_file":"etale-cohomology.tex","source_line":22480,"source_end_line":22492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22480-L22492","statement_sha256":"1022d8be130f58af5552df1d542e53135ab2ee7ac143e2f195224e58ad8b3f20","origin":"The Stacks Project","memory_eligible":false,"source_rank":10227,"rank":10227,"depth":83,"x":1498.16,"y":1043.167,"cluster":"tale-geometry"},{"id":"stacks:0EW8","tag":"0EW8","title":"Comparing h and étale topologies · Lemma 0EW8","summary":"With notation as above. Let F be a sheaf on S_etale. The rule (Sch/S)_h → Sets, (f : X → S) ↦ Γ(X, f_small^-1F) is a sheaf and a fortiori a sheaf on (Sch/S)_etale. In fact this sheaf is equal to π_S^-1F on (Sch/S)_etale and ε_S^-1π_S^-1F on (Sch/S)_h.","statement_latex":"With notation as above.\nLet $\\mathcal{F}$ be a sheaf on $S_\\etale$. The rule\n$$\n(\\Sch/S)_h \\longrightarrow \\textit{Sets},\\quad\n(f : X \\to S) \\longmapsto \\Gamma(X, f_{small}^{-1}\\mathcal{F})\n$$\nis a sheaf and a fortiori a sheaf on $(\\Sch/S)_\\etale$.\nIn fact this sheaf is equal to\n$\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_\\etale$ and\n$\\epsilon_S^{-1}\\pi_S^{-1}\\mathcal{F}$ on $(\\Sch/S)_h$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing h and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EW8","source_file":"etale-cohomology.tex","source_line":22553,"source_end_line":22565,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22553-L22565","statement_sha256":"4349f517728201234d988cda9a753a2d140978ea853872f0e383793a5de6dbdb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10228,"rank":10228,"depth":55,"x":1718.331,"y":1164.985,"cluster":"tale-geometry"},{"id":"stacks:0EW9","tag":"0EW9","title":"Comparing h and étale topologies · Lemma 0EW9","summary":"With notation as above. Let f : X → Y be a morphism of (Sch/S)_h. Then there are commutative diagrams of topoi xymatrix Sh((Sch/X)_h) ar[rr]_f_big, h ar[d]_ε_X & & Sh((Sch/Y)_h) ar[d]^ε_Y Sh((Sch/X)_etale) ar[rr]^f_big, etale & & Sh((Sch/Y)_etale) and xymatrix Sh((Sch/X)_h) ar[rr]_f_big, h ar[d]_a_X & & Sh((Sch/Y)_h) ar[d]^a_Y Sh(X_etale) ar[rr]^f_small & & Sh(Y_etale) with a_X = π_X ∘ ε_X and a_Y = π_X ∘ ε_X.","statement_latex":"With notation as above.\nLet $f : X \\to Y$ be a morphism of $(\\Sch/S)_h$.\nThen there are commutative diagrams of topoi\n$$\n\\xymatrix{\n\\Sh((\\Sch/X)_h) \\ar[rr]_{f_{big, h}} \\ar[d]_{\\epsilon_X} & &\n\\Sh((\\Sch/Y)_h) \\ar[d]^{\\epsilon_Y} \\\\\n\\Sh((\\Sch/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &\n\\Sh((\\Sch/Y)_\\etale)\n}\n$$\nand\n$$\n\\xymatrix{\n\\Sh((\\Sch/X)_h) \\ar[rr]_{f_{big, h}} \\ar[d]_{a_X} & &\n\\Sh((\\Sch/Y)_h) \\ar[d]^{a_Y} \\\\\n\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &\n\\Sh(Y_\\etale)\n}\n$$\nwith $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing h and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EW9","source_file":"etale-cohomology.tex","source_line":22585,"source_end_line":22608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22585-L22608","statement_sha256":"4225a95b02240e68376dc4927ac91576fdd133aead4d4a12e799a02c7f4e9962","origin":"The Stacks Project","memory_eligible":false,"source_rank":10229,"rank":10229,"depth":0,"x":1457.805,"y":1200.118,"cluster":"tale-geometry"},{"id":"stacks:0EWA","tag":"0EWA","title":"Comparing h and étale topologies · Lemma 0EWA","summary":"In Lemma [Tag 0EW9] if f is proper, then we have a_Y^-1 ∘ f_small, * = f_big, h, * ∘ a_X^-1.","statement_latex":"In Lemma \\ref{lemma-push-pull-h-etale} if $f$ is proper, then we have\n$a_Y^{-1} \\circ f_{small, *} = f_{big, h, *} \\circ a_X^{-1}$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing h and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWA","source_file":"etale-cohomology.tex","source_line":22615,"source_end_line":22619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22615-L22619","statement_sha256":"aec11d0a5d1eedef881e934af2bf10c2c9dfa752a7c5623e75dd378db5642da6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10230,"rank":10230,"depth":80,"x":1621.794,"y":1026.241,"cluster":"tale-geometry"},{"id":"stacks:0F0J","tag":"0F0J","title":"Comparing h and étale topologies · Lemma 0F0J","summary":"Consider the comparison morphism ε : (Sch/S)_h → (Sch/S)_etale. Let P denote the class of proper morphisms. For X in (Sch/S)_etale denote A'_X ⊂ Ab((Sch/X)_etale) the full subcategory consisting of sheaves of the form π_X^-1F where F is a torsion abelian sheaf on X_etale Then Cohomology on Sites, Properties ([Tag 0EZ4]), ([Tag 0EZ5]), ([Tag 0EZ6]), ([Tag 0EZ7]), and ([Tag 0EZ8]) of Cohomology on Sites, Situation [Tag 0EZ3] hold.","statement_latex":"Consider the comparison morphism $\\epsilon : (\\Sch/S)_h \\to (\\Sch/S)_\\etale$.\nLet $\\mathcal{P}$ denote the class of proper morphisms.\nFor $X$ in $(\\Sch/S)_\\etale$ denote\n$\\mathcal{A}'_X \\subset \\textit{Ab}((\\Sch/X)_\\etale)$\nthe full subcategory consisting of sheaves of the form\n$\\pi_X^{-1}\\mathcal{F}$ where $\\mathcal{F}$ is a\ntorsion abelian sheaf on $X_\\etale$\nThen Cohomology on Sites, Properties\n(\\ref{sites-cohomology-item-base-change-P}),\n(\\ref{sites-cohomology-item-restriction-A}),\n(\\ref{sites-cohomology-item-A-sheaf}),\n(\\ref{sites-cohomology-item-A-and-P}), and\n(\\ref{sites-cohomology-item-refine-tau-by-P})\nof Cohomology on Sites, Situation\n\\ref{sites-cohomology-situation-compare} hold.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing h and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F0J","source_file":"etale-cohomology.tex","source_line":22647,"source_end_line":22664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22647-L22664","statement_sha256":"b79fff8a1ac690110698aabeca161d1e09835e0cb838be87817cb8e7fcb567e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10231,"rank":10231,"depth":80,"x":1640.713,"y":1247.686,"cluster":"tale-geometry"},{"id":"stacks:0EWF","tag":"0EWF","title":"Comparing h and étale topologies · Lemma 0EWF","summary":"With notation as above. • For X ∈ Ob((Sch/S)_h) and an abelian torsion sheaf F on X_etale we have ε_X, *a_X^-1F = π_X^-1F and R^iε_X, *(a_X^-1F) = 0 for i > 0. • For a proper morphism f : X → Y in (Sch/S)_h and abelian torsion sheaf F on X we have a_Y^-1(R^if_small, *F) = R^if_big, h, *(a_X^-1F) for all i. • For a scheme X and K in D^+(X_etale) with torsion cohomology sheaves the map π_X^-1K → Rε_X, *(a_X^-1K) is an isomorphism. • For a proper morphism f : X → Y of…","statement_latex":"With notation as above.\n\\begin{enumerate}\n\\item For $X \\in \\Ob((\\Sch/S)_{h})$ and an abelian torsion sheaf $\\mathcal{F}$\non $X_\\etale$ we have\n$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$\nand $R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.\n\\item For a proper morphism $f : X \\to Y$ in $(\\Sch/S)_h$\nand abelian torsion sheaf $\\mathcal{F}$ on $X$ we have\n$a_Y^{-1}(R^if_{small, *}\\mathcal{F}) =\nR^if_{big, h, *}(a_X^{-1}\\mathcal{F})$\nfor all $i$.\n\\item For a scheme $X$ and $K$ in $D^+(X_\\etale)$ with torsion\ncohomology sheaves the map\n$\\pi_X^{-1}K \\to R\\epsilon_{X, *}(a_X^{-1}K)$ is an isomorphism.\n\\item For a proper morphism $f : X \\to Y$ of schemes\nand $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves we have\n$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, h, *}(a_X^{-1}K)$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing h and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWF","source_file":"etale-cohomology.tex","source_line":22717,"source_end_line":22737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22717-L22737","statement_sha256":"5728628b6d2796e095d5c80b80d654737125f734b601053d82429bc422d292e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10232,"rank":10232,"depth":81,"x":1448.527,"y":1095.007,"cluster":"tale-geometry"},{"id":"stacks:0EWG","tag":"0EWG","title":"Comparing h and étale topologies · Lemma 0EWG","summary":"Let X be a scheme. For K ∈ D^+(X_etale) with torsion cohomology sheaves the map K → Ra_X, *a_X^-1K is an isomorphism with a_X : Sh((Sch/X)_h) → Sh(X_etale) as above.","statement_latex":"Let $X$ be a scheme. For $K \\in D^+(X_\\etale)$ with torsion cohomology\nsheaves the map\n$$\nK \\longrightarrow Ra_{X, *}a_X^{-1}K\n$$\nis an isomorphism with $a_X : \\Sh((\\Sch/X)_h) \\to \\Sh(X_\\etale)$ as above.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing h and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWG","source_file":"etale-cohomology.tex","source_line":22769,"source_end_line":22777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22769-L22777","statement_sha256":"90dec5fbe844cef278b360c93cf289f4e7d47707338de60d49650c771ea6cc11","origin":"The Stacks Project","memory_eligible":false,"source_rank":10233,"rank":10233,"depth":82,"x":1713.236,"y":1098.542,"cluster":"tale-geometry"},{"id":"stacks:0EWH","tag":"0EWH","title":"Comparing h and étale topologies · Lemma 0EWH","summary":"For a scheme X and a_X : Sh((Sch/X)_h) → Sh(X_etale) as above: • H^q(X_etale, F) = H^q_h(X, a_X^-1F) for a torsion abelian sheaf F on X_etale, • H^q(X_etale, K) = H^q_h(X, a_X^-1K) for K ∈ D^+(X_etale) with torsion cohomology sheaves. Example: if A is a torsion abelian group, then H^q_etale(X, underlineA) = H^q_h(X, underlineA).","statement_latex":"For a scheme $X$ and $a_X : \\Sh((\\Sch/X)_h) \\to \\Sh(X_\\etale)$\nas above:\n\\begin{enumerate}\n\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_h(X, a_X^{-1}\\mathcal{F})$\nfor a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$,\n\\item $H^q(X_\\etale, K) = H^q_h(X, a_X^{-1}K)$\nfor $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves.\n\\end{enumerate}\nExample: if $A$ is a torsion abelian group, then\n$H^q_\\etale(X, \\underline{A}) = H^q_h(X, \\underline{A})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Comparing h and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWH","source_file":"etale-cohomology.tex","source_line":22809,"source_end_line":22821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22809-L22821","statement_sha256":"73c8bee18a9423435223521d8e79f308ab29c75a7d5e48c2256be3950a2748d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10234,"rank":10234,"depth":83,"x":1515.04,"y":1246.257,"cluster":"tale-geometry"},{"id":"stacks:0GEY","tag":"0GEY","title":"Descending étale sheaves · Lemma 0GEY","summary":"Let f : X → Y be a morphism of schemes which has a section. Then the functor Sh(Y_etale) → descent data for étale sheaves wrt (X → Y) sending G in Sh(Y_etale) to the canonical descent datum is an equivalence of categories.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which has a section. Then the\nfunctor\n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}\n$$\nsending $\\mathcal{G}$ in $\\Sh(Y_\\etale)$ to the canonical descent datum\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEY","source_file":"etale-cohomology.tex","source_line":22951,"source_end_line":22962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22951-L22962","statement_sha256":"214a5a24591d635bb9a342513c45d50cf55ceaef7f44354fd41369e7b36579ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":10235,"rank":10235,"depth":0,"x":1542.421,"y":1024.695,"cluster":"tale-geometry"},{"id":"stacks:0GEZ","tag":"0GEZ","title":"Descending étale sheaves · Lemma 0GEZ","summary":"Let f : X → Y be a surjective integral morphism of schemes. The functor Sh(Y_etale) → descent data for étale sheaves wrt (X → Y) is an equivalence of categories.","statement_latex":"Let $f : X \\to Y$ be a surjective integral morphism of schemes.\nThe functor\n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GEZ","source_file":"etale-cohomology.tex","source_line":22971,"source_end_line":22981,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L22971-L22981","statement_sha256":"06b90aafb7abe0050a22c70a69d2962cf4765c942bb8fc77a788fdffe55f7641","origin":"The Stacks Project","memory_eligible":false,"source_rank":10236,"rank":10236,"depth":54,"x":1700.533,"y":1203.752,"cluster":"tale-geometry"},{"id":"stacks:0GF0","tag":"0GF0","title":"Descending étale sheaves · Lemma 0GF0","summary":"Let f : X → Y be a surjective proper morphism of schemes. The functor Sh(Y_etale) → descent data for étale sheaves wrt (X → Y) is an equivalence of categories.","statement_latex":"Let $f : X \\to Y$ be a surjective proper morphism of schemes.\nThe functor\n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GF0","source_file":"etale-cohomology.tex","source_line":23046,"source_end_line":23056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23046-L23056","statement_sha256":"e7bad792874bf7ec13e267e33d6707aba95d66aad1b448069272eb275534f2c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10237,"rank":10237,"depth":55,"x":1439.741,"y":1161.4,"cluster":"tale-geometry"},{"id":"stacks:0GF1","tag":"0GF1","title":"Descending étale sheaves · Lemma 0GF1","summary":"Let f : X → Y be a morphism of schemes. Let Z → Y be a surjective integral morphism of schemes or a surjective proper morphism of schemes. If the functors Sh(Z_etale) → descent data for étale sheaves wrt (X ×_Y Z → Z) and Sh((Z ×_Y Z)_etale) → descent data for étale sheaves wrt (X ×_Y (Z ×_Y Z) → Z ×_Y Z) are equivalences of categories, then Sh(Y_etale) → descent data for étale sheaves wrt (X → Y) is an equivalence.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $Z \\to Y$ be a surjective\nintegral morphism of schemes or a surjective proper morphism of schemes.\nIf the functors\n$$\n\\Sh(Z_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\times_Y Z \\to Z\\}\n$$\nand\n$$\n\\Sh((Z \\times_Y Z)_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\n\\{X \\times_Y (Z \\times_Y Z) \\to Z \\times_Y Z\\}\n$$\nare equivalences of categories, then\n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}\n$$\nis an equivalence.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GF1","source_file":"etale-cohomology.tex","source_line":23067,"source_end_line":23091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23067-L23091","statement_sha256":"8bf624850236ad6e97b8f97f5d7e743ae4425cde0b7b2f975ad948ef025d7dc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10238,"rank":10238,"depth":56,"x":1666.285,"y":1044.557,"cluster":"tale-geometry"},{"id":"stacks:0GF2","tag":"0GF2","title":"Descending étale sheaves · Lemma 0GF2","summary":"Let f : X → Y be a morphism of schemes which is surjective, flat, locally of finite presentation. The functor Sh(Y_etale) → descent data for étale sheaves wrt (X → Y) is an equivalence of categories.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is\nsurjective, flat, locally of finite presentation.\nThe functor\n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GF2","source_file":"etale-cohomology.tex","source_line":23099,"source_end_line":23110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23099-L23110","statement_sha256":"04cf5a6c8a3c3f04cbc73961764bbf197070ed63867a6de521e88285b057ac6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10239,"rank":10239,"depth":57,"x":1593.139,"y":1259.434,"cluster":"tale-geometry"},{"id":"stacks:0GF3","tag":"0GF3","title":"Descending étale sheaves · Lemma 0GF3","summary":"Let (f_i : X_i → X) be an fppf covering of schemes. The functor Sh(X_etale) → descent data for étale sheaves wrt (f_i : X_i → X) is an equivalence of categories.","statement_latex":"Let $\\{f_i : X_i \\to X\\}$ be an fppf covering of schemes.\nThe functor\n$$\n\\Sh(X_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{f_i : X_i \\to X\\}\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GF3","source_file":"etale-cohomology.tex","source_line":23145,"source_end_line":23155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23145-L23155","statement_sha256":"b4dd4524e0d710956e648413b76eee43ddb9d06c6cfd479cb66ef9c04f03b8b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10240,"rank":10240,"depth":58,"x":1474.182,"y":1059.322,"cluster":"tale-geometry"},{"id":"stacks:0GF4","tag":"0GF4","title":"Descending étale sheaves · Lemma 0GF4","summary":"Let f : X' → X be a proper morphism of schemes. Let i : Z → X be a closed immersion. Set E = Z ×_X X'. Picture xymatrix E ar[d]_g ar[r]_j & X' ar[d]^f Z ar[r]^i & X If f is an isomorphism over X setminus Z, then the functor Sh(X_etale) → Sh(X'_etale) ×_Sh(E_etale) Sh(Z_etale) is an equivalence of categories.","statement_latex":"Let $f : X' \\to X$ be a proper morphism of schemes. Let $i : Z \\to X$\nbe a closed immersion. Set $E = Z \\times_X X'$. Picture\n$$\n\\xymatrix{\nE \\ar[d]_g \\ar[r]_j & X' \\ar[d]^f \\\\\nZ \\ar[r]^i & X\n}\n$$\nIf $f$ is an isomorphism over $X \\setminus Z$, then the functor\n$$\n\\Sh(X_\\etale)\n\\longrightarrow\n\\Sh(X'_\\etale) \\times_{\\Sh(E_\\etale)} \\Sh(Z_\\etale)\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GF4","source_file":"etale-cohomology.tex","source_line":23166,"source_end_line":23183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23166-L23183","statement_sha256":"1604f7fd99321e80999ec45f4cfaac70666905c32ae764569ada303cd4ee75df","origin":"The Stacks Project","memory_eligible":false,"source_rank":10241,"rank":10241,"depth":55,"x":1723.022,"y":1139.447,"cluster":"tale-geometry"},{"id":"stacks:0GF5","tag":"0GF5","title":"Descending étale sheaves · Lemma 0GF5","summary":"Let S be a scheme. Then the category fibred in groupoids p : S → (Sch/S)_h whose fibre category over U is the category Sh(U_etale) of sheaves on the small étale site of U is a stack in groupoids.","statement_latex":"Let $S$ be a scheme. Then the category fibred in groupoids\n$$\np : \\mathcal{S} \\longrightarrow (\\Sch/S)_h\n$$\nwhose fibre category over $U$ is the category $\\Sh(U_\\etale)$\nof sheaves on the small \\'etale site of $U$ is a stack in groupoids.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GF5","source_file":"etale-cohomology.tex","source_line":23256,"source_end_line":23264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23256-L23264","statement_sha256":"d9df3a34564299edc31b91010fda8cedbb1ac50a6897bdaa955bb92c24814851","origin":"The Stacks Project","memory_eligible":false,"source_rank":10242,"rank":10242,"depth":62,"x":1474.899,"y":1221.627,"cluster":"tale-geometry"},{"id":"stacks:0EW5","tag":"0EW5","title":"Blow up squares and étale cohomology · Lemma 0EW5","summary":"Let X be a scheme and let Z ⊂ X be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square xymatrix E ar[d]_π ar[r]_j & X' ar[d]^b Z ar[r]^i & X For K ∈ D^+(X_etale) with torsion cohomology sheaves we have a distinguished triangle K → Ri_*(K|_Z) ⊕ Rb_*(K|_X') → Rc_*(K|_E) → K[1] in D(X_etale) where c = i ∘ π = b ∘ j.","statement_latex":"Let $X$ be a scheme and let $Z \\subset X$ be a closed subscheme\ncut out by a quasi-coherent ideal of finite type. Consider the\ncorresponding blow up square\n$$\n\\xymatrix{\nE \\ar[d]_\\pi \\ar[r]_j & X' \\ar[d]^b \\\\\nZ \\ar[r]^i & X\n}\n$$\nFor $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves\nwe have a distinguished triangle\n$$\nK \\to Ri_*(K|_Z) \\oplus Rb_*(K|_{X'}) \\to Rc_*(K|_E) \\to K[1]\n$$\nin $D(X_\\etale)$ where $c = i \\circ \\pi = b \\circ j$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Blow up squares and étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EW5","source_file":"etale-cohomology.tex","source_line":23314,"source_end_line":23331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23314-L23331","statement_sha256":"5cffebb9c4f86140c30448532c768297bd8f24f97d2545b2f7a5999087aa66ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":10243,"rank":10243,"depth":76,"x":1591.867,"y":1020.077,"cluster":"tale-geometry"},{"id":"stacks:0EW3","tag":"0EW3","title":"Blow up squares and étale cohomology · Lemma 0EW3","summary":"Let X be a scheme and let K ∈ D^+(X_etale) have torsion cohomology sheaves. Let Z ⊂ X be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square xymatrix E ar[d] ar[r] & X' ar[d]^b Z ar[r] & X Then there is a canonical long exact sequence H^p_etale(X, K) → H^p_etale(X', K|_X') ⊕ H^p_etale(Z, K|_Z) → H^p_etale(E, K|_E) → H^p + 1_etale(X, K)","statement_latex":"Let $X$ be a scheme and let $K \\in D^+(X_\\etale)$ have\ntorsion cohomology sheaves. Let $Z \\subset X$ be a closed subscheme\ncut out by a quasi-coherent ideal of finite type. Consider the\ncorresponding blow up square\n$$\n\\xymatrix{\nE \\ar[d] \\ar[r] & X' \\ar[d]^b \\\\\nZ \\ar[r] & X\n}\n$$\nThen there is a canonical long exact sequence\n$$\nH^p_\\etale(X, K) \\to\nH^p_\\etale(X', K|_{X'}) \\oplus\nH^p_\\etale(Z, K|_Z) \\to\nH^p_\\etale(E, K|_E) \\to\nH^{p + 1}_\\etale(X, K)\n$$","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Blow up squares and étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EW3","source_file":"etale-cohomology.tex","source_line":23387,"source_end_line":23407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23387-L23407","statement_sha256":"5e7c5c99e68990ef20304bca9ed98d1e77f90ce9b3ebe83d07fb8b6a7459e932","origin":"The Stacks Project","memory_eligible":false,"source_rank":10244,"rank":10244,"depth":77,"x":1667.758,"y":1235.238,"cluster":"tale-geometry"},{"id":"stacks:0EW6","tag":"0EW6","title":"Blow up squares and étale cohomology · Lemma 0EW6","summary":"Let X be a scheme and let Z ⊂ X be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square xymatrix E ar[d]_π ar[r]_j & X' ar[d]^b Z ar[r]^i & X Suppose given • an object K' of D^+(X'_etale) with torsion cohomology sheaves, • an object L of D^+(Z_etale) with torsion cohomology sheaves, and • an isomorphism γ : K'|_E → L|_E. Then there exists an object K of D^+(X_etale) and isomorphisms f : K|_X' → K', g : K|_Z → L…","statement_latex":"Let $X$ be a scheme and let $Z \\subset X$ be a closed subscheme\ncut out by a quasi-coherent ideal of finite type. Consider the\ncorresponding blow up square\n$$\n\\xymatrix{\nE \\ar[d]_\\pi \\ar[r]_j & X' \\ar[d]^b \\\\\nZ \\ar[r]^i & X\n}\n$$\nSuppose given\n\\begin{enumerate}\n\\item an object $K'$ of $D^+(X'_\\etale)$ with torsion cohomology sheaves,\n\\item an object $L$ of $D^+(Z_\\etale)$ with torsion cohomology sheaves, and\n\\item an isomorphism $\\gamma : K'|_E \\to L|_E$.\n\\end{enumerate}\nThen there exists an object $K$ of $D^+(X_\\etale)$\nand isomorphisms $f : K|_{X'} \\to K'$, $g : K|_Z \\to L$ such\nthat $\\gamma = g|_E \\circ f^{-1}|_E$.\nMoreover, given\n\\begin{enumerate}\n\\item an object $M$ of $D^+(X_\\etale)$ with torsion cohomology sheaves,\n\\item a morphism $\\alpha : K' \\to M|_{X'}$ of $D(X'_\\etale)$,\n\\item a morphism $\\beta : L \\to M|_Z$ of $D(Z_\\etale)$,\n\\end{enumerate}\nsuch that\n$$\n\\alpha|_E  = \\beta|_E \\circ \\gamma.\n$$\nThen there exists a morphism $M \\to K$ in $D(X_\\etale)$\nwhose restriction to $X'$ is $a \\circ f$\nand whose restriction to $Z$ is $b \\circ g$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Blow up squares and étale cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EW6","source_file":"etale-cohomology.tex","source_line":23436,"source_end_line":23469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23436-L23469","statement_sha256":"96aa4faa9e90f68bd2005c3256026abb7243c0593b5343196541e96328c817c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10245,"rank":10245,"depth":77,"x":1438.588,"y":1119.552,"cluster":"tale-geometry"},{"id":"stacks:0EWN","tag":"0EWN","title":"Almost blow up squares and the h topology · Lemma 0EWN","summary":"With notation as above, if K is in the essential image of Rε_*, then the maps c^K_X, Z, X', E of Cohomology on Sites, Lemma [Tag 0F16] are quasi-isomorphisms.","statement_latex":"With notation as above, if $K$ is in the essential image\nof $R\\epsilon_*$, then the maps $c^K_{X, Z, X', E}$ of\nCohomology on Sites, Lemma \\ref{sites-cohomology-lemma-c-square}\nare quasi-isomorphisms.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWN","source_file":"etale-cohomology.tex","source_line":23576,"source_end_line":23582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23576-L23582","statement_sha256":"a9bb4932ca75d411f461c35cebd1d0584a55d04e6465a05ae008c5bd5d7ac95b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10246,"rank":10246,"depth":42,"x":1700.814,"y":1074.786,"cluster":"tale-geometry"},{"id":"stacks:0EWQ","tag":"0EWQ","title":"Almost blow up squares and the h topology · Proposition 0EWQ","summary":"Let K be an object of D^+((Sch/S)_fppf). Then K is in the essential image of Rε_* : D((Sch/S)_h) → D((Sch/S)_fppf) if and only if c^K_X, X', Z, E is a quasi-isomorphism for every almost blow up square ([Tag 0EWM]) in (Sch/S)_h with X affine.","statement_latex":"Let $K$ be an object of $D^+((\\Sch/S)_{fppf})$.\nThen $K$ is in the essential image of\n$R\\epsilon_* : D((\\Sch/S)_h) \\to D((\\Sch/S)_{fppf})$\nif and only if $c^K_{X, X', Z, E}$ is a quasi-isomorphism\nfor every almost blow up square (\\ref{equation-almost-blow-up-square})\nin $(\\Sch/S)_h$ with $X$ affine.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Almost blow up squares and the h topology","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWQ","source_file":"etale-cohomology.tex","source_line":23595,"source_end_line":23603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23595-L23603","statement_sha256":"703230ae4661304baee422c83e2c5c83e5bc37134f5fe02dcbe1adad21ae4e4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10247,"rank":10247,"depth":60,"x":1543.328,"y":1256.733,"cluster":"tale-geometry"},{"id":"stacks:0EWR","tag":"0EWR","title":"Almost blow up squares and the h topology · Lemma 0EWR","summary":"Let K be an object of D^+((Sch/S)_fppf). Then K is in the essential image of Rε_* : D((Sch/S)_h) → D((Sch/S)_fppf) if and only if c^K_X, X', Z, E is a quasi-isomorphism for every almost blow up square as in More on Flatness, Examples [Tag 0EVG] and [Tag 0EVH].","statement_latex":"Let $K$ be an object of $D^+((\\Sch/S)_{fppf})$. Then $K$ is in the\nessential image of $R\\epsilon_* : D((\\Sch/S)_h) \\to D((\\Sch/S)_{fppf})$\nif and only if $c^K_{X, X', Z, E}$ is a quasi-isomorphism\nfor every almost blow up square as in\nMore on Flatness, Examples \\ref{flat-example-one-generator} and\n\\ref{flat-example-two-generators}.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Almost blow up squares and the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWR","source_file":"etale-cohomology.tex","source_line":23613,"source_end_line":23621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23613-L23621","statement_sha256":"aad172737780d76963bca5566f26e83aaa6f7fa1686f4d56c2008d12c72f18f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10248,"rank":10248,"depth":61,"x":1513.115,"y":1033.03,"cluster":"tale-geometry"},{"id":"stacks:0EWT","tag":"0EWT","title":"Cohomology of the structure sheaf in the h topology · Lemma 0EWT","summary":"Let p be a prime number. Let S be a scheme over F_p. Consider the sheaf O^perf = colim_F O on (Sch/S)_fppf. Then O^perf is in the essential image of Rε_* : D((Sch/S)_h) → D((Sch/S)_fppf).","statement_latex":"Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$.\nConsider the sheaf $\\mathcal{O}^{perf} = \\colim_F \\mathcal{O}$\non $(\\Sch/S)_{fppf}$. Then $\\mathcal{O}^{perf}$ is in the essential\nimage of $R\\epsilon_* : D((\\Sch/S)_h) \\to D((\\Sch/S)_{fppf})$.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of the structure sheaf in the h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWT","source_file":"etale-cohomology.tex","source_line":23650,"source_end_line":23656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23650-L23656","statement_sha256":"5f9d8a05fd8cde299b30fe7106697e0f9b5eaa08ace4c122af3713161d6ebd9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10249,"rank":10249,"depth":62,"x":1715.449,"y":1180.958,"cluster":"tale-geometry"},{"id":"stacks:0EWU","tag":"0EWU","title":"Cohomology of the structure sheaf in the h topology · Proposition 0EWU","summary":"Let p be a prime number. Let S be a quasi-compact and quasi-separated scheme over F_p. Then H^i((Sch/S)_h, O^h) = colim_F H^i(S, O) Here on the left hand side by O^h we mean the h sheafification of the structure sheaf.","statement_latex":"Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated\nscheme over $\\mathbf{F}_p$. Then\n$$\nH^i((\\Sch/S)_h, \\mathcal{O}^h) =\n\\colim_F H^i(S, \\mathcal{O})\n$$\nHere on the left hand side by $\\mathcal{O}^h$ we mean\nthe h sheafification of the structure sheaf.","area":"Étale Geometry","chapter":"Étale Cohomology","chapter_id":"etale-cohomology","section":"Cohomology of the structure sheaf in the h topology","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EWU","source_file":"etale-cohomology.tex","source_line":23707,"source_end_line":23717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/etale-cohomology.tex#L23707-L23717","statement_sha256":"19bc6df900e7561568415e32b3b4523334513003c7d09a8f42331a7a5eed826b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10250,"rank":10250,"depth":63,"x":1447.08,"y":1186.692,"cluster":"tale-geometry"},{"id":"stacks:07H8","tag":"07H8","title":"Divided power envelope · Lemma 07H8","summary":"Let (A, I, γ) be a divided power ring. Let A → B be a ring map. Let J ⊂ B be an ideal with IB ⊂ J. There exists a homomorphism of divided power rings (A, I, γ) → (D, bar J, bar γ) such that Hom_(A, I, γ)((D, bar J, bar γ), (C, K, δ)) = Hom_(A, I)((B, J), (C, K)) functorially in the divided power algebra (C, K, δ) over (A, I, γ). Here the LHS is morphisms of divided power rings over (A, I, γ) and the RHS is morphisms of (ring, ideal) pairs over (A, I).","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring.\nLet $A \\to B$ be a ring map. Let $J \\subset B$ be an ideal\nwith $IB \\subset J$. There exists a homomorphism of\ndivided power rings\n$$\n(A, I, \\gamma) \\longrightarrow (D, \\bar J, \\bar \\gamma)\n$$\nsuch that\n$$\n\\Hom_{(A, I, \\gamma)}((D, \\bar J, \\bar \\gamma), (C, K, \\delta)) =\n\\Hom_{(A, I)}((B, J), (C, K))\n$$\nfunctorially in the divided power algebra $(C, K, \\delta)$ over\n$(A, I, \\gamma)$. Here the LHS is morphisms of divided\npower rings over $(A, I, \\gamma)$ and the RHS is morphisms of\n(ring, ideal) pairs over $(A, I)$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power envelope","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07H8","source_file":"crystalline.tex","source_line":53,"source_end_line":71,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L53-L71","statement_sha256":"a766011c1d7a2152e3641431fa8e4d79ea94d3f195f2691beff9ccdd8e1d6f40","origin":"The Stacks Project","memory_eligible":false,"source_rank":10251,"rank":10251,"depth":2,"x":2499.898,"y":1008.641,"cluster":"local-crystalline-methods"},{"id":"stacks:07H9","tag":"07H9","title":"Divided power envelope · Definition 07H9","summary":"Let (A, I, γ) be a divided power ring. Let A → B be a ring map. Let J ⊂ B be an ideal with IB ⊂ J. The divided power algebra (D, bar J, barγ) constructed in Lemma [Tag 07H8] is called the divided power envelope of J in B relative to (A, I, γ) and is denoted D_B(J) or D_B, γ(J).","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring.\nLet $A \\to B$ be a ring map. Let $J \\subset B$ be an ideal\nwith $IB \\subset J$. The divided power algebra $(D, \\bar J, \\bar\\gamma)$\nconstructed in Lemma \\ref{lemma-divided-power-envelope}\nis called the {\\it divided power envelope of $J$ in $B$\nrelative to $(A, I, \\gamma)$} and is denoted $D_B(J)$ or $D_{B, \\gamma}(J)$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power envelope","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07H9","source_file":"crystalline.tex","source_line":110,"source_end_line":118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L110-L118","statement_sha256":"c50bbb1b8e855765b278abdad5cd197ad58d2efb5fc449f88b3ec04572768454","origin":"The Stacks Project","memory_eligible":false,"source_rank":10252,"rank":10252,"depth":3,"x":2571.117,"y":1248.337,"cluster":"local-crystalline-methods"},{"id":"stacks:07HB","tag":"07HB","title":"Divided power envelope · Lemma 07HB","summary":"Let (A, I, γ) be a divided power ring. Let φ : B' → B be a surjection of A-algebras with kernel K. Let IB ⊂ J ⊂ B be an ideal. Let J' ⊂ B' be the inverse image of J. Write D_B', γ(J') = (D', bar J', barγ). Then D_B, γ(J) = (D'/K', bar J'/K', barγ) where K' is the ideal generated by the elements barγ_n(k) for n ≥ 1 and k ∈ K.","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring.\nLet $\\varphi : B' \\to B$ be a surjection of $A$-algebras with kernel $K$.\nLet $IB \\subset J \\subset B$ be an ideal. Let $J' \\subset B'$\nbe the inverse image of $J$. Write\n$D_{B', \\gamma}(J') = (D', \\bar J', \\bar\\gamma)$.\nThen $D_{B, \\gamma}(J) = (D'/K', \\bar J'/K', \\bar\\gamma)$\nwhere $K'$ is the ideal generated by the elements $\\bar\\gamma_n(k)$\nfor $n \\geq 1$ and $k \\in K$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power envelope","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HB","source_file":"crystalline.tex","source_line":148,"source_end_line":158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L148-L158","statement_sha256":"c00055ede55c0c9c045789ad143ac7644cdb685d62b01784339c080b45bc37d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10253,"rank":10253,"depth":2,"x":2325.429,"y":1111.768,"cluster":"local-crystalline-methods"},{"id":"stacks:07HC","tag":"07HC","title":"Divided power envelope · Lemma 07HC","summary":"Let (B, I, γ) be a divided power algebra. Let I ⊂ J ⊂ B be an ideal. Let (D, bar J, bar γ) be the divided power envelope of J relative to γ. Choose elements f_t ∈ J, t ∈ T such that J = I + (f_t). Then there exists a surjection Ψ : (Blangle x_t rangle, IBlangle x_t rangle + Blangle x_t rangle_+, δ) → (D, bar J, bar γ) of divided power rings mapping x_t to the image of f_t in D. The kernel of Ψ is generated by the elements x_t - f_t and all δ_n(∑ r_t x_t - r_0) whenever ∑…","statement_latex":"Let $(B, I, \\gamma)$ be a divided power algebra. Let $I \\subset J \\subset B$\nbe an ideal. Let $(D, \\bar J, \\bar \\gamma)$ be the divided power envelope\nof $J$ relative to $\\gamma$. Choose elements $f_t \\in J$, $t \\in T$ such\nthat $J = I + (f_t)$. Then there exists a surjection\n$$\n\\Psi :\n(B\\langle x_t \\rangle, IB\\langle x_t \\rangle + B\\langle x_t \\rangle_+, \\delta)\n\\longrightarrow\n(D, \\bar J, \\bar \\gamma)\n$$\nof divided power rings mapping $x_t$ to the image of $f_t$ in $D$.\nThe kernel of $\\Psi$ is generated by the elements $x_t - f_t$ and\nall\n$$\n\\delta_n\\left(\\sum r_t x_t - r_0\\right)\n$$\nwhenever $\\sum r_t f_t = r_0$ in $B$ for some $r_t \\in B$, $r_0 \\in I$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power envelope","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HC","source_file":"crystalline.tex","source_line":183,"source_end_line":202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L183-L202","statement_sha256":"9f2569ed3f47c7915001d119c31dccbcce58297b979732fbe49e2ffad108b1ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":10254,"rank":10254,"depth":2,"x":2616.912,"y":1072.997,"cluster":"local-crystalline-methods"},{"id":"stacks:07KE","tag":"07KE","title":"Divided power envelope · Lemma 07KE","summary":"Let (A, I, γ) be a divided power ring. Let B be an A-algebra and IB ⊂ J ⊂ B an ideal. Let x_i be a set of variables. Then D_B[x_i], γ(JB[x_i] + (x_i)) = D_B, γ(J) langle x_i rangle","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring.\nLet $B$ be an $A$-algebra and $IB \\subset J \\subset B$ an ideal.\nLet $x_i$ be a set of variables. Then\n$$\nD_{B[x_i], \\gamma}(JB[x_i] + (x_i)) = D_{B, \\gamma}(J) \\langle x_i \\rangle\n$$","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power envelope","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KE","source_file":"crystalline.tex","source_line":258,"source_end_line":266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L258-L266","statement_sha256":"d307cfcec3c9919c624654f2cac649d211276b581a1c583cd26f2f67b31b3afa","origin":"The Stacks Project","memory_eligible":false,"source_rank":10255,"rank":10255,"depth":3,"x":2432.846,"y":1267.308,"cluster":"local-crystalline-methods"},{"id":"stacks:07HD","tag":"07HD","title":"Divided power envelope · Lemma 07HD","summary":"Let (A, I, γ) be a divided power ring. Let B → B' be a homomorphism of A-algebras. Assume that • B/IB → B'/IB' is flat, and • Tor_1^B(B', B/IB) = 0. Then for any ideal IB ⊂ J ⊂ B the canonical map D_B(J) ⊗_B B' → D_B'(JB') is an isomorphism.","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring.\nLet $B \\to B'$ be a homomorphism of $A$-algebras.\nAssume that\n\\begin{enumerate}\n\\item $B/IB \\to B'/IB'$ is flat, and\n\\item $\\text{Tor}_1^B(B', B/IB) = 0$.\n\\end{enumerate}\nThen for any ideal $IB \\subset J \\subset B$ the canonical map\n$$\nD_B(J) \\otimes_B B' \\longrightarrow D_{B'}(JB')\n$$\nis an isomorphism.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power envelope","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HD","source_file":"crystalline.tex","source_line":285,"source_end_line":299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L285-L299","statement_sha256":"29756a228d4efbedc6369b7b0a2d047e3b5ece72cea61ac803d8e3ce280c3526","origin":"The Stacks Project","memory_eligible":false,"source_rank":10256,"rank":10256,"depth":5,"x":2412.28,"y":1019.167,"cluster":"local-crystalline-methods"},{"id":"stacks:07HE","tag":"07HE","title":"Divided power envelope · Lemma 07HE","summary":"Let (B, I, γ) → (B', I', γ') be a homomorphism of divided power rings. Let I ⊂ J ⊂ B and I' ⊂ J' ⊂ B' be ideals. Assume • B/I → B'/I' is flat, and • J' = JB' + I'. Then the canonical map D_B, γ(J) ⊗_B B' → D_B', γ'(J') is an isomorphism.","statement_latex":"Let $(B, I, \\gamma) \\to (B', I', \\gamma')$ be a homomorphism of\ndivided power rings. Let $I \\subset J \\subset B$ and\n$I' \\subset J' \\subset B'$ be ideals. Assume\n\\begin{enumerate}\n\\item $B/I \\to B'/I'$ is flat, and\n\\item $J' = JB' + I'$.\n\\end{enumerate}\nThen the canonical map\n$$\nD_{B, \\gamma}(J) \\otimes_B B' \\longrightarrow D_{B', \\gamma'}(J')\n$$\nis an isomorphism.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power envelope","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HE","source_file":"crystalline.tex","source_line":369,"source_end_line":383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L369-L383","statement_sha256":"090e4b66582b813f37663af92dbedb084cfe72bdc12d38962cfa63299b4b8c0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10257,"rank":10257,"depth":3,"x":2627.352,"y":1190.759,"cluster":"local-crystalline-methods"},{"id":"stacks:07HH","tag":"07HH","title":"Some explicit divided power thickenings · Lemma 07HH","summary":"Let (A, I, γ) be a divided power ring. Let M be an A-module. Let B = A ⊕ M as an A-algebra where M is an ideal of square zero and set J = I ⊕ M. Set δ_n(x + z) = γ_n(x) + γ_n - 1(x)z for x ∈ I and z ∈ M. Then δ is a divided power structure and A → B is a homomorphism of divided power rings from (A, I, γ) to (B, J, δ).","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring. Let $M$ be an $A$-module.\nLet $B = A \\oplus M$ as an $A$-algebra where $M$ is an ideal of square zero\nand set $J = I \\oplus M$. Set\n$$\n\\delta_n(x + z) = \\gamma_n(x) + \\gamma_{n - 1}(x)z\n$$\nfor $x \\in I$ and $z \\in M$.\nThen $\\delta$ is a divided power structure and\n$A \\to B$ is a homomorphism of divided power rings from\n$(A, I, \\gamma)$ to $(B, J, \\delta)$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Some explicit divided power thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HH","source_file":"crystalline.tex","source_line":471,"source_end_line":483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L471-L483","statement_sha256":"fea8b58ab082d7378d18ff2a8b18994120fde70ac108c980ea7fce741cb4061d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10258,"rank":10258,"depth":1,"x":2330.277,"y":1186.259,"cluster":"local-crystalline-methods"},{"id":"stacks:07HI","tag":"07HI","title":"Some explicit divided power thickenings · Lemma 07HI","summary":"Let (A, I, γ) be a divided power ring. Let M, N be A-modules. Let q : M × M → N be an A-bilinear map. Let B = A ⊕ M ⊕ N as an A-algebra with multiplication (x, z, w)· (x', z', w') = (xx', xz' + x'z, xw' + x'w + q(z, z') + q(z', z)) and set J = I ⊕ M ⊕ N. Set δ_n(x, z, w) = (γ_n(x), γ_n - 1(x)z, γ_n - 1(x)w + γ_n - 2(x)q(z, z)) for (x, z, w) ∈ J. Then δ is a divided power structure and A → B is a homomorphism of divided power rings from (A, I, γ) to (B, J, δ).","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring. Let $M$, $N$ be $A$-modules.\nLet $q : M \\times M \\to N$ be an $A$-bilinear map.\nLet $B = A \\oplus M \\oplus N$ as an $A$-algebra with multiplication\n$$\n(x, z, w)\\cdot (x', z', w') = (xx', xz' + x'z, xw' + x'w + q(z, z') + q(z', z))\n$$\nand set $J = I \\oplus M \\oplus N$. Set\n$$\n\\delta_n(x, z, w) = (\\gamma_n(x), \\gamma_{n - 1}(x)z,\n\\gamma_{n - 1}(x)w + \\gamma_{n - 2}(x)q(z, z))\n$$\nfor $(x, z, w) \\in J$.\nThen $\\delta$ is a divided power structure and\n$A \\to B$ is a homomorphism of divided power rings from\n$(A, I, \\gamma)$ to $(B, J, \\delta)$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Some explicit divided power thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HI","source_file":"crystalline.tex","source_line":534,"source_end_line":551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L534-L551","statement_sha256":"ae317c0cc2887f3bf356aae2bc6bec2db3f287517b2853f2ab3738e05704f82e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10259,"rank":10259,"depth":2,"x":2553.325,"y":1020.736,"cluster":"local-crystalline-methods"},{"id":"stacks:07HK","tag":"07HK","title":"Compatibility · Definition 07HK","summary":"Let (A, I, γ) and (B, J, δ) be divided power rings. Let A → B be a ring map. We say δ is compatible with γ if there exists a divided power structure barγ on J + IB such that (A, I, γ) → (B, J + IB, bar γ) and (B, J, δ) → (B, J + IB, bar γ) are homomorphisms of divided power rings.","statement_latex":"Let $(A, I, \\gamma)$ and $(B, J, \\delta)$ be divided power rings.\nLet $A \\to B$ be a ring map. We say\n{\\it $\\delta$ is compatible with $\\gamma$}\nif there exists a divided power structure $\\bar\\gamma$ on\n$J + IB$ such that\n$$\n(A, I, \\gamma) \\to (B, J + IB, \\bar \\gamma)\\quad\\text{and}\\quad\n(B, J, \\delta) \\to (B, J + IB, \\bar \\gamma)\n$$\nare homomorphisms of divided power rings.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Compatibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HK","source_file":"crystalline.tex","source_line":608,"source_end_line":620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L608-L620","statement_sha256":"63e05e2c77b4ecf551de300eac74f3a7eac404500fbfc0082619a6ad3a27dddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10260,"rank":10260,"depth":0,"x":2521.909,"y":1269.764,"cluster":"local-crystalline-methods"},{"id":"stacks:07HM","tag":"07HM","title":"Affine crystalline site · Definition 07HM","summary":"In Situation [Tag 07MD]. • A divided power thickening of C over (A, I, γ) is a homomorphism of divided power algebras (A, I, γ) → (B, J, δ) such that p is nilpotent in B and a ring map C → B/J such that xymatrix B ar[r] & B/J & C ar[u] A ar[uu] ar[r] & A/I ar[u] is commutative. • A homomorphism of divided power thickenings (B, J, δ, C → B/J) → (B', J', δ', C → B'/J') is a homomorphism φ : B → B' of divided power A-algebras such that C → B/J → B'/J' is the given map C →…","statement_latex":"In Situation \\ref{situation-affine}.\n\\begin{enumerate}\n\\item A {\\it divided power thickening} of $C$ over $(A, I, \\gamma)$\nis a homomorphism of divided power algebras $(A, I, \\gamma) \\to (B, J, \\delta)$\nsuch that $p$ is nilpotent in $B$ and a ring map $C \\to B/J$ such that\n$$\n\\xymatrix{\nB \\ar[r] & B/J \\\\\n& C \\ar[u] \\\\\nA \\ar[uu] \\ar[r] & A/I \\ar[u]\n}\n$$\nis commutative.\n\\item A {\\it homomorphism of divided power thickenings}\n$$\n(B, J, \\delta, C \\to B/J) \\longrightarrow (B', J', \\delta', C \\to B'/J')\n$$\nis a homomorphism $\\varphi : B \\to B'$ of divided power $A$-algebras such\nthat $C \\to B/J \\to B'/J'$ is the given map $C \\to B'/J'$.\n\\item We denote $\\text{CRIS}(C/A, I, \\gamma)$ or simply $\\text{CRIS}(C/A)$\nthe category of divided power thickenings of $C$ over $(A, I, \\gamma)$.\n\\item We denote $\\text{Cris}(C/A, I, \\gamma)$ or simply $\\text{Cris}(C/A)$\nthe full subcategory consisting of $(B, J, \\delta, C \\to B/J)$ such that\n$C \\to B/J$ is an isomorphism. We often denote such an object\n$(B \\to C, \\delta)$ with $J = \\Ker(B \\to C)$ being understood.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Affine crystalline site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HM","source_file":"crystalline.tex","source_line":682,"source_end_line":710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L682-L710","statement_sha256":"62a211ed1dd121a535d2770a4ed956cfbcc2e80b0b89a5518c303995c89d99ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":10261,"rank":10261,"depth":0,"x":2344.522,"y":1067.976,"cluster":"local-crystalline-methods"},{"id":"stacks:07HN","tag":"07HN","title":"Affine crystalline site · Lemma 07HN","summary":"In Situation [Tag 07MD]. • CRIS(C/A) has finite products (but not infinite ones), • CRIS(C/A) has all finite nonempty colimits and ([Tag 07KF]) commutes with these, and • Cris(C/A) has all finite nonempty colimits and Cris(C/A) → CRIS(C/A) commutes with them.","statement_latex":"In Situation \\ref{situation-affine}.\n\\begin{enumerate}\n\\item $\\text{CRIS}(C/A)$ has finite products (but not infinite ones),\n\\item $\\text{CRIS}(C/A)$ has all finite nonempty colimits and\n(\\ref{equation-forget-affine}) commutes with these, and\n\\item $\\text{Cris}(C/A)$ has all finite nonempty colimits and\n$\\text{Cris}(C/A) \\to \\text{CRIS}(C/A)$ commutes with them.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Affine crystalline site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HN","source_file":"crystalline.tex","source_line":725,"source_end_line":735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L725-L735","statement_sha256":"47fd14c828575e8b4fb662756adbec743a2df093eba73bf11242ff6178cf5cdb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10262,"rank":10262,"depth":3,"x":2638.078,"y":1116.197,"cluster":"local-crystalline-methods"},{"id":"stacks:07KG","tag":"07KG","title":"Affine crystalline site · Lemma 07KG","summary":"In Situation [Tag 07MD]. Let P → C be a surjection of A-algebras with kernel J. Write D_P, γ(J) = (D, bar J, barγ). Let (D^wedge, J^wedge, barγ^wedge) be the p-adic completion of D, see Divided Power Algebra, Lemma [Tag 07KD]. For every e ≥ 1 set P_e = P/p^eP and J_e ⊂ P_e the image of J and write D_P_e, γ(J_e) = (D_e, bar J_e, barγ). Then for all e large enough we have • p^eD ⊂ bar J and p^eD^wedge ⊂ bar J^wedge are preserved by divided powers, • D^wedge/p^eD^wedge =…","statement_latex":"In Situation \\ref{situation-affine}.\nLet $P \\to C$ be a surjection of $A$-algebras with kernel $J$.\nWrite $D_{P, \\gamma}(J) = (D, \\bar J, \\bar\\gamma)$.\nLet $(D^\\wedge, J^\\wedge, \\bar\\gamma^\\wedge)$ be the\n$p$-adic completion of $D$, see\nDivided Power Algebra, Lemma \\ref{dpa-lemma-extend-to-completion}.\nFor every $e \\geq 1$ set $P_e = P/p^eP$ and $J_e \\subset P_e$\nthe image of $J$ and write\n$D_{P_e, \\gamma}(J_e) = (D_e, \\bar J_e, \\bar\\gamma)$.\nThen for all $e$ large enough we have\n\\begin{enumerate}\n\\item $p^eD \\subset \\bar J$ and $p^eD^\\wedge \\subset \\bar J^\\wedge$\nare preserved by divided powers,\n\\item $D^\\wedge/p^eD^\\wedge = D/p^eD = D_e$ as divided power rings,\n\\item $(D_e, \\bar J_e, \\bar\\gamma)$ is an object of $\\text{Cris}(C/A)$,\n\\item $(D^\\wedge, \\bar J^\\wedge, \\bar\\gamma^\\wedge)$ is equal to\n$\\lim_e (D_e, \\bar J_e, \\bar\\gamma)$, and\n\\item $(D^\\wedge, \\bar J^\\wedge, \\bar\\gamma^\\wedge)$ is an object of\n$\\text{Cris}^\\wedge(C/A)$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Affine crystalline site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KG","source_file":"crystalline.tex","source_line":824,"source_end_line":846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L824-L846","statement_sha256":"bee16bf7c19a255ea40aa1cb8049f60a30678939839e221663b906d1c1f89b04","origin":"The Stacks Project","memory_eligible":false,"source_rank":10263,"rank":10263,"depth":5,"x":2382.418,"y":1247.425,"cluster":"local-crystalline-methods"},{"id":"stacks:07HP","tag":"07HP","title":"Affine crystalline site · Lemma 07HP","summary":"In Situation [Tag 07MD]. Let P be a polynomial algebra over A and let P → C be a surjection of A-algebras with kernel J. With (D_e, bar J_e, barγ) as in Lemma [Tag 07KG]: for every object (B, J_B, δ) of CRIS(C/A) there exists an e and a morphism D_e → B of CRIS(C/A).","statement_latex":"In Situation \\ref{situation-affine}.\nLet $P$ be a polynomial algebra over $A$ and let\n$P \\to C$ be a surjection of $A$-algebras with kernel $J$.\nWith $(D_e, \\bar J_e, \\bar\\gamma)$ as in Lemma \\ref{lemma-list-properties}:\nfor every object $(B, J_B, \\delta)$ of $\\text{CRIS}(C/A)$ there\nexists an $e$ and a morphism $D_e \\to B$ of $\\text{CRIS}(C/A)$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Affine crystalline site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HP","source_file":"crystalline.tex","source_line":863,"source_end_line":871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L863-L871","statement_sha256":"c906fb2f32295ef38c9ef95f03365093865efe14814677b161e7a8a5f5639b80","origin":"The Stacks Project","memory_eligible":false,"source_rank":10264,"rank":10264,"depth":6,"x":2465.543,"y":1005.197,"cluster":"local-crystalline-methods"},{"id":"stacks:07KI","tag":"07KI","title":"Affine crystalline site · Lemma 07KI","summary":"In Situation [Tag 07MD]. Let P be a polynomial algebra over A and let P → C be a surjection of A-algebras with kernel J. Let (D, bar J, barγ) be the p-adic completion of D_P, γ(J). For every object (B → C, δ) of Cris^wedge(C/A) there exists a morphism D → B of Cris^wedge(C/A).","statement_latex":"In Situation \\ref{situation-affine}.\nLet $P$ be a polynomial algebra over $A$ and let\n$P \\to C$ be a surjection of $A$-algebras with kernel $J$.\nLet $(D, \\bar J, \\bar\\gamma)$ be the $p$-adic completion of\n$D_{P, \\gamma}(J)$. For every object $(B \\to C, \\delta)$ of\n$\\text{Cris}^\\wedge(C/A)$ there\nexists a morphism $D \\to B$ of $\\text{Cris}^\\wedge(C/A)$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Affine crystalline site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KI","source_file":"crystalline.tex","source_line":882,"source_end_line":891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L882-L891","statement_sha256":"acdf6061f4d914c078931d8d5fc6a668ba0af2bd0105503d7d3f40417b8bb7e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10265,"rank":10265,"depth":0,"x":2599.258,"y":1231.347,"cluster":"local-crystalline-methods"},{"id":"stacks:07HR","tag":"07HR","title":"Module of differentials · Definition 07HR","summary":"Let A be a ring. Let (B, J, δ) be a divided power ring. Let A → B be a ring map. Let M be an B-module. A divided power A-derivation into M is a map theta : B → M which is additive, annihilates the elements of A, satisfies the Leibniz rule theta(bb') = btheta(b') + b'theta(b) and satisfies theta(δ_n(x)) = δ_n - 1(x)theta(x) for all n ≥ 1 and all x ∈ J.","statement_latex":"Let $A$ be a ring. Let $(B, J, \\delta)$ be a divided power ring.\nLet $A \\to B$ be a ring map. Let $M$ be an $B$-module.\nA {\\it divided power $A$-derivation} into $M$ is a map\n$\\theta : B \\to M$ which is additive, annihilates the elements\nof $A$, satisfies the Leibniz rule\n$\\theta(bb') = b\\theta(b') + b'\\theta(b)$ and satisfies\n$$\n\\theta(\\delta_n(x)) = \\delta_{n - 1}(x)\\theta(x)\n$$\nfor all $n \\geq 1$ and all $x \\in J$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Module of differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HR","source_file":"crystalline.tex","source_line":910,"source_end_line":922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L910-L922","statement_sha256":"ca79227ab6c69e14ab1d2059d000c9790e4e6467c909d5f41af03891b530b3b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10266,"rank":10266,"depth":0,"x":2318.341,"y":1140.313,"cluster":"local-crystalline-methods"},{"id":"stacks:07HS","tag":"07HS","title":"Module of differentials · Lemma 07HS","summary":"Let A be a ring. Let (B, J, δ) be a divided power ring and A → B a ring map. • Consider B[x] with divided power ideal (JB[x], δ') where δ' is the extension of δ to B[x]. Then Ω_B[x]/A, δ' = Ω_B/A, δ ⊗_B B[x] ⊕ B[x]dx. • Consider Blangle x rangle with divided power ideal (JBlangle x rangle + Blangle x rangle_+, δ'). Then Ω_Blangle xrangle/A, δ' = Ω_B/A, δ ⊗_B Blangle x rangle ⊕ Blangle xrangle dx. • Let K ⊂ J be an ideal preserved by δ_n for all n > 0. Set B' = B/K and…","statement_latex":"Let $A$ be a ring. Let $(B, J, \\delta)$ be a divided power ring and\n$A \\to B$ a ring map. \n\\begin{enumerate}\n\\item Consider $B[x]$ with divided power ideal $(JB[x], \\delta')$\nwhere $\\delta'$ is the extension of $\\delta$ to $B[x]$. Then\n$$\n\\Omega_{B[x]/A, \\delta'} =\n\\Omega_{B/A, \\delta} \\otimes_B B[x] \\oplus B[x]\\text{d}x.\n$$\n\\item Consider $B\\langle x \\rangle$ with divided power ideal\n$(JB\\langle x \\rangle + B\\langle x \\rangle_{+}, \\delta')$. Then\n$$\n\\Omega_{B\\langle x\\rangle/A, \\delta'} =\n\\Omega_{B/A, \\delta} \\otimes_B B\\langle x \\rangle \\oplus\nB\\langle x\\rangle \\text{d}x.\n$$\n\\item Let $K \\subset J$ be an ideal preserved by $\\delta_n$ for\nall $n > 0$. Set $B' = B/K$ and denote $\\delta'$ the induced\ndivided power on $J/K$. Then $\\Omega_{B'/A, \\delta'}$ is the quotient\nof $\\Omega_{B/A, \\delta} \\otimes_B B'$ by the $B'$-submodule generated\nby $\\text{d}k$ for $k \\in K$.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Module of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HS","source_file":"crystalline.tex","source_line":938,"source_end_line":962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L938-L962","statement_sha256":"1baeabf8b6c32d9150a1403595ee7b45c958316af9ad4df2c15ffc0f28276f42","origin":"The Stacks Project","memory_eligible":false,"source_rank":10267,"rank":10267,"depth":0,"x":2599.146,"y":1047.892,"cluster":"local-crystalline-methods"},{"id":"stacks:07HT","tag":"07HT","title":"Module of differentials · Lemma 07HT","summary":"Let (A, I, γ) → (B, J, δ) be a homomorphism of divided power rings. Let (B(1), J(1), δ(1)) be the coproduct of (B, J, δ) with itself over (A, I, γ), i.e., such that xymatrix (B, J, δ) ar[r] & (B(1), J(1), δ(1)) (A, I, γ) ar[r] ar[u] & (B, J, δ) ar[u] is cocartesian. Denote K = Ker(B(1) → B). Then K ∩ J(1) ⊂ J(1) is preserved by the divided power structure and Ω_B/A, δ = K/ (K^2 + (K ∩ J(1))^[2]) canonically.","statement_latex":"Let $(A, I, \\gamma) \\to (B, J, \\delta)$ be a homomorphism\nof divided power rings. Let $(B(1), J(1), \\delta(1))$ be the coproduct\nof $(B, J, \\delta)$ with itself over $(A, I, \\gamma)$, i.e.,\nsuch that\n$$\n\\xymatrix{\n(B, J, \\delta) \\ar[r] & (B(1), J(1), \\delta(1)) \\\\\n(A, I, \\gamma) \\ar[r] \\ar[u] & (B, J, \\delta) \\ar[u]\n}\n$$\nis cocartesian. Denote $K = \\Ker(B(1) \\to B)$.\nThen $K \\cap J(1) \\subset J(1)$ is preserved by the divided power\nstructure and\n$$\n\\Omega_{B/A, \\delta} = K/ \\left(K^2 + (K \\cap J(1))^{[2]}\\right)\n$$\ncanonically.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Module of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HT","source_file":"crystalline.tex","source_line":990,"source_end_line":1009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L990-L1009","statement_sha256":"79aba09d6a18549b17778774bbb41f21e5c6459d613da5a75e800de45ab8d21b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10268,"rank":10268,"depth":2,"x":2466.193,"y":1275.743,"cluster":"local-crystalline-methods"},{"id":"stacks:07HV","tag":"07HV","title":"Module of differentials · Lemma 07HV","summary":"In Situation [Tag 07MD]. Let (B, J, δ) be an object of CRIS(C/A). Let (B(1), J(1), δ(1)) be the coproduct of (B, J, δ) with itself in CRIS(C/A). Denote K = Ker(B(1) → B). Then K ∩ J(1) ⊂ J(1) is preserved by the divided power structure and Ω_B/A, δ = K/ (K^2 + (K ∩ J(1))^[2]) canonically.","statement_latex":"In Situation \\ref{situation-affine}.\nLet $(B, J, \\delta)$ be an object of $\\text{CRIS}(C/A)$.\nLet $(B(1), J(1), \\delta(1))$ be the coproduct of $(B, J, \\delta)$\nwith itself in $\\text{CRIS}(C/A)$. Denote\n$K = \\Ker(B(1) \\to B)$. Then $K \\cap J(1) \\subset J(1)$\nis preserved by the divided power structure and\n$$\n\\Omega_{B/A, \\delta} = K/ \\left(K^2 + (K \\cap J(1))^{[2]}\\right)\n$$\ncanonically.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Module of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HV","source_file":"crystalline.tex","source_line":1075,"source_end_line":1087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1075-L1087","statement_sha256":"59c852f979a408d3383c121c9f7719bfd9bb3c7b81234a79dbc3c91358be365a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10269,"rank":10269,"depth":3,"x":2380.861,"y":1031.897,"cluster":"local-crystalline-methods"},{"id":"stacks:07HW","tag":"07HW","title":"Module of differentials · Lemma 07HW","summary":"Let (A, I, γ) be a divided power ring. Let A → B be a ring map and let IB ⊂ J ⊂ B be an ideal. Let D_B, γ(J) = (D, bar J, bar γ) be the divided power envelope. Then we have Ω_D/A, barγ = Ω_B/A ⊗_B D","statement_latex":"Let $(A, I, \\gamma)$ be a divided power ring. Let $A \\to B$ be a ring\nmap and let $IB \\subset J \\subset B$ be an ideal. Let\n$D_{B, \\gamma}(J) = (D, \\bar J, \\bar \\gamma)$ be the divided power envelope.\nThen we have\n$$\n\\Omega_{D/A, \\bar\\gamma} = \\Omega_{B/A} \\otimes_B D\n$$","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Module of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07HW","source_file":"crystalline.tex","source_line":1100,"source_end_line":1109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1100-L1109","statement_sha256":"d20ac26b9e7ad7879e73be5e1e7fd2b50db8be418a764e12df8b7d615a09892d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10270,"rank":10270,"depth":2,"x":2640.292,"y":1163.497,"cluster":"local-crystalline-methods"},{"id":"stacks:07KK","tag":"07KK","title":"Module of differentials · Lemma 07KK","summary":"Let A → B be a ring map and let (J, δ) be a divided power structure on B. Let p be a prime number. Assume that A is a Z_(p)-algebra and that p is nilpotent in B/J. Then we have lim_e Ω_B_e/A, barδ = lim_e Ω_B/A, δ/p^eΩ_B/A, δ = lim_e Ω_B^wedge/A, δ^wedge/p^e Ω_B^wedge/A, δ^wedge see proof for notation and explanation.","statement_latex":"Let $A \\to B$ be a ring map and let $(J, \\delta)$ be a divided power\nstructure on $B$. Let $p$ be a prime number. Assume that $A$ is a\n$\\mathbf{Z}_{(p)}$-algebra and that $p$ is nilpotent in $B/J$. Then\nwe have\n$$\n\\lim_e \\Omega_{B_e/A, \\bar\\delta} =\n\\lim_e \\Omega_{B/A, \\delta}/p^e\\Omega_{B/A, \\delta} =\n\\lim_e \\Omega_{B^\\wedge/A, \\delta^\\wedge}/p^e \\Omega_{B^\\wedge/A, \\delta^\\wedge}\n$$\nsee proof for notation and explanation.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Module of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KK","source_file":"crystalline.tex","source_line":1314,"source_end_line":1326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1314-L1326","statement_sha256":"5483070f887b8b658bae47451acb4af4967105c0ee4deff8c933581b9a33121e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10271,"rank":10271,"depth":5,"x":2342.689,"y":1213.744,"cluster":"local-crystalline-methods"},{"id":"stacks:07I2","tag":"07I2","title":"Divided power schemes · Definition 07I2","summary":"Let C be a site. Let O be a sheaf of rings on C. Let I ⊂ O be a sheaf of ideals. A divided power structure γ on I is a sequence of maps γ_n : I → I, n ≥ 1 such that for any object U of C the triple (O(U), I(U), γ) is a divided power ring.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O}$ be a sheaf of rings\non $\\mathcal{C}$. Let $\\mathcal{I} \\subset \\mathcal{O}$ be a\nsheaf of ideals. A {\\it divided power structure $\\gamma$} on $\\mathcal{I}$\nis a sequence of maps $\\gamma_n : \\mathcal{I} \\to \\mathcal{I}$, $n \\geq 1$\nsuch that for any object $U$ of $\\mathcal{C}$ the triple\n$$\n(\\mathcal{O}(U), \\mathcal{I}(U), \\gamma)\n$$\nis a divided power ring.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07I2","source_file":"crystalline.tex","source_line":1351,"source_end_line":1362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1351-L1362","statement_sha256":"ba082596b1cb499cf24f926dc7f0f038ea44b8d2d51b79c1efeb2238c8487505","origin":"The Stacks Project","memory_eligible":false,"source_rank":10272,"rank":10272,"depth":0,"x":2522.015,"y":1007.498,"cluster":"local-crystalline-methods"},{"id":"stacks:07I3","tag":"07I3","title":"Divided power schemes · Definition 07I3","summary":"A divided power scheme is a triple (S, I, γ) where S is a scheme, I is a quasi-coherent sheaf of ideals, and γ is a divided power structure on I. A morphism of divided power schemes (S, I, γ) → (S', I', γ') is a morphism of schemes f : S → S' such that f^-1I'O_S ⊂ I and such that (O_S'(U'), I'(U'), γ') → (O_S(f^-1U'), I(f^-1U'), γ) is a homomorphism of divided power rings for all U' ⊂ S' open.","statement_latex":"A {\\it divided power scheme} is a triple $(S, \\mathcal{I}, \\gamma)$\nwhere $S$ is a scheme, $\\mathcal{I}$ is a quasi-coherent sheaf of\nideals, and $\\gamma$ is a divided power structure on $\\mathcal{I}$.\nA {\\it morphism of divided power schemes}\n$(S, \\mathcal{I}, \\gamma) \\to (S', \\mathcal{I}', \\gamma')$ is\na morphism of schemes $f : S \\to S'$ such that\n$f^{-1}\\mathcal{I}'\\mathcal{O}_S \\subset \\mathcal{I}$ and such that\n$$\n(\\mathcal{O}_{S'}(U'), \\mathcal{I}'(U'), \\gamma')\n\\longrightarrow\n(\\mathcal{O}_S(f^{-1}U'), \\mathcal{I}(f^{-1}U'), \\gamma)\n$$\nis a homomorphism of divided power rings for all $U' \\subset S'$ open.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07I3","source_file":"crystalline.tex","source_line":1399,"source_end_line":1414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1399-L1414","statement_sha256":"846ab3e92870ffc2dbb1b72d863bd34b7082b2699a516f562d82e7ecb15e180b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10273,"rank":10273,"depth":0,"x":2555.693,"y":1261.737,"cluster":"local-crystalline-methods"},{"id":"stacks:07I4","tag":"07I4","title":"Divided power schemes · Definition 07I4","summary":"A triple (U, T, γ) as above is called a divided power thickening if U → T is a thickening.","statement_latex":"A triple $(U, T, \\gamma)$ as above is called a {\\it divided power thickening}\nif $U \\to T$ is a thickening.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07I4","source_file":"crystalline.tex","source_line":1429,"source_end_line":1433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1429-L1433","statement_sha256":"1a19c32417bda48b5adfb4ddcdad4c04a549a6c2f80a80aadaddf1770dc59675","origin":"The Stacks Project","memory_eligible":false,"source_rank":10274,"rank":10274,"depth":0,"x":2326.045,"y":1093.109,"cluster":"local-crystalline-methods"},{"id":"stacks:07ME","tag":"07ME","title":"Divided power schemes · Lemma 07ME","summary":"Let f : (T, J, δ) → (S, I, γ) and f' : (T', J', δ') → (S, I, γ) be morphisms of divided power schemes. There exists a divided power scheme (T\", J\", δ\") and a cartesian diagram xymatrix T ar[d]_f & T\" ar[d] ar[l] S & T' ar[l]_f' in the category of divided power schemes. The morphsm T\" → T ×_S T' is a closed immersion and the morphism T\"_0 → T_0 ×_S_0 T'_0 is an isomorphism.","statement_latex":"Let $f : (T, \\mathcal{J}, \\delta) \\to (S, \\mathcal{I}, \\gamma)$ and\n$f' : (T', \\mathcal{J}', \\delta') \\to (S, \\mathcal{I}, \\gamma)$\nbe morphisms of divided power schemes. There exists a divided power scheme\n$(T'', \\mathcal{J}'', \\delta'')$ and a cartesian diagram\n$$\n\\xymatrix{\nT \\ar[d]_f & T'' \\ar[d] \\ar[l] \\\\\nS & T' \\ar[l]_{f'}\n}\n$$\nin the category of divided power schemes. The morphsm\n$T'' \\to T \\times_S T'$ is a closed immersion and the morphism\n$T''_0 \\to T_0 \\times_{S_0} T'_0$ is an isomorphism.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ME","source_file":"crystalline.tex","source_line":1439,"source_end_line":1454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1439-L1454","statement_sha256":"6944f89228a2c8e2283587802441842490100d0fa3d5af5b1fe4e6f7052db102","origin":"The Stacks Project","memory_eligible":false,"source_rank":10275,"rank":10275,"depth":5,"x":2631.47,"y":1087.136,"cluster":"local-crystalline-methods"},{"id":"stacks:07I6","tag":"07I6","title":"The big crystalline site · Definition 07I6","summary":"In Situation [Tag 07MF]. • A divided power thickening of X relative to (S, I, γ) is given by a divided power thickening (U, T, δ) over (S, I, γ) and an S-morphism U → X. • A morphism of divided power thickenings of X relative to (S, I, γ) is defined in the obvious manner. The category of divided power thickenings of X relative to (S, I, γ) is denoted CRIS(X/S, I, γ) or simply CRIS(X/S).","statement_latex":"In Situation \\ref{situation-global}.\n\\begin{enumerate}\n\\item A {\\it divided power thickening of $X$ relative to\n$(S, \\mathcal{I}, \\gamma)$} is given by a divided power thickening\n$(U, T, \\delta)$ over $(S, \\mathcal{I}, \\gamma)$\nand an $S$-morphism $U \\to X$.\n\\item A {\\it morphism of divided power thickenings of $X$\nrelative to $(S, \\mathcal{I}, \\gamma)$} is defined in the obvious\nmanner.\n\\end{enumerate}\nThe category of divided power thickenings of $X$ relative to\n$(S, \\mathcal{I}, \\gamma)$ is denoted $\\text{CRIS}(X/S, \\mathcal{I}, \\gamma)$\nor simply $\\text{CRIS}(X/S)$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"The big crystalline site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07I6","source_file":"crystalline.tex","source_line":1525,"source_end_line":1540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1525-L1540","statement_sha256":"d1bc49791baa2a2addb26e6313c39970175185af2d2785d1f63084a262f97ee9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10276,"rank":10276,"depth":0,"x":2410.712,"y":1265.125,"cluster":"local-crystalline-methods"},{"id":"stacks:07I9","tag":"07I9","title":"The big crystalline site · Lemma 07I9","summary":"In Situation [Tag 07MF]. The category CRIS(X/S) has all finite nonempty limits, in particular products of pairs and fibre products. The functor ([Tag 07I7]) commutes with limits.","statement_latex":"In Situation \\ref{situation-global}.\nThe category $\\text{CRIS}(X/S)$ has all finite nonempty limits,\nin particular products of pairs and fibre products.\nThe functor (\\ref{equation-forget}) commutes with limits.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"The big crystalline site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07I9","source_file":"crystalline.tex","source_line":1571,"source_end_line":1577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1571-L1577","statement_sha256":"cb6c40922a402f8fc6cfebd13db99c2f2216c77ebef93ee9581409abff9f2fa9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10277,"rank":10277,"depth":4,"x":2430.393,"y":1008.214,"cluster":"local-crystalline-methods"},{"id":"stacks:07IA","tag":"07IA","title":"The big crystalline site · Lemma 07IA","summary":"In Situation [Tag 07MF]. Let xymatrix (U_3, T_3, δ_3) ar[d] ar[r] & (U_2, T_2, δ_2) ar[d] (U_1, T_1, δ_1) ar[r] & (U, T, δ) be a fibre square in the category of divided power thickenings of X relative to (S, I, γ). If T_2 → T is flat and U_2 = T_2 ×_T U, then T_3 = T_1 ×_T T_2 (as schemes).","statement_latex":"In Situation \\ref{situation-global}. Let\n$$\n\\xymatrix{\n(U_3, T_3, \\delta_3) \\ar[d] \\ar[r] & (U_2, T_2, \\delta_2) \\ar[d] \\\\\n(U_1, T_1, \\delta_1) \\ar[r] & (U, T, \\delta)\n}\n$$\nbe a fibre square in the category of divided power thickenings of\n$X$ relative to $(S, \\mathcal{I}, \\gamma)$. If $T_2 \\to T$ is\nflat and $U_2 = T_2 \\times_T U$, then $T_3 = T_1 \\times_T T_2$ (as schemes).","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"The big crystalline site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IA","source_file":"crystalline.tex","source_line":1585,"source_end_line":1597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1585-L1597","statement_sha256":"73115a6692cd7d66a2cad5130e63e296115c0a0fbeef85279c95ec9c0d3ddefb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10278,"rank":10278,"depth":5,"x":2622.78,"y":1209.137,"cluster":"local-crystalline-methods"},{"id":"stacks:07IB","tag":"07IB","title":"The big crystalline site · Definition 07IB","summary":"In Situation [Tag 07MF]. • A family of morphisms ((U_i, T_i, δ_i) → (U, T, δ)) of divided power thickenings of X/S is a Zariski, étale, smooth, syntomic, or fppf covering if and only if • U_i = U ×_T T_i for all i and • (T_i → T) is a Zariski, étale, smooth, syntomic, or fppf covering. • The big crystalline site of X over (S, I, γ), is the category CRIS(X/S) endowed with the Zariski topology. • The topos of sheaves on CRIS(X/S) is denoted (X/S)_CRIS or sometimes (X/S, I,…","statement_latex":"In Situation \\ref{situation-global}.\n\\begin{enumerate}\n\\item A family of morphisms $\\{(U_i, T_i, \\delta_i) \\to (U, T, \\delta)\\}$\nof divided power thickenings of $X/S$ is a\n{\\it Zariski, \\'etale, smooth, syntomic, or fppf covering}\nif and only if\n\\begin{enumerate}\n\\item $U_i = U \\times_T T_i$ for all $i$ and\n\\item $\\{T_i \\to T\\}$ is a Zariski, \\'etale, smooth, syntomic, or fppf covering.\n\\end{enumerate}\n\\item The {\\it big crystalline site} of $X$ over $(S, \\mathcal{I}, \\gamma)$,\nis the category $\\text{CRIS}(X/S)$ endowed with the Zariski topology.\n\\item The topos of sheaves on $\\text{CRIS}(X/S)$ is denoted\n$(X/S)_{\\text{CRIS}}$ or sometimes\n$(X/S, \\mathcal{I}, \\gamma)_{\\text{CRIS}}$\\footnote{This clashes with\nour convention to denote the topos associated to a site $\\mathcal{C}$\nby $\\Sh(\\mathcal{C})$.}.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"The big crystalline site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IB","source_file":"crystalline.tex","source_line":1611,"source_end_line":1631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1611-L1631","statement_sha256":"93cd5b778d420705843deb1d4b21ebfb90e3e171763d1aaca031b2efc8fb55f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10279,"rank":10279,"depth":0,"x":2318.868,"y":1170.083,"cluster":"local-crystalline-methods"},{"id":"stacks:07IG","tag":"07IG","title":"The crystalline site · Definition 07IG","summary":"In Situation [Tag 07MF]. • The (small) crystalline site of X over (S, I, γ), denoted Cris(X/S, I, γ) or simply Cris(X/S) is the full subcategory of CRIS(X/S) consisting of those (U, T, δ) in CRIS(X/S) such that U → X is an open immersion. It comes endowed with the Zariski topology. • The topos of sheaves on Cris(X/S) is denoted (X/S)_cris or sometimes (X/S, I, γ)_cris by Sh(C)..","statement_latex":"In Situation \\ref{situation-global}.\n\\begin{enumerate}\n\\item The (small) {\\it crystalline site} of $X$ over\n$(S, \\mathcal{I}, \\gamma)$, denoted $\\text{Cris}(X/S, \\mathcal{I}, \\gamma)$\nor simply $\\text{Cris}(X/S)$ is the full subcategory of $\\text{CRIS}(X/S)$\nconsisting of those $(U, T, \\delta)$ in $\\text{CRIS}(X/S)$ such that\n$U \\to X$ is an open immersion. It comes endowed with the Zariski topology.\n\\item The topos of sheaves on $\\text{Cris}(X/S)$ is denoted\n$(X/S)_{\\text{cris}}$ or sometimes\n$(X/S, \\mathcal{I}, \\gamma)_{\\text{cris}}$\\footnote{This clashes with\nour convention to denote the topos associated to a site $\\mathcal{C}$\nby $\\Sh(\\mathcal{C})$.}.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"The crystalline site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IG","source_file":"crystalline.tex","source_line":1739,"source_end_line":1754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1739-L1754","statement_sha256":"2d8de198528611170e71ac532a17d1d9a61b7f37c98e7cba144494ec90b6fcca","origin":"The Stacks Project","memory_eligible":false,"source_rank":10280,"rank":10280,"depth":0,"x":2574.77,"y":1026.211,"cluster":"local-crystalline-methods"},{"id":"stacks:07IJ","tag":"07IJ","title":"The crystalline site · Lemma 07IJ","summary":"Assumptions as in Definition [Tag 07I6]. The inclusion functor Cris(X/S) → CRIS(X/S) commutes with finite nonempty limits, is fully faithful, continuous, and cocontinuous. There are morphisms of topoi (X/S)_cris xrightarrowi (X/S)_CRIS xrightarrowπ (X/S)_cris whose composition is the identity and of which the first is induced by the inclusion functor. Moreover, π_* = i^-1.","statement_latex":"Assumptions as in Definition \\ref{definition-divided-power-thickening-X}.\nThe inclusion functor\n$$\n\\text{Cris}(X/S) \\to \\text{CRIS}(X/S)\n$$\ncommutes with finite nonempty limits, is fully faithful, continuous,\nand cocontinuous. There are morphisms of topoi\n$$\n(X/S)_{\\text{cris}} \\xrightarrow{i} (X/S)_{\\text{CRIS}}\n\\xrightarrow{\\pi} (X/S)_{\\text{cris}}\n$$\nwhose composition is the identity and of which the first is induced\nby the inclusion functor. Moreover, $\\pi_* = i^{-1}$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"The crystalline site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IJ","source_file":"crystalline.tex","source_line":1778,"source_end_line":1793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1778-L1793","statement_sha256":"4ff9a0a248e3b06ab3ab0a1605b0fad0e7a3128e3ee63555b79c9e13730bafb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10281,"rank":10281,"depth":7,"x":2501.657,"y":1277.893,"cluster":"local-crystalline-methods"},{"id":"stacks:07KL","tag":"07KL","title":"The crystalline site · Lemma 07KL","summary":"In Situation [Tag 07MF]. Let X' ⊂ X and S' ⊂ S be open subschemes such that X' maps into S'. Then there is a fully faithful functor Cris(X'/S') → Cris(X/S) which gives rise to a morphism of topoi fitting into the commutative diagram xymatrix (X'/S')_cris ar[r] ar[d]_u_X'/S' & (X/S)_cris ar[d]^u_X/S Sh(X'_Zar) ar[r] & Sh(X_Zar) Moreover, this diagram is an example of localization of morphisms of topoi as in Sites, Lemma [Tag 04H1].","statement_latex":"In Situation \\ref{situation-global}.\nLet $X' \\subset X$ and $S' \\subset S$ be open subschemes such that\n$X'$ maps into $S'$. Then there is a fully faithful functor\n$\\text{Cris}(X'/S') \\to \\text{Cris}(X/S)$\nwhich gives rise to a morphism of topoi fitting into the commutative\ndiagram\n$$\n\\xymatrix{\n(X'/S')_{\\text{cris}} \\ar[r] \\ar[d]_{u_{X'/S'}} &\n(X/S)_{\\text{cris}} \\ar[d]^{u_{X/S}} \\\\\n\\Sh(X'_{Zar}) \\ar[r] & \\Sh(X_{Zar})\n}\n$$\nMoreover, this diagram is an example of localization of morphisms of\ntopoi as in Sites, Lemma \\ref{sites-lemma-localize-morphism-topoi}.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"The crystalline site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KL","source_file":"crystalline.tex","source_line":1861,"source_end_line":1878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L1861-L1878","statement_sha256":"d056b23fa8944a8ac9ea374256d61ffa313a1f8e38eb77392b7ab84ebbcb49c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10282,"rank":10282,"depth":12,"x":2352.945,"y":1050.471,"cluster":"local-crystalline-methods"},{"id":"stacks:07IS","tag":"07IS","title":"Crystals in modules · Definition 07IS","summary":"In Situation [Tag 07MF]. Let C = CRIS(X/S) or C = Cris(X/S). Let F be a sheaf of O_X/S-modules on C. • We say F is locally quasi-coherent if for every object (U, T, δ) of C the restriction F_T is a quasi-coherent O_T-module. • We say F is quasi-coherent if it is quasi-coherent in the sense of Modules on Sites, Definition [Tag 03DL]. • We say F is a crystal in O_X/S-modules if all the comparison maps ([Tag 07IQ]) are isomorphisms.","statement_latex":"In Situation \\ref{situation-global}.\nLet $\\mathcal{C} = \\text{CRIS}(X/S)$ or $\\mathcal{C} = \\text{Cris}(X/S)$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_{X/S}$-modules on $\\mathcal{C}$.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ is {\\it locally quasi-coherent} if for every\nobject $(U, T, \\delta)$ of $\\mathcal{C}$ the restriction $\\mathcal{F}_T$\nis a quasi-coherent $\\mathcal{O}_T$-module.\n\\item We say $\\mathcal{F}$ is {\\it quasi-coherent} if it is quasi-coherent\nin the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local}.\n\\item We say $\\mathcal{F}$ is a {\\it crystal in $\\mathcal{O}_{X/S}$-modules}\nif all the comparison maps (\\ref{equation-comparison-modules}) are\nisomorphisms.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Crystals in modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IS","source_file":"crystalline.tex","source_line":2063,"source_end_line":2079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2063-L2079","statement_sha256":"6825599750d99e9484e7c06be5df795096788c4dfaf7489d83a83ad4eda3e852","origin":"The Stacks Project","memory_eligible":false,"source_rank":10283,"rank":10283,"depth":2,"x":2645.939,"y":1133.914,"cluster":"local-crystalline-methods"},{"id":"stacks:07IT","tag":"07IT","title":"Crystals in modules · Lemma 07IT","summary":"With notation X/S, I, γ, C, F as in Definition [Tag 07IS]. The following are equivalent • F is quasi-coherent, and • F is locally quasi-coherent and a crystal in O_X/S-modules.","statement_latex":"With notation $X/S, \\mathcal{I}, \\gamma, \\mathcal{C}, \\mathcal{F}$\nas in Definition \\ref{definition-modules}. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is quasi-coherent, and\n\\item $\\mathcal{F}$ is locally quasi-coherent and a crystal in\n$\\mathcal{O}_{X/S}$-modules.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Crystals in modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IT","source_file":"crystalline.tex","source_line":2084,"source_end_line":2093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2084-L2093","statement_sha256":"9a3fbf39185b3579d1961187667b01063411c5cf35fdcb5b3e35b3df970c7692","origin":"The Stacks Project","memory_eligible":false,"source_rank":10284,"rank":10284,"depth":3,"x":2362.354,"y":1238.796,"cluster":"local-crystalline-methods"},{"id":"stacks:07IU","tag":"07IU","title":"Crystals in modules · Definition 07IU","summary":"If F satisfies the equivalent conditions of Lemma [Tag 07IT], then we say that F is a crystal in quasi-coherent modules. We say that F is a crystal in finite locally free modules if, in addition, F is finite locally free.","statement_latex":"If $\\mathcal{F}$ satisfies the equivalent conditions of\nLemma \\ref{lemma-crystal-quasi-coherent-modules}, then\nwe say that $\\mathcal{F}$ is a\n{\\it crystal in quasi-coherent modules}.\nWe say that $\\mathcal{F}$ is a {\\it crystal in finite locally free modules}\nif, in addition, $\\mathcal{F}$ is finite locally free.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Crystals in modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IU","source_file":"crystalline.tex","source_line":2144,"source_end_line":2152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2144-L2152","statement_sha256":"776cf181775b7b97d9ab3b9c32ef1597bcc1856231d92d4bdfc3df4af33ecf91","origin":"The Stacks Project","memory_eligible":false,"source_rank":10285,"rank":10285,"depth":4,"x":2487.313,"y":1000.182,"cluster":"local-crystalline-methods"},{"id":"stacks:07IX","tag":"07IX","title":"Sheaf of differentials · Definition 07IX","summary":"In Situation [Tag 07MF] let F be a sheaf of O_X/S-modules on Cris(X/S). An S-derivation D : O_X/S → F is a map of sheaves such that for every object (U, T, δ) of Cris(X/S) the map D : Γ(T, O_T) → Γ(T, F) is a divided power Γ(V, O_V)-derivation where V ⊂ S is any open such that T → S factors through V.","statement_latex":"In Situation \\ref{situation-global} let\n$\\mathcal{F}$ be a sheaf of $\\mathcal{O}_{X/S}$-modules on\n$\\text{Cris}(X/S)$. An\n{\\it $S$-derivation $D : \\mathcal{O}_{X/S} \\to \\mathcal{F}$}\nis a map of sheaves such that for every object $(U, T, \\delta)$ of\n$\\text{Cris}(X/S)$ the map\n$$\nD : \\Gamma(T, \\mathcal{O}_T) \\longrightarrow \\Gamma(T, \\mathcal{F})\n$$\nis a divided power $\\Gamma(V, \\mathcal{O}_V)$-derivation where $V \\subset S$\nis any open such that $T \\to S$ factors through $V$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Sheaf of differentials","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IX","source_file":"crystalline.tex","source_line":2186,"source_end_line":2199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2186-L2199","statement_sha256":"75f82220891e09e06c8261eef39f9aee000992c2b05812de09eeaf89e172677e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10286,"rank":10286,"depth":0,"x":2587.208,"y":1247.412,"cluster":"local-crystalline-methods"},{"id":"stacks:07IY","tag":"07IY","title":"Sheaf of differentials · Lemma 07IY","summary":"Let (T, J, δ) be a divided power scheme. Let T → S be a morphism of schemes. The quotient Ω_T/S → Ω_T/S, δ described above is a quasi-coherent O_T-module. For W ⊂ T affine open mapping into V ⊂ S affine open we have Γ(W, Ω_T/S, δ) = Ω_Γ(W, O_W)/Γ(V, O_V), δ where the right hand side is as constructed in Section [Tag 07HQ].","statement_latex":"Let $(T, \\mathcal{J}, \\delta)$ be a divided power scheme.\nLet $T \\to S$ be a morphism of schemes.\nThe quotient $\\Omega_{T/S} \\to \\Omega_{T/S, \\delta}$\ndescribed above is a quasi-coherent $\\mathcal{O}_T$-module.\nFor $W \\subset T$ affine open mapping into $V \\subset S$ affine open\nwe have\n$$\n\\Gamma(W, \\Omega_{T/S, \\delta}) =\n\\Omega_{\\Gamma(W, \\mathcal{O}_W)/\\Gamma(V, \\mathcal{O}_V), \\delta}\n$$\nwhere the right hand side is\nas constructed in Section \\ref{section-differentials}.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Sheaf of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IY","source_file":"crystalline.tex","source_line":2274,"source_end_line":2288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2274-L2288","statement_sha256":"129e228902c74d1b2b0b375b72cdec327da780a5d5c78d92ff10fd2912c3b64d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10287,"rank":10287,"depth":0,"x":2314.318,"y":1121.601,"cluster":"local-crystalline-methods"},{"id":"stacks:07IZ","tag":"07IZ","title":"Sheaf of differentials · Lemma 07IZ","summary":"In Situation [Tag 07MF]. For (U, T, δ) in Cris(X/S) the restriction (Ω_X/S)_T to T is Ω_T/S, δ and the restriction d_X/S|_T is equal to d_T/S, δ.","statement_latex":"In Situation \\ref{situation-global}.\nFor $(U, T, \\delta)$ in $\\text{Cris}(X/S)$ the restriction\n$(\\Omega_{X/S})_T$ to $T$ is $\\Omega_{T/S, \\delta}$ and the restriction\n$\\text{d}_{X/S}|_T$ is equal to $\\text{d}_{T/S, \\delta}$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Sheaf of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07IZ","source_file":"crystalline.tex","source_line":2294,"source_end_line":2300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2294-L2300","statement_sha256":"7ca09520ba95fddfff2ca60d5a173a6bbe616bdd9e6516afb3eb086d3f7577a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10288,"rank":10288,"depth":0,"x":2617.174,"y":1059.434,"cluster":"local-crystalline-methods"},{"id":"stacks:07J0","tag":"07J0","title":"Sheaf of differentials · Lemma 07J0","summary":"In Situation [Tag 07MF]. For any affine object (U, T, δ) of Cris(X/S) mapping into an affine open V ⊂ S we have Γ((U, T, δ), Ω_X/S) = Ω_Γ(T, O_T)/Γ(V, O_V), δ where the right hand side is as constructed in Section [Tag 07HQ].","statement_latex":"In Situation \\ref{situation-global}.\nFor any affine object $(U, T, \\delta)$ of $\\text{Cris}(X/S)$\nmapping into an affine open $V \\subset S$ we have\n$$\n\\Gamma((U, T, \\delta), \\Omega_{X/S}) =\n\\Omega_{\\Gamma(T, \\mathcal{O}_T)/\\Gamma(V, \\mathcal{O}_V), \\delta}\n$$\nwhere the right hand side is\nas constructed in Section \\ref{section-differentials}.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Sheaf of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07J0","source_file":"crystalline.tex","source_line":2306,"source_end_line":2317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2306-L2317","statement_sha256":"2905e2b798fcce9f3b9dab2f142ebff58600154604aecd73ca56219693d6170e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10289,"rank":10289,"depth":1,"x":2443.584,"y":1277.451,"cluster":"local-crystalline-methods"},{"id":"stacks:07J1","tag":"07J1","title":"Sheaf of differentials · Lemma 07J1","summary":"In Situation [Tag 07MF]. Let (U, T, δ) be an object of Cris(X/S). Let (U(1), T(1), δ(1)) = (U, T, δ) × (U, T, δ) in Cris(X/S). Let K ⊂ O_T(1) be the quasi-coherent sheaf of ideals corresponding to the closed immersion Δ : T → T(1). Then K ⊂ J_T(1) is preserved by the divided structure on J_T(1) and we have (Ω_X/S)_T = K/K^[2]","statement_latex":"In Situation \\ref{situation-global}.\nLet $(U, T, \\delta)$ be an object of $\\text{Cris}(X/S)$.\nLet\n$$\n(U(1), T(1), \\delta(1)) = (U, T, \\delta) \\times (U, T, \\delta)\n$$\nin $\\text{Cris}(X/S)$. Let $\\mathcal{K} \\subset \\mathcal{O}_{T(1)}$\nbe the quasi-coherent sheaf of ideals corresponding to the closed\nimmersion $\\Delta : T \\to T(1)$. Then\n$\\mathcal{K} \\subset \\mathcal{J}_{T(1)}$ is preserved by the\ndivided structure on $\\mathcal{J}_{T(1)}$ and we have\n$$\n(\\Omega_{X/S})_T = \\mathcal{K}/\\mathcal{K}^{[2]}\n$$","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Sheaf of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07J1","source_file":"crystalline.tex","source_line":2324,"source_end_line":2340,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2324-L2340","statement_sha256":"743c28fb72ae6d26c55cbacb8560e26fe631ccd248a2b8309e7093d6e3e6a06e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10290,"rank":10290,"depth":4,"x":2396.193,"y":1017.8,"cluster":"local-crystalline-methods"},{"id":"stacks:07KM","tag":"07KM","title":"Sheaf of differentials · Lemma 07KM","summary":"In Situation [Tag 07MF]. The sheaf of differentials Ω_X/S has the following two properties: • Ω_X/S is locally quasi-coherent, and • for any morphism (U, T, δ) → (U', T', δ') of Cris(X/S) where f : T → T' is a closed immersion the map c_f : f^*(Ω_X/S)_T' → (Ω_X/S)_T is surjective.","statement_latex":"In Situation \\ref{situation-global}.\nThe sheaf of differentials $\\Omega_{X/S}$ has the following two\nproperties:\n\\begin{enumerate}\n\\item $\\Omega_{X/S}$ is locally quasi-coherent, and\n\\item for any morphism $(U, T, \\delta) \\to (U', T', \\delta')$\nof $\\text{Cris}(X/S)$ where $f : T \\to T'$ is a closed immersion\nthe map $c_f : f^*(\\Omega_{X/S})_{T'} \\to (\\Omega_{X/S})_T$ is surjective.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Sheaf of differentials","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KM","source_file":"crystalline.tex","source_line":2357,"source_end_line":2368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2357-L2368","statement_sha256":"11f114374fb737b349ca0b4e63eca66a86be480ad84df54fd8aeb0acea94f03e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10291,"rank":10291,"depth":1,"x":2640.307,"y":1182.618,"cluster":"local-crystalline-methods"},{"id":"stacks:07J6","tag":"07J6","title":"Connections · Lemma 07J6","summary":"In Situation [Tag 07MF]. Let F be a crystal in O_X/S-modules on Cris(X/S). Then F comes equipped with a canonical integrable connection.","statement_latex":"In Situation \\ref{situation-global}.\nLet $\\mathcal{F}$ be a crystal in $\\mathcal{O}_{X/S}$-modules\non $\\text{Cris}(X/S)$. Then $\\mathcal{F}$ comes equipped with a\ncanonical integrable connection.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Connections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07J6","source_file":"crystalline.tex","source_line":2597,"source_end_line":2603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2597-L2603","statement_sha256":"55a28cc674d74f0d1155009df3578bd553829d276c05ab86b2599257c582f712","origin":"The Stacks Project","memory_eligible":false,"source_rank":10292,"rank":10292,"depth":0,"x":2327.292,"y":1199.625,"cluster":"local-crystalline-methods"},{"id":"stacks:07KQ","tag":"07KQ","title":"Cosimplicial algebra · Lemma 07KQ","summary":"Let A_* be a cosimplicial ring. Let φ_*, ψ_* : K_* → M_* be homomorphisms of cosimplicial A_*-modules. • If φ_* and ψ_* are homotopic, then φ_* ⊗ 1, ψ_* ⊗ 1 : K_* ⊗_A_* L_* → M_* ⊗_A_* L_* are homotopic for any cosimplicial A_*-module L_*. • If φ_* and ψ_* are homotopic, then wedge^i(φ_*), wedge^i(ψ_*) : wedge^i(K_*) → wedge^i(M_*) are homotopic. • If φ_* and ψ_* are homotopic, and A_* → B_* is a homomorphism of cosimplicial rings, then φ_* ⊗ 1, ψ_* ⊗ 1 : K_* ⊗_A_* B_* →…","statement_latex":"Let $A_*$ be a cosimplicial ring. Let $\\varphi_*, \\psi_* : K_* \\to M_*$\nbe homomorphisms of cosimplicial $A_*$-modules.\n\\begin{enumerate}\n\\item\n\nIf $\\varphi_*$ and $\\psi_*$ are homotopic, then\n$$\n\\varphi_* \\otimes 1, \\psi_* \\otimes 1 :\nK_* \\otimes_{A_*} L_* \\longrightarrow M_* \\otimes_{A_*} L_*\n$$\nare homotopic for any cosimplicial $A_*$-module $L_*$.\n\\item\n\nIf $\\varphi_*$ and $\\psi_*$ are homotopic, then\n$$\n\\wedge^i(\\varphi_*), \\wedge^i(\\psi_*) :\n\\wedge^i(K_*) \\longrightarrow \\wedge^i(M_*)\n$$\nare homotopic.\n\\item\n\nIf $\\varphi_*$ and $\\psi_*$ are homotopic, and $A_* \\to B_*$\nis a homomorphism of cosimplicial rings, then\n$$\n\\varphi_* \\otimes 1, \\psi_* \\otimes 1 :\nK_* \\otimes_{A_*} B_* \\longrightarrow M_* \\otimes_{A_*} B_*\n$$\nare homotopic as homomorphisms of cosimplicial $B_*$-modules.\n\\item\n\nIf $I_* \\subset A_*$ is a cosimplicial ideal, then the induced\nmaps\n$$\n\\varphi^\\wedge_*, \\psi^\\wedge_* :\nK_*^\\wedge \\longrightarrow M_*^\\wedge\n$$\nbetween completions are homotopic.\n\\item Add more here as needed, for example symmetric powers.\n\\end{enumerate}","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Cosimplicial algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07KQ","source_file":"crystalline.tex","source_line":2736,"source_end_line":2777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2736-L2777","statement_sha256":"3889c595f70be3959cae8f87a6c2db142634a768e54635a58ba53dd48cd3e50f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10293,"rank":10293,"depth":0,"x":2544.752,"y":1009.189,"cluster":"local-crystalline-methods"},{"id":"stacks:07L2","tag":"07L2","title":"Crystals in quasi-coherent modules · Lemma 07L2","summary":"Let D and D(n) be as in ([Tag 07J8]) and ([Tag 07JF]). The coprojection P → P ⊗_A … ⊗_A P, f ↦ f ⊗ 1 ⊗ … ⊗ 1 induces an isomorphism D(n) = lim_e Dlangle xi_i(j) rangle/p^eDlangle xi_i(j) rangle of algebras over D with xi_i(j) = x_i ⊗ 1 ⊗ … ⊗ 1 - 1 ⊗ … ⊗ 1 ⊗ x_i ⊗ 1 ⊗ … ⊗ 1 for j = 1, …, n where the second x_i is placed in the j + 1st slot; recall that D(n) is constructed starting with the n + 1-fold tensor product of P over A.","statement_latex":"Let $D$ and $D(n)$ be as in (\\ref{equation-D}) and (\\ref{equation-Dn}).\nThe coprojection $P \\to P \\otimes_A \\ldots \\otimes_A P$,\n$f \\mapsto f \\otimes 1 \\otimes \\ldots \\otimes 1$\ninduces an isomorphism\n\\begin{equation}\n\nD(n) = \\lim_e D\\langle \\xi_i(j) \\rangle/p^eD\\langle \\xi_i(j) \\rangle\n\\end{equation}\nof algebras over $D$ with\n$$\n\\xi_i(j) = x_i \\otimes 1 \\otimes \\ldots \\otimes 1 -\n1 \\otimes \\ldots \\otimes 1 \\otimes x_i \\otimes 1 \\otimes \\ldots \\otimes 1\n$$\nfor $j = 1, \\ldots, n$ where the second $x_i$ is placed in the $j + 1$st\nslot; recall that $D(n)$ is constructed starting with the\n$n + 1$-fold tensor product of $P$ over $A$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Crystals in quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07L2","source_file":"crystalline.tex","source_line":2913,"source_end_line":2931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2913-L2931","statement_sha256":"4ff467039849e06b1bb99131457a44a3a0bcfd24f72beabd31c5afaeb5583b6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10294,"rank":10294,"depth":4,"x":2537.532,"y":1273.397,"cluster":"local-crystalline-methods"},{"id":"stacks:07L4","tag":"07L4","title":"Crystals in quasi-coherent modules · Lemma 07L4","summary":"Let D and D(n) be as in ([Tag 07J8]) and ([Tag 07JF]). Then (D, bar J, barγ) and (D(n), bar J(n), barγ(n)) are objects of Cris^wedge(C/A), see Remark [Tag 07KH], and D(n) = coprod_j = 0, …, n D in Cris^wedge(C/A).","statement_latex":"Let $D$ and $D(n)$ be as in (\\ref{equation-D}) and (\\ref{equation-Dn}).\nThen $(D, \\bar J, \\bar\\gamma)$ and $(D(n), \\bar J(n), \\bar\\gamma(n))$\nare objects of $\\text{Cris}^\\wedge(C/A)$, see\nRemark \\ref{remark-completed-affine-site}, and\n$$\nD(n) = \\coprod\\nolimits_{j = 0, \\ldots, n} D\n$$\nin $\\text{Cris}^\\wedge(C/A)$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Crystals in quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07L4","source_file":"crystalline.tex","source_line":2943,"source_end_line":2953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2943-L2953","statement_sha256":"6447c8e395850cd2824010fbbc26d393bc5fff214e38ebfbe6e3ba874aabf769","origin":"The Stacks Project","memory_eligible":false,"source_rank":10295,"rank":10295,"depth":0,"x":2330.082,"y":1074.184,"cluster":"local-crystalline-methods"},{"id":"stacks:07JG","tag":"07JG","title":"Crystals in quasi-coherent modules · Lemma 07JG","summary":"In the situation above there is a functor crystals in quasi-coherent O_X/S-modules on Cris(X/S) → pairs (M, nabla) satisfying ([Tag 07JB]), ([Tag 07JC]), ([Tag 07JD]), and ([Tag 07JE])","statement_latex":"In the situation above there is a functor\n$$\n\\begin{matrix}\n\\text{crystals in quasi-coherent} \\\\\n\\mathcal{O}_{X/S}\\text{-modules on }\\text{Cris}(X/S)\n\\end{matrix}\n\\longrightarrow\n\\begin{matrix}\n\\text{pairs }(M, \\nabla)\\text{ satisfying} \\\\\n\\text{(\\ref{item-complete}), (\\ref{item-connection}),\n(\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})}\n\\end{matrix}\n$$","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Crystals in quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JG","source_file":"crystalline.tex","source_line":2996,"source_end_line":3011,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L2996-L3011","statement_sha256":"eb4498010c86f7cd223c75df55fd2b0f4a80ee1c964fff31bb5f4aa0b3876c87","origin":"The Stacks Project","memory_eligible":false,"source_rank":10296,"rank":10296,"depth":7,"x":2643.716,"y":1103.411,"cluster":"local-crystalline-methods"},{"id":"stacks:07JH","tag":"07JH","title":"Crystals in quasi-coherent modules · Proposition 07JH","summary":"The functor crystals in quasi-coherent O_X/S-modules on Cris(X/S) → pairs (M, nabla) satisfying ([Tag 07JB]), ([Tag 07JC]), ([Tag 07JD]), and ([Tag 07JE]) of Lemma [Tag 07JG] is an equivalence of categories.","statement_latex":"The functor\n$$\n\\begin{matrix}\n\\text{crystals in quasi-coherent} \\\\\n\\mathcal{O}_{X/S}\\text{-modules on }\\text{Cris}(X/S)\n\\end{matrix}\n\\longrightarrow\n\\begin{matrix}\n\\text{pairs }(M, \\nabla)\\text{ satisfying} \\\\\n\\text{(\\ref{item-complete}), (\\ref{item-connection}),\n(\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})}\n\\end{matrix}\n$$\nof Lemma \\ref{lemma-crystals-on-affine}\nis an equivalence of categories.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Crystals in quasi-coherent modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JH","source_file":"crystalline.tex","source_line":3108,"source_end_line":3125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3108-L3125","statement_sha256":"c84fbf44501b4cb634142c7d23ce4db2b05dd87eb95646975c4777c325e73cd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10297,"rank":10297,"depth":8,"x":2388.568,"y":1260.053,"cluster":"local-crystalline-methods"},{"id":"stacks:07L5","tag":"07L5","title":"Crystals in quasi-coherent modules · Lemma 07L5","summary":"In Situation [Tag 07MD]. Let A → P' → C be ring maps with A → P' smooth and P' → C surjective with kernel J'. Let D' be the p-adic completion of D_P', γ(J'). Then there are a choice of the data A → P → C as above and homomorphisms of divided power A-algebras a : D → D', b : D' → D compatible with the maps D → C and D' → C such that a ∘ b = id_D'. These maps induce an equivalence of categories of pairs (M, nabla) satisfying ([Tag 07JB]), ([Tag 07JC]), ([Tag 07JD]), and…","statement_latex":"In Situation \\ref{situation-affine}.\nLet $A \\to P' \\to C$ be ring maps with $A \\to P'$ smooth and $P' \\to C$\nsurjective with kernel $J'$. Let $D'$ be the $p$-adic completion of\n$D_{P', \\gamma}(J')$. Then there are a choice of the data $A \\to P \\to C$\nas above and homomorphisms of divided power $A$-algebras\n$$\na : D \\longrightarrow D',\\quad b : D' \\longrightarrow D\n$$\ncompatible with the maps $D \\to C$ and $D' \\to C$ such that\n$a \\circ b = \\text{id}_{D'}$. These maps induce\nan equivalence of categories of pairs $(M, \\nabla)$ satisfying\n(\\ref{item-complete}), (\\ref{item-connection}),\n(\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})\nover $D$ and pairs $(M', \\nabla')$  satisfying\n(\\ref{item-complete}), (\\ref{item-connection}),\n(\\ref{item-integrable}), and\n(\\ref{item-topologically-quasi-nilpotent})\\footnote{This condition\nis tricky to formulate for $(M', \\nabla')$ over $D'$. See proof.} over $D'$.\nIn particular, the equivalence of categories of\nProposition \\ref{proposition-crystals-on-affine}\nalso holds for the corresponding functor towards pairs over $D'$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Crystals in quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07L5","source_file":"crystalline.tex","source_line":3190,"source_end_line":3213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3190-L3213","statement_sha256":"34b1d8b1c4d817ea33d51b3f80cd1176e1939205438fb21afa8917ee7fb52b88","origin":"The Stacks Project","memory_eligible":false,"source_rank":10298,"rank":10298,"depth":9,"x":2450.835,"y":999.389,"cluster":"local-crystalline-methods"},{"id":"stacks:07JJ","tag":"07JJ","title":"General remarks on cohomology · Lemma 07JJ","summary":"In Situation [Tag 07MF]. Let F be a locally quasi-coherent O_X/S-module on Cris(X/S). Then we have H^p((U, T, δ), F) = 0 for all p > 0 and all (U, T, δ) with T or U affine.","statement_latex":"In Situation \\ref{situation-global}.\nLet $\\mathcal{F}$ be a locally quasi-coherent $\\mathcal{O}_{X/S}$-module\non $\\text{Cris}(X/S)$. Then we have\n$$\nH^p((U, T, \\delta), \\mathcal{F}) = 0\n$$\nfor all $p > 0$ and all $(U, T, \\delta)$ with $T$ or $U$ affine.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"General remarks on cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JJ","source_file":"crystalline.tex","source_line":3305,"source_end_line":3314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3305-L3314","statement_sha256":"55d83097897f67b9776b011289b03f204cdc9b458d7e6618fcf15b6302a29323","origin":"The Stacks Project","memory_eligible":false,"source_rank":10299,"rank":10299,"depth":30,"x":2614.778,"y":1227.262,"cluster":"local-crystalline-methods"},{"id":"stacks:07JK","tag":"07JK","title":"General remarks on cohomology · Lemma 07JK","summary":"In Situation [Tag 07MF]. Assume moreover X and S are affine schemes. Consider the full subcategory C ⊂ Cris(X/S) consisting of divided power thickenings (X, T, δ) endowed with the chaotic topology (see Sites, Example [Tag 07GE]). For any locally quasi-coherent O_X/S-module F we have RΓ(C, F|_C) = RΓ(Cris(X/S), F)","statement_latex":"In Situation \\ref{situation-global}.\nAssume moreover $X$ and $S$ are affine schemes.\nConsider the full subcategory $\\mathcal{C} \\subset \\text{Cris}(X/S)$\nconsisting of divided power thickenings $(X, T, \\delta)$\nendowed with the chaotic topology (see\nSites, Example \\ref{sites-example-indiscrete}).\nFor any locally quasi-coherent $\\mathcal{O}_{X/S}$-module $\\mathcal{F}$\nwe have\n$$\nR\\Gamma(\\mathcal{C}, \\mathcal{F}|_\\mathcal{C}) =\nR\\Gamma(\\text{Cris}(X/S), \\mathcal{F})\n$$","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"General remarks on cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JK","source_file":"crystalline.tex","source_line":3339,"source_end_line":3353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3339-L3353","statement_sha256":"084d09407efb38177dc7a8265caf464cfac450f165c16d63c431b660bb0b5de0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10300,"rank":10300,"depth":31,"x":2310.196,"y":1152.148,"cluster":"local-crystalline-methods"},{"id":"stacks:07JL","tag":"07JL","title":"General remarks on cohomology · Lemma 07JL","summary":"In Situation [Tag 07MD]. Set C = (Cris(C/A))^opp and C^wedge = (Cris^wedge(C/A))^opp endowed with the chaotic topology, see Remark [Tag 07KH] for notation. There is a morphism of topoi g : Sh(C) → Sh(C^wedge) such that if F is a sheaf of abelian groups on C, then R^pg_*F(B → C, δ) = ( lim_e F(B_e → C, δ) & if p = 0 R^1lim_e F(B_e → C, δ) & if p = 1 0 & else . where B_e = B/p^eB for e gg 0.","statement_latex":"In Situation \\ref{situation-affine}.\nSet $\\mathcal{C} = (\\text{Cris}(C/A))^{opp}$ and\n$\\mathcal{C}^\\wedge = (\\text{Cris}^\\wedge(C/A))^{opp}$\nendowed with the chaotic topology, see\nRemark \\ref{remark-completed-affine-site} for notation.\nThere is a morphism of topoi\n$$\ng : \\Sh(\\mathcal{C}) \\longrightarrow \\Sh(\\mathcal{C}^\\wedge)\n$$\nsuch that if $\\mathcal{F}$ is a sheaf of abelian groups on\n$\\mathcal{C}$, then\n$$\nR^pg_*\\mathcal{F}(B \\to C, \\delta) =\n\\left\\{\n\\begin{matrix}\n\\lim_e \\mathcal{F}(B_e \\to C, \\delta) & \\text{if }p = 0 \\\\\nR^1\\lim_e \\mathcal{F}(B_e \\to C, \\delta) & \\text{if }p = 1 \\\\\n0 & \\text{else}\n\\end{matrix}\n\\right.\n$$\nwhere $B_e = B/p^eB$ for $e \\gg 0$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"General remarks on cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JL","source_file":"crystalline.tex","source_line":3417,"source_end_line":3441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3417-L3441","statement_sha256":"f5c2517774336882d2432853c9b83ab152b63a30e3c37fe9ef7c4ff8c0646fa1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10301,"rank":10301,"depth":0,"x":2595.609,"y":1034.539,"cluster":"local-crystalline-methods"},{"id":"stacks:07JM","tag":"07JM","title":"General remarks on cohomology · Lemma 07JM","summary":"Let C be a category endowed with the chaotic topology. Let X be an object of C such that every object of C has a morphism towards X. Assume that C has products of pairs. Then for every abelian sheaf F on C the total cohomology RΓ(C, F) is represented by the complex F(X) → F(X × X) → F(X × X × X) → … associated to the cosimplicial abelian group [n] ↦ F(X^n).","statement_latex":"Let $\\mathcal{C}$ be a category endowed with the chaotic topology.\nLet $X$ be an object of $\\mathcal{C}$ such that every object of\n$\\mathcal{C}$ has a morphism towards $X$. Assume that $\\mathcal{C}$\nhas products of pairs.\nThen for every abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}$\nthe total cohomology $R\\Gamma(\\mathcal{C}, \\mathcal{F})$ is represented\nby the complex\n$$\n\\mathcal{F}(X) \\to \\mathcal{F}(X \\times X) \\to\n\\mathcal{F}(X \\times X \\times X) \\to \\ldots\n$$\nassociated to the cosimplicial abelian group $[n] \\mapsto \\mathcal{F}(X^n)$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"General remarks on cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JM","source_file":"crystalline.tex","source_line":3470,"source_end_line":3484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3470-L3484","statement_sha256":"9d0e606acd65baa4974587705fc0c50a500633606329d08389e67fcf0fb67c94","origin":"The Stacks Project","memory_eligible":false,"source_rank":10302,"rank":10302,"depth":23,"x":2479.56,"y":1283.572,"cluster":"local-crystalline-methods"},{"id":"stacks:07L9","tag":"07L9","title":"Cosimplicial preparations · Lemma 07L9","summary":"With notation as in ([Tag 07L0]) the complex Ω_D(0) → Ω_D(1) → Ω_D(2) → … is homotopic to zero as a D(*)-cosimplicial module.","statement_latex":"With notation as in (\\ref{equation-omega-Dn}) the complex\n$$\n\\Omega_{D(0)} \\to \\Omega_{D(1)} \\to \\Omega_{D(2)} \\to \\ldots\n$$\nis homotopic to zero as a $D(*)$-cosimplicial module.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Cosimplicial preparations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07L9","source_file":"crystalline.tex","source_line":3590,"source_end_line":3597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3590-L3597","statement_sha256":"a1a6498c170bc8a2620c4d9cc5bdbd711612c31c3cff185adc7928dfbfdf37bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10303,"rank":10303,"depth":5,"x":2364.702,"y":1033.729,"cluster":"local-crystalline-methods"},{"id":"stacks:07LA","tag":"07LA","title":"Cosimplicial preparations · Lemma 07LA","summary":"With notation as in ([Tag 07JF]) and ([Tag 07L0]), given any cosimplicial module M_* over D(*) and i > 0 the cosimplicial module M_0 ⊗^wedge_D(0) Ω^i_D(0) → M_1 ⊗^wedge_D(1) Ω^i_D(1) → M_2 ⊗^wedge_D(2) Ω^i_D(2) → … is homotopic to zero, where Ω^i_D(n) is the p-adic completion of the ith exterior power of Ω_D(n).","statement_latex":"With notation as in (\\ref{equation-Dn}) and (\\ref{equation-omega-Dn}),\ngiven any cosimplicial module $M_*$ over $D(*)$ and\n$i > 0$ the cosimplicial module\n$$\nM_0 \\otimes^\\wedge_{D(0)} \\Omega^i_{D(0)} \\to\nM_1 \\otimes^\\wedge_{D(1)} \\Omega^i_{D(1)} \\to\nM_2 \\otimes^\\wedge_{D(2)} \\Omega^i_{D(2)} \\to \\ldots\n$$\nis homotopic to zero, where $\\Omega^i_{D(n)}$ is the $p$-adic completion\nof the $i$th exterior power of $\\Omega_{D(n)}$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Cosimplicial preparations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LA","source_file":"crystalline.tex","source_line":3636,"source_end_line":3648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3636-L3648","statement_sha256":"934f08b94d0786451445d47b0b271317a71b934471305703d922fa82d5842faa","origin":"The Stacks Project","memory_eligible":false,"source_rank":10304,"rank":10304,"depth":6,"x":2650.724,"y":1152.958,"cluster":"local-crystalline-methods"},{"id":"stacks:07LC","tag":"07LC","title":"Divided power Poincaré lemma · Lemma 07LC","summary":"Let A be a ring. Let P = Alangle x_i rangle be a divided power polynomial ring over A. For any A-module M the complex 0 → M → M ⊗_A P → M ⊗_A Ω^1_P/A, δ → M ⊗_A Ω^2_P/A, δ → … is exact. Let D be the p-adic completion of P. Let Ω^i_D be the p-adic completion of the ith exterior power of Ω_D/A, δ. For any p-adically complete A-module M the complex 0 → M → M ⊗^wedge_A D → M ⊗^wedge_A Ω^1_D → M ⊗^wedge_A Ω^2_D → … is exact.","statement_latex":"Let $A$ be a ring. Let $P = A\\langle x_i \\rangle$ be a divided\npower polynomial ring over $A$. For any $A$-module $M$ the complex\n$$\n0 \\to M \\to\nM \\otimes_A P \\to\nM \\otimes_A \\Omega^1_{P/A, \\delta} \\to\nM \\otimes_A \\Omega^2_{P/A, \\delta} \\to \\ldots\n$$\nis exact. Let $D$ be the $p$-adic completion of $P$.\nLet $\\Omega^i_D$ be the $p$-adic completion of the $i$th exterior\npower of $\\Omega_{D/A, \\delta}$. For any $p$-adically complete\n$A$-module $M$ the complex\n$$\n0 \\to M \\to\nM \\otimes^\\wedge_A D \\to\nM \\otimes^\\wedge_A \\Omega^1_D \\to\nM \\otimes^\\wedge_A \\Omega^2_D \\to \\ldots\n$$\nis exact.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power Poincaré lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LC","source_file":"crystalline.tex","source_line":3671,"source_end_line":3692,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3671-L3692","statement_sha256":"72ce95ea279b20fd2a59e0284f2165944ecece0ccefefdda0beaadb3242869d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10305,"rank":10305,"depth":1,"x":2343.495,"y":1227.443,"cluster":"local-crystalline-methods"},{"id":"stacks:07LD","tag":"07LD","title":"Divided power Poincaré lemma · Lemma 07LD","summary":"Let A be a ring. Let (B, I, δ) be a divided power ring where B is an A-algebra. Let P = Blangle x_i rangle be a divided power polynomial ring over B with divided power ideal J = IP + Blangle x_i rangle_+ as usual. Let M be a B-module endowed with an integrable connection nabla : M → M ⊗_B Ω^1_B/A, δ. Then the map of de Rham complexes M ⊗_B Ω^*_B/A, δ → M ⊗_P Ω^*_P/A, δ is a quasi-isomorphism. Let D, resp. D' be the p-adic completion of B, resp. P and let Ω^i_D, resp.…","statement_latex":"Let $A$ be a ring. Let $(B, I, \\delta)$ be a divided power ring\nwhere $B$ is an $A$-algebra.\nLet $P = B\\langle x_i \\rangle$ be a divided power polynomial\nring over $B$ with divided power ideal $J = IP + B\\langle x_i \\rangle_{+}$\nas usual. Let $M$ be a $B$-module endowed with an integrable connection\n$\\nabla : M \\to M \\otimes_B \\Omega^1_{B/A, \\delta}$. Then the map of\nde Rham complexes\n$$\nM \\otimes_B \\Omega^*_{B/A, \\delta}\n\\longrightarrow\nM \\otimes_P \\Omega^*_{P/A, \\delta}\n$$\nis a quasi-isomorphism. Let $D$, resp.\\ $D'$ be the $p$-adic completion of\n$B$, resp.\\ $P$ and let $\\Omega^i_D$, resp.\\ $\\Omega^i_{D'}$ be the $p$-adic\ncompletion of $\\Omega^i_{B/A, \\delta}$,\nresp.\\ $\\Omega^i_{P/A, \\delta}$. Let $M$ be a $p$-adically complete\n$D$-module endowed with an integral connection\n$\\nabla : M \\to M \\otimes^\\wedge_D \\Omega^1_D$.\nThen the map of de Rham complexes\n$$\nM \\otimes^\\wedge_D \\Omega^*_D\n\\longrightarrow\nM \\otimes^\\wedge_D \\Omega^*_{D'}\n$$\nis a quasi-isomorphism.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Divided power Poincaré lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LD","source_file":"crystalline.tex","source_line":3729,"source_end_line":3756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3729-L3756","statement_sha256":"832ac75a55f3c23473dff0dad9870dc870e22cedd358749ff1b19ca17a9bf85d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10306,"rank":10306,"depth":2,"x":2510.381,"y":997.862,"cluster":"local-crystalline-methods"},{"id":"stacks:07JN","tag":"07JN","title":"Cohomology in the affine case · Proposition 07JN","summary":"With notations as above assume that • F is locally quasi-coherent, and • for any morphism (U, T, δ) → (U', T', δ') of Cris(X/S) where f : T → T' is a closed immersion the map c_f : f^*F_T' → F_T is surjective. Then the complex M(0) → M(1) → M(2) → … computes RΓ(Cris(X/S), F).","statement_latex":"With notations as above assume that\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is locally quasi-coherent, and\n\\item for any morphism $(U, T, \\delta) \\to (U', T', \\delta')$\nof $\\text{Cris}(X/S)$ where $f : T \\to T'$ is a closed immersion\nthe map $c_f : f^*\\mathcal{F}_{T'} \\to \\mathcal{F}_T$ is surjective.\n\\end{enumerate}\nThen the complex\n$$\nM(0) \\to M(1) \\to M(2) \\to \\ldots\n$$\ncomputes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Cohomology in the affine case","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JN","source_file":"crystalline.tex","source_line":3828,"source_end_line":3842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3828-L3842","statement_sha256":"e2ed28a4b151609dc3682e9a821ba0b82c45d4e906776c5e58a4d2ff907e84de","origin":"The Stacks Project","memory_eligible":false,"source_rank":10307,"rank":10307,"depth":32,"x":2572.032,"y":1262.223,"cluster":"local-crystalline-methods"},{"id":"stacks:07LF","tag":"07LF","title":"Cohomology in the affine case · Lemma 07LF","summary":"Assumptions and notation as in Proposition [Tag 07JN]. Then H^j(Cris(X/S), F ⊗_O_X/S Ω^i_X/S) = 0 for all i > 0 and all j ≥ 0.","statement_latex":"Assumptions and notation as in\nProposition \\ref{proposition-compute-cohomology}.\nThen\n$$\nH^j(\\text{Cris}(X/S), \\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^i_{X/S})\n= 0\n$$\nfor all $i > 0$ and all $j \\geq 0$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Cohomology in the affine case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LF","source_file":"crystalline.tex","source_line":3880,"source_end_line":3890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3880-L3890","statement_sha256":"deafc13cab45a45757cba3234793822aea55042cf3add5efc1694ce35ed8a088","origin":"The Stacks Project","memory_eligible":false,"source_rank":10308,"rank":10308,"depth":33,"x":2313.612,"y":1102.042,"cluster":"local-crystalline-methods"},{"id":"stacks:07LG","tag":"07LG","title":"Cohomology in the affine case · Proposition 07LG","summary":"Assumptions as in Proposition [Tag 07JN] but now assume that F is a crystal in quasi-coherent modules. Let (M, nabla) be the corresponding module with connection over D, see Proposition [Tag 07JH]. Then the complex M ⊗^wedge_D Ω^*_D computes RΓ(Cris(X/S), F).","statement_latex":"Assumptions as in Proposition \\ref{proposition-compute-cohomology}\nbut now assume that $\\mathcal{F}$ is a crystal in quasi-coherent modules.\nLet $(M, \\nabla)$ be the corresponding module with connection over $D$, see\nProposition \\ref{proposition-crystals-on-affine}. Then the complex\n$$\nM \\otimes^\\wedge_D \\Omega^*_D\n$$\ncomputes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Cohomology in the affine case","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LG","source_file":"crystalline.tex","source_line":3926,"source_end_line":3936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3926-L3936","statement_sha256":"5cee3a8af02cc4a0cfb8eadc1bdd45e7a8878a8f57fc0a739af253eff4658a1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10309,"rank":10309,"depth":33,"x":2633.433,"y":1073.483,"cluster":"local-crystalline-methods"},{"id":"stacks:07LH","tag":"07LH","title":"Cohomology in the affine case · Lemma 07LH","summary":"Assumptions as in Proposition [Tag 07LG]. Let A → P' → C be ring maps with A → P' smooth and P' → C surjective with kernel J'. Let D' be the p-adic completion of D_P', γ(J'). Let (M', nabla') be the pair over D' corresponding to F, see Lemma [Tag 07L5]. Then the complex M' ⊗^wedge_D' Ω^*_D' computes RΓ(Cris(X/S), F).","statement_latex":"Assumptions as in Proposition \\ref{proposition-compute-cohomology-crystal}.\nLet $A \\to P' \\to C$ be ring maps with $A \\to P'$ smooth and $P' \\to C$\nsurjective with kernel $J'$. Let $D'$ be the $p$-adic completion of\n$D_{P', \\gamma}(J')$. Let $(M', \\nabla')$ be the pair over $D'$\ncorresponding to $\\mathcal{F}$, see\nLemma \\ref{lemma-crystals-on-affine-smooth}. Then the complex\n$$\nM' \\otimes^\\wedge_{D'} \\Omega^*_{D'}\n$$\ncomputes $R\\Gamma(\\text{Cris}(X/S), \\mathcal{F})$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Cohomology in the affine case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LH","source_file":"crystalline.tex","source_line":3990,"source_end_line":4002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L3990-L4002","statement_sha256":"64a989a166ff6ec136cf6687ded26010fec4f3a1d9b408b89532a631dc9c7f33","origin":"The Stacks Project","memory_eligible":false,"source_rank":10310,"rank":10310,"depth":34,"x":2420.267,"y":1276.302,"cluster":"local-crystalline-methods"},{"id":"stacks:07LM","tag":"07LM","title":"Applications · Proposition 07LM","summary":"In Situation [Tag 07MF]. Let F be a crystal in quasi-coherent modules on Cris(X/S). The truncation map of complexes (F → F ⊗_O_X/S Ω^1_X/S → F ⊗_O_X/S Ω^2_X/S → …) → F[0], while not a quasi-isomorphism, becomes a quasi-isomorphism after applying Ru_X/S, *. In fact, for any i > 0, we have Ru_X/S, *(F ⊗_O_X/S Ω^i_X/S) = 0.","statement_latex":"In Situation \\ref{situation-global}.\nLet $\\mathcal{F}$ be a crystal in quasi-coherent modules on\n$\\text{Cris}(X/S)$. The truncation map of complexes\n$$\n(\\mathcal{F} \\to\n\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^1_{X/S} \\to\n\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^2_{X/S} \\to \\ldots)\n\\longrightarrow \\mathcal{F}[0],\n$$\nwhile not a quasi-isomorphism, becomes a quasi-isomorphism after applying\n$Ru_{X/S, *}$. In fact, for any $i > 0$, we have \n$$\nRu_{X/S, *}(\\mathcal{F} \\otimes_{\\mathcal{O}_{X/S}} \\Omega^i_{X/S}) = 0.\n$$","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Applications","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07LM","source_file":"crystalline.tex","source_line":4260,"source_end_line":4276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L4260-L4276","statement_sha256":"07970217efe8c0cb1fd348665dcdc56c4a7cdbc910bb799762bc3e7994c7aef4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10311,"rank":10311,"depth":34,"x":2414.343,"y":1005.41,"cluster":"local-crystalline-methods"},{"id":"stacks:07Q7","tag":"07Q7","title":"Pulling back along purely inseparable maps · Lemma 07Q7","summary":"In the situation above there exists a map of complexes e_M^bullet : M ⊗_B (Ω')^bullet → M ⊗_B Ω^bullet such that c_M^bullet ∘ e_M^bullet and e_M^bullet ∘ c_M^bullet are homotopic to multiplication by a.","statement_latex":"In the situation above there exists a map of complexes\n$$\ne_M^\\bullet :\nM \\otimes_B (\\Omega')^\\bullet\n\\longrightarrow\nM \\otimes_B \\Omega^\\bullet\n$$\nsuch that $c_M^\\bullet \\circ e_M^\\bullet$\nand $e_M^\\bullet \\circ c_M^\\bullet$ are homotopic to\nmultiplication by $a$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Pulling back along purely inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Q7","source_file":"crystalline.tex","source_line":5052,"source_end_line":5064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L5052-L5064","statement_sha256":"141c32e389ebcf3efa71ec2e3049d5b8a1da2146e04f23a59377f963f5f42521","origin":"The Stacks Project","memory_eligible":false,"source_rank":10312,"rank":10312,"depth":0,"x":2636.868,"y":1202.077,"cluster":"local-crystalline-methods"},{"id":"stacks:07N1","tag":"07N1","title":"Pulling back along purely inseparable maps · Lemma 07N1","summary":"In Situation [Tag 07MD]. Assume D and Ω_D are as in ([Tag 07J8]) and ([Tag 07J9]). Let λ ∈ D. Let D' be the p-adic completion of D[z]langle xi rangle/(xi - (z^p - λ)) and let Ω_D' be the p-adic completion of the module of divided power differentials of D' over A. For any pair (M, nabla) over D satisfying ([Tag 07JB]), ([Tag 07JC]), ([Tag 07JD]), and ([Tag 07JE]) the canonical map of complexes ([Tag 07PY]) c_M^bullet : M ⊗_D^wedge Ω^bullet_D → M ⊗_D^wedge Ω^bullet_D' has…","statement_latex":"In Situation \\ref{situation-affine}. Assume $D$ and $\\Omega_D$ are as in\n(\\ref{equation-D}) and (\\ref{equation-omega-D}).\nLet $\\lambda \\in D$. Let $D'$ be the $p$-adic completion of\n$$\nD[z]\\langle \\xi \\rangle/(\\xi - (z^p - \\lambda))\n$$\nand let $\\Omega_{D'}$ be the $p$-adic completion of the module of\ndivided power differentials of $D'$ over $A$. For any pair $(M, \\nabla)$\nover $D$ satisfying (\\ref{item-complete}), (\\ref{item-connection}),\n(\\ref{item-integrable}), and (\\ref{item-topologically-quasi-nilpotent})\nthe canonical map of complexes (\\ref{equation-base-change-map-complexes})\n$$\nc_M^\\bullet : M \\otimes_D^\\wedge \\Omega^\\bullet_D\n\\longrightarrow\nM \\otimes_D^\\wedge \\Omega^\\bullet_{D'}\n$$\nhas the following property: There exists a map $e_M^\\bullet$\nin the opposite direction such that both $c_M^\\bullet \\circ e_M^\\bullet$\nand $e_M^\\bullet \\circ c_M^\\bullet$ are homotopic to multiplication by $p$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Pulling back along purely inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07N1","source_file":"crystalline.tex","source_line":5177,"source_end_line":5198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L5177-L5198","statement_sha256":"a06ccb2e2e7bda9f28c657081be22fd3b2002c78a99294d8845af8be3a470098","origin":"The Stacks Project","memory_eligible":false,"source_rank":10313,"rank":10313,"depth":3,"x":2314.176,"y":1183.297,"cluster":"local-crystalline-methods"},{"id":"stacks:07Q9","tag":"07Q9","title":"Pulling back along purely inseparable maps · Lemma 07Q9","summary":"Let p be a prime number. Let (S, I, γ) be a divided power scheme over Z_(p) with p ∈ I. We set S_0 = V(I) ⊂ S. Let f : X' → X be an iterated α_p-cover of schemes over S_0 with constant degree q. Let F be any crystal in quasi-coherent sheaves on X and set F' = f_cris^*F. In the distinguished triangle Ru_X/S, *F → f_*Ru_X'/S, *F' → E → Ru_X/S, *F[1] the object E has cohomology sheaves annihilated by q.","statement_latex":"Let $p$ be a prime number. Let $(S, \\mathcal{I}, \\gamma)$ be a divided power\nscheme over $\\mathbf{Z}_{(p)}$ with $p \\in \\mathcal{I}$. We set\n$S_0 = V(\\mathcal{I}) \\subset S$. Let $f : X' \\to X$ be an iterated\n$\\alpha_p$-cover of schemes over $S_0$ with constant degree $q$. Let\n$\\mathcal{F}$ be any crystal in quasi-coherent sheaves on $X$ and set\n$\\mathcal{F}' = f_{\\text{cris}}^*\\mathcal{F}$.\nIn the distinguished triangle\n$$\nRu_{X/S, *}\\mathcal{F}\n\\longrightarrow\nf_*Ru_{X'/S, *}\\mathcal{F}'\n\\longrightarrow\nE\n\\longrightarrow\nRu_{X/S, *}\\mathcal{F}[1]\n$$\nthe object $E$ has cohomology sheaves annihilated by $q$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Pulling back along purely inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Q9","source_file":"crystalline.tex","source_line":5354,"source_end_line":5373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L5354-L5373","statement_sha256":"ec7397600146ae24610c9c4f4ae4ab095bcd2dca53d888b416860ca240e2f83a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10314,"rank":10314,"depth":34,"x":2567.58,"y":1013.805,"cluster":"local-crystalline-methods"},{"id":"stacks:07QA","tag":"07QA","title":"Pulling back along purely inseparable maps · Lemma 07QA","summary":"With notations and assumptions as in Lemma [Tag 07Q9] the map f^* : H^i(Cris(X/S), F) → H^i(Cris(X'/S), F') has kernel and cokernel annihilated by q^i + 1.","statement_latex":"With notations and assumptions as in\nLemma \\ref{lemma-pullback-along-p-power-cover}\nthe map\n$$\nf^* :\nH^i(\\text{Cris}(X/S), \\mathcal{F})\n\\longrightarrow\nH^i(\\text{Cris}(X'/S), \\mathcal{F}')\n$$\nhas kernel and cokernel annihilated by $q^{i + 1}$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Pulling back along purely inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QA","source_file":"crystalline.tex","source_line":5414,"source_end_line":5426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L5414-L5426","statement_sha256":"04f4d117344501d46abb02f9a998e3d04ef9b3bbb8709069884e54db7702c539","origin":"The Stacks Project","memory_eligible":false,"source_rank":10315,"rank":10315,"depth":35,"x":2516.954,"y":1282.948,"cluster":"local-crystalline-methods"},{"id":"stacks:07QB","tag":"07QB","title":"Pulling back along purely inseparable maps · Lemma 07QB","summary":"In the situation above, assume that X → S_0 is smooth of relative dimension d. Then F_X/S_0 is an iterated α_p-cover of degree p^d. Hence Lemmas [Tag 07Q9] and [Tag 07QA] apply to this situation. In particular, for any crystal in quasi-coherent modules G on Cris(X^(1)/S) the map F_X/S_0^* : H^i(Cris(X^(1)/S), G) → H^i(Cris(X/S), F_X/S_0, cris^*G) has kernel and cokernel annihilated by p^d(i + 1).","statement_latex":"In the situation above, assume that $X \\to S_0$ is smooth of relative\ndimension $d$. Then $F_{X/S_0}$ is an iterated $\\alpha_p$-cover\nof degree $p^d$. Hence Lemmas \\ref{lemma-pullback-along-p-power-cover} and\n\\ref{lemma-pullback-along-p-power-cover-cohomology} apply to this\nsituation. In particular, for any crystal in quasi-coherent modules\n$\\mathcal{G}$ on $\\text{Cris}(X^{(1)}/S)$ the map\n$$\nF_{X/S_0}^* : H^i(\\text{Cris}(X^{(1)}/S), \\mathcal{G})\n\\longrightarrow\nH^i(\\text{Cris}(X/S), F_{X/S_0, \\text{cris}}^*\\mathcal{G})\n$$\nhas kernel and cokernel annihilated by $p^{d(i + 1)}$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Pulling back along purely inseparable maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07QB","source_file":"crystalline.tex","source_line":5444,"source_end_line":5458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L5444-L5458","statement_sha256":"4affac7655981accd924567d6f75cad3d4125fe6749549bbbd6b42ddf16fbadd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10316,"rank":10316,"depth":45,"x":2337.599,"y":1055.443,"cluster":"local-crystalline-methods"},{"id":"stacks:07N3","tag":"07N3","title":"Frobenius action on crystalline cohomology · Definition 07N3","summary":"In Situation [Tag 07N2] an F-crystal on X/S (relative to σ) is a pair (E, F_E) given by a crystal in finite locally free O_X/S-modules E together with a map F_E : (F_X)_cris^*E → E An F-crystal is called nondegenerate if there exists an integer i ≥ 0 a map V : E → (F_X)_cris^*E such that V ∘ F_E = p^i id.","statement_latex":"In Situation \\ref{situation-F-crystal} an {\\it $F$-crystal on $X/S$\n(relative to $\\sigma$)} is a pair $(\\mathcal{E}, F_\\mathcal{E})$\ngiven by a crystal in finite locally free $\\mathcal{O}_{X/S}$-modules\n$\\mathcal{E}$ together with a map\n$$\nF_\\mathcal{E} : (F_X)_{\\text{cris}}^*\\mathcal{E} \\longrightarrow \\mathcal{E}\n$$\nAn $F$-crystal is called {\\it nondegenerate} if there exists an integer\n$i \\geq 0$ a map $V : \\mathcal{E} \\to (F_X)_{\\text{cris}}^*\\mathcal{E}$\nsuch that $V \\circ F_{\\mathcal{E}} = p^i \\text{id}$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Frobenius action on crystalline cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07N3","source_file":"crystalline.tex","source_line":5533,"source_end_line":5545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L5533-L5545","statement_sha256":"812d261db04a02d8f3374ae434acdde47377849752407361d4632dee28e9da4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10317,"rank":10317,"depth":0,"x":2653.242,"y":1121.523,"cluster":"local-crystalline-methods"},{"id":"stacks:07N5","tag":"07N5","title":"Frobenius action on crystalline cohomology · Theorem 07N5","summary":"In Situation [Tag 07N2] let (E, F_E) be a nondegenerate F-crystal. Assume A is a p-adically complete Noetherian ring and that X → S_0 is proper smooth. Then the canonical map F_E ∘ (F_X)_cris^* : RΓ(Cris(X/S), E) ⊗^L_A, σ A → RΓ(Cris(X/S), E) becomes an isomorphism after inverting p.","statement_latex":"In Situation \\ref{situation-F-crystal} let $(\\mathcal{E}, F_\\mathcal{E})$\nbe a nondegenerate $F$-crystal. Assume $A$ is a $p$-adically complete\nNoetherian ring and that $X \\to S_0$ is proper smooth. Then\nthe canonical map\n$$\nF_\\mathcal{E} \\circ (F_X)_{\\text{cris}}^* :\nR\\Gamma(\\text{Cris}(X/S), \\mathcal{E}) \\otimes^\\mathbf{L}_{A, \\sigma} A\n\\longrightarrow\nR\\Gamma(\\text{Cris}(X/S), \\mathcal{E})\n$$\nbecomes an isomorphism after inverting $p$.","area":"Local & Crystalline Methods","chapter":"Crystalline Cohomology","chapter_id":"crystalline","section":"Frobenius action on crystalline cohomology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07N5","source_file":"crystalline.tex","source_line":5573,"source_end_line":5586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/crystalline.tex#L5573-L5586","statement_sha256":"ed80e2c65ffba4ce49762e31e0a3ecc9cfcc6c859a7a922b99c3edc52d50faa6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10318,"rank":10318,"depth":46,"x":2366.957,"y":1252.081,"cluster":"local-crystalline-methods"},{"id":"stacks:0968","tag":"0968","title":"Some topology · Lemma 0968","summary":"Let X be a spectral space. Let X_0 ⊂ X be the set of closed points. The following are equivalent • Every open covering of X can be refined by a finite disjoint union decomposition X = coprod U_i with U_i open and closed in X. • The composition X_0 → X → π_0(X) is bijective. Moreover, if X_0 is closed in X and every point of X specializes to a unique point of X_0, then these conditions are satisfied.","statement_latex":"Let $X$ be a spectral space. Let $X_0 \\subset X$ be the set of closed points.\nThe following are equivalent\n\\begin{enumerate}\n\\item Every open covering of $X$ can be refined by a finite\ndisjoint union decomposition $X = \\coprod U_i$ with $U_i$\nopen and closed in $X$.\n\\item The composition $X_0 \\to X \\to \\pi_0(X)$ is bijective.\n\\end{enumerate}\nMoreover, if $X_0$ is closed in $X$ and every point of $X$ specializes\nto a unique point of $X_0$, then these conditions are satisfied.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Some topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0968","source_file":"proetale.tex","source_line":78,"source_end_line":90,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L78-L90","statement_sha256":"d3de8d0bd11cb1b6334d0369924778da265e83902a162410e313931ee1aee5a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10319,"rank":10319,"depth":6,"x":1640.512,"y":1030.062,"cluster":"tale-geometry"},{"id":"stacks:096A","tag":"096A","title":"Some topology · Definition 096A","summary":"A spectral space X is w-local if the set of closed points X_0 is closed and every point of X specializes to a unique closed point. A continuous map f : X → Y of w-local spaces is w-local if it is spectral and maps any closed point of X to a closed point of Y.","statement_latex":"A spectral space $X$ is {\\it w-local} if the set of closed points $X_0$\nis closed and every point of $X$ specializes to a unique closed point.\nA continuous map $f : X \\to Y$ of w-local spaces is {\\it w-local}\nif it is spectral and maps any closed point of $X$ to a closed point of $Y$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Some topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096A","source_file":"proetale.tex","source_line":187,"source_end_line":193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L187-L193","statement_sha256":"88df6186875d7d3fcfc39fb2571da93046166d36cc0418e66410189f04f24b22","origin":"The Stacks Project","memory_eligible":false,"source_rank":10320,"rank":10320,"depth":0,"x":1623.822,"y":1255.493,"cluster":"tale-geometry"},{"id":"stacks:096B","tag":"096B","title":"Some topology · Lemma 096B","summary":"Let X be a w-local spectral space. If Y ⊂ X is closed, then Y is w-local.","statement_latex":"Let $X$ be a w-local spectral space. If $Y \\subset X$ is closed,\nthen $Y$ is w-local.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Some topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096B","source_file":"proetale.tex","source_line":202,"source_end_line":206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L202-L206","statement_sha256":"edddddbd9f8bc4bc24514cf19002281573bc460a1566edc77a234bbf90324a35","origin":"The Stacks Project","memory_eligible":false,"source_rank":10321,"rank":10321,"depth":0,"x":1454.712,"y":1079.656,"cluster":"tale-geometry"},{"id":"stacks:096C","tag":"096C","title":"Some topology · Lemma 096C","summary":"Let X be a spectral space. Let xymatrix Y ar[r] ar[d] & T ar[d] X ar[r] & π_0(X) be a cartesian diagram in the category of topological spaces with T profinite. Then Y is spectral and T = π_0(Y). If moreover X is w-local, then Y is w-local, Y → X is w-local, and the set of closed points of Y is the inverse image of the set of closed points of X.","statement_latex":"Let $X$ be a spectral space. Let\n$$\n\\xymatrix{\nY \\ar[r] \\ar[d] & T \\ar[d] \\\\\nX \\ar[r] & \\pi_0(X)\n}\n$$\nbe a cartesian diagram in the category of topological spaces\nwith $T$ profinite. Then $Y$ is spectral and $T = \\pi_0(Y)$.\nIf moreover $X$ is w-local, then $Y$ is w-local, $Y \\to X$ is w-local,\nand the set of closed points of $Y$ is the inverse image of the\nset of closed points of $X$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Some topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096C","source_file":"proetale.tex","source_line":216,"source_end_line":230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L216-L230","statement_sha256":"e655f1afab6dc6cddccbb01a2337c96f7f6343c2d2869e458d452bf8c58b89ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":10322,"rank":10322,"depth":7,"x":1721.023,"y":1113.384,"cluster":"tale-geometry"},{"id":"stacks:096E","tag":"096E","title":"Local isomorphisms · Definition 096E","summary":"Let φ : A → B be a ring map. • We say A → B is a local isomorphism if for every prime q ⊂ B there exists a g ∈ B, g not ∈ q such that A → B_g induces an open immersion Spec(B_g) → Spec(A). • We say A → B identifies local rings if for every prime q ⊂ B the canonical map A_φ^-1( q) → B_ q is an isomorphism.","statement_latex":"Let $\\varphi : A \\to B$ be a ring map.\n\\begin{enumerate}\n\\item We say $A \\to B$ is a {\\it local isomorphism} if for every prime\n$\\mathfrak q \\subset B$ there exists a $g \\in B$, $g \\not \\in \\mathfrak q$\nsuch that $A \\to B_g$ induces an open immersion $\\Spec(B_g) \\to \\Spec(A)$.\n\\item We say $A \\to B$ {\\it identifies local rings} if for every prime\n$\\mathfrak q \\subset B$ the canonical map\n$A_{\\varphi^{-1}(\\mathfrak q)} \\to B_\\mathfrak q$ is an isomorphism.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Local isomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096E","source_file":"proetale.tex","source_line":277,"source_end_line":288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L277-L288","statement_sha256":"8910bdff564dd3fdb8f57f85d4a753fc723d56f1f502ff766f8f5cabc0c1d7d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10323,"rank":10323,"depth":0,"x":1497.351,"y":1239.723,"cluster":"tale-geometry"},{"id":"stacks:096F","tag":"096F","title":"Local isomorphisms · Lemma 096F","summary":"Let A → B and A → A' be ring maps. Let B' = B ⊗_A A' be the base change of B. • If A → B is a local isomorphism, then A' → B' is a local isomorphism. • If A → B identifies local rings, then A' → B' identifies local rings.","statement_latex":"Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$\nbe the base change of $B$.\n\\begin{enumerate}\n\\item If $A \\to B$ is a local isomorphism, then $A' \\to B'$ is a\nlocal isomorphism.\n\\item If $A \\to B$ identifies local rings, then $A' \\to B'$\nidentifies local rings.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Local isomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096F","source_file":"proetale.tex","source_line":293,"source_end_line":303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L293-L303","statement_sha256":"7532af6779d9e3511c44a513423e914e6532f186b91f61336d5127e93248e18b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10324,"rank":10324,"depth":0,"x":1560.734,"y":1019.475,"cluster":"tale-geometry"},{"id":"stacks:096G","tag":"096G","title":"Local isomorphisms · Lemma 096G","summary":"Let A → B and B → C be ring maps. • If A → B and B → C are local isomorphisms, then A → C is a local isomorphism. • If A → B and B → C identify local rings, then A → C identifies local rings.","statement_latex":"Let $A \\to B$ and $B \\to C$ be ring maps.\n\\begin{enumerate}\n\\item If $A \\to B$ and $B \\to C$ are local isomorphisms, then $A \\to C$\nis a local isomorphism.\n\\item If $A \\to B$ and $B \\to C$ identify local rings, then $A \\to C$\nidentifies local rings.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Local isomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096G","source_file":"proetale.tex","source_line":309,"source_end_line":318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L309-L318","statement_sha256":"e90f65a2883861db1edd1a1afed17e4f9a651dba84f615598e00cf86d5cdbf55","origin":"The Stacks Project","memory_eligible":false,"source_rank":10325,"rank":10325,"depth":0,"x":1691.216,"y":1218.002,"cluster":"tale-geometry"},{"id":"stacks:096H","tag":"096H","title":"Local isomorphisms · Lemma 096H","summary":"Let A be a ring. Let B → C be an A-algebra homomorphism. • If A → B and A → C are local isomorphisms, then B → C is a local isomorphism. • If A → B and A → C identify local rings, then B → C identifies local rings.","statement_latex":"Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra homomorphism.\n\\begin{enumerate}\n\\item If $A \\to B$ and $A \\to C$ are local isomorphisms, then $B \\to C$\nis a local isomorphism.\n\\item If $A \\to B$ and $A \\to C$ identify local rings, then $B \\to C$\nidentifies local rings.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Local isomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096H","source_file":"proetale.tex","source_line":324,"source_end_line":333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L324-L333","statement_sha256":"e363cc7a8080f4f8caac2fb46ab2320f3ff9800e6ecaad3659ce3316f64bcb73","origin":"The Stacks Project","memory_eligible":false,"source_rank":10326,"rank":10326,"depth":0,"x":1435.152,"y":1145.593,"cluster":"tale-geometry"},{"id":"stacks:096I","tag":"096I","title":"Local isomorphisms · Lemma 096I","summary":"Let A → B be a local isomorphism. Then • A → B is étale, • A → B identifies local rings, • A → B is quasi-finite.","statement_latex":"Let $A \\to B$ be a local isomorphism. Then\n\\begin{enumerate}\n\\item $A \\to B$ is \\'etale,\n\\item $A \\to B$ identifies local rings,\n\\item $A \\to B$ is quasi-finite.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Local isomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096I","source_file":"proetale.tex","source_line":339,"source_end_line":347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L339-L347","statement_sha256":"81f9c2bf79ca99cfafea89d269cd8543f50cf1e98eb4578e1bf3218eab71214c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10327,"rank":10327,"depth":0,"x":1682.39,"y":1053.618,"cluster":"tale-geometry"},{"id":"stacks:096J","tag":"096J","title":"Local isomorphisms · Lemma 096J","summary":"Let A → B be a local isomorphism. Then there exist n ≥ 0, g_1, …, g_n ∈ B, f_1, …, f_n ∈ A such that (g_1, …, g_n) = B and A_f_i ≅ B_g_i.","statement_latex":"Let $A \\to B$ be a local isomorphism. Then there exist $n \\geq 0$,\n$g_1, \\ldots, g_n \\in B$, $f_1, \\ldots, f_n \\in A$ such that\n$(g_1, \\ldots, g_n) = B$ and $A_{f_i} \\cong B_{g_i}$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Local isomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096J","source_file":"proetale.tex","source_line":353,"source_end_line":358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L353-L358","statement_sha256":"403857b167631ab6cb33d7cea0cca458a9fb0b4c4498048ffeeec4696a8507af","origin":"The Stacks Project","memory_eligible":false,"source_rank":10328,"rank":10328,"depth":0,"x":1573.96,"y":1261.889,"cluster":"tale-geometry"},{"id":"stacks:096K","tag":"096K","title":"Local isomorphisms · Lemma 096K","summary":"Let p : (Y, O_Y) → (X, O_X) and q : (Z, O_Z) → (X, O_X) be morphisms of locally ringed spaces. If O_Y = p^-1O_X, then Mor_LRS/(X, O_X)((Z, O_Z), (Y, O_Y)) → Mor_Top/X(Z, Y), (f, f^sharp) ↦ f is bijective. Here LRS/(X, O_X) is the category of locally ringed spaces over X and Top/X is the category of topological spaces over X.","statement_latex":"Let $p : (Y, \\mathcal{O}_Y) \\to (X, \\mathcal{O}_X)$ and\n$q : (Z, \\mathcal{O}_Z) \\to (X, \\mathcal{O}_X)$\nbe morphisms of locally ringed spaces.\nIf $\\mathcal{O}_Y = p^{-1}\\mathcal{O}_X$, then\n$$\n\\Mor_{\\text{LRS}/(X, \\mathcal{O}_X)}((Z, \\mathcal{O}_Z), (Y, \\mathcal{O}_Y))\n\\longrightarrow\n\\Mor_{\\textit{Top}/X}(Z, Y),\\quad\n(f, f^\\sharp) \\longmapsto f\n$$\nis bijective. Here $\\text{LRS}/(X, \\mathcal{O}_X)$ is the category of\nlocally ringed spaces over $X$ and $\\textit{Top}/X$ is the category\nof topological spaces over $X$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Local isomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096K","source_file":"proetale.tex","source_line":364,"source_end_line":379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L364-L379","statement_sha256":"90715b04034cfbd58102ecf58aed3c2b86c7c1370c8764cce538be7cbc04d485","origin":"The Stacks Project","memory_eligible":false,"source_rank":10329,"rank":10329,"depth":0,"x":1486.361,"y":1046.621,"cluster":"tale-geometry"},{"id":"stacks:096L","tag":"096L","title":"Local isomorphisms · Lemma 096L","summary":"Let A be a ring. Set X = Spec(A). The functor B ↦ Spec(B) from the category of A-algebras B such that A → B identifies local rings to the category of topological spaces over X is fully faithful.","statement_latex":"Let $A$ be a ring. Set $X = \\Spec(A)$. The functor\n$$\nB \\longmapsto \\Spec(B)\n$$\nfrom the category of $A$-algebras $B$ such that $A \\to B$ identifies\nlocal rings to the category of\ntopological spaces over $X$ is fully faithful.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Local isomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096L","source_file":"proetale.tex","source_line":385,"source_end_line":394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L385-L394","statement_sha256":"18518f6e3ff14beabb1062d6225220a2cc46973a99a9886531631269e41ea4bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10330,"rank":10330,"depth":1,"x":1724.251,"y":1155.735,"cluster":"tale-geometry"},{"id":"stacks:096N","tag":"096N","title":"Ind-Zariski algebra · Definition 096N","summary":"A ring map A → B is said to be ind-Zariski if B can be written as a filtered colimit B = colim B_i with each A → B_i a local isomorphism.","statement_latex":"A ring map $A \\to B$ is said to be {\\it ind-Zariski} if $B$ can be written\nas a filtered colimit $B = \\colim B_i$ with each $A \\to B_i$ a local\nisomorphism.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-Zariski algebra","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096N","source_file":"proetale.tex","source_line":413,"source_end_line":418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L413-L418","statement_sha256":"184a3af0f3cd572f012490d417094e9ca4f16f2e1948126e435c16b4af564da8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10331,"rank":10331,"depth":0,"x":1460.885,"y":1210.303,"cluster":"tale-geometry"},{"id":"stacks:096P","tag":"096P","title":"Ind-Zariski algebra · Lemma 096P","summary":"Let A → B and A → A' be ring maps. Let B' = B ⊗_A A' be the base change of B. If A → B is ind-Zariski, then A' → B' is ind-Zariski.","statement_latex":"Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$\nbe the base change of $B$.\nIf $A \\to B$ is ind-Zariski, then $A' \\to B'$ is ind-Zariski.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-Zariski algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096P","source_file":"proetale.tex","source_line":426,"source_end_line":431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L426-L431","statement_sha256":"adfc090ae71a452af72176b0af5156d9e9a07a99f53243aa40505cc7927d7906","origin":"The Stacks Project","memory_eligible":false,"source_rank":10332,"rank":10332,"depth":0,"x":1611.322,"y":1020.48,"cluster":"tale-geometry"},{"id":"stacks:096Q","tag":"096Q","title":"Ind-Zariski algebra · Lemma 096Q","summary":"Let A → B and B → C be ring maps. If A → B and B → C are ind-Zariski, then A → C is ind-Zariski.","statement_latex":"Let $A \\to B$ and $B \\to C$ be ring maps. If $A \\to B$ and $B \\to C$\nare ind-Zariski, then $A \\to C$ is ind-Zariski.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-Zariski algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096Q","source_file":"proetale.tex","source_line":437,"source_end_line":441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L437-L441","statement_sha256":"f55090c785306daa30eb9f7f1e1c22fcf8924f10ba59abff5dc480bcd6ed6dab","origin":"The Stacks Project","memory_eligible":false,"source_rank":10333,"rank":10333,"depth":0,"x":1653.075,"y":1245.987,"cluster":"tale-geometry"},{"id":"stacks:096R","tag":"096R","title":"Ind-Zariski algebra · Lemma 096R","summary":"Let A be a ring. Let B → C be an A-algebra homomorphism. If A → B and A → C are ind-Zariski, then B → C is ind-Zariski.","statement_latex":"Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra homomorphism.\nIf $A \\to B$ and $A \\to C$ are ind-Zariski, then $B \\to C$\nis ind-Zariski.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-Zariski algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096R","source_file":"proetale.tex","source_line":447,"source_end_line":452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L447-L452","statement_sha256":"e5bbabff95b3946d6a3a56d606964bf604bb040e5e74f1a4c67ccca5d54c25c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10334,"rank":10334,"depth":0,"x":1440.778,"y":1103.283,"cluster":"tale-geometry"},{"id":"stacks:096S","tag":"096S","title":"Ind-Zariski algebra · Lemma 096S","summary":"A filtered colimit of ind-Zariski A-algebras is ind-Zariski over A.","statement_latex":"A filtered colimit of ind-Zariski $A$-algebras is ind-Zariski over $A$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-Zariski algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096S","source_file":"proetale.tex","source_line":458,"source_end_line":461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L458-L461","statement_sha256":"3506f6ef9af2f864d60dc6b936bb53582d26099303dcb0a868ebb207b2048a9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10335,"rank":10335,"depth":0,"x":1712.288,"y":1088.036,"cluster":"tale-geometry"},{"id":"stacks:096T","tag":"096T","title":"Ind-Zariski algebra · Lemma 096T","summary":"Let A → B be ind-Zariski. Then A → B identifies local rings,","statement_latex":"Let $A \\to B$ be ind-Zariski. Then $A \\to B$ identifies local rings,","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-Zariski algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096T","source_file":"proetale.tex","source_line":467,"source_end_line":470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L467-L470","statement_sha256":"a1eb2881d486dff9cfe0363f27e7ea7663106c22116d79baa57125f4e217eba9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10336,"rank":10336,"depth":0,"x":1524.198,"y":1253.467,"cluster":"tale-geometry"},{"id":"stacks:096V","tag":"096V","title":"Constructing w-local affine schemes · Lemma 096V","summary":"Let A be a ring. Set X = Spec(A). Let Z ⊂ X be a locally closed subscheme which is of the form D(f) ∩ V(I) for some f ∈ A and ideal I ⊂ A. Then • there exists a multiplicative subset S ⊂ A such that Spec(S^-1A) maps by a homeomorphism to the set of points of X specializing to Z, • the A-algebra A_Z^sim = S^-1A depends only on the underlying locally closed subset Z ⊂ X, • Z is a closed subscheme of Spec(A_Z^sim), If A → A' is a ring map and Z' ⊂ X' = Spec(A') is a locally…","statement_latex":"Let $A$ be a ring. Set $X = \\Spec(A)$. Let $Z \\subset X$ be a locally closed\nsubscheme which is of the form $D(f) \\cap V(I)$ for some $f \\in A$ and\nideal $I \\subset A$. Then\n\\begin{enumerate}\n\\item there exists a multiplicative subset $S \\subset A$ such that\n$\\Spec(S^{-1}A)$ maps by a homeomorphism to the set of points of $X$\nspecializing to $Z$,\n\\item the $A$-algebra $A_Z^\\sim = S^{-1}A$ depends only on\nthe underlying locally closed subset $Z \\subset X$,\n\\item $Z$ is a closed subscheme of $\\Spec(A_Z^\\sim)$,\n\\end{enumerate}\nIf $A \\to A'$ is a ring map and $Z' \\subset X' = \\Spec(A')$ is a\nlocally closed subscheme of the same form which maps into $Z$,\nthen there is a unique $A$-algebra map\n$A_Z^\\sim \\to (A')_{Z'}^\\sim$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-local affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096V","source_file":"proetale.tex","source_line":492,"source_end_line":509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L492-L509","statement_sha256":"25e837750897c4f2866bdfb8de5a512664a3f9307d733e7317b6d00476dfa87b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10337,"rank":10337,"depth":0,"x":1529.862,"y":1024.58,"cluster":"tale-geometry"},{"id":"stacks:096Y","tag":"096Y","title":"Constructing w-local affine schemes · Lemma 096Y","summary":"Let X = Spec(A) as above. Given any finite stratification X = coprod T_i by constructible subsets, there exists a finite subset E ⊂ A such that the stratification ([Tag 096X]) refines X = coprod T_i.","statement_latex":"Let $X = \\Spec(A)$ as above. Given any finite stratification\n$X = \\coprod T_i$ by constructible subsets, there exists a finite\nsubset $E \\subset A$ such that the stratification (\\ref{equation-stratify})\nrefines $X = \\coprod T_i$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-local affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/096Y","source_file":"proetale.tex","source_line":553,"source_end_line":559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L553-L559","statement_sha256":"e65189de76649965c84f972abf62de75365b35c3162b0777a1eba6ef51be1940","origin":"The Stacks Project","memory_eligible":false,"source_rank":10338,"rank":10338,"depth":0,"x":1709.887,"y":1196.703,"cluster":"tale-geometry"},{"id":"stacks:0975","tag":"0975","title":"Constructing w-local affine schemes · Lemma 0975","summary":"Let X = Spec(A) be an affine scheme. With A → A_w, X_w = Spec(A_w), and Z ⊂ X_w as above. • A → A_w is ind-Zariski and faithfully flat, • X_w → X induces a bijection Z → X, • Z is the set of closed points of X_w, • Z is a reduced scheme, and • every point of X_w specializes to a unique point of Z. In particular, X_w is w-local (Definition [Tag 096A]).","statement_latex":"Let $X = \\Spec(A)$ be an affine scheme. With $A \\to A_w$, $X_w = \\Spec(A_w)$,\nand $Z \\subset X_w$ as above.\n\\begin{enumerate}\n\\item $A \\to A_w$ is ind-Zariski and faithfully flat,\n\\item $X_w \\to X$ induces a bijection $Z \\to X$,\n\\item $Z$ is the set of closed points of $X_w$,\n\\item $Z$ is a reduced scheme, and\n\\item every point of $X_w$ specializes to a unique point of $Z$.\n\\end{enumerate}\nIn particular, $X_w$ is w-local (Definition \\ref{definition-w-local}).","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-local affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0975","source_file":"proetale.tex","source_line":626,"source_end_line":638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L626-L638","statement_sha256":"7469888e38a326c83e9ecf3f2cce323eace19d48394f045fd2d90428a03d7f2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10339,"rank":10339,"depth":3,"x":1438.517,"y":1171.914,"cluster":"tale-geometry"},{"id":"stacks:0977","tag":"0977","title":"Universal property of the construction · Lemma 0977","summary":"Let A be a ring. Let A → A_w be the ring map constructed in Lemma [Tag 0975]. For any ring map A → B such that Spec(B) is w-local, there is a unique factorization A → A_w → B such that Spec(B) → Spec(A_w) is w-local.","statement_latex":"Let $A$ be a ring. Let $A \\to A_w$ be the ring map constructed in\nLemma \\ref{lemma-make-w-local}. For any ring map $A \\to B$ such that\n$\\Spec(B)$ is w-local, there is a unique factorization $A \\to A_w \\to B$\nsuch that $\\Spec(B) \\to \\Spec(A_w)$ is w-local.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-local affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0977","source_file":"proetale.tex","source_line":696,"source_end_line":702,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L696-L702","statement_sha256":"f16a9dd084526efce1541cee083beda90315c1b411e28aa6129af09b399618bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":10340,"rank":10340,"depth":4,"x":1658.723,"y":1036.108,"cluster":"tale-geometry"},{"id":"stacks:0978","tag":"0978","title":"Constructing w-local affine schemes · Lemma 0978","summary":"Let A be a ring such that Spec(A) is profinite. Let A → B be a ring map. Then Spec(B) is profinite in each of the following cases: • if q, q' ⊂ B lie over the same prime of A, then neither q ⊂ q', nor q' ⊂ q, • A → B induces algebraic extensions of residue fields, • A → B is a local isomorphism, • A → B identifies local rings, • A → B is weakly étale, • A → B is quasi-finite, • A → B is unramified, • A → B is étale, • B is a filtered colimit of A-algebras as in (1) --…","statement_latex":"Let $A$ be a ring such that $\\Spec(A)$ is profinite. Let $A \\to B$ be a\nring map. Then $\\Spec(B)$ is profinite in each of the following cases:\n\\begin{enumerate}\n\\item if $\\mathfrak q,\\mathfrak q' \\subset B$ lie over the same\nprime of $A$, then neither $\\mathfrak q \\subset \\mathfrak q'$, nor\n$\\mathfrak q' \\subset \\mathfrak q$,\n\\item $A \\to B$ induces algebraic extensions of residue fields,\n\\item $A \\to B$ is a local isomorphism,\n\\item $A \\to B$ identifies local rings,\n\\item $A \\to B$ is weakly \\'etale,\n\\item $A \\to B$ is quasi-finite,\n\\item $A \\to B$ is unramified,\n\\item $A \\to B$ is \\'etale,\n\\item $B$ is a filtered colimit of $A$-algebras as in (1) -- (8),\n\\item etc.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-local affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0978","source_file":"proetale.tex","source_line":738,"source_end_line":756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L738-L756","statement_sha256":"e6694987ce63c02492890e849225e11359fa44253eb5e79f3990dc17b3e7ad4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10341,"rank":10341,"depth":45,"x":1605.517,"y":1261.37,"cluster":"tale-geometry"},{"id":"stacks:0979","tag":"0979","title":"Constructing w-local affine schemes · Lemma 0979","summary":"Let A be a ring. Let V(I) ⊂ Spec(A) be a closed subset which is a profinite topological space. Then there exists an ind-Zariski ring map A → B such that Spec(B) is w-local, the set of closed points is V(IB), and A/I ≅ B/IB.","statement_latex":"Let $A$ be a ring. Let $V(I) \\subset \\Spec(A)$ be a closed subset\nwhich is a profinite topological space. Then there exists an\nind-Zariski ring map $A \\to B$ such that $\\Spec(B)$ is w-local,\nthe set of closed points is $V(IB)$, and $A/I \\cong B/IB$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-local affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0979","source_file":"proetale.tex","source_line":794,"source_end_line":800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L794-L800","statement_sha256":"5c191b059bfc71fc0e5d3c98e3ff12f6affeb311411997a26c0fcd7f37afab07","origin":"The Stacks Project","memory_eligible":false,"source_rank":10342,"rank":10342,"depth":4,"x":1463.493,"y":1064.926,"cluster":"tale-geometry"},{"id":"stacks:097A","tag":"097A","title":"Constructing w-local affine schemes · Lemma 097A","summary":"Let A be a ring such that X = Spec(A) is w-local. Let I ⊂ A be the radical ideal cutting out the set X_0 of closed points in X. Let A → B be a ring map inducing algebraic extensions on residue fields at primes. Then • every point of Z = V(IB) is a closed point of Spec(B), • there exists an ind-Zariski ring map B → C such that • B/IB → C/IC is an isomorphism, • the space Y = Spec(C) is w-local, • the induced map p : Y → X is w-local, and • p^-1(X_0) is the set of closed…","statement_latex":"Let $A$ be a ring such that $X = \\Spec(A)$ is w-local. Let $I \\subset A$\nbe the radical ideal cutting out the set $X_0$ of closed points in $X$.\nLet $A \\to B$ be a ring map inducing algebraic extensions on residue\nfields at primes. Then\n\\begin{enumerate}\n\\item every point of $Z = V(IB)$ is a closed point of $\\Spec(B)$,\n\\item there exists an ind-Zariski ring map $B \\to C$ such that\n\\begin{enumerate}\n\\item $B/IB \\to C/IC$ is an isomorphism,\n\\item the space $Y = \\Spec(C)$ is w-local,\n\\item the induced map $p : Y \\to X$ is w-local, and\n\\item $p^{-1}(X_0)$ is the set of closed points of $Y$.\n\\end{enumerate}\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-local affine schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097A","source_file":"proetale.tex","source_line":816,"source_end_line":832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L816-L832","statement_sha256":"04e64b401295ed6a09669c183bc3e30766a77d648254036e5d62679264e09b53","origin":"The Stacks Project","memory_eligible":false,"source_rank":10343,"rank":10343,"depth":46,"x":1726.393,"y":1129.242,"cluster":"tale-geometry"},{"id":"stacks:097C","tag":"097C","title":"Identifying local rings versus ind-Zariski · Lemma 097C","summary":"Let A be a ring. Let X = Spec(A). Let T ⊂ π_0(X) be a closed subset. There exists a surjective ind-Zariski ring map A → B such that Spec(B) → Spec(A) induces a homeomorphism of Spec(B) with the inverse image of T in X.","statement_latex":"Let $A$ be a ring. Let $X = \\Spec(A)$. Let $T \\subset \\pi_0(X)$ be a\nclosed subset. There exists a surjective ind-Zariski ring map $A \\to B$\nsuch that $\\Spec(B) \\to \\Spec(A)$ induces a homeomorphism of $\\Spec(B)$\nwith the inverse image of $T$ in $X$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Identifying local rings versus ind-Zariski","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097C","source_file":"proetale.tex","source_line":863,"source_end_line":869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L863-L869","statement_sha256":"0b8ebf91fbc7cf806248994abfefa3ea32520cd1ecc9cdae35464be2d97de8be","origin":"The Stacks Project","memory_eligible":false,"source_rank":10344,"rank":10344,"depth":3,"x":1480.629,"y":1231.069,"cluster":"tale-geometry"},{"id":"stacks:097D","tag":"097D","title":"Identifying local rings versus ind-Zariski · Lemma 097D","summary":"Let A be a ring and let X = Spec(A). Let T be a profinite space and let T → π_0(X) be a continuous map. There exists an ind-Zariski ring map A → B such that with Y = Spec(B) the diagram xymatrix Y ar[r] ar[d] & π_0(Y) ar[d] X ar[r] & π_0(X) is cartesian in the category of topological spaces and such that π_0(Y) = T as spaces over π_0(X).","statement_latex":"Let $A$ be a ring and let $X = \\Spec(A)$. Let $T$ be a profinite space and\nlet $T \\to \\pi_0(X)$ be a continuous map. There exists an\nind-Zariski ring map $A \\to B$ such that with $Y = \\Spec(B)$ the diagram\n$$\n\\xymatrix{\nY \\ar[r] \\ar[d] & \\pi_0(Y) \\ar[d] \\\\\nX \\ar[r] & \\pi_0(X)\n}\n$$\nis cartesian in the category of topological spaces and such that\n$\\pi_0(Y) = T$ as spaces over $\\pi_0(X)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Identifying local rings versus ind-Zariski","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097D","source_file":"proetale.tex","source_line":880,"source_end_line":893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L880-L893","statement_sha256":"29e81b3ebada0a9f636024aaddc5633b8878bad888d966be9e73743fdcf52507","origin":"The Stacks Project","memory_eligible":false,"source_rank":10345,"rank":10345,"depth":8,"x":1580.04,"y":1016.369,"cluster":"tale-geometry"},{"id":"stacks:097E","tag":"097E","title":"Identifying local rings versus ind-Zariski · Lemma 097E","summary":"Let A → B be ring map such that • A → B identifies local rings, • the topological spaces Spec(B), Spec(A) are w-local, • Spec(B) → Spec(A) is w-local, and • π_0(Spec(B)) → π_0(Spec(A)) is bijective. Then A → B is an isomorphism","statement_latex":"Let $A \\to B$ be ring map such that\n\\begin{enumerate}\n\\item $A \\to B$ identifies local rings,\n\\item the topological spaces $\\Spec(B)$, $\\Spec(A)$ are w-local,\n\\item $\\Spec(B) \\to \\Spec(A)$ is w-local, and\n\\item $\\pi_0(\\Spec(B)) \\to \\pi_0(\\Spec(A))$ is bijective.\n\\end{enumerate}\nThen $A \\to B$ is an isomorphism","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Identifying local rings versus ind-Zariski","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097E","source_file":"proetale.tex","source_line":937,"source_end_line":947,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L937-L947","statement_sha256":"fcc443c3b7519d14eb43a309558a2aec5a7caae189620f5837a50ee8b17a9f9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10346,"rank":10346,"depth":5,"x":1679.466,"y":1231.255,"cluster":"tale-geometry"},{"id":"stacks:097F","tag":"097F","title":"Identifying local rings versus ind-Zariski · Lemma 097F","summary":"Let A → B be ring map such that • A → B identifies local rings, • the topological spaces Spec(B), Spec(A) are w-local, and • Spec(B) → Spec(A) is w-local. Then A → B is ind-Zariski.","statement_latex":"Let $A \\to B$ be ring map such that\n\\begin{enumerate}\n\\item $A \\to B$ identifies local rings,\n\\item the topological spaces $\\Spec(B)$, $\\Spec(A)$ are w-local, and\n\\item $\\Spec(B) \\to \\Spec(A)$ is w-local.\n\\end{enumerate}\nThen $A \\to B$ is ind-Zariski.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Identifying local rings versus ind-Zariski","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097F","source_file":"proetale.tex","source_line":967,"source_end_line":976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L967-L976","statement_sha256":"f21e14201d91a882e276ea91d772b5e55e9c5e9ae679c55aecac5ddb72332661","origin":"The Stacks Project","memory_eligible":false,"source_rank":10347,"rank":10347,"depth":9,"x":1433.16,"y":1129.141,"cluster":"tale-geometry"},{"id":"stacks:097G","tag":"097G","title":"Identifying local rings versus ind-Zariski · Proposition 097G","summary":"Let A → B be a ring map which identifies local rings. Then there exists a faithfully flat, ind-Zariski ring map B → B' such that A → B' is ind-Zariski.","statement_latex":"Let $A \\to B$ be a ring map which identifies local rings.\nThen there exists a faithfully flat, ind-Zariski ring map\n$B \\to B'$ such that $A \\to B'$ is ind-Zariski.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Identifying local rings versus ind-Zariski","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097G","source_file":"proetale.tex","source_line":1010,"source_end_line":1015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1010-L1015","statement_sha256":"9829bde4cb4c93c971e0c50b403b2fa9dd81b2e956bc8eb08c64f583df5d6ac4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10348,"rank":10348,"depth":10,"x":1697.097,"y":1064.63,"cluster":"tale-geometry"},{"id":"stacks:09AZ","tag":"09AZ","title":"Identifying local rings versus ind-Zariski · Lemma 09AZ","summary":"Let A be a ring. The following are equivalent • every faithfully flat ring map A → B identifying local rings has a retraction, • every faithfully flat ind-Zariski ring map A → B has a retraction, and • A satisfies • Spec(A) is w-local, and • π_0(Spec(A)) is extremally disconnected.","statement_latex":"Let $A$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item every faithfully flat ring map $A \\to B$ identifying local rings\nhas a retraction,\n\\item every faithfully flat ind-Zariski ring map $A \\to B$ has a retraction, and\n\\item $A$ satisfies\n\\begin{enumerate}\n\\item $\\Spec(A)$ is w-local, and\n\\item $\\pi_0(\\Spec(A))$ is extremally disconnected.\n\\end{enumerate}\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Identifying local rings versus ind-Zariski","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AZ","source_file":"proetale.tex","source_line":1037,"source_end_line":1050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1037-L1050","statement_sha256":"9b99fbc32759b610e83536488ecf6c7f29be95204447b32146da62f89c82d8cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10349,"rank":10349,"depth":47,"x":1554.245,"y":1262.112,"cluster":"tale-geometry"},{"id":"stacks:09B0","tag":"09B0","title":"Identifying local rings versus ind-Zariski · Lemma 09B0","summary":"Let A be a ring. There exists a faithfully flat, ind-Zariski ring map A → B such that B satisfies the equivalent conditions of Lemma [Tag 09AZ].","statement_latex":"Let $A$ be a ring. There exists a faithfully flat, ind-Zariski ring\nmap $A \\to B$ such that $B$ satisfies the equivalent conditions\nof Lemma \\ref{lemma-w-local-extremally-disconnected}.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Identifying local rings versus ind-Zariski","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09B0","source_file":"proetale.tex","source_line":1103,"source_end_line":1108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1103-L1108","statement_sha256":"378c70eab6bf931e72065f9df205239e13fe54cd3b7042cd667462e18b384740","origin":"The Stacks Project","memory_eligible":false,"source_rank":10350,"rank":10350,"depth":48,"x":1500.733,"y":1035.265,"cluster":"tale-geometry"},{"id":"stacks:097I","tag":"097I","title":"Ind-étale algebra · Definition 097I","summary":"A ring map A → B is said to be ind-étale if B can be written as a filtered colimit of étale A-algebras.","statement_latex":"A ring map $A \\to B$ is said to be {\\it ind-\\'etale} if $B$ can be written\nas a filtered colimit of \\'etale $A$-algebras.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-étale algebra","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097I","source_file":"proetale.tex","source_line":1159,"source_end_line":1163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1159-L1163","statement_sha256":"e1446040c4922f0ccb095e2a5d07e229c07f167e411d332a5c227edaabb16c9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10351,"rank":10351,"depth":0,"x":1722.782,"y":1172.276,"cluster":"tale-geometry"},{"id":"stacks:097J","tag":"097J","title":"Ind-étale algebra · Lemma 097J","summary":"Let A → B and A → A' be ring maps. Let B' = B ⊗_A A' be the base change of B. If A → B is ind-étale, then A' → B' is ind-étale.","statement_latex":"Let $A \\to B$ and $A \\to A'$ be ring maps. Let $B' = B \\otimes_A A'$\nbe the base change of $B$.\nIf $A \\to B$ is ind-\\'etale, then $A' \\to B'$ is ind-\\'etale.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-étale algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097J","source_file":"proetale.tex","source_line":1169,"source_end_line":1174,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1169-L1174","statement_sha256":"f0c08cba86b92fca5f92719572fbd777c2a7cc339956732c0e89433f7fcd9d25","origin":"The Stacks Project","memory_eligible":false,"source_rank":10352,"rank":10352,"depth":39,"x":1448.661,"y":1197.26,"cluster":"tale-geometry"},{"id":"stacks:097K","tag":"097K","title":"Ind-étale algebra · Lemma 097K","summary":"Let A → B and B → C be ring maps. If A → B and B → C are ind-étale, then A → C is ind-étale.","statement_latex":"Let $A \\to B$ and $B \\to C$ be ring maps. If $A \\to B$ and $B \\to C$\nare ind-\\'etale, then $A \\to C$ is ind-\\'etale.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-étale algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097K","source_file":"proetale.tex","source_line":1180,"source_end_line":1184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1180-L1184","statement_sha256":"eeb87868eafb3929c7a7357541cd374559919ed64f27dfa346d8a93f09e2882e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10353,"rank":10353,"depth":39,"x":1630.838,"y":1023.166,"cluster":"tale-geometry"},{"id":"stacks:097L","tag":"097L","title":"Ind-étale algebra · Lemma 097L","summary":"A filtered colimit of ind-étale A-algebras is ind-étale over A.","statement_latex":"A filtered colimit of ind-\\'etale $A$-algebras is ind-\\'etale over $A$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-étale algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097L","source_file":"proetale.tex","source_line":1190,"source_end_line":1193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1190-L1193","statement_sha256":"f5c830bc2a9d8cc5204af836c690b8b3071cdb61b3899be1f162dcd235517fde","origin":"The Stacks Project","memory_eligible":false,"source_rank":10354,"rank":10354,"depth":7,"x":1636.51,"y":1255.084,"cluster":"tale-geometry"},{"id":"stacks:097M","tag":"097M","title":"Ind-étale algebra · Lemma 097M","summary":"Let A be a ring. Let B → C be an A-algebra map of ind-étale A-algebras. Then C is an ind-étale B-algebra.","statement_latex":"Let $A$ be a ring. Let $B \\to C$ be an $A$-algebra map of ind-\\'etale\n$A$-algebras. Then $C$ is an ind-\\'etale $B$-algebra.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-étale algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097M","source_file":"proetale.tex","source_line":1199,"source_end_line":1203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1199-L1203","statement_sha256":"09d2d5373e16134e8c89c2b5371bcf94d162a9d75091ec00b84d0f1948e63f01","origin":"The Stacks Project","memory_eligible":false,"source_rank":10355,"rank":10355,"depth":43,"x":1445.684,"y":1087.165,"cluster":"tale-geometry"},{"id":"stacks:097N","tag":"097N","title":"Ind-étale algebra · Lemma 097N","summary":"Let A → B be ind-étale. Then A → B is weakly étale (More on Algebra, Definition [Tag 092B]).","statement_latex":"Let $A \\to B$ be ind-\\'etale. Then $A \\to B$ is weakly \\'etale\n(More on Algebra, Definition \\ref{more-algebra-definition-weakly-etale}).","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-étale algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097N","source_file":"proetale.tex","source_line":1209,"source_end_line":1213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1209-L1213","statement_sha256":"af6d4e374befe068493093a2ae43b579907af4fd56b3e51c7749d2032421b298","origin":"The Stacks Project","memory_eligible":false,"source_rank":10356,"rank":10356,"depth":43,"x":1721.636,"y":1102.718,"cluster":"tale-geometry"},{"id":"stacks:097P","tag":"097P","title":"Ind-étale algebra · Lemma 097P","summary":"Let A be a ring and let I ⊂ A be an ideal. The base change functor ind-étale A-algebras → ind-étale A/I-algebras, C ↦ C/IC has a fully faithful right adjoint v. In particular, given an ind-étale A/I-algebra overlineC there exists an ind-étale A-algebra C = v(overlineC) such that overlineC = C/IC.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be an ideal. The base change functor\n$$\n\\text{ind-\\'etale }A\\text{-algebras}\n\\longrightarrow\n\\text{ind-\\'etale }A/I\\text{-algebras},\\quad\nC \\longmapsto C/IC\n$$\nhas a fully faithful right adjoint $v$. In particular, given\nan ind-\\'etale $A/I$-algebra $\\overline{C}$ there exists\nan ind-\\'etale $A$-algebra $C = v(\\overline{C})$ such that\n$\\overline{C} = C/IC$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Ind-étale algebra","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097P","source_file":"proetale.tex","source_line":1220,"source_end_line":1233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1220-L1233","statement_sha256":"58e76958fff54529120113fbe700760afe1dc8cf73720b84d9c287d51bf1e3e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10357,"rank":10357,"depth":44,"x":1505.486,"y":1247.938,"cluster":"tale-geometry"},{"id":"stacks:097R","tag":"097R","title":"Constructing ind-étale algebras · Lemma 097R","summary":"Given a ring A there exists a faithfully flat ind-étale A-algebra C such that every faithfully flat étale ring map C → B has a retraction.","statement_latex":"Given a ring $A$ there exists a faithfully flat ind-\\'etale $A$-algebra $C$\nsuch that every faithfully flat \\'etale ring map $C \\to B$ has a retraction.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing ind-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097R","source_file":"proetale.tex","source_line":1357,"source_end_line":1361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1357-L1361","statement_sha256":"1c318374bc172585d5c9496091c6cf78e245213eaa577cb0cd9410e81f76ed29","origin":"The Stacks Project","memory_eligible":false,"source_rank":10358,"rank":10358,"depth":38,"x":1548.121,"y":1018.037,"cluster":"tale-geometry"},{"id":"stacks:097U","tag":"097U","title":"Constructing ind-étale algebras · Lemma 097U","summary":"Let A be a ring such that every faithfully flat étale ring map A → B has a retraction. Then the same is true for every quotient ring A/I.","statement_latex":"Let $A$ be a ring such that every faithfully flat \\'etale ring map\n$A \\to B$ has a retraction. Then the same is true for every quotient ring\n$A/I$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing ind-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097U","source_file":"proetale.tex","source_line":1423,"source_end_line":1428,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1423-L1428","statement_sha256":"cbdf47bf151a5077aa9f689c7969b83c067ff8935a36ff5cec6a59aee06f27f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10359,"rank":10359,"depth":7,"x":1701.676,"y":1211.898,"cluster":"tale-geometry"},{"id":"stacks:097V","tag":"097V","title":"Constructing ind-étale algebras · Lemma 097V","summary":"Let A be a ring such that every faithfully flat étale ring map A → B has a retraction. Then every local ring of A at a maximal ideal is strictly henselian.","statement_latex":"Let $A$ be a ring such that every faithfully flat \\'etale ring map\n$A \\to B$ has a retraction. Then every local ring of $A$ at a maximal\nideal is strictly henselian.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing ind-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097V","source_file":"proetale.tex","source_line":1443,"source_end_line":1448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1443-L1448","statement_sha256":"f9aeabce072304ab92604862a03bbce6c97250a341ebebc4560a61cb8c63e318","origin":"The Stacks Project","memory_eligible":false,"source_rank":10360,"rank":10360,"depth":46,"x":1432.352,"y":1156.037,"cluster":"tale-geometry"},{"id":"stacks:097W","tag":"097W","title":"Constructing ind-étale algebras · Lemma 097W","summary":"Let A be a ring such that every faithfully flat étale ring map A → B has a retraction. Let Z ⊂ Spec(A) be a closed subscheme. Let A → A_Z^sim be as constructed in Lemma [Tag 096V]. Then every faithfully flat étale ring map A_Z^sim → C has a retraction.","statement_latex":"Let $A$ be a ring such that every faithfully flat \\'etale ring map\n$A \\to B$ has a retraction. Let $Z \\subset \\Spec(A)$ be a closed subscheme.\nLet $A \\to A_Z^\\sim$ be as constructed in Lemma \\ref{lemma-localization}.\nThen every faithfully flat \\'etale ring map $A_Z^\\sim \\to C$ has\na retraction.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing ind-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097W","source_file":"proetale.tex","source_line":1485,"source_end_line":1492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1485-L1492","statement_sha256":"792b3b437ac516d30a89906c1a66b89dccd766ea9d560d686fba883b67ade218","origin":"The Stacks Project","memory_eligible":false,"source_rank":10361,"rank":10361,"depth":1,"x":1676.047,"y":1044.325,"cluster":"tale-geometry"},{"id":"stacks:097X","tag":"097X","title":"Constructing ind-étale algebras · Lemma 097X","summary":"Let A → B be a ring map inducing algebraic extensions on residue fields. There exists a commutative diagram xymatrix B ar[r] & D A ar[r] ar[u] & C ar[u] with the following properties: • A → C is faithfully flat and ind-étale, • B → D is faithfully flat and ind-étale, • Spec(C) is w-local, • Spec(D) is w-local, • Spec(D) → Spec(C) is w-local, • the set of closed points of Spec(D) is the inverse image of the set of closed points of Spec(C), • the set of closed points of…","statement_latex":"Let $A \\to B$ be a ring map inducing algebraic extensions on residue fields.\nThere exists a commutative diagram\n$$\n\\xymatrix{\nB \\ar[r] & D \\\\\nA \\ar[r] \\ar[u] & C \\ar[u]\n}\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item $A \\to C$ is faithfully flat and ind-\\'etale,\n\\item $B \\to D$ is faithfully flat and ind-\\'etale,\n\\item $\\Spec(C)$ is w-local,\n\\item $\\Spec(D)$ is w-local,\n\\item $\\Spec(D) \\to \\Spec(C)$ is w-local,\n\\item the set of closed points of $\\Spec(D)$ is the inverse image\nof the set of closed points of $\\Spec(C)$,\n\\item the set of closed points of $\\Spec(C)$ surjects onto $\\Spec(A)$,\n\\item the set of closed points of $\\Spec(D)$ surjects onto $\\Spec(B)$,\n\\item for $\\mathfrak m \\subset C$ maximal the local ring\n$C_\\mathfrak m$ is strictly henselian.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing ind-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097X","source_file":"proetale.tex","source_line":1505,"source_end_line":1529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1505-L1529","statement_sha256":"3ee3f2cc515e39f995080df5675a627e7a9cbe7f50938d2a6f8bf8de853b1135","origin":"The Stacks Project","memory_eligible":false,"source_rank":10362,"rank":10362,"depth":47,"x":1586.12,"y":1265.141,"cluster":"tale-geometry"},{"id":"stacks:097Z","tag":"097Z","title":"Weakly étale versus pro-étale · Proposition 097Z","summary":"Let A → B be a weakly étale ring map. Then there exists a faithfully flat, ind-étale ring map B → B' such that A → B' is ind-étale.","statement_latex":"Let $A \\to B$ be a weakly \\'etale ring map.\nThen there exists a faithfully flat, ind-\\'etale ring map\n$B \\to B'$ such that $A \\to B'$ is ind-\\'etale.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Weakly étale versus pro-étale","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/097Z","source_file":"proetale.tex","source_line":1584,"source_end_line":1589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1584-L1589","statement_sha256":"6f8a78a2850f716b6b5ba5c2037b573ef359fc25c39722188393b8db56536062","origin":"The Stacks Project","memory_eligible":false,"source_rank":10363,"rank":10363,"depth":48,"x":1474.775,"y":1051.131,"cluster":"tale-geometry"},{"id":"stacks:0EVN","tag":"0EVN","title":"The V topology and the pro-h topology · Lemma 0EVN","summary":"Let Y be an affine scheme. Let X = lim X_i be a directed limit of affine schemes over Y. The following are equivalent • (X → Y) is a standard V covering (Topologies, Definition [Tag 0ETB]), and • (X_i → Y) is a standard V covering for all i.","statement_latex":"Let $Y$ be an affine scheme. Let $X = \\lim X_i$ be a directed limit\nof affine schemes over $Y$. The following are equivalent\n\\begin{enumerate}\n\\item $\\{X \\to Y\\}$ is a standard V covering\n(Topologies, Definition \\ref{topologies-definition-standard-V-covering}), and\n\\item $\\{X_i \\to Y\\}$ is a standard V covering for all $i$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The V topology and the pro-h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVN","source_file":"proetale.tex","source_line":1669,"source_end_line":1678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1669-L1678","statement_sha256":"54d8db81d82e66141bb72b07ae7ab01c900ddba7d59cb4cd5aa05244b51a4435","origin":"The Stacks Project","memory_eligible":false,"source_rank":10364,"rank":10364,"depth":14,"x":1729.166,"y":1145.828,"cluster":"tale-geometry"},{"id":"stacks:0EVP","tag":"0EVP","title":"The V topology and the pro-h topology · Lemma 0EVP","summary":"Let X → Y be a morphism of affine schemes. The following are equivalent • (X → Y) is a standard V covering (Topologies, Definition [Tag 0ETB]), • X = lim X_i is a directed limit of affine schemes over Y such that (X_i → Y) is a ph covering for each i, and • X = lim X_i is a directed limit of affine schemes over Y such that (X_i → Y) is an h covering for each i.","statement_latex":"Let $X \\to Y$ be a morphism of affine schemes. The following are equivalent\n\\begin{enumerate}\n\\item $\\{X \\to Y\\}$ is a standard V covering\n(Topologies, Definition \\ref{topologies-definition-standard-V-covering}),\n\\item $X = \\lim X_i$ is a directed limit of affine schemes over $Y$\nsuch that $\\{X_i \\to Y\\}$ is a ph covering for each $i$, and\n\\item $X = \\lim X_i$ is a directed limit of affine schemes over $Y$\nsuch that $\\{X_i \\to Y\\}$ is an h covering for each $i$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The V topology and the pro-h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVP","source_file":"proetale.tex","source_line":1720,"source_end_line":1731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1720-L1731","statement_sha256":"382fa5663d5148e90c41023ad4ed191b9507a4807b45e516094f4f49e9d9ea12","origin":"The Stacks Project","memory_eligible":false,"source_rank":10365,"rank":10365,"depth":57,"x":1465.237,"y":1220.401,"cluster":"tale-geometry"},{"id":"stacks:0EVQ","tag":"0EVQ","title":"The V topology and the pro-h topology · Lemma 0EVQ","summary":"Let S be a scheme. Let F be a contravariant functor defined on the category of all schemes over S. If • F satisfies the sheaf property for the h topology, and • F is limit preserving (Limits, Remark [Tag 05LX]), then F satisfies the sheaf property for the V topology.","statement_latex":"Let $S$ be a scheme. Let $F$ be a contravariant functor defined\non the category of all schemes over $S$. If\n\\begin{enumerate}\n\\item $F$ satisfies the sheaf property for the h topology, and\n\\item $F$ is limit preserving\n(Limits, Remark \\ref{limits-remark-limit-preserving}),\n\\end{enumerate}\nthen $F$ satisfies the sheaf property for the V topology.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The V topology and the pro-h topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EVQ","source_file":"proetale.tex","source_line":1770,"source_end_line":1780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1770-L1780","statement_sha256":"62553cd94564cc2bcc9ef8d9ffb7526245520f8b100be543cefe01af7c5bcf08","origin":"The Stacks Project","memory_eligible":false,"source_rank":10366,"rank":10366,"depth":58,"x":1599.983,"y":1015.503,"cluster":"tale-geometry"},{"id":"stacks:0981","tag":"0981","title":"Constructing w-contractible covers · Definition 0981","summary":"Let A be a ring. We say A is w-contractible if every faithfully flat weakly étale ring map A → B has a retraction.","statement_latex":"Let $A$ be a ring. We say $A$ is {\\it w-contractible} if every\nfaithfully flat weakly \\'etale ring map $A \\to B$ has a retraction.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-contractible covers","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0981","source_file":"proetale.tex","source_line":1856,"source_end_line":1860,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1856-L1860","statement_sha256":"517bcd3e65e684fbb4f57cbd923fd8e2ccff21eb7635ed5e63279b30de3d0992","origin":"The Stacks Project","memory_eligible":false,"source_rank":10367,"rank":10367,"depth":0,"x":1665.444,"y":1243.216,"cluster":"tale-geometry"},{"id":"stacks:0982","tag":"0982","title":"Constructing w-contractible covers · Lemma 0982","summary":"Let A be a ring. The following are equivalent • A is w-contractible, • every faithfully flat, ind-étale ring map A → B has a retraction, and • A satisfies • Spec(A) is w-local, • π_0(Spec(A)) is extremally disconnected, and • for every maximal ideal m ⊂ A the local ring A_ m is strictly henselian.","statement_latex":"Let $A$ be a ring. The following are equivalent\n\\begin{enumerate}\n\\item $A$ is w-contractible,\n\\item every faithfully flat, ind-\\'etale ring map $A \\to B$ has\na retraction, and\n\\item $A$ satisfies\n\\begin{enumerate}\n\\item $\\Spec(A)$ is w-local,\n\\item $\\pi_0(\\Spec(A))$ is extremally disconnected, and\n\\item for every maximal ideal $\\mathfrak m \\subset A$ the\nlocal ring $A_\\mathfrak m$ is strictly henselian.\n\\end{enumerate}\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-contractible covers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0982","source_file":"proetale.tex","source_line":1869,"source_end_line":1884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1869-L1884","statement_sha256":"471b504970e41141fbab058715605292f4f53f1e72cf1c7b6e760b608fa5db86","origin":"The Stacks Project","memory_eligible":false,"source_rank":10368,"rank":10368,"depth":49,"x":1433.886,"y":1112.353,"cluster":"tale-geometry"},{"id":"stacks:0983","tag":"0983","title":"Constructing w-contractible covers · Proposition 0983","summary":"For every ring A there exists a faithfully flat, ind-étale ring map A → D such that D is w-contractible.","statement_latex":"For every ring $A$ there exists a faithfully flat, ind-\\'etale ring\nmap $A \\to D$ such that $D$ is w-contractible.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-contractible covers","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0983","source_file":"proetale.tex","source_line":1917,"source_end_line":1921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1917-L1921","statement_sha256":"11cf2030f4c0133ac00598b92faa561abd14a28c54f24b23c95a43656c71909e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10369,"rank":10369,"depth":50,"x":1710.069,"y":1077.432,"cluster":"tale-geometry"},{"id":"stacks:0985","tag":"0985","title":"Constructing w-contractible covers · Lemma 0985","summary":"Let A → B be a quasi-finite and finitely presented ring map. If the residue fields of A are separably algebraically closed and Spec(A) is Hausdorff and extremally disconnected, then Spec(B) is extremally disconnected.","statement_latex":"Let $A \\to B$ be a quasi-finite and finitely presented ring map.\nIf the residue fields of $A$ are separably algebraically closed\nand $\\Spec(A)$ is Hausdorff and extremally disconnected, then $\\Spec(B)$ is\nextremally disconnected.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-contractible covers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0985","source_file":"proetale.tex","source_line":1987,"source_end_line":1993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L1987-L1993","statement_sha256":"76b72921b2f5db214d0b5bec622e9cabf2acdc0cde9935a4477750050fd387ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":10370,"rank":10370,"depth":46,"x":1534.371,"y":1260.028,"cluster":"tale-geometry"},{"id":"stacks:0986","tag":"0986","title":"Constructing w-contractible covers · Lemma 0986","summary":"Let A → B be a finite and finitely presented ring map. If A is w-contractible, so is B.","statement_latex":"Let $A \\to B$ be a finite and finitely presented ring map.\nIf $A$ is w-contractible, so is $B$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-contractible covers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0986","source_file":"proetale.tex","source_line":2027,"source_end_line":2031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2027-L2031","statement_sha256":"0d4ce0243d8f1d18867981334d0dcda7f0fc9f2c46ad068e753ff31671de74cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10371,"rank":10371,"depth":50,"x":1517.077,"y":1025.52,"cluster":"tale-geometry"},{"id":"stacks:0987","tag":"0987","title":"Constructing w-contractible covers · Lemma 0987","summary":"Let A be a ring. Let Z ⊂ Spec(A) be a closed subset of the form Z = V(f_1, …, f_r). Set B = A_Z^sim, see Lemma [Tag 096V]. If A is w-contractible, so is B.","statement_latex":"Let $A$ be a ring. Let $Z \\subset \\Spec(A)$ be a closed subset\nof the form $Z = V(f_1, \\ldots, f_r)$. Set $B = A_Z^\\sim$, see\nLemma \\ref{lemma-localization}. If $A$ is w-contractible, so is $B$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructing w-contractible covers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0987","source_file":"proetale.tex","source_line":2065,"source_end_line":2070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2065-L2070","statement_sha256":"054a23c3862a0da494642b203122cebe0705b630340afff22137948ca631cfdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":10372,"rank":10372,"depth":1,"x":1718.56,"y":1188.747,"cluster":"tale-geometry"},{"id":"stacks:0989","tag":"0989","title":"The pro-étale site · Definition 0989","summary":"Let T be a scheme. A pro-étale covering of T is a family of morphisms (f_i : T_i → T)_i ∈ I of schemes such that each f_i is weakly-étale and such that for every affine open U ⊂ T there exists n ≥ 0, a map a : (1, …, n) → I and affine opens V_j ⊂ T_a(j), j = 1, …, n with ⋃_j = 1^n f_a(j)(V_j) = U.","statement_latex":"Let $T$ be a scheme. A {\\it pro-\\'etale covering of $T$} is a family\nof morphisms $\\{f_i : T_i \\to T\\}_{i \\in I}$ of schemes\nsuch that each $f_i$ is weakly-\\'etale and such that for every affine open\n$U \\subset T$ there exists $n \\geq 0$, a map\n$a : \\{1, \\ldots, n\\} \\to I$ and affine opens\n$V_j \\subset T_{a(j)}$, $j = 1, \\ldots, n$\nwith $\\bigcup_{j = 1}^n f_{a(j)}(V_j) = U$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0989","source_file":"proetale.tex","source_line":2125,"source_end_line":2134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2125-L2134","statement_sha256":"4ccb44a0bf6b1cbad4daa02c8758410dbbcbf9955d6abdaa12baa0c3394917cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10373,"rank":10373,"depth":0,"x":1438.525,"y":1182.708,"cluster":"tale-geometry"},{"id":"stacks:098A","tag":"098A","title":"The pro-étale site · Lemma 098A","summary":"Let T be a scheme. Let (f_i : T_i → T)_i ∈ I be a family of morphisms of schemes with target T. The following are equivalent • (f_i : T_i → T)_i ∈ I is a pro-étale covering, • each f_i is weakly étale and (f_i : T_i → T)_i ∈ I is an fpqc covering, • each f_i is weakly étale and for every affine open U ⊂ T there exist quasi-compact opens U_i ⊂ T_i which are almost all empty, such that U = ⋃ f_i(U_i), • each f_i is weakly étale and there exists an affine open covering T =…","statement_latex":"Let $T$ be a scheme. Let $\\{f_i : T_i \\to T\\}_{i \\in I}$ be a family of\nmorphisms of schemes with target $T$. The following are equivalent\n\\begin{enumerate}\n\\item $\\{f_i : T_i \\to T\\}_{i \\in I}$ is a pro-\\'etale covering,\n\\item each $f_i$ is weakly \\'etale and $\\{f_i : T_i \\to T\\}_{i \\in I}$\nis an fpqc covering,\n\\item each $f_i$ is weakly \\'etale and for every affine open $U \\subset T$\nthere exist quasi-compact opens $U_i \\subset T_i$ which are almost all empty,\nsuch that $U = \\bigcup f_i(U_i)$,\n\\item each $f_i$ is weakly \\'etale and there exists an affine open covering\n$T = \\bigcup_{\\alpha \\in A} U_\\alpha$ and for each $\\alpha \\in A$\nthere exist $i_{\\alpha, 1}, \\ldots, i_{\\alpha, n(\\alpha)} \\in I$\nand quasi-compact opens $U_{\\alpha, j} \\subset T_{i_{\\alpha, j}}$ such that\n$U_\\alpha =\n\\bigcup_{j = 1, \\ldots, n(\\alpha)} f_{i_{\\alpha, j}}(U_{\\alpha, j})$.\n\\end{enumerate}\nIf $T$ is quasi-separated, these are also equivalent to\n\\begin{enumerate}\n\\item[(5)] each $f_i$ is weakly \\'etale, and for every $t \\in T$ there exist\n$i_1, \\ldots, i_n \\in I$ and quasi-compact opens $U_j \\subset T_{i_j}$\nsuch that $\\bigcup_{j = 1, \\ldots, n} f_{i_j}(U_j)$ is a\n(not necessarily open) neighbourhood of $t$ in $T$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098A","source_file":"proetale.tex","source_line":2142,"source_end_line":2167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2142-L2167","statement_sha256":"2d847f02b904dfcb3b516251c621dcb4abe44363e5e42806c5e3cd34630afb0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10374,"rank":10374,"depth":19,"x":1650.028,"y":1028.151,"cluster":"tale-geometry"},{"id":"stacks:098B","tag":"098B","title":"The pro-étale site · Lemma 098B","summary":"Any étale covering and any Zariski covering is a pro-étale covering.","statement_latex":"Any \\'etale covering and any Zariski covering is a pro-\\'etale covering.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098B","source_file":"proetale.tex","source_line":2175,"source_end_line":2178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2175-L2178","statement_sha256":"b84926101c3d3fe08d1a305e70235cba0807429162afc99a3f3288ca33880836","origin":"The Stacks Project","memory_eligible":false,"source_rank":10375,"rank":10375,"depth":39,"x":1618.336,"y":1262.299,"cluster":"tale-geometry"},{"id":"stacks:098C","tag":"098C","title":"The pro-étale site · Lemma 098C","summary":"Let T be a scheme. • If T' → T is an isomorphism then (T' → T) is a pro-étale covering of T. • If (T_i → T)_i∈ I is a pro-étale covering and for each i we have a pro-étale covering (T_ij → T_i)_j∈ J_i, then (T_ij → T)_i ∈ I, j∈ J_i is a pro-étale covering. • If (T_i → T)_i∈ I is a pro-étale covering and T' → T is a morphism of schemes then (T' ×_T T_i → T')_i∈ I is a pro-étale covering.","statement_latex":"Let $T$ be a scheme.\n\\begin{enumerate}\n\\item If $T' \\to T$ is an isomorphism then $\\{T' \\to T\\}$\nis a pro-\\'etale covering of $T$.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a pro-\\'etale covering and for each\n$i$ we have a pro-\\'etale covering $\\{T_{ij} \\to T_i\\}_{j\\in J_i}$, then\n$\\{T_{ij} \\to T\\}_{i \\in I, j\\in J_i}$ is a pro-\\'etale covering.\n\\item If $\\{T_i \\to T\\}_{i\\in I}$ is a pro-\\'etale covering\nand $T' \\to T$ is a morphism of schemes then\n$\\{T' \\times_T T_i \\to T'\\}_{i\\in I}$ is a pro-\\'etale covering.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098C","source_file":"proetale.tex","source_line":2189,"source_end_line":2202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2189-L2202","statement_sha256":"a9ef038f5d91d413c5e823a1c2538d42a04aa1bddbb1e5bbc9e558a9f6c23efd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10376,"rank":10376,"depth":20,"x":1453.292,"y":1071.522,"cluster":"tale-geometry"},{"id":"stacks:098D","tag":"098D","title":"The pro-étale site · Lemma 098D","summary":"Let T be an affine scheme. Let (T_i → T)_i ∈ I be a pro-étale covering of T. Then there exists a pro-étale covering (U_j → T)_j = 1, …, n which is a refinement of (T_i → T)_i ∈ I such that each U_j is an affine scheme. Moreover, we may choose each U_j to be open affine in one of the T_i.","statement_latex":"Let $T$ be an affine scheme. Let $\\{T_i \\to T\\}_{i \\in I}$ be a pro-\\'etale\ncovering of $T$. Then there exists a pro-\\'etale covering\n$\\{U_j \\to T\\}_{j = 1, \\ldots, n}$ which is a refinement\nof $\\{T_i \\to T\\}_{i \\in I}$ such that each $U_j$ is an affine\nscheme. Moreover, we may choose each $U_j$ to be open affine\nin one of the $T_i$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098D","source_file":"proetale.tex","source_line":2215,"source_end_line":2223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2215-L2223","statement_sha256":"a1c1087db2588094a9bf8502c82384b62a33e57472f3f687814261dc98d37d78","origin":"The Stacks Project","memory_eligible":false,"source_rank":10377,"rank":10377,"depth":0,"x":1728.607,"y":1118.582,"cluster":"tale-geometry"},{"id":"stacks:098E","tag":"098E","title":"The pro-étale site · Definition 098E","summary":"Let T be an affine scheme. A standard pro-étale covering of T is a family (f_i : T_i → T)_i = 1, …, n where each T_j is affine, each f_i is weakly étale, and T = ⋃ f_i(T_i).","statement_latex":"Let $T$ be an affine scheme. A {\\it standard pro-\\'etale covering}\nof $T$ is a family $\\{f_i : T_i \\to T\\}_{i = 1, \\ldots, n}$\nwhere each $T_j$ is affine, each $f_i$ is weakly \\'etale, and\n$T = \\bigcup f_i(T_i)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098E","source_file":"proetale.tex","source_line":2232,"source_end_line":2238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2232-L2238","statement_sha256":"c1815f4a9e85fa9c08be2d4fb5c4a89d67a6895958b583fd1344d4ba76640201","origin":"The Stacks Project","memory_eligible":false,"source_rank":10378,"rank":10378,"depth":0,"x":1487.577,"y":1240.188,"cluster":"tale-geometry"},{"id":"stacks:098G","tag":"098G","title":"The pro-étale site · Definition 098G","summary":"A big pro-étale site is any site Sch_proetale as in Sites, Definition [Tag 00VH] constructed as follows: • Choose any set of schemes S_0, and any set of pro-étale coverings Cov_0 among these schemes. • Change the function Bound of Sets, Equation ([Tag 046U]) into Bound(kappa) = max(kappa^2^2^2^kappa, kappa^aleph_0, kappa^+). • As underlying category take any category Sch_α constructed as in Sets, Lemma [Tag 000J] starting with the set S_0 and the function Bound. • Choose…","statement_latex":"A {\\it big pro-\\'etale site} is any site $\\Sch_\\proetale$ as in\nSites, Definition \\ref{sites-definition-site} constructed as follows:\n\\begin{enumerate}\n\\item Choose any set of schemes $S_0$, and any set of pro-\\'etale coverings\n$\\text{Cov}_0$ among these schemes.\n\\item Change the function $Bound$ of\nSets, Equation (\\ref{sets-equation-bound}) into\n$$\nBound(\\kappa) = \\max\\{\\kappa^{2^{2^{2^\\kappa}}}, \\kappa^{\\aleph_0}, \\kappa^+\\}.\n$$\n\\item As underlying category take any category $\\Sch_\\alpha$\nconstructed as in Sets, Lemma \\ref{sets-lemma-construct-category}\nstarting with the set $S_0$ and the function $Bound$.\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\Sch_\\alpha$ and the class of pro-\\'etale coverings,\nand the set $\\text{Cov}_0$ chosen above.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098G","source_file":"proetale.tex","source_line":2247,"source_end_line":2267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2247-L2267","statement_sha256":"eb0a4e11257b1de44ed1a7d745996151811a3ef3b65e975ff1496683ca6205f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10379,"rank":10379,"depth":2,"x":1567.574,"y":1013.59,"cluster":"tale-geometry"},{"id":"stacks:098K","tag":"098K","title":"The pro-étale site · Definition 098K","summary":"Let S be a scheme. Let Sch_proetale be a big pro-étale site containing S. • The big pro-étale site of S, denoted (Sch/S)_proetale, is the site Sch_proetale/S introduced in Sites, Section [Tag 00XZ]. • The small pro-étale site of S, which we denote S_proetale, is the full subcategory of (Sch/S)_proetale whose objects are those U/S such that U → S is weakly étale. A covering of S_proetale is any covering (U_i → U) of (Sch/S)_proetale with U ∈ Ob(S_proetale). • The big…","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale\nsite containing $S$.\n\\begin{enumerate}\n\\item The {\\it big pro-\\'etale site of $S$}, denoted\n$(\\Sch/S)_\\proetale$, is the site $\\Sch_\\proetale/S$\nintroduced in Sites, Section \\ref{sites-section-localize}.\n\\item The {\\it small pro-\\'etale site of $S$}, which we denote\n$S_\\proetale$, is the full subcategory of $(\\Sch/S)_\\proetale$\nwhose objects are those $U/S$ such that $U \\to S$ is weakly \\'etale.\nA covering of $S_\\proetale$ is any covering $\\{U_i \\to U\\}$ of\n$(\\Sch/S)_\\proetale$ with $U \\in \\Ob(S_\\proetale)$.\n\\item The {\\it big affine pro-\\'etale site of $S$}, denoted\n$(\\textit{Aff}/S)_\\proetale$, is the full subcategory of\n$(\\Sch/S)_\\proetale$ whose objects are affine $U/S$.\nA covering of $(\\textit{Aff}/S)_\\proetale$ is any covering\n$\\{U_i \\to U\\}$ of $(\\Sch/S)_\\proetale$ which is a\nstandard pro-\\'etale covering.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098K","source_file":"proetale.tex","source_line":2279,"source_end_line":2299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2279-L2299","statement_sha256":"56a8329bb9bea67918df040ea53365076db518682d75e312aa4b3efb974696b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10380,"rank":10380,"depth":0,"x":1690.898,"y":1226.223,"cluster":"tale-geometry"},{"id":"stacks:098L","tag":"098L","title":"The pro-étale site · Lemma 098L","summary":"Let S be a scheme. Let Sch_proetale be a big pro-étale site containing S. Both S_proetale and (Aff/S)_proetale are sites.","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale site\ncontaining $S$. Both $S_\\proetale$ and $(\\textit{Aff}/S)_\\proetale$ are sites.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098L","source_file":"proetale.tex","source_line":2305,"source_end_line":2309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2305-L2309","statement_sha256":"6ce81c7701d3ad6e677b5ad9aab016f25f41b43f21482580a0f0ece4223ae7ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":10381,"rank":10381,"depth":20,"x":1428.78,"y":1139.346,"cluster":"tale-geometry"},{"id":"stacks:098M","tag":"098M","title":"The pro-étale site · Lemma 098M","summary":"Let S be a scheme. Let Sch_proetale be a big pro-étale site containing S. Let Sch be the category of all schemes. • The categories Sch_proetale, (Sch/S)_proetale, S_proetale, and (Aff/S)_proetale have fibre products agreeing with fibre products in Sch. • The categories Sch_proetale, (Sch/S)_proetale, S_proetale have equalizers agreeing with equalizers in Sch. • The categories (Sch/S)_proetale, and S_proetale both have a final object, namely S/S. • The category…","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale\nsite containing $S$. Let $\\Sch$ be the category of all schemes.\n\\begin{enumerate}\n\\item The categories $\\Sch_\\proetale$, $(\\Sch/S)_\\proetale$,\n$S_\\proetale$, and $(\\textit{Aff}/S)_\\proetale$ have fibre products\nagreeing with fibre products in $\\Sch$.\n\\item The categories $\\Sch_\\proetale$, $(\\Sch/S)_\\proetale$,\n$S_\\proetale$ have equalizers agreeing with equalizers in $\\Sch$.\n\\item The categories $(\\Sch/S)_\\proetale$, and $S_\\proetale$ both have\na final object, namely $S/S$.\n\\item The category $\\Sch_\\proetale$ has a final object agreeing\nwith the final object of $\\Sch$, namely $\\Spec(\\mathbf{Z})$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098M","source_file":"proetale.tex","source_line":2332,"source_end_line":2347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2332-L2347","statement_sha256":"46dd3c1579444fad9f3eae60b52258fe91742d87e2e064f29fbe80cb33342f8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10382,"rank":10382,"depth":3,"x":1692.113,"y":1054.616,"cluster":"tale-geometry"},{"id":"stacks:098N","tag":"098N","title":"The pro-étale site · Lemma 098N","summary":"Let S be a scheme. Let Sch_proetale be a big pro-étale site containing S. The functor (Aff/S)_proetale → (Sch/S)_proetale is a special cocontinuous functor. Hence it induces an equivalence of topoi from Sh((Aff/S)_proetale) to Sh((Sch/S)_proetale).","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale\nsite containing $S$.\nThe functor $(\\textit{Aff}/S)_\\proetale \\to (\\Sch/S)_\\proetale$\nis a special cocontinuous functor. Hence it induces an equivalence\nof topoi from $\\Sh((\\textit{Aff}/S)_\\proetale)$ to\n$\\Sh((\\Sch/S)_\\proetale)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098N","source_file":"proetale.tex","source_line":2393,"source_end_line":2401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2393-L2401","statement_sha256":"1a69839ea0adda135e2cbeb2277e2d8dfb200f4bcabb96b989deeefcadd0cf7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10383,"rank":10383,"depth":8,"x":1565.983,"y":1266.667,"cluster":"tale-geometry"},{"id":"stacks:098P","tag":"098P","title":"The pro-étale site · Lemma 098P","summary":"Let Sch_proetale be a big pro-étale site. Let f : T → S be a morphism in Sch_proetale. The functor T_proetale → (Sch/S)_proetale is cocontinuous and induces a morphism of topoi i_f : Sh(T_proetale) → Sh((Sch/S)_proetale) For a sheaf G on (Sch/S)_proetale we have the formula (i_f^-1G)(U/T) = G(U/S). The functor i_f^-1 also has a left adjoint i_f, ! which commutes with fibre products and equalizers.","statement_latex":"Let $\\Sch_\\proetale$ be a big pro-\\'etale site.\nLet $f : T \\to S$ be a morphism in $\\Sch_\\proetale$.\nThe functor $T_\\proetale \\to (\\Sch/S)_\\proetale$\nis cocontinuous and induces a morphism of topoi\n$$\ni_f :\n\\Sh(T_\\proetale)\n\\longrightarrow\n\\Sh((\\Sch/S)_\\proetale)\n$$\nFor a sheaf $\\mathcal{G}$ on $(\\Sch/S)_\\proetale$\nwe have the formula $(i_f^{-1}\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nThe functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes\nwith fibre products and equalizers.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098P","source_file":"proetale.tex","source_line":2421,"source_end_line":2437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2421-L2437","statement_sha256":"8f536877ee30bff0d16d58cc50f832e0a06c9ff1515bcd52e6bca370ca139e9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10384,"rank":10384,"depth":6,"x":1488.409,"y":1038.572,"cluster":"tale-geometry"},{"id":"stacks:098Q","tag":"098Q","title":"The pro-étale site · Lemma 098Q","summary":"Let S be a scheme. Let Sch_proetale be a big pro-étale site containing S. The inclusion functor S_proetale → (Sch/S)_proetale satisfies the hypotheses of Sites, Lemma [Tag 00XU] and hence induces a morphism of sites π_S : (Sch/S)_proetale → S_proetale and a morphism of topoi i_S : Sh(S_proetale) → Sh((Sch/S)_proetale) such that π_S ∘ i_S = id. Moreover, i_S = i_id_S with i_id_S as in Lemma [Tag 098P]. In particular the functor i_S^-1 = π_S, * is described by the rule…","statement_latex":"Let $S$ be a scheme. Let $\\Sch_\\proetale$ be a big pro-\\'etale\nsite containing $S$.\nThe inclusion functor $S_\\proetale \\to (\\Sch/S)_\\proetale$\nsatisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site}\nand hence induces a morphism of sites\n$$\n\\pi_S : (\\Sch/S)_\\proetale \\longrightarrow S_\\proetale\n$$\nand a morphism of topoi\n$$\ni_S : \\Sh(S_\\proetale) \\longrightarrow \\Sh((\\Sch/S)_\\proetale)\n$$\nsuch that $\\pi_S \\circ i_S = \\text{id}$. Moreover, $i_S = i_{\\text{id}_S}$\nwith $i_{\\text{id}_S}$ as in Lemma \\ref{lemma-put-in-T}. In particular the\nfunctor $i_S^{-1} = \\pi_{S, *}$ is described by the rule\n$i_S^{-1}(\\mathcal{G})(U/S) = \\mathcal{G}(U/S)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098Q","source_file":"proetale.tex","source_line":2454,"source_end_line":2472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2454-L2472","statement_sha256":"56cd6cb18ba46494e86b3b0873826b4e27e5379f8dd519a9ac66f003dabf3cbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10385,"rank":10385,"depth":8,"x":1729.208,"y":1162.836,"cluster":"tale-geometry"},{"id":"stacks:098R","tag":"098R","title":"The pro-étale site · Definition 098R","summary":"In the situation of Lemma [Tag 098Q] the functor i_S^-1 = π_S, * is often called the restriction to the small pro-étale site, and for a sheaf F on the big pro-étale site we denote F|_S_proetale this restriction.","statement_latex":"In the situation of\nLemma \\ref{lemma-at-the-bottom}\nthe functor $i_S^{-1} = \\pi_{S, *}$ is often\ncalled the {\\it restriction to the small pro-\\'etale site}, and for a sheaf\n$\\mathcal{F}$ on the big pro-\\'etale site we denote\n$\\mathcal{F}|_{S_\\proetale}$ this restriction.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098R","source_file":"proetale.tex","source_line":2482,"source_end_line":2490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2482-L2490","statement_sha256":"8da0e3f128ade27e82adbcdcd1176dbc387cb77d936d60106d6b5c094a5a767d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10386,"rank":10386,"depth":9,"x":1451.521,"y":1207.873,"cluster":"tale-geometry"},{"id":"stacks:098S","tag":"098S","title":"The pro-étale site · Lemma 098S","summary":"Let Sch_proetale be a big pro-étale site. Let f : T → S be a morphism in Sch_proetale. The functor u : (Sch/T)_proetale → (Sch/S)_proetale, V/T ↦ V/S is cocontinuous, and has a continuous right adjoint v : (Sch/S)_proetale → (Sch/T)_proetale, (U → S) ↦ (U ×_S T → T). They induce the same morphism of topoi f_big : Sh((Sch/T)_proetale) → Sh((Sch/S)_proetale) We have f_big^-1(G)(U/T) = G(U/S). We have f_big, *(F)(U/S) = F(U ×_S T/T). Also, f_big^-1 has a left adjoint f_big!…","statement_latex":"Let $\\Sch_\\proetale$ be a big pro-\\'etale site.\nLet $f : T \\to S$ be a morphism in $\\Sch_\\proetale$.\nThe functor\n$$\nu : (\\Sch/T)_\\proetale \\longrightarrow (\\Sch/S)_\\proetale, \\quad\nV/T \\longmapsto V/S\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv : (\\Sch/S)_\\proetale \\longrightarrow (\\Sch/T)_\\proetale, \\quad\n(U \\to S) \\longmapsto (U \\times_S T \\to T).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\Sch/T)_\\proetale)\n\\longrightarrow\n\\Sh((\\Sch/S)_\\proetale)\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/T) = \\mathcal{G}(U/S)$.\nWe have $f_{big, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098S","source_file":"proetale.tex","source_line":2507,"source_end_line":2532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2507-L2532","statement_sha256":"72a1a620db4f79ed070edc304f7f41751fcc043e3121e8f6412968ffb3ea38c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10387,"rank":10387,"depth":8,"x":1620.185,"y":1016.962,"cluster":"tale-geometry"},{"id":"stacks:098T","tag":"098T","title":"The pro-étale site · Lemma 098T","summary":"Let Sch_proetale be a big pro-étale site. Let f : T → S be a morphism in Sch_proetale. • We have i_f = f_big ∘ i_T with i_f as in Lemma [Tag 098P] and i_T as in Lemma [Tag 098Q]. • The functor S_proetale → T_proetale, (U → S) ↦ (U ×_S T → T) is continuous and induces a morphism of topoi f_small : Sh(T_proetale) → Sh(S_proetale). We have f_small, *(F)(U/S) = F(U ×_S T/T). • We have a commutative diagram of morphisms of sites xymatrix T_proetale ar[d]_f_small &…","statement_latex":"Let $\\Sch_\\proetale$ be a big pro-\\'etale site.\nLet $f : T \\to S$ be a morphism in $\\Sch_\\proetale$.\n\\begin{enumerate}\n\\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in\nLemma \\ref{lemma-put-in-T} and $i_T$ as in\nLemma \\ref{lemma-at-the-bottom}.\n\\item The functor $S_\\proetale \\to T_\\proetale$,\n$(U \\to S) \\mapsto (U \\times_S T \\to T)$ is continuous and induces\na morphism of topoi\n$$\nf_{small} : \\Sh(T_\\proetale) \\longrightarrow \\Sh(S_\\proetale).\n$$\nWe have $f_{small, *}(\\mathcal{F})(U/S) = \\mathcal{F}(U \\times_S T/T)$.\n\\item We have a commutative diagram of morphisms of sites\n$$\n\\xymatrix{\nT_\\proetale \\ar[d]_{f_{small}} &\n(\\Sch/T)_\\proetale \\ar[d]^{f_{big}} \\ar[l]^{\\pi_T}\\\\\nS_\\proetale &\n(\\Sch/S)_\\proetale \\ar[l]_{\\pi_S}\n}\n$$\nso that $f_{small} \\circ \\pi_T = \\pi_S \\circ f_{big}$ as morphisms of topoi.\n\\item We have $f_{small} = \\pi_S \\circ f_{big} \\circ i_T = \\pi_S \\circ i_f$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098T","source_file":"proetale.tex","source_line":2550,"source_end_line":2577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2550-L2577","statement_sha256":"7f95be8c2a15dd8c28116dbc86e1dde7b7b9a861ea47ba91624c992d947513d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10388,"rank":10388,"depth":21,"x":1649.36,"y":1253.608,"cluster":"tale-geometry"},{"id":"stacks:098U","tag":"098U","title":"The pro-étale site · Lemma 098U","summary":"Given schemes X, Y, Y in Sch_proetale and morphisms f : X → Y, g : Y → Z we have g_big ∘ f_big = (g ∘ f)_big and g_small ∘ f_small = (g ∘ f)_small.","statement_latex":"Given schemes $X$, $Y$, $Y$ in $\\Sch_\\proetale$\nand morphisms $f : X \\to Y$, $g : Y \\to Z$ we have\n$g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and\n$g_{small} \\circ f_{small} = (g \\circ f)_{small}$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098U","source_file":"proetale.tex","source_line":2622,"source_end_line":2628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2622-L2628","statement_sha256":"9c1ca18cdc2620ba541d4bb6807f8105476457f0b6312f6479e53f8d2974237f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10389,"rank":10389,"depth":22,"x":1437.394,"y":1095.553,"cluster":"tale-geometry"},{"id":"stacks:0F61","tag":"0F61","title":"The pro-étale site · Lemma 0F61","summary":"Let Sch_proetale be a big pro-étale site. Consider a cartesian diagram xymatrix T' ar[r]_g' ar[d]_f' & T ar[d]^f S' ar[r]^g & S in Sch_proetale. Then i_g^-1 ∘ f_big, * = f'_small, * ∘ (i_g')^-1 and g_big^-1 ∘ f_big, * = f'_big, * ∘ (g'_big)^-1.","statement_latex":"Let $\\Sch_\\proetale$ be a big pro-\\'etale site. Consider a cartesian diagram\n$$\n\\xymatrix{\nT' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nin $\\Sch_\\proetale$. Then\n$i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$\nand $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F61","source_file":"proetale.tex","source_line":2638,"source_end_line":2650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2638-L2650","statement_sha256":"19c44229e06e99a3ff023a0466cb90ec27952cd0369f1b6ce6afabca7c5c46a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10390,"rank":10390,"depth":9,"x":1720.998,"y":1091.823,"cluster":"tale-geometry"},{"id":"stacks:098V","tag":"098V","title":"The pro-étale site · Lemma 098V","summary":"Let S be a scheme contained in a big pro-étale site Sch_proetale. A sheaf F on the big pro-étale site (Sch/S)_proetale is given by the following data: • for every T/S ∈ Ob((Sch/S)_proetale) a sheaf F_T on T_proetale, • for every f : T' → T in (Sch/S)_proetale a map c_f : f_small^-1F_T → F_T'. These data are subject to the following conditions: • [(a)] given any f : T' → T and g : T\" → T' in (Sch/S)_proetale the composition c_g ∘ g_small^-1c_f is equal to c_f ∘ g, and •…","statement_latex":"Let $S$ be a scheme contained in a big pro-\\'etale site $\\Sch_\\proetale$.\nA sheaf $\\mathcal{F}$ on the big pro-\\'etale site $(\\Sch/S)_\\proetale$\nis given by the following data:\n\\begin{enumerate}\n\\item for every $T/S \\in \\Ob((\\Sch/S)_\\proetale)$ a sheaf\n$\\mathcal{F}_T$ on $T_\\proetale$,\n\\item for every $f : T' \\to T$ in\n$(\\Sch/S)_\\proetale$ a map\n$c_f : f_{small}^{-1}\\mathcal{F}_T \\to \\mathcal{F}_{T'}$.\n\\end{enumerate}\nThese data are subject to the following conditions:\n\\begin{enumerate}\n\\item[(a)] given any $f : T' \\to T$ and $g : T'' \\to T'$ in\n$(\\Sch/S)_\\proetale$ the composition\n$c_g \\circ g_{small}^{-1}c_f$ is equal to $c_{f \\circ g}$, and\n\\item[(b)] if $f : T' \\to T$ in $(\\Sch/S)_\\proetale$\nis weakly \\'etale then $c_f$ is an isomorphism.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098V","source_file":"proetale.tex","source_line":2668,"source_end_line":2688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2668-L2688","statement_sha256":"374b980d6a7cfc64f00c0185779dd8aa642da9dd24637f4ccaaa459701594cc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10391,"rank":10391,"depth":8,"x":1514.727,"y":1255.612,"cluster":"tale-geometry"},{"id":"stacks:098W","tag":"098W","title":"The pro-étale site · Lemma 098W","summary":"Let S be a scheme. Let S_affine, proetale denote the full subcategory of S_proetale consisting of affine objects. A covering of S_affine, proetale will be a standard pro-étale covering, see Definition [Tag 098E]. Then restriction F ↦ F|_S_affine, etale defines an equivalence of topoi Sh(S_proetale) ≅ Sh(S_affine, proetale).","statement_latex":"Let $S$ be a scheme. Let $S_{affine, \\proetale}$ denote the full subcategory\nof $S_\\proetale$ consisting of affine objects. A covering of\n$S_{affine, \\proetale}$ will be a standard pro-\\'etale covering, see\nDefinition \\ref{definition-standard-proetale}.\nThen restriction\n$$\n\\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{affine, \\etale}}\n$$\ndefines an equivalence of topoi\n$\\Sh(S_\\proetale) \\cong \\Sh(S_{affine, \\proetale})$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098W","source_file":"proetale.tex","source_line":2695,"source_end_line":2707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2695-L2707","statement_sha256":"3fc64f959a844d9129e1e671728790fe7b798c543121d6773338800d791957e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10392,"rank":10392,"depth":8,"x":1535.128,"y":1017.627,"cluster":"tale-geometry"},{"id":"stacks:098X","tag":"098X","title":"The pro-étale site · Lemma 098X","summary":"Let S be an affine scheme. Let S_app denote the full subcategory of S_proetale consisting of affine objects U such that O(S) → O(U) is ind-étale. A covering of S_app will be a standard pro-étale covering, see Definition [Tag 098E]. Then restriction F ↦ F|_S_app defines an equivalence of topoi Sh(S_proetale) ≅ Sh(S_app).","statement_latex":"Let $S$ be an affine scheme. Let $S_{app}$ denote the full subcategory\nof $S_\\proetale$ consisting of affine objects $U$ such that\n$\\mathcal{O}(S) \\to \\mathcal{O}(U)$ is ind-\\'etale. A covering of\n$S_{app}$ will be a standard pro-\\'etale covering, see\nDefinition \\ref{definition-standard-proetale}.\nThen restriction\n$$\n\\mathcal{F} \\longmapsto \\mathcal{F}|_{S_{app}}\n$$\ndefines an equivalence of topoi $\\Sh(S_\\proetale) \\cong \\Sh(S_{app})$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098X","source_file":"proetale.tex","source_line":2719,"source_end_line":2731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2719-L2731","statement_sha256":"e5ba02b006c9b96b05a649ee890820365c7c5dcd02a9581cda87d99ef0b67dec","origin":"The Stacks Project","memory_eligible":false,"source_rank":10393,"rank":10393,"depth":49,"x":1711.589,"y":1204.82,"cluster":"tale-geometry"},{"id":"stacks:098Z","tag":"098Z","title":"The pro-étale site · Lemma 098Z","summary":"Let S be a scheme. The topology on each of the pro-étale sites Sch_proetale, S_proetale, (Sch/S)_proetale, S_affine, proetale, and (Aff/S)_proetale is subcanonical.","statement_latex":"Let $S$ be a scheme. The topology on each of the pro-\\'etale sites\n$\\Sch_\\proetale$, $S_\\proetale$, $(\\Sch/S)_\\proetale$,\n$S_{affine, \\proetale}$, and $(\\textit{Aff}/S)_\\proetale$ is subcanonical.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"The pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098Z","source_file":"proetale.tex","source_line":2748,"source_end_line":2753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2748-L2753","statement_sha256":"8f96b6f13e63b60719e1caecab972f7abc44a5d98dea9f5a8e0d36a4ee2e4f60","origin":"The Stacks Project","memory_eligible":false,"source_rank":10394,"rank":10394,"depth":41,"x":1430.738,"y":1166.888,"cluster":"tale-geometry"},{"id":"stacks:098F","tag":"098F","title":"Weakly contractible objects · Lemma 098F","summary":"Let T = Spec(A) be an affine scheme. The following are equivalent • A is w-contractible, and • every pro-étale covering of T can be refined by a Zariski covering of the form T = coprod_i = 1, …, n U_i.","statement_latex":"Let $T = \\Spec(A)$ be an affine scheme. The following are equivalent\n\\begin{enumerate}\n\\item $A$ is w-contractible, and\n\\item every pro-\\'etale covering of $T$ can be refined by\na Zariski covering of the form $T = \\coprod_{i = 1, \\ldots, n} U_i$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Weakly contractible objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098F","source_file":"proetale.tex","source_line":2780,"source_end_line":2788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2780-L2788","statement_sha256":"d530d58d0d2b3cbfafaf3910e8df75dea6f0a3f19089a5c11693230d5923a863","origin":"The Stacks Project","memory_eligible":false,"source_rank":10395,"rank":10395,"depth":1,"x":1668.502,"y":1035.405,"cluster":"tale-geometry"},{"id":"stacks:098H","tag":"098H","title":"Weakly contractible objects · Lemma 098H","summary":"Let Sch_proetale be a big pro-étale site as in Definition [Tag 098G]. Let T = Spec(A) be an affine object of Sch_proetale. The following are equivalent • A is w-contractible, • T is a weakly contractible (Sites, Definition [Tag 090L]) object of Sch_proetale, and • every pro-étale covering of T can be refined by a Zariski covering of the form T = coprod_i = 1, …, n U_i.","statement_latex":"Let $\\Sch_\\proetale$ be a big pro-\\'etale site as in\nDefinition \\ref{definition-big-proetale-site}.\nLet $T = \\Spec(A)$ be an affine object of $\\Sch_\\proetale$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $A$ is w-contractible,\n\\item $T$ is a weakly contractible\n(Sites, Definition \\ref{sites-definition-w-contractible})\nobject of $\\Sch_\\proetale$, and\n\\item every pro-\\'etale covering of $T$ can be refined by\na Zariski covering of the form $T = \\coprod_{i = 1, \\ldots, n} U_i$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Weakly contractible objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098H","source_file":"proetale.tex","source_line":2809,"source_end_line":2823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2809-L2823","statement_sha256":"d4525b03b5777f654c04155b20d8ac2e41e28676c90ad30f4c0625d55db768aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":10396,"rank":10396,"depth":51,"x":1598.865,"y":1267.434,"cluster":"tale-geometry"},{"id":"stacks:098I","tag":"098I","title":"Weakly contractible objects · Lemma 098I","summary":"Let Sch_proetale be a big pro-étale site as in Definition [Tag 098G]. For every object T of Sch_proetale there exists a covering (T_i → T) in Sch_proetale with each T_i affine and the spectrum of a w-contractible ring. In particular, T_i is weakly contractible in Sch_proetale.","statement_latex":"Let $\\Sch_\\proetale$ be a big pro-\\'etale site as in\nDefinition \\ref{definition-big-proetale-site}.\nFor every object $T$ of $\\Sch_\\proetale$ there exists\na covering $\\{T_i \\to T\\}$ in $\\Sch_\\proetale$\nwith each $T_i$ affine and the spectrum of a w-contractible\nring. In particular, $T_i$ is weakly contractible in $\\Sch_\\proetale$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Weakly contractible objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098I","source_file":"proetale.tex","source_line":2869,"source_end_line":2877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2869-L2877","statement_sha256":"e790ae168ba3d80cd09930021f813aaf4f2c381c3430c9731e66290d72196c77","origin":"The Stacks Project","memory_eligible":false,"source_rank":10397,"rank":10397,"depth":52,"x":1463.53,"y":1056.679,"cluster":"tale-geometry"},{"id":"stacks:0990","tag":"0990","title":"Weakly contractible objects · Lemma 0990","summary":"Let S be a scheme. The pro-étale sites S_proetale, (Sch/S)_proetale, S_affine, proetale, and (Aff/S)_proetale and if S is affine S_app have enough (affine) quasi-compact, weakly contractible objects, see Sites, Definition [Tag 090L].","statement_latex":"Let $S$ be a scheme. The pro-\\'etale sites\n$S_\\proetale$, $(\\Sch/S)_\\proetale$, $S_{affine, \\proetale}$, and\n$(\\textit{Aff}/S)_\\proetale$ and if $S$ is affine $S_{app}$\nhave enough (affine) quasi-compact, weakly contractible\nobjects, see Sites, Definition \\ref{sites-definition-w-contractible}.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Weakly contractible objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0990","source_file":"proetale.tex","source_line":2912,"source_end_line":2919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2912-L2919","statement_sha256":"f6dec877cec26799da3fbd85a567c5e89bdcc333aab82ba079041c51e45c4a47","origin":"The Stacks Project","memory_eligible":false,"source_rank":10398,"rank":10398,"depth":53,"x":1732.994,"y":1135.348,"cluster":"tale-geometry"},{"id":"stacks:0F4P","tag":"0F4P","title":"Weakly contractible objects · Lemma 0F4P","summary":"Let S be a scheme. The pro-étale sites Sch_proetale, S_proetale, (Sch/S)_proetale have the following property: for any object U there exists a covering (V → U) with V a weakly contractible object. If U is quasi-compact, then we may choose V affine and weakly contractible.","statement_latex":"Let $S$ be a scheme. The pro-\\'etale sites\n$\\Sch_\\proetale$, $S_\\proetale$, $(\\Sch/S)_\\proetale$\nhave the following property: for any object\n$U$ there exists a covering $\\{V \\to U\\}$ with $V$ a\nweakly contractible object. If $U$ is quasi-compact, then we\nmay choose $V$ affine and weakly contractible.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Weakly contractible objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4P","source_file":"proetale.tex","source_line":2925,"source_end_line":2933,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2925-L2933","statement_sha256":"0c0430ad46d2cd3acfdb11745b9c57de77555a25f4819b4f00c2442afbeec8dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10399,"rank":10399,"depth":53,"x":1470.85,"y":1230.305,"cluster":"tale-geometry"},{"id":"stacks:09A1","tag":"09A1","title":"Weakly contractible hypercoverings · Lemma 09A1","summary":"Let X be a scheme. • For every object U of X_proetale there exists a hypercovering K of U in X_proetale such that each term K_n consists of a single weakly contractible object of X_proetale covering U. • For every quasi-compact and quasi-separated object U of X_proetale there exists a hypercovering K of U in X_proetale such that each term K_n consists of a single affine and weakly contractible object of X_proetale covering U.","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item For every object $U$ of $X_\\proetale$ there exists a hypercovering\n$K$ of $U$ in $X_\\proetale$ such that each term $K_n$ consists of a\nsingle weakly contractible object of $X_\\proetale$ covering $U$.\n\\item For every quasi-compact and quasi-separated object $U$ of $X_\\proetale$\nthere exists a hypercovering $K$ of $U$ in $X_\\proetale$ such that each\nterm $K_n$ consists of a single affine and weakly contractible object of\n$X_\\proetale$ covering $U$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Weakly contractible hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09A1","source_file":"proetale.tex","source_line":2998,"source_end_line":3010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L2998-L3010","statement_sha256":"5d194ac375effcae5b9eea64dda394759dfc12d251f28f177f4d9aba1f99f50d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10400,"rank":10400,"depth":54,"x":1587.87,"y":1011.387,"cluster":"tale-geometry"},{"id":"stacks:09A2","tag":"09A2","title":"Weakly contractible hypercoverings · Lemma 09A2","summary":"Let X be a scheme. Let E ∈ D^+(X_proetale) be represented by a bounded below complex E^bullet of abelian sheaves. Let K be a hypercovering of U ∈ Ob(X_proetale) with K_n = (U_n → U) where U_n is a weakly contractible object of X_proetale. Then RΓ(U, E) = Tot(s(E^bullet(K))) in D(Ab).","statement_latex":"Let $X$ be a scheme. Let $E \\in D^+(X_\\proetale)$ be represented by\na bounded below complex $\\mathcal{E}^\\bullet$ of abelian sheaves.\nLet $K$ be a hypercovering of $U \\in \\Ob(X_\\proetale)$ with\n$K_n = \\{U_n \\to U\\}$ where $U_n$ is a weakly contractible object of\n$X_\\proetale$. Then\n$$\nR\\Gamma(U, E) = \\text{Tot}(s(\\mathcal{E}^\\bullet(K)))\n$$\nin $D(\\textit{Ab})$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Weakly contractible hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09A2","source_file":"proetale.tex","source_line":3065,"source_end_line":3076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3065-L3076","statement_sha256":"b1657eb2cf667f653e7695037204d55d9d7c579267468d169f574ce31f9119b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10401,"rank":10401,"depth":14,"x":1677.691,"y":1239.372,"cluster":"tale-geometry"},{"id":"stacks:09A3","tag":"09A3","title":"Weakly contractible hypercoverings · Lemma 09A3","summary":"Let X be a quasi-compact and quasi-separated scheme. The functor RΓ(X, -) : D^+(X_proetale) → D(Ab) commutes with direct sums and homotopy colimits.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nThe functor $R\\Gamma(X, -) : D^+(X_\\proetale) \\to D(\\textit{Ab})$\ncommutes with direct sums and homotopy colimits.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Weakly contractible hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09A3","source_file":"proetale.tex","source_line":3112,"source_end_line":3117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3112-L3117","statement_sha256":"5d58a2f219c27f99b851ee7dc2037eca1ad56c64a8b9f24defe62d681bd0fa09","origin":"The Stacks Project","memory_eligible":false,"source_rank":10402,"rank":10402,"depth":55,"x":1427.947,"y":1122.145,"cluster":"tale-geometry"},{"id":"stacks:0F4Q","tag":"0F4Q","title":"Compact generation · Lemma 0F4Q","summary":"Let S be a scheme. Let Lambda be a ring. • D(S_proetale) is compactly generated, • D(S_proetale, Lambda) is compactly generated, • D(S_proetale, A) is compactly generated for any sheaf of rings A on S_proetale, • D((Sch/S)_proetale) is compactly generated, • D((Sch/S)_proetale, Lambda) is compactly generated, and • D((Sch/S)_proetale, A) is compactly generated for any sheaf of rings A on (Sch/S)_proetale,","statement_latex":"Let $S$ be a scheme. Let $\\Lambda$ be a ring.\n\\begin{enumerate}\n\\item $D(S_\\proetale)$ is compactly generated,\n\\item $D(S_\\proetale, \\Lambda)$ is compactly generated,\n\\item $D(S_\\proetale, \\mathcal{A})$ is compactly generated\nfor any sheaf of rings $\\mathcal{A}$ on $S_\\proetale$,\n\\item $D((\\Sch/S)_\\proetale)$ is compactly generated,\n\\item $D((\\Sch/S)_\\proetale, \\Lambda)$ is compactly generated, and\n\\item $D((\\Sch/S)_\\proetale, \\mathcal{A})$ is compactly generated\nfor any sheaf of rings $\\mathcal{A}$ on $(\\Sch/S)_\\proetale$,\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Compact generation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4Q","source_file":"proetale.tex","source_line":3185,"source_end_line":3198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3185-L3198","statement_sha256":"28f69e58aae1e06e013b0a4f4387641cac5d78e4d9f9d6f7fabf38c826635ec2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10403,"rank":10403,"depth":53,"x":1706.567,"y":1066.837,"cluster":"tale-geometry"},{"id":"stacks:0F63","tag":"0F63","title":"Comparing topologies · Lemma 0F63","summary":"Let X be a scheme. Let F be a presheaf of sets on X_proetale which sends finite disjoint unions to products. Then F^\\#(W) = F(W) if W is an affine weakly contractible object of X_proetale.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be a presheaf of sets on $X_\\proetale$\nwhich sends finite disjoint unions to products. Then\n$\\mathcal{F}^\\#(W) = \\mathcal{F}(W)$ if $W$ is an affine weakly contractible\nobject of $X_\\proetale$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparing topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F63","source_file":"proetale.tex","source_line":3247,"source_end_line":3253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3247-L3253","statement_sha256":"d44c73e26cd28d82dafe632fd8fe24df3e8e56b873f635300a15f59f2492ad2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10404,"rank":10404,"depth":52,"x":1545.483,"y":1265.853,"cluster":"tale-geometry"},{"id":"stacks:0F64","tag":"0F64","title":"Comparing topologies · Lemma 0F64","summary":"Let f : X → Y be a morphism of schemes. Let F be a sheaf of sets on Y_proetale. If W is an affine weakly contractible object of X_proetale, then f_small^-1F(W) = colim_W → V F(V) where the colimit is over morphisms W → V over Y with V ∈ Y_proetale.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $\\mathcal{F}$ be a sheaf\nof sets on $Y_\\proetale$. If $W$ is an affine weakly contractible\nobject of $X_\\proetale$, then\n$$\nf_{small}^{-1}\\mathcal{F}(W) = \\colim_{W \\to V} \\mathcal{F}(V)\n$$\nwhere the colimit is over morphisms $W \\to V$ over $Y$\nwith $V \\in Y_\\proetale$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparing topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F64","source_file":"proetale.tex","source_line":3282,"source_end_line":3292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3282-L3292","statement_sha256":"80abeff534cc1a79e8708661aa013abe3ff9fa6fc5a68b5a7dc0be3b0c31df63","origin":"The Stacks Project","memory_eligible":false,"source_rank":10405,"rank":10405,"depth":53,"x":1504.193,"y":1027.535,"cluster":"tale-geometry"},{"id":"stacks:0F65","tag":"0F65","title":"Comparing topologies · Lemma 0F65","summary":"Let S be a scheme. Consider the morphism π_S : (Sch/S)_proetale → S_proetale of Lemma [Tag 098Q]. Let F be a sheaf on S_proetale. Then π_S^-1F is given by the rule (π_S^-1F)(T) = Γ(T_proetale, f_small^-1F) where f : T → S. Moreover, π_S^-1F satisfies the sheaf condition with respect to fpqc coverings.","statement_latex":"Let $S$ be a scheme. Consider the morphism\n$$\n\\pi_S : (\\Sch/S)_\\proetale \\longrightarrow S_\\proetale\n$$\nof Lemma \\ref{lemma-at-the-bottom}. Let $\\mathcal{F}$ be a sheaf on\n$S_\\proetale$. Then $\\pi_S^{-1}\\mathcal{F}$ is given by the rule\n$$\n(\\pi_S^{-1}\\mathcal{F})(T) = \\Gamma(T_\\proetale, f_{small}^{-1}\\mathcal{F})\n$$\nwhere $f : T \\to S$. Moreover, $\\pi_S^{-1}\\mathcal{F}$ satisfies the\nsheaf condition with respect to fpqc coverings.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparing topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F65","source_file":"proetale.tex","source_line":3323,"source_end_line":3336,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3323-L3336","statement_sha256":"1bd37bd9c758302045e4c72941bd7eb7f7bea68179b16d6e9132506a604ab92f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10406,"rank":10406,"depth":54,"x":1726.44,"y":1179.943,"cluster":"tale-geometry"},{"id":"stacks:0F67","tag":"0F67","title":"Comparing big and small topoi · Lemma 0F67","summary":"Let S be a scheme. Let T be an object of (Sch/S)_proetale. • If I is injective in Ab((Sch/S)_proetale), then • i_f^-1I is injective in Ab(T_proetale), • I|_S_proetale is injective in Ab(S_proetale), • If I^bullet is a K-injective complex in Ab((Sch/S)_proetale), then • i_f^-1I^bullet is a K-injective complex in Ab(T_proetale), • I^bullet|_S_proetale is a K-injective complex in Ab(S_proetale),","statement_latex":"Let $S$ be a scheme. Let $T$ be an object of $(\\Sch/S)_\\proetale$.\n\\begin{enumerate}\n\\item If $\\mathcal{I}$ is injective in $\\textit{Ab}((\\Sch/S)_\\proetale)$, then\n\\begin{enumerate}\n\\item $i_f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(T_\\proetale)$,\n\\item $\\mathcal{I}|_{S_\\proetale}$ is injective in $\\textit{Ab}(S_\\proetale)$,\n\\end{enumerate}\n\\item If $\\mathcal{I}^\\bullet$ is a K-injective complex\nin $\\textit{Ab}((\\Sch/S)_\\proetale)$, then\n\\begin{enumerate}\n\\item $i_f^{-1}\\mathcal{I}^\\bullet$ is a K-injective complex in\n$\\textit{Ab}(T_\\proetale)$,\n\\item $\\mathcal{I}^\\bullet|_{S_\\proetale}$ is a K-injective complex in\n$\\textit{Ab}(S_\\proetale)$,\n\\end{enumerate}\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F67","source_file":"proetale.tex","source_line":3404,"source_end_line":3422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3404-L3422","statement_sha256":"4c8b020c04e3075a240804a6b60ea38717f77f5512d38eccfadab55c91313406","origin":"The Stacks Project","memory_eligible":false,"source_rank":10407,"rank":10407,"depth":8,"x":1439.8,"y":1193.677,"cluster":"tale-geometry"},{"id":"stacks:0F68","tag":"0F68","title":"Comparing big and small topoi · Lemma 0F68","summary":"Let f : T → S be a morphism of schemes. For K in D((Sch/T)_proetale) we have (Rf_big, *K)|_S_proetale = Rf_small, *(K|_T_proetale) in D(S_proetale). More generally, let S' ∈ Ob((Sch/S)_proetale) with structure morphism g : S' → S. Consider the fibre product xymatrix T' ar[r]_g' ar[d]_f' & T ar[d]^f S' ar[r]^g & S Then for K in D((Sch/T)_proetale) we have i_g^-1(Rf_big, *K) = Rf'_small, *(i_g'^-1K) in D(S'_proetale) and g_big^-1(Rf_big, *K) = Rf'_big, *((g'_big)^-1K) in…","statement_latex":"Let $f : T \\to S$ be a morphism of schemes.\nFor $K$ in $D((\\Sch/T)_\\proetale)$ we have\n$$\n(Rf_{big, *}K)|_{S_\\proetale} = Rf_{small, *}(K|_{T_\\proetale})\n$$\nin $D(S_\\proetale)$. More generally, let $S' \\in \\Ob((\\Sch/S)_\\proetale)$\nwith structure morphism $g : S' \\to S$. Consider the fibre product\n$$\n\\xymatrix{\nT' \\ar[r]_{g'} \\ar[d]_{f'} & T \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nThen for $K$ in $D((\\Sch/T)_\\proetale)$ we have\n$$\ni_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)\n$$\nin $D(S'_\\proetale)$ and\n$$\ng_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)\n$$\nin $D((\\Sch/S')_\\proetale)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F68","source_file":"proetale.tex","source_line":3442,"source_end_line":3466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3442-L3466","statement_sha256":"9302f82d4595008d4494f5c085b05bb647f9e7058311cbf56760e3eda098766c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10408,"rank":10408,"depth":10,"x":1640.256,"y":1020.785,"cluster":"tale-geometry"},{"id":"stacks:0F69","tag":"0F69","title":"Comparing big and small topoi · Lemma 0F69","summary":"Let f : T → S be a morphism of schemes. For K in D(S_proetale) we have H^n_proetale(S, π_S^-1K) = H^n(S_proetale, K) and H^n_proetale(T, π_S^-1K) = H^n(T_proetale, f_small^-1K). For M in D((Sch/S)_proetale) we have H^n_proetale(T, M) = H^n(T_proetale, i_f^-1M).","statement_latex":"Let $f : T \\to S$ be a morphism of schemes. For $K$ in $D(S_\\proetale)$\nwe have\n$$\nH^n_\\proetale(S, \\pi_S^{-1}K) = H^n(S_\\proetale, K)\n$$\nand\n$$\nH^n_\\proetale(T, \\pi_S^{-1}K) = H^n(T_\\proetale, f_{small}^{-1}K).\n$$\nFor $M$ in $D((\\Sch/S)_\\proetale)$ we have\n$$\nH^n_\\proetale(T, M) = H^n(T_\\proetale, i_f^{-1}M).\n$$","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F69","source_file":"proetale.tex","source_line":3488,"source_end_line":3503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3488-L3503","statement_sha256":"2b068fe56fa87e4012bbdc49519e88c4f3e9f807f8883cfe87de2d847f3c16a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10409,"rank":10409,"depth":9,"x":1631.472,"y":1262.182,"cluster":"tale-geometry"},{"id":"stacks:0F6A","tag":"0F6A","title":"Comparing big and small topoi · Lemma 0F6A","summary":"Let S be a scheme. For K ∈ D(S_proetale) the map K → Rπ_S, *π_S^-1K is an isomorphism.","statement_latex":"Let $S$ be a scheme. For $K \\in D(S_\\proetale)$ the map\n$$\nK \\longrightarrow R\\pi_{S, *}\\pi_S^{-1}K\n$$\nis an isomorphism.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6A","source_file":"proetale.tex","source_line":3515,"source_end_line":3522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3515-L3522","statement_sha256":"6d0d718e2632efc8fea5d08bdeddc2a8c6ce52bd93e618f3037598f6c0ed161a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10410,"rank":10410,"depth":0,"x":1443.697,"y":1079.07,"cluster":"tale-geometry"},{"id":"stacks:0992","tag":"0992","title":"Points of the pro-étale site · Lemma 0992","summary":"Let S be a scheme. The pro-étale sites Sch_proetale, S_proetale, (Sch/S)_proetale, S_affine, proetale, and (Aff/S)_proetale have enough points.","statement_latex":"Let $S$ be a scheme. The pro-\\'etale sites $\\Sch_\\proetale$,\n$S_\\proetale$, $(\\Sch/S)_\\proetale$, $S_{affine, \\proetale}$, and\n$(\\textit{Aff}/S)_\\proetale$ have enough points.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Points of the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0992","source_file":"proetale.tex","source_line":3557,"source_end_line":3562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3557-L3562","statement_sha256":"895c121f108a37805028995026c6167da2436c362f1bf999d22f4323105c1bbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10411,"rank":10411,"depth":9,"x":1729.608,"y":1107.565,"cluster":"tale-geometry"},{"id":"stacks:0F6B","tag":"0F6B","title":"Points of the pro-étale site · Lemma 0F6B","summary":"Let S be a scheme and let overlines : Spec(k) → S be a geometric point. The category of pro-étale neighbourhoods of overlines is cofiltered.","statement_latex":"Let $S$ be a scheme and let $\\overline{s} : \\Spec(k) \\to S$ be a\ngeometric point. The category of pro-\\'etale neighbourhoods of\n$\\overline{s}$ is cofiltered.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Points of the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6B","source_file":"proetale.tex","source_line":3590,"source_end_line":3595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3590-L3595","statement_sha256":"f3820003362c564a8d623f5d16f724d10a66ee2df86d76c6d4963effbc29ce6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10412,"rank":10412,"depth":48,"x":1495.708,"y":1248.882,"cluster":"tale-geometry"},{"id":"stacks:0F6C","tag":"0F6C","title":"Points of the pro-étale site · Lemma 0F6C","summary":"Let S be a scheme. Let overlines be a geometric point of S. Let U = (φ_i : S_i → S)_i∈ I be a pro-étale covering. Then there exist i ∈ I and geometric point overlines_i of S_i mapping to overlines.","statement_latex":"Let $S$ be a scheme. Let $\\overline{s}$ be a geometric point of $S$.\nLet $\\mathcal{U} = \\{\\varphi_i : S_i \\to S\\}_{i\\in I}$ be a\npro-\\'etale covering. Then there exist $i \\in I$ and geometric\npoint $\\overline{s}_i$ of $S_i$ mapping to $\\overline{s}$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Points of the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6C","source_file":"proetale.tex","source_line":3608,"source_end_line":3614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3608-L3614","statement_sha256":"2a0f2ea6c3ddbb6aea13c98fa81587222464060840ce647c32cb8e1df6d18b96","origin":"The Stacks Project","memory_eligible":false,"source_rank":10413,"rank":10413,"depth":49,"x":1554.579,"y":1011.796,"cluster":"tale-geometry"},{"id":"stacks:0993","tag":"0993","title":"Points of the pro-étale site · Lemma 0993","summary":"In the situation above the scheme Spec(O_S, overlines^sh) is an object of X_proetale and there is a canonical isomorphism F(Spec(O_S, overlines^sh)) = F_overlines functorial in F.","statement_latex":"In the situation above the scheme $\\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh})$\nis an object of $X_\\proetale$ and there is a canonical isomorphism\n$$\n\\mathcal{F}(\\Spec(\\mathcal{O}_{S, \\overline{s}}^{sh})) =\n\\mathcal{F}_{\\overline{s}}\n$$\nfunctorial in $\\mathcal{F}$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Points of the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0993","source_file":"proetale.tex","source_line":3649,"source_end_line":3658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3649-L3658","statement_sha256":"88f7d09460556da389de420fcd8df574496bb7496ff10c6342a9eb00b1112efc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10414,"rank":10414,"depth":49,"x":1701.926,"y":1220.166,"cluster":"tale-geometry"},{"id":"stacks:0GLZ","tag":"0GLZ","title":"Comparison with the étale site · Lemma 0GLZ","summary":"With notation as above. Let F be a sheaf on X_etale. The rule X_proetale → Sets, (f : Y → X) ↦ Γ(Y_etale, f_etale^-1F) is a sheaf and is equal to ε^-1F. Here f_etale : Y_etale → X_etale is the morphism of small étale sites constructed in Étale Cohomology, Section [Tag 04I0].","statement_latex":"With notation as above. Let $\\mathcal{F}$ be a sheaf on $X_\\etale$.\nThe rule\n$$\nX_\\proetale \\longrightarrow \\textit{Sets},\\quad\n(f : Y \\to X) \\longmapsto \\Gamma(Y_\\etale, f_\\etale^{-1}\\mathcal{F})\n$$\nis a sheaf and is equal to $\\epsilon^{-1}\\mathcal{F}$.\nHere $f_\\etale : Y_\\etale \\to X_\\etale$ is the morphism of\nsmall \\'etale sites constructed in\n\\'Etale Cohomology, Section \\ref{etale-cohomology-section-functoriality}.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLZ","source_file":"proetale.tex","source_line":3759,"source_end_line":3771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3759-L3771","statement_sha256":"ccbe25135c1e10cf9e6db5ebeb15f31d3b577ce0c58bed6901ac713cdf8afd9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10415,"rank":10415,"depth":54,"x":1425.522,"y":1150.078,"cluster":"tale-geometry"},{"id":"stacks:099T","tag":"099T","title":"Comparison with the étale site · Lemma 099T","summary":"Let X be a scheme. For every sheaf F on X_etale the adjunction map F → ε_*ε^-1F is an isomorphism, i.e., ε^-1F(U) = F(U) for U in X_etale.","statement_latex":"Let $X$ be a scheme. For every sheaf $\\mathcal{F}$ on $X_\\etale$\nthe adjunction map $\\mathcal{F} \\to \\epsilon_*\\epsilon^{-1}\\mathcal{F}$ is an\nisomorphism, i.e., $\\epsilon^{-1}\\mathcal{F}(U) = \\mathcal{F}(U)$\nfor $U$ in $X_\\etale$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099T","source_file":"proetale.tex","source_line":3827,"source_end_line":3833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3827-L3833","statement_sha256":"30bad93cf413680df70b30d790121bb2a72b7f96e87c4ae9b18c5e04319f8ad5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10416,"rank":10416,"depth":55,"x":1685.877,"y":1044.85,"cluster":"tale-geometry"},{"id":"stacks:099S","tag":"099S","title":"Comparison with the étale site · Lemma 099S","summary":"Let X be a scheme. Let Y = lim Y_i be the limit of a directed inverse system of quasi-compact and quasi-separated objects of X_proetale with affine transition morphisms. For any sheaf F on X_etale we have ε^-1F(Y) = colim ε^-1F(Y_i) Moreover, if Y_i is in X_etale we have ε^-1F(Y) = colim F(Y_i).","statement_latex":"Let $X$ be a scheme. Let $Y = \\lim Y_i$ be the limit of a directed inverse\nsystem of quasi-compact and quasi-separated objects of $X_\\proetale$\nwith affine transition morphisms. For any sheaf $\\mathcal{F}$ on $X_\\etale$\nwe have\n$$\n\\epsilon^{-1}\\mathcal{F}(Y) = \\colim \\epsilon^{-1}\\mathcal{F}(Y_i)\n$$\nMoreover, if $Y_i$ is in $X_\\etale$ we have\n$\\epsilon^{-1}\\mathcal{F}(Y) = \\colim \\mathcal{F}(Y_i)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099S","source_file":"proetale.tex","source_line":3840,"source_end_line":3851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3840-L3851","statement_sha256":"bc43f9b61fe1de7717e1982595c0be5389cadd35368f5996bc77f71431ee7a22","origin":"The Stacks Project","memory_eligible":false,"source_rank":10417,"rank":10417,"depth":56,"x":1578.444,"y":1270.327,"cluster":"tale-geometry"},{"id":"stacks:099U","tag":"099U","title":"Comparison with the étale site · Lemma 099U","summary":"Let X be an affine scheme. For injective abelian sheaf I on X_etale we have H^p(X_proetale, ε^-1I) = 0 for p > 0.","statement_latex":"Let $X$ be an affine scheme. For injective abelian sheaf $\\mathcal{I}$ on\n$X_\\etale$ we have $H^p(X_\\proetale, \\epsilon^{-1}\\mathcal{I}) = 0$\nfor $p > 0$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099U","source_file":"proetale.tex","source_line":3865,"source_end_line":3870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3865-L3870","statement_sha256":"831c051cac8bd57065cc2447595e6605d63e1f0f1364ae50d387b86e4fc53618","origin":"The Stacks Project","memory_eligible":false,"source_rank":10418,"rank":10418,"depth":57,"x":1476.272,"y":1042.95,"cluster":"tale-geometry"},{"id":"stacks:099V","tag":"099V","title":"Comparison with the étale site · Lemma 099V","summary":"Let X be a scheme. • For an abelian sheaf F on X_etale we have Rε_*(ε^-1F) = F. • For K ∈ D^+(X_etale) the map K → Rε_*ε^-1K is an isomorphism.","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item For an abelian sheaf $\\mathcal{F}$ on $X_\\etale$\nwe have $R\\epsilon_*(\\epsilon^{-1}\\mathcal{F}) = \\mathcal{F}$.\n\\item For $K \\in D^+(X_\\etale)$ the map $K \\to R\\epsilon_*\\epsilon^{-1}K$\nis an isomorphism.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099V","source_file":"proetale.tex","source_line":3952,"source_end_line":3961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3952-L3961","statement_sha256":"978d64d86f2b9d2ede5d6d92150bc80683d4f665b921eaca4550e091dfe19ad5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10419,"rank":10419,"depth":58,"x":1734.636,"y":1152.714,"cluster":"tale-geometry"},{"id":"stacks:099W","tag":"099W","title":"Comparison with the étale site · Lemma 099W","summary":"Let X be a scheme. • For an abelian sheaf F on X_etale we have H^i(X_etale, F) = H^i(X_proetale, ε^-1F) for all i. • For K ∈ D^+(X_etale) we have RΓ(X_etale, K) = RΓ(X_proetale, ε^-1K)","statement_latex":"Let $X$ be a scheme.\n\\begin{enumerate}\n\\item For an abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have\n$$\nH^i(X_\\etale, \\mathcal{F}) =\nH^i(X_\\proetale, \\epsilon^{-1}\\mathcal{F})\n$$\nfor all $i$.\n\\item For $K \\in D^+(X_\\etale)$ we have\n$$\nR\\Gamma(X_\\etale, K) = R\\Gamma(X_\\proetale, \\epsilon^{-1}K)\n$$\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099W","source_file":"proetale.tex","source_line":3985,"source_end_line":4000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L3985-L4000","statement_sha256":"ec2846ab8be1e997fc81bf6a23094d9a367c9f619f5835bcdfedb3918948b654","origin":"The Stacks Project","memory_eligible":false,"source_rank":10420,"rank":10420,"depth":59,"x":1455.666,"y":1218.422,"cluster":"tale-geometry"},{"id":"stacks:099X","tag":"099X","title":"Comparison with the étale site · Lemma 099X","summary":"Let X be a scheme. Let G be a sheaf of (possibly noncommutative) groups on X_etale. We have H^1(X_etale, G) = H^1(X_proetale, ε^-1G) where H^1 is defined as the set of isomorphism classes of torsors (see Cohomology on Sites, Section [Tag 03AG]).","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{G}$ be a sheaf of (possibly\nnoncommutative) groups on $X_\\etale$. We have\n$$\nH^1(X_\\etale, \\mathcal{G}) =\nH^1(X_\\proetale, \\epsilon^{-1}\\mathcal{G})\n$$\nwhere $H^1$ is defined as the set of isomorphism classes of\ntorsors (see\nCohomology on Sites, Section \\ref{sites-cohomology-section-h1-torsors}).","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099X","source_file":"proetale.tex","source_line":4008,"source_end_line":4019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4008-L4019","statement_sha256":"b324a362b1321995df480607a8aa155272d984addbb59805aa0515012ebf32ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":10421,"rank":10421,"depth":57,"x":1608.636,"y":1011.536,"cluster":"tale-geometry"},{"id":"stacks:09B1","tag":"09B1","title":"Comparison with the étale site · Lemma 09B1","summary":"Let X be a scheme. Let Lambda be a ring. • The essential image of the fully faithful functor ε^-1 : Mod(X_etale, Lambda) → Mod(X_proetale, Lambda) is a weak Serre subcategory C. • The functor ε^-1 defines an equivalence of categories of D^+(X_etale, Lambda) with D^+_C(X_proetale, Lambda) with quasi-inverse given by Rε_*.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a ring.\n\\begin{enumerate}\n\\item The essential image of the fully faithful functor\n$\\epsilon^{-1} : \\textit{Mod}(X_\\etale, \\Lambda) \\to\n\\textit{Mod}(X_\\proetale, \\Lambda)$\nis a weak Serre subcategory $\\mathcal{C}$.\n\\item The functor $\\epsilon^{-1}$ defines an equivalence of categories\nof $D^+(X_\\etale, \\Lambda)$ with $D^+_\\mathcal{C}(X_\\proetale, \\Lambda)$\nwith quasi-inverse given by $R\\epsilon_*$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09B1","source_file":"proetale.tex","source_line":4067,"source_end_line":4079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4067-L4079","statement_sha256":"9f9dc0cdbd58c90103a83e716105d0d53799456e1df0f5500d4a4fabcd6612e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10422,"rank":10422,"depth":59,"x":1662.246,"y":1251.05,"cluster":"tale-geometry"},{"id":"stacks:099Y","tag":"099Y","title":"Comparison with the étale site · Lemma 099Y","summary":"Let X be a scheme. Let Lambda be a ring. The functor ε^-1 defines an equivalence of categories ( locally constant sheaves of Lambda-modules on X_etale of finite presentation ) longleftrightarrow ( locally constant sheaves of Lambda-modules on X_proetale of finite presentation )","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a ring.\nThe functor $\\epsilon^{-1}$ defines an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{locally constant sheaves}\\\\\n\\text{of }\\Lambda\\text{-modules on }X_\\etale\\\\\n\\text{of finite presentation}\n\\end{matrix}\n\\right\\}\n\\longleftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{locally constant sheaves}\\\\\n\\text{of }\\Lambda\\text{-modules on }X_\\proetale\\\\\n\\text{of finite presentation}\n\\end{matrix}\n\\right\\}\n$$","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099Y","source_file":"proetale.tex","source_line":4120,"source_end_line":4141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4120-L4141","statement_sha256":"e09f8834c7cd0b63f0eb158a87ae04ea80448bb0e861c15cadc714ed06092445","origin":"The Stacks Project","memory_eligible":false,"source_rank":10423,"rank":10423,"depth":58,"x":1429.949,"y":1104.758,"cluster":"tale-geometry"},{"id":"stacks:099Z","tag":"099Z","title":"Comparison with the étale site · Lemma 099Z","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. Let D_flc(X_etale, Lambda), resp. D_flc(X_proetale, Lambda) be the full subcategory of D(X_etale, Lambda), resp. D(X_proetale, Lambda) consisting of those complexes whose cohomology sheaves are locally constant sheaves of Lambda-modules of finite type. Then ε^-1 : D_flc^+(X_etale, Lambda) → D_flc^+(X_proetale, Lambda) is an equivalence of categories.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nLet $D_{flc}(X_\\etale, \\Lambda)$, resp.\\ $D_{flc}(X_\\proetale, \\Lambda)$\nbe the full subcategory of\n$D(X_\\etale, \\Lambda)$, resp.\\ $D(X_\\proetale, \\Lambda)$\nconsisting of those complexes whose cohomology sheaves are locally\nconstant sheaves of $\\Lambda$-modules of finite type. Then\n$$\n\\epsilon^{-1} :\nD_{flc}^+(X_\\etale, \\Lambda)\n\\longrightarrow\nD_{flc}^+(X_\\proetale, \\Lambda)\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099Z","source_file":"proetale.tex","source_line":4173,"source_end_line":4188,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4173-L4188","statement_sha256":"64f8557212dc80cef981edb5eeb6457df38ce8d90aacb4f2153ef28facaf2d6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10424,"rank":10424,"depth":60,"x":1719.079,"y":1080.805,"cluster":"tale-geometry"},{"id":"stacks:09B2","tag":"09B2","title":"Comparison with the étale site · Lemma 09B2","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. Let K be an object of D(X_proetale, Lambda). Set K_n = K ⊗_Lambda^L underlineLambda/I^n. If K_1 is • in the essential image of ε^-1 :D(X_etale, Lambda/I) → D(X_proetale, Lambda/I), and • has tor amplitude in [a,∞) for some a ∈ Z, then (1) and (2) hold for K_n as an object of D(X_proetale, Lambda/I^n).","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nLet $K$ be an object of $D(X_\\proetale, \\Lambda)$.\nSet $K_n = K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$.\nIf $K_1$ is\n\\begin{enumerate}\n\\item in the essential image of\n$\\epsilon^{-1} :D(X_\\etale, \\Lambda/I) \\to D(X_\\proetale, \\Lambda/I)$, and\n\\item has tor amplitude in $[a,\\infty)$ for some $a \\in \\mathbf{Z}$,\n\\end{enumerate}\nthen (1) and (2) hold for $K_n$ as an object of $D(X_\\proetale, \\Lambda/I^n)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Comparison with the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09B2","source_file":"proetale.tex","source_line":4203,"source_end_line":4215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4203-L4215","statement_sha256":"a0f8594c453ed0ea02ffa44dce1cba2426f9ff18e45f731c7e02d961a5f5fb6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10425,"rank":10425,"depth":60,"x":1525.012,"y":1262.648,"cluster":"tale-geometry"},{"id":"stacks:099M","tag":"099M","title":"Derived completion in the constant Noetherian case · Lemma 099M","summary":"Let C be a site. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. The left adjoint to the inclusion functor D_comp(C, Lambda) → D(C, Lambda) of Algebraic and Formal Geometry, Proposition [Tag 099F] sends K to K^wedge = Rlim(K ⊗_Lambda^L underlineLambda/I^n) In particular, K is derived complete if and only if K = Rlim(K ⊗_Lambda^L underlineLambda/I^n).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\Lambda$ be a Noetherian ring\nand let $I \\subset \\Lambda$ be an ideal. The left adjoint\nto the inclusion functor\n$D_{comp}(\\mathcal{C}, \\Lambda) \\to D(\\mathcal{C}, \\Lambda)$\nof Algebraic and Formal Geometry, Proposition\n\\ref{algebraization-proposition-derived-completion} sends $K$ to\n$$\nK^\\wedge = R\\lim(K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n})\n$$\nIn particular, $K$ is derived complete if and only if\n$K = R\\lim(K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n})$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Derived completion in the constant Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099M","source_file":"proetale.tex","source_line":4304,"source_end_line":4317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4304-L4317","statement_sha256":"2ef6c97ccd8a1584c20f826f2e67214eff1c9bb69eed2eaafc0766a1b53f0646","origin":"The Stacks Project","memory_eligible":false,"source_rank":10426,"rank":10426,"depth":14,"x":1521.879,"y":1018.278,"cluster":"tale-geometry"},{"id":"stacks:099N","tag":"099N","title":"Derived completion in the constant Noetherian case · Lemma 099N","summary":"Let Lambda be a Noetherian ring. Let I ⊂ Lambda be an ideal. Let f : Sh(D) → Sh(C) be a morphism of topoi. Then • Rf_* sends D_comp(D, Lambda) into D_comp(C, Lambda), • the map Rf_* : D_comp(D, Lambda) → D_comp(C, Lambda) has a left adjoint Lf_comp^* : D_comp(C, Lambda) → D_comp(D, Lambda) which is Lf^* followed by derived completion, • Rf_* commutes with derived completion, • for K in D_comp(D, Lambda) we have Rf_*K = Rlim Rf_*(K ⊗^L_Lambda underlineLambda/I^n). • for M…","statement_latex":"Let $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal.\nLet $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.\nThen\n\\begin{enumerate}\n\\item $Rf_*$ sends $D_{comp}(\\mathcal{D}, \\Lambda)$\ninto $D_{comp}(\\mathcal{C}, \\Lambda)$,\n\\item the map $Rf_* : D_{comp}(\\mathcal{D}, \\Lambda) \\to\nD_{comp}(\\mathcal{C}, \\Lambda)$ has a left adjoint\n$Lf_{comp}^* : D_{comp}(\\mathcal{C}, \\Lambda) \\to\nD_{comp}(\\mathcal{D}, \\Lambda)$ which is $Lf^*$ followed by\nderived completion,\n\\item $Rf_*$ commutes with derived completion,\n\\item for $K$ in $D_{comp}(\\mathcal{D}, \\Lambda)$ we have\n$Rf_*K = R\\lim Rf_*(K \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n})$.\n\\item for $M$ in $D_{comp}(\\mathcal{C}, \\Lambda)$ we have\n$Lf^*_{comp}M =\nR\\lim Lf^*(M \\otimes^\\mathbf{L}_\\Lambda \\underline{\\Lambda/I^n})$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Derived completion in the constant Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099N","source_file":"proetale.tex","source_line":4335,"source_end_line":4355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4335-L4355","statement_sha256":"867e911685409fa78630b216d8b19e6d7400364335ded0f06f3085814cfce423","origin":"The Stacks Project","memory_eligible":false,"source_rank":10427,"rank":10427,"depth":21,"x":1720.837,"y":1196.814,"cluster":"tale-geometry"},{"id":"stacks:099Q","tag":"099Q","title":"Derived completion and weakly contractible objects · Proposition 099Q","summary":"Let C be a site. Assume C has enough weakly contractible objects. Let Lambda be a Noetherian ring. Let I ⊂ Lambda be an ideal. • The category of derived complete sheaves Lambda-modules is a weak Serre subcategory of Mod(C, Lambda). • A sheaf F of Lambda-modules satisfies F = lim F/I^nF if and only if F is derived complete and ⋂ I^nF = 0. • The sheaf underlineLambda^wedge is derived complete. • If … → F_3 → F_2 → F_1 is an inverse system of derived complete sheaves of…","statement_latex":"Let $\\mathcal{C}$ be a site. Assume $\\mathcal{C}$ has enough\nweakly contractible objects.\nLet $\\Lambda$ be a Noetherian ring. Let $I \\subset \\Lambda$ be an ideal.\n\\begin{enumerate}\n\\item The category of derived complete sheaves $\\Lambda$-modules is a\nweak Serre subcategory of $\\textit{Mod}(\\mathcal{C}, \\Lambda)$.\n\\item A sheaf $\\mathcal{F}$ of $\\Lambda$-modules satisfies\n$\\mathcal{F} = \\lim \\mathcal{F}/I^n\\mathcal{F}$ if and only if\n$\\mathcal{F}$ is derived complete and $\\bigcap I^n\\mathcal{F} = 0$.\n\\item The sheaf $\\underline{\\Lambda}^\\wedge$ is derived complete.\n\\item If $\\ldots \\to \\mathcal{F}_3 \\to \\mathcal{F}_2 \\to \\mathcal{F}_1$\nis an inverse system of derived complete sheaves of $\\Lambda$-modules,\nthen $\\lim \\mathcal{F}_n$ is derived complete.\n\\item An object $K \\in D(\\mathcal{C}, \\Lambda)$ is derived complete if\nand only if each cohomology sheaf $H^p(K)$ is derived complete.\n\\item An object $K \\in D_{comp}(\\mathcal{C}, \\Lambda)$ is bounded above\nif and only if $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$\nis bounded above.\n\\item An object $K \\in D_{comp}(\\mathcal{C}, \\Lambda)$ is bounded\nif $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ has finite\ntor dimension.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Derived completion and weakly contractible objects","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/099Q","source_file":"proetale.tex","source_line":4458,"source_end_line":4482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4458-L4482","statement_sha256":"88d000e80040e26f49f126c1c0deb676c6bcef073a0045df22024ec655256261","origin":"The Stacks Project","memory_eligible":false,"source_rank":10428,"rank":10428,"depth":15,"x":1430.362,"y":1178.046,"cluster":"tale-geometry"},{"id":"stacks:09B4","tag":"09B4","title":"Cohomology of a point · Lemma 09B4","summary":"Let k be a field. Let G = Gal(k^sep/k) be its absolute Galois group. Further, • let M be a profinite abelian group with a continuous G-action, or • let Lambda be a Noetherian ring and I ⊂ Lambda an ideal an let M be an I-adically complete Lambda-module with continuous G-action. Then there is a canonical sheaf underlineM^wedge on Spec(k)_proetale associated to M such that H^i(Spec(k), underlineM^wedge) = H^i_cont(G, M) as abelian groups or Lambda-modules.","statement_latex":"Let $k$ be a field. Let $G = \\text{Gal}(k^{sep}/k)$ be its absolute\nGalois group. Further, \n\\begin{enumerate}\n\\item let $M$ be a profinite abelian group with a continuous\n$G$-action, or\n\\item let $\\Lambda$ be a Noetherian ring and $I \\subset \\Lambda$ an ideal\nan let $M$ be an $I$-adically complete $\\Lambda$-module with continuous\n$G$-action.\n\\end{enumerate}\nThen there is a canonical sheaf $\\underline{M}^\\wedge$ on\n$\\Spec(k)_\\proetale$ associated to $M$ such that\n$$\nH^i(\\Spec(k), \\underline{M}^\\wedge) = H^i_{cont}(G, M)\n$$\nas abelian groups or $\\Lambda$-modules.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Cohomology of a point","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09B4","source_file":"proetale.tex","source_line":4633,"source_end_line":4650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4633-L4650","statement_sha256":"3781af96f7de33f2d2b85837c6ef92a035ec7fb866a98bf7076bc833674fb6dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10429,"rank":10429,"depth":47,"x":1659.799,"y":1026.962,"cluster":"tale-geometry"},{"id":"stacks:09A6","tag":"09A6","title":"Functoriality of the pro-étale site · Lemma 09A6","summary":"Let f : X → Y be a morphism of schemes which is quasi-compact and quasi-separated. • Let F be a sheaf of sets on X_etale. Then we have f_proetale, *ε^-1F = ε^-1f_etale, *F. • Let F be an abelian sheaf on X_etale. Then we have Rf_proetale, *ε^-1F = ε^-1Rf_etale, *F.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is quasi-compact and\nquasi-separated.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$. Then we have\n$f_{\\proetale, *}\\epsilon^{-1}\\mathcal{F} =\n\\epsilon^{-1}f_{\\etale, *}\\mathcal{F}$.\n\\item Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Then we have\n$Rf_{\\proetale, *}\\epsilon^{-1}\\mathcal{F} =\n\\epsilon^{-1}Rf_{\\etale, *}\\mathcal{F}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Functoriality of the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09A6","source_file":"proetale.tex","source_line":4695,"source_end_line":4707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4695-L4707","statement_sha256":"d327f1b44004b38de58adc406ae8f59b12ea49a37482f178b09bea866314cadb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10430,"rank":10430,"depth":57,"x":1612.081,"y":1268.717,"cluster":"tale-geometry"},{"id":"stacks:09A8","tag":"09A8","title":"Finite morphisms and pro-étale sites · Lemma 09A8","summary":"Let f : Z → X be a finite morphism of schemes which is locally of finite presentation. Then f_proetale, * : Ab(Z_proetale) → Ab(X_proetale) is exact.","statement_latex":"Let $f : Z \\to X$ be a finite morphism of schemes which is\nlocally of finite presentation. Then\n$f_{\\proetale, *} : \\textit{Ab}(Z_\\proetale) \\to \\textit{Ab}(X_\\proetale)$\nis exact.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Finite morphisms and pro-étale sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09A8","source_file":"proetale.tex","source_line":4763,"source_end_line":4769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4763-L4769","statement_sha256":"449b71d65f0235910a0213de4c9fa747a4fa2d20eef9d712cec3f44d2187ee82","origin":"The Stacks Project","memory_eligible":false,"source_rank":10431,"rank":10431,"depth":51,"x":1452.749,"y":1063.239,"cluster":"tale-geometry"},{"id":"stacks:09BK","tag":"09BK","title":"Closed immersions and pro-étale sites · Lemma 09BK","summary":"Let i : Z → X be a closed immersion morphism of affine schemes. Denote X_app and Z_app the sites introduced in Lemma [Tag 098X]. The base change functor u : X_app → Z_app, U ↦ u(U) = U ×_X Z is continuous and has a fully faithful left adjoint v. For V in Z_app the morphism V → v(V) is a closed immersion identifying V with u(v(V)) = v(V) ×_X Z and every point of v(V) specializes to a point of V. The functor v is cocontinuous and sends coverings to coverings.","statement_latex":"Let $i : Z \\to X$ be a closed immersion morphism of affine schemes.\nDenote $X_{app}$ and $Z_{app}$ the sites introduced in\nLemma \\ref{lemma-affine-alternative}.\nThe base change functor\n$$\nu : X_{app} \\to Z_{app},\\quad U \\longmapsto u(U) = U \\times_X Z\n$$\nis continuous and has a fully faithful left adjoint $v$.\nFor $V$ in $Z_{app}$ the morphism $V \\to v(V)$ is a closed immersion\nidentifying $V$ with $u(v(V)) = v(V) \\times_X Z$ and every point of\n$v(V)$ specializes to a point of $V$.\nThe functor $v$ is cocontinuous and sends coverings to coverings.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Closed immersions and pro-étale sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BK","source_file":"proetale.tex","source_line":4800,"source_end_line":4814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4800-L4814","statement_sha256":"9ce6585b0e2ffcdf6b8f0703f9ba0c64b43f4c99a49d06e676e2e5346f505b61","origin":"The Stacks Project","memory_eligible":false,"source_rank":10432,"rank":10432,"depth":50,"x":1735.665,"y":1124.387,"cluster":"tale-geometry"},{"id":"stacks:09BL","tag":"09BL","title":"Closed immersions and pro-étale sites · Lemma 09BL","summary":"Let Z → X be a closed immersion morphism of affine schemes. The corresponding morphism of topoi i = i_proetale is equal to the morphism of topoi associated to the fully faithful cocontinuous functor v : Z_app → X_app of Lemma [Tag 09BK]. It follows that • i^-1F is the sheaf associated to the presheaf V ↦ F(v(V)), • for a weakly contractible object V of Z_app we have i^-1F(V) = F(v(V)), • i^-1 : Sh(X_proetale) → Sh(Z_proetale) has a left adjoint i^Sh_!, • i^-1 :…","statement_latex":"Let $Z \\to X$ be a closed immersion morphism of affine schemes.\nThe corresponding morphism of topoi $i = i_\\proetale$\nis equal to the morphism of topoi\nassociated to the fully faithful cocontinuous functor\n$v : Z_{app} \\to X_{app}$ of Lemma \\ref{lemma-closed-immersion-affines}.\nIt follows that\n\\begin{enumerate}\n\\item $i^{-1}\\mathcal{F}$ is the sheaf associated to the presheaf\n$V \\mapsto \\mathcal{F}(v(V))$,\n\\item for a weakly contractible object $V$ of $Z_{app}$ we have\n$i^{-1}\\mathcal{F}(V) = \\mathcal{F}(v(V))$,\n\\item $i^{-1} : \\Sh(X_\\proetale) \\to \\Sh(Z_\\proetale)$\nhas a left adjoint $i^{Sh}_!$,\n\\item $i^{-1} : \\textit{Ab}(X_\\proetale) \\to \\textit{Ab}(Z_\\proetale)$\nhas a left adjoint $i_!$,\n\\item $\\text{id} \\to i^{-1}i^{Sh}_!$, $\\text{id} \\to i^{-1}i_!$, and\n$i^{-1}i_* \\to \\text{id}$ are isomorphisms, and\n\\item $i_*$, $i^{Sh}_!$ and $i_!$ are fully faithful.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Closed immersions and pro-étale sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BL","source_file":"proetale.tex","source_line":4851,"source_end_line":4872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4851-L4872","statement_sha256":"cbf3713300f9278608ecb0c2592adf82cb216e69bc208a13c3e6ab86f88bdf65","origin":"The Stacks Project","memory_eligible":false,"source_rank":10433,"rank":10433,"depth":54,"x":1477.703,"y":1239.907,"cluster":"tale-geometry"},{"id":"stacks:09AA","tag":"09AA","title":"Closed immersions and pro-étale sites · Lemma 09AA","summary":"Let i : Z → X be a closed immersion of schemes. Then • i_proetale^-1 commutes with limits, • i_proetale, * is fully faithful, and • i_proetale^-1i_proetale, * ≅ id_Sh(Z_proetale).","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes. Then\n\\begin{enumerate}\n\\item $i_\\proetale^{-1}$ commutes with limits,\n\\item $i_{\\proetale, *}$ is fully faithful, and\n\\item $i_\\proetale^{-1}i_{\\proetale, *} \\cong \\text{id}_{\\Sh(Z_\\proetale)}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Closed immersions and pro-étale sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AA","source_file":"proetale.tex","source_line":4944,"source_end_line":4952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4944-L4952","statement_sha256":"ebb05a1e4008b8e4570eda004805256c989ad76a89bc61b334d450e651d8568d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10434,"rank":10434,"depth":55,"x":1575.087,"y":1008.198,"cluster":"tale-geometry"},{"id":"stacks:09AB","tag":"09AB","title":"Closed immersions and pro-étale sites · Lemma 09AB","summary":"Let i : Z → X be an integral universally injective and surjective morphism of schemes. Then i_proetale, * and i_proetale^-1 are quasi-inverse equivalences of categories of pro-étale topoi.","statement_latex":"Let $i : Z \\to X$ be an integral universally injective and surjective morphism\nof schemes. Then\n$i_{\\proetale, *}$ and $i_\\proetale^{-1}$ are quasi-inverse\nequivalences of categories of pro-\\'etale topoi.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Closed immersions and pro-étale sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AB","source_file":"proetale.tex","source_line":4962,"source_end_line":4968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4962-L4968","statement_sha256":"20a200c8a6f8b1d3533cfc434de922fdb5f838a6e95ff7799ca3a639848cb7c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10435,"rank":10435,"depth":50,"x":1689.687,"y":1234.463,"cluster":"tale-geometry"},{"id":"stacks:09AC","tag":"09AC","title":"Closed immersions and pro-étale sites · Lemma 09AC","summary":"Let i : Z → X be a closed immersion of schemes. Let U → X be an object of X_proetale such that • U is affine and weakly contractible, and • every point of U specializes to a point of U ×_X Z. Then i_proetale^-1F(U ×_X Z) = F(U) for all abelian sheaves on X_proetale.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nLet $U \\to X$ be an object of $X_\\proetale$ such that\n\\begin{enumerate}\n\\item $U$ is affine and weakly contractible, and\n\\item every point of $U$ specializes to a point of $U \\times_X Z$.\n\\end{enumerate}\nThen $i_\\proetale^{-1}\\mathcal{F}(U \\times_X Z) = \\mathcal{F}(U)$\nfor all abelian sheaves on $X_\\proetale$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Closed immersions and pro-étale sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AC","source_file":"proetale.tex","source_line":4987,"source_end_line":4997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L4987-L4997","statement_sha256":"87c4e7d47d7c2605a9e4d9ccb3baa60e330252aa5007b870e104b7a38496ccaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":10436,"rank":10436,"depth":55,"x":1423.051,"y":1132.578,"cluster":"tale-geometry"},{"id":"stacks:09BM","tag":"09BM","title":"Closed immersions and pro-étale sites · Lemma 09BM","summary":"Let i : Z → X be a closed immersion of schemes. If X setminus i(Z) is a retrocompact open of X, then i_proetale, * is exact.","statement_latex":"Let $i : Z \\to X$ be a closed immersion of schemes.\nIf $X \\setminus i(Z)$ is a retrocompact open of $X$, then\n$i_{\\proetale, *}$ is exact.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Closed immersions and pro-étale sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BM","source_file":"proetale.tex","source_line":5019,"source_end_line":5024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5019-L5024","statement_sha256":"84d615f6f09191a99132720325c1097509aa7481219ae7569ae8c7f914fe6f93","origin":"The Stacks Project","memory_eligible":false,"source_rank":10437,"rank":10437,"depth":52,"x":1701.78,"y":1056.361,"cluster":"tale-geometry"},{"id":"stacks:09AE","tag":"09AE","title":"Extension by zero · Definition 09AE","summary":"Let j : U → X be a weakly étale morphism of schemes. • The restriction functor j^-1 : Sh(X_proetale) → Sh(U_proetale) has a left adjoint j_!^Sh : Sh(X_proetale) → Sh(U_proetale). • The restriction functor j^-1 : Ab(X_proetale) → Ab(U_proetale) has a left adjoint which is denoted j_! : Ab(U_proetale) → Ab(X_proetale) and called extension by zero. • Let Lambda be a ring. The functor j^-1 : Mod(X_proetale, Lambda) → Mod(U_proetale, Lambda) has a left adjoint j_! :…","statement_latex":"Let $j : U \\to X$ be a weakly \\'etale morphism of schemes.\n\\begin{enumerate}\n\\item The restriction functor\n$j^{-1} : \\Sh(X_\\proetale) \\to \\Sh(U_\\proetale)$\nhas a left adjoint\n$j_!^{Sh} : \\Sh(X_\\proetale) \\to \\Sh(U_\\proetale)$.\n\\item The restriction functor\n$j^{-1} : \\textit{Ab}(X_\\proetale) \\to \\textit{Ab}(U_\\proetale)$\nhas a left adjoint which is denoted\n$j_! : \\textit{Ab}(U_\\proetale) \\to \\textit{Ab}(X_\\proetale)$\nand called {\\it extension by zero}.\n\\item Let $\\Lambda$ be a ring. The functor\n$j^{-1} : \\textit{Mod}(X_\\proetale, \\Lambda) \\to\n\\textit{Mod}(U_\\proetale, \\Lambda)$\nhas a left adjoint\n$j_! : \\textit{Mod}(U_\\proetale, \\Lambda) \\to\n\\textit{Mod}(X_\\proetale, \\Lambda)$\nand called {\\it extension by zero}.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Extension by zero","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AE","source_file":"proetale.tex","source_line":5049,"source_end_line":5070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5049-L5070","statement_sha256":"02aa6b80fdbcf4489cb01786e7e3b68c356cdac814f862f4b4ed4accd2fdb95e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10438,"rank":10438,"depth":0,"x":1557.446,"y":1270.861,"cluster":"tale-geometry"},{"id":"stacks:09AF","tag":"09AF","title":"Extension by zero · Lemma 09AF","summary":"Let j : U → X be an étale morphism of schemes. Let G be an abelian sheaf on U_etale. Then ε^-1 j_!G = j_!ε^-1G as sheaves on X_proetale.","statement_latex":"Let $j : U \\to X$ be an \\'etale morphism of schemes.\nLet $\\mathcal{G}$ be an abelian sheaf on $U_\\etale$.\nThen $\\epsilon^{-1} j_!\\mathcal{G} = j_!\\epsilon^{-1}\\mathcal{G}$\nas sheaves on $X_\\proetale$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AF","source_file":"proetale.tex","source_line":5075,"source_end_line":5081,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5075-L5081","statement_sha256":"2214835eff2661bde4d3779c2609060ec112c94b81738e21dc9e5f28775b2871","origin":"The Stacks Project","memory_eligible":false,"source_rank":10439,"rank":10439,"depth":0,"x":1491.338,"y":1030.636,"cluster":"tale-geometry"},{"id":"stacks:09AG","tag":"09AG","title":"Extension by zero · Lemma 09AG","summary":"Let j : U → X be a weakly étale morphism of schemes. Let i : Z → X be a closed immersion such that U ×_X Z = ∅. Let V → X be an affine object of X_proetale such that every point of V specializes to a point of V_Z = Z ×_X V. Then j_!F(V) = 0 for all abelian sheaves on U_proetale.","statement_latex":"Let $j : U \\to X$ be a weakly \\'etale morphism of schemes.\nLet $i : Z \\to X$ be a closed immersion such that $U \\times_X Z = \\emptyset$.\nLet $V \\to X$ be an affine object of $X_\\proetale$ such that every point\nof $V$ specializes to a point of $V_Z = Z \\times_X V$.\nThen $j_!\\mathcal{F}(V) = 0$ for all abelian sheaves on $U_\\proetale$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AG","source_file":"proetale.tex","source_line":5089,"source_end_line":5096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5089-L5096","statement_sha256":"e35553c5d7cbf4e9212bd8acc358dbccf3ad869ac7906c3bed07232c2defd671","origin":"The Stacks Project","memory_eligible":false,"source_rank":10440,"rank":10440,"depth":1,"x":1733.425,"y":1170.354,"cluster":"tale-geometry"},{"id":"stacks:09BN","tag":"09BN","title":"Extension by zero · Lemma 09BN","summary":"Let j : U → X be an open immersion of schemes. Then id ≅ j^-1j_! and j^-1j_* ≅ id and the functors j_! and j_* are fully faithful.","statement_latex":"Let $j : U \\to X$ be an open immersion of schemes.\nThen $\\text{id} \\cong j^{-1}j_!$ and $j^{-1}j_* \\cong \\text{id}$\nand the functors $j_!$ and $j_*$ are fully faithful.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BN","source_file":"proetale.tex","source_line":5116,"source_end_line":5121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5116-L5121","statement_sha256":"9be7ef112a89716f70dee2c342bd7bffba04d7a48e6efeda54d5a14b477690d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10441,"rank":10441,"depth":9,"x":1442.367,"y":1204.717,"cluster":"tale-geometry"},{"id":"stacks:09AH","tag":"09AH","title":"Extension by zero · Lemma 09AH","summary":"Let X be a scheme. Let Z ⊂ X be a closed subscheme and let U ⊂ X be the complement. Denote i : Z → X and j : U → X the inclusion morphisms. Assume that j is a quasi-compact morphism. For every abelian sheaf on X_proetale there is a canonical short exact sequence 0 → j_!j^-1F → F → i_*i^-1F → 0 on X_proetale where all the functors are for the pro-étale topology.","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme and let\n$U \\subset X$ be the complement. Denote $i : Z \\to X$ and $j : U \\to X$\nthe inclusion morphisms. Assume that $j$ is a quasi-compact morphism.\nFor every abelian sheaf on $X_\\proetale$ there is a canonical short exact\nsequence\n$$\n0 \\to j_!j^{-1}\\mathcal{F} \\to \\mathcal{F} \\to i_*i^{-1}\\mathcal{F} \\to 0\n$$\non $X_\\proetale$ where all the functors are for the pro-\\'etale topology.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AH","source_file":"proetale.tex","source_line":5134,"source_end_line":5145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5134-L5145","statement_sha256":"3f0c8a260a350ee668752bedfb55dcae69c9a765cfb0a649843406f1e08cd2f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10442,"rank":10442,"depth":56,"x":1629.477,"y":1014.1,"cluster":"tale-geometry"},{"id":"stacks:09BP","tag":"09BP","title":"Extension by zero · Lemma 09BP","summary":"Let j : U → X be a quasi-compact open immersion morphism of schemes. The functor j_! : Ab(U_proetale) → Ab(X_proetale) commutes with limits.","statement_latex":"Let $j : U \\to X$ be a quasi-compact open immersion\nmorphism of schemes. The functor\n$j_! : \\textit{Ab}(U_\\proetale) \\to \\textit{Ab}(X_\\proetale)$\ncommutes with limits.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Extension by zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BP","source_file":"proetale.tex","source_line":5184,"source_end_line":5190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5184-L5190","statement_sha256":"0d2743c4c63856c60c12882dca7c211751949f104f2ed0e539194321f5b26139","origin":"The Stacks Project","memory_eligible":false,"source_rank":10443,"rank":10443,"depth":57,"x":1644.803,"y":1260.99,"cluster":"tale-geometry"},{"id":"stacks:09AJ","tag":"09AJ","title":"Constructible sheaves on the pro-étale site · Definition 09AJ","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. A sheaf of Lambda-modules on X_proetale is constructible if for every affine open U ⊂ X there exists a finite decomposition of U into constructible locally closed subschemes U = coprod_i U_i such that F|_U_i is of finite type and locally constant for all i.","statement_latex":"Let $X$ be a scheme.\nLet $\\Lambda$ be a Noetherian ring. A sheaf of $\\Lambda$-modules\non $X_\\proetale$ is {\\it constructible} if for every affine open\n$U \\subset X$ there exists a finite decomposition\nof $U$ into constructible locally closed subschemes\n$U = \\coprod_i U_i$ such that\n$\\mathcal{F}|_{U_i}$ is of finite type and locally constant for all $i$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible sheaves on the pro-étale site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AJ","source_file":"proetale.tex","source_line":5224,"source_end_line":5233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5224-L5233","statement_sha256":"6d5261997f65103103676bb5568af074b092f9fbce39346ef2b482f51d2fa029","origin":"The Stacks Project","memory_eligible":false,"source_rank":10444,"rank":10444,"depth":0,"x":1434.824,"y":1087.521,"cluster":"tale-geometry"},{"id":"stacks:09AK","tag":"09AK","title":"Constructible sheaves on the pro-étale site · Lemma 09AK","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. The functor ε^-1 defines an equivalence of categories ( constructible sheaves of Lambda-modules on X_etale ) longleftrightarrow ( constructible sheaves of Lambda-modules on X_proetale ) between constructible sheaves of Lambda-modules on X_etale and constructible sheaves of Lambda-modules on X_proetale.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nThe functor $\\epsilon^{-1}$ defines an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{constructible sheaves of}\\\\\n\\Lambda\\text{-modules on }X_\\etale\\\\\n\\end{matrix}\n\\right\\}\n\\longleftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{constructible sheaves of}\\\\\n\\Lambda\\text{-modules on }X_\\proetale\\\\\n\\end{matrix}\n\\right\\}\n$$\nbetween constructible sheaves of $\\Lambda$-modules on $X_\\etale$\nand constructible sheaves of $\\Lambda$-modules on $X_\\proetale$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible sheaves on the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AK","source_file":"proetale.tex","source_line":5238,"source_end_line":5259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5238-L5259","statement_sha256":"cdb42c4dc6b7a32bec351c8435641eaa0addecb7a2aec403552d0462b301d54a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10445,"rank":10445,"depth":59,"x":1729.35,"y":1096.293,"cluster":"tale-geometry"},{"id":"stacks:09B5","tag":"09B5","title":"Constructible sheaves on the pro-étale site · Lemma 09B5","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. The category of constructible sheaves of Lambda-modules on X_proetale is a weak Serre subcategory of Mod(X_proetale, Lambda).","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nThe category of constructible sheaves of $\\Lambda$-modules on $X_\\proetale$\nis a weak Serre subcategory of $\\textit{Mod}(X_\\proetale, \\Lambda)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible sheaves on the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09B5","source_file":"proetale.tex","source_line":5342,"source_end_line":5347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5342-L5347","statement_sha256":"b2517c2ff79bb9020192a70a64b7346ff6e027827396096cc2bd7dbcb6a3e573","origin":"The Stacks Project","memory_eligible":false,"source_rank":10446,"rank":10446,"depth":60,"x":1504.972,"y":1257.049,"cluster":"tale-geometry"},{"id":"stacks:09AL","tag":"09AL","title":"Constructible sheaves on the pro-étale site · Lemma 09AL","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. Let D_c(X_etale, Lambda), resp. D_c(X_proetale, Lambda) be the full subcategory of D(X_etale, Lambda), resp. D(X_proetale, Lambda) consisting of those complexes whose cohomology sheaves are constructible sheaves of Lambda-modules. Then ε^-1 : D_c^+(X_etale, Lambda) → D_c^+(X_proetale, Lambda) is an equivalence of categories.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nLet $D_c(X_\\etale, \\Lambda)$, resp.\\ $D_c(X_\\proetale, \\Lambda)$\nbe the full subcategory of\n$D(X_\\etale, \\Lambda)$, resp.\\ $D(X_\\proetale, \\Lambda)$\nconsisting of those complexes whose cohomology sheaves are\nconstructible sheaves of $\\Lambda$-modules. Then\n$$\n\\epsilon^{-1} :\nD_c^+(X_\\etale, \\Lambda)\n\\longrightarrow\nD_c^+(X_\\proetale, \\Lambda)\n$$\nis an equivalence of categories.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible sheaves on the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AL","source_file":"proetale.tex","source_line":5356,"source_end_line":5371,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5356-L5371","statement_sha256":"b9506143f17aa4f2009b8f33d6828405fc312d2430ed6960bd5c733794231055","origin":"The Stacks Project","memory_eligible":false,"source_rank":10447,"rank":10447,"depth":61,"x":1541.172,"y":1011.033,"cluster":"tale-geometry"},{"id":"stacks:09BQ","tag":"09BQ","title":"Constructible sheaves on the pro-étale site · Lemma 09BQ","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. Let K, L ∈ D_c^-(X_proetale, Lambda). Then K ⊗_Lambda^L L is in D_c^-(X_proetale, Lambda).","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nLet $K, L \\in D_c^-(X_\\proetale, \\Lambda)$. Then\n$K \\otimes_\\Lambda^\\mathbf{L} L$ is in $D_c^-(X_\\proetale, \\Lambda)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible sheaves on the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BQ","source_file":"proetale.tex","source_line":5387,"source_end_line":5392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5387-L5392","statement_sha256":"9ac7abd6fa39bf91ed8ddbedbabafdb69b395f709756e5003da5fe96b7fdda09","origin":"The Stacks Project","memory_eligible":false,"source_rank":10448,"rank":10448,"depth":62,"x":1712.428,"y":1213.113,"cluster":"tale-geometry"},{"id":"stacks:09BR","tag":"09BR","title":"Constructible sheaves on the pro-étale site · Lemma 09BR","summary":"Let X be a scheme. Let Lambda be a Noetherian ring. Let I ⊂ Lambda be an ideal. Let K be an object of D(X_proetale, Lambda). Set K_n = K ⊗_Lambda^L underlineLambda/I^n. If K_1 is in D^-_c(X_proetale, Lambda/I), then K_n is in D^-_c(X_proetale, Lambda/I^n) for all n.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring.\nLet $I \\subset \\Lambda$ be an ideal.\nLet $K$ be an object of $D(X_\\proetale, \\Lambda)$.\nSet $K_n = K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$.\nIf $K_1$ is in $D^-_c(X_\\proetale, \\Lambda/I)$, then\n$K_n$ is in $D^-_c(X_\\proetale, \\Lambda/I^n)$ for all $n$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible sheaves on the pro-étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BR","source_file":"proetale.tex","source_line":5403,"source_end_line":5411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5403-L5411","statement_sha256":"4da61c9f91bc0642a6a09f3c4bcb4c19b5f15dd4962ca3c276f26c08656a6655","origin":"The Stacks Project","memory_eligible":false,"source_rank":10449,"rank":10449,"depth":63,"x":1423.453,"y":1161.244,"cluster":"tale-geometry"},{"id":"stacks:09BT","tag":"09BT","title":"Constructible adic sheaves · Definition 09BT","summary":"Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. Let X be a scheme. Let F be a sheaf of Lambda-modules on X_proetale. • We say F is a constructible Lambda-sheaf if F = lim F/I^nF and each F/I^nF is a constructible sheaf of Lambda/I^n-modules. • If F is a constructible Lambda-sheaf, then we say F is lisse if each F/I^nF is locally constant. • We say F is adic lisse if there exists a I-adically complete Lambda-module M with M/IM finite such that F is locally…","statement_latex":"Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal.\nLet $X$ be a scheme. Let $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules\non $X_\\proetale$.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ is a {\\it constructible $\\Lambda$-sheaf}\nif $\\mathcal{F} = \\lim \\mathcal{F}/I^n\\mathcal{F}$ and each\n$\\mathcal{F}/I^n\\mathcal{F}$ is a constructible sheaf of $\\Lambda/I^n$-modules.\n\\item If $\\mathcal{F}$ is a constructible $\\Lambda$-sheaf, then we say\n$\\mathcal{F}$ is {\\it lisse} if each $\\mathcal{F}/I^n\\mathcal{F}$ is\nlocally constant.\n\\item We say $\\mathcal{F}$ is {\\it adic lisse}\\footnote{This may\nbe nonstandard notation.} if there exists a\n$I$-adically complete $\\Lambda$-module $M$ with $M/IM$ finite\nsuch that $\\mathcal{F}$ is locally isomorphic to\n$$\n\\underline{M}^\\wedge = \\lim \\underline{M/I^nM}.\n$$\n\\item We say $\\mathcal{F}$ is\n{\\it adic constructible}\\footnote{This may be nonstandard notation.}\nif for every affine open $U \\subset X$\nthere exists a decomposition $U = \\coprod U_i$ into\nconstructible locally closed subschemes such that $\\mathcal{F}|_{U_i}$\nis adic lisse.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible adic sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BT","source_file":"proetale.tex","source_line":5447,"source_end_line":5473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5447-L5473","statement_sha256":"ef305328cf4cf20ae99f9ae71c348c877865c050df71c29a70a0a3fbc4588b6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10450,"rank":10450,"depth":0,"x":1678.415,"y":1035.439,"cluster":"tale-geometry"},{"id":"stacks:09BU","tag":"09BU","title":"Constructible adic sheaves · Lemma 09BU","summary":"Let X be a Noetherian scheme. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. Let F be a constructible Lambda-sheaf on X_proetale. Then there exists a finite partition X = coprod X_i by locally closed subschemes such that the restriction F|_X_i is lisse.","statement_latex":"Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring and\nlet $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a\nconstructible $\\Lambda$-sheaf on $X_\\proetale$.\nThen there exists a finite partition $X = \\coprod X_i$ by\nlocally closed subschemes such that the restriction $\\mathcal{F}|_{X_i}$\nis lisse.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible adic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BU","source_file":"proetale.tex","source_line":5514,"source_end_line":5522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5514-L5522","statement_sha256":"d03e765ec982514ef90d96c6d2bc1e487a174aba2ef6d0fb89a50c3341fd5674","origin":"The Stacks Project","memory_eligible":false,"source_rank":10451,"rank":10451,"depth":60,"x":1591.523,"y":1273.03,"cluster":"tale-geometry"},{"id":"stacks:09BV","tag":"09BV","title":"Constructible adic sheaves · Lemma 09BV","summary":"Let X be a weakly contractible affine scheme. Let Lambda be a Noetherian ring and I ⊂ Lambda be an ideal. Let F be a sheaf of Lambda-modules on X_proetale such that • F = lim F/I^nF, • F/I^nF is a constant sheaf of Lambda/I^n-modules, • F/IF is of finite type. Then F ≅ underlineM^wedge where M is a finite Lambda^wedge-module.","statement_latex":"Let $X$ be a weakly contractible affine scheme. Let $\\Lambda$ be a Noetherian\nring and $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a sheaf of\n$\\Lambda$-modules on $X_\\proetale$ such that\n\\begin{enumerate}\n\\item $\\mathcal{F} = \\lim \\mathcal{F}/I^n\\mathcal{F}$,\n\\item $\\mathcal{F}/I^n\\mathcal{F}$ is a constant sheaf of\n$\\Lambda/I^n$-modules,\n\\item $\\mathcal{F}/I\\mathcal{F}$ is of finite type.\n\\end{enumerate}\nThen $\\mathcal{F} \\cong \\underline{M}^\\wedge$ where $M$ is\na finite $\\Lambda^\\wedge$-module.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible adic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BV","source_file":"proetale.tex","source_line":5565,"source_end_line":5578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5565-L5578","statement_sha256":"c96f06bc3a40b5b114b897ae4bae109eb2af66e758c050f08f48679a217f47c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10452,"rank":10452,"depth":7,"x":1464.45,"y":1048.385,"cluster":"tale-geometry"},{"id":"stacks:09BW","tag":"09BW","title":"Constructible adic sheaves · Lemma 09BW","summary":"Let X be a connected scheme. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. If F is a lisse constructible Lambda-sheaf on X_proetale, then F is adic lisse.","statement_latex":"Let $X$ be a connected scheme. Let $\\Lambda$ be a Noetherian ring and let\n$I \\subset \\Lambda$ be an ideal. If $\\mathcal{F}$ is a lisse\nconstructible $\\Lambda$-sheaf on $X_\\proetale$, then $\\mathcal{F}$\nis adic lisse.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible adic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BW","source_file":"proetale.tex","source_line":5628,"source_end_line":5634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5628-L5634","statement_sha256":"f9af99307a57cef150981591a8669f56b8ce18dd3bf9f4385fa532838d5bddc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10453,"rank":10453,"depth":59,"x":1738.981,"y":1141.991,"cluster":"tale-geometry"},{"id":"stacks:09BX","tag":"09BX","title":"Constructible adic sheaves · Lemma 09BX","summary":"Let X be a Noetherian scheme. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. Let F be a constructible Lambda-sheaf on X_proetale. Then F is adic constructible.","statement_latex":"Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring and\nlet $I \\subset \\Lambda$ be an ideal. Let $\\mathcal{F}$ be a\nconstructible $\\Lambda$-sheaf on $X_\\proetale$. Then $\\mathcal{F}$\nis adic constructible.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible adic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BX","source_file":"proetale.tex","source_line":5656,"source_end_line":5662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5656-L5662","statement_sha256":"1432b07e60c21ba1c7ef6c40b57e39f4fd38a4ab22920b2e5c92675d090b9cfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10454,"rank":10454,"depth":61,"x":1461.092,"y":1228.799,"cluster":"tale-geometry"},{"id":"stacks:09BY","tag":"09BY","title":"Constructible adic sheaves · Lemma 09BY","summary":"Let X be a scheme. Let Lambda be a ring and let I ⊂ Lambda be a finitely generated ideal. Let F be a sheaf of Lambda-modules on X_proetale. If F is derived complete and F/IF = 0, then F = 0.","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a ring and let\n$I \\subset \\Lambda$ be a finitely generated ideal.\nLet $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules on $X_\\proetale$.\nIf $\\mathcal{F}$ is derived complete and $\\mathcal{F}/I\\mathcal{F} = 0$,\nthen $\\mathcal{F} = 0$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible adic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BY","source_file":"proetale.tex","source_line":5681,"source_end_line":5688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5681-L5688","statement_sha256":"162972992349c79e34879bdc5d059639f2938dbb788d3ad943d96ea03ebd4098","origin":"The Stacks Project","memory_eligible":false,"source_rank":10455,"rank":10455,"depth":16,"x":1596.282,"y":1006.965,"cluster":"tale-geometry"},{"id":"stacks:09BZ","tag":"09BZ","title":"Constructible adic sheaves · Lemma 09BZ","summary":"Let X be a weakly contractible affine scheme. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. Let F be a derived complete sheaf of Lambda-modules on X_proetale with F/IF a locally constant sheaf of Lambda/I-modules of finite type. Then there exists an integer t and a surjective map (underlineLambda^wedge)^⊕ t → F","statement_latex":"Let $X$ be a weakly contractible affine scheme.\nLet $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal.\nLet $\\mathcal{F}$ be a derived complete sheaf of $\\Lambda$-modules\non $X_\\proetale$ with $\\mathcal{F}/I\\mathcal{F}$ a locally\nconstant sheaf of $\\Lambda/I$-modules of finite type.\nThen there exists an integer $t$ and a surjective map\n$$\n(\\underline{\\Lambda}^\\wedge)^{\\oplus t} \\to \\mathcal{F}\n$$","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Constructible adic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09BZ","source_file":"proetale.tex","source_line":5707,"source_end_line":5718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5707-L5718","statement_sha256":"fff6c991da7aa9efff5297f31a9bd8cf04bbc9a59fb326aa252d2383393787f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10456,"rank":10456,"depth":17,"x":1675.038,"y":1247.405,"cluster":"tale-geometry"},{"id":"stacks:09C1","tag":"09C1","title":"A suitable derived category · Definition 09C1","summary":"Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. Let X be a scheme. An object K of D(X_proetale, Lambda) is called constructible if • K is derived complete with respect to I, • K ⊗_Lambda^L underlineLambda/I has constructible cohomology sheaves and locally has finite tor dimension. We denote D_cons(X, Lambda) the full subcategory of constructible K in D(X_proetale, Lambda).","statement_latex":"Let $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal.\nLet $X$ be a scheme. An object $K$ of $D(X_\\proetale, \\Lambda)$ is called\n{\\it constructible} if\n\\begin{enumerate}\n\\item $K$ is derived complete with respect to $I$,\n\\item $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$\nhas constructible cohomology sheaves and locally has finite tor dimension.\n\\end{enumerate}\nWe denote $D_{cons}(X, \\Lambda)$ the full subcategory of constructible\n$K$ in $D(X_\\proetale, \\Lambda)$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"A suitable derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09C1","source_file":"proetale.tex","source_line":5767,"source_end_line":5779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5767-L5779","statement_sha256":"c5bc006829bc36acae264d563d1b5621314e0284bf3d4c77437097e94fbfdf71","origin":"The Stacks Project","memory_eligible":false,"source_rank":10457,"rank":10457,"depth":0,"x":1423.448,"y":1114.712,"cluster":"tale-geometry"},{"id":"stacks:09C2","tag":"09C2","title":"A suitable derived category · Lemma 09C2","summary":"In the situation above suppose K is in D_cons(X, Lambda) and X is quasi-compact. Set K_n = K ⊗_Lambda^L underlineLambda/I^n. There exist a, b such that • K = Rlim K_n and H^i(K) = 0 for i not ∈ [a, b], • each K_n has tor amplitude in [a, b], • each K_n has constructible cohomology sheaves, • each K_n = ε^-1L_n for some L_n ∈ D_ctf(X_etale, Lambda/I^n) (Étale Cohomology, Definition [Tag 03TQ]).","statement_latex":"In the situation above suppose $K$ is in $D_{cons}(X, \\Lambda)$\nand $X$ is quasi-compact. Set\n$K_n = K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$.\nThere exist $a, b$ such that\n\\begin{enumerate}\n\\item $K = R\\lim K_n$ and $H^i(K) = 0$ for $i \\not \\in [a, b]$,\n\\item each $K_n$ has tor amplitude in $[a, b]$,\n\\item each $K_n$ has constructible cohomology sheaves,\n\\item each $K_n = \\epsilon^{-1}L_n$ for some\n$L_n \\in D_{ctf}(X_\\etale, \\Lambda/I^n)$\n(\\'Etale Cohomology, Definition \\ref{etale-cohomology-definition-ctf}).\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"A suitable derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09C2","source_file":"proetale.tex","source_line":5787,"source_end_line":5801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5787-L5801","statement_sha256":"f0753313e89d7cc74b01acbaf4f4c00dc463af0ed95b9a74f9348e271730a9b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10458,"rank":10458,"depth":64,"x":1715.861,"y":1069.771,"cluster":"tale-geometry"},{"id":"stacks:09C3","tag":"09C3","title":"A suitable derived category · Lemma 09C3","summary":"Let X be a weakly contractible affine scheme. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. Let K be an object of D_cons(X, Lambda) such that the cohomology sheaves of K ⊗_Lambda^L underlineLambda/I are locally constant. Then there exists a finite disjoint open covering X = coprod U_i and for each i a finite collection of finite projective Lambda^wedge-modules M^a, …, M^b such that K|_U_i is represented by a complex (underlineM^a)^wedge → … →…","statement_latex":"Let $X$ be a weakly contractible affine scheme. Let $\\Lambda$ be a Noetherian\nring and let $I \\subset \\Lambda$ be an ideal. Let $K$ be an object of\n$D_{cons}(X, \\Lambda)$ such that the cohomology sheaves of\n$K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I}$ are locally\nconstant. Then there exists a finite disjoint open covering\n$X = \\coprod U_i$ and for each $i$ a finite collection of\nfinite projective $\\Lambda^\\wedge$-modules $M^a, \\ldots, M^b$\nsuch that $K|_{U_i}$ is represented by a complex\n$$\n(\\underline{M^a})^\\wedge \\to \\ldots \\to (\\underline{M^b})^\\wedge\n$$\nin $D(U_{i, \\proetale}, \\Lambda)$ for some maps of sheaves of\n$\\Lambda$-modules $(\\underline{M^i})^\\wedge \\to (\\underline{M^{i + 1}})^\\wedge$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"A suitable derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09C3","source_file":"proetale.tex","source_line":5824,"source_end_line":5839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5824-L5839","statement_sha256":"8b0680b02af579ddbe8ed819e9f793667dc869d4f7a8d8beb7bcfd8e525886fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10459,"rank":10459,"depth":65,"x":1536.266,"y":1268.959,"cluster":"tale-geometry"},{"id":"stacks:09C4","tag":"09C4","title":"A suitable derived category · Definition 09C4","summary":"Let X be a scheme. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. Let K ∈ D(X_proetale, Lambda). • We say K is adic lisse if there exists a finite complex of finite projective Lambda^wedge-modules M^bullet such that K is locally isomorphic to underlineM^a^wedge → … → underlineM^b^wedge • We say K is adic constructible if for every affine open U ⊂ X there exists a decomposition U = coprod U_i into constructible locally closed subschemes such that K|_U_i is…","statement_latex":"Let $X$ be a scheme. Let $\\Lambda$ be a Noetherian ring and let\n$I \\subset \\Lambda$ be an ideal. Let $K \\in D(X_\\proetale, \\Lambda)$.\n\\begin{enumerate}\n\\item We say $K$ is {\\it adic lisse}\\footnote{This may be\nnonstandard notation} if there exists a finite complex of finite\nprojective $\\Lambda^\\wedge$-modules $M^\\bullet$ such that\n$K$ is locally isomorphic to\n$$\n\\underline{M^a}^\\wedge \\to \\ldots \\to \\underline{M^b}^\\wedge\n$$\n\\item We say $K$ is {\\it adic constructible}\\footnote{This may be\nnonstandard notation.} if for every affine open $U \\subset X$\nthere exists a decomposition $U = \\coprod U_i$ into\nconstructible locally closed subschemes such that $K|_{U_i}$\nis adic lisse.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"A suitable derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09C4","source_file":"proetale.tex","source_line":5958,"source_end_line":5976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5958-L5976","statement_sha256":"26732f46b241f6bb92a8bbc388dce3d3343156504137f664e73e9c7ac66eca4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10460,"rank":10460,"depth":0,"x":1508.498,"y":1020.016,"cluster":"tale-geometry"},{"id":"stacks:09C5","tag":"09C5","title":"A suitable derived category · Lemma 09C5","summary":"Let X be a weakly contractible affine scheme. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. Let K be an object of D_cons(X, Lambda) such that K ⊗_Lambda^L underlineLambda/I^n is isomorphic in D(X_proetale, Lambda/I^n) to a complex of constant sheaves of Lambda/I^n-modules. Then H^0(X, K ⊗_Lambda^L Lambda/I^n) has the Mittag-Leffler condition.","statement_latex":"Let $X$ be a weakly contractible affine scheme. Let $\\Lambda$ be a Noetherian\nring and let $I \\subset \\Lambda$ be an ideal. Let $K$ be an object of\n$D_{cons}(X, \\Lambda)$ such that\n$K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$\nis isomorphic in $D(X_\\proetale, \\Lambda/I^n)$ to a\ncomplex of constant sheaves of $\\Lambda/I^n$-modules. Then\n$$\nH^0(X, K \\otimes_\\Lambda^\\mathbf{L} \\Lambda/I^n)\n$$\nhas the Mittag-Leffler condition.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"A suitable derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09C5","source_file":"proetale.tex","source_line":5987,"source_end_line":5999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L5987-L5999","statement_sha256":"8c996aaa7672300022986bd70c915c9fb366f626ff5e3cc806ae3d133707364a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10461,"rank":10461,"depth":21,"x":1729.307,"y":1187.933,"cluster":"tale-geometry"},{"id":"stacks:09C6","tag":"09C6","title":"A suitable derived category · Lemma 09C6","summary":"Let X be a connected scheme. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. If K is in D_cons(X, Lambda) such that K ⊗_Lambda underlineLambda/I has locally constant cohomology sheaves, then K is adic lisse (Definition [Tag 09C4]).","statement_latex":"Let $X$ be a connected scheme. Let $\\Lambda$ be a Noetherian ring and let\n$I \\subset \\Lambda$ be an ideal. If $K$ is in $D_{cons}(X, \\Lambda)$\nsuch that $K \\otimes_\\Lambda \\underline{\\Lambda/I}$\nhas locally constant cohomology sheaves, then $K$ is adic lisse\n(Definition \\ref{definition-adic-constructible}).","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"A suitable derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09C6","source_file":"proetale.tex","source_line":6070,"source_end_line":6077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L6070-L6077","statement_sha256":"8ede29e12af32b55b9ece0c4fcb78ea1af03255b2b2f7322e4ba8c301a0fb80d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10462,"rank":10462,"depth":65,"x":1431.263,"y":1189.407,"cluster":"tale-geometry"},{"id":"stacks:09C7","tag":"09C7","title":"A suitable derived category · Proposition 09C7","summary":"Let X be a Noetherian scheme. Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal. Let K be an object of D_cons(X, Lambda). Then K is adic constructible (Definition [Tag 09C4]).","statement_latex":"Let $X$ be a Noetherian scheme. Let $\\Lambda$ be a Noetherian ring and\nlet $I \\subset \\Lambda$ be an ideal. Let $K$ be an object of\n$D_{cons}(X, \\Lambda)$. Then $K$ is adic constructible\n(Definition \\ref{definition-adic-constructible}).","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"A suitable derived category","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09C7","source_file":"proetale.tex","source_line":6154,"source_end_line":6160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L6154-L6160","statement_sha256":"afb5134d2b48e36870c7163b037be64c10c8b60a7fd4e1dbdf488505d829d762","origin":"The Stacks Project","memory_eligible":false,"source_rank":10463,"rank":10463,"depth":66,"x":1649.99,"y":1019.095,"cluster":"tale-geometry"},{"id":"stacks:09C9","tag":"09C9","title":"Proper base change · Theorem 09C9","summary":"Let f : X → Y be a proper morphism of schemes. Let g : Y' → Y be a morphism of schemes giving rise to the base change diagram xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y Let Lambda be a Noetherian ring and let I ⊂ Lambda be an ideal such that Lambda/I is torsion. Let K be an object of D(X_proetale) such that • K is derived complete, and • K ⊗_Lambda^L underlineLambda/I^n is bounded below with cohomology sheaves coming from X_etale, • Lambda/I^n is a perfect…","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes. Let $g : Y' \\to Y$ be\na morphism of schemes giving rise to the base change diagram\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nLet $\\Lambda$ be a Noetherian ring and let $I \\subset \\Lambda$ be an ideal\nsuch that $\\Lambda/I$ is torsion. Let $K$ be an object\nof $D(X_\\proetale)$ such that\n\\begin{enumerate}\n\\item $K$ is derived complete, and\n\\item $K \\otimes_\\Lambda^\\mathbf{L} \\underline{\\Lambda/I^n}$ is\nbounded below with cohomology sheaves coming from $X_\\etale$,\n\\item $\\Lambda/I^n$ is a perfect $\\Lambda$-module\\footnote{This assumption\ncan be removed if $K$ is a constructible complex, see \\cite{BS}.}.\n\\end{enumerate}\nThen the base change map\n$$\nLg_{comp}^*Rf_*K \\longrightarrow Rf'_*L(g')^*_{comp}K\n$$\nis an isomorphism.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Proper base change","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09C9","source_file":"proetale.tex","source_line":6186,"source_end_line":6211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L6186-L6211","statement_sha256":"0c32b15c98c7a04a55d415ec9275a191c730f7816dbf0880661cfecebe60b521","origin":"The Stacks Project","memory_eligible":false,"source_rank":10464,"rank":10464,"depth":75,"x":1625.648,"y":1268.949,"cluster":"tale-geometry"},{"id":"stacks:098J","tag":"098J","title":"Change of partial universe · Lemma 098J","summary":"Let Sch_proetale be a big pro-étale site as in Definition [Tag 098G]. Let T ∈ Ob(Sch_proetale). Let (T_i → T)_i ∈ I be an arbitrary pro-étale covering of T. There exists a covering (U_j → T)_j ∈ J of T in the site Sch_proetale which refines (T_i → T)_i ∈ I.","statement_latex":"Let $\\Sch_\\proetale$ be a big pro-\\'etale site as in\nDefinition \\ref{definition-big-proetale-site}.\nLet $T \\in \\Ob(\\Sch_\\proetale)$.\nLet $\\{T_i \\to T\\}_{i \\in I}$ be an arbitrary pro-\\'etale covering of $T$.\nThere exists a covering $\\{U_j \\to T\\}_{j \\in J}$ of $T$ in the site\n$\\Sch_\\proetale$ which refines $\\{T_i \\to T\\}_{i \\in I}$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Change of partial universe","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098J","source_file":"proetale.tex","source_line":6298,"source_end_line":6306,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L6298-L6306","statement_sha256":"950051d933c784012825a709af7cbad1002bdacc4b69234f3674e9381be7493b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10465,"rank":10465,"depth":53,"x":1442.556,"y":1070.773,"cluster":"tale-geometry"},{"id":"stacks:098Y","tag":"098Y","title":"Change of partial universe · Lemma 098Y","summary":"Let S be a scheme. Let S_proetale ⊂ S_proetale' be two small pro-étale sites of S as constructed in Definition [Tag 098K]. Then the inclusion functor satisfies the assumptions of Sites, Lemma [Tag 00XU]. Hence there exist morphisms of topoi xymatrix Sh(S_proetale) ar[r]^g & Sh(S_proetale') ar[r]^f & Sh(S_proetale) whose composition is isomorphic to the identity and with f_* = g^-1. Moreover, • for F' ∈ Ab(S_proetale') we have H^p(S_proetale', F') = H^p(S_proetale,…","statement_latex":"Let $S$ be a scheme. Let $S_\\proetale \\subset S_\\proetale'$ be\ntwo small pro-\\'etale sites of $S$ as constructed in\nDefinition \\ref{definition-big-small-proetale}. Then the inclusion functor\nsatisfies the assumptions of \nSites, Lemma \\ref{sites-lemma-bigger-site}.\nHence there exist morphisms of topoi\n$$\n\\xymatrix{\n\\Sh(S_\\proetale) \\ar[r]^g &\n\\Sh(S_\\proetale') \\ar[r]^f &\n\\Sh(S_\\proetale)\n}\n$$\nwhose composition is isomorphic to the identity and with $f_* = g^{-1}$.\nMoreover,\n\\begin{enumerate}\n\\item for $\\mathcal{F}' \\in \\textit{Ab}(S_\\proetale')$ we have\n$H^p(S_\\proetale', \\mathcal{F}') = H^p(S_\\proetale, g^{-1}\\mathcal{F}')$,\n\\item for $\\mathcal{F} \\in \\textit{Ab}(S_\\proetale)$ we have\n$$\nH^p(S_\\proetale, \\mathcal{F}) =\nH^p(S_\\proetale', g_*\\mathcal{F}) =\nH^p(S_\\proetale', f^{-1}\\mathcal{F}).\n$$\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Change of partial universe","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/098Y","source_file":"proetale.tex","source_line":6327,"source_end_line":6354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L6327-L6354","statement_sha256":"1989729c02b9d89aca3db1cd97507060443f026638bcf0f47905641d4e177563","origin":"The Stacks Project","memory_eligible":false,"source_rank":10466,"rank":10466,"depth":54,"x":1737.119,"y":1113.041,"cluster":"tale-geometry"},{"id":"stacks:0F4S","tag":"0F4S","title":"Change of partial universe · Lemma 0F4S","summary":"Suppose given big sites Sch_proetale and Sch'_proetale as in Definition [Tag 098G]. Assume that Sch_proetale is contained in Sch'_proetale. The inclusion functor Sch_proetale → Sch'_proetale satisfies the assumptions of Sites, Lemma [Tag 00XU]. There are morphisms of topoi g : Sh(Sch_proetale) & → & Sh(Sch'_proetale) f : Sh(Sch'_proetale) & → & Sh(Sch_proetale) such that f ∘ g ≅ id. For any object S of Sch_proetale the inclusion functor (Sch/S)_proetale →…","statement_latex":"Suppose given big sites $\\Sch_\\proetale$ and $\\Sch'_\\proetale$ as in\nDefinition \\ref{definition-big-proetale-site}.\nAssume that $\\Sch_\\proetale$ is contained in $\\Sch'_\\proetale$.\nThe inclusion functor $\\Sch_\\proetale \\to \\Sch'_\\proetale$ satisfies\nthe assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}.\nThere are morphisms of topoi\n\\begin{eqnarray*}\ng : \\Sh(\\Sch_\\proetale) &\n\\longrightarrow &\n\\Sh(\\Sch'_\\proetale) \\\\\nf : \\Sh(\\Sch'_\\proetale) &\n\\longrightarrow &\n\\Sh(\\Sch_\\proetale)\n\\end{eqnarray*}\nsuch that $f \\circ g \\cong \\text{id}$. For any object $S$\nof $\\Sch_\\proetale$ the inclusion functor\n$(\\Sch/S)_\\proetale \\to (\\Sch'/S)_\\proetale$ satisfies\nthe assumptions of Sites, Lemma \\ref{sites-lemma-bigger-site}\nalso. Hence similarly we obtain morphisms\n\\begin{eqnarray*}\ng : \\Sh((\\Sch/S)_\\proetale) &\n\\longrightarrow &\n\\Sh((\\Sch'/S)_\\proetale) \\\\\nf : \\Sh((\\Sch'/S)_\\proetale) &\n\\longrightarrow &\n\\Sh((\\Sch/S)_\\proetale)\n\\end{eqnarray*}\nwith $f \\circ g \\cong \\text{id}$.","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Change of partial universe","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4S","source_file":"proetale.tex","source_line":6374,"source_end_line":6404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L6374-L6404","statement_sha256":"8e96fbb0530a75cbc07841ba17c4ebda3344210a94195424922a6f9ddbbff757","origin":"The Stacks Project","memory_eligible":false,"source_rank":10467,"rank":10467,"depth":54,"x":1485.764,"y":1249.101,"cluster":"tale-geometry"},{"id":"stacks:0F4T","tag":"0F4T","title":"Change of partial universe · Lemma 0F4T","summary":"Let S be a scheme. Let (Sch/S)_proetale and (Sch'/S)_proetale be two big pro-étale sites of S as in Definition [Tag 098K]. Assume that the first is contained in the second. In this case • for any abelian sheaf F' defined on (Sch'/S)_proetale and any object U of (Sch/S)_proetale we have H^p(U, F'|_(Sch/S)_proetale) = H^p(U, F') In words: the cohomology of F' over U computed in the bigger site agrees with the cohomology of F' restricted to the smaller site over U. • for any…","statement_latex":"Let $S$ be a scheme. Let $(\\Sch/S)_\\proetale$ and $(\\Sch'/S)_\\proetale$ be two\nbig pro-\\'etale sites of $S$ as in\nDefinition \\ref{definition-big-small-proetale}.\nAssume that the first is contained in\nthe second. In this case\n\\begin{enumerate}\n\\item for any abelian sheaf $\\mathcal{F}'$ defined on $(\\Sch'/S)_\\proetale$\nand any object $U$ of $(\\Sch/S)_\\proetale$ we have\n$$\nH^p(U, \\mathcal{F}'|_{(\\Sch/S)_\\proetale}) =\nH^p(U, \\mathcal{F}')\n$$\nIn words: the cohomology of $\\mathcal{F}'$ over $U$ computed in the bigger site\nagrees with the cohomology of $\\mathcal{F}'$ restricted to the smaller site\nover $U$.\n\\item for any abelian sheaf $\\mathcal{F}$ on $(\\Sch/S)_\\proetale$ there is an\nabelian sheaf $\\mathcal{F}'$ on $(\\Sch/S)_\\proetale'$ whose restriction to\n$(\\Sch/S)_\\proetale$ is isomorphic to $\\mathcal{F}$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"Pro-étale Cohomology","chapter_id":"proetale","section":"Change of partial universe","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4T","source_file":"proetale.tex","source_line":6418,"source_end_line":6439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/proetale.tex#L6418-L6439","statement_sha256":"183081fd03d75989d8a6787a21288ec6316949ec73ad42c63968eec64f30b0b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10468,"rank":10468,"depth":55,"x":1561.741,"y":1005.995,"cluster":"tale-geometry"},{"id":"stacks:0H4F","tag":"0H4F","title":"Cycles relative to fields · Lemma 0H4F","summary":"Let K/k be a field extension. Let Z be an integral locally algebraic scheme over k. The multiplicity m_Z', Z_K of an irreducible component Z' ⊂ Z_K is 1 or a power of the characteristic of k.","statement_latex":"Let $K/k$ be a field extension. Let $Z$ be an integral locally algebraic\nscheme over $k$. The multiplicity $m_{Z', Z_K}$ of an irreducible\ncomponent $Z' \\subset Z_K$ is $1$ or a power of the characteristic of $k$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Cycles relative to fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4F","source_file":"relative-cycles.tex","source_line":87,"source_end_line":92,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L87-L92","statement_sha256":"56cf06c24f552740a68ef3f4251fe7b96fd723fb959cc30f6697bdfa771ebd73","origin":"The Stacks Project","memory_eligible":false,"source_rank":10469,"rank":10469,"depth":40,"x":2645.936,"y":679.358,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4G","tag":"0H4G","title":"Cycles relative to fields · Lemma 0H4G","summary":"Let k be a field of characteristic p > 0 with perfect closure k^perf. Let X be an algebraic scheme over k. Let r ≥ 0 be an integer. The cokernel of the injective map Z_r(X) → Z_r(X_k^perf) is a p-power torsion module (More on Algebra, Definition [Tag 05E6]).","statement_latex":"Let $k$ be a field of characteristic $p > 0$ with perfect closure $k^{perf}$.\nLet $X$ be an algebraic scheme over $k$. Let $r \\geq 0$ be an integer.\nThe cokernel of the injective map $Z_r(X) \\to Z_r(X_{k^{perf}})$ is a\n$p$-power torsion module (More on Algebra, Definition\n\\ref{more-algebra-definition-f-power-torsion}).","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Cycles relative to fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4G","source_file":"relative-cycles.tex","source_line":146,"source_end_line":153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L146-L153","statement_sha256":"79196c0f441697d3089f2f3f587a4decb8abdd1064c2d5709e443bd9a6fefb00","origin":"The Stacks Project","memory_eligible":false,"source_rank":10470,"rank":10470,"depth":41,"x":2358.06,"y":774.705,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4I","tag":"0H4I","title":"Specialization of cycles · Lemma 0H4I","summary":"Let R be a discrete valuation ring with fraction field K and residue field kappa. Let X be a scheme locally of finite type over R. Let r ≥ 0. Let F be a coherent O_X-module flat over R. Assume dim(Supp(F_K)) ≤ r. Then dim(Supp(F_kappa)) ≤ r and sp_X/R([F_K]_r) = [F_kappa]_r","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$ and residue field\n$\\kappa$. Let $X$ be a scheme locally of finite type over $R$. Let $r \\geq 0$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module flat over $R$. Assume\n$\\dim(\\text{Supp}(\\mathcal{F}_K)) \\leq r$. Then\n$\\dim(\\text{Supp}(\\mathcal{F}_\\kappa)) \\leq r$ and\n$$\nsp_{X/R}([\\mathcal{F}_K]_r) = [\\mathcal{F}_\\kappa]_r\n$$","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Specialization of cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4I","source_file":"relative-cycles.tex","source_line":198,"source_end_line":208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L198-L208","statement_sha256":"70fb3ad1c3c9eba18033b95a0699cc959ebc7e88cc83910c9ec56cc962187f6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10471,"rank":10471,"depth":41,"x":2493.768,"y":540.864,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4J","tag":"0H4J","title":"Specialization of cycles · Lemma 0H4J","summary":"Let R be a discrete valuation ring with fraction field K and residue field kappa. Let X be a scheme locally of finite type over R. Let r ≥ 0. Let W ⊂ X be a closed subscheme flat over R. Assume dim(W_K) ≤ r. Then dim(W_kappa) ≤ r and sp_X/R([W_K]_r) = [W_kappa]_r","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$ and residue field\n$\\kappa$. Let $X$ be a scheme locally of finite type over $R$. Let $r \\geq 0$.\nLet $W \\subset X$ be a closed subscheme flat over $R$. Assume\n$\\dim(W_K) \\leq r$. Then $\\dim(W_\\kappa) \\leq r$ and\n$$\nsp_{X/R}([W_K]_r) = [W_\\kappa]_r\n$$","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Specialization of cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4J","source_file":"relative-cycles.tex","source_line":241,"source_end_line":250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L241-L250","statement_sha256":"49d5c0233a58ed5c1394954ef4ac422b128467c0295e00dce7ab3e3ebd0e809e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10472,"rank":10472,"depth":42,"x":2581.818,"y":790.497,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4K","tag":"0H4K","title":"Specialization of cycles · Lemma 0H4K","summary":"Let R'/R be an extension of discrete valuation rings inducing fraction field extension K'/K and residue field extension kappa'/kappa (More on Algebra, Definition [Tag 09E4]). Let X be locally of finite type over R. Denote X' = X_R'. Then the diagram xymatrix Z_r(X'_K') ar[rr]_sp_X'/R' & & Z_r(X'_kappa') Z_r(X_K) ar[rr]^sp_X/R ar[u] & & Z_r(X_kappa) ar[u] commutes where r ≥ 0 and the vertical arrows are base change maps.","statement_latex":"Let $R'/R$ be an extension of discrete valuation rings inducing fraction field\nextension $K'/K$ and residue field extension $\\kappa'/\\kappa$\n(More on Algebra, Definition\n\\ref{more-algebra-definition-extension-discrete-valuation-rings}).\nLet $X$ be locally of finite type over $R$. Denote $X' = X_{R'}$.\nThen the diagram\n$$\n\\xymatrix{\nZ_r(X'_{K'}) \\ar[rr]_{sp_{X'/R'}} & & Z_r(X'_{\\kappa'}) \\\\\nZ_r(X_K) \\ar[rr]^{sp_{X/R}} \\ar[u] & & Z_r(X_\\kappa) \\ar[u]\n}\n$$\ncommutes where $r \\geq 0$ and the vertical arrows are base change maps.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Specialization of cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4K","source_file":"relative-cycles.tex","source_line":258,"source_end_line":273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L258-L273","statement_sha256":"aeecc6f4b8794dd08a695aaea2d2f033cbb462163153cfc14cf8b1c3ce8de0c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10473,"rank":10473,"depth":42,"x":2315.932,"y":656.276,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4L","tag":"0H4L","title":"Specialization of cycles · Lemma 0H4L","summary":"Let R be a discrete valuation ring with fraction field K and residue field kappa. Let X be a scheme locally of finite type over R. Let f : X' → X be a morphism which is locally of finite type, flat, and of relative dimension e. Then the diagram xymatrix Z_r + e(X'_K) ar[rr]_sp_X'/R & & Z_r + e(X'_kappa) Z_r(X_K) ar[rr]^sp_X/R ar[u] & & Z_r(X_kappa) ar[u] commutes where r ≥ 0 and the vertical arrows are given by flat pullback.","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$ and residue field\n$\\kappa$. Let $X$ be a scheme locally of finite type over $R$.\nLet $f : X' \\to X$ be a morphism which is locally of finite type, flat,\nand of relative dimension $e$. Then the diagram\n$$\n\\xymatrix{\nZ_{r + e}(X'_K) \\ar[rr]_{sp_{X'/R}} & & Z_{r + e}(X'_\\kappa) \\\\\nZ_r(X_K) \\ar[rr]^{sp_{X/R}} \\ar[u] & & Z_r(X_\\kappa) \\ar[u]\n}\n$$\ncommutes where $r \\geq 0$ and the vertical arrows are given\nby flat pullback.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Specialization of cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4L","source_file":"relative-cycles.tex","source_line":294,"source_end_line":308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L294-L308","statement_sha256":"fafe7a74cedc6277342d5e7988022689f6069b0bfced9d0373a05072b242b4a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10474,"rank":10474,"depth":43,"x":2620.17,"y":604.338,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4M","tag":"0H4M","title":"Specialization of cycles · Lemma 0H4M","summary":"Let R be a discrete valuation ring with fraction field K and residue field kappa. Let f : X → Y be a proper morphism of schemes locally of finite type over R. Then the diagram xymatrix Z_r(X_K) ar[rr]_sp_X/R ar[d] & & Z_r(X_kappa) ar[d] Z_r(Y_K) ar[rr]^sp_Y/R & & Z_r(Y_kappa) commutes where r ≥ 0 and the vertical arrows are given by proper pushforward.","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$ and residue field\n$\\kappa$. Let $f : X \\to Y$ be a proper morphism of schemes locally of\nfinite type over $R$. Then the diagram\n$$\n\\xymatrix{\nZ_r(X_K) \\ar[rr]_{sp_{X/R}} \\ar[d] & & Z_r(X_\\kappa) \\ar[d] \\\\\nZ_r(Y_K) \\ar[rr]^{sp_{Y/R}} & & Z_r(Y_\\kappa)\n}\n$$\ncommutes where $r \\geq 0$ and the vertical arrows are given\nby proper pushforward.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Specialization of cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4M","source_file":"relative-cycles.tex","source_line":324,"source_end_line":337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L324-L337","statement_sha256":"cff4a1e9c143c4a251779f8758710aa0990ded4c595514fddd96a0e99cb841c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10475,"rank":10475,"depth":46,"x":2437.453,"y":815.435,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4P","tag":"0H4P","title":"Families of cycles on fibres · Lemma 0H4P","summary":"We have the following compatibilities between the operations above: (1) base change is functorial, (2) restriction is a combination of base change and (a special case of) flat pullback, (3) flat pullback commutes with base change, (4) flat pullback is functorial, (5) proper pushforward commutes with base change, (6) proper pushforward is functorial, and (7) proper pushforward commutes with flat pullback.","statement_latex":"We have the following compatibilities between the operations above:\n(1) base change is functorial,\n(2) restriction is a combination of base change and (a special case of)\nflat pullback,\n(3) flat pullback commutes with base change,\n(4) flat pullback is functorial,\n(5) proper pushforward commutes with base change,\n(6) proper pushforward is functorial, and\n(7) proper pushforward commutes with flat pullback.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4P","source_file":"relative-cycles.tex","source_line":458,"source_end_line":469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L458-L469","statement_sha256":"76a4ada8fce932a8111ab1207f28f970dd69227f5c0274e3b71f1b14b64fdcc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10476,"rank":10476,"depth":51,"x":2402.399,"y":555.892,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4R","tag":"0H4R","title":"Families of cycles on fibres · Lemma 0H4R","summary":"The construction in Example [Tag 0H4Q] is compatible with base change, restriction, and flat pullback.","statement_latex":"The construction in Example \\ref{example-family-associated-module}\nis compatible with base change, restriction, and flat pullback.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4R","source_file":"relative-cycles.tex","source_line":507,"source_end_line":511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L507-L511","statement_sha256":"8f39571ff09f0fa4722331cd92221832c77d19201bbbb2e8a20ab214bde8ee51","origin":"The Stacks Project","memory_eligible":false,"source_rank":10477,"rank":10477,"depth":29,"x":2637.149,"y":727.52,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4T","tag":"0H4T","title":"Families of cycles on fibres · Lemma 0H4T","summary":"The construction in Example [Tag 0H4S] is compatible with base change, restriction, and flat pullback.","statement_latex":"The construction in Example \\ref{example-family-associated-closed}\nis compatible with base change, restriction, and flat pullback.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4T","source_file":"relative-cycles.tex","source_line":537,"source_end_line":541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L537-L541","statement_sha256":"971337165383406c96a736d875f9921095e2f962c8200b7ce26726e94c334e0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10478,"rank":10478,"depth":30,"x":2325.785,"y":734.172,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4V","tag":"0H4V","title":"Families of cycles on fibres · Lemma 0H4V","summary":"Taking the support as in Remark [Tag 0H4U] is compatible with base change, restriction, and flat pullback.","statement_latex":"Taking the support as in Remark \\ref{remark-supports-family}\nis compatible with base change, restriction, and flat pullback.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4V","source_file":"relative-cycles.tex","source_line":564,"source_end_line":568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L564-L568","statement_sha256":"d9cd3763fc800f8586a4bfe11add652e4cf4690356348e195965266ce82965fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10479,"rank":10479,"depth":0,"x":2550.207,"y":552.449,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4W","tag":"0H4W","title":"Families of cycles on fibres · Lemma 0H4W","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let r ≥ 0 be an integer. Let g : S' → S be a surjective morphism of schemes. Set S\" = S' ×_S S' and let f' : X' → S' and f\" : X\" → S\" be the base changes of f. Let x ∈ X with trdeg_kappa(f(x))(kappa(x)) = r. • There exists an x' ∈ X' mapping to x with trdeg_kappa(f'(x'))(kappa(x')) = r. • If x'_1, x'_2 ∈ X' are both as in (1), then there exists an x\" ∈ X\" with trdeg_kappa(f\"(x\"))(kappa(x\")) = r and…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $r \\geq 0$ be an integer. Let $g : S' \\to S$ be a surjective morphism of\nschemes. Set $S'' = S' \\times_S S'$ and let $f' : X' \\to S'$\nand $f'' : X'' \\to S''$ be the base changes of $f$.\nLet $x \\in X$ with $\\text{trdeg}_{\\kappa(f(x))}(\\kappa(x)) = r$.\n\\begin{enumerate}\n\\item There exists an $x' \\in X'$ mapping to $x$\nwith $\\text{trdeg}_{\\kappa(f'(x'))}(\\kappa(x')) = r$.\n\\item If $x'_1, x'_2 \\in X'$ are both as in (1), then there\nexists an $x'' \\in X''$ with\n$\\text{trdeg}_{\\kappa(f''(x''))}(\\kappa(x'')) = r$ and\n$\\text{pr}_i(x'') = x'_i$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4W","source_file":"relative-cycles.tex","source_line":574,"source_end_line":589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L574-L589","statement_sha256":"a58719f5a3c4a7ec260face1ed1a80e3ffb72ef498b7a1c9167c2e76ab66a6f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10480,"rank":10480,"depth":27,"x":2530.843,"y":813.997,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4X","tag":"0H4X","title":"Families of cycles on fibres · Lemma 0H4X","summary":"Let f : X → S be a morphism of schemes which is locally of finite type. Let r ≥ 0 be an integer. Let g : S' → S be a morphism of schemes and X' = S' ×_S X. Assume that for every s ∈ S there exists a point s' ∈ S' with g(s') = s and such that kappa(s')/kappa(s) is a separable extension of fields. Then • For families α_1 and α_2 of r-cycles on fibres of X/S if g^*α_1 = g^*α_2, then α_1 = α_2. • Given a family α' of r-cycles on fibres of X'/S' if pr_1^*α' = pr_2^*α' as…","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which is locally of finite type.\nLet $r \\geq 0$ be an integer. Let $g : S' \\to S$ be a morphism of\nschemes and $X' = S' \\times_S X$. Assume that for every $s \\in S$ there\nexists a point $s' \\in S'$ with $g(s') = s$ and such that\n$\\kappa(s')/\\kappa(s)$ is a separable extension of fields. Then\n\\begin{enumerate}\n\\item For families $\\alpha_1$ and $\\alpha_2$ of $r$-cycles on fibres of $X/S$\nif $g^*\\alpha_1 = g^*\\alpha_2$, then $\\alpha_1 = \\alpha_2$.\n\\item Given a family $\\alpha'$ of $r$-cycles on fibres of $X'/S'$ if\n$\\text{pr}_1^*\\alpha' = \\text{pr}_2^*\\alpha'$ as families of\n$r$-cycles on fibres of $(S' \\times_S S') \\times_S X / (S' \\times_S S')$,\nthen there is a unique family $\\alpha$ of $r$-cycles on fibres of $X/S$\nsuch that $g^*\\alpha = \\alpha'$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4X","source_file":"relative-cycles.tex","source_line":612,"source_end_line":628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L612-L628","statement_sha256":"ffdd049f1246085efa8687e84e648c9c594051bb326c366a17a8223944550536","origin":"The Stacks Project","memory_eligible":false,"source_rank":10481,"rank":10481,"depth":41,"x":2334.639,"y":609.988,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H4Y","tag":"0H4Y","title":"Families of cycles on fibres · Lemma 0H4Y","summary":"Let g : S' → S be a bijective morphism of schemes which induces isomorphisms of residue fields. Let f : X → S be locally of finite type. Set X' = S' ×_S X. Let r ≥ 0. Then base change by g determines a bijection between the group of families of r-cycles on fibres of X/S and the group of families of r-cycles on fibres of X'/S'.","statement_latex":"Let $g : S' \\to S$ be a bijective morphism of schemes\nwhich induces isomorphisms of residue fields.\nLet $f : X \\to S$ be locally of finite type. Set $X' = S' \\times_S X$.\nLet $r \\geq 0$. Then base change by $g$ determines a bijection\nbetween the group of families of $r$-cycles on fibres of $X/S$ and\nthe group of families of $r$-cycles on fibres of $X'/S'$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H4Y","source_file":"relative-cycles.tex","source_line":703,"source_end_line":711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L703-L711","statement_sha256":"f6783414c14d9613fb8841ee7343a391e0197580e983d1804f62a2189407b194","origin":"The Stacks Project","memory_eligible":false,"source_rank":10482,"rank":10482,"depth":0,"x":2643.617,"y":649.12,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H50","tag":"0H50","title":"Relative cycles · Definition 0H50","summary":"Let S be a locally Noetherian scheme. Let f : X → S be a morphism of schemes which is locally of finite type. Let r ≥ 0 be an integer. A relative r-cycle on X/S is a family α of r-cycles on fibres of X/S such that for every morphism g : S' → S where S' is the spectrum of a discrete valuation ring we have sp_X'/S'(α_eta) = α_0 where sp_X'/S' is as in Section [Tag 0H4H] and α_eta (resp. α_0) is the value of the base change g^*α of α at the generic (resp. closed) point of…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $f : X \\to S$ be a morphism of\nschemes which is locally of finite type. Let $r \\geq 0$ be an integer.\nA {\\it relative $r$-cycle on $X/S$} is a family $\\alpha$ of $r$-cycles\non fibres of $X/S$ such that for every morphism $g : S' \\to S$\nwhere $S'$ is the spectrum of a discrete valuation ring we have\n$$\nsp_{X'/S'}(\\alpha_\\eta) = \\alpha_0\n$$\nwhere $sp_{X'/S'}$ is as in Section \\ref{section-specialization}\nand $\\alpha_\\eta$ (resp.\\ $\\alpha_0$) is the value of the base change\n$g^*\\alpha$ of $\\alpha$ at the generic (resp.\\ closed) point of $S'$.\nThe group of all relative $r$-cycles on $X/S$ is denoted $z(X/S, r)$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H50","source_file":"relative-cycles.tex","source_line":731,"source_end_line":745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L731-L745","statement_sha256":"d19e1df6e42c52424efd376074b7caa86b27849cbdbd16a25836664ef1643598","origin":"The Stacks Project","memory_eligible":false,"source_rank":10483,"rank":10483,"depth":0,"x":2384.11,"y":795.7,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H51","tag":"0H51","title":"Relative cycles · Lemma 0H51","summary":"Let α be a relative r-cycle on X/S as in Definition [Tag 0H50]. Then any restriction, base change, flat pullback, or proper pushforward of α is a relative r-cycle.","statement_latex":"Let $\\alpha$ be a relative $r$-cycle on $X/S$ as in\nDefinition \\ref{definition-relative-cycles}.\nThen any restriction, base change, flat pullback, or proper pushforward\nof $\\alpha$ is a relative $r$-cycle.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H51","source_file":"relative-cycles.tex","source_line":747,"source_end_line":753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L747-L753","statement_sha256":"f3112e1d98777f1d81e2668921acc049f1410566b8629a5200d6b9b2ec228a32","origin":"The Stacks Project","memory_eligible":false,"source_rank":10484,"rank":10484,"depth":47,"x":2457.648,"y":540.165,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H52","tag":"0H52","title":"Relative cycles · Lemma 0H52","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r ≥ 0 be an integer. Let α be a family of r-cycles on fibres of X/S. Let (g_i : S_i → S) be a h covering (More on Flatness, Definition [Tag 0ETS]). Then α is a relative r-cycle if and only if each base change g_i^*α is a relative r-cycle.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r \\geq 0$ be an integer. Let $\\alpha$\nbe a family of $r$-cycles on fibres of $X/S$. Let $\\{g_i : S_i \\to S\\}$\nbe a h covering (More on Flatness, Definition\n\\ref{flat-definition-h-covering}). Then $\\alpha$ is a relative $r$-cycle\nif and only if each base change $g_i^*\\alpha$ is a relative $r$-cycle.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H52","source_file":"relative-cycles.tex","source_line":762,"source_end_line":770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L762-L770","statement_sha256":"6d5b93dd4beef1aa89d0f78114ae96ec115afebb27763b5664e71eb42d9918cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10485,"rank":10485,"depth":57,"x":2609.034,"y":770.499,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H53","tag":"0H53","title":"Relative cycles · Lemma 0H53","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r, e ≥ 0 be integers. Let α be a family of r-cycles on fibres of X/S. Let (f_i : X_i → X) be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension e. Then α is a relative r-cycle if and only if each flat pullback f_i^*α is a relative r-cycle.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r, e \\geq 0$ be integers.\nLet $\\alpha$ be a family of $r$-cycles on fibres of $X/S$.\nLet $\\{f_i : X_i \\to X\\}$ be a jointly surjective family\nof flat morphisms, locally of finite type, and of relative dimension $e$.\nThen $\\alpha$ is a relative $r$-cycle if and only if each flat\npullback $f_i^*\\alpha$ is a relative $r$-cycle.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H53","source_file":"relative-cycles.tex","source_line":804,"source_end_line":813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L804-L813","statement_sha256":"c4c566190b18416a646d19dacc90f77d2cbb023bb103780785d2cda6f00403c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10486,"rank":10486,"depth":48,"x":2311.945,"y":686.489,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H54","tag":"0H54","title":"Relative cycles · Lemma 0H54","summary":"Let S be a locally Noetherian scheme. Let i : X → Y be a closed immersion of schemes locally of finite type over S. Let r ≥ 0. Let α be a family of r-cycles on fibres of X/S. Then α is a relative r-cycle on X/S if and only if i_*α is a relative r-cycle on Y/S.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $i : X \\to Y$ be a closed immersion\nof schemes locally of finite type over $S$. Let $r \\geq 0$.\nLet $\\alpha$ be a family of $r$-cycles on fibres of $X/S$.\nThen $\\alpha$ is a relative $r$-cycle on $X/S$ if and only if\n$i_*\\alpha$ is a relative $r$-cycle on $Y/S$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H54","source_file":"relative-cycles.tex","source_line":843,"source_end_line":850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L843-L850","statement_sha256":"741cde3ecbbfa8450da0987966568884d037c8b732fa0ad0a082b9c37e85a271","origin":"The Stacks Project","memory_eligible":false,"source_rank":10487,"rank":10487,"depth":52,"x":2598.795,"y":579.779,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H55","tag":"0H55","title":"Relative cycles · Lemma 0H55","summary":"Let f : X → S be a morphism of schemes. Assume S is locally Noetherian and f locally of finite type. Let r ≥ 0. Let α and β be relative r-cycles on X/S. The following are equivalent • α = β, and • α_eta = β_eta for any generic point eta ∈ S of an irreducible component of S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian\nand $f$ locally of finite type. Let $r \\geq 0$. Let $\\alpha$ and $\\beta$\nbe relative $r$-cycles on $X/S$. The following are equivalent\n\\begin{enumerate}\n\\item $\\alpha = \\beta$, and\n\\item $\\alpha_\\eta = \\beta_\\eta$ for any generic point $\\eta \\in S$\nof an irreducible component of $S$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H55","source_file":"relative-cycles.tex","source_line":868,"source_end_line":878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L868-L878","statement_sha256":"ae2f71b3171d1f650eb5ba885a5437ac2f7e84c4fa3a4dec72cb6f3cca695148","origin":"The Stacks Project","memory_eligible":false,"source_rank":10488,"rank":10488,"depth":18,"x":2472.992,"y":821.418,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H56","tag":"0H56","title":"Relative cycles · Lemma 0H56","summary":"In the situation of Example [Tag 0H4Q] assume S is locally Noetherian and F is flat over S in dimensions ≥ r (More on Flatness, Definition [Tag 0CWG]). Then [F/X/S]_r is a relative r-cycle on X/S.","statement_latex":"In the situation of Example \\ref{example-family-associated-module}\nassume $S$ is locally Noetherian and\n$\\mathcal{F}$ is flat over $S$ in dimensions $\\geq r$\n(More on Flatness, Definition \\ref{flat-definition-flat-dimension-n}).\nThen $[\\mathcal{F}/X/S]_r$ is a relative $r$-cycle on $X/S$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H56","source_file":"relative-cycles.tex","source_line":897,"source_end_line":904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L897-L904","statement_sha256":"26c44f230254dab1632e161a3ac5125c0f8f253370a099f4332092c11196170f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10489,"rank":10489,"depth":58,"x":2371.359,"y":571.661,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H57","tag":"0H57","title":"Relative cycles · Lemma 0H57","summary":"In the situation of Example [Tag 0H4S] assume S is locally Noetherian and Z is flat over S in dimensions ≥ r. Then [Z/X/S]_r is a relative r-cycle on X/S.","statement_latex":"In the situation of Example \\ref{example-family-associated-closed}\nassume $S$ is locally Noetherian and $Z$ is flat over $S$ in dimensions\n$\\geq r$. Then $[Z/X/S]_r$ is a relative $r$-cycle on $X/S$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H57","source_file":"relative-cycles.tex","source_line":929,"source_end_line":934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L929-L934","statement_sha256":"8bc20d50d144dae2fb805ec4e8b942e65d9619fca51a5e710440e52bf707dac9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10490,"rank":10490,"depth":59,"x":2647.363,"y":698.257,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H59","tag":"0H59","title":"Relative cycles · Lemma 0H59","summary":"Let f : X → S be a finite type morphism of schemes with S Noetherian. Let r ≥ 0. Let α be a relative r-cycle on X/S. Then there is a proper, completely decomposed (More on Morphisms, Definition [Tag 0GTI]) morphism g : S' → S such that g^*α is in the image of ([Tag 0H58]).","statement_latex":"Let $f : X \\to S$ be a finite type morphism of schemes with $S$ Noetherian.\nLet $r \\geq 0$. Let $\\alpha$ be a relative $r$-cycle on $X/S$. Then there is\na proper, completely decomposed\n(More on Morphisms, Definition \\ref{more-morphisms-definition-cd-morphism})\nmorphism $g : S' \\to S$ such that $g^*\\alpha$ is in the image of\n(\\ref{equation-cycle-classes}).","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H59","source_file":"relative-cycles.tex","source_line":959,"source_end_line":967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L959-L967","statement_sha256":"85121ff6f9f17a035128a02352f9acc59be16e533809f9cc8cf470d68106e2f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10491,"rank":10491,"depth":56,"x":2341.802,"y":761.566,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5A","tag":"0H5A","title":"Relative cycles · Lemma 0H5A","summary":"Let f : X → S be a finite type morphism of schemes with S the spectrum of a discrete valuation ring. Let r ≥ 0. Then ([Tag 0H58]) is surjective.","statement_latex":"Let $f : X \\to S$ be a finite type morphism of schemes with $S$\nthe spectrum of a discrete valuation ring. Let $r \\geq 0$.\nThen (\\ref{equation-cycle-classes}) is surjective.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5A","source_file":"relative-cycles.tex","source_line":1016,"source_end_line":1021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1016-L1021","statement_sha256":"054b74e99bf18e3aa535caee97e730069c5c5e22511dfedfb725380ae7264bc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10492,"rank":10492,"depth":57,"x":2516.34,"y":541.331,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5B","tag":"0H5B","title":"Relative cycles · Lemma 0H5B","summary":"Let f : X → S be a morphism of schemes. Let r ≥ 0. Assume S locally Noetherian and f smooth of relative dimension r. Let α ∈ z(X/S, r). Then the support of α is open and closed in X (see proof for a more precise result).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Let $r \\geq 0$. Assume $S$\nlocally Noetherian and $f$ smooth of relative dimension $r$. Let\n$\\alpha \\in z(X/S, r)$. Then the support of $\\alpha$ is open and closed in $X$\n(see proof for a more precise result).","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5B","source_file":"relative-cycles.tex","source_line":1032,"source_end_line":1038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1032-L1038","statement_sha256":"a6c11b261397e51de2d907df2940aa1f97285ecdbca16cdaa38bea2fcb94269c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10493,"rank":10493,"depth":38,"x":2564.782,"y":802.967,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5C","tag":"0H5C","title":"Relative cycles · Lemma 0H5C","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r ≥ 0 and α, β ∈ z(X/S, r). The set E = (s ∈ S : α_s = β_s) is closed in S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r \\geq 0$ and\n$\\alpha, \\beta \\in z(X/S, r)$. The set $E = \\{s \\in S : \\alpha_s = \\beta_s\\}$\nis closed in $S$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5C","source_file":"relative-cycles.tex","source_line":1088,"source_end_line":1094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1088-L1094","statement_sha256":"3d25a31890fc84f9ea51974759561f7fbe856abfce8507c76abc8cc2e7f4e728","origin":"The Stacks Project","memory_eligible":false,"source_rank":10494,"rank":10494,"depth":60,"x":2318.472,"y":637.401,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5D","tag":"0H5D","title":"Relative cycles · Lemma 0H5D","summary":"Let S = lim_i ∈ I S_i be the limit of a directed inverse system of Noetherian schemes with affine transition morphisms. Let 0 ∈ I and let X_0 → S_0 be a finite type morphism of schemes. For i ≥ 0 set X_i = S_i ×_S_0 X_0 and set X = S ×_S_0 X_0. If S is Noetherian too, then z(X/S, r) = colim_i ≥ 0 z(X_i/S_i, r) where the transition maps are given by base change of relative r-cycles.","statement_latex":"Let $S = \\lim_{i \\in I} S_i$ be the limit of a directed inverse system of\nNoetherian schemes with affine transition morphisms.\nLet $0 \\in I$ and let $X_0 \\to S_0$ be a finite type morphism of schemes.\nFor $i \\geq 0$ set $X_i = S_i \\times_{S_0} X_0$ and set\n$X = S \\times_{S_0} X_0$. If $S$ is Noetherian too, then\n$$\nz(X/S, r) = \\colim_{i \\geq 0} z(X_i/S_i, r)\n$$\nwhere the transition maps are given by base change of relative\n$r$-cycles.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5D","source_file":"relative-cycles.tex","source_line":1165,"source_end_line":1177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1165-L1177","statement_sha256":"65181909bdb996bc2dfdbb8e7a2d793e3358f0e790736e8398848547ed086496","origin":"The Stacks Project","memory_eligible":false,"source_rank":10495,"rank":10495,"depth":61,"x":2633.482,"y":619.711,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5E","tag":"0H5E","title":"Relative cycles · Lemma 0H5E","summary":"Let S be a locally Noetherian scheme. Let i : X → X' be a thickening of schemes locally of finite type over S. Let r ≥ 0. Then i_* : z(X/S, r) → z(X'/S, r) is a bijection.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $i : X \\to X'$ be a thickening\nof schemes locally of finite type over $S$. Let $r \\geq 0$.\nThen $i_* : z(X/S, r) \\to z(X'/S, r)$ is a bijection.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5E","source_file":"relative-cycles.tex","source_line":1245,"source_end_line":1250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1245-L1250","statement_sha256":"1bef17bdb2fbc4618ba507ff4e8d590b060cf8cce1fcb917698622655f227994","origin":"The Stacks Project","memory_eligible":false,"source_rank":10496,"rank":10496,"depth":53,"x":2415.258,"y":811.646,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5F","tag":"0H5F","title":"Relative cycles · Lemma 0H5F","summary":"Let S be a locally Noetherian scheme. Let X be a scheme locally of finite type over S. Let r ≥ 0. Let U ⊂ X be an open such that X setminus U has relative dimension < r over S, i.e., dim(X_s setminus U_s) < r for all s ∈ S. Then restriction defines a bijection z(X/S, r) → z(U/S, r).","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $X$ be a scheme locally\nof finite type over $S$. Let $r \\geq 0$. Let $U \\subset X$ be an open\nsuch that $X \\setminus U$ has relative dimension $< r$ over $S$, i.e.,\n$\\dim(X_s \\setminus U_s) < r$ for all $s \\in S$. Then\nrestriction defines a bijection $z(X/S, r) \\to z(U/S, r)$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5F","source_file":"relative-cycles.tex","source_line":1262,"source_end_line":1269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1262-L1269","statement_sha256":"c58fe733c6423421be2964f65bf4ec6c09bc7fd09addc480c31571d467a7a3ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":10497,"rank":10497,"depth":52,"x":2421.829,"y":546.088,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5G","tag":"0H5G","title":"Relative cycles · Lemma 0H5G","summary":"Let g : S' → S be a universal homeomorphism of locally Noetherian schemes which induces isomorphisms of residue fields. Let f : X → S be locally of finite type. Set X' = S' ×_S X. Let r ≥ 0. Then base change by g determines a bijection z(X/S, r) → z(X'/S', r).","statement_latex":"Let $g : S' \\to S$ be a universal homeomorphism of locally Noetherian schemes\nwhich induces isomorphisms of residue fields. Let $f : X \\to S$ be locally of\nfinite type. Set $X' = S' \\times_S X$. Let $r \\geq 0$. Then base change by $g$\ndetermines a bijection $z(X/S, r) \\to z(X'/S', r)$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5G","source_file":"relative-cycles.tex","source_line":1282,"source_end_line":1288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1282-L1288","statement_sha256":"5062c7ece3b78e786b7574fc7da23f1e1645cc2038df88f0f2b1dacec2f3b00a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10498,"rank":10498,"depth":1,"x":2630.697,"y":745.787,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5I","tag":"0H5I","title":"Equidimensional relative cycles · Definition 0H5I","summary":"Let f : X → S be a morphism of schemes. Assume S is locally Noetherian and f is locally of finite type. Let r ≥ 0 be an integer. We say a relative r-cycle α on X/S equidimensional if the support of α (Remark [Tag 0H4U]) is contained in a closed subset W ⊂ X whose relative dimension over S is ≤ r. The group of all equidimensional relative r-cycles on X/S is denoted z_equi(X/S, r).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian\nand $f$ is locally of finite type. Let $r \\geq 0$ be an integer. We say a\nrelative $r$-cycle $\\alpha$ on $X/S$ {\\it equidimensional} if the support\nof $\\alpha$ (Remark \\ref{remark-supports-family})\nis contained in a closed subset $W \\subset X$ whose relative\ndimension over $S$ is $\\leq r$.\nThe group of all equidimensional relative $r$-cycles on $X/S$ is\ndenoted $z_{equi}(X/S, r)$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Equidimensional relative cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5I","source_file":"relative-cycles.tex","source_line":1320,"source_end_line":1330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1320-L1330","statement_sha256":"26c2f363a8f3dba0bf0f25ab2836da2b917d3a9693eaaa918b24635ed58bc9cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":10499,"rank":10499,"depth":0,"x":2315.849,"y":717.027,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5K","tag":"0H5K","title":"Equidimensional relative cycles · Lemma 0H5K","summary":"Let f : X → S be a morphism of schemes. Assume S is locally Noetherian and f is locally of finite type. Let r ≥ 0 be an integer. Let α be a relative r-cycle on X/S. If α is equidimensional, then any restriction, base change, or flat pullback of α is equidimensional.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian\nand $f$ is locally of finite type. Let $r \\geq 0$ be an integer. Let\n$\\alpha$ be a relative $r$-cycle on $X/S$. If $\\alpha$ is equidimensional,\nthen any restriction, base change, or flat pullback of $\\alpha$ is\nequidimensional.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Equidimensional relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5K","source_file":"relative-cycles.tex","source_line":1355,"source_end_line":1362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1355-L1362","statement_sha256":"4c1ba465249b3e1a09bbda46100489afd117cb5ab72bfe6a6bfb3ea61e11749e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10500,"rank":10500,"depth":0,"x":2571.336,"y":559.463,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5L","tag":"0H5L","title":"Equidimensional relative cycles · Lemma 0H5L","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r ≥ 0 be an integer. Let α be a relative r-cycle on X/S. Then to check that α is equidimensional we may work Zariski locally on X and S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r \\geq 0$ be an integer. Let $\\alpha$\nbe a relative $r$-cycle on $X/S$. Then to check that $\\alpha$ is equidimensional\nwe may work Zariski locally on $X$ and $S$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Equidimensional relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5L","source_file":"relative-cycles.tex","source_line":1368,"source_end_line":1374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1368-L1374","statement_sha256":"35fbc14f784fd9fdd3da2b403f56d6d75352a5ae1e84729159fa15a28f25729f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10501,"rank":10501,"depth":0,"x":2509.606,"y":820.815,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5M","tag":"0H5M","title":"Equidimensional relative cycles · Lemma 0H5M","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r ≥ 0 be an integer. Let α be a relative r-cycle on X/S. Let (g_i : S_i → S) be an fppf covering. Then α is equidimensional if and only if each base change g_i^*α is equidimensional.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r \\geq 0$ be an integer. Let $\\alpha$\nbe a relative $r$-cycle on $X/S$. Let $\\{g_i : S_i \\to S\\}$\nbe an fppf covering. Then $\\alpha$ is equidimensional\nif and only if each base change $g_i^*\\alpha$ is equidimensional.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Equidimensional relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5M","source_file":"relative-cycles.tex","source_line":1383,"source_end_line":1390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1383-L1390","statement_sha256":"63e62c51432c281d1250a00c8272a12464be10b607a03da90f49523c6c662440","origin":"The Stacks Project","memory_eligible":false,"source_rank":10502,"rank":10502,"depth":27,"x":2344.827,"y":592.898,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5N","tag":"0H5N","title":"Equidimensional relative cycles · Lemma 0H5N","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r, e ≥ 0 be integers. Let α be a relative r-cycle on X/S. Let (f_i : X_i → X) be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension e. Then α is equidimensional if and only if each flat pullback f_i^*α is equidimensional.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r, e \\geq 0$ be integers.\nLet $\\alpha$ be a relative $r$-cycle on $X/S$.\nLet $\\{f_i : X_i \\to X\\}$ be a jointly surjective family\nof flat morphisms, locally of finite type, and of relative dimension $e$.\nThen $\\alpha$ is equidimensional if and only if each flat\npullback $f_i^*\\alpha$ is equidimensional.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Equidimensional relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5N","source_file":"relative-cycles.tex","source_line":1409,"source_end_line":1418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1409-L1418","statement_sha256":"f9bafa641ba92535de02305e07a2da5871f8e74717f958305f2edcc8f02184c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10503,"rank":10503,"depth":28,"x":2649.847,"y":667.518,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5P","tag":"0H5P","title":"Weightings and relative zero cycles · Lemma 0H5P","summary":"Let S be a locally Noetherian scheme. Let f : X → S be a locally quasi-finite morphism of schemes. • For α ∈ z(X/S, 0) the map w_α : X → Z constructed above is a weighting. • If X is quasi-compact, then given a weighting w : X → Z there exists an integer n > 0 such that nw = w_α for some α ∈ z(X/S, 0). • The integer n in (2) may be chosen to be a power of the prime p if S is a scheme over F_p.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $f : X \\to S$ be a locally\nquasi-finite morphism of schemes.\n\\begin{enumerate}\n\\item For $\\alpha \\in z(X/S, 0)$ the map $w_\\alpha : X \\to \\mathbf{Z}$\nconstructed above is a weighting.\n\\item If $X$ is quasi-compact, then given a weighting\n$w : X \\to \\mathbf{Z}$ there exists an integer $n > 0$ such\nthat $nw = w_\\alpha$ for some $\\alpha \\in z(X/S, 0)$.\n\\item The integer $n$ in (2) may be chosen to be a power of\nthe prime $p$ if $S$ is a scheme over $\\mathbf{F}_p$.\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Weightings and relative zero cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5P","source_file":"relative-cycles.tex","source_line":1504,"source_end_line":1517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1504-L1517","statement_sha256":"45a904e3f2be0aa84d069436fa4666a8f0a1241e514b92df74c2b7d83fc45a4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10504,"rank":10504,"depth":39,"x":2364.708,"y":785.659,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5R","tag":"0H5R","title":"Effective relative cycles · Definition 0H5R","summary":"Let f : X → S be a morphism of schemes. Assume S is locally Noetherian and f is locally of finite type. Let r ≥ 0 be an integer. We say a relative r-cycle α on X/S effective if α_s is an effective cycle (Chow Homology, Definition [Tag 0H47]) for all s ∈ S. The monoid of all effective relative r-cycles on X/S is denoted z^eff(X/S, r).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian\nand $f$ is locally of finite type. Let $r \\geq 0$ be an integer. We say a\nrelative $r$-cycle $\\alpha$ on $X/S$ {\\it effective} if $\\alpha_s$ is an\neffective cycle\n(Chow Homology, Definition \\ref{chow-definition-effective-cycle})\nfor all $s \\in S$. The monoid of all effective relative $r$-cycles\non $X/S$ is denoted $z^{eff}(X/S, r)$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Effective relative cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5R","source_file":"relative-cycles.tex","source_line":1651,"source_end_line":1660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1651-L1660","statement_sha256":"4eedd4b15fe228eb054e708e4487401b407ed461e595b5b95d483a561d9c728f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10505,"rank":10505,"depth":1,"x":2480.046,"y":536.561,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5S","tag":"0H5S","title":"Effective relative cycles · Lemma 0H5S","summary":"Let f : X → S be a morphism of schemes. Assume S is locally Noetherian and f is locally of finite type. Let r ≥ 0 be an integer. Let α be a relative r-cycle on X/S. If α is effective, then any restriction, base change, flat pullback, or proper pushforward of α is effective.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian\nand $f$ is locally of finite type. Let $r \\geq 0$ be an integer. Let\n$\\alpha$ be a relative $r$-cycle on $X/S$. If $\\alpha$ is effective,\nthen any restriction, base change, flat pullback, or proper pushforward\nof $\\alpha$ is effective.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Effective relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5S","source_file":"relative-cycles.tex","source_line":1666,"source_end_line":1673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1666-L1673","statement_sha256":"8f0467f413f2613cb66ec392168934091e8bab26006152c81df95c986c78093b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10506,"rank":10506,"depth":0,"x":2595.402,"y":785.875,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5T","tag":"0H5T","title":"Effective relative cycles · Lemma 0H5T","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r ≥ 0 be an integer. Let α be a relative r-cycle on X/S. Then to check that α is effective we may work Zariski locally on X and S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r \\geq 0$ be an integer. Let $\\alpha$\nbe a relative $r$-cycle on $X/S$. Then to check that $\\alpha$ is effective\nwe may work Zariski locally on $X$ and $S$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Effective relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5T","source_file":"relative-cycles.tex","source_line":1679,"source_end_line":1685,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1679-L1685","statement_sha256":"a862554e97b7277f9bb508cd3a6c54032c41da418af6a33bdd6bd745ff697374","origin":"The Stacks Project","memory_eligible":false,"source_rank":10507,"rank":10507,"depth":0,"x":2309.634,"y":667.401,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5U","tag":"0H5U","title":"Effective relative cycles · Lemma 0H5U","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r ≥ 0 be an integer. Let α be a relative r-cycle on X/S. Let g : S' → S be a surjective morphism. Then α is effective if and only if the base change g^*α is effective.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r \\geq 0$ be an integer. Let $\\alpha$\nbe a relative $r$-cycle on $X/S$. Let $g : S' \\to S$ be a surjective morphism.\nThen $\\alpha$ is effective if and only if the base change $g^*\\alpha$\nis effective.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Effective relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5U","source_file":"relative-cycles.tex","source_line":1691,"source_end_line":1698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1691-L1698","statement_sha256":"b9d135da09f4073eb8b3ba46af66df5deaeae132ec1461755d5215d9007bdc6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10508,"rank":10508,"depth":0,"x":2615.858,"y":592.555,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5V","tag":"0H5V","title":"Effective relative cycles · Lemma 0H5V","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r, e ≥ 0 be integers. Let α be a relative r-cycle on X/S. Let (f_i : X_i → X) be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension e. Then α is effective if and only if each flat pullback f_i^*α is effective.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r, e \\geq 0$ be integers.\nLet $\\alpha$ be a relative $r$-cycle on $X/S$.\nLet $\\{f_i : X_i \\to X\\}$ be a jointly surjective family\nof flat morphisms, locally of finite type, and of relative dimension $e$.\nThen $\\alpha$ is effective if and only if each flat\npullback $f_i^*\\alpha$ is effective.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Effective relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5V","source_file":"relative-cycles.tex","source_line":1704,"source_end_line":1713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1704-L1713","statement_sha256":"23176b099c73d15964089b1f1e14d126d57b57862299397c879297ffaa7170f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10509,"rank":10509,"depth":0,"x":2450.119,"y":821.676,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5W","tag":"0H5W","title":"Effective relative cycles · Lemma 0H5W","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r, e ≥ 0 be integers. Let α be a relative r-cycle on X/S. If α is effective, then Supp(α) is closed in X.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r, e \\geq 0$ be integers.\nLet $\\alpha$ be a relative $r$-cycle on $X/S$.\nIf $\\alpha$ is effective, then $\\text{Supp}(\\alpha)$ is\nclosed in $X$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Effective relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5W","source_file":"relative-cycles.tex","source_line":1719,"source_end_line":1726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1719-L1726","statement_sha256":"1823de8eef53e79d42a027858558e62efe298b5b7c0980f6b2facb8ae859eaca","origin":"The Stacks Project","memory_eligible":false,"source_rank":10510,"rank":10510,"depth":33,"x":2388.034,"y":558.485,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H5X","tag":"0H5X","title":"Effective relative cycles · Lemma 0H5X","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r, e ≥ 0 be integers. Let α be a relative r-cycle on X/S. If α is effective, then α is equidimensional.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r, e \\geq 0$ be integers.\nLet $\\alpha$ be a relative $r$-cycle on $X/S$.\nIf $\\alpha$ is effective, then $\\alpha$ is equidimensional.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Effective relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H5X","source_file":"relative-cycles.tex","source_line":1762,"source_end_line":1768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1762-L1768","statement_sha256":"8cfd294ef5194e16e703819c93708b6741a7d0127e8a4a2790907d6eb0b6870f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10511,"rank":10511,"depth":34,"x":2645.657,"y":717.448,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H61","tag":"0H61","title":"Proper relative cycles · Definition 0H61","summary":"Let f : X → S be a morphism of schemes. Assume S is locally Noetherian and f is locally of finite type. Let r ≥ 0 be an integer. We say a relative r-cycle α on X/S is a proper relative cycle if the support of α (Remark [Tag 0H4U]) is contained in a closed subset W ⊂ X proper over S (Cohomology of Schemes, Definition [Tag 0CYM]). The group of all proper relative r-cycles on X/S is denoted c(X/S, r).","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian\nand $f$ is locally of finite type. Let $r \\geq 0$ be an integer. We say a\nrelative $r$-cycle $\\alpha$ on $X/S$ is a {\\it proper relative cycle}\nif the support of $\\alpha$ (Remark \\ref{remark-supports-family})\nis contained in a closed subset $W \\subset X$ proper over $S$\n(Cohomology of Schemes, Definition \\ref{coherent-definition-proper-over-base}).\nThe group of all proper relative $r$-cycles on $X/S$ is\ndenoted $c(X/S, r)$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Proper relative cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H61","source_file":"relative-cycles.tex","source_line":1879,"source_end_line":1889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1879-L1889","statement_sha256":"7579be8ed8d2814c30e477df189e2e684602f493727f6645431a8dd9a2d7e2d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10512,"rank":10512,"depth":20,"x":2327.619,"y":746.434,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H62","tag":"0H62","title":"Proper relative cycles · Lemma 0H62","summary":"Let f : X → S be a morphism of schemes. Assume S is locally Noetherian and f is locally of finite type. Let r ≥ 0 be an integer. Let α be a relative r-cycle on X/S. If α is proper, then any base change α is proper.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ is locally Noetherian\nand $f$ is locally of finite type. Let $r \\geq 0$ be an integer. Let\n$\\alpha$ be a relative $r$-cycle on $X/S$. If $\\alpha$ is proper,\nthen any base change $\\alpha$ is proper.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Proper relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H62","source_file":"relative-cycles.tex","source_line":1898,"source_end_line":1904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1898-L1904","statement_sha256":"41e71cb6c3da9fb715407c019ee4f0dfeffb9dc393ae6070263875cac595d7e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10513,"rank":10513,"depth":0,"x":2538.983,"y":544.447,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H63","tag":"0H63","title":"Proper relative cycles · Lemma 0H63","summary":"Let f : X → S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r ≥ 0 be an integer. Let α be a relative r-cycle on X/S. Let (g_i : S_i → S) be a h covering. Then α is proper if and only if each base change g_i^*α is proper.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. Assume $S$ locally Noetherian\nand $f$ locally of finite type. Let $r \\geq 0$ be an integer. Let $\\alpha$\nbe a relative $r$-cycle on $X/S$. Let $\\{g_i : S_i \\to S\\}$\nbe a h covering. Then $\\alpha$ is proper\nif and only if each base change $g_i^*\\alpha$ is proper.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Proper relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H63","source_file":"relative-cycles.tex","source_line":1910,"source_end_line":1917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L1910-L1917","statement_sha256":"4f91dd3f8376010146441b1e3cae5fe9ad0d2d0616d04a9f356b95247889e352","origin":"The Stacks Project","memory_eligible":false,"source_rank":10514,"rank":10514,"depth":21,"x":2545.563,"y":813.522,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H66","tag":"0H66","title":"Action on cycles · Lemma 0H66","summary":"The construction above is bilinear, i.e., we have (α_1 + α_2) ∩ β = α_1 ∩ β + α_2 ∩ β and α ∩ (β_1 + β_2) = α ∩ β_1 + α ∩ β_2.","statement_latex":"The construction above is bilinear, i.e., we have\n$(\\alpha_1 + \\alpha_2) \\cap \\beta = \\alpha_1 \\cap \\beta +\n\\alpha_2 \\cap \\beta$ and $\\alpha \\cap (\\beta_1 + \\beta_2) =\n\\alpha \\cap \\beta_1 + \\alpha \\cap \\beta_2$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Action on cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H66","source_file":"relative-cycles.tex","source_line":2017,"source_end_line":2023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2017-L2023","statement_sha256":"94c993f9d40297211fce62725ab1c547b5678833c267f7effe445228754773d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10515,"rank":10515,"depth":0,"x":2324.165,"y":618.7,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H67","tag":"0H67","title":"Action on cycles · Lemma 0H67","summary":"If U ⊂ X and V ⊂ Y are open and f(U) ⊂ V, then (α ∩ β)|_U is equal to α|_U ∩ β|_V.","statement_latex":"If $U \\subset X$ and $V \\subset Y$ are open and $f(U) \\subset V$, then\n$(\\alpha \\cap \\beta)|_U$ is equal to $\\alpha|_U \\cap \\beta|_V$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Action on cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H67","source_file":"relative-cycles.tex","source_line":2029,"source_end_line":2033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2029-L2033","statement_sha256":"d841701c514f5764b7674edb638ff49558f36039b3d80f734c57ace49ba09557","origin":"The Stacks Project","memory_eligible":false,"source_rank":10516,"rank":10516,"depth":0,"x":2644.328,"y":636.745,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H68","tag":"0H68","title":"Action on cycles · Lemma 0H68","summary":"Forming α ∩ β is compatible with flat base change and flat pullback (see proof for elucidation).","statement_latex":"Forming $\\alpha \\cap \\beta$ is compatible with flat base change\nand flat pullback (see proof for elucidation).","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Action on cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H68","source_file":"relative-cycles.tex","source_line":2040,"source_end_line":2044,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2040-L2044","statement_sha256":"37dd62152dc6da12b2d467034b30629b5d773e47f6ae495cc0f95c04fefc2002","origin":"The Stacks Project","memory_eligible":false,"source_rank":10517,"rank":10517,"depth":30,"x":2393.548,"y":805.232,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H69","tag":"0H69","title":"Action on cycles · Lemma 0H69","summary":"Let (S, δ) and f : X → Y be as above. Let F be a coherent O_X-module with dim(Supp(F_y)) ≤ r for all y ∈ Y. Let G be a coherent O_Y-module with dim_δ(Supp(G)) ≤ e. Set α = [F/X/Y]_r (Example [Tag 0H4Q]) and β = [G]_e (Chow Homology, Definition [Tag 02QX]). If F is flat over Y, then α ∩ β = [F ⊗_O_X f^*G]_r + e.","statement_latex":"Let $(S, \\delta)$ and $f : X \\to Y$ be as above.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module\nwith $\\dim(\\text{Supp}(\\mathcal{F}_y)) \\leq r$ for all $y \\in Y$.\nLet $\\mathcal{G}$ be a coherent $\\mathcal{O}_Y$-module\nwith $\\dim_\\delta(\\text{Supp}(\\mathcal{G})) \\leq e$.\nSet $\\alpha = [\\mathcal{F}/X/Y]_r$\n(Example \\ref{example-family-associated-module}) and\n$\\beta = [\\mathcal{G}]_e$ (Chow Homology, Definition\n\\ref{chow-definition-cycle-associated-to-coherent-sheaf}).\nIf $\\mathcal{F}$ is flat over $Y$, then $\\alpha \\cap \\beta =\n[\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}]_{r + e}$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Action on cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H69","source_file":"relative-cycles.tex","source_line":2141,"source_end_line":2154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2141-L2154","statement_sha256":"3946ed527ada7ee61af78c6b4e4b2b6f04e9dfc10681797ee15de443556d87bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10518,"rank":10518,"depth":22,"x":2443.013,"y":538.497,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6A","tag":"0H6A","title":"Action on cycles · Lemma 0H6A","summary":"Let (S, δ) and f : X → Y be as above. Let Z ⊂ X be a closed subscheme of relative dimension ≤ r over Y. Set α = [Z/X/Y]_r (Example [Tag 0H4S]). Let W ⊂ Y be a closed subscheme of δ-dimension ≤ e. Set β = [W]_e (Chow Homology, Definition [Tag 02QU]). If Z is flat over Y, then α ∩ β = [Z ×_Y W]_r + e.","statement_latex":"Let $(S, \\delta)$ and $f : X \\to Y$ be as above. Let $Z \\subset X$\nbe a closed subscheme of relative dimension $\\leq r$ over $Y$.\nSet $\\alpha = [Z/X/Y]_r$ (Example \\ref{example-family-associated-closed}).\nLet $W \\subset Y$ be a closed subscheme of $\\delta$-dimension $\\leq e$.\nSet $\\beta = [W]_e$ (Chow Homology, Definition\n\\ref{chow-definition-cycle-associated-to-closed-subscheme}).\nIf $Z$ is flat over $Y$, then $\\alpha \\cap \\beta = [Z \\times_Y W]_{r + e}$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Action on cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6A","source_file":"relative-cycles.tex","source_line":2214,"source_end_line":2223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2214-L2223","statement_sha256":"89be554f400a3c16f58db8e45323376f5c170706f5ac4d7947c910cd6a11d0f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10519,"rank":10519,"depth":23,"x":2621.17,"y":763.417,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6B","tag":"0H6B","title":"Action on cycles · Lemma 0H6B","summary":"Let (S, δ) be as above. Let xymatrix X' ar[r]_f ar[d] & X ar[d] Y' ar[r]^g & Y be a cartesian diagram of schemes locally of finite type over S with g proper. Let r, e ≥ 0. Let α be a family of r-cycles on the fibres of X/Y. Let β' ∈ Z_e(Y'). Then we have f_*(g^*α ∩ β') = α ∩ g_*β'.","statement_latex":"Let $(S, \\delta)$ be as above. Let\n$$\n\\xymatrix{\nX' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian diagram of schemes locally of finite type over $S$\nwith $g$ proper. Let $r, e \\geq 0$. Let $\\alpha$ be a family of\n$r$-cycles on the fibres of $X/Y$. Let $\\beta' \\in Z_e(Y')$.\nThen we have $f_*(g^*\\alpha \\cap \\beta') = \\alpha \\cap g_*\\beta'$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Action on cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6B","source_file":"relative-cycles.tex","source_line":2251,"source_end_line":2264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2251-L2264","statement_sha256":"7fe81f018465ad536e7cb214ee3b402dfa5939195787670c6980519b55c0a68e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10520,"rank":10520,"depth":33,"x":2308.696,"y":698.607,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6D","tag":"0H6D","title":"Action on chow groups · Lemma 0H6D","summary":"Let (S, δ) be as in Section [Tag 0H65]. Let f : X' → X be a proper morphism of schemes locally of finite type over S. Let (L, s, i : D → X) be as in Chow Homology, Definition [Tag 02T8]. Form the diagram xymatrix D' ar[d]_g ar[r]_i' & X' ar[d]^f D ar[r]^i & X as in Chow Homology, Remark [Tag 0B6Y]. If L|_D ≅ O_D, then i^*f_*α' = g_*(i')^*α' in Z_k(D) for any α' ∈ Z_k + 1(X').","statement_latex":"Let $(S, \\delta)$ be as in Section \\ref{section-action}.\nLet $f : X' \\to X$ be a proper morphism of schemes\nlocally of finite type over $S$.\nLet $(\\mathcal{L}, s, i : D \\to X)$ be as in\nChow Homology, Definition \\ref{chow-definition-gysin-homomorphism}.\nForm the diagram\n$$\n\\xymatrix{\nD' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\\nD \\ar[r]^i & X\n}\n$$\nas in Chow Homology, Remark \\ref{chow-remark-pullback-pairs}.\nIf $\\mathcal{L}|_D \\cong \\mathcal{O}_D$, then\n$i^*f_*\\alpha' = g_*(i')^*\\alpha'$ in $Z_k(D)$\nfor any $\\alpha' \\in Z_{k + 1}(X')$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Action on chow groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6D","source_file":"relative-cycles.tex","source_line":2350,"source_end_line":2368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2350-L2368","statement_sha256":"ee6c5ef2ebc9aec6cee4d683c3c956c9cd6513d51c0c398093449a63697b3aab","origin":"The Stacks Project","memory_eligible":false,"source_rank":10521,"rank":10521,"depth":45,"x":2591.435,"y":568.996,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6E","tag":"0H6E","title":"Action on chow groups · Lemma 0H6E","summary":"Let (S, δ) be as in Section [Tag 0H65]. Let X → Y be a morphism of schemes locally of finite type over S. Let r ≥ 0 and let α ∈ z(X/Y, r) be a relative r-cycle on X/Y. Let (L, s, i : D → Y) be as in Chow Homology, Definition [Tag 02T8]. Form the cartesian diagram xymatrix E ar[d] ar[r]_j & X ar[d] D ar[r]^i & Y See Chow Homology, Remark [Tag 0B6Y]. If L|_D ≅ O_D, then for e ∈ Z the diagram xymatrix Z_e(D) ar[rr]_i^*α ∩ - & & Z_e + r(E) Z_e + 1(Y) ar[u]^i^* ar[rr]^α ∩ - &…","statement_latex":"Let $(S, \\delta)$ be as in Section \\ref{section-action}.\nLet $X \\to Y$ be a morphism of schemes\nlocally of finite type over $S$. Let $r \\geq 0$ and let\n$\\alpha \\in z(X/Y, r)$ be a relative $r$-cycle on $X/Y$.\nLet $(\\mathcal{L}, s, i : D \\to Y)$ be as in\nChow Homology, Definition \\ref{chow-definition-gysin-homomorphism}.\nForm the cartesian diagram\n$$\n\\xymatrix{\nE \\ar[d] \\ar[r]_j & X \\ar[d] \\\\\nD \\ar[r]^i & Y\n}\n$$\nSee Chow Homology, Remark \\ref{chow-remark-pullback-pairs}.\nIf $\\mathcal{L}|_D \\cong \\mathcal{O}_D$, then for $e \\in \\mathbf{Z}$\nthe diagram\n$$\n\\xymatrix{\nZ_e(D) \\ar[rr]_{i^*\\alpha \\cap -} & &\nZ_{e + r}(E) \\\\\nZ_{e + 1}(Y) \\ar[u]^{i^*} \\ar[rr]^{\\alpha \\cap -} & &\nZ_{r + e + 1}(X) \\ar[u]_{j^*}\n}\n$$\ncommutes where the vertical arrows $i^*$ and $j^*$ are the\nGysin maps on cycles as in\nChow Homology, Remark \\ref{chow-remark-gysin-on-cycles}.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Action on chow groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6E","source_file":"relative-cycles.tex","source_line":2388,"source_end_line":2417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2388-L2417","statement_sha256":"3b45e3e28943c8350de46a77a4df9f729a83962331397dc369f84555f7ad6f54","origin":"The Stacks Project","memory_eligible":false,"source_rank":10522,"rank":10522,"depth":57,"x":2487.101,"y":825.19,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6F","tag":"0H6F","title":"Action on chow groups · Proposition 0H6F","summary":"Let (S, δ) be as in Section [Tag 0H65]. Let X → Y be a morphism of schemes locally of finite type over S. Let r ≥ 0 and let α ∈ z(X/Y, r) be a relative r-cycle on X/Y. The rule that to every morphism g : Y' → Y locally of finite type and every e ∈ Z associates the operation g^*α ∩ - : Z_e(Y') → Z_r + e(X') where X' = Y' ×_Y X factors through rational equivalence to define a bivariant class c(α) ∈ A^-r(X → Y).","statement_latex":"Let $(S, \\delta)$ be as in Section \\ref{section-action}. Let $X \\to Y$\nbe a morphism of schemes locally of finite type over $S$. Let $r \\geq 0$\nand let $\\alpha \\in z(X/Y, r)$ be a relative $r$-cycle on $X/Y$.\nThe rule that to every morphism $g : Y' \\to Y$ locally of finite type\nand every $e \\in \\mathbf{Z}$ associates the operation\n$$\ng^*\\alpha \\cap - : Z_e(Y') \\to Z_{r + e}(X')\n$$\nwhere $X' = Y' \\times_Y X$ factors through rational equivalence to\ndefine a bivariant class $c(\\alpha) \\in A^{-r}(X \\to Y)$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Action on chow groups","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6F","source_file":"relative-cycles.tex","source_line":2500,"source_end_line":2512,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2500-L2512","statement_sha256":"2547e0ce9f0c5011cf0d54b8e5ab2e0e42a8e0eedb10df3644fde857bf840ccd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10523,"rank":10523,"depth":58,"x":2357.917,"y":576.893,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6I","tag":"0H6I","title":"Composition of families of cycles on fibres · Lemma 0H6I","summary":"The construction above is bilinear, i.e., we have (α_1 + α_2) ∘ β α_1 ∘ β + α_1 ∘ β and α ∘ (β_1 + β_2) = α ∘ β_1 + α ∘ β_2.","statement_latex":"The construction above is bilinear, i.e., we have\n$(\\alpha_1 + \\alpha_2) \\circ \\beta \\alpha_1 \\circ \\beta +\n\\alpha_1 \\circ \\beta$ and $\\alpha \\circ (\\beta_1 + \\beta_2) =\n\\alpha \\circ \\beta_1 + \\alpha \\circ \\beta_2$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Composition of families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6I","source_file":"relative-cycles.tex","source_line":2567,"source_end_line":2573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2567-L2573","statement_sha256":"e95a2bf98194d2b0dee2f74ffcc89b2f128a97b496e9777d6a21e59a56142c19","origin":"The Stacks Project","memory_eligible":false,"source_rank":10524,"rank":10524,"depth":1,"x":2653.064,"y":686.762,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6J","tag":"0H6J","title":"Composition of families of cycles on fibres · Lemma 0H6J","summary":"If U ⊂ X and V ⊂ Y are open and f(U) ⊂ V, then (α ∘ β)|_U is equal to α|_U ∘ β|_V.","statement_latex":"If $U \\subset X$ and $V \\subset Y$ are open and $f(U) \\subset V$, then\n$(\\alpha \\circ \\beta)|_U$ is equal to $\\alpha|_U \\circ \\beta|_V$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Composition of families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6J","source_file":"relative-cycles.tex","source_line":2580,"source_end_line":2584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2580-L2584","statement_sha256":"030725ca99d8c93d6efd7b512c46866ea8e3b47ba94d3d571414b40f19f6b1de","origin":"The Stacks Project","memory_eligible":false,"source_rank":10525,"rank":10525,"depth":1,"x":2346.85,"y":773.283,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6K","tag":"0H6K","title":"Composition of families of cycles on fibres · Lemma 0H6K","summary":"The formation of α ∘ β is compatible with base change.","statement_latex":"The formation of $\\alpha \\circ \\beta$ is compatible with base change.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Composition of families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6K","source_file":"relative-cycles.tex","source_line":2591,"source_end_line":2594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2591-L2594","statement_sha256":"abbb70bb025ea3801b8f1b96a386049fe7241e8eedef218c0f5d1b2fefa26e98","origin":"The Stacks Project","memory_eligible":false,"source_rank":10526,"rank":10526,"depth":31,"x":2503.184,"y":535.557,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6L","tag":"0H6L","title":"Composition of families of cycles on fibres · Lemma 0H6L","summary":"Let f : X → Y and Y → S be morphisms of schemes, both locally of finite type. Let r, e ≥ 0. Let F be a quasi-coherent O_X-module of finite type, with dim(Supp(F_y)) ≤ r for all y ∈ Y. Let G be a quasi-coherent O_Y-module of finite type, with dim(Supp(G_s)) ≤ e for all s ∈ S. If α = [F/X/Y]_r and β = [G/Y/S]_e (Example [Tag 0H4Q]) and F is flat over Y, then α ∘ β = [F ⊗_O_X f^*G/X/S]_r + e.","statement_latex":"Let $f : X \\to Y$ and $Y \\to S$ be morphisms of schemes, both locally of\nfinite type. Let $r, e \\geq 0$. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module of finite type, with\n$\\dim(\\text{Supp}(\\mathcal{F}_y)) \\leq r$ for all $y \\in Y$.\nLet $\\mathcal{G}$  be a quasi-coherent $\\mathcal{O}_Y$-module of finite\ntype, with $\\dim(\\text{Supp}(\\mathcal{G}_s)) \\leq e$ for all $s \\in S$.\nIf $\\alpha = [\\mathcal{F}/X/Y]_r$ and $\\beta = [\\mathcal{G}/Y/S]_e$\n(Example \\ref{example-family-associated-module}) and $\\mathcal{F}$\nis flat over $Y$, then $\\alpha \\circ \\beta =\n[\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}/X/S]_{r + e}$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Composition of families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6L","source_file":"relative-cycles.tex","source_line":2622,"source_end_line":2634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2622-L2634","statement_sha256":"43edfaf03e33af924fa532533bf6f750005cd2c719f86c1760ba0c8bd8cd5d5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10527,"rank":10527,"depth":23,"x":2579.134,"y":799.752,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6M","tag":"0H6M","title":"Composition of families of cycles on fibres · Lemma 0H6M","summary":"Let f : X → Y and Y → S be morphisms of schemes, both locally of finite type. Let r, e ≥ 0. Let Z ⊂ X be a closed subscheme of relative dimension ≤ r over Y. Let W ⊂ Y be a closed subscheme of relative dimension ≤ e over S. If α = [Z/X/Y]_r and β = [W/Y/S]_e (Example [Tag 0H4S]) and Z is flat over Y, then α ∘ β = [Z ×_Y W/X/S]_r + e.","statement_latex":"Let $f : X \\to Y$ and $Y \\to S$ be morphisms of schemes, both locally of\nfinite type. Let $r, e \\geq 0$. Let $Z \\subset X$ be a closed\nsubscheme of relative dimension $\\leq r$ over $Y$.\nLet $W \\subset Y$ be a closed subscheme of relative dimension $\\leq e$\nover $S$. If $\\alpha = [Z/X/Y]_r$ and $\\beta = [W/Y/S]_e$\n(Example \\ref{example-family-associated-closed}) and $Z$ is flat over $Y$,\nthen $\\alpha \\circ \\beta = [Z \\times_Y W/X/S]_{r + e}$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Composition of families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6M","source_file":"relative-cycles.tex","source_line":2651,"source_end_line":2660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2651-L2660","statement_sha256":"fe9abe2c9fedb027ca9d9c01f2e089fa7fb755d3f1725f805eabff5ae3862fd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10528,"rank":10528,"depth":24,"x":2310.476,"y":647.924,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6N","tag":"0H6N","title":"Composition of families of cycles on fibres · Lemma 0H6N","summary":"Let S be a scheme. Let xymatrix X' ar[r]_f ar[d] & X ar[d] Y' ar[r]^g & Y be a cartesian diagram of schemes locally of finite type over S with g proper. Let r, e ≥ 0. Let α be a family of r-cycles on the fibres of X/Y. Let β' be a family of e-cycles on the fibres of Y'/S. Then we have f_*(g^*(α) ∘ β') = α ∘ g_*β'.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r]_f \\ar[d] & X \\ar[d] \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian diagram of schemes locally of finite type over $S$\nwith $g$ proper. Let $r, e \\geq 0$. Let $\\alpha$ be a family of\n$r$-cycles on the fibres of $X/Y$. Let $\\beta'$ be a family of\n$e$-cycles on the fibres of $Y'/S$. Then we have\n$f_*(g^*(\\alpha) \\circ \\beta') = \\alpha \\circ g_*\\beta'$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Composition of families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6N","source_file":"relative-cycles.tex","source_line":2683,"source_end_line":2697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2683-L2697","statement_sha256":"b875825b2bc033c3926707bece6cc813279d6a069cfce93dd786f28a23957bcc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10529,"rank":10529,"depth":34,"x":2630.908,"y":607.408,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6P","tag":"0H6P","title":"Composition of families of cycles on fibres · Lemma 0H6P","summary":"Let (S, δ) be as in Chow Homology, Situation [Tag 02QL]. Let X → Y → Z be morphisms of schemes locally of finite type over S. Let r, s, e ≥ 0. Then (α ∘ β) ∩ γ = α ∩ (β ∩ γ) in Z_r + s + e(X) where α is a family of r-cycles on fibres of X/Y, β is a family of s-cycles on fibres of Y/Z, and γ ∈ Z_e(Z).","statement_latex":"Let $(S, \\delta)$ be as in Chow Homology, Situation \\ref{chow-situation-setup}.\nLet $X \\to Y \\to Z$ be morphisms of schemes locally of finite type over $S$.\nLet $r, s, e \\geq 0$. Then\n$$\n(\\alpha \\circ \\beta) \\cap \\gamma = \\alpha \\cap (\\beta \\cap \\gamma)\n\\quad\\text{in}\\quad Z_{r + s + e}(X)\n$$\nwhere $\\alpha$ is a family of $r$-cycles on fibres of $X/Y$,\n$\\beta$ is a family of $s$-cycles on fibres of $Y/Z$, and $\\gamma \\in Z_e(Z)$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Composition of families of cycles on fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6P","source_file":"relative-cycles.tex","source_line":2704,"source_end_line":2715,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2704-L2715","statement_sha256":"9577a89e1442201e10d2814e366a82f7fb7b2e7293578842a616057a86ee2257","origin":"The Stacks Project","memory_eligible":false,"source_rank":10530,"rank":10530,"depth":1,"x":2427.061,"y":819.258,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6R","tag":"0H6R","title":"Composition of relative cycles · Lemma 0H6R","summary":"If α and β are relative cycles, then so is α ∘ β.","statement_latex":"If $\\alpha$ and $\\beta$ are relative cycles, then so is $\\alpha \\circ \\beta$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Composition of relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6R","source_file":"relative-cycles.tex","source_line":2786,"source_end_line":2789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2786-L2789","statement_sha256":"f035b96c99a3d4779baca206c5896d204192bc0912efc2b6bb157e5489d01d13","origin":"The Stacks Project","memory_eligible":false,"source_rank":10531,"rank":10531,"depth":60,"x":2406.996,"y":547.178,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6S","tag":"0H6S","title":"Composition of relative cycles · Lemma 0H6S","summary":"Let f : X → Y and g : Y → S be a morphisms of schemes. Assume S locally Noetherian, g locally of finite type and flat of relative dimension e ge 0, and f locally of finite type and flat of relative dimension r ≥ 0. Then [X/X/Y]_r ∘ [Y/Y/S]_e = [X/X/S]_r + e in z(X/S, r + e).","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to S$ be a morphisms of schemes.\nAssume $S$ locally Noetherian, $g$ locally of finite type and\nflat of relative dimension $e \\ge 0$, and $f$ locally of finite type\nand flat of relative dimension $r \\geq 0$. Then\n$[X/X/Y]_r \\circ [Y/Y/S]_e = [X/X/S]_{r + e}$ in $z(X/S, r + e)$.","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Composition of relative cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6S","source_file":"relative-cycles.tex","source_line":2846,"source_end_line":2853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2846-L2853","statement_sha256":"b77b118a09d03e916e741712eb0598fa19209d09749a985d7784e3fa2bbec033","origin":"The Stacks Project","memory_eligible":false,"source_rank":10532,"rank":10532,"depth":25,"x":2640.759,"y":736.557,"cluster":"divisors-intersection-theory"},{"id":"stacks:0H6W","tag":"0H6W","title":"Relative cycles in the non-Noetherian case · Lemma 0H6W","summary":"Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a morphism of finite presentation. Let r ≥ 0 and let α be a family of r-cycles on fibres of X/S. The following are equivalent • there exists a cartesian diagram xymatrix X ar[r] ar[d] & X_0 ar[d] S ar[r] & S_0 where X_0 → S_0 is a finite type morphism of Noetherian schemes and α_0 ∈ z(X_0/S_0, r) such that α is the base change of α_0 by S → S_0 • there exists a completely decomposed proper morphism g :…","statement_latex":"Let $S$ be a quasi-compact and quasi-separated scheme.\nLet $f : X \\to S$ be a morphism of finite presentation.\nLet $r \\geq 0$ and let $\\alpha$ be a family of $r$-cycles on fibres of $X/S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists a cartesian diagram\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d] & X_0 \\ar[d] \\\\\nS \\ar[r] & S_0\n}\n$$\nwhere $X_0 \\to S_0$ is a finite type morphism of Noetherian schemes\nand $\\alpha_0 \\in z(X_0/S_0, r)$ such that $\\alpha$ is the base change\nof $\\alpha_0$ by $S \\to S_0$\n\\item there exists a completely decomposed proper morphism $g : S' \\to S$\nof finite presentation such that $g^*\\alpha$ is in the image of\n(\\ref{equation-cycle-classes-general}).\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Relative Cycles","chapter_id":"relative-cycles","section":"Relative cycles in the non-Noetherian case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H6W","source_file":"relative-cycles.tex","source_line":2937,"source_end_line":2958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/relative-cycles.tex#L2937-L2958","statement_sha256":"70ea079db5ed0fe37417e9ace520e8be5678a986cf127701f3b9d3b2e026de1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10533,"rank":10533,"depth":61,"x":2315.858,"y":729.55,"cluster":"divisors-intersection-theory"},{"id":"stacks:0F6F","tag":"0F6F","title":"Growing sections · Lemma 0F6F","summary":"Let X be a scheme. Let F be an abelian sheaf on X_etale. Let φ : U' → U be a morphism of X_etale. Let Z' ⊂ U' be a closed subscheme such that Z' → U' → U is a closed immersion with image Z ⊂ U. Then there is a canonical bijection (s ∈ F(U) mid Supp(s) ⊂ Z) = (s' ∈ F(U') mid Supp(s') ⊂ Z') which is given by restriction if φ^-1(Z) = Z'.","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.\nLet $\\varphi : U' \\to U$ be a morphism of $X_\\etale$. Let $Z' \\subset U'$ be a\nclosed subscheme such that $Z' \\to U' \\to U$ is a closed immersion\nwith image $Z \\subset U$. Then there is a canonical bijection\n$$\n\\{s \\in \\mathcal{F}(U) \\mid \\text{Supp}(s) \\subset Z\\} =\n\\{s' \\in \\mathcal{F}(U') \\mid \\text{Supp}(s') \\subset Z'\\}\n$$\nwhich is given by restriction if $\\varphi^{-1}(Z) = Z'$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Growing sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6F","source_file":"more-etale.tex","source_line":42,"source_end_line":53,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L42-L53","statement_sha256":"8e40948cde204f7400b3a7a2b55268e16f68b5fba5a1af23f1a3c1495d3209a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10534,"rank":10534,"depth":0,"x":1701.304,"y":1228.508,"cluster":"tale-geometry"},{"id":"stacks:0F6G","tag":"0F6G","title":"Growing sections · Lemma 0F6G","summary":"Let X be a scheme. Let Z ⊂ X be a locally closed subscheme. Let F be an abelian sheaf on X_etale. Given U, U' ⊂ X open containing Z as a closed subscheme, there is a canonical bijection (s ∈ F(U) mid Supp(s) ⊂ Z) = (s ∈ F(U') mid Supp(s) ⊂ Z) which is given by restriction if U' ⊂ U.","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a locally closed subscheme.\nLet $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Given\n$U, U' \\subset X$ open containing $Z$ as a closed subscheme,\nthere is a canonical bijection\n$$\n\\{s \\in \\mathcal{F}(U) \\mid \\text{Supp}(s) \\subset Z\\} =\n\\{s \\in \\mathcal{F}(U') \\mid \\text{Supp}(s) \\subset Z\\}\n$$\nwhich is given by restriction if $U' \\subset U$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Growing sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6G","source_file":"more-etale.tex","source_line":76,"source_end_line":87,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L76-L87","statement_sha256":"eef1a95e5b49792f0c44541ad1d1342043c1a8602de60130e92211b05f619d9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10535,"rank":10535,"depth":1,"x":1419.276,"y":1143.568,"cluster":"tale-geometry"},{"id":"stacks:0F4X","tag":"0F4X","title":"Sections with compact support · Lemma 0F4X","summary":"Let f : X → Y be a morphism of schemes which is locally of finite type. Let F be an abelian sheaf on X_etale. The rule Y_etale → Ab, V ↦ (s ∈ f_*F(V) = F(X_V) mid Supp(s) ⊂ X_V is proper over V) is an abelian subsheaf of f_*F.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is locally of finite type.\nLet $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. The rule\n$$\nY_\\etale \\longrightarrow \\textit{Ab},\\quad\nV \\longmapsto \\{s \\in f_*\\mathcal{F}(V) = \\mathcal{F}(X_V) \\mid\n\\text{Supp}(s) \\subset X_V \\text{ is proper over }V\\}\n$$\nis an abelian subsheaf of $f_*\\mathcal{F}$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4X","source_file":"more-etale.tex","source_line":161,"source_end_line":171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L161-L171","statement_sha256":"d33515bded72b0e82c4531977022fc26a5506171a3b8c9c0e1ae1198b419f96d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10536,"rank":10536,"depth":0,"x":1695.719,"y":1046.111,"cluster":"tale-geometry"},{"id":"stacks:0F4Y","tag":"0F4Y","title":"Sections with compact support · Lemma 0F4Y","summary":"Let j : U → X be a separated étale morphism. Let F be an abelian sheaf on U_etale. The image of the injective map j_!F → j_*F of Étale Cohomology, Lemma [Tag 0F4L] is the subsheaf of Lemma [Tag 0F4X].","statement_latex":"Let $j : U \\to X$ be a separated \\'etale morphism. Let $\\mathcal{F}$\nbe an abelian sheaf on $U_\\etale$. The image of the injective map\n$j_!\\mathcal{F} \\to j_*\\mathcal{F}$ of\n\\'Etale Cohomology, Lemma\n\\ref{etale-cohomology-lemma-shriek-into-star-separated-etale}\nis the subsheaf of Lemma \\ref{lemma-f-shriek-separated}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4Y","source_file":"more-etale.tex","source_line":199,"source_end_line":207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L199-L207","statement_sha256":"d6ab20a85c6a5effb8e78a1ce51b6d1e6c976aaebd3e849b9c74f0dddfb9186d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10537,"rank":10537,"depth":18,"x":1570.167,"y":1274.978,"cluster":"tale-geometry"},{"id":"stacks:0F4Z","tag":"0F4Z","title":"Sections with compact support · Definition 0F4Z","summary":"Let f : X → Y be a morphism of schemes which is separated (!) and locally of finite type. Let F be an abelian sheaf on X_etale. The subsheaf f_!F ⊂ f_*F constructed in Lemma [Tag 0F4X] is called the direct image with compact support.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is separated (!) and\nlocally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on\n$X_\\etale$. The subsheaf $f_!\\mathcal{F} \\subset f_*\\mathcal{F}$\nconstructed in Lemma \\ref{lemma-f-shriek-separated} is called the\n{\\it direct image with compact support}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4Z","source_file":"more-etale.tex","source_line":249,"source_end_line":256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L249-L256","statement_sha256":"142b04bcfec1920c0baca1d551fe4862e0ea9b04d1f9640cb6e82a363d4df62c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10538,"rank":10538,"depth":1,"x":1478.642,"y":1034.824,"cluster":"tale-geometry"},{"id":"stacks:0F51","tag":"0F51","title":"Sections with compact support · Lemma 0F51","summary":"Let f : X → Y be a proper morphism of schemes. Then f_! = f_*.","statement_latex":"Let $f : X \\to Y$ be a proper morphism of schemes.\nThen $f_! = f_*$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F51","source_file":"more-etale.tex","source_line":263,"source_end_line":267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L263-L267","statement_sha256":"58f8a8f08c91390bf95006d12c5d19cdced2623987e43896df05c46421de3b03","origin":"The Stacks Project","memory_eligible":false,"source_rank":10539,"rank":10539,"depth":0,"x":1739.419,"y":1160.054,"cluster":"tale-geometry"},{"id":"stacks:0F52","tag":"0F52","title":"Sections with compact support · Lemma 0F52","summary":"Let Y be a scheme. Let j : X → overlineX be an open immersion of schemes over Y with overlineX proper over Y. Denote f : X → Y and overlinef : overlineX → Y the structure morphisms. For F ∈ Ab(X_etale) there is a canonical isomorphism (see proof) f_!F → overlinef_!j_!F As we have overlinef_! = overlinef_* by Lemma [Tag 0F51] we obtain overlinef_* ∘ j_! = f_! as functors Ab(X_etale) → Ab(Y_etale).","statement_latex":"Let $Y$ be a scheme. Let $j : X \\to \\overline{X}$ be an open\nimmersion of schemes over $Y$ with $\\overline{X}$ proper over $Y$.\nDenote $f : X \\to Y$ and $\\overline{f} : \\overline{X} \\to Y$\nthe structure morphisms. For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$\nthere is a canonical isomorphism (see proof)\n$$\nf_!\\mathcal{F} \\longrightarrow \\overline{f}_!j_!\\mathcal{F}\n$$\nAs we have $\\overline{f}_! = \\overline{f}_*$ by\nLemma \\ref{lemma-proper-f-shriek} we obtain\n$\\overline{f}_* \\circ j_! = f_!$ as functors\n$\\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F52","source_file":"more-etale.tex","source_line":313,"source_end_line":327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L313-L327","statement_sha256":"ce5b3b50677e845f839638f6b68fd57b3f13ec5978e7d747781513359500a1f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10540,"rank":10540,"depth":52,"x":1446.237,"y":1215.718,"cluster":"tale-geometry"},{"id":"stacks:0F72","tag":"0F72","title":"Sections with compact support · Definition 0F72","summary":"Let X be a separated scheme locally of finite type over a field k. Let F be an abelian sheaf on X_etale. We let H^0_c(X, F) ⊂ H^0(X, F) be the set of sections whose support is proper over k. Elements of H^0_c(X, F) are called sections with compact support.","statement_latex":"Let $X$ be a separated scheme locally of finite type over a field $k$.\nLet $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. We let\n$H^0_c(X, \\mathcal{F}) \\subset H^0(X, \\mathcal{F})$ be the\nset of sections whose support is proper over $k$. Elements of\n$H^0_c(X, \\mathcal{F})$ are called {\\it sections with compact support}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F72","source_file":"more-etale.tex","source_line":349,"source_end_line":356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L349-L356","statement_sha256":"8151e6a2ecd48ccfc4a12a9505fd48fa1a7946d1d4febc6eac8da17b05464719","origin":"The Stacks Project","memory_eligible":false,"source_rank":10541,"rank":10541,"depth":0,"x":1617.769,"y":1008.185,"cluster":"tale-geometry"},{"id":"stacks:0F73","tag":"0F73","title":"Sections with compact support · Lemma 0F73","summary":"Let X be a proper scheme over a field k. Then H^0_c(X, F) = H^0(X, F).","statement_latex":"Let $X$ be a proper scheme over a field $k$. Then\n$H^0_c(X, \\mathcal{F}) = H^0(X, \\mathcal{F})$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F73","source_file":"more-etale.tex","source_line":362,"source_end_line":366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L362-L366","statement_sha256":"6eea240260bf9c3b909964cc021d205b709dea1945f9ebbf937079f15fd5c0c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10542,"rank":10542,"depth":0,"x":1658.201,"y":1258.703,"cluster":"tale-geometry"},{"id":"stacks:0F75","tag":"0F75","title":"Sections with compact support · Lemma 0F75","summary":"Let k be a field. Let j : X → overlineX be an open immersion of schemes over k with overlineX proper over k. For F ∈ Ab(X_etale) there is a canonical isomorphism (see proof) H^0_c(X, F) → H^0_c(overlineX, j_!F) = H^0(overlineX, j_!F) where we have the equality on the right by Lemma [Tag 0F73].","statement_latex":"Let $k$ be a field. Let $j : X \\to \\overline{X}$ be an open\nimmersion of schemes over $k$ with $\\overline{X}$ proper over $k$.\nFor $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$\nthere is a canonical isomorphism (see proof)\n$$\nH^0_c(X, \\mathcal{F}) \\longrightarrow\nH^0_c(\\overline{X}, j_!\\mathcal{F}) =\nH^0(\\overline{X}, j_!\\mathcal{F})\n$$\nwhere we have the equality on the right by\nLemma \\ref{lemma-proper-compact-support}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F75","source_file":"more-etale.tex","source_line":384,"source_end_line":397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L384-L397","statement_sha256":"c2166844c2c752731fc0548735e98c08fef77a1a688961f88078b00045bb8555","origin":"The Stacks Project","memory_eligible":false,"source_rank":10543,"rank":10543,"depth":52,"x":1426.783,"y":1096.816,"cluster":"tale-geometry"},{"id":"stacks:0F76","tag":"0F76","title":"Sections with compact support · Lemma 0F76","summary":"Let f : X → Y be a morphism of schemes which is separated and locally of finite type. Let F be an abelian sheaf on X_etale. Then there is a canonical isomorphism (f_!F)_overliney → H^0_c(X_overliney, F|_X_overliney) for any geometric point overliney : Spec(k) → Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is separated and\nlocally of finite type. Let $\\mathcal{F}$ be an abelian sheaf on\n$X_\\etale$. Then there is a canonical isomorphism\n$$\n(f_!\\mathcal{F})_{\\overline{y}}\n\\longrightarrow\nH^0_c(X_{\\overline{y}}, \\mathcal{F}|_{X_{\\overline{y}}})\n$$\nfor any geometric point $\\overline{y} : \\Spec(k) \\to Y$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F76","source_file":"more-etale.tex","source_line":413,"source_end_line":424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L413-L424","statement_sha256":"edc2f92b2a451cf795b90a3c47c13b10ab6860e08a1b2c5717c5384024c83c4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10544,"rank":10544,"depth":61,"x":1727.799,"y":1084.87,"cluster":"tale-geometry"},{"id":"stacks:0F55","tag":"0F55","title":"Sections with compact support · Lemma 0F55","summary":"Consider a cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y of schemes with f separated and locally of finite type. For any abelian sheaf F on X_etale we have f'_!(g')^-1F = g^-1f_!F.","statement_latex":"Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nof schemes with $f$ separated and locally of finite type.\nFor any abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have\n$f'_!(g')^{-1}\\mathcal{F} = g^{-1}f_!\\mathcal{F}$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F55","source_file":"more-etale.tex","source_line":545,"source_end_line":557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L545-L557","statement_sha256":"eb50219ba6e55acee3778d896305117d3ab27a048a3c9944d614b95411c95cea","origin":"The Stacks Project","memory_eligible":false,"source_rank":10545,"rank":10545,"depth":62,"x":1515.31,"y":1264.594,"cluster":"tale-geometry"},{"id":"stacks:0F50","tag":"0F50","title":"Sections with compact support · Lemma 0F50","summary":"Let f : X → Y and g : Y → Z be composable morphisms of schemes which are separated and locally of finite type. Let F be an abelian sheaf on X_etale. Then g_!f_!F = (g ∘ f)_!F as subsheaves of (g ∘ f)_*F.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be composable morphisms of schemes which\nare separated and locally of finite type. Let $\\mathcal{F}$ be an abelian\nsheaf on $X_\\etale$. Then $g_!f_!\\mathcal{F} = (g \\circ f)_!\\mathcal{F}$\nas subsheaves of $(g \\circ f)_*\\mathcal{F}$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F50","source_file":"more-etale.tex","source_line":614,"source_end_line":620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L614-L620","statement_sha256":"46e02ab878fa2de43144a070ed9ec5f8903669fd9e18ef72678ce168db8be6a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10546,"rank":10546,"depth":21,"x":1527.474,"y":1011.34,"cluster":"tale-geometry"},{"id":"stacks:0F54","tag":"0F54","title":"Sections with compact support · Lemma 0F54","summary":"Let f : X → Y be morphism of schemes which is separated and locally of finite type. Let X = ⋃_i ∈ I X_i be an open covering such that for all i, j ∈ I there exists a k with X_i ∪ X_j ⊂ X_k. Denote f_i : X_i → Y the restriction of f. Then f_!F = colim_i ∈ I f_i, !(F|_X_i) functorially in F ∈ Ab(X_etale) where the transition maps are the ones constructed in Remark [Tag 0F53].","statement_latex":"Let $f : X \\to Y$ be morphism of schemes which is separated and\nlocally of finite type. Let $X = \\bigcup_{i \\in I} X_i$ be an\nopen covering such that for all $i, j \\in I$ there exists a $k$\nwith $X_i \\cup X_j \\subset X_k$. Denote $f_i : X_i \\to Y$\nthe restriction of $f$. Then\n$$\nf_!\\mathcal{F} = \\colim_{i \\in I} f_{i, !}(\\mathcal{F}|_{X_i})\n$$\nfunctorially in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$\nwhere the transition maps are the ones constructed in\nRemark \\ref{remark-covariance-f-shriek-separated}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F54","source_file":"more-etale.tex","source_line":707,"source_end_line":720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L707-L720","statement_sha256":"5e28148e63311b2496ec3fa56f1496a409019421823762781bd6e18362b969df","origin":"The Stacks Project","memory_eligible":false,"source_rank":10547,"rank":10547,"depth":4,"x":1722.284,"y":1205.108,"cluster":"tale-geometry"},{"id":"stacks:0F56","tag":"0F56","title":"Sections with compact support · Lemma 0F56","summary":"Let f : X → Y be a morphism of schemes which is separated and locally of finite type. Then functor f_! commutes with direct sums.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is separated and\nlocally of finite type. Then functor $f_!$ commutes with direct sums.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F56","source_file":"more-etale.tex","source_line":735,"source_end_line":739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L735-L739","statement_sha256":"ed89532b65df70ae2117fce8cf9119a8bb83cfe97c806c1d62716080e3ef93f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10548,"rank":10548,"depth":42,"x":1422.627,"y":1172.746,"cluster":"tale-geometry"},{"id":"stacks:0F57","tag":"0F57","title":"Sections with compact support · Lemma 0F57","summary":"Let f : X → Y be a morphism of schemes which is separated and locally quasi-finite. Then • for F in Ab(X_etale) and a geometric point overliney : Spec(k) → Y we have (f_!F)_overliney = bigoplus_f(overlinex) = overliney F_overlinex functorially in F, and • the functor f_! is exact.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is separated and\nlocally quasi-finite. Then\n\\begin{enumerate}\n\\item for $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ and a geometric\npoint $\\overline{y} : \\Spec(k) \\to Y$ we have\n$$\n(f_!\\mathcal{F})_{\\overline{y}} =\n\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}\n$$\nfunctorially in $\\mathcal{F}$, and\n\\item the functor $f_!$ is exact.\n\\end{enumerate}","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with compact support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F57","source_file":"more-etale.tex","source_line":760,"source_end_line":774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L760-L774","statement_sha256":"d292072baef3de1aac0722e4f4ad143fc9b266466e30ad3addcb5a8682007d26","origin":"The Stacks Project","memory_eligible":false,"source_rank":10549,"rank":10549,"depth":62,"x":1669.766,"y":1026.486,"cluster":"tale-geometry"},{"id":"stacks:0F6N","tag":"0F6N","title":"Sections with finite support · Lemma 0F6N","summary":"Let f : X → Y be a separated and locally quasi-finite morphism of schemes. Functorially in F ∈ Ab(X_etale) there is a canonical isomorphism(!) f_p!F → f_!F of abelian presheaves which identifies the sheaf f_!F of Definition [Tag 0F4Z] with the presheaf f_p!F constructed above.","statement_latex":"Let $f : X \\to Y$ be a separated and locally quasi-finite morphism\nof schemes. Functorially in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$\nthere is a canonical isomorphism(!)\n$$\nf_{p!}\\mathcal{F} \\longrightarrow f_!\\mathcal{F}\n$$\nof abelian presheaves which identifies the sheaf\n$f_!\\mathcal{F}$ of Definition \\ref{definition-f-shriek-separated}\nwith the presheaf $f_{p!}\\mathcal{F}$ constructed above.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with finite support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6N","source_file":"more-etale.tex","source_line":901,"source_end_line":912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L901-L912","statement_sha256":"632266170d42c20103ab2d8b128965423423b0fbf94cf9cc85863adce539a20c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10550,"rank":10550,"depth":2,"x":1605.108,"y":1274.721,"cluster":"tale-geometry"},{"id":"stacks:0F6P","tag":"0F6P","title":"Sections with finite support · Lemma 0F6P","summary":"Let f : X → Y be a morphism of schemes which is locally quasi-finite. Let overliney : Spec(k) → Y be a geometric point. Functorially in F in Ab(X_etale) we have (f_p!F)_overliney = bigoplus_f(overlinex) = overliney F_overlinex","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is locally quasi-finite.\nLet $\\overline{y} : \\Spec(k) \\to Y$ be a geometric point.\nFunctorially in $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ we have\n$$\n(f_{p!}\\mathcal{F})_{\\overline{y}} =\n\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}\n$$","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with finite support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6P","source_file":"more-etale.tex","source_line":930,"source_end_line":939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L930-L939","statement_sha256":"219591f112299ed776c144b5b57bbc144c928b5f999c955d497ce77da1ef10fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10551,"rank":10551,"depth":45,"x":1453.069,"y":1054.853,"cluster":"tale-geometry"},{"id":"stacks:0F6Q","tag":"0F6Q","title":"Sections with finite support · Lemma 0F6Q","summary":"Let f = j : U → X be an étale morphism of schemes. Denote j_p! the construction of Étale Cohomology, Equation ([Tag 0F4K]) and denote f_p! the construction above. Functorially in F ∈ Ab(X_etale) there is a canonical map j_p!F → f_p!F of abelian presheaves which identifies the sheaf j_!F = (j_p!F)^\\# of Étale Cohomology, Definition [Tag 03S3] with (f_p!F)^\\#.","statement_latex":"Let $f = j : U \\to X$ be an \\'etale morphism of schemes. Denote $j_{p!}$\nthe construction of \\'Etale Cohomology, Equation\n(\\ref{etale-cohomology-equation-j-p-shriek})\nand denote $f_{p!}$ the construction above. Functorially in\n$\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ there is a canonical map\n$$\nj_{p!}\\mathcal{F} \\longrightarrow f_{p!}\\mathcal{F}\n$$\nof abelian presheaves which identifies the sheaf\n$j_!\\mathcal{F} = (j_{p!}\\mathcal{F})^\\#$ of \\'Etale Cohomology,\nDefinition \\ref{etale-cohomology-definition-extension-zero}\nwith $(f_{p!}\\mathcal{F})^\\#$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with finite support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6Q","source_file":"more-etale.tex","source_line":1057,"source_end_line":1071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1057-L1071","statement_sha256":"9ae08ccda2f024d1089b44943699edb6b2ee1890ac9765c357c0714c40450041","origin":"The Stacks Project","memory_eligible":false,"source_rank":10552,"rank":10552,"depth":51,"x":1742.169,"y":1130.756,"cluster":"tale-geometry"},{"id":"stacks:0F6R","tag":"0F6R","title":"Sections with finite support · Definition 0F6R","summary":"Let f : X → Y be a locally quasi-finite morphism of schemes. We define the direct image with compact support to be the functor f_! : Ab(X_etale) → Ab(Y_etale) defined by the formula f_!F = (f_p!F)^\\#, i.e., f_!F is the sheafification of the presheaf f_p!F constructed above.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.\nWe define the {\\it direct image with compact support} to be the\nfunctor\n$$\nf_! : \\textit{Ab}(X_\\etale) \\longrightarrow \\textit{Ab}(Y_\\etale)\n$$\ndefined by the formula $f_!\\mathcal{F} = (f_{p!}\\mathcal{F})^\\#$,\ni.e., $f_!\\mathcal{F}$ is the sheafification of the presheaf\n$f_{p!}\\mathcal{F}$ constructed above.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with finite support","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6R","source_file":"more-etale.tex","source_line":1134,"source_end_line":1145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1134-L1145","statement_sha256":"ddccf481da076d55aa05b9df2c6a67832a5cce04b0bdd7ad1837b051bbd5658d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10553,"rank":10553,"depth":0,"x":1467.784,"y":1238.895,"cluster":"tale-geometry"},{"id":"stacks:0F5F","tag":"0F5F","title":"Sections with finite support · Lemma 0F5F","summary":"Let f : X → Y be a locally quasi-finite morphism of schemes. Then • for F in Ab(X_etale) and a geometric point overliney : Spec(k) → Y we have (f_!F)_overliney = bigoplus_f(overlinex) = overliney F_overlinex functorially in F, and • the functor f_! : Ab(X_etale) → Ab(Y_etale) is exact and commutes with direct sums.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes. Then\n\\begin{enumerate}\n\\item for $\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ and a geometric\npoint $\\overline{y} : \\Spec(k) \\to Y$ we have\n$$\n(f_!\\mathcal{F})_{\\overline{y}} =\n\\bigoplus\\nolimits_{f(\\overline{x}) = \\overline{y}} \\mathcal{F}_{\\overline{x}}\n$$\nfunctorially in $\\mathcal{F}$, and\n\\item the functor $f_! : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$\nis exact and commutes with direct sums.\n\\end{enumerate}","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with finite support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5F","source_file":"more-etale.tex","source_line":1156,"source_end_line":1170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1156-L1170","statement_sha256":"d8cc269196d308d6d2a5e8ff98d1563681d148a778315b2a87e8e25ff6fdf288","origin":"The Stacks Project","memory_eligible":false,"source_rank":10554,"rank":10554,"depth":51,"x":1583.22,"y":1003.318,"cluster":"tale-geometry"},{"id":"stacks:0F5H","tag":"0F5H","title":"Sections with finite support · Lemma 0F5H","summary":"Let f : X → Y be a locally quasi-finite morphism of schemes. Let X = ⋃_i ∈ I X_i be an open covering. Then there exists an exact complex … → bigoplus_i_0, i_1, i_2 f_i_0i_1i_2, ! F|_X_i_0i_1i_2 → bigoplus_i_0, i_1 f_i_0i_1, ! F|_X_i_0i_1 → bigoplus_i_0 f_i_0, ! F|_X_i_0 → f_!F → 0 functorial in F ∈ Ab(X_etale), see proof for details.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.\nLet $X = \\bigcup_{i \\in I} X_i$ be an open covering. Then there\nexists an exact complex\n$$\n\\ldots \\to\n\\bigoplus\\nolimits_{i_0, i_1, i_2} f_{i_0i_1i_2, !}\n\\mathcal{F}|_{X_{i_0i_1i_2}} \\to\n\\bigoplus\\nolimits_{i_0, i_1} f_{i_0i_1, !} \\mathcal{F}|_{X_{i_0i_1}} \\to\n\\bigoplus\\nolimits_{i_0} f_{i_0, !} \\mathcal{F}|_{X_{i_0}}\n\\to f_!\\mathcal{F} \\to 0\n$$\nfunctorial in $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$, see\nproof for details.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with finite support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5H","source_file":"more-etale.tex","source_line":1229,"source_end_line":1244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1229-L1244","statement_sha256":"e7c8a25151eac9270b9c8c60e822747d3c1d707dc25ade799e8c58ee0757b988","origin":"The Stacks Project","memory_eligible":false,"source_rank":10555,"rank":10555,"depth":52,"x":1687.607,"y":1242.677,"cluster":"tale-geometry"},{"id":"stacks:0F5J","tag":"0F5J","title":"Sections with finite support · Lemma 0F5J","summary":"Consider a cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y of schemes with f locally quasi-finite. There is an isomorphism g^-1f_!F → f'_!(g')^-1F functorial for F in Ab(X_etale) which is compatible with the descriptions of stalks given in Lemma [Tag 0F5F] (see proof for the precise statement).","statement_latex":"Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nof schemes with $f$ locally quasi-finite. There is an isomorphism\n$g^{-1}f_!\\mathcal{F} \\to f'_!(g')^{-1}\\mathcal{F}$ functorial for\n$\\mathcal{F}$ in $\\textit{Ab}(X_\\etale)$ which is compatible with\nthe descriptions of stalks given in Lemma \\ref{lemma-lqf-f-shriek-stalk}\n(see proof for the precise statement).","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with finite support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5J","source_file":"more-etale.tex","source_line":1320,"source_end_line":1334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1320-L1334","statement_sha256":"0c2733c035949a8685f15aa933f8af5e56da4ec97c7d069a50606b24e23635cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10556,"rank":10556,"depth":52,"x":1417.984,"y":1125.338,"cluster":"tale-geometry"},{"id":"stacks:0F79","tag":"0F79","title":"Sections with finite support · Lemma 0F79","summary":"Let f' : X → Y' and g : Y' → Y be composable morphisms of schemes with f' and f = g ∘ f' locally quasi-finite and g separated and locally of finite type. Then there is a canonical isomorphism of functors g_! ∘ f'_! = f_!. This isomorphism is compatible with • [(a)] covariance with respect to open embeddings as in Remarks [Tag 0F53] and [Tag 0F6S], • [(b)] the base change isomorphisms of Lemmas [Tag 0F5J] and [Tag 0F55], and • [(c)] equal to the isomorphism of Lemma [Tag…","statement_latex":"Let $f' : X \\to Y'$ and $g : Y' \\to Y$ be composable morphisms of schemes\nwith $f'$ and $f = g \\circ f'$ locally quasi-finite and $g$ separated and\nlocally of finite type. Then there is a canonical isomorphism of functors\n$g_! \\circ f'_! = f_!$. This isomorphism is compatible with\n\\begin{enumerate}\n\\item[(a)] covariance with respect to open embeddings as in\nRemarks \\ref{remark-covariance-f-shriek-separated} and\n\\ref{remark-covariance-lqf-f-shriek},\n\\item[(b)] the base change isomorphisms of\nLemmas \\ref{lemma-lqf-base-change-f-shriek}\nand \\ref{lemma-base-change-f-shriek-separated}, and\n\\item[(c)] equal to the isomorphism of Lemma \\ref{lemma-f-shriek-composition}\nvia the identifications of Lemma \\ref{lemma-finite-support-f-shriek-separated}\nin case $f'$ is separated.\n\\end{enumerate}","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with finite support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F79","source_file":"more-etale.tex","source_line":1401,"source_end_line":1418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1401-L1418","statement_sha256":"3bf5581448420081c8b4b01ce699ceb071e234e239db0dff5a6b0e66516d88b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10557,"rank":10557,"depth":63,"x":1711.339,"y":1058.83,"cluster":"tale-geometry"},{"id":"stacks:0F6T","tag":"0F6T","title":"Sections with finite support · Lemma 0F6T","summary":"Let f : X → Y and g : Y → Z be composable locally quasi-finite morphisms of schemes. Then there is a canonical isomorphism of functors (g ∘ f)_! → g_! ∘ f_! These isomorphisms satisfy the following properties: • If f and g are separated, then the isomorphism agrees with Lemma [Tag 0F50]. • If g is separated, then the isomorphism agrees with Lemma [Tag 0F79]. • For a geometric point overlinez : Spec(k) → Z the diagram xymatrix ((g ∘ f)_!F)_overlinez ar[d] ar[rr] & &…","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be composable locally quasi-finite\nmorphisms of schemes. Then there is a canonical isomorphism of functors\n$$\n(g \\circ f)_! \\longrightarrow g_! \\circ f_!\n$$\nThese isomorphisms satisfy the following properties:\n\\begin{enumerate}\n\\item If $f$ and $g$ are separated, then the isomorphism agrees\nwith Lemma \\ref{lemma-f-shriek-composition}.\n\\item If $g$ is separated, then the isomorphism agrees with\nLemma \\ref{lemma-lqf-separated-shriek-composition}.\n\\item For a geometric point $\\overline{z} : \\Spec(k) \\to Z$ the diagram\n$$\n\\xymatrix{\n((g \\circ f)_!\\mathcal{F})_{\\overline{z}} \\ar[d] \\ar[rr] & &\n\\bigoplus\\nolimits_{g(f(\\overline{x})) = \\overline{z}}\n\\mathcal{F}_{\\overline{x}} \\ar@{=}[d] \\\\\n(g_!f_!\\mathcal{F})_{\\overline{z}} \\ar[r] &\n\\bigoplus\\nolimits_{g(\\overline{y}) = \\overline{z}}\n(f_!\\mathcal{F})_{\\overline{y}} \\ar[r] &\n\\bigoplus\\nolimits_{g(f(\\overline{x})) = \\overline{z}}\n\\mathcal{F}_{\\overline{x}}\n}\n$$\nis commutative where the horizontal arrows are given by\nLemma \\ref{lemma-lqf-f-shriek-stalk}.\n\\item Let $h : Z \\to T$ be a third locally quasi-finite\nmorphism of schemes. Then the diagram\n$$\n\\xymatrix{\n(h \\circ g \\circ f)_! \\ar[r] \\ar[d] &\n(h \\circ g)_! \\circ f_! \\ar[d] \\\\\nh_! \\circ (g \\circ f)_! \\ar[r] &\nh_! \\circ g_! \\circ f_!\n}\n$$\ncommutes.\n\\item Suppose that we have a diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_c & X \\ar[d]^f \\\\\nY' \\ar[d]_{g'} \\ar[r]_b & Y \\ar[d]^g \\\\\nZ' \\ar[r]^a & Z\n}\n$$\nwith both squares cartesian and $f$ and $g$\nlocally quasi-finite. Then the diagram\n$$\n\\xymatrix{\na^{-1} \\circ (g \\circ f)_! \\ar[d] \\ar[rr] & &\n(g' \\circ f')_! \\circ c^{-1} \\ar[d] \\\\\na^{-1} \\circ g_! \\circ f_! \\ar[r] &\ng'_! \\circ b^{-1} \\circ f_! \\ar[r] &\ng'_! \\circ f'_! \\circ c^{-1}\n}\n$$\ncommutes where the horizontal arrows are those of\nLemma \\ref{lemma-lqf-base-change-f-shriek}.\n\\end{enumerate}","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Sections with finite support","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6T","source_file":"more-etale.tex","source_line":1610,"source_end_line":1671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1610-L1671","statement_sha256":"62571c9a97c5a4a10e543b31e327c7dbaeb082b2660460f92c9900beed576f1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10558,"rank":10558,"depth":64,"x":1548.408,"y":1274.46,"cluster":"tale-geometry"},{"id":"stacks:0GKF","tag":"0GKF","title":"Weightings and trace maps for locally quasi-finite morphisms · Lemma 0GKF","summary":"Let f : X → Y be a locally quasi-finite morphism of schemes. Let Lambda be a ring. Let F be a sheaf of Lambda-modules on X_etale and let G be a sheaf of Lambda-modues on Y_etale. There is a canonical isomorphism can : f_!F ⊗_Lambda G → f_!(F ⊗_Lambda f^-1G) of sheaves of Lambda-modules on Y_etale.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.\nLet $\\Lambda$ be a ring.\nLet $\\mathcal{F}$ be a sheaf of $\\Lambda$-modules on $X_\\etale$\nand let $\\mathcal{G}$ be a sheaf of $\\Lambda$-modues on $Y_\\etale$.\nThere is a canonical isomorphism\n$$\ncan :\nf_!\\mathcal{F} \\otimes_\\Lambda \\mathcal{G}\n\\longrightarrow\nf_!(\\mathcal{F} \\otimes_\\Lambda f^{-1}\\mathcal{G})\n$$\nof sheaves of $\\Lambda$-modules on $Y_\\etale$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Weightings and trace maps for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKF","source_file":"more-etale.tex","source_line":1807,"source_end_line":1821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1807-L1821","statement_sha256":"05b006f1d40a62b364bb66ba50a9fef30a85ee8bbb2e603d5e10d923166c675a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10559,"rank":10559,"depth":52,"x":1495.115,"y":1022.854,"cluster":"tale-geometry"},{"id":"stacks:0GKG","tag":"0GKG","title":"Weightings and trace maps for locally quasi-finite morphisms · Lemma 0GKG","summary":"Let f : X → Y be a locally quasi-finite morphism of schemes. Let w : X → Z be a weighting of f. For any abelian sheaf F on Y there exists a unique trace map Tr_f, w, F : f_!f^-1F → F having the prescribed behaviour on stalks.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.\nLet $w : X \\to \\mathbf{Z}$ be a weighting of $f$. For any abelian sheaf\n$\\mathcal{F}$ on $Y$ there exists a unique trace map\n$\\text{Tr}_{f, w, \\mathcal{F}} : f_!f^{-1}\\mathcal{F} \\to \\mathcal{F}$\nhaving the prescribed behaviour on stalks.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Weightings and trace maps for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKG","source_file":"more-etale.tex","source_line":1888,"source_end_line":1895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1888-L1895","statement_sha256":"eb1da6cb11b250bd700c5943c74a41a43da52ac8bc1cb928950e6f64fba632af","origin":"The Stacks Project","memory_eligible":false,"source_rank":10560,"rank":10560,"depth":53,"x":1736.894,"y":1178.239,"cluster":"tale-geometry"},{"id":"stacks:0GKH","tag":"0GKH","title":"Weightings and trace maps for locally quasi-finite morphisms · Lemma 0GKH","summary":"Let f : X → Y be a locally quasi-finite morphism of schemes. Let w : X → Z be a weighting of f. The trace maps constructed above have the following properties: • Tr_f, w, F is functorial in F, • Tr_f, w, F is compatible with arbitrary base change, • given a ring Lambda and K in D(Y_etale, Lambda) we obtain Tr_f, w, K : f_!f^-1K → K functorial in K and compatible with arbitrary base change.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.\nLet $w : X \\to \\mathbf{Z}$ be a weighting of $f$. The trace maps\nconstructed above have the following properties:\n\\begin{enumerate}\n\\item $\\text{Tr}_{f, w, \\mathcal{F}}$ is functorial in $\\mathcal{F}$,\n\\item $\\text{Tr}_{f, w, \\mathcal{F}}$ is compatible with arbitrary base change,\n\\item given a ring $\\Lambda$ and $K$ in $D(Y_\\etale, \\Lambda)$\nwe obtain $\\text{Tr}_{f, w, K} : f_!f^{-1}K \\to K$ functorial in $K$\nand compatible with arbitrary base change.\n\\end{enumerate}","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Weightings and trace maps for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKH","source_file":"more-etale.tex","source_line":1983,"source_end_line":1995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L1983-L1995","statement_sha256":"d88780a8106158ba8124359cb6b5c8916f4df632fa182736866f5e8bd958e6e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10561,"rank":10561,"depth":54,"x":1433.469,"y":1200.865,"cluster":"tale-geometry"},{"id":"stacks:0GL3","tag":"0GL3","title":"Weightings and trace maps for locally quasi-finite morphisms · Lemma 0GL3","summary":"Let f : X → Y and g : Y → Z be locally quasi-finite morphisms. Let w_f : X → Z be a weighting of f and let w_g : Y → Z be a weighting of g. For K ∈ D(Z_etale, Lambda) the composition (g ∘ f)_!(g ∘ f)^-1K = g_! f_! f^-1 g^-1K xrightarrowg_! Tr_f, w_f, g^-1K g_!g^-1K xrightarrowTr_g, w_g, K K is equal to Tr_g ∘ f, w_g ∘ f, K where w_g ∘ f(x) = w_f(x) w_g(f(x)).","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be locally quasi-finite morphisms.\nLet $w_f : X \\to \\mathbf{Z}$ be a weighting of $f$ and let\n$w_g : Y \\to \\mathbf{Z}$ be a weighting of $g$. For\n$K \\in D(Z_\\etale, \\Lambda)$ the composition\n$$\n(g \\circ f)_!(g \\circ f)^{-1}K =\ng_! f_! f^{-1} g^{-1}K\n\\xrightarrow{g_! \\text{Tr}_{f, w_f, g^{-1}K}}\ng_!g^{-1}K\n\\xrightarrow{\\text{Tr}_{g, w_g, K}}\nK\n$$\nis equal to $\\text{Tr}_{g \\circ f, w_{g \\circ f}, K}$ where\n$w_{g \\circ f}(x) = w_f(x) w_g(f(x))$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Weightings and trace maps for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GL3","source_file":"more-etale.tex","source_line":2043,"source_end_line":2059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2043-L2059","statement_sha256":"becc19127bdb9f780ce51058d2f5e6e8c59b5c1ac0604629e96fac9e90249b54","origin":"The Stacks Project","memory_eligible":false,"source_rank":10562,"rank":10562,"depth":65,"x":1639.139,"y":1011.896,"cluster":"tale-geometry"},{"id":"stacks:0F59","tag":"0F59","title":"Upper shriek for locally quasi-finite morphisms · Lemma 0F59","summary":"Let f : X → Y be a locally quasi-finite morphism of schemes. • The functor f_! : Ab(X_etale) → Ab(Y_etale) has a right adjoint f^! : Ab(Y_etale) → Ab(X_etale). • We have f^!(overliney_*A) = ∏_f(overlinex) = overliney overlinex_*A. • If Lambda is a ring, then the functor f_! : Mod(X_etale, Lambda) → Mod(Y_etale, Lambda) has a right adjoint f^! : Mod(Y_etale, Lambda) → Mod(X_etale, Lambda) which agrees with f^! on underlying abelian sheaves.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.\n\\begin{enumerate}\n\\item The functor $f_! : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$\nhas a right adjoint $f^! : \\textit{Ab}(Y_\\etale) \\to \\textit{Ab}(X_\\etale)$.\n\\item We have\n$f^!(\\overline{y}_*A) = \\prod_{f(\\overline{x}) = \\overline{y}} \\overline{x}_*A$.\n\\item If $\\Lambda$ is a ring, then the functor\n$f_! : \\textit{Mod}(X_\\etale, \\Lambda) \\to \\textit{Mod}(Y_\\etale, \\Lambda)$\nhas a right adjoint\n$f^! : \\textit{Mod}(Y_\\etale, \\Lambda) \\to \\textit{Mod}(X_\\etale, \\Lambda)$\nwhich agrees with $f^!$ on underlying abelian sheaves.\n\\end{enumerate}","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Upper shriek for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F59","source_file":"more-etale.tex","source_line":2154,"source_end_line":2168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2154-L2168","statement_sha256":"c8461dac2991f9b0aaac67bfeb9b339b70b64f858ab601386456c9f9b532a81a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10563,"rank":10563,"depth":52,"x":1639.445,"y":1268.096,"cluster":"tale-geometry"},{"id":"stacks:0F5A","tag":"0F5A","title":"Upper shriek for locally quasi-finite morphisms · Lemma 0F5A","summary":"Let j : U → X be an étale morphism. Then j^! = j^-1.","statement_latex":"Let $j : U \\to X$ be an \\'etale morphism. Then $j^! = j^{-1}$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Upper shriek for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5A","source_file":"more-etale.tex","source_line":2237,"source_end_line":2240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2237-L2240","statement_sha256":"16ef57a4aaef1c2fc836de664547affa784f0d57bfccb8c35bc5db16baf83725","origin":"The Stacks Project","memory_eligible":false,"source_rank":10564,"rank":10564,"depth":52,"x":1433.066,"y":1079.238,"cluster":"tale-geometry"},{"id":"stacks:0F5B","tag":"0F5B","title":"Upper shriek for locally quasi-finite morphisms · Lemma 0F5B","summary":"Let f : X → Y and g : Y → Z be separated and locally quasi-finite morphisms. There is a canonical isomorphism (g ∘ f)^! → f^! ∘ g^!. Given a third locally quasi-finite morphism h : Z → T the diagram xymatrix (h ∘ g ∘ f)^! ar[r] ar[d] & f^! ∘ (h ∘ g)^! ar[d] (g ∘ f)^! ∘ h^! ar[r] & f^! ∘ g^! ∘ h^! commutes.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be separated and locally quasi-finite\nmorphisms. There is a canonical isomorphism $(g \\circ f)^! \\to f^! \\circ g^!$.\nGiven a third locally quasi-finite morphism $h : Z \\to T$\nthe diagram\n$$\n\\xymatrix{\n(h \\circ g \\circ f)^! \\ar[r] \\ar[d] &\nf^! \\circ (h \\circ g)^! \\ar[d] \\\\\n(g \\circ f)^! \\circ h^! \\ar[r] & f^! \\circ g^! \\circ h^!\n}\n$$\ncommutes.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Upper shriek for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5B","source_file":"more-etale.tex","source_line":2254,"source_end_line":2268,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2254-L2268","statement_sha256":"9de7278c417e0f4b7af96f4bc158a3d8bbb6874643bf7b156b88a27ad562c8a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10565,"rank":10565,"depth":65,"x":1737.305,"y":1101.408,"cluster":"tale-geometry"},{"id":"stacks:0F5C","tag":"0F5C","title":"Upper shriek for locally quasi-finite morphisms · Lemma 0F5C","summary":"Let j : U → X and j' : V → U be étale morphisms. The isomorphism (j ∘ j')^-1 = (j')^-1 ∘ j^-1 and the isomorphism (j ∘ j')^! = (j')^! ∘ j^! of Lemma [Tag 0F5B] agree via the isomorphism of Lemma [Tag 0F5A].","statement_latex":"Let $j : U \\to X$ and $j' : V \\to U$ be \\'etale morphisms.\nThe isomorphism $(j \\circ j')^{-1} = (j')^{-1} \\circ j^{-1}$\nand the isomorphism $(j \\circ j')^! = (j')^! \\circ j^!$ of\nLemma \\ref{lemma-upper-shriek-restriction}\nagree via the isomorphism of Lemma \\ref{lemma-etale-upper-shriek}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Upper shriek for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5C","source_file":"more-etale.tex","source_line":2276,"source_end_line":2283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2276-L2283","statement_sha256":"89700932031ad67eda173b891d733c92b0738bea1d5806ac9b68e91a75d81f96","origin":"The Stacks Project","memory_eligible":false,"source_rank":10566,"rank":10566,"depth":66,"x":1494.989,"y":1257.786,"cluster":"tale-geometry"},{"id":"stacks:0F6U","tag":"0F6U","title":"Upper shriek for locally quasi-finite morphisms · Lemma 0F6U","summary":"Consider a cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y of schemes with f locally quasi-finite. For any abelian sheaf F on Y'_etale we have (g')_*(f')^!F = f^!g_*F.","statement_latex":"Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nof schemes with $f$ locally quasi-finite. For any abelian sheaf $\\mathcal{F}$\non $Y'_\\etale$ we have $(g')_*(f')^!\\mathcal{F} = f^!g_*\\mathcal{F}$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Upper shriek for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F6U","source_file":"more-etale.tex","source_line":2289,"source_end_line":2300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2289-L2300","statement_sha256":"e295d1342384a8e0d175fcec2ffb2fafcdfcd8c5dcdad9b47b50d4ebdad4646e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10567,"rank":10567,"depth":53,"x":1547.946,"y":1004.828,"cluster":"tale-geometry"},{"id":"stacks:0F5N","tag":"0F5N","title":"Derived upper shriek for locally quasi-finite morphisms · Lemma 0F5N","summary":"Let f : X → Y be a locally quasi-finite morphism of schemes. Let Lambda be a ring. The functors f_! and f^! of Definition [Tag 0F6R] and Lemma [Tag 0F59] induce adjoint functors f_! : D(X_etale, Lambda) → D(Y_etale, Lambda) and Rf^! : D(Y_etale, Lambda) → D(X_etale, Lambda) on derived categories.","statement_latex":"Let $f : X \\to Y$ be a locally quasi-finite morphism of schemes.\nLet $\\Lambda$ be a ring. The functors $f_!$ and $f^!$ of\nDefinition \\ref{definition-f-shriek-lqf} and\nLemma \\ref{lemma-lqf-f-upper-shriek}\ninduce adjoint functors $f_! : D(X_\\etale, \\Lambda) \\to D(Y_\\etale, \\Lambda)$\nand $Rf^! : D(Y_\\etale, \\Lambda) \\to D(X_\\etale, \\Lambda)$\non derived categories.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F5N","source_file":"more-etale.tex","source_line":2367,"source_end_line":2376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2367-L2376","statement_sha256":"f01a5d08dd5bc6a5c557a2ffaf5663445ccf6d3f51ed121eee520d1f6b52c0ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":10568,"rank":10568,"depth":53,"x":1712.417,"y":1221.535,"cluster":"tale-geometry"},{"id":"stacks:0GKK","tag":"0GKK","title":"Derived upper shriek for locally quasi-finite morphisms · Lemma 0GKK","summary":"Let X be a scheme. Let X = U ∪ V with U and V open. Let Lambda be a ring. Let K ∈ D(X_etale, Lambda). There is a distinguished triangle j_U ∩ V!K|_U ∩ V → j_U!K|_U ⊕ j_V!K|_V → K → j_U ∩ V!K|_U ∩ V[1] in D(X_etale, Lambda) with obvious notation.","statement_latex":"Let $X$ be a scheme. Let $X = U \\cup V$ with $U$ and $V$ open.\nLet $\\Lambda$ be a ring. Let $K \\in D(X_\\etale, \\Lambda)$. There\nis a distinguished triangle\n$$\nj_{U \\cap V!}K|_{U \\cap V} \\to\nj_{U!}K|_U \\oplus j_{V!}K|_V \\to K \\to \nj_{U \\cap V!}K|_{U \\cap V}[1]\n$$\nin $D(X_\\etale, \\Lambda)$ with obvious notation.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKK","source_file":"more-etale.tex","source_line":2403,"source_end_line":2414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2403-L2414","statement_sha256":"5f4441fae51bc8066bb6a05c7bb4b1e5b78e8ea7417e513ec3ec1a41bce32dfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10569,"rank":10569,"depth":0,"x":1416.692,"y":1155.023,"cluster":"tale-geometry"},{"id":"stacks:0GKL","tag":"0GKL","title":"Derived upper shriek for locally quasi-finite morphisms · Lemma 0GKL","summary":"Let X be a scheme. Let Z ⊂ X be a closed subscheme and let U ⊂ X be the complement. Denote i : Z → X and j : U → X the inclusion morphisms. Let Lambda be a ring. Let K ∈ D(X_etale, Lambda). There is a distinguished triangle j_!j^-1K → K → i_*i^-1K → j_!j^-1K[1] in D(X_etale, Lambda).","statement_latex":"Let $X$ be a scheme. Let $Z \\subset X$ be a closed subscheme and let\n$U \\subset X$ be the complement. Denote $i : Z \\to X$ and $j : U \\to X$\nthe inclusion morphisms. Let $\\Lambda$ be a ring.\nLet $K \\in D(X_\\etale, \\Lambda)$. There is a distinguished triangle\n$$\nj_!j^{-1}K \\to K \\to i_*i^{-1}K \\to j_!j^{-1}K[1]\n$$\nin $D(X_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek for locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKL","source_file":"more-etale.tex","source_line":2427,"source_end_line":2437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2427-L2437","statement_sha256":"e4a8bcf1168a13a3d077200c9a2326c3ccaef1577b46b6c646d85606b6ac767e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10570,"rank":10570,"depth":52,"x":1688.405,"y":1036.195,"cluster":"tale-geometry"},{"id":"stacks:0F7B","tag":"0F7B","title":"Preliminaries to derived lower shriek via compactifications · Lemma 0F7B","summary":"Consider a commutative diagram of schemes xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y with f and f' proper and g and g' separated and locally quasi-finite. Let Lambda be a ring. Functorially in K ∈ D(X'_etale, Lambda) there is a canonical map g_!Rf'_*K → Rf_*(g'_!K) in D(Y_etale, Lambda). This map is an isomorphism if (a) K is bounded below and has torsion cohomology sheaves, or (b) Lambda is a torsion ring.","statement_latex":"Consider a commutative diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nwith $f$ and $f'$ proper and $g$ and $g'$ separated and locally quasi-finite.\nLet $\\Lambda$ be a ring. Functorially in $K \\in D(X'_\\etale, \\Lambda)$\nthere is a canonical map\n$$\ng_!Rf'_*K \\longrightarrow Rf_*(g'_!K)\n$$\nin $D(Y_\\etale, \\Lambda)$. This map is an isomorphism if\n(a) $K$ is bounded below and has torsion cohomology sheaves, or\n(b) $\\Lambda$ is a torsion ring.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Preliminaries to derived lower shriek via compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7B","source_file":"more-etale.tex","source_line":2459,"source_end_line":2477,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2459-L2477","statement_sha256":"1e67cbd99a5432e65cfcf286e08e15c3f4adbfc21111a6f6b015e61a17b9196c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10571,"rank":10571,"depth":78,"x":1583.544,"y":1278.138,"cluster":"tale-geometry"},{"id":"stacks:0F7C","tag":"0F7C","title":"Preliminaries to derived lower shriek via compactifications · Lemma 0F7C","summary":"Consider a commutative diagram of schemes xymatrix X' ar[r]_k ar[d]_f' & X ar[d]^f Y' ar[r]_l ar[d]_g' & Y ar[d]^g Z' ar[r]^m & Z with f, f', g and g' proper and k, l, and m separated and locally quasi-finite. Then the isomorphisms of Lemma [Tag 0F7B] for the two squares compose to give the isomorphism for the outer rectangle (see proof for a precise statement).","statement_latex":"Consider a commutative diagram of schemes\n$$\n\\xymatrix{\nX' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]_l \\ar[d]_{g'} & Y \\ar[d]^g \\\\\nZ' \\ar[r]^m & Z\n}\n$$\nwith $f$, $f'$, $g$ and $g'$ proper and\n$k$, $l$, and $m$ separated and locally quasi-finite.\nThen the isomorphisms of Lemma \\ref{lemma-shriek-proper-and-open}\nfor the two squares compose to give the isomorphism\nfor the outer rectangle (see proof for a precise statement).","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Preliminaries to derived lower shriek via compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7C","source_file":"more-etale.tex","source_line":2616,"source_end_line":2631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2616-L2631","statement_sha256":"f07954a4a8587799ff2508fd8a3291a9ef8af888eebbcda2d0d8f0b6fcb2ae28","origin":"The Stacks Project","memory_eligible":false,"source_rank":10572,"rank":10572,"depth":79,"x":1466.232,"y":1040.09,"cluster":"tale-geometry"},{"id":"stacks:0F7D","tag":"0F7D","title":"Preliminaries to derived lower shriek via compactifications · Lemma 0F7D","summary":"Consider a commutative diagram of schemes xymatrix X\" ar[r]_g' ar[d]_f\" & X' ar[r]_g ar[d]_f' & X ar[d]^f Y\" ar[r]^h' & Y' ar[r]^h & Y with f, f', and f\" proper and g, g', h, and h' separated and locally quasi-finite. Then the isomorphisms of Lemma [Tag 0F7B] for the two squares compose to give the isomorphism for the outer rectangle (see proof for a precise statement).","statement_latex":"Consider a commutative diagram of schemes\n$$\n\\xymatrix{\nX'' \\ar[r]_{g'} \\ar[d]_{f''} &\nX' \\ar[r]_g \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nY'' \\ar[r]^{h'} &\nY' \\ar[r]^h &\nY\n}\n$$\nwith $f$, $f'$, and $f''$ proper and\n$g$, $g'$, $h$, and $h'$ separated and locally quasi-finite.\nThen the isomorphisms of Lemma \\ref{lemma-shriek-proper-and-open}\nfor the two squares compose to give the isomorphism\nfor the outer rectangle (see proof for a precise statement).","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Preliminaries to derived lower shriek via compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7D","source_file":"more-etale.tex","source_line":2675,"source_end_line":2693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2675-L2693","statement_sha256":"1d85c7f99f449c8eeebc9dc57c10ecb76e90e396c38a07ced7c9efb3ec574337","origin":"The Stacks Project","memory_eligible":false,"source_rank":10573,"rank":10573,"depth":79,"x":1744.334,"y":1149.123,"cluster":"tale-geometry"},{"id":"stacks:0F7F","tag":"0F7F","title":"Preliminaries to derived lower shriek via compactifications · Lemma 0F7F","summary":"Let b : Y_1 → Y be a morphism of schemes. Consider a commutative diagram of schemes vcenter xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y and let vcenter xymatrix X'_1 ar[r]_g'_1 ar[d]_f'_1 & X_1 ar[d]^f_1 Y'_1 ar[r]^g_1 & Y_1 be the base change by b. Assume f and f' proper and g and g' separated and locally quasi-finite. For a ring Lambda and K in D(X'_etale, Lambda) there is commutative diagram xymatrix b^-1g_!Rf'_*K ar[d] ar[r] & g_1, !(b')^-1Rf'_*K ar[r] &…","statement_latex":"Let $b : Y_1 \\to Y$ be a morphism of schemes. Consider a commutative diagram\nof schemes\n$$\n\\vcenter{\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n}\n\\quad\\text{and let}\\quad\n\\vcenter{\n\\xymatrix{\nX'_1 \\ar[r]_{g'_1} \\ar[d]_{f'_1} & X_1 \\ar[d]^{f_1} \\\\\nY'_1 \\ar[r]^{g_1} & Y_1\n}\n}\n$$\nbe the base change by $b$. Assume $f$ and $f'$ proper and\n$g$ and $g'$ separated and locally quasi-finite.\nFor a ring $\\Lambda$ and $K$ in $D(X'_\\etale, \\Lambda)$\nthere is commutative diagram\n$$\n\\xymatrix{\nb^{-1}g_!Rf'_*K \\ar[d] \\ar[r] &\ng_{1, !}(b')^{-1}Rf'_*K \\ar[r] &\ng_{1, !}Rf'_{1, *}(a')^{-1}K \\ar[d] \\\\\nb^{-1}Rf_*g'_!K \\ar[r] &\nRf_{1, *}a^{-1}g'_!K \\ar[r] &\nRf_{1, *}g'_{1, !}(a')^{-1}K\n}\n$$\nin $D(Y_{1, \\etale}, \\Lambda)$ where $a : X_1 \\to X$, $a' : X'_1 \\to X'$,\n$b' : Y'_1 \\to Y'$ are the projections, the vertical maps are the arrows\nof Lemma \\ref{lemma-shriek-proper-and-open}\nand the horizontal arrows are the base change map\n(from \\'Etale Cohomology, Section\n\\ref{etale-cohomology-section-base-change-preliminaries})\nand the base change map of Lemma \\ref{lemma-base-change-f-shriek-separated}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Preliminaries to derived lower shriek via compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7F","source_file":"more-etale.tex","source_line":2829,"source_end_line":2869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2829-L2869","statement_sha256":"847330ceb33dd9bdc1eeda1ceba84eeff963fa236d5b8b16e4018d988dd8e647","origin":"The Stacks Project","memory_eligible":false,"source_rank":10574,"rank":10574,"depth":79,"x":1451.41,"y":1226.571,"cluster":"tale-geometry"},{"id":"stacks:0F7G","tag":"0F7G","title":"Preliminaries to derived lower shriek via compactifications · Lemma 0F7G","summary":"Consider a commutative diagram of schemes xymatrix X ar[r]_f ar[rd]_g & Y ar[d]^h & Z with f and g locally quasi-finite and h proper. Let Lambda be a ring. Funtorially in K ∈ D(X_etale, Lambda) there is a canonical map g_!K → Rh_*(f_!K) in D(Z_etale, Lambda). This map is an isomorphism if (a) K is bounded below and has torsion cohomology sheaves, or (b) Lambda is a torsion ring.","statement_latex":"Consider a commutative diagram of schemes\n$$\n\\xymatrix{\nX \\ar[r]_f \\ar[rd]_g & Y \\ar[d]^h \\\\\n& Z\n}\n$$\nwith $f$ and $g$ locally quasi-finite and $h$ proper. Let $\\Lambda$ be a ring.\nFuntorially in $K \\in D(X_\\etale, \\Lambda)$ there is a canonical map\n$$\ng_!K \\longrightarrow Rh_*(f_!K)\n$$\nin $D(Z_\\etale, \\Lambda)$. This map is an isomorphism if\n(a) $K$ is bounded below and has torsion cohomology sheaves, or\n(b) $\\Lambda$ is a torsion ring.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Preliminaries to derived lower shriek via compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7G","source_file":"more-etale.tex","source_line":2961,"source_end_line":2978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L2961-L2978","statement_sha256":"876689d7781589e170265653ae9e48504080f7da34fff41f31d31a2953043c11","origin":"The Stacks Project","memory_eligible":false,"source_rank":10575,"rank":10575,"depth":81,"x":1605.217,"y":1003.117,"cluster":"tale-geometry"},{"id":"stacks:0F7I","tag":"0F7I","title":"Derived lower shriek via compactifications · Lemma 0F7I","summary":"Let f : X → Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functors Rf_! constructed above are, up to canonical isomorphism, independent of the choice of the compactification.","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact\nand quasi-separated schemes. The functors $Rf_!$ constructed above\nare, up to canonical isomorphism, independent of the choice of the\ncompactification.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived lower shriek via compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7I","source_file":"more-etale.tex","source_line":3080,"source_end_line":3086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3080-L3086","statement_sha256":"29dcb2127f38efed1159dbeb997afee9e31f9fbd35af33d078157e6dd2e3da18","origin":"The Stacks Project","memory_eligible":false,"source_rank":10576,"rank":10576,"depth":82,"x":1671.537,"y":1255.313,"cluster":"tale-geometry"},{"id":"stacks:0F7J","tag":"0F7J","title":"Derived lower shriek via compactifications · Lemma 0F7J","summary":"Let f : X → Y and g : Y → Z be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then there is a canonical isomorphism Rg_! ∘ Rf_! → R(g ∘ f)_!.","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be separated morphisms of finite type\nof quasi-compact and quasi-separated schemes. Then there is a canonical\nisomorphism $Rg_! \\circ Rf_! \\to R(g \\circ f)_!$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived lower shriek via compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7J","source_file":"more-etale.tex","source_line":3158,"source_end_line":3163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3158-L3163","statement_sha256":"37c1084ce1081cc03646e4406f8a4e76913b3dc8b0b3b52b0ae21979d642e1be","origin":"The Stacks Project","memory_eligible":false,"source_rank":10577,"rank":10577,"depth":83,"x":1419.675,"y":1106.891,"cluster":"tale-geometry"},{"id":"stacks:0F7K","tag":"0F7K","title":"Derived lower shriek via compactifications · Lemma 0F7K","summary":"Let f : X → Y, g : Y → Z, h : Z → T be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then the diagram xymatrix Rh_! ∘ Rg_! ∘ Rf_! ar[r]_γ_C ar[d]^γ_A & R(h ∘ g)_! ∘ Rf_! ar[d]_γ_A + B Rh_! ∘ R(g ∘ f)_! ar[r]^γ_B + C & R(h ∘ g ∘ f)_! of isomorphisms of Lemma [Tag 0F7J] commutes (for the meaning of the γ's see proof).","statement_latex":"Let $f : X \\to Y$, $g : Y \\to Z$, $h : Z \\to T$ be separated morphisms of\nfinite type of quasi-compact and quasi-separated schemes. Then\nthe diagram\n$$\n\\xymatrix{\nRh_! \\circ Rg_! \\circ Rf_! \\ar[r]_{\\gamma_C} \\ar[d]^{\\gamma_A} &\nR(h \\circ g)_! \\circ Rf_! \\ar[d]_{\\gamma_{A + B}} \\\\\nRh_! \\circ R(g \\circ f)_! \\ar[r]^{\\gamma_{B + C}} &\nR(h \\circ g \\circ f)_!\n}\n$$\nof isomorphisms of Lemma \\ref{lemma-shriek-composition} commutes\n(for the meaning of the $\\gamma$'s see proof).","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived lower shriek via compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7K","source_file":"more-etale.tex","source_line":3373,"source_end_line":3388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3373-L3388","statement_sha256":"1922f5580d6842acf9c12eab689afee65becdbce8705ed0a8485862f09362db5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10578,"rank":10578,"depth":84,"x":1724.932,"y":1073.403,"cluster":"tale-geometry"},{"id":"stacks:0F7L","tag":"0F7L","title":"Derived lower shriek via compactifications · Lemma 0F7L","summary":"Consider a cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y of quasi-compact and quasi-separated schemes with f separated and of finite type. Then there is a canonical isomorphism g^-1 ∘ Rf_! → Rf'_! ∘ (g')^-1 Moreover, these isomorphisms are compatible with the isomorphisms of Lemma [Tag 0F7J].","statement_latex":"Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nof quasi-compact and quasi-separated schemes with $f$ separated and\nof finite type. Then there is a canonical isomorphism\n$$\ng^{-1} \\circ Rf_! \\to Rf'_! \\circ (g')^{-1}\n$$\nMoreover, these isomorphisms are compatible with the isomorphisms\nof Lemma \\ref{lemma-shriek-composition}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived lower shriek via compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F7L","source_file":"more-etale.tex","source_line":3428,"source_end_line":3444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3428-L3444","statement_sha256":"b1d81df25ea3eced8674a8f25925ec2928ab1aed84372e7329047c71648548b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10579,"rank":10579,"depth":84,"x":1526.653,"y":1271.424,"cluster":"tale-geometry"},{"id":"stacks:0G29","tag":"0G29","title":"Properties of derived lower shriek · Lemma 0G29","summary":"Let f : X → Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let Lambda be a ring. • Let K_i ∈ D^+_tors(X_etale, Lambda), i ∈ I be a family of objects. Assume given a ∈ Z such that H^n(K_i) = 0 for n < a and i ∈ I. Then Rf_!(bigoplus_i K_i) = bigoplus_i Rf_!K_i. • If Lambda is torsion, then the functor Rf_! : D(X_etale, Lambda) → D(Y_etale, Lambda) commutes with direct sums.","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact\nand quasi-separated schemes. Let $\\Lambda$ be a ring.\n\\begin{enumerate}\n\\item Let $K_i \\in D^+_{tors}(X_\\etale, \\Lambda)$, $i \\in I$ be a family\nof objects. Assume given $a \\in \\mathbf{Z}$ such that\n$H^n(K_i) = 0$ for $n < a$ and $i \\in I$. Then $Rf_!(\\bigoplus_i K_i) =\n\\bigoplus_i Rf_!K_i$.\n\\item If $\\Lambda$ is torsion, then the functor\n$Rf_! : D(X_\\etale, \\Lambda) \\to D(Y_\\etale, \\Lambda)$\ncommutes with direct sums.\n\\end{enumerate}","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Properties of derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G29","source_file":"more-etale.tex","source_line":3543,"source_end_line":3556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3543-L3556","statement_sha256":"b92992d95dbcab19fee10932366fcf81579d036d12f464f65af0dd99c5898f74","origin":"The Stacks Project","memory_eligible":false,"source_rank":10580,"rank":10580,"depth":79,"x":1513.61,"y":1012.745,"cluster":"tale-geometry"},{"id":"stacks:0G2A","tag":"0G2A","title":"Properties of derived lower shriek · Lemma 0G2A","summary":"Let f : X → Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let Lambda be a ring. The functors Rf_! constructed in Section [Tag 0F7H] are bounded in the following sense: There exists an integer N such that for E ∈ D^+_tors(X_etale, Lambda) or E ∈ D(X_etale, Lambda) if Lambda is torsion, we have • H^i(Rf_!(τ_≤ aE) → H^i(Rf_!(E)) is an isomorphism for i ≤ a, • H^i(Rf_!(E)) → H^i(Rf_!(τ_≥ b - NE)) is an isomorphism for i ≥ b, • if H^i(E) =…","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact\nand quasi-separated schemes. Let $\\Lambda$ be a ring. The functors $Rf_!$\nconstructed in Section \\ref{section-derived-lower-shriek-compactification}\nare bounded in the following sense: There exists an integer $N$ such that\nfor $E \\in D^+_{tors}(X_\\etale, \\Lambda)$ or $E \\in D(X_\\etale, \\Lambda)$\nif $\\Lambda$ is torsion, we have\n\\begin{enumerate}\n\\item $H^i(Rf_!(\\tau_{\\leq a}E) \\to H^i(Rf_!(E))$ is an isomorphism\nfor $i \\leq a$,\n\\item $H^i(Rf_!(E)) \\to H^i(Rf_!(\\tau_{\\geq b - N}E))$ is an isomorphism\nfor $i \\geq b$,\n\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some\n$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(Rf_!(E)) = 0$\nfor $i \\not \\in [a, b + N]$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Properties of derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2A","source_file":"more-etale.tex","source_line":3573,"source_end_line":3590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3573-L3590","statement_sha256":"273ebf115543167d5504cf1e148db97f9e8482c628650a4f5bd9f6c5b6223945","origin":"The Stacks Project","memory_eligible":false,"source_rank":10581,"rank":10581,"depth":78,"x":1731.379,"y":1196.198,"cluster":"tale-geometry"},{"id":"stacks:0GKM","tag":"0GKM","title":"Properties of derived lower shriek · Lemma 0GKM","summary":"Let f : X → Y be a quasi-finite separated morphism of quasi-compact and quasi-separated schemes. Then the functors Rf_! constructed in Section [Tag 0F7H] agree with the restriction of the functor f_! : D(X_etale, Lambda) → D(Y_etale, Lambda) constructed in Section [Tag 0F5M] to their common domains of definition.","statement_latex":"Let $f : X \\to Y$ be a quasi-finite separated morphism of quasi-compact and\nquasi-separated schemes. Then the functors $Rf_!$ constructed in\nSection \\ref{section-derived-lower-shriek-compactification}\nagree with the restriction of the functor\n$f_! : D(X_\\etale, \\Lambda) \\to D(Y_\\etale, \\Lambda)$\nconstructed in Section \\ref{section-derived-duality-locally-quasi-finite}\nto their common domains of definition.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Properties of derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKM","source_file":"more-etale.tex","source_line":3621,"source_end_line":3630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3621-L3630","statement_sha256":"c356680a22b6c49f373900db8d655e11d097222409e1301e30d33b89b5a74b43","origin":"The Stacks Project","memory_eligible":false,"source_rank":10582,"rank":10582,"depth":53,"x":1423.09,"y":1184.483,"cluster":"tale-geometry"},{"id":"stacks:0GKN","tag":"0GKN","title":"Properties of derived lower shriek · Lemma 0GKN","summary":"Let f : X → Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let U and V be quasi-compact opens of X such that X = U ∪ V. Denote a : U → Y, b : V → Y and c : U ∩ V → Y the restrictions of f. Let Lambda be a ring. For K in D^+_tors(X_etale, Lambda) or K ∈ D(X_etale, Lambda) if Lambda is torsion, we have a distinguished triangle Rc_!(K|_U ∩ V) → Ra_!(K|_U) ⊕ Rb_!(K|_V) → Rf_!K → Rc_!(K|_U ∩ V)[1] in D(Y_etale, Lambda).","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact and\nquasi-separated schemes. Let $U$ and $V$ be quasi-compact opens of $X$\nsuch that $X = U \\cup V$. Denote $a : U \\to Y$, $b : V \\to Y$ and\n$c : U \\cap V \\to Y$ the restrictions of $f$. Let $\\Lambda$ be a ring.\nFor $K$ in $D^+_{tors}(X_\\etale, \\Lambda)$ or $K \\in D(X_\\etale, \\Lambda)$\nif $\\Lambda$ is torsion, we have a distinguished triangle\n$$\nRc_!(K|_{U \\cap V}) \\to\nRa_!(K|_U) \\oplus Rb_!(K|_V) \\to\nRf_!K \\to \nRc_!(K|_{U \\cap V})[1]\n$$\nin $D(Y_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Properties of derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKN","source_file":"more-etale.tex","source_line":3645,"source_end_line":3660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3645-L3660","statement_sha256":"20e37d6caf19557f8052b2d670ca855459418d89a329ccc64236b706405a9e16","origin":"The Stacks Project","memory_eligible":false,"source_rank":10583,"rank":10583,"depth":84,"x":1659.979,"y":1018.093,"cluster":"tale-geometry"},{"id":"stacks:0GKP","tag":"0GKP","title":"Properties of derived lower shriek · Lemma 0GKP","summary":"Let f : X → Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let U be a quasi-compact open of X with complement Z ⊂ X. Denote g : U → Y and h : Z → Y the restrictions of f. Let Lambda be a ring. For K in D^+_tors(X_etale, Lambda) or K ∈ D(X_etale, Lambda) if Lambda is torsion, we have a distinguished triangle Rg_!(K|_U) → Rf_!K → Rh_!(K|_Z) → Rg_!(K|_U)[1] in D(Y_etale, Lambda).","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact and\nquasi-separated schemes. Let $U$ be a quasi-compact open of $X$ with\ncomplement $Z \\subset X$.  Denote $g : U \\to Y$ and $h : Z \\to Y$\nthe restrictions of $f$. Let $\\Lambda$ be a ring.\nFor $K$ in $D^+_{tors}(X_\\etale, \\Lambda)$ or $K \\in D(X_\\etale, \\Lambda)$\nif $\\Lambda$ is torsion, we have a distinguished triangle\n$$\nRg_!(K|_U) \\to\nRf_!K \\to\nRh_!(K|_Z) \\to\nRg_!(K|_U)[1]\n$$\nin $D(Y_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Properties of derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKP","source_file":"more-etale.tex","source_line":3669,"source_end_line":3684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3669-L3684","statement_sha256":"248fed5ef7befcf7e06c42925d55dc9601bf07667003ff51ba4df868e0e1fcf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10584,"rank":10584,"depth":84,"x":1619.082,"y":1275.354,"cluster":"tale-geometry"},{"id":"stacks:0GKQ","tag":"0GKQ","title":"Properties of derived lower shriek · Lemma 0GKQ","summary":"Let f' : X' → Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let i : X → X' be a thickening and denote f = f' ∘ i. Let Lambda be a ring. For K' in D^+_tors(X'_etale, Lambda) or K' ∈ D(X'_etale, Lambda) if Lambda is torsion, we have Rf_!i^-1K' = Rf'_!K'.","statement_latex":"Let $f' : X' \\to Y$ be a finite type separated morphism of quasi-compact and\nquasi-separated schemes. Let $i : X \\to X'$ be a thickening and\ndenote $f = f' \\circ i$. Let $\\Lambda$ be a ring.\nFor $K'$ in $D^+_{tors}(X'_\\etale, \\Lambda)$ or $K' \\in D(X'_\\etale, \\Lambda)$\nif $\\Lambda$ is torsion, we have $Rf_!i^{-1}K' = Rf'_!K'$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Properties of derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKQ","source_file":"more-etale.tex","source_line":3695,"source_end_line":3702,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3695-L3702","statement_sha256":"d43cf56b2f1bda61c65fcf6cebaa93939138e0ab535c1ea31662e01357ec9204","origin":"The Stacks Project","memory_eligible":false,"source_rank":10585,"rank":10585,"depth":84,"x":1442.252,"y":1062.323,"cluster":"tale-geometry"},{"id":"stacks:0GL5","tag":"0GL5","title":"Properties of derived lower shriek · Lemma 0GL5","summary":"Let f : X → Y be a separated finite type morphism of quasi-compact and quasi-separated schemes. Let Lambda be a torsion ring. Let E ∈ D(X_etale, Lambda) and K ∈ D(Y_etale, Lambda). Then Rf_!E ⊗_Lambda^L K = Rf_!(E ⊗_Lambda^L f^-1K) in D(Y_etale, Lambda).","statement_latex":"Let $f : X \\to Y$ be a separated finite type morphism of quasi-compact\nand quasi-separated schemes. Let $\\Lambda$ be a torsion ring. Let\n$E \\in D(X_\\etale, \\Lambda)$ and $K \\in D(Y_\\etale, \\Lambda)$. Then\n$$\nRf_!E \\otimes_\\Lambda^\\mathbf{L} K =\nRf_!(E \\otimes_\\Lambda^\\mathbf{L} f^{-1}K)\n$$\nin $D(Y_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Properties of derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GL5","source_file":"more-etale.tex","source_line":3713,"source_end_line":3723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3713-L3723","statement_sha256":"14429ac47c66356ce0e4346d7d55c3610b0620ebb7bcbe7d03c8e25e364d9f63","origin":"The Stacks Project","memory_eligible":false,"source_rank":10586,"rank":10586,"depth":80,"x":1744.135,"y":1119.104,"cluster":"tale-geometry"},{"id":"stacks:0G2C","tag":"0G2C","title":"Derived upper shriek · Lemma 0G2C","summary":"Let f : X → Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let Lambda be a torsion coefficient ring. The functor Rf_! : D(X_etale, Lambda) → D(Y_etale, Lambda) has a right adjoint Rf^! : D(Y_etale, Lambda) → D(X_etale, Lambda).","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of quasi-compact\nand quasi-separated schemes. Let $\\Lambda$ be a torsion coefficient ring.\nThe functor\n$Rf_! : D(X_\\etale, \\Lambda) \\to D(Y_\\etale, \\Lambda)$\nhas a right adjoint\n$Rf^! : D(Y_\\etale, \\Lambda) \\to D(X_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2C","source_file":"more-etale.tex","source_line":3841,"source_end_line":3849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3841-L3849","statement_sha256":"d55cd7336f4f4256d0afb2a2eb0f08bd003d5f3109b5ff66c38c349722f365b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10587,"rank":10587,"depth":80,"x":1475.712,"y":1248.606,"cluster":"tale-geometry"},{"id":"stacks:0GL8","tag":"0GL8","title":"Derived upper shriek · Lemma 0GL8","summary":"Let f : X → Y be a separated quasi-finite morphism of quasi-compact and quasi-separated schemes. Let Lambda be a torsion coefficient ring. The functor Rf^! : D(Y_etale, Lambda) → D(X_etale, Lambda) of Lemma [Tag 0G2C] is the same as the functor Rf^! of Lemma [Tag 0F5N].","statement_latex":"Let $f : X \\to Y$ be a separated quasi-finite morphism of quasi-compact\nand quasi-separated schemes. Let $\\Lambda$ be a torsion coefficient ring.\nThe functor $Rf^! : D(Y_\\etale, \\Lambda) \\to D(X_\\etale, \\Lambda)$\nof Lemma \\ref{lemma-upper-shriek-derived} is the same as the functor\n$Rf^!$ of Lemma \\ref{lemma-lqf-shriek-derived}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GL8","source_file":"more-etale.tex","source_line":3857,"source_end_line":3864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3857-L3864","statement_sha256":"35268a627b632d2851de14933b1156cb677d37a1fddc54cfe92c2fbedfad388b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10588,"rank":10588,"depth":81,"x":1569.556,"y":1000.658,"cluster":"tale-geometry"},{"id":"stacks:0GL9","tag":"0GL9","title":"Derived upper shriek · Lemma 0GL9","summary":"Let j : U → X be a separated étale morphism of quasi-compact and quasi-separated schemes. Let Lambda be a torsion coefficient ring. The functor Rj^! : D(X_etale, Lambda) → D(U_etale, Lambda) is equal to j^-1.","statement_latex":"Let $j : U \\to X$ be a separated \\'etale morphism of quasi-compact\nand quasi-separated schemes. Let $\\Lambda$ be a torsion coefficient ring.\nThe functor $Rj^! : D(X_\\etale, \\Lambda) \\to D(U_\\etale, \\Lambda)$\nis equal to $j^{-1}$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GL9","source_file":"more-etale.tex","source_line":3871,"source_end_line":3877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3871-L3877","statement_sha256":"629da5223213d70068c84ba1de4f813aeeb969c07a3d4eb1c312a2e40d48fa41","origin":"The Stacks Project","memory_eligible":false,"source_rank":10589,"rank":10589,"depth":82,"x":1699.826,"y":1236.879,"cluster":"tale-geometry"},{"id":"stacks:0GLA","tag":"0GLA","title":"Derived upper shriek · Lemma 0GLA","summary":"Let f : X → Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let Lambda be a torsion ring. The functor Rf^! sends D^+(Y_etale, Lambda) into D^+(X_etale, Lambda). More precisely, there exists an integer N ≥ 0 such that if K ∈ D(Y_etale, Lambda) has H^i(K) = 0 for i < a then H^i(Rf^!K) = 0 for i < a - N.","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of\nquasi-compact and quasi-separated schemes. Let $\\Lambda$ be a torsion\nring. The functor $Rf^!$ sends $D^+(Y_\\etale, \\Lambda)$ into\n$D^+(X_\\etale, \\Lambda)$. More precisely, there exists an integer\n$N \\geq 0$ such that if $K \\in D(Y_\\etale, \\Lambda)$ has $H^i(K) = 0$\nfor $i < a$ then $H^i(Rf^!K) = 0$ for $i < a - N$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLA","source_file":"more-etale.tex","source_line":3886,"source_end_line":3894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3886-L3894","statement_sha256":"76746824db83f1f7976da4dbabca5cdc6babe3109fdf8ae710d4f78a549f4c57","origin":"The Stacks Project","memory_eligible":false,"source_rank":10590,"rank":10590,"depth":79,"x":1413.638,"y":1136.553,"cluster":"tale-geometry"},{"id":"stacks:0GLC","tag":"0GLC","title":"Derived upper shriek · Lemma 0GLC","summary":"Let f : X → Y be a separated finite type morphism of quasi-compact and quasi-separated schemes. Let Lambda be a torsion ring. For every K ∈ D(Y_etale, Lambda) and L ∈ D(X_etale, Lambda) the map ([Tag 0GLB]) Rf_*RSheafHom_Lambda(L, Rf^!K) → RSheafHom_Lambda(Rf_!L, K) is an isomorphism.","statement_latex":"Let $f : X \\to Y$ be a separated finite type morphism of quasi-compact and\nquasi-separated schemes. Let $\\Lambda$ be a torsion ring.\nFor every $K \\in D(Y_\\etale, \\Lambda)$ and $L \\in D(X_\\etale, \\Lambda)$\nthe map (\\ref{equation-sheafy-trace})\n$$\nRf_*R\\SheafHom_\\Lambda(L, Rf^!K) \\longrightarrow R\\SheafHom_\\Lambda(Rf_!L, K)\n$$\nis an isomorphism.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLC","source_file":"more-etale.tex","source_line":3929,"source_end_line":3939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3929-L3939","statement_sha256":"846480aae57ab7a6301fbb4b8c8275e5dcc19c51a7512ba22faf11845b10ecc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10591,"rank":10591,"depth":81,"x":1705.518,"y":1048.091,"cluster":"tale-geometry"},{"id":"stacks:0GLD","tag":"0GLD","title":"Derived upper shriek · Lemma 0GLD","summary":"Let f : X → Y be a separated finite type morphism of quasi-separated and quasi-compact schemes. Let Lambda be a torsion ring. For every K ∈ D(Y_etale, Lambda) and L ∈ D(X_etale, Lambda) the map ([Tag 0GLB]) induces an isomorphism RHom_X(L, Rf^!K) → RHom_Y(Rf_!L, K) of global derived homs.","statement_latex":"Let $f : X \\to Y$ be a separated finite type morphism of\nquasi-separated and quasi-compact schemes. Let $\\Lambda$ be a torsion ring.\nFor every $K \\in D(Y_\\etale, \\Lambda)$ and $L \\in D(X_\\etale, \\Lambda)$ the map\n(\\ref{equation-sheafy-trace}) induces an isomorphism\n$$\nR\\Hom_X(L, Rf^!K) \\longrightarrow R\\Hom_Y(Rf_!L, K)\n$$\nof global derived homs.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLD","source_file":"more-etale.tex","source_line":3974,"source_end_line":3984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L3974-L3984","statement_sha256":"b17ae3de5ed5da6a0aa15e05fc10d3317207279e885c88a169ec6dcb6d3d69a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10592,"rank":10592,"depth":82,"x":1561.345,"y":1279.076,"cluster":"tale-geometry"},{"id":"stacks:0GLE","tag":"0GLE","title":"Derived upper shriek · Lemma 0GLE","summary":"Consider a cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y of quasi-compact and quasi-separated schemes with f separated and of finite type. Then we have Rf^! ∘ Rg_* = Rg'_* ∘ R(f')^!.","statement_latex":"Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nof quasi-compact and quasi-separated schemes with $f$ separated and\nof finite type. Then we have $Rf^! \\circ Rg_* = Rg'_* \\circ R(f')^!$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Derived upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLE","source_file":"more-etale.tex","source_line":4002,"source_end_line":4013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4002-L4013","statement_sha256":"e6a94e767032aba3eacf9aafcac40bddfddd856371df6eb3397dfe708ddc4290","origin":"The Stacks Project","memory_eligible":false,"source_rank":10593,"rank":10593,"depth":85,"x":1481.858,"y":1026.797,"cluster":"tale-geometry"},{"id":"stacks:0GJZ","tag":"0GJZ","title":"Compactly supported cohomology · Definition 0GJZ","summary":"Let X be a separated scheme of finite type over a field k. Let Lambda be a ring. Let K be an object of D^+_tors(X_etale, Lambda) or of D(X_etale, Lambda) in case Lambda is torsion. The cohomology of K with compact support or the compactly supported cohomology of K is RΓ_c(X, K) = RΓ(Spec(k), Rf_!K) where f : X → Spec(k) is the structure morphism. We will write H^i_c(X, K) = H^i(RΓ_c(X, K)).","statement_latex":"Let $X$ be a separated scheme of finite type over a field $k$.\nLet $\\Lambda$ be a ring. Let $K$ be an object of\n$D^+_{tors}(X_\\etale, \\Lambda)$\nor of $D(X_\\etale, \\Lambda)$ in case $\\Lambda$ is torsion.\nThe {\\it cohomology of $K$ with compact support} or the\n{\\it compactly supported cohomology of $K$} is\n$$\nR\\Gamma_c(X, K) = R\\Gamma(\\Spec(k), Rf_!K)\n$$\nwhere $f : X \\to \\Spec(k)$ is the structure morphism. We will\nwrite  $H^i_c(X, K) = H^i(R\\Gamma_c(X, K))$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Compactly supported cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GJZ","source_file":"more-etale.tex","source_line":4071,"source_end_line":4084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4071-L4084","statement_sha256":"5826799fb99694fa257de655512a38cbc14579c6c11876a5b1f81afeafd7b77a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10594,"rank":10594,"depth":0,"x":1743.498,"y":1167.802,"cluster":"tale-geometry"},{"id":"stacks:0GK0","tag":"0GK0","title":"Compactly supported cohomology · Lemma 0GK0","summary":"Let f : X → Y be a finite type separated morphism of schemes with Y quasi-compact and quasi-separated. Let K be an object of D^+_tors(X_etale, Lambda) or of D(X_etale, Lambda) in case Lambda is torsion. Then there is a canonical isomorphism (Rf_!K)_overliney → RΓ_c(X_overliney, K|_X_overliney) in D(Lambda) for any geometric point overliney : Spec(k) → Y.","statement_latex":"Let $f : X \\to Y$ be a finite type separated morphism of schemes\nwith $Y$ quasi-compact and quasi-separated. Let $K$ be an object of\n$D^+_{tors}(X_\\etale, \\Lambda)$ or of $D(X_\\etale, \\Lambda)$ in case\n$\\Lambda$ is torsion. Then there is a canonical isomorphism\n$$\n(Rf_!K)_{\\overline{y}}\n\\longrightarrow\nR\\Gamma_c(X_{\\overline{y}}, K|_{X_{\\overline{y}}})\n$$\nin $D(\\Lambda)$ for any geometric point $\\overline{y} : \\Spec(k) \\to Y$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Compactly supported cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GK0","source_file":"more-etale.tex","source_line":4092,"source_end_line":4104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4092-L4104","statement_sha256":"d13f9c110640a0c8eda834b41c6b449dfd94cf0b220af1cf45427b407e7c0265","origin":"The Stacks Project","memory_eligible":false,"source_rank":10595,"rank":10595,"depth":85,"x":1436.998,"y":1212.314,"cluster":"tale-geometry"},{"id":"stacks:0GK1","tag":"0GK1","title":"Compactly supported cohomology · Lemma 0GK1","summary":"Let X be a separated scheme of finite type over a field k. If F is a torsion abelian sheaf, then the abelian group H^0_c(X, F) defined in Definition [Tag 0F72] agrees with the abelian group H^0_c(X, F) defined in Definition [Tag 0GJZ].","statement_latex":"Let $X$ be a separated scheme of finite type over a field $k$.\nIf $\\mathcal{F}$ is a torsion abelian sheaf, then the abelian group\n$H^0_c(X, \\mathcal{F})$ defined in Definition \\ref{definition-compact-support}\nagrees with the abelian group $H^0_c(X, \\mathcal{F})$ defined in\nDefinition \\ref{definition-cohomology-compact-support}.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Compactly supported cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GK1","source_file":"more-etale.tex","source_line":4111,"source_end_line":4118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4111-L4118","statement_sha256":"8a9363d9d2301fac2bfe534b27ce3f050666531ffa9181dc26da9dcd7d4de189","origin":"The Stacks Project","memory_eligible":false,"source_rank":10596,"rank":10596,"depth":53,"x":1627.322,"y":1005.456,"cluster":"tale-geometry"},{"id":"stacks:0GKR","tag":"0GKR","title":"Compactly supported cohomology · Lemma 0GKR","summary":"Let k be an algebraically closed field. Let X be a separated scheme of finite type type over k of dimension ≤ 1. Let Lambda be a Noetherian ring. Let F be a constructible sheaf of Lambda-modules on X which is torsion. Then H^q_c(X, F) is a finite Lambda-module.","statement_latex":"Let $k$ be an algebraically closed field. Let $X$ be a separated\nscheme of finite type type over $k$ of dimension $\\leq 1$.\nLet $\\Lambda$ be a Noetherian ring.\nLet $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules\non $X$ which is torsion. Then $H^q_c(X, \\mathcal{F})$ is a\nfinite $\\Lambda$-module.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Compactly supported cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKR","source_file":"more-etale.tex","source_line":4128,"source_end_line":4136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4128-L4136","statement_sha256":"0b62a94a55db64029937f041216c1e90440cb4d53c8055d91437db8c9a0fe019","origin":"The Stacks Project","memory_eligible":false,"source_rank":10597,"rank":10597,"depth":77,"x":1653.344,"y":1266.135,"cluster":"tale-geometry"},{"id":"stacks:0GKU","tag":"0GKU","title":"A constructibility result · Lemma 0GKU","summary":"Let p be a prime number. Let S be a scheme over F_p. Let E be a finite locally free O_S-module viewed as an O_S-module on S_etale. Let F : E → E be a homomorphism of abelian sheaves on S_etale such that F(a e) = a^pF(e) for local sections a, e of O_S, E on S_etale. Then Coker(F - 1 : E → E) is zero and Ker(F - 1 : E → E) is a constructible abelian sheaf on S_etale.","statement_latex":"Let $p$ be a prime number. Let $S$ be a scheme over $\\mathbf{F}_p$.\nLet $\\mathcal{E}$ be a finite locally free\n$\\mathcal{O}_S$-module viewed as an $\\mathcal{O}_S$-module on $S_\\etale$.\nLet $F : \\mathcal{E} \\to \\mathcal{E}$ be a homomorphism of abelian sheaves\non $S_\\etale$ such that $F(a e) = a^pF(e)$ for local sections $a$, $e$\nof $\\mathcal{O}_S$, $\\mathcal{E}$ on $S_\\etale$. Then\n$$\n\\Coker(F - 1 : \\mathcal{E} \\to \\mathcal{E})\n$$\nis zero and\n$$\n\\Ker(F - 1 : \\mathcal{E} \\to \\mathcal{E})\n$$\nis a constructible abelian sheaf on $S_\\etale$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"A constructibility result","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKU","source_file":"more-etale.tex","source_line":4212,"source_end_line":4228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4212-L4228","statement_sha256":"7ddd915356ebafd525f9aceedccecb960434be1a7ae5c301380ce994c32576f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10598,"rank":10598,"depth":0,"x":1424.393,"y":1088.577,"cluster":"tale-geometry"},{"id":"stacks:0GKV","tag":"0GKV","title":"A constructibility result · Lemma 0GKV","summary":"Let f : X → S be a proper smooth morphism of schemes with geometrically connected fibres of dimension 1. Let ℓ be a prime number. Then R^qf_*underlineZ/ℓZ is a constructible.","statement_latex":"Let $f : X \\to S$ be a proper smooth morphism of schemes with\ngeometrically connected fibres of dimension $1$. Let $\\ell$\nbe a prime number. Then $R^qf_*\\underline{\\mathbf{Z}/\\ell\\mathbf{Z}}$\nis a constructible.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"A constructibility result","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKV","source_file":"more-etale.tex","source_line":4279,"source_end_line":4285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4279-L4285","statement_sha256":"f662f83d3c0bbd0353cc2ba2baaa33398587dfa630379bd88fc6d0561c70c1ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":10599,"rank":10599,"depth":82,"x":1736.185,"y":1089.594,"cluster":"tale-geometry"},{"id":"stacks:0GKW","tag":"0GKW","title":"A constructibility result · Lemma 0GKW","summary":"Let f : X → S be a proper smooth morphism of schemes with geometrically connected fibres of dimension 1. Let Lambda be a Noetherian ring. Let M be a finite Lambda-module annihilated by an integer n > 0. Then R^qf_*underlineM is a constructible sheaf of Lambda-modules on S.","statement_latex":"Let $f : X \\to S$ be a proper smooth morphism of schemes with\ngeometrically connected fibres of dimension $1$. Let $\\Lambda$\nbe a Noetherian ring. Let $M$ be a finite $\\Lambda$-module\nannihilated by an integer $n > 0$. Then $R^qf_*\\underline{M}$\nis a constructible sheaf of $\\Lambda$-modules on $S$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"A constructibility result","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKW","source_file":"more-etale.tex","source_line":4365,"source_end_line":4372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4365-L4372","statement_sha256":"205bf3645f34d62cf849e3a89754e43fd5942120e42b31eef693b4338036736f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10600,"rank":10600,"depth":83,"x":1505.323,"y":1265.865,"cluster":"tale-geometry"},{"id":"stacks:0GK3","tag":"0GK3","title":"Complexes with constructible cohomology · Lemma 0GK3","summary":"Let f : X → Y be a morphism of schemes which is locally quasi-finite and of finite presentation. The functor f_! : D(X_etale, Lambda) → D(Y_etale, Lambda) of Lemma [Tag 0F5N] sends D_c(X_etale, Lambda) into D_c(Y_etale, Lambda).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes which is\nlocally quasi-finite and of finite presentation.\nThe functor $f_! : D(X_\\etale, \\Lambda) \\to D(Y_\\etale, \\Lambda)$\nof Lemma \\ref{lemma-lqf-shriek-derived}\nsends $D_c(X_\\etale, \\Lambda)$ into $D_c(Y_\\etale, \\Lambda)$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Complexes with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GK3","source_file":"more-etale.tex","source_line":4469,"source_end_line":4476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4469-L4476","statement_sha256":"3ac4f4b3b834222b61df12e56fa7ba3fc3903245ea52380d8c5705e58d8587e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10601,"rank":10601,"depth":55,"x":1533.822,"y":1004.738,"cluster":"tale-geometry"},{"id":"stacks:0GKX","tag":"0GKX","title":"Complexes with constructible cohomology · Lemma 0GKX","summary":"Let S be a Noetherian affine scheme of finite dimension. Let f : X → S be a separated, affine, smooth morphism of relative dimension 1. Let Lambda be a Noetherian ring which is torsion. Let M be a finite Lambda-module. Then Rf_!underlineM has constructible cohomology sheaves.","statement_latex":"Let $S$ be a Noetherian affine scheme of finite dimension.\nLet $f : X \\to S$ be a separated, affine, smooth morphism\nof relative dimension $1$. Let $\\Lambda$ be a Noetherian ring\nwhich is torsion. Let $M$ be a finite $\\Lambda$-module.\nThen $Rf_!\\underline{M}$ has constructible cohomology sheaves.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Complexes with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKX","source_file":"more-etale.tex","source_line":4511,"source_end_line":4518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4511-L4518","statement_sha256":"28c6918f85c32562c54e418909db586ea4703a1c45a5f211aba0e28a97ae24ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":10602,"rank":10602,"depth":86,"x":1722.907,"y":1213.58,"cluster":"tale-geometry"},{"id":"stacks:0GKY","tag":"0GKY","title":"Complexes with constructible cohomology · Lemma 0GKY","summary":"Let Y be a Noetherian affine scheme of finite dimension. Let Lambda be a Noetherian ring which is torsion. Let F be a finite type, locally constant sheaf of Lambda-modules on an open subscheme U ⊂ A^1_Y. Then Rf_!F has constructible cohomology sheaves where f : U → Y is the structure morphism.","statement_latex":"Let $Y$ be a Noetherian affine scheme of finite dimension.\nLet $\\Lambda$ be a Noetherian ring which is torsion.\nLet $\\mathcal{F}$ be a finite type, locally constant sheaf of\n$\\Lambda$-modules on an open subscheme $U \\subset \\mathbf{A}^1_Y$.\nThen $Rf_!\\mathcal{F}$ has constructible cohomology sheaves where\n$f : U \\to Y$ is the structure morphism.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Complexes with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKY","source_file":"more-etale.tex","source_line":4611,"source_end_line":4619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4611-L4619","statement_sha256":"f4e3f17aa3ffce5ca6abf7807dc4fdd653686dce0815824b478f1e7080169eee","origin":"The Stacks Project","memory_eligible":false,"source_rank":10603,"rank":10603,"depth":87,"x":1415.357,"y":1166.848,"cluster":"tale-geometry"},{"id":"stacks:0GKZ","tag":"0GKZ","title":"Complexes with constructible cohomology · Lemma 0GKZ","summary":"Let Y be an affine scheme. Let Lambda be a Noetherian ring. Let F be a constructible sheaf of Lambda-modules on A^1_Y which is torsion. Then Rf_!F has constructible cohomology sheaves where f : A^1_Y → Y is the structure morphism.","statement_latex":"Let $Y$ be an affine scheme. Let $\\Lambda$ be a Noetherian ring.\nLet $\\mathcal{F}$ be a constructible sheaf of $\\Lambda$-modules on\n$\\mathbf{A}^1_Y$ which is torsion. Then $Rf_!\\mathcal{F}$ has constructible\ncohomology sheaves where $f : \\mathbf{A}^1_Y \\to Y$ is the structure morphism.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Complexes with constructible cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GKZ","source_file":"more-etale.tex","source_line":4651,"source_end_line":4657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4651-L4657","statement_sha256":"86969f13aec05f6a61d24f4398488040d23cc8e55b387fa4e31802fc8c9479b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10604,"rank":10604,"depth":88,"x":1679.872,"y":1026.716,"cluster":"tale-geometry"},{"id":"stacks:0GL0","tag":"0GL0","title":"Complexes with constructible cohomology · Theorem 0GL0","summary":"Let f : X → Y be a separated morphism of finite presentation of quasi-compact and quasi-separated schemes. Let Lambda be a Noetherian ring. Let K be an object of D^+_tors, c(X_etale, Lambda) or of D_c(X_etale, Lambda) in case Lambda is torsion. Then Rf_!K has constructible cohomology sheaves, i.e., Rf_!K is in D^+_tors, c(Y_etale, Lambda) or in D_c(Y_etale, Lambda) in case Lambda is torsion.","statement_latex":"Let $f : X \\to Y$ be a separated morphism of finite presentation\nof quasi-compact and quasi-separated schemes. Let $\\Lambda$ be a\nNoetherian ring. Let $K$ be an object of $D^+_{tors, c}(X_\\etale, \\Lambda)$\nor of $D_c(X_\\etale, \\Lambda)$ in case $\\Lambda$ is torsion.\nThen $Rf_!K$ has constructible cohomology sheaves, i.e.,\n$Rf_!K$ is in $D^+_{tors, c}(Y_\\etale, \\Lambda)$\nor in $D_c(Y_\\etale, \\Lambda)$ in case $\\Lambda$ is torsion.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Complexes with constructible cohomology","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GL0","source_file":"more-etale.tex","source_line":4710,"source_end_line":4719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4710-L4719","statement_sha256":"4d363e5e702267c071d8aa7d9ff405e6c25cb25ff934f1bf463fda2a9bfe66f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10605,"rank":10605,"depth":89,"x":1597.466,"y":1280.284,"cluster":"tale-geometry"},{"id":"stacks:0GLH","tag":"0GLH","title":"Applications · Lemma 0GLH","summary":"Let k be an algebraically closed field. Let X be a finite type separated scheme over k. Let Lambda be a Noetherian ring. Let K be an object of D^+_tors, c(X_etale, Lambda) or of D_c(X_etale, Lambda) in case Lambda is torsion. Then H^i_c(X, K) is a finite Lambda-module for all i ∈ Z.","statement_latex":"Let $k$ be an algebraically closed field.\nLet $X$ be a finite type separated scheme over $k$.\nLet $\\Lambda$ be a Noetherian ring. Let $K$ be an object of\n$D^+_{tors, c}(X_\\etale, \\Lambda)$ or of $D_c(X_\\etale, \\Lambda)$\nin case $\\Lambda$ is torsion. Then\n$H^i_c(X, K)$ is a finite $\\Lambda$-module for all\n$i \\in \\mathbf{Z}$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLH","source_file":"more-etale.tex","source_line":4768,"source_end_line":4777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4768-L4777","statement_sha256":"b8b6126309f95ad77cc7768439dcd937096c16e776a724be651998e3ed291524","origin":"The Stacks Project","memory_eligible":false,"source_rank":10606,"rank":10606,"depth":90,"x":1454.236,"y":1046.413,"cluster":"tale-geometry"},{"id":"stacks:0GLI","tag":"0GLI","title":"Applications · Proposition 0GLI","summary":"Let f : X → S be a smooth proper morphism of schemes. Let Lambda be a Noetherian ring. Let F be a finite type, locally constant sheaf of Lambda-modules on X_etale such that for every geometric point overlinex of X the stalk F_overlinex is annihilated by an integer n > 0 prime to the residue characteristic of overlinex. Then R^if_*F is a finite type, locally constant sheaf of Lambda-modules on S_etale for all i ∈ Z.","statement_latex":"Let $f : X \\to S$ be a smooth proper morphism of schemes.\nLet $\\Lambda$ be a Noetherian ring. Let $\\mathcal{F}$ be a\nfinite type, locally constant sheaf of $\\Lambda$-modules\non $X_\\etale$ such that for every geometric point $\\overline{x}$ of $X$\nthe stalk $\\mathcal{F}_{\\overline{x}}$ is annihilated by an integer\n$n > 0$ prime to the residue characteristic of $\\overline{x}$.\nThen $R^if_*\\mathcal{F}$ is a finite type, locally constant sheaf of\n$\\Lambda$-modules on $S_\\etale$ for all $i \\in \\mathbf{Z}$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"Applications","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLI","source_file":"more-etale.tex","source_line":4785,"source_end_line":4795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4785-L4795","statement_sha256":"7a93cee9359bcce470c24f83b04103a0fd407244856ef7c4fcb899ba69b7ce35","origin":"The Stacks Project","memory_eligible":false,"source_rank":10607,"rank":10607,"depth":90,"x":1748.092,"y":1137.647,"cluster":"tale-geometry"},{"id":"stacks:0GLK","tag":"0GLK","title":"More on derived upper shriek · Lemma 0GLK","summary":"Let f : X → Y be a separated finite type morphism of quasi-compact and quasi-separated schemes. Let Lambda be a torsion ring. Let K ∈ D(Y_etale, Lambda). For n ∈ Z the cohomology sheaf H^n(Rf^!K) restricted to X_affine, etale is the sheaf associated to the presheaf U ↦ Hom_Y(R(U → Y)_!Lambda, K[n]) See discussion above for the functorial nature of R(U → Y)_!Lambda.","statement_latex":"Let $f : X \\to Y$ be a separated finite type morphism of quasi-compact\nand quasi-separated schemes. Let $\\Lambda$ be a torsion ring.\nLet $K \\in D(Y_\\etale, \\Lambda)$. For $n \\in \\mathbf{Z}$ the\ncohomology sheaf $H^n(Rf^!K)$ restricted to $X_{affine, \\etale}$\nis the sheaf associated to the presheaf\n$$\nU \\longmapsto \\Hom_Y(R(U \\to Y)_!\\Lambda, K[n])\n$$\nSee discussion above for the functorial nature of $R(U \\to Y)_!\\Lambda$.","area":"Étale Geometry","chapter":"More Étale Cohomology","chapter_id":"more-etale","section":"More on derived upper shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GLK","source_file":"more-etale.tex","source_line":4919,"source_end_line":4930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/more-etale.tex#L4919-L4930","statement_sha256":"afcb4fc3324a405a632513d83f20715b0d58b60c11387ae3fa9a68ecb397875b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10608,"rank":10608,"depth":0,"x":1457.874,"y":1237.17,"cluster":"tale-geometry"},{"id":"stacks:03SP","tag":"03SP","title":"Frobenii · Lemma 03SP","summary":"Let X be a scheme and g : X → X a morphism. Assume that for all φ : U → X étale, there is an isomorphism xymatrix U ar[rd]_φ ar[rr]^-sim & & U ×_φ, X, g X ar[ld]^pr_2 & X functorial in U. Then g induces the identity on cohomology (for any sheaf).","statement_latex":"Let $X$ be a scheme and $g : X \\to X$ a morphism. Assume that for all\n$\\varphi : U \\to X$ \\'etale, there is an isomorphism\n$$\n\\xymatrix{\nU \\ar[rd]_\\varphi \\ar[rr]^-\\sim & & {U\n\\times_{\\varphi, X, g} X} \\ar[ld]^{\\text{pr}_2} \\\\\n& X\n}\n$$\nfunctorial in $U$. Then $g$ induces the identity on cohomology (for any sheaf).","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SP","source_file":"trace.tex","source_line":82,"source_end_line":94,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L82-L94","statement_sha256":"ecf152f14871be07605e3be7598f68c98b0a88d16de3d6dfb41d90e2fd33530c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10609,"rank":10609,"depth":0,"x":1591.919,"y":998.971,"cluster":"tale-geometry"},{"id":"stacks:03SN","tag":"03SN","title":"The Baffling Theorem · Theorem 03SN","summary":"Let X be a scheme in characteristic p > 0. Then the absolute frobenius induces (by pullback) the trivial map on cohomology, i.e., for all integers j≥ 0, F_X^* : H^j (X, underlineZ/nZ) → H^j (X, underlineZ/nZ) is the identity.","statement_latex":"Let $X$ be a scheme in characteristic $p > 0$. Then the absolute frobenius\ninduces (by pullback) the trivial map on cohomology, i.e., for all\nintegers $j\\geq 0$,\n$$\nF_X^* : H^j (X, \\underline{\\mathbf{Z}/n\\mathbf{Z}}) \\longrightarrow H^j (X,\n\\underline{\\mathbf{Z}/n\\mathbf{Z}})\n$$\nis the identity.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Frobenii","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SN","source_file":"trace.tex","source_line":104,"source_end_line":114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L104-L114","statement_sha256":"25966ca45d9adbca595fa945ca721b160ec6edcbbff6ad9a2ecc3f107c408e76","origin":"The Stacks Project","memory_eligible":false,"source_rank":10610,"rank":10610,"depth":0,"x":1684.683,"y":1250.819,"cluster":"tale-geometry"},{"id":"stacks:03SQ","tag":"03SQ","title":"Frobenii · Definition 03SQ","summary":"Let k be a finite field with q = p^f elements. Let X be a scheme over k. The geometric frobenius of X is the morphism π_X : X → X over Spec(k) which equals F_X^f.","statement_latex":"Let $k$ be a finite field with $q = p^f$ elements. Let $X$ be a scheme\nover $k$. The {\\it geometric frobenius} of $X$ is the morphism\n$\\pi_X : X \\to X$ over $\\Spec(k)$ which equals $F_X^f$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Frobenii","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SQ","source_file":"trace.tex","source_line":146,"source_end_line":151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L146-L151","statement_sha256":"20e8ef9499388997544c789850e137d22a38cf3989f2cbd6df4dc9d964f1d97c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10611,"rank":10611,"depth":0,"x":1413.596,"y":1117.671,"cluster":"tale-geometry"},{"id":"stacks:03SR","tag":"03SR","title":"Frobenii · Lemma 03SR","summary":"Let F be a sheaf on X_etale. Then there are canonical isomorphisms π_X^-1 F ≅ F and F ≅ π_X_*F.","statement_latex":"Let $\\mathcal{F}$ be a sheaf on $X_\\etale$.\nThen there are canonical isomorphisms\n$\\pi_X^{-1} \\mathcal{F} \\cong \\mathcal{F}$ and\n$\\mathcal{F} \\cong {\\pi_X}_*\\mathcal{F}$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SR","source_file":"trace.tex","source_line":160,"source_end_line":166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L160-L166","statement_sha256":"7b14f61d4a5aa99a34f38833874a5d276ce489d7048d2a34c0aeab3674adc11d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10612,"rank":10612,"depth":0,"x":1720.74,"y":1062.0,"cluster":"tale-geometry"},{"id":"stacks:03ST","tag":"03ST","title":"Frobenii · Lemma 03ST","summary":"In the situation above denote α : X → Spec(k) the structure morphism. Consider the stalk (R^jα_*F)_Spec(bar k) endowed with its natural Galois action as in Étale Cohomology, Section [Tag 03QW]. Then the identification (R^jα_*F)_Spec(bar k) ≅ H^j (X_bar k, F|_X_bar k) from Étale Cohomology, Theorem [Tag 03Q9] is an isomorphism of G_k-modules.","statement_latex":"In the situation above denote $\\alpha : X \\to \\Spec(k)$ the structure morphism.\nConsider the stalk $(R^j\\alpha_*\\mathcal{F})_{\\Spec(\\bar k)}$ endowed with its\nnatural Galois action as in\n\\'Etale Cohomology, Section \\ref{etale-cohomology-section-galois-action-stalks}.\nThen the identification\n$$\n(R^j\\alpha_*\\mathcal{F})_{\\Spec(\\bar k)} \\cong H^j (X_{\\bar k},\n\\mathcal{F}|_{X_{\\bar k}})\n$$\nfrom\n\\'Etale Cohomology, Theorem \\ref{etale-cohomology-theorem-higher-direct-images}\nis an isomorphism of $G_k$-modules.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Frobenii","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ST","source_file":"trace.tex","source_line":225,"source_end_line":239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L225-L239","statement_sha256":"fb4fc45bc78940a678ede3c660d3cedc83ed65dbf356d618e31db1c3bfd9a799","origin":"The Stacks Project","memory_eligible":false,"source_rank":10613,"rank":10613,"depth":48,"x":1538.923,"y":1277.454,"cluster":"tale-geometry"},{"id":"stacks:03SU","tag":"03SU","title":"Frobenii · Definition 03SU","summary":"The arithmetic frobenius is the map frob_k : bar k → bar k, x ↦ x^q of G_k.","statement_latex":"The {\\it arithmetic frobenius} is the map\n$\\text{frob}_k : \\bar k \\to \\bar k$, $x \\mapsto x^q$ of $G_k$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Frobenii","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SU","source_file":"trace.tex","source_line":250,"source_end_line":254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L250-L254","statement_sha256":"b6231b95c00ae8a52dc9ed67d2cacd53b420d92157883395d5882434834bfe5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10614,"rank":10614,"depth":0,"x":1499.708,"y":1015.265,"cluster":"tale-geometry"},{"id":"stacks:03SV","tag":"03SV","title":"Frobenii · Theorem 03SV","summary":"Let F be an abelian sheaf on X_etale. Then for all j≥ 0, frob_k acts on the cohomology group H^j(X_bar k, F|_X_bar k) as the inverse of the map π_X^*.","statement_latex":"Let $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$. Then for all\n$j\\geq 0$, $\\text{frob}_k$ acts on the cohomology group $H^j(X_{\\bar k},\n\\mathcal{F}|_{X_{\\bar k}})$ as the inverse of the map $\\pi_X^*$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Frobenii","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SV","source_file":"trace.tex","source_line":256,"source_end_line":261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L256-L261","statement_sha256":"737c3de610051a5503f34237f60f8bb8841485d06508e6b5e5417e56b80ffc81","origin":"The Stacks Project","memory_eligible":false,"source_rank":10615,"rank":10615,"depth":0,"x":1739.605,"y":1186.444,"cluster":"tale-geometry"},{"id":"stacks:03SW","tag":"03SW","title":"Frobenii · Definition 03SW","summary":"If x ∈ X(k) is a rational point and bar x : Spec(bar k) → X the geometric point lying over x, we let π_x : F_bar x → F_bar x denote the action by frob_k^-1 and call it the geometric frobenius","statement_latex":"If $x \\in X(k)$ is a rational point and $\\bar x : \\Spec(\\bar k) \\to X$\nthe geometric point lying over $x$, we let $\\pi_x : \\mathcal{F}_{\\bar x} \\to\n\\mathcal{F}_{\\bar x}$ denote the action by $\\text{frob}_k^{-1}$ and call it the\n{\\it geometric frobenius}\\footnote{This notation is not standard.\nThis operator is denoted $F_x$ in \\cite{SGA4.5}. We will likely change\nthis notation in the future.}","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Frobenii","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03SW","source_file":"trace.tex","source_line":283,"source_end_line":291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L283-L291","statement_sha256":"1067476eeb735bf228b8dadce477756ef755013a623f1c3bb8e171a60c687868","origin":"The Stacks Project","memory_eligible":false,"source_rank":10616,"rank":10616,"depth":0,"x":1424.873,"y":1196.348,"cluster":"tale-geometry"},{"id":"stacks:03T0","tag":"03T0","title":"Traces · Definition 03T0","summary":"The trace of the endomorphism a is the sum of the diagonal entries of a matrix representing it. This defines an additive map Tr : End_Lambda(Lambda^⊕ m) → Lambda^natural.","statement_latex":"The {\\it trace} of the endomorphism $a$ is the sum of the diagonal entries of\na matrix representing it. This defines an additive map $\\text{Tr} :\n\\text{End}_\\Lambda(\\Lambda^{\\oplus m}) \\to \\Lambda^\\natural$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Traces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03T0","source_file":"trace.tex","source_line":407,"source_end_line":412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L407-L412","statement_sha256":"7b9603e945528169ee8f3695d44b15f99f85960ebd5f43d41850553bdd6c83eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10617,"rank":10617,"depth":0,"x":1649.114,"y":1010.354,"cluster":"tale-geometry"},{"id":"stacks:03T5","tag":"03T5","title":"Derived categories · Lemma 03T5","summary":"An object E of D(A) is contained in D^+(A) if and only if H^i(E) =0 for all i ll 0. Similar statements hold for D^- and D^b.","statement_latex":"An object $E$ of $D(\\mathcal{A})$ is contained in $D^+(\\mathcal{A})$ if and\nonly if $H^i(E) =0 $ for all $i \\ll 0$. Similar statements hold for $D^-$ and\n$D^b$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03T5","source_file":"trace.tex","source_line":530,"source_end_line":535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L530-L535","statement_sha256":"f8949abbfa2eb20a3a1e462e86aef1351333b43bffa1d18a7b5ef8af02b693eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10618,"rank":10618,"depth":1,"x":1633.325,"y":1274.892,"cluster":"tale-geometry"},{"id":"stacks:03T6","tag":"03T6","title":"Derived categories · Lemma 03T6","summary":"Morphisms between objects in the derived category. • Let I^bullet ∈ Comp^+(A) with I^n injective for all n ∈ Z. Then Hom_D(A)(K^bullet, I^bullet) = Hom_K(A)(K^bullet, I^bullet). • Let P^bullet ∈ Comp^-(A) with P^n is projective for all n ∈ Z. Then Hom_D(A)(P^bullet, K^bullet) = Hom_K(A)(P^bullet, K^bullet). • If A has enough injectives and I ⊂ A is the additive subcategory of injectives, then D^+(A)≅ K^+(I) (as triangulated categories). • If A has enough projectives and P…","statement_latex":"Morphisms between objects in the derived category.\n\\begin{enumerate}\n\\item\nLet $I^\\bullet \\in \\text{Comp}^+(\\mathcal{A})$ with $I^n$ injective for all\n$n \\in \\mathbf{Z}$. Then\n$$\n\\Hom_{D(\\mathcal{A})}(K^\\bullet, I^\\bullet)\n=\n\\Hom_{K(\\mathcal{A})}(K^\\bullet, I^\\bullet).\n$$\n\\item\nLet $P^\\bullet \\in \\text{Comp}^-(\\mathcal{A})$ with $P^n$ is projective for all\n$n \\in \\mathbf{Z}$. Then\n$$\n\\Hom_{D(\\mathcal{A})}(P^\\bullet, K^\\bullet)\n=\n\\Hom_{K(\\mathcal{A})}(P^\\bullet, K^\\bullet).\n$$\n\\item\nIf $\\mathcal{A}$ has enough injectives and $\\mathcal{I} \\subset \\mathcal{A}$\nis the additive subcategory of injectives, then\n$\nD^+(\\mathcal{A})\\cong K^+(\\mathcal{I})\n$\n(as triangulated categories).\n\\item\nIf $\\mathcal{A}$ has enough projectives and $\\mathcal{P} \\subset \\mathcal{A}$\nis the additive subcategory of projectives, then\n$\nD^-(\\mathcal{A}) \\cong K^-(\\mathcal{P}).\n$\n\\end{enumerate}","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03T6","source_file":"trace.tex","source_line":542,"source_end_line":576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L542-L576","statement_sha256":"6cb0f78cc66d0efe90d5c4262177872977446265982b8c8e46a0cdfff3852ac9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10619,"rank":10619,"depth":0,"x":1432.119,"y":1070.751,"cluster":"tale-geometry"},{"id":"stacks:03T7","tag":"03T7","title":"Derived categories · Definition 03T7","summary":"Let F: A → B be a left exact functor and assume that A has enough injectives. We define the total right derived functor of F as the functor RF: D^+(A) → D^+(B) fitting into the diagram xymatrix D^+(A) ar[r]^RF & D^+(B) K^+( I) ar[u] ar[r]^F & K^+(B). ar[u] This is possible since the left vertical arrow is invertible by the previous lemma. Similarly, let G: A → B be a right exact functor and assume that A has enough projectives. We define the total left derived functor of…","statement_latex":"Let $F: \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor and assume that\n$\\mathcal{A}$ has enough injectives. We define the {\\it total right derived\nfunctor of $F$} as the functor $RF: D^+(\\mathcal{A}) \\to D^+(\\mathcal{B})$\nfitting into the diagram\n$$\n\\xymatrix{\nD^+(\\mathcal{A}) \\ar[r]^{RF} & D^+(\\mathcal{B}) \\\\\nK^+(\\mathcal I) \\ar[u] \\ar[r]^F & K^+(\\mathcal{B}). \\ar[u]\n}\n$$\nThis is possible since the left vertical arrow is invertible by the previous\nlemma. Similarly, let $G: \\mathcal{A} \\to \\mathcal{B}$ be a right exact\nfunctor and assume that $\\mathcal{A}$ has enough projectives. We define the\n{\\it total left derived functor of $G$} as the functor $LG: D^-(\\mathcal{A})\n\\to D^-(\\mathcal{B})$ fitting into the diagram\n$$\n\\xymatrix{\nD^-(\\mathcal{A}) \\ar[r]^{LG} & D^-(\\mathcal{B}) \\\\\nK^-(\\mathcal{P}) \\ar[u] \\ar[r]^G & K^-(\\mathcal{B}). \\ar[u]\n}\n$$\nThis is possible since the left vertical arrow is invertible by the previous\nlemma.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Derived categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03T7","source_file":"trace.tex","source_line":582,"source_end_line":607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L582-L607","statement_sha256":"faca2d5082a2bb7c88077d41d5af78af4dd9c1dd4b49d6e82bd08796791d63ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":10620,"rank":10620,"depth":0,"x":1744.825,"y":1107.133,"cluster":"tale-geometry"},{"id":"stacks:03TA","tag":"03TA","title":"Filtered derived category · Definition 03TA","summary":"Let A be an abelian category. • Let Fil(A) be the category of filtered objects (A, F) of A, where F is a filtration of the form A ⊃ … ⊃ F^n A ⊃ F^n+1A ⊃ … ⊃ 0. This is an additive category. • We denote Fil^f(A) the full subcategory of Fil(A) whose objects (A, F) have finite filtration. This is also an additive category. • An object I ∈ Fil^f(A) is called filtered injective (respectively projective) provided that gr^p(I) = gr_F^p(I) = F^pI/F^p+1I is injective (resp.…","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\n\\begin{enumerate}\n\\item Let $\\text{Fil}(\\mathcal{A})$ be the category of filtered objects\n$(A, F)$ of $\\mathcal{A}$, where $F$ is a filtration of the form\n$$\nA \\supset \\ldots \\supset F^n A \\supset F^{n+1}A \\supset \\ldots\n\\supset 0.\n$$\nThis is an additive category.\n\\item We denote $\\text{Fil}^f(\\mathcal{A})$ the full\nsubcategory of $\\text{Fil}(\\mathcal{A})$ whose objects $(A, F)$ have finite\nfiltration. This is also an additive category.\n\\item An object $I \\in \\text{Fil}^f(\\mathcal{A})$ is called\n{\\it filtered injective} (respectively {\\it projective}) provided\nthat $\\text{gr}^p(I) = \\text{gr}_F^p(I) = F^pI/F^{p+1}I$ is injective\n(resp. projective) in $\\mathcal{A}$ for all $p$.\n\\item The category of complexes\n$\\text{Comp}(\\text{Fil}^f(\\mathcal{A})) \\supset\n\\text{Comp}^+(\\text{Fil}^f(\\mathcal{A}))$\nand its homotopy category\n$K(\\text{Fil}^f(\\mathcal{A})) \\supset K^+(\\text{Fil}^f(\\mathcal A))$\nare defined as before.\n\\item A morphism $\\alpha : K^\\bullet \\to L^\\bullet$ of complexes in\n$\\text{Comp}(\\text{Fil}^f(\\mathcal{A}))$ is called a\n{\\it filtered quasi-isomorphism} provided that\n$$\n\\text{gr}^p(\\alpha): \\text{gr}^p(K^\\bullet) \\to \\text{gr}^p(L^\\bullet)\n$$\nis a quasi-isomorphism for all $p \\in \\mathbf{Z}$.\n\\item We define $DF(\\mathcal{A})$ (resp. $DF^+(\\mathcal{A})$)\nby inverting the filtered quasi-isomorphisms in\n$K(\\text{Fil}^f(\\mathcal{A}))$ (resp. $K^+(\\text{Fil}^f(\\mathcal{A}))$).\n\\end{enumerate}","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Filtered derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TA","source_file":"trace.tex","source_line":626,"source_end_line":661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L626-L661","statement_sha256":"83dbf440abd0860313a17fa182c327ac032c1d76a5a1065401d8e7e46653b8a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10621,"rank":10621,"depth":0,"x":1484.837,"y":1257.828,"cluster":"tale-geometry"},{"id":"stacks:03TB","tag":"03TB","title":"Filtered derived category · Lemma 03TB","summary":"If A has enough injectives, then DF^+(A) ≅ K^+(I), where I is the full additive subcategory of Fil^f(A) consisting of filtered injective objects. Similarly, if A has enough projectives, then DF^-(A) ≅ K^-(P), where P is the full additive subcategory of Fil^f(A) consisting of filtered projective objects.","statement_latex":"If $\\mathcal{A}$ has enough injectives, then $DF^+(\\mathcal{A}) \\cong\nK^+(\\mathcal{I})$, where $\\mathcal{I}$ is the full additive subcategory of\n$\\text{Fil}^f(\\mathcal{A})$ consisting of filtered injective objects.\nSimilarly, if $\\mathcal{A}$ has enough projectives, then $DF^-(\\mathcal{A})\n\\cong K^-(\\mathcal{P})$, where $\\mathcal P$ is the full additive subcategory of\n$\\text{Fil}^f(\\mathcal{A})$ consisting of filtered projective objects.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Filtered derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TB","source_file":"trace.tex","source_line":663,"source_end_line":671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L663-L671","statement_sha256":"8b0acc6f277bc593d201ee899bede9898771fe01f92cf975911a4a09e3e7ea1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10622,"rank":10622,"depth":0,"x":1555.404,"y":999.039,"cluster":"tale-geometry"},{"id":"stacks:03TD","tag":"03TD","title":"Filtered derived functors · Definition 03TD","summary":"Let T: A → B be a left exact functor and assume that A has enough injectives. Define RT: DF^+(A) → D F^+(B) to fit in the diagram xymatrix DF^+(A) ar[r]^RT & DF^+(B) K^+(I) ar[u] ar[r]^T & K^+(Fil^f(B)). ar[u] This is well-defined by the previous lemma. Let G: A → B be a right exact functor and assume that A has enough projectives. Define LG: DF^-(A) → DF^-(B) to fit in the diagram xymatrix DF^-(A) ar[r]^LG & DF^-(B) K^-(P) ar[u] ar[r]^G & K^-(Fil^f(B)). ar[u] Again, this…","statement_latex":"Let $T: \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor and assume that\n$\\mathcal{A}$ has enough injectives. Define $RT: DF^+(\\mathcal{A}) \\to D\nF^+(\\mathcal{B})$ to fit in the diagram\n$$\n\\xymatrix{\nDF^+(\\mathcal{A}) \\ar[r]^{RT} & DF^+(\\mathcal{B}) \\\\\nK^+(\\mathcal{I}) \\ar[u] \\ar[r]^{T \\quad} & K^+(\\text{Fil}^f(\\mathcal{B})).\n\\ar[u]}\n$$\nThis is well-defined by the previous lemma. Let $G: \\mathcal{A} \\to\n\\mathcal{B}$ be a right exact functor and assume that $\\mathcal{A}$ has enough\nprojectives. Define $LG: DF^-(\\mathcal{A}) \\to DF^-(\\mathcal{B})$ to fit in\nthe diagram\n$$\n\\xymatrix{\nDF^-(\\mathcal{A}) \\ar[r]^{LG} & DF^-(\\mathcal{B}) \\\\\nK^-(\\mathcal{P}) \\ar[u] \\ar[r]^{G \\quad} & K^-(\\text{Fil}^f(\\mathcal{B})).\n\\ar[u]}\n$$\nAgain, this is well-defined by the previous lemma.\nThe functors $RT$, resp.\\ $LG$, are called the {\\it filtered derived\nfunctor} of $T$, resp.\\ $G$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Filtered derived functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TD","source_file":"trace.tex","source_line":687,"source_end_line":711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L687-L711","statement_sha256":"943240b8961a3ac4035ce525aa76f07428f00599a340df71ff213d8b3bbf063c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10623,"rank":10623,"depth":0,"x":1711.567,"y":1230.037,"cluster":"tale-geometry"},{"id":"stacks:03TE","tag":"03TE","title":"Filtered derived functors · Proposition 03TE","summary":"In the situation above, we have gr^p ∘ RT = RT ∘ gr^p where the RT on the left is the filtered derived functor while the one on the right is the total derived functor. That is, there is a commuting diagram xymatrix DF^+(A) ar[r]^RT ar[d]_gr^p & DF^+(B) ar[d]^gr^p D^+(A) ar[r]^RT & D^+(B).","statement_latex":"In the situation above, we have\n$$\n\\text{gr}^p \\circ RT = RT \\circ \\text{gr}^p\n$$\nwhere the $RT$ on the left is the filtered derived functor while the one on the\nright is the total derived functor. That is, there is a commuting diagram\n$$\n\\xymatrix{\nDF^+(\\mathcal{A}) \\ar[r]^{RT} \\ar[d]_{\\text{gr}^p} & DF^+(\\mathcal{B})\n\\ar[d]^{\\text{gr}^p}\\\\\nD^+(\\mathcal{A}) \\ar[r]^{RT} & D^+(\\mathcal{B}).}\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Filtered derived functors","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TE","source_file":"trace.tex","source_line":713,"source_end_line":727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L713-L727","statement_sha256":"78baf4ea356e0047dc683ab5b2604ec2e4837a9c74f5e87db5d5027513965177","origin":"The Stacks Project","memory_eligible":false,"source_rank":10624,"rank":10624,"depth":0,"x":1410.484,"y":1148.267,"cluster":"tale-geometry"},{"id":"stacks:03TH","tag":"03TH","title":"Perfectness · Definition 03TH","summary":"We denote by K_perf(Lambda) the category whose objects are bounded complexes of finite projective Lambda-modules, and whose morphisms are morphisms of complexes up to homotopy. The functor K_perf(Lambda)→ D(Lambda) is fully faithful (Derived Categories, Lemma [Tag 064B]). Denote D_perf(Lambda) its essential image. An object of D(Lambda) is called perfect if it is in D_perf(Lambda).","statement_latex":"We denote by $K_{perf}(\\Lambda)$ the category whose objects are bounded\ncomplexes of finite projective $\\Lambda$-modules, and whose morphisms are\nmorphisms of complexes up to homotopy. The functor $K_{perf}(\\Lambda)\\to\nD(\\Lambda)$ is fully faithful (Derived Categories, Lemma\n\\ref{derived-lemma-morphisms-from-projective-complex}).\nDenote $D_{perf}(\\Lambda)$ its essential image.\nAn object of $D(\\Lambda)$ is called {\\it perfect} if it is in\n$D_{perf}(\\Lambda)$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Perfectness","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TH","source_file":"trace.tex","source_line":818,"source_end_line":828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L818-L828","statement_sha256":"86a0438eea97407eb540b0569dede1ba34c6bb092ee9ec0db52dd076395d1f10","origin":"The Stacks Project","memory_eligible":false,"source_rank":10625,"rank":10625,"depth":5,"x":1698.416,"y":1037.659,"cluster":"tale-geometry"},{"id":"stacks:03TI","tag":"03TI","title":"Perfectness · Proposition 03TI","summary":"Let K∈ D_perf(Lambda) and f∈ End_D(Lambda)(K). Then the trace Tr(f)∈ Lambda^natural is well defined.","statement_latex":"Let $K\\in D_{perf}(\\Lambda)$ and $f\\in \\text{End}_{D(\\Lambda)}(K)$. Then the\ntrace $\\text{Tr}(f)\\in \\Lambda^\\natural$ is well defined.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Perfectness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TI","source_file":"trace.tex","source_line":830,"source_end_line":834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L830-L834","statement_sha256":"aa77556aa099600911108c4a0133736e45ab511cd039860f7a33699a119ed7b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10626,"rank":10626,"depth":5,"x":1574.979,"y":1282.736,"cluster":"tale-geometry"},{"id":"stacks:03TK","tag":"03TK","title":"Additivity · Lemma 03TK","summary":"Let K∈ DF_perf(Lambda) and f∈ End_DF(K). Then Tr(f|_K) = ∑_p∈ Z Tr(f|_gr^p K).","statement_latex":"Let $K\\in DF_{\\text{perf}}(\\Lambda)$ and $f\\in\n\\text{End}_{DF}(K)$. Then\n$$\n\\text{Tr}(f|_K) =\n\\sum\\nolimits_{p\\in \\mathbf{Z}} \\text{Tr}(f|_{\\text{gr}^p K}).\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Filtrations and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TK","source_file":"trace.tex","source_line":905,"source_end_line":913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L905-L913","statement_sha256":"7297f86f21677338f685f8b5a4930a2fbc9b372bf988c125b1fbbbaa6752f3f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10627,"rank":10627,"depth":6,"x":1468.856,"y":1031.839,"cluster":"tale-geometry"},{"id":"stacks:03TL","tag":"03TL","title":"Filtrations and perfect complexes · Lemma 03TL","summary":"Let P ∈ Fil^f(Mod_Lambda) be filtered finite projective, and f : P → P an endomorphism in Fil^f(Mod_Lambda). Then Tr(f|_P) = ∑_p Tr(f|_gr^p(P)).","statement_latex":"Let $P \\in \\text{Fil}^f(\\text{Mod}_\\Lambda)$ be filtered finite projective, and\n$f : P \\to P$ an endomorphism in $\\text{Fil}^f(\\text{Mod}_\\Lambda)$. Then\n$$\n\\text{Tr}(f|_P) =\n\\sum\\nolimits_p \\text{Tr}(f|_{\\text{gr}^p(P)}).\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Filtrations and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TL","source_file":"trace.tex","source_line":924,"source_end_line":932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L924-L932","statement_sha256":"3477fd2b0c7ff800768ae2a6dd72cc181788395326b58dd28b1998614457eb7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10628,"rank":10628,"depth":0,"x":1749.03,"y":1156.7,"cluster":"tale-geometry"},{"id":"stacks:03TN","tag":"03TN","title":"Characterizing perfect objects · Definition 03TN","summary":"Let Lambda be a (possibly noncommutative) ring. An object K∈ D(Lambda) has finite Tor-dimension if there exist a, b ∈ Z such that for any right Lambda-module N, we have H^i(N ⊗_Lambda^L K) = 0 for all i not ∈ [a, b].","statement_latex":"Let $\\Lambda$ be a (possibly noncommutative) ring.\nAn object $K\\in D(\\Lambda)$ has {\\it finite $\\text{Tor}$-dimension}\nif there exist $a, b \\in \\mathbf{Z}$ such that for any\nright $\\Lambda$-module $N$, we have\n$H^i(N \\otimes_{\\Lambda}^\\mathbf{L} K) = 0$ for all\n$i \\not \\in [a, b]$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Characterizing perfect objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TN","source_file":"trace.tex","source_line":954,"source_end_line":962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L954-L962","statement_sha256":"e15cf50cfd93cdf450ed50a39d4ab73b4e0865d11b5a28c68b227d559ded0133","origin":"The Stacks Project","memory_eligible":false,"source_rank":10629,"rank":10629,"depth":0,"x":1441.853,"y":1223.644,"cluster":"tale-geometry"},{"id":"stacks:03TO","tag":"03TO","title":"Characterizing perfect objects · Lemma 03TO","summary":"Let Lambda be a left Noetherian ring and K∈ D(Lambda). Then K is perfect if and only if the two following conditions hold: • K has finite Tor-dimension, and • for all i ∈ Z, H^i(K) is a finite Lambda-module.","statement_latex":"Let $\\Lambda$ be a left Noetherian ring and $K\\in D(\\Lambda)$. Then $K$ is\nperfect if and only if the two following conditions hold:\n\\begin{enumerate}\n\\item\n$K$ has finite $\\text{Tor}$-dimension, and\n\\item\nfor all $i \\in \\mathbf{Z}$, $H^i(K)$ is a finite $\\Lambda$-module.\n\\end{enumerate}","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Characterizing perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TO","source_file":"trace.tex","source_line":968,"source_end_line":978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L968-L978","statement_sha256":"8d2b7b3da0ca35792c3b16dc31d6a11d5a9b349c62cf07aba34099cd24943245","origin":"The Stacks Project","memory_eligible":false,"source_rank":10630,"rank":10630,"depth":11,"x":1614.622,"y":999.857,"cluster":"tale-geometry"},{"id":"stacks:03TV","tag":"03TV","title":"Cohomology of nice complexes · Proposition 03TV","summary":"Let X be a projective curve over a field k, Lambda a finite ring and K∈ D_ctf(X, Lambda). Then RΓ(X_bar k, K)∈ D_perf(Lambda).","statement_latex":"Let $X$ be a projective curve over a field $k$, $\\Lambda$ a finite ring and\n$K\\in D_{ctf}(X, \\Lambda)$. Then $R\\Gamma(X_{\\bar k}, K)\\in\nD_{perf}(\\Lambda)$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Cohomology of nice complexes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TV","source_file":"trace.tex","source_line":1014,"source_end_line":1019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1014-L1019","statement_sha256":"654fe39b74d02e9921b0fec8e067448b947deb612d0df7638e2c16b01e462d2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10631,"rank":10631,"depth":12,"x":1667.218,"y":1263.053,"cluster":"tale-geometry"},{"id":"stacks:03TX","tag":"03TX","title":"Lefschetz numbers · Definition 03TX","summary":"Let Lambda be a finite ring, X a projective curve over a finite field k and K ∈ D_ctf(X, Lambda) (for instance K = underlineLambda). There is a canonical map c_K : π_X^-1K → K, and its base change c_K|_X_bar k induces an action denoted π_X^* on the perfect complex RΓ(X_bar k, K|_X_bar k). The global Lefschetz number of K is the trace Tr(π_X^* |_RΓ(X_bar k, K)) of that action. It is an element of Lambda^natural.","statement_latex":"Let $\\Lambda$ be a finite ring, $X$ a projective curve over a finite field $k$\nand $K \\in D_{ctf}(X, \\Lambda)$ (for instance $K = \\underline\\Lambda$).\nThere is a canonical map $c_K : \\pi_X^{-1}K \\to K$, and its base change\n$c_K|_{X_{\\bar k}}$ induces an action denoted $\\pi_X^*$ on the perfect\ncomplex $R\\Gamma(X_{\\bar k}, K|_{X_{\\bar k}})$. The\n{\\it global Lefschetz number} of $K$ is the trace\n$\\text{Tr}(\\pi_X^* |_{R\\Gamma(X_{\\bar k}, K)})$ of that action.\nIt is an element of $\\Lambda^\\natural$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Lefschetz numbers","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TX","source_file":"trace.tex","source_line":1087,"source_end_line":1097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1087-L1097","statement_sha256":"13aa4aa89e91644db5f36088c84e51a149c8886923be7600f76d059a89c3bae2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10632,"rank":10632,"depth":0,"x":1416.64,"y":1098.729,"cluster":"tale-geometry"},{"id":"stacks:03TY","tag":"03TY","title":"Lefschetz numbers · Definition 03TY","summary":"With Lambda, X, k, K as in Definition [Tag 03TX]. Since K∈ D_ctf(X, Lambda), for any geometric point bar x of X, the complex K_bar x is a perfect complex (in D_perf(Lambda)). As we have seen in Section [Tag 03SL], the Frobenius π_X acts on K_bar x. The local Lefschetz number of K is the sum ∑_x∈ X(k) Tr(π_x |_K_overlinex) which is again an element of Lambda^natural.","statement_latex":"With $\\Lambda, X, k, K$ as in\nDefinition \\ref{definition-global-lefschetz-number}.\nSince $K\\in D_{ctf}(X, \\Lambda)$, for any geometric point $\\bar x$ of $X$,\nthe complex $K_{\\bar x}$ is a perfect complex (in $D_{perf}(\\Lambda)$). As we\nhave seen in Section \\ref{section-frobenii}, the Frobenius $\\pi_X$ acts on\n$K_{\\bar x}$. The {\\it local Lefschetz number} of $K$ is the sum\n$$\n\\sum\\nolimits_{x\\in X(k)} \\text{Tr}(\\pi_x |_{K_{\\overline{x}}})\n$$\nwhich is again an element of $\\Lambda^\\natural$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Lefschetz numbers","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TY","source_file":"trace.tex","source_line":1099,"source_end_line":1111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1099-L1111","statement_sha256":"cbc1504bead2af30d3b3ff79bfdcbdf4da0c95c60d8515cdec03173f22cdbab0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10633,"rank":10633,"depth":1,"x":1733.733,"y":1077.704,"cluster":"tale-geometry"},{"id":"stacks:03TZ","tag":"03TZ","title":"Lefschetz Trace Formula · Theorem 03TZ","summary":"Let X be a projective curve over a finite field k, Lambda a finite ring and K ∈ D_ctf(X, Lambda). Then the global and local Lefschetz numbers of K are equal, i.e., Tr(π^*_X |_RΓ(X_bar k, K)) = ∑_x∈ X(k) Tr(π_X |_K_bar x) in Lambda^natural.","statement_latex":"Let $X$ be a projective curve over a finite field $k$, $\\Lambda$ a finite ring\nand $K \\in D_{ctf}(X, \\Lambda)$. Then the global and local Lefschetz numbers\nof $K$ are equal, i.e.,\n\\begin{equation}\n\n\\text{Tr}(\\pi^*_X |_{R\\Gamma(X_{\\bar k}, K)})\n=\n\\sum\\nolimits_{x\\in X(k)} \\text{Tr}(\\pi_X |_{K_{\\bar x}})\n\\end{equation}\nin $\\Lambda^\\natural$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Lefschetz numbers","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03TZ","source_file":"trace.tex","source_line":1116,"source_end_line":1128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1116-L1128","statement_sha256":"c9ac2052927bd724992319e1ff2df5a4a1070f7668c9bd842d0d17a75da5a8fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":10634,"rank":10634,"depth":0,"x":1516.701,"y":1273.242,"cluster":"tale-geometry"},{"id":"stacks:03U1","tag":"03U1","title":"Weil · Theorem 03U1","summary":"Let C be a nonsingular projective curve over an algebraically closed field k, and φ : C → C a k-endomorphism of C distinct from the identity. Let V(φ) = Δ_C · Γ_φ, where Δ_C is the diagonal, Γ_φ is the graph of φ, and the intersection number is taken on C × C. Let J = underlinePicardfunctor^0_C/k be the jacobian of C and denote φ^* : J → J the action induced by φ by taking pullbacks. Then V(φ) = 1 - Tr_J(φ^*) + deg φ.","statement_latex":"Let $C$ be a nonsingular projective curve over an algebraically closed field\n$k$, and $\\varphi : C \\to C$ a $k$-endomorphism of $C$ distinct from the\nidentity. Let $V(\\varphi) = \\Delta_C \\cdot \\Gamma_\\varphi$, where $\\Delta_C$ is\nthe diagonal, $\\Gamma_\\varphi$ is the graph of $\\varphi$, and the intersection\nnumber is taken on $C \\times C$. Let $J = \\underline{\\Picardfunctor}^0_{C/k}$\nbe the jacobian of $C$ and denote $\\varphi^* : J \\to J$ the action induced by\n$\\varphi$ by taking pullbacks. Then\n$$\nV(\\varphi) = 1 - \\text{Tr}_J(\\varphi^*) + \\deg \\varphi.\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Lefschetz numbers","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03U1","source_file":"trace.tex","source_line":1145,"source_end_line":1157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1145-L1157","statement_sha256":"ecf01488c2a682b90a17b1328cdb7aa18739bf24eb729ecb9aada7789fe255a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10635,"rank":10635,"depth":0,"x":1519.493,"y":1005.758,"cluster":"tale-geometry"},{"id":"stacks:03U3","tag":"03U3","title":"Lefschetz numbers · Lemma 03U3","summary":"Consider the situation of Theorem [Tag 03U1] and let ℓ be a prime number invertible in k. Then ∑_i = 0^2 (-1)^i Tr(φ^* |_H^i (C, underlineZ/ℓ^n Z)) = V(φ) mod ℓ^n.","statement_latex":"Consider the situation of\nTheorem \\ref{theorem-weil-trace-formula}\nand let $\\ell$ be a prime number invertible in $k$. Then\n$$\n\\sum\\nolimits_{i = 0}^2\n(-1)^i\n\\text{Tr}(\\varphi^* |_{H^i (C, \\underline{\\mathbf{Z}/\\ell^n \\mathbf{Z}})})\n=\nV(\\varphi) \\mod \\ell^n.\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Lefschetz numbers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03U3","source_file":"trace.tex","source_line":1307,"source_end_line":1319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1307-L1319","statement_sha256":"9eca24c7844bcdfb8a5867a63069d7e6a3320578d7be3c6d2a6833870439af5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10636,"rank":10636,"depth":1,"x":1732.655,"y":1204.69,"cluster":"tale-geometry"},{"id":"stacks:03U5","tag":"03U5","title":"Preliminaries and sorites · Lemma 03U5","summary":"Let e∈ G denote the neutral element. The map Lambda[G] & → & Lambda^natural ∑ λ_g· g & ↦ & λ_e factors through Lambda[G]^natural. We denote varepsilon : Lambda[G]^natural→ Lambda^natural the induced map.","statement_latex":"Let $e\\in G$ denote the neutral element. The map\n$$\n\\begin{matrix}\n\\Lambda[G] & \\longrightarrow & \\Lambda^{\\natural}\\\\\n\\sum \\lambda_g\\cdot g & \\longmapsto & \\lambda_e\n\\end{matrix}\n$$\nfactors through $\\Lambda[G]^\\natural$. We denote\n$\\varepsilon : \\Lambda[G]^\\natural\\to \\Lambda^\\natural$ the induced map.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Preliminaries and sorites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03U5","source_file":"trace.tex","source_line":1401,"source_end_line":1412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1401-L1412","statement_sha256":"201e378a3d8ca4a730ef310ce8446e520b72dfc412a595aedb5de47871db7e69","origin":"The Stacks Project","memory_eligible":false,"source_rank":10637,"rank":10637,"depth":0,"x":1415.322,"y":1178.94,"cluster":"tale-geometry"},{"id":"stacks:03U6","tag":"03U6","title":"Preliminaries and sorites · Definition 03U6","summary":"Let f : P→ P be an endomorphism of a finite projective Lambda[G]-module P. We define Tr_Lambda^G(f; P) := varepsilon(Tr_Lambda[G](f; P)) to be the G-trace of f on P.","statement_latex":"Let $f : P\\to P$ be an\nendomorphism of a finite projective $\\Lambda[G]$-module\n$P$. We define\n$$\n\\text{Tr}_{\\Lambda}^G(f; P) := \\varepsilon\\left(\\text{Tr}_{\\Lambda[G]}(f;\nP)\\right)\n$$\nto be the {\\it $G$-trace of $f$ on $P$}.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Preliminaries and sorites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03U6","source_file":"trace.tex","source_line":1430,"source_end_line":1440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1430-L1440","statement_sha256":"c9ff4944351b696e7dade9b4f55eb7eea40ec8eb53258af868a02fbdffe7e842","origin":"The Stacks Project","memory_eligible":false,"source_rank":10638,"rank":10638,"depth":0,"x":1670.166,"y":1017.776,"cluster":"tale-geometry"},{"id":"stacks:03U7","tag":"03U7","title":"Preliminaries and sorites · Lemma 03U7","summary":"Let f : P→ P be an endomorphism of the finite projective Lambda[G]-module P. Then Tr_Lambda(f; P) = \\# G · Tr_Lambda^G(f; P).","statement_latex":"Let $f : P\\to P$ be an endomorphism of the finite projective\n$\\Lambda[G]$-module $P$. Then\n$$\n\\text{Tr}_{\\Lambda}(f; P) = \\# G \\cdot \\text{Tr}_\\Lambda^G(f; P).\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Preliminaries and sorites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03U7","source_file":"trace.tex","source_line":1442,"source_end_line":1449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1442-L1449","statement_sha256":"71a1ad49f474182e0986c3ae839a25a013f90fbe22773221a40fbedac91b5a08","origin":"The Stacks Project","memory_eligible":false,"source_rank":10639,"rank":10639,"depth":0,"x":1611.819,"y":1281.366,"cluster":"tale-geometry"},{"id":"stacks:03U8","tag":"03U8","title":"Preliminaries and sorites · Lemma 03U8","summary":"The map A→ Lambda defines an A-module structure on Lambda^natural.","statement_latex":"The map $A\\to \\Lambda$ defines an $A$-module structure on $\\Lambda^\\natural$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Preliminaries and sorites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03U8","source_file":"trace.tex","source_line":1460,"source_end_line":1463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1460-L1463","statement_sha256":"9556f1caf02a43a395237626113b6bae4fc89384e131100a31324851fc05942f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10640,"rank":10640,"depth":0,"x":1442.779,"y":1053.766,"cluster":"tale-geometry"},{"id":"stacks:03U9","tag":"03U9","title":"Preliminaries and sorites · Lemma 03U9","summary":"Let P be a finite projective A[G]-module and M a Lambda[G]-module, finite projective as a Lambda-module. Then P ⊗_A M is a finite projective Lambda[G]-module, for the structure induced by the diagonal action of G.","statement_latex":"Let $P$ be a finite projective $A[G]$-module and $M$ a $\\Lambda[G]$-module,\nfinite projective as a $\\Lambda$-module. Then $P \\otimes_A M$ is a finite\nprojective $\\Lambda[G]$-module, for the structure induced by the diagonal\naction of $G$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Preliminaries and sorites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03U9","source_file":"trace.tex","source_line":1469,"source_end_line":1475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1469-L1475","statement_sha256":"ae4b1a7b7cdebe2dc5e68711f15dfa69bf248b2992f2c778b022aa3f41f633ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":10641,"rank":10641,"depth":0,"x":1750.625,"y":1125.717,"cluster":"tale-geometry"},{"id":"stacks:03UA","tag":"03UA","title":"Preliminaries and sorites · Lemma 03UA","summary":"With assumptions as in Lemma [Tag 03U9], let u∈ End_A[G](P) and v∈ End_Lambda[G](M). Then Tr_Lambda^G (u ⊗ v; P ⊗_A M) = Tr_A^G(u; P)· Tr_Lambda(v;M).","statement_latex":"With assumptions as in\nLemma \\ref{lemma-diagonal-action-projective-module},\nlet\n$u\\in \\text{End}_{A[G]}(P)$ and $v\\in \\text{End}_{\\Lambda[G]}(M)$. Then\n$$\n\\text{Tr}_\\Lambda^G \\left(u \\otimes v; P \\otimes_A M\\right) = \\text{Tr}_A^G(u;\nP)\\cdot \\text{Tr}_\\Lambda(v;M).\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Preliminaries and sorites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UA","source_file":"trace.tex","source_line":1498,"source_end_line":1508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1498-L1508","statement_sha256":"ac883ad35ffc83f1477463551f27a51194de633e7c7ea24ae23e0a81a1cd8362","origin":"The Stacks Project","memory_eligible":false,"source_rank":10642,"rank":10642,"depth":1,"x":1465.607,"y":1247.408,"cluster":"tale-geometry"},{"id":"stacks:03UB","tag":"03UB","title":"Preliminaries and sorites · Lemma 03UB","summary":"Let P be a Lambda[Γ]-module, finite and projective as a Lambda[G]-module, and γ ∈ Γ. Then Tr_Lambda(γ, P) = \\# Z_γ · Tr_Lambda^Z_γ(γ, P).","statement_latex":"Let $P$ be a $\\Lambda[\\Gamma]$-module, finite and projective as a\n$\\Lambda[G]$-module, and $\\gamma \\in \\Gamma$. Then\n$$\n\\text{Tr}_{\\Lambda}(\\gamma, P) =\n\\# Z_\\gamma \\cdot \\text{Tr}_\\Lambda^{Z_\\gamma}\\left(\\gamma, P\\right).\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Preliminaries and sorites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UB","source_file":"trace.tex","source_line":1553,"source_end_line":1561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1553-L1561","statement_sha256":"c5ef462cf3b5ae6ab786883b972a939821d536e93d3e09400c40e0f573870d6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10643,"rank":10643,"depth":1,"x":1577.976,"y":995.811,"cluster":"tale-geometry"},{"id":"stacks:03UC","tag":"03UC","title":"Preliminaries and sorites · Lemma 03UC","summary":"Let P be an A[Γ]-module, finite projective as A[G]-module. Let M be a Lambda[Γ]-module, finite projective as a Lambda-module. Then Tr_Lambda^Z_γ(γ, P ⊗_A M) = Tr_A^Z_γ(γ, P)· Tr_Lambda(γ, M).","statement_latex":"Let $P$ be an $A[\\Gamma]$-module, finite projective as $A[G]$-module. Let $M$\nbe a $\\Lambda[\\Gamma]$-module, finite projective as a $\\Lambda$-module. Then\n$$\n\\text{Tr}_{\\Lambda}^{Z_\\gamma}(\\gamma, P \\otimes_A M) =\n\\text{Tr}_A^{Z_\\gamma}(\\gamma, P)\\cdot \\text{Tr}_\\Lambda(\\gamma, M).\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Preliminaries and sorites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UC","source_file":"trace.tex","source_line":1567,"source_end_line":1575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1567-L1575","statement_sha256":"17d9d81b7d6aedbea37a0d4e4fe40b11aa0083d74b50c5708750ce81cdb64f96","origin":"The Stacks Project","memory_eligible":false,"source_rank":10644,"rank":10644,"depth":2,"x":1697.509,"y":1245.232,"cluster":"tale-geometry"},{"id":"stacks:03UD","tag":"03UD","title":"Preliminaries and sorites · Lemma 03UD","summary":"Let P be a Lambda[Γ]-module, finite projective as Lambda[G]-module. Then the coinvariants P_G = Lambda ⊗_Lambda[G] P form a finite projective Lambda-module, endowed with an action of Γ/G = N. Moreover, we have Tr_Lambda(1; P_G) = ∑'_γ ↦ 1 Tr_Lambda^Z_γ(γ, P) where ∑_γ↦ 1' means taking the sum over the G-conjugacy classes in Γ.","statement_latex":"Let $P$ be a $\\Lambda[\\Gamma]$-module, finite projective as\n$\\Lambda[G]$-module. Then the coinvariants\n$P_G = \\Lambda \\otimes_{\\Lambda[G]} P$\nform a finite projective $\\Lambda$-module, endowed with an action of\n$\\Gamma/G = \\mathbf{N}$. Moreover, we have\n$$\n\\text{Tr}_\\Lambda(1; P_G) =\n\\sum\\nolimits'_{\\gamma \\mapsto 1} \\text{Tr}_\\Lambda^{Z_\\gamma}(\\gamma, P)\n$$\nwhere $\\sum_{\\gamma\\mapsto 1}'$ means taking the sum over the $G$-conjugacy\nclasses in $\\Gamma$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Preliminaries and sorites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UD","source_file":"trace.tex","source_line":1581,"source_end_line":1594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1581-L1594","statement_sha256":"db0dcf38187b1ca3d89a9454cd1796b4812aa59b481c350891694c67274e4233","origin":"The Stacks Project","memory_eligible":false,"source_rank":10645,"rank":10645,"depth":0,"x":1408.632,"y":1129.075,"cluster":"tale-geometry"},{"id":"stacks:03UG","tag":"03UG","title":"Proof of the trace formula · Theorem 03UG","summary":"Let k be a finite field and X a finite type, separated scheme of dimension at most 1 over k. Let Lambda be a finite ring whose cardinality is prime to that of k, and K∈ D_ctf(X, Lambda). Then Tr(π_X^* |_RΓ_c(X_bar k, K)) = ∑_x∈ X(k) Tr(π_x |_K_bar x) in Lambda^natural.","statement_latex":"Let $k$ be a finite field and $X$ a finite type, separated scheme of dimension\nat most 1 over $k$. Let $\\Lambda$ be a finite ring whose cardinality is prime\nto that of $k$, and $K\\in D_{ctf}(X, \\Lambda)$. Then\n\\begin{equation}\n\n\\text{Tr}(\\pi_X^* |_{R\\Gamma_c(X_{\\bar k}, K)})\n=\n\\sum\\nolimits_{x\\in X(k)}\n\\text{Tr}(\\pi_x |_{K_{\\bar x}})\n\\end{equation}\nin $\\Lambda^{\\natural}$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Proof of the trace formula","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UG","source_file":"trace.tex","source_line":1682,"source_end_line":1695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1682-L1695","statement_sha256":"634dfd707d130407d208e043e7583296b8e641edd572a8bf9a8a1ac590ed3340","origin":"The Stacks Project","memory_eligible":false,"source_rank":10646,"rank":10646,"depth":0,"x":1715.223,"y":1050.769,"cluster":"tale-geometry"},{"id":"stacks:03UM","tag":"03UM","title":"On l-adic sheaves · Definition 03UM","summary":"Let X be a Noetherian scheme. A Z_ℓ-sheaf on X, or simply an ℓ-adic sheaf F is an inverse system (F_n)_n≥ 1 where • F_n is a constructible Z/ℓ^nZ-module on X_etale, and • the transition maps F_n+1→ F_n induce isomorphisms F_n+1 ⊗_Z/ℓ^n+1Z Z/ℓ^nZ ≅ F_n. We say that F is lisse if each F_n is locally constant. A morphism of such is merely a morphism of inverse systems.","statement_latex":"Let $X$ be a Noetherian scheme. A {\\it $\\mathbf{Z}_\\ell$-sheaf} on $X$, or\nsimply an {\\it $\\ell$-adic sheaf} $\\mathcal{F}$ is an\ninverse system $\\left\\{\\mathcal{F}_n\\right\\}_{n\\geq 1}$ where\n\\begin{enumerate}\n\\item\n$\\mathcal{F}_n$ is a constructible $\\mathbf{Z}/\\ell^n\\mathbf{Z}$-module on\n$X_\\etale$, and\n\\item\nthe transition maps $\\mathcal{F}_{n+1}\\to \\mathcal{F}_n$ induce isomorphisms\n$\\mathcal{F}_{n+1} \\otimes_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}}\n\\mathbf{Z}/\\ell^n\\mathbf{Z} \\cong \\mathcal{F}_n$.\n\\end{enumerate}\nWe say that $\\mathcal{F}$ is {\\it lisse} if each $\\mathcal{F}_n$ is locally\nconstant. A {\\it morphism} of such is merely a morphism of inverse systems.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"On l-adic sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UM","source_file":"trace.tex","source_line":1979,"source_end_line":1995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1979-L1995","statement_sha256":"ad786d639dd400e01ac47a0a149cb37eabc28a36326a48e70ea33fb92153f2f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10647,"rank":10647,"depth":0,"x":1552.03,"y":1282.604,"cluster":"tale-geometry"},{"id":"stacks:03UN","tag":"03UN","title":"On l-adic sheaves · Lemma 03UN","summary":"Let (G_n)_n≥ 1 be an inverse system of constructible Z/ℓ^nZ-modules. Suppose that for all k≥ 1, the maps G_n+1/ℓ^k G_n+1→ G_n /ℓ^k G_n are isomorphisms for all ngg 0 (where the bound possibly depends on k). In other words, assume that the system (G_n/ℓ^kG_n)_n≥ 1 is eventually constant, and call F_k the corresponding sheaf. Then the system (F_k)_k≥ 1 forms a Z_ℓ-sheaf on X.","statement_latex":"Let $\\{\\mathcal{G}_n\\}_{n\\geq 1}$ be an inverse system of constructible\n$\\mathbf{Z}/\\ell^n\\mathbf{Z}$-modules.\nSuppose that for all $k\\geq 1$, the maps\n$$\n\\mathcal{G}_{n+1}/\\ell^k \\mathcal{G}_{n+1}\\to \\mathcal{G}_n /\\ell^k\n\\mathcal{G}_n\n$$\nare isomorphisms for all $n\\gg 0$ (where the bound possibly depends on $k$).\nIn other words, assume that the system\n$\\{\\mathcal{G}_n/\\ell^k\\mathcal{G}_n\\}_{n\\geq 1}$\nis eventually constant, and call $\\mathcal{F}_k$ the corresponding sheaf.\nThen the system $\\left\\{\\mathcal{F}_k\\right\\}_{k\\geq 1}$ forms a\n$\\mathbf{Z}_\\ell$-sheaf on $X$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"On l-adic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UN","source_file":"trace.tex","source_line":1997,"source_end_line":2012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L1997-L2012","statement_sha256":"9ae3b0e3214fce90547275752b7550620495c8a388361ba8b571d005cc810361","origin":"The Stacks Project","memory_eligible":false,"source_rank":10648,"rank":10648,"depth":0,"x":1485.895,"y":1018.909,"cluster":"tale-geometry"},{"id":"stacks:03UO","tag":"03UO","title":"On l-adic sheaves · Lemma 03UO","summary":"The category of Z_ℓ-sheaves on X is abelian.","statement_latex":"The category of $\\mathbf{Z}_\\ell$-sheaves on $X$ is abelian.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"On l-adic sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UO","source_file":"trace.tex","source_line":2018,"source_end_line":2021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2018-L2021","statement_sha256":"f319a3185cb1db1a27025eeb05cee4a12ba166f195dfe285d00833c4d1b5dfd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10649,"rank":10649,"depth":1,"x":1746.86,"y":1175.912,"cluster":"tale-geometry"},{"id":"stacks:03UR","tag":"03UR","title":"On l-adic sheaves · Definition 03UR","summary":"A Z_ℓ-sheaf F is torsion if ℓ^n : F → F is the zero map for some n. The abelian category of Q_ℓ-sheaves on X is the quotient of the abelian category of Z_ℓ-sheaves by the Serre subcategory of torsion sheaves. In other words, its objects are Z_ℓ-sheaves on X, and if F, G are two such, then Hom_Q_ℓ (F, G ) = Hom_Z_ℓ (F, G) ⊗_Z_ℓ Q_ℓ. We denote by F ↦ F ⊗ Q_ℓ the quotient functor (right adjoint to the inclusion). If F = F' ⊗ Q_ℓ where F' is a Z_ℓ-sheaf and bar x is a…","statement_latex":"A $\\mathbf{Z}_\\ell$-sheaf $\\mathcal{F}$ is {\\it torsion} if\n$\\ell^n : \\mathcal{F} \\to \\mathcal{F}$ is the zero map for some $n$.\nThe abelian category\nof $\\mathbf{Q}_\\ell$-sheaves on $X$ is the quotient of the abelian category of\n$\\mathbf{Z}_\\ell$-sheaves by the Serre subcategory of torsion sheaves. In\nother words, its objects are $\\mathbf{Z}_\\ell$-sheaves on $X$, and if\n$\\mathcal{F}, \\mathcal{G}$ are two such, then\n$$\n\\Hom_{\\mathbf{Q}_\\ell} \\left(\\mathcal{F}, \\mathcal{G} \\right) =\n\\Hom_{\\mathbf{Z}_\\ell} \\left(\\mathcal{F}, \\mathcal{G}\\right)\n\\otimes_{\\mathbf{Z}_\\ell} \\mathbf{Q}_\\ell.\n$$\nWe denote by $\\mathcal{F} \\mapsto \\mathcal{F} \\otimes \\mathbf{Q}_\\ell$ the\nquotient functor (right adjoint to the inclusion). If $\\mathcal{F} =\n\\mathcal{F}' \\otimes \\mathbf{Q}_\\ell$ where $\\mathcal{F}'$ is a\n$\\mathbf{Z}_\\ell$-sheaf and $\\bar x$ is a geometric point, then the\n{\\it stalk} of $\\mathcal{F}$ at $\\bar x$ is $\\mathcal{F}_{\\bar x} =\n\\mathcal{F}'_{\\bar x} \\otimes \\mathbf{Q}_\\ell$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"On l-adic sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UR","source_file":"trace.tex","source_line":2084,"source_end_line":2104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2084-L2104","statement_sha256":"fc02b02be87d25c05e3368937d26ac21fea429c9d597e11ceeac866e2bcf0996","origin":"The Stacks Project","memory_eligible":false,"source_rank":10650,"rank":10650,"depth":0,"x":1427.998,"y":1208.236,"cluster":"tale-geometry"},{"id":"stacks:03UT","tag":"03UT","title":"On l-adic sheaves · Definition 03UT","summary":"If X is a separated scheme of finite type over an algebraically closed field k and F = (F_n)_n≥ 1 is a Z_ℓ-sheaf on X, then we define H^i(X, F) := lim_n H^i(X, F_n) and H_c^i(X, F) := lim_n H_c^i(X, F_n). If F = F'⊗ Q_ℓ for a Z_ℓ-sheaf F' then we set H_c^i(X , F) := H_c^i(X, F')⊗_Z_ℓQ_ℓ. We call these the ℓ-adic cohomology of X with coefficients F.","statement_latex":"If $X$ is a separated scheme of finite type over an algebraically closed field\n$k$ and $\\mathcal{F} = \\left\\{\\mathcal{F}_n\\right\\}_{n\\geq 1}$ is a\n$\\mathbf{Z}_\\ell$-sheaf on $X$, then we define\n$$\nH^i(X, \\mathcal{F}) := \\lim_n H^i(X, \\mathcal{F}_n)\n\\quad\\text{and}\\quad\nH_c^i(X, \\mathcal{F}) := \\lim_n H_c^i(X, \\mathcal{F}_n).\n$$\nIf $\\mathcal{F} = \\mathcal{F}'\\otimes \\mathbf{Q}_\\ell$ for a\n$\\mathbf{Z}_\\ell$-sheaf $\\mathcal{F}'$ then we set\n$$\nH_c^i(X , \\mathcal{F}) := H_c^i(X,\n\\mathcal{F}')\\otimes_{\\mathbf{Z}_\\ell}\\mathbf{Q}_\\ell.\n$$\nWe call these the {\\it $\\ell$-adic cohomology} of $X$ with coefficients\n$\\mathcal{F}$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"On l-adic sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UT","source_file":"trace.tex","source_line":2112,"source_end_line":2130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2112-L2130","statement_sha256":"6d1ebead32c73665ebefbfcce327dfee1379f2001baba098df7c4d96663288bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":10651,"rank":10651,"depth":0,"x":1637.242,"y":1003.36,"cluster":"tale-geometry"},{"id":"stacks:03UV","tag":"03UV","title":"L-functions · Definition 03UV","summary":"Let X be a scheme of finite type over a finite field k. Let Lambda be a finite ring of order prime to the characteristic of k and F a constructible flat Lambda-module on X_etale. Then we set L(X, F) := ∏_x∈ |X| det(1 - π_x^*T^deg x |_F_bar x)^-1 ∈ Lambda [[ T ]] where |X| is the set of closed points of X, deg x = [kappa(x): k] and bar x is a geometric point lying over x. This definition clearly generalizes to the case where F is replaced by a K ∈ D_ctf(X, Lambda). We call…","statement_latex":"Let $X$ be a scheme of finite type over a finite field $k$. Let $\\Lambda$ be a\nfinite ring of order prime to the characteristic of $k$ and $\\mathcal{F}$ a\nconstructible flat $\\Lambda$-module on $X_\\etale$. Then we set\n$$\nL(X, \\mathcal{F}) :=\n\\prod\\nolimits_{x\\in |X|}\n\\det(1 - \\pi_x^*T^{\\deg x} |_{\\mathcal{F}_{\\bar x}})^{-1} \\in \\Lambda [[ T ]]\n$$\nwhere $|X|$ is the set of closed points of $X$, $\\deg x = [\\kappa(x): k]$ and\n$\\bar x$ is a geometric point lying over $x$. This definition clearly\ngeneralizes to the case where $\\mathcal{F}$ is replaced by a\n$K \\in D_{ctf}(X, \\Lambda)$. We call this the {\\it $L$-function of\n$\\mathcal{F}$}.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"L-functions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UV","source_file":"trace.tex","source_line":2139,"source_end_line":2154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2139-L2154","statement_sha256":"2111e8a79c114ffb382bd8933254c80380605dd96cb2612070ce83b35c5384e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10652,"rank":10652,"depth":0,"x":1647.71,"y":1273.309,"cluster":"tale-geometry"},{"id":"stacks:03UX","tag":"03UX","title":"L-functions · Definition 03UX","summary":"Now assume that F is a Q_ℓ-sheaf on X. In this case we define L(X, F) := ∏_x ∈ |X| det(1 - π_x^*T^deg x |_F_bar x)^-1 ∈ Q_ℓ[[T]]. Note that this product converges since there are finitely many points of a given degree. We call this the L-function of F.","statement_latex":"Now assume that $\\mathcal{F}$ is a $\\mathbf{Q}_\\ell$-sheaf on $X$.\nIn this case we define\n$$\nL(X, \\mathcal{F}) :=\n\\prod\\nolimits_{x \\in |X|}\n\\det(1 - \\pi_x^*T^{\\deg x} |_{\\mathcal{F}_{\\bar x}})^{-1}\n\\in \\mathbf{Q}_\\ell[[T]].\n$$\nNote that this product converges since there are finitely many points of a\ngiven degree. We call this the {\\it $L$-function of\n$\\mathcal{F}$}.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"L-functions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UX","source_file":"trace.tex","source_line":2162,"source_end_line":2175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2162-L2175","statement_sha256":"11a9641a8dcf506f745623065cad3a6499f458a2e249f494763d9f2ee1d661ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":10653,"rank":10653,"depth":0,"x":1422.784,"y":1080.088,"cluster":"tale-geometry"},{"id":"stacks:03UZ","tag":"03UZ","title":"Finite Coefficients · Theorem 03UZ","summary":"Let X be a scheme of finite type over a finite field k. Let Lambda be a finite ring of order prime to the characteristic of k and F a constructible flat Lambda-module on X_etale. Then L(X, F) = det(1 - π_X^* T |_RΓ_c(X_bar k, F))^-1 ∈ Lambda[[T]].","statement_latex":"Let $X$ be a scheme of finite type over a finite field $k$. Let $\\Lambda$ be a\nfinite ring of order prime to the characteristic of $k$ and $\\mathcal{F}$ a\nconstructible flat $\\Lambda$-module on $X_\\etale$. Then\n$$\nL(X, \\mathcal{F}) =\n\\det(1 - \\pi_X^*\\ T |_{R\\Gamma_c(X_{\\bar k}, \\mathcal{F})})^{-1}\n\\in \\Lambda[[T]].\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Cohomological interpretation","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03UZ","source_file":"trace.tex","source_line":2186,"source_end_line":2196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2186-L2196","statement_sha256":"f8e56429196abb47fc8077e2c6d706a04972d5c04bf7959bfe757e05bddbaef1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10654,"rank":10654,"depth":0,"x":1744.197,"y":1094.946,"cluster":"tale-geometry"},{"id":"stacks:03V0","tag":"03V0","title":"Adic sheaves · Theorem 03V0","summary":"Let X be a scheme of finite type over a finite field k, and F a Q_ℓ-sheaf on X. Then L(X, F) = ∏_i det(1 - π_X^*T |_H_c^i(X_bar k , F))^(-1)^i + 1 ∈ Q_ℓ[[T]].","statement_latex":"Let $X$ be a scheme of finite type over a finite field $k$, and $\\mathcal{F}$ a\n$\\mathbf{Q}_\\ell$-sheaf on $X$. Then\n$$\nL(X, \\mathcal{F}) =\n\\prod\\nolimits_i\n\\det(1 - \\pi_X^*T |_{H_c^i(X_{\\bar k} , \\mathcal{F})})^{(-1)^{i + 1}}\n\\in \\mathbf{Q}_\\ell[[T]].\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Cohomological interpretation","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03V0","source_file":"trace.tex","source_line":2206,"source_end_line":2216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2206-L2216","statement_sha256":"a3a23c0dbce7b4520ffef58152726794f2b7b9140a17a7feba56c829d4185adb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10655,"rank":10655,"depth":0,"x":1495.109,"y":1266.46,"cluster":"tale-geometry"},{"id":"stacks:03V3","tag":"03V3","title":"Cohomological interpretation · Theorem 03V3","summary":"Let X/k be as above, let Lambda be a finite ring with \\#Lambda ∈ k^* and K∈ D_ctf(X, Lambda). Then RΓ_c(X_bar k, K)∈ D_perf(Lambda) and ∑_x∈ X(k)Tr(π_x |_K_bar x) = Tr(π_X^* |_RΓ_c(X_bar k, K )).","statement_latex":"Let $X/k$ be as above, let $\\Lambda$ be a finite ring with $\\#\\Lambda \\in k^*$\nand $K\\in D_{ctf}(X, \\Lambda)$. Then $R\\Gamma_c(X_{\\bar k}, K)\\in\nD_{perf}(\\Lambda)$ and\n$$\n\\sum_{x\\in X(k)}\\text{Tr}\\left(\\pi_x |_{K_{\\bar x}}\\right) =\n\\text{Tr}\\left(\\pi_X^* |_{R\\Gamma_c(X_{\\bar k}, K )}\\right).\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Cohomological interpretation","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03V3","source_file":"trace.tex","source_line":2288,"source_end_line":2297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2288-L2297","statement_sha256":"cac7b72c920477be8e769434ba4a192c1c8c7e3de9a69829c133693c9925a07f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10656,"rank":10656,"depth":0,"x":1540.88,"y":998.505,"cluster":"tale-geometry"},{"id":"stacks:03V2","tag":"03V2","title":"Cohomological interpretation · Theorem 03V2","summary":"Let X be a separated scheme of finite type over a finite field k and F be a Q_ℓ-sheaf on X. Then dim_Q_ℓH_c^i(X_bar k, F) is finite for all i, and is nonzero for 0≤ i ≤ 2 dim X only. Furthermore, we have ∑_x∈ X(k) Tr(π_x |_F_bar x) = ∑_i (-1)^iTr(π_X^* |_H_c^i(X_bar k, F)).","statement_latex":"Let $X$ be a separated scheme of finite type over a finite field $k$ and\n$\\mathcal{F}$ be a $\\mathbf{Q}_\\ell$-sheaf on $X$. Then\n$\\dim_{\\mathbf{Q}_\\ell}H_c^i(X_{\\bar k}, \\mathcal{F})$ is finite for all $i$,\nand is nonzero for $0\\leq i \\leq 2 \\dim X$ only. Furthermore, we have\n$$\n\\sum_{x\\in X(k)} \\text{Tr}\\left(\\pi_x |_{\\mathcal{F}_{\\bar x}}\\right) =\n\\sum_i (-1)^i\\text{Tr}\\left(\\pi_X^* |_{H_c^i(X_{\\bar k}, \\mathcal{F})}\\right).\n$$","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Cohomological interpretation","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03V2","source_file":"trace.tex","source_line":2305,"source_end_line":2315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2305-L2315","statement_sha256":"b1feb12e0edef9ae59c988055f0eef0bfe9a78379b48aeed878cca56977ad56f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10657,"rank":10657,"depth":1,"x":1722.71,"y":1222.183,"cluster":"tale-geometry"},{"id":"stacks:03V4","tag":"03V4","title":"Cohomological interpretation · Lemma 03V4","summary":"Suppose we have K_n∈ D_perf(Z/ℓ^nZ), π_n : K_n→ K_n and isomorphisms φ_n : K_n+1 ⊗^L_Z/ℓ^n+1Z Z/ℓ^nZ → K_n compatible with π_n+1 and π_n. Then • the elements t_n = Tr(π_n |_K_n)∈ Z/ℓ^nZ form an element t_∞ = (t_n) of Z_ℓ, • the Z_ℓ-module H_∞^i = lim_n H^i(k_n) is finite and is nonzero for finitely many i only, and • the operators H^i(π_n): H^i(K_n)→ H^i(K_n) are compatible and define π_∞^i : H_∞^i→ H_∞^i satisfying ∑ (-1)^i Tr( π_∞^i |_H_∞^i ⊗_Z_ℓQ_ℓ) = t_∞.","statement_latex":"Suppose we have\n$K_n\\in D_{perf}(\\mathbf{Z}/\\ell^n\\mathbf{Z})$, $\\pi_n : K_n\\to K_n$\nand isomorphisms\n$\\varphi_n :\nK_{n+1} \\otimes^\\mathbf{L}_{\\mathbf{Z}/\\ell^{n+1}\\mathbf{Z}}\n\\mathbf{Z}/\\ell^n\\mathbf{Z}\n\\to K_n$\ncompatible with $\\pi_{n+1}$ and $\\pi_n$. Then\n\\begin{enumerate}\n\\item\nthe elements $t_n = \\text{Tr}(\\pi_n |_{K_n})\\in \\mathbf{Z}/\\ell^n\\mathbf{Z}$\nform an element $t_\\infty = \\{t_n\\}$ of $\\mathbf{Z}_\\ell$,\n\\item\nthe $\\mathbf{Z}_\\ell$-module $H_\\infty^i = \\lim_n H^i(k_n)$ is finite and\nis nonzero for finitely many $i$ only, and\n\\item\nthe operators $H^i(\\pi_n): H^i(K_n)\\to H^i(K_n)$ are compatible and define\n$\\pi_\\infty^i : H_\\infty^i\\to H_\\infty^i$ satisfying\n$$\n\\sum (-1)^i \\text{Tr}(\n\\pi_\\infty^i |_{H_\\infty^i \\otimes_{\\mathbf{Z}_\\ell}\\mathbf{Q}_\\ell}) =\nt_\\infty.\n$$\n\\end{enumerate}","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Cohomological interpretation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03V4","source_file":"trace.tex","source_line":2367,"source_end_line":2393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2367-L2393","statement_sha256":"cd8b36c44923e6562ac2265f9df6f1b5507c0a645e2f37e5d2f42ccc2761941d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10658,"rank":10658,"depth":2,"x":1408.587,"y":1160.387,"cluster":"tale-geometry"},{"id":"stacks:03V8","tag":"03V8","title":"Constant sheaves · Lemma 03V8","summary":"Let X be a smooth, projective, geometrically irreducible curve over a finite field k. Then • the L-function L(X, Q_ℓ) is a rational function, • the eigenvalues α_1, …, α_2g of π_X^* on H^1(X_bar k, Q_ℓ) are algebraic integers independent of ℓ, • the number of rational points of X on k_n, where [k_n : k] = n, is \\# X(k_n) = 1 - ∑_i = 1^2gα_i^n + q^n, • for each i, |α_i| < q.","statement_latex":"Let $X$ be a smooth, projective, geometrically irreducible\ncurve over a finite field $k$. Then\n\\begin{enumerate}\n\\item the $L$-function $L(X, \\mathbf{Q}_\\ell)$ is a rational function,\n\\item the eigenvalues $\\alpha_1, \\ldots, \\alpha_{2g}$ of $\\pi_X^*$ on\n$H^1(X_{\\bar k}, \\mathbf{Q}_\\ell)$ are algebraic integers\nindependent of $\\ell$,\n\\item the number of rational points of $X$ on $k_n$, where $[k_n : k] = n$, is\n$$\n\\# X(k_n) = 1 - \\sum\\nolimits_{i = 1}^{2g}\\alpha_i^n + q^n,\n$$\n\\item for each $i$, $|\\alpha_i| < q$.\n\\end{enumerate}","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Constant sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03V8","source_file":"trace.tex","source_line":2563,"source_end_line":2578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2563-L2578","statement_sha256":"401a08cbe5d4184c53e670c7d59aa52506997101a6e93570ec973ac11a9a543e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10659,"rank":10659,"depth":2,"x":1690.062,"y":1027.642,"cluster":"tale-geometry"},{"id":"stacks:03VE","tag":"03VE","title":"Fundamental groups · Definition 03VE","summary":"A subgroup of the form Stab(overline y∈ F_overlinex(Y))⊂ π_1(X, overlinex) is called open.","statement_latex":"A subgroup of the form\n$\\text{Stab}(\\overline y\\in F_{\\overline{x}}(Y))\\subset \\pi_1(X, \\overline{x})$\nis called {\\it open}.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Fundamental groups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VE","source_file":"trace.tex","source_line":2858,"source_end_line":2863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2858-L2863","statement_sha256":"287b0e0cb27a485a96307f525432069081021e0229d5052a5520b582be54d687","origin":"The Stacks Project","memory_eligible":false,"source_rank":10660,"rank":10660,"depth":0,"x":1589.202,"y":1285.381,"cluster":"tale-geometry"},{"id":"stacks:03VF","tag":"03VF","title":"Grothendieck · Theorem 03VF","summary":"Let X be a connected scheme. • There is a topology on π_1(X, overlinex) such that the open subgroups form a fundamental system of open nbhds of e∈ π_1(X, overline x). • With topology of (1) the group π_1(X, overlinex) is a profinite group. • The functor schemes finite atop étale over X & → & finite discrete continuous atop π_1(X, overlinex)-sets Y / X& ↦ & F_overlinex(Y) with its natural action is an equivalence of categories.","statement_latex":"Let $X$ be a connected scheme.\n\\begin{enumerate}\n\\item There is a topology on $\\pi_1(X, \\overline{x})$ such that the open\nsubgroups form a fundamental system of open nbhds of $e\\in \\pi_1(X, \\overline\nx)$.\n\\item With topology of (1) the group\n$\\pi_1(X, \\overline{x})$ is a profinite group.\n\\item The functor\n$$\n\\begin{matrix}\n\\text{ schemes finite } \\atop \\text{ \\'etale over }X & \\to &\n\\text{ finite discrete continuous } \\atop \\pi_1(X, \\overline{x})\\text{-sets}\\\\\nY / X& \\mapsto & F_{\\overline{x}}(Y) \\text{ with its natural action}\n\\end{matrix}\n$$\nis an equivalence of categories.\n\\end{enumerate}","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Fundamental groups","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VF","source_file":"trace.tex","source_line":2865,"source_end_line":2884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2865-L2884","statement_sha256":"7cad8a926e49f748a76e9c52428cd2b375285287f36c1fc23807152e739068f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10661,"rank":10661,"depth":0,"x":1456.237,"y":1037.965,"cluster":"tale-geometry"},{"id":"stacks:03VG","tag":"03VG","title":"Fundamental groups · Proposition 03VG","summary":"Let X be an integral normal Noetherian scheme. Let overline y→ X be an algebraic geometric point lying over the generic point eta∈ X. Then π_x(X, overline eta) = Gal(M/kappa(eta)) (kappa(eta), function field of X) where kappa(overline eta)⊃ M⊃ kappa(eta) = k(X) is the max sub-extension such that for every finite sub extension M⊃ L⊃ kappa(eta) the normalization of X in L is finite étale over X.","statement_latex":"Let $X$ be an integral normal Noetherian scheme. Let\n$\\overline y\\to X$ be an algebraic geometric point lying\nover the generic point $\\eta\\in X$. Then\n$$\n\\pi_x(X, \\overline \\eta) = Gal(M/\\kappa(\\eta))\n$$\n($\\kappa(\\eta)$, function field of $X$) where\n$$\n\\kappa(\\overline \\eta)\\supset M\\supset \\kappa(\\eta) = k(X)\n$$\nis the max sub-extension such that for every finite sub extension\n$M\\supset L\\supset \\kappa(\\eta)$ the normalization of $X$ in $L$ is finite\n\\'etale over $X$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Fundamental groups","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VG","source_file":"trace.tex","source_line":2890,"source_end_line":2905,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L2890-L2905","statement_sha256":"cd003d6431f91f8d83387c062051e0c8c945bb8d1e097f0200857ef6d5000dea","origin":"The Stacks Project","memory_eligible":false,"source_rank":10662,"rank":10662,"depth":0,"x":1753.407,"y":1145.016,"cluster":"tale-geometry"},{"id":"stacks:03VH","tag":"03VH","title":"Deligne, Weil II · Theorem 03VH","summary":"For a sheaf F_ρ with ρ satisfying the conclusions of the conjecture above then the eigenvalues of π_X^* on H_c^i(X_overlinek, F_ρ) are algebraic numbers α with absolute values |α|=q^w/2, for w∈ Z, w≤ i Moreover, if X smooth and proj. then w = i.","statement_latex":"For a sheaf\n$\\mathcal{F}_\\rho$ with $\\rho$ satisfying the conclusions of the conjecture\nabove then the eigenvalues of $\\pi_X^*$ on $H_c^i(X_{\\overline{k}},\n\\mathcal{F}_{\\rho})$ are algebraic numbers $\\alpha$ with absolute values\n$$\n|\\alpha|=q^{w/2}, \\text{ for }w\\in \\mathbf{Z},\\ w\\leq i\n$$\nMoreover, if $X$ smooth and proj. then $w = i$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Fundamental groups","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VH","source_file":"trace.tex","source_line":3074,"source_end_line":3084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3074-L3084","statement_sha256":"6e08abfe6c982bb8fb2e9478220f151b231f956f085b1eb9258859a8cb29f02f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10663,"rank":10663,"depth":0,"x":1448.028,"y":1234.747,"cluster":"tale-geometry"},{"id":"stacks:03VK","tag":"03VK","title":"Cohomology of curves, revisited · Lemma 03VK","summary":"There is a canonical isomorphism H_c^2(X_overlinek, F_ρ)=(M)_π_1(X_overlinek, overlineeta)(-1) as Gal(k^^sep/k)-modules.","statement_latex":"There is a canonical isomorphism\n$$\nH_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho)=(M)_{\\pi_1(X_{\\overline{k}},\n\\overline\\eta)}(-1)\n$$\nas $\\text{Gal}(k^{^{sep}}/k)$-modules.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Cohomology of curves, revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VK","source_file":"trace.tex","source_line":3187,"source_end_line":3195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3187-L3195","statement_sha256":"18bcc5c2c461112bceb9298c8e84a41b8346672ab319f27b2cff55bf06e9f372","origin":"The Stacks Project","memory_eligible":false,"source_rank":10664,"rank":10664,"depth":0,"x":1601.132,"y":995.173,"cluster":"tale-geometry"},{"id":"stacks:03VM","tag":"03VM","title":"Cohomology of curves, revisited · Proposition 03VM","summary":"Let X/k as before but X_overlinek≠ P^1_overlinek The functors (M, ρ)↦ H_c^2-i(X_overlinek, F_ρ) are the left derived functor of (M, ρ)↦ H_c^2(X_overlinek, F_ρ) so H_c^2-i(X_overlinek, F_ρ) = H_i(π_1(X_overlinek, overline eta), M)(-1) Moreover, there is a derived version, namely RΓ_c(X_overlinek, F_ρ) = LH_0(π_1(X_overlinek, overline eta), M(-1)) = M(-1) ⊗_Lambda[[π_1(X_overlinek, overline eta)]]^L Lambda in D(Lambda[[widehatZ]]). Similarly, the functors (M, ρ)↦…","statement_latex":"Let $X/k$ as before but $X_{\\overline{k}}\\neq \\mathbf{P}^1_{\\overline{k}}$\nThe functors\n$\n(M, \\rho)\\mapsto H_c^{2-i}(X_{\\overline{k}}, \\mathcal{F}_\\rho)\n$\nare the left derived functor of\n$(M, \\rho)\\mapsto H_c^2(X_{\\overline{k}}, \\mathcal{F}_\\rho)$\nso\n$$\nH_c^{2-i}(X_{\\overline{k}}, \\mathcal{F}_\\rho) =\nH_i(\\pi_1(X_{\\overline{k}}, \\overline \\eta), M)(-1)\n$$\nMoreover, there is a derived version, namely\n$$\nR\\Gamma_c(X_{\\overline{k}}, \\mathcal{F}_\\rho)\n=\nLH_0(\\pi_1(X_{\\overline{k}}, \\overline \\eta), M(-1))\n=\nM(-1)\n\\otimes_{\\Lambda[[\\pi_1(X_{\\overline{k}}, \\overline \\eta)]]}^\\mathbf{L}\n\\Lambda\n$$\nin $D(\\Lambda[[\\widehat{\\mathbf{Z}}]])$.\nSimilarly, the functors\n$(M, \\rho)\\mapsto H^i(X_{\\overline{k}}, \\mathcal{F}_\\rho)$\nare the right derived functor of\n$(M, \\rho)\\mapsto M^{\\pi_1(X_{\\overline{k}}, \\overline \\eta)}$\nso\n$$\nH^i(X_{\\overline{k}}, \\mathcal{F}_\\rho) =\nH^i(\\pi_1(X_{\\overline{k}}, \\overline \\eta), M)\n$$\nMoreover, in this case there is a derived version too.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Cohomology of curves, revisited","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VM","source_file":"trace.tex","source_line":3296,"source_end_line":3331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3296-L3331","statement_sha256":"40947e6f30b2d06d36f80b3b7dca5e367e9323b68c2f6b096df57453b487f6d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10665,"rank":10665,"depth":0,"x":1680.937,"y":1258.847,"cluster":"tale-geometry"},{"id":"stacks:03VS","tag":"03VS","title":"Automorphic forms and sheaves · Definition 03VS","summary":"An unramified cusp form on GL_2(A) with values in Lambda is a function f : GL_2(A) → Lambda such that • f(xγ) = f(x) for all x∈ GL_2(A) and all γ∈ GL_2(K) • f(ux) = f(x) for all x∈ GL_2(A) and all u∈ GL_2(O) • for all x∈ GL_2(A), int_A mod K f (x ( 1 & z 0 & 1 ) ) dz = 0 see [dJ-conjecture] for an explanation of how to make sense out of this for a general ring Lambda in which p is invertible.","statement_latex":"An {\\it unramified cusp form on $\\text{GL}_2(\\mathbf{A})$ with values in\n$\\Lambda$}\\footnote{This is likely nonstandard notation.}\nis a function\n$$\nf : \\text{GL}_2(\\mathbf{A}) \\to \\Lambda\n$$\nsuch that\n\\begin{enumerate}\n\\item $f(x\\gamma) = f(x)$ for all $x\\in \\text{GL}_2(\\mathbf{A})$ and all\n$\\gamma\\in \\text{GL}_2(K)$\n\\item $f(ux) = f(x)$ for all $x\\in \\text{GL}_2(\\mathbf{A})$ and all\n$u\\in \\text{GL}_2(O)$\n\\item for all $x\\in \\text{GL}_2(\\mathbf{A})$,\n$$\n\\int_{\\mathbf{A} \\mod K} f\n\\left(x\n\\left(\n\\begin{matrix}\n1 & z \\\\\n0 & 1\n\\end{matrix}\n\\right)\n\\right) dz = 0\n$$\nsee \\cite[Section 4.1]{dJ-conjecture}\nfor an explanation of how to make sense out\nof this for a general ring $\\Lambda$ in which $p$ is invertible.\n\\end{enumerate}","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Automorphic forms and sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VS","source_file":"trace.tex","source_line":3470,"source_end_line":3500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3470-L3500","statement_sha256":"b325ff19a51a2a0e2352a874e3e5a658e8a2fbd8e3de93be9f6cecd86915de61","origin":"The Stacks Project","memory_eligible":false,"source_rank":10666,"rank":10666,"depth":0,"x":1409.906,"y":1109.622,"cluster":"tale-geometry"},{"id":"stacks:03VT","tag":"03VT","title":"Automorphic forms and sheaves · Theorem 03VT","summary":"Given an eigenform f with values in overlineQ_l and eigenvalues u_v∈ overlineZ_l^* then there exists ρ : π_1(X)→ GL_2(E) continuous, absolutely irreducible where E is a finite extension of Q_ℓ contained in overlineQ_l such that t_v = Tr(ρ(F_v)), and u_v = q_v^-1det(ρ(F_v)) for all places v.","statement_latex":"Given an eigenform $f$ with values in\n$\\overline{\\mathbf{Q}}_l$ and eigenvalues\n$u_v\\in \\overline{\\mathbf{Z}}_l^*$ then there exists\n$$\n\\rho : \\pi_1(X)\\to \\text{GL}_2(E)\n$$\ncontinuous, absolutely irreducible where\n$E$ is a finite extension of $\\mathbf{Q}_\\ell$ contained in\n$\\overline{\\mathbf{Q}}_l$ such that\n$t_v = \\text{Tr}(\\rho(F_v))$, and\n$u_v = q_v^{-1}\\det\\left(\\rho(F_v)\\right)$ for all places $v$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Automorphic forms and sheaves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VT","source_file":"trace.tex","source_line":3555,"source_end_line":3568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3555-L3568","statement_sha256":"5d0b5d51e48c13a425c3a57d1074df4bf314cb495134f8794bb3665fd8343e54","origin":"The Stacks Project","memory_eligible":false,"source_rank":10667,"rank":10667,"depth":0,"x":1729.933,"y":1065.846,"cluster":"tale-geometry"},{"id":"stacks:03VU","tag":"03VU","title":"Automorphic forms and sheaves · Theorem 03VU","summary":"Suppose Q_l ⊂ E finite, and ρ : π_1(X)→ GL_2(E) absolutely irreducible, continuous. Then there exists an eigenform f with values in overlineQ_l whose eigenvalues t_v, u_v satisfy the equalities t_v = Tr(ρ(F_v)) and u_v = q_v^-1det(ρ(F_v)).","statement_latex":"Suppose $\\mathbf{Q}_l \\subset E$ finite, and\n$$\n\\rho : \\pi_1(X)\\to \\text{GL}_2(E)\n$$\nabsolutely irreducible, continuous. Then there exists an eigenform $f$ with\nvalues in $\\overline{\\mathbf{Q}}_l$ whose eigenvalues $t_v$, $u_v$\nsatisfy the equalities\n$t_v = \\text{Tr}(\\rho(F_v))$ and $u_v = q_v^{-1}\\det(\\rho(F_v))$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Automorphic forms and sheaves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VU","source_file":"trace.tex","source_line":3574,"source_end_line":3584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3574-L3584","statement_sha256":"cfdaf7f0fcb711ccfb1c3a6d2f8c228a22bd292020a810f6dc117a25270e01d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10668,"rank":10668,"depth":0,"x":1529.047,"y":1279.83,"cluster":"tale-geometry"},{"id":"stacks:03VW","tag":"03VW","title":"Automorphic forms and sheaves · Proposition 03VW","summary":"If Lambda is Noetherian then C(Lambda) is a finitely generated Lambda-module. Moreover, if Lambda is a field with prime subfield F ⊂ Lambda then C(Lambda)=(C(F))⊗_FLambda compatibly with T_v acting.","statement_latex":"If $\\Lambda$ is Noetherian then $C(\\Lambda)$ is a\nfinitely generated $\\Lambda$-module. Moreover, if $\\Lambda$ is a field with\nprime subfield $\\mathbf{F} \\subset \\Lambda$ then\n$$\nC(\\Lambda)=(C(\\mathbf{F}))\\otimes_{\\mathbf{F}}\\Lambda\n$$\ncompatibly with $T_v$ acting.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Automorphic forms and sheaves","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VW","source_file":"trace.tex","source_line":3626,"source_end_line":3635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3626-L3635","statement_sha256":"d03e3df93f0525697241952e8fc9e789f8446ad2b1bf4bfac80be8835646a65e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10669,"rank":10669,"depth":0,"x":1505.084,"y":1007.909,"cluster":"tale-geometry"},{"id":"stacks:03VX","tag":"03VX","title":"Automorphic forms and sheaves · Lemma 03VX","summary":"Algebraicity of eigenvalues. If Lambda is a field then the eigenvalues t_v for f∈ C(Lambda) are algebraic over the prime subfield F ⊂ Lambda.","statement_latex":"Algebraicity of eigenvalues.\nIf $\\Lambda$ is a field then the eigenvalues $t_v$ for $f\\in\nC(\\Lambda)$ are algebraic over the prime subfield\n$\\mathbf{F} \\subset \\Lambda$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Automorphic forms and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VX","source_file":"trace.tex","source_line":3644,"source_end_line":3650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3644-L3650","statement_sha256":"d331eec2e1b103191e34b25c1ac9277079370890b1f5e34e7ef98bdf8652d963","origin":"The Stacks Project","memory_eligible":false,"source_rank":10670,"rank":10670,"depth":1,"x":1741.551,"y":1194.922,"cluster":"tale-geometry"},{"id":"stacks:03VY","tag":"03VY","title":"Automorphic forms and sheaves · Lemma 03VY","summary":"Switching l. Let E be a number field. Start with ρ : π_1(X)→ SL_2(E_λ) absolutely irreducible continuous, where λ is a place of E not lying above p. Then for any second place λ' of E not lying above p there exists a finite extension E'_λ' and a absolutely irreducible continuous representation ρ': π_1(X)→ SL_2(E'_λ') which is compatible with ρ in the sense that the characteristic polynomials of all Frobenii are the same.","statement_latex":"Switching $l$. Let $E$ be a number field.\nStart with\n$$\n\\rho : \\pi_1(X)\\to SL_2(E_\\lambda)\n$$\nabsolutely irreducible continuous, where $\\lambda$ is a place of $E$\nnot lying above $p$. Then for any second place $\\lambda'$ of $E$\nnot lying above $p$ there exists a finite extension $E'_{\\lambda'}$\nand a absolutely irreducible continuous representation\n$$\n\\rho': \\pi_1(X)\\to SL_2(E'_{\\lambda'})\n$$\nwhich is compatible with $\\rho$ in the sense that the characteristic\npolynomials of all Frobenii are the same.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Automorphic forms and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VY","source_file":"trace.tex","source_line":3659,"source_end_line":3675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3659-L3675","statement_sha256":"98d56f7bf60dd352cc08688f076c53271a1c552b8ee44cf7735758e1a352d72e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10671,"rank":10671,"depth":0,"x":1416.622,"y":1191.197,"cluster":"tale-geometry"},{"id":"stacks:03VZ","tag":"03VZ","title":"Automorphic forms and sheaves · Theorem 03VZ","summary":"The Conjecture holds if n≤ 2.","statement_latex":"The Conjecture holds if $n\\leq 2$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Automorphic forms and sheaves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03VZ","source_file":"trace.tex","source_line":3725,"source_end_line":3728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3725-L3728","statement_sha256":"4284acabb1dc64e54b2ec6ab46edac443bfe2a111a6afc5d94b844554f90b16e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10672,"rank":10672,"depth":0,"x":1659.344,"y":1009.475,"cluster":"tale-geometry"},{"id":"stacks:03W0","tag":"03W0","title":"Automorphic forms and sheaves · Theorem 03W0","summary":"Conjecture holds if l > 2n modulo some unproven things.","statement_latex":"Conjecture holds if $l > 2n$ modulo some unproven things.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Automorphic forms and sheaves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03W0","source_file":"trace.tex","source_line":3734,"source_end_line":3737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3734-L3737","statement_sha256":"a63ea82be9fa0400e91276f3fdad915fd0a908604867c47d408aa66bb4f6b6f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10673,"rank":10673,"depth":0,"x":1626.483,"y":1281.344,"cluster":"tale-geometry"},{"id":"stacks:03W1","tag":"03W1","title":"Automorphic forms and sheaves · Theorem 03W1","summary":"(See [dJ-conjecture]) Suppose ρ_0: π_1(X)→ GL_n(F_l) is a continuous, l≠ p. Assume • Conj. holds for X, • ρ_0 |_π_1(X_overlinek) abs. irred., and • l does not divide n. Then the universal deformation ring R_univ of ρ_0 is finite flat over Z_l.","statement_latex":"(See \\cite[Theorem 3.5]{dJ-conjecture})\nSuppose\n$$\n\\rho_0: \\pi_1(X)\\to \\text{GL}_n(\\mathbf{F}_l)\n$$\nis a continuous, $l\\neq p$. Assume\n\\begin{enumerate}\n\\item Conj. holds for $X$,\n\\item $\\rho_0 |_{\\pi_1(X_{\\overline{k}})}$ abs. irred., and\n\\item $l$ does not divide $n$.\n\\end{enumerate}\nThen the universal deformation ring $R_{\\text{univ}}$ of $\\rho_0$ is\nfinite flat over $\\mathbf{Z}_l$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"Automorphic forms and sheaves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03W1","source_file":"trace.tex","source_line":3748,"source_end_line":3763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3748-L3763","statement_sha256":"1408f56d84748d9d260a271c4628e5d6518994e9da02deaff4a267e76751e35f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10674,"rank":10674,"depth":0,"x":1431.982,"y":1062.109,"cluster":"tale-geometry"},{"id":"stacks:03W5","tag":"03W5","title":"How many primes decompose completely? · Proposition 03W5","summary":"There exists a finite set x_1, …, x_n of closed points of X such that set of all frobenius elements corresponding to these points topologically generate π_1(X).","statement_latex":"There exists a finite set $x_1, \\ldots, x_n$ of closed points of $X$\nsuch that set of {\\bf all} frobenius elements corresponding to these\npoints topologically generate $\\pi_1(X)$.","area":"Étale Geometry","chapter":"The Trace Formula","chapter_id":"trace","section":"How many primes decompose completely?","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03W5","source_file":"trace.tex","source_line":3933,"source_end_line":3938,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/trace.tex#L3933-L3938","statement_sha256":"f46be08882b0e608a138c00a5bb56d8fd290671d394564b90c178bd617917eb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10675,"rank":10675,"depth":0,"x":1751.875,"y":1113.432,"cluster":"tale-geometry"},{"id":"stacks:02W9","tag":"02W9","title":"Representable morphisms of presheaves · Lemma 02W9","summary":"Let S be a scheme contained in Sch_fppf and let X, Y be objects of (Sch/S)_fppf. Let f : X → Y be a morphism of schemes. Then h_f : h_X → h_Y is a representable transformation of functors.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$ and let\n$X$, $Y$ be objects of $(\\Sch/S)_{fppf}$.\nLet $f : X \\to Y$ be a morphism of schemes.\nThen\n$$\nh_f : h_X \\longrightarrow h_Y\n$$\nis a representable transformation of functors.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02W9","source_file":"spaces.tex","source_line":139,"source_end_line":149,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L139-L149","statement_sha256":"29a3fd6f6da401ff30db287b70f1f0f75f7528bb2de8941e26ae60542cbd4ce6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10676,"rank":10676,"depth":0,"x":234.118,"y":1600.0,"cluster":"algebraic-spaces"},{"id":"stacks:02WA","tag":"02WA","title":"Representable morphisms of presheaves · Lemma 02WA","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let a : F → G, b : G → H be representable transformations of functors. Then b ∘ a : F → H is a representable transformation of functors.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$, $b : G \\to H$ be representable transformations of functors.\nThen\n$$\nb \\circ a : F \\longrightarrow H\n$$\nis a representable transformation of functors.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WA","source_file":"spaces.tex","source_line":156,"source_end_line":166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L156-L166","statement_sha256":"3d6e8467870e8d8c22e626887725c35972a6a107bc19bfd14ddd8de87478e770","origin":"The Stacks Project","memory_eligible":false,"source_rank":10677,"rank":10677,"depth":0,"x":224.741,"y":1604.047,"cluster":"algebraic-spaces"},{"id":"stacks:02WB","tag":"02WB","title":"Representable morphisms of presheaves · Lemma 02WB","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let a : F → G be a representable transformation of functors. Let b : H → G be any transformation of functors. Consider the fibre product diagram xymatrix H ×_b, G, a F ar[r]_-b' ar[d]_a' & F ar[d]^a H ar[r]^b & G Then the base change a' is a representable transformation of functors.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$ be a representable transformation of functors.\nLet $b : H \\to G$ be any transformation of functors.\nConsider the fibre product diagram\n$$\n\\xymatrix{\nH \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\\nH \\ar[r]^b & G\n}\n$$\nThen the base change $a'$ is a representable transformation of functors.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WB","source_file":"spaces.tex","source_line":172,"source_end_line":186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L172-L186","statement_sha256":"fd1d1b173c83b3253a991d026d77e030d0a547715c656498ce74abaab867213d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10678,"rank":10678,"depth":0,"x":230.805,"y":1592.295,"cluster":"algebraic-spaces"},{"id":"stacks:02WC","tag":"02WC","title":"Representable morphisms of presheaves · Lemma 02WC","summary":"Let S be a scheme contained in Sch_fppf. Let F_i, G_i : (Sch/S)_fppf^opp → Sets, i = 1, 2. Let a_i : F_i → G_i, i = 1, 2 be representable transformations of functors. Then a_1 × a_2 : F_1 × F_2 → G_1 × G_2 is a representable transformation of functors.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$, $i = 1, 2$.\nLet $a_i : F_i \\to G_i$, $i = 1, 2$\nbe representable transformations of functors.\nThen\n$$\na_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2\n$$\nis a representable transformation of functors.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WC","source_file":"spaces.tex","source_line":192,"source_end_line":203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L192-L203","statement_sha256":"a0820792fece26629557e7ec5af0e28f74f57e3a1e75333e30283bb11a12e65d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10679,"rank":10679,"depth":1,"x":236.629,"y":1607.263,"cluster":"algebraic-spaces"},{"id":"stacks:02WD","tag":"02WD","title":"Representable morphisms of presheaves · Lemma 02WD","summary":"Let S be a scheme contained in Sch_fppf. Let F, G : (Sch/S)_fppf^opp → Sets. Let a : F → G be a representable transformation of functors. If G is a sheaf, then so is F.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$ be a representable transformation of functors.\nIf $G$ is a sheaf, then so is $F$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WD","source_file":"spaces.tex","source_line":216,"source_end_line":222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L216-L222","statement_sha256":"6a3f227e718524a27821683888d6c290292793d227176024c347cf820ff2a4d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10680,"rank":10680,"depth":0,"x":217.835,"y":1598.192,"cluster":"algebraic-spaces"},{"id":"stacks:05L9","tag":"05L9","title":"Representable morphisms of presheaves · Lemma 05L9","summary":"Let S be a scheme contained in Sch_fppf. Let F, G : (Sch/S)_fppf^opp → Sets. Let a : F → G be a representable transformation of functors. Then Δ_F/G : F → F ×_G F is representable.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$ be a representable transformation of functors.\nThen $\\Delta_{F/G} : F \\to F \\times_G F$ is representable.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05L9","source_file":"spaces.tex","source_line":239,"source_end_line":245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L239-L245","statement_sha256":"30ad4d3ba15e9e0807d157032f357786f36584f1bf2c692e6bf126440c110195","origin":"The Stacks Project","memory_eligible":false,"source_rank":10681,"rank":10681,"depth":0,"x":241.524,"y":1593.842,"cluster":"algebraic-spaces"},{"id":"stacks:025V","tag":"025V","title":"Properties of representable morphisms of presheaves · Definition 025V","summary":"With S, and a : F → G representable as above. Let P be a property of morphisms of schemes which • is preserved under any base change, see Schemes, Definition [Tag 01JZ], and • is fppf local on the base, see Descent, Definition [Tag 03YH]. In this case we say that a has property P if for every U ∈ Ob((Sch/S)_fppf) and any xi ∈ G(U) the resulting morphism of schemes V_xi → U has property P.","statement_latex":"With $S$, and $a : F \\to G$ representable as above.\nLet $\\mathcal{P}$ be a property of morphisms of schemes which\n\\begin{enumerate}\n\\item is preserved under any base change,\nsee Schemes, Definition \\ref{schemes-definition-preserved-by-base-change},\nand\n\\item is fppf local on the base, see\nDescent, Definition \\ref{descent-definition-property-morphisms-local}.\n\\end{enumerate}\nIn this case we say that $a$ has {\\it property $\\mathcal{P}$} if for every\n$U \\in \\Ob((\\Sch/S)_{fppf})$ and\nany $\\xi \\in G(U)$ the resulting morphism of schemes\n$V_\\xi \\to U$ has property $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025V","source_file":"spaces.tex","source_line":496,"source_end_line":511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L496-L511","statement_sha256":"c6d569df42ff7410c5c5a7b039b33a15e090bacdf8bef3461f2029f4a4c92224","origin":"The Stacks Project","memory_eligible":false,"source_rank":10682,"rank":10682,"depth":1,"x":226.146,"y":1612.044,"cluster":"algebraic-spaces"},{"id":"stacks:02WJ","tag":"02WJ","title":"Properties of representable morphisms of presheaves · Lemma 02WJ","summary":"Let S, X, Y be objects of Sch_fppf. Let f : X → Y be a morphism of schemes. Let P be as in Definition [Tag 025V]. Then h_X → h_Y has property P if and only if f has property P.","statement_latex":"Let $S$, $X$, $Y$ be objects of $\\Sch_{fppf}$.\nLet $f : X \\to Y$ be a morphism of schemes.\nLet $\\mathcal{P}$ be as in\nDefinition \\ref{definition-relative-representable-property}.\nThen $h_X \\longrightarrow h_Y$ has property $\\mathcal{P}$ if\nand only if $f$ has property $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WJ","source_file":"spaces.tex","source_line":542,"source_end_line":550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L542-L550","statement_sha256":"72b41f5af88b70b36ea93dc556fea8b6ab3ea8b051cc86fd9ec78b063ccbe81e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10683,"rank":10683,"depth":2,"x":222.649,"y":1588.111,"cluster":"algebraic-spaces"},{"id":"stacks:02WK","tag":"02WK","title":"Properties of representable morphisms of presheaves · Lemma 02WK","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let P be a property as in Definition [Tag 025V] which is stable under composition. Let a : F → G, b : G → H be representable transformations of functors. If a and b have property P so does b ∘ a : F → H.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-relative-representable-property}\nwhich is stable under composition.\nLet $a : F \\to G$, $b : G \\to H$ be representable transformations of functors.\nIf $a$ and $b$ have property $\\mathcal{P}$ so does\n$b \\circ a : F \\longrightarrow H$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WK","source_file":"spaces.tex","source_line":558,"source_end_line":568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L558-L568","statement_sha256":"09690b3a28e5d698d725d2629e91b0272777669806ed899751fb3f87792f7be8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10684,"rank":10684,"depth":2,"x":245.948,"y":1604.892,"cluster":"algebraic-spaces"},{"id":"stacks:02WL","tag":"02WL","title":"Properties of representable morphisms of presheaves · Lemma 02WL","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let P be a property as in Definition [Tag 025V]. Let a : F → G be a representable transformation of functors. Let b : H → G be any transformation of functors. Consider the fibre product diagram xymatrix H ×_b, G, a F ar[r]_-b' ar[d]_a' & F ar[d]^a H ar[r]^b & G If a has property P then also the base change a' has property P.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-relative-representable-property}.\nLet $a : F \\to G$ be a representable transformation of functors.\nLet $b : H \\to G$ be any transformation of functors.\nConsider the fibre product diagram\n$$\n\\xymatrix{\nH \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\\nH \\ar[r]^b & G\n}\n$$\nIf $a$ has property $\\mathcal{P}$ then also the base change $a'$\nhas property $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WL","source_file":"spaces.tex","source_line":576,"source_end_line":593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L576-L593","statement_sha256":"bd0cc31a5864631d0146b53b698eda5ebe43d2307b4ae5b06fa5714c87d7cdfe","origin":"The Stacks Project","memory_eligible":false,"source_rank":10685,"rank":10685,"depth":2,"x":213.408,"y":1605.753,"cluster":"algebraic-spaces"},{"id":"stacks:03KD","tag":"03KD","title":"Properties of representable morphisms of presheaves · Lemma 03KD","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let P be a property as in Definition [Tag 025V]. Let a : F → G be a representable transformation of functors. Let b : H → G be any transformation of functors. Consider the fibre product diagram xymatrix H ×_b, G, a F ar[r]_-b' ar[d]_a' & F ar[d]^a H ar[r]^b & G Assume that b induces a surjective map of fppf sheaves H^\\# → G^\\#. In this case, if a' has property P, then also a has property P.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-relative-representable-property}.\nLet $a : F \\to G$ be a representable transformation of functors.\nLet $b : H \\to G$ be any transformation of functors.\nConsider the fibre product diagram\n$$\n\\xymatrix{\nH \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\\nH \\ar[r]^b & G\n}\n$$\nAssume that $b$ induces a surjective map of fppf sheaves $H^\\# \\to G^\\#$.\nIn this case, if $a'$ has property $\\mathcal{P}$, then also $a$\nhas property $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KD","source_file":"spaces.tex","source_line":601,"source_end_line":619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L601-L619","statement_sha256":"abadd30332c54e42f3993c663b37aea2757a3fb0b7d3ac3f12a6fb23d627ea08","origin":"The Stacks Project","memory_eligible":false,"source_rank":10686,"rank":10686,"depth":2,"x":237.998,"y":1585.643,"cluster":"algebraic-spaces"},{"id":"stacks:02WM","tag":"02WM","title":"Properties of representable morphisms of presheaves · Lemma 02WM","summary":"Let S be a scheme contained in Sch_fppf. Let F_i, G_i : (Sch/S)_fppf^opp → Sets, i = 1, 2. Let a_i : F_i → G_i, i = 1, 2 be representable transformations of functors. Let P be a property as in Definition [Tag 025V] which is stable under composition. If a_1 and a_2 have property P so does a_1 × a_2 : F_1 × F_2 → G_1 × G_2.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$,\n$i = 1, 2$.\nLet $a_i : F_i \\to G_i$, $i = 1, 2$ be representable transformations\nof functors.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-relative-representable-property}\nwhich is stable under composition.\nIf $a_1$ and $a_2$ have property $\\mathcal{P}$ so does\n$a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WM","source_file":"spaces.tex","source_line":643,"source_end_line":655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L643-L655","statement_sha256":"373666170749a355e4569e98f328b03e7e50215169406ab513863736943f421d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10687,"rank":10687,"depth":2,"x":235.911,"y":1615.829,"cluster":"algebraic-spaces"},{"id":"stacks:02YO","tag":"02YO","title":"Properties of representable morphisms of presheaves · Lemma 02YO","summary":"Let S be a scheme contained in Sch_fppf. Let F, G : (Sch/S)_fppf^opp → Sets. Let a : F → G be a representable transformation of functors. Let P, P' be properties as in Definition [Tag 025V]. Suppose that for any morphism of schemes f : X → Y we have P(f) ⇒ P'(f). If a has property P then a has property P'.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$ be a representable transformation of functors.\nLet $\\mathcal{P}$, $\\mathcal{P}'$ be properties as in\nDefinition \\ref{definition-relative-representable-property}.\nSuppose that for any morphism of schemes $f : X \\to Y$\nwe have $\\mathcal{P}(f) \\Rightarrow \\mathcal{P}'(f)$.\nIf $a$ has property $\\mathcal{P}$ then\n$a$ has property $\\mathcal{P}'$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YO","source_file":"spaces.tex","source_line":663,"source_end_line":674,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L663-L674","statement_sha256":"3fefeb3f76fa4c0a0baa75610c3ad700ba957e077881a46b6c841e41cde2b72d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10688,"rank":10688,"depth":2,"x":212.186,"y":1591.328,"cluster":"algebraic-spaces"},{"id":"stacks:05VM","tag":"05VM","title":"Properties of representable morphisms of presheaves · Lemma 05VM","summary":"Let S be a scheme. Let F, G : (Sch/S)_fppf^opp → Sets be sheaves. Let a : F → G be representable, flat, locally of finite presentation, and surjective. Then a : F → G is surjective as a map of sheaves.","statement_latex":"Let $S$ be a scheme.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be sheaves.\nLet $a : F \\to G$ be representable, flat,\nlocally of finite presentation, and surjective.\nThen $a : F \\to G$ is surjective as a map of sheaves.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VM","source_file":"spaces.tex","source_line":680,"source_end_line":687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L680-L687","statement_sha256":"bd55b9ec6f74cb84661c67218ae4e2b5fd0441d17d1db8a7fe7b3bb92899d0bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10689,"rank":10689,"depth":1,"x":250.898,"y":1596.141,"cluster":"algebraic-spaces"},{"id":"stacks:025W","tag":"025W","title":"Properties of representable morphisms of presheaves · Lemma 025W","summary":"Let S be a scheme contained in Sch_fppf. Let F be a presheaf of sets on (Sch/S)_fppf. The following are equivalent: • the diagonal F → F × F is representable, • for U ∈ Ob((Sch/S)_fppf) and any a ∈ F(U) the map a : h_U → F is representable, • for every pair U, V ∈ Ob((Sch/S)_fppf) and any a ∈ F(U), b ∈ F(V) the fibre product h_U ×_a, F, b h_V is representable.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the diagonal $F \\to F \\times F$ is representable,\n\\item for $U \\in \\Ob((\\Sch/S)_{fppf})$ and any $a \\in F(U)$\nthe map $a : h_U \\to F$ is representable,\n\\item for every pair $U, V \\in \\Ob((\\Sch/S)_{fppf})$\nand any $a \\in F(U)$, $b \\in F(V)$ the fibre product\n$h_U \\times_{a, F, b} h_V$ is representable.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025W","source_file":"spaces.tex","source_line":704,"source_end_line":717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L704-L717","statement_sha256":"33de5fb1df2f6d4c83a593dc35ea4baf12f5b966e5e851788e6ac404692a0e35","origin":"The Stacks Project","memory_eligible":false,"source_rank":10690,"rank":10690,"depth":4,"x":217.246,"y":1615.239,"cluster":"algebraic-spaces"},{"id":"stacks:0CB7","tag":"0CB7","title":"Properties of representable morphisms of presheaves · Lemma 0CB7","summary":"Let S be a scheme contained in Sch_fppf. Let F be a presheaf of sets on (Sch/S)_fppf. Let P be a property as in Definition [Tag 025V]. If for every U, V ∈ Ob((Sch/S)_fppf) and a ∈ F(U), b ∈ F(V) we have • h_U ×_a, F, b h_V is representable, say by the scheme W, and • the morphism W → U ×_S V corresponding to h_U ×_a, F, b h_V → h_U × h_V has property P, then Δ : F → F × F is representable and has property P.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-relative-representable-property}.\nIf for every $U, V \\in \\Ob((\\Sch/S)_{fppf})$ and $a \\in F(U)$,\n$b \\in F(V)$ we have\n\\begin{enumerate}\n\\item $h_U \\times_{a, F, b} h_V$ is representable, say by the scheme $W$, and\n\\item the morphism $W \\to U \\times_S V$ corresponding to\n$h_U \\times_{a, F, b} h_V \\to h_U \\times h_V$ has property $\\mathcal{P}$,\n\\end{enumerate}\nthen $\\Delta : F \\to F \\times F$ is representable and has\nproperty $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Properties of representable morphisms of presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CB7","source_file":"spaces.tex","source_line":738,"source_end_line":753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L738-L753","statement_sha256":"7e8ce76708df04f58351edc5749c411b8fb4cbfa4284a28051ea1d375fe4ed5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10691,"rank":10691,"depth":5,"x":227.054,"y":1580.9,"cluster":"algebraic-spaces"},{"id":"stacks:025Y","tag":"025Y","title":"Algebraic spaces · Definition 025Y","summary":"Let S be a scheme contained in Sch_fppf. An algebraic space over S is a presheaf F : (Sch/S)^opp_fppf → Sets with the following properties • The presheaf F is a sheaf. • The diagonal morphism F → F × F is representable. • There exists a scheme U ∈ Ob((Sch/S)_fppf) and a map h_U → F which is surjective and étale.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nAn {\\it algebraic space over $S$} is a presheaf\n$$\nF : (\\Sch/S)^{opp}_{fppf} \\longrightarrow \\textit{Sets}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item The presheaf $F$ is a sheaf.\n\\item The diagonal morphism $F  \\to F \\times F$ is representable.\n\\item There exists a scheme $U \\in \\Ob((\\Sch/S)_{fppf})$\nand a map $h_U \\to F$ which is surjective and \\'etale\\footnote{See\nLemma \\ref{lemma-representable-diagonal},\nDefinition \\ref{definition-relative-representable-property}, and\nRemark \\ref{remark-warning}.}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025Y","source_file":"spaces.tex","source_line":803,"source_end_line":820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L803-L820","statement_sha256":"6ba7fc2297a3135fb1199704ab6324356816ec94e2e61fdd105bfa9c79337ace","origin":"The Stacks Project","memory_eligible":false,"source_rank":10692,"rank":10692,"depth":5,"x":248.088,"y":1612.806,"cluster":"algebraic-spaces"},{"id":"stacks:025Z","tag":"025Z","title":"Algebraic spaces · Lemma 025Z","summary":"A scheme is an algebraic space. More precisely, given a scheme T ∈ Ob((Sch/S)_fppf) the representable functor h_T is an algebraic space.","statement_latex":"A scheme is an algebraic space. More precisely,\ngiven a scheme $T \\in \\Ob((\\Sch/S)_{fppf})$\nthe representable functor $h_T$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/025Z","source_file":"spaces.tex","source_line":850,"source_end_line":855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L850-L855","statement_sha256":"04e65753d9ada9e0287feba3f373e3ff18b4b5d121d26425c68e9391f12342a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10693,"rank":10693,"depth":0,"x":205.659,"y":1600.846,"cluster":"algebraic-spaces"},{"id":"stacks:0260","tag":"0260","title":"Algebraic spaces · Definition 0260","summary":"Let F, F' be algebraic spaces over S. A morphism f : F → F' of algebraic spaces over S is a transformation of functors from F to F'.","statement_latex":"Let $F$, $F'$ be algebraic spaces over $S$.\nA {\\it morphism $f : F \\to F'$ of algebraic spaces over $S$}\nis a transformation of functors from $F$ to $F'$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0260","source_file":"spaces.tex","source_line":864,"source_end_line":869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L864-L869","statement_sha256":"d459edeedc08c333a70213e602d20e6f903422706c5825c32b5717a16cd848a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10694,"rank":10694,"depth":0,"x":247.755,"y":1585.158,"cluster":"algebraic-spaces"},{"id":"stacks:02X0","tag":"02X0","title":"Fibre products of algebraic spaces · Lemma 02X0","summary":"Let S be a scheme contained in Sch_fppf. Let F, G be algebraic spaces over S. Then F × G is an algebraic space, and is a product in the category of algebraic spaces over S.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G$ be algebraic spaces over $S$.\nThen $F \\times G$ is an algebraic space, and is a product\nin the category of algebraic spaces over $S$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Fibre products of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02X0","source_file":"spaces.tex","source_line":894,"source_end_line":900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L894-L900","statement_sha256":"20a3f64476d26003f90800936c87747a23aedcdddc8aa9bff5fa28332173a9ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":10695,"rank":10695,"depth":3,"x":228.812,"y":1621.579,"cluster":"algebraic-spaces"},{"id":"stacks:04T9","tag":"04T9","title":"Fibre products of algebraic spaces · Lemma 04T9","summary":"Let S be a scheme contained in Sch_fppf. Let H be a sheaf on (Sch/S)_fppf whose diagonal is representable. Let F, G be algebraic spaces over S. Let F → H, G → H be maps of sheaves. Then F ×_H G is an algebraic space.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $H$ be a sheaf on $(\\Sch/S)_{fppf}$ whose diagonal\nis representable. Let $F, G$ be algebraic spaces over $S$.\nLet $F \\to H$, $G \\to H$ be maps of sheaves.\nThen $F \\times_H G$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Fibre products of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04T9","source_file":"spaces.tex","source_line":913,"source_end_line":920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L913-L920","statement_sha256":"8b3c55d81d9d82d0dbccdea12cedc4ae93f8f38d50d5b47b73f2e1858664c400","origin":"The Stacks Project","memory_eligible":false,"source_rank":10696,"rank":10696,"depth":6,"x":213.106,"y":1582.995,"cluster":"algebraic-spaces"},{"id":"stacks:02X2","tag":"02X2","title":"Fibre products of algebraic spaces · Lemma 02X2","summary":"Let S be a scheme contained in Sch_fppf. Let F → H, G → H be morphisms of algebraic spaces over S. Then F ×_H G is an algebraic space, and is a fibre product in the category of algebraic spaces over S.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F \\to H$, $G \\to H$ be morphisms of algebraic spaces over $S$.\nThen $F \\times_H G$ is an algebraic space, and is a fibre product\nin the category of algebraic spaces over $S$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Fibre products of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02X2","source_file":"spaces.tex","source_line":956,"source_end_line":962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L956-L962","statement_sha256":"00b70c6247d3646f758772531126b42df09acde45d1e3c9d5b086b0b1996246e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10697,"rank":10697,"depth":7,"x":256.762,"y":1603.025,"cluster":"algebraic-spaces"},{"id":"stacks:0F15","tag":"0F15","title":"Glueing algebraic spaces · Lemma 0F15","summary":"Let S ∈ Ob(Sch_fppf). Let F and G be sheaves on (Sch/S)_fppf^opp and denote F amalg G the coproduct in the category of sheaves. The map F → F amalg G is representable by open and closed immersions.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$. Let $F$ and $G$ be sheaves on\n$(\\Sch/S)_{fppf}^{opp}$ and denote $F \\amalg G$ the coproduct\nin the category of sheaves. The map $F \\to F \\amalg G$ is representable by\nopen and closed immersions.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Glueing algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F15","source_file":"spaces.tex","source_line":986,"source_end_line":992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L986-L992","statement_sha256":"3b54352256e502dbce18215036197ee82c2c25a1cde958f6b84b51503b43603e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10698,"rank":10698,"depth":3,"x":207.325,"y":1613.253,"cluster":"algebraic-spaces"},{"id":"stacks:02WO","tag":"02WO","title":"Glueing algebraic spaces · Lemma 02WO","summary":"Let S ∈ Ob(Sch_fppf). Let U ∈ Ob((Sch/S)_fppf). Given a set I and sheaves F_i on Ob((Sch/S)_fppf), if U ≅ coprod_i∈ I F_i as sheaves, then each F_i is representable by an open and closed subscheme U_i and U ≅ coprod U_i as schemes.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$.\nLet $U \\in \\Ob((\\Sch/S)_{fppf})$.\nGiven a set $I$ and sheaves $F_i$ on $\\Ob((\\Sch/S)_{fppf})$,\nif $U \\cong \\coprod_{i\\in I} F_i$\nas sheaves, then each $F_i$ is representable by an open and closed\nsubscheme $U_i$ and $U \\cong \\coprod U_i$ as schemes.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Glueing algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WO","source_file":"spaces.tex","source_line":1011,"source_end_line":1019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1011-L1019","statement_sha256":"54317242231982052e6fb1f416228932bb828404e6737166a5e7f66c31e9df26","origin":"The Stacks Project","memory_eligible":false,"source_rank":10699,"rank":10699,"depth":4,"x":236.196,"y":1576.864,"cluster":"algebraic-spaces"},{"id":"stacks:02WP","tag":"02WP","title":"Glueing algebraic spaces · Lemma 02WP","summary":"Let S ∈ Ob(Sch_fppf). Let F be an algebraic space over S. Given a set I and sheaves F_i on Ob((Sch/S)_fppf), if F ≅ coprod_i∈ I F_i as sheaves, then each F_i is an algebraic space over S.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$.\nLet $F$ be an algebraic space over $S$.\nGiven a set $I$ and sheaves $F_i$ on\n$\\Ob((\\Sch/S)_{fppf})$,\nif $F \\cong \\coprod_{i\\in I} F_i$ as sheaves,\nthen each $F_i$ is an algebraic space over $S$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Glueing algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WP","source_file":"spaces.tex","source_line":1029,"source_end_line":1037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1029-L1037","statement_sha256":"50ad0f4b9091ec5fbd861516df8fa3c70d06875032b849b072fd745a21631492","origin":"The Stacks Project","memory_eligible":false,"source_rank":10700,"rank":10700,"depth":6,"x":244.331,"y":1621.009,"cluster":"algebraic-spaces"},{"id":"stacks:02WQ","tag":"02WQ","title":"Glueing algebraic spaces · Lemma 02WQ","summary":"Let S ∈ Ob(Sch_fppf). Suppose given a set I and algebraic spaces F_i, i ∈ I. Then F = coprod_i ∈ I F_i is an algebraic space provided I, and the F_i are not too \"large\": for example if we can choose surjective étale morphisms U_i → F_i such that coprod_i ∈ I U_i is isomorphic to an object of (Sch/S)_fppf, then F is an algebraic space.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$.\nSuppose given a set $I$ and algebraic spaces $F_i$, $i \\in I$.\nThen $F = \\coprod_{i \\in I} F_i$ is an algebraic space\nprovided $I$, and the $F_i$ are not too ``large'': for example if we\ncan choose surjective \\'etale morphisms $U_i \\to F_i$ such that\n$\\coprod_{i \\in I} U_i$ is isomorphic to an object of\n$(\\Sch/S)_{fppf}$, then $F$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Glueing algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WQ","source_file":"spaces.tex","source_line":1059,"source_end_line":1068,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1059-L1068","statement_sha256":"2e51cd4d87a94af0d8811ee4000f3eaa471ee5a55ae6fa463c8487297e95b727","origin":"The Stacks Project","memory_eligible":false,"source_rank":10701,"rank":10701,"depth":0,"x":201.983,"y":1592.492,"cluster":"algebraic-spaces"},{"id":"stacks:02WR","tag":"02WR","title":"Glueing algebraic spaces · Lemma 02WR","summary":"Let S ∈ Ob(Sch_fppf). Let F be a presheaf of sets on (Sch/S)_fppf. Assume • F is a sheaf, • there exists an index set I and subfunctors F_i ⊂ F such that • each F_i is an algebraic space, • each F_i → F is representable, • each F_i → F is an open immersion (see Definition [Tag 025V]), • the map coprod F_i → F is surjective as a map of sheaves, and • coprod F_i is an algebraic space (set theoretic condition, see Lemma [Tag 02WQ]). Then F is an algebraic space.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$.\nLet $F$ be a presheaf of sets on $(\\Sch/S)_{fppf}$.\nAssume\n\\begin{enumerate}\n\\item $F$ is a sheaf,\n\\item there exists an index set $I$\nand subfunctors $F_i \\subset F$ such that\n\\begin{enumerate}\n\\item each $F_i$ is an algebraic space,\n\\item each $F_i \\to F$ is representable,\n\\item each $F_i \\to F$ is an open immersion (see\nDefinition \\ref{definition-relative-representable-property}),\n\\item the map $\\coprod F_i \\to F$ is surjective as a map of sheaves, and\n\\item $\\coprod F_i$ is an algebraic space (set theoretic condition, see\nLemma \\ref{lemma-coproduct-algebraic-spaces}).\n\\end{enumerate}\n\\end{enumerate}\nThen $F$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Glueing algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WR","source_file":"spaces.tex","source_line":1081,"source_end_line":1101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1081-L1101","statement_sha256":"e5c3d8a2ba77290f35e1a96fd576012c81405b95b7e38d00156ec6516766c5be","origin":"The Stacks Project","memory_eligible":false,"source_rank":10702,"rank":10702,"depth":5,"x":257.215,"y":1589.438,"cluster":"algebraic-spaces"},{"id":"stacks:0262","tag":"0262","title":"Presentations of algebraic spaces · Lemma 0262","summary":"Let F be an algebraic space over S. Let f : U → F be a surjective étale morphism from a scheme to F. Set R = U ×_F U. Then • j : R → U ×_S U defines an equivalence relation on U over S (see Groupoids, Definition [Tag 043C]). • the morphisms s, t : R → U are étale, and • the diagram xymatrix R ar@<1ex>[r] ar@<-1ex>[r] & U ar[r] & F is a coequalizer diagram in Sh((Sch/S)_fppf).","statement_latex":"Let $F$ be an algebraic space over $S$. Let $f : U \\to F$ be a\nsurjective \\'etale morphism from a scheme to $F$. Set $R = U \\times_F U$.\nThen\n\\begin{enumerate}\n\\item $j : R \\to U \\times_S U$ defines an equivalence relation on\n$U$ over $S$ (see\nGroupoids, Definition \\ref{groupoids-definition-equivalence-relation}).\n\\item the morphisms $s, t : R \\to U$ are \\'etale, and\n\\item the diagram\n$$\n\\xymatrix{\nR \\ar@<1ex>[r] \\ar@<-1ex>[r] &\nU \\ar[r] &\nF\n}\n$$\nis a coequalizer diagram in $\\Sh((\\Sch/S)_{fppf})$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Presentations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0262","source_file":"spaces.tex","source_line":1170,"source_end_line":1190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1170-L1190","statement_sha256":"8be26aa4574fa193f7a56f0671ad9a07f56a6366b7d1cb5cb9c9847328f7b541","origin":"The Stacks Project","memory_eligible":false,"source_rank":10703,"rank":10703,"depth":2,"x":218.21,"y":1623.664,"cluster":"algebraic-spaces"},{"id":"stacks:02WS","tag":"02WS","title":"Presentations of algebraic spaces · Definition 02WS","summary":"Let S be a scheme. Let U be a scheme over S. An étale equivalence relation on U over S is an equivalence relation j : R → U ×_S U such that s, t : R → U are étale morphisms of schemes.","statement_latex":"Let $S$ be a scheme. Let $U$ be a scheme over $S$.\nAn {\\it \\'etale equivalence relation} on $U$ over $S$\nis an equivalence relation $j : R \\to U \\times_S U$\nsuch that $s, t : R \\to U$ are \\'etale morphisms of schemes.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Presentations of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WS","source_file":"spaces.tex","source_line":1215,"source_end_line":1221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1215-L1221","statement_sha256":"6db7201ce3674e7948a74bc6d313a7ac486f64a7d0d9452f65002d4caeab3922","origin":"The Stacks Project","memory_eligible":false,"source_rank":10704,"rank":10704,"depth":0,"x":219.478,"y":1575.426,"cluster":"algebraic-spaces"},{"id":"stacks:0263","tag":"0263","title":"Presentations of algebraic spaces · Definition 0263","summary":"Let F be an algebraic space over S. A presentation of F is given by a scheme U over S and an étale equivalence relation R on U over S, and a surjective étale morphism U → F such that R = U ×_F U.","statement_latex":"Let $F$ be an algebraic space over $S$.\nA {\\it presentation} of $F$ is given by a scheme\n$U$ over $S$ and an \\'etale equivalence relation $R$ on $U$ over $S$, and\na surjective \\'etale morphism $U \\to F$ such that $R = U \\times_F U$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Presentations of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0263","source_file":"spaces.tex","source_line":1223,"source_end_line":1229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1223-L1229","statement_sha256":"315a1ff641b436bcac72094968f44f277c7b982e7c5054e571e6307151259db6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10705,"rank":10705,"depth":0,"x":257.999,"y":1612.361,"cluster":"algebraic-spaces"},{"id":"stacks:02WT","tag":"02WT","title":"Algebraic spaces and equivalence relations · Lemma 02WT","summary":"Let S be a scheme. Let U be a scheme over S. Let j = (s, t) : R → U ×_S U be an étale equivalence relation on U over S. Let U' → U be an étale morphism. Let R' be the restriction of R to U', see Groupoids, Definition [Tag 043E]. Then j' : R' → U' ×_S U' is an étale equivalence relation also.","statement_latex":"Let $S$ be a scheme. Let $U$ be a scheme over $S$.\nLet $j = (s, t) : R \\to U \\times_S U$\nbe an \\'etale equivalence relation on $U$ over $S$.\nLet $U' \\to U$ be an \\'etale morphism.\nLet $R'$ be the restriction of $R$ to $U'$, see\nGroupoids, Definition \\ref{groupoids-definition-restrict-relation}.\nThen $j' : R' \\to U' \\times_S U'$ is an \\'etale equivalence\nrelation also.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces and equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WT","source_file":"spaces.tex","source_line":1283,"source_end_line":1293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1283-L1293","statement_sha256":"d630c92fc9ae58f4cc2a8c449711bcc4afc77136765c87c12c800c7ce6dc4df0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10706,"rank":10706,"depth":0,"x":198.9,"y":1606.886,"cluster":"algebraic-spaces"},{"id":"stacks:02WU","tag":"02WU","title":"Algebraic spaces and equivalence relations · Lemma 02WU","summary":"Let S be a scheme. Let U be a scheme over S. Let j = (s, t) : R → U ×_S U be a pre-relation. Let g : U' → U be a morphism. Assume • j is an equivalence relation, • s, t : R → U are surjective, flat and locally of finite presentation, • g is flat and locally of finite presentation. Let R' = R|_U' be the restriction of R to U'. Then U'/R' → U/R is representable, and is an open immersion.","statement_latex":"Let $S$ be a scheme.\nLet $U$ be a scheme over $S$.\nLet $j = (s, t) : R \\to U \\times_S U$ be a pre-relation.\nLet $g : U' \\to U$ be a morphism.\nAssume\n\\begin{enumerate}\n\\item $j$ is an equivalence relation,\n\\item $s, t : R \\to U$ are surjective, flat and\nlocally of finite presentation,\n\\item $g$ is flat and locally of finite presentation.\n\\end{enumerate}\nLet $R' = R|_{U'}$ be the restriction of $R$ to $U'$. Then\n$U'/R' \\to U/R$ is representable, and is an open immersion.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces and equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WU","source_file":"spaces.tex","source_line":1309,"source_end_line":1324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1309-L1324","statement_sha256":"7cc5b674ffa1d217255bd90f15034080d3ccc80623d96e09e46c0acc98b0fb46","origin":"The Stacks Project","memory_eligible":false,"source_rank":10707,"rank":10707,"depth":0,"x":247.677,"y":1576.906,"cluster":"algebraic-spaces"},{"id":"stacks:02WV","tag":"02WV","title":"Algebraic spaces and equivalence relations · Lemma 02WV","summary":"Let S be a scheme. Let U be a scheme over S. Let j = (s, t) : R → U ×_S U be an étale equivalence relation on U over S. If the quotient U/R is an algebraic space, then U → U/R is étale and surjective. Hence (U, R, U → U/R) is a presentation of the algebraic space U/R.","statement_latex":"Let $S$ be a scheme. Let $U$ be a scheme over $S$.\nLet $j = (s, t) : R \\to U \\times_S U$\nbe an \\'etale equivalence relation on $U$ over $S$.\nIf the quotient $U/R$ is an algebraic space, then\n$U \\to U/R$ is \\'etale and surjective. Hence\n$(U, R, U \\to U/R)$ is a presentation of the algebraic\nspace $U/R$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces and equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WV","source_file":"spaces.tex","source_line":1430,"source_end_line":1439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1430-L1439","statement_sha256":"5a4a6c601767f29360b8783a895aa104198e151aeaa229ee6263c1173112fc7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10708,"rank":10708,"depth":0,"x":235.623,"y":1627.485,"cluster":"algebraic-spaces"},{"id":"stacks:0265","tag":"0265","title":"Algebraic spaces and equivalence relations · Lemma 0265","summary":"Let S be a scheme. Let U be a scheme over S. Let j = (s, t) : R → U ×_S U be an étale equivalence relation on U over S. Assume that U is affine. Then the quotient F = U/R is an algebraic space, and U → F is étale and surjective.","statement_latex":"Let $S$ be a scheme.\nLet $U$ be a scheme over $S$.\nLet $j = (s, t) : R \\to U \\times_S U$\nbe an \\'etale equivalence relation on $U$ over $S$.\nAssume that $U$ is affine. Then the quotient $F = U/R$\nis an algebraic space, and $U \\to F$ is \\'etale and surjective.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces and equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0265","source_file":"spaces.tex","source_line":1465,"source_end_line":1473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1465-L1473","statement_sha256":"a20d2ef431fd0ad3fc91fa34a79492db59d2c1c9a4a7882bfec20c5085bd99ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":10709,"rank":10709,"depth":50,"x":203.351,"y":1582.663,"cluster":"algebraic-spaces"},{"id":"stacks:02WW","tag":"02WW","title":"Algebraic spaces and equivalence relations · Theorem 02WW","summary":"Let S be a scheme. Let U be a scheme over S. Let j = (s, t) : R → U ×_S U be an étale equivalence relation on U over S. Then the quotient U/R is an algebraic space, and U → U/R is étale and surjective, in other words (U, R, U → U/R) is a presentation of U/R.","statement_latex":"Let $S$ be a scheme. Let $U$ be a scheme over $S$.\nLet $j = (s, t) : R \\to U \\times_S U$\nbe an \\'etale equivalence relation on $U$ over $S$.\nThen the quotient $U/R$ is an algebraic space,\nand $U \\to U/R$ is \\'etale and surjective, in other words\n$(U, R, U \\to U/R)$ is a presentation of $U/R$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces and equivalence relations","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WW","source_file":"spaces.tex","source_line":1602,"source_end_line":1610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1602-L1610","statement_sha256":"a62cd186e75042e2dca7b3ed7a71e1cf78d07ec69f69ec98cd8f5135bfd18e3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10710,"rank":10710,"depth":51,"x":264.089,"y":1597.628,"cluster":"algebraic-spaces"},{"id":"stacks:0BGQ","tag":"0BGQ","title":"Algebraic spaces, retrofitted · Lemma 0BGQ","summary":"Let S be a scheme contained in Sch_fppf. Let F be a sheaf on (Sch/S)_fppf such that there exists U ∈ Ob((Sch/S)_fppf) and a map U → F which is representable, surjective, and étale. Then F is an algebraic space.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F$ be a sheaf on $(\\Sch/S)_{fppf}$\nsuch that there exists $U \\in \\Ob((\\Sch/S)_{fppf})$ and a map\n$U \\to F$ which is representable, surjective, and \\'etale.\nThen $F$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces, retrofitted","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGQ","source_file":"spaces.tex","source_line":1667,"source_end_line":1674,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1667-L1674","statement_sha256":"7301c6df230c7b42b655cb1d36163bf7f11f5b6e500cd2efc9308280e8012b90","origin":"The Stacks Project","memory_eligible":false,"source_rank":10711,"rank":10711,"depth":52,"x":206.437,"y":1621.395,"cluster":"algebraic-spaces"},{"id":"stacks:0BGR","tag":"0BGR","title":"Algebraic spaces, retrofitted · Lemma 0BGR","summary":"Let S be a scheme contained in Sch_fppf. Let G be an algebraic space over S, let F be a sheaf on (Sch/S)_fppf, and let G → F be a representable transformation of functors which is surjective and étale. Then F is an algebraic space.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$. Let $G$ be an algebraic\nspace over $S$, let $F$ be a sheaf on $(\\Sch/S)_{fppf}$, and let\n$G \\to F$ be a representable transformation of functors which is\nsurjective and \\'etale. Then $F$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces, retrofitted","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGR","source_file":"spaces.tex","source_line":1690,"source_end_line":1696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1690-L1696","statement_sha256":"bf3aae81b9b44d6c58fb2f370e13c09a8ef4035a7e38ff74a692f0dc4479e6c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10712,"rank":10712,"depth":53,"x":230.172,"y":1570.446,"cluster":"algebraic-spaces"},{"id":"stacks:02WY","tag":"02WY","title":"Algebraic spaces, retrofitted · Lemma 02WY","summary":"A functor that admits a representable morphism to an algebraic space is an algebraic space. Let S be a scheme contained in Sch_fppf. Let F be an algebraic space over S. Let G → F be a representable transformation of functors. Then G is an algebraic space.","statement_latex":"\\begin{slogan}\nA functor that admits a representable morphism to an algebraic space is\nan algebraic space.\n\\end{slogan}\nLet $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F$ be an algebraic space over $S$.\nLet $G \\to F$ be a representable transformation of functors.\nThen $G$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces, retrofitted","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WY","source_file":"spaces.tex","source_line":1709,"source_end_line":1719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1709-L1719","statement_sha256":"036d2e52b822eb9f46389b5257e156dee961cbaa865e17dbedad0b01efee4077","origin":"The Stacks Project","memory_eligible":false,"source_rank":10713,"rank":10713,"depth":6,"x":253.961,"y":1622.187,"cluster":"algebraic-spaces"},{"id":"stacks:02WZ","tag":"02WZ","title":"Algebraic spaces, retrofitted · Lemma 02WZ","summary":"Let S be a scheme contained in Sch_fppf. Let F, G be algebraic spaces over S. Let G → F be a representable morphism. Let U ∈ Ob((Sch/S)_fppf), and q : U → F surjective and étale. Set V = G ×_F U. Finally, let P be a property of morphisms of schemes as in Definition [Tag 025V]. Then G → F has property P if and only if V → U has property P.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F$, $G$ be algebraic spaces over $S$.\nLet $G \\to F$ be a representable morphism.\nLet $U \\in \\Ob((\\Sch/S)_{fppf})$, and $q : U \\to F$\nsurjective and \\'etale. Set $V = G \\times_F U$.\nFinally, let $\\mathcal{P}$ be a property of morphisms\nof schemes as in Definition \\ref{definition-relative-representable-property}.\nThen $G \\to F$ has property $\\mathcal{P}$ if and only if\n$V \\to U$ has property $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces, retrofitted","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02WZ","source_file":"spaces.tex","source_line":1747,"source_end_line":1758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1747-L1758","statement_sha256":"61f0dd083ebe67a7950bcd0afa5eac9171dfae35de1dcad8206a9e74e5a0ac54","origin":"The Stacks Project","memory_eligible":false,"source_rank":10714,"rank":10714,"depth":3,"x":194.019,"y":1597.199,"cluster":"algebraic-spaces"},{"id":"stacks:03I2","tag":"03I2","title":"Algebraic spaces, retrofitted · Lemma 03I2","summary":"Let S be a scheme contained in Sch_fppf. Let G → F be a transformation of presheaves on (Sch/S)_fppf. Let P be a property of morphisms of schemes. Assume • P is preserved under any base change, fppf local on the base, and morphisms of type P satisfy descent for fppf coverings, see Descent, Definition [Tag 02W2], • G is a sheaf, • F is an algebraic space, • there exists a U ∈ Ob((Sch/S)_fppf) and a surjective étale morphism U → F such that V = G ×_F U is representable, and…","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $G \\to F$ be a transformation of presheaves on $(\\Sch/S)_{fppf}$.\nLet $\\mathcal{P}$ be a property of morphisms of schemes.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{P}$ is preserved under any base change, fppf local on the\nbase, and morphisms of type $\\mathcal{P}$ satisfy descent for fppf coverings,\nsee Descent, Definition \\ref{descent-definition-descending-types-morphisms},\n\\item $G$ is a sheaf,\n\\item $F$ is an algebraic space,\n\\item there exists a $U \\in \\Ob((\\Sch/S)_{fppf})$\nand a surjective \\'etale morphism $U \\to F$ such that\n$V = G \\times_F U$ is representable, and\n\\item $V \\to U$ has $\\mathcal{P}$.\n\\end{enumerate}\nThen $G$ is an algebraic space, $G \\to F$ is representable and has property\n$\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces, retrofitted","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03I2","source_file":"spaces.tex","source_line":1787,"source_end_line":1806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1787-L1806","statement_sha256":"fab56bc9c429f7b6538415620ce802b042a2c400eff90d48d66333560a1c298f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10715,"rank":10715,"depth":7,"x":259.155,"y":1581.413,"cluster":"algebraic-spaces"},{"id":"stacks:02X1","tag":"02X1","title":"Algebraic spaces, retrofitted · Lemma 02X1","summary":"Let S be a scheme contained in Sch_fppf. Let F, G be algebraic spaces over S. Let a : F → G be a morphism. Given any V ∈ Ob((Sch/S)_fppf) and a surjective étale morphism q : V → G there exists a U ∈ Ob((Sch/S)_fppf) and a commutative diagram xymatrix U ar[d]_p ar[r]_α & V ar[d]^q F ar[r]^a & G with p surjective and étale.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G$ be algebraic spaces over $S$.\nLet $a : F \\to G$ be a morphism.\nGiven any $V \\in \\Ob((\\Sch/S)_{fppf})$\nand a surjective \\'etale morphism $q : V \\to G$ there exists\na $U \\in \\Ob((\\Sch/S)_{fppf})$\nand a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_p \\ar[r]_\\alpha &\nV \\ar[d]^q \\\\\nF \\ar[r]^a & G\n}\n$$\nwith $p$ surjective and \\'etale.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Algebraic spaces, retrofitted","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02X1","source_file":"spaces.tex","source_line":1847,"source_end_line":1864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1847-L1864","statement_sha256":"18e6577021199b30d79b166118db50c754fbe55311fa8c7f3f36d7cba600d083","origin":"The Stacks Project","memory_eligible":false,"source_rank":10716,"rank":10716,"depth":3,"x":223.367,"y":1630.629,"cluster":"algebraic-spaces"},{"id":"stacks:02YU","tag":"02YU","title":"Immersions and Zariski coverings of algebraic spaces · Definition 02YU","summary":"Let S ∈ Ob(Sch_fppf) be a scheme. Let F be an algebraic space over S. • A morphism of algebraic spaces over S is called an open immersion if it is representable, and an open immersion in the sense of Definition [Tag 025V]. • An open subspace of F is a subfunctor F' ⊂ F such that F' is an algebraic space and F' → F is an open immersion. • A morphism of algebraic spaces over S is called a closed immersion if it is representable, and a closed immersion in the sense of…","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.\nLet $F$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item A morphism of algebraic spaces over $S$\nis called an {\\it open immersion} if it is representable, and an open immersion\nin the sense of Definition \\ref{definition-relative-representable-property}.\n\\item An {\\it open subspace} of $F$ is a subfunctor $F' \\subset F$\nsuch that $F'$ is an algebraic space and $F' \\to F$ is an\nopen immersion.\n\\item A morphism of algebraic spaces over $S$\nis called a {\\it closed immersion} if it is representable, and a closed\nimmersion in the sense of\nDefinition \\ref{definition-relative-representable-property}.\n\\item A {\\it closed subspace} of $F$ is a subfunctor $F' \\subset F$\nsuch that $F'$ is an algebraic space and $F' \\to F$ is a\nclosed immersion.\n\\item A morphism of algebraic spaces over $S$\nis called an {\\it immersion} if it is representable, and an immersion\nin the sense of Definition \\ref{definition-relative-representable-property}.\n\\item A {\\it locally closed subspace} of $F$ is a subfunctor $F' \\subset F$\nsuch that $F'$ is an algebraic space and $F' \\to F$ is an\nimmersion.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Immersions and Zariski coverings of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YU","source_file":"spaces.tex","source_line":1923,"source_end_line":1948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1923-L1948","statement_sha256":"f2df9c90b0e4fb301f8cef7eead40185146456a598a58006ea288854507cfab9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10717,"rank":10717,"depth":2,"x":210.019,"y":1573.328,"cluster":"algebraic-spaces"},{"id":"stacks:02YV","tag":"02YV","title":"Immersions and Zariski coverings of algebraic spaces · Lemma 02YV","summary":"Let S ∈ Ob(Sch_fppf) be a scheme. A composition of (closed, resp. open) immersions of algebraic spaces over S is a (closed, resp. open) immersion of algebraic spaces over S.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.\nA composition of (closed, resp.\\ open) immersions of\nalgebraic spaces over $S$ is a (closed, resp.\\ open)\nimmersion of algebraic spaces over $S$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Immersions and Zariski coverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YV","source_file":"spaces.tex","source_line":1961,"source_end_line":1967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1961-L1967","statement_sha256":"a56d31d8a0e439cb0a390c46158b83048366c05c93d3c9a7587ededd9e71276d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10718,"rank":10718,"depth":3,"x":266.615,"y":1608.429,"cluster":"algebraic-spaces"},{"id":"stacks:02YW","tag":"02YW","title":"Immersions and Zariski coverings of algebraic spaces · Lemma 02YW","summary":"Let S ∈ Ob(Sch_fppf) be a scheme. A base change of a (closed, resp. open) immersion of algebraic spaces over S is a (closed, resp. open) immersion of algebraic spaces over S.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.\nA base change of a (closed, resp.\\ open) immersion\nof algebraic spaces over $S$ is a (closed, resp.\\ open)\nimmersion of algebraic spaces over $S$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Immersions and Zariski coverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YW","source_file":"spaces.tex","source_line":1975,"source_end_line":1981,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1975-L1981","statement_sha256":"c5001a9afe520a2a1064fdfd42f57a6d2e77515c54dc3e14fd428f925fa34dfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10719,"rank":10719,"depth":3,"x":195.828,"y":1614.731,"cluster":"algebraic-spaces"},{"id":"stacks:02YX","tag":"02YX","title":"Immersions and Zariski coverings of algebraic spaces · Lemma 02YX","summary":"Let S ∈ Ob(Sch_fppf) be a scheme. Let F be an algebraic space over S. Let F_1, F_2 be locally closed subspaces of F. If F_1 ⊂ F_2 as subfunctors of F, then F_1 is a locally closed subspace of F_2. Similarly for closed and open subspaces.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.\nLet $F$ be an algebraic space over $S$. Let $F_1$, $F_2$ be\nlocally closed subspaces of $F$. If $F_1 \\subset F_2$ as subfunctors\nof $F$, then $F_1$ is a locally closed subspace of $F_2$.\nSimilarly for closed and open subspaces.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Immersions and Zariski coverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YX","source_file":"spaces.tex","source_line":1989,"source_end_line":1996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L1989-L1996","statement_sha256":"4cc9539ec784878447aeb67d0d8fa7035ad3d0af30fef8124bd2baaca9da2e0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10720,"rank":10720,"depth":0,"x":243.504,"y":1569.402,"cluster":"algebraic-spaces"},{"id":"stacks:02YY","tag":"02YY","title":"Immersions and Zariski coverings of algebraic spaces · Definition 02YY","summary":"Let S ∈ Ob(Sch_fppf) be a scheme. Let F be an algebraic space over S. A Zariski covering (F_i ⊂ F)_i ∈ I of F is given by a set I and a collection of open subspaces F_i ⊂ F such that coprod F_i → F is a surjective map of sheaves.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme.\nLet $F$ be an algebraic space over $S$.\nA {\\it Zariski covering} $\\{F_i \\subset F\\}_{i \\in I}$ of $F$\nis given by a set $I$ and a collection of open subspaces\n$F_i \\subset F$ such that $\\coprod F_i \\to F$ is a surjective\nmap of sheaves.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Immersions and Zariski coverings of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YY","source_file":"spaces.tex","source_line":2011,"source_end_line":2019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2011-L2019","statement_sha256":"e58d21d0a73dece289ba619f99a8988d016684b563ee490b03f841b99546e13a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10721,"rank":10721,"depth":0,"x":244.811,"y":1630.562,"cluster":"algebraic-spaces"},{"id":"stacks:02YZ","tag":"02YZ","title":"Immersions and Zariski coverings of algebraic spaces · Definition 02YZ","summary":"Let S ∈ Ob(Sch_fppf) be a scheme. Let F be an algebraic space over S. A small Zariski site F_Zar of an algebraic space F is one of the sites described above.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$ be a scheme. Let $F$ be an algebraic space over\n$S$. A {\\it small Zariski site $F_{Zar}$} of an algebraic space $F$ is one\nof the sites described above.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Immersions and Zariski coverings of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YZ","source_file":"spaces.tex","source_line":2041,"source_end_line":2046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2041-L2046","statement_sha256":"d30da787fe07612c111691e842af3c982a1daa370c512ba80cbebeae8608f7a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10722,"rank":10722,"depth":0,"x":194.114,"y":1585.714,"cluster":"algebraic-spaces"},{"id":"stacks:02X4","tag":"02X4","title":"Separation conditions on algebraic spaces · Lemma 02X4","summary":"Let S be a scheme contained in Sch_fppf. Let F be an algebraic space over S. Let Δ : F → F × F be the diagonal morphism. Then • Δ is locally of finite type, • Δ is a monomorphism, • Δ is separated, and • Δ is locally quasi-finite.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F$ be an algebraic space over $S$.\nLet $\\Delta : F \\to F \\times F$ be the diagonal morphism.\nThen\n\\begin{enumerate}\n\\item $\\Delta$ is locally of finite type,\n\\item $\\Delta$ is a monomorphism,\n\\item $\\Delta$ is separated, and\n\\item $\\Delta$ is locally quasi-finite.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Separation conditions on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02X4","source_file":"spaces.tex","source_line":2079,"source_end_line":2091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2079-L2091","statement_sha256":"f0ab64b21bf92d4b2ff62bfcc26b45858d63d3c2aca84062ed20414e5fac3b1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10723,"rank":10723,"depth":51,"x":268.355,"y":1590.067,"cluster":"algebraic-spaces"},{"id":"stacks:02X5","tag":"02X5","title":"Separation conditions on algebraic spaces · Definition 02X5","summary":"Let S be a scheme contained in Sch_fppf. Let F be an algebraic space over S. Let Δ : F → F × F be the diagonal morphism. • We say F is separated over S if Δ is a closed immersion. • We say F is locally separated over S if Δ is an immersion. • We say F is quasi-separated over S if Δ is quasi-compact. • We say F is Zariski locally quasi-separated over S if there exists a Zariski covering F = ⋃_i ∈ I F_i such that each F_i is quasi-separated.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F$ be an algebraic space over $S$.\nLet $\\Delta : F \\to F \\times F$ be the diagonal morphism.\n\\begin{enumerate}\n\\item We say $F$ is {\\it separated over $S$} if $\\Delta$ is a closed immersion.\n\\item We say $F$ is {\\it locally separated over $S$}\\footnote{In the\nliterature this often refers to quasi-separated and\nlocally separated algebraic spaces.} if $\\Delta$ is an\nimmersion.\n\\item We say $F$ is {\\it quasi-separated over $S$} if $\\Delta$ is quasi-compact.\n\\item We say $F$ is {\\it Zariski locally quasi-separated over $S$}\\footnote{This\ndefinition was suggested by B.\\ Conrad.} if there\nexists a Zariski covering $F = \\bigcup_{i \\in I} F_i$ such that\neach $F_i$ is quasi-separated.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Separation conditions on algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02X5","source_file":"spaces.tex","source_line":2129,"source_end_line":2146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2129-L2146","statement_sha256":"2b924479b345749075eaf9ac91d7d0a6f4f59c8e4eb9531b344609684d29edf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10724,"rank":10724,"depth":0,"x":209.494,"y":1629.392,"cluster":"algebraic-spaces"},{"id":"stacks:02Z2","tag":"02Z2","title":"Examples of algebraic spaces · Lemma 02Z2","summary":"Let U → S be a morphism of Sch_fppf. Let G be an abstract group. Let G → Aut_S(U) be a group homomorphism. Assume • [(*)] if u ∈ U is a point, and g(u) = u for some non-identity element g ∈ G, then g induces a nontrivial automorphism of kappa(u). Then j : R = coprod_g ∈ G U → U ×_S U, (g, x) ↦ (g(x), x) is an étale equivalence relation and hence F = U/R is an algebraic space by Theorem [Tag 02WW].","statement_latex":"Let $U \\to S$ be a morphism of $\\Sch_{fppf}$.\nLet $G$ be an abstract group. Let $G \\to \\text{Aut}_S(U)$\nbe a group homomorphism. Assume\n\\begin{itemize}\n\\item[(*)] if $u \\in U$ is a point, and $g(u) = u$\nfor some non-identity element $g \\in G$, then $g$\ninduces a nontrivial automorphism of $\\kappa(u)$.\n\\end{itemize}\nThen\n$$\nj :\nR = \\coprod\\nolimits_{g \\in G} U\n\\longrightarrow\nU \\times_S U,\n\\quad\n(g, x) \\longmapsto (g(x), x)\n$$\nis an \\'etale equivalence relation and hence\n$$\nF = U/R\n$$\nis an algebraic space by Theorem \\ref{theorem-presentation}.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Examples of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Z2","source_file":"spaces.tex","source_line":2246,"source_end_line":2270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2246-L2270","statement_sha256":"868767bd57463dc9fbe1caea1d77ed0b68e9e6220bd31cdaf80c1e5878151bd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10725,"rank":10725,"depth":52,"x":221.398,"y":1566.35,"cluster":"algebraic-spaces"},{"id":"stacks:02Z3","tag":"02Z3","title":"Examples of algebraic spaces · Definition 02Z3","summary":"Notation U → S, G, R as in Lemma [Tag 02Z2]. If the action of G on U satisfies (*) we say G acts freely on the scheme U. In this case the algebraic space U/R is denoted U/G and is called the quotient of U by G.","statement_latex":"Notation $U \\to S$, $G$, $R$ as in Lemma \\ref{lemma-quotient}.\nIf the action of $G$ on $U$ satisfies $(*)$ we say $G$ {\\it acts freely}\non the scheme $U$. In this case the algebraic space $U/R$ is denoted\n$U/G$ and is called the {\\it quotient of $U$ by $G$}.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Examples of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Z3","source_file":"spaces.tex","source_line":2313,"source_end_line":2319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2313-L2319","statement_sha256":"b2d8025a4b6eb40a55fd8dfd64a68597fa5c3144a347de85d94ac6e9b3efc8b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10726,"rank":10726,"depth":53,"x":263.739,"y":1620.132,"cluster":"algebraic-spaces"},{"id":"stacks:02Z4","tag":"02Z4","title":"Examples of algebraic spaces · Lemma 02Z4","summary":"Notation and assumptions as in Lemma [Tag 02Z2]. Assume G is finite. Then • if U → S is quasi-separated, then U/G is quasi-separated over S, and • if U → S is separated, then U/G is separated over S.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-quotient}.\nAssume $G$ is finite. Then\n\\begin{enumerate}\n\\item if $U \\to S$ is quasi-separated, then $U/G$ is quasi-separated\nover $S$, and\n\\item if $U \\to S$ is separated, then $U/G$ is separated over $S$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Examples of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Z4","source_file":"spaces.tex","source_line":2330,"source_end_line":2339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2330-L2339","statement_sha256":"8d0c6a4ff878f7a1425ee634669bb135b241122a6cc255e339b547c0e6be1331","origin":"The Stacks Project","memory_eligible":false,"source_rank":10727,"rank":10727,"depth":53,"x":188.528,"y":1604.342,"cluster":"algebraic-spaces"},{"id":"stacks:02Z5","tag":"02Z5","title":"Examples of algebraic spaces · Lemma 02Z5","summary":"Notation and assumptions as in Lemma [Tag 02Z2]. If Spec(k) → U/G is a morphism, then there exist • a finite Galois extension k'/k, • a finite subgroup H ⊂ G, • an isomorphism H → Gal(k'/k), and • an H-equivariant morphism Spec(k') → U. Conversely, such data determine a morphism Spec(k) → U/G.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-quotient}.\nIf $\\Spec(k) \\to U/G$ is a morphism, then there exist\n\\begin{enumerate}\n\\item a finite Galois extension $k'/k$,\n\\item a finite subgroup $H \\subset G$,\n\\item an isomorphism $H \\to \\text{Gal}(k'/k)$, and\n\\item an $H$-equivariant morphism $\\Spec(k') \\to U$.\n\\end{enumerate}\nConversely, such data determine a morphism $\\Spec(k) \\to U/G$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Examples of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02Z5","source_file":"spaces.tex","source_line":2354,"source_end_line":2365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2354-L2365","statement_sha256":"adca858e549f94b56b9b02eb4dcff416c561f1990bf47c1fd756a42cb1e3cc4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10728,"rank":10728,"depth":53,"x":257.35,"y":1573.009,"cluster":"algebraic-spaces"},{"id":"stacks:03FP","tag":"03FP","title":"Change of big site · Lemma 03FP","summary":"Suppose given big sites Sch_fppf and Sch'_fppf. Assume that Sch_fppf is contained in Sch'_fppf, see Topologies, Section [Tag 022I]. Let S be an object of Sch_fppf. Let g : Sh((Sch/S)_fppf) → Sh((Sch'/S)_fppf), f : Sh((Sch'/S)_fppf) → Sh((Sch/S)_fppf) be the morphisms of topoi of Topologies, Lemma [Tag 022K]. Let F be a sheaf of sets on (Sch/S)_fppf. Then • if F is representable by a scheme X ∈ Ob((Sch/S)_fppf) over S, then f^-1F is representable too, in fact it is…","statement_latex":"Suppose given big sites $\\Sch_{fppf}$ and $\\Sch'_{fppf}$.\nAssume that $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$,\nsee Topologies, Section \\ref{topologies-section-change-alpha}.\nLet $S$ be an object of $\\Sch_{fppf}$. Let\n\\begin{align*}\ng : \\Sh((\\Sch/S)_{fppf})\n\\longrightarrow\n\\Sh((\\Sch'/S)_{fppf}), \\\\\nf : \\Sh((\\Sch'/S)_{fppf})\n\\longrightarrow\n\\Sh((\\Sch/S)_{fppf})\n\\end{align*}\nbe the morphisms of topoi of\nTopologies, Lemma \\ref{topologies-lemma-change-alpha}.\nLet $F$ be a sheaf of sets on $(\\Sch/S)_{fppf}$. Then\n\\begin{enumerate}\n\\item if $F$ is representable by a scheme\n$X \\in \\Ob((\\Sch/S)_{fppf})$ over $S$,\nthen $f^{-1}F$ is representable too, in fact it is representable by the\nsame scheme $X$, now viewed as an object of $(\\Sch'/S)_{fppf}$, and\n\\item if $F$ is an algebraic space over $S$, then $f^{-1}F$ is an algebraic\nspace over $S$ also.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Change of big site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FP","source_file":"spaces.tex","source_line":2557,"source_end_line":2582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2557-L2582","statement_sha256":"00a45f6b3312765cf764dc682a62c0c349bee6df9dabba544bbb08bd9efe63b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10729,"rank":10729,"depth":52,"x":231.552,"y":1635.757,"cluster":"algebraic-spaces"},{"id":"stacks:04W1","tag":"04W1","title":"Change of big site · Lemma 04W1","summary":"Suppose Sch_fppf is contained in Sch'_fppf. Let S be an object of Sch_fppf. Denote Spaces/S the category of algebraic spaces over S defined using Sch_fppf. Similarly, denote Spaces'/S the category of algebraic spaces over S defined using Sch'_fppf. The construction of Lemma [Tag 03FP] defines a fully faithful functor Spaces/S → Spaces'/S whose essential image consists of those X' ∈ Ob(Spaces'/S) such that there exist U, R ∈ Ob((Sch/S)_fppf) U can grow faster than Bound in…","statement_latex":"Suppose $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$.\nLet $S$ be an object of $\\Sch_{fppf}$. Denote\n$\\textit{Spaces}/S$ the category of algebraic spaces over $S$\ndefined using $\\Sch_{fppf}$. Similarly, denote\n$\\textit{Spaces}'/S$ the category of algebraic spaces over $S$\ndefined using $\\Sch'_{fppf}$. The construction of\nLemma \\ref{lemma-change-big-site}\ndefines a fully faithful functor\n$$\n\\textit{Spaces}/S \\longrightarrow \\textit{Spaces}'/S\n$$\nwhose essential image consists of those $X' \\in \\Ob(\\textit{Spaces}'/S)$\nsuch that there exist $U, R \\in \\Ob((\\Sch/S)_{fppf})$\\footnote{Requiring the\nexistence of $R$ is necessary because of our choice of the function $Bound$ in\nSets, Equation (\\ref{sets-equation-bound}). The size of the fibre product\n$U \\times_{X'} U$ can grow faster than $Bound$ in terms of the size of $U$. We\ncan illustrate this by setting $S = \\Spec(A)$, $U = \\Spec(A[x_i, i \\in I])$ and\n$R = \\coprod_{(\\lambda_i) \\in A^I} \\Spec(A[x_i, y_i]/(x_i - \\lambda_i y_i))$.\nIn this case the size of $R$ grows like $\\kappa^\\kappa$ where $\\kappa$ is the\nsize of $U$.} and morphisms\n$$\nU \\longrightarrow X'\n\\quad\\text{and}\\quad\nR \\longrightarrow U \\times_{X'} U\n$$\nin $\\Sh((\\Sch'/S)_{fppf})$ which are surjective as maps of sheaves\n(for example if the displayed morphisms are surjective and \\'etale).","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Change of big site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04W1","source_file":"spaces.tex","source_line":2608,"source_end_line":2637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2608-L2637","statement_sha256":"2f07058745e5c6e11fecbc47a3884316948b7788ae54ca91c55f177d6c98607f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10730,"rank":10730,"depth":53,"x":199.823,"y":1574.278,"cluster":"algebraic-spaces"},{"id":"stacks:03I4","tag":"03I4","title":"Change of base scheme · Lemma 03I4","summary":"Suppose given a big site Sch_fppf. Let g : S → S' be morphism of Sch_fppf. Let j : (Sch/S)_fppf → (Sch/S')_fppf be the corresponding localization functor. Let F be a sheaf of sets on (Sch/S)_fppf. Then • for a scheme T' over S' we have j_!F(T'/S') = coprod_φ : T' → S F(T' xrightarrowφ S), • if F is representable by a scheme X ∈ Ob((Sch/S)_fppf), then j_!F is representable by j(X) which is X viewed as a scheme over S', and • if F is an algebraic space over S, then j_!F is…","statement_latex":"Suppose given a big site $\\Sch_{fppf}$.\nLet $g : S \\to S'$ be morphism of $\\Sch_{fppf}$.\nLet $j : (\\Sch/S)_{fppf} \\to (\\Sch/S')_{fppf}$ be\nthe corresponding localization functor.\nLet $F$ be a sheaf of sets on $(\\Sch/S)_{fppf}$.\nThen\n\\begin{enumerate}\n\\item for a scheme $T'$ over $S'$ we have\n$j_!F(T'/S') =\n\\coprod\\nolimits_{\\varphi : T' \\to S} F(T' \\xrightarrow{\\varphi} S),$\n\\item if $F$ is representable by a scheme\n$X \\in \\Ob((\\Sch/S)_{fppf})$,\nthen $j_!F$ is representable by $j(X)$ which is\n$X$ viewed as a scheme over $S'$, and\n\\item if $F$ is an algebraic space over $S$, then $j_!F$ is an algebraic\nspace over $S'$, and if $F = U/R$ is a presentation, then\n$j_!F = j(U)/j(R)$ is a presentation.\n\\end{enumerate}\nLet $F'$ be a sheaf of sets on $(\\Sch/S')_{fppf}$. Then\n\\begin{enumerate}\n\\item[(4)] for a scheme $T$ over $S$ we have $j^{-1}F'(T/S) = F'(T/S')$,\n\\item[(5)] if $F'$ is representable by a scheme\n$X' \\in \\Ob((\\Sch/S')_{fppf})$, then\n$j^{-1}F'$ is representable, namely by $X'_S = S \\times_{S'} X'$, and\n\\item[(6)] if $F'$ is an algebraic space, then\n$j^{-1}F'$ is an algebraic space, and if $F' = U'/R'$ is a presentation,\nthen $j^{-1}F' = U'_S/R'_S$ is a presentation.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03I4","source_file":"spaces.tex","source_line":2732,"source_end_line":2762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2732-L2762","statement_sha256":"b2ecb99a072c3721e5342a8a3b62d2d69b7a4f3c6347f16da9ba4220884cb5b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10731,"rank":10731,"depth":52,"x":273.329,"y":1601.861,"cluster":"algebraic-spaces"},{"id":"stacks:03I5","tag":"03I5","title":"Change of base scheme · Definition 03I5","summary":"Let Sch_fppf be a big fppf site. Let S → S' be a morphism of this site. • If F' is an algebraic space over S', then the base change of F' to S is the algebraic space j^-1F' described in Lemma [Tag 03I4]. We denote it F'_S. • If F is an algebraic space over S, then F viewed as an algebraic space over S' is the algebraic space j_!F over S' described in Lemma [Tag 03I4]. We often simply denote this F; if not then we will write j_!F.","statement_latex":"Let $\\Sch_{fppf}$ be a big fppf site.\nLet $S \\to S'$ be a morphism of this site.\n\\begin{enumerate}\n\\item If $F'$ is an algebraic space over $S'$, then the\n{\\it base change of $F'$ to $S$} is the\nalgebraic space $j^{-1}F'$ described in\nLemma \\ref{lemma-change-base-scheme}. We denote it $F'_S$.\n\\item If $F$ is an algebraic space over $S$, then $F$\n{\\it viewed as an algebraic space over $S'$}\nis the algebraic space $j_!F$ over $S'$ described in\nLemma \\ref{lemma-change-base-scheme}. We often simply denote this\n$F$; if not then we will write $j_!F$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Change of base scheme","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03I5","source_file":"spaces.tex","source_line":2815,"source_end_line":2830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2815-L2830","statement_sha256":"60af5eadeca5e57dee0a156a73cd28258952ec12dac999f3cd5dddbfd2acb748","origin":"The Stacks Project","memory_eligible":false,"source_rank":10732,"rank":10732,"depth":53,"x":196.254,"y":1623.421,"cluster":"algebraic-spaces"},{"id":"stacks:04SG","tag":"04SG","title":"Change of base scheme · Lemma 04SG","summary":"Let Sch_fppf be a big fppf site. Let S → S' be a morphism of this site. The construction above give an equivalence of categories ( category of algebraic spaces over S ) ↔ ( category of pairs (F', F' → S) consisting of an algebraic space F' over S' and a morphism F' → S of algebraic spaces over S' )","statement_latex":"Let $\\Sch_{fppf}$ be a big fppf site.\nLet $S \\to S'$ be a morphism of this site.\nThe construction above give an equivalence of\ncategories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{category of algebraic}\\\\\n\\text{spaces over }S\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{category of pairs }(F', F' \\to S)\\text{ consisting}\\\\\n\\text{of an algebraic space }F'\\text{ over }S'\\text{ and a}\\\\\n\\text{morphism }F' \\to S\\text{ of algebraic spaces over }S'\n\\end{matrix}\n\\right\\}\n$$","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SG","source_file":"spaces.tex","source_line":2848,"source_end_line":2870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2848-L2870","statement_sha256":"9f288345971261523ba7d39b76de5b446bce1639c7bae62fc46e44e9c54a0f75","origin":"The Stacks Project","memory_eligible":false,"source_rank":10733,"rank":10733,"depth":53,"x":236.102,"y":1563.262,"cluster":"algebraic-spaces"},{"id":"stacks:04SH","tag":"04SH","title":"Change of base scheme · Lemma 04SH","summary":"Let Sch_fppf be a big fppf site. Let S → S' be a morphism of this site. Let F' be a sheaf on (Sch/S')_fppf. The following are equivalent: • The restriction F'|_(Sch/S)_fppf is an algebraic space over S, and • the sheaf h_S × F' is an algebraic space over S'.","statement_latex":"Let $\\Sch_{fppf}$ be a big fppf site.\nLet $S \\to S'$ be a morphism of this site.\nLet $F'$ be a sheaf on $(\\Sch/S')_{fppf}$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The restriction $F'|_{(\\Sch/S)_{fppf}}$\nis an algebraic space over $S$, and\n\\item the sheaf $h_S \\times F'$ is an algebraic space over $S'$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SH","source_file":"spaces.tex","source_line":2901,"source_end_line":2912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2901-L2912","statement_sha256":"3d01cdef3b89ee050ddfd9a56ba4fe76b65e7abd769b454ce4c8c28ad59b90c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10734,"rank":10734,"depth":54,"x":255.261,"y":1630.817,"cluster":"algebraic-spaces"},{"id":"stacks:03I6","tag":"03I6","title":"Change of base scheme · Lemma 03I6","summary":"Let Sch_fppf be a big fppf site. Let S → S' be a morphism of this site. Let F be an algebraic space over S. Let T be a scheme over S and let f : T → F be a morphism over S. Let f' : T' → F' be the morphism over S' we get from f by applying the equivalence of categories described in Lemma [Tag 04SG]. For any property P as in Definition [Tag 025V] we have P(f') ⇔ P(f).","statement_latex":"Let $\\Sch_{fppf}$ be a big fppf site.\nLet $S \\to S'$ be a morphism of this site.\nLet $F$ be an algebraic space over $S$.\nLet $T$ be a scheme over $S$ and let $f : T \\to F$ be\na morphism over $S$.\nLet $f' : T' \\to F'$ be the morphism over $S'$ we get from\n$f$ by applying the equivalence of categories described in\nLemma \\ref{lemma-category-of-spaces-over-smaller-base-scheme}.\nFor any property $\\mathcal{P}$ as in\nDefinition \\ref{definition-relative-representable-property}\nwe have $\\mathcal{P}(f') \\Leftrightarrow \\mathcal{P}(f)$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces","chapter_id":"spaces","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03I6","source_file":"spaces.tex","source_line":2926,"source_end_line":2939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces.tex#L2926-L2939","statement_sha256":"219ded7f40f55e4fa26ea189921f6b0efb6a24aa7abc5b0f42f2cb02602ac929","origin":"The Stacks Project","memory_eligible":false,"source_rank":10735,"rank":10735,"depth":54,"x":186.223,"y":1591.539,"cluster":"algebraic-spaces"},{"id":"stacks:03BS","tag":"03BS","title":"Separation axioms · Definition 03BS","summary":"(Compare Spaces, Definition [Tag 02X5].) Consider a big fppf site Sch_fppf = (Sch/Spec(Z))_fppf. Let X be an algebraic space over Spec(Z). Let Δ : X → X × X be the diagonal morphism. • We say X is separated if Δ is a closed immersion. • We say X is locally separated if Δ is an immersion. • We say X is quasi-separated if Δ is quasi-compact. • We say X is Zariski locally quasi-separated if there exists a Zariski covering X = ⋃_i ∈ I X_i (see Spaces, Definition [Tag 02YY])…","statement_latex":"(Compare Spaces, Definition \\ref{spaces-definition-separated}.)\nConsider a big fppf site\n$\\Sch_{fppf} = (\\Sch/\\Spec(\\mathbf{Z}))_{fppf}$.\nLet $X$ be an algebraic space over\n$\\Spec(\\mathbf{Z})$. Let $\\Delta : X \\to X \\times X$\nbe the diagonal morphism.\n\\begin{enumerate}\n\\item We say $X$ is {\\it separated} if $\\Delta$ is a closed immersion.\n\\item We say $X$ is {\\it locally separated}\\footnote{In the\nliterature this often refers to quasi-separated and locally\nseparated algebraic spaces.} if $\\Delta$ is an\nimmersion.\n\\item We say $X$ is {\\it quasi-separated} if $\\Delta$ is quasi-compact.\n\\item We say $X$ is {\\it Zariski locally quasi-separated}\\footnote{\nThis notion was suggested by B.\\ Conrad.} if there\nexists a Zariski covering $X = \\bigcup_{i \\in I} X_i$ (see Spaces,\nDefinition \\ref{spaces-definition-Zariski-open-covering}) such that\neach $X_i$ is quasi-separated.\n\\end{enumerate}\nLet $S$ is a scheme contained in $\\Sch_{fppf}$, and let\n$X$ be an algebraic space over $S$. Then we say $X$ is {\\it separated},\n{\\it locally separated}, {\\it quasi-separated}, or\n{\\it Zariski locally quasi-separated}\nif $X$ viewed as an algebraic space over $\\Spec(\\mathbf{Z})$ (see\nSpaces, Definition \\ref{spaces-definition-base-change})\nhas the corresponding property.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Separation axioms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BS","source_file":"spaces-properties.tex","source_line":75,"source_end_line":103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L75-L103","statement_sha256":"183cbf78088259021be7cc15799ed4a1deee9640f352ac9afaf9a7ec59740a57","origin":"The Stacks Project","memory_eligible":false,"source_rank":10736,"rank":10736,"depth":54,"x":269.411,"y":1581.244,"cluster":"algebraic-spaces"},{"id":"stacks:03DY","tag":"03DY","title":"Separation axioms · Lemma 03DY","summary":"Let S be a scheme. Let X be an algebraic space over S. We have the following implications among the separation axioms of Definition [Tag 03BS]: • separated implies all the others, • quasi-separated implies Zariski locally quasi-separated.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nWe have the following implications among the separation axioms\nof Definition \\ref{definition-separated}:\n\\begin{enumerate}\n\\item separated implies all the others,\n\\item quasi-separated implies Zariski locally quasi-separated.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DY","source_file":"spaces-properties.tex","source_line":116,"source_end_line":126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L116-L126","statement_sha256":"123f171d785a87a5b923fad137c5199521240f10d9d0bd63ffc23e2a6a7acb20","origin":"The Stacks Project","memory_eligible":false,"source_rank":10737,"rank":10737,"depth":55,"x":215.907,"y":1636.491,"cluster":"algebraic-spaces"},{"id":"stacks:0AHR","tag":"0AHR","title":"Separation axioms · Lemma 0AHR","summary":"Let S be a scheme. Let X be an algebraic space over S. The following are equivalent • X is a quasi-separated algebraic space, • for U → X, V → X with U, V quasi-compact schemes the fibre product U ×_X V is quasi-compact, • for U → X, V → X with U, V affine the fibre product U ×_X V is quasi-compact.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is a quasi-separated algebraic space,\n\\item for $U \\to X$, $V \\to X$ with $U$, $V$ quasi-compact schemes\nthe fibre product $U \\times_X V$ is quasi-compact,\n\\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine\nthe fibre product $U \\times_X V$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHR","source_file":"spaces-properties.tex","source_line":132,"source_end_line":143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L132-L143","statement_sha256":"163f08e608722597aa072d15a95665fde45128394c74a6a23b03ebd2045607bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10738,"rank":10738,"depth":54,"x":210.894,"y":1564.814,"cluster":"algebraic-spaces"},{"id":"stacks:0AHS","tag":"0AHS","title":"Separation axioms · Lemma 0AHS","summary":"Let S be a scheme. Let X be an algebraic space over S. The following are equivalent • X is a separated algebraic space, • for U → X, V → X with U, V affine the fibre product U ×_X V is affine and O(U) ⊗_Z O(V) → O(U ×_X V) is surjective.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is a separated algebraic space,\n\\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine\nthe fibre product $U \\times_X V$ is affine and\n$$\n\\mathcal{O}(U) \\otimes_\\mathbf{Z} \\mathcal{O}(V)\n\\longrightarrow\n\\mathcal{O}(U \\times_X V)\n$$\nis surjective.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHS","source_file":"spaces-properties.tex","source_line":168,"source_end_line":183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L168-L183","statement_sha256":"469c44d370f7b96e9d436c0852e443a60afd1f7a8e587aaa82eea9b62d01ca2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10739,"rank":10739,"depth":54,"x":272.721,"y":1615.225,"cluster":"algebraic-spaces"},{"id":"stacks:03BU","tag":"03BU","title":"Points of algebraic spaces · Definition 03BU","summary":"Let S be a scheme. Let X be an algebraic space over S. A point of X is an equivalence class of morphisms from spectra of fields into X. The set of points of X is denoted |X|.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nA {\\it point} of $X$ is an equivalence class of morphisms\nfrom spectra of fields into $X$.\nThe set of points of $X$ is denoted $|X|$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BU","source_file":"spaces-properties.tex","source_line":255,"source_end_line":261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L255-L261","statement_sha256":"bbc95f3b0aaf2a97f43d7e375de86b841fc2d1ea058026271e21ecdaeb38d958","origin":"The Stacks Project","memory_eligible":false,"source_rank":10740,"rank":10740,"depth":0,"x":185.913,"y":1613.116,"cluster":"algebraic-spaces"},{"id":"stacks:03BV","tag":"03BV","title":"Points of algebraic spaces · Lemma 03BV","summary":"Let S be a scheme. Let X be a scheme over S. The points of X as a scheme are in canonical 1-1 correspondence with the points of X as an algebraic space.","statement_latex":"Let $S$ be a scheme. Let $X$ be a scheme over $S$.\nThe points of $X$ as a scheme are in canonical 1-1 correspondence\nwith the points of $X$ as an algebraic space.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BV","source_file":"spaces-properties.tex","source_line":269,"source_end_line":274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L269-L274","statement_sha256":"82ffc9ba34e300e9f391f362c5917f7e8ebe034e08d205a3ddad206a5bac06b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10741,"rank":10741,"depth":1,"x":252.13,"y":1565.045,"cluster":"algebraic-spaces"},{"id":"stacks:03H4","tag":"03H4","title":"Points of algebraic spaces · Lemma 03H4","summary":"Let S be a scheme. Let xymatrix Z ×_Y X ar[r] ar[d] & X ar[d] Z ar[r] & Y be a cartesian diagram of algebraic spaces over S. Then the map of sets of points |Z ×_Y X| → |Z| ×_|Y| |X| is surjective.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nZ \\times_Y X \\ar[r] \\ar[d] & X \\ar[d] \\\\\nZ \\ar[r] & Y\n}\n$$\nbe a cartesian diagram of algebraic spaces over $S$. Then the map of sets\nof points\n$$\n|Z \\times_Y X|\n\\longrightarrow\n|Z| \\times_{|Y|} |X|\n$$\nis surjective.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03H4","source_file":"spaces-properties.tex","source_line":280,"source_end_line":297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L280-L297","statement_sha256":"c63c5d262fed5307cc06c3b1f9adf8c3bd3f56067cc333aedf124680f655059f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10742,"rank":10742,"depth":0,"x":241.881,"y":1638.623,"cluster":"algebraic-spaces"},{"id":"stacks:03H5","tag":"03H5","title":"Points of algebraic spaces · Lemma 03H5","summary":"Let S be a scheme. Let X be an algebraic space over S. Let f : T → X be a morphism from a scheme to X. The following are equivalent • f : T → X is surjective (according to Spaces, Definition [Tag 025V]), and • |f| : |T| → |X| is surjective.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $f : T \\to X$ be a morphism from a scheme to $X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f : T \\to X$ is surjective (according to\nSpaces, Definition \\ref{spaces-definition-relative-representable-property}),\nand\n\\item $|f| : |T| \\to |X|$ is surjective.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03H5","source_file":"spaces-properties.tex","source_line":311,"source_end_line":323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L311-L323","statement_sha256":"8f9480c3fe5878f099f965df621cc1b93f97cf69415b7914e2ae6cf660b45c32","origin":"The Stacks Project","memory_eligible":false,"source_rank":10743,"rank":10743,"depth":2,"x":189.882,"y":1578.099,"cluster":"algebraic-spaces"},{"id":"stacks:03BW","tag":"03BW","title":"Points of algebraic spaces · Lemma 03BW","summary":"Let S be a scheme. Let X be an algebraic space over S. Let X = U/R be a presentation of X, see Spaces, Definition [Tag 0263]. Then the image of |R| → |U| × |U| is an equivalence relation and |X| is the quotient of |U| by this equivalence relation.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $X = U/R$ be a presentation of $X$, see\nSpaces, Definition \\ref{spaces-definition-presentation}.\nThen the image of $|R| \\to |U| \\times |U|$ is an equivalence relation\nand $|X|$ is the quotient of $|U|$ by this equivalence relation.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BW","source_file":"spaces-properties.tex","source_line":351,"source_end_line":359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L351-L359","statement_sha256":"c107b27838b6bec1a71d71a6d06b479c201868f469b9bbc552f48bce9133e895","origin":"The Stacks Project","memory_eligible":false,"source_rank":10744,"rank":10744,"depth":3,"x":277.542,"y":1593.337,"cluster":"algebraic-spaces"},{"id":"stacks:03BX","tag":"03BX","title":"Points of algebraic spaces · Lemma 03BX","summary":"Let S be a scheme. There exists a unique topology on the sets of points of algebraic spaces over S with the following properties: • if X is a scheme over S, then the topology on |X| is the usual one (via the identification of Lemma [Tag 03BV]), • for every morphism of algebraic spaces X → Y over S the map |X| → |Y| is continuous, and • for every étale morphism U → X with U a scheme the map of topological spaces |U| → |X| is continuous and open.","statement_latex":"Let $S$ be a scheme. There exists a unique topology on the sets of points\nof algebraic spaces over $S$ with the following properties:\n\\begin{enumerate}\n\\item if $X$ is a scheme over $S$, then the topology on $|X|$ is the usual one\n(via the identification of Lemma \\ref{lemma-scheme-points}),\n\\item for every morphism of algebraic spaces $X \\to Y$ over $S$\nthe map $|X| \\to |Y|$ is continuous, and\n\\item for every \\'etale morphism $U \\to X$ with $U$ a scheme\nthe map of topological spaces $|U| \\to |X|$ is continuous and open.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BX","source_file":"spaces-properties.tex","source_line":378,"source_end_line":390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L378-L390","statement_sha256":"f678aa56eaba5ed3cadd71f2d91176c46786e518f8dc5ab0e06c0c94feab222e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10745,"rank":10745,"depth":38,"x":200.086,"y":1632.121,"cluster":"algebraic-spaces"},{"id":"stacks:03BY","tag":"03BY","title":"Points of algebraic spaces · Definition 03BY","summary":"Let S be a scheme. Let X be an algebraic space over S. The underlying topological space of X is the set of points |X| endowed with the topology constructed in Lemma [Tag 03BX].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe underlying {\\it topological space} of $X$ is the set of points\n$|X|$ endowed with the topology constructed in\nLemma \\ref{lemma-topology-points}.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BY","source_file":"spaces-properties.tex","source_line":459,"source_end_line":465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L459-L465","statement_sha256":"319e56edd33927fff21eab67590ff5d00385643d8a3b74622f947c4a3837059b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10746,"rank":10746,"depth":39,"x":226.201,"y":1559.05,"cluster":"algebraic-spaces"},{"id":"stacks:03BZ","tag":"03BZ","title":"Points of algebraic spaces · Lemma 03BZ","summary":"Let S be a scheme. Let X be an algebraic space over S. • The rule X' ↦ |X'| defines an inclusion preserving bijection between open subspaces X' (see Spaces, Definition [Tag 02YU]) of X, and opens of the topological space |X|. • A family (X_i ⊂ X)_i ∈ I of open subspaces of X is a Zariski covering (see Spaces, Definition [Tag 02YY]) if and only if |X| = ⋃ |X_i|. In other words, the small Zariski site X_Zar of X is canonically identified with a site associated to the…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item The rule $X' \\mapsto |X'|$ defines an inclusion preserving\nbijection between open subspaces $X'$ (see\nSpaces, Definition \\ref{spaces-definition-immersion})\nof $X$, and opens of the topological space $|X|$.\n\\item A family $\\{X_i \\subset X\\}_{i \\in I}$ of open subspaces of $X$\nis a Zariski covering (see\nSpaces, Definition \\ref{spaces-definition-Zariski-open-covering})\nif and only if $|X| = \\bigcup |X_i|$.\n\\end{enumerate}\nIn other words, the small Zariski site $X_{Zar}$ of $X$ is canonically\nidentified with a site associated to the topological space $|X|$ (see\nSites, Example \\ref{sites-example-site-topological}).","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BZ","source_file":"spaces-properties.tex","source_line":472,"source_end_line":489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L472-L489","statement_sha256":"df5e4ee5cdf21a9786c7979951a7645f2fd887b8edce3f0492b60398c21ca85e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10747,"rank":10747,"depth":7,"x":265.984,"y":1628.237,"cluster":"algebraic-spaces"},{"id":"stacks:03IE","tag":"03IE","title":"Points of algebraic spaces · Lemma 03IE","summary":"Let S be a scheme. Let X, Y be algebraic spaces over S. Let X' ⊂ X be an open subspace. Let f : Y → X be a morphism of algebraic spaces over S. Then f factors through X' if and only if |f| : |Y| → |X| factors through |X'| ⊂ |X|.","statement_latex":"Let $S$ be a scheme.\nLet $X$, $Y$ be algebraic spaces over $S$.\nLet $X' \\subset X$ be an open subspace.\nLet $f : Y \\to X$ be a morphism of algebraic spaces over $S$.\nThen $f$ factors through $X'$ if and only if $|f| : |Y| \\to |X|$\nfactors through $|X'| \\subset |X|$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IE","source_file":"spaces-properties.tex","source_line":518,"source_end_line":526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L518-L526","statement_sha256":"6bbdfa1ff38449ac32e23eb6a6ec1ecbea693d105adca2f1d7240038006a8ccc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10748,"rank":10748,"depth":8,"x":180.416,"y":1599.594,"cluster":"algebraic-spaces"},{"id":"stacks:06NF","tag":"06NF","title":"Points of algebraic spaces · Lemma 06NF","summary":"Let S be a scheme. Let X be an algebraic spaces over S. Let U be a scheme and let f : U → X be an étale morphism. Let X' ⊂ X be the open subspace corresponding to the open |f|(|U|) ⊂ |X| via Lemma [Tag 03BZ]. Then f factors through a surjective étale morphism f' : U → X'. Moreover, if R = U ×_X U, then R = U ×_X' U and X' has the presentation X' = U/R.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic spaces over $S$.\nLet $U$ be a scheme and let $f : U \\to X$ be an \\'etale morphism.\nLet $X' \\subset X$ be the open subspace corresponding to\nthe open $|f|(|U|) \\subset |X|$ via\nLemma \\ref{lemma-open-subspaces}.\nThen $f$ factors through a surjective \\'etale morphism $f' : U \\to X'$.\nMoreover, if $R = U \\times_X U$, then $R = U \\times_{X'} U$ and $X'$ has\nthe presentation $X' = U/R$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NF","source_file":"spaces-properties.tex","source_line":535,"source_end_line":545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L535-L545","statement_sha256":"ae7ae7a68a0679e958d84aa6b4fb72de0b60e3a820ba97fdbd48e6bae75c5504","origin":"The Stacks Project","memory_eligible":false,"source_rank":10749,"rank":10749,"depth":9,"x":267.142,"y":1571.974,"cluster":"algebraic-spaces"},{"id":"stacks:0H2X","tag":"0H2X","title":"Points of algebraic spaces · Lemma 0H2X","summary":"Let S be a scheme. Let X be an algebraic space over S. Let p : Spec(K) → X and q : Spec(L) → X be morphisms where K and L are fields. Assume p and q determine the same point of |X| and p is a monomorphism. Then q factors uniquely through p.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $p : \\Spec(K) \\to X$ and $q : \\Spec(L) \\to X$\nbe morphisms where $K$ and $L$ are fields. Assume $p$ and $q$\ndetermine the same point of $|X|$ and $p$ is a monomorphism.\nThen $q$ factors uniquely through $p$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2X","source_file":"spaces-properties.tex","source_line":561,"source_end_line":568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L561-L568","statement_sha256":"6b9586a19e7fd1610a198736ac7dc2c53c8d5f04b3561544164887fbaeb2a130","origin":"The Stacks Project","memory_eligible":false,"source_rank":10750,"rank":10750,"depth":4,"x":225.117,"y":1642.024,"cluster":"algebraic-spaces"},{"id":"stacks:03E1","tag":"03E1","title":"Points of algebraic spaces · Lemma 03E1","summary":"Let S be a scheme. Let X be an algebraic space over S. Consider the map (Spec(k) → X monomorphism where k is a field) → |X| This map is injective.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nConsider the map\n$$\n\\{\\Spec(k) \\to X \\text{ monomorphism where }k\\text{ is a field}\\}\n\\longrightarrow\n|X|\n$$\nThis map is injective.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03E1","source_file":"spaces-properties.tex","source_line":583,"source_end_line":593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L583-L593","statement_sha256":"1d07f155ace7fe0a15c2fa9a426e3560d32f5e08720aabd54f05ee50a2894486","origin":"The Stacks Project","memory_eligible":false,"source_rank":10751,"rank":10751,"depth":5,"x":199.605,"y":1566.017,"cluster":"algebraic-spaces"},{"id":"stacks:03E3","tag":"03E3","title":"Quasi-compact spaces · Definition 03E3","summary":"Let S be a scheme. Let X be an algebraic space over S. We say X is quasi-compact if there exists a surjective étale morphism U → X where U is a quasi-compact scheme.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nWe say $X$ is {\\it quasi-compact} if there exists a surjective\n\\'etale morphism $U \\to X$ where $U$ is a quasi-compact scheme.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-compact spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03E3","source_file":"spaces-properties.tex","source_line":624,"source_end_line":630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L624-L630","statement_sha256":"e97db050db677f5c8f02807f0913ee7cbc74039ab121aea02040bffb45243f4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10752,"rank":10752,"depth":0,"x":280.068,"y":1607.863,"cluster":"algebraic-spaces"},{"id":"stacks:03E4","tag":"03E4","title":"Quasi-compact spaces · Lemma 03E4","summary":"Let S be a scheme. Let X be an algebraic space over S. Then X is quasi-compact if and only if |X| is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThen $X$ is quasi-compact if and only if $|X|$ is quasi-compact.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03E4","source_file":"spaces-properties.tex","source_line":632,"source_end_line":637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L632-L637","statement_sha256":"a7c4ad85843868d604da0dc4867feca9df81e46a94632d1d415c2113d1d19b6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10753,"rank":10753,"depth":3,"x":186.476,"y":1622.759,"cluster":"algebraic-spaces"},{"id":"stacks:040T","tag":"040T","title":"Quasi-compact spaces · Lemma 040T","summary":"A finite disjoint union of quasi-compact algebraic spaces is a quasi-compact algebraic space.","statement_latex":"A finite disjoint union of quasi-compact algebraic spaces is\na quasi-compact algebraic space.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040T","source_file":"spaces-properties.tex","source_line":650,"source_end_line":654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L650-L654","statement_sha256":"3303a322a9ca871022b7347f7d2830c335a1c557e8f5384f5fddffdb382a9834","origin":"The Stacks Project","memory_eligible":false,"source_rank":10754,"rank":10754,"depth":4,"x":243.88,"y":1558.256,"cluster":"algebraic-spaces"},{"id":"stacks:04NN","tag":"04NN","title":"Quasi-compact spaces · Lemma 04NN","summary":"Let S be a scheme. Let X be an algebraic space over S. Every point of |X| has a fundamental system of open quasi-compact neighbourhoods. In particular |X| is locally quasi-compact in the sense of Topology, Definition [Tag 0068].","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nEvery point of $|X|$ has a fundamental system of open\nquasi-compact neighbourhoods.\nIn particular $|X|$ is locally quasi-compact in the sense of\nTopology, Definition \\ref{topology-definition-locally-quasi-compact}.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NN","source_file":"spaces-properties.tex","source_line":668,"source_end_line":676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L668-L676","statement_sha256":"95bf9cbff6a4b4ca3a03a250300f3bb6906fc6fc33821dd86c71b4a57d1ac47e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10755,"rank":10755,"depth":1,"x":253.482,"y":1638.902,"cluster":"algebraic-spaces"},{"id":"stacks:03FX","tag":"03FX","title":"Special coverings · Lemma 03FX","summary":"Let S be a scheme. Let X be an algebraic space over S. There exists a surjective étale morphism U → X where U is a disjoint union of affine schemes. We may in addition assume each of these affines maps into an affine open of S.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThere exists a surjective \\'etale morphism $U \\to X$ where\n$U$ is a disjoint union of affine schemes.\nWe may in addition assume each of these affines\nmaps into an affine open of $S$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Special coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FX","source_file":"spaces-properties.tex","source_line":700,"source_end_line":708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L700-L708","statement_sha256":"6f62b3c8d1d5613ee018e8823d77d15d268300c90bc849deeef1bb0df273927a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10756,"rank":10756,"depth":40,"x":181.097,"y":1584.542,"cluster":"algebraic-spaces"},{"id":"stacks:03FY","tag":"03FY","title":"Special coverings · Lemma 03FY","summary":"Let S be a scheme. Let X be an algebraic space over S. There exists a Zariski covering X = ⋃ X_i such that each algebraic space X_i has a surjective étale covering by an affine scheme. We may in addition assume each X_i maps into an affine open of S.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThere exists a Zariski covering $X = \\bigcup X_i$\nsuch that each algebraic space $X_i$ has a surjective\n\\'etale covering by an affine scheme. We may in addition assume\neach $X_i$ maps into an affine open of $S$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Special coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FY","source_file":"spaces-properties.tex","source_line":725,"source_end_line":733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L725-L733","statement_sha256":"be175e30f6883de0858fcf873d14fe7f45f45526f61c7c9b391c14f89a452b58","origin":"The Stacks Project","memory_eligible":false,"source_rank":10757,"rank":10757,"depth":41,"x":278.79,"y":1583.549,"cluster":"algebraic-spaces"},{"id":"stacks:03H6","tag":"03H6","title":"Special coverings · Lemma 03H6","summary":"Let S be a scheme. Let X be an algebraic space over S. Then X is quasi-compact if and only if there exists an étale surjective morphism U → X with U an affine scheme.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThen $X$ is quasi-compact if and only if\nthere exists an \\'etale surjective morphism $U \\to X$\nwith $U$ an affine scheme.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Special coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03H6","source_file":"spaces-properties.tex","source_line":747,"source_end_line":754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L747-L754","statement_sha256":"68a898a8609bf805d676450101edf9322cc832a6ef9cab28a81d241cb9c6268c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10758,"rank":10758,"depth":41,"x":207.114,"y":1640.058,"cluster":"algebraic-spaces"},{"id":"stacks:03FZ","tag":"03FZ","title":"Special coverings · Lemma 03FZ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a separated scheme and U → X étale. Then U → X is separated, and R = U ×_X U is a separated scheme.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $U$ be a separated scheme and $U \\to X$ \\'etale.\nThen $U \\to X$ is separated, and $R = U \\times_X U$ is a separated scheme.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Special coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FZ","source_file":"spaces-properties.tex","source_line":774,"source_end_line":780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L774-L780","statement_sha256":"184f12c3520d7829b9979f5a631180a5a1d35bd8ccf148663c64dfdf5f4c0520","origin":"The Stacks Project","memory_eligible":false,"source_rank":10759,"rank":10759,"depth":52,"x":214.57,"y":1557.219,"cluster":"algebraic-spaces"},{"id":"stacks:07S4","tag":"07S4","title":"Special coverings · Lemma 07S4","summary":"Let S be a scheme. Let X be an algebraic space over S. If there exists a quasi-separated scheme U and a surjective étale morphism U → X such that either of the projections U ×_X U → U is quasi-compact, then X is quasi-separated.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf there exists a quasi-separated scheme $U$ and a surjective\n\\'etale morphism $U \\to X$ such that either of the projections\n$U \\times_X U \\to U$ is quasi-compact, then $X$ is quasi-separated.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Special coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07S4","source_file":"spaces-properties.tex","source_line":821,"source_end_line":827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L821-L827","statement_sha256":"c5280b440787d6492e0131dac1c5b7753c631b499693f3baeb99cc69ec938787","origin":"The Stacks Project","memory_eligible":false,"source_rank":10760,"rank":10760,"depth":18,"x":276.053,"y":1622.926,"cluster":"algebraic-spaces"},{"id":"stacks:03W7","tag":"03W7","title":"Special coverings · Lemma 03W7","summary":"Let S be a scheme. Let X be an algebraic space over S. The following are equivalent • X is Zariski locally quasi-separated over S, • X is Zariski locally quasi-separated, • there exists a Zariski open covering X = ⋃ X_i such that for each i there exists an affine scheme U_i and a quasi-compact surjective étale morphism U_i → X_i, and • there exists a Zariski open covering X = ⋃ X_i such that for each i there exists an affine scheme U_i which maps into an affine open of S…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is Zariski locally quasi-separated over $S$,\n\\item $X$ is Zariski locally quasi-separated,\n\\item there exists a Zariski open covering $X = \\bigcup X_i$\nsuch that for each $i$ there exists an affine scheme\n$U_i$ and a quasi-compact surjective \\'etale\nmorphism $U_i \\to X_i$, and\n\\item there exists a Zariski open covering $X = \\bigcup X_i$\nsuch that for each $i$ there exists an affine scheme\n$U_i$ which maps into an affine open of $S$ and a quasi-compact\nsurjective \\'etale morphism $U_i \\to X_i$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Special coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03W7","source_file":"spaces-properties.tex","source_line":847,"source_end_line":864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L847-L864","statement_sha256":"639f46c2c4d30fdd343ea676825ec2ec5316c236adb6468274c3d0a2d3bf749a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10761,"rank":10761,"depth":42,"x":177.297,"y":1609.281,"cluster":"algebraic-spaces"},{"id":"stacks:03IJ","tag":"03IJ","title":"Special coverings · Lemma 03IJ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a scheme. Let φ : U → X be an étale morphism such that the projections R = U ×_X U → U are quasi-compact; for example if φ is quasi-compact. Then the fibres of |U| → |X| and |R| → |X| are finite.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $U$ be a scheme. Let $\\varphi : U \\to X$ be an \\'etale morphism such that\nthe projections $R = U \\times_X U \\to U$ are quasi-compact; for example if\n$\\varphi$ is quasi-compact. Then the fibres of\n$$\n|U| \\to |X|\n\\quad\\text{and}\\quad\n|R| \\to |X|\n$$\nare finite.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Special coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IJ","source_file":"spaces-properties.tex","source_line":937,"source_end_line":950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L937-L950","statement_sha256":"965be4a0b5ed03a81dc4b48115c2a9567147f2656c332b52873500329556c21e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10762,"rank":10762,"depth":45,"x":261.582,"y":1563.038,"cluster":"algebraic-spaces"},{"id":"stacks:03E8","tag":"03E8","title":"Properties of Spaces defined by properties of schemes · Lemma 03E8","summary":"Let S be a scheme. Let X be an algebraic space over S. Let P be a property of schemes which is local in the étale topology, see Descent, Definition [Tag 0348]. The following are equivalent • for some scheme U and surjective étale morphism U → X the scheme U has property P, and • for every scheme U and every étale morphism U → X the scheme U has property P. If X is representable this is equivalent to P(X).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\mathcal{P}$ be a property of schemes which is local in the \\'etale\ntopology, see\nDescent, Definition \\ref{descent-definition-property-local}.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some scheme $U$ and surjective \\'etale morphism $U \\to X$\nthe scheme $U$ has property $\\mathcal{P}$, and\n\\item for every scheme $U$ and every \\'etale morphism $U \\to X$\nthe scheme $U$ has property $\\mathcal{P}$.\n\\end{enumerate}\nIf $X$ is representable this is equivalent to $\\mathcal{P}(X)$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Properties of Spaces defined by properties of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03E8","source_file":"spaces-properties.tex","source_line":979,"source_end_line":994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L979-L994","statement_sha256":"2f5a437f162020454b59d580a929078c6aacb84469b28a041bc2db4995dfada6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10763,"rank":10763,"depth":1,"x":236.473,"y":1645.435,"cluster":"algebraic-spaces"},{"id":"stacks:03E6","tag":"03E6","title":"Properties of Spaces defined by properties of schemes · Definition 03E6","summary":"Let P be a property of schemes which is local in the étale topology. Let S be a scheme. Let X be an algebraic space over S. We say X has property P if any of the equivalent conditions of Lemma [Tag 03E8] hold.","statement_latex":"Let $\\mathcal{P}$ be a property of schemes which is\nlocal in the \\'etale topology.\nLet $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nWe say $X$ {\\it has property $\\mathcal{P}$}\nif any of the equivalent conditions of\nLemma \\ref{lemma-type-property}\nhold.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Properties of Spaces defined by properties of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03E6","source_file":"spaces-properties.tex","source_line":1008,"source_end_line":1018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1008-L1018","statement_sha256":"234e92dbab6baba19d39fa106d7853dce516e6fea452f2bb7b6d034f483e5b60","origin":"The Stacks Project","memory_eligible":false,"source_rank":10764,"rank":10764,"depth":2,"x":188.455,"y":1570.001,"cluster":"algebraic-spaces"},{"id":"stacks:04N2","tag":"04N2","title":"Properties of Spaces defined by properties of schemes · Lemma 04N2","summary":"Let P be a property of germs of schemes which is étale local, see Descent, Definition [Tag 04N1]. Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point of X. Consider étale morphisms a : U → X where U is a scheme. The following are equivalent • for any U → X as above and u ∈ U with a(u) = x we have P(U, u), and • for some U → X as above and u ∈ U with a(u) = x we have P(U, u). If X is representable, then this is equivalent to P(X, x).","statement_latex":"Let $\\mathcal{P}$ be a property of germs of schemes which is \\'etale local, see\nDescent, Definition \\ref{descent-definition-local-at-point}.\nLet $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $x \\in |X|$ be a point of $X$.\nConsider \\'etale morphisms $a : U \\to X$ where $U$ is a scheme.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any $U \\to X$ as above and $u \\in U$ with $a(u) = x$ we have\n$\\mathcal{P}(U, u)$, and\n\\item for some $U \\to X$ as above and $u \\in U$ with $a(u) = x$ we have\n$\\mathcal{P}(U, u)$.\n\\end{enumerate}\nIf $X$ is representable, then this is equivalent to $\\mathcal{P}(X, x)$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Properties of Spaces defined by properties of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04N2","source_file":"spaces-properties.tex","source_line":1053,"source_end_line":1069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1053-L1069","statement_sha256":"6eab7f2546de7b3e67dd00ef981788136712a3a7df6aedd90510cee268f61849","origin":"The Stacks Project","memory_eligible":false,"source_rank":10765,"rank":10765,"depth":1,"x":285.067,"y":1598.539,"cluster":"algebraic-spaces"},{"id":"stacks:04RC","tag":"04RC","title":"Properties of Spaces defined by properties of schemes · Definition 04RC","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. Let P be a property of germs of schemes which is étale local. We say X has property P at x if any of the equivalent conditions of Lemma [Tag 04N2] hold.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. Let $\\mathcal{P}$ be a property of germs of schemes which is\n\\'etale local.\nWe say $X$ {\\it has property $\\mathcal{P}$ at $x$} if any of the\nequivalent conditions of\nLemma \\ref{lemma-local-source-target-at-point}\nhold.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Properties of Spaces defined by properties of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RC","source_file":"spaces-properties.tex","source_line":1075,"source_end_line":1084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1075-L1084","statement_sha256":"701ae7b9b502e6b71401b64382d17426f5f37c09ecb2bd041c9d1eeac5be322f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10766,"rank":10766,"depth":2,"x":190.351,"y":1632.503,"cluster":"algebraic-spaces"},{"id":"stacks:0ECT","tag":"0ECT","title":"Constructible sets · Lemma 0ECT","summary":"Let S be a scheme. Let X be an algebraic space over S. Let E ⊂ |X| be a subset. The following are equivalent • for every étale morphism U → X where U is a scheme the inverse image of E in U is a locally constructible subset of U, • for every étale morphism U → X where U is an affine scheme the inverse image of E in U is a constructible subset of U, • for some surjective étale morphism U → X where U is a scheme the inverse image of E in U is a locally constructible subset…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $E \\subset |X|$ be a subset. The following are equivalent\n\\begin{enumerate}\n\\item for every \\'etale morphism $U \\to X$ where $U$ is a\nscheme the inverse image of $E$ in $U$ is a locally constructible\nsubset of $U$,\n\\item for every \\'etale morphism $U \\to X$ where $U$ is an\naffine scheme the inverse image of $E$ in $U$ is a constructible\nsubset of $U$,\n\\item for some surjective \\'etale morphism $U \\to X$ where $U$ is a\nscheme the inverse image of $E$ in $U$ is a locally constructible\nsubset of $U$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECT","source_file":"spaces-properties.tex","source_line":1133,"source_end_line":1148,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1133-L1148","statement_sha256":"506fb74e30ebec49083cf10f7b341ef6f6b635a5d3065f9e9b158d2dc058dddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10767,"rank":10767,"depth":7,"x":233.116,"y":1553.28,"cluster":"algebraic-spaces"},{"id":"stacks:0ECU","tag":"0ECU","title":"Constructible sets · Definition 0ECU","summary":"Let S be a scheme. Let X be an algebraic space over S. Let E ⊂ |X| be a subset. We say E is étale locally constructible if the equivalent conditions of Lemma [Tag 0ECT] are satisfied.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $E \\subset |X|$ be a subset. We say $E$ is\n{\\it \\'etale locally constructible} if the equivalent\nconditions of Lemma \\ref{lemma-locally-constructible} are satisfied.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Constructible sets","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECU","source_file":"spaces-properties.tex","source_line":1175,"source_end_line":1181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1175-L1181","statement_sha256":"c5ce9b2ffbd568a6f4d5f7e3d99c2df73d925e69ea956a92ce87bc989ea45a9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10768,"rank":10768,"depth":8,"x":265.465,"y":1636.415,"cluster":"algebraic-spaces"},{"id":"stacks:04N5","tag":"04N5","title":"Dimension at a point · Definition 04N5","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point of X. We define the dimension of X at x to be the element dim_x(X) ∈ (0, 1, 2, …, ∞) such that dim_x(X) = dim_u(U) for any (equivalently some) pair (a : U → X, u) consisting of an étale morphism a : U → X from a scheme to X and a point u ∈ U with a(u) = x. See Definition [Tag 04RC], Lemma [Tag 04N2], and Descent, Lemma [Tag 04N4].","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $x \\in |X|$ be a point of $X$.\nWe define the {\\it dimension of $X$ at $x$} to be\nthe element $\\dim_x(X) \\in \\{0, 1, 2, \\ldots, \\infty\\}$\nsuch that $\\dim_x(X) = \\dim_u(U)$ for any (equivalently some)\npair $(a : U \\to X, u)$ consisting of an \\'etale morphism $a : U \\to X$\nfrom a scheme to $X$ and a point $u \\in U$ with $a(u) = x$.\nSee\nDefinition \\ref{definition-property-at-point},\nLemma \\ref{lemma-local-source-target-at-point}, and\nDescent, Lemma \\ref{descent-lemma-dimension-at-point-local}.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Dimension at a point","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04N5","source_file":"spaces-properties.tex","source_line":1204,"source_end_line":1218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1204-L1218","statement_sha256":"56a5d62255efcf85f19f9c46200fc2fec96a1164c9b953025084d448ef79f57c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10769,"rank":10769,"depth":46,"x":174.267,"y":1593.235,"cluster":"algebraic-spaces"},{"id":"stacks:04N6","tag":"04N6","title":"Dimension at a point · Definition 04N6","summary":"Let S be a scheme. Let X be an algebraic space over S. The dimension dim(X) of X is defined by the rule dim(X) = sup_x ∈ |X| dim_x(X)","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe {\\it dimension} $\\dim(X)$ of $X$ is defined by the rule\n$$\n\\dim(X) = \\sup\\nolimits_{x \\in |X|} \\dim_x(X)\n$$","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Dimension at a point","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04N6","source_file":"spaces-properties.tex","source_line":1229,"source_end_line":1236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1229-L1236","statement_sha256":"5118c44fc3e8bb0bc2d3d0313838a9e87918cbe37682096c92fd99ca2c052c71","origin":"The Stacks Project","memory_eligible":false,"source_rank":10770,"rank":10770,"depth":0,"x":276.784,"y":1573.222,"cluster":"algebraic-spaces"},{"id":"stacks:0BAM","tag":"0BAM","title":"Dimension of local rings · Lemma 0BAM","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point. Let d ∈ (0, 1, 2, …, ∞). The following are equivalent • for some scheme U and étale morphism a : U → X and point u ∈ U with a(u) = x we have dim(O_U, u) = d, • for any scheme U, any étale morphism a : U → X, and any point u ∈ U with a(u) = x we have dim(O_U, u) = d. If X is a scheme, this is equivalent to dim(O_X, x) = d.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $x \\in |X|$ be a point.\nLet $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some scheme $U$ and \\'etale morphism $a : U \\to X$ and point\n$u \\in U$ with $a(u) = x$ we have $\\dim(\\mathcal{O}_{U, u}) = d$,\n\\item for any scheme $U$, any \\'etale morphism $a : U \\to X$, and any point\n$u \\in U$ with $a(u) = x$ we have $\\dim(\\mathcal{O}_{U, u}) = d$.\n\\end{enumerate}\nIf $X$ is a scheme, this is equivalent to $\\dim(\\mathcal{O}_{X, x}) = d$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Dimension of local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAM","source_file":"spaces-properties.tex","source_line":1259,"source_end_line":1273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1259-L1273","statement_sha256":"65bed2df08be9aa87db91014c576d6decf2958d84b8071900cece8415bc95919","origin":"The Stacks Project","memory_eligible":false,"source_rank":10771,"rank":10771,"depth":15,"x":216.968,"y":1646.535,"cluster":"algebraic-spaces"},{"id":"stacks:04NA","tag":"04NA","title":"Dimension of local rings · Definition 04NA","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point. The dimension of the local ring of X at x is the element d ∈ (0, 1, 2, …, ∞) satisfying the equivalent conditions of Lemma [Tag 0BAM]. In this case we will also say x is a point of codimension d on X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$\nbe a point. The {\\it dimension of the local ring of $X$ at $x$} is\nthe element $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$ satisfying the equivalent\nconditions of Lemma \\ref{lemma-pre-dimension-local-ring}. In this case we\nwill also say {\\it $x$ is a point of codimension $d$ on $X$}.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Dimension of local rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NA","source_file":"spaces-properties.tex","source_line":1281,"source_end_line":1288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1281-L1288","statement_sha256":"1f16b4ad4b089aa602dda1bc130e510df02ecf0c7e0a8eca617c9dd9956185b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10772,"rank":10772,"depth":16,"x":202.043,"y":1558.074,"cluster":"algebraic-spaces"},{"id":"stacks:0BAN","tag":"0BAN","title":"Dimension of local rings · Lemma 0BAN","summary":"Let S be a scheme. Let X be an algebraic space over S. The following quantities are equal: • The dimension of X. • The supremum of the dimensions of the local rings of X. • The supremum of dim_x(X) for x ∈ |X|.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe following quantities are equal:\n\\begin{enumerate}\n\\item The dimension of $X$.\n\\item The supremum of the dimensions of the local rings of $X$.\n\\item The supremum of $\\dim_x(X)$ for $x \\in |X|$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Dimension of local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAN","source_file":"spaces-properties.tex","source_line":1295,"source_end_line":1304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1295-L1304","statement_sha256":"b2329b4167a9b6296c893b25eb147b15f03ba067c20c42b18cbbe8920a1a09c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10773,"rank":10773,"depth":2,"x":284.61,"y":1615.129,"cluster":"algebraic-spaces"},{"id":"stacks:0BAQ","tag":"0BAQ","title":"Generic points · Lemma 0BAQ","summary":"Let S be a scheme and let X be an algebraic space over S. Let x ∈ |X|. Consider étale morphisms a : U → X where U is a scheme. The following are equivalent • x is a point of codimension 0 on X, • for some U → X as above and u ∈ U with a(u) = x, the point u is the generic point of an irreducible component of U, and • for any U → X as above and any u ∈ U mapping to x, the point u is the generic point of an irreducible component of U. If X is representable, this is…","statement_latex":"Let $S$ be a scheme and let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. Consider \\'etale morphisms $a : U \\to X$ where\n$U$ is a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $x$ is a point of codimension $0$ on $X$,\n\\item for some $U \\to X$ as above and $u \\in U$ with $a(u) = x$,\nthe point $u$ is the generic point of an irreducible component of $U$, and\n\\item for any $U \\to X$ as above and any $u \\in U$ mapping to $x$,\nthe point $u$ is the generic point of an irreducible component of $U$.\n\\end{enumerate}\nIf $X$ is representable, this is equivalent to $x$ being a generic\npoint of an irreducible component of $|X|$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Generic points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAQ","source_file":"spaces-properties.tex","source_line":1336,"source_end_line":1350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1336-L1350","statement_sha256":"d704c7d2ef1ff216c5b3c8ad636b25d59e097df22ab9c5b46f3318cb4cd0fcd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10774,"rank":10774,"depth":17,"x":177.298,"y":1619.932,"cluster":"algebraic-spaces"},{"id":"stacks:0BAR","tag":"0BAR","title":"Generic points · Lemma 0BAR","summary":"Let S be a scheme and let X be an algebraic space over S. The set of codimension 0 points of X is dense in |X|.","statement_latex":"Let $S$ be a scheme and let $X$ be an algebraic space over $S$.\nThe set of codimension $0$ points of $X$ is dense in $|X|$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Generic points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAR","source_file":"spaces-properties.tex","source_line":1360,"source_end_line":1364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1360-L1364","statement_sha256":"9835f540ae17c781f1eab6b7682e9030843112cbe2e6fee5439dffddf3b8dbbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10775,"rank":10775,"depth":18,"x":252.948,"y":1555.173,"cluster":"algebraic-spaces"},{"id":"stacks:0ABJ","tag":"0ABJ","title":"Reduced spaces · Lemma 0ABJ","summary":"Let S be a scheme. Let Z → X be an immersion of algebraic spaces. Then |Z| → |X| is a homeomorphism of |Z| onto a locally closed subset of |X|.","statement_latex":"Let $S$ be a scheme. Let $Z \\to X$ be an immersion of algebraic spaces.\nThen $|Z| \\to |X|$ is a homeomorphism of $|Z|$ onto a locally closed subset\nof $|X|$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Reduced spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABJ","source_file":"spaces-properties.tex","source_line":1388,"source_end_line":1393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1388-L1393","statement_sha256":"deb5c21842e0ddc6b828eedd7e7b72fb49a8f71515af9bf3cdb7bc3ea05bf3a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10776,"rank":10776,"depth":1,"x":249.223,"y":1646.306,"cluster":"algebraic-spaces"},{"id":"stacks:07TW","tag":"07TW","title":"Reduced spaces · Lemma 07TW","summary":"Let S be a scheme. Let j : R → U ×_S U be an étale equivalence relation. Let X = U/R be the associated algebraic space (Spaces, Theorem [Tag 02WW]). There is a canonical bijection R-invariant locally closed subschemes Z' of U ↔ locally closed subspaces Z of X Moreover, if Z → X is closed (resp. open) if and only if Z' → U is closed (resp. open).","statement_latex":"Let $S$ be a scheme. Let $j : R \\to U \\times_S U$ be an \\'etale equivalence\nrelation. Let $X = U/R$ be the associated algebraic space\n(Spaces, Theorem \\ref{spaces-theorem-presentation}). There is a\ncanonical bijection\n$$\nR\\text{-invariant locally closed subschemes }Z'\\text{ of }U\n\\leftrightarrow\n\\text{locally closed subspaces }Z\\text{ of }X\n$$\nMoreover, if $Z \\to X$ is closed (resp.\\ open) if and only if\n$Z' \\to U$ is closed (resp.\\ open).","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Reduced spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TW","source_file":"spaces-properties.tex","source_line":1414,"source_end_line":1427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1414-L1427","statement_sha256":"009a493ab4436a024558d22a47aab41a505d6d43bc85e0e61b1094291fe343dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10777,"rank":10777,"depth":52,"x":178.333,"y":1576.647,"cluster":"algebraic-spaces"},{"id":"stacks:03IQ","tag":"03IQ","title":"Reduced spaces · Lemma 03IQ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let T ⊂ |X| be a closed subset. There exists a unique closed subspace Z ⊂ X with the following properties: (a) we have |Z| = T, and (b) Z is reduced.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset.\nThere exists a unique closed subspace $Z \\subset X$ with\nthe following properties: (a) we have $|Z| = T$, and (b) $Z$ is reduced.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Reduced spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IQ","source_file":"spaces-properties.tex","source_line":1466,"source_end_line":1473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1466-L1473","statement_sha256":"ef7c6c986ecd56ab94c1773998cad8801f18400d6242379f3b0c3b5723338b28","origin":"The Stacks Project","memory_eligible":false,"source_rank":10778,"rank":10778,"depth":53,"x":287.156,"y":1587.844,"cluster":"algebraic-spaces"},{"id":"stacks:03JJ","tag":"03JJ","title":"Reduced spaces · Lemma 03JJ","summary":"Let S be a scheme. Let X, Y be algebraic spaces over S. Let Z ⊂ X be a closed subspace. Assume Y is reduced. A morphism f : Y → X factors through Z if and only if f(|Y|) ⊂ |Z|.","statement_latex":"Let $S$ be a scheme.\nLet $X$, $Y$ be algebraic spaces over $S$.\nLet $Z \\subset X$ be a closed subspace.\nAssume $Y$ is reduced.\nA morphism $f : Y \\to X$ factors through $Z$ if and only if\n$f(|Y|) \\subset |Z|$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Reduced spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JJ","source_file":"spaces-properties.tex","source_line":1501,"source_end_line":1509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1501-L1509","statement_sha256":"94c531234d608321f6bd8459f8aa68a71cd433797a48f40d2b81fd626af6fbee","origin":"The Stacks Project","memory_eligible":false,"source_rank":10779,"rank":10779,"depth":3,"x":197.472,"y":1641.596,"cluster":"algebraic-spaces"},{"id":"stacks:047X","tag":"047X","title":"Reduced spaces · Definition 047X","summary":"Let S be a scheme, and let X be an algebraic space over S. Let Z ⊂ |X| be a closed subset. An algebraic space structure on Z is given by a closed subspace Z' of X with |Z'| equal to Z. The reduced induced algebraic space structure on Z is the one constructed in Lemma [Tag 03IQ]. The reduction X_red of X is the reduced induced algebraic space structure on |X|.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nLet $Z \\subset |X|$ be a closed subset.\nAn {\\it algebraic space structure on $Z$} is given by a closed subspace\n$Z'$ of $X$ with $|Z'|$ equal to $Z$.\nThe {\\it reduced induced algebraic space structure}\non $Z$ is the one constructed in\nLemma \\ref{lemma-reduced-closed-subspace}.\nThe {\\it reduction $X_{red}$ of $X$} is the reduced induced algebraic\nspace structure on $|X|$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Reduced spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047X","source_file":"spaces-properties.tex","source_line":1530,"source_end_line":1541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1530-L1541","statement_sha256":"f08c6a9b1484d1d57c93e9d01aa76776a5ca9bc9834b8d65a2f4c8f20820d031","origin":"The Stacks Project","memory_eligible":false,"source_rank":10780,"rank":10780,"depth":54,"x":220.49,"y":1550.635,"cluster":"algebraic-spaces"},{"id":"stacks:03JH","tag":"03JH","title":"The schematic locus · Lemma 03JH","summary":"Let S be a scheme. Let X be an algebraic space over S. There exists a largest open subspace X' ⊂ X which is a scheme.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThere exists a largest open subspace $X' \\subset X$ which is a scheme.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The schematic locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JH","source_file":"spaces-properties.tex","source_line":1564,"source_end_line":1569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1564-L1569","statement_sha256":"46f86d4061625ceba9b6124483f452adbd90122b2eab04140803b6a5f3d49666","origin":"The Stacks Project","memory_eligible":false,"source_rank":10781,"rank":10781,"depth":3,"x":276.933,"y":1631.152,"cluster":"algebraic-spaces"},{"id":"stacks:0BAS","tag":"0BAS","title":"The schematic locus · Lemma 0BAS","summary":"Let S be a scheme. Let X be an algebraic space over S. If there exists a finite, étale, surjective morphism U → X where U is a quasi-separated scheme, then there exists a dense open subspace X' of X which is a scheme. More precisely, every point x ∈ |X| of codimension 0 in X is contained in X'.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf there exists a finite, \\'etale, surjective morphism\n$U \\to X$ where $U$ is a quasi-separated scheme, then\nthere exists a dense open subspace $X'$ of $X$ which is a scheme.\nMore precisely, every point $x \\in |X|$ of codimension $0$ in $X$\nis contained in $X'$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The schematic locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAS","source_file":"spaces-properties.tex","source_line":1599,"source_end_line":1607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1599-L1607","statement_sha256":"ee3b2903e6e1ca783315d38a4386cb21bacadc508a245d6cf8f491c031836123","origin":"The Stacks Project","memory_eligible":false,"source_rank":10782,"rank":10782,"depth":46,"x":170.06,"y":1603.677,"cluster":"algebraic-spaces"},{"id":"stacks:06NH","tag":"06NH","title":"The schematic locus · Proposition 06NH","summary":"Let S be a scheme. Let X be an algebraic space over S. If X is Zariski locally quasi-separated (for example if X is quasi-separated), then there exists a dense open subspace X' of X which is a scheme. More precisely, every point x ∈ |X| of codimension 0 on X is contained in X'.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $X$ is\nZariski locally quasi-separated (for example if $X$ is quasi-separated), then\nthere exists a dense open subspace $X'$ of $X$ which is a scheme.\nMore precisely, every point $x \\in |X|$ of codimension $0$ on $X$\nis contained in $X'$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The schematic locus","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NH","source_file":"spaces-properties.tex","source_line":1646,"source_end_line":1653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1646-L1653","statement_sha256":"34ab48b139ff000b532038180008ab36b81643db55165f50f68bfcdc6e23540e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10783,"rank":10783,"depth":53,"x":271.434,"y":1563.106,"cluster":"algebraic-spaces"},{"id":"stacks:07S6","tag":"07S6","title":"Obtaining a scheme · Proposition 07S6","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume • s, t : R → U finite locally free, • j = (t, s) is an equivalence relation, and • every nonempty closed subset Z of U contains a point u whose R-equivalence class t(s^-1((u))) is contained in an affine open of U((u))) is contained in an affine open of U. Condition (3) holds if E = U, or if every finite type point of U is in E, or if every u ∈ U specializes to a point of E.. Then there exists a…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nAssume\n\\begin{enumerate}\n\\item $s, t : R \\to U$ finite locally free,\n\\item $j = (t, s)$ is an equivalence relation, and\n\\item every nonempty closed subset $Z$ of $U$ contains a point\n$u$ whose $R$-equivalence class $t(s^{-1}(\\{u\\}))$ is contained\nin an affine open of $U$\\footnote{Let $E \\subset U$ be the\nset of points $u$ such that $t(s^{-1}(\\{u\\}))$\nis contained in an affine open of $U$. Condition (3) holds\nif $E = U$, or if every finite type point of $U$ is in $E$, or\nif every $u \\in U$ specializes to a point of $E$.}.\n\\end{enumerate}\nThen there exists a finite locally free morphism $U \\to M$\nof schemes over $S$ such that $R = U \\times_M U$ and such that $M$\nrepresents the quotient sheaf $U/R$ in the fppf topology.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Obtaining a scheme","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07S6","source_file":"spaces-properties.tex","source_line":1703,"source_end_line":1722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1703-L1722","statement_sha256":"12ef7ed987cd9c79145d647951f6b3fb9c04bded21ec65e372c11a926f33c011","origin":"The Stacks Project","memory_eligible":false,"source_rank":10784,"rank":10784,"depth":10,"x":229.112,"y":1650.95,"cluster":"algebraic-spaces"},{"id":"stacks:07S7","tag":"07S7","title":"Obtaining a scheme · Lemma 07S7","summary":"Let S be a scheme. Let G → S be a group scheme. Let X → S be a morphism of schemes. Let a : G ×_S X → X be an action. Assume that • G → S is finite locally free, • the action a is free, • X → S is affine, or quasi-affine, or projective, or quasi-projective, or X is isomorphic to an open subscheme of an affine scheme, or X is isomorphic to an open subscheme of Proj(A) for some graded ring A, or G → S is radicial. Then the fppf quotient sheaf X/G is a scheme and X → X/G is…","statement_latex":"Let $S$ be a scheme. Let $G \\to S$ be a group scheme. Let $X \\to S$ be\na morphism of schemes. Let $a : G \\times_S X \\to X$ be an action. Assume that\n\\begin{enumerate}\n\\item $G \\to S$ is finite locally free,\n\\item the action $a$ is free,\n\\item $X \\to S$ is affine, or quasi-affine, or projective, or\nquasi-projective, or $X$ is isomorphic to an open subscheme of an\naffine scheme, or $X$ is isomorphic to an open subscheme of $\\text{Proj}(A)$\nfor some graded ring $A$, or $G \\to S$ is radicial.\n\\end{enumerate}\nThen the fppf quotient sheaf $X/G$ is a scheme and $X \\to X/G$\nis an fppf $G$-torsor.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Obtaining a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07S7","source_file":"spaces-properties.tex","source_line":1753,"source_end_line":1767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1753-L1767","statement_sha256":"b117ae36eab8aff2fc66a638a0a136bd343852f4d8b1770290b5083843fdc12d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10785,"rank":10785,"depth":22,"x":189.498,"y":1561.753,"cluster":"algebraic-spaces"},{"id":"stacks:0BBM","tag":"0BBM","title":"Obtaining a scheme · Lemma 0BBM","summary":"Notation and assumptions as in Proposition [Tag 07S6]. Then • if U is quasi-separated over S, then U/R is quasi-separated over S, • if U is quasi-separated, then U/R is quasi-separated, • if U is separated over S, then U/R is separated over S, • if U is separated, then U/R is separated, and • add more here. Similar results hold in the setting of Lemma [Tag 07S7].","statement_latex":"Notation and assumptions as in\nProposition \\ref{proposition-finite-flat-equivalence-global}. Then\n\\begin{enumerate}\n\\item if $U$ is quasi-separated over $S$, then $U/R$ is quasi-separated\nover $S$,\n\\item if $U$ is quasi-separated, then $U/R$ is quasi-separated,\n\\item if $U$ is separated over $S$, then $U/R$ is separated over $S$,\n\\item if $U$ is separated, then $U/R$ is separated, and\n\\item add more here.\n\\end{enumerate}\nSimilar results hold in the setting of Lemma \\ref{lemma-quotient-scheme}.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Obtaining a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBM","source_file":"spaces-properties.tex","source_line":1802,"source_end_line":1815,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1802-L1815","statement_sha256":"33a072bc9324be6d3c46b03102353326fa7375ec07b3ea35b721aaccd29d9380","origin":"The Stacks Project","memory_eligible":false,"source_rank":10786,"rank":10786,"depth":55,"x":290.898,"y":1605.245,"cluster":"algebraic-spaces"},{"id":"stacks:06NJ","tag":"06NJ","title":"Points on quasi-separated spaces · Lemma 06NJ","summary":"Let S be a scheme. Let X be a Zariski locally quasi-separated algebraic space over S. Then the topological space |X| is sober (see Topology, Definition [Tag 004X]).","statement_latex":"Let $S$ be a scheme. Let $X$ be a Zariski locally quasi-separated\nalgebraic space over $S$. Then the topological space $|X|$ is sober (see\nTopology, Definition \\ref{topology-definition-generic-point}).","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points on quasi-separated spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NJ","source_file":"spaces-properties.tex","source_line":1862,"source_end_line":1867,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1862-L1867","statement_sha256":"cfc173ddac055b68f9c06a835b347cc4ea5ce1332742e0a1a2eab8ac94add3a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10787,"rank":10787,"depth":54,"x":180.656,"y":1630.826,"cluster":"algebraic-spaces"},{"id":"stacks:0A4G","tag":"0A4G","title":"Points on quasi-separated spaces · Lemma 0A4G","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. The topological space |X| is a spectral space.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. The topological space $|X|$ is a spectral space.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points on quasi-separated spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4G","source_file":"spaces-properties.tex","source_line":1910,"source_end_line":1914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1910-L1914","statement_sha256":"3b9ac065e823eb0af6532f865ceadd54f8208edc53e323f3afa82ab25dc4ef86","origin":"The Stacks Project","memory_eligible":false,"source_rank":10788,"rank":10788,"depth":55,"x":241.648,"y":1549.045,"cluster":"algebraic-spaces"},{"id":"stacks:03DZ","tag":"03DZ","title":"Points on quasi-separated spaces · Lemma 03DZ","summary":"Let S be a scheme. Let k be a field. Let X be an algebraic space over S and assume that there exists a surjective étale morphism Spec(k) → X. If X is quasi-separated, then X ≅ Spec(k') where k/k' is a finite separable extension.","statement_latex":"Let $S$ be a scheme. Let $k$ be a field.\nLet $X$ be an algebraic space over $S$ and assume that there exists\na surjective \\'etale morphism $\\Spec(k) \\to X$.\nIf $X$ is quasi-separated, then $X \\cong \\Spec(k')$\nwhere $k/k'$ is a finite separable extension.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points on quasi-separated spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03DZ","source_file":"spaces-properties.tex","source_line":1944,"source_end_line":1951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L1944-L1951","statement_sha256":"f906594e4354a3d8b9e4f23356d112a04f1ad6ed11a0a8b15b96135182c5c52f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10789,"rank":10789,"depth":45,"x":262.531,"y":1644.378,"cluster":"algebraic-spaces"},{"id":"stacks:03EC","tag":"03EC","title":"Étale morphisms of algebraic spaces · Lemma 03EC","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U, U' be schemes over S. • If U → U' is an étale morphism of schemes, and if U' → X is an étale morphism from U' to X, then the composition U → X is an étale morphism from U to X. • If φ : U → X and φ' : U' → X are étale morphisms towards X, and if chi : U → U' is a morphism of schemes such that φ = φ' ∘ chi, then chi is an étale morphism of schemes. • If chi : U → U' is a surjective étale morphism of schemes and…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $U$, $U'$ be schemes over $S$.\n\\begin{enumerate}\n\\item If $U \\to U'$ is an \\'etale morphism of schemes, and\nif $U' \\to X$ is an \\'etale morphism from $U'$ to $X$, then the\ncomposition $U \\to X$ is an \\'etale morphism from $U$ to $X$.\n\\item If $\\varphi : U \\to X$ and $\\varphi' : U' \\to X$ are\n\\'etale morphisms towards $X$, and if $\\chi : U \\to U'$ is a\nmorphism of schemes such that $\\varphi = \\varphi' \\circ \\chi$,\nthen $\\chi$ is an \\'etale morphism of schemes.\n\\item If $\\chi : U \\to U'$ is a surjective \\'etale morphism\nof schemes and $\\varphi' : U' \\to X$ is a morphism such that\n$\\varphi = \\varphi' \\circ \\chi$ is \\'etale, then $\\varphi'$\nis \\'etale.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale morphisms of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EC","source_file":"spaces-properties.tex","source_line":2019,"source_end_line":2037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2019-L2037","statement_sha256":"a878879102f62451d76a921cdda3e2beae6a70adfecff5c68d479ba1f33e9c04","origin":"The Stacks Project","memory_eligible":false,"source_rank":10790,"rank":10790,"depth":43,"x":170.064,"y":1585.673,"cluster":"algebraic-spaces"},{"id":"stacks:03FR","tag":"03FR","title":"Étale morphisms of algebraic spaces · Definition 03FR","summary":"Let S be a scheme. A morphism f : X → Y between algebraic spaces over S is called étale if and only if for every étale morphism φ : U → X where U is a scheme, the composition f ∘ φ is étale also.","statement_latex":"Let $S$ be a scheme.\nA morphism $f : X \\to Y$ between algebraic spaces over $S$ is\ncalled {\\it \\'etale} if and only if for every \\'etale morphism\n$\\varphi : U \\to X$ where $U$ is a scheme, the composition\n$f \\circ \\varphi$ is \\'etale also.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale morphisms of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FR","source_file":"spaces-properties.tex","source_line":2094,"source_end_line":2101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2094-L2101","statement_sha256":"cdfd7e614e920a15ac42a60f44d0edb6f35db9a9b95beb8245e2fffadbe807c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10791,"rank":10791,"depth":0,"x":285.959,"y":1576.454,"cluster":"algebraic-spaces"},{"id":"stacks:03FS","tag":"03FS","title":"Étale morphisms of algebraic spaces · Lemma 03FS","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is étale, • there exists a surjective étale morphism φ : U → X, where U is a scheme, such that the composition f ∘ φ is étale (as a morphism of algebraic spaces), • there exists a surjective étale morphism ψ : V → Y, where V is a scheme, such that the base change V ×_Y X → V is étale (as a morphism of algebraic spaces), • there exists a commutative diagram xymatrix…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is \\'etale,\n\\item there exists a surjective \\'etale morphism $\\varphi : U \\to X$,\nwhere $U$ is a scheme, such that the composition $f \\circ \\varphi$ is\n\\'etale (as a morphism of algebraic spaces),\n\\item there exists a surjective \\'etale morphism $\\psi : V \\to Y$,\nwhere $V$ is a scheme, such that the base change $V \\times_Y X \\to V$\nis \\'etale (as a morphism of algebraic spaces),\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, and the\nleft vertical arrow is surjective such that the horizontal arrow is \\'etale.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale morphisms of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FS","source_file":"spaces-properties.tex","source_line":2111,"source_end_line":2134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2111-L2134","statement_sha256":"af349b1a6c9f6ae3be049d0618bbf4016814c3b5b392e0546d10c5699693cbe6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10792,"rank":10792,"depth":44,"x":207.576,"y":1649.326,"cluster":"algebraic-spaces"},{"id":"stacks:03FT","tag":"03FT","title":"Étale morphisms of algebraic spaces · Lemma 03FT","summary":"The composition of two étale morphisms of algebraic spaces is étale.","statement_latex":"The composition of two \\'etale morphisms of algebraic spaces\nis \\'etale.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale morphisms of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FT","source_file":"spaces-properties.tex","source_line":2163,"source_end_line":2167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2163-L2167","statement_sha256":"66d40ea765d7e696aac73936d5fcdaa1f2c7af4a1b00875ec349df7aae12aa08","origin":"The Stacks Project","memory_eligible":false,"source_rank":10793,"rank":10793,"depth":0,"x":206.77,"y":1550.694,"cluster":"algebraic-spaces"},{"id":"stacks:03FU","tag":"03FU","title":"Étale morphisms of algebraic spaces · Lemma 03FU","summary":"The base change of an étale morphism of algebraic spaces by any morphism of algebraic spaces is étale.","statement_latex":"The base change of an \\'etale morphism of algebraic spaces\nby any morphism of algebraic spaces is \\'etale.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale morphisms of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FU","source_file":"spaces-properties.tex","source_line":2173,"source_end_line":2177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2173-L2177","statement_sha256":"6a346ee7255b04af9b3e52b08a0de06cffee519250c07cfd24d5b53817b2f42e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10794,"rank":10794,"depth":45,"x":287.019,"y":1623.274,"cluster":"algebraic-spaces"},{"id":"stacks:03FV","tag":"03FV","title":"Étale morphisms of algebraic spaces · Lemma 03FV","summary":"Let S be a scheme. Let X, Y, Z be algebraic spaces. Let g : X → Z, h : Y → Z be étale morphisms and let f : X → Y be a morphism such that h ∘ f = g. Then f is étale.","statement_latex":"Let $S$ be a scheme. Let $X, Y, Z$ be algebraic spaces.\nLet $g : X \\to Z$, $h : Y \\to Z$ be \\'etale morphisms and let\n$f : X \\to Y$ be a morphism such that $h \\circ f = g$.\nThen $f$ is \\'etale.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale morphisms of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FV","source_file":"spaces-properties.tex","source_line":2197,"source_end_line":2203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2197-L2203","statement_sha256":"79c14ebce77d9cf9a30e02fddd1ffc0e96eff0a7ae0db5b44167fa81d236208b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10795,"rank":10795,"depth":44,"x":168.984,"y":1615.256,"cluster":"algebraic-spaces"},{"id":"stacks:03IR","tag":"03IR","title":"Étale morphisms of algebraic spaces · Lemma 03IR","summary":"Let S be a scheme. If X → Y is an étale morphism of algebraic spaces over S, then the associated map |X| → |Y| of topological spaces is open.","statement_latex":"Let $S$ be a scheme.\nIf $X \\to Y$ is an \\'etale morphism of algebraic spaces over $S$,\nthen the associated map $|X| \\to |Y|$ of topological spaces\nis open.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale morphisms of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IR","source_file":"spaces-properties.tex","source_line":2222,"source_end_line":2228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2222-L2228","statement_sha256":"c7b958688bb24883ce6701ea93bd829416d7091fa78779a2d890c9da574bc653","origin":"The Stacks Project","memory_eligible":false,"source_rank":10796,"rank":10796,"depth":45,"x":262.86,"y":1553.938,"cluster":"algebraic-spaces"},{"id":"stacks:03KX","tag":"03KX","title":"Étale morphisms of algebraic spaces · Lemma 03KX","summary":"Let S be a scheme. Let X → Spec(k) be étale morphism over S, where k is a field. Then X is a scheme.","statement_latex":"Let $S$ be a scheme. Let $X \\to \\Spec(k)$\nbe \\'etale morphism over $S$, where $k$ is a field.\nThen $X$ is a scheme.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale morphisms of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KX","source_file":"spaces-properties.tex","source_line":2243,"source_end_line":2248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2243-L2248","statement_sha256":"03da749f351776fc61996e4291bec9a4a420e3e6ec4fff885c7f634f3f6ab49c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10797,"rank":10797,"depth":45,"x":242.864,"y":1652.828,"cluster":"algebraic-spaces"},{"id":"stacks:0APL","tag":"0APL","title":"Gabber · Proposition 0APL","summary":"Let S be a scheme. Let X be an algebraic space over S. Then X satisfies the sheaf property for the fpqc topology.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then\n$X$ satisfies the sheaf property for the fpqc topology.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Spaces and fpqc coverings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APL","source_file":"spaces-properties.tex","source_line":2303,"source_end_line":2307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2303-L2307","statement_sha256":"d2de64d40abffd75ae8cfb2646a39363ff5d28bb1bccddad0f2edee1424d5ac7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10798,"rank":10798,"depth":53,"x":177.819,"y":1568.216,"cluster":"algebraic-spaces"},{"id":"stacks:03ED","tag":"03ED","title":"The étale site of an algebraic space · Definition 03ED","summary":"Let S be a scheme. Let Sch_fppf be a big fppf site containing S, and let Sch_etale be the corresponding big étale site (i.e., having the same underlying category). Let X be an algebraic space over S. The small étale site X_etale of X is defined as follows: • An object of X_etale is a morphism φ : U → X where U ∈ Ob((Sch/S)_etale) is a scheme and φ is an étale morphism, • a morphism (φ : U → X) → (φ' : U' → X) is given by a morphism of schemes chi : U → U' such that φ = φ'…","statement_latex":"Let $S$ be a scheme.\nLet $\\Sch_{fppf}$ be a big fppf site containing $S$,\nand let $\\Sch_\\etale$ be the corresponding big \\'etale site\n(i.e., having the same underlying category).\nLet $X$ be an algebraic space over $S$.\nThe {\\it small \\'etale site $X_\\etale$} of $X$ is defined as follows:\n\\begin{enumerate}\n\\item An object of $X_\\etale$ is a morphism $\\varphi : U \\to X$\nwhere $U \\in \\Ob((\\Sch/S)_\\etale)$ is a scheme and\n$\\varphi$ is an \\'etale morphism,\n\\item a morphism $(\\varphi : U \\to X) \\to (\\varphi' : U' \\to X)$\nis given by a morphism of schemes $\\chi : U \\to U'$ such that\n$\\varphi = \\varphi' \\circ \\chi$, and\n\\item a family of morphisms $\\{(U_i \\to X) \\to (U \\to X)\\}_{i \\in I}$\nof $X_\\etale$ is a covering if and only if $\\{U_i \\to U\\}_{i \\in I}$\nis a covering of $(\\Sch/S)_\\etale$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ED","source_file":"spaces-properties.tex","source_line":2430,"source_end_line":2449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2430-L2449","statement_sha256":"a3b464ac753a930fc8c12ec2b85ea1ca27fd8b83691deb90eef077816d8f9171","origin":"The Stacks Project","memory_eligible":false,"source_rank":10799,"rank":10799,"depth":0,"x":294.297,"y":1593.803,"cluster":"algebraic-spaces"},{"id":"stacks:03G0","tag":"03G0","title":"The étale site of an algebraic space · Definition 03G0","summary":"Let S be a scheme. Let Sch_fppf be a big fppf site containing S, and let Sch_etale be the corresponding big étale site (i.e., having the same underlying category). Let X be an algebraic space over S. The site X_spaces, etale of X is defined as follows: • An object of X_spaces, etale is a morphism φ : U → X where U is an algebraic space over S and φ is an étale morphism of algebraic spaces over S, • a morphism (φ : U → X) → (φ' : U' → X) of X_spaces, etale is given by a…","statement_latex":"Let $S$ be a scheme.\nLet $\\Sch_{fppf}$ be a big fppf site containing $S$,\nand let $\\Sch_\\etale$ be the corresponding big \\'etale site\n(i.e., having the same underlying category).\nLet $X$ be an algebraic space over $S$.\nThe site {\\it $X_{spaces, \\etale}$} of $X$ is defined as follows:\n\\begin{enumerate}\n\\item An object of $X_{spaces, \\etale}$ is a morphism\n$\\varphi : U \\to X$ where $U$ is an algebraic space over $S$ and\n$\\varphi$ is an \\'etale morphism of algebraic spaces over $S$,\n\\item a morphism $(\\varphi : U \\to X) \\to (\\varphi' : U' \\to X)$ of\n$X_{spaces, \\etale}$ is given by a morphism of algebraic spaces\n$\\chi : U \\to U'$ such that $\\varphi = \\varphi' \\circ \\chi$, and\n\\item a family of morphisms\n$\\{\\varphi_i : (U_i \\to X) \\to (U \\to X)\\}_{i \\in I}$\nof $X_{spaces, \\etale}$ is a covering if and only if\n$|U| = \\bigcup \\varphi_i(|U_i|)$.\n\\end{enumerate}\nAs usual we choose a set of coverings of this type, including at least\nthe coverings in $X_\\etale$, as in\nSets, Lemma \\ref{sets-lemma-coverings-site}\nto turn $X_{spaces, \\etale}$ into a site.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03G0","source_file":"spaces-properties.tex","source_line":2467,"source_end_line":2491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2467-L2491","statement_sha256":"8b78f321fcd45556b6cab5eeb0c3c2a08375814cea61f23b11aea9003ffce489","origin":"The Stacks Project","memory_eligible":false,"source_rank":10800,"rank":10800,"depth":2,"x":187.401,"y":1641.218,"cluster":"algebraic-spaces"},{"id":"stacks:03G1","tag":"03G1","title":"The étale site of an algebraic space · Lemma 03G1","summary":"The functor X_etale → X_spaces, etale, U/X ↦ U/X is a special cocontinuous functor (Sites, Definition [Tag 03CG]) and hence induces an equivalence of topoi Sh(X_etale) → Sh(X_spaces, etale).","statement_latex":"The functor\n$$\nX_\\etale \\longrightarrow X_{spaces, \\etale}, \\quad\nU/X \\longmapsto U/X\n$$\nis a special cocontinuous functor\n(Sites, Definition \\ref{sites-definition-special-cocontinuous-functor})\nand hence induces an equivalence of topoi\n$\\Sh(X_\\etale) \\to \\Sh(X_{spaces, \\etale})$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03G1","source_file":"spaces-properties.tex","source_line":2499,"source_end_line":2510,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2499-L2510","statement_sha256":"a8dfbe76ac43e1f9dd3f9a76fca69e19f62b5116c977d4d7705961bed9d93ce8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10801,"rank":10801,"depth":8,"x":228.259,"y":1545.218,"cluster":"algebraic-spaces"},{"id":"stacks:0H01","tag":"0H01","title":"The étale site of an algebraic space · Definition 0H01","summary":"Let S be a scheme. Let Sch_fppf be a big fppf site containing S, and let Sch_etale be the corresponding big étale site (i.e., having the same underlying category). Let X be an algebraic space over S. The site X_affine, etale of X is defined as follows: • An object of X_affine, etale is a morphism φ : U → X where U ∈ Ob((Sch/S)_etale) is an affine scheme and φ is an étale morphism, • a morphism (φ : U → X) → (φ' : U' → X) of X_affine, etale is given by a morphism of…","statement_latex":"Let $S$ be a scheme.\nLet $\\Sch_{fppf}$ be a big fppf site containing $S$,\nand let $\\Sch_\\etale$ be the corresponding big \\'etale site\n(i.e., having the same underlying category).\nLet $X$ be an algebraic space over $S$.\nThe site {\\it $X_{affine, \\etale}$} of $X$ is defined as follows:\n\\begin{enumerate}\n\\item An object of $X_{affine, \\etale}$ is a morphism\n$\\varphi : U \\to X$ where $U \\in \\Ob((\\Sch/S)_\\etale)$ is an affine scheme and\n$\\varphi$ is an \\'etale morphism,\n\\item a morphism $(\\varphi : U \\to X) \\to (\\varphi' : U' \\to X)$ of\n$X_{affine, \\etale}$ is given by a morphism of schemes\n$\\chi : U \\to U'$ such that $\\varphi = \\varphi' \\circ \\chi$, and\n\\item a family of morphisms\n$\\{\\varphi_i : (U_i \\to X) \\to (U \\to X)\\}_{i \\in I}$\nof $X_{affine, \\etale}$ is a covering if and only if\n$\\{U_i \\to U\\}$ is a standard \\'etale covering, see\nTopologies, Definition \\ref{topologies-definition-standard-etale}.\n\\end{enumerate}\nAs usual we choose a set of coverings of this type, as in\nSets, Lemma \\ref{sets-lemma-coverings-site}\nto turn $X_{affine, \\etale}$ into a site.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H01","source_file":"spaces-properties.tex","source_line":2547,"source_end_line":2571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2547-L2571","statement_sha256":"71d700c7b2254aaf35bbef87d1c32fbeb3e9ffd0153dde0e29505a53757fedb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10802,"rank":10802,"depth":2,"x":275.518,"y":1639.563,"cluster":"algebraic-spaces"},{"id":"stacks:04JS","tag":"04JS","title":"The étale site of an algebraic space · Lemma 04JS","summary":"Let S be a scheme. Let X be an algebraic space over S. The functor X_affine, etale → X_etale is special cocontinuous and induces an equivalence of topoi from Sh(X_affine, etale) to Sh(X_etale).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe functor $X_{affine, \\etale} \\to X_\\etale$\nis special cocontinuous and induces an equivalence of topoi from\n$\\Sh(X_{affine, \\etale})$ to $\\Sh(X_\\etale)$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JS","source_file":"spaces-properties.tex","source_line":2573,"source_end_line":2579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2573-L2579","statement_sha256":"bf514da3c3226557766c25e0b6bd440f049784a0d3aaff34ef30160852231e1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10803,"rank":10803,"depth":9,"x":164.364,"y":1596.641,"cluster":"algebraic-spaces"},{"id":"stacks:04JT","tag":"04JT","title":"The étale site of an algebraic space · Definition 04JT","summary":"Let S be a scheme. Let X be an algebraic space over S. The étale topos of X, or more precisely the small étale topos of X is the category Sh(X_etale) of sheaves of sets on X_etale.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe {\\it \\'etale topos} of $X$, or more precisely the\n{\\it small \\'etale topos} of $X$ is the category\n$\\Sh(X_\\etale)$\nof sheaves of sets on $X_\\etale$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JT","source_file":"spaces-properties.tex","source_line":2586,"source_end_line":2593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2586-L2593","statement_sha256":"6ff8b2d6d6b7aa29c6ce0356fd1db52ebd36cd4ea7f4f9a7051f2ebe712f60eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10804,"rank":10804,"depth":0,"x":281.299,"y":1565.098,"cluster":"algebraic-spaces"},{"id":"stacks:03G2","tag":"03G2","title":"The étale site of an algebraic space · Lemma 03G2","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • The continuous functor Y_spaces, etale → X_spaces, etale, V ↦ X ×_Y V induces a morphism of sites f_spaces, etale : X_spaces, etale → Y_spaces, etale. • The rule f ↦ f_spaces, etale is compatible with compositions, in other words (f ∘ g)_spaces, etale = f_spaces, etale ∘ g_spaces, etale (see Sites, Definition [Tag 03CC]). • The morphism of topoi associated to f_spaces, etale induces, via Lemma…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item The continuous functor\n$$\nY_{spaces, \\etale} \\longrightarrow X_{spaces, \\etale}, \\quad\nV \\longmapsto X \\times_Y V\n$$\ninduces a morphism of sites\n$$\nf_{spaces, \\etale} :\nX_{spaces, \\etale}\n\\to\nY_{spaces, \\etale}.\n$$\n\\item The rule $f \\mapsto f_{spaces, \\etale}$ is compatible with\ncompositions, in other words $(f \\circ g)_{spaces, \\etale}\n= f_{spaces, \\etale} \\circ g_{spaces, \\etale}$ (see\nSites, Definition \\ref{sites-definition-composition-morphisms-sites}).\n\\item The morphism of topoi associated to $f_{spaces, \\etale}$\ninduces, via Lemma \\ref{lemma-compare-etale-sites}, a morphism of topoi\n$f_{small} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$\nwhose construction is compatible with compositions.\n\\item If $f$ is a representable morphism of algebraic spaces,\nthen $f_{small}$ comes from a morphism of sites\n$X_\\etale \\to Y_\\etale$,\ncorresponding to the continuous functor $V \\mapsto X \\times_Y V$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03G2","source_file":"spaces-properties.tex","source_line":2608,"source_end_line":2638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2608-L2638","statement_sha256":"001981e69c6a05e7ca8a5d7e8a15708e544ea5d296c2b82392abfc10afef0f10","origin":"The Stacks Project","memory_eligible":false,"source_rank":10805,"rank":10805,"depth":46,"x":220.203,"y":1655.057,"cluster":"algebraic-spaces"},{"id":"stacks:03G3","tag":"03G3","title":"The étale site of an algebraic space · Definition 03G3","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a sheaf of sets on X_etale and let G be a sheaf of sets on Y_etale. An f-map φ : G → F is a collection of maps φ_(U, V, g) : G(V) → F(U) indexed by commutative diagrams xymatrix U ar[d]_g ar[r] & X ar[d]^f V ar[r] & Y where U ∈ X_etale, V ∈ Y_etale such that whenever given an extended diagram xymatrix U' ar[r] ar[d]_g' & U ar[d]_g ar[r] & X ar[d]^f V' ar[r] & V ar[r] & Y with V' → V and U'…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$ and\nlet $\\mathcal{G}$ be a sheaf of sets on $Y_\\etale$.\nAn {\\it $f$-map $\\varphi : \\mathcal{G} \\to \\mathcal{F}$}\nis a collection of maps\n$\\varphi_{(U, V, g)} : \\mathcal{G}(V) \\to \\mathcal{F}(U)$\nindexed by commutative diagrams\n$$\n\\xymatrix{\nU \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\\nV \\ar[r] & Y\n}\n$$\nwhere $U \\in X_\\etale$, $V \\in Y_\\etale$ such that whenever\ngiven an extended diagram\n$$\n\\xymatrix{\nU' \\ar[r] \\ar[d]_{g'} & U \\ar[d]_g \\ar[r] & X \\ar[d]^f \\\\\nV' \\ar[r] & V \\ar[r] & Y\n}\n$$\nwith $V' \\to V$ and $U' \\to U$ \\'etale morphisms of schemes the diagram\n$$\n\\xymatrix{\n\\mathcal{G}(V)\n\\ar[rr]_{\\varphi_{(U, V, g)}}\n\\ar[d]_{\\text{restriction of }\\mathcal{G}} & &\n\\mathcal{F}(U)\n\\ar[d]^{\\text{restriction of }\\mathcal{F}} \\\\\n\\mathcal{G}(V')\n\\ar[rr]^{\\varphi_{(U', V', g')}} & &\n\\mathcal{F}(U')\n}\n$$\ncommutes.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03G3","source_file":"spaces-properties.tex","source_line":2678,"source_end_line":2716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2678-L2716","statement_sha256":"0e6b2b24c4b9ac02d950dc37dbf567ed5674a421357b7076e754c9276f69325f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10806,"rank":10806,"depth":0,"x":192.807,"y":1553.666,"cluster":"algebraic-spaces"},{"id":"stacks:03G4","tag":"03G4","title":"The étale site of an algebraic space · Lemma 03G4","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a sheaf of sets on X_etale and let G be a sheaf of sets on Y_etale. There are canonical bijections between the following three sets: • The set of maps G → f_small, *F. • The set of maps f_small^-1G → F. • The set of f-maps φ : G → F.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a sheaf of sets on $X_\\etale$ and\nlet $\\mathcal{G}$ be a sheaf of sets on $Y_\\etale$.\nThere are canonical bijections between the following three sets:\n\\begin{enumerate}\n\\item The set of maps $\\mathcal{G} \\to f_{small, *}\\mathcal{F}$.\n\\item The set of maps $f_{small}^{-1}\\mathcal{G} \\to \\mathcal{F}$.\n\\item The set of $f$-maps $\\varphi : \\mathcal{G} \\to \\mathcal{F}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03G4","source_file":"spaces-properties.tex","source_line":2718,"source_end_line":2730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2718-L2730","statement_sha256":"15f21035d0f8db0b571a615d615c77af5a8a9878fc8d13e723a4a92c7f88bf2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10807,"rank":10807,"depth":9,"x":294.932,"y":1613.112,"cluster":"algebraic-spaces"},{"id":"stacks:03LP","tag":"03LP","title":"The étale site of an algebraic space · Lemma 03LP","summary":"Let S be a scheme, and let f : X → Y be a morphism of algebraic spaces over S. Assume f is étale. In this case there is a functor j : X_etale → Y_etale, (φ : U → X) ↦ (f ∘ φ : U → Y) which is cocontinuous. The morphism of topoi f_small is the morphism of topoi associated to j, see Sites, Lemma [Tag 00XO]. Moreover, j is continuous as well, hence Sites, Lemma [Tag 00XR] applies. In particular f_small^-1G(U) = G(jU) for all sheaves G on Y_etale.","statement_latex":"Let $S$ be a scheme, and let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is \\'etale. In this case there is a functor\n$$\nj : X_\\etale \\to Y_\\etale, \\quad\n(\\varphi : U \\to X) \\mapsto (f \\circ \\varphi : U \\to Y)\n$$\nwhich is cocontinuous. The morphism of topoi $f_{small}$ is the\nmorphism of topoi associated to $j$, see\nSites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi}.\nMoreover, $j$ is continuous as well, hence\nSites, Lemma \\ref{sites-lemma-when-shriek}\napplies. In particular $f_{small}^{-1}\\mathcal{G}(U) = \\mathcal{G}(jU)$\nfor all sheaves $\\mathcal{G}$ on $Y_\\etale$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LP","source_file":"spaces-properties.tex","source_line":2804,"source_end_line":2819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2804-L2819","statement_sha256":"743543778aa16489d587bd4ef5d279c76a063660b15434bba2c78246e4844596","origin":"The Stacks Project","memory_eligible":false,"source_rank":10808,"rank":10808,"depth":47,"x":171.356,"y":1627.278,"cluster":"algebraic-spaces"},{"id":"stacks:03LR","tag":"03LR","title":"The étale site of an algebraic space · Lemma 03LR","summary":"Let S be a scheme. Let xymatrix X' ar[r] ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian square of algebraic spaces over S. Let F be a sheaf on X_etale. If g is étale, then • f'_small, *(F|_X') = (f_small, *F)|_Y' in Sh(Y'_etale)^-1(G|_Y') = (f_small^-1G)|_X' because of commutativity of the diagram and ([Tag 03LQ]), and • if F is an abelian sheaf, then R^if'_small, *(F|_X') = (R^if_small, *F)|_Y'.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r] \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian square of algebraic spaces over $S$. Let\n$\\mathcal{F}$ be a sheaf on $X_\\etale$. If $g$ is \\'etale, then\n\\begin{enumerate}\n\\item $f'_{small, *}(\\mathcal{F}|_{X'}) = (f_{small, *}\\mathcal{F})|_{Y'}$\nin $\\Sh(Y'_\\etale)$\\footnote{Also\n$(f')_{small}^{-1}(\\mathcal{G}|_{Y'}) = (f_{small}^{-1}\\mathcal{G})|_{X'}$\nbecause of commutativity of the diagram and (\\ref{equation-restrict})}, and\n\\item if $\\mathcal{F}$ is an abelian sheaf, then\n$R^if'_{small, *}(\\mathcal{F}|_{X'}) = (R^if_{small, *}\\mathcal{F})|_{Y'}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LR","source_file":"spaces-properties.tex","source_line":2870,"source_end_line":2889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2870-L2889","statement_sha256":"6b59b0d547e126e580d9bce1883f8e773951a61f5e04586fc625c25bd7b92961","origin":"The Stacks Project","memory_eligible":false,"source_rank":10809,"rank":10809,"depth":9,"x":251.387,"y":1546.409,"cluster":"algebraic-spaces"},{"id":"stacks:03LS","tag":"03LS","title":"The étale site of an algebraic space · Lemma 03LS","summary":"Let S be a scheme. Let X be an algebraic space over S. A sheaf F on X_etale is given by the following data: • for every U ∈ Ob(X_etale) a sheaf F_U on U_etale, • for every f : U' → U in X_etale an isomorphism c_f : f_small^-1F_U → F_U'. These data are subject to the condition that given any f : U' → U and g : U\" → U' in X_etale the composition c_g ∘ g_small^-1 c_f is equal to c_f ∘ g.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nA sheaf $\\mathcal{F}$ on $X_\\etale$ is given by the following data:\n\\begin{enumerate}\n\\item for every $U \\in \\Ob(X_\\etale)$ a sheaf\n$\\mathcal{F}_U$ on $U_\\etale$,\n\\item for every $f : U' \\to U$ in $X_\\etale$ an isomorphism\n$c_f : f_{small}^{-1}\\mathcal{F}_U \\to \\mathcal{F}_{U'}$.\n\\end{enumerate}\nThese data are subject to the condition that given any $f : U' \\to U$\nand $g : U'' \\to U'$ in $X_\\etale$ the composition\n$c_g \\circ g_{small}^{-1} c_f$ is equal to $c_{f \\circ g}$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LS","source_file":"spaces-properties.tex","source_line":2919,"source_end_line":2932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2919-L2932","statement_sha256":"d096bc941f10b6087e4581a9a8dc6a596af6edde2360fd8ded44272779dca15d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10810,"rank":10810,"depth":48,"x":257.427,"y":1651.844,"cluster":"algebraic-spaces"},{"id":"stacks:05YY","tag":"05YY","title":"The étale site of an algebraic space · Lemma 05YY","summary":"With S, φ : U → X, and (U, R, s, t, c, e, i) as above. For any sheaf F on X_etale the sheaf instead of φ_small^-1 and similarly for all the other pullbacks. G = φ^-1F comes equipped with a canonical isomorphism α : t^-1G → s^-1G such that the diagram xymatrix & pr_1^-1t^-1G ar[r]_-pr_1^-1α & pr_1^-1s^-1G ar@=[rd] & pr_0^-1s^-1G ar@=[ru] & & & c^-1s^-1G & pr_0^-1t^-1G ar[lu]^pr_0^-1α ar@=[r] & c^-1t^-1G ar[ru]_c^-1α is a commutative. The functor F ↦ (G, α) defines an…","statement_latex":"With $S$, $\\varphi : U \\to X$, and $(U, R, s, t, c, e, i)$ as above.\nFor any sheaf $\\mathcal{F}$ on $X_\\etale$ the\nsheaf\\footnote{In this lemma\nand its proof we write simply $\\varphi^{-1}$ instead of $\\varphi_{small}^{-1}$\nand similarly for all the other pullbacks.}\n$\\mathcal{G} = \\varphi^{-1}\\mathcal{F}$ comes equipped with a canonical\nisomorphism\n$$\n\\alpha :\nt^{-1}\\mathcal{G}\n\\longrightarrow\ns^{-1}\\mathcal{G}\n$$\nsuch that the diagram\n$$\n\\xymatrix{\n& \\text{pr}_1^{-1}t^{-1}\\mathcal{G} \\ar[r]_-{\\text{pr}_1^{-1}\\alpha} &\n\\text{pr}_1^{-1}s^{-1}\\mathcal{G} \\ar@{=}[rd] & \\\\\n\\text{pr}_0^{-1}s^{-1}\\mathcal{G} \\ar@{=}[ru] & & &\nc^{-1}s^{-1}\\mathcal{G} \\\\\n&\n\\text{pr}_0^{-1}t^{-1}\\mathcal{G} \\ar[lu]^{\\text{pr}_0^{-1}\\alpha} \\ar@{=}[r] &\nc^{-1}t^{-1}\\mathcal{G} \\ar[ru]_{c^{-1}\\alpha}\n}\n$$\nis a commutative. The functor $\\mathcal{F} \\mapsto (\\mathcal{G}, \\alpha)$\ndefines an equivalence of categories between sheaves on\n$X_\\etale$ and pairs $(\\mathcal{G}, \\alpha)$ as above.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The étale site of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YY","source_file":"spaces-properties.tex","source_line":2949,"source_end_line":2979,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L2949-L2979","statement_sha256":"4ec3938d48de41dccf3095fab5acdea93addd7b9c5cf2b5c24e7ef83d78169ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":10811,"rank":10811,"depth":48,"x":167.855,"y":1577.25,"cluster":"algebraic-spaces"},{"id":"stacks:0486","tag":"0486","title":"Points of the small étale site · Definition 0486","summary":"Let S be a scheme. Let X be an algebraic space over S. • A geometric point of X is a morphism overlinex : Spec(k) → X, where k is an algebraically closed field. We often abuse notation and write overlinex = Spec(k). • For every geometric point overlinex we have the corresponding \"image\" point x ∈ |X|. We say that overlinex is a geometric point lying over x.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item A {\\it geometric point} of $X$ is a morphism\n$\\overline{x} : \\Spec(k) \\to X$, where $k$ is an algebraically\nclosed field. We often abuse notation and\nwrite $\\overline{x} = \\Spec(k)$.\n\\item For every geometric point $\\overline{x}$ we have the corresponding\n``image'' point $x \\in |X|$. We say that $\\overline{x}$ is a\n{\\it geometric point lying over $x$}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0486","source_file":"spaces-properties.tex","source_line":3049,"source_end_line":3061,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3049-L3061","statement_sha256":"cb818b1ab97647654c39570c68c0a4ac6aff76b9f5951d9a6af3a6045f782c38","origin":"The Stacks Project","memory_eligible":false,"source_rank":10812,"rank":10812,"depth":0,"x":294.354,"y":1581.446,"cluster":"algebraic-spaces"},{"id":"stacks:04JV","tag":"04JV","title":"Points of the small étale site · Definition 04JV","summary":"Let S be a scheme. Let X be an algebraic space over S. Let overlinex be a geometric point of X. • An étale neighborhood of overlinex of X is a commutative diagram xymatrix & U ar[d]^φ bar x ar[r]^bar x ar[ur]^bar u & X where φ is an étale morphism of algebraic spaces over S. We will use the notation φ : (U, overlineu) → (X, overlinex) to indicate this situation. • A morphism of étale neighborhoods (U, overlineu) → (U', overlineu') is an X-morphism h : U → U' such that…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\overline{x}$ be a geometric point of $X$.\n\\begin{enumerate}\n\\item An {\\it \\'etale neighborhood} of $\\overline{x}$\nof $X$ is a commutative diagram\n$$\n\\xymatrix{\n& U \\ar[d]^\\varphi \\\\\n{\\bar x} \\ar[r]^{\\bar x} \\ar[ur]^{\\bar u} & X\n}\n$$\nwhere $\\varphi$ is an \\'etale morphism of algebraic spaces over $S$.\nWe will use the notation $\\varphi : (U, \\overline{u}) \\to (X, \\overline{x})$\nto indicate this situation.\n\\item A {\\it morphism of \\'etale neighborhoods}\n$(U, \\overline{u}) \\to (U', \\overline{u}')$\nis an $X$-morphism $h : U \\to U'$\nsuch that $\\overline{u}' = h \\circ \\overline{u}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JV","source_file":"spaces-properties.tex","source_line":3069,"source_end_line":3090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3069-L3090","statement_sha256":"2cca497a0fde5b9c9151b949e5727396fd11242c6fddbc44985cf3ebb9adad3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10813,"rank":10813,"depth":0,"x":197.349,"y":1650.38,"cluster":"algebraic-spaces"},{"id":"stacks:04JW","tag":"04JW","title":"Points of the small étale site · Lemma 04JW","summary":"Let S be a scheme. Let X be an algebraic space over S. Let overlinex be a geometric point of X. The category of étale neighborhoods is cofiltered. More precisely: • Let (U_i, overlineu_i)_i = 1, 2 be two étale neighborhoods of overlinex in X. Then there exists a third étale neighborhood (U, overlineu) and morphisms (U, overlineu) → (U_i, overlineu_i), i = 1, 2. • Let h_1, h_2: (U, overlineu) → (U', overlineu') be two morphisms between étale neighborhoods of overlines.…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\overline{x}$ be a geometric point of $X$.\nThe category of \\'etale neighborhoods is cofiltered. More precisely:\n\\begin{enumerate}\n\\item Let $(U_i, \\overline{u}_i)_{i = 1, 2}$ be two \\'etale neighborhoods of\n$\\overline{x}$ in $X$. Then there exists a third \\'etale neighborhood\n$(U, \\overline{u})$ and morphisms\n$(U, \\overline{u}) \\to (U_i, \\overline{u}_i)$, $i = 1, 2$.\n\\item Let $h_1, h_2: (U, \\overline{u}) \\to (U', \\overline{u}')$ be two\nmorphisms between \\'etale neighborhoods of $\\overline{s}$. Then there exist an\n\\'etale neighborhood $(U'', \\overline{u}'')$ and a morphism\n$h : (U'', \\overline{u}'') \\to (U, \\overline{u})$\nwhich equalizes $h_1$ and $h_2$, i.e., such that\n$h_1 \\circ h = h_2 \\circ h$.\n\\end{enumerate}\nMoreover, given any \\'etale neighbourhood\n$(U, \\overline{u}) \\to (X, \\overline{x})$\nthere exists a morphism of \\'etale neighbourhoods\n$(U', \\overline{u}') \\to (U, \\overline{u})$\nwhere $U'$ is a scheme.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JW","source_file":"spaces-properties.tex","source_line":3100,"source_end_line":3122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3100-L3122","statement_sha256":"1c371f42d11f2c1ec33d7919d986d55605ea2bc82b3cf655e88eff8949925e0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10814,"rank":10814,"depth":46,"x":213.503,"y":1544.122,"cluster":"algebraic-spaces"},{"id":"stacks:05VN","tag":"05VN","title":"Points of the small étale site · Lemma 05VN","summary":"Let S be a scheme. Let X be an algebraic space over S. Let overlinex : Spec(k) → X be a geometric point of X lying over x ∈ |X|. Let φ : U → X be an étale morphism of algebraic spaces and let u ∈ |U| with φ(u) = x. Then there exists a geometric point overlineu : Spec(k) → U lying over u with overlinex = φ ∘ overlineu.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\overline{x} : \\Spec(k) \\to X$ be a geometric point of $X$\nlying over $x \\in |X|$. Let $\\varphi : U \\to X$ be an \\'etale morphism\nof algebraic spaces and let $u \\in |U|$ with $\\varphi(u) = x$.\nThen there exists a geometric point\n$\\overline{u} : \\Spec(k) \\to U$ lying over $u$ with\n$\\overline{x} = \\varphi \\circ \\overline{u}$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VN","source_file":"spaces-properties.tex","source_line":3163,"source_end_line":3172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3163-L3172","statement_sha256":"2e68cbb0c11febacc306fad66643c9ac953247e468ff228ffc0b362a93de07dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10815,"rank":10815,"depth":46,"x":287.305,"y":1631.956,"cluster":"algebraic-spaces"},{"id":"stacks:04JX","tag":"04JX","title":"Points of the small étale site · Lemma 04JX","summary":"Let S be a scheme. Let X be an algebraic space over S. Let overlinex be a geometric point of X. Let (U, overlineu) an étale neighborhood of overlinex. Let (φ_i : U_i → U)_i ∈ I be an étale covering in X_spaces, etale. Then there exist i ∈ I and overlineu_i : overlinex → U_i such that φ_i : (U_i, overlineu_i) → (U, overlineu) is a morphism of étale neighborhoods.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\overline{x}$ be a geometric point of $X$.\nLet $(U, \\overline{u})$ an \\'etale neighborhood of $\\overline{x}$.\nLet $\\{\\varphi_i : U_i \\to U\\}_{i \\in I}$ be an \\'etale covering in\n$X_{spaces, \\etale}$.\nThen there exist $i \\in I$ and $\\overline{u}_i : \\overline{x} \\to U_i$\nsuch that $\\varphi_i : (U_i, \\overline{u}_i) \\to (U, \\overline{u})$\nis a morphism of \\'etale neighborhoods.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JX","source_file":"spaces-properties.tex","source_line":3209,"source_end_line":3219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3209-L3219","statement_sha256":"1bc9d6e91d154ac5ddc7693ee89d6af38f9d3b6d18b434bce3e1e086dc2ba90f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10816,"rank":10816,"depth":47,"x":161.804,"y":1608.984,"cluster":"algebraic-spaces"},{"id":"stacks:04JY","tag":"04JY","title":"Points of the small étale site · Definition 04JY","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a presheaf on X_etale. Let overlinex be a geometric point of X. The stalk of F at overlinex is F_bar x = colim_(U, overlineu) F(U) where (U, overlineu) runs over all étale neighborhoods of overlinex in X with U ∈ Ob(X_etale).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a presheaf on $X_\\etale$.\nLet $\\overline{x}$ be a geometric point of $X$.\nThe {\\it stalk} of $\\mathcal{F}$ at $\\overline{x}$ is\n$$\n\\mathcal{F}_{\\bar x}\n=\n\\colim_{(U, \\overline{u})} \\mathcal{F}(U)\n$$\nwhere $(U, \\overline{u})$ runs over all \\'etale neighborhoods\nof $\\overline{x}$ in $X$ with $U \\in \\Ob(X_\\etale)$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04JY","source_file":"spaces-properties.tex","source_line":3230,"source_end_line":3243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3230-L3243","statement_sha256":"e3b859799631a280eefbd68c0c4bf6b8057c4ab8e09fdff70802c3d05f635195","origin":"The Stacks Project","memory_eligible":false,"source_rank":10817,"rank":10817,"depth":0,"x":273.214,"y":1554.52,"cluster":"algebraic-spaces"},{"id":"stacks:04K0","tag":"04K0","title":"Points of the small étale site · Lemma 04K0","summary":"A geometric point of an algebraic space gives a point of its étale topos. Let S be a scheme. Let X be an algebraic space over S. Let overlinex be a geometric point of X. Consider the functor u : X_etale → Sets, U ↦ |U_overlinex| Then u defines a point p of the site X_etale (Sites, Definition [Tag 00Y5]) and its associated stalk functor F ↦ F_p (Sites, Equation [Tag 04EH]) is the functor F ↦ F_overlinex defined above.","statement_latex":"\\begin{slogan}\nA geometric point of an algebraic space gives a point of its \\'etale topos.\n\\end{slogan}\nLet $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\overline{x}$ be a geometric point of $X$.\nConsider the functor\n$$\nu : X_\\etale \\longrightarrow \\textit{Sets}, \\quad\nU \\longmapsto |U_{\\overline{x}}|\n$$\nThen $u$ defines a point $p$ of the site $X_\\etale$\n(Sites, Definition \\ref{sites-definition-point})\nand its associated stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$\n(Sites, Equation \\ref{sites-equation-stalk})\nis the functor $\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$\ndefined above.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04K0","source_file":"spaces-properties.tex","source_line":3288,"source_end_line":3307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3288-L3307","statement_sha256":"97d31b1dc1499b4a0e7b150fbb0a27cd984ca46e3736749be26a0d5fec58684a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10818,"rank":10818,"depth":48,"x":234.725,"y":1658.261,"cluster":"algebraic-spaces"},{"id":"stacks:04K1","tag":"04K1","title":"Points of the small étale site · Lemma 04K1","summary":"Let S be a scheme. Let X be an algebraic space over S. Let overlinex be a geometric point of X. • The stalk functor PAb(X_etale) → Ab, F ↦ F_overlinex is exact. • We have (F^\\#)_overlinex = F_overlinex for any presheaf of sets F on X_etale. • The functor Ab(X_etale) → Ab, F ↦ F_overlinex is exact. • Similarly the functors PSh(X_etale) → Sets and Sh(X_etale) → Sets given by the stalk functor F ↦ F_overlinex are exact (see Categories, Definition [Tag 0034]) and commute with…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\overline{x}$ be a geometric point of $X$.\n\\begin{enumerate}\n\\item The stalk functor\n$\\textit{PAb}(X_\\etale) \\to \\textit{Ab}$,\n$\\mathcal{F}  \\mapsto  \\mathcal{F}_{\\overline{x}}$\nis exact.\n\\item We have $(\\mathcal{F}^\\#)_{\\overline{x}} = \\mathcal{F}_{\\overline{x}}$\nfor any presheaf of sets $\\mathcal{F}$ on $X_\\etale$.\n\\item The functor\n$\\textit{Ab}(X_\\etale) \\to \\textit{Ab}$,\n$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ is exact.\n\\item Similarly the functors\n$\\textit{PSh}(X_\\etale) \\to \\textit{Sets}$ and\n$\\Sh(X_\\etale) \\to \\textit{Sets}$ given by the stalk functor\n$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$ are exact (see\nCategories, Definition \\ref{categories-definition-exact})\nand commute with arbitrary colimits.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04K1","source_file":"spaces-properties.tex","source_line":3335,"source_end_line":3357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3335-L3357","statement_sha256":"20bda4df50741af8b69170bfa7574fadea35558da6ea118921a5e4e765d234cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10819,"rank":10819,"depth":50,"x":179.489,"y":1559.581,"cluster":"algebraic-spaces"},{"id":"stacks:04K2","tag":"04K2","title":"Points of the small étale site · Lemma 04K2","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • The functor f_small^-1 : Ab(Y_etale) → Ab(X_etale) is exact. • The functor f_small^-1 : Sh(Y_etale) → Sh(X_etale) is exact, i.e., it commutes with finite limits and colimits, see Categories, Definition [Tag 0034]. • For any étale morphism V → Y of algebraic spaces we have f_small^-1h_V = h_X ×_Y V. • Let overlinex → X be a geometric point. Let G be a sheaf on Y_etale. Then there is a canonical…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item The functor\n$f_{small}^{-1} :\n\\textit{Ab}(Y_\\etale)\n\\to\n\\textit{Ab}(X_\\etale)$\nis exact.\n\\item The functor\n$f_{small}^{-1} :\n\\Sh(Y_\\etale)\n\\to\n\\Sh(X_\\etale)$\nis exact, i.e., it commutes with finite limits and colimits, see\nCategories, Definition \\ref{categories-definition-exact}.\n\\item For any \\'etale morphism $V \\to Y$ of algebraic spaces\nwe have $f_{small}^{-1}h_V = h_{X \\times_Y V}$.\n\\item Let $\\overline{x} \\to X$ be a geometric point.\nLet $\\mathcal{G}$ be a sheaf on $Y_\\etale$.\nThen there is a canonical identification\n$$\n(f_{small}^{-1}\\mathcal{G})_{\\overline{x}} = \\mathcal{G}_{\\overline{y}}.\n$$\nwhere $\\overline{y} = f \\circ \\overline{x}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04K2","source_file":"spaces-properties.tex","source_line":3376,"source_end_line":3404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3376-L3404","statement_sha256":"1ae3e56577f4376555b95e7f19ae553f489eceb2a407d95be00448a8b93b6ebe","origin":"The Stacks Project","memory_eligible":false,"source_rank":10820,"rank":10820,"depth":51,"x":299.992,"y":1601.147,"cluster":"algebraic-spaces"},{"id":"stacks:04K5","tag":"04K5","title":"Points of the small étale site · Theorem 04K5","summary":"Let S be a scheme. Let X be an algebraic space over S. A map a : F → G of sheaves of sets is injective (resp. surjective) if and only if the map on stalks a_overlinex : F_overlinex → G_overlinex is injective (resp. surjective) for all geometric points of X. A sequence of abelian sheaves on X_etale is exact if and only if it is exact on all stalks at geometric points of S.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nA map $a : \\mathcal{F} \\to \\mathcal{G}$ of sheaves of sets is injective\n(resp.\\ surjective) if and only if the map on stalks\n$a_{\\overline{x}} : \\mathcal{F}_{\\overline{x}} \\to \\mathcal{G}_{\\overline{x}}$\nis injective (resp.\\ surjective) for all geometric points of $X$.\nA sequence of abelian sheaves on $X_\\etale$ is exact\nif and only if it is exact on all stalks at geometric points of $S$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04K5","source_file":"spaces-properties.tex","source_line":3522,"source_end_line":3531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3522-L3531","statement_sha256":"ee51ca7476912dc484b79a8907b1a4ac17e2d80044009d1ed8a0eb99e5f60027","origin":"The Stacks Project","memory_eligible":false,"source_rank":10821,"rank":10821,"depth":52,"x":177.286,"y":1639.002,"cluster":"algebraic-spaces"},{"id":"stacks:04K6","tag":"04K6","title":"Points of the small étale site · Lemma 04K6","summary":"Let S be a scheme. Let X be an algebraic space over S. Let p : Sh(pt) → Sh(X_etale) be a point of the small étale topos of X. Then there exists a geometric point overlinex of X such that the stalk functor F ↦ F_p is isomorphic to the stalk functor F ↦ F_overlinex.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $p : \\Sh(pt) \\to \\Sh(X_\\etale)$\nbe a point of the small \\'etale topos of $X$.\nThen there exists a geometric point $\\overline{x}$ of $X$\nsuch that the stalk functor $\\mathcal{F} \\mapsto \\mathcal{F}_p$\nis isomorphic to the stalk functor\n$\\mathcal{F} \\mapsto \\mathcal{F}_{\\overline{x}}$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Points of the small étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04K6","source_file":"spaces-properties.tex","source_line":3552,"source_end_line":3562,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3552-L3562","statement_sha256":"6d968d4862e48342fd4d95ae916ff34e4eb920fe4d382300836bf4511612c10e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10822,"rank":10822,"depth":52,"x":237.531,"y":1541.129,"cluster":"algebraic-spaces"},{"id":"stacks:04K8","tag":"04K8","title":"Supports of abelian sheaves · Lemma 04K8","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a subsheaf of the final object of the étale topos of X (see Sites, Example [Tag 00W3]). Then there exists a unique open W ⊂ X such that F = h_W.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a subsheaf of the final object of the \\'etale\ntopos of $X$ (see\nSites, Example \\ref{sites-example-singleton-sheaf}).\nThen there exists a unique open\n$W \\subset X$ such that $\\mathcal{F} = h_W$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Supports of abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04K8","source_file":"spaces-properties.tex","source_line":3603,"source_end_line":3611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3603-L3611","statement_sha256":"6a1253c4002a0d3d7702904acc52c5a06e54478101b526607f38974cf6320926","origin":"The Stacks Project","memory_eligible":false,"source_rank":10823,"rank":10823,"depth":46,"x":271.93,"y":1647.845,"cluster":"algebraic-spaces"},{"id":"stacks:04K9","tag":"04K9","title":"Supports of abelian sheaves · Lemma 04K9","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be an abelian sheaf on X_spaces, etale. Let σ ∈ F(U) be a local section. There exists an open subspace W ⊂ U such that • W ⊂ U is the largest open subspace of U such that σ|_W = 0, • for every φ : V → U in X_spaces, etale we have σ|_V = 0 ⇔ φ(V) ⊂ W, • for every geometric point overlineu of U we have (U, overlineu, σ) = 0 in F_overlinex ⇔ overlineu ∈ W where overlinex = (U → X) ∘ overlineu.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be an abelian sheaf on $X_{spaces, \\etale}$.\nLet $\\sigma \\in \\mathcal{F}(U)$ be a local section.\nThere exists an open subspace $W \\subset U$ such that\n\\begin{enumerate}\n\\item $W \\subset U$ is the largest open subspace of $U$ such\nthat $\\sigma|_W = 0$,\n\\item for every $\\varphi : V \\to U$ in $X_{spaces, \\etale}$ we have\n$$\n\\sigma|_V = 0 \\Leftrightarrow \\varphi(V) \\subset W,\n$$\n\\item for every geometric point $\\overline{u}$ of $U$ we have\n$$\n(U, \\overline{u}, \\sigma) = 0\\text{ in }\\mathcal{F}_{\\overline{x}}\n\\Leftrightarrow\n\\overline{u} \\in W\n$$\nwhere $\\overline{x} = (U \\to X) \\circ \\overline{u}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Supports of abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04K9","source_file":"spaces-properties.tex","source_line":3628,"source_end_line":3650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3628-L3650","statement_sha256":"7e9b9b1f224f684af1b7a7d2bdb12f87078687c8725d3848d809bbf6220ed300","origin":"The Stacks Project","memory_eligible":false,"source_rank":10824,"rank":10824,"depth":46,"x":160.372,"y":1588.473,"cluster":"algebraic-spaces"},{"id":"stacks:04KA","tag":"04KA","title":"Supports of abelian sheaves · Definition 04KA","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be an abelian sheaf on X_etale. • The support of F is the set of points x ∈ |X| such that F_overlinex not = 0 for any (some) geometric point overlinex lying over x. • Let σ ∈ F(U) be a section. The support of σ is the closed subset U setminus W, where W ⊂ U is the largest open subset of U on which σ restricts to zero (see Lemma [Tag 04K9]).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.\n\\begin{enumerate}\n\\item The {\\it support of $\\mathcal{F}$} is the set of\npoints $x \\in |X|$ such that $\\mathcal{F}_{\\overline{x}} \\not = 0$\nfor any (some) geometric point $\\overline{x}$ lying over $x$.\n\\item Let $\\sigma \\in \\mathcal{F}(U)$ be a section.\nThe {\\it support of $\\sigma$} is the closed subset $U \\setminus W$, where\n$W \\subset U$ is the largest open subset of $U$ on which $\\sigma$\nrestricts to zero (see\nLemma \\ref{lemma-zero-over-image}).\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Supports of abelian sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KA","source_file":"spaces-properties.tex","source_line":3681,"source_end_line":3696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3681-L3696","statement_sha256":"4574d934c9166e1c6f869c2bff9c1f67bdd22fa01e90cfdb6763244d34a62019","origin":"The Stacks Project","memory_eligible":false,"source_rank":10825,"rank":10825,"depth":47,"x":290.814,"y":1568.888,"cluster":"algebraic-spaces"},{"id":"stacks:04KB","tag":"04KB","title":"Supports of abelian sheaves · Lemma 04KB","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be an abelian sheaf on X_etale. Let U ∈ Ob(X_etale) and σ ∈ F(U). • The support of σ is closed in |X|. • The support of σ + σ' is contained in the union of the supports of σ, σ' ∈ F(X). • If φ : F → G is a map of abelian sheaves on X_etale, then the support of φ(σ) is contained in the support of σ ∈ F(U). • The support of F is the union of the images of the supports of all local sections of F. • If F → G is…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be an abelian sheaf on $X_\\etale$.\nLet $U \\in \\Ob(X_\\etale)$ and $\\sigma \\in \\mathcal{F}(U)$.\n\\begin{enumerate}\n\\item The support of $\\sigma$ is closed in $|X|$.\n\\item The support of $\\sigma + \\sigma'$ is contained in the union of\nthe supports of $\\sigma, \\sigma' \\in \\mathcal{F}(X)$.\n\\item If $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a map of\nabelian sheaves on $X_\\etale$, then the support of $\\varphi(\\sigma)$ is\ncontained in the support of $\\sigma \\in \\mathcal{F}(U)$.\n\\item The support of $\\mathcal{F}$ is the union of the images of the\nsupports of all local sections of $\\mathcal{F}$.\n\\item If $\\mathcal{F} \\to \\mathcal{G}$ is surjective then the support\nof $\\mathcal{G}$ is a subset of the support of $\\mathcal{F}$.\n\\item If $\\mathcal{F} \\to \\mathcal{G}$ is injective then the support\nof $\\mathcal{F}$ is a subset of the support of $\\mathcal{G}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Supports of abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KB","source_file":"spaces-properties.tex","source_line":3698,"source_end_line":3718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3698-L3718","statement_sha256":"49f0734acde8f75fe8f09b436fe73fb28630daf01ab9506829db972dc8c62882","origin":"The Stacks Project","memory_eligible":false,"source_rank":10826,"rank":10826,"depth":47,"x":210.11,"y":1657.64,"cluster":"algebraic-spaces"},{"id":"stacks:04KC","tag":"04KC","title":"Supports of abelian sheaves · Lemma 04KC","summary":"The support of a sheaf of rings on the small étale site of an algebraic space is closed.","statement_latex":"The support of a sheaf of rings on the small \\'etale site of an\nalgebraic space is closed.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Supports of abelian sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KC","source_file":"spaces-properties.tex","source_line":3730,"source_end_line":3734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3730-L3734","statement_sha256":"242d3a7ecdb85ff2527064e427f2b28d2893868fcb9693651ee4c8a31751bfa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10827,"rank":10827,"depth":0,"x":198.209,"y":1546.034,"cluster":"algebraic-spaces"},{"id":"stacks:03G6","tag":"03G6","title":"The structure sheaf of an algebraic space · Lemma 03G6","summary":"Let S be a scheme. Let X be an algebraic space over S. The rule U ↦ Γ(U, O_U) defines a sheaf of rings on X_etale.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe rule $U \\mapsto \\Gamma(U, \\mathcal{O}_U)$ defines\na sheaf of rings on $X_\\etale$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The structure sheaf of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03G6","source_file":"spaces-properties.tex","source_line":3755,"source_end_line":3760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3755-L3760","statement_sha256":"20b44ee1aaea9323e1e56bb0be7d21750f0e5e2ea7e6b546d9c63cd0a62d7c2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10828,"rank":10828,"depth":40,"x":297.059,"y":1621.826,"cluster":"algebraic-spaces"},{"id":"stacks:03G7","tag":"03G7","title":"The structure sheaf of an algebraic space · Definition 03G7","summary":"Let S be a scheme. Let X be an algebraic space over S. The structure sheaf of X is the sheaf of rings O_X on the small étale site X_etale described in Lemma [Tag 03G6].","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThe {\\it structure sheaf} of $X$\nis the sheaf of rings $\\mathcal{O}_X$\non the small \\'etale site $X_\\etale$ described in\nLemma \\ref{lemma-sheaf-condition-holds}.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The structure sheaf of an algebraic space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03G7","source_file":"spaces-properties.tex","source_line":3767,"source_end_line":3775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3767-L3775","statement_sha256":"3ae717b6bb9c2ac91316ea646c5508108ecc28d5c6521618a51ac23cb2c77c22","origin":"The Stacks Project","memory_eligible":false,"source_rank":10829,"rank":10829,"depth":41,"x":162.782,"y":1622.028,"cluster":"algebraic-spaces"},{"id":"stacks:03G8","tag":"03G8","title":"The structure sheaf of an algebraic space · Lemma 03G8","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then there is a canonical map f^sharp : f_small^-1O_Y → O_X such that (f_small, f^sharp) : (Sh(X_etale), O_X) → (Sh(Y_etale), O_Y) is a morphism of ringed topoi. Furthermore, • The construction f ↦ (f_small, f^sharp) is compatible with compositions. • If f is a morphism of schemes, then f^sharp is the map described in Descent, Remark [Tag 070R].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThen there is a canonical map\n$f^\\sharp : f_{small}^{-1}\\mathcal{O}_Y \\to \\mathcal{O}_X$ such that\n$$\n(f_{small}, f^\\sharp) :\n(\\Sh(X_\\etale), \\mathcal{O}_X)\n\\longrightarrow\n(\\Sh(Y_\\etale), \\mathcal{O}_Y)\n$$\nis a morphism of ringed topoi. Furthermore,\n\\begin{enumerate}\n\\item The construction $f \\mapsto (f_{small}, f^\\sharp)$ is compatible with\ncompositions.\n\\item If $f$ is a morphism of schemes, then $f^\\sharp$ is the map described in\nDescent, Remark \\ref{descent-remark-change-topologies-ringed}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The structure sheaf of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03G8","source_file":"spaces-properties.tex","source_line":3797,"source_end_line":3816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3797-L3816","statement_sha256":"c62180fabf17595980eac9f1342991d77c89c9e80e6cceb9d01810a229f52432","origin":"The Stacks Project","memory_eligible":false,"source_rank":10830,"rank":10830,"depth":10,"x":261.954,"y":1545.44,"cluster":"algebraic-spaces"},{"id":"stacks:0BGS","tag":"0BGS","title":"The structure sheaf of an algebraic space · Lemma 0BGS","summary":"Let S be a scheme. Let X be an algebraic space over S. The following are equivalent • X is reduced, • for every x ∈ |X| the local ring of X at x is reduced (Remark [Tag 0BBL]). In this case Γ(X, O_X) is a reduced ring and if f ∈ Γ(X, O_X) has X = V(f), then f = 0.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is reduced,\n\\item for every $x \\in |X|$ the local ring of $X$ at $x$ is\nreduced (Remark \\ref{remark-list-properties-local-ring-local-etale-topology}).\n\\end{enumerate}\nIn this case $\\Gamma(X, \\mathcal{O}_X)$ is a reduced ring and\nif $f \\in \\Gamma(X, \\mathcal{O}_X)$ has $X = V(f)$, then $f = 0$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"The structure sheaf of an algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGS","source_file":"spaces-properties.tex","source_line":3844,"source_end_line":3855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3844-L3855","statement_sha256":"f961a705853b61bc45908eff8faf3b138b9dead340b53e93fb1932372447d40f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10831,"rank":10831,"depth":4,"x":250.378,"y":1658.55,"cluster":"algebraic-spaces"},{"id":"stacks:04KF","tag":"04KF","title":"Stalks of the structure sheaf · Lemma 04KF","summary":"Let S be a scheme. Let X be an algebraic space over S. Let overlinex be a geometric point of X. Let (U, overlineu) be an étale neighbourhood of overlinex where U is a scheme. Then we have O_X, overlinex = O_U, overlineu = O_U, u^sh where the left hand side is the stalk of the structure sheaf of X, and the right hand side is the strict henselization of the local ring of U at the point u at which overlineu is centered.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\overline{x}$ be a geometric point of $X$.\nLet $(U, \\overline{u})$ be an \\'etale neighbourhood of $\\overline{x}$\nwhere $U$ is a scheme. Then we have\n$$\n\\mathcal{O}_{X, \\overline{x}} =\n\\mathcal{O}_{U, \\overline{u}} =\n\\mathcal{O}_{U, u}^{sh}\n$$\nwhere the left hand side is the stalk of the structure sheaf of $X$,\nand the right hand side is the strict henselization of the local ring\nof $U$ at the point $u$ at which $\\overline{u}$ is centered.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Stalks of the structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KF","source_file":"spaces-properties.tex","source_line":3880,"source_end_line":3895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3880-L3895","statement_sha256":"6bd6abadc57fcdcd4d18028a273989b6858d4f7235ccd333b3622674531ac212","origin":"The Stacks Project","memory_eligible":false,"source_rank":10832,"rank":10832,"depth":52,"x":167.691,"y":1568.288,"cluster":"algebraic-spaces"},{"id":"stacks:04KG","tag":"04KG","title":"Stalks of the structure sheaf · Definition 04KG","summary":"Let S be a scheme. Let X be an algebraic space over S. Let overlinex be a geometric point of X lying over the point x ∈ |X|. • The étale local ring of X at overlinex is the stalk of the structure sheaf O_X on X_etale at overlinex. Notation: O_X, overlinex. • The strict henselization of X at overlinex is the scheme Spec(O_X, overlinex).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\overline{x}$ be a geometric point of $X$ lying over the point\n$x \\in |X|$.\n\\begin{enumerate}\n\\item The {\\it \\'etale local ring of $X$ at $\\overline{x}$}\nis the stalk of the structure sheaf $\\mathcal{O}_X$ on $X_\\etale$\nat $\\overline{x}$.\nNotation: $\\mathcal{O}_{X, \\overline{x}}$.\n\\item The {\\it strict henselization of $X$ at $\\overline{x}$}\nis the scheme $\\Spec(\\mathcal{O}_{X, \\overline{x}})$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Stalks of the structure sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KG","source_file":"spaces-properties.tex","source_line":3907,"source_end_line":3921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3907-L3921","statement_sha256":"2431154a782e10c1a32297676d8bc49c04fa336cd90f818bba2d812d65d4d646","origin":"The Stacks Project","memory_eligible":false,"source_rank":10833,"rank":10833,"depth":0,"x":301.674,"y":1587.99,"cluster":"algebraic-spaces"},{"id":"stacks:04KH","tag":"04KH","title":"Stalks of the structure sheaf · Lemma 04KH","summary":"Let S be a scheme. Let X be an algebraic space over S. The small étale site X_etale endowed with its structure sheaf O_X is a locally ringed site, see Modules on Sites, Definition [Tag 04EU].","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThe small \\'etale site $X_\\etale$ endowed with its\nstructure sheaf $\\mathcal{O}_X$ is a locally ringed site, see\nModules on Sites, Definition \\ref{sites-modules-definition-locally-ringed}.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Stalks of the structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KH","source_file":"spaces-properties.tex","source_line":3929,"source_end_line":3936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3929-L3936","statement_sha256":"fab230e0c1ebaff92837c792cfcc832a06e84ce8f2e6f7d063fee4044805e0c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10834,"rank":10834,"depth":53,"x":186.671,"y":1649.681,"cluster":"algebraic-spaces"},{"id":"stacks:04N9","tag":"04N9","title":"Stalks of the structure sheaf · Lemma 04N9","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point. Let d ∈ (0, 1, 2, …, ∞). The following are equivalent • the dimension of the local ring of X at x (Definition [Tag 04NA]) is d, • dim(O_X, overlinex) = d for some geometric point overlinex lying over x, and • dim(O_X, overlinex) = d for any geometric point overlinex lying over x.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$ be a point. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item the dimension of the local ring of $X$ at $x$\n(Definition \\ref{definition-dimension-local-ring}) is $d$,\n\\item $\\dim(\\mathcal{O}_{X, \\overline{x}}) = d$ for some geometric\npoint $\\overline{x}$ lying over $x$, and\n\\item $\\dim(\\mathcal{O}_{X, \\overline{x}}) = d$ for any geometric\npoint $\\overline{x}$ lying over $x$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Stalks of the structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04N9","source_file":"spaces-properties.tex","source_line":3950,"source_end_line":3963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3950-L3963","statement_sha256":"1c03d8000a553605bf80bc76944f0ba49b011555b29117d772a2ee86428c5c68","origin":"The Stacks Project","memory_eligible":false,"source_rank":10835,"rank":10835,"depth":53,"x":221.973,"y":1538.588,"cluster":"algebraic-spaces"},{"id":"stacks:0A4H","tag":"0A4H","title":"Stalks of the structure sheaf · Lemma 0A4H","summary":"Let S be a scheme. Let f : X → Y be an étale morphism of algebraic spaces over S. Let x ∈ X. Then (1) dim_x(X) = dim_f(x)(Y) and (2) the dimension of the local ring of X at x equals the dimension of the local ring of Y at f(x). If f is surjective, then (3) dim(X) = dim(Y).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale morphism\nof algebraic spaces over $S$. Let $x \\in X$. Then\n(1) $\\dim_x(X) = \\dim_{f(x)}(Y)$ and (2) the dimension of\nthe local ring of $X$ at $x$ equals the dimension of\nthe local ring of $Y$ at $f(x)$. If $f$ is surjective, then\n(3) $\\dim(X) = \\dim(Y)$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Stalks of the structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4H","source_file":"spaces-properties.tex","source_line":3976,"source_end_line":3984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3976-L3984","statement_sha256":"cbaa66782bb329c9dae78b75721833a828e314ab1d36931481b4b10c91d2c2da","origin":"The Stacks Project","memory_eligible":false,"source_rank":10836,"rank":10836,"depth":47,"x":285.476,"y":1640.856,"cluster":"algebraic-spaces"},{"id":"stacks:0E01","tag":"0E01","title":"Stalks of the structure sheaf · Lemma 0E01","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point. The following are equivalent • the local ring of X at x is reduced (Remark [Tag 0BBL]), • O_X, overlinex is reduced for some geometric point overlinex lying over x, and • O_X, overlinex is reduced for any geometric point overlinex lying over x.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$ be a point. The following are equivalent\n\\begin{enumerate}\n\\item the local ring of $X$ at $x$ is reduced\n(Remark \\ref{remark-list-properties-local-ring-local-etale-topology}),\n\\item $\\mathcal{O}_{X, \\overline{x}}$ is reduced for some geometric\npoint $\\overline{x}$ lying over $x$, and\n\\item $\\mathcal{O}_{X, \\overline{x}}$ is reduced for any geometric\npoint $\\overline{x}$ lying over $x$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Stalks of the structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E01","source_file":"spaces-properties.tex","source_line":3996,"source_end_line":4008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L3996-L4008","statement_sha256":"cfe5000c0a41e0f366c00a10e264fce9d812f9b30271710a4928f65b4ad84a24","origin":"The Stacks Project","memory_eligible":false,"source_rank":10837,"rank":10837,"depth":53,"x":156.009,"y":1601.356,"cluster":"algebraic-spaces"},{"id":"stacks:06DK","tag":"06DK","title":"Local irreducibility · Lemma 06DK","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point. The following are equivalent • for any scheme U and étale morphism a : U → X and u ∈ U with a(u) = x the local ring O_U, u has a unique minimal prime, • for any scheme U and étale morphism a : U → X and u ∈ U with a(u) = x there is a unique irreducible component of U through u, • for any scheme U and étale morphism a : U → X and u ∈ U with a(u) = x the local ring O_U, u is unibranch, • for any…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $x \\in |X|$ be a point.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and\n$u \\in U$ with $a(u) = x$ the local ring $\\mathcal{O}_{U, u}$ has a\nunique minimal prime,\n\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and\n$u \\in U$ with $a(u) = x$ there is a unique irreducible component of $U$\nthrough $u$,\n\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and\n$u \\in U$ with $a(u) = x$ the local ring $\\mathcal{O}_{U, u}$\nis unibranch,\n\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and\n$u \\in U$ with $a(u) = x$ the local ring $\\mathcal{O}_{U, u}$\nis geometrically unibranch,\n\\item $\\mathcal{O}_{X, \\overline{x}}$ has a unique minimal prime\nfor any geometric point $\\overline{x}$ lying over $x$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DK","source_file":"spaces-properties.tex","source_line":4041,"source_end_line":4063,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4041-L4063","statement_sha256":"01c66eff2942eb09adb386ffe45c9531475fe725a6fc925d5b55d7bf168e6dc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10838,"rank":10838,"depth":55,"x":283.633,"y":1556.884,"cluster":"algebraic-spaces"},{"id":"stacks:06DL","tag":"06DL","title":"Local irreducibility · Definition 06DL","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. We say that X is geometrically unibranch at x if the equivalent conditions of Lemma [Tag 06DK] hold. We say that X is geometrically unibranch if X is geometrically unibranch at every x ∈ |X|.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. We say that $X$ is {\\it geometrically unibranch\nat $x$} if the equivalent conditions of\nLemma \\ref{lemma-irreducible-local-ring}\nhold. We say that $X$ is {\\it geometrically unibranch} if $X$ is\ngeometrically unibranch at every $x \\in |X|$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Local irreducibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DL","source_file":"spaces-properties.tex","source_line":4078,"source_end_line":4086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4078-L4086","statement_sha256":"e25f66e5644cf538ad2df0b1da22895aa7686aa6aa24cbdc95e5123728d9ad8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10839,"rank":10839,"depth":56,"x":225.11,"y":1662.416,"cluster":"algebraic-spaces"},{"id":"stacks:0DQ3","tag":"0DQ3","title":"Local irreducibility · Lemma 0DQ3","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point. Let n ∈ (1, 2, …) be an integer. The following are equivalent • for any scheme U and étale morphism a : U → X and u ∈ U with a(u) = x the number of minimal primes of the local ring O_U, u is ≤ n and for at least one choice of U, a, u it is n, • for any scheme U and étale morphism a : U → X and u ∈ U with a(u) = x the number irreducible components of U passing through u is ≤ n and for at least…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$ be a point. Let $n \\in \\{1, 2, \\ldots\\}$ be an integer.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and\n$u \\in U$ with $a(u) = x$ the number of minimal primes\nof the local ring $\\mathcal{O}_{U, u}$ is $\\leq n$\nand for at least one choice of $U, a, u$ it is $n$,\n\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and\n$u \\in U$ with $a(u) = x$ the number irreducible components of\n$U$ passing through $u$ is $\\leq n$ and for at least one choice\nof $U, a, u$ it is $n$,\n\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$\nwith $a(u) = x$ the number of branches of $U$ at $u$ is $\\leq n$\nand for at least one choice of $U, a, u$ it is $n$,\n\\item for any scheme $U$ and \\'etale morphism $a : U \\to X$ and $u \\in U$\nwith $a(u) = x$ the number of geometric branches of $U$ at $u$ is $n$, and\n\\item the number of minimal prime ideals of\n$\\mathcal{O}_{X, \\overline{x}}$ is $n$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQ3","source_file":"spaces-properties.tex","source_line":4092,"source_end_line":4114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4092-L4114","statement_sha256":"99c9f0e203669b727347f755187721c6ea623a751197b07bbfd46761bcc86dfe","origin":"The Stacks Project","memory_eligible":false,"source_rank":10840,"rank":10840,"depth":55,"x":183.272,"y":1551.053,"cluster":"algebraic-spaces"},{"id":"stacks:0DQ4","tag":"0DQ4","title":"Local irreducibility · Definition 0DQ4","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. The number of geometric branches of X at x is either n ∈ N if the equivalent conditions of Lemma [Tag 0DQ3] hold, or else ∞.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. The {\\it number of geometric branches of $X$ at $x$} is\neither $n \\in \\mathbf{N}$ if the equivalent conditions\nof Lemma \\ref{lemma-nr-branches-local-ring}\nhold, or else $\\infty$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Local irreducibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQ4","source_file":"spaces-properties.tex","source_line":4130,"source_end_line":4138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4130-L4138","statement_sha256":"a7ab240557b3cb1768b7571e1d03835f2d1189446c47657dec2770b6c6fd8c14","origin":"The Stacks Project","memory_eligible":false,"source_rank":10841,"rank":10841,"depth":56,"x":304.041,"y":1609.607,"cluster":"algebraic-spaces"},{"id":"stacks:03EA","tag":"03EA","title":"Noetherian spaces · Definition 03EA","summary":"Let S be a scheme. Let X be an algebraic space over S. We say X is Noetherian if X is quasi-compact, quasi-separated and locally Noetherian.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nWe say $X$ is {\\it Noetherian} if $X$ is quasi-compact, quasi-separated\nand locally Noetherian.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Noetherian spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03EA","source_file":"spaces-properties.tex","source_line":4154,"source_end_line":4159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4154-L4159","statement_sha256":"0507ea0a7ec83896d5965db0fe2c34da2c11a2916e11f2a6d571801615153c60","origin":"The Stacks Project","memory_eligible":false,"source_rank":10842,"rank":10842,"depth":0,"x":167.491,"y":1635.033,"cluster":"algebraic-spaces"},{"id":"stacks:04ZF","tag":"04ZF","title":"Noetherian spaces · Lemma 04ZF","summary":"Let S be a scheme. Let X be an algebraic space over S. • If X is locally Noetherian then |X| is a locally Noetherian topological space. • If X is quasi-compact and locally Noetherian, then |X| is a Noetherian topological space.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $X$ is locally Noetherian then $|X|$ is a locally Noetherian\ntopological space.\n\\item If $X$ is quasi-compact and locally Noetherian, then $|X|$\nis a Noetherian topological space.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZF","source_file":"spaces-properties.tex","source_line":4188,"source_end_line":4197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4188-L4197","statement_sha256":"eaf1bbe7ae25c0e2d8a6f8ea749e76e43509a18b0fc91a41b3998556f5ddff2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10843,"rank":10843,"depth":3,"x":247.974,"y":1538.515,"cluster":"algebraic-spaces"},{"id":"stacks:04ZG","tag":"04ZG","title":"Noetherian spaces · Lemma 04ZG","summary":"Let S be a scheme. Let X be an algebraic space over S. If X is Noetherian, then |X| is a sober Noetherian topological space.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf $X$ is Noetherian, then $|X|$ is a sober Noetherian topological space.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZG","source_file":"spaces-properties.tex","source_line":4216,"source_end_line":4220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4216-L4220","statement_sha256":"f7421f22ca7efb7bb4ddcefc356f9c5e77c4260a132b3bacc745f0a3291103db","origin":"The Stacks Project","memory_eligible":false,"source_rank":10844,"rank":10844,"depth":55,"x":266.298,"y":1655.701,"cluster":"algebraic-spaces"},{"id":"stacks:08AH","tag":"08AH","title":"Noetherian spaces · Lemma 08AH","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let overlinex be a geometric point of X. Then O_X, overlinex is a Noetherian local ring.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\overline{x}$ be a geometric point of $X$. Then\n$\\mathcal{O}_{X, \\overline{x}}$ is a Noetherian local ring.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AH","source_file":"spaces-properties.tex","source_line":4230,"source_end_line":4235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4230-L4235","statement_sha256":"9d4a164686066ec0fa1f65fd71da5f5fc54aeda3f2fb216ae8ff08c1e41fa87c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10845,"rank":10845,"depth":53,"x":158.231,"y":1579.463,"cluster":"algebraic-spaces"},{"id":"stacks:06LQ","tag":"06LQ","title":"Regular algebraic spaces · Lemma 06LQ","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. The following are equivalent • X is regular, and • every étale local ring O_X, overlinex is regular.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is regular, and\n\\item every \\'etale local ring $\\mathcal{O}_{X, \\overline{x}}$ is\nregular.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Regular algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LQ","source_file":"spaces-properties.tex","source_line":4261,"source_end_line":4271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4261-L4271","statement_sha256":"c6af4ef0f92a60ccdc0bb3801420f598fa83f36ff9b5a3bb20758768fa54fdf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10846,"rank":10846,"depth":53,"x":299.64,"y":1574.346,"cluster":"algebraic-spaces"},{"id":"stacks:0AH9","tag":"0AH9","title":"Regular algebraic spaces · Definition 0AH9","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point. We say X is regular at x if O_U, u is a regular local ring for any (equivalently some) pair (a : U → X, u) consisting of an étale morphism a : U → X from a scheme to X and a point u ∈ U with a(u) = x.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$ be a point. We say {\\it $X$ is regular at $x$}\nif $\\mathcal{O}_{U, u}$ is a regular local ring for any\n(equivalently some) pair $(a : U \\to X, u)$ consisting of an\n\\'etale morphism $a : U \\to X$ from a scheme to $X$ and a point\n$u \\in U$ with $a(u) = x$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Regular algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AH9","source_file":"spaces-properties.tex","source_line":4293,"source_end_line":4301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4293-L4301","statement_sha256":"5768ed71a65139a12d2d25600708ca1b7c0ab89104c1828b36e22574f87bd979","origin":"The Stacks Project","memory_eligible":false,"source_rank":10847,"rank":10847,"depth":0,"x":199.19,"y":1658.602,"cluster":"algebraic-spaces"},{"id":"stacks:0AHA","tag":"0AHA","title":"Regular algebraic spaces · Lemma 0AHA","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| be a point. The following are equivalent • X is regular at x, and • the étale local ring O_X, overlinex is regular for any (equivalently some) geometric point overlinex lying over x.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$ be a point. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is regular at $x$, and\n\\item the \\'etale local ring $\\mathcal{O}_{X, \\overline{x}}$ is\nregular for any (equivalently some) geometric point $\\overline{x}$\nlying over $x$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Regular algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHA","source_file":"spaces-properties.tex","source_line":4308,"source_end_line":4318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4308-L4318","statement_sha256":"491c4042c88eb51fadb8d4988279fb76a47b2415b8bdc557ca37a97ae0ffdbf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10848,"rank":10848,"depth":53,"x":205.522,"y":1539.13,"cluster":"algebraic-spaces"},{"id":"stacks:0BGT","tag":"0BGT","title":"Regular algebraic spaces · Lemma 0BGT","summary":"A regular algebraic space is normal.","statement_latex":"A regular algebraic space is normal.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Regular algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGT","source_file":"spaces-properties.tex","source_line":4331,"source_end_line":4334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4331-L4334","statement_sha256":"2adedde049a2b39b2f162894f30e819c083c815b5b8337133c84f23943eb8cdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10849,"rank":10849,"depth":19,"x":297.192,"y":1631.084,"cluster":"algebraic-spaces"},{"id":"stacks:03LV","tag":"03LV","title":"Sheaves of modules on algebraic spaces · Lemma 03LV","summary":"Let S be a scheme. Let f : X → Y be an étale morphism of algebraic spaces over S. Then f^-1O_Y = O_X, and f^*G = f_small^-1G for any sheaf of O_Y-modules G. In particular, f^* : Mod(O_Y) → Mod(O_X) is exact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be an \\'etale morphism of algebraic spaces over $S$.\nThen $f^{-1}\\mathcal{O}_Y = \\mathcal{O}_X$, and\n$f^*\\mathcal{G} = f_{small}^{-1}\\mathcal{G}$ for any sheaf of\n$\\mathcal{O}_Y$-modules $\\mathcal{G}$. In particular,\n$f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$\nis exact.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Sheaves of modules on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LV","source_file":"spaces-properties.tex","source_line":4382,"source_end_line":4391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4382-L4391","statement_sha256":"06b3bf6097afc9c4a463e0945c19f40a92d877adb271e7ffc2994392d827f93e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10850,"rank":10850,"depth":48,"x":155.243,"y":1615.249,"cluster":"algebraic-spaces"},{"id":"stacks:03LX","tag":"03LX","title":"Sheaves of modules on algebraic spaces · Lemma 03LX","summary":"Let S be a scheme. Let xymatrix X' ar[r] ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian square of algebraic spaces over S. Let F ∈ Mod(O_X). If g is étale, then f'_*(F|_X') = (f_*F)|_Y'|_Y') = (f^*G)|_X' by commutativity of the diagram and ([Tag 03LW]) and R^if'_*(F|_X') = (R^if_*F)|_Y' in Mod(O_Y').","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r] \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian square of algebraic spaces over $S$. Let\n$\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O}_X)$. If $g$ is \\'etale, then\n$f'_*(\\mathcal{F}|_{X'}) = (f_*\\mathcal{F})|_{Y'}$\\footnote{Also\n$(f')^*(\\mathcal{G}|_{Y'}) = (f^*\\mathcal{G})|_{X'}$\nby commutativity of the diagram and (\\ref{equation-restrict-modules})} and\n$R^if'_*(\\mathcal{F}|_{X'}) = (R^if_*\\mathcal{F})|_{Y'}$ in\n$\\textit{Mod}(\\mathcal{O}_{Y'})$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Sheaves of modules on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LX","source_file":"spaces-properties.tex","source_line":4421,"source_end_line":4437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4421-L4437","statement_sha256":"df99ef5af89b7c8a49f8598d14932ba529d92317d27da20fb2e717873dffc6be","origin":"The Stacks Project","memory_eligible":false,"source_rank":10851,"rank":10851,"depth":10,"x":272.984,"y":1546.185,"cluster":"algebraic-spaces"},{"id":"stacks:03LY","tag":"03LY","title":"Sheaves of modules on algebraic spaces · Lemma 03LY","summary":"Let S be a scheme. Let X be an algebraic space over S. A sheaf F of O_X-modules is given by the following data: • for every U ∈ Ob(X_etale) a sheaf F_U of O_U-modules on U_etale, • for every f : U' → U in X_etale an isomorphism c_f : f_small^*F_U → F_U'. These data are subject to the condition that given any f : U' → U and g : U\" → U' in X_etale the composition c_g ∘ g_small^*c_f is equal to c_f ∘ g.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nA sheaf $\\mathcal{F}$ of $\\mathcal{O}_X$-modules is given by the following\ndata:\n\\begin{enumerate}\n\\item for every $U \\in \\Ob(X_\\etale)$ a sheaf\n$\\mathcal{F}_U$ of $\\mathcal{O}_U$-modules on $U_\\etale$,\n\\item for every $f : U' \\to U$ in $X_\\etale$ an isomorphism\n$c_f : f_{small}^*\\mathcal{F}_U \\to \\mathcal{F}_{U'}$.\n\\end{enumerate}\nThese data are subject to the condition that given any $f : U' \\to U$\nand $g : U'' \\to U'$ in $X_\\etale$ the composition\n$c_g \\circ g_{small}^*c_f$ is equal to $c_{f \\circ g}$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Sheaves of modules on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LY","source_file":"spaces-properties.tex","source_line":4445,"source_end_line":4459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4445-L4459","statement_sha256":"697dbf53414eeb1a4b49c561f24928aea7969b3b521ddc782afc2c84eca2fb14","origin":"The Stacks Project","memory_eligible":false,"source_rank":10852,"rank":10852,"depth":49,"x":241.614,"y":1664.253,"cluster":"algebraic-spaces"},{"id":"stacks:04M4","tag":"04M4","title":"Étale localization · Lemma 04M4","summary":"Let S be a scheme. Let xymatrix U ar[d]_p ar[r]_g & V ar[d]^q X ar[r]^f & Y be a commutative diagram of algebraic spaces over S with p and q étale. Via the identifications ([Tag 04LZ]) for U → X and V → Y the morphism of ringed topoi (g_spaces, etale, g^sharp) : (Sh(U_spaces, etale), O_U) → (Sh(V_spaces, etale), O_V) is 2-isomorphic to the morphism (f_spaces, etale, c, f_c^sharp) constructed in Modules on Sites, Lemma [Tag 04J1] starting with the morphism of ringed sites…","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nU \\ar[d]_p \\ar[r]_g & V \\ar[d]^q \\\\\nX \\ar[r]^f & Y\n}\n$$\nbe a commutative diagram of algebraic spaces over $S$ with $p$ and $q$ \\'etale.\nVia the identifications\n(\\ref{equation-localize-ringed}) for $U \\to X$ and $V \\to Y$\nthe morphism of ringed topoi\n$$\n(g_{spaces, \\etale}, g^\\sharp) :\n(\\Sh(U_{spaces, \\etale}), \\mathcal{O}_U)\n\\longrightarrow\n(\\Sh(V_{spaces, \\etale}), \\mathcal{O}_V)\n$$\nis $2$-isomorphic to the morphism $(f_{spaces, \\etale, c}, f_c^\\sharp)$\nconstructed in\nModules on Sites,\nLemma \\ref{sites-modules-lemma-relocalize-morphism-ringed-sites}\nstarting with the morphism of ringed sites\n$(f_{spaces, \\etale}, f^\\sharp)$ and\nthe map $c : U \\to V \\times_Y X$ corresponding to $g$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04M4","source_file":"spaces-properties.tex","source_line":4540,"source_end_line":4566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4540-L4566","statement_sha256":"66f6932b970d45b9a50f82fefe43041243d4b5f34f1111ae72ffabe8d0199990","origin":"The Stacks Project","memory_eligible":false,"source_rank":10853,"rank":10853,"depth":45,"x":169.596,"y":1559.096,"cluster":"algebraic-spaces"},{"id":"stacks:04M5","tag":"04M5","title":"Étale localization · Lemma 04M5","summary":"Same notation and assumptions as in Lemma [Tag 04M4] except that we also assume U and V are schemes. Via the identifications ([Tag 04M1]) for U → X and V → Y the morphism of ringed topoi (g_small, g^sharp) : (Sh(U_etale), O_U) → (Sh(V_etale), O_V) is 2-isomorphic to the morphism (f_small, s, f_s^sharp) constructed in Modules on Sites, Lemma [Tag 04J8] starting with (f_small, f^sharp) and the map s : h_U → f_small^-1h_V corresponding to g.","statement_latex":"Same notation and assumptions as in\nLemma \\ref{lemma-relocalize-morphism}\nexcept that we also assume $U$ and $V$ are schemes.\nVia the identifications\n(\\ref{equation-localize-at-scheme-ringed})\nfor $U \\to X$ and $V \\to Y$ the morphism of ringed topoi\n$$\n(g_{small}, g^\\sharp) :\n(\\Sh(U_\\etale), \\mathcal{O}_U)\n\\longrightarrow\n(\\Sh(V_\\etale), \\mathcal{O}_V)\n$$\nis $2$-isomorphic to the morphism $(f_{small, s}, f_s^\\sharp)$\nconstructed in\nModules on Sites,\nLemma \\ref{sites-modules-lemma-relocalize-morphism-ringed-topoi}\nstarting with $(f_{small}, f^\\sharp)$ and\nthe map $s : h_U \\to f_{small}^{-1}h_V$ corresponding to $g$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04M5","source_file":"spaces-properties.tex","source_line":4593,"source_end_line":4613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4593-L4613","statement_sha256":"c98f75bb9a615cab5a0e29d9b9eee4951ec373e789e3a5e4f2a793189fe48790","origin":"The Stacks Project","memory_eligible":false,"source_rank":10854,"rank":10854,"depth":46,"x":307.651,"y":1595.876,"cluster":"algebraic-spaces"},{"id":"stacks:0GF6","tag":"0GF6","title":"Étale localization · Lemma 0GF6","summary":"Let S be a scheme and let Y be an algebraic space over S. Let F be a sheaf of sets on Y_etale. Provided a set theoretic condition is satisfied (see proof) we have • the functor X associated to F above is an algebraic space, • the map X → Y is an étale morphism of algebraic spaces, • via the identification Sh(Y_etale) = Sh(Y_spaces, etale) we have F ≅ h_X, • we have F ≅ f_small, !*. Here * is the final object of the category Sh(X_etale) and f_small, ! exists by Lemma [Tag…","statement_latex":"Let $S$ be a scheme and let $Y$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a sheaf of sets on $Y_\\etale$.\nProvided a set theoretic condition is satisfied (see proof) we have\n\\begin{enumerate}\n\\item the functor $X$ associated to $\\mathcal{F}$ above is an\nalgebraic space,\n\\item the map $X \\to Y$ is an \\'etale morphism of algebraic spaces,\n\\item via the identification $\\Sh(Y_\\etale) = \\Sh(Y_{spaces, \\etale})$\nwe have $\\mathcal{F} \\cong h_X$,\n\\item we have $\\mathcal{F} \\cong f_{small, !}*$. Here $*$ is the final object\nof the category $\\Sh(X_\\etale)$ and $f_{small, !}$ exists by\nLemma \\ref{lemma-etale-morphism-topoi}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Étale localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GF6","source_file":"spaces-properties.tex","source_line":4646,"source_end_line":4661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4646-L4661","statement_sha256":"a928397f7b105d73f62f592be251a03ceb827c09b88bc9deb1a59c058f484f73","origin":"The Stacks Project","memory_eligible":false,"source_rank":10855,"rank":10855,"depth":54,"x":175.908,"y":1647.233,"cluster":"algebraic-spaces"},{"id":"stacks:04KJ","tag":"04KJ","title":"Recovering morphisms · Lemma 04KJ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The morphism of ringed topoi (f_small, f^sharp) associated to f is a morphism of locally ringed topoi, see Modules on Sites, Definition [Tag 04HA].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe morphism of ringed topoi $(f_{small}, f^\\sharp)$\nassociated to $f$ is a morphism of locally ringed topoi, see\nModules on Sites,\nDefinition \\ref{sites-modules-definition-morphism-locally-ringed-topoi}.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Recovering morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KJ","source_file":"spaces-properties.tex","source_line":4756,"source_end_line":4764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4756-L4764","statement_sha256":"4200e2b38d20befb3682e2ce0bbc324bd0f4ccc6ac9367a463d6ed5574aa2bf8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10856,"rank":10856,"depth":54,"x":231.908,"y":1534.297,"cluster":"algebraic-spaces"},{"id":"stacks:04KK","tag":"04KK","title":"Recovering morphisms · Lemma 04KK","summary":"Let S be a scheme. Let X, Y be algebraic spaces over S. Let f : X → Y be a morphism of algebraic spaces over S. Let t be a 2-morphism from (f_small, f^sharp) to itself, see Modules on Sites, Definition [Tag 04IC]. Then t = id.","statement_latex":"Let $S$ be a scheme.\nLet $X$, $Y$ be algebraic spaces over $S$.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $t$ be a $2$-morphism from $(f_{small}, f^\\sharp)$ to itself, see\nModules on Sites,\nDefinition \\ref{sites-modules-definition-2-morphism-ringed-topoi}.\nThen $t = \\text{id}$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Recovering morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KK","source_file":"spaces-properties.tex","source_line":4821,"source_end_line":4830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4821-L4830","statement_sha256":"ff5cad7e03c99714a77c7aa3f25278298a99080e302bf54e3ff57d7a570e9944","origin":"The Stacks Project","memory_eligible":false,"source_rank":10857,"rank":10857,"depth":54,"x":281.57,"y":1649.667,"cluster":"algebraic-spaces"},{"id":"stacks:04M6","tag":"04M6","title":"Recovering morphisms · Lemma 04M6","summary":"Let S be a scheme. Let X, Y be algebraic spaces over S. Any two morphisms a, b : X → Y of algebraic spaces over S for which there exists a 2-isomorphism (a_small, a^sharp) ≅ (b_small, b^sharp) in the 2-category of ringed topoi are equal.","statement_latex":"Let $S$ be a scheme.\nLet $X$, $Y$ be algebraic spaces over $S$.\nAny two morphisms $a, b : X \\to Y$ of algebraic spaces over $S$\nfor which there exists a $2$-isomorphism\n$(a_{small}, a^\\sharp) \\cong (b_{small}, b^\\sharp)$\nin the $2$-category of ringed topoi are equal.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Recovering morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04M6","source_file":"spaces-properties.tex","source_line":4917,"source_end_line":4925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L4917-L4925","statement_sha256":"c105c59c3e1caae1287edcd764c6ba4d3a627d49c0830647388ec865392de74c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10858,"rank":10858,"depth":46,"x":151.819,"y":1592.619,"cluster":"algebraic-spaces"},{"id":"stacks:04KL","tag":"04KL","title":"Recovering morphisms · Theorem 04KL","summary":"Let X, Y be algebraic spaces over Spec(Z). Let (g, g^sharp) : (Sh(X_etale), O_X) → (Sh(Y_etale), O_Y) be a morphism of locally ringed topoi. Then there exists a unique morphism of algebraic spaces f : X → Y such that (g, g^sharp) is isomorphic to (f_small, f^sharp). In other words, the construction Spaces/Spec(Z) → Locally ringed topoi, X → (X_etale, O_X) is fully faithful (morphisms up to 2-isomorphisms on the right hand side).","statement_latex":"Let $X$, $Y$ be algebraic spaces over $\\Spec(\\mathbf{Z})$.\nLet\n$$\n(g, g^\\sharp) :\n(\\Sh(X_\\etale), \\mathcal{O}_X)\n\\longrightarrow\n(\\Sh(Y_\\etale), \\mathcal{O}_Y)\n$$\nbe a morphism of locally ringed topoi. Then there exists a\nunique morphism of algebraic spaces $f : X \\to Y$ such that\n$(g, g^\\sharp)$ is isomorphic to $(f_{small}, f^\\sharp)$.\nIn other words, the construction\n$$\n\\textit{Spaces}/\\Spec(\\mathbf{Z})\n\\longrightarrow \\textit{Locally ringed topoi},\n\\quad\nX \\longrightarrow (X_\\etale, \\mathcal{O}_X)\n$$\nis fully faithful (morphisms up to $2$-isomorphisms on the right hand side).","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Recovering morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04KL","source_file":"spaces-properties.tex","source_line":5026,"source_end_line":5047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5026-L5047","statement_sha256":"6785b627ee03ded3c151bdd360f54922df1222067165d56aff40f8bf5b9154da","origin":"The Stacks Project","memory_eligible":false,"source_rank":10859,"rank":10859,"depth":54,"x":293.758,"y":1560.976,"cluster":"algebraic-spaces"},{"id":"stacks:05YZ","tag":"05YZ","title":"Recovering morphisms · Lemma 05YZ","summary":"Let X, Y be algebraic spaces over Z. If (g, g^sharp) : (Sh(X_etale), O_X) → (Sh(Y_etale), O_Y) is an isomorphism of ringed topoi, then there exists a unique morphism f : X → Y of algebraic spaces such that (g, g^sharp) is isomorphic to (f_small, f^sharp) and moreover f is an isomorphism of algebraic spaces.","statement_latex":"Let $X$, $Y$ be algebraic spaces over $\\mathbf{Z}$. If\n$$\n(g, g^\\sharp) :\n(\\Sh(X_\\etale), \\mathcal{O}_X)\n\\longrightarrow\n(\\Sh(Y_\\etale), \\mathcal{O}_Y)\n$$\nis an isomorphism of ringed topoi, then there exists a unique\nmorphism $f : X \\to Y$ of algebraic spaces such that\n$(g, g^\\sharp)$ is isomorphic to $(f_{small}, f^\\sharp)$\nand moreover $f$ is an isomorphism of algebraic spaces.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Recovering morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YZ","source_file":"spaces-properties.tex","source_line":5165,"source_end_line":5178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5165-L5178","statement_sha256":"bd357be5af645da1f29cd697173fca3156f3c1eac4b6db17c27485cec3274e1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10860,"rank":10860,"depth":55,"x":214.326,"y":1665.129,"cluster":"algebraic-spaces"},{"id":"stacks:03G9","tag":"03G9","title":"Quasi-coherent sheaves on algebraic spaces · Definition 03G9","summary":"Let S be a scheme. Let X be an algebraic space over S. A quasi-coherent O_X-module is a quasi-coherent module on the ringed site (X_etale, O_X) in the sense of Modules on Sites, Definition [Tag 03DL]. The category of quasi-coherent sheaves on X is denoted QCoh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nA {\\it quasi-coherent} $\\mathcal{O}_X$-module\nis a quasi-coherent module on the ringed site\n$(X_\\etale, \\mathcal{O}_X)$ in the sense of\nModules on Sites,\nDefinition \\ref{sites-modules-definition-site-local}.\nThe category of quasi-coherent sheaves on $X$ is denoted\n$\\QCoh(\\mathcal{O}_X)$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-coherent sheaves on algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03G9","source_file":"spaces-properties.tex","source_line":5222,"source_end_line":5232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5222-L5232","statement_sha256":"6a797e743aba0ac10b6bab48b12950053968fb0684e3b2bdb3c34cba7196161f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10861,"rank":10861,"depth":2,"x":189.073,"y":1542.928,"cluster":"algebraic-spaces"},{"id":"stacks:03GA","tag":"03GA","title":"Quasi-coherent sheaves on algebraic spaces · Lemma 03GA","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The pullback functor f^* : Mod(O_Y) → Mod(O_X) preserves quasi-coherent sheaves.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe pullback functor\n$f^* : \\textit{Mod}(\\mathcal{O}_Y) \\to \\textit{Mod}(\\mathcal{O}_X)$\npreserves quasi-coherent sheaves.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-coherent sheaves on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03GA","source_file":"spaces-properties.tex","source_line":5243,"source_end_line":5250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5243-L5250","statement_sha256":"53ff2143c3dd79d6175491ecb1986fb2c1f364232993ea74968be2a32b7c0084","origin":"The Stacks Project","memory_eligible":false,"source_rank":10862,"rank":10862,"depth":11,"x":306.277,"y":1618.911,"cluster":"algebraic-spaces"},{"id":"stacks:03LZ","tag":"03LZ","title":"Quasi-coherent sheaves on algebraic spaces · Lemma 03LZ","summary":"Let S be a scheme. Let X be an algebraic space over S. A quasi-coherent O_X-module F is given by the following data: • for every U ∈ Ob(X_etale) a quasi-coherent O_U-module F_U on U_etale, • for every f : U' → U in X_etale an isomorphism c_f : f_small^*F_U → F_U'. These data are subject to the condition that given any f : U' → U and g : U\" → U' in X_etale the composition c_g ∘ g_small^*c_f is equal to c_f ∘ g.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nA quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$\nis given by the following data:\n\\begin{enumerate}\n\\item for every $U \\in \\Ob(X_\\etale)$ a quasi-coherent\n$\\mathcal{O}_U$-module $\\mathcal{F}_U$ on $U_\\etale$,\n\\item for every $f : U' \\to U$ in $X_\\etale$ an isomorphism\n$c_f : f_{small}^*\\mathcal{F}_U \\to \\mathcal{F}_{U'}$.\n\\end{enumerate}\nThese data are subject to the condition that given any $f : U' \\to U$\nand $g : U'' \\to U'$ in $X_\\etale$ the composition\n$c_g \\circ g_{small}^*c_f$ is equal to $c_{f \\circ g}$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-coherent sheaves on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03LZ","source_file":"spaces-properties.tex","source_line":5266,"source_end_line":5280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5266-L5280","statement_sha256":"124c4ce717d6473c20fca772c5b86b850794532d5d45b8b54237a89bdb2b910d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10863,"rank":10863,"depth":50,"x":158.356,"y":1629.415,"cluster":"algebraic-spaces"},{"id":"stacks:05VP","tag":"05VP","title":"Quasi-coherent sheaves on algebraic spaces · Lemma 05VP","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Let x ∈ |X| be a point and let overlinex be a geometric point lying over x. Finally, let φ : (U, overlineu) → (X, overlinex) be an étale neighbourhood where U is a scheme. Then (φ^*F)_u ⊗_O_U, u O_X, overlinex = F_overlinex where u ∈ U is the image of overlineu.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in |X|$ be a point and let $\\overline{x}$ be a geometric\npoint lying over $x$. Finally, let\n$\\varphi : (U, \\overline{u}) \\to (X, \\overline{x})$\nbe an \\'etale neighbourhood where $U$ is a scheme.\nThen\n$$\n(\\varphi^*\\mathcal{F})_u \\otimes_{\\mathcal{O}_{U, u}}\n\\mathcal{O}_{X, \\overline{x}} =\n\\mathcal{F}_{\\overline{x}}\n$$\nwhere $u \\in U$ is the image of $\\overline{u}$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-coherent sheaves on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VP","source_file":"spaces-properties.tex","source_line":5287,"source_end_line":5303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5287-L5303","statement_sha256":"a1b21f9efa37aa41ee2009b6cff41084a18bb7f62f04737719234f82be83d577","origin":"The Stacks Project","memory_eligible":false,"source_rank":10864,"rank":10864,"depth":53,"x":259.251,"y":1537.493,"cluster":"algebraic-spaces"},{"id":"stacks:05VQ","tag":"05VQ","title":"Quasi-coherent sheaves on algebraic spaces · Lemma 05VQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let G be a quasi-coherent O_Y-module. Let overlinex be a geometric point of X and let overliney = f ∘ overlinex be the image in Y. Then there is a canonical isomorphism (f^*G)_overlinex = G_overliney ⊗_O_Y, overliney O_X, overlinex of the stalk of the pullback with the tensor product of the stalk with the local ring of X at overlinex.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.\nLet $\\overline{x}$ be a geometric point of $X$ and let\n$\\overline{y} = f \\circ \\overline{x}$ be the image in $Y$.\nThen there is a canonical isomorphism\n$$\n(f^*\\mathcal{G})_{\\overline{x}} =\n\\mathcal{G}_{\\overline{y}} \\otimes_{\\mathcal{O}_{Y, \\overline{y}}}\n\\mathcal{O}_{X, \\overline{x}}\n$$\nof the stalk of the pullback with the tensor product of the stalk\nwith the local ring of $X$ at $\\overline{x}$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-coherent sheaves on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VQ","source_file":"spaces-properties.tex","source_line":5330,"source_end_line":5344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5330-L5344","statement_sha256":"e8292a0ccec7a94c3854a5d2e2e67a719ec924080c0c109ebf1cd2603081e82c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10865,"rank":10865,"depth":54,"x":258.772,"y":1662.854,"cluster":"algebraic-spaces"},{"id":"stacks:03M0","tag":"03M0","title":"Quasi-coherent sheaves on algebraic spaces · Lemma 03M0","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a sheaf of O_X-modules. The following are equivalent • F is a quasi-coherent O_X-module, • there exists an étale morphism f : Y → X of algebraic spaces over S with |f| : |Y| → |X| surjective such that f^*F is quasi-coherent on Y, • there exists a scheme U and a surjective étale morphism φ : U → X such that φ^*F is a quasi-coherent O_U-module, and • for every affine scheme U and étale morphism φ : U → X the…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module,\n\\item there exists an \\'etale morphism $f : Y \\to X$ of\nalgebraic spaces over $S$ with $|f| : |Y| \\to |X|$ surjective\nsuch that $f^*\\mathcal{F}$ is quasi-coherent on $Y$,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that $\\varphi^*\\mathcal{F}$ is a quasi-coherent\n$\\mathcal{O}_U$-module, and\n\\item for every affine scheme $U$ and \\'etale morphism $\\varphi : U \\to X$ the\nrestriction $\\varphi^*\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_U$-module.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-coherent sheaves on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03M0","source_file":"spaces-properties.tex","source_line":5391,"source_end_line":5407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5391-L5407","statement_sha256":"1e6b7714ca560a3bcbd1117dc42526d7ac827a7251ede27ee68a9731cb0154b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10866,"rank":10866,"depth":12,"x":158.051,"y":1569.9,"cluster":"algebraic-spaces"},{"id":"stacks:03M1","tag":"03M1","title":"Quasi-coherent sheaves on algebraic spaces · Lemma 03M1","summary":"Let S be a scheme. Let X be an algebraic space over S. The category QCoh(O_X) of quasi-coherent sheaves on X has the following properties: • Any direct sum of quasi-coherent sheaves is quasi-coherent. • Any colimit of quasi-coherent sheaves is quasi-coherent. • The kernel and cokernel of a morphism of quasi-coherent sheaves is quasi-coherent. • Given a short exact sequence of O_X-modules 0 → F_1 → F_2 → F_3 → 0 if two out of three are quasi-coherent so is the third. •…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe category $\\QCoh(\\mathcal{O}_X)$ of quasi-coherent sheaves on $X$\nhas the following properties:\n\\begin{enumerate}\n\\item Any direct sum of quasi-coherent sheaves is quasi-coherent.\n\\item Any colimit of quasi-coherent sheaves is quasi-coherent.\n\\item The kernel and cokernel of a morphism of quasi-coherent sheaves\nis quasi-coherent.\n\\item Given a short exact sequence of $\\mathcal{O}_X$-modules\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nif two out of three are quasi-coherent so is the third.\n\\item Given two quasi-coherent $\\mathcal{O}_X$-modules\nthe tensor product is quasi-coherent.\n\\item Given two quasi-coherent $\\mathcal{O}_X$-modules\n$\\mathcal{F}$, $\\mathcal{G}$ such that $\\mathcal{F}$\nis of finite presentation (see\nSection \\ref{section-properties-modules}),\nthen the internal hom\n$\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$\nis quasi-coherent.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-coherent sheaves on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03M1","source_file":"spaces-properties.tex","source_line":5440,"source_end_line":5463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5440-L5463","statement_sha256":"53955bc0334a40c4e73c1c8d9d721f478029d407eda44aed637148cb7b9f9c75","origin":"The Stacks Project","memory_eligible":false,"source_rank":10867,"rank":10867,"depth":41,"x":307.461,"y":1581.321,"cluster":"algebraic-spaces"},{"id":"stacks:060Q","tag":"060Q","title":"Locally projective modules · Lemma 060Q","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. The following are equivalent • for some scheme U and surjective étale morphism U → X the restriction F|_U is locally projective on U, and • for any scheme U and any étale morphism U → X the restriction F|_U is locally projective on U.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some scheme $U$ and surjective \\'etale morphism\n$U \\to X$ the restriction $\\mathcal{F}|_U$ is locally projective\non $U$, and\n\\item for any scheme $U$ and any \\'etale morphism\n$U \\to X$ the restriction $\\mathcal{F}|_U$ is locally projective\non $U$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Locally projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060Q","source_file":"spaces-properties.tex","source_line":5648,"source_end_line":5661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5648-L5661","statement_sha256":"fc27aff2261e8376c67338b2e7d3ec1dc65be8c7091ccc8e880627d7fd709b74","origin":"The Stacks Project","memory_eligible":false,"source_rank":10868,"rank":10868,"depth":12,"x":187.793,"y":1657.876,"cluster":"algebraic-spaces"},{"id":"stacks:060R","tag":"060R","title":"Locally projective modules · Definition 060R","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. We say F is locally projective if the equivalent conditions of Lemma [Tag 060Q] are satisfied.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nWe say $\\mathcal{F}$ is {\\it locally projective}\nif the equivalent conditions of\nLemma \\ref{lemma-locally-projective}\nare satisfied.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Locally projective modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060R","source_file":"spaces-properties.tex","source_line":5673,"source_end_line":5681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5673-L5681","statement_sha256":"c76d3356d71dc81c27c3802ea2e33b59c9065eea07d776d17a1ca28b1240bad9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10869,"rank":10869,"depth":13,"x":214.541,"y":1533.203,"cluster":"algebraic-spaces"},{"id":"stacks:060S","tag":"060S","title":"Locally projective modules · Lemma 060S","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let G be a quasi-coherent O_Y-module. If G is locally projective on Y, then f^*G is locally projective on X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.\nIf $\\mathcal{G}$ is locally projective on $Y$, then $f^*\\mathcal{G}$\nis locally projective on $X$.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Locally projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060S","source_file":"spaces-properties.tex","source_line":5683,"source_end_line":5690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5683-L5690","statement_sha256":"395c5aadfde6af19a282169d2bd82fb547d1bea86759404cc8d17eae0da89a8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10870,"rank":10870,"depth":12,"x":295.283,"y":1640.587,"cluster":"algebraic-spaces"},{"id":"stacks:03M3","tag":"03M3","title":"Quasi-coherent sheaves and presentations · Proposition 03M3","summary":"With S, φ : U → X, and (U, R, s, t, c) as above. For any quasi-coherent O_X-module F the sheaf φ^*F comes equipped with a canonical isomorphism α : t^*φ^*F → s^*φ^*F which satisfies the conditions of Groupoids, Definition [Tag 03LI] and therefore defines a quasi-coherent sheaf on (U, R, s, t, c). The functor F ↦ (φ^*F, α) defines an equivalence of categories Quasi-coherent O_X-modules longleftrightarrow Quasi-coherent modules on (U, R, s, t, c)","statement_latex":"With $S$, $\\varphi : U \\to X$, and $(U, R, s, t, c)$ as above.\nFor any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the\nsheaf $\\varphi^*\\mathcal{F}$ comes equipped with a canonical\nisomorphism\n$$\n\\alpha : t^*\\varphi^*\\mathcal{F} \\longrightarrow s^*\\varphi^*\\mathcal{F}\n$$\nwhich satisfies the conditions of\nGroupoids, Definition \\ref{groupoids-definition-groupoid-module}\nand therefore defines a quasi-coherent sheaf on $(U, R, s, t, c)$.\nThe functor $\\mathcal{F} \\mapsto (\\varphi^*\\mathcal{F}, \\alpha)$\ndefines an equivalence of categories\n$$\n\\begin{matrix}\n\\text{Quasi-coherent} \\\\\n\\mathcal{O}_X\\text{-modules}\n\\end{matrix}\n\\longleftrightarrow\n\\begin{matrix}\n\\text{Quasi-coherent modules}\\\\\n\\text{on }(U, R, s, t, c)\n\\end{matrix}\n$$","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-coherent sheaves and presentations","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03M3","source_file":"spaces-properties.tex","source_line":5726,"source_end_line":5751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5726-L5751","statement_sha256":"9adcfcc5ecc9b7a48a04a797b7d431cdd8b5b60abf1541aa4413c8c86c677ec8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10871,"rank":10871,"depth":51,"x":149.017,"y":1607.133,"cluster":"algebraic-spaces"},{"id":"stacks:077V","tag":"077V","title":"Quasi-coherent sheaves and presentations · Proposition 077V","summary":"Let S be a scheme. Let X be an algebraic space over S. • The category QCoh(O_X) is a Grothendieck abelian category. Consequently, QCoh(O_X) has enough injectives and all limits. • The inclusion functor QCoh(O_X) → Mod(O_X) has a right adjoint. Q : Mod(O_X) → QCoh(O_X) such that for every quasi-coherent sheaf F the adjunction mapping Q(F) → F is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item The category $\\QCoh(\\mathcal{O}_X)$ is a Grothendieck\nabelian category. Consequently, $\\QCoh(\\mathcal{O}_X)$\nhas enough injectives and all limits.\n\\item The inclusion functor\n$\\QCoh(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_X)$\nhas a right adjoint\\footnote{This functor is sometimes called\nthe {\\it coherator}.}\n$$\nQ : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\QCoh(\\mathcal{O}_X)\n$$\nsuch that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping\n$Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quasi-coherent sheaves and presentations","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077V","source_file":"spaces-properties.tex","source_line":5806,"source_end_line":5823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5806-L5823","statement_sha256":"dffa4be63c9ddd17a71ef02a0a819ce01ab98e44d4972e4ac145234741eaa693","origin":"The Stacks Project","memory_eligible":false,"source_rank":10872,"rank":10872,"depth":52,"x":284.116,"y":1548.659,"cluster":"algebraic-spaces"},{"id":"stacks:05Z1","tag":"05Z1","title":"Morphisms towards schemes · Lemma 05Z1","summary":"Let X be an algebraic space over Z. Let T be an affine scheme. The map Mor(X, T) → Hom(Γ(T, O_T), Γ(X, O_X)) which maps f to f^sharp (on global sections) is bijective.","statement_latex":"Let $X$ be an algebraic space over $\\mathbf{Z}$.\nLet $T$ be an affine scheme.\nThe map\n$$\n\\Mor(X, T)\n\\longrightarrow\n\\Hom(\\Gamma(T, \\mathcal{O}_T), \\Gamma(X, \\mathcal{O}_X))\n$$\nwhich maps $f$ to $f^\\sharp$ (on global sections) is bijective.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Morphisms towards schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Z1","source_file":"spaces-properties.tex","source_line":5902,"source_end_line":5913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5902-L5913","statement_sha256":"a16babfabb3d1844c98e9f59d6c47b74304f1800e460c3f0d7489f87cce7a464","origin":"The Stacks Project","memory_eligible":false,"source_rank":10873,"rank":10873,"depth":11,"x":231.386,"y":1668.738,"cluster":"algebraic-spaces"},{"id":"stacks:071S","tag":"071S","title":"Quotients by free actions · Lemma 071S","summary":"Let S be a scheme. Let X be an algebraic space over S. Let G be an abstract group with a free action on X. Then the quotient sheaf X/G is an algebraic space.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $G$ be an abstract group with a free action on $X$.\nThen the quotient sheaf $X/G$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Properties of Algebraic Spaces","chapter_id":"spaces-properties","section":"Quotients by free actions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/071S","source_file":"spaces-properties.tex","source_line":5967,"source_end_line":5973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-properties.tex#L5967-L5973","statement_sha256":"c4fe34414a126aaa8c42473b6597ab4b042ae797aa0f6e9f4a4048f307b746e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10874,"rank":10874,"depth":52,"x":173.56,"y":1549.975,"cluster":"algebraic-spaces"},{"id":"stacks:03HK","tag":"03HK","title":"Separation axioms · Lemma 03HK","summary":"Let S be a scheme contained in Sch_fppf. Let f : X → Y be a morphism of algebraic spaces over S. Let Δ_X/Y : X → X ×_Y X be the diagonal morphism. Then • Δ_X/Y is representable, • Δ_X/Y is locally of finite type, • Δ_X/Y is a monomorphism, • Δ_X/Y is separated, and • Δ_X/Y is locally quasi-finite.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\Delta_{X/Y} : X \\to X \\times_Y X$ be the diagonal morphism.\nThen\n\\begin{enumerate}\n\\item $\\Delta_{X/Y}$ is representable,\n\\item $\\Delta_{X/Y}$ is locally of finite type,\n\\item $\\Delta_{X/Y}$ is a monomorphism,\n\\item $\\Delta_{X/Y}$ is separated, and\n\\item $\\Delta_{X/Y}$ is locally quasi-finite.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HK","source_file":"spaces-morphisms.tex","source_line":164,"source_end_line":177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L164-L177","statement_sha256":"6bc1d57eb9dbb73a2a91457f513893a88a3d3f9001e19b191943f455698fe9f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10875,"rank":10875,"depth":52,"x":312.051,"y":1604.874,"cluster":"algebraic-spaces"},{"id":"stacks:03HL","tag":"03HL","title":"Separation axioms · Definition 03HL","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let Δ_X/Y : X → X ×_Y X be the diagonal morphism. • We say f is separated if Δ_X/Y is a closed immersion. • We say f is locally separated if Δ_X/Y is an immersion. • We say f is quasi-separated if Δ_X/Y is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\Delta_{X/Y} : X \\to X \\times_Y X$ be the diagonal morphism.\n\\begin{enumerate}\n\\item We say $f$ is {\\it separated} if $\\Delta_{X/Y}$ is a closed immersion.\n\\item We say $f$ is {\\it locally separated}\\footnote{In the literature\nthis term often refers to quasi-separated and locally separated morphisms.}\nif $\\Delta_{X/Y}$ is an immersion.\n\\item We say $f$ is {\\it quasi-separated} if $\\Delta_{X/Y}$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HL","source_file":"spaces-morphisms.tex","source_line":236,"source_end_line":248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L236-L248","statement_sha256":"95761c55baf759c3f09c4a4b1b7af4e0fead938cd48d9274ed63c0bb3d746b50","origin":"The Stacks Project","memory_eligible":false,"source_rank":10876,"rank":10876,"depth":0,"x":165.418,"y":1643.07,"cluster":"algebraic-spaces"},{"id":"stacks:03KK","tag":"03KK","title":"Separation axioms · Lemma 03KK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is separated, then f is locally separated and f is quasi-separated.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. If $f$ is separated, then $f$ is locally separated and\n$f$ is quasi-separated.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KK","source_file":"spaces-morphisms.tex","source_line":257,"source_end_line":262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L257-L262","statement_sha256":"93292fc1128dd5f3d7f6f598b08585b921d38b2fe6fd36d324d60fdcc526ffaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":10877,"rank":10877,"depth":3,"x":243.018,"y":1531.426,"cluster":"algebraic-spaces"},{"id":"stacks:03KL","tag":"03KL","title":"Separation axioms · Lemma 03KL","summary":"All of the separation axioms listed in Definition [Tag 03HL] are stable under base change.","statement_latex":"All of the separation axioms listed in Definition \\ref{definition-separated}\nare stable under base change.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KL","source_file":"spaces-morphisms.tex","source_line":271,"source_end_line":275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L271-L275","statement_sha256":"5a158d3a8714bf864c8033a504acf7105ecee5b553f2e050ec76b63c20e11ddd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10878,"rank":10878,"depth":1,"x":275.658,"y":1658.094,"cluster":"algebraic-spaces"},{"id":"stacks:03KN","tag":"03KN","title":"Separation axioms · Lemma 03KN","summary":"The top arrow of a \"magic diagram\" of algebraic spaces has nice immersion-like properties, and under separatedness hypotheses these get stronger. Let S be a scheme. Let f : X → Z, g : Y → Z and Z → T be morphisms of algebraic spaces over S. Consider the induced morphism i : X ×_Z Y → X ×_T Y. Then • i is representable, locally of finite type, locally quasi-finite, separated and a monomorphism, • if Z → T is locally separated, then i is an immersion, • if Z → T is…","statement_latex":"\\begin{slogan}\nThe top arrow of a ``magic diagram'' of algebraic spaces has nice\nimmersion-like properties, and under separatedness hypotheses\nthese get stronger.\n\\end{slogan}\nLet $S$ be a scheme. Let $f : X \\to Z$, $g : Y \\to Z$ and $Z \\to T$\nbe morphisms of algebraic spaces over $S$. Consider the induced morphism\n$i : X \\times_Z Y \\to X \\times_T Y$. Then\n\\begin{enumerate}\n\\item $i$ is representable, locally of finite type, locally quasi-finite,\nseparated and a monomorphism,\n\\item if $Z \\to T$ is locally separated, then $i$ is an immersion,\n\\item if $Z \\to T$ is separated, then $i$ is a closed immersion, and\n\\item if $Z \\to T$ is quasi-separated, then $i$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KN","source_file":"spaces-morphisms.tex","source_line":288,"source_end_line":305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L288-L305","statement_sha256":"5cb3cc3bfddbc77c835ffdbde2a61189de092c76f8ea2f84b826d1f3067d8feb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10879,"rank":10879,"depth":53,"x":149.418,"y":1583.028,"cluster":"algebraic-spaces"},{"id":"stacks:03KO","tag":"03KO","title":"Separation axioms · Lemma 03KO","summary":"Properties of the graph of a morphism of algebraic spaces as a consequence of separation properties of the target. Let S be a scheme. Let T be an algebraic space over S. Let g : X → Y be a morphism of algebraic spaces over T. Consider the graph i : X → X ×_T Y of g. Then • i is representable, locally of finite type, locally quasi-finite, separated and a monomorphism, • if Y → T is locally separated, then i is an immersion, • if Y → T is separated, then i is a closed…","statement_latex":"\\begin{slogan}\nProperties of the graph of a morphism of algebraic spaces\nas a consequence of separation properties of the target.\n\\end{slogan}\nLet $S$ be a scheme. Let $T$ be an algebraic space over $S$.\nLet $g : X \\to Y$ be a morphism of algebraic spaces over $T$.\nConsider the graph $i : X \\to X \\times_T Y$ of $g$. Then\n\\begin{enumerate}\n\\item $i$ is representable, locally of finite type, locally quasi-finite,\nseparated and a monomorphism,\n\\item if $Y \\to T$ is locally separated, then $i$ is an immersion,\n\\item if $Y \\to T$ is separated, then $i$ is a closed immersion, and\n\\item if $Y \\to T$ is quasi-separated, then $i$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KO","source_file":"spaces-morphisms.tex","source_line":321,"source_end_line":337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L321-L337","statement_sha256":"4fbae898cc097fa6224c4323895ce853ca1b4e923e29b4f5deefa061af72e583","origin":"The Stacks Project","memory_eligible":false,"source_rank":10880,"rank":10880,"depth":54,"x":303.246,"y":1566.71,"cluster":"algebraic-spaces"},{"id":"stacks:03KP","tag":"03KP","title":"Separation axioms · Lemma 03KP","summary":"Let S be a scheme. Let f : X → T be a morphism of algebraic spaces over S. Let s : T → X be a section of f (in a formula f ∘ s = id_T). Then • s is representable, locally of finite type, locally quasi-finite, separated and a monomorphism, • if f is locally separated, then s is an immersion, • if f is separated, then s is a closed immersion, and • if f is quasi-separated, then s is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to T$ be a morphism of algebraic spaces over $S$.\nLet $s : T \\to X$ be a section of $f$ (in a formula\n$f \\circ s = \\text{id}_T$). Then\n\\begin{enumerate}\n\\item $s$ is representable, locally of finite type, locally quasi-finite,\nseparated and a monomorphism,\n\\item if $f$ is locally separated, then $s$ is an immersion,\n\\item if $f$ is separated, then $s$ is a closed immersion, and\n\\item if $f$ is quasi-separated, then $s$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KP","source_file":"spaces-morphisms.tex","source_line":344,"source_end_line":357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L344-L357","statement_sha256":"3efbf1fee65b4bb3b40be4225ac42b9f19e58cd51a1b42b43d2329550474f404","origin":"The Stacks Project","memory_eligible":false,"source_rank":10881,"rank":10881,"depth":55,"x":202.694,"y":1666.269,"cluster":"algebraic-spaces"},{"id":"stacks:03KQ","tag":"03KQ","title":"Separation axioms · Lemma 03KQ","summary":"All of the separation axioms listed in Definition [Tag 03HL] are stable under composition of morphisms.","statement_latex":"All of the separation axioms listed in Definition \\ref{definition-separated}\nare stable under composition of morphisms.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KQ","source_file":"spaces-morphisms.tex","source_line":364,"source_end_line":368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L364-L368","statement_sha256":"c98b841c9d6e5a4abc8870b706e2a40850653ff76dde50baa5bd9319580e978c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10882,"rank":10882,"depth":54,"x":196.763,"y":1535.485,"cluster":"algebraic-spaces"},{"id":"stacks:04ZH","tag":"04ZH","title":"Separation axioms · Lemma 04ZH","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • If Y is separated and f is separated, then X is separated. • If Y is quasi-separated and f is quasi-separated, then X is quasi-separated. • If Y is locally separated and f is locally separated, then X is locally separated. • If Y is separated over S and f is separated, then X is separated over S. • If Y is quasi-separated over S and f is quasi-separated, then X is quasi-separated over S. • If Y…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $Y$ is separated and $f$ is separated, then $X$ is separated.\n\\item If $Y$ is quasi-separated and $f$ is quasi-separated, then\n$X$ is quasi-separated.\n\\item If $Y$ is locally separated and $f$ is locally separated, then\n$X$ is locally separated.\n\\item If $Y$ is separated over $S$ and $f$ is separated, then\n$X$ is separated over $S$.\n\\item If $Y$ is quasi-separated over $S$ and $f$ is quasi-separated, then\n$X$ is quasi-separated over $S$.\n\\item If $Y$ is locally separated over $S$ and $f$ is locally separated, then\n$X$ is locally separated over $S$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZH","source_file":"spaces-morphisms.tex","source_line":386,"source_end_line":403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L386-L403","statement_sha256":"3c8f72bbc6c32694965f17fa2ceedb278fcec2ba6fbdc8b8341ccbe6f2f697dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":10883,"rank":10883,"depth":55,"x":306.572,"y":1628.782,"cluster":"algebraic-spaces"},{"id":"stacks:03KR","tag":"03KR","title":"Separation axioms · Lemma 03KR","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of algebraic spaces over S. • If g ∘ f is separated then so is f. • If g ∘ f is locally separated then so is f. • If g ∘ f is quasi-separated then so is f.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $g \\circ f$ is separated then so is $f$.\n\\item If $g \\circ f$ is locally separated then so is $f$.\n\\item If $g \\circ f$ is quasi-separated then so is $f$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KR","source_file":"spaces-morphisms.tex","source_line":415,"source_end_line":424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L415-L424","statement_sha256":"e360bf58a5a023e0d05cd0332de9ec1c4337e058aaa2186d9a0d7daf0ca9ea1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10884,"rank":10884,"depth":0,"x":150.202,"y":1622.279,"cluster":"algebraic-spaces"},{"id":"stacks:04ZI","tag":"04ZI","title":"Separation axioms · Lemma 04ZI","summary":"Let S be a scheme. Let X be an algebraic space over S. • If X is separated then X is separated over S. • If X is locally separated then X is locally separated over S. • If X is quasi-separated then X is quasi-separated over S. Let f : X → Y be a morphism of algebraic spaces over S. • [(4)] If X is separated over S then f is separated. • [(5)] If X is locally separated over S then f is locally separated. • [(6)] If X is quasi-separated over S then f is quasi-separated.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $X$ is separated then $X$ is separated over $S$.\n\\item If $X$ is locally separated then $X$ is locally separated over $S$.\n\\item If $X$ is quasi-separated then $X$ is quasi-separated over $S$.\n\\end{enumerate}\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item[(4)] If $X$ is separated over $S$ then $f$ is separated.\n\\item[(5)] If $X$ is locally separated over $S$ then $f$ is locally separated.\n\\item[(6)] If $X$ is quasi-separated over $S$ then $f$ is quasi-separated.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZI","source_file":"spaces-morphisms.tex","source_line":440,"source_end_line":454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L440-L454","statement_sha256":"52a245e12a159e55a6e7253762edefd7f8e52821a07df554ed1e54fa6b2aee36","origin":"The Stacks Project","memory_eligible":false,"source_rank":10885,"rank":10885,"depth":55,"x":271.023,"y":1538.146,"cluster":"algebraic-spaces"},{"id":"stacks:03KM","tag":"03KM","title":"Separation axioms · Lemma 03KM","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let P be any of the separation axioms of Definition [Tag 03HL]. The following are equivalent • f is P, • for every scheme Z and morphism Z → Y the base change Z ×_Y X → Z of f is P, • for every affine scheme Z and every morphism Z → Y the base change Z ×_Y X → Z of f is P, • for every affine scheme Z and every morphism Z → Y the algebraic space Z ×_Y X is P (see Properties of Spaces, Definition…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{P}$ be any of the separation\naxioms of Definition \\ref{definition-separated}.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is $\\mathcal{P}$,\n\\item for every scheme $Z$ and morphism $Z \\to Y$ the\nbase change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the\nbase change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the\nalgebraic space $Z \\times_Y X$ is $\\mathcal{P}$ (see\nProperties of Spaces, Definition \\ref{spaces-properties-definition-separated}),\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that the base change $V \\times_Y X \\to V$ has\n$\\mathcal{P}$, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each\nof the morphisms $f^{-1}(Y_i) \\to Y_i$ has $\\mathcal{P}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KM","source_file":"spaces-morphisms.tex","source_line":467,"source_end_line":489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L467-L489","statement_sha256":"b675398c0d6359e260c85f828727b0da8288d89d6c17e3dc5b69170d353dee5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10886,"rank":10886,"depth":55,"x":249.537,"y":1669.051,"cluster":"algebraic-spaces"},{"id":"stacks:03KY","tag":"03KY","title":"Separation axioms · Lemma 03KY","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. • The morphism f is locally separated. • The morphism f is (quasi-)separated in the sense of Definition [Tag 03HL] above if and only if f is (quasi-)separated in the sense of Section [Tag 03HA]. In particular, if f : X → Y is a morphism of schemes over S, then f is (quasi-)separated in the sense of Definition [Tag 03HL] if and only if f is (quasi-)separated as a morphism of schemes.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item The morphism $f$ is locally separated.\n\\item The morphism $f$ is (quasi-)separated in the sense of\nDefinition \\ref{definition-separated}\nabove if and only if $f$ is (quasi-)separated in the sense of\nSection \\ref{section-representable}.\n\\end{enumerate}\nIn particular, if $f : X \\to Y$ is a morphism of schemes over $S$, then\n$f$ is (quasi-)separated in the sense of\nDefinition \\ref{definition-separated}\nif and only if $f$ is (quasi-)separated as a morphism of schemes.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KY","source_file":"spaces-morphisms.tex","source_line":558,"source_end_line":573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L558-L573","statement_sha256":"9eddf8aa1c0472f17242422aac0193741272691ce9937b64583d1082ae26dd60","origin":"The Stacks Project","memory_eligible":false,"source_rank":10887,"rank":10887,"depth":56,"x":159.9,"y":1560.076,"cluster":"algebraic-spaces"},{"id":"stacks:03MD","tag":"03MD","title":"Surjective morphisms · Lemma 03MD","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. Then f is surjective (in the sense of Section [Tag 03HA]) if and only if |f| : |X| → |Y| is surjective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable\nmorphism of algebraic spaces over $S$. Then\n$f$ is surjective (in the sense of Section \\ref{section-representable})\nif and only if $|f| : |X| \\to |Y|$ is surjective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MD","source_file":"spaces-morphisms.tex","source_line":600,"source_end_line":606,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L600-L606","statement_sha256":"75ec2d8c030deed94079e1420551522a0ec46e928c5f898739ab28bc05555e8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10888,"rank":10888,"depth":1,"x":313.993,"y":1589.639,"cluster":"algebraic-spaces"},{"id":"stacks:03ME","tag":"03ME","title":"Surjective morphisms · Definition 03ME","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say f is surjective if the map |f| : |X| → |Y| of associated topological spaces is surjective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. We say $f$ is {\\it surjective}\nif the map $|f| : |X| \\to |Y|$ of associated topological spaces\nis surjective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Surjective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ME","source_file":"spaces-morphisms.tex","source_line":624,"source_end_line":630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L624-L630","statement_sha256":"e6010d460bd2eb99103d9b4b4ee24219dba4592dae07de772a19f2f5dc26a61e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10889,"rank":10889,"depth":0,"x":176.272,"y":1655.428,"cluster":"algebraic-spaces"},{"id":"stacks:03MF","tag":"03MF","title":"Surjective morphisms · Lemma 03MF","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is surjective, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is surjective, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is surjective, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is a surjective morphism, • there exists a scheme U and a surjective étale morphism φ : U →…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is surjective,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is surjective,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is surjective,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is a surjective morphism,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis surjective,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are surjective \\'etale\nsuch that the top horizontal arrow is surjective, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that\neach of the morphisms $f^{-1}(Y_i) \\to Y_i$ is surjective.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MF","source_file":"spaces-morphisms.tex","source_line":632,"source_end_line":660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L632-L660","statement_sha256":"caaee459c2e0d389f953218f5a956a855e576550a7da7e2aafd4af78cfce8e34","origin":"The Stacks Project","memory_eligible":false,"source_rank":10890,"rank":10890,"depth":0,"x":225.033,"y":1528.476,"cluster":"algebraic-spaces"},{"id":"stacks:03MG","tag":"03MG","title":"Surjective morphisms · Lemma 03MG","summary":"The composition of surjective morphisms is surjective.","statement_latex":"The composition of surjective morphisms is surjective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MG","source_file":"spaces-morphisms.tex","source_line":666,"source_end_line":669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L666-L669","statement_sha256":"4dd4fe35ca16f2c73e3e366b1f2912c07a5b5b2a6386e2f44bd252c2338c7567","origin":"The Stacks Project","memory_eligible":false,"source_rank":10891,"rank":10891,"depth":0,"x":291.321,"y":1650.037,"cluster":"algebraic-spaces"},{"id":"stacks:03MH","tag":"03MH","title":"Surjective morphisms · Lemma 03MH","summary":"The base change of a surjective morphism is surjective.","statement_latex":"The base change of a surjective morphism is surjective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MH","source_file":"spaces-morphisms.tex","source_line":675,"source_end_line":678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L675-L678","statement_sha256":"0d5614dbadbf8741512daa6ee61694cffd32e3cab3ae71b919ab1c129a04b3fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":10892,"rank":10892,"depth":1,"x":144.348,"y":1597.894,"cluster":"algebraic-spaces"},{"id":"stacks:03Z1","tag":"03Z1","title":"Open morphisms · Lemma 03Z1","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. The following are equivalent • f is universally open (in the sense of Section [Tag 03HA]), and • for every morphism of algebraic spaces Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is open.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of\nalgebraic spaces over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally open\n(in the sense of Section \\ref{section-representable}), and\n\\item for every morphism of algebraic spaces $Z \\to Y$ the morphism of\ntopological spaces $|Z \\times_Y X| \\to |Z|$ is open.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Z1","source_file":"spaces-morphisms.tex","source_line":703,"source_end_line":713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L703-L713","statement_sha256":"4469924e536cd9fd5c7c178d8d8ccece1126877a4f2dc0f4fe65be5084b1ed4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10893,"rank":10893,"depth":0,"x":295.001,"y":1552.845,"cluster":"algebraic-spaces"},{"id":"stacks:03Z2","tag":"03Z2","title":"Open morphisms · Definition 03Z2","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is open if the map of topological spaces |f| : |X| → |Y| is open. • We say f is universally open if for every morphism of algebraic spaces Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is open, i.e., the base change Z ×_Y X → Z is open.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it open} if the map of topological spaces\n$|f| : |X| \\to |Y|$ is open.\n\\item We say $f$ is {\\it universally open} if for every morphism\nof algebraic spaces $Z \\to Y$ the morphism of topological spaces\n$$\n|Z \\times_Y X| \\to |Z|\n$$\nis open, i.e., the base change $Z \\times_Y X \\to Z$ is open.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Open morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Z2","source_file":"spaces-morphisms.tex","source_line":740,"source_end_line":754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L740-L754","statement_sha256":"b40f79450da2a6bc408f8564cb58c48ac662abd22e47af18e55fd879e5e93063","origin":"The Stacks Project","memory_eligible":false,"source_rank":10894,"rank":10894,"depth":0,"x":219.968,"y":1671.817,"cluster":"algebraic-spaces"},{"id":"stacks:03Z3","tag":"03Z3","title":"Open morphisms · Lemma 03Z3","summary":"The base change of a universally open morphism of algebraic spaces by any morphism of algebraic spaces is universally open.","statement_latex":"The base change of a universally open morphism of algebraic spaces\nby any morphism of algebraic spaces is universally open.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Z3","source_file":"spaces-morphisms.tex","source_line":763,"source_end_line":767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L763-L767","statement_sha256":"89f290211bfa9365ceb5bf3c50a962545127054f83687ad01fea079e1f8b8e6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10895,"rank":10895,"depth":0,"x":179.53,"y":1541.218,"cluster":"algebraic-spaces"},{"id":"stacks:03Z4","tag":"03Z4","title":"Open morphisms · Lemma 03Z4","summary":"The composition of a pair of (universally) open morphisms of algebraic spaces is (universally) open.","statement_latex":"The composition of a pair of (universally) open morphisms of algebraic spaces\nis (universally) open.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Z4","source_file":"spaces-morphisms.tex","source_line":773,"source_end_line":777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L773-L777","statement_sha256":"77bebdda00c703cc265615cded53f17726fb302d1caec3364f9b3a3aa826cc4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10896,"rank":10896,"depth":0,"x":314.677,"y":1614.741,"cluster":"algebraic-spaces"},{"id":"stacks:03Z5","tag":"03Z5","title":"Open morphisms · Lemma 03Z5","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f is universally open, • for every scheme Z and every morphism Z → Y the projection |Z ×_Y X| → |Z| is open, • for every affine scheme Z and every morphism Z → Y the projection |Z ×_Y X| → |Z| is open, and • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is a universally open morphism of algebraic spaces, and • there exists a…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally open,\n\\item for every scheme $Z$ and every morphism $Z \\to Y$\nthe projection $|Z \\times_Y X| \\to |Z|$ is open,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$\nthe projection $|Z \\times_Y X| \\to |Z|$ is open, and\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is a universally open morphism\nof algebraic spaces, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that\neach of the morphisms $f^{-1}(Y_i) \\to Y_i$ is universally open.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Z5","source_file":"spaces-morphisms.tex","source_line":783,"source_end_line":799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L783-L799","statement_sha256":"65919dc157b61792f0187e1c5ba0e144cfcf02340f87527cd263965201bda1fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10897,"rank":10897,"depth":46,"x":155.54,"y":1637.262,"cluster":"algebraic-spaces"},{"id":"stacks:06DN","tag":"06DN","title":"Open morphisms · Lemma 06DN","summary":"Let S be a scheme. Let p : X → Spec(k) be a morphism of algebraic spaces over S where k is a field. Then p : X → Spec(k) is universally open.","statement_latex":"Let $S$ be a scheme. Let $p : X \\to \\Spec(k)$ be a morphism of\nalgebraic spaces over $S$ where $k$ is a field. Then\n$p : X \\to \\Spec(k)$ is universally open.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DN","source_file":"spaces-morphisms.tex","source_line":859,"source_end_line":864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L859-L864","statement_sha256":"424113f547ddc9b37cdcc1d3d9057970aa2c0422ba01681d46d5a5a8e95fae90","origin":"The Stacks Project","memory_eligible":false,"source_rank":10898,"rank":10898,"depth":47,"x":254.997,"y":1530.117,"cluster":"algebraic-spaces"},{"id":"stacks:0CFQ","tag":"0CFQ","title":"Submersive morphisms · Lemma 0CFQ","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. The following are equivalent • f is universally submersive (in the sense of Section [Tag 03HA]), and • for every morphism of algebraic spaces Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is submersive.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of\nalgebraic spaces over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally submersive\n(in the sense of Section \\ref{section-representable}), and\n\\item for every morphism of algebraic spaces $Z \\to Y$ the morphism of\ntopological spaces $|Z \\times_Y X| \\to |Z|$ is submersive.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Submersive morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CFQ","source_file":"spaces-morphisms.tex","source_line":900,"source_end_line":910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L900-L910","statement_sha256":"9fd06e29aa7b6bce8a2b79f7575f8947dc5f3706c6d155d4545d7936cde06363","origin":"The Stacks Project","memory_eligible":false,"source_rank":10899,"rank":10899,"depth":0,"x":267.85,"y":1665.86,"cluster":"algebraic-spaces"},{"id":"stacks:0412","tag":"0412","title":"Submersive morphisms · Definition 0412","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is submersive if the continuous map |X| → |Y| is submersive, see Topology, Definition [Tag 0406]. • We say f is universally submersive if for every morphism of algebraic spaces Y' → Y the base change Y' ×_Y X → Y' is submersive.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it submersive}\\footnote{This is very different\nfrom the notion of a submersion of differential manifolds.}\nif the continuous map $|X| \\to |Y|$ is submersive, see\nTopology, Definition \\ref{topology-definition-submersive}.\n\\item We say $f$ is {\\it universally submersive} if for every\nmorphism of algebraic spaces $Y' \\to Y$ the base change\n$Y' \\times_Y X \\to Y'$ is submersive.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Submersive morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0412","source_file":"spaces-morphisms.tex","source_line":937,"source_end_line":950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L937-L950","statement_sha256":"b25b3db26106422294fe629231b281369c98290c454b7a5ac444ef71b41e904d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10900,"rank":10900,"depth":1,"x":148.948,"y":1572.852,"cluster":"algebraic-spaces"},{"id":"stacks:0CFR","tag":"0CFR","title":"Submersive morphisms · Lemma 0CFR","summary":"The base change of a universally submersive morphism of algebraic spaces by any morphism of algebraic spaces is universally submersive.","statement_latex":"The base change of a universally submersive morphism of algebraic spaces\nby any morphism of algebraic spaces is universally submersive.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Submersive morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CFR","source_file":"spaces-morphisms.tex","source_line":955,"source_end_line":959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L955-L959","statement_sha256":"f7776b96005262fa5499cfd45e8d346850e7aa6aa4e8a00a35f2fce7d85091eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10901,"rank":10901,"depth":0,"x":311.777,"y":1573.97,"cluster":"algebraic-spaces"},{"id":"stacks:0CFS","tag":"0CFS","title":"Submersive morphisms · Lemma 0CFS","summary":"The composition of a pair of (universally) submersive morphisms of algebraic spaces is (universally) submersive.","statement_latex":"The composition of a pair of (universally) submersive morphisms of\nalgebraic spaces is (universally) submersive.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Submersive morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CFS","source_file":"spaces-morphisms.tex","source_line":965,"source_end_line":969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L965-L969","statement_sha256":"778d92c2d50c6179ee422192121aabc3c17c6799695c6ba5fa6f18b20fc4a495","origin":"The Stacks Project","memory_eligible":false,"source_rank":10902,"rank":10902,"depth":0,"x":190.545,"y":1665.74,"cluster":"algebraic-spaces"},{"id":"stacks:03HD","tag":"03HD","title":"Quasi-compact morphisms · Lemma 03HD","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. The following are equivalent: • f is quasi-compact (in the sense of Section [Tag 03HA]), and • for every quasi-compact algebraic space Z and any morphism Z → Y the algebraic space Z ×_Y X is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is quasi-compact\n(in the sense of Section \\ref{section-representable}), and\n\\item for every quasi-compact algebraic space $Z$ and any morphism\n$Z \\to Y$ the algebraic space $Z \\times_Y X$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HD","source_file":"spaces-morphisms.tex","source_line":995,"source_end_line":1006,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L995-L1006","statement_sha256":"cebda6cde9142d8d4f6cbd7a80fd7b85b361ae6c331178f20027e3602007c9c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10903,"rank":10903,"depth":1,"x":206.175,"y":1528.982,"cluster":"algebraic-spaces"},{"id":"stacks:03HE","tag":"03HE","title":"Quasi-compact morphisms · Definition 03HE","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say f is quasi-compact if for every quasi-compact algebraic space Z and morphism Z → Y the fibre product Z ×_Y X is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nWe say $f$ is {\\it quasi-compact} if for every quasi-compact\nalgebraic space $Z$ and morphism $Z \\to Y$ the fibre product\n$Z \\times_Y X$ is quasi-compact.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HE","source_file":"spaces-morphisms.tex","source_line":1035,"source_end_line":1042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1035-L1042","statement_sha256":"49b4b349e4e109c2ea4a639789300caf217c3aedb8248c4527d6bbda3700448f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10904,"rank":10904,"depth":0,"x":304.841,"y":1638.934,"cluster":"algebraic-spaces"},{"id":"stacks:0EMK","tag":"0EMK","title":"Quasi-compact morphisms · Lemma 0EMK","summary":"Let S be a scheme. If f : X → Y is a quasi-compact morphism of algebraic spaces over S, then the underlying map |f| : |X| → |Y| of topological space is quasi-compact.","statement_latex":"Let $S$ be a scheme. If $f : X \\to Y$ is a quasi-compact morphism of\nalgebraic spaces over $S$, then the underlying map\n$|f| : |X| \\to |Y|$ of topological space is quasi-compact.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMK","source_file":"spaces-morphisms.tex","source_line":1049,"source_end_line":1054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1049-L1054","statement_sha256":"bf94c61aae71d1f66462f1ec211c51cd70d609d2561a03f1727c1fd575667d90","origin":"The Stacks Project","memory_eligible":false,"source_rank":10905,"rank":10905,"depth":8,"x":143.317,"y":1613.787,"cluster":"algebraic-spaces"},{"id":"stacks:03HF","tag":"03HF","title":"Quasi-compact morphisms · Lemma 03HF","summary":"The base change of a quasi-compact morphism of algebraic spaces by any morphism of algebraic spaces is quasi-compact.","statement_latex":"The base change of a quasi-compact morphism of algebraic spaces\nby any morphism of algebraic spaces is quasi-compact.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HF","source_file":"spaces-morphisms.tex","source_line":1073,"source_end_line":1077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1073-L1077","statement_sha256":"fce1dc66099bfc284f952c1f54e274156d4399482fa29f88ad56428b378ad7a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":10906,"rank":10906,"depth":0,"x":282.945,"y":1540.519,"cluster":"algebraic-spaces"},{"id":"stacks:03HG","tag":"03HG","title":"Quasi-compact morphisms · Lemma 03HG","summary":"The composition of a pair of quasi-compact morphisms of algebraic spaces is quasi-compact.","statement_latex":"The composition of a pair of quasi-compact morphisms of algebraic spaces\nis quasi-compact.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HG","source_file":"spaces-morphisms.tex","source_line":1083,"source_end_line":1087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1083-L1087","statement_sha256":"43e25df74a14e8c744ee16d6cb382092df95a8df919406fb2d921eca654c77f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10907,"rank":10907,"depth":0,"x":238.81,"y":1674.061,"cluster":"algebraic-spaces"},{"id":"stacks:040W","tag":"040W","title":"Quasi-compact morphisms · Lemma 040W","summary":"The image of a quasi-compact algebraic space under a surjective morphism is quasi-compact. Let S be a scheme. • If X → Y is a surjective morphism of algebraic spaces over S, and X is quasi-compact then Y is quasi-compact. • If xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & Z is a commutative diagram of morphisms of algebraic spaces over S and f is surjective and p is quasi-compact, then q is quasi-compact.","statement_latex":"\\begin{slogan}\nThe image of a quasi-compact algebraic space under a surjective morphism\nis quasi-compact.\n\\end{slogan}\nLet $S$ be a scheme.\n\\begin{enumerate}\n\\item If $X \\to Y$ is a surjective morphism of algebraic spaces over $S$,\nand $X$ is quasi-compact then $Y$ is quasi-compact.\n\\item If\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& Z\n}\n$$\nis a commutative diagram of morphisms of algebraic spaces over $S$\nand $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040W","source_file":"spaces-morphisms.tex","source_line":1093,"source_end_line":1114,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1093-L1114","statement_sha256":"acaabdc19877370d0369a96831b0409738148f14bf6da0300abbbcc27338c1ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":10908,"rank":10908,"depth":4,"x":163.804,"y":1550.282,"cluster":"algebraic-spaces"},{"id":"stacks:04ZJ","tag":"04ZJ","title":"Quasi-compact morphisms · Lemma 04ZJ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let g : Y' → Y be a universally open and surjective morphism of algebraic spaces such that the base change f' : X' → Y' is quasi-compact. Then f is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $g : Y' \\to Y$ be a universally open and surjective morphism of\nalgebraic spaces such that the base change $f' : X' \\to Y'$ is quasi-compact.\nThen $f$ is quasi-compact.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZJ","source_file":"spaces-morphisms.tex","source_line":1133,"source_end_line":1140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1133-L1140","statement_sha256":"926d66d943116f54c8e8b1c3218713eb321b1dda8be9ef14cb11dcf3ad1e9bbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10909,"rank":10909,"depth":5,"x":318.983,"y":1599.099,"cluster":"algebraic-spaces"},{"id":"stacks:03KG","tag":"03KG","title":"Quasi-compact morphisms · Lemma 03KG","summary":"Quasi-compact morphisms of algebraic spaces are preserved under pullback and local on the target. Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is quasi-compact, • for every scheme Z and any morphism Z → Y the morphism of algebraic spaces Z ×_Y X → Z is quasi-compact, • for every affine scheme Z and any morphism Z → Y the algebraic space Z ×_Y X is quasi-compact, • there exists a scheme V and a surjective…","statement_latex":"\\begin{slogan}\nQuasi-compact morphisms of algebraic spaces are preserved under pullback\nand local on the target.\n\\end{slogan}\nLet $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is quasi-compact,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism of\nalgebraic spaces $Z \\times_Y X \\to Z$ is quasi-compact,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the algebraic space $Z \\times_Y X$ is quasi-compact,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is a quasi-compact morphism\nof algebraic spaces, and\n\\item there exists a surjective \\'etale morphism\n$Y' \\to Y$ of algebraic spaces such that $Y' \\times_Y X \\to Y'$\nis a quasi-compact morphism of algebraic spaces, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that\neach of the morphisms $f^{-1}(Y_i) \\to Y_i$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KG","source_file":"spaces-morphisms.tex","source_line":1160,"source_end_line":1184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1160-L1184","statement_sha256":"11ea5ae8a1b139d85d80ae8fd9440a545c94dff2831b9529e1cc3916875ce545","origin":"The Stacks Project","memory_eligible":false,"source_rank":10910,"rank":10910,"depth":42,"x":164.973,"y":1651.264,"cluster":"algebraic-spaces"},{"id":"stacks:03KS","tag":"03KS","title":"Quasi-compact morphisms · Lemma 03KS","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of algebraic spaces over S. If g ∘ f is quasi-compact and g is quasi-separated then f is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.\nIf $g \\circ f$ is quasi-compact and $g$ is quasi-separated\nthen $f$ is quasi-compact.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KS","source_file":"spaces-morphisms.tex","source_line":1235,"source_end_line":1241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1235-L1241","statement_sha256":"82280166f5610c74e48adf234249be340423750ceaf1860871b8b578a8845c46","origin":"The Stacks Project","memory_eligible":false,"source_rank":10911,"rank":10911,"depth":56,"x":236.739,"y":1525.143,"cluster":"algebraic-spaces"},{"id":"stacks:073B","tag":"073B","title":"Quasi-compact morphisms · Lemma 073B","summary":"Let f : X → Y be a morphism of algebraic spaces over a scheme S. • If X is quasi-compact and Y is quasi-separated, then f is quasi-compact. • If X is quasi-compact and quasi-separated and Y is quasi-separated, then f is quasi-compact and quasi-separated. • A fibre product of quasi-compact and quasi-separated algebraic spaces is quasi-compact and quasi-separated.","statement_latex":"Let $f : X \\to Y$ be a morphism of algebraic spaces\nover a scheme $S$.\n\\begin{enumerate}\n\\item If $X$ is quasi-compact and $Y$ is quasi-separated, then $f$ is\nquasi-compact.\n\\item If $X$ is quasi-compact and quasi-separated and $Y$ is quasi-separated,\nthen $f$ is quasi-compact and quasi-separated.\n\\item A fibre product of quasi-compact and quasi-separated algebraic spaces\nis quasi-compact and quasi-separated.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/073B","source_file":"spaces-morphisms.tex","source_line":1256,"source_end_line":1268,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1256-L1268","statement_sha256":"147c0ea1c42374ae4472d3e768d1b997b8345b00092a6ff030ba85119ab2eaf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10912,"rank":10912,"depth":57,"x":285.344,"y":1659.146,"cluster":"algebraic-spaces"},{"id":"stacks:03XD","tag":"03XD","title":"Universally closed morphisms · Lemma 03XD","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. The following are equivalent • f is universally closed (in the sense of Section [Tag 03HA]), and • for every morphism of algebraic spaces Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is closed.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism of\nalgebraic spaces over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed\n(in the sense of Section \\ref{section-representable}), and\n\\item for every morphism of algebraic spaces $Z \\to Y$ the morphism of\ntopological spaces $|Z \\times_Y X| \\to |Z|$ is closed.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XD","source_file":"spaces-morphisms.tex","source_line":1299,"source_end_line":1309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1299-L1309","statement_sha256":"21fac05f97b2c6ae9adc99124c7ee70962417b3997ea95385274e68d64d0177d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10913,"rank":10913,"depth":0,"x":141.442,"y":1587.765,"cluster":"algebraic-spaces"},{"id":"stacks:03HI","tag":"03HI","title":"Universally closed morphisms · Definition 03HI","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is closed if the map of topological spaces |X| → |Y| is closed. • We say f is universally closed if for every morphism of algebraic spaces Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is closed, i.e., the base change Z ×_Y X → Z is closed.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it closed} if the map of topological\nspaces $|X| \\to |Y|$ is closed.\n\\item We say $f$ is {\\it universally closed} if for every morphism\nof algebraic spaces $Z \\to Y$ the morphism of topological spaces\n$$\n|Z \\times_Y X| \\to |Z|\n$$\nis closed, i.e., the base change $Z \\times_Y X \\to Z$ is closed.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally closed morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03HI","source_file":"spaces-morphisms.tex","source_line":1336,"source_end_line":1350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1336-L1350","statement_sha256":"4da6245883fb10bdd5ec6c36c5b63520cf3251003be74a5f25b66dfacb94386b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10914,"rank":10914,"depth":0,"x":305.297,"y":1558.686,"cluster":"algebraic-spaces"},{"id":"stacks:03IS","tag":"03IS","title":"Universally closed morphisms · Lemma 03IS","summary":"The base change of a universally closed morphism of algebraic spaces by any morphism of algebraic spaces is universally closed.","statement_latex":"The base change of a universally closed morphism of algebraic spaces\nby any morphism of algebraic spaces is universally closed.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IS","source_file":"spaces-morphisms.tex","source_line":1352,"source_end_line":1356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1352-L1356","statement_sha256":"9f5f9ee739825e464ad9871464e7fa2e8eee438f43b2c983f9d19f0861e0d3dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10915,"rank":10915,"depth":0,"x":207.654,"y":1673.341,"cluster":"algebraic-spaces"},{"id":"stacks:03IU","tag":"03IU","title":"Universally closed morphisms · Lemma 03IU","summary":"The composition of a pair of (universally) closed morphisms of algebraic spaces is (universally) closed.","statement_latex":"The composition of a pair of (universally) closed morphisms of algebraic spaces\nis (universally) closed.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IU","source_file":"spaces-morphisms.tex","source_line":1362,"source_end_line":1366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1362-L1366","statement_sha256":"67b9acb6e63159118049b3a6ef73248f80a2954fc6de09935128d127b71b7731","origin":"The Stacks Project","memory_eligible":false,"source_rank":10916,"rank":10916,"depth":0,"x":187.41,"y":1533.102,"cluster":"algebraic-spaces"},{"id":"stacks:03IT","tag":"03IT","title":"Universally closed morphisms · Lemma 03IT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f is universally closed, • for every scheme Z and every morphism Z → Y the projection |Z ×_Y X| → |Z| is closed, • for every affine scheme Z and every morphism Z → Y the projection |Z ×_Y X| → |Z| is closed, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is a universally closed morphism of algebraic spaces, and • there exists a…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed,\n\\item for every scheme $Z$ and every morphism $Z \\to Y$\nthe projection $|Z \\times_Y X| \\to |Z|$ is closed,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$\nthe projection $|Z \\times_Y X| \\to |Z|$ is closed,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is a universally closed morphism\nof algebraic spaces, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that\neach of the morphisms $f^{-1}(Y_i) \\to Y_i$ is universally closed.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IT","source_file":"spaces-morphisms.tex","source_line":1372,"source_end_line":1388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1372-L1388","statement_sha256":"0d58d362c2fdffb2a12db2e8f14969d8a65d174a4e82015f22cc54d47aeb00a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10917,"rank":10917,"depth":46,"x":315.378,"y":1625.215,"cluster":"algebraic-spaces"},{"id":"stacks:04XW","tag":"04XW","title":"Universally closed morphisms · Lemma 04XW","summary":"Let S be a scheme. A universally closed morphism of algebraic spaces over S is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nA universally closed morphism of algebraic spaces over $S$ is quasi-compact.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XW","source_file":"spaces-morphisms.tex","source_line":1467,"source_end_line":1471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1467-L1471","statement_sha256":"0fb1c016c3bcec39c420c43c2843be00d08f1fcb3fdf0e28dcb77ef77404747c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10918,"rank":10918,"depth":57,"x":146.596,"y":1629.913,"cluster":"algebraic-spaces"},{"id":"stacks:05Z2","tag":"05Z2","title":"Universally closed morphisms · Lemma 05Z2","summary":"Let S be a scheme. Let B be an algebraic space over S. Let f : X → Y be a surjective universally closed morphism of algebraic spaces over B. • If X is quasi-separated, then Y is quasi-separated. • If X is separated, then Y is separated. • If X is quasi-separated over B, then Y is quasi-separated over B. • If X is separated over B, then Y is separated over B.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $f : X \\to Y$ be a surjective universally closed\nmorphism of algebraic spaces over $B$.\n\\begin{enumerate}\n\\item If $X$ is quasi-separated, then $Y$ is quasi-separated.\n\\item If $X$ is separated, then $Y$ is separated.\n\\item If $X$ is quasi-separated over $B$, then $Y$ is quasi-separated over $B$.\n\\item If $X$ is separated over $B$, then $Y$ is separated over $B$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Z2","source_file":"spaces-morphisms.tex","source_line":1560,"source_end_line":1571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1560-L1571","statement_sha256":"d74b3f7008a96e2a6abf77bd243dde256ce7640dc394ea5f090404b2a79441c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10919,"rank":10919,"depth":58,"x":267.522,"y":1530.476,"cluster":"algebraic-spaces"},{"id":"stacks:042L","tag":"042L","title":"Monomorphisms · Definition 042L","summary":"Let S be a scheme. A morphism of algebraic spaces over S is called a monomorphism if it is an injective map of sheaves, i.e., a monomorphism in the category of sheaves on (Sch/S)_fppf.","statement_latex":"Let $S$ be a scheme.\nA morphism of algebraic spaces over $S$ is called a {\\it monomorphism}\nif it is an injective map of sheaves, i.e., a monomorphism in the category\nof sheaves on $(\\Sch/S)_{fppf}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Monomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042L","source_file":"spaces-morphisms.tex","source_line":1649,"source_end_line":1655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1649-L1655","statement_sha256":"c70d704d424d1f759d80f1e43cabb4dbe13cd507aa360bea983ace96cf1fb03e","origin":"The Stacks Project","memory_eligible":false,"source_rank":10920,"rank":10920,"depth":0,"x":258.299,"y":1672.704,"cluster":"algebraic-spaces"},{"id":"stacks:042M","tag":"042M","title":"Monomorphisms · Lemma 042M","summary":"Let S be a scheme. Let j : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • j is a monomorphism (as in Definition [Tag 042L]), • j is a monomorphism in the category of algebraic spaces over S, and • the diagonal morphism Δ_X/Y : X → X ×_Y X is an isomorphism.","statement_latex":"Let $S$ be a scheme.\nLet $j : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $j$ is a monomorphism (as in Definition \\ref{definition-monomorphism}),\n\\item $j$ is a monomorphism in the category of algebraic spaces over $S$, and\n\\item the diagonal morphism $\\Delta_{X/Y} : X \\to X \\times_Y X$ is\nan isomorphism.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042M","source_file":"spaces-morphisms.tex","source_line":1661,"source_end_line":1672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1661-L1672","statement_sha256":"5b602706372fea0c5840319058e80cd012cef139f3c006485254c5587a355a8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10921,"rank":10921,"depth":8,"x":150.506,"y":1562.371,"cluster":"algebraic-spaces"},{"id":"stacks:042N","tag":"042N","title":"Monomorphisms · Lemma 042N","summary":"A monomorphism of algebraic spaces is separated.","statement_latex":"A monomorphism of algebraic spaces is separated.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042N","source_file":"spaces-morphisms.tex","source_line":1685,"source_end_line":1688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1685-L1688","statement_sha256":"ac958cdd1c4edd063a0db3f09d60aa7a526a8a9e9c548e2b2d59c425ff661c31","origin":"The Stacks Project","memory_eligible":false,"source_rank":10922,"rank":10922,"depth":9,"x":319.057,"y":1582.605,"cluster":"algebraic-spaces"},{"id":"stacks:042O","tag":"042O","title":"Monomorphisms · Lemma 042O","summary":"A composition of monomorphisms is a monomorphism.","statement_latex":"A composition of monomorphisms is a monomorphism.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042O","source_file":"spaces-morphisms.tex","source_line":1695,"source_end_line":1698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1695-L1698","statement_sha256":"fc2d9b90d40afb899acbc38095627bdee0738275df033c611a4b80d3cc9ce849","origin":"The Stacks Project","memory_eligible":false,"source_rank":10923,"rank":10923,"depth":0,"x":178.215,"y":1663.487,"cluster":"algebraic-spaces"},{"id":"stacks:042P","tag":"042P","title":"Monomorphisms · Lemma 042P","summary":"The base change of a monomorphism is a monomorphism.","statement_latex":"The base change of a monomorphism is a monomorphism.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042P","source_file":"spaces-morphisms.tex","source_line":1704,"source_end_line":1707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1704-L1707","statement_sha256":"6afcf6b219373aa1dd57f79208e52771db943d295642b8451bf6e8a7ab3178c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10924,"rank":10924,"depth":0,"x":217.105,"y":1523.65,"cluster":"algebraic-spaces"},{"id":"stacks:042Q","tag":"042Q","title":"Monomorphisms · Lemma 042Q","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f is a monomorphism, • for every scheme Z and morphism Z → Y the base change Z ×_Y X → Z of f is a monomorphism, • for every affine scheme Z and every morphism Z → Y the base change Z ×_Y X → Z of f is a monomorphism, • there exists a scheme V and a surjective étale morphism V → Y such that the base change V ×_Y X → V is a monomorphism, and • there exists a Zariski…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is a monomorphism,\n\\item for every scheme $Z$ and morphism $Z \\to Y$ the\nbase change $Z \\times_Y X \\to Z$ of $f$ is a monomorphism,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the\nbase change $Z \\times_Y X \\to Z$ of $f$ is a monomorphism,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that the base change $V \\times_Y X \\to V$ is a\nmonomorphism, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that each\nof the morphisms $f^{-1}(Y_i) \\to Y_i$ is a monomorphism.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042Q","source_file":"spaces-morphisms.tex","source_line":1713,"source_end_line":1730,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1713-L1730","statement_sha256":"ae9abedaa0c99a0386983d73173f562351cec2ad7f645685434608ec059cf469","origin":"The Stacks Project","memory_eligible":false,"source_rank":10925,"rank":10925,"depth":41,"x":301.048,"y":1649.08,"cluster":"algebraic-spaces"},{"id":"stacks:042R","tag":"042R","title":"Monomorphisms · Lemma 042R","summary":"An immersion of algebraic spaces is a monomorphism. In particular, any immersion is separated.","statement_latex":"An immersion of algebraic spaces is a monomorphism.\nIn particular, any immersion is separated.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042R","source_file":"spaces-morphisms.tex","source_line":1761,"source_end_line":1765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1761-L1765","statement_sha256":"f93e394ee4f8075a96442cf0ddc9a8ba8954a8eb2b05e12e53a1abe36f03b988","origin":"The Stacks Project","memory_eligible":false,"source_rank":10926,"rank":10926,"depth":3,"x":137.96,"y":1604.132,"cluster":"algebraic-spaces"},{"id":"stacks:06MG","tag":"06MG","title":"Monomorphisms · Lemma 06MG","summary":"Let S be a scheme. Let k be a field and let Z → Spec(k) be a monomorphism of algebraic spaces over S. Then either Z = ∅ or Z = Spec(k).","statement_latex":"Let $S$ be a scheme. Let $k$ be a field and let $Z \\to \\Spec(k)$\nbe a monomorphism of algebraic spaces over $S$. Then either\n$Z = \\emptyset$ or $Z = \\Spec(k)$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MG","source_file":"spaces-morphisms.tex","source_line":1781,"source_end_line":1786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1781-L1786","statement_sha256":"bbb3f621a9f555492d6cdc3c661ecfbacc123460f8f5b69503b9a59bd543a878","origin":"The Stacks Project","memory_eligible":false,"source_rank":10927,"rank":10927,"depth":56,"x":294.674,"y":1544.618,"cluster":"algebraic-spaces"},{"id":"stacks:06RV","tag":"06RV","title":"Monomorphisms · Lemma 06RV","summary":"Let S be a scheme. If X → Y is a monomorphism of algebraic spaces over S, then |X| → |Y| is injective.","statement_latex":"Let $S$ be a scheme. If $X \\to Y$ is a monomorphism of algebraic spaces\nover $S$, then $|X| \\to |Y|$ is injective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RV","source_file":"spaces-morphisms.tex","source_line":1805,"source_end_line":1809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1805-L1809","statement_sha256":"aba646dc1e4b74d2bcca0245f9a2e4cb31c7953a5625d91021eb7ab6a86635d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10928,"rank":10928,"depth":0,"x":226.841,"y":1677.687,"cluster":"algebraic-spaces"},{"id":"stacks:03M8","tag":"03M8","title":"Pushforward of quasi-coherent sheaves · Lemma 03M8","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let U → X be a surjective étale morphism from a scheme to X. Set R = U ×_X U and denote t, s : R → U the projection morphisms as usual. Denote a : U → Y and b : R → Y the induced morphisms. For any object F of Mod(O_X) there exists an exact sequence 0 → f_*F → a_*(F|_U) → b_*(F|_R) where the second arrow is the difference t^* - s^*.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $U \\to X$ be a surjective \\'etale morphism from a scheme to $X$.\nSet $R = U \\times_X U$ and denote $t, s : R \\to U$ the projection\nmorphisms as usual. Denote $a : U \\to Y$ and $b : R \\to Y$ the induced\nmorphisms. For any object $\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_X)$\nthere exists an exact sequence\n$$\n0 \\to f_*\\mathcal{F} \\to a_*(\\mathcal{F}|_U) \\to b_*(\\mathcal{F}|_R)\n$$\nwhere the second arrow is the difference $t^* - s^*$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Pushforward of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03M8","source_file":"spaces-morphisms.tex","source_line":1837,"source_end_line":1850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1837-L1850","statement_sha256":"463e66e380084a65a6ad5e882fe40d896b346bec291dc305c8661c8a90987999","origin":"The Stacks Project","memory_eligible":false,"source_rank":10929,"rank":10929,"depth":0,"x":169.738,"y":1540.806,"cluster":"algebraic-spaces"},{"id":"stacks:03M9","tag":"03M9","title":"Pushforward of quasi-coherent sheaves · Lemma 03M9","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is quasi-compact and quasi-separated, then f_* transforms quasi-coherent O_X-modules into quasi-coherent O_Y-modules.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is quasi-compact and quasi-separated, then $f_*$ transforms\nquasi-coherent $\\mathcal{O}_X$-modules into\nquasi-coherent $\\mathcal{O}_Y$-modules.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Pushforward of quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03M9","source_file":"spaces-morphisms.tex","source_line":1906,"source_end_line":1913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1906-L1913","statement_sha256":"16ce318bba51badfc75765fbf645c0efc58b75363d3e524cc8985c5ea0391c30","origin":"The Stacks Project","memory_eligible":false,"source_rank":10930,"rank":10930,"depth":42,"x":322.218,"y":1609.473,"cluster":"algebraic-spaces"},{"id":"stacks:03M4","tag":"03M4","title":"Immersions · Lemma 03M4","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is a closed immersion (resp. open immersion, resp. immersion), • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is a closed immersion (resp. open immersion, resp. immersion), • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is a closed immersion (resp. open immersion, resp. immersion), • there exists a scheme V and a…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is a closed immersion (resp.\\ open immersion, resp.\\ immersion),\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is a closed immersion (resp.\\ open immersion,\nresp.\\ immersion),\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is a closed immersion\n(resp.\\ open immersion, resp.\\ immersion),\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is a closed immersion\n(resp.\\ open immersion, resp.\\ immersion), and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that\neach of the morphisms $f^{-1}(Y_i) \\to Y_i$ is a closed immersion\n(resp.\\ open immersion, resp.\\ immersion).\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03M4","source_file":"spaces-morphisms.tex","source_line":1989,"source_end_line":2008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L1989-L2008","statement_sha256":"66488786bc38af1df3ffa4c6120c0bd3b2f82848d88d8b2d73a0a903d0ea1e5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10931,"rank":10931,"depth":50,"x":154.235,"y":1645.429,"cluster":"algebraic-spaces"},{"id":"stacks:0AGC","tag":"0AGC","title":"Immersions · Lemma 0AGC","summary":"Let S be a scheme. Let Z → Y → X be morphisms of algebraic spaces over S. • If Z → X is representable, locally of finite type, locally quasi-finite, separated, and a monomorphism, then Z → Y is representable, locally of finite type, locally quasi-finite, separated, and a monomorphism. • If Z → X is an immersion and Y → X is locally separated, then Z → Y is an immersion. • If Z → X is a closed immersion and Y → X is separated, then Z → Y is a closed immersion.","statement_latex":"Let $S$ be a scheme.\nLet $Z \\to Y \\to X$ be morphisms of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $Z \\to X$ is representable, locally of finite type, locally\nquasi-finite, separated, and a monomorphism, then $Z \\to Y$ is\nrepresentable, locally of finite type, locally quasi-finite,\nseparated, and a monomorphism.\n\\item If $Z \\to X$ is an immersion and $Y \\to X$ is locally separated,\nthen $Z \\to Y$ is an immersion.\n\\item If $Z \\to X$ is a closed immersion and $Y \\to X$ is separated,\nthen $Z \\to Y$ is a closed immersion.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGC","source_file":"spaces-morphisms.tex","source_line":2044,"source_end_line":2058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2044-L2058","statement_sha256":"c292a91a7d94654a1520ea74b130bc6227bcaac64618686247a0551b7d30bc3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10932,"rank":10932,"depth":56,"x":249.372,"y":1523.362,"cluster":"algebraic-spaces"},{"id":"stacks:04CD","tag":"04CD","title":"Immersions · Lemma 04CD","summary":"Let S be a scheme. Let i : Z → X be an immersion of algebraic spaces over S. Then |i| : |Z| → |X| is a homeomorphism onto a locally closed subset, and i is a closed immersion if and only if the image |i|(|Z|) ⊂ |X| is a closed subset.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic\nspaces over $S$. Then $|i| : |Z| \\to |X|$ is a homeomorphism onto a\nlocally closed subset, and $i$ is a closed immersion if and only if\nthe image $|i|(|Z|) \\subset |X|$ is a closed subset.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CD","source_file":"spaces-morphisms.tex","source_line":2085,"source_end_line":2091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2085-L2091","statement_sha256":"5b7afe01b77544d9c627bbe9a82c6b33eec5761712bbe4dcce50515328c56cb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10933,"rank":10933,"depth":51,"x":277.436,"y":1667.633,"cluster":"algebraic-spaces"},{"id":"stacks:081U","tag":"081U","title":"Immersions · Lemma 081U","summary":"Let S be a scheme. Let Z → X be an immersion of algebraic spaces over S. Assume Z → X is quasi-compact. There exists a factorization Z → overlineZ → X where Z → overlineZ is an open immersion and overlineZ → X is a closed immersion.","statement_latex":"Let $S$ be a scheme. Let $Z \\to X$ be an immersion of algebraic spaces over\n$S$. Assume $Z \\to X$ is quasi-compact.\nThere exists a factorization $Z \\to \\overline{Z} \\to X$ where\n$Z \\to \\overline{Z}$ is an open immersion and $\\overline{Z} \\to X$\nis a closed immersion.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081U","source_file":"spaces-morphisms.tex","source_line":2157,"source_end_line":2164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2157-L2164","statement_sha256":"005c36c9c6e927261843d221e760620aabbc9d574e16d40ecb9cffc388e83240","origin":"The Stacks Project","memory_eligible":false,"source_rank":10934,"rank":10934,"depth":53,"x":140.461,"y":1577.001,"cluster":"algebraic-spaces"},{"id":"stacks:03MB","tag":"03MB","title":"Closed immersions · Lemma 03MB","summary":"Let S be a scheme. Let X be an algebraic space over S. For every closed immersion i : Z → X the sheaf i_*O_Z is a quasi-coherent O_X-module, the map i^sharp : O_X → i_*O_Z is surjective and its kernel is a quasi-coherent sheaf of ideals. The rule Z ↦ Ker(O_X → i_*O_Z) defines an inclusion reversing bijection closed subspaces Z ⊂ X → quasi-coherent sheaves of ideals I ⊂ O_X Moreover, given a closed subscheme Z corresponding to the quasi-coherent sheaf of ideals I ⊂ O_X a…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nFor every closed immersion $i : Z \\to X$ the sheaf\n$i_*\\mathcal{O}_Z$ is a quasi-coherent $\\mathcal{O}_X$-module, the map\n$i^\\sharp : \\mathcal{O}_X \\to i_*\\mathcal{O}_Z$ is surjective and its\nkernel is a quasi-coherent sheaf of ideals. The rule\n$Z \\mapsto \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$\ndefines an inclusion reversing bijection\n$$\n\\begin{matrix}\n\\text{closed subspaces}\\\\\nZ \\subset X\n\\end{matrix}\n\\longrightarrow\n\\begin{matrix}\n\\text{quasi-coherent sheaves}\\\\\n\\text{of ideals }\\mathcal{I} \\subset \\mathcal{O}_X\n\\end{matrix}\n$$\nMoreover, given a closed subscheme $Z$ corresponding to the quasi-coherent\nsheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_X$ a morphism of algebraic\nspaces $h : Y \\to X$ factors through $Z$ if and only if the map\n$h^*\\mathcal{I} \\to h^*\\mathcal{O}_X = \\mathcal{O}_Y$ is zero.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MB","source_file":"spaces-morphisms.tex","source_line":2215,"source_end_line":2240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2215-L2240","statement_sha256":"f99231eef6e63f2a633aef035b42a78afc87944d1662a45b89e8bb56b215b4a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":10935,"rank":10935,"depth":51,"x":314.682,"y":1566.088,"cluster":"algebraic-spaces"},{"id":"stacks:083Q","tag":"083Q","title":"Closed immersions · Definition 083Q","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Let Z ⊂ X be a closed subspace. The inverse image f^-1(Z) of the closed subspace Z is the closed subspace Z ×_X Y of Y.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces\nover $S$. Let $Z \\subset X$ be a closed subspace. The\n{\\it inverse image $f^{-1}(Z)$ of the closed subspace $Z$}\nis the closed subspace $Z \\times_X Y$ of $Y$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083Q","source_file":"spaces-morphisms.tex","source_line":2293,"source_end_line":2299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2293-L2299","statement_sha256":"c9134ab99372ca86ef386ddc7ded74a5f6e742a70613c836918ccaf71351d325","origin":"The Stacks Project","memory_eligible":false,"source_rank":10936,"rank":10936,"depth":0,"x":194.761,"y":1673.196,"cluster":"algebraic-spaces"},{"id":"stacks:04CG","tag":"04CG","title":"Closed immersions · Lemma 04CG","summary":"A closed immersion of algebraic spaces is quasi-compact.","statement_latex":"A closed immersion of algebraic spaces is quasi-compact.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CG","source_file":"spaces-morphisms.tex","source_line":2308,"source_end_line":2311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2308-L2311","statement_sha256":"592d9f2eb4deb733944a32500f2cfe1b184d460f15658665d0566b0dce9932ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":10937,"rank":10937,"depth":3,"x":197.06,"y":1525.891,"cluster":"algebraic-spaces"},{"id":"stacks:04CH","tag":"04CH","title":"Closed immersions · Lemma 04CH","summary":"A closed immersion of algebraic spaces is separated.","statement_latex":"A closed immersion of algebraic spaces is separated.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CH","source_file":"spaces-morphisms.tex","source_line":2321,"source_end_line":2324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2321-L2324","statement_sha256":"b392e6ad59be7eeab37478c14149e724ef715ad029e1b9f78328c38e26445f7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10938,"rank":10938,"depth":3,"x":314.044,"y":1636.024,"cluster":"algebraic-spaces"},{"id":"stacks:04E5","tag":"04E5","title":"Closed immersions · Lemma 04E5","summary":"Let S be a scheme. Let i : Z → X be a closed immersion of algebraic spaces over S. • The functor i_small, * : Sh(Z_etale) → Sh(X_etale) is fully faithful and its essential image is those sheaves of sets F on X_etale whose restriction to X setminus Z is isomorphic to *, and • the functor i_small, * : Ab(Z_etale) → Ab(X_etale) is fully faithful and its essential image is those abelian sheaves on X_etale whose support is contained in |Z|. In both cases i_small^-1 is a left…","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic\nspaces over $S$.\n\\begin{enumerate}\n\\item The functor\n$$\ni_{small, *} :\n\\Sh(Z_\\etale)\n\\longrightarrow\n\\Sh(X_\\etale)\n$$\nis fully faithful and its essential image is those sheaves of sets\n$\\mathcal{F}$ on $X_\\etale$ whose restriction to $X \\setminus Z$ is\nisomorphic to $*$, and\n\\item the functor\n$$\ni_{small, *} :\n\\textit{Ab}(Z_\\etale)\n\\longrightarrow\n\\textit{Ab}(X_\\etale)\n$$\nis fully faithful and its essential image is those abelian sheaves on\n$X_\\etale$ whose support is contained in $|Z|$.\n\\end{enumerate}\nIn both cases $i_{small}^{-1}$ is a left inverse to the functor\n$i_{small, *}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04E5","source_file":"spaces-morphisms.tex","source_line":2334,"source_end_line":2361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2334-L2361","statement_sha256":"74413fa1a803a2fcc3371426b423510bf82941cc8ef1446c9977f3218f345066","origin":"The Stacks Project","memory_eligible":false,"source_rank":10939,"rank":10939,"depth":53,"x":138.886,"y":1621.165,"cluster":"algebraic-spaces"},{"id":"stacks:0DK1","tag":"0DK1","title":"Closed immersions · Lemma 0DK1","summary":"Let S be a scheme. Let i : Z → X be a closed immersion of algebraic spaces over S. Let overlinez be a geometric point of Z with image overlinex in X. Then (i_small, *F)_overlinez = F_overlinex for any sheaf F on Z_etale.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic\nspaces over $S$. Let $\\overline{z}$ be a geometric point of $Z$ with\nimage $\\overline{x}$ in $X$. Then\n$(i_{small, *}\\mathcal{F})_{\\overline{z}} = \\mathcal{F}_{\\overline{x}}$\nfor any sheaf $\\mathcal{F}$ on $Z_\\etale$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DK1","source_file":"spaces-morphisms.tex","source_line":2398,"source_end_line":2405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2398-L2405","statement_sha256":"47ef6c031c4162c03aae3da7c4468718b8d2c62b293b3775b7b68c84cb5ddf23","origin":"The Stacks Project","memory_eligible":false,"source_rank":10940,"rank":10940,"depth":19,"x":280.26,"y":1532.567,"cluster":"algebraic-spaces"},{"id":"stacks:04G0","tag":"04G0","title":"Closed immersions · Lemma 04G0","summary":"Let S be a scheme. Let i : Z → X be a closed immersion of algebraic spaces over S. Let A be a sheaf of rings on X_etale. Let B be a sheaf of rings on Z_etale. Let φ : A → i_small, *B be a homomorphism of sheaves of rings so that we obtain a morphism of ringed topoi f : (Sh(Z_etale), B) → (Sh(X_etale), A). For a sheaf of A-modules F and a sheaf of B-modules G the canonical map F ⊗_A f_*G → f_*(f^*F ⊗_B G). is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic\nspaces over $S$. Let $\\mathcal{A}$ be a sheaf of rings on $X_\\etale$.\nLet $\\mathcal{B}$ be a sheaf of rings on $Z_\\etale$.\nLet $\\varphi : \\mathcal{A} \\to i_{small, *}\\mathcal{B}$\nbe a homomorphism of sheaves of rings so that we obtain a\nmorphism of ringed topoi\n$$\nf : (\\Sh(Z_\\etale), \\mathcal{B}) \\longrightarrow (\\Sh(X_\\etale), \\mathcal{A}).\n$$\nFor a sheaf of $\\mathcal{A}$-modules $\\mathcal{F}$ and a\nsheaf of $\\mathcal{B}$-modules $\\mathcal{G}$ the canonical map\n$$\n\\mathcal{F} \\otimes_\\mathcal{A} f_*\\mathcal{G}\n\\longrightarrow\nf_*(f^*\\mathcal{F} \\otimes_\\mathcal{B} \\mathcal{G}).\n$$\nis an isomorphism.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04G0","source_file":"spaces-morphisms.tex","source_line":2423,"source_end_line":2442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2423-L2442","statement_sha256":"c89bffb498b4b186e3790dafde9b88f2091c2415b5f4eb5fbbf3a177983b39a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10941,"rank":10941,"depth":53,"x":247.199,"y":1678.389,"cluster":"algebraic-spaces"},{"id":"stacks:04CJ","tag":"04CJ","title":"Closed immersions and quasi-coherent sheaves · Lemma 04CJ","summary":"Let S be a scheme. Let i : Z → X be a closed immersion of algebraic spaces over S. Let I ⊂ O_X be the quasi-coherent sheaf of ideals cutting out Z. • For any O_X-module F the adjunction map F → i_*i^*F induces an isomorphism F/IF ≅ i_*i^*F. • The functor i^* is a left inverse to i_*, i.e., for any O_Z-module G the adjunction map i^*i_*G → G is an isomorphism. • The functor i_* : QCoh(O_Z) → QCoh(O_X) is exact, fully faithful, with essential image those quasi-coherent…","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of algebraic\nspaces over $S$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent\nsheaf of ideals cutting out $Z$.\n\\begin{enumerate}\n\\item For any $\\mathcal{O}_X$-module $\\mathcal{F}$ the adjunction map\n$\\mathcal{F} \\to i_*i^*\\mathcal{F}$ induces an isomorphism\n$\\mathcal{F}/\\mathcal{I}\\mathcal{F} \\cong i_*i^*\\mathcal{F}$.\n\\item The functor $i^*$ is a left inverse to $i_*$, i.e., for any\n$\\mathcal{O}_Z$-module $\\mathcal{G}$ the adjunction map\n$i^*i_*\\mathcal{G} \\to \\mathcal{G}$ is an isomorphism.\n\\item The functor\n$$\ni_* :\n\\QCoh(\\mathcal{O}_Z)\n\\longrightarrow\n\\QCoh(\\mathcal{O}_X)\n$$\nis exact, fully faithful, with essential image those quasi-coherent\n$\\mathcal{O}_X$-modules $\\mathcal{F}$ such that $\\mathcal{I}\\mathcal{F} = 0$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CJ","source_file":"spaces-morphisms.tex","source_line":2491,"source_end_line":2513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2491-L2513","statement_sha256":"f8e9ab12b6fc6a4a9c94226f8fa32a414cfdd8543ffc23da1703efaf0af57a6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10942,"rank":10942,"depth":54,"x":154.139,"y":1551.867,"cluster":"algebraic-spaces"},{"id":"stacks:04CK","tag":"04CK","title":"Closed immersions and quasi-coherent sheaves · Lemma 04CK","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Let G ⊂ F be a O_X-submodule. There exists a unique quasi-coherent O_X-submodule G' ⊂ G with the following property: For every quasi-coherent O_X-module H the map Hom_O_X(H, G') → Hom_O_X(H, G) is bijective. In particular G' is the largest quasi-coherent O_X-submodule of F contained in G.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{G} \\subset \\mathcal{F}$ be a $\\mathcal{O}_X$-submodule.\nThere exists a unique quasi-coherent $\\mathcal{O}_X$-submodule\n$\\mathcal{G}' \\subset \\mathcal{G}$ with the following property:\nFor every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{H}$ the map\n$$\n\\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G}')\n\\longrightarrow\n\\Hom_{\\mathcal{O}_X}(\\mathcal{H}, \\mathcal{G})\n$$\nis bijective. In particular $\\mathcal{G}'$ is the largest quasi-coherent\n$\\mathcal{O}_X$-submodule of $\\mathcal{F}$ contained in $\\mathcal{G}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CK","source_file":"spaces-morphisms.tex","source_line":2592,"source_end_line":2607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2592-L2607","statement_sha256":"8fb2c2789d489c7c647ae58452b51f15a60ea3f65d35608acb1fdd298f2e8040","origin":"The Stacks Project","memory_eligible":false,"source_rank":10943,"rank":10943,"depth":42,"x":324.821,"y":1592.433,"cluster":"algebraic-spaces"},{"id":"stacks:04CL","tag":"04CL","title":"Closed immersions and quasi-coherent sheaves · Lemma 04CL","summary":"Let S be a scheme. Let i : Z → X be a closed immersion of algebraic spaces over S. There is a functor i^! : QCoh(O_X) → QCoh(O_Z) which is a right adjoint to i_*. (Compare Modules, Lemma [Tag 01AZ].)","statement_latex":"Let $S$ be a scheme.\nLet $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$.\nThere is a functor\\footnote{This is likely nonstandard notation.}\n$i^! : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Z)$\nwhich is a right adjoint to $i_*$. (Compare\nModules, Lemma \\ref{modules-lemma-i-star-right-adjoint}.)","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CL","source_file":"spaces-morphisms.tex","source_line":2634,"source_end_line":2642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2634-L2642","statement_sha256":"031f783721dd6770f05d406a524930038e001b98d6c3c7c43c42400045d8d949","origin":"The Stacks Project","memory_eligible":false,"source_rank":10944,"rank":10944,"depth":43,"x":166.048,"y":1659.493,"cluster":"algebraic-spaces"},{"id":"stacks:0CYZ","tag":"0CYZ","title":"Closed immersions and quasi-coherent sheaves · Definition 0CYZ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z, Y ⊂ X be closed subspaces corresponding to quasi-coherent ideal sheaves I, J ⊂ O_X. The scheme theoretic intersection of Z and Y is the closed subspace of X cut out by I + J. Then scheme theoretic union of Z and Y is the closed subspace of X cut out by I ∩ J.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z, Y \\subset X$ be closed subspaces\ncorresponding to quasi-coherent ideal sheaves\n$\\mathcal{I}, \\mathcal{J} \\subset \\mathcal{O}_X$.\nThe {\\it scheme theoretic intersection} of $Z$ and $Y$\nis the closed subspace of $X$ cut out by $\\mathcal{I} + \\mathcal{J}$.\nThen {\\it scheme theoretic union} of $Z$ and $Y$\nis the closed subspace of $X$ cut out by\n$\\mathcal{I} \\cap \\mathcal{J}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYZ","source_file":"spaces-morphisms.tex","source_line":2668,"source_end_line":2679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2668-L2679","statement_sha256":"9546bd26561e3f5d7ea96b7abef3d07f85d06c566b16c3115b982916c3dffb16","origin":"The Stacks Project","memory_eligible":false,"source_rank":10945,"rank":10945,"depth":0,"x":229.313,"y":1519.695,"cluster":"algebraic-spaces"},{"id":"stacks:0CZ0","tag":"0CZ0","title":"Closed immersions and quasi-coherent sheaves · Lemma 0CZ0","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z, Y ⊂ X be closed subspaces. Let Z ∩ Y be the scheme theoretic intersection of Z and Y. Then Z ∩ Y → Z and Z ∩ Y → Y are closed immersions and xymatrix Z ∩ Y ar[r] ar[d] & Z ar[d] Y ar[r] & X is a cartesian diagram of algebraic spaces over S, i.e., Z ∩ Y = Z ×_X Y.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z, Y \\subset X$ be closed subspaces.\nLet $Z \\cap Y$ be the scheme theoretic intersection of $Z$ and $Y$.\nThen $Z \\cap Y \\to Z$ and $Z \\cap Y \\to Y$ are closed immersions\nand\n$$\n\\xymatrix{\nZ \\cap Y \\ar[r] \\ar[d] & Z \\ar[d] \\\\\nY \\ar[r] & X\n}\n$$\nis a cartesian diagram of algebraic spaces over $S$, i.e.,\n$Z \\cap Y = Z \\times_X Y$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZ0","source_file":"spaces-morphisms.tex","source_line":2686,"source_end_line":2701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2686-L2701","statement_sha256":"5fce93e67264109a2a40fe7f18211f376bdb4e42f23e7e0c241fde5d3a6480d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10946,"rank":10946,"depth":52,"x":295.204,"y":1658.934,"cluster":"algebraic-spaces"},{"id":"stacks:0CZ1","tag":"0CZ1","title":"Closed immersions and quasi-coherent sheaves · Lemma 0CZ1","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Y, Z ⊂ X be closed subspaces. Let Y ∪ Z be the scheme theoretic union of Y and Z. Let Y ∩ Z be the scheme theoretic intersection of Y and Z. Then Y → Y ∪ Z and Z → Y ∪ Z are closed immersions, there is a short exact sequence 0 → O_Y ∪ Z → O_Y × O_Z → O_Y ∩ Z → 0 of O_Z-modules, and the diagram xymatrix Y ∩ Z ar[r] ar[d] & Y ar[d] Z ar[r] & Y ∪ Z is cocartesian in the category of algebraic spaces over S, i.e., Y ∪…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Y, Z \\subset X$ be closed subspaces.\nLet $Y \\cup Z$ be the scheme theoretic union of $Y$ and $Z$.\nLet $Y \\cap Z$ be the scheme theoretic intersection of $Y$ and $Z$.\nThen $Y \\to Y \\cup Z$ and $Z \\to Y \\cup Z$ are closed immersions,\nthere is a short exact sequence\n$$\n0 \\to \\mathcal{O}_{Y \\cup Z} \\to \\mathcal{O}_Y \\times \\mathcal{O}_Z\n\\to \\mathcal{O}_{Y \\cap Z} \\to 0\n$$\nof $\\mathcal{O}_Z$-modules, and the diagram\n$$\n\\xymatrix{\nY \\cap Z \\ar[r] \\ar[d] & Y \\ar[d] \\\\\nZ \\ar[r] & Y \\cup Z\n}\n$$\nis cocartesian in the category of algebraic spaces over $S$, i.e.,\n$Y \\cup Z = Y \\amalg_{Y \\cap Z} Z$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Closed immersions and quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZ1","source_file":"spaces-morphisms.tex","source_line":2712,"source_end_line":2733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2712-L2733","statement_sha256":"8146a75f5d5347c78acace3c6424ede7d00c84f5e34bd9dbda20a0fe510a343b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10947,"rank":10947,"depth":55,"x":134.352,"y":1593.53,"cluster":"algebraic-spaces"},{"id":"stacks:07TY","tag":"07TY","title":"Supports of modules · Lemma 07TY","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Let U be a scheme and let φ : U → X be an étale morphism. Then Supp(φ^*F) = |φ|^-1(Supp(F)) where the left hand side is the support of φ^*F as a quasi-coherent module on the scheme U.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $U$ be a scheme and let $\\varphi : U \\to X$ be an \\'etale morphism.\nThen\n$$\n\\text{Supp}(\\varphi^*\\mathcal{F}) = |\\varphi|^{-1}(\\text{Supp}(\\mathcal{F}))\n$$\nwhere the left hand side is the support of $\\varphi^*\\mathcal{F}$ as a\nquasi-coherent module on the scheme $U$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Supports of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TY","source_file":"spaces-morphisms.tex","source_line":2799,"source_end_line":2810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2799-L2810","statement_sha256":"8bddf2e7ff59ee668bf5b4634eabc17b34ebeeb2003f2a19c18992041d3b09c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":10948,"rank":10948,"depth":54,"x":305.87,"y":1550.408,"cluster":"algebraic-spaces"},{"id":"stacks:07TZ","tag":"07TZ","title":"Supports of modules · Lemma 07TZ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a finite type quasi-coherent O_X-module. Then • The support of F is closed. • For a geometric point overlinex lying over x ∈ |X| we have x ∈ Supp(F) ⇔ F_overlinex not = 0 ⇔ F_overlinex ⊗_O_X, overlinex kappa(overlinex) not = 0. • For any morphism of algebraic spaces f : Y → X the pullback f^*F is of finite type as well and we have Supp(f^*F) = f^-1(Supp(F)).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nThen\n\\begin{enumerate}\n\\item The support of $\\mathcal{F}$ is closed.\n\\item For a geometric point $\\overline{x}$ lying over $x \\in |X|$ we have\n$$\nx \\in \\text{Supp}(\\mathcal{F})\n\\Leftrightarrow\n\\mathcal{F}_{\\overline{x}} \\not = 0\n\\Leftrightarrow\n\\mathcal{F}_{\\overline{x}} \\otimes_{\\mathcal{O}_{X, \\overline{x}}}\n\\kappa(\\overline{x}) \\not = 0.\n$$\n\\item For any morphism of algebraic spaces $f : Y \\to X$ the pullback\n$f^*\\mathcal{F}$ is of finite type as well and we have\n$\\text{Supp}(f^*\\mathcal{F}) = f^{-1}(\\text{Supp}(\\mathcal{F}))$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Supports of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TZ","source_file":"spaces-morphisms.tex","source_line":2834,"source_end_line":2854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2834-L2854","statement_sha256":"862b9b2eae911ec2fbc0d3419a76b11499c5ef6afc039edbe870b4a6bcd7fb1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10949,"rank":10949,"depth":55,"x":213.906,"y":1679.763,"cluster":"algebraic-spaces"},{"id":"stacks:07U0","tag":"07U0","title":"Supports of modules · Lemma 07U0","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a finite type quasi-coherent O_X-module. There exists a smallest closed subspace i : Z → X such that there exists a quasi-coherent O_Z-module G with i_*G ≅ F. Moreover: • If U is a scheme and φ : U → X is an étale morphism then Z ×_X U is the scheme theoretic support of φ^*F. • The quasi-coherent sheaf G is unique up to unique isomorphism. • The quasi-coherent sheaf G is of finite type. • The support of G and…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nThere exists a smallest closed subspace $i : Z \\to X$ such that there\nexists a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{G}$ with\n$i_*\\mathcal{G} \\cong \\mathcal{F}$. Moreover:\n\\begin{enumerate}\n\\item If $U$ is a scheme and $\\varphi : U \\to X$ is an \\'etale morphism\nthen $Z \\times_X U$ is the scheme theoretic support of $\\varphi^*\\mathcal{F}$.\n\\item The quasi-coherent sheaf $\\mathcal{G}$ is unique up to unique\nisomorphism.\n\\item The quasi-coherent sheaf $\\mathcal{G}$ is of finite type.\n\\item The support of $\\mathcal{G}$ and of $\\mathcal{F}$ is $|Z|$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Supports of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07U0","source_file":"spaces-morphisms.tex","source_line":2896,"source_end_line":2911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2896-L2911","statement_sha256":"2e56e98910a320dffbc653264a3f07f7e14ffc8fa7d35d686923a8a98cb27da6","origin":"The Stacks Project","memory_eligible":false,"source_rank":10950,"rank":10950,"depth":55,"x":177.63,"y":1531.929,"cluster":"algebraic-spaces"},{"id":"stacks:07U1","tag":"07U1","title":"Supports of modules · Definition 07U1","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a finite type quasi-coherent O_X-module. The scheme theoretic support of F is the closed subspace Z ⊂ X constructed in Lemma [Tag 07U0].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nThe {\\it scheme theoretic support of $\\mathcal{F}$} is the closed subspace\n$Z \\subset X$ constructed in Lemma \\ref{lemma-scheme-theoretic-support}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Supports of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07U1","source_file":"spaces-morphisms.tex","source_line":2958,"source_end_line":2964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2958-L2964","statement_sha256":"086b445fbe4f809c27dd02d7ebf0fda0e557bdf6975d67b096f095ae2fc5e191","origin":"The Stacks Project","memory_eligible":false,"source_rank":10951,"rank":10951,"depth":56,"x":323.526,"y":1620.515,"cluster":"algebraic-spaces"},{"id":"stacks:082X","tag":"082X","title":"Scheme theoretic image · Lemma 082X","summary":"The scheme-theoretic image of a morphism of algebraic spaces exists. Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. There exists a closed subspace Z ⊂ Y such that f factors through Z and such that for any other closed subspace Z' ⊂ Y such that f factors through Z' we have Z ⊂ Z'.","statement_latex":"\\begin{slogan}\nThe scheme-theoretic image of a morphism of algebraic spaces exists.\n\\end{slogan}\nLet $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. There exists a closed subspace $Z \\subset Y$ such that $f$ factors\nthrough $Z$ and such that for any other closed subspace $Z' \\subset Y$\nsuch that $f$ factors through $Z'$ we have $Z \\subset Z'$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082X","source_file":"spaces-morphisms.tex","source_line":2983,"source_end_line":2992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L2983-L2992","statement_sha256":"270f5394f09da0860997d91b6281b05f725a9ac34f208ef30c8da450b24bbf33","origin":"The Stacks Project","memory_eligible":false,"source_rank":10952,"rank":10952,"depth":52,"x":144.385,"y":1638.009,"cluster":"algebraic-spaces"},{"id":"stacks:082Y","tag":"082Y","title":"Scheme theoretic image · Definition 082Y","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The scheme theoretic image of f is the smallest closed subspace Z ⊂ Y through which f factors, see Lemma [Tag 082X] above.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. The {\\it scheme theoretic image} of $f$ is the smallest closed\nsubspace $Z \\subset Y$ through which $f$\nfactors, see Lemma \\ref{lemma-scheme-theoretic-image} above.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic image","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082Y","source_file":"spaces-morphisms.tex","source_line":3018,"source_end_line":3024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3018-L3024","statement_sha256":"5d2f27e5aa8e3f9358b786f62deacd15b356a0edffaf7c8afff18f04a7269640","origin":"The Stacks Project","memory_eligible":false,"source_rank":10953,"rank":10953,"depth":53,"x":262.624,"y":1523.256,"cluster":"algebraic-spaces"},{"id":"stacks:082Z","tag":"082Z","title":"Scheme theoretic image · Lemma 082Z","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let Z ⊂ Y be the scheme theoretic image of f. If f is quasi-compact then • the sheaf of ideals I = Ker(O_Y → f_*O_X) is quasi-coherent, • the scheme theoretic image Z is the closed subspace corresponding to I, • for any étale morphism V → Y the scheme theoretic image of X ×_Y V → V is equal to Z ×_Y V, and • the image |f|(|X|) ⊂ |Z| is a dense subset of |Z|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $Z \\subset Y$ be the scheme theoretic image of $f$.\nIf $f$ is quasi-compact then\n\\begin{enumerate}\n\\item the sheaf of ideals\n$\\mathcal{I} = \\Ker(\\mathcal{O}_Y \\to f_*\\mathcal{O}_X)$\nis quasi-coherent,\n\\item the scheme theoretic image $Z$ is the closed subspace\ncorresponding to $\\mathcal{I}$,\n\\item for any \\'etale morphism $V \\to Y$ the scheme theoretic image of\n$X \\times_Y V \\to V$ is equal to $Z \\times_Y V$, and\n\\item the image $|f|(|X|) \\subset |Z|$ is a dense subset of $|Z|$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082Z","source_file":"spaces-morphisms.tex","source_line":3032,"source_end_line":3048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3032-L3048","statement_sha256":"06278d9566a5f6c158fe356d4f4cccc1057c2ac80b4e0e5bbc062a0dd83aac4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10954,"rank":10954,"depth":53,"x":267.726,"y":1675.235,"cluster":"algebraic-spaces"},{"id":"stacks:0830","tag":"0830","title":"Scheme theoretic image · Lemma 0830","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume X is reduced. Then • the scheme theoretic image Z of f is the reduced induced algebraic space structure on overline|f|(|X|), and • for any étale morphism V → Y the scheme theoretic image of X ×_Y V → V is equal to Z ×_Y V.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $X$ is reduced. Then\n\\begin{enumerate}\n\\item the scheme theoretic image $Z$ of $f$ is the reduced induced algebraic\nspace structure on $\\overline{|f|(|X|)}$, and\n\\item for any \\'etale morphism $V \\to Y$ the scheme theoretic image of\n$X \\times_Y V \\to V$ is equal to $Z \\times_Y V$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0830","source_file":"spaces-morphisms.tex","source_line":3069,"source_end_line":3079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3069-L3079","statement_sha256":"0f47d1d40fff8ed8737b540d9b8284800b1e269d52edf7fec0eb5270024868a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":10955,"rank":10955,"depth":4,"x":141.523,"y":1565.869,"cluster":"algebraic-spaces"},{"id":"stacks:089B","tag":"089B","title":"Scheme theoretic image · Lemma 089B","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact morphism of algebraic spaces over S. Let Z be the scheme theoretic image of f. Let z ∈ |Z|. There exists a valuation ring A with fraction field K and a commutative diagram xymatrix Spec(K) ar[rr] ar[d] & & X ar[d] ar[ld] Spec(A) ar[r] & Z ar[r] & Y such that the closed point of Spec(A) maps to z.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a quasi-compact morphism of algebraic spaces over $S$.\nLet $Z$ be the scheme theoretic image of $f$.\nLet $z \\in |Z|$. There exists a valuation ring $A$ with\nfraction field $K$ and a commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[rr] \\ar[d] & & X \\ar[d] \\ar[ld] \\\\\n\\Spec(A) \\ar[r] & Z \\ar[r] & Y\n}\n$$\nsuch that the closed point of $\\Spec(A)$ maps to $z$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089B","source_file":"spaces-morphisms.tex","source_line":3090,"source_end_line":3104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3090-L3104","statement_sha256":"a1f3662cb2d46c16ed17ddb7528ed0b23cc1b44b50494698428b1cf472500dfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10956,"rank":10956,"depth":54,"x":322.852,"y":1574.919,"cluster":"algebraic-spaces"},{"id":"stacks:0CP2","tag":"0CP2","title":"Scheme theoretic image · Lemma 0CP2","summary":"Let S be a scheme. Let xymatrix X_1 ar[d] ar[r]_f_1 & Y_1 ar[d] X_2 ar[r]^f_2 & Y_2 be a commutative diagram of algebraic spaces over S. Let Z_i ⊂ Y_i, i = 1, 2 be the scheme theoretic image of f_i. Then the morphism Y_1 → Y_2 induces a morphism Z_1 → Z_2 and a commutative diagram xymatrix X_1 ar[r] ar[d] & Z_1 ar[d] ar[r] & Y_1 ar[d] X_2 ar[r] & Z_2 ar[r] & Y_2","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX_1 \\ar[d] \\ar[r]_{f_1} & Y_1 \\ar[d] \\\\\nX_2 \\ar[r]^{f_2} & Y_2\n}\n$$\nbe a commutative diagram of algebraic spaces over $S$.\nLet $Z_i \\subset Y_i$, $i = 1, 2$ be\nthe scheme theoretic image of $f_i$. Then the morphism\n$Y_1 \\to Y_2$ induces a morphism $Z_1 \\to Z_2$ and a\ncommutative diagram\n$$\n\\xymatrix{\nX_1 \\ar[r] \\ar[d] & Z_1 \\ar[d] \\ar[r] & Y_1 \\ar[d] \\\\\nX_2 \\ar[r] & Z_2 \\ar[r] & Y_2\n}\n$$","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CP2","source_file":"spaces-morphisms.tex","source_line":3136,"source_end_line":3156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3136-L3156","statement_sha256":"360d9d4cc6882d0d78337c0bc601db5abe313aa799590f482bf40fbdf197b972","origin":"The Stacks Project","memory_eligible":false,"source_rank":10957,"rank":10957,"depth":0,"x":181.616,"y":1671.306,"cluster":"algebraic-spaces"},{"id":"stacks:0CP3","tag":"0CP3","title":"Scheme theoretic image · Lemma 0CP3","summary":"Let S be a scheme. Let f : X → Y be a separated morphism of algebraic spaces over S. Let V ⊂ Y be an open subspace such that V → Y is quasi-compact. Let s : V → X be a morphism such that f ∘ s = id_V. Let Y' be the scheme theoretic image of s. Then Y' → Y is an isomorphism over V.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a separated morphism of algebraic spaces over $S$.\nLet $V \\subset Y$ be an open subspace such that $V \\to Y$ is quasi-compact.\nLet $s : V \\to X$ be a morphism such that $f \\circ s = \\text{id}_V$.\nLet $Y'$ be the scheme theoretic image of $s$.\nThen $Y' \\to Y$ is an isomorphism over $V$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CP3","source_file":"spaces-morphisms.tex","source_line":3165,"source_end_line":3173,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3165-L3173","statement_sha256":"7cf6de35b34811b7b8d3941c9b3631affe3e667a10a29646c640e1ae3e0f136c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10958,"rank":10958,"depth":57,"x":208.297,"y":1519.826,"cluster":"algebraic-spaces"},{"id":"stacks:0832","tag":"0832","title":"Scheme theoretic closure and density · Lemma 0832","summary":"Let S be a scheme. Let W ⊂ S be a scheme theoretically dense open subscheme (Morphisms, Definition [Tag 01RB]). Let f : X → S be a morphism of schemes which is flat, locally of finite presentation, and locally quasi-finite. Then f^-1(W) is scheme theoretically dense in X.","statement_latex":"Let $S$ be a scheme. Let $W \\subset S$ be a scheme theoretically\ndense open subscheme\n(Morphisms, Definition \\ref{morphisms-definition-scheme-theoretically-dense}).\nLet $f : X \\to S$ be a morphism of schemes which is flat, locally of\nfinite presentation, and locally quasi-finite.\nThen $f^{-1}(W)$ is scheme theoretically dense in $X$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0832","source_file":"spaces-morphisms.tex","source_line":3204,"source_end_line":3212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3204-L3212","statement_sha256":"3ed93402230d7f6634de37dda8ca23f661560ac2d791fc59904321273f7227ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":10959,"rank":10959,"depth":48,"x":310.618,"y":1646.886,"cluster":"algebraic-spaces"},{"id":"stacks:0833","tag":"0833","title":"Scheme theoretic closure and density · Lemma 0833","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U ⊂ X be an open subspace. The following are equivalent • for every étale morphism φ : V → X (of algebraic spaces) the scheme theoretic closure of φ^-1(U) in V is equal to V, • there exists a scheme V and a surjective étale morphism φ : V → X such that the scheme theoretic closure of φ^-1(U) in V is equal to V,","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $U \\subset X$ be an open subspace.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every \\'etale morphism $\\varphi : V \\to X$ (of algebraic spaces)\nthe scheme theoretic closure of $\\varphi^{-1}(U)$ in $V$ is equal to $V$,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$\\varphi : V \\to X$ such that the scheme theoretic closure of\n$\\varphi^{-1}(U)$ in $V$ is equal to $V$,\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0833","source_file":"spaces-morphisms.tex","source_line":3241,"source_end_line":3254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3241-L3254","statement_sha256":"da6d29e1674dc0ad1981c55dd53a85897c62baa14b107e7189e9791131afa7a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":10960,"rank":10960,"depth":49,"x":132.68,"y":1611.191,"cluster":"algebraic-spaces"},{"id":"stacks:0834","tag":"0834","title":"Scheme theoretic closure and density · Definition 0834","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U ⊂ X be an open subspace. • The scheme theoretic image of the morphism U → X is called the scheme theoretic closure of U in X. • We say U is scheme theoretically dense in X if the equivalent conditions of Lemma [Tag 0833] are satisfied.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $U \\subset X$ be an open subspace.\n\\begin{enumerate}\n\\item The scheme theoretic image of the morphism $U \\to X$\nis called the {\\it scheme theoretic closure of $U$ in $X$}.\n\\item We say $U$ is {\\it scheme theoretically dense in $X$}\nif the equivalent conditions of\nLemma \\ref{lemma-scheme-theoretically-dense} are satisfied.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic closure and density","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0834","source_file":"spaces-morphisms.tex","source_line":3276,"source_end_line":3288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3276-L3288","statement_sha256":"d028a2a9e0f5afa520be3f63e54e6617f41ad14184a5941398b930353b07a20c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10961,"rank":10961,"depth":50,"x":292.872,"y":1536.416,"cluster":"algebraic-spaces"},{"id":"stacks:0835","tag":"0835","title":"Scheme theoretic closure and density · Lemma 0835","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U ⊂ X be an open subspace. If U → X is quasi-compact, then U is scheme theoretically dense in X if and only if the scheme theoretic closure of U in X is X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U \\subset X$ be an open subspace.\nIf $U \\to X$ is quasi-compact, then $U$\nis scheme theoretically dense in $X$ if and only if the scheme theoretic\nclosure of $U$ in $X$ is $X$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0835","source_file":"spaces-morphisms.tex","source_line":3296,"source_end_line":3303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3296-L3303","statement_sha256":"f8208fd38c701cb6851a236a8cd1176c93de3b9d0b844545550b97340642d067","origin":"The Stacks Project","memory_eligible":false,"source_rank":10962,"rank":10962,"depth":54,"x":234.78,"y":1682.704,"cluster":"algebraic-spaces"},{"id":"stacks:0836","tag":"0836","title":"Scheme theoretic closure and density · Lemma 0836","summary":"Let S be a scheme. Let j : U → X be an open immersion of algebraic spaces over S. Then U is scheme theoretically dense in X if and only if O_X → j_*O_U is injective.","statement_latex":"Let $S$ be a scheme.\nLet $j : U \\to X$ be an open immersion of algebraic spaces over $S$.\nThen $U$ is scheme theoretically dense in $X$ if and only if\n$\\mathcal{O}_X \\to j_*\\mathcal{O}_U$ is injective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0836","source_file":"spaces-morphisms.tex","source_line":3309,"source_end_line":3315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3309-L3315","statement_sha256":"f64c34884de12ec4bfe18109f91d798c4b552ddac97331bb5e073c93839a6df8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10963,"rank":10963,"depth":53,"x":159.847,"y":1541.628,"cluster":"algebraic-spaces"},{"id":"stacks:0837","tag":"0837","title":"Scheme theoretic closure and density · Lemma 0837","summary":"Let S be a scheme. Let X be an algebraic space over S. If U, V are scheme theoretically dense open subspaces of X, then so is U ∩ V.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf $U$, $V$ are scheme theoretically dense\nopen subspaces of $X$, then so is $U \\cap V$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0837","source_file":"spaces-morphisms.tex","source_line":3334,"source_end_line":3339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3334-L3339","statement_sha256":"aee647985d9759f167648ef233088b0f85983e519f294143f34088433a214d41","origin":"The Stacks Project","memory_eligible":false,"source_rank":10964,"rank":10964,"depth":54,"x":328.841,"y":1603.242,"cluster":"algebraic-spaces"},{"id":"stacks:088G","tag":"088G","title":"Scheme theoretic closure and density · Lemma 088G","summary":"Let S be a scheme. Let h : Z → X be an immersion of algebraic spaces over S. Assume either Z → X is quasi-compact or Z is reduced. Let overlineZ ⊂ X be the scheme theoretic image of h. Then the morphism Z → overlineZ is an open immersion which identifies Z with a scheme theoretically dense open subspace of overlineZ. Moreover, Z is topologically dense in overlineZ.","statement_latex":"Let $S$ be a scheme. Let $h : Z \\to X$ be an immersion of algebraic spaces\nover $S$. Assume either $Z \\to X$ is quasi-compact or $Z$ is reduced.\nLet $\\overline{Z} \\subset X$ be the scheme theoretic image of $h$.\nThen the morphism $Z \\to \\overline{Z}$ is an open immersion\nwhich identifies $Z$ with a scheme theoretically dense open\nsubspace of $\\overline{Z}$. Moreover, $Z$ is topologically\ndense in $\\overline{Z}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088G","source_file":"spaces-morphisms.tex","source_line":3351,"source_end_line":3360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3351-L3360","statement_sha256":"232aa6c5772ac279679d8fd944204f86de073718ad87f588838a0c3b9e579608","origin":"The Stacks Project","memory_eligible":false,"source_rank":10965,"rank":10965,"depth":54,"x":154.38,"y":1653.786,"cluster":"algebraic-spaces"},{"id":"stacks:084N","tag":"084N","title":"Scheme theoretic closure and density · Lemma 084N","summary":"Let S be a scheme. Let B be an algebraic space over S. Let f, g : X → Y be morphisms of algebraic spaces over B. Let U ⊂ X be an open subspace such that f|_U = g|_U. If the scheme theoretic closure of U in X is X and Y → B is separated, then f = g.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $f, g : X \\to Y$ be morphisms of algebraic spaces over $B$.\nLet $U \\subset X$ be an open subspace such that\n$f|_U = g|_U$. If the scheme theoretic closure of $U$\nin $X$ is $X$ and $Y \\to B$ is separated, then $f = g$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Scheme theoretic closure and density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084N","source_file":"spaces-morphisms.tex","source_line":3371,"source_end_line":3378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3371-L3378","statement_sha256":"bdaf05038b14b1f96606ed3959171e0c28eb7844185b25f09667851adff7db5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":10966,"rank":10966,"depth":0,"x":242.529,"y":1517.29,"cluster":"algebraic-spaces"},{"id":"stacks:0ABL","tag":"0ABL","title":"Dominant morphisms · Definition 0ABL","summary":"Let S be a scheme. A morphism f : X → Y of algebraic spaces over S is called dominant if the image of |f| : |X| → |Y| is dense in |Y|.","statement_latex":"Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$ is\ncalled {\\it dominant} if the image of $|f| : |X| \\to |Y|$ is dense in $|Y|$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Dominant morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABL","source_file":"spaces-morphisms.tex","source_line":3402,"source_end_line":3406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3402-L3406","statement_sha256":"cb8186c185cbce4c23d3c6727aef8b60fa1650d0fad49a6a329c6fe06f092833","origin":"The Stacks Project","memory_eligible":false,"source_rank":10967,"rank":10967,"depth":0,"x":287.372,"y":1668.214,"cluster":"algebraic-spaces"},{"id":"stacks:03MU","tag":"03MU","title":"Universally injective morphisms · Lemma 03MU","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. Then f is universally injective (in the sense of Section [Tag 03HA]) if and only if for all fields K the map X(K) → Y(K) is injective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable\nmorphism of algebraic spaces over $S$. Then $f$ is universally injective\n(in the sense of Section \\ref{section-representable})\nif and only if for all fields $K$ the map $X(K) \\to Y(K)$ is injective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MU","source_file":"spaces-morphisms.tex","source_line":3426,"source_end_line":3432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3426-L3432","statement_sha256":"f162da5e3d9b978f62d164bfccc8e832560249218fc23bbd54bc916837486f0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":10968,"rank":10968,"depth":3,"x":132.675,"y":1582.224,"cluster":"algebraic-spaces"},{"id":"stacks:040X","tag":"040X","title":"Universally injective morphisms · Lemma 040X","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • the map X(K) → Y(K) is injective for every field K over S • for every morphism Y' → Y of algebraic spaces over S the induced map |Y' ×_Y X| → |Y'| is injective, and • the diagonal morphism X → X ×_Y X is surjective.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the map $X(K) \\to Y(K)$ is injective for every field $K$ over $S$\n\\item for every morphism $Y' \\to Y$ of algebraic spaces over $S$\nthe induced map $|Y' \\times_Y X| \\to |Y'|$ is injective, and\n\\item the diagonal morphism $X \\to X \\times_Y X$ is surjective.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040X","source_file":"spaces-morphisms.tex","source_line":3460,"source_end_line":3471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3460-L3471","statement_sha256":"0d55319ca09f0fcd405e66a754cfa9abbae599e227a071439fde34409097275c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10969,"rank":10969,"depth":54,"x":316.206,"y":1557.812,"cluster":"algebraic-spaces"},{"id":"stacks:03MV","tag":"03MV","title":"Universally injective morphisms · Definition 03MV","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say f is universally injective if for every morphism Y' → Y the induced map |Y' ×_Y X| → |Y'| is injective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. We say $f$ is {\\it universally injective} if\nfor every morphism $Y' \\to Y$ the induced map\n$|Y' \\times_Y X| \\to |Y'|$ is injective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally injective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MV","source_file":"spaces-morphisms.tex","source_line":3554,"source_end_line":3560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3554-L3560","statement_sha256":"bbbcf8b05e720af7747987d4b291ab24d5fb5682562cc139d3859e2b7ee1d6eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10970,"rank":10970,"depth":0,"x":200.309,"y":1680.159,"cluster":"algebraic-spaces"},{"id":"stacks:03MW","tag":"03MW","title":"Universally injective morphisms · Lemma 03MW","summary":"The base change of a universally injective morphism is universally injective.","statement_latex":"The base change of a universally injective morphism is universally injective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MW","source_file":"spaces-morphisms.tex","source_line":3584,"source_end_line":3587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3584-L3587","statement_sha256":"56d81a13d3640a88c9db742569f40b3e1942fab7b93e3c151c1fa76e1fab40bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10971,"rank":10971,"depth":0,"x":187.36,"y":1523.918,"cluster":"algebraic-spaces"},{"id":"stacks:03MX","tag":"03MX","title":"Universally injective morphisms · Lemma 03MX","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is universally injective, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is universally injective, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is universally injective, • there exists a scheme Z and a surjective morphism Z → Y such that Z ×_Y X → Z is universally injective, and • there exists a Zariski covering…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is universally injective,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is universally injective,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is universally injective,\n\\item there exists a scheme $Z$ and a surjective morphism\n$Z \\to Y$ such that $Z \\times_Y X \\to Z$ is universally injective, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that\neach of the morphisms $f^{-1}(Y_i) \\to Y_i$ is universally injective.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MX","source_file":"spaces-morphisms.tex","source_line":3593,"source_end_line":3609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3593-L3609","statement_sha256":"6082d9d788a223f46ecc57a2904a683c230399e347128e671a1c9e26bbecbf65","origin":"The Stacks Project","memory_eligible":false,"source_rank":10972,"rank":10972,"depth":54,"x":322.78,"y":1631.96,"cluster":"algebraic-spaces"},{"id":"stacks:03MY","tag":"03MY","title":"Universally injective morphisms · Lemma 03MY","summary":"A composition of universally injective morphisms is universally injective.","statement_latex":"A composition of universally injective morphisms is universally injective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MY","source_file":"spaces-morphisms.tex","source_line":3638,"source_end_line":3641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3638-L3641","statement_sha256":"1c21b8609561966d60b97f69631b0e3499a19fd31aa3aa9092cb901708df38e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":10973,"rank":10973,"depth":0,"x":135.727,"y":1629.127,"cluster":"algebraic-spaces"},{"id":"stacks:03WE","tag":"03WE","title":"Affine morphisms · Lemma 03WE","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. Then f is affine (in the sense of Section [Tag 03HA]) if and only if for all affine schemes Z and morphisms Z → Y the scheme X ×_Y Z is affine.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism\nof algebraic spaces over $S$. Then\n$f$ is affine (in the sense of Section \\ref{section-representable})\nif and only if for all affine schemes $Z$\nand morphisms $Z \\to Y$ the scheme $X \\times_Y Z$ is affine.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WE","source_file":"spaces-morphisms.tex","source_line":3665,"source_end_line":3672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3665-L3672","statement_sha256":"d7ba6a3d910f041be613702cde690f1ae6174811aa1bc077b7c850ba39edd707","origin":"The Stacks Project","memory_eligible":false,"source_rank":10974,"rank":10974,"depth":1,"x":276.169,"y":1524.906,"cluster":"algebraic-spaces"},{"id":"stacks:03WF","tag":"03WF","title":"Affine morphisms · Definition 03WF","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say f is affine if for every affine scheme Z and morphism Z → Y the algebraic space X ×_Y Z is representable by an affine scheme.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nWe say $f$ is {\\it affine} if for every affine scheme $Z$ and\nmorphism $Z \\to Y$ the algebraic space $X \\times_Y Z$ is representable\nby an affine scheme.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WF","source_file":"spaces-morphisms.tex","source_line":3682,"source_end_line":3689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3682-L3689","statement_sha256":"9fa105d19b9ee05383f3d44285a99f1134d6c8f4183085edca0288f2e460e1ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":10975,"rank":10975,"depth":0,"x":256.388,"y":1681.705,"cluster":"algebraic-spaces"},{"id":"stacks:03WG","tag":"03WG","title":"Affine morphisms · Lemma 03WG","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is representable and affine, • f is affine, • for every affine scheme V and étale morphism V → Y the scheme X ×_Y V is affine, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is affine, and • there exists a Zariski covering Y = ⋃ Y_i such that each of the morphisms f^-1(Y_i) → Y_i is affine.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is representable and affine,\n\\item $f$ is affine,\n\\item for every affine scheme $V$ and \\'etale morphism $V \\to Y$\nthe scheme $X \\times_Y V$ is affine,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is affine, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such\nthat each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is affine.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WG","source_file":"spaces-morphisms.tex","source_line":3691,"source_end_line":3706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3691-L3706","statement_sha256":"83a6e699803db17df0a07dac1fdb09dd486f99713c893e277bc1ef2c9fce9c90","origin":"The Stacks Project","memory_eligible":false,"source_rank":10976,"rank":10976,"depth":46,"x":144.696,"y":1554.65,"cluster":"algebraic-spaces"},{"id":"stacks:03WH","tag":"03WH","title":"Affine morphisms · Lemma 03WH","summary":"The composition of affine morphisms is affine.","statement_latex":"The composition of affine morphisms is affine.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WH","source_file":"spaces-morphisms.tex","source_line":3733,"source_end_line":3736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3733-L3736","statement_sha256":"3b9345b8c80d29a730c4d673b97b2bdb26ff7d431c438e8f0234ed012b727eab","origin":"The Stacks Project","memory_eligible":false,"source_rank":10977,"rank":10977,"depth":0,"x":329.534,"y":1585.012,"cluster":"algebraic-spaces"},{"id":"stacks:03WI","tag":"03WI","title":"Affine morphisms · Lemma 03WI","summary":"The base change of an affine morphism is affine.","statement_latex":"The base change of an affine morphism is affine.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WI","source_file":"spaces-morphisms.tex","source_line":3742,"source_end_line":3745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3742-L3745","statement_sha256":"feae9d2f56c03a76ef9a6156e304d0f427028b609f231f125ea086532b40a895","origin":"The Stacks Project","memory_eligible":false,"source_rank":10978,"rank":10978,"depth":0,"x":168.558,"y":1667.64,"cluster":"algebraic-spaces"},{"id":"stacks:07U2","tag":"07U2","title":"Affine morphisms · Lemma 07U2","summary":"A closed immersion is affine.","statement_latex":"A closed immersion is affine.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07U2","source_file":"spaces-morphisms.tex","source_line":3751,"source_end_line":3754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3751-L3754","statement_sha256":"685ea3d7ef77a07866af0cd475216f20344438e1b575211977861517e5b781eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":10979,"rank":10979,"depth":13,"x":220.897,"y":1515.121,"cluster":"algebraic-spaces"},{"id":"stacks:081V","tag":"081V","title":"Affine morphisms · Lemma 081V","summary":"Let S be a scheme. Let X be an algebraic space over S. There is an anti-equivalence of categories algebraic spaces affine over X longleftrightarrow quasi-coherent sheaves of O_X-algebras which associates to f : Y → X the sheaf f_*O_Y. Moreover, this equivalence is compatible with arbitrary base change.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThere is an anti-equivalence of categories\n$$\n\\begin{matrix}\n\\text{algebraic spaces} \\\\\n\\text{affine over }X\n\\end{matrix}\n\\longleftrightarrow\n\\begin{matrix}\n\\text{quasi-coherent sheaves} \\\\\n\\text{of }\\mathcal{O}_X\\text{-algebras}\n\\end{matrix}\n$$\nwhich associates to $f : Y \\to X$ the sheaf $f_*\\mathcal{O}_Y$.\nMoreover, this equivalence is compatible with arbitrary base change.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081V","source_file":"spaces-morphisms.tex","source_line":3761,"source_end_line":3778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3761-L3778","statement_sha256":"f24218b4def59f4a6a372565d8606e0bb329a3490c3d1fccddd2f260bee6b59f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10980,"rank":10980,"depth":52,"x":305.091,"y":1657.516,"cluster":"algebraic-spaces"},{"id":"stacks:081W","tag":"081W","title":"Affine morphisms · Definition 081W","summary":"Let S be a scheme. Let X be an algebraic space over S. Let A be a quasi-coherent sheaf of O_X-algebras. The relative spectrum of A over X, or simply the spectrum of A over X is the affine morphism underlineSpec(A) → X corresponding to A under the equivalence of categories of Lemma [Tag 081V].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{A}$ be a quasi-coherent sheaf of\n$\\mathcal{O}_X$-algebras. The {\\it relative spectrum of $\\mathcal{A}$ over\n$X$}, or simply the {\\it spectrum of $\\mathcal{A}$ over $X$} is the\naffine morphism $\\underline{\\Spec}(\\mathcal{A}) \\to X$\ncorresponding to $\\mathcal{A}$ under the equivalence of categories of\nLemma \\ref{lemma-affine-equivalence-algebras}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081W","source_file":"spaces-morphisms.tex","source_line":3834,"source_end_line":3843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3834-L3843","statement_sha256":"123cbe12fc8c2a3728dfdc42b56b8e75aac347bbb38b2c15944bea258375cb64","origin":"The Stacks Project","memory_eligible":false,"source_rank":10981,"rank":10981,"depth":53,"x":128.212,"y":1600.197,"cluster":"algebraic-spaces"},{"id":"stacks:08AI","tag":"08AI","title":"Affine morphisms · Lemma 08AI","summary":"Let S be a scheme. Let f : Y → X be an affine morphism of algebraic spaces over S. Let A = f_*O_Y. The functor F ↦ f_*F induces an equivalence of categories ( category of quasi-coherent O_Y-modules ) → ( category of quasi-coherent A-modules ) Moreover, an A-module is quasi-coherent as an O_X-module if and only if it is quasi-coherent as an A-module.","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be an affine morphism of algebraic spaces over $S$.\nLet $\\mathcal{A} = f_*\\mathcal{O}_Y$.\nThe functor $\\mathcal{F} \\mapsto f_*\\mathcal{F}$ induces\nan equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{category of quasi-coherent}\\\\\n\\mathcal{O}_Y\\text{-modules}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{category of quasi-coherent}\\\\\n\\mathcal{A}\\text{-modules}\n\\end{matrix}\n\\right\\}\n$$\nMoreover, an $\\mathcal{A}$-module is\nquasi-coherent as an $\\mathcal{O}_X$-module if and only if\nit is quasi-coherent as an $\\mathcal{A}$-module.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AI","source_file":"spaces-morphisms.tex","source_line":3867,"source_end_line":3892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3867-L3892","statement_sha256":"475633a834419d2926d3475ff7b0a8bd7c62e168fa3b3527c685d106f4624305","origin":"The Stacks Project","memory_eligible":false,"source_rank":10982,"rank":10982,"depth":0,"x":305.02,"y":1542.004,"cluster":"algebraic-spaces"},{"id":"stacks:08GB","tag":"08GB","title":"Affine morphisms · Lemma 08GB","summary":"Let S be a scheme. Let B be an algebraic space over S. Suppose g : X → Y is a morphism of algebraic spaces over B. • If X is affine over B and Δ : Y → Y ×_B Y is affine, then g is affine. • If X is affine over B and Y is separated over B, then g is affine. • A morphism from an affine scheme to an algebraic space with affine diagonal over Z (as in Properties of Spaces, Definition [Tag 03BS]) is affine. • A morphism from an affine scheme to a separated algebraic space is…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nSuppose $g : X \\to Y$ is a morphism of algebraic spaces over $B$.\n\\begin{enumerate}\n\\item If $X$ is affine over $B$ and $\\Delta : Y \\to Y \\times_B Y$ is affine,\nthen $g$ is affine.\n\\item If $X$ is affine over $B$ and $Y$ is separated over $B$,\nthen $g$ is affine.\n\\item A morphism from an affine scheme to an algebraic space with affine\ndiagonal over $\\mathbf{Z}$ (as in Properties of Spaces, Definition\n\\ref{spaces-properties-definition-separated}) is affine.\n\\item A morphism from an affine scheme to a separated algebraic space\nis affine.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GB","source_file":"spaces-morphisms.tex","source_line":3898,"source_end_line":3913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3898-L3913","statement_sha256":"6a07c4126d27fa9bd7af56177fad227d038e3bd53268af3b62bdc38683c366c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":10983,"rank":10983,"depth":55,"x":221.306,"y":1685.471,"cluster":"algebraic-spaces"},{"id":"stacks:09TF","tag":"09TF","title":"Affine morphisms · Lemma 09TF","summary":"Let S be a scheme. Let X be a quasi-separated algebraic space over S. Let A be an Artinian ring. Any morphism Spec(A) → X is affine.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$.\nLet $A$ be an Artinian ring. Any morphism $\\Spec(A) \\to X$ is affine.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TF","source_file":"spaces-morphisms.tex","source_line":3930,"source_end_line":3934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3930-L3934","statement_sha256":"8a6204d60833b5c3e016b3fb8323e9f82d38863051b0344dcbcd78d43b28e1b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10984,"rank":10984,"depth":47,"x":167.578,"y":1531.933,"cluster":"algebraic-spaces"},{"id":"stacks:03WK","tag":"03WK","title":"Quasi-affine morphisms · Lemma 03WK","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. Then f is quasi-affine (in the sense of Section [Tag 03HA]) if and only if for all affine schemes Z and morphisms Z → Y the scheme X ×_Y Z is quasi-affine.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable\nmorphism of algebraic spaces over $S$. Then\n$f$ is quasi-affine (in the sense of Section \\ref{section-representable})\nif and only if for all affine schemes $Z$\nand morphisms $Z \\to Y$ the scheme $X \\times_Y Z$ is quasi-affine.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WK","source_file":"spaces-morphisms.tex","source_line":3968,"source_end_line":3975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3968-L3975","statement_sha256":"0703713bddf7a42f3d4400f95e64bdf2f6c19a99d70b9d0dd4a7e9eeb4238f1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":10985,"rank":10985,"depth":1,"x":330.928,"y":1614.795,"cluster":"algebraic-spaces"},{"id":"stacks:03WL","tag":"03WL","title":"Quasi-affine morphisms · Definition 03WL","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say f is quasi-affine if for every affine scheme Z and morphism Z → Y the algebraic space X ×_Y Z is representable by a quasi-affine scheme.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nWe say $f$ is {\\it quasi-affine} if for every affine scheme $Z$ and\nmorphism $Z \\to Y$ the algebraic space $X \\times_Y Z$ is representable\nby a quasi-affine scheme.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-affine morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WL","source_file":"spaces-morphisms.tex","source_line":3986,"source_end_line":3993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3986-L3993","statement_sha256":"e8ccf204d2dc8829d4995b3a97dd4905803ff2afcf7a5ac2db50c90c2adbb15c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10986,"rank":10986,"depth":0,"x":143.542,"y":1646.433,"cluster":"algebraic-spaces"},{"id":"stacks:03WM","tag":"03WM","title":"Quasi-affine morphisms · Lemma 03WM","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is representable and quasi-affine, • f is quasi-affine, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is quasi-affine, and • there exists a Zariski covering Y = ⋃ Y_i such that each of the morphisms f^-1(Y_i) → Y_i is quasi-affine.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is representable and quasi-affine,\n\\item $f$ is quasi-affine,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is quasi-affine, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such\nthat each of the morphisms $f^{-1}(Y_i) \\to Y_i$ is quasi-affine.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WM","source_file":"spaces-morphisms.tex","source_line":3995,"source_end_line":4008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L3995-L4008","statement_sha256":"7b6e67125f58d1b5e60a67438385dd3fcca39cee8417603a01d516c8ef91eecf","origin":"The Stacks Project","memory_eligible":false,"source_rank":10987,"rank":10987,"depth":47,"x":256.454,"y":1516.571,"cluster":"algebraic-spaces"},{"id":"stacks:03WN","tag":"03WN","title":"Quasi-affine morphisms · Lemma 03WN","summary":"The composition of quasi-affine morphisms is quasi-affine.","statement_latex":"The composition of quasi-affine morphisms is quasi-affine.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WN","source_file":"spaces-morphisms.tex","source_line":4035,"source_end_line":4038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4035-L4038","statement_sha256":"2e96c483637b26537f4c363f84d8762df764bcf8a27e9cf54b7f83a013366ac2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10988,"rank":10988,"depth":0,"x":277.66,"y":1676.651,"cluster":"algebraic-spaces"},{"id":"stacks:03WO","tag":"03WO","title":"Quasi-affine morphisms · Lemma 03WO","summary":"The base change of a quasi-affine morphism is quasi-affine.","statement_latex":"The base change of a quasi-affine morphism is quasi-affine.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03WO","source_file":"spaces-morphisms.tex","source_line":4044,"source_end_line":4047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4044-L4047","statement_sha256":"3194d114c6c502c225c3e4bd26cb9c99418c399c4b4afc344ab5314e7972dea2","origin":"The Stacks Project","memory_eligible":false,"source_rank":10989,"rank":10989,"depth":0,"x":133.063,"y":1570.475,"cluster":"algebraic-spaces"},{"id":"stacks:086S","tag":"086S","title":"Quasi-affine morphisms · Lemma 086S","summary":"Let S be a scheme. A quasi-compact and quasi-separated morphism of algebraic spaces f : Y → X is quasi-affine if and only if the canonical factorization Y → underlineSpec_X(f_*O_Y) (Remark [Tag 081X]) is an open immersion.","statement_latex":"Let $S$ be a scheme.\nA quasi-compact and quasi-separated morphism of algebraic spaces\n$f : Y \\to X$ is quasi-affine if and only if the canonical factorization\n$Y \\to \\underline{\\Spec}_X(f_*\\mathcal{O}_Y)$\n(Remark \\ref{remark-factorization-quasi-compact-quasi-separated})\nis an open immersion.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086S","source_file":"spaces-morphisms.tex","source_line":4053,"source_end_line":4061,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4053-L4061","statement_sha256":"2105a120ff9ee49cf12d760f2536081f08f968404bab51616b4bce9c54cb25bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":10990,"rank":10990,"depth":53,"x":325.373,"y":1566.714,"cluster":"algebraic-spaces"},{"id":"stacks:03MJ","tag":"03MJ","title":"Types of morphisms étale local on source-and-target · Lemma 03MJ","summary":"Let P be a property of morphisms of schemes which is étale local on the source-and-target. Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Consider commutative diagrams xymatrix U ar[d]_a ar[r]_h & V ar[d]^b X ar[r]^f & Y where U and V are schemes and the vertical arrows are étale. The following are equivalent • for any diagram as above the morphism h has property P, and • for some diagram as above with a : U → X surjective the morphism h has…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes\nwhich is \\'etale local on the source-and-target.\nLet $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nConsider commutative diagrams\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwhere $U$ and $V$ are schemes and the vertical arrows are \\'etale.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any diagram as above the morphism $h$ has property\n$\\mathcal{P}$, and\n\\item for some diagram as above with $a : U \\to X$ surjective\nthe morphism $h$ has property $\\mathcal{P}$.\n\\end{enumerate}\nIf $X$ and $Y$ are representable, then this is also equivalent to\n$f$ (as a morphism of schemes) having property $\\mathcal{P}$.\nIf $\\mathcal{P}$ is also preserved under any base change, and\nfppf local on the base, then for representable morphisms $f$ this\nis also equivalent to $f$ having property $\\mathcal{P}$ in the sense\nof Section \\ref{section-representable}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Types of morphisms étale local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MJ","source_file":"spaces-morphisms.tex","source_line":4096,"source_end_line":4123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4096-L4123","statement_sha256":"d7297a22c3e0fbb989587120d37202b9a4d145e4fbaadc6aac7b32f62ff3db2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":10991,"rank":10991,"depth":45,"x":186.373,"y":1678.784,"cluster":"algebraic-spaces"},{"id":"stacks:04RD","tag":"04RD","title":"Types of morphisms étale local on source-and-target · Definition 04RD","summary":"Let S be a scheme. Let P be a property of morphisms of schemes which is étale local on the source-and-target. We say a morphism f : X → Y of algebraic spaces over S has property P if the equivalent conditions of Lemma [Tag 03MJ] hold.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of schemes\nwhich is \\'etale local on the source-and-target.\nWe say a morphism $f : X \\to Y$ of algebraic spaces over $S$\n{\\it has property $\\mathcal{P}$} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Types of morphisms étale local on source-and-target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RD","source_file":"spaces-morphisms.tex","source_line":4189,"source_end_line":4198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4189-L4198","statement_sha256":"2c4fae885cf52122fc3a3c4e1b378fc97a66def0c0b4dd7c56fcd423f6bfc9ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":10992,"rank":10992,"depth":46,"x":198.765,"y":1517.021,"cluster":"algebraic-spaces"},{"id":"stacks:04NC","tag":"04NC","title":"Types of morphisms étale local on source-and-target · Lemma 04NC","summary":"Let Q be a property of morphisms of germs which is étale local on the source-and-target. Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let x ∈ |X| be a point of X. Consider the diagrams xymatrix U ar[d]_a ar[r]_h & V ar[d]^b X ar[r]^f & Y xymatrix u ar[d] ar[r] & v ar[d] x ar[r] & y where U and V are schemes, a, b are étale, and u, v, x, y are points of the corresponding spaces. The following are equivalent • for any diagram as above we have…","statement_latex":"Let $\\mathcal{Q}$ be a property of morphisms of germs which is\n\\'etale local on the source-and-target.\nLet $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $x \\in |X|$ be a point of $X$.\nConsider the diagrams\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n\\quad\\quad\n\\xymatrix{\nu \\ar[d] \\ar[r] & v \\ar[d] \\\\\nx \\ar[r] & y\n}\n$$\nwhere $U$ and $V$ are schemes, $a, b$ are \\'etale, and $u, v, x, y$\nare points of the corresponding spaces.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any diagram as above we have $\\mathcal{Q}((U, u) \\to (V, v))$, and\n\\item for some diagram as above we have $\\mathcal{Q}((U, u) \\to (V, v))$.\n\\end{enumerate}\nIf $X$ and $Y$ are representable, then this is also\nequivalent to $\\mathcal{Q}((X, x) \\to (Y, y))$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Types of morphisms étale local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NC","source_file":"spaces-morphisms.tex","source_line":4227,"source_end_line":4255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4227-L4255","statement_sha256":"d8f57309420ff90df604acce630c83adf52052029a8724bde7e2302b0970a9fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":10993,"rank":10993,"depth":46,"x":319.901,"y":1643.532,"cluster":"algebraic-spaces"},{"id":"stacks:04RE","tag":"04RE","title":"Types of morphisms étale local on source-and-target · Definition 04RE","summary":"Let Q be a property of morphisms of germs of schemes which is étale local on the source-and-target. Let S be a scheme. Given a morphism f : X → Y of algebraic spaces over S and a point x ∈ |X| we say that f has property Q at x if the equivalent conditions of Lemma [Tag 04NC] hold.","statement_latex":"Let $\\mathcal{Q}$ be a property of morphisms of germs\nof schemes which is \\'etale local on the source-and-target.\nLet $S$ be a scheme.\nGiven a morphism $f : X \\to Y$ of algebraic spaces over $S$ and\na point $x \\in |X|$ we say that $f$\n{\\it has property $\\mathcal{Q}$ at $x$} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target-at-point}\nhold.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Types of morphisms étale local on source-and-target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RE","source_file":"spaces-morphisms.tex","source_line":4262,"source_end_line":4272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4262-L4272","statement_sha256":"adcc450027438a3fc808a543f5e4659263e5fbfa93d5090f152353c491523812","origin":"The Stacks Project","memory_eligible":false,"source_rank":10994,"rank":10994,"depth":47,"x":128.544,"y":1618.942,"cluster":"algebraic-spaces"},{"id":"stacks:04RF","tag":"04RF","title":"Types of morphisms étale local on source-and-target · Lemma 04RF","summary":"Let P be a property of morphisms of schemes which is étale local on the source-and-target. Consider the property Q of morphisms of germs associated to P in Descent, Lemma [Tag 04R6]. Then • Q is étale local on the source-and-target. • given a morphism of algebraic spaces f : X → Y and x ∈ |X| the following are equivalent • f has Q at x, and • there is an open neighbourhood X' ⊂ X of x such that X' → Y has P. • given a morphism of algebraic spaces f : X → Y the following…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes\nwhich is \\'etale local on the source-and-target.\nConsider the property $\\mathcal{Q}$ of morphisms\nof germs associated to $\\mathcal{P}$ in\nDescent, Lemma \\ref{descent-lemma-local-source-target-global-implies-local}.\nThen\n\\begin{enumerate}\n\\item $\\mathcal{Q}$ is \\'etale local on the source-and-target.\n\\item given a morphism of algebraic spaces $f : X \\to Y$ and $x \\in |X|$\nthe following are equivalent\n\\begin{enumerate}\n\\item $f$ has $\\mathcal{Q}$ at $x$, and\n\\item there is an open neighbourhood $X' \\subset X$ of $x$\nsuch that $X' \\to Y$ has $\\mathcal{P}$.\n\\end{enumerate}\n\\item given a morphism of algebraic spaces $f : X \\to Y$\nthe following are equivalent:\n\\begin{enumerate}\n\\item $f$ has $\\mathcal{P}$,\n\\item for every $x \\in |X|$ the morphism $f$ has $\\mathcal{Q}$ at $x$.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Types of morphisms étale local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RF","source_file":"spaces-morphisms.tex","source_line":4281,"source_end_line":4305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4281-L4305","statement_sha256":"54b7bd16d0e3ef11d7e27906fdca60d875b9e3e55c9c0b35345f735e053153d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":10995,"rank":10995,"depth":47,"x":289.672,"y":1528.353,"cluster":"algebraic-spaces"},{"id":"stacks:03XF","tag":"03XF","title":"Morphisms of finite type · Definition 03XF","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f locally of finite type if the equivalent conditions of Lemma [Tag 03MJ] hold with P = locally of finite type. • Let x ∈ |X|. We say f is of finite type at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is locally of finite type. • We say f is of finite type if it is locally of finite type and quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$\n{\\it locally of finite type} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with\n$\\mathcal{P} = \\text{locally of finite type}$.\n\\item Let $x \\in |X|$. We say $f$ is of {\\it finite type at $x$}\nif there exists an open neighbourhood $X' \\subset X$ of $x$ such\nthat $f|_{X'} : X' \\to Y$ is locally of finite type.\n\\item We say $f$ is\n{\\it of finite type} if it is locally of finite type and quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XF","source_file":"spaces-morphisms.tex","source_line":4351,"source_end_line":4367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4351-L4367","statement_sha256":"6a17a160eb8ca08c4f28c90e234468979fec3956db507a700db08c0d51a0b754","origin":"The Stacks Project","memory_eligible":false,"source_rank":10996,"rank":10996,"depth":46,"x":243.636,"y":1686.824,"cluster":"algebraic-spaces"},{"id":"stacks:03XG","tag":"03XG","title":"Morphisms of finite type · Lemma 03XG","summary":"The composition of finite type morphisms is of finite type. The same holds for locally of finite type.","statement_latex":"The composition of finite type morphisms is of finite type.\nThe same holds for locally of finite type.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XG","source_file":"spaces-morphisms.tex","source_line":4375,"source_end_line":4379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4375-L4379","statement_sha256":"29114cbfbfd7bef270f56fe8296d511b38800454e0768aee916b29c579651f51","origin":"The Stacks Project","memory_eligible":false,"source_rank":10997,"rank":10997,"depth":6,"x":150.0,"y":1543.628,"cluster":"algebraic-spaces"},{"id":"stacks:03XH","tag":"03XH","title":"Morphisms of finite type · Lemma 03XH","summary":"A base change of a finite type morphism is finite type. The same holds for locally of finite type.","statement_latex":"A base change of a finite type morphism is finite type.\nThe same holds for locally of finite type.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XH","source_file":"spaces-morphisms.tex","source_line":4386,"source_end_line":4390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4386-L4390","statement_sha256":"d2cce72dc29f174023a6766bc2b0fb301a292ad246c29acd7a1c207369745713","origin":"The Stacks Project","memory_eligible":false,"source_rank":10998,"rank":10998,"depth":6,"x":334.483,"y":1596.168,"cluster":"algebraic-spaces"},{"id":"stacks:040Y","tag":"040Y","title":"Morphisms of finite type · Lemma 040Y","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is locally of finite type, • for every x ∈ |X| the morphism f is of finite type at x, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is locally of finite type, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is locally of finite type, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item for every $x \\in |X|$ the morphism $f$ is of finite type at $x$,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is locally of finite type,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally of finite type,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is locally of finite type,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis locally of finite type,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale\nthe top horizontal arrow is locally of finite type,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, $U \\to X$\nis surjective, and the top horizontal arrow is locally of finite type, and\n\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,\nand $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that\neach morphism $X_{ij} \\to Y_i$ is locally of finite type.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040Y","source_file":"spaces-morphisms.tex","source_line":4397,"source_end_line":4436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4397-L4436","statement_sha256":"b95c444ab1d38cf042d55c7ff01615be088e5cc5cfdef56814227a4c0335ef1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":10999,"rank":10999,"depth":41,"x":155.924,"y":1662.207,"cluster":"algebraic-spaces"},{"id":"stacks:04ZK","tag":"04ZK","title":"Morphisms of finite type · Lemma 04ZK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally of finite type and Y is locally Noetherian, then X is locally Noetherian.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is locally of finite type and $Y$ is locally Noetherian,\nthen $X$ is locally Noetherian.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZK","source_file":"spaces-morphisms.tex","source_line":4463,"source_end_line":4469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4463-L4469","statement_sha256":"507d1faa30eb0a824d374a246c02f0faefca4ea1439b0fff2857157f1b953ed3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11000,"rank":11000,"depth":2,"x":234.604,"y":1511.964,"cluster":"algebraic-spaces"},{"id":"stacks:0462","tag":"0462","title":"Morphisms of finite type · Lemma 0462","summary":"Let S be a scheme. Let f : X → Y, g : Y → Z be morphisms of algebraic spaces over S. If g ∘ f : X → Z is locally of finite type, then f : X → Y is locally of finite type.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$, $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.\nIf $g \\circ f : X \\to Z$ is locally of finite type, then $f : X \\to Y$\nis locally of finite type.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0462","source_file":"spaces-morphisms.tex","source_line":4488,"source_end_line":4494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4488-L4494","statement_sha256":"fe5236ad36b7df3c78363a1c013f86608a32fcbc8cbc2a7c6c6596efab9cda28","origin":"The Stacks Project","memory_eligible":false,"source_rank":11001,"rank":11001,"depth":42,"x":297.504,"y":1667.632,"cluster":"algebraic-spaces"},{"id":"stacks:06ED","tag":"06ED","title":"Morphisms of finite type · Lemma 06ED","summary":"An immersion is locally of finite type.","statement_latex":"An immersion is locally of finite type.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ED","source_file":"spaces-morphisms.tex","source_line":4514,"source_end_line":4517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4514-L4517","statement_sha256":"f76ed2ac68b4b6bf58906b1272ce8dbcebfcac4153d67ac089e49a5a6c9a3600","origin":"The Stacks Project","memory_eligible":false,"source_rank":11002,"rank":11002,"depth":3,"x":125.679,"y":1588.415,"cluster":"algebraic-spaces"},{"id":"stacks:0487","tag":"0487","title":"Points and geometric points · Lemma 0487","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type. The following are equivalent: • f is surjective, and • for every algebraically closed field k over S the induced map X(k) → Y(k) is surjective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is locally of finite type. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is surjective, and\n\\item for every algebraically closed field $k$ over $S$ the induced\nmap $X(k) \\to Y(k)$ is surjective.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Points and geometric points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0487","source_file":"spaces-morphisms.tex","source_line":4564,"source_end_line":4573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4564-L4573","statement_sha256":"6e526ce1fb39fbd5ded4b9b8f38bb7da7b6c0ff84e0a2ac8b5fa30974c994515","origin":"The Stacks Project","memory_eligible":false,"source_rank":11003,"rank":11003,"depth":18,"x":316.371,"y":1549.272,"cluster":"algebraic-spaces"},{"id":"stacks:0488","tag":"0488","title":"Points and geometric points · Lemma 0488","summary":"Let S be a scheme. Let X, Y be algebraic spaces over S. • As k ranges over all algebraically closed fields over S the collection of geometric points overliney ∈ Y(k) cover all of |Y|. • As k ranges over all algebraically closed fields over S with |k| ≥ λ(Y) and |k| > λ(X) the geometric points overliney ∈ Y(k) cover all of |Y|. • For any geometric point overlines : Spec(k) → S where k has cardinality > λ(X) the map X(k) → |X_s| is surjective. • Let X → Y be a morphism of…","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$.\n\\begin{enumerate}\n\\item As $k$ ranges over all algebraically closed fields over $S$\nthe collection of geometric points $\\overline{y} \\in Y(k)$ cover all of $|Y|$.\n\\item As $k$ ranges over all algebraically closed fields over $S$ with\n$|k| \\geq \\lambda(Y)$ and $|k| > \\lambda(X)$ the geometric points\n$\\overline{y} \\in Y(k)$ cover all of $|Y|$.\n\\item For any geometric point\n$\\overline{s} : \\Spec(k) \\to S$ where\n$k$ has cardinality $> \\lambda(X)$ the map\n$$\nX(k) \\longrightarrow |X_s|\n$$\nis surjective.\n\\item Let $X \\to Y$ be a morphism of algebraic spaces over $S$.\nFor any geometric point $\\overline{s} : \\Spec(k) \\to S$ where\n$k$ has cardinality $> \\lambda(X)$ the map\n$$\nX(k) \\longrightarrow |X| \\times_{|Y|} Y(k)\n$$\nis surjective.\n\\item Let $X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the map $X \\to Y$ is surjective,\n\\item for all algebraically closed fields $k$ over $S$ with\n$|k| > \\lambda(X)$, and $|k| \\geq \\lambda(Y)$ the map $X(k) \\to Y(k)$\nis surjective.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Points and geometric points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0488","source_file":"spaces-morphisms.tex","source_line":4648,"source_end_line":4680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4648-L4680","statement_sha256":"99578fe5ed56612bd6e3bb8761c8edfa1cd3824c24147fcd715e5167af361801","origin":"The Stacks Project","memory_eligible":false,"source_rank":11004,"rank":11004,"depth":45,"x":207.071,"y":1686.545,"cluster":"algebraic-spaces"},{"id":"stacks:06EF","tag":"06EF","title":"Points of finite type · Lemma 06EF","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. The following are equivalent: • There exists a morphism Spec(k) → X which is locally of finite type and represents x. • There exists a scheme U, a closed point u ∈ U, and an étale morphism φ : U → X such that φ(u) = x.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. The following are equivalent:\n\\begin{enumerate}\n\\item There exists a morphism $\\Spec(k) \\to X$\nwhich is locally of finite type and represents $x$.\n\\item There exists a scheme $U$, a closed point $u \\in U$, and an \\'etale\nmorphism $\\varphi : U \\to X$ such that $\\varphi(u) = x$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EF","source_file":"spaces-morphisms.tex","source_line":4803,"source_end_line":4813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4803-L4813","statement_sha256":"aa9932a926a096bac6609f820024eee60f2497352d7bd4aea65ac22b63985e8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11005,"rank":11005,"depth":10,"x":177.231,"y":1523.057,"cluster":"algebraic-spaces"},{"id":"stacks:06EG","tag":"06EG","title":"Points of finite type · Definition 06EG","summary":"Let S be a scheme. Let X be an algebraic space over S. We say a point x ∈ |X| is a finite type point if the equivalent conditions of Lemma [Tag 06EF] are satisfied. We denote X_ft-pts the set of finite type points of X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nWe say a point $x \\in |X|$ is a {\\it finite type point}\\footnote{This is a\nslight abuse of language as it would perhaps be more correct to say\n``locally finite type point''.} if the equivalent conditions of\nLemma \\ref{lemma-point-finite-type}\nare satisfied. We denote $X_{\\text{ft-pts}}$ the set of finite type points\nof $X$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Points of finite type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EG","source_file":"spaces-morphisms.tex","source_line":4843,"source_end_line":4852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4843-L4852","statement_sha256":"a6a10cdbeec031f4d7d904e4f58f813548838239db21d5395e1bc1ca9f42c795","origin":"The Stacks Project","memory_eligible":false,"source_rank":11006,"rank":11006,"depth":11,"x":330.937,"y":1626.834,"cluster":"algebraic-spaces"},{"id":"stacks:06EH","tag":"06EH","title":"Points of finite type · Lemma 06EH","summary":"Let S be a scheme. Let X be an algebraic space over S. We have X_ft-pts = ⋃_φ : U → X étale |φ|(U_0) where U_0 is the set of closed points of U. Here we may let U range over all schemes étale over X or over all affine schemes étale over X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We have\n$$\nX_{\\text{ft-pts}} =\n\\bigcup\\nolimits_{\\varphi : U \\to X\\text{ \\'etale }} |\\varphi|(U_0)\n$$\nwhere $U_0$ is the set of closed points of $U$.\nHere we may let $U$ range over all schemes \\'etale over $X$ or over all\naffine schemes \\'etale over $X$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EH","source_file":"spaces-morphisms.tex","source_line":4857,"source_end_line":4867,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4857-L4867","statement_sha256":"e1bcced755c6e9812230e28b5c90be1c26f26efb917310067376da3ea2615d42","origin":"The Stacks Project","memory_eligible":false,"source_rank":11007,"rank":11007,"depth":11,"x":133.848,"y":1637.543,"cluster":"algebraic-spaces"},{"id":"stacks:06EI","tag":"06EI","title":"Points of finite type · Lemma 06EI","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally of finite type, then f(X_ft-pts) ⊂ Y_ft-pts.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is locally of finite type, then\n$f(X_{\\text{ft-pts}}) \\subset Y_{\\text{ft-pts}}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EI","source_file":"spaces-morphisms.tex","source_line":4874,"source_end_line":4880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4874-L4880","statement_sha256":"f0ebfffd02ac61363141ac466a2728b326d989c4e0d11dcf345ca3a9964d00a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11008,"rank":11008,"depth":7,"x":270.771,"y":1517.635,"cluster":"algebraic-spaces"},{"id":"stacks:06EJ","tag":"06EJ","title":"Points of finite type · Lemma 06EJ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally of finite type and surjective, then f(X_ft-pts) = Y_ft-pts.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is locally of finite type and surjective, then\n$f(X_{\\text{ft-pts}}) = Y_{\\text{ft-pts}}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EJ","source_file":"spaces-morphisms.tex","source_line":4889,"source_end_line":4895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4889-L4895","statement_sha256":"a8bf7234d2aeeed3d00d527746582a1b28fd03ddc58e05db44af8ee95b0fff1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11009,"rank":11009,"depth":12,"x":266.225,"y":1683.993,"cluster":"algebraic-spaces"},{"id":"stacks:06EK","tag":"06EK","title":"Points of finite type · Lemma 06EK","summary":"Let S be a scheme. Let X be an algebraic space over S. For any locally closed subset T ⊂ |X| we have T not = ∅ ⇒ T ∩ X_ft-pts not = ∅. In particular, for any closed subset T ⊂ |X| we see that T ∩ X_ft-pts is dense in T.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nFor any locally closed subset $T \\subset |X|$ we have\n$$\nT \\not = \\emptyset\n\\Rightarrow\nT \\cap X_{\\text{ft-pts}} \\not = \\emptyset.\n$$\nIn particular, for any closed subset $T \\subset |X|$ we\nsee that $T \\cap X_{\\text{ft-pts}}$ is dense in $T$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EK","source_file":"spaces-morphisms.tex","source_line":4914,"source_end_line":4925,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4914-L4925","statement_sha256":"726cae9bf0f6472b2bff2c58925a3fbb55a67be44f0d1f24008e24c6fa1b1ef1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11010,"rank":11010,"depth":12,"x":135.604,"y":1558.559,"cluster":"algebraic-spaces"},{"id":"stacks:06EL","tag":"06EL","title":"Points of finite type · Lemma 06EL","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. The following are equivalent: • x is a finite type point, • there exists an algebraic space Z whose underlying topological space |Z| is a singleton, and a morphism f : Z → X which is locally of finite type such that (x) = |f|(|Z|), and • there exists an algebraic space Z and a morphism f : Z → X with the following properties: • there is a surjective étale morphism z : Spec(k) → Z where k is a field, • f…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. The following are equivalent:\n\\begin{enumerate}\n\\item $x$ is a finite type point,\n\\item there exists an algebraic space $Z$ whose underlying topological space\n$|Z|$ is a singleton, and a morphism $f : Z \\to X$ which is locally of finite\ntype such that $\\{x\\} = |f|(|Z|)$, and\n\\item there exists an algebraic space $Z$ and a morphism $f : Z \\to X$\nwith the following properties:\n\\begin{enumerate}\n\\item there is a surjective \\'etale morphism\n$z : \\Spec(k) \\to Z$ where $k$ is a field,\n\\item $f$ is locally of finite type,\n\\item $f$ is a monomorphism, and\n\\item $x = f(z)$.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EL","source_file":"spaces-morphisms.tex","source_line":4946,"source_end_line":4965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L4946-L4965","statement_sha256":"28a5be07dda1a128908738c7a1c354fd8a06038af9579c678aec93206239453f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11011,"rank":11011,"depth":52,"x":333.083,"y":1576.962,"cluster":"algebraic-spaces"},{"id":"stacks:0BAU","tag":"0BAU","title":"Nagata spaces · Lemma 0BAU","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If Y is Nagata and f locally of finite type then X is Nagata.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $Y$ is Nagata and $f$ locally of finite type then $X$ is Nagata.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Nagata spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAU","source_file":"spaces-morphisms.tex","source_line":5021,"source_end_line":5026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5021-L5026","statement_sha256":"718f4208d35b6a19f7a32557c888449b4b6071f10dc35b60a9ab882c533c1512","origin":"The Stacks Project","memory_eligible":false,"source_rank":11012,"rank":11012,"depth":34,"x":172.43,"y":1675.591,"cluster":"algebraic-spaces"},{"id":"stacks:0BAV","tag":"0BAV","title":"Nagata spaces · Lemma 0BAV","summary":"The following types of algebraic spaces are Nagata. • Any algebraic space locally of finite type over a Nagata scheme. • Any algebraic space locally of finite type over a field. • Any algebraic space locally of finite type over a Noetherian complete local ring. • Any algebraic space locally of finite type over Z. • Any algebraic space locally of finite type over a Dedekind ring of characteristic zero. • And so on.","statement_latex":"The following types of algebraic spaces are Nagata.\n\\begin{enumerate}\n\\item Any algebraic space locally of finite type over a Nagata scheme.\n\\item Any algebraic space locally of finite type over a field.\n\\item Any algebraic space locally of finite type over a\nNoetherian complete local ring.\n\\item Any algebraic space locally of finite type over $\\mathbf{Z}$.\n\\item Any algebraic space locally of finite type over a Dedekind ring of\ncharacteristic zero.\n\\item And so on.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Nagata spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAV","source_file":"spaces-morphisms.tex","source_line":5038,"source_end_line":5051,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5038-L5051","statement_sha256":"1b75674d345d61f8aba8be3457d6cf8b10579b7dcd9e643a1fdce034ba13b370","origin":"The Stacks Project","memory_eligible":false,"source_rank":11013,"rank":11013,"depth":35,"x":211.636,"y":1511.465,"cluster":"algebraic-spaces"},{"id":"stacks:03XJ","tag":"03XJ","title":"Quasi-finite morphisms · Definition 03XJ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is locally quasi-finite if the equivalent conditions of Lemma [Tag 03MJ] hold with P = locally quasi-finite. • Let x ∈ |X|. We say f is quasi-finite at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is locally quasi-finite. • A morphism of algebraic spaces f : X → Y is quasi-finite if it is locally quasi-finite and quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is\n{\\it locally quasi-finite} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target} hold with\n$\\mathcal{P} = \\text{locally quasi-finite}$.\n\\item Let $x \\in |X|$. We say $f$ is {\\it quasi-finite at $x$}\nif there exists an open neighbourhood $X' \\subset X$ of $x$ such\nthat $f|_{X'} : X' \\to Y$ is locally quasi-finite.\n\\item A morphism of algebraic spaces $f : X \\to Y$ is\n{\\it quasi-finite} if it is locally quasi-finite and quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XJ","source_file":"spaces-morphisms.tex","source_line":5084,"source_end_line":5099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5084-L5099","statement_sha256":"bbbbf0355a93dedd71004f7a7f6721f60e4ca91a477ce9f17999a81f5ee8a437","origin":"The Stacks Project","memory_eligible":false,"source_rank":11014,"rank":11014,"depth":46,"x":314.863,"y":1654.945,"cluster":"algebraic-spaces"},{"id":"stacks:0ABM","tag":"0ABM","title":"Quasi-finite morphisms · Lemma 0ABM","summary":"Let S be a scheme. Let f : X → Y and g : Y' → Y be morphisms of algebraic spaces over S. Denote f' : X' → Y' the base change of f by g. Denote g' : X' → X the projection. Assume f is locally of finite type. Let W ⊂ |X|, resp. W' ⊂ |X'| be the set of points where f, resp. f' is quasi-finite. • W ⊂ |X| and W' ⊂ |X'| are open, • W' = (g')^-1(W), i.e., formation of the locus where f is quasi-finite commutes with base change, • the base change of a locally quasi-finite…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y' \\to Y$ be morphisms\nof algebraic spaces over $S$. Denote $f' : X' \\to Y'$ the base change\nof $f$ by $g$. Denote $g' : X' \\to X$ the projection.\nAssume $f$ is locally of finite type.\nLet $W \\subset |X|$, resp.\\ $W' \\subset |X'|$\nbe the set of points where $f$, resp.\\ $f'$ is quasi-finite.\n\\begin{enumerate}\n\\item $W \\subset |X|$ and $W' \\subset |X'|$ are open,\n\\item $W' = (g')^{-1}(W)$, i.e., formation of the locus where\n$f$ is quasi-finite commutes with base change,\n\\item the base change of a locally quasi-finite morphism is\nlocally quasi-finite, and\n\\item the base change of a quasi-finite morphism is quasi-finite.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABM","source_file":"spaces-morphisms.tex","source_line":5106,"source_end_line":5122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5106-L5122","statement_sha256":"a7933e497a07a1763c5e9fd80664d04b7eac790e58434564345517c15d72d4f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11015,"rank":11015,"depth":48,"x":123.084,"y":1607.649,"cluster":"algebraic-spaces"},{"id":"stacks:03XK","tag":"03XK","title":"Quasi-finite morphisms · Lemma 03XK","summary":"The composition of quasi-finite morphisms is quasi-finite. The same holds for locally quasi-finite.","statement_latex":"The composition of quasi-finite morphisms is quasi-finite.\nThe same holds for locally quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XK","source_file":"spaces-morphisms.tex","source_line":5160,"source_end_line":5164,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5160-L5164","statement_sha256":"abc2dfc8958fb3500a455fe9ef19fb72c43900709f66b2e31cea9f862f1bef34","origin":"The Stacks Project","memory_eligible":false,"source_rank":11016,"rank":11016,"depth":29,"x":302.793,"y":1533.597,"cluster":"algebraic-spaces"},{"id":"stacks:03XL","tag":"03XL","title":"Quasi-finite morphisms · Lemma 03XL","summary":"A base change of a quasi-finite morphism is quasi-finite. The same holds for locally quasi-finite.","statement_latex":"A base change of a quasi-finite morphism is quasi-finite.\nThe same holds for locally quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XL","source_file":"spaces-morphisms.tex","source_line":5171,"source_end_line":5175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5171-L5175","statement_sha256":"6b7f28a633a30bb24769d252a34f80f0e27ef59ddb08929634b952f196f645b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11017,"rank":11017,"depth":49,"x":229.723,"y":1690.4,"cluster":"algebraic-spaces"},{"id":"stacks:06RW","tag":"06RW","title":"Quasi-finite morphisms · Lemma 06RW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces. Assume f is locally of finite type. The following are equivalent • f is locally quasi-finite, • for every morphism Spec(k) → Y where k is a field the space |X_k| is discrete. Here X_k = Spec(k) ×_Y X.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces.\nAssume $f$ is locally of finite type. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite,\n\\item for every morphism $\\Spec(k) \\to Y$ where $k$ is a field\nthe space $|X_k|$ is discrete. Here $X_k = \\Spec(k) \\times_Y X$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RW","source_file":"spaces-morphisms.tex","source_line":5189,"source_end_line":5198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5189-L5198","statement_sha256":"7699707f274b35abeefbff789c2443003883835dd90d849c369a7c45a8e4b873","origin":"The Stacks Project","memory_eligible":false,"source_rank":11018,"rank":11018,"depth":50,"x":157.403,"y":1533.087,"cluster":"algebraic-spaces"},{"id":"stacks:040Z","tag":"040Z","title":"Quasi-finite morphisms · Lemma 040Z","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is locally quasi-finite, • for every x ∈ |X| the morphism f is quasi-finite at x, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is locally quasi-finite, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is locally quasi-finite, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite,\n\\item for every $x \\in |X|$ the morphism $f$ is quasi-finite at $x$,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is locally quasi-finite,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally quasi-finite,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is locally quasi-finite,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis locally quasi-finite,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale\nthe top horizontal arrow is locally quasi-finite,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, and\n$U \\to X$ is surjective such that the top horizontal arrow is\nlocally quasi-finite, and\n\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,\nand $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that\neach morphism $X_{ij} \\to Y_i$ is locally quasi-finite.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040Z","source_file":"spaces-morphisms.tex","source_line":5252,"source_end_line":5292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5252-L5292","statement_sha256":"2b40b6e5ad0714b9d0434f62404d62b09bb48c0bcf47b5240d4d79f916d4f68c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11019,"rank":11019,"depth":0,"x":337.496,"y":1608.159,"cluster":"algebraic-spaces"},{"id":"stacks:03XM","tag":"03XM","title":"Quasi-finite morphisms · Lemma 03XM","summary":"An immersion is locally quasi-finite.","statement_latex":"An immersion is locally quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XM","source_file":"spaces-morphisms.tex","source_line":5298,"source_end_line":5301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5298-L5301","statement_sha256":"8c5be06d07a1e86abbf215f0f9bbb07b97ec106a7c69d0f11d40a9c179704268","origin":"The Stacks Project","memory_eligible":false,"source_rank":11020,"rank":11020,"depth":0,"x":144.05,"y":1655.058,"cluster":"algebraic-spaces"},{"id":"stacks:03XN","tag":"03XN","title":"Quasi-finite morphisms · Lemma 03XN","summary":"Let S be a scheme. Let X → Y → Z be morphisms of algebraic spaces over S. If X → Z is locally quasi-finite, then X → Y is locally quasi-finite.","statement_latex":"Let $S$ be a scheme.\nLet $X \\to Y \\to Z$ be morphisms of algebraic spaces over $S$.\nIf $X \\to Z$ is locally quasi-finite, then $X \\to Y$\nis locally quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XN","source_file":"spaces-morphisms.tex","source_line":5307,"source_end_line":5313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5307-L5313","statement_sha256":"08efa05b2e039dd8799595ecf9ae2c2fef57b986ebb6e3a3ade4b060284b0d92","origin":"The Stacks Project","memory_eligible":false,"source_rank":11021,"rank":11021,"depth":29,"x":249.129,"y":1510.503,"cluster":"algebraic-spaces"},{"id":"stacks:0ABN","tag":"0ABN","title":"Quasi-finite morphisms · Lemma 0ABN","summary":"Let S be a scheme. Let f : X → Y be a finite type morphism of algebraic spaces over S. Let y ∈ |Y|. There are at most finitely many points of |X| lying over y at which f is quasi-finite.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a finite type\nmorphism of algebraic spaces over $S$. Let $y \\in |Y|$.\nThere are at most finitely many\npoints of $|X|$ lying over $y$ at which $f$ is quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABN","source_file":"spaces-morphisms.tex","source_line":5330,"source_end_line":5336,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5330-L5336","statement_sha256":"694ef519e9f066f858d3500881273fc7465eaf29329e4c2abaef16ecfd8e4cb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11022,"rank":11022,"depth":49,"x":287.948,"y":1676.958,"cluster":"algebraic-spaces"},{"id":"stacks:0463","tag":"0463","title":"Quasi-finite morphisms · Lemma 0463","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally of finite type and a monomorphism, then f is separated and locally quasi-finite.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is locally of finite type and a monomorphism, then $f$\nis separated and locally quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0463","source_file":"spaces-morphisms.tex","source_line":5360,"source_end_line":5366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5360-L5366","statement_sha256":"630490c2bf71e91ebbc0ce381f69af30e051846c4065f7d52d1e9edb5fd3b7de","origin":"The Stacks Project","memory_eligible":false,"source_rank":11023,"rank":11023,"depth":45,"x":125.234,"y":1576.1,"cluster":"algebraic-spaces"},{"id":"stacks:03XP","tag":"03XP","title":"Morphisms of finite presentation · Definition 03XP","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is locally of finite presentation if the equivalent conditions of Lemma [Tag 03MJ] hold with P =\"locally of finite presentation\". • Let x ∈ |X|. We say f is of finite presentation at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is locally of finite presentation mean that there is an open neighbourhood X' ⊂ X such that f|_X' is of finite presentation.. • A…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it locally of finite presentation} if\nthe equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with $\\mathcal{P} =$``locally of finite presentation''.\n\\item Let $x \\in |X|$. We say $f$ is of {\\it finite presentation at $x$}\nif there exists an open neighbourhood $X' \\subset X$ of $x$ such\nthat $f|_{X'} : X' \\to Y$ is locally of finite\npresentation\\footnote{It seems awkward to use ``locally of finite presentation\nat $x$'', but the current terminology may be misleading in the sense that\n``of finite presentation at $x$'' does {\\bf not} mean that there is\nan open neighbourhood $X' \\subset X$ such that $f|_{X'}$ is of finite\npresentation.}.\n\\item A morphism of algebraic spaces $f : X \\to Y$ is\n{\\it of finite presentation}\nif it is locally of finite presentation, quasi-compact and\nquasi-separated.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XP","source_file":"spaces-morphisms.tex","source_line":5424,"source_end_line":5446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5424-L5446","statement_sha256":"172d7ecab9928ac79bfd98b451686646eb0b807b14a3210036b0cfd29368f8d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11024,"rank":11024,"depth":46,"x":326.609,"y":1558.118,"cluster":"algebraic-spaces"},{"id":"stacks:03XQ","tag":"03XQ","title":"Morphisms of finite presentation · Lemma 03XQ","summary":"The composition of morphisms of finite presentation is of finite presentation. The same holds for locally of finite presentation.","statement_latex":"The composition of morphisms of finite presentation is of finite presentation.\nThe same holds for locally of finite presentation.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XQ","source_file":"spaces-morphisms.tex","source_line":5452,"source_end_line":5456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5452-L5456","statement_sha256":"decfc9a0b3a2d4a064bc147fe71291aa328d1d06b0b5e7a518698dfcba7c5c17","origin":"The Stacks Project","memory_eligible":false,"source_rank":11025,"rank":11025,"depth":55,"x":192.389,"y":1685.823,"cluster":"algebraic-spaces"},{"id":"stacks:03XR","tag":"03XR","title":"Morphisms of finite presentation · Lemma 03XR","summary":"A base change of a morphism of finite presentation is of finite presentation. The same holds for locally of finite presentation.","statement_latex":"A base change of a morphism of finite presentation is of finite presentation.\nThe same holds for locally of finite presentation.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XR","source_file":"spaces-morphisms.tex","source_line":5466,"source_end_line":5470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5466-L5470","statement_sha256":"1e405b343b356a361b3c841b9edc4016f30a60916843f254bc971093c1458fcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11026,"rank":11026,"depth":17,"x":188.659,"y":1515.255,"cluster":"algebraic-spaces"},{"id":"stacks:0410","tag":"0410","title":"Morphisms of finite presentation · Lemma 0410","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is locally of finite presentation, • for every x ∈ |X| the morphism f is of finite presentation at x, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is locally of finite presentation, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is locally of finite presentation, • there exists a scheme V and a surjective étale…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item for every $x \\in |X|$ the morphism $f$ is of finite presentation at $x$,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is locally of finite presentation,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is locally of finite presentation,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is\nlocally of finite presentation,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis locally of finite presentation,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale\nthe top horizontal arrow is locally of finite presentation,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, and\n$U \\to X$ is surjective such that the top horizontal arrow is\nlocally of finite presentation, and\n\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,\nand $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that\neach morphism $X_{ij} \\to Y_i$ is locally of finite presentation.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0410","source_file":"spaces-morphisms.tex","source_line":5480,"source_end_line":5521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5480-L5521","statement_sha256":"ec3b22edce2af92ae087ff6e72b33a992f2fc94dc7bb7dacd965e9badf72f69f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11027,"rank":11027,"depth":0,"x":328.772,"y":1639.086,"cluster":"algebraic-spaces"},{"id":"stacks:0464","tag":"0464","title":"Morphisms of finite presentation · Lemma 0464","summary":"A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type.","statement_latex":"A morphism which is locally of finite presentation is locally of finite type.\nA morphism of finite presentation is of finite type.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0464","source_file":"spaces-morphisms.tex","source_line":5527,"source_end_line":5531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5527-L5531","statement_sha256":"92df17b1f929423d789954b1427ef46ce17eb80205bc17747f44a7084dc1c0e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11028,"rank":11028,"depth":46,"x":125.589,"y":1627.262,"cluster":"algebraic-spaces"},{"id":"stacks:04ZL","tag":"04ZL","title":"Morphisms of finite presentation · Lemma 04ZL","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is of finite presentation and Y is Noetherian, then X is Noetherian.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is of finite presentation and $Y$ is Noetherian,\nthen $X$ is Noetherian.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZL","source_file":"spaces-morphisms.tex","source_line":5545,"source_end_line":5551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5545-L5551","statement_sha256":"fda9d210d9de21a7bd2b71c1e0ac783427a5c1bbb9579657fabec48aa9a55d2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11029,"rank":11029,"depth":56,"x":285.145,"y":1520.542,"cluster":"algebraic-spaces"},{"id":"stacks:06G4","tag":"06G4","title":"Morphisms of finite presentation · Lemma 06G4","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • If Y is locally Noetherian and f locally of finite type then f is locally of finite presentation. • If Y is locally Noetherian and f of finite type and quasi-separated then f is of finite presentation.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $Y$ is locally Noetherian and $f$ locally of finite type\nthen $f$ is locally of finite presentation.\n\\item If $Y$ is locally Noetherian and $f$ of finite type and quasi-separated\nthen $f$ is of finite presentation.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06G4","source_file":"spaces-morphisms.tex","source_line":5572,"source_end_line":5582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5572-L5582","statement_sha256":"1d6436d27eaf49c18c5a48d9a48707aad4d117336198d18cddde84ad56757622","origin":"The Stacks Project","memory_eligible":false,"source_rank":11030,"rank":11030,"depth":46,"x":253.268,"y":1690.007,"cluster":"algebraic-spaces"},{"id":"stacks:06G5","tag":"06G5","title":"Morphisms of finite presentation · Lemma 06G5","summary":"Let S be a scheme. Let Y be an algebraic space over S which is quasi-compact and quasi-separated. If X is of finite presentation over Y, then X is quasi-compact and quasi-separated.","statement_latex":"Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$ which is\nquasi-compact and quasi-separated. If $X$ is of finite presentation over\n$Y$, then $X$ is quasi-compact and quasi-separated.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06G5","source_file":"spaces-morphisms.tex","source_line":5598,"source_end_line":5603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5598-L5603","statement_sha256":"9c4c94178d2fe1cfa8de0366711b6df3d00a27cd817f0809a773a56d75cfa0e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11031,"rank":11031,"depth":0,"x":140.337,"y":1546.759,"cluster":"algebraic-spaces"},{"id":"stacks:05WT","tag":"05WT","title":"Morphisms of finite presentation · Lemma 05WT","summary":"Let S be a scheme. Let f : X → Y and Y → Z be morphisms of algebraic spaces over S. If X is locally of finite presentation over Z, and Y is locally of finite type over Z, then f is locally of finite presentation.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$.\nIf $X$ is locally of finite presentation over $Z$, and\n$Y$ is locally of finite type over $Z$, then $f$ is locally\nof finite presentation.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WT","source_file":"spaces-morphisms.tex","source_line":5609,"source_end_line":5616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5609-L5616","statement_sha256":"d3a31f91ddf99d0418528d4ed9423ea496791313410ea045f5d74a951fa89dda","origin":"The Stacks Project","memory_eligible":false,"source_rank":11032,"rank":11032,"depth":19,"x":339.082,"y":1588.366,"cluster":"algebraic-spaces"},{"id":"stacks:084P","tag":"084P","title":"Morphisms of finite presentation · Lemma 084P","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S with diagonal Δ : X → X ×_Y X. If f is locally of finite type then Δ is locally of finite presentation. If f is quasi-separated and locally of finite type, then Δ is of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ with diagonal $\\Delta : X \\to X \\times_Y X$. If $f$ is locally of\nfinite type then $\\Delta$ is locally of finite presentation. If $f$ is\nquasi-separated and locally of finite type, then $\\Delta$ is of finite\npresentation.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084P","source_file":"spaces-morphisms.tex","source_line":5629,"source_end_line":5636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5629-L5636","statement_sha256":"b9ef4fa160b18913c28d496084d93f31fd6862de34884ae93bc8b10594cfcb04","origin":"The Stacks Project","memory_eligible":false,"source_rank":11033,"rank":11033,"depth":53,"x":158.822,"y":1670.571,"cluster":"algebraic-spaces"},{"id":"stacks:06CN","tag":"06CN","title":"Morphisms of finite presentation · Lemma 06CN","summary":"An open immersion of algebraic spaces is locally of finite presentation.","statement_latex":"An open immersion of algebraic spaces is locally of finite presentation.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CN","source_file":"spaces-morphisms.tex","source_line":5650,"source_end_line":5653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5650-L5653","statement_sha256":"f166e68160428f9ac8860edb25f48cf819076ef3af95dbb9397c547cba16922c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11034,"rank":11034,"depth":3,"x":225.728,"y":1507.447,"cluster":"algebraic-spaces"},{"id":"stacks:084Q","tag":"084Q","title":"Morphisms of finite presentation · Lemma 084Q","summary":"A closed immersion i : Z → X is of finite presentation if and only if the associated quasi-coherent sheaf of ideals I = Ker(O_X → i_*O_Z) is of finite type (as an O_X-module).","statement_latex":"A closed immersion $i : Z \\to X$ is of finite presentation if and only if\nthe associated quasi-coherent sheaf of ideals\n$\\mathcal{I} = \\Ker(\\mathcal{O}_X \\to i_*\\mathcal{O}_Z)$\nis of finite type (as an $\\mathcal{O}_X$-module).","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084Q","source_file":"spaces-morphisms.tex","source_line":5665,"source_end_line":5671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5665-L5671","statement_sha256":"cf3f5267e6eb69f56edbb497ead9262a5b77daaebdb6a7cc8c88c091ebd77356","origin":"The Stacks Project","memory_eligible":false,"source_rank":11035,"rank":11035,"depth":5,"x":307.686,"y":1665.914,"cluster":"algebraic-spaces"},{"id":"stacks:0ECW","tag":"0ECW","title":"Constructible sets · Lemma 0ECW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let E ⊂ |Y| be a subset. If E is étale locally constructible in Y, then f^-1(E) is étale locally constructible in X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $E \\subset |Y|$ be a subset.\nIf $E$ is \\'etale locally constructible in $Y$, then\n$f^{-1}(E)$ is \\'etale locally constructible in $X$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Constructible sets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECW","source_file":"spaces-morphisms.tex","source_line":5698,"source_end_line":5705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5698-L5705","statement_sha256":"1f49a27a8da462c78d75059472077264b018e3930c2d5834fcb68ecbfcdf98e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11036,"rank":11036,"depth":9,"x":119.559,"y":1595.471,"cluster":"algebraic-spaces"},{"id":"stacks:0ECX","tag":"0ECX","title":"Chevalley's Theorem · Theorem 0ECX","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-compact and locally of finite presentation. Then the image of every étale locally constructible subset of |X| is an étale locally constructible subset of |Y|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is quasi-compact and locally of finite presentation.\nThen the image of every \\'etale locally constructible subset of $|X|$ is\nan \\'etale locally constructible subset of $|Y|$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Constructible sets","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECX","source_file":"spaces-morphisms.tex","source_line":5719,"source_end_line":5726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5719-L5726","statement_sha256":"a79e7933161c17aee91dde3f479e0bb93c064941f57f44fc8fb1e51420ab87b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11037,"rank":11037,"depth":42,"x":315.196,"y":1540.592,"cluster":"algebraic-spaces"},{"id":"stacks:03ML","tag":"03ML","title":"Flat morphisms · Definition 03ML","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is flat if the equivalent conditions of Lemma [Tag 03MJ] with P =\"flat\". • Let x ∈ |X|. We say f is flat at x if the equivalent conditions of Lemma [Tag 04NC] hold with Q =\"induced map local rings is flat\". Note that the second part makes sense by Descent, Lemma [Tag 04ND].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it flat} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target} with\n$\\mathcal{P} =$``flat''.\n\\item Let $x \\in |X|$. We say $f$ is {\\it flat at $x$} if the\nequivalent conditions of\nLemma \\ref{lemma-local-source-target-at-point}\nhold with $\\mathcal{Q} =$``induced map local rings is flat''.\n\\end{enumerate}\nNote that the second part makes sense by\nDescent, Lemma \\ref{descent-lemma-flat-at-point}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ML","source_file":"spaces-morphisms.tex","source_line":5768,"source_end_line":5783,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5768-L5783","statement_sha256":"38e26469dc849a2cfe37c3c1da509074288aea82840e37f40a27a3cbb7e2000b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11038,"rank":11038,"depth":47,"x":214.932,"y":1692.274,"cluster":"algebraic-spaces"},{"id":"stacks:08EW","tag":"08EW","title":"Flat morphisms · Lemma 08EW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then f is flat if and only if f is flat at all points of |X|.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Then $f$ is flat if and only if $f$ is flat at all points of $|X|$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EW","source_file":"spaces-morphisms.tex","source_line":5788,"source_end_line":5792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5788-L5792","statement_sha256":"b54355faf4dcbffa3f658c772b3a41033d065587760c191bd1cd981be8f1555b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11039,"rank":11039,"depth":48,"x":166.821,"y":1523.305,"cluster":"algebraic-spaces"},{"id":"stacks:03MN","tag":"03MN","title":"Flat morphisms · Lemma 03MN","summary":"The composition of flat morphisms is flat.","statement_latex":"The composition of flat morphisms is flat.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MN","source_file":"spaces-morphisms.tex","source_line":5813,"source_end_line":5816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5813-L5816","statement_sha256":"1cef9b5636167763759bac6e58fd08ed733a3d6d6e3faac26fee58c85f9d5eef","origin":"The Stacks Project","memory_eligible":false,"source_rank":11040,"rank":11040,"depth":5,"x":338.41,"y":1620.733,"cluster":"algebraic-spaces"},{"id":"stacks:03MO","tag":"03MO","title":"Flat morphisms · Lemma 03MO","summary":"The base change of a flat morphism is flat.","statement_latex":"The base change of a flat morphism is flat.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MO","source_file":"spaces-morphisms.tex","source_line":5823,"source_end_line":5826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5823-L5826","statement_sha256":"b92b5be92432973f4ae3a890fc14c6e019d752c9702428d20ff93e9a6e64fae3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11041,"rank":11041,"depth":3,"x":133.257,"y":1646.289,"cluster":"algebraic-spaces"},{"id":"stacks:03MM","tag":"03MM","title":"Flat morphisms · Lemma 03MM","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is flat, • for every x ∈ |X| the morphism f is flat at x, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is flat, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is flat, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is flat, • there exists a scheme U and a surjective étale…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item for every $x \\in |X|$ the morphism $f$ is flat at $x$,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is flat,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is flat,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is flat,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis flat,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale\nthe top horizontal arrow is flat,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, and\n$U \\to X$ is surjective such that the top horizontal arrow is flat, and\n\\item there exists a Zariski coverings $Y = \\bigcup Y_i$ and\n$f^{-1}(Y_i) = \\bigcup X_{ij}$ such that\neach morphism $X_{ij} \\to Y_i$ is flat.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MM","source_file":"spaces-morphisms.tex","source_line":5833,"source_end_line":5872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5833-L5872","statement_sha256":"2774f7dde94e065ef76f1697f35321d5f5dc487f357c5c2d44633f5be77c824e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11042,"rank":11042,"depth":0,"x":264.159,"y":1510.853,"cluster":"algebraic-spaces"},{"id":"stacks:042S","tag":"042S","title":"Flat morphisms · Lemma 042S","summary":"A flat morphism locally of finite presentation is universally open.","statement_latex":"A flat morphism locally of finite presentation is universally open.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042S","source_file":"spaces-morphisms.tex","source_line":5878,"source_end_line":5881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5878-L5881","statement_sha256":"f84ddab678262b4e472730a6cb98f1b3051792337dd22c621cf28bc54682a941","origin":"The Stacks Project","memory_eligible":false,"source_rank":11043,"rank":11043,"depth":47,"x":276.564,"y":1685.232,"cluster":"algebraic-spaces"},{"id":"stacks:0413","tag":"0413","title":"Flat morphisms · Lemma 0413","summary":"Let S be a scheme. Let f : X → Y be a flat, quasi-compact, surjective morphism of algebraic spaces over S. A subset T ⊂ |Y| is open (resp. closed) if and only f^-1(|T|) is open (resp. closed) in |X|. In other words f is submersive, and in fact universally submersive.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a flat, quasi-compact, surjective morphism of\nalgebraic spaces over $S$.\nA subset $T \\subset |Y|$ is open (resp.\\ closed) if and only\n$f^{-1}(|T|)$ is open (resp.\\ closed) in $|X|$.\nIn other words $f$ is submersive, and in fact universally submersive.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0413","source_file":"spaces-morphisms.tex","source_line":5905,"source_end_line":5913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5905-L5913","statement_sha256":"b2c820d6a17db105680efcd0fdb4eaa342eed4cc5f6b9f65a5240d21139d8c04","origin":"The Stacks Project","memory_eligible":false,"source_rank":11044,"rank":11044,"depth":42,"x":126.986,"y":1563.524,"cluster":"algebraic-spaces"},{"id":"stacks:04NG","tag":"04NG","title":"Flat morphisms · Lemma 04NG","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let overlinex be a geometric point of X lying over the point x ∈ |X|. Let overliney = f ∘ overlinex. The following are equivalent • f is flat at x, and • the map on étale local rings O_Y, overliney → O_X, overlinex is flat.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\overline{x}$ be a geometric point of $X$ lying over the point\n$x \\in |X|$. Let $\\overline{y} = f \\circ \\overline{x}$. The following\nare equivalent\n\\begin{enumerate}\n\\item $f$ is flat at $x$, and\n\\item the map on \\'etale local rings\n$\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$\nis flat.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NG","source_file":"spaces-morphisms.tex","source_line":5945,"source_end_line":5958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L5945-L5958","statement_sha256":"073ed1b9bdf4c15104048082ad3c0fdc2a77b9eefb41b3822288bb3b1d81ba56","origin":"The Stacks Project","memory_eligible":false,"source_rank":11045,"rank":11045,"depth":53,"x":335.435,"y":1568.402,"cluster":"algebraic-spaces"},{"id":"stacks:073C","tag":"073C","title":"Flat morphisms · Lemma 073C","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then f is flat if and only if the morphism of sites (f_small, f^sharp) : (X_etale, O_X) → (Y_etale, O_Y) associated to f is flat.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThen $f$ is flat if and only if the morphism of sites\n$\n(f_{small}, f^\\sharp) :\n(X_\\etale, \\mathcal{O}_X)\n\\to\n(Y_\\etale, \\mathcal{O}_Y)\n$\nassociated to $f$ is flat.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/073C","source_file":"spaces-morphisms.tex","source_line":6002,"source_end_line":6014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6002-L6014","statement_sha256":"3cae84ded673af9821fd638605000bd6c9e88f0b156212a9321efce458522dc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11046,"rank":11046,"depth":54,"x":177.594,"y":1683.237,"cluster":"algebraic-spaces"},{"id":"stacks:089C","tag":"089C","title":"Flat morphisms · Lemma 089C","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Let F be a finite type quasi-coherent O_X-module with scheme theoretic support Z ⊂ X. If f is flat, then f^-1(Z) is the scheme theoretic support of f^*F.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces\nover $S$. Let $\\mathcal{F}$ be a finite type quasi-coherent\n$\\mathcal{O}_X$-module with scheme theoretic support $Z \\subset X$.\nIf $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of\n$f^*\\mathcal{F}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089C","source_file":"spaces-morphisms.tex","source_line":6027,"source_end_line":6034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6027-L6034","statement_sha256":"d663ce9b6a0970a3bc2fc66430e073c9b50ac5fc5f4c5e907fc31ddf0d1fa6d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11047,"rank":11047,"depth":56,"x":201.669,"y":1508.765,"cluster":"algebraic-spaces"},{"id":"stacks:089D","tag":"089D","title":"Flat morphisms · Lemma 089D","summary":"Let S be a scheme. Let f : X → Y be a flat morphism of algebraic spaces over S. Let V → Y be a quasi-compact open immersion. If V is scheme theoretically dense in Y, then f^-1V is scheme theoretically dense in X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a flat morphism of algebraic spaces over $S$.\nLet $V \\to Y$ be a quasi-compact open immersion. If $V$\nis scheme theoretically dense in $Y$, then $f^{-1}V$\nis scheme theoretically dense in $X$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089D","source_file":"spaces-morphisms.tex","source_line":6045,"source_end_line":6052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6045-L6052","statement_sha256":"c2f34da4c084510f17ef8b068cfd913f847435356986c5b178386e5a694ea216","origin":"The Stacks Project","memory_eligible":false,"source_rank":11048,"rank":11048,"depth":50,"x":324.384,"y":1651.267,"cluster":"algebraic-spaces"},{"id":"stacks:089E","tag":"089E","title":"Flat morphisms · Lemma 089E","summary":"Let S be a scheme. Let f : X → Y be a flat morphism of algebraic spaces over S. Let g : V → Y be a quasi-compact morphism of algebraic spaces. Let Z ⊂ Y be the scheme theoretic image of g and let Z' ⊂ X be the scheme theoretic image of the base change V ×_Y X → X. Then Z' = f^-1Z.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism of algebraic spaces\nover $S$. Let $g : V \\to Y$ be a quasi-compact morphism of algebraic spaces.\nLet $Z \\subset Y$ be the scheme theoretic image of $g$ and let $Z' \\subset X$\nbe the scheme theoretic image of the base change $V \\times_Y X \\to X$.\nThen $Z' = f^{-1}Z$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089E","source_file":"spaces-morphisms.tex","source_line":6064,"source_end_line":6071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6064-L6071","statement_sha256":"9422970cdcd941e2ab6e6fc01574eae3d74677bea8a574fddcc8f8bc85803f2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11049,"rank":11049,"depth":54,"x":119.029,"y":1615.773,"cluster":"algebraic-spaces"},{"id":"stacks:05VU","tag":"05VU","title":"Flat modules · Lemma 05VU","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf on X. Let x ∈ |X|. The following are equivalent • for some commutative diagram xymatrix U ar[d]_a ar[r]_h & V ar[d]^b X ar[r]^f & Y where U and V are schemes, a, b are étale, and u ∈ U mapping to x the module a^*F is flat at u over V, • the stalk F_overlinex is flat over the étale local ring O_Y, overliney where overlinex is any geometric point lying over x and…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $x \\in |X|$. The following are equivalent\n\\begin{enumerate}\n\\item for some commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwhere $U$ and $V$ are schemes, $a, b$ are \\'etale, and\n$u \\in U$ mapping to $x$ the module $a^*\\mathcal{F}$ is flat at $u$ over $V$,\n\\item the stalk $\\mathcal{F}_{\\overline{x}}$ is flat over\nthe \\'etale local ring $\\mathcal{O}_{Y, \\overline{y}}$\nwhere $\\overline{x}$ is any geometric point lying over\n$x$ and $\\overline{y} = f \\circ \\overline{x}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VU","source_file":"spaces-morphisms.tex","source_line":6113,"source_end_line":6133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6113-L6133","statement_sha256":"30de15dcf53877bdef7fcfb4587029c8e368db4229598f3d6ecce733669adde2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11050,"rank":11050,"depth":54,"x":299.235,"y":1525.303,"cluster":"algebraic-spaces"},{"id":"stacks:05VV","tag":"05VV","title":"Flat modules · Definition 05VV","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf on X. • Let x ∈ |X|. We say F is flat at x over Y if the equivalent conditions of Lemma [Tag 05VU] hold. • We say F is flat over Y if F is flat over Y at all x ∈ |X|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\n\\begin{enumerate}\n\\item Let $x \\in |X|$. We say $\\mathcal{F}$ is {\\it flat at $x$ over $Y$}\nif the equivalent conditions of\nLemma \\ref{lemma-flat-at-point}\nhold.\n\\item We say $\\mathcal{F}$ is {\\it flat over $Y$} if $\\mathcal{F}$ is\nflat over $Y$ at all $x \\in |X|$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VV","source_file":"spaces-morphisms.tex","source_line":6179,"source_end_line":6192,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6179-L6192","statement_sha256":"fbbc665d791f24f49d69e8be3f4e3801f0fc67fe98d3d90f45d7e5b20c77ed93","origin":"The Stacks Project","memory_eligible":false,"source_rank":11051,"rank":11051,"depth":55,"x":239.028,"y":1694.49,"cluster":"algebraic-spaces"},{"id":"stacks:05VW","tag":"05VW","title":"Flat modules · Lemma 05VW","summary":"Let S be a scheme. Let xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f Y' ar[r]^g & Y be a cartesian diagram of algebraic spaces over S. Let x' ∈ |X'| with image x ∈ |X|. Let F be a quasi-coherent sheaf on X and denote F' = (g')^*F. • If F is flat at x over Y then F' is flat at x' over Y'. • If g is flat at f'(x') and F' is flat at x' over Y', then F is flat at x over Y. In particular, if F is flat over Y, then F' is flat over Y'.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian diagram of algebraic spaces over $S$. Let $x' \\in |X'|$\nwith image $x \\in |X|$. Let $\\mathcal{F}$ be a quasi-coherent\nsheaf on $X$ and denote $\\mathcal{F}' = (g')^*\\mathcal{F}$.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is flat at $x$ over $Y$\nthen $\\mathcal{F}'$ is flat at $x'$ over $Y'$.\n\\item If $g$ is flat at $f'(x')$ and\n$\\mathcal{F}'$ is flat at $x'$ over $Y'$, then\n$\\mathcal{F}$ is flat at $x$ over $Y$.\n\\end{enumerate}\nIn particular, if $\\mathcal{F}$ is flat over $Y$, then\n$\\mathcal{F}'$ is flat over $Y'$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VW","source_file":"spaces-morphisms.tex","source_line":6199,"source_end_line":6220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6199-L6220","statement_sha256":"e0e1c104559b6d1a337f833ac9eab511b776aa649a17e3c490cd5d994575b51f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11052,"rank":11052,"depth":36,"x":147.249,"y":1535.363,"cluster":"algebraic-spaces"},{"id":"stacks:05VX","tag":"05VX","title":"Flat modules · Lemma 05VX","summary":"Let S be a scheme. Let X → Y → Z be morphisms of algebraic spaces over S. Let F be a quasi-coherent sheaf on X. Let x ∈ |X| with image y ∈ |Y|. • If F is flat at x over Y and Y is flat at y over Z, then F is flat at x over Z. • Let x : Spec(K) → X be a representative of x. If • F is flat at x over Y, • x^*F not = 0, and • F is flat at x over Z, then Y is flat at y over Z. • Let overlinex be a geometric point of X lying over x with image overliney in Y. If F_overlinex is a…","statement_latex":"Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic spaces\nover $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $x \\in |X|$ with image $y \\in |Y|$.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is flat at $x$ over $Y$ and\n$Y$ is flat at $y$ over $Z$, then $\\mathcal{F}$ is flat at\n$x$ over $Z$.\n\\item Let $x : \\Spec(K) \\to X$ be a representative of $x$. If\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat at $x$ over $Y$,\n\\item $x^*\\mathcal{F} \\not = 0$, and\n\\item $\\mathcal{F}$ is flat at $x$ over $Z$,\n\\end{enumerate}\nthen $Y$ is flat at $y$ over $Z$.\n\\item Let $\\overline{x}$ be a geometric point of $X$ lying over $x$\nwith image $\\overline{y}$ in $Y$. If $\\mathcal{F}_{\\overline{x}}$ is a\nfaithfully flat $\\mathcal{O}_{Y, \\overline{y}}$-module and\n$\\mathcal{F}$ is flat at $x$ over $Z$, then\n$Y$ is flat at $y$ over $Z$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VX","source_file":"spaces-morphisms.tex","source_line":6239,"source_end_line":6261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6239-L6261","statement_sha256":"c382df286c4674958490423ad1f12e3c2206c350f02ca5a890641fb58917842c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11053,"rank":11053,"depth":55,"x":343.146,"y":1600.708,"cluster":"algebraic-spaces"},{"id":"stacks:05VY","tag":"05VY","title":"Flat modules · Lemma 05VY","summary":"Let S be a scheme. Let f : X → Y, g : Y → Z be morphisms of algebraic spaces over S. Let G be a quasi-coherent sheaf on Y. Let x ∈ |X| with image y ∈ |Y|. If f is flat at x, then G flat over Z at y ⇔ f^*G flat over Z at x. In particular: If f is surjective and flat, then G is flat over Z, if and only if f^*G is flat over Z.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$, $g : Y \\to Z$ be morphisms of\nalgebraic spaces over $S$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$.\nLet $x \\in |X|$ with image $y \\in |Y|$.\nIf $f$ is flat at $x$, then\n$$\n\\mathcal{G}\\text{ flat over }Z\\text{ at }y\n\\Leftrightarrow\nf^*\\mathcal{G}\\text{ flat over }Z\\text{ at }x.\n$$\nIn particular: If $f$ is surjective and flat, then\n$\\mathcal{G}$ is flat over $Z$, if and only if\n$f^*\\mathcal{G}$ is flat over $Z$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VY","source_file":"spaces-morphisms.tex","source_line":6288,"source_end_line":6302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6288-L6302","statement_sha256":"0136a2d8eff098d24706142bab1747b8dd9964395910ec191454a2098827e3cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11054,"rank":11054,"depth":55,"x":145.889,"y":1663.763,"cluster":"algebraic-spaces"},{"id":"stacks:0CVU","tag":"0CVU","title":"Flat modules · Lemma 0CVU","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Assume f locally finite presentation, F of finite type, X = Supp(F), and F flat over Y. Then f is universally open.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $f$ locally finite presentation, $\\mathcal{F}$ of\nfinite type, $X = \\text{Supp}(\\mathcal{F})$, and\n$\\mathcal{F}$ flat over $Y$. Then $f$ is universally open.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Flat modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVU","source_file":"spaces-morphisms.tex","source_line":6322,"source_end_line":6330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6322-L6330","statement_sha256":"7af722a6c86e7c7e39e793ddcac99ce928d0cb028f2ddd6b8c74018b23e60f31","origin":"The Stacks Project","memory_eligible":false,"source_rank":11055,"rank":11055,"depth":18,"x":240.76,"y":1505.133,"cluster":"algebraic-spaces"},{"id":"stacks:06QS","tag":"06QS","title":"Generic flatness · Proposition 06QS","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf of O_X-modules. Assume • Y is reduced, • f is of finite type, and • F is a finite type O_X-module. Then there exists an open dense subspace W ⊂ Y such that the base change X_W → W of f is flat, locally of finite presentation, and quasi-compact and such that F|_X_W is flat over W and of finite presentation over O_X_W.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.\nAssume\n\\begin{enumerate}\n\\item $Y$ is reduced,\n\\item $f$ is of finite type, and\n\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.\n\\end{enumerate}\nThen there exists an open dense subspace $W \\subset Y$ such that\nthe base change $X_W \\to W$ of $f$ is flat, locally of finite presentation, and\nquasi-compact and such that $\\mathcal{F}|_{X_W}$ is flat over $W$ and of\nfinite presentation over $\\mathcal{O}_{X_W}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Generic flatness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QS","source_file":"spaces-morphisms.tex","source_line":6353,"source_end_line":6368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6353-L6368","statement_sha256":"1230566dd6ba72398e8b2c4ffbc2fb0d4daa1aa0e0b8db99010217fd2a3e4228","origin":"The Stacks Project","memory_eligible":false,"source_rank":11056,"rank":11056,"depth":43,"x":298.444,"y":1676.158,"cluster":"algebraic-spaces"},{"id":"stacks:06QT","tag":"06QT","title":"Generic flatness · Proposition 06QT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf of O_X-modules. Assume • Y is reduced, • f is quasi-separated, • f is of finite type, and • F is a finite type O_X-module. Then there exists an open dense subspace W ⊂ Y such that the base change X_W → W of f is flat and of finite presentation and such that F|_X_W is flat over W and of finite presentation over O_X_W.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.\nAssume\n\\begin{enumerate}\n\\item $Y$ is reduced,\n\\item $f$ is quasi-separated,\n\\item $f$ is of finite type, and\n\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.\n\\end{enumerate}\nThen there exists an open dense subspace $W \\subset Y$ such that\nthe base change $X_W \\to W$ of $f$ is flat and of finite presentation\nand such that $\\mathcal{F}|_{X_W}$ is flat over $W$ and of\nfinite presentation over $\\mathcal{O}_{X_W}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Generic flatness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QT","source_file":"spaces-morphisms.tex","source_line":6435,"source_end_line":6451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6435-L6451","statement_sha256":"64eeb0174a44a721ccd929d01fe2b281fe778220d8b563aad33cd06d70e5d956","origin":"The Stacks Project","memory_eligible":false,"source_rank":11057,"rank":11057,"depth":44,"x":118.142,"y":1582.656,"cluster":"algebraic-spaces"},{"id":"stacks:04NM","tag":"04NM","title":"Relative dimension · Definition 04NM","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let x ∈ |X|. Let d, r ∈ (0, 1, 2, …, ∞). • We say the dimension of the local ring of the fibre of f at x is d if the equivalent conditions of Lemma [Tag 04NC] hold for the property P_d described in Descent, Lemma [Tag 04NJ]. • We say the transcendence degree of x/f(x) is r if the equivalent conditions of Lemma [Tag 04NC] hold for the property P_r described in Descent, Lemma [Tag 04NK]. • We say f…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $x \\in |X|$.\nLet $d, r \\in \\{0, 1, 2, \\ldots, \\infty\\}$.\n\\begin{enumerate}\n\\item We say the\n{\\it dimension of the local ring of the fibre of $f$ at $x$} is $d$\nif the equivalent conditions of\nLemma \\ref{lemma-local-source-target-at-point}\nhold for the property\n$\\mathcal{P}_d$ described in\nDescent, Lemma \\ref{descent-lemma-dimension-local-ring-fibre}.\n\\item We say the\n{\\it transcendence degree of $x/f(x)$} is $r$\nif the equivalent conditions of\nLemma \\ref{lemma-local-source-target-at-point}\nhold for the property\n$\\mathcal{P}_r$ described in\nDescent, Lemma \\ref{descent-lemma-transcendence-degree-at-point}.\n\\item We say\n{\\it $f$ has relative dimension $d$ at $x$}\nif the equivalent conditions of\nLemma \\ref{lemma-local-source-target-at-point}\nhold for the property\n$\\mathcal{P}_d$ described in\nDescent, Lemma \\ref{descent-lemma-dimension-at-point}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NM","source_file":"spaces-morphisms.tex","source_line":6483,"source_end_line":6512,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6483-L6512","statement_sha256":"e526ce90719d10f67df0602dc9f89fc8f8f30bcfcd0daaefd66d94354829b1e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11058,"rank":11058,"depth":47,"x":326.554,"y":1549.253,"cluster":"algebraic-spaces"},{"id":"stacks:06LR","tag":"06LR","title":"Relative dimension · Definition 06LR","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let d ∈ (0, 1, 2, …). • We say f has relative dimension ≤ d if f has relative dimension ≤ d at all x ∈ |X|. • We say f has relative dimension d if f has relative dimension d at all x ∈ |X|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $d \\in \\{0, 1, 2, \\ldots\\}$.\n\\begin{enumerate}\n\\item We say $f$ has {\\it relative dimension $\\leq d$} if\n$f$ has relative dimension $\\leq d$ at all $x \\in |X|$.\n\\item We say $f$ has {\\it relative dimension $d$} if\n$f$ has relative dimension $d$ at all $x \\in |X|$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative dimension","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LR","source_file":"spaces-morphisms.tex","source_line":6548,"source_end_line":6559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6548-L6559","statement_sha256":"62b35403fdbe44098c81e88e679866e01aa96ba9da824fa91a9e1b0f0a7f1f26","origin":"The Stacks Project","memory_eligible":false,"source_rank":11059,"rank":11059,"depth":0,"x":199.573,"y":1692.326,"cluster":"algebraic-spaces"},{"id":"stacks:06RX","tag":"06RX","title":"Relative dimension · Lemma 06RX","summary":"Let S be a scheme. Let X → Y → Z be morphisms of algebraic spaces over S. Let x ∈ |X| and let y ∈ |Y|, z ∈ |Z| be the images. Assume X → Y is locally quasi-finite and Y → Z locally of finite type. Then the transcendence degree of x/z is equal to the transcendence degree of y/z.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic\nspaces over $S$. Let $x \\in |X|$ and let $y \\in |Y|$, $z \\in |Z|$\nbe the images. Assume $X \\to Y$ is locally quasi-finite and $Y \\to Z$\nlocally of finite type. Then the transcendence degree of $x/z$\nis equal to the transcendence degree of $y/z$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RX","source_file":"spaces-morphisms.tex","source_line":6565,"source_end_line":6572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6565-L6572","statement_sha256":"3c2c3a1a86d5e5457477739a9b456d1a803440b4c1bdaf44dc41ae01cf00ce10","origin":"The Stacks Project","memory_eligible":false,"source_rank":11060,"rank":11060,"depth":27,"x":178.124,"y":1514.546,"cluster":"algebraic-spaces"},{"id":"stacks:0ECY","tag":"0ECY","title":"Relative dimension · Lemma 0ECY","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally of finite type, Y is Jacobson (Properties of Spaces, Remark [Tag 03E7]), and x ∈ |X| is a finite type point of X, then the transcendence degree of x/f(x) is 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. If $f$ is locally of finite type, $Y$ is Jacobson\n(Properties of Spaces, Remark\n\\ref{spaces-properties-remark-list-properties-local-etale-topology}),\nand $x \\in |X|$ is a finite type point of $X$,\nthen the transcendence degree of $x/f(x)$ is $0$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECY","source_file":"spaces-morphisms.tex","source_line":6594,"source_end_line":6602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6594-L6602","statement_sha256":"7fd077f684b1ddc1f1bba7bdee4e5c49835419a6f241989b442cf23835e32a9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11061,"rank":11061,"depth":13,"x":337.109,"y":1633.619,"cluster":"algebraic-spaces"},{"id":"stacks:0AFH","tag":"0AFH","title":"Relative dimension · Lemma 0AFH","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian algebraic spaces over S which is flat, locally of finite type and of relative dimension d. For every point x in |X| with image y in |Y| we have dim_x(X) = dim_y(Y) + d.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian\nalgebraic spaces over $S$ which is flat, locally of finite type and of\nrelative dimension $d$. For every point $x$ in $|X|$ with image\n$y$ in $|Y|$ we have $\\dim_x(X) = \\dim_y(Y) + d$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFH","source_file":"spaces-morphisms.tex","source_line":6617,"source_end_line":6623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6617-L6623","statement_sha256":"57280845084c978133cc02a2194fd8932ed15cf34baa8e63f383b39687a7e63d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11062,"rank":11062,"depth":47,"x":123.849,"y":1636.031,"cluster":"algebraic-spaces"},{"id":"stacks:04NQ","tag":"04NQ","title":"Morphisms and dimensions of fibres · Lemma 04NQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let x ∈ |X|. Assume f is locally of finite type. Then we have relative dimension of f at x = dimension of local ring of the fibre of f at x + transcendence degree of x/f(x) where the notation is as in Definition [Tag 04NM].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $x \\in |X|$.\nAssume $f$ is locally of finite type.\nThen we have\n$$\n\\begin{matrix}\n\\text{relative dimension of }f\\text{ at }x \\\\\n= \\\\\n\\text{dimension of local ring of the fibre of }f\\text{ at }x \\\\\n+ \\\\\n\\text{transcendence degree of }x/f(x)\n\\end{matrix}\n$$\nwhere the notation is as in\nDefinition \\ref{definition-dimension-fibre}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NQ","source_file":"spaces-morphisms.tex","source_line":6649,"source_end_line":6667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6649-L6667","statement_sha256":"adb9fad6cff07d6e77929a5a171bd86ec7f942f7423e82735cfbbb461c0e5dd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11063,"rank":11063,"depth":48,"x":279.362,"y":1513.088,"cluster":"algebraic-spaces"},{"id":"stacks:04NR","tag":"04NR","title":"Morphisms and dimensions of fibres · Lemma 04NR","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of algebraic spaces over S. Let x ∈ |X| and set y = f(x). Assume f and g locally of finite type. Then • relative dimension of g ∘ f at x ≤ relative dimension of f at x + relative dimension of g at y • equality holds in (1) if for some morphism Spec(k) → Z from the spectrum of a field in the class of g(f(x)) = g(y) the morphism X_k → Y_k is flat at x, for example if f is flat at x, • transcendence degree of…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.\nLet $x \\in |X|$ and set $y = f(x)$.\nAssume $f$ and $g$ locally of finite type.\nThen\n\\begin{enumerate}\n\\item\n$$\n\\begin{matrix}\n\\text{relative dimension of }g \\circ f\\text{ at }x \\\\\n\\leq \\\\\n\\text{relative dimension of }f\\text{ at }x \\\\\n+ \\\\\n\\text{relative dimension of }g\\text{ at }y\n\\end{matrix}\n$$\n\\item equality holds in (1) if for some morphism $\\Spec(k) \\to Z$\nfrom the spectrum of a field in the class of $g(f(x)) = g(y)$\nthe morphism $X_k \\to Y_k$ is flat at $x$, for example if $f$ is flat at $x$,\n\\item\n$$\n\\begin{matrix}\n\\text{transcendence degree of }x/g(f(x)) \\\\\n= \\\\\n\\text{transcendence degree of }x/f(x) \\\\\n+ \\\\\n\\text{transcendence degree of }f(x)/g(f(x))\n\\end{matrix}\n$$\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NR","source_file":"spaces-morphisms.tex","source_line":6677,"source_end_line":6709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6677-L6709","statement_sha256":"fb1f649c1d7df2060322242ecbbf8e2376db78d310608b152dc52fedd3087bc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11064,"rank":11064,"depth":25,"x":263.536,"y":1692.213,"cluster":"algebraic-spaces"},{"id":"stacks:04NS","tag":"04NS","title":"Morphisms and dimensions of fibres · Lemma 04NS","summary":"Let S be a scheme. Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a fibre product diagram of algebraic spaces over S. Let x' ∈ |X'|. Set x = g'(x'). Assume f locally of finite type. Then • relative dimension of f at x = relative dimension of f' at x' • we have dimension of local ring of the fibre of f' at x' - dimension of local ring of the fibre of f at x = transcendence degree of x/f(x) - transcendence degree of x'/f'(x') and the common value is ≥ 0, •…","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a fibre product diagram of algebraic spaces over $S$.\nLet $x' \\in |X'|$. Set $x = g'(x')$. Assume $f$ locally of finite type.\nThen\n\\begin{enumerate}\n\\item\n$$\n\\begin{matrix}\n\\text{relative dimension of }f\\text{ at }x \\\\\n= \\\\\n\\text{relative dimension of }f'\\text{ at }x'\n\\end{matrix}\n$$\n\\item we have\n$$\n\\begin{matrix}\n\\text{dimension of local ring of the fibre of }f'\\text{ at }x' \\\\\n- \\\\\n\\text{dimension of local ring of the fibre of }f\\text{ at }x \\\\\n= \\\\\n\\text{transcendence degree of }x/f(x) \\\\\n- \\\\\n\\text{transcendence degree of }x'/f'(x')\n\\end{matrix}\n$$\nand the common value is $\\geq 0$,\n\\item given $x$ and $y' \\in |Y'|$ mapping to the same $y \\in |Y|$\nthere exists a choice of $x'$ such that the integer in (2) is $0$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NS","source_file":"spaces-morphisms.tex","source_line":6728,"source_end_line":6765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6728-L6765","statement_sha256":"295fbbb8672df79c43d1cbee511d4e4358129d334d79b203baf5b2742dc4f40e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11065,"rank":11065,"depth":27,"x":130.99,"y":1550.971,"cluster":"algebraic-spaces"},{"id":"stacks:04NT","tag":"04NT","title":"Morphisms and dimensions of fibres · Lemma 04NT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let n ≥ 0. Assume f is locally of finite type. The set W_n = (x ∈ |X| such that the relative dimension of f at x ≤ n) is open in |X|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $n \\geq 0$. Assume $f$ is locally of finite type.\nThe set\n$$\nW_n = \\{x \\in |X|\n\\text{ such that the relative dimension of }f\\text{ at } x \\leq n\\}\n$$\nis open in $|X|$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NT","source_file":"spaces-morphisms.tex","source_line":6787,"source_end_line":6798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6787-L6798","statement_sha256":"2ff240c2036edee53d44f339bd52ace08cd44101a3692a733c4a599bab08b8a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11066,"rank":11066,"depth":32,"x":342.578,"y":1579.948,"cluster":"algebraic-spaces"},{"id":"stacks:04NU","tag":"04NU","title":"Morphisms and dimensions of fibres · Lemma 04NU","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S Let n ≥ 0. Assume f is locally of finite presentation. The open W_n = (x ∈ |X| such that the relative dimension of f at x ≤ n) of Lemma [Tag 04NT] is retrocompact in |X|. (See Topology, Definition [Tag 005A].)","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nLet $n \\geq 0$. Assume $f$ is locally of finite presentation.\nThe open\n$$\nW_n = \\{x \\in |X|\n\\text{ such that the relative dimension of }f\\text{ at } x \\leq n\\}\n$$\nof Lemma \\ref{lemma-openness-bounded-dimension-fibres}\nis retrocompact in $|X|$. (See\nTopology, Definition \\ref{topology-definition-quasi-compact}.)","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NU","source_file":"spaces-morphisms.tex","source_line":6820,"source_end_line":6833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6820-L6833","statement_sha256":"ecb62c657f40d4af778bdadef1255fbe3c2001d8b9a79c05819aebf3d6ed9552","origin":"The Stacks Project","memory_eligible":false,"source_rank":11067,"rank":11067,"depth":33,"x":163.028,"y":1678.765,"cluster":"algebraic-spaces"},{"id":"stacks:04NV","tag":"04NV","title":"Morphisms and dimensions of fibres · Lemma 04NV","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type. Then f is locally quasi-finite if and only if f has relative dimension 0 at each x ∈ |X|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is locally of finite type.\nThen $f$ is locally quasi-finite if and only if $f$ has relative\ndimension $0$ at each $x \\in |X|$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NV","source_file":"spaces-morphisms.tex","source_line":6859,"source_end_line":6866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6859-L6866","statement_sha256":"acdef310640b6492432e3977dd0edfbfd268e99ebe5bf4ad1ed0727c26538bf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11068,"rank":11068,"depth":28,"x":216.026,"y":1503.798,"cluster":"algebraic-spaces"},{"id":"stacks:04NW","tag":"04NW","title":"Morphisms and dimensions of fibres · Lemma 04NW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type. Then there exists a canonical open subspace X' ⊂ X such that f|_X' : X' → Y is locally quasi-finite, and such that the relative dimension of f at any x ∈ |X|, x not ∈ |X'| is ≥ 1. Formation of X' commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is locally of finite type.\nThen there exists a canonical open subspace $X' \\subset X$ such that\n$f|_{X'} : X' \\to Y$ is locally quasi-finite, and such that the\nrelative dimension of $f$ at any $x \\in |X|$, $x \\not \\in |X'|$ is\n$\\geq 1$. Formation of $X'$ commutes with arbitrary base change.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NW","source_file":"spaces-morphisms.tex","source_line":6887,"source_end_line":6896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6887-L6896","statement_sha256":"1c49559dfb4db99e6e63cd717609f181487696c9f197542dafba86e01c513f11","origin":"The Stacks Project","memory_eligible":false,"source_rank":11069,"rank":11069,"depth":33,"x":317.777,"y":1663.088,"cluster":"algebraic-spaces"},{"id":"stacks:06LS","tag":"06LS","title":"Morphisms and dimensions of fibres · Lemma 06LS","summary":"Let S be a scheme. Consider a cartesian diagram xymatrix X ar[d] & F ar[l]^p ar[d] Y & Spec(k) ar[l] where X → Y is a morphism of algebraic spaces over S which is locally of finite type and where k is a field over S. Let z ∈ |F| be such that dim_z(F) = 0. Then, after replacing X by an open subspace containing p(z), the morphism X → Y is locally quasi-finite.","statement_latex":"Let $S$ be a scheme. Consider a cartesian diagram\n$$\n\\xymatrix{\nX \\ar[d] & F \\ar[l]^p \\ar[d] \\\\\nY & \\Spec(k) \\ar[l]\n}\n$$\nwhere $X \\to Y$ is a morphism of algebraic spaces over $S$ which is\nlocally of finite type and where $k$ is a field over $S$.\nLet $z \\in |F|$ be such that $\\dim_z(F) = 0$. Then, after replacing $X$\nby an open subspace containing $p(z)$, the morphism\n$$\nX \\longrightarrow Y\n$$\nis locally quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Morphisms and dimensions of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LS","source_file":"spaces-morphisms.tex","source_line":6905,"source_end_line":6922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6905-L6922","statement_sha256":"ac26bfec5d6ce8494cb9f8e7db7dcac6ee90b83a670f483642cd2837ba2b7b63","origin":"The Stacks Project","memory_eligible":false,"source_rank":11070,"rank":11070,"depth":34,"x":114.397,"y":1603.291,"cluster":"algebraic-spaces"},{"id":"stacks:0BAX","tag":"0BAX","title":"The dimension formula · Lemma 0BAX","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume Y is locally Noetherian and f locally of finite type. Let x ∈ |X| with image y ∈ |Y|. Then we have & the dimension of the local ring of X at x ≤ & the dimension of the local ring of Y at y + E - & the transcendence degree of x/y Here E is the maximum of the transcendence degrees of xi/f(xi) where xi ∈ |X| runs over the points specializing to x at which the local ring of X has dimension 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume $Y$ is locally Noetherian and $f$ locally\nof finite type. Let $x \\in |X|$ with image $y \\in |Y|$.\nThen we have\n\\begin{align*}\n& \\text{the dimension of the local ring of }X\\text{ at }x \\leq \\\\\n& \\text{the dimension of the local ring of }Y\\text{ at }y + E - \\\\\n& \\text{ the transcendence degree of }x/y\n\\end{align*}\nHere $E$ is the maximum of the transcendence degrees of $\\xi/f(\\xi)$\nwhere $\\xi \\in |X|$ runs over the points specializing to $x$ at\nwhich the local ring of $X$ has dimension $0$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"The dimension formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAX","source_file":"spaces-morphisms.tex","source_line":6947,"source_end_line":6961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6947-L6961","statement_sha256":"ebf600da92a3098964449daa245f6c2ff0624b855ad0e71ec752833e3e6b9bc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11071,"rank":11071,"depth":48,"x":312.7,"y":1531.893,"cluster":"algebraic-spaces"},{"id":"stacks:0BAY","tag":"0BAY","title":"The dimension formula · Lemma 0BAY","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume Y is locally Noetherian and f is locally of finite type. Then dim(X) ≤ dim(Y) + E where E is the supremum of the transcendence degrees of xi/f(xi) where xi runs through the points at which the local ring of X has dimension 0.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume\n$Y$ is locally Noetherian and $f$ is locally of finite type.\nThen\n$$\n\\dim(X) \\leq \\dim(Y) + E\n$$\nwhere $E$ is the supremum of the transcendence degrees of\n$\\xi/f(\\xi)$ where $\\xi$ runs through the points at\nwhich the local ring of $X$ has dimension $0$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"The dimension formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BAY","source_file":"spaces-morphisms.tex","source_line":6988,"source_end_line":7000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L6988-L7000","statement_sha256":"117f35c0e29209dfbec59b70edbe64cca007540702a967793ca5cfff4f8954e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11072,"rank":11072,"depth":49,"x":223.78,"y":1697.268,"cluster":"algebraic-spaces"},{"id":"stacks:03Z7","tag":"03Z7","title":"Syntomic morphisms · Definition 03Z7","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is syntomic if the equivalent conditions of Lemma [Tag 03MJ] hold with P =\"syntomic\". • Let x ∈ |X|. We say f is syntomic at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is syntomic.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it syntomic} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with $\\mathcal{P} =$``syntomic''.\n\\item Let $x \\in |X|$. We say $f$ is {\\it syntomic at $x$} if\nthere exists an open neighbourhood $X' \\subset X$ of $x$ such\nthat $f|_{X'} : X' \\to Y$ is syntomic.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Syntomic morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Z7","source_file":"spaces-morphisms.tex","source_line":7030,"source_end_line":7042,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7030-L7042","statement_sha256":"2ac5d800a212360df2a3d6500c5fbc550d576652993395e7e2470d8e25487f06","origin":"The Stacks Project","memory_eligible":false,"source_rank":11073,"rank":11073,"depth":46,"x":156.275,"y":1524.654,"cluster":"algebraic-spaces"},{"id":"stacks:03Z8","tag":"03Z8","title":"Syntomic morphisms · Lemma 03Z8","summary":"The composition of syntomic morphisms is syntomic.","statement_latex":"The composition of syntomic morphisms is syntomic.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Z8","source_file":"spaces-morphisms.tex","source_line":7044,"source_end_line":7047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7044-L7047","statement_sha256":"7e6ae00943a36e89b4e8e664c990f43053bb9a2d73f6005636c0b31e75a91a7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11074,"rank":11074,"depth":38,"x":345.097,"y":1613.743,"cluster":"algebraic-spaces"},{"id":"stacks:03Z9","tag":"03Z9","title":"Syntomic morphisms · Lemma 03Z9","summary":"The base change of a syntomic morphism is syntomic.","statement_latex":"The base change of a syntomic morphism is syntomic.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Z9","source_file":"spaces-morphisms.tex","source_line":7054,"source_end_line":7057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7054-L7057","statement_sha256":"a08291c2f469d1e37fe74a6ab283c9b7c9cee1c80e61932d889aa1fd3e9c88f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11075,"rank":11075,"depth":38,"x":133.959,"y":1655.243,"cluster":"algebraic-spaces"},{"id":"stacks:03ZA","tag":"03ZA","title":"Syntomic morphisms · Lemma 03ZA","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is syntomic, • for every x ∈ |X| the morphism f is syntomic at x, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is syntomic, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is syntomic, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is a syntomic morphism, • there exists a…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is syntomic,\n\\item for every $x \\in |X|$ the morphism $f$ is syntomic at $x$,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is syntomic,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is syntomic,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is a syntomic morphism,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis syntomic,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale\nthe top horizontal arrow is syntomic,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, and\n$U \\to X$ is surjective such that the top horizontal arrow is syntomic, and\n\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,\nand $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that\neach morphism $X_{ij} \\to Y_i$ is syntomic.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZA","source_file":"spaces-morphisms.tex","source_line":7064,"source_end_line":7103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7064-L7103","statement_sha256":"c98c4edeceadff9627f9bb67415925fa5da2078c05faa94b9e2e0b9d736a8d29","origin":"The Stacks Project","memory_eligible":false,"source_rank":11076,"rank":11076,"depth":0,"x":256.426,"y":1504.652,"cluster":"algebraic-spaces"},{"id":"stacks:0DEY","tag":"0DEY","title":"Syntomic morphisms · Lemma 0DEY","summary":"A syntomic morphism is locally of finite presentation.","statement_latex":"A syntomic morphism is locally of finite presentation.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEY","source_file":"spaces-morphisms.tex","source_line":7109,"source_end_line":7112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7109-L7112","statement_sha256":"5500725fe7c5bede429755d81148587bbd6f7e79a7ae97f4902036586e4f5a1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11077,"rank":11077,"depth":1,"x":287.26,"y":1685.408,"cluster":"algebraic-spaces"},{"id":"stacks:0DEZ","tag":"0DEZ","title":"Syntomic morphisms · Lemma 0DEZ","summary":"A syntomic morphism is flat.","statement_latex":"A syntomic morphism is flat.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEZ","source_file":"spaces-morphisms.tex","source_line":7119,"source_end_line":7122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7119-L7122","statement_sha256":"e4c837523da7554db0d37dd511766f900f47b318fb9ce04e2f37e6d79076fda1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11078,"rank":11078,"depth":1,"x":118.959,"y":1569.475,"cluster":"algebraic-spaces"},{"id":"stacks:0DF0","tag":"0DF0","title":"Syntomic morphisms · Lemma 0DF0","summary":"A syntomic morphism is universally open.","statement_latex":"A syntomic morphism is universally open.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Syntomic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DF0","source_file":"spaces-morphisms.tex","source_line":7129,"source_end_line":7132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7129-L7132","statement_sha256":"1497b83dc1319d91a4b5f12fd32016ab26187e4b18501de3e7ec2e1c06fd5bbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":11079,"rank":11079,"depth":48,"x":336.557,"y":1559.452,"cluster":"algebraic-spaces"},{"id":"stacks:03ZC","tag":"03ZC","title":"Smooth morphisms · Definition 03ZC","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is smooth if the equivalent conditions of Lemma [Tag 03MJ] hold with P =\"smooth\". • Let x ∈ |X|. We say f is smooth at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is smooth.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it smooth} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target} hold with\n$\\mathcal{P} =$``smooth''.\n\\item Let $x \\in |X|$. We say $f$ is {\\it smooth at $x$} if there exists\nan open neighbourhood $X' \\subset X$ of $x$ such that $f|_{X'} : X' \\to Y$\nis smooth.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZC","source_file":"spaces-morphisms.tex","source_line":7163,"source_end_line":7175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7163-L7175","statement_sha256":"a21f56826ccdf3888b37e1bef8395b20e3c76033e8c516367c76516113eda331","origin":"The Stacks Project","memory_eligible":false,"source_rank":11080,"rank":11080,"depth":46,"x":183.978,"y":1690.473,"cluster":"algebraic-spaces"},{"id":"stacks:03ZD","tag":"03ZD","title":"Smooth morphisms · Lemma 03ZD","summary":"The composition of smooth morphisms is smooth.","statement_latex":"The composition of smooth morphisms is smooth.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZD","source_file":"spaces-morphisms.tex","source_line":7177,"source_end_line":7180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7177-L7180","statement_sha256":"bd19c8deb448decac76bd06f1603021371747cddd17692e4c3999c0f730881af","origin":"The Stacks Project","memory_eligible":false,"source_rank":11081,"rank":11081,"depth":38,"x":191.133,"y":1507.06,"cluster":"algebraic-spaces"},{"id":"stacks:03ZE","tag":"03ZE","title":"Smooth morphisms · Lemma 03ZE","summary":"The base change of a smooth morphism is smooth.","statement_latex":"The base change of a smooth morphism is smooth.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZE","source_file":"spaces-morphisms.tex","source_line":7187,"source_end_line":7190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7187-L7190","statement_sha256":"a2ecbe7213400f0cf3c11190cbb57c7753209695bdd44a11de9776da003a816c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11082,"rank":11082,"depth":38,"x":333.525,"y":1646.535,"cluster":"algebraic-spaces"},{"id":"stacks:03ZF","tag":"03ZF","title":"Smooth morphisms · Lemma 03ZF","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is smooth, • for every x ∈ |X| the morphism f is smooth at x, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is smooth, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is smooth, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is a smooth morphism, • there exists a scheme U and…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is smooth,\n\\item for every $x \\in |X|$ the morphism $f$ is smooth at $x$,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is smooth,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is smooth,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is a smooth morphism,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis smooth,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale\nthe top horizontal arrow is smooth,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, and $U \\to X$ is\nsurjective such that the top horizontal arrow is smooth, and\n\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,\nand $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that\neach morphism $X_{ij} \\to Y_i$ is smooth.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZF","source_file":"spaces-morphisms.tex","source_line":7197,"source_end_line":7236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7197-L7236","statement_sha256":"e617c5e0d7f06993afab42260d781f7f443b0430de3f68a074ee2421a6f81cbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11083,"rank":11083,"depth":0,"x":116.103,"y":1624.458,"cluster":"algebraic-spaces"},{"id":"stacks:04AJ","tag":"04AJ","title":"Smooth morphisms · Lemma 04AJ","summary":"A smooth morphism of algebraic spaces is locally of finite presentation.","statement_latex":"A smooth morphism of algebraic spaces is locally of finite presentation.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AJ","source_file":"spaces-morphisms.tex","source_line":7242,"source_end_line":7245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7242-L7245","statement_sha256":"e7a010d953797bf74ef8c2db3e5636c42940d21fd9b20cbff93ff29563ab1799","origin":"The Stacks Project","memory_eligible":false,"source_rank":11084,"rank":11084,"depth":46,"x":294.395,"y":1517.238,"cluster":"algebraic-spaces"},{"id":"stacks:06MH","tag":"06MH","title":"Smooth morphisms · Lemma 06MH","summary":"A smooth morphism of algebraic spaces is locally of finite type.","statement_latex":"A smooth morphism of algebraic spaces is locally of finite type.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MH","source_file":"spaces-morphisms.tex","source_line":7257,"source_end_line":7260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7257-L7260","statement_sha256":"e9e4a67bfeb17e6e26d91f47b5b1c82ae7f4a3f1a036b23a3eee72f249cf0aed","origin":"The Stacks Project","memory_eligible":false,"source_rank":11085,"rank":11085,"depth":47,"x":249.094,"y":1697.684,"cluster":"algebraic-spaces"},{"id":"stacks:04TA","tag":"04TA","title":"Smooth morphisms · Lemma 04TA","summary":"A smooth morphism of algebraic spaces is flat.","statement_latex":"A smooth morphism of algebraic spaces is flat.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TA","source_file":"spaces-morphisms.tex","source_line":7268,"source_end_line":7271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7268-L7271","statement_sha256":"b38b96508331ea9cc11003c3523ff5bba307ff9551d4a65139eeef54b5b77cfc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11086,"rank":11086,"depth":46,"x":137.254,"y":1538.731,"cluster":"algebraic-spaces"},{"id":"stacks:06CP","tag":"06CP","title":"Smooth morphisms · Lemma 06CP","summary":"A smooth morphism of algebraic spaces is syntomic.","statement_latex":"A smooth morphism of algebraic spaces is syntomic.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CP","source_file":"spaces-morphisms.tex","source_line":7282,"source_end_line":7285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7282-L7285","statement_sha256":"aef27b5bcbc1ca6678db3cbe230cb34650b304f59159b135a9740313b32b854e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11087,"rank":11087,"depth":46,"x":347.801,"y":1592.544,"cluster":"algebraic-spaces"},{"id":"stacks:0DZI","tag":"0DZI","title":"Smooth morphisms · Lemma 0DZI","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. There is a largest open subspace U ⊂ X such that f|_U : U → Y is smooth. Moreover, formation of this open commutes with base change by • morphisms which are flat and locally of finite presentation, • flat morphisms provided f is locally of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. There is a largest open subspace $U \\subset X$\nsuch that $f|_U : U \\to Y$ is smooth. Moreover, formation of\nthis open commutes with base change by\n\\begin{enumerate}\n\\item morphisms which are flat and\nlocally of finite presentation,\n\\item flat morphisms provided $f$ is locally of finite presentation.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZI","source_file":"spaces-morphisms.tex","source_line":7296,"source_end_line":7307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7296-L7307","statement_sha256":"a25e90f5577ad7fb59170c01b35b1525a0bc88dfc005b2706932cfdb722a6ea8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11088,"rank":11088,"depth":40,"x":149.035,"y":1672.427,"cluster":"algebraic-spaces"},{"id":"stacks:0AFI","tag":"0AFI","title":"Smooth morphisms · Lemma 0AFI","summary":"Let X and Y be locally Noetherian algebraic spaces over a scheme S, and let f : X → Y be a smooth morphism. For every point x ∈ |X| with image y ∈ |Y|, dim_x(X) = dim_y(Y) + dim_x(X_y) where dim_x(X_y) is the relative dimension of f at x as in Definition [Tag 04NM].","statement_latex":"Let $X$ and $Y$ be locally Noetherian algebraic spaces over a scheme\n$S$, and let $f : X \\to Y$ be a smooth morphism.\nFor every point $x \\in |X|$ with image $y \\in |Y|$,\n$$\n\\dim_x(X) = \\dim_y(Y) + \\dim_x(X_y)\n$$\nwhere $\\dim_x(X_y)$ is the relative dimension of $f$ at $x$ as\nin Definition \\ref{definition-dimension-fibre}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFI","source_file":"spaces-morphisms.tex","source_line":7319,"source_end_line":7329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7319-L7329","statement_sha256":"6f85fa8cc83ce2e9d2eec33e63d35244623b5b2041bbda154b1900216ddb1747","origin":"The Stacks Project","memory_eligible":false,"source_rank":11089,"rank":11089,"depth":48,"x":231.46,"y":1500.533,"cluster":"algebraic-spaces"},{"id":"stacks:03ZH","tag":"03ZH","title":"Unramified morphisms · Definition 03ZH","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is unramified if the equivalent conditions of Lemma [Tag 03MJ] hold with P = unramified. • Let x ∈ |X|. We say f is unramified at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is unramified. • We say f is G-unramified if the equivalent conditions of Lemma [Tag 03MJ] hold with P = G-unramified. • Let x ∈ |X|. We say f is G-unramified at x if there exists an…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it unramified} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with $\\mathcal{P} = \\text{unramified}$.\n\\item Let $x \\in |X|$. We say $f$ is {\\it unramified at $x$} if there\nexists an open neighbourhood $X' \\subset X$ of $x$ such that\n$f|_{X'} : X' \\to Y$ is unramified.\n\\item We say $f$ is {\\it G-unramified} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with $\\mathcal{P} = \\text{G-unramified}$.\n\\item Let $x \\in |X|$. We say $f$ is {\\it G-unramified at $x$} if there\nexists an open neighbourhood $X' \\subset X$ of $x$ such that\n$f|_{X'} : X' \\to Y$ is G-unramified.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZH","source_file":"spaces-morphisms.tex","source_line":7359,"source_end_line":7377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7359-L7377","statement_sha256":"33e5e22e493afaec825404a1cb4a4d5c4ad7df8e7e19e3db74de353378b55be1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11090,"rank":11090,"depth":46,"x":309.005,"y":1674.262,"cluster":"algebraic-spaces"},{"id":"stacks:04G1","tag":"04G1","title":"Unramified morphisms · Lemma 04G1","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then f is G-unramified if and only if f is unramified and locally of finite presentation.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThen $f$ is G-unramified if and only if $f$ is unramified and\nlocally of finite presentation.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04G1","source_file":"spaces-morphisms.tex","source_line":7385,"source_end_line":7391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7385-L7391","statement_sha256":"9f695261f53011ed7f67869ad9dc001ea58149efba825af23d681a8e2696c3e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11091,"rank":11091,"depth":46,"x":111.884,"y":1590.058,"cluster":"algebraic-spaces"},{"id":"stacks:03ZI","tag":"03ZI","title":"Unramified morphisms · Lemma 03ZI","summary":"The composition of unramified morphisms is unramified.","statement_latex":"The composition of unramified morphisms is unramified.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZI","source_file":"spaces-morphisms.tex","source_line":7402,"source_end_line":7405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7402-L7405","statement_sha256":"48d704be9aec3ae374190c7fb628f18f90d875388af9d0dc01808bb62aa080cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11092,"rank":11092,"depth":7,"x":325.205,"y":1540.238,"cluster":"algebraic-spaces"},{"id":"stacks:03ZJ","tag":"03ZJ","title":"Unramified morphisms · Lemma 03ZJ","summary":"The base change of an unramified morphism is unramified.","statement_latex":"The base change of an unramified morphism is unramified.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZJ","source_file":"spaces-morphisms.tex","source_line":7412,"source_end_line":7415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7412-L7415","statement_sha256":"ad6e995a299eb368a71803322ccc581ff4b45c22f409469118c5952942fbc1cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11093,"rank":11093,"depth":7,"x":207.83,"y":1698.204,"cluster":"algebraic-spaces"},{"id":"stacks:03ZK","tag":"03ZK","title":"Unramified morphisms · Lemma 03ZK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is unramified, • for every x ∈ |X| the morphism f is unramified at x, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is unramified, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is unramified, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is an unramified morphism, • there…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is unramified,\n\\item for every $x \\in |X|$ the morphism $f$ is unramified at $x$,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is unramified,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is unramified,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is an unramified morphism,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis unramified,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale\nthe top horizontal arrow is unramified,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, and\n$U \\to X$ is surjective such that the top horizontal arrow is unramified, and\n\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,\nand $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that\neach morphism $X_{ij} \\to Y_i$ is unramified.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZK","source_file":"spaces-morphisms.tex","source_line":7422,"source_end_line":7461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7422-L7461","statement_sha256":"6a1a7f589b01116e5b2f3a51228a032101b9c025a9c8b6eba435ef031c4d58cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11094,"rank":11094,"depth":0,"x":167.301,"y":1514.906,"cluster":"algebraic-spaces"},{"id":"stacks:05VZ","tag":"05VZ","title":"Unramified morphisms · Lemma 05VZ","summary":"An unramified morphism of algebraic spaces is locally of finite type.","statement_latex":"An unramified morphism of algebraic spaces is locally of finite type.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05VZ","source_file":"spaces-morphisms.tex","source_line":7468,"source_end_line":7471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7468-L7471","statement_sha256":"81e23abb04d3b1559a4a8f01bd2a11550412fcea649fdc2ab1406f3365fc9cef","origin":"The Stacks Project","memory_eligible":false,"source_rank":11095,"rank":11095,"depth":46,"x":344.798,"y":1627.202,"cluster":"algebraic-spaces"},{"id":"stacks:05W0","tag":"05W0","title":"Unramified morphisms · Lemma 05W0","summary":"If f is unramified at x then f is quasi-finite at x. In particular, an unramified morphism is locally quasi-finite.","statement_latex":"If $f$ is unramified at $x$ then $f$ is quasi-finite at $x$.\nIn particular, an unramified morphism is locally quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05W0","source_file":"spaces-morphisms.tex","source_line":7480,"source_end_line":7484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7480-L7484","statement_sha256":"b064f96553614983d8d52ddc8d5c195449ad775fd185ed2218007908f2078888","origin":"The Stacks Project","memory_eligible":false,"source_rank":11096,"rank":11096,"depth":46,"x":123.35,"y":1645.134,"cluster":"algebraic-spaces"},{"id":"stacks:06CQ","tag":"06CQ","title":"Unramified morphisms · Lemma 06CQ","summary":"An immersion of algebraic spaces is unramified.","statement_latex":"An immersion of algebraic spaces is unramified.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CQ","source_file":"spaces-morphisms.tex","source_line":7493,"source_end_line":7496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7493-L7496","statement_sha256":"874cfb74020d7a16d3cc1ce44555d48d3c19d9c4b9713222b620ce811f49845e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11097,"rank":11097,"depth":6,"x":272.396,"y":1506.094,"cluster":"algebraic-spaces"},{"id":"stacks:05W1","tag":"05W1","title":"Unramified morphisms · Lemma 05W1","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • If f is unramified, then the diagonal morphism Δ_X/Y : X → X ×_Y X is an open immersion. • If f is locally of finite type and Δ_X/Y is an open immersion, then f is unramified.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $f$ is unramified, then the diagonal morphism\n$\\Delta_{X/Y} : X \\to X \\times_Y X$ is an open immersion.\n\\item If $f$ is locally of finite type\nand $\\Delta_{X/Y}$ is an open immersion, then $f$ is unramified.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05W1","source_file":"spaces-morphisms.tex","source_line":7507,"source_end_line":7517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7507-L7517","statement_sha256":"c699ab46c8b600035faea445418e23f1f51ed7e9651f84c040c84268f650a0a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11098,"rank":11098,"depth":53,"x":274.306,"y":1693.41,"cluster":"algebraic-spaces"},{"id":"stacks:05W2","tag":"05W2","title":"Unramified morphisms · Lemma 05W2","summary":"Let S be a scheme. Consider a commutative diagram xymatrix X ar[rr]_f ar[rd]_p & & Y ar[ld]^q & Z of algebraic spaces over S. Assume that X → Z is locally of finite type. Then there exists an open subspace U(f) ⊂ X such that |U(f)| ⊂ |X| is the set of points where f is unramified. Moreover, for any morphism of algebraic spaces Z' → Z, if f' : X' → Y' is the base change of f by Z' → Z, then U(f') is the inverse image of U(f) under the projection X' → X.","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\\n& Z\n}\n$$\nof algebraic spaces over $S$. Assume that $X \\to Z$ is locally of\nfinite type. Then there exists an open subspace $U(f) \\subset X$\nsuch that $|U(f)| \\subset |X|$ is the set of points where $f$ is unramified.\nMoreover, for any morphism of algebraic spaces $Z' \\to Z$, if\n$f' : X' \\to Y'$ is the base change of $f$ by $Z' \\to Z$, then\n$U(f')$ is the inverse image of $U(f)$ under the projection $X' \\to X$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05W2","source_file":"spaces-morphisms.tex","source_line":7577,"source_end_line":7592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7577-L7592","statement_sha256":"b670f59a300a8f60cdbd57a34f7d10cc78f3b082ed2621b0ed39bc6b5e4960d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11099,"rank":11099,"depth":47,"x":122.087,"y":1556.21,"cluster":"algebraic-spaces"},{"id":"stacks:06G6","tag":"06G6","title":"Unramified morphisms · Lemma 06G6","summary":"Let S be a scheme. Let X → Y → Z be morphisms of algebraic spaces over S. If X → Z is unramified, then X → Y is unramified.","statement_latex":"Let $S$ be a scheme.\nLet $X \\to Y \\to Z$ be morphisms of algebraic spaces over $S$.\nIf $X \\to Z$ is unramified, then $X \\to Y$ is unramified.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06G6","source_file":"spaces-morphisms.tex","source_line":7636,"source_end_line":7641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7636-L7641","statement_sha256":"2df01d6465bedf7b40d00804e118e3c3e5b755a0d0193e0a2ce9b5fc8426d7ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":11100,"rank":11100,"depth":20,"x":344.921,"y":1571.024,"cluster":"algebraic-spaces"},{"id":"stacks:04RH","tag":"04RH","title":"Étale morphisms · Definition 04RH","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let x ∈ |X|. We say f is étale at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is étale.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $x \\in |X|$. We say $f$ is {\\it \\'etale at $x$} if there\nexists an open neighbourhood $X' \\subset X$ of $x$ such that\n$f|_{X'} : X' \\to Y$ is \\'etale.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RH","source_file":"spaces-morphisms.tex","source_line":7679,"source_end_line":7686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7679-L7686","statement_sha256":"2d7991aaa165eb07986ce3b6c0b705c8a0f069d5709846b0741152bd979ba984","origin":"The Stacks Project","memory_eligible":false,"source_rank":11101,"rank":11101,"depth":0,"x":168.49,"y":1686.676,"cluster":"algebraic-spaces"},{"id":"stacks:03XT","tag":"03XT","title":"Étale morphisms · Lemma 03XT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is étale, • for every x ∈ |X| the morphism f is étale at x, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is étale, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is étale, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is an étale morphism, • there exists a scheme U and a…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is \\'etale,\n\\item for every $x \\in |X|$ the morphism $f$ is \\'etale at $x$,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is \\'etale,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is \\'etale,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is an \\'etale morphism,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis \\'etale,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale\nthe top horizontal arrow is \\'etale,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, and\n$U \\to X$ surjective such that the top horizontal arrow is \\'etale, and\n\\item there exist Zariski coverings $Y = \\bigcup Y_i$ and\n$f^{-1}(Y_i) = \\bigcup X_{ij}$ such that each morphism\n$X_{ij} \\to Y_i$ is \\'etale.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XT","source_file":"spaces-morphisms.tex","source_line":7688,"source_end_line":7727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7688-L7727","statement_sha256":"e481bff2525d90fb3f070707ff8dc63b87fad8ec6166dc24f2ea1839e028cc0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11102,"rank":11102,"depth":46,"x":205.626,"y":1501.07,"cluster":"algebraic-spaces"},{"id":"stacks:0465","tag":"0465","title":"Étale morphisms · Lemma 0465","summary":"The composition of two étale morphisms of algebraic spaces is étale.","statement_latex":"The composition of two \\'etale morphisms of algebraic spaces\nis \\'etale.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0465","source_file":"spaces-morphisms.tex","source_line":7738,"source_end_line":7742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7738-L7742","statement_sha256":"66d40ea765d7e696aac73936d5fcdaa1f2c7af4a1b00875ec349df7aae12aa08","origin":"The Stacks Project","memory_eligible":false,"source_rank":11103,"rank":11103,"depth":1,"x":327.641,"y":1659.187,"cluster":"algebraic-spaces"},{"id":"stacks:0466","tag":"0466","title":"Étale morphisms · Lemma 0466","summary":"The base change of an étale morphism of algebraic spaces by any morphism of algebraic spaces is étale.","statement_latex":"The base change of an \\'etale morphism of algebraic spaces\nby any morphism of algebraic spaces is \\'etale.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0466","source_file":"spaces-morphisms.tex","source_line":7749,"source_end_line":7753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7749-L7753","statement_sha256":"6a346ee7255b04af9b3e52b08a0de06cffee519250c07cfd24d5b53817b2f42e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11104,"rank":11104,"depth":46,"x":110.267,"y":1611.774,"cluster":"algebraic-spaces"},{"id":"stacks:03XU","tag":"03XU","title":"Étale morphisms · Lemma 03XU","summary":"An étale morphism of algebraic spaces is locally quasi-finite.","statement_latex":"An \\'etale morphism of algebraic spaces is locally quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XU","source_file":"spaces-morphisms.tex","source_line":7760,"source_end_line":7763,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7760-L7763","statement_sha256":"624fe2eefe07db710d76d80a2ab5feb63dbdcf45b0ae16b2cb0fbf49f356d653","origin":"The Stacks Project","memory_eligible":false,"source_rank":11105,"rank":11105,"depth":46,"x":308.911,"y":1523.291,"cluster":"algebraic-spaces"},{"id":"stacks:04XX","tag":"04XX","title":"Étale morphisms · Lemma 04XX","summary":"An étale morphism of algebraic spaces is smooth.","statement_latex":"An \\'etale morphism of algebraic spaces is smooth.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XX","source_file":"spaces-morphisms.tex","source_line":7778,"source_end_line":7781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7778-L7781","statement_sha256":"91119a3b0cf8103f42ed07549db2ebf848b32331d773306eff03403805d88831","origin":"The Stacks Project","memory_eligible":false,"source_rank":11106,"rank":11106,"depth":47,"x":233.503,"y":1701.456,"cluster":"algebraic-spaces"},{"id":"stacks:0467","tag":"0467","title":"Étale morphisms · Lemma 0467","summary":"An étale morphism of algebraic spaces is flat.","statement_latex":"An \\'etale morphism of algebraic spaces is flat.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0467","source_file":"spaces-morphisms.tex","source_line":7790,"source_end_line":7793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7790-L7793","statement_sha256":"b82ff0b165b6aad9deaa5f6a570c91eb21b48df25ed128c2d3840adec16b98b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11107,"rank":11107,"depth":47,"x":145.733,"y":1527.093,"cluster":"algebraic-spaces"},{"id":"stacks:0468","tag":"0468","title":"Étale morphisms · Lemma 0468","summary":"Étale implies locally of finite presentation. An étale morphism of algebraic spaces is locally of finite presentation.","statement_latex":"\\begin{slogan}\n\\'Etale implies locally of finite presentation.\n\\end{slogan}\nAn \\'etale morphism of algebraic spaces is locally of finite presentation.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0468","source_file":"spaces-morphisms.tex","source_line":7802,"source_end_line":7808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7802-L7808","statement_sha256":"199148a29663889ae430116a6f5e7df7f65f7a25b3642cfad026ab4181ee8e5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11108,"rank":11108,"depth":47,"x":350.905,"y":1605.952,"cluster":"algebraic-spaces"},{"id":"stacks:06LT","tag":"06LT","title":"Étale morphisms · Lemma 06LT","summary":"An étale morphism of algebraic spaces is locally of finite type.","statement_latex":"An \\'etale morphism of algebraic spaces is locally of finite type.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LT","source_file":"spaces-morphisms.tex","source_line":7817,"source_end_line":7820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7817-L7820","statement_sha256":"e8941d0e395593f575ebd78ddaf5beb9f5debb757b023b36ca0395bd130eb74b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11109,"rank":11109,"depth":48,"x":135.954,"y":1664.288,"cluster":"algebraic-spaces"},{"id":"stacks:06CR","tag":"06CR","title":"Étale morphisms · Lemma 06CR","summary":"An étale morphism of algebraic spaces is unramified.","statement_latex":"An \\'etale morphism of algebraic spaces is unramified.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CR","source_file":"spaces-morphisms.tex","source_line":7830,"source_end_line":7833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7830-L7833","statement_sha256":"eefd3871c931acf670faa77d5d82e12c8d4c2a43b0ab5b2a28bcaae6832e113e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11110,"rank":11110,"depth":47,"x":247.67,"y":1499.117,"cluster":"algebraic-spaces"},{"id":"stacks:05W3","tag":"05W3","title":"Étale morphisms · Lemma 05W3","summary":"Let S be a scheme. Let X, Y be algebraic spaces étale over an algebraic space Z. Any morphism X → Y over Z is étale.","statement_latex":"Let $S$ be a scheme. Let $X, Y$ be algebraic spaces \\'etale over\nan algebraic space $Z$. Any morphism $X \\to Y$ over $Z$ is \\'etale.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05W3","source_file":"spaces-morphisms.tex","source_line":7842,"source_end_line":7846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7842-L7846","statement_sha256":"7baf2a58175af7bf047dae2f3458e06fdf450956d82c0ba580d2345ae2b4fd4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11111,"rank":11111,"depth":45,"x":298.175,"y":1684.511,"cluster":"algebraic-spaces"},{"id":"stacks:06LU","tag":"06LU","title":"Étale morphisms · Lemma 06LU","summary":"A locally finitely presented, flat, unramified morphism of algebraic spaces is étale.","statement_latex":"A locally finitely presented, flat, unramified morphism of algebraic\nspaces is \\'etale.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LU","source_file":"spaces-morphisms.tex","source_line":7853,"source_end_line":7857,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7853-L7857","statement_sha256":"bb898d8898ae1fe690dfbb93a661a656d4a4cc312c0d1e4ea95d958d6d7aecfe","origin":"The Stacks Project","memory_eligible":false,"source_rank":11112,"rank":11112,"depth":46,"x":111.634,"y":1576.34,"cluster":"algebraic-spaces"},{"id":"stacks:03ZM","tag":"03ZM","title":"Proper morphisms · Definition 03ZM","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say f is proper if f is separated, finite type, and universally closed.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nWe say $f$ is {\\it proper} if $f$ is separated, finite type, and\nuniversally closed.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Proper morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZM","source_file":"spaces-morphisms.tex","source_line":7883,"source_end_line":7889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7883-L7889","statement_sha256":"86398563c040bbcb99ffd04052692d04a7b03ae77b433ca2ea513e224b6ca0fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11113,"rank":11113,"depth":0,"x":336.427,"y":1550.227,"cluster":"algebraic-spaces"},{"id":"stacks:083R","tag":"083R","title":"Proper morphisms · Lemma 083R","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f is proper, • for every scheme Z and every morphism Z → Y the projection Z ×_Y X → Z is proper, • for every affine scheme Z and every morphism Z → Y the projection Z ×_Y X → Z is proper, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is proper, and • there exists a Zariski covering Y = ⋃ Y_i such that each of the morphisms…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item for every scheme $Z$ and every morphism $Z \\to Y$\nthe projection $Z \\times_Y X \\to Z$ is proper,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$\nthe projection $Z \\times_Y X \\to Z$ is proper,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is proper, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that\neach of the morphisms $f^{-1}(Y_i) \\to Y_i$ is proper.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083R","source_file":"spaces-morphisms.tex","source_line":7891,"source_end_line":7906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7891-L7906","statement_sha256":"12f77bab12075fe86ab152c0612dade1f9f2b530ca65886be6a13d8563d28b5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11114,"rank":11114,"depth":56,"x":191.506,"y":1697.2,"cluster":"algebraic-spaces"},{"id":"stacks:04WP","tag":"04WP","title":"Proper morphisms · Lemma 04WP","summary":"A base change of a proper morphism is proper.","statement_latex":"A base change of a proper morphism is proper.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WP","source_file":"spaces-morphisms.tex","source_line":7915,"source_end_line":7918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7915-L7918","statement_sha256":"d30cd8ba74796aa24cebe9df0b79419746fe7185db40fc848efc7e9bf749daa1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11115,"rank":11115,"depth":7,"x":180.164,"y":1506.379,"cluster":"algebraic-spaces"},{"id":"stacks:04XY","tag":"04XY","title":"Proper morphisms · Lemma 04XY","summary":"A composition of proper morphisms is proper.","statement_latex":"A composition of proper morphisms is proper.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XY","source_file":"spaces-morphisms.tex","source_line":7927,"source_end_line":7930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7927-L7930","statement_sha256":"a2215eeaf72a90343a8435f29bbba8328292bda0f1848d568ab048e6c11c78d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11116,"rank":11116,"depth":55,"x":342.161,"y":1640.802,"cluster":"algebraic-spaces"},{"id":"stacks:04XZ","tag":"04XZ","title":"Proper morphisms · Lemma 04XZ","summary":"A closed immersion of algebraic spaces is a proper morphism of algebraic spaces.","statement_latex":"A closed immersion of algebraic spaces is a proper morphism of\nalgebraic spaces.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XZ","source_file":"spaces-morphisms.tex","source_line":7939,"source_end_line":7943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7939-L7943","statement_sha256":"6e145d3e6876e02822f9007682ba97c20d0ef249addf32c894d8011f366bacd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11117,"rank":11117,"depth":15,"x":114.354,"y":1633.593,"cluster":"algebraic-spaces"},{"id":"stacks:04NX","tag":"04NX","title":"Proper morphisms · Lemma 04NX","summary":"Let S be a scheme. Consider a commutative diagram of algebraic spaces xymatrix X ar[rr] ar[rd] & & Y ar[ld] & B & over S. • If X → B is universally closed and Y → B is separated, then the morphism X → Y is universally closed. In particular, the image of |X| in |Y| is closed. • If X → B is proper and Y → B is separated, then the morphism X → Y is proper.","statement_latex":"Let $S$ be a scheme.\nConsider a commutative diagram of algebraic spaces\n$$\n\\xymatrix{\nX \\ar[rr] \\ar[rd] & &\nY \\ar[ld] \\\\\n& B &\n}\n$$\nover $S$.\n\\begin{enumerate}\n\\item If $X \\to B$ is universally closed and $Y \\to B$ is\nseparated, then the morphism $X \\to Y$ is universally closed.\nIn particular, the image of $|X|$ in $|Y|$ is closed.\n\\item If $X \\to B$ is proper and $Y \\to B$ is separated, then\nthe morphism $X \\to Y$ is proper.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NX","source_file":"spaces-morphisms.tex","source_line":7953,"source_end_line":7972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7953-L7972","statement_sha256":"54c0b12e2e04190151083197ba90b6199307cc948ee4753fae2cec2063286961","origin":"The Stacks Project","memory_eligible":false,"source_rank":11118,"rank":11118,"depth":57,"x":288.326,"y":1509.508,"cluster":"algebraic-spaces"},{"id":"stacks:08AJ","tag":"08AJ","title":"Proper morphisms · Lemma 08AJ","summary":"Let S be a scheme. Let B be an algebraic space over S. Let f : X → Y be a morphism of algebraic spaces over B. If X is universally closed over B and f is surjective then Y is universally closed over B. In particular, if also Y is separated and of finite type over B, then Y is proper over B.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $B$.\nIf $X$ is universally closed over $B$ and $f$ is surjective then\n$Y$ is universally closed over $B$. In particular, if also $Y$ is\nseparated and of finite type over $B$, then $Y$ is proper over $B$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AJ","source_file":"spaces-morphisms.tex","source_line":7993,"source_end_line":8000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L7993-L8000","statement_sha256":"412ddafea655939eaef425819ca7f4a586d8fa89fcb3bb11fbd80d3a5b246e96","origin":"The Stacks Project","memory_eligible":false,"source_rank":11119,"rank":11119,"depth":2,"x":259.796,"y":1699.933,"cluster":"algebraic-spaces"},{"id":"stacks:0AGD","tag":"0AGD","title":"Proper morphisms · Lemma 0AGD","summary":"Let S be a scheme. Let xymatrix X ar[rr]_h ar[rd]_f & & Y ar[ld]^g & B be a commutative diagram of morphism of algebraic spaces over S. Assume • X → B is a proper morphism, • Y → B is separated and locally of finite type, Then the scheme theoretic image Z ⊂ Y of h is proper over B and X → Z is surjective.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\\n& B\n}\n$$\nbe a commutative diagram of morphism of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $X \\to B$ is a proper morphism,\n\\item $Y \\to B$ is separated and locally of finite type,\n\\end{enumerate}\nThen the scheme theoretic image $Z \\subset Y$ of $h$\nis proper over $B$ and $X \\to Z$ is surjective.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGD","source_file":"spaces-morphisms.tex","source_line":8014,"source_end_line":8031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8014-L8031","statement_sha256":"56d957ed8a24e1e7d4cf1a127064f53a6d3041bca7be01badc4e790285e37943","origin":"The Stacks Project","memory_eligible":false,"source_rank":11120,"rank":11120,"depth":58,"x":127.553,"y":1543.155,"cluster":"algebraic-spaces"},{"id":"stacks:04Y0","tag":"04Y0","title":"Proper morphisms · Lemma 04Y0","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is separated, • Δ_X/Y : X → X ×_Y X is universally closed, and • Δ_X/Y : X → X ×_Y X is proper.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is separated,\n\\item $\\Delta_{X/Y} : X \\to X \\times_Y X$ is universally closed, and\n\\item $\\Delta_{X/Y} : X \\to X \\times_Y X$ is proper.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Y0","source_file":"spaces-morphisms.tex","source_line":8046,"source_end_line":8056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8046-L8056","statement_sha256":"bde7d63816fe5fda35e261c26322f7441a11128a21535b721cf62824694f6a2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11121,"rank":11121,"depth":53,"x":351.39,"y":1583.768,"cluster":"algebraic-spaces"},{"id":"stacks:03IX","tag":"03IX","title":"Valuative criteria · Definition 03IX","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say f satisfies the uniqueness part of the valuative criterion if given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & Y where A is a valuation ring with field of fractions K, there exists at most one dotted arrow (without requiring existence). We say f satisfies the existence part of the valuative criterion if given any solid diagram as above…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nWe say $f$ {\\it satisfies the uniqueness part of the valuative criterion}\nif given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$, there exists\nat most one dotted arrow (without requiring existence).\nWe say $f$ {\\it satisfies the existence part of the valuative criterion}\nif given any solid diagram as above there exists an extension\n$K'/K$ of fields, a valuation ring $A' \\subset K'$ dominating\n$A$ and a morphism $\\Spec(A') \\to X$ such that the following\ndiagram commutes\n$$\n\\xymatrix{\n\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\\n\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y\n}\n$$\nWe say $f$ {\\it satisfies the valuative criterion}\nif $f$ satisfies both the existence and uniqueness part.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criteria","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IX","source_file":"spaces-morphisms.tex","source_line":8111,"source_end_line":8138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8111-L8138","statement_sha256":"6c79be9cba1770257c95b652dca3702f632dff988960de3f34e68ed951b42ef7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11122,"rank":11122,"depth":0,"x":153.458,"y":1680.938,"cluster":"algebraic-spaces"},{"id":"stacks:03K8","tag":"03K8","title":"Valuative criteria · Lemma 03K8","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is representable. The following are equivalent • f satisfies the existence part of the valuative criterion as in Definition [Tag 03IX], • given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & Y where A is a valuation ring with field of fractions K, there exists a dotted arrow, i.e., f satisfies the existence part of the valuative criterion as…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is representable. The following are equivalent\n\\begin{enumerate}\n\\item $f$ satisfies the existence part of the valuative criterion\nas in Definition \\ref{definition-valuative-criterion},\n\\item given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$, there exists\na dotted arrow, i.e., $f$ satisfies the existence part of the valuative\ncriterion as in\nSchemes, Definition \\ref{schemes-definition-valuative-criterion}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03K8","source_file":"spaces-morphisms.tex","source_line":8169,"source_end_line":8189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8169-L8189","statement_sha256":"5baa20beec6d2a9121cd9c2db143fe4493f75178211badb710207b3a28225b95","origin":"The Stacks Project","memory_eligible":false,"source_rank":11123,"rank":11123,"depth":1,"x":221.343,"y":1496.773,"cluster":"algebraic-spaces"},{"id":"stacks:03KH","tag":"03KH","title":"Valuative criteria · Lemma 03KH","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f satisfies the existence part of the valuative criterion as in Definition [Tag 03IX], • f satisfies the existence part of the valuative criterion as in Definition [Tag 03IX] modified by requiring the extension K'/K to be finite separable.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ satisfies the existence part of the valuative criterion\nas in Definition \\ref{definition-valuative-criterion},\n\\item $f$ satisfies the existence part of the valuative criterion\nas in Definition \\ref{definition-valuative-criterion} modified by\nrequiring the extension $K'/K$ to be finite separable.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KH","source_file":"spaces-morphisms.tex","source_line":8219,"source_end_line":8231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8219-L8231","statement_sha256":"d38cbe56a01b49c338e0af9c106147f0f2038f418d81624ef795a73933d26437","origin":"The Stacks Project","memory_eligible":false,"source_rank":11124,"rank":11124,"depth":7,"x":319.494,"y":1671.284,"cluster":"algebraic-spaces"},{"id":"stacks:0ARH","tag":"0ARH","title":"Valuative criteria · Lemma 0ARH","summary":"Let S be a scheme. Let f : X → Y be a separated morphism of algebraic spaces over S. Suppose given a diagram xymatrix Spec(K') ar[r] ar[d] & Spec(K) ar[r] & X ar[d] Spec(A') ar[r] ar[rru] & Spec(A) ar[r] ar@-->[ru] & Y as in Definition [Tag 03IX] with K ⊂ K' arbitrary. Then the dotted arrow exists making the diagram commute.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a separated morphism of\nalgebraic spaces over $S$. Suppose given a diagram\n$$\n\\xymatrix{\n\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\\n\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nas in Definition \\ref{definition-valuative-criterion} with $K \\subset K'$\narbitrary. Then the dotted arrow exists making the diagram commute.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARH","source_file":"spaces-morphisms.tex","source_line":8303,"source_end_line":8315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8303-L8315","statement_sha256":"0d50f694b85fdeaf572fef3d5e53b0b864e89a5bbaed8c96f550075fd33d630a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11125,"rank":11125,"depth":57,"x":106.549,"y":1598.215,"cluster":"algebraic-spaces"},{"id":"stacks:0A3W","tag":"0A3W","title":"Valuative criteria · Lemma 0A3W","summary":"Let S be a scheme. Let f : X → Y be a separated morphism of algebraic spaces over S. The following are equivalent • f satisfies the existence part of the valuative criterion as in Definition [Tag 03IX], • given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & Y where A is a valuation ring with field of fractions K, there exists a dotted arrow, i.e., f satisfies the existence part of the valuative criterion as in Schemes,…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a separated morphism of\nalgebraic spaces over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ satisfies the existence part of the valuative criterion\nas in Definition \\ref{definition-valuative-criterion},\n\\item given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$, there exists\na dotted arrow, i.e., $f$ satisfies the existence part of the valuative\ncriterion as in\nSchemes, Definition \\ref{schemes-definition-valuative-criterion}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3W","source_file":"spaces-morphisms.tex","source_line":8398,"source_end_line":8417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8398-L8417","statement_sha256":"65df32b2e9406effe248750e95ddb8bada7d884ddf3af27d4d48b8329d17b11d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11126,"rank":11126,"depth":58,"x":322.567,"y":1531.192,"cluster":"algebraic-spaces"},{"id":"stacks:03IY","tag":"03IY","title":"Valuative criteria · Lemma 03IY","summary":"The base change of a morphism of algebraic spaces which satisfies the existence part of (resp. uniqueness part of) the valuative criterion by any morphism of algebraic spaces satisfies the existence part of (resp. uniqueness part of) the valuative criterion.","statement_latex":"The base change of a morphism of algebraic spaces which satisfies the\nexistence part of (resp.\\ uniqueness part of) the valuative criterion\nby any morphism of algebraic spaces satisfies the\nexistence part of (resp.\\ uniqueness part of) the valuative criterion.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IY","source_file":"spaces-morphisms.tex","source_line":8473,"source_end_line":8479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8473-L8479","statement_sha256":"d6a44cd42517e3185da9f179743b959fd1b72c6c72a8cdd3c471ae85f2130802","origin":"The Stacks Project","memory_eligible":false,"source_rank":11127,"rank":11127,"depth":0,"x":217.062,"y":1703.375,"cluster":"algebraic-spaces"},{"id":"stacks:03IZ","tag":"03IZ","title":"Valuative criteria · Lemma 03IZ","summary":"The composition of two morphisms of algebraic spaces which satisfy the (existence part of, resp. uniqueness part of) the valuative criterion satisfies the (existence part of, resp. uniqueness part of) the valuative criterion.","statement_latex":"The composition of two morphisms of algebraic spaces which satisfy the\n(existence part of, resp.\\ uniqueness part of) the valuative criterion\nsatisfies the (existence part of, resp.\\ uniqueness part of) the valuative\ncriterion.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IZ","source_file":"spaces-morphisms.tex","source_line":8511,"source_end_line":8517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8511-L8517","statement_sha256":"7fb1514b5c817f6c17276df66a5958a21b5d7babb47f5092a344cb3c688c614e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11128,"rank":11128,"depth":0,"x":156.329,"y":1516.341,"cluster":"algebraic-spaces"},{"id":"stacks:03KA","tag":"03KA","title":"Valuative criterion for universal closedness · Lemma 03KA","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • f is quasi-compact, and • f satisfies the existence part of the valuative criterion. Then f is universally closed.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $f$ is quasi-compact, and\n\\item $f$ satisfies the existence part of the valuative criterion.\n\\end{enumerate}\nThen $f$ is universally closed.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KA","source_file":"spaces-morphisms.tex","source_line":8580,"source_end_line":8590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8580-L8590","statement_sha256":"957f40e310507e0e80364a0e6b1aac9b75ea5254ce7c45657bda8551c744e65b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11129,"rank":11129,"depth":47,"x":351.732,"y":1619.908,"cluster":"algebraic-spaces"},{"id":"stacks:0A3X","tag":"0A3X","title":"Valuative criterion for universal closedness · Lemma 0A3X","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • If f is quasi-separated and universally closed, then f satisfies the existence part of the valuative criterion. • If f is quasi-compact and quasi-separated, then f is universally closed if and only if the existence part of the valuative criterion holds.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a\nmorphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $f$ is quasi-separated and universally closed, then\n$f$ satisfies the existence part of the valuative criterion.\n\\item If $f$ is quasi-compact and quasi-separated, then\n$f$ is universally closed if and only if the existence part of the\nvaluative criterion holds.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3X","source_file":"spaces-morphisms.tex","source_line":8644,"source_end_line":8655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8644-L8655","statement_sha256":"943f2cde8f33ee035bcb8dbb88291d747855d9e020e9e82e9222a100f47d8c07","origin":"The Stacks Project","memory_eligible":false,"source_rank":11130,"rank":11130,"depth":57,"x":124.113,"y":1654.452,"cluster":"algebraic-spaces"},{"id":"stacks:0A3Y","tag":"0A3Y","title":"Valuative criterion for universal closedness · Lemma 0A3Y","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-compact and separated. Then the following are equivalent • f is universally closed, • the existence part of the valuative criterion holds as in Definition [Tag 03IX], and • given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & Y where A is a valuation ring with field of fractions K, there exists a dotted arrow, i.e., f satisfies the…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is quasi-compact and separated. Then the following\nare equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed,\n\\item the existence part of the valuative criterion holds\nas in Definition \\ref{definition-valuative-criterion}, and\n\\item given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$, there exists\na dotted arrow, i.e., $f$ satisfies the existence part of the valuative\ncriterion as in\nSchemes, Definition \\ref{schemes-definition-valuative-criterion}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3Y","source_file":"spaces-morphisms.tex","source_line":8710,"source_end_line":8731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8710-L8731","statement_sha256":"5da3d2535e5c015c4607358e961e492b27dd1bcdaead6dd1c7ba98fa53efb120","origin":"The Stacks Project","memory_eligible":false,"source_rank":11131,"rank":11131,"depth":59,"x":264.327,"y":1499.657,"cluster":"algebraic-spaces"},{"id":"stacks:089F","tag":"089F","title":"Valuative criterion for universal closedness · Lemma 089F","summary":"Let S be a scheme. Let f : X → Y be a flat morphism of algebraic spaces over S. Let Spec(A) → Y be a morphism where A is a valuation ring. If the closed point of Spec(A) maps to a point of |Y| in the image of |X| → |Y|, then there exists a commutative diagram xymatrix Spec(A') ar[r] ar[d] & X ar[d] Spec(A) ar[r] & Y where A → A' is an extension of valuation rings (More on Algebra, Definition [Tag 0ASG]).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism\nof algebraic spaces over $S$.\nLet $\\Spec(A) \\to Y$ be a morphism where $A$ is a\nvaluation ring. If the closed point of $\\Spec(A)$ maps to a\npoint of $|Y|$ in the image of $|X| \\to |Y|$, then there exists\na commutative diagram\n$$\n\\xymatrix{\n\\Spec(A') \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] & Y\n}\n$$\nwhere $A \\to A'$ is an extension of valuation rings\n(More on Algebra, Definition\n\\ref{more-algebra-definition-extension-valuation-rings}).","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089F","source_file":"spaces-morphisms.tex","source_line":8740,"source_end_line":8757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8740-L8757","statement_sha256":"983a418b7c68990d7cd0f7e4bfad09b70edc19ba9890657f0bbe54f78d9773ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":11132,"rank":11132,"depth":7,"x":285.441,"y":1693.57,"cluster":"algebraic-spaces"},{"id":"stacks:089G","tag":"089G","title":"Valuative criterion for universal closedness · Lemma 089G","summary":"Let S be a scheme. Let f : X → Y and h : U → X be morphisms of algebraic spaces over S. If • f and h are quasi-compact, • |h|(|U|) is dense in |X|, and given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & U ar[r] & X ar[d] Spec(A) ar[rr] ar@-->[rru] & & Y where A is a valuation ring with field of fractions K • [(3)] there exists at most one dotted arrow making the diagram commute, and • [(4)] there exists an extension K'/K of fields, a valuation ring A' ⊂ K'…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $h : U \\to X$ be morphisms of\nalgebraic spaces over $S$. If\n\\begin{enumerate}\n\\item $f$ and $h$ are quasi-compact,\n\\item $|h|(|U|)$ is dense in $|X|$, and\n\\end{enumerate}\ngiven any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r] & X \\ar[d] \\\\\n\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$\n\\begin{enumerate}\n\\item[(3)] there exists at most one dotted arrow making the diagram\ncommute, and\n\\item[(4)] there exists an extension $K'/K$ of fields, a\nvaluation ring $A' \\subset K'$ dominating $A$ and a morphism\n$\\Spec(A') \\to X$ such that the following diagram commutes\n$$\n\\xymatrix{\n\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & U \\ar[r] & X \\ar[d] \\\\\n\\Spec(A') \\ar[r] \\ar[rrru] & \\Spec(A) \\ar[rr] & & Y\n}\n$$\n\\end{enumerate}\nthen $f$ is universally closed. If moreover\n\\begin{enumerate}\n\\item[(5)] $f$ is quasi-separated\n\\end{enumerate}\nthen $f$ is separated and universally closed.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089G","source_file":"spaces-morphisms.tex","source_line":8778,"source_end_line":8812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8778-L8812","statement_sha256":"52da0950433d8ed405855dc862a48a7998e9688f2cb1c1c090dcbd6d0e8e0b20","origin":"The Stacks Project","memory_eligible":false,"source_rank":11133,"rank":11133,"depth":55,"x":113.748,"y":1562.421,"cluster":"algebraic-spaces"},{"id":"stacks:03KU","tag":"03KU","title":"Valuative criterion of separatedness · Lemma 03KU","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is separated, then f satisfies the uniqueness part of the valuative criterion.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is separated, then $f$ satisfies the uniqueness\npart of the valuative criterion.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criterion of separatedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KU","source_file":"spaces-morphisms.tex","source_line":8967,"source_end_line":8973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8967-L8973","statement_sha256":"28b3b3ad46195e0fe8dcbb98960fabb1f17adebd85fe4aabb36c1b15289db2ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":11134,"rank":11134,"depth":1,"x":346.067,"y":1561.705,"cluster":"algebraic-spaces"},{"id":"stacks:03KV","tag":"03KV","title":"Valuative criterion separatedness · Lemma 03KV","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • the morphism f is quasi-separated, and • the morphism f satisfies the uniqueness part of the valuative criterion. Then f is separated.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item the morphism $f$ is quasi-separated, and\n\\item the morphism $f$ satisfies the uniqueness\npart of the valuative criterion.\n\\end{enumerate}\nThen $f$ is separated.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criterion of separatedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KV","source_file":"spaces-morphisms.tex","source_line":8987,"source_end_line":8998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L8987-L8998","statement_sha256":"b5063d387a39481f8e0f839dffa41f33e03156dc2f60145a6010c2efe8d310fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":11135,"rank":11135,"depth":48,"x":175.151,"y":1694.198,"cluster":"algebraic-spaces"},{"id":"stacks:0A3Z","tag":"0A3Z","title":"Valuative criterion of separatedness · Lemma 0A3Z","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-compact and quasi-separated. Then the following are equivalent • f is separated and universally closed, • the valuative criterion holds as in Definition [Tag 03IX], • given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & Y where A is a valuation ring with field of fractions K, there exists a unique dotted arrow, i.e., f satisfies the…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is quasi-compact and quasi-separated. Then the\nfollowing are equivalent\n\\begin{enumerate}\n\\item $f$ is separated and universally closed,\n\\item the valuative criterion holds as in Definition\n\\ref{definition-valuative-criterion},\n\\item given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$, there exists\na unique dotted arrow, i.e., $f$ satisfies the valuative\ncriterion as in\nSchemes, Definition \\ref{schemes-definition-valuative-criterion}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criterion of separatedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A3Z","source_file":"spaces-morphisms.tex","source_line":9030,"source_end_line":9051,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9030-L9051","statement_sha256":"4884faf8651a82c4ce6dd9e69e18d1a90099e3d42e0d24c09e073aa240958cc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11136,"rank":11136,"depth":60,"x":194.656,"y":1499.31,"cluster":"algebraic-spaces"},{"id":"stacks:0A40","tag":"0A40","title":"Valuative criterion for properness · Lemma 0A40","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is of finite type and quasi-separated. Then the following are equivalent • f is proper, • the valuative criterion holds as in Definition [Tag 03IX], • given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & Y where A is a valuation ring with field of fractions K, there exists a unique dotted arrow, i.e., f satisfies the valuative criterion as…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is of finite type and quasi-separated. Then the\nfollowing are equivalent\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item the valuative criterion holds as in Definition\n\\ref{definition-valuative-criterion},\n\\item given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$, there exists\na unique dotted arrow, i.e., $f$ satisfies the valuative\ncriterion as in\nSchemes, Definition \\ref{schemes-definition-valuative-criterion}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Valuative criterion of properness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A40","source_file":"spaces-morphisms.tex","source_line":9075,"source_end_line":9096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9075-L9096","statement_sha256":"c0fa084e742b98132a187266089a4a3d04baa2b9933a6c12122a7ea09346f253","origin":"The Stacks Project","memory_eligible":false,"source_rank":11137,"rank":11137,"depth":61,"x":337.148,"y":1654.25,"cluster":"algebraic-spaces"},{"id":"stacks:03ZO","tag":"03ZO","title":"Integral and finite morphisms · Lemma 03ZO","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. Then f is integral, resp. finite (in the sense of Section [Tag 03HA]), if and only if for all affine schemes Z and morphisms Z → Y the scheme X ×_Y Z is affine and integral, resp. finite, over Z.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable\nmorphism of algebraic spaces over $S$. Then\n$f$ is integral, resp.\\ finite\n(in the sense of Section \\ref{section-representable}),\nif and only if for all affine schemes $Z$\nand morphisms $Z \\to Y$ the scheme $X \\times_Y Z$ is affine and\nintegral, resp.\\ finite, over $Z$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZO","source_file":"spaces-morphisms.tex","source_line":9116,"source_end_line":9125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9116-L9125","statement_sha256":"ecbf2dc28b6956aababb4d0c7e79ae2892b2d70641d867f682d6dd9cfb16b6b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11138,"rank":11138,"depth":1,"x":107.234,"y":1620.818,"cluster":"algebraic-spaces"},{"id":"stacks:03ZP","tag":"03ZP","title":"Integral and finite morphisms · Definition 03ZP","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say that f is integral if for every affine scheme Z and morphisms Z → Y the algebraic space X ×_Y Z is representable by an affine scheme integral over Z. • We say that f is finite if for every affine scheme Z and morphisms Z → Y the algebraic space X ×_Y Z is representable by an affine scheme finite over Z.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say that $f$ is {\\it integral} if for every affine scheme $Z$\nand morphisms $Z \\to Y$ the algebraic space $X \\times_Y Z$ is\nrepresentable by an affine scheme integral over $Z$.\n\\item We say that $f$ is {\\it finite} if for every affine scheme $Z$\nand morphisms $Z \\to Y$ the algebraic space $X \\times_Y Z$ is\nrepresentable by an affine scheme finite over $Z$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZP","source_file":"spaces-morphisms.tex","source_line":9136,"source_end_line":9148,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9136-L9148","statement_sha256":"42e8332c839a9ad7abe536a8b23c8ad16bbe174d117dfb55995d5bc0d75d21db","origin":"The Stacks Project","memory_eligible":false,"source_rank":11139,"rank":11139,"depth":0,"x":303.863,"y":1514.899,"cluster":"algebraic-spaces"},{"id":"stacks:03ZQ","tag":"03ZQ","title":"Integral and finite morphisms · Lemma 03ZQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is representable and integral (resp. finite), • f is integral (resp. finite), • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is integral (resp. finite), and • there exists a Zariski covering Y = ⋃ Y_i such that each of the morphisms f^-1(Y_i) → Y_i is integral (resp. finite).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is representable and integral (resp.\\ finite),\n\\item $f$ is integral (resp.\\ finite),\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is integral (resp. finite), and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that\neach of the morphisms $f^{-1}(Y_i) \\to Y_i$ is integral (resp.\\ finite).\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZQ","source_file":"spaces-morphisms.tex","source_line":9150,"source_end_line":9163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9150-L9163","statement_sha256":"f7f2d1bd1c4e48bd874c18f2c7c6fb488799e81248c2a1be6cda5d190375b0c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11140,"rank":11140,"depth":46,"x":243.985,"y":1704.774,"cluster":"algebraic-spaces"},{"id":"stacks:03ZR","tag":"03ZR","title":"Integral and finite morphisms · Lemma 03ZR","summary":"The composition of integral (resp. finite) morphisms is integral (resp. finite).","statement_latex":"The composition of integral (resp.\\ finite) morphisms is integral\n(resp.\\ finite).","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZR","source_file":"spaces-morphisms.tex","source_line":9194,"source_end_line":9198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9194-L9198","statement_sha256":"5fba3571457b50a8344e9c092f03dc4ac80109dccd067c3e5fd9cac02de529a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11141,"rank":11141,"depth":0,"x":135.331,"y":1530.604,"cluster":"algebraic-spaces"},{"id":"stacks:03ZS","tag":"03ZS","title":"Integral and finite morphisms · Lemma 03ZS","summary":"The base change of an integral (resp. finite) morphism is integral (resp. finite).","statement_latex":"The base change of an integral (resp.\\ finite) morphism is integral\n(resp.\\ finite).","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZS","source_file":"spaces-morphisms.tex","source_line":9204,"source_end_line":9208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9204-L9208","statement_sha256":"7cf1a717c509ced1d3e2963ac7baa38de81f40fdc959abbada4a7201f8bc43c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11142,"rank":11142,"depth":0,"x":355.746,"y":1597.452,"cluster":"algebraic-spaces"},{"id":"stacks:0414","tag":"0414","title":"Integral and finite morphisms · Lemma 0414","summary":"A finite morphism of algebraic spaces is integral. An integral morphism of algebraic spaces which is locally of finite type is finite.","statement_latex":"A finite morphism of algebraic spaces is integral.\nAn integral morphism of algebraic spaces\nwhich is locally of finite type is finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0414","source_file":"spaces-morphisms.tex","source_line":9214,"source_end_line":9219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9214-L9219","statement_sha256":"d91077f1742ec388d60a6424c5d9e07a1cd1525fad0fcd75f0d2ed6373aca8f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11143,"rank":11143,"depth":47,"x":139.231,"y":1673.307,"cluster":"algebraic-spaces"},{"id":"stacks:0415","tag":"0415","title":"Integral and finite morphisms · Lemma 0415","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f is integral, and • f is affine and universally closed.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is integral, and\n\\item $f$ is affine and universally closed.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0415","source_file":"spaces-morphisms.tex","source_line":9229,"source_end_line":9238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9229-L9238","statement_sha256":"570b99ae3dc9a6a1737be357ff9fcd6f60292f61a71958f0fcc4cc54f64a38ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":11144,"rank":11144,"depth":47,"x":237.989,"y":1494.329,"cluster":"algebraic-spaces"},{"id":"stacks:04NY","tag":"04NY","title":"Integral and finite morphisms · Lemma 04NY","summary":"A finite morphism of algebraic spaces is quasi-finite.","statement_latex":"A finite morphism of algebraic spaces is quasi-finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NY","source_file":"spaces-morphisms.tex","source_line":9250,"source_end_line":9253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9250-L9253","statement_sha256":"284fd06b192b3fe01c250859231366319cd4a8d921ff2220fcd0b9b143c6359a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11145,"rank":11145,"depth":43,"x":309.169,"y":1682.539,"cluster":"algebraic-spaces"},{"id":"stacks:04NZ","tag":"04NZ","title":"Integral and finite morphisms · Lemma 04NZ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f is finite, and • f is affine and proper.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over\n$S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is finite, and\n\\item $f$ is affine and proper.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04NZ","source_file":"spaces-morphisms.tex","source_line":9269,"source_end_line":9277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9269-L9277","statement_sha256":"011507125f4f632d84f254ee1b1ec207480b9165dc80a788a30fdf57ec176fd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11146,"rank":11146,"depth":57,"x":105.116,"y":1584.043,"cluster":"algebraic-spaces"},{"id":"stacks:081Y","tag":"081Y","title":"Integral and finite morphisms · Lemma 081Y","summary":"A closed immersion is finite (and a fortiori integral).","statement_latex":"A closed immersion is finite (and a fortiori integral).","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081Y","source_file":"spaces-morphisms.tex","source_line":9287,"source_end_line":9290,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9287-L9290","statement_sha256":"e24b81e4748fb76e21a19ad4b2d5f1561c446f1b1a133f5bc27ee2e265740582","origin":"The Stacks Project","memory_eligible":false,"source_rank":11147,"rank":11147,"depth":0,"x":335.029,"y":1540.843,"cluster":"algebraic-spaces"},{"id":"stacks:0CZ2","tag":"0CZ2","title":"Integral and finite morphisms · Lemma 0CZ2","summary":"Let S be a scheme. Let X_i → Y, i = 1, …, n be finite morphisms of algebraic spaces over S. Then X_1 amalg … amalg X_n → Y is finite too.","statement_latex":"Let $S$ be a scheme.\nLet $X_i \\to Y$, $i = 1, \\ldots, n$ be finite morphisms of\nalgebraic spaces over $S$.\nThen $X_1 \\amalg \\ldots \\amalg X_n \\to Y$ is finite too.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZ2","source_file":"spaces-morphisms.tex","source_line":9296,"source_end_line":9302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9296-L9302","statement_sha256":"60b673162fae14c85627583d77cf824722b5e7908a5c2b2d000de933efd50611","origin":"The Stacks Project","memory_eligible":false,"source_rank":11148,"rank":11148,"depth":1,"x":200.095,"y":1703.325,"cluster":"algebraic-spaces"},{"id":"stacks:081Z","tag":"081Z","title":"Integral and finite morphisms · Lemma 081Z","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of algebraic spaces over S. • If g ∘ f is finite and g separated then f is finite. • If g ∘ f is integral and g separated then f is integral.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $g \\circ f$ is finite and $g$ separated then $f$ is finite.\n\\item If $g \\circ f$ is integral and $g$ separated then $f$ is integral.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/081Z","source_file":"spaces-morphisms.tex","source_line":9310,"source_end_line":9318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9310-L9318","statement_sha256":"c8977a2f735271923234127190d1d20dec24e074ca2baea159875ff19aa21560","origin":"The Stacks Project","memory_eligible":false,"source_rank":11149,"rank":11149,"depth":56,"x":168.897,"y":1506.744,"cluster":"algebraic-spaces"},{"id":"stacks:03ZU","tag":"03ZU","title":"Finite locally free morphisms · Lemma 03ZU","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. Then f is finite locally free (in the sense of Section [Tag 03HA]) if and only if f is affine and the sheaf f_*O_X is a finite locally free O_Y-module.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable morphism\nof algebraic spaces over $S$. Then $f$ is finite locally free\n(in the sense of Section \\ref{section-representable})\nif and only if $f$ is affine and the sheaf $f_*\\mathcal{O}_X$ is\na finite locally free $\\mathcal{O}_Y$-module.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZU","source_file":"spaces-morphisms.tex","source_line":9346,"source_end_line":9353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9346-L9353","statement_sha256":"469407ba6084729a26c99743e23c6aa3a2ca206ab84918be13f91d4fcfa9a73e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11150,"rank":11150,"depth":46,"x":350.174,"y":1634.129,"cluster":"algebraic-spaces"},{"id":"stacks:03ZV","tag":"03ZV","title":"Finite locally free morphisms · Definition 03ZV","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say that f is finite locally free if f is affine and f_*O_X is a finite locally free O_Y-module. In this case we say f is has rank or degree d if the sheaf f_*O_X is finite locally free of rank d.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nWe say that $f$ is {\\it finite locally free} if $f$ is affine\nand $f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_Y$-module.\nIn this case we say $f$ is\nhas {\\it rank} or {\\it degree} $d$\nif the sheaf $f_*\\mathcal{O}_X$ is finite locally free of rank $d$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Finite locally free morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZV","source_file":"spaces-morphisms.tex","source_line":9393,"source_end_line":9402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9393-L9402","statement_sha256":"d2e4669dd5ee80335248d20cce84732ddde37b54082468fc9f91bfaba2297080","origin":"The Stacks Project","memory_eligible":false,"source_rank":11151,"rank":11151,"depth":0,"x":113.82,"y":1643.067,"cluster":"algebraic-spaces"},{"id":"stacks:03ZW","tag":"03ZW","title":"Finite locally free morphisms · Lemma 03ZW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is representable and finite locally free, • f is finite locally free, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is finite locally free, and • there exists a Zariski covering Y = ⋃ Y_i such that each morphism f^-1(Y_i) → Y_i is finite locally free.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is representable and finite locally free,\n\\item $f$ is finite locally free,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is finite locally free, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such that\neach morphism $f^{-1}(Y_i) \\to Y_i$ is finite locally free.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZW","source_file":"spaces-morphisms.tex","source_line":9404,"source_end_line":9417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9404-L9417","statement_sha256":"1806eeb42c2a99c9a204772e41560a99ac00671ce5ca68d5fa648fb01088a094","origin":"The Stacks Project","memory_eligible":false,"source_rank":11152,"rank":11152,"depth":46,"x":281.088,"y":1502.219,"cluster":"algebraic-spaces"},{"id":"stacks:03ZX","tag":"03ZX","title":"Finite locally free morphisms · Lemma 03ZX","summary":"The composition of finite locally free morphisms is finite locally free.","statement_latex":"The composition of finite locally free morphisms is finite locally free.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZX","source_file":"spaces-morphisms.tex","source_line":9445,"source_end_line":9448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9445-L9448","statement_sha256":"8d129f8396628c4e97dbb7df41add6446f55fe710d72087e3705c0c9d6a817de","origin":"The Stacks Project","memory_eligible":false,"source_rank":11153,"rank":11153,"depth":0,"x":271.003,"y":1701.195,"cluster":"algebraic-spaces"},{"id":"stacks:03ZY","tag":"03ZY","title":"Finite locally free morphisms · Lemma 03ZY","summary":"The base change of a finite locally free morphism is finite locally free.","statement_latex":"The base change of a finite locally free morphism is finite locally free.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03ZY","source_file":"spaces-morphisms.tex","source_line":9454,"source_end_line":9457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9454-L9457","statement_sha256":"775dd980ea06b6a90ed2531e79705c90b0955be16a7907f16aa9bf21f155c1c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11154,"rank":11154,"depth":0,"x":118.272,"y":1548.594,"cluster":"algebraic-spaces"},{"id":"stacks:0416","tag":"0416","title":"Finite locally free morphisms · Lemma 0416","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is finite locally free, • f is finite, flat, and locally of finite presentation. If Y is locally Noetherian these are also equivalent to • [(3)] f is finite and flat.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is finite locally free,\n\\item $f$ is finite, flat, and locally of finite presentation.\n\\end{enumerate}\nIf $Y$ is locally Noetherian these are also equivalent to\n\\begin{enumerate}\n\\item[(3)] $f$ is finite and flat.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Finite locally free morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0416","source_file":"spaces-morphisms.tex","source_line":9463,"source_end_line":9476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9463-L9476","statement_sha256":"6d4989a4ab8fd76a08201dfcc654b858583c00e361df6e9d638619d321f71aed","origin":"The Stacks Project","memory_eligible":false,"source_rank":11155,"rank":11155,"depth":47,"x":353.852,"y":1574.483,"cluster":"algebraic-spaces"},{"id":"stacks:0EMM","tag":"0EMM","title":"Rational maps · Definition 0EMM","summary":"Let S be a scheme. Let X, Y be algebraic spaces over S. • Let f : U → Y, g : V → Y be morphisms of algebraic spaces over S defined on dense open subspaces U, V of X. We say that f is equivalent to g if f|_W = g|_W for some dense open subspace W ⊂ U ∩ V. • A rational map from X to Y is an equivalence class for the equivalence relation defined in (1). • Given morphisms X → B and Y → B of algebraic spaces over S we say that a rational map from X to Y is a B-rational map from…","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$.\n\\begin{enumerate}\n\\item Let $f : U \\to Y$, $g : V \\to Y$ be morphisms of algebraic spaces\nover $S$ defined on dense open subspaces $U$, $V$ of $X$. We say that $f$ is\n{\\it equivalent} to $g$ if $f|_W = g|_W$ for some dense open\nsubspace $W \\subset U \\cap V$.\n\\item A {\\it rational map from $X$ to $Y$}\nis an equivalence class for the equivalence relation defined in (1).\n\\item Given morphisms $X \\to B$ and $Y \\to B$ of algebraic spaces over $S$\nwe say that a rational map from $X$ to $Y$ is a\n{\\it $B$-rational map from $X$ to $Y$}\nif there exists a representative $f : U \\to Y$ of the equivalence\nclass which is a morphism over $B$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMM","source_file":"spaces-morphisms.tex","source_line":9505,"source_end_line":9521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9505-L9521","statement_sha256":"51a75d9589af3926d9dd5b802a985315d6ca5982b284a423ce07fb8c370a1d14","origin":"The Stacks Project","memory_eligible":false,"source_rank":11156,"rank":11156,"depth":0,"x":159.121,"y":1689.183,"cluster":"algebraic-spaces"},{"id":"stacks:0EMN","tag":"0EMN","title":"Rational maps · Definition 0EMN","summary":"Let S be a scheme. Let X be an algebraic space over S. A rational function on X is a rational map from X to A^1_S.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A\n{\\it rational function on $X$} is a rational map from $X$ to $\\mathbf{A}^1_S$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMN","source_file":"spaces-morphisms.tex","source_line":9533,"source_end_line":9537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9533-L9537","statement_sha256":"e3f0a45fada47f116700d4778a1e70917268ded8297281d25cdaee130ef83079","origin":"The Stacks Project","memory_eligible":false,"source_rank":11157,"rank":11157,"depth":0,"x":210.526,"y":1493.911,"cluster":"algebraic-spaces"},{"id":"stacks:0EMP","tag":"0EMP","title":"Rational maps · Definition 0EMP","summary":"Let S be a scheme. Let X be an algebraic space over S. The ring of rational functions on X is the ring R(X) whose elements are rational functions with addition and multiplication as just described.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThe {\\it ring of rational functions on $X$}\nis the ring $R(X)$ whose elements are rational functions with\naddition and multiplication as just described.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMP","source_file":"spaces-morphisms.tex","source_line":9565,"source_end_line":9572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9565-L9572","statement_sha256":"42f0389464c735ff4be97df74c9d2c2db1463fbd1d1738706212951606cd7c31","origin":"The Stacks Project","memory_eligible":false,"source_rank":11158,"rank":11158,"depth":0,"x":329.775,"y":1667.247,"cluster":"algebraic-spaces"},{"id":"stacks:0EMQ","tag":"0EMQ","title":"Rational maps · Definition 0EMQ","summary":"Let S be a scheme. Let φ be a rational map between two algebraic spaces X and Y over S. We say φ is defined in a point x ∈ |X| if there exists a representative (U, f) of φ with x ∈ |U|. The domain of definition of φ is the set of all points where φ is defined.","statement_latex":"Let $S$ be a scheme. Let $\\varphi$ be a rational map between two\nalgebraic spaces $X$ and $Y$ over $S$. We say\n$\\varphi$ is {\\it defined in a point $x \\in |X|$} if there exists a\nrepresentative $(U, f)$ of $\\varphi$ with $x \\in |U|$. The\n{\\it domain of definition} of $\\varphi$ is the set of all points\nwhere $\\varphi$ is defined.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMQ","source_file":"spaces-morphisms.tex","source_line":9578,"source_end_line":9586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9578-L9586","statement_sha256":"a0ddceccf86145494cd3c32d4d32b8f54f59d28789cd4c40a3774d4ca9440a10","origin":"The Stacks Project","memory_eligible":false,"source_rank":11159,"rank":11159,"depth":0,"x":102.22,"y":1607.035,"cluster":"algebraic-spaces"},{"id":"stacks:0EMR","tag":"0EMR","title":"Rational maps · Lemma 0EMR","summary":"Let S be a scheme. Let X and Y be algebraic spaces over S. Assume X is reduced and Y is separated over S. Let φ be a rational map from X to Y with domain of definition U ⊂ X. Then there exists a unique morphism f : U → Y of algebraic spaces representing φ.","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$.\nAssume $X$ is reduced and $Y$ is separated over $S$. Let\n$\\varphi$ be a rational map from $X$ to $Y$ with domain of definition\n$U \\subset X$. Then there exists a unique morphism $f : U \\to Y$\nof algebraic spaces representing $\\varphi$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Rational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMR","source_file":"spaces-morphisms.tex","source_line":9594,"source_end_line":9601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9594-L9601","statement_sha256":"8550450f1e2d2e5162d6bc7a424f92a0af2793fe3f6e34a3d61dab5cea1b6c32","origin":"The Stacks Project","memory_eligible":false,"source_rank":11160,"rank":11160,"depth":4,"x":318.655,"y":1522.229,"cluster":"algebraic-spaces"},{"id":"stacks:0EMS","tag":"0EMS","title":"Rational maps · Definition 0EMS","summary":"Let S be a scheme. Let X and Y be algebraic spaces over S. Assume |X| and |Y| are irreducible. A rational map from X to Y is called dominant if any representative f : U → Y is a dominant morphism in the sense of Definition [Tag 0ABL].","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces over $S$.\nAssume $|X|$ and $|Y|$ are irreducible. A rational map from $X$ to $Y$\nis called {\\it dominant} if any representative $f : U \\to Y$ is a dominant\nmorphism in the sense of Definition \\ref{definition-dominant}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMS","source_file":"spaces-morphisms.tex","source_line":9627,"source_end_line":9633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9627-L9633","statement_sha256":"dbaac338c851e8668d0944f78ca50f77e081a006c7950188eb10f94e36d2d808","origin":"The Stacks Project","memory_eligible":false,"source_rank":11161,"rank":11161,"depth":1,"x":227.166,"y":1707.761,"cluster":"algebraic-spaces"},{"id":"stacks:0EMT","tag":"0EMT","title":"Rational maps · Definition 0EMT","summary":"Let S be a scheme. Let X and Y be algebraic spaces over S with |X| and |Y| irreducible. We say X and Y are birational if X and Y are isomorphic in the category of irreducible algebraic spaces over S and dominant rational maps.","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be algebraic spaces\nover $S$ with $|X|$ and $|Y|$ irreducible.\nWe say $X$ and $Y$ are {\\it birational} if $X$ and $Y$ are isomorphic\nin the category of irreducible algebraic spaces over $S$\nand dominant rational maps.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Rational maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMT","source_file":"spaces-morphisms.tex","source_line":9652,"source_end_line":9659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9652-L9659","statement_sha256":"ebb7f93d5e1d00d2c59d889d8479867df22325012ac5f0eee31e85aef124cd7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11162,"rank":11162,"depth":0,"x":145.346,"y":1518.849,"cluster":"algebraic-spaces"},{"id":"stacks:0EMU","tag":"0EMU","title":"Rational maps · Lemma 0EMU","summary":"Let S be a scheme. Let X and Y be algebraic space over S with |X| and |Y| irreducible. Then X and Y are birational if and only if there are nonempty open subspaces U ⊂ X and V ⊂ Y which are isomorphic as algebraic spaces over S.","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ be algebraic\nspace over $S$ with $|X|$ and $|Y|$ irreducible.\nThen $X$ and $Y$ are birational if and only if\nthere are nonempty open subspaces $U \\subset X$ and $V \\subset Y$\nwhich are isomorphic as algebraic spaces over $S$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Rational maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMU","source_file":"spaces-morphisms.tex","source_line":9671,"source_end_line":9678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9671-L9678","statement_sha256":"36bec6ed761d7a6cae7da01b98635efb343ada612297356f5f23554adaa1d42d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11163,"rank":11163,"depth":8,"x":357.811,"y":1611.817,"cluster":"algebraic-spaces"},{"id":"stacks:0820","tag":"0820","title":"Relative normalization of algebraic spaces · Lemma 0820","summary":"Let S be a scheme. Let X be an algebraic space over S. Let A be a quasi-coherent sheaf of O_X-algebras. There exists a quasi-coherent sheaf of O_X-algebras A' ⊂ A such that for any affine object U of X_etale the ring A'(U) ⊂ A(U) is the integral closure of O_X(U) in A(U).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras.\nThere exists a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras\n$\\mathcal{A}' \\subset \\mathcal{A}$ such that\nfor any affine object $U$ of $X_\\etale$ the ring\n$\\mathcal{A}'(U) \\subset \\mathcal{A}(U)$ is\nthe integral closure of $\\mathcal{O}_X(U)$ in $\\mathcal{A}(U)$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0820","source_file":"spaces-morphisms.tex","source_line":9717,"source_end_line":9726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9717-L9726","statement_sha256":"f94c376ea375a22b565f67accc688246c6777285b2c6967d0ef5d7f59d3aaa99","origin":"The Stacks Project","memory_eligible":false,"source_rank":11164,"rank":11164,"depth":46,"x":126.148,"y":1663.873,"cluster":"algebraic-spaces"},{"id":"stacks:0821","tag":"0821","title":"Relative normalization of algebraic spaces · Definition 0821","summary":"Let S be a scheme. Let X be an algebraic space over S. Let A be a quasi-coherent sheaf of O_X-algebras. The integral closure of O_X in A is the quasi-coherent O_X-subalgebra A' ⊂ A constructed in Lemma [Tag 0820] above.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{A}$ be a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras.\nThe {\\it integral closure of $\\mathcal{O}_X$ in $\\mathcal{A}$} is the\nquasi-coherent $\\mathcal{O}_X$-subalgebra $\\mathcal{A}' \\subset \\mathcal{A}$\nconstructed in Lemma \\ref{lemma-integral-closure} above.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0821","source_file":"spaces-morphisms.tex","source_line":9758,"source_end_line":9765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9758-L9765","statement_sha256":"bc9c02c197c8a0c270a6c9510cb2e18d5a541b1d8845df4c418954065e200669","origin":"The Stacks Project","memory_eligible":false,"source_rank":11165,"rank":11165,"depth":47,"x":255.239,"y":1493.866,"cluster":"algebraic-spaces"},{"id":"stacks:0822","tag":"0822","title":"Relative normalization of algebraic spaces · Definition 0822","summary":"Let S be a scheme. Let f : Y → X be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let O' be the integral closure of O_X in f_*O_Y. The normalization of X in Y is the morphism of algebraic spaces ν : X' = underlineSpec_X(O') → X over S. It comes equipped with a natural factorization Y xrightarrowf' X' xrightarrowν X of the initial morphism f.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated\nmorphism of algebraic spaces over $S$. Let $\\mathcal{O}'$ be the integral\nclosure of $\\mathcal{O}_X$ in $f_*\\mathcal{O}_Y$. The {\\it normalization of\n$X$ in $Y$} is the morphism of algebraic spaces\n$$\n\\nu : X' = \\underline{\\Spec}_X(\\mathcal{O}') \\to X\n$$\nover $S$. It comes equipped with a natural factorization\n$$\nY \\xrightarrow{f'} X' \\xrightarrow{\\nu} X\n$$\nof the initial morphism $f$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0822","source_file":"spaces-morphisms.tex","source_line":9775,"source_end_line":9789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9775-L9789","statement_sha256":"75db111e9a218d6af5416fa6bbc15c3d300e6abfa938b94e6b58b81fc77eaaf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11166,"rank":11166,"depth":0,"x":296.805,"y":1692.675,"cluster":"algebraic-spaces"},{"id":"stacks:0ABP","tag":"0ABP","title":"Relative normalization of algebraic spaces · Lemma 0ABP","summary":"Let S be a scheme. Let f : Y → X be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let Y → X' → X be the normalization of X in Y. • If W → X is an étale morphism of algebraic spaces over S, then W ×_X X' is the normalization of W in W ×_X Y. • If Y and X are representable, then X' is representable and is canonically isomorphic to the normalization of the scheme X in the scheme Y as constructed in Morphisms, Section [Tag 035E].","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated\nmorphism of algebraic spaces over $S$. Let $Y \\to X' \\to X$ be the\nnormalization of $X$ in $Y$.\n\\begin{enumerate}\n\\item If $W \\to X$ is an \\'etale morphism of algebraic spaces over $S$,\nthen $W \\times_X X'$ is the normalization of $W$ in $W \\times_X Y$.\n\\item If $Y$ and $X$ are representable, then $X'$ is representable\nand is canonically isomorphic to the normalization of the scheme $X$\nin the scheme $Y$ as constructed in\nMorphisms, Section \\ref{morphisms-section-normalization}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABP","source_file":"spaces-morphisms.tex","source_line":9796,"source_end_line":9809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9796-L9809","statement_sha256":"9e4797b7fea5f99b32115276dc68b3659e6a24bf4bdd7a3f39eedc647c029efa","origin":"The Stacks Project","memory_eligible":false,"source_rank":11167,"rank":11167,"depth":0,"x":106.088,"y":1569.539,"cluster":"algebraic-spaces"},{"id":"stacks:0823","tag":"0823","title":"Relative normalization of algebraic spaces · Lemma 0823","summary":"Let S be a scheme. Let f : Y → X be a quasi-compact and quasi-separated morphism of algebraic spaces over S. The factorization f = ν ∘ f', where ν : X' → X is the normalization of X in Y is characterized by the following two properties: • the morphism ν is integral, and • for any factorization f = π ∘ g, with π : Z → X integral, there exists a commutative diagram xymatrix Y ar[d]_f' ar[r]_g & Z ar[d]^π X' ar[ru]^h ar[r]^ν & X for a unique morphism h : X' → Z. Moreover, in…","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated\nmorphism of algebraic spaces over $S$. The factorization $f = \\nu \\circ f'$,\nwhere $\\nu : X' \\to X$ is the normalization of $X$ in $Y$ is characterized\nby the following two properties:\n\\begin{enumerate}\n\\item the morphism $\\nu$ is integral, and\n\\item for any factorization $f = \\pi \\circ g$, with $\\pi : Z \\to X$\nintegral, there exists a commutative diagram\n$$\n\\xymatrix{\nY \\ar[d]_{f'} \\ar[r]_g & Z \\ar[d]^\\pi \\\\\nX' \\ar[ru]^h \\ar[r]^\\nu & X\n}\n$$\nfor a unique morphism $h : X' \\to Z$.\n\\end{enumerate}\nMoreover, in (2) the morphism $h : X' \\to Z$ is the normalization of\n$Z$ in $Y$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0823","source_file":"spaces-morphisms.tex","source_line":9824,"source_end_line":9844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9824-L9844","statement_sha256":"88637cfd83fc752ffe6310335dd7fb65baee2090daf6ff588feeed265e9db9ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":11168,"rank":11168,"depth":57,"x":345.982,"y":1552.103,"cluster":"algebraic-spaces"},{"id":"stacks:0AYF","tag":"0AYF","title":"Relative normalization of algebraic spaces · Lemma 0AYF","summary":"Let S be a scheme. Let f : Y → X be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let X' → X be the normalization of X in Y. If Y is reduced, so is X'.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and\nquasi-separated morphism of algebraic spaces over $S$.\nLet $X' \\to X$ be the normalization of $X$ in $Y$.\nIf $Y$ is reduced, so is $X'$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYF","source_file":"spaces-morphisms.tex","source_line":9877,"source_end_line":9883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9877-L9883","statement_sha256":"9a7cecec36e74241c2c0ed20d7b962272bedbb8351ff7902aaf61176ea82272a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11169,"rank":11169,"depth":0,"x":182.948,"y":1701.23,"cluster":"algebraic-spaces"},{"id":"stacks:0AYG","tag":"0AYG","title":"Relative normalization of algebraic spaces · Lemma 0AYG","summary":"Let S be a scheme. Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Let X' → X be the normalization of X in Y. If x' ∈ |X'| is a point of codimension 0 (Properties of Spaces, Definition [Tag 04NA]), then x' is the image of some y ∈ |Y| of codimension 0.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a quasi-compact and quasi-separated\nmorphism of schemes. Let $X' \\to X$ be the normalization of $X$ in $Y$.\nIf $x' \\in |X'|$ is a point of codimension $0$\n(Properties of Spaces, Definition\n\\ref{spaces-properties-definition-dimension-local-ring}), then\n$x'$ is the image of some $y \\in |Y|$ of codimension $0$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYG","source_file":"spaces-morphisms.tex","source_line":9890,"source_end_line":9898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9890-L9898","statement_sha256":"922907a33bc416479424d81901b57ffc742e5c497a7967511055070790d3bebd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11170,"rank":11170,"depth":17,"x":183.243,"y":1498.556,"cluster":"algebraic-spaces"},{"id":"stacks:0824","tag":"0824","title":"Relative normalization of algebraic spaces · Lemma 0824","summary":"Let S be a scheme. Let f : Y → X be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Suppose that Y = Y_1 amalg Y_2 is a disjoint union of two algebraic spaces. Write f_i = f|_Y_i. Let X_i' be the normalization of X in Y_i. Then X_1' amalg X_2' is the normalization of X in Y.","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be a quasi-compact and quasi-separated morphism of\nalgebraic spaces over $S$.\nSuppose that $Y = Y_1 \\amalg Y_2$ is a disjoint union of two\nalgebraic spaces.\nWrite $f_i = f|_{Y_i}$. Let $X_i'$ be the normalization of $X$ in $Y_i$.\nThen $X_1' \\amalg X_2'$ is the normalization of $X$ in $Y$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0824","source_file":"spaces-morphisms.tex","source_line":9912,"source_end_line":9921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9912-L9921","statement_sha256":"b0459e416ffd0f24a2581e42868816c7b2eddb07d2610236614097bbd5ba0899","origin":"The Stacks Project","memory_eligible":false,"source_rank":11171,"rank":11171,"depth":0,"x":346.171,"y":1648.32,"cluster":"algebraic-spaces"},{"id":"stacks:0A0Q","tag":"0A0Q","title":"Relative normalization of algebraic spaces · Lemma 0A0Q","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact, quasi-separated and universally closed morphisms of algebraic spaces over S. Then f_*O_X is integral over O_Y. In other words, the normalization of Y in X is equal to the factorization X → underlineSpec_Y(f_*O_X) → Y of Remark [Tag 081X].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a quasi-compact, quasi-separated and\nuniversally closed morphisms of algebraic spaces over $S$.\nThen $f_*\\mathcal{O}_X$ is integral over $\\mathcal{O}_Y$. In other\nwords, the normalization of $Y$ in $X$ is equal to the factorization\n$$\nX \\longrightarrow \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)\n\\longrightarrow Y\n$$\nof Remark \\ref{remark-factorization-quasi-compact-quasi-separated}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0Q","source_file":"spaces-morphisms.tex","source_line":9927,"source_end_line":9939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9927-L9939","statement_sha256":"07aa8c357811959f837c54a8e21cb0a448250f061b7f13c664d11b02220fd1bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11172,"rank":11172,"depth":58,"x":105.356,"y":1630.317,"cluster":"algebraic-spaces"},{"id":"stacks:0825","tag":"0825","title":"Relative normalization of algebraic spaces · Lemma 0825","summary":"Let S be a scheme. Let f : Y → X be an integral morphism of algebraic spaces over S. Then the integral closure of X in Y is equal to Y.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be an integral morphism\nof algebraic spaces over $S$.\nThen the integral closure of $X$ in $Y$ is equal to $Y$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0825","source_file":"spaces-morphisms.tex","source_line":9971,"source_end_line":9976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9971-L9976","statement_sha256":"897163dba830fd2b847308567eab42f5bdbb57b4a8da15692ea2e0537a32e685","origin":"The Stacks Project","memory_eligible":false,"source_rank":11173,"rank":11173,"depth":59,"x":297.596,"y":1506.827,"cluster":"algebraic-spaces"},{"id":"stacks:0BB0","tag":"0BB0","title":"Relative normalization of algebraic spaces · Lemma 0BB0","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that • Y is Nagata, • f is quasi-separated of finite type, • X is reduced. Then the normalization ν : Y' → Y of Y in X is finite.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume that\n\\begin{enumerate}\n\\item $Y$ is Nagata,\n\\item $f$ is quasi-separated of finite type,\n\\item $X$ is reduced.\n\\end{enumerate}\nThen the normalization $\\nu : Y' \\to Y$ of $Y$ in $X$ is finite.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Relative normalization of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BB0","source_file":"spaces-morphisms.tex","source_line":9983,"source_end_line":9993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L9983-L9993","statement_sha256":"7ce188b8e677409735488aad2c35c0f70d10cac6cad1a84f964417072dad9542","origin":"The Stacks Project","memory_eligible":false,"source_rank":11174,"rank":11174,"depth":42,"x":255.107,"y":1707.165,"cluster":"algebraic-spaces"},{"id":"stacks:0BB1","tag":"0BB1","title":"Normalization · Lemma 0BB1","summary":"Let S be a scheme. Let X be an algebraic space over S. The following are equivalent • there is a surjective étale morphism U → X where U is a scheme such that every quasi-compact open of U has finitely many irreducible components, • for every scheme U and every étale morphism U → X every quasi-compact open of U has finitely many irreducible components, • for every quasi-compact algebraic space Y étale over X the set of codimension 0 points of Y (Properties of Spaces,…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item there is a surjective \\'etale morphism $U \\to X$ where $U$\nis a scheme such that every quasi-compact open of $U$ has\nfinitely many irreducible components,\n\\item for every scheme $U$ and every \\'etale morphism\n$U \\to X$ every quasi-compact open of $U$ has finitely many\nirreducible components,\n\\item for every quasi-compact algebraic space $Y$ \\'etale over $X$\nthe set of codimension $0$ points of $Y$ (Properties of Spaces,\nDefinition \\ref{spaces-properties-definition-dimension-local-ring})\nis finite, and\n\\item for every quasi-compact algebraic space $Y$ \\'etale over $X$\nthe space $|Y|$ has finitely many irreducible components.\n\\end{enumerate}\nIf $X$ is representable this means that every quasi-compact open of $X$\nhas finitely many irreducible components.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BB1","source_file":"spaces-morphisms.tex","source_line":10026,"source_end_line":10046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10026-L10046","statement_sha256":"dde73bc63b4c3e579d81eb5c008726092dd5852108cf57bd8088f849021233ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":11175,"rank":11175,"depth":42,"x":125.204,"y":1535.161,"cluster":"algebraic-spaces"},{"id":"stacks:0GMB","tag":"0GMB","title":"Normalization · Lemma 0GMB","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Then X satisfies the equivalent conditions of Lemma [Tag 0BB1].","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space\nover $S$. Then $X$ satisfies the equivalent conditions of\nLemma \\ref{lemma-prepare-normalization}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMB","source_file":"spaces-morphisms.tex","source_line":10075,"source_end_line":10080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10075-L10080","statement_sha256":"e4727e20fe5cf35ba1429b0dad800ef199bd86c59451de49c1f48bd81f0104d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11176,"rank":11176,"depth":43,"x":359.544,"y":1588.336,"cluster":"algebraic-spaces"},{"id":"stacks:0GMC","tag":"0GMC","title":"Normalization · Lemma 0GMC","summary":"Let S be a scheme. Let f : X → Y be a flat morphism of algebraic spaces over S. Then for x ∈ |X| we have: x has codimension 0 in X ⇒ f(x) has codimension 0 in Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a flat morphism of algebraic\nspaces over $S$. Then for $x \\in |X|$ we have: $x$ has codimension $0$ in\n$X \\Rightarrow f(x)$ has codimension $0$ in $Y$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMC","source_file":"spaces-morphisms.tex","source_line":10091,"source_end_line":10096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10091-L10096","statement_sha256":"45f058d24452e55ccdb770ab913d110adb4a15b37dc66e12ddab3ec4287d868c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11177,"rank":11177,"depth":18,"x":143.772,"y":1682.187,"cluster":"algebraic-spaces"},{"id":"stacks:0GMD","tag":"0GMD","title":"Normalization · Lemma 0GMD","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is flat and locally of finite type and assume Y satisfies the equivalent conditions of Lemma [Tag 0BB1]. Then X satisfies the equivalent conditions of Lemma [Tag 0BB1] and for x ∈ |X| we have: x has codimension 0 in X ⇒ f(x) has codimension 0 in Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume $f$ is flat and locally of finite type and\nassume $Y$ satisfies the equivalent conditions of\nLemma \\ref{lemma-prepare-normalization}.\nThen $X$ satisfies the equivalent conditions of\nLemma \\ref{lemma-prepare-normalization} and for $x \\in |X|$ we have:\n$x$ has codimension $0$ in $X \\Rightarrow f(x)$ has codimension $0$ in $Y$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMD","source_file":"spaces-morphisms.tex","source_line":10107,"source_end_line":10116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10107-L10116","statement_sha256":"fc943930b498a39c926d4a6b87f1fa7a2355e612c22f9669d7b864c4d7493ca3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11178,"rank":11178,"depth":44,"x":227.488,"y":1490.362,"cluster":"algebraic-spaces"},{"id":"stacks:07U4","tag":"07U4","title":"Normalization · Lemma 07U4","summary":"Let S be a scheme. For every algebraic space X over S satisfying the equivalent conditions of Lemma [Tag 0BB1] there exists a morphism of algebraic spaces ν_X : X^ν → X with the following properties • if X satisfies the equivalent conditions of Lemma [Tag 0BB1] then X^ν is normal and ν_X is integral, • if X is a scheme such that every quasi-compact open has finitely many irreducible components, then ν_X : X^ν → X is the normalization of X constructed in Morphisms, Section…","statement_latex":"Let $S$ be a scheme. For every algebraic space $X$ over $S$ satisfying\nthe equivalent conditions of Lemma \\ref{lemma-prepare-normalization}\nthere exists a morphism of algebraic spaces\n$$\n\\nu_X : X^\\nu \\longrightarrow X\n$$\nwith the following properties\n\\begin{enumerate}\n\\item if $X$ satisfies the equivalent conditions of\nLemma \\ref{lemma-prepare-normalization} then\n$X^\\nu$ is normal and $\\nu_X$ is integral,\n\\item if $X$ is a scheme such that every quasi-compact open has finitely\nmany irreducible components, then $\\nu_X : X^\\nu \\to X$ is the\nnormalization of $X$ constructed in\nMorphisms, Section \\ref{morphisms-section-normalization},\n\\item if $f : X \\to Y$ is a morphism of algebraic spaces over $S$\nwhich both satisfy the equivalent conditions of\nLemma \\ref{lemma-prepare-normalization} and every codimension $0$\npoint of $X$ is mapped by $f$ to a codimension $0$ point of $Y$, then\nthere is a unique morphism $f^\\nu : X^\\nu \\to Y^\\nu$ of algebraic\nspaces over $S$ such that $\\nu_Y \\circ f^\\nu = f \\circ \\nu_X$, and\n\\item if $f : X \\to Y$ is an \\'etale or smooth morphism of algebraic\nspaces and $Y$ satisfies the equivalent conditions of\nLemma \\ref{lemma-prepare-normalization}, then the hypotheses of (3)\nhold and the morphism $f^\\nu$ induces an isomorphism\n$X^\\nu  \\to X \\times_Y Y^\\nu$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07U4","source_file":"spaces-morphisms.tex","source_line":10138,"source_end_line":10167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10138-L10167","statement_sha256":"3d58e1736e3dab038b551e543612929bb8fd40c51b67fb55c2405030c632df44","origin":"The Stacks Project","memory_eligible":false,"source_rank":11179,"rank":11179,"depth":52,"x":320.108,"y":1679.497,"cluster":"algebraic-spaces"},{"id":"stacks:0BB2","tag":"0BB2","title":"Normalization · Definition 0BB2","summary":"Let S be a scheme. Let X be an algebraic space over S satisfying the equivalent conditions of Lemma [Tag 0BB1]. We define the normalization of X as the morphism ν_X : X^ν → X constructed in Lemma [Tag 07U4].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the\nequivalent conditions of Lemma \\ref{lemma-prepare-normalization}.\nWe define the {\\it normalization} of $X$ as the morphism\n$$\n\\nu_X : X^\\nu \\longrightarrow X\n$$\nconstructed in Lemma \\ref{lemma-normalization}.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Normalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BB2","source_file":"spaces-morphisms.tex","source_line":10334,"source_end_line":10343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10334-L10343","statement_sha256":"b9c0bf9d7fbfeae3108cde9d150915e4e67819a1134d6d66e43553bd5fa7cf54","origin":"The Stacks Project","memory_eligible":false,"source_rank":11180,"rank":11180,"depth":53,"x":99.5,"y":1592.502,"cluster":"algebraic-spaces"},{"id":"stacks:0BB3","tag":"0BB3","title":"Normalization · Lemma 0BB3","summary":"Let S be a scheme. Let X be an algebraic space over S satisfying the equivalent conditions of Lemma [Tag 0BB1]. The normalization morphism ν factors through the reduction X_red and X^ν → X_red is the normalization of X_red.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the\nequivalent conditions of Lemma \\ref{lemma-prepare-normalization}.\nThe normalization morphism $\\nu$ factors through the reduction $X_{red}$\nand $X^\\nu \\to X_{red}$ is the normalization of $X_{red}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BB3","source_file":"spaces-morphisms.tex","source_line":10352,"source_end_line":10358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10352-L10358","statement_sha256":"08ab59f8a5b2c6821f24e873894507c02817dc8b4d67339cbaa1a68aa0af517e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11181,"rank":11181,"depth":43,"x":332.358,"y":1531.414,"cluster":"algebraic-spaces"},{"id":"stacks:0BB4","tag":"0BB4","title":"Normalization · Lemma 0BB4","summary":"Let S be a scheme. Let X be an algebraic space over S satisfying the equivalent conditions of Lemma [Tag 0BB1]. • The normalization X^ν is normal. • The morphism ν : X^ν → X is integral and surjective. • The map |ν| : |X^ν| → |X| induces a bijection between the sets of points of codimension 0 (Properties of Spaces, Definition [Tag 04NA]). • Let Z → X be a morphism. Assume Z is a normal algebraic space and that for z ∈ |Z| we have: z has codimension 0 in Z ⇒ f(z) has…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying the\nequivalent conditions of Lemma \\ref{lemma-prepare-normalization}.\n\\begin{enumerate}\n\\item The normalization $X^\\nu$ is normal.\n\\item The morphism $\\nu : X^\\nu \\to X$ is integral and surjective.\n\\item The map $|\\nu| : |X^\\nu| \\to |X|$ induces a bijection between\nthe sets of points of codimension $0$ (Properties of Spaces,\nDefinition \\ref{spaces-properties-definition-dimension-local-ring}).\n\\item Let $Z \\to X$ be a morphism. Assume $Z$ is a normal algebraic space\nand that for $z \\in |Z|$ we have: $z$ has codimension $0$ in\n$Z \\Rightarrow f(z)$ has codimension $0$ in $X$. Then\nthere exists a unique factorization $Z \\to X^\\nu \\to X$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BB4","source_file":"spaces-morphisms.tex","source_line":10367,"source_end_line":10382,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10367-L10382","statement_sha256":"d6f182d720d8d374ed768871a7a1afec8676a7521ca7c2ab9c360fc5eefa2253","origin":"The Stacks Project","memory_eligible":false,"source_rank":11182,"rank":11182,"depth":53,"x":209.658,"y":1708.76,"cluster":"algebraic-spaces"},{"id":"stacks:0BB5","tag":"0BB5","title":"Normalization · Lemma 0BB5","summary":"Let S be a scheme. Let X be a Nagata algebraic space over S. The normalization ν : X^ν → X is a finite morphism.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Nagata algebraic space over $S$.\nThe normalization $\\nu : X^\\nu \\to X$ is a finite morphism.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BB5","source_file":"spaces-morphisms.tex","source_line":10432,"source_end_line":10436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10432-L10436","statement_sha256":"6a42a5762fb0ca05eadf80d6a719e2dbf2a6d0e5db0ba7287a8bf23775ae4032","origin":"The Stacks Project","memory_eligible":false,"source_rank":11183,"rank":11183,"depth":54,"x":157.468,"y":1508.171,"cluster":"algebraic-spaces"},{"id":"stacks:03XW","tag":"03XW","title":"Separated, locally quasi-finite morphisms · Lemma 03XW","summary":"Let S be a scheme. Consider a commutative diagram xymatrix V' ar[r] ar[rd] & T' ×_T X ar[r] ar[d] & X ar[d] & T' ar[r] & T of algebraic spaces over S. Assume • T' → T is an étale morphism of affine schemes, • X → T is a separated, locally quasi-finite morphism, • V' is an open subspace of T' ×_T X, and • V' → T' is quasi-affine. In this situation the image U of V' in X is a quasi-compact open subspace of X which is representable.","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nV' \\ar[r] \\ar[rd] & T' \\times_T X \\ar[r] \\ar[d] & X \\ar[d] \\\\\n& T' \\ar[r] & T\n}\n$$\nof algebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item $T' \\to T$ is an \\'etale morphism of affine schemes,\n\\item $X \\to T$ is a separated, locally quasi-finite morphism,\n\\item $V'$ is an open subspace of $T' \\times_T X$, and\n\\item $V' \\to T'$ is quasi-affine.\n\\end{enumerate}\nIn this situation the image $U$ of $V'$ in $X$ is a quasi-compact\nopen subspace of $X$ which is representable.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separated, locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XW","source_file":"spaces-morphisms.tex","source_line":10469,"source_end_line":10487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10469-L10487","statement_sha256":"93accb10f7d1d5298c8e9b8d8670b009cf58d2ffc859d00ed565bff6c596b97b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11184,"rank":11184,"depth":50,"x":357.453,"y":1626.582,"cluster":"algebraic-spaces"},{"id":"stacks:03XX","tag":"03XX","title":"Separated, locally quasi-finite morphisms · Proposition 03XX","summary":"Let S be a scheme. Let f : X → T be a morphism of algebraic spaces over S. Assume • T is representable, • f is locally quasi-finite, and • f is separated. Then X is representable.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to T$ be a morphism of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $T$ is representable,\n\\item $f$ is locally quasi-finite, and\n\\item $f$ is separated.\n\\end{enumerate}\nThen $X$ is representable.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Separated, locally quasi-finite morphisms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XX","source_file":"spaces-morphisms.tex","source_line":10589,"source_end_line":10600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10589-L10600","statement_sha256":"279e5f15e1ded6188940bf66f2000042b59d3b7e8fb274ab762c2c0e0678c92c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11185,"rank":11185,"depth":54,"x":114.531,"y":1652.769,"cluster":"algebraic-spaces"},{"id":"stacks:0418","tag":"0418","title":"Applications · Lemma 0418","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally quasi-finite and separated, then f is representable.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. If $f$ is locally quasi-finite and separated, then\n$f$ is representable.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0418","source_file":"spaces-morphisms.tex","source_line":10712,"source_end_line":10717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10712-L10717","statement_sha256":"f1e9852bea8cb556349d45960d9705b33621bd405796b720ae40f4af17579a33","origin":"The Stacks Project","memory_eligible":false,"source_rank":11186,"rank":11186,"depth":55,"x":272.751,"y":1495.469,"cluster":"algebraic-spaces"},{"id":"stacks:05W5","tag":"05W5","title":"Applications · Lemma 05W5","summary":"Universally injective étale maps are open immersions. Let S be a scheme. Let f : X → Y be an étale and universally injective morphism of algebraic spaces over S. Then f is an open immersion.","statement_latex":"\\begin{slogan}\nUniversally injective \\'etale maps are open immersions.\n\\end{slogan}\nLet $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale and universally\ninjective morphism of algebraic spaces over $S$. Then $f$ is an open\nimmersion.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05W5","source_file":"spaces-morphisms.tex","source_line":10728,"source_end_line":10736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10728-L10736","statement_sha256":"4588f386c63e8d9f4a6e500e40a6b8a0e12c402455a52fb9dfdc382d916a5b89","origin":"The Stacks Project","memory_eligible":false,"source_rank":11187,"rank":11187,"depth":55,"x":282.588,"y":1701.435,"cluster":"algebraic-spaces"},{"id":"stacks:0ABR","tag":"0ABR","title":"Zariski's Main Theorem (representable case) · Lemma 0ABR","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is representable, of finite type, and separated. Let Y' be the normalization of Y in X. Picture: xymatrix X ar[rd]_f ar[rr]_f' & & Y' ar[ld]^ν & Y & Then there exists an open subspace U' ⊂ Y' such that • (f')^-1(U') → U' is an isomorphism, and • (f')^-1(U') ⊂ X is the set of points at which f is quasi-finite.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$ which is representable, of finite type, and separated.\nLet $Y'$ be the normalization of $Y$ in $X$. Picture:\n$$\n\\xymatrix{\nX \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\\n& Y &\n}\n$$\nThen there exists an open subspace $U' \\subset Y'$ such that\n\\begin{enumerate}\n\\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and\n\\item $(f')^{-1}(U') \\subset X$ is the set of points at which\n$f$ is quasi-finite.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Zariski's Main Theorem (representable case)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABR","source_file":"spaces-morphisms.tex","source_line":10773,"source_end_line":10790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10773-L10790","statement_sha256":"28a225f6c46180f860b6df3a53ada26ba1a01af5ad1f618e1d687aa7dd37bc5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11188,"rank":11188,"depth":51,"x":109.537,"y":1555.0,"cluster":"algebraic-spaces"},{"id":"stacks:0ABS","tag":"0ABS","title":"Zariski's Main Theorem (representable case) · Lemma 0ABS","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-finite and separated. Let Y' be the normalization of Y in X. Picture: xymatrix X ar[rd]_f ar[rr]_f' & & Y' ar[ld]^ν & Y & Then f' is a quasi-compact open immersion and ν is integral. In particular f is quasi-affine.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is quasi-finite and separated.\nLet $Y'$ be the normalization of $Y$ in $X$.\nPicture:\n$$\n\\xymatrix{\nX \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\\n& Y &\n}\n$$\nThen $f'$ is a quasi-compact open immersion and $\\nu$ is integral.\nIn particular $f$ is quasi-affine.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Zariski's Main Theorem (representable case)","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABS","source_file":"spaces-morphisms.tex","source_line":10829,"source_end_line":10844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10829-L10844","statement_sha256":"722c3146c24edcb44c379d3c8e1fc37fe4acc3ccd589879f50cd2a5dede2f461","origin":"The Stacks Project","memory_eligible":false,"source_rank":11189,"rank":11189,"depth":57,"x":355.135,"y":1564.795,"cluster":"algebraic-spaces"},{"id":"stacks:05Z4","tag":"05Z4","title":"Universal homeomorphisms · Lemma 05Z4","summary":"Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. Then f is a universal homeomorphism (in the sense of Section [Tag 03HA]) if and only if for every morphism of algebraic spaces Z → Y the base change map Z ×_Y X → Z induces a homeomorphism |Z ×_Y X| → |Z|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$.\nThen $f$ is a universal homeomorphism\n(in the sense of Section \\ref{section-representable}) if and only\nif for every morphism of algebraic spaces $Z \\to Y$ the base change\nmap $Z \\times_Y X \\to Z$ induces a homeomorphism\n$|Z \\times_Y X| \\to |Z|$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Z4","source_file":"spaces-morphisms.tex","source_line":10883,"source_end_line":10892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10883-L10892","statement_sha256":"aa494f85d79851380da8ad05201d7502ae82c29a2a7d3a257451b5c43208b402","origin":"The Stacks Project","memory_eligible":false,"source_rank":11190,"rank":11190,"depth":48,"x":165.978,"y":1697.056,"cluster":"algebraic-spaces"},{"id":"stacks:05Z5","tag":"05Z5","title":"Universal homeomorphisms · Definition 05Z5","summary":"Let S be a scheme. A morphism f : X → Y of algebraic spaces over S is called a universal homeomorphism if and only if for every morphism of algebraic spaces Z → Y the base change Z ×_Y X → Z induces a homeomorphism |Z ×_Y X| → |Z|.","statement_latex":"Let $S$ be a scheme.\nA morphism $f : X \\to Y$ of algebraic spaces over $S$\nis called a {\\it universal homeomorphism}\nif and only if for every morphism of algebraic spaces $Z \\to Y$\nthe base change $Z \\times_Y X \\to Z$ induces a homeomorphism\n$|Z \\times_Y X| \\to |Z|$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universal homeomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Z5","source_file":"spaces-morphisms.tex","source_line":10914,"source_end_line":10922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10914-L10922","statement_sha256":"3aed1f564472562efb4a4a730a87b95c5c71cae78740f4d63560c09686aa7c57","origin":"The Stacks Project","memory_eligible":false,"source_rank":11191,"rank":11191,"depth":0,"x":199.13,"y":1492.002,"cluster":"algebraic-spaces"},{"id":"stacks:0CFT","tag":"0CFT","title":"Universal homeomorphisms · Lemma 0CFT","summary":"The base change of a universal homeomorphism of algebraic spaces by any morphism of algebraic spaces is a universal homeomorphism.","statement_latex":"The base change of a universal homeomorphism of algebraic spaces\nby any morphism of algebraic spaces is a universal homeomorphism.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CFT","source_file":"spaces-morphisms.tex","source_line":10962,"source_end_line":10966,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10962-L10966","statement_sha256":"b9bfbf0f2e15250416de38dcaf6d2b19daba8f6ab547af2ddf0c85fc82ee9474","origin":"The Stacks Project","memory_eligible":false,"source_rank":11192,"rank":11192,"depth":0,"x":339.716,"y":1662.179,"cluster":"algebraic-spaces"},{"id":"stacks:0CFU","tag":"0CFU","title":"Universal homeomorphisms · Lemma 0CFU","summary":"The composition of a pair of universal homeomorphisms of algebraic spaces is a universal homeomorphism.","statement_latex":"The composition of a pair of universal homeomorphisms of\nalgebraic spaces is a universal homeomorphism.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CFU","source_file":"spaces-morphisms.tex","source_line":10972,"source_end_line":10976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10972-L10976","statement_sha256":"6260559b4a00b91ae7d287f46b32b910784a136affcf133af8f9e37d2de28755","origin":"The Stacks Project","memory_eligible":false,"source_rank":11193,"rank":11193,"depth":0,"x":98.971,"y":1616.422,"cluster":"algebraic-spaces"},{"id":"stacks:08AK","tag":"08AK","title":"Universal homeomorphisms · Lemma 08AK","summary":"Let S be a scheme. Let X be an algebraic space over S. The canonical closed immersion X_red → X (see Properties of Spaces, Definition [Tag 047X]) is a universal homeomorphism.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe canonical closed immersion $X_{red} \\to X$ (see\nProperties of Spaces, Definition\n\\ref{spaces-properties-definition-reduced-induced-space})\nis a universal homeomorphism.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AK","source_file":"spaces-morphisms.tex","source_line":10982,"source_end_line":10989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10982-L10989","statement_sha256":"98f2022991d4bd0b598a3d3ca014b5ba92491fe7bb7558b7c58f297aa36ac016","origin":"The Stacks Project","memory_eligible":false,"source_rank":11194,"rank":11194,"depth":55,"x":313.492,"y":1513.46,"cluster":"algebraic-spaces"},{"id":"stacks:0AEH","tag":"0AEH","title":"Universal homeomorphisms · Lemma 0AEH","summary":"Let S be a scheme. Let f : Y → X be a universally injective, integral morphism of algebraic spaces over S. • The functor f_small, * : Sh(Y_etale) → Sh(X_etale) is fully faithful and its essential image is those sheaves of sets F on X_etale whose restriction to |X| setminus f(|Y|) is isomorphic to *, and • the functor f_small, * : Ab(Y_etale) → Ab(X_etale) is fully faithful and its essential image is those abelian sheaves on Y_etale whose support is contained in f(|Y|). In…","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a universally injective,\nintegral morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item The functor\n$$\nf_{small, *} : \\Sh(Y_\\etale) \\longrightarrow \\Sh(X_\\etale)\n$$\nis fully faithful and its essential image is those sheaves of sets\n$\\mathcal{F}$ on $X_\\etale$ whose restriction to $|X| \\setminus f(|Y|)$\nis isomorphic to $*$, and\n\\item the functor\n$$\nf_{small, *} : \\textit{Ab}(Y_\\etale) \\longrightarrow \\textit{Ab}(X_\\etale)\n$$\nis fully faithful and its essential image is those abelian sheaves on\n$Y_\\etale$ whose support is contained in $f(|Y|)$.\n\\end{enumerate}\nIn both cases $f_{small}^{-1}$ is a left inverse to the functor $f_{small, *}$.","area":"Algebraic Spaces","chapter":"Morphisms of Algebraic Spaces","chapter_id":"spaces-morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEH","source_file":"spaces-morphisms.tex","source_line":10999,"source_end_line":11019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-morphisms.tex#L10999-L11019","statement_sha256":"7c1b35c0d8ac58e7f22fe6bd6f64e057e34f6d282990a2bf0ff7319db41c1d4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11195,"rank":11195,"depth":54,"x":238.035,"y":1711.293,"cluster":"algebraic-spaces"},{"id":"stacks:03JL","tag":"03JL","title":"Universally bounded fibres · Definition 03JL","summary":"Let S be a scheme. Let X be an algebraic space over S, and let U be a scheme over S. Let f : U → X be a morphism over S. We say the fibres of f are universally bounded if there exists an integer n such that for all fields k and all morphisms Spec(k) → X the fibre product Spec(k) ×_X U is a finite scheme over k whose degree over k is ≤ n.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$, and\nlet $U$ be a scheme over $S$. Let $f : U \\to X$ be a morphism over $S$.\nWe say the {\\it fibres of $f$ are universally bounded}\\footnote{This is\nprobably nonstandard notation.}\nif there exists an integer $n$ such that for all fields\n$k$ and all morphisms $\\Spec(k) \\to X$ the fibre\nproduct $\\Spec(k) \\times_X U$ is a finite scheme over $k$\nwhose degree over $k$ is $\\leq n$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Universally bounded fibres","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JL","source_file":"decent-spaces.tex","source_line":59,"source_end_line":69,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L59-L69","statement_sha256":"1d5817be68863b10566bc99afa7e0e4267b0631b994d451231df199829ee802c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11196,"rank":11196,"depth":0,"x":134.486,"y":1522.42,"cluster":"algebraic-spaces"},{"id":"stacks:03JM","tag":"03JM","title":"Universally bounded fibres · Lemma 03JM","summary":"Let S be a scheme. Let X be an algebraic space over S. Let V → U be a morphism of schemes over S, and let U → X be a morphism from U to X. If the fibres of V → U and U → X are universally bounded, then so are the fibres of V → X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $V \\to U$ be a morphism of schemes over $S$, and let\n$U \\to X$ be a morphism from $U$ to $X$. If the fibres of\n$V \\to U$ and $U \\to X$ are universally bounded, then so\nare the fibres of $V \\to X$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JM","source_file":"decent-spaces.tex","source_line":79,"source_end_line":86,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L79-L86","statement_sha256":"e733c12b445084b2b138825fd053ac8ecf9c4298975977e155fa5f41334e30f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11197,"rank":11197,"depth":14,"x":362.942,"y":1603.012,"cluster":"algebraic-spaces"},{"id":"stacks:03JN","tag":"03JN","title":"Universally bounded fibres · Lemma 03JN","summary":"Let S be a scheme. Let Y → X be a representable morphism of algebraic spaces over S. Let U → X be a morphism from a scheme to X. If the fibres of U → X are universally bounded, then the fibres of U ×_X Y → Y are universally bounded.","statement_latex":"Let $S$ be a scheme.\nLet $Y \\to X$ be a representable morphism of algebraic spaces over $S$.\nLet $U \\to X$ be a morphism from a scheme to $X$.\nIf the fibres of $U \\to X$ are universally bounded, then the fibres\nof $U \\times_X Y \\to Y$ are universally bounded.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JN","source_file":"decent-spaces.tex","source_line":109,"source_end_line":116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L109-L116","statement_sha256":"313a0c7ebffcab373b254ac8494294f992fec1e6d2903f3a0d90767f51f75ff6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11198,"rank":11198,"depth":0,"x":129.454,"y":1673.282,"cluster":"algebraic-spaces"},{"id":"stacks:03JO","tag":"03JO","title":"Universally bounded fibres · Lemma 03JO","summary":"Let S be a scheme. Let g : Y → X be a representable morphism of algebraic spaces over S. Let f : U → X be a morphism from a scheme towards X. Let f' : U ×_X Y → Y be the base change of f. If Im(|f| : |U| → |X|) ⊂ Im(|g| : |Y| → |X|) and f' has universally bounded fibres, then f has universally bounded fibres.","statement_latex":"Let $S$ be a scheme. Let $g : Y \\to X$ be a representable morphism of\nalgebraic spaces over $S$. Let $f : U \\to X$ be a morphism from a scheme\ntowards $X$. Let $f' : U \\times_X Y \\to Y$ be the base change of $f$.\nIf\n$$\n\\Im(|f| : |U| \\to |X|) \\subset \\Im(|g| : |Y| \\to |X|)\n$$\nand $f'$ has universally bounded fibres, then $f$ has universally\nbounded fibres.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JO","source_file":"decent-spaces.tex","source_line":124,"source_end_line":135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L124-L135","statement_sha256":"e01f2a32a7b66f365f01e2f1bb96a3d61d25b2c32d1f6fea856f550e1f364b06","origin":"The Stacks Project","memory_eligible":false,"source_rank":11199,"rank":11199,"depth":1,"x":245.224,"y":1488.807,"cluster":"algebraic-spaces"},{"id":"stacks:03JP","tag":"03JP","title":"Universally bounded fibres · Lemma 03JP","summary":"Let S be a scheme. Let X be an algebraic space over S. Consider a commutative diagram xymatrix U ar[rd]_g ar[rr]_f & & V ar[ld]^h & X & where U and V are schemes. If g has universally bounded fibres, and f is surjective and flat, then also h has universally bounded fibres.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nConsider a commutative diagram\n$$\n\\xymatrix{\nU \\ar[rd]_g \\ar[rr]_f & & V \\ar[ld]^h \\\\\n& X &\n}\n$$\nwhere $U$ and $V$ are schemes. If $g$ has universally bounded fibres,\nand $f$ is surjective and flat, then also $h$ has universally bounded fibres.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JP","source_file":"decent-spaces.tex","source_line":166,"source_end_line":178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L166-L178","statement_sha256":"f03e324dc62656c8c519032d99483a6fb3191125369b9868715491ae91f83303","origin":"The Stacks Project","memory_eligible":false,"source_rank":11200,"rank":11200,"depth":14,"x":308.266,"y":1690.715,"cluster":"algebraic-spaces"},{"id":"stacks:03JQ","tag":"03JQ","title":"Universally bounded fibres · Lemma 03JQ","summary":"Let S be a scheme. Let X be an algebraic space over S, and let U be a scheme over S. Let φ : U → X be a morphism over S. If the fibres of φ are universally bounded, then there exists an integer n such that each fibre of |U| → |X| has at most n elements.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$, and let $U$ be a scheme over $S$.\nLet $\\varphi : U \\to X$ be a morphism over $S$.\nIf the fibres of $\\varphi$ are universally bounded, then there exists an\ninteger $n$ such that each fibre of $|U| \\to |X|$ has at most\n$n$ elements.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Universally bounded fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JQ","source_file":"decent-spaces.tex","source_line":199,"source_end_line":207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L199-L207","statement_sha256":"2ebc71f2b6756d0e27bc69115102692541e3faa66c86f4cb0147773034936fb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11201,"rank":11201,"depth":1,"x":99.216,"y":1577.497,"cluster":"algebraic-spaces"},{"id":"stacks:03JS","tag":"03JS","title":"Finiteness conditions and points · Lemma 03JS","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. The following are equivalent: • there exists a family of schemes U_i and étale morphisms φ_i : U_i → X such that coprod φ_i : coprod U_i → X is surjective, and such that for each i the fibre of |U_i| → |X| over x is finite, and • for every affine scheme U and étale morphism φ : U → X the fibre of |U| → |X| over x is finite.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. The following are equivalent:\n\\begin{enumerate}\n\\item there exists a family of schemes $U_i$ and\n\\'etale morphisms $\\varphi_i : U_i \\to X$ such that\n$\\coprod \\varphi_i : \\coprod U_i \\to X$ is surjective,\nand such that for each $i$ the fibre of\n$|U_i| \\to |X|$ over $x$ is finite, and\n\\item for every affine scheme $U$ and \\'etale morphism $\\varphi : U \\to X$\nthe fibre of $|U| \\to |X|$ over $x$ is finite.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Finiteness conditions and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JS","source_file":"decent-spaces.tex","source_line":267,"source_end_line":280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L267-L280","statement_sha256":"65389498efc553846dda582d25c520a363bdc02758170817777704b195fcd229","origin":"The Stacks Project","memory_eligible":false,"source_rank":11202,"rank":11202,"depth":31,"x":344.641,"y":1542.33,"cluster":"algebraic-spaces"},{"id":"stacks:03JU","tag":"03JU","title":"Finiteness conditions and points · Lemma 03JU","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. The following are equivalent: • there exists a scheme U, an étale morphism φ : U → X, and points u, u' ∈ U mapping to x such that setting R = U ×_X U the fibre of |R| → |U| ×_|X| |U| over (u, u') is finite, • for every scheme U, étale morphism φ : U → X and any points u, u' ∈ U mapping to x setting R = U ×_X U the fibre of |R| → |U| ×_|X| |U| over (u, u') is finite, • there exists a morphism Spec(k) → X…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. The following are equivalent:\n\\begin{enumerate}\n\\item there exists a scheme $U$, an \\'etale morphism\n$\\varphi : U \\to X$, and points $u, u' \\in U$ mapping to\n$x$ such that setting $R = U \\times_X U$ the fibre of\n$$\n|R| \\to |U| \\times_{|X|} |U|\n$$\nover $(u, u')$ is finite,\n\\item for every scheme $U$, \\'etale morphism $\\varphi : U \\to X$ and\nany points $u, u' \\in U$ mapping to\n$x$ setting $R = U \\times_X U$ the fibre of\n$$\n|R| \\to |U| \\times_{|X|} |U|\n$$\nover $(u, u')$ is finite,\n\\item there exists a morphism $\\Spec(k) \\to X$ with $k$ a field\nin the equivalence class of $x$ such that the projections\n$\\Spec(k) \\times_X \\Spec(k) \\to \\Spec(k)$ are\n\\'etale and quasi-compact, and\n\\item there exists a monomorphism $\\Spec(k) \\to X$ with $k$ a field\nin the equivalence class of $x$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Finiteness conditions and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JU","source_file":"decent-spaces.tex","source_line":316,"source_end_line":342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L316-L342","statement_sha256":"43ffff0bbc87e8fc3461c3e868bebe793872b4bf0c318c3b5def3d60cc7fd950","origin":"The Stacks Project","memory_eligible":false,"source_rank":11203,"rank":11203,"depth":52,"x":191.807,"y":1707.675,"cluster":"algebraic-spaces"},{"id":"stacks:040U","tag":"040U","title":"Finiteness conditions and points · Lemma 040U","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. Let U be a scheme and let φ : U → X be an étale morphism. The following are equivalent: • x is in the image of |U| → |X|, and setting R = U ×_X U the fibres of both |U| → |X| and |R| → |X| over x are finite, • there exists a monomorphism Spec(k) → X with k a field in the equivalence class of x, and the fibre product Spec(k) ×_X U is a finite nonempty scheme over k.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$.\nLet $U$ be a scheme and let $\\varphi : U \\to X$ be an \\'etale morphism.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $x$ is in the image of $|U| \\to |X|$, and\nsetting $R = U \\times_X U$ the fibres of both\n$$\n|U| \\longrightarrow |X|\n\\quad\\text{and}\\quad\n|R| \\longrightarrow |X|\n$$\nover $x$ are finite,\n\\item there exists a monomorphism $\\Spec(k) \\to X$ with $k$ a field\nin the equivalence class of $x$, and\nthe fibre product $\\Spec(k) \\times_X U$ is\na finite nonempty scheme over $k$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Finiteness conditions and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/040U","source_file":"decent-spaces.tex","source_line":469,"source_end_line":489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L469-L489","statement_sha256":"e05d880894c34155c80ef32b4ae896fdc73243905f166a02058a9f4b47a0edc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11204,"rank":11204,"depth":53,"x":171.52,"y":1498.837,"cluster":"algebraic-spaces"},{"id":"stacks:03JV","tag":"03JV","title":"Finiteness conditions and points · Lemma 03JV","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. The following are equivalent: • for every affine scheme U, any étale morphism φ : U → X setting R = U ×_X U the fibres of both |U| → |X| and |R| → |X| over x are finite, • there exist schemes U_i and étale morphisms U_i → X such that coprod U_i → X is surjective and for each i, setting R_i = U_i ×_X U_i the fibres of both |U_i| → |X| and |R_i| → |X| over x are finite, • there exists a monomorphism Spec(k)…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. The following are equivalent:\n\\begin{enumerate}\n\\item for every affine scheme $U$, any \\'etale morphism\n$\\varphi : U \\to X$ setting $R = U \\times_X U$ the fibres of both\n$$\n|U| \\longrightarrow |X|\n\\quad\\text{and}\\quad\n|R| \\longrightarrow |X|\n$$\nover $x$ are finite,\n\\item there exist schemes $U_i$ and \\'etale morphisms\n$U_i \\to X$ such that $\\coprod U_i \\to X$ is surjective and for each\n$i$, setting $R_i = U_i \\times_X U_i$ the fibres of both\n$$\n|U_i| \\longrightarrow |X|\n\\quad\\text{and}\\quad\n|R_i| \\longrightarrow |X|\n$$\nover $x$ are finite,\n\\item there exists a monomorphism $\\Spec(k) \\to X$ with $k$ a field\nin the equivalence class of $x$, and for any affine scheme $U$ and \\'etale\nmorphism $U \\to X$ the fibre product $\\Spec(k) \\times_X U$ is\na finite scheme over $k$,\n\\item there exists a quasi-compact monomorphism $\\Spec(k) \\to X$\nwith $k$ a field in the equivalence class of $x$,\n\\item there exists a quasi-compact morphism $\\Spec(k) \\to X$\nwith $k$ a field in the equivalence class of $x$, and\n\\item every morphism $\\Spec(k) \\to X$ with $k$ a field in the\nequivalence class of $x$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Finiteness conditions and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JV","source_file":"decent-spaces.tex","source_line":514,"source_end_line":547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L514-L547","statement_sha256":"3d6e1e12dc2fa3cf9ab99fbcb87fc4343dda525e7b74c3fbf1e8db44c5fc68b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11205,"rank":11205,"depth":54,"x":354.59,"y":1641.451,"cluster":"algebraic-spaces"},{"id":"stacks:03JT","tag":"03JT","title":"Finiteness conditions and points · Lemma 03JT","summary":"Let S be a scheme. Let X be an algebraic space over S. The following are equivalent: • there exist schemes U_i and étale morphisms U_i → X such that coprod U_i → X is surjective and each U_i → X has universally bounded fibres, and • for every affine scheme U and étale morphism φ : U → X the fibres of U → X are universally bounded.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item there exist schemes $U_i$ and \\'etale morphisms\n$U_i \\to X$ such that $\\coprod U_i \\to X$ is surjective and\neach $U_i \\to X$ has universally bounded fibres, and\n\\item for every affine scheme $U$ and \\'etale morphism $\\varphi : U \\to X$\nthe fibres of $U \\to X$ are universally bounded.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Finiteness conditions and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JT","source_file":"decent-spaces.tex","source_line":629,"source_end_line":640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L629-L640","statement_sha256":"68f0c345c890b47291966375b315b8b6321c4318fc0ddf017854879cdec6a014","origin":"The Stacks Project","memory_eligible":false,"source_rank":11206,"rank":11206,"depth":31,"x":104.68,"y":1640.167,"cluster":"algebraic-spaces"},{"id":"stacks:03IH","tag":"03IH","title":"Finiteness conditions and points · Lemma 03IH","summary":"Let S be a scheme. Let X be an algebraic space over S. The following are equivalent: • there exists a Zariski covering X = ⋃ X_i and for each i a scheme U_i and a quasi-compact surjective étale morphism U_i → X_i, and • there exist schemes U_i and étale morphisms U_i → X such that the projections U_i ×_X U_i → U_i are quasi-compact and coprod U_i → X is surjective.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item there exists a Zariski covering $X = \\bigcup X_i$ and for\neach $i$ a scheme $U_i$ and a quasi-compact surjective \\'etale\nmorphism $U_i \\to X_i$, and\n\\item there exist schemes $U_i$ and \\'etale morphisms $U_i \\to X$\nsuch that the projections $U_i \\times_X U_i \\to U_i$ are quasi-compact\nand $\\coprod U_i \\to X$ is surjective.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Finiteness conditions and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IH","source_file":"decent-spaces.tex","source_line":678,"source_end_line":691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L678-L691","statement_sha256":"c29b51108e6d3dc17de2d408193680fea2d644f10880fde5f955e38a69076e70","origin":"The Stacks Project","memory_eligible":false,"source_rank":11207,"rank":11207,"depth":40,"x":290.164,"y":1499.179,"cluster":"algebraic-spaces"},{"id":"stacks:03JX","tag":"03JX","title":"Conditions on algebraic spaces · Lemma 03JX","summary":"Let S be a scheme. Let X be an algebraic space over S. Consider the following conditions on X: • [] (α) For every x ∈ |X|, the equivalent conditions of Lemma [Tag 03JS] hold. • [] (β) For every x ∈ |X|, the equivalent conditions of Lemma [Tag 03JU] hold. • [] (γ) For every x ∈ |X|, the equivalent conditions of Lemma [Tag 03JV] hold. • [] (δ) The equivalent conditions of Lemma [Tag 03JT] hold. • [] (ε) The equivalent conditions of Lemma [Tag 03IH] hold. • [] (zeta) The…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nConsider the following conditions on $X$:\n\\begin{itemize}\n\\item[] $(\\alpha)$ For every $x \\in |X|$, the equivalent conditions of\nLemma \\ref{lemma-U-finite-above-x}\nhold.\n\\item[] $(\\beta)$ For every $x \\in |X|$, the equivalent conditions of\nLemma \\ref{lemma-R-finite-above-x}\nhold.\n\\item[] $(\\gamma)$ For every $x \\in |X|$, the equivalent conditions of\nLemma \\ref{lemma-UR-finite-above-x}\nhold.\n\\item[] $(\\delta)$ The equivalent conditions of\nLemma \\ref{lemma-U-universally-bounded}\nhold.\n\\item[] $(\\epsilon)$ The equivalent conditions of\nLemma \\ref{lemma-characterize-very-reasonable}\nhold.\n\\item[] $(\\zeta)$ The space $X$ is Zariski locally quasi-separated.\n\\item[] $(\\eta)$ The space $X$ is quasi-separated\n\\item[] $(\\theta)$ The space $X$ is representable, i.e., $X$ is a scheme.\n\\item[] $(\\iota)$ The space $X$ is a quasi-separated scheme.\n\\end{itemize}\nWe have\n$$\n\\xymatrix{\n& (\\theta) \\ar@{=>}[rd] & & & &  \\\\\n(\\iota) \\ar@{=>}[ru] \\ar@{=>}[rd] & &\n(\\zeta) \\ar@{=>}[r] &\n(\\epsilon) \\ar@{=>}[r] &\n(\\delta) \\ar@{=>}[r] &\n(\\gamma) \\ar@{<=>}[r] & (\\alpha) + (\\beta) \\\\\n& (\\eta) \\ar@{=>}[ru] & & & &\n}\n$$","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Conditions on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JX","source_file":"decent-spaces.tex","source_line":734,"source_end_line":771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L734-L771","statement_sha256":"3b2fc647a4f20e885f4619c94a50ca6691f18fbb2d1e2f1bf9ef432d53e8e8b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11208,"rank":11208,"depth":55,"x":266.748,"y":1708.582,"cluster":"algebraic-spaces"},{"id":"stacks:03KE","tag":"03KE","title":"Conditions on algebraic spaces · Lemma 03KE","summary":"Let S be a scheme. Let P be one of the properties (α), (β), (γ), (δ), (ε), (zeta), or (theta) of algebraic spaces listed in Lemma [Tag 03JX]. Then if X is an algebraic space over S, and X = ⋃ X_i is a Zariski open covering such that each X_i has P, then X has P.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be one of the properties\n$(\\alpha)$, $(\\beta)$, $(\\gamma)$, $(\\delta)$, $(\\epsilon)$, $(\\zeta)$, or\n$(\\theta)$ of algebraic spaces listed in\nLemma \\ref{lemma-bounded-fibres}.\nThen if $X$ is an algebraic space over $S$, and $X = \\bigcup X_i$ is a\nZariski open covering such that each $X_i$ has $\\mathcal{P}$,\nthen $X$ has $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Conditions on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KE","source_file":"decent-spaces.tex","source_line":827,"source_end_line":837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L827-L837","statement_sha256":"89be18feb0d80d7b95f889441102e6001ade099abed67e6bc2ef2c56122ea73e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11209,"rank":11209,"depth":56,"x":115.479,"y":1540.731,"cluster":"algebraic-spaces"},{"id":"stacks:03KF","tag":"03KF","title":"Conditions on algebraic spaces · Lemma 03KF","summary":"Let S be a scheme. Let P be one of the properties (β), (γ), (δ), (ε), or (theta) of algebraic spaces listed in Lemma [Tag 03JX]. Let X, Y be algebraic spaces over S. Let X → Y be a representable morphism. If Y has property P, so does X.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{P}$ be one of the properties\n$(\\beta)$, $(\\gamma)$, $(\\delta)$, $(\\epsilon)$, or\n$(\\theta)$ of algebraic spaces listed in\nLemma \\ref{lemma-bounded-fibres}.\nLet $X$, $Y$ be algebraic spaces over $S$.\nLet $X \\to Y$ be a representable morphism.\nIf $Y$ has property $\\mathcal{P}$, so does $X$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Conditions on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KF","source_file":"decent-spaces.tex","source_line":908,"source_end_line":917,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L908-L917","statement_sha256":"62a75f38cf60bc0c053243189c4e9f054bf27f5cac2b9c6adb12c5ee0b43219c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11210,"rank":11210,"depth":56,"x":362.229,"y":1578.703,"cluster":"algebraic-spaces"},{"id":"stacks:03I8","tag":"03I8","title":"Reasonable and decent algebraic spaces · Definition 03I8","summary":"Let S be a scheme. Let X be an algebraic space over S. • We say X is decent if for every point x ∈ X the equivalent conditions of Lemma [Tag 03JV] hold, in other words property (γ) of Lemma [Tag 03JX] holds. • We say X is reasonable if the equivalent conditions of Lemma [Tag 03JT] hold, in other words property (δ) of Lemma [Tag 03JX] holds. • We say X is very reasonable if the equivalent conditions of Lemma [Tag 03IH] hold, i.e., property (ε) of Lemma [Tag 03JX] holds.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item We say $X$ is {\\it decent} if for every point $x \\in X$ the equivalent\nconditions of\nLemma \\ref{lemma-UR-finite-above-x}\nhold, in other words property $(\\gamma)$ of\nLemma \\ref{lemma-bounded-fibres}\nholds.\n\\item We say $X$ is {\\it reasonable} if the equivalent conditions of\nLemma \\ref{lemma-U-universally-bounded}\nhold, in other words property $(\\delta)$ of\nLemma \\ref{lemma-bounded-fibres}\nholds.\n\\item We say $X$ is {\\it very reasonable} if the equivalent conditions of\nLemma \\ref{lemma-characterize-very-reasonable}\nhold, i.e., property $(\\epsilon)$ of\nLemma \\ref{lemma-bounded-fibres}\nholds.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reasonable and decent algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03I8","source_file":"decent-spaces.tex","source_line":1044,"source_end_line":1066,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1044-L1066","statement_sha256":"c9213716afe338d1dcdef6714e2b1c9b6d247ddd61c02ff852dbc96286e17b95","origin":"The Stacks Project","memory_eligible":false,"source_rank":11211,"rank":11211,"depth":56,"x":149.549,"y":1690.817,"cluster":"algebraic-spaces"},{"id":"stacks:03K0","tag":"03K0","title":"Reasonable and decent algebraic spaces · Lemma 03K0","summary":"Let S be a scheme. Let X be a quasi-compact reasonable algebraic space. Then there exists a directed system of quasi-compact and quasi-separated algebraic spaces X_i such that X = colim_i X_i (colimit in the category of sheaves). Moreover we can arrange it such that • for every quasi-compact scheme T over S we have colim X_i(T) = X(T), • the transition morphisms X_i → X_i' of the system and the coprojections X_i → X are surjective and étale, and • if X is a scheme, then…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact reasonable algebraic space.\nThen there exists a directed system of quasi-compact and quasi-separated\nalgebraic spaces $X_i$ such that $X = \\colim_i X_i$\n(colimit in the category of sheaves). Moreover we can arrange it such that\n\\begin{enumerate}\n\\item for every quasi-compact scheme $T$ over $S$ we have\n$\\colim X_i(T) = X(T)$,\n\\item the transition morphisms $X_i \\to X_{i'}$ of the system\nand the coprojections $X_i \\to X$ are surjective and \\'etale, and\n\\item if $X$ is a scheme, then the algebraic spaces $X_i$ are schemes\nand the transition morphisms $X_i \\to X_{i'}$\nand the coprojections $X_i \\to X$ are local isomorphisms.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reasonable and decent algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03K0","source_file":"decent-spaces.tex","source_line":1106,"source_end_line":1122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1106-L1122","statement_sha256":"b5d55c35e6f9253f18fa4c2a17763bb9ec9340effbc8c182f488da0d13909e2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11212,"rank":11212,"depth":52,"x":216.278,"y":1487.28,"cluster":"algebraic-spaces"},{"id":"stacks:0ABT","tag":"0ABT","title":"Reasonable and decent algebraic spaces · Lemma 0ABT","summary":"Let S be a scheme. Let X, Y be algebraic spaces over S. Let X → Y be a representable morphism. If Y is decent (resp. reasonable), then so is X.","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$.\nLet $X \\to Y$ be a representable morphism.\nIf $Y$ is decent (resp.\\ reasonable), then so is $X$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reasonable and decent algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABT","source_file":"decent-spaces.tex","source_line":1203,"source_end_line":1208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1203-L1208","statement_sha256":"a42ae0d17592f2bd5d9eaa164ab70a8058df38a12b7a35871a97426cad5621f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11213,"rank":11213,"depth":57,"x":330.856,"y":1675.4,"cluster":"algebraic-spaces"},{"id":"stacks:0ABU","tag":"0ABU","title":"Reasonable and decent algebraic spaces · Lemma 0ABU","summary":"Let S be a scheme. Let X → Y be an étale morphism of algebraic spaces over S. If Y is decent, resp. reasonable, then so is X.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be an \\'etale morphism of\nalgebraic spaces over $S$. If $Y$ is decent, resp.\\ reasonable,\nthen so is $X$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reasonable and decent algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABU","source_file":"decent-spaces.tex","source_line":1214,"source_end_line":1219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1214-L1219","statement_sha256":"8801294a7dbe0872e5eaa197ce29079afd2dbdf3a6757ff669791d11737e23ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":11214,"rank":11214,"depth":0,"x":94.873,"y":1601.63,"cluster":"algebraic-spaces"},{"id":"stacks:03IM","tag":"03IM","title":"Points and specializations · Lemma 03IM","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U → X be an étale morphism from a scheme to X. Assume u, u' ∈ |U| map to the same point x of |X|, and u' leadsto u. If the pair (X, x) satisfies the equivalent conditions of Lemma [Tag 03JS] then u = u'.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $U \\to X$ be an \\'etale morphism from a scheme to $X$.\nAssume $u, u' \\in |U|$ map to the same point $x$ of $|X|$, and\n$u' \\leadsto u$. If the pair $(X, x)$ satisfies the\nequivalent conditions of\nLemma \\ref{lemma-U-finite-above-x}\nthen $u = u'$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points and specializations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IM","source_file":"decent-spaces.tex","source_line":1257,"source_end_line":1267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1257-L1267","statement_sha256":"abfad3a4aa89aedf1d9e3c5b30b0212f7f795397e638471fd61ad73615f706a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11215,"rank":11215,"depth":32,"x":328.419,"y":1522.053,"cluster":"algebraic-spaces"},{"id":"stacks:0H1Q","tag":"0H1Q","title":"Points and specializations · Lemma 0H1Q","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U → X be an étale morphism from a scheme to X. Assume u, u' ∈ |U| map to the same point x of |X|, and u' leadsto u. If X is locally Noetherian, then u = u'.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U \\to X$ be an \\'etale morphism from a scheme to $X$.\nAssume $u, u' \\in |U|$ map to the same point $x$ of $|X|$, and\n$u' \\leadsto u$. If $X$ is locally Noetherian, then $u = u'$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points and specializations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1Q","source_file":"decent-spaces.tex","source_line":1308,"source_end_line":1314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1308-L1314","statement_sha256":"1d38484cc906b0636401d9755ce4535202d99e00f830610489021dfff73e747a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11216,"rank":11216,"depth":16,"x":220.102,"y":1713.425,"cluster":"algebraic-spaces"},{"id":"stacks:03K2","tag":"03K2","title":"Points and specializations · Lemma 03K2","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x, x' ∈ |X| and assume x' leadsto x, i.e., x is a specialization of x'. Assume the pair (X, x') satisfies the equivalent conditions of Lemma [Tag 03JV]. Then for every étale morphism φ : U → X from a scheme U and any u ∈ U with φ(u) = x, exists a point u'∈ U, u' leadsto u with φ(u') = x'.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $x, x' \\in |X|$ and assume $x' \\leadsto x$, i.e., $x$ is a\nspecialization of $x'$.\nAssume the pair $(X, x')$ satisfies the equivalent conditions\nof Lemma \\ref{lemma-UR-finite-above-x}.\nThen for every \\'etale morphism $\\varphi : U \\to X$ from a scheme $U$ and any\n$u \\in U$ with $\\varphi(u) = x$, exists a point $u'\\in U$,\n$u' \\leadsto u$ with $\\varphi(u') = x'$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points and specializations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03K2","source_file":"decent-spaces.tex","source_line":1326,"source_end_line":1337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1326-L1337","statement_sha256":"c058bde056dadc3c980fc02d02241ad18e87e01db5082d47d7415d8399d3c3c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11217,"rank":11217,"depth":55,"x":146.01,"y":1510.665,"cluster":"algebraic-spaces"},{"id":"stacks:0B7W","tag":"0B7W","title":"Points and specializations · Lemma 0B7W","summary":"Let S be a scheme. Let f : Y → X be a flat morphism of algebraic spaces over S. Let x, x' ∈ |X| and assume x' leadsto x, i.e., x is a specialization of x'. Assume the pair (X, x') satisfies the equivalent conditions of Lemma [Tag 03JV] (for example if X is decent, X is quasi-separated, or X is representable). Then for every y ∈ |Y| with f(y) = x, there exists a point y' ∈ |Y|, y' leadsto y with f(y') = x'.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a  flat morphism of algebraic spaces\nover $S$. Let $x, x' \\in |X|$ and assume $x' \\leadsto x$, i.e., $x$ is a\nspecialization of $x'$. Assume the pair $(X, x')$ satisfies the equivalent\nconditions of Lemma \\ref{lemma-UR-finite-above-x} (for example if\n$X$ is decent, $X$ is quasi-separated, or $X$ is representable).\nThen for every $y \\in |Y|$ with $f(y) = x$, there exists a point $y' \\in |Y|$,\n$y' \\leadsto y$ with $f(y') = x'$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points and specializations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7W","source_file":"decent-spaces.tex","source_line":1392,"source_end_line":1401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1392-L1401","statement_sha256":"4dbf147d8b535b160a2bd06c9f64af8217834e4b611711021b5350d781d5446a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11218,"rank":11218,"depth":56,"x":363.894,"y":1618.233,"cluster":"algebraic-spaces"},{"id":"stacks:07S8","tag":"07S8","title":"Stratifying algebraic spaces by schemes · Lemma 07S8","summary":"Let S be a scheme. Let W → X be a morphism of a scheme W to an algebraic space X which is flat, locally of finite presentation, separated, locally quasi-finite with universally bounded fibres. There exist reduced closed subspaces ∅ = Z_-1 ⊂ Z_0 ⊂ Z_1 ⊂ Z_2 ⊂ … ⊂ Z_n = X such that with X_r = Z_r setminus Z_r - 1 the stratification X = coprod_r = 0, …, n X_r is characterized by the following universal property: Given g : T → X the projection W ×_X T → T is finite locally…","statement_latex":"Let $S$ be a scheme. Let $W \\to X$ be a morphism of a scheme $W$\nto an algebraic space $X$ which is flat, locally of finite presentation,\nseparated, locally quasi-finite with universally bounded fibres. There exist\nreduced closed subspaces\n$$\n\\emptyset = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset Z_2 \\subset\n\\ldots \\subset Z_n = X\n$$\nsuch that with $X_r = Z_r \\setminus Z_{r - 1}$ the stratification\n$X = \\coprod_{r = 0, \\ldots, n} X_r$ is characterized by the following\nuniversal property: Given $g : T \\to X$ the projection\n$W \\times_X T \\to T$ is finite locally free of degree $r$ if and only if\n$g(|T|) \\subset |X_r|$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Stratifying algebraic spaces by schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07S8","source_file":"decent-spaces.tex","source_line":1435,"source_end_line":1450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1435-L1450","statement_sha256":"beca7ae1c50ad24bb0b51bf218c15eeadbaee22bb2677850c3c6f244a3ddae52","origin":"The Stacks Project","memory_eligible":false,"source_rank":11219,"rank":11219,"depth":55,"x":116.504,"y":1662.586,"cluster":"algebraic-spaces"},{"id":"stacks:086T","tag":"086T","title":"Stratifying algebraic spaces by schemes · Lemma 086T","summary":"Let S be a scheme. Let W → X be a morphism of a scheme W to an algebraic space X which is flat, locally of finite presentation, separated, and locally quasi-finite. Then there exist open subspaces X = X_0 ⊃ X_1 ⊃ X_2 ⊃ … such that a morphism Spec(k) → X where k is a field factors through X_d if and only if W ×_X Spec(k) has degree ≥ d over k.","statement_latex":"Let $S$ be a scheme. Let $W \\to X$ be a morphism of a scheme $W$ to an\nalgebraic space $X$ which is flat, locally of finite presentation,\nseparated, and locally quasi-finite. Then there\nexist open subspaces\n$$\nX = X_0 \\supset X_1 \\supset X_2 \\supset \\ldots\n$$\nsuch that a morphism $\\Spec(k) \\to X$ where $k$ is a field\nfactors through $X_d$ if and\nonly if $W \\times_X \\Spec(k)$ has degree $\\geq d$ over $k$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Stratifying algebraic spaces by schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086T","source_file":"decent-spaces.tex","source_line":1497,"source_end_line":1509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1497-L1509","statement_sha256":"38bb508814d7cd4608034ebbae17f886f0c552eb1b962b9d58b5dd0f36da7e72","origin":"The Stacks Project","memory_eligible":false,"source_rank":11220,"rank":11220,"depth":53,"x":263.39,"y":1489.35,"cluster":"algebraic-spaces"},{"id":"stacks:0BBN","tag":"0BBN","title":"Stratifying algebraic spaces by schemes · Lemma 0BBN","summary":"Let S be a scheme. Let X be a quasi-compact algebraic space over S. There exist open subspaces … ⊂ U_4 ⊂ U_3 ⊂ U_2 ⊂ U_1 = X with the following properties: • setting T_p = U_p setminus U_p + 1 (with reduced induced subspace structure) there exists a separated scheme V_p and a surjective étale morphism f_p : V_p → U_p such that f_p^-1(T_p) → T_p is an isomorphism, • if x ∈ |X| can be represented by a quasi-compact morphism Spec(k) → X from a field, then x ∈ T_p for some p.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space\nover $S$. There exist open subspaces\n$$\n\\ldots \\subset U_4 \\subset U_3 \\subset U_2 \\subset U_1 = X\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item setting $T_p = U_p \\setminus U_{p + 1}$ (with reduced induced subspace\nstructure) there exists a separated scheme $V_p$ and a surjective \\'etale\nmorphism $f_p : V_p \\to U_p$ such that $f_p^{-1}(T_p) \\to T_p$ is an\nisomorphism,\n\\item if $x \\in |X|$ can be represented by a quasi-compact morphism\n$\\Spec(k) \\to X$ from a field, then $x \\in T_p$ for some $p$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Stratifying algebraic spaces by schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBN","source_file":"decent-spaces.tex","source_line":1532,"source_end_line":1548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1532-L1548","statement_sha256":"652703c8c9a84bf35132518a3ef24594e8d7020275ae479c40bd426936a5cce4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11221,"rank":11221,"depth":56,"x":294.418,"y":1700.628,"cluster":"algebraic-spaces"},{"id":"stacks:07S9","tag":"07S9","title":"Stratifying algebraic spaces by schemes · Lemma 07S9","summary":"Let S be a scheme. Let X be a quasi-compact, reasonable algebraic space over S. There exist an integer n and open subspaces ∅ = U_n + 1 ⊂ U_n ⊂ U_n - 1 ⊂ … ⊂ U_1 = X with the following property: setting T_p = U_p setminus U_p + 1 (with reduced induced subspace structure) there exists a separated scheme V_p and a surjective étale morphism f_p : V_p → U_p such that f_p^-1(T_p) → T_p is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space\nover $S$. There exist an integer $n$ and open subspaces\n$$\n\\emptyset = U_{n + 1} \\subset\nU_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X\n$$\nwith the following property: setting $T_p = U_p \\setminus U_{p + 1}$\n(with reduced induced subspace structure) there exists a separated scheme\n$V_p$ and a surjective \\'etale morphism $f_p : V_p \\to U_p$ such that\n$f_p^{-1}(T_p) \\to T_p$ is an isomorphism.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Stratifying algebraic spaces by schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07S9","source_file":"decent-spaces.tex","source_line":1628,"source_end_line":1640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1628-L1640","statement_sha256":"b6050283ad8227226b1cf1b65292a11432d40a28dfbe950dd00eee9cb64a1ddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11222,"rank":11222,"depth":57,"x":101.462,"y":1562.317,"cluster":"algebraic-spaces"},{"id":"stacks:07SA","tag":"07SA","title":"Stratifying algebraic spaces by schemes · Lemma 07SA","summary":"Let S be a scheme. Let X be a quasi-compact, reasonable algebraic space over S. There exist an integer n and open subspaces ∅ = U_n + 1 ⊂ U_n ⊂ U_n - 1 ⊂ … ⊂ U_1 = X such that each T_p = U_p setminus U_p + 1 (with reduced induced subspace structure) is a scheme.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space\nover $S$. There exist an integer $n$ and open subspaces\n$$\n\\emptyset = U_{n + 1} \\subset\nU_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X\n$$\nsuch that each $T_p = U_p \\setminus U_{p + 1}$ (with reduced induced subspace\nstructure) is a scheme.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Stratifying algebraic spaces by schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SA","source_file":"decent-spaces.tex","source_line":1651,"source_end_line":1661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1651-L1661","statement_sha256":"269713c75a720d68a3634e61c8fc8cbd4c28b615588fc576a674a51c61fcd7d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11223,"rank":11223,"depth":58,"x":355.197,"y":1554.811,"cluster":"algebraic-spaces"},{"id":"stacks:07ST","tag":"07ST","title":"Stratifying algebraic spaces by schemes · Lemma 07ST","summary":"This result is almost identical to [GruRay]. Let X be a quasi-compact and quasi-separated algebraic space over Spec(Z). There exist an integer n and open subspaces ∅ = U_n + 1 ⊂ U_n ⊂ U_n - 1 ⊂ … ⊂ U_1 = X with the following property: setting T_p = U_p setminus U_p + 1 (with reduced induced subspace structure) there exists a quasi-compact separated scheme V_p and a surjective étale morphism f_p : V_p → U_p such that f_p^-1(T_p) → T_p is an isomorphism.","statement_latex":"\\begin{reference}\nThis result is almost identical to \\cite[Proposition 5.7.8]{GruRay}.\n\\end{reference}\nLet $X$ be a quasi-compact and quasi-separated algebraic space over\n$\\Spec(\\mathbf{Z})$. There exist an integer $n$ and open subspaces\n$$\n\\emptyset = U_{n + 1} \\subset\nU_n \\subset U_{n - 1} \\subset \\ldots \\subset U_1 = X\n$$\nwith the following property: setting $T_p = U_p \\setminus U_{p + 1}$\n(with reduced induced subspace structure) there exists a quasi-compact\nseparated scheme $V_p$ and a surjective \\'etale morphism $f_p : V_p \\to U_p$\nsuch that $f_p^{-1}(T_p) \\to T_p$ is an isomorphism.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Stratifying algebraic spaces by schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07ST","source_file":"decent-spaces.tex","source_line":1671,"source_end_line":1686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1671-L1686","statement_sha256":"e9d5b7562f3d47a8a58d871763d912710459adf9bc566c15916e8660acefb754","origin":"The Stacks Project","memory_eligible":false,"source_rank":11224,"rank":11224,"depth":58,"x":173.972,"y":1704.455,"cluster":"algebraic-spaces"},{"id":"stacks:0ECZ","tag":"0ECZ","title":"Stratifying algebraic spaces by schemes · Lemma 0ECZ","summary":"Let S be a scheme. Let X be a quasi-separated algebraic space over S. Let E ⊂ |X| be a subset. Then E is étale locally constructible (Properties of Spaces, Definition [Tag 0ECU]) if and only if E is a locally constructible subset of the topological space |X| (Topology, Definition [Tag 005G]).","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$.\nLet $E \\subset |X|$ be a subset. Then $E$ is \\'etale locally constructible\n(Properties of Spaces, Definition\n\\ref{spaces-properties-definition-locally-constructible})\nif and only if $E$ is a locally constructible subset of\nthe topological space $|X|$\n(Topology, Definition \\ref{topology-definition-constructible}).","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Stratifying algebraic spaces by schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ECZ","source_file":"decent-spaces.tex","source_line":1706,"source_end_line":1715,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1706-L1715","statement_sha256":"f2bed27aa21d8602922d7083894d6e766c7f80d19c341dc53f7992283b3ee0cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":11225,"rank":11225,"depth":59,"x":187.277,"y":1491.088,"cluster":"algebraic-spaces"},{"id":"stacks:0G2D","tag":"0G2D","title":"Integral cover by a scheme · Lemma 0G2D","summary":"Let S be a scheme. Let j : V → Y be a quasi-compact open immersion of algebraic spaces over S. Let π : Z → V be an integral morphism. Then there exists an integral morphism ν : Y' → Y such that Z is V-isomorphic to the inverse image of V in Y'.","statement_latex":"Let $S$ be a scheme. Let $j : V \\to Y$ be a quasi-compact open immersion\nof algebraic spaces over $S$. Let $\\pi : Z \\to V$ be an integral morphism.\nThen there exists an integral morphism $\\nu : Y' \\to Y$ such that\n$Z$ is $V$-isomorphic to the inverse image of $V$ in $Y'$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Integral cover by a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2D","source_file":"decent-spaces.tex","source_line":1784,"source_end_line":1790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1784-L1790","statement_sha256":"8f30efe6db38fa76412bea2221c871f0bddd93dfc829c2e24b4f14248982ac09","origin":"The Stacks Project","memory_eligible":false,"source_rank":11226,"rank":11226,"depth":60,"x":349.193,"y":1656.117,"cluster":"algebraic-spaces"},{"id":"stacks:09YB","tag":"09YB","title":"Integral cover by a scheme · Lemma 09YB","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. • There exists a surjective integral morphism Y → X where Y is a scheme, • given a surjective étale morphism U → X we may choose Y → X such that for every y ∈ Y there is an open neighbourhood V ⊂ Y such that V → X factors through U.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$.\n\\begin{enumerate}\n\\item There exists a surjective integral morphism $Y \\to X$ where $Y$\nis a scheme,\n\\item given a surjective \\'etale morphism $U \\to X$ we may choose\n$Y \\to X$ such that for every $y \\in Y$ there is an open neighbourhood\n$V \\subset Y$ such that $V \\to X$ factors through $U$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Integral cover by a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YB","source_file":"decent-spaces.tex","source_line":1806,"source_end_line":1817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1806-L1817","statement_sha256":"b15aa67132d518c49ec6bab15d3fbb7e84ed457fcdc696fcbb2494b19204750a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11227,"rank":11227,"depth":61,"x":96.863,"y":1626.276,"cluster":"algebraic-spaces"},{"id":"stacks:0GUL","tag":"0GUL","title":"Integral cover by a scheme · Lemma 0GUL","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S such that |X| has finitely many irreducible components. • There exists a surjective integral morphism Y → X where Y is a scheme such that f is finite étale over a quasi-compact dense open U ⊂ X, • given a surjective étale morphism V → X we may choose Y → X such that for every y ∈ Y there is an open neighbourhood W ⊂ Y such that W → X factors through V.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$ such that $|X|$ has finitely many irreducible\ncomponents.\n\\begin{enumerate}\n\\item There exists a surjective integral morphism $Y \\to X$ where $Y$\nis a scheme such that $f$ is finite \\'etale over a quasi-compact\ndense open $U \\subset X$,\n\\item given a surjective \\'etale morphism $V \\to X$ we may choose\n$Y \\to X$ such that for every $y \\in Y$ there is an open neighbourhood\n$W \\subset Y$ such that $W \\to X$ factors through $V$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Integral cover by a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUL","source_file":"decent-spaces.tex","source_line":1891,"source_end_line":1904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L1891-L1904","statement_sha256":"a950a0097f6f9bc3a454cf5c32daa38a253fe1201e56711a3bef8da94593b830","origin":"The Stacks Project","memory_eligible":false,"source_rank":11228,"rank":11228,"depth":62,"x":307.111,"y":1504.995,"cluster":"algebraic-spaces"},{"id":"stacks:03JI","tag":"03JI","title":"Schematic locus · Proposition 03JI","summary":"Let S be a scheme. Let X be an algebraic space over S. If X is reasonable, then there exists a dense open subspace of X which is a scheme.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf $X$ is reasonable, then there exists a dense open subspace\nof $X$ which is a scheme.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Schematic locus","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03JI","source_file":"decent-spaces.tex","source_line":2055,"source_end_line":2060,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2055-L2060","statement_sha256":"08171ef4843d548752d9a01e4136f8d4f265895399a6f1b269f8408d427a3e7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11229,"rank":11229,"depth":57,"x":249.557,"y":1713.91,"cluster":"algebraic-spaces"},{"id":"stacks:086U","tag":"086U","title":"David Rydh · Theorem 086U","summary":"Let S be a scheme. Let X be an algebraic space over S. If X is decent, then there exists a dense open subspace of X which is a scheme.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf $X$ is decent, then there exists a dense open subspace\nof $X$ which is a scheme.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Schematic locus","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086U","source_file":"decent-spaces.tex","source_line":2105,"source_end_line":2110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2105-L2110","statement_sha256":"58261153f176d8cb9f04a7feed1b1c2500d1370d806530422f9f85be93632ec1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11230,"rank":11230,"depth":58,"x":123.882,"y":1527.037,"cluster":"algebraic-spaces"},{"id":"stacks:0BA1","tag":"0BA1","title":"Schematic locus · Lemma 0BA1","summary":"Let S be a scheme. Let X → Y be a surjective finite locally free morphism of algebraic spaces over S. For y ∈ |Y| the following are equivalent • y is in the schematic locus of Y, and • there exists an affine open U ⊂ X containing the preimage of y.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a surjective finite locally free\nmorphism of algebraic spaces over $S$. For $y \\in |Y|$ the following are\nequivalent\n\\begin{enumerate}\n\\item $y$ is in the schematic locus of $Y$, and\n\\item there exists an affine open $U \\subset X$\ncontaining the preimage of $y$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Schematic locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BA1","source_file":"decent-spaces.tex","source_line":2157,"source_end_line":2167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2157-L2167","statement_sha256":"f037db09cf016b0a32f22bd88755863e179ca640ac4b42decc9b499f1a1a1b0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11231,"rank":11231,"depth":48,"x":367.045,"y":1593.582,"cluster":"algebraic-spaces"},{"id":"stacks:06NG","tag":"06NG","title":"Schematic locus · Lemma 06NG","summary":"Let S be a scheme. Let X be an algebraic space over S. If there exists a finite, étale, surjective morphism U → X where U is a scheme, then there exists a dense open subspace of X which is a scheme.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf there exists a finite, \\'etale, surjective morphism\n$U \\to X$ where $U$ is a scheme, then there exists a dense open subspace\nof $X$ which is a scheme.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Schematic locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NG","source_file":"decent-spaces.tex","source_line":2211,"source_end_line":2217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2211-L2217","statement_sha256":"26c4f36dfa2835e38e36940cf3ea3dfeb2a90301e3018bcc53cf8a7c42200b5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11232,"rank":11232,"depth":58,"x":134.022,"y":1682.567,"cluster":"algebraic-spaces"},{"id":"stacks:03K4","tag":"03K4","title":"Residue fields and henselian local rings · Lemma 03K4","summary":"Let S be a scheme. Let X be an algebraic space over S. Consider the map (Spec(k) → X monomorphism where k is a field) → |X| This map is always injective. If X is decent then this map is a bijection.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nConsider the map\n$$\n\\{\\Spec(k) \\to X \\text{ monomorphism where }k\\text{ is a field}\\}\n\\longrightarrow\n|X|\n$$\nThis map is always injective. If $X$ is decent then this map\nis a bijection.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03K4","source_file":"decent-spaces.tex","source_line":2347,"source_end_line":2358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2347-L2358","statement_sha256":"691b1c74efec08f06e3073822656a2b2f5c7c6cce466b234c2a9e0664ed55fae","origin":"The Stacks Project","memory_eligible":false,"source_rank":11233,"rank":11233,"depth":56,"x":234.378,"y":1484.555,"cluster":"algebraic-spaces"},{"id":"stacks:0EMW","tag":"0EMW","title":"Residue fields and henselian local rings · Definition 0EMW","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X|. The residue field of X at x is the unique field kappa(x) which comes equipped with a monomorphism Spec(kappa(x)) → X representing x.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$. The {\\it residue field of $X$ at $x$}\nis the unique field $\\kappa(x)$ which comes equipped with a\nmonomorphism $\\Spec(\\kappa(x)) \\to X$ representing $x$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMW","source_file":"decent-spaces.tex","source_line":2379,"source_end_line":2385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2379-L2385","statement_sha256":"4474b871a884951f7a506c888ee81c66a6531d2436101c537fe321b24c6aaf92","origin":"The Stacks Project","memory_eligible":false,"source_rank":11234,"rank":11234,"depth":0,"x":319.688,"y":1687.688,"cluster":"algebraic-spaces"},{"id":"stacks:0EMX","tag":"0EMX","title":"Residue fields and henselian local rings · Lemma 0EMX","summary":"Let S be a scheme. Let f : X → Y be a morphism of decent algebraic spaces over S. Let x ∈ |X| be a point with image y = f(x) ∈ |Y|. The following are equivalent • f induces an isomorphism kappa(y) → kappa(x), and • the induced morphism Spec(kappa(x)) → Y is a monomorphism.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of decent\nalgebraic spaces over $S$. Let $x \\in |X|$ be a point\nwith image $y = f(x) \\in |Y|$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ induces an isomorphism $\\kappa(y) \\to \\kappa(x)$, and\n\\item the induced morphism $\\Spec(\\kappa(x)) \\to Y$ is a monomorphism.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMX","source_file":"decent-spaces.tex","source_line":2403,"source_end_line":2413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2403-L2413","statement_sha256":"394671a65ae58f061d7bec72f4088793ca64ea2a5a57cb335e39f5e0198165bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11235,"rank":11235,"depth":0,"x":93.23,"y":1586.219,"cluster":"algebraic-spaces"},{"id":"stacks:0BBP","tag":"0BBP","title":"Residue fields and henselian local rings · Lemma 0BBP","summary":"Let S be a scheme. Let X be a decent algebraic space over S. For every point x ∈ |X| there exists an étale morphism (U, u) → (X, x) where U is an affine scheme, u is the only point of U lying over x, and the induced homomorphism kappa(x) → kappa(u) is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nFor every point $x \\in |X|$ there exists an \\'etale morphism\n$$\n(U, u) \\longrightarrow (X, x)\n$$\nwhere $U$ is an affine scheme, $u$ is the only point of $U$ lying\nover $x$, and the induced homomorphism $\\kappa(x) \\to \\kappa(u)$\nis an isomorphism.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBP","source_file":"decent-spaces.tex","source_line":2423,"source_end_line":2433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2423-L2433","statement_sha256":"1365c107e21de7096656106121fe913da20a0b33b38a100b8a8655fded26c337","origin":"The Stacks Project","memory_eligible":false,"source_rank":11236,"rank":11236,"depth":57,"x":342.032,"y":1532.496,"cluster":"algebraic-spaces"},{"id":"stacks:0BGU","tag":"0BGU","title":"Residue fields and henselian local rings · Definition 0BGU","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ X be a point. An elementary étale neighbourhood is an étale morphism (U, u) → (X, x) where U is a scheme, u ∈ U is a point mapping to x, and the morphism u = Spec(kappa(u)) → X is a monomorphism. A morphism of elementary étale neighbourhoods (U, u) → (U', u') is defined as a morphism U → U' over X mapping u to u'.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in X$ be a point. An {\\it elementary \\'etale neighbourhood}\nis an \\'etale morphism $(U, u) \\to (X, x)$ where $U$ is a scheme,\n$u \\in U$ is a point mapping to $x$, and the morphism\n$u = \\Spec(\\kappa(u)) \\to X$ is a monomorphism.\nA {\\it morphism of elementary \\'etale neighbourhoods}\n$(U, u) \\to (U', u')$ is defined as a morphism $U \\to U'$\nover $X$ mapping $u$ to $u'$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGU","source_file":"decent-spaces.tex","source_line":2443,"source_end_line":2453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2443-L2453","statement_sha256":"bef7623e3de407e1a9cdeeb0ac9c6518d0c538d3041d315a22bcddad0e3b2eca","origin":"The Stacks Project","memory_eligible":false,"source_rank":11237,"rank":11237,"depth":0,"x":201.649,"y":1713.445,"cluster":"algebraic-spaces"},{"id":"stacks:0BGV","tag":"0BGV","title":"Residue fields and henselian local rings · Lemma 0BGV","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x be a point of X. The category of elementary étale neighborhoods of (X, x) is cofiltered (see Categories, Definition [Tag 04AZ]).","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x$ be a point of $X$.\nThe category of elementary \\'etale neighborhoods of $(X, x)$\nis cofiltered (see\nCategories, Definition \\ref{categories-definition-codirected}).","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGV","source_file":"decent-spaces.tex","source_line":2459,"source_end_line":2466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2459-L2466","statement_sha256":"3c18f9b87b45a114da2a1e154b489b12fd31f142ede07543349ee09edf32b25b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11238,"rank":11238,"depth":58,"x":159.615,"y":1500.174,"cluster":"algebraic-spaces"},{"id":"stacks:0BGW","tag":"0BGW","title":"Residue fields and henselian local rings · Definition 0BGW","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X|. The henselian local ring of X at x, is O_X, x^h = colim Γ(U, O_U) where the colimit is over the elementary étale neighbourhoods (U, u) → (X, x).","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$. The {\\it henselian local ring of $X$ at $x$}, is\n$$\n\\mathcal{O}_{X, x}^h = \\colim \\Gamma(U, \\mathcal{O}_U)\n$$\nwhere the colimit is over the elementary \\'etale neighbourhoods\n$(U, u) \\to (X, x)$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGW","source_file":"decent-spaces.tex","source_line":2498,"source_end_line":2507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2498-L2507","statement_sha256":"8c3c7062f4a5286356834d28256a9216e7077d56ce11d192b3efcffedab8469d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11239,"rank":11239,"depth":0,"x":362.292,"y":1633.701,"cluster":"algebraic-spaces"},{"id":"stacks:0EMY","tag":"0EMY","title":"Residue fields and henselian local rings · Lemma 0EMY","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X|. Let (U, u) → (X, x) be an elementary étale neighbourhood. Then O_X, x^h = O_U, u^h In words: the henselian local ring of X at x is equal to the henselization O_U, u^h of the local ring O_U, u of U at u.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$. Let $(U, u) \\to (X, x)$ be an elementary\n\\'etale neighbourhood. Then\n$$\n\\mathcal{O}_{X, x}^h = \\mathcal{O}_{U, u}^h\n$$\nIn words: the henselian local ring of $X$ at $x$\nis equal to the henselization $\\mathcal{O}_{U, u}^h$\nof the local ring $\\mathcal{O}_{U, u}$ of $U$ at $u$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMY","source_file":"decent-spaces.tex","source_line":2514,"source_end_line":2525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2514-L2525","statement_sha256":"ed16dea63f02fb524079829eae3b5251626cc8c452c7b1d37c017f0651fabf8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11240,"rank":11240,"depth":59,"x":105.24,"y":1650.258,"cluster":"algebraic-spaces"},{"id":"stacks:0EMZ","tag":"0EMZ","title":"Residue fields and henselian local rings · Lemma 0EMZ","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let overlinex be a geometric point of X lying over x ∈ |X|. The étale local ring O_X, overlinex of X at overlinex (Properties of Spaces, Definition [Tag 04KG]) is the strict henselization of the henselian local ring O_X, x^h of X at x.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $\\overline{x}$ be a geometric point of $X$ lying over $x \\in |X|$. \nThe \\'etale local ring $\\mathcal{O}_{X, \\overline{x}}$ of $X$ at $\\overline{x}$\n(Properties of Spaces, Definition\n\\ref{spaces-properties-definition-etale-local-rings})\nis the strict henselization\nof the henselian local ring $\\mathcal{O}_{X, x}^h$ of $X$ at $x$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EMZ","source_file":"decent-spaces.tex","source_line":2544,"source_end_line":2553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2544-L2553","statement_sha256":"e351bae3fb0bf71c7c550da1691e1bb26b9a3ea570cee98529765d020598d73d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11241,"rank":11241,"depth":60,"x":281.624,"y":1492.055,"cluster":"algebraic-spaces"},{"id":"stacks:0EN0","tag":"0EN0","title":"Residue fields and henselian local rings · Lemma 0EN0","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X|. The residue field of the henselian local ring of X at x (Definition [Tag 0BGW]) is the residue field of X at x (Definition [Tag 0EMW]).","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$. The residue field of the\nhenselian local ring of $X$ at $x$\n(Definition \\ref{definition-henselian-local-ring})\nis the residue field of $X$ at $x$\n(Definition \\ref{definition-residue-field}).","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Residue fields and henselian local rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EN0","source_file":"decent-spaces.tex","source_line":2566,"source_end_line":2574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2566-L2574","statement_sha256":"cbaecec3395f545f8f4421888f6a01b2d9e40fc22c05e6ecb08073207dee187b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11242,"rank":11242,"depth":60,"x":278.781,"y":1708.984,"cluster":"algebraic-spaces"},{"id":"stacks:03K5","tag":"03K5","title":"Points on decent spaces · Lemma 03K5","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let U → X be an étale morphism from a scheme to X. If u, u' ∈ |U| map to the same point of |X|, and u' leadsto u, then u = u'.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $U \\to X$ be an \\'etale morphism from a scheme to $X$.\nIf $u, u' \\in |U|$ map to the same point of $|X|$, and\n$u' \\leadsto u$, then $u = u'$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points on decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03K5","source_file":"decent-spaces.tex","source_line":2647,"source_end_line":2653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2647-L2653","statement_sha256":"3574d205104fbd2289e1222013e9926d11069d8dce309acd17018a77357ebfaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":11243,"rank":11243,"depth":56,"x":106.281,"y":1547.271,"cluster":"algebraic-spaces"},{"id":"stacks:03IL","tag":"03IL","title":"Points on decent spaces · Lemma 03IL","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x, x' ∈ |X| and assume x' leadsto x, i.e., x is a specialization of x'. Then for every étale morphism φ : U → X from a scheme U and any u ∈ U with φ(u) = x, exists a point u'∈ U, u' leadsto u with φ(u') = x'.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x, x' \\in |X|$ and assume $x' \\leadsto x$, i.e., $x$ is a\nspecialization of $x'$. Then for every \\'etale morphism\n$\\varphi : U \\to X$ from a scheme $U$ and any $u \\in U$ with\n$\\varphi(u) = x$, exists a point $u'\\in U$, $u' \\leadsto u$ with\n$\\varphi(u') = x'$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points on decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IL","source_file":"decent-spaces.tex","source_line":2660,"source_end_line":2668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2660-L2668","statement_sha256":"5e95380272de65b4395a53b610d4d06799e06d870aa91864e86a5ce547b330f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11244,"rank":11244,"depth":56,"x":363.746,"y":1568.653,"cluster":"algebraic-spaces"},{"id":"stacks:03K3","tag":"03K3","title":"Points on decent spaces · Lemma 03K3","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Then |X| is Kolmogorov (see Topology, Definition [Tag 004X]).","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nThen $|X|$ is Kolmogorov (see\nTopology, Definition \\ref{topology-definition-generic-point}).","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points on decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03K3","source_file":"decent-spaces.tex","source_line":2675,"source_end_line":2680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2675-L2680","statement_sha256":"0f1dec6f2e7d8441eccd030a1339cb86f4940311a4bebbdb0b0db7c6e88ac098","origin":"The Stacks Project","memory_eligible":false,"source_rank":11245,"rank":11245,"depth":57,"x":156.523,"y":1699.09,"cluster":"algebraic-spaces"},{"id":"stacks:03K6","tag":"03K6","title":"Points on decent spaces · Proposition 03K6","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Then the topological space |X| is sober (see Topology, Definition [Tag 004X]).","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nThen the topological space $|X|$ is sober (see\nTopology, Definition \\ref{topology-definition-generic-point}).","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points on decent spaces","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03K6","source_file":"decent-spaces.tex","source_line":2697,"source_end_line":2702,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2697-L2702","statement_sha256":"c2f0f658a8186f6143d43b230ae0c78a9e40c24d1177a3dab7101c446b2238d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11246,"rank":11246,"depth":59,"x":204.474,"y":1485.142,"cluster":"algebraic-spaces"},{"id":"stacks:0A4J","tag":"0A4J","title":"Points on decent spaces · Lemma 0A4J","summary":"Let S be a scheme. Dimension as defined in Properties of Spaces, Section [Tag 04N3] behaves well on decent algebraic spaces X over S. • If x ∈ |X|, then dim_x(|X|) = dim_x(X), and • dim(|X|) = dim(X).","statement_latex":"Let $S$ be a scheme. Dimension as defined in\nProperties of Spaces, Section \\ref{spaces-properties-section-dimension}\nbehaves well on decent algebraic spaces $X$ over $S$.\n\\begin{enumerate}\n\\item If $x \\in |X|$, then $\\dim_x(|X|) = \\dim_x(X)$, and\n\\item $\\dim(|X|) = \\dim(X)$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points on decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4J","source_file":"decent-spaces.tex","source_line":2727,"source_end_line":2736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2727-L2736","statement_sha256":"d685d9a3ad5bf78a4f024006127ee5c2b750085e461c17e360a93fd8b255d5b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11247,"rank":11247,"depth":57,"x":341.284,"y":1670.27,"cluster":"algebraic-spaces"},{"id":"stacks:0ABW","tag":"0ABW","title":"Points on decent spaces · Lemma 0ABW","summary":"Let S be a scheme. Let X → Y be a locally quasi-finite morphism of algebraic spaces over S. Let x ∈ |X| with image y ∈ |Y|. Then the dimension of the local ring of Y at y is ≥ to the dimension of the local ring of X at x.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a locally quasi-finite morphism\nof algebraic spaces over $S$. Let $x \\in |X|$ with image $y \\in |Y|$.\nThen the dimension of the local ring of $Y$ at $y$ is $\\geq$ to the\ndimension of the local ring of $X$ at $x$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points on decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABW","source_file":"decent-spaces.tex","source_line":2762,"source_end_line":2768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2762-L2768","statement_sha256":"17501d7f4228abf77a50705b16b6b30f33264f6b337a123c6c6a542d8758f10c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11248,"rank":11248,"depth":29,"x":91.313,"y":1611.338,"cluster":"algebraic-spaces"},{"id":"stacks:0ED0","tag":"0ED0","title":"Points on decent spaces · Lemma 0ED0","summary":"Let S be a scheme. Let X → Y be a locally quasi-finite morphism of algebraic spaces over S. Then dim(X) ≤ dim(Y).","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a locally quasi-finite morphism\nof algebraic spaces over $S$. Then $\\dim(X) \\leq \\dim(Y)$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points on decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ED0","source_file":"decent-spaces.tex","source_line":2784,"source_end_line":2788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2784-L2788","statement_sha256":"434a5b70d5cbe03d527649da7566c2fd06ba4ef3c926cbc5a628aa4a434d848e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11249,"rank":11249,"depth":30,"x":323.227,"y":1512.871,"cluster":"algebraic-spaces"},{"id":"stacks:03IK","tag":"03IK","title":"Points on decent spaces · Lemma 03IK","summary":"Let S be a scheme. Let k be a field. Let X be an algebraic space over S and assume that there exists a surjective étale morphism Spec(k) → X. If X is decent, then X ≅ Spec(k') where k/k' is a finite separable extension.","statement_latex":"Let $S$ be a scheme. Let $k$ be a field. Let $X$ be an algebraic space\nover $S$ and assume that there exists a surjective \\'etale morphism\n$\\Spec(k) \\to X$. If $X$ is decent, then $X \\cong \\Spec(k')$\nwhere $k/k'$ is a finite separable extension.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points on decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03IK","source_file":"decent-spaces.tex","source_line":2801,"source_end_line":2807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2801-L2807","statement_sha256":"225c937bc9166acbcfd895e835d840a8307891a12bb180d7ceff4354b2eb235f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11250,"rank":11250,"depth":4,"x":231.324,"y":1717.246,"cluster":"algebraic-spaces"},{"id":"stacks:06QV","tag":"06QV","title":"Reduced singleton spaces · Lemma 06QV","summary":"Let S be a scheme. Let Z be an algebraic space over S. Let k be a field and let Spec(k) → Z be surjective and flat. Then any morphism Spec(k') → Z where k' is a field is surjective and flat.","statement_latex":"Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$.\nLet $k$ be a field and let $\\Spec(k) \\to Z$ be surjective and flat.\nThen any morphism $\\Spec(k') \\to Z$ where $k'$ is a field is\nsurjective and flat.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QV","source_file":"decent-spaces.tex","source_line":2835,"source_end_line":2841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2835-L2841","statement_sha256":"c2833d6b805ee2eb913a00aab6bf703f2b056436fe10bd2d00f90bfd28eaade0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11251,"rank":11251,"depth":56,"x":134.657,"y":1514.223,"cluster":"algebraic-spaces"},{"id":"stacks:06QW","tag":"06QW","title":"Reduced singleton spaces · Lemma 06QW","summary":"Let S be a scheme. Let Z be an algebraic space over S. The following are equivalent • Z is reduced and |Z| is a singleton, • there exists a surjective flat morphism Spec(k) → Z where k is a field, and • there exists a locally of finite type, surjective, flat morphism Spec(k) → Z where k is a field.","statement_latex":"Let $S$ be a scheme.\nLet $Z$ be an algebraic space over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $Z$ is reduced and $|Z|$ is a singleton,\n\\item there exists a surjective flat morphism $\\Spec(k) \\to Z$\nwhere $k$ is a field, and\n\\item there exists a locally of finite type, surjective, flat morphism\n$\\Spec(k) \\to Z$ where $k$ is a field.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QW","source_file":"decent-spaces.tex","source_line":2860,"source_end_line":2871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2860-L2871","statement_sha256":"1a5dfd9b37236a62e45ebea99abafc0d9a01e05d8b4a128d050476d9251ca16c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11252,"rank":11252,"depth":57,"x":369.402,"y":1609.158,"cluster":"algebraic-spaces"},{"id":"stacks:06QX","tag":"06QX","title":"Reduced singleton spaces · Lemma 06QX","summary":"Let S be a scheme. Let Z be an algebraic space over S. The following are equivalent • Z is reduced, locally Noetherian, and |Z| is a singleton, and • there exists a locally finitely presented, surjective, flat morphism Spec(k) → Z where k is a field.","statement_latex":"Let $S$ be a scheme. Let $Z$ be an algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $Z$ is reduced, locally Noetherian, and $|Z|$\nis a singleton, and\n\\item there exists a locally finitely presented, surjective, flat morphism\n$\\Spec(k) \\to Z$ where $k$ is a field.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QX","source_file":"decent-spaces.tex","source_line":2917,"source_end_line":2927,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2917-L2927","statement_sha256":"fb98800b3062f1e49a039976a72b2b2f569fb50ef2e740a387d5dccf655a0499","origin":"The Stacks Project","memory_eligible":false,"source_rank":11253,"rank":11253,"depth":58,"x":119.749,"y":1672.408,"cluster":"algebraic-spaces"},{"id":"stacks:06QY","tag":"06QY","title":"Reduced singleton spaces · Lemma 06QY","summary":"Let S be a scheme. Let Z' → Z be a monomorphism of algebraic spaces over S. Assume there exists a field k and a locally finitely presented, surjective, flat morphism Spec(k) → Z. Then either Z' is empty or Z' = Z.","statement_latex":"Let $S$ be a scheme.\nLet $Z' \\to Z$ be a monomorphism of algebraic spaces over $S$.\nAssume there exists a field $k$ and a locally finitely presented, surjective,\nflat morphism $\\Spec(k) \\to Z$. Then either $Z'$\nis empty or $Z' = Z$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QY","source_file":"decent-spaces.tex","source_line":2966,"source_end_line":2973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2966-L2973","statement_sha256":"8257f2bb3193488777ea6fc057cc0145953c2452265b1ac66f8568cdfd0e6b40","origin":"The Stacks Project","memory_eligible":false,"source_rank":11254,"rank":11254,"depth":57,"x":253.088,"y":1483.95,"cluster":"algebraic-spaces"},{"id":"stacks:06QZ","tag":"06QZ","title":"Reduced singleton spaces · Lemma 06QZ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. Then there exists a unique monomorphism Z → X of algebraic spaces over S such that Z is an algebraic space which satisfies the equivalent conditions of Lemma [Tag 06QX] and such that the image of |Z| → |X| is (x).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. Then there exists a unique monomorphism\n$Z \\to X$ of algebraic spaces\nover $S$ such that $Z$ is an algebraic space which satisfies the equivalent\nconditions of\nLemma \\ref{lemma-unique-point-better}\nand such that the image of $|Z| \\to |X|$ is $\\{x\\}$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QZ","source_file":"decent-spaces.tex","source_line":2997,"source_end_line":3006,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L2997-L3006","statement_sha256":"dd378c614cb4296db053d248e062cf72bb18b152eecf49da4ec9b555393b0401","origin":"The Stacks Project","memory_eligible":false,"source_rank":11255,"rank":11255,"depth":59,"x":306.363,"y":1698.757,"cluster":"algebraic-spaces"},{"id":"stacks:06R0","tag":"06R0","title":"Reduced singleton spaces · Definition 06R0","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. The residual space of X at x is the monomorphism Z_x → X constructed in Lemma [Tag 06QZ].","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$. Let $x \\in |X|$.\nThe\n{\\it residual space of $X$ at $x$}\\footnote{This is nonstandard notation.}\nis the monomorphism $Z_x \\to X$ constructed in\nLemma \\ref{lemma-find-singleton-from-point}.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R0","source_file":"decent-spaces.tex","source_line":3070,"source_end_line":3078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3070-L3078","statement_sha256":"a7bf6ad0cbdbb83b222540a56e3a82c52bb9cd8f2c42edf8b52568721f1d9e6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11256,"rank":11256,"depth":60,"x":94.159,"y":1570.483,"cluster":"algebraic-spaces"},{"id":"stacks:0H1R","tag":"0H1R","title":"Reduced singleton spaces · Lemma 0H1R","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. The residual space Z_x of X at x is isomorphic to the spectrum of a field if and only if x can be represented by a monomorphism Spec(k) → X where k is a field. If X is decent, this holds for all x ∈ |X|.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$.\nThe residual space $Z_x$ of $X$ at $x$ is isomorphic to the spectrum of a field\nif and only if $x$ can be represented by a monomorphism $\\Spec(k) \\to X$\nwhere $k$ is a field. If $X$ is decent, this holds for all $x \\in |X|$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1R","source_file":"decent-spaces.tex","source_line":3091,"source_end_line":3097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3091-L3097","statement_sha256":"1fd21c69ebee9862686aba308676e11dc218db554f579b88dc45c845318df9e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11257,"rank":11257,"depth":58,"x":354.007,"y":1544.639,"cluster":"algebraic-spaces"},{"id":"stacks:06R1","tag":"06R1","title":"Reduced singleton spaces · Lemma 06R1","summary":"A reduced, locally Noetherian singleton algebraic space Z is regular.","statement_latex":"A reduced, locally Noetherian singleton algebraic space $Z$ is regular.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R1","source_file":"decent-spaces.tex","source_line":3116,"source_end_line":3119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3116-L3119","statement_sha256":"7d444ef851029f0e2195223ddd20f1ade4677f5412ce2d79c85f3ccfa5b5dbb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11258,"rank":11258,"depth":59,"x":183.039,"y":1711.28,"cluster":"algebraic-spaces"},{"id":"stacks:0H1S","tag":"0H1S","title":"Reduced singleton spaces · Lemma 0H1S","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Let x ∈ |X| be a point. Assume • |f|(|Y|) is contained in (x) ⊂ |X|, • Y is reduced, and • X is locally Noetherian. Then f factors through the residual space Z_x of X at x.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic\nspaces over $S$. Let $x \\in |X|$ be a point. Assume\n\\begin{enumerate}\n\\item $|f|(|Y|)$ is contained in $\\{x\\} \\subset |X|$,\n\\item $Y$ is reduced, and\n\\item $X$ is locally Noetherian.\n\\end{enumerate}\nThen $f$ factors through the residual space $Z_x$ of $X$ at $x$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1S","source_file":"decent-spaces.tex","source_line":3136,"source_end_line":3146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3136-L3146","statement_sha256":"7422216e4b216bdfad482e68ef38e7da48de21fb04426fe6f07d4a459395ceb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11259,"rank":11259,"depth":60,"x":175.094,"y":1491.206,"cluster":"algebraic-spaces"},{"id":"stacks:0H1T","tag":"0H1T","title":"Reduced singleton spaces · Lemma 0H1T","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Let x ∈ |X| be a point. Assume • |f|(|Y|) is contained in (x) ⊂ |X|, • Y is reduced, and • x can be represented by a quasi-compact monomorphism x : Spec(k) → X where k is a field (for example if X is decent). Then f factors through the residual space Z_x = Spec(k) of X at x.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic\nspaces over $S$. Let $x \\in |X|$ be a point. Assume\n\\begin{enumerate}\n\\item $|f|(|Y|)$ is contained in $\\{x\\} \\subset |X|$,\n\\item $Y$ is reduced, and\n\\item $x$ can be represented by a quasi-compact monomorphism\n$x : \\Spec(k) \\to X$ where $k$ is a field (for example if $X$ is decent).\n\\end{enumerate}\nThen $f$ factors through\nthe residual space $Z_x = \\Spec(k)$ of $X$ at $x$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1T","source_file":"decent-spaces.tex","source_line":3170,"source_end_line":3182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3170-L3182","statement_sha256":"5fcb23aad427a1ae86b066d920959290a6d0113ba6945f1d10ffa88c3324859e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11260,"rank":11260,"depth":59,"x":358.083,"y":1649.109,"cluster":"algebraic-spaces"},{"id":"stacks:0H1U","tag":"0H1U","title":"Reduced singleton spaces · Lemma 0H1U","summary":"Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X| with residual space Z_x ⊂ X. Assume X is locally Noetherian. Then x is a closed point of |X| if and only if the morphism Z_x → X is a closed immersion.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \\in |X|$\nwith residual space $Z_x \\subset X$. Assume $X$ is locally Noetherian.\nThen $x$ is a closed point of $|X|$ if and only if\nthe morphism $Z_x \\to X$ is a closed immersion.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Reduced singleton spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1U","source_file":"decent-spaces.tex","source_line":3233,"source_end_line":3239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3233-L3239","statement_sha256":"6d7d15e8ce44e7771e5da32e140e56c7c28ba29ea6cfaeedd6c305d21639c13a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11261,"rank":11261,"depth":54,"x":95.95,"y":1636.495,"cluster":"algebraic-spaces"},{"id":"stacks:0BB6","tag":"0BB6","title":"Decent spaces · Lemma 0BB6","summary":"Any locally Noetherian decent algebraic space is quasi-separated.","statement_latex":"Any locally Noetherian decent algebraic space is quasi-separated.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BB6","source_file":"decent-spaces.tex","source_line":3271,"source_end_line":3274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3271-L3274","statement_sha256":"5a9fc215aed6894135e00434ae0eb0361bc9c65c370d331185e8032faab27dba","origin":"The Stacks Project","memory_eligible":false,"source_rank":11262,"rank":11262,"depth":55,"x":299.556,"y":1496.94,"cluster":"algebraic-spaces"},{"id":"stacks:047Z","tag":"047Z","title":"Decent spaces · Lemma 047Z","summary":"Let S be a scheme. Let X be a decent algebraic space over S. • If |X| is a singleton then X is a scheme. • If |X| is a singleton and X is reduced, then X ≅ Spec(k) for some field k.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\n\\begin{enumerate}\n\\item If $|X|$ is a singleton then $X$ is a scheme.\n\\item If $|X|$ is a singleton and $X$ is reduced, then\n$X \\cong \\Spec(k)$ for some field $k$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/047Z","source_file":"decent-spaces.tex","source_line":3313,"source_end_line":3321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3313-L3321","statement_sha256":"064796e2f6ee3ac34a2b54c06b79d388c5291bae4422c645b01c3264d4821fc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11263,"rank":11263,"depth":59,"x":261.615,"y":1715.558,"cluster":"algebraic-spaces"},{"id":"stacks:07U5","tag":"07U5","title":"Decent spaces · Lemma 07U5","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Consider a commutative diagram xymatrix Spec(k) ar[rr] ar[rd] & & X ar[ld] & S Assume that the image point s ∈ S of Spec(k) → S is a closed point and that kappa(s) ⊂ k is algebraic. Then the image x of Spec(k) → X is a closed point of |X|.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nConsider a commutative diagram\n$$\n\\xymatrix{\n\\Spec(k) \\ar[rr] \\ar[rd] & & X \\ar[ld] \\\\\n& S\n}\n$$\nAssume that the image point $s \\in S$ of $\\Spec(k) \\to S$ is\na closed point and that $\\kappa(s) \\subset k$ is algebraic.\nThen the image $x$ of $\\Spec(k) \\to X$ is a closed point of $|X|$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07U5","source_file":"decent-spaces.tex","source_line":3355,"source_end_line":3368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3355-L3368","statement_sha256":"291d6d0326c388b2d4e428e9b1be2c0673b535d537d4354fda9909a668dceea8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11264,"rank":11264,"depth":57,"x":113.662,"y":1532.672,"cluster":"algebraic-spaces"},{"id":"stacks:08AL","tag":"08AL","title":"Decent spaces · Lemma 08AL","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Consider a commutative diagram xymatrix Spec(k) ar[rr] ar[rd] & & X ar[ld] & S Assume that the image point s ∈ S of Spec(k) → S is a closed point and that the field extension k/kappa(s) is finite. Then Spec(k) → X is a finite morphism. If kappa(s) = k then Spec(k) → X is a closed immersion.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nConsider a commutative diagram\n$$\n\\xymatrix{\n\\Spec(k) \\ar[rr] \\ar[rd] & & X \\ar[ld] \\\\\n& S\n}\n$$\nAssume that the image point $s \\in S$ of $\\Spec(k) \\to S$ is\na closed point and that the field extension $k/\\kappa(s)$ is finite.\nThen $\\Spec(k) \\to X$ is a finite morphism. If $\\kappa(s) = k$\nthen $\\Spec(k) \\to X$ is a closed immersion.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AL","source_file":"decent-spaces.tex","source_line":3385,"source_end_line":3399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3385-L3399","statement_sha256":"552548367719dbd5cc40a2cf456b353fd568cc92632cdf734c0a2a3ce52fa4a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11265,"rank":11265,"depth":60,"x":370.045,"y":1583.62,"cluster":"algebraic-spaces"},{"id":"stacks:0AHB","tag":"0AHB","title":"Decent spaces · Lemma 0AHB","summary":"Let S be a scheme. Suppose X is a decent algebraic space over S. Let x ∈ |X| be a closed point. Then x can be represented by a closed immersion i : Spec(k) → X from the spectrum of a field.","statement_latex":"Let $S$ be a scheme. Suppose $X$ is a decent algebraic space over $S$.\nLet $x \\in |X|$ be a closed point. Then $x$ can be represented by a\nclosed immersion $i : \\Spec(k) \\to X$ from the spectrum of a field.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Decent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHB","source_file":"decent-spaces.tex","source_line":3418,"source_end_line":3423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3418-L3423","statement_sha256":"1a8f39068c329d4eef4fa241223b8ddf65d9408afba87dc476b71a496acc039d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11266,"rank":11266,"depth":57,"x":139.831,"y":1691.619,"cluster":"algebraic-spaces"},{"id":"stacks:088I","tag":"088I","title":"Locally separated spaces · Lemma 088I","summary":"Let A be a ring. Let k be a field. Let p_n, n ≥ 1 be a sequence of pairwise distinct primes of A. Moreover, for each n let k → kappa( p_n) be an embedding. Then the closure of the image of coprod_n not = m Spec(kappa( p_n) ⊗_k kappa( p_m)) → Spec(A ⊗ A) meets the diagonal.","statement_latex":"Let $A$ be a ring. Let $k$ be a field. Let $\\mathfrak p_n$, $n \\geq 1$\nbe a sequence of pairwise distinct primes of $A$. Moreover, for each\n$n$ let $k \\to \\kappa(\\mathfrak p_n)$ be an embedding. Then the closure\nof the image of\n$$\n\\coprod\\nolimits_{n \\not = m}\n\\Spec(\\kappa(\\mathfrak p_n) \\otimes_k \\kappa(\\mathfrak p_m))\n\\longrightarrow\n\\Spec(A \\otimes A)\n$$\nmeets the diagonal.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Locally separated spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088I","source_file":"decent-spaces.tex","source_line":3457,"source_end_line":3470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3457-L3470","statement_sha256":"c4b9ceef00223d0bda06c41f1ed7c168a7b2e5a0e5636623982480a1e8ab1e2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11267,"rank":11267,"depth":0,"x":222.806,"y":1481.18,"cluster":"algebraic-spaces"},{"id":"stacks:088J","tag":"088J","title":"David Rydh · Lemma 088J","summary":"A locally separated algebraic space is decent.","statement_latex":"A locally separated algebraic space is decent.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Locally separated spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088J","source_file":"decent-spaces.tex","source_line":3500,"source_end_line":3503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3500-L3503","statement_sha256":"d807db17d1b9c7c006fdbf4b8abe94d5d9f13d0cb65ba2d017335ff2fe6faa18","origin":"The Stacks Project","memory_eligible":false,"source_rank":11268,"rank":11268,"depth":55,"x":330.94,"y":1683.603,"cluster":"algebraic-spaces"},{"id":"stacks:03KJ","tag":"03KJ","title":"Valuative criterion · Proposition 03KJ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-compact, and X is decent. Then f is universally closed if and only if the existence part of the valuative criterion holds.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is quasi-compact, and $X$ is decent. Then $f$ is\nuniversally closed if and only if the existence part of the valuative\ncriterion holds.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Valuative criterion","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KJ","source_file":"decent-spaces.tex","source_line":3571,"source_end_line":3577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3571-L3577","statement_sha256":"aefbdaf38f194542417c5540738d48cd1d4325ee28cf2fa09c29722ef259caf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11269,"rank":11269,"depth":57,"x":88.221,"y":1595.625,"cluster":"algebraic-spaces"},{"id":"stacks:03KZ","tag":"03KZ","title":"Relative conditions · Definition 03KZ","summary":"Let S be a scheme. We say an algebraic space X over S has property (β) if X has the corresponding property of Lemma [Tag 03JX]. Let f : X → Y be a morphism of algebraic spaces over S. • We say f has property (β) if for any scheme T and morphism T → Y the fibre product T ×_Y X has property (β). • We say f is decent if for any scheme T and morphism T → Y the fibre product T ×_Y X is a decent algebraic space. • We say f is reasonable if for any scheme T and morphism T → Y…","statement_latex":"Let $S$ be a scheme. We say an algebraic space $X$ over $S$\n{\\it has property $(\\beta)$} if $X$ has the corresponding property of\nLemma \\ref{lemma-bounded-fibres}.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ {\\it has property $(\\beta)$} if for any scheme $T$ and\nmorphism $T \\to Y$ the fibre product $T \\times_Y X$ has property $(\\beta)$.\n\\item We say $f$ is {\\it decent} if for any scheme $T$ and\nmorphism $T \\to Y$ the fibre product $T \\times_Y X$ is a decent\nalgebraic space.\n\\item We say $f$ is {\\it reasonable} if for any scheme $T$ and\nmorphism $T \\to Y$ the fibre product $T \\times_Y X$ is a reasonable\nalgebraic space.\n\\item We say $f$ is {\\it very reasonable} if for any scheme $T$ and\nmorphism $T \\to Y$ the fibre product $T \\times_Y X$ is a very reasonable\nalgebraic space.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03KZ","source_file":"decent-spaces.tex","source_line":3666,"source_end_line":3685,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3666-L3685","statement_sha256":"79472b1830c78274a3d94a530ffc8bdb2c5be43944b45380c92e929594e02231","origin":"The Stacks Project","memory_eligible":false,"source_rank":11270,"rank":11270,"depth":56,"x":338.153,"y":1522.714,"cluster":"algebraic-spaces"},{"id":"stacks:03M5","tag":"03M5","title":"Relative conditions · Lemma 03M5","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We have the following implications among the conditions on f: xymatrix representable ar@=>[rd] & & & & & very reasonable ar@=>[r] & reasonable ar@=>[r] & decent ar@=>[r] & (β) quasi-separated ar@=>[ru] & & & &","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nWe have the following implications among the conditions on $f$:\n$$\n\\xymatrix{\n\\text{representable} \\ar@{=>}[rd] & & & & \\\\\n& \\text{very reasonable} \\ar@{=>}[r] & \\text{reasonable} \\ar@{=>}[r] &\n\\text{decent} \\ar@{=>}[r] & (\\beta) \\\\\n\\text{quasi-separated} \\ar@{=>}[ru] & & & &\n}\n$$","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03M5","source_file":"decent-spaces.tex","source_line":3692,"source_end_line":3705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3692-L3705","statement_sha256":"ba4314d2b4bf92bc2ac5e92de784346735f7f4911c95ea139c4f29ed6d7c4f76","origin":"The Stacks Project","memory_eligible":false,"source_rank":11271,"rank":11271,"depth":56,"x":212.386,"y":1718.455,"cluster":"algebraic-spaces"},{"id":"stacks:0ABX","tag":"0ABX","title":"Relative conditions · Lemma 0ABX","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If X is decent (resp. is reasonable, resp. has property (β) of Lemma [Tag 03JX]), then f is decent (resp. reasonable, resp. has property (β)).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. If $X$ is decent (resp.\\ is reasonable, resp.\\ has property\n$(\\beta)$ of Lemma \\ref{lemma-bounded-fibres}), then $f$ is\ndecent (resp.\\ reasonable, resp.\\ has property $(\\beta)$).","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABX","source_file":"decent-spaces.tex","source_line":3718,"source_end_line":3724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3718-L3724","statement_sha256":"53d00530ca693ab9eda84c1c34da175292d8d83c5f88c939003e3161d034d006","origin":"The Stacks Project","memory_eligible":false,"source_rank":11272,"rank":11272,"depth":58,"x":147.663,"y":1502.579,"cluster":"algebraic-spaces"},{"id":"stacks:03L0","tag":"03L0","title":"Relative conditions · Lemma 03L0","summary":"Having property (β), being decent, or being reasonable is preserved under arbitrary base change.","statement_latex":"Having property $(\\beta)$, being decent, or being reasonable\nis preserved under arbitrary base change.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03L0","source_file":"decent-spaces.tex","source_line":3735,"source_end_line":3739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3735-L3739","statement_sha256":"7baf60674299125b6c3355e7d7a7ea26c4b128f004bfe0673173b02bb9973e10","origin":"The Stacks Project","memory_eligible":false,"source_rank":11273,"rank":11273,"depth":0,"x":369.171,"y":1625.137,"cluster":"algebraic-spaces"},{"id":"stacks:0ABY","tag":"0ABY","title":"Relative conditions · Lemma 0ABY","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let ω ∈ (β, decent, reasonable). Suppose that Y has property (ω) and f : X → Y has (ω). Then X has (ω).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\omega \\in \\{\\beta, decent, reasonable\\}$.\nSuppose that $Y$ has property $(\\omega)$ and $f : X \\to Y$ has $(\\omega)$.\nThen $X$ has $(\\omega)$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABY","source_file":"decent-spaces.tex","source_line":3745,"source_end_line":3752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3745-L3752","statement_sha256":"e98fa84ebcad0900b117edc906e857a7e2c1ebcc484f5588c0b6ee6d44ce1ff0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11274,"rank":11274,"depth":15,"x":107.063,"y":1660.483,"cluster":"algebraic-spaces"},{"id":"stacks:03L1","tag":"03L1","title":"Relative conditions · Lemma 03L1","summary":"Having property (β), being decent, or being reasonable is preserved under compositions.","statement_latex":"Having property $(\\beta)$, being decent, or being reasonable\nis preserved under compositions.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03L1","source_file":"decent-spaces.tex","source_line":3801,"source_end_line":3805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3801-L3805","statement_sha256":"599140225bc9cf01cdac0559eaf960fc4ab31ce6e701c6b4c481379bd2d39459","origin":"The Stacks Project","memory_eligible":false,"source_rank":11275,"rank":11275,"depth":16,"x":272.048,"y":1485.551,"cluster":"algebraic-spaces"},{"id":"stacks:0ABZ","tag":"0ABZ","title":"Relative conditions · Lemma 0ABZ","summary":"Let S be a scheme. Let f : X → Y, g : Z → Y be morphisms of algebraic spaces over S. If X and Z are decent (resp. reasonable, resp. have property (β) of Lemma [Tag 03JX]), then so does X ×_Y Z.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$, $g : Z \\to Y$ be morphisms\nof algebraic spaces over $S$. If $X$ and $Z$ are decent\n(resp.\\ reasonable, resp.\\ have property \n$(\\beta)$ of Lemma \\ref{lemma-bounded-fibres}), then so does $X \\times_Y Z$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABZ","source_file":"decent-spaces.tex","source_line":3821,"source_end_line":3827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3821-L3827","statement_sha256":"0186646ca31a0fc11bf049f5b1073b92bfaf90b0ad9b52caba5bfb592fda7b80","origin":"The Stacks Project","memory_eligible":false,"source_rank":11276,"rank":11276,"depth":59,"x":291.081,"y":1708.34,"cluster":"algebraic-spaces"},{"id":"stacks:03L2","tag":"03L2","title":"Relative conditions · Lemma 03L2","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let P ∈ ((β), decent, reasonable). Assume • f is quasi-compact, • f is étale, • |f| : |X| → |Y| is surjective, and • the algebraic space X has property P. Then Y has property P.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{P} \\in \\{(\\beta), decent, reasonable\\}$.\nAssume\n\\begin{enumerate}\n\\item $f$ is quasi-compact,\n\\item $f$ is \\'etale,\n\\item $|f| : |X| \\to |Y|$ is surjective, and\n\\item the algebraic space $X$ has property $\\mathcal{P}$.\n\\end{enumerate}\nThen $Y$ has property $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03L2","source_file":"decent-spaces.tex","source_line":3838,"source_end_line":3851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3838-L3851","statement_sha256":"f3ae2603b953046d22d2307d0dc30b87dfc1fa08fac5972e175937b0b1d3fab9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11277,"rank":11277,"depth":55,"x":97.728,"y":1554.734,"cluster":"algebraic-spaces"},{"id":"stacks:03L3","tag":"03L3","title":"Relative conditions · Lemma 03L3","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let P ∈ ((β), decent, reasonable, very reasonable). The following are equivalent • f is P, • for every affine scheme Z and every morphism Z → Y the base change Z ×_Y X → Z of f is P, • for every affine scheme Z and every morphism Z → Y the algebraic space Z ×_Y X is P, and • there exists a Zariski covering Y = ⋃ Y_i such that each morphism f^-1(Y_i) → Y_i has P. If P ∈ ((β), decent, reasonable),…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{P} \\in \\{(\\beta), decent, reasonable, very\\ reasonable\\}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is $\\mathcal{P}$,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the\nbase change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the\nalgebraic space $Z \\times_Y X$ is $\\mathcal{P}$, and\n\\item there exists a Zariski covering $Y = \\bigcup Y_i$ such\nthat each morphism $f^{-1}(Y_i) \\to Y_i$ has $\\mathcal{P}$.\n\\end{enumerate}\nIf $\\mathcal{P} \\in \\{(\\beta), decent, reasonable\\}$, then this is also\nequivalent to\n\\begin{enumerate}\n\\item[(5)] there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that the base change $V \\times_Y X \\to V$ has\n$\\mathcal{P}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03L3","source_file":"decent-spaces.tex","source_line":3944,"source_end_line":3966,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L3944-L3966","statement_sha256":"c6eca53ce6d195e8ae6670f52a9f2584bd8a5ff474e63d414ec750432cc523d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11278,"rank":11278,"depth":57,"x":364.045,"y":1558.291,"cluster":"algebraic-spaces"},{"id":"stacks:03M6","tag":"03M6","title":"Relative conditions · Lemma 03M6","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-compact and decent. (For example if f is representable, or quasi-separated, see Lemma [Tag 03M5].) Then f is universally closed if and only if the existence part of the valuative criterion holds.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is quasi-compact and decent.\n(For example if $f$ is representable, or quasi-separated, see\nLemma \\ref{lemma-properties-trivial-implications}.)\nThen $f$ is universally closed if and only if the\nexistence part of the valuative criterion holds.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Relative conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03M6","source_file":"decent-spaces.tex","source_line":4026,"source_end_line":4035,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4026-L4035","statement_sha256":"3e3583a390d6f49a081ced5f976133d1893c229c343e9255af586916479946e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11279,"rank":11279,"depth":58,"x":164.646,"y":1706.903,"cluster":"algebraic-spaces"},{"id":"stacks:0AC5","tag":"0AC5","title":"Points of fibres · Lemma 0AC5","summary":"In the situation of ([Tag 0AC1]) if Z' → Z is a morphism and z' ∈ |Z'| maps to z, then the induced map F_x, z' → F_x, z is surjective.","statement_latex":"In the situation of (\\ref{equation-points-fibres}) if $Z' \\to Z$\nis a morphism and $z' \\in |Z'|$ maps to $z$, then the induced map\n$F_{x, z'} \\to F_{x, z}$ is surjective.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AC5","source_file":"decent-spaces.tex","source_line":4108,"source_end_line":4113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4108-L4113","statement_sha256":"2592e5e9caa576557ac45fd48a0c2be915ecc4dc29d277d13c91b2435e54dad3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11280,"rank":11280,"depth":1,"x":192.192,"y":1483.995,"cluster":"algebraic-spaces"},{"id":"stacks:0AC6","tag":"0AC6","title":"Points of fibres · Lemma 0AC6","summary":"In diagram ([Tag 0AC1]) the set ([Tag 0AC2]) is finite if f is of finite type and f is quasi-finite at x.","statement_latex":"In diagram (\\ref{equation-points-fibres}) the set (\\ref{equation-fibre})\nis finite if $f$ is of finite type and $f$ is quasi-finite at $x$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AC6","source_file":"decent-spaces.tex","source_line":4122,"source_end_line":4126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4122-L4126","statement_sha256":"d8d0b900fd34dcf318b5ed9c9d4fb4bf8b99c150f1e29421b1166e2f736cfc87","origin":"The Stacks Project","memory_eligible":false,"source_rank":11281,"rank":11281,"depth":50,"x":351.265,"y":1664.139,"cluster":"algebraic-spaces"},{"id":"stacks:0AC7","tag":"0AC7","title":"Points of fibres · Lemma 0AC7","summary":"In diagram ([Tag 0AC1]) the set ([Tag 0AC2]) is finite if y can be represented by a monomorphism Spec(k) → Y where k is a field and g is quasi-finite at z. (Special case: Y is decent and g is étale.)","statement_latex":"In diagram (\\ref{equation-points-fibres}) the set (\\ref{equation-fibre})\nis finite if $y$ can be represented by a monomorphism $\\Spec(k) \\to Y$\nwhere $k$ is a field and $g$ is quasi-finite at $z$.\n(Special case: $Y$ is decent and $g$ is \\'etale.)","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AC7","source_file":"decent-spaces.tex","source_line":4137,"source_end_line":4143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4137-L4143","statement_sha256":"dba2c6b6a72e51481998c31da97de2cefac1b89d5b9e90e8130584738f9b89a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11282,"rank":11282,"depth":49,"x":88.889,"y":1621.53,"cluster":"algebraic-spaces"},{"id":"stacks:0AC8","tag":"0AC8","title":"Points of fibres · Lemma 0AC8","summary":"Fibers of field points of algebraic spaces have the expected Zariski topologies. Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let y ∈ |Y| and assume that y is represented by a quasi-compact monomorphism Spec(k) → Y. Then |X_k| → |X| is a homeomorphism onto f^-1((y)) ⊂ |X| with induced topology.","statement_latex":"\\begin{slogan}\nFibers of field points of algebraic spaces have the\nexpected Zariski topologies.\n\\end{slogan}\nLet $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$.\nLet $y \\in |Y|$ and assume that $y$ is represented by a quasi-compact\nmonomorphism $\\Spec(k) \\to Y$. Then $|X_k| \\to |X|$ is a\nhomeomorphism onto $f^{-1}(\\{y\\}) \\subset |X|$ with induced topology.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AC8","source_file":"decent-spaces.tex","source_line":4174,"source_end_line":4185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4174-L4185","statement_sha256":"877f58422749728ca6afad135c6c79545c01bf58b734d69356ca7a54feb010cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11283,"rank":11283,"depth":55,"x":316.809,"y":1503.977,"cluster":"algebraic-spaces"},{"id":"stacks:0AC9","tag":"0AC9","title":"Points of fibres · Lemma 0AC9","summary":"Let X be an algebraic space locally of finite type over a field k. Let x ∈ |X|. Consider the conditions • dim_x(|X|) = 0, • x is closed in |X| and if x' leadsto x in |X| then x' = x, • x is an isolated point of |X|, • dim_x(X) = 0, • X → Spec(k) is quasi-finite at x. Then (2), (3), (4), and (5) are equivalent. If X is decent, then (1) is equivalent to the others.","statement_latex":"Let $X$ be an algebraic space locally of finite type over a field $k$.\nLet $x \\in |X|$. Consider the conditions\n\\begin{enumerate}\n\\item $\\dim_x(|X|) = 0$,\n\\item $x$ is closed in $|X|$ and if $x' \\leadsto x$  in $|X|$ then $x' = x$,\n\\item $x$ is an isolated point of $|X|$,\n\\item $\\dim_x(X) = 0$,\n\\item $X \\to \\Spec(k)$ is quasi-finite at $x$.\n\\end{enumerate}\nThen (2), (3), (4), and (5) are equivalent.\nIf $X$ is decent, then (1) is equivalent to the others.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AC9","source_file":"decent-spaces.tex","source_line":4224,"source_end_line":4237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4224-L4237","statement_sha256":"3b3e5ddae0b95e7bdc8f3e85dc1c389b0bbfc0b37cf45e1520b91fb60f2fa92d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11284,"rank":11284,"depth":58,"x":243.218,"y":1720.159,"cluster":"algebraic-spaces"},{"id":"stacks:0ACA","tag":"0ACA","title":"Points of fibres · Lemma 0ACA","summary":"Let X be an algebraic space locally of finite type over a field k. Consider the conditions • |X| is a finite set, • |X| is a discrete space, • dim(|X|) = 0, • dim(X) = 0, • X → Spec(k) is locally quasi-finite, Then (2), (3), (4), and (5) are equivalent. If X is decent, then (1) implies the others.","statement_latex":"Let $X$ be an algebraic space locally of finite type over a field $k$.\nConsider the conditions\n\\begin{enumerate}\n\\item $|X|$ is a finite set,\n\\item $|X|$ is a discrete space,\n\\item $\\dim(|X|) = 0$,\n\\item $\\dim(X) = 0$,\n\\item $X \\to \\Spec(k)$ is locally quasi-finite,\n\\end{enumerate}\nThen (2), (3), (4), and (5) are equivalent.\nIf $X$ is decent, then (1) implies the others.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ACA","source_file":"decent-spaces.tex","source_line":4279,"source_end_line":4292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4279-L4292","statement_sha256":"4e3626d032fe5fd4aebbcb7c591054115489212081ab461cb3609f0b0afcd202","origin":"The Stacks Project","memory_eligible":false,"source_rank":11285,"rank":11285,"depth":48,"x":123.539,"y":1518.832,"cluster":"algebraic-spaces"},{"id":"stacks:0ACB","tag":"0ACB","title":"Points of fibres · Lemma 0ACB","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let x ∈ |X| with image y ∈ |Y|. Let F = f^-1((y)) with induced topology from |X|. Let k be a field and let Spec(k) → Y be in the equivalence class defining y. Set X_k = Spec(k) ×_Y X. Let tilde x ∈ |X_k| map to x ∈ |X|. Consider the following conditions • dim_x(F) = 0, • x is isolated in F, • x is closed in F and if x' leadsto x in F, then x = x', • dim_tilde…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is locally of finite type. Let $x \\in |X|$ with image\n$y \\in |Y|$. Let $F = f^{-1}(\\{y\\})$ with induced topology from $|X|$.\nLet $k$ be a field and let $\\Spec(k) \\to Y$ be in the\nequivalence class defining $y$. Set $X_k = \\Spec(k) \\times_Y X$.\nLet $\\tilde x \\in |X_k|$ map to $x \\in |X|$.\nConsider the following conditions\n\\begin{enumerate}\n\\item\n\n$\\dim_x(F) = 0$,\n\\item\n\n$x$ is isolated in $F$,\n\\item\n\n$x$ is closed in $F$ and if $x' \\leadsto x$ in $F$, then $x = x'$,\n\\item\n\n$\\dim_{\\tilde x}(|X_k|) = 0$,\n\\item\n\n$\\tilde x$ is isolated in $|X_k|$,\n\\item\n\n$\\tilde x$ is closed in $|X_k|$ and if $\\tilde x' \\leadsto \\tilde x$\nin $|X_k|$, then $\\tilde x = \\tilde x'$,\n\\item\n\n$\\dim_{\\tilde x}(X_k) = 0$,\n\\item\n\n$f$ is quasi-finite at $x$.\n\\end{enumerate}\nThen we have\n$$\n\\xymatrix{\n(\\ref{item-dimension-top-k-fibre}) \\ar@{=>}[r]_{f\\text{ decent}} &\n(\\ref{item-isolated-in-k-fibre}) \\ar@{<=>}[r] &\n(\\ref{item-no-specializations-in-k-fibre}) \\ar@{<=>}[r] &\n(\\ref{item-k-fibre-at-x-dim-0}) \\ar@{<=>}[r] &\n(\\ref{item-quasi-finite-at-x})\n}\n$$\nIf $Y$ is decent, then conditions (\\ref{item-isolated-in-fibre}) and\n(\\ref{item-no-specializations-in-fibre}) are equivalent to each other\nand to conditions\n(\\ref{item-isolated-in-k-fibre}),\n(\\ref{item-no-specializations-in-k-fibre}),\n(\\ref{item-k-fibre-at-x-dim-0}), and\n(\\ref{item-quasi-finite-at-x}).\nIf $Y$ and $X$ are decent, then all conditions are equivalent.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ACB","source_file":"decent-spaces.tex","source_line":4333,"source_end_line":4387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4333-L4387","statement_sha256":"adcbc6a25672dd9669c6fd14077cc4e49d25d4015b7c96e7c521a1ebcb8a7073","origin":"The Stacks Project","memory_eligible":false,"source_rank":11286,"rank":11286,"depth":59,"x":373.891,"y":1599.443,"cluster":"algebraic-spaces"},{"id":"stacks:0ACK","tag":"0ACK","title":"Points of fibres · Lemma 0ACK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let y ∈ |Y|. Let k be a field and let Spec(k) → Y be in the equivalence class defining y. Set X_k = Spec(k) ×_Y X and let F = f^-1((y)) with the induced topology from |X|. Consider the following conditions • F is finite, • F is a discrete topological space, • dim(F) = 0, • |X_k| is a finite set, • |X_k| is a discrete space, • dim(|X_k|) = 0, • dim(X_k) = 0, • f is…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is locally of finite type. Let $y \\in |Y|$. Let $k$ be a field\nand let $\\Spec(k) \\to Y$ be in the equivalence class defining $y$.\nSet $X_k = \\Spec(k) \\times_Y X$ and let $F = f^{-1}(\\{y\\})$ with the\ninduced topology from $|X|$. Consider the following conditions\n\\begin{enumerate}\n\\item\n\n$F$ is finite,\n\\item\n\n$F$ is a discrete topological space,\n\\item\n\n$\\dim(F) = 0$,\n\\item\n\n$|X_k|$ is a finite set,\n\\item\n\n$|X_k|$ is a discrete space,\n\\item\n\n$\\dim(|X_k|) = 0$,\n\\item\n\n$\\dim(X_k) = 0$,\n\\item\n\n$f$ is quasi-finite at all points of $|X|$ lying over $y$.\n\\end{enumerate}\nThen we have\n$$\n\\xymatrix{\n(\\ref{item-fibre-finite}) &\n(\\ref{item-k-fibre-finite}) \\ar@{=>}[l] \\ar@{=>}[r]_{f\\text{ decent}} &\n(\\ref{item-k-fibre-discrete}) \\ar@{<=>}[r] &\n(\\ref{item-k-fibre-no-specializations}) \\ar@{<=>}[r] &\n(\\ref{item-k-fibre-dim-0}) \\ar@{<=>}[r] &\n(\\ref{item-quasi-finite-at-points-fibre})\n}\n$$\nIf $Y$ is decent, then conditions (\\ref{item-fibre-discrete}) and\n(\\ref{item-fibre-no-specializations})\nare equivalent to each other and to conditions (\\ref{item-k-fibre-discrete}),\n(\\ref{item-k-fibre-no-specializations}), (\\ref{item-k-fibre-dim-0}), and\n(\\ref{item-quasi-finite-at-points-fibre}).\nIf $Y$ and $X$ are decent, then (\\ref{item-fibre-finite}) implies\nall the other conditions.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Points of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ACK","source_file":"decent-spaces.tex","source_line":4412,"source_end_line":4463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4412-L4463","statement_sha256":"91f23b9446e71b364607394aa414995394cb994011b0f1247f932d9f57597724","origin":"The Stacks Project","memory_eligible":false,"source_rank":11287,"rank":11287,"depth":56,"x":124.261,"y":1682.123,"cluster":"algebraic-spaces"},{"id":"stacks:06RZ","tag":"06RZ","title":"Monomorphisms · Lemma 06RZ","summary":"Let S be a scheme. Let Y be a disjoint union of spectra of zero dimensional local rings over S. Let f : X → Y be a monomorphism of algebraic spaces over S. Then f is representable, i.e., X is a scheme.","statement_latex":"Let $S$ be a scheme. Let $Y$ be a disjoint union of spectra of\nzero dimensional local rings over $S$.\nLet $f : X \\to Y$ be a monomorphism of algebraic spaces over $S$.\nThen $f$ is representable, i.e., $X$ is a scheme.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RZ","source_file":"decent-spaces.tex","source_line":4508,"source_end_line":4514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4508-L4514","statement_sha256":"ad35a5a1ba60b321a0524ddcdb986ec72bbc62854aa8ee4e0431632a116b0d33","origin":"The Stacks Project","memory_eligible":false,"source_rank":11288,"rank":11288,"depth":46,"x":241.939,"y":1479.349,"cluster":"algebraic-spaces"},{"id":"stacks:0ABV","tag":"0ABV","title":"Generic points · Lemma 0ABV","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X|. The following are equivalent • x is a generic point of an irreducible component of |X|, • for any étale morphism (Y, y) → (X, x) of pointed algebraic spaces, y is a generic point of an irreducible component of |Y|, • for some étale morphism (Y, y) → (X, x) of pointed algebraic spaces, y is a generic point of an irreducible component of |Y|, • the dimension of the local ring of X at x is zero, and •…","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$. The following are equivalent\n\\begin{enumerate}\n\\item $x$ is a generic point of an irreducible component of $|X|$,\n\\item for any \\'etale morphism $(Y, y) \\to (X, x)$ of pointed algebraic\nspaces, $y$ is a generic point of an irreducible component of $|Y|$,\n\\item for some \\'etale morphism $(Y, y) \\to (X, x)$ of pointed algebraic\nspaces, $y$ is a generic point of an irreducible component of $|Y|$,\n\\item the dimension of the local ring of $X$ at $x$ is zero, and\n\\item $x$ is a point of codimension $0$ on $X$\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Generic points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ABV","source_file":"decent-spaces.tex","source_line":4574,"source_end_line":4587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4574-L4587","statement_sha256":"e4216790289b8d4dab59570cf597b7beb70c5428562da36ad90415f20ec91e63","origin":"The Stacks Project","memory_eligible":false,"source_rank":11289,"rank":11289,"depth":60,"x":318.291,"y":1695.817,"cluster":"algebraic-spaces"},{"id":"stacks:0ED1","tag":"0ED1","title":"Generic points · Lemma 0ED1","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let T ⊂ |X| be an irreducible closed subset. Let xi ∈ T be the generic point (Proposition [Tag 03K6]). Then codim(T, |X|) (Topology, Definition [Tag 02I3]) is the dimension of the local ring of X at xi (Properties of Spaces, Definition [Tag 04NA]).","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $T \\subset |X|$ be an irreducible closed subset. Let $\\xi \\in T$\nbe the generic point (Proposition \\ref{proposition-reasonable-sober}).\nThen $\\text{codim}(T, |X|)$\n(Topology, Definition \\ref{topology-definition-codimension})\nis the dimension of the local ring of $X$ at $\\xi$\n(Properties of Spaces, Definition\n\\ref{spaces-properties-definition-dimension-local-ring}).","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Generic points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ED1","source_file":"decent-spaces.tex","source_line":4619,"source_end_line":4629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4619-L4629","statement_sha256":"5e0efca70a89cf0968c42802b8a70ccc2a9f774be5f7658dfb0ac935307204bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11290,"rank":11290,"depth":60,"x":87.73,"y":1579.428,"cluster":"algebraic-spaces"},{"id":"stacks:0BB8","tag":"0BB8","title":"Generic points · Lemma 0BB8","summary":"Let S be a scheme. Let X be an algebraic space over S. Assume • every quasi-compact scheme étale over X has finitely many irreducible components, and • every x ∈ |X| of codimension 0 on X can be represented by a monomorphism Spec(k) → X. Then X is a reasonable algebraic space.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume\n\\begin{enumerate}\n\\item every quasi-compact scheme \\'etale over $X$ has finitely many\nirreducible components, and\n\\item every $x \\in |X|$ of codimension $0$ on $X$ can be represented\nby a monomorphism $\\Spec(k) \\to X$.\n\\end{enumerate}\nThen $X$ is a reasonable algebraic space.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Generic points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BB8","source_file":"decent-spaces.tex","source_line":4649,"source_end_line":4659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4649-L4659","statement_sha256":"c5ef6c7ca9a1e9b5f1cf2b8664d83f79b9d15262fa0399dc6e0126e62843fea5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11291,"rank":11291,"depth":54,"x":351.547,"y":1534.39,"cluster":"algebraic-spaces"},{"id":"stacks:0BB9","tag":"0BB9","title":"Generic points · Lemma 0BB9","summary":"Let S be a scheme. Let X be an algebraic space over S. The following are equivalent • X is decent and |X| has finitely many irreducible components, • every quasi-compact scheme étale over X has finitely many irreducible components, there are finitely many x ∈ |X| of codimension 0 on X, and each of these can be represented by a monomorphism Spec(k) → X, • there exists a dense open X' ⊂ X which is a scheme, X' has finitely many irreducible components with generic points…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is decent and $|X|$ has finitely many irreducible components,\n\\item every quasi-compact scheme \\'etale over $X$ has finitely many\nirreducible components, there are finitely many $x \\in |X|$ of\ncodimension $0$ on $X$, and each of these can be represented\nby a monomorphism $\\Spec(k) \\to X$,\n\\item there exists a dense open $X' \\subset X$ which is\na scheme, $X'$ has finitely many irreducible components\nwith generic points $\\{x'_1, \\ldots, x'_m\\}$, and\nthe morphism $x'_j \\to X$ is quasi-compact for $j = 1, \\ldots, m$.\n\\end{enumerate}\nMoreover, if these conditions hold, then $X$ is reasonable and the\npoints $x'_j \\in |X|$ are the generic points of the irreducible\ncomponents of $|X|$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Generic points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BB9","source_file":"decent-spaces.tex","source_line":4693,"source_end_line":4711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4693-L4711","statement_sha256":"8fdff2397a9bcebbb664d884f64ef6492aa0795b7a4b8b39051c35451937588c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11292,"rank":11292,"depth":59,"x":193.106,"y":1717.442,"cluster":"algebraic-spaces"},{"id":"stacks:0ACZ","tag":"0ACZ","title":"Generically finite morphisms · Lemma 0ACZ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that f is quasi-separated of finite type. Let y ∈ |Y| be a point of codimension 0 on Y. The following are equivalent: • the space |X_k| is finite where Spec(k) → Y represents y, • X → Y is quasi-finite at all points of |X| over y, • there exists an open subspace Y' ⊂ Y with y ∈ |Y'| such that Y' ×_Y X → Y' is finite. If Y is decent these are also equivalent to • [(4)] the set f^-1((y)) is…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume that $f$ is quasi-separated of finite type.\nLet $y \\in |Y|$ be a point of codimension $0$ on $Y$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the space $|X_k|$ is finite where $\\Spec(k) \\to Y$ represents $y$,\n\\item $X \\to Y$ is quasi-finite at all points of $|X|$ over $y$,\n\\item there exists an open subspace $Y' \\subset Y$ with $y \\in |Y'|$\nsuch that $Y' \\times_Y X \\to Y'$ is finite.\n\\end{enumerate}\nIf $Y$ is decent these are also equivalent to\n\\begin{enumerate}\n\\item[(4)] the set $f^{-1}(\\{y\\})$ is finite.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ACZ","source_file":"decent-spaces.tex","source_line":4761,"source_end_line":4777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4761-L4777","statement_sha256":"f72f85c379f6e2122e3287b8a02f7e33d39f4c28430a7a5b0a38e5b429fbe58e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11293,"rank":11293,"depth":57,"x":162.709,"y":1492.381,"cluster":"algebraic-spaces"},{"id":"stacks:0AD0","tag":"0AD0","title":"Generically finite morphisms · Lemma 0AD0","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that f is quasi-separated and locally of finite type and Y quasi-separated. Let y ∈ |Y| be a point of codimension 0 on Y. The following are equivalent: • the set f^-1((y)) is finite, • the space |X_k| is finite where Spec(k) → Y represents y, • there exist open subspaces X' ⊂ X and Y' ⊂ Y with f(X') ⊂ Y', y ∈ |Y'|, and f^-1((y)) ⊂ |X'| such that f|_X' : X' → Y' is finite.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume that $f$ is quasi-separated and locally of finite type\nand $Y$ quasi-separated. Let $y \\in |Y|$ be a point of codimension $0$ on $Y$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the set $f^{-1}(\\{y\\})$ is finite,\n\\item the space $|X_k|$ is finite where $\\Spec(k) \\to Y$ represents $y$,\n\\item there exist open subspaces $X' \\subset X$ and $Y' \\subset Y$\nwith $f(X') \\subset Y'$, $y \\in |Y'|$, and $f^{-1}(\\{y\\}) \\subset |X'|$\nsuch that $f|_{X'} : X' \\to Y'$ is finite.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AD0","source_file":"decent-spaces.tex","source_line":4835,"source_end_line":4848,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4835-L4848","statement_sha256":"521bf76fedf0b809750c894b3029c9f9ed2c9b2eeeccf263a5c66e736ff5de40","origin":"The Stacks Project","memory_eligible":false,"source_rank":11294,"rank":11294,"depth":58,"x":366.271,"y":1641.207,"cluster":"algebraic-spaces"},{"id":"stacks:0BBB","tag":"0BBB","title":"Generically finite morphisms · Lemma 0BBB","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type. Let X^0 ⊂ |X|, resp. Y^0 ⊂ |Y| denote the set of codimension 0 points of X, resp. Y. Let y ∈ Y^0. The following are equivalent • f^-1((y)) ⊂ X^0, • f is quasi-finite at all points lying over y, • f is quasi-finite at all x ∈ X^0 lying over y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is locally of finite type.\nLet $X^0 \\subset |X|$, resp.\\ $Y^0 \\subset |Y|$ denote the set of\ncodimension $0$ points of $X$, resp.\\ $Y$. Let $y \\in Y^0$. The following are\nequivalent\n\\begin{enumerate}\n\\item $f^{-1}(\\{y\\}) \\subset X^0$,\n\\item $f$ is quasi-finite at all points lying over $y$,\n\\item $f$ is quasi-finite at all $x \\in X^0$ lying over $y$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBB","source_file":"decent-spaces.tex","source_line":4867,"source_end_line":4879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4867-L4879","statement_sha256":"2e1a9f2b593debaa4f3c177217df1dc78bd4afdca8621e5fdb12ea51d8b31fd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11295,"rank":11295,"depth":28,"x":96.273,"y":1646.975,"cluster":"algebraic-spaces"},{"id":"stacks:0BBC","tag":"0BBC","title":"Generically finite morphisms · Lemma 0BBC","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type. Let X^0 ⊂ |X|, resp. Y^0 ⊂ |Y| denote the set of codimension 0 points of X, resp. Y. Assume • Y is decent, • X^0 and Y^0 are finite and f^-1(Y^0) = X^0, • either f is quasi-compact or f is separated. Then there exists a dense open V ⊂ Y such that f^-1(V) → V is finite.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is locally of finite type.\nLet $X^0 \\subset |X|$, resp.\\ $Y^0 \\subset |Y|$ denote the set of\ncodimension $0$ points of $X$, resp.\\ $Y$. Assume\n\\begin{enumerate}\n\\item $Y$ is decent,\n\\item $X^0$ and $Y^0$ are finite and $f^{-1}(Y^0) = X^0$,\n\\item either $f$ is quasi-compact or $f$ is separated.\n\\end{enumerate}\nThen there exists a dense open $V \\subset Y$\nsuch that $f^{-1}(V) \\to V$ is finite.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBC","source_file":"decent-spaces.tex","source_line":4893,"source_end_line":4906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4893-L4906","statement_sha256":"9bcf8598f487218baa84b4e68281563e3b554b2db80536b7ac0ecf25e92afab0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11296,"rank":11296,"depth":61,"x":290.88,"y":1489.395,"cluster":"algebraic-spaces"},{"id":"stacks:0ACV","tag":"0ACV","title":"Birational morphisms · Definition 0ACV","summary":"Let S be a scheme. Let X and Y algebraic spaces over S. Assume X and Y are decent and that |X| and |Y| have finitely many irreducible components. We say a morphism f : X → Y is birational if • |f| induces a bijection between the set of generic points of irreducible components of |X| and the set of generic points of the irreducible components of |Y|, and • for every generic point x ∈ |X| of an irreducible component the local ring map O_Y, f(x) → O_X, x is an isomorphism…","statement_latex":"Let $S$ be a scheme. Let $X$ and $Y$ algebraic spaces over $S$.\nAssume $X$ and $Y$ are decent and that $|X|$ and $|Y|$ have finitely many\nirreducible components. We say a morphism $f : X \\to Y$ is\n{\\it birational} if\n\\begin{enumerate}\n\\item $|f|$ induces a bijection between the set of generic points\nof irreducible components of $|X|$ and the set of generic points\nof the irreducible components of $|Y|$, and\n\\item for every generic point $x \\in |X|$ of an irreducible component\nthe local ring map $\\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X, x}$\nis an isomorphism (see clarification below).\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Birational morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ACV","source_file":"decent-spaces.tex","source_line":4972,"source_end_line":4986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L4972-L4986","statement_sha256":"482b5435253ecb7c35c0cc04c1430e69376fbde67e1fe7567b4582745be27774","origin":"The Stacks Project","memory_eligible":false,"source_rank":11297,"rank":11297,"depth":0,"x":274.088,"y":1716.194,"cluster":"algebraic-spaces"},{"id":"stacks:0ACW","tag":"0ACW","title":"Birational morphisms · Lemma 0ACW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which are decent and have finitely many irreducible components. If f is birational then f is dominant.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which\nare decent and have finitely many irreducible components. If $f$ is\nbirational then $f$ is dominant.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ACW","source_file":"decent-spaces.tex","source_line":5016,"source_end_line":5022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5016-L5022","statement_sha256":"d49518ba68cb81a1e758687b3315ceea5676474e43e366cbcf6473f5ae3387f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11298,"rank":11298,"depth":1,"x":103.951,"y":1539.29,"cluster":"algebraic-spaces"},{"id":"stacks:0BBD","tag":"0BBD","title":"Birational morphisms · Lemma 0BBD","summary":"Let S be a scheme. Let f : X → Y be a birational morphism of algebraic spaces over S which are decent and have finitely many irreducible components. If y ∈ Y is the generic point of an irreducible component, then the base change X ×_Y Spec(O_Y, y) → Spec(O_Y, y) is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism of\nalgebraic spaces over $S$ which are decent and have finitely\nmany irreducible components. If $y \\in Y$ is the generic point of\nan irreducible component, then the base change\n$X \\times_Y \\Spec(\\mathcal{O}_{Y, y}) \\to \\Spec(\\mathcal{O}_{Y, y})$\nis an isomorphism.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBD","source_file":"decent-spaces.tex","source_line":5029,"source_end_line":5037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5029-L5037","statement_sha256":"d65b45163f8e2d43d7f672968cb436d5d8006ba0fe1fc7cb25c11194a8f45e15","origin":"The Stacks Project","memory_eligible":false,"source_rank":11299,"rank":11299,"depth":60,"x":371.879,"y":1573.223,"cluster":"algebraic-spaces"},{"id":"stacks:0BBE","tag":"0BBE","title":"Birational morphisms · Lemma 0BBE","summary":"Let S be a scheme. Let f : X → Y be a birational morphism of algebraic spaces over S which are decent and have finitely many irreducible components. Assume one of the following conditions is satisfied • f is locally of finite type and Y reduced (i.e., integral), • f is locally of finite presentation. Then there exist dense opens U ⊂ X and V ⊂ Y such that f(U) ⊂ V and f|_U : U → V is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism of\nalgebraic spaces over $S$ which are decent and have finitely many\nirreducible components. Assume one of the following conditions is satisfied\n\\begin{enumerate}\n\\item $f$ is locally of finite type and $Y$ reduced (i.e., integral),\n\\item $f$ is locally of finite presentation.\n\\end{enumerate}\nThen there exist dense opens $U \\subset X$ and $V \\subset Y$\nsuch that $f(U) \\subset V$ and $f|_U : U \\to V$ is an isomorphism.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBE","source_file":"decent-spaces.tex","source_line":5051,"source_end_line":5062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5051-L5062","statement_sha256":"c744978bf9a084df099795f61eb921442ed90b3afa5cbe54bc0cddf3965fffd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11300,"rank":11300,"depth":60,"x":146.849,"y":1700.329,"cluster":"algebraic-spaces"},{"id":"stacks:0BBF","tag":"0BBF","title":"Birational morphisms · Lemma 0BBF","summary":"Let S be a scheme. Let f : X → Y be a birational morphism of algebraic spaces over S which are decent and have finitely many irreducible components. Assume • either f is quasi-compact or f is separated, and • either f is locally of finite type and Y is reduced or f is locally of finite presentation. Then there exists a dense open V ⊂ Y such that f^-1(V) → V is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism of\nalgebraic spaces over $S$ which are decent and have finitely\nmany irreducible components. Assume\n\\begin{enumerate}\n\\item either $f$ is quasi-compact or $f$ is separated, and\n\\item either $f$ is locally of finite type and $Y$ is reduced or\n$f$ is locally of finite presentation.\n\\end{enumerate}\nThen there exists a dense open $V \\subset Y$\nsuch that $f^{-1}(V) \\to V$ is an isomorphism.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBF","source_file":"decent-spaces.tex","source_line":5070,"source_end_line":5082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5070-L5082","statement_sha256":"a56cc511b8992956dea095d6405d777c5af1ac9d98f351d8ea08a8aff71dcfae","origin":"The Stacks Project","memory_eligible":false,"source_rank":11301,"rank":11301,"depth":62,"x":210.617,"y":1478.743,"cluster":"algebraic-spaces"},{"id":"stacks:0B4D","tag":"0B4D","title":"Birational morphisms · Lemma 0B4D","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which are decent and have finitely many irreducible components. If f is birational and V → Y is an étale morphism with V affine, then X ×_Y V is decent with finitely many irreducible components and X ×_Y V → V is birational.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$ which are decent and have finitely many irreducible\ncomponents. If $f$ is birational and $V \\to Y$ is an \\'etale morphism\nwith $V$ affine, then $X \\times_Y V$ is decent with finitely\nmany irreducible components and $X \\times_Y V \\to V$ is birational.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4D","source_file":"decent-spaces.tex","source_line":5091,"source_end_line":5098,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5091-L5098","statement_sha256":"5875839de8a4f2a040d557b663a73491522b3e3a9c1aacbe9229c98633bf017a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11302,"rank":11302,"depth":61,"x":341.891,"y":1678.476,"cluster":"algebraic-spaces"},{"id":"stacks:0BBG","tag":"0BBG","title":"Birational morphisms · Lemma 0BBG","summary":"Let S be a scheme. Let f : X → Y be a birational morphism between algebraic spaces over S which are decent and have finitely many irreducible components. Then the normalizations X^ν → X and Y^ν → Y exist and there is a commutative diagram xymatrix X^ν ar[r] ar[d] & Y^ν ar[d] X ar[r] & Y of algebraic spaces over S. The morphism X^ν → Y^ν is birational.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a birational morphism between\nalgebraic spaces over $S$ which are decent and have finitely many irreducible\ncomponents. Then the normalizations $X^\\nu \\to X$ and $Y^\\nu \\to Y$ exist\nand there is a commutative diagram\n$$\n\\xymatrix{\nX^\\nu \\ar[r] \\ar[d] &  Y^\\nu \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nof algebraic spaces over $S$. The morphism $X^\\nu \\to Y^\\nu$ is birational.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBG","source_file":"decent-spaces.tex","source_line":5132,"source_end_line":5145,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5132-L5145","statement_sha256":"4d968cad2cf1bf13e4302d397ee4645896a98b0a893c78a492514d7b1fe441d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11303,"rank":11303,"depth":61,"x":84.272,"y":1605.627,"cluster":"algebraic-spaces"},{"id":"stacks:0B4E","tag":"0B4E","title":"Birational morphisms · Lemma 0B4E","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • X and Y are decent and have finitely many irreducible components, • f is integral and birational, • Y is normal, and • X is reduced. Then f is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume\n\\begin{enumerate}\n\\item $X$ and $Y$ are decent and have finitely many irreducible components,\n\\item $f$ is integral and birational,\n\\item $Y$ is normal, and\n\\item $X$ is reduced.\n\\end{enumerate}\nThen $f$ is an isomorphism.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4E","source_file":"decent-spaces.tex","source_line":5184,"source_end_line":5195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5184-L5195","statement_sha256":"ce64992cd254ffd6ab6191340ab931847bd479914def04c60a9dbb7e9e821b03","origin":"The Stacks Project","memory_eligible":false,"source_rank":11304,"rank":11304,"depth":62,"x":333.012,"y":1513.094,"cluster":"algebraic-spaces"},{"id":"stacks:0BBH","tag":"0BBH","title":"Birational morphisms · Lemma 0BBH","summary":"Let S be a scheme. Let f : X → Y be an integral birational morphism of decent algebraic spaces over S which have finitely many irreducible components. Then there exists a factorization Y^ν → X → Y and Y^ν → X is the normalization of X.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an integral birational morphism of\ndecent algebraic spaces over $S$ which have finitely many irreducible\ncomponents. Then there exists a factorization $Y^\\nu \\to X \\to Y$ and\n$Y^\\nu \\to X$ is the normalization of $X$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Birational morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBH","source_file":"decent-spaces.tex","source_line":5214,"source_end_line":5220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5214-L5220","statement_sha256":"f199912dc65d097d14b1976c7cc04860ec631c18568ea5536e7282d93d11fb4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11305,"rank":11305,"depth":63,"x":223.923,"y":1722.63,"cluster":"algebraic-spaces"},{"id":"stacks:0BA3","tag":"0BA3","title":"Jacobson spaces · Lemma 0BA3","summary":"Let S be a scheme. Let X be a Jacobson algebraic space over S. Any algebraic space locally of finite type over X is Jacobson.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$.\nAny algebraic space locally of finite type over $X$ is Jacobson.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BA3","source_file":"decent-spaces.tex","source_line":5249,"source_end_line":5253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5249-L5253","statement_sha256":"91f2e8cdc36d047eb2a19bb023ca7de469704ff14d3d4006532c19aa595444d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11306,"rank":11306,"depth":11,"x":135.793,"y":1506.054,"cluster":"algebraic-spaces"},{"id":"stacks:0BA4","tag":"0BA4","title":"Jacobson spaces · Lemma 0BA4","summary":"Let S be a scheme. Let X be a Jacobson algebraic space over S. For x ∈ X_ft-pts and g : W → X locally of finite type with W a scheme, if x ∈ Im(|g|), then there exists a closed point of W mapping to x.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$.\nFor $x \\in X_{\\text{ft-pts}}$ and $g : W \\to X$ locally of finite type\nwith $W$ a scheme, if $x \\in \\Im(|g|)$, then there exists a closed\npoint of $W$ mapping to $x$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BA4","source_file":"decent-spaces.tex","source_line":5266,"source_end_line":5272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5266-L5272","statement_sha256":"b8dd54cce08e79a7855cec9e659310e5b4676581ba05fd118f85d8202901675d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11307,"rank":11307,"depth":12,"x":375.127,"y":1615.831,"cluster":"algebraic-spaces"},{"id":"stacks:0BA5","tag":"0BA5","title":"Jacobson spaces · Lemma 0BA5","summary":"Let S be a scheme. Let X be a decent Jacobson algebraic space over S. Then X_ft-pts ⊂ |X| is the set of closed points.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent Jacobson algebraic space over $S$.\nThen $X_{\\text{ft-pts}} \\subset |X|$ is the set of closed points.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BA5","source_file":"decent-spaces.tex","source_line":5293,"source_end_line":5297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5293-L5297","statement_sha256":"63f9e4e3269ee61f4b166d6982c71a345fabc6b7ce595ad5c4fa819605182279","origin":"The Stacks Project","memory_eligible":false,"source_rank":11308,"rank":11308,"depth":58,"x":110.162,"y":1670.73,"cluster":"algebraic-spaces"},{"id":"stacks:0BA6","tag":"0BA6","title":"Jacobson spaces · Lemma 0BA6","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Then X is Jacobson if and only if |X| is Jacobson.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nThen $X$ is Jacobson if and only if $|X|$ is Jacobson.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BA6","source_file":"decent-spaces.tex","source_line":5334,"source_end_line":5338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5334-L5338","statement_sha256":"f967395c0d78e744c16abe69d702c74ec3520c345d2dac892b59b7221fde4eb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11309,"rank":11309,"depth":59,"x":261.512,"y":1479.754,"cluster":"algebraic-spaces"},{"id":"stacks:0ED2","tag":"0ED2","title":"Jacobson spaces · Lemma 0ED2","summary":"Let S be a scheme. Let X be a decent locally Noetherian algebraic space over S. Let x ∈ |X|. Then W = (x' ∈ |X| : x' leadsto x, x' not = x) is a Noetherian, spectral, sober, Jacobson topological space.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent locally Noetherian algebraic\nspace over $S$. Let $x \\in |X|$. Then\n$$\nW = \\{x' \\in |X| : x' \\leadsto x,\\ x' \\not = x\\}\n$$\nis a Noetherian, spectral, sober, Jacobson topological space.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Jacobson spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ED2","source_file":"decent-spaces.tex","source_line":5376,"source_end_line":5384,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5376-L5384","statement_sha256":"ae6444eabb5f8f3a21df05a15278bf3ba6f31dfbb2603e8cfb2d55fbcd279e38","origin":"The Stacks Project","memory_eligible":false,"source_rank":11310,"rank":11310,"depth":60,"x":303.519,"y":1706.63,"cluster":"algebraic-spaces"},{"id":"stacks:0DQ6","tag":"0DQ6","title":"Local irreducibility · Lemma 0DQ6","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X| be a point. The following are equivalent • for any elementary étale neighbourhood (U, u) → (X, x) the local ring O_U, u has a unique minimal prime, • for any elementary étale neighbourhood (U, u) → (X, x) there is a unique irreducible component of U through u, • for any elementary étale neighbourhood (U, u) → (X, x) the local ring O_U, u is unibranch, • the henselian local ring O_X, x^h has a unique…","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$ be a point. The following are equivalent\n\\begin{enumerate}\n\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$\nthe local ring $\\mathcal{O}_{U, u}$ has a unique minimal prime,\n\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$\nthere is a unique irreducible component of $U$ through $u$,\n\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$\nthe local ring $\\mathcal{O}_{U, u}$ is unibranch,\n\\item the henselian local ring\n$\\mathcal{O}_{X, x}^h$ has a unique minimal prime.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQ6","source_file":"decent-spaces.tex","source_line":5457,"source_end_line":5471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5457-L5471","statement_sha256":"3a2c74197860d6eff92c85d61dc8fa2567ade8178dc1ba1b4dbccafda7bcc03c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11311,"rank":11311,"depth":55,"x":89.932,"y":1563.061,"cluster":"algebraic-spaces"},{"id":"stacks:0DQ7","tag":"0DQ7","title":"Local irreducibility · Definition 0DQ7","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X|. We say that X is unibranch at x if the equivalent conditions of Lemma [Tag 0DQ6] hold. We say that X is unibranch if X is unibranch at every x ∈ |X|.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$. We say that $X$ is {\\it unibranch at $x$}\nif the equivalent conditions of\nLemma \\ref{lemma-irreducible-local-ring} hold.\nWe say that $X$ is {\\it unibranch} if $X$ is\nunibranch at every $x \\in |X|$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Local irreducibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQ7","source_file":"decent-spaces.tex","source_line":5485,"source_end_line":5493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5485-L5493","statement_sha256":"21dce70993f01e5954262a23c0a040076e6b5eee0bd251819f9c61af08b5fcd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11312,"rank":11312,"depth":56,"x":363.091,"y":1547.721,"cluster":"algebraic-spaces"},{"id":"stacks:0DQ8","tag":"0DQ8","title":"Local irreducibility · Lemma 0DQ8","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X| be a point. Let n ∈ (1, 2, …) be an integer. The following are equivalent • for any elementary étale neighbourhood (U, u) → (X, x) the number of minimal primes of the local ring O_U, u is ≤ n and for at least one choice of (U, u) it is n, • for any elementary étale neighbourhood (U, u) → (X, x) the number irreducible components of U passing through u is ≤ n and for at least one choice of (U, u) it is…","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$ be a point. Let $n \\in \\{1, 2, \\ldots\\}$ be an integer.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$\nthe number of minimal primes of the local ring $\\mathcal{O}_{U, u}$\nis $\\leq n$ and for at least one choice of $(U, u)$ it is $n$,\n\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$\nthe number irreducible components of $U$ passing through $u$ is $\\leq n$\nand for at least one choice of $(U, u)$ it is $n$,\n\\item for any elementary \\'etale neighbourhood $(U, u) \\to (X, x)$\nthe number of branches of $U$ at $u$ is $\\leq n$\nand for at least one choice of $(U, u)$ it is $n$,\n\\item the number of minimal prime ideals of\n$\\mathcal{O}_{X, x}^h$ is $n$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Local irreducibility","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQ8","source_file":"decent-spaces.tex","source_line":5499,"source_end_line":5517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5499-L5517","statement_sha256":"55ac8e6afa031f0e1d26e3820f16646750eac7b6357aa2277d8a1b6faab32b31","origin":"The Stacks Project","memory_eligible":false,"source_rank":11313,"rank":11313,"depth":55,"x":173.859,"y":1714.156,"cluster":"algebraic-spaces"},{"id":"stacks:0DQ9","tag":"0DQ9","title":"Local irreducibility · Definition 0DQ9","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X|. The number of branches of X at x is either n ∈ N if the equivalent conditions of Lemma [Tag 0DQ8] hold, or else ∞.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$. The {\\it number of branches of $X$ at $x$} is\neither $n \\in \\mathbf{N}$ if the equivalent conditions\nof Lemma \\ref{lemma-nr-branches-local-ring}\nhold, or else $\\infty$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Local irreducibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQ9","source_file":"decent-spaces.tex","source_line":5531,"source_end_line":5538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5531-L5538","statement_sha256":"20be2b34be98b0f67790d9f3fcf00d918ccea23200ad9ff3297ac4575b66bdea","origin":"The Stacks Project","memory_eligible":false,"source_rank":11314,"rank":11314,"depth":56,"x":179.558,"y":1483.879,"cluster":"algebraic-spaces"},{"id":"stacks:0ED4","tag":"0ED4","title":"Catenary algebraic spaces · Definition 0ED4","summary":"Let S be a scheme. Let X be a decent algebraic space over S. We say X is catenary if |X| is catenary (Topology, Definition [Tag 02I1]).","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nWe say $X$ is {\\it catenary} if $|X|$ is catenary\n(Topology, Definition \\ref{topology-definition-catenary}).","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Catenary algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ED4","source_file":"decent-spaces.tex","source_line":5554,"source_end_line":5559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5554-L5559","statement_sha256":"53f6cae0f4e3381e0f2f15f4325b25d600c99ea752ca8eea28083a15f0dd8183","origin":"The Stacks Project","memory_eligible":false,"source_rank":11315,"rank":11315,"depth":1,"x":360.676,"y":1657.047,"cluster":"algebraic-spaces"},{"id":"stacks:0ED5","tag":"0ED5","title":"Catenary algebraic spaces · Lemma 0ED5","summary":"Let S be a locally Noetherian and universally catenary scheme. Let δ : S → Z be a dimension function. Let X be a decent algebraic space over S such that the structure morphism X → S is locally of finite type. Let δ_X : |X| → Z be the map sending x to δ(f(x)) plus the transcendence degree of x/f(x). Then δ_X is a dimension function on |X|.","statement_latex":"Let $S$ be a locally Noetherian and universally catenary scheme.\nLet $\\delta : S \\to \\mathbf{Z}$ be a dimension function.\nLet $X$ be a decent algebraic space over $S$ such that\nthe structure morphism $X \\to S$ is locally of\nfinite type. Let $\\delta_X : |X| \\to \\mathbf{Z}$ be the map\nsending $x$ to $\\delta(f(x))$ plus the transcendence degree\nof $x/f(x)$. Then $\\delta_X$ is a dimension function on $|X|$.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Catenary algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ED5","source_file":"decent-spaces.tex","source_line":5565,"source_end_line":5574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5565-L5574","statement_sha256":"dd4614bb8ee8428a419d929a99a69c14f75dd949d3ee8bad15597aceecea98cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":11316,"rank":11316,"depth":57,"x":87.658,"y":1632.107,"cluster":"algebraic-spaces"},{"id":"stacks:0ED6","tag":"0ED6","title":"Catenary algebraic spaces · Lemma 0ED6","summary":"Let S be a locally Noetherian and universally catenary scheme. Let X be an algebraic space over S such that X is decent and such that the structure morphism X → S is locally of finite type. Then X is catenary.","statement_latex":"Let $S$ be a locally Noetherian and universally catenary scheme.\nLet $X$ be an algebraic space over $S$ such that $X$ is decent\nand such that the structure morphism $X \\to S$ is locally of\nfinite type. Then $X$ is catenary.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Catenary algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ED6","source_file":"decent-spaces.tex","source_line":5598,"source_end_line":5604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5598-L5604","statement_sha256":"a591df74e79513fcc1d42f2c72fb8872f8459eee8896a80b408dd71f1eea4cf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11317,"rank":11317,"depth":60,"x":309.201,"y":1495.477,"cluster":"algebraic-spaces"},{"id":"stacks:0ED7","tag":"0ED7","title":"Catenary algebraic spaces · Definition 0ED7","summary":"Let S be a scheme. Let X be a decent and locally Noetherian algebraic space over S. We say X is universally catenary if for every morphism Y → X of algebraic spaces which is locally of finite type and with Y decent, the algebraic space Y is catenary.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent and locally Noetherian\nalgebraic space over $S$. We say $X$ is {\\it universally catenary}\nif for every morphism $Y \\to X$ of algebraic spaces which is \nlocally of finite type and with $Y$ decent, the algebraic space\n$Y$ is catenary.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Catenary algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ED7","source_file":"decent-spaces.tex","source_line":5624,"source_end_line":5631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5624-L5631","statement_sha256":"8653b537f371bff4731977fed398f921afd822fcc9fa5c1a3cb0bf2d3420f4f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11318,"rank":11318,"depth":0,"x":255.672,"y":1722.107,"cluster":"algebraic-spaces"},{"id":"stacks:0ED8","tag":"0ED8","title":"Catenary algebraic spaces · Lemma 0ED8","summary":"Let S be a scheme. Let X be a decent, locally Noetherian, and universally catenary algebraic space over S. Then any decent algebraic space locally of finite type over X is universally catenary.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent, locally Noetherian, and\nuniversally catenary algebraic space over $S$. Then any decent algebraic\nspace locally of finite type over $X$ is universally catenary.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Catenary algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ED8","source_file":"decent-spaces.tex","source_line":5642,"source_end_line":5647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5642-L5647","statement_sha256":"3e95863cd3380834e0c59a8e413f1f640b1178ee79f0a3a7beace29c410dcc72","origin":"The Stacks Project","memory_eligible":false,"source_rank":11319,"rank":11319,"depth":7,"x":112.786,"y":1524.47,"cluster":"algebraic-spaces"},{"id":"stacks:0ED9","tag":"0ED9","title":"Catenary algebraic spaces · Lemma 0ED9","summary":"Let S be a scheme. Let f : Y → X be a surjective finite morphism of decent and locally Noetherian algebraic spaces. Let δ : |X| → Z be a function. If δ ∘ |f| is a dimension function, then δ is a dimension function.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a surjective finite morphism of\ndecent and locally Noetherian algebraic spaces. Let\n$\\delta : |X| \\to \\mathbf{Z}$ be a function. If $\\delta \\circ |f|$ is a\ndimension function, then $\\delta$ is a dimension function.","area":"Algebraic Spaces","chapter":"Decent Algebraic Spaces","chapter_id":"decent-spaces","section":"Catenary algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ED9","source_file":"decent-spaces.tex","source_line":5656,"source_end_line":5662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/decent-spaces.tex#L5656-L5662","statement_sha256":"559447d7e4abaf0467fd3a1db8c5cd41ad0346aa77ae4ba82919311861f0497b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11320,"rank":11320,"depth":58,"x":377.282,"y":1589.176,"cluster":"algebraic-spaces"},{"id":"stacks:0720","tag":"0720","title":"Higher direct images · Lemma 0720","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is quasi-compact and quasi-separated, then R^if_* transforms quasi-coherent O_X-modules into quasi-coherent O_Y-modules.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. If $f$ is quasi-compact and quasi-separated, then $R^if_*$\ntransforms quasi-coherent $\\mathcal{O}_X$-modules into\nquasi-coherent $\\mathcal{O}_Y$-modules.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0720","source_file":"spaces-cohomology.tex","source_line":121,"source_end_line":127,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L121-L127","statement_sha256":"a085dda1406b91a8098b1c72eb8954aa6bb54ab9d73b89aef1aaf908187a07be","origin":"The Stacks Project","memory_eligible":false,"source_rank":11321,"rank":11321,"depth":43,"x":683.848,"y":1600.0,"cluster":"geometry-of-spaces"},{"id":"stacks:08EX","tag":"08EX","title":"Higher direct images · Lemma 08EX","summary":"Let S be a scheme. Let f : X → Y be a quasi-separated and quasi-compact morphism of algebraic spaces over S. For any quasi-coherent O_X-module F and any affine object V of Y_etale we have H^q(V ×_Y X, F) = H^0(V, R^qf_*F) for all q ∈ Z.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-separated and quasi-compact\nmorphism of algebraic spaces over $S$. For any quasi-coherent\n$\\mathcal{O}_X$-module $\\mathcal{F}$ and any affine object $V$ of\n$Y_\\etale$ we have\n$$\nH^q(V \\times_Y X, \\mathcal{F}) = H^0(V, R^qf_*\\mathcal{F})\n$$\nfor all $q \\in \\mathbf{Z}$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08EX","source_file":"spaces-cohomology.tex","source_line":237,"source_end_line":247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L237-L247","statement_sha256":"6d1ea0495b921b1eec02519e9f064ec3e83a1d9a6bb1aa7cd80c34d1e0680e07","origin":"The Stacks Project","memory_eligible":false,"source_rank":11322,"rank":11322,"depth":44,"x":675.085,"y":1603.782,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4K","tag":"0A4K","title":"Finite morphisms · Lemma 0A4K","summary":"Let S be a scheme. Let f : X → Y be an integral (for example finite) morphism of algebraic spaces. Then f_* : Ab(X_etale) → Ab(Y_etale) is an exact functor and R^pf_* = 0 for p > 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an integral (for example finite)\nmorphism of algebraic spaces. Then\n$f_* : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(Y_\\etale)$\nis an exact functor and $R^pf_* = 0$ for $p > 0$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4K","source_file":"spaces-cohomology.tex","source_line":278,"source_end_line":284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L278-L284","statement_sha256":"a4a9d1d63169e103c41c0080eb215fde073e2ed4e0dd3df0cfeed87d3c99b449","origin":"The Stacks Project","memory_eligible":false,"source_rank":11323,"rank":11323,"depth":52,"x":680.752,"y":1592.8,"cluster":"geometry-of-spaces"},{"id":"stacks:0DK3","tag":"0DK3","title":"Finite morphisms · Lemma 0DK3","summary":"Let S be a scheme. Let f : X → Y be a finite morphism of algebraic spaces over S. Let overliney be a geometric point of Y with lifts overlinex_1, …, overlinex_n in X. Then (f_*F)_overliney = ∏_i = 1, …, n F_overlinex_i for any sheaf F on X_etale.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic\nspaces over $S$. Let $\\overline{y}$ be a geometric point of $Y$ with\nlifts $\\overline{x}_1, \\ldots, \\overline{x}_n$ in $X$. Then\n$$\n(f_*\\mathcal{F})_{\\overline{y}} =\n\\prod\\nolimits_{i = 1, \\ldots, n}\n\\mathcal{F}_{\\overline{x}_i}\n$$\nfor any sheaf $\\mathcal{F}$ on $X_\\etale$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DK3","source_file":"spaces-cohomology.tex","source_line":300,"source_end_line":311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L300-L311","statement_sha256":"729c3903febf5171a748774bc48b7b715815f5040ca58047f40cd1549db4f34c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11324,"rank":11324,"depth":52,"x":686.195,"y":1606.787,"cluster":"geometry-of-spaces"},{"id":"stacks:0DK4","tag":"0DK4","title":"Finite morphisms · Lemma 0DK4","summary":"Let S be a scheme. Let π : X → Y be a finite morphism of algebraic spaces over S. Let A be a sheaf of rings on X_etale. Let B be a sheaf of rings on Y_etale. Let φ : B → π_*A be a homomorphism of sheaves of rings so that we obtain a morphism of ringed topoi f = (π, φ) : (Sh(X_etale), A) → (Sh(Y_etale), B). For a sheaf of A-modules F and a sheaf of B-modules G the canonical map G ⊗_B f_*F → f_*(f^*G ⊗_A F). is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $\\pi : X \\to Y$ be a finite morphism of algebraic\nspaces over $S$. Let $\\mathcal{A}$ be a sheaf of rings on $X_\\etale$.\nLet $\\mathcal{B}$ be a sheaf of rings on $Y_\\etale$.\nLet $\\varphi : \\mathcal{B} \\to \\pi_*\\mathcal{A}$\nbe a homomorphism of sheaves of rings so that we obtain a\nmorphism of ringed topoi\n$$\nf = (\\pi, \\varphi) :\n(\\Sh(X_\\etale), \\mathcal{A})\n\\longrightarrow\n(\\Sh(Y_\\etale), \\mathcal{B}).\n$$\nFor a sheaf of $\\mathcal{A}$-modules $\\mathcal{F}$ and a\nsheaf of $\\mathcal{B}$-modules $\\mathcal{G}$ the canonical map\n$$\n\\mathcal{G} \\otimes_\\mathcal{B} f_*\\mathcal{F}\n\\longrightarrow\nf_*(f^*\\mathcal{G} \\otimes_\\mathcal{A} \\mathcal{F}).\n$$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DK4","source_file":"spaces-cohomology.tex","source_line":325,"source_end_line":347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L325-L347","statement_sha256":"1491f831c4d6f0d60f8b06b8125d6de9b91ef4855e09b478ad38225d140a55c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11325,"rank":11325,"depth":53,"x":668.632,"y":1598.311,"cluster":"geometry-of-spaces"},{"id":"stacks:0DK5","tag":"0DK5","title":"Finite morphisms · Lemma 0DK5","summary":"With S, X, Y, π, A, B, φ, and f as in Lemma [Tag 0DK4] we have K ⊗_B^L Rf_*M = Rf_*(Lf^*K ⊗_A^L M) in D(B) for any K ∈ D(B) and M ∈ D(A).","statement_latex":"With $S$, $X$, $Y$, $\\pi$, $\\mathcal{A}$, $\\mathcal{B}$, $\\varphi$, and $f$\nas in Lemma \\ref{lemma-finite-rings} we have\n$$\nK \\otimes_\\mathcal{B}^\\mathbf{L} Rf_*M =\nRf_*(Lf^*K \\otimes_\\mathcal{A}^\\mathbf{L} M)\n$$\nin $D(\\mathcal{B})$ for any $K \\in D(\\mathcal{B})$ and\n$M \\in D(\\mathcal{A})$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DK5","source_file":"spaces-cohomology.tex","source_line":392,"source_end_line":402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L392-L402","statement_sha256":"123ebab46b5b39851a99eedf03b56b7135a14a5840ea59e8e30f7ced77165e58","origin":"The Stacks Project","memory_eligible":false,"source_rank":11326,"rank":11326,"depth":54,"x":690.769,"y":1594.246,"cluster":"geometry-of-spaces"},{"id":"stacks:073E","tag":"073E","title":"Colimits and cohomology · Lemma 073E","summary":"Let S be a scheme. Let X be an algebraic space over S. If X is quasi-compact and quasi-separated, then colim_i H^p(X, F_i) → H^p(X, colim_i F_i) is an isomorphism for every filtered diagram of abelian sheaves on X_etale.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf $X$ is quasi-compact and quasi-separated, then\n$$\n\\colim_i H^p(X, \\mathcal{F}_i)\n\\longrightarrow\nH^p(X, \\colim_i \\mathcal{F}_i)\n$$\nis an isomorphism\nfor every filtered diagram of abelian sheaves on $X_\\etale$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Colimits and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/073E","source_file":"spaces-cohomology.tex","source_line":449,"source_end_line":460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L449-L460","statement_sha256":"5f37dd300b3ae6aef89af33ebb8b05310cf802d295af79aa848dc4b5a4794eff","origin":"The Stacks Project","memory_eligible":false,"source_rank":11327,"rank":11327,"depth":58,"x":676.398,"y":1611.255,"cluster":"geometry-of-spaces"},{"id":"stacks:07U6","tag":"07U6","title":"Colimits and cohomology · Lemma 07U6","summary":"Higher direct images of qcqs morphisms commute with filtered colimits of sheaves. Let S be a scheme. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let F = colim F_i be a filtered colimit of abelian sheaves on X_etale. Then for any p ≥ 0 we have R^pf_*F = colim R^pf_*F_i.","statement_latex":"\\begin{slogan}\nHigher direct images of qcqs morphisms commute with filtered colimits\nof sheaves.\n\\end{slogan}\nLet $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated\nmorphism of algebraic spaces over $S$. Let $\\mathcal{F} = \\colim \\mathcal{F}_i$\nbe a filtered colimit of abelian sheaves on $X_\\etale$.\nThen for any $p \\geq 0$ we have\n$$\nR^pf_*\\mathcal{F} = \\colim R^pf_*\\mathcal{F}_i.\n$$","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Colimits and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07U6","source_file":"spaces-cohomology.tex","source_line":479,"source_end_line":492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L479-L492","statement_sha256":"85f8a8f492b99aac7a9c869567286d1f14fbca62a00a3c8c06b42f06592585dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11328,"rank":11328,"depth":59,"x":673.131,"y":1588.89,"cluster":"geometry-of-spaces"},{"id":"stacks:07U7","tag":"07U7","title":"Colimits and cohomology · Lemma 07U7","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let I be a directed set and let (F_i, φ_ii') be a system over I of O_X-modules. Let G be an O_X-module of finite presentation. Then we have colim_i Hom_X(G, F_i) = Hom_X(G, colim_i F_i). In particular, Hom_X(G, -) commutes with filtered colimits in QCoh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $I$ be a directed set and\nlet $(\\mathcal{F}_i, \\varphi_{ii'})$ be a system over $I$\nof $\\mathcal{O}_X$-modules. Let $\\mathcal{G}$ be an\n$\\mathcal{O}_X$-module of finite presentation. Then we have\n$$\n\\colim_i \\Hom_X(\\mathcal{G}, \\mathcal{F}_i)\n=\n\\Hom_X(\\mathcal{G}, \\colim_i \\mathcal{F}_i).\n$$\nIn particular, $\\Hom_X(\\mathcal{G}, -)$ commutes with filtered\ncolimits in $\\QCoh(\\mathcal{O}_X)$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Colimits and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07U7","source_file":"spaces-cohomology.tex","source_line":517,"source_end_line":531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L517-L531","statement_sha256":"94124343cf2de80cc8adb7120602d60351787573c77bbd8a06a9ba4d047e6a30","origin":"The Stacks Project","memory_eligible":false,"source_rank":11329,"rank":11329,"depth":58,"x":694.904,"y":1604.572,"cluster":"geometry-of-spaces"},{"id":"stacks:0722","tag":"0722","title":"The alternating v Cech complex · Lemma 0722","summary":"Let S be a scheme. Let f_i : U_i → X be étale morphisms of algebraic spaces over S. Then there are isomorphisms f_1, !underlineZ ⊗_Z f_2, !underlineZ → f_12, !underlineZ where f_12 : U_1 ×_X U_2 → X is the structure morphism and (f_1 amalg f_2)_! underlineZ → f_1, !underlineZ ⊕ f_2, !underlineZ","statement_latex":"Let $S$ be a scheme. Let $f_i : U_i \\to X$ be \\'etale morphisms\nof algebraic spaces over $S$. Then there are isomorphisms\n$$\nf_{1, !}\\underline{\\mathbf{Z}} \\otimes_{\\mathbf{Z}}\nf_{2, !}\\underline{\\mathbf{Z}}\n\\longrightarrow\nf_{12, !}\\underline{\\mathbf{Z}}\n$$\nwhere $f_{12} : U_1 \\times_X U_2 \\to X$ is the structure morphism\nand\n$$\n(f_1 \\amalg f_2)_! \\underline{\\mathbf{Z}}\n\\longrightarrow\nf_{1, !}\\underline{\\mathbf{Z}} \\oplus\nf_{2, !}\\underline{\\mathbf{Z}}\n$$","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0722","source_file":"spaces-cohomology.tex","source_line":643,"source_end_line":661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L643-L661","statement_sha256":"a342af9585b05f5672cab848e10c9c796977e46e6c00c7f9a4067e5c486e6b6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11330,"rank":11330,"depth":0,"x":664.495,"y":1605.376,"cluster":"geometry-of-spaces"},{"id":"stacks:0724","tag":"0724","title":"The alternating v Cech complex · Definition 0724","summary":"Let S be a scheme. Let f : U → X be a surjective étale morphism of algebraic spaces over S. Let F be an object of Ab(X_etale). The alternating v Cech complex checkC^bullet_alt(f, F) associated to F and f is the complex Hom(K^0, F) → Hom(K^1, F) → Hom(K^2, F) → … with Hom groups computed in Ab(X_etale).","statement_latex":"Let $S$ be a scheme. Let $f : U \\to X$ be a surjective \\'etale morphism\nof algebraic spaces over $S$. Let $\\mathcal{F}$ be an object of\n$\\textit{Ab}(X_\\etale)$. The\n{\\it alternating {\\v C}ech complex}\\footnote{This may be nonstandard notation}\n$\\check{\\mathcal{C}}^\\bullet_{alt}(f, \\mathcal{F})$\nassociated to $\\mathcal{F}$ and $f$ is the complex\n$$\n\\Hom(K^0, \\mathcal{F}) \\to \\Hom(K^1, \\mathcal{F}) \\to\n\\Hom(K^2, \\mathcal{F}) \\to \\ldots\n$$\nwith Hom groups computed in $\\textit{Ab}(X_\\etale)$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The alternating v Cech complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0724","source_file":"spaces-cohomology.tex","source_line":738,"source_end_line":751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L738-L751","statement_sha256":"edc112804d282bf27f427ae1314c082241ad6f2e7f38cd3dde23c42fa091d31f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11331,"rank":11331,"depth":0,"x":687.474,"y":1586.583,"cluster":"geometry-of-spaces"},{"id":"stacks:0725","tag":"0725","title":"The alternating v Cech complex · Lemma 0725","summary":"Let S be a scheme. Let f : U → X be a surjective étale morphism of algebraic spaces over S. Let F be an object of Ab(X_etale). There exists a canonical map checkC^bullet_alt(f, F) → RΓ(X, F) in D(Ab). Moreover, there is a spectral sequence with E_1-page E_1^p, q = Ext_Ab(X_etale)^q(K^p, F) converging to H^p + q(X, F) where K^p = wedge^p + 1f_!underlineZ.","statement_latex":"Let $S$ be a scheme. Let $f : U \\to X$ be a surjective \\'etale morphism\nof algebraic spaces over $S$. Let $\\mathcal{F}$ be an object of\n$\\textit{Ab}(X_\\etale)$. There exists a canonical map\n$$\n\\check{\\mathcal{C}}^\\bullet_{alt}(f, \\mathcal{F})\n\\longrightarrow\nR\\Gamma(X, \\mathcal{F})\n$$\nin $D(\\textit{Ab})$. Moreover, there is a spectral sequence with $E_1$-page\n$$\nE_1^{p, q} =\n\\Ext_{\\textit{Ab}(X_\\etale)}^q(K^p, \\mathcal{F})\n$$\nconverging to $H^{p + q}(X, \\mathcal{F})$ where\n$K^p = \\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0725","source_file":"spaces-cohomology.tex","source_line":764,"source_end_line":781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L764-L781","statement_sha256":"3a657540952053fa20b1ee14c29a1e752c80f9cd85e00b60318acdf61123b301","origin":"The Stacks Project","memory_eligible":false,"source_rank":11332,"rank":11332,"depth":6,"x":685.523,"y":1614.792,"cluster":"geometry-of-spaces"},{"id":"stacks:0726","tag":"0726","title":"The alternating v Cech complex · Lemma 0726","summary":"Let S be a scheme. Let f : U → X be a surjective, étale, and separated morphism of algebraic spaces over S. For p ≥ 0 set W_p = U ×_X … ×_X U setminus all diagonals where the fibre product has p + 1 factors. There is a free action of S_p + 1 on W_p over X and Hom(K^p, F) = S_p + 1-anti-invariant elements of F(W_p) functorially in F where K^p = wedge^p + 1f_!underlineZ.","statement_latex":"Let $S$ be a scheme. Let $f : U \\to X$ be a surjective, \\'etale, and separated\nmorphism of algebraic spaces over $S$. For $p \\geq 0$ set\n$$\nW_p = U \\times_X \\ldots \\times_X U \\setminus \\text{all diagonals}\n$$\nwhere the fibre product has $p + 1$ factors.\nThere is a free action of $S_{p + 1}$ on $W_p$ over $X$ and\n$$\n\\Hom(K^p, \\mathcal{F}) = S_{p + 1}\\text{-anti-invariant elements of }\n\\mathcal{F}(W_p)\n$$\nfunctorially in $\\mathcal{F}$ where\n$K^p = \\wedge^{p + 1}f_!\\underline{\\mathbf{Z}}$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0726","source_file":"spaces-cohomology.tex","source_line":850,"source_end_line":865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L850-L865","statement_sha256":"ea5431365c15557f7df6c769382897d55312d23f6265b49a5d2f68b345581b4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11333,"rank":11333,"depth":54,"x":663.352,"y":1591.896,"cluster":"geometry-of-spaces"},{"id":"stacks:0727","tag":"0727","title":"The alternating v Cech complex · Lemma 0727","summary":"Let S be a scheme. Let W be an algebraic space over S. Let G be a finite group acting freely on W. Let U = W/G, see Properties of Spaces, Lemma [Tag 071S]. Let chi : G → (+1, -1) be a character. Then there exists a rank 1 locally free sheaf of Z-modules underlineZ(chi) on U_etale such that for every abelian sheaf F on U_etale we have H^0(W, F|_W)^chi = H^0(U, F ⊗_Z underlineZ(chi))","statement_latex":"Let $S$ be a scheme. Let $W$ be an algebraic space over $S$.\nLet $G$ be a finite group acting freely on $W$.\nLet $U = W/G$, see\nProperties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}.\nLet $\\chi : G \\to \\{+1, -1\\}$ be a character.\nThen there exists a rank 1 locally free sheaf of $\\mathbf{Z}$-modules\n$\\underline{\\mathbf{Z}}(\\chi)$ on $U_\\etale$ such that for every\nabelian sheaf $\\mathcal{F}$ on $U_\\etale$ we have\n$$\nH^0(W, \\mathcal{F}|_W)^\\chi =\nH^0(U, \\mathcal{F} \\otimes_{\\mathbf{Z}} \\underline{\\mathbf{Z}}(\\chi))\n$$","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0727","source_file":"spaces-cohomology.tex","source_line":928,"source_end_line":942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L928-L942","statement_sha256":"f3bdc2794057b441096557597e252028e91a77acd7b40e2c04e15bd4742a0907","origin":"The Stacks Project","memory_eligible":false,"source_rank":11334,"rank":11334,"depth":53,"x":699.53,"y":1596.393,"cluster":"geometry-of-spaces"},{"id":"stacks:0728","tag":"0728","title":"The alternating v Cech complex · Lemma 0728","summary":"Let S be a scheme. Let f : U → X be a surjective, étale, and separated morphism of algebraic spaces over S. For p ≥ 0 set W_p = U ×_X … ×_X U setminus all diagonals (with p + 1 factors) as in Lemma [Tag 0726]. Let chi_p : S_p + 1 → (+1, -1) be the sign character. Let U_p = W_p/S_p + 1 and underlineZ(chi_p) be as in Lemma [Tag 0727]. Then the spectral sequence of Lemma [Tag 0725] has E_1-page E_1^p, q = H^q(U_p, F|_U_p ⊗_Z underlineZ(chi_p)) and converges to H^p + q(X, F).","statement_latex":"Let $S$ be a scheme. Let $f : U \\to X$ be a surjective, \\'etale, and\nseparated morphism of algebraic spaces over $S$. For $p \\geq 0$ set\n$$\nW_p = U \\times_X \\ldots \\times_X U \\setminus \\text{all diagonals}\n$$\n(with $p + 1$ factors) as in Lemma \\ref{lemma-compute}.\nLet $\\chi_p : S_{p + 1} \\to \\{+1, -1\\}$ be the sign character.\nLet $U_p = W_p/S_{p + 1}$ and $\\underline{\\mathbf{Z}}(\\chi_p)$ be as in\nLemma \\ref{lemma-twist}.\nThen the spectral sequence of\nLemma \\ref{lemma-alternating-cech-to-cohomology}\nhas $E_1$-page\n$$\nE_1^{p, q} =\nH^q(U_p, \\mathcal{F}|_{U_p} \\otimes_\\mathbf{Z} \\underline{\\mathbf{Z}}(\\chi_p))\n$$\nand converges to $H^{p + q}(X, \\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The alternating v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0728","source_file":"spaces-cohomology.tex","source_line":999,"source_end_line":1018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L999-L1018","statement_sha256":"2d2f8168db1fbdec363b9168494a00372457956b90d0ce80acce46a19898617b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11335,"rank":11335,"depth":55,"x":668.081,"y":1614.241,"cluster":"geometry-of-spaces"},{"id":"stacks:072A","tag":"072A","title":"Higher vanishing for quasi-coherent sheaves · Lemma 072A","summary":"With S, W, G, U, chi as in Lemma [Tag 0727]. If F is a quasi-coherent O_U-module, then so is F ⊗_Z underlineZ(chi).","statement_latex":"With $S$, $W$, $G$, $U$, $\\chi$ as in\nLemma \\ref{lemma-twist}.\nIf $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_U$-module,\nthen so is $\\mathcal{F} \\otimes_{\\mathbf{Z}} \\underline{\\mathbf{Z}}(\\chi)$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Higher vanishing for quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072A","source_file":"spaces-cohomology.tex","source_line":1087,"source_end_line":1093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1087-L1093","statement_sha256":"4cb1b7cb93e41bfe5b9e4514eccbf5820fbd32eef6b2e0e5c34945c633a20ebc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11336,"rank":11336,"depth":54,"x":677.247,"y":1582.151,"cluster":"geometry-of-spaces"},{"id":"stacks:072B","tag":"072B","title":"Higher vanishing for quasi-coherent sheaves · Proposition 072B","summary":"Let S be a scheme. Let X be an algebraic space over S. Assume X is quasi-compact and separated. Let U be an affine scheme, and let f : U → X be a surjective étale morphism. Let d be an upper bound for the size of the fibres of |U| → |X|. Then for any quasi-coherent O_X-module F we have H^q(X, F) = 0 for q ≥ d.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nAssume $X$ is quasi-compact and separated.\nLet $U$ be an affine scheme, and let\n$f : U \\to X$ be a surjective \\'etale morphism.\nLet $d$ be an upper bound for the size of the fibres of\n$|U| \\to |X|$. Then for any quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$\nwe have $H^q(X, \\mathcal{F}) = 0$ for $q \\geq d$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Higher vanishing for quasi-coherent sheaves","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072B","source_file":"spaces-cohomology.tex","source_line":1115,"source_end_line":1124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1115-L1124","statement_sha256":"79455e96794d9aadd976e3273cadf40b6be14f532e3bc8b08b9bc8a72086d70c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11337,"rank":11337,"depth":56,"x":696.904,"y":1611.967,"cluster":"geometry-of-spaces"},{"id":"stacks:072C","tag":"072C","title":"Higher vanishing for quasi-coherent sheaves · Lemma 072C","summary":"Let S be a scheme. Let X be an algebraic space over S. Assume X is quasi-compact and quasi-separated. Then we can choose • an affine scheme U, • a surjective étale morphism f : U → X, • an integer d bounding the degrees of the fibres of U → X, • for every p = 0, 1, …, d a surjective étale morphism V_p → U_p from an affine scheme V_p where U_p is as in Lemma [Tag 0728], and • an integer d_p bounding the degree of the fibres of V_p → U_p. Moreover, whenever we have (1) --…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nAssume $X$ is quasi-compact and quasi-separated.\nThen we can choose\n\\begin{enumerate}\n\\item an affine scheme $U$,\n\\item a surjective \\'etale morphism $f : U \\to X$,\n\\item an integer $d$ bounding the degrees of the fibres of $U \\to X$,\n\\item for every $p = 0, 1, \\ldots, d$ a surjective \\'etale morphism\n$V_p \\to U_p$ from an affine scheme $V_p$ where $U_p$ is as in\nLemma \\ref{lemma-alternating-spectral-sequence}, and\n\\item an integer $d_p$ bounding the degree of the fibres of $V_p \\to U_p$.\n\\end{enumerate}\nMoreover, whenever we have (1) -- (5), then for any quasi-coherent\n$\\mathcal{O}_X$-module $\\mathcal{F}$ we have $H^q(X, \\mathcal{F}) = 0$ for\n$q \\geq \\max(d_p + p)$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Higher vanishing for quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072C","source_file":"spaces-cohomology.tex","source_line":1167,"source_end_line":1184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1167-L1184","statement_sha256":"4b4c37ef64417a8b48fde61b20b112ca67d186a8735c67e02cfb1392f46ca9db","origin":"The Stacks Project","memory_eligible":false,"source_rank":11338,"rank":11338,"depth":57,"x":657.253,"y":1600.79,"cluster":"geometry-of-spaces"},{"id":"stacks:073G","tag":"073G","title":"Vanishing for higher direct images · Lemma 073G","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that • f is quasi-compact and quasi-separated, and • Y is quasi-compact. Then there exists an integer n(X → Y) such that for any algebraic space Y', any morphism Y' → Y and any quasi-coherent sheaf F' on X' = Y' ×_Y X the higher direct images R^if'_*F' are zero for i ≥ n(X → Y).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a\nmorphism of algebraic spaces over $S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is quasi-compact and quasi-separated, and\n\\item $Y$ is quasi-compact.\n\\end{enumerate}\nThen there exists an integer $n(X \\to Y)$ such that\nfor any algebraic space $Y'$, any morphism $Y' \\to Y$\nand any quasi-coherent sheaf $\\mathcal{F}'$ on $X' = Y' \\times_Y X$\nthe higher direct images $R^if'_*\\mathcal{F}'$ are zero for\n$i \\geq n(X \\to Y)$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing for higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/073G","source_file":"spaces-cohomology.tex","source_line":1248,"source_end_line":1262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1248-L1262","statement_sha256":"53d4d3f3c5f593851759ce246333e9ac576787c43fe6950fd348a22b2cdc7efb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11339,"rank":11339,"depth":58,"x":696.592,"y":1586.13,"cluster":"geometry-of-spaces"},{"id":"stacks:073H","tag":"073H","title":"Vanishing for higher direct images · Lemma 073H","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S. Then R^if_*F = 0 for i > 0 and any quasi-coherent O_X-module F.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an affine\nmorphism of algebraic spaces over $S$. Then\n$R^if_*\\mathcal{F} = 0$ for $i > 0$ and any quasi-coherent\n$\\mathcal{O}_X$-module $\\mathcal{F}$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing for higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/073H","source_file":"spaces-cohomology.tex","source_line":1304,"source_end_line":1310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1304-L1310","statement_sha256":"02c4582a1c86aadf2cc610454af69388b4cff6726e4e70dc3e213d31d45a7c47","origin":"The Stacks Project","memory_eligible":false,"source_rank":11340,"rank":11340,"depth":25,"x":678.89,"y":1620.166,"cluster":"geometry-of-spaces"},{"id":"stacks:0D2U","tag":"0D2U","title":"Vanishing for higher direct images · Lemma 0D2U","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Then H^i(X, F) = H^i(Y, f_*F) for all i ≥ 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an affine\nmorphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $H^i(X, \\mathcal{F}) = H^i(Y, f_*\\mathcal{F})$ for all $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing for higher direct images","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2U","source_file":"spaces-cohomology.tex","source_line":1318,"source_end_line":1324,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1318-L1324","statement_sha256":"b2a69f6fd1d60cf50f76f408d65da23df31fc7b4984717fb7761abac477bd699","origin":"The Stacks Project","memory_eligible":false,"source_rank":11341,"rank":11341,"depth":26,"x":664.212,"y":1584.108,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4M","tag":"0A4M","title":"Cohomology with support in a closed subspace · Lemma 0A4M","summary":"Let S be a scheme. Let i : Z → X be a closed immersion of algebraic spaces over S. Let I be an injective abelian sheaf on X_etale. Then H_Z(I) is an injective abelian sheaf on Z_etale.","statement_latex":"Let $S$ be a scheme.\nLet $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$.\nLet $\\mathcal{I}$ be an injective abelian sheaf on $X_\\etale$.\nThen $\\mathcal{H}_Z(\\mathcal{I})$ is an injective abelian sheaf\non $Z_\\etale$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Cohomology with support in a closed subspace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4M","source_file":"spaces-cohomology.tex","source_line":1405,"source_end_line":1412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1405-L1412","statement_sha256":"efd4f08bca218f565f6c6dba221b8b66834982c3d3d6414ecdc88fc47a302230","origin":"The Stacks Project","memory_eligible":false,"source_rank":11342,"rank":11342,"depth":53,"x":705.009,"y":1602.827,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4N","tag":"0A4N","title":"Cohomology with support in a closed subspace · Lemma 0A4N","summary":"Let S be a scheme. Let i : Z → X be a closed immersion of algebraic spaces over S. Let G be an injective abelian sheaf on Z_etale. Then H^p_Z(i_*G) = 0 for p > 0.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of\nalgebraic spaces over $S$. Let $\\mathcal{G}$ be an injective abelian\nsheaf on $Z_\\etale$. Then $\\mathcal{H}^p_Z(i_*\\mathcal{G}) = 0$ for $p > 0$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Cohomology with support in a closed subspace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4N","source_file":"spaces-cohomology.tex","source_line":1441,"source_end_line":1446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1441-L1446","statement_sha256":"dbbd6bcfd219ef1fc093e13d179efe2db6caf0b59d5798f7ecbcf43526c94b3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11343,"rank":11343,"depth":53,"x":658.81,"y":1612.385,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4P","tag":"0A4P","title":"Cohomology with support in a closed subspace · Lemma 0A4P","summary":"Let S be a scheme. Let f : X → Y be an étale morphism of algebraic spaces over S. Let Z ⊂ Y be a closed subspace such that f^-1(Z) → Z is an isomorphism of algebraic spaces. Let F be an abelian sheaf on X. Then H^q_Z(F) = H^q_f^-1(Z)(f^-1F) as abelian sheaves on Z = f^-1(Z) and we have H^q_Z(Y, F) = H^q_f^-1(Z)(X, f^-1F).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an \\'etale morphism of\nalgebraic spaces over $S$. Let $Z \\subset Y$ be a closed subspace\nsuch that $f^{-1}(Z) \\to Z$ is an isomorphism of algebraic spaces.\nLet $\\mathcal{F}$ be an abelian sheaf on $X$. Then\n$$\n\\mathcal{H}^q_Z(\\mathcal{F}) = \\mathcal{H}^q_{f^{-1}(Z)}(f^{-1}\\mathcal{F})\n$$\nas abelian sheaves on $Z = f^{-1}(Z)$ and we\nhave $H^q_Z(Y, \\mathcal{F}) = H^q_{f^{-1}(Z)}(X, f^{-1}\\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Cohomology with support in a closed subspace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4P","source_file":"spaces-cohomology.tex","source_line":1456,"source_end_line":1467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1456-L1467","statement_sha256":"2f0c671e878581bf61fdbb876239669d781f255395015f98cb562b1421d65603","origin":"The Stacks Project","memory_eligible":false,"source_rank":11344,"rank":11344,"depth":0,"x":685.79,"y":1578.379,"cluster":"geometry-of-spaces"},{"id":"stacks:0AEI","tag":"0AEI","title":"Cohomology with support in a closed subspace · Lemma 0AEI","summary":"Let S be a scheme. Let i : Z → X be a closed immersion of algebraic spaces over S. The map Ri_* = i_* : D(Z_etale) → D(X_etale) induces an equivalence D(Z_etale) → D_|Z|(X_etale) with quasi-inverse i^-1|_D_Z(X_etale) = RH_Z|_D_|Z|(X_etale)","statement_latex":"Let $S$ be a scheme.\nLet $i : Z \\to X$ be a closed immersion of algebraic spaces over $S$.\nThe map $Ri_* = i_* : D(Z_\\etale) \\to D(X_\\etale)$\ninduces an equivalence $D(Z_\\etale) \\to D_{|Z|}(X_\\etale)$ with quasi-inverse\n$$\ni^{-1}|_{D_Z(X_\\etale)} = R\\mathcal{H}_Z|_{D_{|Z|}(X_\\etale)}\n$$","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Cohomology with support in a closed subspace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEI","source_file":"spaces-cohomology.tex","source_line":1484,"source_end_line":1493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1484-L1493","statement_sha256":"f9fc29708c2879ab2bed547cb349efcde47791e56225e8b3f8e39b8ed4090683","origin":"The Stacks Project","memory_eligible":false,"source_rank":11345,"rank":11345,"depth":54,"x":693.393,"y":1619.633,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4R","tag":"0A4R","title":"Vanishing above the dimension · Lemma 0A4R","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Assume dim(X) ≤ d for some integer d. Let F be a quasi-coherent sheaf F on X. • H^q(X, F) = 0 for q > d, • H^d(X, F) → H^d(U, F) is surjective for any quasi-compact open U ⊂ X, • H^q_Z(X, F) = 0 for q > d for any closed subspace Z ⊂ X whose complement is quasi-compact.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Assume $\\dim(X) \\leq d$ for some integer $d$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf $\\mathcal{F}$ on $X$.\n\\begin{enumerate}\n\\item $H^q(X, \\mathcal{F}) = 0$ for $q > d$,\n\\item $H^d(X, \\mathcal{F}) \\to H^d(U, \\mathcal{F})$ is surjective\nfor any quasi-compact open $U \\subset X$,\n\\item $H^q_Z(X, \\mathcal{F}) = 0$ for $q > d$ for any closed subspace\n$Z \\subset X$ whose complement is quasi-compact.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing above the dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4R","source_file":"spaces-cohomology.tex","source_line":1552,"source_end_line":1564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1552-L1564","statement_sha256":"dd876c6dc45b87e9c7fb1da130451c4f5e83b4bbb20403bdd934b501d4cb1540","origin":"The Stacks Project","memory_eligible":false,"source_rank":11346,"rank":11346,"depth":59,"x":653.818,"y":1592.984,"cluster":"geometry-of-spaces"},{"id":"stacks:07U8","tag":"07U8","title":"Cohomology and base change, I · Lemma 07U8","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. In this case f_*F ≅ Rf_*F is a quasi-coherent sheaf, and for every diagram ([Tag 073J]) we have g^*f_*F = f'_*(g')^*F.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic\nspaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIn this case $f_*\\mathcal{F} \\cong Rf_*\\mathcal{F}$ is a quasi-coherent\nsheaf, and for every diagram (\\ref{equation-base-change-diagram})\nwe have\n$$\ng^*f_*\\mathcal{F} = f'_*(g')^*\\mathcal{F}.\n$$","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Cohomology and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07U8","source_file":"spaces-cohomology.tex","source_line":1684,"source_end_line":1694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1684-L1694","statement_sha256":"53c8b2eb353f1285eef4104d299c512fd0044a93a12c8e8abdc82f67bcb0f126","origin":"The Stacks Project","memory_eligible":false,"source_rank":11347,"rank":11347,"depth":26,"x":705.432,"y":1590.13,"cluster":"geometry-of-spaces"},{"id":"stacks:073K","tag":"073K","title":"Flat base change · Lemma 073K","summary":"Let S be a scheme. Consider a cartesian diagram of algebraic spaces xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f Y' ar[r]^g & Y over S. Let F be a quasi-coherent O_X-module with pullback F' = (g')^*F. Assume that g is flat and that f is quasi-compact and quasi-separated. For any i ≥ 0 • the base change map of Cohomology on Sites, Lemma [Tag 0736] is an isomorphism g^*R^if_*F → R^if'_*F', • if Y = Spec(A) and Y' = Spec(B), then H^i(X, F) ⊗_A B = H^i(X', F').","statement_latex":"Let $S$ be a scheme. Consider a cartesian diagram of algebraic spaces\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nover $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nwith pullback $\\mathcal{F}' = (g')^*\\mathcal{F}$.\nAssume that $g$ is flat and that $f$ is quasi-compact and quasi-separated.\nFor any $i \\geq 0$\n\\begin{enumerate}\n\\item the base change map of\nCohomology on Sites, Lemma\n\\ref{sites-cohomology-lemma-base-change-map-flat-case}\nis an isomorphism\n$$\ng^*R^if_*\\mathcal{F} \\longrightarrow R^if'_*\\mathcal{F}',\n$$\n\\item if $Y = \\Spec(A)$ and $Y' = \\Spec(B)$, then\n$H^i(X, \\mathcal{F}) \\otimes_A B = H^i(X', \\mathcal{F}')$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Cohomology and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/073K","source_file":"spaces-cohomology.tex","source_line":1704,"source_end_line":1729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1704-L1729","statement_sha256":"40e021e8a1075ca85b7ac12474f0381a3be8a67031e6d59c26ce388f94a78172","origin":"The Stacks Project","memory_eligible":false,"source_rank":11348,"rank":11348,"depth":59,"x":668.982,"y":1622.115,"cluster":"geometry-of-spaces"},{"id":"stacks:0HAP","tag":"0HAP","title":"Cohomology and base change, I · Lemma 0HAP","summary":"Let f : X → Y be a quasi-compact, separated, étale morphism of algebraic spaces. Then for any quasi-coherent O_X-module F the map f^*f_*F → F is split.","statement_latex":"Let $f : X \\to Y$ be a quasi-compact, separated, \\'etale morphism\nof algebraic spaces. Then for any quasi-coherent $\\mathcal{O}_X$-module\n$\\mathcal{F}$ the map $f^*f_*\\mathcal{F} \\to \\mathcal{F}$ is split.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Cohomology and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAP","source_file":"spaces-cohomology.tex","source_line":1804,"source_end_line":1809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1804-L1809","statement_sha256":"dfda35df15bc7e06b02e6f664c01cc537fe6c4a7c8edb9d07f0a9b5e85ee78ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":11349,"rank":11349,"depth":27,"x":670.167,"y":1577.035,"cluster":"geometry-of-spaces"},{"id":"stacks:07UA","tag":"07UA","title":"Coherent modules on locally Noetherian algebraic spaces · Definition 07UA","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. A quasi-coherent module F on X is called coherent if F is a coherent O_X-module on the site X_etale in the sense of Modules on Sites, Definition [Tag 03DL].","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nA quasi-coherent module $\\mathcal{F}$ on $X$ is called {\\it coherent}\nif $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module on the site\n$X_\\etale$ in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local}.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent modules on locally Noetherian algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UA","source_file":"spaces-cohomology.tex","source_line":1849,"source_end_line":1856,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1849-L1856","statement_sha256":"3605249b10e073ffc462a2bba44ab6e5d4851a54421028bbaf28ebba151889a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11350,"rank":11350,"depth":2,"x":706.165,"y":1611.551,"cluster":"geometry-of-spaces"},{"id":"stacks:07UB","tag":"07UB","title":"Coherent modules on locally Noetherian algebraic spaces · Lemma 07UB","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let F be an O_X-module. The following are equivalent • F is coherent, • F is a quasi-coherent, finite type O_X-module, • F is a finitely presented O_X-module, • for any étale morphism φ : U → X where U is a scheme the pullback φ^*F is a coherent module on U, and • there exists a surjective étale morphism φ : U → X where U is a scheme such that the pullback φ^*F is a coherent module on U. In…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is coherent,\n\\item $\\mathcal{F}$ is a quasi-coherent, finite type $\\mathcal{O}_X$-module,\n\\item $\\mathcal{F}$ is a finitely presented $\\mathcal{O}_X$-module,\n\\item for any \\'etale morphism $\\varphi : U \\to X$ where $U$ is a scheme\nthe pullback $\\varphi^*\\mathcal{F}$ is a coherent module on $U$, and\n\\item there exists a surjective \\'etale morphism $\\varphi : U \\to X$\nwhere $U$ is a scheme such that the pullback $\\varphi^*\\mathcal{F}$ is\na coherent module on $U$.\n\\end{enumerate}\nIn particular $\\mathcal{O}_X$ is coherent, any invertible\n$\\mathcal{O}_X$-module is coherent, and more generally any\nfinite locally free $\\mathcal{O}_X$-module is coherent.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent modules on locally Noetherian algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UB","source_file":"spaces-cohomology.tex","source_line":1876,"source_end_line":1895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1876-L1895","statement_sha256":"e402364cca77a8c5d22151e49e5fbd195fe9e5a188c030715cd36f6cfd9649c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11351,"rank":11351,"depth":8,"x":650.937,"y":1606.435,"cluster":"geometry-of-spaces"},{"id":"stacks:07UC","tag":"07UC","title":"Coherent modules on locally Noetherian algebraic spaces · Lemma 07UC","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. The category of coherent O_X-modules is abelian. More precisely, the kernel and cokernel of a map of coherent O_X-modules are coherent. Any extension of coherent sheaves is coherent.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nThe category of coherent $\\mathcal{O}_X$-modules is abelian. More precisely,\nthe kernel and cokernel of a map of coherent $\\mathcal{O}_X$-modules are\ncoherent. Any extension of coherent sheaves is coherent.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent modules on locally Noetherian algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UC","source_file":"spaces-cohomology.tex","source_line":1908,"source_end_line":1914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1908-L1914","statement_sha256":"96c57cedc078da85377b93233b0eb3a2bc643029edbae4d3b94d5f870d0b953e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11352,"rank":11352,"depth":9,"x":696.52,"y":1578.419,"cluster":"geometry-of-spaces"},{"id":"stacks:07UD","tag":"07UD","title":"Coherent modules on locally Noetherian algebraic spaces · Lemma 07UD","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let F be a coherent O_X-module. Any quasi-coherent submodule of F is coherent. Any quasi-coherent quotient module of F is coherent.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nAny quasi-coherent submodule of $\\mathcal{F}$ is coherent.\nAny quasi-coherent quotient module of $\\mathcal{F}$ is coherent.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent modules on locally Noetherian algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UD","source_file":"spaces-cohomology.tex","source_line":1934,"source_end_line":1941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1934-L1941","statement_sha256":"1e5ecb308d7fa82d47c4e524fb8781af55abede788a24e5dd76b166e6d7c2ecf","origin":"The Stacks Project","memory_eligible":false,"source_rank":11353,"rank":11353,"depth":9,"x":685.255,"y":1625.685,"cluster":"geometry-of-spaces"},{"id":"stacks:07UE","tag":"07UE","title":"Coherent modules on locally Noetherian algebraic spaces · Lemma 07UE","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S,. Let F, G be coherent O_X-modules. The O_X-modules F ⊗_O_X G and SheafHom_O_X(F, G) are coherent.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian algebraic space over $S$,.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.\nThe $\\mathcal{O}_X$-modules $\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{G}$\nand $\\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$ are\ncoherent.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent modules on locally Noetherian algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UE","source_file":"spaces-cohomology.tex","source_line":1957,"source_end_line":1965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1957-L1965","statement_sha256":"3762e67d4a5aed5f56d7117b6d941ddd8241e96877b407d1d83191c466f9b2a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11354,"rank":11354,"depth":9,"x":655.096,"y":1583.799,"cluster":"geometry-of-spaces"},{"id":"stacks:07UF","tag":"07UF","title":"Coherent modules on locally Noetherian algebraic spaces · Lemma 07UF","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let F, G be coherent O_X-modules. Let φ : G → F be a homomorphism of O_X-modules. Let overlinex be a geometric point of X lying over x ∈ |X|. • If F_overlinex = 0 then there exists an open neighbourhood X' ⊂ X of x such that F|_X' = 0. • If φ_overlinex : G_overlinex → F_overlinex is injective, then there exists an open neighbourhood X' ⊂ X of x such that φ|_X' is injective. • If φ_overlinex :…","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.\nLet $\\varphi : \\mathcal{G} \\to \\mathcal{F}$ be a homomorphism\nof $\\mathcal{O}_X$-modules. Let $\\overline{x}$ be a geometric point of $X$\nlying over $x \\in |X|$.\n\\begin{enumerate}\n\\item If $\\mathcal{F}_{\\overline{x}} = 0$ then there exists an open\nneighbourhood $X' \\subset X$ of $x$ such that $\\mathcal{F}|_{X'} = 0$.\n\\item If $\\varphi_{\\overline{x}} : \\mathcal{G}_{\\overline{x}} \\to\n\\mathcal{F}_{\\overline{x}}$ is injective, then there exists an open\nneighbourhood $X' \\subset X$ of $x$ such that $\\varphi|_{X'}$ is injective.\n\\item If $\\varphi_{\\overline{x}} : \\mathcal{G}_{\\overline{x}} \\to\n\\mathcal{F}_{\\overline{x}}$ is surjective, then there exists an open\nneighbourhood $X' \\subset X$ of $x$ such that $\\varphi|_{X'}$ is surjective.\n\\item If $\\varphi_{\\overline{x}} : \\mathcal{G}_{\\overline{x}} \\to\n\\mathcal{F}_{\\overline{x}}$ is bijective, then there exists an open\nneighbourhood $X' \\subset X$ of $x$ such that $\\varphi|_{X'}$ is an isomorphism.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent modules on locally Noetherian algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UF","source_file":"spaces-cohomology.tex","source_line":1973,"source_end_line":1993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L1973-L1993","statement_sha256":"9ad120c00d5918762d8c5e9cc4b8ec9d15667cf982fbeebef607f355dd3c504f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11355,"rank":11355,"depth":54,"x":711.857,"y":1597.783,"cluster":"geometry-of-spaces"},{"id":"stacks:07UG","tag":"07UG","title":"Coherent modules on locally Noetherian algebraic spaces · Lemma 07UG","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let F be a coherent O_X-module. Let i : Z → X be the scheme theoretic support of F and G the quasi-coherent O_Z-module such that i_*G = F, see Morphisms of Spaces, Definition [Tag 07U1]. Then G is a coherent O_Z-module.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let $i : Z \\to X$\nbe the scheme theoretic support of $\\mathcal{F}$ and $\\mathcal{G}$\nthe quasi-coherent $\\mathcal{O}_Z$-module such that\n$i_*\\mathcal{G} = \\mathcal{F}$, see\nMorphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-scheme-theoretic-support}.\nThen $\\mathcal{G}$ is a coherent $\\mathcal{O}_Z$-module.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent modules on locally Noetherian algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UG","source_file":"spaces-cohomology.tex","source_line":2017,"source_end_line":2027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2017-L2027","statement_sha256":"e7df5c17b39665c74a84aca5d7c414e8bddc3370dbce955c985e3399aea08317","origin":"The Stacks Project","memory_eligible":false,"source_rank":11356,"rank":11356,"depth":57,"x":657.98,"y":1619.994,"cluster":"geometry-of-spaces"},{"id":"stacks:08AM","tag":"08AM","title":"Coherent modules on locally Noetherian algebraic spaces · Lemma 08AM","summary":"Let S be a scheme. Let i : Z → X be a closed immersion of locally Noetherian algebraic spaces over S. Let I ⊂ O_X be the quasi-coherent sheaf of ideals cutting out Z. The functor i_* induces an equivalence between the category of coherent O_X-modules annihilated by I and the category of coherent O_Z-modules.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be a closed immersion of locally\nNoetherian algebraic spaces over $S$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be the quasi-coherent sheaf of ideals\ncutting out $Z$. The functor $i_*$ induces an equivalence between the\ncategory of coherent $\\mathcal{O}_X$-modules annihilated by $\\mathcal{I}$\nand the category of coherent $\\mathcal{O}_Z$-modules.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent modules on locally Noetherian algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AM","source_file":"spaces-cohomology.tex","source_line":2041,"source_end_line":2049,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2041-L2049","statement_sha256":"8b9ac0daf28f8f5c7b909f1ae3527ed47235a341b0b22f9c7603398fb9ca3552","origin":"The Stacks Project","memory_eligible":false,"source_rank":11357,"rank":11357,"depth":55,"x":680.16,"y":1572.382,"cluster":"geometry-of-spaces"},{"id":"stacks:07UH","tag":"07UH","title":"Coherent modules on locally Noetherian algebraic spaces · Lemma 07UH","summary":"Let S be a scheme. Let f : X → Y be a finite morphism of algebraic spaces over S with Y locally Noetherian. Let F be a coherent O_X-module. Assume f is finite and Y locally Noetherian. Then R^pf_*F = 0 for p > 0 and f_*F is coherent.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic\nspaces over $S$ with $Y$ locally Noetherian. Let $\\mathcal{F}$ be a\ncoherent $\\mathcal{O}_X$-module. Assume $f$ is finite and $Y$ locally\nNoetherian. Then $R^pf_*\\mathcal{F} = 0$ for $p > 0$ and\n$f_*\\mathcal{F}$ is coherent.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent modules on locally Noetherian algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UH","source_file":"spaces-cohomology.tex","source_line":2066,"source_end_line":2073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2066-L2073","statement_sha256":"70859d5951a18244b9aee2d57ee4b2460c2fc1caaf97e8b9a833d04932e31a57","origin":"The Stacks Project","memory_eligible":false,"source_rank":11358,"rank":11358,"depth":26,"x":702.392,"y":1620.734,"cluster":"geometry-of-spaces"},{"id":"stacks:07UJ","tag":"07UJ","title":"Coherent sheaves on Noetherian spaces · Lemma 07UJ","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let F be a coherent O_X-module. The ascending chain condition holds for quasi-coherent submodules of F. In other words, given any sequence F_1 ⊂ F_2 ⊂ … ⊂ F of quasi-coherent submodules, then F_n = F_n + 1 = … for some n ≥ 0.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThe ascending chain condition holds for quasi-coherent submodules\nof $\\mathcal{F}$. In other words, given any sequence\n$$\n\\mathcal{F}_1 \\subset \\mathcal{F}_2 \\subset \\ldots \\subset \\mathcal{F}\n$$\nof quasi-coherent submodules, then\n$\\mathcal{F}_n = \\mathcal{F}_{n + 1} = \\ldots $ for some $n \\geq 0$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent sheaves on Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UJ","source_file":"spaces-cohomology.tex","source_line":2095,"source_end_line":2106,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2095-L2106","statement_sha256":"82fa567f755dd8806c2143a4866bacb89f574255241e7e06b3876d3474d9ce9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11359,"rank":11359,"depth":42,"x":646.376,"y":1597.382,"cluster":"geometry-of-spaces"},{"id":"stacks:07UK","tag":"07UK","title":"Coherent sheaves on Noetherian spaces · Lemma 07UK","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let F be a coherent sheaf on X. Let I ⊂ O_X be a quasi-coherent sheaf of ideals corresponding to a closed subspace Z ⊂ X. Then there is some n ≥ 0 such that I^nF = 0 if and only if Supp(F) ⊂ Z (set theoretically).","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$. Let\n$\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals\ncorresponding to a closed subspace $Z \\subset X$. Then there is some\n$n \\geq 0$ such that $\\mathcal{I}^n\\mathcal{F} = 0$ if and only if\n$\\text{Supp}(\\mathcal{F}) \\subset Z$ (set theoretically).","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent sheaves on Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UK","source_file":"spaces-cohomology.tex","source_line":2121,"source_end_line":2129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2121-L2129","statement_sha256":"1376092972ca783f24516f953e160529363e0fce7683de6b2371869e394bb83f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11360,"rank":11360,"depth":42,"x":707.245,"y":1582.63,"cluster":"geometry-of-spaces"},{"id":"stacks:07UL","tag":"07UL","title":"Artin-Rees · Lemma 07UL","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let F be a coherent sheaf on X. Let G ⊂ F be a quasi-coherent subsheaf. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Then there exists a c ≥ 0 such that for all n ≥ c we have I^n - c(I^cF ∩ G) = I^nF ∩ G","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$. Let\n$\\mathcal{G} \\subset \\mathcal{F}$ be a quasi-coherent subsheaf.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of\nideals. Then there exists a $c \\geq 0$ such that for all $n \\geq c$ we\nhave\n$$\n\\mathcal{I}^{n - c}(\\mathcal{I}^c\\mathcal{F} \\cap \\mathcal{G})\n=\n\\mathcal{I}^n\\mathcal{F} \\cap \\mathcal{G}\n$$","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent sheaves on Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UL","source_file":"spaces-cohomology.tex","source_line":2144,"source_end_line":2157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2144-L2157","statement_sha256":"a4c4e1765c14c24db325e42e8407012a322e8eee8fb48ad1de32a1f3cc226763","origin":"The Stacks Project","memory_eligible":false,"source_rank":11361,"rank":11361,"depth":42,"x":673.801,"y":1628.623,"cluster":"geometry-of-spaces"},{"id":"stacks:07UM","tag":"07UM","title":"Coherent sheaves on Noetherian spaces · Lemma 07UM","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let F be a quasi-coherent O_X-module. Let G be a coherent O_X-module. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Denote Z ⊂ X the corresponding closed subspace and set U = X setminus Z. There is a canonical isomorphism colim_n Hom_O_X(I^nG, F) → Hom_O_U(G|_U, F|_U). In particular we have an isomorphism colim_n Hom_O_X(I^n, F) → Γ(U, F).","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of\nideals. Denote $Z \\subset X$ the corresponding closed subspace and\nset $U = X \\setminus Z$. There is a canonical isomorphism\n$$\n\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n\\mathcal{G}, \\mathcal{F})\n\\longrightarrow\n\\Hom_{\\mathcal{O}_U}(\\mathcal{G}|_U, \\mathcal{F}|_U).\n$$\nIn particular we have an isomorphism\n$$\n\\colim_n \\Hom_{\\mathcal{O}_X}(\\mathcal{I}^n, \\mathcal{F})\n\\longrightarrow\n\\Gamma(U, \\mathcal{F}).\n$$","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Coherent sheaves on Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UM","source_file":"spaces-cohomology.tex","source_line":2171,"source_end_line":2190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2171-L2190","statement_sha256":"3122d883b79fbfa57e78c3fcd950718be118e2ae5fc09df2fbf4e47bfbda8ec8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11362,"rank":11362,"depth":42,"x":661.327,"y":1575.075,"cluster":"geometry-of-spaces"},{"id":"stacks:07UP","tag":"07UP","title":"Devissage of coherent sheaves · Lemma 07UP","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let F be a coherent sheaf on X. Suppose that Supp(F) = Z ∪ Z' with Z, Z' closed. Then there exists a short exact sequence of coherent sheaves 0 → G' → F → G → 0 with Supp(G') ⊂ Z' and Supp(G) ⊂ Z.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$. Suppose that\n$\\text{Supp}(\\mathcal{F}) = Z \\cup Z'$ with $Z$, $Z'$ closed.\nThen there exists a short exact sequence of coherent sheaves\n$$\n0 \\to \\mathcal{G}' \\to \\mathcal{F} \\to \\mathcal{G} \\to 0\n$$\nwith $\\text{Supp}(\\mathcal{G}') \\subset Z'$ and\n$\\text{Supp}(\\mathcal{G}) \\subset Z$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UP","source_file":"spaces-cohomology.tex","source_line":2217,"source_end_line":2228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2217-L2228","statement_sha256":"4b28f1940bcc508bcb5827ae3ae4b665b07d6326b600a219cb82e704fe60e3ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":11363,"rank":11363,"depth":54,"x":714.217,"y":1607.877,"cluster":"geometry-of-spaces"},{"id":"stacks:07UQ","tag":"07UQ","title":"Devissage of coherent sheaves · Lemma 07UQ","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let F be a coherent sheaf on X. Assume that the scheme theoretic support of F is a reduced Z ⊂ X with |Z| irreducible. Then there exist an integer r > 0, a nonzero sheaf of ideals I ⊂ O_Z, and an injective map of coherent sheaves i_*(I^⊕ r) → F whose cokernel is supported on a proper closed subspace of Z.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$. Assume that the scheme\ntheoretic support of $\\mathcal{F}$ is a reduced $Z \\subset X$ with\n$|Z|$ irreducible. Then there exist an integer $r > 0$, a nonzero\nsheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$, and an injective\nmap of coherent sheaves\n$$\ni_*\\left(\\mathcal{I}^{\\oplus r}\\right) \\to \\mathcal{F}\n$$\nwhose cokernel is supported on a proper closed subspace of $Z$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UQ","source_file":"spaces-cohomology.tex","source_line":2260,"source_end_line":2272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2260-L2272","statement_sha256":"50239cf3cc84f1e2d701247a2de649d2c1191b13db7b0f982e23dce7cd744336","origin":"The Stacks Project","memory_eligible":false,"source_rank":11364,"rank":11364,"depth":58,"x":648.066,"y":1613.766,"cluster":"geometry-of-spaces"},{"id":"stacks:07UR","tag":"07UR","title":"Devissage of coherent sheaves · Lemma 07UR","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let F be a coherent sheaf on X. There exists a filtration 0 = F_0 ⊂ F_1 ⊂ … ⊂ F_m = F by coherent subsheaves such that for each j = 1, …, m there exists a reduced closed subspace Z_j ⊂ X with |Z_j| irreducible and a sheaf of ideals I_j ⊂ O_Z_j such that F_j/F_j - 1 ≅ (Z_j → X)_* I_j","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$. There exists a filtration\n$$\n0 = \\mathcal{F}_0 \\subset \\mathcal{F}_1 \\subset\n\\ldots \\subset \\mathcal{F}_m = \\mathcal{F}\n$$\nby coherent subsheaves such that for each $j = 1, \\ldots, m$\nthere exists a reduced closed subspace $Z_j \\subset X$ with $|Z_j|$\nirreducible and a sheaf of ideals $\\mathcal{I}_j \\subset \\mathcal{O}_{Z_j}$\nsuch that\n$$\n\\mathcal{F}_j/\\mathcal{F}_{j - 1}\n\\cong (Z_j \\to X)_* \\mathcal{I}_j\n$$","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UR","source_file":"spaces-cohomology.tex","source_line":2303,"source_end_line":2319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2303-L2319","statement_sha256":"c4435485211efdd8bba0d365f41903ac0eaddea0359e3bc2dc76872bdef1e146","origin":"The Stacks Project","memory_eligible":false,"source_rank":11365,"rank":11365,"depth":59,"x":692.62,"y":1571.406,"cluster":"geometry-of-spaces"},{"id":"stacks:07US","tag":"07US","title":"Devissage of coherent sheaves · Lemma 07US","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let P be a property of coherent sheaves on X. Assume • For any short exact sequence of coherent sheaves 0 → F_1 → F → F_2 → 0 if F_i, i = 1, 2 have property P then so does F. • For every reduced closed subspace Z ⊂ X with |Z| irreducible and every quasi-coherent sheaf of ideals I ⊂ O_Z we have P for i_*I. Then property P holds for every coherent sheaf on X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume\n\\begin{enumerate}\n\\item For any short exact sequence of coherent sheaves\n$$\n0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0\n$$\nif $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$\nthen so does $\\mathcal{F}$.\n\\item For every reduced closed subspace $Z \\subset X$ with $|Z|$ irreducible\nand every quasi-coherent sheaf of ideals $\\mathcal{I} \\subset \\mathcal{O}_Z$\nwe have $\\mathcal{P}$ for $i_*\\mathcal{I}$.\n\\end{enumerate}\nThen property $\\mathcal{P}$ holds for every coherent sheaf on $X$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07US","source_file":"spaces-cohomology.tex","source_line":2398,"source_end_line":2414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2398-L2414","statement_sha256":"76ccd0da1d70f054fcc5fd30e07d280a3d0f487d0db958b261b0cc48dea0d42a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11366,"rank":11366,"depth":60,"x":693.841,"y":1628.56,"cluster":"geometry-of-spaces"},{"id":"stacks:07UT","tag":"07UT","title":"Devissage of coherent sheaves · Lemma 07UT","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let P be a property of coherent sheaves on X. Assume • For any short exact sequence of coherent sheaves 0 → F_1 → F → F_2 → 0 if F_i, i = 1, 2 have property P then so does F. • If P holds for F^⊕ r for some r ≥ 1, then it holds for F. • For every reduced closed subspace i : Z → X with |Z| irreducible there exists a coherent sheaf G on Z such that • Supp(G) = Z, • for every nonzero quasi-coherent sheaf of…","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume\n\\begin{enumerate}\n\\item For any short exact sequence of coherent sheaves\n$$\n0 \\to \\mathcal{F}_1 \\to \\mathcal{F} \\to \\mathcal{F}_2 \\to 0\n$$\nif $\\mathcal{F}_i$, $i = 1, 2$ have property $\\mathcal{P}$\nthen so does $\\mathcal{F}$.\n\\item If $\\mathcal{P}$ holds for $\\mathcal{F}^{\\oplus r}$ for\nsome $r \\geq 1$, then it holds for $\\mathcal{F}$.\n\\item For every reduced closed subspace $i : Z \\to X$ with\n$|Z|$ irreducible there exists a coherent sheaf $\\mathcal{G}$ on $Z$\nsuch that\n\\begin{enumerate}\n\\item $\\text{Supp}(\\mathcal{G}) = Z$,\n\\item for every nonzero quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_Z$ there exists a quasi-coherent\nsubsheaf $\\mathcal{G}' \\subset \\mathcal{I}\\mathcal{G}$ such that\n$\\text{Supp}(\\mathcal{G}/\\mathcal{G}')$ is proper closed in $|Z|$\nand such that $\\mathcal{P}$ holds for $i_*\\mathcal{G}'$.\n\\end{enumerate}\n\\end{enumerate}\nThen property $\\mathcal{P}$ holds for every coherent sheaf on $X$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UT","source_file":"spaces-cohomology.tex","source_line":2434,"source_end_line":2460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2434-L2460","statement_sha256":"447ac3cd78b3eedde0bf84e78dd791b32f5ead1163a4bb4578b9340eea835a23","origin":"The Stacks Project","memory_eligible":false,"source_rank":11367,"rank":11367,"depth":59,"x":646.464,"y":1586.65,"cluster":"geometry-of-spaces"},{"id":"stacks:08AN","tag":"08AN","title":"Devissage of coherent sheaves · Lemma 08AN","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let P be a property of coherent sheaves on X. Assume • For any short exact sequence of coherent sheaves on X if two out of three have property P so does the third. • If P holds for F^⊕ r for some r ≥ 1, then it holds for F. • For every reduced closed subspace i : Z → X with |Z| irreducible there exists a coherent sheaf G on X whose scheme theoretic support is Z such that P holds for G. Then property P holds…","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{P}$ be a property of coherent sheaves on $X$. Assume\n\\begin{enumerate}\n\\item For any short exact sequence of coherent sheaves on $X$\nif two out of three have property $\\mathcal{P}$ so does the third.\n\\item If $\\mathcal{P}$ holds  for $\\mathcal{F}^{\\oplus r}$ for\nsome $r \\geq 1$, then it holds for $\\mathcal{F}$.\n\\item For every reduced closed subspace $i : Z \\to X$ with\n$|Z|$ irreducible there exists a coherent sheaf $\\mathcal{G}$ on $X$\nwhose scheme theoretic support is $Z$ such that $\\mathcal{P}$ holds for\n$\\mathcal{G}$.\n\\end{enumerate}\nThen property $\\mathcal{P}$ holds for every coherent sheaf on $X$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Devissage of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AN","source_file":"spaces-cohomology.tex","source_line":2561,"source_end_line":2576,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2561-L2576","statement_sha256":"20331409108001deccd8b9ffa3666a036bcb337678a2f744b3441e66acffdce7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11368,"rank":11368,"depth":61,"x":715.843,"y":1590.717,"cluster":"geometry-of-spaces"},{"id":"stacks:07UV","tag":"07UV","title":"Limits of coherent modules · Lemma 07UV","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Every quasi-coherent O_X-module is the filtered colimit of its coherent submodules.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nEvery quasi-coherent $\\mathcal{O}_X$-module is the filtered colimit\nof its coherent submodules.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Limits of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UV","source_file":"spaces-cohomology.tex","source_line":2648,"source_end_line":2653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2648-L2653","statement_sha256":"4419a340acb0520acd26d200c20bce96400835f554656fe853c213e02e9bcdb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11369,"rank":11369,"depth":52,"x":660.836,"y":1627.467,"cluster":"geometry-of-spaces"},{"id":"stacks:07UW","tag":"07UW","title":"Limits of coherent modules · Lemma 07UW","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S with Y Noetherian. Then every quasi-coherent O_X-module is a filtered colimit of finitely presented O_X-modules.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic\nspaces over $S$ with $Y$ Noetherian. Then every quasi-coherent\n$\\mathcal{O}_X$-module is a filtered colimit of finitely presented\n$\\mathcal{O}_X$-modules.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Limits of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UW","source_file":"spaces-cohomology.tex","source_line":2680,"source_end_line":2686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2680-L2686","statement_sha256":"7b64e735a52704abada280c4386c35468f2cb1128345f81673c9d732e01064fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":11370,"rank":11370,"depth":59,"x":671.961,"y":1568.554,"cluster":"geometry-of-spaces"},{"id":"stacks:07UZ","tag":"07UZ","title":"Vanishing of cohomology · Lemma 07UZ","summary":"In Situation [Tag 07UY] for an A-module M we have p_*(M ⊗_A O_X) = widetildeM and Γ(X, M ⊗_A O_X) = M.","statement_latex":"In Situation \\ref{situation-vanishing} for an $A$-module $M$ we have\n$p_*(M \\otimes_A \\mathcal{O}_X) = \\widetilde{M}$ and\n$\\Gamma(X, M \\otimes_A \\mathcal{O}_X) = M$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07UZ","source_file":"spaces-cohomology.tex","source_line":2791,"source_end_line":2796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2791-L2796","statement_sha256":"a93c45e10d678f56be2029a674057fd8fc51d6413f08e00abb467b4e899fcacd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11371,"rank":11371,"depth":59,"x":711.529,"y":1618.813,"cluster":"geometry-of-spaces"},{"id":"stacks:07V0","tag":"07V0","title":"Vanishing of cohomology · Lemma 07V0","summary":"In Situation [Tag 07UY]. • Given an affine morphism X' → X of algebraic spaces, we have H^1(X', F') = 0 for every quasi-coherent O_X'-module F'. • Given an A-algebra A' setting X' = X ×_Spec(A) Spec(A') the morphism X' → X is affine and Γ(X', O_X') = A'.","statement_latex":"In Situation \\ref{situation-vanishing}.\n\\begin{enumerate}\n\\item Given an affine morphism $X' \\to X$ of algebraic spaces, we have\n$H^1(X', \\mathcal{F}') = 0$ for every quasi-coherent\n$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$.\n\\item Given an $A$-algebra $A'$ setting $X' = X \\times_{\\Spec(A)} \\Spec(A')$\nthe morphism $X' \\to X$ is affine and $\\Gamma(X', \\mathcal{O}_{X'}) = A'$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07V0","source_file":"spaces-cohomology.tex","source_line":2823,"source_end_line":2833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2823-L2833","statement_sha256":"1e0d70a30d6bf9917ce35a0debe4d65f8a18de60392dc7fc3a718c98bd4dcaac","origin":"The Stacks Project","memory_eligible":false,"source_rank":11372,"rank":11372,"depth":60,"x":641.244,"y":1604.057,"cluster":"geometry-of-spaces"},{"id":"stacks:07V1","tag":"07V1","title":"Vanishing of cohomology · Lemma 07V1","summary":"In Situation [Tag 07UY]. Let Z_0, Z_1 ⊂ |X| be disjoint closed subsets. Then there exists an a ∈ A such that Z_0 ⊂ V(a) and Z_1 ⊂ V(a - 1).","statement_latex":"In Situation \\ref{situation-vanishing}. Let $Z_0, Z_1 \\subset |X|$\nbe disjoint closed subsets. Then there exists an $a \\in A$ such that\n$Z_0 \\subset V(a)$ and $Z_1 \\subset V(a - 1)$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07V1","source_file":"spaces-cohomology.tex","source_line":2854,"source_end_line":2859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2854-L2859","statement_sha256":"1180563ca27166ac2051fa6358306cd59cee217b72f5f1abdb72080a61c6f086","origin":"The Stacks Project","memory_eligible":false,"source_rank":11373,"rank":11373,"depth":55,"x":705.559,"y":1574.777,"cluster":"geometry-of-spaces"},{"id":"stacks:07V4","tag":"07V4","title":"Vanishing of cohomology · Lemma 07V4","summary":"In Situation [Tag 07UY] the morphism p : X → Spec(A) is universally injective.","statement_latex":"In Situation \\ref{situation-vanishing} the morphism $p : X \\to \\Spec(A)$ is\nuniversally injective.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07V4","source_file":"spaces-cohomology.tex","source_line":2879,"source_end_line":2883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2879-L2883","statement_sha256":"b3718edc098f1f71f9b68f607c5256bb39949349b64098a27f4b54756e66f1c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11374,"rank":11374,"depth":61,"x":681.451,"y":1633.415,"cluster":"geometry-of-spaces"},{"id":"stacks:07V5","tag":"07V5","title":"Vanishing of cohomology · Lemma 07V5","summary":"In Situation [Tag 07UY] the morphism p : X → Spec(A) is separated.","statement_latex":"In Situation \\ref{situation-vanishing} the morphism $p : X \\to \\Spec(A)$ is\nseparated.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07V5","source_file":"spaces-cohomology.tex","source_line":2914,"source_end_line":2918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2914-L2918","statement_sha256":"bb60126310a5c484c8424814fd6a2597e0cb7b3823c0196f6f7c392ba553c770","origin":"The Stacks Project","memory_eligible":false,"source_rank":11375,"rank":11375,"depth":62,"x":651.799,"y":1575.962,"cluster":"geometry-of-spaces"},{"id":"stacks:07V6","tag":"07V6","title":"Vanishing of cohomology · Proposition 07V6","summary":"Serre's criterion for affineness in the setting of algebraic spaces. A quasi-compact and quasi-separated algebraic space is affine if and only if all higher cohomology groups of quasi-coherent sheaves vanish. More precisely, any algebraic space as in Situation [Tag 07UY] is an affine scheme.","statement_latex":"\\begin{slogan}\nSerre's criterion for affineness in the setting of algebraic spaces.\n\\end{slogan}\nA quasi-compact and quasi-separated algebraic space is affine\nif and only if all higher cohomology groups of quasi-coherent sheaves\nvanish. More precisely, any algebraic space as in\nSituation \\ref{situation-vanishing} is an affine scheme.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing of cohomology","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07V6","source_file":"spaces-cohomology.tex","source_line":2942,"source_end_line":2951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2942-L2951","statement_sha256":"1a1ac0488d8ffd88c454d500b9491abc8bea3edddb64f8745621d6c87e0fcc2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11376,"rank":11376,"depth":63,"x":720.491,"y":1601.739,"cluster":"geometry-of-spaces"},{"id":"stacks:0D2V","tag":"0D2V","title":"Vanishing of cohomology · Lemma 0D2V","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Assume that for every coherent O_X-module F we have H^1(X, F) = 0. Then X is an affine scheme.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nAssume that for every coherent $\\mathcal{O}_X$-module\n$\\mathcal{F}$ we have $H^1(X, \\mathcal{F}) = 0$.\nThen $X$ is an affine scheme.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2V","source_file":"spaces-cohomology.tex","source_line":2995,"source_end_line":3001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L2995-L3001","statement_sha256":"fc6fe861e67ffabeb8defab441c5eebc7123bd3e8311c82ffb8d5c6a34e1496c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11377,"rank":11377,"depth":64,"x":648.464,"y":1621.887,"cluster":"geometry-of-spaces"},{"id":"stacks:0D2W","tag":"0D2W","title":"Vanishing of cohomology · Lemma 0D2W","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let L be an invertible O_X-module. Assume that for every coherent O_X-module F there exists an n ≥ 1 such that H^1(X, F ⊗_O_X L^⊗ n) = 0. Then X is a scheme and L is ample on X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume that for every coherent $\\mathcal{O}_X$-module\n$\\mathcal{F}$ there exists an $n \\geq 1$ such that\n$H^1(X, \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes n}) = 0$.\nThen $X$ is a scheme and $\\mathcal{L}$ is ample on $X$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Vanishing of cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2W","source_file":"spaces-cohomology.tex","source_line":3011,"source_end_line":3019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3011-L3019","statement_sha256":"c765085eca0698c59805a00c1b697b63342f177b559663dcc33b6b63b7270c2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11378,"rank":11378,"depth":64,"x":685.702,"y":1565.668,"cluster":"geometry-of-spaces"},{"id":"stacks:0GF7","tag":"0GF7","title":"Finite morphisms and affines · Lemma 0GF7","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Assume f is finite, surjective and X locally Noetherian. Let i : Z → X be a closed immersion. Denote i' : Z' → Y the inverse image of Z (Morphisms of Spaces, Section [Tag 03MA]) and f' : Z' → Z the induced morphism. Then G = f'_*O_Z' is a coherent O_Z-module whose support is Z.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is finite, surjective and $X$ locally Noetherian.\nLet $i : Z \\to X$ be a closed immersion. Denote $i' : Z' \\to Y$\nthe inverse image of $Z$\n(Morphisms of Spaces, Section \\ref{spaces-morphisms-section-closed-immersions})\nand $f' : Z' \\to Z$\nthe induced morphism. Then $\\mathcal{G} = f'_*\\mathcal{O}_{Z'}$\nis a coherent $\\mathcal{O}_Z$-module whose support is $Z$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Finite morphisms and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GF7","source_file":"spaces-cohomology.tex","source_line":3133,"source_end_line":3143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3133-L3143","statement_sha256":"a0ce518f290cc46dc6cc9dca518d5e294bab3aa1ac88129cff27e5768608bc4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11379,"rank":11379,"depth":56,"x":703.606,"y":1628.798,"cluster":"geometry-of-spaces"},{"id":"stacks:0GF8","tag":"0GF8","title":"Finite morphisms and affines · Lemma 0GF8","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf on Y. Let I be a quasi-coherent sheaf of ideals on X. If f is affine then If_*F = f_*(f^-1IF) (with notation as explained in the proof).","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces\nover $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $Y$.\nLet $\\mathcal{I}$ be a quasi-coherent sheaf of ideals on $X$.\nIf $f$ is affine then\n$\\mathcal{I}f_*\\mathcal{F} = f_*(f^{-1}\\mathcal{I}\\mathcal{F})$\n(with notation as explained in the proof).","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Finite morphisms and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GF8","source_file":"spaces-cohomology.tex","source_line":3168,"source_end_line":3176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3168-L3176","statement_sha256":"e7d46a3e157106dd27d5d354869ce69545c9aac1cb87f55904483b51e3b7eb2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11380,"rank":11380,"depth":1,"x":639.09,"y":1592.093,"cluster":"geometry-of-spaces"},{"id":"stacks:07VP","tag":"07VP","title":"Finite morphisms and affines · Lemma 07VP","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Assume • f finite, • f surjective, • Y affine, and • X Noetherian. Then X is affine.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic\nspaces over $S$. Assume\n\\begin{enumerate}\n\\item $f$ finite,\n\\item $f$ surjective,\n\\item $Y$ affine, and\n\\item $X$ Noetherian.\n\\end{enumerate}\nThen $X$ is affine.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Finite morphisms and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VP","source_file":"spaces-cohomology.tex","source_line":3199,"source_end_line":3210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3199-L3210","statement_sha256":"d13c7a21d372056624e94ffd6e86e540be3d79fd326d51a41c6f388c1a569dc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11381,"rank":11381,"depth":64,"x":716.83,"y":1582.472,"cluster":"geometry-of-spaces"},{"id":"stacks:089J","tag":"089J","title":"A weak version of Chow's lemma · Lemma 089J","summary":"Let A be a ring. Let X be an algebraic space over Spec(A) whose structure morphism X → Spec(A) is separated of finite type. Then there exists a proper surjective morphism X' → X where X' is a scheme which is H-quasi-projective over Spec(A).","statement_latex":"Let $A$ be a ring. Let $X$ be an algebraic space over $\\Spec(A)$\nwhose structure morphism $X \\to \\Spec(A)$ is separated of finite type.\nThen there exists a proper surjective morphism $X' \\to X$\nwhere $X'$ is a scheme which is H-quasi-projective over $\\Spec(A)$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"A weak version of Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089J","source_file":"spaces-cohomology.tex","source_line":3280,"source_end_line":3286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3280-L3286","statement_sha256":"d97c6fe0c7ff8e418fff09aedd8a826326353243818d3e1d9a6937cda2979835","origin":"The Stacks Project","memory_eligible":false,"source_rank":11382,"rank":11382,"depth":56,"x":666.83,"y":1634.101,"cluster":"geometry-of-spaces"},{"id":"stacks:0ARJ","tag":"0ARJ","title":"Noetherian valuative criterion · Lemma 0ARJ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • Y is locally Noetherian, • f is locally of finite type and quasi-separated, • for every commutative diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & Y where A is a discrete valuation ring and K its fraction field, there is at most one dotted arrow making the diagram commute. Then f is separated.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume\n\\begin{enumerate}\n\\item $Y$ is locally Noetherian,\n\\item $f$ is locally of finite type and quasi-separated,\n\\item for every commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $A$ is a discrete valuation ring and $K$ its fraction field,\nthere is at most one dotted arrow making the diagram commute.\n\\end{enumerate}\nThen $f$ is separated.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARJ","source_file":"spaces-cohomology.tex","source_line":3414,"source_end_line":3432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3414-L3432","statement_sha256":"36ba3ed66fd3eab81c6685879d1e4b31c79242adad73b569e18bc45b9cda9d33","origin":"The Stacks Project","memory_eligible":false,"source_rank":11383,"rank":11383,"depth":53,"x":662.145,"y":1567.118,"cluster":"geometry-of-spaces"},{"id":"stacks:0ARK","tag":"0ARK","title":"Noetherian valuative criterion · Lemma 0ARK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • Y is locally Noetherian, • f is of finite type and quasi-separated, • for every commutative diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] ar@-->[ru] & Y where A is a discrete valuation ring and K its fraction field, there is a unique dotted arrow making the diagram commute. Then f is proper.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume\n\\begin{enumerate}\n\\item $Y$ is locally Noetherian,\n\\item $f$ is of finite type and quasi-separated,\n\\item for every commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d] \\\\\n\\Spec(A) \\ar[r] \\ar@{-->}[ru] & Y\n}\n$$\nwhere $A$ is a discrete valuation ring and $K$ its fraction field,\nthere is a unique dotted arrow making the diagram commute.\n\\end{enumerate}\nThen $f$ is proper.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARK","source_file":"spaces-cohomology.tex","source_line":3471,"source_end_line":3489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3471-L3489","statement_sha256":"f34a43cb7773fa15ebd8176d94271bf37da9f2ffe176cc7bf0fda4531759d611","origin":"The Stacks Project","memory_eligible":false,"source_rank":11384,"rank":11384,"depth":57,"x":719.923,"y":1614.227,"cluster":"geometry-of-spaces"},{"id":"stacks:08AQ","tag":"08AQ","title":"Higher direct images of coherent sheaves · Lemma 08AQ","summary":"Let S be a scheme. Consider a commutative diagram xymatrix X ar[r]_i ar[rd]_f & P^n_Y ar[d] & Y of algebraic spaces over S. Assume i is a closed immersion and Y Noetherian. Set L = i^*O_P^n_Y(1). Let F be a coherent module on X. Then there exists an integer d_0 such that for all d ≥ d_0 we have R^pf_*(F ⊗_O_X L^⊗ d) = 0 for all p > 0.","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[r]_i \\ar[rd]_f & \\mathbf{P}^n_Y \\ar[d] \\\\\n& Y\n}\n$$\nof algebraic spaces over $S$. Assume $i$ is a closed immersion\nand $Y$ Noetherian. Set $\\mathcal{L} = i^*\\mathcal{O}_{\\mathbf{P}^n_Y}(1)$.\nLet $\\mathcal{F}$ be a coherent module on $X$.\nThen there exists an integer $d_0$ such that for all $d \\geq d_0$ we have\n$R^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d}) = 0$\nfor all $p > 0$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Higher direct images of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AQ","source_file":"spaces-cohomology.tex","source_line":3591,"source_end_line":3606,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3591-L3606","statement_sha256":"60c8f840a83f3b2fac0d580a37e040fc86ba6f70766541c85226023feadd647f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11385,"rank":11385,"depth":30,"x":638.8,"y":1612.257,"cluster":"geometry-of-spaces"},{"id":"stacks:08AR","tag":"08AR","title":"Higher direct images of coherent sheaves · Lemma 08AR","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of algebraic spaces over S with Y locally Noetherian. Let F be a coherent O_X-module. Then R^if_*F is a coherent O_Y-module for all i ≥ 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism\nof algebraic spaces over $S$ with $Y$ locally Noetherian.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThen $R^if_*\\mathcal{F}$ is a coherent $\\mathcal{O}_Y$-module\nfor all $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Higher direct images of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AR","source_file":"spaces-cohomology.tex","source_line":3617,"source_end_line":3624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3617-L3624","statement_sha256":"26333081778f29fbcc8d1f21b7e6b46daa1053dd631db751567c0b15124dcbdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":11386,"rank":11386,"depth":62,"x":700.681,"y":1567.334,"cluster":"geometry-of-spaces"},{"id":"stacks:08AS","tag":"08AS","title":"Higher direct images of coherent sheaves · Lemma 08AS","summary":"Let A be a Noetherian ring. Let f : X → Spec(A) be a proper morphism of algebraic spaces. Let F be a coherent O_X-module. Then H^i(X, F) is finite A-module for all i ≥ 0.","statement_latex":"Let $A$ be a Noetherian ring.\nLet $f : X \\to \\Spec(A)$ be a proper morphism of algebraic spaces.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThen $H^i(X, \\mathcal{F})$ is finite $A$-module for all $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Higher direct images of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AS","source_file":"spaces-cohomology.tex","source_line":3794,"source_end_line":3800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3794-L3800","statement_sha256":"bb5d99167f7783b681a9b00f5324593d9474162a1946118e1ab9c97ea94fbb10","origin":"The Stacks Project","memory_eligible":false,"source_rank":11387,"rank":11387,"depth":63,"x":691.103,"y":1636.094,"cluster":"geometry-of-spaces"},{"id":"stacks:08AT","tag":"08AT","title":"Higher direct images of coherent sheaves · Lemma 08AT","summary":"Let A be a Noetherian ring. Let B be a finitely generated graded A-algebra. Let f : X → Spec(A) be a proper morphism of algebraic spaces. Set B = f^*widetilde B. Let F be a quasi-coherent graded B-module of finite type. For every p ≥ 0 the graded B-module H^p(X, F) is a finite B-module.","statement_latex":"Let $A$ be a Noetherian ring.\nLet $B$ be a finitely generated graded $A$-algebra.\nLet $f : X \\to \\Spec(A)$ be a proper morphism of algebraic spaces.\nSet $\\mathcal{B} = f^*\\widetilde B$.\nLet $\\mathcal{F}$ be a quasi-coherent\ngraded $\\mathcal{B}$-module of finite type.\nFor every $p \\geq 0$ the graded $B$-module $H^p(X, \\mathcal{F})$\nis a finite $B$-module.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Higher direct images of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AT","source_file":"spaces-cohomology.tex","source_line":3819,"source_end_line":3829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3819-L3829","statement_sha256":"a7a8eb9b9ae8d95eaeedfa1946ec3cd08b1ea63d64574ad06c3009c06d4102a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11388,"rank":11388,"depth":64,"x":642.509,"y":1579.533,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFA","tag":"0GFA","title":"Ample invertible sheaves and cohomology · Lemma 0GFA","summary":"Let R be a Noetherian ring. Let X be a proper algebraic space over R. Let L be an invertible O_X-module. The following are equivalent • X is a scheme and L is ample on X, • for every coherent O_X-module F there exists an n_0 ≥ 0 such that H^p(X, F ⊗ L^⊗ n) = 0 for all n ≥ n_0 and p > 0, and • for every coherent O_X-module F there exists an n ≥ 1 such that H^1(X, F ⊗ L^⊗ n) = 0.","statement_latex":"Let $R$ be a Noetherian ring. Let $X$ be a proper algebraic space\nover $R$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is a scheme and $\\mathcal{L}$ is ample on $X$,\n\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists\nan $n_0 \\geq 0$ such that\n$H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$ for all $n \\geq n_0$\nand $p > 0$, and\n\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ there exists\nan $n \\geq 1$ such that\n$H^1(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFA","source_file":"spaces-cohomology.tex","source_line":3891,"source_end_line":3906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3891-L3906","statement_sha256":"3d51e188b8afc922ebb117994a75b411d557066697abafe76d6c1bf4780109ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":11389,"rank":11389,"depth":65,"x":724.428,"y":1593.774,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFB","tag":"0GFB","title":"Ample invertible sheaves and cohomology · Lemma 0GFB","summary":"Let R be a Noetherian ring. Let f : Y → X be a morphism of algebraic spaces proper over R. Let L be an invertible O_X-module. Assume f is finite and surjective. The following are equivalent • X is a scheme and L is ample, and • Y is a scheme and f^*L is ample.","statement_latex":"Let $R$ be a Noetherian ring. Let $f : Y \\to X$ be a morphism of\nalgebraic spaces proper over $R$. Let $\\mathcal{L}$ be an\ninvertible $\\mathcal{O}_X$-module. Assume $f$ is finite and surjective.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is a scheme and $\\mathcal{L}$ is ample, and\n\\item $Y$ is a scheme and $f^*\\mathcal{L}$ is ample.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Ample invertible sheaves and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFB","source_file":"spaces-cohomology.tex","source_line":3916,"source_end_line":3926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L3916-L3926","statement_sha256":"5ddf35cee4aecfdc1ba061552bef69dd360eb3968fcf79b0d49a42d50ba152e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11390,"rank":11390,"depth":66,"x":652.045,"y":1630.017,"cluster":"geometry-of-spaces"},{"id":"stacks:08AW","tag":"08AW","title":"The theorem on formal functions · Lemma 08AW","summary":"In Situation [Tag 08AV]. Set B = bigoplus_n ≥ 0 I^n. Then for every p ≥ 0 the graded B-module bigoplus_n ≥ 0 H^p(X, I^nF) is a finite B-module.","statement_latex":"In Situation \\ref{situation-formal-functions}.\nSet $B = \\bigoplus_{n \\geq 0} I^n$.\nThen for every $p \\geq 0$ the graded $B$-module\n$\\bigoplus_{n \\geq 0} H^p(X, I^n\\mathcal{F})$ is\na finite $B$-module.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AW","source_file":"spaces-cohomology.tex","source_line":4039,"source_end_line":4046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4039-L4046","statement_sha256":"1b0bff613d21fc968b24d05866e6e21615e389777f5009b864cbb4aa5c43fca2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11391,"rank":11391,"depth":65,"x":676.449,"y":1561.732,"cluster":"geometry-of-spaces"},{"id":"stacks:08AX","tag":"08AX","title":"The theorem on formal functions · Lemma 08AX","summary":"In Situation [Tag 08AV]. For every p ≥ 0 there exists an integer c ≥ 0 such that • the multiplication map I^n - c ⊗ H^p(X, I^cF) → H^p(X, I^nF) is surjective for all n ≥ c, and • the image of H^p(X, I^n + mF) → H^p(X, I^nF) is contained in the submodule I^m - c H^p(X, I^nF) for all n ≥ 0, m ≥ c.","statement_latex":"In Situation \\ref{situation-formal-functions}.\nFor every $p \\geq 0$ there exists an integer $c \\geq 0$ such that\n\\begin{enumerate}\n\\item the multiplication map\n$I^{n - c} \\otimes H^p(X, I^c\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$\nis surjective for all $n \\geq c$, and\n\\item the image of $H^p(X, I^{n + m}\\mathcal{F}) \\to H^p(X, I^n\\mathcal{F})$\nis contained in the submodule $I^{m - c} H^p(X, I^n\\mathcal{F})$\nfor all $n \\geq 0$, $m \\geq c$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AX","source_file":"spaces-cohomology.tex","source_line":4055,"source_end_line":4067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4055-L4067","statement_sha256":"8072de68967e7d61bf6c02d7546ba2e235611205ea234c97d9229b56deac5799","origin":"The Stacks Project","memory_eligible":false,"source_rank":11392,"rank":11392,"depth":66,"x":713.628,"y":1626.388,"cluster":"geometry-of-spaces"},{"id":"stacks:08AY","tag":"08AY","title":"The theorem on formal functions · Lemma 08AY","summary":"In Situation [Tag 08AV]. Fix p ≥ 0. • There exists a c_1 ≥ 0 such that for all n ≥ c_1 we have Ker( H^p(X, F) → H^p(X, F/I^nF) ) ⊂ I^n - c_1H^p(X, F). • The inverse system (H^p(X, F/I^nF))_n ∈ N satisfies the Mittag-Leffler condition (see Homology, Definition [Tag 02N0]). • In fact for any p and n there exists a c_2(n) ≥ n such that Im(H^p(X, F/I^kF) → H^p(X, F/I^nF)) = Im(H^p(X, F) → H^p(X, F/I^nF)) for all k ≥ c_2(n).","statement_latex":"In Situation \\ref{situation-formal-functions}.\nFix $p \\geq 0$.\n\\begin{enumerate}\n\\item There exists a $c_1 \\geq 0$ such that for all $n \\geq c_1$\nwe have\n$$\n\\Ker(\nH^p(X, \\mathcal{F}) \\to H^p(X, \\mathcal{F}/I^n\\mathcal{F})\n)\n\\subset\nI^{n - c_1}H^p(X, \\mathcal{F}).\n$$\n\\item The inverse system\n$$\n\\left(H^p(X, \\mathcal{F}/I^n\\mathcal{F})\\right)_{n \\in \\mathbf{N}}\n$$\nsatisfies the Mittag-Leffler condition (see\nHomology, Definition \\ref{homology-definition-Mittag-Leffler}).\n\\item In fact for any $p$ and $n$ there exists a $c_2(n) \\geq n$\nsuch that\n$$\n\\Im(H^p(X, \\mathcal{F}/I^k\\mathcal{F})\n\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))\n=\n\\Im(H^p(X, \\mathcal{F})\n\\to H^p(X, \\mathcal{F}/I^n\\mathcal{F}))\n$$\nfor all $k \\geq c_2(n)$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AY","source_file":"spaces-cohomology.tex","source_line":4093,"source_end_line":4124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4093-L4124","statement_sha256":"266dbb1d52b2e56da25a6d9c3c0001aead73eb058b6a06f622a5a79dea864bd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11393,"rank":11393,"depth":67,"x":633.663,"y":1599.62,"cluster":"geometry-of-spaces"},{"id":"stacks:08AZ","tag":"08AZ","title":"Theorem on formal functions · Theorem 08AZ","summary":"In Situation [Tag 08AV]. Fix p ≥ 0. The system of maps H^p(X, F)/I^nH^p(X, F) → H^p(X, F/I^nF) define an isomorphism of limits H^p(X, F)^wedge → lim_n H^p(X, F/I^nF) where the left hand side is the completion of the A-module H^p(X, F) with respect to the ideal I, see Algebra, Section [Tag 00M9]. Moreover, this is in fact a homeomorphism for the limit topologies.","statement_latex":"In Situation \\ref{situation-formal-functions}. Fix $p \\geq 0$.\nThe system of maps\n$$\nH^p(X, \\mathcal{F})/I^nH^p(X, \\mathcal{F})\n\\longrightarrow\nH^p(X, \\mathcal{F}/I^n\\mathcal{F})\n$$\ndefine an isomorphism of limits\n$$\nH^p(X, \\mathcal{F})^\\wedge\n\\longrightarrow\n\\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F})\n$$\nwhere the left hand side is the completion of the $A$-module\n$H^p(X, \\mathcal{F})$ with respect to the ideal $I$, see\nAlgebra, Section \\ref{algebra-section-completion}.\nMoreover, this is in fact a homeomorphism for the limit topologies.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The theorem on formal functions","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08AZ","source_file":"spaces-cohomology.tex","source_line":4207,"source_end_line":4226,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4207-L4226","statement_sha256":"ab814e3354ce6dc0952219954c05a20d303f2281a6a2a002ac9fa71d3c25827e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11394,"rank":11394,"depth":68,"x":714.71,"y":1573.809,"cluster":"geometry-of-spaces"},{"id":"stacks:08B0","tag":"08B0","title":"The theorem on formal functions · Lemma 08B0","summary":"Let A be a ring. Let I ⊂ A be an ideal. Assume A is Noetherian and complete with respect to I. Let f : X → Spec(A) be a proper morphism of algebraic spaces. Let F be a coherent sheaf on X. Then H^p(X, F) = lim_n H^p(X, F/I^nF) for all p ≥ 0.","statement_latex":"Let $A$ be a ring. Let $I \\subset A$ be an ideal. Assume $A$ is\nNoetherian and complete with respect to $I$.\nLet $f : X \\to \\Spec(A)$ be a proper morphism of algebraic spaces.\nLet $\\mathcal{F}$ be a coherent sheaf on $X$. Then\n$$\nH^p(X, \\mathcal{F}) = \\lim_n H^p(X, \\mathcal{F}/I^n\\mathcal{F})\n$$\nfor all $p \\geq 0$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08B0","source_file":"spaces-cohomology.tex","source_line":4263,"source_end_line":4273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4263-L4273","statement_sha256":"cce1429a90ebf084eec6385ffcdf0597628b9dab48ac8494fd078febd8648046","origin":"The Stacks Project","memory_eligible":false,"source_rank":11395,"rank":11395,"depth":69,"x":675.437,"y":1639.271,"cluster":"geometry-of-spaces"},{"id":"stacks:08B1","tag":"08B1","title":"The theorem on formal functions · Lemma 08B1","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S and let F be a quasi-coherent sheaf on X. Assume • Y locally Noetherian, • f proper, and • F coherent. Let overliney be a geometric point of Y. Consider the \"infinitesimal neighbourhoods\" xymatrix X_n = Spec(O_Y, overliney/ m_overliney^n) ×_Y X ar[r]_-i_n ar[d]_f_n & X ar[d]^f Spec(O_Y, overliney/ m_overliney^n) ar[r]^-c_n & Y of the fibre X_1 = X_overliney and set F_n = i_n^*F. Then we have…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ and let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$. Assume\n\\begin{enumerate}\n\\item $Y$ locally Noetherian,\n\\item $f$ proper, and\n\\item $\\mathcal{F}$ coherent.\n\\end{enumerate}\nLet $\\overline{y}$ be a geometric point of $Y$.\nConsider the ``infinitesimal neighbourhoods''\n$$\n\\xymatrix{\nX_n =\n\\Spec(\\mathcal{O}_{Y, \\overline{y}}/\\mathfrak m_{\\overline{y}}^n) \\times_Y X\n\\ar[r]_-{i_n} \\ar[d]_{f_n} &\nX \\ar[d]^f \\\\\n\\Spec(\\mathcal{O}_{Y, \\overline{y}}/\\mathfrak m_{\\overline{y}}^n)\n\\ar[r]^-{c_n} & Y\n}\n$$\nof the fibre $X_1 = X_{\\overline{y}}$ and set\n$\\mathcal{F}_n = i_n^*\\mathcal{F}$. Then we have\n$$\n\\left(R^pf_*\\mathcal{F}\\right)_{\\overline{y}}^\\wedge\n\\cong\n\\lim_n H^p(X_n, \\mathcal{F}_n)\n$$\nas $\\mathcal{O}_{Y, \\overline{y}}^\\wedge$-modules.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08B1","source_file":"spaces-cohomology.tex","source_line":4285,"source_end_line":4314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4285-L4314","statement_sha256":"af18a601e49a67c369cc168fba7d49e562af28171784a45f0f5fad1e317ceb58","origin":"The Stacks Project","memory_eligible":false,"source_rank":11396,"rank":11396,"depth":69,"x":651.596,"y":1568.242,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4S","tag":"0A4S","title":"The theorem on formal functions · Lemma 0A4S","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let overliney be a geometric point of Y. Assume • Y locally Noetherian, • f is proper, and • X_overliney has discrete underlying topological space. Then for any coherent sheaf F on X we have (R^pf_*F)_overliney = 0 for all p > 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $\\overline{y}$ be a geometric point of $Y$.\nAssume\n\\begin{enumerate}\n\\item $Y$ locally Noetherian,\n\\item $f$ is proper, and\n\\item $X_{\\overline{y}}$ has discrete underlying topological space.\n\\end{enumerate}\nThen for any coherent sheaf $\\mathcal{F}$ on $X$ we have\n$(R^pf_*\\mathcal{F})_{\\overline{y}} = 0$ for all $p > 0$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4S","source_file":"spaces-cohomology.tex","source_line":4363,"source_end_line":4375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4363-L4375","statement_sha256":"691bc9d0e8497801193655eb067e5a816e822bfaccff1f53549d4fdc3c770492","origin":"The Stacks Project","memory_eligible":false,"source_rank":11397,"rank":11397,"depth":70,"x":726.789,"y":1607.348,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4T","tag":"0A4T","title":"The theorem on formal functions · Lemma 0A4T","summary":"For proper maps, stalks of higher direct images are trivial in degrees larger than the dimension of the fibre. Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let overliney be a geometric point of Y. Assume • Y locally Noetherian, • f is proper, and • dim(X_overliney) = d. Then for any coherent sheaf F on X we have (R^pf_*F)_overliney = 0 for all p > d.","statement_latex":"\\begin{slogan}\nFor proper maps, stalks of higher direct images are trivial in degrees\nlarger than the dimension of the fibre.\n\\end{slogan}\nLet $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\overline{y}$ be a geometric point of $Y$.\nAssume\n\\begin{enumerate}\n\\item $Y$ locally Noetherian,\n\\item $f$ is proper, and\n\\item $\\dim(X_{\\overline{y}}) = d$.\n\\end{enumerate}\nThen for any coherent sheaf $\\mathcal{F}$ on $X$ we have\n$(R^pf_*\\mathcal{F})_{\\overline{y}} = 0$ for all $p > d$.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"The theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4T","source_file":"spaces-cohomology.tex","source_line":4408,"source_end_line":4425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4408-L4425","statement_sha256":"77989aa7a9d0ea5875fc37b11fb2b3e0b3a766d468d0869c42a4090dfd78b0d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11398,"rank":11398,"depth":70,"x":639.326,"y":1621.268,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4V","tag":"0A4V","title":"Applications of the theorem on formal functions · Lemma 0A4V","summary":"(For a more general version see More on Morphisms of Spaces, Lemma [Tag 0A4X]). Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume Y is locally Noetherian. The following are equivalent • f is finite, and • f is proper and |X_k| is a discrete space for every morphism Spec(k) → Y where k is a field.","statement_latex":"(For a more general version see\nMore on Morphisms of Spaces, Lemma\n\\ref{spaces-more-morphisms-lemma-characterize-finite}).\nLet $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $Y$ is locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is finite, and\n\\item $f$ is proper and $|X_k|$ is a discrete space\nfor every morphism $\\Spec(k) \\to Y$ where $k$ is a field.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Applications of the theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4V","source_file":"spaces-cohomology.tex","source_line":4457,"source_end_line":4471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4457-L4471","statement_sha256":"1a1192b0f313043251349910a0645d6ea9c00b14d2d11005e912d20a04c09829","origin":"The Stacks Project","memory_eligible":false,"source_rank":11399,"rank":11399,"depth":71,"x":692.971,"y":1560.99,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4W","tag":"0A4W","title":"Applications of the theorem on formal functions · Lemma 0A4W","summary":"(For a more general version see More on Morphisms of Spaces, Lemma [Tag 0A4Y]). Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let overliney be a geometric point of Y. Assume • Y is locally Noetherian, • f is proper, and • |X_overliney| is finite. Then there exists an open neighbourhood V ⊂ Y of overliney such that f|_f^-1(V) : f^-1(V) → V is finite.","statement_latex":"(For a more general version see\nMore on Morphisms of Spaces, Lemma\n\\ref{spaces-more-morphisms-lemma-proper-finite-fibre-finite-in-neighbourhood}).\nLet $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $\\overline{y}$ be a geometric point of $Y$.\nAssume\n\\begin{enumerate}\n\\item $Y$ is locally Noetherian,\n\\item $f$ is proper, and\n\\item $|X_{\\overline{y}}|$ is finite.\n\\end{enumerate}\nThen there exists an open neighbourhood $V \\subset Y$ of $\\overline{y}$\nsuch that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite.","area":"Geometry of Spaces","chapter":"Cohomology of Algebraic Spaces","chapter_id":"spaces-cohomology","section":"Applications of the theorem on formal functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4W","source_file":"spaces-cohomology.tex","source_line":4523,"source_end_line":4538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-cohomology.tex#L4523-L4538","statement_sha256":"7e4f9fc29aee0ef7c09bb199241079f5304d24550c263d9b2580e53987d22dce","origin":"The Stacks Project","memory_eligible":false,"source_rank":11400,"rank":11400,"depth":72,"x":701.944,"y":1636.355,"cluster":"geometry-of-spaces"},{"id":"stacks:049J","tag":"049J","title":"Morphisms of finite presentation · Definition 049J","summary":"Let S be a scheme. • A functor F : (Sch/S)_fppf^opp → Sets is said to be limit preserving or locally of finite presentation if for every affine scheme T over S which is a limit T = lim T_i of a directed inverse system of affine schemes T_i over S, we have F(T) = colim F(T_i). We sometimes say that F is locally of finite presentation over S. • Let F, G : (Sch/S)_fppf^opp → Sets. A transformation of functors a : F → G is limit preserving or locally of finite presentation if…","statement_latex":"Let $S$ be a scheme.\n\\begin{enumerate}\n\\item A functor $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$\nis said to be {\\it limit preserving} or {\\it locally of finite presentation} if\nfor every affine scheme $T$ over $S$ which is a limit $T = \\lim T_i$\nof a directed inverse system of affine schemes $T_i$ over $S$, we have\n$$\nF(T) = \\colim F(T_i).\n$$\nWe sometimes say that $F$ is {\\it locally of finite presentation over $S$}.\n\\item Let $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nA transformation of functors $a : F \\to G$\nis {\\it limit preserving} or {\\it locally of finite presentation}\nif for every scheme $T$ over $S$ and every $y \\in G(T)$ the functor\n$$\nF_y : (\\Sch/T)_{fppf}^{opp} \\longrightarrow \\textit{Sets}, \\quad\nT'/T \\longmapsto \\{x \\in F(T') \\mid a(x) = y|_{T'}\\}\n$$\nis locally of finite presentation over $T$\\footnote{The characterization (2) in\nLemma \\ref{lemma-characterize-relative-limit-preserving}\nmay be easier to parse.}. We sometimes say that\n$F$ is {\\it relatively limit preserving} over $G$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049J","source_file":"spaces-limits.tex","source_line":69,"source_end_line":94,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L69-L94","statement_sha256":"7857c349f3e66b25d97a246501030fbe96bc26e040e37ab33c16c661c9d3780f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11401,"rank":11401,"depth":0,"x":634.3,"y":1585.555,"cluster":"geometry-of-spaces"},{"id":"stacks:06BC","tag":"06BC","title":"Morphisms of finite presentation · Lemma 06BC","summary":"Let S be a scheme. Let a : F → G be a transformation of functors (Sch/S)_fppf^opp → Sets. The following are equivalent • a : F → G is limit preserving, and • for every affine scheme T over S which is a limit T = lim T_i of a directed inverse system of affine schemes T_i over S the diagram of sets xymatrix colim_i F(T_i) ar[r] ar[d]_a & F(T) ar[d]^a colim_i G(T_i) ar[r] & G(T) is a fibre product diagram.","statement_latex":"Let $S$ be a scheme. Let $a : F \\to G$ be a transformation of functors\n$(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $a : F \\to G$ is limit preserving, and\n\\item for every affine scheme $T$ over $S$ which is a\nlimit $T = \\lim T_i$ of a directed inverse system of affine\nschemes $T_i$ over $S$ the diagram of sets\n$$\n\\xymatrix{\n\\colim_i F(T_i) \\ar[r] \\ar[d]_a & F(T) \\ar[d]^a \\\\\n\\colim_i G(T_i) \\ar[r] & G(T)\n}\n$$\nis a fibre product diagram.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BC","source_file":"spaces-limits.tex","source_line":133,"source_end_line":151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L133-L151","statement_sha256":"52454c2ed0338d544f48e7f145677dbfe470806794587231f475363ec214dc3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11402,"rank":11402,"depth":1,"x":725.595,"y":1584.626,"cluster":"geometry-of-spaces"},{"id":"stacks:049L","tag":"049L","title":"Morphisms of finite presentation · Lemma 049L","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let a : F → G, b : G → H be transformations of functors. If a and b are limit preserving, then b ∘ a : F → H is limit preserving.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$, $b : G \\to H$ be transformations of functors.\nIf $a$ and $b$ are limit preserving, then\n$$\nb \\circ a : F \\longrightarrow H\n$$\nis limit preserving.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049L","source_file":"spaces-limits.tex","source_line":181,"source_end_line":191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L181-L191","statement_sha256":"73c012e36734ed7687b2f52c7f05d1def7cb0cfb1e40cb93c1f3fe4c03cf6177","origin":"The Stacks Project","memory_eligible":false,"source_rank":11403,"rank":11403,"depth":2,"x":658.613,"y":1637.435,"cluster":"geometry-of-spaces"},{"id":"stacks:0GDY","tag":"0GDY","title":"Morphisms of finite presentation · Lemma 0GDY","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let a : F → G, b : G → H be transformations of functors. If b ∘ a and b are limit preserving, then a is limit preserving.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$, $b : G \\to H$ be transformations of functors.\nIf $b \\circ a$ and $b$ are limit preserving, then $a$\nis limit preserving.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDY","source_file":"spaces-limits.tex","source_line":208,"source_end_line":215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L208-L215","statement_sha256":"4b9f5a67c040541eb78b57ca07545b6139deab8641e0b2710631fcbd00b400d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11404,"rank":11404,"depth":2,"x":665.58,"y":1560.021,"cluster":"geometry-of-spaces"},{"id":"stacks:049M","tag":"049M","title":"Morphisms of finite presentation · Lemma 049M","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let a : F → G, b : H → G be transformations of functors. Consider the fibre product diagram xymatrix H ×_b, G, a F ar[r]_-b' ar[d]_a' & F ar[d]^a H ar[r]^b & G If a is limit preserving, then the base change a' is limit preserving.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$, $b : H \\to G$ be transformations of functors.\nConsider the fibre product diagram\n$$\n\\xymatrix{\nH \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\\nH \\ar[r]^b & G\n}\n$$\nIf $a$ is limit preserving, then the base change $a'$ is limit preserving.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049M","source_file":"spaces-limits.tex","source_line":233,"source_end_line":246,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L233-L246","statement_sha256":"f72bdabdfa95cf5dacbf558e78512b35ec5c6f348364ffc77e1f4eb7df2993f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11405,"rank":11405,"depth":0,"x":723.037,"y":1621.424,"cluster":"geometry-of-spaces"},{"id":"stacks:0GDZ","tag":"0GDZ","title":"Morphisms of finite presentation · Lemma 0GDZ","summary":"Let S be a scheme contained in Sch_fppf. Let E, F, G, H : (Sch/S)_fppf^opp → Sets. Let a : F → G, b : H → G, and c : G → E be transformations of functors. If c, c ∘ a, and c ∘ b are limit preserving, then F ×_G H → E is too.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $E, F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$, $b : H \\to G$, and $c : G \\to E$\nbe transformations of functors. If $c$, $c \\circ a$, and $c \\circ b$\nare limit preserving, then $F \\times_G H \\to E$ is too.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDZ","source_file":"spaces-limits.tex","source_line":252,"source_end_line":259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L252-L259","statement_sha256":"bc8b4228da3d4b8339511bbad926da56242159b1208fc3418af3b017ef171d08","origin":"The Stacks Project","memory_eligible":false,"source_rank":11406,"rank":11406,"depth":2,"x":630.748,"y":1608.673,"cluster":"geometry-of-spaces"},{"id":"stacks:049O","tag":"049O","title":"Morphisms of finite presentation · Lemma 049O","summary":"Let S be a scheme contained in Sch_fppf. Let F : (Sch/S)_fppf^opp → Sets be a functor. If F is limit preserving then its sheafification F^\\# is limit preserving.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.\nIf $F$ is limit preserving then its sheafification $F^\\#$ is limit preserving.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049O","source_file":"spaces-limits.tex","source_line":288,"source_end_line":293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L288-L293","statement_sha256":"c99e1700546ee0595933c4510045aa687317d003a6858c0140d1e0cf3c9f9d06","origin":"The Stacks Project","memory_eligible":false,"source_rank":11407,"rank":11407,"depth":22,"x":709.513,"y":1565.458,"cluster":"geometry-of-spaces"},{"id":"stacks:049P","tag":"049P","title":"Morphisms of finite presentation · Lemma 049P","summary":"Let S be a scheme. Let F : (Sch/S)_fppf^opp → Sets be a functor. Assume that • F is a sheaf, and • there exists an fppf covering (U_j → S)_j ∈ J such that F|_(Sch/U_j)_fppf is limit preserving. Then F is limit preserving.","statement_latex":"Let $S$ be a scheme.\nLet $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item $F$ is a sheaf, and\n\\item there exists an fppf covering $\\{U_j \\to S\\}_{j \\in J}$ such that\n$F|_{(\\Sch/U_j)_{fppf}}$ is limit preserving.\n\\end{enumerate}\nThen $F$ is limit preserving.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049P","source_file":"spaces-limits.tex","source_line":379,"source_end_line":390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L379-L390","statement_sha256":"6e44cffc9b9c030b952198b18153f064c1e4055f8e50b4100bde1d5c0aec60da","origin":"The Stacks Project","memory_eligible":false,"source_rank":11408,"rank":11408,"depth":22,"x":686.049,"y":1642.459,"cluster":"geometry-of-spaces"},{"id":"stacks:049Q","tag":"049Q","title":"Morphisms of finite presentation · Lemma 049Q","summary":"Let S be a scheme contained in Sch_fppf. Let F, G : (Sch/S)_fppf^opp → Sets be functors. If a : F → G is a transformation which is limit preserving, then the induced transformation of sheaves F^\\# → G^\\# is limit preserving.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be functors.\nIf $a : F \\to G$ is a transformation which is limit preserving,\nthen the induced transformation of sheaves\n$F^\\# \\to G^\\#$ is limit preserving.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049Q","source_file":"spaces-limits.tex","source_line":441,"source_end_line":448,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L441-L448","statement_sha256":"c9ed3f38ce1e08015747d1a17e45200539cc2398d5cad17676db68b42ecd2095","origin":"The Stacks Project","memory_eligible":false,"source_rank":11409,"rank":11409,"depth":23,"x":641.176,"y":1571.965,"cluster":"geometry-of-spaces"},{"id":"stacks:04AK","tag":"04AK","title":"Morphisms of finite presentation · Proposition 04AK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • The morphism f is a morphism of algebraic spaces which is locally of finite presentation, see Morphisms of Spaces, Definition [Tag 03XP]. • The morphism f : X → Y is limit preserving as a transformation of functors, see Definition [Tag 049J].","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. The following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is a morphism of algebraic spaces which is\nlocally of finite presentation, see\nMorphisms of Spaces,\nDefinition \\ref{spaces-morphisms-definition-locally-finite-presentation}.\n\\item The morphism $f : X \\to Y$ is limit preserving as\na transformation of functors, see\nDefinition \\ref{definition-locally-finite-presentation}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AK","source_file":"spaces-limits.tex","source_line":476,"source_end_line":489,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L476-L489","statement_sha256":"26ffb40f5e9b5bb456c2c9b4dfe5a33a1397987550be8b985b734d0bec5698cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":11410,"rank":11410,"depth":48,"x":731.46,"y":1598.635,"cluster":"geometry-of-spaces"},{"id":"stacks:0CM6","tag":"0CM6","title":"Morphisms of finite presentation · Lemma 0CM6","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If for every directed limit T = lim_i ∈ I T_i of affine schemes over S the map colim X(T_i) → X(T) ×_Y(T) colim Y(T_i) is surjective, then f is locally of finite presentation. In other words, in Proposition [Tag 04AK] part (2) it suffices to check surjectivity in the criterion of Lemma [Tag 06BC].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf for every directed limit $T = \\lim_{i \\in I} T_i$\nof affine schemes over $S$ the map\n$$\n\\colim X(T_i) \\longrightarrow X(T) \\times_{Y(T)} \\colim Y(T_i)\n$$\nis surjective, then $f$ is locally of finite presentation.\nIn other words, in\nProposition \\ref{proposition-characterize-locally-finite-presentation}\npart (2) it suffices to check surjectivity in the criterion of\nLemma \\ref{lemma-characterize-relative-limit-preserving}.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CM6","source_file":"spaces-limits.tex","source_line":584,"source_end_line":598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L584-L598","statement_sha256":"8e7f6cce4d3564cf528aa6ae786fd0beace5c3f51674985463e985f89bc9a884","origin":"The Stacks Project","memory_eligible":false,"source_rank":11411,"rank":11411,"depth":49,"x":642.947,"y":1630.374,"cluster":"geometry-of-spaces"},{"id":"stacks:07SF","tag":"07SF","title":"Limits of algebraic spaces · Lemma 07SF","summary":"Let S be a scheme. Let I be a directed set. Let (X_i, f_ii') be an inverse system over I in the category of algebraic spaces over S. If the morphisms f_ii' : X_i → X_i' are affine, then the limit X = lim_i X_i (as an fppf sheaf) is an algebraic space. Moreover, • each of the morphisms f_i : X → X_i is affine, • for any i ∈ I and any morphism of algebraic spaces T → X_i we have X ×_X_i T = lim_i' ≥ i X_i' ×_X_i T. as algebraic spaces over S.","statement_latex":"Let $S$ be a scheme. Let $I$ be a directed set.\nLet $(X_i, f_{ii'})$ be an inverse system over $I$\nin the category of algebraic spaces over $S$.\nIf the morphisms $f_{ii'} : X_i \\to X_{i'}$ are affine, then the\nlimit $X = \\lim_i X_i$ (as an fppf sheaf) is an algebraic space.\nMoreover,\n\\begin{enumerate}\n\\item each of the morphisms $f_i : X \\to X_i$ is affine,\n\\item for any $i \\in I$ and any morphism of algebraic spaces\n$T \\to X_i$ we have\n$$\nX \\times_{X_i} T = \\lim_{i' \\geq i} X_{i'} \\times_{X_i} T.\n$$\nas algebraic spaces over $S$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Limits of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SF","source_file":"spaces-limits.tex","source_line":660,"source_end_line":677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L660-L677","statement_sha256":"283f91e6923c6924554eac7b3cc2e870137c0bb3b1b4d6c08b808ea44f5290a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11412,"rank":11412,"depth":52,"x":682.912,"y":1556.34,"cluster":"geometry-of-spaces"},{"id":"stacks:07SH","tag":"07SH","title":"Limits of algebraic spaces · Lemma 07SH","summary":"Let S be a scheme. Let I be a directed set. Let (X_i, f_ii') be an inverse system over I of algebraic spaces over S with affine transition maps. Let X = lim_i X_i. Let 0 ∈ I. Suppose that T → X_0 is a morphism of algebraic spaces. Then T ×_X_0 X = lim_i ≥ 0 T ×_X_0 X_i as algebraic spaces over S.","statement_latex":"Let $S$ be a scheme. Let $I$ be a directed set.\nLet $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces\nover $S$ with affine transition maps.\nLet $X = \\lim_i X_i$. Let $0 \\in I$. Suppose that $T \\to X_0$ is a\nmorphism of algebraic spaces. Then\n$$\nT \\times_{X_0} X = \\lim_{i \\geq 0} T \\times_{X_0} X_i\n$$\nas algebraic spaces over $S$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Limits of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SH","source_file":"spaces-limits.tex","source_line":734,"source_end_line":745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L734-L745","statement_sha256":"0e6c0564263f2dcc79829787f78a2a069b687a2775e25598c64d938279e66730","origin":"The Stacks Project","memory_eligible":false,"source_rank":11413,"rank":11413,"depth":53,"x":713.142,"y":1634.03,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUH","tag":"0CUH","title":"Limits of algebraic spaces · Lemma 0CUH","summary":"Let S be a scheme. Let I be a directed set. Let (X_i, f_i'i) → (Y_i, g_i'i) be a morphism of inverse systems over I of algebraic spaces over S. Assume • the morphisms f_i'i : X_i' → X_i are affine, • the morphisms g_i'i : Y_i' → Y_i are affine, • the morphisms X_i → Y_i are closed immersions. Then lim X_i → lim Y_i is a closed immersion.","statement_latex":"Let $S$ be a scheme. Let $I$ be a directed set.\nLet $(X_i, f_{i'i}) \\to (Y_i, g_{i'i})$ be a morphism\nof inverse systems over $I$ of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item the morphisms $f_{i'i} : X_{i'} \\to X_i$ are affine,\n\\item the morphisms $g_{i'i} : Y_{i'} \\to Y_i$ are affine,\n\\item the morphisms $X_i \\to Y_i$ are closed immersions.\n\\end{enumerate}\nThen $\\lim X_i \\to \\lim Y_i$ is a closed immersion.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Limits of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUH","source_file":"spaces-limits.tex","source_line":754,"source_end_line":766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L754-L766","statement_sha256":"d2d517e094e112e423d9e4c4fe0511966d61a467aa8e313581c92abb8c744077","origin":"The Stacks Project","memory_eligible":false,"source_rank":11414,"rank":11414,"depth":53,"x":627.917,"y":1593.678,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUI","tag":"0CUI","title":"Limits of algebraic spaces · Lemma 0CUI","summary":"Let S be a scheme. Let I be a directed set. Let (X_i, f_i'i) be an inverse systems over I of algebraic spaces over S. If X_i is reduced for all i, then X is reduced.","statement_latex":"Let $S$ be a scheme. Let $I$ be a directed set.\nLet $(X_i, f_{i'i})$ be an inverse systems over $I$\nof algebraic spaces over $S$. If $X_i$ is reduced\nfor all $i$, then $X$ is reduced.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Limits of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUI","source_file":"spaces-limits.tex","source_line":784,"source_end_line":790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L784-L790","statement_sha256":"f776716b7f6fa31fa3e3904f2c429f02b2483d8ce511110fb9730c0b7cb238cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":11415,"rank":11415,"depth":53,"x":723.72,"y":1574.976,"cluster":"geometry-of-spaces"},{"id":"stacks:0CP4","tag":"0CP4","title":"Limits of algebraic spaces · Lemma 0CP4","summary":"Let S be a scheme. Let X → Y be a morphism of algebraic spaces over S. The equivalent conditions (1) and (2) of Proposition [Tag 04AK] are also equivalent to • [(3)] for every directed limit T = lim T_i of quasi-compact and quasi-separated algebraic spaces T_i over S with affine transition morphisms the diagram of sets xymatrix colim_i Mor(T_i, X) ar[r] ar[d] & Mor(T, X) ar[d] colim_i Mor(T_i, Y) ar[r] & Mor(T, Y) is a fibre product diagram.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a morphism of algebraic spaces\nover $S$. The equivalent conditions (1) and (2) of\nProposition \\ref{proposition-characterize-locally-finite-presentation}\nare also equivalent to\n\\begin{enumerate}\n\\item[(3)] for every directed limit $T = \\lim T_i$ of quasi-compact\nand quasi-separated algebraic spaces $T_i$ over $S$ with affine\ntransition morphisms the diagram of sets\n$$\n\\xymatrix{\n\\colim_i \\Mor(T_i, X) \\ar[r] \\ar[d] & \\Mor(T, X) \\ar[d] \\\\\n\\colim_i \\Mor(T_i, Y) \\ar[r] & \\Mor(T, Y)\n}\n$$\nis a fibre product diagram.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Limits of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CP4","source_file":"spaces-limits.tex","source_line":806,"source_end_line":824,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L806-L824","statement_sha256":"2dd8032077d39b05d743ed2697a7ce5194a427a68aaf486946defff7b24b7848","origin":"The Stacks Project","memory_eligible":false,"source_rank":11416,"rank":11416,"depth":54,"x":667.821,"y":1643.487,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUJ","tag":"0CUJ","title":"Descending properties · Lemma 0CUJ","summary":"Let S be a scheme. Let X = lim_i ∈ I X_i be the limit of a directed inverse system of algebraic spaces over S with affine transition morphisms (Lemma [Tag 07SF]). If each X_i is decent (for example quasi-separated or locally separated) then |X| = lim_i |X_i| as sets.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed\ninverse system of algebraic spaces over $S$ with affine transition morphisms\n(Lemma \\ref{lemma-directed-inverse-system-has-limit}). If each $X_i$\nis decent (for example quasi-separated or locally separated)\nthen $|X| = \\lim_i |X_i|$ as sets.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUJ","source_file":"spaces-limits.tex","source_line":875,"source_end_line":882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L875-L882","statement_sha256":"a4418b46f58179f7c4a60a6608a78812031e990856d96f04eaf01dd7d754462c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11417,"rank":11417,"depth":57,"x":653.874,"y":1560.82,"cluster":"geometry-of-spaces"},{"id":"stacks:086V","tag":"086V","title":"Descending properties · Lemma 086V","summary":"With same notation and assumptions as in Lemma [Tag 0CUJ] we have |X| = lim_i |X_i| as topological spaces.","statement_latex":"With same notation and assumptions as in Lemma \\ref{lemma-inverse-limit-sets}\nwe have $|X| = \\lim_i |X_i|$ as topological spaces.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086V","source_file":"spaces-limits.tex","source_line":938,"source_end_line":942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L938-L942","statement_sha256":"5996b7d03c8ce97966f6c043e64069338ca92eed5b03bb4868e3b0a3db2f714e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11418,"rank":11418,"depth":58,"x":731.034,"y":1614.138,"cluster":"geometry-of-spaces"},{"id":"stacks:086W","tag":"086W","title":"Descending properties · Lemma 086W","summary":"Let S be a scheme. Let X = lim_i ∈ I X_i be the limit of a directed inverse system of algebraic spaces over S with affine transition morphisms (Lemma [Tag 07SF]). If each X_i is quasi-compact and nonempty, then |X| is nonempty.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed\ninverse system of algebraic spaces over $S$ with affine transition morphisms\n(Lemma \\ref{lemma-directed-inverse-system-has-limit}). If each $X_i$\nis quasi-compact and nonempty, then $|X|$ is nonempty.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086W","source_file":"spaces-limits.tex","source_line":971,"source_end_line":977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L971-L977","statement_sha256":"cddf7db299f4571cb20bacdb18c47b26d57d4a08c11b77f2150454fd858a1cd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11419,"rank":11419,"depth":53,"x":630.749,"y":1618.627,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUK","tag":"0CUK","title":"Descending properties · Lemma 0CUK","summary":"Let S be a scheme. Let X = lim_i ∈ I X_i be the limit of a directed inverse system of algebraic spaces over S with affine transition morphisms (Lemma [Tag 07SF]). Let x ∈ |X| with images x_i ∈ |X_i|. If each X_i is decent, then overline(x) = lim_i overline(x_i) as sets and as algebraic spaces if endowed with reduced induced scheme structure.","statement_latex":"Let $S$ be a scheme. Let $X = \\lim_{i \\in I} X_i$ be the limit of a directed\ninverse system of algebraic spaces over $S$ with affine transition morphisms\n(Lemma \\ref{lemma-directed-inverse-system-has-limit}).\nLet $x \\in |X|$ with images $x_i \\in |X_i|$. If each $X_i$ is decent,\nthen $\\overline{\\{x\\}} = \\lim_i \\overline{\\{x_i\\}}$ as sets\nand as algebraic spaces if endowed with reduced induced scheme structure.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUK","source_file":"spaces-limits.tex","source_line":988,"source_end_line":996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L988-L996","statement_sha256":"75d7fbbba65f9d0e2575d0da5093bf3ad230ec39dc8f8457d3b5085281bd1145","origin":"The Stacks Project","memory_eligible":false,"source_rank":11420,"rank":11420,"depth":59,"x":701.445,"y":1558.109,"cluster":"geometry-of-spaces"},{"id":"stacks:07SI","tag":"07SI","title":"Descending properties · Lemma 07SI","summary":"Notation and assumptions as in Situation [Tag 084R]. Suppose that F_0 is a quasi-coherent sheaf on X_0. Set F_i = f_0i^*F_0 for i ≥ 0 and set F = f_0^*F_0. Then Γ(X, F) = colim_i ≥ 0 Γ(X_i, F_i)","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent}.\nSuppose that $\\mathcal{F}_0$ is a quasi-coherent sheaf on $X_0$.\nSet $\\mathcal{F}_i = f_{0i}^*\\mathcal{F}_0$ for $i \\geq 0$ and set\n$\\mathcal{F} = f_0^*\\mathcal{F}_0$. Then\n$$\n\\Gamma(X, \\mathcal{F}) = \\colim_{i \\geq 0} \\Gamma(X_i, \\mathcal{F}_i)\n$$","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SI","source_file":"spaces-limits.tex","source_line":1041,"source_end_line":1050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1041-L1050","statement_sha256":"a18d56ca6744e8cdd5016ecc8b62de04a06647b99ac385ff398392c84683dc1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11421,"rank":11421,"depth":53,"x":697.964,"y":1643.274,"cluster":"geometry-of-spaces"},{"id":"stacks:0827","tag":"0827","title":"Descending properties · Lemma 0827","summary":"Notation and assumptions as in Situation [Tag 084R]. For any quasi-compact open subspace U ⊂ X there exists an i and a quasi-compact open U_i ⊂ X_i whose inverse image in X is U.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent}.\nFor any quasi-compact open subspace $U \\subset X$ there exists an $i$\nand a quasi-compact open $U_i \\subset X_i$ whose inverse image in $X$ is $U$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0827","source_file":"spaces-limits.tex","source_line":1077,"source_end_line":1082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1077-L1082","statement_sha256":"54d7fc6fecdc7de6a417dd04e762ac3ebb65df9403b4c234721e192d9b2ce485","origin":"The Stacks Project","memory_eligible":false,"source_rank":11422,"rank":11422,"depth":53,"x":631.717,"y":1578.176,"cluster":"geometry-of-spaces"},{"id":"stacks:084S","tag":"084S","title":"Descending properties · Lemma 084S","summary":"Notation and assumptions as in Situation [Tag 084R]. Let f_0 : Y_0 → Z_0 be a morphism of algebraic spaces over X_0. Assume (a) Y_0 → X_0 and Z_0 → X_0 are representable, (b) Y_0, Z_0 quasi-compact and quasi-separated, (c) f_0 locally of finite presentation, and (d) Y_0 ×_X_0 X → Z_0 ×_X_0 X an isomorphism. Then there exists an i ≥ 0 such that Y_0 ×_X_0 X_i → Z_0 ×_X_0 X_i is an isomorphism.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent}.\nLet $f_0 : Y_0 \\to Z_0$ be a morphism of algebraic spaces over $X_0$.\nAssume (a) $Y_0 \\to X_0$ and $Z_0 \\to X_0$ are representable, (b)\n$Y_0$, $Z_0$ quasi-compact and quasi-separated, (c)\n$f_0$ locally of finite presentation, and\n(d) $Y_0 \\times_{X_0} X \\to Z_0 \\times_{X_0} X$ an isomorphism.\nThen there exists an $i \\geq 0$ such that\n$Y_0 \\times_{X_0} X_i \\to Z_0 \\times_{X_0} X_i$ is an isomorphism.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084S","source_file":"spaces-limits.tex","source_line":1095,"source_end_line":1105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1095-L1105","statement_sha256":"81470b1b26f914e156a477e7f472dbda94e9585ebb436c44f04a2da0f4f0cc30","origin":"The Stacks Project","memory_eligible":false,"source_rank":11423,"rank":11423,"depth":41,"x":733.413,"y":1588.64,"cluster":"geometry-of-spaces"},{"id":"stacks:084T","tag":"084T","title":"Descending properties · Lemma 084T","summary":"Notation and assumptions as in Situation [Tag 084R]. If X is separated, then X_i is separated for some i ∈ I.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent}.\nIf $X$ is separated, then $X_i$ is separated for some $i \\in I$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084T","source_file":"spaces-limits.tex","source_line":1118,"source_end_line":1122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1118-L1122","statement_sha256":"5683554fb37badc3a786e7f9ed23eb45911bfc087183f036acce5bf70d6e5b12","origin":"The Stacks Project","memory_eligible":false,"source_rank":11424,"rank":11424,"depth":27,"x":649.603,"y":1638.872,"cluster":"geometry-of-spaces"},{"id":"stacks:07SQ","tag":"07SQ","title":"Descending properties · Lemma 07SQ","summary":"Notation and assumptions as in Situation [Tag 084R]. If X is affine, then there exists an i such that X_i is affine.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent}.\nIf $X$ is affine, then there exists an $i$ such that $X_i$ is affine.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SQ","source_file":"spaces-limits.tex","source_line":1152,"source_end_line":1156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1152-L1156","statement_sha256":"bb071d48bd9217f156e17006b9ce0479cbc09dacf65360dc4f9c78ed953cb503","origin":"The Stacks Project","memory_eligible":false,"source_rank":11425,"rank":11425,"depth":54,"x":671.112,"y":1553.868,"cluster":"geometry-of-spaces"},{"id":"stacks:07SR","tag":"07SR","title":"Descending properties · Lemma 07SR","summary":"Notation and assumptions as in Situation [Tag 084R]. If X is a scheme, then there exists an i such that X_i is a scheme.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent}.\nIf $X$ is a scheme, then there exists an $i$ such that $X_i$ is a scheme.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SR","source_file":"spaces-limits.tex","source_line":1217,"source_end_line":1221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1217-L1221","statement_sha256":"e24e719056f07027f611d4f4693ebc496a6904e1b2555fc73f85a3dcfaab3e68","origin":"The Stacks Project","memory_eligible":false,"source_rank":11426,"rank":11426,"depth":55,"x":723.859,"y":1629.112,"cluster":"geometry-of-spaces"},{"id":"stacks:0828","tag":"0828","title":"Descending properties · Lemma 0828","summary":"Let S be a scheme. Let B be an algebraic space over S. Let X = lim X_i be a directed limit of algebraic spaces over B with affine transition morphisms. Let Y → X be a morphism of algebraic spaces over B. • If Y → X is a closed immersion, X_i quasi-compact, and Y → B locally of finite type, then Y → X_i is a closed immersion for i large enough. • If Y → X is an immersion, X_i quasi-separated, Y → B locally of finite type, and Y quasi-compact, then Y → X_i is an immersion…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $X = \\lim X_i$ be a directed limit of\nalgebraic spaces over $B$ with affine transition morphisms.\nLet $Y \\to X$ be a morphism of algebraic spaces over $B$.\n\\begin{enumerate}\n\\item If $Y \\to X$ is a closed immersion, $X_i$ quasi-compact, and\n$Y \\to B$ locally of finite type, then $Y \\to X_i$ is a closed immersion\nfor $i$ large enough.\n\\item If $Y \\to X$ is an immersion, $X_i$ quasi-separated, $Y \\to B$\nlocally of finite type, and $Y$ quasi-compact, then $Y \\to X_i$ is an\nimmersion for $i$ large enough.\n\\item If $Y \\to X$ is an isomorphism, $X_i$ quasi-compact,\n$X_i \\to B$ locally of finite type, the transition morphisms\n$X_{i'} \\to X_i$ are closed immersions, and $Y \\to B$ is locally\nof finite presentation, then $Y \\to X_i$ is an isomorphism for $i$\nlarge enough.\n\\item If $Y \\to X$ is a monomorphism, $X_i$ quasi-separated,\n$Y \\to B$ locally of finite type, and $Y$ quasi-compact, then\n$Y \\to X_i$ is a monomorphism for $i$ large enough.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0828","source_file":"spaces-limits.tex","source_line":1234,"source_end_line":1256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1234-L1256","statement_sha256":"0f63d2d85a5f68ec2f27a1f459ee3f3840a04eb6bfe63ac4ceccd5d3f02dd788","origin":"The Stacks Project","memory_eligible":false,"source_rank":11427,"rank":11427,"depth":54,"x":623.986,"y":1603.436,"cluster":"geometry-of-spaces"},{"id":"stacks:086X","tag":"086X","title":"Descending properties · Lemma 086X","summary":"Let S be a scheme. Let Y be an algebraic space over S. Let X = lim X_i be a directed limit of algebraic spaces over Y with affine transition morphisms. Assume • Y is quasi-separated, • X_i is quasi-compact and quasi-separated, • the morphism X → Y is separated. Then X_i → Y is separated for all i large enough.","statement_latex":"Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$.\nLet $X = \\lim X_i$ be a directed limit of algebraic spaces over $Y$\nwith affine transition morphisms. Assume\n\\begin{enumerate}\n\\item $Y$ is quasi-separated,\n\\item $X_i$ is quasi-compact and quasi-separated,\n\\item the morphism $X \\to Y$ is separated.\n\\end{enumerate}\nThen $X_i \\to Y$ is separated for all $i$ large enough.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086X","source_file":"spaces-limits.tex","source_line":1323,"source_end_line":1334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1323-L1334","statement_sha256":"eba684e4128319726e902c67efaf866f2ade40ea518126614d50e371d985a10c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11428,"rank":11428,"depth":28,"x":718.72,"y":1565.522,"cluster":"geometry-of-spaces"},{"id":"stacks:0A0R","tag":"0A0R","title":"Descending properties · Lemma 0A0R","summary":"Let S be a scheme. Let Y be an algebraic space over S. Let X = lim X_i be a directed limit of algebraic spaces over Y with affine transition morphisms. Assume • Y quasi-compact and quasi-separated, • X_i quasi-compact and quasi-separated, • X → Y affine. Then X_i → Y is affine for i large enough.","statement_latex":"Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$.\nLet $X = \\lim X_i$ be a directed limit of algebraic spaces over $Y$\nwith affine transition morphisms. Assume\n\\begin{enumerate}\n\\item $Y$ quasi-compact and quasi-separated,\n\\item $X_i$ quasi-compact and quasi-separated,\n\\item $X \\to Y$ affine.\n\\end{enumerate}\nThen $X_i \\to Y$ is affine for $i$ large enough.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0R","source_file":"spaces-limits.tex","source_line":1353,"source_end_line":1364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1353-L1364","statement_sha256":"ff3657d38e6b0d2229f2d2d948ee64a3b89319e050c6d64b2b445bfd4d56617c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11429,"rank":11429,"depth":55,"x":679.17,"y":1647.613,"cluster":"geometry-of-spaces"},{"id":"stacks:0A0S","tag":"0A0S","title":"Descending properties · Lemma 0A0S","summary":"Let S be a scheme. Let Y be an algebraic space over S. Let X = lim X_i be a directed limit of algebraic spaces over Y with affine transition morphisms. Assume • Y quasi-compact and quasi-separated, • X_i quasi-compact and quasi-separated, • the transition morphisms X_i' → X_i are finite, • X_i → Y locally of finite type • X → Y integral. Then X_i → Y is finite for i large enough.","statement_latex":"Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$.\nLet $X = \\lim X_i$ be a directed limit of algebraic spaces\nover $Y$ with affine transition morphisms. Assume\n\\begin{enumerate}\n\\item $Y$ quasi-compact and quasi-separated,\n\\item $X_i$ quasi-compact and quasi-separated,\n\\item the transition morphisms $X_{i'} \\to X_i$ are finite,\n\\item $X_i \\to Y$ locally of finite type\n\\item $X \\to Y$ integral.\n\\end{enumerate}\nThen $X_i \\to Y$ is finite for $i$ large enough.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0S","source_file":"spaces-limits.tex","source_line":1374,"source_end_line":1387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1374-L1387","statement_sha256":"81af509b8e9fe2e412953c139d5417d942f0340ac91a2c0a545a4fa65beba424","origin":"The Stacks Project","memory_eligible":false,"source_rank":11430,"rank":11430,"depth":56,"x":642.15,"y":1564.257,"cluster":"geometry-of-spaces"},{"id":"stacks:0A0T","tag":"0A0T","title":"Descending properties · Lemma 0A0T","summary":"Let S be a scheme. Let Y be an algebraic space over S. Let X = lim X_i be a directed limit of algebraic spaces over Y with affine transition morphisms. Assume • Y quasi-compact and quasi-separated, • X_i quasi-compact and quasi-separated, • the transition morphisms X_i' → X_i are closed immersions, • X_i → Y locally of finite type • X → Y is a closed immersion. Then X_i → Y is a closed immersion for i large enough.","statement_latex":"Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$.\nLet $X = \\lim X_i$ be a directed limit of algebraic spaces\nover $Y$ with affine transition morphisms. Assume\n\\begin{enumerate}\n\\item $Y$ quasi-compact and quasi-separated,\n\\item $X_i$ quasi-compact and quasi-separated,\n\\item the transition morphisms $X_{i'} \\to X_i$ are closed immersions,\n\\item $X_i \\to Y$ locally of finite type\n\\item $X \\to Y$ is a closed immersion.\n\\end{enumerate}\nThen $X_i \\to Y$ is a closed immersion for $i$ large enough.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0T","source_file":"spaces-limits.tex","source_line":1399,"source_end_line":1412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1399-L1412","statement_sha256":"0783fe5ac26c682442a02bf5c33051e7b169a967f3a5479c5c74877a0f77ffb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11431,"rank":11431,"depth":56,"x":736.91,"y":1604.901,"cluster":"geometry-of-spaces"},{"id":"stacks:07SL","tag":"07SL","title":"Descending properties of morphisms · Lemma 07SL","summary":"With notation and assumptions as in Situation [Tag 084W]. If • f is étale, • f_0 is locally of finite presentation, then f_i is étale for some i ≥ 0.","statement_latex":"With notation and assumptions as in\nSituation \\ref{situation-descent-property}. If\n\\begin{enumerate}\n\\item $f$ is \\'etale,\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is \\'etale for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SL","source_file":"spaces-limits.tex","source_line":1459,"source_end_line":1468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1459-L1468","statement_sha256":"c563a276fc0cb2881e93be397dde326f749d1f4b05a0f65ee68e80bbb8b1489c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11432,"rank":11432,"depth":47,"x":633.888,"y":1628.807,"cluster":"geometry-of-spaces"},{"id":"stacks:0CN2","tag":"0CN2","title":"Descending properties of morphisms · Lemma 0CN2","summary":"With notation and assumptions as in Situation [Tag 084W]. If • f is smooth, • f_0 is locally of finite presentation, then f_i is smooth for some i ≥ 0.","statement_latex":"With notation and assumptions as in\nSituation \\ref{situation-descent-property}. If\n\\begin{enumerate}\n\\item $f$ is smooth,\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is smooth for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CN2","source_file":"spaces-limits.tex","source_line":1509,"source_end_line":1518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1509-L1518","statement_sha256":"910472e831bb1f5fbd96542af14a28af3bda87c8986fb6e94c1b11169e95f1b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11433,"rank":11433,"depth":47,"x":690.885,"y":1552.382,"cluster":"geometry-of-spaces"},{"id":"stacks:07SN","tag":"07SN","title":"Descending properties of morphisms · Lemma 07SN","summary":"With notation and assumptions as in Situation [Tag 084W]. If • f is surjective, • f_0 is locally of finite presentation, then f_i is surjective for some i ≥ 0.","statement_latex":"With notation and assumptions as in\nSituation \\ref{situation-descent-property}. If\n\\begin{enumerate}\n\\item $f$ is surjective,\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is surjective for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SN","source_file":"spaces-limits.tex","source_line":1558,"source_end_line":1567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1558-L1567","statement_sha256":"bf8641b7a2009748800cde11b59170e8e1c63110917d57163bf38138c3ede00c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11434,"rank":11434,"depth":25,"x":710.4,"y":1641.471,"cluster":"geometry-of-spaces"},{"id":"stacks:084X","tag":"084X","title":"Descending properties of morphisms · Lemma 084X","summary":"Notation and assumptions as in Situation [Tag 084W]. If • f is universally injective, • f_0 is locally of finite type, then f_i is universally injective for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}. If\n\\begin{enumerate}\n\\item $f$ is universally injective,\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen $f_i$ is universally injective for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084X","source_file":"spaces-limits.tex","source_line":1608,"source_end_line":1616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1608-L1616","statement_sha256":"ea4a4bdd59ff18bd51c96cfb659012ea5d451c61dfffa054a67114011e6679fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11435,"rank":11435,"depth":55,"x":623.989,"y":1586.611,"cluster":"geometry-of-spaces"},{"id":"stacks:084Y","tag":"084Y","title":"Descending properties of morphisms · Lemma 084Y","summary":"Notation and assumptions as in Situation [Tag 084W]. If f is affine, then f_i is affine for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}. If\n$f$ is affine, then $f_i$ is affine for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084Y","source_file":"spaces-limits.tex","source_line":1631,"source_end_line":1635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1631-L1635","statement_sha256":"e65e6e98552eff2b4bb3b68296071defc9f091e219ce6d41e2551741ff041615","origin":"The Stacks Project","memory_eligible":false,"source_rank":11436,"rank":11436,"depth":55,"x":732.294,"y":1577.996,"cluster":"geometry-of-spaces"},{"id":"stacks:084Z","tag":"084Z","title":"Descending properties of morphisms · Lemma 084Z","summary":"Notation and assumptions as in Situation [Tag 084W]. If • f is finite, • f_0 is locally of finite type, then f_i is finite for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}. If\n\\begin{enumerate}\n\\item $f$ is finite,\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen $f_i$ is finite for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084Z","source_file":"spaces-limits.tex","source_line":1647,"source_end_line":1655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1647-L1655","statement_sha256":"cd854414c216cfcd8a9b96c9a8d0f3131c00ded494abfd0de243ac9af792bf5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11437,"rank":11437,"depth":55,"x":659.044,"y":1646.096,"cluster":"geometry-of-spaces"},{"id":"stacks:0850","tag":"0850","title":"Descending properties of morphisms · Lemma 0850","summary":"Notation and assumptions as in Situation [Tag 084W]. If • f is a closed immersion, • f_0 is locally of finite type, then f_i is a closed immersion for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}. If\n\\begin{enumerate}\n\\item $f$ is a closed immersion,\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen $f_i$ is a closed immersion for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0850","source_file":"spaces-limits.tex","source_line":1669,"source_end_line":1677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1669-L1677","statement_sha256":"9bf16797df2a63ecfe14e741e96ebd61f6665bafb2bcb595c329fc164c1a644e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11438,"rank":11438,"depth":55,"x":658.291,"y":1553.923,"cluster":"geometry-of-spaces"},{"id":"stacks:0851","tag":"0851","title":"Descending properties of morphisms · Lemma 0851","summary":"Notation and assumptions as in Situation [Tag 084W]. If f is separated, then f_i is separated for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nIf $f$ is separated, then $f_i$ is separated for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0851","source_file":"spaces-limits.tex","source_line":1693,"source_end_line":1697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1693-L1697","statement_sha256":"b00e7306b8ca4d76165dc43c10c3be901d45c25cde97a7d697c3705d91e6123b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11439,"rank":11439,"depth":56,"x":733.285,"y":1621.75,"cluster":"geometry-of-spaces"},{"id":"stacks:0852","tag":"0852","title":"Descending properties of morphisms · Lemma 0852","summary":"Notation and assumptions as in Situation [Tag 084W]. If • f is an isomorphism, • f_0 is locally of finite presentation, then f_i is an isomorphism for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}. If\n\\begin{enumerate}\n\\item $f$ is an isomorphism,\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $f_i$ is an isomorphism for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0852","source_file":"spaces-limits.tex","source_line":1707,"source_end_line":1715,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1707-L1715","statement_sha256":"5cd5f1ea30efa59e3e25933cb1384510f87d6a5dd21b24ca75820c3c23b0025e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11440,"rank":11440,"depth":56,"x":622.98,"y":1614.257,"cluster":"geometry-of-spaces"},{"id":"stacks:07SM","tag":"07SM","title":"Descending properties of morphisms · Lemma 07SM","summary":"Notation and assumptions as in Situation [Tag 084W]. If • f is a monomorphism, • f_0 is locally of finite type, then f_i is a monomorphism for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}. If\n\\begin{enumerate}\n\\item $f$ is a monomorphism,\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen $f_i$ is a monomorphism for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SM","source_file":"spaces-limits.tex","source_line":1728,"source_end_line":1736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1728-L1736","statement_sha256":"20083a0f15bb2f1dfd68169333c8540cdef90a7e0d363d317238dcbf8bbb09ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":11441,"rank":11441,"depth":57,"x":710.708,"y":1556.955,"cluster":"geometry-of-spaces"},{"id":"stacks:08K0","tag":"08K0","title":"Descending properties of morphisms · Lemma 08K0","summary":"Notation and assumptions as in Situation [Tag 084W]. Let F_0 be a quasi-coherent O_X_0-module and denote F_i the pullback to X_i and F the pullback to X. If • F is flat over Y, • F_0 is of finite presentation, and • f_0 is locally of finite presentation, then F_i is flat over Y_i for some i ≥ 0. In particular, if f_0 is locally of finite presentation and f is flat, then f_i is flat for some i ≥ 0.","statement_latex":"Notation and assumptions as in Situation \\ref{situation-descent-property}.\nLet $\\mathcal{F}_0$ be a quasi-coherent $\\mathcal{O}_{X_0}$-module\nand denote $\\mathcal{F}_i$ the pullback to $X_i$ and $\\mathcal{F}$\nthe pullback to $X$. If\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat over $Y$,\n\\item $\\mathcal{F}_0$ is of finite presentation, and\n\\item $f_0$ is locally of finite presentation,\n\\end{enumerate}\nthen $\\mathcal{F}_i$ is flat over $Y_i$ for some $i \\geq 0$.\nIn particular, if $f_0$ is locally of finite presentation and\n$f$ is flat, then $f_i$ is flat for some $i \\geq 0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08K0","source_file":"spaces-limits.tex","source_line":1751,"source_end_line":1765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1751-L1765","statement_sha256":"7b4f71b4eb40808bd92a26e660da5959052b52c6b590d9ea6e8ac10d3160f8bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11442,"rank":11442,"depth":48,"x":692.022,"y":1649.368,"cluster":"geometry-of-spaces"},{"id":"stacks:08K1","tag":"08K1","title":"Descending properties of morphisms · Lemma 08K1","summary":"Assumptions and notation as in Situation [Tag 084W]. If • f is proper, and • f_0 is locally of finite type, then there exists an i such that f_i is proper.","statement_latex":"Assumptions and notation as in Situation \\ref{situation-descent-property}.\nIf\n\\begin{enumerate}\n\\item $f$ is proper, and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen there exists an $i$ such that $f_i$ is proper.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08K1","source_file":"spaces-limits.tex","source_line":1805,"source_end_line":1814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1805-L1814","statement_sha256":"7ecb3857300076caf33d686fb4bc302eb8d31616c048a502017636360d7a0350","origin":"The Stacks Project","memory_eligible":false,"source_rank":11443,"rank":11443,"depth":58,"x":631.237,"y":1570.298,"cluster":"geometry-of-spaces"},{"id":"stacks:0D4K","tag":"0D4K","title":"Descending properties of morphisms · Lemma 0D4K","summary":"Assumptions and notation as in Situation [Tag 084W]. Let d ≥ 0. If • f has relative dimension ≤ d (Morphisms of Spaces, Definition [Tag 06LR]), and • f_0 is locally of finite type, then there exists an i such that f_i has relative dimension ≤ d.","statement_latex":"Assumptions and notation as in Situation \\ref{situation-descent-property}.\nLet $d \\geq 0$. If\n\\begin{enumerate}\n\\item $f$ has relative dimension $\\leq d$\n(Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-relative-dimension}), and\n\\item $f_0$ is locally of finite type,\n\\end{enumerate}\nthen there exists an $i$ such that $f_i$ has relative dimension $\\leq d$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4K","source_file":"spaces-limits.tex","source_line":1855,"source_end_line":1866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1855-L1866","statement_sha256":"57d128d579e86a8bb17fba76f53d3081ab02f57338cb14ca06dce23edf31cfc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11444,"rank":11444,"depth":48,"x":740.086,"y":1594.209,"cluster":"geometry-of-spaces"},{"id":"stacks:07SK","tag":"07SK","title":"Descending relative objects · Lemma 07SK","summary":"Let S be a scheme. Let I be a directed set. Let (X_i, f_ii') be an inverse system over I of algebraic spaces over S. Assume • the morphisms f_ii' : X_i → X_i' are affine, • the spaces X_i are quasi-compact and quasi-separated. Let X = lim_i X_i. Then the category of algebraic spaces of finite presentation over X is the colimit over I of the categories of algebraic spaces of finite presentation over X_i.","statement_latex":"Let $S$ be a scheme. Let $I$ be a directed set.\nLet $(X_i, f_{ii'})$ be an inverse system over $I$ of algebraic spaces\nover $S$. Assume\n\\begin{enumerate}\n\\item the morphisms $f_{ii'} : X_i \\to X_{i'}$ are affine,\n\\item the spaces $X_i$ are quasi-compact and quasi-separated.\n\\end{enumerate}\nLet $X = \\lim_i X_i$. Then the category of algebraic spaces\nof finite presentation over $X$ is the colimit over $I$ of the\ncategories of algebraic spaces of finite presentation over $X_i$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending relative objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SK","source_file":"spaces-limits.tex","source_line":1930,"source_end_line":1942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L1930-L1942","statement_sha256":"92de5f8880f68104c3c2fd1c8adc704c0157dae105fc724c08e7e65554537863","origin":"The Stacks Project","memory_eligible":false,"source_rank":11445,"rank":11445,"depth":53,"x":640.191,"y":1638.518,"cluster":"geometry-of-spaces"},{"id":"stacks:07V7","tag":"07V7","title":"Descending relative objects · Lemma 07V7","summary":"With notation and assumptions as in Lemma [Tag 07SK]. The category of O_X-modules of finite presentation is the colimit over I of the categories O_X_i-modules of finite presentation.","statement_latex":"With notation and assumptions as in\nLemma \\ref{lemma-descend-finite-presentation}.\nThe category of $\\mathcal{O}_X$-modules of finite presentation is the\ncolimit over $I$ of the categories $\\mathcal{O}_{X_i}$-modules of finite\npresentation.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending relative objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07V7","source_file":"spaces-limits.tex","source_line":2012,"source_end_line":2019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2012-L2019","statement_sha256":"3c94df1e8ce080b8fd6ca6d263c53903e2a5c43cb8d3a58c74349c1b5f996725","origin":"The Stacks Project","memory_eligible":false,"source_rank":11446,"rank":11446,"depth":54,"x":678.373,"y":1548.805,"cluster":"geometry-of-spaces"},{"id":"stacks:0D2X","tag":"0D2X","title":"Descending relative objects · Lemma 0D2X","summary":"With notation and assumptions as in Lemma [Tag 07SK]. Then • any finite locally free O_X-module is the pullback of a finite locally free O_X_i-module for some i, • any invertible O_X-module is the pullback of an invertible O_X_i-module for some i.","statement_latex":"With notation and assumptions as in\nLemma \\ref{lemma-descend-finite-presentation}. Then\n\\begin{enumerate}\n\\item any finite locally free $\\mathcal{O}_X$-module is the pullback\nof a finite locally free $\\mathcal{O}_{X_i}$-module for some $i$,\n\\item any invertible $\\mathcal{O}_X$-module is the pullback of an invertible\n$\\mathcal{O}_{X_i}$-module for some $i$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending relative objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2X","source_file":"spaces-limits.tex","source_line":2047,"source_end_line":2057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2047-L2057","statement_sha256":"650a05d65e506d629e813c03b8b003c04217f5a63dbd173c76938c3eeed75b42","origin":"The Stacks Project","memory_eligible":false,"source_rank":11447,"rank":11447,"depth":55,"x":722.537,"y":1636.972,"cluster":"geometry-of-spaces"},{"id":"stacks:07SU","tag":"07SU","title":"Absolute Noetherian approximation · Proposition 07SU","summary":"Our proof follows closely the proof given in [CLO]. Let X be a quasi-compact and quasi-separated algebraic space over Spec(Z). There exist a directed set I and an inverse system of algebraic spaces (X_i, f_ii') over I such that • the transition morphisms f_ii' are affine • each X_i is quasi-separated and of finite type over Z, and • X = lim X_i.","statement_latex":"\\begin{reference}\nOur proof follows closely the proof given in \\cite[Theorem 1.2.2]{CLO}.\n\\end{reference}\nLet $X$ be a quasi-compact and quasi-separated algebraic space over\n$\\Spec(\\mathbf{Z})$. There exist a directed set $I$\nand an inverse system of algebraic spaces $(X_i, f_{ii'})$ over $I$\nsuch that\n\\begin{enumerate}\n\\item the transition morphisms $f_{ii'}$ are affine\n\\item each $X_i$ is quasi-separated and of finite type over\n$\\mathbf{Z}$, and\n\\item $X = \\lim X_i$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Absolute Noetherian approximation","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SU","source_file":"spaces-limits.tex","source_line":2110,"source_end_line":2125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2110-L2125","statement_sha256":"a2ddddc96d6d34093f395840016d5e3c67aef29e5c078b0c8851fa4fff37e4c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11448,"rank":11448,"depth":59,"x":618.662,"y":1596.861,"cluster":"geometry-of-spaces"},{"id":"stacks:07V9","tag":"07V9","title":"Applications · Lemma 07V9","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Every quasi-coherent O_X-module is a filtered colimit of finitely presented O_X-modules.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic\nspace over $S$. Every quasi-coherent $\\mathcal{O}_X$-module is a\nfiltered colimit of finitely presented $\\mathcal{O}_X$-modules.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07V9","source_file":"spaces-limits.tex","source_line":2274,"source_end_line":2279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2274-L2279","statement_sha256":"f1b799af7642afa8d0196ebdfc48bec4343da5b4fd19dfb999924b61ef90f69d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11449,"rank":11449,"depth":60,"x":727.94,"y":1567.383,"cluster":"geometry-of-spaces"},{"id":"stacks:0829","tag":"0829","title":"Applications · Lemma 0829","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let F be a quasi-coherent O_X-module. Then F is the directed colimit of its finite type quasi-coherent submodules.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $\\mathcal{F}$ is the directed colimit of its finite type\nquasi-coherent submodules.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0829","source_file":"spaces-limits.tex","source_line":2301,"source_end_line":2308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2301-L2308","statement_sha256":"4716b5b54c2aeeef584c8c54f4d564c8f437b6d7f333e542a903bc2c04c735b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11450,"rank":11450,"depth":61,"x":670.844,"y":1651.451,"cluster":"geometry-of-spaces"},{"id":"stacks:086Y","tag":"086Y","title":"Applications · Lemma 086Y","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let F be a finite type quasi-coherent O_X-module. Then we can write F = lim F_i where each F_i is an O_X-module of finite presentation and all transition maps F_i → F_i' surjective.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $\\mathcal{F}$ be a finite type\nquasi-coherent $\\mathcal{O}_X$-module. Then we can write\n$\\mathcal{F} = \\lim \\mathcal{F}_i$ where each $\\mathcal{F}_i$ is an\n$\\mathcal{O}_X$-module of finite presentation and all transition maps\n$\\mathcal{F}_i \\to \\mathcal{F}_{i'}$ surjective.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086Y","source_file":"spaces-limits.tex","source_line":2325,"source_end_line":2333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2325-L2333","statement_sha256":"32e81d92927d662c65bbd336fe8f0810b4a4ceba3ebba7665ff6be6ba397883b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11451,"rank":11451,"depth":62,"x":645.243,"y":1556.7,"cluster":"geometry-of-spaces"},{"id":"stacks:082A","tag":"082A","title":"Applications · Lemma 082A","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let A be a quasi-coherent O_X-algebra. Then A is a directed colimit of finitely presented quasi-coherent O_X-algebras.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$.\nLet $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra.\nThen $\\mathcal{A}$ is a directed colimit of finitely presented\nquasi-coherent $\\mathcal{O}_X$-algebras.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082A","source_file":"spaces-limits.tex","source_line":2362,"source_end_line":2369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2362-L2369","statement_sha256":"e2029070756863e38a7c68f0f4dc225127369c01546952c71aab28381a4bf01f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11452,"rank":11452,"depth":62,"x":740.679,"y":1612.253,"cluster":"geometry-of-spaces"},{"id":"stacks:082B","tag":"082B","title":"Applications · Lemma 082B","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let A be a quasi-coherent O_X-algebra. Then A is the directed colimit of its finite type quasi-coherent O_X-subalgebras.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic\nspace over $S$. Let $\\mathcal{A}$ be a quasi-coherent $\\mathcal{O}_X$-algebra.\nThen $\\mathcal{A}$ is the directed colimit of its finite type\nquasi-coherent $\\mathcal{O}_X$-subalgebras.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082B","source_file":"spaces-limits.tex","source_line":2400,"source_end_line":2406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2400-L2406","statement_sha256":"c92726a98f87cad3708a60946692df2b35e60b1347051324543717774bd0b13c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11453,"rank":11453,"depth":62,"x":625.196,"y":1625.492,"cluster":"geometry-of-spaces"},{"id":"stacks:086Z","tag":"086Z","title":"Applications · Lemma 086Z","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let A be a finite quasi-coherent O_X-algebra. Then A = colim A_i is a directed colimit of finite and finitely presented quasi-coherent O_X-algebras with surjective transition maps.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $\\mathcal{A}$ be a finite quasi-coherent\n$\\mathcal{O}_X$-algebra. Then $\\mathcal{A} = \\colim \\mathcal{A}_i$\nis a directed colimit of finite and finitely presented quasi-coherent\n$\\mathcal{O}_X$-algebras with surjective transition maps.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086Z","source_file":"spaces-limits.tex","source_line":2421,"source_end_line":2428,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2421-L2428","statement_sha256":"3bb8e09752f90c5279d8d990023822156c76c83962d46b6f4159e49a4f9694d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11454,"rank":11454,"depth":63,"x":699.986,"y":1549.919,"cluster":"geometry-of-spaces"},{"id":"stacks:082C","tag":"082C","title":"Applications · Lemma 082C","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let A be an integral quasi-coherent O_X-algebra. Then • A is the directed colimit of its finite quasi-coherent O_X-subalgebras, and • A is a directed colimit of finite and finitely presented O_X-algebras.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $\\mathcal{A}$ be an integral quasi-coherent\n$\\mathcal{O}_X$-algebra. Then\n\\begin{enumerate}\n\\item $\\mathcal{A}$ is the directed colimit of its finite\nquasi-coherent $\\mathcal{O}_X$-subalgebras, and\n\\item $\\mathcal{A}$ is a directed colimit of finite and finitely presented\n$\\mathcal{O}_X$-algebras.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082C","source_file":"spaces-limits.tex","source_line":2461,"source_end_line":2472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2461-L2472","statement_sha256":"50997fee4946e0b1a5a9d6b53ec9d5a2d83cd0362a61add01cc07b2e696aba85","origin":"The Stacks Project","memory_eligible":false,"source_rank":11455,"rank":11455,"depth":64,"x":705.631,"y":1648.449,"cluster":"geometry-of-spaces"},{"id":"stacks:0853","tag":"0853","title":"Applications · Lemma 0853","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let U ⊂ X be a quasi-compact open. Let F be a quasi-coherent O_X-module. Let G ⊂ F|_U be a quasi-coherent O_U-submodule which is of finite type. Then there exists a quasi-coherent submodule G' ⊂ F which is of finite type such that G'|_U = G.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $U \\subset X$ be a quasi-compact open.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{G} \\subset \\mathcal{F}|_U$ be a quasi-coherent\n$\\mathcal{O}_U$-submodule which is of finite type. Then\nthere exists a quasi-coherent submodule $\\mathcal{G}' \\subset \\mathcal{F}$\nwhich is of finite type such that $\\mathcal{G}'|_U = \\mathcal{G}$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0853","source_file":"spaces-limits.tex","source_line":2534,"source_end_line":2544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2534-L2544","statement_sha256":"8171518ebff1a34cfd82e06841e423d8695b6256c393e43f0270d5fac49ded13","origin":"The Stacks Project","memory_eligible":false,"source_rank":11456,"rank":11456,"depth":62,"x":621.925,"y":1578.74,"cluster":"geometry-of-spaces"},{"id":"stacks:0GS3","tag":"0GS3","title":"Relative approximation · Lemma 0GS3","summary":"Let f : X → Y be a morphism of quasi-compact and quasi-separated algebraic spaces over Z. Then there exists a direct set I and an inverse system (f_i : X_i → Y_i) of morphisms algebraic spaces over I, such that the transition morphisms X_i → X_i' and Y_i → Y_i' are affine, such that X_i and Y_i are quasi-separated and of finite type over Z, and such that (X → Y) = lim (X_i → Y_i).","statement_latex":"Let $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated\nalgebraic spaces over $\\mathbf{Z}$.\nThen there exists a direct set $I$ and an inverse system $(f_i : X_i \\to Y_i)$\nof morphisms algebraic spaces over $I$, such that the transition morphisms\n$X_i \\to X_{i'}$ and $Y_i \\to Y_{i'}$ are affine, such that $X_i$\nand $Y_i$ are quasi-separated and of finite type over $\\mathbf{Z}$, and\nsuch that $(X \\to Y) = \\lim (X_i \\to Y_i)$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Relative approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GS3","source_file":"spaces-limits.tex","source_line":2578,"source_end_line":2587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2578-L2587","statement_sha256":"1f8b7ab49b6ea09d239b61a73624ea35b0403415041ace8142a56465307f3146","origin":"The Stacks Project","memory_eligible":false,"source_rank":11457,"rank":11457,"depth":60,"x":740.14,"y":1582.661,"cluster":"geometry-of-spaces"},{"id":"stacks:09NS","tag":"09NS","title":"Relative approximation · Lemma 09NS","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that • X is quasi-compact and quasi-separated, and • Y is quasi-separated. Then X = lim X_i is a limit of a directed inverse system of algebraic spaces X_i of finite presentation over Y with affine transition morphisms over Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume that\n\\begin{enumerate}\n\\item $X$ is quasi-compact and quasi-separated, and\n\\item $Y$ is quasi-separated.\n\\end{enumerate}\nThen $X = \\lim X_i$ is a limit of a directed inverse system of algebraic spaces\n$X_i$ of finite presentation over $Y$ with affine transition morphisms\nover $Y$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Relative approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09NS","source_file":"spaces-limits.tex","source_line":2641,"source_end_line":2652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2641-L2652","statement_sha256":"14602238b0058ba6ba03458c0a95bc326279052bb32e0cf668e2853cb58c5da9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11458,"rank":11458,"depth":61,"x":649.487,"y":1647.081,"cluster":"geometry-of-spaces"},{"id":"stacks:0870","tag":"0870","title":"Finite type closed in finite presentation · Lemma 0870","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S. If Y quasi-compact and quasi-separated, then X is a directed limit X = lim X_i with each X_i affine and of finite presentation over Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic\nspaces over $S$. If $Y$ quasi-compact and\nquasi-separated, then $X$ is a directed limit $X = \\lim X_i$\nwith each $X_i$ affine and of finite presentation over $Y$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0870","source_file":"spaces-limits.tex","source_line":2699,"source_end_line":2705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2699-L2705","statement_sha256":"8d0042d20af62ae8ea376aa69ed9fea2f0d0bb79f3182e5cfafd6769298d4ed2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11459,"rank":11459,"depth":63,"x":664.583,"y":1547.782,"cluster":"geometry-of-spaces"},{"id":"stacks:09YA","tag":"09YA","title":"Finite type closed in finite presentation · Lemma 09YA","summary":"Let S be a scheme. Let f : X → Y be an integral morphism of algebraic spaces over S. Assume Y quasi-compact and quasi-separated. Then X can be written as a directed limit X = lim X_i where X_i are finite and of finite presentation over Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism of algebraic\nspaces over $S$. Assume $Y$ quasi-compact and quasi-separated.\nThen $X$ can be written as a directed limit $X = \\lim X_i$\nwhere $X_i$ are finite and of finite presentation over $Y$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YA","source_file":"spaces-limits.tex","source_line":2721,"source_end_line":2727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2721-L2727","statement_sha256":"3e8d038638e1952fff6c573578d35c83560353d489967e0ffb0b5023fd5cb592","origin":"The Stacks Project","memory_eligible":false,"source_rank":11460,"rank":11460,"depth":65,"x":733.552,"y":1629.863,"cluster":"geometry-of-spaces"},{"id":"stacks:07VR","tag":"07VR","title":"Finite type closed in finite presentation · Lemma 07VR","summary":"Let S be a scheme. Let f : X → Y be a finite morphism of algebraic spaces over S. Assume Y quasi-compact and quasi-separated. Then X can be written as a directed limit X = lim X_i where the transition maps are closed immersions and the objects X_i are finite and of finite presentation over Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a finite morphism of algebraic\nspaces over $S$. Assume $Y$ quasi-compact and quasi-separated.\nThen $X$ can be written as a directed limit $X = \\lim X_i$\nwhere the transition maps are closed immersions and the objects\n$X_i$ are finite and of finite presentation over $Y$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VR","source_file":"spaces-limits.tex","source_line":2743,"source_end_line":2750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2743-L2750","statement_sha256":"62976df1a4fad4cecafccf2759e07dd948d5da022ed2e778be5aca6c852f4dd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11461,"rank":11461,"depth":64,"x":616.271,"y":1608.395,"cluster":"geometry-of-spaces"},{"id":"stacks:0A0U","tag":"0A0U","title":"Finite type closed in finite presentation · Lemma 0A0U","summary":"Closed immersions of qcqs algebraic spaces can be approximated by finitely presented closed immersions. Let S be a scheme. Let f : X → Y be a closed immersion of algebraic spaces over S. Assume Y quasi-compact and quasi-separated. Then X can be written as a directed limit X = lim X_i where the transition maps are closed immersions and the morphisms X_i → Y are closed immersions of finite presentation.","statement_latex":"\\begin{slogan}\nClosed immersions of qcqs algebraic spaces can be approximated\nby finitely presented closed immersions.\n\\end{slogan}\nLet $S$ be a scheme. Let $f : X \\to Y$ be a closed immersion of algebraic\nspaces over $S$. Assume $Y$ quasi-compact and quasi-separated.\nThen $X$ can be written as a directed limit $X = \\lim X_i$\nwhere the transition maps are closed immersions and the morphisms\n$X_i \\to Y$ are closed immersions of finite presentation.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0U","source_file":"spaces-limits.tex","source_line":2766,"source_end_line":2777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2766-L2777","statement_sha256":"0a6dda68c263346bf3f11599275f6410a0ee15671c0521b139fd46780f72491c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11462,"rank":11462,"depth":62,"x":720.384,"y":1557.498,"cluster":"geometry-of-spaces"},{"id":"stacks:0871","tag":"0871","title":"Finite type closed in finite presentation · Lemma 0871","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • f is locally of finite type and quasi-affine, and • Y is quasi-compact and quasi-separated. Then there exists a morphism of finite presentation f' : X' → Y and a closed immersion X → X' over Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume\n\\begin{enumerate}\n\\item $f$ is locally of finite type and quasi-affine, and\n\\item $Y$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nThen there exists a morphism of finite presentation\n$f' : X' \\to Y$ and a closed immersion $X \\to X'$ over $Y$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0871","source_file":"spaces-limits.tex","source_line":2790,"source_end_line":2800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2790-L2800","statement_sha256":"c08f6f38dbd448c6a16d98c9202c1f0ad3b127cc615a7a9386a6093ac3e12bd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11463,"rank":11463,"depth":64,"x":684.416,"y":1654.445,"cluster":"geometry-of-spaces"},{"id":"stacks:0872","tag":"0872","title":"Finite type closed in finite presentation · Lemma 0872","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume: • f is of locally of finite type. • X is quasi-compact and quasi-separated, and • Y is quasi-compact and quasi-separated. Then there exists a morphism of finite presentation f' : X' → Y and a closed immersion X → X' of algebraic spaces over Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume:\n\\begin{enumerate}\n\\item $f$ is of locally of finite type.\n\\item $X$ is quasi-compact and quasi-separated, and\n\\item $Y$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nThen there exists a morphism of finite presentation\n$f' : X' \\to Y$ and a closed immersion $X \\to X'$ of\nalgebraic spaces over $Y$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0872","source_file":"spaces-limits.tex","source_line":2818,"source_end_line":2830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2818-L2830","statement_sha256":"53907243194f2572f4c4a810ac6e9b9c05ebbaea13daecd9e106a914aaae752c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11464,"rank":11464,"depth":65,"x":632.797,"y":1562.228,"cluster":"geometry-of-spaces"},{"id":"stacks:0873","tag":"0873","title":"Finite type closed in finite presentation · Proposition 0873","summary":"Let S be a scheme. f : X → Y be a morphism of algebraic spaces over S. Assume • f is of finite type and separated, and • Y is quasi-compact and quasi-separated. Then there exists a separated morphism of finite presentation f' : X' → Y and a closed immersion X → X' over Y.","statement_latex":"Let $S$ be a scheme. $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume\n\\begin{enumerate}\n\\item $f$ is of finite type and separated, and\n\\item $Y$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nThen there exists a separated morphism of finite presentation\n$f' : X' \\to Y$ and a closed immersion $X \\to X'$ over $Y$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite type closed in finite presentation","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0873","source_file":"spaces-limits.tex","source_line":2873,"source_end_line":2883,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2873-L2883","statement_sha256":"5938f1beb2a9b6c0aa7d9a03f052add69b1cdad133a689745b0071b0914f682d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11465,"rank":11465,"depth":66,"x":745.408,"y":1601.072,"cluster":"geometry-of-spaces"},{"id":"stacks:0A0W","tag":"0A0W","title":"Approximating proper morphisms · Lemma 0A0W","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of algebraic spaces over S with Y quasi-compact and quasi-separated. Then X = lim X_i is a directed limit of algebraic spaces X_i proper and of finite presentation over Y and with transition morphisms and morphisms X → X_i closed immersions.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic\nspaces over $S$ with $Y$ quasi-compact and quasi-separated. Then\n$X = \\lim X_i$ is a directed limit of algebraic spaces $X_i$\nproper and of finite presentation over $Y$ and with transition\nmorphisms and morphisms $X \\to X_i$ closed immersions.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Approximating proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0W","source_file":"spaces-limits.tex","source_line":2919,"source_end_line":2926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2919-L2926","statement_sha256":"b24ac211c6a630e45f9e38e3564fb3407676514f5c5b1e13ceedf970218a6983","origin":"The Stacks Project","memory_eligible":false,"source_rank":11466,"rank":11466,"depth":67,"x":630.739,"y":1636.448,"cluster":"geometry-of-spaces"},{"id":"stacks:0A0X","tag":"0A0X","title":"Approximating proper morphisms · Lemma 0A0X","summary":"Let f : X → Y be a proper morphism of algebraic spaces over Z with Y quasi-compact and quasi-separated. Then there exists a directed set I, an inverse system (f_i : X_i → Y_i) of morphisms of algebraic spaces over I, such that the transition morphisms X_i → X_i' and Y_i → Y_i' are affine, such that f_i is proper and of finite presentation, such that Y_i is of finite presentation over Z, and such that (X → Y) = lim (X_i → Y_i).","statement_latex":"Let $f : X \\to Y$ be a proper morphism of algebraic spaces over $\\mathbf{Z}$\nwith $Y$ quasi-compact and quasi-separated. Then there exists a directed\nset $I$, an inverse system $(f_i : X_i \\to Y_i)$ of morphisms of algebraic\nspaces over $I$, such that the transition morphisms $X_i \\to X_{i'}$\nand $Y_i \\to Y_{i'}$ are affine, such that $f_i$ is proper and of\nfinite presentation, such that $Y_i$ is of finite presentation over\n$\\mathbf{Z}$, and such that $(X \\to Y) = \\lim (X_i \\to Y_i)$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Approximating proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0X","source_file":"spaces-limits.tex","source_line":2967,"source_end_line":2976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L2967-L2976","statement_sha256":"ad4207bea4ccb8cc7e6fa0f868b9a34b6c1917ad1a333c80e8c7c366f04b7340","origin":"The Stacks Project","memory_eligible":false,"source_rank":11467,"rank":11467,"depth":68,"x":687.038,"y":1544.985,"cluster":"geometry-of-spaces"},{"id":"stacks:08K2","tag":"08K2","title":"Approximating proper morphisms · Lemma 08K2","summary":"Assumptions and notation as in Situation [Tag 084W]. Let F_0 be a quasi-coherent O_X_0-module. Denote F and F_i the pullbacks of F_0 to X and X_i. Assume • f_0 is locally of finite type, • F_0 is of finite type, • the scheme theoretic support of F is proper over Y. Then the scheme theoretic support of F_i is proper over Y_i for some i.","statement_latex":"Assumptions and notation as in Situation \\ref{situation-descent-property}.\nLet $\\mathcal{F}_0$ be a quasi-coherent $\\mathcal{O}_{X_0}$-module.\nDenote $\\mathcal{F}$ and $\\mathcal{F}_i$ the pullbacks of\n$\\mathcal{F}_0$ to $X$ and $X_i$. Assume\n\\begin{enumerate}\n\\item $f_0$ is locally of finite type,\n\\item $\\mathcal{F}_0$ is of finite type,\n\\item the scheme theoretic support of $\\mathcal{F}$ is proper over $Y$.\n\\end{enumerate}\nThen the scheme theoretic support of $\\mathcal{F}_i$ is proper over $Y_i$\nfor some $i$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Approximating proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08K2","source_file":"spaces-limits.tex","source_line":3006,"source_end_line":3019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3006-L3019","statement_sha256":"27edaacebf561f7ceb44250e69e5fdbbc500c3fb83b930eeea2791e809869c27","origin":"The Stacks Project","memory_eligible":false,"source_rank":11468,"rank":11468,"depth":59,"x":719.184,"y":1644.712,"cluster":"geometry-of-spaces"},{"id":"stacks:088L","tag":"088L","title":"Embedding into affine space · Lemma 088L","summary":"Let S be a scheme. Let f : U → X be a morphism of algebraic spaces over S. Assume U is an affine scheme, f is locally of finite type, and X quasi-separated and locally separated. Then there exists an immersion U → A^n_X over X.","statement_latex":"Let $S$ be a scheme. Let $f : U \\to X$ be a morphism of algebraic\nspaces over $S$. Assume $U$ is an affine scheme, $f$ is locally of\nfinite type, and $X$ quasi-separated and locally separated.\nThen there exists an immersion $U \\to \\mathbf{A}^n_X$ over $X$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Embedding into affine space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088L","source_file":"spaces-limits.tex","source_line":3051,"source_end_line":3057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3051-L3057","statement_sha256":"7a382eef26235630866460ffb786387401d9f5e7e9b268f4e6acc621701b2c4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11469,"rank":11469,"depth":55,"x":614.932,"y":1589.228,"cluster":"geometry-of-spaces"},{"id":"stacks:088N","tag":"088N","title":"Embedding into affine space · Lemma 088N","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Assume X Noetherian and f of finite presentation. Then there exists a dense open V ⊂ Y and an immersion V → A^n_X.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic\nspaces over $S$. Assume $X$ Noetherian and $f$ of finite presentation.\nThen there exists a dense open $V \\subset Y$ and an immersion\n$V \\to \\mathbf{A}^n_X$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Embedding into affine space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088N","source_file":"spaces-limits.tex","source_line":3088,"source_end_line":3094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3088-L3094","statement_sha256":"51ae1b99c24412871f31aab52cc03c09da7a868a2e0d30752cd73ee0302ea71a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11470,"rank":11470,"depth":57,"x":736.832,"y":1570.925,"cluster":"geometry-of-spaces"},{"id":"stacks:0855","tag":"0855","title":"Sections with support in a closed subset · Lemma 0855","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space. Let U ⊂ X be an open subspace. The following are equivalent: • U → X is quasi-compact, • U is quasi-compact, and • there exists a finite type quasi-coherent sheaf of ideals I ⊂ O_X such that |X| setminus |U| = |V(I)|.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space.\nLet $U \\subset X$ be an open subspace. The following are equivalent:\n\\begin{enumerate}\n\\item $U \\to X$ is quasi-compact,\n\\item $U$ is quasi-compact, and\n\\item there exists a finite type quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_X$ such that\n$|X| \\setminus |U| = |V(\\mathcal{I})|$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0855","source_file":"spaces-limits.tex","source_line":3165,"source_end_line":3177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3165-L3177","statement_sha256":"9a34e6f478e23646294f6db19ca963694ae290e61cec421c10f0a604223976ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":11471,"rank":11471,"depth":62,"x":661.413,"y":1653.865,"cluster":"geometry-of-spaces"},{"id":"stacks:0856","tag":"0856","title":"Sections with support in a closed subset · Lemma 0856","summary":"Let S be a scheme. Let X be an algebraic space over S. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let F be a quasi-coherent O_X-module. Consider the sheaf of O_X-modules F' which associates to every object U of X_etale the module F'(U) = (s ∈ F(U) mid Is = 0) Assume I is of finite type. Then • F' is a quasi-coherent sheaf of O_X-modules, • for affine U in X_etale we have F'(U) = (s ∈ F(U) mid I(U)s = 0), and • F'_x = (s ∈ F_x mid I_x s = 0).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nConsider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$\nwhich associates to every object $U$ of $X_\\etale$ the module\n$$\n\\mathcal{F}'(U)\n=\n\\{s \\in \\mathcal{F}(U) \\mid\n\\mathcal{I}s = 0\\}\n$$\nAssume $\\mathcal{I}$ is of finite type. Then\n\\begin{enumerate}\n\\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules,\n\\item for affine $U$ in $X_\\etale$ we have\n$\\mathcal{F}'(U) = \\{s \\in \\mathcal{F}(U) \\mid \\mathcal{I}(U)s = 0\\}$, and\n\\item $\\mathcal{F}'_x = \\{s \\in \\mathcal{F}_x \\mid \\mathcal{I}_x s = 0\\}$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0856","source_file":"spaces-limits.tex","source_line":3216,"source_end_line":3236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3216-L3236","statement_sha256":"61522bf4256cc41ea0d235fc4e093881b315c73c88a875d99ac74845731a25fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11472,"rank":11472,"depth":5,"x":650.291,"y":1549.568,"cluster":"geometry-of-spaces"},{"id":"stacks:0857","tag":"0857","title":"Sections with support in a closed subset · Definition 0857","summary":"Let S be a scheme. Let X be an algebraic space over S. Let I ⊂ O_X be a quasi-coherent sheaf of ideals of finite type. Let F be a quasi-coherent O_X-module. The subsheaf F' ⊂ F defined in Lemma [Tag 0856] above is called the subsheaf of sections annihilated by I.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent\nsheaf of ideals of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in\nLemma \\ref{lemma-sections-annihilated-by-ideal} above is called\nthe {\\it subsheaf of sections annihilated by $\\mathcal{I}$}.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Sections with support in a closed subset","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0857","source_file":"spaces-limits.tex","source_line":3246,"source_end_line":3255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3246-L3255","statement_sha256":"47b2f3364213d20e614420b65f7f635defa494d3895c8dc63c9a509a57a6c233","origin":"The Stacks Project","memory_eligible":false,"source_rank":11473,"rank":11473,"depth":6,"x":742.667,"y":1620.397,"cluster":"geometry-of-spaces"},{"id":"stacks:0858","tag":"0858","title":"Sections with support in a closed subset · Lemma 0858","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let I ⊂ O_Y be a quasi-coherent sheaf of ideals of finite type. Let F be a quasi-coherent O_X-module. Let F' ⊂ F be the subsheaf of sections annihilated by f^-1IO_X. Then f_*F' ⊂ f_*F is the subsheaf of sections annihilated by I.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a quasi-compact and quasi-separated morphism\nof algebraic spaces over $S$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_Y$ be a quasi-coherent\nsheaf of ideals of finite type. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$\nbe the subsheaf of sections annihilated by $f^{-1}\\mathcal{I}\\mathcal{O}_X$.\nThen $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf\nof sections annihilated by $\\mathcal{I}$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0858","source_file":"spaces-limits.tex","source_line":3257,"source_end_line":3268,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3257-L3268","statement_sha256":"1becc8184e2efe2e55c47a3f52f721b26aaea54e4fe69da86b8f69ae43d0d39d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11474,"rank":11474,"depth":43,"x":617.184,"y":1620.585,"cluster":"geometry-of-spaces"},{"id":"stacks:0859","tag":"0859","title":"Sections with support in a closed subset · Lemma 0859","summary":"Let S be a scheme. Let X be an algebraic space over S. Let T ⊂ |X| be a closed subset and let U ⊂ X be the open subspace such that T amalg |U| = |X|. Let F be a quasi-coherent O_X-module. Consider the sheaf of O_X-modules F' which associates to every object φ : W → X of X_etale the module F'(W) = (s ∈ F(W) mid the support of s is contained in |φ|^-1(T)) If U → X is quasi-compact, then • for W affine there exist a finitely generated ideal I ⊂ O_X(W) such that |φ|^-1(T) =…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset and let $U \\subset X$ be\nthe open subspace such that $T \\amalg |U| = |X|$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nConsider the sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}'$\nwhich associates to every object $\\varphi : W \\to X$ of\n$X_\\etale$ the module\n$$\n\\mathcal{F}'(W)\n=\n\\{s \\in \\mathcal{F}(W) \\mid\n\\text{the support of }s\\text{ is contained in }|\\varphi|^{-1}(T)\\}\n$$\nIf $U \\to X$ is quasi-compact, then\n\\begin{enumerate}\n\\item for $W$ affine there exist a finitely generated\nideal $I \\subset \\mathcal{O}_X(W)$ such that $|\\varphi|^{-1}(T) = V(I)$,\n\\item for $W$ and $I$ as in (1) we have\n$\\mathcal{F}'(W) = \\{x \\in \\mathcal{F}(W) \\mid\nI^nx = 0 \\text{ for some } n\\}$,\n\\item $\\mathcal{F}'$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-modules.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0859","source_file":"spaces-limits.tex","source_line":3284,"source_end_line":3308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3284-L3308","statement_sha256":"ee1c95ac63aab6443b57030d2c43fcb56b82df5072fc45359a0e93cb24f500c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11475,"rank":11475,"depth":15,"x":709.862,"y":1549.014,"cluster":"geometry-of-spaces"},{"id":"stacks:085A","tag":"085A","title":"Sections with support in a closed subset · Definition 085A","summary":"Let S be a scheme. Let X be an algebraic space over S. Let T ⊂ |X| be a closed subset whose complement corresponds to an open subspace U ⊂ X with quasi-compact inclusion morphism U → X. Let F be a quasi-coherent O_X-module. The quasi-coherent subsheaf F' ⊂ F defined in Lemma [Tag 0859] above is called the subsheaf of sections supported on T.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset whose complement\ncorresponds to an open subspace $U \\subset X$\nwith quasi-compact inclusion morphism $U \\to X$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe quasi-coherent subsheaf $\\mathcal{F}' \\subset \\mathcal{F}$ defined in\nLemma \\ref{lemma-sections-supported-on-closed-subset} above is called\nthe {\\it subsheaf of sections supported on $T$}.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Sections with support in a closed subset","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085A","source_file":"spaces-limits.tex","source_line":3318,"source_end_line":3328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3318-L3328","statement_sha256":"63ef9d74e9a399912a60c36ba7f963c558309c2700a38dff0d961423dee8d4e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11476,"rank":11476,"depth":16,"x":699.043,"y":1654.716,"cluster":"geometry-of-spaces"},{"id":"stacks:085B","tag":"085B","title":"Sections with support in a closed subset · Lemma 085B","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let T ⊂ |Y| be a closed subset. Assume |Y| setminus T corresponds to an open subspace V ⊂ Y such that V → Y is quasi-compact. Let F be a quasi-coherent O_X-module. Let F' ⊂ F be the subsheaf of sections supported on |f|^-1T. Then f_*F' ⊂ f_*F is the subsheaf of sections supported on T.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a quasi-compact and quasi-separated morphism\nof algebraic spaces over $S$. Let $T \\subset |Y|$ be a closed subset.\nAssume $|Y| \\setminus T$ corresponds to an open subspace $V \\subset Y$\nsuch that $V \\to Y$ is quasi-compact. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Let $\\mathcal{F}' \\subset \\mathcal{F}$\nbe the subsheaf of sections supported on $|f|^{-1}T$.\nThen $f_*\\mathcal{F}' \\subset f_*\\mathcal{F}$ is the subsheaf\nof sections supported on $T$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Sections with support in a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085B","source_file":"spaces-limits.tex","source_line":3330,"source_end_line":3341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3330-L3341","statement_sha256":"25e4b669b5adb503a7ef2359867f5c1e7ebe7b3139f2dc5954bf5a5d2201c929","origin":"The Stacks Project","memory_eligible":false,"source_rank":11477,"rank":11477,"depth":16,"x":621.772,"y":1570.365,"cluster":"geometry-of-spaces"},{"id":"stacks:07VS","tag":"07VS","title":"Characterizing affine spaces · Lemma 07VS","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that f is surjective and finite, and assume that X is affine. Then Y is affine.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume that $f$ is surjective and finite, and assume that $X$\nis affine. Then $Y$ is affine.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Characterizing affine spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VS","source_file":"spaces-limits.tex","source_line":3374,"source_end_line":3379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3374-L3379","statement_sha256":"e6af099c521f07f3cae8f6fe06c8861783998d6668164f1cdec4089662f26853","origin":"The Stacks Project","memory_eligible":false,"source_rank":11478,"rank":11478,"depth":65,"x":746.98,"y":1588.777,"cluster":"geometry-of-spaces"},{"id":"stacks:07VT","tag":"07VT","title":"Characterizing affine spaces · Proposition 07VT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that X is affine and f is surjective and universally closed. Then Y is affine.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume that $X$ is affine and $f$ is surjective and universally\nclosed\\footnote{An integral morphism is universally closed, see\nMorphisms of Spaces, Lemma\n\\ref{spaces-morphisms-lemma-integral-universally-closed}.}. Then $Y$ is affine.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Characterizing affine spaces","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VT","source_file":"spaces-limits.tex","source_line":3423,"source_end_line":3430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3423-L3430","statement_sha256":"4faf244a536c2e9f5c5202ed3e81b6131c6335b3b8cb77be9eb0610a3b707788","origin":"The Stacks Project","memory_eligible":false,"source_rank":11479,"rank":11479,"depth":66,"x":639.508,"y":1646.428,"cluster":"geometry-of-spaces"},{"id":"stacks:07VU","tag":"07VU","title":"Characterizing affine spaces · Lemma 07VU","summary":"[CLO] Let S be a scheme. Let X be an algebraic space over S. If X_red is a scheme, then X is a scheme.","statement_latex":"\\begin{reference}\n\\cite[3.1.12]{CLO}\n\\end{reference}\nLet $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf $X_{red}$ is a scheme, then $X$ is a scheme.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Characterizing affine spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VU","source_file":"spaces-limits.tex","source_line":3481,"source_end_line":3488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3481-L3488","statement_sha256":"5acdbd99fbdc25ada7cf39885d1181114c964ac8c2c2aca6b9e2681d9b78bc91","origin":"The Stacks Project","memory_eligible":false,"source_rank":11480,"rank":11480,"depth":67,"x":672.499,"y":1542.61,"cluster":"geometry-of-spaces"},{"id":"stacks:07VV","tag":"07VV","title":"Characterizing affine spaces · Lemma 07VV","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is integral and induces a bijection |X| → |Y|. Then X is a scheme if and only if Y is a scheme.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is integral and induces a bijection $|X| \\to |Y|$.\nThen $X$ is a scheme if and only if $Y$ is a scheme.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Characterizing affine spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VV","source_file":"spaces-limits.tex","source_line":3500,"source_end_line":3505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3500-L3505","statement_sha256":"4de741e680a45d62d6dc848513633f7e35dfa38bddb83d1d71dd915f5fbde5c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11481,"rank":11481,"depth":67,"x":731.843,"y":1638.18,"cluster":"geometry-of-spaces"},{"id":"stacks:08B2","tag":"08B2","title":"Characterizing affine spaces · Lemma 08B2","summary":"Let S be a scheme. Let f : X → B and B' → B be morphisms of algebraic spaces over S. Assume • B' → B is a closed immersion, • |B'| → |B| is bijective, • X ×_B B' → B' is a closed immersion, and • X → B is of finite type or B' → B is of finite presentation. Then f : X → B is a closed immersion.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to B$ and $B' \\to B$ be morphisms of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $B' \\to B$ is a closed immersion,\n\\item $|B'| \\to |B|$ is bijective,\n\\item $X \\times_B B' \\to B'$ is a closed immersion, and\n\\item $X \\to B$ is of finite type or $B' \\to B$ is of finite presentation.\n\\end{enumerate}\nThen $f : X \\to B$ is a closed immersion.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Characterizing affine spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08B2","source_file":"spaces-limits.tex","source_line":3520,"source_end_line":3532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3520-L3532","statement_sha256":"69ff2767f11912324691a089fac678127d18c4ac819ec8f186dff1e597b9866c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11482,"rank":11482,"depth":68,"x":610.855,"y":1601.267,"cluster":"geometry-of-spaces"},{"id":"stacks:09YC","tag":"09YC","title":"Finite cover by a scheme · Proposition 09YC","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. • There exists a surjective finite morphism Y → X of finite presentation where Y is a scheme, • given a surjective étale morphism U → X we may choose Y → X such that for every y ∈ Y there is an open neighbourhood V ⊂ Y such that V → X factors through U.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$.\n\\begin{enumerate}\n\\item There exists a surjective finite morphism $Y \\to X$\nof finite presentation where $Y$ is a scheme,\n\\item given a surjective \\'etale morphism $U \\to X$ we may choose\n$Y \\to X$ such that for every $y \\in Y$ there is an open neighbourhood\n$V \\subset Y$ such that $V \\to X$ factors through $U$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite cover by a scheme","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09YC","source_file":"spaces-limits.tex","source_line":3568,"source_end_line":3579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3568-L3579","statement_sha256":"a552683e62e52dd879b90507603db43dc61f1c80fe41729fafcddb97f977db2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11483,"rank":11483,"depth":66,"x":730.121,"y":1559.708,"cluster":"geometry-of-spaces"},{"id":"stacks:0GUM","tag":"0GUM","title":"Finite cover by a scheme · Lemma 0GUM","summary":"Let S be a scheme. Let f : X → Y be an integral morphism of algebraic spaces over S. Assume Y quasi-compact and quasi-separated. Let V ⊂ Y be a quasi-compact open subspace such that f^-1(V) → V is finite and of finite presentation. Then X can be written as a directed limit X = lim X_i where f_i : X_i → Y are finite and of finite presentation such that f^-1(V) → f_i^-1(V) is an isomorphism for all i.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism of algebraic\nspaces over $S$. Assume $Y$ quasi-compact and quasi-separated.\nLet $V \\subset Y$ be a quasi-compact open subspace such that\n$f^{-1}(V) \\to V$ is finite and of finite presentation.\nThen $X$ can be written as a directed limit $X = \\lim X_i$\nwhere $f_i : X_i \\to Y$ are finite and of finite presentation\nsuch that $f^{-1}(V) \\to f_i^{-1}(V)$ is an isomorphism for all $i$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite cover by a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUM","source_file":"spaces-limits.tex","source_line":3600,"source_end_line":3609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3600-L3609","statement_sha256":"9d44e5ef4e60e3fe14bee20b02e6630273b5c2e9d3570d562214ea66888bdfef","origin":"The Stacks Project","memory_eligible":false,"source_rank":11484,"rank":11484,"depth":67,"x":675.43,"y":1658.328,"cluster":"geometry-of-spaces"},{"id":"stacks:0GUN","tag":"0GUN","title":"Finite cover by a scheme · Lemma 0GUN","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S such that |X| has finitely many irreducible components. • There exists a surjective finite morphism f : Y → X of finite presentation where Y is a scheme such that f is finite étale over a quasi-compact dense open U ⊂ X, • given a surjective étale morphism V → X we may choose Y → X such that for every y ∈ Y there is an open neighbourhood W ⊂ Y such that W → X factors through V.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$ such that $|X|$ has finitely many irreducible\ncomponents.\n\\begin{enumerate}\n\\item There exists a surjective finite morphism $f : Y \\to X$\nof finite presentation where $Y$ is a scheme such that $f$\nis finite \\'etale over a quasi-compact dense open $U \\subset X$,\n\\item given a surjective \\'etale morphism $V \\to X$ we may choose\n$Y \\to X$ such that for every $y \\in Y$ there is an open neighbourhood\n$W \\subset Y$ such that $W \\to X$ factors through $V$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite cover by a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUN","source_file":"spaces-limits.tex","source_line":3693,"source_end_line":3706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3693-L3706","statement_sha256":"d991f0ec39a186ee9c3da5fbd8750f30eea3653dc363e803805446533b61b4b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11485,"rank":11485,"depth":68,"x":636.332,"y":1554.258,"cluster":"geometry-of-spaces"},{"id":"stacks:0GUP","tag":"0GUP","title":"Finite cover by a scheme · Lemma 0GUP","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. There exists a t ≥ 0 and closed subspaces X ⊃ Z_0 ⊃ Z_1 ⊃ … ⊃ Z_t = ∅ such that Z_i → X is of finite presentation, Z_0 ⊂ X is a thickening, and for each i = 0, … t - 1 there exists a scheme Y_i, a surjective, finite, and finitely presented morphism Y_i → Z_i which is finite étale over Z_i setminus Z_i + 1.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. There exists a $t \\geq 0$ and closed\nsubspaces\n$$\nX \\supset Z_0 \\supset Z_1 \\supset \\ldots \\supset Z_t = \\emptyset\n$$\nsuch that $Z_i \\to X$ is of finite presentation,\n$Z_0 \\subset X$ is a thickening, and for each $i = 0, \\ldots t - 1$\nthere exists a scheme $Y_i$, a surjective, finite, and finitely\npresented morphism $Y_i \\to Z_i$ which is finite \\'etale over\n$Z_i \\setminus Z_{i + 1}$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Finite cover by a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUP","source_file":"spaces-limits.tex","source_line":3732,"source_end_line":3745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3732-L3745","statement_sha256":"96f3945f54fc29be2163ce0d46ebcf1d2f28176737ce4edfdcc4dec3c95711e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11486,"rank":11486,"depth":69,"x":749.192,"y":1608.978,"cluster":"geometry-of-spaces"},{"id":"stacks:0B7Y","tag":"0B7Y","title":"Obtaining schemes · Lemma 0B7Y","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. If X is not a scheme, then there exists a closed subspace Z ⊂ X such that Z is not a scheme, but every proper closed subspace Z' ⊂ Z is a scheme.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. If $X$ is not a scheme, then there exists\na closed subspace $Z \\subset X$ such that $Z$ is not a scheme, but\nevery proper closed subspace $Z' \\subset Z$ is a scheme.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Obtaining schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7Y","source_file":"spaces-limits.tex","source_line":3802,"source_end_line":3808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3802-L3808","statement_sha256":"22a526007a753de458a2a36edccc902647fbec6145ce3d764c39f1c3e523ac78","origin":"The Stacks Project","memory_eligible":false,"source_rank":11487,"rank":11487,"depth":56,"x":621.585,"y":1632.739,"cluster":"geometry-of-spaces"},{"id":"stacks:0B7Z","tag":"0B7Z","title":"Obtaining schemes · Lemma 0B7Z","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Assume that every proper closed subspace Z ⊂ X is a scheme, but X is not a scheme. Then X is reduced and irreducible.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Assume that every proper closed subspace\n$Z \\subset X$ is a scheme, but $X$ is not a scheme. Then $X$ is reduced\nand irreducible.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Obtaining schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B7Z","source_file":"spaces-limits.tex","source_line":3830,"source_end_line":3836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3830-L3836","statement_sha256":"7c667f4c8c4fe4bab77db628d6f4dc200a8667c3bbd125e85cc288dec9ae8667","origin":"The Stacks Project","memory_eligible":false,"source_rank":11488,"rank":11488,"depth":68,"x":696.797,"y":1542.542,"cluster":"geometry-of-spaces"},{"id":"stacks:0B80","tag":"0B80","title":"Obtaining schemes · Lemma 0B80","summary":"Let f: X → S be a quasi-compact and quasi-separated morphism from an algebraic space to a scheme S. If for every x ∈ |X| with image s = f(x) ∈ S the algebraic space X ×_S Spec(O_S,s) is a scheme, then X is a scheme.","statement_latex":"Let $f: X \\to S$ be a quasi-compact and quasi-separated morphism from an\nalgebraic space to a scheme $S$. If for every $x \\in |X|$ with image\n$s = f(x) \\in S$ the algebraic space $X \\times_S \\Spec(\\mathcal{O}_{S,s})$\nis a scheme, then $X$ is a scheme.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Obtaining schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B80","source_file":"spaces-limits.tex","source_line":3872,"source_end_line":3878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3872-L3878","statement_sha256":"0c49b8e5e25ee17a82a89f711ec0fed0b1ba8fb9ec72e57a8ca3d65e596761dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11489,"rank":11489,"depth":56,"x":713.92,"y":1652.053,"cluster":"geometry-of-spaces"},{"id":"stacks:0B81","tag":"0B81","title":"Obtaining schemes · Lemma 0B81","summary":"Let φ : X → Spec(A) be a quasi-compact and quasi-separated morphism from an algebraic space to an affine scheme. If X is not a scheme, then there exists an ideal I ⊂ A such that the base change X_A/I is not a scheme, but for every I ⊂ I', I not = I' the base change X_A/I' is a scheme.","statement_latex":"Let $\\varphi : X \\to \\Spec(A)$ be a quasi-compact and quasi-separated\nmorphism from an algebraic space to an affine scheme.\nIf $X$ is not a scheme, then there exists an ideal $I \\subset A$\nsuch that the base change $X_{A/I}$ is not a scheme, but\nfor every $I \\subset I'$, $I \\not = I'$ the base change\n$X_{A/I'}$ is a scheme.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Obtaining schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B81","source_file":"spaces-limits.tex","source_line":3897,"source_end_line":3905,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3897-L3905","statement_sha256":"14dc8c3a9dd2d5eedbeb75ad8ee712336f893908dbc21e3e41aa218b53a0eab3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11490,"rank":11490,"depth":56,"x":612.931,"y":1580.808,"cluster":"geometry-of-spaces"},{"id":"stacks:0E8Z","tag":"0E8Z","title":"Glueing in closed fibres · Lemma 0E8Z","summary":"Let S = U ∪ W be an open covering of a scheme. Then the functor FP_S → FP_U ×_FP_U ∩ W FP_W given by base change is an equivalence where FP_T is the category of algebraic spaces of finite presentation over the scheme T.","statement_latex":"Let $S = U \\cup W$ be an open covering of a scheme. Then the functor\n$$\nFP_S \\longrightarrow FP_U \\times_{FP_{U \\cap W}} FP_W\n$$\ngiven by base change is an equivalence where $FP_T$\nis the category of algebraic spaces of finite presentation over\nthe scheme $T$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Glueing in closed fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8Z","source_file":"spaces-limits.tex","source_line":3936,"source_end_line":3945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3936-L3945","statement_sha256":"0163a81e9bbb0d1f56060fd665488eb4a4fc14134ecaebbfc47cc19b7629fc34","origin":"The Stacks Project","memory_eligible":false,"source_rank":11491,"rank":11491,"depth":56,"x":745.079,"y":1576.026,"cluster":"geometry-of-spaces"},{"id":"stacks:0E90","tag":"0E90","title":"Glueing in closed fibres · Lemma 0E90","summary":"Let S be a scheme. Let s ∈ S be a closed point such that U = S setminus (s) → S is quasi-compact. With V = Spec(O_S, s) setminus (s) there is an equivalence of categories FP_S → FP_U ×_FP_V FP_Spec(O_S, s) where FP_T is the category of algebraic spaces of finite presentation over T.","statement_latex":"Let $S$ be a scheme. Let $s \\in S$ be a closed point such that\n$U = S \\setminus \\{s\\} \\to S$ is quasi-compact. With\n$V = \\Spec(\\mathcal{O}_{S, s}) \\setminus \\{s\\}$ there is\nan equivalence of categories\n$$\nFP_S \\longrightarrow FP_U \\times_{FP_V} FP_{\\Spec(\\mathcal{O}_{S, s})}\n$$\nwhere $FP_T$ is the category of algebraic spaces of finite presentation\nover $T$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Glueing in closed fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E90","source_file":"spaces-limits.tex","source_line":3980,"source_end_line":3991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L3980-L3991","statement_sha256":"89bf6b4a64a892ae8f3aad64c2d99bc17c1f34c400493946f09fa562d94e25a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11492,"rank":11492,"depth":57,"x":651.208,"y":1654.764,"cluster":"geometry-of-spaces"},{"id":"stacks:0E91","tag":"0E91","title":"Glueing in closed fibres · Lemma 0E91","summary":"Let S be a scheme. Let U ⊂ S be a retrocompact open. Let s ∈ S be a point in the complement of U. With V = Spec(O_S, s) ∩ U there is an equivalence of categories colim_s ∈ U' ⊃ U open FP_U' → FP_U ×_FP_V FP_Spec(O_S, s) where FP_T is the category of algebraic spaces of finite presentation over T.","statement_latex":"Let $S$ be a scheme. Let $U \\subset S$ be a retrocompact open.\nLet $s \\in S$ be a point in the complement of $U$. With\n$V = \\Spec(\\mathcal{O}_{S, s}) \\cap U$ there is\nan equivalence of categories\n$$\n\\colim_{s \\in U' \\supset U\\text{ open}} FP_{U'}\n\\longrightarrow\nFP_U \\times_{FP_V} FP_{\\Spec(\\mathcal{O}_{S, s})}\n$$\nwhere $FP_T$  is the category of algebraic spaces of finite presentation\nover $T$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Glueing in closed fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E91","source_file":"spaces-limits.tex","source_line":4022,"source_end_line":4035,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4022-L4035","statement_sha256":"b520a754899f439901fea9027bd403387508506e3f9260af519896d2503c35ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":11493,"rank":11493,"depth":57,"x":657.125,"y":1543.116,"cluster":"geometry-of-spaces"},{"id":"stacks:0E92","tag":"0E92","title":"Glueing in closed fibres · Lemma 0E92","summary":"Let S be a scheme. Let s_1, …, s_n ∈ S be pairwise distinct closed points such that U = S setminus (s_1, …, s_n) → S is quasi-compact. With S_i = Spec(O_S, s_i) and U_i = S_i setminus (s_i) there is an equivalence of categories FP_S → FP_U ×_(FP_U_1 × … × FP_U_n) (FP_S_1 × … × FP_S_n) where FP_T is the category of algebraic spaces of finite presentation over T.","statement_latex":"Let $S$ be a scheme. Let $s_1, \\ldots, s_n \\in S$ be\npairwise distinct closed points such that\n$U = S \\setminus \\{s_1, \\ldots, s_n\\} \\to S$ is quasi-compact. With\n$S_i = \\Spec(\\mathcal{O}_{S, s_i})$ and $U_i = S_i \\setminus \\{s_i\\}$\nthere is an equivalence of categories\n$$\nFP_S \\longrightarrow\nFP_U \\times_{(FP_{U_1} \\times \\ldots \\times FP_{U_n})}\n(FP_{S_1} \\times \\ldots \\times FP_{S_n})\n$$\nwhere $FP_T$ is the category of algebraic spaces of finite presentation\nover $T$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Glueing in closed fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E92","source_file":"spaces-limits.tex","source_line":4069,"source_end_line":4083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4069-L4083","statement_sha256":"1ed3dd4fd14d5a15a811a146d1f5cd81e6056e72bb36f7d3d9050cd0460e1dd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11494,"rank":11494,"depth":58,"x":742.792,"y":1629.049,"cluster":"geometry-of-spaces"},{"id":"stacks:0BGY","tag":"0BGY","title":"Application to modifications · Lemma 0BGY","summary":"Let S be a scheme. Consider a separated étale morphism f : V → W of algebraic spaces over S. Assume there exists a closed subspace T ⊂ W such that f^-1T → T is an isomorphism. Then, with W^0 = W setminus T and V^0 = f^-1W^0 the base change functor ( g : X → W morphism of algebraic spaces g^-1(W^0) → W^0 is an isomorphism ) → ( h : Y → V morphism of algebraic spaces h^-1(V^0) → V^0 is an isomorphism ) is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Consider a separated \\'etale morphism\n$f : V \\to W$ of algebraic spaces over $S$.\nAssume there exists a\nclosed subspace $T \\subset W$ such that $f^{-1}T \\to T$ is\nan isomorphism. Then, with $W^0 = W \\setminus T$ and\n$V^0 = f^{-1}W^0$ the base change functor\n$$\n\\left\\{\n\\begin{matrix}\ng : X \\to W\\text{ morphism of algebraic spaces} \\\\\ng^{-1}(W^0) \\to W^0\\text{ is an isomorphism}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\nh : Y \\to V\\text{ morphism of algebraic spaces} \\\\\nh^{-1}(V^0) \\to V^0\\text{ is an isomorphism}\n\\end{matrix}\n\\right\\}\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Application to modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGY","source_file":"spaces-limits.tex","source_line":4119,"source_end_line":4143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4119-L4143","statement_sha256":"e8c199ef4b8603410451a91894d9bc1ce9b04d097ff48200d5038197b783630f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11495,"rank":11495,"depth":56,"x":610.139,"y":1614.25,"cluster":"geometry-of-spaces"},{"id":"stacks:0BGZ","tag":"0BGZ","title":"Application to modifications · Lemma 0BGZ","summary":"Notation and assumptions as in Lemma [Tag 0BGY]. Let g : X → W correspond to h : Y → V via the equivalence. Then g is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, and add more here if and only if h is so.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-excision-modifications}.\nLet $g : X \\to W$ correspond to $h : Y \\to V$ via the equivalence.\nThen $g$ is quasi-compact, quasi-separated, separated, locally of finite\npresentation, of finite presentation, locally of finite type, of finite type,\nproper, integral, finite, and add more here if and only if\n$h$ is so.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Application to modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BGZ","source_file":"spaces-limits.tex","source_line":4202,"source_end_line":4210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4202-L4210","statement_sha256":"3ddc082e0cc9dcbd2b77c4e5970b6c06a046a3aeb190d1efe94e71a7469f6add","origin":"The Stacks Project","memory_eligible":false,"source_rank":11496,"rank":11496,"depth":57,"x":720.169,"y":1549.709,"cluster":"geometry-of-spaces"},{"id":"stacks:0BH0","tag":"0BH0","title":"Application to modifications · Lemma 0BH0","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X| be a closed point such that U = X setminus (x) → X is quasi-compact. With V = Spec(O_X, x^h) setminus ( m_x^h) the base change functor ( f : Y → X of finite presentation f^-1(U) → U is an isomorphism ) → ( g : Y → Spec(O_X, x^h) of finite presentation g^-1(V) → V is an isomorphism ) is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$ be a closed point such that $U = X \\setminus \\{x\\} \\to X$\nis quasi-compact. With\n$V = \\Spec(\\mathcal{O}_{X, x}^h) \\setminus \\{\\mathfrak m_x^h\\}$\nthe base change functor\n$$\n\\left\\{\n\\begin{matrix}\nf : Y \\to X\\text{ of finite presentation} \\\\\nf^{-1}(U) \\to U\\text{ is an isomorphism}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\ng : Y \\to \\Spec(\\mathcal{O}_{X, x}^h)\\text{ of finite presentation} \\\\\ng^{-1}(V) \\to V\\text{ is an isomorphism}\n\\end{matrix}\n\\right\\}\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Application to modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BH0","source_file":"spaces-limits.tex","source_line":4238,"source_end_line":4261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4238-L4261","statement_sha256":"c932e773221c19e175cf967ed6b5d1d115613a46be41e6222b0315952947b1cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11497,"rank":11497,"depth":59,"x":690.854,"y":1660.045,"cluster":"geometry-of-spaces"},{"id":"stacks:0CM8","tag":"0CM8","title":"Universally closed morphisms · Lemma 0CM8","summary":"Let S be a scheme. Let f : X → Y and g : Z → Y be morphisms of algebraic spaces over S. Let z ∈ |Z| and let T ⊂ |X ×_Y Z| be a closed subset with z not ∈ Im(T → |Z|). If f is quasi-compact, then there exists an étale neighbourhood (V, v) → (Z, z), a commutative diagram xymatrix V ar[d] ar[r]_a & Z' ar[d]^b Z ar[r]^g & Y, and a closed subset T' ⊂ |X ×_Y Z'| such that • the morphism b : Z' → Y is locally of finite presentation, • with z' = a(v) we have z' not ∈ Im(T' →…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be\nmorphisms of algebraic spaces over $S$. Let $z \\in |Z|$ and let\n$T \\subset |X \\times_Y Z|$ be a closed subset\nwith $z \\not \\in \\Im(T \\to |Z|)$.\nIf $f$ is quasi-compact, then there exists\nan \\'etale neighbourhood $(V, v) \\to (Z, z)$,\na commutative diagram\n$$\n\\xymatrix{\nV \\ar[d] \\ar[r]_a & Z' \\ar[d]^b \\\\\nZ \\ar[r]^g & Y,\n}\n$$\nand a closed subset $T' \\subset |X \\times_Y Z'|$ such that\n\\begin{enumerate}\n\\item the morphism $b : Z' \\to Y$ is locally of finite presentation,\n\\item with $z' = a(v)$ we have $z' \\not \\in \\Im(T' \\to |Z'|)$, and\n\\item the inverse image of $T$ in $|X \\times_Y V|$\nmaps into $T'$ via $|X \\times_Y V| \\to |X \\times_Y Z'|$.\n\\end{enumerate}\nMoreover, we may assume $V$ and $Z'$ are affine schemes and if $Z$\nis a scheme we may assume $V$ is an affine open neighbourhood of $z$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CM8","source_file":"spaces-limits.tex","source_line":4311,"source_end_line":4335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4311-L4335","statement_sha256":"4320c52d6034325477df648f8c1cbe233548c7ac67f15035182ed70685debc85","origin":"The Stacks Project","memory_eligible":false,"source_rank":11498,"rank":11498,"depth":1,"x":623.552,"y":1561.775,"cluster":"geometry-of-spaces"},{"id":"stacks:0CM9","tag":"0CM9","title":"Universally closed morphisms · Lemma 0CM9","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact morphism of algebraic spaces over S. The following are equivalent • f is universally closed, • for every morphism Z → Y which is locally of finite presentation the map |X ×_Y Z| → |Z| is closed, and • there exists a scheme V and a surjective étale morphism V → Y such that |A^n × (X ×_Y V)| → |A^n × V| is closed for all n ≥ 0.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a quasi-compact morphism of algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed,\n\\item for every morphism $Z \\to Y$ which is locally of finite presentation\nthe map $|X \\times_Y Z| \\to |Z|$ is closed, and\n\\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$\nsuch that $|\\mathbf{A}^n \\times (X \\times_Y V)| \\to |\\mathbf{A}^n \\times V|$\nis closed for all $n \\geq 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CM9","source_file":"spaces-limits.tex","source_line":4411,"source_end_line":4424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4411-L4424","statement_sha256":"d9fbb74e8d20df47afd9c598468c63cb80a86874e7dc206f5bd28c2d328b4696","origin":"The Stacks Project","memory_eligible":false,"source_rank":11499,"rank":11499,"depth":47,"x":752.566,"y":1596.146,"cluster":"geometry-of-spaces"},{"id":"stacks:0CMA","tag":"0CMA","title":"Universally closed morphisms · Lemma 0CMA","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f separated and of finite type. The following are equivalent • The morphism f is proper. • For any morphism Y → Z which is locally of finite presentation the map |X ×_Y Z| → |Z| is closed, and • there exists a scheme V and a surjective étale morphism V → Y such that |A^n × (X ×_Y V)| → |A^n × V| is closed for all n ≥ 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a\nmorphism of algebraic spaces over $S$.\nAssume $f$ separated and of finite type.\nThe following are equivalent\n\\begin{enumerate}\n\\item The morphism $f$ is proper.\n\\item For any morphism $Y \\to Z$ which is locally of finite presentation\nthe map $|X \\times_Y Z| \\to |Z|$ is closed, and\n\\item there exists a scheme $V$ and a surjective \\'etale morphism $V \\to Y$\nsuch that $|\\mathbf{A}^n \\times (X \\times_Y V)| \\to |\\mathbf{A}^n \\times V|$\nis closed for all $n \\geq 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMA","source_file":"spaces-limits.tex","source_line":4517,"source_end_line":4531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4517-L4531","statement_sha256":"99e6f7dd512d3c7f0a771cc71eab550ddb8bfdcfc6acadb788e0b8f4ba066756","origin":"The Stacks Project","memory_eligible":false,"source_rank":11500,"rank":11500,"depth":48,"x":629.451,"y":1644.14,"cluster":"geometry-of-spaces"},{"id":"stacks:0CMC","tag":"0CMC","title":"Noetherian valuative criterion · Lemma 0CMC","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f finite type and Y locally Noetherian. Let y ∈ |Y| be a point in the closure of the image of |f|. Then there exists a commutative diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d]^f Spec(A) ar[r] & Y where A is a discrete valuation ring and K is its field of fractions mapping the closed point of Spec(A) to y. Moreover, we can assume that the point x ∈ |X| corresponding to Spec(K) → X is a…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ finite type and $Y$ locally Noetherian.\nLet $y \\in |Y|$ be a point in the closure of the image of $|f|$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[r] & Y\n}\n$$\nwhere $A$ is a discrete valuation ring and $K$ is its field of fractions\nmapping the closed point of $\\Spec(A)$ to $y$. Moreover, we can assume\nthat the point $x \\in |X|$ corresponding to $\\Spec(K) \\to X$ is a\ncodimension $0$ point\\footnote{See discussion in\nProperties of Spaces, Section \\ref{spaces-properties-section-generic-points}.}\nand that $K$ is the residue field of a point\non a scheme \\'etale over $X$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMC","source_file":"spaces-limits.tex","source_line":4554,"source_end_line":4573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4554-L4573","statement_sha256":"bbb62257635eed4f27ce589846c3b2d6248504392fd51a73d422d9d7d63a5273","origin":"The Stacks Project","memory_eligible":false,"source_rank":11501,"rank":11501,"depth":21,"x":681.783,"y":1538.6,"cluster":"geometry-of-spaces"},{"id":"stacks:0H1W","tag":"0H1W","title":"Noetherian valuative criterion · Lemma 0H1W","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-separated and locally of finite type and Y is locally Noetherian. The following are equivalent: • The morphism f is separated. • For any diagram ([Tag 0H1V]) there is at most one dotted arrow. • For all diagrams ([Tag 0H1V]) with A a discrete valuation ring there is at most one dotted arrow. • For all diagrams ([Tag 0H1V]) where A is a discrete valuation ring and where the image…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is quasi-separated and locally of finite type and\n$Y$ is locally Noetherian. The following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is separated.\n\\item For any diagram (\\ref{equation-valuative}) there is at most\none dotted arrow.\n\\item For all diagrams (\\ref{equation-valuative}) with $A$ a discrete\nvaluation ring there is at most one dotted arrow.\n\\item For all diagrams (\\ref{equation-valuative}) where $A$ is a discrete\nvaluation ring and where the image of $\\Spec(K) \\to X$ is a point of\ncodimension $0$ on $X$ there is at most one dotted arrow.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1W","source_file":"spaces-limits.tex","source_line":4611,"source_end_line":4626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4611-L4626","statement_sha256":"ea2c82ae8305e2a3bf00e81ddc14108232f59cfff1a11392930ef186d1493da7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11502,"rank":11502,"depth":53,"x":728.193,"y":1646.414,"cluster":"geometry-of-spaces"},{"id":"stacks:0H1X","tag":"0H1X","title":"Noetherian valuative criterion · Lemma 0H1X","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-separated and of finite type and Y is locally Noetherian. The following are equivalent: • f is proper, • f satisfies the valuative criterion, see Morphisms of Spaces, Definition [Tag 03IX], • for any diagram ([Tag 0H1V]) there exists exactly one dotted arrow, • for all diagrams ([Tag 0H1V]) with A a discrete valuation ring there exists exactly one dotted arrow, and • for all…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is quasi-separated and of finite type and $Y$\nis locally Noetherian. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item $f$ satisfies the valuative criterion, see\nMorphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-valuative-criterion},\n\\item for any diagram (\\ref{equation-valuative}) there exists exactly\none dotted arrow,\n\\item for all diagrams (\\ref{equation-valuative}) with $A$ a discrete\nvaluation ring there exists exactly one dotted arrow, and\n\\item for all diagrams (\\ref{equation-valuative}) where $A$ is a discrete\nvaluation ring and where the image of $\\Spec(K) \\to X$ is a point of\ncodimension $0$ on $X$ there exists exactly one dotted arrow\\footnote{There is\na sharper formulation where in the existence part one only requires\nthe dotted arrow exists after an extension of discrete valuation rings.}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1X","source_file":"spaces-limits.tex","source_line":4679,"source_end_line":4699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4679-L4699","statement_sha256":"361573a0a6b42cc66ad531121d76dd1b3484a8be45e64290a6debab8e0cc2be0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11503,"rank":11503,"depth":62,"x":606.939,"y":1593.102,"cluster":"geometry-of-spaces"},{"id":"stacks:0H1Y","tag":"0H1Y","title":"Noetherian valuative criterion · Lemma 0H1Y","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume Y is locally Noetherian and f is of finite type. Then the following are equivalent • f is universally closed, • f satisfies the existence part of the valuative criterion, • there exists a scheme V and a surjective étale morphism V → Y such that |A^n × X ×_Y V| → |A^n × V| is closed for all n ≥ 0, • for all diagrams ([Tag 0H1V]) with A a discrete valuation ring there there exists a finite…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume $Y$ is locally Noetherian and $f$ is\nof finite type. Then the following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed,\n\\item $f$ satisfies the existence part of the valuative criterion,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that\n$|\\mathbf{A}^n \\times X \\times_Y V| \\to |\\mathbf{A}^n \\times V|$ is closed\nfor all $n \\geq 0$,\n\\item for all diagrams (\\ref{equation-valuative}) with $A$ a discrete\nvaluation ring there there exists a finite separable extension $K'/K$\nof fields, a discrete valuation ring $A' \\subset K'$ dominating $A$, and\na morphism $\\Spec(A') \\to X$ such that the following diagram commutes\n$$\n\\xymatrix{\n\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\\n\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y\n}\n$$\n\\item for all diagrams (\\ref{equation-valuative}) with $A$ a discrete\nvaluation ring there there exists a field extension $K'/K$,\na valuation ring $A' \\subset K'$ dominating $A$, and\na morphism $\\Spec(A') \\to X$ such that the following diagram commutes\n$$\n\\xymatrix{\n\\Spec(K') \\ar[r] \\ar[d] & \\Spec(K) \\ar[r] & X \\ar[d] \\\\\n\\Spec(A') \\ar[r] \\ar[rru] & \\Spec(A) \\ar[r] & Y\n}\n$$\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1Y","source_file":"spaces-limits.tex","source_line":4769,"source_end_line":4802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4769-L4802","statement_sha256":"dfb1d5729ae5e1a7957e1823d5c4a884f50edf4b734173c143470a7500907248","origin":"The Stacks Project","memory_eligible":false,"source_rank":11504,"rank":11504,"depth":63,"x":739.582,"y":1563.531,"cluster":"geometry-of-spaces"},{"id":"stacks:0CMD","tag":"0CMD","title":"Refined Noetherian valuative criteria · Lemma 0CMD","summary":"Let S be a scheme. Let f : X → Y and h : U → X be morphisms of algebraic spaces over S. Assume that Y is locally Noetherian, that f and h are of finite type, that f is separated, and that the image of |h| : |U| → |X| is dense in |X|. If given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & U ar[r]^h & X ar[d]^f Spec(A) ar[rr] ar@-->[rru] & & Y where A is a discrete valuation ring with field of fractions K, there exists a dotted arrow making the diagram…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $h : U \\to X$ be\nmorphisms of algebraic spaces over $S$. Assume that $Y$ is\nlocally Noetherian, that $f$ and $h$ are of finite type,\nthat $f$ is separated, and that the image of $|h| : |U| \\to |X|$\nis dense in $|X|$. If given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y\n}\n$$\nwhere $A$ is a discrete valuation ring with field of fractions $K$, there\nexists a dotted arrow making the diagram commute, then $f$ is proper.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Refined Noetherian valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMD","source_file":"spaces-limits.tex","source_line":4887,"source_end_line":4902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4887-L4902","statement_sha256":"467a5cb79a889af0ba904d6ce9f57d1efb4a510135d51a2425d6d57f6da0ac10","origin":"The Stacks Project","memory_eligible":false,"source_rank":11505,"rank":11505,"depth":62,"x":665.353,"y":1660.864,"cluster":"geometry-of-spaces"},{"id":"stacks:0CME","tag":"0CME","title":"Refined Noetherian valuative criteria · Lemma 0CME","summary":"Let S be a scheme. Let f : X → Y and h : U → X be morphisms of algebraic spaces over S. Assume that Y is locally Noetherian, that f is locally of finite type and quasi-separated, that h is of finite type, and that the image of |h| : |U| → |X| is dense in |X|. If given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & U ar[r]^h & X ar[d]^f Spec(A) ar[rr] ar@-->[rru] & & Y where A is a discrete valuation ring with field of fractions K, there exists at most one…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $h : U \\to X$ be\nmorphisms of algebraic spaces over $S$. Assume that $Y$ is\nlocally Noetherian, that $f$ is locally of finite type and quasi-separated,\nthat $h$ is of finite type, and that the image of $|h| : |U| \\to |X|$\nis dense in $|X|$.\nIf given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y\n}\n$$\nwhere $A$ is a discrete valuation ring with field of fractions $K$, there\nexists at most one dotted arrow making the diagram commute, then $f$ is\nseparated.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Refined Noetherian valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CME","source_file":"spaces-limits.tex","source_line":4953,"source_end_line":4970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4953-L4970","statement_sha256":"010ee9ae851a8c2510609e1b9d1439e1e918e1d0ffd467282f5dcf47b6e49582","origin":"The Stacks Project","memory_eligible":false,"source_rank":11506,"rank":11506,"depth":63,"x":641.754,"y":1546.666,"cluster":"geometry-of-spaces"},{"id":"stacks:0CMF","tag":"0CMF","title":"Refined Noetherian valuative criteria · Lemma 0CMF","summary":"Let S be a scheme. Let f : X → Y and h : U → X be morphisms of algebraic spaces over S. Assume that Y is locally Noetherian, that f and h are of finite type, that f is quasi-separated, and that h(U) is dense in X. If given any commutative solid diagram xymatrix Spec(K) ar[r] ar[d] & U ar[r]^h & X ar[d]^f Spec(A) ar[rr] ar@-->[rru] & & Y where A is a discrete valuation ring with field of fractions K, there exists a unique dotted arrow making the diagram commute, then f is…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $h : U \\to X$ be morphisms of algebraic spaces over $S$.\nAssume that $Y$ is locally Noetherian, that $f$ and $h$ are of finite type,\nthat $f$ is quasi-separated, and\nthat $h(U)$ is dense in $X$. If given any commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & U \\ar[r]^h & X \\ar[d]^f \\\\\n\\Spec(A) \\ar[rr] \\ar@{-->}[rru] & & Y\n}\n$$\nwhere $A$ is a discrete valuation ring with field of fractions $K$, there\nexists a unique dotted arrow making the diagram commute, then $f$ is proper.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Refined Noetherian valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMF","source_file":"spaces-limits.tex","source_line":4991,"source_end_line":5006,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L4991-L5006","statement_sha256":"e25b294d60a75b65ebb5779abb05d005d63c1ba1cc03f03d927b44052b60305f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11507,"rank":11507,"depth":64,"x":751.282,"y":1617.673,"cluster":"geometry-of-spaces"},{"id":"stacks:0CP7","tag":"0CP7","title":"Descending finite type spaces · Lemma 0CP7","summary":"In Situation [Tag 0CP6]. Let X → B be a quasi-separated and finite type morphism of algebraic spaces. Then there exists an i ∈ I and a diagram vcenter xymatrix X ar[r] ar[d] & W ar[d] B ar[r] & B_i such that W → B_i is of finite type and such that the induced morphism X → B ×_B_i W is a closed immersion.","statement_latex":"In Situation \\ref{situation-limit-noetherian}.\nLet $X \\to B$ be a quasi-separated and finite type\nmorphism of algebraic spaces.\nThen there exists an $i \\in I$ and a diagram\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\nX \\ar[r] \\ar[d] & W \\ar[d] \\\\\nB \\ar[r] & B_i\n}\n}\n\\end{equation}\nsuch that $W \\to B_i$ is of finite type and such that\nthe induced morphism $X \\to B \\times_{B_i} W$ is a closed\nimmersion.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending finite type spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CP7","source_file":"spaces-limits.tex","source_line":5040,"source_end_line":5058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L5040-L5058","statement_sha256":"eff741f281212c341f7ec2b28b4557f21dd94da6b03254ea2bc16ed7d8ee75f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11508,"rank":11508,"depth":66,"x":613.048,"y":1627.488,"cluster":"geometry-of-spaces"},{"id":"stacks:0CP9","tag":"0CP9","title":"Descending finite type spaces · Lemma 0CP9","summary":"In Situation [Tag 0CP6]. Let X → B be a quasi-separated and finite type morphism of algebraic spaces. Given i ∈ I and a diagram vcenter xymatrix X ar[r] ar[d] & W ar[d] B ar[r] & B_i as in ([Tag 0CP8]) for i' ≥ i let X_i' be the scheme theoretic image of X → B_i' ×_B_i W. Then X = lim_i' ≥ i X_i'.","statement_latex":"In Situation \\ref{situation-limit-noetherian}.\nLet $X \\to B$ be a quasi-separated and finite type morphism\nof algebraic spaces. Given $i \\in I$ and a diagram\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[r] \\ar[d] & W \\ar[d] \\\\\nB \\ar[r] & B_i\n}\n}\n$$\nas in (\\ref{equation-good-diagram}) for $i' \\geq i$ let\n$X_{i'}$ be the scheme theoretic image of $X \\to B_{i'} \\times_{B_i} W$.\nThen $X = \\lim_{i' \\geq i} X_{i'}$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending finite type spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CP9","source_file":"spaces-limits.tex","source_line":5069,"source_end_line":5085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L5069-L5085","statement_sha256":"92325203d83e2e8b9fe5788be1b622ef159bc251f2d7d8fdabf74fe013e7f86e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11509,"rank":11509,"depth":67,"x":707.336,"y":1541.587,"cluster":"geometry-of-spaces"},{"id":"stacks:0CPA","tag":"0CPA","title":"Descending finite type spaces · Lemma 0CPA","summary":"In Situation [Tag 0CP6]. Let f : X → Y be a morphism of algebraic spaces quasi-separated and of finite type over B. Let vcenter xymatrix X ar[r] ar[d] & W ar[d] B ar[r] & B_i_1 and vcenter xymatrix Y ar[r] ar[d] & V ar[d] B ar[r] & B_i_2 be diagrams as in ([Tag 0CP8]). Let X = lim_i ≥ i_1 X_i and Y = lim_i ≥ i_2 Y_i be the corresponding limit descriptions as in Lemma [Tag 0CP9]. Then there exists an i_0 ≥ max(i_1, i_2) and a morphism (f_i)_i ≥ i_0 : (X_i)_i ≥ i_0 →…","statement_latex":"In Situation \\ref{situation-limit-noetherian}.\nLet $f : X \\to Y$ be a morphism of algebraic spaces quasi-separated\nand of finite type over $B$. Let\n$$\n\\vcenter{\n\\xymatrix{\nX \\ar[r] \\ar[d] & W \\ar[d] \\\\\nB \\ar[r] & B_{i_1}\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\nY \\ar[r] \\ar[d] & V \\ar[d] \\\\\nB \\ar[r] & B_{i_2}\n}\n}\n$$\nbe diagrams as in (\\ref{equation-good-diagram}). Let\n$X = \\lim_{i \\geq i_1} X_i$ and\n$Y = \\lim_{i \\geq i_2} Y_i$ be the corresponding\nlimit descriptions as in Lemma \\ref{lemma-limit-from-good-diagram}.\nThen there exists an $i_0 \\geq \\max(i_1, i_2)$ and a morphism\n$$\n(f_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}\n$$\nof inverse systems over $(B_i)_{i \\geq i_0}$ such that\nsuch that $f = \\lim_{i \\geq i_0} f_i$.\nIf $(g_i)_{i \\geq i_0} : (X_i)_{i \\geq i_0} \\to (Y_i)_{i \\geq i_0}$\nis a second morphism of inverse systems over $(B_i)_{i \\geq i_0}$ such that\nsuch that $f = \\lim_{i \\geq i_0} g_i$\nthen $f_i = g_i$ for all $i \\gg i_0$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending finite type spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPA","source_file":"spaces-limits.tex","source_line":5106,"source_end_line":5140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L5106-L5140","statement_sha256":"d9ffb83be435fba52ecb4c3fa9eb2f104a137d2430816c3ad66c8ef42e1255d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11510,"rank":11510,"depth":68,"x":706.888,"y":1658.738,"cluster":"geometry-of-spaces"},{"id":"stacks:0CPC","tag":"0CPC","title":"Descending finite type spaces · Lemma 0CPC","summary":"Notation and assumptions as in Lemma [Tag 0CPA]. If f is flat and of finite presentation, then there exists an i_3 > i_0 such that for i ≥ i_3 we have f_i is flat, X_i = Y_i ×_Y_i_3 X_i_3, and X = Y ×_Y_i_3 X_i_3.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.\nIf $f$ is flat and of finite presentation, then\nthere exists an $i_3 > i_0$ such that for $i \\geq i_3$ we have\n$f_i$ is flat, $X_i = Y_i \\times_{Y_{i_3}} X_{i_3}$, and\n$X = Y \\times_{Y_{i_3}} X_{i_3}$.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending finite type spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPC","source_file":"spaces-limits.tex","source_line":5206,"source_end_line":5213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L5206-L5213","statement_sha256":"f4a1ed9c2ea146829edb3d33668601042b2d81fe07590dd746efe55459548015","origin":"The Stacks Project","memory_eligible":false,"source_rank":11511,"rank":11511,"depth":69,"x":612.763,"y":1571.871,"cluster":"geometry-of-spaces"},{"id":"stacks:0CPD","tag":"0CPD","title":"Descending finite type spaces · Lemma 0CPD","summary":"Notation and assumptions as in Lemma [Tag 0CPA]. If f is smooth, then there exists an i_3 > i_0 such that for i ≥ i_3 we have f_i is smooth.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.\nIf $f$ is smooth, then there exists an $i_3 > i_0$ such that for\n$i \\geq i_3$ we have $f_i$ is smooth.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending finite type spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPD","source_file":"spaces-limits.tex","source_line":5244,"source_end_line":5249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L5244-L5249","statement_sha256":"ad38dc7cc328f8f02e4ade211c9ccaa9001ab35d1c85757b7c488ca1d75e9fd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11512,"rank":11512,"depth":70,"x":752.388,"y":1582.545,"cluster":"geometry-of-spaces"},{"id":"stacks:0CPE","tag":"0CPE","title":"Descending finite type spaces · Lemma 0CPE","summary":"Notation and assumptions as in Lemma [Tag 0CPA]. If f is proper, then there exists an i_3 ≥ i_0 such that for i ≥ i_3 we have f_i is proper.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-morphism-good-diagram}.\nIf $f$ is proper, then there exists an $i_3 \\geq i_0$ such that for\n$i \\geq i_3$ we have $f_i$ is proper.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending finite type spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPE","source_file":"spaces-limits.tex","source_line":5256,"source_end_line":5261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L5256-L5261","statement_sha256":"fd618d138e0bab5ebb489a7f1130a9c5c0fe75600e6c7b6b7e12bddd1c65e6b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11513,"rank":11513,"depth":69,"x":640.558,"y":1654.086,"cluster":"geometry-of-spaces"},{"id":"stacks:0CPF","tag":"0CPF","title":"Descending finite type spaces · Lemma 0CPF","summary":"In Situation [Tag 0CP6] suppose that we have a cartesian diagram xymatrix X^1 ar[r]_p ar[d]_q & X^3 ar[d]^a X^2 ar[r]^b & X^4 of algebraic spaces quasi-separated and of finite type over B. For each j = 1, 2, 3, 4 choose i_j ∈ I and a diagram xymatrix X^j ar[r] ar[d] & W^j ar[d] B ar[r] & B_i_j as in ([Tag 0CP8]). Let X^j = lim_i ≥ i_j X^j_i be the corresponding limit descriptions as in Lemma [Tag 0CPA]. Let (a_i)_i ≥ i_5, (b_i)_i ≥ i_6, (p_i)_i ≥ i_7, and (q_i)_i ≥ i_8 be…","statement_latex":"In Situation \\ref{situation-limit-noetherian} suppose that we have a\ncartesian diagram\n$$\n\\xymatrix{\nX^1 \\ar[r]_p \\ar[d]_q & X^3 \\ar[d]^a \\\\\nX^2 \\ar[r]^b & X^4\n}\n$$\nof algebraic spaces quasi-separated and of finite type over $B$.\nFor each $j = 1, 2, 3, 4$ choose $i_j \\in I$ and a diagram\n$$\n\\xymatrix{\nX^j \\ar[r] \\ar[d] & W^j \\ar[d] \\\\\nB \\ar[r] & B_{i_j}\n}\n$$\nas in (\\ref{equation-good-diagram}). Let\n$X^j = \\lim_{i \\geq i_j} X^j_i$ be the corresponding limit descriptions\nas in Lemma \\ref{lemma-morphism-good-diagram}.\nLet $(a_i)_{i \\geq i_5}$, $(b_i)_{i \\geq i_6}$, $(p_i)_{i \\geq i_7}$, and\n$(q_i)_{i \\geq i_8}$ be the corresponding morphisms of inverse systems\nconstructed in Lemma \\ref{lemma-morphism-good-diagram}. Then there exists an\n$i_9 \\geq \\max(i_5, i_6, i_7, i_8)$ such that for $i \\geq i_9$ we have\n$a_i \\circ p_i = b_i \\circ q_i$ and such that\n$$\n(q_i, p_i) : X^1_i \\longrightarrow X^2_i \\times_{b_i, X^4_i, a_i} X^3_i\n$$\nis a closed immersion.\nIf $a$ and $b$ are flat and of finite presentation, then there exists an\n$i_{10} \\geq \\max(i_5, i_6, i_7, i_8, i_9)$ such that for $i \\geq i_{10}$\nthe last displayed morphism is an isomorphism.","area":"Geometry of Spaces","chapter":"Limits of Algebraic Spaces","chapter_id":"spaces-limits","section":"Descending finite type spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPF","source_file":"spaces-limits.tex","source_line":5282,"source_end_line":5315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-limits.tex#L5282-L5315","statement_sha256":"72e759a639bb28be50a4dc0a063b12d78b1a5aa90d86acabc50de037a22b7db7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11514,"rank":11514,"depth":70,"x":665.553,"y":1537.577,"cluster":"geometry-of-spaces"},{"id":"stacks:0CTW","tag":"0CTW","title":"Associated and weakly associated points · Lemma 0CTW","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Let x ∈ |X|. The following are equivalent • for some étale morphism f : U → X with U a scheme and u ∈ U mapping to x, the point u is weakly associated to f^*F, • for every étale morphism f : U → X with U a scheme and u ∈ U mapping to x, the point u is weakly associated to f^*F, • the maximal ideal of O_X, overlinex is a weakly associated prime of the stalk F_overlinex. If X is…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in |X|$. The following are equivalent\n\\begin{enumerate}\n\\item for some \\'etale morphism $f : U \\to X$ with $U$ a scheme\nand $u \\in U$ mapping to $x$, the point $u$ is weakly associated\nto $f^*\\mathcal{F}$,\n\\item for every \\'etale morphism $f : U \\to X$ with $U$ a scheme\nand $u \\in U$ mapping to $x$, the point $u$ is weakly associated\nto $f^*\\mathcal{F}$,\n\\item the maximal ideal of $\\mathcal{O}_{X, \\overline{x}}$\nis a weakly associated prime of the stalk $\\mathcal{F}_{\\overline{x}}$.\n\\end{enumerate}\nIf $X$ is locally Noetherian, then these are also equivalent to\n\\begin{enumerate}\n\\item[(4)] for some \\'etale morphism $f : U \\to X$ with $U$ a scheme\nand $u \\in U$ mapping to $x$, the point $u$ is associated\nto $f^*\\mathcal{F}$,\n\\item[(5)] for every \\'etale morphism $f : U \\to X$ with $U$ a scheme\nand $u \\in U$ mapping to $x$, the point $u$ is associated\nto $f^*\\mathcal{F}$,\n\\item[(6)] the maximal ideal of $\\mathcal{O}_{X, \\overline{x}}$\nis an associated prime of the stalk $\\mathcal{F}_{\\overline{x}}$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTW","source_file":"spaces-divisors.tex","source_line":49,"source_end_line":75,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L49-L75","statement_sha256":"a717360569d6116a86d9caccc4eb821ee127041c4bd56b329aeae476ee93d8f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11515,"rank":11515,"depth":54,"x":741.007,"y":1637.929,"cluster":"geometry-of-spaces"},{"id":"stacks:0CTX","tag":"0CTX","title":"Associated and weakly associated points · Definition 0CTX","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent sheaf on X. Let x ∈ |X|. • We say x is weakly associated to F if the equivalent conditions (1), (2), and (3) of Lemma [Tag 0CTW] are satisfied. • We denote WeakAss(F) the set of weakly associated points of F. • The weakly associated points of X are the weakly associated points of O_X. If X is locally Noetherian we will say x is associated to F if and only if x is weakly associated to F and we…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $x \\in |X|$.\n\\begin{enumerate}\n\\item We say $x$ is {\\it weakly associated} to $\\mathcal{F}$\nif the equivalent conditions (1), (2), and (3) of\nLemma \\ref{lemma-associated} are satisfied.\n\\item We denote $\\text{WeakAss}(\\mathcal{F})$ the set of weakly associated\npoints of $\\mathcal{F}$.\n\\item The {\\it weakly associated points of $X$} are the weakly associated\npoints of $\\mathcal{O}_X$.\n\\end{enumerate}\nIf $X$ is locally Noetherian we will say\n{\\it $x$ is associated to $\\mathcal{F}$}\nif and only if $x$ is weakly associated to $\\mathcal{F}$ and we set\n$\\text{Ass}(\\mathcal{F}) = \\text{WeakAss}(\\mathcal{F})$.\nFinally (still assuming $X$ is locally Noetherian),\nwe will say {\\it $x$ is an associated point of $X$} if and only if\n$x$ is a weakly associated point of $X$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTX","source_file":"spaces-divisors.tex","source_line":110,"source_end_line":131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L110-L131","statement_sha256":"dc017baf8e30b970c193c1761358680ec758998343970a1ef4195da556e5d4b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11516,"rank":11516,"depth":55,"x":604.321,"y":1606.666,"cluster":"geometry-of-spaces"},{"id":"stacks:0CTY","tag":"0CTY","title":"Associated and weakly associated points · Lemma 0CTY","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Then WeakAss(F) ⊂ Supp(F).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $\\text{WeakAss}(\\mathcal{F}) \\subset \\text{Supp}(\\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTY","source_file":"spaces-divisors.tex","source_line":136,"source_end_line":141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L136-L141","statement_sha256":"267b42fd570ef5b5178ac42b3002eea92407fa0b9a4ac7050d617360ca0818c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11517,"rank":11517,"depth":48,"x":730.572,"y":1552.021,"cluster":"geometry-of-spaces"},{"id":"stacks:0CTZ","tag":"0CTZ","title":"Associated and weakly associated points · Lemma 0CTZ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let 0 → F_1 → F_2 → F_3 → 0 be a short exact sequence of quasi-coherent sheaves on X. Then WeakAss(F_2) ⊂ WeakAss(F_1) ∪ WeakAss(F_3) and WeakAss(F_1) ⊂ WeakAss(F_2).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nbe a short exact sequence of quasi-coherent sheaves on $X$.\nThen\n$\\text{WeakAss}(\\mathcal{F}_2) \\subset\n\\text{WeakAss}(\\mathcal{F}_1) \\cup \\text{WeakAss}(\\mathcal{F}_3)$\nand\n$\\text{WeakAss}(\\mathcal{F}_1) \\subset \\text{WeakAss}(\\mathcal{F}_2)$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTZ","source_file":"spaces-divisors.tex","source_line":149,"source_end_line":159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L149-L159","statement_sha256":"455654bed814481dbb35b130829a66719e1cee2b17739653294137daabe18957","origin":"The Stacks Project","memory_eligible":false,"source_rank":11518,"rank":11518,"depth":2,"x":681.295,"y":1664.236,"cluster":"geometry-of-spaces"},{"id":"stacks:0CU0","tag":"0CU0","title":"Associated and weakly associated points · Lemma 0CU0","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Then F = (0) ⇔ WeakAss(F) = ∅","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen\n$$\n\\mathcal{F} = (0) \\Leftrightarrow \\text{WeakAss}(\\mathcal{F}) = \\emptyset\n$$","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CU0","source_file":"spaces-divisors.tex","source_line":172,"source_end_line":180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L172-L180","statement_sha256":"a562a4342b25daef88a3a77cc4f25423e575eb61bd1ec371482cea70b6325f8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11519,"rank":11519,"depth":4,"x":627.256,"y":1553.252,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUL","tag":"0CUL","title":"Associated and weakly associated points · Lemma 0CUL","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Let x ∈ |X|. If • x ∈ Supp(F) • x is a codimension 0 point of X (Properties of Spaces, Definition [Tag 04NA]). Then x ∈ WeakAss(F). If F is a finite type O_X-module with scheme theoretic support Z (Morphisms of Spaces, Definition [Tag 07U1]) and x is a codimension 0 point of Z, then x ∈ WeakAss(F).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in |X|$. If\n\\begin{enumerate}\n\\item $x \\in \\text{Supp}(\\mathcal{F})$\n\\item $x$ is a codimension $0$ point of $X$\n(Properties of Spaces, Definition\n\\ref{spaces-properties-definition-dimension-local-ring}).\n\\end{enumerate}\nThen $x \\in \\text{WeakAss}(\\mathcal{F})$. If $\\mathcal{F}$\nis a finite type $\\mathcal{O}_X$-module with scheme theoretic support $Z$\n(Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-scheme-theoretic-support})\nand $x$ is a codimension $0$ point of $Z$, then\n$x \\in \\text{WeakAss}(\\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUL","source_file":"spaces-divisors.tex","source_line":190,"source_end_line":207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L190-L207","statement_sha256":"27078b4df34cfca3c31b5d073746933ac73560441f851f906ee1c0dc0a2e1b3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11520,"rank":11520,"depth":57,"x":756.677,"y":1604.555,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUM","tag":"0CUM","title":"Associated and weakly associated points · Lemma 0CUM","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Let x ∈ |X|. If • X is decent (for example quasi-separated or locally separated), • x ∈ Supp(F) • x is not a specialization of another point in Supp(F). Then x ∈ WeakAss(F).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in |X|$. If\n\\begin{enumerate}\n\\item $X$ is decent (for example quasi-separated or locally separated),\n\\item $x \\in \\text{Supp}(\\mathcal{F})$\n\\item $x$ is not a specialization of another point in\n$\\text{Supp}(\\mathcal{F})$.\n\\end{enumerate}\nThen $x \\in \\text{WeakAss}(\\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUM","source_file":"spaces-divisors.tex","source_line":224,"source_end_line":236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L224-L236","statement_sha256":"93cdb4473a5892dcd5b59cd3bec5e6d6c8c9672ca1ad51fd7e01a2d9242aec11","origin":"The Stacks Project","memory_eligible":false,"source_rank":11521,"rank":11521,"depth":58,"x":619.647,"y":1640.249,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUN","tag":"0CUN","title":"Associated and weakly associated points · Lemma 0CUN","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let F be a coherent O_X-module. Then Ass(F) ∩ W is finite for every quasi-compact open W ⊂ |X|.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nThen $\\text{Ass}(\\mathcal{F}) \\cap W$ is finite for\nevery quasi-compact open $W \\subset |X|$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUN","source_file":"spaces-divisors.tex","source_line":255,"source_end_line":261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L255-L261","statement_sha256":"241edabeab259e633fcfb253675191c8d7286b913a2c3d36531e8b623326f8e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11522,"rank":11522,"depth":9,"x":692.166,"y":1535.917,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUP","tag":"0CUP","title":"Associated and weakly associated points · Lemma 0CUP","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. If U → X is an étale morphism such that WeakAss(F) ⊂ Im(|U| → |X|), then Γ(X, F) → Γ(U, F) is injective.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $U \\to X$ is an \\'etale morphism such that\n$\\text{WeakAss}(\\mathcal{F}) \\subset \\Im(|U| \\to |X|)$, then\n$\\Gamma(X, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})$ is injective.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUP","source_file":"spaces-divisors.tex","source_line":270,"source_end_line":277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L270-L277","statement_sha256":"9eeb25e8c9c4635d1b31639c41d4cdd525d2e227df4133d69d15ad7b648e0ad5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11523,"rank":11523,"depth":5,"x":722.667,"y":1654.29,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUQ","tag":"0CUQ","title":"Associated and weakly associated points · Lemma 0CUQ","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Let y ∈ |Y| be a point which is not in the image of |f|. Then y is not weakly associated to f_*F.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated\nmorphism of algebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Let $y \\in |Y|$ be a point which is not in the\nimage of $|f|$. Then $y$ is not weakly associated to $f_*\\mathcal{F}$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUQ","source_file":"spaces-divisors.tex","source_line":293,"source_end_line":299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L293-L299","statement_sha256":"40d3a2327b39022b2d53903681958be4a1bbb0eb0a103924d4eed5159096f590","origin":"The Stacks Project","memory_eligible":false,"source_rank":11524,"rank":11524,"depth":43,"x":604.696,"y":1584.139,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUR","tag":"0CUR","title":"Associated and weakly associated points · Lemma 0CUR","summary":"Let S be a scheme. Let X be an algebraic space over S. Let φ : F → G be a map of quasi-coherent O_X-modules. Assume that for every x ∈ |X| at least one of the following happens • F_overlinex → G_overlinex is injective, or • x not ∈ WeakAss(F). Then φ is injective.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a map of\nquasi-coherent $\\mathcal{O}_X$-modules. Assume that for every\n$x \\in |X|$ at least one of the following happens\n\\begin{enumerate}\n\\item $\\mathcal{F}_{\\overline{x}} \\to \\mathcal{G}_{\\overline{x}}$\nis injective, or\n\\item $x \\not \\in \\text{WeakAss}(\\mathcal{F})$.\n\\end{enumerate}\nThen $\\varphi$ is injective.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUR","source_file":"spaces-divisors.tex","source_line":319,"source_end_line":331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L319-L331","statement_sha256":"c2c337761b959e8132d161bc4a0cd182c17b3ada66091f78c65541d4fab3f060","origin":"The Stacks Project","memory_eligible":false,"source_rank":11525,"rank":11525,"depth":5,"x":748.449,"y":1568.89,"cluster":"geometry-of-spaces"},{"id":"stacks:0EN1","tag":"0EN1","title":"Associated and weakly associated points · Lemma 0EN1","summary":"Let S be a scheme. Let X be a reduced algebraic space over S. Then the weakly associated point of X are exactly the codimension 0 points of X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a reduced algebraic space over $S$.\nThen the weakly associated point of $X$ are exactly the codimension $0$\npoints of $X$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Associated and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EN1","source_file":"spaces-divisors.tex","source_line":338,"source_end_line":343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L338-L343","statement_sha256":"ea07632ecf96f36f6c6d461f9a97d88c9eb955eed32d90689faa4dd5d03d8096","origin":"The Stacks Project","memory_eligible":false,"source_rank":11526,"rank":11526,"depth":18,"x":654.483,"y":1661.929,"cluster":"geometry-of-spaces"},{"id":"stacks:0CU2","tag":"0CU2","title":"Morphisms and weakly associated points · Lemma 0CU2","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Then we have WeakAss_S(f_*F) ⊂ f(WeakAss_X(F))","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen we have\n$$\n\\text{WeakAss}_S(f_*\\mathcal{F}) \\subset f(\\text{WeakAss}_X(\\mathcal{F}))\n$$","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CU2","source_file":"spaces-divisors.tex","source_line":364,"source_end_line":373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L364-L373","statement_sha256":"5a7d27334508122bc16b9dea2faed514353a3796d8acaec1a188e885cbeefadd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11527,"rank":11527,"depth":3,"x":648.94,"y":1539.71,"cluster":"geometry-of-spaces"},{"id":"stacks:0CU8","tag":"0CU8","title":"Morphisms and weakly associated points · Lemma 0CU8","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. If X is locally Noetherian, then we have WeakAss_Y(f_*F) = f(WeakAss_X(F))","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be an affine morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $X$ is locally Noetherian, then we have\n$$\n\\text{WeakAss}_Y(f_*\\mathcal{F}) =\nf(\\text{WeakAss}_X(\\mathcal{F}))\n$$","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CU8","source_file":"spaces-divisors.tex","source_line":383,"source_end_line":393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L383-L393","statement_sha256":"5e095b68e635fb7f26572c7436f727341385709a60cd08c6ce0a3b320ea36465","origin":"The Stacks Project","memory_eligible":false,"source_rank":11528,"rank":11528,"depth":10,"x":751.557,"y":1626.897,"cluster":"geometry-of-spaces"},{"id":"stacks:0CU9","tag":"0CU9","title":"Morphisms and weakly associated points · Lemma 0CU9","summary":"Let S be a scheme. Let f : X → Y be a finite morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Then WeakAss(f_*F) = f(WeakAss(F)).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a finite morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $\\text{WeakAss}(f_*\\mathcal{F}) = f(\\text{WeakAss}(\\mathcal{F}))$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CU9","source_file":"spaces-divisors.tex","source_line":404,"source_end_line":410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L404-L410","statement_sha256":"fe5f860e423750b200ced6ceaa6f351c7b5fe5c16a3f1fdb996bae8c26dc94c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11529,"rank":11529,"depth":11,"x":605.428,"y":1620.82,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUA","tag":"0CUA","title":"Morphisms and weakly associated points · Lemma 0CUA","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let G be a quasi-coherent O_Y-module. Let x ∈ |X| and y = f(x) ∈ |Y|. If • y ∈ WeakAss_S(G), • f is flat at x, and • the dimension of the local ring of the fibre of f at x is zero (Morphisms of Spaces, Definition [Tag 04NM]), then x ∈ WeakAss(f^*G).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.\nLet $x \\in |X|$ and $y = f(x) \\in |Y|$. If\n\\begin{enumerate}\n\\item $y \\in \\text{WeakAss}_S(\\mathcal{G})$,\n\\item $f$ is flat at $x$, and\n\\item the dimension of the local ring of the fibre of $f$ at $x$\nis zero (Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-dimension-fibre}),\n\\end{enumerate}\nthen $x \\in \\text{WeakAss}(f^*\\mathcal{G})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUA","source_file":"spaces-divisors.tex","source_line":420,"source_end_line":433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L420-L433","statement_sha256":"0ace28da9e64fe4c6098ea1c6cd8efed15075f7861a19a6e96a8d7011c57a2f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11530,"rank":11530,"depth":48,"x":718.337,"y":1542.197,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUS","tag":"0CUS","title":"Morphisms and weakly associated points · Lemma 0CUS","summary":"Let K/k be a field extension. Let X be an algebraic space over k. Let F be a quasi-coherent O_X-module. Let y ∈ X_K with image x ∈ X. If y is a weakly associated point of the pullback F_K, then x is a weakly associated point of F.","statement_latex":"Let $K/k$ be a field extension. Let $X$ be an algebraic space over $k$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $y \\in X_K$ with image $x \\in X$. If $y$ is a weakly\nassociated point of the pullback $\\mathcal{F}_K$, then $x$\nis a weakly associated point of $\\mathcal{F}$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUS","source_file":"spaces-divisors.tex","source_line":450,"source_end_line":457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L450-L457","statement_sha256":"9a29d6a91b29d3f6ffb07c3c48e130e3166001fd84f2b4ef518ef699ed20ab8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11531,"rank":11531,"depth":12,"x":698.258,"y":1664.528,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUT","tag":"0CUT","title":"Morphisms and weakly associated points · Lemma 0CUT","summary":"Let S be a scheme. Let f : X → Y be a finite flat morphism of algebraic spaces. Let G be a quasi-coherent O_Y-module. Let x ∈ |X| be a point with image y ∈ |Y|. Then x ∈ WeakAss(g^*G) ⇔ y ∈ WeakAss(G)","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a finite flat morphism of algebraic spaces.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.\nLet $x \\in |X|$ be a point with image $y \\in |Y|$. Then\n$$\nx \\in \\text{WeakAss}(g^*\\mathcal{G})\n\\Leftrightarrow\ny \\in \\text{WeakAss}(\\mathcal{G})\n$$","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUT","source_file":"spaces-divisors.tex","source_line":466,"source_end_line":477,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L466-L477","statement_sha256":"be434f0ea2a31c2d2e4d0ffa8434a45eff91823020f27665e04f2a88bf81baf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11532,"rank":11532,"depth":12,"x":614.491,"y":1562.69,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUU","tag":"0CUU","title":"Morphisms and weakly associated points · Lemma 0CUU","summary":"Let S be a scheme. Let f : X → Y be an étale morphism of algebraic spaces. Let G be a quasi-coherent O_Y-module. Let x ∈ |X| be a point with image y ∈ |Y|. Then x ∈ WeakAss(f^*G) ⇔ y ∈ WeakAss(G)","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be an \\'etale morphism of algebraic spaces.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.\nLet $x \\in |X|$ be a point with image $y \\in |Y|$. Then\n$$\nx \\in \\text{WeakAss}(f^*\\mathcal{G})\n\\Leftrightarrow\ny \\in \\text{WeakAss}(\\mathcal{G})\n$$","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Morphisms and weakly associated points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUU","source_file":"spaces-divisors.tex","source_line":485,"source_end_line":496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L485-L496","statement_sha256":"f58e6edd6cf30d0e2432a1cfe28a180f5407adc515a253e9437cdaee3ad457e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11533,"rank":11533,"depth":45,"x":758.492,"y":1590.318,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUW","tag":"0CUW","title":"Relative weak assassin · Lemma 0CUW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let y ∈ |Y|. The following are equivalent • for some scheme V, point v ∈ V, and étale morphism V → Y mapping v to y, the algebraic space X_v is locally Noetherian, • for every scheme V, point v ∈ V, and étale morphism V → Y mapping v to y, the algebraic space X_v is locally Noetherian, and • there exists a field k and a morphism Spec(k) → Y representing y such that X_k is locally Noetherian. If…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $y \\in |Y|$. The following are equivalent\n\\begin{enumerate}\n\\item for some scheme $V$, point $v \\in V$, and \\'etale morphism $V \\to Y$\nmapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian,\n\\item for every scheme $V$, point $v \\in V$, and \\'etale morphism $V \\to Y$\nmapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian, and\n\\item there exists a field $k$ and a morphism $\\Spec(k) \\to Y$ representing\n$y$ such that $X_k$ is locally Noetherian.\n\\end{enumerate}\nIf there exists a field $k_0$ and a monomorphism $\\Spec(k_0) \\to Y$\nrepresenting $y$, then these are also equivalent to\n\\begin{enumerate}\n\\item[(4)] the algebraic space $X_{k_0}$ is locally Noetherian.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUW","source_file":"spaces-divisors.tex","source_line":517,"source_end_line":534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L517-L534","statement_sha256":"1177e3783afd0c5723e8860a4f7477bd88a2a36630e6bbc7ed57bc9f10bba734","origin":"The Stacks Project","memory_eligible":false,"source_rank":11534,"rank":11534,"depth":45,"x":629.791,"y":1651.798,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUX","tag":"0CUX","title":"Relative weak assassin · Definition 0CUX","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let y ∈ |Y|. We say the fibre of f over y is locally Noetherian if the equivalent conditions (1), (2), and (3) of Lemma [Tag 0CUW] are satisfied. We say the fibres of f are locally Noetherian if this holds for every y ∈ |Y|.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $y \\in |Y|$. We say {\\it the fibre of $f$ over $y$ is\nlocally Noetherian} if the equivalent conditions (1), (2), and (3)\nof Lemma \\ref{lemma-locally-noetherian-fibre} are satisfied.\nWe say {\\it the fibres of $f$ are locally Noetherian} if this\nholds for every $y \\in |Y|$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUX","source_file":"spaces-divisors.tex","source_line":573,"source_end_line":581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L573-L581","statement_sha256":"fa3c92e88d9d3b099d7a3624c5bd4871534e4fbcec5df17fa35d699ebc0e1ab6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11535,"rank":11535,"depth":46,"x":675.358,"y":1533.161,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUY","tag":"0CUY","title":"Relative weak assassin · Lemma 0CUY","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally of finite type, then the fibres of f are locally Noetherian.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. If $f$ is locally of finite type, then \nthe fibres of $f$ are locally Noetherian.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUY","source_file":"spaces-divisors.tex","source_line":587,"source_end_line":592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L587-L592","statement_sha256":"f9c90d38d2610a5bde138e4ce72b58df4db2c4d6230e0d0d39a0c4e4e9534ae2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11536,"rank":11536,"depth":3,"x":737.305,"y":1646.76,"cluster":"geometry-of-spaces"},{"id":"stacks:0CUZ","tag":"0CUZ","title":"Relative weak assassin · Lemma 0CUZ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let x ∈ |X| and y = f(x) ∈ |Y|. Let F be a quasi-coherent O_X-module. Consider commutative diagrams xymatrix X ar[d] & X ×_Y V ar[d] ar[l] & X_v ar[d] ar[l] Y & V ar[l] & v ar[l] xymatrix X ar[d] & U ar[d] ar[l] & U_v ar[d] ar[l] Y & V ar[l] & v ar[l] xymatrix x ar@|->[d] & x' ar@|->[d] ar@|->[l] & u ar@|->[ld] ar@|->[l] y & v ar@|->[l] where V and U are schemes, V → Y and U → X ×_Y V are étale, v…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $x \\in |X|$ and $y = f(x) \\in |Y|$.\nLet $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module. Consider commutative diagrams\n$$\n\\xymatrix{\nX \\ar[d] & X \\times_Y V \\ar[d] \\ar[l] & X_v \\ar[d] \\ar[l] \\\\\nY & V \\ar[l] & v \\ar[l]\n}\n\\quad\n\\xymatrix{\nX \\ar[d] & U \\ar[d] \\ar[l] & U_v \\ar[d] \\ar[l] \\\\\nY & V \\ar[l] & v \\ar[l]\n}\n\\quad\n\\xymatrix{\nx \\ar@{|->}[d] &\nx' \\ar@{|->}[d] \\ar@{|->}[l] &\nu \\ar@{|->}[ld] \\ar@{|->}[l] \\\\\ny &\nv \\ar@{|->}[l]\n}\n$$\nwhere $V$ and $U$ are schemes, $V \\to Y$ and $U \\to X \\times_Y V$\nare \\'etale, $v \\in V$, $x' \\in |X_v|$, $u \\in U$ are points\nrelated as in the last diagram.\nDenote $\\mathcal{F}|_{X_v}$ and $\\mathcal{F}|_{U_v}$\nthe pullbacks of $\\mathcal{F}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some $V, v, x'$ as above $x'$ is a weakly associated\npoint of $\\mathcal{F}|_{X_v}$,\n\\item for every $V \\to Y, v, x'$ as above $x'$ is a weakly associated\npoint of $\\mathcal{F}|_{X_v}$,\n\\item for some $U, V, u, v$ as above $u$ is a weakly associated\npoint of $\\mathcal{F}|_{U_v}$,\n\\item for every $U, V, u, v$ as above $u$ is a weakly associated\npoint of $\\mathcal{F}|_{U_v}$,\n\\item for some field $k$ and morphism $\\Spec(k) \\to Y$ representing $y$\nand some $t \\in |X_k|$ mapping to $x$, the point $t$ is a weakly\nassociated point of $\\mathcal{F}|_{X_k}$.\n\\end{enumerate}\nIf there exists a field $k_0$ and a monomorphism $\\Spec(k_0) \\to Y$\nrepresenting $y$, then these are also equivalent to\n\\begin{enumerate}\n\\item[(6)] $x_0$ is a weakly associated point of $\\mathcal{F}|_{X_{k_0}}$\nwhere $x_0 \\in |X_{k_0}|$ is the unique point mapping to $x$.\n\\end{enumerate}\nIf the fibre of $f$ over $y$ is locally Noetherian, then in\nconditions (1), (2), (3), (4), and (6) we may replace\n``weakly associated'' with ``associated''.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CUZ","source_file":"spaces-divisors.tex","source_line":600,"source_end_line":653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L600-L653","statement_sha256":"f5c3272e6bdcaa8bcc2471b18e6768b8fce588ed79f60cc5b26fc08a42696fa8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11537,"rank":11537,"depth":46,"x":599.957,"y":1598.031,"cluster":"geometry-of-spaces"},{"id":"stacks:0CV0","tag":"0CV0","title":"Relative weak assassin · Definition 0CV0","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. The relative weak assassin of F in X over Y is the set WeakAss_X/Y(F) ⊂ |X| consisting of those x ∈ |X| such that the equivalent conditions of Lemma [Tag 0CUZ] are satisfied. If the fibres of f are locally Noetherian (Definition [Tag 0CUX]) then we use the notation Ass_X/Y(F).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThe {\\it relative weak assassin of $\\mathcal{F}$ in $X$ over $Y$}\nis the set $\\text{WeakAss}_{X/Y}(\\mathcal{F}) \\subset |X|$\nconsisting of those $x \\in |X|$ such that the equivalent conditions of\nLemma \\ref{lemma-relative-assassin} are satisfied.\nIf the fibres of $f$ are locally Noetherian\n(Definition \\ref{definition-locally-Noetherian-fibre})\nthen we use the notation $\\text{Ass}_{X/Y}(\\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CV0","source_file":"spaces-divisors.tex","source_line":693,"source_end_line":704,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L693-L704","statement_sha256":"08cf79fb0143c7982bcc216870af5f9059548fdf62474b5aef071676318ee29e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11538,"rank":11538,"depth":47,"x":740.744,"y":1555.933,"cluster":"geometry-of-spaces"},{"id":"stacks:0CV1","tag":"0CV1","title":"Relative weak assassin · Lemma 0CV1","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Let G be a quasi-coherent O_Y-module. Assume • F is flat over Y, • X and Y are locally Noetherian, and • the fibres of f are locally Noetherian. Then Ass_X(F ⊗_O_X f^*G) = (x ∈ Ass_X/Y(F) such that f(x) ∈ Ass_Y(G) )","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat over $Y$,\n\\item $X$ and $Y$ are locally Noetherian, and\n\\item the fibres of $f$ are locally Noetherian.\n\\end{enumerate}\nThen\n$$\n\\text{Ass}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) =\n\\{x \\in \\text{Ass}_{X/Y}(\\mathcal{F})\\text{ such that }\nf(x) \\in \\text{Ass}_Y(\\mathcal{G}) \\}\n$$","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CV1","source_file":"spaces-divisors.tex","source_line":710,"source_end_line":728,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L710-L728","statement_sha256":"2ee2aecc3647eca74dd20599bc4bf1c34878abc8300af8f0680c1f6a6b2afb9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11539,"rank":11539,"depth":14,"x":670.625,"y":1667.114,"cluster":"geometry-of-spaces"},{"id":"stacks:0CV2","tag":"0CV2","title":"Relative weak assassin · Lemma 0CV2","summary":"Let S be a scheme. Let xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f Y' ar[r]^g & Y be a cartesian diagram of algebraic spaces over S. Let F be a quasi-coherent O_X-module and set F' = (g')^*F. If f is locally of finite type, then • x' ∈ Ass_X'/Y'(F') ⇒ g'(x') ∈ Ass_X/Y(F) • if x ∈ Ass_X/Y(F), then given y' ∈ |Y'| with f(x) = g(y'), there exists an x' ∈ Ass_X'/Y'(F') with g'(x') = x and f'(x') = y'.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian diagram of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nand set $\\mathcal{F}' = (g')^*\\mathcal{F}$.\nIf $f$ is locally of finite type, then\n\\begin{enumerate}\n\\item $x' \\in \\text{Ass}_{X'/Y'}(\\mathcal{F}')\n\\Rightarrow g'(x') \\in \\text{Ass}_{X/Y}(\\mathcal{F})$\n\\item if $x \\in \\text{Ass}_{X/Y}(\\mathcal{F})$, then given\n$y' \\in |Y'|$ with $f(x) = g(y')$, there exists an\n$x' \\in \\text{Ass}_{X'/Y'}(\\mathcal{F}')$\nwith $g'(x') = x$ and $f'(x') = y'$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CV2","source_file":"spaces-divisors.tex","source_line":741,"source_end_line":762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L741-L762","statement_sha256":"21b5f977835308432db78339b5efce50b74ef709084fafe1eace84f3f4ad6038","origin":"The Stacks Project","memory_eligible":false,"source_rank":11540,"rank":11540,"depth":15,"x":632.835,"y":1545.067,"cluster":"geometry-of-spaces"},{"id":"stacks:0CV3","tag":"0CV3","title":"Relative weak assassin · Lemma 0CV3","summary":"With notation and assumptions as in Lemma [Tag 0CV2]. Assume g is locally quasi-finite, or more generally that for every y' ∈ |Y'| the transcendence degree of y'/g(y') is 0. Then Ass_X'/Y'(F') is the inverse image of Ass_X/Y(F).","statement_latex":"With notation and assumptions as in\nLemma \\ref{lemma-base-change-relative-assassin}.\nAssume $g$ is locally quasi-finite, or more generally that\nfor every $y' \\in |Y'|$ the transcendence degree of $y'/g(y')$ is $0$.\nThen $\\text{Ass}_{X'/Y'}(\\mathcal{F}')$ is the inverse image of\n$\\text{Ass}_{X/Y}(\\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CV3","source_file":"spaces-divisors.tex","source_line":803,"source_end_line":811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L803-L811","statement_sha256":"357bb73be8addf0b0937504072679d68e898d8851d8d1a7477bce44f60d63c0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11541,"rank":11541,"depth":48,"x":759.132,"y":1613.775,"cluster":"geometry-of-spaces"},{"id":"stacks:0CV4","tag":"0CV4","title":"Relative weak assassin · Lemma 0CV4","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let i : Z → X be a finite morphism. Let G be a quasi-coherent O_Z-module. Then WeakAss_X/Y(i_*G) = i(WeakAss_Z/Y(G)).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $i : Z \\to X$ be a finite morphism.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Z$-module.\nThen $\\text{WeakAss}_{X/Y}(i_*\\mathcal{G}) =\ni(\\text{WeakAss}_{Z/Y}(\\mathcal{G}))$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CV4","source_file":"spaces-divisors.tex","source_line":840,"source_end_line":848,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L840-L848","statement_sha256":"a418b4e9d4ebc77bd96ae06b6af2a4d677e956065dd52013f4a218339029e7fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11542,"rank":11542,"depth":27,"x":610.416,"y":1634.822,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVV","tag":"0CVV","title":"Relative weak assassin · Lemma 0CVV","summary":"Let Y be a scheme. Let X be an algebraic space of finite presentation over Y. Let F be a quasi-coherent O_X-module of finite presentation. Let U ⊂ X be an open subspace such that U → Y is quasi-compact. Then the set E = (y ∈ Y mid Ass_X_y(F_y) ⊂ |U_y|) is locally constructible in Y.","statement_latex":"Let $Y$ be a scheme. Let $X$ be an algebraic space of finite presentation\nover $Y$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nof finite presentation. Let $U \\subset X$ be an open subspace\nsuch that $U \\to Y$ is quasi-compact. Then the set\n$$\nE = \\{y \\in Y \\mid \\text{Ass}_{X_y}(\\mathcal{F}_y) \\subset |U_y|\\}\n$$\nis locally constructible in $Y$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative weak assassin","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVV","source_file":"spaces-divisors.tex","source_line":856,"source_end_line":866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L856-L866","statement_sha256":"e7c7d3cf76c4ee6e5106a80180cc8c23452de496bf53d0591565ddd029058e1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11543,"rank":11543,"depth":28,"x":703.36,"y":1534.694,"cluster":"geometry-of-spaces"},{"id":"stacks:0CZ4","tag":"0CZ4","title":"Fitting ideals · Lemma 0CZ4","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a finite type, quasi-coherent O_Y-module. Then f^-1Fit_i(F) · O_X = Fit_i(f^*F).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_Y$-module.\nThen\n$f^{-1}\\text{Fit}_i(\\mathcal{F}) \\cdot \\mathcal{O}_X =\n\\text{Fit}_i(f^*\\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZ4","source_file":"spaces-divisors.tex","source_line":935,"source_end_line":943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L935-L943","statement_sha256":"abca106c7f687467a9b2c690449bc1e9ed7e3dcc9d31776027ebd36ad1124311","origin":"The Stacks Project","memory_eligible":false,"source_rank":11544,"rank":11544,"depth":5,"x":715.371,"y":1661.547,"cluster":"geometry-of-spaces"},{"id":"stacks:0CZ5","tag":"0CZ5","title":"Fitting ideals · Lemma 0CZ5","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a finitely presented O_X-module. Then Fit_r(F) is a quasi-coherent ideal of finite type.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module.\nThen $\\text{Fit}_r(\\mathcal{F})$ is a quasi-coherent ideal of finite type.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZ5","source_file":"spaces-divisors.tex","source_line":951,"source_end_line":956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L951-L956","statement_sha256":"d4fe23da53aa2317c8978f68c7904fa4d9c7c83bf5689f3f82db377ba09295c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11545,"rank":11545,"depth":5,"x":604.256,"y":1574.63,"cluster":"geometry-of-spaces"},{"id":"stacks:0CZ6","tag":"0CZ6","title":"Fitting ideals · Lemma 0CZ6","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a finite type, quasi-coherent O_X-module. Let Z_0 ⊂ X be the closed subspace cut out by Fit_0(F). Let Z ⊂ X be the scheme theoretic support of F. Then • Z ⊂ Z_0 ⊂ X as closed subspaces, • |Z| = |Z_0| = Supp(F) as closed subsets of |X|, • there exists a finite type, quasi-coherent O_Z_0-module G_0 with (Z_0 → X)_*G_0 = F.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module.\nLet $Z_0 \\subset X$ be the closed subspace cut out by\n$\\text{Fit}_0(\\mathcal{F})$.\nLet $Z \\subset X$ be the scheme theoretic support of $\\mathcal{F}$.\nThen\n\\begin{enumerate}\n\\item $Z \\subset Z_0 \\subset X$ as closed subspaces,\n\\item $|Z| = |Z_0| = \\text{Supp}(\\mathcal{F})$ as closed subsets of $|X|$,\n\\item there exists a finite type, quasi-coherent $\\mathcal{O}_{Z_0}$-module\n$\\mathcal{G}_0$ with\n$$\n(Z_0 \\to X)_*\\mathcal{G}_0 = \\mathcal{F}.\n$$\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZ6","source_file":"spaces-divisors.tex","source_line":964,"source_end_line":981,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L964-L981","statement_sha256":"486191b5c92a55dc151794b385cced15909b83f80636b3b6f9ec852440e7f9f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11546,"rank":11546,"depth":57,"x":756.422,"y":1575.675,"cluster":"geometry-of-spaces"},{"id":"stacks:0CZ7","tag":"0CZ7","title":"Fitting ideals · Lemma 0CZ7","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a finite type, quasi-coherent O_X-module. Let x ∈ |X|. Then F can be generated by r elements in an étale neighbourhood of x if and only if Fit_r(F)_overlinex = O_X, overlinex.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent\n$\\mathcal{O}_X$-module. Let $x \\in |X|$. Then $\\mathcal{F}$ can be\ngenerated by $r$ elements in an \\'etale neighbourhood of $x$ if and only\nif $\\text{Fit}_r(\\mathcal{F})_{\\overline{x}} = \\mathcal{O}_{X, \\overline{x}}$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZ7","source_file":"spaces-divisors.tex","source_line":997,"source_end_line":1004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L997-L1004","statement_sha256":"01c5940c999761d4a890aa15c0835ab77e317f491ab0dfd72763425892ac143a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11547,"rank":11547,"depth":6,"x":643.129,"y":1661.435,"cluster":"geometry-of-spaces"},{"id":"stacks:0CZ8","tag":"0CZ8","title":"Fitting ideals · Lemma 0CZ8","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a finite type, quasi-coherent O_X-module. Let r ≥ 0. The following are equivalent • F is finite locally free of rank r • Fit_r - 1(F) = 0 and Fit_r(F) = O_X, and • Fit_k(F) = 0 for k < r and Fit_k(F) = O_X for k ≥ r.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent\n$\\mathcal{O}_X$-module. Let $r \\geq 0$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is finite locally free of rank $r$\n\\item $\\text{Fit}_{r - 1}(\\mathcal{F}) = 0$ and\n$\\text{Fit}_r(\\mathcal{F}) = \\mathcal{O}_X$, and\n\\item $\\text{Fit}_k(\\mathcal{F}) = 0$ for $k < r$ and\n$\\text{Fit}_k(\\mathcal{F}) = \\mathcal{O}_X$ for $k \\geq r$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZ8","source_file":"spaces-divisors.tex","source_line":1017,"source_end_line":1029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1017-L1029","statement_sha256":"3634bbff6f72c3230c8605f320701a5bcd5a70da9bd4d26bdfa557db607e0f1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11548,"rank":11548,"depth":7,"x":657.736,"y":1533.633,"cluster":"geometry-of-spaces"},{"id":"stacks:0CZ9","tag":"0CZ9","title":"Fitting ideals · Lemma 0CZ9","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a finite type, quasi-coherent O_X-module. The closed subspaces X = Z_-1 ⊃ Z_0 ⊃ Z_1 ⊃ Z_2 … defined by the Fitting ideals of F have the following properties • The intersection ⋂ Z_r is empty. • The functor (Sch/X)^opp → Sets defined by the rule T ↦ ( (*) & if F_T is locally generated by ≤ r sections ∅ & otherwise . is representable by the open subspace X setminus Z_r. • The functor F_r : (Sch/X)^opp → Sets…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent\n$\\mathcal{O}_X$-module. The closed subspaces\n$$\nX = Z_{-1} \\supset Z_0 \\supset Z_1 \\supset Z_2 \\ldots\n$$\ndefined by the Fitting ideals of $\\mathcal{F}$ have the following\nproperties\n\\begin{enumerate}\n\\item The intersection $\\bigcap Z_r$ is empty.\n\\item The functor $(\\Sch/X)^{opp} \\to \\textit{Sets}$ defined by the rule\n$$\nT \\longmapsto\n\\left\\{\n\\begin{matrix}\n\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ is locally generated by }\n\\leq r\\text{ sections} \\\\\n\\emptyset & \\text{otherwise}\n\\end{matrix}\n\\right.\n$$\nis representable by the open subspace $X \\setminus Z_r$.\n\\item The functor $F_r : (\\Sch/X)^{opp} \\to \\textit{Sets}$ defined by the rule\n$$\nT \\longmapsto\n\\left\\{\n\\begin{matrix}\n\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ locally free rank }r\\\\\n\\emptyset & \\text{otherwise}\n\\end{matrix}\n\\right.\n$$\nis representable by the locally closed subspace $Z_{r - 1} \\setminus Z_r$\nof $X$.\n\\end{enumerate}\nIf $\\mathcal{F}$ is of finite presentation, then\n$Z_r \\to X$, $X \\setminus Z_r \\to X$, and $Z_{r - 1} \\setminus Z_r \\to X$\nare of finite presentation.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZ9","source_file":"spaces-divisors.tex","source_line":1037,"source_end_line":1077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1037-L1077","statement_sha256":"1aecc99adbe4e8db8d05362a419b8f4f6ae11db86a252912e6251d52b15e67cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11549,"rank":11549,"depth":18,"x":749.94,"y":1636.384,"cluster":"geometry-of-spaces"},{"id":"stacks:0CZA","tag":"0CZA","title":"Fitting ideals · Lemma 0CZA","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be an O_X-module of finite presentation. Let X = Z_-1 ⊂ Z_0 ⊂ Z_1 ⊂ … be as in Lemma [Tag 0CZ9]. Set X_r = Z_r - 1 setminus Z_r. Then X' = coprod_r ≥ 0 X_r represents the functor F_flat : Sch/X → Sets, T ↦ ( (*) & if F_T flat over T ∅ & otherwise . Moreover, F|_X_r is locally free of rank r and the morphisms X_r → X and X' → X are of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module\nof finite presentation. Let $X = Z_{-1} \\subset Z_0 \\subset Z_1 \\subset \\ldots$\nbe as in Lemma \\ref{lemma-locally-free-rank-r-pullback}.\nSet $X_r = Z_{r - 1} \\setminus Z_r$.\nThen $X' = \\coprod_{r \\geq 0} X_r$ represents the functor\n$$\nF_{flat} : \\Sch/X \\longrightarrow \\textit{Sets},\\quad\\quad\nT \\longmapsto\n\\left\\{\n\\begin{matrix}\n\\{*\\} & \\text{if }\\mathcal{F}_T\\text{ flat over }T\\\\\n\\emptyset & \\text{otherwise}\n\\end{matrix}\n\\right.\n$$\nMoreover, $\\mathcal{F}|_{X_r}$ is locally free of rank $r$ and the\nmorphisms $X_r \\to X$ and $X' \\to X$ are of finite presentation.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Fitting ideals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZA","source_file":"spaces-divisors.tex","source_line":1085,"source_end_line":1105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1085-L1105","statement_sha256":"a47ace597852dff49f8363815ed3dd08145b08a96014a400ca1fa56889b40e01","origin":"The Stacks Project","memory_eligible":false,"source_rank":11550,"rank":11550,"depth":19,"x":598.993,"y":1612.884,"cluster":"geometry-of-spaces"},{"id":"stacks:083B","tag":"083B","title":"Effective Cartier divisors · Definition 083B","summary":"Let S be a scheme. Let X be an algebraic space over S. • A locally principal closed subspace of X is a closed subspace whose sheaf of ideals is locally generated by 1 element. • An effective Cartier divisor on X is a closed subspace D ⊂ X such that the ideal sheaf I_D ⊂ O_X is an invertible O_X-module.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item A {\\it locally principal closed subspace} of $X$ is a closed subspace\nwhose sheaf of ideals is locally generated by $1$ element.\n\\item An {\\it effective Cartier divisor} on $X$ is a closed subspace\n$D \\subset X$ such that the ideal sheaf $\\mathcal{I}_D \\subset \\mathcal{O}_X$\nis an invertible $\\mathcal{O}_X$-module.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083B","source_file":"spaces-divisors.tex","source_line":1141,"source_end_line":1151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1141-L1151","statement_sha256":"28456c456abecfabc9fe92eed4c99087a49c2cbf32250e7b1a5ffe909d7799e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11551,"rank":11551,"depth":0,"x":729.478,"y":1544.415,"cluster":"geometry-of-spaces"},{"id":"stacks:083C","tag":"083C","title":"Effective Cartier divisors · Lemma 083C","summary":"Let S be a scheme. Let X be an algebraic space over S. Let D ⊂ X be a closed subspace. The following are equivalent: • The subspace D is an effective Cartier divisor on X. • For some scheme U and surjective étale morphism U → X the inverse image D ×_X U is an effective Cartier divisor on U. • For every scheme U and every étale morphism U → X the inverse image D ×_X U is an effective Cartier divisor on U. • For every x ∈ |D| there exists an étale morphism (U, u) → (X, x)…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $D \\subset X$ be a closed subspace.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The subspace $D$ is an effective Cartier divisor on $X$.\n\\item For some scheme $U$ and surjective \\'etale morphism $U \\to X$\nthe inverse image $D \\times_X U$ is an effective Cartier divisor on $U$.\n\\item For every scheme $U$ and every \\'etale morphism $U \\to X$\nthe inverse image $D \\times_X U$ is an effective Cartier divisor on $U$.\n\\item For every $x \\in |D|$ there exists an \\'etale morphism\n$(U, u) \\to (X, x)$ of pointed algebraic spaces such that $U = \\Spec(A)$\nand $D \\times_X U = \\Spec(A/(f))$ with $f \\in A$ not a zerodivisor.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083C","source_file":"spaces-divisors.tex","source_line":1160,"source_end_line":1175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1160-L1175","statement_sha256":"fd485e0e05540a087466f1cdecc1ae4487dc269708d903cca79afcc1027a544f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11552,"rank":11552,"depth":1,"x":688.233,"y":1669.211,"cluster":"geometry-of-spaces"},{"id":"stacks:083D","tag":"083D","title":"Effective Cartier divisors · Lemma 083D","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z ⊂ X be a locally principal closed subspace. Let U = X setminus Z. Then U → X is an affine morphism.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z \\subset X$ be a locally principal closed\nsubspace. Let $U = X \\setminus Z$. Then $U \\to X$ is an affine morphism.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083D","source_file":"spaces-divisors.tex","source_line":1197,"source_end_line":1202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1197-L1202","statement_sha256":"21f8da3ca84b726fe0441e137b8ddd2917418d59867e1e8d465b7b9a682cb63e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11553,"rank":11553,"depth":47,"x":618.139,"y":1553.538,"cluster":"geometry-of-spaces"},{"id":"stacks:083S","tag":"083S","title":"Effective Cartier divisors · Lemma 083S","summary":"Let S be a scheme. Let X be an algebraic space over S. Let D ⊂ X be an effective Cartier divisor. Let U = X setminus D. Then U → X is an affine morphism and U is scheme theoretically dense in X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $D \\subset X$ be an effective Cartier divisor.\nLet $U = X \\setminus D$. Then $U \\to X$ is an affine morphism and $U$\nis scheme theoretically dense in $X$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083S","source_file":"spaces-divisors.tex","source_line":1214,"source_end_line":1220,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1214-L1220","statement_sha256":"676daaadb97cd55b640eb28690c4257ab9170a036bc1f17b8f60cef2201763e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11554,"rank":11554,"depth":51,"x":763.155,"y":1599.158,"cluster":"geometry-of-spaces"},{"id":"stacks:083T","tag":"083T","title":"Effective Cartier divisors · Lemma 083T","summary":"Let S be a scheme. Let X be an algebraic space over S. Let D ⊂ X be an effective Cartier divisor. Let x ∈ |D|. If dim_x(X) < ∞, then dim_x(D) < dim_x(X).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $D \\subset X$ be an effective Cartier divisor.\nLet $x \\in |D|$.\nIf $\\dim_x(X) < \\infty$, then $\\dim_x(D) < \\dim_x(X)$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083T","source_file":"spaces-divisors.tex","source_line":1232,"source_end_line":1238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1232-L1238","statement_sha256":"7abdfc85afdde271c13d474659611e7f3aa681d85444d8cb6aec34cb6417729e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11555,"rank":11555,"depth":47,"x":619.232,"y":1647.907,"cluster":"geometry-of-spaces"},{"id":"stacks:083U","tag":"083U","title":"Effective Cartier divisors · Definition 083U","summary":"Let S be a scheme. Let X be an algebraic space over S. Given effective Cartier divisors D_1, D_2 on X we set D = D_1 + D_2 equal to the closed subspace of X corresponding to the quasi-coherent sheaf of ideals I_D_1I_D_2 ⊂ O_S. We call this the sum of the effective Cartier divisors D_1 and D_2.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nGiven effective Cartier divisors\n$D_1$, $D_2$ on $X$ we set $D = D_1 + D_2$ equal to the\nclosed subspace of $X$ corresponding to the quasi-coherent\nsheaf of ideals\n$\\mathcal{I}_{D_1}\\mathcal{I}_{D_2} \\subset \\mathcal{O}_S$.\nWe call this the {\\it sum of the effective Cartier divisors\n$D_1$ and $D_2$}.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083U","source_file":"spaces-divisors.tex","source_line":1250,"source_end_line":1260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1250-L1260","statement_sha256":"b32dc70e24721d341015cc0b29aa9d57718569361d37f1e9df21f205b0692dd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11556,"rank":11556,"depth":0,"x":686.298,"y":1530.046,"cluster":"geometry-of-spaces"},{"id":"stacks:083V","tag":"083V","title":"Effective Cartier divisors · Lemma 083V","summary":"The sum of two effective Cartier divisors is an effective Cartier divisor.","statement_latex":"The sum of two effective Cartier divisors is an effective\nCartier divisor.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083V","source_file":"spaces-divisors.tex","source_line":1267,"source_end_line":1271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1267-L1271","statement_sha256":"1da1775ac3b845d3907ce767039e7a5cdb5d961c64577292b80a7a4d8da9fb33","origin":"The Stacks Project","memory_eligible":false,"source_rank":11557,"rank":11557,"depth":0,"x":731.72,"y":1655.272,"cluster":"geometry-of-spaces"},{"id":"stacks:083W","tag":"083W","title":"Effective Cartier divisors · Lemma 083W","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z, Y be two closed subspaces of X with ideal sheaves I and J. If IJ defines an effective Cartier divisor D ⊂ X, then Z and Y are effective Cartier divisors and D = Z + Y.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z, Y$ be two closed subspaces of $X$\nwith ideal sheaves $\\mathcal{I}$ and $\\mathcal{J}$. If $\\mathcal{I}\\mathcal{J}$\ndefines an effective Cartier divisor $D \\subset X$, then $Z$ and $Y$\nare effective Cartier divisors and $D = Z + Y$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083W","source_file":"spaces-divisors.tex","source_line":1279,"source_end_line":1286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1279-L1286","statement_sha256":"a8e120aee176fd80b42b98851e9c28e6a2b0b99f18efb4ac46546663aad23f5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11558,"rank":11558,"depth":7,"x":597.241,"y":1588.566,"cluster":"geometry-of-spaces"},{"id":"stacks:083X","tag":"083X","title":"Effective Cartier divisors · Lemma 083X","summary":"Let S be a scheme. Let f : X' → X be a morphism of algebraic spaces over S. Let Z ⊂ X be a locally principal closed subspace. Then the inverse image f^-1(Z) is a locally principal closed subspace of X'.","statement_latex":"Let $S$ be a scheme.\nLet $f : X' \\to X$ be a morphism of algebraic spaces over $S$.\nLet $Z \\subset X$ be a locally principal closed subspace.\nThen the inverse image $f^{-1}(Z)$ is a locally principal closed\nsubspace of $X'$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083X","source_file":"spaces-divisors.tex","source_line":1300,"source_end_line":1307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1300-L1307","statement_sha256":"e60afc0519ab365e5b81172030e3b9fd4eccd12cb80db2812938883de1fdf7c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11559,"rank":11559,"depth":0,"x":750.366,"y":1561.392,"cluster":"geometry-of-spaces"},{"id":"stacks:083Y","tag":"083Y","title":"Effective Cartier divisors · Definition 083Y","summary":"Let S be a scheme. Let f : X' → X be a morphism of algebraic spaces over S. Let D ⊂ X be an effective Cartier divisor. We say the pullback of D by f is defined if the closed subspace f^-1(D) ⊂ X' is an effective Cartier divisor. In this case we denote it either f^*D or f^-1(D) and we call it the pullback of the effective Cartier divisor.","statement_latex":"Let $S$ be a scheme.\nLet $f : X' \\to X$ be a morphism of algebraic spaces over $S$.\nLet $D \\subset X$\nbe an effective Cartier divisor. We say the {\\it pullback of\n$D$ by $f$ is defined} if the closed subspace $f^{-1}(D) \\subset X'$\nis an effective Cartier divisor. In this case we denote it either\n$f^*D$ or $f^{-1}(D)$ and we call it the\n{\\it pullback of the effective Cartier divisor}.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083Y","source_file":"spaces-divisors.tex","source_line":1313,"source_end_line":1323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1313-L1323","statement_sha256":"630ef7dc99c659289f5dfc5881524f5f7f43b9837ecdbcd4bbd1f0a562f6b51f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11560,"rank":11560,"depth":0,"x":659.118,"y":1668.538,"cluster":"geometry-of-spaces"},{"id":"stacks:083Z","tag":"083Z","title":"Effective Cartier divisors · Lemma 083Z","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let D ⊂ Y be an effective Cartier divisor. The pullback of D by f is defined in each of the following cases: • f(x) not ∈ |D| for any weakly associated point x of X, • f is flat, and • add more here as needed.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $D \\subset Y$ be an effective Cartier divisor.\nThe pullback of $D$ by $f$ is defined in each of the following cases:\n\\begin{enumerate}\n\\item $f(x) \\not \\in |D|$ for any weakly associated point $x$ of $X$,\n\\item $f$ is flat, and\n\\item add more here as needed.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083Z","source_file":"spaces-divisors.tex","source_line":1329,"source_end_line":1340,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1329-L1340","statement_sha256":"754e377e186daea18e168e9ef2a620def82c132c21fd929af3e869967526637d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11561,"rank":11561,"depth":5,"x":640.2,"y":1537.483,"cluster":"geometry-of-spaces"},{"id":"stacks:0840","tag":"0840","title":"Effective Cartier divisors · Lemma 0840","summary":"Let S be a scheme. Let f : X' → X be a morphism of algebraic spaces over S. Let D_1, D_2 be effective Cartier divisors on X. If the pullbacks of D_1 and D_2 are defined then the pullback of D = D_1 + D_2 is defined and f^*D = f^*D_1 + f^*D_2.","statement_latex":"Let $S$ be a scheme.\nLet $f : X' \\to X$ be a morphism of algebraic spaces over $S$.\nLet $D_1$, $D_2$ be effective Cartier divisors on $X$.\nIf the pullbacks of $D_1$ and $D_2$ are defined then the\npullback of $D = D_1 + D_2$ is defined and\n$f^*D = f^*D_1 + f^*D_2$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0840","source_file":"spaces-divisors.tex","source_line":1347,"source_end_line":1355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1347-L1355","statement_sha256":"c9b3992d624645b4672f5583fc2a352d3ce7b45699519fda2d84b3d7c7beead5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11562,"rank":11562,"depth":0,"x":759.786,"y":1623.564,"cluster":"geometry-of-spaces"},{"id":"stacks:0841","tag":"0841","title":"Effective Cartier divisors and invertible sheaves · Definition 0841","summary":"Let S be a scheme. Let X be an algebraic space over S and let D ⊂ X be an effective Cartier divisor with ideal sheaf I_D. • The invertible sheaf O_X(D) associated to D is defined by O_X(D) = SheafHom_O_X(I_D, O_X) = I_D^⊗ -1. • The canonical section, usually denoted 1 or 1_D, is the global section of O_X(D) corresponding to the inclusion mapping I_D → O_X. • We write O_X(-D) = O_X(D)^⊗ -1 = I_D. • Given a second effective Cartier divisor D' ⊂ X we define O_X(D - D') =…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$\nand let $D \\subset X$ be an effective Cartier divisor with ideal\nsheaf $\\mathcal{I}_D$.\n\\begin{enumerate}\n\\item The {\\it invertible sheaf $\\mathcal{O}_X(D)$ associated to $D$}\nis defined by\n$$\n\\mathcal{O}_X(D) =\n\\SheafHom_{\\mathcal{O}_X}(\\mathcal{I}_D, \\mathcal{O}_X) =\n\\mathcal{I}_D^{\\otimes -1}.\n$$\n\\item The canonical section, usually denoted $1$ or $1_D$, is the\nglobal section of $\\mathcal{O}_X(D)$ corresponding to\nthe inclusion mapping $\\mathcal{I}_D \\to \\mathcal{O}_X$.\n\\item We write\n$\\mathcal{O}_X(-D) = \\mathcal{O}_X(D)^{\\otimes -1} = \\mathcal{I}_D$.\n\\item Given a second effective Cartier divisor $D' \\subset X$ we define\n$\\mathcal{O}_X(D - D') =\n\\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(-D')$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0841","source_file":"spaces-divisors.tex","source_line":1372,"source_end_line":1394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1372-L1394","statement_sha256":"786a71da32c86c54744a6ac87feab821e57a67b5c1bfa528340d7f23d904b53e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11563,"rank":11563,"depth":0,"x":602.058,"y":1627.954,"cluster":"geometry-of-spaces"},{"id":"stacks:0B4F","tag":"0B4F","title":"Effective Cartier divisors and invertible sheaves · Lemma 0B4F","summary":"Let S be a scheme. Let X be an algebraic space over S. Let D ⊂ X be an effective Cartier divisor. Then for the conormal sheaf we have C_D/X = I_D|D = O_X(D)^⊗ -1|_D.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $D \\subset X$ be an effective Cartier divisor.\nThen for the conormal sheaf we have $\\mathcal{C}_{D/X} = \\mathcal{I}_D|D =\n\\mathcal{O}_X(D)^{\\otimes -1}|_D$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4F","source_file":"spaces-divisors.tex","source_line":1415,"source_end_line":1421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1415-L1421","statement_sha256":"b46cfb0b6bb7301c0bad263ac4e3a6b50cbd0e76db7b03d5c590437d4243c7ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":11564,"rank":11564,"depth":0,"x":715.064,"y":1535.029,"cluster":"geometry-of-spaces"},{"id":"stacks:0842","tag":"0842","title":"Effective Cartier divisors and invertible sheaves · Lemma 0842","summary":"Let S be a scheme. Let X be an algebraic space over S. Let D_1, D_2 be effective Cartier divisors on X. Let D = D_1 + D_2. Then there is a unique isomorphism O_X(D_1) ⊗_O_X O_X(D_2) → O_X(D) which maps 1_D_1 ⊗ 1_D_2 to 1_D.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $D_1$, $D_2$ be effective Cartier divisors on $X$.\nLet $D = D_1 + D_2$.\nThen there is a unique isomorphism\n$$\n\\mathcal{O}_X(D_1) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D_2)\n\\longrightarrow\n\\mathcal{O}_X(D)\n$$\nwhich maps $1_{D_1} \\otimes 1_{D_2}$ to $1_D$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0842","source_file":"spaces-divisors.tex","source_line":1427,"source_end_line":1439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1427-L1439","statement_sha256":"0e3e4c2b6d5f897d39c30b99c86886f80a167eeb61bc72b4a76665edff642112","origin":"The Stacks Project","memory_eligible":false,"source_rank":11565,"rank":11565,"depth":0,"x":706.446,"y":1667.943,"cluster":"geometry-of-spaces"},{"id":"stacks:0843","tag":"0843","title":"Effective Cartier divisors and invertible sheaves · Definition 0843","summary":"Let S be a scheme. Let X be an algebraic space over S. Let L be an invertible sheaf on X. A global section s ∈ Γ(X, L) is called a regular section if the map O_X → L, f ↦ fs is injective.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{L}$ be an invertible sheaf on $X$.\nA global section $s \\in \\Gamma(X, \\mathcal{L})$ is called a\n{\\it regular section} if the map $\\mathcal{O}_X \\to \\mathcal{L}$,\n$f \\mapsto fs$ is injective.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0843","source_file":"spaces-divisors.tex","source_line":1445,"source_end_line":1452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1445-L1452","statement_sha256":"059000cb2a672b16751ca3f6832ecb4bb31f9d8d2e88601f7d239f3d33f79b1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11566,"rank":11566,"depth":0,"x":605.712,"y":1564.835,"cluster":"geometry-of-spaces"},{"id":"stacks:0844","tag":"0844","title":"Effective Cartier divisors and invertible sheaves · Lemma 0844","summary":"Let S be a scheme. Let X be an algebraic space over S. Let f ∈ Γ(X, O_X). The following are equivalent: • f is a regular section, and • for any x ∈ X the image f ∈ O_X, overlinex is not a zerodivisor. • for any affine U = Spec(A) étale over X the restriction f|_U is a nonzerodivisor of A, and • there exists a scheme U and a surjective étale morphism U → X such that f|_U is a regular section of O_U.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $f \\in \\Gamma(X, \\mathcal{O}_X)$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is a regular section, and\n\\item for any $x \\in X$ the image $f \\in \\mathcal{O}_{X, \\overline{x}}$\nis not a zerodivisor.\n\\item for any affine $U = \\Spec(A)$ \\'etale over $X$\nthe restriction $f|_U$ is a nonzerodivisor of $A$, and\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$U \\to X$ such that $f|_U$ is a regular section of $\\mathcal{O}_U$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0844","source_file":"spaces-divisors.tex","source_line":1454,"source_end_line":1469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1454-L1469","statement_sha256":"1f280a4fcac29936c19b187b9b7588017ad5d5f44cb47fb2125abbaa2f1a86c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11567,"rank":11567,"depth":0,"x":763.225,"y":1583.745,"cluster":"geometry-of-spaces"},{"id":"stacks:0845","tag":"0845","title":"Effective Cartier divisors and invertible sheaves · Definition 0845","summary":"Let S be a scheme. Let X be an algebraic space over S. Let L be an invertible sheaf. Let s ∈ Γ(X, L). The zero scheme of s is the closed subspace Z(s) ⊂ X defined by the quasi-coherent sheaf of ideals I ⊂ O_X which is the image of the map s : L^⊗ -1 → O_X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{L}$ be an invertible sheaf.\nLet $s \\in \\Gamma(X, \\mathcal{L})$.\nThe {\\it zero scheme} of $s$ is the closed subspace $Z(s) \\subset X$\ndefined by the quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_X$ which is the image of the\nmap $s : \\mathcal{L}^{\\otimes -1} \\to \\mathcal{O}_X$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0845","source_file":"spaces-divisors.tex","source_line":1483,"source_end_line":1492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1483-L1492","statement_sha256":"2abda222ad99e47dca9266d190bb141fc3e829189f89c2a4c8ebbf0eee28b2bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11568,"rank":11568,"depth":0,"x":631.607,"y":1659.329,"cluster":"geometry-of-spaces"},{"id":"stacks:0846","tag":"0846","title":"Effective Cartier divisors and invertible sheaves · Lemma 0846","summary":"Let S be a scheme. Let X be an algebraic space over S. Let L be an invertible O_X-module. Let s ∈ Γ(X, L). • Consider closed immersions i : Z → X such that i^*s ∈ Γ(Z, i^*L)) is zero ordered by inclusion. The zero scheme Z(s) is the maximal element of this ordered set. • For any morphism of algebraic spaces f : Y → X over S we have f^*s = 0 in Γ(Y, f^*L) if and only if f factors through Z(s). • The zero scheme Z(s) is a locally principal closed subspace of X. • The zero…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$.\n\\begin{enumerate}\n\\item Consider closed immersions $i : Z \\to X$ such that\n$i^*s \\in \\Gamma(Z, i^*\\mathcal{L}))$ is zero\nordered by inclusion. The zero scheme $Z(s)$ is the\nmaximal element of this ordered set.\n\\item For any morphism of algebraic spaces $f : Y \\to X$ over $S$\nwe have $f^*s = 0$ in $\\Gamma(Y, f^*\\mathcal{L})$ if and only if\n$f$ factors through $Z(s)$.\n\\item The zero scheme $Z(s)$ is a locally principal closed subspace of $X$.\n\\item The zero scheme $Z(s)$ is an effective Cartier divisor on $X$\nif and only if $s$ is a regular section of $\\mathcal{L}$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0846","source_file":"spaces-divisors.tex","source_line":1494,"source_end_line":1511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1494-L1511","statement_sha256":"ad17778c22be12247d9967a4f4b5b5663b30010e6a9d68582b0ecd3ba4376ec8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11569,"rank":11569,"depth":0,"x":667.95,"y":1528.65,"cluster":"geometry-of-spaces"},{"id":"stacks:0847","tag":"0847","title":"Effective Cartier divisors and invertible sheaves · Lemma 0847","summary":"Let S be a scheme. Let X be an algebraic space over S. • If D ⊂ X is an effective Cartier divisor, then the canonical section 1_D of O_X(D) is regular. • Conversely, if s is a regular section of the invertible sheaf L, then there exists a unique effective Cartier divisor D = Z(s) ⊂ X and a unique isomorphism O_X(D) → L which maps 1_D to s. The constructions D ↦ (O_X(D), 1_D) and (L, s) ↦ Z(s) give mutually inverse maps ( effective Cartier divisors on X ) ↔ ( pairs (L, s)…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $D \\subset X$ is an effective Cartier divisor, then\nthe canonical section $1_D$ of $\\mathcal{O}_X(D)$ is regular.\n\\item Conversely, if $s$ is a regular section of the invertible\nsheaf $\\mathcal{L}$, then there exists a unique effective\nCartier divisor $D = Z(s) \\subset X$ and a unique isomorphism\n$\\mathcal{O}_X(D) \\to \\mathcal{L}$ which maps $1_D$ to $s$.\n\\end{enumerate}\nThe constructions\n$D \\mapsto (\\mathcal{O}_X(D), 1_D)$ and $(\\mathcal{L}, s) \\mapsto Z(s)$\ngive mutually inverse maps\n$$\n\\left\\{\n\\begin{matrix}\n\\text{effective Cartier divisors on }X\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{pairs }(\\mathcal{L}, s)\\text{ consisting of an invertible}\\\\\n\\mathcal{O}_X\\text{-module and a regular global section}\n\\end{matrix}\n\\right\\}\n$$","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors and invertible sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0847","source_file":"spaces-divisors.tex","source_line":1517,"source_end_line":1545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1517-L1545","statement_sha256":"d3d6464902b6ba0763de2beb35edde7d83eced4ee65e3716d3dc2441f9cd4599","origin":"The Stacks Project","memory_eligible":false,"source_rank":11570,"rank":11570,"depth":0,"x":746.395,"y":1645.866,"cluster":"geometry-of-spaces"},{"id":"stacks:0B4G","tag":"0B4G","title":"Effective Cartier divisors on Noetherian spaces · Lemma 0B4G","summary":"Let S be a scheme and let X be a locally Noetherian algebraic space over S. Let D ⊂ X be an effective Cartier divisor. If X is (S_k), then D is (S_k - 1).","statement_latex":"Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space\nover $S$. Let $D \\subset X$ be an effective Cartier divisor. If $X$ is\n$(S_k)$, then $D$ is $(S_{k - 1})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors on Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4G","source_file":"spaces-divisors.tex","source_line":1562,"source_end_line":1567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1562-L1567","statement_sha256":"1c18074fc7b983e3f3040b5ecddc97c42d4350faab2fb86916c02b875453bc66","origin":"The Stacks Project","memory_eligible":false,"source_rank":11571,"rank":11571,"depth":15,"x":593.987,"y":1603.861,"cluster":"geometry-of-spaces"},{"id":"stacks:0B4H","tag":"0B4H","title":"Effective Cartier divisors on Noetherian spaces · Lemma 0B4H","summary":"Let S be a scheme and let X be a locally Noetherian normal algebraic space over S. Let D ⊂ X be an effective Cartier divisor. Then D is (S_1).","statement_latex":"Let $S$ be a scheme and let $X$ be a locally Noetherian normal\nalgebraic space over $S$. Let $D \\subset X$ be an\neffective Cartier divisor. Then $D$ is $(S_1)$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors on Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4H","source_file":"spaces-divisors.tex","source_line":1579,"source_end_line":1584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1579-L1584","statement_sha256":"882ea3bcc2dcb1ad203926fa2fbc048f7c34ec00ead53b04a79f87a64096b988","origin":"The Stacks Project","memory_eligible":false,"source_rank":11572,"rank":11572,"depth":19,"x":740.438,"y":1548.245,"cluster":"geometry-of-spaces"},{"id":"stacks:0DML","tag":"0DML","title":"Effective Cartier divisors on Noetherian spaces · Lemma 0DML","summary":"Let S be a scheme. Let X be a regular Noetherian separated algebraic space over S. Let U ⊂ X be a dense affine open. Then there exists an effective Cartier divisor D ⊂ X with U = X setminus D.","statement_latex":"Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space\nover $S$. Let $U \\subset X$ be a dense affine open. Then there exists an\neffective Cartier divisor $D \\subset X$ with $U = X \\setminus D$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors on Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DML","source_file":"spaces-divisors.tex","source_line":1600,"source_end_line":1605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1600-L1605","statement_sha256":"edff14b418f40ede4230f304a4ae19ad6de27fd91ec7b95568ebdb8ef4c8dbce","origin":"The Stacks Project","memory_eligible":false,"source_rank":11573,"rank":11573,"depth":55,"x":677.047,"y":1672.6,"cluster":"geometry-of-spaces"},{"id":"stacks:0DMM","tag":"0DMM","title":"Effective Cartier divisors on Noetherian spaces · Lemma 0DMM","summary":"Let S be a scheme. Let X be a regular Noetherian separated algebraic space over S. Then every invertible O_X-module is isomorphic to O_X(D - D') = O_X(D) ⊗_O_X O_X(D')^⊗ -1 for some effective Cartier divisors D, D' in X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space\nover $S$. Then every invertible $\\mathcal{O}_X$-module is isomorphic to\n$$\n\\mathcal{O}_X(D - D') =\n\\mathcal{O}_X(D) \\otimes_{\\mathcal{O}_X} \\mathcal{O}_X(D')^{\\otimes -1}\n$$\nfor some effective Cartier divisors $D, D'$ in $X$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors on Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMM","source_file":"spaces-divisors.tex","source_line":1626,"source_end_line":1635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1626-L1635","statement_sha256":"e1c7500cb4e5592fe5389d2dbd3fe1887c8c21f4b0cfda468850d94285332e37","origin":"The Stacks Project","memory_eligible":false,"source_rank":11574,"rank":11574,"depth":54,"x":623.685,"y":1544.683,"cluster":"geometry-of-spaces"},{"id":"stacks:0DMC","tag":"0DMC","title":"Effective Cartier divisors on Noetherian spaces · Lemma 0DMC","summary":"Let R be a valuation ring with fraction field K. Let X be an algebraic space over R such that X → Spec(R) is smooth. For every effective Cartier divisor D ⊂ X_K there exists an effective Cartier divisor D' ⊂ X with D'_K = D.","statement_latex":"Let $R$ be a valuation ring with fraction field $K$.\nLet $X$ be an algebraic space over $R$ such that $X \\to \\Spec(R)$\nis smooth. For every effective Cartier divisor $D \\subset X_K$\nthere exists an effective Cartier divisor $D' \\subset X$\nwith $D'_K = D$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Effective Cartier divisors on Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMC","source_file":"spaces-divisors.tex","source_line":1665,"source_end_line":1672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1665-L1672","statement_sha256":"2ee775f92afdbf66936db6f00c6f66b8a549344c68443ee4bad5efd5e23d6e03","origin":"The Stacks Project","memory_eligible":false,"source_rank":11575,"rank":11575,"depth":62,"x":766.178,"y":1608.853,"cluster":"geometry-of-spaces"},{"id":"stacks:0EPN","tag":"0EPN","title":"Relative effective Cartier divisors · Lemma 0EPN","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let D ⊂ X be a closed subspace. Assume • D is an effective Cartier divisor, and • D → Y is a flat morphism. Then for every morphism of schemes g : Y' → Y the pullback (g')^-1D is an effective Cartier divisor on X' = Y' ×_Y X where g' : X' → X is the projection.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $S$. Let $D \\subset X$ be a closed subspace.\nAssume\n\\begin{enumerate}\n\\item $D$ is an effective Cartier divisor, and\n\\item $D \\to Y$ is a flat morphism.\n\\end{enumerate}\nThen for every morphism of schemes $g : Y' \\to Y$ the pullback\n$(g')^{-1}D$ is an effective Cartier divisor on $X' = Y' \\times_Y X$\nwhere $g' : X' \\to X$ is the projection.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPN","source_file":"spaces-divisors.tex","source_line":1745,"source_end_line":1757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1745-L1757","statement_sha256":"fc60ee14882b3d242f438cb83eb600bb73c0d71b6a22ade15b02bc04a1d77d58","origin":"The Stacks Project","memory_eligible":false,"source_rank":11576,"rank":11576,"depth":2,"x":609.197,"y":1642.454,"cluster":"geometry-of-spaces"},{"id":"stacks:0EPP","tag":"0EPP","title":"Relative effective Cartier divisors · Definition 0EPP","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. A relative effective Cartier divisor on X/Y is an effective Cartier divisor D ⊂ X such that D → Y is a flat morphism of algebraic spaces.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nA {\\it relative effective Cartier divisor} on $X/Y$ is an\neffective Cartier divisor $D \\subset X$ such that $D \\to Y$\nis a flat morphism of algebraic spaces.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative effective Cartier divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPP","source_file":"spaces-divisors.tex","source_line":1769,"source_end_line":1776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1769-L1776","statement_sha256":"609144f83e4e174dda23c842f75c29ccf537d40f3fd1b7d73fc99c92db5901fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11577,"rank":11577,"depth":0,"x":698.103,"y":1528.382,"cluster":"geometry-of-spaces"},{"id":"stacks:0EN3","tag":"0EN3","title":"Meromorphic functions and sections · Definition 0EN3","summary":"Let S be a scheme. Let X be an algebraic space over S. The sheaf of meromorphic functions on X is the sheaf K_X on X_etale associated to the presheaf displayed above. A meromorphic function on X is a global section of K_X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe {\\it sheaf of meromorphic functions on $X$} is\nthe sheaf {\\it $\\mathcal{K}_X$} on $X_\\etale$ associated to the presheaf\ndisplayed above. A {\\it meromorphic function} on $X$\nis a global section of $\\mathcal{K}_X$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EN3","source_file":"spaces-divisors.tex","source_line":1814,"source_end_line":1821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1814-L1821","statement_sha256":"152395ca9c19a119051eac44ce2a82a52022089c79020f991b8c0a880b881716","origin":"The Stacks Project","memory_eligible":false,"source_rank":11578,"rank":11578,"depth":0,"x":724.33,"y":1663.204,"cluster":"geometry-of-spaces"},{"id":"stacks:0EN4","tag":"0EN4","title":"Meromorphic functions and sections · Lemma 0EN4","summary":"Let S be a scheme. Let X be an algebraic space over S. For U affine and étale over X the set S_X(U) is the set of nonzerodivisors in O_X(U).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nFor $U$ affine and \\'etale over $X$ the set\n$\\mathcal{S}_X(U)$ is the set of nonzerodivisors in\n$\\mathcal{O}_X(U)$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EN4","source_file":"spaces-divisors.tex","source_line":1838,"source_end_line":1844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1838-L1844","statement_sha256":"41429dd64e621168ee8b4816566bf4d5017e6e8644a6fff2e07a6f30e447be34","origin":"The Stacks Project","memory_eligible":false,"source_rank":11579,"rank":11579,"depth":1,"x":596.325,"y":1578.507,"cluster":"geometry-of-spaces"},{"id":"stacks:0EN5","tag":"0EN5","title":"Meromorphic functions and sections · Definition 0EN5","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a sheaf of O_X-modules on X_etale. • We denote K_X(F) the sheaf of K_X-modules which is the sheafification of the presheaf U ↦ S(U)^-1F(U). Equivalently K_X(F) = F ⊗_O_X K_X (see above). • A meromorphic section of F is a global section of K_X(F).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_X$-modules\non $X_\\etale$.\n\\begin{enumerate}\n\\item We denote $\\mathcal{K}_X(\\mathcal{F})$ the sheaf of\n$\\mathcal{K}_X$-modules which is the sheafification of the presheaf\n$U \\mapsto \\mathcal{S}(U)^{-1}\\mathcal{F}(U)$. Equivalently\n$\\mathcal{K}_X(\\mathcal{F}) =\n\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{K}_X$ (see above).\n\\item A {\\it meromorphic section of $\\mathcal{F}$}\nis a global section of $\\mathcal{K}_X(\\mathcal{F})$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EN5","source_file":"spaces-divisors.tex","source_line":1858,"source_end_line":1872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1858-L1872","statement_sha256":"915fa29f81003b33a4a1245f596dea809cfa195d303d738d5c736380d917c891","origin":"The Stacks Project","memory_eligible":false,"source_rank":11580,"rank":11580,"depth":0,"x":759.136,"y":1568.309,"cluster":"geometry-of-spaces"},{"id":"stacks:0EN6","tag":"0EN6","title":"Meromorphic functions and sections · Lemma 0EN6","summary":"Let S be a scheme. Let X be an algebraic space over S. Assume • [(a)] every weakly associated point of X is a point of codimension 0, and • [(b)] X satisfies the equivalent conditions of Morphisms of Spaces, Lemma [Tag 0BB1]. Then • K_X is a quasi-coherent sheaf of O_X-algebras, • for U ∈ X_etale affine K_X(U) is the total ring of fractions of O_X(U), • for a geometric point overlinex the set S_overlinex the set of nonzerodivisors of O_X, overlinex, and • for a geometric…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume\n\\begin{enumerate}\n\\item[(a)] every weakly associated point of $X$\nis a point of codimension $0$, and\n\\item[(b)] $X$ satisfies the equivalent conditions\nof Morphisms of Spaces, Lemma\n\\ref{spaces-morphisms-lemma-prepare-normalization}.\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item $\\mathcal{K}_X$ is a quasi-coherent sheaf of $\\mathcal{O}_X$-algebras,\n\\item for $U \\in X_\\etale$ affine $\\mathcal{K}_X(U)$\nis the total ring of fractions of $\\mathcal{O}_X(U)$,\n\\item for a geometric point $\\overline{x}$ the set\n$\\mathcal{S}_{\\overline{x}}$\nthe set of nonzerodivisors of $\\mathcal{O}_{X, \\overline{x}}$, and\n\\item for a geometric point $\\overline{x}$ the ring\n$\\mathcal{K}_{X, \\overline{x}}$ is the total ring of fractions of\n$\\mathcal{O}_{X, \\overline{x}}$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EN6","source_file":"spaces-divisors.tex","source_line":1891,"source_end_line":1913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1891-L1913","statement_sha256":"552d37b8bc27c531e78a021d4108424ca9f2f41275c6cb4485a1ce2908a9ae40","origin":"The Stacks Project","memory_eligible":false,"source_rank":11581,"rank":11581,"depth":51,"x":647.069,"y":1668.402,"cluster":"geometry-of-spaces"},{"id":"stacks:0EN7","tag":"0EN7","title":"Meromorphic functions and sections · Lemma 0EN7","summary":"Let S be a scheme. Let X be an algebraic space over S. Assume • [(a)] every weakly associated point of X is a point of codimension 0, and • [(b)] X satisfies the equivalent conditions of Morphisms of Spaces, Lemma [Tag 0BB1]. • [(c)] X is representable by a scheme X_0 (awkward but temporary notation). Then the sheaf of meromorphic functions K_X is the quasi-coherent sheaf of O_X-algebras associated to the quasi-coherent sheaf of meromorphic functions K_X_0.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume\n\\begin{enumerate}\n\\item[(a)] every weakly associated point of $X$\nis a point of codimension $0$, and\n\\item[(b)] $X$ satisfies the equivalent conditions\nof Morphisms of Spaces, Lemma\n\\ref{spaces-morphisms-lemma-prepare-normalization}.\n\\item[(c)] $X$ is representable by a scheme $X_0$\n(awkward but temporary notation).\n\\end{enumerate}\nThen the sheaf of meromorphic functions $\\mathcal{K}_X$\nis the quasi-coherent sheaf of $\\mathcal{O}_X$-algebras\nassociated to the quasi-coherent sheaf of meromorphic\nfunctions $\\mathcal{K}_{X_0}$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EN7","source_file":"spaces-divisors.tex","source_line":1981,"source_end_line":1997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L1981-L1997","statement_sha256":"58e1beb83bcfcef57063e51bcddc3d254a17c96fce0d455969b33cfe3497fe5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11582,"rank":11582,"depth":52,"x":649.218,"y":1530.744,"cluster":"geometry-of-spaces"},{"id":"stacks:0EN8","tag":"0EN8","title":"Meromorphic functions and sections · Definition 0EN8","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say that pullbacks of meromorphic functions are defined for f if for every commutative diagram xymatrix U ar[r] ar[d] & X ar[d] V ar[r] & Y with U ∈ X_etale and V ∈ Y_etale and any section s ∈ S_Y(V) the pullback f^sharp(s) ∈ O_X(U) is an element of S_X(U).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism\nof algebraic spaces over $S$. We say that {\\it pullbacks of meromorphic\nfunctions are defined for $f$} if for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[r] \\ar[d] & X \\ar[d] \\\\\nV \\ar[r] & Y\n}\n$$\nwith $U \\in X_\\etale$ and $V \\in Y_\\etale$ and any\nsection $s \\in \\mathcal{S}_Y(V)$ the pullback\n$f^\\sharp(s) \\in \\mathcal{O}_X(U)$ is an element\nof $\\mathcal{S}_X(U)$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EN8","source_file":"spaces-divisors.tex","source_line":2009,"source_end_line":2024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2009-L2024","statement_sha256":"93441a2cb65995c8d02a82f1cc32b652f919f8ec9db8bbb83dc97283e7b7b07a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11583,"rank":11583,"depth":0,"x":758.54,"y":1633.665,"cluster":"geometry-of-spaces"},{"id":"stacks:0EN9","tag":"0EN9","title":"Meromorphic functions and sections · Lemma 0EN9","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Pullbacks of meromorphic sections are defined in each of the following cases • weakly associated points of X are mapped to points of codimension 0 on Y, • f is flat, • add more here as needed.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Pullbacks of meromorphic sections are defined\nin each of the following cases\n\\begin{enumerate}\n\\item weakly associated points of $X$ are mapped\nto points of codimension $0$ on $Y$,\n\\item $f$ is flat,\n\\item add more here as needed.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EN9","source_file":"spaces-divisors.tex","source_line":2042,"source_end_line":2053,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2042-L2053","statement_sha256":"fb36e9d6b9fe3d6b464abeddfe5e7c8d9846c7cd2766f089d77fb05d88128370","origin":"The Stacks Project","memory_eligible":false,"source_rank":11584,"rank":11584,"depth":56,"x":594.854,"y":1619.779,"cluster":"geometry-of-spaces"},{"id":"stacks:0ENA","tag":"0ENA","title":"Meromorphic functions and sections · Lemma 0ENA","summary":"Let S be a scheme. Let X be an algebraic space over S. Assume • [(a)] every weakly associated point of X is a point of codimension 0, and • [(b)] X satisfies the equivalent conditions of Morphisms of Spaces, Lemma [Tag 0BB1], • [(c)] every codimension 0 point of X can be represented by a monomorphism Spec(k) → X. Let X^0 ⊂ |X| be the set of codimension 0 points of X. Then we have K_X = bigoplus_eta ∈ X^0 j_eta, *O_X, eta = ∏_eta ∈ X^0 j_eta, *O_X, eta where j_eta :…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Assume\n\\begin{enumerate}\n\\item[(a)] every weakly associated point of $X$\nis a point of codimension $0$, and\n\\item[(b)] $X$ satisfies the equivalent conditions\nof Morphisms of Spaces, Lemma\n\\ref{spaces-morphisms-lemma-prepare-normalization},\n\\item[(c)] every codimension $0$ point of $X$ can be represented\nby a monomorphism $\\Spec(k) \\to X$.\n\\end{enumerate}\nLet $X^0 \\subset |X|$ be the set of codimension $0$ points of $X$.\nThen we have\n$$\n\\mathcal{K}_X =\n\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta} =\n\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{O}_{X, \\eta}\n$$\nwhere $j_\\eta : \\Spec(\\mathcal{O}_{X, \\eta}) \\to X$ is the canonical map\nof Schemes, Section \\ref{schemes-section-points}; this makes sense because\n$X^0$ is contained in the schematic locus of $X$. Similarly,\nfor every quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$\nwe obtain the formula\n$$\n\\mathcal{K}_X(\\mathcal{F}) =\n\\bigoplus\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta =\n\\prod\\nolimits_{\\eta \\in X^0} j_{\\eta, *}\\mathcal{F}_\\eta\n$$\nfor the sheaf of meromorphic sections of $\\mathcal{F}$.\nFinally, the ring of rational functions of $X$ is the ring of meromorphic\nfunctions on $X$, in a formula: $R(X) = \\Gamma(X, \\mathcal{K}_X)$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENA","source_file":"spaces-divisors.tex","source_line":2066,"source_end_line":2098,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2066-L2098","statement_sha256":"c5b786a08063f176333625365e59c5fdb5ee0927f6d2edaa9ae7820354395438","origin":"The Stacks Project","memory_eligible":false,"source_rank":11585,"rank":11585,"depth":61,"x":726.968,"y":1536.983,"cluster":"geometry-of-spaces"},{"id":"stacks:0ENB","tag":"0ENB","title":"Meromorphic functions and sections · Definition 0ENB","summary":"Let S be a scheme. Let X be an algebraic space over S. Let L be an invertible O_X-module. A meromorphic section s of L is said to be regular if the induced map K_X → K_X(L) is injective.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nA meromorphic section $s$ of $\\mathcal{L}$ is said to be {\\it regular}\nif the induced map $\\mathcal{K}_X \\to \\mathcal{K}_X(\\mathcal{L})$\nis injective.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENB","source_file":"spaces-divisors.tex","source_line":2168,"source_end_line":2176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2168-L2176","statement_sha256":"92190d948b1e197a847a437f231a1a109a909cd8525563e748269ff7c8fe7569","origin":"The Stacks Project","memory_eligible":false,"source_rank":11586,"rank":11586,"depth":0,"x":696.073,"y":1673.255,"cluster":"geometry-of-spaces"},{"id":"stacks:0ENC","tag":"0ENC","title":"Meromorphic functions and sections · Lemma 0ENC","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that pullbacks of meromorphic functions are defined for f (see Definition [Tag 0EN8]). • Let F be a sheaf of O_Y-modules. There is a canonical pullback map f^* : Γ(Y, K_Y(F)) → Γ(X, K_X(f^*F)) for meromorphic sections of F. • Let L be an invertible O_X-module. A regular meromorphic section s of L pulls back to a regular meromorphic section f^*s of f^*L.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume that pullbacks of meromorphic functions are defined\nfor $f$ (see\nDefinition \\ref{definition-pullback-meromorphic-sections}).\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_Y$-modules.\nThere is a canonical pullback map\n$f^* : \\Gamma(Y, \\mathcal{K}_Y(\\mathcal{F})) \\to\n\\Gamma(X, \\mathcal{K}_X(f^*\\mathcal{F}))$\nfor meromorphic sections of $\\mathcal{F}$.\n\\item Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nA regular meromorphic section $s$ of $\\mathcal{L}$ pulls back\nto a regular meromorphic section $f^*s$ of $f^*\\mathcal{L}$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENC","source_file":"spaces-divisors.tex","source_line":2181,"source_end_line":2198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2181-L2198","statement_sha256":"bfd9030b6606e1b26ff80cdd6aefb883c66781a4af604840a12e2ede5faffc87","origin":"The Stacks Project","memory_eligible":false,"source_rank":11587,"rank":11587,"depth":1,"x":609.107,"y":1555.02,"cluster":"geometry-of-spaces"},{"id":"stacks:0EPQ","tag":"0EPQ","title":"Meromorphic functions and sections · Lemma 0EPQ","summary":"Let S be a scheme. Let X be an algebraic space over S satisfying (a), (b), and (c) of Lemma [Tag 0ENA]. Then every invertible O_X-module L has a regular meromorphic section.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$\nsatisfying (a), (b), and (c) of\nLemma \\ref{lemma-compute-meromorphic}.\nThen every invertible $\\mathcal{O}_X$-module $\\mathcal{L}$ has\na regular meromorphic section.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Meromorphic functions and sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPQ","source_file":"spaces-divisors.tex","source_line":2204,"source_end_line":2211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2204-L2211","statement_sha256":"7023a95727ec9ded9c389cd7eaf32201304027e586ff22752de33dd8e3251f45","origin":"The Stacks Project","memory_eligible":false,"source_rank":11588,"rank":11588,"depth":62,"x":768.611,"y":1592.929,"cluster":"geometry-of-spaces"},{"id":"stacks:084B","tag":"084B","title":"Relative Proj · Lemma 084B","summary":"In Situation [Tag 0849]. The functor F above is an algebraic space. For any morphism g : Z → X where Z is a scheme there is a canonical isomorphism underlineProj_Z(g^*A) = Z ×_X F compatible with further base change.","statement_latex":"In Situation \\ref{situation-relative-proj}. The functor $F$ above is an\nalgebraic space. For any morphism $g : Z \\to X$ where $Z$ is a scheme\nthere is a canonical isomorphism\n$\\underline{\\text{Proj}}_Z(g^*\\mathcal{A}) = Z \\times_X F$\ncompatible with further base change.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084B","source_file":"spaces-divisors.tex","source_line":2308,"source_end_line":2315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2308-L2315","statement_sha256":"099d044d811d2d4bdab3475c8967e6c1ed7800e4d0070901a982b7cab533417a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11589,"rank":11589,"depth":19,"x":620.236,"y":1655.597,"cluster":"geometry-of-spaces"},{"id":"stacks:084C","tag":"084C","title":"Relative Proj · Definition 084C","summary":"Let S be a scheme. Let X be an algebraic space over S. Let A be a quasi-coherent sheaf of graded O_X-algebras. The relative homogeneous spectrum of A over X, or the homogeneous spectrum of A over X, or the relative Proj of A over X is the algebraic space F over X of Lemma [Tag 084B]. We denote it π : underlineProj_X(A) → X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{A}$ be a quasi-coherent sheaf of\ngraded $\\mathcal{O}_X$-algebras. The\n{\\it relative homogeneous spectrum of $\\mathcal{A}$ over $X$},\nor the {\\it homogeneous spectrum of $\\mathcal{A}$ over $X$}, or the\n{\\it relative Proj of $\\mathcal{A}$ over $X$} is the algebraic space\n$F$ over $X$ of Lemma \\ref{lemma-relative-proj}.\nWe denote it $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084C","source_file":"spaces-divisors.tex","source_line":2339,"source_end_line":2349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2339-L2349","statement_sha256":"4b3ac20b75268df21b99368b2c4eaec264125180871d83e8c7bedadc4aac21c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11590,"rank":11590,"depth":20,"x":679.358,"y":1524.954,"cluster":"geometry-of-spaces"},{"id":"stacks:084D","tag":"084D","title":"Relative Proj · Lemma 084D","summary":"In Situation [Tag 0849]. The relative Proj comes equipped with a quasi-coherent sheaf of Z-graded algebras bigoplus_n ∈ Z O_underlineProj_X(A)(n) and a canonical homomorphism of graded algebras ψ : π^*A → bigoplus_n ≥ 0 O_underlineProj_X(A)(n) whose base change to any scheme over X agrees with Constructions, Lemma [Tag 01NR].","statement_latex":"In Situation \\ref{situation-relative-proj}. The relative Proj comes\nequipped with a quasi-coherent sheaf of $\\mathbf{Z}$-graded algebras\n$\\bigoplus_{n \\in \\mathbf{Z}}\n\\mathcal{O}_{\\underline{\\text{Proj}}_X(\\mathcal{A})}(n)$\nand a canonical homomorphism of graded algebras\n$$\n\\psi :\n\\pi^*\\mathcal{A}\n\\longrightarrow\n\\bigoplus\\nolimits_{n \\geq 0}\n\\mathcal{O}_{\\underline{\\text{Proj}}_X(\\mathcal{A})}(n)\n$$\nwhose base change to any scheme over $X$ agrees with\nConstructions, Lemma \\ref{constructions-lemma-glue-relative-proj-twists}.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084D","source_file":"spaces-divisors.tex","source_line":2395,"source_end_line":2411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2395-L2411","statement_sha256":"c6c44e77d2f12150e6d8399d1b6ba05ab188e06b6718b69ea9bc0affe53d35c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11591,"rank":11591,"depth":52,"x":740.934,"y":1655.074,"cluster":"geometry-of-spaces"},{"id":"stacks:085C","tag":"085C","title":"Relative Proj · Lemma 085C","summary":"Let S be a scheme. Let g : X' → X be a morphism of algebraic spaces over S and let A be a quasi-coherent sheaf of graded O_X-algebras. Then there is a canonical isomorphism r : underlineProj_X'(g^*A) → X' ×_X underlineProj_X(A) as well as a corresponding isomorphism theta : r^*pr_2^*(bigoplus_d ∈ Z O_underlineProj_X(A)(d)) → bigoplus_d ∈ Z O_underlineProj_X'(g^*A)(d) of Z-graded O_underlineProj_X'(g^*A)-algebras.","statement_latex":"Let $S$ be a scheme. Let $g : X' \\to X$ be a morphism of algebraic spaces\nover $S$ and let $\\mathcal{A}$ be a quasi-coherent sheaf\nof graded $\\mathcal{O}_X$-algebras. Then there is a canonical isomorphism\n$$\nr :\n\\underline{\\text{Proj}}_{X'}(g^*\\mathcal{A})\n\\longrightarrow\nX' \\times_X \\underline{\\text{Proj}}_X(\\mathcal{A})\n$$\nas well as a corresponding isomorphism\n$$\n\\theta :\nr^*\\text{pr}_2^*\\left(\\bigoplus\\nolimits_{d \\in \\mathbf{Z}}\n\\mathcal{O}_{\\underline{\\text{Proj}}_X(\\mathcal{A})}(d)\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}}\n\\mathcal{O}_{\\underline{\\text{Proj}}_{X'}(g^*\\mathcal{A})}(d)\n$$\nof $\\mathbf{Z}$-graded\n$\\mathcal{O}_{\\underline{\\text{Proj}}_{X'}(g^*\\mathcal{A})}$-algebras.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085C","source_file":"spaces-divisors.tex","source_line":2443,"source_end_line":2465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2443-L2465","statement_sha256":"bcca46bccf0ec64149a990193d7dc0c64ea331d635b5032bafdfe35caca8883b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11592,"rank":11592,"depth":14,"x":590.616,"y":1593.954,"cluster":"geometry-of-spaces"},{"id":"stacks:084E","tag":"084E","title":"Relative Proj · Lemma 084E","summary":"In Situation [Tag 0849] the morphism π : underlineProj_X(A) → X is separated.","statement_latex":"In Situation \\ref{situation-relative-proj} the morphism\n$\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$\nis separated.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084E","source_file":"spaces-divisors.tex","source_line":2485,"source_end_line":2490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2485-L2490","statement_sha256":"23e8c67f281553bddb041b95981561848fb25ebff5e8f479a558f4a5397471e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11593,"rank":11593,"depth":56,"x":750.901,"y":1553.656,"cluster":"geometry-of-spaces"},{"id":"stacks:084F","tag":"084F","title":"Relative Proj · Lemma 084F","summary":"In Situation [Tag 0849]. If one of the following holds • A is of finite type as a sheaf of A_0-algebras, • A is generated by A_1 as an A_0-algebra and A_1 is a finite type A_0-module, • there exists a finite type quasi-coherent A_0-submodule F ⊂ A_+ such that A_+/FA is a locally nilpotent sheaf of ideals of A/FA, then π : underlineProj_X(A) → X is quasi-compact.","statement_latex":"In Situation \\ref{situation-relative-proj}. If one of the following holds\n\\begin{enumerate}\n\\item $\\mathcal{A}$ is of finite type as a sheaf of\n$\\mathcal{A}_0$-algebras,\n\\item $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ as an\n$\\mathcal{A}_0$-algebra and $\\mathcal{A}_1$ is a finite type\n$\\mathcal{A}_0$-module,\n\\item there exists a finite type quasi-coherent $\\mathcal{A}_0$-submodule\n$\\mathcal{F} \\subset \\mathcal{A}_{+}$ such that\n$\\mathcal{A}_{+}/\\mathcal{F}\\mathcal{A}$ is a locally nilpotent\nsheaf of ideals of $\\mathcal{A}/\\mathcal{F}\\mathcal{A}$,\n\\end{enumerate}\nthen $\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is quasi-compact.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084F","source_file":"spaces-divisors.tex","source_line":2499,"source_end_line":2514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2499-L2514","statement_sha256":"823416196091082ad220ad95a317a700f77c282f549b28c0ffb486481a8504d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11594,"rank":11594,"depth":43,"x":664.96,"y":1674.539,"cluster":"geometry-of-spaces"},{"id":"stacks:084G","tag":"084G","title":"Relative Proj · Lemma 084G","summary":"In Situation [Tag 0849]. If A is of finite type as a sheaf of O_X-algebras, then π : underlineProj_X(A) → X is of finite type.","statement_latex":"In Situation \\ref{situation-relative-proj}.\nIf $\\mathcal{A}$ is of finite type as a sheaf of\n$\\mathcal{O}_X$-algebras, then\n$\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is of finite type.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084G","source_file":"spaces-divisors.tex","source_line":2523,"source_end_line":2529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2523-L2529","statement_sha256":"a41ddcc56e4c8a8c0f66f4e09abaeed917d1fb7017d8e70e90498ee14d45a725","origin":"The Stacks Project","memory_eligible":false,"source_rank":11595,"rank":11595,"depth":42,"x":631.06,"y":1536.387,"cluster":"geometry-of-spaces"},{"id":"stacks:084H","tag":"084H","title":"Relative Proj · Lemma 084H","summary":"In Situation [Tag 0849]. If O_X → A_0 is an integral algebra map_X in A_0, see Morphisms of Spaces, Definition [Tag 0821], equals A_0. and A is of finite type as an A_0-algebra, then π : underlineProj_X(A) → X is universally closed.","statement_latex":"In Situation \\ref{situation-relative-proj}. If\n$\\mathcal{O}_X \\to \\mathcal{A}_0$\nis an integral algebra map\\footnote{In other words, the integral\nclosure of $\\mathcal{O}_X$ in $\\mathcal{A}_0$, see\nMorphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-integral-closure}, equals\n$\\mathcal{A}_0$.} and $\\mathcal{A}$ is of finite type as an\n$\\mathcal{A}_0$-algebra, then\n$\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ is universally closed.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084H","source_file":"spaces-divisors.tex","source_line":2538,"source_end_line":2549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2538-L2549","statement_sha256":"58c1177b7073a614c9e2df7b4bf57239db32315143d51f0561bd1744bb46914f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11596,"rank":11596,"depth":48,"x":767.401,"y":1619.172,"cluster":"geometry-of-spaces"},{"id":"stacks:084I","tag":"084I","title":"Relative Proj · Lemma 084I","summary":"In Situation [Tag 0849]. The following conditions are equivalent • A_0 is a finite type O_X-module and A is of finite type as an A_0-algebra, • A_0 is a finite type O_X-module and A is of finite type as an O_X-algebra. If these conditions hold, then π : underlineProj_X(A) → X is proper.","statement_latex":"In Situation \\ref{situation-relative-proj}.\nThe following conditions are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_X$-module\nand $\\mathcal{A}$ is of finite type as an $\\mathcal{A}_0$-algebra,\n\\item $\\mathcal{A}_0$ is a finite type $\\mathcal{O}_X$-module \nand $\\mathcal{A}$ is of finite type as an $\\mathcal{O}_X$-algebra.\n\\end{enumerate}\nIf these conditions hold, then\n$\\pi : \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$\nis proper.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084I","source_file":"spaces-divisors.tex","source_line":2559,"source_end_line":2572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2559-L2572","statement_sha256":"9fcd352a50bad23b00e85715ce32426a9abfe1bdb4b615a80f86286304c42c6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11597,"rank":11597,"depth":57,"x":599.992,"y":1635.52,"cluster":"geometry-of-spaces"},{"id":"stacks:085D","tag":"085D","title":"Relative Proj · Lemma 085D","summary":"Let S be a scheme. Let X be an algebraic space over S. Let A be a quasi-coherent sheaf of graded O_X-modules generated as an A_0-algebra by A_1. With P = underlineProj_X(A) we have • P represents the functor F_1 which associates to T over S the set of isomorphism classes of triples (f, L, ψ), where f : T → X is a morphism over S, L is an invertible O_T-module, and ψ : f^*A → bigoplus_n ≥ 0 L^⊗ n is a map of graded O_T-algebras inducing a surjection f^*A_1 → L, • the…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{A}$ be a quasi-coherent sheaf of graded $\\mathcal{O}_X$-modules\ngenerated as an $\\mathcal{A}_0$-algebra by $\\mathcal{A}_1$.\nWith $P = \\underline{\\text{Proj}}_X(\\mathcal{A})$ we have\n\\begin{enumerate}\n\\item $P$ represents the functor $F_1$ which associates to\n$T$ over $S$ the set of isomorphism classes of\ntriples $(f, \\mathcal{L}, \\psi)$, where $f : T \\to X$ is a morphism\nover $S$, $\\mathcal{L}$ is an invertible $\\mathcal{O}_T$-module, and\n$\\psi : f^*\\mathcal{A} \\to \\bigoplus_{n \\geq 0} \\mathcal{L}^{\\otimes n}$\nis a map of graded $\\mathcal{O}_T$-algebras inducing a surjection\n$f^*\\mathcal{A}_1 \\to \\mathcal{L}$,\n\\item the canonical map $\\pi^*\\mathcal{A}_1 \\to \\mathcal{O}_P(1)$ is\nsurjective, and\n\\item each $\\mathcal{O}_P(n)$ is invertible\nand the multiplication maps induce isomorphisms\n$\\mathcal{O}_P(n) \\otimes_{\\mathcal{O}_P} \\mathcal{O}_P(m) =\n\\mathcal{O}_P(n + m)$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085D","source_file":"spaces-divisors.tex","source_line":2582,"source_end_line":2603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2582-L2603","statement_sha256":"6fbd73df36ecab27a9a1bb2ad2ccf81e6050268a90ed5e2fa942ff34fb08aa67","origin":"The Stacks Project","memory_eligible":false,"source_rank":11598,"rank":11598,"depth":18,"x":710.487,"y":1528.282,"cluster":"geometry-of-spaces"},{"id":"stacks:085F","tag":"085F","title":"Functoriality of relative proj · Lemma 085F","summary":"Let S be a scheme. Let X be an algebraic space over S. Let ψ : A → B be a map of quasi-coherent graded O_X-algebras. Set P = underlineProj_X(A) → X and Q = underlineProj_X(B) → X. There is a canonical open subspace U(ψ) ⊂ Q and a canonical morphism of algebraic spaces r_ψ : U(ψ) → P over X and a map of Z-graded O_U(ψ)-algebras theta = theta_ψ : r_ψ^*( bigoplus_d ∈ Z O_P(d) ) → bigoplus_d ∈ Z O_U(ψ)(d). The triple (U(ψ), r_ψ, theta) is characterized by the property that…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\psi : \\mathcal{A} \\to \\mathcal{B}$ be a map of\nquasi-coherent graded $\\mathcal{O}_X$-algebras. Set\n$P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ and\n$Q = \\underline{\\text{Proj}}_X(\\mathcal{B}) \\to X$.\nThere is a canonical open subspace\n$U(\\psi) \\subset Q$ and a canonical morphism of\nalgebraic spaces\n$$\nr_\\psi :\nU(\\psi)\n\\longrightarrow\nP\n$$\nover $X$ and a map of $\\mathbf{Z}$-graded $\\mathcal{O}_{U(\\psi)}$-algebras\n$$\n\\theta = \\theta_\\psi :\nr_\\psi^*\\left(\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_P(d)\n\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\in \\mathbf{Z}} \\mathcal{O}_{U(\\psi)}(d).\n$$\nThe triple $(U(\\psi), r_\\psi, \\theta)$ is characterized by the property\nthat for any scheme $W$ \\'etale over $X$ the triple\n$$\n(U(\\psi) \\times_X W,\\quad\nr_\\psi|_{U(\\psi) \\times_X W} : U(\\psi) \\times_X W \\to  P \\times_X W,\\quad\n\\theta|_{U(\\psi) \\times_X W})\n$$\nis equal to the triple associated to $\\psi : \\mathcal{A}|_W \\to \\mathcal{B}|_W$\nof Constructions, Lemma \\ref{constructions-lemma-morphism-relative-proj}.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Functoriality of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085F","source_file":"spaces-divisors.tex","source_line":2624,"source_end_line":2658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2624-L2658","statement_sha256":"3daece166e08a5f3f04658fa294a72c89cda269abcd330d10eeaf738b713ec39","origin":"The Stacks Project","memory_eligible":false,"source_rank":11599,"rank":11599,"depth":21,"x":715.256,"y":1670.308,"cluster":"geometry-of-spaces"},{"id":"stacks:085G","tag":"085G","title":"Functoriality of relative proj · Lemma 085G","summary":"Let S be a scheme. Let X be an algebraic space over S. Let A, B, and C be quasi-coherent graded O_X-algebras. Set P = underlineProj_X(A), Q = underlineProj_X(B) and R = underlineProj_X(C). Let φ : A → B, ψ : B → C be graded O_X-algebra maps. Then we have U(ψ ∘ φ) = r_φ^-1(U(ψ)) and r_ψ ∘ φ = r_φ ∘ r_ψ|_U(ψ ∘ φ). In addition we have theta_ψ ∘ r_ψ^*theta_φ = theta_ψ ∘ φ with obvious notation.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$ be\nquasi-coherent graded $\\mathcal{O}_X$-algebras.\nSet $P = \\underline{\\text{Proj}}_X(\\mathcal{A})$,\n$Q = \\underline{\\text{Proj}}_X(\\mathcal{B})$ and\n$R = \\underline{\\text{Proj}}_X(\\mathcal{C})$.\nLet $\\varphi : \\mathcal{A} \\to \\mathcal{B}$,\n$\\psi : \\mathcal{B} \\to \\mathcal{C}$ be graded $\\mathcal{O}_X$-algebra maps.\nThen we have\n$$\nU(\\psi \\circ \\varphi) = r_\\varphi^{-1}(U(\\psi))\n\\quad\n\\text{and}\n\\quad\nr_{\\psi \\circ \\varphi}\n=\nr_\\varphi \\circ r_\\psi|_{U(\\psi \\circ \\varphi)}.\n$$\nIn addition we have\n$$\n\\theta_\\psi \\circ r_\\psi^*\\theta_\\varphi\n=\n\\theta_{\\psi \\circ \\varphi}\n$$\nwith obvious notation.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Functoriality of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085G","source_file":"spaces-divisors.tex","source_line":2666,"source_end_line":2693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2666-L2693","statement_sha256":"e777f57b628bc3421847a0dc4bf7344f88ea7ee923b446bfbbc5ccd93da5bd92","origin":"The Stacks Project","memory_eligible":false,"source_rank":11600,"rank":11600,"depth":0,"x":597.317,"y":1568.105,"cluster":"geometry-of-spaces"},{"id":"stacks:085H","tag":"085H","title":"Functoriality of relative proj · Lemma 085H","summary":"With hypotheses and notation as in Lemma [Tag 085F] above. Assume A_d → B_d is surjective for d gg 0. Then • U(ψ) = Q, • r_ψ : Q → R is a closed immersion, and • the maps theta : r_ψ^*O_P(n) → O_Q(n) are surjective but not isomorphisms in general (even if A → B is surjective).","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}\nabove. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for\n$d \\gg 0$. Then\n\\begin{enumerate}\n\\item $U(\\psi) = Q$,\n\\item $r_\\psi : Q \\to R$ is a closed immersion, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_P(n) \\to \\mathcal{O}_Q(n)$\nare surjective but not isomorphisms in general (even if\n$\\mathcal{A} \\to \\mathcal{B}$ is surjective).\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Functoriality of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085H","source_file":"spaces-divisors.tex","source_line":2699,"source_end_line":2711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2699-L2711","statement_sha256":"a2e3c7208efe28e43a731bcf575f14b1b09c281b3591652132ee22cfe5335138","origin":"The Stacks Project","memory_eligible":false,"source_rank":11601,"rank":11601,"depth":22,"x":766.771,"y":1576.562,"cluster":"geometry-of-spaces"},{"id":"stacks:085I","tag":"085I","title":"Functoriality of relative proj · Lemma 085I","summary":"With hypotheses and notation as in Lemma [Tag 085F] above. Assume A_d → B_d is an isomorphism for all d gg 0. Then • U(ψ) = Q, • r_ψ : Q → P is an isomorphism, and • the maps theta : r_ψ^*O_P(n) → O_Q(n) are isomorphisms.","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}\nabove. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is an isomorphism for all\n$d \\gg 0$. Then\n\\begin{enumerate}\n\\item $U(\\psi) = Q$,\n\\item $r_\\psi : Q \\to P$ is an isomorphism, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_P(n) \\to \\mathcal{O}_Q(n)$\nare isomorphisms.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Functoriality of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085I","source_file":"spaces-divisors.tex","source_line":2720,"source_end_line":2731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2720-L2731","statement_sha256":"7cddc572d6a6a073bb131fe3d3212d76bb31b64d7b3d297799be169651481076","origin":"The Stacks Project","memory_eligible":false,"source_rank":11602,"rank":11602,"depth":22,"x":634.785,"y":1666.636,"cluster":"geometry-of-spaces"},{"id":"stacks:085J","tag":"085J","title":"Functoriality of relative proj · Lemma 085J","summary":"With hypotheses and notation as in Lemma [Tag 085F] above. Assume A_d → B_d is surjective for d gg 0 and that A is generated by A_1 over A_0. Then • U(ψ) = Q, • r_ψ : Q → P is a closed immersion, and • the maps theta : r_ψ^*O_P(n) → O_Q(n) are isomorphisms.","statement_latex":"With hypotheses and notation as in Lemma \\ref{lemma-morphism-relative-proj}\nabove. Assume $\\mathcal{A}_d \\to \\mathcal{B}_d$ is surjective for $d \\gg 0$\nand that $\\mathcal{A}$ is generated by $\\mathcal{A}_1$ over $\\mathcal{A}_0$.\nThen\n\\begin{enumerate}\n\\item $U(\\psi) = Q$,\n\\item $r_\\psi : Q \\to P$ is a closed immersion, and\n\\item the maps $\\theta : r_\\psi^*\\mathcal{O}_P(n) \\to \\mathcal{O}_Q(n)$\nare isomorphisms.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Functoriality of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085J","source_file":"spaces-divisors.tex","source_line":2740,"source_end_line":2752,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2740-L2752","statement_sha256":"2298bf29ae8523fe4f3bc086a10689b40361deff2e226927df649c9e7e2f9d8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11603,"rank":11603,"depth":22,"x":659.718,"y":1525.076,"cluster":"geometry-of-spaces"},{"id":"stacks:0D2Z","tag":"0D2Z","title":"Invertible sheaves and morphisms into relative Proj · Lemma 0D2Z","summary":"With assumptions and notation as above. The morphism ψ induces a canonical morphism of algebraic spaces over Y r_L, ψ : U(ψ) → underlineProj_Y(A) together with a map of graded O_U(ψ)-algebras theta : r_L, ψ^*( bigoplus_d ≥ 0 O_underlineProj_Y(A)(d) ) → bigoplus_d ≥ 0 L^⊗ d|_U(ψ) characterized by the following properties: • For V → Y étale and d ≥ 0 the diagram xymatrix A_d(V) ar[d]_ψ ar[r]_ψ & Γ(V ×_Y X, L^⊗ d) ar[d]^restrict Γ(V ×_Y underlineProj_Y(A),…","statement_latex":"With assumptions and notation as above. The morphism\n$\\psi$ induces a canonical morphism of algebraic spaces over $Y$\n$$\nr_{\\mathcal{L}, \\psi} :\nU(\\psi) \\longrightarrow \\underline{\\text{Proj}}_Y(\\mathcal{A})\n$$\ntogether with a map of graded $\\mathcal{O}_{U(\\psi)}$-algebras\n$$\n\\theta :\nr_{\\mathcal{L}, \\psi}^*\\left(\n\\bigoplus\\nolimits_{d \\geq 0}\n\\mathcal{O}_{\\underline{\\text{Proj}}_Y(\\mathcal{A})}(d)\n\\right)\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\geq 0} \\mathcal{L}^{\\otimes d}|_{U(\\psi)}\n$$\ncharacterized by the following properties:\n\\begin{enumerate}\n\\item For $V \\to Y$ \\'etale and $d \\geq 0$ the diagram\n$$\n\\xymatrix{\n\\mathcal{A}_d(V) \\ar[d]_{\\psi} \\ar[r]_{\\psi} &\n\\Gamma(V \\times_Y X, \\mathcal{L}^{\\otimes d}) \\ar[d]^{restrict} \\\\\n\\Gamma(V \\times_Y \\underline{\\text{Proj}}_Y(\\mathcal{A}),\n\\mathcal{O}_{\\underline{\\text{Proj}}_Y(\\mathcal{A})}(d)) \\ar[r]^-\\theta &\n\\Gamma(V \\times_Y U(\\psi), \\mathcal{L}^{\\otimes d})\n}\n$$\nis commutative.\n\\item For any $d \\geq 1$ and any morphism $W \\to X$ where $W$ is a scheme\nsuch that $\\psi|_W : f^*\\mathcal{A}_d|_W \\to \\mathcal{L}^{\\otimes d}|_W$\nis surjective we have (a) $W \\to X$ factors through $U(\\psi)$ and\n(b) composition of $W \\to U(\\psi)$ with $r_{\\mathcal{L}, \\psi}$\nagrees with the morphism $W \\to \\underline{\\text{Proj}}_Y(\\mathcal{A})$\nwhich exists by the construction of $\\underline{\\text{Proj}}_Y(\\mathcal{A})$,\nsee Definition \\ref{definition-relative-proj}.\n\\item Consider a commutative diagram\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nwhere $X'$ and $Y'$ are schemes, set $\\mathcal{A}' = g^*\\mathcal{A}$\nand $\\mathcal{L}' = (g')^*\\mathcal{L}$ and denote\n$\\psi' : (f')^*\\mathcal{A} \\to \\bigoplus_{d \\geq 0} (\\mathcal{L}')^{\\otimes d}$\nthe pullback of $\\psi$. Let $U(\\psi')$, $r_{\\psi', \\mathcal{L}'}$,\nand $\\theta'$ be the open, morphism, and homomorphism constructed\nin Constructions, Lemma \\ref{lemma-invertible-map-into-relative-proj}.\nThen $U(\\psi') = (g')^{-1}(U(\\psi))$\nand $r_{\\psi', \\mathcal{L}'}$ agrees with the base change\nof $r_{\\psi, \\mathcal{L}}$ via the isomorphism\n$\\underline{\\text{Proj}}_{Y'}(\\mathcal{A}') =\nY' \\times_Y \\underline{\\text{Proj}}_Y(\\mathcal{A})$\nof Lemma \\ref{lemma-relative-proj-base-change}.\nMoreover, $\\theta'$ is the pullback of $\\theta$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Invertible sheaves and morphisms into relative Proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2Z","source_file":"spaces-divisors.tex","source_line":2796,"source_end_line":2855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2796-L2855","statement_sha256":"7a51a1c6f2f9a02ea3bd9a7542413b0436b7c959151cbbd43b11e9d358fd7dba","origin":"The Stacks Project","memory_eligible":false,"source_rank":11604,"rank":11604,"depth":21,"x":755.338,"y":1643.816,"cluster":"geometry-of-spaces"},{"id":"stacks:0D31","tag":"0D31","title":"Relatively ample sheaves · Definition 0D31","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let L be an invertible O_X-module. We say L is relatively ample, or f-relatively ample, or ample on X/Y, or f-ample if f : X → Y is representable and for every morphism Z → Y where Z is a scheme, the pullback L_Z of L to X_Z = Z ×_Y X is ample on X_Z/Z as in Morphisms, Definition [Tag 01VH].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nWe say $\\mathcal{L}$ is {\\it relatively ample}, or {\\it $f$-relatively ample},\nor {\\it ample on $X/Y$}, or {\\it $f$-ample} if $f : X \\to Y$\nis representable and for every morphism $Z \\to Y$\nwhere $Z$ is a scheme, the pullback $\\mathcal{L}_Z$ of $\\mathcal{L}$\nto $X_Z = Z \\times_Y X$ is ample on $X_Z/Z$ as in\nMorphisms, Definition \\ref{morphisms-definition-relatively-ample}.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relatively ample sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D31","source_file":"spaces-divisors.tex","source_line":2907,"source_end_line":2918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2907-L2918","statement_sha256":"eb665969c6ca2905d08bafe1d3dacedbe6b0049a944d94bebf28800a122646e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11605,"rank":11605,"depth":1,"x":589.053,"y":1610.458,"cluster":"geometry-of-spaces"},{"id":"stacks:0D32","tag":"0D32","title":"Relatively ample sheaves · Lemma 0D32","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let L be an invertible O_X-module. Assume Y is a scheme. The following are equivalent • L is ample on X/Y in the sense of Definition [Tag 0D31], and • X is a scheme and L is ample on X/Y in the sense of Morphisms, Definition [Tag 01VH].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume $Y$ is a scheme. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample on $X/Y$ in the sense of\nDefinition \\ref{definition-relatively-ample}, and\n\\item $X$ is a scheme and $\\mathcal{L}$ is ample on $X/Y$\nin the sense of\nMorphisms, Definition \\ref{morphisms-definition-relatively-ample}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D32","source_file":"spaces-divisors.tex","source_line":2925,"source_end_line":2938,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2925-L2938","statement_sha256":"e0a5cc829a58f34078ff5ee094c8a149f1a861b3f3ee430185ef329340de493f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11606,"rank":11606,"depth":24,"x":738.754,"y":1540.58,"cluster":"geometry-of-spaces"},{"id":"stacks:0D33","tag":"0D33","title":"Relatively ample sheaves · Lemma 0D33","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let L be an invertible O_X-module. Let Y' → Y be a morphism of algebraic spaces over S. Let f' : X' → Y' be the base change of f and denote L' the pullback of L to X'. If L is f-ample, then L' is f'-ample.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $Y' \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $f' : X' \\to Y'$ be the base change of $f$ and denote\n$\\mathcal{L}'$ the pullback of $\\mathcal{L}$ to $X'$.\nIf $\\mathcal{L}$ is $f$-ample, then $\\mathcal{L}'$ is $f'$-ample.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D33","source_file":"spaces-divisors.tex","source_line":2947,"source_end_line":2956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2947-L2956","statement_sha256":"f67d040145db1e247689caed4cdfafce3a9a76a1e5447c99f4402d55e19436dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11607,"rank":11607,"depth":0,"x":684.467,"y":1677.287,"cluster":"geometry-of-spaces"},{"id":"stacks:0D34","tag":"0D34","title":"Relatively ample sheaves · Lemma 0D34","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If there exists an f-ample invertible sheaf, then f is representable, quasi-compact, and separated.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf there exists an $f$-ample invertible sheaf, then\n$f$ is representable, quasi-compact, and separated.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D34","source_file":"spaces-divisors.tex","source_line":2963,"source_end_line":2969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2963-L2969","statement_sha256":"31d8b2f13c810302f635bdcdff52c1beafe514dbd348532f11645620da1bdad8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11608,"rank":11608,"depth":20,"x":614.441,"y":1545.45,"cluster":"geometry-of-spaces"},{"id":"stacks:0D35","tag":"0D35","title":"Relatively ample sheaves · Lemma 0D35","summary":"Let V → U be a surjective étale morphism of affine schemes. Let X be an algebraic space over U. Let L be an invertible O_X-module. Let Y = V ×_U X and let N be the pullback of L to Y. The following are equivalent • L is ample on X/U, and • N is ample on Y/V.","statement_latex":"Let $V \\to U$ be a surjective \\'etale morphism of affine schemes.\nLet $X$ be an algebraic space over $U$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $Y = V \\times_U X$ and let $\\mathcal{N}$\nbe the pullback of $\\mathcal{L}$ to $Y$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample on $X/U$, and\n\\item $\\mathcal{N}$ is ample on $Y/V$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D35","source_file":"spaces-divisors.tex","source_line":2979,"source_end_line":2991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L2979-L2991","statement_sha256":"aaa55b923f4d84a1ce3db9ea46c3816e03d1d68314a4716152a4cb4b69dc8ee5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11609,"rank":11609,"depth":56,"x":772.368,"y":1603.03,"cluster":"geometry-of-spaces"},{"id":"stacks:0D36","tag":"0D36","title":"Relatively ample sheaves · Lemma 0D36","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let L be an invertible O_X-module. The following are equivalent • L is ample on X/Y, • for every scheme Z and every morphism Z → Y the algebraic space X_Z = Z ×_Y X is a scheme and the pullback L_Z is ample on X_Z/Z, • for every affine scheme Z and every morphism Z → Y the algebraic space X_Z = Z ×_Y X is a scheme and the pullback L_Z is ample on X_Z/Z, • there exists a scheme V and a surjective…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample on $X/Y$,\n\\item for every scheme $Z$ and every morphism $Z \\to Y$\nthe algebraic space $X_Z = Z \\times_Y X$ is a scheme\nand the pullback $\\mathcal{L}_Z$ is ample on $X_Z/Z$,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$\nthe algebraic space $X_Z = Z \\times_Y X$ is a scheme\nand the pullback $\\mathcal{L}_Z$ is ample on $X_Z/Z$,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that the algebraic space $X_V = V \\times_Y X$ is a scheme\nand the pullback $\\mathcal{L}_V$ is ample on $X_V/V$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D36","source_file":"spaces-divisors.tex","source_line":3045,"source_end_line":3062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3045-L3062","statement_sha256":"657683f7998f808e04b07f3fd132e55089b991a417367b59e55575d5b07c41b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11610,"rank":11610,"depth":57,"x":609.333,"y":1650.263,"cluster":"geometry-of-spaces"},{"id":"stacks:0GUQ","tag":"0GUQ","title":"Relatively ample sheaves · Lemma 0GUQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then f is quasi-affine if and only if O_X is f-relatively ample.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Then $f$ is quasi-affine if and only if $\\mathcal{O}_X$\nis $f$-relatively ample.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relatively ample sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUQ","source_file":"spaces-divisors.tex","source_line":3088,"source_end_line":3093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3088-L3093","statement_sha256":"d1d8db5fdbe2dc6a30c7e23efd0fe4cc99d4ee946357e5f993f729d3f9b7b1f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11611,"rank":11611,"depth":22,"x":691.708,"y":1522.707,"cluster":"geometry-of-spaces"},{"id":"stacks:0D38","tag":"0D38","title":"Relative ampleness and cohomology · Lemma 0D38","summary":"Let R be a Noetherian ring. Let X be an algebraic space over R such that the structure morphism f : X → Spec(R) is proper. Let L be an invertible O_X-module. The following are equivalent • L is ample on X/R (Definition [Tag 0D31]), • for every coherent O_X-module F there exists an n_0 ≥ 0 such that H^p(X, F ⊗ L^⊗ n) = 0 for all n ≥ n_0 and p > 0.","statement_latex":"Let $R$ be a Noetherian ring. Let $X$ be an algebraic space over $R$\nsuch that the structure morphism $f : X \\to \\Spec(R)$ is proper.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample on $X/R$\n(Definition \\ref{definition-relatively-ample}),\n\\item for every coherent $\\mathcal{O}_X$-module $\\mathcal{F}$\nthere exists an $n_0 \\geq 0$ such that\n$H^p(X, \\mathcal{F} \\otimes \\mathcal{L}^{\\otimes n}) = 0$\nfor all $n \\geq n_0$ and $p > 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative ampleness and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D38","source_file":"spaces-divisors.tex","source_line":3114,"source_end_line":3128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3114-L3128","statement_sha256":"d4f2b5222be412cd3fddd718d9f7d61f427c58d15a1838c35090643f0a3beb4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11612,"rank":11612,"depth":65,"x":733.614,"y":1663.747,"cluster":"geometry-of-spaces"},{"id":"stacks:0D39","tag":"0D39","title":"Relative ampleness and cohomology · Lemma 0D39","summary":"Let Y be a Noetherian scheme. Let X be an algebraic space over Y such that the structure morphism f : X → Y is proper. Let L be an invertible O_X-module. Let F be a coherent O_X-module. Let y ∈ Y be a point such that X_y is a scheme and L_y is ample on X_y. Then there exists a d_0 such that for all d ≥ d_0 we have R^pf_*(F ⊗_O_X L^⊗ d)_y = 0 for p > 0 and the map f_*(F ⊗_O_X L^⊗ d)_y → H^0(X_y, F_y ⊗_O_X_y L_y^⊗ d) is surjective.","statement_latex":"Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$\nsuch that the structure morphism $f : X \\to Y$ is proper.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nLet $y \\in Y$ be a point such that $X_y$ is a scheme and\n$\\mathcal{L}_y$ is ample on $X_y$.\nThen there exists a $d_0$ such that for all $d \\geq d_0$ we have\n$$\nR^pf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y = 0\n\\text{ for }p > 0\n$$\nand the map\n$$\nf_*(\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes d})_y\n\\longrightarrow\nH^0(X_y, \\mathcal{F}_y \\otimes_{\\mathcal{O}_{X_y}} \\mathcal{L}_y^{\\otimes d})\n$$\nis surjective.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative ampleness and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D39","source_file":"spaces-divisors.tex","source_line":3139,"source_end_line":3159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3139-L3159","statement_sha256":"c2f4fcd5e1054a19c6b857715893cad5b76be25d03c75ce49f4ff000518b2d25","origin":"The Stacks Project","memory_eligible":false,"source_rank":11613,"rank":11613,"depth":69,"x":589.049,"y":1583.388,"cluster":"geometry-of-spaces"},{"id":"stacks:0D3A","tag":"0D3A","title":"Relative ampleness and cohomology · Lemma 0D3A","summary":"(For a more general version see Descent on Spaces, Lemma [Tag 0D3D]). Let Y be a Noetherian scheme. Let X be an algebraic space over Y such that the structure morphism f : X → Y is proper. Let L be an invertible O_X-module. Let y ∈ Y be a point such that X_y is a scheme and L_y is ample on X_y. Then there is an open neighbourhood V ⊂ Y of y such that L|_f^-1(V) is ample on f^-1(V)/V (as in Definition [Tag 0D31]).","statement_latex":"(For a more general version see\nDescent on Spaces, Lemma \\ref{spaces-descent-lemma-ample-in-neighbourhood}).\nLet $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$\nsuch that the structure morphism $f : X \\to Y$ is proper.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $y \\in Y$ be a point such that $X_y$ is a scheme and\n$\\mathcal{L}_y$ is ample on $X_y$.\nThen there is an open neighbourhood $V \\subset Y$\nof $y$ such that $\\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$\n(as in Definition \\ref{definition-relatively-ample}).","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Relative ampleness and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3A","source_file":"spaces-divisors.tex","source_line":3267,"source_end_line":3279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3267-L3279","statement_sha256":"090656739055866c2a47ff57c946316cd72d88869039800e3bf9da403500b317","origin":"The Stacks Project","memory_eligible":false,"source_rank":11614,"rank":11614,"depth":73,"x":760.561,"y":1560.575,"cluster":"geometry-of-spaces"},{"id":"stacks:085L","tag":"085L","title":"Closed subspaces of relative proj · Lemma 085L","summary":"Let S be a scheme. Let X be an algebraic space over S. Let A be a quasi-coherent graded O_X-algebra. Let π : P = underlineProj_X(A) → X be the relative Proj of A. Let i : Z → P be a closed subspace. Denote I ⊂ A the kernel of the canonical map A → bigoplus_d ≥ 0 π_*((i_*O_Z)(d)) If π is quasi-compact, then there is an isomorphism Z = underlineProj_X(A/I).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_X$-algebra. Let\n$\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ be the relative\nProj of $\\mathcal{A}$. Let $i : Z \\to P$ be a closed subspace. Denote\n$\\mathcal{I} \\subset \\mathcal{A}$ the kernel of the canonical map\n$$\n\\mathcal{A}\n\\longrightarrow\n\\bigoplus\\nolimits_{d \\geq 0} \\pi_*\\left((i_*\\mathcal{O}_Z)(d)\\right)\n$$\nIf $\\pi$ is quasi-compact, then there is an isomorphism\n$Z = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{I})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Closed subspaces of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085L","source_file":"spaces-divisors.tex","source_line":3356,"source_end_line":3370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3356-L3370","statement_sha256":"589b90da9547f045bf74f0398618e2f67c886a0bfb8e04e9bd290ae0436ac6d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11615,"rank":11615,"depth":57,"x":652.254,"y":1674.909,"cluster":"geometry-of-spaces"},{"id":"stacks:085M","tag":"085M","title":"Closed subspaces of relative proj · Lemma 085M","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let A be a quasi-coherent graded O_X-algebra. Let π : P = underlineProj_X(A) → X be the relative Proj of A. Let i : Z → P be a closed subscheme. If π is quasi-compact and i of finite presentation, then there exists a d > 0 and a quasi-coherent finite type O_X-submodule F ⊂ A_d such that Z = underlineProj_X(A/FA).","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$.\nLet $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_X$-algebra. Let\n$\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ be the relative\nProj of $\\mathcal{A}$. Let $i : Z \\to P$ be a closed subscheme.\nIf $\\pi$ is quasi-compact and $i$ of finite presentation, then there exists\na $d > 0$ and a quasi-coherent finite type $\\mathcal{O}_X$-submodule\n$\\mathcal{F} \\subset \\mathcal{A}_d$ such that\n$Z = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Closed subspaces of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085M","source_file":"spaces-divisors.tex","source_line":3399,"source_end_line":3410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3399-L3410","statement_sha256":"fd477175f96c3ed7cb84c394aa0306544c0269f2115ecee926d5985c3ed276eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11616,"rank":11616,"depth":62,"x":640.153,"y":1528.9,"cluster":"geometry-of-spaces"},{"id":"stacks:085N","tag":"085N","title":"Closed subspaces of relative proj · Lemma 085N","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let A be a quasi-coherent graded O_X-algebra. Let π : P = underlineProj_X(A) → X be the relative Proj of A. Let i : Z → X be a closed subspace. Let U ⊂ X be an open. Assume that • π is quasi-compact, • i of finite presentation, • |U| ∩ |π|(|i|(|Z|)) = ∅, • U is quasi-compact, • A_n is a finite type O_X-module for all n. Then there exists a d > 0 and a quasi-coherent finite type…","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$.\nLet $\\mathcal{A}$ be a quasi-coherent graded $\\mathcal{O}_X$-algebra.\nLet $\\pi : P = \\underline{\\text{Proj}}_X(\\mathcal{A}) \\to X$ be the relative\nProj of $\\mathcal{A}$. Let $i : Z \\to X$ be a closed subspace.\nLet $U \\subset X$ be an open. Assume that\n\\begin{enumerate}\n\\item $\\pi$ is quasi-compact,\n\\item $i$ of finite presentation,\n\\item $|U| \\cap |\\pi|(|i|(|Z|)) = \\emptyset$,\n\\item $U$ is quasi-compact,\n\\item $\\mathcal{A}_n$ is a finite type $\\mathcal{O}_X$-module for all $n$.\n\\end{enumerate}\nThen there exists a $d > 0$ and a quasi-coherent finite type\n$\\mathcal{O}_X$-submodule $\\mathcal{F} \\subset \\mathcal{A}_d$ with (a)\n$Z = \\underline{\\text{Proj}}_X(\\mathcal{A}/\\mathcal{F}\\mathcal{A})$\nand (b) the support of $\\mathcal{A}_d/\\mathcal{F}$ is disjoint from $U$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Closed subspaces of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085N","source_file":"spaces-divisors.tex","source_line":3440,"source_end_line":3459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3440-L3459","statement_sha256":"6649f744b229bb1b203f98b7c7f57ebcc76a0b74c9d1c061d56ac1a0b1dd8258","origin":"The Stacks Project","memory_eligible":false,"source_rank":11617,"rank":11617,"depth":63,"x":766.703,"y":1629.867,"cluster":"geometry-of-spaces"},{"id":"stacks:0B4I","tag":"0B4I","title":"Closed subspaces of relative proj · Lemma 0B4I","summary":"Let S be a scheme and let X be an algebraic space over S. Let E be a quasi-coherent O_X-module. There is a bijection ( sections σ of the morphism P(E) → X ) ↔ ( surjections E → L where L is an invertible O_X-module ) In this case σ is a closed immersion and there is a canonical isomorphism Ker(E → L) ⊗_O_X L^⊗ -1 → C_σ(X)/P(E) Both the bijection and isomorphism are compatible with base change.","statement_latex":"Let $S$ be a scheme and let $X$ be an algebraic space over $S$.\nLet $\\mathcal{E}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThere is a bijection\n$$\n\\left\\{\n\\begin{matrix}\n\\text{sections }\\sigma\\text{ of the } \\\\\n\\text{morphism } \\mathbf{P}(\\mathcal{E}) \\to X\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{surjections }\\mathcal{E} \\to \\mathcal{L}\\text{ where} \\\\\n\\mathcal{L}\\text{ is an invertible }\\mathcal{O}_X\\text{-module}\n\\end{matrix}\n\\right\\}\n$$\nIn this case $\\sigma$ is a closed immersion and there is a canonical\nisomorphism\n$$\n\\Ker(\\mathcal{E} \\to \\mathcal{L})\n\\otimes_{\\mathcal{O}_X} \\mathcal{L}^{\\otimes -1}\n\\longrightarrow\n\\mathcal{C}_{\\sigma(X)/\\mathbf{P}(\\mathcal{E})}\n$$\nBoth the bijection and isomorphism are compatible with base change.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Closed subspaces of relative proj","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4I","source_file":"spaces-divisors.tex","source_line":3495,"source_end_line":3524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3495-L3524","statement_sha256":"279a81f9d87c30dd6a526bb7be285aeee9c41c778e345d732b2a0f82500bcfd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11618,"rank":11618,"depth":20,"x":591.901,"y":1627.219,"cluster":"geometry-of-spaces"},{"id":"stacks:085Q","tag":"085Q","title":"Blowing up · Definition 085Q","summary":"Let S be a scheme. Let X be an algebraic space over S. Let I ⊂ O_X be a quasi-coherent sheaf of ideals, and let Z ⊂ X be the closed subspace corresponding to I (Morphisms of Spaces, Lemma [Tag 03MB]). The blowing up of X along Z, or the blowing up of X in the ideal sheaf I is the morphism b : underlineProj_X (bigoplus_n ≥ 0 I^n) → X The exceptional divisor of the blowup is the inverse image b^-1(Z). Sometimes Z is called the center of the blowup.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf\nof ideals, and let $Z \\subset X$ be the closed subspace corresponding\nto $\\mathcal{I}$\n(Morphisms of Spaces, Lemma\n\\ref{spaces-morphisms-lemma-closed-immersion-ideals}).\nThe {\\it blowing up of $X$ along $Z$}, or the\n{\\it blowing up of $X$ in the ideal sheaf $\\mathcal{I}$} is\nthe morphism\n$$\nb :\n\\underline{\\text{Proj}}_X\n\\left(\\bigoplus\\nolimits_{n \\geq 0} \\mathcal{I}^n\\right)\n\\longrightarrow\nX\n$$\nThe {\\it exceptional divisor} of the blowup is the inverse image\n$b^{-1}(Z)$. Sometimes $Z$ is called the {\\it center} of the blowup.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085Q","source_file":"spaces-divisors.tex","source_line":3544,"source_end_line":3564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3544-L3564","statement_sha256":"f2bd9040e854e6b06cc9f7a34764cb51d8fd1de92951f1801a18b6864a0aeb12","origin":"The Stacks Project","memory_eligible":false,"source_rank":11619,"rank":11619,"depth":52,"x":723.145,"y":1529.824,"cluster":"geometry-of-spaces"},{"id":"stacks:085R","tag":"085R","title":"Blowing up · Lemma 085R","summary":"Let S be a scheme. Let X be an algebraic space over S. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let U = Spec(A) be an affine scheme étale over X and let I ⊂ A be the ideal corresponding to I|_U. If X' → X is the blowup of X in I, then there is a canonical isomorphism U ×_X X' = Proj(bigoplus_d ≥ 0 I^d) of schemes over U, where the right hand side is the homogeneous spectrum of the Rees algebra of I in A. Moreover, U ×_X X' has an affine open covering by spectra of…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a\nquasi-coherent sheaf of ideals. Let $U = \\Spec(A)$ be an affine scheme\n\\'etale over $X$ and let $I \\subset A$ be the ideal corresponding to\n$\\mathcal{I}|_U$. If $X' \\to X$ is the blowup of $X$ in $\\mathcal{I}$,\nthen there is a canonical isomorphism\n$$\nU \\times_X X' = \\text{Proj}(\\bigoplus\\nolimits_{d \\geq 0} I^d)\n$$\nof schemes over $U$, where the right hand side is\nthe homogeneous spectrum of the Rees algebra of $I$ in $A$.\nMoreover, $U \\times_X X'$ has an affine open covering by\nspectra of the affine blowup algebras $A[\\frac{I}{a}]$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085R","source_file":"spaces-divisors.tex","source_line":3581,"source_end_line":3596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3581-L3596","statement_sha256":"a97d75a23a1164403def4e3ae032895503761da75f02abfe1d3ca9877b40425e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11620,"rank":11620,"depth":20,"x":704.66,"y":1676.354,"cluster":"geometry-of-spaces"},{"id":"stacks:085S","tag":"085S","title":"Blowing up · Lemma 085S","summary":"Let S be a scheme. Let X_1 → X_2 be a flat morphism of algebraic spaces over S. Let Z_2 ⊂ X_2 be a closed subspace. Let Z_1 be the inverse image of Z_2 in X_1. Let X'_i be the blowup of Z_i in X_i. Then there exists a cartesian diagram xymatrix X_1' ar[r] ar[d] & X_2' ar[d] X_1 ar[r] & X_2 of algebraic spaces over S.","statement_latex":"Let $S$ be a scheme.\nLet $X_1 \\to X_2$ be a flat morphism of algebraic spaces over $S$.\nLet $Z_2 \\subset X_2$ be a closed subspace.\nLet $Z_1$ be the inverse image of $Z_2$ in $X_1$.\nLet $X'_i$ be the blowup of $Z_i$ in $X_i$. Then there exists a cartesian\ndiagram\n$$\n\\xymatrix{\nX_1' \\ar[r] \\ar[d] & X_2' \\ar[d] \\\\\nX_1 \\ar[r] & X_2\n}\n$$\nof algebraic spaces over $S$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085S","source_file":"spaces-divisors.tex","source_line":3606,"source_end_line":3621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3606-L3621","statement_sha256":"f6e2c3836d656eb331535e13cd08c40f60c6e626c8951d8e7aa31afbdb79a5c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11621,"rank":11621,"depth":15,"x":600.283,"y":1557.62,"cluster":"geometry-of-spaces"},{"id":"stacks:085T","tag":"085T","title":"Blowing up · Lemma 085T","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z ⊂ X be a closed subspace. The blowing up b : X' → X of Z in X has the following properties: • b|_b^-1(X setminus Z) : b^-1(X setminus Z) → X setminus Z is an isomorphism, • the exceptional divisor E = b^-1(Z) is an effective Cartier divisor on X', • there is a canonical isomorphism O_X'(-1) = O_X'(E)","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z \\subset X$ be a closed subspace.\nThe blowing up $b : X' \\to X$ of $Z$ in $X$\nhas the following properties:\n\\begin{enumerate}\n\\item $b|_{b^{-1}(X \\setminus Z)} : b^{-1}(X \\setminus Z) \\to X \\setminus Z$\nis an isomorphism,\n\\item the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor\non $X'$,\n\\item there is a canonical isomorphism\n$\\mathcal{O}_{X'}(-1) = \\mathcal{O}_{X'}(E)$\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085T","source_file":"spaces-divisors.tex","source_line":3638,"source_end_line":3652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3638-L3652","statement_sha256":"3eae7f38041e097d849caf0ba444bebc2fa4958639b8f6030cc5c91deaf36a9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11622,"rank":11622,"depth":20,"x":773.015,"y":1585.994,"cluster":"geometry-of-spaces"},{"id":"stacks:085U","tag":"085U","title":"Universal property blowing up · Lemma 085U","summary":"Blow up a closed subset to make it Cartier. Let S be a scheme. Let X be an algebraic space over S. Let Z ⊂ X be a closed subspace. Let C be the full subcategory of (Spaces/X) consisting of Y → X such that the inverse image of Z is an effective Cartier divisor on Y. Then the blowing up b : X' → X of Z in X is a final object of C.","statement_latex":"\\begin{slogan}\nBlow up a closed subset to make it Cartier.\n\\end{slogan}\nLet $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $Z \\subset X$ be a closed subspace.\nLet $\\mathcal{C}$ be the full subcategory of $(\\textit{Spaces}/X)$ consisting\nof $Y \\to X$ such that the inverse image of $Z$ is an effective\nCartier divisor on $Y$. Then the blowing up $b : X' \\to X$ of $Z$ in $X$\nis a final object of $\\mathcal{C}$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085U","source_file":"spaces-divisors.tex","source_line":3665,"source_end_line":3677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3665-L3677","statement_sha256":"0fe43d0a0b8d97b417798bbe11d00586d2f09278b957f1186896058cdf711375","origin":"The Stacks Project","memory_eligible":false,"source_rank":11623,"rank":11623,"depth":57,"x":622.582,"y":1663.21,"cluster":"geometry-of-spaces"},{"id":"stacks:085V","tag":"085V","title":"Blowing up · Lemma 085V","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z ⊂ X be an effective Cartier divisor. The blowup of X in Z is the identity morphism of X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z \\subset X$ be an effective Cartier divisor.\nThe blowup of $X$ in $Z$ is the identity morphism of $X$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085V","source_file":"spaces-divisors.tex","source_line":3706,"source_end_line":3711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3706-L3711","statement_sha256":"2d3a1b6815e6d3dbde15dd2f45e090a991e14b5a70cffced7808691bd773b5ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":11624,"rank":11624,"depth":58,"x":671.494,"y":1520.68,"cluster":"geometry-of-spaces"},{"id":"stacks:085W","tag":"085W","title":"Blowing up · Lemma 085W","summary":"Let S be a scheme. Let X be an algebraic space over S. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. If X is reduced, then the blowup X' of X in I is reduced.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a\nquasi-coherent sheaf of ideals. If $X$ is reduced, then the\nblowup $X'$ of $X$ in $\\mathcal{I}$ is reduced.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085W","source_file":"spaces-divisors.tex","source_line":3718,"source_end_line":3724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3718-L3724","statement_sha256":"f2e94eef9b7dd68ade8a88c3e5addff9926386bec765e9d0274e16f0749ff2f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11625,"rank":11625,"depth":16,"x":750.173,"y":1653.75,"cluster":"geometry-of-spaces"},{"id":"stacks:0BH1","tag":"0BH1","title":"Blowing up · Lemma 0BH1","summary":"Let S be a scheme. Let X be an algebraic space over S. Let b : X' → X be the blowup of X in a closed subspace. If X satisfies the equivalent conditions of Morphisms of Spaces, Lemma [Tag 0BB1] then so does X'.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let\n$b : X' \\to X$ be the blowup of $X$ in a closed subspace. If\n$X$ satisfies the equivalent conditions of\nMorphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}\nthen so does $X'$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BH1","source_file":"spaces-divisors.tex","source_line":3736,"source_end_line":3743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3736-L3743","statement_sha256":"930cc8139ae9646309ac2d440a762788d06ca18f2eb5fac623c31cdf2489bf7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11626,"rank":11626,"depth":43,"x":584.878,"y":1600.184,"cluster":"geometry-of-spaces"},{"id":"stacks:085X","tag":"085X","title":"Blowing up · Lemma 085X","summary":"Let S be a scheme. Let X be an algebraic space over S. Let b : X' → X be a blowup of X in a closed subspace. For any effective Cartier divisor D on X the pullback b^-1D is defined (see Definition [Tag 083Y]).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $b : X' \\to X$ be a blowup of $X$ in a closed subspace.\nFor any effective Cartier divisor $D$ on $X$ the pullback\n$b^{-1}D$ is defined (see Definition\n\\ref{definition-pullback-effective-Cartier-divisor}).","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085X","source_file":"spaces-divisors.tex","source_line":3752,"source_end_line":3759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3752-L3759","statement_sha256":"3249a2e11b3f1491d167c1fd0354f525a6f44a3c559fdf758a25247bd0f3df5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11627,"rank":11627,"depth":21,"x":750.107,"y":1545.802,"cluster":"geometry-of-spaces"},{"id":"stacks:085Y","tag":"085Y","title":"Blowing up · Lemma 085Y","summary":"Let S be a scheme. Let X be an algebraic space over S. Let I ⊂ O_X and J be quasi-coherent sheaves of ideals. Let b : X' → X be the blowing up of X in I. Let b' : X\" → X' be the blowing up of X' in b^-1J O_X'. Then X\" → X is canonically isomorphic to the blowing up of X in IJ.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ and $\\mathcal{J}$ be\nquasi-coherent sheaves of ideals. Let $b : X' \\to X$ be the blowing up\nof $X$ in $\\mathcal{I}$. Let $b' : X'' \\to X'$ be the blowing up of\n$X'$ in $b^{-1}\\mathcal{J} \\mathcal{O}_{X'}$. Then $X'' \\to X$\nis canonically isomorphic to the blowing up of $X$ in $\\mathcal{I}\\mathcal{J}$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085Y","source_file":"spaces-divisors.tex","source_line":3772,"source_end_line":3780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3772-L3780","statement_sha256":"23b974207358ed5c995789096d53101d10f1fba24e3df2038ca6f5e9e7aaf292","origin":"The Stacks Project","memory_eligible":false,"source_rank":11628,"rank":11628,"depth":58,"x":671.876,"y":1679.873,"cluster":"geometry-of-spaces"},{"id":"stacks:085Z","tag":"085Z","title":"Blowing up · Lemma 085Z","summary":"Let S be a scheme. Let X be an algebraic space over S. Let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let b : X' → X be the blowing up of X in the ideal sheaf I. If I is of finite type, then b : X' → X is a proper morphism.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent\nsheaf of ideals. Let $b : X' \\to X$ be the blowing up of $X$\nin the ideal sheaf $\\mathcal{I}$. If $\\mathcal{I}$ is of finite type, then\n$b : X' \\to X$ is a proper morphism.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/085Z","source_file":"spaces-divisors.tex","source_line":3807,"source_end_line":3814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3807-L3814","statement_sha256":"a06094657e85af28a5520646684e7b75ac4761b3b006869a0c043359e526300e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11629,"rank":11629,"depth":57,"x":621.666,"y":1536.391,"cluster":"geometry-of-spaces"},{"id":"stacks:0860","tag":"0860","title":"Blowing up · Lemma 0860","summary":"Let S be a scheme and let X be an algebraic space over S. Assume X is quasi-compact and quasi-separated. Let Z ⊂ X be a closed subspace of finite presentation. Let b : X' → X be the blowing up with center Z. Let Z' ⊂ X' be a closed subspace of finite presentation. Let X\" → X' be the blowing up with center Z'. There exists a closed subspace Y ⊂ X of finite presentation, such that • |Y| = |Z| ∪ |b|(|Z'|), and • the composition X\" → X is isomorphic to the blowing up of X in Y.","statement_latex":"Let $S$ be a scheme and let $X$ be an algebraic space over $S$.\nAssume $X$ is quasi-compact and quasi-separated.\nLet $Z \\subset X$ be a closed subspace of finite presentation.\nLet $b : X' \\to X$ be the blowing up with center $Z$.\nLet $Z' \\subset X'$ be a closed subspace of finite presentation.\nLet $X'' \\to X'$ be the blowing up with center $Z'$.\nThere exists a closed subspace $Y \\subset X$ of finite presentation,\nsuch that\n\\begin{enumerate}\n\\item $|Y| = |Z| \\cup |b|(|Z'|)$, and\n\\item the composition $X'' \\to X$ is isomorphic to the blowing up\nof $X$ in $Y$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Blowing up","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0860","source_file":"spaces-divisors.tex","source_line":3830,"source_end_line":3845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3830-L3845","statement_sha256":"e5c529492f60911207e6fceb2d74d0dc3f4cac4ffc2b3f2080c8fb4754835714","origin":"The Stacks Project","memory_eligible":false,"source_rank":11630,"rank":11630,"depth":64,"x":774.318,"y":1613.826,"cluster":"geometry-of-spaces"},{"id":"stacks:0862","tag":"0862","title":"Strict transform · Definition 0862","summary":"With Z ⊂ B and f : X → B as above. • Given a quasi-coherent O_X-module F the strict transform of F with respect to the blowup of B in Z is the quotient F' of pr_X^*F by the submodule of sections supported on |pr_B'^-1E|. • The strict transform of X is the closed subspace X' ⊂ X ×_B B' cut out by the quasi-coherent ideal of sections of O_X ×_B B' supported on |pr_B'^-1E|.","statement_latex":"With $Z \\subset B$ and $f : X \\to B$ as above.\n\\begin{enumerate}\n\\item Given a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$\nthe {\\it strict transform} of $\\mathcal{F}$ with respect to the blowup\nof $B$ in $Z$ is the quotient $\\mathcal{F}'$ of $\\text{pr}_X^*\\mathcal{F}$\nby the submodule of sections supported on $|\\text{pr}_{B'}^{-1}E|$.\n\\item The {\\it strict transform} of $X$ is the closed subspace\n$X' \\subset X \\times_B B'$ cut out by the quasi-coherent ideal of\nsections of $\\mathcal{O}_{X \\times_B B'}$ supported on\n$|\\text{pr}_{B'}^{-1}E|$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0862","source_file":"spaces-divisors.tex","source_line":3961,"source_end_line":3974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3961-L3974","statement_sha256":"61ce25ee2d8792346dbba83caeda9b53352c0b0355ea7908a00747944fe33ab7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11631,"rank":11631,"depth":0,"x":599.204,"y":1643.392,"cluster":"geometry-of-spaces"},{"id":"stacks:0863","tag":"0863","title":"Étale localization and strict transform · Lemma 0863","summary":"In the situation of Definition [Tag 0862]. Let xymatrix U ar[r] ar[d] & X ar[d] V ar[r] & B be a commutative diagram of morphisms with U and V schemes and étale horizontal arrows. Let V' → V be the blowup of V in Z ×_B V. Then • V' = V ×_B B' and the maps V' → B' and U ×_V V' → X ×_B B' are étale, • the strict transform U' of U relative to V' → V is equal to X' ×_X U where X' is the strict transform of X relative to B' → B, and • for a quasi-coherent O_X-module F the…","statement_latex":"In the situation of Definition \\ref{definition-strict-transform}.\nLet\n$$\n\\xymatrix{\nU \\ar[r] \\ar[d] & X \\ar[d] \\\\\nV \\ar[r] & B\n}\n$$\nbe a commutative diagram of morphisms with $U$ and $V$ schemes and\n\\'etale horizontal arrows. Let $V' \\to V$ be the blowup of $V$\nin $Z \\times_B V$. Then\n\\begin{enumerate}\n\\item $V' = V \\times_B B'$ and the maps\n$V' \\to B'$ and $U \\times_V V' \\to X \\times_B B'$ are \\'etale,\n\\item the strict transform $U'$ of $U$ relative to $V' \\to V$\nis equal to $X' \\times_X U$ where $X'$ is the strict transform of $X$\nrelative to $B' \\to B$, and\n\\item for a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the\nrestriction of the strict transform $\\mathcal{F}'$ to\n$U \\times_V V'$ is the strict transform of $\\mathcal{F}|_U$ relative\nto $V' \\to V$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0863","source_file":"spaces-divisors.tex","source_line":3981,"source_end_line":4005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L3981-L4005","statement_sha256":"d4e126211f2048ea085528154c5f33951077ca8fea6d9822707e6181e9d85f17","origin":"The Stacks Project","memory_eligible":false,"source_rank":11632,"rank":11632,"depth":16,"x":704.722,"y":1522.034,"cluster":"geometry-of-spaces"},{"id":"stacks:0864","tag":"0864","title":"Strict transform · Lemma 0864","summary":"In the situation of Definition [Tag 0862]. • The strict transform X' of X is the blowup of X in the closed subspace f^-1Z of X. • For a quasi-coherent O_X-module F the strict transform F' is canonically isomorphic to the pushforward along X' → X ×_B B' of the strict transform of F relative to the blowing up X' → X.","statement_latex":"In the situation of Definition \\ref{definition-strict-transform}.\n\\begin{enumerate}\n\\item The strict transform $X'$ of $X$ is the blowup of $X$ in the closed\nsubspace $f^{-1}Z$ of $X$.\n\\item For a quasi-coherent $\\mathcal{O}_X$-module $\\mathcal{F}$ the\nstrict transform $\\mathcal{F}'$ is canonically isomorphic to\nthe pushforward along $X' \\to X \\times_B B'$ of the strict transform of\n$\\mathcal{F}$ relative to the blowing up $X' \\to X$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0864","source_file":"spaces-divisors.tex","source_line":4018,"source_end_line":4029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4018-L4029","statement_sha256":"44a340ebd219f063f3680266b24cad9e9dcbbc87c6715ababd8028f13f6b8098","origin":"The Stacks Project","memory_eligible":false,"source_rank":11633,"rank":11633,"depth":58,"x":724.539,"y":1671.632,"cluster":"geometry-of-spaces"},{"id":"stacks:0865","tag":"0865","title":"Strict transform · Lemma 0865","summary":"In the situation of Definition [Tag 0862]. • If X is flat over B at all points lying over Z, then the strict transform of X is equal to the base change X ×_B B'. • Let F be a quasi-coherent O_X-module. If F is flat over B at all points lying over Z, then the strict transform F' of F is equal to the pullback pr_X^*F.","statement_latex":"In the situation of Definition \\ref{definition-strict-transform}.\n\\begin{enumerate}\n\\item If $X$ is flat over $B$ at all points lying over $Z$, then\nthe strict transform of $X$ is equal to the base change $X \\times_B B'$.\n\\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $\\mathcal{F}$ is flat over $B$ at all points lying over $Z$, then\nthe strict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is equal to the\npullback $\\text{pr}_X^*\\mathcal{F}$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0865","source_file":"spaces-divisors.tex","source_line":4051,"source_end_line":4062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4051-L4062","statement_sha256":"98f091c8af17f9d82fab0645723a0c19ef40cd3312ec84912a27053e09521ec3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11634,"rank":11634,"depth":17,"x":589.411,"y":1572.409,"cluster":"geometry-of-spaces"},{"id":"stacks:0866","tag":"0866","title":"Strict transform · Lemma 0866","summary":"Let S be a scheme. Let B be an algebraic space over S. Let Z ⊂ B be a closed subspace. Let b : B' → B be the blowing up of Z in B. Let g : X → Y be an affine morphism of spaces over B. Let F be a quasi-coherent sheaf on X. Let g' : X ×_B B' → Y ×_B B' be the base change of g. Let F' be the strict transform of F relative to b. Then g'_*F' is the strict transform of g_*F.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $Z \\subset B$ be a closed subspace.\nLet $b : B' \\to B$ be the blowing up of $Z$ in $B$. Let\n$g : X \\to Y$ be an affine morphism of spaces over $B$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $g' : X \\times_B B' \\to Y \\times_B B'$ be the base change\nof $g$. Let $\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$\nrelative to $b$. Then $g'_*\\mathcal{F}'$ is the strict transform\nof $g_*\\mathcal{F}$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0866","source_file":"spaces-divisors.tex","source_line":4071,"source_end_line":4082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4071-L4082","statement_sha256":"4cd01d8814ee92a99391a1930a5d383aaa9b7a0d39061dc092fd87ebafd60903","origin":"The Stacks Project","memory_eligible":false,"source_rank":11635,"rank":11635,"depth":27,"x":769.127,"y":1568.894,"cluster":"geometry-of-spaces"},{"id":"stacks:0867","tag":"0867","title":"Strict transform · Lemma 0867","summary":"Let S be a scheme. Let B be an algebraic space over S. Let Z ⊂ B be a closed subspace. Let D ⊂ B be an effective Cartier divisor. Let Z' ⊂ B be the closed subspace cut out by the product of the ideal sheaves of Z and D. Let B' → B be the blowup of B in Z. • The blowup of B in Z' is isomorphic to B' → B. • Let f : X → B be a morphism of algebraic spaces and let F be a quasi-coherent O_X-module. If the subsheaf of F of sections supported on |f^-1D| is zero, then the strict…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $Z \\subset B$ be a closed subspace.\nLet $D \\subset B$ be an effective Cartier divisor.\nLet $Z' \\subset B$ be the closed subspace cut out by the product\nof the ideal sheaves of $Z$ and $D$.\nLet $B' \\to B$ be the blowup of $B$ in $Z$.\n\\begin{enumerate}\n\\item The blowup of $B$ in $Z'$ is isomorphic to $B' \\to B$.\n\\item Let $f : X \\to B$ be a morphism of algebraic spaces and let $\\mathcal{F}$\nbe a quasi-coherent $\\mathcal{O}_X$-module. If the subsheaf of $\\mathcal{F}$ of\nsections supported on $|f^{-1}D|$ is zero, then the\nstrict transform of $\\mathcal{F}$ relative to the blowing up\nin $Z$ agrees with the strict transform of $\\mathcal{F}$ relative\nto the blowing up of $B$ in $Z'$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0867","source_file":"spaces-divisors.tex","source_line":4090,"source_end_line":4107,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4090-L4107","statement_sha256":"513f45cec7bb8f6dda953b77b90bb63fbb927be00fb49f82f9b1f8d0da1c8aa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11636,"rank":11636,"depth":23,"x":639.23,"y":1673.625,"cluster":"geometry-of-spaces"},{"id":"stacks:0868","tag":"0868","title":"Strict transform · Lemma 0868","summary":"Let S be a scheme. Let B be an algebraic space over S. Let Z ⊂ B be a closed subspace. Let b : B' → B be the blowing up with center Z. Let Z' ⊂ B' be a closed subspace. Let B\" → B' be the blowing up with center Z'. Let Y ⊂ B be a closed subscheme such that |Y| = |Z| ∪ |b|(|Z'|) and the composition B\" → B is isomorphic to the blowing up of B in Y. In this situation, given any scheme X over B and F ∈ QCoh(O_X) we have • the strict transform of F with respect to the blowing…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $Z \\subset B$ be a closed subspace.\nLet $b : B' \\to B$ be the blowing up with center $Z$.\nLet $Z' \\subset B'$ be a closed subspace.\nLet $B'' \\to B'$ be the blowing up with center $Z'$.\nLet $Y \\subset B$ be a closed subscheme such that\n$|Y| = |Z| \\cup |b|(|Z'|)$ and the composition $B'' \\to B$\nis isomorphic to the blowing up of $B$ in $Y$.\nIn this situation, given any scheme $X$ over $B$ and\n$\\mathcal{F} \\in \\QCoh(\\mathcal{O}_X)$ we have\n\\begin{enumerate}\n\\item the strict transform of $\\mathcal{F}$ with respect to the blowing\nup of $B$ in $Y$ is equal to the strict transform with respect to the\nblowup $B'' \\to B'$ in $Z'$ of the strict transform of $\\mathcal{F}$\nwith respect to the blowup $B' \\to B$ of $B$ in $Z$, and\n\\item the strict transform of $X$ with respect to the blowing\nup of $B$ in $Y$ is equal to the strict transform with respect to the\nblowup $B'' \\to B'$ in $Z'$ of the strict transform of $X$\nwith respect to the blowup $B' \\to B$ of $B$ in $Z$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0868","source_file":"spaces-divisors.tex","source_line":4115,"source_end_line":4137,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4115-L4137","statement_sha256":"75bc91e47af069c76ef79bc8f0e668b50a5153888e037ebbafb01ba7c8707da1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11637,"rank":11637,"depth":24,"x":650.811,"y":1522.455,"cluster":"geometry-of-spaces"},{"id":"stacks:0869","tag":"0869","title":"Strict transform · Lemma 0869","summary":"In the situation of Definition [Tag 0862]. Suppose that 0 → F_1 → F_2 → F_3 → 0 is an exact sequence of quasi-coherent sheaves on X which remains exact after any base change T → B. Then the strict transforms of F_i' relative to any blowup B' → B form a short exact sequence 0 → F'_1 → F'_2 → F'_3 → 0 too.","statement_latex":"In the situation of Definition \\ref{definition-strict-transform}.\nSuppose that\n$$\n0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0\n$$\nis an exact sequence of quasi-coherent sheaves on $X$ which remains\nexact after any base change $T \\to B$. Then the strict transforms of\n$\\mathcal{F}_i'$ relative to any blowup $B' \\to B$\nform a short exact sequence\n$0 \\to \\mathcal{F}'_1 \\to \\mathcal{F}'_2 \\to \\mathcal{F}'_3 \\to 0$ too.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0869","source_file":"spaces-divisors.tex","source_line":4145,"source_end_line":4157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4145-L4157","statement_sha256":"ca949c9212a0b736315557e37edd924f39db6ea1f975716cfdadadb1fd5de7d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11638,"rank":11638,"depth":28,"x":764.014,"y":1640.681,"cluster":"geometry-of-spaces"},{"id":"stacks:0D0P","tag":"0D0P","title":"Strict transform · Lemma 0D0P","summary":"Let S be a scheme. Let B be an algebraic space over S. Let F be a finite type quasi-coherent O_B-module. Let Z_k ⊂ S be the closed subscheme cut out by Fit_k(F), see Section [Tag 0CZ3]. Let B' → B be the blowup of B in Z_k and let F' be the strict transform of F. Then F' can locally be generated by ≤ k sections.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_B$-module.\nLet $Z_k \\subset S$ be the closed subscheme cut out by\n$\\text{Fit}_k(\\mathcal{F})$, see Section \\ref{section-fitting-ideals}.\nLet $B' \\to B$ be the blowup of $B$ in $Z_k$ and let\n$\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$.\nThen $\\mathcal{F}'$ can locally be generated by $\\leq k$\nsections.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0P","source_file":"spaces-divisors.tex","source_line":4165,"source_end_line":4175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4165-L4175","statement_sha256":"34a5057bd88849f464ce1e26df861d0a48eb262aa551b7b8965f820e2db6e61d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11639,"rank":11639,"depth":17,"x":585.188,"y":1617.701,"cluster":"geometry-of-spaces"},{"id":"stacks:0D0Q","tag":"0D0Q","title":"Strict transform · Lemma 0D0Q","summary":"Let S be a scheme. Let B be an algebraic space over S. Let F be a finite type quasi-coherent O_B-module. Let Z_k ⊂ S be the closed subscheme cut out by Fit_k(F), see Section [Tag 0CZ3]. Assume that F is locally free of rank k on B setminus Z_k. Let B' → B be the blowup of B in Z_k and let F' be the strict transform of F. Then F' is locally free of rank k.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_B$-module.\nLet $Z_k \\subset S$ be the closed subscheme cut out by\n$\\text{Fit}_k(\\mathcal{F})$, see Section \\ref{section-fitting-ideals}.\nAssume that $\\mathcal{F}$ is locally free of rank $k$ on $B \\setminus Z_k$.\nLet $B' \\to B$ be the blowup of $B$ in $Z_k$ and let\n$\\mathcal{F}'$ be the strict transform of $\\mathcal{F}$.\nThen $\\mathcal{F}'$ is locally free of rank $k$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Strict transform","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0Q","source_file":"spaces-divisors.tex","source_line":4183,"source_end_line":4193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4183-L4193","statement_sha256":"0ea9a44cf3ee1aeaed839e78d98abec1d07d5b9ba410270a372726723f20b7dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11640,"rank":11640,"depth":17,"x":735.764,"y":1533.045,"cluster":"geometry-of-spaces"},{"id":"stacks:086B","tag":"086B","title":"Admissible blowups · Definition 086B","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U ⊂ X be an open subspace. A morphism X' → X is called a U-admissible blowup if there exists a closed immersion Z → X of finite presentation with Z disjoint from U such that X' is isomorphic to the blowup of X in Z.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U \\subset X$ be an open subspace. A morphism\n$X' \\to X$ is called a {\\it $U$-admissible blowup} if there exists a\nclosed immersion $Z \\to X$ of finite presentation with $Z$ disjoint from\n$U$ such that $X'$ is isomorphic to the blowup of $X$ in $Z$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Admissible blowups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086B","source_file":"spaces-divisors.tex","source_line":4220,"source_end_line":4227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4220-L4227","statement_sha256":"27fc77c87b6b38e889f782ba3aba39c863b98f4bb332335749ad0c890d4b9e85","origin":"The Stacks Project","memory_eligible":false,"source_rank":11641,"rank":11641,"depth":0,"x":692.743,"y":1681.138,"cluster":"geometry-of-spaces"},{"id":"stacks:086C","tag":"086C","title":"Admissible blowups · Lemma 086C","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let U ⊂ X be a quasi-compact open subspace. Let b : X' → X be a U-admissible blowup. Let X\" → X' be a U-admissible blowup. Then the composition X\" → X is a U-admissible blowup.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $U \\subset X$ be a quasi-compact open subspace.\nLet $b : X' \\to X$ be a $U$-admissible blowup.\nLet $X'' \\to X'$ be a $U$-admissible blowup.\nThen the composition $X'' \\to X$ is a $U$-admissible blowup.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Admissible blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086C","source_file":"spaces-divisors.tex","source_line":4243,"source_end_line":4251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4243-L4251","statement_sha256":"53f3eac2f641a2f0bd519f2d724cffb854ca4aaded95f940f741b2df733d209c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11642,"rank":11642,"depth":65,"x":605.24,"y":1547.32,"cluster":"geometry-of-spaces"},{"id":"stacks:086D","tag":"086D","title":"Admissible blowups · Lemma 086D","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space. Let U, V ⊂ X be quasi-compact open subspaces. Let b : V' → V be a U ∩ V-admissible blowup. Then there exists a U-admissible blowup X' → X whose restriction to V is V'.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space.\nLet $U, V \\subset X$ be quasi-compact open subspaces.\nLet $b : V' \\to V$ be a $U \\cap V$-admissible blowup.\nThen there exists a $U$-admissible blowup $X' \\to X$\nwhose restriction to $V$ is $V'$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Admissible blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086D","source_file":"spaces-divisors.tex","source_line":4258,"source_end_line":4266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4258-L4266","statement_sha256":"2862a2268d8b9481ecf223fb46fe86904b239b9b2b74ac7cb0bcf5a41dd1bf59","origin":"The Stacks Project","memory_eligible":false,"source_rank":11643,"rank":11643,"depth":63,"x":777.64,"y":1596.419,"cluster":"geometry-of-spaces"},{"id":"stacks:086E","tag":"086E","title":"Admissible blowups · Lemma 086E","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let U ⊂ X be a quasi-compact open subspace. Let b_i : X_i → X, i = 1, …, n be U-admissible blowups. There exists a U-admissible blowup b : X' → X such that (a) b factors as X' → X_i → X for i = 1, …, n and (b) each of the morphisms X' → X_i is a U-admissible blowup.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $U \\subset X$ be a quasi-compact open subspace.\nLet $b_i : X_i \\to X$, $i = 1, \\ldots, n$ be $U$-admissible blowups.\nThere exists a $U$-admissible blowup $b : X' \\to X$ such that\n(a) $b$ factors as $X' \\to X_i \\to X$ for $i = 1, \\ldots, n$ and\n(b) each of the morphisms $X' \\to X_i$ is a $U$-admissible blowup.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Admissible blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086E","source_file":"spaces-divisors.tex","source_line":4282,"source_end_line":4291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4282-L4291","statement_sha256":"a64d70acb8f0c0a3b92a466a1cc1edc8dcbb2f696c68a5161fb595899c8f44d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11644,"rank":11644,"depth":59,"x":610.776,"y":1658.133,"cluster":"geometry-of-spaces"},{"id":"stacks:086F","tag":"086F","title":"Admissible blowups · Lemma 086F","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let U, V be quasi-compact disjoint open subspaces of X. Then there exist a U ∪ V-admissible blowup b : X' → X such that X' is a disjoint union of open subspaces X' = X'_1 amalg X'_2 with b^-1(U) ⊂ X'_1 and b^-1(V) ⊂ X'_2.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $U, V$ be quasi-compact disjoint open subspaces of $X$.\nThen there exist a $U \\cup V$-admissible blowup $b : X' \\to X$\nsuch that $X'$ is a disjoint union of open subspaces\n$X' = X'_1 \\amalg X'_2$ with $b^{-1}(U) \\subset X'_1$ and\n$b^{-1}(V) \\subset X'_2$.","area":"Geometry of Spaces","chapter":"Divisors on Algebraic Spaces","chapter_id":"spaces-divisors","section":"Admissible blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086F","source_file":"spaces-divisors.tex","source_line":4303,"source_end_line":4312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-divisors.tex#L4303-L4312","statement_sha256":"69f367d2d1340ca482096dce0ac8d4d0ee0057cfdc9210519263219a9be7fa85","origin":"The Stacks Project","memory_eligible":false,"source_rank":11645,"rank":11645,"depth":63,"x":684.303,"y":1517.729,"cluster":"geometry-of-spaces"},{"id":"stacks:0AD1","tag":"0AD1","title":"Generically finite morphisms · Lemma 0AD1","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type and Y is locally Noetherian. Let y ∈ |Y| be a point of codimension ≤ 1 on Y. Let X^0 ⊂ |X| be the set of points of codimension 0 on X. Assume in addition one of the following conditions is satisfied • for every x ∈ X^0 the transcendence degree of x/f(x) is 0, • for every x ∈ X^0 with f(x) leadsto y the transcendence degree of x/f(x) is 0, • f is quasi-finite at…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is locally of finite type and $Y$ is locally Noetherian.\nLet $y \\in |Y|$ be a point of codimension $\\leq 1$ on $Y$.\nLet $X^0 \\subset |X|$ be the set of points of codimension $0$ on $X$.\nAssume in addition one of the following conditions is satisfied\n\\begin{enumerate}\n\\item for every $x \\in X^0$ the transcendence degree of $x/f(x)$ is $0$,\n\\item for every $x \\in X^0$ with $f(x) \\leadsto y$\nthe transcendence degree of $x/f(x)$ is $0$,\n\\item $f$ is quasi-finite at every $x \\in X^0$,\n\\item $f$ is quasi-finite at a dense set of points of $|X|$,\n\\item add more here.\n\\end{enumerate}\nThen $f$ is quasi-finite at every point of $X$ lying over $y$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AD1","source_file":"spaces-over-fields.tex","source_line":52,"source_end_line":68,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L52-L68","statement_sha256":"91fa9927a28aa9dd42e8d917b2f2db5abffac85bcae8746a9781035937cfc239","origin":"The Stacks Project","memory_eligible":false,"source_rank":11646,"rank":11646,"depth":48,"x":130.025,"y":1691.622,"cluster":"algebraic-spaces"},{"id":"stacks:0AD2","tag":"0AD2","title":"Generically finite morphisms · Lemma 0AD2","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is proper and Y is locally Noetherian. Let y ∈ Y be a point of codimension ≤ 1 in Y. Let X^0 ⊂ |X| be the set of points of codimension 0 on X. Assume in addition one of the following conditions is satisfied • for every x ∈ X^0 the transcendence degree of x/f(x) is 0, • for every x ∈ X^0 with f(x) leadsto y the transcendence degree of x/f(x) is 0, • f is quasi-finite at every x ∈ X^0, • f…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is proper and $Y$ is locally Noetherian.\nLet $y \\in Y$ be a point of codimension $\\leq 1$ in $Y$.\nLet $X^0 \\subset |X|$ be the set of points of codimension $0$ on $X$.\nAssume in addition one of the\nfollowing conditions is satisfied\n\\begin{enumerate}\n\\item for every $x \\in X^0$ the transcendence degree of $x/f(x)$ is $0$,\n\\item for every $x \\in X^0$ with $f(x) \\leadsto y$ the transcendence degree\nof $x/f(x)$ is $0$,\n\\item $f$ is quasi-finite at every $x \\in X^0$,\n\\item $f$ is quasi-finite at a dense set of points of $|X|$,\n\\item add more here.\n\\end{enumerate}\nThen there exists an open subspace $Y' \\subset Y$ containing $y$ such that\n$Y' \\times_Y X \\to Y'$ is finite.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AD2","source_file":"spaces-over-fields.tex","source_line":123,"source_end_line":141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L123-L141","statement_sha256":"9877266b58cdba9d090403d93c1b12ca4c0b3c7f4951a54461618b431ffad02a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11647,"rank":11647,"depth":73,"x":230.04,"y":1475.618,"cluster":"algebraic-spaces"},{"id":"stacks:0BBQ","tag":"0BBQ","title":"Generically finite morphisms · Lemma 0BBQ","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let f : Y → X be a birational proper morphism of algebraic spaces with Y reduced. Let U ⊂ X be the maximal open over which f is an isomorphism. Then U contains • every point of codimension 0 in X, • every x ∈ |X| of codimension 1 on X such that the local ring of X at x is normal (Properties of Spaces, Remark [Tag 0BBL]), and • every x ∈ |X| such that the fibre of |Y| → |X| over x is finite and such that the…","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $f : Y \\to X$ be a birational proper morphism of algebraic spaces\nwith $Y$ reduced.\nLet $U \\subset X$ be the maximal open over which $f$ is an isomorphism.\nThen $U$ contains\n\\begin{enumerate}\n\\item every point of codimension $0$ in $X$,\n\\item every $x \\in |X|$ of codimension $1$ on $X$ such that the local ring of\n$X$ at $x$ is normal (Properties of Spaces, Remark\n\\ref{spaces-properties-remark-list-properties-local-ring-local-etale-topology}),\nand\n\\item every $x \\in |X|$ such that the fibre of $|Y| \\to |X|$ over $x$ is\nfinite and such that the local ring of $X$ at $x$ is normal.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Generically finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BBQ","source_file":"spaces-over-fields.tex","source_line":156,"source_end_line":172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L156-L172","statement_sha256":"4bf57ad06691e2f6281f07bc27a2993b6b91c8685872b43a02d5c38fc61d7166","origin":"The Stacks Project","memory_eligible":false,"source_rank":11648,"rank":11648,"depth":74,"x":330.07,"y":1691.809,"cluster":"algebraic-spaces"},{"id":"stacks:0AD4","tag":"0AD4","title":"Integral algebraic spaces · Definition 0AD4","summary":"Let S be a scheme. We say an algebraic space X over S is integral if it is reduced, decent, and |X| is irreducible.","statement_latex":"Let $S$ be a scheme. We say an algebraic space $X$ over $S$ is\n{\\it integral} if it is reduced, decent, and $|X|$ is irreducible.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Integral algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AD4","source_file":"spaces-over-fields.tex","source_line":220,"source_end_line":224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L220-L224","statement_sha256":"e96408fb75d9a224d961281b2347a3622a350d8342032ad030ffed72baaff682","origin":"The Stacks Project","memory_eligible":false,"source_rank":11649,"rank":11649,"depth":0,"x":82.269,"y":1589.075,"cluster":"algebraic-spaces"},{"id":"stacks:0END","tag":"0END","title":"Integral algebraic spaces · Lemma 0END","summary":"Let S be a scheme. Let X be an integral algebraic space over S. Let eta ∈ |X| be the generic point of X. There are canonical identifications R(X) = O_X, eta^h = kappa(eta) where R(X) is the ring of rational functions defined in Morphisms of Spaces, Definition [Tag 0EMP], kappa(eta) is the residue field defined in Decent Spaces, Definition [Tag 0EMW], and O_X, eta^h is the henselian local ring defined in Decent Spaces, Definition [Tag 0BGU]. In particular, these rings are…","statement_latex":"Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$.\nLet $\\eta \\in |X|$ be the generic point of $X$.\nThere are canonical identifications\n$$\nR(X) = \\mathcal{O}_{X, \\eta}^h = \\kappa(\\eta)\n$$\nwhere $R(X)$ is the ring of rational functions defined in\nMorphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-ring-of-rational-functions},\n$\\kappa(\\eta)$ is the residue field defined in\nDecent Spaces, Definition \\ref{decent-spaces-definition-residue-field},\nand $\\mathcal{O}_{X, \\eta}^h$ is the henselian local ring defined in\nDecent Spaces, Definition\n\\ref{decent-spaces-definition-elemenary-etale-neighbourhood}.\nIn particular, these rings are fields.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0END","source_file":"spaces-over-fields.tex","source_line":238,"source_end_line":255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L238-L255","statement_sha256":"2f6af4566c7cedee32e859382098d699458bba78faf7393b11c51eaabeb926d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11650,"rank":11650,"depth":3,"x":347.808,"y":1524.173,"cluster":"algebraic-spaces"},{"id":"stacks:0ENE","tag":"0ENE","title":"Integral algebraic spaces · Definition 0ENE","summary":"Let S be a scheme. Let X be an integral algebraic space over S. The function field, or the field of rational functions of X is the field R(X) of Lemma [Tag 0END].","statement_latex":"Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$.\nThe {\\it function field}, or the {\\it field of rational functions}\nof $X$ is the field $R(X)$ of\nLemma \\ref{lemma-integral-algebraic-space-rational-functions}.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Integral algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENE","source_file":"spaces-over-fields.tex","source_line":271,"source_end_line":277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L271-L277","statement_sha256":"e3557991dac1e29d34820714bd4dae911b433fc5cd7808b0b97eec0e1708bc61","origin":"The Stacks Project","memory_eligible":false,"source_rank":11651,"rank":11651,"depth":4,"x":204.089,"y":1722.853,"cluster":"algebraic-spaces"},{"id":"stacks:0BH2","tag":"0BH2","title":"Integral algebraic spaces · Lemma 0BH2","summary":"Let S be a scheme. Let X be an integral algebraic space over S. Then Γ(X, O_X) is a domain.","statement_latex":"Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$.\nThen $\\Gamma(X, \\mathcal{O}_X)$ is a domain.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BH2","source_file":"spaces-over-fields.tex","source_line":282,"source_end_line":286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L282-L286","statement_sha256":"2321a4327403936bd63f66c6683441fafd57f518ccf236d6ba9b2af5e9b86e80","origin":"The Stacks Project","memory_eligible":false,"source_rank":11652,"rank":11652,"depth":5,"x":150.252,"y":1494.629,"cluster":"algebraic-spaces"},{"id":"stacks:0AYH","tag":"0AYH","title":"Integral algebraic spaces · Lemma 0AYH","summary":"Let S be a scheme. Let X be a normal integral algebraic space over S. For every x ∈ |X| there exists a normal integral affine scheme U and an étale morphism U → X such that x is in the image.","statement_latex":"Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$.\nFor every $x \\in |X|$ there exists a normal integral affine scheme $U$\nand an \\'etale morphism $U \\to X$ such that $x$ is in the image.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYH","source_file":"spaces-over-fields.tex","source_line":299,"source_end_line":304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L299-L304","statement_sha256":"e9e87e438cf16076bfb0166d11e117d2ca3db29703418a39640ba4a51a75dd29","origin":"The Stacks Project","memory_eligible":false,"source_rank":11653,"rank":11653,"depth":61,"x":373.649,"y":1632.472,"cluster":"algebraic-spaces"},{"id":"stacks:0BH3","tag":"0BH3","title":"Integral algebraic spaces · Lemma 0BH3","summary":"Let S be a scheme. Let X be a normal integral algebraic space over S. Then Γ(X, O_X) is a normal domain.","statement_latex":"Let $S$ be a scheme. Let $X$ be a normal integral algebraic space over $S$.\nThen $\\Gamma(X, \\mathcal{O}_X)$ is a normal domain.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BH3","source_file":"spaces-over-fields.tex","source_line":321,"source_end_line":325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L321-L325","statement_sha256":"491d8f631d55da98e626b5fb4643e1ab964b5cf390f7665f1d467c4e22173194","origin":"The Stacks Project","memory_eligible":false,"source_rank":11654,"rank":11654,"depth":62,"x":97.864,"y":1657.608,"cluster":"algebraic-spaces"},{"id":"stacks:0ENF","tag":"0ENF","title":"Integral algebraic spaces · Lemma 0ENF","summary":"Let S be a scheme. Let X be a decent algebraic space over S. There are canonical bijections between the following sets: • the set of points of X, i.e., |X|, • the set of irreducible closed subsets of |X|, • the set of integral closed subspaces of X. The bijection from (1) to (2) sends x to overline(x). The bijection from (3) to (2) sends Z to |Z|.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nThere are canonical bijections between the following sets:\n\\begin{enumerate}\n\\item the set of points of $X$, i.e., $|X|$,\n\\item the set of irreducible closed subsets of $|X|$,\n\\item the set of integral closed subspaces of $X$.\n\\end{enumerate}\nThe bijection from (1) to (2) sends $x$ to $\\overline{\\{x\\}}$.\nThe bijection from (3) to (2) sends $Z$ to $|Z|$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENF","source_file":"spaces-over-fields.tex","source_line":348,"source_end_line":359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L348-L359","statement_sha256":"7515183330a323553fb7ef5de9abb1c3b0b5739c85e00a501da6ad16d5eddaba","origin":"The Stacks Project","memory_eligible":false,"source_rank":11655,"rank":11655,"depth":60,"x":281.146,"y":1482.456,"cluster":"algebraic-spaces"},{"id":"stacks:0AD5","tag":"0AD5","title":"Morphisms between integral algebraic spaces · Lemma 0AD5","summary":"Let S be a scheme. Let X, Y be integral algebraic spaces over S Let x ∈ |X| and y ∈ |Y| be the generic points. Let f : X → Y be locally of finite type. Assume f is dominant (Morphisms of Spaces, Definition [Tag 0ABL]). The following are equivalent: • the transcendence degree of x/y is 0, • the extension kappa(x)/kappa(y) (see proof) is finite, • there exist nonempty affine opens U ⊂ X and V ⊂ Y such that f(U) ⊂ V and f|_U : U → V is finite, • f is quasi-finite at x, and •…","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be integral algebraic spaces over $S$\nLet $x \\in |X|$ and $y \\in |Y|$ be the generic points. Let $f : X \\to Y$\nbe locally of finite type. Assume $f$ is dominant\n(Morphisms of Spaces, Definition \\ref{spaces-morphisms-definition-dominant}).\nThe following are equivalent:\n\\begin{enumerate}\n\\item the transcendence degree of $x/y$ is $0$,\n\\item the extension $\\kappa(x)/\\kappa(y)$ (see proof) is finite,\n\\item there exist nonempty affine opens $U \\subset X$ and $V \\subset Y$\nsuch that $f(U) \\subset V$ and $f|_U : U \\to V$ is finite,\n\\item $f$ is quasi-finite at $x$, and\n\\item $x$ is the only point of $|X|$ mapping to $y$.\n\\end{enumerate}\nIf $f$ is separated or if $f$ is quasi-compact, then these are\nalso equivalent to\n\\begin{enumerate}\n\\item[(6)] there exists a nonempty affine open $V \\subset Y$ such\nthat $f^{-1}(V) \\to V$ is finite.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Morphisms between integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AD5","source_file":"spaces-over-fields.tex","source_line":386,"source_end_line":407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L386-L407","statement_sha256":"1f9bb1dcd52b2f5e9e9911f2d11fa04228afd28a4ad660449b5e20b6c9712281","origin":"The Stacks Project","memory_eligible":false,"source_rank":11656,"rank":11656,"depth":62,"x":286.853,"y":1715.782,"cluster":"algebraic-spaces"},{"id":"stacks:0AD6","tag":"0AD6","title":"Morphisms between integral algebraic spaces · Definition 0AD6","summary":"Let S be a scheme. Let X and Y be integral algebraic spaces over S. Let f : X → Y be locally of finite type and dominant. Assume any of the equivalent conditions (1) -- (5) of Lemma [Tag 0AD5]. Let x ∈ |X| and y ∈ |Y| be the generic points. Then the positive integer deg(X/Y) = [kappa(x) : kappa(y)] is called the degree of X over Y.","statement_latex":"Let $S$ be a scheme.\nLet $X$ and $Y$ be integral algebraic spaces over $S$.\nLet $f : X \\to Y$ be locally of finite type and dominant.\nAssume any of the equivalent conditions (1) -- (5) of\nLemma \\ref{lemma-finite-degree}. Let $x \\in |X|$ and $y \\in |Y|$\nbe the generic points. Then the positive integer\n$$\n\\deg(X/Y) = [\\kappa(x) : \\kappa(y)]\n$$\nis called the {\\it degree of $X$ over $Y$}.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Morphisms between integral algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AD6","source_file":"spaces-over-fields.tex","source_line":449,"source_end_line":461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L449-L461","statement_sha256":"23690ebcdcfce740796bdb3ad0b5749932dcac99fec1f5c50f7eac041326def6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11657,"rank":11657,"depth":63,"x":94.868,"y":1546.844,"cluster":"algebraic-spaces"},{"id":"stacks:0ENH","tag":"0ENH","title":"Morphisms between integral algebraic spaces · Lemma 0ENH","summary":"Let S be a scheme. Let X, Y, Z be integral algebraic spaces over S. Let f : X → Y and g : Y → Z be dominant morphisms locally of finite type. Assume any of the equivalent conditions (1) -- (5) of Lemma [Tag 0AD5] hold for f and g. Then deg(X/Z) = deg(X/Y) deg(Y/Z).","statement_latex":"Let $S$ be a scheme.\nLet $X$, $Y$, $Z$ be integral algebraic spaces over $S$.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be dominant morphisms locally\nof finite type. Assume any of the equivalent conditions\n(1) -- (5) of Lemma \\ref{lemma-finite-degree} hold for $f$ and $g$. Then\n$$\n\\deg(X/Z) = \\deg(X/Y) \\deg(Y/Z).\n$$","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Morphisms between integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENH","source_file":"spaces-over-fields.tex","source_line":463,"source_end_line":473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L463-L473","statement_sha256":"4d03c5edd2aee17a6421d66635e784cdc06eb5f62f423714fae7b4871fbe79d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11658,"rank":11658,"depth":63,"x":372.496,"y":1562.492,"cluster":"algebraic-spaces"},{"id":"stacks:0EE5","tag":"0EE5","title":"Weil divisors · Lemma 0EE5","summary":"Let S be a scheme and let X be a locally Noetherian algebraic space over S. If T ⊂ |X| is a closed subset, then the collection of irreducible components of T is locally finite.","statement_latex":"Let $S$ be a scheme and let $X$ be a locally Noetherian\nalgebraic space over $S$. If $T \\subset |X|$ is a closed subset,\nthen the collection of irreducible components of $T$ is locally finite.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Weil divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EE5","source_file":"spaces-over-fields.tex","source_line":507,"source_end_line":512,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L507-L512","statement_sha256":"82eec4714f167b238a4b38ec184516acb515b8f48adbf61c830f8a23efcb7026","origin":"The Stacks Project","memory_eligible":false,"source_rank":11659,"rank":11659,"depth":4,"x":155.034,"y":1708.594,"cluster":"algebraic-spaces"},{"id":"stacks:0ENJ","tag":"0ENJ","title":"Weil divisors · Definition 0ENJ","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. • A prime divisor is an integral closed subspace Z ⊂ X of codimension 1, i.e., the generic point of |Z| is a point of codimension 1 on X. • A Weil divisor is a formal sum D = ∑ n_Z Z where the sum is over prime divisors of X and the collection (|Z| : n_Z not = 0) is locally finite in |X| (Topology, Definition [Tag 0BDS]). The group of all Weil divisors on X is denoted Div(X).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian integral algebraic space over $S$.\n\\begin{enumerate}\n\\item A {\\it prime divisor} is an integral closed subspace $Z \\subset X$\nof codimension $1$, i.e., the generic point of $|Z|$ is a point\nof codimension $1$ on $X$.\n\\item A {\\it Weil divisor} is a formal sum $D = \\sum n_Z Z$ where\nthe sum is over prime divisors of $X$ and the collection\n$\\{|Z| : n_Z \\not = 0\\}$ is locally finite in $|X|$\n(Topology, Definition \\ref{topology-definition-locally-finite}).\n\\end{enumerate}\nThe group of all Weil divisors on $X$ is denoted $\\text{Div}(X)$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Weil divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENJ","source_file":"spaces-over-fields.tex","source_line":536,"source_end_line":550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L536-L550","statement_sha256":"357cea110897e3caf8592010b5b03227a0d160ef08efa29c4c1fad5bbc2a02f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11660,"rank":11660,"depth":1,"x":197.927,"y":1477.296,"cluster":"algebraic-spaces"},{"id":"stacks:0ENK","tag":"0ENK","title":"Weil divisors · Lemma 0ENK","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. Let Z ⊂ X be a prime divisor and let xi ∈ |Z| be the generic point. Then the henselian local ring O_X, xi^h is a reduced 1-dimensional Noetherian local ring and there is a canonical injective map R(X) → Q(O_X, xi^h) from the function field R(X) of X into the total ring of fractions.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space\nover $S$. Let $Z \\subset X$ be a prime divisor and let $\\xi \\in |Z|$ be\nthe generic point. Then the henselian local ring $\\mathcal{O}_{X, \\xi}^h$\nis a reduced $1$-dimensional Noetherian local ring and\nthere is a canonical injective map\n$$\nR(X) \\longrightarrow Q(\\mathcal{O}_{X, \\xi}^h)\n$$\nfrom the function field $R(X)$ of $X$ into the total ring of fractions.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Weil divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENK","source_file":"spaces-over-fields.tex","source_line":562,"source_end_line":573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L562-L573","statement_sha256":"1acd0e2b6d8fa353616a3da8ce42df7695548feb3f8ffce27d437fb7ba4c7712","origin":"The Stacks Project","memory_eligible":false,"source_rank":11661,"rank":11661,"depth":48,"x":352.415,"y":1672.334,"cluster":"algebraic-spaces"},{"id":"stacks:0ENL","tag":"0ENL","title":"Weil divisors · Definition 0ENL","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. Let f ∈ R(X)^*. For every prime divisor Z ⊂ X we define the order of vanishing of f along Z as the integer ord_Z(f) = length_O_X, xi^h (O_X, xi^h/a O_X, xi^h) - length_O_X, xi^h (O_X, xi^h/b O_X, xi^h) where a, b ∈ O_X, xi^h are nonzerodivisors such that the image of f in Q(O_X, xi^h) (Lemma [Tag 0ENK]) is equal to a/b. This is well defined by Algebra, Lemma [Tag 02MC].","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic\nspace over $S$. Let $f \\in R(X)^*$. For every prime divisor\n$Z \\subset X$ we define the {\\it order of vanishing of $f$ along $Z$}\nas the integer\n$$\n\\text{ord}_Z(f) =\n\\text{length}_{\\mathcal{O}_{X, \\xi}^h}\n(\\mathcal{O}_{X, \\xi}^h/a \\mathcal{O}_{X, \\xi}^h) -\n\\text{length}_{\\mathcal{O}_{X, \\xi}^h}\n(\\mathcal{O}_{X, \\xi}^h/b \\mathcal{O}_{X, \\xi}^h)\n$$\nwhere $a, b \\in \\mathcal{O}_{X, \\xi}^h$ are nonzerodivisors\nsuch that the image of $f$ in $Q(\\mathcal{O}_{X, \\xi}^h)$\n(Lemma \\ref{lemma-order-vanishing}) is equal to $a/b$.\nThis is well defined by\nAlgebra, Lemma \\ref{algebra-lemma-ord-additive}.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Weil divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENL","source_file":"spaces-over-fields.tex","source_line":620,"source_end_line":638,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L620-L638","statement_sha256":"32fa3d98cb17c44d024732b91568a408f552ac25f42177793d68d6cb2fa5ec5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11662,"rank":11662,"depth":49,"x":81.455,"y":1616.135,"cluster":"algebraic-spaces"},{"id":"stacks:0ENM","tag":"0ENM","title":"Weil divisors · Lemma 0ENM","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. Let f ∈ R(X)^*. If the prime divisor Z ⊂ X meets the schematic locus of X, then the order of vanishing ord_Z(f) of Definition [Tag 0ENL] agrees with the order of vanishing of Divisors, Definition [Tag 02RJ].","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic\nspace over $S$. Let $f \\in R(X)^*$. If the prime divisor\n$Z \\subset X$ meets the schematic locus of $X$, then the order\nof vanishing $\\text{ord}_Z(f)$ of Definition \\ref{definition-order-vanishing}\nagrees with the order of vanishing of\nDivisors, Definition \\ref{divisors-definition-order-vanishing}.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Weil divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENM","source_file":"spaces-over-fields.tex","source_line":659,"source_end_line":667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L659-L667","statement_sha256":"dd81433c88bef8ae2c242aab9ea5b38a458169c032942c63db12a4b1d82a4060","origin":"The Stacks Project","memory_eligible":false,"source_rank":11663,"rank":11663,"depth":60,"x":326.631,"y":1503.744,"cluster":"algebraic-spaces"},{"id":"stacks:0ENN","tag":"0ENN","title":"Weil divisors · Lemma 0ENN","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. Let f ∈ R(X)^*. Then the collections (Z ⊂ X mid Z a prime divisor with generic point xi and f not in O_X, xi) and (Z ⊂ X mid Z a prime divisor and ord_Z(f) not = 0) are locally finite in X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space\nover $S$. Let $f \\in R(X)^*$. Then the collections\n$$\n\\{Z \\subset X \\mid Z\\text{ a prime divisor with generic point }\\xi\n\\text{ and }f\\text{ not in }\\mathcal{O}_{X, \\xi}\\}\n$$\nand\n$$\n\\{Z \\subset X \\mid Z \\text{ a prime divisor and }\\text{ord}_Z(f) \\not = 0\\}\n$$\nare locally finite in $X$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Weil divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENN","source_file":"spaces-over-fields.tex","source_line":687,"source_end_line":700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L687-L700","statement_sha256":"9e0d61e75e59ffc9ac84ab7051a5af2080f1b6cd79f39fdccb61bdc4c82cc0ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":11664,"rank":11664,"depth":5,"x":236.157,"y":1725.901,"cluster":"algebraic-spaces"},{"id":"stacks:0ENP","tag":"0ENP","title":"Weil divisors · Definition 0ENP","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. Let f ∈ R(X)^*. The principal Weil divisor associated to f is the Weil divisor div(f) = div_X(f) = ∑ ord_Z(f) [Z] where the sum is over prime divisors and ord_Z(f) is as in Definition [Tag 0ENL]. This makes sense by Lemma [Tag 0ENN].","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian integral algebraic space over $S$.\nLet $f \\in R(X)^*$.\nThe {\\it principal Weil divisor associated to $f$} is the Weil divisor\n$$\n\\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z]\n$$\nwhere the sum is over prime divisors and $\\text{ord}_Z(f)$ is as in\nDefinition \\ref{definition-order-vanishing}. This makes sense\nby Lemma \\ref{lemma-divisor-locally-finite}.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Weil divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENP","source_file":"spaces-over-fields.tex","source_line":713,"source_end_line":725,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L713-L725","statement_sha256":"16408db5a7082f568f8220bd985d29300519f96f3568aad1f3871620220978fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11665,"rank":11665,"depth":50,"x":124.137,"y":1510.591,"cluster":"algebraic-spaces"},{"id":"stacks:0ENQ","tag":"0ENQ","title":"Weil divisors · Lemma 0ENQ","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. Let f, g ∈ R(X)^*. Then div_X(fg) = div_X(f) + div_X(g) as Weil divisors on X.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian integral algebraic space over $S$.\nLet $f, g \\in R(X)^*$. Then\n$$\n\\text{div}_X(fg) = \\text{div}_X(f) + \\text{div}_X(g)\n$$\nas Weil divisors on $X$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Weil divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENQ","source_file":"spaces-over-fields.tex","source_line":727,"source_end_line":736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L727-L736","statement_sha256":"da2d923fe21b6335287086b367ef17aa5bb5d8903853bce17ff62646c731c6b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11666,"rank":11666,"depth":0,"x":380.072,"y":1605.863,"cluster":"algebraic-spaces"},{"id":"stacks:0ENR","tag":"0ENR","title":"Weil divisors · Definition 0ENR","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. The Weil divisor class group of X is the quotient of the group of Weil divisors by the subgroup of principal Weil divisors. Notation: Cl(X).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian integral algebraic space over $S$. The\n{\\it Weil divisor class group} of $X$ is the quotient of\nthe group of Weil divisors by the subgroup of principal Weil divisors.\nNotation: $\\text{Cl}(X)$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Weil divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENR","source_file":"spaces-over-fields.tex","source_line":747,"source_end_line":754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L747-L754","statement_sha256":"3ee38d2e63101b2b3593fbfffb9716a41d68218ded32042da423bf66a9c229f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11667,"rank":11667,"depth":0,"x":114.54,"y":1680.89,"cluster":"algebraic-spaces"},{"id":"stacks:0EPR","tag":"0EPR","title":"The Weil divisor class associated to an invertible module · Definition 0EPR","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic algebraic space over S. Let L be an invertible O_X-module. Let s ∈ Γ(X, K_X(L)) be a regular meromorphic section of L. For every prime divisor Z ⊂ X with generic point xi ∈ |Z| we define the order of vanishing of s along Z as the integer ord_Z, L(s) = length_O_X, xi^h (O_X, xi^h/a O_X, xi^h) - length_O_X, xi^h (O_X, xi^h/b O_X, xi^h) where a, b ∈ O_X, xi^h are nonzerodivisors such that the element s/s_xi…","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral\nalgebraic algebraic space over $S$. Let $\\mathcal{L}$ be an\ninvertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{K}_X(\\mathcal{L}))$\nbe a regular meromorphic section of $\\mathcal{L}$.\nFor every prime divisor $Z \\subset X$ with generic point $\\xi \\in |Z|$\nwe define the\n{\\it order of vanishing of $s$ along $Z$}\nas the integer\n$$\n\\text{ord}_{Z, \\mathcal{L}}(s) =\n\\text{length}_{\\mathcal{O}_{X, \\xi}^h}\n(\\mathcal{O}_{X, \\xi}^h/a \\mathcal{O}_{X, \\xi}^h) -\n\\text{length}_{\\mathcal{O}_{X, \\xi}^h}\n(\\mathcal{O}_{X, \\xi}^h/b \\mathcal{O}_{X, \\xi}^h)\n$$\nwhere $a, b \\in \\mathcal{O}_{X, \\xi}^h$ are nonzerodivisors\nsuch that the element $s/s_\\xi$ of $Q(\\mathcal{O}_{X, \\xi}^h)$\nconstructed above is equal to $a/b$.\nThis is well defined by the above and\nAlgebra, Lemma \\ref{algebra-lemma-ord-additive}.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"The Weil divisor class associated to an invertible module","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPR","source_file":"spaces-over-fields.tex","source_line":891,"source_end_line":914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L891-L914","statement_sha256":"051e0e930a4b78cd6ef4af07cbe3e43c18e97ec91c8f13c4789aad9947ad41c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11668,"rank":11668,"depth":11,"x":250.104,"y":1474.747,"cluster":"algebraic-spaces"},{"id":"stacks:0EPS","tag":"0EPS","title":"The Weil divisor class associated to an invertible module · Lemma 0EPS","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. Let L be an invertible O_X-module. Let s ∈ K_X(L) be a regular (i.e., nonzero) meromorphic section of L. Then the sets (Z ⊂ X mid Z a prime divisor with generic point xi and s not in L_xi) and (Z ⊂ X mid Z is a prime divisor and ord_Z, L(s) not = 0) are locally finite in X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space\nover $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\mathcal{K}_X(\\mathcal{L})$ be a\nregular (i.e., nonzero) meromorphic section of $\\mathcal{L}$. Then the sets\n$$\n\\{Z \\subset X \\mid Z \\text{ a prime divisor with generic point }\\xi\n\\text{ and }s\\text{ not in }\\mathcal{L}_\\xi\\}\n$$\nand\n$$\n\\{Z \\subset X \\mid Z \\text{ is a prime divisor and }\n\\text{ord}_{Z, \\mathcal{L}}(s) \\not = 0\\}\n$$\nare locally finite in $X$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"The Weil divisor class associated to an invertible module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPS","source_file":"spaces-over-fields.tex","source_line":923,"source_end_line":939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L923-L939","statement_sha256":"99c2ef51895cd8628a23a936bf9fb62bbcebc422b8e1193eff89cfccf8510805","origin":"The Stacks Project","memory_eligible":false,"source_rank":11669,"rank":11669,"depth":5,"x":315.963,"y":1703.842,"cluster":"algebraic-spaces"},{"id":"stacks:0EPT","tag":"0EPT","title":"The Weil divisor class associated to an invertible module · Lemma 0EPT","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S Let L be an invertible O_X-module. Let s, s' ∈ K_X(L) be nonzero meromorphic sections of L. Then f = s/s' is an element of R(X)^* and we have ∑ ord_Z, L(s)[Z] = ∑ ord_Z, L(s')[Z] + div(f) as Weil divisors.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space\nover $S$ Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s, s' \\in \\mathcal{K}_X(\\mathcal{L})$ be nonzero\nmeromorphic sections of $\\mathcal{L}$. Then $f = s/s'$\nis an element of $R(X)^*$ and we have\n$$\n\\sum \\text{ord}_{Z, \\mathcal{L}}(s)[Z]\n=\n\\sum \\text{ord}_{Z, \\mathcal{L}}(s')[Z]\n+\n\\text{div}(f)\n$$\nas Weil divisors.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"The Weil divisor class associated to an invertible module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPT","source_file":"spaces-over-fields.tex","source_line":950,"source_end_line":965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L950-L965","statement_sha256":"181c445db87018feeaf04d2a4f03d1f9b9e1431c5810047c89dab0581b505b42","origin":"The Stacks Project","memory_eligible":false,"source_rank":11670,"rank":11670,"depth":6,"x":82.998,"y":1572.186,"cluster":"algebraic-spaces"},{"id":"stacks:0EPU","tag":"0EPU","title":"The Weil divisor class associated to an invertible module · Definition 0EPU","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. Let L be an invertible O_X-module. • For any nonzero meromorphic section s of L we define the Weil divisor associated to s as div_L(s) = ∑ ord_Z, L(s) [Z] ∈ Div(X) where the sum is over prime divisors. This is well defined by Lemma [Tag 0EPS]. • We define Weil divisor class associated to L as the image of div_L(s) in Cl(X) where s is any nonzero meromorphic section of L over X. This is well…","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space\nover $S$. Let $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$\nwe define the {\\it Weil divisor associated to $s$} as\n$$\n\\text{div}_\\mathcal{L}(s) =\n\\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z] \\in \\text{Div}(X)\n$$\nwhere the sum is over prime divisors. This is well defined by\nLemma \\ref{lemma-divisor-meromorphic-locally-finite}.\n\\item We define {\\it Weil divisor class associated to $\\mathcal{L}$}\nas the image of $\\text{div}_\\mathcal{L}(s)$ in $\\text{Cl}(X)$\nwhere $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over $X$.\nThis is well defined by\nLemma \\ref{lemma-divisor-meromorphic-well-defined}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"The Weil divisor class associated to an invertible module","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPU","source_file":"spaces-over-fields.tex","source_line":973,"source_end_line":992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L973-L992","statement_sha256":"a36b2301035be6e4e5fa0d1ea81905bf7028a18fc2bc87e0c12b2451e40308e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11671,"rank":11671,"depth":7,"x":360.859,"y":1537.052,"cluster":"algebraic-spaces"},{"id":"stacks:0EPV","tag":"0EPV","title":"The Weil divisor class associated to an invertible module · Lemma 0EPV","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. Let L, N be invertible O_X-modules. Let s, resp. t be a nonzero meromorphic section of L, resp. N. Then st is a nonzero meromorphic section of L ⊗_O_X N and div_L ⊗ N(st) = div_L(s) + div_N(t) in Div(X). In particular, the Weil divisor class of L ⊗_O_X N is the sum of the Weil divisor classes of L and N.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space\nover $S$. Let $\\mathcal{L}$, $\\mathcal{N}$ be invertible\n$\\mathcal{O}_X$-modules. Let $s$, resp.\\ $t$ be a nonzero meromorphic section\nof $\\mathcal{L}$, resp.\\ $\\mathcal{N}$. Then $st$ is a nonzero\nmeromorphic section of $\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ and\n$$\n\\text{div}_{\\mathcal{L} \\otimes \\mathcal{N}}(st)\n=\n\\text{div}_\\mathcal{L}(s) + \\text{div}_\\mathcal{N}(t)\n$$\nin $\\text{Div}(X)$. In particular, the Weil divisor class of\n$\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}$ is the sum\nof the Weil divisor classes of $\\mathcal{L}$ and $\\mathcal{N}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"The Weil divisor class associated to an invertible module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPV","source_file":"spaces-over-fields.tex","source_line":997,"source_end_line":1012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L997-L1012","statement_sha256":"b4b0c2b10c48fde74bc4370c197b9beaaf706981089fcb315947e56354b058fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11672,"rank":11672,"depth":11,"x":184.094,"y":1720.757,"cluster":"algebraic-spaces"},{"id":"stacks:0EPX","tag":"0EPX","title":"The Weil divisor class associated to an invertible module · Lemma 0EPX","summary":"Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S. If X is normal, then the map ([Tag 0EPW]) Pic(X) → Cl(X) is injective.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space\nover $S$. If $X$ is normal, then the map (\\ref{equation-c1})\n$\\Pic(X) \\to \\text{Cl}(X)$ is injective.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"The Weil divisor class associated to an invertible module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPX","source_file":"spaces-over-fields.tex","source_line":1045,"source_end_line":1050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1045-L1050","statement_sha256":"fe536ad7ef21b8250b6f9cf6fb3305f5de5297b38d1c85fa27803a36c845e761","origin":"The Stacks Project","memory_eligible":false,"source_rank":11673,"rank":11673,"depth":61,"x":166.695,"y":1484.825,"cluster":"algebraic-spaces"},{"id":"stacks:0AD8","tag":"0AD8","title":"Modifications and alterations · Definition 0AD8","summary":"Let S be a scheme. Let X be an integral algebraic space over S. A modification of X is a birational proper morphism f : X' → X of algebraic spaces over S with X' integral.","statement_latex":"Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. A\n{\\it modification of $X$} is a birational proper morphism\n$f : X' \\to X$ of algebraic spaces over $S$ with $X'$ integral.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Modifications and alterations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AD8","source_file":"spaces-over-fields.tex","source_line":1137,"source_end_line":1142,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1137-L1142","statement_sha256":"d6fcce98b151ce9150644f6feb3237ac884e43928dc2a1fef9a736356d61231d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11674,"rank":11674,"depth":0,"x":369.401,"y":1649.043,"cluster":"algebraic-spaces"},{"id":"stacks:0AD9","tag":"0AD9","title":"Modifications and alterations · Lemma 0AD9","summary":"Let f : X' → X be a modification as in Definition [Tag 0AD8]. There exists a nonempty open U ⊂ X such that f^-1(U) → U is an isomorphism.","statement_latex":"Let $f : X' \\to X$ be a modification as in\nDefinition \\ref{definition-modification}.\nThere exists a nonempty open $U \\subset X$ such that $f^{-1}(U) \\to U$\nis an isomorphism.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Modifications and alterations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AD9","source_file":"spaces-over-fields.tex","source_line":1148,"source_end_line":1154,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1148-L1154","statement_sha256":"11c8a7e6ad388b5d876cd40460a9a28f612ec0d5def484c1da4ce1e63b851580","origin":"The Stacks Project","memory_eligible":false,"source_rank":11675,"rank":11675,"depth":63,"x":87.666,"y":1642.967,"cluster":"algebraic-spaces"},{"id":"stacks:0ADA","tag":"0ADA","title":"Modifications and alterations · Definition 0ADA","summary":"Let S be a scheme. Let X be an integral algebraic space over S. An alteration of X is a proper dominant morphism f : Y → X of algebraic spaces over S with Y integral such that f^-1(U) → U is finite for some nonempty open U ⊂ X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$.\nAn {\\it alteration of $X$} is a proper dominant morphism $f : Y \\to X$\nof algebraic spaces over $S$ with $Y$ integral such that $f^{-1}(U) \\to U$\nis finite for some nonempty open $U \\subset X$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Modifications and alterations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADA","source_file":"spaces-over-fields.tex","source_line":1170,"source_end_line":1176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1170-L1176","statement_sha256":"3268b6cd9b5190ce4fc4aecd4eea9a8881849b17fcde4d951400d72fee67c297","origin":"The Stacks Project","memory_eligible":false,"source_rank":11676,"rank":11676,"depth":0,"x":300.453,"y":1487.472,"cluster":"algebraic-spaces"},{"id":"stacks:0ADB","tag":"0ADB","title":"Modifications and alterations · Lemma 0ADB","summary":"Let S be a scheme. Let f : X → Y be a proper dominant morphism of integral algebraic spaces over S. Then f is an alteration if and only if any of the equivalent conditions (1) -- (6) of Lemma [Tag 0AD5] hold.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper dominant morphism of\nintegral algebraic spaces over $S$. Then $f$ is an alteration\nif and only if any of the equivalent conditions (1) -- (6) of\nLemma \\ref{lemma-finite-degree} hold.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Modifications and alterations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADB","source_file":"spaces-over-fields.tex","source_line":1184,"source_end_line":1190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1184-L1190","statement_sha256":"c3fbc2efd5452edaa8dfeca989248808c56ca6916bfaa2b557355c122542efa2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11677,"rank":11677,"depth":63,"x":268.569,"y":1723.041,"cluster":"algebraic-spaces"},{"id":"stacks:0DMN","tag":"0DMN","title":"Modifications and alterations · Lemma 0DMN","summary":"Let S be a scheme. Let f : X → Y be a proper surjective morphism of algebraic spaces over S. Assume Y is integral. Then there exists an integral closed subspace X' ⊂ X such that f' = f|_X' : X' → Y is an alteration.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper surjective morphism of\nalgebraic spaces over $S$. Assume $Y$ is integral. Then\nthere exists an integral closed subspace $X' \\subset X$ such that\n$f' = f|_{X'} : X' \\to Y$ is an alteration.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Modifications and alterations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMN","source_file":"spaces-over-fields.tex","source_line":1196,"source_end_line":1202,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1196-L1202","statement_sha256":"5105ee5db31db2c0fd82d24dccd56648eb515835a1b1997fa6af6450a1a8cc48","origin":"The Stacks Project","memory_eligible":false,"source_rank":11678,"rank":11678,"depth":64,"x":102.522,"y":1531.106,"cluster":"algebraic-spaces"},{"id":"stacks:06LZ","tag":"06LZ","title":"Schematic locus · Lemma 06LZ","summary":"Let S be a scheme. Let X be an algebraic space over S. Assume X satisfies at least one of the following conditions • X is quasi-separated and dim(X) = 0, • X is locally of finite type over a field k and dim(X) = 0, • X is Noetherian and dim(X) = 0, or • add more here. Then X is a separated scheme and any quasi-compact open of X is affine.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nAssume $X$ satisfies at least one of the following conditions\n\\begin{enumerate}\n\\item $X$ is quasi-separated and $\\dim(X) = 0$,\n\\item $X$ is locally of finite type over a field $k$ and $\\dim(X) = 0$,\n\\item $X$ is Noetherian and $\\dim(X) = 0$, or\n\\item add more here.\n\\end{enumerate}\nThen $X$ is a separated scheme and any quasi-compact open of $X$ is affine.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LZ","source_file":"spaces-over-fields.tex","source_line":1268,"source_end_line":1279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1268-L1279","statement_sha256":"bb62533f7890d3a2394d2d0b05377b1be85cbb7bc96fae3105b48b710dc75317","origin":"The Stacks Project","memory_eligible":false,"source_rank":11679,"rank":11679,"depth":68,"x":379.509,"y":1578.452,"cluster":"algebraic-spaces"},{"id":"stacks:0ADC","tag":"0ADC","title":"Schematic locus · Lemma 0ADC","summary":"Let S be a scheme. Let X be a quasi-separated algebraic space over S. Let x ∈ |X|. The following are equivalent • x is a point of codimension 0 on X, • the local ring of X at x has dimension 0, and • x is a generic point of an irreducible component of |X|. If true, then there exists an open subspace of X containing x which is a scheme.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$.\nLet $x \\in |X|$. The following are equivalent\n\\begin{enumerate}\n\\item $x$ is a point of codimension $0$ on $X$,\n\\item the local ring of $X$ at $x$ has dimension $0$, and\n\\item $x$ is a generic point of an irreducible component of $|X|$.\n\\end{enumerate}\nIf true, then there exists an open subspace of $X$\ncontaining $x$ which is a scheme.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADC","source_file":"spaces-over-fields.tex","source_line":1380,"source_end_line":1391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1380-L1391","statement_sha256":"aa521bfb2bc5418598034c580e322ae12b31ed8bc3251d4ae8fdbd771ee6f223","origin":"The Stacks Project","memory_eligible":false,"source_rank":11680,"rank":11680,"depth":61,"x":137.016,"y":1700.797,"cluster":"algebraic-spaces"},{"id":"stacks:0ADD","tag":"0ADD","title":"Schematic locus · Lemma 0ADD","summary":"Separated algebraic spaces are schemes in codimension 1. Let S be a scheme. Let X be an algebraic space over S. Let x ∈ |X|. If X is separated, locally Noetherian, and the dimension of the local ring of X at x is ≤ 1 (Properties of Spaces, Definition [Tag 04NA]), then there exists an open subspace of X containing x which is a scheme.","statement_latex":"\\begin{slogan}\nSeparated algebraic spaces are schemes in codimension 1.\n\\end{slogan}\nLet $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $x \\in |X|$. If $X$ is separated, locally Noetherian, and\nthe dimension of the local ring of $X$ at $x$ is $\\leq 1$\n(Properties of Spaces, Definition\n\\ref{spaces-properties-definition-dimension-local-ring}),\nthen there exists an open subspace of $X$ containing $x$ which is a scheme.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADD","source_file":"spaces-over-fields.tex","source_line":1442,"source_end_line":1453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1442-L1453","statement_sha256":"43000b03a48d7f2c659014d279bf1b1936fba159e0e58fa855b850bba954e64d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11681,"rank":11681,"depth":67,"x":217.498,"y":1472.822,"cluster":"algebraic-spaces"},{"id":"stacks:0B83","tag":"0B83","title":"Schematic locus and field extension · Lemma 0B83","summary":"Let k be a field. Let X be an algebraic space over k. If there exists a purely inseparable field extension k'/k such that X_k' is a scheme, then X is a scheme.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic space over $k$.\nIf there exists a purely inseparable field extension $k'/k$\nsuch that $X_{k'}$ is a scheme, then $X$ is a scheme.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus and field extension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B83","source_file":"spaces-over-fields.tex","source_line":1577,"source_end_line":1582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1577-L1582","statement_sha256":"93567d38837bd2ad2c7c9172a235dc862b7de2fdf69ecca32543cc5f6cdb94cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11682,"rank":11682,"depth":68,"x":341.571,"y":1686.747,"cluster":"algebraic-spaces"},{"id":"stacks:0B84","tag":"0B84","title":"Schematic locus and field extension · Lemma 0B84","summary":"Let k be a field with algebraic closure overlinek. Let X be a quasi-separated algebraic space over k. • If there exists a field extension K/k such that X_K is a scheme, then X_overlinek is a scheme. • If X is quasi-compact and there exists a field extension K/k such that X_K is a scheme, then X_k' is a scheme for some finite separable extension k' of k.","statement_latex":"Let $k$ be a field with algebraic closure $\\overline{k}$.\nLet $X$ be a quasi-separated algebraic space over $k$.\n\\begin{enumerate}\n\\item If there exists a field extension $K/k$ such that\n$X_K$ is a scheme, then $X_{\\overline{k}}$ is a scheme.\n\\item If $X$ is quasi-compact and there exists a field extension\n$K/k$ such that $X_K$ is a scheme, then $X_{k'}$\nis a scheme for some finite separable extension $k'$ of $k$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus and field extension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B84","source_file":"spaces-over-fields.tex","source_line":1591,"source_end_line":1602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1591-L1602","statement_sha256":"f398df5a3caca0fa2d0d3a40c453dca9840586397ce63f64627938cf53a2e988","origin":"The Stacks Project","memory_eligible":false,"source_rank":11683,"rank":11683,"depth":69,"x":77.861,"y":1599.342,"cluster":"algebraic-spaces"},{"id":"stacks:0B86","tag":"0B86","title":"Schematic locus and field extension · Lemma 0B86","summary":"Let k'/k be a finite Galois extension with Galois group G. Let X be an algebraic space over k. Then G acts freely on the algebraic space X_k' and X = X_k'/G in the sense of Properties of Spaces, Lemma [Tag 071S].","statement_latex":"Let $k'/k$ be a finite Galois extension with Galois group $G$.\nLet $X$ be an algebraic space over $k$. Then $G$ acts freely on the\nalgebraic space $X_{k'}$ and $X = X_{k'}/G$ in the sense of\nProperties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus and field extension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B86","source_file":"spaces-over-fields.tex","source_line":1624,"source_end_line":1630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1624-L1630","statement_sha256":"240f5b994067591ddc8f586b7a2dcbd6ebdd5a38c0047ed59581255d18c673a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11684,"rank":11684,"depth":53,"x":342.794,"y":1514.097,"cluster":"algebraic-spaces"},{"id":"stacks:0B87","tag":"0B87","title":"Schematic locus and field extension · Lemma 0B87","summary":"Let S be a scheme. Let X be an algebraic space over S and let G be a finite group acting freely on X. Set Y = X/G as in Properties of Spaces, Lemma [Tag 071S]. For y ∈ |Y| the following are equivalent • y is in the schematic locus of Y, and • there exists an affine open U ⊂ X containing the preimage of y.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and\nlet $G$ be a finite group acting freely on $X$. Set $Y = X/G$ as\nin Properties of Spaces, Lemma \\ref{spaces-properties-lemma-quotient}.\nFor $y \\in |Y|$ the following are equivalent\n\\begin{enumerate}\n\\item $y$ is in the schematic locus of $Y$, and\n\\item there exists an affine open $U \\subset X$\ncontaining the preimage of $y$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus and field extension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B87","source_file":"spaces-over-fields.tex","source_line":1637,"source_end_line":1648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1637-L1648","statement_sha256":"c1e857704cc8e6390cc6bea70b4498af0c88c22686ad89c8c6c73531aa3303ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":11685,"rank":11685,"depth":53,"x":215.898,"y":1727.436,"cluster":"algebraic-spaces"},{"id":"stacks:0B85","tag":"0B85","title":"Schematic locus and field extension · Lemma 0B85","summary":"Let k be a field. Let X be a quasi-separated algebraic space over k. If there exists a purely transcendental field extension K/k such that X_K is a scheme, then X is a scheme.","statement_latex":"Let $k$ be a field. Let $X$ be a quasi-separated\nalgebraic space over $k$. If there exists a purely transcendental\nfield extension $K/k$ such that $X_K$ is a scheme, then\n$X$ is a scheme.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus and field extension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B85","source_file":"spaces-over-fields.tex","source_line":1660,"source_end_line":1666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1660-L1666","statement_sha256":"deff555c2cb2cbbd946aba4704e21fd299c8e032aa176f115499f4fd77c3e3eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11686,"rank":11686,"depth":62,"x":137.853,"y":1497.957,"cluster":"algebraic-spaces"},{"id":"stacks:0B88","tag":"0B88","title":"Schematic locus and field extension · Lemma 0B88","summary":"Let k be a field with algebraic closure overlinek. Let X be an algebraic space over k such that • X is decent and locally of finite type over k, • X_overlinek is a scheme, and • any finite set of overlinek-rational points of X_overlinek is contained in an affine. Then X is a scheme.","statement_latex":"Let $k$ be a field with algebraic closure $\\overline{k}$. Let $X$\nbe an algebraic space over $k$ such that\n\\begin{enumerate}\n\\item $X$ is decent and locally of finite type over $k$,\n\\item $X_{\\overline{k}}$ is a scheme, and\n\\item any finite set of $\\overline{k}$-rational points of $X_{\\overline{k}}$\nis contained in an affine.\n\\end{enumerate}\nThen $X$ is a scheme.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus and field extension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B88","source_file":"spaces-over-fields.tex","source_line":1738,"source_end_line":1749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1738-L1749","statement_sha256":"5b108d26596b26eb583e0090a5e9f5f8466d9d111e4d35ed9f2e23113c4a9dbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":11687,"rank":11687,"depth":70,"x":380.115,"y":1622.975,"cluster":"algebraic-spaces"},{"id":"stacks:06S0","tag":"06S0","title":"Schematic locus and field extension · Lemma 06S0","summary":"Let k be a field. Let X be an algebraic space over k. The following are equivalent • X is locally quasi-finite over k, • X is locally of finite type over k and has dimension 0, • X is a scheme and is locally quasi-finite over k, • X is a scheme and is locally of finite type over k and has dimension 0, and • X is a disjoint union of spectra of Artinian local k-algebras A over k with dim_k(A) < ∞.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic space over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is locally quasi-finite over $k$,\n\\item $X$ is locally of finite type over $k$ and has dimension $0$,\n\\item $X$ is a scheme and is locally quasi-finite over $k$,\n\\item $X$ is a scheme and is locally of finite type over $k$ and has\ndimension $0$, and\n\\item $X$ is a disjoint union of spectra of Artinian local $k$-algebras\n$A$ over $k$ with $\\dim_k(A) < \\infty$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus and field extension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06S0","source_file":"spaces-over-fields.tex","source_line":1816,"source_end_line":1829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1816-L1829","statement_sha256":"63ef62707ea630681a4d1311a55494e7534d857bd1f8a8777edae389f2cca6fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11688,"rank":11688,"depth":69,"x":100.741,"y":1668.285,"cluster":"algebraic-spaces"},{"id":"stacks:06S1","tag":"06S1","title":"Schematic locus and field extension · Lemma 06S1","summary":"Let k be a field. Let f : X → Y be a monomorphism of algebraic spaces over k. If Y is locally quasi-finite over k so is X.","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a monomorphism of algebraic spaces\nover $k$. If $Y$ is locally quasi-finite over $k$ so is $X$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Schematic locus and field extension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06S1","source_file":"spaces-over-fields.tex","source_line":1844,"source_end_line":1848,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1844-L1848","statement_sha256":"ee0e9396d307b9b9fb5be83d94f4bcff162e327380954d4de51e76c534b08a6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11689,"rank":11689,"depth":70,"x":270.429,"y":1476.215,"cluster":"algebraic-spaces"},{"id":"stacks:0DMQ","tag":"0DMQ","title":"Geometrically reduced algebraic spaces · Definition 0DMQ","summary":"Let k be a field. Let X be an algebraic space over k. • Let x ∈ |X| be a point. We say X is geometrically reduced at x if O_X, overlinex is geometrically reduced over k. • We say X is geometrically reduced over k if X is geometrically reduced at every point of X.","statement_latex":"Let $k$ be a field.\nLet $X$ be an algebraic space over $k$.\n\\begin{enumerate}\n\\item Let $x \\in |X|$ be a point. We say $X$ is\n{\\it geometrically reduced at $x$} if $\\mathcal{O}_{X, \\overline{x}}$\nis geometrically reduced over $k$.\n\\item We say $X$ is {\\it geometrically reduced} over $k$\nif $X$ is geometrically reduced at every point of $X$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically reduced algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMQ","source_file":"spaces-over-fields.tex","source_line":1889,"source_end_line":1900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1889-L1900","statement_sha256":"37da5786fa73eda2f146501714d3ab309cf54edf46b35f18372e7f7d4df8c904","origin":"The Stacks Project","memory_eligible":false,"source_rank":11690,"rank":11690,"depth":0,"x":299.782,"y":1714.298,"cluster":"algebraic-spaces"},{"id":"stacks:0DMR","tag":"0DMR","title":"Geometrically reduced algebraic spaces · Lemma 0DMR","summary":"Let k be a field. Let X be an algebraic space over k. Let x ∈ |X|. The following are equivalent • X is geometrically reduced at x, • for some étale neighbourhood (U, u) → (X, x) where U is a scheme, U is geometrically reduced at u, • for any étale neighbourhood (U, u) → (X, x) where U is a scheme, U is geometrically reduced at u.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic space over $k$.\nLet $x \\in |X|$. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically reduced at $x$,\n\\item for some \\'etale neighbourhood $(U, u) \\to (X, x)$\nwhere $U$ is a scheme, $U$ is geometrically reduced at $u$,\n\\item for any \\'etale neighbourhood $(U, u) \\to (X, x)$\nwhere $U$ is a scheme, $U$ is geometrically reduced at $u$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically reduced algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMR","source_file":"spaces-over-fields.tex","source_line":1911,"source_end_line":1922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1911-L1922","statement_sha256":"97f74db3d1748d570395d271ebb1e470461dea2a12bf1bffa8efa2ce6c5c2a17","origin":"The Stacks Project","memory_eligible":false,"source_rank":11691,"rank":11691,"depth":53,"x":86.528,"y":1555.283,"cluster":"algebraic-spaces"},{"id":"stacks:0DMS","tag":"0DMS","title":"Geometrically reduced algebraic spaces · Lemma 0DMS","summary":"Let k be a field. Let X be an algebraic space over k. The following are equivalent • X is geometrically reduced, • for some surjective étale morphism U → X where U is a scheme, U is geometrically reduced, • for any étale morphism U → X where U is a scheme, U is geometrically reduced.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic space over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically reduced,\n\\item for some surjective \\'etale morphism $U \\to X$ where $U$\nis a scheme, $U$ is geometrically reduced,\n\\item for any \\'etale morphism $U \\to X$\nwhere $U$ is a scheme, $U$ is geometrically reduced.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically reduced algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMS","source_file":"spaces-over-fields.tex","source_line":1953,"source_end_line":1964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1953-L1964","statement_sha256":"1e3ea1e6aa1c7e844b238c65e7d7e9f972e2f2ef85f90144bfbfaf60c980faea","origin":"The Stacks Project","memory_eligible":false,"source_rank":11692,"rank":11692,"depth":54,"x":371.854,"y":1551.53,"cluster":"algebraic-spaces"},{"id":"stacks:0E02","tag":"0E02","title":"Geometrically reduced algebraic spaces · Lemma 0E02","summary":"Let X be an algebraic space over a perfect field k (for example k has characteristic zero). • For x ∈ |X|, if O_X, overlinex is reduced, then X is geometrically reduced at x. • If X is reduced, then X is geometrically reduced over k.","statement_latex":"Let $X$ be an algebraic space over a perfect field $k$ (for example\n$k$ has characteristic zero).\n\\begin{enumerate}\n\\item For $x \\in |X|$, if $\\mathcal{O}_{X, \\overline{x}}$ is\nreduced, then $X$ is geometrically reduced at $x$.\n\\item If $X$ is reduced, then $X$ is geometrically reduced over $k$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically reduced algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E02","source_file":"spaces-over-fields.tex","source_line":1974,"source_end_line":1983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1974-L1983","statement_sha256":"ad254138476a8458904d5c2f7bb47d7d5a8edfa13bcf54c66ba58f4e578a61bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11693,"rank":11693,"depth":7,"x":164.331,"y":1716.314,"cluster":"algebraic-spaces"},{"id":"stacks:0E03","tag":"0E03","title":"Geometrically reduced algebraic spaces · Lemma 0E03","summary":"Let k be a field of characteristic p > 0. Let X be an algebraic space over k. The following are equivalent • X is geometrically reduced over k, • X_k' is reduced for every field extension k'/k, • X_k' is reduced for every finite purely inseparable field extension k'/k, • X_k^1/p is reduced, • X_k^perf is reduced, and • X_bar k is reduced.","statement_latex":"Let $k$ be a field of characteristic $p > 0$. Let $X$ be an algebraic space\nover $k$. The following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically reduced over $k$,\n\\item $X_{k'}$ is reduced for every field extension $k'/k$,\n\\item $X_{k'}$ is reduced for every\nfinite purely inseparable field extension $k'/k$,\n\\item $X_{k^{1/p}}$ is reduced,\n\\item $X_{k^{perf}}$ is reduced, and\n\\item $X_{\\bar k}$ is reduced.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically reduced algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E03","source_file":"spaces-over-fields.tex","source_line":1993,"source_end_line":2006,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L1993-L2006","statement_sha256":"d25c2daf9f661b37e97138612d94788e812b681e657dbdab41d0ac7b14872e3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11694,"rank":11694,"depth":55,"x":184.856,"y":1476.884,"cluster":"algebraic-spaces"},{"id":"stacks:0E04","tag":"0E04","title":"Geometrically reduced algebraic spaces · Lemma 0E04","summary":"Let k be a field. Let X be an algebraic space over k. Let k'/k be a field extension. Let x ∈ |X| be a point and let x' ∈ |X_k'| be a point lying over x. The following are equivalent • X is geometrically reduced at x, • X_k' is geometrically reduced at x'. In particular, X is geometrically reduced over k if and only if X_k' is geometrically reduced over k'.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic space over $k$.\nLet $k'/k$ be a field extension. Let $x \\in |X|$ be a point and let\n$x' \\in |X_{k'}|$ be a point lying over $x$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically reduced at $x$,\n\\item $X_{k'}$ is geometrically reduced at $x'$.\n\\end{enumerate}\nIn particular, $X$ is geometrically reduced over $k$ if and only if\n$X_{k'}$ is geometrically reduced over $k'$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically reduced algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E04","source_file":"spaces-over-fields.tex","source_line":2015,"source_end_line":2027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2015-L2027","statement_sha256":"cad7ae76a5c502c91b4e26631f7cad79da2cbdcc6ffa4d8c6085e520bdfc6a56","origin":"The Stacks Project","memory_eligible":false,"source_rank":11695,"rank":11695,"depth":54,"x":362.388,"y":1665.213,"cluster":"algebraic-spaces"},{"id":"stacks:0E05","tag":"0E05","title":"Geometrically reduced algebraic spaces · Lemma 0E05","summary":"Let k be a field. Let f : X → Y be a morphism of algebraic spaces over k. Let x ∈ |X| be a point with image y ∈ |Y|. • if f is étale at x, then X is geometrically reduced at x ⇔ Y is geometrically reduced at y, • if f is surjective étale, then X is geometrically reduced ⇔ Y is geometrically reduced.","statement_latex":"Let $k$ be a field. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $k$. Let $x \\in |X|$ be a point\nwith image $y \\in |Y|$.\n\\begin{enumerate}\n\\item if $f$ is \\'etale at $x$, then\n$X$ is geometrically reduced at $x$ $\\Leftrightarrow$\n$Y$ is geometrically reduced at $y$,\n\\item if $f$ is surjective \\'etale, then\n$X$ is geometrically reduced $\\Leftrightarrow$\n$Y$ is geometrically reduced.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically reduced algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E05","source_file":"spaces-over-fields.tex","source_line":2039,"source_end_line":2052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2039-L2052","statement_sha256":"d8fc530c6b4b068025a9e63264f1a5c1e38b25c251c0c8ad049ee34ab52d549e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11696,"rank":11696,"depth":0,"x":79.831,"y":1627.052,"cluster":"algebraic-spaces"},{"id":"stacks:0A0Z","tag":"0A0Z","title":"Geometrically connected algebraic spaces · Definition 0A0Z","summary":"Let X be an algebraic space over the field k. We say X is geometrically connected over k if the base change X_k' is connected for every field extension k' of k.","statement_latex":"Let $X$ be an algebraic space over the field $k$. We say $X$ is\n{\\it geometrically connected} over $k$ if the base change $X_{k'}$\nis connected for every field extension $k'$ of $k$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically connected algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A0Z","source_file":"spaces-over-fields.tex","source_line":2074,"source_end_line":2079,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2074-L2079","statement_sha256":"11188d1bf902ec32ee34759bff21e0706a0ddbed122b4f97d27299ff87f3e14c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11697,"rank":11697,"depth":0,"x":319.04,"y":1494.77,"cluster":"algebraic-spaces"},{"id":"stacks:0A10","tag":"0A10","title":"Geometrically connected algebraic spaces · Lemma 0A10","summary":"Let X be an algebraic space over the field k. Let k'/k be a field extension. Then X is geometrically connected over k if and only if X_k' is geometrically connected over k'.","statement_latex":"Let $X$ be an algebraic space over the field $k$.\nLet $k'/k$ be a field extension.\nThen $X$ is geometrically connected over $k$ if and only if\n$X_{k'}$ is geometrically connected over $k'$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically connected algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A10","source_file":"spaces-over-fields.tex","source_line":2085,"source_end_line":2091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2085-L2091","statement_sha256":"dd00a5cadc59fcfcfbc37775f5d91c3a9f5523a8b8d65b1f37fb2241839ff559","origin":"The Stacks Project","memory_eligible":false,"source_rank":11698,"rank":11698,"depth":2,"x":248.98,"y":1728.207,"cluster":"algebraic-spaces"},{"id":"stacks:0A11","tag":"0A11","title":"Geometrically connected algebraic spaces · Lemma 0A11","summary":"Let k be a field. Let X, Y be algebraic spaces over k. Assume X is geometrically connected over k. Then the projection morphism p : X ×_k Y → Y induces a bijection between connected components.","statement_latex":"Let $k$ be a field. Let $X$, $Y$ be algebraic spaces over $k$.\nAssume $X$ is geometrically connected over $k$.\nThen the projection morphism\n$$\np : X \\times_k Y \\longrightarrow Y\n$$\ninduces a bijection between connected components.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically connected algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A11","source_file":"spaces-over-fields.tex","source_line":2106,"source_end_line":2115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2106-L2115","statement_sha256":"3bc6aea9eca26e8c8e31da9d52488f91fca282509d4f1aeb0003850a9ef6a94f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11699,"rank":11699,"depth":48,"x":112.822,"y":1516.173,"cluster":"algebraic-spaces"},{"id":"stacks:0A12","tag":"0A12","title":"Geometrically connected algebraic spaces · Lemma 0A12","summary":"Let k'/k be an extension of fields. Let X be an algebraic space over k. Assume k separably algebraically closed. Then the morphism X_k' → X induces a bijection of connected components. In particular, X is geometrically connected over k if and only if X is connected.","statement_latex":"Let $k'/k$ be an extension of fields. Let $X$ be an algebraic space\nover $k$. Assume $k$ separably algebraically closed. Then the morphism\n$X_{k'} \\to X$ induces a bijection of connected components. In particular,\n$X$ is geometrically connected over $k$ if and only if $X$ is connected.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically connected algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A12","source_file":"spaces-over-fields.tex","source_line":2132,"source_end_line":2138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2132-L2138","statement_sha256":"ebe235ee7459a6c6f73a7d7d51e7282f33e3d80e5e5a0c0734d3645345fd236f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11700,"rank":11700,"depth":49,"x":383.923,"y":1595.32,"cluster":"algebraic-spaces"},{"id":"stacks:0A13","tag":"0A13","title":"Geometrically connected algebraic spaces · Lemma 0A13","summary":"Let k be a field. Let X be an algebraic space over k. Let overlinek be a separable algebraic closure of k. Then X is geometrically connected if and only if the base change X_overlinek is connected.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic space over $k$.\nLet $\\overline{k}$ be a separable algebraic closure of $k$.\nThen $X$ is geometrically connected if and only if the base change\n$X_{\\overline{k}}$ is connected.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically connected algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A13","source_file":"spaces-over-fields.tex","source_line":2151,"source_end_line":2157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2151-L2157","statement_sha256":"c070126009f9df243abfbd213972fa966e79aaa8791c5cf96bc95d99fed6b0dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11701,"rank":11701,"depth":50,"x":120.187,"y":1690.853,"cluster":"algebraic-spaces"},{"id":"stacks:0A15","tag":"0A15","title":"Geometrically connected algebraic spaces · Lemma 0A15","summary":"Let k be a field. Let X be an algebraic space over k. Let overlinek be a (possibly infinite) Galois extension of k. Let V ⊂ X_overlinek be a quasi-compact open. Then • there exists a finite subextension overlinek/k'/k and a quasi-compact open V' ⊂ X_k' such that V = (V')_overlinek, • there exists an open subgroup H ⊂ Gal(overlinek/k) such that σ(V) = V for all σ ∈ H.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic space over $k$.\nLet $\\overline{k}$ be a (possibly infinite) Galois extension of $k$.\nLet $V \\subset X_{\\overline{k}}$ be a quasi-compact open.\nThen\n\\begin{enumerate}\n\\item there exists a finite subextension $\\overline{k}/k'/k$\nand a quasi-compact open $V' \\subset X_{k'}$ such that\n$V = (V')_{\\overline{k}}$,\n\\item there exists an open subgroup $H \\subset \\text{Gal}(\\overline{k}/k)$\nsuch that $\\sigma(V) = V$ for all $\\sigma \\in H$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically connected algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A15","source_file":"spaces-over-fields.tex","source_line":2201,"source_end_line":2214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2201-L2214","statement_sha256":"22e25707d05f24888ba0c547ffcaccfd2c8fa6bc5817870840b3bef9554ed81e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11702,"rank":11702,"depth":3,"x":237.918,"y":1470.606,"cluster":"algebraic-spaces"},{"id":"stacks:0A16","tag":"0A16","title":"Geometrically connected algebraic spaces · Lemma 0A16","summary":"Let k be a field. Let overlinek/k be a (possibly infinite) Galois extension. Let X be an algebraic space over k. Let overlineT ⊂ |X_overlinek| have the following properties • overlineT is a closed subset of |X_overlinek|, • for every σ ∈ Gal(overlinek/k) we have σ(overlineT) = overlineT. Then there exists a closed subset T ⊂ |X| whose inverse image in |X_k'| is overlineT.","statement_latex":"Let $k$ be a field. Let $\\overline{k}/k$ be a (possibly infinite)\nGalois extension. Let $X$ be an algebraic space over $k$. Let\n$\\overline{T} \\subset |X_{\\overline{k}}|$ have the following properties\n\\begin{enumerate}\n\\item $\\overline{T}$ is a closed subset of $|X_{\\overline{k}}|$,\n\\item for every $\\sigma \\in \\text{Gal}(\\overline{k}/k)$\nwe have $\\sigma(\\overline{T}) = \\overline{T}$.\n\\end{enumerate}\nThen there exists a closed subset $T \\subset |X|$ whose inverse image\nin $|X_{k'}|$ is $\\overline{T}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically connected algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A16","source_file":"spaces-over-fields.tex","source_line":2227,"source_end_line":2239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2227-L2239","statement_sha256":"e9bdb3141e6b010d5b03e5c2ff44784f725f974482fce9d90f785b7ae7a2fffd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11703,"rank":11703,"depth":1,"x":328.284,"y":1699.975,"cluster":"algebraic-spaces"},{"id":"stacks:0A17","tag":"0A17","title":"Geometrically connected algebraic spaces · Lemma 0A17","summary":"Let k be a field. Let X be an algebraic space over k. The following are equivalent • X is geometrically connected, • for every finite separable field extension k'/k the algebraic space X_k' is connected.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic space over $k$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is geometrically connected,\n\\item for every finite separable field extension $k'/k$\nthe algebraic space $X_{k'}$ is connected.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically connected algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A17","source_file":"spaces-over-fields.tex","source_line":2257,"source_end_line":2266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2257-L2266","statement_sha256":"a86ba7decb4907ad219e4eeb75966dc5ec4bb348bdf07d458e6aff4600b842d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11704,"rank":11704,"depth":51,"x":77.024,"y":1582.036,"cluster":"algebraic-spaces"},{"id":"stacks:0DMU","tag":"0DMU","title":"Geometrically irreducible algebraic spaces · Definition 0DMU","summary":"Let k be a field. Let X be a decent algebraic space over k. We say X is geometrically irreducible if the topological space |X_k'| is irreducible for any field extension k' of k.","statement_latex":"Let $k$ be a field. \nLet $X$ be a decent algebraic space over $k$.\nWe say $X$ is {\\it geometrically irreducible} if the\ntopological space $|X_{k'}|$ is\nirreducible\\footnote{An irreducible space is nonempty.}\nfor any field extension $k'$ of $k$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically irreducible algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMU","source_file":"spaces-over-fields.tex","source_line":2373,"source_end_line":2381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2373-L2381","statement_sha256":"aa6ce8457a86d2b9039b48b704e2f68cc44455ebdc2a2218e70604aeffa52687","origin":"The Stacks Project","memory_eligible":false,"source_rank":11705,"rank":11705,"depth":0,"x":357.336,"y":1526.393,"cluster":"algebraic-spaces"},{"id":"stacks:0DMW","tag":"0DMW","title":"Geometrically integral algebraic spaces · Definition 0DMW","summary":"Let X be an algebraic space over the field k. We say X is geometrically integral over k if the algebraic space X_k' is integral (Definition [Tag 0AD4]) for every field extension k' of k.","statement_latex":"Let $X$ be an algebraic space over the field $k$. We say $X$ is\n{\\it geometrically integral} over $k$ if the algebraic space\n$X_{k'}$ is integral (Definition \\ref{definition-integral-algebraic-space})\nfor every field extension $k'$ of $k$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically integral algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMW","source_file":"spaces-over-fields.tex","source_line":2401,"source_end_line":2407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2401-L2407","statement_sha256":"e1fcc92844df07bb1ec1f2e075378efb6d8f411fc4465e800d0c51eb20ea37a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11706,"rank":11706,"depth":1,"x":195.273,"y":1726.617,"cluster":"algebraic-spaces"},{"id":"stacks:0DMX","tag":"0DMX","title":"Geometrically integral algebraic spaces · Lemma 0DMX","summary":"Let k be a field. Let X be a decent algebraic space over k. Then X is geometrically integral over k if and only if X is both geometrically reduced and geometrically irreducible over k.","statement_latex":"Let $k$ be a field. Let $X$ be a decent algebraic space over $k$.\nThen $X$ is geometrically integral over $k$ if and only if\n$X$ is both geometrically reduced and geometrically irreducible\nover $k$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMX","source_file":"spaces-over-fields.tex","source_line":2414,"source_end_line":2420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2414-L2420","statement_sha256":"d189bbf6af41c6aef27028bf8175c23a5111768d08fb581e639efe83ed4ce4fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11707,"rank":11707,"depth":0,"x":153.733,"y":1486.852,"cluster":"algebraic-spaces"},{"id":"stacks:0DMY","tag":"0DMY","title":"Geometrically integral algebraic spaces · Lemma 0DMY","summary":"Let k be a field. Let X be a proper algebraic space over k. • A = H^0(X, O_X) is a finite dimensional k-algebra, • A = ∏_i = 1, …, n A_i is a product of Artinian local k-algebras, one factor for each connected component of |X|, • if X is reduced, then A = ∏_i = 1, …, n k_i is a product of fields, each a finite extension of k, • if X is geometrically reduced, then k_i is finite separable over k, • if X is geometrically connected, then A is geometrically irreducible over k,…","statement_latex":"Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.\n\\begin{enumerate}\n\\item $A = H^0(X, \\mathcal{O}_X)$ is a finite dimensional $k$-algebra,\n\\item $A = \\prod_{i = 1, \\ldots, n} A_i$ is a product of Artinian\nlocal $k$-algebras, one factor for each connected component of $|X|$,\n\\item if $X$ is reduced, then $A = \\prod_{i = 1, \\ldots, n} k_i$\nis a product of fields, each a finite extension of $k$,\n\\item if $X$ is geometrically reduced, then $k_i$ is finite separable\nover $k$,\n\\item if $X$ is geometrically connected, then $A$ is geometrically\nirreducible over $k$,\n\\item if $X$ is geometrically irreducible, then $A$ is geometrically\nirreducible over $k$,\n\\item if $X$ is geometrically reduced and connected, then $A = k$, and\n\\item if $X$ is geometrically integral, then $A = k$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMY","source_file":"spaces-over-fields.tex","source_line":2428,"source_end_line":2446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2428-L2446","statement_sha256":"adae5f7970a3d85c8cbe9bbf2ea17f8a694bb073c55b7dccacddd920bf9799b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11708,"rank":11708,"depth":64,"x":377.33,"y":1640.185,"cluster":"algebraic-spaces"},{"id":"stacks:0DMZ","tag":"0DMZ","title":"Geometrically integral algebraic spaces · Lemma 0DMZ","summary":"Let k be a field. Let X be a proper integral algebraic space over k. Let L be an invertible O_X-module. If H^0(X, L) and H^0(X, L^⊗ - 1) are both nonzero, then L ≅ O_X.","statement_latex":"Let $k$ be a field. Let $X$ be a proper integral algebraic space over $k$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nIf $H^0(X, \\mathcal{L})$ and $H^0(X, \\mathcal{L}^{\\otimes - 1})$\nare both nonzero, then $\\mathcal{L} \\cong \\mathcal{O}_X$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Geometrically integral algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMZ","source_file":"spaces-over-fields.tex","source_line":2507,"source_end_line":2513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2507-L2513","statement_sha256":"080725f6d1c6f6e81d6029cac1f6c72ea737c1bc028dab7d5a060c906cb8964d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11709,"rank":11709,"depth":65,"x":88.949,"y":1654.003,"cluster":"algebraic-spaces"},{"id":"stacks:0EDB","tag":"0EDB","title":"Dimension · Lemma 0EDB","summary":"Let S be a scheme. Let f : X → Y be an integral morphism of algebraic spaces. Then dim(X) ≤ dim(Y). If f is surjective then dim(X) = dim(Y).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism\nof algebraic spaces. Then $\\dim(X) \\leq \\dim(Y)$.\nIf $f$ is surjective then $\\dim(X) = \\dim(Y)$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDB","source_file":"spaces-over-fields.tex","source_line":2575,"source_end_line":2580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2575-L2580","statement_sha256":"68850293755fe2ca1b557e05002950fa68d54c2272886602d7306209929250d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11710,"rank":11710,"depth":48,"x":290.622,"y":1480.061,"cluster":"algebraic-spaces"},{"id":"stacks:0EDC","tag":"0EDC","title":"Dimension · Lemma 0EDC","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that • Y is locally Noetherian, • X and Y are integral algebraic spaces, • f is dominant, and • f is locally of finite type. If x ∈ |X| and y ∈ |Y| are the generic points, then dim(X) ≤ dim(Y) + transcendence degree of x/y. If f is proper, then equality holds.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume that\n\\begin{enumerate}\n\\item $Y$ is locally Noetherian,\n\\item $X$ and $Y$ are integral algebraic spaces,\n\\item $f$ is dominant, and\n\\item $f$ is locally of finite type.\n\\end{enumerate}\nIf $x \\in |X|$ and $y \\in |Y|$ are the generic points, then\n$$\n\\dim(X) \\leq \\dim(Y) + \\text{transcendence degree of }x/y.\n$$\nIf $f$ is proper, then equality holds.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Dimension","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDC","source_file":"spaces-over-fields.tex","source_line":2593,"source_end_line":2608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2593-L2608","statement_sha256":"f28a6ed2a3e498ebea3931f9c0a9fc92ae08549d43e35c611cf95a92588aac82","origin":"The Stacks Project","memory_eligible":false,"source_rank":11711,"rank":11711,"depth":67,"x":281.785,"y":1722.924,"cluster":"algebraic-spaces"},{"id":"stacks:06M1","tag":"06M1","title":"Spaces smooth over fields · Lemma 06M1","summary":"Let k be a field. Let X be an algebraic space smooth over k. Then X is a regular algebraic space.","statement_latex":"Let $k$ be a field.\nLet $X$ be an algebraic space smooth over $k$.\nThen $X$ is a regular algebraic space.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Spaces smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06M1","source_file":"spaces-over-fields.tex","source_line":2683,"source_end_line":2688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2683-L2688","statement_sha256":"39153eabc1731fe9f13b3b393f800f393025a03282c6d10e4a0ec449e9fc15d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11712,"rank":11712,"depth":48,"x":92.869,"y":1538.7,"cluster":"algebraic-spaces"},{"id":"stacks:07W4","tag":"07W4","title":"Spaces smooth over fields · Lemma 07W4","summary":"Let k be a field. Let X be an algebraic space smooth over Spec(k). The set of x ∈ |X| which are image of morphisms Spec(k') → X with k' ⊃ k finite separable is dense in |X|.","statement_latex":"Let $k$ be a field. Let $X$ be an algebraic space smooth over $\\Spec(k)$.\nThe set of $x \\in |X|$ which are image of morphisms $\\Spec(k') \\to X$\nwith $k' \\supset k$ finite separable is dense in $|X|$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Spaces smooth over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07W4","source_file":"spaces-over-fields.tex","source_line":2704,"source_end_line":2709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2704-L2709","statement_sha256":"77db05702c0c3c95b4d8ab51e3d1eed44ff608471b9994ddd3b5b6945fa6dc9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11713,"rank":11713,"depth":48,"x":380.516,"y":1567.368,"cluster":"algebraic-spaces"},{"id":"stacks:0DN1","tag":"0DN1","title":"Euler characteristics · Definition 0DN1","summary":"Let k be a field. Let X be a proper algebraic over k. Let F be a coherent O_X-module. In this situation the Euler characteristic of F is the integer chi(X, F) = ∑_i (-1)^i dim_k H^i(X, F). For justification of the formula see below.","statement_latex":"Let $k$ be a field. Let $X$ be a proper algebraic over $k$. Let $\\mathcal{F}$\nbe a coherent $\\mathcal{O}_X$-module. In this situation the\n{\\it Euler characteristic of $\\mathcal{F}$} is the integer\n$$\n\\chi(X, \\mathcal{F}) = \\sum\\nolimits_i (-1)^i \\dim_k H^i(X, \\mathcal{F}).\n$$\nFor justification of the formula see below.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Euler characteristics","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DN1","source_file":"spaces-over-fields.tex","source_line":2737,"source_end_line":2746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2737-L2746","statement_sha256":"ff1b6071e31f90d70794c455f83aae8c203bef2c660880f117eecc9a437f26cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11714,"rank":11714,"depth":0,"x":145.196,"y":1709.543,"cluster":"algebraic-spaces"},{"id":"stacks:0DN2","tag":"0DN2","title":"Euler characteristics · Lemma 0DN2","summary":"Let k be a field. Let X be a proper algebraic space over k. Let 0 → F_1 → F_2 → F_3 → 0 be a short exact sequence of coherent modules on X. Then chi(X, F_2) = chi(X, F_1) + chi(X, F_3)","statement_latex":"Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.\nLet $0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nbe a short exact sequence of coherent modules on $X$. Then\n$$\n\\chi(X, \\mathcal{F}_2) = \\chi(X, \\mathcal{F}_1) + \\chi(X, \\mathcal{F}_3)\n$$","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Euler characteristics","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DN2","source_file":"spaces-over-fields.tex","source_line":2759,"source_end_line":2767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2759-L2767","statement_sha256":"a2131b431386e8f114da717845dce6b0fac788a7e880d2156e4ad7da6c7c7913","origin":"The Stacks Project","memory_eligible":false,"source_rank":11715,"rank":11715,"depth":0,"x":204.424,"y":1471.017,"cluster":"algebraic-spaces"},{"id":"stacks:0EDD","tag":"0EDD","title":"Euler characteristics · Lemma 0EDD","summary":"Let k be a field. Let f : Y → X be a morphism of algebraic spaces proper over k. Let G be a coherent O_Y-module. Then chi(Y, G) = ∑ (-1)^i chi(X, R^if_*G)","statement_latex":"Let $k$ be a field. Let $f : Y \\to X$ be a morphism of\nalgebraic spaces proper over $k$. Let $\\mathcal{G}$ be a\ncoherent $\\mathcal{O}_Y$-module. Then\n$$\n\\chi(Y, \\mathcal{G}) = \\sum (-1)^i \\chi(X, R^if_*\\mathcal{G})\n$$","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Euler characteristics","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDD","source_file":"spaces-over-fields.tex","source_line":2784,"source_end_line":2792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2784-L2792","statement_sha256":"735d64eef5523f4b227b72c6e1cd6993546b5fd2d6aea829316032cdbc55cf5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11716,"rank":11716,"depth":63,"x":352.667,"y":1680.652,"cluster":"algebraic-spaces"},{"id":"stacks:0DN4","tag":"0DN4","title":"Numerical intersections · Lemma 0DN4","summary":"Let k be a field. Let X be a proper algebraic space over k. Let F be a coherent O_X-module. Let L_1, …, L_r be invertible O_X-modules. The map (n_1, …, n_r) ↦ chi(X, F ⊗ L_1^⊗ n_1 ⊗ … ⊗ L_r^⊗ n_r) is a numerical polynomial in n_1, …, n_r of total degree at most the dimension of the scheme theoretic support of F.","statement_latex":"Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules.\nThe map\n$$\n(n_1, \\ldots, n_r) \\longmapsto\n\\chi(X, \\mathcal{F} \\otimes\n\\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes\n\\mathcal{L}_r^{\\otimes n_r})\n$$\nis a numerical polynomial in $n_1, \\ldots, n_r$ of total degree at\nmost the dimension of the scheme theoretic support of $\\mathcal{F}$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DN4","source_file":"spaces-over-fields.tex","source_line":2835,"source_end_line":2849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L2835-L2849","statement_sha256":"5ca6ba1e0f5a409769974948cb7339bdfea437f4fe576135063a1c9eaf1f161d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11717,"rank":11717,"depth":67,"x":74.584,"y":1610.139,"cluster":"algebraic-spaces"},{"id":"stacks:0EDE","tag":"0EDE","title":"Numerical intersections · Lemma 0EDE","summary":"Let k be a field. Let X be a proper algebraic space over k. Let F be a coherent O_X-module. Let L_1, …, L_r be invertible O_X-modules. Let d = dim(Supp(F)). Let Z_i ⊂ X be the irreducible components of Supp(F) of dimension d. Let overlinex_i be a geometric generic point of Z_i and set m_i = length_O_X, overlinex_i (F_overlinex_i). Then chi(X, F ⊗ L_1^⊗ n_1 ⊗ … ⊗ L_r^⊗ n_r) - ∑_i m_i chi(Z_i, L_1^⊗ n_1 ⊗ … ⊗ L_r^⊗ n_r|_Z_i) is a numerical polynomial in n_1, …, n_r of total…","statement_latex":"Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let\n$\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module. Let\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_r$ be invertible $\\mathcal{O}_X$-modules.\nLet $d = \\dim(\\text{Supp}(\\mathcal{F}))$.\nLet $Z_i \\subset X$ be the irreducible components\nof $\\text{Supp}(\\mathcal{F})$ of dimension $d$. Let $\\overline{x}_i$\nbe a geometric generic point of $Z_i$ and set\n$m_i = \\text{length}_{\\mathcal{O}_{X, \\overline{x}_i}}\n(\\mathcal{F}_{\\overline{x}_i})$.\nThen\n$$\n\\chi(X, \\mathcal{F} \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes\n\\mathcal{L}_r^{\\otimes n_r}) -\n\\sum\\nolimits_i\nm_i\\ \\chi(Z_i, \\mathcal{L}_1^{\\otimes n_1} \\otimes \\ldots \\otimes\n\\mathcal{L}_r^{\\otimes n_r}|_{Z_i})\n$$\nis a numerical polynomial in $n_1, \\ldots, n_r$ of total degree $< d$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDE","source_file":"spaces-over-fields.tex","source_line":3017,"source_end_line":3037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L3017-L3037","statement_sha256":"0deeb02ca1504a67eaa38ece9447298ffafcd64da6d62d1a778346fb07e2fb9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11718,"rank":11718,"depth":68,"x":336.52,"y":1504.272,"cluster":"algebraic-spaces"},{"id":"stacks:0EDF","tag":"0EDF","title":"Numerical intersections · Definition 0EDF","summary":"Let k be a field. Let X be a proper algebraic space over k. Let i : Z → X be a closed subspace of dimension d. Let L_1, …, L_d be invertible O_X-modules. We define the intersection number (L_1 … L_d · Z) as the coefficient of n_1 … n_d in the numerical polynomial chi(X, i_*O_Z ⊗ L_1^⊗ n_1 ⊗ … ⊗ L_d^⊗ n_d) = chi(Z, L_1^⊗ n_1 ⊗ … ⊗ L_d^⊗ n_d|_Z) In the special case that L_1 = … = L_d = L we write (L^d · Z).","statement_latex":"Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let\n$i : Z \\to X$ be a closed subspace of dimension $d$. Let\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible\n$\\mathcal{O}_X$-modules. We define the {\\it intersection number}\n$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$\nas the coefficient of $n_1 \\ldots n_d$ in the numerical polynomial\n$$\n\\chi(X, i_*\\mathcal{O}_Z \\otimes \\mathcal{L}_1^{\\otimes n_1} \\otimes\n\\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}) =\n\\chi(Z, \\mathcal{L}_1^{\\otimes n_1} \\otimes\n\\ldots \\otimes \\mathcal{L}_d^{\\otimes n_d}|_Z)\n$$\nIn the special\ncase that $\\mathcal{L}_1 = \\ldots = \\mathcal{L}_d = \\mathcal{L}$\nwe write $(\\mathcal{L}^d \\cdot Z)$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Numerical intersections","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDF","source_file":"spaces-over-fields.tex","source_line":3130,"source_end_line":3147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L3130-L3147","statement_sha256":"579c5f8edf3e264640d53ec9d3218ccf4343f9a6d85ce4c2e387560ab46d5462","origin":"The Stacks Project","memory_eligible":false,"source_rank":11719,"rank":11719,"depth":0,"x":228.435,"y":1731.119,"cluster":"algebraic-spaces"},{"id":"stacks:0EDG","tag":"0EDG","title":"Numerical intersections · Lemma 0EDG","summary":"In the situation of Definition [Tag 0EDF] the intersection number (L_1 … L_d · Z) is an integer.","statement_latex":"In the situation of Definition \\ref{definition-intersection-number}\nthe intersection number\n$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$\nis an integer.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDG","source_file":"spaces-over-fields.tex","source_line":3157,"source_end_line":3163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L3157-L3163","statement_sha256":"6d13db06b5785650261fc58d22116b8dd7b2f7a0946763372b23663730d44722","origin":"The Stacks Project","memory_eligible":false,"source_rank":11720,"rank":11720,"depth":1,"x":125.642,"y":1502.361,"cluster":"algebraic-spaces"},{"id":"stacks:0EDH","tag":"0EDH","title":"Numerical intersections · Lemma 0EDH","summary":"In the situation of Definition [Tag 0EDF] the intersection number (L_1 … L_d · Z) is additive: if L_i = L_i' ⊗ L_i\", then we have (L_1 … L_i … L_d · Z) = (L_1 … L_i' … L_d · Z) + (L_1 … L_i\" … L_d · Z)","statement_latex":"In the situation of Definition \\ref{definition-intersection-number}\nthe intersection number\n$(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z)$\nis additive: if $\\mathcal{L}_i = \\mathcal{L}_i' \\otimes \\mathcal{L}_i''$,\nthen we have\n$$\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_i \\cdots \\mathcal{L}_d \\cdot Z) =\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_i' \\cdots \\mathcal{L}_d \\cdot Z) +\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_i'' \\cdots \\mathcal{L}_d \\cdot Z)\n$$","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDH","source_file":"spaces-over-fields.tex","source_line":3173,"source_end_line":3185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L3173-L3185","statement_sha256":"0223f05c50f05970e9ab2c26e340e714ea336e966c856ce12225d9c16c5cf554","origin":"The Stacks Project","memory_eligible":false,"source_rank":11721,"rank":11721,"depth":68,"x":385.575,"y":1612.791,"cluster":"algebraic-spaces"},{"id":"stacks:0EDI","tag":"0EDI","title":"Numerical intersections · Lemma 0EDI","summary":"In the situation of Definition [Tag 0EDF] let Z_i ⊂ Z be the irreducible components of dimension d. Let m_i = length_O_X, overlinex_i (O_Z, overlinex_i) where overlinex_i is a geometric generic point of Z_i. Then (L_1 … L_d · Z) = ∑ m_i(L_1 … L_d · Z_i)","statement_latex":"In the situation of Definition \\ref{definition-intersection-number}\nlet $Z_i \\subset Z$ be the irreducible components of dimension $d$. Let\n$m_i = \\text{length}_{\\mathcal{O}_{X, \\overline{x}_i}}\n(\\mathcal{O}_{Z, \\overline{x}_i})$\nwhere $\\overline{x}_i$ is a geometric generic point of $Z_i$. Then\n$$\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =\n\\sum m_i(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z_i)\n$$","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDI","source_file":"spaces-over-fields.tex","source_line":3201,"source_end_line":3212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L3201-L3212","statement_sha256":"a7865773c2d161decd8ae10681f23ba27b3a76cbca08f41cd4a2b65f493d1f9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11722,"rank":11722,"depth":69,"x":104.911,"y":1678.898,"cluster":"algebraic-spaces"},{"id":"stacks:0EDJ","tag":"0EDJ","title":"Numerical intersections · Lemma 0EDJ","summary":"Let k be a field. Let f : Y → X be a morphism of algebraic spaces proper over k. Let Z ⊂ Y be an integral closed subspace of dimension d and let L_1, …, L_d be invertible O_X-modules. Then (f^*L_1 … f^*L_d · Z) = deg(f|_Z : Z → f(Z)) (L_1 … L_d · f(Z)) where deg(Z → f(Z)) is as in Definition [Tag 0AD6] or 0 if dim(f(Z)) < d.","statement_latex":"Let $k$ be a field. Let $f : Y \\to X$ be a morphism of\nalgebraic spaces proper over $k$.\nLet $Z \\subset Y$ be an integral closed subspace of dimension $d$ and let\n$\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible $\\mathcal{O}_X$-modules.\nThen\n$$\n(f^*\\mathcal{L}_1 \\cdots f^*\\mathcal{L}_d \\cdot Z) =\n\\deg(f|_Z : Z \\to f(Z)) (\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot f(Z))\n$$\nwhere $\\deg(Z \\to f(Z))$ is as in Definition \\ref{definition-degree}\nor $0$ if $\\dim(f(Z)) < d$.","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDJ","source_file":"spaces-over-fields.tex","source_line":3219,"source_end_line":3232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L3219-L3232","statement_sha256":"e1a67bf8e32f311bc6db2b6ca7189e06f3e26eae248fa60c1ef5249828bb7cf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11723,"rank":11723,"depth":69,"x":258.81,"y":1470.756,"cluster":"algebraic-spaces"},{"id":"stacks:0EDK","tag":"0EDK","title":"Numerical intersections · Lemma 0EDK","summary":"Let k be a field. Let X be a proper algebraic space over k. Let Z ⊂ X be a closed subspace of dimension d. Let L_1, …, L_d be invertible O_X-modules. Assume there exists an effective Cartier divisor D ⊂ Z such that L_1|_Z ≅ O_Z(D). Then (L_1 … L_d · Z) = (L_2 … L_d · D)","statement_latex":"Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.\nLet $Z \\subset X$ be a closed subspace of dimension $d$.\nLet $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$\nbe invertible $\\mathcal{O}_X$-modules. Assume there exists an\neffective Cartier divisor $D \\subset Z$ such that\n$\\mathcal{L}_1|_Z \\cong \\mathcal{O}_Z(D)$. Then\n$$\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =\n(\\mathcal{L}_2 \\cdots \\mathcal{L}_d \\cdot D)\n$$","area":"Algebraic Spaces","chapter":"Algebraic Spaces over Fields","chapter_id":"spaces-over-fields","section":"Numerical intersections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDK","source_file":"spaces-over-fields.tex","source_line":3290,"source_end_line":3302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-over-fields.tex#L3290-L3302","statement_sha256":"318dcbbe9ee2654cff96edeefc4b7f101802118e8257452afef8b2fab005d68a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11724,"rank":11724,"depth":1,"x":312.746,"y":1711.725,"cluster":"algebraic-spaces"},{"id":"stacks:041G","tag":"041G","title":"Zariski topology · Definition 041G","summary":"Let S be a scheme, and let X be an algebraic space over S. A Zariski covering of X is a family of morphisms (f_i : X_i → X)_i ∈ I of algebraic spaces over S such that each f_i is an open immersion and such that |X| = ⋃_i ∈ I |f_i|(|X_i|), i.e., the morphisms are jointly surjective.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nA {\\it Zariski covering of $X$} is a family of morphisms\n$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$\nsuch that each $f_i$ is an open immersion\nand such that\n$$\n|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),\n$$\ni.e., the morphisms are jointly surjective.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Zariski topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041G","source_file":"spaces-topologies.tex","source_line":115,"source_end_line":126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L115-L126","statement_sha256":"a58fe3adcd3014c09a4a0c1b59a058f21fa0183da791feb8d95cef234ab2e937","origin":"The Stacks Project","memory_eligible":false,"source_rank":11725,"rank":11725,"depth":0,"x":79.038,"y":1564.544,"cluster":"algebraic-spaces"},{"id":"stacks:041H","tag":"041H","title":"Zariski topology · Lemma 041H","summary":"Let S be a scheme. Let X be an algebraic space over S. • If X' → X is an isomorphism then (X' → X) is a Zariski covering of X. • If (X_i → X)_i∈ I is a Zariski covering and for each i we have a Zariski covering (X_ij → X_i)_j∈ J_i, then (X_ij → X)_i ∈ I, j∈ J_i is a Zariski covering. • If (X_i → X)_i∈ I is a Zariski covering and X' → X is a morphism of algebraic spaces then (X' ×_X X_i → X')_i∈ I is a Zariski covering.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$\nis a Zariski covering of $X$.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a Zariski covering and for each\n$i$ we have a Zariski covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then\n$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a Zariski covering.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a Zariski covering\nand $X' \\to X$ is a morphism of algebraic spaces then\n$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a Zariski covering.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Zariski topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041H","source_file":"spaces-topologies.tex","source_line":133,"source_end_line":147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L133-L147","statement_sha256":"19ca86ecbca4bfe843f8de4cf3290f4d0b453fd81fecf612c392d70c5f1ef9f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11726,"rank":11726,"depth":0,"x":369.923,"y":1540.445,"cluster":"algebraic-spaces"},{"id":"stacks:041E","tag":"041E","title":"Étale topology · Definition 041E","summary":"Let S be a scheme, and let X be an algebraic space over S. An étale covering of X is a family of morphisms (f_i : X_i → X)_i ∈ I of algebraic spaces over S such that each f_i is étale and such that |X| = ⋃_i ∈ I |f_i|(|X_i|), i.e., the morphisms are jointly surjective.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nAn {\\it \\'etale covering of $X$} is a family of morphisms\n$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$\nsuch that each $f_i$ is \\'etale\nand such that\n$$\n|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),\n$$\ni.e., the morphisms are jointly surjective.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041E","source_file":"spaces-topologies.tex","source_line":171,"source_end_line":182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L171-L182","statement_sha256":"51663ace620d77cd36413bc83e0426bd87f251926f6049e09145d1787c78e964","origin":"The Stacks Project","memory_eligible":false,"source_rank":11727,"rank":11727,"depth":0,"x":174.678,"y":1723.393,"cluster":"algebraic-spaces"},{"id":"stacks:0DF1","tag":"0DF1","title":"Étale topology · Lemma 0DF1","summary":"Any Zariski covering is an étale covering.","statement_latex":"Any Zariski covering is an \\'etale covering.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DF1","source_file":"spaces-topologies.tex","source_line":190,"source_end_line":193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L190-L193","statement_sha256":"47a8295bd1ca74773f6bbc6724a7cfcc6634e1a6b91d97ef7542ee73b10ce144","origin":"The Stacks Project","memory_eligible":false,"source_rank":11728,"rank":11728,"depth":3,"x":171.526,"y":1477.539,"cluster":"algebraic-spaces"},{"id":"stacks:041F","tag":"041F","title":"Étale topology · Lemma 041F","summary":"Let S be a scheme. Let X be an algebraic space over S. • If X' → X is an isomorphism then (X' → X) is a étale covering of X. • If (X_i → X)_i∈ I is a étale covering and for each i we have a étale covering (X_ij → X_i)_j∈ J_i, then (X_ij → X)_i ∈ I, j∈ J_i is a étale covering. • If (X_i → X)_i∈ I is a étale covering and X' → X is a morphism of algebraic spaces then (X' ×_X X_i → X')_i∈ I is a étale covering.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$\nis a \\'etale covering of $X$.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a \\'etale covering and for each\n$i$ we have a \\'etale covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then\n$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a \\'etale covering.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a \\'etale covering\nand $X' \\to X$ is a morphism of algebraic spaces then\n$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a \\'etale covering.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041F","source_file":"spaces-topologies.tex","source_line":204,"source_end_line":218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L204-L218","statement_sha256":"f0943c055dd4191da413790ab6d2c812b42e86bdad4bda238a231b670e52d957","origin":"The Stacks Project","memory_eligible":false,"source_rank":11729,"rank":11729,"depth":0,"x":371.692,"y":1657.159,"cluster":"algebraic-spaces"},{"id":"stacks:0CFV","tag":"0CFV","title":"Étale topology · Lemma 0CFV","summary":"Let S be a scheme. Let X be an algebraic space over S. Let (X_i → X)_i ∈ I be a smooth covering of X. Then there exists an étale covering (U_j → X)_j ∈ J of X which refines (X_i → X)_i ∈ I.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\{X_i \\to X\\}_{i \\in I}$ be a smooth covering of $X$.\nThen there exists an \\'etale covering $\\{U_j \\to X\\}_{j \\in J}$\nof $X$ which refines $\\{X_i \\to X\\}_{i \\in I}$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CFV","source_file":"spaces-topologies.tex","source_line":229,"source_end_line":235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L229-L235","statement_sha256":"3fc125d9e79362ac1c06ad24ea1474bf63311d0668daa0ecfd4f87a936af3b71","origin":"The Stacks Project","memory_eligible":false,"source_rank":11730,"rank":11730,"depth":48,"x":79.453,"y":1638.277,"cluster":"algebraic-spaces"},{"id":"stacks:0DBX","tag":"0DBX","title":"Étale topology · Definition 0DBX","summary":"Let S be a scheme. A big étale site (Spaces/S)_etale is any site constructed as follows: • Choose a big étale site (Sch/S)_etale as in Topologies, Section [Tag 0214]. • As underlying category take the category Spaces/S of algebraic spaces over S (see discussion in Section [Tag 03Y6] why this is a set). • Choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Spaces/S and the class of étale coverings of Definition [Tag 041E].","statement_latex":"Let $S$ be a scheme. A big \\'etale site {\\it $(\\textit{Spaces}/S)_\\etale$}\nis any site constructed as follows:\n\\begin{enumerate}\n\\item Choose a big \\'etale site $(\\Sch/S)_\\etale$ as in\nTopologies, Section \\ref{topologies-section-etale}.\n\\item As underlying category take the category $\\textit{Spaces}/S$\nof algebraic spaces over $S$ (see discussion in\nSection \\ref{section-procedure} why this is a set).\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\textit{Spaces}/S$ and the class of \\'etale coverings\nof Definition \\ref{definition-etale-covering}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBX","source_file":"spaces-topologies.tex","source_line":249,"source_end_line":264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L249-L264","statement_sha256":"30520953268a35ad02f3a5e6af7a4bf55d0409ce68109a4700f727e9af3af6df","origin":"The Stacks Project","memory_eligible":false,"source_rank":11731,"rank":11731,"depth":2,"x":310.283,"y":1486.276,"cluster":"algebraic-spaces"},{"id":"stacks:0DBY","tag":"0DBY","title":"Étale topology · Definition 0DBY","summary":"Let S be a scheme. Let (Spaces/S)_etale be as in Definition [Tag 0DBX]. Let X be an algebraic space over S, i.e., an object of (Spaces/S)_etale. Then the big étale site (Spaces/X)_etale of X is the localization of the site (Spaces/S)_etale at X introduced in Sites, Section [Tag 00XZ].","statement_latex":"Let $S$ be a scheme. Let $(\\textit{Spaces}/S)_\\etale$ be as in\nDefinition \\ref{definition-big-etale-site}.\nLet $X$ be an algebraic space over $S$, i.e., an object of\n$(\\textit{Spaces}/S)_\\etale$. Then the big \\'etale site\n{\\it $(\\textit{Spaces}/X)_\\etale$} of $X$\nis the localization of the site $(\\textit{Spaces}/S)_\\etale$\nat $X$ introduced in Sites, Section \\ref{sites-section-localize}.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBY","source_file":"spaces-topologies.tex","source_line":270,"source_end_line":279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L270-L279","statement_sha256":"af45c2ce8341e9c9815092d085f4106836e0935fe722bccae4974d08d9225584","origin":"The Stacks Project","memory_eligible":false,"source_rank":11732,"rank":11732,"depth":3,"x":262.276,"y":1729.499,"cluster":"algebraic-spaces"},{"id":"stacks:0DF2","tag":"0DF2","title":"Étale topology · Lemma 0DF2","summary":"Let S be a scheme. Let f : Y → X be a morphism of (Spaces/S)_etale. The inclusion functor Y_spaces, etale → (Spaces/X)_etale is cocontinuous and induces a morphism of topoi i_f : Sh(Y_etale) → Sh((Spaces/X)_etale) For a sheaf G on (Spaces/X)_etale we have the formula (i_f^-1G)(U/Y) = G(U/X). The functor i_f^-1 also has a left adjoint i_f, ! which commutes with fibre products and equalizers.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of\n$(\\textit{Spaces}/S)_\\etale$. The inclusion functor\n$Y_{spaces, \\etale} \\to (\\textit{Spaces}/X)_\\etale$\nis cocontinuous and induces a morphism of topoi\n$$\ni_f :\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\Sh((\\textit{Spaces}/X)_\\etale)\n$$\nFor a sheaf $\\mathcal{G}$ on $(\\textit{Spaces}/X)_\\etale$\nwe have the formula $(i_f^{-1}\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$.\nThe functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes\nwith fibre products and equalizers.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DF2","source_file":"spaces-topologies.tex","source_line":293,"source_end_line":309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L293-L309","statement_sha256":"5861d1adb349f83ee745c814e9824b27ce7757992d2e03ae536ecf3e9777189d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11733,"rank":11733,"depth":6,"x":101.976,"y":1522.772,"cluster":"algebraic-spaces"},{"id":"stacks:0DF3","tag":"0DF3","title":"Étale topology · Lemma 0DF3","summary":"Let S be a scheme. Let X be an object of (Spaces/S)_etale. The inclusion functor X_spaces, etale → (Spaces/X)_etale satisfies the hypotheses of Sites, Lemma [Tag 00XU] and hence induces a morphism of sites π_X : (Spaces/X)_etale → X_spaces, etale and a morphism of topoi i_X : Sh(X_etale) → Sh((Spaces/X)_etale) such that π_X ∘ i_X = id. Moreover, i_X = i_id_X with i_id_X as in Lemma [Tag 0DF2]. In particular the functor i_X^-1 = π_X, * is described by the rule…","statement_latex":"Let $S$ be a scheme. Let $X$ be an object of $(\\textit{Spaces}/S)_\\etale$.\nThe inclusion functor $X_{spaces, \\etale} \\to (\\textit{Spaces}/X)_\\etale$\nsatisfies the hypotheses of Sites, Lemma \\ref{sites-lemma-bigger-site}\nand hence induces a morphism of sites\n$$\n\\pi_X :\n(\\textit{Spaces}/X)_\\etale\n\\longrightarrow\nX_{spaces, \\etale}\n$$\nand a morphism of topoi\n$$\ni_X :\n\\Sh(X_\\etale)\n\\longrightarrow\n\\Sh((\\textit{Spaces}/X)_\\etale)\n$$\nsuch that $\\pi_X \\circ i_X = \\text{id}$. Moreover, $i_X = i_{\\text{id}_X}$\nwith $i_{\\text{id}_X}$ as in Lemma \\ref{lemma-put-in-T-etale}.\nIn particular the functor $i_X^{-1} = \\pi_{X, *}$ is described by the rule\n$i_X^{-1}(\\mathcal{G})(U/X) = \\mathcal{G}(U/X)$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DF3","source_file":"spaces-topologies.tex","source_line":325,"source_end_line":348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L325-L348","statement_sha256":"e58104719311f5500fa7a8ac78e0c3ad685ffc1081f38fa4ea885e648e4d3580","origin":"The Stacks Project","memory_eligible":false,"source_rank":11734,"rank":11734,"depth":8,"x":386.61,"y":1584.292,"cluster":"algebraic-spaces"},{"id":"stacks:0DF4","tag":"0DF4","title":"Étale topology · Definition 0DF4","summary":"In the situation of Lemma [Tag 0DF3] the functor i_X^-1 = π_X, * is often called the restriction to the small étale site, and for a sheaf F on the big étale site we often denote F|_X_etale this restriction.","statement_latex":"In the situation of Lemma \\ref{lemma-at-the-bottom-etale}\nthe functor $i_X^{-1} = \\pi_{X, *}$ is often\ncalled the {\\it restriction to the small \\'etale site}, and for a sheaf\n$\\mathcal{F}$ on the big \\'etale site we often denote\n$\\mathcal{F}|_{X_\\etale}$ this restriction.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DF4","source_file":"spaces-topologies.tex","source_line":359,"source_end_line":366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L359-L366","statement_sha256":"e23963314f03960e0a372942bce6ea0f78f2f8cab2713934d0f99a2cb8d97ce7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11735,"rank":11735,"depth":9,"x":127.082,"y":1700.513,"cluster":"algebraic-spaces"},{"id":"stacks:0DF5","tag":"0DF5","title":"Étale topology · Lemma 0DF5","summary":"Let S be a scheme. Let f : Y → X be a morphism in (Spaces/S)_etale. The functor u : (Spaces/Y)_etale → (Spaces/X)_etale, V/Y ↦ V/X is cocontinuous, and has a continuous right adjoint v : (Spaces/X)_etale → (Spaces/Y)_etale, (U → X) ↦ (U ×_X Y → Y). They induce the same morphism of topoi f_big : Sh((Spaces/Y)_etale) → Sh((Spaces/X)_etale) We have f_big^-1(G)(U/Y) = G(U/X). We have f_big, *(F)(U/X) = F(U ×_X Y/Y). Also, f_big^-1 has a left adjoint f_big! which commutes with…","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism in\n$(\\textit{Spaces}/S)_\\etale$. The functor\n$$\nu :\n(\\textit{Spaces}/Y)_\\etale\n\\longrightarrow\n(\\textit{Spaces}/X)_\\etale,\n\\quad\nV/Y \\longmapsto V/X\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv :\n(\\textit{Spaces}/X)_\\etale\n\\longrightarrow\n(\\textit{Spaces}/Y)_\\etale,\n\\quad\n(U \\to X) \\longmapsto (U \\times_X Y \\to Y).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\textit{Spaces}/Y)_\\etale)\n\\longrightarrow\n\\Sh((\\textit{Spaces}/X)_\\etale)\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$.\nWe have $f_{big, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DF5","source_file":"spaces-topologies.tex","source_line":390,"source_end_line":422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L390-L422","statement_sha256":"11c28b9960812421237df6b58a150fe76f39ecc3bf7ba6ca75fa95135bb71b67","origin":"The Stacks Project","memory_eligible":false,"source_rank":11736,"rank":11736,"depth":8,"x":225.057,"y":1467.397,"cluster":"algebraic-spaces"},{"id":"stacks:0DF6","tag":"0DF6","title":"Étale topology · Lemma 0DF6","summary":"Let S be a scheme. Let f : Y → X be a morphism in (Spaces/S)_etale. • We have i_f = f_big ∘ i_T with i_f as in Lemma [Tag 0DF2] and i_T as in Lemma [Tag 0DF3]. • The functor X_spaces, etale → T_spaces, etale, (U → X) ↦ (U ×_X Y → Y) is continuous and induces a morphism of sites f_spaces, etale : Y_spaces, etale → X_spaces, etale The corresponding morphism of small étale topoi is denoted f_small : Sh(Y_etale) → Sh(X_etale) We have f_small, *(F)(U/X) = F(U ×_X Y/Y). • We…","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism in\n$(\\textit{Spaces}/S)_\\etale$.\n\\begin{enumerate}\n\\item We have $i_f = f_{big} \\circ i_T$ with $i_f$ as in\nLemma \\ref{lemma-put-in-T-etale} and $i_T$ as in\nLemma \\ref{lemma-at-the-bottom-etale}.\n\\item The functor $X_{spaces, \\etale} \\to T_{spaces, \\etale}$,\n$(U \\to X) \\mapsto (U \\times_X Y \\to Y)$ is continuous and induces\na morphism of sites\n$$\nf_{spaces, \\etale} : Y_{spaces, \\etale} \\longrightarrow X_{spaces, \\etale}\n$$\nThe corresponding morphism of small \\'etale topoi is denoted\n$$\nf_{small} : \\Sh(Y_\\etale) \\to \\Sh(X_\\etale)\n$$\nWe have $f_{small, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$.\n\\item We have a commutative diagram of morphisms of sites\n$$\n\\xymatrix{\nY_{spaces, \\etale} \\ar[d]_{f_{spaces, \\etale}} &\n(\\textit{Spaces}/Y)_\\etale \\ar[d]^{f_{big}} \\ar[l]^-{\\pi_Y}\\\\\nX_{spaces, \\etale} &\n(\\textit{Spaces}/X)_\\etale \\ar[l]_-{\\pi_X}\n}\n$$\nso that $f_{small} \\circ \\pi_Y = \\pi_X \\circ f_{big}$ as morphisms of topoi.\n\\item We have $f_{small} = \\pi_X \\circ f_{big} \\circ i_Y = \\pi_X \\circ i_f$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DF6","source_file":"spaces-topologies.tex","source_line":441,"source_end_line":472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L441-L472","statement_sha256":"e86ec243266a7ff7eb6e924152d4fc897c483a86ad66616fc0c8e48ea011d233","origin":"The Stacks Project","memory_eligible":false,"source_rank":11737,"rank":11737,"depth":47,"x":340.353,"y":1695.037,"cluster":"algebraic-spaces"},{"id":"stacks:0DF7","tag":"0DF7","title":"Étale topology · Lemma 0DF7","summary":"Let S be a scheme. Given morphisms f : X → Y, g : Y → Z in (Spaces/S)_etale we have g_big ∘ f_big = (g ∘ f)_big and g_small ∘ f_small = (g ∘ f)_small.","statement_latex":"Let $S$ be a scheme. Given morphisms $f : X \\to Y$, $g : Y \\to Z$\nin $(\\textit{Spaces}/S)_\\etale$ we have\n$g_{big} \\circ f_{big} = (g \\circ f)_{big}$ and\n$g_{small} \\circ f_{small} = (g \\circ f)_{small}$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DF7","source_file":"spaces-topologies.tex","source_line":511,"source_end_line":517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L511-L517","statement_sha256":"f74dc96cae283182a2d7318e58aa43ac17887e64dae73d6df95b157e0726bd80","origin":"The Stacks Project","memory_eligible":false,"source_rank":11738,"rank":11738,"depth":48,"x":72.098,"y":1592.533,"cluster":"algebraic-spaces"},{"id":"stacks:0DF8","tag":"0DF8","title":"Étale topology · Lemma 0DF8","summary":"Let S be a scheme. Consider a cartesian diagram xymatrix Y' ar[r]_g' ar[d]_f' & Y ar[d]^f X' ar[r]^g & X in (Spaces/S)_etale. Then i_g^-1 ∘ f_big, * = f'_small, * ∘ (i_g')^-1 and g_big^-1 ∘ f_big, * = f'_big, * ∘ (g'_big)^-1.","statement_latex":"Let $S$ be a scheme. Consider a cartesian diagram\n$$\n\\xymatrix{\nY' \\ar[r]_{g'} \\ar[d]_{f'} & Y \\ar[d]^f \\\\\nX' \\ar[r]^g & X\n}\n$$\nin $(\\textit{Spaces}/S)_\\etale$. Then\n$i_g^{-1} \\circ f_{big, *} = f'_{small, *} \\circ (i_{g'})^{-1}$\nand $g_{big}^{-1} \\circ f_{big, *} = f'_{big, *} \\circ (g'_{big})^{-1}$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DF8","source_file":"spaces-topologies.tex","source_line":527,"source_end_line":539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L527-L539","statement_sha256":"d8d74860d591e833aaa7fc1fd5c1989673fcb233463d7cd5ed1a64120d3a6d7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11739,"rank":11739,"depth":9,"x":352.52,"y":1515.853,"cluster":"algebraic-spaces"},{"id":"stacks:041C","tag":"041C","title":"Smooth topology · Definition 041C","summary":"Let S be a scheme, and let X be an algebraic space over S. A smooth covering of X is a family of morphisms (f_i : X_i → X)_i ∈ I of algebraic spaces over S such that each f_i is smooth and such that |X| = ⋃_i ∈ I |f_i|(|X_i|), i.e., the morphisms are jointly surjective.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nA {\\it smooth covering of $X$} is a family of morphisms\n$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$\nsuch that each $f_i$ is smooth\nand such that\n$$\n|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),\n$$\ni.e., the morphisms are jointly surjective.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Smooth topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041C","source_file":"spaces-topologies.tex","source_line":617,"source_end_line":628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L617-L628","statement_sha256":"c3c38bd9e7ad8abda4fa2ce67ed72b9e55a32aeca32144dd8e9011ea54edc2d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11740,"rank":11740,"depth":0,"x":207.309,"y":1731.655,"cluster":"algebraic-spaces"},{"id":"stacks:0DFA","tag":"0DFA","title":"Smooth topology · Lemma 0DFA","summary":"Any étale covering is a smooth covering, and a fortiori, any Zariski covering is a smooth covering.","statement_latex":"Any \\'etale covering is a smooth covering, and a fortiori,\nany Zariski covering is a smooth covering.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFA","source_file":"spaces-topologies.tex","source_line":636,"source_end_line":640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L636-L640","statement_sha256":"2d2d62ba9e8d349f7ab0f12d7f22cdc1ddba522c9e5ec88ed7bbf94c700fb34d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11741,"rank":11741,"depth":48,"x":140.8,"y":1489.972,"cluster":"algebraic-spaces"},{"id":"stacks:041D","tag":"041D","title":"Smooth topology · Lemma 041D","summary":"Let S be a scheme. Let X be an algebraic space over S. • If X' → X is an isomorphism then (X' → X) is a smooth covering of X. • If (X_i → X)_i∈ I is a smooth covering and for each i we have a smooth covering (X_ij → X_i)_j∈ J_i, then (X_ij → X)_i ∈ I, j∈ J_i is a smooth covering. • If (X_i → X)_i∈ I is a smooth covering and X' → X is a morphism of algebraic spaces then (X' ×_X X_i → X')_i∈ I is a smooth covering.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$\nis a smooth covering of $X$.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a smooth covering and for each\n$i$ we have a smooth covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then\n$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a smooth covering.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a smooth covering\nand $X' \\to X$ is a morphism of algebraic spaces then\n$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a smooth covering.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Smooth topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041D","source_file":"spaces-topologies.tex","source_line":649,"source_end_line":663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L649-L663","statement_sha256":"1601a60c5d36ee4a62d356007977424b914c3d2f20044ec99797d084a43b2ace","origin":"The Stacks Project","memory_eligible":false,"source_rank":11742,"rank":11742,"depth":0,"x":384.357,"y":1630.538,"cluster":"algebraic-spaces"},{"id":"stacks:041A","tag":"041A","title":"Syntomic topology · Definition 041A","summary":"Let S be a scheme, and let X be an algebraic space over S. A syntomic covering of X is a family of morphisms (f_i : X_i → X)_i ∈ I of algebraic spaces over S such that each f_i is syntomic and such that |X| = ⋃_i ∈ I |f_i|(|X_i|), i.e., the morphisms are jointly surjective.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nA {\\it syntomic covering of $X$} is a family of morphisms\n$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$\nsuch that each $f_i$ is syntomic\nand such that\n$$\n|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),\n$$\ni.e., the morphisms are jointly surjective.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Syntomic topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041A","source_file":"spaces-topologies.tex","source_line":689,"source_end_line":700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L689-L700","statement_sha256":"f6ffcbf12131200ba617805601c3defa6161f1d7b219d2aadf9c87f108d0064c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11743,"rank":11743,"depth":0,"x":91.531,"y":1665.11,"cluster":"algebraic-spaces"},{"id":"stacks:0DFB","tag":"0DFB","title":"Syntomic topology · Lemma 0DFB","summary":"Any smooth covering is a syntomic covering, and a fortiori, any étale or Zariski covering is a syntomic covering.","statement_latex":"Any smooth covering is a syntomic covering, and a fortiori,\nany \\'etale or Zariski covering is a syntomic covering.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFB","source_file":"spaces-topologies.tex","source_line":708,"source_end_line":712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L708-L712","statement_sha256":"9f4e6825cface159b8a24ff0c03cdda6028bcccc1054e4f3a32b07a594e80891","origin":"The Stacks Project","memory_eligible":false,"source_rank":11744,"rank":11744,"depth":49,"x":279.778,"y":1473.336,"cluster":"algebraic-spaces"},{"id":"stacks:041B","tag":"041B","title":"Syntomic topology · Lemma 041B","summary":"Let S be a scheme. Let X be an algebraic space over S. • If X' → X is an isomorphism then (X' → X) is a syntomic covering of X. • If (X_i → X)_i∈ I is a syntomic covering and for each i we have a syntomic covering (X_ij → X_i)_j∈ J_i, then (X_ij → X)_i ∈ I, j∈ J_i is a syntomic covering. • If (X_i → X)_i∈ I is a syntomic covering and X' → X is a morphism of algebraic spaces then (X' ×_X X_i → X')_i∈ I is a syntomic covering.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$\nis a syntomic covering of $X$.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a syntomic covering and for each\n$i$ we have a syntomic covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then\n$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a syntomic covering.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a syntomic covering\nand $X' \\to X$ is a morphism of algebraic spaces then\n$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a syntomic covering.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Syntomic topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041B","source_file":"spaces-topologies.tex","source_line":721,"source_end_line":735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L721-L735","statement_sha256":"12b4f504bd8f296bced75e69446414ca7e8f16b05139477136155fc2bbb4ba00","origin":"The Stacks Project","memory_eligible":false,"source_rank":11745,"rank":11745,"depth":0,"x":295.197,"y":1721.724,"cluster":"algebraic-spaces"},{"id":"stacks:03Y8","tag":"03Y8","title":"Fppf topology · Definition 03Y8","summary":"Let S be a scheme, and let X be an algebraic space over S. An fppf covering of X is a family of morphisms (f_i : X_i → X)_i ∈ I of algebraic spaces over S such that each f_i is flat and locally of finite presentation and such that |X| = ⋃_i ∈ I |f_i|(|X_i|), i.e., the morphisms are jointly surjective.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nAn {\\it fppf covering of $X$} is a family of morphisms\n$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$\nsuch that each $f_i$ is flat and locally of finite presentation\nand such that\n$$\n|X| = \\bigcup\\nolimits_{i \\in I} |f_i|(|X_i|),\n$$\ni.e., the morphisms are jointly surjective.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fppf topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Y8","source_file":"spaces-topologies.tex","source_line":761,"source_end_line":772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L761-L772","statement_sha256":"5429a07d19185c23745bf92e076daa2a20e0e8c27e7ee7a996442c1b8f14fbf8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11746,"rank":11746,"depth":0,"x":83.943,"y":1547.202,"cluster":"algebraic-spaces"},{"id":"stacks:0DFC","tag":"0DFC","title":"Fppf topology · Lemma 0DFC","summary":"Any syntomic covering is an fppf covering, and a fortiori, any smooth, étale, or Zariski covering is an fppf covering.","statement_latex":"Any syntomic covering is an fppf covering, and a fortiori,\nany smooth, \\'etale, or Zariski covering is an fppf covering.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFC","source_file":"spaces-topologies.tex","source_line":780,"source_end_line":784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L780-L784","statement_sha256":"1cbb5fdf87cfa8ffea86d7a97776a7db2502567f79b82673af57aa4bb317bab3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11747,"rank":11747,"depth":50,"x":380.256,"y":1556.027,"cluster":"algebraic-spaces"},{"id":"stacks:03Y9","tag":"03Y9","title":"Fppf topology · Lemma 03Y9","summary":"Let S be a scheme. Let X be an algebraic space over S. • If X' → X is an isomorphism then (X' → X) is an fppf covering of X. • If (X_i → X)_i∈ I is an fppf covering and for each i we have an fppf covering (X_ij → X_i)_j∈ J_i, then (X_ij → X)_i ∈ I, j∈ J_i is an fppf covering. • If (X_i → X)_i∈ I is an fppf covering and X' → X is a morphism of algebraic spaces then (X' ×_X X_i → X')_i∈ I is an fppf covering.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$\nis an fppf covering of $X$.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fppf covering and for each\n$i$ we have an fppf covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then\n$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is an fppf covering.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fppf covering\nand $X' \\to X$ is a morphism of algebraic spaces then\n$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is an fppf covering.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Y9","source_file":"spaces-topologies.tex","source_line":795,"source_end_line":809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L795-L809","statement_sha256":"13f59a0db6365ef94733b19750b0c75fc689bc97c18c3fe7a8304028a35fa26a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11748,"rank":11748,"depth":0,"x":154.516,"y":1717.76,"cluster":"algebraic-spaces"},{"id":"stacks:042T","tag":"042T","title":"Fppf topology · Lemma 042T","summary":"Let S be a scheme, and let X be an algebraic space over S. Suppose that U = (f_i : X_i → X)_i ∈ I is an fppf covering of X. Then there exists a refinement V = (g_i : T_i → X) of U which is an fppf covering such that each T_i is a scheme.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nSuppose that $\\mathcal{U} = \\{f_i : X_i \\to X\\}_{i \\in I}$ is an\nfppf covering of $X$. Then there exists a refinement\n$\\mathcal{V} = \\{g_i : T_i \\to X\\}$ of $\\mathcal{U}$ which is an\nfppf covering such that each $T_i$ is a scheme.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042T","source_file":"spaces-topologies.tex","source_line":815,"source_end_line":822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L815-L822","statement_sha256":"bec0ee0d5df020309d769bfd5973945827e5d78876e3c90ab8251dfbaf68449f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11749,"rank":11749,"depth":0,"x":190.936,"y":1470.25,"cluster":"algebraic-spaces"},{"id":"stacks:0469","tag":"0469","title":"Fppf topology · Lemma 0469","summary":"Let S be a scheme. Let (f_i : X_i → X)_i ∈ I be an fppf covering of algebraic spaces over S. Then the map of sheaves coprod X_i → X is surjective.","statement_latex":"Let $S$ be a scheme.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fppf covering of algebraic\nspaces over $S$. Then the map of sheaves\n$$\n\\coprod X_i \\longrightarrow X\n$$\nis surjective.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0469","source_file":"spaces-topologies.tex","source_line":829,"source_end_line":838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L829-L838","statement_sha256":"faa49c9c60b910a1265202d932dea40b5db2d09649e69a2c785c709773af3642","origin":"The Stacks Project","memory_eligible":false,"source_rank":11750,"rank":11750,"depth":2,"x":363.232,"y":1673.557,"cluster":"algebraic-spaces"},{"id":"stacks:0DBV","tag":"0DBV","title":"Fppf topology · Definition 0DBV","summary":"Let S be a scheme. A big fppf site (Spaces/S)_fppf is any site constructed as follows: • Choose a big fppf site (Sch/S)_fppf as in Topologies, Section [Tag 021L]. • As underlying category take the category Spaces/S of algebraic spaces over S (see discussion in Section [Tag 03Y6] why this is a set). • Choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Spaces/S and the class of fppf coverings of Definition [Tag 03Y8].","statement_latex":"Let $S$ be a scheme. A big fppf site {\\it $(\\textit{Spaces}/S)_{fppf}$}\nis any site constructed as follows:\n\\begin{enumerate}\n\\item Choose a big fppf site $(\\Sch/S)_{fppf}$ as in\nTopologies, Section \\ref{topologies-section-fppf}.\n\\item As underlying category take the category $\\textit{Spaces}/S$\nof algebraic spaces over $S$ (see discussion in\nSection \\ref{section-procedure} why this is a set).\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\textit{Spaces}/S$ and the class of fppf coverings\nof Definition \\ref{definition-fppf-covering}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fppf topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBV","source_file":"spaces-topologies.tex","source_line":848,"source_end_line":863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L848-L863","statement_sha256":"fce4ec98f11fcb85673bf4a41e09b9cd45803ea2f317293e248b32ed6bc56ef7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11751,"rank":11751,"depth":2,"x":72.502,"y":1621.373,"cluster":"algebraic-spaces"},{"id":"stacks:0DBW","tag":"0DBW","title":"Fppf topology · Definition 0DBW","summary":"Let S be a scheme. Let (Spaces/S)_fppf be as in Definition [Tag 0DBV]. Let X be an algebraic space over S, i.e., an object of (Spaces/S)_fppf. Then the big fppf site (Spaces/X)_fppf of X is the localization of the site (Spaces/S)_fppf at X introduced in Sites, Section [Tag 00XZ].","statement_latex":"Let $S$ be a scheme. Let $(\\textit{Spaces}/S)_{fppf}$ be as in\nDefinition \\ref{definition-big-fppf-site}.\nLet $X$ be an algebraic space over $S$, i.e., an object of\n$(\\textit{Spaces}/S)_{fppf}$. Then the big fppf site\n{\\it $(\\textit{Spaces}/X)_{fppf}$} of $X$\nis the localization of the site $(\\textit{Spaces}/S)_{fppf}$\nat $X$ introduced in Sites, Section \\ref{sites-section-localize}.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fppf topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBW","source_file":"spaces-topologies.tex","source_line":869,"source_end_line":878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L869-L878","statement_sha256":"afca3d78963a51aaf84f6dcf789eff298a6b8d0a9082bfc84d5054f95e92c028","origin":"The Stacks Project","memory_eligible":false,"source_rank":11752,"rank":11752,"depth":3,"x":329.013,"y":1494.804,"cluster":"algebraic-spaces"},{"id":"stacks:0DFD","tag":"0DFD","title":"Fppf topology · Lemma 0DFD","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. The functor u : (Spaces/Y)_fppf → (Spaces/X)_fppf, V/Y ↦ V/X is cocontinuous, and has a continuous right adjoint v : (Spaces/X)_fppf → (Spaces/Y)_fppf, (U → Y) ↦ (U ×_X Y → Y). They induce the same morphism of topoi f_big : Sh((Spaces/Y)_fppf) → Sh((Spaces/X)_fppf) We have f_big^-1(G)(U/Y) = G(U/X). We have f_big, *(F)(U/X) = F(U ×_X Y/Y). Also, f_big^-1 has a left adjoint f_big! which commutes…","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be a morphism of algebraic spaces over $S$.\nThe functor\n$$\nu : (\\textit{Spaces}/Y)_{fppf} \\longrightarrow (\\textit{Spaces}/X)_{fppf},\n\\quad\nV/Y \\longmapsto V/X\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv : (\\textit{Spaces}/X)_{fppf} \\longrightarrow (\\textit{Spaces}/Y)_{fppf},\n\\quad\n(U \\to Y) \\longmapsto (U \\times_X Y \\to Y).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\textit{Spaces}/Y)_{fppf})\n\\longrightarrow\n\\Sh((\\textit{Spaces}/X)_{fppf})\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$.\nWe have $f_{big, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFD","source_file":"spaces-topologies.tex","source_line":884,"source_end_line":911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L884-L911","statement_sha256":"8b9e787e60f380b8efd7547d73ade26da7d85b8c016a349ad0a5508748e739e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11753,"rank":11753,"depth":8,"x":241.593,"y":1733.838,"cluster":"algebraic-spaces"},{"id":"stacks:0DFE","tag":"0DFE","title":"Fppf topology · Lemma 0DFE","summary":"Let S be a scheme. Given morphisms f : X → Y, g : Y → Z of algebraic spaces over S we have g_big ∘ f_big = (g ∘ f)_big.","statement_latex":"Let $S$ be a scheme. Given morphisms $f : X \\to Y$, $g : Y \\to Z$\nof algebraic spaces over $S$ we have\n$g_{big} \\circ f_{big} = (g \\circ f)_{big}$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFE","source_file":"spaces-topologies.tex","source_line":928,"source_end_line":933,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L928-L933","statement_sha256":"b4e1cd6607ea3e0bdf9b5271819fd5c5172629ba174271a482de3fcc9f56cd40","origin":"The Stacks Project","memory_eligible":false,"source_rank":11754,"rank":11754,"depth":9,"x":113.748,"y":1507.829,"cluster":"algebraic-spaces"},{"id":"stacks:0DFG","tag":"0DFG","title":"The ph topology · Definition 0DFG","summary":"Let S be a scheme and let X be an algebraic space over S. A ph covering of X is a family of morphisms (X_i → X)_i ∈ I of algebraic spaces over S such that f_i is locally of finite type and such that for every U → X with U affine there exists a standard ph covering (U_j → U)_j = 1, …, m refining the family (X_i ×_X U → U)_i ∈ I.","statement_latex":"Let $S$ be a scheme and let $X$ be an algebraic space over $S$.\nA {\\it ph covering of $X$} is a family\nof morphisms $\\{X_i \\to X\\}_{i \\in I}$ of algebraic spaces over $S$\nsuch that $f_i$ is locally of finite type and such that for every\n$U \\to X$ with $U$ affine there exists a standard ph covering\n$\\{U_j \\to U\\}_{j = 1, \\ldots, m}$ refining the family\n$\\{X_i \\times_X U \\to U\\}_{i \\in I}$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFG","source_file":"spaces-topologies.tex","source_line":956,"source_end_line":965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L956-L965","statement_sha256":"79fbd4765765e38b4be5940abee9e5f23324fa8f99b72228a4f4aa8b0bbef622","origin":"The Stacks Project","memory_eligible":false,"source_rank":11755,"rank":11755,"depth":0,"x":389.947,"y":1602.003,"cluster":"algebraic-spaces"},{"id":"stacks:0DFH","tag":"0DFH","title":"The ph topology · Lemma 0DFH","summary":"Any fppf covering is a ph covering, and a fortiori, any syntomic, smooth, étale or Zariski covering is a ph covering.","statement_latex":"Any fppf covering is a ph covering, and a fortiori,\nany syntomic, smooth, \\'etale or Zariski covering is a ph covering.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFH","source_file":"spaces-topologies.tex","source_line":974,"source_end_line":978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L974-L978","statement_sha256":"6b186aea8facca11b1fce31cb2eafaa222fb980a77729b5446e4d3346ceda24e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11756,"rank":11756,"depth":51,"x":110.37,"y":1689.338,"cluster":"algebraic-spaces"},{"id":"stacks:0DFI","tag":"0DFI","title":"The ph topology · Lemma 0DFI","summary":"Let S be a scheme. Let f : Y → X be a surjective proper morphism of algebraic spaces over S. Then (Y → X) is a ph covering.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a surjective proper morphism\nof algebraic spaces over $S$. Then $\\{Y \\to X\\}$ is a ph covering.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFI","source_file":"spaces-topologies.tex","source_line":996,"source_end_line":1000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L996-L1000","statement_sha256":"bab380680e128b1caa5209d8cef04ee8f772d2ce1b83c43b4fa58c7cdd0435e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11757,"rank":11757,"depth":57,"x":246.38,"y":1466.157,"cluster":"algebraic-spaces"},{"id":"stacks:0DFJ","tag":"0DFJ","title":"The ph topology · Lemma 0DFJ","summary":"Let S be a scheme. Let X be an algebraic space over S. • If X' → X is an isomorphism then (X' → X) is a ph covering of X. • If (X_i → X)_i∈ I is a ph covering and for each i we have a ph covering (X_ij → X_i)_j∈ J_i, then (X_ij → X)_i ∈ I, j∈ J_i is a ph covering. • If (X_i → X)_i∈ I is a ph covering and X' → X is a morphism of algebraic spaces then (X' ×_X X_i → X')_i∈ I is a ph covering.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$\nis a ph covering of $X$.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a ph covering and for each\n$i$ we have a ph covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then\n$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is a ph covering.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is a ph covering\nand $X' \\to X$ is a morphism of algebraic spaces then\n$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is a ph covering.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFJ","source_file":"spaces-topologies.tex","source_line":1013,"source_end_line":1026,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1013-L1026","statement_sha256":"2cde3d9427f9955aa8ba9d6fe252efac14bd14216b4684bf5a8cbf1b9254c7d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11758,"rank":11758,"depth":31,"x":325.615,"y":1708.057,"cluster":"algebraic-spaces"},{"id":"stacks:0DFK","tag":"0DFK","title":"The ph topology · Definition 0DFK","summary":"Let S be a scheme. A big ph site (Spaces/S)_ph is any site constructed as follows: • Choose a big ph site (Sch/S)_ph as in Topologies, Section [Tag 0DBC]. • As underlying category take the category Spaces/S of algebraic spaces over S (see discussion in Section [Tag 03Y6] why this is a set). • Choose any set of coverings as in Sets, Lemma [Tag 000X] starting with the category Spaces/S and the class of ph coverings of Definition [Tag 0DFG].","statement_latex":"Let $S$ be a scheme. A big ph site {\\it $(\\textit{Spaces}/S)_{ph}$}\nis any site constructed as follows:\n\\begin{enumerate}\n\\item Choose a big ph site $(\\Sch/S)_{ph}$ as in\nTopologies, Section \\ref{topologies-section-ph}.\n\\item As underlying category take the category $\\textit{Spaces}/S$\nof algebraic spaces over $S$ (see discussion in\nSection \\ref{section-procedure} why this is a set).\n\\item Choose any set of coverings as in\nSets, Lemma \\ref{sets-lemma-coverings-site} starting with the\ncategory $\\textit{Spaces}/S$ and the class of ph coverings\nof Definition \\ref{definition-ph-covering}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFK","source_file":"spaces-topologies.tex","source_line":1058,"source_end_line":1073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1058-L1073","statement_sha256":"700cfeac0d5b1e6f52e4c9798dd576524af4ca43d7e18c1845dd20b7e060fece","origin":"The Stacks Project","memory_eligible":false,"source_rank":11759,"rank":11759,"depth":2,"x":72.498,"y":1574.558,"cluster":"algebraic-spaces"},{"id":"stacks:0DFL","tag":"0DFL","title":"The ph topology · Definition 0DFL","summary":"Let S be a scheme. Let (Spaces/S)_ph be as in Definition [Tag 0DFK]. Let X be an algebraic space over S, i.e., an object of (Spaces/S)_ph. Then the big ph site (Spaces/X)_ph of X is the localization of the site (Spaces/S)_ph at X introduced in Sites, Section [Tag 00XZ].","statement_latex":"Let $S$ be a scheme. Let $(\\textit{Spaces}/S)_{ph}$ be as in\nDefinition \\ref{definition-big-ph-site}.\nLet $X$ be an algebraic space over $S$, i.e., an object of\n$(\\textit{Spaces}/S)_{ph}$. Then the big ph site\n{\\it $(\\textit{Spaces}/X)_{ph}$} of $X$\nis the localization of the site $(\\textit{Spaces}/S)_{ph}$\nat $X$ introduced in Sites, Section \\ref{sites-section-localize}.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFL","source_file":"spaces-topologies.tex","source_line":1079,"source_end_line":1088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1079-L1088","statement_sha256":"48a5b9fd4bc18de15e0f288e9d58aa33671a857dbbfdd101f36f3cb81d2a847f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11760,"rank":11760,"depth":3,"x":366.686,"y":1529.345,"cluster":"algebraic-spaces"},{"id":"stacks:0DFM","tag":"0DFM","title":"The ph topology · Lemma 0DFM","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a presheaf on (Spaces/X)_ph. Then F is a sheaf if and only if • F satisfies the sheaf condition for étale coverings, and • if f : V → U is a proper surjective morphism of (Spaces/X)_ph, then F(U) maps bijectively to the equalizer of the two maps F(V) → F(V ×_U V).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a presheaf on $(\\textit{Spaces}/X)_{ph}$.\nThen $\\mathcal{F}$ is a sheaf if and only if\n\\begin{enumerate}\n\\item $\\mathcal{F}$ satisfies the sheaf condition for \\'etale coverings, and\n\\item if $f : V \\to U$ is a proper surjective morphism of\n$(\\textit{Spaces}/X)_{ph}$, then\n$\\mathcal{F}(U)$ maps bijectively to the equalizer\nof the two maps $\\mathcal{F}(V) \\to \\mathcal{F}(V \\times_U V)$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFM","source_file":"spaces-topologies.tex","source_line":1093,"source_end_line":1105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1093-L1105","statement_sha256":"368cc9e4fb30d31ab41803317a7c1fc6003e82b01252af745f14b78f680a5c70","origin":"The Stacks Project","memory_eligible":false,"source_rank":11761,"rank":11761,"depth":0,"x":186.0,"y":1729.742,"cluster":"algebraic-spaces"},{"id":"stacks:0DFN","tag":"0DFN","title":"The ph topology · Lemma 0DFN","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. The functor u : (Spaces/Y)_ph → (Spaces/X)_ph, V/Y ↦ V/X is cocontinuous, and has a continuous right adjoint v : (Spaces/X)_ph → (Spaces/Y)_ph, (U → Y) ↦ (U ×_X Y → Y). They induce the same morphism of topoi f_big : Sh((Spaces/Y)_ph) → Sh((Spaces/X)_ph) We have f_big^-1(G)(U/Y) = G(U/X). We have f_big, *(F)(U/X) = F(U ×_X Y/Y). Also, f_big^-1 has a left adjoint f_big! which commutes with fibre…","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be a morphism of algebraic spaces over $S$.\nThe functor\n$$\nu : (\\textit{Spaces}/Y)_{ph} \\longrightarrow (\\textit{Spaces}/X)_{ph},\n\\quad\nV/Y \\longmapsto V/X\n$$\nis cocontinuous, and has a continuous right adjoint\n$$\nv : (\\textit{Spaces}/X)_{ph} \\longrightarrow (\\textit{Spaces}/Y)_{ph},\n\\quad\n(U \\to Y) \\longmapsto (U \\times_X Y \\to Y).\n$$\nThey induce the same morphism of topoi\n$$\nf_{big} :\n\\Sh((\\textit{Spaces}/Y)_{ph})\n\\longrightarrow\n\\Sh((\\textit{Spaces}/X)_{ph})\n$$\nWe have $f_{big}^{-1}(\\mathcal{G})(U/Y) = \\mathcal{G}(U/X)$.\nWe have $f_{big, *}(\\mathcal{F})(U/X) = \\mathcal{F}(U \\times_X Y/Y)$.\nAlso, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with\nfibre products and equalizers.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFN","source_file":"spaces-topologies.tex","source_line":1139,"source_end_line":1166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1139-L1166","statement_sha256":"a3f4ee15a9fe8f7396e8c5f092a8b3ee910248721a918072aaf8397ba746db70","origin":"The Stacks Project","memory_eligible":false,"source_rank":11762,"rank":11762,"depth":8,"x":158.064,"y":1479.287,"cluster":"algebraic-spaces"},{"id":"stacks:0DFP","tag":"0DFP","title":"The ph topology · Lemma 0DFP","summary":"Let S be a scheme. Given morphisms f : X → Y, g : Y → Z of algebraic spaces over S we have g_big ∘ f_big = (g ∘ f)_big.","statement_latex":"Let $S$ be a scheme. Given morphisms $f : X \\to Y$, $g : Y \\to Z$\nof algebraic spaces over $S$ we have\n$g_{big} \\circ f_{big} = (g \\circ f)_{big}$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFP","source_file":"spaces-topologies.tex","source_line":1183,"source_end_line":1188,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1183-L1188","statement_sha256":"b4e1cd6607ea3e0bdf9b5271819fd5c5172629ba174271a482de3fcc9f56cd40","origin":"The Stacks Project","memory_eligible":false,"source_rank":11763,"rank":11763,"depth":9,"x":380.214,"y":1648.224,"cluster":"algebraic-spaces"},{"id":"stacks:0DK6","tag":"0DK6","title":"The ph topology · Lemma 0DK6","summary":"Let S be a scheme. Let X be an algebraic space over S. Let P be a property of objects in (Spaces/X)_fppf such that whenever (U_i → U) is a covering in (Spaces/X)_fppf, then P(U_i_0 ×_U … ×_U U_i_p) for all p ≥ 0, i_0, …, i_p ∈ I ⇒ P(U) If P(U) for all U affine and flat, locally of finite presentation over X, then P(X).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $P$ be a property of objects in $(\\textit{Spaces}/X)_{fppf}$\nsuch that whenever $\\{U_i \\to U\\}$ is a covering in\n$(\\textit{Spaces}/X)_{fppf}$, then\n$$\nP(U_{i_0} \\times_U \\ldots \\times_U U_{i_p})\n\\text{ for all }\np \\geq 0,\\ i_0, \\ldots, i_p \\in I\n\\Rightarrow P(U)\n$$\nIf $P(U)$ for all $U$ affine and flat, locally of finite presentation over $X$,\nthen $P(X)$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"The ph topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DK6","source_file":"spaces-topologies.tex","source_line":1196,"source_end_line":1210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1196-L1210","statement_sha256":"f27d9cfba896d67b086d7c5eaa321a328abf4009f2c1188f913dd1776571fcf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11764,"rank":11764,"depth":0,"x":80.36,"y":1649.707,"cluster":"algebraic-spaces"},{"id":"stacks:03MQ","tag":"03MQ","title":"Fpqc topology · Definition 03MQ","summary":"Let S be a scheme, and let X be an algebraic space over S. An fpqc covering of X is a family of morphisms (f_i : X_i → X)_i ∈ I of algebraic spaces such that each f_i is flat and such that for every affine scheme Z and morphism h : Z → X there exists a standard fpqc covering (g_j : Z_j → Z)_j = 1, …, m which refines the family (X_i ×_X Z → Z)_i ∈ I.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nAn {\\it fpqc covering of $X$} is a family of morphisms\n$\\{f_i : X_i \\to X\\}_{i \\in I}$ of algebraic spaces\nsuch that each $f_i$ is flat and such that for every affine scheme\n$Z$ and morphism $h : Z \\to X$ there exists a standard fpqc covering\n$\\{g_j : Z_j \\to Z\\}_{j = 1, \\ldots, m}$ which refines the family\n$\\{X_i \\times_X Z \\to Z\\}_{i \\in I}$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fpqc topology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MQ","source_file":"spaces-topologies.tex","source_line":1249,"source_end_line":1258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1249-L1258","statement_sha256":"099cf84e8a385419bf4bbbf125687cd4649ff2a991e8d72db0d46d8f2e01a333","origin":"The Stacks Project","memory_eligible":false,"source_rank":11765,"rank":11765,"depth":0,"x":300.415,"y":1478.361,"cluster":"algebraic-spaces"},{"id":"stacks:0DFQ","tag":"0DFQ","title":"Fpqc topology · Lemma 0DFQ","summary":"Any fppf covering is an fpqc covering, and a fortiori, any syntomic, smooth, étale or Zariski covering is an fpqc covering.","statement_latex":"Any fppf covering is an fpqc covering, and a fortiori,\nany syntomic, smooth, \\'etale or Zariski covering is an fpqc covering.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFQ","source_file":"spaces-topologies.tex","source_line":1267,"source_end_line":1271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1267-L1271","statement_sha256":"e0f0782f45db2d061347be14fe60e7c0ae1e725e3ff285df26deb4e369189698","origin":"The Stacks Project","memory_eligible":false,"source_rank":11766,"rank":11766,"depth":56,"x":275.925,"y":1729.732,"cluster":"algebraic-spaces"},{"id":"stacks:03MR","tag":"03MR","title":"Fpqc topology · Lemma 03MR","summary":"Let S be a scheme. Let X be an algebraic space over S. • If X' → X is an isomorphism then (X' → X) is an fpqc covering of X. • If (X_i → X)_i∈ I is an fpqc covering and for each i we have an fpqc covering (X_ij → X_i)_j∈ J_i, then (X_ij → X)_i ∈ I, j∈ J_i is an fpqc covering. • If (X_i → X)_i∈ I is an fpqc covering and X' → X is a morphism of algebraic spaces then (X' ×_X X_i → X')_i∈ I is an fpqc covering.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item If $X' \\to X$ is an isomorphism then $\\{X' \\to X\\}$\nis an fpqc covering of $X$.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fpqc covering and for each\n$i$ we have an fpqc covering $\\{X_{ij} \\to X_i\\}_{j\\in J_i}$, then\n$\\{X_{ij} \\to X\\}_{i \\in I, j\\in J_i}$ is an fpqc covering.\n\\item If $\\{X_i \\to X\\}_{i\\in I}$ is an fpqc covering\nand $X' \\to X$ is a morphism of algebraic spaces then\n$\\{X' \\times_X X_i \\to X'\\}_{i\\in I}$ is an fpqc covering.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MR","source_file":"spaces-topologies.tex","source_line":1305,"source_end_line":1319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1305-L1319","statement_sha256":"c17dd90dbf05a59ffa71891efcc0ab84086dae69b855506444f3da4997dfaed0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11767,"rank":11767,"depth":14,"x":91.721,"y":1530.353,"cluster":"algebraic-spaces"},{"id":"stacks:03MS","tag":"03MS","title":"Fpqc topology · Lemma 03MS","summary":"Let S be a scheme, and let X be an algebraic space over S. Suppose that (f_i : X_i → X)_i ∈ I is a family of morphisms of algebraic spaces with target X. Let U → X be a surjective étale morphism from a scheme towards X. Then (f_i : X_i → X)_i ∈ I is an fpqc covering of X if and only if (U ×_X X_i → U)_i ∈ I is an fpqc covering of U.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nSuppose that $\\{f_i : X_i \\to X\\}_{i \\in I}$ is a family of morphisms of\nalgebraic spaces with target $X$. Let $U \\to X$ be a surjective\n\\'etale morphism from a scheme towards $X$. Then\n$\\{f_i : X_i \\to X\\}_{i \\in I}$ is an fpqc covering of $X$ if and only\nif $\\{U \\times_X X_i \\to U\\}_{i \\in I}$ is an fpqc covering of $U$.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03MS","source_file":"spaces-topologies.tex","source_line":1351,"source_end_line":1359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1351-L1359","statement_sha256":"2e11ed58765db83c184fee4f1e28e5aa69b7dc0a5609414b27689c3d84f659a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11768,"rank":11768,"depth":15,"x":388.073,"y":1572.877,"cluster":"algebraic-spaces"},{"id":"stacks:0419","tag":"0419","title":"Fpqc topology · Lemma 0419","summary":"Let S be a scheme, and let X be an algebraic space over S. Suppose that U = (f_i : X_i → X)_i ∈ I is an fpqc covering of X. Then there exists a refinement V = (g_i : T_i → X) of U which is an fpqc covering such that each T_i is a scheme.","statement_latex":"Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.\nSuppose that $\\mathcal{U} = \\{f_i : X_i \\to X\\}_{i \\in I}$ is an\nfpqc covering of $X$. Then there exists a refinement\n$\\mathcal{V} = \\{g_i : T_i \\to X\\}$ of $\\mathcal{U}$ which is an\nfpqc covering such that each $T_i$ is a scheme.","area":"Algebraic Spaces","chapter":"Topologies on Algebraic Spaces","chapter_id":"spaces-topologies","section":"Fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0419","source_file":"spaces-topologies.tex","source_line":1379,"source_end_line":1386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-topologies.tex#L1379-L1386","statement_sha256":"d1fb6e441d8d2ff47bd0cfabc40291078585ec162e0341a2a7c011d180814172","origin":"The Stacks Project","memory_eligible":false,"source_rank":11769,"rank":11769,"depth":0,"x":135.192,"y":1709.763,"cluster":"algebraic-spaces"},{"id":"stacks:04W3","tag":"04W3","title":"Descent data for quasi-coherent sheaves · Definition 04W3","summary":"Let S be a scheme. Let (f_i : X_i → X)_i ∈ I be a family of morphisms of algebraic spaces over S with fixed target X. • A descent datum (F_i, φ_ij) for quasi-coherent sheaves with respect to the given family is given by a quasi-coherent sheaf F_i on X_i for each i ∈ I, an isomorphism of quasi-coherent O_X_i ×_X X_j-modules φ_ij : pr_0^*F_i → pr_1^*F_j for each pair (i, j) ∈ I^2 such that for every triple of indices (i, j, k) ∈ I^3 the diagram xymatrix pr_0^*F_i…","statement_latex":"Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}_{i \\in I}$ be a family\nof morphisms of algebraic spaces over $S$ with fixed target $X$.\n\\begin{enumerate}\n\\item A {\\it descent datum $(\\mathcal{F}_i, \\varphi_{ij})$\nfor quasi-coherent sheaves} with respect to the given family\nis given by a quasi-coherent sheaf $\\mathcal{F}_i$ on $X_i$ for\neach $i \\in I$, an isomorphism of quasi-coherent\n$\\mathcal{O}_{X_i \\times_X X_j}$-modules\n$\\varphi_{ij} : \\text{pr}_0^*\\mathcal{F}_i \\to \\text{pr}_1^*\\mathcal{F}_j$\nfor each pair $(i, j) \\in I^2$\nsuch that for every triple of indices $(i, j, k) \\in I^3$ the\ndiagram\n$$\n\\xymatrix{\n\\text{pr}_0^*\\mathcal{F}_i \\ar[rd]_{\\text{pr}_{01}^*\\varphi_{ij}}\n\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} & &\n\\text{pr}_2^*\\mathcal{F}_k \\\\\n& \\text{pr}_1^*\\mathcal{F}_j \\ar[ru]_{\\text{pr}_{12}^*\\varphi_{jk}} &\n}\n$$\nof $\\mathcal{O}_{X_i \\times_X X_j \\times_X X_k}$-modules\ncommutes. This is called the {\\it cocycle condition}.\n\\item A {\\it morphism $\\psi : (\\mathcal{F}_i, \\varphi_{ij}) \\to\n(\\mathcal{F}'_i, \\varphi'_{ij})$ of descent data} is given\nby a family $\\psi = (\\psi_i)_{i\\in I}$ of morphisms of\n$\\mathcal{O}_{X_i}$-modules $\\psi_i : \\mathcal{F}_i \\to \\mathcal{F}'_i$\nsuch that all the diagrams\n$$\n\\xymatrix{\n\\text{pr}_0^*\\mathcal{F}_i \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\text{pr}_0^*\\psi_i}\n& \\text{pr}_1^*\\mathcal{F}_j \\ar[d]^{\\text{pr}_1^*\\psi_j} \\\\\n\\text{pr}_0^*\\mathcal{F}'_i \\ar[r]^{\\varphi'_{ij}} &\n\\text{pr}_1^*\\mathcal{F}'_j \\\\\n}\n$$\ncommute.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for quasi-coherent sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04W3","source_file":"spaces-descent.tex","source_line":60,"source_end_line":99,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L60-L99","statement_sha256":"782192be3a521e23bc83d03124d84aa8d6254df6ba77122687b332b55dc7d8c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11770,"rank":11770,"depth":0,"x":211.63,"y":1465.181,"cluster":"algebraic-spaces"},{"id":"stacks:04W4","tag":"04W4","title":"Descent data for quasi-coherent sheaves · Lemma 04W4","summary":"Let S be a scheme. Let U = (U_i → U)_i ∈ I and V = (V_j → V)_j ∈ J be families of morphisms of algebraic spaces over S with fixed targets. Let (g, α : I → J, (g_i)) : U → V be a morphism of families of maps with fixed target, see Sites, Definition [Tag 00VT]. Let (F_j, φ_jj') be a descent datum for quasi-coherent sheaves with respect to the family (V_j → V)_j ∈ J. Then • The system (g_i^*F_α(i), (g_i × g_i')^*φ_α(i)α(i')) is a descent datum with respect to the family (U_i…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{U} = \\{U_i \\to U\\}_{i \\in I}$ and\n$\\mathcal{V} = \\{V_j \\to V\\}_{j \\in J}$\nbe families of morphisms of algebraic spaces over $S$ with fixed targets.\nLet $(g, \\alpha : I \\to J, (g_i)) : \\mathcal{U} \\to \\mathcal{V}$\nbe a morphism of families of maps with fixed target, see\nSites, Definition \\ref{sites-definition-morphism-coverings}.\nLet $(\\mathcal{F}_j, \\varphi_{jj'})$ be a descent\ndatum for quasi-coherent sheaves with respect to the\nfamily $\\{V_j \\to V\\}_{j \\in J}$. Then\n\\begin{enumerate}\n\\item The system\n$$\n\\left(g_i^*\\mathcal{F}_{\\alpha(i)},\n(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')}\\right)\n$$\nis a descent datum with respect to the family $\\{U_i \\to U\\}_{i \\in I}$.\n\\item This construction is functorial in the descent datum\n$(\\mathcal{F}_j, \\varphi_{jj'})$.\n\\item Given a second morphism\n$(g', \\alpha' : I \\to J, (g'_i))$\nof families of maps with fixed target\nwith $g = g'$ there exists a functorial isomorphism of descent data\n$$\n(g_i^*\\mathcal{F}_{\\alpha(i)},\n(g_i \\times g_{i'})^*\\varphi_{\\alpha(i)\\alpha(i')})\n\\cong\n((g'_i)^*\\mathcal{F}_{\\alpha'(i)},\n(g'_i \\times g'_{i'})^*\\varphi_{\\alpha'(i)\\alpha'(i')}).\n$$\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04W4","source_file":"spaces-descent.tex","source_line":101,"source_end_line":134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L101-L134","statement_sha256":"322d9985c7197a9d990da268c649a691c32a57fde2e1790c8446cb6464a39a3b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11771,"rank":11771,"depth":1,"x":352.04,"y":1689.046,"cluster":"algebraic-spaces"},{"id":"stacks:04W5","tag":"04W5","title":"Descent data for quasi-coherent sheaves · Definition 04W5","summary":"Let S be a scheme. Let (U_i → U)_i ∈ I be a family of morphisms of algebraic spaces over S with fixed target. • Let F be a quasi-coherent O_U-module. We call the unique descent on F datum with respect to the covering (U → U) the trivial descent datum. • The pullback of the trivial descent datum to (U_i → U) is called the canonical descent datum. Notation: (F|_U_i, can). • A descent datum (F_i, φ_ij) for quasi-coherent sheaves with respect to the given family is said to be…","statement_latex":"Let $S$ be a scheme.\nLet $\\{U_i \\to U\\}_{i \\in I}$ be a family of morphisms of algebraic\nspaces over $S$ with fixed target.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_U$-module.\nWe call the unique descent on $\\mathcal{F}$ datum with respect to the covering\n$\\{U \\to U\\}$ the {\\it trivial descent datum}.\n\\item The pullback of the trivial descent datum to\n$\\{U_i \\to U\\}$ is called the {\\it canonical descent datum}.\nNotation: $(\\mathcal{F}|_{U_i}, can)$.\n\\item A descent datum $(\\mathcal{F}_i, \\varphi_{ij})$\nfor quasi-coherent sheaves with respect to the given family\nis said to be {\\it effective} if there exists a quasi-coherent\nsheaf $\\mathcal{F}$ on $U$ such that $(\\mathcal{F}_i, \\varphi_{ij})$\nis isomorphic to $(\\mathcal{F}|_{U_i}, can)$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for quasi-coherent sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04W5","source_file":"spaces-descent.tex","source_line":151,"source_end_line":169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L151-L169","statement_sha256":"95ec9200c5103671ac3f1e9b0be03fdea22f4b86fce2cfe2f022f4778a9d5cda","origin":"The Stacks Project","memory_eligible":false,"source_rank":11772,"rank":11772,"depth":0,"x":68.3,"y":1603.59,"cluster":"algebraic-spaces"},{"id":"stacks:04W6","tag":"04W6","title":"Descent data for quasi-coherent sheaves · Lemma 04W6","summary":"Let S be a scheme. Let U be an algebraic space over S. Let (U_i → U) be a Zariski covering of U, see Topologies on Spaces, Definition [Tag 041G]. Any descent datum on quasi-coherent sheaves for the family U = (U_i → U) is effective. Moreover, the functor from the category of quasi-coherent O_U-modules to the category of descent data with respect to (U_i → U) is fully faithful.","statement_latex":"Let $S$ be a scheme. Let $U$ be an algebraic space over $S$.\nLet $\\{U_i \\to U\\}$ be a Zariski covering of $U$, see\nTopologies on Spaces,\nDefinition \\ref{spaces-topologies-definition-zariski-covering}.\nAny descent datum on quasi-coherent sheaves\nfor the family $\\mathcal{U} = \\{U_i \\to U\\}$ is\neffective. Moreover, the functor from the category of\nquasi-coherent $\\mathcal{O}_U$-modules to the category\nof descent data with respect to $\\{U_i \\to U\\}$ is fully faithful.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04W6","source_file":"spaces-descent.tex","source_line":171,"source_end_line":182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L171-L182","statement_sha256":"03833524e03e8217176a393fe86e37b90cb5689262a86fb0aff4486754c32f2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11773,"rank":11773,"depth":1,"x":346.421,"y":1505.541,"cluster":"algebraic-spaces"},{"id":"stacks:04W8","tag":"04W8","title":"Fpqc descent of quasi-coherent sheaves · Proposition 04W8","summary":"Let S be a scheme. Let (X_i → X) be an fpqc covering of algebraic spaces over S, see Topologies on Spaces, Definition [Tag 03MQ]. Any descent datum on quasi-coherent sheaves for (X_i → X) is effective. Moreover, the functor from the category of quasi-coherent O_X-modules to the category of descent data with respect to (X_i → X) is fully faithful.","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to X\\}$ be an fpqc covering of algebraic spaces over $S$, see\nTopologies on Spaces,\nDefinition \\ref{spaces-topologies-definition-fpqc-covering}.\nAny descent datum on quasi-coherent sheaves\nfor $\\{X_i \\to X\\}$ is effective.\nMoreover, the functor from the category of\nquasi-coherent $\\mathcal{O}_X$-modules to the category\nof descent data with respect to $\\{X_i \\to X\\}$ is fully faithful.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Fpqc descent of quasi-coherent sheaves","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04W8","source_file":"spaces-descent.tex","source_line":207,"source_end_line":218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L207-L218","statement_sha256":"ea453d6d1db1d8803d37aac3daede756000ca4367dd38003a77a75a1a2798bc9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11774,"rank":11774,"depth":51,"x":220.108,"y":1735.798,"cluster":"algebraic-spaces"},{"id":"stacks:0H04","tag":"0H04","title":"Quasi-coherent modules and affines · Lemma 0H04","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a presheaf of O_X-modules on X_affine, etale. The following are equivalent • for every morphism U → U' of X_affine, etale the map F(U') ⊗_O_X(U') O_X(U) → F(U) is an isomorphism, • F is a quasi-coherent module on the ringed site (X_affine, etale, O_X) in the sense of Modules on Sites, Definition [Tag 03DL], • F corresponds to a quasi-coherent module on X via the equivalence ([Tag 0H03]),","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_X$-modules\non $X_{affine, \\etale}$. The following are equivalent\n\\begin{enumerate}\n\\item for every morphism $U \\to U'$ of $X_{affine, \\etale}$ the map\n$\\mathcal{F}(U') \\otimes_{\\mathcal{O}_X(U')} \\mathcal{O}_X(U)\n\\to \\mathcal{F}(U)$ is an isomorphism,\n\\item $\\mathcal{F}$ is a quasi-coherent module on the ringed site\n$(X_{affine, \\etale}, \\mathcal{O}_X)$ in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local},\n\\item $\\mathcal{F}$ corresponds to a quasi-coherent module on $X$\nvia the equivalence (\\ref{equation-alternative-small-ringed}),\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Quasi-coherent modules and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H04","source_file":"spaces-descent.tex","source_line":305,"source_end_line":320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L305-L320","statement_sha256":"4573510fd7e1b849276c35037f797bc466e6758ccb94ec8ef4ca2c53e13f0e37","origin":"The Stacks Project","memory_eligible":false,"source_rank":11775,"rank":11775,"depth":13,"x":128.026,"y":1494.186,"cluster":"algebraic-spaces"},{"id":"stacks:060U","tag":"060U","title":"Descent of finiteness properties of modules · Lemma 060U","summary":"Let X be an algebraic space over a scheme S. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is a finite type O_X_i-module. Then F is a finite type O_X-module.","statement_latex":"Let $X$ be an algebraic space over a scheme $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is a finite type $\\mathcal{O}_{X_i}$-module.\nThen $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060U","source_file":"spaces-descent.tex","source_line":429,"source_end_line":436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L429-L436","statement_sha256":"2c7d52c224de3fb486a0259a4974f619c5c17d076df594b89249dc967a871050","origin":"The Stacks Project","memory_eligible":false,"source_rank":11776,"rank":11776,"depth":4,"x":390.387,"y":1620.176,"cluster":"algebraic-spaces"},{"id":"stacks:060V","tag":"060V","title":"Descent of finiteness properties of modules · Lemma 060V","summary":"Let X be an algebraic space over a scheme S. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is an O_X_i-module of finite presentation. Then F is an O_X-module of finite presentation.","statement_latex":"Let $X$ be an algebraic space over a scheme $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is an $\\mathcal{O}_{X_i}$-module of finite\npresentation. Then $\\mathcal{F}$ is an $\\mathcal{O}_X$-module\nof finite presentation.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060V","source_file":"spaces-descent.tex","source_line":444,"source_end_line":452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L444-L452","statement_sha256":"ecab82838774239ee3157906f1a1b0b62b57ad03f396b9d74476ba30258b3f50","origin":"The Stacks Project","memory_eligible":false,"source_rank":11777,"rank":11777,"depth":4,"x":95.424,"y":1676.177,"cluster":"algebraic-spaces"},{"id":"stacks:060W","tag":"060W","title":"Descent of finiteness properties of modules · Lemma 060W","summary":"Let X be an algebraic space over a scheme S. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is a flat O_X_i-module. Then F is a flat O_X-module.","statement_latex":"Let $X$ be an algebraic space over a scheme $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is a flat $\\mathcal{O}_{X_i}$-module.\nThen $\\mathcal{F}$ is a flat $\\mathcal{O}_X$-module.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060W","source_file":"spaces-descent.tex","source_line":460,"source_end_line":467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L460-L467","statement_sha256":"130a4492763252d0bfe4698a5f0af92082cc7d69c39ef852d2c592d782f8a04c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11778,"rank":11778,"depth":4,"x":267.998,"y":1467.384,"cluster":"algebraic-spaces"},{"id":"stacks:060X","tag":"060X","title":"Descent of finiteness properties of modules · Lemma 060X","summary":"Let X be an algebraic space over a scheme S. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is a finite locally free O_X_i-module. Then F is a finite locally free O_X-module.","statement_latex":"Let $X$ be an algebraic space over a scheme $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is a finite locally free $\\mathcal{O}_{X_i}$-module.\nThen $\\mathcal{F}$ is a finite locally free $\\mathcal{O}_X$-module.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060X","source_file":"spaces-descent.tex","source_line":475,"source_end_line":482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L475-L482","statement_sha256":"8994f3c03b6150c73d17ee02adeabd709b55e503a2b1e0c3cb89857b5b2a008f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11779,"rank":11779,"depth":6,"x":308.676,"y":1719.424,"cluster":"algebraic-spaces"},{"id":"stacks:060Y","tag":"060Y","title":"Descent of finiteness properties of modules · Lemma 060Y","summary":"Let X be an algebraic space over a scheme S. Let F be a quasi-coherent O_X-module. Let (f_i : X_i → X)_i ∈ I be an fpqc covering such that each f_i^*F is a locally projective O_X_i-module. Then F is a locally projective O_X-module.","statement_latex":"Let $X$ be an algebraic space over a scheme $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $\\{f_i : X_i \\to X\\}_{i \\in I}$ be an fpqc covering such that\neach $f_i^*\\mathcal{F}$ is a locally projective $\\mathcal{O}_{X_i}$-module.\nThen $\\mathcal{F}$ is a locally projective $\\mathcal{O}_X$-module.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060Y","source_file":"spaces-descent.tex","source_line":496,"source_end_line":503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L496-L503","statement_sha256":"dec4901cd2c0c1e144cc7dfbacae5f8a09d623a756e997d8e73e7d19e6273206","origin":"The Stacks Project","memory_eligible":false,"source_rank":11780,"rank":11780,"depth":12,"x":75.853,"y":1556.554,"cluster":"algebraic-spaces"},{"id":"stacks:060Z","tag":"060Z","title":"Descent of finiteness properties of modules · Lemma 060Z","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Assume f is a finite morphism. Then F is an O_X-module of finite type if and only if f_*F is an O_Y-module of finite type.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $f$ is a finite morphism.\nThen $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type\nif and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite\ntype.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060Z","source_file":"spaces-descent.tex","source_line":515,"source_end_line":524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L515-L524","statement_sha256":"ae00a3d40c9d30ba1732be33de703c2453f6ff6ba43a7d10bfecf065a9ae4fd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11781,"rank":11781,"depth":5,"x":378.696,"y":1544.535,"cluster":"algebraic-spaces"},{"id":"stacks:0610","tag":"0610","title":"Descent of finiteness properties of modules · Lemma 0610","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Assume f is finite and of finite presentation. Then F is an O_X-module of finite presentation if and only if f_*F is an O_Y-module of finite presentation.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $f$ is finite and of finite presentation.\nThen $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation\nif and only if $f_*\\mathcal{F}$ is an $\\mathcal{O}_Y$-module of finite\npresentation.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness properties of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0610","source_file":"spaces-descent.tex","source_line":537,"source_end_line":546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L537-L546","statement_sha256":"1bbfbbd79cf2dbbab9090609008c56b22f37ffba0a9571e6f9928e29b30c032f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11782,"rank":11782,"depth":8,"x":164.917,"y":1725.35,"cluster":"algebraic-spaces"},{"id":"stacks:04P1","tag":"04P1","title":"Fpqc coverings · Lemma 04P1","summary":"Let S be a scheme. Let (f_i : T_i → T)_i ∈ I be an fpqc covering of algebraic spaces over S. Suppose that for each i we have an open subspace W_i ⊂ T_i such that for all i, j ∈ I we have pr_0^-1(W_i) = pr_1^-1(W_j) as open subspaces of T_i ×_T T_j. Then there exists a unique open subspace W ⊂ T such that W_i = f_i^-1(W) for each i.","statement_latex":"Let $S$ be a scheme.\nLet $\\{f_i : T_i \\to T\\}_{i \\in I}$ be an fpqc covering\nof algebraic spaces over $S$.\nSuppose that for each $i$ we have an open subspace $W_i \\subset T_i$\nsuch that for all $i, j \\in I$ we have\n$\\text{pr}_0^{-1}(W_i) = \\text{pr}_1^{-1}(W_j)$ as open\nsubspaces of $T_i \\times_T T_j$. Then there exists a unique open subspace\n$W \\subset T$ such that $W_i = f_i^{-1}(W)$ for each $i$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Fpqc coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04P1","source_file":"spaces-descent.tex","source_line":573,"source_end_line":583,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L573-L583","statement_sha256":"5b5ab1c1ed9b7c32212bbad7a5ef3c11c160f65492453487310017d158f3d380","origin":"The Stacks Project","memory_eligible":false,"source_rank":11783,"rank":11783,"depth":5,"x":177.155,"y":1470.559,"cluster":"algebraic-spaces"},{"id":"stacks:04P2","tag":"04P2","title":"Fpqc coverings · Lemma 04P2","summary":"Let S be a scheme. Let (T_i → T) be an fpqc covering of algebraic spaces over S, see Topologies on Spaces, Definition [Tag 03MQ]. Then given an algebraic space B over S the sequence xymatrix Mor_S(T, B) ar[r] & ∏_i Mor_S(T_i, B) ar@<1ex>[r] ar@<-1ex>[r] & ∏_i, j Mor_S(T_i ×_T T_j, B) is an equalizer diagram. In other words, every representable functor on the category of algebraic spaces over S satisfies the sheaf condition for fpqc coverings.","statement_latex":"Let $S$ be a scheme. Let $\\{T_i \\to T\\}$ be an fpqc covering of algebraic\nspaces over $S$, see Topologies on Spaces, Definition\n\\ref{spaces-topologies-definition-fpqc-covering}.\nThen given an algebraic space $B$ over $S$ the sequence\n$$\n\\xymatrix{\n\\Mor_S(T, B) \\ar[r] &\n\\prod\\nolimits_i \\Mor_S(T_i, B) \\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\prod\\nolimits_{i, j} \\Mor_S(T_i \\times_T T_j, B)\n}\n$$\nis an equalizer diagram.\nIn other words, every representable functor on the category of\nalgebraic spaces over $S$ satisfies the sheaf condition for\nfpqc coverings.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Fpqc coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04P2","source_file":"spaces-descent.tex","source_line":602,"source_end_line":619,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L602-L619","statement_sha256":"fe43cb9679a0e5ca3974e944093d07aa47647fa1a05b35144212cbf8c5f09271","origin":"The Stacks Project","memory_eligible":false,"source_rank":11784,"rank":11784,"depth":54,"x":373.148,"y":1665.503,"cluster":"algebraic-spaces"},{"id":"stacks:06NR","tag":"06NR","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 06NR","summary":"Let S be a scheme. Let X → Y → Z be morphism of algebraic spaces. Let P be one of the following properties of morphisms of algebraic spaces over S: flat, locally finite type, locally finite presentation. Assume that X → Z has P and that X → Y is a surjection of sheaves on (Sch/S)_fppf. Then Y → Z is P.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphism of algebraic spaces.\nLet $P$ be one of the following properties of morphisms of algebraic spaces\nover $S$:\nflat, locally finite type, locally finite presentation.\nAssume that $X \\to Z$ has $P$ and that\n$X \\to Y$ is a surjection of sheaves on $(\\Sch/S)_{fppf}$.\nThen $Y \\to Z$ is $P$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06NR","source_file":"spaces-descent.tex","source_line":664,"source_end_line":673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L664-L673","statement_sha256":"39998fb9736843ce27f90b53211682e3001d23beafcd59a0759817fb796f8e4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11785,"rank":11785,"depth":48,"x":71.672,"y":1632.945,"cluster":"algebraic-spaces"},{"id":"stacks:0AHC","tag":"0AHC","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 0AHC","summary":"Let S be a scheme. Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & B be a commutative diagram of morphisms of algebraic spaces over S. Assume that f is surjective, flat, and locally of finite presentation and assume that p is locally of finite presentation (resp. locally of finite type). Then q is locally of finite presentation (resp. locally of finite type).","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& B\n}\n$$\nbe a commutative diagram of morphisms of algebraic spaces over $S$.\nAssume that $f$ is surjective, flat, and locally of finite presentation\nand assume that $p$ is locally of finite presentation (resp.\\ locally\nof finite type). Then $q$ is locally of finite presentation\n(resp.\\ locally of finite type).","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHC","source_file":"spaces-descent.tex","source_line":701,"source_end_line":716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L701-L716","statement_sha256":"f6641403b114b1445869546ff73392368423a87f85830e065fa101e0e609ba8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11786,"rank":11786,"depth":49,"x":320.311,"y":1485.797,"cluster":"algebraic-spaces"},{"id":"stacks:0AHD","tag":"0AHD","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 0AHD","summary":"Let S be a scheme. Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & B be a commutative diagram of morphisms of algebraic spaces over S. Assume that • f is surjective, and syntomic (resp. smooth, resp. étale), • p is syntomic (resp. smooth, resp. étale). Then q is syntomic (resp. smooth, resp. étale).","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& B\n}\n$$\nbe a commutative diagram of morphisms of algebraic spaces over $S$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is surjective, and syntomic (resp.\\ smooth, resp.\\ \\'etale),\n\\item $p$ is syntomic (resp.\\ smooth, resp.\\ \\'etale).\n\\end{enumerate}\nThen $q$ is syntomic (resp.\\ smooth, resp.\\ \\'etale).","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHD","source_file":"spaces-descent.tex","source_line":738,"source_end_line":755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L738-L755","statement_sha256":"6d8c9552bfea6784f279426c51585143c67f6d16bf4dc09bb3910779434423ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":11787,"rank":11787,"depth":48,"x":255.26,"y":1735.539,"cluster":"algebraic-spaces"},{"id":"stacks:0AHE","tag":"0AHE","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 0AHE","summary":"Let S be a scheme. Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & B be a commutative diagram of morphisms of algebraic spaces over S. Assume that • f is surjective, flat, and locally of finite presentation, • p is smooth (resp. étale). Then q is smooth (resp. étale).","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& B\n}\n$$\nbe a commutative diagram of morphisms of algebraic spaces over $S$. Assume that\n\\begin{enumerate}\n\\item $f$ is surjective, flat, and locally of finite presentation,\n\\item $p$ is smooth (resp.\\ \\'etale).\n\\end{enumerate}\nThen $q$ is smooth (resp.\\ \\'etale).","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHE","source_file":"spaces-descent.tex","source_line":777,"source_end_line":793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L777-L793","statement_sha256":"8a9989a66a1f7d5067e01bba26c33ddb7f7cc632dbcdfba4b2ab7622c11688d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11788,"rank":11788,"depth":48,"x":102.298,"y":1514.336,"cluster":"algebraic-spaces"},{"id":"stacks:0AHF","tag":"0AHF","title":"Descent of finiteness and smoothness properties of morphisms · Lemma 0AHF","summary":"Let S be a scheme. Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & B be a commutative diagram of morphisms of algebraic spaces over S. Assume that • f is surjective, flat, and locally of finite presentation, • p is syntomic. Then both q and f are syntomic.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & &\nY \\ar[dl]^q \\\\\n& B\n}\n$$\nbe a commutative diagram of morphisms of algebraic spaces over $S$. Assume that\n\\begin{enumerate}\n\\item $f$ is surjective, flat, and locally of finite presentation,\n\\item $p$ is syntomic.\n\\end{enumerate}\nThen both $q$ and $f$ are syntomic.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent of finiteness and smoothness properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHF","source_file":"spaces-descent.tex","source_line":811,"source_end_line":827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L811-L827","statement_sha256":"034e7d57f9746b8e581ab2c6034aecd37d346e42cc4863697ff8fc8d74ea8998","origin":"The Stacks Project","memory_eligible":false,"source_rank":11789,"rank":11789,"depth":48,"x":393.154,"y":1590.7,"cluster":"algebraic-spaces"},{"id":"stacks:06DQ","tag":"06DQ","title":"Descending properties of spaces · Lemma 06DQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let x ∈ |X|. If f is flat at x and X is geometrically unibranch at x, then Y is geometrically unibranch at f(x).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $x \\in |X|$.\nIf $f$ is flat at $x$ and $X$ is geometrically unibranch at $x$, then $Y$ is\ngeometrically unibranch at $f(x)$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DQ","source_file":"spaces-descent.tex","source_line":859,"source_end_line":866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L859-L866","statement_sha256":"ad9125c52ce6fd6a74b255e716bb8c575c865ef73522cab15a101fc043d18c81","origin":"The Stacks Project","memory_eligible":false,"source_rank":11790,"rank":11790,"depth":54,"x":117.102,"y":1699.496,"cluster":"algebraic-spaces"},{"id":"stacks:06MI","tag":"06MI","title":"Descending properties of spaces · Lemma 06MI","summary":"A flat and surjective morphism of algebraic spaces with a reduced source has a reduced target. Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is flat and surjective and X is reduced, then Y is reduced.","statement_latex":"\\begin{slogan}\nA flat and surjective morphism of algebraic spaces with a reduced source\nhas a reduced target.\n\\end{slogan}\nLet $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is flat and surjective and $X$ is reduced, then $Y$ is reduced.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MI","source_file":"spaces-descent.tex","source_line":879,"source_end_line":888,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L879-L888","statement_sha256":"432d98b74f4cd1398995ef99ff7cf8d0fbb63416e60208a7356906d2a6d74579","origin":"The Stacks Project","memory_eligible":false,"source_rank":11791,"rank":11791,"depth":3,"x":233.239,"y":1462.488,"cluster":"algebraic-spaces"},{"id":"stacks:06MJ","tag":"06MJ","title":"Descending properties of spaces · Lemma 06MJ","summary":"Let f : X → Y be a morphism of algebraic spaces. If f is locally of finite presentation, flat, and surjective and X is locally Noetherian, then Y is locally Noetherian.","statement_latex":"Let $f : X \\to Y$ be a morphism of algebraic spaces.\nIf $f$ is locally of finite presentation, flat, and surjective and\n$X$ is locally Noetherian, then $Y$ is locally Noetherian.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MJ","source_file":"spaces-descent.tex","source_line":903,"source_end_line":908,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L903-L908","statement_sha256":"4203161c802ec2bda47fe16000f7952d8b6aff6ffa7402ca1043a627de1646fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11792,"rank":11792,"depth":38,"x":338.261,"y":1703.3,"cluster":"algebraic-spaces"},{"id":"stacks:06MK","tag":"06MK","title":"Descending properties of spaces · Lemma 06MK","summary":"Let f : X → Y be a morphism of algebraic spaces. If f is locally of finite presentation, flat, and surjective and X is regular, then Y is regular.","statement_latex":"Let $f : X \\to Y$ be a morphism of algebraic spaces.\nIf $f$ is locally of finite presentation, flat, and surjective and\n$X$ is regular, then $Y$ is regular.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MK","source_file":"spaces-descent.tex","source_line":924,"source_end_line":929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L924-L929","statement_sha256":"c1d8ec0e180cf55ded5a2bf8dd756a5b71af04dd8a70942a92219ca83b733249","origin":"The Stacks Project","memory_eligible":false,"source_rank":11793,"rank":11793,"depth":39,"x":67.0,"y":1585.249,"cluster":"algebraic-spaces"},{"id":"stacks:0GB3","tag":"0GB3","title":"Descending properties of spaces · Lemma 0GB3","summary":"Let f : X → Y be a smooth morphism of algebraic spaces. If Y is reduced, then X is reduced. If f is surjective and X is reduced, then Y is reduced.","statement_latex":"Let $f : X \\to Y$ be a smooth morphism of algebraic spaces.\nIf $Y$ is reduced, then $X$ is reduced. If $f$ is surjective\nand $X$ is reduced, then $Y$ is reduced.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GB3","source_file":"spaces-descent.tex","source_line":947,"source_end_line":952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L947-L952","statement_sha256":"7e61676f8826c55c0d50f6521a725ee04d1501cbef1fb556c4a7530258c1588c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11794,"rank":11794,"depth":38,"x":362.137,"y":1518.337,"cluster":"algebraic-spaces"},{"id":"stacks:03YH","tag":"03YH","title":"Descending properties of morphisms · Definition 03YH","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S. Let τ ∈ (fpqc, fppf, syntomic, smooth, etale). We say P is τ local on the base, or τ local on the target, or local on the base for the τ-topology if for any τ-covering (Y_i → Y)_i ∈ I of algebraic spaces and any morphism of algebraic spaces f : X → Y we have f has P ⇔ each Y_i ×_Y X → Y_i has P.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.\nLet $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale\\}$.\nWe say $\\mathcal{P}$ is {\\it $\\tau$ local on the base}, or\n{\\it $\\tau$ local on the target}, or\n{\\it local on the base for the $\\tau$-topology} if for any\n$\\tau$-covering $\\{Y_i \\to Y\\}_{i \\in I}$ of algebraic spaces\nand any morphism of algebraic spaces $f : X \\to Y$ we\nhave\n$$\nf \\text{ has }\\mathcal{P}\n\\Leftrightarrow\n\\text{each }Y_i \\times_Y X \\to Y_i\\text{ has }\\mathcal{P}.\n$$","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YH","source_file":"spaces-descent.tex","source_line":991,"source_end_line":1007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L991-L1007","statement_sha256":"90423f488a9b09f91181dfd348c8cf5aef28a3b63f9be6329c0b43e7777f2e7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11795,"rank":11795,"depth":0,"x":198.216,"y":1735.277,"cluster":"algebraic-spaces"},{"id":"stacks:06EM","tag":"06EM","title":"Descending properties of morphisms · Lemma 06EM","summary":"Let S be a scheme. Let τ ∈ (fpqc, fppf, syntomic, smooth, etale). Let P be a property of morphisms of algebraic spaces over S which is τ local on the target. Let f : X → Y have property P. For any morphism Y' → Y which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. étale, the base change f' : Y' ×_Y X → Y' of f has property P.","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{fpqc, fppf, syntomic, smooth, \\etale\\}$.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$\nwhich is $\\tau$ local on the target. Let $f : X \\to Y$ have property\n$\\mathcal{P}$. For any morphism $Y' \\to Y$ which is\nflat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic,\nresp.\\ \\'etale, the base change\n$f' : Y' \\times_Y X \\to Y'$ of $f$ has property $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EM","source_file":"spaces-descent.tex","source_line":1017,"source_end_line":1027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1017-L1027","statement_sha256":"1321a007a177f83f0e199321638c6a6412545e3b3354a8c4d8614e8db9c57062","origin":"The Stacks Project","memory_eligible":false,"source_rank":11796,"rank":11796,"depth":0,"x":144.6,"y":1482.142,"cluster":"algebraic-spaces"},{"id":"stacks:06R2","tag":"06R2","title":"Descending properties of morphisms · Lemma 06R2","summary":"Let S be a scheme. Let τ ∈ (fppf, syntomic, smooth, etale). Let P be a property of morphisms of algebraic spaces over S which is τ local on the target. For any morphism of algebraic spaces f : X → Y over S there exists a largest open subspace W(f) ⊂ Y such that the restriction X_W(f) → W(f) has P. Moreover, • if g : Y' → Y is a morphism of algebraic spaces which is flat and locally of finite presentation, syntomic, smooth, or étale and the base change f' : X_Y' → Y' has…","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{fppf, syntomic, smooth, \\etale\\}$.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$\nwhich is $\\tau$ local on the target. For any morphism of algebraic spaces\n$f : X \\to Y$ over $S$ there exists a largest open subspace\n$W(f) \\subset Y$ such that the restriction $X_{W(f)} \\to W(f)$ has\n$\\mathcal{P}$. Moreover,\n\\begin{enumerate}\n\\item if $g : Y' \\to Y$ is a morphism of algebraic spaces which is\nflat and locally of finite presentation, syntomic, smooth, or \\'etale\nand the base change $f' : X_{Y'} \\to Y'$ has $\\mathcal{P}$, then\n$g$ factors through $W(f)$,\n\\item if $g : Y' \\to Y$ is flat and locally of finite presentation,\nsyntomic, smooth, or \\'etale, then $W(f') = g^{-1}(W(f))$, and\n\\item if $\\{g_i : Y_i \\to Y\\}$ is a $\\tau$-covering, then\n$g_i^{-1}(W(f)) = W(f_i)$, where $f_i$ is the base change of $f$\nby $Y_i \\to Y$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R2","source_file":"spaces-descent.tex","source_line":1042,"source_end_line":1062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1042-L1062","statement_sha256":"970e19f0b18c7da912615a5078b4ef73a2ebe48cecbe0f85819cf22ef0724b77","origin":"The Stacks Project","memory_eligible":false,"source_rank":11797,"rank":11797,"depth":48,"x":387.846,"y":1638.472,"cluster":"algebraic-spaces"},{"id":"stacks:041J","tag":"041J","title":"Descending properties of morphisms · Lemma 041J","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S. Assume • if X_i → Y_i, i = 1, 2 have property P so does X_1 amalg X_2 → Y_1 amalg Y_2, • a morphism of algebraic spaces f : X → Y has property P if and only if for every affine scheme Z and morphism Z → Y the base change Z ×_Y X → Z of f has property P, and • for any surjective flat morphism of affine schemes Z' → Z over S and a morphism f : X → Z from an algebraic space to Z we have f' : Z'…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{P}$ be a property of morphisms of\nalgebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item if $X_i \\to Y_i$, $i = 1, 2$ have property $\\mathcal{P}$ so\ndoes $X_1 \\amalg X_2 \\to Y_1 \\amalg Y_2$,\n\\item a morphism of algebraic spaces $f : X \\to Y$ has property\n$\\mathcal{P}$ if and only if for every affine scheme $Z$ and\nmorphism $Z \\to Y$ the base change $Z \\times_Y X \\to Z$ of $f$\nhas property $\\mathcal{P}$, and\n\\item for any surjective flat morphism of affine schemes\n$Z' \\to Z$ over $S$ and a morphism $f : X \\to Z$ from an algebraic space\nto $Z$ we have\n$$\nf' : Z' \\times_Z X \\to Z'\\text{ has }\\mathcal{P}\n\\Rightarrow\nf\\text{ has }\\mathcal{P}.\n$$\n\\end{enumerate}\nThen $\\mathcal{P}$ is fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041J","source_file":"spaces-descent.tex","source_line":1094,"source_end_line":1115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1094-L1115","statement_sha256":"0d1bcbfb95940df7c7c1cadf54ae4d742bcc689e68a54d7d602df9ca2330e49d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11798,"rank":11798,"depth":1,"x":82.579,"y":1661.235,"cluster":"algebraic-spaces"},{"id":"stacks:041L","tag":"041L","title":"Descending properties of morphisms in the fpqc topology · Lemma 041L","summary":"Let S be a scheme. The property P(f) =\"f is quasi-compact\" is fpqc local on the base on algebraic spaces over S.","statement_latex":"Let $S$ be a scheme.\nThe property $\\mathcal{P}(f) =$``$f$ is quasi-compact''\nis fpqc local on the base on algebraic spaces over $S$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041L","source_file":"spaces-descent.tex","source_line":1162,"source_end_line":1167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1162-L1167","statement_sha256":"3e5125851adf2d84c15f7e56e1ce396236f697138802a2af1d1ca9fb2d430103","origin":"The Stacks Project","memory_eligible":false,"source_rank":11799,"rank":11799,"depth":43,"x":289.498,"y":1471.119,"cluster":"algebraic-spaces"},{"id":"stacks:041N","tag":"041N","title":"Descending properties of morphisms in the fpqc topology · Lemma 041N","summary":"Let S be a scheme. The property P(f) =\"f is quasi-separated\" is fpqc local on the base on algebraic spaces over S.","statement_latex":"Let $S$ be a scheme.\nThe property $\\mathcal{P}(f) =$``$f$ is quasi-separated''\nis fpqc local on the base on algebraic spaces over $S$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041N","source_file":"spaces-descent.tex","source_line":1210,"source_end_line":1215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1210-L1215","statement_sha256":"aced3818a4e50e284790c3c352612364863a159b9510bd4faf1fe7f5bb442d6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11800,"rank":11800,"depth":44,"x":289.806,"y":1728.874,"cluster":"algebraic-spaces"},{"id":"stacks:041O","tag":"041O","title":"Descending properties of morphisms in the fpqc topology · Lemma 041O","summary":"Let S be a scheme. The property P(f) =\"f is universally closed\" is fpqc local on the base on algebraic spaces over S.","statement_latex":"Let $S$ be a scheme.\nThe property $\\mathcal{P}(f) =$``$f$ is universally closed''\nis fpqc local on the base on algebraic spaces over $S$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041O","source_file":"spaces-descent.tex","source_line":1245,"source_end_line":1250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1245-L1250","statement_sha256":"fbab2fd5bcfaf349f29d98caec0b3b514843383270ca123c2dbbda22f898591a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11801,"rank":11801,"depth":47,"x":82.174,"y":1538.869,"cluster":"algebraic-spaces"},{"id":"stacks:041P","tag":"041P","title":"Descending properties of morphisms in the fpqc topology · Lemma 041P","summary":"Let S be a scheme. The property P(f) =\"f is universally open\" is fpqc local on the base on algebraic spaces over S.","statement_latex":"Let $S$ be a scheme.\nThe property $\\mathcal{P}(f) =$``$f$ is universally open''\nis fpqc local on the base on algebraic spaces over $S$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041P","source_file":"spaces-descent.tex","source_line":1284,"source_end_line":1289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1284-L1289","statement_sha256":"0aafc9b68cb15d507f00c560e880da7e70e8ad01b06720344c5bed74747e170c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11802,"rank":11802,"depth":48,"x":388.26,"y":1561.174,"cluster":"algebraic-spaces"},{"id":"stacks:0CFW","tag":"0CFW","title":"Descending properties of morphisms in the fpqc topology · Lemma 0CFW","summary":"The property P(f) =\"f is universally submersive\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is universally submersive''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CFW","source_file":"spaces-descent.tex","source_line":1296,"source_end_line":1300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1296-L1300","statement_sha256":"96fdb5e7c220df003df94ef12262d8740e34fccd9ee547b9092b117ddb968732","origin":"The Stacks Project","memory_eligible":false,"source_rank":11803,"rank":11803,"depth":48,"x":144.473,"y":1718.502,"cluster":"algebraic-spaces"},{"id":"stacks:041Q","tag":"041Q","title":"Descending properties of morphisms in the fpqc topology · Lemma 041Q","summary":"The property P(f) =\"f is surjective\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is surjective''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041Q","source_file":"spaces-descent.tex","source_line":1307,"source_end_line":1311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1307-L1311","statement_sha256":"3e15938fd4adf118c868b89240c680f1ad264443e154f43d8eb3fdfae7565b11","origin":"The Stacks Project","memory_eligible":false,"source_rank":11804,"rank":11804,"depth":1,"x":197.751,"y":1464.007,"cluster":"algebraic-spaces"},{"id":"stacks:041R","tag":"041R","title":"Descending properties of morphisms in the fpqc topology · Lemma 041R","summary":"The property P(f) =\"f is universally injective\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is universally injective''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041R","source_file":"spaces-descent.tex","source_line":1318,"source_end_line":1322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1318-L1322","statement_sha256":"545b119c25a7022b4c0600163bf79756c521a5e569655859a8f56cbc62727020","origin":"The Stacks Project","memory_eligible":false,"source_rank":11805,"rank":11805,"depth":54,"x":363.222,"y":1682.03,"cluster":"algebraic-spaces"},{"id":"stacks:0CFX","tag":"0CFX","title":"Descending properties of morphisms in the fpqc topology · Lemma 0CFX","summary":"The property P(f) =\"f is a universal homeomorphism\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a universal homeomorphism''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CFX","source_file":"spaces-descent.tex","source_line":1359,"source_end_line":1363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1359-L1363","statement_sha256":"9a15771b46f44c805fd97d06884f04a5095479f836d15f7cd2a0bf62842aeb93","origin":"The Stacks Project","memory_eligible":false,"source_rank":11806,"rank":11806,"depth":55,"x":65.7,"y":1615.115,"cluster":"algebraic-spaces"},{"id":"stacks:041S","tag":"041S","title":"Descending properties of morphisms in the fpqc topology · Lemma 041S","summary":"The property P(f) =\"f is locally of finite type\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is locally of finite type''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041S","source_file":"spaces-descent.tex","source_line":1385,"source_end_line":1389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1385-L1389","statement_sha256":"e6d8618cd13ceeae8b842a0f5ed2db263ee3a12c1eeac36e8d870f0b1d07e135","origin":"The Stacks Project","memory_eligible":false,"source_rank":11807,"rank":11807,"depth":42,"x":339.063,"y":1495.564,"cluster":"algebraic-spaces"},{"id":"stacks:041T","tag":"041T","title":"Descending properties of morphisms in the fpqc topology · Lemma 041T","summary":"The property P(f) =\"f is locally of finite presentation\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is locally of finite presentation''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041T","source_file":"spaces-descent.tex","source_line":1414,"source_end_line":1418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1414-L1418","statement_sha256":"748eb336627c26fed014da30630d2168a63b50bfffb1264d3bcc76b8c66e27e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11808,"rank":11808,"depth":38,"x":233.565,"y":1738.977,"cluster":"algebraic-spaces"},{"id":"stacks:041U","tag":"041U","title":"Descending properties of morphisms in the fpqc topology · Lemma 041U","summary":"The property P(f) =\"f is of finite type\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is of finite type''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041U","source_file":"spaces-descent.tex","source_line":1445,"source_end_line":1449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1445-L1449","statement_sha256":"cc476dc1ead67397ba86401264147629e9af75fcf4d9a77130973107df003a50","origin":"The Stacks Project","memory_eligible":false,"source_rank":11809,"rank":11809,"depth":44,"x":115.541,"y":1499.483,"cluster":"algebraic-spaces"},{"id":"stacks:041V","tag":"041V","title":"Descending properties of morphisms in the fpqc 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topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041X","source_file":"spaces-descent.tex","source_line":1497,"source_end_line":1501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1497-L1501","statement_sha256":"043281993c402db64118b4e3e04749413800e5c0fab81e113cfd0f03340d4896","origin":"The Stacks Project","memory_eligible":false,"source_rank":11812,"rank":11812,"depth":54,"x":255.37,"y":1462.286,"cluster":"algebraic-spaces"},{"id":"stacks:041Y","tag":"041Y","title":"Descending properties of morphisms in the fpqc topology · Lemma 041Y","summary":"The property P(f) =\"f is an isomorphism\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is an isomorphism''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041Y","source_file":"spaces-descent.tex","source_line":1545,"source_end_line":1549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1545-L1549","statement_sha256":"b95c624baa3a7cf8751946208ad9999eb8f600aa34918af5811a8d50afc8f7be","origin":"The Stacks Project","memory_eligible":false,"source_rank":11813,"rank":11813,"depth":55,"x":322.093,"y":1716.013,"cluster":"algebraic-spaces"},{"id":"stacks:041Z","tag":"041Z","title":"Descending properties of morphisms in the fpqc topology · Lemma 041Z","summary":"The property P(f) =\"f is affine\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is affine''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/041Z","source_file":"spaces-descent.tex","source_line":1556,"source_end_line":1560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1556-L1560","statement_sha256":"939661cde60c5ee60ad3b70d29d46f5fb213f7c0af2da1db6274c2e779ff0943","origin":"The Stacks Project","memory_eligible":false,"source_rank":11814,"rank":11814,"depth":56,"x":68.702,"y":1566.69,"cluster":"algebraic-spaces"},{"id":"stacks:0420","tag":"0420","title":"Descending properties of morphisms in the fpqc topology · Lemma 0420","summary":"The property P(f) =\"f is a closed immersion\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a closed immersion''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0420","source_file":"spaces-descent.tex","source_line":1600,"source_end_line":1604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1600-L1604","statement_sha256":"a7ac62d40c48a3defa81f2d71d5ea6d4de7800ad399237d6367d3eeb8a07fa49","origin":"The Stacks Project","memory_eligible":false,"source_rank":11815,"rank":11815,"depth":57,"x":375.812,"y":1532.998,"cluster":"algebraic-spaces"},{"id":"stacks:0421","tag":"0421","title":"Descending properties of morphisms in the fpqc topology · Lemma 0421","summary":"The property P(f) =\"f is separated\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is separated''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0421","source_file":"spaces-descent.tex","source_line":1630,"source_end_line":1634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1630-L1634","statement_sha256":"97c027685279b12307797d4294039b28bcd17764757362b3047abcc2b3aad1d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11816,"rank":11816,"depth":58,"x":176.329,"y":1732.222,"cluster":"algebraic-spaces"},{"id":"stacks:0422","tag":"0422","title":"Descending properties of morphisms in the fpqc topology · Lemma 0422","summary":"The property P(f) =\"f is proper\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is proper''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0422","source_file":"spaces-descent.tex","source_line":1665,"source_end_line":1669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1665-L1669","statement_sha256":"8fd6c941df459479d8528bc6500071620cc7b42f013a6b217946769545670682","origin":"The Stacks Project","memory_eligible":false,"source_rank":11817,"rank":11817,"depth":59,"x":163.207,"y":1471.972,"cluster":"algebraic-spaces"},{"id":"stacks:0423","tag":"0423","title":"Descending properties of morphisms in the fpqc topology · Lemma 0423","summary":"The property P(f) =\"f is quasi-affine\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is quasi-affine''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0423","source_file":"spaces-descent.tex","source_line":1678,"source_end_line":1682,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1678-L1682","statement_sha256":"c48ab1474052b839b4ee47386a2a5e2eb8a6f95e4f86478568a86dadbf09a809","origin":"The Stacks Project","memory_eligible":false,"source_rank":11818,"rank":11818,"depth":56,"x":382.298,"y":1656.539,"cluster":"algebraic-spaces"},{"id":"stacks:0424","tag":"0424","title":"Descending properties of morphisms in the fpqc topology · Lemma 0424","summary":"The property P(f) =\"f is a quasi-compact immersion\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a quasi-compact immersion''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0424","source_file":"spaces-descent.tex","source_line":1722,"source_end_line":1726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1722-L1726","statement_sha256":"70f8ab0149083ef1e2d90820453c421f6c131839bab864c3710e59952efa2e90","origin":"The Stacks Project","memory_eligible":false,"source_rank":11819,"rank":11819,"depth":57,"x":72.137,"y":1644.753,"cluster":"algebraic-spaces"},{"id":"stacks:0425","tag":"0425","title":"Descending properties of morphisms in the fpqc topology · Lemma 0425","summary":"The property P(f) =\"f is integral\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is integral''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0425","source_file":"spaces-descent.tex","source_line":1753,"source_end_line":1757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1753-L1757","statement_sha256":"39fcce1122ccba9dcb993fa69faad528be9ad30e97e9a432de8a3e1b6eb3b438","origin":"The Stacks Project","memory_eligible":false,"source_rank":11820,"rank":11820,"depth":57,"x":310.464,"y":1477.352,"cluster":"algebraic-spaces"},{"id":"stacks:0426","tag":"0426","title":"Descending properties of morphisms in the fpqc topology · Lemma 0426","summary":"The property P(f) =\"f is finite\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is finite''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0426","source_file":"spaces-descent.tex","source_line":1769,"source_end_line":1773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1769-L1773","statement_sha256":"d865de4420fc7cb74376474f1ecc7ff14f5484acff58654f61f21722f4bebe51","origin":"The Stacks Project","memory_eligible":false,"source_rank":11821,"rank":11821,"depth":58,"x":269.32,"y":1736.176,"cluster":"algebraic-spaces"},{"id":"stacks:0427","tag":"0427","title":"Descending properties of morphisms in the fpqc topology · Lemma 0427","summary":"The properties P(f) =\"f is locally quasi-finite\" and P(f) =\"f is quasi-finite\" are fpqc local on the base.","statement_latex":"The properties\n$\\mathcal{P}(f) =$``$f$ is locally quasi-finite''\nand\n$\\mathcal{P}(f) =$``$f$ is quasi-finite''\nare fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0427","source_file":"spaces-descent.tex","source_line":1784,"source_end_line":1791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1784-L1791","statement_sha256":"1052253810aa83233ef4fa4eecf46ecd3a27bb3ea7e602098691b8b910afa73c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11822,"rank":11822,"depth":44,"x":91.416,"y":1521.851,"cluster":"algebraic-spaces"},{"id":"stacks:0428","tag":"0428","title":"Descending properties of morphisms in the fpqc topology · Lemma 0428","summary":"The property P(f) =\"f is syntomic\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is syntomic''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0428","source_file":"spaces-descent.tex","source_line":1818,"source_end_line":1822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1818-L1822","statement_sha256":"653b3b3e36bf6d6cccf7f1ed62482c7c762411b98926b425ee9fdde7fc07eb3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11823,"rank":11823,"depth":39,"x":395.132,"y":1578.978,"cluster":"algebraic-spaces"},{"id":"stacks:0429","tag":"0429","title":"Descending properties of morphisms in the fpqc topology · Lemma 0429","summary":"The property P(f) =\"f is smooth\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is smooth''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0429","source_file":"spaces-descent.tex","source_line":1847,"source_end_line":1851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1847-L1851","statement_sha256":"2f19af09181afd64e555f0277fd9a5987daf5c69de8a177848dee18beccb7c0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11824,"rank":11824,"depth":41,"x":125.079,"y":1709.265,"cluster":"algebraic-spaces"},{"id":"stacks:042A","tag":"042A","title":"Descending properties of morphisms in the fpqc topology · Lemma 042A","summary":"The property P(f) =\"f is unramified\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is unramified''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042A","source_file":"spaces-descent.tex","source_line":1876,"source_end_line":1880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1876-L1880","statement_sha256":"62b59b4cafac053bd273347280008f7a19ab71928b943b2c2d3a7418f5f50d8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11825,"rank":11825,"depth":41,"x":219.493,"y":1459.812,"cluster":"algebraic-spaces"},{"id":"stacks:042B","tag":"042B","title":"Descending properties of morphisms in the fpqc topology · Lemma 042B","summary":"The property P(f) =\"f is étale\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is \\'etale''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042B","source_file":"spaces-descent.tex","source_line":1905,"source_end_line":1909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1905-L1909","statement_sha256":"01cf90a9fd8b4a49b70f1f0ca7779a5d32b45bdafa4d0e346accef2b0f6fe22b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11826,"rank":11826,"depth":47,"x":350.553,"y":1697.467,"cluster":"algebraic-spaces"},{"id":"stacks:042C","tag":"042C","title":"Descending properties of morphisms in the fpqc topology · Lemma 042C","summary":"The property P(f) =\"f is finite locally free\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is finite locally free''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042C","source_file":"spaces-descent.tex","source_line":1934,"source_end_line":1938,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1934-L1938","statement_sha256":"c7f8269ad0346cc80adeb536e3693150fcca4f13e3e8c3a50830545250d1362e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11827,"rank":11827,"depth":59,"x":62.628,"y":1596.532,"cluster":"algebraic-spaces"},{"id":"stacks:042D","tag":"042D","title":"Descending properties of morphisms in the fpqc topology · Lemma 042D","summary":"The property P(f) =\"f is a monomorphism\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a monomorphism''\nis fpqc local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fpqc topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042D","source_file":"spaces-descent.tex","source_line":1950,"source_end_line":1954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L1950-L1954","statement_sha256":"fd73e68bbabba7d4f91b64914a6c339dd3f0552869985dfa5ab698aef1319a70","origin":"The Stacks Project","memory_eligible":false,"source_rank":11828,"rank":11828,"depth":56,"x":356.28,"y":1507.532,"cluster":"algebraic-spaces"},{"id":"stacks:042U","tag":"042U","title":"Descending properties of morphisms in the fppf topology · Lemma 042U","summary":"The property P(f) =\"f is an immersion\" is fppf local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is an immersion''\nis fppf local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042U","source_file":"spaces-descent.tex","source_line":2030,"source_end_line":2034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2030-L2034","statement_sha256":"6a0d06d8a1ded3398a9cfb2d856ab1063eac535f99a912ceb32945d63e90371e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11829,"rank":11829,"depth":59,"x":211.232,"y":1739.92,"cluster":"algebraic-spaces"},{"id":"stacks:042F","tag":"042F","title":"Descending properties of morphisms in the fppf topology · Lemma 042F","summary":"The property P(f) =\"f is locally separated\" is fppf local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is locally separated''\nis fppf local on the base.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descending properties of morphisms in the fppf topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/042F","source_file":"spaces-descent.tex","source_line":2078,"source_end_line":2082,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2078-L2082","statement_sha256":"309689aa4405dc743c8712234ffc504a7532b19ded9a726bb640064b70048084","origin":"The Stacks Project","memory_eligible":false,"source_rank":11830,"rank":11830,"depth":60,"x":131.262,"y":1486.109,"cluster":"algebraic-spaces"},{"id":"stacks:0D3C","tag":"0D3C","title":"Application of descent of properties of morphisms · Lemma 0D3C","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let L be an invertible O_X-module. Let (g_i : Y_i → Y)_i ∈ I be an fpqc covering. Let f_i : X_i → Y_i be the base change of f and let L_i be the pullback of L to X_i. The following are equivalent • L is ample on X/Y, and • L_i is ample on X_i/Y_i for every i ∈ I.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $\\{g_i : Y_i \\to Y\\}_{i \\in I}$ be an fpqc covering.\nLet $f_i : X_i \\to Y_i$ be the base change of $f$ and let $\\mathcal{L}_i$\nbe the pullback of $\\mathcal{L}$ to $X_i$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{L}$ is ample on $X/Y$, and\n\\item $\\mathcal{L}_i$ is ample on $X_i/Y_i$\nfor every $i \\in I$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Application of descent of properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3C","source_file":"spaces-descent.tex","source_line":2124,"source_end_line":2138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2124-L2138","statement_sha256":"c890d2ed3a9f21c9d5cc9036f7d08687dd3e6c158659b2d12de08182c17619b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11831,"rank":11831,"depth":60,"x":394.491,"y":1627.971,"cluster":"algebraic-spaces"},{"id":"stacks:0D3D","tag":"0D3D","title":"Application of descent of properties of morphisms · Lemma 0D3D","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of algebraic spaces over S. Let L be an invertible O_X-module. There exists an open subspace V ⊂ Y characterized by the following property: A morphism Y' → Y of algebraic spaces factors through V if and only if the pullback L' of L to X' = Y' ×_Y X is ample on X'/Y' (as in Divisors on Spaces, Definition [Tag 0D31]).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThere exists an open subspace $V \\subset Y$ characterized by\nthe following property:\nA morphism $Y' \\to Y$ of algebraic spaces factors\nthrough $V$ if and only if the pullback $\\mathcal{L}'$\nof $\\mathcal{L}$ to $X' = Y' \\times_Y X$ is ample on $X'/Y'$\n(as in Divisors on Spaces, Definition\n\\ref{spaces-divisors-definition-relatively-ample}).","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Application of descent of properties of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3D","source_file":"spaces-descent.tex","source_line":2209,"source_end_line":2221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2209-L2221","statement_sha256":"57ab3ed7b48d942688c1731a6dcce3b08d0d080d843a221e2f5332cc8b83f285","origin":"The Stacks Project","memory_eligible":false,"source_rank":11832,"rank":11832,"depth":74,"x":86.13,"y":1672.753,"cluster":"algebraic-spaces"},{"id":"stacks:06EP","tag":"06EP","title":"Properties of morphisms local on the source · Definition 06EP","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S. Let τ ∈ (fpqc, linebreak[0] fppf, linebreak[0] syntomic, linebreak[0] smooth, linebreak[0] etale). We say P is τ local on the source, or local on the source for the τ-topology if for any morphism f : X → Y of algebraic spaces over S, and any τ-covering (X_i → X)_i ∈ I of algebraic spaces we have f has P ⇔ each X_i → Y has P.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.\nLet $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0]\nsmooth, \\linebreak[0] \\etale\\}$. We say $\\mathcal{P}$ is\n{\\it $\\tau$ local on the source}, or\n{\\it local on the source for the $\\tau$-topology} if for\nany morphism $f : X \\to Y$ of algebraic spaces over $S$, and any\n$\\tau$-covering $\\{X_i \\to X\\}_{i \\in I}$ of algebraic spaces we have\n$$\nf \\text{ has }\\mathcal{P}\n\\Leftrightarrow\n\\text{each }X_i \\to Y\\text{ has }\\mathcal{P}.\n$$","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local on the source","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EP","source_file":"spaces-descent.tex","source_line":2318,"source_end_line":2333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2318-L2333","statement_sha256":"e1b3b6918081c53616a3a65cd320b44f23e5f4e113d20f5bc83e6bde81f8f75c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11833,"rank":11833,"depth":0,"x":277.61,"y":1464.639,"cluster":"algebraic-spaces"},{"id":"stacks:06EQ","tag":"06EQ","title":"Properties of morphisms local on the source · Lemma 06EQ","summary":"Let S be a scheme. Let τ ∈ (fpqc, linebreak[0] fppf, linebreak[0] syntomic, linebreak[0] smooth, linebreak[0] etale). Let P be a property of morphisms of algebraic spaces over S which is τ local on the source. Let f : X → Y have property P. For any morphism a : X' → X which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. smooth, resp. étale, the composition f ∘ a : X' → Y has property P.","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0]\nsmooth, \\linebreak[0] \\etale\\}$.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$\nwhich is $\\tau$ local on the source. Let $f : X \\to Y$ have property\n$\\mathcal{P}$. For any morphism $a : X' \\to X$ which is\nflat, resp.\\ flat and locally of finite presentation, resp.\\ syntomic,\nresp.\\ smooth, resp.\\ \\'etale, the composition $f \\circ a : X' \\to Y$ has\nproperty $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EQ","source_file":"spaces-descent.tex","source_line":2343,"source_end_line":2354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2343-L2354","statement_sha256":"7dc563693f19010d9d277a5558bf7a8a4563d1fa1921890c357dac2759721e04","origin":"The Stacks Project","memory_eligible":false,"source_rank":11834,"rank":11834,"depth":0,"x":303.789,"y":1726.901,"cluster":"algebraic-spaces"},{"id":"stacks:06ER","tag":"06ER","title":"Properties of morphisms local on the source · Lemma 06ER","summary":"Let S be a scheme. Let τ ∈ (fpqc, linebreak[0] fppf, linebreak[0] syntomic, linebreak[0] smooth, linebreak[0] etale). Suppose that P is a property of morphisms of schemes over S which is étale local on the source-and-target. Denote P_spaces the corresponding property of morphisms of algebraic spaces over S, see Morphisms of Spaces, Definition [Tag 04RD]. If P is local on the source for the τ-topology, then P_spaces is local on the source for the τ-topology.","statement_latex":"Let $S$ be a scheme.\nLet $\\tau \\in \\{fpqc, \\linebreak[0] fppf, \\linebreak[0] syntomic, \\linebreak[0]\nsmooth, \\linebreak[0] \\etale\\}$.\nSuppose that $\\mathcal{P}$ is a property of morphisms of schemes over $S$\nwhich is \\'etale local on the source-and-target. Denote $\\mathcal{P}_{spaces}$\nthe corresponding property of morphisms of algebraic spaces over $S$, see\nMorphisms of Spaces, Definition \\ref{spaces-morphisms-definition-P}.\nIf $\\mathcal{P}$ is local on the source for the $\\tau$-topology, then\n$\\mathcal{P}_{spaces}$ is local on the source for the $\\tau$-topology.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ER","source_file":"spaces-descent.tex","source_line":2361,"source_end_line":2372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2361-L2372","statement_sha256":"e28241e787f71b531886de70f2d68f897ad9c8527458a7d3bbc3a068a8933fcc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11835,"rank":11835,"depth":47,"x":73.448,"y":1548.265,"cluster":"algebraic-spaces"},{"id":"stacks:06ET","tag":"06ET","title":"Properties of morphisms local in the fpqc topology on the source · Lemma 06ET","summary":"The property P(f)=\"f is flat\" is fpqc local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is flat'' is fpqc local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the fpqc topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ET","source_file":"spaces-descent.tex","source_line":2417,"source_end_line":2420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2417-L2420","statement_sha256":"e4d2afff810aacef13a02a4587dbcf293c471d182edbf795b908b606d5066287","origin":"The Stacks Project","memory_eligible":false,"source_rank":11836,"rank":11836,"depth":48,"x":387.134,"y":1549.288,"cluster":"algebraic-spaces"},{"id":"stacks:06EV","tag":"06EV","title":"Properties of morphisms local in the fppf topology on the source · Lemma 06EV","summary":"The property P(f)=\"f is locally of finite presentation\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is locally of finite presentation''\nis fppf local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the fppf topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EV","source_file":"spaces-descent.tex","source_line":2446,"source_end_line":2450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2446-L2450","statement_sha256":"364c47c877e8745394e6833ee79a416b51899399917e3dae84e90bef0d1bfe92","origin":"The Stacks Project","memory_eligible":false,"source_rank":11837,"rank":11837,"depth":48,"x":154.87,"y":1726.629,"cluster":"algebraic-spaces"},{"id":"stacks:06EW","tag":"06EW","title":"Properties of morphisms local in the fppf topology on the source · Lemma 06EW","summary":"The property P(f)=\"f is locally of finite type\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is locally of finite type''\nis fppf local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the fppf topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EW","source_file":"spaces-descent.tex","source_line":2463,"source_end_line":2467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2463-L2467","statement_sha256":"b2107398f2bbfc7a852bc596680e83c5a5a00bacf6f1cc93e27b815d7ebfe93e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11838,"rank":11838,"depth":48,"x":183.542,"y":1463.917,"cluster":"algebraic-spaces"},{"id":"stacks:06EX","tag":"06EX","title":"Properties of morphisms local in the fppf topology on the source · Lemma 06EX","summary":"The property P(f)=\"f is open\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is open''\nis fppf local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the fppf topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EX","source_file":"spaces-descent.tex","source_line":2479,"source_end_line":2483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2479-L2483","statement_sha256":"04bad7246fa67b43820259ed3e28d78c5ce5a5c7e95b662a1fabb6b56797779a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11839,"rank":11839,"depth":48,"x":373.775,"y":1674.027,"cluster":"algebraic-spaces"},{"id":"stacks:06EY","tag":"06EY","title":"Properties of morphisms local in the fppf topology on the source · Lemma 06EY","summary":"The property P(f)=\"f is universally open\" is fppf local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is universally open''\nis fppf local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the fppf topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EY","source_file":"spaces-descent.tex","source_line":2494,"source_end_line":2498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2494-L2498","statement_sha256":"c5579e9331b059ed8085050d3f43c0ceeb274eeb7cdbb116f78dbc0c020edf6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11840,"rank":11840,"depth":48,"x":64.358,"y":1627.011,"cluster":"algebraic-spaces"},{"id":"stacks:06F0","tag":"06F0","title":"Properties of morphisms local in the syntomic topology on the source · Lemma 06F0","summary":"The property P(f)=\"f is syntomic\" is syntomic local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is syntomic''\nis syntomic local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the syntomic topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06F0","source_file":"spaces-descent.tex","source_line":2517,"source_end_line":2521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2517-L2521","statement_sha256":"faac328264a93af43121a9b5bb48fbde7a0cab54cc027b0809abbe335f3e02ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":11841,"rank":11841,"depth":48,"x":330.479,"y":1486.028,"cluster":"algebraic-spaces"},{"id":"stacks:06F2","tag":"06F2","title":"Properties of morphisms local in the smooth topology on the source · Lemma 06F2","summary":"The property P(f)=\"f is smooth\" is smooth local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is smooth''\nis smooth local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the smooth topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06F2","source_file":"spaces-descent.tex","source_line":2541,"source_end_line":2545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2541-L2545","statement_sha256":"602b67e004a79b5229ec1bc89fcccdf05d00f0678190e33428d46affbe100fd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11842,"rank":11842,"depth":48,"x":247.572,"y":1741.136,"cluster":"algebraic-spaces"},{"id":"stacks:06F4","tag":"06F4","title":"Properties of morphisms local in the étale topology on the source · Lemma 06F4","summary":"The property P(f)=\"f is étale\" is étale local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is \\'etale''\nis \\'etale local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the étale topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06F4","source_file":"spaces-descent.tex","source_line":2564,"source_end_line":2568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2564-L2568","statement_sha256":"25d4615ae4106b28cbb32ae528c96a1455fdb6c92cf7d022e20f6bf85a1e1263","origin":"The Stacks Project","memory_eligible":false,"source_rank":11843,"rank":11843,"depth":48,"x":103.473,"y":1505.845,"cluster":"algebraic-spaces"},{"id":"stacks:06F5","tag":"06F5","title":"Properties of morphisms local in the étale topology on the source · Lemma 06F5","summary":"The property P(f)=\"f is locally quasi-finite\" is étale local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is locally quasi-finite''\nis \\'etale local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the étale topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06F5","source_file":"spaces-descent.tex","source_line":2580,"source_end_line":2584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2580-L2584","statement_sha256":"c6ccf25d56e4c6b0f91c610c797ee9a52cd051bd325e9ec7276fb67f6d63c84a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11844,"rank":11844,"depth":48,"x":399.113,"y":1597.632,"cluster":"algebraic-spaces"},{"id":"stacks:06F6","tag":"06F6","title":"Properties of morphisms local in the étale topology on the source · Lemma 06F6","summary":"The property P(f)=\"f is unramified\" is étale local on the source.","statement_latex":"The property $\\mathcal{P}(f)=$``$f$ is unramified''\nis \\'etale local on the source.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms local in the étale topology on the source","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06F6","source_file":"spaces-descent.tex","source_line":2596,"source_end_line":2600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2596-L2600","statement_sha256":"49678ba32b186b8d706c98b177083d3605e29ad2ff7ab027b60291dcbc387483","origin":"The Stacks Project","memory_eligible":false,"source_rank":11845,"rank":11845,"depth":48,"x":107.133,"y":1697.76,"cluster":"algebraic-spaces"},{"id":"stacks:06F8","tag":"06F8","title":"Properties of morphisms smooth local on source-and-target · Definition 06F8","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S. We say P is smooth local on source-and-target if • (stable under precomposing with smooth maps) if f : X → Y is smooth and g : Y → Z has P, then g ∘ f has P, • (stable under smooth base change) if f : X → Y has P and Y' → Y is smooth, then the base change f' : Y' ×_Y X → Y' has P, and • (locality) given a morphism f : X → Y the following are equivalent • f has P, • for every x ∈ |X| there…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.\nWe say $\\mathcal{P}$ is {\\it smooth local on source-and-target} if\n\\begin{enumerate}\n\\item (stable under precomposing with smooth maps)\nif $f : X \\to Y$ is smooth and $g : Y \\to Z$ has $\\mathcal{P}$,\nthen $g \\circ f$ has $\\mathcal{P}$,\n\\item (stable under smooth base change)\nif $f : X \\to Y$ has $\\mathcal{P}$ and $Y' \\to Y$ is smooth, then\nthe base change $f' : Y' \\times_Y X \\to Y'$ has $\\mathcal{P}$, and\n\\item (locality) given a morphism $f : X \\to Y$ the following are\nequivalent\n\\begin{enumerate}\n\\item $f$ has $\\mathcal{P}$,\n\\item for every $x \\in |X|$ there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwith smooth vertical arrows and $u \\in |U|$ with $a(u) = x$ such that\n$h$ has $\\mathcal{P}$.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms smooth local on source-and-target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06F8","source_file":"spaces-descent.tex","source_line":2641,"source_end_line":2668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2641-L2668","statement_sha256":"57201a121efa7349773bb4445dcbe0a0cdc19f94baff8fa088cc497795ff3275","origin":"The Stacks Project","memory_eligible":false,"source_rank":11846,"rank":11846,"depth":0,"x":241.991,"y":1458.114,"cluster":"algebraic-spaces"},{"id":"stacks:06F9","tag":"06F9","title":"Properties of morphisms smooth local on source-and-target · Lemma 06F9","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S which is smooth local on source-and-target. Then • P is smooth local on the source, • P is smooth local on the target, • P is stable under postcomposing with smooth morphisms: if f : X → Y has P and g : Y → Z is smooth, then g ∘ f has P.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$\nwhich is smooth local on source-and-target. Then\n\\begin{enumerate}\n\\item $\\mathcal{P}$ is smooth local on the source,\n\\item $\\mathcal{P}$ is smooth local on the target,\n\\item $\\mathcal{P}$ is stable under postcomposing with smooth morphisms:\nif $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is smooth, then\n$g \\circ f$ has $\\mathcal{P}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms smooth local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06F9","source_file":"spaces-descent.tex","source_line":2675,"source_end_line":2687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2675-L2687","statement_sha256":"1ea94c8d171e6df0d577c5e5c5176ad907264d253d32a6ebf24da52f1a98bc50","origin":"The Stacks Project","memory_eligible":false,"source_rank":11847,"rank":11847,"depth":1,"x":335.319,"y":1711.492,"cluster":"algebraic-spaces"},{"id":"stacks:06FA","tag":"06FA","title":"Properties of morphisms smooth local on source-and-target · Lemma 06FA","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S which is smooth local on source-and-target. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • [(a)] f has property P, • [(b)] for every x ∈ |X| there exists a smooth morphism of pairs a : (U, u) → (X, x), a smooth morphism b : V → Y, and a morphism h : U → V such that f ∘ a = b ∘ h and h has P, • [(c)] for some commutative diagram xymatrix U ar[d]_a ar[r]_h…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$\nwhich is smooth local on source-and-target. Let $f : X \\to Y$ be a morphism\nof algebraic spaces over $S$. The following are equivalent:\n\\begin{enumerate}\n\\item[(a)] $f$ has property $\\mathcal{P}$,\n\\item[(b)] for every $x \\in |X|$ there exists a smooth morphism of pairs\n$a : (U, u) \\to (X, x)$, a smooth morphism $b : V \\to Y$, and\na morphism $h : U \\to V$ such that $f \\circ a = b \\circ h$ and\n$h$ has $\\mathcal{P}$,\n\\item[(c)] for some commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwith $a$, $b$ smooth and $a$ surjective the morphism $h$ has $\\mathcal{P}$,\n\\item[(d)] for any commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwith $b$ smooth and $U \\to X \\times_Y V$ smooth\nthe morphism $h$ has $\\mathcal{P}$,\n\\item[(e)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ such\nthat each base change $Y_i \\times_Y X \\to Y_i$ has $\\mathcal{P}$,\n\\item[(f)] there exists a smooth covering $\\{X_i \\to X\\}_{i \\in I}$ such\nthat each composition $X_i \\to Y$ has $\\mathcal{P}$,\n\\item[(g)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ and\nfor each $i \\in I$ a smooth covering\n$\\{X_{ij} \\to Y_i \\times_Y X\\}_{j \\in J_i}$ such that each morphism\n$X_{ij} \\to Y_i$ has $\\mathcal{P}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms smooth local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FA","source_file":"spaces-descent.tex","source_line":2732,"source_end_line":2770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2732-L2770","statement_sha256":"4044cbdd327ae0f92568b9802f67063f2f447afbb4748a790c2a1484153c1da0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11848,"rank":11848,"depth":2,"x":62.586,"y":1577.535,"cluster":"algebraic-spaces"},{"id":"stacks:06FB","tag":"06FB","title":"Properties of morphisms smooth local on source-and-target · Lemma 06FB","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S. Assume • P is smooth local on the source, • P is smooth local on the target, and • P is stable under postcomposing with smooth morphisms: if f : X → Y has P and Y → Z is a smooth morphism then X → Z has P. Then P is smooth local on the source-and-target.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{P}$ is smooth local on the source,\n\\item $\\mathcal{P}$ is smooth local on the target, and\n\\item $\\mathcal{P}$ is stable under postcomposing with smooth morphisms:\nif $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\to Z$ is a smooth morphism\nthen $X \\to Z$ has $\\mathcal{P}$.\n\\end{enumerate}\nThen $\\mathcal{P}$ is smooth local on the source-and-target.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms smooth local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FB","source_file":"spaces-descent.tex","source_line":2802,"source_end_line":2815,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2802-L2815","statement_sha256":"635302bb5bc129185a2f1c96f0d8eb9b86d29ba9e0b965406cde6873e6ef8c51","origin":"The Stacks Project","memory_eligible":false,"source_rank":11849,"rank":11849,"depth":1,"x":371.595,"y":1521.526,"cluster":"algebraic-spaces"},{"id":"stacks:0CFZ","tag":"0CFZ","title":"Properties of morphisms étale-smooth local on source-and-target · Definition 0CFZ","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S. We say P is étale-smooth local on source-and-target if • (stable under precomposing with étale maps) if f : X → Y is étale and g : Y → Z has P, then g ∘ f has P, • (stable under smooth base change) if f : X → Y has P and Y' → Y is smooth, then the base change f' : Y' ×_Y X → Y' has P, and • (locality) given a morphism f : X → Y the following are equivalent • f has P, • for every x ∈ |X| there…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.\nWe say $\\mathcal{P}$ is {\\it \\'etale-smooth local on source-and-target} if\n\\begin{enumerate}\n\\item (stable under precomposing with \\'etale maps)\nif $f : X \\to Y$ is \\'etale and $g : Y \\to Z$ has $\\mathcal{P}$,\nthen $g \\circ f$ has $\\mathcal{P}$,\n\\item (stable under smooth base change)\nif $f : X \\to Y$ has $\\mathcal{P}$ and $Y' \\to Y$ is smooth, then\nthe base change $f' : Y' \\times_Y X \\to Y'$ has $\\mathcal{P}$, and\n\\item (locality) given a morphism $f : X \\to Y$ the following are\nequivalent\n\\begin{enumerate}\n\\item $f$ has $\\mathcal{P}$,\n\\item for every $x \\in |X|$ there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwith $b$ smooth and $U \\to X \\times_Y V$ \\'etale\nand $u \\in |U|$ with $a(u) = x$ such that\n$h$ has $\\mathcal{P}$.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms étale-smooth local on source-and-target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CFZ","source_file":"spaces-descent.tex","source_line":2907,"source_end_line":2935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2907-L2935","statement_sha256":"c10c5477bfc8bb78ec21eeed74274141bdd90923817c274f2e85a725307e47f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11850,"rank":11850,"depth":0,"x":188.673,"y":1738.288,"cluster":"algebraic-spaces"},{"id":"stacks:0CG0","tag":"0CG0","title":"Properties of morphisms étale-smooth local on source-and-target · Lemma 0CG0","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S which is étale-smooth local on source-and-target. Then • P is étale local on the source, • P is smooth local on the target, • P is stable under postcomposing with étale morphisms: if f : X → Y has P and g : Y → Z is étale, then g ∘ f has P, and • P has a permanence property: given f : X → Y and g : Y → Z étale such that g ∘ f has P, then f has P.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$\nwhich is \\'etale-smooth local on source-and-target. Then\n\\begin{enumerate}\n\\item $\\mathcal{P}$ is \\'etale local on the source,\n\\item $\\mathcal{P}$ is smooth local on the target,\n\\item $\\mathcal{P}$ is stable under postcomposing with \\'etale morphisms:\nif $f : X \\to Y$ has $\\mathcal{P}$ and $g : Y \\to Z$ is \\'etale, then\n$g \\circ f$ has $\\mathcal{P}$, and\n\\item $\\mathcal{P}$ has a permanence property: given $f : X \\to Y$ and\n$g : Y \\to Z$ \\'etale such that $g \\circ f$ has $\\mathcal{P}$, then\n$f$ has $\\mathcal{P}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms étale-smooth local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CG0","source_file":"spaces-descent.tex","source_line":2943,"source_end_line":2958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L2943-L2958","statement_sha256":"c2920acea85a44b7147dde57aca64dd02c6881fc399147902d211834753ad4ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":11851,"rank":11851,"depth":46,"x":149.22,"y":1474.508,"cluster":"algebraic-spaces"},{"id":"stacks:0CG1","tag":"0CG1","title":"Properties of morphisms étale-smooth local on source-and-target · Lemma 0CG1","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S which is etale-smooth local on source-and-target. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • [(a)] f has property P, • [(b)] for every x ∈ |X| there exists a smooth morphism b : V → Y, an étale morphism a : U → V ×_Y X, and a point u ∈ |U| mapping to x such that U → V has P, • [(c)] for some commutative diagram xymatrix U ar[d]_a ar[r]_h & V ar[d]^b…","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$\nwhich is etale-smooth local on source-and-target.\nLet $f : X \\to Y$ be a morphism\nof algebraic spaces over $S$. The following are equivalent:\n\\begin{enumerate}\n\\item[(a)] $f$ has property $\\mathcal{P}$,\n\\item[(b)] for every $x \\in |X|$ there exists a smooth morphism $b : V \\to Y$,\nan \\'etale morphism $a : U \\to V \\times_Y X$, and a point $u \\in |U|$\nmapping to $x$ such that $U \\to V$ has $\\mathcal{P}$,\n\\item[(c)] for some commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwith $b$ smooth, $U \\to V \\times_Y X$ \\'etale, and $a$ surjective\nthe morphism $h$ has $\\mathcal{P}$,\n\\item[(d)] for any commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwith $b$ smooth and $U \\to X \\times_Y V$ \\'etale, the morphism $h$\nhas $\\mathcal{P}$,\n\\item[(e)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ such\nthat each base change $Y_i \\times_Y X \\to Y_i$ has $\\mathcal{P}$,\n\\item[(f)] there exists an \\'etale covering $\\{X_i \\to X\\}_{i \\in I}$ such\nthat each composition $X_i \\to Y$ has $\\mathcal{P}$,\n\\item[(g)] there exists a smooth covering $\\{Y_i \\to Y\\}_{i \\in I}$ and\nfor each $i \\in I$ an \\'etale covering\n$\\{X_{ij} \\to Y_i \\times_Y X\\}_{j \\in J_i}$ such that each morphism\n$X_{ij} \\to Y_i$ has $\\mathcal{P}$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms étale-smooth local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CG1","source_file":"spaces-descent.tex","source_line":3020,"source_end_line":3059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3020-L3059","statement_sha256":"ff6c1d029973ade7797e05b09301191987c7ca1fd7bf753505aab770d11b278b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11852,"rank":11852,"depth":47,"x":390.574,"y":1646.726,"cluster":"algebraic-spaces"},{"id":"stacks:0CG2","tag":"0CG2","title":"Properties of morphisms étale-smooth local on source-and-target · Lemma 0CG2","summary":"Let S be a scheme. Let P be a property of morphisms of algebraic spaces over S. Assume • P is étale local on the source, • P is smooth local on the target, and • P is stable under postcomposing with open immersions: if f : X → Y has P and Y ⊂ Z is an open embedding then X → Z has P. Then P is étale-smooth local on the source-and-target.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{P}$ is \\'etale local on the source,\n\\item $\\mathcal{P}$ is smooth local on the target, and\n\\item $\\mathcal{P}$ is stable under postcomposing with open immersions:\nif $f : X \\to Y$ has $\\mathcal{P}$ and $Y \\subset Z$ is an open embedding\nthen $X \\to Z$ has $\\mathcal{P}$.\n\\end{enumerate}\nThen $\\mathcal{P}$ is \\'etale-smooth local on the source-and-target.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Properties of morphisms étale-smooth local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CG2","source_file":"spaces-descent.tex","source_line":3092,"source_end_line":3105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3092-L3105","statement_sha256":"f1beb7a483a7a53d639e67721316f8dcfb4574967ea90ea7ce3a60ed30ad2806","origin":"The Stacks Project","memory_eligible":false,"source_rank":11853,"rank":11853,"depth":49,"x":73.931,"y":1656.691,"cluster":"algebraic-spaces"},{"id":"stacks:0ADG","tag":"0ADG","title":"Descent data for spaces over spaces · Definition 0ADG","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. • Let V → Y be a morphism of algebraic spaces. A descent datum for V/Y/X is an isomorphism φ : V ×_X Y → Y ×_X V of algebraic spaces over Y ×_X Y satisfying the cocycle condition that the diagram xymatrix V ×_X Y ×_X Y ar[rd]^φ_01 ar[rr]_φ_02 & & Y ×_X Y ×_X V & Y ×_X V ×_X Y ar[ru]^φ_12 commutes (with obvious notation). • We also say that the pair (V/Y, φ) is a descent datum relative to Y → X. • A…","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces\nover $S$.\n\\begin{enumerate}\n\\item Let $V \\to Y$ be a morphism of algebraic spaces.\nA {\\it descent datum for $V/Y/X$} is an isomorphism\n$\\varphi : V \\times_X Y \\to Y \\times_X V$ of algebraic spaces over\n$Y \\times_X Y$ satisfying the {\\it cocycle condition} that the diagram\n$$\n\\xymatrix{\nV \\times_X Y \\times_X Y \\ar[rd]^{\\varphi_{01}} \\ar[rr]_{\\varphi_{02}} &\n&\nY \\times_X Y \\times_X V\\\\\n&\nY \\times_X V \\times_X Y \\ar[ru]^{\\varphi_{12}}\n}\n$$\ncommutes (with obvious notation).\n\\item We also say that the pair $(V/Y, \\varphi)$ is\na {\\it descent datum relative to $Y \\to X$}.\n\\item A {\\it morphism $f : (V/Y, \\varphi) \\to (V'/Y, \\varphi')$ of\ndescent data relative to $Y \\to X$} is a morphism\n$f : V \\to V'$ of algebraic spaces over $Y$ such that\nthe diagram\n$$\n\\xymatrix{\nV \\times_X Y \\ar[r]_{\\varphi} \\ar[d]_{f \\times \\text{id}_Y} &\nY \\times_X V \\ar[d]^{\\text{id}_Y \\times f} \\\\\nV' \\times_X Y \\ar[r]^{\\varphi'} & Y \\times_X V'\n}\n$$\ncommutes.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for spaces over spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADG","source_file":"spaces-descent.tex","source_line":3218,"source_end_line":3252,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3218-L3252","statement_sha256":"a3d30dad0948db57ca107576feda3e7082205e2e4e23917fca9b8d921af6c1b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11854,"rank":11854,"depth":0,"x":299.533,"y":1469.566,"cluster":"algebraic-spaces"},{"id":"stacks:0ADI","tag":"0ADI","title":"Descent data for spaces over spaces · Definition 0ADI","summary":"Let S be a scheme. Let (X_i → X)_i ∈ I be a family of morphisms of algebraic spaces over S with fixed target X. • A descent datum (V_i, φ_ij) relative to the family (X_i → X) is given by an algebraic space V_i over X_i for each i ∈ I, an isomorphism φ_ij : V_i ×_X X_j → X_i ×_X V_j of algebraic spaces over X_i ×_X X_j for each pair (i, j) ∈ I^2 such that for every triple of indices (i, j, k) ∈ I^3 the diagram xymatrix V_i ×_X X_j ×_X X_k ar[rd]^pr_01^*φ_ij…","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to X\\}_{i \\in I}$ be a family of morphisms\nof algebraic spaces over $S$ with fixed target $X$.\n\\begin{enumerate}\n\\item A {\\it descent datum $(V_i, \\varphi_{ij})$ relative to the\nfamily $\\{X_i \\to X\\}$} is given by an algebraic space $V_i$ over $X_i$\nfor each $i \\in I$, an isomorphism\n$\\varphi_{ij} : V_i \\times_X X_j \\to X_i \\times_X V_j$\nof algebraic spaces over $X_i \\times_X X_j$ for each pair $(i, j) \\in I^2$\nsuch that for every triple of indices $(i, j, k) \\in I^3$\nthe diagram\n$$\n\\xymatrix{\nV_i \\times_X X_j \\times_X X_k\n\\ar[rd]^{\\text{pr}_{01}^*\\varphi_{ij}}\n\\ar[rr]_{\\text{pr}_{02}^*\\varphi_{ik}} &\n&\nX_i \\times_X X_j \\times_X V_k\\\\\n&\nX_i \\times_X V_j \\times_X X_k\n\\ar[ru]^{\\text{pr}_{12}^*\\varphi_{jk}}\n}\n$$\nof algebraic spaces over $X_i \\times_X X_j \\times_X X_k$ commutes\n(with obvious notation).\n\\item A {\\it morphism\n$\\psi : (V_i, \\varphi_{ij}) \\to (V'_i, \\varphi'_{ij})$\nof descent data} is given by a family $\\psi = (\\psi_i)_{i \\in I}$\nof morphisms $\\psi_i : V_i \\to V'_i$ of algebraic spaces over $X_i$\nsuch that all the diagrams\n$$\n\\xymatrix{\nV_i \\times_X X_j \\ar[r]_{\\varphi_{ij}} \\ar[d]_{\\psi_i \\times \\text{id}} &\nX_i \\times_X V_j \\ar[d]^{\\text{id} \\times \\psi_j} \\\\\nV'_i \\times_X X_j \\ar[r]^{\\varphi'_{ij}} & X_i \\times_X V'_j\n}\n$$\ncommute.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for spaces over spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADI","source_file":"spaces-descent.tex","source_line":3285,"source_end_line":3326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3285-L3326","statement_sha256":"0cd7cbdf23456d082f30d7ddc022ad2ea3842a006506cd60c2308870116e015a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11855,"rank":11855,"depth":0,"x":283.649,"y":1735.711,"cluster":"algebraic-spaces"},{"id":"stacks:0ADK","tag":"0ADK","title":"Descent data for spaces over spaces · Lemma 0ADK","summary":"Let S be a scheme. Let (X_i → X)_i ∈ I be a family of morphisms of algebraic spaces over S with fixed target X. Set Y = coprod_i ∈ I X_i. There is a canonical equivalence of categories category of descent data relative to the family (X_i → X)_i ∈ I → category of descent data relative to Y/X which maps (V_i, φ_ij) to (V, φ) with V = coprod_i∈ I V_i and φ = coprod φ_ij.","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to X\\}_{i \\in I}$ be a family of morphisms\nof algebraic spaces over $S$ with fixed target $X$.\nSet $Y = \\coprod_{i \\in I} X_i$.\nThere is a canonical equivalence of categories\n$$\n\\begin{matrix}\n\\text{category of descent data } \\\\\n\\text{relative to the family } \\{X_i \\to X\\}_{i \\in I}\n\\end{matrix}\n\\longrightarrow\n\\begin{matrix}\n\\text{ category of descent data} \\\\\n\\text{ relative to } Y/X\n\\end{matrix}\n$$\nwhich maps $(V_i, \\varphi_{ij})$ to $(V, \\varphi)$ with\n$V = \\coprod_{i\\in I} V_i$ and $\\varphi = \\coprod \\varphi_{ij}$.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for spaces over spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADK","source_file":"spaces-descent.tex","source_line":3355,"source_end_line":3375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3355-L3375","statement_sha256":"ba382180ece001c59aa93a7bb2fc662669c05a84ef8530a6bdfcba62e4c5ba10","origin":"The Stacks Project","memory_eligible":false,"source_rank":11856,"rank":11856,"depth":0,"x":81.221,"y":1530.331,"cluster":"algebraic-spaces"},{"id":"stacks:0ADL","tag":"0ADL","title":"Descent data for spaces over spaces · Lemma 0ADL","summary":"Pullback of descent data. Let S be a scheme. • Let xymatrix Y' ar[r]_f ar[d]_a' & Y ar[d]^a X' ar[r]^h & X be a commutative diagram of algebraic spaces over S. The construction (V → Y, φ) ↦ f^*(V → Y, φ) = (V' → Y', φ') where V' = Y' ×_Y V and where φ' is defined as the composition xymatrix V' ×_X' Y' ar@=[r] & (Y' ×_Y V) ×_X' Y' ar@=[r] & (Y' ×_X' Y') ×_Y ×_X Y (V ×_X Y) ar[d]^id × φ Y' ×_X' V' ar@=[r] & Y' ×_X' (Y' ×_Y V) & (Y' ×_X Y') ×_Y ×_X Y (Y ×_X V) ar@=[l]…","statement_latex":"Pullback of descent data. Let $S$ be a scheme.\n\\begin{enumerate}\n\\item Let\n$$\n\\xymatrix{\nY' \\ar[r]_f \\ar[d]_{a'} & Y \\ar[d]^a \\\\\nX' \\ar[r]^h & X\n}\n$$\nbe a commutative diagram of algebraic spaces over $S$.\nThe construction\n$$\n(V \\to Y, \\varphi) \\longmapsto f^*(V \\to Y, \\varphi) = (V' \\to Y', \\varphi')\n$$\nwhere $V' = Y' \\times_Y V$ and where\n$\\varphi'$ is defined as the composition\n$$\n\\xymatrix{\nV' \\times_{X'} Y' \\ar@{=}[r] &\n(Y' \\times_Y V) \\times_{X'} Y' \\ar@{=}[r] &\n(Y' \\times_{X'} Y') \\times_{Y \\times_X Y} (V \\times_X Y)\n\\ar[d]^{\\text{id} \\times \\varphi} \\\\\nY' \\times_{X'} V' \\ar@{=}[r] &\nY' \\times_{X'} (Y' \\times_Y V) &\n(Y' \\times_X Y') \\times_{Y \\times_X Y} (Y \\times_X V) \\ar@{=}[l]\n}\n$$\ndefines a functor from the category of descent data\nrelative to $Y \\to X$ to the category of descent data\nrelative to $Y' \\to X'$.\n\\item Given two morphisms $f_i : Y' \\to Y$, $i = 0, 1$ making the\ndiagram commute the functors $f_0^*$ and $f_1^*$ are\ncanonically isomorphic.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for spaces over spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADL","source_file":"spaces-descent.tex","source_line":3385,"source_end_line":3421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3385-L3421","statement_sha256":"227cea61eb69085488383ab4b5a2b5a3753c4624249ebda0e2d699fce937916e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11857,"rank":11857,"depth":0,"x":395.826,"y":1566.934,"cluster":"algebraic-spaces"},{"id":"stacks:0ADM","tag":"0ADM","title":"Descent data for spaces over spaces · Definition 0ADM","summary":"With S, X, X', Y, Y', f, a, a', h as in Lemma [Tag 0ADL] the functor (V, φ) ↦ f^*(V, φ) constructed in that lemma is called the pullback functor on descent data.","statement_latex":"With $S, X, X', Y, Y', f, a, a', h$ as in Lemma \\ref{lemma-pullback}\nthe functor\n$$\n(V, \\varphi) \\longmapsto f^*(V, \\varphi)\n$$\nconstructed in that lemma is called the {\\it pullback functor} on descent data.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for spaces over spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADM","source_file":"spaces-descent.tex","source_line":3450,"source_end_line":3458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3450-L3458","statement_sha256":"90bbb3bb926827bfeab12046c705c52c1ebce519be9c39867861536953d7e6bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":11858,"rank":11858,"depth":1,"x":134.259,"y":1718.543,"cluster":"algebraic-spaces"},{"id":"stacks:0ADN","tag":"0ADN","title":"Descent data for spaces over spaces · Lemma 0ADN","summary":"Let S be a scheme. Let U' = (X'_i → X')_i ∈ I' and U = (X_j → X)_i ∈ I be families of morphisms with fixed target. Let α : I' → I, g : X' → X and g_i : X'_i → X_α(i) be a morphism of families of maps with fixed target, see Sites, Definition [Tag 00VT]. • Let (V_i, φ_ij) be a descent datum relative to the family U. The system ( g_i^*V_α(i), (g_i × g_j)^*φ_α(i) α(j) ) (with notation as in Remark [Tag 0ADJ]) is a descent datum relative to U'. • This construction defines a…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{U}' = \\{X'_i \\to X'\\}_{i \\in I'}$ and\n$\\mathcal{U} = \\{X_j \\to X\\}_{i \\in I}$ be families of morphisms with\nfixed target. Let $\\alpha : I' \\to I$, $g : X' \\to X$ and\n$g_i : X'_i \\to X_{\\alpha(i)}$ be a morphism of families\nof maps with fixed target, see\nSites, Definition \\ref{sites-definition-morphism-coverings}.\n\\begin{enumerate}\n\\item Let $(V_i, \\varphi_{ij})$ be a descent datum relative to the\nfamily $\\mathcal{U}$. The system\n$$\n\\left(\ng_i^*V_{\\alpha(i)}, (g_i \\times g_j)^*\\varphi_{\\alpha(i) \\alpha(j)}\n\\right)\n$$\n(with notation as in Remark \\ref{remark-easier-family})\nis a descent datum relative to $\\mathcal{U}'$.\n\\item This construction defines a functor between the category of\ndescent data relative to $\\mathcal{U}$ and the category of\ndescent data relative to $\\mathcal{U}'$.\n\\item Given a second $\\beta : I' \\to I$, $h : X' \\to X$ and\n$h'_i : X'_i \\to X_{\\beta(i)}$ morphism of families\nof maps with fixed target, then if $g = h$ the two resulting functors\nbetween descent data are canonically isomorphic.\n\\item These functors agree, via Lemma \\ref{lemma-family-is-one},\nwith the pullback functors constructed in Lemma \\ref{lemma-pullback}.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for spaces over spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADN","source_file":"spaces-descent.tex","source_line":3460,"source_end_line":3488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3460-L3488","statement_sha256":"c112ef81e0887198c56384938d5c5fc4cc9ecd205cfb36f082e3f5388e2c6d2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11859,"rank":11859,"depth":1,"x":205.255,"y":1458.183,"cluster":"algebraic-spaces"},{"id":"stacks:0ADP","tag":"0ADP","title":"Descent data for spaces over spaces · Definition 0ADP","summary":"With U' = (X'_i → X')_i ∈ I', U = (X_i → X)_i ∈ I, α : I' → I, g : X' → X, and g_i : X'_i → X_α(i) as in Lemma [Tag 0ADN] the functor (V_i, φ_ij) ↦ (g_i^*V_α(i), (g_i × g_j)^*φ_α(i) α(j)) constructed in that lemma is called the pullback functor on descent data.","statement_latex":"With $\\mathcal{U}' = \\{X'_i \\to X'\\}_{i \\in I'}$,\n$\\mathcal{U} = \\{X_i \\to X\\}_{i \\in I}$, $\\alpha : I' \\to I$,\n$g : X' \\to X$, and $g_i : X'_i \\to X_{\\alpha(i)}$ as in\nLemma \\ref{lemma-pullback-family} the functor\n$$\n(V_i, \\varphi_{ij}) \\longmapsto\n(g_i^*V_{\\alpha(i)}, (g_i \\times g_j)^*\\varphi_{\\alpha(i) \\alpha(j)})\n$$\nconstructed in that lemma\nis called the {\\it pullback functor} on descent data.","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for spaces over spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADP","source_file":"spaces-descent.tex","source_line":3495,"source_end_line":3507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3495-L3507","statement_sha256":"f494f0e488fc0f27aa909da986667e0d9781c48df00040bb7d4d559f81a75864","origin":"The Stacks Project","memory_eligible":false,"source_rank":11860,"rank":11860,"depth":2,"x":362.366,"y":1690.583,"cluster":"algebraic-spaces"},{"id":"stacks:0ADQ","tag":"0ADQ","title":"Descent data for spaces over spaces · Definition 0ADQ","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. • Given an algebraic space U over X we have the trivial descent datum of U relative to id : X → X, namely the identity morphism on U. • By Lemma [Tag 0ADL] we get a canonical descent datum on Y ×_X U relative to Y → X by pulling back the trivial descent datum via f. We often denote (Y ×_X U, can) this descent datum. • A descent datum (V, φ) relative to Y/X is called effective if (V, φ) is…","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces over\n$S$.\n\\begin{enumerate}\n\\item Given an algebraic space $U$ over $X$ we have the\n{\\it trivial descent datum} of $U$ relative to $\\text{id} : X \\to X$, namely\nthe identity morphism on $U$.\n\\item By Lemma \\ref{lemma-pullback} we get a\n{\\it canonical descent datum} on $Y \\times_X U$\nrelative to $Y \\to X$ by pulling back the trivial\ndescent datum via $f$. We often\ndenote $(Y \\times_X U, can)$ this descent datum.\n\\item A descent datum $(V, \\varphi)$ relative to $Y/X$\nis called {\\it effective} if $(V, \\varphi)$\nis isomorphic to the canonical descent datum\n$(Y \\times_X U, can)$ for some algebraic space $U$ over $X$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for spaces over spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADQ","source_file":"spaces-descent.tex","source_line":3518,"source_end_line":3536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3518-L3536","statement_sha256":"c567c92d8bfdbb63841ef08a0d4497083eb5a78c4502d69f19a87bf06811334e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11861,"rank":11861,"depth":1,"x":59.455,"y":1608.318,"cluster":"algebraic-spaces"},{"id":"stacks:0ADR","tag":"0ADR","title":"Descent data for spaces over spaces · Definition 0ADR","summary":"Let S be a scheme. Let (X_i → X) be a family of morphisms of algebraic spaces over S with fixed target X. • Given an algebraic space U over X we have a canonical descent datum on the family of algebraic spaces X_i ×_X U by pulling back the trivial descent datum for U relative to (id : S → S). We denote this descent datum (X_i ×_X U, can). • A descent datum (V_i, φ_ij) relative to (X_i → S) is called effective if there exists an algebraic space U over X such that (V_i,…","statement_latex":"Let $S$ be a scheme.\nLet $\\{X_i \\to X\\}$ be a family of morphisms of algebraic spaces over $S$\nwith fixed target $X$.\n\\begin{enumerate}\n\\item  Given an algebraic space $U$ over $X$\nwe have a {\\it canonical descent datum} on the family of\nalgebraic spaces $X_i \\times_X U$ by pulling back the trivial\ndescent datum for $U$ relative to $\\{\\text{id} : S \\to S\\}$.\nWe denote this descent datum $(X_i \\times_X U, can)$.\n\\item A descent datum $(V_i, \\varphi_{ij})$\nrelative to $\\{X_i \\to S\\}$ is called {\\it effective}\nif there exists an algebraic space $U$ over $X$ such that\n$(V_i, \\varphi_{ij})$ is isomorphic to $(X_i \\times_X U, can)$.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data for spaces over spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADR","source_file":"spaces-descent.tex","source_line":3555,"source_end_line":3571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3555-L3571","statement_sha256":"517ae8401aae23d5b6dc984a8afce59c59b094961314446b0b358684a1beced7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11862,"rank":11862,"depth":0,"x":349.135,"y":1497.038,"cluster":"algebraic-spaces"},{"id":"stacks:0ADT","tag":"0ADT","title":"Descent data in terms of sheaves · Lemma 0ADT","summary":"Let S be a scheme. Let (X_i → X)_i ∈ I be an fppf covering of algebraic spaces over S (Topologies on Spaces, Definition [Tag 03Y8]). There is an equivalence of categories ( descent data (V_i, φ_ij) relative to (X_i → X) ) ↔ ( sheaves F on (Sch/S)_fppf endowed with a map F → X such that each X_i ×_X F is an algebraic space ). Moreover, • the algebraic space X_i ×_X F on the right hand side corresponds to V_i on the left hand side, and • the sheaf F is an algebraic space if…","statement_latex":"Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf\ncovering of algebraic spaces over $S$ (Topologies on Spaces,\nDefinition \\ref{spaces-topologies-definition-fppf-covering}).\nThere is an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{descent data }(V_i, \\varphi_{ij})\\\\\n\\text{relative to }\\{X_i \\to X\\}\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{sheaves }F\\text{ on }(\\Sch/S)_{fppf}\\text{ endowed}\\\\\n\\text{with a map }F \\to X\\text{ such that each}\\\\\nX_i \\times_X F\\text{ is an algebraic space}\n\\end{matrix}\n\\right\\}.\n$$\nMoreover,\n\\begin{enumerate}\n\\item the algebraic space $X_i \\times_X F$ on the right hand side\ncorresponds to $V_i$ on the left hand side, and\n\\item the sheaf $F$ is an algebraic space\\footnote{We will see\nlater that this is always the case if $I$ is not too large, see\nBootstrap, Lemma \\ref{bootstrap-lemma-descend-algebraic-space}.}\nif and only if the\ncorresponding descent datum $(X_i, \\varphi_{ij})$ is effective.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Descent and Algebraic Spaces","chapter_id":"spaces-descent","section":"Descent data in terms of sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADT","source_file":"spaces-descent.tex","source_line":3586,"source_end_line":3618,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-descent.tex#L3586-L3618","statement_sha256":"4c8ebf5f794bc57359b63aaafbe7b1bb5d36278493a8ce5a300df5a752725471","origin":"The Stacks Project","memory_eligible":false,"source_rank":11863,"rank":11863,"depth":1,"x":224.948,"y":1743.603,"cluster":"algebraic-spaces"},{"id":"stacks:08GE","tag":"08GE","title":"Generalities · Lemma 08GE","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Given an étale morphism V → Y, set U = V ×_Y X and denote g : U → V the projection morphism. Then (Rf_*E)|_V = Rg_*(E|_U) for E in D(O_X).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Given an \\'etale morphism $V \\to Y$, set $U = V \\times_Y X$\nand denote $g : U \\to V$ the projection morphism. Then\n$(Rf_*E)|_V = Rg_*(E|_U)$ for $E$ in $D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Generalities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GE","source_file":"spaces-perfect.tex","source_line":57,"source_end_line":63,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L57-L63","statement_sha256":"06af5cbcf70c696d2f178a92f5c7c119db5cd14c9ba351b30df9dddeef5482e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11864,"rank":11864,"depth":11,"x":119.853,"y":590.534,"cluster":"derived-categories"},{"id":"stacks:08GF","tag":"08GF","title":"Generalities · Definition 08GF","summary":"Let S be a scheme. Let X be an algebraic space over S. Let E be an object of D(O_X). Let T ⊂ |X| be a closed subset. We say E is supported on T if the cohomology sheaves H^i(E) are supported on T.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $E$ be an object of $D(\\mathcal{O}_X)$.\nLet $T \\subset |X|$ be a closed subset.\nWe say $E$ is {\\it supported on $T$} if the\ncohomology sheaves $H^i(E)$ are supported on $T$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Generalities","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GF","source_file":"spaces-perfect.tex","source_line":76,"source_end_line":83,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L76-L83","statement_sha256":"5db31bd2bbbd55541696286106ddc83c384c76a115ba4fd12952159be87f3897","origin":"The Stacks Project","memory_eligible":false,"source_rank":11865,"rank":11865,"depth":0,"x":383.311,"y":683.474,"cluster":"derived-categories"},{"id":"stacks:08H8","tag":"08H8","title":"Derived category of quasi-coherent modules on the small étale site · Lemma 08H8","summary":"The morphism ε of ([Tag 08H7]) is a flat morphism of ringed sites. In particular the functor ε^* : Mod(O_X) → Mod(O_etale) is exact. Moreover, if ε^*F = 0, then F = 0.","statement_latex":"The morphism $\\epsilon$ of (\\ref{equation-epsilon})\nis a flat morphism of ringed sites. In particular the functor\n$\\epsilon^* : \\textit{Mod}(\\mathcal{O}_X) \\to\n\\textit{Mod}(\\mathcal{O}_\\etale)$ is exact.\nMoreover, if $\\epsilon^*\\mathcal{F} = 0$, then $\\mathcal{F} = 0$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules on the small étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08H8","source_file":"spaces-perfect.tex","source_line":112,"source_end_line":119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L112-L119","statement_sha256":"36c5cf7d9b5267102225d35eb6237d0da2d904b7dba121a2926401251874ca4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11866,"rank":11866,"depth":47,"x":114.049,"y":764.51,"cluster":"derived-categories"},{"id":"stacks:071Q","tag":"071Q","title":"Derived category of quasi-coherent modules on the small étale site · Lemma 071Q","summary":"Let X be a scheme. The functor ε^* : D_QCoh(O_X) → D_QCoh(O_etale) defined above is an equivalence.","statement_latex":"Let $X$ be a scheme. The functor\n$\\epsilon^* : D_\\QCoh(\\mathcal{O}_X) \\to\nD_\\QCoh(\\mathcal{O}_\\etale)$\ndefined above is an equivalence.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules on the small étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/071Q","source_file":"spaces-perfect.tex","source_line":184,"source_end_line":190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L184-L190","statement_sha256":"f01db368a6b875fec5759d29468bf350453078c0068281014f853df5d764d2ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":11867,"rank":11867,"depth":40,"x":247.556,"y":551.771,"cluster":"derived-categories"},{"id":"stacks:071X","tag":"071X","title":"Derived category of quasi-coherent modules · Definition 071X","summary":"Let S be a scheme. Let X be an algebraic space over S. The derived category of O_X-modules with quasi-coherent cohomology sheaves is denoted D_QCoh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThe {\\it derived category of $\\mathcal{O}_X$-modules with\nquasi-coherent cohomology sheaves} is denoted\n$D_\\QCoh(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/071X","source_file":"spaces-perfect.tex","source_line":276,"source_end_line":282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L276-L282","statement_sha256":"e69971d5d7751248ee6140ee24e63cca657d900ec329845b553150cea1dafcbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11868,"rank":11868,"depth":0,"x":320.257,"y":784.614,"cluster":"derived-categories"},{"id":"stacks:08F2","tag":"08F2","title":"Derived category of quasi-coherent modules · Lemma 08F2","summary":"Let S be a scheme. Let X be an algebraic space over S. Let E be an object of D(O_X). The following are equivalent • E is in D_QCoh(O_X), • for every étale morphism φ : U → X where U is an affine scheme φ^*E is an object of D_QCoh(O_U), • for every étale morphism φ : U → X where U is a scheme φ^*E is an object of D_QCoh(O_U), • there exists a surjective étale morphism φ : U → X where U is a scheme such that φ^*E is an object of D_QCoh(O_U), and • there exists a surjective…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $E$ be an object of $D(\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item $E$ is in $D_\\QCoh(\\mathcal{O}_X)$,\n\\item for every \\'etale morphism $\\varphi : U \\to X$ where $U$ is an\naffine scheme $\\varphi^*E$ is an object of\n$D_\\QCoh(\\mathcal{O}_U)$,\n\\item for every \\'etale morphism $\\varphi : U \\to X$ where $U$ is a scheme\n$\\varphi^*E$ is an object of\n$D_\\QCoh(\\mathcal{O}_U)$,\n\\item there exists a surjective \\'etale morphism $\\varphi : U \\to X$\nwhere $U$ is a scheme such that $\\varphi^*E$ is an object of\n$D_\\QCoh(\\mathcal{O}_U)$, and\n\\item there exists a surjective \\'etale morphism of algebraic spaces\n$f : Y \\to X$ such that $Lf^*E$ is an object of\n$D_\\QCoh(\\mathcal{O}_Y)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08F2","source_file":"spaces-perfect.tex","source_line":317,"source_end_line":336,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L317-L336","statement_sha256":"f9e998c7f57e7db62f0400577c05afe927b63d7d04c55584bacd90d715a21279","origin":"The Stacks Project","memory_eligible":false,"source_rank":11869,"rank":11869,"depth":13,"x":79.178,"y":654.049,"cluster":"derived-categories"},{"id":"stacks:08F3","tag":"08F3","title":"Derived category of quasi-coherent modules · Lemma 08F3","summary":"Let S be a scheme. Let X be an algebraic space over S. Then D_QCoh(O_X) has direct sums.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nThen $D_\\QCoh(\\mathcal{O}_X)$ has direct sums.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08F3","source_file":"spaces-perfect.tex","source_line":344,"source_end_line":348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L344-L348","statement_sha256":"039f06c49833385d94d6f845d29a1431a2259ba3e40ac711a86fd0d23960c610","origin":"The Stacks Project","memory_eligible":false,"source_rank":11870,"rank":11870,"depth":42,"x":362.205,"y":613.494,"cluster":"derived-categories"},{"id":"stacks:0D3E","tag":"0D3E","title":"Derived category of quasi-coherent modules · Lemma 0D3E","summary":"Let S be a scheme. Let X be an algebraic space over S. Let (K_n) be an inverse system of D_QCoh(O_X) with derived limit K = Rlim K_n in D(O_X). Assume H^q(K_n + 1) → H^q(K_n) is surjective for all q ∈ Z and n ≥ 1. Then • H^q(K) = lim H^q(K_n), • Rlim H^q(K_n) = lim H^q(K_n), and • for every affine open U ⊂ X we have H^p(U, lim H^q(K_n)) = 0 for p > 0.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $(K_n)$ be an inverse system of\n$D_\\QCoh(\\mathcal{O}_X)$ with derived limit\n$K = R\\lim K_n$ in $D(\\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \\to H^q(K_n)$\nis surjective for all $q \\in \\mathbf{Z}$ and $n \\geq 1$.\nThen\n\\begin{enumerate}\n\\item $H^q(K) = \\lim H^q(K_n)$,\n\\item $R\\lim H^q(K_n) = \\lim H^q(K_n)$, and\n\\item for every affine open $U \\subset X$ we have\n$H^p(U, \\lim H^q(K_n)) = 0$ for $p > 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3E","source_file":"spaces-perfect.tex","source_line":367,"source_end_line":381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L367-L381","statement_sha256":"2bc7d3500490c4f13dfed90a553b8cd5d00ad83fe324ac9e7bf427f613b28470","origin":"The Stacks Project","memory_eligible":false,"source_rank":11871,"rank":11871,"depth":24,"x":185.955,"y":804.172,"cluster":"derived-categories"},{"id":"stacks:08F4","tag":"08F4","title":"Derived category of quasi-coherent modules · Lemma 08F4","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. The functor Lf^* sends D_QCoh(O_X) into D_QCoh(O_Y).","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be a morphism of algebraic spaces over $S$.\nThe functor $Lf^*$ sends $D_\\QCoh(\\mathcal{O}_X)$\ninto $D_\\QCoh(\\mathcal{O}_Y)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08F4","source_file":"spaces-perfect.tex","source_line":402,"source_end_line":408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L402-L408","statement_sha256":"6ce3c1b94dce822b759ae39c20bef4e692cffe51b4f6d63528d493a7dbd92cf3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11872,"rank":11872,"depth":28,"x":162.56,"y":563.337,"cluster":"derived-categories"},{"id":"stacks:08F5","tag":"08F5","title":"Derived category of quasi-coherent modules · Lemma 08F5","summary":"Let S be a scheme. Let X be an algebraic space over S. For objects K, L of D_QCoh(O_X) the derived tensor product K ⊗^L L is in D_QCoh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nFor objects $K, L$ of $D_\\QCoh(\\mathcal{O}_X)$\nthe derived tensor product $K \\otimes^\\mathbf{L} L$ is in\n$D_\\QCoh(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08F5","source_file":"spaces-perfect.tex","source_line":427,"source_end_line":433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L427-L433","statement_sha256":"3cc63c5915a44ead0a768e07669190ac3a7f2baaabf7b17d027dffd9f03fe152","origin":"The Stacks Project","memory_eligible":false,"source_rank":11873,"rank":11873,"depth":28,"x":373.679,"y":727.802,"cluster":"derived-categories"},{"id":"stacks:08F6","tag":"08F6","title":"Derived category of quasi-coherent modules · Lemma 08F6","summary":"Let S be a scheme. Let X be an algebraic space over S. Let E be an object of D_QCoh(O_X). Then the map E → Rlim τ_≥ -nE of Derived Categories, Remark [Tag 0H72] is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an\nobject of $D_\\QCoh(\\mathcal{O}_X)$. Then the map\n$E \\to R\\lim \\tau_{\\geq -n}E$ of\nDerived Categories, Remark\n\\ref{derived-remark-map-into-derived-limit-truncations}\nis an isomorphism\\footnote{In particular,\n$E$ has a K-injective representative, see\nDerived Categories, Lemma \\ref{derived-lemma-difficulty-K-injectives}.}.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08F6","source_file":"spaces-perfect.tex","source_line":451,"source_end_line":461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L451-L461","statement_sha256":"8088c687ae6a3dbfc92025d5c1270f1be8e57a318feb92d942b806fa08ba1705","origin":"The Stacks Project","memory_eligible":false,"source_rank":11874,"rank":11874,"depth":24,"x":85.479,"y":726.321,"cluster":"derived-categories"},{"id":"stacks:08F7","tag":"08F7","title":"Derived category of quasi-coherent modules · Lemma 08F7","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F : Mod(O_X) → Ab be a functor and N ≥ 0 an integer. Assume that • F is left exact, • F commutes with countable direct products, • R^pF(F) = 0 for all p ≥ N and F quasi-coherent. Then for E ∈ D_QCoh(O_X) • H^i(RF(τ_≤ aE) → H^i(RF(E)) is an isomorphism for i ≤ a, • H^i(RF(E)) → H^i(RF(τ_≥ b - N + 1E)) is an isomorphism for i ≥ b, • if H^i(E) = 0 for i not ∈ [a, b] for some -∞ ≤ a ≤ b ≤ ∞, then H^i(RF(E)) = 0 for i…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $F : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Ab}$\nbe a functor and $N \\geq 0$ an integer. Assume that\n\\begin{enumerate}\n\\item $F$ is left exact,\n\\item $F$ commutes with countable direct products,\n\\item $R^pF(\\mathcal{F}) = 0$ for all $p \\geq N$ and $\\mathcal{F}$\nquasi-coherent.\n\\end{enumerate}\nThen for $E \\in D_\\QCoh(\\mathcal{O}_X)$\n\\begin{enumerate}\n\\item $H^i(RF(\\tau_{\\leq a}E) \\to H^i(RF(E))$ is an isomorphism\nfor $i \\leq a$,\n\\item $H^i(RF(E)) \\to H^i(RF(\\tau_{\\geq b - N + 1}E))$ is an isomorphism\nfor $i \\geq b$,\n\\item if $H^i(E) = 0$ for $i \\not \\in [a, b]$ for some\n$-\\infty \\leq a \\leq b \\leq \\infty$, then $H^i(RF(E)) = 0$\nfor $i \\not \\in [a, b + N - 1]$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08F7","source_file":"spaces-perfect.tex","source_line":479,"source_end_line":500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L479-L500","statement_sha256":"8ce724cf5b38dadc28185b23517911b78095410fb9594eeb47110182da0e9200","origin":"The Stacks Project","memory_eligible":false,"source_rank":11875,"rank":11875,"depth":25,"x":299.381,"y":563.732,"cluster":"derived-categories"},{"id":"stacks:08FA","tag":"08FA","title":"Total direct image · Lemma 08FA","summary":"Let S be a scheme. Let f : X → Y be a quasi-separated and quasi-compact morphism of algebraic spaces over S. • The functor Rf_* sends D_QCoh(O_X) into D_QCoh(O_Y). • If Y is quasi-compact, there exists an integer N = N(X, Y, f) such that for an object E of D_QCoh(O_X) with H^m(E) = 0 for m > 0 we have H^m(Rf_*E) = 0 for m ≥ N. • In fact, if Y is quasi-compact we can find N = N(X, Y, f) such that for every morphism of algebraic spaces Y' → Y the same conclusion holds for…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-separated and quasi-compact\nmorphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item The functor $Rf_*$ sends $D_\\QCoh(\\mathcal{O}_X)$\ninto $D_\\QCoh(\\mathcal{O}_Y)$.\n\\item If $Y$ is quasi-compact, there exists an integer $N = N(X, Y, f)$\nsuch that for an object $E$ of $D_\\QCoh(\\mathcal{O}_X)$\nwith $H^m(E) = 0$ for $m > 0$ we have\n$H^m(Rf_*E) = 0$ for $m \\geq N$.\n\\item In fact, if $Y$ is quasi-compact we can find $N = N(X, Y, f)$\nsuch that for every morphism of algebraic spaces $Y' \\to Y$\nthe same conclusion holds for the functor $R(f')_*$\nwhere $f' : X' \\to Y'$ is the base change of $f$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Total direct image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FA","source_file":"spaces-perfect.tex","source_line":572,"source_end_line":588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L572-L588","statement_sha256":"eec47e15964ac6f309b923322a71c4680df09ff2bc1a76c2225ceec07415c6d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11876,"rank":11876,"depth":59,"x":272.378,"y":805.218,"cluster":"derived-categories"},{"id":"stacks:08FB","tag":"08FB","title":"Total direct image · Lemma 08FB","summary":"Let S be a scheme. Let f : X → Y be a quasi-separated and quasi-compact morphism of algebraic spaces over S. Then Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) commutes with direct sums.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-separated and\nquasi-compact morphism of algebraic spaces over $S$. Then\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$\ncommutes with direct sums.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Total direct image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08FB","source_file":"spaces-perfect.tex","source_line":645,"source_end_line":651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L645-L651","statement_sha256":"bc797f302959bf8e368164a8ddfae5e18312dfea41f5b2f088aeb47c8eb49fc6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11877,"rank":11877,"depth":60,"x":97.934,"y":611.65,"cluster":"derived-categories"},{"id":"stacks:08II","tag":"08II","title":"Total direct image · Lemma 08II","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S. Then Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) reflects isomorphisms.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic\nspaces over $S$. Then\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$\nreflects isomorphisms.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Total direct image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08II","source_file":"spaces-perfect.tex","source_line":747,"source_end_line":753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L747-L753","statement_sha256":"f3bdca5b7a8c2ff90118bdc909b98c3ae303dcdc63a729eb476ef44c6d61e1f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11878,"rank":11878,"depth":41,"x":382.489,"y":655.44,"cluster":"derived-categories"},{"id":"stacks:08IJ","tag":"08IJ","title":"Total direct image · Lemma 08IJ","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S. For E in D_QCoh(O_Y) we have Rf_* Lf^* E = E ⊗^L_O_Y f_*O_X.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of algebraic\nspaces over $S$. For $E$ in $D_\\QCoh(\\mathcal{O}_Y)$ we have\n$Rf_* Lf^* E = E \\otimes^\\mathbf{L}_{\\mathcal{O}_Y} f_*\\mathcal{O}_X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Total direct image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IJ","source_file":"spaces-perfect.tex","source_line":767,"source_end_line":772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L767-L772","statement_sha256":"31535b8530a5fda0b04c15017424d13854710bcdee397f84633fb9056cdf52eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11879,"rank":11879,"depth":41,"x":137.223,"y":784.731,"cluster":"derived-categories"},{"id":"stacks:0CZC","tag":"0CZC","title":"Being proper over a base · Lemma 0CZC","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let T ⊂ |X| be a closed subset. The following are equivalent • the morphism Z → Y is proper if Z is the reduced induced algebraic space structure on T (Properties of Spaces, Definition [Tag 047X]), • for some closed subspace Z ⊂ X with |Z| = T the morphism Z → Y is proper, and • for any closed subspace Z ⊂ X with |Z| = T the morphism Z → Y is proper.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is locally of finite type. Let $T \\subset |X|$ be a closed\nsubset. The following are equivalent\n\\begin{enumerate}\n\\item the morphism $Z \\to Y$ is proper if $Z$ is the reduced\ninduced algebraic space structure on $T$\n(Properties of Spaces, Definition\n\\ref{spaces-properties-definition-reduced-induced-space}),\n\\item for some closed subspace $Z \\subset X$ with $|Z| = T$\nthe morphism $Z \\to Y$ is proper, and\n\\item for any closed subspace $Z \\subset X$ with $|Z| = T$ the morphism\n$Z \\to Y$ is proper.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZC","source_file":"spaces-perfect.tex","source_line":814,"source_end_line":829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L814-L829","statement_sha256":"aea87d74ef6664dace6e42a6bf7f9677a8b727db46d19c9e3a72c763ff4a4813","origin":"The Stacks Project","memory_eligible":false,"source_rank":11880,"rank":11880,"depth":59,"x":214.176,"y":550.01,"cluster":"derived-categories"},{"id":"stacks:0CZD","tag":"0CZD","title":"Being proper over a base · Definition 0CZD","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let T ⊂ |X| be a closed subset. We say T is proper over Y if the equivalent conditions of Lemma [Tag 0CZC] are satisfied.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is locally of finite type.\nLet $T \\subset |X|$ be a closed subset.\nWe say {\\it $T$ is proper over $Y$}\nif the equivalent conditions of Lemma \\ref{lemma-closed-proper-over-base}\nare satisfied.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZD","source_file":"spaces-perfect.tex","source_line":868,"source_end_line":877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L868-L877","statement_sha256":"1cee1b4c7ec370ab90e8a0d1e9366c65df46ed759333c90398b3ed9a063f1119","origin":"The Stacks Project","memory_eligible":false,"source_rank":11881,"rank":11881,"depth":60,"x":346.308,"y":766.953,"cluster":"derived-categories"},{"id":"stacks:0CZE","tag":"0CZE","title":"Being proper over a base · Lemma 0CZE","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let T' ⊂ T ⊂ |X| be closed subsets. If T is proper over Y, then the same is true for T'.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is locally of finite type.\nLet $T' \\subset T \\subset |X|$ be closed subsets.\nIf $T$ is proper over $Y$, then the same is true for $T'$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZE","source_file":"spaces-perfect.tex","source_line":885,"source_end_line":892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L885-L892","statement_sha256":"38e71ad3356f39b62b2bfd2eaf375fab7b39d5670aece49f90ef380ae42afa06","origin":"The Stacks Project","memory_eligible":false,"source_rank":11882,"rank":11882,"depth":0,"x":74.17,"y":681.88,"cluster":"derived-categories"},{"id":"stacks:0CZF","tag":"0CZF","title":"Being proper over a base · Lemma 0CZF","summary":"Let S be a scheme. Consider a cartesian diagram of algebraic spaces over S xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f Y' ar[r]^g & Y with f locally of finite type. If T is a closed subset of |X| proper over Y, then |g'|^-1(T) is a closed subset of |X'| proper over Y'.","statement_latex":"Let $S$ be a scheme.\nConsider a cartesian diagram of algebraic spaces over $S$\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nwith $f$ locally of finite type.\nIf $T$ is a closed subset of $|X|$ proper over $Y$, then\n$|g'|^{-1}(T)$ is a closed subset of $|X'|$ proper over $Y'$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZF","source_file":"spaces-perfect.tex","source_line":898,"source_end_line":911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L898-L911","statement_sha256":"99104559a32ed6f46fb0784f1981dbf020d33d8b42c82abc23935fd210fb98de","origin":"The Stacks Project","memory_eligible":false,"source_rank":11883,"rank":11883,"depth":8,"x":343.497,"y":590.11,"cluster":"derived-categories"},{"id":"stacks:0CZG","tag":"0CZG","title":"Being proper over a base · Lemma 0CZG","summary":"Let S be a scheme. Let B be an algebraic space over S. Let f : X → Y be a morphism of algebraic spaces which are locally of finite type over B. • If Y is separated over B and T ⊂ |X| is a closed subset proper over B, then |f|(T) is a closed subset of |Y| proper over B. • If f is universally closed and T ⊂ |X| is a closed subset proper over B, then |f|(T) is a closed subset of Y proper over B. • If f is proper and T ⊂ |Y| is a closed subset proper over B, then |f|^-1(T) is…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $f : X \\to Y$ be a morphism of algebraic spaces which\nare locally of finite type over $B$.\n\\begin{enumerate}\n\\item If $Y$ is separated over $B$ and $T \\subset |X|$ is a closed subset\nproper over $B$, then $|f|(T)$ is a closed subset of $|Y|$ proper over $B$.\n\\item If $f$ is universally closed and $T \\subset |X|$ is a\nclosed subset proper over $B$, then $|f|(T)$ is a closed subset\nof $Y$ proper over $B$.\n\\item If $f$ is proper and $T \\subset |Y|$ is a closed subset\nproper over $B$, then $|f|^{-1}(T)$ is a closed subset of $|X|$\nproper over $B$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZG","source_file":"spaces-perfect.tex","source_line":925,"source_end_line":940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L925-L940","statement_sha256":"fc09eb6a268c48614dcd52668a4ce8b3b4eaff6c3e486c1da114f4f39d03a992","origin":"The Stacks Project","memory_eligible":false,"source_rank":11884,"rank":11884,"depth":59,"x":218.585,"y":810.803,"cluster":"derived-categories"},{"id":"stacks:0CZH","tag":"0CZH","title":"Being proper over a base · Lemma 0CZH","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let T_i ⊂ |X|, i = 1, …, n be closed subsets. If T_i, i = 1, …, n are proper over Y, then the same is true for T_1 ∪ … ∪ T_n.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is locally of finite type.\nLet $T_i \\subset |X|$, $i = 1, \\ldots, n$ be closed subsets.\nIf $T_i$, $i = 1, \\ldots, n$ are proper over $Y$, then the same is\ntrue for $T_1 \\cup \\ldots \\cup T_n$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZH","source_file":"spaces-perfect.tex","source_line":992,"source_end_line":1000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L992-L1000","statement_sha256":"6829437f83b0bbae29ef39d94639faf15da5b8e5018c832c598d377f3140da74","origin":"The Stacks Project","memory_eligible":false,"source_rank":11885,"rank":11885,"depth":60,"x":133.142,"y":576.978,"cluster":"derived-categories"},{"id":"stacks:0CZI","tag":"0CZI","title":"Being proper over a base · Lemma 0CZI","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let F be a finite type, quasi-coherent O_X-module. The following are equivalent • the support of F is proper over Y, • the scheme theoretic support of F (Morphisms of Spaces, Definition [Tag 07U1]) is proper over Y, and • there exists a closed subspace Z ⊂ X and a finite type, quasi-coherent O_Z-module G such that (a) Z → Y is proper, and (b) (Z → X)_*G = F.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $S$ which is locally of finite type.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent\n$\\mathcal{O}_X$-module. The following are equivalent\n\\begin{enumerate}\n\\item the support of $\\mathcal{F}$ is proper over $Y$,\n\\item the scheme theoretic support of $\\mathcal{F}$\n(Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-scheme-theoretic-support})\nis proper over $Y$, and\n\\item there exists a closed subspace $Z \\subset X$ and\na finite type, quasi-coherent $\\mathcal{O}_Z$-module\n$\\mathcal{G}$ such that (a) $Z \\to Y$ is proper, and (b)\n$(Z \\to X)_*\\mathcal{G} = \\mathcal{F}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZI","source_file":"spaces-perfect.tex","source_line":1030,"source_end_line":1047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1030-L1047","statement_sha256":"ab76c1010e6dd416ea7a2d5a5e0682e0278ec44c0aec1022f36e01aea7d3a0d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11886,"rank":11886,"depth":61,"x":384.408,"y":701.026,"cluster":"derived-categories"},{"id":"stacks:0CZJ","tag":"0CZJ","title":"Being proper over a base · Lemma 0CZJ","summary":"Let S be a scheme. Consider a cartesian diagram of algebraic spaces over S xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f Y' ar[r]^g & Y with f locally of finite type. Let F be a finite type, quasi-coherent O_X-module. If the support of F is proper over Y, then the support of (g')^*F is proper over Y'.","statement_latex":"Let $S$ be a scheme.\nConsider a cartesian diagram of algebraic spaces over $S$\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nwith $f$ locally of finite type. Let $\\mathcal{F}$ be a\nfinite type, quasi-coherent $\\mathcal{O}_X$-module.\nIf the support of $\\mathcal{F}$ is proper over $Y$, then\nthe support of $(g')^*\\mathcal{F}$ is proper over $Y'$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZJ","source_file":"spaces-perfect.tex","source_line":1066,"source_end_line":1080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1066-L1080","statement_sha256":"9f62609d294b0d8bf0d4c610cbc9079920575919a5f40db2175a6196afa44467","origin":"The Stacks Project","memory_eligible":false,"source_rank":11887,"rank":11887,"depth":56,"x":99.116,"y":752.175,"cluster":"derived-categories"},{"id":"stacks:0CZK","tag":"0CZK","title":"Being proper over a base · Lemma 0CZK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let F, G be finite type, quasi-coherent O_X-module. • If the supports of F, G are proper over Y, then the same is true for F ⊕ G, for any extension of G by F, for Im(u) and Coker(u) given any O_X-module map u : F → G, and for any quasi-coherent quotient of F or G. • If Y is locally Noetherian, then the category of coherent O_X-modules with support proper over Y is a…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is locally of finite type. Let $\\mathcal{F}$, $\\mathcal{G}$\nbe finite type, quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If the supports of $\\mathcal{F}$, $\\mathcal{G}$\nare proper over $Y$, then the same is true\nfor $\\mathcal{F} \\oplus \\mathcal{G}$, for any extension\nof $\\mathcal{G}$ by $\\mathcal{F}$, for $\\Im(u)$ and $\\Coker(u)$\ngiven any $\\mathcal{O}_X$-module map $u : \\mathcal{F} \\to \\mathcal{G}$,\nand for any quasi-coherent quotient of $\\mathcal{F}$ or $\\mathcal{G}$.\n\\item If $Y$ is locally Noetherian, then the category of\ncoherent $\\mathcal{O}_X$-modules with support proper over\n$Y$ is a Serre subcategory (Homology, Definition\n\\ref{homology-definition-serre-subcategory})\nof the abelian category of\ncoherent $\\mathcal{O}_X$-modules.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZK","source_file":"spaces-perfect.tex","source_line":1092,"source_end_line":1112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1092-L1112","statement_sha256":"79a78df0db6e03e4e5e96e55e9f37399eb2f9d89ad45839539b0475394bf6e04","origin":"The Stacks Project","memory_eligible":false,"source_rank":11888,"rank":11888,"depth":61,"x":268.505,"y":552.397,"cluster":"derived-categories"},{"id":"stacks:08GC","tag":"08GC","title":"Being proper over a base · Lemma 08GC","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type and Y locally Noetherian. Let F be a coherent O_X-module with support proper over Y. Then R^pf_*F is a coherent O_Y-module for all p ≥ 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is locally of finite type and $Y$ locally Noetherian.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module with support\nproper over $Y$. Then $R^pf_*\\mathcal{F}$ is a coherent\n$\\mathcal{O}_Y$-module for all $p \\geq 0$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Being proper over a base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GC","source_file":"spaces-perfect.tex","source_line":1137,"source_end_line":1144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1137-L1144","statement_sha256":"b33d3dbc8ccab09ecae3c25287d89cc2ea37aaf05b7011ef272d9c1c58bd970e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11889,"rank":11889,"depth":63,"x":304.288,"y":796.045,"cluster":"derived-categories"},{"id":"stacks:08GK","tag":"08GK","title":"Derived category of coherent modules · Lemma 08GK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type and Y is Noetherian. Let E be an object of D^b_Coh(O_X) such that the support of H^i(E) is proper over Y for all i. Then Rf_*E is an object of D^b_Coh(O_Y).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian.\nLet $E$ be an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the\nsupport of $H^i(E)$ is proper over $Y$ for all $i$.\nThen $Rf_*E$ is an object of $D^b_{\\textit{Coh}}(\\mathcal{O}_Y)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GK","source_file":"spaces-perfect.tex","source_line":1193,"source_end_line":1200,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1193-L1200","statement_sha256":"06ec5a3f67e4008003e152d795dced65fcbd2a3b03b9619470e83deee3585b7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11890,"rank":11890,"depth":64,"x":81.769,"y":636.544,"cluster":"derived-categories"},{"id":"stacks:0D0R","tag":"0D0R","title":"Derived category of coherent modules · Lemma 0D0R","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type and Y is Noetherian. Let E be an object of D^+_Coh(O_X) such that the support of H^i(E) is proper over S for all i. Then Rf_*E is an object of D^+_Coh(O_Y).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian.\nLet $E$ be an object of\n$D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ such that the support of $H^i(E)$\nis proper over $S$ for all $i$.\nThen $Rf_*E$ is an object of $D^+_{\\textit{Coh}}(\\mathcal{O}_Y)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0R","source_file":"spaces-perfect.tex","source_line":1219,"source_end_line":1227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1219-L1227","statement_sha256":"33304ec8a17f89d6e4277c70b7ac95947d957a2fb5d05df4e542d698cdca374e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11891,"rank":11891,"depth":65,"x":374.378,"y":627.887,"cluster":"derived-categories"},{"id":"stacks:0D0S","tag":"0D0S","title":"Derived category of coherent modules · Lemma 0D0S","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. If L is in D^+_Coh(O_X) and K in D^-_Coh(O_X), then RSheafHom(K, L) is in D^+_Coh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nIf $L$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$\nand $K$ in $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, then\n$R\\SheafHom(K, L)$ is in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0S","source_file":"spaces-perfect.tex","source_line":1239,"source_end_line":1245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1239-L1245","statement_sha256":"b4421138e280e27e6052340461d9fea2737d6254c2c95d20da59d0ec80898305","origin":"The Stacks Project","memory_eligible":false,"source_rank":11892,"rank":11892,"depth":31,"x":165.388,"y":800.458,"cluster":"derived-categories"},{"id":"stacks:0D0T","tag":"0D0T","title":"Derived category of coherent modules · Lemma 0D0T","summary":"Let A be a Noetherian ring. Let X be a proper algebraic space over A. For L in D^+_Coh(O_X) and K in D^-_Coh(O_X), the A-modules Ext_O_X^n(K, L) are finite.","statement_latex":"Let $A$ be a Noetherian ring. Let $X$ be a proper algebraic space over $A$.\nFor $L$ in $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and $K$ in\n$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$, the $A$-modules\n$\\Ext_{\\mathcal{O}_X}^n(K, L)$ are finite.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived category of coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0T","source_file":"spaces-perfect.tex","source_line":1255,"source_end_line":1261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1255-L1261","statement_sha256":"0d87c87e7c594f1af356e5442d61cab93f7604326cf5eec8dfc78cb7c9633f5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11893,"rank":11893,"depth":66,"x":180.731,"y":554.402,"cluster":"derived-categories"},{"id":"stacks:08GM","tag":"08GM","title":"Induction principle · Definition 08GM","summary":"Let S be a scheme. A commutative diagram xymatrix U ×_W V ar[r] ar[d] & V ar[d]^f U ar[r]^j & W of algebraic spaces over S is called an elementary distinguished square if • U is an open subspace of W and j is the inclusion morphism, • f is étale, and • setting T = W setminus U (with reduced induced subspace structure) the morphism f^-1(T) → T is an isomorphism. We will indicate this by saying: \"Let (U ⊂ W, f : V → W) be an elementary distinguished square.\"","statement_latex":"Let $S$ be a scheme. A commutative diagram\n$$\n\\xymatrix{\nU \\times_W V \\ar[r] \\ar[d] & V \\ar[d]^f \\\\\nU \\ar[r]^j & W\n}\n$$\nof algebraic spaces over $S$ is called an {\\it elementary distinguished square}\nif\n\\begin{enumerate}\n\\item $U$ is an open subspace of $W$ and $j$ is the inclusion morphism,\n\\item $f$ is \\'etale, and\n\\item setting $T = W \\setminus U$ (with reduced induced\nsubspace structure) the morphism $f^{-1}(T) \\to T$ is an isomorphism.\n\\end{enumerate}\nWe will indicate this by saying: ``Let $(U \\subset W, f : V \\to W)$\nbe an elementary distinguished square.''","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Induction principle","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GM","source_file":"spaces-perfect.tex","source_line":1295,"source_end_line":1314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1295-L1314","statement_sha256":"c2a4555a6536c13d5ab901bb2eb0a44949ba30fee7f1e55f111a7981def71ca3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11894,"rank":11894,"depth":0,"x":367.455,"y":744.715,"cluster":"derived-categories"},{"id":"stacks:08GN","tag":"08GN","title":"Induction principle · Lemma 08GN","summary":"Let S be a scheme. Let (U ⊂ W, f : V → W) be an elementary distinguished square of algebraic spaces over S. • If V' ⊂ V and U ⊂ U' ⊂ W are open subspaces and W' = U' ∪ f(V') then (U' ⊂ W', f|_V' : V' → W') is an elementary distinguished square. • If p : W' → W is a morphism of algebraic spaces, then (p^-1(U) ⊂ W', V ×_W W' → W') is an elementary distinguished square. • If S' → S is a morphism of schemes, then (S' ×_S U ⊂ S' ×_S W, S' ×_S V → S' ×_S W) is an elementary…","statement_latex":"Let $S$ be a scheme. Let $(U \\subset W, f : V \\to W)$ be an elementary\ndistinguished square of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $V' \\subset V$ and\n$U \\subset U' \\subset W$ are open subspaces and $W' = U' \\cup f(V')$\nthen $(U' \\subset W', f|_{V'} : V' \\to W')$ is an elementary distinguished\nsquare.\n\\item If $p : W' \\to W$ is a morphism of algebraic spaces, then\n$(p^{-1}(U) \\subset W', V \\times_W W' \\to W')$ is an elementary distinguished\nsquare.\n\\item If $S' \\to S$ is a morphism of schemes, then\n$(S' \\times_S U \\subset S' \\times_S W, S' \\times_S V \\to S' \\times_S W)$\nis an elementary distinguished square.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Induction principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GN","source_file":"spaces-perfect.tex","source_line":1323,"source_end_line":1339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1323-L1339","statement_sha256":"64ad24b120078341455c33841a74cab1e122d6262ea15b02f370ac16d008b93c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11895,"rank":11895,"depth":0,"x":76.465,"y":710.302,"cluster":"derived-categories"},{"id":"stacks:08GP","tag":"08GP","title":"Induction principle · Lemma 08GP","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let P be a property of the quasi-compact and quasi-separated objects of X_spaces, etale. Assume that • P holds for every affine object of X_spaces, etale, • for every elementary distinguished square (U ⊂ W, f : V → W) such that • W is a quasi-compact and quasi-separated object of X_spaces, etale, • U is quasi-compact, • V is affine, and • P holds for U, V, and U ×_W V, then P holds for…","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $P$ be a property of the quasi-compact\nand quasi-separated objects of $X_{spaces, \\etale}$. Assume that\n\\begin{enumerate}\n\\item $P$ holds for every affine object of $X_{spaces, \\etale}$,\n\\item for every elementary distinguished square $(U \\subset W, f : V \\to W)$\nsuch that\n\\begin{enumerate}\n\\item $W$ is a quasi-compact and quasi-separated object of\n$X_{spaces, \\etale}$,\n\\item $U$ is quasi-compact,\n\\item $V$ is affine, and\n\\item $P$ holds for $U$, $V$, and $U \\times_W V$,\n\\end{enumerate}\nthen $P$ holds for $W$.\n\\end{enumerate}\nThen $P$ holds for every quasi-compact and quasi-separated object\nof $X_{spaces, \\etale}$ and in particular for $X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Induction principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GP","source_file":"spaces-perfect.tex","source_line":1345,"source_end_line":1365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1345-L1365","statement_sha256":"dfd8383a7f15c6c883f0a8ad6993eac16aff8697ece761d14dd9259b501d6647","origin":"The Stacks Project","memory_eligible":false,"source_rank":11896,"rank":11896,"depth":59,"x":318.925,"y":570.44,"cluster":"derived-categories"},{"id":"stacks:08GQ","tag":"08GQ","title":"Induction principle · Lemma 08GQ","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let B ⊂ Ob(X_spaces, etale). Let P be a property of the elements of B. Assume that • every W ∈ B is quasi-compact and quasi-separated, • if W ∈ B and U ⊂ W is quasi-compact open, then U ∈ B, • if V ∈ Ob(X_spaces, etale) is affine, then (a) V ∈ B and (b) P holds for V, • for every elementary distinguished square (U ⊂ W, f : V → W) such that • W ∈ B, • U is quasi-compact, • V is affine,…","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let\n$\\mathcal{B} \\subset \\Ob(X_{spaces, \\etale})$.\nLet $P$ be a property of the elements of $\\mathcal{B}$.\nAssume that\n\\begin{enumerate}\n\\item every $W \\in \\mathcal{B}$ is quasi-compact and quasi-separated,\n\\item if $W \\in \\mathcal{B}$ and $U \\subset W$ is quasi-compact open, then\n$U \\in \\mathcal{B}$,\n\\item if $V \\in \\Ob(X_{spaces, \\etale})$ is affine, then\n(a) $V \\in \\mathcal{B}$ and (b) $P$ holds for $V$,\n\\item for every elementary distinguished square $(U \\subset W, f : V \\to W)$\nsuch that\n\\begin{enumerate}\n\\item $W \\in \\mathcal{B}$,\n\\item $U$ is quasi-compact,\n\\item $V$ is affine, and\n\\item $P$ holds for $U$, $V$, and $U \\times_W V$,\n\\end{enumerate}\nthen $P$ holds for $W$.\n\\end{enumerate}\nThen $P$ holds for every $W \\in \\mathcal{B}$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Induction principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GQ","source_file":"spaces-perfect.tex","source_line":1425,"source_end_line":1449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1425-L1449","statement_sha256":"3753f509519076cb40bfac14b2c0eab3644b23f2090092b8feb9f738a2ae271a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11897,"rank":11897,"depth":60,"x":252.553,"y":811.362,"cluster":"derived-categories"},{"id":"stacks:09IT","tag":"09IT","title":"Induction principle · Lemma 09IT","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let W ⊂ X be a quasi-compact open subspace. Let P be a property of quasi-compact open subspaces of X. Assume that • P holds for W, and • for every elementary distinguished square (W_1 ⊂ W_2, f : V → W_2) where such that • W_1, W_2 are quasi-compact open subspaces of X, • W ⊂ W_1, • V is affine, and • P holds for W_1, then P holds for W_2. Then P holds for X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $W \\subset X$ be a quasi-compact open\nsubspace. Let $P$ be a property of quasi-compact open subspaces of $X$.\nAssume that\n\\begin{enumerate}\n\\item $P$ holds for $W$, and\n\\item for every elementary distinguished square\n$(W_1 \\subset W_2, f : V \\to W_2)$ where \nsuch that\n\\begin{enumerate}\n\\item $W_1$, $W_2$ are quasi-compact open subspaces of $X$,\n\\item $W \\subset W_1$,\n\\item $V$ is affine, and\n\\item $P$ holds for $W_1$,\n\\end{enumerate}\nthen $P$ holds for $W_2$.\n\\end{enumerate}\nThen $P$ holds for $X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Induction principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IT","source_file":"spaces-perfect.tex","source_line":1489,"source_end_line":1509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1489-L1509","statement_sha256":"fe0645716af34ce088c52f3ef51a3f041239a9fcd351e9fed2541db3ae013e30","origin":"The Stacks Project","memory_eligible":false,"source_rank":11898,"rank":11898,"depth":61,"x":107.624,"y":595.859,"cluster":"derived-categories"},{"id":"stacks:08GV","tag":"08GV","title":"Mayer-Vietoris · Lemma 08GV","summary":"Let S be a scheme. Let (U ⊂ X, V → X) be an elementary distinguished square of algebraic spaces over S. • For a sheaf of O_X-modules F we have a short exact sequence 0 → j_U ×_X V!F|_U ×_X V → j_U!F|_U ⊕ j_V!F|_V → F → 0 • For an object E of D(O_X) we have a distinguished triangle j_U ×_X V!E|_U ×_X V → j_U!E|_U ⊕ j_V!E|_V → E → j_U ×_X V!E|_U ×_X V[1] in D(O_X).","statement_latex":"Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary\ndistinguished square of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item For a sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$\nwe have a short exact sequence\n$$\n0 \\to j_{U \\times_X V!}\\mathcal{F}|_{U \\times_X V} \\to\nj_{U!}\\mathcal{F}|_U \\oplus j_{V!}\\mathcal{F}|_V \\to \\mathcal{F} \\to 0\n$$\n\\item For an object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished\ntriangle\n$$\nj_{U \\times_X V!}E|_{U \\times_X V} \\to\nj_{U!}E|_U \\oplus j_{V!}E|_V \\to E \\to \nj_{U \\times_X V!}E|_{U \\times_X V}[1]\n$$\nin $D(\\mathcal{O}_X)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GV","source_file":"spaces-perfect.tex","source_line":1605,"source_end_line":1625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1605-L1625","statement_sha256":"d3eec50ac0296437d47db57a2c3aeaacaa6bca994a991cfbc594974b73c62b83","origin":"The Stacks Project","memory_eligible":false,"source_rank":11899,"rank":11899,"depth":53,"x":388.042,"y":672.599,"cluster":"derived-categories"},{"id":"stacks:08GW","tag":"08GW","title":"Mayer-Vietoris · Lemma 08GW","summary":"Let S be a scheme. Let (U ⊂ X, V → X) be an elementary distinguished square of algebraic spaces over S. • For every sheaf of O_X-modules F we have a short exact sequence 0 → F → j_U, *F|_U ⊕ j_V, *F|_V → j_U ×_X V, *F|_U ×_X V → 0 • For any object E of D(O_X) we have a distinguished triangle E → Rj_U, *E|_U ⊕ Rj_V, *E|_V → Rj_U ×_X V, *E|_U ×_X V → E[1] in D(O_X).","statement_latex":"Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary\ndistinguished square of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item For every sheaf of $\\mathcal{O}_X$-modules $\\mathcal{F}$\nwe have a short exact sequence\n$$\n0 \\to \\mathcal{F} \\to\nj_{U, *}\\mathcal{F}|_U \\oplus j_{V, *}\\mathcal{F}|_V \\to\nj_{U \\times_X V, *}\\mathcal{F}|_{U \\times_X V} \\to 0\n$$\n\\item For any object $E$ of $D(\\mathcal{O}_X)$ we have a distinguished\ntriangle\n$$\nE \\to \nRj_{U, *}E|_U \\oplus Rj_{V, *}E|_V \\to\nRj_{U \\times_X V, *}E|_{U \\times_X V} \\to\nE[1]\n$$\nin $D(\\mathcal{O}_X)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GW","source_file":"spaces-perfect.tex","source_line":1671,"source_end_line":1693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1671-L1693","statement_sha256":"2484e7497127f5db3aab114e7e4b11db19273da920f5c743f27319e82b034a10","origin":"The Stacks Project","memory_eligible":false,"source_rank":11900,"rank":11900,"depth":54,"x":119.317,"y":775.218,"cluster":"derived-categories"},{"id":"stacks:08JK","tag":"08JK","title":"Mayer-Vietoris · Lemma 08JK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let (U ⊂ X, V → X) be an elementary distinguished square. Denote a = f|_U : U → Y, b = f|_V : V → Y, and c = f|_U ×_X V : U ×_X V → Y the restrictions. For every object E of D(O_X) there exists a distinguished triangle Rf_*E → Ra_*(E|_U) ⊕ Rb_*(E|_V) → Rc_*(E|_U ×_X V) → Rf_*E[1] in D(O_Y). This triangle is functorial in E.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $(U \\subset X, V \\to X)$ be an elementary distinguished square.\nDenote $a = f|_U : U \\to Y$, $b = f|_V : V \\to Y$, and\n$c = f|_{U \\times_X V} : U \\times_X V \\to Y$ the restrictions.\nFor every object $E$ of $D(\\mathcal{O}_X)$ there exists a\ndistinguished triangle\n$$\nRf_*E \\to\nRa_*(E|_U) \\oplus Rb_*(E|_V) \\to\nRc_*(E|_{U \\times_X V}) \\to\nRf_*E[1]\n$$\nin $D(\\mathcal{O}_Y)$. This triangle is functorial in $E$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JK","source_file":"spaces-perfect.tex","source_line":1761,"source_end_line":1776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1761-L1776","statement_sha256":"bdc4fdf82de61e35005ce993e5e73b72618efc0d14abac760a1c6219de4614c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11901,"rank":11901,"depth":55,"x":235.049,"y":546.867,"cluster":"derived-categories"},{"id":"stacks:08H9","tag":"08H9","title":"Mayer-Vietoris · Lemma 08H9","summary":"Let S be a scheme. Let (U ⊂ X, V → X) be an elementary distinguished square of algebraic spaces over S. For objects E, F of D(O_X) we have a Mayer-Vietoris sequence xymatrix & … ar[r] & Ext^-1(E_U ×_X V, F_U ×_X V) ar[lld] Hom(E, F) ar[r] & Hom(E_U, F_U) ⊕ Hom(E_V, F_V) ar[r] & Hom(E_U ×_X V, F_U ×_X V) where the subscripts denote restrictions to the relevant opens and the Hom's are taken in the relevant derived categories.","statement_latex":"Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary\ndistinguished square of algebraic spaces over $S$.\nFor objects $E$, $F$ of $D(\\mathcal{O}_X)$ we have a\nMayer-Vietoris sequence\n$$\n\\xymatrix{\n& \\ldots \\ar[r] &\n\\Ext^{-1}(E_{U \\times_X V}, F_{U \\times_X V}) \\ar[lld] \\\\\n\\Hom(E, F) \\ar[r] &\n\\Hom(E_U, F_U) \\oplus\n\\Hom(E_V, F_V) \\ar[r] &\n\\Hom(E_{U \\times_X V}, F_{U \\times_X V})\n}\n$$\nwhere the subscripts denote restrictions to the relevant opens\nand the $\\Hom$'s are taken in the relevant derived categories.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08H9","source_file":"spaces-perfect.tex","source_line":1820,"source_end_line":1838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1820-L1838","statement_sha256":"a7a3a2873fbf90c8c6a9b4077c98cf8bc7d9e3a420941408af86db1c80f3f2ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":11902,"rank":11902,"depth":54,"x":333.429,"y":781.123,"cluster":"derived-categories"},{"id":"stacks:0CRS","tag":"0CRS","title":"Mayer-Vietoris · Lemma 0CRS","summary":"Let S be a scheme. Let (U ⊂ X, V → X) be an elementary distinguished square of algebraic spaces over S. For an object E of D(O_X) we have a distinguished triangle RΓ(X, E) → RΓ(U, E) ⊕ RΓ(V, E) → RΓ(U ×_X V, E) → RΓ(X, E)[1] and in particular a long exact cohomology sequence … → H^n(X, E) → H^n(U, E) ⊕ H^n(V, E) → H^n(U ×_X V, E) → H^n + 1(X, E) → … The construction of the distinguished triangle and the long exact sequence is functorial in E.","statement_latex":"Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary\ndistinguished square of algebraic spaces over $S$. For an object $E$\nof $D(\\mathcal{O}_X)$ we have a distinguished triangle\n$$\nR\\Gamma(X, E) \\to R\\Gamma(U, E) \\oplus R\\Gamma(V, E) \\to\nR\\Gamma(U \\times_X V, E) \\to R\\Gamma(X, E)[1]\n$$\nand in particular a long exact cohomology sequence\n$$\n\\ldots \\to\nH^n(X, E) \\to\nH^n(U, E) \\oplus H^n(V, E) \\to\nH^n(U \\times_X V, E) \\to\nH^{n + 1}(X, E) \\to \\ldots\n$$\nThe construction of the distinguished triangle and the\nlong exact sequence is functorial in $E$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRS","source_file":"spaces-perfect.tex","source_line":1850,"source_end_line":1869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1850-L1869","statement_sha256":"f68a2766df96c23d8daac9e4fd7a60b543c44822599c69f6801647019476e236","origin":"The Stacks Project","memory_eligible":false,"source_rank":11903,"rank":11903,"depth":55,"x":72.275,"y":664.108,"cluster":"derived-categories"},{"id":"stacks:08HA","tag":"08HA","title":"Mayer-Vietoris · Lemma 08HA","summary":"Let S be a scheme. Let j : U → X be a étale morphism of algebraic spaces over S. Given an étale morphism V → Y, set W = V ×_X U and denote j_W : W → V the projection morphism. Then (j_!E)|_V = j_W!(E|_W) for E in D(O_U).","statement_latex":"Let $S$ be a scheme. Let $j : U \\to X$ be a \\'etale morphism of algebraic\nspaces over $S$. Given an \\'etale morphism $V \\to Y$, set $W = V \\times_X U$\nand denote $j_W : W \\to V$ the projection morphism. Then\n$(j_!E)|_V = j_{W!}(E|_W)$ for $E$ in $D(\\mathcal{O}_U)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HA","source_file":"spaces-perfect.tex","source_line":1901,"source_end_line":1907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1901-L1907","statement_sha256":"2fba91edbebd4d7bb4aac15f2acf7e2266661b882c154a9b4417f19a6f94415f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11904,"rank":11904,"depth":2,"x":359.197,"y":602.154,"cluster":"derived-categories"},{"id":"stacks:08GG","tag":"08GG","title":"Mayer-Vietoris · Lemma 08GG","summary":"Let S be a scheme. Let (U ⊂ X, j : V → X) be an elementary distinguished square of algebraic spaces over S. Set T = |X| setminus |U|. • If E is an object of D(O_X) supported on T, then (a) E → Rj_*(E|_V) and (b) j_!(E|_V) → E are isomorphisms. • If F is an object of D(O_V) supported on j^-1T, then (a) F → (j_!F)|_V, (b) (Rj_*F)|_V → F, and (c) j_!F → Rj_*F are isomorphisms.","statement_latex":"Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an elementary\ndistinguished square of algebraic spaces over $S$. Set\n$T = |X| \\setminus |U|$.\n\\begin{enumerate}\n\\item If $E$ is an object of $D(\\mathcal{O}_X)$ supported on $T$, then\n(a) $E \\to Rj_*(E|_V)$ and (b) $j_!(E|_V) \\to E$ are isomorphisms.\n\\item If $F$ is an object of $D(\\mathcal{O}_V)$ supported on $j^{-1}T$, then\n(a) $F \\to (j_!F)|_V$, (b) $(Rj_*F)|_V \\to F$, and (c)\n$j_!F \\to Rj_*F$ are isomorphisms.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GG","source_file":"spaces-perfect.tex","source_line":1917,"source_end_line":1929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1917-L1929","statement_sha256":"c5946e5fe7ac8f804726c71759670323f28bb415747e5d582851479714b70c53","origin":"The Stacks Project","memory_eligible":false,"source_rank":11905,"rank":11905,"depth":55,"x":197.305,"y":810.826,"cluster":"derived-categories"},{"id":"stacks:08HB","tag":"08HB","title":"Mayer-Vietoris · Lemma 08HB","summary":"Let S be a scheme. Let (U ⊂ X, V → X) be an elementary distinguished square of algebraic spaces over S. Suppose given • an object A of D(O_U), • an object B of D(O_V), and • an isomorphism c : A|_U ×_X V → B|_U ×_X V. Then there exists an object F of D(O_X) and isomorphisms f : F|_U → A, g : F|_V → B such that c = g|_U ×_X V ∘ f^-1|_U ×_X V. Moreover, given • an object E of D(O_X), • a morphism a : A → E|_U of D(O_U), • a morphism b : B → E|_V of D(O_V), such that a|_U…","statement_latex":"Let $S$ be a scheme. Let $(U \\subset X, V \\to X)$ be an elementary\ndistinguished square of algebraic spaces over $S$. Suppose given\n\\begin{enumerate}\n\\item an object $A$ of $D(\\mathcal{O}_U)$,\n\\item an object $B$ of $D(\\mathcal{O}_V)$, and\n\\item an isomorphism $c : A|_{U \\times_X V} \\to B|_{U \\times_X V}$.\n\\end{enumerate}\nThen there exists an object $F$ of $D(\\mathcal{O}_X)$\nand isomorphisms $f : F|_U \\to A$, $g : F|_V \\to B$ such\nthat $c = g|_{U \\times_X V} \\circ f^{-1}|_{U \\times_X V}$.\nMoreover, given\n\\begin{enumerate}\n\\item an object $E$ of $D(\\mathcal{O}_X)$,\n\\item a morphism $a : A \\to E|_U$ of $D(\\mathcal{O}_U)$,\n\\item a morphism $b : B \\to E|_V$ of $D(\\mathcal{O}_V)$,\n\\end{enumerate}\nsuch that\n$$\na|_{U \\times_X V}  = b|_{U \\times_X V} \\circ c.\n$$\nThen there exists a morphism $F \\to E$ in $D(\\mathcal{O}_X)$\nwhose restriction to $U$ is $a \\circ f$\nand whose restriction to $V$ is $b \\circ g$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Mayer-Vietoris","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HB","source_file":"spaces-perfect.tex","source_line":1988,"source_end_line":2013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L1988-L2013","statement_sha256":"d4b1028cccba7ef83d43eaa3ff827142682f92170bc7149a9e4e3fc88d211713","origin":"The Stacks Project","memory_eligible":false,"source_rank":11906,"rank":11906,"depth":56,"x":148.832,"y":564.88,"cluster":"derived-categories"},{"id":"stacks:08GY","tag":"08GY","title":"The coherator · Lemma 08GY","summary":"Let S be a scheme. Let f : X → Y be an affine morphism of algebraic spaces over S. Then f_* defines a derived functor f_* : D(QCoh(O_X)) → D(QCoh(O_Y)). This functor has the property that xymatrix D(QCoh(O_X)) ar[d]_f_* ar[r] & D_QCoh(O_X) ar[d]^Rf_* D(QCoh(O_Y)) ar[r] & D_QCoh(O_Y) commutes.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an affine morphism of\nalgebraic spaces over $S$. Then $f_*$ defines a derived functor\n$f_* : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$.\nThis functor has the property that\n$$\n\\xymatrix{\nD(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{f_*} \\ar[r] &\nD_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\\nD(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &\nD_\\QCoh(\\mathcal{O}_Y)\n}\n$$\ncommutes.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GY","source_file":"spaces-perfect.tex","source_line":2121,"source_end_line":2136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2121-L2136","statement_sha256":"ccc3197693a1db26aedb9e21999c38bf5d4942af9e3eea99f9f9292cc5a49a52","origin":"The Stacks Project","memory_eligible":false,"source_rank":11907,"rank":11907,"depth":33,"x":382.561,"y":718.865,"cluster":"derived-categories"},{"id":"stacks:08GZ","tag":"08GZ","title":"The coherator · Lemma 08GZ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-compact, quasi-separated, and flat. Then, denoting Φ : D(QCoh(O_X)) → D(QCoh(O_Y)) the right derived functor of f_* : QCoh(O_X) → QCoh(O_Y) we have RQ_Y ∘ Rf_* = Φ ∘ RQ_X.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume $f$ is quasi-compact, quasi-separated, and flat.\nThen, denoting\n$$\n\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))\n$$\nthe right derived functor of\n$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$\nwe have $RQ_Y \\circ Rf_* = \\Phi \\circ RQ_X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08GZ","source_file":"spaces-perfect.tex","source_line":2160,"source_end_line":2171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2160-L2171","statement_sha256":"bc231ce6383955c89aaf512ae928477b614f0697a78eefccf523328cf6cb3a74","origin":"The Stacks Project","memory_eligible":false,"source_rank":11908,"rank":11908,"depth":53,"x":86.125,"y":737.959,"cluster":"derived-categories"},{"id":"stacks:08H0","tag":"08H0","title":"The coherator · Lemma 08H0","summary":"Let S be a scheme. Let X be an affine algebraic space over S. Set A = Γ(X, O_X). Then • Q_X : Mod(O_X) → QCoh(O_X) is the functor which sends F to the quasi-coherent O_X-module associated to the A-module Γ(X, F), • RQ_X : D(O_X) → D(QCoh(O_X)) is the functor which sends E to the complex of quasi-coherent O_X-modules associated to the object RΓ(X, E) of D(A), • restricted to D_QCoh(O_X) the functor RQ_X defines a quasi-inverse to ([Tag 08F1]).","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine algebraic space over $S$.\nSet $A = \\Gamma(X, \\mathcal{O}_X)$. Then\n\\begin{enumerate}\n\\item $Q_X : \\textit{Mod}(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_X)$\nis the functor\nwhich sends $\\mathcal{F}$ to the quasi-coherent $\\mathcal{O}_X$-module\nassociated to the $A$-module $\\Gamma(X, \\mathcal{F})$,\n\\item $RQ_X : D(\\mathcal{O}_X) \\to D(\\QCoh(\\mathcal{O}_X))$\nis the functor which sends $E$ to the complex of quasi-coherent\n$\\mathcal{O}_X$-modules associated to the object $R\\Gamma(X, E)$ of $D(A)$,\n\\item restricted to $D_\\QCoh(\\mathcal{O}_X)$ the functor\n$RQ_X$ defines a quasi-inverse to (\\ref{equation-compare}).\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08H0","source_file":"spaces-perfect.tex","source_line":2223,"source_end_line":2238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2223-L2238","statement_sha256":"6c52956c4a9f89f21a2cac17d79fd0895b1e68ed12bc7d3a5c0adc404a77b914","origin":"The Stacks Project","memory_eligible":false,"source_rank":11909,"rank":11909,"depth":41,"x":289.534,"y":555.517,"cluster":"derived-categories"},{"id":"stacks:09TG","tag":"09TG","title":"The coherator · Lemma 09TG","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Suppose that for every étale morphism j : V → W with W ⊂ X quasi-compact open and V affine the right derived functor Φ : D(QCoh(O_U)) → D(QCoh(O_W)) of the left exact functor j_* : QCoh(O_V) → QCoh(O_W) fits into a commutative diagram xymatrix D(QCoh(O_V)) ar[d]_Φ ar[r]_i_V & D_QCoh(O_V) ar[d]^Rj_* D(QCoh(O_W)) ar[r]^i_W & D_QCoh(O_W) Then the functor ([Tag 08F1]) D(QCoh(O_X)) →…","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Suppose that for every \\'etale morphism\n$j : V \\to W$ with $W \\subset X$ quasi-compact open and $V$ affine\nthe right derived functor\n$$\n\\Phi : D(\\QCoh(\\mathcal{O}_U)) \\to D(\\QCoh(\\mathcal{O}_W))\n$$\nof the left exact functor\n$j_* : \\QCoh(\\mathcal{O}_V) \\to \\QCoh(\\mathcal{O}_W)$\nfits into a commutative diagram\n$$\n\\xymatrix{\nD(\\QCoh(\\mathcal{O}_V)) \\ar[d]_\\Phi \\ar[r]_{i_V} &\nD_\\QCoh(\\mathcal{O}_V) \\ar[d]^{Rj_*} \\\\\nD(\\QCoh(\\mathcal{O}_W)) \\ar[r]^{i_W} &\nD_\\QCoh(\\mathcal{O}_W)\n}\n$$\nThen the functor (\\ref{equation-compare})\n$$\nD(\\QCoh(\\mathcal{O}_X))\n\\longrightarrow\nD_\\QCoh(\\mathcal{O}_X)\n$$\nis an equivalence with quasi-inverse given by $RQ_X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TG","source_file":"spaces-perfect.tex","source_line":2272,"source_end_line":2299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2272-L2299","statement_sha256":"70cc2601d73e792dd910ec9900d8c2024ffcb3eb94e5ced2eb81b074dc69308d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11910,"rank":11910,"depth":62,"x":286.255,"y":805.681,"cluster":"derived-categories"},{"id":"stacks:08H1","tag":"08H1","title":"The coherator · Proposition 08H1","summary":"Let S be a scheme. Let X be a quasi-compact algebraic space over S with affine diagonal over Z (as in Properties of Spaces, Definition [Tag 03BS]). Then the functor ([Tag 08F1]) D(QCoh(O_X)) → D_QCoh(O_X) is an equivalence with quasi-inverse given by RQ_X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space over $S$\nwith affine diagonal over $\\mathbf{Z}$ (as in Properties of Spaces, Definition\n\\ref{spaces-properties-definition-separated}). Then the functor\n(\\ref{equation-compare})\n$$\nD(\\QCoh(\\mathcal{O}_X))\n\\longrightarrow\nD_\\QCoh(\\mathcal{O}_X)\n$$\nis an equivalence with quasi-inverse given by $RQ_X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08H1","source_file":"spaces-perfect.tex","source_line":2471,"source_end_line":2483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2471-L2483","statement_sha256":"4c9351a0d7a1b00fe6fd07ea0a07672a4195a2ff0d13ef4125818ec86af916b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11911,"rank":11911,"depth":63,"x":87.326,"y":619.193,"cluster":"derived-categories"},{"id":"stacks:0CSR","tag":"0CSR","title":"The coherator · Lemma 0CSR","summary":"Let S be a scheme and let f : X → Y be a morphism of algebraic spaces over S. Assume X and Y are quasi-compact and have affine diagonal over Z (as in Properties of Spaces, Definition [Tag 03BS]). Then, denoting Φ : D(QCoh(O_X)) → D(QCoh(O_Y)) the right derived functor of f_* : QCoh(O_X) → QCoh(O_Y) the diagram xymatrix D(QCoh(O_X)) ar[d]_Φ ar[r] & D_QCoh(O_X) ar[d]^Rf_* D(QCoh(O_Y)) ar[r] & D_QCoh(O_Y) is commutative.","statement_latex":"Let $S$ be a scheme and let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume $X$ and $Y$ are quasi-compact and have affine diagonal\nover $\\mathbf{Z}$ (as in Properties of Spaces, Definition\n\\ref{spaces-properties-definition-separated}). Then, denoting\n$$\n\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))\n$$\nthe right derived functor of\n$f_* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_Y)$\nthe diagram\n$$\n\\xymatrix{\nD(\\QCoh(\\mathcal{O}_X)) \\ar[d]_\\Phi \\ar[r] &\nD_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\\nD(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &\nD_\\QCoh(\\mathcal{O}_Y)\n}\n$$\nis commutative.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSR","source_file":"spaces-perfect.tex","source_line":2496,"source_end_line":2517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2496-L2517","statement_sha256":"9e81072d6490814ec9f1c804c250944483c8ded7d3dc84eb331e84406056a50e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11912,"rank":11912,"depth":64,"x":384.237,"y":643.851,"cluster":"derived-categories"},{"id":"stacks:09TI","tag":"09TI","title":"The coherator for Noetherian spaces · Lemma 09TI","summary":"Let S be a Noetherian affine scheme. Every injective object of QCoh(O_S) is a filtered colimit colim_i F_i of quasi-coherent sheaves of the form F_i = (Z_i → S)_*G_i where Z_i is the spectrum of an Artinian ring and G_i is a coherent module on Z_i.","statement_latex":"Let $S$ be a Noetherian affine scheme. Every injective object of\n$\\QCoh(\\mathcal{O}_S)$ is a filtered colimit $\\colim_i \\mathcal{F}_i$\nof quasi-coherent sheaves of the form\n$$\n\\mathcal{F}_i = (Z_i \\to S)_*\\mathcal{G}_i\n$$\nwhere $Z_i$ is the spectrum of an Artinian ring and $\\mathcal{G}_i$\nis a coherent module on $Z_i$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator for Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TI","source_file":"spaces-perfect.tex","source_line":2613,"source_end_line":2623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2613-L2623","statement_sha256":"309ce8b7309dff92516b0ae50cbcb7b38575cd3fcb43c15310b6fb2d8842296b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11913,"rank":11913,"depth":10,"x":145.265,"y":794.272,"cluster":"derived-categories"},{"id":"stacks:09TJ","tag":"09TJ","title":"The coherator for Noetherian spaces · Lemma 09TJ","summary":"Let S be an affine scheme. Let X be a Noetherian algebraic space over S. Every injective object of QCoh(O_X) is a direct summand of a filtered colimit colim_i F_i of quasi-coherent sheaves of the form F_i = (Z_i → X)_*G_i where Z_i is the spectrum of an Artinian ring and G_i is a coherent module on Z_i.","statement_latex":"Let $S$ be an affine scheme. Let $X$ be a Noetherian algebraic space\nover $S$. Every injective object of $\\QCoh(\\mathcal{O}_X)$ is\na direct summand of a filtered colimit $\\colim_i \\mathcal{F}_i$\nof quasi-coherent sheaves of the form\n$$\n\\mathcal{F}_i = (Z_i \\to X)_*\\mathcal{G}_i\n$$\nwhere $Z_i$ is the spectrum of an Artinian ring and $\\mathcal{G}_i$\nis a coherent module on $Z_i$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator for Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TJ","source_file":"spaces-perfect.tex","source_line":2650,"source_end_line":2661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2650-L2661","statement_sha256":"67d23c4a68f26818bb8c536aefb9bdf450943f094674a21a3b64c1db1097d455","origin":"The Stacks Project","memory_eligible":false,"source_rank":11914,"rank":11914,"depth":42,"x":200.563,"y":547.544,"cluster":"derived-categories"},{"id":"stacks:09TK","tag":"09TK","title":"The coherator for Noetherian spaces · Lemma 09TK","summary":"Let S be a scheme. Let f : X → Y be a flat, quasi-compact, and quasi-separated morphism of algebraic spaces over S. If J is an injective object of QCoh(O_X), then f_*J is an injective object of QCoh(O_Y).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a flat, quasi-compact, and\nquasi-separated morphism of algebraic spaces over $S$. If\n$\\mathcal{J}$ is an injective object of $\\QCoh(\\mathcal{O}_X)$,\nthen $f_*\\mathcal{J}$ is an injective object of\n$\\QCoh(\\mathcal{O}_Y)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator for Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TK","source_file":"spaces-perfect.tex","source_line":2680,"source_end_line":2687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2680-L2687","statement_sha256":"c885104bf22567e273c24e85788df9fa21033c76e8c1a5f9b02ea8bc191c497d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11915,"rank":11915,"depth":43,"x":358.334,"y":761.037,"cluster":"derived-categories"},{"id":"stacks:09TL","tag":"09TL","title":"The coherator for Noetherian spaces · Lemma 09TL","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. If J is an injective object of QCoh(O_X), then • H^p(U, J|_U) = 0 for p > 0 and for every quasi-compact and quasi-separated algebraic space U étale over X, • for any morphism f : X → Y of algebraic spaces over S with Y quasi-separated we have R^pf_*J = 0 for p > 0.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. If\n$\\mathcal{J}$ is an injective object of $\\QCoh(\\mathcal{O}_X)$,\nthen\n\\begin{enumerate}\n\\item $H^p(U, \\mathcal{J}|_U) = 0$ for $p > 0$ and for\nevery quasi-compact and quasi-separated algebraic space $U$ \\'etale over $X$,\n\\item for any morphism $f : X \\to Y$ of algebraic spaces over $S$\nwith $Y$ quasi-separated we have $R^pf_*\\mathcal{J} = 0$ for $p > 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator for Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TL","source_file":"spaces-perfect.tex","source_line":2699,"source_end_line":2710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2699-L2710","statement_sha256":"ebfb3a8b1dac80abcd0c03f7c74929a9d894fe200eed7fd3440168828637287d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11916,"rank":11916,"depth":59,"x":70.065,"y":693.076,"cluster":"derived-categories"},{"id":"stacks:09TM","tag":"09TM","title":"The coherator for Noetherian spaces · Lemma 09TM","summary":"Let S be a scheme. Let f : X → Y be a morphism of Noetherian algebraic spaces over S. Then f_* on quasi-coherent sheaves has a right derived extension Φ : D(QCoh(O_X)) → D(QCoh(O_Y)) such that the diagram xymatrix D(QCoh(O_X)) ar[d]_Φ ar[r] & D_QCoh(O_X) ar[d]^Rf_* D(QCoh(O_Y)) ar[r] & D_QCoh(O_Y) commutes.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of Noetherian algebraic spaces over $S$.\nThen $f_*$ on quasi-coherent sheaves has a right derived\nextension\n$\\Phi : D(\\QCoh(\\mathcal{O}_X)) \\to D(\\QCoh(\\mathcal{O}_Y))$\nsuch that the diagram\n$$\n\\xymatrix{\nD(\\QCoh(\\mathcal{O}_X)) \\ar[d]_{\\Phi} \\ar[r] &\nD_\\QCoh(\\mathcal{O}_X) \\ar[d]^{Rf_*} \\\\\nD(\\QCoh(\\mathcal{O}_Y)) \\ar[r] &\nD_\\QCoh(\\mathcal{O}_Y)\n}\n$$\ncommutes.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator for Noetherian spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TM","source_file":"spaces-perfect.tex","source_line":2748,"source_end_line":2765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2748-L2765","statement_sha256":"5ffb559bc6aa17eb21d0de39eb4716e8c364f563bb9ba4338bf82413876dfb8f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11917,"rank":11917,"depth":60,"x":337.51,"y":579.522,"cluster":"derived-categories"},{"id":"stacks:09TN","tag":"09TN","title":"The coherator for Noetherian spaces · Proposition 09TN","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Then the functor ([Tag 08F1]) D(QCoh(O_X)) → D_QCoh(O_X) is an equivalence with quasi-inverse given by RQ_X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nThen the functor (\\ref{equation-compare})\n$$\nD(\\QCoh(\\mathcal{O}_X))\n\\longrightarrow\nD_\\QCoh(\\mathcal{O}_X)\n$$\nis an equivalence with quasi-inverse given by $RQ_X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator for Noetherian spaces","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TN","source_file":"spaces-perfect.tex","source_line":2817,"source_end_line":2827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2817-L2827","statement_sha256":"e98aa91ba85836b26b573b1d8a0cfe1a282f9d054e342684886e6fd540dba083","origin":"The Stacks Project","memory_eligible":false,"source_rank":11918,"rank":11918,"depth":63,"x":231.527,"y":815.21,"cluster":"derived-categories"},{"id":"stacks:08HD","tag":"08HD","title":"Pseudo-coherent and perfect complexes · Lemma 08HD","summary":"Let X be a scheme. Let F be an O_X-module. The following are equivalent • F is of finite type as an O_X-module, and • ε^*F is of finite type as an O_etale-module on the small étale site of X. Here ε is as in ([Tag 08H7]).","statement_latex":"Let $X$ be a scheme. Let $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is of finite type as an $\\mathcal{O}_X$-module, and\n\\item $\\epsilon^*\\mathcal{F}$ is of finite type as an\n$\\mathcal{O}_\\etale$-module on the small \\'etale site of $X$.\n\\end{enumerate}\nHere $\\epsilon$ is as in (\\ref{equation-epsilon}).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HD","source_file":"spaces-perfect.tex","source_line":2863,"source_end_line":2873,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2863-L2873","statement_sha256":"c456ea6600f61e0bd03aaf4e202454673b2f3e6a95358ca52b36c9937ac1a7dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11919,"rank":11919,"depth":38,"x":120.049,"y":581.081,"cluster":"derived-categories"},{"id":"stacks:08HE","tag":"08HE","title":"Pseudo-coherent and perfect complexes · Lemma 08HE","summary":"Let X be a scheme. Let E be an object of D(O_X). The following are equivalent • E is m-pseudo-coherent, and • ε^*E is m-pseudo-coherent on the small étale site of X. Here ε is as in ([Tag 08H7]).","statement_latex":"Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $E$ is $m$-pseudo-coherent, and\n\\item $\\epsilon^*E$ is $m$-pseudo-coherent on the small \\'etale site of $X$.\n\\end{enumerate}\nHere $\\epsilon$ is as in (\\ref{equation-epsilon}).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HE","source_file":"spaces-perfect.tex","source_line":2915,"source_end_line":2924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2915-L2924","statement_sha256":"769ffac1556271669762915a299309424e2fe235141f0771ebae25f4e8a7e9cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11920,"rank":11920,"depth":48,"x":390.76,"y":690.562,"cluster":"derived-categories"},{"id":"stacks:08HF","tag":"08HF","title":"Pseudo-coherent and perfect complexes · Lemma 08HF","summary":"Let X be a scheme. Let E be an object of D(O_X). Then • E has tor amplitude in [a, b] if and only if ε^*E has tor amplitude in [a, b]. • E has finite tor dimension if and only if ε^*E has finite tor dimension. Here ε is as in ([Tag 08H7]).","statement_latex":"Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$.\nThen\n\\begin{enumerate}\n\\item $E$ has tor amplitude in $[a, b]$ if and only if\n$\\epsilon^*E$ has tor amplitude in $[a, b]$.\n\\item $E$ has finite tor dimension if and only if $\\epsilon^*E$ has finite\ntor dimension.\n\\end{enumerate}\nHere $\\epsilon$ is as in (\\ref{equation-epsilon}).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HF","source_file":"spaces-perfect.tex","source_line":2974,"source_end_line":2985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L2974-L2985","statement_sha256":"2daa7019daea7a5c849fc1360104452c25dfc25341cf0106fe770f4d6edc741c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11921,"rank":11921,"depth":48,"x":102.857,"y":763.502,"cluster":"derived-categories"},{"id":"stacks:0DK7","tag":"0DK7","title":"Pseudo-coherent and perfect complexes · Lemma 0DK7","summary":"Let f : X → Y be a morphism of schemes. Let E be an object of D(O_X). Then • E as an object of D(f^-1O_Y) has tor amplitude in [a, b] if and only if ε^*E has tor amplitude in [a, b] as an object of D(f_small^-1O_Y_etale). • E locally has finite tor dimension as an object of D(f^-1O_Y) if and only if ε^*E locally has finite tor dimension as an object of D(f_small^-1O_Y_etale). Here ε is as in ([Tag 08H7]).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $E$ be an object\nof $D(\\mathcal{O}_X)$. Then\n\\begin{enumerate}\n\\item $E$ as an object of $D(f^{-1}\\mathcal{O}_Y)$ has tor amplitude in\n$[a, b]$ if and only if $\\epsilon^*E$ has tor amplitude in $[a, b]$\nas an object of $D(f_{small}^{-1}\\mathcal{O}_{Y_\\etale})$.\n\\item $E$ locally has finite tor dimension as an object of\n$D(f^{-1}\\mathcal{O}_Y)$ if and only if $\\epsilon^*E$\nlocally has finite tor dimension as an object of\n$D(f_{small}^{-1}\\mathcal{O}_{Y_\\etale})$.\n\\end{enumerate}\nHere $\\epsilon$ is as in (\\ref{equation-epsilon}).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DK7","source_file":"spaces-perfect.tex","source_line":3007,"source_end_line":3021,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3007-L3021","statement_sha256":"c74d898afcb301952b269581f9232d6b28841e685c8d9ec5540a477693dc04dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":11922,"rank":11922,"depth":52,"x":256.626,"y":546.17,"cluster":"derived-categories"},{"id":"stacks:08HG","tag":"08HG","title":"Pseudo-coherent and perfect complexes · Lemma 08HG","summary":"Let X be a scheme. Let E be an object of D(O_X). Then E is a perfect object of D(O_X) if and only if ε^*E is a perfect object of D(O_etale). Here ε is as in ([Tag 08H7]).","statement_latex":"Let $X$ be a scheme. Let $E$ be an object of $D(\\mathcal{O}_X)$.\nThen $E$ is a perfect object of $D(\\mathcal{O}_X)$ if and only if\n$\\epsilon^*E$ is a perfect object of $D(\\mathcal{O}_\\etale)$.\nHere $\\epsilon$ is as in (\\ref{equation-epsilon}).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HG","source_file":"spaces-perfect.tex","source_line":3070,"source_end_line":3076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3070-L3076","statement_sha256":"b07468fd0a7d75a676dc2d8f4f8914545bdfe6f9bb76145fc910e3bb9c7e3b34","origin":"The Stacks Project","memory_eligible":false,"source_rank":11923,"rank":11923,"depth":49,"x":318.063,"y":793.888,"cluster":"derived-categories"},{"id":"stacks:08JL","tag":"08JL","title":"Pseudo-coherent and perfect complexes · Lemma 08JL","summary":"Let S be a scheme. Let X be an algebraic space over S. If E is an m-pseudo-coherent object of D(O_X), then H^i(E) is a quasi-coherent O_X-module for i > m. If E is pseudo-coherent, then E is an object of D_QCoh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nIf $E$ is an $m$-pseudo-coherent object of $D(\\mathcal{O}_X)$,\nthen $H^i(E)$ is a quasi-coherent $\\mathcal{O}_X$-module for $i > m$.\nIf $E$ is pseudo-coherent, then $E$ is an object of\n$D_\\QCoh(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JL","source_file":"spaces-perfect.tex","source_line":3090,"source_end_line":3097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3090-L3097","statement_sha256":"32154c589bba603b838918d4751e42dc6e4ae17b4d04e6f8e25c854ff697e6ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":11924,"rank":11924,"depth":0,"x":73.347,"y":645.961,"cluster":"derived-categories"},{"id":"stacks:08IK","tag":"08IK","title":"Pseudo-coherent and perfect complexes · Lemma 08IK","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let E be an object of D_QCoh(O_X). For m ∈ Z the following are equivalent • H^i(E) is coherent for i ≥ m and zero for i gg 0, and • E is m-pseudo-coherent. In particular, E is pseudo-coherent if and only if E is an object of D^-_Coh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. For\n$m \\in \\mathbf{Z}$ the following are equivalent\n\\begin{enumerate}\n\\item $H^i(E)$ is coherent for $i \\geq m$ and zero for $i \\gg 0$, and\n\\item $E$ is $m$-pseudo-coherent.\n\\end{enumerate}\nIn particular, $E$ is pseudo-coherent if and only if $E$ is an object\nof $D^-_{\\textit{Coh}}(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IK","source_file":"spaces-perfect.tex","source_line":3106,"source_end_line":3117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3106-L3117","statement_sha256":"867bc22093cf08afddbeaf381e6503274fc114bedbd522778be1f91900bd0211","origin":"The Stacks Project","memory_eligible":false,"source_rank":11925,"rank":11925,"depth":49,"x":373.007,"y":616.157,"cluster":"derived-categories"},{"id":"stacks:08IL","tag":"08IL","title":"Pseudo-coherent and perfect complexes · Lemma 08IL","summary":"Let S be a scheme. Let X be a quasi-separated algebraic space over S. Let E be an object of D_QCoh(O_X). Let a ≤ b. The following are equivalent • E has tor amplitude in [a, b], and • for all F in QCoh(O_X) we have H^i(E ⊗_O_X^L F) = 0 for i not ∈ [a, b].","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$.\nLet $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Let $a \\leq b$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $E$ has tor amplitude in $[a, b]$, and\n\\item for all $\\mathcal{F}$ in $\\QCoh(\\mathcal{O}_X)$\nwe have $H^i(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{F}) = 0$\nfor $i \\not \\in [a, b]$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IL","source_file":"spaces-perfect.tex","source_line":3151,"source_end_line":3162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3151-L3162","statement_sha256":"963e7b7538d2a5256bdb213e75ad42bf03cf4ce32437ded09a9fb549f030bb98","origin":"The Stacks Project","memory_eligible":false,"source_rank":11926,"rank":11926,"depth":48,"x":175.844,"y":808.33,"cluster":"derived-categories"},{"id":"stacks:08JP","tag":"08JP","title":"Pseudo-coherent and perfect complexes · Lemma 08JP","summary":"Let X be a scheme. Let E, F be objects of D(O_X). Assume either • E is pseudo-coherent and F lies in D^+(O_X), or • E is perfect and F arbitrary, then there is a canonical isomorphism ε^*RSheafHom(E, F) → RSheafHom(ε^*E, ε^*F) Here ε is as in ([Tag 08H7]).","statement_latex":"Let $X$ be a scheme. Let $E, F$ be objects of $D(\\mathcal{O}_X)$.\nAssume either\n\\begin{enumerate}\n\\item $E$ is pseudo-coherent and $F$ lies in $D^+(\\mathcal{O}_X)$, or\n\\item $E$ is perfect and $F$ arbitrary,\n\\end{enumerate}\nthen there is a canonical isomorphism\n$$\n\\epsilon^*R\\SheafHom(E, F) \\longrightarrow R\\SheafHom(\\epsilon^*E, \\epsilon^*F)\n$$\nHere $\\epsilon$ is as in (\\ref{equation-epsilon}).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JP","source_file":"spaces-perfect.tex","source_line":3208,"source_end_line":3221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3208-L3221","statement_sha256":"905d30ad2627dcb76a932b1d3f43c6b2c155fd8755b091dfa10255a67f41e56c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11927,"rank":11927,"depth":48,"x":166.682,"y":554.537,"cluster":"derived-categories"},{"id":"stacks:0A8A","tag":"0A8A","title":"Pseudo-coherent and perfect complexes · Lemma 0A8A","summary":"Let S be a scheme. Let X be an algebraic space over S. Let L, K be objects of D(O_X). If either • L in D^+_QCoh(O_X) and K is pseudo-coherent, • L in D_QCoh(O_X) and K is perfect, then RSheafHom(K, L) is in D_QCoh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $L, K$ be objects of $D(\\mathcal{O}_X)$.\nIf either\n\\begin{enumerate}\n\\item $L$ in $D^+_\\QCoh(\\mathcal{O}_X)$ and $K$ is pseudo-coherent,\n\\item $L$ in $D_\\QCoh(\\mathcal{O}_X)$ and $K$ is perfect,\n\\end{enumerate}\nthen $R\\SheafHom(K, L)$ is in $D_\\QCoh(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A8A","source_file":"spaces-perfect.tex","source_line":3281,"source_end_line":3291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3281-L3291","statement_sha256":"0a0c92a9e068ed773ed1002b478ac7a7f0f28391b8b9f7e0e557287479d5004e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11928,"rank":11928,"depth":50,"x":377.707,"y":736.633,"cluster":"derived-categories"},{"id":"stacks:0E4Q","tag":"0E4Q","title":"Pseudo-coherent and perfect complexes · Lemma 0E4Q","summary":"Let S be a scheme. Let X be an algebraic space over S. Let K, L, M be objects of D_QCoh(O_X). The map K ⊗_O_X^L RSheafHom(M, L) → RSheafHom(M, K ⊗_O_X^L L) of Cohomology on Sites, Lemma [Tag 0BYU] is an isomorphism in the following cases • M perfect, or • K is perfect, or • M is pseudo-coherent, L ∈ D^+(O_X), and K has finite tor dimension.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nLet $K, L, M$ be objects of $D_\\QCoh(\\mathcal{O}_X)$.\nThe map\n$$\nK \\otimes_{\\mathcal{O}_X}^\\mathbf{L} R\\SheafHom(M, L)\n\\longrightarrow\nR\\SheafHom(M, K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L)\n$$\nof Cohomology on Sites, Lemma\n\\ref{sites-cohomology-lemma-internal-hom-diagonal-better}\nis an isomorphism in the following cases\n\\begin{enumerate}\n\\item $M$ perfect, or\n\\item $K$ is perfect, or\n\\item $M$ is pseudo-coherent, $L \\in D^+(\\mathcal{O}_X)$, and $K$ has finite\ntor dimension.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Pseudo-coherent and perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4Q","source_file":"spaces-perfect.tex","source_line":3303,"source_end_line":3323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3303-L3323","statement_sha256":"4c55c971f2119570a6e4d856ac15b3d324f8cd7790ae00d51db9626640d2d716","origin":"The Stacks Project","memory_eligible":false,"source_rank":11929,"rank":11929,"depth":49,"x":75.412,"y":722.087,"cluster":"derived-categories"},{"id":"stacks:08HI","tag":"08HI","title":"Approximation by perfect complexes · Definition 08HI","summary":"Let S be a scheme. Let X be an algebraic space over S. Consider triples (T, E, m) where • T ⊂ |X| is a closed subset, • E is an object of D_QCoh(O_X), and • m ∈ Z. We say approximation holds for the triple (T, E, m) if there exists a perfect object P of D(O_X) supported on T and a map α : P → E which induces isomorphisms H^i(P) → H^i(E) for i > m and a surjection H^m(P) → H^m(E).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nConsider triples $(T, E, m)$ where\n\\begin{enumerate}\n\\item $T \\subset |X|$ is a closed subset,\n\\item $E$ is an object of $D_\\QCoh(\\mathcal{O}_X)$, and\n\\item $m \\in \\mathbf{Z}$.\n\\end{enumerate}\nWe say {\\it approximation holds for the triple} $(T, E, m)$ if\nthere exists a perfect object $P$ of $D(\\mathcal{O}_X)$ supported on $T$\nand a map $\\alpha : P \\to E$ which induces isomorphisms $H^i(P) \\to H^i(E)$\nfor $i > m$ and a surjection $H^m(P) \\to H^m(E)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Approximation by perfect complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HI","source_file":"spaces-perfect.tex","source_line":3357,"source_end_line":3370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3357-L3370","statement_sha256":"85c7263ee663a58b6cd026009bdd913ad117ddc2b04a5f2d4f55ce8ac50ba5f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11930,"rank":11930,"depth":0,"x":310.212,"y":561.15,"cluster":"derived-categories"},{"id":"stacks:08HJ","tag":"08HJ","title":"Approximation by perfect complexes · Definition 08HJ","summary":"Let S be a scheme. Let X be an algebraic space over S. We say approximation by perfect complexes holds on X if for any closed subset T ⊂ |X| such that the morphism X setminus T → X is quasi-compact there exists an integer r such that for every triple (T, E, m) as in Definition [Tag 08HI] with • E is (m - r)-pseudo-coherent, and • H^i(E) is supported on T for i ≥ m - r approximation holds.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nWe say {\\it approximation by perfect complexes holds}\non $X$ if for any closed subset $T \\subset |X|$ such that\nthe morphism $X \\setminus T \\to X$ is quasi-compact\nthere exists an integer $r$ such that for every triple $(T, E, m)$ as in\nDefinition \\ref{definition-approximation-holds} with\n\\begin{enumerate}\n\\item $E$ is $(m - r)$-pseudo-coherent, and\n\\item $H^i(E)$ is supported on $T$ for $i \\geq m - r$\n\\end{enumerate}\napproximation holds.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Approximation by perfect complexes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HJ","source_file":"spaces-perfect.tex","source_line":3377,"source_end_line":3390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3377-L3390","statement_sha256":"37757043c8abd1e225811473ce4c440e87768a4212524b8b1f68abd56b48a85b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11931,"rank":11931,"depth":1,"x":266.459,"y":813.263,"cluster":"derived-categories"},{"id":"stacks:08HK","tag":"08HK","title":"Approximation by perfect complexes · Lemma 08HK","summary":"Let S be a scheme. Let (U ⊂ X, j : V → X) be an elementary distinguished square of algebraic space over S. Let E be a perfect object of D(O_V) supported on j^-1(T) where T = |X| setminus |U|. Then Rj_*E is a perfect object of D(O_X).","statement_latex":"Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an\nelementary distinguished square of algebraic space over $S$.\nLet $E$ be a perfect object of $D(\\mathcal{O}_V)$ supported on\n$j^{-1}(T)$ where $T = |X| \\setminus |U|$. Then $Rj_*E$ is a\nperfect object of $D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Approximation by perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HK","source_file":"spaces-perfect.tex","source_line":3392,"source_end_line":3399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3392-L3399","statement_sha256":"27c23e59adb7866b4cae4c78da85d96a07a43f4cc8fed9156cd447ff11f3dc25","origin":"The Stacks Project","memory_eligible":false,"source_rank":11932,"rank":11932,"depth":56,"x":95.838,"y":602.357,"cluster":"derived-categories"},{"id":"stacks:08HL","tag":"08HL","title":"Approximation by perfect complexes · Lemma 08HL","summary":"Let S be a scheme. Let (U ⊂ X, j : V → X) be an elementary distinguished square of algebraic spaces over S. Let T be a closed subset of |X| setminus |U| and let (T, E, m) be a triple as in Definition [Tag 08HI]. If • approximation holds for (j^-1T, E|_V, m), and • the sheaves H^i(E) for i ≥ m are supported on T, then approximation holds for (T, E, m).","statement_latex":"Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an elementary\ndistinguished square of algebraic spaces over $S$. Let $T$ be a closed\nsubset of $|X| \\setminus |U|$ and let $(T, E, m)$ be a triple as in\nDefinition \\ref{definition-approximation-holds}. If\n\\begin{enumerate}\n\\item approximation holds for $(j^{-1}T, E|_V, m)$, and\n\\item the sheaves $H^i(E)$ for $i \\geq m$ are supported on $T$,\n\\end{enumerate}\nthen approximation holds for $(T, E, m)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Approximation by perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HL","source_file":"spaces-perfect.tex","source_line":3410,"source_end_line":3421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3410-L3421","statement_sha256":"3f6248e40ca82f72151b264047de07e4fab9ac2f0b5feb564f999d2d221faa34","origin":"The Stacks Project","memory_eligible":false,"source_rank":11933,"rank":11933,"depth":57,"x":391.501,"y":661.11,"cluster":"derived-categories"},{"id":"stacks:08HM","tag":"08HM","title":"Approximation by perfect complexes · Lemma 08HM","summary":"Let S be a scheme. Let X be an algebraic space over S which is representable by an affine scheme. Then approximation holds for every triple (T, E, m) as in Definition [Tag 08HI] such that there exists an integer r ≥ 0 with • E is m-pseudo-coherent, • H^i(E) is supported on T for i ≥ m - r + 1, • X setminus T is the union of r affine opens. In particular, approximation by perfect complexes holds for affine schemes.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ which is\nrepresentable by an affine scheme. Then approximation holds for every\ntriple $(T, E, m)$ as in Definition \\ref{definition-approximation-holds}\nsuch that there exists an integer $r \\geq 0$ with\n\\begin{enumerate}\n\\item $E$ is $m$-pseudo-coherent,\n\\item $H^i(E)$ is supported on $T$ for $i \\geq m - r + 1$,\n\\item $X \\setminus T$ is the union of $r$ affine opens.\n\\end{enumerate}\nIn particular, approximation by perfect complexes holds for affine schemes.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Approximation by perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HM","source_file":"spaces-perfect.tex","source_line":3433,"source_end_line":3445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3433-L3445","statement_sha256":"42edcc8a5fa663abe31f52fbbbdc6c20f0763dddcbef4a9f5e96f37ebae5b8d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11934,"rank":11934,"depth":50,"x":126.016,"y":785.656,"cluster":"derived-categories"},{"id":"stacks:08HN","tag":"08HN","title":"Approximation by perfect complexes · Lemma 08HN","summary":"Let S be a scheme. Let (U ⊂ X, j : V → X) be an elementary distinguished square of algebraic spaces over S. Assume U quasi-compact, V affine, and U ×_X V quasi-compact. If approximation by perfect complexes holds on U, then approximation by perfect complexes holds on X.","statement_latex":"Let $S$ be a scheme. Let $(U \\subset X, j : V \\to X)$ be an\nelementary distinguished square of algebraic spaces over $S$.\nAssume $U$ quasi-compact, $V$ affine, and $U \\times_X V$ quasi-compact.\nIf approximation by perfect complexes holds on $U$,\nthen approximation by perfect complexes holds on $X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Approximation by perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HN","source_file":"spaces-perfect.tex","source_line":3467,"source_end_line":3474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3467-L3474","statement_sha256":"915a810712b4c299d548066dae2567ab03a2a5f5c991b3e21f4ce806914cb658","origin":"The Stacks Project","memory_eligible":false,"source_rank":11935,"rank":11935,"depth":58,"x":221.704,"y":542.975,"cluster":"derived-categories"},{"id":"stacks:08HP","tag":"08HP","title":"Approximation by perfect complexes · Theorem 08HP","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Then approximation by perfect complexes holds on X.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nThen approximation by perfect complexes holds on $X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Approximation by perfect complexes","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08HP","source_file":"spaces-perfect.tex","source_line":3568,"source_end_line":3573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3568-L3573","statement_sha256":"6839952e8429187ba5a57c785361a62c45197a35a31ed4eda3794f0865609de2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11936,"rank":11936,"depth":60,"x":346.405,"y":776.412,"cluster":"derived-categories"},{"id":"stacks:09IV","tag":"09IV","title":"Generating derived categories · Lemma 09IV","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let W ⊂ X be a quasi-compact open. Let T ⊂ |X| be a closed subset such that X setminus T → X is a quasi-compact morphism. Let E be an object of D_QCoh(O_X). Let α : P → E|_W be a map where P is a perfect object of D(O_W) supported on T ∩ W. Then there exists a map β : R → E where R is a perfect object of D(O_X) supported on T such that P is a direct summand of R|_W in D(O_W) compatible…","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $W \\subset X$ be a quasi-compact open.\nLet $T \\subset |X|$ be a closed subset such that\n$X \\setminus T \\to X$ is a quasi-compact morphism.\nLet $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$.\nLet $\\alpha : P \\to E|_W$ be a map where $P$ is a perfect object of\n$D(\\mathcal{O}_W)$ supported on $T \\cap W$. Then there exists a map\n$\\beta : R \\to E$ where $R$ is a perfect object of $D(\\mathcal{O}_X)$\nsupported on $T$ such that $P$ is a direct summand of $R|_W$ in\n$D(\\mathcal{O}_W)$ compatible $\\alpha$ and $\\beta|_W$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Generating derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IV","source_file":"spaces-perfect.tex","source_line":3596,"source_end_line":3608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3596-L3608","statement_sha256":"1f85411f4a08106060c7e5c4731df9740c6cf898c30653a7e4abacfe682fbee2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11937,"rank":11937,"depth":62,"x":66.499,"y":674.954,"cluster":"derived-categories"},{"id":"stacks:09IX","tag":"09IX","title":"Generating derived categories · Lemma 09IX","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let W be a quasi-compact open subspace of X. Let P be a perfect object of D(O_W). Then P is a direct summand of the restriction of a perfect object of D(O_X).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $W$ be a quasi-compact open subspace of $X$.\nLet $P$ be a perfect object of $D(\\mathcal{O}_W)$.\nThen $P$ is a direct summand of the restriction of a perfect\nobject of $D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Generating derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IX","source_file":"spaces-perfect.tex","source_line":3659,"source_end_line":3667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3659-L3667","statement_sha256":"8f8e3359b8108c57fc0eeabe142638b50319df3f0e64a30d621a8276b717a77a","origin":"The Stacks Project","memory_eligible":false,"source_rank":11938,"rank":11938,"depth":63,"x":354.723,"y":590.873,"cluster":"derived-categories"},{"id":"stacks:09IY","tag":"09IY","title":"Generating derived categories · Theorem 09IY","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. The category D_QCoh(O_X) can be generated by a single perfect object. More precisely, there exists a perfect object P of D(O_X) such that for E ∈ D_QCoh(O_X) the following are equivalent • E = 0, and • Hom_D(O_X)(P[n], E) = 0 for all n ∈ Z.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. The category\n$D_\\QCoh(\\mathcal{O}_X)$ can be generated by a single\nperfect object. More precisely, there exists a perfect object\n$P$ of $D(\\mathcal{O}_X)$ such that for \n$E \\in D_\\QCoh(\\mathcal{O}_X)$ the following are equivalent\n\\begin{enumerate}\n\\item $E = 0$, and\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[n], E) = 0$ for all $n \\in \\mathbf{Z}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Generating derived categories","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09IY","source_file":"spaces-perfect.tex","source_line":3673,"source_end_line":3685,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3673-L3685","statement_sha256":"f15280ffab2335bbd11d54c9f63cc4814d37d5dbc7257cce34aed03fb5654bbe","origin":"The Stacks Project","memory_eligible":false,"source_rank":11939,"rank":11939,"depth":64,"x":209.688,"y":816.603,"cluster":"derived-categories"},{"id":"stacks:0AEC","tag":"0AEC","title":"Generating derived categories · Lemma 0AEC","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let T ⊂ |X| be a closed subset such that |X| setminus T is quasi-compact. With notation as above, the category D_QCoh, T(O_X) is generated by a single perfect object.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $T \\subset |X|$ be a\nclosed subset such that $|X| \\setminus T$ is quasi-compact. With notation\nas above, the category $D_{\\QCoh, T}(\\mathcal{O}_X)$ is generated by a\nsingle perfect object.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Generating derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEC","source_file":"spaces-perfect.tex","source_line":3779,"source_end_line":3786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3779-L3786","statement_sha256":"f0bb89ff91b8d4458515430b44196d4eb15dc476501cf8f9f0f80d82d9a7e004","origin":"The Stacks Project","memory_eligible":false,"source_rank":11940,"rank":11940,"depth":63,"x":135.048,"y":567.653,"cluster":"derived-categories"},{"id":"stacks:09M8","tag":"09M8","title":"Compact and perfect objects · Proposition 09M8","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. An object of D_QCoh(O_X) is compact if and only if it is perfect.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nAn object of $D_\\QCoh(\\mathcal{O}_X)$ is compact\nif and only if it is perfect.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Compact and perfect objects","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09M8","source_file":"spaces-perfect.tex","source_line":3878,"source_end_line":3884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3878-L3884","statement_sha256":"7bd28bee3b95c91f6d84a99cd5c70d86e38eb5c06bb58c2252cab39158420217","origin":"The Stacks Project","memory_eligible":false,"source_rank":11941,"rank":11941,"depth":61,"x":390.494,"y":708.988,"cluster":"derived-categories"},{"id":"stacks:0AED","tag":"0AED","title":"Compact and perfect objects · Lemma 0AED","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let T ⊂ |X| be a closed subset such that |X| setminus T is quasi-compact. An object of D_QCoh, T(O_X) is compact if and only if it is perfect as an object of D(O_X).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset such that $|X| \\setminus T$\nis quasi-compact. An object of $D_{\\QCoh, T}(\\mathcal{O}_X)$ is compact\nif and only if it is perfect as an object of $D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Compact and perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AED","source_file":"spaces-perfect.tex","source_line":3949,"source_end_line":3956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L3949-L3956","statement_sha256":"cc6e1050d233c7325da97e86914540c3dd671478318635f0f0c36de1668a922d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11942,"rank":11942,"depth":64,"x":88.228,"y":749.749,"cluster":"derived-categories"},{"id":"stacks:0AEE","tag":"0AEE","title":"Compact and perfect objects · Lemma 0AEE","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let T ⊂ |X| be a closed subset such that the complement U ⊂ X is quasi-compact. Let α : P → E be a morphism of D_QCoh(O_X) with either • P is perfect and E supported on T, or • P pseudo-coherent, E supported on T, and E bounded below. Then there exists a perfect complex of O_X-modules I and a map I → O_X[0] such that I ⊗^L P → E is zero and such that I|_U → O_U[0] is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $T \\subset |X|$\nbe a closed subset such that the complement $U \\subset X$ is quasi-compact.\nLet $\\alpha : P \\to E$ be a morphism of $D_\\QCoh(\\mathcal{O}_X)$ with\neither\n\\begin{enumerate}\n\\item $P$ is perfect and $E$ supported on $T$, or\n\\item $P$ pseudo-coherent, $E$ supported on $T$, and $E$ bounded below.\n\\end{enumerate}\nThen there exists a perfect complex of $\\mathcal{O}_X$-modules $I$\nand a map $I \\to \\mathcal{O}_X[0]$ such that\n$I \\otimes^\\mathbf{L} P \\to E$ is zero and such that\n$I|_U \\to \\mathcal{O}_U[0]$ is an\nisomorphism.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Compact and perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEE","source_file":"spaces-perfect.tex","source_line":4000,"source_end_line":4016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4000-L4016","statement_sha256":"67d4a66ba5442881a61b88145b394a104f95a3208e2af405aaff561e8d4f9620","origin":"The Stacks Project","memory_eligible":false,"source_rank":11943,"rank":11943,"depth":64,"x":278.49,"y":548.016,"cluster":"derived-categories"},{"id":"stacks:09MA","tag":"09MA","title":"Derived categories as module categories · Lemma 09MA","summary":"Let S be a scheme. Let X be an algebraic space over S. Let K^bullet be a complex of O_X-modules whose cohomology sheaves are quasi-coherent. Let (E, d) = Hom_Comp^dg(O_X)(K^bullet, K^bullet) be the endomorphism differential graded algebra. Then the functor - ⊗_E^L K^bullet : D(E, d) → D(O_X) of Differential Graded Algebra, Lemma [Tag 09LX] has image contained in D_QCoh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $K^\\bullet$\nbe a complex of $\\mathcal{O}_X$-modules whose cohomology sheaves are\nquasi-coherent. Let\n$(E, d) = \\Hom_{\\text{Comp}^{dg}(\\mathcal{O}_X)}(K^\\bullet, K^\\bullet)$\nbe the endomorphism differential graded algebra. Then the functor\n$$\n- \\otimes_E^\\mathbf{L} K^\\bullet :\nD(E, \\text{d}) \\longrightarrow D(\\mathcal{O}_X)\n$$\nof\nDifferential Graded Algebra, Lemma\n\\ref{dga-lemma-tensor-with-complex-derived}\nhas image contained in $D_\\QCoh(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived categories as module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MA","source_file":"spaces-perfect.tex","source_line":4076,"source_end_line":4091,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4076-L4091","statement_sha256":"ddde8148d3dfdedab1e5cb307e27613331d7ff0717bde47d4a2b33fdc687cd6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11944,"rank":11944,"depth":15,"x":300.44,"y":804.939,"cluster":"derived-categories"},{"id":"stacks:09MB","tag":"09MB","title":"Derived categories as module categories · Lemma 09MB","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let K be a perfect object of D(O_X). Then • there exist integers a ≤ b such that Hom_D(O_X)(K, L) = 0 for L ∈ D_QCoh(O_X) with H^i(L) = 0 for i ∈ [a, b], and • if L is bounded, then Ext^n_D(O_X)(K, L) is zero for all but finitely many n.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $K$ be a perfect object of $D(\\mathcal{O}_X)$. Then\n\\begin{enumerate}\n\\item there exist integers $a \\leq b$ such that\n$\\Hom_{D(\\mathcal{O}_X)}(K, L) = 0$ for $L \\in D_\\QCoh(\\mathcal{O}_X)$\nwith $H^i(L) = 0$ for $i \\in [a, b]$, and\n\\item if $L$ is bounded, then $\\Ext^n_{D(\\mathcal{O}_X)}(K, L)$\nis zero for all but finitely many $n$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived categories as module categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MB","source_file":"spaces-perfect.tex","source_line":4110,"source_end_line":4122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4110-L4122","statement_sha256":"ba16346806aa873a5dc9679ae66ac70771ec0b4a24b464a402abaf9fc3e73966","origin":"The Stacks Project","memory_eligible":false,"source_rank":11945,"rank":11945,"depth":58,"x":77.463,"y":627.799,"cluster":"derived-categories"},{"id":"stacks:09MC","tag":"09MC","title":"Derived categories as module categories · Theorem 09MC","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Then there exist a differential graded algebra (E, d) with only a finite number of nonzero cohomology groups H^i(E) such that D_QCoh(O_X) is equivalent to D(E, d).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nThen there exist a differential graded algebra $(E, \\text{d})$\nwith only a finite number of nonzero cohomology groups $H^i(E)$\nsuch that $D_\\QCoh(\\mathcal{O}_X)$ is equivalent\nto $D(E, \\text{d})$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Derived categories as module categories","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09MC","source_file":"spaces-perfect.tex","source_line":4176,"source_end_line":4184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4176-L4184","statement_sha256":"80fd7b8f1fe8a70853b0bd95bb1d489118d8ad4f1c04af833198f166747e5650","origin":"The Stacks Project","memory_eligible":false,"source_rank":11946,"rank":11946,"depth":65,"x":384.582,"y":631.9,"cluster":"derived-categories"},{"id":"stacks:0DKB","tag":"0DKB","title":"Characterizing pseudo-coherent complexes, I · Lemma 0DKB","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let K ∈ D(O_X). The following are equivalent • K is pseudo-coherent, and • K = hocolim K_n where K_n is perfect and τ_≥ -nK_n → τ_≥ -nK is an isomorphism for all n.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $K \\in D(\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item $K$ is pseudo-coherent, and\n\\item $K = \\text{hocolim} K_n$ where\n$K_n$ is perfect and $\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$\nis an isomorphism for all $n$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Characterizing pseudo-coherent complexes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKB","source_file":"spaces-perfect.tex","source_line":4319,"source_end_line":4330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4319-L4330","statement_sha256":"7859a460dc4c487a610ebd4440460d053d2bfbf8405632949937398a42fef099","origin":"The Stacks Project","memory_eligible":false,"source_rank":11947,"rank":11947,"depth":61,"x":154.633,"y":803.281,"cluster":"derived-categories"},{"id":"stacks:0DKC","tag":"0DKC","title":"Characterizing pseudo-coherent complexes, I · Lemma 0DKC","summary":"Let X be a quasi-compact and quasi-separated scheme. Let T ⊂ X be a closed subset such that X setminus T is quasi-compact. Let K ∈ D(O_X) supported on T. The following are equivalent • K is pseudo-coherent, and • K = hocolim K_n where K_n is perfect, supported on T, and τ_≥ -nK_n → τ_≥ -nK is an isomorphism for all n.","statement_latex":"Let $X$ be a quasi-compact and quasi-separated scheme.\nLet $T \\subset X$ be a closed subset such that $X \\setminus T$\nis quasi-compact. Let $K \\in D(\\mathcal{O}_X)$ supported on $T$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $K$ is pseudo-coherent, and\n\\item $K = \\text{hocolim} K_n$ where\n$K_n$ is perfect, supported on $T$, and\n$\\tau_{\\geq -n}K_n \\to \\tau_{\\geq -n}K$ is an isomorphism for all $n$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Characterizing pseudo-coherent complexes, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKC","source_file":"spaces-perfect.tex","source_line":4382,"source_end_line":4394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4382-L4394","statement_sha256":"77b40fb1d2d6f253675c13c1857db1543d219e5e6cd97f9873d1ca0107299ad7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11948,"rank":11948,"depth":62,"x":186.4,"y":546.221,"cluster":"derived-categories"},{"id":"stacks:0CR4","tag":"0CR4","title":"The coherator revisited · Lemma 0CR4","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. The inclusion functor D_QCoh(O_X) → D(O_X) has a right adjoint.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nThe inclusion functor $D_\\QCoh(\\mathcal{O}_X) \\to D(\\mathcal{O}_X)$\nhas a right adjoint.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CR4","source_file":"spaces-perfect.tex","source_line":4436,"source_end_line":4442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4436-L4442","statement_sha256":"c3c635b10bb2cb5126f89137ad2e300b54311a0dae8588e09f247b07e59716a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":11949,"rank":11949,"depth":60,"x":369.844,"y":753.966,"cluster":"derived-categories"},{"id":"stacks:0CR5","tag":"0CR5","title":"The coherator revisited · Lemma 0CR5","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. If the right adjoints DQ_X and DQ_Y of the inclusion functors D_QCoh → D exist for X and Y, then Rf_* ∘ DQ_X = DQ_Y ∘ Rf_*","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a quasi-compact and quasi-separated\nmorphism of algebraic spaces over $S$.\nIf the right adjoints $DQ_X$ and $DQ_Y$\nof the inclusion functors $D_\\QCoh \\to D$ exist for $X$ and $Y$, then\n$$\nRf_* \\circ DQ_X = DQ_Y \\circ Rf_*\n$$","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CR5","source_file":"spaces-perfect.tex","source_line":4528,"source_end_line":4538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4528-L4538","statement_sha256":"98f035f76c5f4328178737fbc905693fab49ddb6b266d70517f0a68c1a251e37","origin":"The Stacks Project","memory_eligible":false,"source_rank":11950,"rank":11950,"depth":60,"x":67.269,"y":704.829,"cluster":"derived-categories"},{"id":"stacks:0CSS","tag":"0CSS","title":"The coherator revisited · Lemma 0CSS","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. The functor DQ_X of Lemma [Tag 0CR4] has the following boundedness property: there exists an integer N = N(X) such that, if K in D(O_X) with H^i(U, K) = 0 for U affine étale over X and i not ∈ [a, b], then the cohomology sheaves H^i(DQ_X(K)) are zero for i not ∈ [a, b + N].","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nThe functor $DQ_X$ of Lemma \\ref{lemma-better-coherator}\nhas the following boundedness property:\nthere exists an integer $N = N(X)$ such that, if\n$K$ in $D(\\mathcal{O}_X)$ with\n$H^i(U, K) = 0$ for $U$ affine \\'etale over $X$ and $i \\not \\in [a, b]$, then\nthe cohomology sheaves $H^i(DQ_X(K))$ are zero for\n$i \\not \\in [a, b + N]$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The coherator revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSS","source_file":"spaces-perfect.tex","source_line":4569,"source_end_line":4580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4569-L4580","statement_sha256":"cf2a18bf135379639680c76287df624fc0e9129844b01ce3c6dc20e6ce463d97","origin":"The Stacks Project","memory_eligible":false,"source_rank":11951,"rank":11951,"depth":61,"x":330.109,"y":569.265,"cluster":"derived-categories"},{"id":"stacks:08IN","tag":"08IN","title":"Cohomology and base change, IV · Lemma 08IN","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. For E in D_QCoh(O_X) and K in D_QCoh(O_Y) we have Rf_*(E) ⊗_O_Y^L K = Rf_*(E ⊗_O_X^L Lf^*K)","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and quasi-separated\nmorphism of algebraic spaces over $S$. For $E$ in\n$D_\\QCoh(\\mathcal{O}_X)$ and\n$K$ in $D_\\QCoh(\\mathcal{O}_Y)$ we have\n$$\nRf_*(E) \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} K =\nRf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} Lf^*K)\n$$","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IN","source_file":"spaces-perfect.tex","source_line":4665,"source_end_line":4675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4665-L4675","statement_sha256":"d54c66548a69e81b849972c80d5a4c7113673a167cc71977a0d7696b44e98486","origin":"The Stacks Project","memory_eligible":false,"source_rank":11952,"rank":11952,"depth":61,"x":245.244,"y":818.569,"cluster":"derived-categories"},{"id":"stacks:08IP","tag":"08IP","title":"Cohomology and base change, IV · Definition 08IP","summary":"Let S be a scheme. Let B be an algebraic space over S. Let X, Y be algebraic spaces over B. We say X and Y are Tor independent over B if and only if for every commutative diagram xymatrix Spec(k) ar[d]_overliney ar[dr]_overlineb ar[r]_-overlinex & X ar[d] Y ar[r] & B of geometric points the rings O_X, overlinex and O_Y, overliney are Tor independent over O_B, overlineb (see More on Algebra, Definition [Tag 0660]).","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $X$, $Y$ be algebraic spaces over $B$. We say $X$ and\n$Y$ are {\\it Tor independent over $B$} if and only if for every\ncommutative diagram\n$$\n\\xymatrix{\n\\Spec(k) \\ar[d]_{\\overline{y}} \\ar[dr]_{\\overline{b}} \\ar[r]_-{\\overline{x}} &\nX \\ar[d] \\\\\nY \\ar[r] & B\n}\n$$\nof geometric points the rings\n$\\mathcal{O}_{X, \\overline{x}}$ and $\\mathcal{O}_{Y, \\overline{y}}$\nare Tor independent over $\\mathcal{O}_{B, \\overline{b}}$ (see\nMore on Algebra, Definition \\ref{more-algebra-definition-tor-independent}).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, IV","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IP","source_file":"spaces-perfect.tex","source_line":4724,"source_end_line":4741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4724-L4741","statement_sha256":"dee8e7349217df1435fdb174ebb1b78ea8ef9c72f4fcdfcda4c2465bfee39052","origin":"The Stacks Project","memory_eligible":false,"source_rank":11953,"rank":11953,"depth":1,"x":107.228,"y":586.396,"cluster":"derived-categories"},{"id":"stacks:08IQ","tag":"08IQ","title":"Cohomology and base change, IV · Lemma 08IQ","summary":"Let S be a scheme. Let B be an algebraic space over S. Let X, Y be algebraic spaces over B. The following are equivalent • X and Y are Tor independent over B, • for every commutative diagram xymatrix U ar[d] ar[r] & W ar[d] & V ar[d] ar[l] X ar[r] & B & Y ar[l] with étale vertical arrows U and V are Tor independent over W, • for some commutative diagram as in (2) with (a) W → B étale surjective, (b) U → X ×_B W étale surjective, (c) V → Y ×_B W étale surjective, the…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $X$, $Y$ be algebraic spaces over $B$. The following are equivalent\n\\begin{enumerate}\n\\item $X$ and $Y$ are Tor independent over $B$,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & W \\ar[d] & V \\ar[d] \\ar[l] \\\\\nX \\ar[r] & B & Y \\ar[l]\n}\n$$\nwith \\'etale vertical arrows $U$ and $V$ are Tor independent over $W$,\n\\item for some commutative diagram as in (2) with (a) $W \\to B$ \\'etale\nsurjective, (b) $U \\to X \\times_B W$ \\'etale surjective, (c)\n$V \\to Y \\times_B W$ \\'etale surjective, the spaces $U$ and $V$ are Tor\nindependent over $W$, and\n\\item for some commutative diagram as in (3) with $U$, $V$, $W$ schemes,\nthe schemes $U$ and $V$ are Tor independent over $W$ in the sense of\nDerived Categories of Schemes, Definition\n\\ref{perfect-definition-tor-independent}.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IQ","source_file":"spaces-perfect.tex","source_line":4747,"source_end_line":4770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4747-L4770","statement_sha256":"10b86ece2be0737eee949164c572c862cff45f74937c613a0228a15f8d67325c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11954,"rank":11954,"depth":53,"x":395.936,"y":679.358,"cluster":"derived-categories"},{"id":"stacks:08IR","tag":"08IR","title":"Cohomology and base change, IV · Lemma 08IR","summary":"Let S be a scheme. Let g : Y' → Y be a morphism of algebraic spaces over S. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Consider the base change diagram xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y If X and Y' are Tor independent over Y, then for all E ∈ D_QCoh(O_X) we have Rf'_*L(g')^*E = Lg^*Rf_*E.","statement_latex":"Let $S$ be a scheme. Let $g : Y' \\to Y$ be a morphism of algebraic spaces over\n$S$. Let $f : X \\to Y$ be a quasi-compact and quasi-separated morphism of\nalgebraic spaces over $S$. Consider the base change diagram\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nY' \\ar[r]^g &\nY\n}\n$$\nIf $X$ and $Y'$ are Tor independent over $Y$, then for all\n$E \\in D_\\QCoh(\\mathcal{O}_X)$ we have\n$Rf'_*L(g')^*E = Lg^*Rf_*E$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IR","source_file":"spaces-perfect.tex","source_line":4862,"source_end_line":4878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4862-L4878","statement_sha256":"c952cb21fc286ec8726e358218c8709957194dc18417244553e33d594d493627","origin":"The Stacks Project","memory_eligible":false,"source_rank":11955,"rank":11955,"depth":62,"x":108.06,"y":774.705,"cluster":"derived-categories"},{"id":"stacks:0E4S","tag":"0E4S","title":"Cohomology and base change, IV · Lemma 0E4S","summary":"Let g : S' → S be a morphism of affine schemes. Consider a cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S of quasi-compact and quasi-separated algebraic spaces. Assume g and f Tor independent. Write S = Spec(R) and S' = Spec(R'). For M, K ∈ D(O_X) the canonical map RHom_X(M, K) ⊗^L_R R' → RHom_X'(L(g')^*M, L(g')^*K) in D(R') is an isomorphism in the following two cases • M ∈ D(O_X) is perfect and K ∈ D_QCoh(X), or • M ∈ D(O_X) is…","statement_latex":"Let $g : S' \\to S$ be a morphism of affine schemes.\nConsider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nS' \\ar[r]^g & S\n}\n$$\nof quasi-compact and quasi-separated algebraic spaces. Assume $g$ and $f$\nTor independent. Write $S = \\Spec(R)$ and $S' = \\Spec(R')$. For\n$M, K \\in D(\\mathcal{O}_X)$ the canonical map\n$$\nR\\Hom_X(M, K) \\otimes^\\mathbf{L}_R R'\n\\longrightarrow\nR\\Hom_{X'}(L(g')^*M, L(g')^*K)\n$$\nin $D(R')$ is an isomorphism in the following two cases\n\\begin{enumerate}\n\\item $M \\in D(\\mathcal{O}_X)$ is perfect and $K \\in D_\\QCoh(X)$, or\n\\item $M \\in D(\\mathcal{O}_X)$ is pseudo-coherent,\n$K \\in D_\\QCoh^+(X)$, and $R'$ has finite tor dimension over $R$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4S","source_file":"spaces-perfect.tex","source_line":4923,"source_end_line":4947,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L4923-L4947","statement_sha256":"b32cc1555ae01b9b2a086eb117f86ef01aa513a562a5e49b7ade7e86a4b2945f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11956,"rank":11956,"depth":63,"x":243.768,"y":540.864,"cluster":"derived-categories"},{"id":"stacks:0DKD","tag":"0DKD","title":"Cohomology and base change, IV · Lemma 0DKD","summary":"Let S be a scheme. Consider a cartesian square of algebraic spaces xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y over S. Assume g and f Tor independent. • If E ∈ D(O_X) has tor amplitude in [a, b] as a complex of f^-1O_Y-modules, then L(g')^*E has tor amplitude in [a, b] as a complex of f^-1O_Y'-modules. • If G is an O_X-module flat over Y, then L(g')^*G = (g')^*G.","statement_latex":"Let $S$ be a scheme. Consider a cartesian square of algebraic spaces\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nover $S$. Assume $g$ and $f$ Tor independent.\n\\begin{enumerate}\n\\item If $E \\in D(\\mathcal{O}_X)$ has tor amplitude\nin $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_Y$-modules,\nthen $L(g')^*E$ has tor amplitude\nin $[a, b]$ as a complex of $f^{-1}\\mathcal{O}_{Y'}$-modules.\n\\item If $\\mathcal{G}$ is an $\\mathcal{O}_X$-module flat\nover $Y$, then $L(g')^*\\mathcal{G} = (g')^*\\mathcal{G}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, IV","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKD","source_file":"spaces-perfect.tex","source_line":5016,"source_end_line":5034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5016-L5034","statement_sha256":"4ad8f20beb1d50eb71b6241fc170b79ecaff2ac244826bbe045d5fdf05e50dd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11957,"rank":11957,"depth":53,"x":331.818,"y":790.497,"cluster":"derived-categories"},{"id":"stacks:0DKG","tag":"0DKG","title":"Cohomology and base change, V · Lemma 0DKG","summary":"Let S be a scheme. Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian diagram of algebraic spaces over S. Let K ∈ D_QCoh(O_X) and let L(g')^*K → K' be a map in D_QCoh(O_X'). The following are equivalent • for any x' ∈ X' and i ∈ Z the map ([Tag 0DKF]) is an isomorphism, • for any commutative diagram xymatrix & U ar[d] ar[rd]^a V' ar[r] ar[rd]^c & V ar[rd]^b & X ar[d]^f & Y' ar[r]^g & Y with a, b, c étale, U, V, V' schemes, and with U' = V' ×_V U…","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nY' \\ar[r]^g &\nY\n}\n$$\nbe a cartesian diagram of algebraic spaces over $S$.\nLet $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$\nbe a map in $D_\\QCoh(\\mathcal{O}_{X'})$. The following are equivalent\n\\begin{enumerate}\n\\item for any $x' \\in X'$ and $i \\in \\mathbf{Z}$ the map (\\ref{equation-bc})\nis an isomorphism,\n\\item for any commutative diagram\n$$\n\\xymatrix{\n& U \\ar[d] \\ar[rd]^a \\\\\nV' \\ar[r] \\ar[rd]^c & V \\ar[rd]^b & X \\ar[d]^f \\\\\n& Y' \\ar[r]^g & Y\n}\n$$\nwith $a, b, c$ \\'etale, $U, V, V'$ schemes, and with $U' = V' \\times_V U$\nthe equivalent conditions of\nDerived Categories of Schemes, Lemma\n\\ref{perfect-lemma-single-complex-base-change-condition}\nhold for $(U \\to X)^*K$ and $(U' \\to X')^*K'$, and\n\\item there is some diagram as in (2) with $U' \\to X'$ surjective.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKG","source_file":"spaces-perfect.tex","source_line":5147,"source_end_line":5179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5147-L5179","statement_sha256":"fcc1a3d808278dc06934412d8a31d8245b52ec6b3296575dc942434726e5150f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11958,"rank":11958,"depth":54,"x":65.932,"y":656.276,"cluster":"derived-categories"},{"id":"stacks:0DKH","tag":"0DKH","title":"Cohomology and base change, V · Lemma 0DKH","summary":"Let S be a scheme. Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian diagram of algebraic spaces over S. Let K ∈ D_QCoh(O_X) and let L(g')^*K → K' be a map in D_QCoh(O_X'). If • the equivalent conditions of Lemma [Tag 0DKG] hold, and • f is quasi-compact and quasi-separated, then the composition Lg^*Rf_*K → Rf'_*L(g')^*K → Rf'_*K' is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nY' \\ar[r]^g &\nY\n}\n$$\nbe a cartesian diagram of algebraic spaces over $S$.\nLet $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$\nbe a map in $D_\\QCoh(\\mathcal{O}_{X'})$. If\n\\begin{enumerate}\n\\item the equivalent conditions of\nLemma \\ref{lemma-single-complex-base-change-condition} hold, and\n\\item $f$ is quasi-compact and quasi-separated,\n\\end{enumerate}\nthen the composition $Lg^*Rf_*K \\to Rf'_*L(g')^*K \\to Rf'_*K'$\nis an isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKH","source_file":"spaces-perfect.tex","source_line":5270,"source_end_line":5291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5270-L5291","statement_sha256":"bbde6f0af12b9b45433765ded736242e6ec8ef148f87c7f474401384fdbae4ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":11959,"rank":11959,"depth":60,"x":370.17,"y":604.338,"cluster":"derived-categories"},{"id":"stacks:0DKI","tag":"0DKI","title":"Cohomology and base change, V · Lemma 0DKI","summary":"Let S be a scheme. Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f S' ar[r]^g & S be a cartesian diagram of algebraic spaces over S. Let K ∈ D_QCoh(O_X) and let L(g')^*K → K' be a map in D_QCoh(O_X'). If the equivalent conditions of Lemma [Tag 0DKG] hold, then • for E ∈ D_QCoh(O_X) the equivalent conditions of Lemma [Tag 0DKG] hold for L(g')^*(E ⊗^L K) → L(g')^*E ⊗^L K', • if E in D(O_X) is perfect the equivalent conditions of Lemma [Tag 0DKG] hold for L(g')^*RSheafHom(E,…","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nS' \\ar[r]^g &\nS\n}\n$$\nbe a cartesian diagram of algebraic spaces over $S$.\nLet $K \\in D_\\QCoh(\\mathcal{O}_X)$ and let $L(g')^*K \\to K'$\nbe a map in $D_\\QCoh(\\mathcal{O}_{X'})$. If the equivalent conditions of\nLemma \\ref{lemma-single-complex-base-change-condition} hold, then\n\\begin{enumerate}\n\\item for $E \\in D_\\QCoh(\\mathcal{O}_X)$ the equivalent\nconditions of Lemma \\ref{lemma-single-complex-base-change-condition} hold\nfor $L(g')^*(E \\otimes^\\mathbf{L} K) \\to L(g')^*E \\otimes^\\mathbf{L} K'$,\n\\item if $E$ in $D(\\mathcal{O}_X)$ is perfect the equivalent conditions of\nLemma \\ref{lemma-single-complex-base-change-condition} hold for\n$L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$, and\n\\item if $K$ is bounded below and $E$ in $D(\\mathcal{O}_X)$\npseudo-coherent the equivalent conditions of\nLemma \\ref{lemma-single-complex-base-change-condition} hold for\n$L(g')^*R\\SheafHom(E, K) \\to R\\SheafHom(L(g')^*E, K')$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKI","source_file":"spaces-perfect.tex","source_line":5349,"source_end_line":5376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5349-L5376","statement_sha256":"02751f95d9fc48e8610c2ec16a9d0355132d3e275769c30baa9b9d44c068163f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11960,"rank":11960,"depth":55,"x":187.453,"y":815.435,"cluster":"derived-categories"},{"id":"stacks:0A1K","tag":"0A1K","title":"Cohomology and base change, V · Lemma 0A1K","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let E ∈ D_QCoh(O_X). Let G^bullet be a bounded above complex of quasi-coherent O_X-modules flat over Y. Then formation of Rf_*(E ⊗^L_O_X G^bullet) commutes with arbitrary base change (see proof for precise statement).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and\nquasi-separated morphism of algebraic spaces over $S$.\nLet $E \\in D_\\QCoh(\\mathcal{O}_X)$.\nLet $\\mathcal{G}^\\bullet$ be a bounded above complex of\nquasi-coherent $\\mathcal{O}_X$-modules flat over $Y$.\nThen formation of\n$$\nRf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)\n$$\ncommutes with arbitrary base change (see proof for precise statement).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1K","source_file":"spaces-perfect.tex","source_line":5407,"source_end_line":5419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5407-L5419","statement_sha256":"2fea068297e49ceaad897c88a535734b2702b684d13389a77382a6fb37e22bbf","origin":"The Stacks Project","memory_eligible":false,"source_rank":11961,"rank":11961,"depth":61,"x":152.399,"y":555.892,"cluster":"derived-categories"},{"id":"stacks:08JQ","tag":"08JQ","title":"Cohomology and base change, V · Lemma 08JQ","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let E be an object of D(O_X). Let G^bullet be a complex of quasi-coherent O_X-modules. If • E is perfect, G^bullet is a bounded above, and G^n is flat over Y, or • E is pseudo-coherent, G^bullet is bounded, and G^n is flat over Y, then formation of Rf_*RSheafHom(E, G^bullet) commutes with arbitrary base change (see proof for precise statement).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact and\nquasi-separated morphism of algebraic spaces over $S$. Let $E$\nbe an object of $D(\\mathcal{O}_X)$. Let $\\mathcal{G}^\\bullet$\nbe a complex of quasi-coherent $\\mathcal{O}_X$-modules. If\n\\begin{enumerate}\n\\item $E$ is perfect, $\\mathcal{G}^\\bullet$ is a bounded above,\nand $\\mathcal{G}^n$ is flat over $Y$, or\n\\item $E$ is pseudo-coherent, $\\mathcal{G}^\\bullet$ is bounded,\nand $\\mathcal{G}^n$ is flat over $Y$,\n\\end{enumerate}\nthen formation of\n$$\nRf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)\n$$\ncommutes with arbitrary base change (see proof for precise statement).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, V","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JQ","source_file":"spaces-perfect.tex","source_line":5457,"source_end_line":5474,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5457-L5474","statement_sha256":"15a879e0d4243a8dc29dfa02c92f6f098060abb8f27d0e6cadd22087dfea41b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11962,"rank":11962,"depth":62,"x":387.149,"y":727.52,"cluster":"derived-categories"},{"id":"stacks:08IS","tag":"08IS","title":"Producing perfect complexes · Lemma 08IS","summary":"Let S be a scheme. Let Y be a Noetherian algebraic space over S. Let f : X → Y be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let E ∈ D(O_X) such that • E ∈ D^b_Coh(O_X), • the support of H^i(E) is proper over Y for all i, • E has finite tor dimension as an object of D(f^-1O_Y). Then Rf_*E is a perfect object of D(O_Y).","statement_latex":"Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$.\nLet $f : X \\to Y$ be a morphism of algebraic spaces which is locally of\nfinite type and quasi-separated. Let $E \\in D(\\mathcal{O}_X)$ such that\n\\begin{enumerate}\n\\item $E \\in D^b_{\\textit{Coh}}(\\mathcal{O}_X)$,\n\\item the support of $H^i(E)$ is proper over $Y$ for all $i$,\n\\item $E$ has finite tor dimension as an object of $D(f^{-1}\\mathcal{O}_Y)$.\n\\end{enumerate}\nThen $Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Producing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IS","source_file":"spaces-perfect.tex","source_line":5528,"source_end_line":5539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5528-L5539","statement_sha256":"b738a5a6a5b7e74928d2c79b10f31fce7a8d71a8321bb7638ba6d25e6eb7064e","origin":"The Stacks Project","memory_eligible":false,"source_rank":11963,"rank":11963,"depth":65,"x":75.785,"y":734.172,"cluster":"derived-categories"},{"id":"stacks:0DKJ","tag":"0DKJ","title":"Producing perfect complexes · Lemma 0DKJ","summary":"Let S be a scheme. Let B be a Noetherian algebraic space over S. Let f : X → B be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let E ∈ D(O_X) be perfect. Let G^bullet be a bounded complex of coherent O_X-modules flat over B with support proper over B. Then K = Rf_*(E ⊗^L_O_X G^bullet) is a perfect object of D(O_B).","statement_latex":"Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$.\nLet $f : X \\to B$ be a morphism of algebraic spaces which is locally of\nfinite type and quasi-separated. Let $E \\in D(\\mathcal{O}_X)$ be perfect.\nLet $\\mathcal{G}^\\bullet$ be a bounded complex of coherent\n$\\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then\n$K = Rf_*(E \\otimes^\\mathbf{L}_{\\mathcal{O}_X} \\mathcal{G}^\\bullet)$\nis a perfect object of $D(\\mathcal{O}_B)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Producing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKJ","source_file":"spaces-perfect.tex","source_line":5576,"source_end_line":5585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5576-L5585","statement_sha256":"59c856eed29d72ff585bd8fe7a8fb8febc8660a6909034495489b5d410869ca5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11964,"rank":11964,"depth":66,"x":300.207,"y":552.449,"cluster":"derived-categories"},{"id":"stacks:0DKK","tag":"0DKK","title":"Producing perfect complexes · Lemma 0DKK","summary":"Let S be a scheme. Let B be a Noetherian algebraic space over S. Let f : X → B be a morphism of algebraic spaces which is locally of finite type and quasi-separated. Let E ∈ D(O_X) be perfect. Let G^bullet be a bounded complex of coherent O_X-modules flat over B with support proper over B. Then K = Rf_*RSheafHom(E, G) is a perfect object of D(O_B).","statement_latex":"Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$.\nLet $f : X \\to B$ be a morphism of algebraic spaces which is locally of\nfinite type and quasi-separated. Let $E \\in D(\\mathcal{O}_X)$ be perfect.\nLet $\\mathcal{G}^\\bullet$ be a bounded complex of coherent\n$\\mathcal{O}_X$-modules flat over $B$ with support proper over $B$. Then\n$K = Rf_*R\\SheafHom(E, \\mathcal{G})$ is a perfect object of $D(\\mathcal{O}_B)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Producing perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKK","source_file":"spaces-perfect.tex","source_line":5602,"source_end_line":5610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5602-L5610","statement_sha256":"2fbb7ccc97551d5a7dde145ba9c18a62e46c39f1c876530553cb8f317c016ae4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11965,"rank":11965,"depth":67,"x":280.843,"y":813.997,"cluster":"derived-categories"},{"id":"stacks:0A1M","tag":"0A1M","title":"A projection formula for Ext · Lemma 0A1M","summary":"Assumptions and notation as in Lemma [Tag 0DKJ]. Then there are functorial isomorphisms H^i(B, K ⊗^L_O_B F) → H^i(X, E ⊗^L_O_X (G^bullet ⊗_O_X f^*F)) for F quasi-coherent on B compatible with boundary maps (see proof).","statement_latex":"Assumptions and notation as in Lemma \\ref{lemma-tensor-perfect}.\nThen there are functorial isomorphisms\n$$\nH^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F})\n\\longrightarrow\nH^i(X, E \\otimes^\\mathbf{L}_{\\mathcal{O}_X}\n(\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F}))\n$$\nfor $\\mathcal{F}$ quasi-coherent on $B$\ncompatible with boundary maps (see proof).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"A projection formula for Ext","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1M","source_file":"spaces-perfect.tex","source_line":5646,"source_end_line":5658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5646-L5658","statement_sha256":"22709bdb9d69c30ee22af51bee15fe92a36e0c875484065daafb82a01c01cfb4","origin":"The Stacks Project","memory_eligible":false,"source_rank":11966,"rank":11966,"depth":67,"x":84.639,"y":609.988,"cluster":"derived-categories"},{"id":"stacks:08JN","tag":"08JN","title":"A projection formula for Ext · Lemma 08JN","summary":"Assumption and notation as in Lemma [Tag 0DKK]. Then there are functorial isomorphisms H^i(B, K ⊗^L_O_B F) → Ext^i_O_X(E, G^bullet ⊗_O_X f^*F) for F quasi-coherent on B compatible with boundary maps (see proof).","statement_latex":"Assumption and notation as in Lemma \\ref{lemma-ext-perfect}.\nThen there are functorial isomorphisms\n$$\nH^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F})\n\\longrightarrow\n\\Ext^i_{\\mathcal{O}_X}(E,\n\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})\n$$\nfor $\\mathcal{F}$ quasi-coherent on $B$\ncompatible with boundary maps (see proof).","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"A projection formula for Ext","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JN","source_file":"spaces-perfect.tex","source_line":5728,"source_end_line":5740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5728-L5740","statement_sha256":"6b3b4ca5da730ac84b4a36e9d89cbf28c0614aa014a02098bf9883e6a765794d","origin":"The Stacks Project","memory_eligible":false,"source_rank":11967,"rank":11967,"depth":68,"x":393.617,"y":649.12,"cluster":"derived-categories"},{"id":"stacks:08JR","tag":"08JR","title":"A projection formula for Ext · Lemma 08JR","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S, E ∈ D(O_X), and F^bullet a complex of O_X-modules. Assume • B is Noetherian, • f is locally of finite type and quasi-separated, • E ∈ D^-_Coh(O_X), • G^bullet is a bounded complex of coherent O_X-module flat over B with support proper over B. Then the following two statements are true • [(A)] for every m ∈ Z there exists a perfect object K of D(O_B) and functorial maps α^i_F : Ext^i_O_X(E, G^bullet…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces\nover $S$, $E \\in D(\\mathcal{O}_X)$, and $\\mathcal{F}^\\bullet$ a complex\nof $\\mathcal{O}_X$-modules. Assume\n\\begin{enumerate}\n\\item $B$ is Noetherian,\n\\item $f$ is locally of finite type and quasi-separated,\n\\item $E \\in D^-_{\\textit{Coh}}(\\mathcal{O}_X)$,\n\\item $\\mathcal{G}^\\bullet$ is a bounded complex of coherent\n$\\mathcal{O}_X$-module flat over $B$ with support proper over $B$.\n\\end{enumerate}\nThen the following two statements are true\n\\begin{enumerate}\n\\item[(A)] for every $m \\in \\mathbf{Z}$ there exists a perfect object $K$\nof $D(\\mathcal{O}_B)$ and functorial maps\n$$\n\\alpha^i_\\mathcal{F} :\n\\Ext^i_{\\mathcal{O}_X}(E,\n\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})\n\\longrightarrow\nH^i(B, K \\otimes^\\mathbf{L}_{\\mathcal{O}_B} \\mathcal{F})\n$$\nfor $\\mathcal{F}$ quasi-coherent on $B$\ncompatible with boundary maps (see proof)\nsuch that $\\alpha^i_\\mathcal{F}$ is an isomorphism for $i \\leq m$, and\n\\item[(B)] there exists a pseudo-coherent $L \\in D(\\mathcal{O}_B)$\nand functorial isomorphisms\n$$\n\\Ext^i_{\\mathcal{O}_B}(L, \\mathcal{F}) \\longrightarrow\n\\Ext^i_{\\mathcal{O}_X}(E,\n\\mathcal{G}^\\bullet \\otimes_{\\mathcal{O}_X} f^*\\mathcal{F})\n$$\nfor $\\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"A projection formula for Ext","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JR","source_file":"spaces-perfect.tex","source_line":5792,"source_end_line":5827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5792-L5827","statement_sha256":"37e614ec70edf54878b8fdd0fe4fb6637eededc509bc9d3eea8b8fe27a3bb9db","origin":"The Stacks Project","memory_eligible":false,"source_rank":11968,"rank":11968,"depth":69,"x":134.11,"y":795.7,"cluster":"derived-categories"},{"id":"stacks:09RI","tag":"09RI","title":"Limits and derived categories · Lemma 09RI","summary":"In Situation [Tag 09RH]. Let E_0 and K_0 be objects of D(O_X_0). Set E_i = Lf_i0^*E_0 and K_i = Lf_i0^*K_0 for i ≥ 0 and set E = Lf_0^*E_0 and K = Lf_0^*K_0. Then the map colim_i ≥ 0 Hom_D(O_X_i)(E_i, K_i) → Hom_D(O_X)(E, K) is an isomorphism if either • E_0 is perfect and K_0 ∈ D_QCoh(O_X_0), or • E_0 is pseudo-coherent and K_0 ∈ D_QCoh(O_X_0) has finite tor dimension.","statement_latex":"In Situation \\ref{situation-descent}. Let $E_0$ and $K_0$ be objects of\n$D(\\mathcal{O}_{X_0})$. Set $E_i = Lf_{i0}^*E_0$ and $K_i = Lf_{i0}^*K_0$\nfor $i \\geq 0$ and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map\n$$\n\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{X_i})}(E_i, K_i)\n\\longrightarrow\n\\Hom_{D(\\mathcal{O}_X)}(E, K)\n$$\nis an isomorphism if either\n\\begin{enumerate}\n\\item $E_0$ is perfect and $K_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$, or\n\\item $E_0$ is pseudo-coherent and\n$K_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$ has finite tor dimension.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Limits and derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RI","source_file":"spaces-perfect.tex","source_line":5966,"source_end_line":5982,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L5966-L5982","statement_sha256":"a30b9501c2f0dbcb320130e3845f12d9cec052e5c633db8e477e498de98b73a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":11969,"rank":11969,"depth":60,"x":207.648,"y":540.165,"cluster":"derived-categories"},{"id":"stacks:09RJ","tag":"09RJ","title":"Limits and derived categories · Lemma 09RJ","summary":"In Situation [Tag 09RH] the category of perfect objects of D(O_X) is the colimit of the categories of perfect objects of D(O_X_i).","statement_latex":"In Situation \\ref{situation-descent} the category of perfect\nobjects of $D(\\mathcal{O}_X)$ is the colimit of the categories\nof perfect objects of $D(\\mathcal{O}_{X_i})$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Limits and derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RJ","source_file":"spaces-perfect.tex","source_line":6011,"source_end_line":6016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6011-L6016","statement_sha256":"b055bddccc23f7ddb78cc4f05d6fec3b1d6c0fe52edfbb179eb95366ead52ca5","origin":"The Stacks Project","memory_eligible":false,"source_rank":11970,"rank":11970,"depth":61,"x":359.034,"y":770.499,"cluster":"derived-categories"},{"id":"stacks:0A1P","tag":"0A1P","title":"Cohomology and base change, VI · Lemma 0A1P","summary":"Let S be a scheme. Let f : X → Y be a morphism of finite presentation between algebraic spaces over S. Let E ∈ D(O_X) be a perfect object. Let G^bullet be a bounded complex of finitely presented O_X-modules, flat over Y, with support proper over Y. Then K = Rf_*(E ⊗_O_X^L G^bullet) is a perfect object of D(O_Y) and its formation commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of finite presentation\nbetween algebraic spaces over $S$. Let $E \\in D(\\mathcal{O}_X)$ be a perfect\nobject. Let $\\mathcal{G}^\\bullet$ be a bounded complex of finitely presented\n$\\mathcal{O}_X$-modules, flat over $Y$, with support proper over $Y$. Then\n$$\nK = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)\n$$\nis a perfect object of $D(\\mathcal{O}_Y)$ and its formation\ncommutes with arbitrary base change.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1P","source_file":"spaces-perfect.tex","source_line":6090,"source_end_line":6101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6090-L6101","statement_sha256":"951b3ae2e4b05463d313804be00e2f93b04fccdd071ec88fccb3b244fcd6e7b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11971,"rank":11971,"depth":67,"x":61.945,"y":686.489,"cluster":"derived-categories"},{"id":"stacks:0CTL","tag":"0CTL","title":"Cohomology and base change, VI · Lemma 0CTL","summary":"Let S be a scheme. Let f : X → Y be a morphism of finite presentation between algebraic spaces over S. Let E ∈ D(O_X) be a pseudo-coherent object. Let G^bullet be a bounded above complex of finitely presented O_X-modules, flat over Y, with support proper over Y. Then K = Rf_*(E ⊗_O_X^L G^bullet) is a pseudo-coherent object of D(O_Y) and its formation commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of finite presentation\nbetween algebraic spaces over $S$.\nLet $E \\in D(\\mathcal{O}_X)$ be a pseudo-coherent object.\nLet $\\mathcal{G}^\\bullet$ be a bounded above complex of\nfinitely presented $\\mathcal{O}_X$-modules,\nflat over $Y$, with support proper over $Y$. Then\n$$\nK = Rf_*(E \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{G}^\\bullet)\n$$\nis a pseudo-coherent object of $D(\\mathcal{O}_Y)$ and its formation\ncommutes with arbitrary base change.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTL","source_file":"spaces-perfect.tex","source_line":6156,"source_end_line":6170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6156-L6170","statement_sha256":"0881be8762e3d3fdc9ac5fcfef142d599f74ec3456deb61518c8ad090e25f319","origin":"The Stacks Project","memory_eligible":false,"source_rank":11972,"rank":11972,"depth":68,"x":348.795,"y":579.779,"cluster":"derived-categories"},{"id":"stacks:0CTM","tag":"0CTM","title":"Cohomology and base change, VI · Lemma 0CTM","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of finite presentation of algebraic spaces over S. • Let E ∈ D(O_X) be perfect and f flat. Then Rf_*E is a perfect object of D(O_Y) and its formation commutes with arbitrary base change. • Let G be an O_X-module of finite presentation, flat over S. Then Rf_*G is a perfect object of D(O_Y) and its formation commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper\nmorphism of finite presentation of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item Let $E \\in D(\\mathcal{O}_X)$ be perfect and $f$ flat. Then\n$Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$ and its formation\ncommutes with arbitrary base change.\n\\item Let $\\mathcal{G}$ be an $\\mathcal{O}_X$-module of finite presentation,\nflat over $S$. Then $Rf_*\\mathcal{G}$ is a perfect object of\n$D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTM","source_file":"spaces-perfect.tex","source_line":6224,"source_end_line":6236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6224-L6236","statement_sha256":"a66240dfda2dd22517e185816d007e51fdd973716309ea89c03436fcb68c4fb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":11973,"rank":11973,"depth":68,"x":222.992,"y":821.418,"cluster":"derived-categories"},{"id":"stacks:0CTN","tag":"0CTN","title":"Cohomology and base change, VI · Lemma 0CTN","summary":"Let S be a scheme. Let f : X → Y be a flat proper morphism of finite presentation of algebraic spaces over S. Let E ∈ D(O_X) be pseudo-coherent. Then Rf_*E is a pseudo-coherent object of D(O_Y) and its formation commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a flat proper\nmorphism of finite presentation of algebraic spaces over $S$.\nLet $E \\in D(\\mathcal{O}_X)$\nbe pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of\n$D(\\mathcal{O}_Y)$ and its formation commutes with arbitrary base change.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTN","source_file":"spaces-perfect.tex","source_line":6246,"source_end_line":6253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6246-L6253","statement_sha256":"d74cb5d8a8e1e2ec0237bbc9c89da40d08b7250228a063fe6dc2e58b2faafaa6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11974,"rank":11974,"depth":0,"x":121.359,"y":571.661,"cluster":"derived-categories"},{"id":"stacks:0D3F","tag":"0D3F","title":"Cohomology and base change, VI · Lemma 0D3F","summary":"Let R be a ring. Let X be an algebraic space and let f : X → Spec(R) be proper, flat, and of finite presentation. Let (M_n) be an inverse system of R-modules with surjective transition maps. Then the canonical map O_X ⊗_R (lim M_n) → lim O_X ⊗_R M_n induces an isomorphism from the source to DQ_X applied to the target.","statement_latex":"Let $R$ be a ring. Let $X$ be an algebraic space and let\n$f : X \\to \\Spec(R)$ be proper, flat, and\nof finite presentation. Let $(M_n)$ be an inverse\nsystem of $R$-modules with surjective transition maps.\nThen the canonical map\n$$\n\\mathcal{O}_X \\otimes_R (\\lim M_n)\n\\longrightarrow\n\\lim \\mathcal{O}_X \\otimes_R M_n\n$$\ninduces an isomorphism from the source to $DQ_X$ applied to the target.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3F","source_file":"spaces-perfect.tex","source_line":6269,"source_end_line":6282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6269-L6282","statement_sha256":"60ffe9b9841062a488cea392ef5a3fe76f2c461519afe7bf579e14123002cc54","origin":"The Stacks Project","memory_eligible":false,"source_rank":11975,"rank":11975,"depth":69,"x":397.363,"y":698.257,"cluster":"derived-categories"},{"id":"stacks:0CWH","tag":"0CWH","title":"Cohomology and base change, VI · Lemma 0CWH","summary":"Let A be a ring. Let X be an algebraic space over A which is quasi-compact and quasi-separated. Let K ∈ D^-_QCoh(O_X). If RΓ(X, E ⊗^L K) is pseudo-coherent in D(A) for every perfect E in D(O_X), then RΓ(X, E ⊗^L K) is pseudo-coherent in D(A) for every pseudo-coherent E in D(O_X).","statement_latex":"Let $A$ be a ring. Let $X$ be an algebraic space over $A$ which is\nquasi-compact and quasi-separated. Let $K \\in D^-_\\QCoh(\\mathcal{O}_X)$.\nIf $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent\nin $D(A)$ for every perfect $E$ in $D(\\mathcal{O}_X)$,\nthen $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent\nin $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWH","source_file":"spaces-perfect.tex","source_line":6326,"source_end_line":6334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6326-L6334","statement_sha256":"0f9ed0eb3e03ea734428fd6c4117d4948af947a689668ec20cdd85c26ac14b37","origin":"The Stacks Project","memory_eligible":false,"source_rank":11976,"rank":11976,"depth":61,"x":91.802,"y":761.566,"cluster":"derived-categories"},{"id":"stacks:0A1R","tag":"0A1R","title":"Cohomology and base change, VI · Lemma 0A1R","summary":"Let S be a scheme. Let f : X → Y be a morphism of finite presentation between algebraic spaces over S. Let E ∈ D(O_X) be a perfect object. Let G^bullet be a bounded complex of finitely presented O_X-modules, flat over Y, with support proper over Y. Then K = Rf_*RSheafHom(E, G^bullet) is a perfect object of D(O_Y) and its formation commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of finite presentation\nbetween algebraic spaces over $S$.\nLet $E \\in D(\\mathcal{O}_X)$ be a perfect object. Let $\\mathcal{G}^\\bullet$\nbe a bounded complex of finitely presented $\\mathcal{O}_X$-modules,\nflat over $Y$, with support proper over $Y$. Then\n$$\nK = Rf_*R\\SheafHom(E, \\mathcal{G}^\\bullet)\n$$\nis a perfect object of $D(\\mathcal{O}_Y)$ and its formation\ncommutes with arbitrary base change.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Cohomology and base change, VI","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1R","source_file":"spaces-perfect.tex","source_line":6366,"source_end_line":6379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6366-L6379","statement_sha256":"62acf0205ae94a269651f66cfeb70a34f7bf90b1d195d58ac0f7e526668f4069","origin":"The Stacks Project","memory_eligible":false,"source_rank":11977,"rank":11977,"depth":69,"x":266.34,"y":541.331,"cluster":"derived-categories"},{"id":"stacks:0D1Y","tag":"0D1Y","title":"Perfect complexes · Lemma 0D1Y","summary":"Let S be a scheme. Let X be an algebraic space over S. Let E ∈ D(O_X) be pseudo-coherent (for example perfect). For any i ∈ Z consider the function β_i : |X| → (0, 1, 2, …) defined above. Then we have • formation of β_i commutes with arbitrary base change, • the functions β_i are upper semi-continuous, and • the level sets of β_i are étale locally constructible.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $E \\in D(\\mathcal{O}_X)$ be pseudo-coherent (for example perfect).\nFor any $i \\in \\mathbf{Z}$ consider the function\n$$\n\\beta_i : |X| \\longrightarrow \\{0, 1, 2, \\ldots\\}\n$$\ndefined above. Then we have\n\\begin{enumerate}\n\\item formation of $\\beta_i$ commutes with arbitrary base change,\n\\item the functions $\\beta_i$ are upper semi-continuous, and\n\\item the level sets of $\\beta_i$ are \\'etale locally constructible.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1Y","source_file":"spaces-perfect.tex","source_line":6449,"source_end_line":6463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6449-L6463","statement_sha256":"531a9d118c05c385929721ea2093e3fb169940ed15aed7c93c0eb12f215bbabf","origin":"The Stacks Project","memory_eligible":false,"source_rank":11978,"rank":11978,"depth":49,"x":314.782,"y":802.967,"cluster":"derived-categories"},{"id":"stacks:0E0R","tag":"0E0R","title":"Perfect complexes · Lemma 0E0R","summary":"Let Y be a scheme and let X be an algebraic space over Y such that the structure morphism f : X → Y is flat, proper, and of finite presentation. Let F be an O_X-module of finite presentation, flat over Y. For fixed i ∈ Z consider the function β_i : |Y| → (0, 1, 2, …), y ↦ dim_kappa(y) H^i(X_y, F_y) Then we have • formation of β_i commutes with arbitrary base change, • the functions β_i are upper semi-continuous, and • the level sets of β_i are locally constructible in Y.","statement_latex":"Let $Y$ be a scheme and let $X$ be an algebraic space over $Y$\nsuch that the structure morphism $f : X \\to Y$\nis flat, proper, and of finite presentation.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,\nflat over $Y$. For fixed $i \\in \\mathbf{Z}$ consider the function\n$$\n\\beta_i : |Y| \\to \\{0, 1, 2, \\ldots\\},\\quad\ny \\longmapsto \\dim_{\\kappa(y)} H^i(X_y, \\mathcal{F}_y)\n$$\nThen we have\n\\begin{enumerate}\n\\item formation of $\\beta_i$ commutes with arbitrary base change,\n\\item the functions $\\beta_i$ are upper semi-continuous, and\n\\item the level sets of $\\beta_i$ are locally constructible in $Y$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0R","source_file":"spaces-perfect.tex","source_line":6477,"source_end_line":6494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6477-L6494","statement_sha256":"5dd4a8d25067531e82045c3e5053d840add2e61b932c68b50773680dc0eaf3da","origin":"The Stacks Project","memory_eligible":false,"source_rank":11979,"rank":11979,"depth":69,"x":68.472,"y":637.401,"cluster":"derived-categories"},{"id":"stacks:0D1Z","tag":"0D1Z","title":"Perfect complexes · Lemma 0D1Z","summary":"Let S be a scheme. Let X be an algebraic space over S. Let E ∈ D(O_X) be perfect. The function chi_E : |X| → Z, x ↦ ∑ (-1)^i β_i(x) is locally constant on X.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $E \\in D(\\mathcal{O}_X)$ be perfect. The function\n$$\n\\chi_E : |X| \\longrightarrow \\mathbf{Z},\\quad\nx \\longmapsto \\sum (-1)^i \\beta_i(x)\n$$\nis locally constant on $X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1Z","source_file":"spaces-perfect.tex","source_line":6508,"source_end_line":6517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6508-L6517","statement_sha256":"9d409e5c42b09c5ad313df7fc61e828ecbca70a72ddee88511e8784a21c71730","origin":"The Stacks Project","memory_eligible":false,"source_rank":11980,"rank":11980,"depth":8,"x":383.482,"y":619.711,"cluster":"derived-categories"},{"id":"stacks:0D20","tag":"0D20","title":"Perfect complexes · Lemma 0D20","summary":"Let S be a scheme. Let X be an algebraic space over S. Let E ∈ D(O_X) be perfect. Given i, r ∈ Z, there exists an open subspace U ⊂ X characterized by the following • E|_U ≅ H^i(E|_U)[-i] and H^i(E|_U) is a locally free O_U-module of rank r, • a morphism f : Y → X factors through U if and only if Lf^*E is isomorphic to a locally free module of rank r placed in degree i.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $E \\in D(\\mathcal{O}_X)$ be perfect.\nGiven $i, r \\in \\mathbf{Z}$, there exists an\nopen subspace $U \\subset X$ characterized by the following\n\\begin{enumerate}\n\\item $E|_U \\cong H^i(E|_U)[-i]$ and $H^i(E|_U)$ is a locally free\n$\\mathcal{O}_U$-module of rank $r$,\n\\item a morphism $f : Y \\to X$ factors through $U$ if and only if\n$Lf^*E$ is isomorphic to a locally free module of rank $r$\nplaced in degree $i$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D20","source_file":"spaces-perfect.tex","source_line":6525,"source_end_line":6538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6525-L6538","statement_sha256":"7ab964a0eb45b99cc8b89844ced93e8d9fe35bc73984c1a73c598864e93b9c2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":11981,"rank":11981,"depth":34,"x":165.258,"y":811.646,"cluster":"derived-categories"},{"id":"stacks:0E6A","tag":"0E6A","title":"Perfect complexes · Lemma 0E6A","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is proper, flat, and of finite presentation. Let F be an O_X-module of finite presentation, flat over Y. Fix i, r ∈ Z. Then there exists an open subspace V ⊂ Y with the following property: A morphism T → Y factors through V if and only if Rf_T, *F_T is isomorphic to a finite locally free module of rank r placed in degree i.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is proper, flat, and of finite presentation.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite presentation,\nflat over $Y$. Fix $i, r \\in \\mathbf{Z}$.\nThen there exists an open subspace\n$V \\subset Y$ with the following property:\nA morphism $T \\to Y$ factors through $V$ if and only if\n$Rf_{T, *}\\mathcal{F}_T$ is isomorphic to a\nfinite locally free module of rank $r$ placed in degree $i$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6A","source_file":"spaces-perfect.tex","source_line":6547,"source_end_line":6559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6547-L6559","statement_sha256":"4066d0e9d532b9221ceb1ce0abfe5df4af7ba2f1ed592488cc61b41868fb2b26","origin":"The Stacks Project","memory_eligible":false,"source_rank":11982,"rank":11982,"depth":69,"x":171.829,"y":546.088,"cluster":"derived-categories"},{"id":"stacks:0D21","tag":"0D21","title":"Perfect complexes · Lemma 0D21","summary":"Let S be a scheme. Let X be an algebraic space over S. Let E ∈ D(O_X) be perfect of tor-amplitude in [a, b] for some a, b ∈ Z. Let r ≥ 0. Then there exists a locally closed subspace j : Z → X characterized by the following • H^a(Lj^*E) is a locally free O_Z-module of rank r, and • a morphism f : Y → X factors through Z if and only if for all morphisms g : Y' → Y the O_Y'-module H^a(L(f ∘ g)^*E) is locally free of rank r. Moreover, j : Z → X is of finite presentation and…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $E \\in D(\\mathcal{O}_X)$ be perfect\nof tor-amplitude in $[a, b]$ for some $a, b \\in \\mathbf{Z}$.\nLet $r \\geq 0$.\nThen there exists a locally closed subspace $j : Z \\to X$\ncharacterized by the following\n\\begin{enumerate}\n\\item $H^a(Lj^*E)$ is a locally free $\\mathcal{O}_Z$-module of rank $r$, and\n\\item a morphism $f : Y \\to X$ factors through $Z$ if and only if\nfor all morphisms $g : Y' \\to Y$ the $\\mathcal{O}_{Y'}$-module\n$H^a(L(f \\circ g)^*E)$ is locally free of rank $r$.\n\\end{enumerate}\nMoreover, $j : Z \\to X$ is of finite presentation and we have\n\\begin{enumerate}\n\\item[(3)] if $f : Y \\to X$ factors as $Y \\xrightarrow{g} Z \\to X$, then\n$H^a(Lf^*E) = g^*H^a(Lj^*E)$,\n\\item[(4)] if $\\beta_a(x) \\leq r$ for all $x \\in |X|$, then $j$ is\na closed immersion and given $f : Y \\to X$ the following are equivalent\n\\begin{enumerate}\n\\item $f : Y \\to X$ factors through $Z$,\n\\item $H^0(Lf^*E)$ is a locally free $\\mathcal{O}_Y$-module of rank $r$,\n\\end{enumerate}\nand if $r = 1$ these are also equivalent to\n\\begin{enumerate}\n\\item[(c)] $\\mathcal{O}_Y \\to \\SheafHom_{\\mathcal{O}_Y}(H^0(Lf^*E), H^0(Lf^*E))$\nis injective.\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D21","source_file":"spaces-perfect.tex","source_line":6570,"source_end_line":6600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6570-L6600","statement_sha256":"466323de802e48868d4bb498bf2e6feb264e5720885eb254ef97d81b8f9ee1fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":11983,"rank":11983,"depth":34,"x":380.697,"y":745.787,"cluster":"derived-categories"},{"id":"stacks:0E6B","tag":"0E6B","title":"Perfect complexes · Lemma 0E6B","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • f is proper, flat, and of finite presentation, and • for a morphism Spec(k) → Y where k is a field, we have k = H^0(X_k, O_X_k). Then we have • [(a)] f_*O_X = O_Y and this holds after any base change, • [(b)] étale locally on Y we have Rf_*O_X = O_Y ⊕ P in D(O_Y) where P is perfect of tor amplitude in [1, ∞).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item $f$ is proper, flat, and of finite presentation, and\n\\item for a morphism $\\Spec(k) \\to Y$ where $k$ is a field, we have\n$k = H^0(X_k, \\mathcal{O}_{X_k})$.\n\\end{enumerate}\nThen we have\n\\begin{enumerate}\n\\item[(a)] $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and\nthis holds after any base change,\n\\item[(b)] \\'etale locally on $Y$ we have\n$$\nRf_*\\mathcal{O}_X = \\mathcal{O}_Y \\oplus P\n$$\nin $D(\\mathcal{O}_Y)$\nwhere $P$ is perfect of tor amplitude in $[1, \\infty)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6B","source_file":"spaces-perfect.tex","source_line":6609,"source_end_line":6629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6609-L6629","statement_sha256":"13bc6e5f934c8640bd7ba56b9f60c6a2d2b417310fbb4cd2a0aa387829955411","origin":"The Stacks Project","memory_eligible":false,"source_rank":11984,"rank":11984,"depth":69,"x":65.849,"y":717.027,"cluster":"derived-categories"},{"id":"stacks:0E0S","tag":"0E0S","title":"Perfect complexes · Lemma 0E0S","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • f is proper, flat, and of finite presentation, and • the geometric fibres of f are reduced and connected. Then f_*O_X = O_Y and this holds after any base change.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item $f$ is proper, flat, and of finite presentation, and\n\\item the geometric fibres of $f$ are reduced and connected.\n\\end{enumerate}\nThen $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and this holds\nafter any base change.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Perfect complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0S","source_file":"spaces-perfect.tex","source_line":6679,"source_end_line":6689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6679-L6689","statement_sha256":"7e04ce6165d13473aee80f6e571d7d96a9a24e267a82eaaff851067aded092eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":11985,"rank":11985,"depth":70,"x":321.336,"y":559.463,"cluster":"derived-categories"},{"id":"stacks:0CRU","tag":"0CRU","title":"Other applications · Lemma 0CRU","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let K be an object of D_QCoh(O_X) such that the cohomology sheaves H^i(K) have countable sets of sections over affine schemes étale over X. Then for any quasi-compact and quasi-separated étale morphism U → X and any perfect object E in D(O_X) the sets H^i(U, K ⊗^L E), Ext^i(E|_U, K|_U) are countable.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$\nsuch that the cohomology sheaves $H^i(K)$ have countable\nsets of sections over affine schemes \\'etale over $X$.\nThen for any quasi-compact and quasi-separated \\'etale morphism $U \\to X$\nand any perfect object $E$ in $D(\\mathcal{O}_X)$\nthe sets\n$$\nH^i(U, K \\otimes^\\mathbf{L} E),\\quad \\Ext^i(E|_U, K|_U)\n$$\nare countable.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Other applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRU","source_file":"spaces-perfect.tex","source_line":6714,"source_end_line":6727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6714-L6727","statement_sha256":"bf90644fd156a9227476ba4ecfe490757f1e7d2db6dc21b2cbdb47e4903280e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":11986,"rank":11986,"depth":60,"x":259.606,"y":820.815,"cluster":"derived-categories"},{"id":"stacks:0CRV","tag":"0CRV","title":"Other applications · Lemma 0CRV","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Assume the sets of sections of O_X over affines étale over X are countable. Let K be an object of D_QCoh(O_X). The following are equivalent • K = hocolim E_n with E_n a perfect object of D(O_X), and • the cohomology sheaves H^i(K) have countable sets of sections over affines étale over X.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nAssume the sets of sections of $\\mathcal{O}_X$ over affines \\'etale over $X$\nare countable. Let $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The\nfollowing are equivalent\n\\begin{enumerate}\n\\item $K = \\text{hocolim} E_n$ with $E_n$ a perfect object of\n$D(\\mathcal{O}_X)$, and\n\\item the cohomology sheaves $H^i(K)$ have countable\nsets of sections over affines \\'etale over $X$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Other applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRV","source_file":"spaces-perfect.tex","source_line":6751,"source_end_line":6764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6751-L6764","statement_sha256":"70376ecf93b9f4faf38d63a4cbb6509789599cde02f37508f181433abaa445f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":11987,"rank":11987,"depth":66,"x":94.827,"y":592.898,"cluster":"derived-categories"},{"id":"stacks:0CRW","tag":"0CRW","title":"Other applications · Lemma 0CRW","summary":"Let A be a ring. Let f : U → X be a flat morphism of algebraic spaces of finite presentation over A. Then • there exists an inverse system of perfect objects L_n of D(O_X) such that RΓ(U, Lf^*K) = hocolim RHom_X(L_n, K) in D(A) functorially in K in D_QCoh(O_X), and • there exists a system of perfect objects E_n of D(O_X) such that RΓ(U, Lf^*K) = hocolim RΓ(X, E_n ⊗^L K) in D(A) functorially in K in D_QCoh(O_X).","statement_latex":"Let $A$ be a ring. Let $f : U \\to X$ be a flat morphism of algebraic spaces\nof finite presentation over $A$. Then\n\\begin{enumerate}\n\\item there exists an inverse system of perfect objects $L_n$ of\n$D(\\mathcal{O}_X)$ such that\n$$\nR\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Hom_X(L_n, K)\n$$\nin $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$, and\n\\item there exists a system of perfect objects $E_n$ of\n$D(\\mathcal{O}_X)$ such that\n$$\nR\\Gamma(U, Lf^*K) = \\text{hocolim}\\ R\\Gamma(X, E_n \\otimes^\\mathbf{L} K)\n$$\nin $D(A)$ functorially in $K$ in $D_\\QCoh(\\mathcal{O}_X)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Other applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRW","source_file":"spaces-perfect.tex","source_line":6805,"source_end_line":6823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6805-L6823","statement_sha256":"b88e8727741b4dfaa9bfe136b8eb862b6c44e24be8f5d3a9ccb385c959099e5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11988,"rank":11988,"depth":67,"x":399.847,"y":667.518,"cluster":"derived-categories"},{"id":"stacks:0GUS","tag":"0GUS","title":"The resolution property · Definition 0GUS","summary":"Let S be a scheme. Let X be an algebraic space over S. We say X has the resolution property if every quasi-coherent O_X-module of finite type is the quotient of a finite locally free O_X-module.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nWe say $X$ has the {\\it resolution property} if every quasi-coherent\n$\\mathcal{O}_X$-module of finite type is the quotient of a\nfinite locally free $\\mathcal{O}_X$-module.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The resolution property","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUS","source_file":"spaces-perfect.tex","source_line":6893,"source_end_line":6899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6893-L6899","statement_sha256":"d75822ea5aada47bdfd318196725a2c8e6906bed7e4368339f40090b68b95499","origin":"The Stacks Project","memory_eligible":false,"source_rank":11989,"rank":11989,"depth":0,"x":114.708,"y":785.659,"cluster":"derived-categories"},{"id":"stacks:0GUT","tag":"0GUT","title":"The resolution property · Lemma 0GUT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • Y is quasi-compact and quasi-separated and has the resolution property, • there exists an f-ample invertible module on X (Divisors on Spaces, Definition [Tag 0D31]). Then X has the resolution property.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item $Y$ is quasi-compact and quasi-separated and has the resolution property,\n\\item there exists an $f$-ample invertible module on $X$\n(Divisors on Spaces, Definition\n\\ref{spaces-divisors-definition-relatively-ample}).\n\\end{enumerate}\nThen $X$ has the resolution property.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUT","source_file":"spaces-perfect.tex","source_line":6912,"source_end_line":6923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6912-L6923","statement_sha256":"510f7fbd6f91cd02a640dc15dd45de6f0377b3e1077886df765bf26b536ec0c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":11990,"rank":11990,"depth":62,"x":230.046,"y":536.561,"cluster":"derived-categories"},{"id":"stacks:0GUU","tag":"0GUU","title":"The resolution property · Lemma 0GUU","summary":"Let S be a scheme. Let f : X → Y be an affine or quasi-affine morphism of algebraic spaces over S with Y quasi-compact and quasi-separated. If Y has the resolution property, so does X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be an affine or quasi-affine morphism of\nalgebraic spaces over $S$ with\n$Y$ quasi-compact and quasi-separated.\nIf $Y$ has the resolution property, so does $X$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUU","source_file":"spaces-perfect.tex","source_line":6959,"source_end_line":6966,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6959-L6966","statement_sha256":"d5f2c1cf1065ace40b13864367648b5b585e7dcf88b50c27972184b70467db05","origin":"The Stacks Project","memory_eligible":false,"source_rank":11991,"rank":11991,"depth":63,"x":345.402,"y":785.875,"cluster":"derived-categories"},{"id":"stacks:0GUV","tag":"0GUV","title":"The resolution property · Lemma 0GUV","summary":"Let S be a scheme. Let f : X → Y be a surjective finite locally free morphism of algebraic spaces over S. If X has the resolution property, so does Y.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a surjective finite locally free morphism of\nalgebraic spaces over $S$.\nIf $X$ has the resolution property, so does $Y$.","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUV","source_file":"spaces-perfect.tex","source_line":6977,"source_end_line":6983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L6977-L6983","statement_sha256":"1e92219e6177f9187da2b7f492ec1e9c429e69472890d7c1942931803d71f216","origin":"The Stacks Project","memory_eligible":false,"source_rank":11992,"rank":11992,"depth":0,"x":59.634,"y":667.401,"cluster":"derived-categories"},{"id":"stacks:0GFF","tag":"0GFF","title":"Detecting Boundedness · Lemma 0GFF","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let P ∈ D_perf(O_X) and E ∈ D_QCoh(O_X). Let a ∈ Z. The following are equivalent • Hom_D(O_X)(P[-i], E) = 0 for i gg 0, and • Hom_D(O_X)(P[-i], τ_≥ a E) = 0 for i gg 0.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$.\nLet $a \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\gg 0$, and\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\geq a} E) = 0$ for $i \\gg 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Detecting Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFF","source_file":"spaces-perfect.tex","source_line":7035,"source_end_line":7045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L7035-L7045","statement_sha256":"686672e320bcc9ca6deb5b23443bac6b84904fb395e8d6e0f26593be81a4c452","origin":"The Stacks Project","memory_eligible":false,"source_rank":11993,"rank":11993,"depth":59,"x":365.858,"y":592.555,"cluster":"derived-categories"},{"id":"stacks:0GFG","tag":"0GFG","title":"Detecting Boundedness · Lemma 0GFG","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let P ∈ D_perf(O_X) and E ∈ D_QCoh(O_X). Let a ∈ Z. The following are equivalent • Hom_D(O_X)(P[-i], E) = 0 for i ll 0, and • Hom_D(O_X)(P[-i], τ_≤ a E) = 0 for i ll 0.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let\n$P \\in D_{perf}(\\mathcal{O}_X)$ and $E \\in D_{\\QCoh}(\\mathcal{O}_X)$.\nLet $a \\in \\mathbf{Z}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], E) = 0$ for $i \\ll 0$, and\n\\item $\\Hom_{D(\\mathcal{O}_X)}(P[-i], \\tau_{\\leq a} E) = 0$ for $i \\ll 0$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Detecting Boundedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFG","source_file":"spaces-perfect.tex","source_line":7058,"source_end_line":7068,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L7058-L7068","statement_sha256":"183361fbad8ccfe235a8903f7b9e2e264a57ce3ebc199cc7951f58e8bebe3f90","origin":"The Stacks Project","memory_eligible":false,"source_rank":11994,"rank":11994,"depth":59,"x":200.119,"y":821.676,"cluster":"derived-categories"},{"id":"stacks:0GFH","tag":"0GFH","title":"Detecting Boundedness · Proposition 0GFH","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let G ∈ D_perf(O_X) be a perfect complex which generates D_QCoh (O_X). Let E ∈ D_QCoh (O_X). The following are equivalent • E ∈ D^-_QCoh (O_X), • Hom_D(O_X)(G[-i], E) = 0 for i gg 0, • Ext^i_X(G, E) = 0 for i gg 0, • RHom_X(G, E) is in D^-(Z), • H^i(X, G^vee ⊗_O_X^L E) = 0 for i gg 0, • RΓ(X, G^vee ⊗_O_X^L E) is in D^-(Z), • for every perfect object P of D(O_X) • the assertions (2),…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates\n$D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $E \\in D^-_\\QCoh (\\mathcal{O}_X)$,\n\\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\gg 0$,\n\\item $\\Ext^i_X(G, E) = 0$ for $i \\gg 0$,\n\\item $R\\Hom_X(G, E)$ is in $D^-(\\mathbf{Z})$,\n\\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$\nfor $i \\gg 0$,\n\\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$\nis in $D^-(\\mathbf{Z})$,\n\\item for every perfect object $P$ of $D(\\mathcal{O}_X)$\n\\begin{enumerate}\n\\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and\n\\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\gg 0$,\n\\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$\nis in $D^-(\\mathbf{Z})$.\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Detecting Boundedness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFH","source_file":"spaces-perfect.tex","source_line":7081,"source_end_line":7105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L7081-L7105","statement_sha256":"2d6ffe49764d6c05737d9e885330fbfdbcc56e502fd908265ad993304196d22c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11995,"rank":11995,"depth":65,"x":138.034,"y":558.485,"cluster":"derived-categories"},{"id":"stacks:0GFI","tag":"0GFI","title":"Detecting Boundedness · Proposition 0GFI","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let G ∈ D_perf(O_X) be a perfect complex which generates D_QCoh (O_X). Let E ∈ D_QCoh (O_X). The following are equivalent • E ∈ D^+_QCoh (O_X), • Hom_D(O_X)(G[-i], E) = 0 for i ll 0, • Ext^i_X(G, E) = 0 for i ll 0, • RHom_X(G, E) is in D^+(Z), • H^i(X, G^vee ⊗_O_X^L E) = 0 for i ll 0, • RΓ(X, G^vee ⊗_O_X^L E) is in D^+(Z), • for every perfect object P of D(O_X) • the assertions (2),…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $G \\in D_{perf}(\\mathcal{O}_X)$ be a perfect complex which generates\n$D_\\QCoh (\\mathcal{O}_X)$. Let $E \\in D_\\QCoh (\\mathcal{O}_X)$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $E \\in D^+_\\QCoh (\\mathcal{O}_X)$,\n\\item $\\Hom_{D(\\mathcal{O}_X)}(G[-i], E) = 0$ for $i \\ll 0$,\n\\item $\\Ext^i_X(G, E) = 0$ for $i \\ll 0$,\n\\item $R\\Hom_X(G, E)$ is in $D^+(\\mathbf{Z})$,\n\\item $H^i(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$\nfor $i \\ll 0$,\n\\item $R\\Gamma(X, G^\\vee \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$\nis in $D^+(\\mathbf{Z})$,\n\\item for every perfect object $P$ of $D(\\mathcal{O}_X)$\n\\begin{enumerate}\n\\item the assertions (2), (3), (4) hold with $G$ replaced by $P$, and\n\\item $H^i(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E) = 0$ for $i \\ll 0$,\n\\item $R\\Gamma(X, P \\otimes_{\\mathcal{O}_X}^\\mathbf{L} E)$\nis in $D^+(\\mathbf{Z})$.\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Detecting Boundedness","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFI","source_file":"spaces-perfect.tex","source_line":7204,"source_end_line":7228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L7204-L7228","statement_sha256":"7223007670cf3a0c20b2581c9770410ad333ce0056f43fdccdda53fbc665094f","origin":"The Stacks Project","memory_eligible":false,"source_rank":11996,"rank":11996,"depth":65,"x":395.657,"y":717.448,"cluster":"derived-categories"},{"id":"stacks:0H06","tag":"0H06","title":"Quasi-coherent objects in the derived category · Lemma 0H06","summary":"In the situation above there are canonical exact equivalences between the following triangulated categories • D_QCoh(O_X), • D_QCoh(X_affine, etale, O_X), • D_QCoh(X_affine, O_X), and • mathitQC(X_affine, O_X).","statement_latex":"In the situation above there are canonical exact equivalences between\nthe following triangulated categories\n\\begin{enumerate}\n\\item $D_\\QCoh(\\mathcal{O}_X)$,\n\\item $D_\\QCoh(X_{affine, \\etale}, \\mathcal{O}_X)$,\n\\item $D_\\QCoh(X_{affine}, \\mathcal{O}_X)$, and\n\\item $\\mathit{QC}(X_{affine}, \\mathcal{O}_X)$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Spaces","chapter_id":"spaces-perfect","section":"Quasi-coherent objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H06","source_file":"spaces-perfect.tex","source_line":7379,"source_end_line":7389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-perfect.tex#L7379-L7389","statement_sha256":"853aaeba37256d0b0cd20ee6428a17c2a546d2335e223b06a9090241609591d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":11997,"rank":11997,"depth":42,"x":77.619,"y":746.434,"cluster":"derived-categories"},{"id":"stacks:0481","tag":"0481","title":"Radicial morphisms · Definition 0481","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say f is radicial if for any morphism Spec(K) → Y where K is a field the reduction (Spec(K) ×_Y X)_red is either empty or representable by the spectrum of a purely inseparable field extension of K.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. We say $f$ is {\\it radicial} if for any morphism\n$\\Spec(K) \\to Y$ where $K$ is a field the reduction\n$(\\Spec(K) \\times_Y X)_{red}$ is either empty or\nrepresentable by the spectrum of a purely inseparable field extension of $K$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Radicial morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0481","source_file":"spaces-more-morphisms.tex","source_line":58,"source_end_line":65,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L58-L65","statement_sha256":"6c1152cb22dd70bd224f1eba0435560d3d2afbc3ad3cd995b8cf054255d4909c","origin":"The Stacks Project","memory_eligible":false,"source_rank":11998,"rank":11998,"depth":0,"x":743.083,"y":1663.203,"cluster":"geometry-of-spaces"},{"id":"stacks:0482","tag":"0482","title":"Radicial morphisms · Lemma 0482","summary":"A radicial morphism of algebraic spaces is universally injective.","statement_latex":"A radicial morphism of algebraic spaces is universally injective.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Radicial morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0482","source_file":"spaces-more-morphisms.tex","source_line":67,"source_end_line":70,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L67-L70","statement_sha256":"bfd3a901de7c4fe6c935f6473639cc177634fad2eba8a5d8d5d8a859851090f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":11999,"rank":11999,"depth":55,"x":582.511,"y":1589.174,"cluster":"geometry-of-spaces"},{"id":"stacks:0484","tag":"0484","title":"Radicial morphisms · Lemma 0484","summary":"Let S be a scheme. Let f : X → Y be a universally injective morphism of algebraic spaces over S. • If f is decent then f is radicial. • If f is quasi-separated then f is radicial. • If f is locally separated then f is radicial.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a universally injective\nmorphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $f$ is decent then $f$ is radicial.\n\\item If $f$ is quasi-separated then $f$ is radicial.\n\\item If $f$ is locally separated then $f$ is radicial.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Radicial morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0484","source_file":"spaces-more-morphisms.tex","source_line":101,"source_end_line":110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L101-L110","statement_sha256":"43a26a6655d7747aa939045376a40849f9f462f6ad201e41597d050cf491b51d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12000,"rank":12000,"depth":60,"x":760.715,"y":1552.594,"cluster":"geometry-of-spaces"},{"id":"stacks:0AGE","tag":"0AGE","title":"Radicial morphisms · Lemma 0AGE","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • f is locally of finite type, • for every étale morphism V → Y the map |X ×_Y V| → |V| is injective. Then f is universally injective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item for every \\'etale morphism $V \\to Y$ the map $|X \\times_Y V| \\to |V|$\nis injective.\n\\end{enumerate}\nThen $f$ is universally injective.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Radicial morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGE","source_file":"spaces-more-morphisms.tex","source_line":212,"source_end_line":222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L212-L222","statement_sha256":"54b147e85590ff7d9f381e7351bfaba7b071b11794e886f902b05ff5d08135bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12001,"rank":12001,"depth":70,"x":658.572,"y":1680.878,"cluster":"geometry-of-spaces"},{"id":"stacks:0B8A","tag":"0B8A","title":"David Rydh · Lemma 0B8A","summary":"A flat monomorphism of algebraic spaces is representable by schemes.","statement_latex":"A flat monomorphism of algebraic spaces is representable by schemes.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8A","source_file":"spaces-more-morphisms.tex","source_line":286,"source_end_line":289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L286-L289","statement_sha256":"b73b63d46af187f8e66e2eb58069c4ffb30a1fb24198f50e82d1c431fb46929e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12002,"rank":12002,"depth":57,"x":630.687,"y":1528.096,"cluster":"geometry-of-spaces"},{"id":"stacks:0B8B","tag":"0B8B","title":"Monomorphisms · Lemma 0B8B","summary":"Let S be a scheme. Let f : X → Y be a quasi-compact monomorphism of algebraic spaces such that for every T → Y the map O_T → f_T,*O_X ×_Y T is injective. Then f is an isomorphism (and hence representable by schemes).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a quasi-compact monomorphism\nof algebraic spaces such that for every $T \\to Y$ the map\n$$\n\\mathcal{O}_T \\to f_{T,*}\\mathcal{O}_{X \\times_Y T}\n$$\nis injective. Then $f$ is an isomorphism (and hence representable by schemes).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8B","source_file":"spaces-more-morphisms.tex","source_line":318,"source_end_line":326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L318-L326","statement_sha256":"240add02b4bc5ae9e4162111b7aac8240402066953f80bbe7134f9c0ba6648f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12003,"rank":12003,"depth":56,"x":774.327,"y":1625.077,"cluster":"geometry-of-spaces"},{"id":"stacks:0B8C","tag":"0B8C","title":"Monomorphisms · Lemma 0B8C","summary":"A quasi-compact flat surjective monomorphism of algebraic spaces is an isomorphism.","statement_latex":"A quasi-compact flat surjective monomorphism of algebraic spaces\nis an isomorphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8C","source_file":"spaces-more-morphisms.tex","source_line":362,"source_end_line":366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L362-L366","statement_sha256":"929ebf06b435defa520af998effbbe04e2531b81e7698d8c74a0569c6c00ef3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12004,"rank":12004,"depth":57,"x":590.145,"y":1635.084,"cluster":"geometry-of-spaces"},{"id":"stacks:04CN","tag":"04CN","title":"Conormal sheaf of an immersion · Definition 04CN","summary":"Let i : Z → X be an immersion. The conormal sheaf C_Z/X of Z in X or the conormal sheaf of i is the quasi-coherent O_Z-module I/I^2 described above.","statement_latex":"Let $i : Z \\to X$ be an immersion. The {\\it conormal sheaf\n$\\mathcal{C}_{Z/X}$ of $Z$ in $X$} or the {\\it conormal sheaf of $i$}\nis the quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{I}/\\mathcal{I}^2$\ndescribed above.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Conormal sheaf of an immersion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CN","source_file":"spaces-more-morphisms.tex","source_line":409,"source_end_line":415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L409-L415","statement_sha256":"b65ba2839067ebb5eacf80822d3d143508576abedf4b5ee4f67888e852eae21c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12005,"rank":12005,"depth":0,"x":718.101,"y":1523.03,"cluster":"geometry-of-spaces"},{"id":"stacks:04CO","tag":"04CO","title":"Conormal sheaf of an immersion · Lemma 04CO","summary":"Let S be a scheme. Let i : Z → X be an immersion. Let φ : U → X be an étale morphism where U is a scheme. Set Z_U = U ×_X Z which is a locally closed subscheme of U. Then C_Z/X|_Z_U = C_Z_U/U canonically and functorially in U.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion.\nLet $\\varphi : U \\to X$ be an \\'etale morphism where $U$ is a scheme.\nSet $Z_U = U \\times_X Z$ which is a locally closed subscheme of $U$.\nThen\n$$\n\\mathcal{C}_{Z/X}|_{Z_U} = \\mathcal{C}_{Z_U/U}\n$$\ncanonically and functorially in $U$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Conormal sheaf of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CO","source_file":"spaces-more-morphisms.tex","source_line":431,"source_end_line":441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L431-L441","statement_sha256":"f60975f49969e9e88d3d7d08182918bf6f11dbbbf2a7817a35f9ed0b0bf36f10","origin":"The Stacks Project","memory_eligible":false,"source_rank":12006,"rank":12006,"depth":52,"x":713.853,"y":1678.492,"cluster":"geometry-of-spaces"},{"id":"stacks:04CP","tag":"04CP","title":"Conormal sheaf of an immersion · Lemma 04CP","summary":"Let S be a scheme. Let xymatrix Z ar[r]_i ar[d]_f & X ar[d]^g Z' ar[r]^i' & X' be a commutative diagram of algebraic spaces over S. Assume i, i' immersions. There is a canonical map of O_Z-modules f^*C_Z'/X' → C_Z/X","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\\nZ' \\ar[r]^{i'} & X'\n}\n$$\nbe a commutative diagram of algebraic spaces over $S$.\nAssume $i$, $i'$ immersions. There is a canonical map\nof $\\mathcal{O}_Z$-modules\n$$\nf^*\\mathcal{C}_{Z'/X'}\n\\longrightarrow\n\\mathcal{C}_{Z/X}\n$$","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Conormal sheaf of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CP","source_file":"spaces-more-morphisms.tex","source_line":454,"source_end_line":471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L454-L471","statement_sha256":"c4648a21dab898c38e171c2167c3959ce704622c773dbede7f9fbda27d7a56f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12007,"rank":12007,"depth":52,"x":591.786,"y":1561.273,"cluster":"geometry-of-spaces"},{"id":"stacks:04G2","tag":"04G2","title":"Conormal sheaf of an immersion · Lemma 04G2","summary":"Let S be a scheme. The conormal sheaf of Definition [Tag 04CN], and its functoriality of Lemma [Tag 04CP] satisfy the following properties: • If Z → X is an immersion of schemes over S, then the conormal sheaf agrees with the one from Morphisms, Definition [Tag 01R2]. • If in Lemma [Tag 04CP] all the spaces are schemes, then the map f^*C_Z'/X' → C_Z/X is the same as the one constructed in Morphisms, Lemma [Tag 01R4]. • Given a commutative diagram xymatrix Z ar[r]_i…","statement_latex":"Let $S$ be a scheme. The conormal sheaf of\nDefinition \\ref{definition-conormal-sheaf}, and its functoriality of\nLemma \\ref{lemma-conormal-functorial}\nsatisfy the following properties:\n\\begin{enumerate}\n\\item If $Z \\to X$ is an immersion of schemes over $S$, then the conormal\nsheaf agrees with the one from\nMorphisms, Definition \\ref{morphisms-definition-conormal-sheaf}.\n\\item If in\nLemma \\ref{lemma-conormal-functorial}\nall the spaces are schemes, then the map\n$f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ is the same\nas the one constructed in\nMorphisms, Lemma \\ref{morphisms-lemma-conormal-functorial}.\n\\item Given a commutative diagram\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\\nZ' \\ar[r]^{i'} \\ar[d]_{f'} & X' \\ar[d]^{g'} \\\\\nZ'' \\ar[r]^{i''} & X''\n}\n$$\nthen the map $(f' \\circ f)^*\\mathcal{C}_{Z''/X''} \\to \\mathcal{C}_{Z/X}$\nis the same as the composition of\n$f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$\nwith the pullback by $f$ of\n$(f')^*\\mathcal{C}_{Z''/X''} \\to \\mathcal{C}_{Z'/X'}$\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Conormal sheaf of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04G2","source_file":"spaces-more-morphisms.tex","source_line":519,"source_end_line":549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L519-L549","statement_sha256":"93e76b5d79adc52923bc4728360f83a8c8fd6e3f9314df70f5b6fefb061b8837","origin":"The Stacks Project","memory_eligible":false,"source_rank":12008,"rank":12008,"depth":53,"x":776.332,"y":1578.471,"cluster":"geometry-of-spaces"},{"id":"stacks:04CQ","tag":"04CQ","title":"Conormal sheaf of an immersion · Lemma 04CQ","summary":"Let S be a scheme. Let xymatrix Z ar[r]_i ar[d]_f & X ar[d]^g Z' ar[r]^i' & X' be a fibre product diagram of algebraic spaces over S. Assume i, i' immersions. Then the canonical map f^*C_Z'/X' → C_Z/X of Lemma [Tag 04CP] is surjective. If g is flat, then it is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\\nZ' \\ar[r]^{i'} & X'\n}\n$$\nbe a fibre product diagram of algebraic spaces over $S$. Assume\n$i$, $i'$ immersions. Then the canonical map\n$f^*\\mathcal{C}_{Z'/X'} \\to \\mathcal{C}_{Z/X}$ of\nLemma \\ref{lemma-conormal-functorial}\nis surjective. If $g$ is flat, then it is an isomorphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Conormal sheaf of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CQ","source_file":"spaces-more-morphisms.tex","source_line":556,"source_end_line":570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L556-L570","statement_sha256":"8bba74291adcad447e942298da24986df4b4edd080e531fd0978200769a67b1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12009,"rank":12009,"depth":54,"x":626.201,"y":1670.64,"cluster":"geometry-of-spaces"},{"id":"stacks:06BD","tag":"06BD","title":"Conormal sheaf of an immersion · Lemma 06BD","summary":"Let S be a scheme. Let Z → Y → X be immersions of algebraic spaces. Then there is a canonical exact sequence i^*C_Y/X → C_Z/X → C_Z/Y → 0 where the maps come from Lemma [Tag 04CP] and i : Z → Y is the first morphism.","statement_latex":"Let $S$ be a scheme.\nLet $Z \\to Y \\to X$ be immersions of algebraic spaces.\nThen there is a canonical exact sequence\n$$\ni^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nwhere the maps come from\nLemma \\ref{lemma-conormal-functorial}\nand $i : Z \\to Y$ is the first morphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Conormal sheaf of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BD","source_file":"spaces-more-morphisms.tex","source_line":590,"source_end_line":603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L590-L603","statement_sha256":"514e67043fa9cb49dd1188c57df9c67bb46de50ece4a7fd79b56afa01e7bdaeb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12010,"rank":12010,"depth":54,"x":662.839,"y":1517.263,"cluster":"geometry-of-spaces"},{"id":"stacks:09RN","tag":"09RN","title":"The normal cone of an immersion · Definition 09RN","summary":"Let i : Z → X be an immersion. The conormal algebra C_Z/X, * of Z in X or the conormal algebra of i is the quasi-coherent sheaf of graded O_Z-algebras bigoplus_n ≥ 0 I^n/I^n + 1 described above.","statement_latex":"Let $i : Z \\to X$ be an immersion. The {\\it conormal algebra\n$\\mathcal{C}_{Z/X, *}$ of $Z$ in $X$} or the {\\it conormal algebra of $i$}\nis the quasi-coherent sheaf of graded $\\mathcal{O}_Z$-algebras\n$\\bigoplus_{n \\geq 0} \\mathcal{I}^n/\\mathcal{I}^{n + 1}$ described above.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The normal cone of an immersion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RN","source_file":"spaces-more-morphisms.tex","source_line":653,"source_end_line":659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L653-L659","statement_sha256":"b9e69e9ca8b80ba8ca3f8dac56262d78dcdda4a4afe1cee9ed4127ae86e5e142","origin":"The Stacks Project","memory_eligible":false,"source_rank":12011,"rank":12011,"depth":0,"x":759.305,"y":1651.346,"cluster":"geometry-of-spaces"},{"id":"stacks:09RR","tag":"09RR","title":"The normal cone of an immersion · Lemma 09RR","summary":"Let S be a scheme. Let i : Z → X be an immersion of algebraic spaces over S. Let φ : U → X be an étale morphism where U is a scheme. Set Z_U = U ×_X Z which is a locally closed subscheme of U. Then C_Z/X, *|_Z_U = C_Z_U/U, * canonically and functorially in U.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces\nover $S$. Let $\\varphi : U \\to X$ be an \\'etale morphism where $U$ is a\nscheme. Set $Z_U = U \\times_X Z$ which is a locally closed subscheme of $U$.\nThen\n$$\n\\mathcal{C}_{Z/X, *}|_{Z_U} = \\mathcal{C}_{Z_U/U, *}\n$$\ncanonically and functorially in $U$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The normal cone of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RR","source_file":"spaces-more-morphisms.tex","source_line":679,"source_end_line":689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L679-L689","statement_sha256":"d9c1e07178f299cc060662acb22d692e48769fefd6808b5bf93904ddaca1cfd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12012,"rank":12012,"depth":52,"x":580.086,"y":1607.148,"cluster":"geometry-of-spaces"},{"id":"stacks:09RS","tag":"09RS","title":"The normal cone of an immersion · Lemma 09RS","summary":"Let S be a scheme. Let xymatrix Z ar[r]_i ar[d]_f & X ar[d]^g Z' ar[r]^i' & X' be a commutative diagram of algebraic spaces over S. Assume i, i' immersions. There is a canonical map of graded O_Z-algebras f^*C_Z'/X', * → C_Z/X, *","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\\nZ' \\ar[r]^{i'} & X'\n}\n$$\nbe a commutative diagram of algebraic spaces over $S$.\nAssume $i$, $i'$ immersions. There is a canonical map\nof graded $\\mathcal{O}_Z$-algebras\n$$\nf^*\\mathcal{C}_{Z'/X', *}\n\\longrightarrow\n\\mathcal{C}_{Z/X, *}\n$$","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The normal cone of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RS","source_file":"spaces-more-morphisms.tex","source_line":703,"source_end_line":720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L703-L720","statement_sha256":"5c38ad2fbb048fac5c06b33325eabe7995e1d91eca5f68c2dee9b348fe642921","origin":"The Stacks Project","memory_eligible":false,"source_rank":12013,"rank":12013,"depth":52,"x":748.026,"y":1537.945,"cluster":"geometry-of-spaces"},{"id":"stacks:09RT","tag":"09RT","title":"The normal cone of an immersion · Lemma 09RT","summary":"Let S be a scheme. Let xymatrix Z ar[r]_i ar[d]_f & X ar[d]^g Z' ar[r]^i' & X' be a cartesian square of algebraic spaces over S with i, i' immersions. Then the canonical map f^*C_Z'/X', * → C_Z/X, * of Lemma [Tag 09RS] is surjective. If g is flat, then it is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[d]_f & X \\ar[d]^g \\\\\nZ' \\ar[r]^{i'} & X'\n}\n$$\nbe a cartesian square of algebraic spaces over $S$ with\n$i$, $i'$ immersions. Then the canonical map\n$f^*\\mathcal{C}_{Z'/X', *} \\to \\mathcal{C}_{Z/X, *}$ of\nLemma \\ref{lemma-conormal-algebra-functorial}\nis surjective. If $g$ is flat, then it is an isomorphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The normal cone of an immersion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RT","source_file":"spaces-more-morphisms.tex","source_line":770,"source_end_line":784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L770-L784","statement_sha256":"ced4a8904718b790b8487574965d532eda05b9e4893081fa4418c1ea7994e6dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12014,"rank":12014,"depth":53,"x":679.741,"y":1684.479,"cluster":"geometry-of-spaces"},{"id":"stacks:09RU","tag":"09RU","title":"The normal cone of an immersion · Definition 09RU","summary":"Let S be a scheme. Let i : Z → X be an immersion of algebraic spaces over S. The normal cone C_ZX of Z in X is C_ZX = underlineSpec_Z(C_Z/X, *) see Morphisms of Spaces, Definition [Tag 081W]. The normal bundle of Z in X is the vector bundle N_ZX = underlineSpec_Z(Sym(C_Z/X))","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces\nover $S$. The {\\it normal cone $C_ZX$} of $Z$ in $X$ is\n$$\nC_ZX = \\underline{\\Spec}_Z(\\mathcal{C}_{Z/X, *})\n$$\nsee Morphisms of Spaces,\nDefinition \\ref{spaces-morphisms-definition-relative-spec}. The\n{\\it normal bundle} of $Z$ in $X$ is the vector bundle\n$$\nN_ZX = \\underline{\\Spec}_Z(\\text{Sym}(\\mathcal{C}_{Z/X}))\n$$","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The normal cone of an immersion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RU","source_file":"spaces-more-morphisms.tex","source_line":803,"source_end_line":816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L803-L816","statement_sha256":"a4ed9c36698722b0035e75e472feba70d1182d2549f1086c9717bc05d66b2915","origin":"The Stacks Project","memory_eligible":false,"source_rank":12015,"rank":12015,"depth":54,"x":612.157,"y":1537.469,"cluster":"geometry-of-spaces"},{"id":"stacks:04CS","tag":"04CS","title":"Sheaf of differentials of a morphism · Lemma 04CS","summary":"Let f : X → Y be a morphism of schemes. Let f_small : X_etale → Y_etale be the associated morphism of small étale sites, see Descent, Remark [Tag 070R]. Then there is a canonical isomorphism (Ω_X/Y)^a = Ω_X_etale/Y_etale compatible with universal derivations. Here the first module is the sheaf on X_etale associated to the quasi-coherent O_X-module Ω_X/Y, see Morphisms, Definition [Tag 01UQ], and the second module is the one from Modules on Sites, Definition [Tag 04BN].","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let\n$f_{small} : X_\\etale \\to Y_\\etale$ be the associated\nmorphism of small \\'etale sites, see\nDescent, Remark \\ref{descent-remark-change-topologies-ringed}.\nThen there is a canonical isomorphism\n$$\n(\\Omega_{X/Y})^a = \\Omega_{X_\\etale/Y_\\etale}\n$$\ncompatible with universal derivations. Here the first module\nis the sheaf on $X_\\etale$ associated\nto the quasi-coherent $\\mathcal{O}_X$-module $\\Omega_{X/Y}$, see\nMorphisms, Definition \\ref{morphisms-definition-sheaf-differentials},\nand the second module is the one from\nModules on Sites,\nDefinition \\ref{sites-modules-definition-module-differentials}.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CS","source_file":"spaces-more-morphisms.tex","source_line":860,"source_end_line":877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L860-L877","statement_sha256":"ecd2ef5118139f88f2034c7197c248fc39a017ba3f0b640644c44e95857fe75c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12016,"rank":12016,"depth":19,"x":780.456,"y":1607.625,"cluster":"geometry-of-spaces"},{"id":"stacks:04CT","tag":"04CT","title":"Sheaf of differentials of a morphism · Definition 04CT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The sheaf of differentials Ω_X/Y of X over Y is sheaf of differentials (Modules on Sites, Definition [Tag 04BQ]) for the morphism of ringed topoi (f_small, f^sharp) : (X_etale, O_X) → (Y_etale, O_Y) of Properties of Spaces, Lemma [Tag 03G8]. The universal Y-derivation will be denoted d_X/Y : O_X → Ω_X/Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. The {\\it sheaf of differentials $\\Omega_{X/Y}$ of $X$ over $Y$}\nis sheaf of differentials\n(Modules on Sites,\nDefinition \\ref{sites-modules-definition-sheaf-differentials})\nfor the morphism of ringed topoi\n$$\n(f_{small}, f^\\sharp) :\n(X_\\etale, \\mathcal{O}_X)\n\\to\n(Y_\\etale, \\mathcal{O}_Y)\n$$\nof\nProperties of Spaces,\nLemma \\ref{spaces-properties-lemma-morphism-ringed-topoi}.\nThe {\\it universal $Y$-derivation} will be denoted\n$\\text{d}_{X/Y} : \\mathcal{O}_X \\to \\Omega_{X/Y}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CT","source_file":"spaces-more-morphisms.tex","source_line":948,"source_end_line":967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L948-L967","statement_sha256":"1459e1eaad4ccdecc806ab72bb90134438232aa848a95da384fc6b239d03fb47","origin":"The Stacks Project","memory_eligible":false,"source_rank":12017,"rank":12017,"depth":11,"x":599.679,"y":1651.452,"cluster":"geometry-of-spaces"},{"id":"stacks:04CU","tag":"04CU","title":"Sheaf of differentials of a morphism · Lemma 04CU","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Consider any commutative diagram xymatrix U ar[d]_a ar[r]_ψ & V ar[d]^b X ar[r]^f & Y where the vertical arrows are étale morphisms of algebraic spaces. Then Ω_X/Y|_U_etale = Ω_U/V In particular, if U, V are schemes, then this is equal to the usual sheaf of differentials of the morphism of schemes U → V.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Consider any commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_\\psi & V \\ar[d]^b \\\\\nX \\ar[r]^f & Y\n}\n$$\nwhere the vertical arrows are \\'etale morphisms of algebraic spaces. Then\n$$\n\\Omega_{X/Y}|_{U_\\etale} = \\Omega_{U/V}\n$$\nIn particular, if $U$, $V$ are schemes, then this is equal to the usual\nsheaf of differentials of the morphism of schemes $U \\to V$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CU","source_file":"spaces-more-morphisms.tex","source_line":980,"source_end_line":996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L980-L996","statement_sha256":"f92bec8e9eb5d4c950d95a441408ec5ecfaeede06ba2c1ee23466cb7c36eb998","origin":"The Stacks Project","memory_eligible":false,"source_rank":12018,"rank":12018,"depth":48,"x":697.876,"y":1516.364,"cluster":"geometry-of-spaces"},{"id":"stacks:04CV","tag":"04CV","title":"Sheaf of differentials of a morphism · Lemma 04CV","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then Ω_X/Y is a quasi-coherent O_X-module.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Then $\\Omega_{X/Y}$ is a quasi-coherent $\\mathcal{O}_X$-module.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CV","source_file":"spaces-more-morphisms.tex","source_line":1016,"source_end_line":1020,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1016-L1020","statement_sha256":"d00eef6c2dde5d7a1fad093aaf4787901b0d791016f989cb61dc311d84838d76","origin":"The Stacks Project","memory_eligible":false,"source_rank":12019,"rank":12019,"depth":49,"x":734.153,"y":1671.917,"cluster":"geometry-of-spaces"},{"id":"stacks:04CX","tag":"04CX","title":"Sheaf of differentials of a morphism · Lemma 04CX","summary":"Let S be a scheme. Let xymatrix X' ar[d] ar[r]_f & X ar[d] Y' ar[r] & Y be a commutative diagram of algebraic spaces. The map f^sharp : O_X → f_*O_X' composed with the map f_*d_X'/Y' : f_*O_X' → f_*Ω_X'/Y' is a Y-derivation. Hence we obtain a canonical map of O_X-modules Ω_X/Y → f_*Ω_X'/Y', and by adjointness of f_* and f^* a canonical O_X'-module homomorphism c_f : f^*Ω_X/Y → Ω_X'/Y'. It is uniquely characterized by the property that f^*d_X/Y(t) maps to d_X'/Y'(f^* t)…","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\\nY' \\ar[r] & Y\n}\n$$\nbe a commutative diagram of algebraic spaces. The map\n$f^\\sharp : \\mathcal{O}_X \\to f_*\\mathcal{O}_{X'}$ composed with the map\n$f_*\\text{d}_{X'/Y'} : f_*\\mathcal{O}_{X'} \\to f_*\\Omega_{X'/Y'}$ is a\n$Y$-derivation. Hence we obtain a canonical map of $\\mathcal{O}_X$-modules\n$\\Omega_{X/Y} \\to f_*\\Omega_{X'/Y'}$, and by\nadjointness of $f_*$ and $f^*$ a\ncanonical $\\mathcal{O}_{X'}$-module homomorphism\n$$\nc_f : f^*\\Omega_{X/Y} \\longrightarrow \\Omega_{X'/Y'}.\n$$\nIt is uniquely characterized by the property that\n$f^*\\text{d}_{X/Y}(t)$ maps to $\\text{d}_{X'/Y'}(f^* t)$\nfor any local section $t$ of $\\mathcal{O}_X$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CX","source_file":"spaces-more-morphisms.tex","source_line":1061,"source_end_line":1083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1061-L1083","statement_sha256":"b2fd8b0d7caf64f033e243814320b1cea9b44552bbc05698c57f22d7db9e6263","origin":"The Stacks Project","memory_eligible":false,"source_rank":12020,"rank":12020,"depth":1,"x":582.096,"y":1577.665,"cluster":"geometry-of-spaces"},{"id":"stacks:05Z7","tag":"05Z7","title":"Sheaf of differentials of a morphism · Lemma 05Z7","summary":"Let S be a scheme. Let xymatrix X\" ar[d] ar[r]_g & X' ar[d] ar[r]_f & X ar[d] Y\" ar[r] & Y' ar[r] & Y be a commutative diagram of algebraic spaces over S. Then we have c_f ∘ g = c_g ∘ g^* c_f as maps (f ∘ g)^*Ω_X/Y → Ω_X\"/Y\".","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX'' \\ar[d] \\ar[r]_g & X' \\ar[d] \\ar[r]_f & X \\ar[d] \\\\\nY'' \\ar[r] & Y' \\ar[r] & Y\n}\n$$\nbe a commutative diagram of algebraic spaces over $S$. Then we have\n$$\nc_{f \\circ g} = c_g \\circ g^* c_f\n$$\nas maps $(f \\circ g)^*\\Omega_{X/Y} \\to \\Omega_{X''/Y''}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Z7","source_file":"spaces-more-morphisms.tex","source_line":1091,"source_end_line":1105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1091-L1105","statement_sha256":"a2f6b0973c517c71f2ffb3e086988272ec1b3833f555172473b23b50a4137f96","origin":"The Stacks Project","memory_eligible":false,"source_rank":12021,"rank":12021,"depth":0,"x":770.282,"y":1560.861,"cluster":"geometry-of-spaces"},{"id":"stacks:05Z8","tag":"05Z8","title":"Sheaf of differentials of a morphism · Lemma 05Z8","summary":"Let S be a scheme. Let f : X → Y, g : Y → B be morphisms of algebraic spaces over S. Then there is a canonical exact sequence f^*Ω_Y/B → Ω_X/B → Ω_X/Y → 0 where the maps come from applications of Lemma [Tag 04CX].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$, $g : Y \\to B$ be morphisms of algebraic spaces over $S$.\nThen there is a canonical exact sequence\n$$\nf^*\\Omega_{Y/B} \\to \\Omega_{X/B} \\to \\Omega_{X/Y} \\to 0\n$$\nwhere the maps come from applications of\nLemma \\ref{lemma-functoriality-differentials}.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Z8","source_file":"spaces-more-morphisms.tex","source_line":1112,"source_end_line":1122,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1112-L1122","statement_sha256":"d0b1adfdaf7407a7b4d0ad64084773da72978ffe50e4123e4417ebacd29654dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12022,"rank":12022,"depth":49,"x":644.852,"y":1680.202,"cluster":"geometry-of-spaces"},{"id":"stacks:05Z9","tag":"05Z9","title":"Sheaf of differentials of a morphism · Lemma 05Z9","summary":"Let S be a scheme. If X → Y is an immersion of algebraic spaces over S then Ω_X/S is zero.","statement_latex":"Let $S$ be a scheme. If $X \\to Y$ is an immersion\nof algebraic spaces over $S$ then $\\Omega_{X/S}$ is zero.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Z9","source_file":"spaces-more-morphisms.tex","source_line":1131,"source_end_line":1135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1131-L1135","statement_sha256":"029a36f88fb64d856c1c966cdb51e7934da44f8c52cf25c0b80e96ce5db1dd36","origin":"The Stacks Project","memory_eligible":false,"source_rank":12023,"rank":12023,"depth":49,"x":641.367,"y":1520.805,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZA","tag":"05ZA","title":"Sheaf of differentials of a morphism · Lemma 05ZA","summary":"Let S be a scheme. Let B be an algebraic space over S. Let i : Z → X be an immersion of algebraic spaces over B. There is a canonical exact sequence C_Z/X → i^*Ω_X/B → Ω_Z/B → 0 where the first arrow is induced by d_X/B and the second arrow comes from Lemma [Tag 04CX].","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $i : Z \\to X$ be an immersion of algebraic spaces over $B$.\nThere is a canonical exact sequence\n$$\n\\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0\n$$\nwhere the first arrow is induced by $\\text{d}_{X/B}$\nand the second arrow comes from\nLemma \\ref{lemma-functoriality-differentials}.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZA","source_file":"spaces-more-morphisms.tex","source_line":1144,"source_end_line":1155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1144-L1155","statement_sha256":"a1ac29ba62781abbc9956752730f35b6766a47421870d6384d69de32da713ef2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12024,"rank":12024,"depth":55,"x":772.303,"y":1636.527,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZB","tag":"05ZB","title":"Sheaf of differentials of a morphism · Lemma 05ZB","summary":"Let S be a scheme. Let B be an algebraic space over S. Let i : Z → X be an immersion of algebraic spaces over B, and assume i (étale locally) has a left inverse. Then the canonical sequence 0 → C_Z/X → i^*Ω_X/B → Ω_Z/B → 0 of Lemma [Tag 05ZA] is (étale locally) split exact.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $i : Z \\to X$ be an immersion of algebraic spaces over $B$, and\nassume $i$ (\\'etale locally) has a left inverse. Then the canonical\nsequence\n$$\n0 \\to \\mathcal{C}_{Z/X} \\to i^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0\n$$\nof\nLemma \\ref{lemma-differentials-relative-immersion}\nis (\\'etale locally) split exact.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZB","source_file":"spaces-more-morphisms.tex","source_line":1179,"source_end_line":1191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1179-L1191","statement_sha256":"b1f1147b9ff4e960b11459c9a6afa22f64e24d311ff44501556b6a490ba167ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":12025,"rank":12025,"depth":56,"x":582.427,"y":1625.476,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZC","tag":"05ZC","title":"Sheaf of differentials of a morphism · Lemma 05ZC","summary":"Let S be a scheme. Let X → Y be a morphism of algebraic spaces over S. Let g : Y' → Y be a morphism of algebraic spaces over S. Let X' = X_Y' be the base change of X. Denote g' : X' → X the projection. Then the map (g')^*Ω_X/Y → Ω_X'/Y' of Lemma [Tag 04CX] is an isomorphism.","statement_latex":"Let $S$ be a scheme.\nLet $X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $g : Y' \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $X' = X_{Y'}$ be the base change of $X$.\nDenote $g' : X' \\to X$ the projection.\nThen the map\n$$\n(g')^*\\Omega_{X/Y} \\to \\Omega_{X'/Y'}\n$$\nof\nLemma \\ref{lemma-functoriality-differentials}\nis an isomorphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZC","source_file":"spaces-more-morphisms.tex","source_line":1203,"source_end_line":1217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1203-L1217","statement_sha256":"0f1ec27de3f6ba7e9fbdf1fa8ae18f7597ea0571c0920d29ea4e11c1e567bc41","origin":"The Stacks Project","memory_eligible":false,"source_rank":12026,"rank":12026,"depth":49,"x":731.533,"y":1525.745,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZD","tag":"05ZD","title":"Sheaf of differentials of a morphism · Lemma 05ZD","summary":"Let S be a scheme. Let f : X → B and g : Y → B be morphisms of algebraic spaces over S with the same target. Let p : X ×_B Y → X and q : X ×_B Y → Y be the projection morphisms. The maps from Lemma [Tag 04CX] p^*Ω_X/B ⊕ q^*Ω_Y/B → Ω_X ×_B Y/B give an isomorphism.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to B$ and $g : Y \\to B$ be morphisms of algebraic spaces\nover $S$ with the same target.\nLet $p : X \\times_B Y \\to X$ and $q : X \\times_B Y \\to Y$ be the\nprojection morphisms. The maps from\nLemma \\ref{lemma-functoriality-differentials}\n$$\np^*\\Omega_{X/B} \\oplus q^*\\Omega_{Y/B}\n\\longrightarrow\n\\Omega_{X \\times_B Y/B}\n$$\ngive an isomorphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZD","source_file":"spaces-more-morphisms.tex","source_line":1226,"source_end_line":1240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1226-L1240","statement_sha256":"fb9e3c9c1738573bffc9f84aa717496c1416c8f3686b92b35876f5bd32785e4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12027,"rank":12027,"depth":49,"x":701.744,"y":1684.112,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZE","tag":"05ZE","title":"Sheaf of differentials of a morphism · Lemma 05ZE","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally of finite type, then Ω_X/Y is a finite type O_X-module.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is locally of finite type, then $\\Omega_{X/Y}$ is\na finite type $\\mathcal{O}_X$-module.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZE","source_file":"spaces-more-morphisms.tex","source_line":1249,"source_end_line":1255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1249-L1255","statement_sha256":"f217488f1afc5ac138ed515bb446e78fc9908fd5d6290bd05dff95c0fbb90fe5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12028,"rank":12028,"depth":49,"x":596.209,"y":1550.246,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZF","tag":"05ZF","title":"Sheaf of differentials of a morphism · Lemma 05ZF","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally of finite presentation, then Ω_X/Y is an O_X-module of finite presentation.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is locally of finite presentation, then $\\Omega_{X/Y}$ is\nan $\\mathcal{O}_X$-module of finite presentation.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZF","source_file":"spaces-more-morphisms.tex","source_line":1264,"source_end_line":1270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1264-L1270","statement_sha256":"1d96e9322be8183798393d5654e266753012c49fb91fd01b1b5b4c601b05da39","origin":"The Stacks Project","memory_eligible":false,"source_rank":12029,"rank":12029,"depth":49,"x":781.938,"y":1589.128,"cluster":"geometry-of-spaces"},{"id":"stacks:0CK5","tag":"0CK5","title":"Sheaf of differentials of a morphism · Lemma 0CK5","summary":"Let S be a scheme. Let f : X → Y be a smooth morphism of algebraic spaces over S. Then the module of differentials Ω_X/Y is finite locally free.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a smooth morphism of algebraic spaces over $S$.\nThen the module of differentials $\\Omega_{X/Y}$\nis finite locally free.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Sheaf of differentials of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CK5","source_file":"spaces-more-morphisms.tex","source_line":1279,"source_end_line":1285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1279-L1285","statement_sha256":"413472eac3ea96dade0b97b63350090e7bd574ed5bb13feff64550a8a0105812","origin":"The Stacks Project","memory_eligible":false,"source_rank":12030,"rank":12030,"depth":49,"x":613.484,"y":1665.95,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZH","tag":"05ZH","title":"Topological invariance of the étale site · Theorem 05ZH","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is integral, universally injective and surjective. The functor V ↦ V_X = X ×_Y V defines an equivalence of categories Y_spaces, etale → X_spaces, etale.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is integral, universally injective and surjective.\nThe functor\n$$\nV \\longmapsto V_X = X \\times_Y V\n$$\ndefines an equivalence of categories\n$Y_{spaces, \\etale} \\to X_{spaces, \\etale}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Topological invariance of the étale site","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZH","source_file":"spaces-more-morphisms.tex","source_line":1315,"source_end_line":1326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1315-L1326","statement_sha256":"ad636255450d67781f73ff9f84f704872c9fba46d4978f09152ca7eb03e432a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12031,"rank":12031,"depth":56,"x":676.008,"y":1513.508,"cluster":"geometry-of-spaces"},{"id":"stacks:07VW","tag":"07VW","title":"Topological invariance of the étale site · Lemma 07VW","summary":"With assumption and notation as in Theorem [Tag 05ZH] the equivalence of categories Y_spaces, etale → X_spaces, etale restricts to equivalences of categories Y_etale → X_etale and Y_affine, etale → X_affine, etale.","statement_latex":"With assumption and notation as in\nTheorem \\ref{theorem-topological-invariance}\nthe equivalence of categories\n$Y_{spaces, \\etale} \\to X_{spaces, \\etale}$\nrestricts to equivalences of categories\n$Y_\\etale \\to X_\\etale$ and $Y_{affine, \\etale} \\to X_{affine, \\etale}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Topological invariance of the étale site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VW","source_file":"spaces-more-morphisms.tex","source_line":1414,"source_end_line":1422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1414-L1422","statement_sha256":"c970cdf79f2aa85138cf8ed179e4f387e5531613fbda7b5dc30478eff474ca19","origin":"The Stacks Project","memory_eligible":false,"source_rank":12032,"rank":12032,"depth":68,"x":752.598,"y":1661.597,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZK","tag":"05ZK","title":"Thickenings · Definition 05ZK","summary":"Thickenings. Let S be a scheme. • We say an algebraic space X' is a thickening of an algebraic space X if X is a closed subspace of X' and the associated topological spaces are equal. • We say X' is a first order thickening of X if X is a closed subspace of X' and the quasi-coherent sheaf of ideals I ⊂ O_X' defining X has square zero. • We say X' is a finite order thickening of X if X is a closed subspace of X and the quasi-coherent sheaf of ideals I ⊂ O_X' defining X is…","statement_latex":"Thickenings. Let $S$ be a scheme.\n\\begin{enumerate}\n\\item We say an algebraic space $X'$ is a {\\it thickening} of an algebraic\nspace $X$ if $X$ is a closed subspace of $X'$ and the associated topological\nspaces are equal.\n\\item We say $X'$ is a {\\it first order thickening} of $X$ if\n$X$ is a closed subspace of $X'$ and the quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_{X'}$ defining $X$ has square zero.\n\\item We say $X'$ is a {\\it finite order thickening} of $X$\nif $X$ is a closed subspace of $X$ and the quasi-coherent sheaf of ideals\n$\\mathcal{I} \\subset \\mathcal{O}_{X'}$ defining $X$ is nilpotent, i.e.,\nthere exists an integer $n \\geq 0$ such that $\\mathcal{I}^{n + 1} = 0$.\n\\item We say $X'$ is an {\\it $n$th order thickening} of $X$\nif $X$ is a closed subspace of $X$ and $\\mathcal{I}^{n + 1} = 0$\nwhere $\\mathcal{I} \\subset \\mathcal{O}_{X'}$ is the\nquasi-coherent sheaf of ideals defining $X$.\n\\item Given two thickenings $X \\subset X'$ and $Y \\subset Y'$ a\n{\\it morphism of thickenings} is a morphism $f' : X' \\to Y'$ such that\n$f(X) \\subset Y$, i.e., such that $f'|_X$ factors through the closed\nsubspace $Y$. In this situation we set $f = f'|_X : X \\to Y$ and we say\nthat $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ is a morphism of\nthickenings.\n\\item Let $B$ be an algebraic space. We similarly define\n{\\it thickenings over $B$}, and\n{\\it morphisms of thickenings over $B$}. This means that the spaces\n$X, X', Y, Y'$ above are algebraic spaces endowed with a structure\nmorphism to $B$, and that the morphisms\n$X \\to X'$, $Y \\to Y'$ and $f' : X' \\to Y'$ are morphisms over $B$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZK","source_file":"spaces-more-morphisms.tex","source_line":1507,"source_end_line":1538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1507-L1538","statement_sha256":"2991881b52652df9f3ec8b35f9bc63f605fc8fc6c00003606a6f28c6c383b36d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12033,"rank":12033,"depth":0,"x":576.792,"y":1595.767,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZN","tag":"05ZN","title":"Thickenings · Lemma 05ZN","summary":"Let S be a scheme. Let B be an algebraic space over S. Let X ⊂ X' and Y ⊂ Y' be thickenings of algebraic spaces over B. Let f : X → Y be a morphism of algebraic spaces over B. Given any map of O_B-algebras α : f_spaces, etale^-1O_Y' → O_X' such that xymatrix f_spaces, etale^-1O_Y ar[r]_-f^sharp ar[r] & O_X f_spaces, etale^-1O_Y' ar[r]^-α ar[u]^i_Y^sharp & O_X' ar[u]_i_X^sharp commutes, there exists a unique morphism of (f, f') of thickenings over B such that α = (f')^sharp.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $X \\subset X'$ and $Y \\subset Y'$ be thickenings\nof algebraic spaces over $B$. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $B$. Given any map of $\\mathcal{O}_B$-algebras\n$$\n\\alpha : f_{spaces, \\etale}^{-1}\\mathcal{O}_{Y'} \\to \\mathcal{O}_{X'}\n$$\nsuch that\n$$\n\\xymatrix{\nf_{spaces, \\etale}^{-1}\\mathcal{O}_Y \\ar[r]_-{f^\\sharp} \\ar[r] &\n\\mathcal{O}_X \\\\\nf_{spaces, \\etale}^{-1}\\mathcal{O}_{Y'} \\ar[r]^-\\alpha\n\\ar[u]^{i_Y^\\sharp} &\n\\mathcal{O}_{X'} \\ar[u]_{i_X^\\sharp}\n}\n$$\ncommutes, there exists a unique morphism of $(f, f')$ of\nthickenings over $B$ such that $\\alpha = (f')^\\sharp$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZN","source_file":"spaces-more-morphisms.tex","source_line":1600,"source_end_line":1621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1600-L1621","statement_sha256":"d269e59aa24f22eaae15af1fd873b165956847c30b57567bb1069b7a9ca44a44","origin":"The Stacks Project","memory_eligible":false,"source_rank":12034,"rank":12034,"depth":55,"x":759.616,"y":1544.483,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZP","tag":"05ZP","title":"Thickenings · Lemma 05ZP","summary":"Let S be a scheme. Let X ⊂ X' be a thickening of algebraic spaces over S. For any open subspace U ⊂ X there exists a unique open subspace U' ⊂ X' such that U = X ×_X' U'.","statement_latex":"Let $S$ be a scheme. Let $X \\subset X'$ be a thickening\nof algebraic spaces over $S$. For any open subspace $U \\subset X$ there\nexists a unique open subspace $U' \\subset X'$ such that\n$U = X \\times_{X'} U'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZP","source_file":"spaces-more-morphisms.tex","source_line":1648,"source_end_line":1654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1648-L1654","statement_sha256":"8824643ca7338b1e50f6ebada04b5e6f3ce2aba02cbf2320b3e4ff8c92134e83","origin":"The Stacks Project","memory_eligible":false,"source_rank":12035,"rank":12035,"depth":56,"x":665.919,"y":1686.231,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZQ","tag":"05ZQ","title":"Thickenings · Lemma 05ZQ","summary":"Let S be a scheme. Let X ⊂ X' be a thickening of algebraic spaces over S. Let U be an affine object of X_spaces, etale. Then Γ(U, O_X') → Γ(U, O_X) is surjective where we think of O_X' as a sheaf on X_spaces, etale via ([Tag 05ZM]).","statement_latex":"Let $S$ be a scheme. Let $X \\subset X'$ be a thickening\nof algebraic spaces over $S$. Let $U$ be an affine object of\n$X_{spaces, \\etale}$. Then\n$$\n\\Gamma(U, \\mathcal{O}_{X'}) \\to \\Gamma(U, \\mathcal{O}_X)\n$$\nis surjective where we think of $\\mathcal{O}_{X'}$ as a sheaf on\n$X_{spaces, \\etale}$ via (\\ref{equation-fundamental-equivalence}).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZQ","source_file":"spaces-more-morphisms.tex","source_line":1684,"source_end_line":1694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1684-L1694","statement_sha256":"4a7d52ae720d02b6a6688b9c75cd0b5438d99257530f69cbfe1afe0a61deed82","origin":"The Stacks Project","memory_eligible":false,"source_rank":12036,"rank":12036,"depth":66,"x":620.959,"y":1528.327,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZR","tag":"05ZR","title":"Thickenings · Lemma 05ZR","summary":"Let S be a scheme. Let X ⊂ X' be a thickening of algebraic spaces over S. If X is (representable by) a scheme, then so is X'.","statement_latex":"Let $S$ be a scheme. Let $X \\subset X'$ be a thickening of algebraic spaces\nover $S$. If $X$ is (representable by) a scheme, then so is $X'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZR","source_file":"spaces-more-morphisms.tex","source_line":1729,"source_end_line":1733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1729-L1733","statement_sha256":"f03983ad50b8c7ae9ada007e94f7e884d10d4382ad4be7364f490abc876369ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":12037,"rank":12037,"depth":68,"x":781.31,"y":1619.375,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZS","tag":"05ZS","title":"Thickenings · Lemma 05ZS","summary":"Let S be a scheme. Let X ⊂ X' be a thickening of algebraic spaces over S. The functor V' ↦ V = X ×_X' V' defines an equivalence of categories X'_etale → X_etale.","statement_latex":"Let $S$ be a scheme. Let $X \\subset X'$ be a thickening of algebraic spaces\nover $S$. The functor\n$$\nV' \\longmapsto V = X \\times_{X'} V'\n$$\ndefines an equivalence of categories\n$X'_\\etale \\to X_\\etale$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZS","source_file":"spaces-more-morphisms.tex","source_line":1816,"source_end_line":1825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1816-L1825","statement_sha256":"fd3384d5e078459ae213ebcd1296494ec5d571b6c3e6af7f1019c7d7ecabf686","origin":"The Stacks Project","memory_eligible":false,"source_rank":12038,"rank":12038,"depth":69,"x":589.593,"y":1643.257,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZT","tag":"05ZT","title":"Thickenings · Lemma 05ZT","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. Consider a short exact sequence 0 → I → A → O_X → 0 of sheaves on X_etale where A is a sheaf of f^-1O_B-algebras, A → O_X is a surjection of sheaves of f^-1O_B-algebras, and I is its kernel. If • I is an ideal of square zero in A, and • I is quasi-coherent as an O_X-module then there exists a first order thickening X ⊂ X' over B and an isomorphism O_X' → A of f^-1O_B-algebras compatible with the…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to B$ be a morphism of algebraic spaces over $S$.\nConsider a short exact sequence\n$$\n0 \\to \\mathcal{I} \\to \\mathcal{A} \\to \\mathcal{O}_X \\to 0\n$$\nof sheaves on $X_\\etale$ where $\\mathcal{A}$ is a sheaf of\n$f^{-1}\\mathcal{O}_B$-algebras, $\\mathcal{A} \\to \\mathcal{O}_X$ is a surjection\nof sheaves of $f^{-1}\\mathcal{O}_B$-algebras, and $\\mathcal{I}$ is its kernel.\nIf\n\\begin{enumerate}\n\\item $\\mathcal{I}$ is an ideal of square zero in $\\mathcal{A}$, and\n\\item $\\mathcal{I}$ is quasi-coherent as an $\\mathcal{O}_X$-module\n\\end{enumerate}\nthen there exists a first order thickening\n$X \\subset X'$ over $B$ and an isomorphism\n$\\mathcal{O}_{X'} \\to \\mathcal{A}$ of $f^{-1}\\mathcal{O}_B$-algebras\ncompatible with the surjections to $\\mathcal{O}_X$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZT","source_file":"spaces-more-morphisms.tex","source_line":1839,"source_end_line":1859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1839-L1859","statement_sha256":"dbe4a4cb54f4025e1dcb6f96f536033f5611c99c603917a82e6356c92260b752","origin":"The Stacks Project","memory_eligible":false,"source_rank":12039,"rank":12039,"depth":69,"x":711.922,"y":1516.692,"cluster":"geometry-of-spaces"},{"id":"stacks:09ZX","tag":"09ZX","title":"Thickenings · Lemma 09ZX","summary":"Let S be a scheme. Let Y ⊂ Y' be a thickening of algebraic spaces over S. Let X' → Y' be a morphism and set X = Y ×_Y' X'. Then (X ⊂ X') → (Y ⊂ Y') is a morphism of thickenings. If Y ⊂ Y' is a first (resp. finite order) thickening, then X ⊂ X' is a first (resp. finite order) thickening.","statement_latex":"Let $S$ be a scheme. Let $Y \\subset Y'$ be a thickening of algebraic spaces\nover $S$. Let $X' \\to Y'$ be a morphism and set $X = Y \\times_{Y'} X'$.\nThen $(X \\subset X') \\to (Y \\subset Y')$\nis a morphism of thickenings. If $Y \\subset Y'$ is a first\n(resp.\\ finite order) thickening, then $X \\subset X'$ is a first\n(resp.\\ finite order) thickening.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZX","source_file":"spaces-more-morphisms.tex","source_line":1941,"source_end_line":1949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1941-L1949","statement_sha256":"b68332ae9925093e1b9c524014f031ef02496354270254b12eb5a8fa2f951aad","origin":"The Stacks Project","memory_eligible":false,"source_rank":12040,"rank":12040,"depth":0,"x":723.514,"y":1679.65,"cluster":"geometry-of-spaces"},{"id":"stacks:0BPH","tag":"0BPH","title":"Thickenings · Lemma 0BPH","summary":"Let S be a scheme. If X ⊂ X' and X' ⊂ X\" are thickenings of algebraic spaces over S, then so is X ⊂ X\".","statement_latex":"Let $S$ be a scheme. If $X \\subset X'$ and $X' \\subset X''$ are\nthickenings of algebraic spaces over $S$, then so is $X \\subset X''$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPH","source_file":"spaces-more-morphisms.tex","source_line":1955,"source_end_line":1959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1955-L1959","statement_sha256":"e49f3ef980c022dda66369dcc29e3a41464bc7d0f802c84860407d6c1d8e1808","origin":"The Stacks Project","memory_eligible":false,"source_rank":12041,"rank":12041,"depth":0,"x":583.732,"y":1565.913,"cluster":"geometry-of-spaces"},{"id":"stacks:0BPI","tag":"0BPI","title":"Thickenings · Lemma 0BPI","summary":"The property of being a thickening is fpqc local. Similarly for first order thickenings.","statement_latex":"The property of being a thickening is fpqc local.\nSimilarly for first order thickenings.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPI","source_file":"spaces-more-morphisms.tex","source_line":1965,"source_end_line":1969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L1965-L1969","statement_sha256":"8d5f04e731cbd3c9274dc55ccbb72a192c785df2f2ad9e1e970110c337b8a4c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12042,"rank":12042,"depth":58,"x":778.53,"y":1570.472,"cluster":"geometry-of-spaces"},{"id":"stacks:09ZY","tag":"09ZY","title":"Morphisms of thickenings · Lemma 09ZY","summary":"Let S be a scheme. Let (f, f') : (X ⊂ X') → (Y ⊂ Y') be a morphism of thickenings of algebraic spaces over S. Then • f is an affine morphism if and only if f' is an affine morphism, • f is a surjective morphism if and only if f' is a surjective morphism, • f is quasi-compact if and only if f' quasi-compact, • f is universally closed if and only if f' is universally closed, • f is integral if and only if f' is integral, • f is (quasi-)separated if and only if f' is…","statement_latex":"Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\subset Y')$\nbe a morphism of thickenings of algebraic spaces over $S$. Then\n\\begin{enumerate}\n\\item $f$ is an affine morphism if and only if $f'$ is an affine morphism,\n\\item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism,\n\\item $f$ is quasi-compact if and only if $f'$ quasi-compact,\n\\item $f$ is universally closed if and only if $f'$ is universally closed,\n\\item $f$ is integral if and only if $f'$ is integral,\n\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,\n\\item $f$ is universally injective if and only if $f'$ is universally injective,\n\\item $f$ is universally open if and only if $f'$ is universally open,\n\\item $f$ is representable if and only if $f'$ is representable, and\n\\item add more here.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Morphisms of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZY","source_file":"spaces-more-morphisms.tex","source_line":2000,"source_end_line":2016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2000-L2016","statement_sha256":"f648754ae28170ff35e6f905b447fd10913d8dfd684be9b896cf247f90e38377","origin":"The Stacks Project","memory_eligible":false,"source_rank":12043,"rank":12043,"depth":69,"x":631.026,"y":1677.785,"cluster":"geometry-of-spaces"},{"id":"stacks:09ZZ","tag":"09ZZ","title":"Morphisms of thickenings · Lemma 09ZZ","summary":"Let S be a scheme. Let (f, f') : (X ⊂ X') → (Y ⊂ Y') be a morphism of thickenings of algebraic spaces over S such that X = Y ×_Y' X'. If X ⊂ X' is a finite order thickening, then • f is a closed immersion if and only if f' is a closed immersion, • f is locally of finite type if and only if f' is locally of finite type, • f is locally quasi-finite if and only if f' is locally quasi-finite, • f is locally of finite type of relative dimension d if and only if f' is locally…","statement_latex":"Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ be a\nmorphism of thickenings of algebraic spaces over $S$ such that\n$X = Y \\times_{Y'} X'$. If $X \\subset X'$ is a finite order thickening, then\n\\begin{enumerate}\n\\item $f$ is a closed immersion if and only if $f'$ is a closed immersion,\n\\item $f$ is locally of finite type if and only if $f'$ is\nlocally of finite type,\n\\item $f$ is locally quasi-finite if and only if $f'$ is locally\nquasi-finite,\n\\item $f$ is locally of finite type of relative dimension $d$ if and\nonly if $f'$ is locally of finite type of relative dimension $d$,\n\\item $\\Omega_{X/Y} = 0$ if and only if $\\Omega_{X'/Y'} = 0$,\n\\item $f$ is unramified if and only if $f'$ is unramified,\n\\item $f$ is proper if and only if $f'$ is proper,\n\\item $f$ is a finite morphism if and only if $f'$ is an finite morphism,\n\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,\n\\item $f$ is an immersion if and only if $f'$ is an immersion, and\n\\item add more here.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Morphisms of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZZ","source_file":"spaces-more-morphisms.tex","source_line":2043,"source_end_line":2064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2043-L2064","statement_sha256":"9e73ba19e73e992b56783c7d2f6709d76a61ba1692c972a4dc11b031a0b94f53","origin":"The Stacks Project","memory_eligible":false,"source_rank":12044,"rank":12044,"depth":70,"x":653.524,"y":1514.741,"cluster":"geometry-of-spaces"},{"id":"stacks:0BPJ","tag":"0BPJ","title":"Morphisms of thickenings · Lemma 0BPJ","summary":"Let S be a scheme. Let (f, f') : (X ⊂ X') → (Y → Y') be a morphism of thickenings of algebraic spaces over S. Assume f and f' are locally of finite type and X = Y ×_Y' X'. Then • f is locally quasi-finite if and only if f' is locally quasi-finite, • f is finite if and only if f' is finite, • f is a closed immersion if and only if f' is a closed immersion, • Ω_X/Y = 0 if and only if Ω_X'/Y' = 0, • f is unramified if and only if f' is unramified, • f is a monomorphism if…","statement_latex":"Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\to Y')$ be a\nmorphism of thickenings of algebraic spaces over $S$. Assume $f$ and $f'$\nare locally of finite type and $X = Y \\times_{Y'} X'$. Then\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,\n\\item $f$ is finite if and only if $f'$ is finite,\n\\item $f$ is a closed immersion if and only if $f'$ is a closed immersion,\n\\item $\\Omega_{X/Y} = 0$ if and only if $\\Omega_{X'/Y'} = 0$,\n\\item $f$ is unramified if and only if $f'$ is unramified,\n\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,\n\\item $f$ is an immersion if and only if $f'$ is an immersion,\n\\item $f$ is proper if and only if $f'$ is proper, and\n\\item add more here.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Morphisms of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPJ","source_file":"spaces-more-morphisms.tex","source_line":2155,"source_end_line":2171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2155-L2171","statement_sha256":"7c0ff71fa8e4d16d03a50fd8d589c353166e8bf731eceb74fd98c916ebb1cc1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12045,"rank":12045,"depth":70,"x":768.203,"y":1647.909,"cluster":"geometry-of-spaces"},{"id":"stacks:0DNM","tag":"0DNM","title":"Picard groups of thickenings · Lemma 0DNM","summary":"Let S be a scheme. Let X ⊂ X' be a first order thickening of algebraic spaces over S with ideal sheaf I. Then there is a canonical exact sequence xymatrix 0 ar[r] & H^0(X, I) ar[r] & H^0(X', O_X'^*) ar[r] & H^0(X, O^*_X) ar `r[d] `d[l] `l[llld] `d[dll] [dll] & H^1(X, I) ar[r] & Pic(X') ar[r] & Pic(X) ar `r[d] `d[l] `l[llld] `d[dll] [dll] & H^2(X, I) ar[r] & … ar[r] & … of abelian groups.","statement_latex":"Let $S$ be a scheme. Let $X \\subset X'$ be a first order thickening\nof algebraic spaces over $S$ with ideal sheaf $\\mathcal{I}$.\nThen there is a canonical exact sequence\n$$\n\\xymatrix{\n0 \\ar[r] &\nH^0(X, \\mathcal{I}) \\ar[r] &\nH^0(X', \\mathcal{O}_{X'}^*) \\ar[r] &\nH^0(X, \\mathcal{O}^*_X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\\n& H^1(X, \\mathcal{I}) \\ar[r] &\n\\Pic(X') \\ar[r] &\n\\Pic(X) \\ar `r[d] `d[l] `l[llld] `d[dll] [dll] \\\\\n& H^2(X, \\mathcal{I}) \\ar[r] & \\ldots \\ar[r] & \\ldots\n}\n$$\nof abelian groups.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Picard groups of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNM","source_file":"spaces-more-morphisms.tex","source_line":2264,"source_end_line":2282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2264-L2282","statement_sha256":"fe0d7bd17bdfb2e9f5f52a708a804e6854bf148b0cb0f636f8bf8fd1fe5daf46","origin":"The Stacks Project","memory_eligible":false,"source_rank":12046,"rank":12046,"depth":70,"x":576.297,"y":1614.74,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZV","tag":"05ZV","title":"Infinitesimal neighbourhoods · Definition 05ZV","summary":"Let i : Z → X be an immersion of algebraic spaces. • The first order infinitesimal neighbourhood of Z in X is the first order thickening Z ⊂ Z_1 over X described above. • The nth order infinitesimal neighbourhood of Z in X is the nth order thickening Z ⊂ Z_n over X described above.","statement_latex":"Let $i : Z \\to X$ be an immersion of algebraic spaces.\n\\begin{enumerate}\n\\item The {\\it first order infinitesimal neighbourhood} of $Z$ in $X$ is\nthe first order thickening $Z \\subset Z_1$ over $X$ described above.\n\\item The {\\it $n$th order infinitesimal neighbourhood} of $Z$ in $X$ is\nthe $n$th order thickening $Z \\subset Z_n$ over $X$ described above.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal neighbourhoods","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZV","source_file":"spaces-more-morphisms.tex","source_line":2324,"source_end_line":2333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2324-L2333","statement_sha256":"db8fde3793e3e8500c49b365436e8b62bcd56f0f7438cfafa8cccc4762c385b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12047,"rank":12047,"depth":0,"x":744.701,"y":1530.195,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZW","tag":"05ZW","title":"Infinitesimal neighbourhoods · Lemma 05ZW","summary":"Let i : Z → X be an immersion of algebraic spaces. • The first order infinitesimal neighbourhood Z' of Z in X has the following universal property: Given any commutative diagram xymatrix Z ar[d]_i & T ar[l]^a ar[d] X & T' ar[l]_b where T ⊂ T' is a first order thickening over X, there exists a unique morphism (a', a) : (T ⊂ T') → (Z ⊂ Z') of thickenings over X. • For n ≥ 1 the nth order infinitesimal neighbourhood Z_n of Z in X has the following universal property: Given…","statement_latex":"Let $i : Z \\to X$ be an immersion of algebraic spaces.\n\\begin{enumerate}\n\\item The first order infinitesimal neighbourhood $Z'$ of $Z$ in $X$\nhas the following universal property:\nGiven any commutative diagram\n$$\n\\xymatrix{\nZ \\ar[d]_i & T \\ar[l]^a \\ar[d] \\\\\nX & T' \\ar[l]_b\n}\n$$\nwhere $T \\subset T'$ is a first order thickening over $X$, there exists\na unique morphism $(a', a) : (T \\subset T') \\to (Z \\subset Z')$ of\nthickenings over $X$.\n\\item For $n \\geq 1$ the $n$th order infinitesimal neighbourhood\n$Z_n$ of $Z$ in $X$ has the following universal property:\nGiven any commutative diagram\n$$\n\\xymatrix{\nZ \\ar[d]_i & T \\ar[l]^a \\ar[d] \\\\\nX & T' \\ar[l]_b\n}\n$$\nwhere $T \\subset T'$ is an $n$th order thickening over $X$, there exists\na unique morphism $(a', a) : (T \\subset T') \\to (Z \\subset Z_n)$ of\nthickenings over $X$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal neighbourhoods","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZW","source_file":"spaces-more-morphisms.tex","source_line":2340,"source_end_line":2369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2340-L2369","statement_sha256":"14bc73bf230c36a0ddc2f7e3a9b32692e4405c613ae050dd4f910df71fdc4073","origin":"The Stacks Project","memory_eligible":false,"source_rank":12048,"rank":12048,"depth":52,"x":688.437,"y":1688.301,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZX","tag":"05ZX","title":"Infinitesimal neighbourhoods · Lemma 05ZX","summary":"Let i : Z → X be an immersion of algebraic spaces. Let Z ⊂ Z' be the first order infinitesimal neighbourhood of Z in X. Then the diagram xymatrix Z ar[r] ar[d] & Z' ar[d] Z ar[r] & X induces a map of conormal sheaves C_Z/X → C_Z/Z' by Lemma [Tag 04CP]. This map is an isomorphism.","statement_latex":"Let $i : Z \\to X$ be an immersion of algebraic spaces.\nLet $Z \\subset Z'$ be the first order infinitesimal neighbourhood\nof $Z$ in $X$. Then the diagram\n$$\n\\xymatrix{\nZ \\ar[r] \\ar[d] & Z' \\ar[d] \\\\\nZ \\ar[r] & X\n}\n$$\ninduces a map of conormal sheaves\n$\\mathcal{C}_{Z/X} \\to \\mathcal{C}_{Z/Z'}$ by\nLemma \\ref{lemma-conormal-functorial}.\nThis map is an isomorphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal neighbourhoods","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZX","source_file":"spaces-more-morphisms.tex","source_line":2394,"source_end_line":2409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2394-L2409","statement_sha256":"2366e2af98724c91045b3d36886a170ab80722b95a37b54bf2a0500645c6aa69","origin":"The Stacks Project","memory_eligible":false,"source_rank":12049,"rank":12049,"depth":53,"x":602.668,"y":1539.596,"cluster":"geometry-of-spaces"},{"id":"stacks:049S","tag":"049S","title":"Formally smooth, étale, unramified transformations · Definition 049S","summary":"Let S be a scheme. Let a : F → G be a transformation of functors F, G : (Sch/S)_fppf^opp → Sets. Consider commutative solid diagrams of the form xymatrix F ar[d]_a & T ar[d]^i ar[l] G & T' ar[l] ar@-->[lu] where T and T' are affine schemes and i is a closed immersion defined by an ideal of square zero. • We say a is formally smooth if given any solid diagram as above there exists a dotted arrow making the diagram commute. • We say a is formally étale if given any solid…","statement_latex":"Let $S$ be a scheme.\nLet $a : F \\to G$ be a transformation of functors\n$F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nConsider commutative solid diagrams of the form\n$$\n\\xymatrix{\nF \\ar[d]_a & T \\ar[d]^i \\ar[l] \\\\\nG & T' \\ar[l] \\ar@{-->}[lu]\n}\n$$\nwhere $T$ and $T'$ are affine schemes and $i$ is a closed immersion\ndefined by an ideal of square zero.\n\\begin{enumerate}\n\\item We say $a$ is {\\it formally smooth} if given any solid\ndiagram as above there exists a dotted arrow making the diagram\ncommute\\footnote{This is just one possible definition that one can\nmake here. Another slightly weaker condition would be to require that\nthe dotted arrow exists fppf locally on $T'$. This weaker notion\nhas in some sense better formal properties.}.\n\\item We say $a$ is {\\it formally \\'etale} if given any solid\ndiagram as above there exists exactly one dotted arrow making the diagram\ncommute.\n\\item We say $a$ is {\\it formally unramified} if given any solid\ndiagram as above there exists at most one dotted arrow making the diagram\ncommute.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth, étale, unramified transformations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049S","source_file":"spaces-more-morphisms.tex","source_line":2442,"source_end_line":2470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2442-L2470","statement_sha256":"54346406c6d1d9c7a7011e9dddd006b76850f57150afb66cd734add190d2ff75","origin":"The Stacks Project","memory_eligible":false,"source_rank":12050,"rank":12050,"depth":0,"x":785.736,"y":1600.662,"cluster":"geometry-of-spaces"},{"id":"stacks:04G4","tag":"04G4","title":"Formally smooth, étale, unramified transformations · Lemma 04G4","summary":"Let S be a scheme. Let a : F → G be a transformation of functors F, G : (Sch/S)_fppf^opp → Sets. Then a is formally étale if and only if a is both formally smooth and formally unramified.","statement_latex":"Let $S$ be a scheme.\nLet $a : F \\to G$ be a transformation of functors\n$F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nThen $a$ is formally \\'etale if and only if $a$ is both formally\nsmooth and formally unramified.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth, étale, unramified transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04G4","source_file":"spaces-more-morphisms.tex","source_line":2472,"source_end_line":2479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2472-L2479","statement_sha256":"73213c125982099fd82df387b0731df16d40ef6cc6159bd526957ea1013996bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12051,"rank":12051,"depth":0,"x":601.397,"y":1659.587,"cluster":"geometry-of-spaces"},{"id":"stacks:049T","tag":"049T","title":"Formally smooth, étale, unramified transformations · Lemma 049T","summary":"Composition. • A composition of formally smooth transformations of functors is formally smooth. • A composition of formally étale transformations of functors is formally étale. • A composition of formally unramified transformations of functors is formally unramified.","statement_latex":"Composition.\n\\begin{enumerate}\n\\item A composition of formally smooth transformations of functors is formally\nsmooth.\n\\item A composition of formally \\'etale transformations of functors is formally\n\\'etale.\n\\item A composition of formally unramified transformations of functors is\nformally unramified.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth, étale, unramified transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049T","source_file":"spaces-more-morphisms.tex","source_line":2485,"source_end_line":2496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2485-L2496","statement_sha256":"6b4bf2afbd95cccf0ac71e782a5aef0d0f1884ffc18dce74d9270c68724bbe01","origin":"The Stacks Project","memory_eligible":false,"source_rank":12052,"rank":12052,"depth":0,"x":690.056,"y":1511.346,"cluster":"geometry-of-spaces"},{"id":"stacks:049U","tag":"049U","title":"Formally smooth, étale, unramified transformations · Lemma 049U","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let a : F → G, b : H → G be transformations of functors. Consider the fibre product diagram xymatrix H ×_b, G, a F ar[r]_-b' ar[d]_a' & F ar[d]^a H ar[r]^b & G • If a is formally smooth, then the base change a' is formally smooth. • If a is formally étale, then the base change a' is formally étale. • If a is formally unramified, then the base change a' is formally unramified.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$, $b : H \\to G$ be transformations of functors.\nConsider the fibre product diagram\n$$\n\\xymatrix{\nH \\times_{b, G, a} F \\ar[r]_-{b'} \\ar[d]_{a'} & F \\ar[d]^a \\\\\nH \\ar[r]^b & G\n}\n$$\n\\begin{enumerate}\n\\item If $a$ is formally smooth, then the base change $a'$ is\nformally smooth.\n\\item If $a$ is formally \\'etale, then the base change $a'$ is\nformally \\'etale.\n\\item If $a$ is formally unramified, then the base change $a'$ is\nformally unramified.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth, étale, unramified transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049U","source_file":"spaces-more-morphisms.tex","source_line":2502,"source_end_line":2522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2502-L2522","statement_sha256":"2b6197f700360c81e4adb7fa26f0d84b50e2ab5e4203ec0ad69dff3223b30b27","origin":"The Stacks Project","memory_eligible":false,"source_rank":12053,"rank":12053,"depth":0,"x":743.961,"y":1671.17,"cluster":"geometry-of-spaces"},{"id":"stacks:04AL","tag":"04AL","title":"Formally smooth, étale, unramified transformations · Lemma 04AL","summary":"Let S be a scheme. Let F, G : (Sch/S)_fppf^opp → Sets. Let a : F → G be a representable transformation of functors. • If a is smooth then a is formally smooth. • If a is étale, then a is formally étale. • If a is unramified, then a is formally unramified.","statement_latex":"Let $S$ be a scheme.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$ be a representable transformation of functors.\n\\begin{enumerate}\n\\item If $a$ is smooth then $a$ is formally smooth.\n\\item If $a$ is \\'etale, then $a$ is formally \\'etale.\n\\item If $a$ is unramified, then $a$ is formally unramified.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth, étale, unramified transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AL","source_file":"spaces-more-morphisms.tex","source_line":2528,"source_end_line":2538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2528-L2538","statement_sha256":"ccbc4ba9eae2ce47c72a6dd26cb152e9b6fe5ee5bab9b982092911c0da91cf40","origin":"The Stacks Project","memory_eligible":false,"source_rank":12054,"rank":12054,"depth":40,"x":575.468,"y":1583.792,"cluster":"geometry-of-spaces"},{"id":"stacks:04CY","tag":"04CY","title":"Formally smooth, étale, unramified transformations · Lemma 04CY","summary":"Let S be a scheme contained in Sch_fppf. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let a : F → G, b : G → H be transformations of functors. Assume that a is representable, surjective, and étale. • If b is formally smooth, then b ∘ a is formally smooth. • If b is formally étale, then b ∘ a is formally étale. • If b is formally unramified, then b ∘ a is formally unramified. Conversely, consider a solid commutative diagram xymatrix G ar[d]_b & T ar[d]^i ar[l] H & T' ar[l]…","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$, $b : G \\to H$ be transformations of functors.\nAssume that $a$ is representable, surjective, and \\'etale.\n\\begin{enumerate}\n\\item If $b$ is formally smooth, then $b \\circ a$ is formally smooth.\n\\item If $b$ is formally \\'etale, then $b \\circ a$ is formally \\'etale.\n\\item If $b$ is formally unramified, then $b \\circ a$ is formally unramified.\n\\end{enumerate}\nConversely, consider a solid commutative diagram\n$$\n\\xymatrix{\nG \\ar[d]_b & T \\ar[d]^i \\ar[l] \\\\\nH & T' \\ar[l] \\ar@{-->}[lu]\n}\n$$\nwith $T'$ an affine scheme over $S$\nand $i : T \\to T'$ a closed immersion defined by an ideal of square zero.\n\\begin{enumerate}\n\\item[(4)] If $b \\circ a$ is formally smooth, then for every $t \\in T$\nthere exists an \\'etale morphism of affines $U' \\to T'$ and a morphism\n$U' \\to G$ such that\n$$\n\\xymatrix{\nG \\ar[d]_b & T \\ar[l] & T \\times_{T'} U' \\ar[d] \\ar[l]\\\\\nH & T' \\ar[l] & U' \\ar[llu] \\ar[l]\n}\n$$\ncommutes and $t$ is in the image of $U' \\to T'$.\n\\item[(5)] If $b \\circ a$ is formally unramified, then there exists at most\none dotted arrow in the diagram above, i.e., $b$ is formally unramified.\n\\item[(6)] If $b \\circ a$ is formally \\'etale, then there exists exactly one\ndotted arrow in the diagram above, i.e., $b$ is formally \\'etale.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth, étale, unramified transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04CY","source_file":"spaces-more-morphisms.tex","source_line":2566,"source_end_line":2602,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2566-L2602","statement_sha256":"85fe8ab105973a1bec37b597b5f506d10e6a104b0d08c02fb07989d203451de7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12055,"rank":12055,"depth":41,"x":770.23,"y":1552.576,"cluster":"geometry-of-spaces"},{"id":"stacks:04G5","tag":"04G5","title":"Formally smooth, étale, unramified transformations · Lemma 04G5","summary":"Let S be a scheme. Let F, G, H : (Sch/S)_fppf^opp → Sets. Let a : F → G, b : G → H be transformations of functors. Assume b is formally unramified. • If b ∘ a is formally unramified then a is formally unramified. • If b ∘ a is formally étale then a is formally étale. • If b ∘ a is formally smooth then a is formally smooth.","statement_latex":"Let $S$ be a scheme.\nLet $F, G, H : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$, $b : G \\to H$ be transformations of functors.\nAssume $b$ is formally unramified.\n\\begin{enumerate}\n\\item If $b \\circ a$ is formally unramified then $a$ is formally unramified.\n\\item If $b \\circ a$ is formally \\'etale then $a$ is formally \\'etale.\n\\item If $b \\circ a$ is formally smooth then $a$ is formally smooth.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth, étale, unramified transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04G5","source_file":"spaces-more-morphisms.tex","source_line":2665,"source_end_line":2676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2665-L2676","statement_sha256":"3522aae19a4353a8954805d0ae703ba8562756ed8bac684b30527514c279105c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12056,"rank":12056,"depth":0,"x":651.566,"y":1686.279,"cluster":"geometry-of-spaces"},{"id":"stacks:04G7","tag":"04G7","title":"Formally unramified morphisms · Definition 04G7","summary":"Let S be a scheme. A morphism f : X → Y of algebraic spaces over S is said to be formally unramified if it is formally unramified as a transformation of functors as in Definition [Tag 049S].","statement_latex":"Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$\nis said to be {\\it formally unramified} if it is formally unramified as a\ntransformation of functors as in\nDefinition \\ref{definition-formally-smooth-etale-unramified}.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally unramified morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04G7","source_file":"spaces-more-morphisms.tex","source_line":2707,"source_end_line":2713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2707-L2713","statement_sha256":"ae7e20a1bd4a27e0d83cbeff8fd116a517b67016cadc5ac7690098a3ebf64f6d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12057,"rank":12057,"depth":1,"x":631.521,"y":1520.143,"cluster":"geometry-of-spaces"},{"id":"stacks:04G8","tag":"04G8","title":"Formally unramified morphisms · Lemma 04G8","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is formally unramified, • for every diagram xymatrix U ar[d] ar[r]_ψ & V ar[d] X ar[r]^f & Y where U and V are schemes and the vertical arrows are étale the morphism of schemes ψ is formally unramified (as in More on Morphisms, Definition [Tag 02H8]), and • for one such diagram with surjective vertical arrows the morphism ψ is formally unramified.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over\n$S$. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is formally unramified,\n\\item for every diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\\nX \\ar[r]^f & Y\n}\n$$\nwhere $U$ and $V$ are schemes and the vertical arrows are \\'etale\nthe morphism of schemes $\\psi$ is formally unramified (as in\nMore on Morphisms,\nDefinition \\ref{more-morphisms-definition-formally-unramified}), and\n\\item for one such diagram with surjective vertical arrows the morphism\n$\\psi$ is formally unramified.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04G8","source_file":"spaces-more-morphisms.tex","source_line":2723,"source_end_line":2743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2723-L2743","statement_sha256":"352e08092dce5eb88c607cb88f95daa16a2ecca5164360bc71a1bdca440bdb56","origin":"The Stacks Project","memory_eligible":false,"source_rank":12058,"rank":12058,"depth":42,"x":780.094,"y":1631.418,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZY","tag":"05ZY","title":"Formally unramified morphisms · Lemma 05ZY","summary":"Let S be a scheme. If f : X → Y is a formally unramified morphism of algebraic spaces over S, then given any solid commutative diagram xymatrix X ar[d]_f & T ar[d]^i ar[l] Y & T' ar[l] ar@-->[lu] where T ⊂ T' is a first order thickening of algebraic spaces over S there exists at most one dotted arrow making the diagram commute. In other words, in Definition [Tag 04G7] the condition that T be an affine scheme may be dropped.","statement_latex":"Let $S$ be a scheme.\nIf $f : X \\to Y$ is a formally unramified morphism of algebraic spaces\nover $S$, then given any solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & T \\ar[d]^i \\ar[l] \\\\\nY & T' \\ar[l] \\ar@{-->}[lu]\n}\n$$\nwhere $T \\subset T'$ is a first order thickening of algebraic spaces\nover $S$ there exists at most one dotted arrow making the diagram commute.\nIn other words, in\nDefinition \\ref{definition-formally-unramified}\nthe condition that $T$ be an affine scheme may be dropped.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZY","source_file":"spaces-more-morphisms.tex","source_line":2765,"source_end_line":2781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2765-L2781","statement_sha256":"3b1f002a8f1e9fc685190dc0f2d86e8f8a1b2e4e7691b04b214a9daaac2cd547","origin":"The Stacks Project","memory_eligible":false,"source_rank":12059,"rank":12059,"depth":41,"x":580.801,"y":1633.672,"cluster":"geometry-of-spaces"},{"id":"stacks:05ZZ","tag":"05ZZ","title":"Formally unramified morphisms · Lemma 05ZZ","summary":"A composition of formally unramified morphisms is formally unramified.","statement_latex":"A composition of formally unramified morphisms is formally unramified.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05ZZ","source_file":"spaces-more-morphisms.tex","source_line":2791,"source_end_line":2794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2791-L2794","statement_sha256":"614aceced4d470e43169fec0aca479e1c7d6f8a43e123bf816fa05e7b0098151","origin":"The Stacks Project","memory_eligible":false,"source_rank":12060,"rank":12060,"depth":0,"x":726.129,"y":1518.78,"cluster":"geometry-of-spaces"},{"id":"stacks:0600","tag":"0600","title":"Formally unramified morphisms · Lemma 0600","summary":"A base change of a formally unramified morphism is formally unramified.","statement_latex":"A base change of a formally unramified morphism is formally unramified.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0600","source_file":"spaces-more-morphisms.tex","source_line":2800,"source_end_line":2803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2800-L2803","statement_sha256":"3844a93bbe7fa5f67f213b19add0972e29d48b7a9c27c9d7b8740e139ce35282","origin":"The Stacks Project","memory_eligible":false,"source_rank":12061,"rank":12061,"depth":0,"x":711.34,"y":1686.174,"cluster":"geometry-of-spaces"},{"id":"stacks:04G9","tag":"04G9","title":"Formally unramified morphisms · Lemma 04G9","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is formally unramified, and • Ω_X/Y = 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over\n$S$. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is formally unramified, and\n\\item $\\Omega_{X/Y} = 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04G9","source_file":"spaces-more-morphisms.tex","source_line":2809,"source_end_line":2817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2809-L2817","statement_sha256":"351548e388ffdfb1ba7042006730043276d4fd1f893e061636a3eac6502d6edc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12062,"rank":12062,"depth":49,"x":587.474,"y":1554.182,"cluster":"geometry-of-spaces"},{"id":"stacks:04GA","tag":"04GA","title":"Formally unramified morphisms · Lemma 04GA","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • The morphism f is unramified, • the morphism f is locally of finite type and Ω_X/Y = 0, and • the morphism f is locally of finite type and formally unramified.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is unramified,\n\\item the morphism $f$ is locally of finite type and $\\Omega_{X/Y} = 0$, and\n\\item the morphism $f$ is locally of finite type and formally unramified.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GA","source_file":"spaces-more-morphisms.tex","source_line":2828,"source_end_line":2838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2828-L2838","statement_sha256":"7e324a62205ea6c4e73558b3f3b4b00c3066ce65832028af913e4e7ff8699aef","origin":"The Stacks Project","memory_eligible":false,"source_rank":12063,"rank":12063,"depth":50,"x":785.205,"y":1581.261,"cluster":"geometry-of-spaces"},{"id":"stacks:05W6","tag":"05W6","title":"Formally unramified morphisms · Lemma 05W6","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is unramified and a monomorphism, • f is unramified and universally injective, • f is locally of finite type and a monomorphism, • f is universally injective, locally of finite type, and formally unramified. Moreover, in this case f is also representable, separated, and locally quasi-finite.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is unramified and a monomorphism,\n\\item $f$ is unramified and universally injective,\n\\item $f$ is locally of finite type and a monomorphism,\n\\item $f$ is universally injective, locally of finite type, and\nformally unramified.\n\\end{enumerate}\nMoreover, in this case $f$ is also representable, separated, and\nlocally quasi-finite.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05W6","source_file":"spaces-more-morphisms.tex","source_line":2866,"source_end_line":2880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2866-L2880","statement_sha256":"90f0992657931255317c9153b052231c01022e65f4a668caa2674fab9af4a912","origin":"The Stacks Project","memory_eligible":false,"source_rank":12064,"rank":12064,"depth":56,"x":617.414,"y":1673.606,"cluster":"geometry-of-spaces"},{"id":"stacks:05W8","tag":"05W8","title":"Formally unramified morphisms · Lemma 05W8","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is a closed immersion, • f is universally closed, unramified, and a monomorphism, • f is universally closed, unramified, and universally injective, • f is universally closed, locally of finite type, and a monomorphism, • f is universally closed, universally injective, locally of finite type, and formally unramified.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is a closed immersion,\n\\item $f$ is universally closed, unramified, and a monomorphism,\n\\item $f$ is universally closed, unramified, and universally injective,\n\\item $f$ is universally closed, locally of finite type, and a monomorphism,\n\\item $f$ is universally closed, universally injective, locally of\nfinite type, and formally unramified.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05W8","source_file":"spaces-more-morphisms.tex","source_line":2905,"source_end_line":2918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2905-L2918","statement_sha256":"d0467e442ba74978bee68ed197faf822eaffd73622255a1137a391d11fc76a11","origin":"The Stacks Project","memory_eligible":false,"source_rank":12065,"rank":12065,"depth":57,"x":666.941,"y":1510.098,"cluster":"geometry-of-spaces"},{"id":"stacks:0603","tag":"0603","title":"Universal first order thickenings · Lemma 0603","summary":"Let S be a scheme. Let h : Z → X be a morphism of algebraic spaces over S. Let Z ⊂ Z' be a first order thickening over X. The following are equivalent • Z ⊂ Z' is a universal first order thickening, • for any diagram ([Tag 0602]) with T' a scheme a unique dotted arrow exists making the diagram commute, and • for any diagram ([Tag 0602]) with T' an affine scheme a unique dotted arrow exists making the diagram commute.","statement_latex":"Let $S$ be a scheme.\nLet $h : Z \\to X$ be a morphism of algebraic spaces over $S$.\nLet $Z \\subset Z'$ be a first order thickening over $X$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $Z \\subset Z'$ is a universal first order thickening,\n\\item for any diagram (\\ref{equation-universal-first-order-thickening})\nwith $T'$ a scheme a unique dotted arrow exists making the diagram commute, and\n\\item for any diagram (\\ref{equation-universal-first-order-thickening})\nwith $T'$ an affine scheme a unique dotted arrow exists making the\ndiagram commute.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0603","source_file":"spaces-more-morphisms.tex","source_line":2963,"source_end_line":2977,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L2963-L2977","statement_sha256":"e1a950ef99dcea99d3bf09039ff5722a40085e8284ab5f36b9442825b32e9b75","origin":"The Stacks Project","memory_eligible":false,"source_rank":12066,"rank":12066,"depth":1,"x":762.028,"y":1658.956,"cluster":"geometry-of-spaces"},{"id":"stacks:0604","tag":"0604","title":"Universal first order thickenings · Lemma 0604","summary":"Let S be a scheme. Let Z → Y → X be morphisms of algebraic spaces over S. If Z ⊂ Z' is a universal first order thickening of Z over Y and Y → X is formally étale, then Z ⊂ Z' is a universal first order thickening of Z over X.","statement_latex":"Let $S$ be a scheme.\nLet $Z \\to Y \\to X$ be morphisms of algebraic spaces over $S$.\nIf $Z \\subset Z'$ is a universal first order thickening of\n$Z$ over $Y$ and $Y \\to X$ is formally \\'etale, then $Z \\subset Z'$ is\na universal first order thickening of $Z$ over $X$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0604","source_file":"spaces-more-morphisms.tex","source_line":3003,"source_end_line":3010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3003-L3010","statement_sha256":"f22097fe4b53015ed5462fc701668e59e229d65de076e8452e117cbf1868bfe3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12067,"rank":12067,"depth":2,"x":571.968,"y":1603.075,"cluster":"geometry-of-spaces"},{"id":"stacks:0605","tag":"0605","title":"Universal first order thickenings · Lemma 0605","summary":"Let S be a scheme. Let Z → Y → X be morphisms of algebraic spaces over S. Assume Z → Y is étale. • If Y ⊂ Y' is a universal first order thickening of Y over X, then the unique étale morphism Z' → Y' such that Z = Y ×_Y' Z' (see Theorem [Tag 05ZH]) is a universal first order thickening of Z over X. • If Z → Y is surjective and (Z ⊂ Z') → (Y ⊂ Y') is an étale morphism of first order thickenings over X and Z' is a universal first order thickening of Z over X, then Y' is a…","statement_latex":"Let $S$ be a scheme.\nLet $Z \\to Y \\to X$ be morphisms of algebraic spaces over $S$.\nAssume $Z \\to Y$ is \\'etale.\n\\begin{enumerate}\n\\item If $Y \\subset Y'$ is a universal first order thickening of\n$Y$ over $X$, then the unique \\'etale morphism $Z' \\to Y'$ such\nthat $Z = Y \\times_{Y'} Z'$ (see\nTheorem \\ref{theorem-topological-invariance})\nis a universal first order thickening of $Z$ over $X$.\n\\item If $Z \\to Y$ is surjective and\n$(Z \\subset Z') \\to (Y \\subset Y')$ is an \\'etale morphism\nof first order thickenings over $X$ and $Z'$ is a universal first\norder thickening of $Z$ over $X$, then $Y'$ is a universal first\norder thickening of $Y$ over $X$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0605","source_file":"spaces-more-morphisms.tex","source_line":3024,"source_end_line":3041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3024-L3041","statement_sha256":"a410f9185824eb7e55d42c5faa2f2e6cda044c836cc5f3629e6c5bfae1ed71b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12068,"rank":12068,"depth":57,"x":757.284,"y":1536.353,"cluster":"geometry-of-spaces"},{"id":"stacks:0606","tag":"0606","title":"Universal first order thickenings · Lemma 0606","summary":"Let S be a scheme. Let h : Z → X be a formally unramified morphism of algebraic spaces over S. There exists a universal first order thickening Z ⊂ Z' of Z over X.","statement_latex":"Let $S$ be a scheme.\nLet $h : Z \\to X$ be a formally unramified morphism of algebraic\nspaces over $S$.\nThere exists a universal first order thickening $Z \\subset Z'$ of\n$Z$ over $X$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0606","source_file":"spaces-more-morphisms.tex","source_line":3071,"source_end_line":3078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3071-L3078","statement_sha256":"9de46e2031b6f4f844b78437789ac99149ed8e8e53a8d0676eb2ede4f30c5ad4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12069,"rank":12069,"depth":58,"x":674.188,"y":1690.898,"cluster":"geometry-of-spaces"},{"id":"stacks:0607","tag":"0607","title":"Universal first order thickenings · Definition 0607","summary":"Let S be a scheme. Let h : Z → X be a formally unramified morphism of algebraic spaces over S. • The universal first order thickening of Z over X is the thickening Z ⊂ Z' constructed in Lemma [Tag 0606]. • The conormal sheaf of Z over X is the conormal sheaf of Z in its universal first order thickening Z' over X. We often denote the conormal sheaf C_Z/X in this situation.","statement_latex":"Let $S$ be a scheme.\nLet $h : Z \\to X$ be a formally unramified morphism of\nalgebraic spaces over $S$.\n\\begin{enumerate}\n\\item The {\\it universal first order thickening} of $Z$ over $X$\nis the thickening $Z \\subset Z'$ constructed in\nLemma \\ref{lemma-universal-thickening}.\n\\item The {\\it conormal sheaf of $Z$ over $X$} is the conormal sheaf\nof $Z$ in its universal first order thickening $Z'$ over $X$.\n\\end{enumerate}\nWe often denote the conormal sheaf $\\mathcal{C}_{Z/X}$ in this situation.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0607","source_file":"spaces-more-morphisms.tex","source_line":3127,"source_end_line":3140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3127-L3140","statement_sha256":"b5b1e7c91b33b1cdd1b21741248e359d2c36bf993079a462ddc588bb9aaafcbf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12070,"rank":12070,"depth":59,"x":611.103,"y":1529.588,"cluster":"geometry-of-spaces"},{"id":"stacks:0608","tag":"0608","title":"Universal first order thickenings · Lemma 0608","summary":"Let S be a scheme. Let i : Z → X be an immersion of algebraic spaces over S. Then • i is formally unramified, • the universal first order thickening of Z over X is the first order infinitesimal neighbourhood of Z in X of Definition [Tag 05ZV], • the conormal sheaf of i in the sense of Definition [Tag 04CN] agrees with the conormal sheaf of i in the sense of Definition [Tag 0607].","statement_latex":"Let $S$ be a scheme.\nLet $i : Z \\to X$ be an immersion of algebraic spaces over $S$. Then\n\\begin{enumerate}\n\\item $i$ is formally unramified,\n\\item the universal first order thickening of $Z$ over $X$ is the first order\ninfinitesimal neighbourhood of $Z$ in $X$ of\nDefinition \\ref{definition-first-order-infinitesimal-neighbourhood},\n\\item the conormal sheaf of $i$ in the sense of\nDefinition \\ref{definition-conormal-sheaf}\nagrees with the conormal sheaf of $i$ in the sense of\nDefinition \\ref{definition-universal-thickening}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0608","source_file":"spaces-more-morphisms.tex","source_line":3152,"source_end_line":3166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3152-L3166","statement_sha256":"3bf559bef57f5fec32d6e0b952fc24bdada6197aa7745858216c08189e63f7ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":12071,"rank":12071,"depth":60,"x":787.559,"y":1612.843,"cluster":"geometry-of-spaces"},{"id":"stacks:0609","tag":"0609","title":"Universal first order thickenings · Lemma 0609","summary":"Let S be a scheme. Let Z → X be a formally unramified morphism of algebraic spaces over S. Then the universal first order thickening Z' is formally unramified over X.","statement_latex":"Let $S$ be a scheme.\nLet $Z \\to X$ be a formally unramified morphism of algebraic spaces over $S$.\nThen the universal first order thickening $Z'$ is formally\nunramified over $X$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0609","source_file":"spaces-more-morphisms.tex","source_line":3184,"source_end_line":3190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3184-L3190","statement_sha256":"191207acbc59d65146f92d5d15dcbc3c568a0d3438dfd2f033d59ae31f4e2883","origin":"The Stacks Project","memory_eligible":false,"source_rank":12072,"rank":12072,"depth":0,"x":590.249,"y":1651.625,"cluster":"geometry-of-spaces"},{"id":"stacks:060A","tag":"060A","title":"Universal first order thickenings · Lemma 060A","summary":"Let S be a scheme Consider a commutative diagram of algebraic spaces over S xymatrix Z ar[r]_h ar[d]_f & X ar[d]^g W ar[r]^h' & Y with h and h' formally unramified. Let Z ⊂ Z' be the universal first order thickening of Z over X. Let W ⊂ W' be the universal first order thickening of W over Y. There exists a canonical morphism (f, f') : (Z, Z') → (W, W') of thickenings over Y which fits into the following commutative diagram xymatrix & & & Z' ar[ld] ar[d]^f' Z ar[rr]…","statement_latex":"Let $S$ be a scheme\nConsider a commutative diagram of algebraic spaces over $S$\n$$\n\\xymatrix{\nZ \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\\nW \\ar[r]^{h'} & Y\n}\n$$\nwith $h$ and $h'$ formally unramified. Let $Z \\subset Z'$ be the universal\nfirst order thickening of $Z$ over $X$. Let $W \\subset W'$ be the universal\nfirst order thickening of $W$ over $Y$. There exists a canonical morphism\n$(f, f') : (Z, Z') \\to (W, W')$ of thickenings over $Y$ which fits into\nthe following commutative diagram\n$$\n\\xymatrix{\n& & & Z' \\ar[ld] \\ar[d]^{f'} \\\\\nZ \\ar[rr] \\ar[d]_f \\ar[rrru] & & X \\ar[d] & W' \\ar[ld] \\\\\nW \\ar[rrru]|!{[rr];[rruu]}\\hole \\ar[rr] & & Y\n}\n$$\nIn particular the morphism $(f, f')$ of thickenings induces a morphism\nof conormal sheaves $f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060A","source_file":"spaces-more-morphisms.tex","source_line":3217,"source_end_line":3241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3217-L3241","statement_sha256":"9b356fb0ddae26697268bba365c13d00e5487ddadccb35cf251485f8cdaf65d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12073,"rank":12073,"depth":53,"x":704.696,"y":1510.896,"cluster":"geometry-of-spaces"},{"id":"stacks:060B","tag":"060B","title":"Universal first order thickenings · Lemma 060B","summary":"Let S be a scheme. Let xymatrix Z ar[r]_h ar[d]_f & X ar[d]^g W ar[r]^h' & Y be a fibre product diagram of algebraic spaces over S with h' formally unramified. Then h is formally unramified and if W ⊂ W' is the universal first order thickening of W over Y, then Z = X ×_Y W ⊂ X ×_Y W' is the universal first order thickening of Z over X. In particular the canonical map f^*C_W/Y → C_Z/X of Lemma [Tag 060A] is surjective.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nZ \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\\nW \\ar[r]^{h'} & Y\n}\n$$\nbe a fibre product diagram of algebraic spaces over $S$ with\n$h'$ formally unramified. Then $h$ is formally unramified and if\n$W \\subset W'$ is the universal first order thickening of $W$ over $Y$,\nthen $Z = X \\times_Y W \\subset X \\times_Y W'$ is the universal\nfirst order thickening of $Z$ over $X$. In particular the canonical map\n$f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ of\nLemma \\ref{lemma-universal-thickening-functorial}\nis surjective.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060B","source_file":"spaces-more-morphisms.tex","source_line":3250,"source_end_line":3267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3250-L3267","statement_sha256":"2fe0bf5d4cfc8e010976d77a84cf4f1e05964f22bfe3643d93efdf76bd769e28","origin":"The Stacks Project","memory_eligible":false,"source_rank":12074,"rank":12074,"depth":55,"x":733.51,"y":1679.814,"cluster":"geometry-of-spaces"},{"id":"stacks:060C","tag":"060C","title":"Universal first order thickenings · Lemma 060C","summary":"Let S be a scheme. Let xymatrix Z ar[r]_h ar[d]_f & X ar[d]^g W ar[r]^h' & Y be a fibre product diagram of algebraic spaces over S with h' formally unramified and g flat. In this case the corresponding map Z' → W' of universal first order thickenings is flat, and f^*C_W/Y → C_Z/X is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nZ \\ar[r]_h \\ar[d]_f & X \\ar[d]^g \\\\\nW \\ar[r]^{h'} & Y\n}\n$$\nbe a fibre product diagram of algebraic spaces over $S$ with\n$h'$ formally unramified and $g$ flat. In this case the corresponding\nmap $Z' \\to W'$ of universal first order thickenings is flat, and\n$f^*\\mathcal{C}_{W/Y} \\to \\mathcal{C}_{Z/X}$ is an isomorphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060C","source_file":"spaces-more-morphisms.tex","source_line":3280,"source_end_line":3293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3280-L3293","statement_sha256":"447eac3d02aff97836e4532836930cf535725a40b1c29aae7de9ea62c994d83b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12075,"rank":12075,"depth":56,"x":576.231,"y":1571.474,"cluster":"geometry-of-spaces"},{"id":"stacks:060D","tag":"060D","title":"Universal first order thickenings · Lemma 060D","summary":"Taking the universal first order thickenings commutes with étale localization. More precisely, let h : Z → X be a formally unramified morphism of algebraic spaces over a base scheme S. Let xymatrix V ar[d] ar[r] & U ar[d] Z ar[r] & X be a commutative diagram with étale vertical arrows. Let Z' be the universal first order thickening of Z over X. Then V → U is formally unramified and the universal first order thickening V' of V over U is étale over Z'. In particular,…","statement_latex":"Taking the universal first order thickenings commutes with \\'etale\nlocalization. More precisely, let $h : Z \\to X$ be a formally unramified\nmorphism of algebraic spaces over a base scheme $S$.\nLet\n$$\n\\xymatrix{\nV \\ar[d] \\ar[r] & U \\ar[d] \\\\\nZ \\ar[r] & X\n}\n$$\nbe a commutative diagram with \\'etale vertical arrows.\nLet $Z'$ be the universal first order thickening of $Z$ over $X$.\nThen $V \\to U$ is formally unramified and the universal first\norder thickening $V'$ of $V$ over $U$ is \\'etale over $Z'$.\nIn particular, $\\mathcal{C}_{Z/X}|_V = \\mathcal{C}_{V/U}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060D","source_file":"spaces-more-morphisms.tex","source_line":3308,"source_end_line":3325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3308-L3325","statement_sha256":"cc4c6c2ac047d59fca1d6c9ff6df09bc14fe2307eafa7601fda0e3401c43469e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12076,"rank":12076,"depth":58,"x":779.579,"y":1562.107,"cluster":"geometry-of-spaces"},{"id":"stacks:060E","tag":"060E","title":"Universal first order thickenings · Lemma 060E","summary":"Let S be a scheme. Let B be an algebraic space over S. Let h : Z → X be a formally unramified morphism of algebraic spaces over B. Let Z ⊂ Z' be the universal first order thickening of Z over X with structure morphism h' : Z' → X. The canonical map dh' : (h')^*Ω_X/B → Ω_Z'/B induces an isomorphism h^*Ω_X/B → Ω_Z'/B ⊗ O_Z.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $h : Z \\to X$ be a formally unramified morphism of algebraic spaces\nover $B$. Let $Z \\subset Z'$ be the universal first order thickening of $Z$\nover $X$ with structure morphism $h' : Z' \\to X$. The canonical map\n$$\n\\text{d}h' : (h')^*\\Omega_{X/B} \\to \\Omega_{Z'/B}\n$$\ninduces an isomorphism\n$h^*\\Omega_{X/B} \\to \\Omega_{Z'/B} \\otimes \\mathcal{O}_Z$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060E","source_file":"spaces-more-morphisms.tex","source_line":3336,"source_end_line":3347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3336-L3347","statement_sha256":"6177247794d9825cc6c4c2afb9bc1927e3e597dd7f445e06d90c7e5245f18d38","origin":"The Stacks Project","memory_eligible":false,"source_rank":12077,"rank":12077,"depth":59,"x":636.992,"y":1684.548,"cluster":"geometry-of-spaces"},{"id":"stacks:060F","tag":"060F","title":"Universal first order thickenings · Lemma 060F","summary":"Let S be a scheme. Let B be an algebraic space over S. Let h : Z → X be a formally unramified morphism of algebraic spaces over B. There is a canonical exact sequence C_Z/X → h^*Ω_X/B → Ω_Z/B → 0. The first arrow is induced by d_Z'/B where Z' is the universal first order neighbourhood of Z over X.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $h : Z \\to X$ be a formally unramified morphism of algebraic\nspaces over $B$.\nThere is a canonical exact sequence\n$$\n\\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0.\n$$\nThe first arrow is induced by $\\text{d}_{Z'/B}$ where\n$Z'$ is the universal first order neighbourhood of $Z$ over $X$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060F","source_file":"spaces-more-morphisms.tex","source_line":3365,"source_end_line":3376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3365-L3376","statement_sha256":"4d5b78b0a8775b9c70223228209c767480218d39d3a1584d730a8883d32c38ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":12078,"rank":12078,"depth":60,"x":643.678,"y":1513.147,"cluster":"geometry-of-spaces"},{"id":"stacks:06BE","tag":"06BE","title":"Universal first order thickenings · Lemma 06BE","summary":"Let S be a scheme. Let xymatrix Z ar[r]_i ar[rd]_j & X ar[d] & Y be a commutative diagram of algebraic spaces over S where i and j are formally unramified. Then there is a canonical exact sequence C_Z/Y → C_Z/X → i^*Ω_X/Y → 0 where the first arrow comes from Lemma [Tag 060A] and the second from Lemma [Tag 060F].","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[rd]_j & X \\ar[d] \\\\\n& Y\n}\n$$\nbe a commutative diagram of algebraic spaces over $S$\nwhere $i$ and $j$ are formally unramified. Then there is a\ncanonical exact sequence\n$$\n\\mathcal{C}_{Z/Y} \\to\n\\mathcal{C}_{Z/X} \\to\ni^*\\Omega_{X/Y} \\to 0\n$$\nwhere the first arrow comes from\nLemma \\ref{lemma-universal-thickening-functorial}\nand the second from\nLemma \\ref{lemma-universally-unramified-differentials-sequence}.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BE","source_file":"spaces-more-morphisms.tex","source_line":3391,"source_end_line":3412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3391-L3412","statement_sha256":"cb3b19d916e7c23e3d864ff5f4e9a98607b998286bebeac153c0a5cbbb5c0083","origin":"The Stacks Project","memory_eligible":false,"source_rank":12079,"rank":12079,"depth":61,"x":776.745,"y":1643.487,"cluster":"geometry-of-spaces"},{"id":"stacks:06BF","tag":"06BF","title":"Universal first order thickenings · Lemma 06BF","summary":"Let S be a scheme. Let Z → Y → X be formally unramified morphisms of algebraic spaces over S. • If Z ⊂ Z' is the universal first order thickening of Z over X and Y ⊂ Y' is the universal first order thickening of Y over X, then there is a morphism Z' → Y' and Y ×_Y' Z' is the universal first order thickening of Z over Y. • There is a canonical exact sequence i^*C_Y/X → C_Z/X → C_Z/Y → 0 where the maps come from Lemma [Tag 060A] and i : Z → Y is the first morphism.","statement_latex":"Let $S$ be a scheme.\nLet $Z \\to Y \\to X$ be formally unramified morphisms of\nalgebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $Z \\subset Z'$ is the universal first order thickening of $Z$\nover $X$ and $Y \\subset Y'$ is the universal first order thickening of $Y$\nover $X$, then there is a morphism $Z' \\to Y'$ and $Y \\times_{Y'} Z'$ is\nthe universal first order thickening of $Z$ over $Y$.\n\\item There is a canonical exact sequence\n$$\ni^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nwhere the maps come from\nLemma \\ref{lemma-universal-thickening-functorial}\nand $i : Z \\to Y$ is the first morphism.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Universal first order thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BF","source_file":"spaces-more-morphisms.tex","source_line":3425,"source_end_line":3445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3425-L3445","statement_sha256":"cbba94d1ebcb1d7455b86fe22778305e6605bf7d8888b7d7fd121af7f0c6a48d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12080,"rank":12080,"depth":55,"x":573.562,"y":1622.856,"cluster":"geometry-of-spaces"},{"id":"stacks:04GC","tag":"04GC","title":"Formally étale morphisms · Definition 04GC","summary":"Let S be a scheme. A morphism f : X → Y of algebraic spaces over S is said to be formally étale if it is formally étale as a transformation of functors as in Definition [Tag 049S].","statement_latex":"Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$\nis said to be {\\it formally \\'etale} if it is formally \\'etale as a\ntransformation of functors as in\nDefinition \\ref{definition-formally-smooth-etale-unramified}.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GC","source_file":"spaces-more-morphisms.tex","source_line":3490,"source_end_line":3496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3490-L3496","statement_sha256":"c3e1212017512263ab4252c25844cf9749485d89a4fe26187df418ae83848d91","origin":"The Stacks Project","memory_eligible":false,"source_rank":12081,"rank":12081,"depth":1,"x":740.177,"y":1522.658,"cluster":"geometry-of-spaces"},{"id":"stacks:04GD","tag":"04GD","title":"Formally étale morphisms · Lemma 04GD","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is formally étale, • for every diagram xymatrix U ar[d] ar[r]_ψ & V ar[d] X ar[r]^f & Y where U and V are schemes and the vertical arrows are étale the morphism of schemes ψ is formally étale (as in More on Morphisms, Definition [Tag 02HG]), and • for one such diagram with surjective vertical arrows the morphism ψ is formally étale.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over\n$S$. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is formally \\'etale,\n\\item for every diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\\nX \\ar[r]^f & Y\n}\n$$\nwhere $U$ and $V$ are schemes and the vertical arrows are \\'etale\nthe morphism of schemes $\\psi$ is formally \\'etale (as in\nMore on Morphisms,\nDefinition \\ref{more-morphisms-definition-formally-etale}), and\n\\item for one such diagram with surjective vertical arrows the morphism\n$\\psi$ is formally \\'etale.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04GD","source_file":"spaces-more-morphisms.tex","source_line":3503,"source_end_line":3523,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3503-L3523","statement_sha256":"5dfb9d2f1a436895a24318f35fbe6415b9738003cd63f052118a2058e72f009d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12082,"rank":12082,"depth":42,"x":697.844,"y":1691.287,"cluster":"geometry-of-spaces"},{"id":"stacks:0611","tag":"0611","title":"Formally étale morphisms · Lemma 0611","summary":"Let S be a scheme. Let f : X → Y be a formally étale morphism of algebraic spaces over S. Then given any solid commutative diagram xymatrix X ar[d]_f & T ar[d]^i ar[l]_a Y & T' ar[l] ar@-->[lu] where T ⊂ T' is a first order thickening of algebraic spaces over Y there exists exactly one dotted arrow making the diagram commute. In other words, in Definition [Tag 04GC] the condition that T be affine may be dropped.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a formally \\'etale morphism of algebraic spaces over $S$.\nThen given any solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & T \\ar[d]^i \\ar[l]_a \\\\\nY & T' \\ar[l] \\ar@{-->}[lu]\n}\n$$\nwhere $T \\subset T'$ is a first order thickening of algebraic spaces\nover $Y$ there exists exactly one dotted arrow making the diagram commute.\nIn other words, in\nDefinition \\ref{definition-formally-etale}\nthe condition that $T$ be affine may be dropped.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0611","source_file":"spaces-more-morphisms.tex","source_line":3545,"source_end_line":3561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3545-L3561","statement_sha256":"b6d7a22088944774f99f504dd4e84c6380fac0f1d255e1f95c031697995498a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12083,"rank":12083,"depth":42,"x":593.328,"y":1542.743,"cluster":"geometry-of-spaces"},{"id":"stacks:0612","tag":"0612","title":"Formally étale morphisms · Lemma 0612","summary":"A composition of formally étale morphisms is formally étale.","statement_latex":"A composition of formally \\'etale morphisms is formally \\'etale.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0612","source_file":"spaces-more-morphisms.tex","source_line":3575,"source_end_line":3578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3575-L3578","statement_sha256":"97e4898911c64421df2522e9a3bd38ce2fc5ff0024401b1b868c447141a76e96","origin":"The Stacks Project","memory_eligible":false,"source_rank":12084,"rank":12084,"depth":0,"x":790.086,"y":1593.032,"cluster":"geometry-of-spaces"},{"id":"stacks:0613","tag":"0613","title":"Formally étale morphisms · Lemma 0613","summary":"A base change of a formally étale morphism is formally étale.","statement_latex":"A base change of a formally \\'etale morphism is formally \\'etale.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0613","source_file":"spaces-more-morphisms.tex","source_line":3584,"source_end_line":3587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3584-L3587","statement_sha256":"120023d88b68c4f8199602d927a051b5242f3ceb43b1a701d42b2a641658a485","origin":"The Stacks Project","memory_eligible":false,"source_rank":12085,"rank":12085,"depth":0,"x":604.337,"y":1667.684,"cluster":"geometry-of-spaces"},{"id":"stacks:0614","tag":"0614","title":"Formally étale morphisms · Lemma 0614","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S The following are equivalent: • f is formally étale, • f is formally unramified and the universal first order thickening of X over Y is equal to X, • f is formally unramified and C_X/Y = 0, and • Ω_X/Y = 0 and C_X/Y = 0.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is formally \\'etale,\n\\item $f$ is formally unramified and the universal first order thickening\nof $X$ over $Y$ is equal to $X$,\n\\item $f$ is formally unramified and $\\mathcal{C}_{X/Y} = 0$, and\n\\item $\\Omega_{X/Y} = 0$ and $\\mathcal{C}_{X/Y} = 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0614","source_file":"spaces-more-morphisms.tex","source_line":3593,"source_end_line":3605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3593-L3605","statement_sha256":"4a789d08f4afbb124eb7a4a326066db3f96304ed2ee34f2a196aae4c4d993d30","origin":"The Stacks Project","memory_eligible":false,"source_rank":12086,"rank":12086,"depth":60,"x":681.365,"y":1507.047,"cluster":"geometry-of-spaces"},{"id":"stacks:0615","tag":"0615","title":"Formally étale morphisms · Lemma 0615","summary":"An unramified flat morphism is formally étale.","statement_latex":"An unramified flat morphism is formally \\'etale.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0615","source_file":"spaces-more-morphisms.tex","source_line":3625,"source_end_line":3628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3625-L3628","statement_sha256":"de9d7886ca617fb74c3cd68b36c06cdce8b43c4fbfa7b0419e26588d273d18ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":12087,"rank":12087,"depth":43,"x":753.831,"y":1669.398,"cluster":"geometry-of-spaces"},{"id":"stacks:0616","tag":"0616","title":"Formally étale morphisms · Lemma 0616","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • The morphism f is étale, and • the morphism f is locally of finite presentation and formally étale.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is \\'etale, and\n\\item the morphism $f$ is locally of finite presentation and\nformally \\'etale.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0616","source_file":"spaces-more-morphisms.tex","source_line":3640,"source_end_line":3650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3640-L3650","statement_sha256":"dfe17662ca4b0fa3ac5d718fff9d6c1abecd90ce3931cf8021f17774996b5f2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12088,"rank":12088,"depth":47,"x":569.619,"y":1590.709,"cluster":"geometry-of-spaces"},{"id":"stacks:0618","tag":"0618","title":"Infinitesimal deformations of maps · Lemma 0618","summary":"Let S be a scheme. Let B be an algebraic space over S. Let X ⊂ X' and Y ⊂ Y' be two first order thickenings of algebraic spaces over B. Let (a, a'), (b, b') : (X ⊂ X') → (Y ⊂ Y') be two morphisms of thickenings over B. Assume that • a = b, and • the two maps a^*C_Y/Y' → C_X/X' (Lemma [Tag 04CP]) are equal. Then the map (a')^sharp - (b')^sharp factors as O_Y' → O_Y xrightarrowD a_*C_X/X' → a_*O_X' where D is an O_B-derivation.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $X \\subset X'$ and $Y \\subset Y'$ be two first order thickenings\nof algebraic spaces over $B$.\nLet $(a, a'), (b, b') : (X \\subset X') \\to (Y \\subset Y')$\nbe two morphisms of thickenings over $B$. Assume that\n\\begin{enumerate}\n\\item $a = b$, and\n\\item the two maps $a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$\n(Lemma \\ref{lemma-conormal-functorial})\nare equal.\n\\end{enumerate}\nThen the map $(a')^\\sharp - (b')^\\sharp$ factors as\n$$\n\\mathcal{O}_{Y'} \\to \\mathcal{O}_Y \\xrightarrow{D}\na_*\\mathcal{C}_{X/X'} \\to a_*\\mathcal{O}_{X'}\n$$\nwhere $D$ is an $\\mathcal{O}_B$-derivation.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0618","source_file":"spaces-more-morphisms.tex","source_line":3688,"source_end_line":3707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3688-L3707","statement_sha256":"23164dedfe58545ec3174b9b43d06e221226d155e001c4fa287a72203071e5b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12089,"rank":12089,"depth":53,"x":768.97,"y":1544.152,"cluster":"geometry-of-spaces"},{"id":"stacks:04D0","tag":"04D0","title":"Infinitesimal deformations of maps · Lemma 04D0","summary":"Let S be a scheme. Let B be an algebraic space over S. Let (a, a') : (X ⊂ X') → (Y ⊂ Y') be a morphism of first order thickenings over B. Let theta : a^*Ω_Y/B → C_X/X' be an O_X-linear map. Then there exists a unique morphism of pairs (b, b') : (X ⊂ X') → (Y ⊂ Y') such that (1) and (2) of Lemma [Tag 0618] hold and the derivation D and theta are related by Equation ([Tag 0619]).","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $(a, a') : (X \\subset X') \\to (Y \\subset Y')$\nbe a morphism of first order thickenings over $B$.\nLet\n$$\n\\theta : a^*\\Omega_{Y/B} \\to \\mathcal{C}_{X/X'}\n$$\nbe an $\\mathcal{O}_X$-linear map. Then there exists a unique morphism of pairs\n$(b, b') : (X \\subset X') \\to (Y \\subset Y')$ such that\n(1) and (2) of\nLemma \\ref{lemma-difference-derivation}\nhold and the derivation $D$ and $\\theta$ are related by\nEquation (\\ref{equation-D}).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04D0","source_file":"spaces-more-morphisms.tex","source_line":3747,"source_end_line":3762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3747-L3762","statement_sha256":"de560e32cd857c90c06238626958498e646a0ea6a5822403e53aedbe9e5295ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":12090,"rank":12090,"depth":56,"x":659.282,"y":1691.773,"cluster":"geometry-of-spaces"},{"id":"stacks:061A","tag":"061A","title":"Infinitesimal deformations of maps · Lemma 061A","summary":"Let S be a scheme. Let B be an algebraic space over S. Let X ⊂ X' and Y ⊂ Y' be first order thickenings over B. Assume given a morphism a : X → Y and a map A : a^*C_Y/Y' → C_X/X' of O_X-modules. For an object U' of (X')_spaces, etale with U = X ×_X' U' consider morphisms a' : U' → Y' such that • a' is a morphism over B, • a'|_U = a|_U, and • the induced map a^*C_Y/Y'|_U → C_X/X'|_U is the restriction of A to U. Then the rule U' ↦ (a' : U' → Y' such that (1), (2), (3)…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $X \\subset X'$ and $Y \\subset Y'$ be first order thickenings\nover $B$. Assume given a morphism $a : X \\to Y$ and a map\n$A : a^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$ of\n$\\mathcal{O}_X$-modules. For an object $U'$ of\n$(X')_{spaces, \\etale}$ with $U = X \\times_{X'} U'$\nconsider morphisms $a' : U' \\to Y'$ such that\n\\begin{enumerate}\n\\item $a'$ is a morphism over $B$,\n\\item $a'|_U = a|_U$, and\n\\item the induced map\n$a^*\\mathcal{C}_{Y/Y'}|_U \\to \\mathcal{C}_{X/X'}|_U$\nis the restriction of $A$ to $U$.\n\\end{enumerate}\nThen the rule\n\\begin{equation}\n\nU' \\mapsto\n\\{a' : U' \\to Y'\\text{ such that (1), (2), (3) hold.}\\}\n\\end{equation}\ndefines a sheaf of sets on $(X')_{spaces, \\etale}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061A","source_file":"spaces-more-morphisms.tex","source_line":3790,"source_end_line":3813,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3790-L3813","statement_sha256":"2f5c9178285c66e5241f33b865f616fb7f895e70d46304cdbccacbf992b6bdc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12091,"rank":12091,"depth":55,"x":621.407,"y":1520.479,"cluster":"geometry-of-spaces"},{"id":"stacks:061C","tag":"061C","title":"Infinitesimal deformations of maps · Lemma 061C","summary":"Same notation and assumptions as in Lemma [Tag 061A]. We identify sheaves on X and X' via ([Tag 05ZL]). There is an action of the sheaf SheafHom_O_X(a^*Ω_Y/B, C_X/X') on the sheaf ([Tag 061B]). Moreover, the action is simply transitive for any object U' of (X')_spaces, etale over which the sheaf ([Tag 061B]) has a section.","statement_latex":"Same notation and assumptions as in Lemma \\ref{lemma-sheaf}.\nWe identify sheaves on $X$ and $X'$ via\n(\\ref{equation-equivalence-etale-spaces}).\nThere is an action of the sheaf\n$$\n\\SheafHom_{\\mathcal{O}_X}(a^*\\Omega_{Y/B}, \\mathcal{C}_{X/X'})\n$$\non the sheaf (\\ref{equation-sheaf}). Moreover, the action\nis simply transitive for any object $U'$ of $(X')_{spaces, \\etale}$\nover which the sheaf (\\ref{equation-sheaf}) has a section.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061C","source_file":"spaces-more-morphisms.tex","source_line":3826,"source_end_line":3838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3826-L3838","statement_sha256":"6e1b02a50919317a81e0df17b52fa0c5282d81b3be809972b2bd8d92bc35f9c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12092,"rank":12092,"depth":57,"x":787.28,"y":1625.42,"cluster":"geometry-of-spaces"},{"id":"stacks:0CK8","tag":"0CK8","title":"Infinitesimal deformations of maps · Lemma 0CK8","summary":"Let S be a scheme. Consider a commutative diagram of first order thickenings vcenter xymatrix (T_2 ⊂ T_2') ar[d]_(h, h') ar[rr]_(a_2, a_2') & & (X_2 ⊂ X_2') ar[d]^(f, f') (T_1 ⊂ T_1') ar[rr]^(a_1, a_1') & & (X_1 ⊂ X_1') and a commutative diagram vcenter xymatrix X_2' ar[r] ar[d] & B_2 ar[d] X_1' ar[r] & B_1 of algebraic spaces over S with X_2 → X_1 and B_2 → B_1 étale. For any O_T_1-linear map theta_1 : a_1^*Ω_X_1/B_1 → C_T_1/T'_1 let theta_2 be the composition xymatrix…","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram of first order\nthickenings\n$$\n\\vcenter{\n\\xymatrix{\n(T_2 \\subset T_2') \\ar[d]_{(h, h')} \\ar[rr]_{(a_2, a_2')} & &\n(X_2 \\subset X_2') \\ar[d]^{(f, f')} \\\\\n(T_1 \\subset T_1') \\ar[rr]^{(a_1, a_1')} & &\n(X_1 \\subset X_1')\n}\n}\n\\quad\n\\begin{matrix}\n\\text{and a commutative} \\\\\n\\text{diagram}\n\\end{matrix}\n\\quad\n\\vcenter{\n\\xymatrix{\nX_2' \\ar[r] \\ar[d] & B_2 \\ar[d] \\\\\nX_1' \\ar[r] & B_1\n}\n}\n$$\nof algebraic spaces over $S$\nwith $X_2 \\to X_1$ and $B_2 \\to B_1$ \\'etale.\nFor any $\\mathcal{O}_{T_1}$-linear map\n$\\theta_1 : a_1^*\\Omega_{X_1/B_1} \\to \\mathcal{C}_{T_1/T'_1}$ let\n$\\theta_2$ be the composition\n$$\n\\xymatrix{\na_2^*\\Omega_{X_2/B_2} \\ar@{=}[r] &\nh^*a_1^*\\Omega_{X_1/B_1} \\ar[r]^-{h^*\\theta_1} &\nh^*\\mathcal{C}_{T_1/T'_1} \\ar[r] &\n\\mathcal{C}_{T_2/T'_2}\n}\n$$\n(equality sign is explained in the proof). Then the diagram\n$$\n\\xymatrix{\nT_2' \\ar[rr]_{\\theta_2 \\cdot a_2'} \\ar[d] & & X'_2 \\ar[d] \\\\\nT_1' \\ar[rr]^{\\theta_1 \\cdot a_1'} & & X'_1\n}\n$$\ncommutes where the actions $\\theta_2 \\cdot a_2'$ and $\\theta_1 \\cdot a_1'$\nare as in Remark \\ref{remark-action-by-derivations}.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CK8","source_file":"spaces-more-morphisms.tex","source_line":3900,"source_end_line":3948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3900-L3948","statement_sha256":"bead45fd29c711486f1b2f406c3eb536497f77284690706956a4cc58286a67de","origin":"The Stacks Project","memory_eligible":false,"source_rank":12093,"rank":12093,"depth":56,"x":580.335,"y":1642.178,"cluster":"geometry-of-spaces"},{"id":"stacks:06BH","tag":"06BH","title":"Infinitesimal deformations of algebraic spaces · Lemma 06BH","summary":"Let S be a scheme. Let (f, f') : (X ⊂ X') → (Y ⊂ Y') be a morphism of first order thickenings of algebraic spaces over S. Assume that f is flat. Then the following are equivalent • f' is flat and X = Y ×_Y' X', and • the canonical map f^*C_Y/Y' → C_X/X' is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $(f, f') : (X \\subset X') \\to (Y \\subset Y')$ be a\nmorphism of first order thickenings of algebraic spaces over $S$. Assume that\n$f$ is flat. Then the following are equivalent\n\\begin{enumerate}\n\\item $f'$ is flat and $X = Y \\times_{Y'} X'$, and\n\\item the canonical map $f^*\\mathcal{C}_{Y/Y'} \\to \\mathcal{C}_{X/X'}$\nis an isomorphism.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BH","source_file":"spaces-more-morphisms.tex","source_line":3990,"source_end_line":4000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L3990-L4000","statement_sha256":"70fdea3eadf3b1baf37570882d99c6c20d3598ea035a3d5765f9436ad2e9504a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12094,"rank":12094,"depth":5,"x":719.62,"y":1512.244,"cluster":"geometry-of-spaces"},{"id":"stacks:0CG5","tag":"0CG5","title":"Infinitesimal deformations of algebraic spaces · Lemma 0CG5","summary":"Let S be a scheme. Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (B ⊂ B') of thickenings of algebraic spaces over S. Assume • X' is flat over B', • f is flat, • B ⊂ B' is a finite order thickening, and • X = B ×_B' X' and Y = B ×_B' Y'. Then f' is flat and Y' is flat over B' at all points in the image of f'.","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\n(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\\n& (B \\subset B')\n}\n$$\nof thickenings of algebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item $X'$ is flat over $B'$,\n\\item $f$ is flat,\n\\item $B \\subset B'$ is a finite order thickening, and\n\\item $X = B \\times_{B'} X'$ and $Y = B \\times_{B'} Y'$.\n\\end{enumerate}\nThen $f'$ is flat and $Y'$ is flat over $B'$ at all points in\nthe image of $f'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CG5","source_file":"spaces-more-morphisms.tex","source_line":4023,"source_end_line":4041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4023-L4041","statement_sha256":"44a2afa7c26cd9c5fccc64881b334983e9a19691161f0c49b5346a204afdb078","origin":"The Stacks Project","memory_eligible":false,"source_rank":12095,"rank":12095,"depth":4,"x":721.404,"y":1687.293,"cluster":"geometry-of-spaces"},{"id":"stacks:0CG6","tag":"0CG6","title":"Infinitesimal deformations of algebraic spaces · Lemma 0CG6","summary":"Let S be a scheme. Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (B ⊂ B') of thickenings of algebraic spaces over S. Assume B ⊂ B' is a finite order thickening, X' flat over B', X = B ×_B' X', and Y = B ×_B' Y'. Then • f is representable if and only if f' is representable, • f is flat if and only if f' is flat, • f is an isomorphism if and only if f' is an isomorphism, • f is an open immersion if and only if f' is an open…","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\n(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\\n& (B \\subset B')\n}\n$$\nof thickenings of algebraic spaces over $S$. Assume $B \\subset B'$\nis a finite order thickening, $X'$ flat over $B'$, $X = B \\times_{B'} X'$,\nand $Y = B \\times_{B'} Y'$. Then\n\\begin{enumerate}\n\\item $f$ is representable if and only if $f'$ is representable,\n\n\\item $f$ is flat if and only if $f'$ is flat,\n\n\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,\n\n\\item $f$ is an open immersion if and only if $f'$ is an open immersion,\n\n\\item $f$ is quasi-compact if and only if $f'$ is quasi-compact,\n\n\\item $f$ is universally closed if and only if $f'$ is universally closed,\n\n\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,\n\n\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,\n\n\\item $f$ is surjective if and only if $f'$ is surjective,\n\n\\item $f$ is universally injective if and only if $f'$ is universally injective,\n\n\\item $f$ is affine if and only if $f'$ is affine,\n\n\\item\n\n$f$ is locally of finite type if and only if $f'$ is locally of finite type,\n\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,\n\n\\item\n\n$f$ is locally of finite presentation if and only if $f'$ is locally of\nfinite presentation,\n\\item\n\n$f$ is locally of finite type of relative dimension $d$ if and only if\n$f'$ is locally of finite type of relative dimension $d$,\n\\item $f$ is universally open if and only if $f'$ is universally open,\n\n\\item $f$ is syntomic if and only if $f'$ is syntomic,\n\n\\item $f$ is smooth if and only if $f'$ is smooth,\n\n\\item $f$ is unramified if and only if $f'$ is unramified,\n\n\\item $f$ is \\'etale if and only if $f'$ is \\'etale,\n\n\\item $f$ is proper if and only if $f'$ is proper,\n\n\\item $f$ is integral if and only if $f'$ is integral,\n\n\\item $f$ is finite if and only if $f'$ is finite,\n\n\\item\n\n$f$ is finite locally free (of rank $d$) if and only if $f'$\nis finite locally free (of rank $d$), and\n\\item add more here.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CG6","source_file":"spaces-more-morphisms.tex","source_line":4070,"source_end_line":4140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4070-L4140","statement_sha256":"0b7fe1e5e71204914b31f048b834d5d2f416de9dfbe44a718a1bae2f17478f16","origin":"The Stacks Project","memory_eligible":false,"source_rank":12096,"rank":12096,"depth":71,"x":579.154,"y":1559.078,"cluster":"geometry-of-spaces"},{"id":"stacks:0CGW","tag":"0CGW","title":"Infinitesimal deformations of algebraic spaces · Lemma 0CGW","summary":"Let S be a scheme. Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (B ⊂ B') of thickenings of algebraic spaces over S. Assume • Y' → B' is locally of finite type, • X' → B' is flat and locally of finite presentation, • f is flat, and • X = B ×_B' X' and Y = B ×_B' Y'. Then f' is flat and for all y' ∈ |Y'| in the image of |f'| the morphism Y' → B' is flat at y'.","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\n(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\\n& (B \\subset B')\n}\n$$\nof thickenings of algebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item $Y' \\to B'$ is locally of finite type,\n\\item $X' \\to B'$ is flat and locally of finite presentation,\n\\item $f$ is flat, and\n\\item $X = B \\times_{B'} X'$ and $Y = B \\times_{B'} Y'$.\n\\end{enumerate}\nThen $f'$ is flat and for all $y' \\in |Y'|$ in the image of $|f'|$\nthe morphism $Y' \\to B'$ is flat at $y'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CGW","source_file":"spaces-more-morphisms.tex","source_line":4247,"source_end_line":4265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4247-L4265","statement_sha256":"c678796a95d0aef37a39adbb4468125405653a13632b6d37ad6e3f515d8c277a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12097,"rank":12097,"depth":15,"x":787.395,"y":1572.921,"cluster":"geometry-of-spaces"},{"id":"stacks:0CGX","tag":"0CGX","title":"Infinitesimal deformations of algebraic spaces · Lemma 0CGX","summary":"Let S be a scheme. Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (B ⊂ B') of thickenings of algebraic spaces over S. Assume Y' → B' locally of finite type, X' → B' flat and locally of finite presentation, X = B ×_B' X', and Y = B ×_B' Y'. Then • f is representable if and only if f' is representable, • f is flat if and only if f' is flat, • f is an isomorphism if and only if f' is an isomorphism, • f is an open immersion if…","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\n(X \\subset X') \\ar[rr]_{(f, f')} \\ar[rd] & & (Y \\subset Y') \\ar[ld] \\\\\n& (B \\subset B')\n}\n$$\nof thickenings of algebraic spaces over $S$.\nAssume $Y' \\to B'$ locally of finite type,\n$X' \\to B'$ flat and locally of finite presentation,\n$X = B \\times_{B'} X'$, and $Y = B \\times_{B'} Y'$. Then\n\\begin{enumerate}\n\\item $f$ is representable if and only if $f'$ is representable,\n\n\\item $f$ is flat if and only if $f'$ is flat,\n\n\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,\n\n\\item $f$ is an open immersion if and only if $f'$ is an open immersion,\n\n\\item $f$ is quasi-compact if and only if $f'$ is quasi-compact,\n\n\\item $f$ is universally closed if and only if $f'$ is universally closed,\n\n\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,\n\n\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,\n\n\\item $f$ is surjective if and only if $f'$ is surjective,\n\n\\item $f$ is universally injective if and only if $f'$ is universally injective,\n\n\\item $f$ is affine if and only if $f'$ is affine,\n\n\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,\n\n\\item\n\n$f$ is locally of finite type of relative dimension $d$ if and only if\n$f'$ is locally of finite type of relative dimension $d$,\n\\item $f$ is universally open if and only if $f'$ is universally open,\n\n\\item $f$ is syntomic if and only if $f'$ is syntomic,\n\n\\item $f$ is smooth if and only if $f'$ is smooth,\n\n\\item $f$ is unramified if and only if $f'$ is unramified,\n\n\\item $f$ is \\'etale if and only if $f'$ is \\'etale,\n\n\\item $f$ is proper if and only if $f'$ is proper,\n\n\\item $f$ is finite if and only if $f'$ is finite,\n\n\\item\n\n$f$ is finite locally free (of rank $d$) if and only if $f'$\nis finite locally free (of rank $d$), and\n\\item add more here.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Infinitesimal deformations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CGX","source_file":"spaces-more-morphisms.tex","source_line":4294,"source_end_line":4356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4294-L4356","statement_sha256":"1a4778186664fc5ef303dd156560861f491a418367a55d2301c717ec286260a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12098,"rank":12098,"depth":71,"x":622.518,"y":1680.999,"cluster":"geometry-of-spaces"},{"id":"stacks:060G","tag":"060G","title":"Formally smooth morphisms · Definition 060G","summary":"Let S be a scheme. A morphism f : X → Y of algebraic spaces over S is said to be formally smooth if it is formally smooth as a transformation of functors as in Definition [Tag 049S].","statement_latex":"Let $S$ be a scheme. A morphism $f : X \\to Y$ of algebraic spaces over $S$\nis said to be {\\it formally smooth} if it is formally smooth as a\ntransformation of functors as in\nDefinition \\ref{definition-formally-smooth-etale-unramified}.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060G","source_file":"spaces-more-morphisms.tex","source_line":4490,"source_end_line":4496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4490-L4496","statement_sha256":"d5d7d7cfed2b393439c36e289f9d68ecee9b05d981ba4184e4149794be34e2e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12099,"rank":12099,"depth":1,"x":657.223,"y":1507.549,"cluster":"geometry-of-spaces"},{"id":"stacks:061E","tag":"061E","title":"Formally smooth morphisms · Lemma 061E","summary":"A composition of formally smooth morphisms is formally smooth.","statement_latex":"A composition of formally smooth morphisms is formally smooth.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061E","source_file":"spaces-more-morphisms.tex","source_line":4518,"source_end_line":4521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4518-L4521","statement_sha256":"eeb2f93084340d9b800c583c6b4dc9f4ffe91ffb67ca174042638bbe3f3e88eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12100,"rank":12100,"depth":0,"x":771.247,"y":1655.311,"cluster":"geometry-of-spaces"},{"id":"stacks:061F","tag":"061F","title":"Formally smooth morphisms · Lemma 061F","summary":"A base change of a formally smooth morphism is formally smooth.","statement_latex":"A base change of a formally smooth morphism is formally smooth.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061F","source_file":"spaces-more-morphisms.tex","source_line":4527,"source_end_line":4530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4527-L4530","statement_sha256":"772e0042117a12f8e84c911fe9f5eda6db78b586794db10b65ae09215ceb5a2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12101,"rank":12101,"depth":2,"x":568.108,"y":1611.003,"cluster":"geometry-of-spaces"},{"id":"stacks:061G","tag":"061G","title":"Formally smooth morphisms · Lemma 061G","summary":"Let f : X → S be a morphism of schemes. Then f is formally étale if and only if f is formally smooth and formally unramified.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nThen $f$ is formally \\'etale if and only if\n$f$ is formally smooth and formally unramified.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061G","source_file":"spaces-more-morphisms.tex","source_line":4538,"source_end_line":4543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4538-L4543","statement_sha256":"b09728323f018acb37507ec49dc318d2ea8ac8e1049f7caaa02dcda5bf628c41","origin":"The Stacks Project","memory_eligible":false,"source_rank":12102,"rank":12102,"depth":0,"x":753.743,"y":1528.315,"cluster":"geometry-of-spaces"},{"id":"stacks:061H","tag":"061H","title":"Formally smooth morphisms · Lemma 061H","summary":"Let S be a scheme. Let xymatrix U ar[d] ar[r]_ψ & V ar[d] X ar[r]^f & Y be a commutative diagram of morphisms of algebraic spaces over S. If the vertical arrows are étale and f is formally smooth, then ψ is formally smooth.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\\nX \\ar[r]^f & Y\n}\n$$\nbe a commutative diagram of morphisms of algebraic spaces over $S$.\nIf the vertical arrows are \\'etale and $f$ is formally smooth, then\n$\\psi$ is formally smooth.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061H","source_file":"spaces-more-morphisms.tex","source_line":4553,"source_end_line":4565,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4553-L4565","statement_sha256":"6429acd00f6d45a19464d735bd676179958fb6106b7fcf31b0e0088f5c4cc36e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12103,"rank":12103,"depth":41,"x":683.274,"y":1694.812,"cluster":"geometry-of-spaces"},{"id":"stacks:04AM","tag":"04AM","title":"Infinitesimal lifting criterion · Lemma 04AM","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • The morphism f is smooth. • The morphism f is locally of finite presentation, and formally smooth.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is smooth.\n\\item The morphism $f$ is locally of finite presentation, and\nformally smooth.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AM","source_file":"spaces-more-morphisms.tex","source_line":4586,"source_end_line":4596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4586-L4596","statement_sha256":"4764c1483f98ca8aa090f8eb66dc4cfd5220afcf49b6898f786c5d35f2760ace","origin":"The Stacks Project","memory_eligible":false,"source_rank":12104,"rank":12104,"depth":58,"x":601.251,"y":1531.868,"cluster":"geometry-of-spaces"},{"id":"stacks:0CHJ","tag":"0CHJ","title":"Formally smooth morphisms · Lemma 0CHJ","summary":"Let S be a scheme. Consider a commutative diagram xymatrix X ar[d] & T ar[l] ar[d] Y & T' ar[l] of algebraic spaces over S where X → Y is smooth and T → T' is a thickening. Then there exists an étale covering (T'_i → T') such that we can find the dotted arrow in xymatrix X ar[d] & T ar[l] ar[d] & T ×_T' T'_i ar[l] ar[d] Y & T' ar[l] & T'_i ar[l] ar@..>[llu] making the diagram commute (for all i).","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[d] & T \\ar[l] \\ar[d] \\\\\nY & T' \\ar[l]\n}\n$$\nof algebraic spaces over $S$ where $X \\to Y$ is smooth\nand $T \\to T'$ is a thickening. Then there exists an\n\\'etale covering $\\{T'_i \\to T'\\}$ such that we\ncan find the dotted arrow in\n$$\n\\xymatrix{\nX \\ar[d] & T \\ar[l] \\ar[d] & T \\times_{T'} T'_i \\ar[l] \\ar[d] \\\\\nY & T' \\ar[l] & T'_i \\ar[l] \\ar@{..>}[llu]\n}\n$$\nmaking the diagram commute (for all $i$).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHJ","source_file":"spaces-more-morphisms.tex","source_line":4713,"source_end_line":4733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4713-L4733","statement_sha256":"6287a6cbb2644322821af7c3a9a13be52faf853d5aaf702e0cf4bd5fb8035287","origin":"The Stacks Project","memory_eligible":false,"source_rank":12105,"rank":12105,"depth":57,"x":792.986,"y":1605.562,"cluster":"geometry-of-spaces"},{"id":"stacks:061I","tag":"061I","title":"Formally smooth morphisms · Lemma 061I","summary":"Let S be a scheme. Let f : X → Y be a formally smooth morphism of algebraic spaces over S. Then Ω_X/Y is locally projective on X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a formally smooth morphism of algebraic spaces over $S$.\nThen $\\Omega_{X/Y}$ is locally projective on $X$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061I","source_file":"spaces-more-morphisms.tex","source_line":4768,"source_end_line":4773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4768-L4773","statement_sha256":"fa21aae10ce33fc6c26088ee4d0d2c9859a54e07a07d153a851b4ffd6367037a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12106,"rank":12106,"depth":42,"x":592.113,"y":1660.078,"cluster":"geometry-of-spaces"},{"id":"stacks:061J","tag":"061J","title":"Formally smooth morphisms · Lemma 061J","summary":"Let T be an affine scheme. Let F, G be quasi-coherent O_T-modules on T_etale. Consider the internal hom sheaf H = SheafHom_O_T(F, G) on T_etale. If F is locally projective, then H^1(T_etale, H) = 0.","statement_latex":"Let $T$ be an affine scheme.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be quasi-coherent\n$\\mathcal{O}_T$-modules on $T_\\etale$.\nConsider the internal hom sheaf\n$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_T}(\\mathcal{F}, \\mathcal{G})$\non $T_\\etale$.\nIf $\\mathcal{F}$ is locally projective, then\n$H^1(T_\\etale, \\mathcal{H}) = 0$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061J","source_file":"spaces-more-morphisms.tex","source_line":4802,"source_end_line":4812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4802-L4812","statement_sha256":"e2bca3a791b135bfff84bcb1162f3e728b7bd92316877be5f4f1cf80a359f81c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12107,"rank":12107,"depth":26,"x":696.512,"y":1505.724,"cluster":"geometry-of-spaces"},{"id":"stacks:061K","tag":"061K","title":"Formally smooth morphisms · Lemma 061K","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is formally smooth, • for every diagram xymatrix U ar[d] ar[r]_ψ & V ar[d] X ar[r]^f & Y where U and V are schemes and the vertical arrows are étale the morphism of schemes ψ is formally smooth (as in More on Morphisms, Definition [Tag 02H8]), and • for one such diagram with surjective vertical arrows the morphism ψ is formally smooth.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over\n$S$. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is formally smooth,\n\\item for every diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r]_\\psi & V \\ar[d] \\\\\nX \\ar[r]^f & Y\n}\n$$\nwhere $U$ and $V$ are schemes and the vertical arrows are \\'etale\nthe morphism of schemes $\\psi$ is formally smooth (as in\nMore on Morphisms,\nDefinition \\ref{more-morphisms-definition-formally-unramified}), and\n\\item for one such diagram with surjective vertical arrows the morphism\n$\\psi$ is formally smooth.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/061K","source_file":"spaces-more-morphisms.tex","source_line":4836,"source_end_line":4857,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4836-L4857","statement_sha256":"26e7f85be4a990c4a2efc59b05d2006999e4091793ed93def45f8f43d34dde72","origin":"The Stacks Project","memory_eligible":false,"source_rank":12108,"rank":12108,"depth":59,"x":743.71,"y":1678.976,"cluster":"geometry-of-spaces"},{"id":"stacks:06CS","tag":"06CS","title":"Formally smooth morphisms · Lemma 06CS","summary":"The property P(f) =\"f is formally smooth\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is formally smooth''\nis fpqc local on the base.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CS","source_file":"spaces-more-morphisms.tex","source_line":4937,"source_end_line":4941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4937-L4941","statement_sha256":"eb2520f3eb84535305fba7dddcbf0586d0e463c2e10b01acf44249d3e1a9be0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12109,"rank":12109,"depth":60,"x":569.386,"y":1577.89,"cluster":"geometry-of-spaces"},{"id":"stacks:06BI","tag":"06BI","title":"Formally smooth morphisms · Lemma 06BI","summary":"Let S be a scheme. Let f : X → Y, g : Y → Z be morphisms of algebraic spaces over S. Assume f is formally smooth. Then 0 → f^*Ω_Y/Z → Ω_X/Z → Ω_X/Y → 0 Lemma [Tag 05Z8] is short exact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$, $g : Y \\to Z$ be morphisms of algebraic spaces over $S$.\nAssume $f$ is formally smooth. Then\n$$\n0 \\to f^*\\Omega_{Y/Z} \\to \\Omega_{X/Z} \\to \\Omega_{X/Y} \\to 0\n$$\nLemma \\ref{lemma-triangle-differentials}\nis short exact.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BI","source_file":"spaces-more-morphisms.tex","source_line":4965,"source_end_line":4975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4965-L4975","statement_sha256":"704705a365847c2ed3aef0b7d16902bfd96bbf09071af10e83e8b4bd60efb991","origin":"The Stacks Project","memory_eligible":false,"source_rank":12110,"rank":12110,"depth":60,"x":779.457,"y":1553.487,"cluster":"geometry-of-spaces"},{"id":"stacks:06BJ","tag":"06BJ","title":"Formally smooth morphisms · Lemma 06BJ","summary":"Let S be a scheme. Let B be an algebraic space over S. Let h : Z → X be a formally unramified morphism of algebraic spaces over B. Assume that Z is formally smooth over B. Then the canonical exact sequence 0 → C_Z/X → h^*Ω_X/B → Ω_Z/B → 0 of Lemma [Tag 060F] is short exact.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $h : Z \\to X$ be a formally unramified morphism of algebraic spaces\nover $B$.\nAssume that $Z$ is formally smooth over $B$. Then the\ncanonical exact sequence\n$$\n0 \\to \\mathcal{C}_{Z/X} \\to h^*\\Omega_{X/B} \\to \\Omega_{Z/B} \\to 0\n$$\nof\nLemma \\ref{lemma-universally-unramified-differentials-sequence}\nis short exact.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BJ","source_file":"spaces-more-morphisms.tex","source_line":4985,"source_end_line":4998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L4985-L4998","statement_sha256":"f8e0a412689462effacbcbe6a9b61692990eb9146c3f73710322e9d934f309bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12111,"rank":12111,"depth":61,"x":644.027,"y":1690.834,"cluster":"geometry-of-spaces"},{"id":"stacks:06BK","tag":"06BK","title":"Formally smooth morphisms · Lemma 06BK","summary":"Let S be a scheme. Let xymatrix Z ar[r]_i ar[rd]_j & X ar[d]^f & Y be a commutative diagram of algebraic spaces over S where i and j are formally unramified and f is formally smooth. Then the canonical exact sequence 0 → C_Z/Y → C_Z/X → i^*Ω_X/Y → 0 of Lemma [Tag 06BE] is exact and locally split.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nZ \\ar[r]_i \\ar[rd]_j & X \\ar[d]^f \\\\\n& Y\n}\n$$\nbe a commutative diagram of algebraic spaces over $S$\nwhere $i$ and $j$ are formally unramified and $f$ is formally smooth.\nThen the canonical exact sequence\n$$\n0 \\to\n\\mathcal{C}_{Z/Y} \\to\n\\mathcal{C}_{Z/X} \\to\ni^*\\Omega_{X/Y} \\to 0\n$$\nof\nLemma \\ref{lemma-two-unramified-morphisms}\nis exact and locally split.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BK","source_file":"spaces-more-morphisms.tex","source_line":5015,"source_end_line":5036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5015-L5036","statement_sha256":"e5f5a4bfeeee5cfca95359bf2cf2688c70e447395b4c7eca85335d1d0b91e698","origin":"The Stacks Project","memory_eligible":false,"source_rank":12112,"rank":12112,"depth":62,"x":633.428,"y":1512.51,"cluster":"geometry-of-spaces"},{"id":"stacks:0APN","tag":"0APN","title":"Smoothness over a Noetherian base · Lemma 0APN","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let x ∈ |X|. Assume that Y is locally Noetherian and f locally of finite type. The following are equivalent: • f is smooth at x, • for every solid commutative diagram xymatrix X ar[d]_f & Spec(B) ar[d]^i ar[l]^-α Y & Spec(B') ar[l]_-β ar@-->[lu] where B' → B is a surjection of local rings with Ker(B' → B) of square zero, and α mapping the closed point of Spec(B) to x there exists a dotted arrow…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $x \\in |X|$.\nAssume that $Y$ is locally Noetherian and $f$ locally of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is smooth at $x$,\n\\item for every solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\\nY & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu]\n}\n$$\nwhere $B' \\to B$ is a surjection of local rings with\n$\\Ker(B' \\to B)$ of square zero, and $\\alpha$ mapping the\nclosed point of $\\Spec(B)$ to $x$ there exists\na dotted arrow making the diagram commute, and\n\\item same as in (2) but with $B' \\to B$ ranging over small\nextensions (see Algebra, Definition \\ref{algebra-definition-small-extension}).\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Smoothness over a Noetherian base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APN","source_file":"spaces-more-morphisms.tex","source_line":5095,"source_end_line":5117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5095-L5117","statement_sha256":"ea804eb9eea32ccc5be149c839fe582e2db9c2d8e697f96b90e5e293a48800a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12113,"rank":12113,"depth":59,"x":784.815,"y":1638.13,"cluster":"geometry-of-spaces"},{"id":"stacks:0APP","tag":"0APP","title":"Smoothness over a Noetherian base · Lemma 0APP","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume Y is locally Noetherian and f locally of finite type. The following are equivalent: • f is smooth, • for every solid commutative diagram xymatrix X ar[d]_f & Spec(B) ar[d]^i ar[l]^-α Y & Spec(B') ar[l]_-β ar@-->[lu] where B' → B is a small extension of Artinian local rings and β of finite type (!) there exists a dotted arrow making the diagram commute.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $Y$ is locally Noetherian and $f$ locally of finite type.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is smooth,\n\\item for every solid commutative diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & \\Spec(B) \\ar[d]^i \\ar[l]^-\\alpha \\\\\nY & \\Spec(B') \\ar[l]_-{\\beta} \\ar@{-->}[lu]\n}\n$$\nwhere $B' \\to B$ is a small extension of Artinian local rings\nand $\\beta$ of finite type (!) there exists a dotted arrow making\nthe diagram commute.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Smoothness over a Noetherian base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APP","source_file":"spaces-more-morphisms.tex","source_line":5160,"source_end_line":5178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5160-L5178","statement_sha256":"9db45f0b69b27f2ad84842d0e71c2ed6d4886c4acfcfe696294ebac260f97940","origin":"The Stacks Project","memory_eligible":false,"source_rank":12114,"rank":12114,"depth":60,"x":571.928,"y":1631.393,"cluster":"geometry-of-spaces"},{"id":"stacks:0APQ","tag":"0APQ","title":"Smoothness over a Noetherian base · Lemma 0APQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is locally of finite type and Y locally Noetherian. Let Z ⊂ Y be a closed subspace with nth infinitesimal neighbourhood Z_n ⊂ Y. Set X_n = Z_n ×_Y X. • If X_n → Z_n is smooth for all n, then f is smooth at every point of f^-1(Z). • If X_n → Z_n is étale for all n, then f is étale at every point of f^-1(Z).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $S$. Assume $f$ is\nlocally of finite type and $Y$ locally Noetherian.\nLet $Z \\subset Y$ be a closed subspace with $n$th infinitesimal\nneighbourhood $Z_n \\subset Y$. Set $X_n = Z_n \\times_Y X$.\n\\begin{enumerate}\n\\item If $X_n \\to Z_n$ is smooth for all $n$, then $f$\nis smooth at every point of $f^{-1}(Z)$.\n\\item If $X_n \\to Z_n$ is \\'etale for all $n$, then $f$\nis \\'etale at every point of $f^{-1}(Z)$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Smoothness over a Noetherian base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APQ","source_file":"spaces-more-morphisms.tex","source_line":5230,"source_end_line":5243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5230-L5243","statement_sha256":"14a5d5fb81d98e0714ec74811347e18c70d86d806f89b92d04290fd6f07ad720","origin":"The Stacks Project","memory_eligible":false,"source_rank":12115,"rank":12115,"depth":60,"x":734.506,"y":1515.435,"cluster":"geometry-of-spaces"},{"id":"stacks:0D0V","tag":"0D0V","title":"The naive cotangent complex · Definition 0D0V","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The naive cotangent complex of f is the complex defined in Modules on Sites, Definition [Tag 08U0] for the morphism of ringed topoi f_small between the small étale sites of X and Y, see Properties of Spaces, Lemma [Tag 03G8]. Notation: NL_f or NL_X/Y.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe {\\it naive cotangent complex of $f$}\nis the complex defined in Modules on Sites, Definition\n\\ref{sites-modules-definition-cotangent-complex-morphism-ringed-topoi}\nfor the morphism of ringed topoi $f_{small}$ between the\nsmall \\'etale sites of $X$ and $Y$, see\nProperties of Spaces, Lemma\n\\ref{spaces-properties-lemma-morphism-ringed-topoi}.\nNotation: $\\NL_f$ or $\\NL_{X/Y}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The naive cotangent complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0V","source_file":"spaces-more-morphisms.tex","source_line":5277,"source_end_line":5289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5277-L5289","statement_sha256":"50663e55da1afc52e710583f9901b24ea36743c6367103ae67476462726f23da","origin":"The Stacks Project","memory_eligible":false,"source_rank":12116,"rank":12116,"depth":11,"x":707.844,"y":1693.388,"cluster":"geometry-of-spaces"},{"id":"stacks:0D0W","tag":"0D0W","title":"The naive cotangent complex · Lemma 0D0W","summary":"Let S be a scheme. Consider a commutative diagram xymatrix U ar[d]_p ar[r]_g & V ar[d]^q X ar[r]^f & Y of algebraic spaces over S with p and q étale. Then there is a canonical identification NL_X/Y|_U_etale = NL_U/V in D(O_U).","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_p \\ar[r]_g & V \\ar[d]^q \\\\\nX \\ar[r]^f & Y\n}\n$$\nof algebraic spaces over $S$ with $p$ and $q$ \\'etale.\nThen there is a canonical identification\n$\\NL_{X/Y}|_{U_\\etale} = \\NL_{U/V}$ in $D(\\mathcal{O}_U)$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0W","source_file":"spaces-more-morphisms.tex","source_line":5296,"source_end_line":5308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5296-L5308","statement_sha256":"f97cfde8673a6a8b75edd94deb838739eadd746a700457f538b471925dd029b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12117,"rank":12117,"depth":49,"x":584.262,"y":1546.878,"cluster":"geometry-of-spaces"},{"id":"stacks:0D0X","tag":"0D0X","title":"The naive cotangent complex · Lemma 0D0X","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume X and Y representable by schemes X_0 and Y_0. Then there is a canonical identification NL_X/Y = ε^*NL_X_0/Y_0 in D(O_X) where ε is as in Derived Categories of Spaces, Section [Tag 071P] and NL_X_0/Y_0 is as in More on Morphisms, Definition [Tag 0D0H].","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$.\nThen there is a canonical identification\n$\\NL_{X/Y} = \\epsilon^*\\NL_{X_0/Y_0}$ in $D(\\mathcal{O}_X)$\nwhere $\\epsilon$ is as in Derived Categories of Spaces, Section\n\\ref{spaces-perfect-section-derived-quasi-coherent-etale}\nand $\\NL_{X_0/Y_0}$ is as in\nMore on Morphisms, Definition\n\\ref{more-morphisms-definition-netherlander}.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0X","source_file":"spaces-more-morphisms.tex","source_line":5323,"source_end_line":5334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5323-L5334","statement_sha256":"9766a4aded4f0bef9beb6139fec02ef0e3bf6935e49d626a1cd670161175b119","origin":"The Stacks Project","memory_eligible":false,"source_rank":12118,"rank":12118,"depth":53,"x":793.44,"y":1584.831,"cluster":"geometry-of-spaces"},{"id":"stacks:0D0Y","tag":"0D0Y","title":"The naive cotangent complex · Lemma 0D0Y","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The cohomology sheaves of the complex NL_X/Y are quasi-coherent, zero outside degrees -1, 0 and equal to Ω_X/Y in degree 0.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $S$. The cohomology sheaves\nof the complex $\\NL_{X/Y}$ are quasi-coherent, zero outside\ndegrees $-1$, $0$ and equal to $\\Omega_{X/Y}$ in degree $0$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0Y","source_file":"spaces-more-morphisms.tex","source_line":5397,"source_end_line":5403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5397-L5403","statement_sha256":"a26aa254a2bb6123cf6997a26b174ae36ed628110d1f2176c261f6ae31793d29","origin":"The Stacks Project","memory_eligible":false,"source_rank":12119,"rank":12119,"depth":54,"x":608.471,"y":1675.637,"cluster":"geometry-of-spaces"},{"id":"stacks:0D0Z","tag":"0D0Z","title":"The naive cotangent complex · Lemma 0D0Z","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is locally of finite presentation, then NL_X/Y is étale locally on X quasi-isomorphic to a complex … → 0 → F^-1 → F^0 → 0 → … of quasi-coherent O_X-modules with F^0 of finite presentation and F^-1 of finite type.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is locally of finite\npresentation, then $\\NL_{X/Y}$ is \\'etale locally on $X$\nquasi-isomorphic to a complex\n$$\n\\ldots \\to 0 \\to \\mathcal{F}^{-1} \\to \\mathcal{F}^0 \\to 0 \\to \\ldots\n$$\nof quasi-coherent $\\mathcal{O}_X$-modules\nwith $\\mathcal{F}^0$ of finite presentation\nand $\\mathcal{F}^{-1}$ of finite type.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D0Z","source_file":"spaces-more-morphisms.tex","source_line":5425,"source_end_line":5438,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5425-L5438","statement_sha256":"ae7597f899e6952c5ce24f1a28555d461b852ddb4bf1e4a32082e8b6d4449eaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12120,"rank":12120,"depth":54,"x":671.91,"y":1503.533,"cluster":"geometry-of-spaces"},{"id":"stacks:0D10","tag":"0D10","title":"The naive cotangent complex · Lemma 0D10","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f is formally smooth, • H^-1(NL_X/Y) = 0 and H^0(NL_X/Y) = Ω_X/Y is locally projective.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is formally smooth,\n\\item $H^{-1}(\\NL_{X/Y}) = 0$ and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$\nis locally projective.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D10","source_file":"spaces-more-morphisms.tex","source_line":5450,"source_end_line":5460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5450-L5460","statement_sha256":"41c15a576f662b46c7484af7b87d5d0342558c6cda850872a21fbc6e47589ea7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12121,"rank":12121,"depth":60,"x":763.633,"y":1666.616,"cluster":"geometry-of-spaces"},{"id":"stacks:0D11","tag":"0D11","title":"The naive cotangent complex · Lemma 0D11","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent • f is formally étale, • H^-1(NL_X/Y) = H^0(NL_X/Y) = 0.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is formally \\'etale,\n\\item $H^{-1}(\\NL_{X/Y}) = H^0(\\NL_{X/Y}) = 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D11","source_file":"spaces-more-morphisms.tex","source_line":5471,"source_end_line":5478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5471-L5478","statement_sha256":"b573ba341023d30111ba0e110ec764e3b6ba17c880f6a8a858e030ae25d3ac58","origin":"The Stacks Project","memory_eligible":false,"source_rank":12122,"rank":12122,"depth":61,"x":564.634,"y":1598.332,"cluster":"geometry-of-spaces"},{"id":"stacks:0D12","tag":"0D12","title":"The naive cotangent complex · Lemma 0D12","summary":"Let f : X → Y be a morphism of schemes. The following are equivalent • f is smooth, and • f is locally of finite presentation, H^-1(NL_X/Y) = 0, and H^0(NL_X/Y) = Ω_X/Y is finite locally free.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is smooth, and\n\\item $f$ is locally of finite presentation,\n$H^{-1}(\\NL_{X/Y}) = 0$, and $H^0(\\NL_{X/Y}) = \\Omega_{X/Y}$\nis finite locally free.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D12","source_file":"spaces-more-morphisms.tex","source_line":5494,"source_end_line":5503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5494-L5503","statement_sha256":"b544269a7e01eee4438a309518f1935cac6e1bf73a7c9311dc39702ff58a86ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":12123,"rank":12123,"depth":60,"x":766.505,"y":1535.698,"cluster":"geometry-of-spaces"},{"id":"stacks:05WV","tag":"05WV","title":"Openness of the flat locus · Theorem 05WV","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf on X. Assume f is locally of finite presentation and that F is an O_X-module which is locally of finite presentation. Then (x ∈ |X| : F is flat over Y at x) is open in |X|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nAssume $f$ is locally of finite presentation and that\n$\\mathcal{F}$ is an $\\mathcal{O}_X$-module which is\nlocally of finite presentation. Then\n$$\n\\{x \\in |X| : \\mathcal{F}\\text{ is flat over }Y\\text{ at }x\\}\n$$\nis open in $|X|$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Openness of the flat locus","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WV","source_file":"spaces-more-morphisms.tex","source_line":5534,"source_end_line":5546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5534-L5546","statement_sha256":"23c9ba5af55467f7420e0b56c7c053596b5a35e0dbf41bbb4e87ed45a03cc789","origin":"The Stacks Project","memory_eligible":false,"source_rank":12124,"rank":12124,"depth":56,"x":667.909,"y":1696.605,"cluster":"geometry-of-spaces"},{"id":"stacks:05WW","tag":"05WW","title":"Openness of the flat locus · Lemma 05WW","summary":"Let S be a scheme. Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian diagram of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Assume g is flat, f is locally of finite presentation, and F is locally of finite presentation. Then (x' ∈ |X'| : (g')^*F is flat over Y' at x') is the inverse image of the open subset of Theorem [Tag 05WV] under the continuous map |g'| : |X'| → |X|.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian diagram of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume $g$ is flat, $f$ is locally of finite presentation,\nand $\\mathcal{F}$ is locally of finite presentation.\nThen\n$$\n\\{x' \\in |X'| : (g')^*\\mathcal{F}\\text{ is flat over }Y'\\text{ at }x'\\}\n$$\nis the inverse image of the open subset of\nTheorem \\ref{theorem-openness-flatness}\nunder the continuous map $|g'| : |X'| \\to |X|$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Openness of the flat locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WW","source_file":"spaces-more-morphisms.tex","source_line":5571,"source_end_line":5591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5571-L5591","statement_sha256":"fbc0b5089c1f796e02821648e0043ee3218d98689294a00bbec7ecc976bb4cea","origin":"The Stacks Project","memory_eligible":false,"source_rank":12125,"rank":12125,"depth":57,"x":611.154,"y":1521.82,"cluster":"geometry-of-spaces"},{"id":"stacks:05WY","tag":"05WY","title":"Crit\\`ere de platitude par fibres · Lemma 05WY","summary":"In the situation above the following are equivalent • Pick a geometric point overlinex of X lying over x. Set overliney = f ∘ overlinex and overlinez = g ∘ overlinex. Then the module F_overlinex/ m_overlinezF_overlinex is flat over O_Y, overliney/ m_overlinezO_Y, overliney. • Pick a morphism x : Spec(K) → X in the equivalence class of x. Set z = g ∘ x, X_z = Spec(K) ×_z, Z X, Y_z = Spec(K) ×_z, Z Y, and F_z the pullback of F to X_z. Then F_z is flat at x over Y_z (as…","statement_latex":"In the situation above the following are equivalent\n\\begin{enumerate}\n\\item Pick a geometric point $\\overline{x}$ of $X$ lying over $x$.\nSet $\\overline{y} = f \\circ \\overline{x}$ and\n$\\overline{z} = g \\circ \\overline{x}$. Then the module\n$\\mathcal{F}_{\\overline{x}}/\n\\mathfrak m_{\\overline{z}}\\mathcal{F}_{\\overline{x}}$\nis flat over\n$\\mathcal{O}_{Y, \\overline{y}}/\n\\mathfrak m_{\\overline{z}}\\mathcal{O}_{Y, \\overline{y}}$.\n\\item Pick a morphism $x : \\Spec(K) \\to X$ in the equivalence class of\n$x$. Set $z = g \\circ x$, $X_z = \\Spec(K) \\times_{z, Z} X$,\n$Y_z = \\Spec(K) \\times_{z, Z} Y$, and $\\mathcal{F}_z$ the pullback\nof $\\mathcal{F}$ to $X_z$. Then $\\mathcal{F}_z$ is flat at $x$ over\n$Y_z$ (as defined in Morphisms of Spaces,\nDefinition \\ref{spaces-morphisms-definition-flat-module}).\n\\item Pick a commutative diagram\n$$\n\\xymatrix{\n& & & U \\ar[llld]_a \\ar[rr] \\ar[dr] & & V \\ar[llld]_>>>>>>>b \\ar[dl] \\\\\nX \\ar[rr]_f \\ar[dr]_g & & Y \\ar[dl]^h &  & W \\ar[llld]_c \\\\\n& Z\n}\n$$\nwhere $U, V, W$ are schemes, and $a, b, c$ are \\'etale,\nand a point $u \\in U$ mapping to $x$. Let $w \\in W$ be the image of\n$u$. Let $\\mathcal{F}_w$ be the pullback of $\\mathcal{F}$ to\nthe fibre $U_w$ of $U \\to W$ at $w$. Then $\\mathcal{F}_w$\nis flat over $V_w$ at $u$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WY","source_file":"spaces-more-morphisms.tex","source_line":5631,"source_end_line":5663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5631-L5663","statement_sha256":"33de7afe383457df996dbd93b2c4f43392cf48eb0c9f6a3024c58d8ddbe38cb2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12126,"rank":12126,"depth":56,"x":793.76,"y":1618.604,"cluster":"geometry-of-spaces"},{"id":"stacks:05WZ","tag":"05WZ","title":"Crit\\`ere de platitude par fibres · Definition 05WZ","summary":"Let S be a scheme. Let X → Y → Z be morphisms of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Let x ∈ |X| be a point and denote z ∈ |Z| its image. • We say the restriction of F to its fibre over z is flat at x over the fibre of Y over z if the equivalent conditions of Lemma [Tag 05WY] are satisfied. • We say the fibre of X over z is flat at x over the fibre of Y over z if the equivalent conditions of Lemma [Tag 05WY] hold with F = O_X. • We say the fibre…","statement_latex":"Let $S$ be a scheme. Let $X \\to Y \\to Z$ be morphisms of algebraic\nspaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in |X|$ be a point and denote $z \\in |Z|$ its image.\n\\begin{enumerate}\n\\item We say {\\it the restriction of $\\mathcal{F}$ to its fibre over $z$\nis flat at $x$ over the fibre of $Y$ over $z$} if the equivalent conditions of\nLemma \\ref{lemma-flat-on-fibres-at-point}\nare satisfied.\n\\item We say {\\it the fibre of $X$ over $z$ is flat at $x$ over the fibre of\n$Y$ over $z$} if the equivalent conditions of\nLemma \\ref{lemma-flat-on-fibres-at-point}\nhold with $\\mathcal{F} = \\mathcal{O}_X$.\n\\item We say {\\it the fibre of $X$ over $z$ is flat over the fibre of $Y$\nover $z$} if for all $x \\in |X|$ lying over $z$ the fibre of $X$ over $z$\nis flat at $x$ over the fibre of $Y$ over $z$\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WZ","source_file":"spaces-more-morphisms.tex","source_line":5789,"source_end_line":5807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5789-L5807","statement_sha256":"37c4fc0a2c8efe89b934368c6ee41a39da9a84f029c596ce128c60443e2c9362","origin":"The Stacks Project","memory_eligible":false,"source_rank":12127,"rank":12127,"depth":57,"x":581.048,"y":1650.886,"cluster":"geometry-of-spaces"},{"id":"stacks:05X0","tag":"05X0","title":"Crit\\`ere de platitude par fibres · Theorem 05X0","summary":"Let S be a scheme. Let f : X → Y and Y → Z be morphisms of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Assume • X is locally of finite presentation over Z, • F an O_X-module of finite presentation, and • Y is locally of finite type over Z. Let x ∈ |X| and let y ∈ |Y| and z ∈ |Z| be the images of x. If F_overlinex not = 0, then the following are equivalent: • F is flat over Z at x and the restriction of F to its fibre over z is flat at x over the fibre…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite presentation over $Z$,\n\\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation, and\n\\item $Y$ is locally of finite type over $Z$.\n\\end{enumerate}\nLet $x \\in |X|$ and let $y \\in |Y|$ and $z \\in |Z|$ be the images of\n$x$. If $\\mathcal{F}_{\\overline{x}} \\not = 0$, then the following are\nequivalent:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat over $Z$ at $x$ and\nthe restriction of $\\mathcal{F}$ to its fibre over $z$\nis flat at $x$ over the fibre of $Y$ over $z$, and\n\\item $Y$ is flat over $Z$ at $y$ and $\\mathcal{F}$ is\nflat over $Y$ at $x$.\n\\end{enumerate}\nMoreover, the set of points $x$ where (1) and (2) hold is open in\n$\\text{Supp}(\\mathcal{F})$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05X0","source_file":"spaces-more-morphisms.tex","source_line":5814,"source_end_line":5837,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5814-L5837","statement_sha256":"3538595c1034560b0578d078924b735f2347a4a5e966aa30ae368e603092541e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12128,"rank":12128,"depth":57,"x":712.079,"y":1506.228,"cluster":"geometry-of-spaces"},{"id":"stacks:05X1","tag":"05X1","title":"Crit\\`ere de platitude par fibres · Lemma 05X1","summary":"Let S be a scheme. Let f : X → Y and Y → Z be a morphism of algebraic spaces over S. Assume • X is locally of finite presentation over Z, • X is flat over Z, • for every z ∈ |Z| the fibre of X over z is flat over the fibre of Y over z, and • Y is locally of finite type over Z. Then f is flat. If f is also surjective, then Y is flat over Z.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $Y \\to Z$ be a morphism of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite presentation over $Z$,\n\\item $X$ is flat over $Z$,\n\\item for every $z \\in |Z|$ the fibre of $X$ over $z$\nis flat over the fibre of $Y$ over $z$, and\n\\item $Y$ is locally of finite type over $Z$.\n\\end{enumerate}\nThen $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $Z$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05X1","source_file":"spaces-more-morphisms.tex","source_line":5851,"source_end_line":5864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5851-L5864","statement_sha256":"42c205ea8ae0013cb8a632d7c126c86accc821c4c511b2f34c4f1449817066f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12129,"rank":12129,"depth":58,"x":731.81,"y":1687.442,"cluster":"geometry-of-spaces"},{"id":"stacks:05X2","tag":"05X2","title":"Crit\\`ere de platitude par fibres · Lemma 05X2","summary":"Let S be a scheme. Let f : X → Y and Y → Z be morphisms of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Assume • X is locally of finite presentation over Z, • F an O_X-module of finite presentation, • F is flat over Z, and • Y is locally of finite type over Z. Then the set A = (x ∈ |X| : F flat at x over Y). is open in |X| and its formation commutes with arbitrary base change: If Z' → Z is a morphism of algebraic spaces, and A' is the set of points of X'…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $Y \\to Z$ be morphisms of\nalgebraic spaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite presentation over $Z$,\n\\item $\\mathcal{F}$ an $\\mathcal{O}_X$-module of finite presentation,\n\\item $\\mathcal{F}$ is flat over $Z$, and\n\\item $Y$ is locally of finite type over $Z$.\n\\end{enumerate}\nThen the set\n$$\nA = \\{x \\in |X| : \\mathcal{F} \\text{ flat at }x \\text{ over }Y\\}.\n$$\nis open in $|X|$ and its formation commutes with arbitrary base change:\nIf $Z' \\to Z$ is a morphism of algebraic spaces, and $A'$ is the set of\npoints of $X' = X \\times_Z Z'$ where $\\mathcal{F}' = \\mathcal{F} \\times_Z Z'$\nis flat over $Y' = Y \\times_Z Z'$, then $A'$ is the inverse image of\n$A$ under the continuous map $|X'| \\to |X|$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05X2","source_file":"spaces-more-morphisms.tex","source_line":5871,"source_end_line":5892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5871-L5892","statement_sha256":"8bf36b240ab919087619a68cb684770d1f563615ac770a3a8456df6f62fbc692","origin":"The Stacks Project","memory_eligible":false,"source_rank":12130,"rank":12130,"depth":57,"x":571.361,"y":1564.882,"cluster":"geometry-of-spaces"},{"id":"stacks:05X3","tag":"05X3","title":"Crit\\`ere de platitude par fibres · Lemma 05X3","summary":"Let S be a scheme. Let f : X → Y and Y → Z be a morphism of algebraic spaces over S. Assume • X is locally of finite presentation over Z, • X is flat over Z, and • Y is locally of finite type over Z. Then the set (x ∈ |X| : X flat at x over Y). is open in |X| and its formation commutes with arbitrary base change Z' → Z.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $Y \\to Z$ be a morphism of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $X$ is locally of finite presentation over $Z$,\n\\item $X$ is flat over $Z$, and\n\\item $Y$ is locally of finite type over $Z$.\n\\end{enumerate}\nThen the set\n$$\n\\{x \\in |X| : X\\text{ flat at }x \\text{ over }Y\\}.\n$$\nis open in $|X|$ and its formation commutes with arbitrary base change\n$Z' \\to Z$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05X3","source_file":"spaces-more-morphisms.tex","source_line":5916,"source_end_line":5932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5916-L5932","statement_sha256":"83d9bb40ad9ad36dbea5a3cbe713dec7f021f424abcae8df6e5048c8d678d156","origin":"The Stacks Project","memory_eligible":false,"source_rank":12131,"rank":12131,"depth":58,"x":788.466,"y":1564.213,"cluster":"geometry-of-spaces"},{"id":"stacks:0CZS","tag":"0CZS","title":"Crit\\`ere de platitude par fibres · Lemma 0CZS","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite presentation. Let F be a finitely presented O_X-module. Let x ∈ |X| with image y ∈ |Y|. If F is flat at x over Y, then the following are equivalent • (F_overliney)_overlinex is a flat O_X_overliney, overlinex-module, • (F_overliney)_overlinex is a free O_X_overliney, overlinex-module, • F_overliney is finite free in an étale neighbourhood of overlinex in X_overliney, and •…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is locally of finite presentation.\nLet $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module.\nLet $x \\in |X|$ with image $y \\in |Y|$. If $\\mathcal{F}$ is flat at $x$\nover $Y$, then the following are equivalent\n\\begin{enumerate}\n\\item $(\\mathcal{F}_{\\overline{y}})_{\\overline{x}}$ is a flat\n$\\mathcal{O}_{X_{\\overline{y}}, \\overline{x}}$-module,\n\\item $(\\mathcal{F}_{\\overline{y}})_{\\overline{x}}$ is a free\n$\\mathcal{O}_{X_{\\overline{y}}, \\overline{x}}$-module,\n\\item $\\mathcal{F}_{\\overline{y}}$ is finite free in an\n\\'etale neighbourhood of $\\overline{x}$ in $X_{\\overline{y}}$, and\n\\item $\\mathcal{F}$ is finite free in an \\'etale neighbourhood of $x$ in $X$.\n\\end{enumerate}\nHere $\\overline{x}$ is a geometric point of $X$ lying over $x$\nand $\\overline{y} = f \\circ \\overline{x}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZS","source_file":"spaces-more-morphisms.tex","source_line":5939,"source_end_line":5957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5939-L5957","statement_sha256":"60346ffdf19f61a46b8b0e59269c469620cf34042d0de7f1f0a631e4057bfa9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12132,"rank":12132,"depth":57,"x":628.743,"y":1688.029,"cluster":"geometry-of-spaces"},{"id":"stacks:0CZT","tag":"0CZT","title":"Crit\\`ere de platitude par fibres · Lemma 0CZT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite presentation. Let F be a finitely presented O_X-module flat over Y. Then the set (x ∈ |X| : F free in an étale neighbourhood of x) is open in |X| and its formation commutes with arbitrary base change Y' → Y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is locally of finite presentation.\nLet $\\mathcal{F}$ be a finitely presented $\\mathcal{O}_X$-module\nflat over $Y$. Then the set\n$$\n\\{x \\in |X| : \\mathcal{F}\\text{ free in an \\'etale neighbourhood of }x\\}\n$$\nis open in $|X|$ and its formation commutes with arbitrary base change\n$Y' \\to Y$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Crit\\`ere de platitude par fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CZT","source_file":"spaces-more-morphisms.tex","source_line":5988,"source_end_line":5999,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L5988-L5999","statement_sha256":"479de9d287b29ac17c71ee329adfbc1bfd980950d2791cce6c775222c4778d04","origin":"The Stacks Project","memory_eligible":false,"source_rank":12133,"rank":12133,"depth":58,"x":646.971,"y":1505.904,"cluster":"geometry-of-spaces"},{"id":"stacks:0APR","tag":"0APR","title":"Flatness over a Noetherian base · Theorem 0APR","summary":"Let S be a scheme. Let f : X → Y and Y → Z be morphisms of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Assume • X, Y, Z locally Noetherian, and • F a coherent O_X-module. Let x ∈ |X| and let y ∈ |Y| and z ∈ |Z| be the images of x. If F_overlinex not = 0, then the following are equivalent: • F is flat over Z at x and the restriction of F to its fibre over z is flat at x over the fibre of Y over z, and • Y is flat over Z at y and F is flat over Y at x.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $Y \\to Z$ be morphisms of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $X$, $Y$, $Z$ locally Noetherian, and\n\\item $\\mathcal{F}$ a coherent $\\mathcal{O}_X$-module.\n\\end{enumerate}\nLet $x \\in |X|$ and let $y \\in |Y|$ and $z \\in |Z|$ be the images of\n$x$. If $\\mathcal{F}_{\\overline{x}} \\not = 0$, then the following are\nequivalent:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat over $Z$ at $x$ and\nthe restriction of $\\mathcal{F}$ to its fibre over $z$\nis flat at $x$ over the fibre of $Y$ over $z$, and\n\\item $Y$ is flat over $Z$ at $y$ and $\\mathcal{F}$ is\nflat over $Y$ at $x$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Flatness over a Noetherian base","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APR","source_file":"spaces-more-morphisms.tex","source_line":6031,"source_end_line":6051,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6031-L6051","statement_sha256":"62e8014bd12693ddd3046dd87d5f2b32a1cb2956513d6d390794c1b4593cc43b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12134,"rank":12134,"depth":57,"x":780.131,"y":1650.697,"cluster":"geometry-of-spaces"},{"id":"stacks:0APS","tag":"0APS","title":"Flatness over a Noetherian base · Lemma 0APS","summary":"Let S be a scheme. Let f : X → Y and Y → Z be a morphism of algebraic spaces over S. Assume • X, Y, Z locally Noetherian, • X is flat over Z, • for every z ∈ |Z| the fibre of X over z is flat over the fibre of Y over z. Then f is flat. If f is also surjective, then Y is flat over Z.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $Y \\to Z$ be a morphism of algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $X$, $Y$, $Z$ locally Noetherian,\n\\item $X$ is flat over $Z$,\n\\item for every $z \\in |Z|$ the fibre of $X$ over $z$\nis flat over the fibre of $Y$ over $z$.\n\\end{enumerate}\nThen $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $Z$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Flatness over a Noetherian base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APS","source_file":"spaces-more-morphisms.tex","source_line":6065,"source_end_line":6077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6065-L6077","statement_sha256":"6765bc2a60369be68eeffbec835aea5cdbf113f0d7444c5aafd6887d69f36c2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12135,"rank":12135,"depth":58,"x":565.274,"y":1619.454,"cluster":"geometry-of-spaces"},{"id":"stacks:08VP","tag":"08VP","title":"Flatness over a Noetherian base · Lemma 08VP","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let X be an algebraic space locally of finite presentation over S = Spec(A). For n ≥ 1 set S_n = Spec(A/I^n) and X_n = S_n ×_S X. Let F be coherent O_X-module. If for every n ≥ 1 the pullback F_n of F to X is flat over S_n, then the (open) locus where F is flat over X contains the inverse image of V(I) under X → S.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nLet $X$ be an algebraic space locally of finite presentation over\n$S = \\Spec(A)$. For $n \\geq 1$ set $S_n = \\Spec(A/I^n)$ and\n$X_n = S_n \\times_S X$. Let $\\mathcal{F}$ be coherent $\\mathcal{O}_X$-module.\nIf for every $n \\geq 1$ the pullback $\\mathcal{F}_n$ of $\\mathcal{F}$ to $X$\nis flat over $S_n$, then the (open) locus where $\\mathcal{F}$\nis flat over $X$ contains the inverse image of $V(I)$ under $X \\to S$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Flatness over a Noetherian base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VP","source_file":"spaces-more-morphisms.tex","source_line":6088,"source_end_line":6097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6088-L6097","statement_sha256":"c615c05bd7620b8cf930f69c9757801f8f5afe28f0660efba6a7f9e333c1c56a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12136,"rank":12136,"depth":57,"x":749.025,"y":1520.473,"cluster":"geometry-of-spaces"},{"id":"stacks:082E","tag":"082E","title":"Normalization revisited · Lemma 082E","summary":"Let S be a scheme. Let f : Y → X be a smooth morphism of algebraic spaces over S. Let A be a quasi-coherent sheaf of O_X-algebras. The integral closure of O_Y in f^*A is equal to f^*A' where A' ⊂ A is the integral closure of O_X in A.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a smooth morphism of\nalgebraic spaces over $S$. Let $\\mathcal{A}$ be a quasi-coherent\nsheaf of $\\mathcal{O}_X$-algebras. The integral closure\nof $\\mathcal{O}_Y$ in $f^*\\mathcal{A}$ is equal to $f^*\\mathcal{A}'$\nwhere $\\mathcal{A}' \\subset \\mathcal{A}$ is the integral closure of\n$\\mathcal{O}_X$ in $\\mathcal{A}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Normalization revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082E","source_file":"spaces-more-morphisms.tex","source_line":6121,"source_end_line":6129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6121-L6129","statement_sha256":"d7705b299830f935b28418db86186806e0e695f46042898c5da02e2fe866f6f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12137,"rank":12137,"depth":48,"x":693.069,"y":1697.912,"cluster":"geometry-of-spaces"},{"id":"stacks:082F","tag":"082F","title":"Normalization commutes with smooth base change · Lemma 082F","summary":"Let S be a scheme. Let xymatrix Y_2 ar[r] ar[d] & Y_1 ar[d]^f X_2 ar[r]^φ & X_1 be a fibre square of algebraic spaces over S. Assume f is quasi-compact and quasi-separated and φ is smooth. Let Y_i → X_i' → X_i be the normalization of X_i in Y_i. Then X_2' ≅ X_2 ×_X_1 X_1'.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nY_2 \\ar[r] \\ar[d] & Y_1 \\ar[d]^f \\\\\nX_2 \\ar[r]^\\varphi & X_1\n}\n$$\nbe a fibre square of algebraic spaces over $S$. Assume $f$ is quasi-compact\nand quasi-separated and $\\varphi$ is smooth.\nLet $Y_i \\to X_i' \\to X_i$ be the normalization of $X_i$ in $Y_i$.\nThen $X_2' \\cong X_2 \\times_{X_1} X_1'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Normalization revisited","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082F","source_file":"spaces-more-morphisms.tex","source_line":6140,"source_end_line":6153,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6140-L6153","statement_sha256":"d4f279db979dcfd6f6d483e8f7f814422d686ae6fbb51b422be39c618904f1ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":12138,"rank":12138,"depth":60,"x":591.531,"y":1535.149,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0U","tag":"0E0U","title":"Cohen-Macaulay morphisms · Lemma 0E0U","summary":"The property of morphisms of germs of schemes & P((X, x) → (S, s)) = & the local ring O_X_s, x of the fibre is Noetherian and Cohen-Macaulay is étale local on the source-and-target (Descent, Definition [Tag 04NB]).","statement_latex":"The property of morphisms of germs of schemes\n\\begin{align*}\n& \\mathcal{P}((X, x) \\to (S, s)) = \\\\\n& \\text{the local ring }\n\\mathcal{O}_{X_s, x}\n\\text{ of the fibre is Noetherian and Cohen-Macaulay}\n\\end{align*}\nis \\'etale local on the source-and-target (Descent, Definition\n\\ref{descent-definition-local-source-target-at-point}).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0U","source_file":"spaces-more-morphisms.tex","source_line":6192,"source_end_line":6203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6192-L6203","statement_sha256":"19b17d670c31f36fa33ffe66db35b8bf1369d9deb2463ea556bc8dba92a15e33","origin":"The Stacks Project","memory_eligible":false,"source_rank":12139,"rank":12139,"depth":50,"x":797.511,"y":1597.619,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0V","tag":"0E0V","title":"Cohen-Macaulay morphisms · Definition 0E0V","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume the fibres of f are locally Noetherian (Divisors on Spaces, Definition [Tag 0CUX]). • Let x ∈ |X|, and y = f(x). We say that f is Cohen-Macaulay at x if f is flat at x and the equivalent conditions of Morphisms of Spaces, Lemma [Tag 04NC] hold for the property P described in Lemma [Tag 0E0U]. • We say f is a Cohen-Macaulay morphism if f is Cohen-Macaulay at every point of X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume the fibres of $f$ are locally Noetherian\n(Divisors on Spaces, Definition\n\\ref{spaces-divisors-definition-locally-Noetherian-fibre}).\n\\begin{enumerate}\n\\item Let $x \\in |X|$, and $y = f(x)$. We say that $f$ is\n{\\it Cohen-Macaulay at $x$} if $f$ is flat at $x$ and\nthe equivalent conditions of\nMorphisms of Spaces, Lemma\n\\ref{spaces-morphisms-lemma-local-source-target-at-point}\nhold for the property $\\mathcal{P}$ described in\nLemma \\ref{lemma-CM-local-ring-fibre}.\n\\item We say $f$ is a {\\it Cohen-Macaulay morphism} if $f$ is\nCohen-Macaulay at every point of $X$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Cohen-Macaulay morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0V","source_file":"spaces-more-morphisms.tex","source_line":6217,"source_end_line":6235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6217-L6235","statement_sha256":"891ca91a0bf3df6f1f1ce8f2d02cca1a24b7f2d812e07f39e44f941f1b179904","origin":"The Stacks Project","memory_eligible":false,"source_rank":12140,"rank":12140,"depth":51,"x":595.175,"y":1668.506,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0W","tag":"0E0W","title":"Cohen-Macaulay morphisms · Lemma 0E0W","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume the fibres of f are locally Noetherian. The following are equivalent • f is Cohen-Macaulay, • f is flat and for some surjective étale morphism V → Y where V is a scheme, the fibres of X_V → V are Cohen-Macaulay algebraic spaces, and • f is flat and for any étale morphism V → Y where V is a scheme, the fibres of X_V → V are Cohen-Macaulay algebraic spaces. Given x ∈ |X| with image y ∈ |Y| the…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume the fibres of $f$ are locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is Cohen-Macaulay,\n\\item $f$ is flat and for some surjective \\'etale morphism $V \\to Y$\nwhere $V$ is a scheme, the fibres of $X_V \\to V$\nare Cohen-Macaulay algebraic spaces, and\n\\item $f$ is flat and for any \\'etale morphism $V \\to Y$\nwhere $V$ is a scheme, the fibres of $X_V \\to V$\nare Cohen-Macaulay algebraic spaces.\n\\end{enumerate}\nGiven $x \\in |X|$ with image $y \\in |Y|$ the following are\nequivalent\n\\begin{enumerate}\n\\item[(a)] $f$ is Cohen-Macaulay at $x$, and\n\\item[(b)] $\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$\nis flat and\n$\\mathcal{O}_{X, \\overline{x}}/\n\\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}}$ is Cohen-Macaulay.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0W","source_file":"spaces-more-morphisms.tex","source_line":6240,"source_end_line":6263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6240-L6263","statement_sha256":"f00c50f177dde008296be6def784acbb462edf5ca2f072fec4c1641c9401c8b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12141,"rank":12141,"depth":54,"x":687.465,"y":1501.25,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0X","tag":"0E0X","title":"Cohen-Macaulay morphisms · Lemma 0E0X","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of algebraic spaces over S. Assume that the fibres of f, g, and g ∘ f are locally Noetherian. Let x ∈ |X| with images y ∈ |Y| and z ∈ |Z|. • If f is Cohen-Macaulay at x and g is Cohen-Macaulay at f(x), then g ∘ f is Cohen-Macaulay at x. • If f and g are Cohen-Macaulay, then g ∘ f is Cohen-Macaulay. • If g ∘ f is Cohen-Macaulay at x and f is flat at x, then f is Cohen-Macaulay at x and g is Cohen-Macaulay at f(x).…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces\nover $S$. Assume that the\nfibres of $f$, $g$, and $g \\circ f$ are locally Noetherian.\nLet $x \\in |X|$ with images $y \\in |Y|$ and $z \\in |Z|$.\n\\begin{enumerate}\n\\item If $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay\nat $f(x)$, then $g \\circ f$ is Cohen-Macaulay at $x$.\n\\item If $f$ and $g$ are Cohen-Macaulay, then $g \\circ f$ is Cohen-Macaulay.\n\\item If $g \\circ f$ is Cohen-Macaulay at $x$ and $f$ is flat at $x$,\nthen $f$ is Cohen-Macaulay at $x$ and $g$ is Cohen-Macaulay at $f(x)$.\n\\item If $f \\circ g$ is Cohen-Macaulay and $f$ is flat, then\n$f$ is Cohen-Macaulay and $g$ is Cohen-Macaulay at every point in\nthe image of $f$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0X","source_file":"spaces-more-morphisms.tex","source_line":6310,"source_end_line":6327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6310-L6327","statement_sha256":"19afbdc24c96ce6c53ddded16b9691113ecbcef38f3ae46f16254de16621ac56","origin":"The Stacks Project","memory_eligible":false,"source_rank":12142,"rank":12142,"depth":55,"x":753.985,"y":1677.133,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0Y","tag":"0E0Y","title":"Cohen-Macaulay morphisms · Lemma 0E0Y","summary":"Let S be a scheme. Let f : X → Y be a flat morphism of locally Noetherian algebraic spaces over S. If X is Cohen-Macaulay, then f is Cohen-Macaulay and O_Y, f(overlinex) is Cohen-Macaulay for all x ∈ |X|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a flat morphism of locally Noetherian\nalgebraic spaces over $S$.\nIf $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and\n$\\mathcal{O}_{Y, f(\\overline{x})}$ is Cohen-Macaulay for all $x \\in |X|$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0Y","source_file":"spaces-more-morphisms.tex","source_line":6351,"source_end_line":6358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6351-L6358","statement_sha256":"14f0a7b1f360d8f78b93cc5515531458058e01efa33eb5469951f663bcfcfdd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12143,"rank":12143,"depth":56,"x":563.295,"y":1585.088,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0Z","tag":"0E0Z","title":"Cohen-Macaulay morphisms · Lemma 0E0Z","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume the fibres of f are locally Noetherian. Let Y' → Y be locally of finite type. Let f' : X' → Y' be the base change of f. Let x' ∈ |X'| be a point with image x ∈ |X|. • If f is Cohen-Macaulay at x, then f' : X' → Y' is Cohen-Macaulay at x'. • If f is flat at x and f' is Cohen-Macaulay at x', then f is Cohen-Macaulay at x. • If Y' → Y is flat at f'(x') and f' is Cohen-Macaulay at x', then f is…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume the fibres of $f$ are locally Noetherian.\nLet $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$\nbe the base change of $f$.\nLet $x' \\in |X'|$ be a point with image $x \\in |X|$.\n\\begin{enumerate}\n\\item If $f$ is Cohen-Macaulay at $x$, then\n$f' : X' \\to Y'$ is Cohen-Macaulay at $x'$.\n\\item If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$\nis Cohen-Macaulay at $x$.\n\\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at\n$x'$, then $f$ is Cohen-Macaulay at $x$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0Z","source_file":"spaces-more-morphisms.tex","source_line":6367,"source_end_line":6383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6367-L6383","statement_sha256":"5fb6426da110078b486d4ac95fdc04ec154dd159404d2ca57ea02df2273eff11","origin":"The Stacks Project","memory_eligible":false,"source_rank":12144,"rank":12144,"depth":33,"x":778.15,"y":1544.717,"cluster":"geometry-of-spaces"},{"id":"stacks:0E10","tag":"0E10","title":"Cohen-Macaulay morphisms · Lemma 0E10","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and locally of finite presentation. Let W = (x ∈ |X| : f is Cohen-Macaulay at x) Then W is open in |X| and the formation of W commutes with arbitrary base change of f: For any morphism g : Y' → Y, consider the base change f' : X' → Y' of f and the projection g' : X' → X. Then the corresponding set W' for the morphism f' is equal to W' = (g')^-1(W).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is flat and locally of finite presentation. Let\n$$\nW = \\{x \\in |X| : f\\text{ is Cohen-Macaulay at }x\\}\n$$\nThen $W$ is open in $|X|$ and the formation of $W$\ncommutes with arbitrary base change of $f$:\nFor any morphism $g : Y' \\to Y$, consider\nthe base change $f' : X' \\to Y'$ of $f$ and the\nprojection $g' : X' \\to X$. Then the corresponding\nset $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E10","source_file":"spaces-more-morphisms.tex","source_line":6402,"source_end_line":6416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6402-L6416","statement_sha256":"cc62291df5a3598071d3d9a58fe79023d6dc4cdf34875510c9a11f8aa0e680cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12145,"rank":12145,"depth":37,"x":652.053,"y":1696.558,"cluster":"geometry-of-spaces"},{"id":"stacks:0E11","tag":"0E11","title":"Cohen-Macaulay morphisms · Lemma 0E11","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume that f is locally of finite presentation and Cohen-Macaulay. Then there exist open and closed subschemes X_d ⊂ X such that X = coprod_d ≥ 0 X_d and f|_X_d : X_d → Y has relative dimension d.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a\nmorphism of algebraic spaces over $S$. Assume that\n$f$ is locally of finite presentation and Cohen-Macaulay.\nThen there exist open and closed subschemes $X_d \\subset X$\nsuch that $X = \\coprod_{d \\geq 0} X_d$ and $f|_{X_d} : X_d \\to Y$\nhas relative dimension $d$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E11","source_file":"spaces-more-morphisms.tex","source_line":6435,"source_end_line":6443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6435-L6443","statement_sha256":"34711eed438f1fc7384f113f7950cc0a81b17f54645497b6e664ffee4e76263f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12146,"rank":12146,"depth":37,"x":622.899,"y":1512.852,"cluster":"geometry-of-spaces"},{"id":"stacks:0E13","tag":"0E13","title":"Gorenstein morphisms · Lemma 0E13","summary":"The property of morphisms of germs of schemes & P((X, x) → (S, s)) = & the local ring O_X_s, x of the fibre is Noetherian and Gorenstein is étale local on the source-and-target (Descent, Definition [Tag 04NB]).","statement_latex":"The property of morphisms of germs of schemes\n\\begin{align*}\n& \\mathcal{P}((X, x) \\to (S, s)) = \\\\\n& \\text{the local ring }\n\\mathcal{O}_{X_s, x}\n\\text{ of the fibre is Noetherian and Gorenstein}\n\\end{align*}\nis \\'etale local on the source-and-target (Descent, Definition\n\\ref{descent-definition-local-source-target-at-point}).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E13","source_file":"spaces-more-morphisms.tex","source_line":6483,"source_end_line":6494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6483-L6494","statement_sha256":"e125fe6c4fe1484b8d4430f0e6b557578ef521e6718b001e904181057f007be4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12147,"rank":12147,"depth":46,"x":792.303,"y":1631.894,"cluster":"geometry-of-spaces"},{"id":"stacks:0E14","tag":"0E14","title":"Gorenstein morphisms · Definition 0E14","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume the fibres of f are locally Noetherian (Divisors on Spaces, Definition [Tag 0CUX]). • Let x ∈ |X|, and y = f(x). We say that f is Gorenstein at x if f is flat at x and the equivalent conditions of Morphisms of Spaces, Lemma [Tag 04NC] hold for the property P described in Lemma [Tag 0E13]. • We say f is a Gorenstein morphism if f is Gorenstein at every point of X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume the fibres of $f$ are locally Noetherian\n(Divisors on Spaces, Definition\n\\ref{spaces-divisors-definition-locally-Noetherian-fibre}).\n\\begin{enumerate}\n\\item Let $x \\in |X|$, and $y = f(x)$. We say that $f$ is\n{\\it Gorenstein at $x$} if $f$ is flat at $x$ and\nthe equivalent conditions of\nMorphisms of Spaces, Lemma\n\\ref{spaces-morphisms-lemma-local-source-target-at-point}\nhold for the property $\\mathcal{P}$ described in\nLemma \\ref{lemma-gorenstein-local-ring-fibre}.\n\\item We say $f$ is a {\\it Gorenstein morphism} if $f$ is\nGorenstein at every point of $X$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Gorenstein morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E14","source_file":"spaces-more-morphisms.tex","source_line":6514,"source_end_line":6532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6514-L6532","statement_sha256":"b7b0a597af12dc9243e7181c4df43c4bee801bbefbe9bd487a065197c5e054af","origin":"The Stacks Project","memory_eligible":false,"source_rank":12148,"rank":12148,"depth":47,"x":571.429,"y":1640.247,"cluster":"geometry-of-spaces"},{"id":"stacks:0E15","tag":"0E15","title":"Gorenstein morphisms · Lemma 0E15","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume the fibres of f are locally Noetherian. The following are equivalent • f is Gorenstein, • f is flat and for some surjective étale morphism V → Y where V is a scheme, the fibres of X_V → V are Gorenstein algebraic spaces, and • f is flat and for any étale morphism V → Y where V is a scheme, the fibres of X_V → V are Gorenstein algebraic spaces. Given x ∈ |X| with image y ∈ |Y| the following…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume the fibres of $f$ are locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is Gorenstein,\n\\item $f$ is flat and for some surjective \\'etale morphism $V \\to Y$\nwhere $V$ is a scheme, the fibres of $X_V \\to V$\nare Gorenstein algebraic spaces, and\n\\item $f$ is flat and for any \\'etale morphism $V \\to Y$\nwhere $V$ is a scheme, the fibres of $X_V \\to V$\nare Gorenstein algebraic spaces.\n\\end{enumerate}\nGiven $x \\in |X|$ with image $y \\in |Y|$ the following are\nequivalent\n\\begin{enumerate}\n\\item[(a)] $f$ is Gorenstein at $x$, and\n\\item[(b)] $\\mathcal{O}_{Y, \\overline{y}} \\to \\mathcal{O}_{X, \\overline{x}}$\nis flat and\n$\\mathcal{O}_{X, \\overline{x}}/\n\\mathfrak m_{\\overline{y}}\\mathcal{O}_{X, \\overline{x}}$ is Gorenstein.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E15","source_file":"spaces-more-morphisms.tex","source_line":6537,"source_end_line":6560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6537-L6560","statement_sha256":"f58164c173322f41b2e353d5bd0666b291f54a9f042476cbd6a88cca2218405c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12149,"rank":12149,"depth":54,"x":727.742,"y":1508.623,"cluster":"geometry-of-spaces"},{"id":"stacks:0E16","tag":"0E16","title":"Gorenstein morphisms · Lemma 0E16","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of algebraic spaces over S. Assume that the fibres of f, g, and g ∘ f are locally Noetherian. Let x ∈ |X| with images y ∈ |Y| and z ∈ |Z|. • If f is Gorenstein at x and g is Gorenstein at f(x), then g ∘ f is Gorenstein at x. • If f and g are Gorenstein, then g ∘ f is Gorenstein. • If g ∘ f is Gorenstein at x and f is flat at x, then f is Gorenstein at x and g is Gorenstein at f(x). • If f ∘ g is Gorenstein and f…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of algebraic spaces\nover $S$. Assume that the\nfibres of $f$, $g$, and $g \\circ f$ are locally Noetherian.\nLet $x \\in |X|$ with images $y \\in |Y|$ and $z \\in |Z|$.\n\\begin{enumerate}\n\\item If $f$ is Gorenstein at $x$ and $g$ is Gorenstein\nat $f(x)$, then $g \\circ f$ is Gorenstein at $x$.\n\\item If $f$ and $g$ are Gorenstein, then $g \\circ f$ is Gorenstein.\n\\item If $g \\circ f$ is Gorenstein at $x$ and $f$ is flat at $x$,\nthen $f$ is Gorenstein at $x$ and $g$ is Gorenstein at $f(x)$.\n\\item If $f \\circ g$ is Gorenstein and $f$ is flat, then\n$f$ is Gorenstein and $g$ is Gorenstein at every point in\nthe image of $f$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E16","source_file":"spaces-more-morphisms.tex","source_line":6611,"source_end_line":6628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6611-L6628","statement_sha256":"2d0e4ccb20f71c5ddbfbb4fe84f0f2f60b064a063a198aabd4127feb93e6b449","origin":"The Stacks Project","memory_eligible":false,"source_rank":12150,"rank":12150,"depth":55,"x":718.318,"y":1694.567,"cluster":"geometry-of-spaces"},{"id":"stacks:0E17","tag":"0E17","title":"Gorenstein morphisms · Lemma 0E17","summary":"Let S be a scheme. Let f : X → Y be a flat morphism of locally Noetherian algebraic spaces over S. If X is Gorenstein, then f is Gorenstein and O_Y, f(overlinex) is Gorenstein for all x ∈ |X|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a flat morphism of locally Noetherian\nalgebraic spaces over $S$.\nIf $X$ is Gorenstein, then $f$ is Gorenstein and\n$\\mathcal{O}_{Y, f(\\overline{x})}$ is Gorenstein for all $x \\in |X|$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E17","source_file":"spaces-more-morphisms.tex","source_line":6653,"source_end_line":6660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6653-L6660","statement_sha256":"73e92b0782bd402f952ff59cb9a24eff5c6ec6077cbd447e323f68a3e239cdb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12151,"rank":12151,"depth":56,"x":575.59,"y":1551.961,"cluster":"geometry-of-spaces"},{"id":"stacks:0E18","tag":"0E18","title":"Gorenstein morphisms · Lemma 0E18","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume the fibres of f are locally Noetherian. Let Y' → Y be locally of finite type. Let f' : X' → Y' be the base change of f. Let x' ∈ |X'| be a point with image x ∈ |X|. • If f is Gorenstein at x, then f' : X' → Y' is Gorenstein at x'. • If f is flat at x and f' is Gorenstein at x', then f is Gorenstein at x. • If Y' → Y is flat at f'(x') and f' is Gorenstein at x', then f is Gorenstein at x.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume the fibres of $f$ are locally Noetherian.\nLet $Y' \\to Y$ be locally of finite type. Let $f' : X' \\to Y'$\nbe the base change of $f$.\nLet $x' \\in |X'|$ be a point with image $x \\in |X|$.\n\\begin{enumerate}\n\\item If $f$ is Gorenstein at $x$, then\n$f' : X' \\to Y'$ is Gorenstein at $x'$.\n\\item If $f$ is flat at $x$ and $f'$ is Gorenstein at $x'$, then $f$\nis Gorenstein at $x$.\n\\item If $Y' \\to Y$ is flat at $f'(x')$ and $f'$ is Gorenstein at\n$x'$, then $f$ is Gorenstein at $x$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E18","source_file":"spaces-more-morphisms.tex","source_line":6669,"source_end_line":6685,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6669-L6685","statement_sha256":"c01ab274446e3204a9a5a57275410245186f2bdae95d1800fa91df0558aae1b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12152,"rank":12152,"depth":44,"x":795.741,"y":1576.154,"cluster":"geometry-of-spaces"},{"id":"stacks:0E19","tag":"0E19","title":"Gorenstein morphisms · Lemma 0E19","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and locally of finite presentation. Let W = (x ∈ |X| : f is Gorenstein at x) Then W is open in |X| and the formation of W commutes with arbitrary base change of f: For any morphism g : Y' → Y, consider the base change f' : X' → Y' of f and the projection g' : X' → X. Then the corresponding set W' for the morphism f' is equal to W' = (g')^-1(W).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is flat and locally of finite presentation. Let\n$$\nW = \\{x \\in |X| : f\\text{ is Gorenstein at }x\\}\n$$\nThen $W$ is open in $|X|$ and the formation of $W$\ncommutes with arbitrary base change of $f$:\nFor any morphism $g : Y' \\to Y$, consider\nthe base change $f' : X' \\to Y'$ of $f$ and the\nprojection $g' : X' \\to X$. Then the corresponding\nset $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Gorenstein morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E19","source_file":"spaces-more-morphisms.tex","source_line":6704,"source_end_line":6718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6704-L6718","statement_sha256":"df3edf32d43831cd84bb255b738f587c31d25986d2556bbe5d0891398ac6de94","origin":"The Stacks Project","memory_eligible":false,"source_rank":12153,"rank":12153,"depth":68,"x":613.763,"y":1683.343,"cluster":"geometry-of-spaces"},{"id":"stacks:06LW","tag":"06LW","title":"Slicing Cohen-Macaulay morphisms · Lemma 06LW","summary":"Let S be a scheme. Consider a cartesian diagram xymatrix X ar[d] & F ar[l]^p ar[d] Y & Spec(k) ar[l] where X → Y is a morphism of algebraic spaces over S which is flat and locally of finite presentation, and where k is a field over S. Let f_1, …, f_r ∈ Γ(X, O_X) and z ∈ |F| such that f_1, …, f_r map to a regular sequence in the local ring O_F, overlinez. Then, after replacing X by an open subspace containing p(z), the morphism V(f_1, …, f_r) → Y is flat and locally of…","statement_latex":"Let $S$ be a scheme. Consider a cartesian diagram\n$$\n\\xymatrix{\nX \\ar[d] & F \\ar[l]^p \\ar[d] \\\\\nY & \\Spec(k) \\ar[l]\n}\n$$\nwhere $X \\to Y$ is a morphism of algebraic spaces over $S$\nwhich is flat and locally of finite presentation, and where\n$k$ is a field over $S$. Let $f_1, \\ldots, f_r \\in \\Gamma(X, \\mathcal{O}_X)$\nand $z \\in |F|$ such that $f_1, \\ldots, f_r$ map to a regular sequence\nin the local ring $\\mathcal{O}_{F, \\overline{z}}$.\nThen, after replacing $X$ by an open subspace containing $p(z)$, the morphism\n$$\nV(f_1, \\ldots, f_r) \\longrightarrow Y\n$$\nis flat and locally of finite presentation.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Slicing Cohen-Macaulay morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06LW","source_file":"spaces-more-morphisms.tex","source_line":6766,"source_end_line":6785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6766-L6785","statement_sha256":"d0217333a8c2196b2e873f736fc9a4cab4f7726026c05e4f95b1d02861349118","origin":"The Stacks Project","memory_eligible":false,"source_rank":12154,"rank":12154,"depth":56,"x":661.802,"y":1500.859,"cluster":"geometry-of-spaces"},{"id":"stacks:0E07","tag":"0E07","title":"Reduced fibres · Lemma 0E07","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let y ∈ |Y|. The following are equivalent • for some morphism Spec(k) → Y in the equivalence class of y the algebraic space X_k is geometrically reduced over k, • for every morphism Spec(k) → Y in the equivalence class of y the algebraic space X_k is geometrically reduced over k, • for every morphism Spec(k) → Y in the equivalence class of y the algebraic space X_k is reduced.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a\nmorphism of algebraic spaces over $S$. Let $y \\in |Y|$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some morphism $\\Spec(k) \\to Y$ in the equivalence class\nof $y$ the algebraic space $X_k$ is geometrically reduced over $k$,\n\\item for every morphism $\\Spec(k) \\to Y$ in the equivalence class\nof $y$ the algebraic space $X_k$ is geometrically reduced over $k$,\n\\item for every morphism $\\Spec(k) \\to Y$ in the equivalence class\nof $y$ the algebraic space $X_k$ is reduced.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E07","source_file":"spaces-more-morphisms.tex","source_line":6880,"source_end_line":6893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6880-L6893","statement_sha256":"818cca97c8d6fceb1334fd943bff85e80bd9236c1ade8ed06bef4868708e1ba5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12155,"rank":12155,"depth":55,"x":773.24,"y":1662.843,"cluster":"geometry-of-spaces"},{"id":"stacks:0E08","tag":"0E08","title":"Reduced fibres · Definition 0E08","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let y ∈ |Y|. We say the fibre of f : X → Y at y is geometrically reduced if the equivalent conditions of Lemma [Tag 0E07] hold.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a\nmorphism of algebraic spaces over $S$. Let $y \\in |Y|$.\nWe say {\\it the fibre of $f : X \\to Y$ at $y$ is geometrically reduced}\nif the equivalent conditions of\nLemma \\ref{lemma-geometrically-reduced-fibre} hold.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Reduced fibres","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E08","source_file":"spaces-more-morphisms.tex","source_line":6903,"source_end_line":6910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6903-L6910","statement_sha256":"c499454420e7acca976166ad41b24cad42328b9613b7ea5f2cb81276ab778569","origin":"The Stacks Project","memory_eligible":false,"source_rank":12156,"rank":12156,"depth":56,"x":560.589,"y":1606.574,"cluster":"geometry-of-spaces"},{"id":"stacks:0E09","tag":"0E09","title":"Reduced fibres · Lemma 0E09","summary":"Let S be a scheme. Let f : X → Y and g : Y' → Y be morphisms of algebraic spaces over S. Denote f' : X' → Y' the base change of f by g. Then (y' ∈ |Y'| : the fibre of f' : X' → Y' at y' is geometrically reduced) = g^-1((y ∈ |Y| : the fibre of f : X → Y at y is geometrically reduced)).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y' \\to Y$\nbe morphisms of algebraic spaces over $S$. Denote\n$f' : X' \\to Y'$ the base change of $f$ by $g$. Then\n\\begin{align*}\n\\{y' \\in |Y'| :\n\\text{the fibre of }f' : X' \\to Y'\\text{ at }y'\n\\text{ is geometrically reduced}\\} \\\\\n= g^{-1}(\\{y \\in |Y| :\n\\text{the fibre of }f : X \\to Y\\text{ at }y\n\\text{ is geometrically reduced}\\}).\n\\end{align*}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E09","source_file":"spaces-more-morphisms.tex","source_line":6915,"source_end_line":6928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6915-L6928","statement_sha256":"bc0058aa0f515173a2e5b5762096eb80016c123670d433786e1b173f960def73","origin":"The Stacks Project","memory_eligible":false,"source_rank":12157,"rank":12157,"depth":0,"x":762.849,"y":1527.322,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0A","tag":"0E0A","title":"Reduced fibres · Lemma 0E0A","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is quasi-compact and locally of finite presentation. Then the set E = (y ∈ |Y| : the fibre of f : X → Y at y is geometrically reduced) is étale locally constructible.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is quasi-compact and\nlocally of finite presentation. Then the set\n$$\nE = \\{y \\in |Y| : \\text{the fibre of }f : X \\to Y\\text{ at }y\n\\text{ is geometrically reduced}\\}\n$$\nis \\'etale locally constructible.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0A","source_file":"spaces-more-morphisms.tex","source_line":6938,"source_end_line":6948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6938-L6948","statement_sha256":"39e77274e1c0e37fad7fe687affca425c10e3dd01bae56f4598afa5055351d3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12158,"rank":12158,"depth":47,"x":677.352,"y":1700.704,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0B","tag":"0E0B","title":"Reduced fibres · Lemma 0E0B","summary":"Let X be an algebraic space over a discrete valuation ring R whose structure morphism X → Spec(R) is proper and flat. If the special fibre is reduced, then both X and the generic fibre X_eta are reduced.","statement_latex":"Let $X$ be an algebraic space over a discrete valuation ring $R$\nwhose structure morphism $X \\to \\Spec(R)$ is proper and flat.\nIf the special fibre is reduced, then\nboth $X$ and the generic fibre $X_\\eta$ are reduced.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0B","source_file":"spaces-more-morphisms.tex","source_line":6970,"source_end_line":6976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L6970-L6976","statement_sha256":"51223b7ead1119d4c8bcb82f4d07c967c4557d72330e344a02b7cbf602574811","origin":"The Stacks Project","memory_eligible":false,"source_rank":12159,"rank":12159,"depth":4,"x":600.89,"y":1524.163,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0C","tag":"0E0C","title":"Reduced fibres · Lemma 0E0C","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is flat, proper, and of finite presentation, then the set E = (y ∈ |Y| : the fibre of f : X → Y at y is geometrically reduced) is open in |Y|.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. If $f$ is flat, proper, and of finite presentation,\nthen the set\n$$\nE = \\{y \\in |Y| : \\text{the fibre of }f : X \\to Y\\text{ at }y\n\\text{ is geometrically reduced}\\}\n$$\nis open in $|Y|$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Reduced fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0C","source_file":"spaces-more-morphisms.tex","source_line":7004,"source_end_line":7014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7004-L7014","statement_sha256":"9c4b8479e38778d240724ef2dbfd1a6c297e855431ec063be4a28863c17cf361","origin":"The Stacks Project","memory_eligible":false,"source_rank":12160,"rank":12160,"depth":59,"x":799.44,"y":1611.043,"cluster":"geometry-of-spaces"},{"id":"stacks:0E1B","tag":"0E1B","title":"Connected components of fibres · Lemma 0E1B","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let n_X/Y : |Y| → (0, 1, 2, 3, …, ∞) be the function which associates to y ∈ Y the number of connected components of X_k where Spec(k) → Y is in the equivalence class of y with k algebraically closed. This is well defined and if g : Y' → Y is a morphism then n_X'/Y' = n_X/Y ∘ g where X' → Y' is the base change of f.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Let\n$$\nn_{X/Y} : |Y| \\to \\{0, 1, 2, 3, \\ldots, \\infty\\}\n$$\nbe the function which associates to $y \\in Y$ the number of connected\ncomponents of $X_k$ where $\\Spec(k) \\to Y$ is in the equivalence\nclass of $y$ with $k$ algebraically closed.\nThis is well defined and if $g : Y' \\to Y$ is a morphism then\n$$\nn_{X'/Y'} = n_{X/Y} \\circ g\n$$\nwhere $X' \\to Y'$ is the base change of $f$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Connected components of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1B","source_file":"spaces-more-morphisms.tex","source_line":7069,"source_end_line":7084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7069-L7084","statement_sha256":"bba2203a579a2bc810fbf46a89332e7faa3e55dc527ddf200bb60a874be4b758","origin":"The Stacks Project","memory_eligible":false,"source_rank":12161,"rank":12161,"depth":50,"x":582.949,"y":1659.69,"cluster":"geometry-of-spaces"},{"id":"stacks:0D4M","tag":"0D4M","title":"Dimension of fibres · Lemma 0D4M","summary":"Let S be a scheme. Let f : X → Y be a finite type morphism of algebraic spaces over S. Let y ∈ |Y|. The following quantities are the same • d = -∞ if y is not in the image of |f| and otherwise the minimal integer d such that f has relative dimension ≤ d at every x ∈ |X| mapping to y, • the dimension of the algebraic space X_k = Spec(k) ×_Y X for any morphism Spec(k) → Y in the equivalence class defining y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a finite type morphism of\nalgebraic spaces over $S$. Let $y \\in |Y|$. The following quantities\nare the same\n\\begin{enumerate}\n\\item $d = -\\infty$ if $y$ is not in the image of $|f|$ and\notherwise the minimal integer $d$ such that $f$ has relative dimension $\\leq d$\nat every $x \\in |X|$ mapping to $y$,\n\\item the dimension of the algebraic space $X_k = \\Spec(k) \\times_Y X$\nfor any morphism $\\Spec(k) \\to Y$ in the equivalence class defining $y$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4M","source_file":"spaces-more-morphisms.tex","source_line":7123,"source_end_line":7135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7123-L7135","statement_sha256":"006dd2513b964358c56a9342a1c49f1dc8efa39b621e5dddce21887d1f4fcc22","origin":"The Stacks Project","memory_eligible":false,"source_rank":12162,"rank":12162,"depth":48,"x":703.586,"y":1500.817,"cluster":"geometry-of-spaces"},{"id":"stacks:0D4N","tag":"0D4N","title":"Dimension of fibres · Lemma 0D4N","summary":"Let S be a scheme. Let f : X → Y be a finite type morphism of algebraic spaces over S. Let n_X/Y : |Y| → (-∞, 0, 1, 2, 3, …) be the function which associates to y ∈ |Y| the integer discussed in Lemma [Tag 0D4M]. If g : Y' → Y is a morphism then n_X'/Y' = n_X/Y ∘ |g| where X' → Y' is the base change of f.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a finite type morphism of\nalgebraic spaces over $S$. Let\n$$\nn_{X/Y} : |Y| \\to \\{-\\infty, 0, 1, 2, 3, \\ldots\\}\n$$\nbe the function which associates to $y \\in |Y|$ the\ninteger discussed in Lemma \\ref{lemma-dimension-fibre}.\nIf $g : Y' \\to Y$ is a morphism then\n$$\nn_{X'/Y'} = n_{X/Y} \\circ |g|\n$$\nwhere $X' \\to Y'$ is the base change of $f$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4N","source_file":"spaces-more-morphisms.tex","source_line":7182,"source_end_line":7196,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7182-L7196","statement_sha256":"f08db59b9c4ead5ff6efec89a00a5fcce27deeed37e754614e69c70331d6ba87","origin":"The Stacks Project","memory_eligible":false,"source_rank":12163,"rank":12163,"depth":49,"x":742.43,"y":1686.606,"cluster":"geometry-of-spaces"},{"id":"stacks:0D4P","tag":"0D4P","title":"Dimension of fibres · Lemma 0D4P","summary":"Let S be a scheme. Let f : X → Y be a flat morphism of finite presentation of algebraic spaces over S. Let n_X/Y be the function on Y giving the dimension of fibres of f introduced in Lemma [Tag 0D4N]. Then n_X/Y is lower semi-continuous.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a flat morphism of finite presentation of\nalgebraic spaces over $S$. Let\n$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$\nintroduced in Lemma \\ref{lemma-base-change-dimension-fibres}.\nThen $n_{X/Y}$ is lower semi-continuous.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4P","source_file":"spaces-more-morphisms.tex","source_line":7203,"source_end_line":7211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7203-L7211","statement_sha256":"40b42cf44c7c28d23d687974c56abfd5150100f593a74898044e34332612db67","origin":"The Stacks Project","memory_eligible":false,"source_rank":12164,"rank":12164,"depth":50,"x":564.203,"y":1571.534,"cluster":"geometry-of-spaces"},{"id":"stacks:0D4Q","tag":"0D4Q","title":"Dimension of fibres · Lemma 0D4Q","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of algebraic spaces over S. Let n_X/Y be the function on Y giving the dimension of fibres of f introduced in Lemma [Tag 0D4N]. Then n_X/Y is upper semi-continuous.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$. Let\n$n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$\nintroduced in Lemma \\ref{lemma-base-change-dimension-fibres}.\nThen $n_{X/Y}$ is upper semi-continuous.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4Q","source_file":"spaces-more-morphisms.tex","source_line":7226,"source_end_line":7233,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7226-L7233","statement_sha256":"94da31925c50b8f2ea731ba7f2067c388ccd2cb736a88d9cf431dc63f4f54f02","origin":"The Stacks Project","memory_eligible":false,"source_rank":12165,"rank":12165,"depth":50,"x":788.386,"y":1555.24,"cluster":"geometry-of-spaces"},{"id":"stacks:0D4R","tag":"0D4R","title":"Dimension of fibres · Lemma 0D4R","summary":"Let S be a scheme. Let f : X → Y be a proper, flat, finitely presented morphism of algebraic spaces over S. Let n_X/Y be the function on Y giving the dimension of fibres of f introduced in Lemma [Tag 0D4N]. Then n_X/Y is locally constant.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper, flat, finitely presented\nmorphism of algebraic spaces over $S$.\nLet $n_{X/Y}$ be the function on $Y$ giving the dimension of fibres of $f$\nintroduced in Lemma \\ref{lemma-base-change-dimension-fibres}.\nThen $n_{X/Y}$ is locally constant.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Dimension of fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4R","source_file":"spaces-more-morphisms.tex","source_line":7246,"source_end_line":7253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7246-L7253","statement_sha256":"e1b27c7d58c064bccd06392b6921e8a13fee70aa70a5e2584d44e9b08cf09303","origin":"The Stacks Project","memory_eligible":false,"source_rank":12166,"rank":12166,"depth":51,"x":636.029,"y":1694.6,"cluster":"geometry-of-spaces"},{"id":"stacks:0EDM","tag":"0EDM","title":"Catenary algebraic spaces · Lemma 0EDM","summary":"Let S be a scheme. Let f : X → Y be an integral morphism of algebraic spaces over S. Let y ∈ |Y| be a point which can be represented by a closed immersion y : Spec(k) → Y. Then there exists a factorization X → X' → Y of f such that • X' → Y is integral, • X → X' is an isomorphism over X' setminus X'_y, • X'_y has a unique point x' with kappa(x') = k. Moreover, if f is finite and Y is locally Noetherian, then X' → Y is finite.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be an integral morphism\nof algebraic spaces over $S$.\nLet $y \\in |Y|$ be a point which can be represented by a closed immersion\n$y : \\Spec(k) \\to Y$. Then there exists\na factorization $X \\to X' \\to Y$ of $f$ such that\n\\begin{enumerate}\n\\item $X' \\to Y$ is integral,\n\\item $X \\to X'$ is an isomorphism over $X' \\setminus X'_y$,\n\\item $X'_y$ has a unique point $x'$ with $\\kappa(x') = k$.\n\\end{enumerate}\nMoreover, if $f$ is finite and $Y$ is locally Noetherian, then\n$X' \\to Y$ is finite.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Catenary algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDM","source_file":"spaces-more-morphisms.tex","source_line":7273,"source_end_line":7287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7273-L7287","statement_sha256":"d008fbdd8bec8d6bfb758be9fe8ffeaa675d828313f9346abb8e3a329b57f30e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12167,"rank":12167,"depth":43,"x":636.306,"y":1505.199,"cluster":"geometry-of-spaces"},{"id":"stacks:0EDN","tag":"0EDN","title":"Catenary algebraic spaces · Lemma 0EDN","summary":"Let S be a scheme. Let B be an algebraic space over S. Let δ : |B| → Z be a function. Assume B is decent, locally Noetherian, and universally catenary and δ is a dimension function. If X is a decent algebraic space over B whose structure morphism f : X → B is locally of finite type we define δ_X : |X| → Z by the rule δ_X(x) = δ(f(x)) + transcendence degreeof x/f(x) (Morphisms of Spaces, Definition [Tag 04NM]). Then δ_X is a dimension function.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $\\delta : |B| \\to \\mathbf{Z}$ be a function.\nAssume $B$ is decent, locally Noetherian, and\nuniversally catenary and $\\delta$ is a dimension function.\nIf $X$ is a decent algebraic space over $B$ whose structure morphism\n$f : X \\to B$ is locally of finite type we define\n$\\delta_X : |X| \\to \\mathbf{Z}$ by the rule\n$$\n\\delta_X(x) = \\delta(f(x)) + \\text{transcendence degreeof }x/f(x)\n$$\n(Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-dimension-fibre}).\nThen $\\delta_X$ is a dimension function.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Catenary algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDN","source_file":"spaces-more-morphisms.tex","source_line":7348,"source_end_line":7363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7348-L7363","statement_sha256":"258d574a4e30fadc21ef2d0e8725c12929a67aa17a16efa8f886028d6d1fdf94","origin":"The Stacks Project","memory_eligible":false,"source_rank":12168,"rank":12168,"depth":67,"x":788.563,"y":1645.156,"cluster":"geometry-of-spaces"},{"id":"stacks:082H","tag":"082H","title":"Étale localization of morphisms · Lemma 082H","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let y ∈ |Y|. Let x_1, …, x_n ∈ |X| mapping to y. Assume that • f is locally of finite type, • f is separated, • f is quasi-finite at x_1, …, x_n, and • f is quasi-compact or Y is decent. Then there exists an étale morphism (U, u) → (Y, y) of pointed algebraic spaces and a decomposition U ×_Y X = W amalg V into open and closed subspaces such that the morphism V → U is finite, every point of the…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $y \\in |Y|$. Let $x_1, \\ldots, x_n \\in |X|$ mapping to $y$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item $f$ is separated,\n\\item $f$ is quasi-finite at $x_1, \\ldots, x_n$, and\n\\item $f$ is quasi-compact or $Y$ is decent.\n\\end{enumerate}\nThen there exists an \\'etale morphism $(U, u) \\to (Y, y)$\nof pointed algebraic spaces and a decomposition\n$$\nU \\times_Y X = W \\amalg V\n$$\ninto open and closed subspaces such that the morphism $V \\to U$ is finite,\nevery point of the fibre of $|V| \\to |U|$ over $u$ maps to an $x_i$,\nand the fibre of $|W| \\to |U|$ over $u$ contains no point mapping to an\n$x_i$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Étale localization of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082H","source_file":"spaces-more-morphisms.tex","source_line":7471,"source_end_line":7492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7471-L7492","statement_sha256":"c41ddc076656c645def6f6e3970cfaa9cd5bcafcc1daf0167026f85210332f08","origin":"The Stacks Project","memory_eligible":false,"source_rank":12169,"rank":12169,"depth":58,"x":563.519,"y":1628.332,"cluster":"geometry-of-spaces"},{"id":"stacks:0ADU","tag":"0ADU","title":"Étale localization of morphisms · Lemma 0ADU","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let x ∈ |X| with image y ∈ |Y|. Assume that • f is locally of finite type, • f is separated, and • f is quasi-finite at x. Then there exists an étale morphism (U, u) → (Y, y) of pointed algebraic spaces and a decomposition U ×_Y X = W amalg V into open and closed subspaces such that the morphism V → U is finite and there exists a point v ∈ |V| which maps to x in |X| and u in |U|.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $x \\in |X|$ with image $y \\in |Y|$. Assume that\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item $f$ is separated, and\n\\item $f$ is quasi-finite at $x$.\n\\end{enumerate}\nThen there exists an \\'etale morphism $(U, u) \\to (Y, y)$\nof pointed algebraic spaces and a decomposition\n$$\nU \\times_Y X = W \\amalg V\n$$\ninto open and closed subspaces such that the morphism $V \\to U$ is finite\nand there exists a point $v \\in |V|$ which maps to $x$ in $|X|$\nand $u$ in $|U|$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Étale localization of morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADU","source_file":"spaces-more-morphisms.tex","source_line":7543,"source_end_line":7561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7543-L7561","statement_sha256":"06ebeef0c289a96c053eee0f8d00b0f57bacf3e385037386cd8d6187c705301e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12170,"rank":12170,"depth":59,"x":743.169,"y":1512.929,"cluster":"geometry-of-spaces"},{"id":"stacks:082I","tag":"082I","title":"Zariski's Main Theorem · Lemma 082I","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is of finite type and separated. Let Y' be the normalization of Y in X. Picture: xymatrix X ar[rd]_f ar[rr]_f' & & Y' ar[ld]^ν & Y & Then there exists an open subspace U' ⊂ Y' such that • (f')^-1(U') → U' is an isomorphism, and • (f')^-1(U') ⊂ X is the set of points at which f is quasi-finite.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$ which is of finite type and separated.\nLet $Y'$ be the normalization of $Y$ in $X$. Picture:\n$$\n\\xymatrix{\nX \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\\n& Y &\n}\n$$\nThen there exists an open subspace $U' \\subset Y'$ such that\n\\begin{enumerate}\n\\item $(f')^{-1}(U') \\to U'$ is an isomorphism, and\n\\item $(f')^{-1}(U') \\subset X$ is the set of points at which\n$f$ is quasi-finite.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082I","source_file":"spaces-more-morphisms.tex","source_line":7592,"source_end_line":7609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7592-L7609","statement_sha256":"e658b1358a17565366df7c939343fa3f17178e63946d9b869f383d2c09f767d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12171,"rank":12171,"depth":61,"x":703.463,"y":1700.147,"cluster":"geometry-of-spaces"},{"id":"stacks:082J","tag":"082J","title":"Zariski's Main Theorem · Lemma 082J","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-finite and separated. Let Y' be the normalization of Y in X. Picture: xymatrix X ar[rd]_f ar[rr]_f' & & Y' ar[ld]^ν & Y & Then f' is a quasi-compact open immersion and ν is integral. In particular f is quasi-affine.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is quasi-finite and separated.\nLet $Y'$ be the normalization of $Y$ in $X$.\nPicture:\n$$\n\\xymatrix{\nX \\ar[rd]_f \\ar[rr]_{f'} & & Y' \\ar[ld]^\\nu \\\\\n& Y &\n}\n$$\nThen $f'$ is a quasi-compact open immersion and $\\nu$ is integral.\nIn particular $f$ is quasi-affine.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082J","source_file":"spaces-more-morphisms.tex","source_line":7723,"source_end_line":7738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7723-L7738","statement_sha256":"722c3146c24edcb44c379d3c8e1fc37fe4acc3ccd589879f50cd2a5dede2f461","origin":"The Stacks Project","memory_eligible":false,"source_rank":12172,"rank":12172,"depth":62,"x":582.067,"y":1539.407,"cluster":"geometry-of-spaces"},{"id":"stacks:082K","tag":"082K","title":"Zariski's Main Theorem · Lemma 082K","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is quasi-finite and separated and assume that Y is quasi-compact and quasi-separated. Then there exists a factorization xymatrix X ar[rd]_f ar[rr]_j & & T ar[ld]^π & Y & where j is a quasi-compact open immersion and π is finite.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is quasi-finite and separated and assume that\n$Y$ is quasi-compact and quasi-separated. Then there exists\na factorization\n$$\n\\xymatrix{\nX \\ar[rd]_f \\ar[rr]_j & & T \\ar[ld]^\\pi \\\\\n& Y &\n}\n$$\nwhere $j$ is a quasi-compact open immersion and $\\pi$ is finite.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082K","source_file":"spaces-more-morphisms.tex","source_line":7754,"source_end_line":7768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7754-L7768","statement_sha256":"cb36cb18114553d1fefc65466866ce8917beb216edd68fbfc90d04de39bf1180","origin":"The Stacks Project","memory_eligible":false,"source_rank":12173,"rank":12173,"depth":65,"x":801.06,"y":1589.1,"cluster":"geometry-of-spaces"},{"id":"stacks:0874","tag":"0874","title":"Zariski's Main Theorem · Lemma 0874","summary":"With notation and hypotheses as in Lemma [Tag 082K]. Assume moreover that f is locally of finite presentation. Then we can choose the factorization such that T is finite and of finite presentation over Y.","statement_latex":"With notation and hypotheses as in\nLemma \\ref{lemma-quasi-finite-separated-pass-through-finite}.\nAssume moreover that $f$ is locally of finite presentation. Then we can\nchoose the factorization such that $T$ is finite and of\nfinite presentation over $Y$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Zariski's Main Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0874","source_file":"spaces-more-morphisms.tex","source_line":7805,"source_end_line":7812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7805-L7812","statement_sha256":"94a2f0c6b680f0cf9be4ae36dcfd573e0704ed07ce578040db321f9f62a7028b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12174,"rank":12174,"depth":66,"x":599.419,"y":1676.804,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4X","tag":"0A4X","title":"Applications of Zariski's Main Theorem, I · Lemma 0A4X","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is finite, • f is proper and locally quasi-finite, • f is proper and |X_k| is a discrete space for every morphism Spec(k) → Y where k is a field, • f is universally closed, separated, locally of finite type and |X_k| is a discrete space for every morphism Spec(k) → Y where k is a field.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is finite,\n\\item $f$ is proper and locally quasi-finite,\n\\item $f$ is proper and $|X_k|$ is a discrete space for every morphism\n$\\Spec(k) \\to Y$ where $k$ is a field,\n\\item $f$ is universally closed, separated, locally of finite type\nand $|X_k|$ is a discrete space for every morphism $\\Spec(k) \\to Y$\nwhere $k$ is a field.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Applications of Zariski's Main Theorem, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4X","source_file":"spaces-more-morphisms.tex","source_line":7847,"source_end_line":7860,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7847-L7860","statement_sha256":"77887201305e31fb8c94d0c0fdb2a2bee861963cafa51771c007216d7f9be3ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":12175,"rank":12175,"depth":59,"x":677.653,"y":1497.542,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4Y","tag":"0A4Y","title":"Applications of Zariski's Main Theorem, I · Lemma 0A4Y","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let y ∈ |Y|. Assume • f is proper, and • f is quasi-finite at all x ∈ |X| lying over y (Decent Spaces, Lemma [Tag 0ACK]). Then there exists an open neighbourhood V ⊂ Y of y such that f|_f^-1(V) : f^-1(V) → V is finite.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $y \\in |Y|$. Assume\n\\begin{enumerate}\n\\item $f$ is proper, and\n\\item $f$ is quasi-finite at all $x \\in |X|$ lying over $y$\n(Decent Spaces, Lemma \\ref{decent-spaces-lemma-conditions-on-fibre-and-qf}).\n\\end{enumerate}\nThen there exists an open neighbourhood $V \\subset Y$ of $y$\nsuch that $f|_{f^{-1}(V)} : f^{-1}(V) \\to V$ is finite.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Applications of Zariski's Main Theorem, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4Y","source_file":"spaces-more-morphisms.tex","source_line":7905,"source_end_line":7916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7905-L7916","statement_sha256":"b2f9d6e088d400704758ede56b9365cde8aa0bfb5a6b1234c8480d9bc9f8e8bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12176,"rank":12176,"depth":60,"x":764.206,"y":1674.291,"cluster":"geometry-of-spaces"},{"id":"stacks:0AEJ","tag":"0AEJ","title":"Applications of Zariski's Main Theorem, I · Lemma 0AEJ","summary":"Collapsing a fibre of a proper family forces nearby ones to collapse too. Let S be a scheme. Let xymatrix X ar[rr]_h ar[rd]_f & & Y ar[ld]^g & B be a commutative diagram of morphism of algebraic spaces over S. Let b ∈ B and let Spec(k) → B be a morphism in the equivalence class of b. Assume • X → B is a proper morphism, • Y → B is separated and locally of finite type, • one of the following is true • the image of |X_k| → |Y_k| is finite, • the image of |f|^-1((b)) in |Y|…","statement_latex":"\\begin{slogan}\nCollapsing a fibre of a proper family forces nearby ones to collapse too.\n\\end{slogan}\nLet $S$ be a scheme. Let\n$$\n\\xymatrix{\nX \\ar[rr]_h \\ar[rd]_f & & Y \\ar[ld]^g \\\\\n& B\n}\n$$\nbe a commutative diagram of morphism of algebraic spaces over $S$.\nLet $b \\in B$ and let $\\Spec(k) \\to B$ be a morphism in the equivalence\nclass of $b$. Assume\n\\begin{enumerate}\n\\item $X \\to B$ is a proper morphism,\n\\item $Y \\to B$ is separated and locally of finite type,\n\\item one of the following is true\n\\begin{enumerate}\n\\item the image of $|X_k| \\to |Y_k|$ is finite,\n\\item the image of $|f|^{-1}(\\{b\\})$ in $|Y|$ is finite\nand $B$ is decent.\n\\end{enumerate}\n\\end{enumerate}\nThen there is an open\nsubspace $B' \\subset B$ containing $b$ such that $X_{B'} \\to Y_{B'}$\nfactors through a closed subspace $Z \\subset Y_{B'}$ finite over $B'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Applications of Zariski's Main Theorem, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEJ","source_file":"spaces-more-morphisms.tex","source_line":7930,"source_end_line":7958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7930-L7958","statement_sha256":"f269f066394d0559460f43535e3dc938d4be56a61e8d15c1d0ff342acfc49558","origin":"The Stacks Project","memory_eligible":false,"source_rank":12177,"rank":12177,"depth":61,"x":558.046,"y":1592.993,"cluster":"geometry-of-spaces"},{"id":"stacks:0A19","tag":"0A19","title":"Stein factorization · Lemma 0A19","summary":"Let S be a scheme. Let f : X → Y be a universally closed and quasi-separated morphism of algebraic spaces over S. There exists a factorization xymatrix X ar[rr]_f' ar[rd]_f & & Y' ar[dl]^π & Y & with the following properties: • the morphism f' is universally closed, quasi-compact, quasi-separated, and surjective, • the morphism π : Y' → Y is integral, • we have f'_*O_X = O_Y', • we have Y' = underlineSpec_Y(f_*O_X), and • Y' is the normalization of Y in X as defined in…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a universally closed and\nquasi-separated morphism of algebraic spaces over $S$.\nThere exists a factorization\n$$\n\\xymatrix{\nX \\ar[rr]_{f'} \\ar[rd]_f & & Y' \\ar[dl]^\\pi \\\\\n& Y &\n}\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item the morphism $f'$ is universally closed, quasi-compact, quasi-separated,\nand surjective,\n\\item the morphism $\\pi : Y' \\to Y$ is integral,\n\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{Y'}$,\n\\item we have $Y' = \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)$, and\n\\item $Y'$ is the normalization of $Y$ in $X$ as defined in\nMorphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-normalization-X-in-Y}.\n\\end{enumerate}\nFormation of the factorization $f = \\pi \\circ f'$ commutes with flat\nbase change.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A19","source_file":"spaces-more-morphisms.tex","source_line":7993,"source_end_line":8017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L7993-L8017","statement_sha256":"852f2569de02efd93208f446db625bc7805a7592b41cb639b2a3c1589a49e027","origin":"The Stacks Project","memory_eligible":false,"source_rank":12178,"rank":12178,"depth":59,"x":775.654,"y":1535.906,"cluster":"geometry-of-spaces"},{"id":"stacks:0E1C","tag":"0E1C","title":"Stein factorization · Lemma 0E1C","summary":"In Lemma [Tag 0A19] assume in addition that f is locally of finite type and Y affine. Then for y ∈ Y the fibre π^-1((y)) = (y_1, …, y_n) is finite and the field extensions kappa(y_i)/kappa(y) are finite.","statement_latex":"In Lemma \\ref{lemma-stein-universally-closed} assume in addition that\n$f$ is locally of finite type and $Y$ affine. Then for $y \\in Y$ the fibre\n$\\pi^{-1}(\\{y\\}) = \\{y_1, \\ldots, y_n\\}$ is finite and the field extensions\n$\\kappa(y_i)/\\kappa(y)$ are finite.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1C","source_file":"spaces-more-morphisms.tex","source_line":8060,"source_end_line":8066,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8060-L8066","statement_sha256":"2c13dbe77fd08f9d217a7fa34004bbfb2e16500215a8413114b0960df4499e88","origin":"The Stacks Project","memory_eligible":false,"source_rank":12179,"rank":12179,"depth":60,"x":660.991,"y":1701.637,"cluster":"geometry-of-spaces"},{"id":"stacks:0A1A","tag":"0A1A","title":"Stein factorization · Lemma 0A1A","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let overliney be a geometric point of Y. Then X_overliney is connected, if and only if for every étale neighbourhood (V, overlinev) → (Y, overliney) where V is a scheme the base change X_V → V has connected fibre X_v.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $\\overline{y}$ be a geometric point of $Y$. Then\n$X_{\\overline{y}}$ is connected, if and only if for every \\'etale\nneighbourhood $(V, \\overline{v}) \\to (Y, \\overline{y})$ where $V$\nis a scheme the base change $X_V \\to V$ has connected fibre $X_v$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1A","source_file":"spaces-more-morphisms.tex","source_line":8098,"source_end_line":8105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8098-L8105","statement_sha256":"04e16fa75aafd1c939edf84fd7b0abeac7d838a69bb517b65676dd26a68e48da","origin":"The Stacks Project","memory_eligible":false,"source_rank":12180,"rank":12180,"depth":52,"x":612.218,"y":1514.185,"cluster":"geometry-of-spaces"},{"id":"stacks:0A1B","tag":"0A1B","title":"Stein factorization; Noetherian case · Theorem 0A1B","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of algebraic spaces over S with Y locally Noetherian. There exists a factorization xymatrix X ar[rr]_f' ar[rd]_f & & Y' ar[dl]^π & Y & with the following properties: • the morphism f' is proper with connected geometric fibres, • the morphism π : Y' → Y is finite, • we have f'_*O_X = O_Y', • we have Y' = underlineSpec_Y(f_*O_X), and • Y' is the normalization of Y in X, see Morphisms, Definition [Tag 035H].","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic\nspaces over $S$ with $Y$ locally Noetherian.\nThere exists a factorization\n$$\n\\xymatrix{\nX \\ar[rr]_{f'} \\ar[rd]_f & & Y' \\ar[dl]^\\pi \\\\\n& Y &\n}\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item the morphism $f'$ is proper with connected geometric fibres,\n\\item the morphism $\\pi : Y' \\to Y$ is finite,\n\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{Y'}$,\n\\item we have $Y' = \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)$, and\n\\item $Y'$ is the normalization of $Y$ in $X$, see\nMorphisms, Definition \\ref{morphisms-definition-normalization-X-in-Y}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1B","source_file":"spaces-more-morphisms.tex","source_line":8126,"source_end_line":8146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8126-L8146","statement_sha256":"ea7a39d62a9ed906296709c08fad32ec0c97846ceae43d3914557e3f56920e87","origin":"The Stacks Project","memory_eligible":false,"source_rank":12181,"rank":12181,"depth":70,"x":799.106,"y":1624.841,"cluster":"geometry-of-spaces"},{"id":"stacks:0A1C","tag":"0A1C","title":"Stein factorization; general case · Theorem 0A1C","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of algebraic spaces over S. There exists a factorization xymatrix X ar[rr]_f' ar[rd]_f & & Y' ar[dl]^π & Y & with the following properties: • the morphism f' is proper with connected geometric fibres, • the morphism π : Y' → Y is integral, • we have f'_*O_X = O_Y', • we have Y' = underlineSpec_Y(f_*O_X), and • Y' is the normalization of Y in X (Morphisms of Spaces, Definition [Tag 0822]).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of algebraic\nspaces over $S$. There exists a factorization\n$$\n\\xymatrix{\nX \\ar[rr]_{f'} \\ar[rd]_f & & Y' \\ar[dl]^\\pi \\\\\n& Y &\n}\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item the morphism $f'$ is proper with connected geometric fibres,\n\\item the morphism $\\pi : Y' \\to Y$ is integral,\n\\item we have $f'_*\\mathcal{O}_X = \\mathcal{O}_{Y'}$,\n\\item we have $Y' = \\underline{\\Spec}_Y(f_*\\mathcal{O}_X)$, and\n\\item $Y'$ is the normalization of $Y$ in $X$ (Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-normalization-X-in-Y}).\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A1C","source_file":"spaces-more-morphisms.tex","source_line":8196,"source_end_line":8215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8196-L8215","statement_sha256":"3b1ff4ab62659a57a1578a2b933f93e8bddb473b5c2fd7cc7d2b6d5183798b04","origin":"The Stacks Project","memory_eligible":false,"source_rank":12182,"rank":12182,"depth":71,"x":572.093,"y":1649.313,"cluster":"geometry-of-spaces"},{"id":"stacks:0AYI","tag":"0AYI","title":"Stein factorization · Lemma 0AYI","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • f is proper, • Y is integral (Spaces over Fields, Definition [Tag 0AD4]) with generic point xi, • Y is normal, • X is reduced, • every generic point of an irreducible component of |X| maps to xi, • we have H^0(X_xi, O) = kappa(xi). Then f_*O_X = O_Y and f has geometrically connected fibres.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item $Y$ is integral (Spaces over Fields, Definition\n\\ref{spaces-over-fields-definition-integral-algebraic-space})\nwith generic point $\\xi$,\n\\item $Y$ is normal,\n\\item $X$ is reduced,\n\\item every generic point of an irreducible component of $|X|$ maps to $\\xi$,\n\\item we have $H^0(X_\\xi, \\mathcal{O}) = \\kappa(\\xi)$.\n\\end{enumerate}\nThen $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and $f$\nhas geometrically connected fibres.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYI","source_file":"spaces-more-morphisms.tex","source_line":8286,"source_end_line":8302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8286-L8302","statement_sha256":"db2f9d62f6197e858cf0398384efbde2725a59cbecb9530835c0d2612796bd30","origin":"The Stacks Project","memory_eligible":false,"source_rank":12183,"rank":12183,"depth":72,"x":719.951,"y":1502.314,"cluster":"geometry-of-spaces"},{"id":"stacks:0E1D","tag":"0E1D","title":"Stein factorization · Lemma 0E1D","summary":"Let S be a scheme. Let X → Y be a morphism of algebraic spaces over S. If f is proper, flat, and of finite presentation, then the function n_X/Y : |Y| → Z counting the number of geometric connected components of fibres of f (Lemma [Tag 0E1B]) is lower semi-continuous.","statement_latex":"Let $S$ be a scheme.\nLet $X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is proper, flat, and of finite presentation, then the function\n$n_{X/Y} : |Y| \\to \\mathbf{Z}$ counting the number of geometric\nconnected components of fibres of $f$\n(Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components})\nis lower semi-continuous.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1D","source_file":"spaces-more-morphisms.tex","source_line":8349,"source_end_line":8358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8349-L8358","statement_sha256":"00dbed4208802e0c0bfb8d8eabd3050b2e2177a8715584aa66d39e17d6eb0966","origin":"The Stacks Project","memory_eligible":false,"source_rank":12184,"rank":12184,"depth":72,"x":729.144,"y":1694.792,"cluster":"geometry-of-spaces"},{"id":"stacks:0E1E","tag":"0E1E","title":"Stein factorization · Lemma 0E1E","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume • f is proper, flat, and of finite presentation, and • the geometric fibres of f are reduced. Then the function n_X/S : |Y| → Z counting the numbers of geometric connected components of fibres of f (Lemma [Tag 0E1B]) is locally constant.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item $f$ is proper, flat, and of finite presentation, and\n\\item the geometric fibres of $f$ are reduced.\n\\end{enumerate}\nThen the function $n_{X/S} : |Y| \\to \\mathbf{Z}$\ncounting the numbers of geometric connected components\nof fibres of $f$\n(Lemma \\ref{lemma-base-change-fibres-nr-geometrically-connected-components})\nis locally constant.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1E","source_file":"spaces-more-morphisms.tex","source_line":8386,"source_end_line":8399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8386-L8399","statement_sha256":"16c923284353f45375b456bfeaa37eabcc83e29a4b6fc06d3cfacd2f08d0310c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12185,"rank":12185,"depth":73,"x":567.426,"y":1557.947,"cluster":"geometry-of-spaces"},{"id":"stacks:0E0D","tag":"0E0D","title":"Stein factorization · Lemma 0E0D","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of algebraic spaces over S. Let X → Y' → Y be the Stein factorization of f (Theorem [Tag 0A1C]). If f is of finite presentation, flat, with geometrically reduced fibres (Definition [Tag 0E08]), then Y' → Y is finite étale.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a proper morphism of algebraic spaces over $S$.\nLet $X \\to Y' \\to Y$ be the Stein factorization of $f$\n(Theorem \\ref{theorem-stein-factorization-general}).\nIf $f$ is of finite presentation, flat, with geometrically\nreduced fibres (Definition \\ref{definition-geometrically-reduced-fibre}),\nthen $Y' \\to Y$ is finite \\'etale.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0D","source_file":"spaces-more-morphisms.tex","source_line":8425,"source_end_line":8434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8425-L8434","statement_sha256":"2485ba3c43a5a2f76d0323a6fe835f25d5847756509b48af6e88c23a7de6ce9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12186,"rank":12186,"depth":74,"x":796.94,"y":1567.1,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWI","tag":"0CWI","title":"Stein factorization · Lemma 0CWI","summary":"Let (A, I) be a henselian pair. Let X be an algebraic space separated and of finite type over A. Set X_0 = X ×_Spec(A) Spec(A/I). Let Y ⊂ X_0 be an open and closed subspace such that Y → Spec(A/I) is proper. Then there exists an open and closed subspace W ⊂ X which is proper over A with W ×_Spec(A) Spec(A/I) = Y.","statement_latex":"Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space\nseparated and of finite type over $A$. Set\n$X_0 = X \\times_{\\Spec(A)} \\Spec(A/I)$.\nLet $Y \\subset X_0$ be an open and closed subspace such that\n$Y \\to \\Spec(A/I)$ is proper. Then there exists an open and closed\nsubspace $W \\subset X$ which is proper over $A$ with\n$W \\times_{\\Spec(A)} \\Spec(A/I) = Y$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Stein factorization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWI","source_file":"spaces-more-morphisms.tex","source_line":8482,"source_end_line":8491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8482-L8491","statement_sha256":"0e606e01e8ef38d99509d9e458f80a10761a25b2c2ac2f0ebbfdc2ada0ae6f58","origin":"The Stacks Project","memory_eligible":false,"source_rank":12187,"rank":12187,"depth":58,"x":620.171,"y":1690.7,"cluster":"geometry-of-spaces"},{"id":"stacks:0876","tag":"0876","title":"Extending properties from an open · Lemma 0876","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Let V ⊂ Y be an open subspace. Assume • f is locally of finite presentation, • F is of finite type and flat over Y, • V → Y is quasi-compact and scheme theoretically dense, • F|_f^-1V is of finite presentation. Then F is of finite presentation.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $V \\subset Y$ be an open subspace. Assume\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $\\mathcal{F}$ is of finite type and flat over $Y$,\n\\item $V \\to Y$ is quasi-compact and scheme theoretically dense,\n\\item $\\mathcal{F}|_{f^{-1}V}$ is of finite presentation.\n\\end{enumerate}\nThen $\\mathcal{F}$ is of finite presentation.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0876","source_file":"spaces-more-morphisms.tex","source_line":8540,"source_end_line":8553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8540-L8553","statement_sha256":"da8363f9db324fe5212ec67ce15656ace8e0f282c703f2ef5b986dacaaf4b621","origin":"The Stacks Project","memory_eligible":false,"source_rank":12188,"rank":12188,"depth":54,"x":651.151,"y":1499.075,"cluster":"geometry-of-spaces"},{"id":"stacks:0877","tag":"0877","title":"Extending properties from an open · Lemma 0877","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let V ⊂ Y be an open subspace. Assume • f is locally of finite type and flat, • V → Y is quasi-compact and scheme theoretically dense, • f|_f^-1V : f^-1V → V is locally of finite presentation. Then f is of locally of finite presentation.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $V \\subset Y$ be an open subspace.\nAssume\n\\begin{enumerate}\n\\item $f$ is locally of finite type and flat,\n\\item $V \\to Y$ is quasi-compact and scheme theoretically dense,\n\\item $f|_{f^{-1}V} : f^{-1}V \\to V$ is locally of finite presentation.\n\\end{enumerate}\nThen $f$ is of locally of finite presentation.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0877","source_file":"spaces-more-morphisms.tex","source_line":8568,"source_end_line":8580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8568-L8580","statement_sha256":"753d36bd559dea119000e321e8421a5f8bdd67672e21077d9600abc4548426ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":12189,"rank":12189,"depth":55,"x":782.531,"y":1658.106,"cluster":"geometry-of-spaces"},{"id":"stacks:0878","tag":"0878","title":"Extending properties from an open · Lemma 0878","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and locally of finite type. Let V ⊂ Y be an open subspace such that |V| ⊂ |Y| is dense and such that X_V → V has relative dimension ≤ d. If also either • f is locally of finite presentation, or • V → Y is quasi-compact, then f : X → Y has relative dimension ≤ d.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is flat and locally of finite type. Let $V \\subset Y$ be an\nopen subspace such that $|V| \\subset |Y|$ is dense and such that $X_V \\to V$\nhas relative dimension $\\leq d$. If also either\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation, or\n\\item $V \\to Y$ is quasi-compact,\n\\end{enumerate}\nthen $f : X \\to Y$ has relative dimension $\\leq d$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0878","source_file":"spaces-more-morphisms.tex","source_line":8590,"source_end_line":8601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8590-L8601","statement_sha256":"3e7a06c69f5b51736539bf4c9a967841c9f65f028234ada18f7e1bf34447522d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12190,"rank":12190,"depth":56,"x":557.552,"y":1615.346,"cluster":"geometry-of-spaces"},{"id":"stacks:0B4J","tag":"0B4J","title":"Extending properties from an open · Lemma 0B4J","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and proper. Let V → Y be an open subspace with |V| ⊂ |Y| dense such that X_V → V is finite. If also either f is locally of finite presentation or V → Y is quasi-compact, then f is finite.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is flat and proper. Let $V \\to Y$ be an open subspace\nwith $|V| \\subset |Y|$ dense such that $X_V \\to V$ is finite. If also\neither $f$ is locally of finite presentation or $V \\to Y$ is quasi-compact,\nthen $f$ is finite.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4J","source_file":"spaces-more-morphisms.tex","source_line":8624,"source_end_line":8631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8624-L8631","statement_sha256":"31b4bf95cf56b55b5b1371ebb23718a52659d8764e0051b5dce372323d2d6ba7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12191,"rank":12191,"depth":57,"x":758.024,"y":1519.128,"cluster":"geometry-of-spaces"},{"id":"stacks:0879","tag":"0879","title":"Extending properties from an open · Lemma 0879","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let V ⊂ Y be an open subspace. If • f is separated, locally of finite type, and flat, • f^-1(V) → V is an isomorphism, and • V → Y is quasi-compact and scheme theoretically dense, then f is an open immersion.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $V \\subset Y$ be an open subspace. If\n\\begin{enumerate}\n\\item $f$ is separated, locally of finite type, and flat,\n\\item $f^{-1}(V) \\to V$ is an isomorphism, and\n\\item $V \\to Y$ is quasi-compact and scheme theoretically dense,\n\\end{enumerate}\nthen $f$ is an open immersion.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Extending properties from an open","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0879","source_file":"spaces-more-morphisms.tex","source_line":8648,"source_end_line":8659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8648-L8659","statement_sha256":"5b67ffe2359116907943d9647113aa07517983fc73918272a5d6c79f1b629f55","origin":"The Stacks Project","memory_eligible":false,"source_rank":12192,"rank":12192,"depth":57,"x":687.509,"y":1704.005,"cluster":"geometry-of-spaces"},{"id":"stacks:087B","tag":"087B","title":"Blowing up and flatness · Lemma 087B","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let φ : W → X be a quasi-compact separated étale morphism. Let U ⊂ X be a quasi-compact open subspace. Let I ⊂ O_W be a finite type quasi-coherent sheaf of ideals such that V(I) ∩ φ^-1(U) = ∅. Then there exists a finite type quasi-coherent sheaf of ideals J ⊂ O_X such that • V(J) ∩ U = ∅, and • φ^-1(J)O_W = I I' for some finite type quasi-coherent ideal I' ⊂ O_W.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $\\varphi : W \\to X$ be a quasi-compact\nseparated \\'etale morphism. Let $U \\subset X$ be a quasi-compact open\nsubspace. Let $\\mathcal{I} \\subset \\mathcal{O}_W$ be a finite type\nquasi-coherent sheaf of ideals such that\n$V(\\mathcal{I}) \\cap \\varphi^{-1}(U) = \\emptyset$.\nThen there exists a finite type quasi-coherent sheaf of ideals\n$\\mathcal{J} \\subset \\mathcal{O}_X$ such that\n\\begin{enumerate}\n\\item $V(\\mathcal{J}) \\cap U = \\emptyset$, and\n\\item $\\varphi^{-1}(\\mathcal{J})\\mathcal{O}_W = \\mathcal{I} \\mathcal{I}'$\nfor some finite type quasi-coherent ideal\n$\\mathcal{I}' \\subset \\mathcal{O}_W$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087B","source_file":"spaces-more-morphisms.tex","source_line":8697,"source_end_line":8713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8697-L8713","statement_sha256":"4a2b8ab2ef717aea44d3fe2a5df3f1fbb2e2376eca66f4513d728d40c4dfcf1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12193,"rank":12193,"depth":67,"x":590.742,"y":1527.501,"cluster":"geometry-of-spaces"},{"id":"stacks:087C","tag":"087C","title":"Blowing up and flatness · Theorem 087C","summary":"Let S be a scheme. Let B be a quasi-compact and quasi-separated algebraic space over S. Let X be an algebraic space over B. Let F be a quasi-coherent module on X. Let U ⊂ B be a quasi-compact open subspace. Assume • X is quasi-compact, • X is locally of finite presentation over B, • F is a module of finite type, • F_U is of finite presentation, and • F_U is flat over U. Then there exists a U-admissible blowup B' → B such that the strict transform F' of F is an O_X ×_B…","statement_latex":"Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $X$ be an algebraic space over $B$.\nLet $\\mathcal{F}$ be a quasi-coherent module on $X$.\nLet $U \\subset B$ be a quasi-compact open subspace. Assume\n\\begin{enumerate}\n\\item $X$ is quasi-compact,\n\\item $X$ is locally of finite presentation over $B$,\n\\item $\\mathcal{F}$ is a module of finite type,\n\\item $\\mathcal{F}_U$ is of finite presentation, and\n\\item $\\mathcal{F}_U$ is flat over $U$.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $B' \\to B$ such that the\nstrict transform $\\mathcal{F}'$ of $\\mathcal{F}$ is an\n$\\mathcal{O}_{X \\times_B B'}$-module of finite presentation and\nflat over $B'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Blowing up and flatness","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087C","source_file":"spaces-more-morphisms.tex","source_line":8777,"source_end_line":8794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8777-L8794","statement_sha256":"940a5078242472c42103e0a1f859e68eacd0b65fc0e22fcfe2958ce3562387b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12194,"rank":12194,"depth":68,"x":804.236,"y":1602.815,"cluster":"geometry-of-spaces"},{"id":"stacks:087E","tag":"087E","title":"Applications · Lemma 087E","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. Let U ⊂ B be an open subspace. Assume • B is quasi-compact and quasi-separated, • U is quasi-compact, • f : X → B is of finite type and quasi-separated, and • f^-1(U) → U is flat and locally of finite presentation. Then there exists a U-admissible blowup B' → B such that the strict transform X' of X is flat and of finite presentation over B'.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic\nspaces over $S$. Let $U \\subset B$ be an open subspace. Assume\n\\begin{enumerate}\n\\item $B$ is quasi-compact and quasi-separated,\n\\item $U$ is quasi-compact,\n\\item $f : X \\to B$ is of finite type and quasi-separated, and\n\\item $f^{-1}(U) \\to U$ is flat and locally of finite presentation.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $B' \\to B$ such that\nthe strict transform $X'$ of $X$ is flat and of finite presentation\nover $B'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087E","source_file":"spaces-more-morphisms.tex","source_line":8860,"source_end_line":8873,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8860-L8873","statement_sha256":"8b88942cff0c49640288fbecc70f82f2f15e6af5858ce34a44b44a8d1b685506","origin":"The Stacks Project","memory_eligible":false,"source_rank":12195,"rank":12195,"depth":69,"x":586.039,"y":1668.483,"cluster":"geometry-of-spaces"},{"id":"stacks:0B4K","tag":"0B4K","title":"Applications · Lemma 0B4K","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. Let U ⊂ B be an open subspace. Assume • B is quasi-compact and quasi-separated, • U is quasi-compact, • f : X → B is proper, and • f^-1(U) → U is finite locally free. Then there exists a U-admissible blowup B' → B such that the strict transform X' of X is finite locally free over B'.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic\nspaces over $S$. Let $U \\subset B$ be an open subspace. Assume\n\\begin{enumerate}\n\\item $B$ is quasi-compact and quasi-separated,\n\\item $U$ is quasi-compact,\n\\item $f : X \\to B$ is proper, and\n\\item $f^{-1}(U) \\to U$ is finite locally free.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $B' \\to B$ such that\nthe strict transform $X'$ of $X$ is finite locally free over $B'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B4K","source_file":"spaces-more-morphisms.tex","source_line":8912,"source_end_line":8924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8912-L8924","statement_sha256":"45c7003ef94a702714a98c14f501b88026cbe1774fd3542e95850112f8bf083a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12196,"rank":12196,"depth":70,"x":694.227,"y":1496.089,"cluster":"geometry-of-spaces"},{"id":"stacks:0GUW","tag":"0GUW","title":"Applications · Lemma 0GUW","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. Let U ⊂ B be an open subspace. Assume • B is quasi-compact and quasi-separated, • U is quasi-compact, • f : X → B is proper, and • f^-1(U) → U is an isomorphism. Then there exists a U-admissible blowup B' → B such that the strict transform X' of X maps isomorphically to B'.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic\nspaces over $S$. Let $U \\subset B$ be an open subspace. Assume\n\\begin{enumerate}\n\\item $B$ is quasi-compact and quasi-separated,\n\\item $U$ is quasi-compact,\n\\item $f : X \\to B$ is proper, and\n\\item $f^{-1}(U) \\to U$ is an isomorphism.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $B' \\to B$ such that\nthe strict transform $X'$ of $X$ maps isomorphically to $B'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUW","source_file":"spaces-more-morphisms.tex","source_line":8936,"source_end_line":8948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8936-L8948","statement_sha256":"82e5ac6c2a3814bead39ac3b5d61d6b8386236f3f2b2f616b752d97d842d54ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":12197,"rank":12197,"depth":70,"x":753.14,"y":1684.774,"cluster":"geometry-of-spaces"},{"id":"stacks:087F","tag":"087F","title":"Applications · Lemma 087F","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. Let U ⊂ B be an open subspace. Assume • B quasi-compact and quasi-separated, • U is quasi-compact, • f is of finite type • f^-1(U) → U is an isomorphism. Then there exists a U-admissible blowup B' → B such that U is scheme theoretically dense in B' and such that the strict transform X' of X maps isomorphically to an open subspace of B'.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces\nover $S$. Let $U \\subset B$ be an open subspace. Assume\n\\begin{enumerate}\n\\item $B$ quasi-compact and quasi-separated,\n\\item $U$ is quasi-compact,\n\\item $f$ is of finite type\n\\item $f^{-1}(U) \\to U$ is an isomorphism.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $B' \\to B$ such that $U$\nis scheme theoretically dense in $B'$ and such that the strict\ntransform $X'$ of $X$ maps isomorphically to an open subspace of $B'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087F","source_file":"spaces-more-morphisms.tex","source_line":8961,"source_end_line":8974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L8961-L8974","statement_sha256":"242164b704a66c5c0ea882662fba49ee683c247d8d0300c1c0cd6bf9653bb9d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12198,"rank":12198,"depth":71,"x":557.781,"y":1578.971,"cluster":"geometry-of-spaces"},{"id":"stacks:087G","tag":"087G","title":"Applications · Lemma 087G","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. Let U ⊂ B be an open subspace. Assume • B is quasi-compact and quasi-separated, • U is quasi-compact, • f : X → B is proper, • f^-1(U) → U us an isomorphism. Then there exists a U-admissible blowup B' → B which dominates X, i.e., such that there exists a factorization B' → X → B of the blowup morphism.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces\nover $S$. Let $U \\subset B$ be an open subspace. Assume\n\\begin{enumerate}\n\\item $B$ is quasi-compact and quasi-separated,\n\\item $U$ is quasi-compact,\n\\item $f : X \\to B$ is proper,\n\\item $f^{-1}(U) \\to U$ us an isomorphism.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $B' \\to B$ which dominates $X$,\ni.e., such that there exists a factorization $B' \\to X \\to B$\nof the blowup morphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087G","source_file":"spaces-more-morphisms.tex","source_line":9003,"source_end_line":9016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9003-L9016","statement_sha256":"aff1dfcef1e887a6db0effae396c49882d6c02f1870e840144f5faac5b46d8e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12199,"rank":12199,"depth":71,"x":787.133,"y":1546.107,"cluster":"geometry-of-spaces"},{"id":"stacks:0CPI","tag":"0CPI","title":"Applications · Lemma 0CPI","summary":"Let S be a scheme. Let X, Y be algebraic spaces over S. Let U ⊂ W ⊂ Y be open subspaces. Let f : X → W and let s : U → X be morphisms such that f ∘ s = id_U. Assume • f is proper, • Y is quasi-compact and quasi-separated, and • U and W are quasi-compact. Then there exists a U-admissible blowup b : Y' → Y and a morphism s' : b^-1(W) → X extending s with f ∘ s' = b|_b^-1(W).","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$.\nLet $U \\subset W \\subset Y$ be open subspaces.\nLet $f : X \\to W$ and let $s : U \\to X$ be morphisms\nsuch that $f \\circ s = \\text{id}_U$. Assume\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item $Y$ is quasi-compact and quasi-separated, and\n\\item $U$ and $W$ are quasi-compact.\n\\end{enumerate}\nThen there exists a $U$-admissible blowup $b : Y' \\to Y$ and a morphism\n$s' : b^{-1}(W) \\to X$ extending $s$ with $f \\circ s' = b|_{b^{-1}(W)}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPI","source_file":"spaces-more-morphisms.tex","source_line":9024,"source_end_line":9037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9024-L9037","statement_sha256":"ac5a0162e19c4949334981a96b1002c57f15974d7cc6d2baedcf8c892e52b999","origin":"The Stacks Project","memory_eligible":false,"source_rank":12200,"rank":12200,"depth":72,"x":644.308,"y":1700.623,"cluster":"geometry-of-spaces"},{"id":"stacks:088Q","tag":"088Q","title":"Chow's lemma · Lemma 088Q","summary":"Let S be a scheme. Let Y be a quasi-compact and quasi-separated algebraic space over S. Let U → X_1 and U → X_2 be open immersions of algebraic spaces over Y and assume U, X_1, X_2 of finite type and separated over Y. Then there exists a commutative diagram xymatrix X_1' ar[d] ar[r] & X & X_2' ar[l] ar[d] X_1 & U ar[l] ar[lu] ar[u] ar[ru] ar[r] & X_2 of algebraic spaces over Y where X_i' → X_i is a U-admissible blowup, X_i' → X is an open immersion, and X is separated and…","statement_latex":"Let $S$ be a scheme. Let $Y$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $U \\to X_1$ and $U \\to X_2$ be open immersions\nof algebraic spaces over $Y$ and assume $U$, $X_1$, $X_2$ of finite\ntype and separated over $Y$. Then there exists a commutative diagram\n$$\n\\xymatrix{\nX_1' \\ar[d] \\ar[r] & X & X_2' \\ar[l] \\ar[d] \\\\\nX_1 & U \\ar[l] \\ar[lu] \\ar[u] \\ar[ru] \\ar[r] & X_2\n}\n$$\nof algebraic spaces over $Y$ where $X_i' \\to X_i$ is a $U$-admissible\nblowup, $X_i' \\to X$ is an open immersion, and $X$ is separated and finite\ntype over $Y$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088Q","source_file":"spaces-more-morphisms.tex","source_line":9083,"source_end_line":9098,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9083-L9098","statement_sha256":"59280861c7c85a61ba15065f07a93d473bf5862026145b0e260bcae844f9c657","origin":"The Stacks Project","memory_eligible":false,"source_rank":12201,"rank":12201,"depth":72,"x":625.35,"y":1505.462,"cluster":"geometry-of-spaces"},{"id":"stacks:088R","tag":"088R","title":"Chow's lemma · Lemma 088R","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let U ⊂ X be an open subspace. Assume • U is quasi-compact, • Y is quasi-compact and quasi-separated, • there exists an immersion U → P^n_Y over Y, • f is of finite type and separated. Then there exists a commutative diagram xymatrix & U ar[ld] ar[d] ar[rd] ar[rrd] X ar[rd] & X' ar[l] ar[d] ar[r] & Z' ar[ld] ar[r] & Z ar[ld] & Y & P^n_Y ar[l] where the arrows with source U are open immersions, X' →…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $U \\subset X$ be an open subspace. Assume\n\\begin{enumerate}\n\\item $U$ is quasi-compact,\n\\item $Y$ is quasi-compact and quasi-separated,\n\\item there exists an immersion $U \\to \\mathbf{P}^n_Y$ over $Y$,\n\\item $f$ is of finite type and separated.\n\\end{enumerate}\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n& U \\ar[ld] \\ar[d] \\ar[rd] \\ar[rrd] \\\\\nX \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & Z' \\ar[ld] \\ar[r] & Z \\ar[ld] \\\\\n& Y & \\mathbf{P}^n_Y \\ar[l]\n}\n$$\nwhere\nthe arrows with source $U$ are open immersions,\n$X' \\to X$ is a $U$-admissible blowup,\n$X' \\to Z'$ is an open immersion,\n$Z' \\to Y$ is a proper and representable morphism of algebraic spaces.\nMore precisely, $Z' \\to Z$ is a $U$-admissible blowup\nand $Z \\to \\mathbf{P}^n_Y$ is a closed immersion.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088R","source_file":"spaces-more-morphisms.tex","source_line":9170,"source_end_line":9195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9170-L9195","statement_sha256":"bdb07230f6531614de17a2eefc2ee35bb17bf9503f47c94837927a0cd5e76ed1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12202,"rank":12202,"depth":73,"x":796.431,"y":1638.737,"cluster":"geometry-of-spaces"},{"id":"stacks:088S","tag":"088S","title":"Chow's lemma · Lemma 088S","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f separated, of finite type, and Y Noetherian. Then there exists a dense open subspace U ⊂ X and a commutative diagram xymatrix & U ar[ld] ar[d] ar[rd] ar[rrd] X ar[rd] & X' ar[l] ar[d] ar[r] & Z' ar[ld] ar[r] & Z ar[ld] & Y & P^n_Y ar[l] where the arrows with source U are open immersions, X' → X is a U-admissible blowup, X' → Z' is an open immersion, Z' → Y is a proper and representable…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ separated, of finite type, and $Y$ Noetherian.\nThen there exists a dense open subspace $U \\subset X$ and\na commutative diagram\n$$\n\\xymatrix{\n& U \\ar[ld] \\ar[d] \\ar[rd] \\ar[rrd] \\\\\nX \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & Z' \\ar[ld] \\ar[r] & Z \\ar[ld] \\\\\n& Y & \\mathbf{P}^n_Y \\ar[l]\n}\n$$\nwhere\nthe arrows with source $U$ are open immersions,\n$X' \\to X$ is a $U$-admissible blowup,\n$X' \\to Z'$ is an open immersion,\n$Z' \\to Y$ is a proper and representable morphism of algebraic spaces.\nMore precisely, $Z' \\to Z$ is a $U$-admissible blowup\nand $Z \\to \\mathbf{P}^n_Y$ is a closed immersion.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088S","source_file":"spaces-more-morphisms.tex","source_line":9224,"source_end_line":9244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9224-L9244","statement_sha256":"6b2c7d0fff467a9381ab91cd00cc6ef3ee3220fd7b3cadb9685f1261d87b0786","origin":"The Stacks Project","memory_eligible":false,"source_rank":12203,"rank":12203,"depth":74,"x":562.887,"y":1637.536,"cluster":"geometry-of-spaces"},{"id":"stacks:088U","tag":"088U","title":"Chow's lemma · Lemma 088U","summary":"[Kn] Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f separated of finite type, and Y separated and Noetherian. Then there exists a commutative diagram xymatrix X ar[rd] & X' ar[l] ar[d] ar[r] & P^n_Y ar[ld] & Y where X' → X is a U-admissible blowup for some dense open U ⊂ X and the morphism X' → P^n_Y is an immersion.","statement_latex":"\\begin{reference}\n\\cite[IV Theorem 3.1]{Kn}\n\\end{reference}\nLet $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ separated of finite type, and $Y$ separated and\nNoetherian. Then there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & \\mathbf{P}^n_Y \\ar[ld] \\\\\n& Y\n}\n$$\nwhere $X' \\to X$ is a $U$-admissible blowup for some dense open\n$U \\subset X$ and the morphism $X' \\to \\mathbf{P}^n_Y$ is an immersion.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Chow's lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088U","source_file":"spaces-more-morphisms.tex","source_line":9274,"source_end_line":9290,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9274-L9290","statement_sha256":"52b66859cdb000479acefb161ea4300cdb35c128488541956382e66826be5947","origin":"The Stacks Project","memory_eligible":false,"source_rank":12204,"rank":12204,"depth":75,"x":736.223,"y":1505.782,"cluster":"geometry-of-spaces"},{"id":"stacks:089L","tag":"089L","title":"Variants of Chow's Lemma · Lemma 089L","summary":"Let S be a scheme. Let Y be a quasi-compact and quasi-separated algebraic space over S. Let f : X → Y be a separated morphism of finite type. Then there exists a commutative diagram xymatrix X ar[rd] & X' ar[l] ar[d] ar[r] & overlineX' ar[ld] & Y where X' → X is proper surjective, X' → overlineX' is an open immersion, and overlineX' → Y is proper and representable morphism of algebraic spaces.","statement_latex":"Let $S$ be a  scheme. Let $Y$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $f : X \\to Y$ be a separated morphism of\nfinite type. Then there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & \\overline{X}' \\ar[ld] \\\\\n& Y\n}\n$$\nwhere $X' \\to X$ is proper surjective,\n$X' \\to \\overline{X}'$ is an open immersion, and\n$\\overline{X}' \\to Y$ is proper and representable morphism\nof algebraic spaces.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Variants of Chow's Lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089L","source_file":"spaces-more-morphisms.tex","source_line":9383,"source_end_line":9398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9383-L9398","statement_sha256":"cd3c4cc5ce9243e26d244697c955a2993a67749f7fec973e6b4ddddd59c3fe63","origin":"The Stacks Project","memory_eligible":false,"source_rank":12205,"rank":12205,"depth":75,"x":714.341,"y":1701.471,"cluster":"geometry-of-spaces"},{"id":"stacks:089M","tag":"089M","title":"Variants of Chow's Lemma · Lemma 089M","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f separated of finite type, and Y separated and quasi-compact. Then there exists a commutative diagram xymatrix X ar[rd] & X' ar[l] ar[d] ar[r] & P^n_Y ar[ld] & Y where X' → X is proper surjective morphism and the morphism X' → P^n_Y is an immersion.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ separated of finite type, and $Y$ separated and\nquasi-compact. Then there exists a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rd] & X' \\ar[l] \\ar[d] \\ar[r] & \\mathbf{P}^n_Y \\ar[ld] \\\\\n& Y\n}\n$$\nwhere $X' \\to X$ is proper surjective morphism\nand the morphism $X' \\to \\mathbf{P}^n_Y$ is an immersion.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Variants of Chow's Lemma","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089M","source_file":"spaces-more-morphisms.tex","source_line":9431,"source_end_line":9444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9431-L9444","statement_sha256":"3dcbb97f397fe7ec3263fb1479ceb691c04c67d0aba98c20c850fbd2f988fe7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12206,"rank":12206,"depth":76,"x":572.98,"y":1544.612,"cluster":"geometry-of-spaces"},{"id":"stacks:089P","tag":"089P","title":"Grothendieck's existence theorem · Lemma 089P","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S and let I ⊂ O_X be a quasi-coherent sheaf of ideals. • The category Coh(X, I) is abelian. • Exactness in Coh(X, I) can be checked étale locally. • For any flat morphism f : X' → X of Noetherian algebraic spaces the functor f^* : Coh(X, I) → Coh(X', f^-1IO_X') is exact.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and\nlet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.\n\\begin{enumerate}\n\\item The category $\\textit{Coh}(X, \\mathcal{I})$ is abelian.\n\\item Exactness in $\\textit{Coh}(X, \\mathcal{I})$\ncan be checked \\'etale locally.\n\\item For any flat morphism $f : X' \\to X$ of Noetherian algebraic spaces\nthe functor $f^* : \\textit{Coh}(X, \\mathcal{I}) \\to\n\\textit{Coh}(X', f^{-1}\\mathcal{I}\\mathcal{O}_{X'})$ is exact.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's existence theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/089P","source_file":"spaces-more-morphisms.tex","source_line":9562,"source_end_line":9574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9562-L9574","statement_sha256":"8b9bc08d7c21ac13ba096dce531317281791383b23174617143f1515b741c7bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12207,"rank":12207,"depth":11,"x":803.57,"y":1580.098,"cluster":"geometry-of-spaces"},{"id":"stacks:08B3","tag":"08B3","title":"Grothendieck's existence theorem · Lemma 08B3","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S and let I ⊂ O_X be a quasi-coherent sheaf of ideals. A map (F_n) → (G_n) is surjective in Coh(X, I) if and only if F_1 → G_1 is surjective.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$\nand let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf\nof ideals. A map $(\\mathcal{F}_n) \\to (\\mathcal{G}_n)$ is surjective in\n$\\textit{Coh}(X, \\mathcal{I})$ if and only if\n$\\mathcal{F}_1 \\to \\mathcal{G}_1$ is surjective.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's existence theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08B3","source_file":"spaces-more-morphisms.tex","source_line":9614,"source_end_line":9621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9614-L9621","statement_sha256":"4f7ab42e4031d4e428d50d41f8c869f05ce92e81c0411b59bdb50410d8d898dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12208,"rank":12208,"depth":12,"x":604.818,"y":1684.869,"cluster":"geometry-of-spaces"},{"id":"stacks:08B5","tag":"08B5","title":"Grothendieck's existence theorem · Lemma 08B5","summary":"The functor ([Tag 08B4]) is exact.","statement_latex":"The functor (\\ref{equation-completion-functor}) is exact.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's existence theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08B5","source_file":"spaces-more-morphisms.tex","source_line":9643,"source_end_line":9646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9643-L9646","statement_sha256":"3d84ad28c8ce1221d5e389cd6fb6d0c0c70c60cbc1f625d1ea281346a2c1ea5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12209,"rank":12209,"depth":12,"x":667.177,"y":1494.662,"cluster":"geometry-of-spaces"},{"id":"stacks:08B6","tag":"08B6","title":"Grothendieck's existence theorem · Lemma 08B6","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S and let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let F, G be coherent O_X-modules. Set H = SheafHom_O_X(F, G). Then lim H^0(X, H/I^nH) = Mor_Coh(X, I) (F^wedge, G^wedge).","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and\nlet $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules. Set\n$\\mathcal{H} = \\SheafHom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{G})$.\nThen\n$$\n\\lim H^0(X, \\mathcal{H}/\\mathcal{I}^n\\mathcal{H}) =\n\\Mor_{\\textit{Coh}(X, \\mathcal{I})}\n(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge).\n$$","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's existence theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08B6","source_file":"spaces-more-morphisms.tex","source_line":9655,"source_end_line":9667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9655-L9667","statement_sha256":"1f5af06aa9ea5f21f405ebe8f537160d81244cd6c4fbf8ec36f16d6930a37c28","origin":"The Stacks Project","memory_eligible":false,"source_rank":12210,"rank":12210,"depth":15,"x":774.251,"y":1670.462,"cluster":"geometry-of-spaces"},{"id":"stacks:08B9","tag":"08B9","title":"Grothendieck's existence theorem · Lemma 08B9","summary":"In Situation [Tag 08B7]. Let F, G be coherent O_X-modules. Assume that the intersection of the supports of F and G is proper over Spec(A). Then the map Mor_Coh(O_X)(F, G) → Mor_Coh(X, I) (F^wedge, G^wedge) coming from ([Tag 08B4]) is a bijection. In particular, ([Tag 08B8]) is fully faithful.","statement_latex":"In Situation \\ref{situation-existence}.\nLet $\\mathcal{F}$, $\\mathcal{G}$ be coherent $\\mathcal{O}_X$-modules.\nAssume that the intersection of the supports of\n$\\mathcal{F}$ and $\\mathcal{G}$ is proper over $\\Spec(A)$. Then the map\n$$\n\\Mor_{\\textit{Coh}(\\mathcal{O}_X)}(\\mathcal{F}, \\mathcal{G})\n\\longrightarrow\n\\Mor_{\\textit{Coh}(X, \\mathcal{I})}\n(\\mathcal{F}^\\wedge, \\mathcal{G}^\\wedge)\n$$\ncoming from (\\ref{equation-completion-functor}) is a bijection.\nIn particular, (\\ref{equation-completion-functor-proper-over-A})\nis fully faithful.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's existence theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08B9","source_file":"spaces-more-morphisms.tex","source_line":9717,"source_end_line":9732,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9717-L9732","statement_sha256":"90fc17ee322066dcc687cac287e9d872889324111eff467802e377ab9c889735","origin":"The Stacks Project","memory_eligible":false,"source_rank":12211,"rank":12211,"depth":70,"x":553.723,"y":1601.524,"cluster":"geometry-of-spaces"},{"id":"stacks:08BB","tag":"08BB","title":"Grothendieck's existence theorem · Lemma 08BB","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S and let I ⊂ O_X be a quasi-coherent sheaf of ideals. Let G be a coherent O_X-module, (F_n) an object of Coh(X, I), and α : (F_n) → G^wedge a map whose kernel and cokernel are annihilated by a power of I. Then there exists a unique (up to unique isomorphism) triple (F, a, β) where • F is a coherent O_X-module, • a : F → G is an O_X-module map whose kernel and cokernel are annihilated by a power of I, • β :…","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let\n$\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent sheaf of ideals.\nLet $\\mathcal{G}$ be a coherent $\\mathcal{O}_X$-module, $(\\mathcal{F}_n)$\nan object of $\\textit{Coh}(X, \\mathcal{I})$, and\n$\\alpha : (\\mathcal{F}_n) \\to \\mathcal{G}^\\wedge$\na map whose kernel and cokernel are annihilated by a power of $\\mathcal{I}$.\nThen there exists a unique (up to unique isomorphism) triple\n$(\\mathcal{F}, a, \\beta)$ where\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a coherent $\\mathcal{O}_X$-module,\n\\item $a : \\mathcal{F} \\to \\mathcal{G}$ is an $\\mathcal{O}_X$-module map\nwhose kernel and cokernel are annihilated by a power of $\\mathcal{I}$,\n\\item $\\beta : (\\mathcal{F}_n) \\to \\mathcal{F}^\\wedge$ is an isomorphism, and\n\\item $\\alpha = a^\\wedge \\circ \\beta$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's existence theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BB","source_file":"spaces-more-morphisms.tex","source_line":9800,"source_end_line":9817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9800-L9817","statement_sha256":"53c22c6958ac6c75b92eb44e45466c3fd0203ec2ffeb908cfa707f553db85ac0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12212,"rank":12212,"depth":22,"x":771.973,"y":1527.158,"cluster":"geometry-of-spaces"},{"id":"stacks:08BC","tag":"08BC","title":"Grothendieck's existence theorem · Lemma 08BC","summary":"In Situation [Tag 08B7]. Let K ⊂ O_X be a quasi-coherent sheaf of ideals. Let X_e ⊂ X be the closed subspace cut out by K^e. Let I_e = IO_X_e. Let (F_n) be an object of Coh_support proper over A(X, I). Assume • the functor Coh_support proper over A(O_X_e) → Coh_support proper over A(X_e, I_e) is an equivalence for all e ≥ 1, and • there exists an object H of Coh_support proper over A(O_X) and a map α : (F_n) → H^wedge whose kernel and cokernel are annihilated by a power…","statement_latex":"In Situation \\ref{situation-existence}. Let $\\mathcal{K} \\subset \\mathcal{O}_X$\nbe a quasi-coherent sheaf of ideals. Let $X_e \\subset X$ be the closed subspace\ncut out by $\\mathcal{K}^e$. Let $\\mathcal{I}_e = \\mathcal{I}\\mathcal{O}_{X_e}$.\nLet $(\\mathcal{F}_n)$ be an object of\n$\\textit{Coh}_{\\text{support proper over } A}(X, \\mathcal{I})$.\nAssume\n\\begin{enumerate}\n\\item the functor\n$\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_{X_e})\n\\to \\textit{Coh}_{\\text{support proper over } A}(X_e, \\mathcal{I}_e)$\nis an equivalence for all $e \\geq 1$, and\n\\item there exists an object $\\mathcal{H}$ of\n$\\textit{Coh}_{\\text{support proper over } A}(\\mathcal{O}_X)$ and a map\n$\\alpha : (\\mathcal{F}_n) \\to \\mathcal{H}^\\wedge$ whose\nkernel and cokernel are annihilated by a power of $\\mathcal{K}$.\n\\end{enumerate}\nThen $(\\mathcal{F}_n)$ is in the essential image of\n(\\ref{equation-completion-functor-proper-over-A}).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's existence theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BC","source_file":"spaces-more-morphisms.tex","source_line":9827,"source_end_line":9847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9827-L9847","statement_sha256":"fc01ab376436e036b293c86278da60a427f49bbb4a306facfd689b80a5175355","origin":"The Stacks Project","memory_eligible":false,"source_rank":12213,"rank":12213,"depth":61,"x":670.75,"y":1705.996,"cluster":"geometry-of-spaces"},{"id":"stacks:08BD","tag":"08BD","title":"Grothendieck's existence theorem · Lemma 08BD","summary":"Let S be a scheme. Let f : X → Y be a representable proper morphism of Noetherian algebraic spaces over S. Let J, K ⊂ O_Y be quasi-coherent sheaves of ideals. Assume f is an isomorphism over V = Y setminus V(K). Set I = f^-1J O_X. Let (G_n) be an object of Coh(Y, J), let F be a coherent O_X-module, and let β : (f^*G_n) → F^wedge be an isomorphism in Coh(X, I). Then there exists a map α : (G_n) → (f_*F)^wedge in Coh(Y, J) whose kernel and cokernel are annihilated by a…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a representable\nproper morphism of Noetherian algebraic spaces over $S$. Let\n$\\mathcal{J}, \\mathcal{K} \\subset \\mathcal{O}_Y$\nbe quasi-coherent sheaves of ideals.\nAssume $f$ is an isomorphism over $V = Y \\setminus V(\\mathcal{K})$.\nSet $\\mathcal{I} = f^{-1}\\mathcal{J} \\mathcal{O}_X$.\nLet $(\\mathcal{G}_n)$ be an object of $\\textit{Coh}(Y, \\mathcal{J})$,\nlet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module, and let\n$\\beta : (f^*\\mathcal{G}_n)  \\to \\mathcal{F}^\\wedge$ be an isomorphism in\n$\\textit{Coh}(X, \\mathcal{I})$. Then there exists a map\n$$\n\\alpha :\n(\\mathcal{G}_n)\n\\longrightarrow\n(f_*\\mathcal{F})^\\wedge\n$$\nin $\\textit{Coh}(Y, \\mathcal{J})$ whose kernel and cokernel\nare annihilated by a power of $\\mathcal{K}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's existence theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BD","source_file":"spaces-more-morphisms.tex","source_line":9940,"source_end_line":9960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9940-L9960","statement_sha256":"4930be417db57da2fd6308f7ffeff7f20bc6a32fa5ef07332e1f97cf18d3a4e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12214,"rank":12214,"depth":63,"x":601.511,"y":1516.516,"cluster":"geometry-of-spaces"},{"id":"stacks:08BE","tag":"08BE","title":"Grothendieck's existence theorem · Theorem 08BE","summary":"In Situation [Tag 08B7] the functor ([Tag 08B8]) is an equivalence.","statement_latex":"In Situation \\ref{situation-existence} the functor\n(\\ref{equation-completion-functor-proper-over-A})\nis an equivalence.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's existence theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BE","source_file":"spaces-more-morphisms.tex","source_line":9989,"source_end_line":9994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L9989-L9994","statement_sha256":"88969e281da496867360572240efea9e64b6eb1ae97e43d778d81dffb686785a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12215,"rank":12215,"depth":76,"x":805.125,"y":1617.039,"cluster":"geometry-of-spaces"},{"id":"stacks:08BG","tag":"08BG","title":"Grothendieck's algebraization theorem · Lemma 08BG","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Write S = Spec(A) and S_n = Spec(A/I^n). Let X → S be a morphism of algebraic spaces that is separated and of finite type. For n ≥ 1 we set X_n = X ×_S S_n. Suppose given a commutative diagram xymatrix Z_1 ar[r] ar[d] & Z_2 ar[r] ar[d] & Z_3 ar[r] ar[d] & … X_1 ar[r]^i_1 & X_2 ar[r]^i_2 & X_3 ar[r] & … of algebraic spaces with cartesian squares. Assume that • Z_1 → X_1 is a closed immersion, and • Z_1 → S_1…","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nWrite $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.\nLet $X \\to S$ be a morphism of algebraic spaces that is separated\nand of finite type.\nFor $n \\geq 1$ we set $X_n = X \\times_S S_n$.\nSuppose given a commutative diagram\n$$\n\\xymatrix{\nZ_1 \\ar[r] \\ar[d] & Z_2 \\ar[r] \\ar[d] & Z_3 \\ar[r] \\ar[d] & \\ldots \\\\\nX_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots\n}\n$$\nof algebraic spaces with cartesian squares. Assume that\n\\begin{enumerate}\n\\item $Z_1 \\to X_1$ is a closed immersion, and\n\\item $Z_1 \\to S_1$ is proper.\n\\end{enumerate}\nThen there exists a closed immersion of algebraic spaces $Z \\to X$ such that\n$Z_n = Z \\times_S S_n$ for all $n \\geq 1$. Moreover, $Z$ is proper over $S$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's algebraization theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BG","source_file":"spaces-more-morphisms.tex","source_line":10145,"source_end_line":10166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10145-L10166","statement_sha256":"9c1d2b7aaa32feab3fe2ce75e0d99a9c8cc7bcddf8d3947bf8660db07646ed80","origin":"The Stacks Project","memory_eligible":false,"source_rank":12216,"rank":12216,"depth":69,"x":573.937,"y":1658.488,"cluster":"geometry-of-spaces"},{"id":"stacks:0A01","tag":"0A01","title":"Grothendieck's algebraization theorem · Lemma 0A01","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Write S = Spec(A) and S_n = Spec(A/I^n). Let X → S be a morphism of algebraic spaces that is separated and of finite type. For n ≥ 1 we set X_n = X ×_S S_n. Suppose given a commutative diagram xymatrix Y_1 ar[r] ar[d] & Y_2 ar[r] ar[d] & Y_3 ar[r] ar[d] & … X_1 ar[r]^i_1 & X_2 ar[r]^i_2 & X_3 ar[r] & … of algebraic spaces with cartesian squares. Assume that • Y_1 → X_1 is a finite morphism, and • Y_1 → S_1 is…","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nWrite $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$.\nLet $X \\to S$ be a morphism of algebraic spaces that is separated\nand of finite type.\nFor $n \\geq 1$ we set $X_n = X \\times_S S_n$.\nSuppose given a commutative diagram\n$$\n\\xymatrix{\nY_1 \\ar[r] \\ar[d] & Y_2 \\ar[r] \\ar[d] & Y_3 \\ar[r] \\ar[d] & \\ldots \\\\\nX_1 \\ar[r]^{i_1} & X_2 \\ar[r]^{i_2} & X_3 \\ar[r] & \\ldots\n}\n$$\nof algebraic spaces with cartesian squares. Assume that\n\\begin{enumerate}\n\\item $Y_1 \\to X_1$ is a finite morphism, and\n\\item $Y_1 \\to S_1$ is proper.\n\\end{enumerate}\nThen there exists a finite morphism of algebraic spaces $Y \\to X$ such that\n$Y_n = Y \\times_S S_n$ for all $n \\geq 1$. Moreover, $Y$ is proper over $S$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's algebraization theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A01","source_file":"spaces-more-morphisms.tex","source_line":10193,"source_end_line":10214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10193-L10214","statement_sha256":"6e34ba193bbda5b89bccbd29c8e31064e9bf893d60ae848266de4a48ac415c66","origin":"The Stacks Project","memory_eligible":false,"source_rank":12217,"rank":12217,"depth":77,"x":711.203,"y":1496.597,"cluster":"geometry-of-spaces"},{"id":"stacks:0A4Z","tag":"0A4Z","title":"Grothendieck's algebraization theorem · Lemma 0A4Z","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Write S = Spec(A) and S_n = Spec(A/I^n). Let X, Y be algebraic spaces over S. For n ≥ 1 we set X_n = X ×_S S_n and Y_n = Y ×_S S_n. Suppose given a compatible system of commutative diagrams xymatrix & & X_n + 1 ar[rd] ar[rr]_g_n + 1 & & Y_n + 1 ar[ld] X_n ar[rru] ar[rd] ar[rr]_g_n & & Y_n ar[rru] ar[ld] & S_n + 1 & S_n ar[rru] Assume that • X → S is proper, and • Y → S is separated of finite type. Then there…","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nWrite $S = \\Spec(A)$ and $S_n = \\Spec(A/I^n)$. Let $X$, $Y$ be algebraic\nspaces over $S$. For $n \\geq 1$ we set $X_n = X \\times_S S_n$ and\n$Y_n = Y \\times_S S_n$. Suppose given a compatible system of\ncommutative diagrams\n$$\n\\xymatrix{\n& & X_{n + 1} \\ar[rd] \\ar[rr]_{g_{n + 1}} & & Y_{n + 1} \\ar[ld] \\\\\nX_n \\ar[rru] \\ar[rd] \\ar[rr]_{g_n} & & Y_n \\ar[rru] \\ar[ld] & S_{n + 1} \\\\\n& S_n \\ar[rru]\n}\n$$\nAssume that\n\\begin{enumerate}\n\\item $X \\to S$ is proper, and\n\\item $Y \\to S$ is separated of finite type.\n\\end{enumerate}\nThen there exists a unique morphism of algebraic spaces $g : X \\to Y$\nover $S$ such that $g_n$ is the base change of $g$ to $S_n$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's algebraization theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0A4Z","source_file":"spaces-more-morphisms.tex","source_line":10238,"source_end_line":10259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10238-L10259","statement_sha256":"78ea7e77461495cc3fc18bed1bba23f95fa812be6aaa9c989841a37d3a47beec","origin":"The Stacks Project","memory_eligible":false,"source_rank":12218,"rank":12218,"depth":77,"x":740.199,"y":1694.037,"cluster":"geometry-of-spaces"},{"id":"stacks:0E7R","tag":"0E7R","title":"Grothendieck's algebraization theorem · Lemma 0E7R","summary":"Let (A, m, kappa) be a complete local Noetherian ring. Set S = Spec(A) and S_n = Spec(A/ m^n). Consider a commutative diagram xymatrix X_1 ar[r]_i_1 ar[d] & X_2 ar[r]_i_2 ar[d] & X_3 ar[r] ar[d] & … S_1 ar[r] & S_2 ar[r] & S_3 ar[r] & … of algebraic spaces with cartesian squares. If dim(X_1) ≤ 1, then there exists a projective morphism of schemes X → S and isomorphisms X_n ≅ X ×_S S_n compatible with i_n.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a complete local Noetherian ring.\nSet $S = \\Spec(A)$ and $S_n = \\Spec(A/\\mathfrak m^n)$.\nConsider a commutative diagram\n$$\n\\xymatrix{\nX_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] &\n\\ldots \\\\\nS_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots\n}\n$$\nof algebraic spaces with cartesian squares. If $\\dim(X_1) \\leq 1$,\nthen there exists a projective morphism of schemes $X \\to S$\nand isomorphisms $X_n \\cong X \\times_S S_n$ compatible with $i_n$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's algebraization theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7R","source_file":"spaces-more-morphisms.tex","source_line":10338,"source_end_line":10353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10338-L10353","statement_sha256":"712c2ac478d54cfbf85b8b5d010def1391acf511e070b24d5af7c1746ca95943","origin":"The Stacks Project","memory_eligible":false,"source_rank":12219,"rank":12219,"depth":69,"x":559.88,"y":1564.785,"cluster":"geometry-of-spaces"},{"id":"stacks:0AE7","tag":"0AE7","title":"Grothendieck's algebraization theorem · Lemma 0AE7","summary":"Let (A, m, kappa) be a complete Noetherian local ring. Let X be an algebraic space over Spec(A). If X → Spec(A) is proper and dim(X_kappa) ≤ 1, then X is a scheme projective over A.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a complete Noetherian local ring.\nLet $X$ be an algebraic space over $\\Spec(A)$.\nIf $X \\to \\Spec(A)$ is proper and $\\dim(X_\\kappa) \\leq 1$, then\n$X$ is a scheme projective over $A$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Grothendieck's algebraization theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AE7","source_file":"spaces-more-morphisms.tex","source_line":10395,"source_end_line":10401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10395-L10401","statement_sha256":"3d4bdae281fdb083b27e4ade61bbab7160c54f92ed71cee06f456c412d8e0cf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12220,"rank":12220,"depth":78,"x":796.998,"y":1557.77,"cluster":"geometry-of-spaces"},{"id":"stacks:06BM","tag":"06BM","title":"Regular immersions · Lemma 06BM","summary":"Let P be a property of morphisms of schemes which is étale local on the target. Let S be a scheme. Let f : X → Y be a representable morphism of algebraic spaces over S. Consider commutative diagrams xymatrix X ×_Y V ar[d] ar[r] & V ar[d] X ar[r]^f & Y where V is a scheme and V → Y is étale. The following are equivalent • for any diagram as above the projection X ×_Y V → V has property P, and • for some diagram as above with V → Y surjective the projection X ×_Y V → V has…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of schemes which is \\'etale\nlocal on the target. Let $S$ be a scheme.\nLet $f : X \\to Y$ be a representable morphism of algebraic spaces over $S$.\nConsider commutative diagrams\n$$\n\\xymatrix{\nX \\times_Y V \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r]^f & Y\n}\n$$\nwhere $V$ is a scheme and $V \\to Y$ is \\'etale.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any diagram as above the projection $X \\times_Y V \\to V$\nhas property $\\mathcal{P}$, and\n\\item for some diagram as above with $V \\to Y$ surjective\nthe projection $X \\times_Y V \\to V$ has property $\\mathcal{P}$.\n\\end{enumerate}\nIf $X$ and $Y$ are representable, then this is also equivalent to\n$f$ (as a morphism of schemes) having property $\\mathcal{P}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BM","source_file":"spaces-more-morphisms.tex","source_line":10453,"source_end_line":10475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10453-L10475","statement_sha256":"f61916e1df86c726e05c986c1f401431e51aec598af2d3db77e0ff22c58b920a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12221,"rank":12221,"depth":1,"x":627.641,"y":1697.614,"cluster":"geometry-of-spaces"},{"id":"stacks:06BN","tag":"06BN","title":"Regular immersions · Definition 06BN","summary":"Let S be a scheme. Let i : X → Y be a morphism of algebraic spaces over S. • We say i is a Koszul-regular immersion if i is representable and the equivalent conditions of Lemma [Tag 06BM] hold with P(f) =\"f is a Koszul-regular immersion\". • We say i is an H_1-regular immersion if i is representable and the equivalent conditions of Lemma [Tag 06BM] hold with P(f) =\"f is an H_1-regular immersion\". • We say i is a quasi-regular immersion if i is representable and the…","statement_latex":"Let $S$ be a scheme. Let $i : X \\to Y$ be a morphism of algebraic\nspaces over $S$.\n\\begin{enumerate}\n\\item We say $i$ is a {\\it Koszul-regular immersion} if $i$ is representable\nand the equivalent conditions of\nLemma \\ref{lemma-representable-etale-local-target}\nhold with $\\mathcal{P}(f) =$``$f$ is a Koszul-regular immersion''.\n\\item We say $i$ is an {\\it $H_1$-regular immersion} if $i$ is representable\nand the equivalent conditions of\nLemma \\ref{lemma-representable-etale-local-target}\nhold with $\\mathcal{P}(f) =$``$f$ is an $H_1$-regular immersion''.\n\\item We say $i$ is a {\\it quasi-regular immersion} if $i$ is representable\nand the equivalent conditions of\nLemma \\ref{lemma-representable-etale-local-target}\nhold with $\\mathcal{P}(f) =$``$f$ is a quasi-regular immersion''.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BN","source_file":"spaces-more-morphisms.tex","source_line":10529,"source_end_line":10547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10529-L10547","statement_sha256":"9928861b6b8cf251cda622bd7ca88d7c5fc2d48709639c54027d02e041af436d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12222,"rank":12222,"depth":2,"x":640.075,"y":1498.221,"cluster":"geometry-of-spaces"},{"id":"stacks:06BP","tag":"06BP","title":"Regular immersions · Lemma 06BP","summary":"Let S be a scheme. Let i : Z → X be an immersion of algebraic spaces over S. We have the following implications: i is Koszul-regular ⇒ i is H_1-regular ⇒ i is quasi-regular.","statement_latex":"Let $S$ be a scheme.\nLet $i : Z \\to X$ be an immersion of algebraic spaces over $S$.\nWe have the following implications:\n$i$ is Koszul-regular $\\Rightarrow$\n$i$ is $H_1$-regular $\\Rightarrow$\n$i$ is quasi-regular.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BP","source_file":"spaces-more-morphisms.tex","source_line":10549,"source_end_line":10557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10549-L10557","statement_sha256":"b2cefcbddd365e1acc7c740c725d08986758eb8f49de1791709eb52e42c4a1b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12223,"rank":12223,"depth":8,"x":791.387,"y":1652.442,"cluster":"geometry-of-spaces"},{"id":"stacks:09RW","tag":"09RW","title":"Regular immersions · Lemma 09RW","summary":"Let S be a scheme. Let i : Z → X be an immersion of algebraic spaces over S. Assume X is locally Noetherian. Then i is Koszul-regular ⇔ i is H_1-regular ⇔ i is quasi-regular.","statement_latex":"Let $S$ be a scheme.\nLet $i : Z \\to X$ be an immersion of algebraic spaces over $S$.\nAssume $X$ is locally Noetherian. Then\n$i$ is Koszul-regular $\\Leftrightarrow$\n$i$ is $H_1$-regular $\\Leftrightarrow$\n$i$ is quasi-regular.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RW","source_file":"spaces-more-morphisms.tex","source_line":10564,"source_end_line":10572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10564-L10572","statement_sha256":"3c0f44f95e25d593b49d9eb043ed9170228bc3a73a35c2a1bf59d7696281bca4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12224,"rank":12224,"depth":16,"x":555.582,"y":1624.555,"cluster":"geometry-of-spaces"},{"id":"stacks:09RX","tag":"09RX","title":"Regular immersions · Lemma 09RX","summary":"Regular immersions are stable under flat base change. Let S be a scheme. Let i : Z → X be a Koszul-regular, H_1-regular, or quasi-regular immersion of algebraic spaces over S. Let X' → X be a flat morphism of algebraic spaces over S. Then the base change i' : Z ×_X X' → X' is a Koszul-regular, H_1-regular, or quasi-regular immersion.","statement_latex":"\\begin{slogan}\nRegular immersions are stable under flat base change.\n\\end{slogan}\nLet $S$ be a scheme. Let $i : Z \\to X$ be a Koszul-regular,\n$H_1$-regular, or quasi-regular immersion of algebraic spaces over $S$.\nLet $X' \\to X$ be a flat morphism of algebraic spaces over $S$.\nThen the base change $i' : Z \\times_X X' \\to X'$ is a Koszul-regular,\n$H_1$-regular, or quasi-regular immersion.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RX","source_file":"spaces-more-morphisms.tex","source_line":10583,"source_end_line":10593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10583-L10593","statement_sha256":"de1da9fe74338c7de85d73e488ce085e4368c1777f3aed2154910fd15c39b967","origin":"The Stacks Project","memory_eligible":false,"source_rank":12225,"rank":12225,"depth":15,"x":752.061,"y":1511.217,"cluster":"geometry-of-spaces"},{"id":"stacks:09RY","tag":"09RY","title":"Regular immersions · Lemma 09RY","summary":"Let S be a scheme. Let i : Z → X be an immersion of algebraic spaces over S. Then i is a quasi-regular immersion if and only if the following conditions are satisfied • i is locally of finite presentation, • the conormal sheaf C_Z/X is finite locally free, and • the map ([Tag 09RQ]) is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to X$ be an immersion of algebraic spaces\nover $S$. Then $i$ is a quasi-regular immersion if and only if the following\nconditions are satisfied\n\\begin{enumerate}\n\\item $i$ is locally of finite presentation,\n\\item the conormal sheaf $\\mathcal{C}_{Z/X}$ is finite locally free, and\n\\item the map (\\ref{equation-conormal-algebra-quotient}) is an isomorphism.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RY","source_file":"spaces-more-morphisms.tex","source_line":10604,"source_end_line":10614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10604-L10614","statement_sha256":"a2e1c8b7df17c91b71680de1f8f6ad3acf652cb134443df7739c60942677eb49","origin":"The Stacks Project","memory_eligible":false,"source_rank":12226,"rank":12226,"depth":53,"x":698.276,"y":1706.45,"cluster":"geometry-of-spaces"},{"id":"stacks:09RZ","tag":"09RZ","title":"Regular immersions · Lemma 09RZ","summary":"Let S be a scheme. Let Z → Y → X be immersions of algebraic spaces over S. Assume that Z → Y is H_1-regular. Then the canonical sequence of Lemma [Tag 06BD] 0 → i^*C_Y/X → C_Z/X → C_Z/Y → 0 is exact and (étale) locally split.","statement_latex":"Let $S$ be a scheme. Let $Z \\to Y \\to X$ be immersions of algebraic spaces\nover $S$. Assume that $Z \\to Y$ is $H_1$-regular. Then the canonical\nsequence of Lemma \\ref{lemma-transitivity-conormal}\n$$\n0 \\to i^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nis exact and (\\'etale) locally split.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09RZ","source_file":"spaces-more-morphisms.tex","source_line":10624,"source_end_line":10635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10624-L10635","statement_sha256":"074eae6395badc137a4f280d02ffd11361f183eb9513aab7fe17e05a3e06a8f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12227,"rank":12227,"depth":55,"x":580.832,"y":1531.816,"cluster":"geometry-of-spaces"},{"id":"stacks:09S0","tag":"09S0","title":"Regular immersions · Lemma 09S0","summary":"Let S be a scheme. Let i : Z → Y and j : Y → X be immersions of algebraic spaces over S. • If i and j are Koszul-regular immersions, so is j ∘ i. • If i and j are H_1-regular immersions, so is j ∘ i. • If i is an H_1-regular immersion and j is a quasi-regular immersion, then j ∘ i is a quasi-regular immersion.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of\nalgebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $i$ and $j$ are Koszul-regular immersions, so is $j \\circ i$.\n\\item If $i$ and $j$ are $H_1$-regular immersions, so is $j \\circ i$.\n\\item If $i$ is an $H_1$-regular immersion and $j$ is a quasi-regular\nimmersion, then $j \\circ i$ is a quasi-regular immersion.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09S0","source_file":"spaces-more-morphisms.tex","source_line":10655,"source_end_line":10665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10655-L10665","statement_sha256":"802694e00dc2c83d44e2a11793bdbdf4d69fd818986c24f3e057b0ee3eb57b88","origin":"The Stacks Project","memory_eligible":false,"source_rank":12228,"rank":12228,"depth":8,"x":808.07,"y":1594.002,"cluster":"geometry-of-spaces"},{"id":"stacks:09S1","tag":"09S1","title":"Regular immersions · Lemma 09S1","summary":"Let S be a scheme. Let i : Z → Y and j : Y → X be immersions of algebraic spaces over S. Assume j is locally of finite presentation and that the sequence 0 → i^*C_Y/X → C_Z/X → C_Z/Y → 0 of Lemma [Tag 06BD] is exact and locally split. • If j ∘ i is a quasi-regular immersion, so is i. • If j ∘ i is a H_1-regular immersion, so is i. • If both j and j ∘ i are Koszul-regular immersions, so is i.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of\nalgebraic spaces over $S$. Assume $j$ is locally of finite presentation\nand that the sequence\n$$\n0 \\to i^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nof Lemma \\ref{lemma-transitivity-conormal} is exact and locally split.\n\\begin{enumerate}\n\\item If $j \\circ i$ is a quasi-regular immersion, so is $i$.\n\\item If $j \\circ i$ is a $H_1$-regular immersion, so is $i$.\n\\item If both $j$ and $j \\circ i$ are Koszul-regular immersions, so is $i$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09S1","source_file":"spaces-more-morphisms.tex","source_line":10672,"source_end_line":10688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10672-L10688","statement_sha256":"698f941691a2b48730a749c6581bc722cb82bc9af3ef52bdc0bf13869bdcfb5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12229,"rank":12229,"depth":55,"x":590.308,"y":1677.16,"cluster":"geometry-of-spaces"},{"id":"stacks:09S2","tag":"09S2","title":"Regular immersions · Lemma 09S2","summary":"Let S be a scheme. Let i : Z → Y and j : Y → X be immersions of algebraic spaces over S. Assume X is locally Noetherian. The following are equivalent • i and j are Koszul regular immersions, • i and j ∘ i are Koszul regular immersions, • j ∘ i is a Koszul regular immersion and the conormal sequence 0 → i^*C_Y/X → C_Z/X → C_Z/Y → 0 is exact and locally split.","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to Y$ and $j : Y \\to X$ be immersions of\nalgebraic spaces over $S$. Assume $X$ is locally Noetherian.\nThe following are equivalent\n\\begin{enumerate}\n\\item $i$ and $j$ are Koszul regular immersions,\n\\item $i$ and $j \\circ i$ are Koszul regular immersions,\n\\item $j \\circ i$ is a Koszul regular immersion and the conormal sequence\n$$\n0 \\to i^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nis exact and locally split.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Regular immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09S2","source_file":"spaces-more-morphisms.tex","source_line":10695,"source_end_line":10711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10695-L10711","statement_sha256":"b371020f2ba9d0ed7c762fc65c67bbbf83c1ca02cd62336814bcdf289ae594a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12230,"rank":12230,"depth":16,"x":684.091,"y":1492.115,"cluster":"geometry-of-spaces"},{"id":"stacks:0CSX","tag":"0CSX","title":"Relative pseudo-coherence · Lemma 0CSX","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let m ∈ Z. Let E ∈ D_QCoh(O_X). With notation as explained in Remark [Tag 0CSW] the following are equivalent: • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y where U, V are schemes and the vertical arrows are étale, the complex E|_U is m-pseudo-coherent relative to V, • for some commutative diagram as in (1) with U → X surjective, the…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is locally of finite type. Let $m \\in \\mathbf{Z}$.\nLet $E \\in D_\\QCoh(\\mathcal{O}_X)$. With notation as explained in\nRemark \\ref{remark-match-relative-pseudo-coherence}\nthe following are equivalent:\n\\begin{enumerate}\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale, the complex\n$E|_U$ is $m$-pseudo-coherent relative to $V$,\n\\item for some commutative diagram as in (1) with $U \\to X$\nsurjective, the complex $E|_U$ is $m$-pseudo-coherent relative to $V$,\n\\item for every commutative diagram as in (1) with $U$ and $V$\naffine the complex $R\\Gamma(U, E)$ of $\\mathcal{O}_X(U)$-modules\nis $m$-pseudo-coherent relative to $\\mathcal{O}_Y(V)$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSX","source_file":"spaces-more-morphisms.tex","source_line":10761,"source_end_line":10784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10761-L10784","statement_sha256":"d6e1f828a6280739eebb54da4a8d10e7b9a14a5b0c28ce82a5700959ec3dbc78","origin":"The Stacks Project","memory_eligible":false,"source_rank":12231,"rank":12231,"depth":43,"x":763.814,"y":1681.946,"cluster":"geometry-of-spaces"},{"id":"stacks:0CSY","tag":"0CSY","title":"Relative pseudo-coherence · Definition 0CSY","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let E be an object of D_QCoh(O_X). Let F be a quasi-coherent O_X-module. Fix m ∈ Z. • We say E is m-pseudo-coherent relative to Y if the equivalent conditions of Lemma [Tag 0CSX] are satisfied. • We say E is pseudo-coherent relative to Y if E is m-pseudo-coherent relative to Y for all m ∈ Z. • We say F is m-pseudo-coherent relative to Y if F viewed as an object of…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $S$ which is locally of finite type.\nLet $E$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. Let $\\mathcal{F}$ be a\nquasi-coherent $\\mathcal{O}_X$-module. Fix $m \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item We say $E$ is {\\it $m$-pseudo-coherent relative to $Y$}\nif the equivalent conditions of\nLemma \\ref{lemma-qcoh-relative-pseudo-coherence-characterize} are satisfied.\n\\item We say $E$ is {\\it pseudo-coherent relative to $Y$}\nif $E$ is $m$-pseudo-coherent relative to $Y$ for all $m \\in \\mathbf{Z}$.\n\\item We say $\\mathcal{F}$ is {\\it $m$-pseudo-coherent relative to $Y$} if\n$\\mathcal{F}$ viewed as an object of $D_\\QCoh(\\mathcal{O}_X)$ is\n$m$-pseudo-coherent relative to $Y$.\n\\item We say $\\mathcal{F}$ is {\\it pseudo-coherent relative to $Y$} if\n$\\mathcal{F}$ viewed as an object of $D_\\QCoh(\\mathcal{O}_X)$ is\npseudo-coherent relative to $Y$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relative pseudo-coherence","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CSY","source_file":"spaces-more-morphisms.tex","source_line":10833,"source_end_line":10852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10833-L10852","statement_sha256":"008239baa6030d3279d7aa7eb20d29e2aaa3d4481aff0b295b2574fb198e48e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12232,"rank":12232,"depth":44,"x":552.187,"y":1587.122,"cluster":"geometry-of-spaces"},{"id":"stacks:0DII","tag":"0DII","title":"Relative pseudo-coherence · Lemma 0DII","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let E in D_QCoh(O_X). If f is flat and locally of finite presentation, then the following are equivalent • E is pseudo-coherent relative to Y, and • E is pseudo-coherent on X.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $S$. Let $E$ in $D_\\QCoh(\\mathcal{O}_X)$.\nIf $f$ is flat and locally of finite presentation, then\nthe following are equivalent\n\\begin{enumerate}\n\\item $E$ is pseudo-coherent relative to $Y$, and\n\\item $E$ is pseudo-coherent on $X$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relative pseudo-coherence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DII","source_file":"spaces-more-morphisms.tex","source_line":10859,"source_end_line":10869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10859-L10869","statement_sha256":"ed5acce4a3beb65f72989949e9604dab353fbf7c5334ddfe9ed0534c3fa90215","origin":"The Stacks Project","memory_eligible":false,"source_rank":12233,"rank":12233,"depth":37,"x":784.695,"y":1536.917,"cluster":"geometry-of-spaces"},{"id":"stacks:06BR","tag":"06BR","title":"Pseudo-coherent morphisms · Definition 06BR","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is pseudo-coherent if the equivalent conditions of Morphisms of Spaces, Lemma [Tag 03MJ] hold with P =\"pseudo-coherent\". • Let x ∈ |X|. We say f is pseudo-coherent at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is pseudo-coherent.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it pseudo-coherent} if the equivalent conditions of\nMorphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target}\nhold with $\\mathcal{P} =$``pseudo-coherent''.\n\\item Let $x \\in |X|$. We say $f$ is {\\it pseudo-coherent at $x$} if\nthere exists an open neighbourhood $X' \\subset X$ of $x$ such\nthat $f|_{X'} : X' \\to Y$ is pseudo-coherent.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Pseudo-coherent morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BR","source_file":"spaces-more-morphisms.tex","source_line":10919,"source_end_line":10931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10919-L10931","statement_sha256":"5b2965926e028025dc3f266f7355d444a116dc7d8537508709ca37385b2166bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12234,"rank":12234,"depth":46,"x":653.506,"y":1706.015,"cluster":"geometry-of-spaces"},{"id":"stacks:06BS","tag":"06BS","title":"Pseudo-coherent morphisms · Lemma 06BS","summary":"A flat base change of a pseudo-coherent morphism is pseudo-coherent.","statement_latex":"A flat base change of a pseudo-coherent morphism is pseudo-coherent.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BS","source_file":"spaces-more-morphisms.tex","source_line":10937,"source_end_line":10940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10937-L10940","statement_sha256":"e2655591472dc15ff273008f78e93acbd65b35010e7c0cb0e0a423f7b4a156b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12235,"rank":12235,"depth":13,"x":614.225,"y":1506.712,"cluster":"geometry-of-spaces"},{"id":"stacks:06BT","tag":"06BT","title":"Pseudo-coherent morphisms · Lemma 06BT","summary":"A composition of pseudo-coherent morphisms is pseudo-coherent.","statement_latex":"A composition of pseudo-coherent morphisms is pseudo-coherent.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BT","source_file":"spaces-more-morphisms.tex","source_line":10948,"source_end_line":10951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10948-L10951","statement_sha256":"8dd45db9e1c9db1916e77409525932f528ec4270489da0b63715819d5ce83519","origin":"The Stacks Project","memory_eligible":false,"source_rank":12236,"rank":12236,"depth":14,"x":803.628,"y":1631.494,"cluster":"geometry-of-spaces"},{"id":"stacks:06BU","tag":"06BU","title":"Pseudo-coherent morphisms · Lemma 06BU","summary":"A pseudo-coherent morphism is locally of finite presentation.","statement_latex":"A pseudo-coherent morphism is locally of finite presentation.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BU","source_file":"spaces-more-morphisms.tex","source_line":10959,"source_end_line":10962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10959-L10962","statement_sha256":"6c1b6570e88dc9c4a15994774025e848d5c7aaf43b168827173378c7cbd75ee1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12237,"rank":12237,"depth":0,"x":563.411,"y":1646.967,"cluster":"geometry-of-spaces"},{"id":"stacks:06BV","tag":"06BV","title":"Pseudo-coherent morphisms · Lemma 06BV","summary":"A flat morphism which is locally of finite presentation is pseudo-coherent.","statement_latex":"A flat morphism which is locally of finite presentation is pseudo-coherent.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BV","source_file":"spaces-more-morphisms.tex","source_line":10968,"source_end_line":10971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10968-L10971","statement_sha256":"9ea99dfec6791cdb392e352a671a3d2cf267d7c770dce2e5a653813fff50ad19","origin":"The Stacks Project","memory_eligible":false,"source_rank":12238,"rank":12238,"depth":37,"x":728.243,"y":1499.125,"cluster":"geometry-of-spaces"},{"id":"stacks:06BW","tag":"06BW","title":"Pseudo-coherent morphisms · Lemma 06BW","summary":"Let f : X → Y be a morphism of algebraic spaces pseudo-coherent over a base algebraic space B. Then f is pseudo-coherent.","statement_latex":"Let $f : X \\to Y$ be a morphism of algebraic spaces pseudo-coherent\nover a base algebraic space $B$. Then $f$ is pseudo-coherent.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BW","source_file":"spaces-more-morphisms.tex","source_line":10979,"source_end_line":10983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10979-L10983","statement_sha256":"daf44b1577dcb243970a1b8a01a91ba6383d1b1f067909a9deb7487dc792c1d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12239,"rank":12239,"depth":14,"x":725.587,"y":1701.846,"cluster":"geometry-of-spaces"},{"id":"stacks:06BX","tag":"06BX","title":"Pseudo-coherent morphisms · Lemma 06BX","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If Y is locally Noetherian, then f is pseudo-coherent if and only if f is locally of finite type.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. If $Y$ is locally Noetherian, then $f$ is pseudo-coherent if\nand only if $f$ is locally of finite type.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Pseudo-coherent morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BX","source_file":"spaces-more-morphisms.tex","source_line":10991,"source_end_line":10996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L10991-L10996","statement_sha256":"36f64f35f0c4b79e2865c388124080c199414cc39e5ade7016ec570553af68d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12240,"rank":12240,"depth":14,"x":564.384,"y":1550.725,"cluster":"geometry-of-spaces"},{"id":"stacks:06BZ","tag":"06BZ","title":"Perfect morphisms · Definition 06BZ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is perfect if the equivalent conditions of Morphisms of Spaces, Lemma [Tag 03MJ] hold with P =\"perfect\". • Let x ∈ |X|. We say f is perfect at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is perfect.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it perfect} if the equivalent conditions of\nMorphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target}\nhold with $\\mathcal{P} =$``perfect''.\n\\item Let $x \\in |X|$. We say $f$ is {\\it perfect at $x$} if\nthere exists an open neighbourhood $X' \\subset X$ of $x$ such\nthat $f|_{X'} : X' \\to Y$ is perfect.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Perfect morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06BZ","source_file":"spaces-more-morphisms.tex","source_line":11036,"source_end_line":11048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11036-L11048","statement_sha256":"234d533bff57c570d2d47ecf517de42c081bcfd9f5f31ee04ceab8aa955d4b07","origin":"The Stacks Project","memory_eligible":false,"source_rank":12241,"rank":12241,"depth":46,"x":804.987,"y":1570.706,"cluster":"geometry-of-spaces"},{"id":"stacks:06C0","tag":"06C0","title":"Perfect morphisms · Lemma 06C0","summary":"A flat base change of a perfect morphism is perfect.","statement_latex":"A flat base change of a perfect morphism is perfect.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06C0","source_file":"spaces-more-morphisms.tex","source_line":11055,"source_end_line":11058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11055-L11058","statement_sha256":"5c36f0a840c7e616f858c0e90e07205247f5659a225749218abbe4b55fafaa23","origin":"The Stacks Project","memory_eligible":false,"source_rank":12242,"rank":12242,"depth":14,"x":611.335,"y":1692.601,"cluster":"geometry-of-spaces"},{"id":"stacks:06C1","tag":"06C1","title":"Perfect morphisms · Lemma 06C1","summary":"A composition of perfect morphisms is perfect.","statement_latex":"A composition of perfect morphisms is perfect.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06C1","source_file":"spaces-more-morphisms.tex","source_line":11066,"source_end_line":11069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11066-L11069","statement_sha256":"3df13a64d8e5ce1b0a7662dfde924216f75d61286d67386bc178d8f0001a7709","origin":"The Stacks Project","memory_eligible":false,"source_rank":12243,"rank":12243,"depth":15,"x":656.145,"y":1492.664,"cluster":"geometry-of-spaces"},{"id":"stacks:06C2","tag":"06C2","title":"Perfect morphisms · Lemma 06C2","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f is flat and perfect, and • f is flat and locally of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over\n$S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is flat and perfect, and\n\\item $f$ is flat and locally of finite presentation.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06C2","source_file":"spaces-more-morphisms.tex","source_line":11077,"source_end_line":11085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11077-L11085","statement_sha256":"66417d2aeb1e752bb75b86b3df6c69314062084ac54bce096e28b95f3cc99c0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12244,"rank":12244,"depth":37,"x":783.996,"y":1665.668,"cluster":"geometry-of-spaces"},{"id":"stacks:0E4U","tag":"0E4U","title":"Perfect morphisms · Lemma 0E4U","summary":"Let S be a scheme. Let Y be a Noetherian algebraic space over S. Let f : X → Y be a perfect proper morphism of algebraic spaces. Let E ∈ D(O_X) be perfect. Then Rf_*E is a perfect object of D(O_Y).","statement_latex":"Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$.\nLet $f : X \\to Y$ be a perfect proper morphism of algebraic spaces.\nLet $E \\in D(\\mathcal{O}_X)$ be perfect. Then\n$Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Perfect morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4U","source_file":"spaces-more-morphisms.tex","source_line":11093,"source_end_line":11099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11093-L11099","statement_sha256":"a97b749f04eec5a230eb5e1e178afa4a79a06d9931c62621dc07d64be6f335eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12245,"rank":12245,"depth":66,"x":550.396,"y":1610.596,"cluster":"geometry-of-spaces"},{"id":"stacks:06C4","tag":"06C4","title":"Local complete intersection morphisms · Definition 06C4","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. • We say f is a Koszul morphism, or that f is a local complete intersection morphism if the equivalent conditions of Morphisms of Spaces, Lemma [Tag 03MJ] hold with P(f) =\"f is a local complete intersection morphism\". • Let x ∈ |X|. We say f is Koszul at x if there exists an open neighbourhood X' ⊂ X of x such that f|_X' : X' → Y is a local complete intersection morphism.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is a {\\it Koszul morphism}, or that $f$ is a\n{\\it local complete intersection morphism} if the equivalent conditions of\nMorphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target}\nhold with $\\mathcal{P}(f) =$``$f$ is a local complete intersection morphism''.\n\\item Let $x \\in |X|$. We say $f$ is {\\it Koszul at $x$} if\nthere exists an open neighbourhood $X' \\subset X$ of $x$ such\nthat $f|_{X'} : X' \\to Y$ is a local complete intersection morphism.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06C4","source_file":"spaces-more-morphisms.tex","source_line":11152,"source_end_line":11165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11152-L11165","statement_sha256":"7836a55eab0f449d5a9144371ec752af18b4731e56677fdfb2c8ebcc44858fc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12246,"rank":12246,"depth":46,"x":767.121,"y":1518.577,"cluster":"geometry-of-spaces"},{"id":"stacks:06C5","tag":"06C5","title":"Local complete intersection morphisms · Lemma 06C5","summary":"Let S be a scheme. Let f : X → Y be a local complete intersection morphism of algebraic spaces over S. Let P be an algebraic space smooth over Y. Let U → X be an étale morphism of algebraic spaces and let i : U → P an immersion of algebraic spaces over Y. Picture: xymatrix X ar[rd] & U ar[l] ar[d] ar[r]_i & P ar[ld] & Y Then i is a Koszul-regular immersion of algebraic spaces.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a local complete intersection morphism\nof algebraic spaces over $S$.\nLet $P$ be an algebraic space smooth over $Y$.\nLet $U \\to X$ be an \\'etale morphism of algebraic spaces\nand let $i : U \\to P$ an immersion of algebraic spaces over $Y$.\nPicture:\n$$\n\\xymatrix{\nX \\ar[rd] & U \\ar[l] \\ar[d] \\ar[r]_i & P \\ar[ld] \\\\\n& Y\n}\n$$\nThen $i$ is a Koszul-regular immersion of algebraic spaces.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06C5","source_file":"spaces-more-morphisms.tex","source_line":11171,"source_end_line":11187,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11171-L11187","statement_sha256":"f39e7c1230078b548eb19f5380abb22f5b17e65ad81554c2fcb76ed9e2bebdc2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12247,"rank":12247,"depth":47,"x":681.237,"y":1709.568,"cluster":"geometry-of-spaces"},{"id":"stacks:06C6","tag":"06C6","title":"Local complete intersection morphisms · Lemma 06C6","summary":"Let S be a scheme. Let f : X → Y be a local complete intersection morphism of algebraic spaces over S. Then • f is locally of finite presentation, • f is pseudo-coherent, and • f is perfect.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a local complete intersection\nmorphism of algebraic spaces over $S$. Then\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $f$ is pseudo-coherent, and\n\\item $f$ is perfect.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06C6","source_file":"spaces-more-morphisms.tex","source_line":11207,"source_end_line":11216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11207-L11216","statement_sha256":"649432a3b8190fc7d3027cae946d111d10fc7130feaf3013503aed884fc05161","origin":"The Stacks Project","memory_eligible":false,"source_rank":12248,"rank":12248,"depth":38,"x":590.901,"y":1519.841,"cluster":"geometry-of-spaces"},{"id":"stacks:06C7","tag":"06C7","title":"Local complete intersection morphisms · Lemma 06C7","summary":"A flat base change of a local complete intersection morphism is a local complete intersection morphism.","statement_latex":"A flat base change of a local complete intersection morphism is a\nlocal complete intersection morphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06C7","source_file":"spaces-more-morphisms.tex","source_line":11227,"source_end_line":11231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11227-L11231","statement_sha256":"4939fc818834b6cbedb15f33ca90b78fd08862c8bca0b3a1e103a45fc1ba6036","origin":"The Stacks Project","memory_eligible":false,"source_rank":12249,"rank":12249,"depth":17,"x":810.273,"y":1608.559,"cluster":"geometry-of-spaces"},{"id":"stacks:06C8","tag":"06C8","title":"Local complete intersection morphisms · Lemma 06C8","summary":"A composition of local complete intersection morphisms is a local complete intersection morphism.","statement_latex":"A composition of local complete intersection morphisms is a\nlocal complete intersection morphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06C8","source_file":"spaces-more-morphisms.tex","source_line":11239,"source_end_line":11243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11239-L11243","statement_sha256":"669d2ce0855477e79c8ef0168ff0fe0edd7da5a7701d202289fcbf1ee1b5a237","origin":"The Stacks Project","memory_eligible":false,"source_rank":12250,"rank":12250,"depth":39,"x":576.969,"y":1667.666,"cluster":"geometry-of-spaces"},{"id":"stacks:06C9","tag":"06C9","title":"Local complete intersection morphisms · Lemma 06C9","summary":"Syntomic equals flat plus lci (for algebraic spaces). Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent • f is flat and a local complete intersection morphism, and • f is syntomic.","statement_latex":"\\begin{slogan}\nSyntomic equals flat plus lci (for algebraic spaces).\n\\end{slogan}\nLet $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is flat and a local complete intersection morphism, and\n\\item $f$ is syntomic.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06C9","source_file":"spaces-more-morphisms.tex","source_line":11251,"source_end_line":11263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11251-L11263","statement_sha256":"ac0eee070f1c53e27d3b050db85654f27d3f3d89313079c2bbcaaccf10d5f900","origin":"The Stacks Project","memory_eligible":false,"source_rank":12251,"rank":12251,"depth":39,"x":701.576,"y":1491.551,"cluster":"geometry-of-spaces"},{"id":"stacks:0CHK","tag":"0CHK","title":"Local complete intersection morphisms · Lemma 0CHK","summary":"Let S be a scheme. A Koszul-regular immersion of algebraic spaces over S is a local complete intersection morphism.","statement_latex":"Let $S$ be a scheme. A Koszul-regular immersion of algebraic spaces\nover $S$ is a local complete intersection morphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHK","source_file":"spaces-more-morphisms.tex","source_line":11271,"source_end_line":11275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11271-L11275","statement_sha256":"5935c510986e75f0a5d5ba9e6063789eb4530be7b7104ad27c7e5f22deb65117","origin":"The Stacks Project","memory_eligible":false,"source_rank":12252,"rank":12252,"depth":47,"x":751.362,"y":1692.29,"cluster":"geometry-of-spaces"},{"id":"stacks:0CHL","tag":"0CHL","title":"Local complete intersection morphisms · Lemma 0CHL","summary":"Let S be a scheme. Let xymatrix X ar[rr]_f ar[rd] & & Y ar[ld] & Z be a commutative diagram of morphisms of algebraic spaces over S. Assume Y → Z is smooth and X → Z is a local complete intersection morphism. Then f : X → Y is a local complete intersection morphism.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd] & & Y \\ar[ld] \\\\\n& Z\n}\n$$\nbe a commutative diagram of morphisms of algebraic spaces over $S$.\nAssume $Y \\to Z$ is smooth and $X \\to Z$ is a\nlocal complete intersection morphism.\nThen $f : X \\to Y$ is a local complete intersection morphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHL","source_file":"spaces-more-morphisms.tex","source_line":11289,"source_end_line":11302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11289-L11302","statement_sha256":"121eba55baf152c75f4ae01f87a9b85b69314c4894febb8d68f8e3646b33d597","origin":"The Stacks Project","memory_eligible":false,"source_rank":12253,"rank":12253,"depth":1,"x":553.055,"y":1572.416,"cluster":"geometry-of-spaces"},{"id":"stacks:0CHM","tag":"0CHM","title":"Local complete intersection morphisms · Lemma 0CHM","summary":"The property P(f) =\"f is a local complete intersection morphism\" is fpqc local on the base.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection\nmorphism'' is fpqc local on the base.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHM","source_file":"spaces-more-morphisms.tex","source_line":11315,"source_end_line":11319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11315-L11319","statement_sha256":"5f34d21469d19928941013d81445ed615cc6b3e5314be16bb4834b2a3a852d06","origin":"The Stacks Project","memory_eligible":false,"source_rank":12254,"rank":12254,"depth":40,"x":795.886,"y":1548.265,"cluster":"geometry-of-spaces"},{"id":"stacks:0CHN","tag":"0CHN","title":"Local complete intersection morphisms · Lemma 0CHN","summary":"The property P(f) =\"f is a local complete intersection morphism\" is syntomic local on the source.","statement_latex":"The property $\\mathcal{P}(f) =$``$f$ is a local complete intersection\nmorphism'' is syntomic local on the source.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHN","source_file":"spaces-more-morphisms.tex","source_line":11370,"source_end_line":11374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11370-L11374","statement_sha256":"ce4e46da168bdbd136158335dbd39aa3f1b3b74fcf604b753e610e632e2c24bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12255,"rank":12255,"depth":48,"x":636.115,"y":1703.992,"cluster":"geometry-of-spaces"},{"id":"stacks:06CA","tag":"06CA","title":"Local complete intersection morphisms · Lemma 06CA","summary":"Let S be a scheme. Consider a commutative diagram xymatrix X ar[rr]_f ar[rd]_p & & Y ar[ld]^q & Z of algebraic spaces over S. Assume that both p and q are flat and locally of finite presentation. Then there exists an open subspace U(f) ⊂ X such that |U(f)| ⊂ |X| is the set of points where f is Koszul. Moreover, for any morphism of algebraic spaces Z' → Z, if f' : X' → Y' is the base change of f by Z' → Z, then U(f') is the inverse image of U(f) under the projection X' → X.","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\\n& Z\n}\n$$\nof algebraic spaces over $S$. Assume that both $p$ and $q$\nare flat and locally of finite presentation.\nThen there exists an open subspace $U(f) \\subset X$\nsuch that $|U(f)| \\subset |X|$ is the set of points where $f$ is Koszul.\nMoreover, for any morphism of algebraic spaces $Z' \\to Z$, if\n$f' : X' \\to Y'$ is the base change of $f$ by $Z' \\to Z$, then\n$U(f')$ is the inverse image of $U(f)$ under the projection $X' \\to X$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CA","source_file":"spaces-more-morphisms.tex","source_line":11382,"source_end_line":11398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11382-L11398","statement_sha256":"66c7f0e07587438f535d27b6fc622c71705cf968eb48c502f0bb9f99d905c85f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12256,"rank":12256,"depth":47,"x":628.69,"y":1498.331,"cluster":"geometry-of-spaces"},{"id":"stacks:06CB","tag":"06CB","title":"Local complete intersection morphisms · Lemma 06CB","summary":"Let S be a scheme. Let f : X → Y be a local complete intersection morphism of algebraic spaces over S. Then f is unramified if and only if f is formally unramified and in this case the conormal sheaf C_X/Y is finite locally free on X.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a local complete intersection\nmorphism of algebraic spaces over $S$. Then $f$ is unramified if and only\nif $f$ is formally unramified and in this case the conormal sheaf\n$\\mathcal{C}_{X/Y}$ is finite locally free on $X$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CB","source_file":"spaces-more-morphisms.tex","source_line":11440,"source_end_line":11446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11440-L11446","statement_sha256":"ab57564892207d92a9572ec32e22b9563af0453b7f9cff0f073cbdc8ffdee16f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12257,"rank":12257,"depth":59,"x":799.695,"y":1645.893,"cluster":"geometry-of-spaces"},{"id":"stacks:06CC","tag":"06CC","title":"Local complete intersection morphisms · Lemma 06CC","summary":"Let S be a scheme. Let Z → Y → X be formally unramified morphisms of algebraic spaces over S. Assume that Z → Y is a local complete intersection morphism. The exact sequence 0 → i^*C_Y/X → C_Z/X → C_Z/Y → 0 of Lemma [Tag 06BD] is short exact.","statement_latex":"Let $S$ be a scheme. Let $Z \\to Y \\to X$ be formally unramified morphisms\nof algebraic spaces over $S$. Assume that $Z \\to Y$ is a local complete\nintersection morphism. The exact sequence\n$$\n0 \\to i^*\\mathcal{C}_{Y/X} \\to\n\\mathcal{C}_{Z/X} \\to\n\\mathcal{C}_{Z/Y} \\to 0\n$$\nof\nLemma \\ref{lemma-transitivity-conormal}\nis short exact.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CC","source_file":"spaces-more-morphisms.tex","source_line":11457,"source_end_line":11470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11457-L11470","statement_sha256":"a54130c93a08344e670e924e951e8cf493d48375e0e31833f436a2c756deb598","origin":"The Stacks Project","memory_eligible":false,"source_rank":12258,"rank":12258,"depth":59,"x":554.729,"y":1634.105,"cluster":"geometry-of-spaces"},{"id":"stacks:05X8","tag":"05X8","title":"When is a morphism an isomorphism? · Lemma 05X8","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd]_p & & Y ar[ld]^q & Z of algebraic spaces. Assume that p is locally of finite type and closed. Then there exists an open subspace W ⊂ Z such that a morphism Z' → Z factors through W if and only if the base change f_Z' : X_Z' → Y_Z' is unramified.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\\n& Z\n}\n$$\nof algebraic spaces. Assume that $p$ is locally of finite type and closed.\nThen there exists an open subspace $W \\subset Z$\nsuch that a morphism $Z' \\to Z$ factors through $W$ if and only if the\nbase change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is unramified.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"When is a morphism an isomorphism?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05X8","source_file":"spaces-more-morphisms.tex","source_line":11528,"source_end_line":11541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11528-L11541","statement_sha256":"685314c04c1dbf1a7e9de6c54cfd5a8e53bfe3a009f3a31a08f2c111084bf7f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12259,"rank":12259,"depth":48,"x":745.0,"y":1503.689,"cluster":"geometry-of-spaces"},{"id":"stacks:05X9","tag":"05X9","title":"When is a morphism an isomorphism? · Lemma 05X9","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd]_p & & Y ar[ld]^q & Z of algebraic spaces. Assume that • p is locally of finite type, • p is closed, and • p_2 : X ×_Y X → Z is closed. Then there exists an open subspace W ⊂ Z such that a morphism Z' → Z factors through W if and only if the base change f_Z' : X_Z' → Y_Z' is unramified and universally injective.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\\n& Z\n}\n$$\nof algebraic spaces. Assume that\n\\begin{enumerate}\n\\item $p$ is locally of finite type,\n\\item $p$ is closed, and\n\\item $p_2 : X \\times_Y X \\to Z$ is closed.\n\\end{enumerate}\nThen there exists an open subspace $W \\subset Z$\nsuch that a morphism $Z' \\to Z$ factors through $W$ if and only if the\nbase change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is unramified and universally\ninjective.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"When is a morphism an isomorphism?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05X9","source_file":"spaces-more-morphisms.tex","source_line":11565,"source_end_line":11584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11565-L11584","statement_sha256":"b31b269bf4ab0d90a6a8cdb461dd7c57353f5bdce7ef0e126684119efa92a497","origin":"The Stacks Project","memory_eligible":false,"source_rank":12260,"rank":12260,"depth":55,"x":709.545,"y":1707.99,"cluster":"geometry-of-spaces"},{"id":"stacks:05XA","tag":"05XA","title":"When is a morphism an isomorphism? · Lemma 05XA","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd]_p & & Y ar[ld]^q & Z of algebraic spaces. Assume that • p is locally of finite type, • p is universally closed, and • q : Y → Z is separated. Then there exists an open subspace W ⊂ Z such that a morphism Z' → Z factors through W if and only if the base change f_Z' : X_Z' → Y_Z' is a closed immersion.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\\n& Z\n}\n$$\nof algebraic spaces. Assume that\n\\begin{enumerate}\n\\item $p$ is locally of finite type,\n\\item $p$ is universally closed, and\n\\item $q : Y \\to Z$ is separated.\n\\end{enumerate}\nThen there exists an open subspace $W \\subset Z$\nsuch that a morphism $Z' \\to Z$ factors through $W$ if and only if the\nbase change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is a closed immersion.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"When is a morphism an isomorphism?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XA","source_file":"spaces-more-morphisms.tex","source_line":11641,"source_end_line":11659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11641-L11659","statement_sha256":"f3a94ea9bff2b4a097740c85cee0cba057daf5d6da9222f83ca9bb9a127cb58c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12261,"rank":12261,"depth":58,"x":571.281,"y":1537.082,"cluster":"geometry-of-spaces"},{"id":"stacks:05XB","tag":"05XB","title":"When is a morphism an isomorphism? · Lemma 05XB","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd]_p & & Y ar[ld]^q & Z of algebraic spaces. Assume that • p is locally of finite presentation, • p is flat, • p is closed, and • q is locally of finite type. Then there exists an open subspace W ⊂ Z such that a morphism Z' → Z factors through W if and only if the base change f_Z' : X_Z' → Y_Z' is flat.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\\n& Z\n}\n$$\nof algebraic spaces. Assume that\n\\begin{enumerate}\n\\item $p$ is locally of finite presentation,\n\\item $p$ is flat,\n\\item $p$ is closed, and\n\\item $q$ is locally of finite type.\n\\end{enumerate}\nThen there exists an open subspace $W \\subset Z$\nsuch that a morphism $Z' \\to Z$ factors through $W$ if and only if the\nbase change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is flat.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"When is a morphism an isomorphism?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XB","source_file":"spaces-more-morphisms.tex","source_line":11686,"source_end_line":11705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11686-L11705","statement_sha256":"cb5a98058ea1cabdf5f7f2b117c2baf1314bc75ec323f5c3e70b5b2a52110509","origin":"The Stacks Project","memory_eligible":false,"source_rank":12262,"rank":12262,"depth":59,"x":810.873,"y":1584.693,"cluster":"geometry-of-spaces"},{"id":"stacks:05XC","tag":"05XC","title":"When is a morphism an isomorphism? · Lemma 05XC","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd]_p & & Y ar[ld]^q & Z of algebraic spaces. Assume that • p is locally of finite presentation, • p is flat, • p is closed, • q is locally of finite type, and • q is closed. Then there exists an open subspace W ⊂ Z such that a morphism Z' → Z factors through W if and only if the base change f_Z' : X_Z' → Y_Z' is surjective and flat.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\\n& Z\n}\n$$\nof algebraic spaces. Assume that\n\\begin{enumerate}\n\\item $p$ is locally of finite presentation,\n\\item $p$ is flat,\n\\item $p$ is closed,\n\\item $q$ is locally of finite type, and\n\\item $q$ is closed.\n\\end{enumerate}\nThen there exists an open subspace $W \\subset Z$\nsuch that a morphism $Z' \\to Z$ factors through $W$ if and only if the\nbase change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is surjective and flat.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"When is a morphism an isomorphism?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XC","source_file":"spaces-more-morphisms.tex","source_line":11728,"source_end_line":11748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11728-L11748","statement_sha256":"42ca41bcaa08968848c88b7badf90f8b4d817b821194fadb069b3b4a133367bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12263,"rank":12263,"depth":60,"x":595.736,"y":1685.618,"cluster":"geometry-of-spaces"},{"id":"stacks:05XD","tag":"05XD","title":"When is a morphism an isomorphism? · Lemma 05XD","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd]_p & & Y ar[ld]^q & Z of algebraic spaces. Assume that • p is locally of finite presentation, • p is flat, • p is universally closed, • q is locally of finite type, • q is closed, and • q is separated. Then there exists an open subspace W ⊂ Z such that a morphism Z' → Z factors through W if and only if the base change f_Z' : X_Z' → Y_Z' is an isomorphism.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\\n& Z\n}\n$$\nof algebraic spaces. Assume that\n\\begin{enumerate}\n\\item $p$ is locally of finite presentation,\n\\item $p$ is flat,\n\\item $p$ is universally closed,\n\\item $q$ is locally of finite type,\n\\item $q$ is closed, and\n\\item $q$ is separated.\n\\end{enumerate}\nThen there exists an open subspace $W \\subset Z$\nsuch that a morphism $Z' \\to Z$ factors through $W$ if and only if the\nbase change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is an isomorphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"When is a morphism an isomorphism?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XD","source_file":"spaces-more-morphisms.tex","source_line":11791,"source_end_line":11812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11791-L11812","statement_sha256":"36e008551dd7150f9185326091777c42fffd783df9f4dedd1177f9339524fbfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12264,"rank":12264,"depth":61,"x":673.277,"y":1488.962,"cluster":"geometry-of-spaces"},{"id":"stacks:06CE","tag":"06CE","title":"When is a morphism an isomorphism? · Lemma 06CE","summary":"Consider a commutative diagram xymatrix X ar[rr]_f ar[rd]_p & & Y ar[ld]^q & Z of algebraic spaces. Assume that • p is flat and locally of finite presentation, • p is closed, and • q is flat and locally of finite presentation, Then there exists an open subspace W ⊂ Z such that a morphism Z' → Z factors through W if and only if the base change f_Z' : X_Z' → Y_Z' is a local complete intersection morphism.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nX \\ar[rr]_f \\ar[rd]_p & & Y \\ar[ld]^q \\\\\n& Z\n}\n$$\nof algebraic spaces. Assume that\n\\begin{enumerate}\n\\item $p$ is flat and locally of finite presentation,\n\\item $p$ is closed, and\n\\item $q$ is flat and locally of finite presentation,\n\\end{enumerate}\nThen there exists an open subspace $W \\subset Z$\nsuch that a morphism $Z' \\to Z$ factors through $W$ if and only if the\nbase change $f_{Z'} : X_{Z'} \\to Y_{Z'}$ is a local complete intersection\nmorphism.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"When is a morphism an isomorphism?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CE","source_file":"spaces-more-morphisms.tex","source_line":11835,"source_end_line":11854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11835-L11854","statement_sha256":"d24a609e81b68a2bea4abf3dc468dd4dc6b29121ab1475c30e9a1ba9df129e62","origin":"The Stacks Project","memory_eligible":false,"source_rank":12265,"rank":12265,"depth":48,"x":774.329,"y":1678.128,"cluster":"geometry-of-spaces"},{"id":"stacks:0CTQ","tag":"0CTQ","title":"Characterizing pseudo-coherent complexes, II · Lemma 0CTQ","summary":"Let S be a scheme. Consider a commutative diagram of algebraic spaces xymatrix Z' ar[d] ar[r] & Y' ar[d] X' ar[r] & B' over S. Let B → B' be a morphism. Denote by X and Y the base changes of X' and Y' to B. Assume Y' → B' and Z' → X' are flat. Then X ×_B Y and Z' are Tor independent over X' ×_B' Y'.","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram of algebraic spaces\n$$\n\\xymatrix{\nZ' \\ar[d] \\ar[r] & Y' \\ar[d] \\\\\nX' \\ar[r] & B'\n}\n$$\nover $S$.\nLet $B \\to B'$ be a morphism. Denote by $X$ and $Y$ the base\nchanges of $X'$ and $Y'$ to $B$.\nAssume $Y' \\to B'$ and $Z' \\to X'$ are flat.\nThen $X \\times_B Y$ and $Z'$ are Tor independent over $X' \\times_{B'} Y'$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Characterizing pseudo-coherent complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTQ","source_file":"spaces-more-morphisms.tex","source_line":11968,"source_end_line":11982,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L11968-L11982","statement_sha256":"f347318c69a3f1e3ead56d7459def36c486d52d140070ecb0fef62367200c294","origin":"The Stacks Project","memory_eligible":false,"source_rank":12266,"rank":12266,"depth":54,"x":547.506,"y":1595.911,"cluster":"geometry-of-spaces"},{"id":"stacks:0CTR","tag":"0CTR","title":"Derived Chow's lemma · Lemma 0CTR","summary":"Let A be a ring. Let X be a separated algebraic space of finite presentation over A. Let x ∈ |X|. Then there exist an n ≥ 0, a closed subspace Z ⊂ X ×_A P^n_A, a point z ∈ |Z|, an open V ⊂ P^n_A, and an object E in D(O_X ×_A P^n_A) such that • Z → X ×_A P^n_A is of finite presentation, • c : Z → P^n_A is a closed immersion over V, set W = c^-1(V), • the restriction of b : Z → X to W is étale, z ∈ W, and b(z) = x, • E|_X ×_A V ≅ (b, c)_*O_Z|_X ×_A V, • E is pseudo-coherent…","statement_latex":"Let $A$ be a ring. Let $X$ be a separated algebraic space\nof finite presentation over $A$. Let $x \\in |X|$. Then there exist\nan $n \\geq 0$,\na closed subspace $Z \\subset X \\times_A \\mathbf{P}^n_A$,\na point $z \\in |Z|$,\nan open $V \\subset \\mathbf{P}^n_A$, and\nan object $E$ in $D(\\mathcal{O}_{X \\times_A \\mathbf{P}^n_A})$ such that\n\\begin{enumerate}\n\\item $Z \\to X \\times_A \\mathbf{P}^n_A$ is of finite presentation,\n\\item $c : Z \\to \\mathbf{P}^n_A$ is a closed immersion over $V$,\nset $W = c^{-1}(V)$,\n\\item the restriction of $b : Z \\to X$ to $W$ is \\'etale,\n$z \\in W$, and $b(z) = x$,\n\\item $E|_{X \\times_A V} \\cong\n(b, c)_*\\mathcal{O}_Z|_{X \\times_A V}$,\n\\item $E$ is pseudo-coherent and supported on $Z$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Characterizing pseudo-coherent complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTR","source_file":"spaces-more-morphisms.tex","source_line":12012,"source_end_line":12031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12012-L12031","statement_sha256":"3358656b3baf3f28ef87f49b0a00e237a8e624106b5ba7f9dcdb61dc0b23f754","origin":"The Stacks Project","memory_eligible":false,"source_rank":12267,"rank":12267,"depth":63,"x":781.07,"y":1527.776,"cluster":"geometry-of-spaces"},{"id":"stacks:0CTS","tag":"0CTS","title":"Characterizing pseudo-coherent complexes, II · Lemma 0CTS","summary":"Let X/A, x ∈ |X|, and n, Z, z, V, E be as in Lemma [Tag 0CTR]. For any K ∈ D_QCoh(O_X) we have Rq_*(Lp^*K ⊗^L E)|_V = R(W → V)_*K|_W where p : X ×_A P^n_A → X and q : X ×_A P^n_A → P^n_A are the projections and where the morphism W → V is the finitely presented closed immersion c|_W : W → V.","statement_latex":"Let $X/A$, $x \\in |X|$, and\n$n, Z, z, V, E$ be as in Lemma \\ref{lemma-derived-chow}.\nFor any $K \\in D_\\QCoh(\\mathcal{O}_X)$ we have\n$$\nRq_*(Lp^*K \\otimes^\\mathbf{L} E)|_V = R(W \\to V)_*K|_W\n$$\nwhere $p : X \\times_A \\mathbf{P}^n_A \\to X$ and\n$q : X \\times_A \\mathbf{P}^n_A \\to \\mathbf{P}^n_A$ are\nthe projections and where the morphism $W \\to V$ is\nthe finitely presented closed immersion $c|_W : W \\to V$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Characterizing pseudo-coherent complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTS","source_file":"spaces-more-morphisms.tex","source_line":12114,"source_end_line":12126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12114-L12126","statement_sha256":"3816a9a643889bdff6b76e3e167b7ecefff36cf012dfb99da9ee1aca10bb33b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12268,"rank":12268,"depth":64,"x":663.54,"y":1710.697,"cluster":"geometry-of-spaces"},{"id":"stacks:0CTT","tag":"0CTT","title":"Characterizing pseudo-coherent complexes, II · Lemma 0CTT","summary":"Let A be a ring. Let X be an algebraic space separated and of finite presentation over A. Let K ∈ D_QCoh(O_X). If RΓ(X, E ⊗^L K) is pseudo-coherent in D(A) for every pseudo-coherent E in D(O_X), then K is pseudo-coherent relative to A (Definition [Tag 0CSY]).","statement_latex":"Let $A$ be a ring. Let $X$ be an algebraic space separated and\nof finite presentation over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$.\nIf $R\\Gamma(X, E \\otimes^\\mathbf{L} K)$ is pseudo-coherent\nin $D(A)$ for every pseudo-coherent $E$ in $D(\\mathcal{O}_X)$,\nthen $K$ is pseudo-coherent relative to $A$\n(Definition \\ref{definition-relative-pseudo-coherence}).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Characterizing pseudo-coherent complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CTT","source_file":"spaces-more-morphisms.tex","source_line":12164,"source_end_line":12172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12164-L12172","statement_sha256":"c1fac7e0c0fd3014295fd0d87350c73face349f2e2be8c18321c15f1d1b4b8ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":12269,"rank":12269,"depth":65,"x":603.055,"y":1508.959,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFJ","tag":"0GFJ","title":"Characterizing pseudo-coherent complexes, II · Lemma 0GFJ","summary":"Let A be a ring. Let X be an algebraic space separated and of finite presentation over A. Let K ∈ D_QCoh(O_X). If R Γ (X, E ⊗ ^L K) is pseudo-coherent in D(A) for every perfect E ∈ D(O_X), then K is pseudo-coherent relative to A.","statement_latex":"Let $A$ be a ring. Let $X$ be an algebraic space separated and\nof finite presentation over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$. If \n$R \\Gamma (X, E \\otimes ^{\\mathbf{L}} K)$ is\npseudo-coherent in $D(A)$ for every perfect \n$E \\in D(\\mathcal{O}_X)$, then $K$ is pseudo-coherent\nrelative to $A$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Characterizing pseudo-coherent complexes, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFJ","source_file":"spaces-more-morphisms.tex","source_line":12229,"source_end_line":12237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12229-L12237","statement_sha256":"d940fa9826443f2f70f840998fe55e59d63152ec903faf708aa965efd4c7d9e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12270,"rank":12270,"depth":66,"x":810.057,"y":1623.491,"cluster":"geometry-of-spaces"},{"id":"stacks:0DKN","tag":"0DKN","title":"Relatively perfect objects · Definition 0DKN","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and locally of finite presentation. An object E of D(O_X) is perfect relative to Y or Y-perfect if E is pseudo-coherent (Cohomology on Sites, Definition [Tag 08FT]) and E locally has finite tor dimension as an object of D(f^-1O_Y) (Cohomology on Sites, Definition [Tag 08FZ]).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is flat and locally of finite presentation.\nAn object $E$ of $D(\\mathcal{O}_X)$ is {\\it perfect relative to $Y$} or\n{\\it $Y$-perfect} if $E$ is pseudo-coherent\n(Cohomology on Sites, Definition\n\\ref{sites-cohomology-definition-pseudo-coherent}) and\n$E$ locally has finite tor dimension as an object of\n$D(f^{-1}\\mathcal{O}_Y)$\n(Cohomology on Sites, Definition\n\\ref{sites-cohomology-definition-tor-amplitude}).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKN","source_file":"spaces-more-morphisms.tex","source_line":12270,"source_end_line":12282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12270-L12282","statement_sha256":"7053c22bd1aac88b3b0712f09b063d8dc8cf08d5327813d2bffc73a06534a6ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":12271,"rank":12271,"depth":1,"x":565.114,"y":1656.522,"cluster":"geometry-of-spaces"},{"id":"stacks:0DKQ","tag":"0DKQ","title":"Relatively perfect objects · Lemma 0DKQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and locally of finite presentation. Let E ∈ D_QCoh(O_X). The following are equivalent: • E is Y-perfect, • for every commutative diagram xymatrix U ar[d] ar[r]_g & V ar[d] X ar[r]^f & Y where U, V are schemes and the vertical arrows are étale, the complex E|_U is V-perfect in the sense of Derived Categories of Schemes, Definition [Tag 0DI0], • for some commutative diagram as in (2)…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is flat and locally of finite presentation.\nLet $E \\in D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent:\n\\begin{enumerate}\n\\item $E$ is $Y$-perfect,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r]_g & V \\ar[d] \\\\\nX \\ar[r]^f & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale, the complex\n$E|_U$ is $V$-perfect in the sense of Derived Categories of Schemes,\nDefinition \\ref{perfect-definition-relatively-perfect},\n\\item for some commutative diagram as in (2) with $U \\to X$\nsurjective, the complex $E|_U$ is $V$-perfect in the sense of\nDerived Categories of Schemes,\nDefinition \\ref{perfect-definition-relatively-perfect},\n\\item for every commutative diagram as in (2) with $U$ and $V$\naffine the complex $R\\Gamma(U, E)$ is $\\mathcal{O}_Y(V)$-perfect.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKQ","source_file":"spaces-more-morphisms.tex","source_line":12312,"source_end_line":12336,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12312-L12336","statement_sha256":"95c4e986e9c7dc48475db7ee5bccbce48fcad3b7e5882cc822ed1d8cd71d434f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12272,"rank":12272,"depth":53,"x":719.294,"y":1493.046,"cluster":"geometry-of-spaces"},{"id":"stacks:0DKR","tag":"0DKR","title":"Relatively perfect objects · Lemma 0DKR","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and locally of finite presentation. The full subcategory of D(O_X) consisting of Y-perfect objects is a saturated triangulated subcategory.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $S$ which is flat and locally of finite presentation.\nThe full subcategory of $D(\\mathcal{O}_X)$ consisting of $Y$-perfect objects is\na saturated\\footnote{Derived Categories, Definition\n\\ref{derived-definition-saturated}.} triangulated subcategory.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKR","source_file":"spaces-more-morphisms.tex","source_line":12366,"source_end_line":12373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12366-L12373","statement_sha256":"af346e8a06e4cdf21e52fd880ebfeac5e7795ae1e45ce1d7c8000722e9f56167","origin":"The Stacks Project","memory_eligible":false,"source_rank":12273,"rank":12273,"depth":10,"x":737.081,"y":1701.245,"cluster":"geometry-of-spaces"},{"id":"stacks:0DKS","tag":"0DKS","title":"Relatively perfect objects · Lemma 0DKS","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and locally of finite presentation. A perfect object of D(O_X) is Y-perfect. If K, M ∈ D(O_X), then K ⊗_O_X^L M is Y-perfect if K is perfect and M is Y-perfect.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is flat and locally of finite presentation.\nA perfect object of $D(\\mathcal{O}_X)$ is $Y$-perfect.\nIf $K, M \\in D(\\mathcal{O}_X)$, then $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} M$\nis $Y$-perfect if $K$ is perfect and $M$ is $Y$-perfect.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKS","source_file":"spaces-more-morphisms.tex","source_line":12383,"source_end_line":12391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12383-L12391","statement_sha256":"0f4dd682bf21c8f6669168b5115148d6fd0de2a061535c81b74a6e9e59a480da","origin":"The Stacks Project","memory_eligible":false,"source_rank":12274,"rank":12274,"depth":54,"x":556.391,"y":1557.699,"cluster":"geometry-of-spaces"},{"id":"stacks:0DKT","tag":"0DKT","title":"Relatively perfect objects · Lemma 0DKT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and locally of finite presentation. Let g : Y' → Y be a morphism of algebraic spaces over S. Set X' = Y' ×_Y X and denote g' : X' → X the projection. If K ∈ D(O_X) is Y-perfect, then L(g')^*K is Y'-perfect.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is flat and locally of finite presentation.\nLet $g : Y' \\to Y$ be a morphism of algebraic spaces over $S$.\nSet $X' = Y' \\times_Y X$ and denote $g' : X' \\to X$ the projection.\nIf $K \\in D(\\mathcal{O}_X)$ is $Y$-perfect, then $L(g')^*K$\nis $Y'$-perfect.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKT","source_file":"spaces-more-morphisms.tex","source_line":12401,"source_end_line":12410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12401-L12410","statement_sha256":"f14d55687494d6f2ee04cfb874be68f5f19c17220366b11b1933bd251e2541e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12275,"rank":12275,"depth":54,"x":805.266,"y":1561.022,"cluster":"geometry-of-spaces"},{"id":"stacks:0DKV","tag":"0DKV","title":"Relatively perfect objects · Lemma 0DKV","summary":"In Situation [Tag 0DKU]. Let K_0 and L_0 be objects of D(O_X_0). Set K_i = Lf_i0^*K_0 and L_i = Lf_i0^*L_0 for i ≥ 0 and set K = Lf_0^*K_0 and L = Lf_0^*L_0. Then the map colim_i ≥ 0 Hom_D(O_X_i)(K_i, L_i) → Hom_D(O_X)(K, L) is an isomorphism if K_0 is pseudo-coherent and L_0 ∈ D_QCoh(O_X_0) has (locally) finite tor dimension as an object of D((X_0 → Y_0)^-1O_Y_0)","statement_latex":"In Situation \\ref{situation-relative-descent}.\nLet $K_0$ and $L_0$ be objects of $D(\\mathcal{O}_{X_0})$.\nSet $K_i = Lf_{i0}^*K_0$ and $L_i = Lf_{i0}^*L_0$ for $i \\geq 0$\nand set $K = Lf_0^*K_0$ and $L = Lf_0^*L_0$. Then the map\n$$\n\\colim_{i \\geq 0} \\Hom_{D(\\mathcal{O}_{X_i})}(K_i, L_i)\n\\longrightarrow\n\\Hom_{D(\\mathcal{O}_X)}(K, L)\n$$\nis an isomorphism if $K_0$ is pseudo-coherent and\n$L_0 \\in D_\\QCoh(\\mathcal{O}_{X_0})$ has (locally)\nfinite tor dimension as an object of\n$D((X_0 \\to Y_0)^{-1}\\mathcal{O}_{Y_0})$","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKV","source_file":"spaces-more-morphisms.tex","source_line":12434,"source_end_line":12449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12434-L12449","statement_sha256":"b1c091f06a5f17ec69fec04f495d8b46d27117c9f3590a94fac0726bcd0ed604","origin":"The Stacks Project","memory_eligible":false,"source_rank":12276,"rank":12276,"depth":60,"x":618.926,"y":1699.901,"cluster":"geometry-of-spaces"},{"id":"stacks:0DKW","tag":"0DKW","title":"Relatively perfect objects · Lemma 0DKW","summary":"In Situation [Tag 0DKU] the category of Y-perfect objects of D(O_X) is the colimit of the categories of Y_i-perfect objects of D(O_X_i).","statement_latex":"In Situation \\ref{situation-relative-descent} the category of\n$Y$-perfect objects of $D(\\mathcal{O}_X)$ is the colimit of the categories\nof $Y_i$-perfect objects of $D(\\mathcal{O}_{X_i})$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKW","source_file":"spaces-more-morphisms.tex","source_line":12498,"source_end_line":12503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12498-L12503","statement_sha256":"4e4aa4970189c93b559ac2edad1b2e9cee23d4d9d00ea9bca325c66aa3c9befa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12277,"rank":12277,"depth":61,"x":644.668,"y":1491.592,"cluster":"geometry-of-spaces"},{"id":"stacks:0DKX","tag":"0DKX","title":"Relatively perfect objects · Lemma 0DKX","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat, proper, and of finite presentation. Let E ∈ D(O_X) be Y-perfect. Then Rf_*E is a perfect object of D(O_Y) and its formation commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is flat, proper, and\nof finite presentation. Let $E \\in D(\\mathcal{O}_X)$ be $Y$-perfect.\nThen $Rf_*E$ is a perfect object of $D(\\mathcal{O}_Y)$\nand its formation commutes with arbitrary base change.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKX","source_file":"spaces-more-morphisms.tex","source_line":12577,"source_end_line":12585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12577-L12585","statement_sha256":"b7c4ba456d13fcb5b53dbe9650c2004bd780f94c2e0403943c4e2fe9b923ba7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12278,"rank":12278,"depth":66,"x":793.323,"y":1659.939,"cluster":"geometry-of-spaces"},{"id":"stacks:0DKY","tag":"0DKY","title":"Relatively perfect objects · Lemma 0DKY","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let E, K ∈ D(O_X). Assume • Y is quasi-compact and quasi-separated, • f is proper, flat, and of finite presentation, • E is Y-perfect, • K is pseudo-coherent. Then there exists a pseudo-coherent L ∈ D(O_Y) such that Rf_*RSheafHom(K, E) = RSheafHom(L, O_Y) and the same is true after arbitrary base change: given vcenter xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y cartesian, then we have…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $E, K \\in D(\\mathcal{O}_X)$.\nAssume\n\\begin{enumerate}\n\\item $Y$ is quasi-compact and quasi-separated,\n\\item $f$ is proper, flat, and of finite presentation,\n\\item $E$ is $Y$-perfect,\n\\item $K$ is pseudo-coherent.\n\\end{enumerate}\nThen there exists a pseudo-coherent $L \\in D(\\mathcal{O}_Y)$ such that\n$$\nRf_*R\\SheafHom(K, E) = R\\SheafHom(L, \\mathcal{O}_Y)\n$$\nand the same is true after arbitrary base change: given\n$$\n\\vcenter{\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} &\nX \\ar[d]^f \\\\\nY' \\ar[r]^g &\nY\n}\n}\n\\quad\\quad\n\\begin{matrix}\n\\text{cartesian, then we have } \\\\\nRf'_*R\\SheafHom(L(g')^*K, L(g')^*E) \\\\\n= R\\SheafHom(Lg^*L, \\mathcal{O}_{Y'})\n\\end{matrix}\n$$","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DKY","source_file":"spaces-more-morphisms.tex","source_line":12614,"source_end_line":12647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12614-L12647","statement_sha256":"fcc010f1a3ebdad450f3463a8c438ff7850f8a3926d1bdd5b4afc8658e8ebffc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12279,"rank":12279,"depth":67,"x":548.13,"y":1620.12,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFK","tag":"0GFK","title":"Relatively perfect objects · Lemma 0GFK","summary":"Let S be a scheme. Let X be an algebraic space over S such that the structure morphism f : X → S is flat and locally of finite presentation. Let E be a pseudo-coherent object of D(O_X). The following are equivalent • E is S-perfect, and • E is locally bounded below and for every point s ∈ S the object L(X_s → X)^*E of D(O_X_s) is locally bounded below.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ such that\nthe structure morphism $f : X \\to S$ is flat and\nlocally of finite presentation. Let $E$ be a pseudo-coherent\nobject of $D(\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item $E$ is $S$-perfect, and\n\\item $E$ is locally bounded below and for every point $s \\in S$\nthe object $L(X_s \\to X)^*E$ of $D(\\mathcal{O}_{X_s})$\nis locally bounded below.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFK","source_file":"spaces-more-morphisms.tex","source_line":12758,"source_end_line":12770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12758-L12770","statement_sha256":"cd76f070ac0fd5f0539afb3c255de44b7438a014ebfce99e2e9c83e7ea09a286","origin":"The Stacks Project","memory_eligible":false,"source_rank":12280,"rank":12280,"depth":54,"x":761.123,"y":1510.266,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFL","tag":"0GFL","title":"Relatively perfect objects · Lemma 0GFL","summary":"Let A be a ring. Let X be an algebraic space separated, of finite presentation, and flat over A. Let K ∈ D_QCoh(O_X). If R Γ (X, E ⊗^L K) is perfect in D(A) for every perfect E ∈ D(O_X), then K is Spec(A)-perfect.","statement_latex":"Let $A$ be a ring. Let $X$ be an algebraic space separated, of\nfinite presentation, and flat over $A$. Let $K \\in D_\\QCoh(\\mathcal{O}_X)$.\nIf  $R \\Gamma (X, E \\otimes^\\mathbf{L} K)$ is perfect in\n$D(A)$ for every perfect $E \\in D(\\mathcal{O}_X)$, then $K$ is\n$\\Spec(A)$-perfect.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Relatively perfect objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFL","source_file":"spaces-more-morphisms.tex","source_line":12780,"source_end_line":12787,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12780-L12787","statement_sha256":"aa54117781d6963f095118d13c4a195a3b96fade158b89687d78b9fb525509bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12281,"rank":12281,"depth":67,"x":692.353,"y":1712.29,"cluster":"geometry-of-spaces"},{"id":"stacks:0D23","tag":"0D23","title":"Theorem of the cube · Lemma 0D23","summary":"Let S be a scheme. Let f : X → Y be a flat, proper morphism of finite presentation of algebraic spaces over S. Let E be a finite locally free O_X-module. For a morphism g : Y' → Y consider the base change diagram xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f Y' ar[r]^g & Y Assume O_Y' → f'_*O_X' is an isomorphism for all g : Y' → Y. Then there exists an immersion j : Z → Y of finite presentation such that a morphism g : Y' → Y factors through Z if and only if there exists a…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a flat, proper morphism of finite presentation\nof algebraic spaces over $S$.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module.\nFor a morphism $g : Y' \\to Y$ consider the base change diagram\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nAssume $\\mathcal{O}_{Y'} \\to f'_*\\mathcal{O}_{X'}$ is an\nisomorphism for all $g : Y' \\to Y$.\nThen there exists an immersion $j : Z \\to Y$ of finite presentation\nsuch that a morphism $g : Y' \\to Y$ factors through $Z$ if and only if\nthere exists a finite locally free $\\mathcal{O}_{Y'}$-module $\\mathcal{N}$\nwith $(f')^*\\mathcal{N} \\cong (g')^*\\mathcal{L}$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Theorem of the cube","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D23","source_file":"spaces-more-morphisms.tex","source_line":12854,"source_end_line":12873,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12854-L12873","statement_sha256":"aa7193f8e81325ede092751d1fbbebe4a89c57546eca19427f7baed030482d24","origin":"The Stacks Project","memory_eligible":false,"source_rank":12282,"rank":12282,"depth":69,"x":580.511,"y":1524.148,"cluster":"geometry-of-spaces"},{"id":"stacks:0DL1","tag":"0DL1","title":"Descent of finiteness properties of complexes · Lemma 0DL1","summary":"Let S be a scheme. Let (f_i : X_i → X) be an fpqc covering of algebraic spaces over S. Let E ∈ D_QCoh(O_X). Let m ∈ Z. Then E is m-pseudo-coherent if and only if each Lf_i^*E is m-pseudo-coherent.","statement_latex":"Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}$ be an fpqc covering of\nalgebraic spaces over $S$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$.\nLet $m \\in \\mathbf{Z}$. Then $E$ is $m$-pseudo-coherent if and only if each\n$Lf_i^*E$ is $m$-pseudo-coherent.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Descent of finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DL1","source_file":"spaces-more-morphisms.tex","source_line":12984,"source_end_line":12990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L12984-L12990","statement_sha256":"979438d7ff4b9c9f3a8918d180e667f94b901f012ba4df1cd47b32068d0d997d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12283,"rank":12283,"depth":49,"x":814.467,"y":1599.48,"cluster":"geometry-of-spaces"},{"id":"stacks:0DL2","tag":"0DL2","title":"Descent of finiteness properties of complexes · Lemma 0DL2","summary":"Let S be a scheme. Let (g_i : Y_i → Y) be an fpqc covering of algebraic spaces over S. Let f : X → Y be a morphism of algebraic spaces and set X_i = Y_i ×_Y X with projections f_i : X_i → Y_i and g'_i : X_i → X. Let E ∈ D_QCoh(O_X). Let a, b ∈ Z. Then the following are equivalent • E has tor amplitude in [a, b] as an object of D(f^-1O_Y), and • L(g'_i)^*E has tor amplitude in [a, b] as a object of D(f_i^-1O_Y_i) for all i. Also true if \"tor amplitude in [a, b]\" is…","statement_latex":"Let $S$ be a scheme. Let $\\{g_i : Y_i \\to Y\\}$ be an fpqc covering of\nalgebraic spaces over $S$. Let $f : X \\to Y$ be a morphism of algebraic\nspaces and set $X_i = Y_i \\times_Y X$ with projections $f_i : X_i \\to Y_i$\nand $g'_i : X_i \\to X$. Let $E \\in D_\\QCoh(\\mathcal{O}_X)$.\nLet $a, b \\in \\mathbf{Z}$. Then the following are equivalent\n\\begin{enumerate}\n\\item $E$ has tor amplitude in $[a, b]$ as an object of\n$D(f^{-1}\\mathcal{O}_Y)$, and\n\\item $L(g'_i)^*E$ has tor amplitude in $[a, b]$ as a object of\n$D(f_i^{-1}\\mathcal{O}_{Y_i})$ for all $i$.\n\\end{enumerate}\nAlso true if ``tor amplitude in $[a, b]$'' is replaced by\n``locally finite tor dimension''.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Descent of finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DL2","source_file":"spaces-more-morphisms.tex","source_line":13020,"source_end_line":13035,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13020-L13035","statement_sha256":"805fb9872f875d15734f33f34a11cecc3f5948ab211f00898e1a29f5f83b6ed5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12284,"rank":12284,"depth":54,"x":581.186,"y":1676.745,"cluster":"geometry-of-spaces"},{"id":"stacks:0DL3","tag":"0DL3","title":"Descent of finiteness properties of complexes · Lemma 0DL3","summary":"Let S be a scheme. Let i : X → X' be a finite order thickening of algebraic spaces. Let K' ∈ D(O_X') be an object such that K = Li^*K' is pseudo-coherent. Then K' is pseudo-coherent.","statement_latex":"Let $S$ be a scheme. Let $i : X \\to X'$ be a finite order thickening of\nalgebraic spaces. Let $K' \\in D(\\mathcal{O}_{X'})$ be an object such that\n$K = Li^*K'$ is pseudo-coherent. Then $K'$ is pseudo-coherent.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Descent of finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DL3","source_file":"spaces-more-morphisms.tex","source_line":13080,"source_end_line":13085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13080-L13085","statement_sha256":"f94c7535443177fade4d60bd4bfa6f2bc7a2ba546332b3932fc03ab16890c6bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12285,"rank":12285,"depth":60,"x":691.157,"y":1487.25,"cluster":"geometry-of-spaces"},{"id":"stacks:0DL4","tag":"0DL4","title":"Descent of finiteness properties of complexes · Lemma 0DL4","summary":"Let S be a scheme. Consider a cartesian diagram xymatrix X ar[r]_i ar[d]_f & X' ar[d]^f' Y ar[r]^j & Y' of algebraic spaces over S. Assume X' → Y' is flat and locally of finite presentation and Y → Y' is a finite order thickening. Let E' ∈ D(O_X'). If E = Li^*(E') is Y-perfect, then E' is Y'-perfect.","statement_latex":"Let $S$ be a scheme. Consider a cartesian diagram\n$$\n\\xymatrix{\nX \\ar[r]_i \\ar[d]_f & X' \\ar[d]^{f'} \\\\\nY \\ar[r]^j & Y'\n}\n$$\nof algebraic spaces over $S$. Assume $X' \\to Y'$ is flat and locally\nof finite presentation and $Y \\to Y'$ is a finite order thickening.\nLet $E' \\in D(\\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect,\nthen $E'$ is $Y'$-perfect.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Descent of finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DL4","source_file":"spaces-more-morphisms.tex","source_line":13138,"source_end_line":13151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13138-L13151","statement_sha256":"560d216d70e1cd122a0d87b053b568dafe490285a624e82bd9d9ba8d140d4f6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12286,"rank":12286,"depth":61,"x":762.509,"y":1689.542,"cluster":"geometry-of-spaces"},{"id":"stacks:0DL5","tag":"0DL5","title":"Descent of finiteness properties of complexes · Lemma 0DL5","summary":"Let (R, I) be a pair consisting of a ring and an ideal I contained in the Jacobson radical. Set S = Spec(R) and S_0 = Spec(R/I). Let X be an algebraic space over R whose structure morphism f : X → S is proper, flat, and of finite presentation. Denote X_0 = S_0 ×_S X. Let E ∈ D(O_X) be pseudo-coherent. If the derived restriction E_0 of E to X_0 is S_0-perfect, then E is S-perfect.","statement_latex":"Let $(R, I)$ be a pair consisting of a ring and an ideal $I$\ncontained in the Jacobson radical. Set $S = \\Spec(R)$ and $S_0 = \\Spec(R/I)$.\nLet $X$ be an algebraic space over $R$ whose structure morphism\n$f : X \\to S$ is proper, flat, and of finite presentation.\nDenote $X_0 = S_0 \\times_S X$. Let $E \\in D(\\mathcal{O}_X)$\nbe pseudo-coherent. If the derived restriction $E_0$ of $E$\nto $X_0$ is $S_0$-perfect, then $E$ is $S$-perfect.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Descent of finiteness properties of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DL5","source_file":"spaces-more-morphisms.tex","source_line":13170,"source_end_line":13179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13170-L13179","statement_sha256":"6dac6c0a73d1743a2eafd7b7faab167da7d454d0d87e4abac81796dfe3576214","origin":"The Stacks Project","memory_eligible":false,"source_rank":12287,"rank":12287,"depth":57,"x":547.047,"y":1580.775,"cluster":"geometry-of-spaces"},{"id":"stacks:0DSE","tag":"0DSE","title":"Families of nodal curves · Definition 0DSE","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. We say f is at-worst-nodal of relative dimension 1 if the equivalent conditions of Morphisms of Spaces, Lemma [Tag 03MJ] hold with P =\"at-worst-nodal of relative dimension 1\".","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nWe say $f$ is {\\it at-worst-nodal of relative dimension $1$}\nif the equivalent conditions of\nMorphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-local-source-target}\nhold with $\\mathcal{P} =$``at-worst-nodal of relative dimension $1$''.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Families of nodal curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSE","source_file":"spaces-more-morphisms.tex","source_line":13296,"source_end_line":13304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13296-L13304","statement_sha256":"7c7adfb8996df88af187a2f6766f52555dfbff0bb00579cc69b9be2f3a51bc56","origin":"The Stacks Project","memory_eligible":false,"source_rank":12288,"rank":12288,"depth":46,"x":793.587,"y":1538.687,"cluster":"geometry-of-spaces"},{"id":"stacks:0DSF","tag":"0DSF","title":"Families of nodal curves · Lemma 0DSF","summary":"The property of being at-worst-nodal of relative dimension 1 is preserved under base change.","statement_latex":"The property of being at-worst-nodal of relative dimension $1$\nis preserved under base change.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSF","source_file":"spaces-more-morphisms.tex","source_line":13306,"source_end_line":13310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13306-L13310","statement_sha256":"26e18884c2d6aa743a5708348b674b3a43efbddb8eea0cc32ffc6eba9f7a5798","origin":"The Stacks Project","memory_eligible":false,"source_rank":12289,"rank":12289,"depth":39,"x":645.522,"y":1709.75,"cluster":"geometry-of-spaces"},{"id":"stacks:0DSG","tag":"0DSG","title":"Families of nodal curves · Lemma 0DSG","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The following are equivalent: • f is at-worst-nodal of relative dimension 1, • for every scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is at-worst-nodal of relative dimension 1, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is at-worst-nodal of relative dimension 1, • there exists a scheme V and a surjective étale morphism V → Y such that V ×_Y X → V is…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is at-worst-nodal of relative dimension $1$,\n\\item for every scheme $Z$ and any morphism $Z \\to Y$ the morphism\n$Z \\times_Y X \\to Z$ is at-worst-nodal of relative dimension $1$,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to Y$ the morphism $Z \\times_Y X \\to Z$ is\nat-worst-nodal of relative dimension $1$,\n\\item there exists a scheme $V$ and a surjective \\'etale morphism\n$V \\to Y$ such that $V \\times_Y X \\to V$ is\nat-worst-nodal of relative dimension $1$,\n\\item there exists a scheme $U$ and a surjective \\'etale morphism\n$\\varphi : U \\to X$ such that the composition $f \\circ \\varphi$\nis at-worst-nodal of relative dimension $1$,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes and the vertical arrows are \\'etale\nthe top horizontal arrow is at-worst-nodal of relative dimension $1$,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are schemes, the vertical arrows are \\'etale, and\n$U \\to X$ is surjective such that the top horizontal arrow is\nat-worst-nodal of relative dimension $1$, and\n\\item there exist Zariski coverings $Y = \\bigcup_{i \\in I} Y_i$,\nand $f^{-1}(Y_i) = \\bigcup X_{ij}$ such that\neach morphism $X_{ij} \\to Y_i$ is\nat-worst-nodal of relative dimension $1$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSG","source_file":"spaces-more-morphisms.tex","source_line":13319,"source_end_line":13361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13319-L13361","statement_sha256":"48a301a40656d4d469b36d5fd97d527b447623bbb94133ecfdc53f4cc5b7b828","origin":"The Stacks Project","memory_eligible":false,"source_rank":12290,"rank":12290,"depth":0,"x":617.116,"y":1499.429,"cluster":"geometry-of-spaces"},{"id":"stacks:0DSH","tag":"0DSH","title":"Families of nodal curves · Lemma 0DSH","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is flat and locally of finite presentation. Then there is a maximal open subspace X' ⊂ X such that f|_X' : X' → Y is at-worst-nodal of relative dimension 1. Moreover, formation of X' commutes with arbitrary base change.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is flat and locally of finite presentation. Then there\nis a maximal open subspace $X' \\subset X$ such that $f|_{X'} : X' \\to Y$\nis at-worst-nodal of relative dimension $1$. Moreover, formation\nof $X'$ commutes with arbitrary base change.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Families of nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSH","source_file":"spaces-more-morphisms.tex","source_line":13371,"source_end_line":13379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13371-L13379","statement_sha256":"8dc79ab53014c288b01209c9dfe5270ee4484e7403c37fc912162296dc65aefd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12291,"rank":12291,"depth":45,"x":807.347,"y":1638.508,"cluster":"geometry-of-spaces"},{"id":"stacks:0GUZ","tag":"0GUZ","title":"The resolution property · Lemma 0GUZ","summary":"In Situation [Tag 0GUY], assume X is Noetherian. Then for any coherent O_X-module F there exist r ≥ 0, integers n_1, …, n_r ≥ 0, and a surjection bigoplus_i = 1, …, r π_*(I^n_i) → F of O_X-modules.","statement_latex":"In Situation \\ref{situation-descend-resolution-property}, assume\n$X$ is Noetherian. Then for any coherent $\\mathcal{O}_X$-module\n$\\mathcal{F}$ there exist $r \\geq 0$,\nintegers $n_1, \\ldots, n_r \\geq 0$, and a surjection\n$$\n\\bigoplus\\nolimits_{i = 1, \\ldots, r} \\pi_*(\\mathcal{I}^{n_i})\n\\longrightarrow \\mathcal{F}\n$$\nof $\\mathcal{O}_X$-modules.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GUZ","source_file":"spaces-more-morphisms.tex","source_line":13439,"source_end_line":13450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13439-L13450","statement_sha256":"f19747034d5f2312ad7a0e2409163b18506dd56a563a6e59d7e0dee7a2506208","origin":"The Stacks Project","memory_eligible":false,"source_rank":12292,"rank":12292,"depth":60,"x":555.031,"y":1643.899,"cluster":"geometry-of-spaces"},{"id":"stacks:0GV0","tag":"0GV0","title":"The resolution property · Lemma 0GV0","summary":"In Situation [Tag 0GUY], assume X is Noetherian. Then X has the resolution property if and only if π_*I is the quotient of a finite locally free O_X-module.","statement_latex":"In Situation \\ref{situation-descend-resolution-property}, assume\n$X$ is Noetherian. Then $X$ has the resolution property if and only\nif $\\pi_*\\mathcal{I}$ is the quotient of a finite locally free\n$\\mathcal{O}_X$-module.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GV0","source_file":"spaces-more-morphisms.tex","source_line":13537,"source_end_line":13543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13537-L13543","statement_sha256":"9af0e528df106b8ea3c06ffb444265bfe6789701edbd3ea0e1446828e0d842a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12293,"rank":12293,"depth":61,"x":736.893,"y":1496.638,"cluster":"geometry-of-spaces"},{"id":"stacks:0GV1","tag":"0GV1","title":"The resolution property · Lemma 0GV1","summary":"In Situation [Tag 0GUY], the algebraic space X has the resolution property if and only if π_*I is the quotient of a finite locally free O_X-module.","statement_latex":"In Situation \\ref{situation-descend-resolution-property},\nthe algebraic space $X$ has the resolution property\nif and only if $\\pi_*\\mathcal{I}$ is the quotient of a\nfinite locally free $\\mathcal{O}_X$-module.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GV1","source_file":"spaces-more-morphisms.tex","source_line":13565,"source_end_line":13571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13565-L13571","statement_sha256":"173950571529c88689e5146c3b4a7702b1d712dfb5cfad8398b368516c6514bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12294,"rank":12294,"depth":65,"x":721.201,"y":1708.585,"cluster":"geometry-of-spaces"},{"id":"stacks:0GV2","tag":"0GV2","title":"The resolution property · Lemma 0GV2","summary":"Let S be a scheme. Let X = lim X_i be a limit of a direct system of quasi-compact and quasi-separated algebraic spaces over S with affine transition morphisms. Then X has the resolution property if and only if X_i has the resolution properties for some i.","statement_latex":"Let $S$ be a scheme.\nLet $X = \\lim X_i$ be a limit of a direct system of quasi-compact\nand quasi-separated algebraic spaces over $S$ with affine transition morphisms.\nThen $X$ has the resolution property if and only if $X_i$ has\nthe resolution properties for some $i$.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GV2","source_file":"spaces-more-morphisms.tex","source_line":13711,"source_end_line":13718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13711-L13718","statement_sha256":"0c782f02b9cfe43bdf6b688eb26009f10bdd29969c4a2e4171b246c3417f2c8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12295,"rank":12295,"depth":67,"x":562.206,"y":1543.266,"cluster":"geometry-of-spaces"},{"id":"stacks:0GV3","tag":"0GV3","title":"The resolution property · Lemma 0GV3","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space with the resolution property. Then X has affine diagonal over Z (as in Properties of Spaces, Definition [Tag 03BS]).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space with\nthe resolution property. Then $X$ has affine diagonal over $\\mathbf{Z}$\n(as in Properties of Spaces, Definition\n\\ref{spaces-properties-definition-separated}).","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"The resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GV3","source_file":"spaces-more-morphisms.tex","source_line":13770,"source_end_line":13777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13770-L13777","statement_sha256":"ae0b0c3de47b69e9c9260e027f4df1e5666e353f67f66df9d9abb5577989fd32","origin":"The Stacks Project","memory_eligible":false,"source_rank":12296,"rank":12296,"depth":67,"x":812.587,"y":1574.977,"cluster":"geometry-of-spaces"},{"id":"stacks:0GV5","tag":"0GV5","title":"Blowing up and the resolution property · Lemma 0GV5","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Assume that |X| has finitely many irreducible components. There exists a dense quasi-compact open U ⊂ X and a U-admissible blowing up X' → X such that the algebraic space X' has the resolution property.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Assume that $|X|$ has finitely many irreducible\ncomponents. There exists a dense quasi-compact open $U \\subset X$\nand a $U$-admissible blowing up $X' \\to X$ such that the algebraic space\n$X'$ has the resolution property.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Blowing up and the resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GV5","source_file":"spaces-more-morphisms.tex","source_line":13832,"source_end_line":13839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13832-L13839","statement_sha256":"50d33935154fdc30897865c0268ef0d483ffab7086de0eb261fe3847c85df7ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":12297,"rank":12297,"depth":71,"x":602.295,"y":1693.758,"cluster":"geometry-of-spaces"},{"id":"stacks:0GV6","tag":"0GV6","title":"Blowing up and the resolution property · Lemma 0GV6","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. There exists a t ≥ 0 and closed subspaces X ⊃ Z_0 ⊃ Z_1 ⊃ … ⊃ Z_t = ∅ such that Z_i → X is of finite presentation, Z_0 ⊂ X is a thickening, and for each i = 0, … t - 1 there exists a (Z_i setminus Z_i - 1)-admissible blowing up Z'_i → Z_i such that Z'_i has the resolution property.","statement_latex":"Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. There exists a $t \\geq 0$ and closed\nsubspaces\n$$\nX \\supset Z_0 \\supset Z_1 \\supset \\ldots \\supset Z_t = \\emptyset\n$$\nsuch that $Z_i \\to X$ is of finite presentation,\n$Z_0 \\subset X$ is a thickening, and for each $i = 0, \\ldots t - 1$\nthere exists a $(Z_i \\setminus Z_{i - 1})$-admissible\nblowing up $Z'_i \\to Z_i$ such that $Z'_i$ has the resolution property.","area":"Geometry of Spaces","chapter":"More on Morphisms of Spaces","chapter_id":"spaces-more-morphisms","section":"Blowing up and the resolution property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GV6","source_file":"spaces-more-morphisms.tex","source_line":13880,"source_end_line":13892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-morphisms.tex#L13880-L13892","statement_sha256":"02f1b5f4c02198b4eec22ee1fefeb8322f6b9abde5e30d0ff4fada9e5a753f84","origin":"The Stacks Project","memory_eligible":false,"source_rank":12298,"rank":12298,"depth":72,"x":661.887,"y":1486.684,"cluster":"geometry-of-spaces"},{"id":"stacks:0CV8","tag":"0CV8","title":"Impurities · Definition 0CV8","summary":"In Situation [Tag 0CV6] we say a diagram ([Tag 0CV7]) defines an impurity of F above y if xi ∈ Ass_X_T/T(F_T) and t not ∈ f_T(overline(xi)). We will indicate this by saying \"let (g : T → Y, t' leadsto t, xi) be an impurity of F above y\".","statement_latex":"In\nSituation \\ref{situation-pre-pure}\nwe say a diagram (\\ref{equation-impurity}) defines an\n{\\it impurity of $\\mathcal{F}$ above $y$}\nif $\\xi \\in \\text{Ass}_{X_T/T}(\\mathcal{F}_T)$ and\n$t \\not \\in f_T(\\overline{\\{\\xi\\}})$. We will indicate\nthis by saying ``let $(g : T \\to Y, t' \\leadsto t, \\xi)$ be\nan impurity of $\\mathcal{F}$ above $y$''.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Impurities","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CV8","source_file":"spaces-flat.tex","source_line":80,"source_end_line":90,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L80-L90","statement_sha256":"7933e6ca8b6c50d06cf7d4a3710e76f6557b4f49328b808774328058c04d3d55","origin":"The Stacks Project","memory_eligible":false,"source_rank":12299,"rank":12299,"depth":0,"x":784.563,"y":1673.336,"cluster":"geometry-of-spaces"},{"id":"stacks:0CV9","tag":"0CV9","title":"Impurities · Lemma 0CV9","summary":"In Situation [Tag 0CV6]. Let (g : T → S, t' leadsto t, xi) be an impurity of F above y. Assume T = lim_i ∈ I T_i is a directed limit of affine schemes over Y. Then for some i the triple (T_i → Y, t'_i leadsto t_i, xi_i) is an impurity of F above y.","statement_latex":"In Situation \\ref{situation-pre-pure}.\nLet $(g : T \\to S, t' \\leadsto t, \\xi)$ be an impurity of\n$\\mathcal{F}$ above $y$. Assume $T = \\lim_{i \\in I} T_i$ is a directed limit\nof affine schemes over $Y$. Then for some $i$ the triple\n$(T_i \\to Y, t'_i \\leadsto t_i, \\xi_i)$ is an impurity of\n$\\mathcal{F}$ above $y$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Impurities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CV9","source_file":"spaces-flat.tex","source_line":101,"source_end_line":109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L101-L109","statement_sha256":"2fc3a3923956653446fb7cee87392df9e4c25e01cdb72d95dcf0f577e908a472","origin":"The Stacks Project","memory_eligible":false,"source_rank":12300,"rank":12300,"depth":60,"x":543.816,"y":1605.259,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVA","tag":"0CVA","title":"Impurities · Lemma 0CVA","summary":"In Situation [Tag 0CV6]. Let (Y_1, y_1) → (Y, y) be a morphism of pointed algebraic spaces over S. Assume Y_1 → Y is flat at y_1. If (T → Y, t' leadsto t, xi) is an impurity of F above y, then there exists an impurity (T_1 → Y_1, t_1' leadsto t_1, xi_1) of the pullback F_1 of F to X_1 = Y_1 ×_Y X over y_1 such that T_1 is étale over Y_1 ×_Y T.","statement_latex":"In Situation \\ref{situation-pre-pure}.\nLet $(Y_1, y_1) \\to (Y, y)$ be a morphism of pointed\nalgebraic spaces over $S$. Assume $Y_1 \\to Y$ is flat at $y_1$.\nIf $(T \\to Y, t' \\leadsto t, \\xi)$ is an impurity of\n$\\mathcal{F}$ above $y$, then there exists an impurity\n$(T_1 \\to Y_1, t_1' \\leadsto t_1, \\xi_1)$ of the pullback\n$\\mathcal{F}_1$ of $\\mathcal{F}$ to $X_1 = Y_1 \\times_Y X$\nover $y_1$ such that $T_1$ is \\'etale over $Y_1 \\times_Y T$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Impurities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVA","source_file":"spaces-flat.tex","source_line":148,"source_end_line":158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L148-L158","statement_sha256":"6780417c81321737b36628c6b9f1dae6577a90ecb9605e9d09d7ebb38ededc35","origin":"The Stacks Project","memory_eligible":false,"source_rank":12301,"rank":12301,"depth":16,"x":776.266,"y":1518.785,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVB","tag":"0CVB","title":"Impurities · Lemma 0CVB","summary":"In Situation [Tag 0CV6]. Let overliney be a geometric point lying over y. Let O = O_Y, overliney be the étale local ring of Y at overliney. Denote Y^sh = Spec(O), X^sh = X ×_Y Y^sh, and F^sh the pullback of F to X^sh. The following are equivalent • there exists an impurity (Y^sh → Y, y' leadsto overliney, xi) of F above y, • every point of Ass_X^sh/Y^sh(F^sh) specializes to a point of the closed fibre X_overliney, • there exists an impurity (T → Y, t' leadsto t, xi) of F…","statement_latex":"In Situation \\ref{situation-pre-pure}. Let $\\overline{y}$ be a geometric\npoint lying over $y$. Let $\\mathcal{O} = \\mathcal{O}_{Y, \\overline{y}}$\nbe the \\'etale local ring of $Y$ at $\\overline{y}$. Denote\n$Y^{sh} = \\Spec(\\mathcal{O})$, $X^{sh} = X \\times_Y Y^{sh}$, and\n$\\mathcal{F}^{sh}$ the pullback of $\\mathcal{F}$ to $X^{sh}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists an impurity\n$(Y^{sh} \\to Y, y' \\leadsto \\overline{y}, \\xi)$\nof $\\mathcal{F}$ above $y$,\n\\item every point of $\\text{Ass}_{X^{sh}/Y^{sh}}(\\mathcal{F}^{sh})$\nspecializes to a point of the closed fibre $X_{\\overline{y}}$,\n\\item there exists an impurity $(T \\to Y, t' \\leadsto t, \\xi)$\nof $\\mathcal{F}$ above $y$ such that $(T, t) \\to (Y, y)$ is an\n\\'etale neighbourhood, and\n\\item there exists an impurity $(T \\to Y, t' \\leadsto t, \\xi)$\nof $\\mathcal{F}$ above $y$ such that $T \\to Y$ is quasi-finite at $t$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Impurities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVB","source_file":"spaces-flat.tex","source_line":184,"source_end_line":204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L184-L204","statement_sha256":"2f0479f0965dc3d9968809eaffa494f93763050267129ac453c7ca858967d894","origin":"The Stacks Project","memory_eligible":false,"source_rank":12302,"rank":12302,"depth":61,"x":674.321,"y":1714.599,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVD","tag":"0CVD","title":"Relatively pure modules · Definition 0CVD","summary":"In Situation [Tag 0CV6]. • We say F is pure above y if none of the equivalent conditions of Lemma [Tag 0CVB] hold. • We say F is universally pure above y if there does not exist any impurity of F above y. • We say that X is pure above y if O_X is pure above y. • We say F is universally Y-pure, or universally pure relative to Y if F is universally pure above y for every y ∈ |Y|. • We say F is Y-pure, or pure relative to Y if F is pure above y for every y ∈ |Y|. • We say…","statement_latex":"In Situation \\ref{situation-pre-pure}.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ is {\\it pure above $y$} if {\\bf none} of the\nequivalent conditions of Lemma \\ref{lemma-pure-along-X-y} hold.\n\\item We say $\\mathcal{F}$ is {\\it universally pure above $y$}\nif there does not exist any impurity of $\\mathcal{F}$ above $y$.\n\\item We say that $X$ is {\\it pure above $y$} if $\\mathcal{O}_X$\nis pure above $y$.\n\\item We say $\\mathcal{F}$ is {\\it universally $Y$-pure}, or\n{\\it universally pure relative to $Y$} if $\\mathcal{F}$ is universally\npure above $y$ for every $y \\in |Y|$.\n\\item We say $\\mathcal{F}$ is {\\it $Y$-pure}, or\n{\\it pure relative to $Y$} if $\\mathcal{F}$ is pure above $y$\nfor every $y \\in |Y|$.\n\\item We say that $X$ is {\\it $Y$-pure} or {\\it pure relative to $Y$}\nif $\\mathcal{O}_X$ is pure relative to $Y$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Relatively pure modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVD","source_file":"spaces-flat.tex","source_line":266,"source_end_line":285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L266-L285","statement_sha256":"eed220dea5e57f09dbeca553d7bbaa5c4217434af9987383d8da01a522b31deb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12303,"rank":12303,"depth":62,"x":591.963,"y":1512.207,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVE","tag":"0CVE","title":"Relatively pure modules · Lemma 0CVE","summary":"In Situation [Tag 0CV6]. • F is universally pure above y, and • for every morphism (Y', y') → (Y, y) of pointed algebraic spaces the pullback F_Y' is pure above y'. In particular, F is universally pure relative to Y if and only if every base change F_Y' of F is pure relative to Y'.","statement_latex":"In Situation \\ref{situation-pre-pure}.\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is universally pure above $y$, and\n\\item for every morphism $(Y', y') \\to (Y, y)$ of pointed algebraic spaces\nthe pullback $\\mathcal{F}_{Y'}$ is pure above $y'$.\n\\end{enumerate}\nIn particular, $\\mathcal{F}$ is universally pure relative to $Y$ if and\nonly if every base change $\\mathcal{F}_{Y'}$ of $\\mathcal{F}$ is\npure relative to $Y'$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVE","source_file":"spaces-flat.tex","source_line":290,"source_end_line":301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L290-L301","statement_sha256":"391a55a121e6d78f76d36ce8a908181fe0a18b60998a6b62cbfeb6adbd2cfba0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12304,"rank":12304,"depth":0,"x":815.623,"y":1614.794,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVF","tag":"0CVF","title":"Relatively pure modules · Lemma 0CVF","summary":"In Situation [Tag 0CV6]. Let (Y', y') → (Y, y) be a morphism of pointed algebraic spaces. If Y' → Y is quasi-finite at y' and F is pure above y, then F_Y' is pure above y'.","statement_latex":"In Situation \\ref{situation-pre-pure}.\nLet $(Y', y') \\to (Y, y)$ be a morphism of pointed algebraic spaces.\nIf $Y' \\to Y$ is quasi-finite at $y'$ and $\\mathcal{F}$ is pure above $y$,\nthen $\\mathcal{F}_{Y'}$ is pure above $y'$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVF","source_file":"spaces-flat.tex","source_line":307,"source_end_line":313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L307-L313","statement_sha256":"f4e6f0b539dbfa102f3b9dc981d42bb902be3fd52a1f2c898d26cef5071272a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12305,"rank":12305,"depth":30,"x":568.01,"y":1666.097,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVG","tag":"0CVG","title":"Relatively pure modules · Lemma 0CVG","summary":"In Situation [Tag 0CV6]. Let (Y_1, y_1) → (Y, y) be a morphism of pointed algebraic spaces. Assume Y_1 → Y is flat at y_1. • If F_Y_1 is pure above y_1, then F is pure above y. • If F_Y_1 is universally pure above y_1, then F is universally pure above y.","statement_latex":"In Situation \\ref{situation-pre-pure}.\nLet $(Y_1, y_1) \\to (Y, y)$ be a morphism of pointed algebraic spaces.\nAssume $Y_1 \\to Y$ is flat at $y_1$.\n\\begin{enumerate}\n\\item If $\\mathcal{F}_{Y_1}$ is pure above $y_1$,\nthen $\\mathcal{F}$ is pure above $y$.\n\\item If $\\mathcal{F}_{Y_1}$ is universally pure above $y_1$,\nthen $\\mathcal{F}$ is universally pure above $y$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVG","source_file":"spaces-flat.tex","source_line":328,"source_end_line":339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L328-L339","statement_sha256":"b2acdb54d497e7b8d4dcf544e7a859bb8753159c94cbe3aac9344a7d9871a94a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12306,"rank":12306,"depth":47,"x":709.448,"y":1487.629,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVH","tag":"0CVH","title":"Relatively pure modules · Lemma 0CVH","summary":"In Situation [Tag 0CV6]. Let i : Z → X be a closed immersion and assume that F = i_*G for some finite type, quasi-coherent sheaf G on Z. Then G is (universally) pure above y if and only if F is (universally) pure above y.","statement_latex":"In Situation \\ref{situation-pre-pure}. Let $i : Z \\to X$ be a closed immersion\nand assume that $\\mathcal{F} = i_*\\mathcal{G}$ for some\nfinite type, quasi-coherent sheaf $\\mathcal{G}$ on $Z$.\nThen $\\mathcal{G}$ is (universally) pure above $y$\nif and only if $\\mathcal{F}$ is (universally) pure above $y$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVH","source_file":"spaces-flat.tex","source_line":359,"source_end_line":366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L359-L366","statement_sha256":"df9ab405072a2a0cea65a78de68a5aa1d613478acdd39563bf7ae4bec3a3beb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12307,"rank":12307,"depth":28,"x":748.704,"y":1699.647,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVI","tag":"0CVI","title":"Relatively pure modules · Lemma 0CVI","summary":"In Situation [Tag 0CV6]. • If the support of F is proper over Y, then F is universally pure relative to Y. • If f is proper, then F is universally pure relative to Y. • If f is proper, then X is universally pure relative to Y.","statement_latex":"In Situation \\ref{situation-pre-pure}.\n\\begin{enumerate}\n\\item If the support of $\\mathcal{F}$ is proper over $Y$, then\n$\\mathcal{F}$ is universally pure relative to $Y$.\n\\item If $f$ is proper, then\n$\\mathcal{F}$ is universally pure relative to $Y$.\n\\item If $f$ is proper, then $X$ is universally pure relative to $Y$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Relatively pure modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVI","source_file":"spaces-flat.tex","source_line":373,"source_end_line":383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L373-L383","statement_sha256":"e3c8255eb689be39381cb87dff933bb54505cb13d52459ffe24b270640d05c2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12308,"rank":12308,"depth":57,"x":549.105,"y":1565.48,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWJ","tag":"0CWJ","title":"Flat finite type modules · Lemma 0CWJ","summary":"Let S be a scheme. Let X → Y be a finite type morphism of algebraic spaces over S. Let F be a finite type quasi-coherent O_X-module. Let y ∈ |Y| be a point. There exists an étale morphism (Y', y') → (Y, y) with Y' an affine scheme and étale morphisms h_i : W_i → X_Y', i = 1, …, n such that for each i there exists a complete dévissage of F_i/W_i/Y' over y', where F_i is the pullback of F to W_i and such that |(X_Y')_y'| ⊂ ⋃ h_i(W_i).","statement_latex":"Let $S$ be a scheme.\nLet $X \\to Y$ be a finite type morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $y \\in |Y|$ be a point. There exists an \\'etale morphism\n$(Y', y') \\to (Y, y)$ with $Y'$ an affine scheme and \\'etale morphisms\n$h_i : W_i \\to X_{Y'}$, $i = 1, \\ldots, n$ such that for each\n$i$ there exists a complete d\\'evissage of $\\mathcal{F}_i/W_i/Y'$ over $y'$,\nwhere $\\mathcal{F}_i$ is the pullback of $\\mathcal{F}$ to $W_i$\nand such that $|(X_{Y'})_{y'}| \\subset \\bigcup h_i(W_i)$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finite type modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWJ","source_file":"spaces-flat.tex","source_line":422,"source_end_line":433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L422-L433","statement_sha256":"5458a34db62a7229e240d2e71b95531cd82e58d3accd48a601ab26014b40fb1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12309,"rank":12309,"depth":41,"x":804.375,"y":1551.145,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWK","tag":"0CWK","title":"Flat finite type modules · Lemma 0CWK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let y ∈ |Y| and F = f^-1((y)) ⊂ |X|. Then the set (x ∈ F mid F flat over Y at x) is open in F.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $y \\in |Y|$ and $F = f^{-1}(\\{y\\}) \\subset |X|$. Then the set\n$$\n\\{x \\in F \\mid \\mathcal{F} \\text{ flat over }Y\\text{ at }x\\}\n$$\nis open in $F$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finite type modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWK","source_file":"spaces-flat.tex","source_line":445,"source_end_line":456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L445-L456","statement_sha256":"44054029e8fed4e51ff5ca8b7e80ea20c4d2a3a3f2025392892c2d74069f14b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12310,"rank":12310,"depth":53,"x":627.536,"y":1706.68,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVK","tag":"0CVK","title":"Flat finite type modules · Lemma 0CVK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let x ∈ |X| with image y ∈ |Y|. Let F be a finite type quasi-coherent sheaf on X. Let G be a quasi-coherent sheaf on Y. If F is flat at x over Y, then x ∈ WeakAss_X(F ⊗_O_X f^*G) ⇔ y ∈ WeakAss_Y(G) and x ∈ Ass_X/Y(F).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is\nlocally of finite type. Let $x \\in |X|$ with image $y \\in |Y|$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$.\nLet $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$.\nIf $\\mathcal{F}$ is flat at $x$ over $Y$, then\n$$\nx \\in \\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G})\n\\Leftrightarrow\ny \\in \\text{WeakAss}_Y(\\mathcal{G})\n\\text{ and }\nx \\in \\text{Ass}_{X/Y}(\\mathcal{F}).\n$$","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finite type modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVK","source_file":"spaces-flat.tex","source_line":467,"source_end_line":482,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L467-L482","statement_sha256":"0500a802500edec6b8cbcbc8f41cda386a451675beca015024ce8be7b2c3271a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12311,"rank":12311,"depth":56,"x":632.861,"y":1491.484,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVL","tag":"0CVL","title":"Flat finite type modules · Lemma 0CVL","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let F be a finite type quasi-coherent sheaf on X which is flat over Y. Let G be a quasi-coherent sheaf on Y. Then we have WeakAss_X(F ⊗_O_X f^*G) = Ass_X/Y(F) ∩ |f|^-1(WeakAss_Y(G))","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is locally of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$\nwhich is flat over $Y$. Let $\\mathcal{G}$ be a quasi-coherent sheaf on $Y$.\nThen we have\n$$\n\\text{WeakAss}_X(\\mathcal{F} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{G}) =\n\\text{Ass}_{X/Y}(\\mathcal{F}) \\cap\n|f|^{-1}(\\text{WeakAss}_Y(\\mathcal{G}))\n$$","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finite type modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVL","source_file":"spaces-flat.tex","source_line":515,"source_end_line":527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L515-L527","statement_sha256":"b56887a5c972e617fb6a9686fab92d8285f2ea4b4e3b670b585cceae7ebb2ddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12312,"rank":12312,"depth":57,"x":802.118,"y":1653.311,"cluster":"geometry-of-spaces"},{"id":"stacks:0DLR","tag":"0DLR","title":"Flat finite type modules · Theorem 0DLR","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Assume • X → Y is locally of finite presentation, • F is an O_X-module of finite type, and • the set of weakly associated points of Y is locally finite in Y. Then U = (x ∈ |X| : F flat at x over Y) is open in X and F|_U is an O_U-module of finite presentation and flat over Y.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nAssume\n\\begin{enumerate}\n\\item $X \\to Y$ is locally of finite presentation,\n\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite type, and\n\\item the set of weakly associated points of $Y$ is locally finite in $Y$.\n\\end{enumerate}\nThen $U = \\{x \\in |X| : \\mathcal{F}\\text{ flat at }x\\text{ over }Y\\}$\nis open in $X$ and $\\mathcal{F}|_U$ is an $\\mathcal{O}_U$-module\nof finite presentation and flat over $Y$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finite type modules","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLR","source_file":"spaces-flat.tex","source_line":534,"source_end_line":548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L534-L548","statement_sha256":"206f7d422b70d540b7db9462dec47d5244e2aa4da55b0e7b1035097c2ca83a85","origin":"The Stacks Project","memory_eligible":false,"source_rank":12313,"rank":12313,"depth":57,"x":546.98,"y":1630.005,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVW","tag":"0CVW","title":"Flat finite type modules · Lemma 0CVW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf on X. Let y ∈ |Y|. Set F = f^-1((y)) ⊂ |X|. Assume that • f is of finite type, • F is of finite type, and • F is flat over Y at all x ∈ F. Then there exists an étale morphism (Y', y') → (Y, y) where Y' is a scheme and a commutative diagram of algebraic spaces xymatrix X ar[d] & X' ar[l]^g ar[d] Y & Spec(O_Y', y') ar[l] such that X' → X ×_Y Spec(O_Y', y') is étale,…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$. Assume that\n\\begin{enumerate}\n\\item $f$ is of finite type,\n\\item $\\mathcal{F}$ is of finite type, and\n\\item $\\mathcal{F}$ is flat over $Y$ at all $x \\in F$.\n\\end{enumerate}\nThen there exists an \\'etale morphism $(Y', y') \\to (Y, y)$\nwhere $Y'$ is a scheme and a commutative diagram of algebraic spaces\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\\nY & \\Spec(\\mathcal{O}_{Y', y'}) \\ar[l]\n}\n$$\nsuch that $X' \\to X \\times_Y \\Spec(\\mathcal{O}_{Y', y'})$\nis \\'etale, $|X'_{y'}| \\to F$ is surjective, $X'$ is affine,\nand $\\Gamma(X', g^*\\mathcal{F})$ is a free $\\mathcal{O}_{Y', y'}$-module.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finite type modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVW","source_file":"spaces-flat.tex","source_line":563,"source_end_line":584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L563-L584","statement_sha256":"47d65d6418640b1468d150fae11b93dacc3720d961979793710310b615e3fc8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12314,"rank":12314,"depth":56,"x":754.014,"y":1502.322,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWL","tag":"0CWL","title":"Flat finite type modules · Theorem 0CWL","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let x ∈ |X| with image y ∈ |Y|. Set F = f^-1((y)) ⊂ |X|. Consider the conditions • F is flat at x over Y, and • for every x' ∈ F ∩ Ass_X/Y(F) which specializes to x we have that F is flat at x' over Y. Then we always have (2) ⇒ (1). If X and Y are decent, then (1) ⇒ (2).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$\nwhich is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $x \\in |X|$ with image $y \\in |Y|$.\nSet $F = f^{-1}(\\{y\\}) \\subset |X|$.\nConsider the conditions\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is flat at $x$ over $Y$, and\n\\item for every $x' \\in F \\cap \\text{Ass}_{X/Y}(\\mathcal{F})$ which\nspecializes to $x$ we have that $\\mathcal{F}$ is flat at $x'$ over $Y$.\n\\end{enumerate}\nThen we always have (2) $\\Rightarrow$ (1). If $X$ and $Y$ are\ndecent, then (1) $\\Rightarrow$ (2).","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finite type modules","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWL","source_file":"spaces-flat.tex","source_line":617,"source_end_line":633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L617-L633","statement_sha256":"42d434bb436894fe6a328189d694651455cbac33cdb2a3f4efc86e58fd7a01b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12315,"rank":12315,"depth":64,"x":703.991,"y":1714.11,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWM","tag":"0CWM","title":"Flat finite type modules · Lemma 0CWM","summary":"Let S be a local scheme with closed point s. Let f : X → S be a morphism from an algebraic space X to S which is locally of finite type. Let F be a finite type quasi-coherent O_X-module. Assume that • every point of Ass_X/S(F) specializes to a point of the closed fibre X_s is pure along X_s, or if f is proper., • F is flat over S at every point of X_s. Then F is flat over S.","statement_latex":"Let $S$ be a local scheme with closed point $s$.\nLet $f : X \\to S$ be a morphism from an algebraic space $X$ to $S$\nwhich is locally of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nAssume that\n\\begin{enumerate}\n\\item every point of $\\text{Ass}_{X/S}(\\mathcal{F})$ specializes\nto a point of the closed fibre $X_s$\\footnote{For example this holds if\n$f$ is finite type and $\\mathcal{F}$ is pure along $X_s$, or\nif $f$ is proper.},\n\\item $\\mathcal{F}$ is flat over $S$ at every point of $X_s$.\n\\end{enumerate}\nThen $\\mathcal{F}$ is flat over $S$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finite type modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWM","source_file":"spaces-flat.tex","source_line":672,"source_end_line":687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L672-L687","statement_sha256":"63d0a37ce15a69250272c8f57a97d55167cf96718bd24f34041f4286be978781","origin":"The Stacks Project","memory_eligible":false,"source_rank":12316,"rank":12316,"depth":65,"x":570.462,"y":1529.417,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVY","tag":"0CVY","title":"Flat finitely presented modules · Proposition 0CVY","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf on X. Let x ∈ |X| with image y ∈ |Y|. Assume that • f is locally of finite presentation, • F is of finite presentation, and • F is flat at x over Y. Then there exists a commutative diagram of pointed schemes xymatrix (X, x) ar[d] & (X', x') ar[l]^g ar[d] (Y, y) & (Y', y') ar[l] whose horizontal arrows are étale such that X', Y' are affine and such that Γ(X', g^*F) is…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $x \\in |X|$ with image $y \\in |Y|$.\nAssume that\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $\\mathcal{F}$ is of finite presentation, and\n\\item $\\mathcal{F}$ is flat at $x$ over $Y$.\n\\end{enumerate}\nThen there exists a commutative diagram of pointed schemes\n$$\n\\xymatrix{\n(X, x) \\ar[d] & (X', x') \\ar[l]^g \\ar[d] \\\\\n(Y, y) & (Y', y') \\ar[l]\n}\n$$\nwhose horizontal arrows are \\'etale such that $X'$, $Y'$\nare affine and such that\n$\\Gamma(X', g^*\\mathcal{F})$ is a projective\n$\\Gamma(Y', \\mathcal{O}_{Y'})$-module.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finitely presented modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVY","source_file":"spaces-flat.tex","source_line":706,"source_end_line":729,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L706-L729","statement_sha256":"fbd304a451d1e342e4a1984f9b825ea9d61f4061632326173a72894bc21f543a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12317,"rank":12317,"depth":53,"x":817.636,"y":1589.885,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVZ","tag":"0CVZ","title":"Flat finitely presented modules · Lemma 0CVZ","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf on X. Let y ∈ |Y|. Set F = f^-1((y)) ⊂ |X|. Assume that • f is of finite presentation, • F is of finite presentation, and • F is flat over Y at all x ∈ F. Then there exists a commutative diagram of algebraic spaces xymatrix X ar[d] & X' ar[l]^g ar[d] Y & Y' ar[l]_h such that h and g are étale, there is a point y' ∈ |Y'| mapping to y, we have F ⊂ g(|X'|), the…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $X$.\nLet $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$. Assume that\n\\begin{enumerate}\n\\item $f$ is of finite presentation,\n\\item $\\mathcal{F}$ is of finite presentation, and\n\\item $\\mathcal{F}$ is flat over $Y$ at all $x \\in F$.\n\\end{enumerate}\nThen there exists a commutative diagram of algebraic spaces\n$$\n\\xymatrix{\nX \\ar[d] & X' \\ar[l]^g \\ar[d] \\\\\nY & Y' \\ar[l]_h\n}\n$$\nsuch that $h$ and $g$ are \\'etale, there is a point\n$y' \\in |Y'|$ mapping to $y$, we have $F \\subset g(|X'|)$,\nthe algebraic spaces $X'$, $Y'$ are affine, and\n$\\Gamma(X', g^*\\mathcal{F})$ is a projective\n$\\Gamma(Y', \\mathcal{O}_{Y'})$-module.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flat finitely presented modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVZ","source_file":"spaces-flat.tex","source_line":738,"source_end_line":761,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L738-L761","statement_sha256":"0feec4e9410d6dcbfb24763ac88b84bfebdd9758706c08e902a100c2c9013fdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12318,"rank":12318,"depth":54,"x":586.573,"y":1685.621,"cluster":"geometry-of-spaces"},{"id":"stacks:0CW1","tag":"0CW1","title":"A criterion for purity · Lemma 0CW1","summary":"Let S be a scheme. Let X be a decent algebraic space locally of finite type over S. Let F be a finite type, quasi-coherent O_X-module. Let s ∈ S such that F is flat over S at all points of X_s. Let x' ∈ Ass_X/S(F). If the closure of (x') in |X| meets |X_s|, then the closure meets Ass_X/S(F) ∩ |X_s|.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space\nlocally of finite type over $S$.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module.\nLet $s \\in S$ such that $\\mathcal{F}$ is flat over $S$ at all points of $X_s$.\nLet $x' \\in \\text{Ass}_{X/S}(\\mathcal{F})$. If\nthe closure of $\\{x'\\}$ in $|X|$ meets $|X_s|$, then the closure\nmeets $\\text{Ass}_{X/S}(\\mathcal{F}) \\cap |X_s|$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CW1","source_file":"spaces-flat.tex","source_line":780,"source_end_line":789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L780-L789","statement_sha256":"b7bae15ef7e15f80146f84da419e2372b9d736fc173ec04872bcaf9bb186a3d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12319,"rank":12319,"depth":57,"x":680.038,"y":1483.764,"cluster":"geometry-of-spaces"},{"id":"stacks:0CW2","tag":"0CW2","title":"A criterion for purity · Lemma 0CW2","summary":"Let Y be an algebraic space over a scheme S. Let g : X' → X be a morphism of algebraic spaces over Y with X locally of finite type over Y. Let F be a quasi-coherent O_X-module. If Ass_X/Y(F) ⊂ g(|X'|), then for any morphism Z → Y we have Ass_X_Z/Z(F_Z) ⊂ g_Z(|X'_Z|).","statement_latex":"Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \\to X$ be a\nmorphism of algebraic spaces over $Y$ with $X$ locally of finite type over $Y$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nIf $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$, then for any morphism\n$Z \\to Y$ we have $\\text{Ass}_{X_Z/Z}(\\mathcal{F}_Z) \\subset g_Z(|X'_Z|)$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CW2","source_file":"spaces-flat.tex","source_line":813,"source_end_line":820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L813-L820","statement_sha256":"dfa5f77a4481abdd1f904fbd1ebf43a9a37feeef4b34c665ee8d3e67d990ce85","origin":"The Stacks Project","memory_eligible":false,"source_rank":12320,"rank":12320,"depth":16,"x":773.516,"y":1685.797,"cluster":"geometry-of-spaces"},{"id":"stacks:0CW3","tag":"0CW3","title":"A criterion for purity · Lemma 0CW3","summary":"Let Y be an algebraic space over a scheme S. Let g : X' → X be an étale morphism of algebraic spaces over Y. Assume the structure morphisms X' → Y and X → Y are decent and of finite type. Let F be a finite type, quasi-coherent O_X-module. Let y ∈ |Y|. Set F = f^-1((y)) ⊂ |X|. • If Ass_X/Y(F) ⊂ g(|X'|) and g^*F is (universally) pure above y, then F is (universally) pure above y. • If F is pure above y, g(|X'|) contains F, and Y is affine local with closed point y, then…","statement_latex":"Let $Y$ be an algebraic space over a scheme $S$. Let $g : X' \\to X$ be an\n\\'etale morphism of algebraic spaces over $Y$. Assume the structure\nmorphisms $X' \\to Y$ and $X \\to Y$ are decent and of finite type.\nLet $\\mathcal{F}$ be a finite type, quasi-coherent $\\mathcal{O}_X$-module.\nLet $y \\in |Y|$. Set $F = f^{-1}(\\{y\\}) \\subset |X|$.\n\\begin{enumerate}\n\\item If $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$\nand $g^*\\mathcal{F}$ is (universally) pure above $y$, then\n$\\mathcal{F}$ is (universally) pure above $y$.\n\\item If $\\mathcal{F}$ is pure above $y$, $g(|X'|)$ contains $F$, and\n$Y$ is affine local with closed point $y$, then\n$\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$.\n\\item If $\\mathcal{F}$ is pure above $y$, $\\mathcal{F}$ is flat\nat all points of $F$, $g(|X'|)$ contains\n$\\text{Ass}_{X/Y}(\\mathcal{F}) \\cap F$, and $Y$ is affine local\nwith closed point $y$, then\n$\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$.\n\\item Add more here.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CW3","source_file":"spaces-flat.tex","source_line":832,"source_end_line":853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L832-L853","statement_sha256":"fd124933ed607b803681be125d95d2d5b7003de280e3dc4648310b52225fc813","origin":"The Stacks Project","memory_eligible":false,"source_rank":12321,"rank":12321,"depth":58,"x":541.944,"y":1589.791,"cluster":"geometry-of-spaces"},{"id":"stacks:0CW4","tag":"0CW4","title":"A criterion for purity · Lemma 0CW4","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. Let y ∈ |Y|. Assume • f is decent and of finite type, • F is of finite type, • F is flat over Y at all points lying over y, and • F is pure above y. Then F is universally pure above y.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $y \\in |Y|$.\nAssume\n\\begin{enumerate}\n\\item $f$ is decent and of finite type,\n\\item $\\mathcal{F}$ is of finite type,\n\\item $\\mathcal{F}$ is flat over $Y$ at all points lying over $y$, and\n\\item $\\mathcal{F}$ is pure above $y$.\n\\end{enumerate}\nThen $\\mathcal{F}$ is universally pure above $y$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CW4","source_file":"spaces-flat.tex","source_line":889,"source_end_line":903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L889-L903","statement_sha256":"fb68d46c2a12e25b8bd4bcc1b411c8e872ace0f06204f703e2ed5782f3d77cd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12322,"rank":12322,"depth":59,"x":790.093,"y":1529.139,"cluster":"geometry-of-spaces"},{"id":"stacks:0CW5","tag":"0CW5","title":"A criterion for purity · Lemma 0CW5","summary":"Let S be a scheme. Let f : X → Y be a decent, finite type morphism of algebraic spaces over S. Let F be a finite type quasi-coherent O_X-module. Assume F is flat over Y. In this case F is pure relative to Y if and only if F is universally pure relative to Y.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a decent, finite type morphism of algebraic\nspaces over $S$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nAssume $\\mathcal{F}$ is flat over $Y$. In this case\n$\\mathcal{F}$ is pure relative to $Y$ if and only if $\\mathcal{F}$\nis universally pure relative to $Y$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CW5","source_file":"spaces-flat.tex","source_line":928,"source_end_line":937,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L928-L937","statement_sha256":"98218531ea04a911061f485edfe5a06bc8c3c38240fdfa94bc1eb09dec917272","origin":"The Stacks Project","memory_eligible":false,"source_rank":12323,"rank":12323,"depth":60,"x":655.786,"y":1714.807,"cluster":"geometry-of-spaces"},{"id":"stacks:0CW6","tag":"0CW6","title":"A criterion for purity · Lemma 0CW6","summary":"Let Y be an algebraic space over a scheme S. Let g : X' → X be a flat morphism of algebraic spaces over Y with X locally of finite type over Y. Let F be a finite type quasi-coherent O_X-module which is flat over Y. If Ass_X/Y(F) ⊂ g(|X'|) then the canonical map F → g_*g^*F is injective, and remains injective after any base change.","statement_latex":"Let $Y$ be an algebraic space over a scheme $S$.\nLet $g : X' \\to X$ be a flat morphism of algebraic spaces over $Y$\nwith $X$ locally of finite type over $Y$.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module\nwhich is flat over $Y$. If $\\text{Ass}_{X/Y}(\\mathcal{F}) \\subset g(|X'|)$\nthen the canonical map\n$$\n\\mathcal{F} \\longrightarrow g_*g^*\\mathcal{F}\n$$\nis injective, and remains injective after any base change.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"A criterion for purity","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CW6","source_file":"spaces-flat.tex","source_line":945,"source_end_line":957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L945-L957","statement_sha256":"7fbb2168ef1c68995dd46abf0d5aafc068397c9c7f60e6ce0b4a748b92eac0c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12324,"rank":12324,"depth":58,"x":605.475,"y":1501.53,"cluster":"geometry-of-spaces"},{"id":"stacks:083H","tag":"083H","title":"Flattening functors · Lemma 083H","summary":"In Situation [Tag 083F]. Each of the functors F_iso, F_inj, F_surj, F_zero satisfies the sheaf property for the fpqc topology.","statement_latex":"In Situation \\ref{situation-iso}.\nEach of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$\nsatisfies the sheaf property for the fpqc topology.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083H","source_file":"spaces-flat.tex","source_line":1051,"source_end_line":1056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1051-L1056","statement_sha256":"ebcccee9959f2deba549877945b276999a97d379416438c02fb6cd0430a2a285","origin":"The Stacks Project","memory_eligible":false,"source_rank":12325,"rank":12325,"depth":15,"x":814.241,"y":1630.346,"cluster":"geometry-of-spaces"},{"id":"stacks:083I","tag":"083I","title":"Flattening functors · Lemma 083I","summary":"In Situation [Tag 083F] let X' → X be a flat morphism of algebraic spaces. Denote u' : F' → G' the pullback of u to X'. Denote F'_iso, F'_inj, F'_surj, F'_zero the functors on Sch/B associated to u'. • If G is of finite type and the image of |X'| → |X| contains the support of G, then F_surj = F'_surj and F_zero = F'_zero. • If F is of finite type and the image of |X'| → |X| contains the support of F, then F_inj = F'_inj and F_zero = F'_zero. • If F and G are of finite…","statement_latex":"In Situation \\ref{situation-iso} let $X' \\to X$ be a flat morphism\nof algebraic spaces. Denote $u' : \\mathcal{F}' \\to \\mathcal{G}'$\nthe pullback of $u$ to $X'$. Denote $F'_{iso}$, $F'_{inj}$, $F'_{surj}$,\n$F'_{zero}$ the functors on $\\Sch/B$ associated to $u'$.\n\\begin{enumerate}\n\\item If $\\mathcal{G}$ is of finite type and the image of $|X'| \\to |X|$\ncontains the support of $\\mathcal{G}$, then $F_{surj} = F'_{surj}$\nand $F_{zero} = F'_{zero}$.\n\\item If $\\mathcal{F}$ is of finite type and the image of $|X'| \\to |X|$\ncontains the support of $\\mathcal{F}$, then $F_{inj} = F'_{inj}$\nand $F_{zero} = F'_{zero}$.\n\\item If $\\mathcal{F}$ and $\\mathcal{G}$ are of finite type and the image of\n$|X'| \\to |X|$ contains the supports of $\\mathcal{F}$ and $\\mathcal{G}$,\nthen $F_{iso} = F'_{iso}$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083I","source_file":"spaces-flat.tex","source_line":1073,"source_end_line":1090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1073-L1090","statement_sha256":"94b50861ba601873780fc51547b29fef39063e1b2149eca7337b25db3cf08973","origin":"The Stacks Project","memory_eligible":false,"source_rank":12326,"rank":12326,"depth":56,"x":556.518,"y":1653.835,"cluster":"geometry-of-spaces"},{"id":"stacks:083J","tag":"083J","title":"Flattening functors · Lemma 083J","summary":"In Situation [Tag 083F]. • If G is of finite type and the scheme theoretic support of G is quasi-compact over B, then F_surj is limit preserving. • If F of finite type and the scheme theoretic support of F is quasi-compact over B, then F_zero is limit preserving. • If F is of finite type, G is of finite presentation, and the scheme theoretic supports of F and G are quasi-compact over B, then F_iso is limit preserving.","statement_latex":"In Situation \\ref{situation-iso}.\n\\begin{enumerate}\n\\item If $\\mathcal{G}$ is of finite type and the scheme theoretic support\nof $\\mathcal{G}$ is quasi-compact over $B$, then $F_{surj}$ is limit\npreserving.\n\\item If $\\mathcal{F}$ of finite type and the scheme theoretic support\nof $\\mathcal{F}$ is quasi-compact over $B$, then\n$F_{zero}$ is limit preserving.\n\\item If $\\mathcal{F}$ is of finite type,\n$\\mathcal{G}$ is of finite presentation, and the\nscheme theoretic supports of $\\mathcal{F}$ and $\\mathcal{G}$ are\nquasi-compact over $B$, then $F_{iso}$ is limit preserving.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083J","source_file":"spaces-flat.tex","source_line":1117,"source_end_line":1132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1117-L1132","statement_sha256":"0daf27c4aade012fa395e1c90b560efd3940c0c2dc26fda61008013153408fd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12327,"rank":12327,"depth":56,"x":727.797,"y":1490.154,"cluster":"geometry-of-spaces"},{"id":"stacks:0CVM","tag":"0CVM","title":"Flattening functors · Lemma 0CVM","summary":"In Situation [Tag 083F] suppose given an exact sequence F xrightarrowu G xrightarrowv H → 0 Then we have F_v, iso = F_u, zero with obvious notation.","statement_latex":"In Situation \\ref{situation-iso} suppose given an exact sequence\n$$\n\\mathcal{F} \\xrightarrow{u} \\mathcal{G} \\xrightarrow{v} \\mathcal{H} \\to 0\n$$\nThen we have $F_{v, iso} = F_{u, zero}$ with obvious notation.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CVM","source_file":"spaces-flat.tex","source_line":1202,"source_end_line":1209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1202-L1209","statement_sha256":"ad8407c25dc6eed06faf896568e0b6450417592ecc30bebd872d93baa9a59c92","origin":"The Stacks Project","memory_eligible":false,"source_rank":12328,"rank":12328,"depth":0,"x":733.13,"y":1708.2,"cluster":"geometry-of-spaces"},{"id":"stacks:0CW7","tag":"0CW7","title":"Flattening functors · Lemma 0CW7","summary":"In Situation [Tag 083F] suppose given an affine morphism i : Z → X and a quasi-coherent O_Z-module H such that G = i_*H. Let v : i^*F → H be the map adjoint to u. Then • F_v, zero = F_u, zero, and • if i is a closed immersion, then F_v, surj = F_u, surj.","statement_latex":"In Situation \\ref{situation-iso} suppose given an affine morphism\n$i : Z \\to X$ and a quasi-coherent $\\mathcal{O}_Z$-module $\\mathcal{H}$\nsuch that $\\mathcal{G} = i_*\\mathcal{H}$. Let\n$v : i^*\\mathcal{F} \\to \\mathcal{H}$ be the map adjoint to $u$.\nThen\n\\begin{enumerate}\n\\item $F_{v, zero} = F_{u, zero}$, and\n\\item if $i$ is a closed immersion, then $F_{v, surj} = F_{u, surj}$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CW7","source_file":"spaces-flat.tex","source_line":1218,"source_end_line":1229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1218-L1229","statement_sha256":"84bb4e21930ce3e576be7e2f389604a9ade3a2a3a283ac2740730ad3255fcc81","origin":"The Stacks Project","memory_eligible":false,"source_rank":12329,"rank":12329,"depth":55,"x":553.718,"y":1550.325,"cluster":"geometry-of-spaces"},{"id":"stacks:0CW8","tag":"0CW8","title":"Flattening functors · Lemma 0CW8","summary":"In Situation [Tag 083F] suppose given an affine morphism g : X → X'. Set u' = f_*u : f_*F → f_*G. Then F_u, iso = F_u', iso, F_u, inj = F_u', inj, F_u, surj = F_u', surj, and F_u, zero = F_u', zero.","statement_latex":"In Situation \\ref{situation-iso} suppose given an affine morphism\n$g : X \\to X'$. Set $u' = f_*u : f_*\\mathcal{F} \\to f_*\\mathcal{G}$.\nThen $F_{u, iso} = F_{u', iso}$, $F_{u, inj} = F_{u', inj}$,\n$F_{u, surj} = F_{u', surj}$, and $F_{u, zero} = F_{u', zero}$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CW8","source_file":"spaces-flat.tex","source_line":1253,"source_end_line":1259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1253-L1259","statement_sha256":"25929ebdeea4067f02d65e3469a10ae44396db68063c197da33205ed481a34d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12330,"rank":12330,"depth":27,"x":813.164,"y":1564.948,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWZ","tag":"0CWZ","title":"Flattening functors · Lemma 0CWZ","summary":"In Situation [Tag 0CWX]. • The functor F_flat satisfies the sheaf property for the fpqc topology. • If f is quasi-compact and locally of finite presentation and F is of finite presentation, then the functor F_flat is limit preserving.","statement_latex":"In Situation \\ref{situation-flat}.\n\\begin{enumerate}\n\\item The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology.\n\\item If $f$ is quasi-compact and locally of finite presentation\nand $\\mathcal{F}$ is of finite presentation, then the functor\n$F_{flat}$ is limit preserving.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWZ","source_file":"spaces-flat.tex","source_line":1302,"source_end_line":1311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1302-L1311","statement_sha256":"c1c71af5251b3e2db81917859dc1458adfbcc3ead92fdec6832d60afee613358","origin":"The Stacks Project","memory_eligible":false,"source_rank":12331,"rank":12331,"depth":49,"x":609.943,"y":1701.482,"cluster":"geometry-of-spaces"},{"id":"stacks:083K","tag":"083K","title":"Making a map zero · Lemma 083K","summary":"In Situation [Tag 0CWA]. Let T → S be a quasi-compact morphism of schemes such that the base change u_T is zero. Then exists a closed subscheme Z ⊂ S such that (a) T → S factors through Z and (b) the base change u_Z is zero. If F is a finite type O_X-module and the scheme theoretic support of F is quasi-compact, then we can take Z → S of finite presentation.","statement_latex":"In Situation \\ref{situation-somewhat-closed}. Let $T \\to S$\nbe a quasi-compact morphism of schemes such that the base change $u_T$ is\nzero. Then exists a closed subscheme $Z \\subset S$ such that\n(a) $T \\to S$ factors through $Z$ and (b) the base change $u_Z$ is zero.\nIf $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module and\nthe scheme theoretic support of $\\mathcal{F}$ is quasi-compact,\nthen we can take $Z \\to S$ of finite presentation.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Making a map zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083K","source_file":"spaces-flat.tex","source_line":1349,"source_end_line":1358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1349-L1358","statement_sha256":"d45becb006028c031cb3babce75d0150e263042b22e36dbb8be3ab873eed4d20","origin":"The Stacks Project","memory_eligible":false,"source_rank":12332,"rank":12332,"depth":57,"x":650.028,"y":1485.332,"cluster":"geometry-of-spaces"},{"id":"stacks:083L","tag":"083L","title":"Making a map zero · Lemma 083L","summary":"Let A be a ring. Let u : M → N be a map of A-modules. If N is projective as an A-module, then there exists an ideal I ⊂ A such that for any ring map φ : A → B the following are equivalent • u ⊗ 1 : M ⊗_A B → N ⊗_A B is zero, and • φ(I) = 0.","statement_latex":"Let $A$ be a ring. Let $u : M \\to N$ be a map of $A$-modules.\nIf $N$ is projective as an $A$-module, then there exists an ideal\n$I \\subset A$ such that for any ring map $\\varphi : A \\to B$\nthe following are equivalent\n\\begin{enumerate}\n\\item $u \\otimes 1 : M \\otimes_A B \\to N \\otimes_A B$ is zero, and\n\\item $\\varphi(I) = 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Making a map zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083L","source_file":"spaces-flat.tex","source_line":1409,"source_end_line":1419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1409-L1419","statement_sha256":"32a5deac9654b2fdc69fe8b15e206ba89a9d7f8ee0b5996949ae43c55e1a9f31","origin":"The Stacks Project","memory_eligible":false,"source_rank":12333,"rank":12333,"depth":0,"x":794.398,"y":1667.597,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWB","tag":"0CWB","title":"Making a map zero · Lemma 0CWB","summary":"In Situation [Tag 0CWA]. Let T ⊂ S be a subset. Let s ∈ S be in the closure of T. For t ∈ T, let u_t be the pullback of u to X_t and let u_s be the pullback of u to X_s. If X is locally of finite presentation over S, G is of finite presentation is finitely presented relative to S, but this notion hasn't yet been defined in the setting of algebraic spaces. The definition for schemes is given in More on Morphisms, Section [Tag 05GX]., and u_t = 0 for all t ∈ T, then u_s = 0.","statement_latex":"In Situation \\ref{situation-somewhat-closed}.\nLet $T \\subset S$ be a subset. Let $s \\in S$ be in the closure of $T$.\nFor $t \\in T$, let $u_t$ be the pullback of $u$ to $X_t$\nand let $u_s$ be the pullback of $u$ to $X_s$.\nIf $X$ is locally of finite presentation over $S$,\n$\\mathcal{G}$ is of finite presentation\\footnote{It would\nsuffice if $X$ is locally of finite type over $S$\nand $\\mathcal{G}$ is finitely presented relative to $S$,\nbut this notion hasn't yet been defined in the setting\nof algebraic spaces. The definition for schemes is\ngiven in More on Morphisms, Section\n\\ref{more-morphisms-section-finite-type-finite-presentation}.}, and\n$u_t = 0$ for all $t \\in T$, then $u_s = 0$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Making a map zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWB","source_file":"spaces-flat.tex","source_line":1430,"source_end_line":1445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1430-L1445","statement_sha256":"97d36888d77fbb855ecaa9d78a077d23f705d0c1093c8dab8d274ce0abd0d4dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12334,"rank":12334,"depth":54,"x":541.183,"y":1615.078,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWC","tag":"0CWC","title":"Making a map zero · Lemma 0CWC","summary":"In Situation [Tag 083F]. Assume • f is of finite presentation, and • G is of finite presentation, flat over B, and pure relative to B. Then F_zero is an algebraic space and F_zero → B is a closed immersion. If F is of finite type, then F_zero → B is of finite presentation.","statement_latex":"In Situation \\ref{situation-iso}. Assume\n\\begin{enumerate}\n\\item $f$ is of finite presentation, and\n\\item $\\mathcal{G}$ is of finite presentation,\nflat over $B$, and pure relative to $B$.\n\\end{enumerate}\nThen $F_{zero}$ is an algebraic space and $F_{zero} \\to B$\nis a closed immersion. If $\\mathcal{F}$ is of finite type, then\n$F_{zero} \\to B$ is of finite presentation.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Making a map zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWC","source_file":"spaces-flat.tex","source_line":1489,"source_end_line":1500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1489-L1500","statement_sha256":"1b82838bcecfb5e11f7d594232f070581096629f76ac1a691a78a3ff46d1cd1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12335,"rank":12335,"depth":61,"x":770.302,"y":1510.048,"cluster":"geometry-of-spaces"},{"id":"stacks:083M","tag":"083M","title":"Making a map zero · Lemma 083M","summary":"In Situation [Tag 083F]. Assume • f is locally of finite presentation, • G is an O_X-module of finite presentation flat over B, • the support of G is proper over B. Then the functor F_zero is an algebraic space and F_zero → B is a closed immersion. If F is of finite type, then F_zero → B is of finite presentation.","statement_latex":"In Situation \\ref{situation-iso}. Assume\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $\\mathcal{G}$ is an $\\mathcal{O}_X$-module of finite presentation\nflat over $B$,\n\\item the support of $\\mathcal{G}$ is proper over $B$.\n\\end{enumerate}\nThen the functor $F_{zero}$ is an algebraic space and $F_{zero} \\to B$\nis a closed immersion. If $\\mathcal{F}$ is of finite type, then\n$F_{zero} \\to B$ is of finite presentation.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Making a map zero","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083M","source_file":"spaces-flat.tex","source_line":1572,"source_end_line":1584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1572-L1584","statement_sha256":"9a6260a45067b629f3c6062a73a247b7f98c2e896ec631024592b1904b40de40","origin":"The Stacks Project","memory_eligible":false,"source_rank":12336,"rank":12336,"depth":62,"x":685.754,"y":1717.655,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWD","tag":"0CWD","title":"Flattening a map · Theorem 0CWD","summary":"In Situation [Tag 083F] assume • f is of finite presentation, • F is of finite presentation, flat over B, and pure relative to B, and • u is surjective. Then F_iso is representable by a closed immersion Z → B. Moreover Z → S is of finite presentation if G is of finite presentation.","statement_latex":"In Situation \\ref{situation-iso} assume\n\\begin{enumerate}\n\\item $f$ is of finite presentation,\n\\item $\\mathcal{F}$ is of finite presentation, flat over $B$, and\npure relative to $B$, and\n\\item $u$ is surjective.\n\\end{enumerate}\nThen $F_{iso}$ is representable by a closed immersion $Z \\to B$.\nMoreover $Z \\to S$ is of finite presentation if $\\mathcal{G}$ is\nof finite presentation.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening a map","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWD","source_file":"spaces-flat.tex","source_line":1641,"source_end_line":1653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1641-L1653","statement_sha256":"be6ec5345b576e5324e1cea64f373d911f24f677d349f6fcf237fb39c44a8aee","origin":"The Stacks Project","memory_eligible":false,"source_rank":12337,"rank":12337,"depth":62,"x":581.07,"y":1516.447,"cluster":"geometry-of-spaces"},{"id":"stacks:083N","tag":"083N","title":"Flattening a map · Lemma 083N","summary":"In Situation [Tag 083F]. Assume • f is locally of finite presentation, • F is locally of finite presentation and flat over B, • the support of F is proper over B, and • u is surjective. Then the functor F_iso is an algebraic space and F_iso → B is a closed immersion. If G is of finite presentation, then F_iso → B is of finite presentation.","statement_latex":"In Situation \\ref{situation-iso}. Assume\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $\\mathcal{F}$ is locally of finite presentation and flat over $B$,\n\\item the support of $\\mathcal{F}$ is proper over $B$, and\n\\item $u$ is surjective.\n\\end{enumerate}\nThen the functor $F_{iso}$ is an algebraic space and $F_{iso} \\to B$\nis a closed immersion. If $\\mathcal{G}$ is of finite presentation, then\n$F_{iso} \\to B$ is of finite presentation.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening a map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/083N","source_file":"spaces-flat.tex","source_line":1667,"source_end_line":1679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1667-L1679","statement_sha256":"62d5c0fa9e0a04dd8c20065558e05230433d35eb9445beab87d599ca238d2a26","origin":"The Stacks Project","memory_eligible":false,"source_rank":12338,"rank":12338,"depth":63,"x":820.243,"y":1605.479,"cluster":"geometry-of-spaces"},{"id":"stacks:09TP","tag":"09TP","title":"Flattening a map · Lemma 09TP","summary":"In Situation [Tag 083F]. Assume • f is locally of finite presentation, • G is of finite type, • the support of G is proper over B. Then F_surj is an algebraic space and F_surj → B is an open immersion.","statement_latex":"In Situation \\ref{situation-iso}. Assume\n\\begin{enumerate}\n\\item $f$ is locally of finite presentation,\n\\item $\\mathcal{G}$ is of finite type,\n\\item the support of $\\mathcal{G}$ is proper over $B$.\n\\end{enumerate}\nThen $F_{surj}$ is an algebraic space and $F_{surj} \\to B$\nis an open immersion.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening a map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TP","source_file":"spaces-flat.tex","source_line":1696,"source_end_line":1706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1696-L1706","statement_sha256":"dc6f2217cfd8740e9d5c5b171d0a3ad0dc4f2d947d47b3ad73c89f8ec4aa6b1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12339,"rank":12339,"depth":55,"x":572.102,"y":1675.592,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWP","tag":"0CWP","title":"Flattening in the local case · Lemma 0CWP","summary":"Let S be the spectrum of a henselian local ring with closed point s. Let X → S be a morphism of algebraic spaces which is locally of finite type. Let F be a finite type quasi-coherent O_X-module. Let E ⊂ |X_s| be a subset. There exists a closed subscheme Z ⊂ S with the following property: for any morphism of pointed schemes (T, t) → (S, s) the following are equivalent • F_T is flat over T at all points of |X_t| which map to a point of E ⊂ |X_s|, and • Spec(O_T, t) → S…","statement_latex":"Let $S$ be the spectrum of a henselian local ring with closed point $s$.\nLet $X \\to S$ be a morphism of algebraic spaces which is\nlocally of finite type.\nLet $\\mathcal{F}$ be a finite type quasi-coherent $\\mathcal{O}_X$-module.\nLet $E \\subset |X_s|$ be a subset. There exists a closed subscheme\n$Z \\subset S$ with the following property: for any morphism of pointed\nschemes $(T, t) \\to (S, s)$ the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}_T$ is flat over $T$ at all points of\n$|X_t|$ which map to a point of $E \\subset |X_s|$, and\n\\item $\\Spec(\\mathcal{O}_{T, t}) \\to S$ factors through $Z$.\n\\end{enumerate}\nMoreover, if $X \\to S$ is locally of finite presentation,\n$\\mathcal{F}$ is of finite presentation, and $E \\subset |X_s|$ is\nclosed and quasi-compact, then $Z \\to S$ is of finite presentation.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Flattening in the local case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWP","source_file":"spaces-flat.tex","source_line":1737,"source_end_line":1754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1737-L1754","statement_sha256":"84cb0b86e42603e8c3b9d842c01e60518a59dbd64e683a42cbbc1b650e9107bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12340,"rank":12340,"depth":60,"x":698.787,"y":1482.95,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWR","tag":"0CWR","title":"Universal flattening · Definition 0CWR","summary":"Let S be a scheme. Let X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. We say that the universal flattening of F exists if the functor F_flat defined in Situation [Tag 0CWX] is an algebraic space. We say that the universal flattening of X exists if the universal flattening of O_X exists.","statement_latex":"Let $S$ be a scheme.\nLet $X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nWe say that the {\\it universal flattening of $\\mathcal{F}$ exists}\nif the functor $F_{flat}$ defined in Situation \\ref{situation-flat}\nis an algebraic space.\nWe say that the {\\it universal flattening of $X$ exists}\nif the universal flattening of $\\mathcal{O}_X$ exists.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Universal flattening","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWR","source_file":"spaces-flat.tex","source_line":1792,"source_end_line":1802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1792-L1802","statement_sha256":"893fde2ad5eb8ab27c07c690bb18ece0590ab1f4d42dbfd48a26c14db04d6497","origin":"The Stacks Project","memory_eligible":false,"source_rank":12341,"rank":12341,"depth":0,"x":760.333,"y":1697.042,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWS","tag":"0CWS","title":"Universal flattening · Lemma 0CWS","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces which is locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let n ≥ 0. The following are equivalent • for some commutative diagram xymatrix U ar[d]_φ ar[r] & V ar[d] X ar[r] & Y with surjective, étale vertical arrows where U and V are schemes, the sheaf φ^*F is flat over V in dimensions ≥ n (More on Flatness, Definition [Tag 0CWG]), • for every commutative diagram xymatrix U…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces which is\nlocally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite\ntype. Let $n \\geq 0$. The following are equivalent\n\\begin{enumerate}\n\\item for some commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_\\varphi \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith surjective, \\'etale vertical arrows where $U$ and $V$ are\nschemes, the sheaf $\\varphi^*\\mathcal{F}$ is flat over $V$\nin dimensions $\\geq n$ (More on Flatness, Definition\n\\ref{flat-definition-flat-dimension-n}),\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_\\varphi \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith \\'etale vertical arrows where $U$ and $V$ are schemes,\nthe sheaf $\\varphi^*\\mathcal{F}$ is flat over $V$ in dimensions $\\geq n$, and\n\\item for $x \\in |X|$ such that $\\mathcal{F}$ is not flat at $x$\nover $Y$ the transcendence degree of $x/f(x)$ is $< n$ (Morphisms of Spaces,\nDefinition \\ref{spaces-morphisms-definition-dimension-fibre}).\n\\end{enumerate}\nIf this is true, then it remains true after any base change $Y' \\to Y$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Universal flattening","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWS","source_file":"spaces-flat.tex","source_line":1812,"source_end_line":1845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1812-L1845","statement_sha256":"1609af2cea54e4f81249f1154afb31a8f5df837053712c19481aac4ee4b9f0b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12342,"rank":12342,"depth":55,"x":542.626,"y":1574.007,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWT","tag":"0CWT","title":"Universal flattening · Definition 0CWT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let F be a quasi-coherent O_X-module of finite type. Let n ≥ 0. We say F is flat over Y in dimensions ≥ n if the equivalent conditions of Lemma [Tag 0CWS] are satisfied.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$ which is locally of finite type.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module of finite type.\nLet $n \\geq 0$.\nWe say {\\it $\\mathcal{F}$ is flat over $Y$ in dimensions $\\geq n$}\nif the equivalent conditions of Lemma \\ref{lemma-pre-flat-dimension-n}\nare satisfied.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Universal flattening","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWT","source_file":"spaces-flat.tex","source_line":1856,"source_end_line":1865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1856-L1865","statement_sha256":"c63fa488e69ef53fb45fdb74fd437a5539cb923d42ea0c40bf0a52ae732113dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12343,"rank":12343,"depth":56,"x":802.289,"y":1541.175,"cluster":"geometry-of-spaces"},{"id":"stacks:0CWW","tag":"0CWW","title":"Universal flattening · Lemma 0CWW","summary":"In Situation [Tag 0CWU]. • The functor F_n satisfies the sheaf property for the fpqc topology. • If f is quasi-compact and locally of finite presentation and F is of finite presentation, then the functor F_n is limit preserving.","statement_latex":"In Situation \\ref{situation-flat-dimension-n}.\n\\begin{enumerate}\n\\item The functor $F_n$ satisfies the sheaf property for the fpqc topology.\n\\item If $f$ is quasi-compact and locally of finite presentation\nand $\\mathcal{F}$ is of finite presentation, then the functor $F_n$ is\nlimit preserving.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Universal flattening","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CWW","source_file":"spaces-flat.tex","source_line":1904,"source_end_line":1913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1904-L1913","statement_sha256":"44c38a946f8ed466dbf217a90bd372241667166476f8d925fbc8dd10de2120bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12344,"rank":12344,"depth":56,"x":637.101,"y":1712.848,"cluster":"geometry-of-spaces"},{"id":"stacks:0CX0","tag":"0CX0","title":"Universal flattening · Lemma 0CX0","summary":"In Situation [Tag 0CWU]. Let h : X' → X be an étale morphism. Set F' = h^*F and f' = f ∘ h. Let F_n' be ([Tag 0CWV]) associated to (f' : X' → Y, F'). Then F_n is a subfunctor of F_n' and if h(X') ⊃ Ass_X/Y(F), then F_n = F'_n.","statement_latex":"In Situation \\ref{situation-flat-dimension-n}.\nLet $h : X' \\to X$ be an \\'etale morphism.\nSet $\\mathcal{F}' = h^*\\mathcal{F}$ and $f' = f \\circ h$.\nLet $F_n'$ be (\\ref{equation-flat-dimension-n})\nassociated to $(f' : X' \\to Y, \\mathcal{F}')$.\nThen $F_n$ is a subfunctor of $F_n'$ and if\n$h(X') \\supset \\text{Ass}_{X/Y}(\\mathcal{F})$, then $F_n = F'_n$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Universal flattening","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CX0","source_file":"spaces-flat.tex","source_line":1951,"source_end_line":1960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1951-L1960","statement_sha256":"592ba5f4de6d5fd5ee180f4fc7eaa34f3190e8dbba8b32fb35f3785e0599fac3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12345,"rank":12345,"depth":65,"x":620.841,"y":1492.367,"cluster":"geometry-of-spaces"},{"id":"stacks:0CX1","tag":"0CX1","title":"Universal flattening · Theorem 0CX1","summary":"In Situation [Tag 0CWU]. Assume moreover that f is of finite presentation, that F is an O_X-module of finite presentation, and that F is pure relative to Y. Then F_n is an algebraic space and F_n → Y is a monomorphism of finite presentation.","statement_latex":"In Situation \\ref{situation-flat-dimension-n}.\nAssume moreover that $f$ is of finite presentation, that\n$\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation,\nand that $\\mathcal{F}$ is pure relative to $Y$.\nThen $F_n$ is an algebraic space and\n$F_n \\to Y$ is a monomorphism of finite presentation.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Universal flattening","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CX1","source_file":"spaces-flat.tex","source_line":1978,"source_end_line":1986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L1978-L1986","statement_sha256":"01ab8082d7577cc7fbb6f9b9236a1630047808bd6427151eb407d2f6660cb04d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12346,"rank":12346,"depth":66,"x":810.272,"y":1645.831,"cluster":"geometry-of-spaces"},{"id":"stacks:0CX2","tag":"0CX2","title":"Universal flattening · Lemma 0CX2","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module. • If f is of finite presentation, F is an O_X-module of finite presentation, and F is pure relative to Y, then there exists a universal flattening Y' → Y of F. Moreover Y' → Y is a monomorphism of finite presentation. • If f is of finite presentation and X is pure relative to Y, then there exists a universal flattening Y' → Y of X. Moreover Y' → Y is a…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item If $f$ is of finite presentation, $\\mathcal{F}$ is an\n$\\mathcal{O}_X$-module of finite presentation, and $\\mathcal{F}$ is\npure relative to $Y$, then there exists a universal flattening\n$Y' \\to Y$ of $\\mathcal{F}$. Moreover $Y' \\to Y$ is a monomorphism\nof finite presentation.\n\\item If $f$ is of finite presentation and $X$ is pure relative to $Y$,\nthen there exists a universal flattening $Y' \\to Y$ of $X$.\nMoreover $Y' \\to Y$ is a monomorphism of finite presentation.\n\\item If $f$ is proper and of finite presentation and $\\mathcal{F}$ is an\n$\\mathcal{O}_X$-module of finite presentation, then there exists a\nuniversal flattening $Y' \\to Y$ of $\\mathcal{F}$. Moreover $Y' \\to Y$ is\na monomorphism of finite presentation.\n\\item If $f$ is proper and of finite presentation\nthen there exists a universal flattening $Y' \\to Y$ of $X$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Universal flattening","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CX2","source_file":"spaces-flat.tex","source_line":2096,"source_end_line":2117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2096-L2117","statement_sha256":"ee67e459dc1a1b5dcf58e853735f1f43c6ef23856d12ffffec79441d58dc3763","origin":"The Stacks Project","memory_eligible":false,"source_rank":12347,"rank":12347,"depth":67,"x":546.987,"y":1640.153,"cluster":"geometry-of-spaces"},{"id":"stacks:0CX5","tag":"0CX5","title":"Grothendieck's Existence Theorem · Lemma 0CX5","summary":"In Situation [Tag 0CX4] consider K = Rlim_D_QCoh(O_X)(F_n) = DQ_X(Rlim_D(O_X)F_n) Then K is in D^b_QCoh(O_X) and in fact K has nonzero cohomology sheaves only in degrees ≥ 0.","statement_latex":"In Situation \\ref{situation-existence} consider\n$$\nK = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(\\mathcal{F}_n) =\nDQ_X(R\\lim_{D(\\mathcal{O}_X)}\\mathcal{F}_n)\n$$\nThen $K$ is in $D^b_{\\QCoh}(\\mathcal{O}_X)$ and in fact\n$K$ has nonzero cohomology sheaves only in degrees $\\geq 0$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CX5","source_file":"spaces-flat.tex","source_line":2170,"source_end_line":2179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2170-L2179","statement_sha256":"bfc446ef1d1213acdb9d46359f3598a38e53ed4662a458eae3215451e21758cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12348,"rank":12348,"depth":0,"x":745.839,"y":1494.842,"cluster":"geometry-of-spaces"},{"id":"stacks:0CX6","tag":"0CX6","title":"Grothendieck's Existence Theorem · Lemma 0CX6","summary":"In Situation [Tag 0CX4] let K be as in Lemma [Tag 0CX5]. For any perfect object E of D(O_X) we have • M = RΓ(X, K ⊗^L E) is a perfect object of D(A) and there is a canonical isomorphism RΓ(X_n, F_n ⊗^L E|_X_n) = M ⊗_A^L A_n in D(A_n), • N = RHom_X(E, K) is a perfect object of D(A) and there is a canonical isomorphism RHom_X_n(E|_X_n, F_n) = N ⊗_A^L A_n in D(A_n). In both statements E|_X_n denotes the derived pullback of E to X_n.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}. For any perfect\nobject $E$ of $D(\\mathcal{O}_X)$ we have\n\\begin{enumerate}\n\\item $M = R\\Gamma(X, K \\otimes^\\mathbf{L} E)$ is a perfect object of $D(A)$\nand there is a canonical isomorphism\n$R\\Gamma(X_n, \\mathcal{F}_n \\otimes^\\mathbf{L} E|_{X_n}) =\nM \\otimes_A^\\mathbf{L} A_n$\nin $D(A_n)$,\n\\item $N = R\\Hom_X(E, K)$ is a perfect object of $D(A)$\nand there is a canonical isomorphism\n$R\\Hom_{X_n}(E|_{X_n}, \\mathcal{F}_n) = N \\otimes_A^\\mathbf{L} A_n$\nin $D(A_n)$.\n\\end{enumerate}\nIn both statements $E|_{X_n}$ denotes the derived pullback\nof $E$ to $X_n$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CX6","source_file":"spaces-flat.tex","source_line":2187,"source_end_line":2205,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2187-L2205","statement_sha256":"5422291292a943c30c77c29f1d5b1aa71fd319627ee9af54127069deb5e8736b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12349,"rank":12349,"depth":70,"x":716.043,"y":1714.983,"cluster":"geometry-of-spaces"},{"id":"stacks:0CX7","tag":"0CX7","title":"Grothendieck's Existence Theorem · Lemma 0CX7","summary":"In Situation [Tag 0CX4] let K be as in Lemma [Tag 0CX5]. Then K is pseudo-coherent relative to A.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}. Then $K$\nis pseudo-coherent relative to $A$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CX7","source_file":"spaces-flat.tex","source_line":2237,"source_end_line":2242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2237-L2242","statement_sha256":"f18ccde9cc5bbedd6d3a748219a851a3df7112b1d709d5ebd35258a36e5f8b6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12350,"rank":12350,"depth":71,"x":560.871,"y":1535.618,"cluster":"geometry-of-spaces"},{"id":"stacks:0CX8","tag":"0CX8","title":"Grothendieck's Existence Theorem · Lemma 0CX8","summary":"In Situation [Tag 0CX4] let K be as in Lemma [Tag 0CX5]. For any étale morphism U → X with U quasi-compact and quasi-separated we have RΓ(U, K) ⊗_A^L A_n = RΓ(U_n, F_n) in D(A_n) where U_n = U ×_X X_n.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}. For any\n\\'etale morphism $U \\to X$ with $U$ quasi-compact and quasi-separated we have\n$$\nR\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n =\nR\\Gamma(U_n, \\mathcal{F}_n)\n$$\nin $D(A_n)$ where $U_n = U \\times_X X_n$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CX8","source_file":"spaces-flat.tex","source_line":2254,"source_end_line":2264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2254-L2264","statement_sha256":"1e5ee9123ce8e81128fbc1640dfdde4ba767360b3cbf378ad5f021edffe35204","origin":"The Stacks Project","memory_eligible":false,"source_rank":12351,"rank":12351,"depth":71,"x":819.718,"y":1579.863,"cluster":"geometry-of-spaces"},{"id":"stacks:0CX9","tag":"0CX9","title":"Grothendieck's Existence Theorem · Lemma 0CX9","summary":"In Situation [Tag 0CX4] let K be as in Lemma [Tag 0CX5]. Denote X_0 ⊂ |X| the closed subset consisting of points lying over the closed subset Spec(A_1) = Spec(A_2) = … of Spec(A). There exists an open subspace W ⊂ X containing X_0 such that • H^i(K)|_W is zero unless i = 0, • F = H^0(K)|_W is of finite presentation, and • F_n = F ⊗_O_X O_X_n.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}.\nDenote $X_0 \\subset |X|$ the closed subset\nconsisting of points lying over the closed subset\n$\\Spec(A_1) = \\Spec(A_2) = \\ldots$ of $\\Spec(A)$.\nThere exists an open subspace $W \\subset X$ containing $X_0$\nsuch that\n\\begin{enumerate}\n\\item $H^i(K)|_W$ is zero unless $i = 0$,\n\\item $\\mathcal{F} = H^0(K)|_W$ is of finite presentation, and\n\\item $\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CX9","source_file":"spaces-flat.tex","source_line":2290,"source_end_line":2304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2290-L2304","statement_sha256":"8050a2a1639b0e99de04deea45258961c0d1b8692e1df8e8551437952e4fd4ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":12352,"rank":12352,"depth":72,"x":593.106,"y":1694.195,"cluster":"geometry-of-spaces"},{"id":"stacks:0CXA","tag":"0CXA","title":"Grothendieck's Existence Theorem · Lemma 0CXA","summary":"In Situation [Tag 0CX4] let K be as in Lemma [Tag 0CX5]. Let W ⊂ X be as in Lemma [Tag 0CX9]. Set F = H^0(K)|_W. Then, after possibly shrinking the open W, the support of F is proper over A.","statement_latex":"In Situation \\ref{situation-existence} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be}. Let $W \\subset X$\nbe as in Lemma \\ref{lemma-finitely-presented}.\nSet $\\mathcal{F} = H^0(K)|_W$. Then, after possibly shrinking the open $W$,\nthe support of $\\mathcal{F}$ is proper over $A$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXA","source_file":"spaces-flat.tex","source_line":2355,"source_end_line":2362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2355-L2362","statement_sha256":"f2df6c91cdcae8c430223dcd42e6a12e7c299912a2d6bd7e7ee9321c6c12539c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12353,"rank":12353,"depth":73,"x":668.317,"y":1481.151,"cluster":"geometry-of-spaces"},{"id":"stacks:0CXB","tag":"0CXB","title":"Grothendieck Existence Theorem · Theorem 0CXB","summary":"In Situation [Tag 0CX4] there exists a finitely presented O_X-module F, flat over A, with support proper over A, such that F_n = F ⊗_O_X O_X_n for all n compatibly with the maps φ_n.","statement_latex":"In Situation \\ref{situation-existence}\nthere exists a finitely presented $\\mathcal{O}_X$-module\n$\\mathcal{F}$, flat over $A$, with support proper over $A$,\nsuch that\n$\\mathcal{F}_n = \\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{O}_{X_n}$\nfor all $n$ compatibly with the maps $\\varphi_n$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXB","source_file":"spaces-flat.tex","source_line":2384,"source_end_line":2392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2384-L2392","statement_sha256":"78e804dc6a303fe2cf05352875db642b640ec4dd15de3088e7c29df0317129d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12354,"rank":12354,"depth":74,"x":784.264,"y":1681.066,"cluster":"geometry-of-spaces"},{"id":"stacks:0DIL","tag":"0DIL","title":"Grothendieck's Existence Theorem, bis · Lemma 0DIL","summary":"In Situation [Tag 0DIK] consider K = Rlim_D_QCoh(O_X)(K_n) = DQ_X(Rlim_D(O_X) K_n) Then K is in D^-_QCoh(O_X).","statement_latex":"In Situation \\ref{situation-existence-derived} consider\n$$\nK = R\\lim_{D_\\QCoh(\\mathcal{O}_X)}(K_n) =\nDQ_X(R\\lim_{D(\\mathcal{O}_X)} K_n)\n$$\nThen $K$ is in $D^-_{\\QCoh}(\\mathcal{O}_X)$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIL","source_file":"spaces-flat.tex","source_line":2477,"source_end_line":2485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2477-L2485","statement_sha256":"48415e60a796a77757247839754afb7e76a7e0de48a70804318d3653b843c290","origin":"The Stacks Project","memory_eligible":false,"source_rank":12355,"rank":12355,"depth":62,"x":537.825,"y":1599.385,"cluster":"geometry-of-spaces"},{"id":"stacks:0DIM","tag":"0DIM","title":"Grothendieck's Existence Theorem, bis · Lemma 0DIM","summary":"In Situation [Tag 0DIK] let K be as in Lemma [Tag 0DIL]. For any perfect object E of D(O_X) the cohomology M = RΓ(X, K ⊗^L E) is a pseudo-coherent object of D(A) and there is a canonical isomorphism RΓ(X_n, K_n ⊗^L E|_X_n) = M ⊗_A^L A_n in D(A_n). Here E|_X_n denotes the derived pullback of E to X_n.","statement_latex":"In Situation \\ref{situation-existence-derived} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be-derived}. For any perfect\nobject $E$ of $D(\\mathcal{O}_X)$ the cohomology\n$$\nM = R\\Gamma(X, K \\otimes^\\mathbf{L} E)\n$$\nis a pseudo-coherent object of $D(A)$ and there is a canonical isomorphism\n$$\nR\\Gamma(X_n, K_n \\otimes^\\mathbf{L} E|_{X_n}) = M \\otimes_A^\\mathbf{L} A_n\n$$\nin $D(A_n)$. Here $E|_{X_n}$ denotes the derived pullback of $E$ to $X_n$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIM","source_file":"spaces-flat.tex","source_line":2535,"source_end_line":2548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2535-L2548","statement_sha256":"82112153cf4b6f8a3db00729eac278bdd792be9328abb30d04d248cdb1c0c852","origin":"The Stacks Project","memory_eligible":false,"source_rank":12356,"rank":12356,"depth":63,"x":785.407,"y":1519.723,"cluster":"geometry-of-spaces"},{"id":"stacks:0DIN","tag":"0DIN","title":"Grothendieck's Existence Theorem, bis · Lemma 0DIN","summary":"In Situation [Tag 0DIK] let K be as in Lemma [Tag 0DIL]. Then K is pseudo-coherent on X.","statement_latex":"In Situation \\ref{situation-existence-derived} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be-derived}. Then $K$\nis pseudo-coherent on $X$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIN","source_file":"spaces-flat.tex","source_line":2576,"source_end_line":2581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2576-L2581","statement_sha256":"69642dcddcaa5b1cfaab4a6c9a118d748a1c5aa8ab02938ae8723e240e061ac5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12357,"rank":12357,"depth":66,"x":666.822,"y":1719.09,"cluster":"geometry-of-spaces"},{"id":"stacks:0DIP","tag":"0DIP","title":"Grothendieck's Existence Theorem, bis · Lemma 0DIP","summary":"In Situation [Tag 0DIK] let K be as in Lemma [Tag 0DIL]. For any étale morphism U → X with U quasi-compact and quasi-separated we have RΓ(U, K) ⊗_A^L A_n = RΓ(U_n, K_n) in D(A_n) where U_n = U ×_X X_n.","statement_latex":"In Situation \\ref{situation-existence-derived} let $K$ be as in\nLemma \\ref{lemma-compute-what-it-should-be-derived}. For any\n\\'etale morphism $U \\to X$ with $U$ quasi-compact and quasi-separated we have\n$$\nR\\Gamma(U, K) \\otimes_A^\\mathbf{L} A_n =\nR\\Gamma(U_n, K_n)\n$$\nin $D(A_n)$ where $U_n = U \\times_X X_n$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem, bis","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIP","source_file":"spaces-flat.tex","source_line":2599,"source_end_line":2609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2599-L2609","statement_sha256":"c1abbb577c3e89687aa99151cec9bf5b03292387455e4d2c07eb85e97680bd7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12358,"rank":12358,"depth":68,"x":593.888,"y":1504.639,"cluster":"geometry-of-spaces"},{"id":"stacks:0DIQ","tag":"0DIQ","title":"Derived Grothendieck Existence Theorem · Theorem 0DIQ","summary":"In Situation [Tag 0DIK] there exists a pseudo-coherent K in D(O_X) such that K_n = K ⊗_O_X^L O_X_n for all n compatibly with the maps φ_n.","statement_latex":"In Situation \\ref{situation-existence-derived}\nthere exists a pseudo-coherent $K$ in $D(\\mathcal{O}_X)$\nsuch that $K_n = K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} \\mathcal{O}_{X_n}$\nfor all $n$ compatibly with the maps $\\varphi_n$.","area":"Geometry of Spaces","chapter":"Flatness on Algebraic Spaces","chapter_id":"spaces-flat","section":"Grothendieck's Existence Theorem, bis","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIQ","source_file":"spaces-flat.tex","source_line":2635,"source_end_line":2641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-flat.tex#L2635-L2641","statement_sha256":"53bda896e354ad77fc31f6da8df4706510971116ebfba9d51b1799f4d824b54f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12359,"rank":12359,"depth":69,"x":820.283,"y":1621.47,"cluster":"geometry-of-spaces"},{"id":"stacks:043C","tag":"043C","title":"Equivalence relations · Definition 043C","summary":"Let B → S as in Section [Tag 043A]. Let U be an algebraic space over B. • A pre-relation on U over B is any morphism j : R → U ×_B U of algebraic spaces over B. In this case we set t = pr_0 ∘ j and s = pr_1 ∘ j, so that j = (t, s). • A relation on U over B is a monomorphism j : R → U ×_B U of algebraic spaces over B. • A pre-equivalence relation is a pre-relation j : R → U ×_B U such that the image of j : R(T) → U(T) × U(T) is an equivalence relation for all schemes T…","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $U$ be an algebraic space over $B$.\n\\begin{enumerate}\n\\item A {\\it pre-relation} on $U$ over $B$ is any morphism\n$j : R \\to U \\times_B U$ of algebraic spaces over $B$.\nIn this case we set\n$t = \\text{pr}_0 \\circ j$ and $s = \\text{pr}_1 \\circ j$, so\nthat $j = (t, s)$.\n\\item A {\\it relation} on $U$ over $B$ is a monomorphism\n$j : R \\to U \\times_B U$ of algebraic spaces over $B$.\n\\item A {\\it pre-equivalence relation} is a pre-relation\n$j : R \\to U \\times_B U$ such that the image of\n$j : R(T) \\to U(T) \\times U(T)$ is an equivalence relation for\nall schemes $T$ over $B$.\n\\item We say a morphism $R \\to U \\times_B U$ of algebraic spaces over $B$\nis an {\\it equivalence relation on $U$ over $B$}\nif and only if for every $T$ over $B$ the $T$-valued\npoints of $R$ define an equivalence relation\non the set of $T$-valued points of $U$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Equivalence relations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043C","source_file":"spaces-groupoids.tex","source_line":118,"source_end_line":140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L118-L140","statement_sha256":"ef37b7b0862a1b9bb3014438b05db0272d412369d94ded399be18891464c4c51","origin":"The Stacks Project","memory_eligible":false,"source_rank":12360,"rank":12360,"depth":0,"x":1220.378,"y":1067.344,"cluster":"groupoids-quotients"},{"id":"stacks:043D","tag":"043D","title":"Equivalence relations · Lemma 043D","summary":"Let B → S as in Section [Tag 043A]. Let U be an algebraic space over B. Let j : R → U ×_B U be a pre-relation. Let g : U' → U be a morphism of algebraic spaces over B. Finally, set R' = (U' ×_B U')×_U ×_B U R xrightarrowj' U' ×_B U' Then j' is a pre-relation on U' over B. If j is a relation, then j' is a relation. If j is a pre-equivalence relation, then j' is a pre-equivalence relation. If j is an equivalence relation, then j' is an equivalence relation.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $U$ be an algebraic space over $B$.\nLet $j : R \\to U \\times_B U$ be a pre-relation.\nLet $g : U' \\to U$ be a morphism of algebraic spaces over $B$.\nFinally, set\n$$\nR' = (U' \\times_B U')\\times_{U \\times_B U} R\n\\xrightarrow{j'}\nU' \\times_B U'\n$$\nThen $j'$ is a pre-relation on $U'$ over $B$.\nIf $j$ is a relation, then $j'$ is a relation.\nIf $j$ is a pre-equivalence relation, then $j'$ is a pre-equivalence relation.\nIf $j$ is an equivalence relation, then $j'$ is an equivalence relation.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043D","source_file":"spaces-groupoids.tex","source_line":146,"source_end_line":162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L146-L162","statement_sha256":"f342ea21fa83b483652d7eccffbf3cdd8a4fcfa03785103a539f0607a8b62fc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12361,"rank":12361,"depth":0,"x":1121.759,"y":1245.178,"cluster":"groupoids-quotients"},{"id":"stacks:043E","tag":"043E","title":"Equivalence relations · Definition 043E","summary":"Let B → S as in Section [Tag 043A]. Let U be an algebraic space over B. Let j : R → U ×_B U be a pre-relation. Let g : U' → U be a morphism of algebraic spaces over B. The pre-relation j' : R' → U' ×_B U' of Lemma [Tag 043D] is called the restriction, or pullback of the pre-relation j to U'. In this situation we sometimes write R' = R|_U'.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $U$ be an algebraic space over $B$.\nLet $j : R \\to U \\times_B U$ be a pre-relation.\nLet $g : U' \\to U$ be a morphism of algebraic spaces over $B$.\nThe pre-relation $j' : R' \\to U' \\times_B U'$ of\nLemma \\ref{lemma-restrict-relation} is called\nthe {\\it restriction}, or {\\it pullback} of the pre-relation $j$ to $U'$.\nIn this situation we sometimes write $R' = R|_{U'}$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Equivalence relations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043E","source_file":"spaces-groupoids.tex","source_line":168,"source_end_line":178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L168-L178","statement_sha256":"0de2dda2e8f0becb22719b73a2adac6929293e14c984591be03cbca07b5e0a71","origin":"The Stacks Project","memory_eligible":false,"source_rank":12362,"rank":12362,"depth":1,"x":1051.258,"y":1057.518,"cluster":"groupoids-quotients"},{"id":"stacks:043F","tag":"043F","title":"Equivalence relations · Lemma 043F","summary":"Let B → S as in Section [Tag 043A]. Let j : R → U ×_B U be a pre-relation of algebraic spaces over B. Consider the relation on |U| defined by the rule x sim y ⇔ ∃ r ∈ |R| : t(r) = x, s(r) = y. If j is a pre-equivalence relation then this is an equivalence relation.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $j : R \\to U \\times_B U$ be a pre-relation of algebraic spaces over $B$.\nConsider the relation on $|U|$ defined by the rule\n$$\nx \\sim y\n\\Leftrightarrow\n\\exists\\ r \\in |R| :\nt(r) = x,\ns(r) = y.\n$$\nIf $j$ is a pre-equivalence relation then this is an equivalence relation.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043F","source_file":"spaces-groupoids.tex","source_line":180,"source_end_line":193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L180-L193","statement_sha256":"e8b793fef1b0490e12d853761235427bf93309f42dfbdda0be7c500c1c635e1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12363,"rank":12363,"depth":0,"x":1254.768,"y":1156.189,"cluster":"groupoids-quotients"},{"id":"stacks:043H","tag":"043H","title":"Group algebraic spaces · Definition 043H","summary":"Let B → S as in Section [Tag 043A]. • A group algebraic space over B is a pair (G, m), where G is an algebraic space over B and m : G ×_B G → G is a morphism of algebraic spaces over B with the following property: For every scheme T over B the pair (G(T), m) is a group. • A morphism ψ : (G, m) → (G', m') of group algebraic spaces over B is a morphism ψ : G → G' of algebraic spaces over B such that for every T/B the induced map ψ : G(T) → G'(T) is a homomorphism of groups.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\n\\begin{enumerate}\n\\item A {\\it group algebraic space over $B$} is a pair $(G, m)$, where\n$G$ is an algebraic space over $B$ and $m : G \\times_B G \\to G$ is\na morphism of algebraic spaces over $B$ with the following property:\nFor every scheme $T$ over $B$ the pair $(G(T), m)$ is a group.\n\\item A {\\it morphism $\\psi : (G, m) \\to (G', m')$ of\ngroup algebraic spaces over $B$}\nis a morphism $\\psi : G \\to G'$ of algebraic spaces over $B$ such that for\nevery $T/B$ the induced map $\\psi : G(T) \\to G'(T)$ is a homomorphism\nof groups.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Group algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043H","source_file":"spaces-groupoids.tex","source_line":235,"source_end_line":249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L235-L249","statement_sha256":"5c4ae5a5dae78e4aca1a2f1119ccc2b1883bcc2d89aabcd9f7c5bf8a42a32f22","origin":"The Stacks Project","memory_eligible":false,"source_rank":12364,"rank":12364,"depth":0,"x":1024.665,"y":1199.035,"cluster":"groupoids-quotients"},{"id":"stacks:043I","tag":"043I","title":"Group algebraic spaces · Lemma 043I","summary":"Let B → S as in Section [Tag 043A]. Let (G, m) be a group algebraic space over B. Let B' → B be a morphism of algebraic spaces. The pullback (G_B', m_B') is a group algebraic space over B'.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(G, m)$ be a group algebraic space over $B$.\nLet $B' \\to B$ be a morphism of algebraic spaces.\nThe pullback $(G_{B'}, m_{B'})$ is a group algebraic space over $B'$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Group algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043I","source_file":"spaces-groupoids.tex","source_line":274,"source_end_line":280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L274-L280","statement_sha256":"66915bf4556e2a4fdd5203f5654f7330895be7b918fae9d06bb3b41fe63cfcf5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12365,"rank":12365,"depth":0,"x":1160.288,"y":1036.391,"cluster":"groupoids-quotients"},{"id":"stacks:06P6","tag":"06P6","title":"Properties of group algebraic spaces · Lemma 06P6","summary":"Let S be a scheme. Let B be an algebraic space over S. Let G be a group algebraic space over B. Then G → B is separated (resp. quasi-separated, resp. locally separated) if and only if the identity morphism e : B → G is a closed immersion (resp. quasi-compact, resp. an immersion).","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $G$ be a group algebraic space over $B$.\nThen $G \\to B$ is separated (resp.\\ quasi-separated, resp.\\ locally separated)\nif and only if the identity morphism $e : B \\to G$ is a closed immersion\n(resp.\\ quasi-compact, resp.\\ an immersion).","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Properties of group algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06P6","source_file":"spaces-groupoids.tex","source_line":300,"source_end_line":307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L300-L307","statement_sha256":"c727f2b19fa063d3f18fd3f388804f402266f6752aa4c1888993aae7a25599d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12366,"rank":12366,"depth":56,"x":1191.166,"y":1233.863,"cluster":"groupoids-quotients"},{"id":"stacks:0DSI","tag":"0DSI","title":"Properties of group algebraic spaces · Lemma 0DSI","summary":"Let S be a scheme. Let B be an algebraic space over S. Let G be a group algebraic space over B. Assume G → B is locally of finite type. Then G → B is unramified (resp. locally quasi-finite) if and only if G → B is unramified (resp. quasi-finite) at e(b) for all b ∈ |B|.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $G$ be a group algebraic space over $B$. Assume $G \\to B$\nis locally of finite type. Then\n$G \\to B$ is unramified (resp.\\ locally quasi-finite)\nif and only if $G \\to B$ is unramified (resp.\\ quasi-finite)\nat $e(b)$ for all $b \\in |B|$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Properties of group algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSI","source_file":"spaces-groupoids.tex","source_line":334,"source_end_line":342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L334-L342","statement_sha256":"eb8e854206ab579b660bdb6eb17158823d1003e57600e2302deb19b172587982","origin":"The Stacks Project","memory_eligible":false,"source_rank":12367,"rank":12367,"depth":49,"x":1009.061,"y":1105.392,"cluster":"groupoids-quotients"},{"id":"stacks:0DSJ","tag":"0DSJ","title":"Properties of group algebraic spaces · Lemma 0DSJ","summary":"Let S be a scheme. Let B be an algebraic space over S. Let G be a group algebraic space over B. Assume G → B is locally of finite type. • There exists a maximal open subspace U ⊂ B such that G_U → U is unramified and formation of U commutes with base change. • There exists a maximal open subspace U ⊂ B such that G_U → U is locally quasi-finite and formation of U commutes with base change.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $G$ be a group algebraic space over $B$. Assume $G \\to B$\nis locally of finite type.\n\\begin{enumerate}\n\\item There exists a maximal open subspace $U \\subset B$\nsuch that $G_U \\to U$ is unramified and formation of $U$\ncommutes with base change.\n\\item There exists a maximal open subspace $U \\subset B$\nsuch that $G_U \\to U$ is locally quasi-finite and formation of $U$\ncommutes with base change.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Properties of group algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSJ","source_file":"spaces-groupoids.tex","source_line":365,"source_end_line":378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L365-L378","statement_sha256":"8841b5d08e6e0dcb7dffdd8942637f1ee81067d025353b7195c222dd1f393b98","origin":"The Stacks Project","memory_eligible":false,"source_rank":12368,"rank":12368,"depth":50,"x":1247.351,"y":1096.769,"cluster":"groupoids-quotients"},{"id":"stacks:043Q","tag":"043Q","title":"Actions of group algebraic spaces · Definition 043Q","summary":"Let B → S as in Section [Tag 043A]. Let (G, m) be a group algebraic space over B. Let X be an algebraic space over B. • An action of G on the algebraic space X/B is a morphism a : G ×_B X → X over B such that for every scheme T over B the map a : G(T) × X(T) → X(T) defines the structure of a G(T)-set on X(T). • Suppose that X, Y are algebraic spaces over B each endowed with an action of G. An equivariant or more precisely a G-equivariant morphism ψ : X → Y is a morphism…","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(G, m)$ be a group algebraic space over $B$.\nLet $X$ be an algebraic space over $B$.\n\\begin{enumerate}\n\\item An {\\it action of $G$ on the algebraic space $X/B$} is\na morphism $a : G \\times_B X \\to X$ over $B$ such that\nfor every scheme $T$ over $B$ the map $a : G(T) \\times X(T) \\to X(T)$\ndefines the structure of a $G(T)$-set on $X(T)$.\n\\item Suppose that $X$, $Y$ are algebraic spaces over $B$ each endowed\nwith an action of $G$. An {\\it equivariant} or more precisely\na {\\it $G$-equivariant} morphism $\\psi : X \\to Y$\nis a morphism of algebraic spaces over $B$ such\nthat for every $T$ over $B$ the map $\\psi : X(T) \\to Y(T)$ is\na morphism of $G(T)$-sets.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Actions of group algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043Q","source_file":"spaces-groupoids.tex","source_line":517,"source_end_line":534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L517-L534","statement_sha256":"95cf35dfab289382f57c57e55f84cc0d29748238ed6339c6a87bf88fe6342ed1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12369,"rank":12369,"depth":0,"x":1078.081,"y":1238.752,"cluster":"groupoids-quotients"},{"id":"stacks:06P8","tag":"06P8","title":"Actions of group algebraic spaces · Definition 06P8","summary":"Let B → S, G → B, and X → B as in Definition [Tag 043Q]. Let a : G ×_B X → X be an action of G on X/B. We say the action is free if for every scheme T over B the action a : G(T) × X(T) → X(T) is a free action of the group G(T) on the set X(T).","statement_latex":"Let $B \\to S$, $G \\to B$, and $X \\to B$ as in\nDefinition \\ref{definition-action-group-space}.\nLet $a : G \\times_B X \\to X$ be an action of $G$ on $X/B$.\nWe say the action is {\\it free} if for every scheme $T$ over $B$\nthe action $a : G(T) \\times X(T) \\to X(T)$ is a free action of\nthe group $G(T)$ on the set $X(T)$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Actions of group algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06P8","source_file":"spaces-groupoids.tex","source_line":561,"source_end_line":569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L561-L569","statement_sha256":"c024177457c09402782f58939eaf255c303920d4d7334c5c9bb102ff0e1efc90","origin":"The Stacks Project","memory_eligible":false,"source_rank":12370,"rank":12370,"depth":1,"x":1088.752,"y":1037.426,"cluster":"groupoids-quotients"},{"id":"stacks:06P9","tag":"06P9","title":"Actions of group algebraic spaces · Lemma 06P9","summary":"Situation as in Definition [Tag 06P8], The action a is free if and only if G ×_B X → X ×_B X, (g, x) ↦ (a(g, x), x) is a monomorphism of algebraic spaces.","statement_latex":"Situation as in\nDefinition \\ref{definition-free-action},\nThe action $a$ is free if and only if\n$$\nG \\times_B X \\to X \\times_B X, \\quad (g, x) \\mapsto (a(g, x), x)\n$$\nis a monomorphism of algebraic spaces.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Actions of group algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06P9","source_file":"spaces-groupoids.tex","source_line":571,"source_end_line":580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L571-L580","statement_sha256":"5a8c86b8d7d5e8c279af4e13ca5baeacbb74ed4ebf87537ab352b77d124e2ba1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12371,"rank":12371,"depth":2,"x":1243.227,"y":1192.381,"cluster":"groupoids-quotients"},{"id":"stacks:04TW","tag":"04TW","title":"Principal homogeneous spaces · Definition 04TW","summary":"Let S be a scheme. Let B be an algebraic space over S. Let (G, m) be a group algebraic space over B. Let X be an algebraic space over B, and let a : G ×_B X → X be an action of G on X. • We say X is a pseudo G-torsor or that X is formally principally homogeneous under G if the induced morphism G ×_B X → X ×_B X, (g, x) ↦ (a(g, x), x) is an isomorphism. • A pseudo G-torsor X is called trivial if there exists an G-equivariant isomorphism G → X over B where G acts on G by…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $(G, m)$ be a group algebraic space over $B$.\nLet $X$ be an algebraic space over $B$, and let\n$a : G \\times_B X \\to X$ be an action of $G$ on $X$.\n\\begin{enumerate}\n\\item We say $X$ is a {\\it pseudo $G$-torsor} or that $X$ is\n{\\it formally principally homogeneous under $G$} if the induced\nmorphism $G \\times_B X \\to X \\times_B X$,\n$(g, x) \\mapsto (a(g, x), x)$ is an isomorphism.\n\\item A pseudo $G$-torsor $X$ is called {\\it trivial} if there exists\nan $G$-equivariant isomorphism $G \\to X$ over $B$ where $G$ acts on\n$G$ by left multiplication.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Principal homogeneous spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TW","source_file":"spaces-groupoids.tex","source_line":602,"source_end_line":617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L602-L617","statement_sha256":"2b3fea0d4b88d1e15a6d29e6988b25a519eb7e890d402aafd28f0e68bef7ef61","origin":"The Stacks Project","memory_eligible":false,"source_rank":12372,"rank":12372,"depth":0,"x":1004.026,"y":1165.696,"cluster":"groupoids-quotients"},{"id":"stacks:04TX","tag":"04TX","title":"Principal homogeneous spaces · Lemma 04TX","summary":"In the situation of Definition [Tag 04TW]. • The algebraic space X is a pseudo G-torsor if and only if for every scheme T over B the set X(T) is either empty or the action of the group G(T) on X(T) is simply transitive. • A pseudo G-torsor X is trivial if and only if the morphism X → B has a section.","statement_latex":"In the situation of\nDefinition \\ref{definition-pseudo-torsor}.\n\\begin{enumerate}\n\\item The algebraic space $X$ is a pseudo $G$-torsor if and only if for\nevery scheme $T$ over $B$ the set $X(T)$ is either empty or the action\nof the group $G(T)$ on $X(T)$ is simply transitive.\n\\item A pseudo $G$-torsor $X$ is trivial if and only if the morphism\n$X \\to B$ has a section.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Principal homogeneous spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TX","source_file":"spaces-groupoids.tex","source_line":624,"source_end_line":635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L624-L635","statement_sha256":"a83fe34a5937176391c12cdb6a66a9a3a7b1b152d26c036c27c886c1a0ee0a24","origin":"The Stacks Project","memory_eligible":false,"source_rank":12373,"rank":12373,"depth":1,"x":1202.432,"y":1049.315,"cluster":"groupoids-quotients"},{"id":"stacks:04TY","tag":"04TY","title":"Principal homogeneous spaces · Definition 04TY","summary":"Let S be a scheme. Let B be an algebraic space over S. Let (G, m) be a group algebraic space over B. Let X be a pseudo G-torsor over B. • We say X is a principal homogeneous space, or more precisely a principal homogeneous G-space over B if there exists a fpqc covering (B_i → B)_i ∈ I such that each X_B_i → B_i has a section (i.e., is a trivial pseudo G_B_i-torsor). • Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). We say X is a G-torsor in the τ topology, or a τ…","statement_latex":"Let $S$ be a scheme.\nLet $B$ be an algebraic space over $S$.\nLet $(G, m)$ be a group algebraic space over $B$.\nLet $X$ be a pseudo $G$-torsor over $B$.\n\\begin{enumerate}\n\\item We say $X$ is a\n{\\it principal homogeneous space}, or more precisely a\n{\\it principal homogeneous $G$-space over $B$}\nif there exists a fpqc covering\\footnote{The default type of torsor in\nGroupoids, Definition \\ref{groupoids-definition-principal-homogeneous-space}\nis a pseudo torsor which is trivial on an fpqc covering.\nSince $G$, as an algebraic space, can be seen a sheaf of groups\nthere already is a notion of a $G$-torsor which corresponds\nto fppf-torsor, see\nLemma \\ref{lemma-torsor}.\nHence we use ``principal homogeneous space'' for a pseudo torsor which\nis fpqc locally trivial, and we try to avoid using the word torsor in\nthis situation.}\n$\\{B_i \\to B\\}_{i \\in I}$ such that each\n$X_{B_i} \\to B_i$ has a section (i.e., is a trivial pseudo $G_{B_i}$-torsor).\n\\item Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nWe say $X$ is a {\\it $G$-torsor in the $\\tau$ topology}, or a\n{\\it $\\tau$ $G$-torsor}, or simply a {\\it $\\tau$ torsor}\nif there exists a $\\tau$ covering $\\{B_i \\to B\\}_{i \\in I}$\nsuch that each $X_{B_i} \\to B_i$ has a section.\n\\item If $X$ is a principal homogeneous $G$-space over $B$,\nthen we say that it is\n{\\it quasi-isotrivial} if it is a torsor for the \\'etale topology.\n\\item If $X$ is a principal homogeneous $G$-space over $B$,\nthen we say that it is\n{\\it locally trivial} if it is a torsor for the Zariski topology.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Principal homogeneous spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TY","source_file":"spaces-groupoids.tex","source_line":641,"source_end_line":675,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L641-L675","statement_sha256":"a7eaebca8aa9249b3b1f76e5dc700192c67f443a34b4c7310264228e7d7de779","origin":"The Stacks Project","memory_eligible":false,"source_rank":12374,"rank":12374,"depth":1,"x":1149.571,"y":1248.275,"cluster":"groupoids-quotients"},{"id":"stacks:04TZ","tag":"04TZ","title":"Principal homogeneous spaces · Lemma 04TZ","summary":"Let S be a scheme. Let (G, m) be a group algebraic space over S. Let X be an algebraic space over S, and let a : G ×_S X → X be an action of G on X. Then X is a G-torsor in the fppf-topology in the sense of Definition [Tag 04TY] if and only if X is a G-torsor on (Sch/S)_fppf in the sense of Cohomology on Sites, Definition [Tag 03AH].","statement_latex":"Let $S$ be a scheme.\nLet $(G, m)$ be a group algebraic space over $S$.\nLet $X$ be an algebraic space over $S$, and let\n$a : G \\times_S X \\to X$ be an action of $G$ on $X$.\nThen\n$X$ is a $G$-torsor in the $fppf$-topology in the sense of\nDefinition \\ref{definition-principal-homogeneous-space}\nif and only if\n$X$ is a $G$-torsor on $(\\Sch/S)_{fppf}$\nin the sense of\nCohomology on Sites, Definition \\ref{sites-cohomology-definition-torsor}.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Principal homogeneous spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TZ","source_file":"spaces-groupoids.tex","source_line":684,"source_end_line":697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L684-L697","statement_sha256":"4f5c728e2203875c50ed2cf8a253ee67e2b290841ac1d502a12614f54fb4ab28","origin":"The Stacks Project","memory_eligible":false,"source_rank":12375,"rank":12375,"depth":2,"x":1028.212,"y":1071.072,"cluster":"groupoids-quotients"},{"id":"stacks:0DSK","tag":"0DSK","title":"Principal homogeneous spaces · Lemma 0DSK","summary":"Let S be a scheme. Let B be an algebraic space over S. Let G be a group algebraic space over B. Let X be a pseudo G-torsor over B. Assume G and X locally of finite type over B. • If G → B is unramified, then X → B is unramified. • If G → B is locally quasi-finite, then X → B is locally quasi-finite.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $G$ be a group algebraic space over $B$.\nLet $X$ be a pseudo $G$-torsor over $B$.\nAssume $G$ and $X$ locally of finite type over $B$.\n\\begin{enumerate}\n\\item If $G \\to B$ is unramified, then $X \\to B$ is unramified.\n\\item If $G \\to B$ is locally quasi-finite, then $X \\to B$ is\nlocally quasi-finite.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Principal homogeneous spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSK","source_file":"spaces-groupoids.tex","source_line":703,"source_end_line":714,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L703-L714","statement_sha256":"fe209ad8c79a08ca46b1a0a259f3800556f3ec68aeadfb90e8884ccba0554fe1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12376,"rank":12376,"depth":49,"x":1260.851,"y":1133.05,"cluster":"groupoids-quotients"},{"id":"stacks:043T","tag":"043T","title":"Equivariant quasi-coherent sheaves · Definition 043T","summary":"Let B → S as in Section [Tag 043A]. Let (G, m) be a group algebraic space over B, and let a : G ×_B X → X be an action of G on the algebraic space X over B. An G-equivariant quasi-coherent O_X-module, or simply a equivariant quasi-coherent O_X-module, is a pair (F, α), where F is a quasi-coherent O_X-module, and α is a O_G ×_B X-module map α : a^*F → pr_1^*F where pr_1 : G ×_B X → X is the projection such that • the diagram xymatrix (1_G × a)^*pr_2^*F ar[r]_-pr_12^*α &…","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(G, m)$ be a group algebraic space over $B$, and\nlet $a : G \\times_B X \\to X$ be an action of $G$\non the algebraic space $X$ over $B$.\nAn {\\it $G$-equivariant quasi-coherent $\\mathcal{O}_X$-module},\nor simply a {\\it equivariant quasi-coherent $\\mathcal{O}_X$-module},\nis a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent\n$\\mathcal{O}_X$-module, and $\\alpha$ is a $\\mathcal{O}_{G \\times_B X}$-module\nmap\n$$\n\\alpha : a^*\\mathcal{F} \\longrightarrow \\text{pr}_1^*\\mathcal{F}\n$$\nwhere $\\text{pr}_1 : G \\times_B X \\to X$ is the projection\nsuch that\n\\begin{enumerate}\n\\item the diagram\n$$\n\\xymatrix{\n(1_G \\times a)^*\\text{pr}_2^*\\mathcal{F} \\ar[r]_-{\\text{pr}_{12}^*\\alpha} &\n\\text{pr}_2^*\\mathcal{F} \\\\\n(1_G \\times a)^*a^*\\mathcal{F} \\ar[u]^{(1_G \\times a)^*\\alpha} \\ar@{=}[r] &\n(m \\times 1_X)^*a^*\\mathcal{F} \\ar[u]_{(m \\times 1_X)^*\\alpha}\n}\n$$\nis a commutative in the category of\n$\\mathcal{O}_{G \\times_B G \\times_B X}$-modules, and\n\\item the pullback\n$$\n(e \\times 1_X)^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F}\n$$\nis the identity map.\n\\end{enumerate}\nFor explanation compare with the relevant diagrams of\nEquation (\\ref{equation-action}).","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Equivariant quasi-coherent sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043T","source_file":"spaces-groupoids.tex","source_line":752,"source_end_line":788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L752-L788","statement_sha256":"7a48d6d7eda1b5ba18f11d89fa3b872c8f08966be3fc9081fae82b07764fd628","origin":"The Stacks Project","memory_eligible":false,"source_rank":12377,"rank":12377,"depth":0,"x":1038.849,"y":1219.593,"cluster":"groupoids-quotients"},{"id":"stacks:043U","tag":"043U","title":"Equivariant quasi-coherent sheaves · Lemma 043U","summary":"Let B → S as in Section [Tag 043A]. Let G be a group algebraic space over B. Let f : X → Y be a G-equivariant morphism between algebraic spaces over B endowed with G-actions. Then pullback f^* given by (F, α) ↦ (f^*F, (1_G × f)^*α) defines a functor from the category of quasi-coherent G-equivariant sheaves on Y to the category of quasi-coherent G-equivariant sheaves on X.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $G$ be a group algebraic space over $B$.\nLet $f : X \\to Y$ be a $G$-equivariant morphism between\nalgebraic spaces over $B$ endowed with $G$-actions.\nThen pullback $f^*$ given by\n$(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, (1_G \\times f)^*\\alpha)$\ndefines a functor from the category of quasi-coherent $G$-equivariant sheaves\non $Y$ to the category of quasi-coherent $G$-equivariant sheaves on $X$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Equivariant quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043U","source_file":"spaces-groupoids.tex","source_line":795,"source_end_line":805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L795-L805","statement_sha256":"61647bd9262e841f3cb9798c0e822b246a3f2a4699250bd1b3440ef601f7e906","origin":"The Stacks Project","memory_eligible":false,"source_rank":12378,"rank":12378,"depth":0,"x":1133.216,"y":1029.283,"cluster":"groupoids-quotients"},{"id":"stacks:043W","tag":"043W","title":"Groupoids in algebraic spaces · Definition 043W","summary":"Let B → S as in Section [Tag 043A]. • A groupoid in algebraic spaces over B is a quintuple (U, R, s, t, c) where U and R are algebraic spaces over B, and s, t : R → U and c : R ×_s, U, t R → R are morphisms of algebraic spaces over B with the following property: For any scheme T over B the quintuple (U(T), R(T), s, t, c) is a groupoid category. • A morphism f : (U, R, s, t, c) → (U', R', s', t', c') of groupoids in algebraic spaces over B is given by morphisms of…","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\n\\begin{enumerate}\n\\item A {\\it groupoid in algebraic spaces over $B$} is a\nquintuple $(U, R, s, t, c)$ where\n$U$ and $R$ are algebraic spaces over $B$, and\n$s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$\nare morphisms of algebraic spaces over $B$ with the\nfollowing property: For any scheme $T$ over $B$ the quintuple\n$$\n(U(T), R(T), s, t, c)\n$$\nis a groupoid category.\n\\item A {\\it morphism\n$f : (U, R, s, t, c) \\to (U', R', s', t', c')$\nof groupoids in algebraic spaces over $B$} is given by morphisms\nof algebraic spaces $f : U \\to U'$ and $f : R \\to R'$ over $B$\nwith the following property:  For any scheme\n$T$ over $B$ the maps $f$ define a functor from the\ngroupoid category $(U(T), R(T), s, t, c)$ to the\ngroupoid category $(U'(T), R'(T), s', t', c')$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Groupoids in algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043W","source_file":"spaces-groupoids.tex","source_line":823,"source_end_line":846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L823-L846","statement_sha256":"b93ff448de2df444dd07d35b560b290fd4e4606310656740a9ab704ce3f6d88f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12379,"rank":12379,"depth":0,"x":1216.902,"y":1223.694,"cluster":"groupoids-quotients"},{"id":"stacks:043X","tag":"043X","title":"Groupoids in algebraic spaces · Lemma 043X","summary":"Let B → S as in Section [Tag 043A]. Given a groupoid in algebraic spaces (U, R, s, t, c) over B the morphism j : R → U ×_B U is a pre-equivalence relation.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nGiven a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$\nthe morphism $j : R \\to U \\times_B U$ is a pre-equivalence\nrelation.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Groupoids in algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043X","source_file":"spaces-groupoids.tex","source_line":870,"source_end_line":876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L870-L876","statement_sha256":"712e5d09a03e01ab6fcb2725e057d97b19417fb9f9e69b06568394e940b6ebad","origin":"The Stacks Project","memory_eligible":false,"source_rank":12380,"rank":12380,"depth":0,"x":998.256,"y":1127.561,"cluster":"groupoids-quotients"},{"id":"stacks:043Y","tag":"043Y","title":"Groupoids in algebraic spaces · Lemma 043Y","summary":"Let B → S as in Section [Tag 043A]. Given an equivalence relation j : R → U ×_B U over B there is a unique way to extend it to a groupoid in algebraic spaces (U, R, s, t, c) over B.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nGiven an equivalence relation $j : R \\to U \\times_B U$ over $B$\nthere is a unique way to extend it to a groupoid in algebraic spaces\n$(U, R, s, t, c)$ over $B$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Groupoids in algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043Y","source_file":"spaces-groupoids.tex","source_line":883,"source_end_line":889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L883-L889","statement_sha256":"6030d90e195986bfb6e068762464a2dc61701a3c9f57140ac5f75840daa75ad8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12381,"rank":12381,"depth":0,"x":1237.439,"y":1074.239,"cluster":"groupoids-quotients"},{"id":"stacks:043Z","tag":"043Z","title":"Groupoids in algebraic spaces · Lemma 043Z","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. In the commutative diagram xymatrix & U & R ar[d]_s ar[ru]^t & R ×_s, U, t R ar[l]^-pr_0 ar[d]^pr_1 ar[r]_-c & R ar[d]^s ar[lu]_t U & R ar[l]_t ar[r]^s & U the two lower squares are fibre product squares. Moreover, the triangle on top (which is really a square) is also cartesian.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nIn the commutative diagram\n$$\n\\xymatrix{\n& U & \\\\\nR \\ar[d]_s \\ar[ru]^t &\nR \\times_{s, U, t} R\n\\ar[l]^-{\\text{pr}_0} \\ar[d]^{\\text{pr}_1} \\ar[r]_-c &\nR \\ar[d]^s \\ar[lu]_t \\\\\nU & R \\ar[l]_t \\ar[r]^s & U\n}\n$$\nthe two lower squares are fibre product squares.\nMoreover, the triangle on top (which is really a square)\nis also cartesian.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Groupoids in algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/043Z","source_file":"spaces-groupoids.tex","source_line":896,"source_end_line":914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L896-L914","statement_sha256":"da1d0e29f3e1b99058b7650d246f33d0cb624dc91f2cea05938dd1b0caa72b74","origin":"The Stacks Project","memory_eligible":false,"source_rank":12382,"rank":12382,"depth":0,"x":1103.588,"y":1249.75,"cluster":"groupoids-quotients"},{"id":"stacks:0450","tag":"0450","title":"Groupoids in algebraic spaces · Lemma 0450","summary":"Let B → S be as in Section [Tag 043A]. Let (U, R, s, t, c, e, i) be a groupoid in algebraic spaces over B. The diagram xymatrix R ×_t, U, t R ar@<1ex>[r]^-pr_1 ar@<-1ex>[r]_-pr_0 ar[d]_pr_0 × c ∘ (i, 1) & R ar[r]^t ar[d]^id_R & U ar[d]^id_U R ×_s, U, t R ar@<1ex>[r]^-c ar@<-1ex>[r]_-pr_0 ar[d]_pr_1 & R ar[r]^t ar[d]^s & U R ar@<1ex>[r]^s ar@<-1ex>[r]_t & U is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian.","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$.\nThe diagram\n\\begin{equation}\n\n\\xymatrix{\nR \\times_{t, U, t} R\n\\ar@<1ex>[r]^-{\\text{pr}_1} \\ar@<-1ex>[r]_-{\\text{pr}_0}\n\\ar[d]_{\\text{pr}_0 \\times c \\circ (i, 1)} &\nR \\ar[r]^t \\ar[d]^{\\text{id}_R} &\nU \\ar[d]^{\\text{id}_U} \\\\\nR \\times_{s, U, t} R\n\\ar@<1ex>[r]^-c \\ar@<-1ex>[r]_-{\\text{pr}_0} \\ar[d]_{\\text{pr}_1} &\nR \\ar[r]^t \\ar[d]^s &\nU \\\\\nR \\ar@<1ex>[r]^s \\ar@<-1ex>[r]_t &\nU\n}\n\\end{equation}\nis commutative. The two top rows are isomorphic via the vertical maps given.\nThe two lower left squares are cartesian.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Groupoids in algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0450","source_file":"spaces-groupoids.tex","source_line":922,"source_end_line":945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L922-L945","statement_sha256":"e62485dff2b31fec565483f37a0ac967fe2a405bac93c5162c26d46ab1efe109","origin":"The Stacks Project","memory_eligible":false,"source_rank":12383,"rank":12383,"depth":1,"x":1061.034,"y":1043.827,"cluster":"groupoids-quotients"},{"id":"stacks:0DTA","tag":"0DTA","title":"Groupoids in algebraic spaces · Lemma 0DTA","summary":"Let B → S be as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let B' → B be a morphism of algebraic spaces. Then the base changes U' = B' ×_B U, R' = B' ×_B R endowed with the base changes s', t', c' of the morphisms s, t, c form a groupoid in algebraic spaces (U', R', s', t', c') over B' and the projections determine a morphism (U', R', s', t', c') → (U, R, s, t, c) of groupoids in algebraic spaces over B.","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $B' \\to B$ be a morphism of algebraic spaces.\nThen the base changes $U' = B' \\times_B U$,\n$R' = B' \\times_B R$ endowed with the base changes $s'$, $t'$, $c'$\nof the morphisms $s, t, c$ form a groupoid in algebraic spaces\n$(U', R', s', t', c')$ over $B'$ and the projections\ndetermine a morphism\n$(U', R', s', t', c') \\to (U, R, s, t, c)$\nof groupoids in algebraic spaces over $B$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Groupoids in algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTA","source_file":"spaces-groupoids.tex","source_line":960,"source_end_line":972,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L960-L972","statement_sha256":"cd4c43c4b441716bc3184aa8f4497ec7602d1d87dc316a67b6be40c9fcb8205d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12384,"rank":12384,"depth":0,"x":1258.536,"y":1171.868,"cluster":"groupoids-quotients"},{"id":"stacks:0441","tag":"0441","title":"Quasi-coherent sheaves on groupoids · Definition 0441","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. A quasi-coherent module on (U, R, s, t, c) is a pair (F, α), where F is a quasi-coherent O_U-module, and α is a O_R-module map α : t^*F → s^*F such that • the diagram xymatrix & pr_1^*t^*F ar[r]_-pr_1^*α & pr_1^*s^*F ar@=[rd] & pr_0^*s^*F ar@=[ru] & & & c^*s^*F & pr_0^*t^*F ar[lu]^pr_0^*α ar@=[r] & c^*t^*F ar[ru]_c^*α is a commutative in the category of O_R ×_s, U, t…","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nA {\\it quasi-coherent module on $(U, R, s, t, c)$}\nis a pair $(\\mathcal{F}, \\alpha)$, where $\\mathcal{F}$ is a quasi-coherent\n$\\mathcal{O}_U$-module, and $\\alpha$ is a $\\mathcal{O}_R$-module\nmap\n$$\n\\alpha : t^*\\mathcal{F} \\longrightarrow s^*\\mathcal{F}\n$$\nsuch that\n\\begin{enumerate}\n\\item the diagram\n$$\n\\xymatrix{\n& \\text{pr}_1^*t^*\\mathcal{F} \\ar[r]_-{\\text{pr}_1^*\\alpha} &\n\\text{pr}_1^*s^*\\mathcal{F} \\ar@{=}[rd] & \\\\\n\\text{pr}_0^*s^*\\mathcal{F} \\ar@{=}[ru] & & & c^*s^*\\mathcal{F} \\\\\n& \\text{pr}_0^*t^*\\mathcal{F} \\ar[lu]^{\\text{pr}_0^*\\alpha} \\ar@{=}[r] &\nc^*t^*\\mathcal{F} \\ar[ru]_{c^*\\alpha}\n}\n$$\nis a commutative in the category of\n$\\mathcal{O}_{R \\times_{s, U, t} R}$-modules, and\n\\item the pullback\n$$\ne^*\\alpha : \\mathcal{F} \\longrightarrow \\mathcal{F}\n$$\nis the identity map.\n\\end{enumerate}\nCompare with the commutative diagrams of Lemma \\ref{lemma-diagram}.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0441","source_file":"spaces-groupoids.tex","source_line":990,"source_end_line":1022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L990-L1022","statement_sha256":"7ff8e66d9226cf88bc0bdd63c283531ddf39273744751eacb70f88065dc56728","origin":"The Stacks Project","memory_eligible":false,"source_rank":12385,"rank":12385,"depth":1,"x":1009.272,"y":1189.567,"cluster":"groupoids-quotients"},{"id":"stacks:077W","tag":"077W","title":"Quasi-coherent sheaves on groupoids · Lemma 077W","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. If (F, α) is a quasi-coherent module on (U, R, s, t, c) then α is an isomorphism.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nIf $(\\mathcal{F}, \\alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$\nthen $\\alpha$ is an isomorphism.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077W","source_file":"spaces-groupoids.tex","source_line":1030,"source_end_line":1036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1030-L1036","statement_sha256":"40292a8a0c4a12d41e53c42bfd2455a72c8b734b56251d29c1214aa790720105","origin":"The Stacks Project","memory_eligible":false,"source_rank":12386,"rank":12386,"depth":2,"x":1179.292,"y":1034.669,"cluster":"groupoids-quotients"},{"id":"stacks:0442","tag":"0442","title":"Quasi-coherent sheaves on groupoids · Lemma 0442","summary":"Let B → S as in Section [Tag 043A]. Consider a morphism f : (U, R, s, t, c) → (U', R', s', t', c') of groupoid in algebraic spaces over B. Then pullback f^* given by (F, α) ↦ (f^*F, f^*α) defines a functor from the category of quasi-coherent sheaves on (U', R', s', t', c') to the category of quasi-coherent sheaves on (U, R, s, t, c).","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nConsider a morphism\n$f : (U, R, s, t, c) \\to (U', R', s', t', c')$\nof groupoid in algebraic spaces over $B$. Then pullback $f^*$ given by\n$$\n(\\mathcal{F}, \\alpha) \\mapsto (f^*\\mathcal{F}, f^*\\alpha)\n$$\ndefines a functor from the category of quasi-coherent sheaves on\n$(U', R', s', t', c')$ to the category of quasi-coherent sheaves on\n$(U, R, s, t, c)$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0442","source_file":"spaces-groupoids.tex","source_line":1049,"source_end_line":1061,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1049-L1061","statement_sha256":"e24d798477dbcd205e867d99a97b05e538861f79703a8a3a5bbcea3e27acf1ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":12387,"rank":12387,"depth":0,"x":1178.484,"y":1245.917,"cluster":"groupoids-quotients"},{"id":"stacks:0GPM","tag":"0GPM","title":"Quasi-coherent sheaves on groupoids · Lemma 0GPM","summary":"Let B → S as in Section [Tag 043A]. Consider a morphism f : (U, R, s, t, c) → (U', R', s', t', c') of groupoids in algebraic spaces over B. Assume that • f : U → U' is quasi-compact and quasi-separated, • the square xymatrix R ar[d]_t ar[r]_f & R' ar[d]^t' U ar[r]^f & U' is cartesian, and • s' and t' are flat. Then pushforward f_* given by (F, α) ↦ (f_*F, f_*α) defines a functor from the category of quasi-coherent sheaves on (U, R, s, t, c) to the category of…","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nConsider a morphism\n$f : (U, R, s, t, c) \\to (U', R', s', t', c')$\nof groupoids in algebraic spaces over $B$. Assume that\n\\begin{enumerate}\n\\item $f : U \\to U'$ is quasi-compact and quasi-separated,\n\\item the square\n$$\n\\xymatrix{\nR \\ar[d]_t \\ar[r]_f & R' \\ar[d]^{t'} \\\\\nU \\ar[r]^f & U'\n}\n$$\nis cartesian, and\n\\item $s'$ and $t'$ are flat.\n\\end{enumerate}\nThen pushforward $f_*$ given by\n$$\n(\\mathcal{F}, \\alpha) \\mapsto (f_*\\mathcal{F}, f_*\\alpha)\n$$\ndefines a functor from the category of quasi-coherent sheaves on\n$(U, R, s, t, c)$ to the category of quasi-coherent sheaves on\n$(U', R', s', t', c')$ which is right adjoint to pullback as defined in\nLemma \\ref{lemma-pullback}.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPM","source_file":"spaces-groupoids.tex","source_line":1067,"source_end_line":1093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1067-L1093","statement_sha256":"e767a246a0046da955fcdff11f7c3e2212d071a58dd06f836e60aa13d5f7a57c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12388,"rank":12388,"depth":60,"x":1008.757,"y":1089.277,"cluster":"groupoids-quotients"},{"id":"stacks:077X","tag":"077X","title":"Quasi-coherent sheaves on groupoids · Lemma 077X","summary":"Let B → S be as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. The category of quasi-coherent modules on (U, R, s, t, c) has colimits.","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nThe category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077X","source_file":"spaces-groupoids.tex","source_line":1128,"source_end_line":1133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1128-L1133","statement_sha256":"9680b8c07180c95c47942cb2660a51fd82a98f251ec85ba5af2c1e222fc9dde1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12389,"rank":12389,"depth":42,"x":1260.531,"y":1108.524,"cluster":"groupoids-quotients"},{"id":"stacks:06VZ","tag":"06VZ","title":"Quasi-coherent sheaves on groupoids · Lemma 06VZ","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. If s, t are flat, then the category of quasi-coherent modules on (U, R, s, t, c) is abelian.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nIf $s$, $t$ are flat, then the category of quasi-coherent modules on\n$(U, R, s, t, c)$ is abelian.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quasi-coherent sheaves on groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06VZ","source_file":"spaces-groupoids.tex","source_line":1150,"source_end_line":1156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1150-L1156","statement_sha256":"e61b0585bbe0efa2987b750068fc5021448cacb309818d91595225e3c0eca811","origin":"The Stacks Project","memory_eligible":false,"source_rank":12390,"rank":12390,"depth":0,"x":1058.877,"y":1237.528,"cluster":"groupoids-quotients"},{"id":"stacks:0GPP","tag":"0GPP","title":"Colimits of quasi-coherent modules · Lemma 0GPP","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Assume s, t are flat, quasi-compact, and quasi-separated. For any quasi-coherent module G on U, there exists a canonical isomorphism α : t^*s_*t^*G → s^*s_*t^*G which turns (s_*t^*G, α) into a quasi-coherent module on (U, R, s, t, c). This construction defines a functor QCoh(O_U) → QCoh(U, R, s, t, c) which is a right adjoint to the forgetful functor (F, β) ↦ F.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nAssume $s, t$ are flat, quasi-compact, and quasi-separated.\nFor any quasi-coherent module $\\mathcal{G}$ on $U$, there exists\na canonical isomorphism\n$\\alpha : t^*s_*t^*\\mathcal{G} \\to s^*s_*t^*\\mathcal{G}$\nwhich turns $(s_*t^*\\mathcal{G}, \\alpha)$ into a quasi-coherent module\non $(U, R, s, t, c)$. This construction defines a functor\n$$\n\\QCoh(\\mathcal{O}_U) \\longrightarrow \\QCoh(U, R, s, t, c)\n$$\nwhich is a right adjoint to the forgetful functor\n$(\\mathcal{F}, \\beta) \\mapsto \\mathcal{F}$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Colimits of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPP","source_file":"spaces-groupoids.tex","source_line":1194,"source_end_line":1209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1194-L1209","statement_sha256":"a306122e3102270734b5bc71bb2011959a4fb0d315758ae3ea97c558d8147548","origin":"The Stacks Project","memory_eligible":false,"source_rank":12391,"rank":12391,"depth":61,"x":1103.949,"y":1027.439,"cluster":"groupoids-quotients"},{"id":"stacks:0GPR","tag":"0GPR","title":"Colimits of quasi-coherent modules · Lemma 0GPR","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Let F be a quasi-coherent O_X-module, let G be a quasi-coherent O_Y-module, and let φ : G → f^*F be a module map. Assume • φ is injective, • f is quasi-compact, quasi-separated, flat, and surjective, • X, Y are locally Noetherian, and • G is a coherent O_Y-module. Then F ∩ f_*G defined as the pullback xymatrix F ar[r] & f_*f^*F F ∩ f_*G ar[u] ar[r] & f_*G ar[u] is a coherent O_X-module.","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Let $\\mathcal{F}$\nbe a quasi-coherent $\\mathcal{O}_X$-module, let $\\mathcal{G}$\nbe a quasi-coherent $\\mathcal{O}_Y$-module, and let\n$\\varphi : \\mathcal{G} \\to f^*\\mathcal{F}$ be a module map. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is injective,\n\\item $f$ is quasi-compact, quasi-separated, flat, and surjective,\n\\item $X$, $Y$ are locally Noetherian, and\n\\item $\\mathcal{G}$ is a coherent $\\mathcal{O}_Y$-module.\n\\end{enumerate}\nThen $\\mathcal{F} \\cap f_*\\mathcal{G}$ defined as the pullback\n$$\n\\xymatrix{\n\\mathcal{F} \\ar[r] & f_*f^*\\mathcal{F} \\\\\n\\mathcal{F} \\cap f_*\\mathcal{G} \\ar[u] \\ar[r] &\nf_*\\mathcal{G} \\ar[u]\n}\n$$\nis a coherent $\\mathcal{O}_X$-module.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Colimits of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPR","source_file":"spaces-groupoids.tex","source_line":1333,"source_end_line":1355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1333-L1355","statement_sha256":"6bcc073db8514225658cd6dbcbc298c88f560d3bf7c1cc71928d212c70a69f51","origin":"The Stacks Project","memory_eligible":false,"source_rank":12392,"rank":12392,"depth":60,"x":1240.009,"y":1208.394,"cluster":"groupoids-quotients"},{"id":"stacks:0GPS","tag":"0GPS","title":"Colimits of quasi-coherent modules · Lemma 0GPS","summary":"Let B → S be as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Assume that • U, R are Noetherian, • s, t are flat, quasi-compact, and quasi-separated. Then every quasi-coherent module (F, α) on (U, R, s, t, c) is a filtered colimit of coherent modules.","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nAssume that\n\\begin{enumerate}\n\\item $U$, $R$ are Noetherian,\n\\item $s, t$ are flat, quasi-compact, and quasi-separated.\n\\end{enumerate}\nThen every quasi-coherent module $(\\mathcal{F}, \\alpha)$ on $(U, R, s, t, c)$\nis a filtered colimit of coherent modules.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Colimits of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPS","source_file":"spaces-groupoids.tex","source_line":1386,"source_end_line":1397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1386-L1397","statement_sha256":"39a79787d99ca33ed3e761b0fc15da8d4a697cd6f0dec322a71c1bd563dcb696","origin":"The Stacks Project","memory_eligible":false,"source_rank":12393,"rank":12393,"depth":62,"x":993.534,"y":1152.02,"cluster":"groupoids-quotients"},{"id":"stacks:077Z","tag":"077Z","title":"Crystals in quasi-coherent sheaves · Lemma 077Z","summary":"In the situation above, if all the morphisms f_φ are flat, then there exists a cardinal kappa such that every object ((F_i)_i ∈ I, (α_φ)_φ ∈ Φ) of CQC(X) is the directed colimit of its kappa-generated submodules.","statement_latex":"In the situation above, if all the morphisms $f_\\phi$ are flat, then there\nexists a cardinal $\\kappa$ such that every object\n$(\\{\\mathcal{F}_i\\}_{i \\in I}, \\{\\alpha_\\phi\\}_{\\phi \\in \\Phi})$\nof $\\textit{CQC}(X)$ is the directed colimit of its\n$\\kappa$-generated submodules.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Crystals in quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077Z","source_file":"spaces-groupoids.tex","source_line":1495,"source_end_line":1502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1495-L1502","statement_sha256":"be8adf22cfae6bb24c0b7a24f63b2f1db05226cda510ea1a64370b734d3abba4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12394,"rank":12394,"depth":14,"x":1221.192,"y":1053.484,"cluster":"groupoids-quotients"},{"id":"stacks:0780","tag":"0780","title":"Crystals in quasi-coherent sheaves · Lemma 0780","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. If s, t are flat, then there exists a set T and a family of objects (F_t, α_t)_t ∈ T of QCoh(U, R, s, t, c) such that every object (F, α) is the directed colimit of its submodules isomorphic to one of the objects (F_t, α_t).","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nIf $s$, $t$ are flat, then there exists a set $T$ and a family of objects\n$(\\mathcal{F}_t, \\alpha_t)_{t \\in T}$ of $\\QCoh(U, R, s, t, c)$\nsuch that every object $(\\mathcal{F}, \\alpha)$ is the directed colimit\nof its submodules isomorphic to one of the objects $(\\mathcal{F}_t, \\alpha_t)$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Crystals in quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0780","source_file":"spaces-groupoids.tex","source_line":1605,"source_end_line":1613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1605-L1613","statement_sha256":"8f37112856c363428880b4c304ace0dab19ed6e07cb41fd143fbc5cf87fcef26","origin":"The Stacks Project","memory_eligible":false,"source_rank":12395,"rank":12395,"depth":52,"x":1132.336,"y":1255.831,"cluster":"groupoids-quotients"},{"id":"stacks:0444","tag":"0444","title":"Groupoids and group spaces · Lemma 0444","summary":"Let B → S as in Section [Tag 043A]. Let (G, m) be a group algebraic space over B with identity e_G and inverse i_G. Let X be an algebraic space over B and let a : G ×_B X → X be an action of G on X over B. Then we get a groupoid in algebraic spaces (U, R, s, t, c, e, i) over B in the following manner: • We set U = X, and R = G ×_B X. • We set s : R → U equal to (g, x) ↦ x. • We set t : R → U equal to (g, x) ↦ a(g, x). • We set c : R ×_s, U, t R → R equal to ((g, x), (g',…","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(G, m)$ be a group algebraic space over $B$ with\nidentity $e_G$ and inverse $i_G$.\nLet $X$ be an algebraic space over $B$ and let $a : G \\times_B X \\to X$\nbe an action of $G$ on $X$ over $B$.\nThen we get a groupoid in algebraic spaces $(U, R, s, t, c, e, i)$ over $B$\nin the following manner:\n\\begin{enumerate}\n\\item We set $U = X$, and $R = G \\times_B X$.\n\\item We set $s : R \\to U$ equal to $(g, x) \\mapsto x$.\n\\item We set $t : R \\to U$ equal to $(g, x) \\mapsto a(g, x)$.\n\\item We set $c : R \\times_{s, U, t} R \\to R$ equal to\n$((g, x), (g', x')) \\mapsto (m(g, g'), x')$.\n\\item We set $e : U \\to R$ equal to $x \\mapsto (e_G(x), x)$.\n\\item We set $i : R \\to R$ equal to $(g, x) \\mapsto (i_G(g), a(g, x))$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Groupoids and group spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0444","source_file":"spaces-groupoids.tex","source_line":1708,"source_end_line":1726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1708-L1726","statement_sha256":"d46e653807bf9f426ba2e74ccc70e659e87c6da74c9ac7caa3d1cda64d043d68","origin":"The Stacks Project","memory_eligible":false,"source_rank":12396,"rank":12396,"depth":0,"x":1034.892,"y":1055.703,"cluster":"groupoids-quotients"},{"id":"stacks:0445","tag":"0445","title":"Groupoids and group spaces · Lemma 0445","summary":"Let B → S as in Section [Tag 043A]. Let (G, m) be a group algebraic space over B. Let X be an algebraic space over B and let a : G ×_B X → X be an action of G on X over B. Let (U, R, s, t, c) be the groupoid in algebraic spaces constructed in Lemma [Tag 0444]. The rule (F, α) ↦ (F, α) defines an equivalence of categories between G-equivariant O_X-modules and the category of quasi-coherent modules on (U, R, s, t, c).","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(G, m)$ be a group algebraic space over $B$.\nLet $X$ be an algebraic space over $B$ and let $a : G \\times_B X \\to X$\nbe an action of $G$ on $X$ over $B$. Let $(U, R, s, t, c)$ be\nthe groupoid in algebraic spaces constructed in\nLemma \\ref{lemma-groupoid-from-action}.\nThe rule\n$(\\mathcal{F}, \\alpha) \\mapsto (\\mathcal{F}, \\alpha)$ defines\nan equivalence of categories between $G$-equivariant\n$\\mathcal{O}_X$-modules and the category of quasi-coherent\nmodules on $(U, R, s, t, c)$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Groupoids and group spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0445","source_file":"spaces-groupoids.tex","source_line":1734,"source_end_line":1747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1734-L1747","statement_sha256":"735e4b61bcf80f8b552da446769cae7cfad0e81902c4e801bd48c16f1fec7854","origin":"The Stacks Project","memory_eligible":false,"source_rank":12397,"rank":12397,"depth":2,"x":1268.265,"y":1148.213,"cluster":"groupoids-quotients"},{"id":"stacks:0447","tag":"0447","title":"The stabilizer group algebraic space · Lemma 0447","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. The algebraic space G defined by the cartesian square xymatrix G ar[r] ar[d] & R ar[d]^j = (t, s) U ar[r]^-Δ & U ×_B U is a group algebraic space over U with composition law m induced by the composition law c.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nThe algebraic space $G$ defined by the cartesian square\n$$\n\\xymatrix{\nG \\ar[r] \\ar[d] & R \\ar[d]^{j = (t, s)} \\\\\nU \\ar[r]^-{\\Delta} & U \\times_B U\n}\n$$\nis a group algebraic space over $U$ with composition law\n$m$ induced by the composition law $c$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"The stabilizer group algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0447","source_file":"spaces-groupoids.tex","source_line":1771,"source_end_line":1784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1771-L1784","statement_sha256":"24e12fc936bbcc50c38741bd5388ac8828027ec348e735101c4f9e3f57ac5790","origin":"The Stacks Project","memory_eligible":false,"source_rank":12398,"rank":12398,"depth":0,"x":1021.171,"y":1212.578,"cluster":"groupoids-quotients"},{"id":"stacks:0448","tag":"0448","title":"The stabilizer group algebraic space · Definition 0448","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. The group algebraic space j^-1(Δ_U/B) → U is called the stabilizer of the groupoid in algebraic spaces (U, R, s, t, c).","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nThe group algebraic space $j^{-1}(\\Delta_{U/B}) \\to U$ is called the\n{\\it stabilizer of the groupoid in algebraic spaces $(U, R, s, t, c)$}.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"The stabilizer group algebraic space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0448","source_file":"spaces-groupoids.tex","source_line":1797,"source_end_line":1803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1797-L1803","statement_sha256":"6fd92ce26636994c8ce4251ea735757f7f3420de1f67639b853c7e38830c156e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12399,"rank":12399,"depth":0,"x":1151.937,"y":1024.445,"cluster":"groupoids-quotients"},{"id":"stacks:0449","tag":"0449","title":"The stabilizer group algebraic space · Lemma 0449","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B, and let G/U be its stabilizer. Denote R_t/U the algebraic space R seen as an algebraic space over U via the morphism t : R → U. There is a canonical left action a : G ×_U R_t → R_t induced by the composition law c.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$, and let\n$G/U$ be its stabilizer. Denote $R_t/U$ the algebraic space $R$ seen as an\nalgebraic space over $U$ via the morphism $t : R \\to U$. There is a\ncanonical left action\n$$\na : G \\times_U R_t \\longrightarrow R_t\n$$\ninduced by the composition law $c$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"The stabilizer group algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0449","source_file":"spaces-groupoids.tex","source_line":1810,"source_end_line":1821,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1810-L1821","statement_sha256":"bc6d1eb7d7f48f58cdc8d4322306da8c02df390615a04355c37bb2a2d418aae2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12400,"rank":12400,"depth":0,"x":1206.938,"y":1237.896,"cluster":"groupoids-quotients"},{"id":"stacks:044B","tag":"044B","title":"Restricting groupoids · Lemma 044B","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let g : U' → U be a morphism of algebraic spaces. Consider the following diagram xymatrix R' ar[d] ar[r] ar@/_3pc/[dd]_t' ar@/^1pc/[rr]^s'& R ×_s, U U' ar[r] ar[d] & U' ar[d]^g U' ×_U, t R ar[d] ar[r] & R ar[r]^s ar[d]_t & U U' ar[r]^g & U where all the squares are fibre product squares. Then there is a canonical composition law c' : R' ×_s', U', t' R' → R' such that (U', R',…","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $g : U' \\to U$ be a morphism of algebraic spaces.\nConsider the following diagram\n$$\n\\xymatrix{\nR' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'}&\nR \\times_{s, U} U' \\ar[r] \\ar[d] &\nU' \\ar[d]^g \\\\\nU' \\times_{U, t} R \\ar[d] \\ar[r] &\nR \\ar[r]^s \\ar[d]_t &\nU \\\\\nU' \\ar[r]^g &\nU\n}\n$$\nwhere all the squares are fibre product squares. Then there is a\ncanonical composition law $c' : R' \\times_{s', U', t'} R' \\to R'$\nsuch that $(U', R', s', t', c')$ is a groupoid in algebraic spaces over\n$B$ and such that $U' \\to U$, $R' \\to R$ defines a morphism\n$(U', R', s', t', c') \\to (U, R, s, t, c)$ of groupoids in algebraic spaces\nover $B$. Moreover, for any scheme $T$ over $B$ the functor of groupoids\n$$\n(U'(T), R'(T), s', t', c') \\to (U(T), R(T), s, t, c)\n$$\nis the restriction (see\nGroupoids, Section \\ref{groupoids-section-restrict-groupoid})\nof $(U(T), R(T), s, t, c)$ via the map $U'(T) \\to U(T)$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044B","source_file":"spaces-groupoids.tex","source_line":1842,"source_end_line":1872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1842-L1872","statement_sha256":"6a3222beb89e4262ab0a0972cbac9cdd87742a78bb801567736c2cfc753e5a4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12401,"rank":12401,"depth":0,"x":994.21,"y":1111.4,"cluster":"groupoids-quotients"},{"id":"stacks:044C","tag":"044C","title":"Restricting groupoids · Definition 044C","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let g : U' → U be a morphism of algebraic spaces over B. The morphism of groupoids in algebraic spaces (U', R', s', t', c') → (U, R, s, t, c) constructed in Lemma [Tag 044B] is called the restriction of (U, R, s, t, c) to U'. We sometime use the notation R' = R|_U' in this case.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $g : U' \\to U$ be a morphism of algebraic spaces over $B$.\nThe morphism of groupoids in algebraic spaces\n$(U', R', s', t', c') \\to (U, R, s, t, c)$\nconstructed in Lemma \\ref{lemma-restrict-groupoid} is called\nthe {\\it restriction of $(U, R, s, t, c)$ to $U'$}.\nWe sometime use the notation $R' = R|_{U'}$ in this case.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Restricting groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044C","source_file":"spaces-groupoids.tex","source_line":1878,"source_end_line":1888,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1878-L1888","statement_sha256":"237ed3f6e1399c7fe0d813959e794a6063c16ffc85b7c20f6b64dd9608124a66","origin":"The Stacks Project","memory_eligible":false,"source_rank":12402,"rank":12402,"depth":1,"x":1253.428,"y":1083.903,"cluster":"groupoids-quotients"},{"id":"stacks:044D","tag":"044D","title":"Restricting groupoids · Lemma 044D","summary":"The notions of restricting groupoids and (pre-)equivalence relations defined in Definitions [Tag 044C] and [Tag 043E] agree via the constructions of Lemmas [Tag 043X] and [Tag 043Y].","statement_latex":"The notions of restricting groupoids and\n(pre-)equivalence relations defined in Definitions\n\\ref{definition-restrict-groupoid} and \\ref{definition-restrict-relation}\nagree via the constructions of\nLemmas \\ref{lemma-groupoid-pre-equivalence} and\n\\ref{lemma-equivalence-groupoid}.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044D","source_file":"spaces-groupoids.tex","source_line":1890,"source_end_line":1898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1890-L1898","statement_sha256":"2978cdbbb53407d68e006e6dbdce4ce944befe062e6e95aa9383ac0c713de09f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12403,"rank":12403,"depth":2,"x":1083.987,"y":1251.671,"cluster":"groupoids-quotients"},{"id":"stacks:044F","tag":"044F","title":"Invariant subspaces · Definition 044F","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over the base B. • We say an open subspace W ⊂ U is R-invariant if t(s^-1(W)) ⊂ W. • A locally closed subspace Z ⊂ U is called R-invariant if t^-1(Z) = s^-1(Z) as locally closed subspaces of R. • A monomorphism of algebraic spaces T → U is R-invariant if T ×_U, t R = R ×_s, U T as algebraic spaces over R.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over the base $B$.\n\\begin{enumerate}\n\\item We say an open subspace $W \\subset U$ is {\\it $R$-invariant} if\n$t(s^{-1}(W)) \\subset W$.\n\\item A locally closed subspace $Z \\subset U$ is called {\\it $R$-invariant}\nif $t^{-1}(Z) = s^{-1}(Z)$ as locally closed subspaces of $R$.\n\\item A monomorphism of algebraic spaces $T \\to U$ is {\\it $R$-invariant}\nif $T \\times_{U, t} R = R \\times_{s, U} T$ as algebraic spaces over $R$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Invariant subspaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044F","source_file":"spaces-groupoids.tex","source_line":1922,"source_end_line":1934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1922-L1934","statement_sha256":"750d3e17d58f841a10bc9e4554446196d3b8891e03aa8716da275abdc91c05aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12404,"rank":12404,"depth":0,"x":1073.992,"y":1031.285,"cluster":"groupoids-quotients"},{"id":"stacks:044G","tag":"044G","title":"Invariant subspaces · Lemma 044G","summary":"Let B → S as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. • If s and t are open, then for every open W ⊂ U the open s(t^-1(W)) is R-invariant. • If s and t are open and quasi-compact, then U has an open covering consisting of R-invariant quasi-compact open subspaces.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\n\\begin{enumerate}\n\\item If $s$ and $t$ are open, then for every open $W \\subset U$\nthe open $s(t^{-1}(W))$ is $R$-invariant.\n\\item If $s$ and $t$ are open and quasi-compact, then $U$ has an open\ncovering consisting of $R$-invariant quasi-compact open subspaces.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Invariant subspaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044G","source_file":"spaces-groupoids.tex","source_line":1944,"source_end_line":1954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L1944-L1954","statement_sha256":"7dde539cd3f9f5e35998c40716611e655d56ea54c0fe067047f7bb0c66ed4775","origin":"The Stacks Project","memory_eligible":false,"source_rank":12405,"rank":12405,"depth":1,"x":1259.033,"y":1188.5,"cluster":"groupoids-quotients"},{"id":"stacks:044J","tag":"044J","title":"Quotient sheaves · Definition 044J","summary":"Let B → S and the pre-relation j : R → U ×_B U be as above. In this setting the quotient sheaf U/R associated to j is the sheafification of the presheaf ([Tag 044I]) on (Sch/S)_fppf. If j : R → U ×_B U comes from the action of a group algebraic space G over B on U as in Lemma [Tag 0444] then we denote the quotient sheaf U/G.","statement_latex":"Let $B \\to S$ and the pre-relation $j : R \\to U \\times_B U$ be as above.\nIn this setting the {\\it quotient sheaf $U/R$} associated\nto $j$ is the sheafification of the presheaf\n(\\ref{equation-quotient-presheaf}) on $(\\Sch/S)_{fppf}$.\nIf $j : R \\to U \\times_B U$ comes from the action of a\ngroup algebraic space $G$ over $B$ on $U$ as in\nLemma \\ref{lemma-groupoid-from-action}\nthen we denote the quotient sheaf $U/G$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044J","source_file":"spaces-groupoids.tex","source_line":2023,"source_end_line":2033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2023-L2033","statement_sha256":"4bf8b026639ca939464bfe77d426bfcf54389bb67985fc065d6d6f097a543519","origin":"The Stacks Project","memory_eligible":false,"source_rank":12406,"rank":12406,"depth":1,"x":995.53,"y":1177.542,"cluster":"groupoids-quotients"},{"id":"stacks:044L","tag":"044L","title":"Quotient sheaves · Definition 044L","summary":"In the situation of Definition [Tag 044J]. We say that the pre-relation j has a quotient representable by an algebraic space if the sheaf U/R is an algebraic space. We say that the pre-relation j has a representable quotient if the sheaf U/R is representable by a scheme. We will say a groupoid in algebraic spaces (U, R, s, t, c) over B has a representable quotient (resp. quotient representable by an algebraic space) if the quotient U/R with j = (t, s) is representable…","statement_latex":"In the situation of Definition \\ref{definition-quotient-sheaf}.\nWe say that the pre-relation $j$ has a\n{\\it quotient representable by an algebraic space}\nif the sheaf $U/R$ is an algebraic space.\nWe say that the pre-relation $j$ has a\n{\\it representable quotient}\nif the sheaf $U/R$ is representable by a scheme.\nWe will say a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$ has a\n{\\it representable quotient}\n(resp.\\ {\\it quotient representable by an algebraic space})\nif the quotient $U/R$ with $j = (t, s)$ is representable (resp.\\ an\nalgebraic space).","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044L","source_file":"spaces-groupoids.tex","source_line":2070,"source_end_line":2084,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2070-L2084","statement_sha256":"de1dac57562d2fc87e1e8dd117f63e1b5c9de949cc3fc290527a3aa3e34ab159","origin":"The Stacks Project","memory_eligible":false,"source_rank":12407,"rank":12407,"depth":2,"x":1199.129,"y":1035.769,"cluster":"groupoids-quotients"},{"id":"stacks:044M","tag":"044M","title":"Quotient sheaves · Lemma 044M","summary":"In the situation of Definition [Tag 044J]. Assume there is an algebraic space M over S, and a morphism U → M such that • the morphism U → M equalizes s, t, • the map U → M is a surjection of sheaves, and • the induced map (t, s) : R → U ×_M U is a surjection of sheaves. In this case M represents the quotient sheaf U/R.","statement_latex":"In the situation of Definition \\ref{definition-quotient-sheaf}.\nAssume there is an algebraic space $M$ over $S$,\nand a morphism $U \\to M$ such that\n\\begin{enumerate}\n\\item the morphism $U \\to M$ equalizes $s, t$,\n\\item the map $U \\to M$ is a surjection of sheaves, and\n\\item the induced map $(t, s) : R \\to U \\times_M U$ is a\nsurjection of sheaves.\n\\end{enumerate}\nIn this case $M$ represents the quotient sheaf $U/R$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044M","source_file":"spaces-groupoids.tex","source_line":2096,"source_end_line":2108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2096-L2108","statement_sha256":"25c390ebba3c79e93879d9b0250c1a40b6ac76c31aac489353695a925cc345b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12408,"rank":12408,"depth":2,"x":1162.923,"y":1256.358,"cluster":"groupoids-quotients"},{"id":"stacks:046O","tag":"046O","title":"Quotient sheaves · Lemma 046O","summary":"Let S be a scheme. Let B be an algebraic space over S. Let j : R → U ×_B U be a pre-equivalence relation over B. For a scheme S' over S and a, b ∈ U(S') the following are equivalent: • a and b map to the same element of (U/R)(S'), and • there exists an fppf covering (f_i : S_i → S') of S' and morphisms r_i : S_i → R such that a ∘ f_i = s ∘ r_i and b ∘ f_i = t ∘ r_i. In other words, in this case the map of sheaves R → U ×_U/R U is surjective.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-equivalence relation over $B$.\nFor a scheme $S'$ over $S$ and $a, b \\in U(S')$ the following are equivalent:\n\\begin{enumerate}\n\\item $a$ and $b$ map to the same element of $(U/R)(S')$, and\n\\item there exists an fppf covering $\\{f_i : S_i \\to S'\\}$ of $S'$\nand morphisms $r_i : S_i \\to R$ such that\n$a \\circ f_i = s \\circ r_i$ and $b \\circ f_i = t \\circ r_i$.\n\\end{enumerate}\nIn other words, in this case the map of sheaves\n$$\nR \\longrightarrow U \\times_{U/R} U\n$$\nis surjective.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046O","source_file":"spaces-groupoids.tex","source_line":2121,"source_end_line":2137,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2121-L2137","statement_sha256":"bcd7f2b0b2e933097e8673f9297b0cdf0070bc722aa0a39aac3029334e48d7d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12409,"rank":12409,"depth":1,"x":1011.874,"y":1072.723,"cluster":"groupoids-quotients"},{"id":"stacks:046P","tag":"046P","title":"Quotient sheaves · Lemma 046P","summary":"Let S be a scheme. Let B be an algebraic space over S. Let j : R → U ×_B U be a pre-equivalence relation over B and g : U' → U a morphism of algebraic spaces over B. Let j' : R' → U' ×_B U' be the restriction of j to U'. The map of quotient sheaves U'/R' → U/R is injective. If U' → U is surjective as a map of sheaves, for example if (g : U' → U) is an fppf covering (see Topologies on Spaces, Definition [Tag 03Y8]), then U'/R' → U/R is an isomorphism of sheaves.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-equivalence relation over $B$\nand $g : U' \\to U$ a morphism of algebraic spaces over $B$.\nLet $j' : R' \\to U' \\times_B U'$ be the restriction of $j$ to $U'$.\nThe map of quotient sheaves\n$$\nU'/R' \\longrightarrow U/R\n$$\nis injective. If $U' \\to U$ is surjective as a map of sheaves, for\nexample if $\\{g : U' \\to U\\}$ is an fppf covering (see\nTopologies on Spaces,\nDefinition \\ref{spaces-topologies-definition-fppf-covering}),\nthen $U'/R' \\to U/R$ is an isomorphism of sheaves.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046P","source_file":"spaces-groupoids.tex","source_line":2147,"source_end_line":2162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2147-L2162","statement_sha256":"ebf90e2cda59e1885141207d1a43e0dc3ff5dd20068c20036587ee57ad239c99","origin":"The Stacks Project","memory_eligible":false,"source_rank":12410,"rank":12410,"depth":3,"x":1271.537,"y":1122.541,"cluster":"groupoids-quotients"},{"id":"stacks:044N","tag":"044N","title":"Quotient sheaves · Lemma 044N","summary":"Let S be a scheme. Let B be an algebraic space over S. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let g : U' → U a morphism of algebraic spaces over B. Let (U', R', s', t', c') be the restriction of (U, R, s, t, c) to U'. The map of quotient sheaves U'/R' → U/R is injective. If the composition xymatrix U' ×_g, U, t R ar[r]_-pr_1 ar@/^3ex/[rr]^h & R ar[r]_s & U is a surjection of fppf sheaves then the map is bijective. This holds for example if (h : U'…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $g : U' \\to U$ a morphism of algebraic spaces over $B$.\nLet $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ to $U'$.\nThe map of quotient sheaves\n$$\nU'/R' \\longrightarrow U/R\n$$\nis injective. If the composition\n$$\n\\xymatrix{\nU' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h\n& R \\ar[r]_s & U\n}\n$$\nis a surjection of fppf sheaves then the map is bijective.\nThis holds for example if $\\{h : U' \\times_{g, U, t} R \\to U\\}$ is an\n$fppf$-covering, or if $U' \\to U$ is a surjection of sheaves, or if\n$\\{g : U' \\to U\\}$ is a covering in the fppf topology.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044N","source_file":"spaces-groupoids.tex","source_line":2188,"source_end_line":2209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2188-L2209","statement_sha256":"2b8f7fe5f5fe18be818927297fd10bc1f86b439dcca7a10e5c487afb27ff0285","origin":"The Stacks Project","memory_eligible":false,"source_rank":12411,"rank":12411,"depth":4,"x":1039.462,"y":1233.403,"cluster":"groupoids-quotients"},{"id":"stacks:044Q","tag":"044Q","title":"Quotient stacks · Definition 044Q","summary":"Quotient stacks. Let B → S be as above. • Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. The quotient stack p : [U/R] → (Sch/S)_fppf of (U, R, s, t, c) is the stackification (see Stacks, Lemma [Tag 02ZP]) of the category fibred in groupoids [U/_ pR] over (Sch/S)_fppf associated to ([Tag 044P]). • Let (G, m) be a group algebraic space over B. Let a : G ×_B X → X be an action of G on an algebraic space over B. The quotient stack p : [X/G] → (Sch/S)_fppf is…","statement_latex":"Quotient stacks. Let $B \\to S$ be as above.\n\\begin{enumerate}\n\\item Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nThe {\\it quotient stack}\n$$\np : [U/R] \\longrightarrow (\\Sch/S)_{fppf}\n$$\nof $(U, R, s, t, c)$ is the stackification (see\nStacks, Lemma \\ref{stacks-lemma-stackify-groupoids})\nof the category fibred in groupoids $[U/_{\\!p}R]$ over\n$(\\Sch/S)_{fppf}$ associated to\n(\\ref{equation-quotient-stack}).\n\\item Let $(G, m)$ be a group algebraic space over $B$.\nLet $a : G \\times_B X \\to X$ be an action of $G$ on an algebraic space\nover $B$. The {\\it quotient stack}\n$$\np : [X/G] \\longrightarrow (\\Sch/S)_{fppf}\n$$\nis the quotient stack associated to the groupoid in algebraic spaces\n$(X, G \\times_B X, s, t, c)$ over $B$ of\nLemma \\ref{lemma-groupoid-from-action}.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044Q","source_file":"spaces-groupoids.tex","source_line":2284,"source_end_line":2308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2284-L2308","statement_sha256":"7b2daf85b4d5585f819379c7069d7f25e86e52ba230a7cff5712d7281a16b14a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12412,"rank":12412,"depth":6,"x":1121.63,"y":1019.474,"cluster":"groupoids-quotients"},{"id":"stacks:044R","tag":"044R","title":"Quotient stacks · Lemma 044R","summary":"Assume B → S and (U, R, s, t, c) as in Definition [Tag 044Q] (1). There are canonical 1-morphisms π : S_U → [U/R], and [U/R] → S_B of stacks in groupoids over (Sch/S)_fppf. The composition S_U → S_B is the 1-morphism associated to the structure morphism U → B.","statement_latex":"Assume $B \\to S$ and $(U, R, s, t, c)$ as in\nDefinition \\ref{definition-quotient-stack} (1).\nThere are canonical $1$-morphisms\n$\\pi : \\mathcal{S}_U \\to [U/R]$, and $[U/R] \\to \\mathcal{S}_B$\nof stacks in groupoids over $(\\Sch/S)_{fppf}$.\nThe composition $\\mathcal{S}_U \\to \\mathcal{S}_B$ is the $1$-morphism\nassociated to the structure morphism $U \\to B$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044R","source_file":"spaces-groupoids.tex","source_line":2321,"source_end_line":2330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2321-L2330","statement_sha256":"e741ce188419a862f92d42272b3ad5d1844b0939762115a5d55e11f1d38abfc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12413,"rank":12413,"depth":7,"x":1233.333,"y":1224.319,"cluster":"groupoids-quotients"},{"id":"stacks:044S","tag":"044S","title":"Quotient stacks · Lemma 044S","summary":"Assumptions and notation as in Lemma [Tag 044R]. There exists a canonical 2-morphism α : π ∘ s → π ∘ t making the diagram xymatrix S_R ar[r]_s ar[d]_t & S_U ar[d]^π S_U ar[r]^-π & [U/R] 2-commutative.","statement_latex":"Assumptions and notation as in Lemma \\ref{lemma-quotient-stack-arrows}.\nThere exists a canonical $2$-morphism\n$\\alpha : \\pi \\circ s \\to \\pi \\circ t$ making the diagram\n$$\n\\xymatrix{\n\\mathcal{S}_R \\ar[r]_s \\ar[d]_t & \\mathcal{S}_U \\ar[d]^\\pi \\\\\n\\mathcal{S}_U \\ar[r]^-\\pi & [U/R]\n}\n$$\n$2$-commutative.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044S","source_file":"spaces-groupoids.tex","source_line":2366,"source_end_line":2378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2366-L2378","statement_sha256":"1a4cf311b1760001b6d9fa4fdb9d7daf7bbca026d5c75ee5efa375c8f7654323","origin":"The Stacks Project","memory_eligible":false,"source_rank":12414,"rank":12414,"depth":8,"x":985.666,"y":1136.45,"cluster":"groupoids-quotients"},{"id":"stacks:046Q","tag":"046Q","title":"Functoriality of quotient stacks · Lemma 046Q","summary":"Let S be a scheme. Let B be an algebraic space over S. Let f : (U, R, s, t, c) → (U', R', s', t', c') be a morphism of groupoids in algebraic spaces over B. Then f induces a canonical 1-morphism of quotient stacks [f] : [U/R] → [U'/R'].","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $f : (U, R, s, t, c) \\to (U', R', s', t', c')$ be a morphism of\ngroupoids in algebraic spaces over $B$.\nThen $f$ induces a canonical $1$-morphism of quotient stacks\n$$\n[f] : [U/R] \\longrightarrow [U'/R'].\n$$","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Functoriality of quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046Q","source_file":"spaces-groupoids.tex","source_line":2430,"source_end_line":2439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2430-L2439","statement_sha256":"107b390cec416dee8b59d16036981a0e8b0f65d84aaed0040dab6ad01401812e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12415,"rank":12415,"depth":7,"x":1239.534,"y":1060.538,"cluster":"groupoids-quotients"},{"id":"stacks:04Y4","tag":"04Y4","title":"Functoriality of quotient stacks · Lemma 04Y4","summary":"Notation and assumption as in Lemma [Tag 046Q]. Let (U\", R\", s\", t\", c\") be the groupoid in algebraic spaces over B constructed above. There is a 2-commutative square xymatrix [U\"/R\"] ar[d] ar[r]_[g] & [U/R] ar[d]^[f] S_U' ar[r] & [U'/R'] which identifies [U\"/R\"] with the 2-fibre product.","statement_latex":"Notation and assumption as in\nLemma \\ref{lemma-quotient-stack-functorial}.\nLet $(U'', R'', s'', t'', c'')$ be the groupoid in algebraic spaces over $B$\nconstructed above.\nThere is a $2$-commutative square\n$$\n\\xymatrix{\n[U''/R''] \\ar[d] \\ar[r]_{[g]} & [U/R] \\ar[d]^{[f]} \\\\\n\\mathcal{S}_{U'} \\ar[r] & [U'/R']\n}\n$$\nwhich identifies $[U''/R'']$ with the $2$-fibre product.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Functoriality of quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Y4","source_file":"spaces-groupoids.tex","source_line":2483,"source_end_line":2497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2483-L2497","statement_sha256":"4b6963c6571a5d04ef268f82606a70643b7f8a26d80ae553f352d7f756a9529e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12416,"rank":12416,"depth":8,"x":1113.094,"y":1261.021,"cluster":"groupoids-quotients"},{"id":"stacks:044V","tag":"044V","title":"The 2-cartesian square of a quotient stack · Lemma 044V","summary":"Assume B → S, (U, R, s, t, c) and π : S_U → [U/R] are as in Lemma [Tag 044R]. Let S' be a scheme over S. Let x, y ∈ Ob([U/R]_S') be objects of the quotient stack over S'. If x = π(x') and y = π(y') for some morphisms x', y' : S' → U, then mathitIsom(x, y) = S' ×_(y', x'), U ×_S U R as sheaves over S'.","statement_latex":"Assume $B \\to S$, $(U, R, s, t, c)$ and $\\pi : \\mathcal{S}_U \\to [U/R]$\nare as in\nLemma \\ref{lemma-quotient-stack-arrows}.\nLet $S'$ be a scheme over $S$.\nLet $x, y \\in \\Ob([U/R]_{S'})$ be objects of the\nquotient stack over $S'$. If $x = \\pi(x')$ and $y = \\pi(y')$ for\nsome morphisms $x', y' : S' \\to U$, then\n$$\n\\mathit{Isom}(x, y) = S' \\times_{(y', x'), U \\times_S U} R\n$$\nas sheaves over $S'$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"The 2-cartesian square of a quotient stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044V","source_file":"spaces-groupoids.tex","source_line":2561,"source_end_line":2574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2561-L2574","statement_sha256":"9b686930724c7471fbb08c8a3be8e1c43acdb0d228dcf67bc6bdcc9ffe74bfb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12417,"rank":12417,"depth":8,"x":1044.952,"y":1040.945,"cluster":"groupoids-quotients"},{"id":"stacks:04M9","tag":"04M9","title":"The 2-cartesian square of a quotient stack · Lemma 04M9","summary":"Assume B → S, (U, R, s, t, c), and π : S_U → [U/R] are as in Lemma [Tag 044R]. The 2-commutative square xymatrix S_R ar[r]_s ar[d]_t & S_U ar[d]^π S_U ar[r]^-π & [U/R] of Lemma [Tag 044S] is a 2-fibre product of stacks in groupoids of (Sch/S)_fppf.","statement_latex":"Assume $B \\to S$, $(U, R, s, t, c)$, and $\\pi : \\mathcal{S}_U \\to [U/R]$\nare as in\nLemma \\ref{lemma-quotient-stack-arrows}.\nThe $2$-commutative square\n$$\n\\xymatrix{\n\\mathcal{S}_R \\ar[r]_s \\ar[d]_t & \\mathcal{S}_U \\ar[d]^\\pi \\\\\n\\mathcal{S}_U \\ar[r]^-\\pi & [U/R]\n}\n$$\nof\nLemma \\ref{lemma-quotient-stack-2-arrow}\nis a $2$-fibre product of stacks in groupoids of $(\\Sch/S)_{fppf}$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"The 2-cartesian square of a quotient stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04M9","source_file":"spaces-groupoids.tex","source_line":2589,"source_end_line":2604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2589-L2604","statement_sha256":"536f845aaa4fc2e048945034b4f8436c5c0916f972f114e8fbff6f02e5530fc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12418,"rank":12418,"depth":9,"x":1272.691,"y":1164.84,"cluster":"groupoids-quotients"},{"id":"stacks:044W","tag":"044W","title":"The 2-cartesian square of a quotient stack · Lemma 044W","summary":"Assume B → S and (U, R, s, t, c) are as in Definition [Tag 044Q] (1). For any scheme T over S and objects x, y of [U/R] over T the sheaf mathitIsom(x, y) on (Sch/T)_fppf has the following property: There exists a fppf covering (T_i → T)_i ∈ I such that mathitIsom(x, y)|_(Sch/T_i)_fppf is representable by an algebraic space.","statement_latex":"Assume $B \\to S$ and $(U, R, s, t, c)$ are as in\nDefinition \\ref{definition-quotient-stack} (1).\nFor any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$\nthe sheaf $\\mathit{Isom}(x, y)$ on $(\\Sch/T)_{fppf}$ has\nthe following property: There exists a fppf covering\n$\\{T_i \\to T\\}_{i \\in I}$ such that\n$\\mathit{Isom}(x, y)|_{(\\Sch/T_i)_{fppf}}$\nis representable by an algebraic space.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"The 2-cartesian square of a quotient stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044W","source_file":"spaces-groupoids.tex","source_line":2624,"source_end_line":2634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2624-L2634","statement_sha256":"1755c0435f7d704d52e281f6168e68370fb6e4391f32b0a229f241cd22cfa40e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12419,"rank":12419,"depth":9,"x":1004.525,"y":1202.791,"cluster":"groupoids-quotients"},{"id":"stacks:044T","tag":"044T","title":"The 2-coequalizer property of a quotient stack · Lemma 044T","summary":"Assumptions and notation as in Lemmas [Tag 044R] and [Tag 044S]. The vertical composition of xymatrix@C=15pc S_R ×_s, U, t R ruppertwocell^π ∘ s ∘ pr_1 = π ∘ s ∘ c α star id_pr_1 ar[r]_(.3)π ∘ t ∘ pr_1 = π ∘ s ∘ pr_0 rlowertwocell_π ∘ t ∘ pr_0 = π ∘ t ∘ c α star id_pr_0 & [U/R] is the 2-morphism α star id_c. In a formula α star id_c = (α star id_pr_0) ∘ (α star id_pr_1) .","statement_latex":"Assumptions and notation as in\nLemmas \\ref{lemma-quotient-stack-arrows} and\n\\ref{lemma-quotient-stack-2-arrow}.\nThe vertical composition of\n$$\n\\xymatrix@C=15pc{\n\\mathcal{S}_{R \\times_{s, U, t} R}\n\\ruppertwocell^{\\pi \\circ s \\circ \\text{pr}_1 = \\pi \\circ s \\circ c}{\\ \\ \\ \\ \\ \\ \\alpha \\star \\text{id}_{\\text{pr}_1}}\n\\ar[r]_(.3){\\pi \\circ t \\circ \\text{pr}_1 = \\pi \\circ s \\circ \\text{pr}_0}\n\\rlowertwocell_{\\pi \\circ t \\circ \\text{pr}_0 = \\pi \\circ t \\circ c}{\\ \\ \\ \\ \\ \\ \\alpha \\star \\text{id}_{\\text{pr}_0}}\n&\n[U/R]\n}\n$$\nis the $2$-morphism $\\alpha \\star \\text{id}_c$. In a formula\n$\\alpha \\star \\text{id}_c =\n(\\alpha \\star \\text{id}_{\\text{pr}_0})\n\\circ\n(\\alpha \\star \\text{id}_{\\text{pr}_1})\n$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"The 2-coequalizer property of a quotient stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044T","source_file":"spaces-groupoids.tex","source_line":2667,"source_end_line":2689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2667-L2689","statement_sha256":"d2e130b04411b8b657216eae5f259bee1684d5f2e548a6921dddf68ffb6b6092","origin":"The Stacks Project","memory_eligible":false,"source_rank":12420,"rank":12420,"depth":9,"x":1172.122,"y":1022.239,"cluster":"groupoids-quotients"},{"id":"stacks:044U","tag":"044U","title":"The 2-coequalizer property of a quotient stack · Lemma 044U","summary":"Assumptions and notation as in Lemmas [Tag 044R] and [Tag 044S]. The 2-commutative diagram of Lemma [Tag 044S] is a 2-coequalizer in the following sense: Given • a stack in groupoids X over (Sch/S)_fppf, • a 1-morphism f : S_U → X, and • a 2-arrow β : f ∘ s → f ∘ t such that β star id_c = (β star id_pr_0) ∘ (β star id_pr_1) then there exists a 1-morphism [U/R] → X which makes the diagram xymatrix S_R ar[r]_s ar[d]^t & S_U ar[d] ar[ddr]^f S_U ar[r] ar[rrd]_f & [U/R] ar[rd]…","statement_latex":"Assumptions and notation as in\nLemmas \\ref{lemma-quotient-stack-arrows} and\n\\ref{lemma-quotient-stack-2-arrow}.\nThe $2$-commutative diagram of Lemma \\ref{lemma-quotient-stack-2-arrow}\nis a $2$-coequalizer in the following sense:\nGiven\n\\begin{enumerate}\n\\item a stack in groupoids $\\mathcal{X}$ over $(\\Sch/S)_{fppf}$,\n\\item a $1$-morphism $f : \\mathcal{S}_U \\to \\mathcal{X}$, and\n\\item a $2$-arrow $\\beta : f \\circ s \\to f \\circ t$\n\\end{enumerate}\nsuch that\n$$\n\\beta \\star \\text{id}_c\n=\n(\\beta \\star \\text{id}_{\\text{pr}_0})\n\\circ\n(\\beta \\star \\text{id}_{\\text{pr}_1})\n$$\nthen there exists a $1$-morphism $[U/R] \\to \\mathcal{X}$ which makes the\ndiagram\n$$\n\\xymatrix{\n\\mathcal{S}_R \\ar[r]_s \\ar[d]^t & \\mathcal{S}_U \\ar[d] \\ar[ddr]^f \\\\\n\\mathcal{S}_U \\ar[r] \\ar[rrd]_f & [U/R] \\ar[rd] \\\\\n& & \\mathcal{X}\n}\n$$\n$2$-commute.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"The 2-coequalizer property of a quotient stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044U","source_file":"spaces-groupoids.tex","source_line":2719,"source_end_line":2750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2719-L2750","statement_sha256":"1e65fcef8826b0ec8cfc073c4f950abf2b2abc7501ebc1649e989693890af4bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12421,"rank":12421,"depth":9,"x":1193.781,"y":1250.982,"cluster":"groupoids-quotients"},{"id":"stacks:044X","tag":"044X","title":"Explicit description of quotient stacks · Lemma 044X","summary":"Assume B → S and (U, R, s, t, c) are as in Definition [Tag 044Q] (1). Let π : S_U → [U/R] be as in Lemma [Tag 044R]. Let T be a scheme over S. • for every object x of the fibre category [U/R]_T there exists an fppf covering (f_i : T_i → T)_i ∈ I such that f_i^*x ≅ π(u_i) for some u_i ∈ U(T_i), • the composition of the isomorphisms π(u_i ∘ pr_0) = pr_0^*π(u_i) ≅ pr_0^*f_i^*x ≅ pr_1^*f_j^*x ≅ pr_1^*π(u_j) = π(u_j ∘ pr_1) are of the form π(r_ij) for certain morphisms r_ij :…","statement_latex":"Assume $B \\to S$ and $(U, R, s, t, c)$ are as in\nDefinition \\ref{definition-quotient-stack} (1).\nLet $\\pi : \\mathcal{S}_U \\to [U/R]$ be as in\nLemma \\ref{lemma-quotient-stack-arrows}.\nLet $T$ be a scheme over $S$.\n\\begin{enumerate}\n\\item for every object $x$ of the fibre category $[U/R]_T$\nthere exists an fppf covering $\\{f_i : T_i \\to T\\}_{i \\in I}$ such that\n$f_i^*x \\cong \\pi(u_i)$ for some $u_i \\in U(T_i)$,\n\\item the composition of the isomorphisms\n$$\n\\pi(u_i \\circ \\text{pr}_0)\n=\n\\text{pr}_0^*\\pi(u_i)\n\\cong\n\\text{pr}_0^*f_i^*x\n\\cong\n\\text{pr}_1^*f_j^*x\n\\cong\n\\text{pr}_1^*\\pi(u_j)\n=\n\\pi(u_j \\circ \\text{pr}_1)\n$$\nare of the form $\\pi(r_{ij})$ for certain morphisms\n$r_{ij} : T_i \\times_T T_j \\to R$,\n\\item the system $(u_i, r_{ij})$ forms\na $[U/R]$-descent datum as defined above,\n\\item any $[U/R]$-descent datum $(u_i, r_{ij})$ arises in this manner,\n\\item if $x$ corresponds to $(u_i, r_{ij})$ as above, and\n$y \\in \\Ob([U/R]_T)$ corresponds to $(u'_i, r'_{ij})$\nthen there is a canonical bijection\n$$\n\\Mor_{[U/R]_T}(x, y)\n\\longleftrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{morphisms }(u_i, r_{ij}) \\to (u'_i, r'_{ij})\\\\\n\\text{of }[U/R]\\text{-descent data}\n\\end{matrix}\n\\right\\}\n$$\n\\item this correspondence is compatible with refinements of fppf coverings.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Explicit description of quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044X","source_file":"spaces-groupoids.tex","source_line":2890,"source_end_line":2935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2890-L2935","statement_sha256":"318c84e46c99a78f9e039a5b09972a7464d5b7509c5f1667557aa7c305feba13","origin":"The Stacks Project","memory_eligible":false,"source_rank":12422,"rank":12422,"depth":9,"x":993.418,"y":1094.253,"cluster":"groupoids-quotients"},{"id":"stacks:046S","tag":"046S","title":"Restriction and quotient stacks · Lemma 046S","summary":"Notation and assumption as in Lemma [Tag 046Q]. The morphism of quotient stacks [f] : [U/R] → [U'/R'] is fully faithful if and only if R is the restriction of R' via the morphism f : U → U'.","statement_latex":"Notation and assumption as in\nLemma \\ref{lemma-quotient-stack-functorial}.\nThe morphism of quotient stacks\n$$\n[f] : [U/R] \\longrightarrow [U'/R']\n$$\nis fully faithful if and only if $R$ is the restriction of\n$R'$ via the morphism $f : U \\to U'$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Restriction and quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046S","source_file":"spaces-groupoids.tex","source_line":2961,"source_end_line":2971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L2961-L2971","statement_sha256":"7cc85838f3084df4ee4df3d726e80ee0a180701cb1951b947d1a2fa74e62b6f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12423,"rank":12423,"depth":10,"x":1267.804,"y":1096.137,"cluster":"groupoids-quotients"},{"id":"stacks:046T","tag":"046T","title":"Restriction and quotient stacks · Lemma 046T","summary":"Notation and assumption as in Lemma [Tag 046Q]. The morphism of quotient stacks [f] : [U/R] → [U'/R'] is an equivalence if and only if • (U, R, s, t, c) is the restriction of (U', R', s', t', c') via f : U → U', and • the map xymatrix U ×_f, U', t' R' ar[r]_-pr_1 ar@/^3ex/[rr]^h & R' ar[r]_s' & U' is a surjection of sheaves. Part (2) holds for example if (h : U ×_f, U', t' R' → U') is an fppf covering, or if f : U → U' is a surjection of sheaves, or if (f : U → U') is an…","statement_latex":"Notation and assumption as in\nLemma \\ref{lemma-quotient-stack-functorial}.\nThe morphism of quotient stacks\n$$\n[f] : [U/R] \\longrightarrow [U'/R']\n$$\nis an equivalence if and only if\n\\begin{enumerate}\n\\item $(U, R, s, t, c)$ is the restriction of $(U', R', s', t', c')$\nvia $f : U \\to U'$, and\n\\item the map\n$$\n\\xymatrix{\nU \\times_{f, U', t'} R' \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h\n& R' \\ar[r]_{s'} & U'\n}\n$$\nis a surjection of sheaves.\n\\end{enumerate}\nPart (2) holds for example if $\\{h : U \\times_{f, U', t'} R' \\to U'\\}$\nis an fppf covering, or if $f : U \\to U'$ is a surjection of sheaves, or if\n$\\{f : U \\to U'\\}$ is an fppf covering.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Restriction and quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046T","source_file":"spaces-groupoids.tex","source_line":3001,"source_end_line":3025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L3001-L3025","statement_sha256":"62ee4154d8717765bbbf75f33a00c5aaf4507f5903fdbabe38b3f20efd0f1050","origin":"The Stacks Project","memory_eligible":false,"source_rank":12424,"rank":12424,"depth":11,"x":1063.513,"y":1250.78,"cluster":"groupoids-quotients"},{"id":"stacks:04ZN","tag":"04ZN","title":"Restriction and quotient stacks · Lemma 04ZN","summary":"Notation and assumption as in Lemma [Tag 046Q]. Assume that xymatrix R ar[d]_s ar[r]_f & R' ar[d]^s' U ar[r]^f & U' is cartesian. Then xymatrix S_U ar[d] ar[r] & [U/R] ar[d]^[f] S_U' ar[r] & [U'/R'] is a 2-fibre product square.","statement_latex":"Notation and assumption as in\nLemma \\ref{lemma-quotient-stack-functorial}.\nAssume that\n$$\n\\xymatrix{\nR \\ar[d]_s \\ar[r]_f & R' \\ar[d]^{s'} \\\\\nU \\ar[r]^f & U'\n}\n$$\nis cartesian. Then\n$$\n\\xymatrix{\n\\mathcal{S}_U \\ar[d] \\ar[r] & [U/R] \\ar[d]^{[f]} \\\\\n\\mathcal{S}_{U'} \\ar[r] & [U'/R']\n}\n$$\nis a $2$-fibre product square.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Restriction and quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZN","source_file":"spaces-groupoids.tex","source_line":3094,"source_end_line":3113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L3094-L3113","statement_sha256":"275cd169bf84559ad41fb276f02170168ca1070e12bbed9b33dfb180585f4299","origin":"The Stacks Project","memory_eligible":false,"source_rank":12425,"rank":12425,"depth":12,"x":1089.853,"y":1020.326,"cluster":"groupoids-quotients"},{"id":"stacks:06PB","tag":"06PB","title":"Inertia and quotient stacks · Lemma 06PB","summary":"Assume B → S and (U, R, s, t, c) as in Definition [Tag 044Q] (1). Let G/U be the stabilizer group algebraic space of the groupoid (U, R, s, t, c, e, i), see Definition [Tag 0448]. Set R' = R ×_s, U G and set • s' : R' → G, (r, g) ↦ g, • t' : R' → G, (r, g) ↦ c(r, c(g, i(r))), • c' : R' ×_s', G, t' R' → R', ((r_1, g_1), (r_2, g_2) ↦ (c(r_1, r_2), g_1). Then (G, R', s', t', c') is a groupoid in algebraic spaces over B and I_[U/R] = [G/ R']. i.e., the associated quotient…","statement_latex":"Assume $B \\to S$ and $(U, R, s, t, c)$ as in\nDefinition \\ref{definition-quotient-stack} (1).\nLet $G/U$ be the stabilizer group algebraic space of the groupoid\n$(U, R, s, t, c, e, i)$, see\nDefinition \\ref{definition-stabilizer-groupoid}.\nSet $R' = R \\times_{s, U} G$ and set\n\\begin{enumerate}\n\\item $s' : R' \\to G$, $(r, g) \\mapsto g$,\n\\item $t' : R' \\to G$, $(r, g) \\mapsto c(r, c(g, i(r)))$,\n\\item $c' : R' \\times_{s', G, t'} R' \\to R'$,\n$((r_1, g_1), (r_2, g_2) \\mapsto (c(r_1, r_2), g_1)$.\n\\end{enumerate}\nThen $(G, R', s', t', c')$ is a groupoid in algebraic spaces over $B$\nand\n$$\n\\mathcal{I}_{[U/R]} = [G/ R'].\n$$\ni.e., the associated quotient stack is the inertia stack of $[U/R]$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Inertia and quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PB","source_file":"spaces-groupoids.tex","source_line":3206,"source_end_line":3226,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L3206-L3226","statement_sha256":"0eee27e2ff2db0e03a48abfc78355163cc3614a542338d484f12a76f7f54e37d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12426,"rank":12426,"depth":7,"x":1256.116,"y":1205.608,"cluster":"groupoids-quotients"},{"id":"stacks:06PC","tag":"06PC","title":"Inertia and quotient stacks · Lemma 06PC","summary":"Assume B → S and (U, R, s, t, c) as in Definition [Tag 044Q] (1). Let G/U be the stabilizer group algebraic space of the groupoid (U, R, s, t, c, e, i), see Definition [Tag 0448]. There is a canonical 2-cartesian diagram xymatrix S_G ar[r] ar[d] & S_U ar[d] I_[U/R] ar[r] & [U/R] of stacks in groupoids of (Sch/S)_fppf.","statement_latex":"Assume $B \\to S$ and $(U, R, s, t, c)$ as in\nDefinition \\ref{definition-quotient-stack} (1).\nLet $G/U$ be the stabilizer group algebraic space of the groupoid\n$(U, R, s, t, c, e, i)$, see\nDefinition \\ref{definition-stabilizer-groupoid}.\nThere is a canonical $2$-cartesian diagram\n$$\n\\xymatrix{\n\\mathcal{S}_G \\ar[r] \\ar[d] & \\mathcal{S}_U \\ar[d] \\\\\n\\mathcal{I}_{[U/R]} \\ar[r] & [U/R]\n}\n$$\nof stacks in groupoids of $(\\Sch/S)_{fppf}$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Inertia and quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PC","source_file":"spaces-groupoids.tex","source_line":3262,"source_end_line":3277,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L3262-L3277","statement_sha256":"52dade4635367e97ee7283a555838ae1097c7f2bf65755ea39941a0a5f30a026","origin":"The Stacks Project","memory_eligible":false,"source_rank":12427,"rank":12427,"depth":13,"x":983.928,"y":1163.233,"cluster":"groupoids-quotients"},{"id":"stacks:06PE","tag":"06PE","title":"Gerbes and quotient stacks · Lemma 06PE","summary":"Notation and assumption as in Lemma [Tag 046Q]. The morphism of quotient stacks [f] : [U/R] → [U'/R'] turns [U/R] into a gerbe over [U'/R'] if f : U → U' and R → R'|_U are surjective maps of fppf sheaves. Here R'|_U is the restriction of R' to U via f : U → U'.","statement_latex":"Notation and assumption as in\nLemma \\ref{lemma-quotient-stack-functorial}.\nThe morphism of quotient stacks\n$$\n[f] : [U/R] \\longrightarrow [U'/R']\n$$\nturns $[U/R]$ into a gerbe over $[U'/R']$ if $f : U \\to U'$ and\n$R \\to R'|_U$ are surjective maps of fppf sheaves. Here $R'|_U$ is\nthe restriction of $R'$ to $U$ via $f : U \\to U'$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Gerbes and quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PE","source_file":"spaces-groupoids.tex","source_line":3307,"source_end_line":3318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L3307-L3318","statement_sha256":"3961935143b04926dbcf26b1c755832d0f9ced063000b41cf1426f2547aa8859","origin":"The Stacks Project","memory_eligible":false,"source_rank":12428,"rank":12428,"depth":10,"x":1219.219,"y":1039.768,"cluster":"groupoids-quotients"},{"id":"stacks:06PF","tag":"06PF","title":"Gerbes and quotient stacks · Lemma 06PF","summary":"Let S be a scheme. Let B be an algebraic space over S. Let G be a group algebraic space over B. Endow B with the trivial action of G. The morphism [B/G] → S_B (Lemma [Tag 044R]) turns [B/G] into a gerbe over B.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let\n$G$ be a group algebraic space over $B$. Endow $B$ with the trivial\naction of $G$. The morphism\n$$\n[B/G] \\longrightarrow \\mathcal{S}_B\n$$\n(Lemma \\ref{lemma-quotient-stack-arrows})\nturns $[B/G]$ into a gerbe over $B$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Gerbes and quotient stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PF","source_file":"spaces-groupoids.tex","source_line":3348,"source_end_line":3358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L3348-L3358","statement_sha256":"6fbca3073c9989570ccdeb9bf8e3c94f344bf9ed5ffc9d5cedd40886753a592b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12429,"rank":12429,"depth":11,"x":1144.847,"y":1264.802,"cluster":"groupoids-quotients"},{"id":"stacks:04WX","tag":"04WX","title":"Quotient stacks and change of big site · Lemma 04WX","summary":"Suppose given big sites Sch_fppf and Sch'_fppf. Assume that Sch_fppf is contained in Sch'_fppf, see Topologies, Section [Tag 022I]. Let S ∈ Ob(Sch_fppf). Let B, U, R ∈ Sh((Sch/S)_fppf) be algebraic spaces, and let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let f : (Sch'/S)_fppf → (Sch/S)_fppf the morphism of sites corresponding to the inclusion functor u : Sch_fppf → Sch'_fppf. Then we have a canonical equivalence [f^-1U/f^-1R] → f^-1[U/R] of stacks in…","statement_latex":"Suppose given big sites $\\Sch_{fppf}$ and $\\Sch'_{fppf}$.\nAssume that $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$,\nsee Topologies, Section \\ref{topologies-section-change-alpha}.\nLet $S \\in \\Ob(\\Sch_{fppf})$.\nLet $B, U, R \\in \\Sh((\\Sch/S)_{fppf})$ be algebraic spaces,\nand let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $f : (\\Sch'/S)_{fppf} \\to (\\Sch/S)_{fppf}$ the morphism\nof sites corresponding to the inclusion functor\n$u : \\Sch_{fppf} \\to \\Sch'_{fppf}$.\nThen we have a canonical equivalence\n$$\n[f^{-1}U/f^{-1}R]\n\\longrightarrow\nf^{-1}[U/R]\n$$\nof stacks in groupoids over $(\\Sch'/S)_{fppf}$.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Quotient stacks and change of big site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WX","source_file":"spaces-groupoids.tex","source_line":3382,"source_end_line":3400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L3382-L3400","statement_sha256":"64058d043b3909870da8a70cd9fd0fc987925ed81c1b6b67fc5bb38416da0b55","origin":"The Stacks Project","memory_eligible":false,"source_rank":12430,"rank":12430,"depth":54,"x":1018.452,"y":1056.219,"cluster":"groupoids-quotients"},{"id":"stacks:0454","tag":"0454","title":"Separation conditions · Lemma 0454","summary":"Let B → S be as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let G → U be the stabilizer group algebraic space. The commutative diagram xymatrix R ar[d]^Δ_R/U ×_B U ar[rrr]_f ↦ (f, s(f)) & & & R ×_s, U U ar[d] ar[r] & U ar[d] R ×_(U ×_B U) R ar[rrr]^(f, g) ↦ (f, f^-1 ∘ g) & & & R ×_s, U G ar[r] & G the two left horizontal arrows are isomorphisms and the right square is a fibre product square.","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $G \\to U$ be the stabilizer group algebraic space.\nThe commutative diagram\n$$\n\\xymatrix{\nR \\ar[d]^{\\Delta_{R/U \\times_B U}} \\ar[rrr]_{f \\mapsto (f, s(f))} & & &\nR \\times_{s, U} U \\ar[d] \\ar[r] & U \\ar[d] \\\\\nR \\times_{(U \\times_B U)} R \\ar[rrr]^{(f, g) \\mapsto (f, f^{-1} \\circ g)} & & &\nR \\times_{s, U} G \\ar[r] & G\n}\n$$\nthe two left horizontal arrows are isomorphisms\nand the right square is a fibre product square.","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Separation conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0454","source_file":"spaces-groupoids.tex","source_line":3509,"source_end_line":3525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L3509-L3525","statement_sha256":"5f9b86651d97ae8b1774ec54cd47d8e785047f0f560b71f6ec966bed5eddb973","origin":"The Stacks Project","memory_eligible":false,"source_rank":12431,"rank":12431,"depth":0,"x":1279.947,"y":1138.481,"cluster":"groupoids-quotients"},{"id":"stacks:0455","tag":"0455","title":"Separation conditions · Lemma 0455","summary":"Let B → S be as in Section [Tag 043A]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let G → U be the stabilizer group algebraic space. • The following are equivalent • j : R → U ×_B U is separated, • G → U is separated, and • e : U → G is a closed immersion. • The following are equivalent • j : R → U ×_B U is locally separated, • G → U is locally separated, and • e : U → G is an immersion. • The following are equivalent • j : R → U ×_B U is…","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $G \\to U$ be the stabilizer group algebraic space.\n\\begin{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $j : R \\to U \\times_B U$ is separated,\n\\item $G \\to U$ is separated, and\n\\item $e : U \\to G$ is a closed immersion.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $j : R \\to U \\times_B U$ is locally separated,\n\\item $G \\to U$ is locally separated, and\n\\item $e : U \\to G$ is an immersion.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $j : R \\to U \\times_B U$ is quasi-separated,\n\\item $G \\to U$ is quasi-separated, and\n\\item $e : U \\to G$ is quasi-compact.\n\\end{enumerate}\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Groupoids in Algebraic Spaces","chapter_id":"spaces-groupoids","section":"Separation conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0455","source_file":"spaces-groupoids.tex","source_line":3533,"source_end_line":3558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-groupoids.tex#L3533-L3558","statement_sha256":"b976c0a986e7c71230fe3807e01bf5d571df4800f78d609e8abf35d2024ac7df","origin":"The Stacks Project","memory_eligible":false,"source_rank":12432,"rank":12432,"depth":56,"x":1020.422,"y":1226.386,"cluster":"groupoids-quotients"},{"id":"stacks:0CKA","tag":"0CKA","title":"Local structure · Lemma 0CKA","summary":"The map I/I^2 → J/J^2 induced by c is the composition I/I^2 xrightarrow(1, 1) I/I^2 ⊕ I/I^2 → J/J^2 where the second arrow comes from the equality J = (I ⊗ B + B ⊗ I)C. The map i : B → B induces the map -1 : I/I^2 → I/I^2.","statement_latex":"The map $I/I^2 \\to J/J^2$ induced by $c$ is the composition\n$$\nI/I^2 \\xrightarrow{(1, 1)} I/I^2 \\oplus I/I^2 \\to J/J^2\n$$\nwhere the second arrow comes from the equality\n$J = (I \\otimes B + B \\otimes I)C$.\nThe map $i : B \\to B$ induces the map $-1 : I/I^2 \\to I/I^2$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Local structure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKA","source_file":"spaces-more-groupoids.tex","source_line":161,"source_end_line":170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L161-L170","statement_sha256":"9247651f3f3a0e217bbbfe7d0ea7984036451d1130b94ca6546790f507dc179a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12433,"rank":12433,"depth":47,"x":1141.356,"y":1013.858,"cluster":"groupoids-quotients"},{"id":"stacks:0CKD","tag":"0CKD","title":"Groupoid of sections · Lemma 0CKD","summary":"In the situation discussed in this section, let δ ∈ Γ_0 and f = t ∘ δ : U → U. If s, t are flat, then the canonical map C_U_0/U → C_U_0/U induced by f (More on Morphisms of Spaces, Lemma [Tag 04CP]) is the identity map.","statement_latex":"In the situation discussed in this section, let $\\delta \\in \\Gamma_0$\nand $f = t \\circ \\delta : U \\to U$. If $s, t$ are flat, then the\ncanonical map $\\mathcal{C}_{U_0/U} \\to \\mathcal{C}_{U_0/U}$ induced by $f$\n(More on Morphisms of Spaces, Lemma\n\\ref{spaces-more-morphisms-lemma-conormal-functorial})\nis the identity map.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Groupoid of sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKD","source_file":"spaces-more-groupoids.tex","source_line":286,"source_end_line":294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L286-L294","statement_sha256":"71c05b9d112d3787e1c1b381a4404cc05246afb334daaf19764f4720e3701f7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12434,"rank":12434,"depth":54,"x":1223.262,"y":1239.668,"cluster":"groupoids-quotients"},{"id":"stacks:0CKF","tag":"0CKF","title":"Groupoid of sections · Lemma 0CKF","summary":"The bijection ([Tag 0CKE]) is an isomorphism of groups.","statement_latex":"The bijection (\\ref{equation-isomorphism}) is an isomorphism\nof groups.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Groupoid of sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKF","source_file":"spaces-more-groupoids.tex","source_line":370,"source_end_line":374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L370-L374","statement_sha256":"ad9aa05efdd9ecee6cc69290b59448736779ad691b1510dc116c1f55a2fb5ded","origin":"The Stacks Project","memory_eligible":false,"source_rank":12435,"rank":12435,"depth":57,"x":980.769,"y":1119.382,"cluster":"groupoids-quotients"},{"id":"stacks:044Z","tag":"044Z","title":"Properties of groupoids · Lemma 044Z","summary":"Let B → S be as in Section [Tag 04P7]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let τ ∈ (fppf, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic). Let P be a property of morphisms of algebraic spaces which is τ-local on the target (Descent on Spaces, Definition [Tag 03YH]). Assume (s : R → U) and (t : R → U) are coverings for the τ-topology. Let W ⊂ U be the maximal open subspace such that s^-1(W) → W has property P. Then W is R-invariant…","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet\n$\\tau \\in \\{fppf, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic\\}$.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces\nwhich is $\\tau$-local on the target\n(Descent on Spaces,\nDefinition \\ref{spaces-descent-definition-property-morphisms-local}).\nAssume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings for the\n$\\tau$-topology. Let $W \\subset U$ be the maximal open subspace such that\n$s^{-1}(W) \\to W$ has property $\\mathcal{P}$.\nThen $W$ is $R$-invariant\n(Groupoids in Spaces,\nDefinition \\ref{spaces-groupoids-definition-invariant-open}).","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/044Z","source_file":"spaces-more-groupoids.tex","source_line":446,"source_end_line":463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L446-L463","statement_sha256":"0f5a5f5a67fe085a483692247d8872fd43aeca6d9c81c6786d7c218996713ece","origin":"The Stacks Project","memory_eligible":false,"source_rank":12436,"rank":12436,"depth":49,"x":1256.885,"y":1070.381,"cluster":"groupoids-quotients"},{"id":"stacks:06R4","tag":"06R4","title":"Properties of groupoids · Lemma 06R4","summary":"Let B → S be as in Section [Tag 04P7]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let G → U be its stabilizer group algebraic space. Let τ ∈ (fppf, linebreak[0] etale, linebreak[0] smooth, linebreak[0] syntomic). Let P be a property of morphisms of algebraic spaces which is τ-local on the target. Assume (s : R → U) and (t : R → U) are coverings for the τ-topology. Let W ⊂ U be the maximal open subspace such that G_W → W has property P. Then W is…","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $G \\to U$ be its stabilizer group algebraic space.\nLet\n$\\tau \\in \\{fppf, \\linebreak[0] \\etale, \\linebreak[0]\nsmooth, \\linebreak[0] syntomic\\}$.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces\nwhich is $\\tau$-local on the target.\nAssume $\\{s : R \\to U\\}$ and $\\{t : R \\to U\\}$ are coverings for the\n$\\tau$-topology. Let $W \\subset U$ be the maximal open subspace such that\n$G_W \\to W$ has property $\\mathcal{P}$.\nThen $W$ is $R$-invariant (see\nGroupoids in Spaces,\nDefinition \\ref{spaces-groupoids-definition-invariant-open}).","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R4","source_file":"spaces-more-groupoids.tex","source_line":478,"source_end_line":494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L478-L494","statement_sha256":"b828c599a0c98ca3a2259043a906fee026fa958538ea4fc61b5c4a16d7b59603","origin":"The Stacks Project","memory_eligible":false,"source_rank":12437,"rank":12437,"depth":49,"x":1092.345,"y":1263.589,"cluster":"groupoids-quotients"},{"id":"stacks:0452","tag":"0452","title":"Comparing fibres · Lemma 0452","summary":"Let B → S be as in Section [Tag 04P7]. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let K be a field and let r, r' : Spec(K) → R be morphisms such that t ∘ r = t ∘ r' : Spec(K) → U. Set u = s ∘ r, u' = s ∘ r' and denote F_u = Spec(K) ×_u, U, s R and F_u' = Spec(K) ×_u', U, s R the fibre products. Then F_u ≅ F_u' as algebraic spaces over K.","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-notation}.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $K$ be a field and let $r, r' : \\Spec(K) \\to R$\nbe morphisms such that $t \\circ r = t \\circ r' : \\Spec(K) \\to U$.\nSet $u = s \\circ r$, $u' = s \\circ r'$ and denote\n$F_u = \\Spec(K) \\times_{u, U, s} R$ and\n$F_{u'} = \\Spec(K) \\times_{u', U, s} R$ the fibre products.\nThen $F_u \\cong F_{u'}$ as algebraic spaces over $K$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Comparing fibres","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0452","source_file":"spaces-more-groupoids.tex","source_line":521,"source_end_line":531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L521-L531","statement_sha256":"ec961a9777bb48bcce28c8d935fd5e1b0cef5c2979d4ca8bacb4235ebcad307d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12438,"rank":12438,"depth":0,"x":1058.232,"y":1027.269,"cluster":"groupoids-quotients"},{"id":"stacks:04RP","tag":"04RP","title":"Restricting groupoids · Lemma 04RP","summary":"Let S be a scheme. Let B be an algebraic space over S. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let g : U' → U be a morphism of algebraic spaces over B. Let (U', R', s', t', c') be the restriction of (U, R, s, t, c) via g. • If s, t are locally of finite type and g is locally of finite type, then s', t' are locally of finite type. • If s, t are locally of finite presentation and g is locally of finite presentation, then s', t' are locally of finite…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $g : U' \\to U$ be a morphism of algebraic spaces over $B$.\nLet $(U', R', s', t', c')$ be the restriction of\n$(U, R, s, t, c)$ via $g$.\n\\begin{enumerate}\n\\item If $s, t$ are locally of finite type and $g$ is locally of finite\ntype, then $s', t'$ are locally of finite type.\n\\item If $s, t$ are locally of finite presentation and $g$ is locally of finite\npresentation, then $s', t'$ are locally of finite presentation.\n\\item If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat.\n\\item Add more here.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Restricting groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RP","source_file":"spaces-more-groupoids.tex","source_line":584,"source_end_line":599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L584-L599","statement_sha256":"bdfc5a215d576291ac3181c7460ee86f79b298610c60c5b58089dc619d1a0e69","origin":"The Stacks Project","memory_eligible":false,"source_rank":12439,"rank":12439,"depth":56,"x":1273.871,"y":1182.49,"cluster":"groupoids-quotients"},{"id":"stacks:06E0","tag":"06E0","title":"Properties of groups over fields and groupoids on fields · Lemma 06E0","summary":"In Situation [Tag 06DY] the composition morphism c : R ×_s, U, t R → R is flat and universally open. In Situation [Tag 06DX] the group law m : G ×_k G → G is flat and universally open.","statement_latex":"In\nSituation \\ref{situation-groupoid-on-field}\nthe composition morphism $c : R \\times_{s, U, t} R \\to R$ is flat and\nuniversally open.\nIn\nSituation \\ref{situation-group-over-field}\nthe group law $m : G \\times_k G \\to G$ is flat and\nuniversally open.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groups over fields and groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06E0","source_file":"spaces-more-groupoids.tex","source_line":662,"source_end_line":672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L662-L672","statement_sha256":"300554c798d8f7ff7a884fcbc625c7a4d6aef71240fc964825552ebb90351c75","origin":"The Stacks Project","memory_eligible":false,"source_rank":12440,"rank":12440,"depth":48,"x":989.455,"y":1190.407,"cluster":"groupoids-quotients"},{"id":"stacks:08BH","tag":"08BH","title":"Properties of groups over fields and groupoids on fields · Lemma 08BH","summary":"In Situation [Tag 06DY] assume R is a decent space. Then R is a separated algebraic space. In Situation [Tag 06DX] assume that G is a decent algebraic space. Then G is separated algebraic space.","statement_latex":"In Situation \\ref{situation-groupoid-on-field}\nassume $R$ is a decent space. Then $R$ is a separated algebraic space.\nIn Situation \\ref{situation-group-over-field} assume that\n$G$ is a decent algebraic space. Then $G$ is separated algebraic space.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groups over fields and groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08BH","source_file":"spaces-more-groupoids.tex","source_line":691,"source_end_line":697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L691-L697","statement_sha256":"478a2b0015861ccab0fe62926db817ee82f25961134ec21a1ba04c05db919c95","origin":"The Stacks Project","memory_eligible":false,"source_rank":12441,"rank":12441,"depth":61,"x":1193.228,"y":1022.843,"cluster":"groupoids-quotients"},{"id":"stacks:06E1","tag":"06E1","title":"Properties of groups over fields and groupoids on fields · Lemma 06E1","summary":"In Situation [Tag 06DY]. Let k'/k be a field extension, U' = Spec(k') and let (U', R', s', t', c') be the restriction of (U, R, s, t, c) via U' → U. In the defining diagram xymatrix R' ar[d] ar[r] ar@/_3pc/[dd]_t' ar@/^1pc/[rr]^s' ar@..>[rd] & R ×_s, U U' ar[r] ar[d] & U' ar[d] U' ×_U, t R ar[d] ar[r] & R ar[r]^s ar[d]_t & U U' ar[r] & U all the morphisms are surjective, flat, and universally open. The dotted arrow R' → R is in addition affine.","statement_latex":"In\nSituation \\ref{situation-groupoid-on-field}.\nLet $k'/k$ be a field extension, $U' = \\Spec(k')$\nand let $(U', R', s', t', c')$ be the restriction of\n$(U, R, s, t, c)$ via $U' \\to U$. In the defining diagram\n$$\n\\xymatrix{\nR' \\ar[d] \\ar[r] \\ar@/_3pc/[dd]_{t'} \\ar@/^1pc/[rr]^{s'} \\ar@{..>}[rd] &\nR \\times_{s, U} U' \\ar[r] \\ar[d] &\nU' \\ar[d] \\\\\nU' \\times_{U, t} R \\ar[d] \\ar[r] &\nR \\ar[r]^s \\ar[d]_t &\nU \\\\\nU' \\ar[r] &\nU\n}\n$$\nall the morphisms are surjective, flat, and universally open.\nThe dotted arrow $R' \\to R$ is in addition affine.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groups over fields and groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06E1","source_file":"spaces-more-groupoids.tex","source_line":719,"source_end_line":740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L719-L740","statement_sha256":"60626fb93c1fd1efb85ac25d428a0fc04a373420c1a039af18727ec95c310197","origin":"The Stacks Project","memory_eligible":false,"source_rank":12442,"rank":12442,"depth":10,"x":1177.687,"y":1262.515,"cluster":"groupoids-quotients"},{"id":"stacks:06E2","tag":"06E2","title":"Properties of groups over fields and groupoids on fields · Lemma 06E2","summary":"In Situation [Tag 06DY]. For any point r ∈ |R| there exist • a field extension k'/k with k' algebraically closed, • a point r' : Spec(k') → R' where (U', R', s', t', c') is the restriction of (U, R, s, t, c) via Spec(k') → Spec(k) such that • the point r' maps to r under the morphism R' → R, and • the maps s' ∘ r', t' ∘ r' : Spec(k') → Spec(k') are automorphisms.","statement_latex":"In\nSituation \\ref{situation-groupoid-on-field}.\nFor any point $r \\in |R|$ there exist\n\\begin{enumerate}\n\\item a field extension $k'/k$ with $k'$ algebraically closed,\n\\item a point $r' : \\Spec(k') \\to R'$ where\n$(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$\nvia $\\Spec(k') \\to \\Spec(k)$\n\\end{enumerate}\nsuch that\n\\begin{enumerate}\n\\item the point $r'$ maps to $r$ under the morphism $R' \\to R$, and\n\\item the maps\n$s' \\circ r', t' \\circ r' : \\Spec(k') \\to \\Spec(k')$\nare automorphisms.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groups over fields and groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06E2","source_file":"spaces-more-groupoids.tex","source_line":759,"source_end_line":777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L759-L777","statement_sha256":"0420c0119142c19144cc99793a1261fbbf431ec37afe7ff647d3bef5d002cbda","origin":"The Stacks Project","memory_eligible":false,"source_rank":12443,"rank":12443,"depth":1,"x":996.043,"y":1076.59,"cluster":"groupoids-quotients"},{"id":"stacks:06E3","tag":"06E3","title":"Properties of groups over fields and groupoids on fields · Lemma 06E3","summary":"In Situation [Tag 06DY]. If r : Spec(k) → R is a morphism such that s ∘ r, t ∘ r are automorphisms of Spec(k), then the map R → R, x ↦ c(r, x) is an automorphism R → R which maps e to r.","statement_latex":"In\nSituation \\ref{situation-groupoid-on-field}.\nIf $r : \\Spec(k) \\to R$ is a morphism such that\n$s \\circ r, t \\circ r$ are automorphisms of $\\Spec(k)$, then the map\n$$\nR \\longrightarrow R, \\quad\nx \\longmapsto c(r, x)\n$$\nis an automorphism $R \\to R$ which maps $e$ to $r$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groups over fields and groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06E3","source_file":"spaces-more-groupoids.tex","source_line":797,"source_end_line":808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L797-L808","statement_sha256":"7ebccbf7ce6a1563e2c0ab9e08bd64d9106eb009c993fa53621e063b25ef5cbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12444,"rank":12444,"depth":1,"x":1280.072,"y":1110.688,"cluster":"groupoids-quotients"},{"id":"stacks:06E4","tag":"06E4","title":"Properties of groups over fields and groupoids on fields · Lemma 06E4","summary":"In Situation [Tag 06DY] the algebraic space R is geometrically unibranch. In Situation [Tag 06DX] the algebraic space G is geometrically unibranch.","statement_latex":"In\nSituation \\ref{situation-groupoid-on-field}\nthe algebraic space $R$ is geometrically unibranch. In\nSituation \\ref{situation-group-over-field}\nthe algebraic space $G$ is geometrically unibranch.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groups over fields and groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06E4","source_file":"spaces-more-groupoids.tex","source_line":816,"source_end_line":823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L816-L823","statement_sha256":"ccbbc3e50547a84ceac08fa8a87ecc2f77a1eaad4e3cc649686738edbeb6d033","origin":"The Stacks Project","memory_eligible":false,"source_rank":12445,"rank":12445,"depth":55,"x":1042.737,"y":1246.983,"cluster":"groupoids-quotients"},{"id":"stacks:06FD","tag":"06FD","title":"Properties of groups over fields and groupoids on fields · Lemma 06FD","summary":"In Situation [Tag 06DY] assume s, t are locally of finite type. For all r ∈ |R| • dim(R) = dim_r(R), • the transcendence degree of r over Spec(k) via s equals the transcendence degree of r over Spec(k) via t, and • if the transcendence degree mentioned in (2) is 0, then dim(R) = dim(O_R, overliner).","statement_latex":"In\nSituation \\ref{situation-groupoid-on-field}\nassume $s, t$ are locally of finite type.\nFor all $r \\in |R|$\n\\begin{enumerate}\n\\item $\\dim(R) = \\dim_r(R)$,\n\\item the transcendence degree of $r$ over $\\Spec(k)$\nvia $s$ equals the transcendence degree of $r$ over $\\Spec(k)$\nvia $t$, and\n\\item if the transcendence degree mentioned in (2) is $0$, then\n$\\dim(R) = \\dim(\\mathcal{O}_{R, \\overline{r}})$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groups over fields and groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FD","source_file":"spaces-more-groupoids.tex","source_line":889,"source_end_line":903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L889-L903","statement_sha256":"7838bf6e5fb231a0e0edff6914779764d31c9c19e41ec359bda3d75750033a81","origin":"The Stacks Project","memory_eligible":false,"source_rank":12446,"rank":12446,"depth":28,"x":1108.271,"y":1011.341,"cluster":"groupoids-quotients"},{"id":"stacks:06FE","tag":"06FE","title":"Properties of groups over fields and groupoids on fields · Lemma 06FE","summary":"In Situation [Tag 06DX] assume G locally of finite type. For all g ∈ |G| • dim(G) = dim_g(G), • if the transcendence degree of g over k is 0, then dim(G) = dim(O_G, overlineg).","statement_latex":"In\nSituation \\ref{situation-group-over-field}\nassume $G$ locally of finite type.\nFor all $g \\in |G|$\n\\begin{enumerate}\n\\item $\\dim(G) = \\dim_g(G)$,\n\\item if the transcendence degree of $g$ over $k$ is $0$, then\n$\\dim(G) = \\dim(\\mathcal{O}_{G, \\overline{g}})$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groups over fields and groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FE","source_file":"spaces-more-groupoids.tex","source_line":942,"source_end_line":953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L942-L953","statement_sha256":"1d0284d3ff64f7f4ca0ca9b15cd39727118315ff2ba2f3434132a2dc48c1cfef","origin":"The Stacks Project","memory_eligible":false,"source_rank":12447,"rank":12447,"depth":29,"x":1249.724,"y":1222.706,"cluster":"groupoids-quotients"},{"id":"stacks:06FF","tag":"06FF","title":"Properties of groups over fields and groupoids on fields · Lemma 06FF","summary":"In Situation [Tag 06DY] assume s, t are locally of finite type. Let G = Spec(k) ×_Δ, Spec(k) ×_B Spec(k), t × s R be the stabilizer group algebraic space. Then we have dim(R) = dim(G).","statement_latex":"In\nSituation \\ref{situation-groupoid-on-field}\nassume $s, t$ are locally of finite type.\nLet\n$G = \\Spec(k)\n\\times_{\\Delta, \\Spec(k) \\times_B \\Spec(k), t \\times s} R$\nbe the stabilizer group algebraic space.\nThen we have $\\dim(R) = \\dim(G)$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Properties of groups over fields and groupoids on fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FF","source_file":"spaces-more-groupoids.tex","source_line":961,"source_end_line":971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L961-L971","statement_sha256":"431aa8c5473f4fdbc2d2c78c73202e4d18e49cbf626bd83cc893182e8895e512","origin":"The Stacks Project","memory_eligible":false,"source_rank":12448,"rank":12448,"depth":48,"x":974.901,"y":1146.962,"cluster":"groupoids-quotients"},{"id":"stacks:0B8E","tag":"0B8E","title":"Group algebraic spaces over fields · Lemma 0B8E","summary":"Let k be a field with algebraic closure overlinek. Let G be a group algebraic space over k which is separated. Then G_overlinek is a scheme.","statement_latex":"Let $k$ be a field with algebraic closure $\\overline{k}$.\nLet $G$ be a group algebraic space over $k$\nwhich is separated\\footnote{It is enough to assume $G$ is decent,\ne.g., locally separated or quasi-separated by\nLemma \\ref{lemma-group-scheme-over-field-separated}.}.\nThen $G_{\\overline{k}}$ is a scheme.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Group algebraic spaces over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8E","source_file":"spaces-more-groupoids.tex","source_line":1040,"source_end_line":1048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1040-L1048","statement_sha256":"c431a7bebe15044ebd06cbcfea032d19c87f2b1bbca37ce3546b560c91adbba7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12449,"rank":12449,"depth":70,"x":1238.983,"y":1046.675,"cluster":"groupoids-quotients"},{"id":"stacks:0B8F","tag":"0B8F","title":"Group algebraic spaces over fields · Lemma 0B8F","summary":"Let k be a field. Let G be a group algebraic space over k. If G is separated and locally of finite type over k, then G is a scheme.","statement_latex":"Let $k$ be a field. Let $G$ be a group algebraic space over $k$.\nIf $G$ is separated and locally of finite type over $k$,\nthen $G$ is a scheme.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Group algebraic spaces over fields","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8F","source_file":"spaces-more-groupoids.tex","source_line":1078,"source_end_line":1083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1078-L1083","statement_sha256":"9202d59d5489fec894725eec9f3eff84cf30e735a0d2ea723808ffbf54d6df47","origin":"The Stacks Project","memory_eligible":false,"source_rank":12450,"rank":12450,"depth":71,"x":1124.676,"y":1270.913,"cluster":"groupoids-quotients"},{"id":"stacks:0B8G","tag":"0B8G","title":"Group algebraic spaces over fields · Proposition 0B8G","summary":"Let k be a field. Let G be a group algebraic space over k. If G is separated, then G is a scheme.","statement_latex":"Let $k$ be a field. Let $G$ be a group algebraic space over $k$.\nIf $G$ is separated, then $G$ is a scheme.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Group algebraic spaces over fields","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B8G","source_file":"spaces-more-groupoids.tex","source_line":1093,"source_end_line":1097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1093-L1097","statement_sha256":"c83d4dd08d82c6c386ad8a0f6d594845d09a3d31fa471463bf0d9dde932e44c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12451,"rank":12451,"depth":72,"x":1028.451,"y":1040.251,"cluster":"groupoids-quotients"},{"id":"stacks:0AEL","tag":"0AEL","title":"No rational curves on groups · Lemma 0AEL","summary":"Let S be a scheme. Let B be an algebraic space over S. Let f : X → Y and g : X → Z be morphisms of algebraic spaces over B. Assume • Y → B is separated, • g is surjective, flat, and locally of finite presentation, • there is a scheme theoretically dense open V ⊂ Z such that f|_g^-1(V) : g^-1(V) → Y factors through V. Then f factors through g.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $f : X \\to Y$ and $g : X \\to Z$ be morphisms of algebraic\nspaces over $B$. Assume\n\\begin{enumerate}\n\\item $Y \\to B$ is separated,\n\\item $g$ is surjective, flat, and locally of finite presentation,\n\\item there is a scheme theoretically dense open $V \\subset Z$\nsuch that $f|_{g^{-1}(V)} : g^{-1}(V) \\to Y$ factors through $V$.\n\\end{enumerate}\nThen $f$ factors through $g$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"No rational curves on groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEL","source_file":"spaces-more-groupoids.tex","source_line":1183,"source_end_line":1195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1183-L1195","statement_sha256":"e3e04d943a524d96b51c0ef56bef8899060cb740d559cac5e31baae3bd47b2b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12452,"rank":12452,"depth":51,"x":1285.398,"y":1155.963,"cluster":"groupoids-quotients"},{"id":"stacks:0AEM","tag":"0AEM","title":"No rational curves on groups · Lemma 0AEM","summary":"A morphism from a nonempty product of projective lines over a field to a separated finite type algebraic space over a field factors as a finite morphism after a projection to a product of projective lines. Let k be a field. Let n ≥ 1 and let (P^1_k)^n be the n-fold self product over Spec(k). Let f : (P^1_k)^n → Z be a morphism of algebraic spaces over k. If Z is separated of finite type over k, then f factors as (P^1_k)^n xrightarrowprojection (P^1_k)^m xrightarrowfinite Z.","statement_latex":"\\begin{slogan}\nA morphism from a nonempty product of projective lines over a field to\na separated finite type algebraic space over a field factors as a\nfinite morphism after a projection to a product of projective lines.\n\\end{slogan}\nLet $k$ be a field. Let $n \\geq 1$ and let $(\\mathbf{P}^1_k)^n$\nbe the $n$-fold self product over $\\Spec(k)$. Let\n$f : (\\mathbf{P}^1_k)^n \\to Z$ be a morphism of algebraic spaces over $k$.\nIf $Z$ is separated of finite type over $k$, then $f$ factors as\n$$\n(\\mathbf{P}^1_k)^n \\xrightarrow{projection}\n(\\mathbf{P}^1_k)^m \\xrightarrow{finite} Z.\n$$","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"No rational curves on groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEM","source_file":"spaces-more-groupoids.tex","source_line":1208,"source_end_line":1223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1208-L1223","statement_sha256":"bcf960658a9bb94b06b5c8d867fc2ddab70dfa9619d87261659e9a1785299610","origin":"The Stacks Project","memory_eligible":false,"source_rank":12453,"rank":12453,"depth":62,"x":1002.328,"y":1216.554,"cluster":"groupoids-quotients"},{"id":"stacks:0AEN","tag":"0AEN","title":"No rational curves on groups · Lemma 0AEN","summary":"No complete rational curves on groups. Let k be a field. Let G be a separated group algebraic space locally of finite type over k. There does not exist a nonconstant morphism f : P^1_k → G over Spec(k).","statement_latex":"\\begin{slogan}\nNo complete rational curves on groups.\n\\end{slogan}\nLet $k$ be a field. Let $G$ be a separated group algebraic space locally\nof finite type over $k$. There does not exist a nonconstant\nmorphism $f : \\mathbf{P}^1_k \\to G$ over $\\Spec(k)$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"No rational curves on groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEN","source_file":"spaces-more-groupoids.tex","source_line":1265,"source_end_line":1273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1265-L1273","statement_sha256":"8868e885369ef01520416b42b32d3577344e628c481e7af82e04c077a0114c76","origin":"The Stacks Project","memory_eligible":false,"source_rank":12454,"rank":12454,"depth":63,"x":1162.644,"y":1010.857,"cluster":"groupoids-quotients"},{"id":"stacks:04PE","tag":"04PE","title":"The finite part of a morphism · Lemma 04PE","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then we have • The presheaf (X/Y)_fin satisfies the sheaf condition for the fppf topology. • If T is an algebraic space over S, then there is a canonical bijection Mor_Sh((Sch/S)_fppf)(T, (X/Y)_fin) = ((a, Z) satisfying [Tag 04PC])","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThen we have\n\\begin{enumerate}\n\\item The presheaf $(X/Y)_{fin}$ satisfies the sheaf condition for\nthe fppf topology.\n\\item If $T$ is an algebraic space over $S$, then there is a\ncanonical bijection\n$$\n\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, (X/Y)_{fin})\n=\n\\{(a, Z)\\text{ satisfying \\ref{equation-finite-conditions}}\\}\n$$\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PE","source_file":"spaces-more-groupoids.tex","source_line":1332,"source_end_line":1348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1332-L1348","statement_sha256":"03474fc0ac664c1c475345c5808b7b6adc44af24a7c3fdb3835f6f8a0f46c95e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12455,"rank":12455,"depth":59,"x":1209.936,"y":1253.97,"cluster":"groupoids-quotients"},{"id":"stacks:04PF","tag":"04PF","title":"The finite part of a morphism · Lemma 04PF","summary":"Let S be a scheme. Consider a commutative diagram xymatrix X' ar[rr]_j ar[rd] & & X ar[ld] & Y of algebraic spaces over S. If j is an open immersion, then there is a canonical injective map of sheaves j : (X'/Y)_fin → (X/Y)_fin.","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nX' \\ar[rr]_j \\ar[rd] & & X \\ar[ld] \\\\\n& Y\n}\n$$\nof algebraic spaces over $S$. If $j$ is an open immersion, then\nthere is a canonical injective map of sheaves\n$j : (X'/Y)_{fin} \\to (X/Y)_{fin}$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PF","source_file":"spaces-more-groupoids.tex","source_line":1410,"source_end_line":1422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1410-L1422","statement_sha256":"9bbc04838b004188facffcb1b738ab30d1c6e3014bae1c5c6bd2cff0ee2ea669","origin":"The Stacks Project","memory_eligible":false,"source_rank":12456,"rank":12456,"depth":0,"x":979.116,"y":1101.243,"cluster":"groupoids-quotients"},{"id":"stacks:04PG","tag":"04PG","title":"The finite part of a morphism · Lemma 04PG","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is locally of finite type. Let X' ⊂ X be the maximal open subspace over which f is locally quasi-finite, see Morphisms of Spaces, Lemma [Tag 04NW]. Then (X/Y)_fin = (X'/Y)_fin.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which is\nlocally of finite type.\nLet $X' \\subset X$ be the maximal open subspace over which $f$ is\nlocally quasi-finite, see\nMorphisms of Spaces,\nLemma \\ref{spaces-morphisms-lemma-locally-finite-type-quasi-finite-part}.\nThen $(X/Y)_{fin} = (X'/Y)_{fin}$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PG","source_file":"spaces-more-groupoids.tex","source_line":1429,"source_end_line":1439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1429-L1439","statement_sha256":"23e957b8317ea66a14a62f2ac259732dfe5bbf77ab3d5698a8d294aa325a36de","origin":"The Stacks Project","memory_eligible":false,"source_rank":12457,"rank":12457,"depth":44,"x":1272.699,"y":1082.854,"cluster":"groupoids-quotients"},{"id":"stacks:04PH","tag":"04PH","title":"The finite part of a morphism · Lemma 04PH","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let T be an algebraic space over S, and let (a, Z) be a pair as in [Tag 04PC]. If f is separated, then Z is closed in T ×_Y X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $T$ be an algebraic space over $S$, and let $(a, Z)$ be\na pair as in \\ref{equation-finite-conditions}.\nIf $f$ is separated, then $Z$ is closed in $T \\times_Y X$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PH","source_file":"spaces-more-groupoids.tex","source_line":1454,"source_end_line":1461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1454-L1461","statement_sha256":"25edd847512500592fdc36e36294b4215f3ff922b3af685888f82250e6254452","origin":"The Stacks Project","memory_eligible":false,"source_rank":12458,"rank":12458,"depth":58,"x":1070.618,"y":1263.344,"cluster":"groupoids-quotients"},{"id":"stacks:04PK","tag":"04PK","title":"The finite part of a morphism · Lemma 04PK","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The diagonal of (X/Y)_fin → Y (X/Y)_fin → (X/Y)_fin ×_Y (X/Y)_fin is representable (by schemes) and an open immersion and the \"absolute\" diagonal (X/Y)_fin → (X/Y)_fin × (X/Y)_fin is representable (by schemes).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThe diagonal of $(X/Y)_{fin} \\to Y$\n$$\n(X/Y)_{fin} \\longrightarrow (X/Y)_{fin} \\times_Y (X/Y)_{fin}\n$$\nis representable (by schemes) and an open immersion and the ``absolute''\ndiagonal\n$$\n(X/Y)_{fin} \\longrightarrow (X/Y)_{fin} \\times (X/Y)_{fin}\n$$\nis representable (by schemes).","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04PK","source_file":"spaces-more-groupoids.tex","source_line":1526,"source_end_line":1540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1526-L1540","statement_sha256":"c3fae8ffa40d0c86bebb2781e6cc88d7f4f458dd7c7a94e97ae7ddee01e9ed4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12459,"rank":12459,"depth":58,"x":1074.493,"y":1015.117,"cluster":"groupoids-quotients"},{"id":"stacks:04QE","tag":"04QE","title":"The finite part of a morphism · Lemma 04QE","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Suppose that U is a scheme, U → Y is an étale morphism and Z ⊂ U ×_Y X is an open subspace finite over U. Then the induced morphism U → (X/Y)_fin is étale.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nSuppose that $U$ is a scheme, $U \\to Y$ is an \\'etale morphism and\n$Z \\subset U \\times_Y X$ is an open subspace finite over $U$.\nThen the induced morphism $U \\to (X/Y)_{fin}$ is \\'etale.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QE","source_file":"spaces-more-groupoids.tex","source_line":1572,"source_end_line":1579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1572-L1579","statement_sha256":"c3416c06c5f81e03514429a0d14c86397170e38c777d8ec5f228e4b30a0836a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12460,"rank":12460,"depth":59,"x":1271.624,"y":1200.705,"cluster":"groupoids-quotients"},{"id":"stacks:04QF","tag":"04QF","title":"The finite part of a morphism · Lemma 04QF","summary":"Let S be a scheme. Let xymatrix X' ar[d] ar[r] & X ar[d] Y' ar[r] & Y be a fibre product square of algebraic spaces over S. Then xymatrix (X'/Y')_fin ar[d] ar[r] & (X/Y)_fin ar[d] Y' ar[r] & Y is a fibre product square of sheaves on (Sch/S)_fppf.","statement_latex":"Let $S$ be a scheme.\nLet\n$$\n\\xymatrix{\nX' \\ar[d] \\ar[r] & X \\ar[d] \\\\\nY' \\ar[r] & Y\n}\n$$\nbe a fibre product square of algebraic spaces over $S$. Then\n$$\n\\xymatrix{\n(X'/Y')_{fin} \\ar[d] \\ar[r] & (X/Y)_{fin} \\ar[d] \\\\\nY' \\ar[r] & Y\n}\n$$\nis a fibre product square of sheaves on $(\\Sch/S)_{fppf}$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QF","source_file":"spaces-more-groupoids.tex","source_line":1604,"source_end_line":1622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1604-L1622","statement_sha256":"3e974e8d759da765764904b8c7efb09afa98beace7c1daa8f2882af215ae7c2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12461,"rank":12461,"depth":0,"x":976.463,"y":1175.665,"cluster":"groupoids-quotients"},{"id":"stacks:04QG","tag":"04QG","title":"The finite part of a morphism · Lemma 04QG","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. If f is separated and locally quasi-finite, then there exists a scheme U étale over Y and a surjective étale morphism U → (X/Y)_fin over Y.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $f$ is separated and locally quasi-finite, then there exists a\nscheme $U$ \\'etale over $Y$ and a surjective \\'etale morphism\n$U \\to (X/Y)_{fin}$ over $Y$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QG","source_file":"spaces-more-groupoids.tex","source_line":1630,"source_end_line":1637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1630-L1637","statement_sha256":"5fb49355b353a2d6f82d69120f09058b55ed7a2229610f1bdb8844e7f0ea7006","origin":"The Stacks Project","memory_eligible":false,"source_rank":12462,"rank":12462,"depth":60,"x":1214.693,"y":1026.368,"cluster":"groupoids-quotients"},{"id":"stacks:04QH","tag":"04QH","title":"The finite part of a morphism · Proposition 04QH","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is separated and locally of finite type. Then (X/Y)_fin is an algebraic space. Moreover, the morphism (X/Y)_fin → Y is étale.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which\nis separated and locally of finite type. Then $(X/Y)_{fin}$\nis an algebraic space. Moreover, the morphism\n$(X/Y)_{fin} \\to Y$ is \\'etale.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QH","source_file":"spaces-more-groupoids.tex","source_line":1794,"source_end_line":1801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1794-L1801","statement_sha256":"066ad824845978e8db2eb7e2dabd01cb03c4b5ed50e6cfeca84f6199080ae595","origin":"The Stacks Project","memory_eligible":false,"source_rank":12463,"rank":12463,"depth":61,"x":1158.982,"y":1272.094,"cluster":"groupoids-quotients"},{"id":"stacks:04RI","tag":"04RI","title":"The finite part of a morphism · Lemma 04RI","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which is separated, flat, and locally of finite presentation. In this case • (X/Y)_fin → Y is separated, representable, and étale, and • if Y is a scheme, then (X/Y)_fin is (representable by) a scheme.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which\nis separated, flat, and locally of finite presentation.\nIn this case\n\\begin{enumerate}\n\\item $(X/Y)_{fin} \\to Y$ is separated, representable, and \\'etale, and\n\\item if $Y$ is a scheme, then $(X/Y)_{fin}$ is (representable by) a scheme.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RI","source_file":"spaces-more-groupoids.tex","source_line":1864,"source_end_line":1874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1864-L1874","statement_sha256":"f4457902d60a98c415a76893bfa82a1d826411ad3fd4ca15ccbda942ec2b51e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12464,"rank":12464,"depth":62,"x":1002.165,"y":1058.891,"cluster":"groupoids-quotients"},{"id":"stacks:04RR","tag":"04RR","title":"The finite part of a morphism · Lemma 04RR","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let σ : Y → X be a section of f. Consider the transformation of functors t : (X/Y, σ)_fin → (X/Y)_fin. defined above. Then • t is representable by open immersions, • if f is separated, then t is representable by open and closed immersions, • if (X/Y)_fin is an algebraic space, then (X/Y, σ)_fin is an algebraic space and an open subspace of (X/Y)_fin, and • if (X/Y)_fin is a scheme, then (X/Y,…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nLet $\\sigma : Y \\to X$ be a section of $f$. Consider the\ntransformation of functors\n$$\nt : (X/Y, \\sigma)_{fin} \\longrightarrow (X/Y)_{fin}.\n$$\ndefined above. Then\n\\begin{enumerate}\n\\item $t$ is representable by open immersions,\n\\item if $f$ is separated, then $t$ is representable by open\nand closed immersions,\n\\item if $(X/Y)_{fin}$ is an algebraic space, then\n$(X/Y, \\sigma)_{fin}$ is an algebraic space and\nan open subspace of $(X/Y)_{fin}$, and\n\\item if $(X/Y)_{fin}$ is a scheme, then $(X/Y, \\sigma)_{fin}$ is an\nopen subscheme of it.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RR","source_file":"spaces-more-groupoids.tex","source_line":1950,"source_end_line":1970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L1950-L1970","statement_sha256":"df5160205e73b24489fac5c266ae5c6c841c1e74ccbb1fdfcd9778f99a714627","origin":"The Stacks Project","memory_eligible":false,"source_rank":12465,"rank":12465,"depth":0,"x":1289.785,"y":1127.249,"cluster":"groupoids-quotients"},{"id":"stacks:04RU","tag":"04RU","title":"The finite part of a groupoid · Lemma 04RU","summary":"Let S be a scheme. Let B be an algebraic space over S. Let (U, R, s, t, c, e, i) be a groupoid in algebraic spaces over B. Assume the morphisms s, t are separated and locally of finite type. There exists a canonical morphism (U', Z_univ, s', t', c', e', i') → (U, R, s, t, c, e, i) of groupoids in algebraic spaces over B where • g : U' → U is identified with (R_s/U, e)_fin → U, and • Z_univ ⊂ R ×_s, U, g U' is the universal open (and closed) subspace finite over U' which…","statement_latex":"Let $S$ be a scheme.\nLet $B$ be an algebraic space over $S$.\nLet $(U, R, s, t, c, e, i)$ be a groupoid in algebraic spaces over $B$.\nAssume the morphisms $s, t$ are separated and locally of finite type.\nThere exists a canonical morphism\n$$\n(U', Z_{univ}, s', t', c', e', i')\n\\longrightarrow\n(U, R, s, t, c, e, i)\n$$\nof groupoids in algebraic spaces over $B$ where\n\\begin{enumerate}\n\\item $g : U' \\to U$ is identified with $(R_s/U, e)_{fin} \\to U$, and\n\\item $Z_{univ} \\subset R \\times_{s, U, g} U'$ is the universal\nopen (and closed) subspace finite over $U'$ which contains the base\nchange of the unit $e$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"The finite part of a groupoid","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RU","source_file":"spaces-more-groupoids.tex","source_line":2138,"source_end_line":2157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2138-L2157","statement_sha256":"8a90956676af2668f4250fd3cfaa0e406179c0ccd0cd83dcebeaa4b08e114a0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12466,"rank":12466,"depth":0,"x":1022.233,"y":1240.253,"cluster":"groupoids-quotients"},{"id":"stacks:04RK","tag":"04RK","title":"Étale localization of groupoid schemes · Definition 04RK","summary":"Let S be a scheme. Let B be an algebraic space over S Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let u ∈ |U| be a point. • We say R is strongly split over u if there exists an open subspace P ⊂ R such that • (U, P, s|_P, t|_P, c|_P ×_s, U, t P) is a groupoid in algebraic spaces over B, • s|_P, t|_P are finite, and • (r ∈ |R| : s(r) = u, t(r) = u) ⊂ |P|. The choice of such a P will be called a strong splitting of R over u. • We say R is split over u if…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $u \\in |U|$ be a point.\n\\begin{enumerate}\n\\item We say $R$ is {\\it strongly split over $u$} if there exists an open\nsubspace $P \\subset R$ such that\n\\begin{enumerate}\n\\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a\ngroupoid in algebraic spaces over $B$,\n\\item $s|_P$, $t|_P$ are finite, and\n\\item $\\{r \\in |R| : s(r) = u, t(r) = u\\} \\subset |P|$.\n\\end{enumerate}\nThe choice of such a $P$ will be called a\n{\\it strong splitting of $R$ over $u$}.\n\\item We say $R$ is {\\it split over $u$} if there exists an open\nsubspace $P \\subset R$ such that\n\\begin{enumerate}\n\\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a\ngroupoid in algebraic spaces over $B$,\n\\item $s|_P$, $t|_P$ are finite, and\n\\item $\\{g \\in |G| : g\\text{ maps to }u\\} \\subset |P|$ where\n$G \\to U$ is the stabilizer.\n\\end{enumerate}\nThe choice of such a $P$ will be called a\n{\\it splitting of $R$ over $u$}.\n\\item We say $R$ is {\\it quasi-split over $u$} if there exists an open\nsubspace $P \\subset R$ such that\n\\begin{enumerate}\n\\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a\ngroupoid in algebraic spaces over $B$,\n\\item $s|_P$, $t|_P$ are finite, and\n\\item $e(u) \\in |P|$\\footnote{This condition is implied by (a).}.\n\\end{enumerate}\nThe choice of such a $P$ will be called a {\\it quasi-splitting of $R$ over $u$}.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RK","source_file":"spaces-more-groupoids.tex","source_line":2182,"source_end_line":2219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2182-L2219","statement_sha256":"8e03e6a9b68ee535e6b47f6ec824715ca51191ae0a407727ef40eb8f2e415caa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12467,"rank":12467,"depth":0,"x":1128.843,"y":1004.677,"cluster":"groupoids-quotients"},{"id":"stacks:03FM","tag":"03FM","title":"Existence of strong splitting · Lemma 03FM","summary":"In Situation [Tag 04RL] there exists an algebraic space U', an étale morphism U' → U, and a point u' : Spec(kappa(u)) → U' lying over u : Spec(kappa(u)) → U such that the restriction R' = R|_U' of R to U' is strongly split over u'.","statement_latex":"In Situation \\ref{situation-strong-splitting}\nthere exists an algebraic space $U'$, an \\'etale morphism\n$U' \\to U$, and a point $u' : \\Spec(\\kappa(u)) \\to U'$\nlying over $u : \\Spec(\\kappa(u)) \\to U$\nsuch that the restriction $R' = R|_{U'}$ of $R$ to $U'$\nis strongly split over $u'$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03FM","source_file":"spaces-more-groupoids.tex","source_line":2296,"source_end_line":2304,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2296-L2304","statement_sha256":"126e44ff33565d206e0eb6f0286a7b8c1b4840c15bf15ce971eecc3110129117","origin":"The Stacks Project","memory_eligible":false,"source_rank":12468,"rank":12468,"depth":1,"x":1239.877,"y":1239.31,"cluster":"groupoids-quotients"},{"id":"stacks:0DTC","tag":"0DTC","title":"Existence of splitting · Lemma 0DTC","summary":"In Situation [Tag 0DTB] there exists an algebraic space U', an étale morphism U' → U, and a point u' : Spec(kappa(u)) → U' lying over u : Spec(kappa(u)) → U such that the restriction R' = R|_U' of R to U' is split over u'.","statement_latex":"In Situation \\ref{situation-splitting}\nthere exists an algebraic space $U'$, an \\'etale morphism\n$U' \\to U$, and a point $u' : \\Spec(\\kappa(u)) \\to U'$\nlying over $u : \\Spec(\\kappa(u)) \\to U$\nsuch that the restriction $R' = R|_{U'}$ of $R$ to $U'$\nis split over $u'$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTC","source_file":"spaces-more-groupoids.tex","source_line":2365,"source_end_line":2373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2365-L2373","statement_sha256":"6fbb33b6be418c8014d034f73a1186b30acd643bc4ceaf83e8406c955de8b979","origin":"The Stacks Project","memory_eligible":false,"source_rank":12469,"rank":12469,"depth":1,"x":968.822,"y":1129.097,"cluster":"groupoids-quotients"},{"id":"stacks:04RW","tag":"04RW","title":"Existence of quasi-splitting · Lemma 04RW","summary":"In Situation [Tag 04RV] there exists an algebraic space U', an étale morphism U' → U, and a point u' : Spec(kappa(u)) → U' lying over u : Spec(kappa(u)) → U such that the restriction R' = R|_U' of R to U' is quasi-split over u'.","statement_latex":"In Situation \\ref{situation-quasi-splitting}\nthere exists an algebraic space $U'$, an \\'etale morphism\n$U' \\to U$, and a point $u' : \\Spec(\\kappa(u)) \\to U'$\nlying over $u : \\Spec(\\kappa(u)) \\to U$\nsuch that the restriction $R' = R|_{U'}$ of $R$ to $U'$\nis quasi-split over $u'$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RW","source_file":"spaces-more-groupoids.tex","source_line":2425,"source_end_line":2433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2425-L2433","statement_sha256":"835ac9403e76169b77f55854f4023e0113832cdd335733fb856a337e01a6159e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12470,"rank":12470,"depth":1,"x":1257.85,"y":1056.432,"cluster":"groupoids-quotients"},{"id":"stacks:04RX","tag":"04RX","title":"Étale localization of groupoid schemes · Lemma 04RX","summary":"In Situation [Tag 04RL] assume in addition that s, t are flat and locally of finite presentation. Then there exists a scheme U', a separated étale morphism U' → U, and a point u' ∈ U' lying over u with kappa(u) = kappa(u') such that the restriction R' = R|_U' of R to U' is strongly split over u'.","statement_latex":"In Situation \\ref{situation-strong-splitting} assume in addition that\n$s, t$ are flat and locally of finite presentation.\nThen there exists a scheme $U'$, a separated \\'etale morphism\n$U' \\to U$, and a point $u' \\in U'$\nlying over $u$ with $\\kappa(u) = \\kappa(u')$\nsuch that the restriction $R' = R|_{U'}$ of $R$ to $U'$\nis strongly split over $u'$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RX","source_file":"spaces-more-groupoids.tex","source_line":2471,"source_end_line":2480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2471-L2480","statement_sha256":"9947776fb1a6d20b0f0c285ace1bc9ee645e16d088cc183340ec65270bb10664","origin":"The Stacks Project","memory_eligible":false,"source_rank":12471,"rank":12471,"depth":63,"x":1102.88,"y":1274.41,"cluster":"groupoids-quotients"},{"id":"stacks:0DTD","tag":"0DTD","title":"Étale localization of groupoid schemes · Lemma 0DTD","summary":"In Situation [Tag 0DTB] assume in addition that s, t are flat and locally of finite presentation. Then there exists a scheme U', a separated étale morphism U' → U, and a point u' ∈ U' lying over u with kappa(u) = kappa(u') such that the restriction R' = R|_U' of R to U' is split over u'.","statement_latex":"In Situation \\ref{situation-splitting} assume in addition that\n$s, t$ are flat and locally of finite presentation.\nThen there exists a scheme $U'$, a separated \\'etale morphism\n$U' \\to U$, and a point $u' \\in U'$\nlying over $u$ with $\\kappa(u) = \\kappa(u')$\nsuch that the restriction $R' = R|_{U'}$ of $R$ to $U'$\nis split over $u'$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTD","source_file":"spaces-more-groupoids.tex","source_line":2491,"source_end_line":2500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2491-L2500","statement_sha256":"49bceaa90a5feb5a94498956106f3ab99f91187066ad24a6702f98c5660359ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":12472,"rank":12472,"depth":63,"x":1041.75,"y":1025.293,"cluster":"groupoids-quotients"},{"id":"stacks:04RY","tag":"04RY","title":"Étale localization of groupoid schemes · Lemma 04RY","summary":"In Situation [Tag 04RV] assume in addition that s, t are flat and locally of finite presentation. Then there exists a scheme U', a separated étale morphism U' → U, and a point u' ∈ U' lying over u with kappa(u) = kappa(u') such that the restriction R' = R|_U' of R to U' is quasi-split over u'.","statement_latex":"In Situation \\ref{situation-quasi-splitting} assume in addition that\n$s, t$ are flat and locally of finite presentation.\nThen there exists a scheme $U'$, a separated \\'etale morphism\n$U' \\to U$, and a point $u' \\in U'$ lying over $u$ with\n$\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of\n$R$ to $U'$ is quasi-split over $u'$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RY","source_file":"spaces-more-groupoids.tex","source_line":2511,"source_end_line":2519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2511-L2519","statement_sha256":"4201421ef0931ec8004b8b2270eb0f0461f8119734c41fa86327ea554f552838","origin":"The Stacks Project","memory_eligible":false,"source_rank":12473,"rank":12473,"depth":63,"x":1287.602,"y":1174.57,"cluster":"groupoids-quotients"},{"id":"stacks:04RZ","tag":"04RZ","title":"Étale localization of groupoid schemes · Lemma 04RZ","summary":"In Situation [Tag 04RL] assume in addition that s, t are flat and locally of finite presentation and that U is affine. Then there exists an affine scheme U', an étale morphism U' → U, and a point u' ∈ U' lying over u with kappa(u) = kappa(u') such that the restriction R' = R|_U' of R to U' is strongly split over u'.","statement_latex":"In Situation \\ref{situation-strong-splitting} assume in addition that\n$s, t$ are flat and locally of finite presentation and that $U$ is affine.\nThen there exists an affine scheme $U'$, an \\'etale morphism\n$U' \\to U$, and a point $u' \\in U'$ lying over $u$ with\n$\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of\n$R$ to $U'$ is strongly split over $u'$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04RZ","source_file":"spaces-more-groupoids.tex","source_line":2534,"source_end_line":2542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2534-L2542","statement_sha256":"80688d4bad6dd623d4b6224fdd056a8d3a25dd254f07957d93fd65f384545128","origin":"The Stacks Project","memory_eligible":false,"source_rank":12474,"rank":12474,"depth":64,"x":985.728,"y":1204.05,"cluster":"groupoids-quotients"},{"id":"stacks:0DTE","tag":"0DTE","title":"Étale localization of groupoid schemes · Lemma 0DTE","summary":"In Situation [Tag 0DTB] assume in addition that s, t are flat and locally of finite presentation and that U is affine. Then there exists an affine scheme U', an étale morphism U' → U, and a point u' ∈ U' lying over u with kappa(u) = kappa(u') such that the restriction R' = R|_U' of R to U' is split over u'.","statement_latex":"In Situation \\ref{situation-splitting} assume in addition that\n$s, t$ are flat and locally of finite presentation and that $U$ is affine.\nThen there exists an affine scheme $U'$, an \\'etale morphism\n$U' \\to U$, and a point $u' \\in U'$ lying over $u$ with\n$\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of\n$R$ to $U'$ is split over $u'$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTE","source_file":"spaces-more-groupoids.tex","source_line":2589,"source_end_line":2597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2589-L2597","statement_sha256":"255a1c25bcf8ef8a257017495645980066928faf4b5d81391330e8b2986a4933","origin":"The Stacks Project","memory_eligible":false,"source_rank":12475,"rank":12475,"depth":65,"x":1184.975,"y":1010.677,"cluster":"groupoids-quotients"},{"id":"stacks:04S0","tag":"04S0","title":"Étale localization of groupoid schemes · Lemma 04S0","summary":"In Situation [Tag 04RV] assume in addition that s, t are flat and locally of finite presentation and that U is affine. Then there exists an affine scheme U', an étale morphism U' → U, and a point u' ∈ U' lying over u with kappa(u) = kappa(u') such that the restriction R' = R|_U' of R to U' is quasi-split over u'.","statement_latex":"In Situation \\ref{situation-quasi-splitting} assume in addition that\n$s, t$ are flat and locally of finite presentation and that $U$ is affine.\nThen there exists an affine scheme $U'$, an \\'etale morphism\n$U' \\to U$, and a point $u' \\in U'$ lying over $u$ with\n$\\kappa(u) = \\kappa(u')$ such that the restriction $R' = R|_{U'}$ of\n$R$ to $U'$ is quasi-split over $u'$.","area":"Groupoids & Quotients","chapter":"More on Groupoids in Spaces","chapter_id":"spaces-more-groupoids","section":"Étale localization of groupoid schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04S0","source_file":"spaces-more-groupoids.tex","source_line":2609,"source_end_line":2617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-groupoids.tex#L2609-L2617","statement_sha256":"c9bb98702852a4084d6daa8746ab7f276cc6ddba3d00fefa60b9951c06432753","origin":"The Stacks Project","memory_eligible":false,"source_rank":12476,"rank":12476,"depth":65,"x":1193.573,"y":1266.78,"cluster":"groupoids-quotients"},{"id":"stacks:02YQ","tag":"02YQ","title":"Morphisms representable by algebraic spaces · Definition 02YQ","summary":"Let S be a scheme contained in Sch_fppf. Let F, G be presheaves on Sch_fppf/S. We say a morphism a : F → G is representable by algebraic spaces if for every U ∈ Ob((Sch/S)_fppf) and any xi : U → G the fiber product U ×_xi, G F is an algebraic space.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F$, $G$ be presheaves on $\\Sch_{fppf}/S$.\nWe say a morphism $a : F \\to G$ is\n{\\it representable by algebraic spaces}\nif for every $U \\in \\Ob((\\Sch/S)_{fppf})$ and\nany $\\xi : U \\to G$ the fiber product $U \\times_{\\xi, G} F$\nis an algebraic space.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YQ","source_file":"bootstrap.tex","source_line":62,"source_end_line":71,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L62-L71","statement_sha256":"3d23007700dc83817f27f18586c6af4dd7c9ce1e1e4e65280906ced5f0b1d692","origin":"The Stacks Project","memory_eligible":false,"source_rank":12477,"rank":12477,"depth":0,"x":118.181,"y":1491.182,"cluster":"algebraic-spaces"},{"id":"stacks:03BN","tag":"03BN","title":"Morphisms representable by algebraic spaces · Lemma 03BN","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then f is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThen $f$ is representable by algebraic spaces.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03BN","source_file":"bootstrap.tex","source_line":76,"source_end_line":81,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L76-L81","statement_sha256":"48a8a938896812b45169d851441d68bc4155f920c5e9cd7b403e1758ec2e3c9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12478,"rank":12478,"depth":8,"x":400.057,"y":1616.801,"cluster":"algebraic-spaces"},{"id":"stacks:03Y0","tag":"03Y0","title":"Morphisms representable by algebraic spaces · Lemma 03Y0","summary":"A base change of a representable by algebraic spaces morphism of presheaves is representable by algebraic spaces. Let S be a scheme. Let xymatrix G' ×_G F ar[r] ar[d]^a' & F ar[d]^a G' ar[r] & G be a fibre square of presheaves on (Sch/S)_fppf. If a is representable by algebraic spaces so is a'.","statement_latex":"\\begin{slogan}\nA base change of a representable by algebraic spaces morphism of\npresheaves is representable by algebraic spaces.\n\\end{slogan}\nLet $S$ be a scheme. Let\n$$\n\\xymatrix{\nG' \\times_G F \\ar[r] \\ar[d]^{a'} & F \\ar[d]^a \\\\\nG' \\ar[r] & G\n}\n$$\nbe a fibre square of presheaves on $(\\Sch/S)_{fppf}$.\nIf $a$ is representable by algebraic spaces so is $a'$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Y0","source_file":"bootstrap.tex","source_line":89,"source_end_line":104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L89-L104","statement_sha256":"3350adae3b692c876046019cbad0a9b0ca0452dc17391909000815179cc33661","origin":"The Stacks Project","memory_eligible":false,"source_rank":12479,"rank":12479,"depth":0,"x":91.014,"y":1684.152,"cluster":"algebraic-spaces"},{"id":"stacks:02YR","tag":"02YR","title":"Morphisms representable by algebraic spaces · Lemma 02YR","summary":"Let S be a scheme contained in Sch_fppf. Let F, G : (Sch/S)_fppf^opp → Sets. Let a : F → G be representable by algebraic spaces. If G is a sheaf, then so is F.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$ be representable by algebraic spaces.\nIf $G$ is a sheaf, then so is $F$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YR","source_file":"bootstrap.tex","source_line":110,"source_end_line":116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L110-L116","statement_sha256":"7da77099a765370e5592cb584329a8c9c4740e50889430eba6d8f3e1cf112482","origin":"The Stacks Project","memory_eligible":false,"source_rank":12480,"rank":12480,"depth":1,"x":264.832,"y":1459.006,"cluster":"algebraic-spaces"},{"id":"stacks:05LA","tag":"05LA","title":"Morphisms representable by algebraic spaces · Lemma 05LA","summary":"Let S be a scheme contained in Sch_fppf. Let F, G : (Sch/S)_fppf^opp → Sets. Let a : F → G be representable by algebraic spaces. Then Δ_F/G : F → F ×_G F is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$ be representable by algebraic spaces.\nThen $\\Delta_{F/G} : F \\to F \\times_G F$ is representable by\nalgebraic spaces.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LA","source_file":"bootstrap.tex","source_line":134,"source_end_line":141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L134-L141","statement_sha256":"f697622c1fe96917cb5eea84b6ef62f599dea713e5d735f6946d56e6eac0705c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12481,"rank":12481,"depth":1,"x":317.748,"y":1723.801,"cluster":"algebraic-spaces"},{"id":"stacks:02YS","tag":"02YS","title":"Morphisms representable by algebraic spaces · Lemma 02YS","summary":"Let S be a scheme contained in Sch_fppf. Let F, G : (Sch/S)_fppf^opp → Sets. Let a : F → G be representable by algebraic spaces. If G is an algebraic space, then so is F.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$ be representable by algebraic spaces.\nIf $G$ is an algebraic space, then so is $F$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02YS","source_file":"bootstrap.tex","source_line":175,"source_end_line":181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L175-L181","statement_sha256":"75593e47e1c22fe7cff05edc6ee0389b7df5b27452b3b469ced19a12a67abf78","origin":"The Stacks Project","memory_eligible":false,"source_rank":12482,"rank":12482,"depth":52,"x":65.648,"y":1558.478,"cluster":"algebraic-spaces"},{"id":"stacks:03XY","tag":"03XY","title":"Morphisms representable by algebraic spaces · Lemma 03XY","summary":"Let S be a scheme. Let a : F → G be a map of presheaves on (Sch/S)_fppf. Suppose a : F → G is representable by algebraic spaces. If X is an algebraic space over S, and X → G is a map of presheaves then X ×_G F is an algebraic space.","statement_latex":"Let $S$ be a scheme.\nLet $a : F \\to G$ be a map of presheaves on $(\\Sch/S)_{fppf}$.\nSuppose $a : F \\to G$ is representable by algebraic spaces.\nIf $X$ is an algebraic space over $S$, and $X \\to G$ is a map of presheaves\nthen $X \\times_G F$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XY","source_file":"bootstrap.tex","source_line":223,"source_end_line":230,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L223-L230","statement_sha256":"64df21429fbaa9f7d85d7f533ff3a11b9496f399ffe6f5ab017dd45434f9a01d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12483,"rank":12483,"depth":53,"x":384.667,"y":1537.325,"cluster":"algebraic-spaces"},{"id":"stacks:03Y1","tag":"03Y1","title":"Morphisms representable by algebraic spaces · Lemma 03Y1","summary":"Let S be a scheme. Let xymatrix F ar[r]^a & G ar[r]^b & H be maps of presheaves on (Sch/S)_fppf. If a and b are representable by algebraic spaces, so is b ∘ a.","statement_latex":"Let $S$ be a scheme.\nLet\n$$\n\\xymatrix{\nF \\ar[r]^a & G \\ar[r]^b & H\n}\n$$\nbe maps of presheaves on $(\\Sch/S)_{fppf}$.\nIf $a$ and $b$ are representable by algebraic spaces, so is\n$b \\circ a$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Y1","source_file":"bootstrap.tex","source_line":239,"source_end_line":251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L239-L251","statement_sha256":"d6778a7ca1384692c016ff31e936f454e9d0ae8c37aa7b7eea25d57c837c6dda","origin":"The Stacks Project","memory_eligible":false,"source_rank":12484,"rank":12484,"depth":54,"x":166.317,"y":1734.051,"cluster":"algebraic-spaces"},{"id":"stacks:046D","tag":"046D","title":"Morphisms representable by algebraic spaces · Lemma 046D","summary":"Let S be a scheme. Let F_i, G_i : (Sch/S)_fppf^opp → Sets, i = 1, 2. Let a_i : F_i → G_i, i = 1, 2 be representable by algebraic spaces. Then a_1 × a_2 : F_1 × F_2 → G_1 × G_2 is a representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme.\nLet $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$, $i = 1, 2$.\nLet $a_i : F_i \\to G_i$, $i = 1, 2$\nbe representable by algebraic spaces.\nThen\n$$\na_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2\n$$\nis a representable by algebraic spaces.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046D","source_file":"bootstrap.tex","source_line":261,"source_end_line":272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L261-L272","statement_sha256":"be4483f281d9f8b296f715a6c3f7cb638b485919b2948e2a40ebeaf315456c2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12485,"rank":12485,"depth":55,"x":169.125,"y":1464.943,"cluster":"algebraic-spaces"},{"id":"stacks:0AMN","tag":"0AMN","title":"Morphisms representable by algebraic spaces · Lemma 0AMN","summary":"Let S be a scheme. Let a : F → G and b : G → H be transformations of functors (Sch/S)_fppf^opp → Sets. Assume • Δ : G → G ×_H G is representable by algebraic spaces, and • b ∘ a : F → H is representable by algebraic spaces. Then a is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $a : F \\to G$ and $b : G \\to H$ be\ntransformations of functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nAssume\n\\begin{enumerate}\n\\item $\\Delta : G \\to G \\times_H G$ is representable\nby algebraic spaces, and\n\\item $b \\circ a : F \\to H$ is representable by algebraic spaces.\n\\end{enumerate}\nThen $a$ is representable by algebraic spaces.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMN","source_file":"bootstrap.tex","source_line":285,"source_end_line":296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L285-L296","statement_sha256":"28ce7fc65ebfeddeb81f13fdbb14a6a906fc49d2f4e65ddfdc7a39d7c1f5ebc1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12486,"rank":12486,"depth":54,"x":383.582,"y":1665.083,"cluster":"algebraic-spaces"},{"id":"stacks:07WE","tag":"07WE","title":"Morphisms representable by algebraic spaces · Lemma 07WE","summary":"Let S ∈ Ob(Sch_fppf). Let F be a presheaf of sets on (Sch/S)_fppf. Assume • F is a sheaf for the Zariski topology on (Sch/S)_fppf, • there exists an index set I and subfunctors F_i ⊂ F such that • each F_i is an fppf sheaf, • each F_i → F is representable by algebraic spaces, • coprod F_i → F becomes surjective after fppf sheafification. Then F is an fppf sheaf.","statement_latex":"Let $S \\in \\Ob(\\Sch_{fppf})$. Let $F$ be a presheaf of sets on\n$(\\Sch/S)_{fppf}$. Assume\n\\begin{enumerate}\n\\item $F$ is a sheaf for the Zariski topology on $(\\Sch/S)_{fppf}$,\n\\item there exists an index set $I$ and subfunctors $F_i \\subset F$ such that\n\\begin{enumerate}\n\\item each $F_i$ is an fppf sheaf,\n\\item each $F_i \\to F$ is representable by algebraic spaces,\n\\item $\\coprod F_i \\to F$ becomes surjective after fppf sheafification.\n\\end{enumerate}\n\\end{enumerate}\nThen $F$ is an fppf sheaf.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WE","source_file":"bootstrap.tex","source_line":307,"source_end_line":321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L307-L321","statement_sha256":"1cbda551680fe4f5d2b6c0be7d1219023adc3723b9fa4a31297f2b0fc2c7370c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12487,"rank":12487,"depth":18,"x":64.322,"y":1639.177,"cluster":"algebraic-spaces"},{"id":"stacks:03XZ","tag":"03XZ","title":"Properties of maps of presheaves representable by algebraic spaces · Definition 03XZ","summary":"Let S be a scheme. Let a : F → G be a map of presheaves on (Sch/S)_fppf which is representable by algebraic spaces. Let P be a property of morphisms of algebraic spaces which • is preserved under any base change, and • is fppf local on the base, see Descent on Spaces, Definition [Tag 03YH]. In this case we say that a has property P if for every scheme U and xi : U → G the resulting morphism of algebraic spaces U ×_G F → U has property P.","statement_latex":"Let $S$ be a scheme. Let $a : F \\to G$ be a map of presheaves on\n$(\\Sch/S)_{fppf}$ which is representable by algebraic spaces.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces which\n\\begin{enumerate}\n\\item is preserved under any base change, and\n\\item is fppf local on the base, see\nDescent on Spaces,\nDefinition \\ref{spaces-descent-definition-property-morphisms-local}.\n\\end{enumerate}\nIn this case we say that $a$ has {\\it property $\\mathcal{P}$} if for every\nscheme $U$ and $\\xi : U \\to G$ the resulting morphism of algebraic spaces\n$U \\times_G F \\to U$ has property $\\mathcal{P}$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Properties of maps of presheaves representable by algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03XZ","source_file":"bootstrap.tex","source_line":378,"source_end_line":392,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L378-L392","statement_sha256":"f657cf34bebce80c210c5cd1040f1b1b736e6edd93284792cd4e3f2dd88095ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":12488,"rank":12488,"depth":1,"x":320.714,"y":1477.034,"cluster":"algebraic-spaces"},{"id":"stacks:046F","tag":"046F","title":"Properties of maps of presheaves representable by algebraic spaces · Lemma 046F","summary":"Let S be a scheme. Let P be a property as in Definition [Tag 03XZ]. Let xymatrix G' ×_G F ar[r] ar[d]^a' & F ar[d]^a G' ar[r] & G be a fibre square of presheaves on (Sch/S)_fppf. If a is representable by algebraic spaces and has P so does a'.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-property-transformation}.\nLet\n$$\n\\xymatrix{\nG' \\times_G F \\ar[r] \\ar[d]^{a'} & F \\ar[d]^a \\\\\nG' \\ar[r] & G\n}\n$$\nbe a fibre square of presheaves on $(\\Sch/S)_{fppf}$.\nIf $a$ is representable by algebraic spaces and has $\\mathcal{P}$\nso does $a'$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Properties of maps of presheaves representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046F","source_file":"bootstrap.tex","source_line":509,"source_end_line":524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L509-L524","statement_sha256":"400193365fd6b1a549879899489edf4a7034d6211d42f298ac4cc08072e5cc05","origin":"The Stacks Project","memory_eligible":false,"source_rank":12489,"rank":12489,"depth":2,"x":262.012,"y":1742.225,"cluster":"algebraic-spaces"},{"id":"stacks:046G","tag":"046G","title":"Properties of maps of presheaves representable by algebraic spaces · Lemma 046G","summary":"Let S be a scheme. Let P be a property as in Definition [Tag 03XZ], and assume P is stable under composition. Let xymatrix F ar[r]^a & G ar[r]^b & H be maps of presheaves on (Sch/S)_fppf. If a, b are representable by algebraic spaces and has P so does b ∘ a.","statement_latex":"Let $S$ be a scheme.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-property-transformation},\nand assume $\\mathcal{P}$ is stable under composition.\nLet\n$$\n\\xymatrix{\nF \\ar[r]^a & G \\ar[r]^b & H\n}\n$$\nbe maps of presheaves on $(\\Sch/S)_{fppf}$.\nIf $a$, $b$ are representable by algebraic spaces and has\n$\\mathcal{P}$ so does $b \\circ a$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Properties of maps of presheaves representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046G","source_file":"bootstrap.tex","source_line":530,"source_end_line":545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L530-L545","statement_sha256":"139ed1c017e6fe9369e12a2362e226598e2ad8fdcfa65b9e6ef5af2421279395","origin":"The Stacks Project","memory_eligible":false,"source_rank":12490,"rank":12490,"depth":55,"x":91.946,"y":1513.243,"cluster":"algebraic-spaces"},{"id":"stacks:046H","tag":"046H","title":"Properties of maps of presheaves representable by algebraic spaces · Lemma 046H","summary":"Let S be a scheme. Let F_i, G_i : (Sch/S)_fppf^opp → Sets, i = 1, 2. Let a_i : F_i → G_i, i = 1, 2 be representable by algebraic spaces. Let P be a property as in Definition [Tag 03XZ] which is stable under composition. If a_1 and a_2 have property P so does a_1 × a_2 : F_1 × F_2 → G_1 × G_2.","statement_latex":"Let $S$ be a scheme.\nLet $F_i, G_i : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$,\n$i = 1, 2$.\nLet $a_i : F_i \\to G_i$, $i = 1, 2$ be representable by algebraic spaces.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-property-transformation}\nwhich is stable under composition.\nIf $a_1$ and $a_2$ have property $\\mathcal{P}$ so does\n$a_1 \\times a_2 : F_1 \\times F_2 \\longrightarrow G_1 \\times G_2$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Properties of maps of presheaves representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046H","source_file":"bootstrap.tex","source_line":553,"source_end_line":564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L553-L564","statement_sha256":"b2f2d61cbc29d1fd26943c099f42aeb211351b0d9a171f0833926addb80ea7dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12491,"rank":12491,"depth":56,"x":401.661,"y":1585.63,"cluster":"algebraic-spaces"},{"id":"stacks:0AM1","tag":"0AM1","title":"Properties of maps of presheaves representable by algebraic spaces · Lemma 0AM1","summary":"Let S be a scheme. Let F, G : (Sch/S)_fppf^opp → Sets. Let a : F → G be a transformation of functors representable by algebraic spaces. Let P, P' be properties as in Definition [Tag 03XZ]. Suppose that for any morphism f : X → Y of algebraic spaces over S we have P(f) ⇒ P'(f). If a has property P, then a has property P'.","statement_latex":"Let $S$ be a scheme.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nLet $a : F \\to G$ be a transformation of functors representable by\nalgebraic spaces.\nLet $\\mathcal{P}$, $\\mathcal{P}'$ be properties as in\nDefinition \\ref{definition-property-transformation}.\nSuppose that for any morphism $f : X \\to Y$ of algebraic spaces over $S$\nwe have $\\mathcal{P}(f) \\Rightarrow \\mathcal{P}'(f)$.\nIf $a$ has property $\\mathcal{P}$, then\n$a$ has property $\\mathcal{P}'$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Properties of maps of presheaves representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AM1","source_file":"bootstrap.tex","source_line":572,"source_end_line":584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L572-L584","statement_sha256":"7c285d3d7396df6d882c359c87956af538cdf370bbb67ba721abd7207dd9ac37","origin":"The Stacks Project","memory_eligible":false,"source_rank":12492,"rank":12492,"depth":2,"x":114.912,"y":1708.06,"cluster":"algebraic-spaces"},{"id":"stacks:04S1","tag":"04S1","title":"Properties of maps of presheaves representable by algebraic spaces · Lemma 04S1","summary":"Let S be a scheme. Let F, G : (Sch/S)_fppf^opp → Sets be sheaves. Let a : F → G be representable by algebraic spaces, flat, locally of finite presentation, and surjective. Then a : F → G is surjective as a map of sheaves.","statement_latex":"Let $S$ be a scheme.\nLet $F, G : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be sheaves.\nLet $a : F \\to G$ be representable by algebraic spaces, flat,\nlocally of finite presentation, and surjective.\nThen $a : F \\to G$ is surjective as a map of sheaves.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Properties of maps of presheaves representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04S1","source_file":"bootstrap.tex","source_line":590,"source_end_line":597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L590-L597","statement_sha256":"98b6a572ab0263b0c0be02c9efc428bca6defb029e65a08f9a63c155f47462cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12493,"rank":12493,"depth":1,"x":227.964,"y":1454.935,"cluster":"algebraic-spaces"},{"id":"stacks:03Y2","tag":"03Y2","title":"Bootstrapping the diagonal · Lemma 03Y2","summary":"The diagonal of a presheaf is representable by algebraic spaces if and only if every map from a scheme to the presheaf is representable by algebraic spaces. Let S be a scheme. If F is a presheaf on (Sch/S)_fppf. The following are equivalent: • Δ_F : F → F × F is representable by algebraic spaces, • for every scheme T any map T → F is representable by algebraic spaces, and • for every algebraic space X any map X → F is representable by algebraic spaces.","statement_latex":"\\begin{slogan}\nThe diagonal of a presheaf is representable by algebraic spaces if and only if\nevery map from a scheme to the presheaf is representable by algebraic spaces.\n\\end{slogan}\nLet $S$ be a scheme.\nIf $F$ is a presheaf on $(\\Sch/S)_{fppf}$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\Delta_F : F \\to F \\times F$ is representable by algebraic spaces,\n\\item for every scheme $T$ any map $T \\to F$ is representable by algebraic\nspaces, and\n\\item for every algebraic space $X$ any map $X \\to F$ is representable\nby algebraic spaces.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Bootstrapping the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Y2","source_file":"bootstrap.tex","source_line":629,"source_end_line":645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L629-L645","statement_sha256":"4623ea3da2992c118af3f8b0ceffa4caf658a3c1d28cb880459d1c409fe8eadd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12494,"rank":12494,"depth":54,"x":348.223,"y":1705.871,"cluster":"algebraic-spaces"},{"id":"stacks:046J","tag":"046J","title":"Bootstrapping the diagonal · Lemma 046J","summary":"Let S be a scheme. Let xymatrix E ar[r]_a ar[d]_f & F ar[d]^g H ar[r]^b & G be a cartesian diagram of sheaves on (Sch/S)_fppf, so E = H ×_G F. If • g is representable by algebraic spaces, surjective, flat, and locally of finite presentation, and • a is representable by algebraic spaces, separated, and locally quasi-finite then b is representable (by schemes) as well as separated and locally quasi-finite.","statement_latex":"Let $S$ be a scheme.\nLet\n$$\n\\xymatrix{\nE \\ar[r]_a \\ar[d]_f & F \\ar[d]^g \\\\\nH \\ar[r]^b & G\n}\n$$\nbe a cartesian diagram of sheaves on $(\\Sch/S)_{fppf}$, so\n$E = H \\times_G F$. If\n\\begin{enumerate}\n\\item $g$ is representable by algebraic spaces, surjective, flat, and\nlocally of finite presentation, and\n\\item $a$ is representable by algebraic spaces, separated, and\nlocally quasi-finite\n\\end{enumerate}\nthen $b$ is representable (by schemes) as well as separated and\nlocally quasi-finite.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Bootstrapping the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046J","source_file":"bootstrap.tex","source_line":677,"source_end_line":697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L677-L697","statement_sha256":"a0a433f2fae560f47aacbdfe14529813703529d292f5aa41ff2a8f7a80e96193","origin":"The Stacks Project","memory_eligible":false,"source_rank":12495,"rank":12495,"depth":55,"x":57.592,"y":1589.009,"cluster":"algebraic-spaces"},{"id":"stacks:046K","tag":"046K","title":"Bootstrapping the diagonal · Lemma 046K","summary":"Let S be a scheme. Let F : (Sch/S)_fppf^opp → Sets be a functor. Assume that • the presheaf F is a sheaf, • there exists an algebraic space X and a map X → F which is representable by algebraic spaces, surjective, flat and locally of finite presentation. Then Δ_F is representable (by schemes).","statement_latex":"Let $S$ be a scheme.\nLet $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item the presheaf $F$ is a sheaf,\n\\item there exists an algebraic space $X$ and a map $X \\to F$\nwhich is representable by algebraic spaces, surjective, flat and\nlocally of finite presentation.\n\\end{enumerate}\nThen $\\Delta_F$ is representable (by schemes).","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Bootstrapping the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046K","source_file":"bootstrap.tex","source_line":746,"source_end_line":758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L746-L758","statement_sha256":"52ecac09507cb97bb2d16212fcc0c317852152f3e9ceab9038dda63dd2e9fa1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12496,"rank":12496,"depth":57,"x":366.044,"y":1510.227,"cluster":"algebraic-spaces"},{"id":"stacks:0AHV","tag":"0AHV","title":"Bootstrapping the diagonal · Lemma 0AHV","summary":"Let S be a scheme. Let F : (Sch/S)_fppf^opp → Sets be a functor. Let X be a scheme and let X → F be representable by algebraic spaces and locally quasi-finite. Then X → F is representable (by schemes).","statement_latex":"Let $S$ be a scheme. Let $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a\nfunctor. Let $X$ be a scheme and let $X \\to F$ be representable by algebraic\nspaces and locally quasi-finite. Then $X \\to F$ is representable\n(by schemes).","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Bootstrapping the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHV","source_file":"bootstrap.tex","source_line":829,"source_end_line":835,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L829-L835","statement_sha256":"9ad5da50e307cd57c8134c575cda05950ad98187632703569958dee9cae0e213","origin":"The Stacks Project","memory_eligible":false,"source_rank":12497,"rank":12497,"depth":55,"x":201.86,"y":1743.47,"cluster":"algebraic-spaces"},{"id":"stacks:03Y3","tag":"03Y3","title":"Bootstrap · Theorem 03Y3","summary":"Let S be a scheme. Let F : (Sch/S)_fppf^opp → Sets be a functor. Assume that • the presheaf F is a sheaf, • the diagonal morphism F → F × F is representable by algebraic spaces, and • there exists an algebraic space X and a map X → F which is surjective, and étale. or assume that • [(a)] the presheaf F is a sheaf, and • [(b)] there exists an algebraic space X and a map X → F which is representable by algebraic spaces, surjective, and étale. Then F is an algebraic space.","statement_latex":"Let $S$ be a scheme.\nLet $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.\nAssume that\n\\begin{enumerate}\n\\item the presheaf $F$ is a sheaf,\n\\item the diagonal morphism $F  \\to F \\times F$ is representable by\nalgebraic spaces, and\n\\item there exists an algebraic space $X$\nand a map $X \\to F$ which is surjective, and \\'etale.\n\\end{enumerate}\nor assume that\n\\begin{enumerate}\n\\item[(a)] the presheaf $F$ is a sheaf, and\n\\item[(b)] there exists an algebraic space $X$ and a map $X \\to F$\nwhich is representable by algebraic spaces, surjective, and \\'etale.\n\\end{enumerate}\nThen $F$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Bootstrap","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03Y3","source_file":"bootstrap.tex","source_line":902,"source_end_line":921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L902-L921","statement_sha256":"7516235a75d39ca405a1b2765df0ab3c6d95a58928d49d1a31d74aa36873b783","origin":"The Stacks Project","memory_eligible":false,"source_rank":12498,"rank":12498,"depth":58,"x":135.324,"y":1478.174,"cluster":"algebraic-spaces"},{"id":"stacks:046M","tag":"046M","title":"Finding opens · Lemma 046M","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' → U be a morphism. Assume • the composition xymatrix U' ×_g, U, t R ar[r]_-pr_1 ar@/^3ex/[rr]^h & R ar[r]_s & U has an open image W ⊂ U, and • the resulting map h : U' ×_g, U, t R → W defines a surjection of sheaves in the fppf topology. Let R' = R|_U' be the restriction of R to U'. Then the map of quotient sheaves U'/R' → U/R in the fppf topology is representable, and is an open immersion.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $g : U' \\to U$ be a morphism.\nAssume\n\\begin{enumerate}\n\\item the composition\n$$\n\\xymatrix{\nU' \\times_{g, U, t} R \\ar[r]_-{\\text{pr}_1} \\ar@/^3ex/[rr]^h\n& R \\ar[r]_s & U\n}\n$$\nhas an open image $W \\subset U$, and\n\\item the resulting map $h : U' \\times_{g, U, t} R \\to W$\ndefines a surjection of sheaves in the fppf topology.\n\\end{enumerate}\nLet $R' = R|_{U'}$ be the restriction of $R$ to $U'$. Then the map\nof quotient sheaves\n$$\nU'/R' \\to U/R\n$$\nin the fppf topology is representable, and is an open immersion.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Finding opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/046M","source_file":"bootstrap.tex","source_line":989,"source_end_line":1013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L989-L1013","statement_sha256":"0c7a895e7d27fc9285d73995b29d592f5bcb6b00cc7a3687403cb63a49c11a3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12499,"rank":12499,"depth":5,"x":397.873,"y":1636.131,"cluster":"algebraic-spaces"},{"id":"stacks:0489","tag":"0489","title":"Slicing equivalence relations · Lemma 0489","summary":"Let S be a scheme. Let j : R → U ×_S U be an equivalence relation on schemes over S. Assume s, t : R → U are flat and locally of finite presentation. Then there exists an equivalence relation j' : R' → U'×_S U' on schemes over S, and an isomorphism U'/R' → U/R induced by a morphism U' → U which maps R' into R such that s', t' : R → U are flat, locally of finite presentation and locally quasi-finite.","statement_latex":"Let $S$ be a scheme.\nLet $j : R \\to U \\times_S U$ be an equivalence relation on schemes over $S$.\nAssume $s, t : R \\to U$ are flat and locally of finite presentation.\nThen there exists an equivalence relation $j' : R' \\to U'\\times_S U'$\non schemes over $S$, and an isomorphism\n$$\nU'/R' \\longrightarrow U/R\n$$\ninduced by a morphism $U' \\to U$ which maps $R'$ into $R$ such that\n$s', t' : R \\to U$ are flat, locally of finite presentation\nand locally quasi-finite.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Slicing equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0489","source_file":"bootstrap.tex","source_line":1120,"source_end_line":1133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1120-L1133","statement_sha256":"b6081e869ee256105eb3004d0632ec699ccfe3b28dcdd5064c9f5f6c9364cc34","origin":"The Stacks Project","memory_eligible":false,"source_rank":12500,"rank":12500,"depth":45,"x":77.075,"y":1668.65,"cluster":"algebraic-spaces"},{"id":"stacks:04S4","tag":"04S4","title":"Quotient by a subgroupoid · Lemma 04S4","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let P → R be monomorphism of schemes. Assume that • (U, P, s|_P, t|_P, c|_P ×_s, U, tP) is a groupoid scheme, • s|_P, t|_P : P → U are finite locally free, • j|_P : P → U ×_S U is a monomorphism. • U is affine, and • j : R → U ×_S U is separated and locally quasi-finite, Then U/P is representable by an affine scheme overlineU, the quotient morphism U → overlineU is finite locally free, and P = U…","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid scheme over $S$.\nLet $P \\to R$ be monomorphism of schemes. Assume that\n\\begin{enumerate}\n\\item $(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t}P})$ is a groupoid scheme,\n\\item $s|_P, t|_P : P \\to U$ are finite locally free,\n\\item $j|_P : P \\to U \\times_S U$ is a monomorphism.\n\\item $U$ is affine, and\n\\item $j : R \\to U \\times_S U$ is separated and locally quasi-finite,\n\\end{enumerate}\nThen $U/P$ is representable by an affine scheme $\\overline{U}$, the\nquotient morphism $U \\to \\overline{U}$ is finite locally free, and\n$P = U \\times_{\\overline{U}} U$. Moreover, $R$ is the restriction of a\ngroupoid scheme\n$(\\overline{U}, \\overline{R}, \\overline{s}, \\overline{t}, \\overline{c})$\non $\\overline{U}$ via the quotient morphism $U \\to \\overline{U}$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Quotient by a subgroupoid","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04S4","source_file":"bootstrap.tex","source_line":1300,"source_end_line":1318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1300-L1318","statement_sha256":"5b8960bd30410ebb2359f78001d94d2f30fd5e3a68383f3f6715e8390accbb44","origin":"The Stacks Project","memory_eligible":false,"source_rank":12501,"rank":12501,"depth":50,"x":287.589,"y":1462.53,"cluster":"algebraic-spaces"},{"id":"stacks:04S6","tag":"04S6","title":"Final bootstrap · Theorem 04S6","summary":"Let S be a scheme. Let F : (Sch/S)_fppf^opp → Sets be a functor. Any one of the following conditions implies that F is an algebraic space: • F = U/R where (U, R, s, t, c) is a groupoid in algebraic spaces over S such that s, t are flat and locally of finite presentation, and j = (t, s) : R → U ×_S U is an equivalence relation, • F = U/R where (U, R, s, t, c) is a groupoid scheme over S such that s, t are flat and locally of finite presentation, and j = (t, s) : R → U ×_S…","statement_latex":"Let $S$ be a scheme.\nLet $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.\nAny one of the following conditions implies that $F$ is an algebraic space:\n\\begin{enumerate}\n\\item $F = U/R$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces\nover $S$ such that $s, t$ are flat and locally of finite presentation, and\n$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation,\n\\item $F = U/R$ where $(U, R, s, t, c)$ is a groupoid scheme\nover $S$ such that $s, t$ are flat and locally of finite presentation, and\n$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation,\n\\item $F$ is a sheaf and there exists an algebraic space $U$ and a morphism\n$U \\to F$ which is representable by algebraic spaces,\nsurjective, flat and locally of finite presentation,\n\\item $F$ is a sheaf and there exists a scheme $U$ and a morphism\n$U \\to F$ which is representable by algebraic spaces or schemes,\nsurjective, flat and locally of finite presentation,\n\\item $F$ is a sheaf, $\\Delta_F$ is representable by algebraic spaces,\nand there exists an algebraic space $U$ and a morphism $U \\to F$ which is\nsurjective, flat, and locally of finite presentation, or\n\\item $F$ is a sheaf, $\\Delta_F$ is representable,\nand there exists a scheme $U$ and a morphism $U \\to F$ which is\nsurjective, flat, and locally of finite presentation.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Final bootstrap","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04S6","source_file":"bootstrap.tex","source_line":1445,"source_end_line":1470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1445-L1470","statement_sha256":"0cf71e1af87a65aa0175f689e5191b6955cd26b1c99e9577a8fea2d5e5ed373b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12502,"rank":12502,"depth":66,"x":298.121,"y":1734.118,"cluster":"algebraic-spaces"},{"id":"stacks:04SK","tag":"04SK","title":"Applications · Lemma 04SK","summary":"The definition of an algebraic space is fppf local. Let S be a scheme. Let F : (Sch/S)_fppf^opp → Sets be a functor. Let (S_i → S)_i ∈ I be a covering of (Sch/S)_fppf. Assume that • F is a sheaf, • each F_i = h_S_i × F is an algebraic space, and • coprod_i ∈ I F_i is an algebraic space (see Spaces, Lemma [Tag 02WQ]). Then F is an algebraic space.","statement_latex":"\\begin{slogan}\nThe definition of an algebraic space is fppf local.\n\\end{slogan}\nLet $S$ be a scheme.\nLet $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.\nLet $\\{S_i \\to S\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$.\nAssume that\n\\begin{enumerate}\n\\item $F$ is a sheaf,\n\\item each $F_i = h_{S_i} \\times F$ is an algebraic space, and\n\\item $\\coprod_{i \\in I} F_i$ is an algebraic space (see\nSpaces, Lemma \\ref{spaces-lemma-coproduct-algebraic-spaces}).\n\\end{enumerate}\nThen $F$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SK","source_file":"bootstrap.tex","source_line":1673,"source_end_line":1689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1673-L1689","statement_sha256":"aeef4ef0ad6438e6513446937b248e15eb0005f441a2dcc4688ba59b78095bc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12503,"rank":12503,"depth":67,"x":71.829,"y":1539.724,"cluster":"algebraic-spaces"},{"id":"stacks:04U0","tag":"04U0","title":"Applications · Lemma 04U0","summary":"Let S be a scheme. Let F : (Sch/S)_fppf^opp → Sets be a functor. Let (S_i → S)_i ∈ I be a covering of (Sch/S)_fppf. Assume that • F is a sheaf, • each F_i = h_S_i × F is an algebraic space, and • the morphisms F_i → S_i are of finite type. Then F is an algebraic space.","statement_latex":"Let $S$ be a scheme.\nLet $F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor.\nLet $\\{S_i \\to S\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$.\nAssume that\n\\begin{enumerate}\n\\item $F$ is a sheaf,\n\\item each $F_i = h_{S_i} \\times F$ is an algebraic space, and\n\\item the morphisms $F_i \\to S_i$ are of finite type.\n\\end{enumerate}\nThen $F$ is an algebraic space.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04U0","source_file":"bootstrap.tex","source_line":1706,"source_end_line":1718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1706-L1718","statement_sha256":"6b7019580f888d830ce97aa097a660cca68258e2ed727444854847882c3fc05b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12504,"rank":12504,"depth":68,"x":395.194,"y":1554.672,"cluster":"algebraic-spaces"},{"id":"stacks:0ADV","tag":"0ADV","title":"Applications · Lemma 0ADV","summary":"Fppf descent data for algebraic spaces are effective. Let S be a scheme. Let (X_i → X)_i ∈ I be an fppf covering of algebraic spaces over S. • If I is countable, then any descent datum for algebraic spaces relative to (X_i → X) is effective. • Any descent datum (Y_i, φ_ij) relative to (X_i → X)_i ∈ I (Descent on Spaces, Definition [Tag 0ADI]) with Y_i → X_i of finite type is effective.","statement_latex":"\\begin{slogan}\nFppf descent data for algebraic spaces are effective.\n\\end{slogan}\nLet $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf\ncovering of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $I$ is countable\\footnote{The restriction on countablility can be\nignored by those who do not care about set theoretical issues. We can allow\nlarger index sets here if we can bound the size of the algebraic spaces\nwhich we are descending. See for example\nLemma \\ref{lemma-locally-algebraic-space-finite-type}.}, then any\ndescent datum for algebraic spaces relative to $\\{X_i \\to X\\}$ is effective.\n\\item Any descent datum $(Y_i, \\varphi_{ij})$ relative to\n$\\{X_i \\to X\\}_{i \\in I}$ (Descent on Spaces, Definition\n\\ref{spaces-descent-definition-descent-datum-for-family-of-morphisms})\nwith $Y_i \\to X_i$ of finite type\nis effective.\n\\end{enumerate}","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ADV","source_file":"bootstrap.tex","source_line":1777,"source_end_line":1797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1777-L1797","statement_sha256":"ce10bfe152ce444f9e564ce9588b0774d1c4c757955ec138a46370419fa1ee35","origin":"The Stacks Project","memory_eligible":false,"source_rank":12505,"rank":12505,"depth":69,"x":144.594,"y":1727.228,"cluster":"algebraic-spaces"},{"id":"stacks:0AMP","tag":"0AMP","title":"Applications · Lemma 0AMP","summary":"Let S be a scheme. Let a : F → G and b : G → H be transformations of functors (Sch/S)_fppf^opp → Sets. Assume • F, G, H are sheaves, • a : F → G is representable by algebraic spaces, flat, locally of finite presentation, and surjective, and • b ∘ a : F → H is representable by algebraic spaces. Then b is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $a : F \\to G$ and $b : G \\to H$ be\ntransformations of functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nAssume\n\\begin{enumerate}\n\\item $F, G, H$ are sheaves,\n\\item $a : F \\to G$ is representable by algebraic spaces, flat,\nlocally of finite presentation, and surjective, and\n\\item $b \\circ a : F \\to H$ is representable by algebraic spaces.\n\\end{enumerate}\nThen $b$ is representable by algebraic spaces.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMP","source_file":"bootstrap.tex","source_line":1847,"source_end_line":1859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1847-L1859","statement_sha256":"407b6d46f1ae91435d5930f84d731ab07920ae32188cacbde4b3811e2fe1b2e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12506,"rank":12506,"depth":67,"x":190.643,"y":1457.646,"cluster":"algebraic-spaces"},{"id":"stacks:04TB","tag":"04TB","title":"Applications · Lemma 04TB","summary":"Assume B → S and (U, R, s, t, c) are as in Groupoids in Spaces, Definition [Tag 044Q] (1). For any scheme T over S and objects x, y of [U/R] over T the sheaf mathitIsom(x, y) on (Sch/T)_fppf is an algebraic space.","statement_latex":"Assume $B \\to S$ and $(U, R, s, t, c)$ are as in\nGroupoids in Spaces,\nDefinition \\ref{spaces-groupoids-definition-quotient-stack} (1).\nFor any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$\nthe sheaf $\\mathit{Isom}(x, y)$ on $(\\Sch/T)_{fppf}$\nis an algebraic space.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TB","source_file":"bootstrap.tex","source_line":1872,"source_end_line":1880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1872-L1880","statement_sha256":"1f8efa6c49d081327ddefce8c008706030730713ab03309a718c455030fed8b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12507,"rank":12507,"depth":68,"x":373.576,"y":1682.682,"cluster":"algebraic-spaces"},{"id":"stacks:06PG","tag":"06PG","title":"Applications · Lemma 06PG","summary":"Let S be a scheme. Consider an algebraic space F of the form F = U/R where (U, R, s, t, c) is a groupoid in algebraic spaces over S such that s, t are flat and locally of finite presentation, and j = (t, s) : R → U ×_S U is an equivalence relation. Then U → F is surjective, flat, and locally of finite presentation.","statement_latex":"Let $S$ be a scheme. Consider an algebraic space $F$ of the form $F = U/R$\nwhere $(U, R, s, t, c)$ is a groupoid in algebraic spaces\nover $S$ such that $s, t$ are flat and locally of finite presentation, and\n$j = (t, s) : R \\to U \\times_S U$ is an equivalence relation.\nThen $U \\to F$ is surjective, flat, and locally of finite presentation.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PG","source_file":"bootstrap.tex","source_line":1949,"source_end_line":1956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1949-L1956","statement_sha256":"832a6d7d01d9b8df2e9aabe0ffc2465eb7185aef667b9d626556a89a30c35916","origin":"The Stacks Project","memory_eligible":false,"source_rank":12508,"rank":12508,"depth":39,"x":57.546,"y":1620.511,"cluster":"algebraic-spaces"},{"id":"stacks:06PH","tag":"06PH","title":"Applications · Lemma 06PH","summary":"Let S be a scheme. Let X → B be a morphism of algebraic spaces over S. Let G be a group algebraic space over B and let a : G ×_B X → X be an action of G on X over B. If • a is a free action, and • G → B is flat and locally of finite presentation, then X/G (see Groupoids in Spaces, Definition [Tag 044J]) is an algebraic space, the morphism X → X/G is surjective, flat, and locally of finite presentation, and X is an fppf G-torsor over X/G.","statement_latex":"Let $S$ be a scheme. Let $X \\to B$ be a morphism of algebraic spaces over\n$S$. Let $G$ be a group algebraic space over $B$ and let\n$a : G \\times_B X \\to X$ be an action of $G$ on $X$ over $B$.\nIf\n\\begin{enumerate}\n\\item $a$ is a free action, and\n\\item $G \\to B$ is flat and locally of finite presentation,\n\\end{enumerate}\nthen $X/G$ (see\nGroupoids in Spaces, Definition\n\\ref{spaces-groupoids-definition-quotient-sheaf})\nis an algebraic space, the morphism $X \\to X/G$ is\nsurjective, flat, and locally of finite presentation, and\n$X$ is an fppf $G$-torsor over $X/G$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PH","source_file":"bootstrap.tex","source_line":1984,"source_end_line":2000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L1984-L2000","statement_sha256":"6d3f32eed8c0131c13560e94c3567c9b3e7960bd6398b35869dfc0aed11037e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12509,"rank":12509,"depth":67,"x":340.73,"y":1486.96,"cluster":"algebraic-spaces"},{"id":"stacks:04U1","tag":"04U1","title":"Applications · Lemma 04U1","summary":"Let (S_i → S)_i ∈ I be a covering of (Sch/S)_fppf. Let G be a group algebraic space over S, and denote G_i = G_S_i the base changes. Suppose given • for each i ∈ I an fppf G_i-torsor X_i over S_i, and • for each i, j ∈ I a G_S_i ×_S S_j-equivariant isomorphism φ_ij : X_i ×_S S_j → S_i ×_S X_j satisfying the cocycle condition over every S_i ×_S S_j ×_S S_j. Then there exists an fppf G-torsor X over S whose base change to S_i is isomorphic to X_i such that we recover the…","statement_latex":"Let $\\{S_i \\to S\\}_{i \\in I}$ be a covering of $(\\Sch/S)_{fppf}$.\nLet $G$ be a group algebraic space over $S$, and denote\n$G_i = G_{S_i}$ the base changes. Suppose given\n\\begin{enumerate}\n\\item for each $i \\in I$ an fppf $G_i$-torsor $X_i$ over $S_i$,\nand\n\\item for each $i, j \\in I$ a $G_{S_i \\times_S S_j}$-equivariant isomorphism\n$\\varphi_{ij} : X_i \\times_S S_j \\to S_i \\times_S X_j$ satisfying the cocycle\ncondition over every $S_i \\times_S S_j \\times_S S_j$.\n\\end{enumerate}\nThen there exists an fppf $G$-torsor $X$ over $S$\nwhose base change to $S_i$ is isomorphic to $X_i$ such that we\nrecover the descent datum $\\varphi_{ij}$.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04U1","source_file":"bootstrap.tex","source_line":2026,"source_end_line":2041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L2026-L2041","statement_sha256":"bf287c12cc6064c1cf83827209c8fd5452f557590654eb1710be32ecec8fe6d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12510,"rank":12510,"depth":68,"x":239.258,"y":1746.263,"cluster":"algebraic-spaces"},{"id":"stacks:076M","tag":"076M","title":"Algebraic spaces in the étale topology · Lemma 076M","summary":"Denote the common underlying category of Sch_fppf and Sch_etale by Sch_α (see Topologies, Remark [Tag 03FF]). Let S be an object of Sch_α. Let F : (Sch_α/S)^opp → Sets be a presheaf with the following properties: • F is a sheaf for the étale topology, • the diagonal Δ : F → F × F is representable, and • there exists U ∈ Ob(Sch_α/S) and U → F which is surjective and étale. Then F is an algebraic space in the sense of Algebraic Spaces, Definition [Tag 025Y].","statement_latex":"Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by\n$\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}).\nLet $S$ be an object of $\\Sch_\\alpha$. Let\n$$\nF : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets}\n$$\nbe a presheaf with the following properties:\n\\begin{enumerate}\n\\item $F$ is a sheaf for the \\'etale topology,\n\\item the diagonal $\\Delta : F \\to F \\times F$ is representable, and\n\\item there exists $U \\in \\Ob(\\Sch_\\alpha/S)$\nand $U \\to F$ which is surjective and \\'etale.\n\\end{enumerate}\nThen $F$ is an algebraic space in the sense of\nAlgebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Algebraic spaces in the étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/076M","source_file":"bootstrap.tex","source_line":2118,"source_end_line":2135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L2118-L2135","statement_sha256":"1ebf20fd6f6368d4febc980448f30ea4429c336f38173a7d68ff828e1b8bf2ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":12511,"rank":12511,"depth":52,"x":105.486,"y":1497.346,"cluster":"algebraic-spaces"},{"id":"stacks:0BH4","tag":"0BH4","title":"Algebraic spaces in the étale topology · Lemma 0BH4","summary":"Denote the common underlying category of Sch_fppf and Sch_etale by Sch_α (see Topologies, Remark [Tag 03FF]). Let S be an object of Sch_α. Let F : (Sch_α/S)^opp → Sets be a presheaf with the following properties: • F is a sheaf for the étale topology, • there exists an algebraic space U over S and a map U → F which is representable by algebraic spaces, surjective, and étale. Then F is an algebraic space in the sense of Algebraic Spaces, Definition [Tag 025Y].","statement_latex":"Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by\n$\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}).\nLet $S$ be an object of $\\Sch_\\alpha$. Let\n$$\nF : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets}\n$$\nbe a presheaf with the following properties:\n\\begin{enumerate}\n\\item $F$ is a sheaf for the \\'etale topology,\n\\item there exists an algebraic space $U$ over $S$\nand a map $U \\to F$ which is representable by\nalgebraic spaces, surjective, and \\'etale.\n\\end{enumerate}\nThen $F$ is an algebraic space in the sense of\nAlgebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Algebraic spaces in the étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BH4","source_file":"bootstrap.tex","source_line":2184,"source_end_line":2201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L2184-L2201","statement_sha256":"84021bd695511e44f6150b815ccdfce1f08e89cf83f4d34ba1ef95b21171b497","origin":"The Stacks Project","memory_eligible":false,"source_rank":12512,"rank":12512,"depth":67,"x":404.46,"y":1605.046,"cluster":"algebraic-spaces"},{"id":"stacks:07WF","tag":"07WF","title":"Algebraic spaces in the étale topology · Lemma 07WF","summary":"Denote the common underlying category of Sch_fppf and Sch_etale by Sch_α (see Topologies, Remark [Tag 03FF]). Let S be an object of Sch_α. F : (Sch_α/S)^opp → Sets be a presheaf with the following properties: • F is a sheaf for the étale topology, • the diagonal Δ : F → F × F is representable by algebraic spaces, and • there exists U ∈ Ob(Sch_α/S) and U → F which is surjective and smooth. Then F is an algebraic space in the sense of Algebraic Spaces, Definition [Tag 025Y].","statement_latex":"Denote the common underlying category of $\\Sch_{fppf}$\nand $\\Sch_\\etale$ by $\\Sch_\\alpha$ (see\nTopologies, Remark \\ref{topologies-remark-choice-sites}). Let $S$ be an object\nof $\\Sch_\\alpha$. \n$$\nF : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets}\n$$\nbe a presheaf with the following properties:\n\\begin{enumerate}\n\\item $F$ is a sheaf for the \\'etale topology,\n\\item the diagonal $\\Delta : F \\to F \\times F$ is representable\nby algebraic spaces, and\n\\item there exists $U \\in \\Ob(\\Sch_\\alpha/S)$\nand $U \\to F$ which is surjective and smooth.\n\\end{enumerate}\nThen $F$ is an algebraic space in the sense of\nAlgebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Algebraic spaces in the étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WF","source_file":"bootstrap.tex","source_line":2233,"source_end_line":2252,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L2233-L2252","statement_sha256":"bcda2b7dbd02918f24a82963eedd22436261e9dec6c8eab1622083f86402ba23","origin":"The Stacks Project","memory_eligible":false,"source_rank":12513,"rank":12513,"depth":67,"x":97.227,"y":1695.323,"cluster":"algebraic-spaces"},{"id":"stacks:0GE0","tag":"0GE0","title":"Algebraic spaces in the étale topology · Lemma 0GE0","summary":"Denote the common underlying category of Sch_fppf and Sch_etale by Sch_α (see Topologies, Remark [Tag 03FF]). Let S be an object of Sch_α. Let F : (Sch_α/S)^opp → Sets be a presheaf with the following properties: • F is a sheaf for the étale topology, • there exists an algebraic space U over S and a map U → F which is representable by algebraic spaces, surjective, and smooth. Then F is an algebraic space in the sense of Algebraic Spaces, Definition [Tag 025Y].","statement_latex":"Denote the common underlying category of $\\Sch_{fppf}$ and $\\Sch_\\etale$ by\n$\\Sch_\\alpha$ (see Topologies, Remark \\ref{topologies-remark-choice-sites}).\nLet $S$ be an object of $\\Sch_\\alpha$. Let\n$$\nF : (\\Sch_\\alpha/S)^{opp} \\longrightarrow \\textit{Sets}\n$$\nbe a presheaf with the following properties:\n\\begin{enumerate}\n\\item $F$ is a sheaf for the \\'etale topology,\n\\item there exists an algebraic space $U$ over $S$\nand a map $U \\to F$ which is representable by\nalgebraic spaces, surjective, and smooth.\n\\end{enumerate}\nThen $F$ is an algebraic space in the sense of\nAlgebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}.","area":"Algebraic Spaces","chapter":"Bootstrap","chapter_id":"bootstrap","section":"Algebraic spaces in the étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GE0","source_file":"bootstrap.tex","source_line":2298,"source_end_line":2315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/bootstrap.tex#L2298-L2315","statement_sha256":"e9f0ed4cd1dbb896105e82f5f349222b2cbc6558336eea8cad7774bc5ca84e25","origin":"The Stacks Project","memory_eligible":false,"source_rank":12514,"rank":12514,"depth":68,"x":251.26,"y":1454.294,"cluster":"algebraic-spaces"},{"id":"stacks:07SX","tag":"07SX","title":"Colimits of algebraic spaces · Lemma 07SX","summary":"Let S be a scheme. Let I → (Sch/S)_fppf, i ↦ X_i be a diagram of schemes over S as above. Assume that • X = colim X_i exists in the category of schemes, • coprod X_i → X is surjective, • if U → X is étale and U_i = X_i ×_X U, then U = colim U_i in the category of schemes, and • every object (U_i → X_i) of lim X_i, etale with U_i → X_i separated is in the essential image of the functor X_etale → lim X_i, etale. Then X = colim X_i in the category of algebraic spaces over S…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$\nbe a diagram of schemes over $S$ as above. Assume that\n\\begin{enumerate}\n\\item $X = \\colim X_i$ exists in the category of schemes,\n\\item $\\coprod X_i \\to X$ is surjective,\n\\item if $U \\to X$ is \\'etale and $U_i = X_i \\times_X U$, then\n$U = \\colim U_i$ in the category of schemes, and\n\\item every object $(U_i \\to X_i)$ of $\\lim X_{i, \\etale}$\nwith $U_i \\to X_i$ separated is in the essential image of\nthe functor $X_\\etale \\to \\lim X_{i, \\etale}$.\n\\end{enumerate}\nThen $X = \\colim X_i$ in the category of algebraic spaces over $S$ also.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Colimits of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SX","source_file":"spaces-pushouts.tex","source_line":116,"source_end_line":130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L116-L130","statement_sha256":"5f68d7702caab793a6cf984909d09a040ffa419c867c260c8339751c4ffbb4fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12515,"rank":12515,"depth":0,"x":559.206,"y":1663.813,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFQ","tag":"0GFQ","title":"Colimits of algebraic spaces · Lemma 0GFQ","summary":"Let S be a scheme. Let B be an algebraic space over S. Let I → (Sch/S)_fppf, i ↦ X_i be a diagram of algebraic spaces over B. Let (X, X_i → X) be a cocone for the diagram in the category of algebraic spaces over B (Categories, Remark [Tag 0G2U]). If there exists a fpqc covering (U_a → X)_a ∈ A such that • for all a ∈ A we have U_a = colim X_i ×_X U_a in the category of algebraic spaces over B, and • for all a, b ∈ A we have U_a ×_X U_b = colim X_i ×_X U_a ×_X U_b in the…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$\nbe a diagram of algebraic spaces over $B$. Let $(X, X_i \\to X)$\nbe a cocone for the diagram in the category of algebraic spaces over $B$\n(Categories, Remark \\ref{categories-remark-cones-and-cocones}).\nIf there exists a fpqc covering $\\{U_a \\to X\\}_{a \\in A}$ such that\n\\begin{enumerate}\n\\item for all $a \\in A$ we have\n$U_a = \\colim X_i \\times_X U_a$\nin the category of algebraic spaces over $B$, and\n\\item for all $a, b \\in A$ we have\n$U_a \\times_X U_b = \\colim X_i \\times_X U_a \\times_X U_b$\nin the category of algebraic spaces over $B$,\n\\end{enumerate}\nthen $X = \\colim X_i$ in the category of algebraic spaces over $B$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Colimits of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFQ","source_file":"spaces-pushouts.tex","source_line":164,"source_end_line":181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L164-L181","statement_sha256":"0852bd3c7d08c128fc5413c2e27fb042a607e40371a9b1f41d0e95c434630e70","origin":"The Stacks Project","memory_eligible":false,"source_rank":12516,"rank":12516,"depth":55,"x":717.781,"y":1484.322,"cluster":"geometry-of-spaces"},{"id":"stacks:0GHL","tag":"0GHL","title":"Colimits of algebraic spaces · Lemma 0GHL","summary":"Let S be a scheme. Let B be an algebraic space over S. Let I → (Sch/S)_fppf, i ↦ X_i be a diagram of algebraic spaces over B. Let (X, X_i → X) be a cocone for the diagram in the category of algebraic spaces over B (Categories, Remark [Tag 0G2U]). Assume that • the base change functor X_spaces, étale → lim X_i, spaces, etale, sending U to U_i = X_i ×_X U is an equivalence, • given • B' affine and étale over B, • Z an affine scheme over B', • U → X ×_B B' an étale morphism…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$\nbe a diagram of algebraic spaces over $B$. Let $(X, X_i \\to X)$\nbe a cocone for the diagram in the category of algebraic spaces over $B$\n(Categories, Remark \\ref{categories-remark-cones-and-cocones}).\nAssume that\n\\begin{enumerate}\n\\item the base change functor\n$X_{spaces, \\'etale} \\to \\lim X_{i, spaces, \\etale}$,\nsending $U$ to $U_i = X_i \\times_X U$ is an equivalence,\n\\item given\n\\begin{enumerate}\n\\item $B'$ affine and \\'etale over $B$,\n\\item $Z$ an affine scheme over $B'$,\n\\item $U \\to X \\times_B B'$ an \\'etale morphism of algebraic spaces\nwith $U$ affine,\n\\item $f_i : U_i \\to Z$ a cocone over $B'$ of the diagram\n$i \\mapsto U_i = U \\times_X X_i$,\n\\end{enumerate}\nthere exists a unique morphism $f : U \\to Z$ over $B'$\nsuch that $f_i$ equals the composition $U_i \\to U \\to Z$.\n\\end{enumerate}\nThen $X = \\colim X_i$ in the category of all algebraic spaces over $B$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Colimits of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHL","source_file":"spaces-pushouts.tex","source_line":232,"source_end_line":257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L232-L257","statement_sha256":"2fc7467258fb7eea1a762486cfa51b6851969cde20313bd5a8ff8b92eccecdf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12517,"rank":12517,"depth":56,"x":745.212,"y":1706.812,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFP","tag":"0GFP","title":"Colimits of algebraic spaces · Lemma 0GFP","summary":"Let S be a scheme. Let B be an algebraic space over S. Let I → (Sch/S)_fppf, i ↦ X_i be a diagram of algebraic spaces over B. Assume that • each X_i is separated over B, • X = colim X_i exists in the category of algebraic spaces separated over B, • coprod X_i → X is surjective, • if U → X is an étale separated morphism of algebraic spaces and U_i = X_i ×_X U, then U = colim U_i in the category of algebraic spaces separated over B, and • every object (U_i → X_i) of lim…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $\\mathcal{I} \\to (\\Sch/S)_{fppf}$, $i \\mapsto X_i$\nbe a diagram of algebraic spaces over $B$. Assume that\n\\begin{enumerate}\n\\item each $X_i$ is separated over $B$,\n\\item $X = \\colim X_i$ exists in the category of\nalgebraic spaces separated over $B$,\n\\item $\\coprod X_i \\to X$ is surjective,\n\\item if $U \\to X$ is an \\'etale separated morphism of algebraic spaces and\n$U_i = X_i \\times_X U$, then $U = \\colim U_i$ in\nthe category of algebraic spaces separated over $B$, and\n\\item every object $(U_i \\to X_i)$ of $\\lim X_{i, spaces, \\etale}$\nwith $U_i \\to X_i$ separated is of the form $U_i = X_i \\times_X U$\nfor some \\'etale separated morphism of algebraic spaces $U \\to X$.\n\\end{enumerate}\nThen $X = \\colim X_i$ in the category of all algebraic spaces over $B$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Colimits of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFP","source_file":"spaces-pushouts.tex","source_line":339,"source_end_line":357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L339-L357","statement_sha256":"78a591c36731c0d42c7ae5319645d810a3f2628099a5320da19346bbb9fc7d71","origin":"The Stacks Project","memory_eligible":false,"source_rank":12518,"rank":12518,"depth":57,"x":545.924,"y":1558.211,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFS","tag":"0GFS","title":"Descending étale sheaves · Lemma 0GFS","summary":"Let S be a scheme. Let (f_i : X_i → X) be an étale covering of algebraic spaces. The functor Sh(X_etale) → descent data for étale sheaves wrt (f_i : X_i → X) is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Let $\\{f_i : X_i \\to X\\}$ be an \\'etale covering of\nalgebraic spaces. The functor\n$$\n\\Sh(X_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{f_i : X_i \\to X\\}\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFS","source_file":"spaces-pushouts.tex","source_line":487,"source_end_line":497,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L487-L497","statement_sha256":"b2f5e3eeaee46516525315c9db61a6da6a2ebbb4862990f30fb9d382d2d79f64","origin":"The Stacks Project","memory_eligible":false,"source_rank":12519,"rank":12519,"depth":0,"x":812.564,"y":1554.705,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFT","tag":"0GFT","title":"Descending étale sheaves · Lemma 0GFT","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let (Y_i → Y)_i ∈ I be an étale covering of algebraic spaces. If for each i ∈ I the functor Sh(Y_i, etale) → descent data for étale sheaves wrt (X ×_Y Y_i → Y_i) is an equivalence of categories and for each i, j ∈ I the functor Sh((Y_i ×_Y Y_j)_etale) → descent data for étale sheaves wrt (X ×_Y Y_i ×_Y Y_j → Y_i ×_Y Y_j) is an equivalence of categories, then Sh(Y_etale) → descent data for étale…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $S$. Let $\\{Y_i \\to Y\\}_{i \\in I}$ be an \\'etale\ncovering of algebraic spaces. If for each $i \\in I$ the functor\n$$\n\\Sh(Y_{i, \\etale})\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\times_Y Y_i \\to Y_i\\}\n$$\nis an equivalence of categories and for each $i, j \\in I$ the functor\n$$\n\\Sh((Y_i \\times_Y Y_j)_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\n\\{X \\times_Y Y_i \\times_Y Y_j \\to Y_i \\times_Y Y_j\\}\n$$\nis an equivalence of categories, then\n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFT","source_file":"spaces-pushouts.tex","source_line":511,"source_end_line":535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L511-L535","statement_sha256":"c60c3054a7b57bb92ceb6963fec0cc113a3afd020e9faeb65495950ccb73fb7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12520,"rank":12520,"depth":1,"x":618.632,"y":1708.696,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFU","tag":"0GFU","title":"Descending étale sheaves · Lemma 0GFU","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is representable (by schemes) and f has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation Then Sh(Y_etale) → descent data for étale sheaves wrt (X → Y) is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $f$ is representable (by schemes) and $f$\nhas one of the following properties:\nsurjective and integral,\nsurjective and proper, or\nsurjective and flat and locally of finite presentation\nThen \n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFU","source_file":"spaces-pushouts.tex","source_line":542,"source_end_line":557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L542-L557","statement_sha256":"04e17583952e857b591be35411bdcb3775a03b50b92c423f5ba42398d46d3554","origin":"The Stacks Project","memory_eligible":false,"source_rank":12521,"rank":12521,"depth":58,"x":637.812,"y":1484.947,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFV","tag":"0GFV","title":"Descending étale sheaves · Lemma 0GFV","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Let π : X' → X be a morphism of algebraic spaces. Assume • f ∘ π is representable (by schemes), • f ∘ π has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. Then Sh(Y_etale) → descent data for étale sheaves wrt (X → Y) is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Let $\\pi : X' \\to X$ be a morphism of algebraic spaces. Assume\n\\begin{enumerate}\n\\item $f \\circ \\pi$ is representable (by schemes),\n\\item $f \\circ \\pi$ has one of the following properties:\nsurjective and integral,\nsurjective and proper, or\nsurjective and flat and locally of finite presentation.\n\\end{enumerate}\nThen \n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFV","source_file":"spaces-pushouts.tex","source_line":574,"source_end_line":592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L574-L592","statement_sha256":"0b2248dddf7cfb65c65a846be8ecfc865c7ebb9d1f2ecdc303c754908e61f15a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12522,"rank":12522,"depth":59,"x":803.718,"y":1660.943,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFW","tag":"0GFW","title":"Descending étale sheaves · Lemma 0GFW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S which has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation. Then the functor Sh(Y_etale) → descent data for étale sheaves wrt (X → Y) is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces over $S$ which has one of the following\nproperties: surjective and integral, surjective and proper, or\nsurjective and flat and locally of finite presentation. Then the functor\n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{X \\to Y\\}\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFW","source_file":"spaces-pushouts.tex","source_line":599,"source_end_line":611,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L599-L611","statement_sha256":"ee5e03c3ac4ed5c92847b79abe1e2063da063938b11d447ef2e0e8d7c5395fdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12523,"rank":12523,"depth":67,"x":539.666,"y":1625.28,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFX","tag":"0GFX","title":"Descending étale sheaves · Lemma 0GFX","summary":"Let S be a scheme. Let (f_i : X_i → X) be an fppf covering of algebraic spaces over S. The functor Sh(X_etale) → descent data for étale sheaves wrt (f_i : X_i → X) is an equivalence of categories.","statement_latex":"Let $S$ be a scheme.\nLet $\\{f_i : X_i \\to X\\}$ be an fppf covering of algebraic spaces over $S$.\nThe functor\n$$\n\\Sh(X_\\etale)\n\\longrightarrow\n\\text{descent data for \\'etale sheaves wrt }\\{f_i : X_i \\to X\\}\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFX","source_file":"spaces-pushouts.tex","source_line":648,"source_end_line":659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L648-L659","statement_sha256":"32c6c424d2fd834dc149a564616459cb9dc3f3360fd9f730a7b024cafd503577","origin":"The Stacks Project","memory_eligible":false,"source_rank":12524,"rank":12524,"depth":68,"x":763.209,"y":1501.662,"cluster":"geometry-of-spaces"},{"id":"stacks:0GFY","tag":"0GFY","title":"Descending étale sheaves · Lemma 0GFY","summary":"Let S be a scheme. Let f : Y' → Y be a proper morphism of algebraic spaces over S. Let i : Z → Y be a closed immersion. Set E = Z ×_Y Y'. Picture xymatrix E ar[d]_g ar[r]_j & Y' ar[d]^f Z ar[r]^i & Y If f is an isomorphism over Y setminus Z, then the functor Sh(Y_etale) → Sh(Y'_etale) ×_Sh(E_etale) Sh(Z_etale) is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Let $f : Y' \\to Y$ be a proper morphism of\nalgebraic spaces over $S$. Let $i : Z \\to Y$\nbe a closed immersion. Set $E = Z \\times_Y Y'$. Picture\n$$\n\\xymatrix{\nE \\ar[d]_g \\ar[r]_j & Y' \\ar[d]^f \\\\\nZ \\ar[r]^i & Y\n}\n$$\nIf $f$ is an isomorphism over $Y \\setminus Z$, then the functor\n$$\n\\Sh(Y_\\etale)\n\\longrightarrow\n\\Sh(Y'_\\etale) \\times_{\\Sh(E_\\etale)} \\Sh(Z_\\etale)\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Descending étale sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GFY","source_file":"spaces-pushouts.tex","source_line":670,"source_end_line":688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L670-L688","statement_sha256":"f3b01380702a61251d8dc5edea3d796f134607a7b5dd6c49754330f6d53334c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12525,"rank":12525,"depth":68,"x":697.737,"y":1719.811,"cluster":"geometry-of-spaces"},{"id":"stacks:0GG0","tag":"0GG0","title":"Descending étale morphisms of algebraic spaces · Lemma 0GG0","summary":"Let S be a scheme. Let f : X → Y be a proper surjective morphism of algebraic spaces over S. Any descent datum (U/X, φ) relative to f (Descent on Spaces, Definition [Tag 0ADG]) with U étale over X is effective (Descent on Spaces, Definition [Tag 0ADQ]). More precisely, there exists an étale morphism V → Y of algebraic spaces whose corresponding canonical descent datum is isomorphic to (U/X, φ).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper surjective morphism\nof algebraic spaces over $S$. Any descent datum $(U/X, \\varphi)$ relative to $f$\n(Descent on Spaces, Definition \\ref{spaces-descent-definition-descent-datum})\nwith $U$ \\'etale over $X$ is effective\n(Descent on Spaces, Definition \\ref{spaces-descent-definition-effective}).\nMore precisely, there exists an \\'etale morphism $V \\to Y$ of algebraic spaces\nwhose corresponding canonical descent datum is isomorphic to $(U/X, \\varphi)$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Descending étale morphisms of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GG0","source_file":"spaces-pushouts.tex","source_line":748,"source_end_line":757,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L748-L757","statement_sha256":"fe58313323d8a57e09a15b650449f499d86d7c9f5bbabb66af2d74971fa90fff","origin":"The Stacks Project","memory_eligible":false,"source_rank":12526,"rank":12526,"depth":68,"x":570.496,"y":1521.663,"cluster":"geometry-of-spaces"},{"id":"stacks:0GG1","tag":"0GG1","title":"Descending étale morphisms of algebraic spaces · Lemma 0GG1","summary":"Let S be a scheme. Let f : Y' → Y be a proper morphism of algebraic spaces over S. Let i : Z → Y be a closed immersion. Set E = Z ×_Y Y'. Picture xymatrix E ar[d]_g ar[r]_j & Y' ar[d]^f Z ar[r]^i & Y If f is an isomorphism over Y setminus Z, then the functor Y_spaces, etale → Y'_spaces, etale ×_E_spaces, etale Z_spaces, etale is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Let $f : Y' \\to Y$ be a proper morphism of\nalgebraic spaces over $S$. Let $i : Z \\to Y$\nbe a closed immersion. Set $E = Z \\times_Y Y'$. Picture\n$$\n\\xymatrix{\nE \\ar[d]_g \\ar[r]_j & Y' \\ar[d]^f \\\\\nZ \\ar[r]^i & Y\n}\n$$\nIf $f$ is an isomorphism over $Y \\setminus Z$, then the functor\n$$\nY_{spaces, \\etale}\n\\longrightarrow\nY'_{spaces, \\etale} \\times_{E_{spaces, \\etale}} Z_{spaces, \\etale}\n$$\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Descending étale morphisms of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GG1","source_file":"spaces-pushouts.tex","source_line":779,"source_end_line":797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L779-L797","statement_sha256":"613eed1587328c3e8f5039850e74307ab9e1cd2266e511242ec472914d8bc233","origin":"The Stacks Project","memory_eligible":false,"source_rank":12527,"rank":12527,"depth":69,"x":823.843,"y":1595.627,"cluster":"geometry-of-spaces"},{"id":"stacks:07SY","tag":"07SY","title":"Pushouts along thickenings and affine morphisms · Lemma 07SY","summary":"Let S be a scheme. Let X → X' be a thickening of schemes over S and let X → Y be an affine morphism of schemes over S. Let Y' = Y amalg_X X' be the pushout in the category of schemes (see More on Morphisms, Lemma [Tag 07RT]). Then Y' is also a pushout in the category of algebraic spaces over S.","statement_latex":"Let $S$ be a scheme. Let $X \\to X'$ be a thickening of schemes\nover $S$ and let $X \\to Y$ be an affine morphism of schemes over $S$.\nLet $Y' = Y \\amalg_X X'$ be the pushout in the category of schemes (see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}).\nThen $Y'$ is also a pushout in the category of algebraic spaces over $S$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Pushouts along thickenings and affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07SY","source_file":"spaces-pushouts.tex","source_line":841,"source_end_line":848,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L841-L848","statement_sha256":"b86c52b79f68297301686eb3fe5e42fd9cfd9947282c20f2aeed90fd6eeb2e2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12528,"rank":12528,"depth":45,"x":577.379,"y":1684.903,"cluster":"geometry-of-spaces"},{"id":"stacks:07VX","tag":"07VX","title":"Pushouts along thickenings and affine morphisms · Lemma 07VX","summary":"Let S be a scheme. Let X → X' be a thickening of algebraic spaces over S and let X → Y be an affine morphism of algebraic spaces over S. Then there exists a pushout xymatrix X ar[r] ar[d]_f & X' ar[d]^f' Y ar[r] & Y amalg_X X' in the category of algebraic spaces over S. Moreover Y' = Y amalg_X X' is a thickening of Y and O_Y' = O_Y ×_f_*O_X f'_*O_X' as sheaves on Y_etale = (Y')_etale.","statement_latex":"Let $S$ be a scheme. Let $X \\to X'$ be a thickening of algebraic spaces\nover $S$ and let $X \\to Y$ be an affine morphism of algebraic spaces over $S$.\nThen there exists a pushout\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d]_f\n&\nX' \\ar[d]^{f'}\n\\\\\nY \\ar[r]\n&\nY \\amalg_X X'\n}\n$$\nin the category of algebraic spaces over $S$. Moreover $Y' = Y \\amalg_X X'$\nis a thickening of $Y$ and\n$$\n\\mathcal{O}_{Y'} = \\mathcal{O}_Y \\times_{f_*\\mathcal{O}_X} f'_*\\mathcal{O}_{X'}\n$$\nas sheaves on $Y_\\etale = (Y')_\\etale$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Pushouts along thickenings and affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VX","source_file":"spaces-pushouts.tex","source_line":858,"source_end_line":880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L858-L880","statement_sha256":"6834dbb51d0e24e597d9304a1271d2ee9d9c5e2251f011b9e6cffe7399e5478b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12529,"rank":12529,"depth":70,"x":687.4,"y":1479.08,"cluster":"geometry-of-spaces"},{"id":"stacks:07VY","tag":"07VY","title":"Pushouts along thickenings and affine morphisms · Lemma 07VY","summary":"Let S be a base scheme. Let X → X' be a thickening of algebraic spaces over S and let X → Y be an affine morphism of algebraic spaces over S. Let Y' = Y amalg_X X' be the pushout (see Lemma [Tag 07VX]). Base change gives a functor F : (Spaces/Y') → (Spaces/Y) ×_(Spaces/Y') (Spaces/X') given by V' ↦ (V' ×_Y' Y, V' ×_Y' X', 1) which sends (Sch/Y') into (Sch/Y) ×_(Sch/Y') (Sch/X'). The functor F has a left adjoint G : (Spaces/Y) ×_(Spaces/Y') (Spaces/X') → (Spaces/Y') which…","statement_latex":"Let $S$ be a base scheme. Let $X \\to X'$ be a thickening of algebraic spaces\nover $S$ and let $X \\to Y$ be an affine morphism of algebraic spaces over $S$.\nLet $Y' = Y \\amalg_X X'$ be the pushout (see\nLemma \\ref{lemma-pushout-along-thickening}). Base change gives a functor\n$$\nF :\n(\\textit{Spaces}/Y')\n\\longrightarrow\n(\\textit{Spaces}/Y) \\times_{(\\textit{Spaces}/Y')} (\\textit{Spaces}/X')\n$$\ngiven by $V' \\longmapsto (V' \\times_{Y'} Y, V' \\times_{Y'} X', 1)$ which\nsends $(\\Sch/Y')$ into $(\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')$.\nThe functor $F$ has a left adjoint\n$$\nG :\n(\\textit{Spaces}/Y) \\times_{(\\textit{Spaces}/Y')} (\\textit{Spaces}/X')\n\\longrightarrow\n(\\textit{Spaces}/Y')\n$$\nwhich sends the triple $(V, U', \\varphi)$ to the pushout\n$V \\amalg_{(V \\times_Y X)} U'$ in the category of algebraic spaces over $S$.\nThe functor $G$ sends $(\\Sch/Y) \\times_{(\\Sch/Y')} (\\Sch/X')$ into $(\\Sch/Y')$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Pushouts along thickenings and affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VY","source_file":"spaces-pushouts.tex","source_line":977,"source_end_line":1001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L977-L1001","statement_sha256":"06b9f752c46c4f8a5de937232926de6ce62386e96ad8de59b892fb1b62cae0cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12530,"rank":12530,"depth":71,"x":771.848,"y":1693.428,"cluster":"geometry-of-spaces"},{"id":"stacks:07VZ","tag":"07VZ","title":"Pushouts along thickenings and affine morphisms · Lemma 07VZ","summary":"Let S be a scheme. Let xymatrix A ar[r] ar[d] & C ar[d] ar[r] & E ar[d] B ar[r] & D ar[r] & F be a commutative diagram of algebraic spaces over S. Assume that A, B, C, D and A, B, E, F form cartesian squares and that B → D is surjective étale. Then C, D, E, F is a cartesian square.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nA \\ar[r] \\ar[d] & C \\ar[d] \\ar[r] & E \\ar[d] \\\\\nB \\ar[r] & D \\ar[r] & F\n}\n$$\nbe a commutative diagram of algebraic spaces over $S$.\nAssume that $A, B, C, D$ and $A, B, E, F$ form cartesian squares\nand that $B \\to D$ is surjective \\'etale.\nThen $C, D, E, F$ is a cartesian square.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Pushouts along thickenings and affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07VZ","source_file":"spaces-pushouts.tex","source_line":1057,"source_end_line":1070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1057-L1070","statement_sha256":"83eeb6476fbc8d8820839d2a308c154970dceb35b2b652fd9c092f3ca8a0ef74","origin":"The Stacks Project","memory_eligible":false,"source_rank":12531,"rank":12531,"depth":0,"x":537.042,"y":1583.213,"cluster":"geometry-of-spaces"},{"id":"stacks:07W0","tag":"07W0","title":"Pushouts along thickenings and affine morphisms · Lemma 07W0","summary":"In the situation of Lemma [Tag 07VY] the functor F ∘ G is isomorphic to the identity functor.","statement_latex":"In the situation of Lemma \\ref{lemma-categories-spaces-over-pushout}\nthe functor $F \\circ G$ is isomorphic to the identity functor.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Pushouts along thickenings and affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07W0","source_file":"spaces-pushouts.tex","source_line":1076,"source_end_line":1080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1076-L1080","statement_sha256":"42f74af3fa25424551411cf328a627e114e0e0b5bc8263914696b2f964414af5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12532,"rank":12532,"depth":72,"x":798.997,"y":1531.214,"cluster":"geometry-of-spaces"},{"id":"stacks:08KV","tag":"08KV","title":"Pushouts along thickenings and affine morphisms · Lemma 08KV","summary":"Let S be a base scheme. Let X → X' be a thickening of algebraic spaces over S and let X → Y be an affine morphism of algebraic spaces over S. Let Y' = Y amalg_X X' be the pushout (see Lemma [Tag 07VX]). Let V' → Y' be a morphism of algebraic spaces over S. Set V = Y ×_Y' V', U' = X' ×_Y' V', and U = X ×_Y' V'. There is an equivalence of categories between • quasi-coherent O_V'-modules flat over Y', and • the category of triples (G, F', φ) where • G is a quasi-coherent…","statement_latex":"Let $S$ be a base scheme.\nLet $X \\to X'$ be a thickening of algebraic spaces over $S$\nand let $X \\to Y$ be an affine morphism of algebraic spaces over $S$.\nLet $Y' = Y \\amalg_X X'$ be the pushout\n(see Lemma \\ref{lemma-pushout-along-thickening}).\nLet $V' \\to Y'$ be a morphism of algebraic spaces over $S$. Set\n$V = Y \\times_{Y'} V'$, $U' = X' \\times_{Y'} V'$, and $U = X \\times_{Y'} V'$.\nThere is an equivalence of categories between\n\\begin{enumerate}\n\\item quasi-coherent $\\mathcal{O}_{V'}$-modules flat over $Y'$, and\n\\item the category of triples $(\\mathcal{G}, \\mathcal{F}', \\varphi)$ where\n\\begin{enumerate}\n\\item $\\mathcal{G}$ is a quasi-coherent $\\mathcal{O}_V$-module flat over $Y$,\n\\item $\\mathcal{F}'$ is a quasi-coherent $\\mathcal{O}_{U'}$-module flat\nover $X$, and\n\\item $\\varphi : (U \\to V)^*\\mathcal{G} \\to (U \\to U')^*\\mathcal{F}'$\nis an isomorphism of $\\mathcal{O}_U$-modules.\n\\end{enumerate}\n\\end{enumerate}\nThe equivalence maps $\\mathcal{G}'$ to\n$((V \\to V')^*\\mathcal{G}', (U' \\to V')^*\\mathcal{G}', can)$.\nSuppose $\\mathcal{G}'$ corresponds to the triple\n$(\\mathcal{G}, \\mathcal{F}', \\varphi)$. Then\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{G}'$ is a finite type $\\mathcal{O}_{V'}$-module if and\nonly if $\\mathcal{G}$ and $\\mathcal{F}'$ are finite type\n$\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules.\n\\item[(b)] if $V' \\to Y'$ is locally of finite presentation, then\n$\\mathcal{G}'$ is an $\\mathcal{O}_{V'}$-module of finite\npresentation if and only if $\\mathcal{G}$ and $\\mathcal{F}'$ are\n$\\mathcal{O}_Y$ and $\\mathcal{O}_{U'}$-modules of finite presentation.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Pushouts along thickenings and affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KV","source_file":"spaces-pushouts.tex","source_line":1191,"source_end_line":1225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1191-L1225","statement_sha256":"be66c5d76dc8e2f6fb950d328c27af74e730aa4a235f3bbdf8ce83bde2fd514e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12533,"rank":12533,"depth":71,"x":647.547,"y":1718.325,"cluster":"geometry-of-spaces"},{"id":"stacks:07W3","tag":"07W3","title":"Pushouts along thickenings and affine morphisms · Lemma 07W3","summary":"In the situation of Lemma [Tag 07W0]. If V' = G(V, U', φ) for some triple (V, U', φ), then • V' → Y' is locally of finite type if and only if V → Y and U' → X' are locally of finite type, • V' → Y' is flat if and only if V → Y and U' → X' are flat, • V' → Y' is flat and locally of finite presentation if and only if V → Y and U' → X' are flat and locally of finite presentation, • V' → Y' is smooth if and only if V → Y and U' → X' are smooth, • V' → Y' is étale if and only…","statement_latex":"In the situation of\nLemma \\ref{lemma-equivalence-categories-spaces-over-pushout}.\nIf $V' = G(V, U', \\varphi)$ for some triple $(V, U', \\varphi)$, then\n\\begin{enumerate}\n\\item $V' \\to Y'$ is locally of finite type if and only if $V \\to Y$ and\n$U' \\to X'$ are locally of finite type,\n\\item $V' \\to Y'$ is flat if and only if $V \\to Y$ and $U' \\to X'$ are flat,\n\\item $V' \\to Y'$ is flat and locally of finite presentation if and only if\n$V \\to Y$ and $U' \\to X'$ are flat and locally of finite presentation,\n\\item $V' \\to Y'$ is smooth if and only if $V \\to Y$ and $U' \\to X'$ are smooth,\n\\item $V' \\to Y'$ is \\'etale if and only if $V \\to Y$ and $U' \\to X'$\nare \\'etale, and\n\\item add more here as needed.\n\\end{enumerate}\nIf $W'$ is flat over $Y'$, then the adjunction mapping\n$G(F(W')) \\to W'$ is an isomorphism. Hence $F$ and $G$ define mutually\nquasi-inverse functors between the category of spaces flat over $Y'$\nand the category of triples $(V, U', \\varphi)$ with $V \\to Y$\nand $U' \\to X'$ flat.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Pushouts along thickenings and affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07W3","source_file":"spaces-pushouts.tex","source_line":1253,"source_end_line":1274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1253-L1274","statement_sha256":"c295b7bff6e5f4ab043fff8dab1f156ace3ef702f49cab7dfd4e570c326dcca4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12534,"rank":12534,"depth":73,"x":608.728,"y":1494.262,"cluster":"geometry-of-spaces"},{"id":"stacks:0EDP","tag":"0EDP","title":"Pushouts along closed immersions and integral morphisms · Lemma 0EDP","summary":"In More on Morphisms, Situation [Tag 0ECI] let Y amalg_Z X be the pushout in the category of schemes (More on Morphisms, Proposition [Tag 0E25]). Then Y amalg_Z X is also a pushout in the category of algebraic spaces over S.","statement_latex":"In More on Morphisms, Situation\n\\ref{more-morphisms-situation-pushout-along-closed-immersion-and-integral}\nlet $Y \\amalg_Z X$ be the pushout in the category of schemes\n(More on Morphisms, Proposition\n\\ref{more-morphisms-proposition-pushout-along-closed-immersion-and-integral}).\nThen $Y \\amalg_Z X$\nis also a pushout in the category of algebraic spaces over $S$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Pushouts along closed immersions and integral morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDP","source_file":"spaces-pushouts.tex","source_line":1324,"source_end_line":1333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1324-L1333","statement_sha256":"9ec0911099f1a0cb53a4eccc16c0e20c8622f5ac372aa312aa1a72e676d307e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12535,"rank":12535,"depth":52,"x":817.681,"y":1637.553,"cluster":"geometry-of-spaces"},{"id":"stacks:0DL7","tag":"0DL7","title":"Pushouts and derived categories · Lemma 0DL7","summary":"Let S be a scheme. Consider a pushout xymatrix X ar[r]_i ar[d]_f & X' ar[d]^f' Y ar[r]^j & Y' in the category of algebraic spaces over S as in Lemma [Tag 07VX]. Assume i is a thickening. Then the essential image of the functor D(O_Y') → D(O_Y) ×_D(O_X) D(O_X') contains every triple (M, K', α) where M ∈ D(O_Y) and K' ∈ D(O_X') are pseudo-coherent.","statement_latex":"Let $S$ be a scheme. Consider a pushout\n$$\n\\xymatrix{\nX \\ar[r]_i \\ar[d]_f & X' \\ar[d]^{f'}\n\\\\\nY \\ar[r]^j & Y'\n}\n$$\nin the category of algebraic spaces over $S$\nas in Lemma \\ref{lemma-pushout-along-thickening}.\nAssume $i$ is a thickening. Then the essential\nimage of the functor\\footnote{All functors given by derived pullback.}\n$$\nD(\\mathcal{O}_{Y'}) \\longrightarrow\nD(\\mathcal{O}_Y) \\times_{D(\\mathcal{O}_X)} D(\\mathcal{O}_{X'})\n$$\ncontains every triple $(M, K', \\alpha)$ where $M \\in D(\\mathcal{O}_Y)$\nand $K' \\in D(\\mathcal{O}_{X'})$ are pseudo-coherent.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Pushouts and derived categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DL7","source_file":"spaces-pushouts.tex","source_line":1364,"source_end_line":1384,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1364-L1384","statement_sha256":"ea868e1d9c2ac23c5219dc83d6abe91e9c0742e2f5b8fbc84667582e52cd4df2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12536,"rank":12536,"depth":71,"x":548.186,"y":1650.467,"cluster":"geometry-of-spaces"},{"id":"stacks:0DVI","tag":"0DVI","title":"Constructing elementary distinguished squares · Lemma 0DVI","summary":"Let S be a scheme. Let (U ⊂ W, f : V → W) be an elementary distinguished square. Then xymatrix U ×_W V ar[r] ar[d] & V ar[d]^f U ar[r] & W is a pushout in the category of algebraic spaces over S.","statement_latex":"Let $S$ be a scheme. Let $(U \\subset W, f : V \\to W)$ be\nan elementary distinguished square. Then\n$$\n\\xymatrix{\nU \\times_W V \\ar[r] \\ar[d] &\nV \\ar[d]^f \\\\\nU \\ar[r] & W\n}\n$$\nis a pushout in the category of algebraic spaces over $S$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Constructing elementary distinguished squares","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVI","source_file":"spaces-pushouts.tex","source_line":1506,"source_end_line":1518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1506-L1518","statement_sha256":"2df9f2b42449ff45e64e5cc85423d4f93a7fb458ba5f44d1a9144a1dd5735b90","origin":"The Stacks Project","memory_eligible":false,"source_rank":12537,"rank":12537,"depth":0,"x":736.652,"y":1487.916,"cluster":"geometry-of-spaces"},{"id":"stacks:0DVJ","tag":"0DVJ","title":"Constructing elementary distinguished squares · Lemma 0DVJ","summary":"Let S be a scheme. Let V, U be algebraic spaces over S. Let V' ⊂ V be an open subspace and let f' : V' → U be a separated étale morphism of algebraic spaces over S. Then there exists a pushout xymatrix V' ar[r] ar[d] & V ar[d]^f U ar[r] & W in the category of algebraic spaces over S and moreover (U ⊂ W, f : V → W) is an elementary distinguished square.","statement_latex":"Let $S$ be a scheme. Let $V$, $U$ be algebraic spaces over $S$.\nLet $V' \\subset V$ be an open subspace and let $f' : V' \\to U$ be a\nseparated \\'etale morphism of algebraic spaces over $S$.\nThen there exists a pushout\n$$\n\\xymatrix{\nV' \\ar[r] \\ar[d] &\nV \\ar[d]^f \\\\\nU \\ar[r] & W\n}\n$$\nin the category of algebraic spaces over $S$ and moreover\n$(U \\subset W, f : V \\to W)$ is an elementary distinguished square.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Constructing elementary distinguished squares","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVJ","source_file":"spaces-pushouts.tex","source_line":1546,"source_end_line":1561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1546-L1561","statement_sha256":"b79ab11761dfc0bd85a32da9e1afb37045257b2655c96bd2771dff6bfbb1258c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12538,"rank":12538,"depth":67,"x":728.394,"y":1714.873,"cluster":"geometry-of-spaces"},{"id":"stacks:0AEQ","tag":"0AEQ","title":"Formal glueing of quasi-coherent modules · Lemma 0AEQ","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Let Z ⊂ X closed subspace such that f^-1Z → Z is integral and universally injective. Let overliney be a geometric point of Y and overlinex = f(overliney). We have (Rf_*Q)_overlinex = Q_overliney in D(Ab) for any object Q of D(Y_etale) supported on |f^-1Z|.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces\nover $S$. Let $Z \\subset X$ closed subspace such that $f^{-1}Z \\to Z$ is\nintegral and universally injective. Let $\\overline{y}$ be a geometric point\nof $Y$ and $\\overline{x} = f(\\overline{y})$. We have\n$$\n(Rf_*Q)_{\\overline{x}} = Q_{\\overline{y}}\n$$\nin $D(\\textit{Ab})$ for any object $Q$ of $D(Y_\\etale)$ supported\non $|f^{-1}Z|$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEQ","source_file":"spaces-pushouts.tex","source_line":1622,"source_end_line":1633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1622-L1633","statement_sha256":"6a265a51afa4a0861f0c165c7a06498f0fa02cd5d22954a4344dc0d8f0a8be06","origin":"The Stacks Project","memory_eligible":false,"source_rank":12539,"rank":12539,"depth":55,"x":551.85,"y":1542.715,"cluster":"geometry-of-spaces"},{"id":"stacks:0AER","tag":"0AER","title":"Formal glueing of quasi-coherent modules · Lemma 0AER","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Let Z ⊂ X closed subspace such that f^-1Z → Z is integral and universally injective. Let overliney be a geometric point of Y and overlinex = f(overliney). Let G be an abelian sheaf on Y. Then the map of two term complexes (f_*G_overlinex → (f ∘ j')_*(G|_V)_overlinex) → (G_overliney → j'_*(G|_V)_overliney) induces an isomorphism on kernels and an injection on cokernels. Here V = Y setminus f^-1Z and…","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces\nover $S$. Let $Z \\subset X$ closed subspace such that $f^{-1}Z \\to Z$ is\nintegral and universally injective. Let $\\overline{y}$ be a geometric point\nof $Y$ and $\\overline{x} = f(\\overline{y})$. Let $\\mathcal{G}$\nbe an abelian sheaf on $Y$. Then the map of two term complexes\n$$\n\\left(f_*\\mathcal{G}_{\\overline{x}} \\to\n(f \\circ j')_*(\\mathcal{G}|_V)_{\\overline{x}}\\right)\n\\longrightarrow\n\\left(\\mathcal{G}_{\\overline{y}} \\to j'_*(\\mathcal{G}|_V)_{\\overline{y}}\\right)\n$$\ninduces an isomorphism on kernels and an injection on cokernels.\nHere $V = Y \\setminus f^{-1}Z$ and $j' : V \\to Y$ is the inclusion.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AER","source_file":"spaces-pushouts.tex","source_line":1664,"source_end_line":1679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1664-L1679","statement_sha256":"2428571537d8494db02fca1419b9dd25fb085bf41c840456cde2acd4b28f81b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12540,"rank":12540,"depth":56,"x":820.659,"y":1569.505,"cluster":"geometry-of-spaces"},{"id":"stacks:0AES","tag":"0AES","title":"Formal glueing of quasi-coherent modules · Lemma 0AES","summary":"Let S be a scheme. Let X be an algebraic space over S. Let f : Y → X be a quasi-compact and quasi-separated morphism. Let overlinex be a geometric point of X and let Spec(O_X, overlinex) → X be the canonical morphism. For a quasi-coherent module G on Y we have f_*G_overlinex = Γ(Y ×_X Spec(O_X, overlinex), p^*F) where p : Y ×_X Spec(O_X, overlinex) → Y is the projection.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $f : Y \\to X$ be a quasi-compact and quasi-separated morphism.\nLet $\\overline{x}$ be a geometric point of $X$ and let\n$\\Spec(\\mathcal{O}_{X, \\overline{x}}) \\to X$\nbe the canonical morphism. For a quasi-coherent module\n$\\mathcal{G}$ on $Y$ we have\n$$\nf_*\\mathcal{G}_{\\overline{x}} =\n\\Gamma(Y \\times_X \\Spec(\\mathcal{O}_{X, \\overline{x}}), p^*\\mathcal{F})\n$$\nwhere $p : Y \\times_X \\Spec(\\mathcal{O}_{X, \\overline{x}}) \\to Y$\nis the projection.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AES","source_file":"spaces-pushouts.tex","source_line":1713,"source_end_line":1727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1713-L1727","statement_sha256":"cbccb77ddbbb6924acb45e164e8d4bcb54c547f081f16cb355feeef2ed0a6267","origin":"The Stacks Project","memory_eligible":false,"source_rank":12541,"rank":12541,"depth":60,"x":600.75,"y":1702.369,"cluster":"geometry-of-spaces"},{"id":"stacks:0AET","tag":"0AET","title":"Formal glueing of quasi-coherent modules · Lemma 0AET","summary":"Let S be a scheme. Let X be an algebraic space over S. Let i : Z → X be a closed immersion of finite presentation. Let Q ∈ D_QCoh(O_X) be supported on |Z|. Let overlinex be a geometric point of X and let I_overlinex ⊂ O_X, overlinex be the stalk of the ideal sheaf of Z. Then the cohomology modules H^n(Q_overlinex) are I_overlinex-power torsion (see More on Algebra, Definition [Tag 05E6]).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $i : Z \\to X$ be a closed immersion of finite presentation.\nLet $Q \\in D_\\QCoh(\\mathcal{O}_X)$ be supported on $|Z|$.\nLet $\\overline{x}$ be a geometric point of $X$ and let\n$I_{\\overline{x}} \\subset \\mathcal{O}_{X, \\overline{x}}$ be the stalk of\nthe ideal sheaf of $Z$. Then the cohomology modules\n$H^n(Q_{\\overline{x}})$ are $I_{\\overline{x}}$-power torsion\n(see More on Algebra, Definition\n\\ref{more-algebra-definition-f-power-torsion}).","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AET","source_file":"spaces-pushouts.tex","source_line":1739,"source_end_line":1750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1739-L1750","statement_sha256":"d3f1b2c381d5e7eb6f5aa90313eeeaf79b49ea4ddc5dcc7db7812cd49c4e4d1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12542,"rank":12542,"depth":41,"x":656.099,"y":1479.464,"cluster":"geometry-of-spaces"},{"id":"stacks:0AEU","tag":"0AEU","title":"Formal glueing of quasi-coherent modules · Lemma 0AEU","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Let Z ⊂ X be a closed subspace. Assume f^-1Z → Z is an isomorphism and that f is flat in every point of f^-1Z. For any Q in D_QCoh(O_Y) supported on |f^-1Z| we have Lf^*Rf_*Q = Q.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of algebraic spaces\nover $S$. Let $Z \\subset X$ be a closed subspace. Assume $f^{-1}Z \\to Z$\nis an isomorphism and that $f$ is flat in every point of $f^{-1}Z$. For any\n$Q$ in $D_\\QCoh(\\mathcal{O}_Y)$ supported on $|f^{-1}Z|$ we have\n$Lf^*Rf_*Q = Q$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEU","source_file":"spaces-pushouts.tex","source_line":1776,"source_end_line":1783,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1776-L1783","statement_sha256":"939fabb4bea2886545b5e6a70efdfa483fe026094b892096df45793b9ba7dbaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12543,"rank":12543,"depth":56,"x":794.633,"y":1675.37,"cluster":"geometry-of-spaces"},{"id":"stacks:0AEY","tag":"0AEY","title":"Formal glueing of quasi-coherent modules · Lemma 0AEY","summary":"In Situation [Tag 0AEV]. The functor ([Tag 0AEX]) is right adjoint to the functor ([Tag 0AEW]).","statement_latex":"In Situation \\ref{situation-formal-glueing}.\nThe functor (\\ref{equation-reverse}) is right adjoint to\nthe functor (\\ref{equation-formal-glueing-modules}).","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEY","source_file":"spaces-pushouts.tex","source_line":1874,"source_end_line":1879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1874-L1879","statement_sha256":"bc5bdf2c21552a3bfacf20fe0bf49ff314d542c4da369adfc3063d135b5873ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":12544,"rank":12544,"depth":0,"x":534.762,"y":1609.475,"cluster":"geometry-of-spaces"},{"id":"stacks:0AEZ","tag":"0AEZ","title":"Formal glueing of quasi-coherent modules · Lemma 0AEZ","summary":"In Situation [Tag 0AEV]. Let X' → X be a flat morphism of algebraic spaces. Set Z' = X' ×_X Z and Y' = X' ×_X Y. The pullbacks QCoh(O_X) → QCoh(O_X') and QCoh(Y → X, Z) → QCoh(Y' → X', Z') are compatible with the functors ([Tag 0AEX]) and [Tag 0AEW]).","statement_latex":"In Situation \\ref{situation-formal-glueing}.\nLet $X' \\to X$ be a flat morphism of algebraic spaces.\nSet $Z' = X' \\times_X Z$ and $Y' = X' \\times_X Y$.\nThe pullbacks $\\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_{X'})$\nand $\\QCoh(Y \\to X, Z) \\to \\QCoh(Y' \\to X', Z')$ are compatible\nwith the functors (\\ref{equation-reverse}) and\n\\ref{equation-formal-glueing-modules}).","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AEZ","source_file":"spaces-pushouts.tex","source_line":1886,"source_end_line":1895,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1886-L1895","statement_sha256":"8166072f930d24bae506bef4fc6b1952ceb793cdf97b9b376a2251cc49668d19","origin":"The Stacks Project","memory_eligible":false,"source_rank":12545,"rank":12545,"depth":60,"x":779.544,"y":1510.541,"cluster":"geometry-of-spaces"},{"id":"stacks:0AF0","tag":"0AF0","title":"Formal glueing of quasi-coherent modules · Proposition 0AF0","summary":"In Situation [Tag 0AEV] the functor ([Tag 0AEW]) is an equivalence with quasi-inverse given by ([Tag 0AEX]).","statement_latex":"In Situation \\ref{situation-formal-glueing} the functor\n(\\ref{equation-formal-glueing-modules}) is an equivalence\nwith quasi-inverse given by (\\ref{equation-reverse}).","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AF0","source_file":"spaces-pushouts.tex","source_line":1905,"source_end_line":1910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L1905-L1910","statement_sha256":"93e69625d68b4430f5f4588fd7b3e924bdd6a2c86fabdfb54a9df39cb6c5b489","origin":"The Stacks Project","memory_eligible":false,"source_rank":12546,"rank":12546,"depth":61,"x":678.537,"y":1722.532,"cluster":"geometry-of-spaces"},{"id":"stacks:0AFJ","tag":"0AFJ","title":"Formal glueing of quasi-coherent modules · Lemma 0AFJ","summary":"In Situation [Tag 0AEV] the functor Rf_* induces an equivalence between D_QCoh, |f^-1Z|(O_Y) and D_QCoh, |Z|(O_X) with quasi-inverse given by Lf^*.","statement_latex":"In Situation \\ref{situation-formal-glueing} the functor\n$Rf_*$ induces an equivalence between $D_{\\QCoh, |f^{-1}Z|}(\\mathcal{O}_Y)$\nand $D_{\\QCoh, |Z|}(\\mathcal{O}_X)$ with quasi-inverse given by\n$Lf^*$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFJ","source_file":"spaces-pushouts.tex","source_line":2026,"source_end_line":2032,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2026-L2032","statement_sha256":"6de32f7e7b3a5ce710d6d1eea841247e5b5e4757778c0aa47238dd27d2647686","origin":"The Stacks Project","memory_eligible":false,"source_rank":12547,"rank":12547,"depth":60,"x":582.476,"y":1508.755,"cluster":"geometry-of-spaces"},{"id":"stacks:0AF1","tag":"0AF1","title":"Formal glueing of quasi-coherent modules · Lemma 0AF1","summary":"In Situation [Tag 0AEV] there exists an fpqc covering (X_i → X)_i ∈ I refining the family (U → X, Y → X).","statement_latex":"In Situation \\ref{situation-formal-glueing} there exists an\nfpqc covering $\\{X_i \\to X\\}_{i \\in I}$ refining the\nfamily $\\{U \\to X, Y \\to X\\}$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AF1","source_file":"spaces-pushouts.tex","source_line":2054,"source_end_line":2059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2054-L2059","statement_sha256":"47dd098431a5d67e02386a90e7f16b91e801fd57639bd42196828f6a11776ed2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12548,"rank":12548,"depth":19,"x":825.387,"y":1611.953,"cluster":"geometry-of-spaces"},{"id":"stacks:0AF4","tag":"0AF4","title":"Formal glueing of algebraic spaces · Lemma 0AF4","summary":"In Situation [Tag 0AEV] the functor ([Tag 0AF3]) restricts to an equivalence • from the category of algebraic spaces affine over X to the full subcategory of Spaces(Y → X, Z) consisting of (U' ← V' → Y') with U' → U, V' → V, and Y' → Y affine, • from the category of closed immersions X' → X to the full subcategory of Spaces(Y → X, Z) consisting of (U' ← V' → Y') with U' → U, V' → V, and Y' → Y closed immersions, and • same statement as in (2) for finite morphisms.","statement_latex":"In Situation \\ref{situation-formal-glueing} the functor\n(\\ref{equation-formal-glueing-spaces}) restricts to an\nequivalence\n\\begin{enumerate}\n\\item from the category of algebraic spaces affine over $X$\nto the full subcategory of $\\textit{Spaces}(Y \\to X, Z)$ consisting\nof $(U' \\leftarrow V' \\rightarrow Y')$ with $U' \\to U$, $V' \\to V$,\nand $Y' \\to Y$ affine,\n\\item from the category of closed immersions $X' \\to X$\nto the full subcategory of $\\textit{Spaces}(Y \\to X, Z)$ consisting\nof $(U' \\leftarrow V' \\rightarrow Y')$ with $U' \\to U$, $V' \\to V$,\nand $Y' \\to Y$ closed immersions, and\n\\item same statement as in (2) for finite morphisms.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AF4","source_file":"spaces-pushouts.tex","source_line":2108,"source_end_line":2124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2108-L2124","statement_sha256":"e8026e3a7b613ea7d1792d09c3b91c33e77d8c1033d7d337ead9e3a10e001c58","origin":"The Stacks Project","memory_eligible":false,"source_rank":12549,"rank":12549,"depth":62,"x":563.103,"y":1673.731,"cluster":"geometry-of-spaces"},{"id":"stacks:0AF5","tag":"0AF5","title":"Formal glueing of algebraic spaces · Lemma 0AF5","summary":"In Situation [Tag 0AEV] the functor ([Tag 0AF3]) reflects isomorphisms.","statement_latex":"In Situation \\ref{situation-formal-glueing} the functor\n(\\ref{equation-formal-glueing-spaces}) reflects isomorphisms.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AF5","source_file":"spaces-pushouts.tex","source_line":2158,"source_end_line":2162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2158-L2162","statement_sha256":"9cf54d5e31dc9e6e85f2fdc7a171f3061382d857381b171c0d754704bc9abb95","origin":"The Stacks Project","memory_eligible":false,"source_rank":12550,"rank":12550,"depth":56,"x":706.923,"y":1479.221,"cluster":"geometry-of-spaces"},{"id":"stacks:0AF6","tag":"0AF6","title":"Formal glueing of algebraic spaces · Lemma 0AF6","summary":"In Situation [Tag 0AEV] the functor ([Tag 0AF3]) is fully faithful on algebraic spaces separated over X. More precisely, it induces a bijection Mor_X(X'_1, X'_2) → Mor_Spaces(Y → X, Z)(F(X'_1), F(X'_2)) whenever X'_2 → X is separated.","statement_latex":"In Situation \\ref{situation-formal-glueing} the functor\n(\\ref{equation-formal-glueing-spaces}) is fully faithful\non algebraic spaces separated over $X$. More precisely, it induces\na bijection\n$$\n\\Mor_X(X'_1, X'_2)\n\\longrightarrow\n\\Mor_{\\textit{Spaces}(Y \\to X, Z)}(F(X'_1), F(X'_2))\n$$\nwhenever $X'_2 \\to X$ is separated.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Formal glueing of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AF6","source_file":"spaces-pushouts.tex","source_line":2175,"source_end_line":2187,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2175-L2187","statement_sha256":"014e5fc859dc84fb1b33a999a08198d9197290d5426334061b6f774e0622b2f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12551,"rank":12551,"depth":63,"x":757.326,"y":1704.408,"cluster":"geometry-of-spaces"},{"id":"stacks:0F9P","tag":"0F9P","title":"Glueing and the Beauville-Laszlo theorem · Lemma 0F9P","summary":"Let (R → R', f) be a glueing pair, see above. Let Y be an algebraic space over X. The following are equivalent • there exists an étale covering (Y_i → Y)_i ∈ I with Y_i affine and Γ(Y_i, O_Y_i) glueable as an R-module, • for every étale morphism W → Y with W affine Γ(W, O_W) is a glueable R-module.","statement_latex":"Let $(R \\to R', f)$ be a glueing pair, see above. Let $Y$ be an algebraic\nspace over $X$. The following are equivalent\n\\begin{enumerate}\n\\item there exists an \\'etale covering $\\{Y_i \\to Y\\}_{i \\in I}$\nwith $Y_i$ affine and $\\Gamma(Y_i, \\mathcal{O}_{Y_i})$\nglueable as an $R$-module,\n\\item for every \\'etale morphism $W \\to Y$ with $W$ affine\n$\\Gamma(W, \\mathcal{O}_W)$ is a glueable $R$-module.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Glueing and the Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9P","source_file":"spaces-pushouts.tex","source_line":2270,"source_end_line":2281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2270-L2281","statement_sha256":"2c3383bef522a1f9f332869cc1f2c7c7d729270356c278b2f22406ae0dc1bdb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12552,"rank":12552,"depth":1,"x":538.924,"y":1566.866,"cluster":"geometry-of-spaces"},{"id":"stacks:0F9Q","tag":"0F9Q","title":"Glueing and the Beauville-Laszlo theorem · Lemma 0F9Q","summary":"Let (R → R', f) be a glueing pair, see above. The functor ([Tag 0F9N]) restricts to an equivalence between the category of affine Y/X which are glueable for (R → R', f) and the full subcategory of objects (V, V', Y') of Spaces(U ← U' → X') with V, V', Y' affine.","statement_latex":"Let $(R \\to R', f)$ be a glueing pair, see above.\nThe functor (\\ref{equation-beauville-laszlo-glueing-spaces})\nrestricts to an equivalence between the category of affine\n$Y/X$ which are glueable for $(R \\to R', f)$ and the\nfull subcategory of objects $(V, V', Y')$ of\n$\\textit{Spaces}(U \\leftarrow U' \\to X')$\nwith $V$, $V'$, $Y'$ affine.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Glueing and the Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9Q","source_file":"spaces-pushouts.tex","source_line":2329,"source_end_line":2338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2329-L2338","statement_sha256":"66f1c4be09ebb9e553d3cf95d43807aedd1d029bdd21964c06c36e4835bfc650","origin":"The Stacks Project","memory_eligible":false,"source_rank":12553,"rank":12553,"depth":11,"x":810.76,"y":1544.346,"cluster":"geometry-of-spaces"},{"id":"stacks:0F9R","tag":"0F9R","title":"Glueing and the Beauville-Laszlo theorem · Lemma 0F9R","summary":"Let P be one of the following properties of morphisms: \"finite\", \"closed immersion\", \"flat\", \"finite type\", \"flat and finite presentation\", \"étale\". Under the equivalence of Lemma [Tag 0F9Q] the morphisms having P correspond to morphisms of triples whose components have P.","statement_latex":"Let $P$ be one of the following properties of morphisms:\n``finite'', ``closed immersion'', ``flat'', ``finite type'',\n``flat and finite presentation'', ``\\'etale''.\nUnder the equivalence of Lemma \\ref{lemma-glueing-affines}\nthe morphisms having $P$ correspond to morphisms of triples\nwhose components have $P$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Glueing and the Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9R","source_file":"spaces-pushouts.tex","source_line":2364,"source_end_line":2372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2364-L2372","statement_sha256":"6b09724ac5715dc16d9181fed5a9ba03d5de5bbf184d1f8d02818bbb4f51a38e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12554,"rank":12554,"depth":46,"x":628.301,"y":1715.312,"cluster":"geometry-of-spaces"},{"id":"stacks:0F9S","tag":"0F9S","title":"Glueing and the Beauville-Laszlo theorem · Lemma 0F9S","summary":"Let (R → R', f) be a glueing pair, see above. The functor ([Tag 0F9N]) is faithful on the full subcategory of algebraic spaces Y/X glueable for (R → R', f).","statement_latex":"Let $(R \\to R', f)$ be a glueing pair, see above.\nThe functor (\\ref{equation-beauville-laszlo-glueing-spaces})\nis faithful on the full subcategory of\nalgebraic spaces $Y/X$ glueable for $(R \\to R', f)$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Glueing and the Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9S","source_file":"spaces-pushouts.tex","source_line":2430,"source_end_line":2436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2430-L2436","statement_sha256":"ff4951c6427ad544ffbf746a69ce3c1c6bd34ad2b305ae4a7f9f097a88114943","origin":"The Stacks Project","memory_eligible":false,"source_rank":12555,"rank":12555,"depth":53,"x":625.355,"y":1485.559,"cluster":"geometry-of-spaces"},{"id":"stacks:0F9T","tag":"0F9T","title":"Glueing and the Beauville-Laszlo theorem · Lemma 0F9T","summary":"Let (R → R', f) be a glueing pair, see above. The functor ([Tag 0F9N]) is fully faithful on the full subcategory of algebraic spaces Y/X which are (a) glueable for (R → R', f) and (b) have affine diagonal Y → Y ×_X Y.","statement_latex":"Let $(R \\to R', f)$ be a glueing pair, see above.\nThe functor (\\ref{equation-beauville-laszlo-glueing-spaces})\nis fully faithful on the full subcategory of\nalgebraic spaces $Y/X$ which are (a) glueable for $(R \\to R', f)$\nand (b) have affine diagonal $Y \\to Y \\times_X Y$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Glueing and the Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9T","source_file":"spaces-pushouts.tex","source_line":2468,"source_end_line":2475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2468-L2475","statement_sha256":"ffb01196c5f32232d2b95b72cf2e12d7403441dc99f42a9f6c9be5d7b605a0d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12556,"rank":12556,"depth":54,"x":812.412,"y":1653.415,"cluster":"geometry-of-spaces"},{"id":"stacks:0F9U","tag":"0F9U","title":"Glueing and the Beauville-Laszlo theorem · Lemma 0F9U","summary":"Let (R → R', f) be a glueing pair, see above. Any object (V, V', Y') of Spaces(U ← U' → X') with V, V', Y' quasi-affine is isomorphic to the image under the functor ([Tag 0F9N]) of a separated algebraic space Y over X.","statement_latex":"Let $(R \\to R', f)$ be a glueing pair, see above. Any object\n$(V, V', Y')$ of $\\textit{Spaces}(U \\leftarrow U' \\to X')$\nwith $V$, $V'$, $Y'$ quasi-affine is isomorphic to\nthe image under the functor (\\ref{equation-beauville-laszlo-glueing-spaces})\nof a separated algebraic space $Y$ over $X$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Glueing and the Beauville-Laszlo theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F9U","source_file":"spaces-pushouts.tex","source_line":2540,"source_end_line":2547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2540-L2547","statement_sha256":"f5281b6aaba381c769d683561724fdcedf89559104d475ac44d6e420e24ae9d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12557,"rank":12557,"depth":67,"x":539.312,"y":1635.77,"cluster":"geometry-of-spaces"},{"id":"stacks:0AGG","tag":"0AGG","title":"Coequalizers and glueing · Lemma 0AGG","summary":"Let S be a scheme. Let g : Y → X be a morphism of algebraic spaces over S. Assume X is locally Noetherian, and g is proper. Let R = Y ×_X Y with projection morphisms t, s : R → Y. There exists a coequalizer X' of s, t : R → Y in the category of algebraic spaces over S. Moreover • The morphism X' → X is finite. • The morphism Y → X' is proper. • The morphism Y → X' is surjective. • The morphism X' → X is universally injective. • If g is surjective, the morphism X' → X is a…","statement_latex":"Let $S$ be a scheme. Let\n$$\ng : Y \\longrightarrow X\n$$\nbe a morphism of algebraic spaces over $S$. Assume $X$ is locally Noetherian,\nand $g$ is proper. Let $R = Y \\times_X Y$ with projection morphisms\n$t, s : R \\to Y$. There exists a coequalizer $X'$ of $s, t : R \\to Y$\nin the category of algebraic spaces over $S$. Moreover\n\\begin{enumerate}\n\\item The morphism $X' \\to X$ is finite.\n\\item The morphism $Y \\to X'$ is proper.\n\\item The morphism $Y \\to X'$ is surjective.\n\\item The morphism $X' \\to X$ is universally injective.\n\\item If $g$ is surjective, the morphism $X' \\to X$\nis a universal homeomorphism.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Coequalizers and glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGG","source_file":"spaces-pushouts.tex","source_line":2645,"source_end_line":2663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2645-L2663","statement_sha256":"35adbc4e048cd28dd210927fb383d94410d9fe33edea934ec39de6f29072100e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12558,"rank":12558,"depth":71,"x":755.025,"y":1493.724,"cluster":"geometry-of-spaces"},{"id":"stacks:0AGI","tag":"0AGI","title":"Coequalizers and glueing · Lemma 0AGI","summary":"In Situation [Tag 0AGH] let Y = X' amalg Z and R = Y ×_X Y with projections t, s : R → Y. There exists a coequalizer X_1 of s, t : R → Y in the category of algebraic spaces over S. The morphism X_1 → X is a finite universal homeomorphism, an isomorphism over U, and Z → X lifts to X_1.","statement_latex":"In Situation \\ref{situation-coequalizer-glue} let $Y = X' \\amalg Z$ and\n$R = Y \\times_X Y$ with projections $t, s : R \\to Y$. There exists a\ncoequalizer $X_1$ of $s, t : R \\to Y$ in the category of algebraic spaces\nover $S$. The morphism $X_1 \\to X$ is a finite universal homeomorphism,\nan isomorphism over $U$, and $Z \\to X$ lifts to $X_1$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Coequalizers and glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGI","source_file":"spaces-pushouts.tex","source_line":2811,"source_end_line":2818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2811-L2818","statement_sha256":"3368cd97c3d1fe95bbdfde145b306f2a07b9a9015354050cf7430939a99445dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12559,"rank":12559,"depth":72,"x":710.162,"y":1721.018,"cluster":"geometry-of-spaces"},{"id":"stacks:0AGK","tag":"0AGK","title":"Coequalizers and glueing · Lemma 0AGK","summary":"In Situation [Tag 0AGH] assume X quasi-compact. In ([Tag 0AGJ]) for all n large enough, there exists an m such that X_n → X_n + m factors through a closed immersion X → X_n + m.","statement_latex":"In Situation \\ref{situation-coequalizer-glue} assume $X$ quasi-compact.\nIn (\\ref{equation-system-coequalizers}) for all $n$ large enough, there\nexists an $m$ such that $X_n \\to X_{n + m}$ factors through a\nclosed immersion $X \\to X_{n + m}$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Coequalizers and glueing","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGK","source_file":"spaces-pushouts.tex","source_line":2849,"source_end_line":2855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L2849-L2855","statement_sha256":"773f7cb828c449ac74573398e54a31e7551ca140a098c670b7e31030fe22962b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12560,"rank":12560,"depth":68,"x":560.361,"y":1527.83,"cluster":"geometry-of-spaces"},{"id":"stacks:0F45","tag":"0F45","title":"Compactifications · Lemma 0F45","summary":"Let S be a scheme. Let X → Y be a morphism of algebraic spaces over S. If (U ⊂ X, f : V → X) is an elementary distinguished square such that U → Y and V → Y are separated and U ×_X V → U ×_Y V is closed, then X → Y is separated.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a morphism of algebraic spaces over $S$.\nIf $(U \\subset X, f : V \\to X)$ is an elementary distinguished square\nsuch that $U \\to Y$ and $V \\to Y$ are separated and\n$U \\times_X V \\to U \\times_Y V$ is closed, then $X \\to Y$ is separated.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F45","source_file":"spaces-pushouts.tex","source_line":3001,"source_end_line":3007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L3001-L3007","statement_sha256":"58e112df55c2da3de0edc358fcd9825756051ab23a27a97559395d4c054569db","origin":"The Stacks Project","memory_eligible":false,"source_rank":12561,"rank":12561,"depth":0,"x":826.354,"y":1585.321,"cluster":"geometry-of-spaces"},{"id":"stacks:0F46","tag":"0F46","title":"Compactifications · Lemma 0F46","summary":"Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let U ⊂ X be a quasi-compact open. • If Z_1, Z_2 ⊂ X are closed subspaces of finite presentation such that Z_1 ∩ Z_2 ∩ U = ∅, then there exists a U-admissible blowing up X' → X such that the strict transforms of Z_1 and Z_2 are disjoint. • If T_1, T_2 ⊂ |U| are disjoint constructible closed subsets, then there is a U-admissible blowing up X' → X such that the closures of T_1 and T_2…","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $U \\subset X$ be a quasi-compact open.\n\\begin{enumerate}\n\\item If $Z_1, Z_2 \\subset X$ are closed subspaces of finite\npresentation such that $Z_1 \\cap Z_2 \\cap U = \\emptyset$, then\nthere exists a $U$-admissible blowing up $X' \\to X$\nsuch that the strict transforms of $Z_1$ and $Z_2$ are disjoint.\n\\item If $T_1, T_2 \\subset |U|$ are disjoint constructible closed\nsubsets, then there is a $U$-admissible blowing up $X' \\to X$\nsuch that the closures of $T_1$ and $T_2$ are disjoint.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F46","source_file":"spaces-pushouts.tex","source_line":3022,"source_end_line":3036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L3022-L3036","statement_sha256":"d31d7f5746dada9669aaf233329d488ef19060fa140aa0ebfadb1eaeb9fa9333","origin":"The Stacks Project","memory_eligible":false,"source_rank":12562,"rank":12562,"depth":64,"x":583.822,"y":1693.931,"cluster":"geometry-of-spaces"},{"id":"stacks:0F47","tag":"0F47","title":"Compactifications · Lemma 0F47","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of quasi-compact and quasi-separated algebraic spaces over S. Let V ⊂ Y be a quasi-compact open and U = f^-1(V). Let T ⊂ |V| be a closed subset such that f|_U : U → V is an isomorphism over an open neighbourhood of T in V. Then there exists a V-admissible blowing up Y' → Y such that the strict transform f' : X' → Y' of f is an isomorphism over an open neighbourhood of the closure of T in |Y'|.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper morphism of quasi-compact\nand quasi-separated algebraic spaces over $S$. Let $V \\subset Y$ be a\nquasi-compact open and $U = f^{-1}(V)$. Let $T \\subset |V|$ be a closed subset\nsuch that $f|_U : U \\to V$ is an isomorphism over an open neighbourhood of $T$\nin $V$. Then there exists a $V$-admissible blowing up $Y' \\to Y$ such that\nthe strict transform $f' : X' \\to Y'$ of $f$ is an isomorphism over an open\nneighbourhood of the closure of $T$ in $|Y'|$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F47","source_file":"spaces-pushouts.tex","source_line":3079,"source_end_line":3088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L3079-L3088","statement_sha256":"f67e3a3122bbd2e1f8e1761c1223699f7ff4de8584b7de3149b42f9130c04c3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12563,"rank":12563,"depth":72,"x":675.381,"y":1476.082,"cluster":"geometry-of-spaces"},{"id":"stacks:0F48","tag":"0F48","title":"Compactifications · Lemma 0F48","summary":"Let S be a scheme. Consider a diagram xymatrix X ar[d]_f & U ar[l] ar[d]_f|_U & A ar[d] ar[l] Y & V ar[l] & B ar[l] of quasi-compact and quasi-separated algebraic spaces over S. Assume • f is proper, • V is a quasi-compact open of Y, U = f^-1(V), • B ⊂ V and A ⊂ U are closed subspaces, • f|_A : A → B is an isomorphism, and f is étale at every point of A. Then there exists a V-admissible blowing up Y' → Y such that the strict transform f' : X' → Y' satisfies: for every…","statement_latex":"Let $S$ be a scheme. Consider a diagram\n$$\n\\xymatrix{\nX \\ar[d]_f & U \\ar[l] \\ar[d]_{f|_U} & A \\ar[d] \\ar[l] \\\\\nY & V \\ar[l] & B \\ar[l]\n}\n$$\nof quasi-compact and quasi-separated algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item $V$ is a quasi-compact open of $Y$, $U = f^{-1}(V)$,\n\\item $B \\subset V$ and $A \\subset U$ are closed subspaces,\n\\item $f|_A : A \\to B$ is an isomorphism, and\n$f$ is \\'etale at every point of $A$.\n\\end{enumerate}\nThen there exists a $V$-admissible blowing up $Y' \\to Y$ such that the strict\ntransform $f' : X' \\to Y'$ satisfies: for every geometric point\n$\\overline{a}$ of the closure of $|A|$ in $|X'|$\nthere exists a quotient $\\mathcal{O}_{X', \\overline{a}} \\to \\mathcal{O}$\nsuch that $\\mathcal{O}_{Y', f'(\\overline{a})} \\to \\mathcal{O}$\nis finite flat.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F48","source_file":"spaces-pushouts.tex","source_line":3122,"source_end_line":3146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L3122-L3146","statement_sha256":"0745684752177874a1e429ffc0a7eeba61d5acbede650ec1e58a535e3689a436","origin":"The Stacks Project","memory_eligible":false,"source_rank":12564,"rank":12564,"depth":0,"x":783.126,"y":1688.813,"cluster":"geometry-of-spaces"},{"id":"stacks:0F49","tag":"0F49","title":"Compactifications · Lemma 0F49","summary":"Let S be a scheme. Let X → B and Y → B be morphisms of algebraic spaces over S. Let U ⊂ X be an open subspace. Let V → X ×_B Y be a quasi-compact morphism whose composition with the first projection maps into U. Let Z ⊂ X ×_B Y be the scheme theoretic image of V → X ×_B Y. Let X' → X be a U-admissible blowup. Then the scheme theoretic image of V → X' ×_B Y is the strict transform of Z with respect to the blowing up.","statement_latex":"Let $S$ be a scheme. Let $X \\to B$ and $Y \\to B$ be morphisms of\nalgebraic spaces over $S$. Let $U \\subset X$ be an open subspace.\nLet $V \\to X \\times_B Y$ be a quasi-compact morphism\nwhose composition with the first projection maps into $U$.\nLet $Z \\subset X \\times_B Y$ be the scheme theoretic image of\n$V \\to X \\times_B Y$. Let $X' \\to X$ be a $U$-admissible blowup.\nThen the scheme theoretic image of $V \\to X' \\times_B Y$ is the\nstrict transform of $Z$ with respect to the blowing up.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F49","source_file":"spaces-pushouts.tex","source_line":3208,"source_end_line":3218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L3208-L3218","statement_sha256":"c7579d10af56122cdf3a5341d9dad358bd39813b6291c953db3e8419e97b5f8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12565,"rank":12565,"depth":57,"x":532.439,"y":1593.022,"cluster":"geometry-of-spaces"},{"id":"stacks:0F4A","tag":"0F4A","title":"Compactifications · Lemma 0F4A","summary":"Let S be a scheme. Let B be a quasi-compact and quasi-separated algebraic space over S. Let U be an algebraic space of finite type and separated over B. Let V → U be an étale morphism. If V has a compactification V ⊂ Y over B, then there exists a V-admissible blowing up Y' → Y and an open V ⊂ V' ⊂ Y' such that V → U extends to a proper morphism V' → U.","statement_latex":"Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $U$ be an algebraic space\nof finite type and separated over $B$. Let $V \\to U$ be an \\'etale morphism.\nIf $V$ has a compactification $V \\subset Y$ over $B$, then there\nexists a $V$-admissible blowing up $Y' \\to Y$ and an\nopen $V \\subset V' \\subset Y'$ such that $V \\to U$\nextends to a proper morphism $V' \\to U$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4A","source_file":"spaces-pushouts.tex","source_line":3241,"source_end_line":3250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L3241-L3250","statement_sha256":"64b6777ed8c70707933c19e4b1c48dad2ac211c2b740147230cd0dc445c42da1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12566,"rank":12566,"depth":72,"x":794.496,"y":1521.364,"cluster":"geometry-of-spaces"},{"id":"stacks:0F4B","tag":"0F4B","title":"Compactifications · Lemma 0F4B","summary":"Let B be an algebraic space of finite type over Z. Let U be an algebraic space of finite type and separated over B. Let (U_2 ⊂ U, f : U_1 → U) be an elementary distinguished square. Assume U_1 and U_2 have compactifications over B and U_1 ×_U U_2 → U has dense image. Then U has a compactification over B.","statement_latex":"Let $B$ be an algebraic space of finite type over $\\mathbf{Z}$.\nLet $U$ be an algebraic space of finite type and separated over $B$.\nLet $(U_2 \\subset U, f : U_1 \\to U)$ be an\nelementary distinguished square. Assume $U_1$ and $U_2$ have\ncompactifications over $B$ and $U_1 \\times_U U_2 \\to U$ has dense image.\nThen $U$ has a compactification over $B$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4B","source_file":"spaces-pushouts.tex","source_line":3275,"source_end_line":3283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L3275-L3283","statement_sha256":"1a3b70d95a747e6be9a7a1c58dc3ac43ef2dc13fe1f86c960b98eadf4f8b916d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12567,"rank":12567,"depth":73,"x":658.795,"y":1723.033,"cluster":"geometry-of-spaces"},{"id":"stacks:0F4C","tag":"0F4C","title":"Compactifications · Lemma 0F4C","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let U ⊂ X be a proper dense open subspace. Then there exists an affine scheme V and an étale morphism V → X such that • the open subspace W = U ∪ Im(V → X) is strictly larger than U, • (U ⊂ W, V → W) is a distinguished square, and • U ×_W V → U has dense image.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $U \\subset X$ be a proper dense open subspace. Then there exists an\naffine scheme $V$ and an \\'etale morphism $V \\to X$ such that\n\\begin{enumerate}\n\\item the open subspace $W = U \\cup \\Im(V \\to X)$ is strictly larger\nthan $U$,\n\\item $(U \\subset W, V \\to W)$ is a distinguished square, and\n\\item $U \\times_W V \\to U$ has dense image.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Compactifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4C","source_file":"spaces-pushouts.tex","source_line":3540,"source_end_line":3551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L3540-L3551","statement_sha256":"c2128dd715497024a58b43ce64bcc8cbc55fbd0c4ebbc3b64262e51f79e532e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12568,"rank":12568,"depth":59,"x":596.642,"y":1497.178,"cluster":"geometry-of-spaces"},{"id":"stacks:0F4D","tag":"0F4D","title":"Compactifications · Theorem 0F4D","summary":"[CLO] Let S be a scheme. Let B be a quasi-compact and quasi-separated algebraic space over S. Let X → B be a separated, finite type morphism. Then X has a compactification over B.","statement_latex":"\\begin{reference}\n\\cite{CLO}\n\\end{reference}\nLet $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated\nalgebraic space over $S$. Let $X \\to B$ be a separated, finite type morphism.\nThen $X$ has a compactification over $B$.","area":"Geometry of Spaces","chapter":"Pushouts of Algebraic Spaces","chapter_id":"spaces-pushouts","section":"Compactifications","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F4D","source_file":"spaces-pushouts.tex","source_line":3581,"source_end_line":3589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-pushouts.tex#L3581-L3589","statement_sha256":"3e3a99dc16135d0a0e9793e8c4200b2fe981394d91afe379a1de0d44bd3eb46f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12569,"rank":12569,"depth":74,"x":824.248,"y":1628.538,"cluster":"geometry-of-spaces"},{"id":"stacks:0EDW","tag":"0EDW","title":"Setup · Lemma 0EDW","summary":"In Situation [Tag 0EDT] assume B is Jacobson and that δ(b) = 0 for every closed point b of |B|. Let X/B be good. If Z ⊂ X is an integral closed subspace with generic point xi ∈ |Z|, then the following integers are the same: • δ(xi) = δ_X/B(xi), • dim(|Z|), • codim((z), |Z|) for z ∈ |Z| closed, • the dimension of the local ring of Z at z for z ∈ |Z| closed, and • dim(O_Z, overlinez) for z ∈ |Z| closed.","statement_latex":"In Situation \\ref{situation-setup} assume $B$ is Jacobson\nand that $\\delta(b) = 0$ for every closed point $b$ of $|B|$.\nLet $X/B$ be good. If $Z \\subset X$ is an integral closed subspace\nwith generic point $\\xi \\in |Z|$, then the following integers are the same:\n\\begin{enumerate}\n\\item $\\delta(\\xi) = \\delta_{X/B}(\\xi)$,\n\\item $\\dim(|Z|)$,\n\\item $\\text{codim}(\\{z\\}, |Z|)$ for $z \\in |Z|$ closed,\n\\item the dimension of the local ring of $Z$ at $z$ for\n$z \\in |Z|$ closed, and\n\\item $\\dim(\\mathcal{O}_{Z, \\overline{z}})$ for $z \\in |Z|$ closed.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Setup","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDW","source_file":"spaces-chow.tex","source_line":110,"source_end_line":124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L110-L124","statement_sha256":"ded014ac05854d6547f5c93bf08422faa92663b0da39e4e60d1bdf726a95f730","origin":"The Stacks Project","memory_eligible":false,"source_rank":12570,"rank":12570,"depth":61,"x":550.6,"y":1660.846,"cluster":"geometry-of-spaces"},{"id":"stacks:0EDX","tag":"0EDX","title":"Setup · Definition 0EDX","summary":"In Situation [Tag 0EDT] for any good X/B and any irreducible closed subset T ⊂ |X| we define dim_δ(T) = δ(xi) where xi ∈ T is the generic point of T. We will call this the δ-dimension of T. If T ⊂ |X| is any closed subset, then we define dim_δ(T) as the supremum of the δ-dimensions of the irreducible components of T. If Z is a closed subspace of X, then we set dim_δ(Z) = dim_δ(|Z|).","statement_latex":"In Situation \\ref{situation-setup} for any good $X/B$\nand any irreducible closed subset $T \\subset |X|$ we define\n$$\n\\dim_\\delta(T) = \\delta(\\xi)\n$$\nwhere $\\xi \\in T$ is the generic point of $T$.\nWe will call this the {\\it $\\delta$-dimension of $T$}.\nIf $T \\subset |X|$ is any closed subset, then we define\n$\\dim_\\delta(T)$ as the supremum of the $\\delta$-dimensions\nof the irreducible components of $T$.\nIf $Z$ is a closed subspace of $X$, then we set\n$\\dim_\\delta(Z) = \\dim_\\delta(|Z|)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Setup","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDX","source_file":"spaces-chow.tex","source_line":171,"source_end_line":185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L171-L185","statement_sha256":"c2441d2b58cabe35ed31424320f451c85beb1cc1dd8c72fe25e9c47592e50f9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12571,"rank":12571,"depth":0,"x":726.518,"y":1481.631,"cluster":"geometry-of-spaces"},{"id":"stacks:0EDZ","tag":"0EDZ","title":"Cycles · Definition 0EDZ","summary":"In Situation [Tag 0EDT] let X/B be good. Let k ∈ Z. • A cycle on X is a formal sum α = ∑ n_Z [Z] where the sum is over integral closed subspaces Z ⊂ X, each n_Z ∈ Z, and (|Z|; n_Z not = 0) is a locally finite collection of subsets of |X| (Topology, Definition [Tag 0BDS]). • A k-cycle on X is a cycle α = ∑ n_Z [Z] where n_Z not = 0 ⇒ dim_δ(Z) = k. • The abelian group of all k-cycles on X is denoted Z_k(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $k \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item A {\\it cycle on $X$} is a formal sum\n$$\n\\alpha = \\sum n_Z [Z]\n$$\nwhere the sum is over integral closed subspaces $Z \\subset X$,\neach $n_Z \\in \\mathbf{Z}$, and\n$\\{|Z|; n_Z \\not = 0\\}$ is a locally finite\ncollection of subsets of $|X|$\n(Topology, Definition \\ref{topology-definition-locally-finite}).\n\\item A {\\it $k$-cycle} on $X$ is\na cycle\n$$\n\\alpha = \\sum n_Z [Z]\n$$\nwhere $n_Z \\not = 0 \\Rightarrow \\dim_\\delta(Z) = k$.\n\\item The abelian group of all $k$-cycles on $X$ is denoted $Z_k(X)$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EDZ","source_file":"spaces-chow.tex","source_line":210,"source_end_line":232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L210-L232","statement_sha256":"7163d08d5f7c7a97c6e0bab9fff999249b3eb6ad38a36855fe7c8367422ca381","origin":"The Stacks Project","memory_eligible":false,"source_rank":12572,"rank":12572,"depth":1,"x":740.927,"y":1713.753,"cluster":"geometry-of-spaces"},{"id":"stacks:0EE1","tag":"0EE1","title":"Multiplicities · Lemma 0EE1","summary":"Let S be a scheme and let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Let x ∈ |X|. Let d ∈ (0, 1, 2, …, ∞). The following are equivalent • length_O_X, overlinex F_overlinex = d • for some étale morphism U → X with U a scheme and u ∈ U mapping to x we have length_O_U, u (F|_U)_u = d • for any étale morphism U → X with U a scheme and u ∈ U mapping to x we have length_O_U, u (F|_U)_u = d","statement_latex":"Let $S$ be a scheme and let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in |X|$. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item\n$\\text{length}_{\\mathcal{O}_{X, \\overline{x}}} \\mathcal{F}_{\\overline{x}} = d$\n\\item for some \\'etale morphism $U \\to X$ with $U$ a scheme\nand $u \\in U$ mapping to $x$ we have\n$\\text{length}_{\\mathcal{O}_{U, u}} (\\mathcal{F}|_U)_u = d$\n\\item for any \\'etale morphism $U \\to X$ with $U$ a scheme\nand $u \\in U$ mapping to $x$ we have\n$\\text{length}_{\\mathcal{O}_{U, u}} (\\mathcal{F}|_U)_u = d$\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EE1","source_file":"spaces-chow.tex","source_line":254,"source_end_line":270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L254-L270","statement_sha256":"7da054eb178465fe0b2d058a4e79fd4491ef1cfe319bf3b6a49354234980f07c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12573,"rank":12573,"depth":54,"x":543.508,"y":1550.66,"cluster":"geometry-of-spaces"},{"id":"stacks:0EE2","tag":"0EE2","title":"Multiplicities · Definition 0EE2","summary":"Let S be a scheme and let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. Let x ∈ |X|. Let d ∈ (0, 1, 2, …, ∞). We say F has length d at x if the equivalent conditions of Lemma [Tag 0EE1] are satisfied.","statement_latex":"Let $S$ be a scheme and let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nLet $x \\in |X|$. Let $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$.\nWe say {\\it $\\mathcal{F}$ has length $d$ at $x$}\nif the equivalent conditions of Lemma \\ref{lemma-length}\nare satisfied.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Multiplicities","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EE2","source_file":"spaces-chow.tex","source_line":293,"source_end_line":301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L293-L301","statement_sha256":"5c2dc788a91f530cd557a19b528fb2ce2299cef2d5048f26d63946c5720fda5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12574,"rank":12574,"depth":55,"x":820.416,"y":1558.907,"cluster":"geometry-of-spaces"},{"id":"stacks:0EE3","tag":"0EE3","title":"Multiplicities · Lemma 0EE3","summary":"Let S be a scheme. Let i : Y → X be a closed immersion of algebraic spaces over S. Let G be a quasi-coherent O_Y-module. Let y ∈ |Y| with image x ∈ |X|. Let d ∈ (0, 1, 2, …, ∞). The following are equivalent • G has length d at y, and • i_*G has length d at x.","statement_latex":"Let $S$ be a scheme. Let $i : Y \\to X$ be a closed immersion of\nalgebraic spaces over $S$. Let $\\mathcal{G}$ be a quasi-coherent\n$\\mathcal{O}_Y$-module. Let $y \\in |Y|$ with image $x \\in |X|$.\nLet $d \\in \\{0, 1, 2, \\ldots, \\infty\\}$. The following are\nequivalent\n\\begin{enumerate}\n\\item $\\mathcal{G}$ has length $d$ at $y$, and\n\\item $i_*\\mathcal{G}$ has length $d$ at $x$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EE3","source_file":"spaces-chow.tex","source_line":303,"source_end_line":314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L303-L314","statement_sha256":"c2e6fd4cc523f7b26b7e77234823771015be5833f9b2195ab1a7267ac20ae5db","origin":"The Stacks Project","memory_eligible":false,"source_rank":12575,"rank":12575,"depth":4,"x":609.459,"y":1710.048,"cluster":"geometry-of-spaces"},{"id":"stacks:0EE4","tag":"0EE4","title":"Multiplicities · Lemma 0EE4","summary":"Let S be a scheme and let X be a locally Noetherian algebraic space over S. Let F be a coherent O_X-module. Let x ∈ |X|. The following are equivalent • for some étale morphism U → X with U a scheme and u ∈ U mapping to x we have u is a generic point of an irreducible component of Supp(F|_U), • for any étale morphism U → X with U a scheme and u ∈ U mapping to x we have u is a generic point of an irreducible component of Supp(F|_U), • the length of F at x is finite and…","statement_latex":"Let $S$ be a scheme and let $X$ be a\nlocally Noetherian algebraic space over $S$.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\nLet $x \\in |X|$. The following are equivalent\n\\begin{enumerate}\n\\item for some \\'etale morphism $U \\to X$ with $U$ a scheme\nand $u \\in U$ mapping to $x$ we have $u$ is a generic point\nof an irreducible component of $\\text{Supp}(\\mathcal{F}|_U)$,\n\\item for any \\'etale morphism $U \\to X$ with $U$ a scheme\nand $u \\in U$ mapping to $x$ we have $u$ is a generic point\nof an irreducible component of $\\text{Supp}(\\mathcal{F}|_U)$,\n\\item the length of $\\mathcal{F}$ at $x$ is finite and nonzero.\n\\end{enumerate}\nIf $X$ is decent (equivalently quasi-separated) then these are\nalso equivalent to\n\\begin{enumerate}\n\\item[(4)] $x$ is a generic point of an irreducible component of\n$\\text{Supp}(\\mathcal{F})$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EE4","source_file":"spaces-chow.tex","source_line":341,"source_end_line":362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L341-L362","statement_sha256":"cc3ddb0cd8840cdc6b379a91c0cb1ddfe40c7bbe6fe20fbabcdf24b1694cfa90","origin":"The Stacks Project","memory_eligible":false,"source_rank":12576,"rank":12576,"depth":60,"x":643.494,"y":1478.748,"cluster":"geometry-of-spaces"},{"id":"stacks:0EE6","tag":"0EE6","title":"Multiplicities · Lemma 0EE6","summary":"In Situation [Tag 0EDT] let X/B be good. Let T ⊂ |X| be a closed subset and t ∈ T. If dim_δ(T) ≤ k and δ(t) = k, then t is a generic point of an irreducible component of T.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $T \\subset |X|$ be a closed subset and $t \\in T$.\nIf $\\dim_\\delta(T) \\leq k$ and $\\delta(t) = k$, then\n$t$ is a generic point of an irreducible component of $T$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EE6","source_file":"spaces-chow.tex","source_line":408,"source_end_line":414,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L408-L414","statement_sha256":"12f61c260c71a0a4421b4b89d387c96961d81ff3d6181ebf3d41901724d8cdeb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12577,"rank":12577,"depth":0,"x":804.507,"y":1668.74,"cluster":"geometry-of-spaces"},{"id":"stacks:0EE9","tag":"0EE9","title":"Cycle associated to a closed subspace · Definition 0EE9","summary":"In Situation [Tag 0EDT] let X/B be good. Let Y ⊂ X be a closed subspace. • For an irreducible component Z ⊂ Y with generic point xi the length of O_Y at xi (Definition [Tag 0EE2]) is called the multiplicity of Z in Y. By Lemma [Tag 0EE4] applied to O_Y on Y this is a positive integer. • Assume dim_δ(Y) ≤ k. The k-cycle associated to Y is [Y]_k = ∑ m_Z, Y[Z] where the sum is over the irreducible components Z of Y of δ-dimension k and m_Z, Y is the multiplicity of Z in Y.…","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $Y \\subset X$ be a closed subspace.\n\\begin{enumerate}\n\\item For an irreducible component $Z \\subset Y$ with generic point $\\xi$\nthe length of $\\mathcal{O}_Y$ at $\\xi$\n(Definition \\ref{definition-length-at-x}) is called the\n{\\it multiplicity of $Z$ in $Y$}.\nBy Lemma \\ref{lemma-length-finite} applied to $\\mathcal{O}_Y$\non $Y$ this is a positive integer.\n\\item Assume $\\dim_\\delta(Y) \\leq k$.\nThe {\\it $k$-cycle associated to $Y$} is\n$$\n[Y]_k = \\sum m_{Z, Y}[Z]\n$$\nwhere the sum is over the irreducible components $Z$ of $Y$\nof $\\delta$-dimension $k$ and $m_{Z, Y}$ is the multiplicity\nof $Z$ in $Y$.\nThis is a $k$-cycle by Spaces over Fields, Lemma\n\\ref{spaces-over-fields-lemma-components-locally-finite}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Cycle associated to a closed subspace","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EE9","source_file":"spaces-chow.tex","source_line":449,"source_end_line":471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L449-L471","statement_sha256":"83d0f47cde331768f4fdf6f3d1b28a4932c31119f442a0dc128f2d14a3a877c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12578,"rank":12578,"depth":61,"x":532.817,"y":1619.973,"cluster":"geometry-of-spaces"},{"id":"stacks:0EEB","tag":"0EEB","title":"Cycle associated to a coherent sheaf · Definition 0EEB","summary":"In Situation [Tag 0EDT] let X/B be good. Let F be a coherent O_X-module. • For an integral closed subspace Z ⊂ X with generic point xi such that |Z| is an irreducible component of Supp(F) the length of F at xi (Definition [Tag 0EE2]) is called the multiplicity of Z in F. By Lemma [Tag 0EE4] this is a positive integer. • Assume dim_δ(Supp(F)) ≤ k. The k-cycle associated to F is [F]_k = ∑ m_Z, F[Z] where the sum is over the integral closed subspaces Z ⊂ X corresponding to…","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item For an integral closed subspace $Z \\subset X$ with generic point $\\xi$\nsuch that $|Z|$ is an irreducible component of $\\text{Supp}(\\mathcal{F})$\nthe length of $\\mathcal{F}$ at $\\xi$ (Definition \\ref{definition-length-at-x})\nis called the {\\it multiplicity of $Z$ in $\\mathcal{F}$}.\nBy Lemma \\ref{lemma-length-finite} this is a positive integer.\n\\item Assume $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.\nThe {\\it $k$-cycle associated to $\\mathcal{F}$} is\n$$\n[\\mathcal{F}]_k = \\sum m_{Z, \\mathcal{F}}[Z]\n$$\nwhere the sum is over the integral closed subspaces $Z \\subset X$\ncorresponding to irreducible components of\n$\\text{Supp}(\\mathcal{F})$ of $\\delta$-dimension $k$\nand $m_{Z, \\mathcal{F}}$ is the multiplicity of $Z$ in $\\mathcal{F}$.\nThis is a $k$-cycle by Spaces over Fields, Lemma\n\\ref{spaces-over-fields-lemma-components-locally-finite}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Cycle associated to a coherent sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEB","source_file":"spaces-chow.tex","source_line":491,"source_end_line":513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L491-L513","statement_sha256":"676207f7808fb9dd02b902e6e1ab577f05e0d9f1cc1019462cdb4cfd00a43391","origin":"The Stacks Project","memory_eligible":false,"source_rank":12579,"rank":12579,"depth":61,"x":772.528,"y":1501.693,"cluster":"geometry-of-spaces"},{"id":"stacks:0EEC","tag":"0EEC","title":"Cycle associated to a coherent sheaf · Lemma 0EEC","summary":"In Situation [Tag 0EDT] let X/B be good. Let F be a coherent O_X-module with dim_δ(Supp(F)) ≤ k. Let Z be an integral closed subspace of X with dim_δ(Z) = k. Let xi ∈ |Z| be the generic point. Then the coefficient of Z in [F]_k is the length of F at xi.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{F}$ be a coherent $\\mathcal{O}_X$-module\nwith $\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.\nLet $Z$ be an integral closed subspace of $X$ with $\\dim_\\delta(Z) = k$.\nLet $\\xi \\in |Z|$ be the generic point.\nThen the coefficient of $Z$ in $[\\mathcal{F}]_k$\nis the length of $\\mathcal{F}$ at $\\xi$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Cycle associated to a coherent sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEC","source_file":"spaces-chow.tex","source_line":523,"source_end_line":532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L523-L532","statement_sha256":"7df799d6cb6e5e9a93e2e98ab3712070d4bc180b4bde0525e8d881990cde39b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12580,"rank":12580,"depth":62,"x":690.834,"y":1725.073,"cluster":"geometry-of-spaces"},{"id":"stacks:0EED","tag":"0EED","title":"Cycle associated to a coherent sheaf · Lemma 0EED","summary":"In Situation [Tag 0EDT] let X/B be good. Let Y ⊂ X be a closed subspace. If dim_δ(Y) ≤ k, then [Y]_k = [i_*O_Y]_k where i : Y → X is the inclusion morphism.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $Y \\subset X$ be a closed subspace.\nIf $\\dim_\\delta(Y) \\leq k$, then $[Y]_k = [i_*\\mathcal{O}_Y]_k$\nwhere $i : Y \\to X$ is the inclusion morphism.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Cycle associated to a coherent sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EED","source_file":"spaces-chow.tex","source_line":544,"source_end_line":550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L544-L550","statement_sha256":"f50dc8946d92ddc69064f63794a56ee5ab1aabc3f43974231f3376b08610792c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12581,"rank":12581,"depth":5,"x":571.361,"y":1513.865,"cluster":"geometry-of-spaces"},{"id":"stacks:0EEE","tag":"0EEE","title":"Cycle associated to a coherent sheaf · Lemma 0EEE","summary":"In Situation [Tag 0EDT] let X/B be good. Let 0 → F → G → H → 0 be a short exact sequence of coherent O_X-modules. Assume that the δ-dimension of the supports of F, G, and H are ≤ k. Then [G]_k = [F]_k + [H]_k.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $0 \\to \\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H} \\to 0$\nbe a short exact sequence of coherent $\\mathcal{O}_X$-modules.\nAssume that the $\\delta$-dimension of the supports\nof $\\mathcal{F}$, $\\mathcal{G}$, and $\\mathcal{H}$ are $\\leq k$.\nThen $[\\mathcal{G}]_k = [\\mathcal{F}]_k + [\\mathcal{H}]_k$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Cycle associated to a coherent sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEE","source_file":"spaces-chow.tex","source_line":564,"source_end_line":572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L564-L572","statement_sha256":"eb227698012b1d9c1009e93ef6ef8466ebff6c2590db2b53b6aed171885d8611","origin":"The Stacks Project","memory_eligible":false,"source_rank":12582,"rank":12582,"depth":63,"x":829.472,"y":1601.872,"cluster":"geometry-of-spaces"},{"id":"stacks:0EEG","tag":"0EEG","title":"Preparation for proper pushforward · Lemma 0EEG","summary":"In Situation [Tag 0EDT] let X,Y/B be good and let f : X → Y be a morphism over B. If Z ⊂ X is an integral closed subspace, then there exists a unique integral closed subspace Z' ⊂ Y such that there is a commutative diagram xymatrix Z ar[r] ar[d] & X ar[d]^f Z' ar[r] & Y with Z → Z' dominant. If f is proper, then Z → Z' is proper and surjective.","statement_latex":"In Situation \\ref{situation-setup} let $X,Y/B$ be good and let $f : X \\to Y$\nbe a morphism over $B$. If $Z \\subset X$ is an integral closed subspace, then\nthere exists a unique integral closed subspace $Z' \\subset Y$ such that there\nis a commutative diagram\n$$\n\\xymatrix{\nZ \\ar[r] \\ar[d] & X \\ar[d]^f \\\\\nZ' \\ar[r] & Y\n}\n$$\nwith $Z \\to Z'$ dominant. If $f$ is proper, then $Z \\to Z'$ is proper\nand surjective.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Preparation for proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEG","source_file":"spaces-chow.tex","source_line":602,"source_end_line":616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L602-L616","statement_sha256":"d0ff57207c4a8fd4ff3c80b91799b9f7722bc944778decb10fcb51cd34da624f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12583,"rank":12583,"depth":58,"x":568.205,"y":1683.487,"cluster":"geometry-of-spaces"},{"id":"stacks:0ENY","tag":"0ENY","title":"Preparation for proper pushforward · Lemma 0ENY","summary":"In Situation [Tag 0EDT] let X, Y/B be good and let f : X → Y be a morphism over B. Assume X, Y integral and dim_δ(X) = dim_δ(Y). Then either f factors through a proper closed subspace of Y, or f is dominant and the extension of function fields R(X) / R(Y) is finite.","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good and let\n$f : X \\to Y$ be a morphism over $B$.\nAssume $X$, $Y$ integral and $\\dim_\\delta(X) = \\dim_\\delta(Y)$.\nThen either $f$ factors through a proper closed subspace\nof $Y$, or $f$ is dominant and the extension of function fields\n$R(X) / R(Y)$ is finite.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Preparation for proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENY","source_file":"spaces-chow.tex","source_line":668,"source_end_line":676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L668-L676","statement_sha256":"8bd33ed7b0f883d350d1b9fd0baf2a39c4f76435bfce1543b9df2ee8431843dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12584,"rank":12584,"depth":63,"x":695.308,"y":1474.923,"cluster":"geometry-of-spaces"},{"id":"stacks:0ENZ","tag":"0ENZ","title":"Preparation for proper pushforward · Lemma 0ENZ","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a morphism over B. Assume f is quasi-compact, and (T_i)_i ∈ I is a locally finite collection of closed subsets of |X|. Then (overline|f|(T_i))_i ∈ I is a locally finite collection of closed subsets of |Y|.","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a morphism over $B$.\nAssume $f$ is quasi-compact, and $\\{T_i\\}_{i \\in I}$ is a locally\nfinite collection of closed subsets of $|X|$.\nThen $\\{\\overline{|f|(T_i)}\\}_{i \\in I}$ is a locally finite\ncollection of closed subsets of $|Y|$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Preparation for proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ENZ","source_file":"spaces-chow.tex","source_line":690,"source_end_line":698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L690-L698","statement_sha256":"de5b57bd5b1fe4d3efe5c95ed8109bb6b2260819a49c15c8836e565f4064c930","origin":"The Stacks Project","memory_eligible":false,"source_rank":12585,"rank":12585,"depth":9,"x":769.353,"y":1700.981,"cluster":"geometry-of-spaces"},{"id":"stacks:0EP1","tag":"0EP1","title":"Proper pushforward · Definition 0EP1","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a morphism over B. Assume f is proper. • Let Z ⊂ X be an integral closed subspace with dim_δ(Z) = k. Let Z' ⊂ Y be the image of Z as in Lemma [Tag 0EEG]. We define f_*[Z] = ( 0 & if & dim_δ(Z')< k, deg(Z/Z') [Z'] & if & dim_δ(Z') = k. . The degree of Z over Z' is defined and finite if dim_δ(Z') = dim_δ(Z) by Lemma [Tag 0ENY] and Spaces over Fields, Definition [Tag 0AD6]. • Let α = ∑ n_Z [Z] be a k-cycle on X.…","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a morphism over $B$.\nAssume $f$ is proper.\n\\begin{enumerate}\n\\item Let $Z \\subset X$ be an integral closed subspace\nwith $\\dim_\\delta(Z) = k$. Let $Z' \\subset Y$ be the\nimage of $Z$ as in Lemma \\ref{lemma-proper-image}.\nWe define\n$$\nf_*[Z] =\n\\left\\{\n\\begin{matrix}\n0 & \\text{if} & \\dim_\\delta(Z')< k, \\\\\n\\deg(Z/Z') [Z'] & \\text{if} & \\dim_\\delta(Z') = k.\n\\end{matrix}\n\\right.\n$$\nThe degree of $Z$ over $Z'$ is defined and finite if\n$\\dim_\\delta(Z') = \\dim_\\delta(Z)$ by Lemma \\ref{lemma-equal-dimension} and\nSpaces over Fields, Definition \\ref{spaces-over-fields-definition-degree}.\n\\item Let $\\alpha = \\sum n_Z [Z]$ be a $k$-cycle on $X$. The\n{\\it pushforward} of $\\alpha$ as the sum\n$$\nf_* \\alpha = \\sum n_Z f_*[Z]\n$$\nwhere each $f_*[Z]$ is defined as above. The sum is locally finite\nby Lemma \\ref{lemma-quasi-compact-locally-finite} above.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Proper pushforward","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EP1","source_file":"spaces-chow.tex","source_line":732,"source_end_line":762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L732-L762","statement_sha256":"0534b510a2ecc906100e2897d58b3dea1bc02e1448afe49725ef48242d2562fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":12586,"rank":12586,"depth":64,"x":532.813,"y":1576.224,"cluster":"geometry-of-spaces"},{"id":"stacks:0EP2","tag":"0EP2","title":"Proper pushforward · Lemma 0EP2","summary":"In Situation [Tag 0EDT] let X, Y, Z/B be good. Let f : X → Y and g : Y → Z be proper morphisms over B. Then g_* ∘ f_* = (g ∘ f)_* as maps Z_k(X) → Z_k(Z).","statement_latex":"In Situation \\ref{situation-setup} let $X, Y, Z/B$ be good.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be proper morphisms over $B$.\nThen $g_* \\circ f_* = (g \\circ f)_*$ as maps $Z_k(X) \\to Z_k(Z)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EP2","source_file":"spaces-chow.tex","source_line":774,"source_end_line":779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L774-L779","statement_sha256":"d0cbb77b4c1dc1015f5aa92a5c8a400ef56309660e4006fb75d93556ebc8bd48","origin":"The Stacks Project","memory_eligible":false,"source_rank":12587,"rank":12587,"depth":64,"x":807.735,"y":1533.972,"cluster":"geometry-of-spaces"},{"id":"stacks:0EP3","tag":"0EP3","title":"Proper pushforward · Lemma 0EP3","summary":"In Situation [Tag 0EDT] let f : X → Y be a proper morphism of good algebraic spaces over B. • Let Z ⊂ X be a closed subspace with dim_δ(Z) ≤ k. Then f_*[Z]_k = [f_* O_Z]_k. • Let F be a coherent sheaf on X such that dim_δ(Supp(F)) ≤ k. Then f_*[F]_k = [f_* F]_k. Note that the statement makes sense since f_*F and f_*O_Z are coherent O_Y-modules by Cohomology of Spaces, Lemma [Tag 08AR].","statement_latex":"In Situation \\ref{situation-setup} let $f : X \\to Y$ be a proper morphism\nof good algebraic spaces over $B$.\n\\begin{enumerate}\n\\item Let $Z \\subset X$ be a closed subspace with $\\dim_\\delta(Z) \\leq k$.\nThen\n$$\nf_*[Z]_k = [f_*{\\mathcal O}_Z]_k.\n$$\n\\item Let $\\mathcal{F}$ be a coherent sheaf on $X$ such that\n$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$. Then\n$$\nf_*[\\mathcal{F}]_k = [f_*{\\mathcal F}]_k.\n$$\n\\end{enumerate}\nNote that the statement makes sense since $f_*\\mathcal{F}$ and\n$f_*\\mathcal{O}_Z$ are coherent $\\mathcal{O}_Y$-modules by\nCohomology of Spaces, Lemma\n\\ref{spaces-cohomology-lemma-proper-pushforward-coherent}.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Proper pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EP3","source_file":"spaces-chow.tex","source_line":803,"source_end_line":823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L803-L823","statement_sha256":"abde6924737101bd7a373eaec0757c21a50cbf5785cf4a4f758322a5df843cd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12588,"rank":12588,"depth":74,"x":638.882,"y":1721.245,"cluster":"geometry-of-spaces"},{"id":"stacks:0EP5","tag":"0EP5","title":"Preparation for flat pullback · Lemma 0EP5","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a morphism over B. Assume f is flat of relative dimension r. For any closed subset T ⊂ |Y| we have dim_δ(|f|^-1(T)) = dim_δ(T) + r. provided |f|^-1(T) is nonempty. If Z ⊂ Y is an integral closed subscheme and Z' ⊂ f^-1(Z) is an irreducible component, then Z' dominates Z and dim_δ(Z') = dim_δ(Z) + r.","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a morphism over $B$.\nAssume $f$ is flat of relative dimension $r$.\nFor any closed subset $T \\subset |Y|$ we have\n$$\n\\dim_\\delta(|f|^{-1}(T)) = \\dim_\\delta(T) + r.\n$$\nprovided $|f|^{-1}(T)$ is nonempty.\nIf $Z \\subset Y$ is an integral closed subscheme and\n$Z' \\subset f^{-1}(Z)$ is an irreducible component, then\n$Z'$ dominates $Z$ and $\\dim_\\delta(Z') = \\dim_\\delta(Z) + r$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Preparation for flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EP5","source_file":"spaces-chow.tex","source_line":931,"source_end_line":944,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L931-L944","statement_sha256":"91fe9e3a7da1a593a3b23ac62301793d0a6c9e8335c634e50e7e04d682337137","origin":"The Stacks Project","memory_eligible":false,"source_rank":12589,"rank":12589,"depth":54,"x":612.775,"y":1487.193,"cluster":"geometry-of-spaces"},{"id":"stacks:0EP6","tag":"0EP6","title":"Preparation for flat pullback · Lemma 0EP6","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a morphism over B. Assume (T_i)_i ∈ I is a locally finite collection of closed subsets of |Y|. Then (|f|^-1(T_i))_i ∈ I is a locally finite collection of closed subsets of X.","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a morphism over $B$.\nAssume $\\{T_i\\}_{i \\in I}$ is a locally\nfinite collection of closed subsets of $|Y|$.\nThen $\\{|f|^{-1}(T_i)\\}_{i \\in I}$ is a locally finite\ncollection of closed subsets of $X$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Preparation for flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EP6","source_file":"spaces-chow.tex","source_line":980,"source_end_line":988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L980-L988","statement_sha256":"ffc951bd0e8420d1493a36d637f017e88d55b0d42479151fd72760c8480d3eac","origin":"The Stacks Project","memory_eligible":false,"source_rank":12590,"rank":12590,"depth":0,"x":820.376,"y":1645.066,"cluster":"geometry-of-spaces"},{"id":"stacks:0EP8","tag":"0EP8","title":"Flat pullback · Definition 0EP8","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a morphism over B. Assume f is flat of relative dimension r. • Let Z ⊂ Y be an integral closed subspace of δ-dimension k. We define f^*[Z] to be the (k+r)-cycle on X associated to the scheme theoretic inverse image f^*[Z] = [f^-1(Z)]_k+r. This makes sense since dim_δ(f^-1(Z)) = k + r by Lemma [Tag 0EP5]. • Let α = ∑ n_i [Z_i] be a k-cycle on Y. The flat pullback of α by f is the sum f^* α = ∑ n_i f^*[Z_i] where…","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a morphism over $B$.\nAssume $f$ is flat of relative dimension $r$.\n\\begin{enumerate}\n\\item Let $Z \\subset Y$ be an integral closed subspace of\n$\\delta$-dimension $k$. We define $f^*[Z]$ to be the\n$(k+r)$-cycle on $X$ associated to the scheme theoretic inverse image\n$$\nf^*[Z] = [f^{-1}(Z)]_{k+r}.\n$$\nThis makes sense since $\\dim_\\delta(f^{-1}(Z)) = k + r$\nby Lemma \\ref{lemma-flat-inverse-image-dimension}.\n\\item Let $\\alpha = \\sum n_i [Z_i]$ be\na $k$-cycle on $Y$. The {\\it flat pullback of $\\alpha$ by $f$}\nis the sum\n$$\nf^* \\alpha = \\sum n_i f^*[Z_i]\n$$\nwhere each $f^*[Z_i]$ is defined as above.\nThe sum is locally finite by Lemma \\ref{lemma-inverse-image-locally-finite}.\n\\item We denote $f^* : Z_k(Y) \\to Z_{k + r}(X)$ the map of abelian\ngroups so obtained.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Flat pullback","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EP8","source_file":"spaces-chow.tex","source_line":1044,"source_end_line":1069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1044-L1069","statement_sha256":"3aea2dfa8d40f7b23941da122b4b61dadc144289aa839ea33adbf1acdd71914c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12591,"rank":12591,"depth":55,"x":540.16,"y":1646.451,"cluster":"geometry-of-spaces"},{"id":"stacks:0EP9","tag":"0EP9","title":"Flat pullback · Lemma 0EP9","summary":"In Situation [Tag 0EDT] let X/B be good. Let U ⊂ X be an open subspace. Let Y be the reduced closed subspace of X with |Y| = |X| setminus |U| and denote i : Y → X the inclusion morphism. For every k ∈ Z the sequence xymatrix Z_k(Y) ar[r]^i_* & Z_k(X) ar[r]^j^* & Z_k(U) ar[r] & 0 is an exact complex of abelian groups.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $U \\subset X$ be an open subspace. Let $Y$ be the\nreduced closed subspace of $X$ with $|Y| = |X| \\setminus |U|$\nand denote $i : Y \\to X$ the inclusion morphism.\nFor every $k \\in \\mathbf{Z}$ the sequence\n$$\n\\xymatrix{\nZ_k(Y) \\ar[r]^{i_*} & Z_k(X) \\ar[r]^{j^*} & Z_k(U) \\ar[r] & 0\n}\n$$\nis an exact complex of abelian groups.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EP9","source_file":"spaces-chow.tex","source_line":1088,"source_end_line":1101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1088-L1101","statement_sha256":"1182942de6e90a32350fd52933b6858aefad52b36c6eb332a26de9cd45e455a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12592,"rank":12592,"depth":0,"x":745.803,"y":1486.327,"cluster":"geometry-of-spaces"},{"id":"stacks:0EPY","tag":"0EPY","title":"Flat pullback · Lemma 0EPY","summary":"In Situation [Tag 0EDT] let f : X → Y be an étale morphism of good algebraic spaces over B. If Z ⊂ Y is an integral closed subspace, then f^*[Z] = ∑ [Z'] where the sum is over the irreducible components (Remark [Tag 0EE8]) of f^-1(Z).","statement_latex":"In Situation \\ref{situation-setup} let $f : X \\to Y$ be an \\'etale\nmorphism of good algebraic spaces over $B$. If $Z \\subset Y$ is an integral\nclosed subspace, then $f^*[Z] = \\sum [Z']$ where the sum is over the\nirreducible components (Remark \\ref{remark-irreducible-component})\nof $f^{-1}(Z)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPY","source_file":"spaces-chow.tex","source_line":1114,"source_end_line":1121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1114-L1121","statement_sha256":"acdf089326aee6067a140a8c881b7afc74cce8c93d38019013540814c15456a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12593,"rank":12593,"depth":0,"x":722.918,"y":1721.236,"cluster":"geometry-of-spaces"},{"id":"stacks:0EPA","tag":"0EPA","title":"Flat pullback · Lemma 0EPA","summary":"In Situation [Tag 0EDT] let X, Y, Z/B be good. Let f : X → Y and g : Y → Z be flat morphisms of relative dimensions r and s over B. Then g ∘ f is flat of relative dimension r + s and f^* ∘ g^* = (g ∘ f)^* as maps Z_k(Z) → Z_k + r + s(X).","statement_latex":"In Situation \\ref{situation-setup} let $X, Y, Z/B$ be good.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be flat morphisms of relative dimensions\n$r$ and $s$ over $B$. Then $g \\circ f$ is flat of relative dimension\n$r + s$ and\n$$\nf^* \\circ g^* = (g \\circ f)^*\n$$\nas maps $Z_k(Z) \\to Z_{k + r + s}(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPA","source_file":"spaces-chow.tex","source_line":1129,"source_end_line":1139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1129-L1139","statement_sha256":"7504b294f25c11fd0b6748eb19180e9faf3871108a40a8365c0edd7039c30781","origin":"The Stacks Project","memory_eligible":false,"source_rank":12594,"rank":12594,"depth":55,"x":550.777,"y":1534.914,"cluster":"geometry-of-spaces"},{"id":"stacks:0EPB","tag":"0EPB","title":"Flat pullback · Lemma 0EPB","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a flat morphism of relative dimension r. • Let Z ⊂ Y be a closed subspace with dim_δ(Z) ≤ k. Then we have dim_δ(f^-1(Z)) ≤ k + r and [f^-1(Z)]_k + r = f^*[Z]_k in Z_k + r(X). • Let F be a coherent sheaf on Y with dim_δ(Supp(F)) ≤ k. Then we have dim_δ(Supp(f^*F)) ≤ k + r and f^*[ F]_k = [f^* F]_k+r in Z_k + r(X).","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\n\\begin{enumerate}\n\\item Let $Z \\subset Y$ be a closed subspace with\n$\\dim_\\delta(Z) \\leq k$. Then we have\n$\\dim_\\delta(f^{-1}(Z)) \\leq k + r$\nand $[f^{-1}(Z)]_{k + r} = f^*[Z]_k$ in $Z_{k + r}(X)$.\n\\item Let $\\mathcal{F}$ be a coherent sheaf on $Y$ with\n$\\dim_\\delta(\\text{Supp}(\\mathcal{F})) \\leq k$.\nThen we have $\\dim_\\delta(\\text{Supp}(f^*\\mathcal{F})) \\leq k + r$\nand\n$$\nf^*[{\\mathcal F}]_k = [f^*{\\mathcal F}]_{k+r}\n$$\nin $Z_{k + r}(X)$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Flat pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPB","source_file":"spaces-chow.tex","source_line":1188,"source_end_line":1206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1188-L1206","statement_sha256":"9141d741ee3aded8f7e388efe27817ceea74dd64de11871b9db1957fe81156d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12595,"rank":12595,"depth":61,"x":827.72,"y":1574.653,"cluster":"geometry-of-spaces"},{"id":"stacks:0EPD","tag":"0EPD","title":"Push and pull · Lemma 0EPD","summary":"In Situation [Tag 0EDT] let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a fibre product diagram of good algebraic spaces over B. Assume f : X → Y proper and g : Y' → Y flat of relative dimension r. Then also f' is proper and g' is flat of relative dimension r. For any k-cycle α on X we have g^*f_*α = f'_*(g')^*α in Z_k + r(Y').","statement_latex":"In Situation \\ref{situation-setup} let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a fibre product diagram of good algebraic spaces over $B$.\nAssume $f : X \\to Y$ proper and $g : Y' \\to Y$ flat of relative dimension $r$.\nThen also $f'$ is proper and $g'$ is flat of relative dimension $r$.\nFor any $k$-cycle $\\alpha$ on $X$ we have\n$$\ng^*f_*\\alpha = f'_*(g')^*\\alpha\n$$\nin $Z_{k + r}(Y')$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPD","source_file":"spaces-chow.tex","source_line":1280,"source_end_line":1297,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1280-L1297","statement_sha256":"7a9a5f2a9233c43bd3a808ff5ddd52341d66e1ebc0e6d1a572e2c9e4fd3f1c6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12596,"rank":12596,"depth":75,"x":591.401,"y":1702.575,"cluster":"geometry-of-spaces"},{"id":"stacks:0EPE","tag":"0EPE","title":"Push and pull · Lemma 0EPE","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a finite locally free morphism of degree d (see Morphisms of Spaces, Definition [Tag 03ZV]). Then f is both proper and flat of relative dimension 0, and f_*f^*α = dα for every α ∈ Z_k(Y).","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a finite locally free morphism\nof degree $d$ (see\nMorphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-finite-locally-free}).\nThen $f$ is both proper and flat of relative dimension $0$, and\n$$\nf_*f^*\\alpha = d\\alpha\n$$\nfor every $\\alpha \\in Z_k(Y)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPE","source_file":"spaces-chow.tex","source_line":1318,"source_end_line":1330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1318-L1330","statement_sha256":"e87367914d39c13df00b8eb7f960f9704aec1c01d84624569aa16d127ac5055d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12597,"rank":12597,"depth":75,"x":662.833,"y":1474.01,"cluster":"geometry-of-spaces"},{"id":"stacks:0EPZ","tag":"0EPZ","title":"Preparation for principal divisors · Lemma 0EPZ","summary":"In Situation [Tag 0EDT] let X/B be good. Assume X is integral. • If Z ⊂ X is an integral closed subspace, then the following are equivalent: • Z is a prime divisor, • |Z| has codimension 1 in |X|, and • dim_δ(Z) = dim_δ(X) - 1. • If Z is an irreducible component of an effective Cartier divisor on X, then dim_δ(Z) = dim_δ(X) - 1.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is\nintegral.\n\\begin{enumerate}\n\\item If $Z \\subset X$ is an integral closed subspace, then\nthe following are equivalent:\n\\begin{enumerate}\n\\item $Z$ is a prime divisor,\n\\item $|Z|$ has codimension $1$ in $|X|$, and\n\\item $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$.\n\\end{enumerate}\n\\item If $Z$ is an irreducible component of an effective Cartier\ndivisor on $X$, then $\\dim_\\delta(Z) = \\dim_\\delta(X) - 1$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Preparation for principal divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EPZ","source_file":"spaces-chow.tex","source_line":1375,"source_end_line":1390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1375-L1390","statement_sha256":"6c835b60dc2834852e25087033afa2578074292720bb7ff32811ca7caafc2614","origin":"The Stacks Project","memory_eligible":false,"source_rank":12598,"rank":12598,"depth":61,"x":794.048,"y":1683.214,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQ1","tag":"0EQ1","title":"Principal divisors · Definition 0EQ1","summary":"In Situation [Tag 0EDT] let X/B be good. Assume X is integral with dim_δ(X) = n. Let f ∈ R(X)^*. The principal divisor associated to f is the (n - 1)-cycle div(f) = div_X(f) = ∑ ord_Z(f) [Z] defined in Spaces over Fields, Definition [Tag 0ENP]. This makes sense because prime divisors have δ-dimension n - 1 by Lemma [Tag 0EPZ].","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good. Assume $X$ is\nintegral with $\\dim_\\delta(X) = n$.\nLet $f \\in R(X)^*$. The {\\it principal divisor associated to $f$}\nis the $(n - 1)$-cycle\n$$\n\\text{div}(f) = \\text{div}_X(f) = \\sum \\text{ord}_Z(f) [Z]\n$$\ndefined in Spaces over Fields, Definition\n\\ref{spaces-over-fields-definition-principal-divisor}.\nThis makes sense because prime divisors have $\\delta$-dimension $n - 1$ by\nLemma \\ref{lemma-divisor-delta-dimension}.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Principal divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQ1","source_file":"spaces-chow.tex","source_line":1437,"source_end_line":1450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1437-L1450","statement_sha256":"fbc064d5e1d4e0fd22b514997378fc4bf6fd25b6c8e04695877c18e52edebc6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12599,"rank":12599,"depth":62,"x":528.89,"y":1603.355,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQ2","tag":"0EQ2","title":"Principal divisors · Lemma 0EQ2","summary":"In Situation [Tag 0EDT] let f : X → Y be an étale morphism of good algebraic spaces over B. Assume Y is integral. Let g ∈ R(Y)^*. As cycles on X we have f^*(div_Y(g)) = ∑_X' (X' → X)_*div_X'(g ∘ f|_X') where the sum is over the irreducible components of X (Remark [Tag 0EE8]).","statement_latex":"In Situation \\ref{situation-setup} let $f : X \\to Y$ be an \\'etale\nmorphism of good algebraic spaces over $B$. Assume $Y$ is integral.\nLet $g \\in R(Y)^*$. As cycles on $X$ we have\n$$\nf^*(\\text{div}_Y(g)) =\n\\sum\\nolimits_{X'} (X' \\to X)_*\\text{div}_{X'}(g \\circ f|_{X'})\n$$\nwhere the sum is over the irreducible components of $X$\n(Remark \\ref{remark-irreducible-component}).","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Principal divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQ2","source_file":"spaces-chow.tex","source_line":1462,"source_end_line":1473,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1462-L1473","statement_sha256":"7a91128b81ef2f162f2a8ab6abe85851ea02d71d19a8e49c8d9c48028552192c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12600,"rank":12600,"depth":59,"x":788.797,"y":1511.727,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQ4","tag":"0EQ4","title":"Principal divisors and pushforward · Lemma 0EQ4","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Assume X, Y are integral and n = dim_δ(X) = dim_δ(Y). Let p : X → Y be a dominant proper morphism. Let f ∈ R(X)^*. Set g = Nm_R(X)/R(Y)(f). Then we have p_*div(f) = div(g).","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nAssume $X$, $Y$ are integral and $n = \\dim_\\delta(X) = \\dim_\\delta(Y)$.\nLet $p : X \\to Y$ be a dominant proper morphism.\nLet $f \\in R(X)^*$. Set\n$$\ng = \\text{Nm}_{R(X)/R(Y)}(f).\n$$\nThen we have\n$p_*\\text{div}(f) = \\text{div}(g)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Principal divisors and pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQ4","source_file":"spaces-chow.tex","source_line":1601,"source_end_line":1612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1601-L1612","statement_sha256":"b60331720a3ef609f596a702ad4d8ff75e9cd310caf5690ee1c149f0a712ffbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12601,"rank":12601,"depth":76,"x":670.756,"y":1726.904,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQ6","tag":"0EQ6","title":"Rational equivalence · Definition 0EQ6","summary":"In Situation [Tag 0EDT] let X/B be good. Let k ∈ Z. • Given any locally finite collection (W_j ⊂ X) of integral closed subspaces with dim_δ(W_j) = k + 1, and any f_j ∈ R(W_j)^* we may consider ∑ (i_j)_*div(f_j) ∈ Z_k(X) where i_j : W_j → X is the inclusion morphism. This makes sense as the morphism coprod i_j : coprod W_j → X is proper. • We say that α ∈ Z_k(X) is rationally equivalent to zero if α is a cycle of the form displayed above. • We say α, β ∈ Z_k(X) are…","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $k \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item Given any locally finite collection $\\{W_j \\subset X\\}$\nof integral closed subspaces with $\\dim_\\delta(W_j) = k + 1$,\nand any $f_j \\in R(W_j)^*$ we may consider\n$$\n\\sum (i_j)_*\\text{div}(f_j) \\in Z_k(X)\n$$\nwhere $i_j : W_j \\to X$ is the inclusion morphism.\nThis makes sense as the morphism\n$\\coprod i_j : \\coprod W_j \\to X$ is proper.\n\\item We say that $\\alpha \\in Z_k(X)$ is {\\it rationally equivalent to zero}\nif $\\alpha$ is a cycle of the form displayed above.\n\\item We say $\\alpha, \\beta \\in Z_k(X)$ are\n{\\it rationally equivalent} and we write $\\alpha \\sim_{rat} \\beta$\nif $\\alpha - \\beta$ is rationally equivalent to zero.\n\\item We define\n$$\n\\CH_k(X) = Z_k(X) / \\sim_{rat}\n$$\nto be the {\\it Chow group of $k$-cycles on $X$}. This is sometimes called\nthe {\\it Chow group of $k$-cycles modulo rational equivalence on $X$}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Rational equivalence","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQ6","source_file":"spaces-chow.tex","source_line":1700,"source_end_line":1726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1700-L1726","statement_sha256":"bed405fe3d4c4538e21aa55897c5a44d0cbba36af3c3ec080a8b0d0778d95285","origin":"The Stacks Project","memory_eligible":false,"source_rank":12602,"rank":12602,"depth":0,"x":584.704,"y":1501.116,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQ7","tag":"0EQ7","title":"Rational equivalence · Lemma 0EQ7","summary":"In Situation [Tag 0EDT] let X/B be good. Let U ⊂ X be an open subspace. Let Y be the reduced closed subspace of X with |Y| = |X| setminus |U| and denote i : Y → X the inclusion morphism. Let k ∈ Z. Suppose α, β ∈ Z_k(X). If α|_U sim_rat β|_U then there exist a cycle γ ∈ Z_k(Y) such that α sim_rat β + i_*γ. In other words, the sequence xymatrix CH_k(Y) ar[r]^i_* & CH_k(X) ar[r]^j^* & CH_k(U) ar[r] & 0 is an exact complex of abelian groups.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $U \\subset X$ be an open subspace. Let $Y$ be the\nreduced closed subspace of $X$ with $|Y| = |X| \\setminus |U|$\nand denote $i : Y \\to X$ the inclusion morphism. Let $k \\in \\mathbf{Z}$.\nSuppose $\\alpha, \\beta \\in Z_k(X)$. If $\\alpha|_U \\sim_{rat} \\beta|_U$\nthen there exist a cycle $\\gamma \\in Z_k(Y)$ such that\n$$\n\\alpha \\sim_{rat} \\beta + i_*\\gamma.\n$$\nIn other words, the sequence\n$$\n\\xymatrix{\n\\CH_k(Y) \\ar[r]^{i_*} & \\CH_k(X) \\ar[r]^{j^*} & \\CH_k(U) \\ar[r] & 0\n}\n$$\nis an exact complex of abelian groups.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQ7","source_file":"spaces-chow.tex","source_line":1733,"source_end_line":1751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1733-L1751","statement_sha256":"18f21103422e993705b9b3a32846094c3313eb509ebad7ebb4eee866a9c07ea4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12603,"rank":12603,"depth":1,"x":829.883,"y":1618.855,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQA","tag":"0EQA","title":"Rational equivalence and push and pull · Lemma 0EQA","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Assume Y integral with dim_δ(Y) = k. Let f : X → Y be a flat morphism of relative dimension r. Then for g ∈ R(Y)^* we have f^*div_Y(g) = ∑ m_X', X (X' → X)_*div_X'(g ∘ f|_X') as (k + r - 1)-cycles on X where the sum is over the irreducible components X' of X and m_X', X is the multiplicity of X' in X.","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nAssume $Y$ integral with $\\dim_\\delta(Y) = k$.\nLet $f : X \\to Y$ be a flat morphism of\nrelative dimension $r$. Then for $g \\in R(Y)^*$ we have\n$$\nf^*\\text{div}_Y(g) =\n\\sum m_{X', X} (X' \\to X)_*\\text{div}_{X'}(g \\circ f|_{X'})\n$$\nas $(k + r - 1)$-cycles on $X$ where the sum is over the irreducible\ncomponents $X'$ of $X$ and $m_{X', X}$ is the multiplicity of $X'$ in $X$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Rational equivalence and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQA","source_file":"spaces-chow.tex","source_line":1821,"source_end_line":1833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1821-L1833","statement_sha256":"64c7ffcb6eb9f0a17e7e8e193963144a5485bea4d2ead520bdb1ff3382066d0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12604,"rank":12604,"depth":76,"x":554.238,"y":1671.188,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQB","tag":"0EQB","title":"Rational equivalence and push and pull · Lemma 0EQB","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a flat morphism of relative dimension r. Let α sim_rat β be rationally equivalent k-cycles on Y. Then f^*α sim_rat f^*β as (k + r)-cycles on X.","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $\\alpha \\sim_{rat} \\beta$ be rationally equivalent $k$-cycles\non $Y$. Then $f^*\\alpha \\sim_{rat} f^*\\beta$ as $(k + r)$-cycles on $X$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Rational equivalence and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQB","source_file":"spaces-chow.tex","source_line":1916,"source_end_line":1922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1916-L1922","statement_sha256":"f9b2349cff2d21004a9efc32ff96552554fd99273336bcab45b966eae9867783","origin":"The Stacks Project","memory_eligible":false,"source_rank":12605,"rank":12605,"depth":77,"x":715.509,"y":1476.069,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQC","tag":"0EQC","title":"Rational equivalence and push and pull · Lemma 0EQC","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let p : X → Y be a proper morphism. Suppose α, β ∈ Z_k(X) are rationally equivalent. Then p_*α is rationally equivalent to p_*β.","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $p : X \\to Y$ be a proper morphism.\nSuppose $\\alpha, \\beta \\in Z_k(X)$ are rationally equivalent.\nThen $p_*\\alpha$ is rationally equivalent to $p_*\\beta$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Rational equivalence and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQC","source_file":"spaces-chow.tex","source_line":1958,"source_end_line":1964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L1958-L1964","statement_sha256":"f9c7d658590649653b292543e78c316da4286169e947d77b6e5fc7fa2a39ae27","origin":"The Stacks Project","memory_eligible":false,"source_rank":12606,"rank":12606,"depth":77,"x":753.523,"y":1711.603,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQE","tag":"0EQE","title":"The divisor associated to an invertible sheaf · Definition 0EQE","summary":"In Situation [Tag 0EDT] let X/B be good. Assume X is integral and n = dim_δ(X). Let L be an invertible O_X-module. • For any nonzero meromorphic section s of L we define the Weil divisor associated to s is the (n - 1)-cycle div_L(s) = ∑ ord_Z, L(s) [Z] defined in Spaces over Fields, Definition [Tag 0EPU]. This makes sense because Weil divisors have δ-dimension n - 1 by Lemma [Tag 0EPZ]. • We define Weil divisor associated to L as c_1(L) ∩ [X] = class of div_L(s) ∈ CH_n -…","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nAssume $X$ is integral and $n = \\dim_\\delta(X)$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item For any nonzero meromorphic section $s$ of $\\mathcal{L}$\nwe define the {\\it Weil divisor associated to $s$} is the\n$(n - 1)$-cycle\n$$\n\\text{div}_\\mathcal{L}(s) =\n\\sum \\text{ord}_{Z, \\mathcal{L}}(s) [Z]\n$$\ndefined in Spaces over Fields, Definition\n\\ref{spaces-over-fields-definition-divisor-invertible-sheaf}.\nThis makes sense because Weil divisors have $\\delta$-dimension $n - 1$\nby Lemma \\ref{lemma-divisor-delta-dimension}.\n\\item We define {\\it Weil divisor associated to $\\mathcal{L}$} as\n$$\nc_1(\\mathcal{L}) \\cap [X] =\n\\text{class of }\\text{div}_\\mathcal{L}(s) \\in \\CH_{n - 1}(X)\n$$\nwhere $s$ is any nonzero meromorphic section of $\\mathcal{L}$ over\n$X$. This is well defined by\nSpaces over Fields, Lemma\n\\ref{spaces-over-fields-lemma-divisor-meromorphic-well-defined}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The divisor associated to an invertible sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQE","source_file":"spaces-chow.tex","source_line":2076,"source_end_line":2103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2076-L2103","statement_sha256":"0fc7602936359040d4d4cba4eccaba661972a50161ed548fd4e0b7b567e633d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12607,"rank":12607,"depth":62,"x":535.948,"y":1559.399,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQF","tag":"0EQF","title":"The divisor associated to an invertible sheaf · Lemma 0EQF","summary":"In Situation [Tag 0EDT] let X/B be good. Assume X is integral and n = dim_δ(X). Let L be an invertible O_X-module. Let s ∈ Γ(X, L) be a nonzero global section. Then div_L(s) = [Z(s)]_n - 1 in Z_n - 1(X) and c_1(L) ∩ [X] = [Z(s)]_n - 1 in CH_n - 1(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nAssume $X$ is integral and $n = \\dim_\\delta(X)$.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $s \\in \\Gamma(X, \\mathcal{L})$ be a nonzero global section.\nThen\n$$\n\\text{div}_\\mathcal{L}(s) = [Z(s)]_{n - 1}\n$$\nin $Z_{n - 1}(X)$ and\n$$\nc_1(\\mathcal{L}) \\cap [X] = [Z(s)]_{n - 1}\n$$\nin $\\CH_{n - 1}(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The divisor associated to an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQF","source_file":"spaces-chow.tex","source_line":2110,"source_end_line":2125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2110-L2125","statement_sha256":"fb0be17999d3631b077608358ea6ff317ee8ec108930330852b72a666f83176f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12608,"rank":12608,"depth":12,"x":818.958,"y":1548.167,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQG","tag":"0EQG","title":"The divisor associated to an invertible sheaf · Lemma 0EQG","summary":"In Situation [Tag 0EDT] let X/B be good. Let L be an invertible O_X-module. The morphism q : T = underlineSpec( bigoplus_n ∈ Z L^⊗ n) → X has the following properties: • q is surjective, smooth, affine, of relative dimension 1, • there is an isomorphism α : q^*L ≅ O_T, • formation of (q : T → X, α) commutes with base change, • q^* : Z_k(X) → Z_k + 1(T) is injective, • if Z ⊂ X is an integral closed subspace, then q^-1(Z) ⊂ T is an integral closed subspace, • if Z ⊂ X is a…","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThe morphism\n$$\nq :\nT = \\underline{\\Spec}\\left(\n\\bigoplus\\nolimits_{n \\in \\mathbf{Z}} \\mathcal{L}^{\\otimes n}\\right)\n\\longrightarrow\nX\n$$\nhas the following properties:\n\\begin{enumerate}\n\\item $q$ is surjective, smooth, affine, of relative dimension $1$,\n\\item there is an isomorphism $\\alpha : q^*\\mathcal{L} \\cong \\mathcal{O}_T$,\n\\item formation of $(q : T \\to X, \\alpha)$ commutes with base change,\n\\item $q^* : Z_k(X) \\to Z_{k + 1}(T)$ is injective,\n\\item if $Z \\subset X$ is an integral closed subspace, then\n$q^{-1}(Z) \\subset T$ is an integral closed subspace,\n\\item if $Z \\subset X$ is a closed subspace of $X$\nof $\\delta$-dimension $\\leq k$, then $q^{-1}(Z)$ is a closed subspace of $T$\nof $\\delta$-dimension $\\leq k + 1$ and\n$q^*[Z]_k = [q^{-1}(Z)]_{k + 1}$,\n\\item if $\\xi' \\in |T|$ is the generic point of the fibre of $|T| \\to |X|$\nover $\\xi$, then the ring map\n$\\mathcal{O}_{X, \\xi}^h \\to \\mathcal{O}_{T, \\xi'}^h$ is flat,\nwe have $\\mathfrak m_{\\xi'}^h = \\mathfrak m_\\xi^h \\mathcal{O}_{T, \\xi'}^h$, and\nthe residue field extension is purely transcendental of\ntranscendence degree $1$, and\n\\item add more here as needed.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The divisor associated to an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQG","source_file":"spaces-chow.tex","source_line":2167,"source_end_line":2199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2167-L2199","statement_sha256":"f729c3e3e9cb1b5c7cc99b5f026b0880b766736b9ebdab10c8d1df3017ee4131","origin":"The Stacks Project","memory_eligible":false,"source_rank":12609,"rank":12609,"depth":1,"x":619.179,"y":1717.141,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQH","tag":"0EQH","title":"The divisor associated to an invertible sheaf · Lemma 0EQH","summary":"In Situation [Tag 0EDT] let X/B be good. Let L be an invertible O_X-module. Assume X is integral. Let s be a nonzero meromorphic section of L. Let q : T → X be the morphism of Lemma [Tag 0EQG]. Then q^*div_L(s) = div_T(q^*(s)) where we view the pullback q^*(s) as a nonzero meromorphic function on T using the isomorphism q^*L → O_T","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nAssume $X$ is integral. Let $s$ be a nonzero meromorphic\nsection of $\\mathcal{L}$. Let $q : T \\to X$ be the morphism\nof Lemma \\ref{lemma-Gm-torsor}. Then\n$$\nq^*\\text{div}_\\mathcal{L}(s) = \\text{div}_T(q^*(s))\n$$\nwhere we view the pullback $q^*(s)$\nas a nonzero meromorphic function on $T$\nusing the isomorphism $q^*\\mathcal{L} \\to \\mathcal{O}_T$","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The divisor associated to an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQH","source_file":"spaces-chow.tex","source_line":2259,"source_end_line":2272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2259-L2272","statement_sha256":"644c357c99a28797ecc3288cdfec0ebc8065815cd58c97d34e5afd9cb33fc746","origin":"The Stacks Project","memory_eligible":false,"source_rank":12610,"rank":12610,"depth":5,"x":630.616,"y":1479.036,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQJ","tag":"0EQJ","title":"Intersecting with an invertible sheaf · Definition 0EQJ","summary":"In Situation [Tag 0EDT] let X/B be good. Let L be an invertible O_X-module. We define, for every integer k, an operation c_1(L) ∩ - : Z_k + 1(X) → CH_k(X) called intersection with the first Chern class of L. • Given an integral closed subspace i : W → X with dim_δ(W) = k + 1 we define c_1(L) ∩ [W] = i_*(c_1(i^*L) ∩ [W]) where the right hand side is defined in Definition [Tag 0EQE]. • For a general (k + 1)-cycle α = ∑ n_i [W_i] we set c_1(L) ∩ α = ∑ n_i c_1(L) ∩ [W_i]","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nWe define, for every integer $k$, an operation\n$$\nc_1(\\mathcal{L}) \\cap - :\nZ_{k + 1}(X) \\to \\CH_k(X)\n$$\ncalled {\\it intersection with the first Chern class of $\\mathcal{L}$}.\n\\begin{enumerate}\n\\item Given an integral closed subspace $i : W \\to X$ with\n$\\dim_\\delta(W) = k + 1$ we define\n$$\nc_1(\\mathcal{L}) \\cap [W] = i_*(c_1({i^*\\mathcal{L}}) \\cap [W])\n$$\nwhere the right hand side is defined in\nDefinition \\ref{definition-divisor-invertible-sheaf}.\n\\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_i [W_i]$ we set\n$$\nc_1(\\mathcal{L}) \\cap \\alpha = \\sum n_i c_1(\\mathcal{L}) \\cap [W_i]\n$$\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQJ","source_file":"spaces-chow.tex","source_line":2351,"source_end_line":2374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2351-L2374","statement_sha256":"c7b11636b600a11b241024d1fde1939563379e5acfee213ae7cf1bf52d54d3d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12611,"rank":12611,"depth":63,"x":813.773,"y":1661.213,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQK","tag":"0EQK","title":"Intersecting with an invertible sheaf · Lemma 0EQK","summary":"In Situation [Tag 0EDT] let X/B be good. Let L, N be an invertible sheaves on X. Then c_1(L) ∩ α + c_1(N) ∩ α = c_1(L ⊗_O_X N) ∩ α in CH_k(X) for every α ∈ Z_k - 1(X). Moreover, c_1(O_X) ∩ α = 0 for all α.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$, $\\mathcal{N}$ be an invertible sheaves on $X$.\nThen\n$$\nc_1(\\mathcal{L}) \\cap \\alpha  + c_1(\\mathcal{N}) \\cap \\alpha =\nc_1(\\mathcal{L} \\otimes_{\\mathcal{O}_X} \\mathcal{N}) \\cap \\alpha\n$$\nin $\\CH_k(X)$ for every $\\alpha \\in Z_{k - 1}(X)$. Moreover,\n$c_1(\\mathcal{O}_X) \\cap \\alpha = 0$ for all $\\alpha$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQK","source_file":"spaces-chow.tex","source_line":2396,"source_end_line":2407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2396-L2407","statement_sha256":"2bebd9b1d3749ad3aeafa97fdc3dcd542cc3b34b2a499d59ae1c34bd441bb8dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12612,"rank":12612,"depth":12,"x":532.041,"y":1630.788,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQL","tag":"0EQL","title":"Intersecting with an invertible sheaf · Lemma 0EQL","summary":"In Situation [Tag 0EDT] let Y/B be good. Let L be an invertible O_Y-module. Let s ∈ Γ(Y, L) be a regular section and assume dim_δ(Y) ≤ k + 1. Write [Y]_k + 1 = ∑ n_i[Y_i] where Y_i ⊂ Y are the irreducible components of Y of δ-dimension k + 1. Set s_i = s|_Y_i ∈ Γ(Y_i, L|_Y_i). Then [Z(s)]_k = ∑ n_i[Z(s_i)]_k as k-cycles on Y.","statement_latex":"In Situation \\ref{situation-setup} let $Y/B$ be good.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_Y$-module.\nLet $s \\in \\Gamma(Y, \\mathcal{L})$ be a regular section and\nassume $\\dim_\\delta(Y) \\leq k + 1$.\nWrite $[Y]_{k + 1} = \\sum n_i[Y_i]$ where $Y_i \\subset Y$ are the\nirreducible components of $Y$ of $\\delta$-dimension $k + 1$.\nSet $s_i = s|_{Y_i} \\in \\Gamma(Y_i, \\mathcal{L}|_{Y_i})$. Then\n\\begin{equation}\n\n[Z(s)]_k =  \\sum n_i[Z(s_i)]_k\n\\end{equation}\nas $k$-cycles on $Y$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQL","source_file":"spaces-chow.tex","source_line":2423,"source_end_line":2437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2423-L2437","statement_sha256":"08c8585db22be2548582de134b1ada8f813c9ec1eb853a973192060780654012","origin":"The Stacks Project","memory_eligible":false,"source_rank":12613,"rank":12613,"depth":16,"x":764.396,"y":1493.276,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQN","tag":"0EQN","title":"Intersecting with an invertible sheaf · Lemma 0EQN","summary":"In Situation [Tag 0EDT] let X/B be good. Let L be an invertible O_X-module. Let Y ⊂ X be a closed subscheme with dim_δ(Y) ≤ k + 1 and let s ∈ Γ(Y, L|_Y) be a regular section. Then c_1(L) ∩ [Y]_k + 1 = [Z(s)]_k in CH_k(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet $Y \\subset X$ be a closed subscheme with\n$\\dim_\\delta(Y) \\leq k + 1$ and let $s \\in \\Gamma(Y, \\mathcal{L}|_Y)$\nbe a regular section. Then\n$$\nc_1(\\mathcal{L}) \\cap [Y]_{k + 1} = [Z(s)]_k\n$$\nin $\\CH_k(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQN","source_file":"spaces-chow.tex","source_line":2469,"source_end_line":2480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2469-L2480","statement_sha256":"849bce1ec1876478477d1d3c9c96f943e803bc44acabc508c1c9cd21381a85bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12614,"rank":12614,"depth":17,"x":703.606,"y":1726.663,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQQ","tag":"0EQQ","title":"Intersecting with an invertible sheaf and push and pull · Lemma 0EQQ","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a flat morphism of relative dimension r. Let L be an invertible sheaf on Y. Assume Y is integral and n = dim_δ(Y). Let s be a nonzero meromorphic section of L. Then we have f^*div_L(s) = ∑ n_idiv_f^*L|_X_i(s_i) in Z_n + r - 1(X). Here the sum is over the irreducible components X_i ⊂ X of δ-dimension n + r, the section s_i = f|_X_i^*(s) is the pullback of s, and n_i = m_X_i, X is the multiplicity of X_i in X.","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $\\mathcal{L}$ be an invertible sheaf on $Y$.\nAssume $Y$ is integral and $n = \\dim_\\delta(Y)$.\nLet $s$ be a nonzero meromorphic section of $\\mathcal{L}$.\nThen we have\n$$\nf^*\\text{div}_\\mathcal{L}(s) = \\sum n_i\\text{div}_{f^*\\mathcal{L}|_{X_i}}(s_i)\n$$\nin $Z_{n + r - 1}(X)$. Here the sum is over the irreducible\ncomponents $X_i \\subset X$ of $\\delta$-dimension $n + r$,\nthe section $s_i = f|_{X_i}^*(s)$ is the pullback of $s$, and\n$n_i = m_{X_i, X}$ is the multiplicity of $X_i$ in $X$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQQ","source_file":"spaces-chow.tex","source_line":2517,"source_end_line":2532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2517-L2532","statement_sha256":"4ced226bb231698bd4de66aae872712131159e6b48295e4342f9b575c5a8e265","origin":"The Stacks Project","memory_eligible":false,"source_rank":12615,"rank":12615,"depth":77,"x":560.661,"y":1519.946,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQR","tag":"0EQR","title":"Intersecting with an invertible sheaf and push and pull · Lemma 0EQR","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a flat morphism of relative dimension r. Let L be an invertible sheaf on Y. Let α be a k-cycle on Y. Then f^*(c_1(L) ∩ α) = c_1(f^*L) ∩ f^*α in CH_k + r - 1(X).","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $\\mathcal{L}$ be an invertible sheaf on $Y$.\nLet $\\alpha$ be a $k$-cycle on $Y$.\nThen\n$$\nf^*(c_1(\\mathcal{L}) \\cap \\alpha) = c_1(f^*\\mathcal{L}) \\cap f^*\\alpha\n$$\nin $\\CH_{k + r - 1}(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQR","source_file":"spaces-chow.tex","source_line":2585,"source_end_line":2596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2585-L2596","statement_sha256":"4f7640a6e1f2f91487bce82374f1560e841aec572404ca23c1ff751c1364b4bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12616,"rank":12616,"depth":78,"x":832.469,"y":1591.309,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQS","tag":"0EQS","title":"Intersecting with an invertible sheaf and push and pull · Lemma 0EQS","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a proper morphism. Let L be an invertible sheaf on Y. Assume X, Y integral, f dominant, and dim_δ(X) = dim_δ(Y). Let s be a nonzero meromorphic section s of L on Y. Then f_*(div_f^*L(f^*s)) = [R(X) : R(Y)]div_L(s). as cycles on Y. In particular f_*(c_1(f^*L) ∩ [X]) = c_1(L) ∩ f_*[Y].","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a proper morphism.\nLet $\\mathcal{L}$ be an invertible sheaf on $Y$.\nAssume $X$, $Y$ integral, $f$ dominant, and $\\dim_\\delta(X) = \\dim_\\delta(Y)$.\nLet $s$ be a nonzero meromorphic section $s$ of $\\mathcal{L}$ on $Y$.\nThen\n$$\nf_*\\left(\\text{div}_{f^*\\mathcal{L}}(f^*s)\\right) =\n[R(X) : R(Y)]\\text{div}_\\mathcal{L}(s).\n$$\nas cycles on $Y$. In particular\n$$\nf_*(c_1(f^*\\mathcal{L}) \\cap [X]) = c_1(\\mathcal{L}) \\cap f_*[Y].\n$$","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQS","source_file":"spaces-chow.tex","source_line":2641,"source_end_line":2657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2641-L2657","statement_sha256":"870ed4a56b443adc3e3b9f1450c6ba631992510429b68d60f1d4a08d036c473d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12617,"rank":12617,"depth":78,"x":574.496,"y":1692.98,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQT","tag":"0EQT","title":"Intersecting with an invertible sheaf and push and pull · Lemma 0EQT","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let p : X → Y be a proper morphism. Let α ∈ Z_k + 1(X). Let L be an invertible sheaf on Y. Then p_*(c_1(p^*L) ∩ α) = c_1(L) ∩ p_*α in CH_k(Y).","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $p : X \\to Y$ be a proper morphism.\nLet $\\alpha \\in Z_{k + 1}(X)$.\nLet $\\mathcal{L}$ be an invertible sheaf on $Y$.\nThen\n$$\np_*(c_1(p^*\\mathcal{L}) \\cap \\alpha) = c_1(\\mathcal{L}) \\cap p_*\\alpha\n$$\nin $\\CH_k(Y)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf and push and pull","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQT","source_file":"spaces-chow.tex","source_line":2715,"source_end_line":2726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2715-L2726","statement_sha256":"8a9c2832ef04cbe7e19f5283c6238d970048603c75b15c63ce2854e7f893ed03","origin":"The Stacks Project","memory_eligible":false,"source_rank":12618,"rank":12618,"depth":79,"x":683.027,"y":1471.494,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQV","tag":"0EQV","title":"Key formula · Lemma 0EQV","summary":"In the situation above the cycle ∑ (Z_i → X)_*( ord_B_i(f_i) div_N|_Z_i(t_i|_Z_i) - ord_B_i(g_i) div_L|_Z_i(s_i|_Z_i) ) is equal to the cycle ∑ (Z_i → X)_*div(∂_B_i(f_i, g_i))","statement_latex":"In the situation above the cycle\n$$\n\\sum\n(Z_i \\to X)_*\\left(\n\\text{ord}_{B_i}(f_i) \\text{div}_{\\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) -\n\\text{ord}_{B_i}(g_i) \\text{div}_{\\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \\right)\n$$\nis equal to the cycle\n$$\n\\sum (Z_i \\to X)_*\\text{div}(\\partial_{B_i}(f_i, g_i))\n$$","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The key formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQV","source_file":"spaces-chow.tex","source_line":2867,"source_end_line":2880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L2867-L2880","statement_sha256":"746c89adfa9cbc1576d90c3efa67e11aa1bf5d5faab462a9125f2a32f993af1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12619,"rank":12619,"depth":78,"x":781.17,"y":1696.535,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQX","tag":"0EQX","title":"Intersecting with an invertible sheaf and rational equivalence · Lemma 0EQX","summary":"In Situation [Tag 0EDT] let X/B be good. Assume X integral and dim_δ(X) = n. Let L, N be invertible on X. Choose a nonzero meromorphic section s of L and a nonzero meromorphic section t of N. Set α = div_L(s) and β = div_N(t). Then c_1(N) ∩ α = c_1(L) ∩ β in CH_n - 2(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nAssume $X$ integral and $\\dim_\\delta(X) = n$.\nLet $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$.\nChoose a nonzero meromorphic section $s$ of $\\mathcal{L}$\nand a nonzero meromorphic section $t$ of $\\mathcal{N}$.\nSet $\\alpha = \\text{div}_\\mathcal{L}(s)$ and\n$\\beta = \\text{div}_\\mathcal{N}(t)$.\nThen\n$$\nc_1(\\mathcal{N}) \\cap \\alpha\n=\nc_1(\\mathcal{L}) \\cap \\beta\n$$\nin $\\CH_{n - 2}(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQX","source_file":"spaces-chow.tex","source_line":3086,"source_end_line":3102,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3086-L3102","statement_sha256":"590a760d011f88f0b65eaca3cbea94349b2e83327278522469d8e4314fd8da86","origin":"The Stacks Project","memory_eligible":false,"source_rank":12620,"rank":12620,"depth":79,"x":527.675,"y":1586.215,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQY","tag":"0EQY","title":"Intersecting with an invertible sheaf and rational equivalence · Lemma 0EQY","summary":"In Situation [Tag 0EDT] let X/B be good. Let L be invertible on X. The operation α ↦ c_1(L) ∩ α factors through rational equivalence to give an operation c_1(L) ∩ - : CH_k + 1(X) → CH_k(X)","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$ be invertible on $X$.\nThe operation $\\alpha \\mapsto c_1(\\mathcal{L}) \\cap \\alpha$\nfactors through rational equivalence to give an operation\n$$\nc_1(\\mathcal{L}) \\cap - : \\CH_{k + 1}(X) \\to \\CH_k(X)\n$$","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQY","source_file":"spaces-chow.tex","source_line":3109,"source_end_line":3118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3109-L3118","statement_sha256":"e8f3223c63f59ed4ae39bc918f577b49dc118be11c526a8ddd20a1708357290f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12621,"rank":12621,"depth":80,"x":803.483,"y":1523.686,"cluster":"geometry-of-spaces"},{"id":"stacks:0EQZ","tag":"0EQZ","title":"Intersecting with an invertible sheaf and rational equivalence · Lemma 0EQZ","summary":"In Situation [Tag 0EDT] let X/B be good. Let L, N be invertible on X. For any α ∈ CH_k + 2(X) we have c_1(L) ∩ c_1(N) ∩ α = c_1(N) ∩ c_1(L) ∩ α as elements of CH_k(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$, $\\mathcal{N}$ be invertible on $X$.\nFor any $\\alpha \\in \\CH_{k + 2}(X)$ we have\n$$\nc_1(\\mathcal{L}) \\cap c_1(\\mathcal{N}) \\cap \\alpha\n=\nc_1(\\mathcal{N}) \\cap c_1(\\mathcal{L}) \\cap \\alpha\n$$\nas elements of $\\CH_k(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with an invertible sheaf and rational equivalence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EQZ","source_file":"spaces-chow.tex","source_line":3166,"source_end_line":3177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3166-L3177","statement_sha256":"36a777a4b6760affe74fddd02f72d9d232c4013206917dd514e01cd7c8ba99f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12622,"rank":12622,"depth":80,"x":650.297,"y":1726.417,"cluster":"geometry-of-spaces"},{"id":"stacks:0ER1","tag":"0ER1","title":"Intersecting with effective Cartier divisors · Definition 0ER1","summary":"In Situation [Tag 0EDT] let X/B be good. Let (L, s) be a pair consisting of an invertible sheaf and a global section s ∈ Γ(X, L). Let D = Z(s) be the vanishing locus of s, and denote i : D → X the closed immersion. We define, for every integer k, a (refined) Gysin homomorphism i^* : Z_k + 1(X) → CH_k(D). by the following rules: • Given an integral closed subspace W ⊂ X with dim_δ(W) = k + 1 we define • if W not ⊂ D, then i^*[W] = [D ∩ W]_k as a k-cycle on D, and • if W ⊂…","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $(\\mathcal{L}, s)$ be a pair consisting of an invertible\nsheaf and a global section $s \\in \\Gamma(X, \\mathcal{L})$.\nLet $D = Z(s)$ be the vanishing locus of $s$, and\ndenote $i : D \\to X$ the closed immersion.\nWe define, for every integer $k$, a (refined) {\\it Gysin homomorphism}\n$$\ni^* : Z_{k + 1}(X) \\to \\CH_k(D).\n$$\nby the following rules:\n\\begin{enumerate}\n\\item Given an integral closed subspace $W \\subset X$ with\n$\\dim_\\delta(W) = k + 1$ we define\n\\begin{enumerate}\n\\item if $W \\not \\subset D$, then $i^*[W] = [D \\cap W]_k$ as a\n$k$-cycle on $D$, and\n\\item if $W \\subset D$, then\n$i^*[W] = i'_*(c_1(\\mathcal{L}|_W) \\cap [W])$,\nwhere $i' : W \\to D$ is the induced closed immersion.\n\\end{enumerate}\n\\item For a general $(k + 1)$-cycle $\\alpha = \\sum n_j[W_j]$\nwe set\n$$\ni^*\\alpha = \\sum n_j i^*[W_j]\n$$\n\\item If $D$ is an effective Cartier divisor, then we denote\n$D \\cdot \\alpha = i_*i^*\\alpha$ the pushforward of\nthe class to a class on $X$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with effective Cartier divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ER1","source_file":"spaces-chow.tex","source_line":3230,"source_end_line":3261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3230-L3261","statement_sha256":"d88175c0a413d4dabd0f72c76ffcdc898d21bf7577d511a3e1cab335926c8754","origin":"The Stacks Project","memory_eligible":false,"source_rank":12623,"rank":12623,"depth":0,"x":600.193,"y":1489.861,"cluster":"geometry-of-spaces"},{"id":"stacks:0ER4","tag":"0ER4","title":"Intersecting with effective Cartier divisors · Lemma 0ER4","summary":"In Situation [Tag 0EDT] let X/B be good. Let (L, s, i : D → X) be as in Definition [Tag 0ER1]. Let α be a (k + 1)-cycle on X. Then i_*i^*α = c_1(L) ∩ α in CH_k(X). In particular, if D is an effective Cartier divisor, then D · α = c_1(O_X(D)) ∩ α.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good. Let\n$(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}. Let $\\alpha$ be a\n$(k + 1)$-cycle on $X$. Then $i_*i^*\\alpha = c_1(\\mathcal{L}) \\cap \\alpha$\nin $\\CH_k(X)$. In particular, if $D$ is an effective Cartier divisor, then\n$D \\cdot \\alpha = c_1(\\mathcal{O}_X(D)) \\cap \\alpha$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ER4","source_file":"spaces-chow.tex","source_line":3294,"source_end_line":3302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3294-L3302","statement_sha256":"8081aab3183c27fb09902ed929c9acab2104e3f18ae95734d0c2f641701560aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12624,"rank":12624,"depth":64,"x":827.509,"y":1635.952,"cluster":"geometry-of-spaces"},{"id":"stacks:0ER5","tag":"0ER5","title":"Intersecting with effective Cartier divisors · Lemma 0ER5","summary":"In Situation [Tag 0EDT]. Let f : X' → X be a proper morphism of good algebraic spaces over B. Let (L, s, i : D → X) be as in Definition [Tag 0ER1]. Form the diagram xymatrix D' ar[d]_g ar[r]_i' & X' ar[d]^f D ar[r]^i & X as in Remark [Tag 0ER3]. For any (k + 1)-cycle α' on X' we have i^*f_*α' = g_*(i')^*α' in CH_k(D) (this makes sense as f_* is defined on the level of cycles).","statement_latex":"In Situation \\ref{situation-setup}. Let $f : X' \\to X$ be a proper morphism\nof good algebraic spaces over $B$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}.\nForm the diagram\n$$\n\\xymatrix{\nD' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\\nD \\ar[r]^i & X\n}\n$$\nas in Remark \\ref{remark-pullback-pairs}.\nFor any $(k + 1)$-cycle $\\alpha'$ on $X'$ we have\n$i^*f_*\\alpha' = g_*(i')^*\\alpha'$ in $\\CH_k(D)$\n(this makes sense as $f_*$ is defined on the level of cycles).","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ER5","source_file":"spaces-chow.tex","source_line":3327,"source_end_line":3343,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3327-L3343","statement_sha256":"861c58a3eaaa7414eaca251d919f397ae236b6c77489d6aa145d51af3fccbaec","origin":"The Stacks Project","memory_eligible":false,"source_rank":12625,"rank":12625,"depth":79,"x":542.234,"y":1657.224,"cluster":"geometry-of-spaces"},{"id":"stacks:0ER6","tag":"0ER6","title":"Intersecting with effective Cartier divisors · Lemma 0ER6","summary":"In Situation [Tag 0EDT]. Let f : X' → X be a flat morphism of relative dimension r of good algebraic spaces over B. Let (L, s, i : D → X) be as in Definition [Tag 0ER1]. Form the diagram xymatrix D' ar[d]_g ar[r]_i' & X' ar[d]^f D ar[r]^i & X as in Remark [Tag 0ER3]. For any (k + 1)-cycle α on X we have (i')^*f^*α = g^*i^*α' in CH_k + r(D) (this makes sense as f^* is defined on the level of cycles).","statement_latex":"In Situation \\ref{situation-setup}. Let $f : X' \\to X$\nbe a flat morphism of relative dimension $r$ of\ngood algebraic spaces over $B$. Let $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}. Form the diagram\n$$\n\\xymatrix{\nD' \\ar[d]_g \\ar[r]_{i'} & X' \\ar[d]^f \\\\\nD \\ar[r]^i & X\n}\n$$\nas in Remark \\ref{remark-pullback-pairs}.\nFor any $(k + 1)$-cycle $\\alpha$ on $X$ we have\n$(i')^*f^*\\alpha = g^*i^*\\alpha'$ in $\\CH_{k + r}(D)$\n(this makes sense as $f^*$ is defined on the level of cycles).","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ER6","source_file":"spaces-chow.tex","source_line":3365,"source_end_line":3381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3365-L3381","statement_sha256":"c30a33b50fce22ce3605837b2963894afc5e513c70ebf6768f8db268ad44afe8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12626,"rank":12626,"depth":79,"x":735.602,"y":1479.559,"cluster":"geometry-of-spaces"},{"id":"stacks:0ER7","tag":"0ER7","title":"Intersecting with effective Cartier divisors · Lemma 0ER7","summary":"In Situation [Tag 0EDT] let X/B be good. Let (L, s, i : D → X) be as in Definition [Tag 0ER1]. Let Z ⊂ X be a closed subscheme such that dim_δ(Z) ≤ k + 1 and such that D ∩ Z is an effective Cartier divisor on Z. Then i^*([Z]_k + 1) = [D ∩ Z]_k.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}.\nLet $Z \\subset X$ be a closed subscheme such\nthat $\\dim_\\delta(Z) \\leq k + 1$ and such that\n$D \\cap Z$ is an effective Cartier divisor on $Z$. Then\n$i^*([Z]_{k + 1}) = [D \\cap Z]_k$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Intersecting with effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ER7","source_file":"spaces-chow.tex","source_line":3399,"source_end_line":3408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3399-L3408","statement_sha256":"bbd76c02bda81726cf22b313c4b80e471e2e57cc5e899f07253fb4cc4cc68cc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12627,"rank":12627,"depth":17,"x":735.889,"y":1720.434,"cluster":"geometry-of-spaces"},{"id":"stacks:0ER9","tag":"0ER9","title":"Gysin homomorphisms · Lemma 0ER9","summary":"In Situation [Tag 0EDT] let X/B be good. Assume X integral and n = dim_δ(X). Let i : D → X be an effective Cartier divisor. Let N be an invertible O_X-module and let t be a nonzero meromorphic section of N. Then i^*div_N(t) = c_1(N) ∩ [D]_n - 1 in CH_n - 2(D).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nAssume $X$ integral and $n = \\dim_\\delta(X)$.\nLet $i : D \\to X$ be an effective Cartier divisor.\nLet $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module\nand let $t$ be a nonzero meromorphic section of $\\mathcal{N}$.\nThen $i^*\\text{div}_\\mathcal{N}(t) = c_1(\\mathcal{N}) \\cap [D]_{n - 1}$\nin $\\CH_{n - 2}(D)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ER9","source_file":"spaces-chow.tex","source_line":3444,"source_end_line":3453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3444-L3453","statement_sha256":"52c51e2b992a75e795a5b54b2cdd64717af1e3e61b5d0ec091d905d0ccbca68e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12628,"rank":12628,"depth":79,"x":541.855,"y":1542.872,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERA","tag":"0ERA","title":"Gysin homomorphisms · Lemma 0ERA","summary":"In Situation [Tag 0EDT] let X/B be good. Let (L, s, i : D → X) be as in Definition [Tag 0ER1]. The Gysin homomorphism factors through rational equivalence to give a map i^* : CH_k + 1(X) → CH_k(D).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $(\\mathcal{L}, s, i : D \\to X)$ be as in\nDefinition \\ref{definition-gysin-homomorphism}.\nThe Gysin homomorphism factors through rational equivalence to\ngive a map $i^* : \\CH_{k + 1}(X) \\to \\CH_k(D)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERA","source_file":"spaces-chow.tex","source_line":3512,"source_end_line":3519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3512-L3519","statement_sha256":"644a7d6774ee436a994bbb636a1d2ec8c483ec684dd95985042920a7d3717b1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12629,"rank":12629,"depth":81,"x":827.895,"y":1563.717,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERB","tag":"0ERB","title":"Gysin homomorphisms · Lemma 0ERB","summary":"In Situation [Tag 0EDT] let X/B be good. Let (L, s, i : D → X) be a triple as in Definition [Tag 0ER1]. Let N be an invertible O_X-module. Then i^*(c_1(N) ∩ α) = c_1(i^*N) ∩ i^*α in CH_k - 2(D) for all α ∈ CH_k(Z).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $(\\mathcal{L}, s, i : D \\to X)$\nbe a triple as in Definition \\ref{definition-gysin-homomorphism}.\nLet $\\mathcal{N}$ be an invertible $\\mathcal{O}_X$-module.\nThen $i^*(c_1(\\mathcal{N}) \\cap \\alpha) = c_1(i^*\\mathcal{N}) \\cap i^*\\alpha$\nin $\\CH_{k - 2}(D)$ for all $\\alpha \\in \\CH_k(Z)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERB","source_file":"spaces-chow.tex","source_line":3553,"source_end_line":3561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3553-L3561","statement_sha256":"ea23f521b25acd3e05cc34d9c3142d4851725342b03de27bd69eccf1a1f7f54e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12630,"rank":12630,"depth":82,"x":600.074,"y":1710.741,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERC","tag":"0ERC","title":"Gysin homomorphisms · Lemma 0ERC","summary":"In Situation [Tag 0EDT] let X/B be good. Let (L, s, i : D → X) and (L', s', i' : D' → X) be two triples as in Definition [Tag 0ER1]. Then the diagram xymatrix CH_k(X) ar[r]_i^* ar[d]_(i')^* & CH_k - 1(D) ar[d] CH_k - 1(D') ar[r] & CH_k - 2(D ∩ D') commutes where each of the maps is a Gysin map.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $(\\mathcal{L}, s, i : D \\to X)$ and\n$(\\mathcal{L}', s', i' : D' \\to X)$ be two triples as in\nDefinition \\ref{definition-gysin-homomorphism}. Then the diagram\n$$\n\\xymatrix{\n\\CH_k(X) \\ar[r]_{i^*} \\ar[d]_{(i')^*} & \\CH_{k - 1}(D) \\ar[d] \\\\\n\\CH_{k - 1}(D') \\ar[r] & \\CH_{k - 2}(D \\cap D')\n}\n$$\ncommutes where each of the maps is a Gysin map.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Gysin homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERC","source_file":"spaces-chow.tex","source_line":3571,"source_end_line":3584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3571-L3584","statement_sha256":"cfcb014f9aa78121f1547a3b99f744ff31a8f313282f00e54d9fdc24ef04d94f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12631,"rank":12631,"depth":79,"x":649.863,"y":1472.913,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERE","tag":"0ERE","title":"Relative effective Cartier divisors · Lemma 0ERE","summary":"In Situation [Tag 0EDT]. Let X, Y/B be good. Let p : X → Y be a flat morphism of relative dimension r. Let i : D → X be a relative effective Cartier divisor (Divisors on Spaces, Definition [Tag 0EPP]). Let L = O_X(D). For any α ∈ CH_k + 1(Y) we have i^*p^*α = (p|_D)^*α in CH_k + r(D) and c_1(L) ∩ p^*α = i_* ((p|_D)^*α) in CH_k + r(X).","statement_latex":"In Situation \\ref{situation-setup}. Let $X, Y/B$ be good.\nLet $p : X \\to Y$ be a flat morphism of relative dimension $r$.\nLet $i : D \\to X$ be a relative effective Cartier divisor\n(Divisors on Spaces, Definition\n\\ref{spaces-divisors-definition-relative-effective-Cartier-divisor}).\nLet $\\mathcal{L} = \\mathcal{O}_X(D)$.\nFor any $\\alpha \\in \\CH_{k + 1}(Y)$ we have\n$$\ni^*p^*\\alpha = (p|_D)^*\\alpha\n$$\nin $\\CH_{k + r}(D)$ and\n$$\nc_1(\\mathcal{L}) \\cap p^*\\alpha = i_* ((p|_D)^*\\alpha)\n$$\nin $\\CH_{k + r}(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Relative effective Cartier divisors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERE","source_file":"spaces-chow.tex","source_line":3671,"source_end_line":3688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3671-L3688","statement_sha256":"d6fd8d31d1f3e28e8a460038184dcdc8fed2d28859e6188d050e29a288ce658d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12632,"rank":12632,"depth":65,"x":804.497,"y":1676.658,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERG","tag":"0ERG","title":"Affine bundles · Lemma 0ERG","summary":"In Situation [Tag 0EDT] let X, Y/B be good. Let f : X → Y be a quasi-compact flat morphism over B of relative dimension r. Assume that for every y ∈ Y we have X_y ≅ A^r_kappa(y). Then f^* : CH_k(Y) → CH_k + r(X) is surjective for all k ∈ Z.","statement_latex":"In Situation \\ref{situation-setup} let $X, Y/B$ be good.\nLet $f : X \\to Y$ be a quasi-compact flat morphism over $B$\nof relative dimension $r$. Assume that for every $y \\in Y$ we have\n$X_y \\cong \\mathbf{A}^r_{\\kappa(y)}$.\nThen $f^* : \\CH_k(Y) \\to \\CH_{k + r}(X)$ is surjective for all\n$k \\in \\mathbf{Z}$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Affine bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERG","source_file":"spaces-chow.tex","source_line":3730,"source_end_line":3738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3730-L3738","statement_sha256":"d3e28a3eb167531e9433d3bd89ef91294174308d908ad9c82d3d92f9aa5c4580","origin":"The Stacks Project","memory_eligible":false,"source_rank":12633,"rank":12633,"depth":59,"x":526.46,"y":1614.125,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERH","tag":"0ERH","title":"Affine bundles · Lemma 0ERH","summary":"In Situation [Tag 0EDT] let X/B be good. Let L be an invertible O_X-module. Let p : L = underlineSpec(Sym^*(L)) → X be the associated vector bundle over X. Then p^* : CH_k(X) → CH_k + 1(L) is an isomorphism for all k.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nLet\n$$\np :\nL = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{L}))\n\\longrightarrow\nX\n$$\nbe the associated vector bundle over $X$.\nThen $p^* : \\CH_k(X) \\to \\CH_{k + 1}(L)$ is an isomorphism for all $k$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Affine bundles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERH","source_file":"spaces-chow.tex","source_line":3799,"source_end_line":3812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3799-L3812","statement_sha256":"b8c4025c0ebf4842c77a53bbb4baad8e25a7eb795434e44c7d2a896c3b990269","origin":"The Stacks Project","memory_eligible":false,"source_rank":12634,"rank":12634,"depth":66,"x":781.92,"y":1502.404,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERJ","tag":"0ERJ","title":"Bivariant intersection theory · Definition 0ERJ","summary":"Similar to [F] In Situation [Tag 0EDT] let f : X → Y be a morphism of good algebraic spaces over B. Let p ∈ Z. A bivariant class c of degree p for f is given by a rule which assigns to every morphism Y' → Y of good algebraic spaces over B and every k a map c ∩ - : CH_k(Y') → CH_k - p(X') where X' = Y' ×_Y X, satisfying the following conditions • if Y\" → Y' is a proper morphism, then c ∩ (Y\" → Y')_*α\" = (X\" → X')_*(c ∩ α\") for all α\" on Y\", • if Y\" → Y' a morphism of good…","statement_latex":"\\begin{reference}\nSimilar to \\cite[Definition 17.1]{F}\n\\end{reference}\nIn Situation \\ref{situation-setup} let $f : X \\to Y$ be a morphism of\ngood algebraic spaces over $B$. Let $p \\in \\mathbf{Z}$.\nA {\\it bivariant class $c$ of degree $p$ for $f$} is given by a rule\nwhich assigns to every morphism $Y' \\to Y$ of good algebraic spaces over $B$\nand every $k$ a map\n$$\nc \\cap - : \\CH_k(Y') \\longrightarrow \\CH_{k - p}(X')\n$$\nwhere $X' = Y' \\times_Y X$, satisfying the following conditions\n\\begin{enumerate}\n\\item if $Y'' \\to Y'$ is a proper morphism, then\n$c \\cap (Y'' \\to Y')_*\\alpha'' = (X'' \\to X')_*(c \\cap \\alpha'')$\nfor all $\\alpha''$ on $Y''$,\n\\item if $Y'' \\to Y'$ a morphism of good algebraic spaces over $B$\nwhich is flat of relative dimension $r$, then\n$c \\cap (Y'' \\to Y')^*\\alpha' = (X'' \\to X')^*(c \\cap \\alpha')$\nfor all $\\alpha'$ on $Y'$,\n\\item if $(\\mathcal{L}', s', i' : D' \\to Y')$ is as in\nDefinition \\ref{definition-gysin-homomorphism}\nwith pullback $(\\mathcal{N}', t', j' : E' \\to X')$ to $X'$,\nthen we have $c \\cap (i')^*\\alpha' = (j')^*(c \\cap \\alpha')$\nfor all $\\alpha'$ on $Y'$.\n\\end{enumerate}\nThe collection of all bivariant classes of degree $p$ for $f$ is\ndenoted $A^p(X \\to Y)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Bivariant intersection theory","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERJ","source_file":"spaces-chow.tex","source_line":3859,"source_end_line":3889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3859-L3889","statement_sha256":"b881fecf3c8e8be64965033b3da3a165b53537d543fae0066920c65f77b71d78","origin":"The Stacks Project","memory_eligible":false,"source_rank":12635,"rank":12635,"depth":1,"x":683.332,"y":1729.876,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERK","tag":"0ERK","title":"Bivariant intersection theory · Definition 0ERK","summary":"In Situation [Tag 0EDT] let X/B be good. The Chow cohomology of X is the graded Z-algebra A^*(X) whose degree p component is A^p(X → X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good. The {\\it Chow cohomology}\nof $X$ is the graded $\\mathbf{Z}$-algebra $A^*(X)$ whose degree\n$p$ component is $A^p(X \\to X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Bivariant intersection theory","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERK","source_file":"spaces-chow.tex","source_line":3904,"source_end_line":3909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3904-L3909","statement_sha256":"129d9d7e8bdf011cb079a234fc65252e27508bb0ba072282856ec6906ea5924d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12636,"rank":12636,"depth":0,"x":573.037,"y":1506.066,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERM","tag":"0ERM","title":"Bivariant intersection theory · Lemma 0ERM","summary":"In Situation [Tag 0EDT] let X/B be good. Let L be an invertible O_X-module. Then the rule that to f : X' → X assigns c_1(f^*L) ∩ - : CH_k(X') → CH_k - 1(X') is a bivariant class of degree 1.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen the rule that to $f : X' \\to X$ assigns\n$c_1(f^*\\mathcal{L}) \\cap - : \\CH_k(X') \\to \\CH_{k - 1}(X')$\nis a bivariant class of degree $1$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Bivariant intersection theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERM","source_file":"spaces-chow.tex","source_line":3926,"source_end_line":3933,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3926-L3933","statement_sha256":"330bcf58139bb7ad9b8029a8bfe79fc5549ba7ae2a3cc08a19c3e89b12533e44","origin":"The Stacks Project","memory_eligible":false,"source_rank":12637,"rank":12637,"depth":83,"x":834.504,"y":1608.577,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERN","tag":"0ERN","title":"Bivariant intersection theory · Lemma 0ERN","summary":"In Situation [Tag 0EDT] let f : X → Y be a morphism of good algebraic spaces over B which is flat of relative dimension r. Then the rule that to Y' → Y assigns (f')^* : CH_k(Y') → CH_k + r(X') where X' = X ×_Y Y' is a bivariant class of degree -r.","statement_latex":"In Situation \\ref{situation-setup} let $f : X \\to Y$ be a morphism\nof good algebraic spaces over $B$ which is flat of relative dimension $r$.\nThen the rule that to $Y' \\to Y$ assigns\n$(f')^* : \\CH_k(Y') \\to \\CH_{k + r}(X')$ where $X' = X \\times_Y Y'$\nis a bivariant class of degree $-r$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Bivariant intersection theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERN","source_file":"spaces-chow.tex","source_line":3942,"source_end_line":3949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3942-L3949","statement_sha256":"2b76b8cc4ec0224fabea73520027049750f79cf2022fca4d4df472b2d1753eea","origin":"The Stacks Project","memory_eligible":false,"source_rank":12638,"rank":12638,"depth":80,"x":559.101,"y":1681.393,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERP","tag":"0ERP","title":"Bivariant intersection theory · Lemma 0ERP","summary":"In Situation [Tag 0EDT] let X/B be good. Let (L, s, i : D → X) be a triple as in Definition [Tag 0ER1]. Then the rule that to f : X' → X assigns (i')^* : CH_k(X') → CH_k - 1(D') where D' = D ×_X X' is a bivariant class of degree 1.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $(\\mathcal{L}, s, i : D \\to X)$ be a triple as in\nDefinition \\ref{definition-gysin-homomorphism}.\nThen the rule that to $f : X' \\to X$ assigns\n$(i')^* : \\CH_k(X') \\to \\CH_{k - 1}(D')$ where $D' = D \\times_X X'$\nis a bivariant class of degree $1$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Bivariant intersection theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERP","source_file":"spaces-chow.tex","source_line":3959,"source_end_line":3967,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3959-L3967","statement_sha256":"b3622b0eb2987ab6dd0116f6290a2dca60b6376aca7ecf7fbb7f26cb3d81ecaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12639,"rank":12639,"depth":82,"x":703.709,"y":1471.305,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERQ","tag":"0ERQ","title":"Bivariant intersection theory · Lemma 0ERQ","summary":"In Situation [Tag 0EDT] let f : X → Y and g : Y → Z be morphisms of good algebraic spaces over B. Let c ∈ A^p(X → Z) and assume f is proper. Then the rule that to X' → X assigns α ↦ f_*(c ∩ α) is a bivariant class of degree p.","statement_latex":"In Situation \\ref{situation-setup} let $f : X \\to Y$ and\n$g : Y \\to Z$ be morphisms of good algebraic spaces over $B$.\nLet $c \\in A^p(X \\to Z)$ and assume $f$ is proper.\nThen the rule that to $X' \\to X$ assigns\n$\\alpha \\longmapsto f_*(c \\cap \\alpha)$\nis a bivariant class of degree $p$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Bivariant intersection theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERQ","source_file":"spaces-chow.tex","source_line":3976,"source_end_line":3984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3976-L3984","statement_sha256":"003b0aee94f7fb95f65ede309cca14f31a5a829fd0ffc0e84250686b61627578","origin":"The Stacks Project","memory_eligible":false,"source_rank":12640,"rank":12640,"depth":80,"x":766.062,"y":1708.415,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERR","tag":"0ERR","title":"Bivariant intersection theory · Lemma 0ERR","summary":"In Situation [Tag 0EDT] let X/B be good. Let L be an invertible O_X-module. Then c_1(L) ∈ A^1(X) commutes with every element c ∈ A^p(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen $c_1(\\mathcal{L}) \\in A^1(X)$ commutes with every\nelement $c \\in A^p(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Bivariant intersection theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERR","source_file":"spaces-chow.tex","source_line":3995,"source_end_line":4001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L3995-L4001","statement_sha256":"47c38f7274d79e82f523857d4e9483765aed013b111f19fb13a815f9fa7b0823","origin":"The Stacks Project","memory_eligible":false,"source_rank":12641,"rank":12641,"depth":79,"x":529.265,"y":1568.871,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERS","tag":"0ERS","title":"Bivariant intersection theory · Lemma 0ERS","summary":"In Situation [Tag 0EDT] let X/B be good. Let c ∈ A^p(X). Then c is zero if and only if c ∩ [Y] = 0 in CH_*(Y) for every integral algebraic space Y locally of finite type over X.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good. Let $c \\in A^p(X)$.\nThen $c$ is zero if and only if $c \\cap [Y] = 0$ in $\\CH_*(Y)$\nfor every integral algebraic space $Y$ locally of finite type over $X$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Bivariant intersection theory","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERS","source_file":"spaces-chow.tex","source_line":4036,"source_end_line":4041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4036-L4041","statement_sha256":"b11721421739c4392bef122de0cdbe09550025e6c19604c21da120b223550fbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12642,"rank":12642,"depth":0,"x":816.263,"y":1537.386,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERU","tag":"0ERU","title":"Projective space bundle formula · Lemma 0ERU","summary":"In Situation [Tag 0EDT] let X/B be good. Let E be a finite locally free O_X-module E of rank r. Let (π : P → X, O_P(1)) be the projective bundle associated to E. For any α ∈ CH_k(X) the element π_*( c_1(O_P(1))^s ∩ π^*α ) ∈ CH_k + r - 1 - s(X) is 0 if s < r - 1 and is equal to α when s = r - 1.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module\n$\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$\nbe the projective bundle associated to $\\mathcal{E}$.\nFor any $\\alpha \\in \\CH_k(X)$ the element\n$$\n\\pi_*\\left(\nc_1(\\mathcal{O}_P(1))^s \\cap \\pi^*\\alpha\n\\right)\n\\in\n\\CH_{k + r - 1 - s}(X)\n$$\nis $0$ if $s < r - 1$ and is equal to $\\alpha$ when $s = r - 1$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Projective space bundle formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERU","source_file":"spaces-chow.tex","source_line":4103,"source_end_line":4118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4103-L4118","statement_sha256":"23fe0d44a7b70a1a396fa2b65789c55d4668503278e522957a2656d44014db25","origin":"The Stacks Project","memory_eligible":false,"source_rank":12643,"rank":12643,"depth":84,"x":629.844,"y":1723.562,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERV","tag":"0ERV","title":"Projective space bundle formula · Lemma 0ERV","summary":"Let (S, δ) be as in Situation [Tag 0EDT]. Let X be locally of finite type over S. Let E be a finite locally free O_X-module E of rank r. Let (π : P → X, O_P(1)) be the projective bundle associated to E. The map bigoplus_i = 0^r - 1 CH_k + i(X) → CH_k + r - 1(P), (α_0, …, α_r-1) ↦ π^*α_0 + c_1(O_P(1)) ∩ π^*α_1 + … + c_1(O_P(1))^r - 1 ∩ π^*α_r-1 is an isomorphism.","statement_latex":"Let $(S, \\delta)$ be as in Situation \\ref{situation-setup}.\nLet $X$ be locally of finite type over $S$.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module\n$\\mathcal{E}$ of rank $r$. Let $(\\pi : P \\to X, \\mathcal{O}_P(1))$\nbe the projective bundle associated to $\\mathcal{E}$.\nThe map\n$$\n\\bigoplus\\nolimits_{i = 0}^{r - 1}\n\\CH_{k + i}(X)\n\\longrightarrow\n\\CH_{k + r - 1}(P),\n$$\n$$\n(\\alpha_0, \\ldots, \\alpha_{r-1})\n\\longmapsto\n\\pi^*\\alpha_0 +\nc_1(\\mathcal{O}_P(1)) \\cap \\pi^*\\alpha_1\n+ \\ldots +\nc_1(\\mathcal{O}_P(1))^{r - 1} \\cap \\pi^*\\alpha_{r-1}\n$$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Projective space bundle formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERV","source_file":"spaces-chow.tex","source_line":4163,"source_end_line":4186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4163-L4186","statement_sha256":"7de2dd72e28da674df0bc00d18a8e22c739b0b5dcf75ffdb07a24cc294afa394","origin":"The Stacks Project","memory_eligible":false,"source_rank":12644,"rank":12644,"depth":85,"x":617.582,"y":1480.357,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERW","tag":"0ERW","title":"Projective space bundle formula · Lemma 0ERW","summary":"In Situation [Tag 0EDT] let X/B be good. Let E be a finite locally free sheaf of rank r on X. Let p : E = underlineSpec(Sym^*(E)) → X be the associated vector bundle over X. Then p^* : CH_k(X) → CH_k + r(E) is an isomorphism for all k.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.\nLet\n$$\np :\nE = \\underline{\\Spec}(\\text{Sym}^*(\\mathcal{E}))\n\\longrightarrow\nX\n$$\nbe the associated vector bundle over $X$.\nThen $p^* : \\CH_k(X) \\to \\CH_{k + r}(E)$ is an isomorphism for all $k$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Projective space bundle formula","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERW","source_file":"spaces-chow.tex","source_line":4297,"source_end_line":4310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4297-L4310","statement_sha256":"370df82755b50d54a1a84c09c4261ca40986c534d18c7208213f1ee13aab52a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12645,"rank":12645,"depth":86,"x":822.324,"y":1652.837,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERY","tag":"0ERY","title":"The Chern classes of a vector bundle · Lemma 0ERY","summary":"In Situation [Tag 0EDT] let X/B be good. Let E be a finite locally free sheaf of rank r on X. Let (π : P → X, O_P(1)) be the projective space bundle associated to E. For every morphism X' → X of good algebraic spaces over B there are unique maps c_i(E) ∩ - : CH_k(X') → CH_k - i(X'), i = 0, …, r such that for α ∈ CH_k(X') we have c_0(E) ∩ α = α and ∑_i = 0, …, r (-1)^i c_1(O_P'(1))^i ∩ (π')^*(c_r - i(E) ∩ α) = 0 where π' : P' → X' is the base change of π. Moreover, these…","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.\nLet $(\\pi : P \\to X, \\mathcal{O}_P(1))$ be the projective space\nbundle associated to $\\mathcal{E}$. For every\nmorphism $X' \\to X$ of good algebraic spaces over $B$\nthere are unique maps\n$$\nc_i(\\mathcal{E}) \\cap - : \\CH_k(X') \\longrightarrow \\CH_{k - i}(X'),\\quad\ni = 0, \\ldots, r\n$$\nsuch that for $\\alpha \\in \\CH_k(X')$ we have\n$c_0(\\mathcal{E}) \\cap \\alpha = \\alpha$ and\n$$\n\\sum\\nolimits_{i = 0, \\ldots, r}\n(-1)^i c_1(\\mathcal{O}_{P'}(1))^i \\cap\n(\\pi')^*\\left(c_{r - i}(\\mathcal{E}) \\cap \\alpha\\right) = 0\n$$\nwhere $\\pi' : P' \\to X'$ is the base change of $\\pi$.\nMoreover, these maps define a bivariant class\n$c_i(\\mathcal{E})$ of degree $i$ on $X$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The Chern classes of a vector bundle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERY","source_file":"spaces-chow.tex","source_line":4379,"source_end_line":4401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4379-L4401","statement_sha256":"11ef02aefd92ef78645a290d1f1d8b41d8085e10196106660743650b8511450f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12646,"rank":12646,"depth":86,"x":532.475,"y":1641.822,"cluster":"geometry-of-spaces"},{"id":"stacks:0ERZ","tag":"0ERZ","title":"The Chern classes of a vector bundle · Definition 0ERZ","summary":"In Situation [Tag 0EDT] let X/B be good. Let E be a finite locally free sheaf of rank r on X. For i = 0, …, r the ith Chern class of E is the bivariant class c_i(E) ∈ A^i(X) of degree i constructed in Lemma [Tag 0ERY]. The total Chern class of E is the formal sum c(E) = c_0(E) + c_1(E) + … + c_r(E) which is viewed as a nonhomogeneous bivariant class on X.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$.\nFor $i = 0, \\ldots, r$ the {\\it $i$th Chern class of $\\mathcal{E}$}\nis the bivariant class $c_i(\\mathcal{E}) \\in A^i(X)$ of degree $i$\nconstructed in Lemma \\ref{lemma-segre-classes}.\nThe {\\it total Chern class of $\\mathcal{E}$}\nis the formal sum\n$$\nc(\\mathcal{E}) =\nc_0(\\mathcal{E}) + c_1(\\mathcal{E}) + \\ldots + c_r(\\mathcal{E})\n$$\nwhich is viewed as a nonhomogeneous bivariant class on $X$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The Chern classes of a vector bundle","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ERZ","source_file":"spaces-chow.tex","source_line":4439,"source_end_line":4453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4439-L4453","statement_sha256":"36a192ad5495f9ccfdaf8230fae9811a6740dcc36c03ece1c1e11d0eb0d970f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12647,"rank":12647,"depth":87,"x":755.195,"y":1485.385,"cluster":"geometry-of-spaces"},{"id":"stacks:0ES0","tag":"0ES0","title":"The Chern classes of a vector bundle · Lemma 0ES0","summary":"In Situation [Tag 0EDT] let X/B be good. Let L be an invertible O_X-module. The first Chern class of L on X of Definition [Tag 0ERZ] is equal to the bivariant class of Lemma [Tag 0ERM].","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThe first Chern class of $\\mathcal{L}$ on $X$ of\nDefinition \\ref{definition-chern-classes}\nis equal to the bivariant class of Lemma \\ref{lemma-cap-c1-bivariant}.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The Chern classes of a vector bundle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ES0","source_file":"spaces-chow.tex","source_line":4461,"source_end_line":4468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4461-L4468","statement_sha256":"2115c37a366879db9ad6316552edffdf858c7a87d2cbf51abd57a5be8862552f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12648,"rank":12648,"depth":88,"x":716.745,"y":1727.258,"cluster":"geometry-of-spaces"},{"id":"stacks:0ES1","tag":"0ES1","title":"The Chern classes of a vector bundle · Lemma 0ES1","summary":"In Situation [Tag 0EDT] let X/B be good. Let E be a locally free O_X-module of rank r. Then c_j(L) ∈ A^j(X) commutes with every element c ∈ A^p(X). In particular, if F is a second locally free O_X-module on X of rank s, then c_i(E) ∩ c_j(F) ∩ α = c_j(F) ∩ c_i(E) ∩ α as elements of CH_k - i - j(X) for all α ∈ CH_k(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$ be a locally free $\\mathcal{O}_X$-module of rank $r$.\nThen $c_j(\\mathcal{L}) \\in A^j(X)$ commutes with every\nelement $c \\in A^p(X)$. In particular, if $\\mathcal{F}$ is a\nsecond locally free $\\mathcal{O}_X$-module on $X$ of rank $s$, then\n$$\nc_i(\\mathcal{E}) \\cap c_j(\\mathcal{F}) \\cap \\alpha\n=\nc_j(\\mathcal{F}) \\cap c_i(\\mathcal{E}) \\cap \\alpha\n$$\nas elements of $\\CH_{k - i - j}(X)$ for all $\\alpha \\in \\CH_k(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The Chern classes of a vector bundle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ES1","source_file":"spaces-chow.tex","source_line":4490,"source_end_line":4503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4490-L4503","statement_sha256":"b93116ddfdbf20f48ab9411829045470d8495d3f1c1c21d0048b30fb0c0a4aa3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12649,"rank":12649,"depth":87,"x":550.492,"y":1526.969,"cluster":"geometry-of-spaces"},{"id":"stacks:0ES4","tag":"0ES4","title":"Polynomial relations among Chern classes · Lemma 0ES4","summary":"In Situation [Tag 0EDT] let X/B be good. Let E be a finite locally free sheaf of rank r on X. Let L be an invertible sheaf on X. Then we have c_i( E ⊗ L) = ∑_j = 0^i binomr - i + jj c_i - j( E) c_1( L)^j in A^*(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$ be a finite locally free sheaf of\nrank $r$ on $X$. Let $\\mathcal{L}$ be an invertible\nsheaf on $X$. Then we have\n\\begin{equation}\n\nc_i({\\mathcal E} \\otimes {\\mathcal L})\n=\n\\sum\\nolimits_{j = 0}^i\n\\binom{r - i + j}{j} c_{i - j}({\\mathcal E}) c_1({\\mathcal L})^j\n\\end{equation}\nin $A^*(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Polynomial relations among Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ES4","source_file":"spaces-chow.tex","source_line":4610,"source_end_line":4624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4610-L4624","statement_sha256":"fdc17c5432c64bd9cfa143eb71723f51d6dafbb0baacf705c944cc3f55137db3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12650,"rank":12650,"depth":87,"x":834.317,"y":1580.354,"cluster":"geometry-of-spaces"},{"id":"stacks:0ES7","tag":"0ES7","title":"Additivity of Chern classes · Lemma 0ES7","summary":"In Situation [Tag 0EDT] let X/B be good. Let E, F be finite locally free sheaves on X of ranks r, r - 1 which fit into a short exact sequence 0 → O_X → E → F → 0 Then we have c_r(E) = 0, c_j(E) = c_j(F), j = 0, …, r - 1 in A^*(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves\non $X$ of ranks $r$, $r - 1$ which fit into a short\nexact sequence\n$$\n0 \\to \\mathcal{O}_X \\to \\mathcal{E} \\to \\mathcal{F} \\to 0\n$$\nThen we have\n$$\nc_r(\\mathcal{E}) = 0, \\quad\nc_j(\\mathcal{E}) = c_j(\\mathcal{F}), \\quad j = 0, \\ldots, r - 1\n$$\nin $A^*(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Additivity of Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ES7","source_file":"spaces-chow.tex","source_line":4651,"source_end_line":4666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4651-L4666","statement_sha256":"e9174471b74bbfd4a7271262a85791ce30109c07e60ceb575b94242c4103c146","origin":"The Stacks Project","memory_eligible":false,"source_rank":12651,"rank":12651,"depth":87,"x":581.95,"y":1702.11,"cluster":"geometry-of-spaces"},{"id":"stacks:0ES8","tag":"0ES8","title":"Additivity of Chern classes · Lemma 0ES8","summary":"In Situation [Tag 0EDT] let X/B be good. Let E, F be finite locally free sheaves on X of ranks r, r - 1 which fit into a short exact sequence 0 → L → E → F → 0 where L is an invertible sheaf. Then c(E) = c(L) c(F) in A^*(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$, $\\mathcal{F}$ be finite locally free sheaves\non $X$ of ranks $r$, $r - 1$ which fit into a short\nexact sequence\n$$\n0 \\to \\mathcal{L} \\to \\mathcal{E} \\to \\mathcal{F} \\to 0\n$$\nwhere $\\mathcal{L}$ is an invertible sheaf.\nThen\n$$\nc(\\mathcal{E}) = c(\\mathcal{L}) c(\\mathcal{F})\n$$\nin $A^*(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Additivity of Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ES8","source_file":"spaces-chow.tex","source_line":4678,"source_end_line":4693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4678-L4693","statement_sha256":"923e42874daa37d6c7243b09d0b2e934fb1c0c4b065eacd3e35eca0527675e79","origin":"The Stacks Project","memory_eligible":false,"source_rank":12652,"rank":12652,"depth":88,"x":670.181,"y":1468.993,"cluster":"geometry-of-spaces"},{"id":"stacks:0ES9","tag":"0ES9","title":"Additivity of Chern classes · Lemma 0ES9","summary":"In Situation [Tag 0EDT] let X/B be good. Suppose that E sits in an exact sequence 0 → E_1 → E → E_2 → 0 of finite locally free sheaves E_i of rank r_i. The total Chern classes satisfy c(E) = c(E_1) c(E_2) in A^*(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nSuppose that $\\mathcal{E}$ sits in an exact sequence\n$$\n0\n\\to\n\\mathcal{E}_1\n\\to\n\\mathcal{E}\n\\to\n\\mathcal{E}_2\n\\to\n0\n$$\nof finite locally free sheaves $\\mathcal{E}_i$ of rank $r_i$.\nThe total Chern classes satisfy\n$$\nc(\\mathcal{E}) = c(\\mathcal{E}_1) c(\\mathcal{E}_2)\n$$\nin $A^*(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Additivity of Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ES9","source_file":"spaces-chow.tex","source_line":4704,"source_end_line":4725,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4704-L4725","statement_sha256":"0e193e34b41a0ff273904c12b5c4f4f3553262c001887b29ad3bc91d65117c47","origin":"The Stacks Project","memory_eligible":false,"source_rank":12653,"rank":12653,"depth":89,"x":792.658,"y":1691.084,"cluster":"geometry-of-spaces"},{"id":"stacks:0ESA","tag":"0ESA","title":"Additivity of Chern classes · Lemma 0ESA","summary":"In Situation [Tag 0EDT] let X/B be good. Let L_i, i = 1, …, r be invertible O_X-modules. Let E be a locally free rank O_X-module endowed with a filtration 0 = E_0 ⊂ E_1 ⊂ E_2 ⊂ … ⊂ E_r = E such that E_i/E_i - 1 ≅ L_i. Set c_1( L_i) = x_i. Then c(E) = ∏_i = 1^r (1 + x_i) in A^*(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet ${\\mathcal L}_i$, $i = 1, \\ldots, r$ be invertible\n$\\mathcal{O}_X$-modules.\nLet $\\mathcal{E}$ be a locally free rank\n$\\mathcal{O}_X$-module endowed with a filtration\n$$\n0 = \\mathcal{E}_0 \\subset \\mathcal{E}_1 \\subset \\mathcal{E}_2\n\\subset \\ldots \\subset \\mathcal{E}_r = \\mathcal{E}\n$$\nsuch that $\\mathcal{E}_i/\\mathcal{E}_{i - 1} \\cong \\mathcal{L}_i$.\nSet $c_1({\\mathcal L}_i) = x_i$. Then\n$$\nc(\\mathcal{E})\n=\n\\prod\\nolimits_{i = 1}^r (1 + x_i)\n$$\nin $A^*(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Additivity of Chern classes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESA","source_file":"spaces-chow.tex","source_line":4736,"source_end_line":4755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4736-L4755","statement_sha256":"8496175db6ffbe954857d7cef72ecf42c206e76e28a15ec0b6d7371814ca707f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12654,"rank":12654,"depth":89,"x":523.589,"y":1596.759,"cluster":"geometry-of-spaces"},{"id":"stacks:0ESC","tag":"0ESC","title":"The splitting principle · Lemma 0ESC","summary":"In Situation [Tag 0EDT] let X/B be good. Let E_i be a finite collection of locally free O_X-modules of rank r_i. There exists a projective flat morphism π : P → X of relative dimension d such that • for any morphism f : Y → X of good algebraic spaces over B the map π_Y^* : CH_*(Y) → CH_* + d(Y ×_X P) is injective, and • each π^*E_i has a filtration whose successive quotients L_i, 1, …, L_i, r_i are invertible O_P-modules.","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}_i$ be a finite collection of locally free\n$\\mathcal{O}_X$-modules of rank $r_i$. There exists a projective\nflat morphism $\\pi : P \\to X$ of relative dimension $d$ such that\n\\begin{enumerate}\n\\item for any morphism $f : Y \\to X$ of good algebraic spaces\nover $B$ the map\n$\\pi_Y^* : \\CH_*(Y) \\to \\CH_{* + d}(Y \\times_X P)$ is injective, and\n\\item each $\\pi^*\\mathcal{E}_i$ has a filtration\nwhose successive quotients $\\mathcal{L}_{i, 1}, \\ldots, \\mathcal{L}_{i, r_i}$\nare invertible ${\\mathcal O}_P$-modules.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The splitting principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESC","source_file":"spaces-chow.tex","source_line":4786,"source_end_line":4800,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4786-L4800","statement_sha256":"93f43ec1aef3bcc9c71b884ed4a5f04e9ae329dce988efe677da663e633c6033","origin":"The Stacks Project","memory_eligible":false,"source_rank":12655,"rank":12655,"depth":86,"x":798.01,"y":1513.588,"cluster":"geometry-of-spaces"},{"id":"stacks:0ESD","tag":"0ESD","title":"The splitting principle · Lemma 0ESD","summary":"In Situation [Tag 0EDT] let X/B be good. Let E be a finite locally free O_X-module with dual E^vee. Then c_i(E^vee) = (-1)^i c_i(E) in A^i(X).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$ be a finite locally free $\\mathcal{O}_X$-module\nwith dual $\\mathcal{E}^\\vee$. Then\n$$\nc_i(\\mathcal{E}^\\vee) = (-1)^i c_i(\\mathcal{E})\n$$\nin $A^i(X)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The splitting principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESD","source_file":"spaces-chow.tex","source_line":4829,"source_end_line":4838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4829-L4838","statement_sha256":"f60d5137f7187328c73c3f0de819e41759ef9c1b5664b9fe4dd411faba3f1006","origin":"The Stacks Project","memory_eligible":false,"source_rank":12656,"rank":12656,"depth":90,"x":662.461,"y":1730.757,"cluster":"geometry-of-spaces"},{"id":"stacks:0ESE","tag":"0ESE","title":"The splitting principle · Lemma 0ESE","summary":"In Situation [Tag 0EDT] let X/B be good. Let E and F be a finite locally free O_X-modules of ranks r and s. Then we have c_1(E ⊗ F) = r c_1(F) + s c_1(E) c_2(E ⊗ F) = r^2 c_2(F) + rs c_1(F)c_1(E) + s^2 c_2(E) and so on (see proof).","statement_latex":"In Situation \\ref{situation-setup} let $X/B$ be good.\nLet $\\mathcal{E}$ and $\\mathcal{F}$ be a finite locally free\n$\\mathcal{O}_X$-modules of ranks $r$ and $s$. Then we have\n$$\nc_1(\\mathcal{E} \\otimes \\mathcal{F})\n=\nr c_1(\\mathcal{F}) + s c_1(\\mathcal{E})\n$$\n$$\nc_2(\\mathcal{E} \\otimes \\mathcal{F})\n=\nr^2 c_2(\\mathcal{F}) +\nrs c_1(\\mathcal{F})c_1(\\mathcal{E}) +\ns^2 c_2(\\mathcal{E})\n$$\nand so on (see proof).","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"The splitting principle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESE","source_file":"spaces-chow.tex","source_line":4876,"source_end_line":4894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4876-L4894","statement_sha256":"6e0328cb1133545b64802d47902659613fc50b3975d854484c9b84e665f66086","origin":"The Stacks Project","memory_eligible":false,"source_rank":12657,"rank":12657,"depth":91,"x":587.729,"y":1493.568,"cluster":"geometry-of-spaces"},{"id":"stacks:0ESG","tag":"0ESG","title":"Degrees of zero cycles · Definition 0ESG","summary":"Let k be a field. Let p : X → Spec(k) be a proper morphism of algebraic spaces. The degree of a zero cycle on X is given by proper pushforward p_* : CH_0(X) → CH_0(Spec(k)) → Z (Lemma [Tag 0EQC]) composed with the natural isomorphism CH_0(Spec(k)) → Z which maps [Spec(k)] to 1. Notation: deg(α).","statement_latex":"Let $k$ be a field. Let $p : X \\to \\Spec(k)$ be a proper morphism of\nalgebraic spaces. The {\\it degree of a zero cycle} on $X$ is given by\nproper pushforward\n$$\np_* : \\CH_0(X) \\longrightarrow \\CH_0(\\Spec(k)) \\longrightarrow \\mathbf{Z}\n$$\n(Lemma \\ref{lemma-proper-pushforward-rational-equivalence})\ncomposed with the natural isomorphism $\\CH_0(\\Spec(k)) \\to \\mathbf{Z}$\nwhich maps $[\\Spec(k)]$ to $1$. Notation: $\\deg(\\alpha)$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Degrees of zero cycles","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESG","source_file":"spaces-chow.tex","source_line":4963,"source_end_line":4974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4963-L4974","statement_sha256":"273bed8012992bb0d7bb1eb013b707157a949de58960028f6df141666864c6e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12658,"rank":12658,"depth":78,"x":833.719,"y":1626.139,"cluster":"geometry-of-spaces"},{"id":"stacks:0ESH","tag":"0ESH","title":"Degrees of zero cycles · Lemma 0ESH","summary":"Let k be a field. Let X be a proper algebraic space over k. Let α = ∑ n_i[Z_i] be in Z_0(X). Then deg(α) = ∑ n_ideg(Z_i) where deg(Z_i) is the degree of Z_i → Spec(k), i.e., deg(Z_i) = dim_k Γ(Z_i, O_Z_i).","statement_latex":"Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.\nLet $\\alpha = \\sum n_i[Z_i]$ be in $Z_0(X)$. Then\n$$\n\\deg(\\alpha) = \\sum n_i\\deg(Z_i)\n$$\nwhere $\\deg(Z_i)$ is the degree of $Z_i \\to \\Spec(k)$, i.e.,\n$\\deg(Z_i) = \\dim_k \\Gamma(Z_i, \\mathcal{O}_{Z_i})$.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Degrees of zero cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESH","source_file":"spaces-chow.tex","source_line":4979,"source_end_line":4988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4979-L4988","statement_sha256":"e6fcf78effc17a06f664a1885e943270e9e7c86eb6faf7574223518b48b2cd11","origin":"The Stacks Project","memory_eligible":false,"source_rank":12659,"rank":12659,"depth":65,"x":545.552,"y":1667.988,"cluster":"geometry-of-spaces"},{"id":"stacks:0ESI","tag":"0ESI","title":"Degrees of zero cycles · Lemma 0ESI","summary":"Let k be a field. Let X be a proper algebraic space over k. Let Z ⊂ X be a closed subspace of dimension d. Let L_1, …, L_d be invertible O_X-modules. Then (L_1 … L_d · Z) = deg( c_1(L_1) ∩ … ∩ c_1(L_1) ∩ [Z]_d) where the left hand side is defined in Spaces over Fields, Definition [Tag 0EDF].","statement_latex":"Let $k$ be a field. Let $X$ be a proper algebraic space over $k$.\nLet $Z \\subset X$ be a closed subspace of dimension $d$.\nLet $\\mathcal{L}_1, \\ldots, \\mathcal{L}_d$ be invertible\n$\\mathcal{O}_X$-modules. Then\n$$\n(\\mathcal{L}_1 \\cdots \\mathcal{L}_d \\cdot Z) =\n\\deg(\nc_1(\\mathcal{L}_1) \\cap \\ldots \\cap c_1(\\mathcal{L}_1) \\cap [Z]_d)\n$$\nwhere the left hand side is defined in\nSpaces over Fields, Definition\n\\ref{spaces-over-fields-definition-intersection-number}.","area":"Geometry of Spaces","chapter":"Chow Groups of Spaces","chapter_id":"spaces-chow","section":"Degrees of zero cycles","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ESI","source_file":"spaces-chow.tex","source_line":4995,"source_end_line":5009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-chow.tex#L4995-L5009","statement_sha256":"4a6bd13d5f0f3ef7d47e7a239a39ab49512fba3bbaf195ae2a7e9ccac536d6ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":12660,"rank":12660,"depth":80,"x":724.492,"y":1473.504,"cluster":"geometry-of-spaces"},{"id":"stacks:048E","tag":"048E","title":"Invariant morphisms · Definition 048E","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j = (t, s) : R → U ×_B U be a pre-relation of algebraic spaces over B. We say a morphism φ : U → X of algebraic spaces over B is R-invariant if the diagram xymatrix R ar[r]_s ar[d]_t & U ar[d]^φ U ar[r]^φ & X is commutative. If j : R → U ×_B U comes from the action of a group algebraic space G on U over B as in Groupoids in Spaces, Lemma [Tag 0444], then we say that φ is G-invariant.","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j = (t, s) : R \\to U \\times_B U$ be a pre-relation of algebraic\nspaces over $B$. We say a morphism $\\phi : U \\to X$ of algebraic spaces\nover $B$ is {\\it $R$-invariant} if the diagram\n$$\n\\xymatrix{\nR \\ar[r]_s \\ar[d]_t & U \\ar[d]^\\phi \\\\\nU \\ar[r]^\\phi & X\n}\n$$\nis commutative. If $j : R \\to U \\times_B U$ comes from the action\nof a group algebraic space $G$ on $U$ over $B$ as in\nGroupoids in Spaces, Lemma \\ref{spaces-groupoids-lemma-groupoid-from-action},\nthen we say that $\\phi$ is {\\it $G$-invariant}.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Invariant morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048E","source_file":"groupoids-quotients.tex","source_line":43,"source_end_line":59,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L43-L59","statement_sha256":"91fdb4dcbf21896a98c96a3ef530106572a5e44868ed16b2c3d909247c289837","origin":"The Stacks Project","memory_eligible":false,"source_rank":12661,"rank":12661,"depth":1,"x":980.905,"y":1082.486,"cluster":"groupoids-quotients"},{"id":"stacks:048F","tag":"048F","title":"Invariant morphisms · Lemma 048F","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j = (t, s) : R → U ×_B U be a pre-relation of algebraic spaces over B. A morphism of algebraic spaces φ : U → X is R-invariant if and only if it factors as U → U/R → X.","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j = (t, s) : R \\to U \\times_B U$ be a pre-relation of algebraic\nspaces over $B$. A morphism of algebraic spaces $\\phi : U \\to X$ is\n$R$-invariant if and only if it factors as\n$U \\to U/R \\to X$.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Invariant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048F","source_file":"groupoids-quotients.tex","source_line":66,"source_end_line":73,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L66-L73","statement_sha256":"43804450d8eacb68103ec2dc87e72309ed5bd0edd6f07356e0eb5e57bff49f96","origin":"The Stacks Project","memory_eligible":false,"source_rank":12662,"rank":12662,"depth":0,"x":1286.467,"y":1097.736,"cluster":"groupoids-quotients"},{"id":"stacks:048G","tag":"048G","title":"Invariant morphisms · Lemma 048G","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j = (t, s) : R → U ×_B U be a pre-relation of algebraic spaces over B. Let U → X be an R-invariant morphism of algebraic spaces over B. Let X' → X be any morphism of algebraic spaces. • Setting U' = X' ×_X U, R' = X' ×_X R we obtain a pre-relation j' : R' → U' ×_B U'. • If j is a relation, then j' is a relation. • If j is a pre-equivalence relation, then j' is a pre-equivalence relation. • If j is an…","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j = (t, s) : R \\to U \\times_B U$ be a pre-relation of algebraic\nspaces over $B$. Let $U \\to X$ be an $R$-invariant morphism of algebraic\nspaces over $B$. Let $X' \\to X$ be any morphism of algebraic spaces.\n\\begin{enumerate}\n\\item Setting $U' = X' \\times_X U$, $R' = X' \\times_X R$ we obtain\na pre-relation $j' : R' \\to U' \\times_B U'$.\n\\item If $j$ is a relation, then $j'$ is a relation.\n\\item If $j$ is a pre-equivalence relation, then $j'$ is a\npre-equivalence relation.\n\\item If $j$ is an equivalence relation, then $j'$ is an equivalence\nrelation.\n\\item If $j$ comes from a groupoid in algebraic spaces\n$(U, R, s, t, c)$ over $B$, then\n\\begin{enumerate}\n\\item $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $X$, and\n\\item $j'$ comes from the base change $(U', R', s', t', c')$\nof this groupoid to $X'$, see\nGroupoids in Spaces, Lemma\n\\ref{spaces-groupoids-lemma-base-change-groupoid}.\n\\end{enumerate}\n\\item If $j$ comes from the action of a group algebraic space $G/B$ on $U$\nas in Groupoids in Spaces, Lemma\n\\ref{spaces-groupoids-lemma-groupoid-from-action}\nthen $j'$ comes from the induced action of $G$ on $U'$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Invariant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048G","source_file":"groupoids-quotients.tex","source_line":80,"source_end_line":108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L80-L108","statement_sha256":"0011b063a91f5f2f3e034436bffd5cf4fb79eb2873b955031722739310ebad41","origin":"The Stacks Project","memory_eligible":false,"source_rank":12663,"rank":12663,"depth":1,"x":1048.468,"y":1260.159,"cluster":"groupoids-quotients"},{"id":"stacks:048H","tag":"048H","title":"Invariant morphisms · Definition 048H","summary":"In the situation of Lemma [Tag 048G] we call j' : R' → U' ×_B U' the base change of the pre-relation j to X'. We say it is a flat base change if X' → X is a flat morphism of algebraic spaces.","statement_latex":"In the situation of Lemma \\ref{lemma-base-change-on-invariant}\nwe call $j' : R' \\to U' \\times_B U'$ the {\\it base change} of the pre-relation\n$j$ to $X'$. We say it is a {\\it flat base change} if $X' \\to X$ is a flat\nmorphism of algebraic spaces.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Invariant morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048H","source_file":"groupoids-quotients.tex","source_line":123,"source_end_line":129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L123-L129","statement_sha256":"1b4c67e9d1d6d5ca42f4c017c79dd4b0605868beefe9a44079fe349e6cb3aab8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12664,"rank":12664,"depth":2,"x":1093.428,"y":1004.897,"cluster":"groupoids-quotients"},{"id":"stacks:0DTF","tag":"0DTF","title":"Invariant morphisms · Lemma 0DTF","summary":"In the situation of Lemma [Tag 048G] there is an isomorphism of sheaves U'/R' = X' ×_X U/R For the construction of quotient sheaves, see Groupoids in Spaces, Section [Tag 044H].","statement_latex":"In the situation of Lemma \\ref{lemma-base-change-on-invariant}\nthere is an isomorphism of sheaves\n$$\nU'/R' = X' \\times_X U/R\n$$\nFor the construction of quotient sheaves, see\nGroupoids in Spaces, Section \\ref{spaces-groupoids-section-quotient-sheaves}.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Invariant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTF","source_file":"groupoids-quotients.tex","source_line":135,"source_end_line":144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L135-L144","statement_sha256":"40d2939da4982310a5c6065aefb65c41385e7e5a6a50e4619f045bb2265cec6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12665,"rank":12665,"depth":2,"x":1265.851,"y":1219.009,"cluster":"groupoids-quotients"},{"id":"stacks:0DTG","tag":"0DTG","title":"Invariant morphisms · Lemma 0DTG","summary":"Let S be a scheme. Let B be an algebraic space over S. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let U → X be an R-invariant morphism of algebraic spaces over B. Let g : X' → X be a morphism of algebraic spaces over B and let (U', R', s', t', c') be the base change as in Lemma [Tag 048G]. Then xymatrix [U'/R'] ar[r] ar[d] & [U/R] ar[d] S_X' ar[r] & S_X is a 2-fibre product of stacks in groupoids over (Sch/S)_fppf. For the construction of quotient…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nLet $U \\to X$ be an $R$-invariant morphism of algebraic spaces over\n$B$. Let $g : X' \\to X$ be a morphism of algebraic spaces over $B$\nand let $(U', R', s', t', c')$ be the base change as in\nLemma \\ref{lemma-base-change-on-invariant}. Then\n$$\n\\xymatrix{\n[U'/R'] \\ar[r] \\ar[d] & [U/R] \\ar[d] \\\\\n\\mathcal{S}_{X'} \\ar[r] & \\mathcal{S}_X\n}\n$$\nis a $2$-fibre product of stacks in groupoids over $(\\Sch/S)_{fppf}$.\nFor the construction of quotient stacks and the morphisms in this\ndiagram, see\nGroupoids in Spaces, Section \\ref{spaces-groupoids-section-stacks}.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Invariant morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTG","source_file":"groupoids-quotients.tex","source_line":164,"source_end_line":182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L164-L182","statement_sha256":"96bf895c3fd8bf3425c6ece9bc86324c3a8e5139269f307b1451d4851c0c49c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12666,"rank":12666,"depth":10,"x":966.004,"y":1158.858,"cluster":"groupoids-quotients"},{"id":"stacks:048J","tag":"048J","title":"Categorical quotients · Definition 048J","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j = (t, s) : R → U ×_B U be pre-relation in algebraic spaces over B. • We say a morphism φ : U → X of algebraic spaces over B is a categorical quotient if it is R-invariant, and for every R-invariant morphism ψ : U → Y of algebraic spaces over B there exists a unique morphism chi : X → Y such that ψ = φ ∘ chi. • Let C be a full subcategory of the category of algebraic spaces over B. Assume U, R are objects of…","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j = (t, s) : R \\to U \\times_B U$ be pre-relation in algebraic spaces\nover $B$.\n\\begin{enumerate}\n\\item We say a morphism $\\phi : U \\to X$ of algebraic spaces over $B$\nis a {\\it categorical quotient} if it is $R$-invariant, and\nfor every $R$-invariant morphism $\\psi : U \\to Y$ of algebraic spaces over $B$\nthere exists a unique morphism $\\chi : X \\to Y$ such that\n$\\psi = \\phi \\circ \\chi$.\n\\item Let $\\mathcal{C}$ be a full subcategory of the category of algebraic\nspaces over $B$. Assume $U$, $R$ are objects of $\\mathcal{C}$.\nIn this situation we say\na morphism $\\phi : U \\to X$ of algebraic spaces over $B$\nis a {\\it categorical quotient in $\\mathcal{C}$}\nif $X \\in \\Ob(\\mathcal{C})$, and $\\phi$ is $R$-invariant,\nand for every $R$-invariant morphism\n$\\psi : U \\to Y$ with $Y \\in \\Ob(\\mathcal{C})$\nthere exists a unique morphism $\\chi : X \\to Y$ such\nthat $\\psi = \\phi \\circ \\chi$.\n\\item If $B = S$ and $\\mathcal{C}$ is the category of schemes over $S$,\nthen we say $U \\to X$ is a\n{\\it categorical quotient in the category of schemes}, or simply a\n{\\it categorical quotient in schemes}.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Categorical quotients","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048J","source_file":"groupoids-quotients.tex","source_line":279,"source_end_line":305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L279-L305","statement_sha256":"c424ea64365fa70ccd1b2e8cda8834d485b3dc5037e63dd8f9bcf3585a44f040","origin":"The Stacks Project","memory_eligible":false,"source_rank":12667,"rank":12667,"depth":0,"x":1235.947,"y":1032.853,"cluster":"groupoids-quotients"},{"id":"stacks:048K","tag":"048K","title":"Categorical quotients · Lemma 048K","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j : R → U ×_B U be a pre-relation in algebraic spaces over B. If a categorical quotient in the category of algebraic spaces over B exists, then it is unique up to unique isomorphism. Similarly for categorical quotients in full subcategories of Spaces/B.","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation in algebraic spaces over $B$.\nIf a categorical quotient in the category of algebraic spaces\nover $B$ exists, then it is unique up to unique isomorphism.\nSimilarly for categorical quotients in full subcategories of\n$\\textit{Spaces}/B$.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Categorical quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048K","source_file":"groupoids-quotients.tex","source_line":317,"source_end_line":325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L317-L325","statement_sha256":"f606a536a7e684dae9a44b5baccdad36b65f42abd5e1c3ff07e6f01a1c70dbe1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12668,"rank":12668,"depth":0,"x":1138.055,"y":1279.365,"cluster":"groupoids-quotients"},{"id":"stacks:048L","tag":"048L","title":"Categorical quotients · Definition 048L","summary":"Let S be a scheme, and let B be an algebraic space over S. Let C be a full subcategory of the category of algebraic spaces over B closed under fibre products. Let j = (t, s) : R → U ×_B U be pre-relation in C, and let U → X be an R-invariant morphism with X ∈ Ob(C). • We say U → X is a universal categorical quotient in C if for every morphism X' → X in C the morphism U' = X' ×_X U → X' is the categorical quotient in C of the base change j' : R' → U' of j. • We say U → X…","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $\\mathcal{C}$ be a full subcategory of the category of algebraic\nspaces over $B$ closed under fibre products.\nLet $j = (t, s) : R \\to U \\times_B U$ be pre-relation in\n$\\mathcal{C}$, and let $U \\to X$ be an $R$-invariant morphism with\n$X \\in \\Ob(\\mathcal{C})$.\n\\begin{enumerate}\n\\item We say $U \\to X$ is a {\\it universal categorical quotient}\nin $\\mathcal{C}$ if for every morphism $X' \\to X$ in $\\mathcal{C}$\nthe morphism $U' = X' \\times_X U \\to X'$ is the categorical quotient in\n$\\mathcal{C}$ of the base change $j' : R' \\to U'$ of $j$.\n\\item We say $U \\to X$ is a {\\it uniform categorical quotient}\nin $\\mathcal{C}$ if for every flat morphism $X' \\to X$ in $\\mathcal{C}$\nthe morphism $U' = X' \\times_X U \\to X'$ is the categorical quotient in\n$\\mathcal{C}$ of the base change $j' : R' \\to U'$ of $j$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Categorical quotients","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048L","source_file":"groupoids-quotients.tex","source_line":359,"source_end_line":377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L359-L377","statement_sha256":"3c4ab41a9eed92ef3398a3373e84f46639b11e0adad440245a9a1c91b028ca80","origin":"The Stacks Project","memory_eligible":false,"source_rank":12669,"rank":12669,"depth":0,"x":1011.783,"y":1041.636,"cluster":"groupoids-quotients"},{"id":"stacks:049W","tag":"049W","title":"Categorical quotients · Lemma 049W","summary":"In the situation of Definition [Tag 048J]. If φ : U → X is a categorical quotient and U is reduced, then X is reduced. The same holds for categorical quotients in a category of spaces C listed in Example [Tag 049V].","statement_latex":"In the situation of\nDefinition \\ref{definition-categorical}.\nIf $\\phi : U \\to X$ is a categorical quotient and $U$ is reduced,\nthen $X$ is reduced. The same holds for categorical quotients in\na category of spaces $\\mathcal{C}$ listed in\nExample \\ref{example-categories}.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Categorical quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049W","source_file":"groupoids-quotients.tex","source_line":379,"source_end_line":387,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L379-L387","statement_sha256":"e2a340b571e54771b0f625d730e67a4107c64995852e453ae8f3c2b142b2147f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12670,"rank":12670,"depth":4,"x":1296.558,"y":1145.463,"cluster":"groupoids-quotients"},{"id":"stacks:048N","tag":"048N","title":"Quotients as orbit spaces · Definition 048N","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j : R → U ×_B U be a pre-relation over B. If u ∈ |U|, then the orbit, or more precisely the R-orbit of u is O_u = ( u' ∈ |U| : ∃ n ≥ 1, ∃ u_0, …, u_n ∈ |U| such that u_0 = u and u_n = u' and for all i ∈ (0, …, n - 1) either u_i = u_i + 1 or ∃ r ∈ |R|, s(r) = u_i, t(r) = u_i + 1 or ∃ r ∈ |R|, t(r) = u_i, s(r) = u_i + 1 )","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation over $B$.\nIf $u \\in |U|$, then the {\\it orbit}, or more precisely the\n{\\it $R$-orbit} of $u$ is\n$$\nO_u =\n\\left\\{\nu' \\in |U|\\ :\n\\begin{matrix}\n\\exists n \\geq 1, \\ \\exists u_0, \\ldots, u_n \\in |U|\\text{ such that }\nu_0 = u \\text{ and } u_n = u' \\\\\n\\text{and for all }i \\in \\{0, \\ldots, n - 1\\}\\text{ either }\nu_i = u_{i + 1}\\text{ or } \\\\\n\\exists r \\in |R|, \\ s(r) = u_i, t(r) = u_{i + 1}\n\\text{ or } \\\\\n\\exists r \\in |R|, \\ t(r) = u_i, s(r) = u_{i + 1}\n\\end{matrix}\n\\right\\}\n$$","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048N","source_file":"groupoids-quotients.tex","source_line":424,"source_end_line":445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L424-L445","statement_sha256":"483282ce71463292ccc2fe07a457ea922f2ba050cd391e2ddb54f508bf970528","origin":"The Stacks Project","memory_eligible":false,"source_rank":12671,"rank":12671,"depth":0,"x":1002.572,"y":1230.636,"cluster":"groupoids-quotients"},{"id":"stacks:048O","tag":"048O","title":"Quotients as orbit spaces · Lemma 048O","summary":"Let B → S as in Section [Tag 048C]. Let j : R → U ×_B U be a pre-equivalence relation of algebraic spaces over B. Then O_u = (u' ∈ |U| such that ∃ r ∈ |R|, s(r) = u, t(r) = u').","statement_latex":"Let $B \\to S$ as in Section \\ref{section-conventions-notation}.\nLet $j : R \\to U \\times_B U$ be a pre-equivalence relation\nof algebraic spaces over $B$. Then\n$$\nO_u =\n\\{u' \\in |U| \\text{ such that } \\exists r \\in |R|, \\ s(r) = u, \\ t(r) = u'\\}.\n$$","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048O","source_file":"groupoids-quotients.tex","source_line":454,"source_end_line":463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L454-L463","statement_sha256":"e78c762b9925db8632c25b0023df28a795400e475c983fc719898aa2f73a53c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12672,"rank":12672,"depth":1,"x":1151.113,"y":1000.624,"cluster":"groupoids-quotients"},{"id":"stacks:048P","tag":"048P","title":"Quotients as orbit spaces · Lemma 048P","summary":"In the situation of Definition [Tag 048N]. Let φ : U → X be an R-invariant morphism of algebraic spaces over B. Then |φ| : |U| → |X| is constant on the orbits.","statement_latex":"In the situation of Definition \\ref{definition-orbit}.\nLet $\\phi : U \\to X$ be an $R$-invariant morphism of algebraic spaces over\n$B$. Then $|\\phi| : |U| \\to |X|$ is constant on the orbits.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048P","source_file":"groupoids-quotients.tex","source_line":474,"source_end_line":479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L474-L479","statement_sha256":"13c14e92a7cd7e9fe482548dc6a930cf762cffe086d20ca200dfd7ddc5f0b1c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12673,"rank":12673,"depth":1,"x":1226.678,"y":1254.949,"cluster":"groupoids-quotients"},{"id":"stacks:048Q","tag":"048Q","title":"Quotients as orbit spaces · Definition 048Q","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j : R → U ×_B U be a pre-relation over B. Let Spec(k) → B be a geometric point of B. • We say overlineu, overlineu' ∈ U(k) are weakly R-equivalent if they are in the same equivalence class for the equivalence relation generated by the relation j(R(k)) ⊂ U(k) × U(k). • We say overlineu, overlineu' ∈ U(k) are R-equivalent if for some overfield k ⊂ Ω the images in U(Ω) are weakly R-equivalent. • The weak orbit,…","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation over $B$.\nLet $\\Spec(k) \\to B$ be a geometric point of $B$.\n\\begin{enumerate}\n\\item We say $\\overline{u}, \\overline{u}' \\in U(k)$ are\n{\\it weakly $R$-equivalent} if they are in the same equivalence class\nfor the equivalence relation generated by the relation\n$j(R(k)) \\subset U(k) \\times U(k)$.\n\\item We say $\\overline{u}, \\overline{u}' \\in U(k)$ are\n{\\it $R$-equivalent} if for some overfield $k \\subset \\Omega$\nthe images in $U(\\Omega)$ are weakly $R$-equivalent.\n\\item The {\\it weak orbit}, or more precisely the {\\it weak $R$-orbit}\nof $\\overline{u} \\in U(k)$ is set of all\nelements of $U(k)$ which are weakly $R$-equivalent to $\\overline{u}$.\n\\item The {\\it orbit}, or more precisely the {\\it $R$-orbit}\nof $\\overline{u} \\in U(k)$ is set of all\nelements of $U(k)$ which are $R$-equivalent to $\\overline{u}$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048Q","source_file":"groupoids-quotients.tex","source_line":536,"source_end_line":556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L536-L556","statement_sha256":"9f872e268a0a13736ec813fdfe800767abc58699857cab844ede199a2cc2156b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12674,"rank":12674,"depth":0,"x":965.995,"y":1110.045,"cluster":"groupoids-quotients"},{"id":"stacks:048R","tag":"048R","title":"Quotients as orbit spaces · Lemma 048R","summary":"Let S be a scheme, and let B be an algebraic space over S. Let Spec(k) → B be a geometric point of B. Let j : R → U ×_B U be a pre-equivalence relation over B. In this case the weak orbit of overlineu ∈ U(k) is simply ( overlineu' ∈ U(k) such that ∃ overliner ∈ R(k), s(overliner) = overlineu, t(overliner) = overlineu' ) and the orbit of overlineu ∈ U(k) is ( overlineu' ∈ U(k) : ∃ field extension K/k, ∃ r ∈ R(K), s(r) = overlineu, t(r) = overlineu')","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $\\Spec(k) \\to B$ be a geometric point of $B$.\nLet $j : R \\to U \\times_B U$ be a pre-equivalence relation over $B$.\nIn this case the weak orbit of $\\overline{u} \\in U(k)$ is simply\n$$\n\\{\n\\overline{u}' \\in U(k)\n\\text{ such that }\n\\exists \\overline{r} \\in R(k),\n\\ s(\\overline{r}) = \\overline{u},\n\\ t(\\overline{r}) = \\overline{u}'\n\\}\n$$\nand the orbit of $\\overline{u} \\in U(k)$ is\n$$\n\\{\n\\overline{u}' \\in U(k) :\n\\exists\\text{ field extension }K/k, \\ \\exists\\ r \\in R(K),\n\\ s(r) = \\overline{u}, \\ t(r) = \\overline{u}'\\}\n$$","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048R","source_file":"groupoids-quotients.tex","source_line":563,"source_end_line":585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L563-L585","statement_sha256":"04c9a2857cc08d8578bf1ab1f4cf48dd98bf9d2b0031361261b6f6cba2b249ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":12675,"rank":12675,"depth":0,"x":1275.268,"y":1068.908,"cluster":"groupoids-quotients"},{"id":"stacks:048T","tag":"048T","title":"Quotients as orbit spaces · Lemma 048T","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j : R → U ×_B U be a pre-relation over B. Then j_∞ : R_∞ → U ×_B U is a pre-equivalence relation over B. Moreover • φ : U → X is R-invariant if and only if it is R_∞-invariant, • the canonical map of quotient sheaves U/R → U/R_∞ (see Groupoids in Spaces, Section [Tag 044H]) is an isomorphism, • weak R-orbits agree with weak R_∞-orbits, • R-orbits agree with R_∞-orbits, • if s, t are locally of finite type,…","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation over $B$.\nThen $j_\\infty : R_\\infty \\to U \\times_B U$ is a\npre-equivalence relation over $B$. Moreover\n\\begin{enumerate}\n\\item $\\phi : U \\to X$ is $R$-invariant if and only if it is\n$R_\\infty$-invariant,\n\\item the canonical map of quotient sheaves $U/R \\to U/R_\\infty$ (see\nGroupoids in Spaces, Section \\ref{spaces-groupoids-section-quotient-sheaves})\nis an isomorphism,\n\\item weak $R$-orbits agree with weak $R_\\infty$-orbits,\n\\item $R$-orbits agree with $R_\\infty$-orbits,\n\\item if $s, t$ are locally of finite type, then $s_\\infty$, $t_\\infty$\nare locally of finite type,\n\\item add more here as needed.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048T","source_file":"groupoids-quotients.tex","source_line":648,"source_end_line":666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L648-L666","statement_sha256":"3140878679e7113ad04c6762e59a9c177132ec4f21f186e45e0eae1d12891f82","origin":"The Stacks Project","memory_eligible":false,"source_rank":12676,"rank":12676,"depth":0,"x":1079.968,"y":1275.077,"cluster":"groupoids-quotients"},{"id":"stacks:048U","tag":"048U","title":"Quotients as orbit spaces · Lemma 048U","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j : R → U ×_B U be a pre-relation over B. Let Spec(k) → B be a geometric point of B. • If s, t : R → U are locally of finite type then weak R-equivalence on U(k) agrees with R-equivalence, and weak R-orbits agree with R-orbits on U(k). • If k has sufficiently large cardinality then weak R-equivalence on U(k) agrees with R-equivalence, and weak R-orbits agree with R-orbits on U(k).","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation over $B$.\nLet $\\Spec(k) \\to B$ be a geometric point of $B$.\n\\begin{enumerate}\n\\item If $s, t : R \\to U$ are locally of finite type\nthen weak $R$-equivalence on $U(k)$ agrees with $R$-equivalence, and\nweak $R$-orbits agree with $R$-orbits on $U(k)$.\n\\item If $k$ has sufficiently large cardinality then weak $R$-equivalence\non $U(k)$ agrees with $R$-equivalence, and weak $R$-orbits agree\nwith $R$-orbits on $U(k)$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048U","source_file":"groupoids-quotients.tex","source_line":674,"source_end_line":687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L674-L687","statement_sha256":"6a5dd2a609c04ed2b6a633359355d7fd632af3f7d2e8f6e6c409ff4797a56aa5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12677,"rank":12677,"depth":46,"x":1058.147,"y":1011.792,"cluster":"groupoids-quotients"},{"id":"stacks:048V","tag":"048V","title":"Quotients as orbit spaces · Definition 048V","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j : R → U ×_B U be a pre-relation over B. • We say φ : U → X is set-theoretically R-invariant if and only if the map U(k) → X(k) equalizes the two maps s, t : R(k) → U(k) for every algebraically closed field k over B. • We say φ : U → X separates orbits, or separates R-orbits if it is set-theoretically R-invariant and φ(overlineu) = φ(overlineu') in X(k) implies that overlineu, overlineu' ∈ U(k) are in the…","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation over $B$.\n\\begin{enumerate}\n\\item We say $\\phi : U \\to X$ is {\\it set-theoretically $R$-invariant}\nif and only if the map $U(k) \\to X(k)$ equalizes the two maps\n$s, t : R(k) \\to U(k)$ for every algebraically closed field $k$\nover $B$.\n\\item We say $\\phi : U \\to X$ {\\it separates orbits}, or\n{\\it separates $R$-orbits} if it is set-theoretically $R$-invariant and\n$\\phi(\\overline{u}) = \\phi(\\overline{u}')$ in $X(k)$ implies that\n$\\overline{u}, \\overline{u}' \\in U(k)$ are in the same orbit\nfor every algebraically closed field $k$ over $B$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048V","source_file":"groupoids-quotients.tex","source_line":735,"source_end_line":750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L735-L750","statement_sha256":"06b71d071efe9f44d092d5476bf39161d4c6b4ce1779d98d9abea8d58dfdff8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12678,"rank":12678,"depth":0,"x":1286.345,"y":1193.857,"cluster":"groupoids-quotients"},{"id":"stacks:048W","tag":"048W","title":"Quotients as orbit spaces · Lemma 048W","summary":"In the situation of Definition [Tag 048V]. A morphism φ : U → X is set-theoretically R-invariant if and only if for any algebraically closed field k over B the map U(k) → X(k) is constant on orbits.","statement_latex":"In the situation of Definition \\ref{definition-set-theoretically-invariant}.\nA morphism $\\phi : U \\to X$ is set-theoretically $R$-invariant if and\nonly if for any algebraically closed field $k$ over $B$ the map\n$U(k) \\to X(k)$ is constant on orbits.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048W","source_file":"groupoids-quotients.tex","source_line":760,"source_end_line":766,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L760-L766","statement_sha256":"0fbf44101b64aefee45ca5b300ad0faf6f03424398b6945dccbb8179b835ca53","origin":"The Stacks Project","memory_eligible":false,"source_rank":12679,"rank":12679,"depth":1,"x":971.14,"y":1189.082,"cluster":"groupoids-quotients"},{"id":"stacks:048X","tag":"048X","title":"Quotients as orbit spaces · Lemma 048X","summary":"In the situation of Definition [Tag 048V]. An invariant morphism is set-theoretically invariant.","statement_latex":"In the situation of Definition \\ref{definition-set-theoretically-invariant}.\nAn invariant morphism is set-theoretically invariant.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048X","source_file":"groupoids-quotients.tex","source_line":773,"source_end_line":777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L773-L777","statement_sha256":"893090c94db494e7948cee2b86b72d65adb745113d5e6f7e8b3b4f86940c46b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12680,"rank":12680,"depth":1,"x":1207.8,"y":1013.457,"cluster":"groupoids-quotients"},{"id":"stacks:048Y","tag":"048Y","title":"Quotients as orbit spaces · Lemma 048Y","summary":"In the situation of Definition [Tag 048V]. Let φ : U → X be a morphism of algebraic spaces over B. Assume • φ is set-theoretically R-invariant, • R is reduced, and • X is locally separated over B. Then φ is R-invariant.","statement_latex":"In the situation of Definition \\ref{definition-set-theoretically-invariant}.\nLet $\\phi : U \\to X$ be a morphism of algebraic spaces over $B$.\nAssume\n\\begin{enumerate}\n\\item $\\phi$ is set-theoretically $R$-invariant,\n\\item $R$ is reduced, and\n\\item $X$ is locally separated over $B$.\n\\end{enumerate}\nThen $\\phi$ is $R$-invariant.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/048Y","source_file":"groupoids-quotients.tex","source_line":783,"source_end_line":794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L783-L794","statement_sha256":"d72611ff56ccc17cfee6f03cca5e68544a0ea9c301fcbf5343fb519e0bd5182e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12681,"rank":12681,"depth":2,"x":1174.467,"y":1277.683,"cluster":"groupoids-quotients"},{"id":"stacks:0490","tag":"0490","title":"Quotients as orbit spaces · Definition 0490","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j : R → U ×_B U be a pre-relation over B. • We say j is a set-theoretic pre-equivalence relation if for all algebraically closed fields k over B the relation sim_R on U(k) defined by overlineu sim_R overlineu' ⇔ ∃ field extension K/k, ∃ r ∈ R(K), s(r) = overlineu, t(r) = overlineu' is an equivalence relation. • We say j is a set-theoretic equivalence relation if j is universally injective and a set-theoretic…","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation over $B$.\n\\begin{enumerate}\n\\item We say $j$ is a {\\it set-theoretic pre-equivalence relation} if\nfor all algebraically closed fields $k$ over $B$ the relation\n$\\sim_R$ on $U(k)$ defined by\n$$\n\\overline{u} \\sim_R \\overline{u}'\n\\Leftrightarrow\n\\begin{matrix}\n\\exists\\text{ field extension }K/k, \\ \\exists\\ r \\in R(K), \\\\\ns(r) = \\overline{u}, \\ t(r) = \\overline{u}'\n\\end{matrix}\n$$\nis an equivalence relation.\n\\item We say $j$ is a {\\it set-theoretic equivalence relation}\nif $j$ is universally injective and a set-theoretic pre-equivalence\nrelation.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0490","source_file":"groupoids-quotients.tex","source_line":847,"source_end_line":868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L847-L868","statement_sha256":"7a9d7bd2501faa6b88162f0f58fff21342a6d5840934bd9307a606356217ffe6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12682,"rank":12682,"depth":0,"x":986.253,"y":1063.581,"cluster":"groupoids-quotients"},{"id":"stacks:0491","tag":"0491","title":"Quotients as orbit spaces · Lemma 0491","summary":"In the situation of Definition [Tag 0490]. The following are equivalent: • The morphism j is a set-theoretic pre-equivalence relation. • The subset j(|R|) ⊂ |U ×_B U| contains the image of |j'| for any of the morphisms j' as in Equation ([Tag 048S]). • For every algebraically closed field k over B of sufficiently large cardinality the subset j(R(k)) ⊂ U(k) × U(k) is an equivalence relation. If s, t are locally of finite type these are also equivalent to • [(4)] For every…","statement_latex":"In the situation of Definition \\ref{definition-set-theoretic-equivalence}.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $j$ is a set-theoretic pre-equivalence relation.\n\\item The subset $j(|R|) \\subset |U \\times_B U|$ contains the image of\n$|j'|$ for any of the morphisms $j'$ as in Equation (\\ref{equation-list}).\n\\item For every algebraically closed field $k$ over $B$ of sufficiently large\ncardinality the subset $j(R(k)) \\subset U(k) \\times U(k)$ is an equivalence\nrelation.\n\\end{enumerate}\nIf $s, t$ are locally of finite type these are also equivalent to\n\\begin{enumerate}\n\\item[(4)] For every algebraically closed field $k$ over $B$\nthe subset $j(R(k)) \\subset U(k) \\times U(k)$ is an equivalence relation.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0491","source_file":"groupoids-quotients.tex","source_line":873,"source_end_line":890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L873-L890","statement_sha256":"467f611a65049978fd7eab8c82a1b531df613423b69c51f5ba625ea4b9ec558b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12683,"rank":12683,"depth":46,"x":1297.726,"y":1114.744,"cluster":"groupoids-quotients"},{"id":"stacks:049X","tag":"049X","title":"Quotients as orbit spaces · Lemma 049X","summary":"In the situation of Definition [Tag 0490]. The following are equivalent: • The morphism j is a set-theoretic equivalence relation. • The morphism j is universally injective and j(|R|) ⊂ |U ×_B U| contains the image of |j'| for any of the morphisms j' as in Equation ([Tag 048S]). • For every algebraically closed field k over B of sufficiently large cardinality the map j : R(k) → U(k) × U(k) is injective and its image is an equivalence relation. If j is decent, or locally…","statement_latex":"In the situation of Definition \\ref{definition-set-theoretic-equivalence}.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $j$ is a set-theoretic equivalence relation.\n\\item The morphism $j$ is universally injective and\n$j(|R|) \\subset |U \\times_B U|$ contains the image of\n$|j'|$ for any of the morphisms $j'$ as in Equation (\\ref{equation-list}).\n\\item For every algebraically closed field $k$ over $B$ of sufficiently large\ncardinality the map $j : R(k) \\to U(k) \\times U(k)$ is injective and\nits image is an equivalence relation.\n\\end{enumerate}\nIf $j$ is decent, or locally separated, or quasi-separated\nthese are also equivalent to\n\\begin{enumerate}\n\\item[(4)] For every algebraically closed field $k$ over $B$\nthe map $j : R(k) \\to U(k) \\times U(k)$ is injective and its image\nis an equivalence relation.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049X","source_file":"groupoids-quotients.tex","source_line":1005,"source_end_line":1025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1005-L1025","statement_sha256":"3681107c7885e06d05d5318b5b9d650e90ac0943bec0080d7fa813fedce673a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12684,"rank":12684,"depth":61,"x":1026.463,"y":1253.981,"cluster":"groupoids-quotients"},{"id":"stacks:0492","tag":"0492","title":"Quotients as orbit spaces · Lemma 0492","summary":"Let S be a scheme, and let B be an algebraic space over S. Let j : R → U ×_B U be a pre-relation over B. • If j is a pre-equivalence relation, then j is a set-theoretic pre-equivalence relation. This holds in particular when j comes from a groupoid in algebraic spaces, or from an action of a group algebraic space on U. • If j is an equivalence relation, then j is a set-theoretic equivalence relation.","statement_latex":"Let $S$ be a scheme, and let $B$ be an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation over $B$.\n\\begin{enumerate}\n\\item If $j$ is a pre-equivalence relation, then $j$ is a\nset-theoretic pre-equivalence relation. This holds in particular\nwhen $j$ comes from a groupoid in algebraic spaces, or from an\naction of a group algebraic space on $U$.\n\\item If $j$ is an equivalence relation, then $j$ is a\nset-theoretic equivalence relation.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0492","source_file":"groupoids-quotients.tex","source_line":1055,"source_end_line":1067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1055-L1067","statement_sha256":"ecf4b935d0f338b709d9fb8d80fb1999dbbd971cf632e001893d6e5634c22535","origin":"The Stacks Project","memory_eligible":false,"source_rank":12685,"rank":12685,"depth":0,"x":1114.661,"y":996.97,"cluster":"groupoids-quotients"},{"id":"stacks:049Y","tag":"049Y","title":"Quotients as orbit spaces · Lemma 049Y","summary":"Let B → S be as in Section [Tag 048C]. Let j : R → U ×_B U be a pre-relation. Let φ : U → X be a morphism of algebraic spaces over B. Consider the diagram xymatrix (U ×_X U) ×_(U ×_B U) R ar[d]^q ar[r]_-p & R ar[d]^j U ×_X U ar[r]^c & U ×_B U Then we have: • The morphism φ is set-theoretically invariant if and only if p is surjective. • If j is a set-theoretic pre-equivalence relation then φ separates orbits if and only if p and q are surjective. • If p and q are…","statement_latex":"Let $B \\to S$ be as in Section \\ref{section-conventions-notation}.\nLet $j : R \\to U \\times_B U$ be a pre-relation.\nLet $\\phi : U \\to X$ be a morphism of algebraic spaces over $B$.\nConsider the diagram\n$$\n\\xymatrix{\n(U \\times_X U) \\times_{(U \\times_B U)} R \\ar[d]^q \\ar[r]_-p & R \\ar[d]^j \\\\\nU \\times_X U \\ar[r]^c & U \\times_B U\n}\n$$\nThen we have:\n\\begin{enumerate}\n\\item The morphism $\\phi$ is set-theoretically invariant if and only\nif $p$ is surjective.\n\\item If $j$ is a set-theoretic pre-equivalence relation then\n$\\phi$ separates orbits if and only if $p$ and $q$ are surjective.\n\\item If $p$ and $q$ are surjective, then $j$ is a set-theoretic\npre-equivalence relation (and $\\phi$ separates orbits).\n\\item If $\\phi$ is $R$-invariant and $j$ is a set-theoretic pre-equivalence\nrelation, then $\\phi$ separates orbits if and only if the induced morphism\n$R \\to U \\times_X U$ is surjective.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049Y","source_file":"groupoids-quotients.tex","source_line":1073,"source_end_line":1097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1073-L1097","statement_sha256":"e2df06280e192f607f131e363d9921edd827e3147680e7303bce85bd84adb5dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12686,"rank":12686,"depth":46,"x":1256.537,"y":1236.922,"cluster":"groupoids-quotients"},{"id":"stacks:0493","tag":"0493","title":"Quotients as orbit spaces · Definition 0493","summary":"Let B → S as in Section [Tag 048C]. Let j : R → U ×_B U be a pre-relation. We say φ : U → X is an orbit space for R if • φ is R-invariant, • φ separates R-orbits, and • φ is surjective.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-conventions-notation}.\nLet $j : R \\to U \\times_B U$ be a pre-relation.\nWe say $\\phi : U \\to X$ is an {\\it orbit space for $R$} if\n\\begin{enumerate}\n\\item $\\phi$ is $R$-invariant,\n\\item $\\phi$ separates $R$-orbits, and\n\\item $\\phi$ is surjective.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0493","source_file":"groupoids-quotients.tex","source_line":1136,"source_end_line":1146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1136-L1146","statement_sha256":"27ff0b4440ce7d504116b70ea2e785660333e231490274aa46f15beaf67c4d4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12687,"rank":12687,"depth":0,"x":958.474,"y":1140.332,"cluster":"groupoids-quotients"},{"id":"stacks:049Z","tag":"049Z","title":"Quotients as orbit spaces · Lemma 049Z","summary":"Let B → S as in Section [Tag 048C]. Let j : R → U ×_B U be a set-theoretic pre-equivalence relation. A morphism φ : U → X is an orbit space for R if and only if • φ ∘ s = φ ∘ t, i.e., φ is invariant, • the induced morphism (t, s) : R → U ×_X U is surjective, and • the morphism φ : U → X is surjective. This characterization applies for example if j is a pre-equivalence relation, or comes from a groupoid in algebraic spaces over B, or comes from the action of a group…","statement_latex":"Let $B \\to S$ as in Section \\ref{section-conventions-notation}.\nLet $j : R \\to U \\times_B U$ be a set-theoretic pre-equivalence\nrelation. A morphism $\\phi : U \\to X$ is an orbit space for $R$ if and only if\n\\begin{enumerate}\n\\item $\\phi \\circ s = \\phi \\circ t$, i.e., $\\phi$ is invariant,\n\\item the induced morphism $(t, s) : R \\to U \\times_X U$ is surjective, and\n\\item the morphism $\\phi : U \\to X$ is surjective.\n\\end{enumerate}\nThis characterization applies for example if $j$ is a pre-equivalence relation,\nor comes from a groupoid in algebraic spaces over $B$, or comes from the action\nof a group algebraic space over $B$ on $U$.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/049Z","source_file":"groupoids-quotients.tex","source_line":1156,"source_end_line":1169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1156-L1169","statement_sha256":"6a6951f558b2b0ce9b6d506062cc0f03a3b18613e54ca85b3c7652954ca317bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12688,"rank":12688,"depth":47,"x":1256.417,"y":1042.27,"cluster":"groupoids-quotients"},{"id":"stacks:04A0","tag":"04A0","title":"Quotients as orbit spaces · Lemma 04A0","summary":"Let B → S as in Section [Tag 048C]. Let j = (t, s) : R → U ×_B U be a pre-relation. Assume R, U are locally of finite type over B. Let φ : U → X be an R-invariant morphism of algebraic spaces over B. Then φ is an orbit space for R if and only if the natural map U(k)/big(equivalence relation generated by j(R(k))big) → X(k) is bijective for all algebraically closed fields k over B.","statement_latex":"Let $B \\to S$ as in Section \\ref{section-conventions-notation}.\nLet $j = (t, s) : R \\to U \\times_B U$ be a pre-relation.\nAssume $R, U$ are locally of finite type over $B$.\nLet $\\phi : U \\to X$ be an $R$-invariant morphism of algebraic spaces over $B$.\nThen $\\phi$ is an orbit space for $R$ if and only if the natural map\n$$\nU(k)/\\big(\\text{equivalence relation generated by }j(R(k))\\big)\n\\longrightarrow\nX(k)\n$$\nis bijective for all algebraically closed fields $k$ over $B$.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Quotients as orbit spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04A0","source_file":"groupoids-quotients.tex","source_line":1182,"source_end_line":1195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1182-L1195","statement_sha256":"8e64d7b53202273cba058cef6100e14b8088f28d4f8b08852ff43b41345c21fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12689,"rank":12689,"depth":47,"x":1115.35,"y":1284.028,"cluster":"groupoids-quotients"},{"id":"stacks:04A2","tag":"04A2","title":"Coarse quotients · Definition 04A2","summary":"Let S be a scheme and B an algebraic space over S. Let j : R → U ×_B U be a pre-relation. A morphism φ : U → X of algebraic spaces over B is called a coarse quotient if • φ is a categorical quotient, and • φ is an orbit space. If S = B, U, R are all schemes, then we say a morphism of schemes φ : U → X is a coarse quotient in schemes if • φ is a categorical quotient in schemes, and • φ is an orbit space.","statement_latex":"Let $S$ be a scheme and $B$ an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation.\nA morphism $\\phi : U \\to X$ of algebraic spaces over $B$\nis called a {\\it coarse quotient} if\n\\begin{enumerate}\n\\item $\\phi$ is a categorical quotient, and\n\\item $\\phi$ is an orbit space.\n\\end{enumerate}\nIf $S = B$, $U$, $R$ are all schemes, then we say a morphism of schemes\n$\\phi : U \\to X$ is a {\\it coarse quotient in schemes} if\n\\begin{enumerate}\n\\item $\\phi$ is a categorical quotient in schemes, and\n\\item $\\phi$ is an orbit space.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Coarse quotients","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04A2","source_file":"groupoids-quotients.tex","source_line":1242,"source_end_line":1258,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1242-L1258","statement_sha256":"3d39eb6238fc8a9b22a0708bf4ff15e5bb1cff20abc381a0ff9f8d47c885cb06","origin":"The Stacks Project","memory_eligible":false,"source_rank":12690,"rank":12690,"depth":0,"x":1024.811,"y":1025.3,"cluster":"groupoids-quotients"},{"id":"stacks:04A4","tag":"04A4","title":"Topological properties · Definition 04A4","summary":"Let S be a scheme and B an algebraic space over S. Let j : R → U ×_B U be a pre-relation. Let φ : U → X be an R-invariant morphism of algebraic spaces over B. • The morphism φ is submersive. • For any R-invariant closed subset Z ⊂ |U| the image φ(Z) is closed in |X|. • Condition ([Tag 04A6]) holds and for any pair of R-invariant closed subsets Z_1, Z_2 ⊂ |U| we have φ(Z_1 ∩ Z_2) = φ(Z_1) ∩ φ(Z_2) • The morphism (t, s) : R → U ×_X U is universally submersive. For each of…","statement_latex":"Let $S$ be a scheme and $B$ an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation.\nLet $\\phi : U \\to X$ be an $R$-invariant morphism of algebraic spaces over $B$.\n\\begin{enumerate}\n\\item\n\nThe morphism $\\phi$ is submersive.\n\\item\n\nFor any $R$-invariant closed subset $Z \\subset |U|$ the image\n$\\phi(Z)$ is closed in $|X|$.\n\\item\n\nCondition (\\ref{item-invariant-closed}) holds and for any pair of\n$R$-invariant closed subsets $Z_1, Z_2 \\subset |U|$ we have\n$$\n\\phi(Z_1 \\cap Z_2) = \\phi(Z_1) \\cap \\phi(Z_2)\n$$\n\\item The morphism $(t, s) : R \\to U \\times_X U$ is universally submersive.\n\n\\end{enumerate}\nFor each of these properties we can also require them to hold after any\nflat base change, or after any base change, see\nDefinition \\ref{definition-base-change}. In this case we say condition\n(\\ref{item-submersive}),\n(\\ref{item-invariant-closed}),\n(\\ref{item-intersect-invariant-closed}), or\n(\\ref{item-strong}) holds {\\it uniformly} or {\\it universally}.","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Topological properties","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04A4","source_file":"groupoids-quotients.tex","source_line":1299,"source_end_line":1329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1299-L1329","statement_sha256":"c90053c74dad655e22eac2c92547db880a2c25f874bd96ab0b26c16bb3d13559","origin":"The Stacks Project","memory_eligible":false,"source_rank":12691,"rank":12691,"depth":3,"x":1300.075,"y":1164.931,"cluster":"groupoids-quotients"},{"id":"stacks:04AA","tag":"04AA","title":"Invariant functions · Definition 04AA","summary":"Let S be a scheme and B an algebraic space over S. Let j : R → U ×_B U be a pre-relation. Let φ : U → X be an R-invariant morphism. Denote φ' = φ ∘ s = φ ∘ t : R → X. • We denote (φ_*O_U)^R the O_X-sub-algebra of φ_*O_U which is the equalizer of the two maps xymatrix φ_*O_U ar@<1ex>[rr]^φ_*s^sharp ar@<-1ex>[rr]_φ_*t^sharp & & φ'_*O_R on X_etale. We sometimes call this the sheaf of R-invariant functions on X. • We say the functions on X are the R-invariant functions on U…","statement_latex":"Let $S$ be a scheme and $B$ an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation.\nLet $\\phi : U \\to X$ be an $R$-invariant morphism.\nDenote $\\phi' = \\phi \\circ s = \\phi \\circ t : R \\to X$.\n\\begin{enumerate}\n\\item We denote $(\\phi_*\\mathcal{O}_U)^R$ the $\\mathcal{O}_X$-sub-algebra\nof $\\phi_*\\mathcal{O}_U$ which is the equalizer of the two maps\n$$\n\\xymatrix{\n\\phi_*\\mathcal{O}_U\n\\ar@<1ex>[rr]^{\\phi_*s^\\sharp}\n\\ar@<-1ex>[rr]_{\\phi_*t^\\sharp}\n& &\n\\phi'_*\\mathcal{O}_R\n}\n$$\non $X_\\etale$. We sometimes call this the\n{\\it sheaf of $R$-invariant functions on $X$}.\n\\item We say {\\it the functions on $X$ are the $R$-invariant functions on\n$U$} if the natural map $\\mathcal{O}_X \\to (\\phi_*\\mathcal{O}_U)^R$\nis an isomorphism.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Invariant functions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AA","source_file":"groupoids-quotients.tex","source_line":1346,"source_end_line":1370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1346-L1370","statement_sha256":"626ecbf803112c9e2b63acd5c4a4a3c7807f5c5fb41de6fdd5149154335db0ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":12692,"rank":12692,"depth":0,"x":984.309,"y":1218.245,"cluster":"groupoids-quotients"},{"id":"stacks:04AC","tag":"04AC","title":"Good quotients · Definition 04AC","summary":"Let S be a scheme and B an algebraic space over S. Let j : R → U ×_B U be a pre-relation. A morphism φ : U → X of algebraic spaces over B is called a good quotient if • φ is invariant, • φ is affine, • φ is surjective, • condition ([Tag 04A7]) holds universally, and • the functions on X are the R-invariant functions on U.","statement_latex":"Let $S$ be a scheme and $B$ an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation.\nA morphism $\\phi : U \\to X$ of algebraic spaces over $B$\nis called a {\\it good quotient} if\n\\begin{enumerate}\n\\item $\\phi$ is invariant,\n\\item $\\phi$ is affine,\n\\item $\\phi$ is surjective,\n\\item condition (\\ref{item-intersect-invariant-closed}) holds universally, and\n\\item the functions on $X$ are the $R$-invariant functions on $U$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Good quotients","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AC","source_file":"groupoids-quotients.tex","source_line":1405,"source_end_line":1418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1405-L1418","statement_sha256":"ae127c18e14f81d8abca94a084e26c71159ca16e12c390b7431c209177c07fd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12693,"rank":12693,"depth":4,"x":1174.579,"y":999.412,"cluster":"groupoids-quotients"},{"id":"stacks:04AE","tag":"04AE","title":"Geometric quotients · Definition 04AE","summary":"Let S be a scheme and B an algebraic space over S. Let j : R → U ×_B U be a pre-relation. A morphism φ : U → X of algebraic spaces over B is called a geometric quotient if • φ is an orbit space, • condition ([Tag 04A5]) holds universally, i.e., φ is universally submersive, and • the functions on X are the R-invariant functions on U.","statement_latex":"Let $S$ be a scheme and $B$ an algebraic space over $S$.\nLet $j : R \\to U \\times_B U$ be a pre-relation.\nA morphism $\\phi : U \\to X$ of algebraic spaces over $B$\nis called a {\\it geometric quotient} if\n\\begin{enumerate}\n\\item $\\phi$ is an orbit space,\n\\item condition (\\ref{item-submersive}) holds universally, i.e.,\n$\\phi$ is universally submersive, and\n\\item the functions on $X$ are the $R$-invariant functions on $U$.\n\\end{enumerate}","area":"Groupoids & Quotients","chapter":"Quotients of Groupoids","chapter_id":"groupoids-quotients","section":"Geometric quotients","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AE","source_file":"groupoids-quotients.tex","source_line":1440,"source_end_line":1452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/groupoids-quotients.tex#L1440-L1452","statement_sha256":"990f9708ccdeeb7572f5782bedb2a214ced092218355016944df61e16f1bbad5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12694,"rank":12694,"depth":4,"x":1210.313,"y":1269.167,"cluster":"groupoids-quotients"},{"id":"stacks:0DFV","tag":"0DFV","title":"Transporting results from schemes · Lemma 0DFV","summary":"Let S be a scheme. Let τ ∈ (etale, fppf, ph) (add more here). The inclusion functor (Sch/S)_τ → (Spaces/S)_τ is a special cocontinuous functor (Sites, Definition [Tag 03CG]) and hence identifies topoi.","statement_latex":"Let $S$ be a scheme. Let $\\tau \\in \\{\\etale, fppf, ph\\}$ (add more here).\nThe inclusion functor\n$$\n(\\Sch/S)_\\tau \\longrightarrow (\\textit{Spaces}/S)_\\tau\n$$\nis a special cocontinuous functor\n(Sites, Definition \\ref{sites-definition-special-cocontinuous-functor})\nand hence identifies topoi.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Transporting results from schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFV","source_file":"spaces-more-cohomology.tex","source_line":116,"source_end_line":126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L116-L126","statement_sha256":"5a5ee85e0f8e0a3b232d511a1add39e29ed3cc8e3e8816fa74c13afd4e1b9bcd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12695,"rank":12695,"depth":8,"x":748.957,"y":1718.59,"cluster":"geometry-of-spaces"},{"id":"stacks:0DFX","tag":"0DFX","title":"Proper base change · Lemma 0DFX","summary":"Let S be a scheme. Let f : Y → X be a surjective proper morphism of algebraic spaces over S. Let F be a sheaf on X_etale. Then F → f_*f^-1F is injective with image the equalizer of the two maps f_*f^-1F → g_*g^-1F where g is the structure morphism g : Y ×_X Y → X.","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a surjective proper morphism\nof algebraic spaces over $S$. Let $\\mathcal{F}$ be a sheaf on $X_\\etale$.\nThen $\\mathcal{F} \\to f_*f^{-1}\\mathcal{F}$ is injective with\nimage the equalizer of the two maps\n$f_*f^{-1}\\mathcal{F} \\to g_*g^{-1}\\mathcal{F}$ where\n$g$ is the structure morphism $g : Y \\times_X Y \\to X$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFX","source_file":"spaces-more-cohomology.tex","source_line":148,"source_end_line":156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L148-L156","statement_sha256":"2d8dcd388673d24cd4a02ce62992fde64c5a2ddf2a646511fd3d0cc001f216dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12696,"rank":12696,"depth":57,"x":533.701,"y":1551.653,"cluster":"geometry-of-spaces"},{"id":"stacks:0DFY","tag":"0DFY","title":"Proper base change · Lemma 0DFY","summary":"Let (A, I) be a henselian pair. Let X be an algebraic space over A such that the structure morphism f : X → Spec(A) is proper. Let i : X_0 → X be the inclusion of X ×_Spec(A) Spec(A/I). For any sheaf F on X_etale we have Γ(X, F) = Γ(X_0, i^-1F).","statement_latex":"Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space over $A$\nsuch that the structure morphism $f : X \\to \\Spec(A)$ is proper.\nLet $i : X_0 \\to X$ be the inclusion of $X \\times_{\\Spec(A)} \\Spec(A/I)$.\nFor any sheaf $\\mathcal{F}$ on $X_\\etale$ we\nhave $\\Gamma(X, \\mathcal{F}) = \\Gamma(X_0, i^{-1}\\mathcal{F})$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFY","source_file":"spaces-more-cohomology.tex","source_line":188,"source_end_line":195,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L188-L195","statement_sha256":"5aaee9036ea699a864c1ebe49ef3c47c6134773d3af96d52c745d48ccb854746","origin":"The Stacks Project","memory_eligible":false,"source_rank":12697,"rank":12697,"depth":58,"x":826.843,"y":1552.609,"cluster":"geometry-of-spaces"},{"id":"stacks:0DFZ","tag":"0DFZ","title":"Proper base change · Lemma 0DFZ","summary":"Let A be a henselian local ring. Let X be an algebraic space over A such that f : X → Spec(A) is a proper morphism. Let X_0 ⊂ X be the fibre of f over the closed point. For any sheaf F on X_etale we have Γ(X, F) = Γ(X_0, F|_X_0).","statement_latex":"Let $A$ be a henselian local ring. Let $X$ be an algebraic space\nover $A$ such that $f : X \\to \\Spec(A)$\nis a proper morphism. Let $X_0 \\subset X$ be the fibre of\n$f$ over the closed point. For any sheaf $\\mathcal{F}$ on $X_\\etale$ we\nhave $\\Gamma(X, \\mathcal{F}) = \\Gamma(X_0, \\mathcal{F}|_{X_0})$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DFZ","source_file":"spaces-more-cohomology.tex","source_line":220,"source_end_line":227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L220-L227","statement_sha256":"554889cbca70a1f48a0044bd87890782caeb9ff206bfac3fbac962db94d76925","origin":"The Stacks Project","memory_eligible":false,"source_rank":12698,"rank":12698,"depth":59,"x":609.79,"y":1718.336,"cluster":"geometry-of-spaces"},{"id":"stacks:0DG0","tag":"0DG0","title":"Proper base change · Lemma 0DG0","summary":"Let S be a scheme. Let f : X → Y and g : Y' → Y be a morphisms of algebraic spaces over S. Assume f is proper. Set X' = Y' ×_Y X with projections f' : X' → Y' and g' : X' → X. Let F be any sheaf on X_etale. Then g^-1f_*F = f'_*(g')^-1F.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y' \\to Y$\nbe a morphisms of algebraic spaces over $S$. Assume $f$ is proper.\nSet $X' = Y' \\times_Y X$ with projections $f' : X' \\to Y'$ and $g' : X' \\to X$.\nLet $\\mathcal{F}$ be any sheaf on $X_\\etale$. Then\n$g^{-1}f_*\\mathcal{F} = f'_*(g')^{-1}\\mathcal{F}$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DG0","source_file":"spaces-more-cohomology.tex","source_line":233,"source_end_line":240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L233-L240","statement_sha256":"1bf71679760c9ba8165fddef2c7ded68ba1be301c6b9581f32a398a761b37004","origin":"The Stacks Project","memory_eligible":false,"source_rank":12699,"rank":12699,"depth":58,"x":636.584,"y":1472.829,"cluster":"geometry-of-spaces"},{"id":"stacks:0DG1","tag":"0DG1","title":"Proper base change · Lemma 0DG1","summary":"Let S be a scheme. Let f : Y → X be a proper morphism of algebraic spaces over S. Let overlinex → X be a geometric point. For any sheaf F on Y_etale the canonical map (f_*F)_overlinex → Γ(Y_overlinex, F_overlinex) is bijective.","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be a proper morphism of algebraic spaces over $S$. Let\n$\\overline{x} \\to X$ be a geometric point.\nFor any sheaf $\\mathcal{F}$ on $Y_\\etale$\nthe canonical map\n$$\n(f_*\\mathcal{F})_{\\overline{x}} \\longrightarrow\n\\Gamma(Y_{\\overline{x}}, \\mathcal{F}_{\\overline{x}})\n$$\nis bijective.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DG1","source_file":"spaces-more-cohomology.tex","source_line":296,"source_end_line":308,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L296-L308","statement_sha256":"4e0f081b29ae801d31f52f1c0bc59073648dcb364602d926fba85b939718e2c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12700,"rank":12700,"depth":59,"x":814.359,"y":1669.179,"cluster":"geometry-of-spaces"},{"id":"stacks:0DG2","tag":"0DG2","title":"Proper base change · Theorem 0DG2","summary":"Let S be a scheme. Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian square of algebraic spaces over S. Assume f is proper. Let F be an abelian torsion sheaf on X_etale. Then the base change map g^-1Rf_*F → Rf'_*(g')^-1F is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian square of algebraic spaces over $S$.\nAssume $f$ is proper.\nLet $\\mathcal{F}$ be an abelian torsion sheaf on $X_\\etale$.\nThen the base change map\n$$\ng^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F}\n$$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Proper base change","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DG2","source_file":"spaces-more-cohomology.tex","source_line":314,"source_end_line":331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L314-L331","statement_sha256":"efb633651195ee579d6b02c9a4ad4c7b527184ba896654c050d23185c8e42691","origin":"The Stacks Project","memory_eligible":false,"source_rank":12701,"rank":12701,"depth":60,"x":525.206,"y":1625.242,"cluster":"geometry-of-spaces"},{"id":"stacks:0DG3","tag":"0DG3","title":"Proper base change · Lemma 0DG3","summary":"Let S be a scheme. Let xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian square of algebraic spaces over S. Assume f is proper. Let E ∈ D^+(X_etale) have torsion cohomology sheaves. Then the base change map g^-1Rf_*E → Rf'_*(g')^-1E is an isomorphism.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nbe a cartesian square of algebraic spaces over $S$. Assume $f$ is proper.\nLet $E \\in D^+(X_\\etale)$ have torsion cohomology sheaves.\nThen the base change map $g^{-1}Rf_*E \\to Rf'_*(g')^{-1}E$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DG3","source_file":"spaces-more-cohomology.tex","source_line":398,"source_end_line":411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L398-L411","statement_sha256":"f1885fb294a284c4892c88cbaf48844e8af79e30342fba930c027026ed241344","origin":"The Stacks Project","memory_eligible":false,"source_rank":12702,"rank":12702,"depth":61,"x":773.898,"y":1493.492,"cluster":"geometry-of-spaces"},{"id":"stacks:0DG4","tag":"0DG4","title":"Proper base change · Lemma 0DG4","summary":"Let S be a scheme. Let f : X → Y be a proper morphism of algebraic spaces. Let overliney → Y be a geometric point. • For a torsion abelian sheaf F on X_etale we have (R^nf_*F)_overliney = H^n_etale(X_overliney, F_overliney). • For E ∈ D^+(X_etale) with torsion cohomology sheaves we have (R^nf_*E)_overliney = H^n_etale(X_overliney, E_overliney).","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a proper morphism of algebraic spaces.\nLet $\\overline{y} \\to Y$ be a geometric point.\n\\begin{enumerate}\n\\item For a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$ we have\n$(R^nf_*\\mathcal{F})_{\\overline{y}} =\nH^n_\\etale(X_{\\overline{y}}, \\mathcal{F}_{\\overline{y}})$.\n\\item For $E \\in D^+(X_\\etale)$ with torsion cohomology sheaves we have\n$(R^nf_*E)_{\\overline{y}} = H^n_\\etale(X_{\\overline{y}}, E_{\\overline{y}})$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DG4","source_file":"spaces-more-cohomology.tex","source_line":428,"source_end_line":440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L428-L440","statement_sha256":"b50b89af59938d917a4d46318ff3ea9d60f436182fc405d583fc8781b5010213","origin":"The Stacks Project","memory_eligible":false,"source_rank":12703,"rank":12703,"depth":62,"x":696.421,"y":1731.893,"cluster":"geometry-of-spaces"},{"id":"stacks:0DG5","tag":"0DG5","title":"Proper base change · Lemma 0DG5","summary":"Let k'/k be an extension of separably closed fields. Let X be a proper algebraic space over k. Let F be a torsion abelian sheaf on X. Then the map H^q_etale(X, F) → H^q_etale(X_k', F|_X_k') is an isomorphism for q ≥ 0.","statement_latex":"Let $k'/k$ be an extension of separably closed fields.\nLet $X$ be a proper algebraic space over $k$.\nLet $\\mathcal{F}$ be a torsion abelian sheaf on $X$.\nThen the map $H^q_\\etale(X, \\mathcal{F}) \\to\nH^q_\\etale(X_{k'}, \\mathcal{F}|_{X_{k'}})$ is an isomorphism\nfor $q \\geq 0$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Proper base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DG5","source_file":"spaces-more-cohomology.tex","source_line":451,"source_end_line":459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L451-L459","statement_sha256":"dd6b74c8f6d35fb911e4a4888750a76212705bf0e97a9c9bea0e963bb606575c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12704,"rank":12704,"depth":61,"x":561.759,"y":1512.011,"cluster":"geometry-of-spaces"},{"id":"stacks:0DG7","tag":"0DG7","title":"Comparing big and small topoi · Lemma 0DG7","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a sheaf on X_etale. Then π_X^-1F is given by the rule (π_X^-1F)(Y) = Γ(Y_etale, f_small^-1F) for f : Y → X in (Spaces/X)_etale. Moreover, π_Y^-1F satisfies the sheaf condition with respect to smooth, syntomic, fppf, fpqc, and ph coverings.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a sheaf on $X_\\etale$. Then\n$\\pi_X^{-1}\\mathcal{F}$ is given by the rule\n$$\n(\\pi_X^{-1}\\mathcal{F})(Y) = \\Gamma(Y_\\etale, f_{small}^{-1}\\mathcal{F})\n$$\nfor $f : Y \\to X$ in $(\\textit{Spaces}/X)_\\etale$.\nMoreover, $\\pi_Y^{-1}\\mathcal{F}$ satisfies the\nsheaf condition with respect to smooth, syntomic, fppf, fpqc, and ph coverings.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DG7","source_file":"spaces-more-cohomology.tex","source_line":530,"source_end_line":541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L530-L541","statement_sha256":"fde63c75d6a95861f74e00f0cef754de1ce895de911d47630f789c07f4cdc3cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12705,"rank":12705,"depth":58,"x":838.037,"y":1597.787,"cluster":"geometry-of-spaces"},{"id":"stacks:0DG8","tag":"0DG8","title":"Comparing big and small topoi · Lemma 0DG8","summary":"Let S be a scheme. Let Y → X be a morphism of (Spaces/S)_etale. • If I is injective in Ab((Spaces/X)_etale), then • i_f^-1I is injective in Ab(Y_etale), • I|_X_etale is injective in Ab(X_etale), • If I^bullet is a K-injective complex in Ab((Spaces/X)_etale), then • i_f^-1I^bullet is a K-injective complex in Ab(Y_etale), • I^bullet|_X_etale is a K-injective complex in Ab(X_etale), The corresponding statements for modules do not hold.","statement_latex":"Let $S$ be a scheme.\nLet $Y \\to X$ be a morphism of $(\\textit{Spaces}/S)_\\etale$.\n\\begin{enumerate}\n\\item If $\\mathcal{I}$ is injective in\n$\\textit{Ab}((\\textit{Spaces}/X)_\\etale)$, then\n\\begin{enumerate}\n\\item $i_f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(Y_\\etale)$,\n\\item $\\mathcal{I}|_{X_\\etale}$ is injective in $\\textit{Ab}(X_\\etale)$,\n\\end{enumerate}\n\\item If $\\mathcal{I}^\\bullet$ is a K-injective complex\nin $\\textit{Ab}((\\textit{Spaces}/X)_\\etale)$, then\n\\begin{enumerate}\n\\item $i_f^{-1}\\mathcal{I}^\\bullet$ is a K-injective complex in\n$\\textit{Ab}(Y_\\etale)$,\n\\item $\\mathcal{I}^\\bullet|_{X_\\etale}$ is a K-injective complex in\n$\\textit{Ab}(X_\\etale)$,\n\\end{enumerate}\n\\end{enumerate}\nThe corresponding statements for modules do not hold.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DG8","source_file":"spaces-more-cohomology.tex","source_line":582,"source_end_line":603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L582-L603","statement_sha256":"a61baa874e963a21e8d1591f6b6456c3dba2ffe53287877b370d596a99aa04dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12706,"rank":12706,"depth":11,"x":565.179,"y":1691.358,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGA","tag":"0DGA","title":"Comparing big and small topoi · Lemma 0DGA","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. • For K in D((Spaces/Y)_etale) we have (Rf_big, *K)|_X_etale = Rf_small, *(K|_Y_etale) in D(X_etale). • For K in D((Spaces/Y)_etale, O) we have (Rf_big, *K)|_X_etale = Rf_small, *(K|_Y_etale) in D(Mod(X_etale, O_X)). More generally, let g : X' → X be an object of (Spaces/X)_etale. Consider the fibre product xymatrix Y' ar[r]_g' ar[d]_f' & Y ar[d]^f X' ar[r]^g & X Then • [(3)] For K in…","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be a morphism of algebraic spaces over $S$.\n\\begin{enumerate}\n\\item For $K$ in $D((\\textit{Spaces}/Y)_\\etale)$ we have\n$\n(Rf_{big, *}K)|_{X_\\etale} = Rf_{small, *}(K|_{Y_\\etale})\n$\nin $D(X_\\etale)$.\n\\item For $K$ in $D((\\textit{Spaces}/Y)_\\etale, \\mathcal{O})$ we have\n$\n(Rf_{big, *}K)|_{X_\\etale} = Rf_{small, *}(K|_{Y_\\etale})\n$\nin $D(\\textit{Mod}(X_\\etale, \\mathcal{O}_X))$.\n\\end{enumerate}\nMore generally, let $g : X' \\to X$ be an object of\n$(\\textit{Spaces}/X)_\\etale$. Consider the fibre product\n$$\n\\xymatrix{\nY' \\ar[r]_{g'} \\ar[d]_{f'} & Y \\ar[d]^f \\\\\nX' \\ar[r]^g & X\n}\n$$\nThen\n\\begin{enumerate}\n\\item[(3)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale)$ we have\n$i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$\nin $D(X'_\\etale)$.\n\\item[(4)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale, \\mathcal{O})$ we have\n$i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$\nin $D(\\textit{Mod}(X'_\\etale, \\mathcal{O}_{X'}))$.\n\\item[(5)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale)$ we have\n$g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$\nin $D((\\textit{Spaces}/X')_\\etale)$.\n\\item[(6)] For $K$ in $D((\\textit{Spaces}/Y)_\\etale, \\mathcal{O})$ we have\n$g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$\nin $D(\\textit{Mod}(X'_\\etale, \\mathcal{O}_{X'}))$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGA","source_file":"spaces-more-cohomology.tex","source_line":664,"source_end_line":703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L664-L703","statement_sha256":"efc77b99477aa3062079d6dbdda29c883d9e251bc7e6dec539846aa44e8074ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":12707,"rank":12707,"depth":24,"x":691.206,"y":1467.406,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGB","tag":"0DGB","title":"Comparing big and small topoi · Lemma 0DGB","summary":"Let S be a scheme. Let f : Y → X be a morphism of algebraic spaces over S. Then • For K in D(X_etale) we have H^n_etale(X, π_X^-1K) = H^n(X_etale, K). • For K in D(X_etale, O_X) we have H^n_etale(X, Lπ_X^*K) = H^n(X_etale, K). • For K in D(X_etale) we have H^n_etale(Y, π_X^-1K) = H^n(Y_etale, f_small^-1K). • For K in D(X_etale, O_X) we have H^n_etale(Y, Lπ_X^*K) = H^n(Y_etale, Lf_small^*K). • For M in D((Spaces/X)_etale) we have H^n_etale(Y, M) = H^n(Y_etale, i_f^-1M). •…","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be a morphism of algebraic spaces over $S$. Then\n\\begin{enumerate}\n\\item For $K$ in $D(X_\\etale)$ we have\n$H^n_\\etale(X, \\pi_X^{-1}K) = H^n(X_\\etale, K)$.\n\\item For $K$ in $D(X_\\etale, \\mathcal{O}_X)$ we have\n$H^n_\\etale(X, L\\pi_X^*K) = H^n(X_\\etale, K)$.\n\\item For $K$ in $D(X_\\etale)$ we have\n$H^n_\\etale(Y, \\pi_X^{-1}K) = H^n(Y_\\etale, f_{small}^{-1}K)$.\n\\item For $K$ in $D(X_\\etale, \\mathcal{O}_X)$ we have\n$H^n_\\etale(Y, L\\pi_X^*K) = H^n(Y_\\etale, Lf_{small}^*K)$.\n\\item For $M$ in $D((\\textit{Spaces}/X)_\\etale)$ we have\n$H^n_\\etale(Y, M) = H^n(Y_\\etale, i_f^{-1}M)$.\n\\item For $M$ in $D((\\textit{Spaces}/X)_\\etale, \\mathcal{O})$ we have\n$H^n_\\etale(Y, M) = H^n(Y_\\etale, i_f^*M)$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGB","source_file":"spaces-more-cohomology.tex","source_line":746,"source_end_line":764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L746-L764","statement_sha256":"c2ab907f60915426dc6e08279dc95e9154a6f51fdb3f6c79247f618b6b594c95","origin":"The Stacks Project","memory_eligible":false,"source_rank":12708,"rank":12708,"depth":26,"x":778.421,"y":1704.19,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGC","tag":"0DGC","title":"Comparing big and small topoi · Lemma 0DGC","summary":"Let S be a scheme. Let X be an algebraic space over S. For K ∈ D(X_etale) the map K → Rπ_X, *π_X^-1K is an isomorphism where π_X : Sh((Spaces/X)_etale) → Sh(X_etale) is as above.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nFor $K \\in D(X_\\etale)$ the map\n$$\nK \\longrightarrow R\\pi_{X, *}\\pi_X^{-1}K\n$$\nis an isomorphism where\n$\\pi_X : \\Sh((\\textit{Spaces}/X)_\\etale) \\to \\Sh(X_\\etale)$ is as above.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGC","source_file":"spaces-more-cohomology.tex","source_line":780,"source_end_line":789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L780-L789","statement_sha256":"14d874567641e7bd6f1efd05913bbac99279f56699bd492ea1b8bc6bbb556904","origin":"The Stacks Project","memory_eligible":false,"source_rank":12709,"rank":12709,"depth":0,"x":523.55,"y":1579.006,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGD","tag":"0DGD","title":"Comparing big and small topoi · Lemma 0DGD","summary":"Let S be a scheme. Let f : Y → X be a proper morphism of algebraic spaces over S. Then we have • π_X^-1 ∘ f_small, * = f_big, * ∘ π_Y^-1 as functors Sh(Y_etale) → Sh((Spaces/X)_etale), • π_X^-1Rf_small, *K = Rf_big, *π_Y^-1K for K in D^+(Y_etale) whose cohomology sheaves are torsion, and • π_X^-1Rf_small, *K = Rf_big, *π_Y^-1K for all K in D(Y_etale) if f is finite.","statement_latex":"Let $S$ be a scheme.\nLet $f : Y \\to X$ be a proper morphism of algebraic spaces over $S$.\nThen we have\n\\begin{enumerate}\n\\item $\\pi_X^{-1} \\circ f_{small, *} = f_{big, *} \\circ \\pi_Y^{-1}$\nas functors $\\Sh(Y_\\etale) \\to \\Sh((\\textit{Spaces}/X)_\\etale)$,\n\\item $\\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_Y^{-1}K$\nfor $K$ in $D^+(Y_\\etale)$ whose cohomology sheaves are torsion, and\n\\item $\\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\\pi_Y^{-1}K$\nfor all $K$ in $D(Y_\\etale)$ if $f$ is finite.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing big and small topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGD","source_file":"spaces-more-cohomology.tex","source_line":797,"source_end_line":810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L797-L810","statement_sha256":"a758315dc8512e52721b0af05a7b52c25d107fe6d2d7a8e3f5252f279415ec52","origin":"The Stacks Project","memory_eligible":false,"source_rank":12710,"rank":12710,"depth":62,"x":812.322,"y":1526.665,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGF","tag":"0DGF","title":"Comparing fppf and étale topologies · Lemma 0DGF","summary":"Let S be a scheme. Let X be an algebraic space over S. • For F ∈ Sh(X_etale) we have ε_X, *a_X^-1F = π_X^-1F and a_X, *a_X^-1F = F. • For F ∈ Ab(X_etale) we have R^iε_X, *(a_X^-1F) = 0 for i > 0.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item For $\\mathcal{F} \\in \\Sh(X_\\etale)$ we have\n$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$\nand $a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$.\n\\item For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ we have\n$R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGF","source_file":"spaces-more-cohomology.tex","source_line":927,"source_end_line":937,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L927-L937","statement_sha256":"fbf725a0c31553d97af161b4f3beb3d68035a4068e46f06affa94e55b6fca5f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12711,"rank":12711,"depth":82,"x":641.38,"y":1729.232,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGG","tag":"0DGG","title":"Comparing fppf and étale topologies · Lemma 0DGG","summary":"Let S be a scheme. Let X be an algebraic space over S. For K ∈ D^+(X_etale) the maps π_X^-1K → Rε_X, *a_X^-1K and K → Ra_X, *a_X^-1K are isomorphisms with a_X : Sh((Spaces/X)_fppf) → Sh(X_etale) as above.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nFor $K \\in D^+(X_\\etale)$ the maps\n$$\n\\pi_X^{-1}K \\longrightarrow R\\epsilon_{X, *}a_X^{-1}K\n\\quad\\text{and}\\quad\nK \\longrightarrow Ra_{X, *}a_X^{-1}K\n$$\nare isomorphisms with\n$a_X : \\Sh((\\textit{Spaces}/X)_{fppf}) \\to \\Sh(X_\\etale)$ as above.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGG","source_file":"spaces-more-cohomology.tex","source_line":1001,"source_end_line":1012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1001-L1012","statement_sha256":"ee489589799f28c455496757c777e0602425bd3aa4da19f21d197f4f9d317258","origin":"The Stacks Project","memory_eligible":false,"source_rank":12712,"rank":12712,"depth":83,"x":604.51,"y":1482.726,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGH","tag":"0DGH","title":"Comparing fppf and étale topologies · Lemma 0DGH","summary":"Let S be a scheme and let X be an algebraic space over S. With a_X : Sh((Spaces/X)_fppf) → Sh(X_etale) as above: • H^q(X_etale, F) = H^q_fppf(X, a_X^-1F) for an abelian sheaf F on X_etale, • H^q(X_etale, K) = H^q_fppf(X, a_X^-1K) for K ∈ D^+(X_etale). Example: if A is an abelian group, then H^q_etale(X, underlineA) = H^q_fppf(X, underlineA).","statement_latex":"Let $S$ be a scheme and let $X$ be an algebraic space over $S$.\nWith $a_X : \\Sh((\\textit{Spaces}/X)_{fppf}) \\to \\Sh(X_\\etale)$\nas above:\n\\begin{enumerate}\n\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{fppf}(X, a_X^{-1}\\mathcal{F})$\nfor an abelian sheaf $\\mathcal{F}$ on $X_\\etale$,\n\\item $H^q(X_\\etale, K) = H^q_{fppf}(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$.\n\\end{enumerate}\nExample: if $A$ is an abelian group, then\n$H^q_\\etale(X, \\underline{A}) = H^q_{fppf}(X, \\underline{A})$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGH","source_file":"spaces-more-cohomology.tex","source_line":1044,"source_end_line":1056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1044-L1056","statement_sha256":"12ab4f38ba12de719ddc275b676b7b36c2fff4ae41ec833521a6ae0667c753f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12713,"rank":12713,"depth":84,"x":830.058,"y":1643.666,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGI","tag":"0DGI","title":"Comparing fppf and étale topologies · Lemma 0DGI","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then there are commutative diagrams of topoi xymatrix Sh((Spaces/X)_fppf) ar[rr]_f_big, fppf ar[d]_ε_X & & Sh((Spaces/Y)_fppf) ar[d]^ε_Y Sh((Spaces/X)_etale) ar[rr]^f_big, etale & & Sh((Spaces/Y)_etale) and xymatrix Sh((Spaces/X)_fppf) ar[rr]_f_big, fppf ar[d]_a_X & & Sh((Spaces/Y)_fppf) ar[d]^a_Y Sh(X_etale) ar[rr]^f_small & & Sh(Y_etale) with a_X = π_X ∘ ε_X and a_Y = π_X ∘ ε_X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThen there are commutative diagrams of topoi\n$$\n\\xymatrix{\n\\Sh((\\textit{Spaces}/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{\\epsilon_X} & &\n\\Sh((\\textit{Spaces}/Y)_{fppf}) \\ar[d]^{\\epsilon_Y} \\\\\n\\Sh((\\textit{Spaces}/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &\n\\Sh((\\textit{Spaces}/Y)_\\etale)\n}\n$$\nand\n$$\n\\xymatrix{\n\\Sh((\\textit{Spaces}/X)_{fppf}) \\ar[rr]_{f_{big, fppf}} \\ar[d]_{a_X} & &\n\\Sh((\\textit{Spaces}/Y)_{fppf}) \\ar[d]^{a_Y} \\\\\n\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &\n\\Sh(Y_\\etale)\n}\n$$\nwith $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGI","source_file":"spaces-more-cohomology.tex","source_line":1063,"source_end_line":1086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1063-L1086","statement_sha256":"07d14baa92c551845db8e563afb2e08ffd3d0477861cc3a285c8713f3f46e8e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12714,"rank":12714,"depth":0,"x":534.152,"y":1652.978,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGJ","tag":"0DGJ","title":"Comparing fppf and étale topologies · Lemma 0DGJ","summary":"In Lemma [Tag 0DGI] if f is proper, then we have • a_Y^-1 ∘ f_small, * = f_big, fppf, * ∘ a_X^-1, and • a_Y^-1(Rf_small, *K) = Rf_big, fppf, *(a_X^-1K) for K in D^+(X_etale) with torsion cohomology sheaves.","statement_latex":"In Lemma \\ref{lemma-push-pull-fppf-etale} if $f$ is proper, then we have\n\\begin{enumerate}\n\\item $a_Y^{-1} \\circ f_{small, *} = f_{big, fppf, *} \\circ a_X^{-1}$, and\n\\item\n$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$\nfor $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGJ","source_file":"spaces-more-cohomology.tex","source_line":1095,"source_end_line":1104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1095-L1104","statement_sha256":"d56361e56b112a401b9102e089806fca4ac51ac77e9bbdeb1295a6b10d88cfc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12715,"rank":12715,"depth":84,"x":744.979,"y":1478.108,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGK","tag":"0DGK","title":"Comparing fppf and étale topologies · Lemma 0DGK","summary":"In Lemma [Tag 0DGI] if f is finite, then a_Y^-1(Rf_small, *K) = Rf_big, fppf, *(a_X^-1K) for K in D^+(X_etale).","statement_latex":"In Lemma \\ref{lemma-push-pull-fppf-etale} if $f$ is finite, then\n$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$\nfor $K$ in $D^+(X_\\etale)$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGK","source_file":"spaces-more-cohomology.tex","source_line":1177,"source_end_line":1182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1177-L1182","statement_sha256":"cfea605df35e8fb0ecc5c8e05a097b1ee8ff50043157d27b1aa97e9001a012ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":12716,"rank":12716,"depth":82,"x":730.135,"y":1726.823,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGL","tag":"0DGL","title":"Comparing fppf and étale topologies · Lemma 0DGL","summary":"In Lemma [Tag 0DGI] assume f is flat, locally of finite presentation, and surjective. Then the functor Sh(Y_etale) → ( (G, H, α) middle| G ∈ Sh(X_etale), H ∈ Sh((Sch/Y)_fppf), α : a_X^-1G → f_big, fppf^-1H an isomorphism ) sending F to (f_small^-1F, a_Y^-1F, can) is an equivalence.","statement_latex":"In Lemma \\ref{lemma-push-pull-fppf-etale} assume\n$f$ is flat, locally of finite presentation, and surjective.\nThen the functor\n$$\n\\Sh(Y_\\etale) \\longrightarrow\n\\left\\{\n(\\mathcal{G}, \\mathcal{H}, \\alpha)\n\\middle|\n\\begin{matrix}\n\\mathcal{G} \\in \\Sh(X_\\etale),\\ \\mathcal{H} \\in \\Sh((\\Sch/Y)_{fppf}), \\\\\n\\alpha : a_X^{-1}\\mathcal{G} \\to f_{big, fppf}^{-1}\\mathcal{H}\n\\text{ an isomorphism}\n\\end{matrix}\n\\right\\}\n$$\nsending $\\mathcal{F}$ to\n$(f_{small}^{-1}\\mathcal{F}, a_Y^{-1}\\mathcal{F}, can)$ is an equivalence.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGL","source_file":"spaces-more-cohomology.tex","source_line":1199,"source_end_line":1218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1199-L1218","statement_sha256":"f44f90fa8e650da72981ac0aba3804ffdf73f06260e741f1a953300c7fc94cd8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12717,"rank":12717,"depth":83,"x":540.965,"y":1534.894,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGN","tag":"0DGN","title":"Comparing fppf and étale topologies: modules · Lemma 0DGN","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F be a quasi-coherent O_X-module. • The rule F^a : (Spaces/X)_etale → Ab, (f : Y → X) ↦ Γ(Y, f^*F) satisfies the sheaf condition for fpqc and a fortiori fppf and étale coverings, • F^a = π_X^*F on (Spaces/X)_etale, • F^a = a_X^*F on (Spaces/X)_fppf, • the rule F ↦ F^a defines an equivalence between quasi-coherent O_X-modules and quasi-coherent modules on ((Spaces/X)_etale, O), • the rule F ↦ F^a defines an…","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\n\\begin{enumerate}\n\\item The rule\n$$\n\\mathcal{F}^a : (\\textit{Spaces}/X)_\\etale \\longrightarrow \\textit{Ab},\\quad\n(f : Y \\to X) \\longmapsto \\Gamma(Y, f^*\\mathcal{F})\n$$\nsatisfies the sheaf condition for fpqc and a fortiori\nfppf and \\'etale coverings,\n\\item $\\mathcal{F}^a = \\pi_X^*\\mathcal{F}$ on $(\\textit{Spaces}/X)_\\etale$,\n\\item $\\mathcal{F}^a = a_X^*\\mathcal{F}$ on $(\\textit{Spaces}/X)_{fppf}$,\n\\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines\nan equivalence between quasi-coherent $\\mathcal{O}_X$-modules\nand quasi-coherent modules on\n$((\\textit{Spaces}/X)_\\etale, \\mathcal{O})$,\n\\item the rule $\\mathcal{F} \\mapsto \\mathcal{F}^a$ defines\nan equivalence between quasi-coherent $\\mathcal{O}_X$-modules\nand quasi-coherent modules on\n$((\\textit{Spaces}/X)_{fppf}, \\mathcal{O})$,\n\\item we have $\\epsilon_{X, *}a_X^*\\mathcal{F} = \\pi_X^*\\mathcal{F}$\nand $a_{X, *}a_X^*\\mathcal{F} = \\mathcal{F}$,\n\\item we have $R^i\\epsilon_{X, *}(a_X^*\\mathcal{F}) = 0$\nand $R^ia_{X, *}(a_X^*\\mathcal{F}) = 0$ for $i > 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGN","source_file":"spaces-more-cohomology.tex","source_line":1302,"source_end_line":1329,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1302-L1329","statement_sha256":"ce65d9499226f0038668dc3829f00297d18e99f4d811476639d3490dae4b0edc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12718,"rank":12718,"depth":52,"x":834.966,"y":1569.099,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGP","tag":"0DGP","title":"Comparing fppf and étale topologies: modules · Lemma 0DGP","summary":"Let S be a scheme. Let X be an algebraic space over S. For F a quasi-coherent O_X-module the maps π_X^*F → Rε_X, *(a_X^*F) and F → Ra_X, *(a_X^*F) are isomorphisms.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nFor $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module the maps\n$$\n\\pi_X^*\\mathcal{F} \\longrightarrow R\\epsilon_{X, *}(a_X^*\\mathcal{F})\n\\quad\\text{and}\\quad\n\\mathcal{F} \\longrightarrow Ra_{X, *}(a_X^*\\mathcal{F})\n$$\nare isomorphisms.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGP","source_file":"spaces-more-cohomology.tex","source_line":1381,"source_end_line":1391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1381-L1391","statement_sha256":"4ba843e174402e2cf64daff3da283703979417a173448169a3216fd2324359d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12719,"rank":12719,"depth":53,"x":590.53,"y":1710.78,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGQ","tag":"0DGQ","title":"Comparing fppf and étale topologies: modules · Lemma 0DGQ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let F_1 → F_2 → F_3 be a complex of quasi-coherent O_X-modules. Set H_etale = Ker(π_X^*F_2 → π_X^*F_3)/ Im(π_X^*F_1 → π_X^*F_2) on (Spaces/X)_etale and set H_fppf = Ker(a_X^*F_2 → a_X^*F_3)/ Im(a_X^*F_1 → a_X^*F_2) on (Spaces/X)_fppf. Then H_etale = ε_X, *H_fppf and H^p_etale(U, H_etale) = H^p_fppf(U, H_fppf) = 0 for p > 0 and any affine object U of (Spaces/X)_etale.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $\\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3$\nbe a complex of quasi-coherent $\\mathcal{O}_X$-modules.\nSet\n$$\n\\mathcal{H}_\\etale =\n\\Ker(\\pi_X^*\\mathcal{F}_2 \\to \\pi_X^*\\mathcal{F}_3)/\n\\Im(\\pi_X^*\\mathcal{F}_1 \\to \\pi_X^*\\mathcal{F}_2)\n$$\non $(\\textit{Spaces}/X)_\\etale$ and set\n$$\n\\mathcal{H}_{fppf} =\n\\Ker(a_X^*\\mathcal{F}_2 \\to a_X^*\\mathcal{F}_3)/\n\\Im(a_X^*\\mathcal{F}_1 \\to a_X^*\\mathcal{F}_2)\n$$\non $(\\textit{Spaces}/X)_{fppf}$.\nThen $\\mathcal{H}_\\etale = \\epsilon_{X, *}\\mathcal{H}_{fppf}$\nand\n$$\nH^p_\\etale(U, \\mathcal{H}_\\etale) = H^p_{fppf}(U, \\mathcal{H}_{fppf}) = 0\n$$\nfor $p > 0$ and any affine object $U$ of $(\\textit{Spaces}/X)_\\etale$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGQ","source_file":"spaces-more-cohomology.tex","source_line":1398,"source_end_line":1422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1398-L1422","statement_sha256":"aa7ba0985fdc1599afbb97d899770a8b04232d4efee18a60acf7f95c90df347b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12720,"rank":12720,"depth":0,"x":656.875,"y":1467.471,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGR","tag":"0DGR","title":"Comparing fppf and étale topologies: modules · Lemma 0DGR","summary":"Let S be a scheme. Let X be an algebraic space over S. For K ∈ D_QCoh(O_X) the maps Lπ_X^*K → Rε_X, *(La_X^*K) and K → Ra_X, *(La_X^*K) are isomorphisms. Here a_X : Sh((Spaces/X)_fppf) → Sh(X_etale) is as above.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nFor $K \\in D_\\QCoh(\\mathcal{O}_X)$ the maps\n$$\nL\\pi_X^*K \\longrightarrow R\\epsilon_{X, *}(La_X^*K)\n\\quad\\text{and}\\quad\nK \\longrightarrow Ra_{X, *}(La_X^*K)\n$$\nare isomorphisms. Here\n$a_X : \\Sh((\\textit{Spaces}/X)_{fppf}) \\to \\Sh(X_\\etale)$ is as above.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing fppf and étale topologies: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGR","source_file":"spaces-more-cohomology.tex","source_line":1488,"source_end_line":1499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1488-L1499","statement_sha256":"d551d29a70edf2743bafafe9b470f4ce36b8ea54d29368767866d7db4d54df7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12721,"rank":12721,"depth":41,"x":803.697,"y":1684.651,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGT","tag":"0DGT","title":"Comparing ph and étale topologies · Lemma 0DGT","summary":"Let S be a scheme. Let X be an algebraic space over S. • For F ∈ Sh(X_etale) we have ε_X, *a_X^-1F = π_X^-1F and a_X, *a_X^-1F = F. • For F ∈ Ab(X_etale) torsion we have R^iε_X, *(a_X^-1F) = 0 for i > 0.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\n\\begin{enumerate}\n\\item For $\\mathcal{F} \\in \\Sh(X_\\etale)$ we have\n$\\epsilon_{X, *}a_X^{-1}\\mathcal{F} = \\pi_X^{-1}\\mathcal{F}$\nand $a_{X, *}a_X^{-1}\\mathcal{F} = \\mathcal{F}$.\n\\item For $\\mathcal{F} \\in \\textit{Ab}(X_\\etale)$ torsion we have\n$R^i\\epsilon_{X, *}(a_X^{-1}\\mathcal{F}) = 0$ for $i > 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGT","source_file":"spaces-more-cohomology.tex","source_line":1671,"source_end_line":1681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1671-L1681","statement_sha256":"92f18111d3b6de2002e7885abe2dd70f99da8305a211a2c0b98f1a515e469921","origin":"The Stacks Project","memory_eligible":false,"source_rank":12722,"rank":12722,"depth":82,"x":520.624,"y":1607.773,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGU","tag":"0DGU","title":"Comparing ph and étale topologies · Lemma 0DGU","summary":"Let S be a scheme. Let X be an algebraic space over S. For K ∈ D^+(X_etale) with torsion cohomology sheaves the maps π_X^-1K → Rε_X, *a_X^-1K and K → Ra_X, *a_X^-1K are isomorphisms with a_X : Sh((Spaces/X)_ph) → Sh(X_etale) as above.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nFor $K \\in D^+(X_\\etale)$ with torsion cohomology sheaves the maps\n$$\n\\pi_X^{-1}K \\longrightarrow R\\epsilon_{X, *}a_X^{-1}K\n\\quad\\text{and}\\quad\nK \\longrightarrow Ra_{X, *}a_X^{-1}K\n$$\nare isomorphisms with\n$a_X : \\Sh((\\textit{Spaces}/X)_{ph}) \\to \\Sh(X_\\etale)$ as above.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGU","source_file":"spaces-more-cohomology.tex","source_line":1745,"source_end_line":1756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1745-L1756","statement_sha256":"e39f57cea97184853b8865cb07e261eb148a9b113589ebf07d532c128054b49f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12723,"rank":12723,"depth":83,"x":791.333,"y":1503.781,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGV","tag":"0DGV","title":"Comparing ph and étale topologies · Lemma 0DGV","summary":"Let S be a scheme and let X be an algebraic space over S. With a_X : Sh((Spaces/X)_ph) → Sh(X_etale) as above: • H^q(X_etale, F) = H^q_ph(X, a_X^-1F) for a torsion abelian sheaf F on X_etale, • H^q(X_etale, K) = H^q_ph(X, a_X^-1K) for K ∈ D^+(X_etale) with torsion cohomology sheaves Example: if A is a torsion abelian group, then H^q_etale(X, underlineA) = H^q_ph(X, underlineA).","statement_latex":"Let $S$ be a scheme and let $X$ be an algebraic space over $S$.\nWith $a_X : \\Sh((\\textit{Spaces}/X)_{ph}) \\to \\Sh(X_\\etale)$\nas above:\n\\begin{enumerate}\n\\item $H^q(X_\\etale, \\mathcal{F}) = H^q_{ph}(X, a_X^{-1}\\mathcal{F})$\nfor a torsion abelian sheaf $\\mathcal{F}$ on $X_\\etale$,\n\\item $H^q(X_\\etale, K) = H^q_{ph}(X, a_X^{-1}K)$ for $K \\in D^+(X_\\etale)$\nwith torsion cohomology sheaves\n\\end{enumerate}\nExample: if $A$ is a torsion abelian group, then\n$H^q_\\etale(X, \\underline{A}) = H^q_{ph}(X, \\underline{A})$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGV","source_file":"spaces-more-cohomology.tex","source_line":1790,"source_end_line":1803,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1790-L1803","statement_sha256":"f54322f21ae3ac1d869b88034737b28eca18a0017e41989211b98894aff2f922","origin":"The Stacks Project","memory_eligible":false,"source_rank":12724,"rank":12724,"depth":84,"x":675.279,"y":1734.198,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGW","tag":"0DGW","title":"Comparing ph and étale topologies · Lemma 0DGW","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Then there are commutative diagrams of topoi xymatrix Sh((Spaces/X)_ph) ar[rr]_f_big, ph ar[d]_ε_X & & Sh((Spaces/Y)_ph) ar[d]^ε_Y Sh((Spaces/X)_etale) ar[rr]^f_big, etale & & Sh((Spaces/Y)_etale) and xymatrix Sh((Spaces/X)_ph) ar[rr]_f_big, ph ar[d]_a_X & & Sh((Spaces/Y)_ph) ar[d]^a_Y Sh(X_etale) ar[rr]^f_small & & Sh(Y_etale) with a_X = π_X ∘ ε_X and a_Y = π_X ∘ ε_X.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nThen there are commutative diagrams of topoi\n$$\n\\xymatrix{\n\\Sh((\\textit{Spaces}/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{\\epsilon_X} & &\n\\Sh((\\textit{Spaces}/Y)_{ph}) \\ar[d]^{\\epsilon_Y} \\\\\n\\Sh((\\textit{Spaces}/X)_\\etale) \\ar[rr]^{f_{big, \\etale}} & &\n\\Sh((\\textit{Spaces}/Y)_\\etale)\n}\n$$\nand\n$$\n\\xymatrix{\n\\Sh((\\textit{Spaces}/X)_{ph}) \\ar[rr]_{f_{big, ph}} \\ar[d]_{a_X} & &\n\\Sh((\\textit{Spaces}/Y)_{ph}) \\ar[d]^{a_Y} \\\\\n\\Sh(X_\\etale) \\ar[rr]^{f_{small}} & &\n\\Sh(Y_\\etale)\n}\n$$\nwith $a_X = \\pi_X \\circ \\epsilon_X$ and $a_Y = \\pi_X \\circ \\epsilon_X$.","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGW","source_file":"spaces-more-cohomology.tex","source_line":1810,"source_end_line":1833,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1810-L1833","statement_sha256":"bb9472723aba376e3cad8df0fcd70b2841a56860a983fc3d23f1aca63234d34c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12725,"rank":12725,"depth":0,"x":575.504,"y":1498.309,"cluster":"geometry-of-spaces"},{"id":"stacks:0DGX","tag":"0DGX","title":"Comparing ph and étale topologies · Lemma 0DGX","summary":"In Lemma [Tag 0DGW] if f is proper, then we have • a_Y^-1 ∘ f_small, * = f_big, ph, * ∘ a_X^-1, and • a_Y^-1(Rf_small, *K) = Rf_big, ph, *(a_X^-1K) for K in D^+(X_etale) with torsion cohomology sheaves.","statement_latex":"In Lemma \\ref{lemma-push-pull-ph-etale} if $f$ is proper, then we have\n\\begin{enumerate}\n\\item $a_Y^{-1} \\circ f_{small, *} = f_{big, ph, *} \\circ a_X^{-1}$, and\n\\item\n$a_Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_X^{-1}K)$\nfor $K$ in $D^+(X_\\etale)$ with torsion cohomology sheaves.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"More on Cohomology of Spaces","chapter_id":"spaces-more-cohomology","section":"Comparing ph and étale topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DGX","source_file":"spaces-more-cohomology.tex","source_line":1842,"source_end_line":1851,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-more-cohomology.tex#L1842-L1851","statement_sha256":"2746d1dca565cc296808784e6e4f7a21472d99f7847391a60197d61ccd131f7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12726,"rank":12726,"depth":84,"x":838.919,"y":1615.701,"cluster":"geometry-of-spaces"},{"id":"stacks:09VL","tag":"09VL","title":"Simplicial topological spaces · Lemma 09VL","summary":"Let X be a simplicial space. Then X_Zar as defined above is a site.","statement_latex":"Let $X$ be a simplicial space. Then $X_{Zar}$\nas defined above is a site.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VL","source_file":"spaces-simplicial.tex","source_line":101,"source_end_line":105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L101-L105","statement_sha256":"2df2e715e194228e0d0d6e47e4eb8fc28de2f75b9b53987a7e3b5781aadbc6e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12727,"rank":12727,"depth":0,"x":550.117,"y":1678.641,"cluster":"geometry-of-spaces"},{"id":"stacks:09VM","tag":"09VM","title":"Simplicial topological spaces · Lemma 09VM","summary":"Let X be a simplicial space. There is an equivalence of categories between • Sh(X_Zar), and • category of systems (F_n, F(φ)) described above.","statement_latex":"Let $X$ be a simplicial space. There is an equivalence of\ncategories between\n\\begin{enumerate}\n\\item $\\Sh(X_{Zar})$, and\n\\item category of systems $(\\mathcal{F}_n, \\mathcal{F}(\\varphi))$\ndescribed above.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VM","source_file":"spaces-simplicial.tex","source_line":134,"source_end_line":143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L134-L143","statement_sha256":"9f7da10ad5483ce9761043c0fb9ff132c1506710bb0cdc2fb3de22e92e93f725","origin":"The Stacks Project","memory_eligible":false,"source_rank":12728,"rank":12728,"depth":0,"x":712.551,"y":1468.239,"cluster":"geometry-of-spaces"},{"id":"stacks:09VN","tag":"09VN","title":"Simplicial topological spaces · Lemma 09VN","summary":"Let f : Y → X be a morphism of simplicial spaces. Then the functor u : X_Zar → Y_Zar which associates to the open U ⊂ X_n the open f_n^-1(U) ⊂ Y_n defines a morphism of sites f_Zar : Y_Zar → X_Zar.","statement_latex":"Let $f : Y \\to X$ be a morphism of simplicial spaces.\nThen the functor $u : X_{Zar} \\to Y_{Zar}$\nwhich associates to the open $U \\subset X_n$ the open\n$f_n^{-1}(U) \\subset Y_n$ defines a morphism of sites\n$f_{Zar} : Y_{Zar} \\to X_{Zar}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VN","source_file":"spaces-simplicial.tex","source_line":149,"source_end_line":156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L149-L156","statement_sha256":"b2cd620844ae1b6a9cdeb61eef5b3e288dc91b892498cab29bde55e929becd97","origin":"The Stacks Project","memory_eligible":false,"source_rank":12729,"rank":12729,"depth":5,"x":762.001,"y":1715.693,"cluster":"geometry-of-spaces"},{"id":"stacks:09VP","tag":"09VP","title":"Simplicial topological spaces · Lemma 09VP","summary":"Let f : Y → X be a morphism of simplicial spaces. In terms of the description of sheaves in Lemma [Tag 09VM] the morphism f_Zar of Lemma [Tag 09VN] can be described as follows. • If G is a sheaf on Y, then (f_Zar, *G)_n = f_n, *G_n. • If F is a sheaf on X, then (f_Zar^-1F)_n = f_n^-1F_n.","statement_latex":"Let $f : Y \\to X$ be a morphism of simplicial spaces. In terms of the\ndescription of sheaves in\nLemma \\ref{lemma-describe-sheaves-simplicial-site} the\nmorphism $f_{Zar}$ of Lemma \\ref{lemma-simplicial-space-site-functorial}\ncan be described as follows.\n\\begin{enumerate}\n\\item If $\\mathcal{G}$ is a sheaf on $Y$, then\n$(f_{Zar, *}\\mathcal{G})_n = f_{n, *}\\mathcal{G}_n$.\n\\item If $\\mathcal{F}$ is a sheaf on $X$, then\n$(f_{Zar}^{-1}\\mathcal{F})_n = f_n^{-1}\\mathcal{F}_n$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09VP","source_file":"spaces-simplicial.tex","source_line":196,"source_end_line":209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L196-L209","statement_sha256":"379cba07d6c0816e0bbddf486b73ff4fe0523c6a650fdb412d7661873423e3e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12730,"rank":12730,"depth":6,"x":526.412,"y":1561.198,"cluster":"geometry-of-spaces"},{"id":"stacks:09W0","tag":"09W0","title":"Simplicial topological spaces · Lemma 09W0","summary":"Let X be a simplicial space. The functor X_n, Zar → X_Zar, U ↦ U is continuous and cocontinuous. The associated morphism of topoi g_n : Sh(X_n) → Sh(X_Zar) satisfies • g_n^-1 associates to the sheaf F on X the sheaf F_n on X_n, • g_n^-1 : Sh(X_Zar) → Sh(X_n) has a left adjoint g^Sh_n!, • g^Sh_n! commutes with finite connected limits, • g_n^-1 : Ab(X_Zar) → Ab(X_n) has a left adjoint g_n!, and • g_n! is exact.","statement_latex":"Let $X$ be a simplicial space. The functor\n$X_{n, Zar} \\to X_{Zar}$, $U \\mapsto U$ is continuous\nand cocontinuous. The associated morphism of\ntopoi $g_n : \\Sh(X_n) \\to \\Sh(X_{Zar})$ satisfies\n\\begin{enumerate}\n\\item $g_n^{-1}$ associates to the sheaf $\\mathcal{F}$ on $X$\nthe sheaf $\\mathcal{F}_n$ on $X_n$,\n\\item $g_n^{-1} : \\Sh(X_{Zar}) \\to \\Sh(X_n)$ has a left adjoint $g^{Sh}_{n!}$,\n\\item $g^{Sh}_{n!}$ commutes with finite connected limits,\n\\item $g_n^{-1} : \\textit{Ab}(X_{Zar}) \\to \\textit{Ab}(X_n)$\nhas a left adjoint $g_{n!}$, and\n\\item $g_{n!}$ is exact.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09W0","source_file":"spaces-simplicial.tex","source_line":238,"source_end_line":253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L238-L253","statement_sha256":"19b8ef2def55e3fc6e5c170e857602bb25b56e6809a1a8be2f8c17f331bb46e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12731,"rank":12731,"depth":6,"x":824.538,"y":1541.43,"cluster":"geometry-of-spaces"},{"id":"stacks:09W1","tag":"09W1","title":"Simplicial topological spaces · Lemma 09W1","summary":"Let X be a simplicial space. If I is an injective abelian sheaf on X_Zar, then I_n is an injective abelian sheaf on X_n.","statement_latex":"Let $X$ be a simplicial space. If $\\mathcal{I}$ is an injective abelian\nsheaf on $X_{Zar}$, then $\\mathcal{I}_n$ is an injective abelian sheaf\non $X_n$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09W1","source_file":"spaces-simplicial.tex","source_line":274,"source_end_line":279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L274-L279","statement_sha256":"a58133cd4911cfea5bb2a21b85ddd2b244fdb2fa2fb4e6fd13bf0d33d589b1df","origin":"The Stacks Project","memory_eligible":false,"source_rank":12732,"rank":12732,"depth":8,"x":620.488,"y":1725.272,"cluster":"geometry-of-spaces"},{"id":"stacks:09W2","tag":"09W2","title":"Simplicial topological spaces · Lemma 09W2","summary":"Let f : Y → X be a morphism of simplicial spaces. Then xymatrix Sh(Y_n) ar[d] ar[r]_f_n & Sh(X_n) ar[d] Sh(Y_Zar) ar[r]^f_Zar & Sh(X_Zar) is a commutative diagram of topoi.","statement_latex":"Let $f : Y \\to X$ be a morphism of simplicial spaces. Then\n$$\n\\xymatrix{\n\\Sh(Y_n) \\ar[d] \\ar[r]_{f_n} & \\Sh(X_n) \\ar[d] \\\\\n\\Sh(Y_{Zar}) \\ar[r]^{f_{Zar}} & \\Sh(X_{Zar})\n}\n$$\nis a commutative diagram of topoi.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09W2","source_file":"spaces-simplicial.tex","source_line":288,"source_end_line":298,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L288-L298","statement_sha256":"bf8e2222b9b6f29ebd36d0bb0b45c2f3d43370d2a4b3fa839241691143570303","origin":"The Stacks Project","memory_eligible":false,"source_rank":12733,"rank":12733,"depth":7,"x":623.112,"y":1473.788,"cluster":"geometry-of-spaces"},{"id":"stacks:09W4","tag":"09W4","title":"Simplicial topological spaces · Lemma 09W4","summary":"Let Y be a simplicial space and let a : Y → X be an augmentation (Simplicial, Definition [Tag 018G]). Let a_n : Y_n → X be the corresponding morphisms of topological spaces. There is a canonical morphism of topoi a : Sh(Y_Zar) → Sh(X) with the following properties: • a^-1F is the sheaf restricting to a_n^-1F on Y_n, • a_m ∘ Y(φ) = a_n for all φ : [m] → [n], • a ∘ g_n = a_n as morphisms of topoi with g_n as in Lemma [Tag 09W0], • a_*G for G ∈ Sh(Y_Zar) is the equalizer of…","statement_latex":"Let $Y$ be a simplicial space and let $a : Y \\to X$ be an augmentation\n(Simplicial, Definition \\ref{simplicial-definition-augmentation}).\nLet $a_n : Y_n \\to X$ be the corresponding morphisms of topological spaces.\nThere is a canonical morphism of topoi\n$$\na : \\Sh(Y_{Zar}) \\to \\Sh(X)\n$$\nwith the following properties:\n\\begin{enumerate}\n\\item $a^{-1}\\mathcal{F}$ is the sheaf restricting to $a_n^{-1}\\mathcal{F}$\non $Y_n$,\n\\item $a_m \\circ Y(\\varphi) = a_n$ for all $\\varphi : [m] \\to [n]$,\n\\item $a \\circ g_n = a_n$ as morphisms of topoi with\n$g_n$ as in Lemma \\ref{lemma-restriction-to-components},\n\\item $a_*\\mathcal{G}$ for $\\mathcal{G} \\in \\Sh(Y_{Zar})$\nis the equalizer of the two maps\n$a_{0, *}\\mathcal{G}_0 \\to a_{1, *}\\mathcal{G}_1$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09W4","source_file":"spaces-simplicial.tex","source_line":306,"source_end_line":326,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L306-L326","statement_sha256":"d804a6dc24c635f3ee218f4848a907e13d9803d16f169c37e6a1b6e0e74efb2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12734,"rank":12734,"depth":7,"x":823.524,"y":1660.821,"cluster":"geometry-of-spaces"},{"id":"stacks:09W5","tag":"09W5","title":"Simplicial topological spaces · Lemma 09W5","summary":"Let X be a simplicial topological space. The complex of abelian presheaves on X_Zar … → Z_X_2 → Z_X_1 → Z_X_0 with boundary ∑ (-1)^i d^n_i is a resolution of the constant presheaf Z.","statement_latex":"Let $X$ be a simplicial topological space. The complex of\nabelian presheaves on $X_{Zar}$\n$$\n\\ldots \\to \\mathbf{Z}_{X_2} \\to \\mathbf{Z}_{X_1} \\to \\mathbf{Z}_{X_0}\n$$\nwith boundary $\\sum (-1)^i d^n_i$ is a resolution\nof the constant presheaf $\\mathbf{Z}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09W5","source_file":"spaces-simplicial.tex","source_line":372,"source_end_line":381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L372-L381","statement_sha256":"012e57367b8937b56e8eba07c420049b535d28d435e9232976ddbec6e9482db1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12735,"rank":12735,"depth":6,"x":525.172,"y":1636.611,"cluster":"geometry-of-spaces"},{"id":"stacks:09W6","tag":"09W6","title":"Simplicial topological spaces · Lemma 09W6","summary":"Let X be a simplicial topological space. Let F be an abelian sheaf on X. There is a spectral sequence (E_r, d_r)_r ≥ 0 with E_1^p, q = H^q(X_p, F_p) converging to H^p + q(X_Zar, F). This spectral sequence is functorial in F.","statement_latex":"Let $X$ be a simplicial topological space. Let $\\mathcal{F}$ be an abelian\nsheaf on $X$. There is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_1^{p, q} = H^q(X_p, \\mathcal{F}_p)\n$$\nconverging to $H^{p + q}(X_{Zar}, \\mathcal{F})$.\nThis spectral sequence is functorial in $\\mathcal{F}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09W6","source_file":"spaces-simplicial.tex","source_line":406,"source_end_line":415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L406-L415","statement_sha256":"6985de81645a4a9eec27eac8e6460839c6308378b71f51e9a58da0c6470e0c6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12736,"rank":12736,"depth":9,"x":764.773,"y":1485.087,"cluster":"geometry-of-spaces"},{"id":"stacks:0D84","tag":"0D84","title":"Simplicial topological spaces · Lemma 0D84","summary":"Let X be a simplicial space and let a : X → Y be an augmentation. Let F be an abelian sheaf on X_Zar. Then R^na_*F is the sheaf associated to the presheaf V ↦ H^n((X ×_Y V)_Zar, F|_(X ×_Y V)_Zar)","statement_latex":"Let $X$ be a simplicial space and let $a : X \\to Y$\nbe an augmentation. Let $\\mathcal{F}$ be an abelian sheaf\non $X_{Zar}$. Then $R^na_*\\mathcal{F}$ is the sheaf associated\nto the presheaf\n$$\nV \\longmapsto H^n((X \\times_Y V)_{Zar}, \\mathcal{F}|_{(X \\times_Y V)_{Zar}})\n$$","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D84","source_file":"spaces-simplicial.tex","source_line":451,"source_end_line":460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L451-L460","statement_sha256":"52c3282d935ab07f680cb84c7e727ef1251977b47fb1358470f46a91dbfa79c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12737,"rank":12737,"depth":10,"x":709.916,"y":1732.91,"cluster":"geometry-of-spaces"},{"id":"stacks:09W3","tag":"09W3","title":"Simplicial topological spaces · Lemma 09W3","summary":"Let X be a topological space. Let X_bullet be the constant simplicial topological space with value X. The functor X_bullet, Zar → X_Zar, U ↦ U is continuous and cocontinuous and defines a morphism of topoi g : Sh(X_bullet, Zar) → Sh(X) as well as a left adjoint g_! to g^-1. We have • g^-1 associates to a sheaf on X the constant cosimplicial sheaf on X, • g_! associates to a sheaf F on X_bullet, Zar the sheaf F_0, and • g_* associates to a sheaf F on X_bullet, Zar the…","statement_latex":"Let $X$ be a topological space. Let $X_\\bullet$ be the constant\nsimplicial topological space with value $X$. The functor\n$$\nX_{\\bullet, Zar} \\longrightarrow X_{Zar},\\quad\nU \\longmapsto U\n$$\nis continuous and cocontinuous and defines a morphism of\ntopoi $g : \\Sh(X_{\\bullet, Zar}) \\to \\Sh(X)$ as well as a left adjoint\n$g_!$ to $g^{-1}$. We have\n\\begin{enumerate}\n\\item $g^{-1}$ associates to a sheaf on $X$ the constant cosimplicial\nsheaf on $X$,\n\\item $g_!$ associates to a sheaf $\\mathcal{F}$ on $X_{\\bullet, Zar}$ the\nsheaf $\\mathcal{F}_0$, and\n\\item $g_*$ associates to a sheaf $\\mathcal{F}$ on $X_{\\bullet, Zar}$ the\nequalizer of the two maps $\\mathcal{F}_0 \\to \\mathcal{F}_1$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial topological spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09W3","source_file":"spaces-simplicial.tex","source_line":503,"source_end_line":522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L503-L522","statement_sha256":"1a3324390e5605f5531c549691d36537e2217f4aedd491b3f080185978521c32","origin":"The Stacks Project","memory_eligible":false,"source_rank":12738,"rank":12738,"depth":6,"x":550.987,"y":1518.924,"cluster":"geometry-of-spaces"},{"id":"stacks:09WC","tag":"09WC","title":"Simplicial sites and topoi · Lemma 09WC","summary":"Let C be a simplicial object in the category of sites. With notation as above we construct a site C_total as follows. • An object of C_total is an object U of C_n for some n, • a morphism (φ, f) : U → V of C_total is given by a map φ : [m] → [n] with U ∈ Ob(C_n), V ∈ Ob(C_m) and a morphism f : U → u_φ(V) of C_n, and • a covering ((id, f_i) : U_i → U) in C_total is given by an n and a covering (f_i : U_i → U) of C_n.","statement_latex":"Let $\\mathcal{C}$ be a simplicial object in the category of sites.\nWith notation as above we construct a site $\\mathcal{C}_{total}$ as follows.\n\\begin{enumerate}\n\\item An object of $\\mathcal{C}_{total}$ is an object $U$ of\n$\\mathcal{C}_n$ for some $n$,\n\\item a morphism $(\\varphi, f) : U \\to V$ of $\\mathcal{C}_{total}$\nis given by a map $\\varphi : [m] \\to [n]$ with\n$U \\in \\Ob(\\mathcal{C}_n)$, $V \\in \\Ob(\\mathcal{C}_m)$\nand a morphism $f : U \\to u_\\varphi(V)$ of $\\mathcal{C}_n$, and\n\\item a covering $\\{(\\text{id}, f_i) :  U_i \\to U\\}$ in $\\mathcal{C}_{total}$\nis given by an $n$ and a covering $\\{f_i : U_i \\to U\\}$\nof $\\mathcal{C}_n$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial sites and topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WC","source_file":"spaces-simplicial.tex","source_line":579,"source_end_line":594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L579-L594","statement_sha256":"e1225de26ee7ece9f8e7b61fde6bb224fa989113e91950de5b9129a91fa39e37","origin":"The Stacks Project","memory_eligible":false,"source_rank":12739,"rank":12739,"depth":0,"x":840.419,"y":1586.572,"cluster":"geometry-of-spaces"},{"id":"stacks:09WD","tag":"09WD","title":"Simplicial sites and topoi · Lemma 09WD","summary":"Let C be a simplicial object in the category whose objects are sites and whose morphisms are cocontinuous functors. With notation as above, assume the functors u_φ : C_n → C_m have property P of Sites, Remark [Tag 09W7]. Then we can construct a site C_total as follows. • An object of C_total is an object U of C_n for some n, • a morphism (φ, f) : U → V of C_total is given by a map φ : [m] → [n] with U ∈ Ob(C_n), V ∈ Ob(C_m) and a morphism f : u_φ(U) → V of C_m, and • a…","statement_latex":"Let $\\mathcal{C}$ be a simplicial object in the category whose objects are\nsites and whose morphisms are cocontinuous functors. With notation as above,\nassume the functors $u_\\varphi : \\mathcal{C}_n \\to \\mathcal{C}_m$\nhave property $P$ of Sites, Remark \\ref{sites-remark-cartesian-cocontinuous}.\nThen we can construct a site $\\mathcal{C}_{total}$ as follows.\n\\begin{enumerate}\n\\item An object of $\\mathcal{C}_{total}$ is an object $U$ of\n$\\mathcal{C}_n$ for some $n$,\n\\item a morphism $(\\varphi, f) : U \\to V$ of $\\mathcal{C}_{total}$\nis given by a map $\\varphi : [m] \\to [n]$ with\n$U \\in \\Ob(\\mathcal{C}_n)$, $V \\in \\Ob(\\mathcal{C}_m)$\nand a morphism $f : u_\\varphi(U) \\to V$ of $\\mathcal{C}_m$, and\n\\item a covering $\\{(\\text{id}, f_i) :  U_i \\to U\\}$ in $\\mathcal{C}_{total}$\nis given by an $n$ and a covering $\\{f_i : U_i \\to U\\}$\nof $\\mathcal{C}_n$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial sites and topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WD","source_file":"spaces-simplicial.tex","source_line":626,"source_end_line":644,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L626-L644","statement_sha256":"cc5d5b74ccf4e307b857e77544f62af8fac45b6a4155a0f6c426bea8ee90710a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12740,"rank":12740,"depth":0,"x":572.449,"y":1700.983,"cluster":"geometry-of-spaces"},{"id":"stacks:09WF","tag":"09WF","title":"Simplicial sites and topoi · Lemma 09WF","summary":"In Situation [Tag 09WE] there is an equivalence of categories between • Sh(C_total), and • the category of systems (F_n, F(φ)) described above. In particular, the topos Sh(C_total) only depends on the topoi Sh(C_n) and the morphisms of topoi f_φ.","statement_latex":"In Situation \\ref{situation-simplicial-site} there is an equivalence of\ncategories between\n\\begin{enumerate}\n\\item $\\Sh(\\mathcal{C}_{total})$, and\n\\item the category of systems $(\\mathcal{F}_n, \\mathcal{F}(\\varphi))$\ndescribed above.\n\\end{enumerate}\nIn particular, the topos $\\Sh(\\mathcal{C}_{total})$ only depends on\nthe topoi $\\Sh(\\mathcal{C}_n)$ and the morphisms of topoi $f_\\varphi$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial sites and topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WF","source_file":"spaces-simplicial.tex","source_line":735,"source_end_line":746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L735-L746","statement_sha256":"4cdcc7afb07c82baa69742faa472a8e8aa92276a62a1d0d746d930609cbb312a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12741,"rank":12741,"depth":0,"x":678.098,"y":1464.436,"cluster":"geometry-of-spaces"},{"id":"stacks:09WG","tag":"09WG","title":"Simplicial sites and topoi · Lemma 09WG","summary":"In Situation [Tag 09WE] the functor C_n → C_total, U ↦ U is continuous and cocontinuous. The associated morphism of topoi g_n : Sh(C_n) → Sh(C_total) satisfies • g_n^-1 associates to the sheaf F on C_total the sheaf F_n on C_n, • g_n^-1 : Sh(C_total) → Sh(C_n) has a left adjoint g^Sh_n!, • for G in Sh(C_n) the restriction of g_n!^ShG to C_m is coprod_φ : [n] → [m] f_φ^-1G, • g_n!^Sh commutes with finite connected limits, • g_n^-1 : Ab(C_total) → Ab(C_n) has a left adjoint…","statement_latex":"In Situation \\ref{situation-simplicial-site} the functor\n$\\mathcal{C}_n \\to \\mathcal{C}_{total}$, $U \\mapsto U$ is continuous\nand cocontinuous. The associated morphism of\ntopoi $g_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{C}_{total})$ satisfies\n\\begin{enumerate}\n\\item $g_n^{-1}$ associates to the sheaf $\\mathcal{F}$ on $\\mathcal{C}_{total}$\nthe sheaf $\\mathcal{F}_n$ on $\\mathcal{C}_n$,\n\\item $g_n^{-1} : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{C}_n)$\nhas a left adjoint $g^{Sh}_{n!}$,\n\\item for $\\mathcal{G}$ in $\\Sh(\\mathcal{C}_n)$ the restriction of\n$g_{n!}^{Sh}\\mathcal{G}$ to $\\mathcal{C}_m$ is\n$\\coprod\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}\\mathcal{G}$,\n\\item $g_{n!}^{Sh}$ commutes with finite connected limits,\n\\item $g_n^{-1} : \\textit{Ab}(\\mathcal{C}_{total}) \\to\n\\textit{Ab}(\\mathcal{C}_n)$ has a left adjoint $g_{n!}$,\n\\item for $\\mathcal{G}$ in $\\textit{Ab}(\\mathcal{C}_n)$ the restriction of\n$g_{n!}\\mathcal{G}$ to $\\mathcal{C}_m$ is\n$\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^{-1}\\mathcal{G}$, and\n\\item $g_{n!}$ is exact.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial sites and topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WG","source_file":"spaces-simplicial.tex","source_line":752,"source_end_line":774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L752-L774","statement_sha256":"a04c1de1c9a28267b1ff6418f217e03eacf5afd226af11b9191391ecfca44e9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12742,"rank":12742,"depth":6,"x":790.48,"y":1698.937,"cluster":"geometry-of-spaces"},{"id":"stacks:09WH","tag":"09WH","title":"Simplicial sites and topoi · Lemma 09WH","summary":"An injective abelian sheaf on a simplicial site is injective on each component In Situation [Tag 09WE]. If I is injective in Ab(C_total), then I_n is injective in Ab(C_n). If I^bullet is a K-injective complex in Ab(C_total), then I_n^bullet is K-injective in Ab(C_n).","statement_latex":"\\begin{slogan}\nAn injective abelian sheaf on a simplicial site is injective on each component\n\\end{slogan}\nIn Situation \\ref{situation-simplicial-site}.\nIf $\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{C}_{total})$,\nthen $\\mathcal{I}_n$ is injective in $\\textit{Ab}(\\mathcal{C}_n)$.\nIf $\\mathcal{I}^\\bullet$ is a K-injective complex in\n$\\textit{Ab}(\\mathcal{C}_{total})$,\nthen $\\mathcal{I}_n^\\bullet$ is K-injective in $\\textit{Ab}(\\mathcal{C}_n)$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial sites and topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WH","source_file":"spaces-simplicial.tex","source_line":868,"source_end_line":879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L868-L879","statement_sha256":"e126bebbd3f3bd820fa672c4e7f0d138dd90552d10de394a643330e9ce312814","origin":"The Stacks Project","memory_eligible":false,"source_rank":12743,"rank":12743,"depth":8,"x":518.883,"y":1589.729,"cluster":"geometry-of-spaces"},{"id":"stacks:0D70","tag":"0D70","title":"Augmentations of simplicial sites · Lemma 0D70","summary":"In Situation [Tag 09WE] let a_0 be an augmentation towards a site D as in Remark [Tag 0D6Z]. Then a_0 induces • a morphism of topoi a_n : Sh(C_n) → Sh(D) for all n ≥ 0, • a morphism of topoi a : Sh(C_total) → Sh(D) such that • for all φ : [m] → [n] we have a_m ∘ f_φ = a_n, • if g_n : Sh(C_n) → Sh(C_total) is as in Lemma [Tag 09WG], then a ∘ g_n = a_n, and • a_*F for F ∈ Sh(C_total) is the equalizer of the two maps a_0, *F_0 → a_1, *F_1.","statement_latex":"In Situation \\ref{situation-simplicial-site} let $a_0$ be an\naugmentation towards a site $\\mathcal{D}$ as in\nRemark \\ref{remark-augmentation-site}. Then $a_0$ induces\n\\begin{enumerate}\n\\item a morphism of topoi $a_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{D})$\nfor all $n \\geq 0$,\n\\item a morphism of topoi $a : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{D})$\n\\end{enumerate}\nsuch that\n\\begin{enumerate}\n\\item for all $\\varphi : [m] \\to [n]$ we have $a_m \\circ f_\\varphi = a_n$,\n\\item if $g_n : \\Sh(\\mathcal{C}_n) \\to \\Sh(\\mathcal{C}_{total})$\nis as in Lemma \\ref{lemma-restriction-to-components-site}, then\n$a \\circ g_n = a_n$, and\n\\item $a_*\\mathcal{F}$ for $\\mathcal{F} \\in \\Sh(\\mathcal{C}_{total})$\nis the equalizer of the two maps\n$a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Augmentations of simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D70","source_file":"spaces-simplicial.tex","source_line":923,"source_end_line":943,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L923-L943","statement_sha256":"b827b5954195596cb48669cb6a1a6da2a77a579d6e546b18e25e3d8b81f4d53c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12744,"rank":12744,"depth":7,"x":807.135,"y":1516.107,"cluster":"geometry-of-spaces"},{"id":"stacks:0D96","tag":"0D96","title":"Morphisms of simplicial sites · Lemma 0D96","summary":"Let C_n, f_φ, u_φ and C'_n, f'_φ, u'_φ be as in Situation [Tag 09WE]. Let h be a morphism between simplicial sites as in Remark [Tag 0D95]. Then we obtain a morphism of topoi h_total : Sh(C_total) → Sh(C'_total) and commutative diagrams xymatrix Sh(C_n) ar[d]_g_n ar[r]_h_n & Sh(C'_n) ar[d]^g'_n Sh(C_total) ar[r]^h_total & Sh(C'_total) Moreover, we have (g'_n)^-1 ∘ h_total, * = h_n, * ∘ g_n^-1.","statement_latex":"Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and\n$\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in\nSituation \\ref{situation-simplicial-site}.\nLet $h$ be a morphism between simplicial sites as in\nRemark \\ref{remark-morphism-simplicial-sites}.\nThen we obtain a morphism of topoi\n$$\nh_{total} : \\Sh(\\mathcal{C}_{total}) \\to \\Sh(\\mathcal{C}'_{total})\n$$\nand commutative diagrams\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C}_n) \\ar[d]_{g_n} \\ar[r]_{h_n} &\n\\Sh(\\mathcal{C}'_n) \\ar[d]^{g'_n} \\\\\n\\Sh(\\mathcal{C}_{total}) \\ar[r]^{h_{total}} &\n\\Sh(\\mathcal{C}'_{total})\n}\n$$\nMoreover, we have $(g'_n)^{-1} \\circ h_{total, *} = h_{n, *} \\circ g_n^{-1}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Morphisms of simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D96","source_file":"spaces-simplicial.tex","source_line":1055,"source_end_line":1076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1055-L1076","statement_sha256":"30a891973472cabd9eedce3b9a601c19ba556999472af66e07163ab80fa7721e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12745,"rank":12745,"depth":7,"x":653.703,"y":1734.074,"cluster":"geometry-of-spaces"},{"id":"stacks:0D97","tag":"0D97","title":"Morphisms of simplicial sites · Lemma 0D97","summary":"With notation and hypotheses as in Lemma [Tag 0D96]. For K ∈ D(C_total) we have (g'_n)^-1Rh_total, *K = Rh_n, *g_n^-1K.","statement_latex":"With notation and hypotheses as in Lemma \\ref{lemma-morphism-simplicial-sites}.\nFor $K \\in D(\\mathcal{C}_{total})$ we have\n$(g'_n)^{-1}Rh_{total, *}K = Rh_{n, *}g_n^{-1}K$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Morphisms of simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D97","source_file":"spaces-simplicial.tex","source_line":1135,"source_end_line":1140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1135-L1140","statement_sha256":"ee98c0a3c9f97a12c66e04c124607f0863a5c71e17c2cd2548d1eb485a3cdd2f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12746,"rank":12746,"depth":9,"x":591.524,"y":1486.153,"cluster":"geometry-of-spaces"},{"id":"stacks:0D99","tag":"0D99","title":"Morphisms of simplicial sites · Lemma 0D99","summary":"Let C_n, f_φ, u_φ, D, a_0, C'_n, f'_φ, u'_φ, D', a'_0, and h_n, n ≥ -1 be as in Remark [Tag 0D98]. Then we obtain a commutative diagram xymatrix Sh(C_total) ar[d]_a ar[r]_h_total & Sh(C'_total) ar[d]^a' Sh(D) ar[r]^h_-1 & Sh(D')","statement_latex":"Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi, \\mathcal{D}, a_0$,\n$\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi, \\mathcal{D}', a'_0$, and\n$h_n$, $n \\geq -1$ be as in\nRemark \\ref{remark-morphism-augmentation-simplicial-sites}.\nThen we obtain a commutative diagram\n$$\n\\xymatrix{\n\\Sh(\\mathcal{C}_{total}) \\ar[d]_a \\ar[r]_{h_{total}} &\n\\Sh(\\mathcal{C}'_{total}) \\ar[d]^{a'} \\\\\n\\Sh(\\mathcal{D}) \\ar[r]^{h_{-1}} &\n\\Sh(\\mathcal{D}')\n}\n$$","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Morphisms of simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D99","source_file":"spaces-simplicial.tex","source_line":1184,"source_end_line":1199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1184-L1199","statement_sha256":"7f1db0c5880f70feecf3f905e4c4cd3938a02a3295dfe0b8aee0ff378d553688","origin":"The Stacks Project","memory_eligible":false,"source_rank":12747,"rank":12747,"depth":8,"x":836.879,"y":1633.764,"cluster":"geometry-of-spaces"},{"id":"stacks:0D72","tag":"0D72","title":"Ringed simplicial sites · Lemma 0D72","summary":"In Situation [Tag 09WE]. Let O be a sheaf of rings on C_total. There is a canonical morphism of ringed topoi g_n : (Sh(C_n), O_n) → (Sh(C_total), O) agreeing with the morphism g_n of Lemma [Tag 09WG] on underlying topoi. The functor g_n^* : Mod(O) → Mod(O_n) has a left adjoint g_n!. For G in Mod(O_n)-modules the restriction of g_n!G to C_m is bigoplus_φ : [n] → [m] f_φ^*G where f_φ : (Sh(C_m), O_m) → (Sh(C_n), O_n) is the morphism of ringed topoi agreeing with the…","statement_latex":"In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$\nbe a sheaf of rings on $\\mathcal{C}_{total}$.\nThere is a canonical morphism of ringed topoi\n$g_n : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\to\n(\\Sh(\\mathcal{C}_{total}), \\mathcal{O})$\nagreeing with the morphism $g_n$ of\nLemma \\ref{lemma-restriction-to-components-site} on underlying topoi.\nThe functor\n$g_n^* : \\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}_n)$\nhas a left adjoint $g_{n!}$.\nFor $\\mathcal{G}$ in $\\textit{Mod}(\\mathcal{O}_n)$-modules the\nrestriction of $g_{n!}\\mathcal{G}$ to $\\mathcal{C}_m$ is\n$$\n\\bigoplus\\nolimits_{\\varphi : [n] \\to [m]} f_\\varphi^*\\mathcal{G}\n$$\nwhere $f_\\varphi : (\\Sh(\\mathcal{C}_m), \\mathcal{O}_m) \\to\n(\\Sh(\\mathcal{C}_n), \\mathcal{O}_n)$ is the morphism of ringed topoi\nagreeing with the previously defined $f_\\varphi$ on topoi and\nusing the map\n$\\mathcal{O}(\\varphi) : f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$\non sheaves of rings.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D72","source_file":"spaces-simplicial.tex","source_line":1241,"source_end_line":1264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1241-L1264","statement_sha256":"48b5955f2918f1a7b268a72e7e59ee7c93c8317ad20345f76b0b183de07ee575","origin":"The Stacks Project","memory_eligible":false,"source_rank":12748,"rank":12748,"depth":7,"x":537.09,"y":1664.154,"cluster":"geometry-of-spaces"},{"id":"stacks:0D73","tag":"0D73","title":"Ringed simplicial sites · Lemma 0D73","summary":"In Situation [Tag 09WE]. Let O be a sheaf of rings on C_total. If I is injective in Mod(O), then I_n is a totally acyclic sheaf on C_n.","statement_latex":"In Situation \\ref{situation-simplicial-site}.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$.\nIf $\\mathcal{I}$ is injective in $\\textit{Mod}(\\mathcal{O})$, then\n$\\mathcal{I}_n$ is a totally acyclic sheaf on $\\mathcal{C}_n$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D73","source_file":"spaces-simplicial.tex","source_line":1300,"source_end_line":1306,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1300-L1306","statement_sha256":"86cd95a0209541a81f70adec8110a74cd884a4f897e34d5c04580579c16e9134","origin":"The Stacks Project","memory_eligible":false,"source_rank":12749,"rank":12749,"depth":25,"x":733.818,"y":1471.533,"cluster":"geometry-of-spaces"},{"id":"stacks:0D74","tag":"0D74","title":"Ringed simplicial sites · Lemma 0D74","summary":"With assumptions as in Lemma [Tag 0D72] the functor g_n! : Mod(O_n) → Mod(O) is exact if the maps f_φ^-1O_n → O_m are flat for all φ : [n] → [m].","statement_latex":"With assumptions as in\nLemma \\ref{lemma-restriction-module-to-components-site} the functor\n$g_{n!} : \\textit{Mod}(\\mathcal{O}_n) \\to \\textit{Mod}(\\mathcal{O})$\nis exact if the maps $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$\nare flat for all $\\varphi : [n] \\to [m]$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D74","source_file":"spaces-simplicial.tex","source_line":1315,"source_end_line":1322,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1315-L1322","statement_sha256":"e102356dc8430e0bdb05de01c86668c28967956d3e6e1df1db07ac978cb21bd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12750,"rank":12750,"depth":8,"x":743.66,"y":1725.335,"cluster":"geometry-of-spaces"},{"id":"stacks:0D75","tag":"0D75","title":"Ringed simplicial sites · Lemma 0D75","summary":"In Situation [Tag 09WE]. Let O be a sheaf of rings on C_total such that f_φ^-1O_n → O_m is flat for all φ : [n] → [m]. If I is injective in Mod(O), then I_n is injective in Mod(O_n).","statement_latex":"In Situation \\ref{situation-simplicial-site}.\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$\nsuch that $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$\nis flat for all $\\varphi : [n] \\to [m]$.\nIf $\\mathcal{I}$ is injective in $\\textit{Mod}(\\mathcal{O})$, then\n$\\mathcal{I}_n$ is injective in $\\textit{Mod}(\\mathcal{O}_n)$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D75","source_file":"spaces-simplicial.tex","source_line":1341,"source_end_line":1349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1341-L1349","statement_sha256":"9f11c7d24a70307ba6a1b0702f11fc1ffbefa4e4fb053fa893e8f66be46f00d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12751,"rank":12751,"depth":9,"x":532.188,"y":1543.672,"cluster":"geometry-of-spaces"},{"id":"stacks:0DH0","tag":"0DH0","title":"Morphisms of ringed simplicial sites · Lemma 0DH0","summary":"Let C_n, f_φ, u_φ and C'_n, f'_φ, u'_φ be as in Situation [Tag 09WE]. Let O and O' be a sheaf of rings on C_total and C'_total. Let (h, h^sharp) be a morphism between simplicial sites as in Remark [Tag 0DGZ]. Then we obtain a morphism of ringed topoi h_total : (Sh(C_total), O) → (Sh(C'_total), O') and commutative diagrams xymatrix (Sh(C_n), O_n) ar[d]_g_n ar[r]_h_n & (Sh(C'_n), O'_n) ar[d]^g'_n (Sh(C_total), O) ar[r]^h_total & (Sh(C'_total), O') of ringed topoi where g_n…","statement_latex":"Let $\\mathcal{C}_n, f_\\varphi, u_\\varphi$ and\n$\\mathcal{C}'_n, f'_\\varphi, u'_\\varphi$ be as in\nSituation \\ref{situation-simplicial-site}.\nLet $\\mathcal{O}$ and $\\mathcal{O}'$\nbe a sheaf of rings on $\\mathcal{C}_{total}$ and $\\mathcal{C}'_{total}$.\nLet $(h, h^\\sharp)$ be a morphism between simplicial sites as in\nRemark \\ref{remark-morphism-simplicial-sites-modules}.\nThen we obtain a morphism of ringed topoi\n$$\nh_{total} :\n(\\Sh(\\mathcal{C}_{total}), \\mathcal{O})\n\\to\n(\\Sh(\\mathcal{C}'_{total}), \\mathcal{O}')\n$$\nand commutative diagrams\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\ar[d]_{g_n} \\ar[r]_{h_n} &\n(\\Sh(\\mathcal{C}'_n), \\mathcal{O}'_n) \\ar[d]^{g'_n} \\\\\n(\\Sh(\\mathcal{C}_{total}), \\mathcal{O}) \\ar[r]^{h_{total}} &\n(\\Sh(\\mathcal{C}'_{total}), \\mathcal{O}')\n}\n$$\nof ringed topoi where $g_n$ and $g'_n$ are as in\nLemma \\ref{lemma-restriction-module-to-components-site}.\nMoreover, we have\n$(g'_n)^* \\circ h_{total, *} = h_{n, *} \\circ g_n^*$\nas functor $\\textit{Mod}(\\mathcal{O}) \\to \\textit{Mod}(\\mathcal{O}'_n)$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Morphisms of ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DH0","source_file":"spaces-simplicial.tex","source_line":1386,"source_end_line":1416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1386-L1416","statement_sha256":"ba7091262c27f0d13dd3cb5b457cf917da0c57067d54a96b97f7b8e561a7c860","origin":"The Stacks Project","memory_eligible":false,"source_rank":12752,"rank":12752,"depth":8,"x":834.376,"y":1557.641,"cluster":"geometry-of-spaces"},{"id":"stacks:0DH1","tag":"0DH1","title":"Morphisms of ringed simplicial sites · Lemma 0DH1","summary":"With notation and hypotheses as in Lemma [Tag 0DH0]. For K ∈ D(O) we have (g'_n)^*Rh_total, *K = Rh_n, *g_n^*K.","statement_latex":"With notation and hypotheses as in\nLemma \\ref{lemma-morphism-simplicial-sites-modules}.\nFor $K \\in D(\\mathcal{O})$ we have\n$(g'_n)^*Rh_{total, *}K = Rh_{n, *}g_n^*K$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Morphisms of ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DH1","source_file":"spaces-simplicial.tex","source_line":1438,"source_end_line":1444,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1438-L1444","statement_sha256":"4634de208a01f3583e303a198a23e6da9feeb695f8a114f3d95887c494f2ae62","origin":"The Stacks Project","memory_eligible":false,"source_rank":12753,"rank":12753,"depth":23,"x":600.187,"y":1718.896,"cluster":"geometry-of-spaces"},{"id":"stacks:09WI","tag":"09WI","title":"Cohomology on simplicial sites · Lemma 09WI","summary":"In Situation [Tag 09WE] and with notation as above there is a complex … → g_2!Z → g_1!Z → g_0!Z of abelian sheaves on C_total which forms a resolution of the constant sheaf with value Z on C_total.","statement_latex":"In Situation \\ref{situation-simplicial-site} and with notation as above\nthere is a complex\n$$\n\\ldots \\to g_{2!}\\mathbf{Z} \\to g_{1!}\\mathbf{Z} \\to g_{0!}\\mathbf{Z}\n$$\nof abelian sheaves on $\\mathcal{C}_{total}$ which forms a resolution of\nthe constant sheaf with value $\\mathbf{Z}$ on $\\mathcal{C}_{total}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology on simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WI","source_file":"spaces-simplicial.tex","source_line":1493,"source_end_line":1502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1493-L1502","statement_sha256":"94530358c6b76f20abb72d679901a480f209ded5e26728a9bc359b9d65b94fef","origin":"The Stacks Project","memory_eligible":false,"source_rank":12754,"rank":12754,"depth":7,"x":643.22,"y":1466.969,"cluster":"geometry-of-spaces"},{"id":"stacks:0D77","tag":"0D77","title":"Cohomology on simplicial sites · Lemma 0D77","summary":"In Situation [Tag 09WE]. Let F be an abelian sheaf on C_total there is a canonical complex 0 → Γ(C_total, F) → Γ(C_0, F_0) → Γ(C_1, F_1) → Γ(C_2, F_2) → … which is exact in degrees -1, 0 and exact everywhere if F is injective.","statement_latex":"In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{F}$ be an abelian\nsheaf on $\\mathcal{C}_{total}$ there is a canonical complex\n$$\n0 \\to \\Gamma(\\mathcal{C}_{total}, \\mathcal{F}) \\to\n\\Gamma(\\mathcal{C}_0, \\mathcal{F}_0) \\to\n\\Gamma(\\mathcal{C}_1, \\mathcal{F}_1) \\to\n\\Gamma(\\mathcal{C}_2, \\mathcal{F}_2) \\to \\ldots\n$$\nwhich is exact in degrees $-1, 0$ and exact everywhere\nif $\\mathcal{F}$ is injective.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology on simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D77","source_file":"spaces-simplicial.tex","source_line":1534,"source_end_line":1546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1534-L1546","statement_sha256":"fed5159ebe387e4d6f40e1a3aa222b09cb67426613acd195fac7021ebd6b3512","origin":"The Stacks Project","memory_eligible":false,"source_rank":12755,"rank":12755,"depth":8,"x":814.173,"y":1677.267,"cluster":"geometry-of-spaces"},{"id":"stacks:09WJ","tag":"09WJ","title":"Cohomology on simplicial sites · Lemma 09WJ","summary":"In Situation [Tag 09WE]. For K in D^+(C_total) there is a spectral sequence (E_r, d_r)_r ≥ 0 with E_1^p, q = H^q(C_p, K_p), d_1^p, q : E_1^p, q → E_1^p + 1, q converging to H^p + q(C_total, K). This spectral sequence is functorial in K.","statement_latex":"In Situation \\ref{situation-simplicial-site}. For $K$ in\n$D^+(\\mathcal{C}_{total})$ there is a spectral sequence\n$(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_1^{p, q} = H^q(\\mathcal{C}_p, K_p),\\quad\nd_1^{p, q} : E_1^{p, q} \\to E_1^{p + 1, q}\n$$\nconverging to $H^{p + q}(\\mathcal{C}_{total}, K)$.\nThis spectral sequence is functorial in $K$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology on simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WJ","source_file":"spaces-simplicial.tex","source_line":1559,"source_end_line":1570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1559-L1570","statement_sha256":"43fb71b1501cde420f15bd2d901b16820aa700047cda024c1b3669484d1fa4aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12756,"rank":12756,"depth":9,"x":518.84,"y":1619.168,"cluster":"geometry-of-spaces"},{"id":"stacks:0H0W","tag":"0H0W","title":"Cohomology on simplicial sites · Lemma 0H0W","summary":"In Situation [Tag 09WE]. Let K be an object of D(C_total). • If H^-p(C_p, K_p) = 0 for all p ≥ 0, then H^0(C_total, K) = 0. • If RΓ(C_p, K_p) = 0 for all p ≥ 0, then RΓ(C_total, K) = 0.","statement_latex":"In Situation \\ref{situation-simplicial-site}. Let $K$ be an object of\n$D(\\mathcal{C}_{total})$.\n\\begin{enumerate}\n\\item If $H^{-p}(\\mathcal{C}_p, K_p) = 0$ for all $p \\geq 0$, then\n$H^0(\\mathcal{C}_{total}, K) = 0$.\n\\item If $R\\Gamma(\\mathcal{C}_p, K_p) = 0$\nfor all $p \\geq 0$, then $R\\Gamma(\\mathcal{C}_{total}, K) = 0$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology on simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0W","source_file":"spaces-simplicial.tex","source_line":1643,"source_end_line":1653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1643-L1653","statement_sha256":"a6edc9bf39d0a350a3a9acf1388568f2efdea83692e4180fbf17b08a29c55e7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12757,"rank":12757,"depth":1,"x":783.478,"y":1494.363,"cluster":"geometry-of-spaces"},{"id":"stacks:0DBZ","tag":"0DBZ","title":"Cohomology on simplicial sites · Lemma 0DBZ","summary":"Let C be as in Situation [Tag 09WE]. Let U ∈ Ob(C_n). Let F ∈ Ab(C_total). Then H^p(U, F) = H^p(U, g_n^-1F) where on the left hand side U is viewed as an object of C_total.","statement_latex":"Let $\\mathcal{C}$ be as in Situation \\ref{situation-simplicial-site}.\nLet $U \\in \\Ob(\\mathcal{C}_n)$. Let\n$\\mathcal{F} \\in \\textit{Ab}(\\mathcal{C}_{total})$.\nThen $H^p(U, \\mathcal{F}) = H^p(U, g_n^{-1}\\mathcal{F})$\nwhere on the left hand side $U$ is viewed as an object of $\\mathcal{C}_{total}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology on simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DBZ","source_file":"spaces-simplicial.tex","source_line":1665,"source_end_line":1672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1665-L1672","statement_sha256":"10da5ff9b4ecdb5e7f5318d40df6768a0d300a243b231ba2cacb83a3ded56a5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12758,"rank":12758,"depth":9,"x":688.651,"y":1736.684,"cluster":"geometry-of-spaces"},{"id":"stacks:0D78","tag":"0D78","title":"Cohomology and augmentations of simplicial sites · Lemma 0D78","summary":"In Situation [Tag 09WE] let a_0 be an augmentation towards a site D as in Remark [Tag 0D6Z]. For any abelian sheaf G on D there is an exact complex … → g_2!(a_2^-1G) → g_1!(a_1^-1G) → g_0!(a_0^-1G) → a^-1G → 0 of abelian sheaves on C_total.","statement_latex":"In Situation \\ref{situation-simplicial-site} let\n$a_0$ be an augmentation towards a site $\\mathcal{D}$\nas in Remark \\ref{remark-augmentation-site}.\nFor any abelian sheaf $\\mathcal{G}$ on $\\mathcal{D}$ \nthere is an exact complex\n$$\n\\ldots \\to\ng_{2!}(a_2^{-1}\\mathcal{G}) \\to\ng_{1!}(a_1^{-1}\\mathcal{G}) \\to\ng_{0!}(a_0^{-1}\\mathcal{G}) \\to\na^{-1}\\mathcal{G} \\to 0\n$$\nof abelian sheaves on $\\mathcal{C}_{total}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology and augmentations of simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D78","source_file":"spaces-simplicial.tex","source_line":1711,"source_end_line":1726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1711-L1726","statement_sha256":"35d00d94763bcf72fbe36e67f62680869b44c32d5362c7e845c781940e92b39c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12759,"rank":12759,"depth":8,"x":563.64,"y":1504.069,"cluster":"geometry-of-spaces"},{"id":"stacks:0D79","tag":"0D79","title":"Cohomology and augmentations of simplicial sites · Lemma 0D79","summary":"In Situation [Tag 09WE] let a_0 be an augmentation towards a site D as in Remark [Tag 0D6Z]. For an abelian sheaf F on C_total there is a canonical complex 0 → a_*F → a_0, *F_0 → a_1, *F_1 → a_2, *F_2 → … on D which is exact in degrees -1, 0 and exact everywhere if F is injective.","statement_latex":"In Situation \\ref{situation-simplicial-site} let\n$a_0$ be an augmentation towards a site $\\mathcal{D}$\nas in Remark \\ref{remark-augmentation-site}.\nFor an abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}_{total}$\nthere is a canonical complex\n$$\n0 \\to a_*\\mathcal{F} \\to a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1 \\to\na_{2, *}\\mathcal{F}_2 \\to \\ldots\n$$\non $\\mathcal{D}$ which is exact in degrees $-1, 0$ and\nexact everywhere if $\\mathcal{F}$ is injective.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology and augmentations of simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D79","source_file":"spaces-simplicial.tex","source_line":1770,"source_end_line":1783,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1770-L1783","statement_sha256":"fca25109ecb457763c4956bf8eeccbe9f98aa14591d1b15fcc0742f376103bf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12760,"rank":12760,"depth":9,"x":843.034,"y":1604.716,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7A","tag":"0D7A","title":"Cohomology and augmentations of simplicial sites · Lemma 0D7A","summary":"In Situation [Tag 09WE] let a_0 be an augmentation towards a site D as in Remark [Tag 0D6Z]. For any K in D^+(C_total) there is a spectral sequence (E_r, d_r)_r ≥ 0 with E_1^p, q = R^qa_p, * K_p, d_1^p, q : E_1^p, q → E_1^p + 1, q converging to R^p + qa_*K. This spectral sequence is functorial in K.","statement_latex":"In Situation \\ref{situation-simplicial-site} let\n$a_0$ be an augmentation towards a site $\\mathcal{D}$\nas in Remark \\ref{remark-augmentation-site}.\nFor any $K$ in $D^+(\\mathcal{C}_{total})$ there is a spectral\nsequence \n$(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_1^{p, q} = R^qa_{p, *} K_p,\\quad\nd_1^{p, q} : E_1^{p, q} \\to E_1^{p + 1, q}\n$$\nconverging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology and augmentations of simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7A","source_file":"spaces-simplicial.tex","source_line":1802,"source_end_line":1815,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1802-L1815","statement_sha256":"b6a61cc92e97e6a3f06dfab1138590f30f5fa7bf2022db68590b279db3c9f925","origin":"The Stacks Project","memory_eligible":false,"source_rank":12761,"rank":12761,"depth":10,"x":555.922,"y":1689.08,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9B","tag":"0D9B","title":"Cohomology on ringed simplicial sites · Lemma 0D9B","summary":"In Situation [Tag 09WE] let O be a sheaf of rings on C_total. There is a complex … → g_2!O_2 → g_1!O_1 → g_0!O_0 of O-modules which forms a resolution of O. Here g_n! is as in Lemma [Tag 0D72].","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$\nbe a sheaf of rings on $\\mathcal{C}_{total}$. There is a complex\n$$\n\\ldots \\to g_{2!}\\mathcal{O}_2 \\to g_{1!}\\mathcal{O}_1 \\to g_{0!}\\mathcal{O}_0\n$$\nof $\\mathcal{O}$-modules which forms a resolution of\n$\\mathcal{O}$.\nHere $g_{n!}$ is as in Lemma \\ref{lemma-restriction-module-to-components-site}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology on ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9B","source_file":"spaces-simplicial.tex","source_line":1874,"source_end_line":1884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1874-L1884","statement_sha256":"7fd2db56d1898c09dee7616ab3e987c3acbc259fecd8dd5ec344f68d021e674a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12762,"rank":12762,"depth":8,"x":699.868,"y":1463.836,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9C","tag":"0D9C","title":"Cohomology on ringed simplicial sites · Lemma 0D9C","summary":"In Situation [Tag 09WE] let O be a sheaf of rings. Let F be a sheaf of O-modules. There is a canonical complex 0 → Γ(C_total, F) → Γ(C_0, F_0) → Γ(C_1, F_1) → Γ(C_2, F_2) → … which is exact in degrees -1, 0 and exact everywhere if F is an injective O-module.","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$\nbe a sheaf of rings. Let $\\mathcal{F}$ be a\nsheaf of $\\mathcal{O}$-modules. There is a canonical complex\n$$\n0 \\to \\Gamma(\\mathcal{C}_{total}, \\mathcal{F}) \\to\n\\Gamma(\\mathcal{C}_0, \\mathcal{F}_0) \\to\n\\Gamma(\\mathcal{C}_1, \\mathcal{F}_1) \\to\n\\Gamma(\\mathcal{C}_2, \\mathcal{F}_2) \\to \\ldots\n$$\nwhich is exact in degrees $-1, 0$ and exact everywhere\nif $\\mathcal{F}$ is an injective $\\mathcal{O}$-module.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology on ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9C","source_file":"spaces-simplicial.tex","source_line":1917,"source_end_line":1930,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1917-L1930","statement_sha256":"27594a92b7a2f9929ec2a30c1defd10dff3b797e971376a071a3cc0f5e9dce93","origin":"The Stacks Project","memory_eligible":false,"source_rank":12763,"rank":12763,"depth":9,"x":774.899,"y":1711.738,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7E","tag":"0D7E","title":"Cohomology on ringed simplicial sites · Lemma 0D7E","summary":"In Situation [Tag 09WE] let O be a sheaf of rings. For K in D^+(O) there is a spectral sequence (E_r, d_r)_r ≥ 0 with E_1^p, q = H^q(C_p, K_p), d_1^p, q : E_1^p, q → E_1^p + 1, q converging to H^p + q(C_total, K). This spectral sequence is functorial in K.","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$\nbe a sheaf of rings. For $K$ in $D^+(\\mathcal{O})$\nthere is a spectral sequence $(E_r, d_r)_{r \\geq 0}$ with\n$$\nE_1^{p, q} = H^q(\\mathcal{C}_p, K_p),\\quad\nd_1^{p, q} : E_1^{p, q} \\to E_1^{p + 1, q}\n$$\nconverging to $H^{p + q}(\\mathcal{C}_{total}, K)$.\nThis spectral sequence is functorial in $K$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology on ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7E","source_file":"spaces-simplicial.tex","source_line":1943,"source_end_line":1954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1943-L1954","statement_sha256":"c176121f8c8bf8cad9bfca1e66b98556eb4a426cac795f9e9cd4de054b52666c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12764,"rank":12764,"depth":26,"x":520.081,"y":1571.439,"cluster":"geometry-of-spaces"},{"id":"stacks:0DH2","tag":"0DH2","title":"Cohomology on ringed simplicial sites · Lemma 0DH2","summary":"In Situation [Tag 09WE] let O be a sheaf of rings. Let U ∈ Ob(C_n). Let F ∈ Mod(O). Then H^p(U, F) = H^p(U, g_n^*F) where on the left hand side U is viewed as an object of C_total.","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$\nbe a sheaf of rings. Let $U \\in \\Ob(\\mathcal{C}_n)$. Let\n$\\mathcal{F} \\in \\textit{Mod}(\\mathcal{O})$.\nThen $H^p(U, \\mathcal{F}) = H^p(U, g_n^*\\mathcal{F})$\nwhere on the left hand side $U$ is viewed as an object of\n$\\mathcal{C}_{total}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology on ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DH2","source_file":"spaces-simplicial.tex","source_line":1990,"source_end_line":1998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L1990-L1998","statement_sha256":"350fa0dd3c48ed7ff0d486f967b0d8062721d6a732ded5eba416db92a805a228","origin":"The Stacks Project","memory_eligible":false,"source_rank":12765,"rank":12765,"depth":26,"x":820.965,"y":1530.281,"cluster":"geometry-of-spaces"},{"id":"stacks:0DH3","tag":"0DH3","title":"Cohomology and augmentations of ringed simplicial sites · Lemma 0DH3","summary":"With notation as above. The morphism a : (Sh(C_total), O) → (Sh(D), O_D) is flat if and only if a_n : (Sh(C_n), O_n) → (Sh(D), O_D) is flat for n ≥ 0.","statement_latex":"With notation as above. The morphism\n$a : (\\Sh(\\mathcal{C}_{total}), \\mathcal{O}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nis flat if and only if\n$a_n : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$\nis flat for $n \\geq 0$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology and augmentations of ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DH3","source_file":"spaces-simplicial.tex","source_line":2068,"source_end_line":2077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2068-L2077","statement_sha256":"73ac922fd6defb471e60d00e73ac5044d0eeb58a8affce6b9544146631e97e44","origin":"The Stacks Project","memory_eligible":false,"source_rank":12766,"rank":12766,"depth":0,"x":632.095,"y":1731.466,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7C","tag":"0D7C","title":"Cohomology and augmentations of ringed simplicial sites · Lemma 0D7C","summary":"With notation as above. For a O_D-module G there is an exact complex … → g_2!(a_2^*G) → g_1!(a_1^*G) → g_0!(a_0^*G) → a^*G → 0 of sheaves of O-modules on C_total. Here g_n! is as in Lemma [Tag 0D72].","statement_latex":"With notation as above. For a $\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{G}$\nthere is an exact complex\n$$\n\\ldots \\to\ng_{2!}(a_2^*\\mathcal{G}) \\to\ng_{1!}(a_1^*\\mathcal{G}) \\to\ng_{0!}(a_0^*\\mathcal{G}) \\to\na^*\\mathcal{G} \\to 0\n$$\nof sheaves of $\\mathcal{O}$-modules on $\\mathcal{C}_{total}$.\nHere $g_{n!}$ is as in Lemma \\ref{lemma-restriction-module-to-components-site}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology and augmentations of ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7C","source_file":"spaces-simplicial.tex","source_line":2108,"source_end_line":2121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2108-L2121","statement_sha256":"577da80d07163293f83ccb3c6597ace0a63587023c55469ea33ed62cd08386de","origin":"The Stacks Project","memory_eligible":false,"source_rank":12767,"rank":12767,"depth":9,"x":609.565,"y":1475.81,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7D","tag":"0D7D","title":"Cohomology and augmentations of ringed simplicial sites · Lemma 0D7D","summary":"With notation as above. For an O-module F on C_total there is a canonical complex 0 → a_*F → a_0, *F_0 → a_1, *F_1 → a_2, *F_2 → … of O_D-modules which is exact in degrees -1, 0. If F is an injective O-module, then the complex is exact in all degrees and remains exact on applying the functor Hom_O_D(G, -) for any O_D-module G.","statement_latex":"With notation as above.\nFor an $\\mathcal{O}$-module $\\mathcal{F}$ on $\\mathcal{C}_{total}$\nthere is a canonical complex\n$$\n0 \\to a_*\\mathcal{F} \\to a_{0, *}\\mathcal{F}_0 \\to a_{1, *}\\mathcal{F}_1 \\to\na_{2, *}\\mathcal{F}_2 \\to \\ldots\n$$\nof $\\mathcal{O}_\\mathcal{D}$-modules which is exact in degrees $-1, 0$.\nIf $\\mathcal{F}$ is an injective $\\mathcal{O}$-module, then the complex\nis exact in all degrees and remains exact on applying the functor\n$\\Hom_{\\mathcal{O}_\\mathcal{D}}(\\mathcal{G}, -)$ for any\n$\\mathcal{O}_\\mathcal{D}$-module $\\mathcal{G}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology and augmentations of ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7D","source_file":"spaces-simplicial.tex","source_line":2141,"source_end_line":2155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2141-L2155","statement_sha256":"2c089faab4b6cbd0610dc5fe2ba7e74229bbe595c1365115db173853efba0f65","origin":"The Stacks Project","memory_eligible":false,"source_rank":12768,"rank":12768,"depth":10,"x":831.888,"y":1651.637,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7F","tag":"0D7F","title":"Cohomology and augmentations of ringed simplicial sites · Lemma 0D7F","summary":"With notation as above for any K in D^+(O) there is a spectral sequence (E_r, d_r)_r ≥ 0 in Mod(O_D) with E_1^p, q = R^qa_p, * K_p converging to R^p + qa_*K. This spectral sequence is functorial in K.","statement_latex":"With notation as above for any $K$ in $D^+(\\mathcal{O})$ there is a spectral\nsequence $(E_r, d_r)_{r \\geq 0}$ in $\\textit{Mod}(\\mathcal{O}_\\mathcal{D})$\nwith\n$$\nE_1^{p, q} = R^qa_{p, *} K_p\n$$\nconverging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomology and augmentations of ringed simplicial sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7F","source_file":"spaces-simplicial.tex","source_line":2177,"source_end_line":2186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2177-L2186","statement_sha256":"a3bbd474ca0f72223eda1fccfc5464fbbdfd7791f2238a8a70e9b41eda26243c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12769,"rank":12769,"depth":26,"x":526.394,"y":1648.134,"cluster":"geometry-of-spaces"},{"id":"stacks:07TF","tag":"07TF","title":"Cartesian sheaves and modules · Definition 07TF","summary":"In Situation [Tag 09WE]. • A sheaf F of sets or of abelian groups on C_total is cartesian if the maps F(φ) : f_φ^-1F_m → F_n are isomorphisms for all φ : [m] → [n]. • If O is a sheaf of rings on C_total, then a sheaf F of O-modules is cartesian if the maps f_φ^*F_m → F_n are isomorphisms for all φ : [m] → [n]. • An object K of D(C_total) is cartesian if the maps f_φ^-1K_m → K_n are isomorphisms for all φ : [m] → [n]. • If O is a sheaf of rings on C_total, then an object K…","statement_latex":"In Situation \\ref{situation-simplicial-site}.\n\\begin{enumerate}\n\\item A sheaf $\\mathcal{F}$ of sets or of abelian groups on\n$\\mathcal{C}_{total}$ is {\\it cartesian} if the maps\n$\\mathcal{F}(\\varphi) : f_\\varphi^{-1}\\mathcal{F}_m \\to \\mathcal{F}_n$\nare isomorphisms for all $\\varphi : [m] \\to [n]$.\n\\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$,\nthen a sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules is\n{\\it cartesian} if  the maps $f_\\varphi^*\\mathcal{F}_m \\to \\mathcal{F}_n$\nare isomorphisms for all $\\varphi : [m] \\to [n]$.\n\\item An object $K$ of $D(\\mathcal{C}_{total})$ is {\\it cartesian} if the maps\n$f_\\varphi^{-1}K_m \\to K_n$\nare isomorphisms for all $\\varphi : [m] \\to [n]$.\n\\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$, then\nan object $K$ of $D(\\mathcal{O})$ is {\\it cartesian} if the maps\n$Lf_\\varphi^*K_m \\to K_n$\nare isomorphisms for all $\\varphi : [m] \\to [n]$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cartesian sheaves and modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TF","source_file":"spaces-simplicial.tex","source_line":2229,"source_end_line":2249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2229-L2249","statement_sha256":"afedea97411932b58c7a46bca2daf96ee13e12ab2f70f4f052fad074f26aaf63","origin":"The Stacks Project","memory_eligible":false,"source_rank":12770,"rank":12770,"depth":0,"x":754.598,"y":1477.282,"cluster":"geometry-of-spaces"},{"id":"stacks:07TG","tag":"07TG","title":"Cartesian sheaves and modules · Lemma 07TG","summary":"In Situation [Tag 09WE]. • A sheaf F of sets or abelian groups is cartesian if and only if the maps (f_δ^n_j)^-1F_n - 1 → F_n are isomorphisms. • An object K of D(C_total) is cartesian if and only if the maps (f_δ^n_j)^-1K_n - 1 → K_n are isomorphisms. • If O is a sheaf of rings on C_total a sheaf F of O-modules is cartesian if and only if the maps (f_δ^n_j)^*F_n - 1 → F_n are isomorphisms. • If O is a sheaf of rings on C_total an object K of D(O) is cartesian if and only…","statement_latex":"In Situation \\ref{situation-simplicial-site}.\n\\begin{enumerate}\n\\item A sheaf $\\mathcal{F}$ of sets or abelian groups is cartesian\nif and only if the maps\n$(f_{\\delta^n_j})^{-1}\\mathcal{F}_{n - 1} \\to \\mathcal{F}_n$\nare isomorphisms.\n\\item An object $K$ of $D(\\mathcal{C}_{total})$ is cartesian\nif and only if the maps\n$(f_{\\delta^n_j})^{-1}K_{n - 1} \\to K_n$\nare isomorphisms.\n\\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$\na sheaf $\\mathcal{F}$ of $\\mathcal{O}$-modules is cartesian\nif and only if the maps\n$(f_{\\delta^n_j})^*\\mathcal{F}_{n - 1} \\to \\mathcal{F}_n$\nare isomorphisms.\n\\item If $\\mathcal{O}$ is a sheaf of rings on $\\mathcal{C}_{total}$\nan object $K$ of $D(\\mathcal{O})$ is cartesian\nif and only if the maps\n$L(f_{\\delta^n_j})^*K_{n - 1} \\to K_n$\nare isomorphisms.\n\\item Add more here.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cartesian sheaves and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TG","source_file":"spaces-simplicial.tex","source_line":2256,"source_end_line":2280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2256-L2280","statement_sha256":"d2f7859c863ec935974f5af1a65860234c179fb23b7cb73859725d40eeca571f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12771,"rank":12771,"depth":19,"x":723.702,"y":1732.889,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7H","tag":"0D7H","title":"Cartesian sheaves and modules · Lemma 0D7H","summary":"In Situation [Tag 09WE] let a_0 be an augmentation towards a site D as in Remark [Tag 0D6Z]. • The pullback a^-1G of a sheaf of sets or abelian groups on D is cartesian. • The pullback a^-1K of an object K of D(D) is cartesian. Let O be a sheaf of rings on C_total and O_D a sheaf of rings on D and a^sharp : O_D → a_*O a morphism as in Section [Tag 0D9D]. • [(3)] The pullback a^*F of a sheaf of O_D-modules is cartesian. • [(4)] The derived pullback La^*K of an object K of…","statement_latex":"In Situation \\ref{situation-simplicial-site} let\n$a_0$ be an augmentation towards a site $\\mathcal{D}$ as in\nRemark \\ref{remark-augmentation-site}.\n\\begin{enumerate}\n\\item The pullback $a^{-1}\\mathcal{G}$ of a sheaf of sets or abelian groups\non $\\mathcal{D}$ is cartesian.\n\\item The pullback $a^{-1}K$ of an object $K$ of $D(\\mathcal{D})$\nis cartesian.\n\\end{enumerate}\nLet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$ and\n$\\mathcal{O}_\\mathcal{D}$ a sheaf of rings on $\\mathcal{D}$\nand $a^\\sharp : \\mathcal{O}_\\mathcal{D} \\to a_*\\mathcal{O}$ a\nmorphism as in\nSection \\ref{section-cohomology-augmentation-ringed-simplicial-sites}.\n\\begin{enumerate}\n\\item[(3)] The pullback $a^*\\mathcal{F}$ of a sheaf of\n$\\mathcal{O}_\\mathcal{D}$-modules is cartesian.\n\\item[(4)] The derived pullback $La^*K$ of an object\n$K$ of $D(\\mathcal{O}_\\mathcal{D})$ is cartesian.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cartesian sheaves and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7H","source_file":"spaces-simplicial.tex","source_line":2312,"source_end_line":2334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2312-L2334","statement_sha256":"dd393c164baaf914c5cfe0edbe3c3deb1e7177676422c5a61ce7e6d4861d7d95","origin":"The Stacks Project","memory_eligible":false,"source_rank":12772,"rank":12772,"depth":8,"x":540.835,"y":1526.769,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7I","tag":"0D7I","title":"Cartesian sheaves and modules · Lemma 0D7I","summary":"In Situation [Tag 09WE]. The category of cartesian sheaves of sets (resp. abelian groups) is equivalent to the category of pairs (F, α) where F is a sheaf of sets (resp. abelian groups) on C_0 and α : (f_δ_1^1)^-1F → (f_δ_0^1)^-1F is an isomorphism of sheaves of sets (resp. abelian groups) on C_1 such that (f_δ^2_1)^-1α = (f_δ^2_0)^-1α ∘ (f_δ^2_2)^-1α as maps of sheaves on C_2.","statement_latex":"In Situation \\ref{situation-simplicial-site}.\nThe category of cartesian sheaves of sets (resp.\\ abelian groups)\nis equivalent to the category of pairs $(\\mathcal{F}, \\alpha)$\nwhere $\\mathcal{F}$ is a sheaf of sets (resp.\\ abelian groups)\non $\\mathcal{C}_0$ and\n$$\n\\alpha :\n(f_{\\delta_1^1})^{-1}\\mathcal{F}\n\\longrightarrow (f_{\\delta_0^1})^{-1}\\mathcal{F}\n$$\nis an isomorphism of sheaves of sets (resp.\\ abelian groups)\non $\\mathcal{C}_1$ such that\n$(f_{\\delta^2_1})^{-1}\\alpha =\n(f_{\\delta^2_0})^{-1}\\alpha \\circ (f_{\\delta^2_2})^{-1}\\alpha$\nas maps of sheaves on $\\mathcal{C}_2$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cartesian sheaves and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7I","source_file":"spaces-simplicial.tex","source_line":2343,"source_end_line":2360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2343-L2360","statement_sha256":"21d90b6d7fec04c5bfe7586a533d32707d99e414d8cf206295463512284d3ba4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12773,"rank":12773,"depth":1,"x":841.594,"y":1575.021,"cluster":"geometry-of-spaces"},{"id":"stacks:07TH","tag":"07TH","title":"Cartesian sheaves and modules · Lemma 07TH","summary":"In Situation [Tag 09WE] let O be a sheaf of rings on C_total. The category of cartesian O-modules is equivalent to the category of pairs (F, α) where F is a O_0-module and α : (f_δ_1^1)^*F → (f_δ_0^1)^*F is an isomorphism of O_1-modules such that (f_δ^2_1)^*α = (f_δ^2_0)^*α ∘ (f_δ^2_2)^*α as O_2-module maps.","statement_latex":"In Situation \\ref{situation-simplicial-site}\nlet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$.\nThe category of cartesian $\\mathcal{O}$-modules\nis equivalent to the category of pairs $(\\mathcal{F}, \\alpha)$\nwhere $\\mathcal{F}$ is a $\\mathcal{O}_0$-module\nand\n$$\n\\alpha :\n(f_{\\delta_1^1})^*\\mathcal{F}\n\\longrightarrow (f_{\\delta_0^1})^*\\mathcal{F}\n$$\nis an isomorphism of $\\mathcal{O}_1$-modules such that\n$(f_{\\delta^2_1})^*\\alpha =\n(f_{\\delta^2_0})^*\\alpha \\circ (f_{\\delta^2_2})^*\\alpha$\nas $\\mathcal{O}_2$-module maps.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cartesian sheaves and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TH","source_file":"spaces-simplicial.tex","source_line":2425,"source_end_line":2442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2425-L2442","statement_sha256":"691126380ca79e77dbdfc29182daabfeb4b0dfe306a3a7040ad874c5fd659518","origin":"The Stacks Project","memory_eligible":false,"source_rank":12774,"rank":12774,"depth":2,"x":580.878,"y":1710.169,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7J","tag":"0D7J","title":"Cartesian sheaves and modules · Lemma 0D7J","summary":"In Situation [Tag 09WE]. • The full subcategory of cartesian abelian sheaves forms a weak Serre subcategory of Ab(C_total). Colimits of systems of cartesian abelian sheaves are cartesian. • Let O be a sheaf of rings on C_total such that the morphisms f_δ^n_j : (Sh(C_n), O_n) → (Sh(C_n - 1), O_n - 1) are flat. The full subcategory of cartesian O-modules forms a weak Serre subcategory of Mod(O). Colimits of systems of cartesian O-modules are cartesian.","statement_latex":"In Situation \\ref{situation-simplicial-site}.\n\\begin{enumerate}\n\\item The full subcategory of cartesian abelian sheaves forms a\nweak Serre subcategory of $\\textit{Ab}(\\mathcal{C}_{total})$.\nColimits of systems of cartesian abelian sheaves are cartesian.\n\\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$\nsuch that the morphisms\n$$\nf_{\\delta^n_j} : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n)\n\\to (\\Sh(\\mathcal{C}_{n - 1}), \\mathcal{O}_{n - 1})\n$$\nare flat. The full subcategory of cartesian $\\mathcal{O}$-modules forms a\nweak Serre subcategory of $\\textit{Mod}(\\mathcal{O})$.\nColimits of systems of cartesian $\\mathcal{O}$-modules are cartesian.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cartesian sheaves and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7J","source_file":"spaces-simplicial.tex","source_line":2451,"source_end_line":2468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2451-L2468","statement_sha256":"3d77d7bf7bcc349b351a4b42a65f936420cff1d87f5d3e7c9a0e5dcd879c46e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12775,"rank":12775,"depth":20,"x":664.488,"y":1462.447,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7L","tag":"0D7L","title":"Cartesian sheaves and modules · Lemma 0D7L","summary":"In Situation [Tag 09WE]. • An object K of D(C_total) is cartesian if and only if H^q(K) is a cartesian abelian sheaf for all q. • Let O be a sheaf of rings on C_total such that the morphisms f_δ^n_j : (Sh(C_n), O_n) → (Sh(C_n - 1), O_n - 1) are flat. Then an object K of D(O) is cartesian if and only if H^q(K) is a cartesian O-module for all q.","statement_latex":"In Situation \\ref{situation-simplicial-site}.\n\\begin{enumerate}\n\\item An object $K$ of $D(\\mathcal{C}_{total})$ is cartesian if and only\nif $H^q(K)$ is a cartesian abelian sheaf for all $q$.\n\\item Let $\\mathcal{O}$ be a sheaf\nof rings on $\\mathcal{C}_{total}$ such that the morphisms\n$f_{\\delta^n_j} : (\\Sh(\\mathcal{C}_n), \\mathcal{O}_n)\n\\to (\\Sh(\\mathcal{C}_{n - 1}), \\mathcal{O}_{n - 1})$ are flat.\nThen an object $K$ of $D(\\mathcal{O})$ is cartesian if and only\nif $H^q(K)$ is a cartesian $\\mathcal{O}$-module for all $q$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cartesian sheaves and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7L","source_file":"spaces-simplicial.tex","source_line":2511,"source_end_line":2524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2511-L2524","statement_sha256":"450a19838881b5a0b2cf63c1218a426ac55a4588a258568ae6c5d26eab11512d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12776,"rank":12776,"depth":20,"x":802.119,"y":1692.676,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9E","tag":"0D9E","title":"Cartesian sheaves and modules · Lemma 0D9E","summary":"In Situation [Tag 09WE]. • An object K of D(C_total) is cartesian if and only the canonical map g_n!K_n → g_n!Z ⊗^L_Z K is an isomorphism for all n. • Let O be a sheaf of rings on C_total such that the morphisms f_φ^-1O_n → O_m are flat for all φ : [n] → [m]. Then an object K of D(O) is cartesian if and only if the canonical map g_n!K_n → g_n!O_n ⊗^L_O K is an isomorphism for all n.","statement_latex":"In Situation \\ref{situation-simplicial-site}.\n\\begin{enumerate}\n\\item An object $K$ of $D(\\mathcal{C}_{total})$ is cartesian if and only\nthe canonical map\n$$\ng_{n!}K_n \\longrightarrow\ng_{n!}\\mathbf{Z} \\otimes^\\mathbf{L}_\\mathbf{Z} K\n$$\nis an isomorphism for all $n$.\n\\item Let $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$\nsuch that the morphisms $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$\nare flat for all $\\varphi : [n] \\to [m]$. Then an object $K$ of\n$D(\\mathcal{O})$ is cartesian if and only if the canonical map\n$$\ng_{n!}K_n \\longrightarrow\ng_{n!}\\mathcal{O}_n \\otimes^\\mathbf{L}_\\mathcal{O} K\n$$\nis an isomorphism for all $n$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cartesian sheaves and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9E","source_file":"spaces-simplicial.tex","source_line":2534,"source_end_line":2555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2534-L2555","statement_sha256":"e0fffc4119209860df765f5792fe75ca6721a9338a5ef4445e7bed8c1885a03c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12777,"rank":12777,"depth":9,"x":515.337,"y":1600.956,"cluster":"geometry-of-spaces"},{"id":"stacks:0D7M","tag":"0D7M","title":"Cartesian sheaves and modules · Lemma 0D7M","summary":"In Situation [Tag 09WE] let O be a sheaf of rings on C_total. Let F be a sheaf of O-modules. Then F is quasi-coherent in the sense of Modules on Sites, Definition [Tag 03DL] if and only if F is cartesian and F_n is a quasi-coherent O_n-module for all n.","statement_latex":"In Situation \\ref{situation-simplicial-site}\nlet $\\mathcal{O}$ be a sheaf of rings on $\\mathcal{C}_{total}$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nThen $\\mathcal{F}$ is quasi-coherent in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local}\nif and only if $\\mathcal{F}$ is cartesian\nand $\\mathcal{F}_n$ is a quasi-coherent $\\mathcal{O}_n$-module for all $n$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cartesian sheaves and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D7M","source_file":"spaces-simplicial.tex","source_line":2599,"source_end_line":2608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2599-L2608","statement_sha256":"7a9624ab6f2a1a0fcadc970219cc6f7fb5981339bf78b6e7bb0315459d6848fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12778,"rank":12778,"depth":11,"x":800.715,"y":1505.812,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9G","tag":"0D9G","title":"Simplicial systems of the derived category · Definition 0D9G","summary":"In Situation [Tag 09WE]. A simplicial system of the derived category consists of the following data • for every n an object K_n of D(C_n), • for every φ : [m] → [n] a map K_φ : f_φ^-1K_m → K_n in D(C_n) subject to the condition that K_φ ∘ ψ = K_φ ∘ f_φ^-1K_ψ : f_φ ∘ ψ^-1K_l = f_φ^-1 f_ψ^-1K_l → K_n for any morphisms φ : [m] → [n] and ψ : [l] → [m] of Δ. We say the simplicial system is cartesian if the maps K_φ are isomorphisms for all φ. Given two simplicial systems of…","statement_latex":"In Situation \\ref{situation-simplicial-site}. A\n{\\it simplicial system of the derived category}\nconsists of the following data\n\\begin{enumerate}\n\\item for every $n$ an object $K_n$ of $D(\\mathcal{C}_n)$,\n\\item for every $\\varphi : [m] \\to [n]$ a map\n$K_\\varphi : f_\\varphi^{-1}K_m \\to K_n$ in $D(\\mathcal{C}_n)$\n\\end{enumerate}\nsubject to the condition that\n$$\nK_{\\varphi \\circ \\psi} = K_\\varphi \\circ f_\\varphi^{-1}K_\\psi :\nf_{\\varphi \\circ \\psi}^{-1}K_l = f_\\varphi^{-1} f_\\psi^{-1}K_l\n\\longrightarrow\nK_n\n$$\nfor any morphisms $\\varphi : [m] \\to [n]$ and $\\psi : [l] \\to [m]$ of $\\Delta$.\nWe say the simplicial system is {\\it cartesian} if the maps $K_\\varphi$\nare isomorphisms for all $\\varphi$.\nGiven two simplicial systems of the derived category\nthere is an obvious notion of a\n{\\it morphism of simplicial systems of the derived category}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9G","source_file":"spaces-simplicial.tex","source_line":2696,"source_end_line":2719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2696-L2719","statement_sha256":"efd55e3d7cb6d6e1b8dccc7d3d4a801e371f7bbab73ebabed3b7f109c79d30ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":12779,"rank":12779,"depth":0,"x":666.723,"y":1738.022,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9H","tag":"0D9H","title":"Simplicial systems of the derived category · Lemma 0D9H","summary":"In Situation [Tag 09WE]. If K ∈ D(C_total) is an object, then (K_n, K(φ)) is a simplicial system of the derived category. If K is cartesian, so is the system.","statement_latex":"In Situation \\ref{situation-simplicial-site}.\nIf $K \\in D(\\mathcal{C}_{total})$ is an object,\nthen $(K_n, K(\\varphi))$ is a simplicial system of the derived category.\nIf $K$ is cartesian, so is the system.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9H","source_file":"spaces-simplicial.tex","source_line":2727,"source_end_line":2733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2727-L2733","statement_sha256":"b45fbfc12156220888c274cf90ebd2637d5fa965e87bbd18329b2cc01770a408","origin":"The Stacks Project","memory_eligible":false,"source_rank":12780,"rank":12780,"depth":0,"x":578.744,"y":1490.634,"cluster":"geometry-of-spaces"},{"id":"stacks:0GME","tag":"0GME","title":"Simplicial systems of the derived category · Lemma 0GME","summary":"In Situation [Tag 09WE] suppose given K_0 ∈ D(C_0) and an isomorphism α : f_δ_1^1^-1K_0 → f_δ_0^1^-1K_0 satisfying the cocycle condition. Set τ^n_i : [0] → [n], 0 ↦ i and set K_n = f_τ^n_n^-1K_0. Then the K_n form a cartesian simplicial system of the derived category.","statement_latex":"In Situation \\ref{situation-simplicial-site}\nsuppose given $K_0 \\in D(\\mathcal{C}_0)$ and an isomorphism\n$$\n\\alpha :\nf_{\\delta_1^1}^{-1}K_0\n\\longrightarrow\nf_{\\delta_0^1}^{-1}K_0\n$$\nsatisfying the cocycle condition. Set\n$\\tau^n_i : [0] \\to [n]$, $0 \\mapsto i$ and\nset $K_n = f_{\\tau^n_n}^{-1}K_0$. Then the $K_n$\nform a cartesian simplicial system of the derived category.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GME","source_file":"spaces-simplicial.tex","source_line":2739,"source_end_line":2753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2739-L2753","statement_sha256":"79d119ff3e6fb478ece495d97662ba126c3c1fea6782ce1538162f69a99150ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":12781,"rank":12781,"depth":2,"x":842.699,"y":1623.202,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9I","tag":"0D9I","title":"Simplicial systems of the derived category · Lemma 0D9I","summary":"In Situation [Tag 09WE]. Let K be an object of D(C_total). Set X_n = (g_n!Z) ⊗^L_Z K and Y_n = (g_n!Z → … → g_0!Z)[-n] ⊗^L_Z K as objects of D(C_total) where the maps are as in Lemma [Tag 09WI]. With the evident canonical maps Y_n → X_n and Y_0 → Y_1[1] → Y_2[2] → … we have • the distinguished triangles Y_n → X_n → Y_n - 1 → Y_n[1] define a Postnikov system (Derived Categories, Definition [Tag 0D7Z]) for … → X_2 → X_1 → X_0, • K = hocolim Y_n[n] in D(C_total).","statement_latex":"In Situation \\ref{situation-simplicial-site}. Let $K$ be\nan object of $D(\\mathcal{C}_{total})$. Set\n$$\nX_n = (g_{n!}\\mathbf{Z})\n\\otimes^\\mathbf{L}_\\mathbf{Z} K\n\\quad\\text{and}\\quad\nY_n =\n(g_{n!}\\mathbf{Z} \\to \\ldots \\to g_{0!}\\mathbf{Z})[-n]\n\\otimes^\\mathbf{L}_\\mathbf{Z} K\n$$\nas objects of $D(\\mathcal{C}_{total})$ where the maps are\nas in Lemma \\ref{lemma-simplicial-resolution-Z-site}.\nWith the evident canonical maps $Y_n \\to X_n$ and\n$Y_0 \\to Y_1[1] \\to Y_2[2] \\to \\ldots$ we have\n\\begin{enumerate}\n\\item the distinguished triangles $Y_n \\to X_n \\to Y_{n - 1} \\to Y_n[1]$\ndefine a Postnikov system\n(Derived Categories, Definition \\ref{derived-definition-postnikov-system})\nfor $\\ldots \\to X_2 \\to X_1 \\to X_0$,\n\\item $K = \\text{hocolim} Y_n[n]$ in $D(\\mathcal{C}_{total})$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9I","source_file":"spaces-simplicial.tex","source_line":2784,"source_end_line":2807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2784-L2807","statement_sha256":"8720faa4391b43cece51833a41b0e69629f9e37e7d6e4718597e67f312d7a5cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12782,"rank":12782,"depth":8,"x":541.298,"y":1675.25,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9J","tag":"0D9J","title":"Simplicial systems of the derived category · Lemma 0D9J","summary":"In Situation [Tag 09WE]. If K, K' ∈ D(C_total). Assume • K is cartesian, • Hom(K_i[i], K'_i) = 0 for i > 0, and • Hom(K_i[i + 1], K'_i) = 0 for i ≥ 0. Then any map K → K' which induces the zero map K_0 → K'_0 is zero.","statement_latex":"In Situation \\ref{situation-simplicial-site}.\nIf $K, K' \\in D(\\mathcal{C}_{total})$.\nAssume\n\\begin{enumerate}\n\\item $K$ is cartesian,\n\\item $\\Hom(K_i[i], K'_i) = 0$ for $i > 0$, and\n\\item $\\Hom(K_i[i + 1], K'_i) = 0$ for $i \\geq 0$.\n\\end{enumerate}\nThen any map $K \\to K'$ which induces the zero map $K_0 \\to K'_0$ is zero.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9J","source_file":"spaces-simplicial.tex","source_line":2828,"source_end_line":2839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2828-L2839","statement_sha256":"d0c8aaf7ed513719ea24d42e557049f573ddf9cc0f83c8a825d4a46bc19ef04f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12783,"rank":12783,"depth":10,"x":721.785,"y":1465.739,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9K","tag":"0D9K","title":"Simplicial systems of the derived category · Lemma 0D9K","summary":"In Situation [Tag 09WE]. If K, K' ∈ D(C_total). Assume • K is cartesian, • Hom(K_i[i - 1], K'_i) = 0 for i > 1. Then any map (K_n → K'_n) between the associated simplicial systems of K and K' comes from a map K → K' in D(C_total).","statement_latex":"In Situation \\ref{situation-simplicial-site}.\nIf $K, K' \\in D(\\mathcal{C}_{total})$.\nAssume\n\\begin{enumerate}\n\\item $K$ is cartesian,\n\\item $\\Hom(K_i[i - 1], K'_i) = 0$ for $i > 1$.\n\\end{enumerate}\nThen any map $\\{K_n \\to K'_n\\}$ between the associated simplicial systems \nof $K$ and $K'$ comes from a map $K \\to K'$ in $D(\\mathcal{C}_{total})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9K","source_file":"spaces-simplicial.tex","source_line":2885,"source_end_line":2896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2885-L2896","statement_sha256":"81b09ef916fc07b3b452089872d734b5ac99b70af712c32a785d0915e3fd015d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12784,"rank":12784,"depth":10,"x":757.198,"y":1722.775,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9L","tag":"0D9L","title":"Simplicial systems of the derived category · Lemma 0D9L","summary":"In Situation [Tag 09WE]. Let (K_n, K_φ) be a simplicial system of the derived category. Assume • (K_n, K_φ) is cartesian, • Hom(K_i[t], K_i) = 0 for i ≥ 0 and t > 0. Then there exists a cartesian object K of D(C_total) whose associated simplicial system is isomorphic to (K_n, K_φ).","statement_latex":"In Situation \\ref{situation-simplicial-site}. Let\n$(K_n, K_\\varphi)$ be a simplicial system of the derived category.\nAssume\n\\begin{enumerate}\n\\item $(K_n, K_\\varphi)$ is cartesian,\n\\item $\\Hom(K_i[t], K_i) = 0$ for $i \\geq 0$ and $t > 0$.\n\\end{enumerate}\nThen there exists a cartesian object $K$ of $D(\\mathcal{C}_{total})$\nwhose associated simplicial system is isomorphic to $(K_n, K_\\varphi)$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9L","source_file":"spaces-simplicial.tex","source_line":2957,"source_end_line":2968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L2957-L2968","statement_sha256":"23e0eb7af718c805617481b4e8ffc16ec2c016312b802280751a5b0d632049d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12785,"rank":12785,"depth":10,"x":524.262,"y":1553.247,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9N","tag":"0D9N","title":"Simplicial systems of the derived category: modules · Definition 0D9N","summary":"In Situation [Tag 09WE]. Let O be a sheaf of rings on C_total. A simplicial system of the derived category of modules consists of the following data • for every n an object K_n of D(O_n), • for every φ : [m] → [n] a map K_φ : Lf_φ^*K_m → K_n in D(O_n) subject to the condition that K_φ ∘ ψ = K_φ ∘ Lf_φ^*K_ψ : Lf_φ ∘ ψ^*K_l = Lf_φ^* Lf_ψ^*K_l → K_n for any morphisms φ : [m] → [n] and ψ : [l] → [m] of Δ. We say the simplicial system is cartesian if the maps K_φ are…","statement_latex":"In Situation \\ref{situation-simplicial-site}. Let $\\mathcal{O}$\nbe a sheaf of rings on $\\mathcal{C}_{total}$. A\n{\\it simplicial system of the derived category of modules}\nconsists of the following data\n\\begin{enumerate}\n\\item for every $n$ an object $K_n$ of $D(\\mathcal{O}_n)$,\n\\item for every $\\varphi : [m] \\to [n]$ a map\n$K_\\varphi : Lf_\\varphi^*K_m \\to K_n$ in $D(\\mathcal{O}_n)$\n\\end{enumerate}\nsubject to the condition that\n$$\nK_{\\varphi \\circ \\psi} = K_\\varphi \\circ Lf_\\varphi^*K_\\psi :\nLf_{\\varphi \\circ \\psi}^*K_l = Lf_\\varphi^* Lf_\\psi^*K_l\n\\longrightarrow\nK_n\n$$\nfor any morphisms $\\varphi : [m] \\to [n]$ and $\\psi : [l] \\to [m]$ of $\\Delta$.\nWe say the simplicial system is {\\it cartesian} if the maps $K_\\varphi$\nare isomorphisms for all $\\varphi$.\nGiven two simplicial systems of the derived category\nthere is an obvious notion of a\n{\\it morphism of simplicial systems of the derived category of modules}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category: modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9N","source_file":"spaces-simplicial.tex","source_line":3139,"source_end_line":3163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3139-L3163","statement_sha256":"879c4e234578b2471c07bcdd2b614c0c170989925c5d7f8cc17c9f2c0f8fe4e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12786,"rank":12786,"depth":0,"x":832.516,"y":1546.078,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9P","tag":"0D9P","title":"Simplicial systems of the derived category: modules · Lemma 0D9P","summary":"In Situation [Tag 09WE] let O be a sheaf of rings on C_total. If K ∈ D(O) is an object, then (K_n, K(φ)) is a simplicial system of the derived category of modules. If K is cartesian, so is the system.","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a\nsheaf of rings on $\\mathcal{C}_{total}$.\nIf $K \\in D(\\mathcal{O})$ is an object, then $(K_n, K(\\varphi))$\nis a simplicial system of the derived category of modules.\nIf $K$ is cartesian, so is the system.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9P","source_file":"spaces-simplicial.tex","source_line":3171,"source_end_line":3178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3171-L3178","statement_sha256":"e3a50d66c65a7d20345f219ffe9f9f57f1257bba69290797edd25515a2c7d53f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12787,"rank":12787,"depth":0,"x":610.864,"y":1726.368,"cluster":"geometry-of-spaces"},{"id":"stacks:0GMF","tag":"0GMF","title":"Simplicial systems of the derived category: modules · Lemma 0GMF","summary":"In Situation [Tag 09WE] let O be a sheaf of rings on C_total. Suppose given K_0 ∈ D(O_0) and an isomorphism α : L(f_δ_1^1)^*K_0 → L(f_δ_0^1)^*K_0 satisfying the cocycle condition. Set τ^n_i : [0] → [n], 0 ↦ i and set K_n = Lf_τ^n_n^*K_0. The objects K_n form the members of a cartesian simplicial system of the derived category of modules.","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be a\nsheaf of rings on $\\mathcal{C}_{total}$.\nSuppose given $K_0 \\in D(\\mathcal{O}_0)$ and an isomorphism\n$$\n\\alpha :\nL(f_{\\delta_1^1})^*K_0\n\\longrightarrow\nL(f_{\\delta_0^1})^*K_0\n$$\nsatisfying the cocycle condition. Set $\\tau^n_i : [0] \\to [n]$, $0 \\mapsto i$\nand set $K_n = Lf_{\\tau^n_n}^*K_0$. The objects $K_n$ form the members of a \ncartesian simplicial system of the derived category of modules.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMF","source_file":"spaces-simplicial.tex","source_line":3184,"source_end_line":3198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3184-L3198","statement_sha256":"67cb40e0f7893b2078e7c6d49de19eedd10b4b3298ccbcb28164add96cabd131","origin":"The Stacks Project","memory_eligible":false,"source_rank":12788,"rank":12788,"depth":3,"x":629.332,"y":1467.52,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9Q","tag":"0D9Q","title":"Simplicial systems of the derived category: modules · Lemma 0D9Q","summary":"In Situation [Tag 09WE] let O be a sheaf of rings on C_total. Let K be an object of D(C_total). Set X_n = (g_n!O_n) ⊗^L_O K and Y_n = (g_n!O_n → … → g_0!O_0)[-n] ⊗^L_O K as objects of D(O) where the maps are as in Lemma [Tag 09WI]. With the evident canonical maps Y_n → X_n and Y_0 → Y_1[1] → Y_2[2] → … we have • the distinguished triangles Y_n → X_n → Y_n - 1 → Y_n[1] define a Postnikov system (Derived Categories, Definition [Tag 0D7Z]) for … → X_2 → X_1 → X_0, • K =…","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$\nbe a sheaf of rings on $\\mathcal{C}_{total}$. Let $K$ be\nan object of $D(\\mathcal{C}_{total})$. Set\n$$\nX_n = (g_{n!}\\mathcal{O}_n)\n\\otimes^\\mathbf{L}_\\mathcal{O} K\n\\quad\\text{and}\\quad\nY_n =\n(g_{n!}\\mathcal{O}_n \\to \\ldots \\to g_{0!}\\mathcal{O}_0)[-n]\n\\otimes^\\mathbf{L}_\\mathcal{O} K\n$$\nas objects of $D(\\mathcal{O})$ where the maps are\nas in Lemma \\ref{lemma-simplicial-resolution-Z-site}.\nWith the evident canonical maps $Y_n \\to X_n$ and\n$Y_0 \\to Y_1[1] \\to Y_2[2] \\to \\ldots$ we have\n\\begin{enumerate}\n\\item the distinguished triangles $Y_n \\to X_n \\to Y_{n - 1} \\to Y_n[1]$\ndefine a Postnikov system\n(Derived Categories, Definition \\ref{derived-definition-postnikov-system})\nfor $\\ldots \\to X_2 \\to X_1 \\to X_0$,\n\\item $K = \\text{hocolim} Y_n[n]$ in $D(\\mathcal{O})$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9Q","source_file":"spaces-simplicial.tex","source_line":3230,"source_end_line":3254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3230-L3254","statement_sha256":"073de1124831f069b3b5d8642dce8bff66503475e6b7cc2596b85c414f0982be","origin":"The Stacks Project","memory_eligible":false,"source_rank":12789,"rank":12789,"depth":9,"x":823.973,"y":1668.974,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9R","tag":"0D9R","title":"Simplicial systems of the derived category: modules · Lemma 0D9R","summary":"In Situation [Tag 09WE] let O be a sheaf of rings on C_total. If K, K' ∈ D(O). Assume • f_φ^-1O_n → O_m is flat for φ : [m] → [n], • K is cartesian, • Hom(K_i[i], K'_i) = 0 for i > 0, and • Hom(K_i[i + 1], K'_i) = 0 for i ≥ 0. Then any map K → K' which induces the zero map K_0 → K'_0 is zero.","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be\na sheaf of rings on $\\mathcal{C}_{total}$.\nIf $K, K' \\in D(\\mathcal{O})$.\nAssume\n\\begin{enumerate}\n\\item $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for\n$\\varphi : [m] \\to [n]$,\n\\item $K$ is cartesian,\n\\item $\\Hom(K_i[i], K'_i) = 0$ for $i > 0$, and\n\\item $\\Hom(K_i[i + 1], K'_i) = 0$ for $i \\geq 0$.\n\\end{enumerate}\nThen any map $K \\to K'$ which induces the zero map $K_0 \\to K'_0$ is zero.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9R","source_file":"spaces-simplicial.tex","source_line":3275,"source_end_line":3289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3275-L3289","statement_sha256":"78a575a5e6b753ef6e76aa5d0be552a719a721289abd64f87ea5f7af00aee744","origin":"The Stacks Project","memory_eligible":false,"source_rank":12790,"rank":12790,"depth":11,"x":518.286,"y":1630.85,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9S","tag":"0D9S","title":"Simplicial systems of the derived category: modules · Lemma 0D9S","summary":"In Situation [Tag 09WE] let O be a sheaf of rings on C_total. If K, K' ∈ D(O). Assume • f_φ^-1O_n → O_m is flat for φ : [m] → [n], • K is cartesian, • Hom(K_i[i - 1], K'_i) = 0 for i > 1. Then any map (K_n → K'_n) between the associated simplicial systems of K and K' comes from a map K → K' in D(O).","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be\na sheaf of rings on $\\mathcal{C}_{total}$.\nIf $K, K' \\in D(\\mathcal{O})$.\nAssume\n\\begin{enumerate}\n\\item $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for\n$\\varphi : [m] \\to [n]$,\n\\item $K$ is cartesian,\n\\item $\\Hom(K_i[i - 1], K'_i) = 0$ for $i > 1$.\n\\end{enumerate}\nThen any map $\\{K_n \\to K'_n\\}$ between the associated simplicial systems \nof $K$ and $K'$ comes from a map $K \\to K'$ in $D(\\mathcal{O})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9S","source_file":"spaces-simplicial.tex","source_line":3298,"source_end_line":3312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3298-L3312","statement_sha256":"6f8705c94aae9bd0cb53643607b439b6bf616b5106ba30a01afd8ad478bb4658","origin":"The Stacks Project","memory_eligible":false,"source_rank":12791,"rank":12791,"depth":11,"x":774.486,"y":1485.432,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9T","tag":"0D9T","title":"Simplicial systems of the derived category: modules · Lemma 0D9T","summary":"In Situation [Tag 09WE] let O be a sheaf of rings on C_total. Let (K_n, K_φ) be a simplicial system of the derived category of modules. Assume • f_φ^-1O_n → O_m is flat for φ : [m] → [n], • (K_n, K_φ) is cartesian, • Hom(K_i[t], K_i) = 0 for i ≥ 0 and t > 0. Then there exists a cartesian object K of D(O) whose associated simplicial system is isomorphic to (K_n, K_φ).","statement_latex":"In Situation \\ref{situation-simplicial-site} let $\\mathcal{O}$ be\na sheaf of rings on $\\mathcal{C}_{total}$. Let\n$(K_n, K_\\varphi)$ be a simplicial system of the derived category\nof modules. Assume\n\\begin{enumerate}\n\\item $f_\\varphi^{-1}\\mathcal{O}_n \\to \\mathcal{O}_m$ is flat for\n$\\varphi : [m] \\to [n]$,\n\\item $(K_n, K_\\varphi)$ is cartesian,\n\\item $\\Hom(K_i[t], K_i) = 0$ for $i \\geq 0$ and $t > 0$.\n\\end{enumerate}\nThen there exists a cartesian object $K$ of $D(\\mathcal{O})$\nwhose associated simplicial system is isomorphic to $(K_n, K_\\varphi)$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Simplicial systems of the derived category: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9T","source_file":"spaces-simplicial.tex","source_line":3321,"source_end_line":3335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3321-L3335","statement_sha256":"49fbe99cbad444401dc293911e38213bfe01f3767b39b712be47d322181bd085","origin":"The Stacks Project","memory_eligible":false,"source_rank":12792,"rank":12792,"depth":11,"x":702.471,"y":1738.166,"cluster":"geometry-of-spaces"},{"id":"stacks:0D85","tag":"0D85","title":"The site associated to a semi-representable object · Lemma 0D85","summary":"Let C be a site. • For K in SR(C) the functor j : C/K → C is continuous, cocontinuous, and has property P of Sites, Remark [Tag 09W7]. • For f : K → L in SR(C) the functor v : C/K → C/L (see above) is continuous, cocontinuous, and has property P of Sites, Remark [Tag 09W7].","statement_latex":"Let $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item For $K$ in $\\text{SR}(\\mathcal{C})$ the functor\n$j : \\mathcal{C}/K \\to \\mathcal{C}$ is continuous,\ncocontinuous, and has property P of\nSites, Remark \\ref{sites-remark-cartesian-cocontinuous}.\n\\item For $f : K \\to L$ in $\\text{SR}(\\mathcal{C})$\nthe functor $v : \\mathcal{C}/K \\to \\mathcal{C}/L$ (see above)\nis continuous, cocontinuous, and has property P of\nSites, Remark \\ref{sites-remark-cartesian-cocontinuous}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"The site associated to a semi-representable object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D85","source_file":"spaces-simplicial.tex","source_line":3464,"source_end_line":3477,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3464-L3477","statement_sha256":"b0764543bfdc6552aadb806e9b13c182ae88646268a475b3c93625d1cb2058d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12793,"rank":12793,"depth":0,"x":552.255,"y":1510.824,"cluster":"geometry-of-spaces"},{"id":"stacks:0D86","tag":"0D86","title":"The site associated to a semi-representable object · Lemma 0D86","summary":"Let C be a site and K in SR(C). For F in Sh(C) we have j_*j^-1F = SheafHom(F(K)^\\#, F) where F is as in Hypercoverings, Definition [Tag 01G1].","statement_latex":"Let $\\mathcal{C}$ be a site and $K$ in $\\text{SR}(\\mathcal{C})$.\nFor $\\mathcal{F}$ in $\\Sh(\\mathcal{C})$ we have\n$$\nj_*j^{-1}\\mathcal{F} = \\SheafHom(F(K)^\\#, \\mathcal{F})\n$$\nwhere $F$ is as in\nHypercoverings, Definition \\ref{hypercovering-definition-SR-F}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"The site associated to a semi-representable object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D86","source_file":"spaces-simplicial.tex","source_line":3490,"source_end_line":3499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3490-L3499","statement_sha256":"230f984aa916612f75302e9dddf44a0b0b5057d4dc70d197ed0aa98bd2bb3054","origin":"The Stacks Project","memory_eligible":false,"source_rank":12794,"rank":12794,"depth":9,"x":845.996,"y":1593.268,"cluster":"geometry-of-spaces"},{"id":"stacks:0D87","tag":"0D87","title":"The site associated to a semi-representable object · Lemma 0D87","summary":"Let C be a site. • For K in SR(C) the functor j_! gives an equivalence Sh(C/K) → Sh(C)/F(K)^\\# where F is as in Hypercoverings, Definition [Tag 01G1]. • The functor j^-1 : Sh(C) → Sh(C/K) corresponds via the identification of (1) with F ↦ (F × F(K)^\\# → F(K)^\\#). • For f : K → L in SR(C) the functor f^-1 corresponds via the identifications of (1) to the functor Sh(C)/F(L)^\\# → Sh(C)/F(K)^\\#, (G → F(L)^\\#) ↦ (G ×_F(L)^\\# F(K)^\\# → F(K)^\\#).","statement_latex":"Let $\\mathcal{C}$ be a site.\n\\begin{enumerate}\n\\item For $K$ in $\\text{SR}(\\mathcal{C})$ the functor $j_!$\ngives an equivalence $\\Sh(\\mathcal{C}/K) \\to \\Sh(\\mathcal{C})/F(K)^\\#$\nwhere $F$ is as in\nHypercoverings, Definition \\ref{hypercovering-definition-SR-F}.\n\\item The functor $j^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh(\\mathcal{C}/K)$\ncorresponds via the identification of (1) with\n$\\mathcal{F} \\mapsto (\\mathcal{F} \\times F(K)^\\# \\to F(K)^\\#)$.\n\\item For $f : K \\to L$ in $\\text{SR}(\\mathcal{C})$ the functor\n$f^{-1}$ corresponds via the identifications of (1) to the functor\n$\\Sh(\\mathcal{C})/F(L)^\\# \\to \\Sh(\\mathcal{C})/F(K)^\\#$,\n$(\\mathcal{G} \\to F(L)^\\#) \\mapsto\n(\\mathcal{G} \\times_{F(L)^\\#} F(K)^\\# \\to F(K)^\\#)$.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"The site associated to a semi-representable object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D87","source_file":"spaces-simplicial.tex","source_line":3518,"source_end_line":3535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3518-L3535","statement_sha256":"b47fdf7c3a94181b6707cfb7587fb22639e5842575b50715aa5c35ab089610ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":12795,"rank":12795,"depth":9,"x":562.95,"y":1699.205,"cluster":"geometry-of-spaces"},{"id":"stacks:0D88","tag":"0D88","title":"The site associated to a semi-representable object · Lemma 0D88","summary":"Let C be a site. For K in SR(C) the functor j^-1 sends injective abelian sheaves to injective abelian sheaves. Similarly, the functor j^-1 sends K-injective complexes of abelian sheaves to K-injective complexes of abelian sheaves.","statement_latex":"Let $\\mathcal{C}$ be a site. For $K$ in $\\text{SR}(\\mathcal{C})$\nthe functor $j^{-1}$ sends injective abelian sheaves to injective\nabelian sheaves. Similarly, the functor $j^{-1}$ sends K-injective\ncomplexes of abelian sheaves to K-injective complexes of\nabelian sheaves.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"The site associated to a semi-representable object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D88","source_file":"spaces-simplicial.tex","source_line":3557,"source_end_line":3564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3557-L3564","statement_sha256":"c182e046028b41ccc6dac008353e79f69a0899970236bbf8b4039dde2cba8c38","origin":"The Stacks Project","memory_eligible":false,"source_rank":12796,"rank":12796,"depth":11,"x":686.536,"y":1460.36,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8B","tag":"0D8B","title":"The site associate to a simplicial semi-representable object · Lemma 0D8B","summary":"Let C be a site. Let K be a simplicial object of SR(C). The localization functor j_0 : C/K_0 → C defines an augmentation a_0 : Sh(C/K_0) → Sh(C), as in case (B) of Remark [Tag 0D6Z]. The corresponding morphisms of topoi a_n : Sh(C/K_n) → Sh(C), a : Sh((C/K)_total) → Sh(C) of Lemma [Tag 0D70] are equal to the morphisms of topoi associated to the continuous and cocontinuous localization functors j_n : C/K_n → C and j_total : (C/K)_total → C.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $K$ be a simplicial object of\n$\\text{SR}(\\mathcal{C})$. The localization functor\n$j_0 : \\mathcal{C}/K_0 \\to \\mathcal{C}$ defines an augmentation\n$a_0 : \\Sh(\\mathcal{C}/K_0) \\to \\Sh(\\mathcal{C})$, as in case (B) of\nRemark \\ref{remark-augmentation-site}.\nThe corresponding morphisms of topoi\n$$\na_n : \\Sh(\\mathcal{C}/K_n) \\longrightarrow \\Sh(\\mathcal{C}),\\quad\na : \\Sh((\\mathcal{C}/K)_{total}) \\longrightarrow \\Sh(\\mathcal{C})\n$$\nof Lemma \\ref{lemma-augmentation-site}\nare equal to the morphisms of topoi associated to the\ncontinuous and cocontinuous localization functors\n$j_n : \\mathcal{C}/K_n \\to \\mathcal{C}$ and\n$j_{total} : (\\mathcal{C}/K)_{total} \\to \\mathcal{C}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"The site associate to a simplicial semi-representable object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8B","source_file":"spaces-simplicial.tex","source_line":3739,"source_end_line":3756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3739-L3756","statement_sha256":"0f539f53127ab886b601f3d3726dc83d848889990078bf2fed6b87034e940892","origin":"The Stacks Project","memory_eligible":false,"source_rank":12797,"rank":12797,"depth":8,"x":787.53,"y":1706.732,"cluster":"geometry-of-spaces"},{"id":"stacks:09WM","tag":"09WM","title":"The site associate to a simplicial semi-representable object · Lemma 09WM","summary":"With assumption and notation as in Lemma [Tag 0D8B] we have the following properties: • there is a functor a^Sh_! : Sh((C/K)_total) → Sh(C) left adjoint to a^-1 : Sh(C) → Sh((C/K)_total), • there is a functor a_! : Ab((C/K)_total) → Ab(C) left adjoint to a^-1 : Ab(C) → Ab((C/K)_total), • the functor a^-1 associates to F in Sh(C) the sheaf on (C/K)_total which in degree n is equal to a_n^-1F, • the functor a_* associates to G in Ab((C/K)_total) the equalizer of the two…","statement_latex":"With assumption and notation as in\nLemma \\ref{lemma-augmentation-simplicial-semi-representable}\nwe have the following properties:\n\\begin{enumerate}\n\\item there is a functor\n$a^{Sh}_! : \\Sh((\\mathcal{C}/K)_{total}) \\to \\Sh(\\mathcal{C})$\nleft adjoint to $a^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh((\\mathcal{C}/K)_{total})$,\n\\item there is a functor\n$a_! : \\textit{Ab}((\\mathcal{C}/K)_{total}) \\to \\textit{Ab}(\\mathcal{C})$\nleft adjoint to\n$a^{-1} : \\textit{Ab}(\\mathcal{C}) \\to \\textit{Ab}((\\mathcal{C}/K)_{total})$,\n\\item the functor $a^{-1}$ associates to\n$\\mathcal{F}$ in $\\Sh(\\mathcal{C})$ the sheaf on $(\\mathcal{C}/K)_{total}$\nwhich in degree $n$ is equal to $a_n^{-1}\\mathcal{F}$,\n\\item the functor $a_*$ associates to $\\mathcal{G}$ in\n$\\textit{Ab}((\\mathcal{C}/K)_{total})$ the equalizer of the two maps\n$j_{0, *}\\mathcal{G}_0 \\to j_{1, *}\\mathcal{G}_1$,\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"The site associate to a simplicial semi-representable object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WM","source_file":"spaces-simplicial.tex","source_line":3765,"source_end_line":3785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3765-L3785","statement_sha256":"1d1f1db6627f26ed00c3487cd98abb5272a5d6b5c305c02f66020b8554e8adb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12798,"rank":12798,"depth":9,"x":514.793,"y":1582.303,"cluster":"geometry-of-spaces"},{"id":"stacks:0DC0","tag":"0DC0","title":"The site associate to a simplicial semi-representable object · Lemma 0DC0","summary":"Let C be a site. Let K be a simplicial object of SR(C). Let U/U_n, i be an object of C/K_n. Let F ∈ Ab((C/K)_total). Then H^p(U, F) = H^p(U, F_n, i) where • on the left hand side U is viewed as an object of C_total, and • on the right hand side F_n, i is the ith component of the sheaf F_n on C/K_n in the decomposition Sh(C/K_n) = ∏ Sh(C/U_n, i) of Section [Tag 09WK].","statement_latex":"Let $\\mathcal{C}$ be a site. Let $K$ be a simplicial object of\n$\\text{SR}(\\mathcal{C})$. Let $U/U_{n, i}$ be an object of\n$\\mathcal{C}/K_n$. Let\n$\\mathcal{F} \\in \\textit{Ab}((\\mathcal{C}/K)_{total})$.\nThen\n$$\nH^p(U, \\mathcal{F}) = H^p(U, \\mathcal{F}_{n, i})\n$$\nwhere\n\\begin{enumerate}\n\\item on the left hand side $U$ is viewed as an object of\n$\\mathcal{C}_{total}$, and\n\\item on the right hand side $\\mathcal{F}_{n, i}$ is the $i$th\ncomponent of the sheaf $\\mathcal{F}_n$ on $\\mathcal{C}/K_n$\nin the decomposition $\\Sh(\\mathcal{C}/K_n) = \\prod \\Sh(\\mathcal{C}/U_{n, i})$\nof Section \\ref{section-semi-representable}.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"The site associate to a simplicial semi-representable object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DC0","source_file":"spaces-simplicial.tex","source_line":3798,"source_end_line":3817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3798-L3817","statement_sha256":"e27a52004346adea8da2178d831407861b7d1f3ef19a723e8c652b278b26d187","origin":"The Stacks Project","memory_eligible":false,"source_rank":12799,"rank":12799,"depth":10,"x":816.122,"y":1519.265,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8E","tag":"0D8E","title":"Cohomological descent for hypercoverings · Lemma 0D8E","summary":"Let C be a site with equalizers and fibre products. Let K be a hypercovering. Then • a^-1 : Sh(C) → Sh((C/K)_total) is fully faithful with essential image the cartesian sheaves of sets, • a^-1 : Ab(C) → Ab((C/K)_total) is fully faithful with essential image the cartesian sheaves of abelian groups. In both cases a_* provides the quasi-inverse functor.","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $K$ be a hypercovering. Then\n\\begin{enumerate}\n\\item $a^{-1} : \\Sh(\\mathcal{C}) \\to \\Sh((\\mathcal{C}/K)_{total})$\nis fully faithful with essential image the cartesian sheaves of sets,\n\\item $a^{-1} : \\textit{Ab}(\\mathcal{C}) \\to\n\\textit{Ab}((\\mathcal{C}/K)_{total})$\nis fully faithful with essential image the cartesian sheaves\nof abelian groups.\n\\end{enumerate}\nIn both cases $a_*$ provides the quasi-inverse functor.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8E","source_file":"spaces-simplicial.tex","source_line":3949,"source_end_line":3962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L3949-L3962","statement_sha256":"62babd9c62692095a046f6eeec07a305754b92de5c01746f2b15998dfb0237d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12800,"rank":12800,"depth":10,"x":644.532,"y":1736.841,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8F","tag":"0D8F","title":"Cohomological descent for hypercoverings · Lemma 0D8F","summary":"Let C be a site with equalizers and fibre products. Let K be a hypercovering. The v Cech complex of Lemma [Tag 0D79] associated to a^-1F a_0, *a_0^-1F → a_1, *a_1^-1F → a_2, *a_2^-1F → … is equal to the complex SheafHom(s(Z_F(K)^\\#), F). Here s(Z_F(K)^\\#) is as in Hypercoverings, Definition [Tag 01GB].","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $K$ be a hypercovering. The {\\v C}ech complex\nof Lemma \\ref{lemma-augmentation-cech-complex} associated to\n$a^{-1}\\mathcal{F}$\n$$\na_{0, *}a_0^{-1}\\mathcal{F} \\to a_{1, *}a_1^{-1}\\mathcal{F} \\to\na_{2, *}a_2^{-1}\\mathcal{F} \\to \\ldots\n$$\nis equal to the complex $\\SheafHom(s(\\mathbf{Z}_{F(K)}^\\#), \\mathcal{F})$.\nHere $s(\\mathbf{Z}_{F(K)}^\\#)$ is as in\nHypercoverings, Definition \\ref{hypercovering-definition-homology}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8F","source_file":"spaces-simplicial.tex","source_line":4136,"source_end_line":4149,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4136-L4149","statement_sha256":"92f081ab4d46fcb281b54a0b4fa0951e91af28b2a35271aa570fbf41f99b402d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12801,"rank":12801,"depth":10,"x":596.066,"y":1478.908,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8G","tag":"0D8G","title":"Cohomological descent for hypercoverings · Lemma 0D8G","summary":"Let C be a site with equalizers and fibre products. Let K be a hypercovering. For E ∈ D(C) the map E → Ra_*a^-1E is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $K$ be a hypercovering. For\n$E \\in D(\\mathcal{C})$ the map\n$$\nE \\longrightarrow Ra_*a^{-1}E\n$$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8G","source_file":"spaces-simplicial.tex","source_line":4167,"source_end_line":4176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4167-L4176","statement_sha256":"4beafe1cb6d31005be278cba148786be2418afdc37b6f85a2405036bbed1443e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12802,"rank":12802,"depth":12,"x":839.351,"y":1641.687,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8H","tag":"0D8H","title":"Cohomological descent for hypercoverings · Lemma 0D8H","summary":"Let C be a site with equalizers and fibre products. Let K be a hypercovering. Then we have a canonical isomorphism RΓ(C, E) = RΓ((C/K)_total, a^-1E) for E ∈ D(C).","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $K$ be a hypercovering.\nThen we have a canonical isomorphism\n$$\nR\\Gamma(\\mathcal{C}, E) =\nR\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}E)\n$$\nfor $E \\in D(\\mathcal{C})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8H","source_file":"spaces-simplicial.tex","source_line":4350,"source_end_line":4360,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4350-L4360","statement_sha256":"21bc19b90f3bdbdac9fb87a5602f426dca24e86147a800d34c2981d69cbb8635","origin":"The Stacks Project","memory_eligible":false,"source_rank":12803,"rank":12803,"depth":13,"x":528.896,"y":1659.711,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8I","tag":"0D8I","title":"Cohomological descent for hypercoverings · Lemma 0D8I","summary":"Let C be a site with equalizers and fibre products. Let K be a hypercovering. Let A ⊂ Ab((C/K)_total) denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor a^-1 defines an equivalence D^+(C) → D_A^+((C/K)_total) with quasi-inverse Ra_*.","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $K$ be a hypercovering.\nLet $\\mathcal{A} \\subset \\textit{Ab}((\\mathcal{C}/K)_{total})$\ndenote the weak Serre subcategory of cartesian abelian sheaves.\nThen the functor $a^{-1}$ defines an equivalence\n$$\nD^+(\\mathcal{C}) \\longrightarrow D_\\mathcal{A}^+((\\mathcal{C}/K)_{total})\n$$\nwith quasi-inverse $Ra_*$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8I","source_file":"spaces-simplicial.tex","source_line":4369,"source_end_line":4380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4369-L4380","statement_sha256":"71a131e2d82950e8bfb1ee8422e68fa96c0689439ec0bb7b0e8578c39b2fd516","origin":"The Stacks Project","memory_eligible":false,"source_rank":12804,"rank":12804,"depth":22,"x":743.436,"y":1470.164,"cluster":"geometry-of-spaces"},{"id":"stacks:0D9Z","tag":"0D9Z","title":"Cohomological descent for hypercoverings: modules · Lemma 0D9Z","summary":"Let C be a site with equalizers and fibre products. Let O_C be a sheaf of rings. Let K be a hypercovering. With notation as above a^* : Mod(O_C) → Mod(O) is fully faithful with essential image the cartesian O-modules. The functor a_* provides the quasi-inverse.","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $K$ be a hypercovering. With notation as above\n$$\na^* : \\textit{Mod}(\\mathcal{O}_\\mathcal{C}) \\to \\textit{Mod}(\\mathcal{O})\n$$\nis fully faithful with essential image the cartesian $\\mathcal{O}$-modules.\nThe functor $a_*$ provides the quasi-inverse.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D9Z","source_file":"spaces-simplicial.tex","source_line":4474,"source_end_line":4484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4474-L4484","statement_sha256":"d3b297ab51d260b31f1ed7e1eb5f1fa1f3f39fe3488f67c72ed7a9a3252e48a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12805,"rank":12805,"depth":11,"x":737.663,"y":1731.801,"cluster":"geometry-of-spaces"},{"id":"stacks:0DA0","tag":"0DA0","title":"Cohomological descent for hypercoverings: modules · Lemma 0DA0","summary":"Let C be a site with equalizers and fibre products. Let O_C be a sheaf of rings. Let K be a hypercovering. For E ∈ D(O_C) the map E → Ra_*La^*E is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $K$ be a hypercovering. For\n$E \\in D(\\mathcal{O}_\\mathcal{C})$ the map\n$$\nE \\longrightarrow Ra_*La^*E\n$$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DA0","source_file":"spaces-simplicial.tex","source_line":4492,"source_end_line":4502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4492-L4502","statement_sha256":"5bba7b8d2c24fc4666a0e0f3313541cb934dcec7d782401848a6382a6841a5e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12806,"rank":12806,"depth":23,"x":531.414,"y":1535.499,"cluster":"geometry-of-spaces"},{"id":"stacks:0DA1","tag":"0DA1","title":"Cohomological descent for hypercoverings: modules · Lemma 0DA1","summary":"Let C be a site with equalizers and fibre products. Let O_C be a sheaf of rings. Let K be a hypercovering. Then we have a canonical isomorphism RΓ(C, E) = RΓ((C/K)_total, La^*E) for E ∈ D(O_C).","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $K$ be a hypercovering.\nThen we have a canonical isomorphism\n$$\nR\\Gamma(\\mathcal{C}, E) =\nR\\Gamma((\\mathcal{C}/K)_{total}, La^*E)\n$$\nfor $E \\in D(\\mathcal{O}_\\mathcal{C})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DA1","source_file":"spaces-simplicial.tex","source_line":4514,"source_end_line":4525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4514-L4525","statement_sha256":"9cf2cde4a0bbf1bd0c36f40782de822dbed6f9d533861aa8094a065157cf9193","origin":"The Stacks Project","memory_eligible":false,"source_rank":12807,"rank":12807,"depth":24,"x":841.517,"y":1563.232,"cluster":"geometry-of-spaces"},{"id":"stacks:0DA2","tag":"0DA2","title":"Cohomological descent for hypercoverings: modules · Lemma 0DA2","summary":"Let C be a site with equalizers and fibre products. Let O_C be a sheaf of rings. Let K be a hypercovering. Let A ⊂ Mod(O) denote the weak Serre subcategory of cartesian O-modules. Then the functor La^* defines an equivalence D^+(O_C) → D_A^+(O) with quasi-inverse Ra_*.","statement_latex":"Let $\\mathcal{C}$ be a site with equalizers and fibre products.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $K$ be a hypercovering.\nLet $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$\ndenote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules.\nThen the functor $La^*$ defines an equivalence\n$$\nD^+(\\mathcal{O}_\\mathcal{C}) \\longrightarrow D_\\mathcal{A}^+(\\mathcal{O})\n$$\nwith quasi-inverse $Ra_*$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DA2","source_file":"spaces-simplicial.tex","source_line":4536,"source_end_line":4548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4536-L4548","statement_sha256":"45e07aca796875be3455dec705bd3b0f098bf815cf1172208f7b0231cb5ac64f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12808,"rank":12808,"depth":24,"x":590.422,"y":1718.821,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8K","tag":"0D8K","title":"Cohomological descent for hypercoverings of an object · Lemma 0D8K","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let K be a hypercovering of X. Then • a^-1 : Sh(C/X) → Sh((C/K)_total) is fully faithful with essential image the cartesian sheaves of sets, • a^-1 : Ab(C/X) → Ab((C/K)_total) is fully faithful with essential image the cartesian sheaves of abelian groups. In both cases a_* provides the quasi-inverse functor.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $K$ be a hypercovering of $X$. Then\n\\begin{enumerate}\n\\item $a^{-1} : \\Sh(\\mathcal{C}/X) \\to \\Sh((\\mathcal{C}/K)_{total})$\nis fully faithful with essential image the cartesian sheaves of sets,\n\\item $a^{-1} : \\textit{Ab}(\\mathcal{C}/X) \\to\n\\textit{Ab}((\\mathcal{C}/K)_{total})$\nis fully faithful with essential image the cartesian sheaves\nof abelian groups.\n\\end{enumerate}\nIn both cases $a_*$ provides the quasi-inverse functor.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings of an object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8K","source_file":"spaces-simplicial.tex","source_line":4599,"source_end_line":4612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4599-L4612","statement_sha256":"cc8078a53a85c4bfbb1ec9deb0d4628849ebaf81e078a3f0ef5d16e4fdb4c572","origin":"The Stacks Project","memory_eligible":false,"source_rank":12809,"rank":12809,"depth":11,"x":650.486,"y":1461.484,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8L","tag":"0D8L","title":"Cohomological descent for hypercoverings of an object · Lemma 0D8L","summary":"Let C be a site with fibre product and X ∈ Ob(C). Let K be a hypercovering of X. For E ∈ D(C/X) the map E → Ra_*a^-1E is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$.\nLet $K$ be a hypercovering of $X$. For\n$E \\in D(\\mathcal{C}/X)$ the map\n$$\nE \\longrightarrow Ra_*a^{-1}E\n$$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings of an object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8L","source_file":"spaces-simplicial.tex","source_line":4621,"source_end_line":4630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4621-L4630","statement_sha256":"f28a5a3a8454b5ed1ae300e4aa1acf0b410556074c1bbd7d175dee62fee16940","origin":"The Stacks Project","memory_eligible":false,"source_rank":12810,"rank":12810,"depth":13,"x":813.221,"y":1685.436,"cluster":"geometry-of-spaces"},{"id":"stacks:09X7","tag":"09X7","title":"Cohomological descent for hypercoverings of an object · Lemma 09X7","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let K be a hypercovering of X. Then we have a canonical isomorphism RΓ(X, E) = RΓ((C/K)_total, a^-1E) for E ∈ D(C/X).","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $K$ be a hypercovering of $X$.\nThen we have a canonical isomorphism\n$$\nR\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/K)_{total}, a^{-1}E)\n$$\nfor $E \\in D(\\mathcal{C}/X)$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings of an object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09X7","source_file":"spaces-simplicial.tex","source_line":4639,"source_end_line":4648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4639-L4648","statement_sha256":"a49c5fa67f090f3140e706786be8bb6e4e7140be7440312ada6f856925597a98","origin":"The Stacks Project","memory_eligible":false,"source_rank":12811,"rank":12811,"depth":14,"x":512.975,"y":1612.6,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8M","tag":"0D8M","title":"Cohomological descent for hypercoverings of an object · Lemma 0D8M","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let K be a hypercovering of X. Let A ⊂ Ab((C/K)_total) denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor a^-1 defines an equivalence D^+(C/X) → D_A^+((C/K)_total) with quasi-inverse Ra_*.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $K$ be a hypercovering of $X$.\nLet $\\mathcal{A} \\subset \\textit{Ab}((\\mathcal{C}/K)_{total})$\ndenote the weak Serre subcategory of cartesian abelian sheaves.\nThen the functor $a^{-1}$ defines an equivalence\n$$\nD^+(\\mathcal{C}/X) \\longrightarrow D_\\mathcal{A}^+((\\mathcal{C}/K)_{total})\n$$\nwith quasi-inverse $Ra_*$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings of an object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8M","source_file":"spaces-simplicial.tex","source_line":4656,"source_end_line":4667,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4656-L4667","statement_sha256":"80b732642a4edefffd9926de6b495dbc8d134ee4e1898e1ca055f4aae8044124","origin":"The Stacks Project","memory_eligible":false,"source_rank":12812,"rank":12812,"depth":23,"x":793.086,"y":1495.882,"cluster":"geometry-of-spaces"},{"id":"stacks:0DA4","tag":"0DA4","title":"Cohomological descent for hypercoverings of an object: modules · Lemma 0DA4","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let O_C be a sheaf of rings. Let K be a hypercovering of X. With notation as above a^* : Mod(O_X) → Mod(O) is fully faithful with essential image the cartesian O-modules. The functor a_* provides the quasi-inverse.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $K$ be a hypercovering of $X$. With notation as above\n$$\na^* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O})\n$$\nis fully faithful with essential image the cartesian $\\mathcal{O}$-modules.\nThe functor $a_*$ provides the quasi-inverse.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings of an object: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DA4","source_file":"spaces-simplicial.tex","source_line":4707,"source_end_line":4717,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4707-L4717","statement_sha256":"4f2db179c890e50778e95471671d3eed516c4065407b429de1cacc767dcb1ae9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12813,"rank":12813,"depth":12,"x":680.341,"y":1741.013,"cluster":"geometry-of-spaces"},{"id":"stacks:0DA5","tag":"0DA5","title":"Cohomological descent for hypercoverings of an object: modules · Lemma 0DA5","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let O_C be a sheaf of rings. Let K be a hypercovering of X. For E ∈ D(O_X) the map E → Ra_*La^*E is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $K$ be a hypercovering of $X$. For\n$E \\in D(\\mathcal{O}_X)$ the map\n$$\nE \\longrightarrow Ra_*La^*E\n$$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings of an object: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DA5","source_file":"spaces-simplicial.tex","source_line":4726,"source_end_line":4736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4726-L4736","statement_sha256":"e93e7d957ddf610907fa73418cbe2f901b402467ae39151692cba418dd7200eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12814,"rank":12814,"depth":24,"x":566.292,"y":1496.16,"cluster":"geometry-of-spaces"},{"id":"stacks:0DA6","tag":"0DA6","title":"Cohomological descent for hypercoverings of an object: modules · Lemma 0DA6","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let O_C be a sheaf of rings. Let K be a hypercovering of X. Then we have a canonical isomorphism RΓ(X, E) = RΓ((C/K)_total, La^*E) for E ∈ D(O_C).","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $K$ be a hypercovering of $X$.\nThen we have a canonical isomorphism\n$$\nR\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/K)_{total}, La^*E)\n$$\nfor $E \\in D(\\mathcal{O}_\\mathcal{C})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings of an object: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DA6","source_file":"spaces-simplicial.tex","source_line":4745,"source_end_line":4755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4745-L4755","statement_sha256":"05511884200ca4c0a2dcbd222cf2a8e37d5d826f69a0bf9faed77e11f38f6254","origin":"The Stacks Project","memory_eligible":false,"source_rank":12815,"rank":12815,"depth":25,"x":847.436,"y":1612.056,"cluster":"geometry-of-spaces"},{"id":"stacks:0DA7","tag":"0DA7","title":"Cohomological descent for hypercoverings of an object: modules · Lemma 0DA7","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let O_C be a sheaf of rings. Let K be a hypercovering of X. Let A ⊂ Mod(O) denote the weak Serre subcategory of cartesian O-modules. Then the functor La^* defines an equivalence D^+(O_X) → D_A^+(O) with quasi-inverse Ra_*.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $K$ be a hypercovering of $X$.\nLet $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$\ndenote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules.\nThen the functor $La^*$ defines an equivalence\n$$\nD^+(\\mathcal{O}_X) \\longrightarrow D_\\mathcal{A}^+(\\mathcal{O})\n$$\nwith quasi-inverse $Ra_*$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Cohomological descent for hypercoverings of an object: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DA7","source_file":"spaces-simplicial.tex","source_line":4764,"source_end_line":4776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4764-L4776","statement_sha256":"134feb9106193bdf52966cd787668ff2fd5b9eaa0bba32499a2af98dd42e354f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12816,"rank":12816,"depth":25,"x":546.774,"y":1686.161,"cluster":"geometry-of-spaces"},{"id":"stacks:0DA8","tag":"0DA8","title":"Hypercovering by a simplicial object of the site · Lemma 0DA8","summary":"Let C be a site with fibre product and X ∈ Ob(C). Let a : U → X be a hypercovering of X in C as defined above. Then • a^-1 : Sh(C/X) → Sh((C/U)_total) is fully faithful with essential image the cartesian sheaves of sets, • a^-1 : Ab(C/X) → Ab((C/U)_total) is fully faithful with essential image the cartesian sheaves of abelian groups. In both cases a_* provides the quasi-inverse functor.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$.\nLet $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above.\nThen\n\\begin{enumerate}\n\\item $a^{-1} : \\Sh(\\mathcal{C}/X) \\to \\Sh((\\mathcal{C}/U)_{total})$\nis fully faithful with essential image the cartesian sheaves of sets,\n\\item $a^{-1} : \\textit{Ab}(\\mathcal{C}/X) \\to\n\\textit{Ab}((\\mathcal{C}/U)_{total})$\nis fully faithful with essential image the cartesian sheaves\nof abelian groups.\n\\end{enumerate}\nIn both cases $a_*$ provides the quasi-inverse functor.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Hypercovering by a simplicial object of the site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DA8","source_file":"spaces-simplicial.tex","source_line":4847,"source_end_line":4861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4847-L4861","statement_sha256":"2efe304b9e1fffa737678f880417e48a6b6e22a0efe9e34b89eea25014851ec8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12817,"rank":12817,"depth":12,"x":708.965,"y":1460.801,"cluster":"geometry-of-spaces"},{"id":"stacks:0D8N","tag":"0D8N","title":"Hypercovering by a simplicial object of the site · Lemma 0D8N","summary":"Let C be a site with fibre product and X ∈ Ob(C). Let a : U → X be a hypercovering of X in C as defined above. For E ∈ D(C/X) the map E → Ra_*a^-1E is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$.\nLet $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above.\nFor $E \\in D(\\mathcal{C}/X)$ the map\n$$\nE \\longrightarrow Ra_*a^{-1}E\n$$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Hypercovering by a simplicial object of the site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D8N","source_file":"spaces-simplicial.tex","source_line":4868,"source_end_line":4877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4868-L4877","statement_sha256":"f3f67b968a25f0b5faf5cc2bbd96c027d742a68fac706c9936c18c85dd1681b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12818,"rank":12818,"depth":14,"x":770.627,"y":1719.139,"cluster":"geometry-of-spaces"},{"id":"stacks:09X9","tag":"09X9","title":"Hypercovering by a simplicial object of the site · Lemma 09X9","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let a : U → X be a hypercovering of X in C as defined above. Then we have a canonical isomorphism RΓ(X, E) = RΓ((C/U)_total, a^-1E) for E ∈ D(C/X).","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above.\nThen we have a canonical isomorphism\n$$\nR\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/U)_{total}, a^{-1}E)\n$$\nfor $E \\in D(\\mathcal{C}/X)$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Hypercovering by a simplicial object of the site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09X9","source_file":"spaces-simplicial.tex","source_line":4884,"source_end_line":4893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4884-L4893","statement_sha256":"50c2cb983e61cfd95079385cad2d053217a36fee586823483c933d7953ffa058","origin":"The Stacks Project","memory_eligible":false,"source_rank":12819,"rank":12819,"depth":15,"x":517.283,"y":1563.555,"cluster":"geometry-of-spaces"},{"id":"stacks:0DA9","tag":"0DA9","title":"Hypercovering by a simplicial object of the site · Lemma 0DA9","summary":"Let C be a site with fibre product and X ∈ Ob(C). Let a : U → X be a hypercovering of X in C as defined above. Let A ⊂ Ab((C/U)_total) denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor a^-1 defines an equivalence D^+(C/X) → D_A^+((C/U)_total) with quasi-inverse Ra_*.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre product and $X \\in \\Ob(\\mathcal{C})$.\nLet $a : U \\to X$ be a hypercovering of $X$ in $\\mathcal{C}$ as defined above.\nLet $\\mathcal{A} \\subset \\textit{Ab}((\\mathcal{C}/U)_{total})$\ndenote the weak Serre subcategory of cartesian abelian sheaves.\nThen the functor $a^{-1}$ defines an equivalence\n$$\nD^+(\\mathcal{C}/X) \\longrightarrow D_\\mathcal{A}^+((\\mathcal{C}/U)_{total})\n$$\nwith quasi-inverse $Ra_*$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Hypercovering by a simplicial object of the site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DA9","source_file":"spaces-simplicial.tex","source_line":4900,"source_end_line":4911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4900-L4911","statement_sha256":"c50efb79da5e817da33e93a81eac87c258cf97481a18ec162de8d83b5fbe5edf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12820,"rank":12820,"depth":24,"x":829.368,"y":1534.512,"cluster":"geometry-of-spaces"},{"id":"stacks:09WL","tag":"09WL","title":"Hypercovering by a simplicial object of the site · Lemma 09WL","summary":"Let U be a simplicial object of a site C with fibre products. • C/U has the structure of a simplicial object in the category whose objects are sites and whose morphisms are morphisms of sites, • the construction of Lemma [Tag 09WC] applied to the structure in (1) reproduces the site (C/U)_total above, • if a : U → X is an augmentation, then a_0 : C/U_0 → C/X is an augmentation as in Remark [Tag 0D6Z] part (A) and gives the same morphism of topoi a : Sh((C/U)_total) →…","statement_latex":"Let $U$ be a simplicial object of a site $\\mathcal{C}$\nwith fibre products.\n\\begin{enumerate}\n\\item $\\mathcal{C}/U$ has the structure of a simplicial object\nin the category whose objects are sites and\nwhose morphisms are morphisms of sites,\n\\item the construction of Lemma \\ref{lemma-simplicial-site-site}\napplied to the structure in (1)\nreproduces the site $(\\mathcal{C}/U)_{total}$ above,\n\\item if $a : U \\to X$ is an augmentation, then\n$a_0 : \\mathcal{C}/U_0 \\to \\mathcal{C}/X$ is an augmentation\nas in Remark \\ref{remark-augmentation-site} part (A) and gives the\nsame morphism of topoi\n$a : \\Sh((\\mathcal{C}/U)_{total}) \\to \\Sh(\\mathcal{C}/X)$\nas the one above.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Hypercovering by a simplicial object of the site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09WL","source_file":"spaces-simplicial.tex","source_line":4918,"source_end_line":4936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4918-L4936","statement_sha256":"28650ca4499ad0a74bc4c9b51a040de0668b4a736aac7210a85164bdeb64e4fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12821,"rank":12821,"depth":10,"x":622.493,"y":1733.111,"cluster":"geometry-of-spaces"},{"id":"stacks:0DAB","tag":"0DAB","title":"Hypercovering by a simplicial object of the site: modules · Lemma 0DAB","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let O_C be a sheaf of rings. Let U be a hypercovering of X in C. With notation as above a^* : Mod(O_X) → Mod(O) is fully faithful with essential image the cartesian O-modules. The functor a_* provides the quasi-inverse.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $U$ be a hypercovering of $X$ in $\\mathcal{C}$. With notation as above\n$$\na^* : \\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O})\n$$\nis fully faithful with essential image the cartesian $\\mathcal{O}$-modules.\nThe functor $a_*$ provides the quasi-inverse.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Hypercovering by a simplicial object of the site: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAB","source_file":"spaces-simplicial.tex","source_line":4991,"source_end_line":5001,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L4991-L5001","statement_sha256":"65e0c6c51781543baf02db13ee85610e67b004b5795e7f2aaf46cc29ac9b939d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12822,"rank":12822,"depth":13,"x":615.328,"y":1469.15,"cluster":"geometry-of-spaces"},{"id":"stacks:0DAC","tag":"0DAC","title":"Hypercovering by a simplicial object of the site: modules · Lemma 0DAC","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let O_C be a sheaf of rings. Let U be a hypercovering of X in C. For E ∈ D(O_X) the map E → Ra_*La^*E is an isomorphism.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $U$ be a hypercovering of $X$ in $\\mathcal{C}$. For\n$E \\in D(\\mathcal{O}_X)$ the map\n$$\nE \\longrightarrow Ra_*La^*E\n$$\nis an isomorphism.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Hypercovering by a simplicial object of the site: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAC","source_file":"spaces-simplicial.tex","source_line":5008,"source_end_line":5018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5008-L5018","statement_sha256":"9a0fd22eb9f14f05d8edcf51ceb462256bb625d999b26a8f63f3c2ab69a317d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12823,"rank":12823,"depth":25,"x":832.991,"y":1659.82,"cluster":"geometry-of-spaces"},{"id":"stacks:0DAD","tag":"0DAD","title":"Hypercovering by a simplicial object of the site: modules · Lemma 0DAD","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let O_C be a sheaf of rings. Let U be a hypercovering of X in C. Then we have a canonical isomorphism RΓ(X, E) = RΓ((C/U)_total, La^*E) for E ∈ D(O_C).","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $U$ be a hypercovering of $X$ in $\\mathcal{C}$.\nThen we have a canonical isomorphism\n$$\nR\\Gamma(X, E) = R\\Gamma((\\mathcal{C}/U)_{total}, La^*E)\n$$\nfor $E \\in D(\\mathcal{O}_\\mathcal{C})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Hypercovering by a simplicial object of the site: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAD","source_file":"spaces-simplicial.tex","source_line":5025,"source_end_line":5035,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5025-L5035","statement_sha256":"5ce34299c4435182fcfcdb559d22f82608cb9d93c986841f7bd50d8790c17f8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12824,"rank":12824,"depth":26,"x":519.001,"y":1642.721,"cluster":"geometry-of-spaces"},{"id":"stacks:0DAE","tag":"0DAE","title":"Hypercovering by a simplicial object of the site: modules · Lemma 0DAE","summary":"Let C be a site with fibre products and X ∈ Ob(C). Let O_C be a sheaf of rings. Let U be a hypercovering of X in C. Let A ⊂ Mod(O) denote the weak Serre subcategory of cartesian O-modules. Then the functor La^* defines an equivalence D^+(O_X) → D_A^+(O) with quasi-inverse Ra_*.","statement_latex":"Let $\\mathcal{C}$ be a site with fibre products and $X \\in \\Ob(\\mathcal{C})$.\nLet $\\mathcal{O}_\\mathcal{C}$ be a sheaf of rings.\nLet $U$ be a hypercovering of $X$ in $\\mathcal{C}$.\nLet $\\mathcal{A} \\subset \\textit{Mod}(\\mathcal{O})$\ndenote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules.\nThen the functor $La^*$ defines an equivalence\n$$\nD^+(\\mathcal{O}_X) \\longrightarrow D_\\mathcal{A}^+(\\mathcal{O})\n$$\nwith quasi-inverse $Ra_*$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Hypercovering by a simplicial object of the site: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAE","source_file":"spaces-simplicial.tex","source_line":5042,"source_end_line":5054,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5042-L5054","statement_sha256":"c6ebe849765668e0519c0de2855a51f80fb4b8074fd1ad627be68276fee6826b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12825,"rank":12825,"depth":26,"x":764.405,"y":1477.082,"cluster":"geometry-of-spaces"},{"id":"stacks:0DC7","tag":"0DC7","title":"Unbounded cohomological descent for hypercoverings · Lemma 0DC7","summary":"Let (C, O_C) be a ringed site. Assume given weak Serre subcategories A_U ⊂ Mod(O_U) satisfying conditions ([Tag 0DC2]), ([Tag 0DC3]), and ([Tag 0DC4]) above. Assume C has equalizers and fibre products and let K be a hypercovering. Let ((C/K)_total, O) be as in Remark [Tag 0D9W]. Let A_total ⊂ Mod(O) denote the weak Serre subcategory of cartesian O-modules F whose restriction F_n is in A_K_n for all n (as defined above). Then the functor La^* defines an equivalence…","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O}_\\mathcal{C})$ be a ringed site.\nAssume given weak Serre subcategories\n$\\mathcal{A}_U \\subset \\textit{Mod}(\\mathcal{O}_U)$\nsatisfying conditions (\\ref{item-restriction}),\n(\\ref{item-local}), and (\\ref{item-bounded-dimension}) above.\nAssume $\\mathcal{C}$ has equalizers and fibre products and\nlet $K$ be a hypercovering.\nLet $((\\mathcal{C}/K)_{total}, \\mathcal{O})$ be as in\nRemark \\ref{remark-augmentation-ringed}.\nLet $\\mathcal{A}_{total} \\subset \\textit{Mod}(\\mathcal{O})$\ndenote the weak Serre subcategory of cartesian $\\mathcal{O}$-modules\n$\\mathcal{F}$ whose restriction $\\mathcal{F}_n$ is in\n$\\mathcal{A}_{K_n}$ for all $n$ (as defined above).\nThen the functor $La^*$ defines an equivalence\n$$\nD_\\mathcal{A}(\\mathcal{O}_\\mathcal{C})\n\\longrightarrow\nD_{\\mathcal{A}_{total}}(\\mathcal{O})\n$$\nwith quasi-inverse $Ra_*$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Unbounded cohomological descent for hypercoverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DC7","source_file":"spaces-simplicial.tex","source_line":5214,"source_end_line":5236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5214-L5236","statement_sha256":"1146a94c740646213e3a7304e2e88b03cbd50ee59529e9414d7606126e0adfa8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12826,"rank":12826,"depth":25,"x":716.626,"y":1738.6,"cluster":"geometry-of-spaces"},{"id":"stacks:0DCA","tag":"0DCA","title":"Glueing complexes · Lemma 0DCA","summary":"In Situation [Tag 0DC9]. Assume negative self-exts of E_U in D(O_u(U)) are zero. Let L be a simplicial object of SR(B). Consider the simplicial object K = u(L) of SR(C) and let ((C/K)_total, O) be as in Remark [Tag 0D9W]. There exists a cartesian object E of D(O) such that writing L_n = (U_n, i)_i ∈ I_n the restriction of E to D(O_C/u(U_n, i)) is E_U_n, i compatibly (see proof for details). Moreover, E is unique up to unique isomorphism.","statement_latex":"In Situation \\ref{situation-locally-given}.\nAssume negative self-exts of $E_U$ in $D(\\mathcal{O}_{u(U)})$ are zero.\nLet $L$ be a simplicial object of $\\text{SR}(\\mathcal{B})$.\nConsider the simplicial object $K = u(L)$ of $\\text{SR}(\\mathcal{C})$\nand let $((\\mathcal{C}/K)_{total}, \\mathcal{O})$ be as in\nRemark \\ref{remark-augmentation-ringed}.\nThere exists a cartesian object $E$ of $D(\\mathcal{O})$\nsuch that writing $L_n = \\{U_{n, i}\\}_{i \\in I_n}$\nthe restriction of $E$ to $D(\\mathcal{O}_{\\mathcal{C}/u(U_{n, i})})$\nis $E_{U_{n, i}}$ compatibly (see proof for details).\nMoreover, $E$ is unique up to unique isomorphism.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Glueing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCA","source_file":"spaces-simplicial.tex","source_line":5370,"source_end_line":5383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5370-L5383","statement_sha256":"68f85c01b415ae739372843caef9203719308133bf43824c5ca4ce780748434d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12827,"rank":12827,"depth":12,"x":541.465,"y":1518.541,"cluster":"geometry-of-spaces"},{"id":"stacks:0DCB","tag":"0DCB","title":"BBD glueing lemma · Lemma 0DCB","summary":"In Situation [Tag 0DC9]. Assume • C has equalizers and fibre products, • there is a morphism of sites f : C → D given by a continuous functor u : D → C such that • D has equalizers and fibre products and u commutes with them, • B is a full subcategory of D and u : B → C is the restriction of u, • every object of D has a covering whose members are objects of B, • for all U in B all negative self-exts of E_U in D(O_u(U)) are zero, and • there exists a t ∈ Z such that…","statement_latex":"In Situation \\ref{situation-locally-given}. Assume\n\\begin{enumerate}\n\\item $\\mathcal{C}$ has equalizers and fibre products,\n\\item there is a morphism of sites $f : \\mathcal{C} \\to \\mathcal{D}$\ngiven by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$\nsuch that\n\\begin{enumerate}\n\\item $\\mathcal{D}$ has equalizers and fibre products and $u$\ncommutes with them,\n\\item $\\mathcal{B}$ is a full subcategory of $\\mathcal{D}$\nand $u : \\mathcal{B} \\to \\mathcal{C}$ is the restriction of $u$,\n\\item every object of $\\mathcal{D}$ has a covering whose members\nare objects of $\\mathcal{B}$,\n\\end{enumerate}\n\\item for all $U$ in $\\mathcal{B}$ all negative self-exts of $E_U$\nin $D(\\mathcal{O}_{u(U)})$ are zero, and\n\\item there exists a $t \\in \\mathbf{Z}$ such that $H^i(E_U) = 0$ for $i < t$\nand $U \\in \\Ob(\\mathcal{B})$.\n\\end{enumerate}\nThen there exists a solution unique up to unique isomorphism.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Glueing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCB","source_file":"spaces-simplicial.tex","source_line":5412,"source_end_line":5434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5412-L5434","statement_sha256":"9f11ef3d9acce22be910e1d1d681adaee054745b0314687d0c1aeb45b5edee2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12828,"rank":12828,"depth":25,"x":847.744,"y":1581.45,"cluster":"geometry-of-spaces"},{"id":"stacks:0DCC","tag":"0DCC","title":"Unbounded BBD glueing lemma · Lemma 0DCC","summary":"In Situation [Tag 0DC9]. Assume • C has equalizers and fibre products, • there is a morphism of sites f : C → D given by a continuous functor u : D → C such that • D has equalizers and fibre products and u commutes with them, • B is a full subcategory of D and u : B → C is the restriction of u, • every object of D has a covering whose members are objects of B, • all negative self-exts of E_U in D(O_u(U)) are zero, and • there exist weak Serre subcategories A_U ⊂ Mod(O_U)…","statement_latex":"In Situation \\ref{situation-locally-given}. Assume\n\\begin{enumerate}\n\\item $\\mathcal{C}$ has equalizers and fibre products,\n\\item there is a morphism of sites $f : \\mathcal{C} \\to \\mathcal{D}$\ngiven by a continuous functor $u : \\mathcal{D} \\to \\mathcal{C}$\nsuch that\n\\begin{enumerate}\n\\item $\\mathcal{D}$ has equalizers and fibre products and $u$\ncommutes with them,\n\\item $\\mathcal{B}$ is a full subcategory of $\\mathcal{D}$\nand $u : \\mathcal{B} \\to \\mathcal{C}$ is the restriction of $u$,\n\\item every object of $\\mathcal{D}$ has a covering whose members\nare objects of $\\mathcal{B}$,\n\\end{enumerate}\n\\item all negative self-exts of $E_U$ in $D(\\mathcal{O}_{u(U)})$ are zero, and\n\\item there exist weak Serre subcategories\n$\\mathcal{A}_U \\subset \\textit{Mod}(\\mathcal{O}_U)$ for all\n$U \\in \\Ob(\\mathcal{C})$ satisfying conditions (\\ref{item-restriction}),\n(\\ref{item-local}), and (\\ref{item-bounded-dimension}),\n\\item $E_U \\in D_{\\mathcal{A}_U}(\\mathcal{O}_U)$.\n\\end{enumerate}\nThen there exists a solution unique up to unique isomorphism.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Glueing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCC","source_file":"spaces-simplicial.tex","source_line":5491,"source_end_line":5515,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5491-L5515","statement_sha256":"21d6c0d496427ee198b8d43229b9b821aadbf99c9f3ae5f58afef9db3391711e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12829,"rank":12829,"depth":26,"x":571.172,"y":1708.915,"cluster":"geometry-of-spaces"},{"id":"stacks:0GMG","tag":"0GMG","title":"Glueing complexes · Lemma 0GMG","summary":"Email of Martin Olsson dated Sep 9, 2021. Let (C, O_C) be a ringed site. Assume C has fibre products. Let (U_i → X)_i ∈ I be a covering in C. For i ∈ I let E_i be an object of D(O_U_i) and for i, j ∈ I let ρ_ij : E_i|_C/U_ij → E_j|_C/U_ij be an isomorphism in D(O_U_ij) where U_ij = U_i ×_X U_j. Assume • the ρ_ij satisfy the cocycle condition on U_i ×_X U_j ×_X U_k for all i, j, k ∈ I, • SheafExt^p_O_U_i(E_i, E_i) = 0 for all p < 0 and i ∈ I, and • there exists a t ∈ Z…","statement_latex":"\\begin{reference}\nEmail of Martin Olsson dated Sep 9, 2021.\n\\end{reference}\nLet $(\\mathcal{C}, \\mathcal{O}_\\mathcal{C})$ be a ringed site.\nAssume $\\mathcal{C}$ has fibre products.\nLet $\\{U_i \\to X\\}_{i \\in I}$ be a covering in $\\mathcal{C}$.\nFor $i \\in I$ let $E_i$ be an object of $D(\\mathcal{O}_{U_i})$ and\nfor $i, j \\in I$ let\n$$\n\\rho_{ij} : E_i|_{\\mathcal{C}/U_{ij}} \\longrightarrow E_j|_{\\mathcal{C}/U_{ij}}\n$$\nbe an isomorphism in $D(\\mathcal{O}_{U_{ij}})$ where\n$U_{ij} = U_i \\times_X U_j$. Assume\n\\begin{enumerate}\n\\item the $\\rho_{ij}$ satisfy the cocycle condition on\n$U_i \\times_X U_j \\times_X U_k$ for all $i, j, k \\in I$,\n\\item $\\SheafExt^p_{\\mathcal{O}_{U_i}}(E_i, E_i) = 0$\nfor all $p < 0$ and $i \\in I$, and\n\\item there exists a $t \\in \\mathbf{Z}$ such that\n$H^p(E_i) = 0$ for $p < t$ and all $i \\in I$.\n\\end{enumerate}\nThen there exists a unique pair $(E, \\rho_i)$ where $E$ is an object of \n$D(\\mathcal{O}_X)$ and $\\rho_i : E|_{U_i} \\to E_i$ are isomorphisms in\n$D(\\mathcal{O}_{U_i})$ compatible with the $\\rho_{ij}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Glueing complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMG","source_file":"spaces-simplicial.tex","source_line":5529,"source_end_line":5555,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5529-L5555","statement_sha256":"2de7d0289896a07cee4fa37631ae5e2507dc8e8b92c04d165cb83a41d7a721c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12830,"rank":12830,"depth":26,"x":672.658,"y":1457.866,"cluster":"geometry-of-spaces"},{"id":"stacks:0DAF","tag":"0DAF","title":"Proper hypercoverings in topology · Lemma 0DAF","summary":"Let U be a simplicial object of LC and let a : U → X be an augmentation. There is a commutative diagram xymatrix Sh((LC_qc/U)_total) ar[r]_-h ar[d]_a_qc & Sh(U_Zar) ar[d]^a Sh(LC_qc/X) ar[r]^-h_-1 & Sh(X) where the left vertical arrow is defined in Section [Tag 09X8] and the right vertical arrow is defined in Lemma [Tag 09W4].","statement_latex":"Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$\nbe an augmentation. There is a commutative diagram\n$$\n\\xymatrix{\n\\Sh((\\textit{LC}_{qc}/U)_{total}) \\ar[r]_-h \\ar[d]_{a_{qc}} &\n\\Sh(U_{Zar}) \\ar[d]^a \\\\\n\\Sh(\\textit{LC}_{qc}/X) \\ar[r]^-{h_{-1}} &\n\\Sh(X)\n}\n$$\nwhere the left vertical arrow is defined in\nSection \\ref{section-hypercovering}\nand the right vertical arrow is defined in\nLemma \\ref{lemma-augmentation}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings in topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAF","source_file":"spaces-simplicial.tex","source_line":5692,"source_end_line":5708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5692-L5708","statement_sha256":"91a56776ac5ad0f7f85530015bc767988548f7d4d4e5ff1b7d588729e9e373b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12831,"rank":12831,"depth":11,"x":799.773,"y":1700.692,"cluster":"geometry-of-spaces"},{"id":"stacks:0DAG","tag":"0DAG","title":"Proper hypercoverings in topology · Lemma 0DAG","summary":"Let U be a simplicial object of LC and let a : U → X be an augmentation. If a : U → X gives a proper hypercovering of X, then a^-1 : Sh(X) → Sh(U_Zar) and a^-1 : Ab(X) → Ab(U_Zar) are fully faithful with essential image the cartesian sheaves and quasi-inverse given by a_*. Here a : Sh(U_Zar) → Sh(X) is as in Lemma [Tag 09W4].","statement_latex":"Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ gives a proper hypercovering of $X$,\nthen\n$$\na^{-1} : \\Sh(X) \\to \\Sh(U_{Zar})\n\\quad\\text{and}\\quad\na^{-1} : \\textit{Ab}(X) \\to \\textit{Ab}(U_{Zar})\n$$\nare fully faithful with essential image the cartesian sheaves and\nquasi-inverse given by $a_*$. Here $a : \\Sh(U_{Zar}) \\to \\Sh(X)$ is as in\nLemma \\ref{lemma-augmentation}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings in topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAG","source_file":"spaces-simplicial.tex","source_line":5730,"source_end_line":5743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5730-L5743","statement_sha256":"ab6aefec7f86ae89403f3934b1b5f0c71f0f90048730b2d9b34e34eea0203926","origin":"The Stacks Project","memory_eligible":false,"source_rank":12832,"rank":12832,"depth":30,"x":510.624,"y":1593.711,"cluster":"geometry-of-spaces"},{"id":"stacks:09XS","tag":"09XS","title":"Proper hypercoverings in topology · Lemma 09XS","summary":"Let U be a simplicial object of LC and let a : U → X be an augmentation. If a : U → X gives a proper hypercovering of X, then for K ∈ D^+(X) K → Ra_*(a^-1K) is an isomorphism where a : Sh(U_Zar) → Sh(X) is as in Lemma [Tag 09W4].","statement_latex":"Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ gives a proper hypercovering of $X$,\nthen for $K \\in D^+(X)$\n$$\nK \\to Ra_*(a^{-1}K)\n$$\nis an isomorphism where $a : \\Sh(U_{Zar}) \\to \\Sh(X)$ is as in\nLemma \\ref{lemma-augmentation}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings in topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XS","source_file":"spaces-simplicial.tex","source_line":5824,"source_end_line":5834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5824-L5834","statement_sha256":"8188d5398c32f7839c066b2d7a7f878ac5c480039f40efb3f8d9f244d9fe0135","origin":"The Stacks Project","memory_eligible":false,"source_rank":12833,"rank":12833,"depth":32,"x":810.017,"y":1508.485,"cluster":"geometry-of-spaces"},{"id":"stacks:09XC","tag":"09XC","title":"Proper hypercoverings in topology · Lemma 09XC","summary":"Let U be a simplicial object of LC and let a : U → X be an augmentation. If U is a proper hypercovering of X, then RΓ(X, K) = RΓ(U_Zar, a^-1K) for K ∈ D^+(X) where a : Sh(U_Zar) → Sh(X) is as in Lemma [Tag 09W4].","statement_latex":"Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$\nbe an augmentation. If $U$ is a proper hypercovering of $X$, then\n$$\nR\\Gamma(X, K) = R\\Gamma(U_{Zar}, a^{-1}K)\n$$\nfor $K \\in D^+(X)$ where $a : \\Sh(U_{Zar}) \\to \\Sh(X)$\nis as in Lemma \\ref{lemma-augmentation}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings in topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XC","source_file":"spaces-simplicial.tex","source_line":5867,"source_end_line":5876,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5867-L5876","statement_sha256":"29fd21fac566d7d1d43e7444a0f24d71da0321ff9a2a1e4a38cec215ce5c065d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12834,"rank":12834,"depth":33,"x":657.711,"y":1741.327,"cluster":"geometry-of-spaces"},{"id":"stacks:0DAH","tag":"0DAH","title":"Proper hypercoverings in topology · Lemma 0DAH","summary":"Let U be a simplicial object of LC and let a : U → X be an augmentation. Let A ⊂ Ab(U_Zar) denote the weak Serre subcategory of cartesian abelian sheaves. If U is a proper hypercovering of X, then the functor a^-1 defines an equivalence D^+(X) → D_A^+(U_Zar) with quasi-inverse Ra_* where a : Sh(U_Zar) → Sh(X) is as in Lemma [Tag 09W4].","statement_latex":"Let $U$ be a simplicial object of $\\textit{LC}$ and let $a : U \\to X$\nbe an augmentation.\nLet $\\mathcal{A} \\subset \\textit{Ab}(U_{Zar})$\ndenote the weak Serre subcategory of cartesian abelian sheaves.\nIf $U$ is a proper hypercovering of $X$, then\nthe functor $a^{-1}$ defines an equivalence\n$$\nD^+(X) \\longrightarrow D_\\mathcal{A}^+(U_{Zar})\n$$\nwith quasi-inverse $Ra_*$ where $a : \\Sh(U_{Zar}) \\to \\Sh(X)$\nis as in Lemma \\ref{lemma-augmentation}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings in topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DAH","source_file":"spaces-simplicial.tex","source_line":5885,"source_end_line":5898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5885-L5898","statement_sha256":"152204592f55c5200398af87342b571c7b6aae45e5e2fe0b5a29fda1821308ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":12835,"rank":12835,"depth":33,"x":582.737,"y":1483.083,"cluster":"geometry-of-spaces"},{"id":"stacks:09XB","tag":"09XB","title":"Proper hypercoverings in topology · Lemma 09XB","summary":"Let U be a simplicial object of LC and let a : U → X be an augmentation. Let F be an abelian sheaf on X. Let F_n be the pullback to U_n. If U is a proper hypercovering of X, then there exists a canonical spectral sequence E_1^p, q = H^q(U_p, F_p) converging to H^p + q(X, F).","statement_latex":"Let $U$ be a simplicial object of $\\textit{LC}$ and let\n$a : U \\to X$ be an augmentation. Let $\\mathcal{F}$ be an abelian sheaf\non $X$. Let $\\mathcal{F}_n$ be the pullback to $U_n$.\nIf $U$ is a proper hypercovering of $X$, then\nthere exists a canonical spectral sequence\n$$\nE_1^{p, q} = H^q(U_p, \\mathcal{F}_p)\n$$\nconverging to $H^{p + q}(X, \\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings in topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09XB","source_file":"spaces-simplicial.tex","source_line":5910,"source_end_line":5921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L5910-L5921","statement_sha256":"41ac54ceefed73faf36f7fa71e084339753a7638070bf675187e55658f32c1ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":12836,"rank":12836,"depth":34,"x":845.823,"y":1631.039,"cluster":"geometry-of-spaces"},{"id":"stacks:0249","tag":"0249","title":"Descent in terms of simplicial schemes · Definition 0249","summary":"Let a : Y → X be a morphism of simplicial schemes. We say a is cartesian, or that Y is cartesian over X, if for every morphism φ : [n] → [m] of Δ the corresponding diagram xymatrix Y_m ar[r]_a ar[d]_Y(φ) & X_m ar[d]^X(φ) Y_n ar[r]^a & X_n is a fibre square in the category of schemes.","statement_latex":"Let $a : Y \\to X$ be a morphism of simplicial schemes.\nWe say $a$ is {\\it cartesian}, or that {\\it $Y$ is cartesian over $X$},\nif for every morphism $\\varphi : [n] \\to [m]$ of $\\Delta$ the corresponding\ndiagram\n$$\n\\xymatrix{\nY_m \\ar[r]_a \\ar[d]_{Y(\\varphi)} & X_m \\ar[d]^{X(\\varphi)}\\\\\nY_n \\ar[r]^{a} & X_n\n}\n$$\nis a fibre square in the category of schemes.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Descent in terms of simplicial schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0249","source_file":"spaces-simplicial.tex","source_line":6106,"source_end_line":6119,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6106-L6119","statement_sha256":"7949108a2b00879a2c68ec885b8e17f8cf965ca27098d4db87257a2c93fb784f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12837,"rank":12837,"depth":0,"x":532.692,"y":1671.24,"cluster":"geometry-of-spaces"},{"id":"stacks:07TC","tag":"07TC","title":"Descent in terms of simplicial schemes · Lemma 07TC","summary":"Let X be a simplicial scheme. The category of simplicial schemes cartesian over X is equivalent to the category of pairs (V, φ) where V is a scheme over X_0 and φ : V ×_X_0, d^1_1 X_1 → X_1 ×_d^1_0, X_0 V is an isomorphism over X_1 such that (s_0^0)^*φ = id_V and such that (d^2_1)^*φ = (d^2_0)^*φ ∘ (d^2_2)^*φ as morphisms of schemes over X_2.","statement_latex":"Let $X$ be a simplicial scheme. The category of simplicial schemes cartesian\nover $X$ is equivalent to the category of pairs $(V, \\varphi)$\nwhere $V$ is a scheme over $X_0$ and\n$$\n\\varphi :\nV \\times_{X_0, d^1_1} X_1\n\\longrightarrow\nX_1 \\times_{d^1_0, X_0} V\n$$\nis an isomorphism over $X_1$ such that\n$(s_0^0)^*\\varphi = \\text{id}_V$ and such that\n$$\n(d^2_1)^*\\varphi = (d^2_0)^*\\varphi \\circ (d^2_2)^*\\varphi\n$$\nas morphisms of schemes over $X_2$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Descent in terms of simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TC","source_file":"spaces-simplicial.tex","source_line":6127,"source_end_line":6144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6127-L6144","statement_sha256":"15284b6777f3fe9138befa209ba29e7bf22b44a53cff3807dcb9a424b4080fb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12838,"rank":12838,"depth":0,"x":731.359,"y":1463.816,"cluster":"geometry-of-spaces"},{"id":"stacks:024A","tag":"024A","title":"Descent in terms of simplicial schemes · Definition 024A","summary":"Let f : X → S be a morphism of schemes. The simplicial scheme associated to f, denoted (X/S)_bullet, is the functor Δ^opp → Sch, [n] ↦ X ×_S … ×_S X described in Simplicial, Example [Tag 016E].","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. The {\\it simplicial scheme\nassociated to $f$}, denoted $(X/S)_\\bullet$, is the functor\n$\\Delta^{opp} \\to \\Sch$, $[n] \\mapsto X \\times_S \\ldots \\times_S X$\ndescribed in\nSimplicial, Example \\ref{simplicial-example-fibre-products-simplicial-object}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Descent in terms of simplicial schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024A","source_file":"spaces-simplicial.tex","source_line":6208,"source_end_line":6215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6208-L6215","statement_sha256":"1ba2cedc610b2c2628ff4e6f5416fad0824770bb25b4eb0f595b2cc72bc87586","origin":"The Stacks Project","memory_eligible":false,"source_rank":12839,"rank":12839,"depth":0,"x":751.679,"y":1729.626,"cluster":"geometry-of-spaces"},{"id":"stacks:024B","tag":"024B","title":"Descent in terms of simplicial schemes · Lemma 024B","summary":"Let f : X → S be a morphism of schemes. Let π : Y → (X/S)_bullet be a cartesian morphism of simplicial schemes. Set V = Y_0 considered as a scheme over X. The morphisms d^1_0, d^1_1 : Y_1 → Y_0 and the morphism π_1 : Y_1 → X ×_S X induce isomorphisms xymatrix V ×_S X & & Y_1 ar[ll]_-(d^1_1, pr_1 ∘ π_1) ar[rr]^-(pr_0 ∘ π_1, d^1_0) & & X ×_S V. Denote φ : V ×_S X → X ×_S V the resulting isomorphism. Then the pair (V, φ) is a descent datum relative to X → S.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\pi : Y \\to (X/S)_\\bullet$ be a cartesian morphism\nof simplicial schemes.\nSet $V = Y_0$ considered as a scheme over $X$.\nThe morphisms $d^1_0, d^1_1 : Y_1 \\to Y_0$ and the morphism\n$\\pi_1 : Y_1 \\to X \\times_S X$ induce isomorphisms\n$$\n\\xymatrix{\nV \\times_S X & &\nY_1 \\ar[ll]_-{(d^1_1, \\text{pr}_1 \\circ \\pi_1)}\n\\ar[rr]^-{(\\text{pr}_0 \\circ \\pi_1, d^1_0)} & &\nX \\times_S V.\n}\n$$\nDenote $\\varphi : V \\times_S X \\to X \\times_S V$ the\nresulting isomorphism.\nThen the pair $(V, \\varphi)$ is a descent datum relative\nto $X \\to S$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Descent in terms of simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024B","source_file":"spaces-simplicial.tex","source_line":6224,"source_end_line":6244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6224-L6244","statement_sha256":"c90f43ab560d7947b2c5fae0856590b9d9015212902f380c8ecc03d53634fa64","origin":"The Stacks Project","memory_eligible":false,"source_rank":12840,"rank":12840,"depth":1,"x":522.826,"y":1545.061,"cluster":"geometry-of-spaces"},{"id":"stacks:024C","tag":"024C","title":"Descent in terms of simplicial schemes · Lemma 024C","summary":"Let f : X → S be a morphism of schemes. The construction category of cartesian schemes over (X/S)_bullet → category of descent data relative to X/S of Lemma [Tag 024B] is an equivalence of categories.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes. The construction\n$$\n\\begin{matrix}\n\\text{category of cartesian } \\\\\n\\text{schemes over } (X/S)_\\bullet\n\\end{matrix}\n\\longrightarrow\n\\begin{matrix}\n\\text{ category of descent data} \\\\\n\\text{ relative to } X/S\n\\end{matrix}\n$$\nof Lemma \\ref{lemma-cartesian-over}\nis an equivalence of categories.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Descent in terms of simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024C","source_file":"spaces-simplicial.tex","source_line":6254,"source_end_line":6270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6254-L6270","statement_sha256":"531bbe1d289e8ab23179c2396a21db8f3e45a3c75b4326abe5e0c414c57b0960","origin":"The Stacks Project","memory_eligible":false,"source_rank":12841,"rank":12841,"depth":2,"x":840.156,"y":1551.303,"cluster":"geometry-of-spaces"},{"id":"stacks:07TI","tag":"07TI","title":"Quasi-coherent modules on simplicial schemes · Lemma 07TI","summary":"Let f : V → U be a morphism of simplicial schemes. Given a quasi-coherent module F on U_Zar the pullback f^*F is a quasi-coherent module on V_Zar.","statement_latex":"Let $f : V \\to U$ be a morphism of simplicial schemes. Given a\nquasi-coherent module $\\mathcal{F}$ on $U_{Zar}$ the pullback\n$f^*\\mathcal{F}$ is a quasi-coherent module on $V_{Zar}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Quasi-coherent modules on simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TI","source_file":"spaces-simplicial.tex","source_line":6316,"source_end_line":6321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6316-L6321","statement_sha256":"49e806c02c2373c8c59b7dc4d1d6ffaa215b7c2abcf7c8c25ff21e982e67b2f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12842,"rank":12842,"depth":12,"x":601.025,"y":1726.847,"cluster":"geometry-of-spaces"},{"id":"stacks:07TJ","tag":"07TJ","title":"Quasi-coherent modules on simplicial schemes · Lemma 07TJ","summary":"Let f : V → U be a cartesian morphism of simplicial schemes. Assume the morphisms d^n_j : U_n → U_n - 1 are flat and the morphisms V_n → U_n are quasi-compact and quasi-separated. For a quasi-coherent module G on V_Zar the pushforward f_*G is a quasi-coherent module on U_Zar.","statement_latex":"Let $f : V \\to U$ be a cartesian morphism of simplicial schemes.\nAssume the morphisms $d^n_j : U_n \\to U_{n - 1}$ are\nflat and the morphisms $V_n \\to U_n$ are quasi-compact and quasi-separated.\nFor a quasi-coherent module $\\mathcal{G}$ on $V_{Zar}$\nthe pushforward $f_*\\mathcal{G}$ is a quasi-coherent module on $U_{Zar}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Quasi-coherent modules on simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TJ","source_file":"spaces-simplicial.tex","source_line":6335,"source_end_line":6342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6335-L6342","statement_sha256":"545d3db81dbacd164845b39a6a259298c6b6a1cd1bad2e897f67910d278afdc6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12843,"rank":12843,"depth":30,"x":636.207,"y":1461.585,"cluster":"geometry-of-spaces"},{"id":"stacks:07TK","tag":"07TK","title":"Quasi-coherent modules on simplicial schemes · Lemma 07TK","summary":"Let f : V → U be a cartesian morphism of simplicial schemes. Assume the morphisms d^n_j : U_n → U_n - 1 are flat and the morphisms V_n → U_n are quasi-compact and quasi-separated. Then f^* and f_* form an adjoint pair of functors between the categories of quasi-coherent modules on U_Zar and V_Zar.","statement_latex":"Let $f : V \\to U$ be a cartesian morphism of\nsimplicial schemes. Assume the morphisms $d^n_j : U_n \\to U_{n - 1}$ are\nflat and the morphisms $V_n \\to U_n$ are quasi-compact and quasi-separated.\nThen $f^*$ and $f_*$ form an adjoint pair of functors\nbetween the categories of quasi-coherent modules on $U_{Zar}$ and $V_{Zar}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Quasi-coherent modules on simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TK","source_file":"spaces-simplicial.tex","source_line":6365,"source_end_line":6372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6365-L6372","statement_sha256":"9a201fbf43199d4a78102fd6c03ea7adbe804e342ba514e8574214571bb471fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":12844,"rank":12844,"depth":31,"x":823.672,"y":1677.253,"cluster":"geometry-of-spaces"},{"id":"stacks:07TL","tag":"07TL","title":"Quasi-coherent modules on simplicial schemes · Lemma 07TL","summary":"Let f : X → S be a morphism of schemes which has a section. Let (X/S)_bullet be the simplicial scheme associated to X → S, see Definition [Tag 024A]. Then pullback defines an equivalence between the category of quasi-coherent O_S-modules and the category of quasi-coherent modules on ((X/S)_bullet)_Zar.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes which has a\nsection\\footnote{In fact, it would be enough to assume that $f$\nhas fpqc locally on $S$ a section, since we have descent of\nquasi-coherent modules by Descent,\nSection \\ref{descent-section-fpqc-descent-quasi-coherent}.}.\nLet $(X/S)_\\bullet$ be the simplicial\nscheme associated to $X \\to S$, see\nDefinition \\ref{definition-fibre-products-simplicial-scheme}.\nThen pullback defines an equivalence between the category of\nquasi-coherent $\\mathcal{O}_S$-modules and the category of\nquasi-coherent modules on $((X/S)_\\bullet)_{Zar}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Quasi-coherent modules on simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TL","source_file":"spaces-simplicial.tex","source_line":6381,"source_end_line":6394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6381-L6394","statement_sha256":"7406165b8a376190870771c95ed167c78c61d2adc7a3e47ac743f39ec79b6781","origin":"The Stacks Project","memory_eligible":false,"source_rank":12845,"rank":12845,"depth":3,"x":511.852,"y":1624.57,"cluster":"geometry-of-spaces"},{"id":"stacks:07TN","tag":"07TN","title":"Groupoids and simplicial schemes · Lemma 07TN","summary":"Let (U, R, s, t, c, e, i) be a groupoid scheme over S. There exists a simplicial scheme X over S with the following properties • X_0 = U, X_1 = R, X_2 = R ×_s, U, t R, • s_0^0 = e : X_0 → X_1, • d^1_0 = s : X_1 → X_0, d^1_1 = t : X_1 → X_0, • s_0^1 = (e ∘ t, 1) : X_1 → X_2, s_1^1 = (1, e ∘ t) : X_1 → X_2, • d^2_0 = pr_1 : X_2 → X_1, d^2_1 = c : X_2 → X_1, d^2_2 = pr_0, and • X = cosk_2 sk_2 X. For all n we have X_n = R ×_s, U, t … ×_s, U, t R with n factors. The map d^n_j…","statement_latex":"Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$.\nThere exists a simplicial scheme $X$ over $S$\nwith the following properties\n\\begin{enumerate}\n\\item $X_0 = U$, $X_1 = R$, $X_2 = R \\times_{s, U, t} R$,\n\\item $s_0^0 = e : X_0 \\to X_1$,\n\\item $d^1_0 = s : X_1 \\to X_0$, $d^1_1 = t : X_1 \\to X_0$,\n\\item $s_0^1 = (e \\circ t, 1) : X_1 \\to X_2$,\n$s_1^1 = (1, e \\circ t) : X_1 \\to X_2$,\n\\item $d^2_0 = \\text{pr}_1 : X_2 \\to X_1$,\n$d^2_1 = c : X_2 \\to X_1$,\n$d^2_2 = \\text{pr}_0$, and\n\\item $X = \\text{cosk}_2 \\text{sk}_2 X$.\n\\end{enumerate}\nFor all $n$ we have $X_n = R \\times_{s, U, t} \\ldots \\times_{s, U, t} R$\nwith $n$ factors. The map $d^n_j : X_n \\to X_{n - 1}$ is given on\nfunctors of points by\n$$\n(r_1, \\ldots, r_n) \\longmapsto (r_1, \\ldots, c(r_j, r_{j + 1}), \\ldots, r_n)\n$$\nfor $1 \\leq j \\leq n - 1$ whereas\n$d^n_0(r_1, \\ldots, r_n) = (r_2, \\ldots, r_n)$ and\n$d^n_n(r_1, \\ldots, r_n) = (r_1, \\ldots, r_{n - 1})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Groupoids and simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TN","source_file":"spaces-simplicial.tex","source_line":6433,"source_end_line":6458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6433-L6458","statement_sha256":"9d39a60770c179fd9a84e08daea2329efc51fbf8f71fde7cf315fd8ba733e07d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12846,"rank":12846,"depth":3,"x":784.282,"y":1486.416,"cluster":"geometry-of-spaces"},{"id":"stacks:07TP","tag":"07TP","title":"Groupoids and simplicial schemes · Lemma 07TP","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let X be the simplicial scheme over S constructed in Lemma [Tag 07TN]. Then the category of quasi-coherent modules on (U, R, s, t, c) is equivalent to the category of quasi-coherent modules on X_Zar.","statement_latex":"Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme\nover $S$. Let $X$ be the simplicial scheme over $S$ constructed\nin Lemma \\ref{lemma-groupoid-simplicial}.\nThen the category of quasi-coherent modules on $(U, R, s, t, c)$\nis equivalent to the category of quasi-coherent modules on $X_{Zar}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Groupoids and simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TP","source_file":"spaces-simplicial.tex","source_line":6498,"source_end_line":6505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6498-L6505","statement_sha256":"b61fb9a45c42e0e2c03f733973d18ea362b4aeb124f119e76344c78bf0ee6fa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12847,"rank":12847,"depth":12,"x":694.452,"y":1742.997,"cluster":"geometry-of-spaces"},{"id":"stacks:07TQ","tag":"07TQ","title":"Groupoids and simplicial schemes · Lemma 07TQ","summary":"Let (U, R, s, t, c) be a groupoid scheme over a scheme S. Let X be the simplicial scheme over S constructed in Lemma [Tag 07TN]. Let (R/U)_bullet be the simplicial scheme associated to s : R → U, see Definition [Tag 024A]. There exists a cartesian morphism t_bullet : (R/U)_bullet → X of simplicial schemes with low degree morphisms given by xymatrix R ×_s, U, s R ×_s, U, s R ar@<3ex>[r]_-pr_12 ar@<0ex>[r]_-pr_02 ar@<-3ex>[r]_-pr_01 ar[dd]_(r_0, r_1, r_2) ↦ (r_0 ∘ r_1^-1,…","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.\nLet $X$ be the simplicial scheme over $S$ constructed\nin Lemma \\ref{lemma-groupoid-simplicial}.\nLet $(R/U)_\\bullet$ be the simplicial\nscheme associated to $s : R \\to U$, see\nDefinition \\ref{definition-fibre-products-simplicial-scheme}.\nThere exists a cartesian morphism $t_\\bullet : (R/U)_\\bullet \\to X$\nof simplicial schemes with low degree morphisms given by\n$$\n\\xymatrix{\nR \\times_{s, U, s} R \\times_{s, U, s} R\n\\ar@<3ex>[r]_-{\\text{pr}_{12}}\n\\ar@<0ex>[r]_-{\\text{pr}_{02}}\n\\ar@<-3ex>[r]_-{\\text{pr}_{01}}\n\\ar[dd]_{(r_0, r_1, r_2) \\mapsto (r_0 \\circ r_1^{-1}, r_1 \\circ r_2^{-1})} &\nR \\times_{s, U, s} R\n\\ar@<1ex>[r]_-{\\text{pr}_1} \\ar@<-2ex>[r]_-{\\text{pr}_0}\n\\ar[dd]_{(r_0, r_1) \\mapsto r_0 \\circ r_1^{-1}} &\nR \\ar[dd]^t\n\\\\\n\\\\\nR \\times_{s, U, t} R\n\\ar@<3ex>[r]_{\\text{pr}_1}\n\\ar@<0ex>[r]_c\n\\ar@<-3ex>[r]_{\\text{pr}_0} &\nR \\ar@<1ex>[r]_s \\ar@<-2ex>[r]_t &\nU\n}\n$$","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Groupoids and simplicial schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07TQ","source_file":"spaces-simplicial.tex","source_line":6519,"source_end_line":6550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6519-L6550","statement_sha256":"db12e49a29e7a6f5d9c46087fb8bcaf9c116ab249e78d89398093ea407b25816","origin":"The Stacks Project","memory_eligible":false,"source_rank":12848,"rank":12848,"depth":4,"x":554.288,"y":1502.709,"cluster":"geometry-of-spaces"},{"id":"stacks:024E","tag":"024E","title":"Descent data give equivalence relations · Lemma 024E","summary":"Let f : X → S be a morphism of schemes. Let π : Y → (X/S)_bullet be a cartesian morphism of simplicial schemes, see Definitions [Tag 0249] and [Tag 024A]. Then the morphism j = (d^1_1, d^1_0) : Y_1 → Y_0 ×_S Y_0 defines an equivalence relation on Y_0 over S, see Groupoids, Definition [Tag 022P].","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $\\pi : Y \\to (X/S)_\\bullet$ be a cartesian morphism of simplicial\nschemes, see Definitions \\ref{definition-cartesian-morphism} and\n\\ref{definition-fibre-products-simplicial-scheme}.\nThen the morphism\n$$\nj = (d^1_1, d^1_0) : Y_1 \\to Y_0 \\times_S Y_0\n$$\ndefines an equivalence relation on $Y_0$ over $S$,\nsee Groupoids, Definition \\ref{groupoids-definition-equivalence-relation}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Descent data give equivalence relations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024E","source_file":"spaces-simplicial.tex","source_line":6581,"source_end_line":6593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6581-L6593","statement_sha256":"47b3c9ca1249982f5219dcd1b018e3eac03d3e6b9a1cc8d6f53fb171d49db488","origin":"The Stacks Project","memory_eligible":false,"source_rank":12849,"rank":12849,"depth":1,"x":851.019,"y":1600.409,"cluster":"geometry-of-spaces"},{"id":"stacks:024G","tag":"024G","title":"An example case · Lemma 024G","summary":"Let X → S be a morphism of schemes. Suppose Y → (X/S)_bullet is a cartesian morphism of simplicial schemes. For y ∈ Y_0 a point define T_y = (y' ∈ Y_0 mid ∃ y_1 ∈ Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y') as a subset of Y_0. Then y ∈ T_y and T_y ∩ T_y' not = ∅ ⇒ T_y = T_y'.","statement_latex":"Let $X \\to S$ be a morphism of schemes. Suppose $Y \\to (X/S)_\\bullet$\nis a cartesian morphism of simplicial schemes. For $y \\in Y_0$ a point define\n$$\nT_y = \\{y' \\in Y_0 \\mid \\exists\\ y_1 \\in Y_1:\nd^1_1(y_1) = y, d^1_0(y_1) = y'\\}\n$$\nas a subset of $Y_0$. Then $y \\in T_y$ and\n$T_y \\cap T_{y'} \\not = \\emptyset \\Rightarrow T_y = T_{y'}$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"An example case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024G","source_file":"spaces-simplicial.tex","source_line":6671,"source_end_line":6681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6671-L6681","statement_sha256":"38982ea8ef5e11fe7f1c82b6baa8a8ea5cd7170f2b436d4cf99776d1c490cbb9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12850,"rank":12850,"depth":2,"x":553.503,"y":1696.786,"cluster":"geometry-of-spaces"},{"id":"stacks:024H","tag":"024H","title":"An example case · Lemma 024H","summary":"Let X → S be a morphism of schemes. Suppose Y → (X/S)_bullet is a cartesian morphism of simplicial schemes. Let y ∈ Y_0 be a point. If X → S is quasi-compact, then T_y = (y' ∈ Y_0 mid ∃ y_1 ∈ Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y') is a quasi-compact subset of Y_0.","statement_latex":"Let $X \\to S$ be a morphism of schemes.\nSuppose $Y \\to (X/S)_\\bullet$ is a cartesian morphism of simplicial schemes.\nLet $y \\in Y_0$ be a point. If $X \\to S$ is quasi-compact, then\n$$\nT_y = \\{y' \\in Y_0 \\mid \\exists\\ y_1 \\in Y_1:\nd^1_1(y_1) = y, d^1_0(y_1) = y'\\}\n$$\nis a quasi-compact subset of $Y_0$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"An example case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024H","source_file":"spaces-simplicial.tex","source_line":6689,"source_end_line":6699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6689-L6699","statement_sha256":"f8700ef2aa627cc059077d9c1bf45ab0ca83c8015c3bc6dc427087c52a8a7281","origin":"The Stacks Project","memory_eligible":false,"source_rank":12851,"rank":12851,"depth":0,"x":695.452,"y":1456.785,"cluster":"geometry-of-spaces"},{"id":"stacks:024I","tag":"024I","title":"An example case · Lemma 024I","summary":"Let X → S be a quasi-compact flat surjective morphism. Let (V, φ) be a descent datum relative to X → S. If V is a disjoint union of spectra of Artinian rings, then (V, φ) is effective.","statement_latex":"Let $X \\to S$ be a quasi-compact flat surjective morphism.\nLet $(V, \\varphi)$ be a descent datum relative\nto $X \\to S$. If $V$ is a disjoint union of\nspectra of Artinian rings, then $(V, \\varphi)$ is effective.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"An example case","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/024I","source_file":"spaces-simplicial.tex","source_line":6716,"source_end_line":6722,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L6716-L6722","statement_sha256":"106c53b8cd7a51615980fbc4400a6e61d3964a92b9cd688058377daad3655be7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12852,"rank":12852,"depth":46,"x":783.826,"y":1714.428,"cluster":"geometry-of-spaces"},{"id":"stacks:0DH5","tag":"0DH5","title":"Fppf hypercoverings of algebraic spaces · Lemma 0DH5","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. There is a commutative diagram xymatrix Sh((Spaces/U)_fppf, total) ar[r]_-h ar[d]_a_fppf & Sh(U_etale) ar[d]^a Sh((Spaces/X)_fppf) ar[r]^-h_-1 & Sh(X_etale) where the left vertical arrow is defined in Section [Tag 09X8] and the right vertical arrow is defined in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$.\nLet $a : U \\to X$ be an augmentation. There is a commutative diagram\n$$\n\\xymatrix{\n\\Sh((\\textit{Spaces}/U)_{fppf, total}) \\ar[r]_-h \\ar[d]_{a_{fppf}} &\n\\Sh(U_\\etale) \\ar[d]^a \\\\\n\\Sh((\\textit{Spaces}/X)_{fppf}) \\ar[r]^-{h_{-1}} &\n\\Sh(X_\\etale)\n}\n$$\nwhere the left vertical arrow is defined in\nSection \\ref{section-hypercovering}\nand the right vertical arrow is defined in\nSection \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DH5","source_file":"spaces-simplicial.tex","source_line":7015,"source_end_line":7032,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7015-L7032","statement_sha256":"b1b0fca893c1d28f481ffe238395aa9732bd80839da1f0ec2cfc3364c614a1e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12853,"rank":12853,"depth":11,"x":511.339,"y":1574.524,"cluster":"geometry-of-spaces"},{"id":"stacks:0DH6","tag":"0DH6","title":"Fppf hypercoverings of algebraic spaces · Lemma 0DH6","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. If a : U → X is an fppf hypercovering of X, then a^-1 : Sh(X_etale) → Sh(U_etale) and a^-1 : Ab(X_etale) → Ab(U_etale) are fully faithful with essential image the cartesian sheaves and quasi-inverse given by a_*. Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$,\nthen\n$$\na^{-1} : \\Sh(X_\\etale) \\to \\Sh(U_\\etale)\n\\quad\\text{and}\\quad\na^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(U_\\etale)\n$$\nare fully faithful with essential image the cartesian sheaves and\nquasi-inverse given by $a_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DH6","source_file":"spaces-simplicial.tex","source_line":7069,"source_end_line":7083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7069-L7083","statement_sha256":"d9ee6d8a7202d37ab44dd8062a9be58bb97a258e3a6208b556ce1d914ce306ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":12854,"rank":12854,"depth":84,"x":824.925,"y":1523.047,"cluster":"geometry-of-spaces"},{"id":"stacks:0DH7","tag":"0DH7","title":"Fppf hypercoverings of algebraic spaces · Lemma 0DH7","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. If a : U → X is an fppf hypercovering of X, then for K ∈ D^+(X_etale) K → Ra_*(a^-1K) is an isomorphism. Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$,\nthen for $K \\in D^+(X_\\etale)$\n$$\nK \\to Ra_*(a^{-1}K)\n$$\nis an isomorphism. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DH7","source_file":"spaces-simplicial.tex","source_line":7144,"source_end_line":7155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7144-L7155","statement_sha256":"4ec6ef5dfbe6c7bd3d4315dd5e0b87607588339d2c5a14db3940b58e04f40f12","origin":"The Stacks Project","memory_eligible":false,"source_rank":12855,"rank":12855,"depth":84,"x":634.996,"y":1739.045,"cluster":"geometry-of-spaces"},{"id":"stacks:0DH8","tag":"0DH8","title":"Fppf hypercoverings of algebraic spaces · Lemma 0DH8","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. If a : U → X is an fppf hypercovering of X, then RΓ(X_etale, K) = RΓ(U_etale, a^-1K) for K ∈ D^+(X_etale). Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then\n$$\nR\\Gamma(X_\\etale, K) = R\\Gamma(U_\\etale, a^{-1}K)\n$$\nfor $K \\in D^+(X_\\etale)$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DH8","source_file":"spaces-simplicial.tex","source_line":7188,"source_end_line":7198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7188-L7198","statement_sha256":"66a2c35b16b60ea3ceeb3e34529cfb6164783924ad02b45f6c282f9a5a419472","origin":"The Stacks Project","memory_eligible":false,"source_rank":12856,"rank":12856,"depth":85,"x":601.331,"y":1471.872,"cluster":"geometry-of-spaces"},{"id":"stacks:0DH9","tag":"0DH9","title":"Fppf hypercoverings of algebraic spaces · Lemma 0DH9","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. Let A ⊂ Ab(U_etale) denote the weak Serre subcategory of cartesian abelian sheaves. If U is an fppf hypercovering of X, then the functor a^-1 defines an equivalence D^+(X_etale) → D_A^+(U_etale) with quasi-inverse Ra_*. Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation.\nLet $\\mathcal{A} \\subset \\textit{Ab}(U_\\etale)$\ndenote the weak Serre subcategory of cartesian abelian sheaves.\nIf $U$ is an fppf hypercovering of $X$, then\nthe functor $a^{-1}$ defines an equivalence\n$$\nD^+(X_\\etale) \\longrightarrow D_\\mathcal{A}^+(U_\\etale)\n$$\nwith quasi-inverse $Ra_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DH9","source_file":"spaces-simplicial.tex","source_line":7207,"source_end_line":7221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7207-L7221","statement_sha256":"3b7503307bd685518711b3c50ca16a14dbee28a7456e969ccead411511b70317","origin":"The Stacks Project","memory_eligible":false,"source_rank":12857,"rank":12857,"depth":85,"x":841.124,"y":1649.864,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHA","tag":"0DHA","title":"Fppf hypercoverings of algebraic spaces · Lemma 0DHA","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. Let F be an abelian sheaf on X_etale. Let F_n be the pullback to U_n, etale. If U is an fppf hypercovering of X, then there exists a canonical spectral sequence E_1^p, q = H^q_etale(U_p, F_p) converging to H^p + q_etale(X, F).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. Let $\\mathcal{F}$ be an abelian sheaf\non $X_\\etale$. Let $\\mathcal{F}_n$ be the pullback to $U_{n, \\etale}$.\nIf $U$ is an fppf hypercovering of $X$, then\nthere exists a canonical spectral sequence\n$$\nE_1^{p, q} = H^q_\\etale(U_p, \\mathcal{F}_p)\n$$\nconverging to $H^{p + q}_\\etale(X, \\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHA","source_file":"spaces-simplicial.tex","source_line":7233,"source_end_line":7245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7233-L7245","statement_sha256":"d9e9fbeb93cbdafe65cf3ac47ee431d148b5aa23861fb92292e692ffd8244416","origin":"The Stacks Project","memory_eligible":false,"source_rank":12858,"rank":12858,"depth":86,"x":521.014,"y":1654.682,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHC","tag":"0DHC","title":"Fppf hypercoverings of algebraic spaces: modules · Lemma 0DHC","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. There is a commutative diagram xymatrix (Sh((Spaces/U)_fppf, total), O_big, total) ar[r]_-h ar[d]_a_fppf & (Sh(U_etale), O_U) ar[d]^a (Sh((Spaces/X)_fppf), O_big) ar[r]^-h_-1 & (Sh(X_etale), O_X) of ringed topoi where the left vertical arrow is defined in Section [Tag 0DAA] and the right vertical arrow is defined in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$.\nLet $a : U \\to X$ be an augmentation. There is a commutative diagram\n$$\n\\xymatrix{\n(\\Sh((\\textit{Spaces}/U)_{fppf, total}), \\mathcal{O}_{big, total})\n\\ar[r]_-h \\ar[d]_{a_{fppf}} &\n(\\Sh(U_\\etale), \\mathcal{O}_U) \\ar[d]^a \\\\\n(\\Sh((\\textit{Spaces}/X)_{fppf}), \\mathcal{O}_{big}) \\ar[r]^-{h_{-1}} &\n(\\Sh(X_\\etale), \\mathcal{O}_X)\n}\n$$\nof ringed topoi where the left vertical arrow is defined in\nSection \\ref{section-hypercovering-modules}\nand the right vertical arrow is defined in\nSection \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHC","source_file":"spaces-simplicial.tex","source_line":7269,"source_end_line":7287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7269-L7287","statement_sha256":"2ab5e8098526b6c2e680d5874f4d3738945ca78806402f99643e724de617f8b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12859,"rank":12859,"depth":12,"x":753.295,"y":1469.403,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHD","tag":"0DHD","title":"Fppf hypercoverings of algebraic spaces: modules · Lemma 0DHD","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. If a : U → X is an fppf hypercovering of X, then a^* : QCoh(O_X) → QCoh(O_U) is an equivalence fully faithful with quasi-inverse given by a_*. Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$,\nthen\n$$\na^* : \\QCoh(\\mathcal{O}_X) \\to \\QCoh(\\mathcal{O}_U)\n$$\nis an equivalence fully faithful with quasi-inverse given by $a_*$.\nHere $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHD","source_file":"spaces-simplicial.tex","source_line":7326,"source_end_line":7338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7326-L7338","statement_sha256":"5adf6398cef64257923012080758706c9402702406cf4ed011e969ab954a3abe","origin":"The Stacks Project","memory_eligible":false,"source_rank":12860,"rank":12860,"depth":53,"x":731.0,"y":1737.956,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHE","tag":"0DHE","title":"Fppf hypercoverings of algebraic spaces: modules · Lemma 0DHE","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. If a : U → X is an fppf hypercovering of X, then for F a quasi-coherent O_X-module the map F → Ra_*(a^*F) is an isomorphism. Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$,\nthen for $\\mathcal{F}$ a quasi-coherent $\\mathcal{O}_X$-module\nthe map\n$$\n\\mathcal{F} \\to Ra_*(a^*\\mathcal{F})\n$$\nis an isomorphism. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHE","source_file":"spaces-simplicial.tex","source_line":7410,"source_end_line":7422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7410-L7422","statement_sha256":"de187b0d66707470bc3d946cd3c2d4a02ce02b2a5e97935d5332477cfdb3f87d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12861,"rank":12861,"depth":54,"x":531.382,"y":1527.177,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHF","tag":"0DHF","title":"Fppf hypercoverings of algebraic spaces: modules · Lemma 0DHF","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. Assume a : U → X is an fppf hypercovering of X. Then QCoh(O_U) is a weak Serre subcategory of Mod(O_U) and a^* : D_QCoh(O_X) → D_QCoh(O_U) is an equivalence of categories with quasi-inverse given by Ra_*. Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. Assume $a : U \\to X$ is an fppf hypercovering of $X$.\nThen $\\QCoh(\\mathcal{O}_U)$ is a weak Serre subcategory of\n$\\textit{Mod}(\\mathcal{O}_U)$ and\n$$\na^* : D_\\QCoh(\\mathcal{O}_X) \\longrightarrow D_\\QCoh(\\mathcal{O}_U)\n$$\nis an equivalence of categories with quasi-inverse given by\n$Ra_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHF","source_file":"spaces-simplicial.tex","source_line":7458,"source_end_line":7471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7458-L7471","statement_sha256":"4cdb00b90fea986c3b8b0274ef1b8e156d56facfbd2281783827f26757a4af4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12862,"rank":12862,"depth":55,"x":848.231,"y":1569.354,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHG","tag":"0DHG","title":"Fppf hypercoverings of algebraic spaces: modules · Lemma 0DHG","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. If a : U → X is an fppf hypercovering of X, then RΓ(X_etale, K) = RΓ(U_etale, a^*K) for K ∈ D_QCoh(O_X). Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ is an fppf hypercovering of $X$, then\n$$\nR\\Gamma(X_\\etale, K) = R\\Gamma(U_\\etale, a^*K)\n$$\nfor $K \\in D_\\QCoh(\\mathcal{O}_X)$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHG","source_file":"spaces-simplicial.tex","source_line":7548,"source_end_line":7558,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7548-L7558","statement_sha256":"ff2a05e7aaebfbb35d5efbdea0134d9f754942922d4c8797e36d31bacb7ec29e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12863,"rank":12863,"depth":56,"x":580.546,"y":1718.113,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHH","tag":"0DHH","title":"Fppf hypercoverings of algebraic spaces: modules · Lemma 0DHH","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. Let F be quasi-coherent O_X-module. Let F_n be the pullback to U_n, etale. If U is an fppf hypercovering of X, then there exists a canonical spectral sequence E_1^p, q = H^q_etale(U_p, F_p) converging to H^p + q_etale(X, F).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. Let $\\mathcal{F}$ be quasi-coherent\n$\\mathcal{O}_X$-module. Let $\\mathcal{F}_n$ be the pullback to\n$U_{n, \\etale}$. If $U$ is an fppf hypercovering of $X$, then\nthere exists a canonical spectral sequence\n$$\nE_1^{p, q} = H^q_\\etale(U_p, \\mathcal{F}_p)\n$$\nconverging to $H^{p + q}_\\etale(X, \\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf hypercoverings of algebraic spaces: modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHH","source_file":"spaces-simplicial.tex","source_line":7567,"source_end_line":7579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7567-L7579","statement_sha256":"e7ae9f43f9230cce3b3aa53899d0dc4130014ba443722608d5f48c758c378242","origin":"The Stacks Project","memory_eligible":false,"source_rank":12864,"rank":12864,"depth":57,"x":658.342,"y":1456.404,"cluster":"geometry-of-spaces"},{"id":"stacks:0DL9","tag":"0DL9","title":"Fppf descent of complexes · Lemma 0DL9","summary":"Let X be an algebraic space over a scheme S. Let K, E ∈ D_QCoh(O_X). Let a : U → X be an fppf hypercovering. Assume that for all n ≥ 0 we have Ext_O_U_n^i(La_n^*K, La_n^*E) = 0 for i < 0 Then we have • Ext_O_X^i(K, E) = 0 for i < 0, and • there is an exact sequence 0 → Hom_O_X(K, E) → Hom_O_U_0(La_0^*K, La_0^*E) → Hom_O_U_1(La_1^*K, La_1^*E)","statement_latex":"Let $X$ be an algebraic space over a scheme $S$.\nLet $K, E \\in D_\\QCoh(\\mathcal{O}_X)$.\nLet $a : U \\to X$ be an fppf hypercovering.\nAssume that for all $n \\geq 0$ we have\n$$\n\\Ext_{\\mathcal{O}_{U_n}}^i(La_n^*K, La_n^*E) = 0\n\\text{ for } i < 0\n$$\nThen we have\n\\begin{enumerate}\n\\item $\\Ext_{\\mathcal{O}_X}^i(K, E) = 0$ for $i < 0$, and\n\\item there is an exact sequence\n$$\n0\n\\to\n\\Hom_{\\mathcal{O}_X}(K, E)\n\\to\n\\Hom_{\\mathcal{O}_{U_0}}(La_0^*K, La_0^*E)\n\\to\n\\Hom_{\\mathcal{O}_{U_1}}(La_1^*K, La_1^*E)\n$$\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf descent of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DL9","source_file":"spaces-simplicial.tex","source_line":7601,"source_end_line":7625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7601-L7625","statement_sha256":"a3224849e2f16f7934168b517ef5d527915c0868309b4389a6e548108868cfdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12865,"rank":12865,"depth":56,"x":811.508,"y":1693.641,"cluster":"geometry-of-spaces"},{"id":"stacks:0DLA","tag":"0DLA","title":"Fppf descent of complexes · Lemma 0DLA","summary":"Let X be an algebraic space over a scheme S. Let a : U → X be an fppf hypercovering. Suppose given K_0 ∈ D_QCoh(U_0) and an isomorphism α : L(f_δ_1^1)^*K_0 → L(f_δ_0^1)^*K_0 satisfying the cocycle condition on U_1. Set τ^n_i : [0] → [n], 0 ↦ i and set K_n = Lf_τ^n_n^*K_0. Assume Ext^i_O_U_n(K_n, K_n) = 0 for i < 0. Then there exists an object K ∈ D_QCoh(O_X) and an isomorphism La_0^*K → K compatible with α.","statement_latex":"Let $X$ be an algebraic space over a scheme $S$.\nLet $a : U \\to X$ be an fppf hypercovering.\nSuppose given $K_0 \\in D_\\QCoh(U_0)$ and an isomorphism\n$$\n\\alpha :\nL(f_{\\delta_1^1})^*K_0\n\\longrightarrow\nL(f_{\\delta_0^1})^*K_0\n$$\nsatisfying the cocycle condition on $U_1$. Set\n$\\tau^n_i : [0] \\to [n]$, $0 \\mapsto i$ and\nset $K_n = Lf_{\\tau^n_n}^*K_0$.\nAssume $\\Ext^i_{\\mathcal{O}_{U_n}}(K_n, K_n) = 0$ for $i < 0$.\nThen there exists an object $K \\in D_\\QCoh(\\mathcal{O}_X)$\nand an isomorphism $La_0^*K \\to K$ compatible with $\\alpha$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Fppf descent of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLA","source_file":"spaces-simplicial.tex","source_line":7661,"source_end_line":7678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7661-L7678","statement_sha256":"68add87713075f119e7ca6c4f3e498551bfc16da6486f6d67d432c3b7b95008b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12866,"rank":12866,"depth":56,"x":507.642,"y":1605.574,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHJ","tag":"0DHJ","title":"Proper hypercoverings of algebraic spaces · Lemma 0DHJ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. There is a commutative diagram xymatrix Sh((Spaces/U)_ph, total) ar[r]_-h ar[d]_a_ph & Sh(U_etale) ar[d]^a Sh((Spaces/X)_ph) ar[r]^-h_-1 & Sh(X_etale) where the left vertical arrow is defined in Section [Tag 09X8] and the right vertical arrow is defined in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$.\nLet $a : U \\to X$ be an augmentation. There is a commutative diagram\n$$\n\\xymatrix{\n\\Sh((\\textit{Spaces}/U)_{ph, total}) \\ar[r]_-h \\ar[d]_{a_{ph}} &\n\\Sh(U_\\etale) \\ar[d]^a \\\\\n\\Sh((\\textit{Spaces}/X)_{ph}) \\ar[r]^-{h_{-1}} &\n\\Sh(X_\\etale)\n}\n$$\nwhere the left vertical arrow is defined in\nSection \\ref{section-hypercovering}\nand the right vertical arrow is defined in\nSection \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHJ","source_file":"spaces-simplicial.tex","source_line":7755,"source_end_line":7772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7755-L7772","statement_sha256":"1008288b4f4156a2ccf1b4e49c020d5c651a9baa126b162fc29bbd99df88e2ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":12867,"rank":12867,"depth":11,"x":802.67,"y":1498.041,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHK","tag":"0DHK","title":"Proper hypercoverings of algebraic spaces · Lemma 0DHK","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. If a : U → X is a proper hypercovering of X, then a^-1 : Sh(X_etale) → Sh(U_etale) and a^-1 : Ab(X_etale) → Ab(U_etale) are fully faithful with essential image the cartesian sheaves and quasi-inverse given by a_*. Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ is a proper hypercovering of $X$,\nthen\n$$\na^{-1} : \\Sh(X_\\etale) \\to \\Sh(U_\\etale)\n\\quad\\text{and}\\quad\na^{-1} : \\textit{Ab}(X_\\etale) \\to \\textit{Ab}(U_\\etale)\n$$\nare fully faithful with essential image the cartesian sheaves and\nquasi-inverse given by $a_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHK","source_file":"spaces-simplicial.tex","source_line":7809,"source_end_line":7823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7809-L7823","statement_sha256":"a8b68afb134f27fa3dc17769aedcfbc0417d5292b6e0ab1eb2c37dba1941f0c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12868,"rank":12868,"depth":85,"x":671.534,"y":1744.858,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHL","tag":"0DHL","title":"Proper hypercoverings of algebraic spaces · Lemma 0DHL","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. If a : U → X is a proper hypercovering of X, then for K ∈ D^+(X_etale) K → Ra_*(a^-1K) is an isomorphism. Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ is a proper hypercovering of $X$,\nthen for $K \\in D^+(X_\\etale)$\n$$\nK \\to Ra_*(a^{-1}K)\n$$\nis an isomorphism. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHL","source_file":"spaces-simplicial.tex","source_line":7913,"source_end_line":7924,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7913-L7924","statement_sha256":"014db99e11c5909d8a954fbbf85291b97f838ce1f7482b1f63a3005c7863bfeb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12869,"rank":12869,"depth":84,"x":569.7,"y":1488.327,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHM","tag":"0DHM","title":"Proper hypercoverings of algebraic spaces · Lemma 0DHM","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. If a : U → X is a proper hypercovering of X, then RΓ(X_etale, K) = RΓ(U_etale, a^-1K) for K ∈ D^+(X_etale). Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. If $a : U \\to X$ is a proper hypercovering of $X$, then\n$$\nR\\Gamma(X_\\etale, K) = R\\Gamma(U_\\etale, a^{-1}K)\n$$\nfor $K \\in D^+(X_\\etale)$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHM","source_file":"spaces-simplicial.tex","source_line":7959,"source_end_line":7969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7959-L7969","statement_sha256":"341a5782004edf4fb8411027ded6011a09885484c39a2fe73e078886029c55b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12870,"rank":12870,"depth":85,"x":851.219,"y":1619.768,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHN","tag":"0DHN","title":"Proper hypercoverings of algebraic spaces · Lemma 0DHN","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. Let A ⊂ Ab(U_etale) denote the weak Serre subcategory of cartesian abelian sheaves. If U is a proper hypercovering of X, then the functor a^-1 defines an equivalence D^+(X_etale) → D_A^+(U_etale) with quasi-inverse Ra_*. Here a : Sh(U_etale) → Sh(X_etale) is as in Section [Tag 0DE7].","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation.\nLet $\\mathcal{A} \\subset \\textit{Ab}(U_\\etale)$\ndenote the weak Serre subcategory of cartesian abelian sheaves.\nIf $U$ is a proper hypercovering of $X$, then\nthe functor $a^{-1}$ defines an equivalence\n$$\nD^+(X_\\etale) \\longrightarrow D_\\mathcal{A}^+(U_\\etale)\n$$\nwith quasi-inverse $Ra_*$. Here $a : \\Sh(U_\\etale) \\to \\Sh(X_\\etale)$\nis as in Section \\ref{section-simplicial-algebraic-spaces}.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHN","source_file":"spaces-simplicial.tex","source_line":7978,"source_end_line":7992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L7978-L7992","statement_sha256":"696097e83c4c143e2e7218044a65076b49dfa7d83354522f291ed733c198deb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12871,"rank":12871,"depth":86,"x":537.781,"y":1682.616,"cluster":"geometry-of-spaces"},{"id":"stacks:0DHP","tag":"0DHP","title":"Proper hypercoverings of algebraic spaces · Lemma 0DHP","summary":"Let S be a scheme. Let X be an algebraic space over S. Let U be a simplicial algebraic space over S. Let a : U → X be an augmentation. Let F be an abelian sheaf on X_etale. Let F_n be the pullback to U_n, etale. If U is a ph hypercovering of X, then there exists a canonical spectral sequence E_1^p, q = H^q_etale(U_p, F_p) converging to H^p + q_etale(X, F).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $U$ be a simplicial algebraic space over $S$. Let $a : U \\to X$\nbe an augmentation. Let $\\mathcal{F}$ be an abelian sheaf\non $X_\\etale$. Let $\\mathcal{F}_n$ be the pullback to $U_{n, \\etale}$.\nIf $U$ is a ph hypercovering of $X$, then\nthere exists a canonical spectral sequence\n$$\nE_1^{p, q} = H^q_\\etale(U_p, \\mathcal{F}_p)\n$$\nconverging to $H^{p + q}_\\etale(X, \\mathcal{F})$.","area":"Geometry of Spaces","chapter":"Simplicial Spaces","chapter_id":"spaces-simplicial","section":"Proper hypercoverings of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DHP","source_file":"spaces-simplicial.tex","source_line":8004,"source_end_line":8016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-simplicial.tex#L8004-L8016","statement_sha256":"5c75e70c2551c992871e747b052532366be793910607a260d6fafb5fef8f8427","origin":"The Stacks Project","memory_eligible":false,"source_rank":12872,"rank":12872,"depth":86,"x":718.451,"y":1458.315,"cluster":"geometry-of-spaces"},{"id":"stacks:0E4Y","tag":"0E4Y","title":"Dualizing complexes on algebraic spaces · Lemma 0E4Y","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let K be an object of D_QCoh(O_X). The following are equivalent • For every étale morphism U → X where U is a scheme the restriction K|_U is a dualizing complex for U (as discussed above). • There exists a surjective étale morphism U → X where U is a scheme such that K|_U is a dualizing complex for U.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nLet $K$ be an object of $D_\\QCoh(\\mathcal{O}_X)$. The following are equivalent\n\\begin{enumerate}\n\\item For every \\'etale morphism $U \\to X$ where $U$ is a scheme\nthe restriction $K|_U$ is a dualizing complex for $U$ (as discussed above).\n\\item There exists a surjective \\'etale morphism $U \\to X$ where $U$ is a\nscheme such that $K|_U$ is a dualizing complex for $U$.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Dualizing complexes on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4Y","source_file":"spaces-duality.tex","source_line":55,"source_end_line":65,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L55-L65","statement_sha256":"0a80d73d3518c4e3b1b7f683be6c19e8a6dfcc63fc734650f554a1d6fef536f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12873,"rank":12873,"depth":42,"x":1861.648,"y":1161.099,"cluster":"duality-cohomology"},{"id":"stacks:0E4Z","tag":"0E4Z","title":"Dualizing complexes on algebraic spaces · Definition 0E4Z","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. An object K of D_QCoh(O_X) is called a dualizing complex if K satisfies the equivalent conditions of Lemma [Tag 0E4Y].","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian algebraic space over $S$.\nAn object $K$ of $D_\\QCoh(\\mathcal{O}_X)$ is called a\n{\\it dualizing complex} if $K$ satisfies the equivalent conditions of\nLemma \\ref{lemma-equivalent-definitions}.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Dualizing complexes on algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E4Z","source_file":"spaces-duality.tex","source_line":80,"source_end_line":87,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L80-L87","statement_sha256":"801b5ab97fc21d48dfb1d515fa4ccc2a17f17873ce05c556587e2ccf618dd10b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12874,"rank":12874,"depth":43,"x":2137.274,"y":1028.81,"cluster":"duality-cohomology"},{"id":"stacks:0E50","tag":"0E50","title":"Dualizing complexes on algebraic spaces · Lemma 0E50","summary":"Let A be a Noetherian ring and let X = Spec(A). Let O_etale be the structure sheaf of X on the small étale site of X. Let K, L be objects of D(A). If K ∈ D_Coh(A) and L has finite injective dimension, then ε^*widetildeRHom_A(K, L) = RSheafHom_O_etale(ε^*widetildeK, ε^*widetildeL) in D(O_etale) where ε : (X_etale, O_etale) → (X, O_X) is as in Derived Categories of Spaces, Section [Tag 071P].","statement_latex":"Let $A$ be a Noetherian ring and let $X = \\Spec(A)$. Let\n$\\mathcal{O}_\\etale$ be the structure sheaf of $X$ on the\nsmall \\'etale site of $X$. Let $K, L$ be objects of $D(A)$.\nIf $K \\in D_{\\textit{Coh}}(A)$ and $L$ has finite injective\ndimension, then\n$$\n\\epsilon^*\\widetilde{R\\Hom_A(K, L)} =\nR\\SheafHom_{\\mathcal{O}_\\etale}(\\epsilon^*\\widetilde{K},\n\\epsilon^*\\widetilde{L})\n$$\nin $D(\\mathcal{O}_\\etale)$ where\n$\\epsilon : (X_\\etale, \\mathcal{O}_\\etale) \\to (X, \\mathcal{O}_X)$\nis as in Derived Categories of Spaces, Section\n\\ref{spaces-perfect-section-derived-quasi-coherent-etale}.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Dualizing complexes on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E50","source_file":"spaces-duality.tex","source_line":89,"source_end_line":105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L89-L105","statement_sha256":"cd4fc1453f2e2c65d29700cdd0e98ebe00429bfc4a3cda54e7ec7c1501376bb0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12875,"rank":12875,"depth":42,"x":2040.324,"y":1282.994,"cluster":"duality-cohomology"},{"id":"stacks:0E51","tag":"0E51","title":"Dualizing complexes on algebraic spaces · Lemma 0E51","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let K be a dualizing complex on X. Then K is an object of D_Coh(O_X) and D = RSheafHom_O_X(-, K) induces an anti-equivalence D : D_Coh(O_X) → D_Coh(O_X) which comes equipped with a canonical isomorphism id → D ∘ D. If X is quasi-compact, then D exchanges D^+_Coh(O_X) and D^-_Coh(O_X) and induces an equivalence D^b_Coh(O_X) → D^b_Coh(O_X).","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nLet $K$ be a dualizing complex on $X$.\nThen $K$ is an object of $D_{\\textit{Coh}}(\\mathcal{O}_X)$\nand $D = R\\SheafHom_{\\mathcal{O}_X}(-, K)$ induces an anti-equivalence\n$$\nD :\nD_{\\textit{Coh}}(\\mathcal{O}_X)\n\\longrightarrow\nD_{\\textit{Coh}}(\\mathcal{O}_X)\n$$\nwhich comes equipped with a canonical isomorphism\n$\\text{id} \\to D \\circ D$. If $X$ is quasi-compact, then\n$D$ exchanges $D^+_{\\textit{Coh}}(\\mathcal{O}_X)$ and\n$D^-_{\\textit{Coh}}(\\mathcal{O}_X)$ and induces an equivalence\n$D^b_{\\textit{Coh}}(\\mathcal{O}_X) \\to D^b_{\\textit{Coh}}(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Dualizing complexes on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E51","source_file":"spaces-duality.tex","source_line":173,"source_end_line":190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L173-L190","statement_sha256":"1b6bf77816c30fc0d7297c976cce77cb6938ba4acebac0083f431d7419d756e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12876,"rank":12876,"depth":43,"x":1907.28,"y":1040.322,"cluster":"duality-cohomology"},{"id":"stacks:0E52","tag":"0E52","title":"Dualizing complexes on algebraic spaces · Lemma 0E52","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. If K and K' are dualizing complexes on X, then K' is isomorphic to K ⊗_O_X^L L for some invertible object L of D(O_X).","statement_latex":"Let $S$ be a scheme.\nLet $X$ be a locally Noetherian algebraic space over $S$.\nIf $K$ and $K'$ are dualizing complexes on $X$, then $K'$\nis isomorphic to $K \\otimes_{\\mathcal{O}_X}^\\mathbf{L} L$\nfor some invertible object $L$ of $D(\\mathcal{O}_X)$.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Dualizing complexes on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E52","source_file":"spaces-duality.tex","source_line":246,"source_end_line":253,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L246-L253","statement_sha256":"3361bb4a2335fd2c6f7ef6f65fead498e5459b7641c718cf1bed1c162d586dd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12877,"rank":12877,"depth":43,"x":2200.81,"y":1143.87,"cluster":"duality-cohomology"},{"id":"stacks:0E53","tag":"0E53","title":"Dualizing complexes on algebraic spaces · Lemma 0E53","summary":"Let S be a scheme. Let X be a locally Noetherian quasi-separated algebraic space over S. Let ω_X^bullet be a dualizing complex on X. Then X the function |X| → Z defined by x ↦ δ(x) such that ω_X, overlinex^bullet[-δ(x)] is a normalized dualizing complex over O_X, overlinex is a dimension function on |X|.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian\nquasi-separated algebraic space over $S$.\nLet $\\omega_X^\\bullet$ be a dualizing complex on $X$. Then $X$ the function\n$|X| \\to \\mathbf{Z}$ defined by\n$$\nx \\longmapsto \\delta(x)\\text{ such that }\n\\omega_{X, \\overline{x}}^\\bullet[-\\delta(x)]\n\\text{ is a normalized dualizing complex over }\n\\mathcal{O}_{X, \\overline{x}}\n$$\nis a dimension function on $|X|$.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Dualizing complexes on algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E53","source_file":"spaces-duality.tex","source_line":268,"source_end_line":281,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L268-L281","statement_sha256":"72eba845f31864a169f8f2212149490d48914b8213e14ccb9ee620a330513de3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12878,"rank":12878,"depth":57,"x":1900.814,"y":1234.156,"cluster":"duality-cohomology"},{"id":"stacks:0E55","tag":"0E55","title":"Right adjoint of pushforward · Lemma 0E55","summary":"This is almost the same as [Neeman-Grothendieck]. Let S be a scheme. Let f : X → Y be a morphism between quasi-separated and quasi-compact algebraic spaces over S. The functor Rf_* : D_QCoh(X) → D_QCoh(Y) has a right adjoint.","statement_latex":"\\begin{reference}\nThis is almost the same as \\cite[Example 4.2]{Neeman-Grothendieck}.\n\\end{reference}\nLet $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism between quasi-separated and quasi-compact\nalgebraic spaces over $S$. The functor $Rf_* : D_\\QCoh(X) \\to D_\\QCoh(Y)$\nhas a right adjoint.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E55","source_file":"spaces-duality.tex","source_line":319,"source_end_line":328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L319-L328","statement_sha256":"8fe99aabda2e7bc0927be5af639183c3845d43cb214c9ba08331e0ee8ccd6c4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12879,"rank":12879,"depth":65,"x":2049.56,"y":997.134,"cluster":"duality-cohomology"},{"id":"stacks:0E56","tag":"0E56","title":"Right adjoint of pushforward · Lemma 0E56","summary":"Notation and assumptions as in Lemma [Tag 0E55]. Let a : D_QCoh(O_Y) → D_QCoh(O_X) be the right adjoint to Rf_*. Then a maps D^+_QCoh(O_Y) into D^+_QCoh(O_X). In fact, there exists an integer N such that H^i(K) = 0 for i ≤ c implies H^i(a(K)) = 0 for i ≤ c - N.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-twisted-inverse-image}.\nLet $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$ be the right\nadjoint to $Rf_*$. Then $a$ maps\n$D^+_\\QCoh(\\mathcal{O}_Y)$ into $D^+_\\QCoh(\\mathcal{O}_X)$.\nIn fact, there exists an integer $N$ such that\n$H^i(K) = 0$ for $i \\leq c$ implies $H^i(a(K)) = 0$ for $i \\leq c - N$.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E56","source_file":"spaces-duality.tex","source_line":349,"source_end_line":357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L349-L357","statement_sha256":"cca152b5555014238e03c49865e6c3511d7cbcd7022afebecda9153e20842b0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12880,"rank":12880,"depth":66,"x":2130.559,"y":1256.555,"cluster":"duality-cohomology"},{"id":"stacks:0E58","tag":"0E58","title":"Right adjoint of pushforward · Lemma 0E58","summary":"Let S be a scheme. Let f : X → Y be a morphism of quasi-compact and quasi-separated algebraic spaces over S. Let a be the right adjoint to Rf_* : D_QCoh(O_X) → D_QCoh(O_Y). Let L ∈ D_QCoh(O_X) and K ∈ D_QCoh(O_Y). Then the map ([Tag 0E57]) Rf_*RSheafHom_O_X(L, a(K)) → RSheafHom_O_Y(Rf_*L, K) becomes an isomorphism after applying the functor DQ_Y : D(O_Y) → D_QCoh(O_Y) discussed in Derived Categories of Spaces, Section [Tag 0CR3].","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated\nalgebraic spaces over $S$.\nLet $a$ be the right adjoint to\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$.\nLet $L \\in D_\\QCoh(\\mathcal{O}_X)$ and\n$K \\in D_\\QCoh(\\mathcal{O}_Y)$.\nThen the map (\\ref{equation-sheafy-trace})\n$$\nRf_*R\\SheafHom_{\\mathcal{O}_X}(L, a(K))\n\\longrightarrow\nR\\SheafHom_{\\mathcal{O}_Y}(Rf_*L, K)\n$$\nbecomes an isomorphism after applying the functor\n$DQ_Y : D(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_Y)$\ndiscussed in Derived Categories of Spaces, Section\n\\ref{spaces-perfect-section-better-coherator}.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E58","source_file":"spaces-duality.tex","source_line":400,"source_end_line":419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L400-L419","statement_sha256":"0f8b84ef71885e0c8d454f8cc4ca33cc2bf29c65fb27902b72217f15d8359b90","origin":"The Stacks Project","memory_eligible":false,"source_rank":12881,"rank":12881,"depth":62,"x":1861.963,"y":1111.087,"cluster":"duality-cohomology"},{"id":"stacks:0E59","tag":"0E59","title":"Right adjoint of pushforward · Lemma 0E59","summary":"Let S be a scheme. Let f : X → Y be a morphism of quasi-separated and quasi-compact algebraic spaces over S. For all L ∈ D_QCoh(O_X) and K ∈ D_QCoh(O_Y) ([Tag 0E57]) induces an isomorphism RHom_X(L, a(K)) → RHom_Y(Rf_*L, K) of global derived homs.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of quasi-separated and quasi-compact\nalgebraic spaces over $S$.\nFor all $L \\in D_\\QCoh(\\mathcal{O}_X)$ and $K \\in D_\\QCoh(\\mathcal{O}_Y)$\n(\\ref{equation-sheafy-trace}) induces an isomorphism\n$R\\Hom_X(L, a(K)) \\to R\\Hom_Y(Rf_*L, K)$ of global derived homs.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E59","source_file":"spaces-duality.tex","source_line":509,"source_end_line":517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L509-L517","statement_sha256":"c0049ca3f1505d51e28e4d920707d21c34f0f37f7f2c0054c182a28b203b36dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12882,"rank":12882,"depth":63,"x":2177.296,"y":1065.903,"cluster":"duality-cohomology"},{"id":"stacks:0E5C","tag":"0E5C","title":"Right adjoint of pushforward and base change, I · Lemma 0E5C","summary":"In diagram ([Tag 0E5B]) the map a ∘ Rg_* ← Rg'_* ∘ a' is an isomorphism.","statement_latex":"In diagram (\\ref{equation-base-change}) the map\n$a \\circ Rg_* \\leftarrow Rg'_* \\circ a'$ is an isomorphism.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5C","source_file":"spaces-duality.tex","source_line":584,"source_end_line":588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L584-L588","statement_sha256":"aec20ca0e1a7ceffd2c8072dbd8007b601ef495070c7b48b1089be308c3a6b2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12883,"rank":12883,"depth":63,"x":1980.928,"y":1278.346,"cluster":"duality-cohomology"},{"id":"stacks:0E5E","tag":"0E5E","title":"Right adjoint of pushforward and base change, I · Lemma 0E5E","summary":"Let S be a scheme. Consider a commutative diagram xymatrix X' ar[r]_k ar[d]_f' & X ar[d]^f Y' ar[r]^l ar[d]_g' & Y ar[d]^g Z' ar[r]^m & Z of quasi-compact and quasi-separated algebraic spaces over S where both diagrams are cartesian and where f and l as well as g and m are Tor independent. Then the maps ([Tag 0E5D]) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nX' \\ar[r]_k \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^l \\ar[d]_{g'} & Y \\ar[d]^g \\\\\nZ' \\ar[r]^m & Z\n}\n$$\nof quasi-compact and quasi-separated algebraic spaces over $S$ where\nboth diagrams are cartesian and where $f$ and $l$\nas well as $g$ and $m$ are Tor independent.\nThen the maps (\\ref{equation-base-change-map})\nfor the two squares compose to give the base\nchange map for the outer rectangle (see proof for a precise statement).","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5E","source_file":"spaces-duality.tex","source_line":653,"source_end_line":669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L653-L669","statement_sha256":"a5188ed72c57d42df7f801a31c21f3a18aa4aabf300c87b58b4ddd93327d43e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12884,"rank":12884,"depth":66,"x":1954.862,"y":1010.021,"cluster":"duality-cohomology"},{"id":"stacks:0E5F","tag":"0E5F","title":"Right adjoint of pushforward and base change, I · Lemma 0E5F","summary":"Let S be a scheme. Consider a commutative diagram xymatrix X\" ar[r]_g' ar[d]_f\" & X' ar[r]_g ar[d]_f' & X ar[d]^f Y\" ar[r]^h' & Y' ar[r]^h & Y of quasi-compact and quasi-separated algebraic spaces over S where both diagrams are cartesian and where f and h as well as f' and h' are Tor independent. Then the maps ([Tag 0E5D]) for the two squares compose to give the base change map for the outer rectangle (see proof for a precise statement).","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nX'' \\ar[r]_{g'} \\ar[d]_{f''} & X' \\ar[r]_g \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY'' \\ar[r]^{h'} & Y' \\ar[r]^h & Y\n}\n$$\nof quasi-compact and quasi-separated algebraic spaces over $S$ where\nboth diagrams are cartesian and where $f$ and $h$\nas well as $f'$ and $h'$ are Tor independent.\nThen the maps (\\ref{equation-base-change-map})\nfor the two squares compose to give the base\nchange map for the outer rectangle (see proof for a precise statement).","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward and base change, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5F","source_file":"spaces-duality.tex","source_line":790,"source_end_line":805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L790-L805","statement_sha256":"9d16b842c1e6c663bed7b7bddbb5c07bdd25b3e04497ee97e4556de593b09129","origin":"The Stacks Project","memory_eligible":false,"source_rank":12885,"rank":12885,"depth":66,"x":2190.079,"y":1193.258,"cluster":"duality-cohomology"},{"id":"stacks:0E5I","tag":"0E5I","title":"Right adjoint of pushforward and base change, II · Lemma 0E5I","summary":"In diagram ([Tag 0E5B]) assume in addition g : Y' → Y is a morphism of affine schemes and f : X → Y is proper. Then the base change map ([Tag 0E5D]) induces an isomorphism L(g')^*a(K) → a'(Lg^*K) in the following cases • for all K ∈ D_QCoh(O_X) if f is flat of finite presentation, • for all K ∈ D_QCoh(O_X) if f is perfect and Y Noetherian, • for K ∈ D_QCoh^+(O_X) if g has finite Tor dimension and Y Noetherian.","statement_latex":"In diagram (\\ref{equation-base-change}) assume in addition\n$g : Y' \\to Y$ is a morphism of affine schemes and $f : X \\to Y$ is proper.\nThen the base change map (\\ref{equation-base-change-map}) induces an\nisomorphism\n$$\nL(g')^*a(K) \\longrightarrow a'(Lg^*K)\n$$\nin the following cases\n\\begin{enumerate}\n\\item for all $K \\in D_\\QCoh(\\mathcal{O}_X)$ if $f$\nis flat of finite presentation,\n\\item for all $K \\in D_\\QCoh(\\mathcal{O}_X)$ if $f$\nis perfect and $Y$ Noetherian,\n\\item for $K \\in D_\\QCoh^+(\\mathcal{O}_X)$ if $g$ has finite Tor dimension\nand $Y$ Noetherian.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward and base change, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5I","source_file":"spaces-duality.tex","source_line":999,"source_end_line":1017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L999-L1017","statement_sha256":"7fe3bdb9ffb3aa85351642a2ac3da44a5be19871821dc1e24fa5ad74d7002ff6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12886,"rank":12886,"depth":69,"x":1868.983,"y":1191.608,"cluster":"duality-cohomology"},{"id":"stacks:0E5L","tag":"0E5L","title":"Trace map and base change · Lemma 0E5L","summary":"Suppose we have a diagram ([Tag 0E5B]). Then the maps 1 star Tr_f : Lg^* ∘ Rf_* ∘ a → Lg^* and Tr_f' star 1 : Rf'_* ∘ a' ∘ Lg^* → Lg^* agree via the base change maps β : Lg^* ∘ Rf_* → Rf'_* ∘ L(g')^* (Cohomology on Sites, Remark [Tag 07A7]) and α : L(g')^* ∘ a → a' ∘ Lg^* ([Tag 0E5D]). More precisely, the diagram xymatrix Lg^* ∘ Rf_* ∘ a ar[d]_β star 1 ar[r]_-1 star Tr_f & Lg^* Rf'_* ∘ L(g')^* ∘ a ar[r]^1 star α & Rf'_* ∘ a' ∘ Lg^* ar[u]_Tr_f' star 1 of transformations of…","statement_latex":"Suppose we have a diagram (\\ref{equation-base-change}).\nThen the maps\n$1 \\star \\text{Tr}_f : Lg^* \\circ Rf_* \\circ a \\to Lg^*$ and\n$\\text{Tr}_{f'} \\star 1 : Rf'_* \\circ a' \\circ Lg^* \\to Lg^*$\nagree via the base change maps\n$\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$\n(Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change})\nand $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$\n(\\ref{equation-base-change-map}).\nMore precisely, the diagram\n$$\n\\xymatrix{\nLg^* \\circ Rf_* \\circ a\n\\ar[d]_{\\beta \\star 1} \\ar[r]_-{1 \\star \\text{Tr}_f} &\nLg^* \\\\\nRf'_* \\circ L(g')^* \\circ a \\ar[r]^{1 \\star \\alpha} &\nRf'_* \\circ a' \\circ Lg^* \\ar[u]_{\\text{Tr}_{f'} \\star 1}\n}\n$$\nof transformations of functors commutes.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward and trace maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5L","source_file":"spaces-duality.tex","source_line":1233,"source_end_line":1255,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1233-L1255","statement_sha256":"140f29ab295be8ccfc3b919d3fb783f48cf78f432bcdacd179268655c0444585","origin":"The Stacks Project","memory_eligible":false,"source_rank":12887,"rank":12887,"depth":64,"x":2107.301,"y":1010.461,"cluster":"duality-cohomology"},{"id":"stacks:0E5M","tag":"0E5M","title":"Right adjoint of pushforward and trace maps · Lemma 0E5M","summary":"Suppose we have a diagram ([Tag 0E5B]). Then the maps 1 star eta_f : L(g')^* → L(g')^* ∘ a ∘ Rf_* and eta_f' star 1 : L(g')^* → a' ∘ Rf'_* ∘ L(g')^* agree via the base change maps β : Lg^* ∘ Rf_* → Rf'_* ∘ L(g')^* (Cohomology on Sites, Remark [Tag 07A7]) and α : L(g')^* ∘ a → a' ∘ Lg^* ([Tag 0E5D]). More precisely, the diagram xymatrix L(g')^* ar[r]_-1 star eta_f ar[d]_eta_f' star 1 & L(g')^* ∘ a ∘ Rf_* ar[d]^α a' ∘ Rf'_* ∘ L(g')^* & a' ∘ Lg^* ∘ Rf_* ar[l]_-β of…","statement_latex":"Suppose we have a diagram (\\ref{equation-base-change}). Then the maps\n$1 \\star \\eta_f : L(g')^* \\to L(g')^* \\circ a \\circ Rf_*$ and\n$\\eta_{f'} \\star 1 : L(g')^* \\to a' \\circ Rf'_* \\circ L(g')^*$\nagree via the base change maps\n$\\beta : Lg^* \\circ Rf_* \\to Rf'_* \\circ L(g')^*$\n(Cohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change})\nand $\\alpha : L(g')^* \\circ a \\to a' \\circ Lg^*$\n(\\ref{equation-base-change-map}).\nMore precisely, the diagram\n$$\n\\xymatrix{\nL(g')^* \\ar[r]_-{1 \\star \\eta_f} \\ar[d]_{\\eta_{f'} \\star 1} &\nL(g')^* \\circ a \\circ Rf_* \\ar[d]^\\alpha \\\\\na' \\circ Rf'_* \\circ L(g')^* &\na' \\circ Lg^* \\circ Rf_* \\ar[l]_-\\beta\n}\n$$\nof transformations of functors commutes.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward and trace maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5M","source_file":"spaces-duality.tex","source_line":1324,"source_end_line":1344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1324-L1344","statement_sha256":"97b94cca84c720066ba8e29408a6d5f653bd6e29fef39e1972fdb688ed0aba61","origin":"The Stacks Project","memory_eligible":false,"source_rank":12888,"rank":12888,"depth":65,"x":2077.215,"y":1279.511,"cluster":"duality-cohomology"},{"id":"stacks:0E5Q","tag":"0E5Q","title":"Right adjoint of pushforward and pullback · Lemma 0E5Q","summary":"Let S be a scheme. Let f : X → Y be a morphism of quasi-compact and quasi-separated algebraic spaces over S. The map Lf^*K ⊗^L_O_X a(L) → a(K ⊗_O_Y^L L) defined above for K, L ∈ D_QCoh(O_Y) is an isomorphism if K is perfect. In particular, ([Tag 0E5P]) is an isomorphism if K is perfect.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of quasi-compact and quasi-separated\nalgebraic spaces over $S$. The map\n$Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(L) \\to\na(K \\otimes_{\\mathcal{O}_Y}^\\mathbf{L} L)$\ndefined above for $K, L \\in D_\\QCoh(\\mathcal{O}_Y)$\nis an isomorphism if $K$ is perfect. In particular,\n(\\ref{equation-compare-with-pullback}) is an isomorphism if $K$ is perfect.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward and pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5Q","source_file":"spaces-duality.tex","source_line":1440,"source_end_line":1450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1440-L1450","statement_sha256":"a588a8c56433ed426460bfbf4f525e1f86e20bb73949cb4775d64b08be7cfa7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12889,"rank":12889,"depth":62,"x":1882.859,"y":1063.848,"cluster":"duality-cohomology"},{"id":"stacks:0E5R","tag":"0E5R","title":"Right adjoint of pushforward and pullback · Lemma 0E5R","summary":"Suppose we have a diagram ([Tag 0E5B]). Let K ∈ D_QCoh(O_Y). The diagram xymatrix L(g')^*(Lf^*K ⊗^L_O_X a(O_Y)) ar[r] ar[d] & L(g')^*a(K) ar[d] L(f')^*Lg^*K ⊗_O_X'^L a'(O_Y') ar[r] & a'(Lg^*K) commutes where the horizontal arrows are the maps ([Tag 0E5P]) for K and Lg^*K and the vertical maps are constructed using Cohomology on Sites, Remark [Tag 07A7] and ([Tag 0E5D]).","statement_latex":"Suppose we have a diagram (\\ref{equation-base-change}).\nLet $K \\in D_\\QCoh(\\mathcal{O}_Y)$. The diagram\n$$\n\\xymatrix{\nL(g')^*(Lf^*K \\otimes^\\mathbf{L}_{\\mathcal{O}_X} a(\\mathcal{O}_Y))\n\\ar[r] \\ar[d] & L(g')^*a(K) \\ar[d] \\\\\nL(f')^*Lg^*K \\otimes_{\\mathcal{O}_{X'}}^\\mathbf{L} a'(\\mathcal{O}_{Y'})\n\\ar[r] & a'(Lg^*K)\n}\n$$\ncommutes where the horizontal arrows are the maps\n(\\ref{equation-compare-with-pullback}) for $K$ and $Lg^*K$\nand the vertical maps are constructed using\nCohomology on Sites, Remark \\ref{sites-cohomology-remark-base-change} and\n(\\ref{equation-base-change-map}).","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward and pullback","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5R","source_file":"spaces-duality.tex","source_line":1476,"source_end_line":1493,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1476-L1493","statement_sha256":"a9ae9aaec879a398dfb15617ddad349ed6fae2323ae1d7c0f26c920392146956","origin":"The Stacks Project","memory_eligible":false,"source_rank":12890,"rank":12890,"depth":66,"x":2199.894,"y":1112.637,"cluster":"duality-cohomology"},{"id":"stacks:0E5T","tag":"0E5T","title":"Right adjoint of pushforward for proper flat morphisms · Lemma 0E5T","summary":"Let S be a scheme. Let Y be a quasi-compact and quasi-separated algebraic space over S. Let f : X → Y be a morphism of algebraic spaces which is proper, flat, and of finite presentation. Let a be the right adjoint for Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) of Lemma [Tag 0E55]. Then a commutes with direct sums.","statement_latex":"Let $S$ be a scheme.\nLet $Y$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $f : X \\to Y$ be a morphism of algebraic spaces which is proper, flat, and\nof finite presentation.\nLet $a$ be the right adjoint for\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of\nLemma \\ref{lemma-twisted-inverse-image}. Then $a$ commutes with direct sums.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5T","source_file":"spaces-duality.tex","source_line":1547,"source_end_line":1556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1547-L1556","statement_sha256":"b3903f432cee4e1c885be9e44a93891b367074b72b3f5aaebc9e449184a85aae","origin":"The Stacks Project","memory_eligible":false,"source_rank":12891,"rank":12891,"depth":69,"x":1926.633,"y":1256.686,"cluster":"duality-cohomology"},{"id":"stacks:0E5U","tag":"0E5U","title":"Right adjoint of pushforward for proper flat morphisms · Lemma 0E5U","summary":"Let S be a scheme. Let Y be a quasi-compact and quasi-separated algebraic space over S. Let f : X → Y be a morphism of algebraic spaces which is proper, flat, and of finite presentation. The map ([Tag 0E5P]) is an isomorphism for every object K of D_QCoh(O_Y).","statement_latex":"Let $S$ be a scheme.\nLet $Y$ be a quasi-compact and quasi-separated algebraic space over $S$.\nLet $f : X \\to Y$ be a morphism of algebraic spaces which is proper, flat, and\nof finite presentation.\nThe map (\\ref{equation-compare-with-pullback}) is an isomorphism\nfor every object $K$ of $D_\\QCoh(\\mathcal{O}_Y)$.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5U","source_file":"spaces-duality.tex","source_line":1583,"source_end_line":1591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1583-L1591","statement_sha256":"98e6db13780ded1e6965f1d7449062fffae35574d8a68bd83b12e6488952a532","origin":"The Stacks Project","memory_eligible":false,"source_rank":12892,"rank":12892,"depth":70,"x":2012.37,"y":995.173,"cluster":"duality-cohomology"},{"id":"stacks:0E5V","tag":"0E5V","title":"Right adjoint of pushforward for proper flat morphisms · Lemma 0E5V","summary":"Let Y be an affine scheme. Let f : X → Y be a morphism of algebraic spaces which is proper, flat, and of finite presentation. Let a be the right adjoint for Rf_* : D_QCoh(O_X) → D_QCoh(O_Y) of Lemma [Tag 0E55]. Then • a(O_Y) is a Y-perfect object of D(O_X), • Rf_*a(O_Y) has vanishing cohomology sheaves in positive degrees, • O_X → RSheafHom_O_X(a(O_Y), a(O_Y)) is an isomorphism.","statement_latex":"Let $Y$ be an affine scheme. Let $f : X \\to Y$ be a morphism of\nalgebraic spaces which is proper, flat, and of finite presentation.\nLet $a$ be the right adjoint for\n$Rf_* : D_\\QCoh(\\mathcal{O}_X) \\to D_\\QCoh(\\mathcal{O}_Y)$ of\nLemma \\ref{lemma-twisted-inverse-image}.\nThen\n\\begin{enumerate}\n\\item $a(\\mathcal{O}_Y)$ is a $Y$-perfect object of $D(\\mathcal{O}_X)$,\n\\item $Rf_*a(\\mathcal{O}_Y)$ has vanishing cohomology sheaves\nin positive degrees,\n\\item $\\mathcal{O}_X \\to\nR\\SheafHom_{\\mathcal{O}_X}(a(\\mathcal{O}_Y), a(\\mathcal{O}_Y))$\nis an isomorphism.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Right adjoint of pushforward for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5V","source_file":"spaces-duality.tex","source_line":1612,"source_end_line":1628,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1612-L1628","statement_sha256":"c66a3239e0185230238aed66f1ae36a3a641bdb20d5097e92918d620d66e62ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":12893,"rank":12893,"depth":69,"x":2159.584,"y":1236.878,"cluster":"duality-cohomology"},{"id":"stacks:0E5X","tag":"0E5X","title":"Relative dualizing complexes for proper flat morphisms · Definition 0E5X","summary":"Let S be a scheme. Let f : X → Y be a proper, flat morphism of algebraic spaces over S which is of finite presentation. A relative dualizing complex for X/Y is a pair (ω_X/Y^bullet, τ) consisting of a Y-perfect object ω_X/Y^bullet of D(O_X) and a map τ : Rf_*ω_X/Y^bullet → O_Y such that for any cartesian square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y where Y' is an affine scheme the pair (L(g')^*ω_X/Y^bullet, Lg^*τ) is isomorphic to the pair (a'(O_Y'),…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a proper, flat morphism\nof algebraic spaces over $S$ which is of finite presentation.\nA {\\it relative dualizing complex} for $X/Y$ is a pair\n$(\\omega_{X/Y}^\\bullet, \\tau)$ consisting of a\n$Y$-perfect object $\\omega_{X/Y}^\\bullet$ of $D(\\mathcal{O}_X)$\nand a map\n$$\n\\tau : Rf_*\\omega_{X/Y}^\\bullet \\longrightarrow \\mathcal{O}_Y\n$$\nsuch that for any cartesian square\n$$\n\\xymatrix{\nX' \\ar[r]_{g'} \\ar[d]_{f'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nwhere $Y'$ is an affine scheme the pair\n$(L(g')^*\\omega_{X/Y}^\\bullet, Lg^*\\tau)$\nis isomorphic to the pair\n$(a'(\\mathcal{O}_{Y'}), \\text{Tr}_{f', \\mathcal{O}_{Y'}})$\nstudied in Sections\n\\ref{section-twisted-inverse-image},\n\\ref{section-base-change-map},\n\\ref{section-base-change-II},\n\\ref{section-trace},\n\\ref{section-compare-with-pullback}, and\n\\ref{section-proper-flat}.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Relative dualizing complexes for proper flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5X","source_file":"spaces-duality.tex","source_line":1760,"source_end_line":1789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1760-L1789","statement_sha256":"5f87a87767f894a75f29eca3772db1d2eb8f50ec36770ed13d26e269989ad441","origin":"The Stacks Project","memory_eligible":false,"source_rank":12894,"rank":12894,"depth":0,"x":1856.383,"y":1142.095,"cluster":"duality-cohomology"},{"id":"stacks:0E5Y","tag":"0E5Y","title":"Relative dualizing complexes for proper flat morphisms · Lemma 0E5Y","summary":"Let S be a scheme. Let X → Y be a proper, flat morphism of algebraic spaces which is of finite presentation. If (ω_X/Y^bullet, τ) is a relative dualizing complex, then O_X → RSheafHom_O_X(ω_X/Y^bullet, ω_X/Y^bullet) is an isomorphism and Rf_*ω_X/Y^bullet has vanishing cohomology sheaves in positive degrees.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of\nalgebraic spaces which is of finite presentation.\nIf $(\\omega_{X/Y}^\\bullet, \\tau)$ is a relative dualizing complex,\nthen  $\\mathcal{O}_X \\to\nR\\SheafHom_{\\mathcal{O}_X}(\\omega_{X/Y}^\\bullet, \\omega_{X/Y}^\\bullet)$\nis an isomorphism and $Rf_*\\omega_{X/Y}^\\bullet$ has vanishing cohomology\nsheaves in positive degrees.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Relative dualizing complexes for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5Y","source_file":"spaces-duality.tex","source_line":1836,"source_end_line":1845,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1836-L1845","statement_sha256":"d234ce11d948128fa4871fe7a6e2a7f18db83c7921da325c9a1f03654d8a8634","origin":"The Stacks Project","memory_eligible":false,"source_rank":12895,"rank":12895,"depth":70,"x":2156.453,"y":1039.85,"cluster":"duality-cohomology"},{"id":"stacks:0E5Z","tag":"0E5Z","title":"Relative dualizing complexes for proper flat morphisms · Lemma 0E5Z","summary":"Let S be a scheme. Let X → Y be a proper, flat morphism of algebraic spaces which is of finite presentation. If (ω_j^bullet, τ_j), j = 1, 2 are two relative dualizing complexes on X/Y, then there is a unique isomorphism (ω_1^bullet, τ_1) → (ω_2^bullet, τ_2).","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of\nalgebraic spaces which is of finite presentation.\nIf $(\\omega_j^\\bullet, \\tau_j)$, $j = 1, 2$\nare two relative dualizing complexes on $X/Y$,\nthen there is a unique isomorphism\n$(\\omega_1^\\bullet, \\tau_1) \\to (\\omega_2^\\bullet, \\tau_2)$.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Relative dualizing complexes for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E5Z","source_file":"spaces-duality.tex","source_line":1853,"source_end_line":1861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1853-L1861","statement_sha256":"ac489125a79fa510d3fa87105bd99f019db543583f6877e4e86d065356ecd8c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":12896,"rank":12896,"depth":71,"x":2017.282,"y":1285.733,"cluster":"duality-cohomology"},{"id":"stacks:0E60","tag":"0E60","title":"Relative dualizing complexes for proper flat morphisms · Lemma 0E60","summary":"Let S be a scheme. Let X → Y be a proper, flat morphism of algebraic spaces which is of finite presentation. Let (ω^bullet, τ) be a pair consisting of a Y-perfect object of D(O_X) and a map τ : Rf_*ω^bullet → O_Y. Assume we have cartesian diagrams xymatrix X_i ar[r]_g_i' ar[d]_f_i & X ar[d]^f Y_i ar[r]^g_i & Y with Y_i affine such that (g_i : Y_i → Y) is an étale covering and isomorphisms of pairs (ω^bullet|_X_i, τ|_Y_i) → (a_i(O_Y_i), Tr_f_i, O_Y_i) as in Definition [Tag…","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of\nalgebraic spaces which is of finite presentation.\nLet $(\\omega^\\bullet, \\tau)$ be a pair consisting\nof a $Y$-perfect object of $D(\\mathcal{O}_X)$ and a map\n$\\tau : Rf_*\\omega^\\bullet \\to \\mathcal{O}_Y$.\nAssume we have cartesian diagrams\n$$\n\\xymatrix{\nX_i \\ar[r]_{g_i'} \\ar[d]_{f_i} & X \\ar[d]^f \\\\\nY_i \\ar[r]^{g_i} & Y\n}\n$$\nwith $Y_i$ affine such that $\\{g_i : Y_i \\to Y\\}$ is an \\'etale covering\nand isomorphisms of pairs $(\\omega^\\bullet|_{X_i}, \\tau|_{Y_i})\n\\to (a_i(\\mathcal{O}_{Y_i}), \\text{Tr}_{f_i, \\mathcal{O}_{Y_i}})$\nas in Definition \\ref{definition-relative-dualizing-proper-flat}.\nThen $(\\omega^\\bullet, \\tau)$ is a relative dualizing complex for $X$ over $Y$.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Relative dualizing complexes for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E60","source_file":"spaces-duality.tex","source_line":1923,"source_end_line":1942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L1923-L1942","statement_sha256":"8ecbf058d20585776483d23efe61cedbf3b5b6dbf4f2c0baff8c943120131b8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12897,"rank":12897,"depth":72,"x":1922.086,"y":1025.219,"cluster":"duality-cohomology"},{"id":"stacks:0E61","tag":"0E61","title":"Relative dualizing complexes for proper flat morphisms · Lemma 0E61","summary":"Let S be a scheme. Let X → Y be a proper, flat morphism of algebraic spaces which is of finite presentation. There exists a relative dualizing complex (ω_X/Y^bullet, τ).","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a proper, flat morphism of\nalgebraic spaces which is of finite presentation.\nThere exists a relative dualizing complex $(\\omega_{X/Y}^\\bullet, \\tau)$.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Relative dualizing complexes for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E61","source_file":"spaces-duality.tex","source_line":2011,"source_end_line":2016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L2011-L2016","statement_sha256":"70ad110c189fd6df7d380d9c5961eaadd197f6c221abaa4167d96ab15f42f55e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12898,"rank":12898,"depth":73,"x":2202.033,"y":1163.426,"cluster":"duality-cohomology"},{"id":"stacks:0E6C","tag":"0E6C","title":"Relative dualizing complexes for proper flat morphisms · Lemma 0E6C","summary":"Let S be a scheme. Consider a cartesian square xymatrix X' ar[d]_f' ar[r]_g' & X ar[d]^f Y' ar[r]^g & Y of algebraic spaces over S. Assume X → Y is proper, flat, and of finite presentation. Let (ω_X/Y^bullet, τ) be a relative dualizing complex for f. Then (L(g')^*ω_X/Y^bullet, Lg^*τ) is a relative dualizing complex for f'.","statement_latex":"Let $S$ be a scheme. Consider a cartesian square\n$$\n\\xymatrix{\nX' \\ar[d]_{f'} \\ar[r]_{g'} & X \\ar[d]^f \\\\\nY' \\ar[r]^g & Y\n}\n$$\nof algebraic spaces over $S$. Assume $X \\to Y$ is proper, flat, and\nof finite presentation. Let $(\\omega_{X/Y}^\\bullet, \\tau)$ be a\nrelative dualizing complex for $f$. Then\n$(L(g')^*\\omega_{X/Y}^\\bullet, Lg^*\\tau)$ is a relative dualizing\ncomplex for $f'$.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Relative dualizing complexes for proper flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6C","source_file":"spaces-duality.tex","source_line":2099,"source_end_line":2113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L2099-L2113","statement_sha256":"d988247d29152974aa4f9cf79cb5330f65810654148c3ae023846b6476c1f6ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":12899,"rank":12899,"depth":55,"x":1884.176,"y":1220.414,"cluster":"duality-cohomology"},{"id":"stacks:0E6E","tag":"0E6E","title":"Comparison with the case of schemes · Lemma 0E6E","summary":"Let S be a scheme. Let f : X → Y be a morphism of quasi-compact and quasi-separated algebraic spaces over S. Assume X and Y are representable and let f_0 : X_0 → Y_0 be a morphism of schemes representing f (awkward but temporary notation). Let a : D_QCoh(O_Y) → D_QCoh(O_X) be the right adjoint of Rf_* from Lemma [Tag 0E55]. Let a_0 : D_QCoh(O_Y_0) → D_QCoh(O_X_0) be the right adjoint of Rf_* from Duality for Schemes, Lemma [Tag 0A9E]. Then xymatrix D_QCoh(O_X_0)…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nquasi-compact and quasi-separated algebraic spaces over $S$.\nAssume $X$ and $Y$ are representable and let $f_0 : X_0 \\to Y_0$ be a\nmorphism of schemes representing $f$ (awkward but temporary notation).\nLet $a : D_\\QCoh(\\mathcal{O}_Y) \\to D_\\QCoh(\\mathcal{O}_X)$\nbe the right adjoint of $Rf_*$ from Lemma \\ref{lemma-twisted-inverse-image}.\nLet $a_0 : D_\\QCoh(\\mathcal{O}_{Y_0}) \\to D_\\QCoh(\\mathcal{O}_{X_0})$\nbe the right adjoint of $Rf_*$ from\nDuality for Schemes, Lemma \\ref{duality-lemma-twisted-inverse-image}.\nThen \n$$\n\\xymatrix{\nD_\\QCoh(\\mathcal{O}_{X_0})\n\\ar@{=}[rrrrrr]_{\\text{Derived Categories of Spaces, Lemma\n\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}}\n& & & & & &\nD_\\QCoh(\\mathcal{O}_X) \\\\\nD_\\QCoh(\\mathcal{O}_{Y_0}) \\ar[u]^{a_0}\n\\ar@{=}[rrrrrr]^{\\text{Derived Categories of Spaces, Lemma\n\\ref{spaces-perfect-lemma-derived-quasi-coherent-small-etale-site}}}\n& & & & & &\nD_\\QCoh(\\mathcal{O}_Y) \\ar[u]_a\n}\n$$\nis commutative.","area":"Duality & Cohomology","chapter":"Duality for Spaces","chapter_id":"spaces-duality","section":"Comparison with the case of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6E","source_file":"spaces-duality.tex","source_line":2136,"source_end_line":2163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-duality.tex#L2136-L2163","statement_sha256":"7a295574761fb8643d8a1b255c3af913bbcbf5b880939d5b1343ce8774f5a980","origin":"The Stacks Project","memory_eligible":false,"source_rank":12900,"rank":12900,"depth":66,"x":2072.9,"y":997.832,"cluster":"duality-cohomology"},{"id":"stacks:0AI1","tag":"0AI1","title":"Formal schemes \\`a la EGA · Lemma 0AI1","summary":"Choose a category of schemes Sch_α as in Sets, Lemma [Tag 000J]. Given a formal scheme X let h_ X : (Sch_α)^opp → Sets, h_ X(S) = Mor_Formal Schemes(S, X) be its functor of points. Then we have Mor_Formal Schemes( X, Y) = Mor_PSh(Sch_α)(h_ X, h_ Y) provided the size of X is not too large.","statement_latex":"Choose a category of schemes $\\Sch_\\alpha$\nas in Sets, Lemma \\ref{sets-lemma-construct-category}.\nGiven a formal scheme $\\mathfrak X$ let\n$$\nh_\\mathfrak X : (\\Sch_\\alpha)^{opp} \\longrightarrow \\textit{Sets},\\quad\nh_\\mathfrak X(S) = \\Mor_{\\textit{Formal Schemes}}(S, \\mathfrak X)\n$$\nbe its functor of points. Then we have\n$$\n\\Mor_{\\textit{Formal Schemes}}(\\mathfrak X, \\mathfrak Y) =\n\\Mor_{\\textit{PSh}(\\Sch_\\alpha)}(h_\\mathfrak X, h_\\mathfrak Y)\n$$\nprovided the size of $\\mathfrak X$ is not too large.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Formal schemes \\`a la EGA","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AI1","source_file":"formal-spaces.tex","source_line":305,"source_end_line":320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L305-L320","statement_sha256":"d3112a2b495b350bb2cd2520bcf9c38dcc6bc028e09b86834144cb2525bd8bc5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12901,"rank":12901,"depth":2,"x":1116.449,"y":1676.152,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AI2","tag":"0AI2","title":"Formal schemes \\`a la EGA · Lemma 0AI2","summary":"Formal schemes are fpqc sheaves Let X be a formal scheme. The functor of points h_ X (see Lemma [Tag 0AI1]) satisfies the sheaf condition for fpqc coverings.","statement_latex":"\\begin{slogan}\nFormal schemes are fpqc sheaves\n\\end{slogan}\nLet $\\mathfrak X$ be a formal scheme. The functor of points\n$h_\\mathfrak X$ (see Lemma \\ref{lemma-fully-faithful})\nsatisfies the sheaf condition for fpqc coverings.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Formal schemes \\`a la EGA","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AI2","source_file":"formal-spaces.tex","source_line":429,"source_end_line":437,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L429-L437","statement_sha256":"4891c54ecacf0704ecf21b9dacf559ab01e67e0dcb63b16a06417eaf992faa68","origin":"The Stacks Project","memory_eligible":false,"source_rank":12902,"rank":12902,"depth":8,"x":1078.557,"y":1535.912,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AMS","tag":"0AMS","title":"Topological rings and modules · Lemma 0AMS","summary":"Let R be a topological ring. Let M be a linearly topologized R-module and let M_λ, λ ∈ Lambda be a fundamental system of open submodules. Let N ⊂ M be a submodule. The closure of N is ⋂_λ ∈ Lambda (N + M_λ).","statement_latex":"Let $R$ be a topological ring. Let $M$ be a linearly topologized\n$R$-module and let $M_\\lambda$, $\\lambda \\in \\Lambda$ be a fundamental\nsystem of open submodules. Let $N \\subset M$ be a submodule.\nThe closure of $N$ is $\\bigcap_{\\lambda \\in \\Lambda} (N + M_\\lambda)$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMS","source_file":"formal-spaces.tex","source_line":704,"source_end_line":710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L704-L710","statement_sha256":"602f3f5319e361d44100d9066a7d8232365b4e90ed6d0107e3341d89cb728317","origin":"The Stacks Project","memory_eligible":false,"source_rank":12903,"rank":12903,"depth":0,"x":1219.811,"y":1618.136,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARZ","tag":"0ARZ","title":"Topological rings and modules · Lemma 0ARZ","summary":"Let R be a topological ring. Let M be a linearly topologized R-module. Let N ⊂ M be a submodule. Then • 0 → N^wedge → M^wedge → (M/N)^wedge is exact, and • N^wedge is the closure of the image of N → M^wedge.","statement_latex":"Let $R$ be a topological ring. Let $M$ be a linearly topologized\n$R$-module. Let $N \\subset M$ be a submodule. Then\n\\begin{enumerate}\n\\item $0 \\to N^\\wedge \\to M^\\wedge \\to (M/N)^\\wedge$ is exact, and\n\\item $N^\\wedge$ is the closure of the image of $N \\to M^\\wedge$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARZ","source_file":"formal-spaces.tex","source_line":730,"source_end_line":738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L730-L738","statement_sha256":"dbefea50e0723170cd69c2e54a1e5ffcc03db9784dff1818f1bd6a1a6955e293","origin":"The Stacks Project","memory_eligible":false,"source_rank":12904,"rank":12904,"depth":0,"x":1048.885,"y":1637.73,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AMT","tag":"0AMT","title":"Topological rings and modules · Lemma 0AMT","summary":"Let R be a topological ring. Let M be a complete, linearly topologized R-module. Let N ⊂ M be a closed submodule. If M has a countable fundamental system of neighbourhoods of 0, then M/N is complete and the map M → M/N is open.","statement_latex":"Let $R$ be a topological ring. Let $M$ be a complete, linearly topologized\n$R$-module. Let $N \\subset M$ be a closed submodule. If $M$ has a\ncountable fundamental system of neighbourhoods of $0$, then\n$M/N$ is complete and the map $M \\to M/N$ is open.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMT","source_file":"formal-spaces.tex","source_line":754,"source_end_line":760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L754-L760","statement_sha256":"cddd97be1baa02a3794791d6f60cc8c0ea17486d62883507c164b17c93ed00e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12905,"rank":12905,"depth":2,"x":1159.582,"y":1525.876,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AS0","tag":"0AS0","title":"Topological rings and modules · Lemma 0AS0","summary":"[Ma] Let R be a topological ring. Let M be a linearly topologized R-module. Let N ⊂ M be a submodule. Assume M has a countable fundamental system of neighbourhoods of 0. Then • 0 → N^wedge → M^wedge → (M/N)^wedge → 0 is exact, • N^wedge is the closure of the image of N → M^wedge, • M^wedge → (M/N)^wedge is open.","statement_latex":"\\begin{reference}\n\\cite[Theorem 8.1]{Ma}\n\\end{reference}\nLet $R$ be a topological ring. Let $M$ be a linearly topologized\n$R$-module. Let $N \\subset M$ be a submodule. Assume $M$ has a\ncountable fundamental system of neighbourhoods of $0$. Then\n\\begin{enumerate}\n\\item $0 \\to N^\\wedge \\to M^\\wedge \\to (M/N)^\\wedge \\to 0$ is exact,\n\\item $N^\\wedge$ is the closure of the image of $N \\to M^\\wedge$,\n\\item $M^\\wedge \\to (M/N)^\\wedge$ is open.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AS0","source_file":"formal-spaces.tex","source_line":780,"source_end_line":793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L780-L793","statement_sha256":"4b11e176e9b2f8353950179add384baad7f9d3f939fee9ce2e4d70ffb84b75b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12906,"rank":12906,"depth":3,"x":1167.936,"y":1671.709,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0F1S","tag":"0F1S","title":"Topological rings and modules · Lemma 0F1S","summary":"Let R be a topological ring. Let M be a topological R-module. Let I ⊂ R be a finitely generated ideal. Assume M has an open submodule whose topology is I-adic. Then M^wedge has an open submodule whose topology is I-adic and we have M^wedge/I^n M^wedge = M/I^nM for all n ≥ 1.","statement_latex":"Let $R$ be a topological ring. Let $M$ be a topological $R$-module.\nLet $I \\subset R$ be a finitely generated ideal. Assume $M$\nhas an open submodule whose topology is $I$-adic. Then $M^\\wedge$\nhas an open submodule whose topology is $I$-adic and we have\n$M^\\wedge/I^n M^\\wedge = M/I^nM$ for all $n \\geq 1$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1S","source_file":"formal-spaces.tex","source_line":807,"source_end_line":814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L807-L814","statement_sha256":"761709878062c040c9fb75564a685a7eb84bd0aec8a077c0aba69cda1e4da787","origin":"The Stacks Project","memory_eligible":false,"source_rank":12907,"rank":12907,"depth":5,"x":1044.044,"y":1568.533,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AMU","tag":"0AMU","title":"Topological rings and modules · Definition 0AMU","summary":"Compare with [EGA] Let R be a topological ring. Let M and N be linearly topologized R-modules. The tensor product of M and N is the (usual) tensor product M ⊗_R N endowed with the linear topology defined by declaring Im(M_μ ⊗_R N + M ⊗_R N_ν → M ⊗_R N) to be a fundamental system of open submodules, where M_μ ⊂ M and N_ν ⊂ N run through fundamental systems of open submodules in M and N. The completed tensor product M widehat⊗_R N = lim M ⊗_R N/(M_μ ⊗_R N + M ⊗_R N_ν) = lim…","statement_latex":"\\begin{reference}\nCompare with \\cite[0, Section 7.7]{EGA}\n\\end{reference}\nLet $R$ be a topological ring. Let $M$ and $N$ be linearly\ntopologized $R$-modules. The {\\it tensor product} of $M$ and $N$\nis the (usual) tensor product $M \\otimes_R N$ endowed\nwith the linear topology defined by declaring\n$$\n\\Im(M_\\mu \\otimes_R N + M \\otimes_R N_\\nu \\longrightarrow M \\otimes_R N)\n$$\nto be a fundamental system of open submodules, where\n$M_\\mu \\subset M$ and $N_\\nu \\subset N$ run through fundamental\nsystems of open submodules in $M$ and $N$.\nThe {\\it completed tensor product}\n$$\nM \\widehat{\\otimes}_R N =\n\\lim M \\otimes_R N/(M_\\mu \\otimes_R N + M \\otimes_R N_\\nu) =\n\\lim M/M_\\mu \\otimes_R N/N_\\nu\n$$\nis the completion of the tensor product.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMU","source_file":"formal-spaces.tex","source_line":838,"source_end_line":860,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L838-L860","statement_sha256":"3e786308a2712a55edc2a528c68bc04a7ec1895b61cf2a82a87e7c6e96cac40c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12908,"rank":12908,"depth":0,"x":1219.012,"y":1574.336,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AMV","tag":"0AMV","title":"Topological rings and modules · Definition 0AMV","summary":"Let A be a linearly topologized ring. • An element f ∈ A is called topologically nilpotent if f^n → 0 as n → ∞. • A weak ideal of definition for A is an open ideal I ⊂ A consisting entirely of topologically nilpotent elements. • We say A is weakly pre-admissible if A has a weak ideal of definition. • We say A is weakly admissible if A is weakly pre-admissible and complete.","statement_latex":"Let $A$ be a linearly topologized ring.\n\\begin{enumerate}\n\\item An element $f \\in A$ is called {\\it topologically nilpotent}\nif $f^n \\to 0$ as $n \\to \\infty$.\n\\item A {\\it weak ideal of definition} for $A$ is an open ideal\n$I \\subset A$ consisting entirely of topologically nilpotent elements.\n\\item We say $A$ is {\\it weakly pre-admissible} if $A$ has a weak\nideal of definition.\n\\item We say $A$ is {\\it weakly admissible} if $A$ is weakly pre-admissible\nand complete\\footnote{By our conventions this includes separated.}.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMV","source_file":"formal-spaces.tex","source_line":873,"source_end_line":886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L873-L886","statement_sha256":"64de47006ec2a6f6fe5a1dff975ee427f63c2b7f30e0828028df38a1d013ab11","origin":"The Stacks Project","memory_eligible":false,"source_rank":12909,"rank":12909,"depth":0,"x":1084.838,"y":1669.684,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DCZ","tag":"0DCZ","title":"Topological rings and modules · Lemma 0DCZ","summary":"Let A be a weakly admissible topological ring. Let I ⊂ A be a weak ideal of definition. Then (A, I) is a henselian pair.","statement_latex":"Let $A$ be a weakly admissible topological ring. Let $I \\subset A$\nbe a weak ideal of definition. Then $(A, I)$ is a henselian pair.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DCZ","source_file":"formal-spaces.tex","source_line":898,"source_end_line":902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L898-L902","statement_sha256":"87734ba9fe6a72a8105abe34a057a89b5808fd6d8bf069314b00f01b1529dfa4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12910,"rank":12910,"depth":47,"x":1107.181,"y":1522.712,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AMW","tag":"0AMW","title":"Topological rings and modules · Lemma 0AMW","summary":"Let B be a linearly topologized ring. The set of topologically nilpotent elements of B is a closed, radical ideal of B. Let φ : A → B be a continuous map of linearly topologized rings. • If f ∈ A is topologically nilpotent, then φ(f) is topologically nilpotent. • If I ⊂ A consists of topologically nilpotent elements, then the closure of φ(I)B consists of topologically nilpotent elements.","statement_latex":"Let $B$ be a linearly topologized ring. The set of topologically nilpotent\nelements of $B$ is a closed, radical ideal of $B$.\nLet $\\varphi : A \\to B$ be a continuous map of linearly topologized rings.\n\\begin{enumerate}\n\\item If $f \\in A$ is topologically nilpotent, then $\\varphi(f)$ is\ntopologically nilpotent.\n\\item If $I \\subset A$ consists of topologically nilpotent elements,\nthen the closure of $\\varphi(I)B$ consists of topologically nilpotent\nelements.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMW","source_file":"formal-spaces.tex","source_line":916,"source_end_line":928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L916-L928","statement_sha256":"fefabb61f8cbc322062d5324c28b0df6640ee7dd1cff5690b9f9b2bd3235efd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12911,"rank":12911,"depth":1,"x":1209.262,"y":1644.201,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AMZ","tag":"0AMZ","title":"Topological rings and modules · Lemma 0AMZ","summary":"Let A → B be a continuous map of linearly topologized rings. Let I ⊂ A be an ideal. The closure of IB is the kernel of B → B widehat⊗_A A/I.","statement_latex":"Let $A \\to B$ be a continuous map of linearly topologized rings.\nLet $I \\subset A$ be an ideal. The closure of $IB$\nis the kernel of $B \\to B \\widehat{\\otimes}_A A/I$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMZ","source_file":"formal-spaces.tex","source_line":947,"source_end_line":952,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L947-L952","statement_sha256":"884491f2940af1ebe799d9159282e6ddeed6491163787a90187f7524f6ac04ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":12912,"rank":12912,"depth":1,"x":1035.675,"y":1612.426,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GB4","tag":"0GB4","title":"Topological rings and modules · Lemma 0GB4","summary":"Let B → A and B → C be continuous homomorphisms of linearly topologized rings. • If A and C are weakly pre-admissible, then A widehat⊗_B C is weakly admissible. • If A and C are pre-admissible, then A widehat⊗_B C is admissible. • If A and C have a countable fundamental system of open ideals, then A widehat⊗_B C has a countable fundamental system of open ideals. • If A and C are pre-adic and have finitely generated ideals of definition, then A widehat⊗_B C is adic and has…","statement_latex":"Let $B \\to A$ and $B \\to C$ be continuous homomorphisms of\nlinearly topologized rings.\n\\begin{enumerate}\n\\item If $A$ and $C$ are weakly pre-admissible, then\n$A \\widehat{\\otimes}_B C$ is weakly admissible.\n\\item If $A$ and $C$ are pre-admissible, then\n$A \\widehat{\\otimes}_B C$ is admissible.\n\\item If $A$ and $C$ have a countable fundamental system of open\nideals, then $A \\widehat{\\otimes}_B C$ has a countable fundamental\nsystem of open ideals.\n\\item If $A$ and $C$ are pre-adic and have finitely generated ideals\nof definition, then $A \\widehat{\\otimes}_B C$ is adic and has\na finitely generated ideal of definition.\n\\item If $A$ and $C$ are pre-adic Noetherian rings and\n$B/\\mathfrak b \\to A/\\mathfrak a$ is of finite type\nwhere $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$\nare the ideals of topologically nilpotent elements, then\n$A \\widehat{\\otimes}_B C$ is adic Noetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Topological rings and modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GB4","source_file":"formal-spaces.tex","source_line":969,"source_end_line":990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L969-L990","statement_sha256":"ebac94ff73666ea48b868d0756e3a82395dc049e2d1b7618e80b26b2966dc572","origin":"The Stacks Project","memory_eligible":false,"source_rank":12913,"rank":12913,"depth":6,"x":1189.772,"y":1537.093,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AMX","tag":"0AMX","title":"Taut ring maps · Definition 0AMX","summary":"Let φ : A → B be a continuous map of linearly topologized rings. We say φ is taut if for every open ideal I ⊂ A the closure of the ideal φ(I)B is open and these closures form a fundamental system of open ideals.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous map of linearly topologized rings.\nWe say $\\varphi$ is {\\it taut}\\footnote{This is nonstandard notation.\nThe definition generalizes to modules, by saying a linearly topologized\n$A$-module $M$ is $A$-taut if for every open ideal $I \\subset A$ the closure\nof $IM$ in $M$ is open and these closures form a fundamental system of\nneighbourhoods of $0$ in $M$.}\nif for every open ideal $I \\subset A$ the closure of the ideal $\\varphi(I)B$\nis open and these closures form a fundamental system of open ideals.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMX","source_file":"formal-spaces.tex","source_line":1082,"source_end_line":1092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1082-L1092","statement_sha256":"d75d0cb5c2b1ac05b6534e74e5b755d862869a97d314529617ef395f236ee0ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":12914,"rank":12914,"depth":0,"x":1136.536,"y":1680.584,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AMY","tag":"0AMY","title":"Taut ring maps · Lemma 0AMY","summary":"Let φ : A → B be a continuous map of weakly admissible topological rings. The following are equivalent • φ is taut, • for every weak ideal of definition I ⊂ A the closure of φ(I)B is a weak ideal of definition of B and these form a fundamental system of weak ideals of definition of B.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous map of weakly admissible topological\nrings. The following are equivalent\n\\begin{enumerate}\n\\item $\\varphi$ is taut,\n\\item for every weak ideal of definition $I \\subset A$ the closure of\n$\\varphi(I)B$ is a weak ideal of definition of $B$ and these form a\nfundamental system of weak ideals of definition of $B$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMY","source_file":"formal-spaces.tex","source_line":1101,"source_end_line":1111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1101-L1111","statement_sha256":"c08bfd2edeac659047473a14b0c61f1e4fdae9388d0e7e08595ca6638e720162","origin":"The Stacks Project","memory_eligible":false,"source_rank":12915,"rank":12915,"depth":2,"x":1060.135,"y":1544.093,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GX2","tag":"0GX2","title":"Taut ring maps · Lemma 0GX2","summary":"Let A be a linearly topologized ring. The map A → A^wedge from A to its completion is taut.","statement_latex":"Let $A$ be a linearly topologized ring. The map $A \\to A^\\wedge$\nfrom $A$ to its completion is taut.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GX2","source_file":"formal-spaces.tex","source_line":1124,"source_end_line":1128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1124-L1128","statement_sha256":"bb6e2d1477a536c305c6cb4cb99c6a5f3d3ebfff530c86b1e1101e58c70a1f3c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12916,"rank":12916,"depth":2,"x":1226.81,"y":1601.587,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GX3","tag":"0GX3","title":"Taut ring maps · Lemma 0GX3","summary":"Let A → B and B → C be continuous homomorphisms of linearly topologized rings. If A → B and B → C are taut, then A → C is taut.","statement_latex":"Let $A \\to B$ and $B \\to C$ be continuous homomorphisms of\nlinearly topologized rings. If $A \\to B$ and $B \\to C$ are taut, then\n$A \\to C$ is taut.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GX3","source_file":"formal-spaces.tex","source_line":1139,"source_end_line":1144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1139-L1144","statement_sha256":"074d1afc541114037630f5049df3bd49a486e0e8105d7d962cb3c84307663bda","origin":"The Stacks Project","memory_eligible":false,"source_rank":12917,"rank":12917,"depth":0,"x":1057.088,"y":1653.946,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GX4","tag":"0GX4","title":"Taut ring maps · Lemma 0GX4","summary":"Let A → B and B → C be continuous homomorphisms of linearly topologized rings. If A → C is taut, then B → C is taut.","statement_latex":"Let $A \\to B$ and $B \\to C$ be continuous homomorphisms of\nlinearly topologized rings. If $A \\to C$ is taut, then\n$B \\to C$ is taut.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GX4","source_file":"formal-spaces.tex","source_line":1151,"source_end_line":1156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1151-L1156","statement_sha256":"aed04d8bc82405b26fd14cc7063b1d58f0e077792d8432270fd152eb17aa1804","origin":"The Stacks Project","memory_eligible":false,"source_rank":12918,"rank":12918,"depth":0,"x":1140.417,"y":1518.572,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GX5","tag":"0GX5","title":"Taut ring maps · Lemma 0GX5","summary":"Let A → B and A → C be continuous homomorphisms of linearly topologized rings. If A → B is taut, then C → B widehat⊗_A C is taut.","statement_latex":"Let $A \\to B$ and $A \\to C$ be continuous homomorphisms of\nlinearly topologized rings. If $A \\to B$ is taut, then\n$C \\to B \\widehat{\\otimes}_A C$ is taut.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GX5","source_file":"formal-spaces.tex","source_line":1170,"source_end_line":1175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1170-L1175","statement_sha256":"9fccf29fbe1ef4c6892a1d534173d26458de636b3df77934f74a925fba116e6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12919,"rank":12919,"depth":0,"x":1187.996,"y":1666.178,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GX6","tag":"0GX6","title":"Taut ring maps · Lemma 0GX6","summary":"Let φ : A → B be a continuous homomorphism of linearly topologized rings. If φ is taut and A has a countable fundamental system of open ideals, then B has a countable fundamental system of open ideals.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous homomorphism of\nlinearly topologized rings. If $\\varphi$ is taut and $A$\nhas a countable fundamental system of open ideals, then\n$B$ has a countable fundamental system of open ideals.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GX6","source_file":"formal-spaces.tex","source_line":1192,"source_end_line":1198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1192-L1198","statement_sha256":"980c51be7978ec1a5dbefd03df9b827381efeaf0852c3b6f6fee5d8123fda836","origin":"The Stacks Project","memory_eligible":false,"source_rank":12920,"rank":12920,"depth":0,"x":1033.693,"y":1584.057,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GX7","tag":"0GX7","title":"Taut ring maps · Lemma 0GX7","summary":"Let φ : A → B be a continuous homomorphism of linearly topologized rings. If φ is taut and A is weakly pre-admissible, then B is weakly pre-admissible.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous homomorphism of\nlinearly topologized rings. If $\\varphi$ is taut and $A$\nis weakly pre-admissible, then $B$ is weakly pre-admissible.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GX7","source_file":"formal-spaces.tex","source_line":1204,"source_end_line":1209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1204-L1209","statement_sha256":"8e18e657d862aecdb3c7f6084c009985f73e288cdb1b1cf7ba2077d34849a3ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":12921,"rank":12921,"depth":2,"x":1214.116,"y":1556.967,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GX8","tag":"0GX8","title":"Taut ring maps · Lemma 0GX8","summary":"Let φ : A → B be a continuous homomorphism of linearly topologized rings. If φ is taut and A is pre-admissible, then B is pre-admissible.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous homomorphism of\nlinearly topologized rings. If $\\varphi$ is taut and $A$\nis pre-admissible, then $B$ is pre-admissible.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GX8","source_file":"formal-spaces.tex","source_line":1218,"source_end_line":1223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1218-L1223","statement_sha256":"ca87b8f21e646ff259af185936fe69876f4763bd0b6e923bf66d1eb6d3c816f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12922,"rank":12922,"depth":0,"x":1102.489,"y":1679.725,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0APT","tag":"0APT","title":"Taut ring maps · Lemma 0APT","summary":"Let φ : A → B be a continuous homomorphism of linearly topologized rings. Assume • φ is taut and has dense image, • A is complete and has a countable fundamental system of open ideals, and • B is separated. Then φ is surjective and open, B is complete, and B = A/K for some closed ideal K ⊂ A.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous homomorphism of\nlinearly topologized rings. Assume\n\\begin{enumerate}\n\\item $\\varphi$ is taut and has dense image,\n\\item $A$ is complete and has a countable fundamental system of\nopen ideals, and\n\\item $B$ is separated.\n\\end{enumerate}\nThen $\\varphi$ is surjective and open, $B$ is complete, and $B = A/K$ for\nsome closed ideal $K \\subset A$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APT","source_file":"formal-spaces.tex","source_line":1237,"source_end_line":1249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1237-L1249","statement_sha256":"1683679116d31349cb982c7a6a703d17eed08b4a76512c5318c5026954d6bbf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12923,"rank":12923,"depth":3,"x":1086.029,"y":1525.356,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBR","tag":"0GBR","title":"Adic ring maps · Definition 0GBR","summary":"Let A and B be pre-adic topological rings. A ring homomorphism φ : A → B is adic if there exists an ideal of definition I ⊂ A such that the topology on B is the I-adic topology.","statement_latex":"Let $A$ and $B$ be pre-adic topological rings. A ring homomorphism\n$\\varphi : A \\to B$ is {\\it adic}\\footnote{This may be nonstandard terminology.}\nif there exists an ideal of definition $I \\subset A$ such that\nthe topology on $B$ is the $I$-adic topology.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Adic ring maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBR","source_file":"formal-spaces.tex","source_line":1274,"source_end_line":1280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1274-L1280","statement_sha256":"0d17624bfdd230af7decefdcb42323931546a50a4369b24933e04519db64f57f","origin":"The Stacks Project","memory_eligible":false,"source_rank":12924,"rank":12924,"depth":0,"x":1222.753,"y":1630.189,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXA","tag":"0GXA","title":"Adic ring maps · Lemma 0GXA","summary":"Let A → B and B → C be continuous homomorphisms of pre-adic rings. If A → B and B → C are adic, then A → C is adic.","statement_latex":"Let $A \\to B$ and $B \\to C$ be continuous homomorphisms of\npre-adic rings. If $A \\to B$ and $B \\to C$ are adic, then\n$A \\to C$ is adic.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Adic ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXA","source_file":"formal-spaces.tex","source_line":1287,"source_end_line":1292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1287-L1292","statement_sha256":"9aa0f04d61c6cae73d6b1b45533166188fb59e20f7556174c4e126fa545fecf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12925,"rank":12925,"depth":0,"x":1037.027,"y":1630.468,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXB","tag":"0GXB","title":"Adic ring maps · Lemma 0GXB","summary":"Let A → B and B → C be continuous homomorphisms of pre-adic rings. If A → C is adic, then B → C is adic.","statement_latex":"Let $A \\to B$ and $B \\to C$ be continuous homomorphisms of\npre-adic rings. If $A \\to C$ is adic, then\n$B \\to C$ is adic.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Adic ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXB","source_file":"formal-spaces.tex","source_line":1298,"source_end_line":1303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1298-L1303","statement_sha256":"072697e0d7683c8bbf136bb939ac99474a760c95bb70e34830487c9a590c0f38","origin":"The Stacks Project","memory_eligible":false,"source_rank":12926,"rank":12926,"depth":0,"x":1174.198,"y":1524.535,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXC","tag":"0GXC","title":"Adic ring maps · Lemma 0GXC","summary":"Let φ : A → B be a continuous homomorphism between pre-adic topological rings. If φ is adic, then φ is taut.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous homomorphism between\npre-adic topological rings. If $\\varphi$ is adic, then $\\varphi$ is taut.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Adic ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXC","source_file":"formal-spaces.tex","source_line":1316,"source_end_line":1320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1316-L1320","statement_sha256":"5ee3f124c7e9f94cc8c4a493dcfa79fd8aa97f34ac92c9259a0df4de6c25f000","origin":"The Stacks Project","memory_eligible":false,"source_rank":12927,"rank":12927,"depth":0,"x":1158.186,"y":1680.985,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0APU","tag":"0APU","title":"Adic ring maps · Lemma 0APU","summary":"Let φ : A → B be a continuous map of linearly topologized rings. If φ is taut, A is pre-adic and has a finitely generated ideal of definition, and B is complete, then B is adic and has a finitely generated ideal of definition and the ring map φ is adic.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous map of linearly topologized rings.\nIf $\\varphi$ is taut, $A$ is pre-adic and has a finitely generated ideal\nof definition, and $B$ is complete, then $B$ is adic and has a finitely\ngenerated ideal of definition and the ring map $\\varphi$ is adic.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Adic ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APU","source_file":"formal-spaces.tex","source_line":1339,"source_end_line":1345,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1339-L1345","statement_sha256":"56f527aa14c1513b302a8687791529bc1191ef6a7c6961e0b61a09a98bca0a2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12928,"rank":12928,"depth":5,"x":1043.817,"y":1556.137,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXE","tag":"0GXE","title":"Weakly adic rings · Definition 0GXE","summary":"[Gabber-Ramero] Let A be a linearly topologized ring. • We say A is weakly pre-adic. if there exists an ideal I ⊂ A such that the closure of I^n is open for all n ≥ 0 and these closures form a fundamental system of open ideals. • We say A is weakly adic if A is weakly pre-adic and complete.","statement_latex":"\\begin{reference}\n\\cite[Definition 8.3.8]{Gabber-Ramero}\n\\end{reference}\nLet $A$ be a linearly topologized ring.\n\\begin{enumerate}\n\\item We say $A$ is {\\it weakly pre-adic}\\footnote{In \\cite{Gabber-Ramero} the\nauthors say $A$ is {\\it $c$-adic}.} if there exists an ideal\n$I \\subset A$ such that the closure of $I^n$ is open for all $n \\geq 0$\nand these closures form a fundamental system of open ideals.\n\\item We say $A$ is {\\it weakly adic} if $A$ is weakly pre-adic\nand complete\\footnote{By our conventions this includes separated.}.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Weakly adic rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXE","source_file":"formal-spaces.tex","source_line":1377,"source_end_line":1391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1377-L1391","statement_sha256":"6306b2c09f0ce49756ce00667dcf1ca0da71aafc498fa47dad8ab2bed92cedd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12929,"rank":12929,"depth":0,"x":1229.137,"y":1583.389,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXF","tag":"0GXF","title":"Weakly adic rings · Lemma 0GXF","summary":"Let A be a linearly topologized ring. The following are equivalent • A is weakly pre-adic, • there exists a taut continuous ring map A' → A where A' is a pre-adic topological ring, and • A is pre-admissible and there exists an ideal of definition I such that the closure of I^n is open for all n ≥ 1, and • A is pre-admissible and for every ideal of definition I the closure of I^n is open for all n ≥ 1. The completion of a weakly pre-adic ring is weakly adic. If A is weakly…","statement_latex":"Let $A$ be a linearly topologized ring. The following are equivalent\n\\begin{enumerate}\n\\item $A$ is weakly pre-adic,\n\\item there exists a taut continuous ring map $A' \\to A$\nwhere $A'$ is a pre-adic topological ring, and\n\\item $A$ is pre-admissible and there exists an ideal of definition $I$\nsuch that the closure of $I^n$ is open for all $n \\geq 1$, and\n\\item $A$ is pre-admissible and for every ideal of definition $I$\nthe closure of $I^n$ is open for all $n \\geq 1$.\n\\end{enumerate}\nThe completion of a weakly pre-adic ring is weakly adic.\nIf $A$ is weakly adic, then $A$ is admissible and has a countable\nfundamental system of open ideals.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Weakly adic rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXF","source_file":"formal-spaces.tex","source_line":1415,"source_end_line":1430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1415-L1430","statement_sha256":"bfda3a4953a0553ab717a0922ebfcf75866d5e580d93e2b4e639ff80936d5731","origin":"The Stacks Project","memory_eligible":false,"source_rank":12930,"rank":12930,"depth":3,"x":1070.068,"y":1668.717,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXG","tag":"0GXG","title":"Weakly adic rings · Lemma 0GXG","summary":"Let A be a complete linearly topologized ring. Let I ⊂ A be a finitely generated ideal such that the closure of I^n is open for all n ≥ 0 and these closures form a fundamental system of open ideals. Then A is adic and has a finitely generated ideal of definition.","statement_latex":"Let $A$ be a complete linearly topologized ring. Let $I \\subset A$ be a\nfinitely generated ideal such that the closure of $I^n$ is open for all\n$n \\geq 0$ and these closures form a fundamental system of open ideals.\nThen $A$ is adic and has a finitely generated ideal of definition.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Weakly adic rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXG","source_file":"formal-spaces.tex","source_line":1481,"source_end_line":1487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1481-L1487","statement_sha256":"244fc69e23cf2b395de56b930a446c723ad9a07e8dae0ad3619f01fff5104e74","origin":"The Stacks Project","memory_eligible":false,"source_rank":12931,"rank":12931,"depth":6,"x":1118.898,"y":1515.058,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXH","tag":"0GXH","title":"Weakly adic rings · Lemma 0GXH","summary":"Let A be a weakly adic topological ring. Let I be an ideal of definition such that I/I_2 is a finitely generated module where I_2 is the closure of I^2. Then A is adic and has a finitely generated ideal of definition.","statement_latex":"Let $A$ be a weakly adic topological ring. Let $I$ be an\nideal of definition such that $I/I_2$ is a finitely generated\nmodule where $I_2$ is the closure of $I^2$.\nThen $A$ is adic and has a finitely generated ideal of definition.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Weakly adic rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXH","source_file":"formal-spaces.tex","source_line":1496,"source_end_line":1502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1496-L1502","statement_sha256":"af72078365bcaba18874f961c8ad9945294544681da1d3b9826ca7072441ae87","origin":"The Stacks Project","memory_eligible":false,"source_rank":12932,"rank":12932,"depth":7,"x":1206.733,"y":1656.511,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXI","tag":"0GXI","title":"Weakly adic rings · Lemma 0GXI","summary":"Let φ : A → B be a continuous homomorphism of linearly topologized rings. If φ is taut and A is weakly pre-adic, then B is weakly pre-adic.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous homomorphism of\nlinearly topologized rings. If $\\varphi$ is taut and $A$\nis weakly pre-adic, then $B$ is weakly pre-adic.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Weakly adic rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXI","source_file":"formal-spaces.tex","source_line":1523,"source_end_line":1528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1523-L1528","statement_sha256":"17f585ef5343a01bc0b0b5751195c06b2b2afff4268d42e61955b3b13805b392","origin":"The Stacks Project","memory_eligible":false,"source_rank":12933,"rank":12933,"depth":0,"x":1027.659,"y":1601.876,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXJ","tag":"0GXJ","title":"Weakly adic rings · Lemma 0GXJ","summary":"Let B → A and B → C be continuous homomorphisms of linearly topologized rings. If A and C are weakly pre-adic, then A widehat⊗_B C is weakly adic.","statement_latex":"Let $B \\to A$ and $B \\to C$ be continuous homomorphisms of\nlinearly topologized rings. If $A$ and $C$ are weakly pre-adic, then\n$A \\widehat{\\otimes}_B C$ is weakly adic.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Weakly adic rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXJ","source_file":"formal-spaces.tex","source_line":1538,"source_end_line":1543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1538-L1543","statement_sha256":"db6436d51986166ad41858913f259859e573ee7f33a642116bae0f43e86b5ce8","origin":"The Stacks Project","memory_eligible":false,"source_rank":12934,"rank":12934,"depth":7,"x":1204.183,"y":1540.364,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXL","tag":"0GXL","title":"Descending properties · Lemma 0GXL","summary":"In the situation above, if B has a countable fundamental system of open ideals, then A has a countable fundamental system of open ideals.","statement_latex":"In the situation above, if $B$ has a countable fundamental system of\nopen ideals, then $A$ has a countable fundamental system of open ideals.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXL","source_file":"formal-spaces.tex","source_line":1583,"source_end_line":1587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1583-L1587","statement_sha256":"2c9c1eb3073addaca81157c1fcec143ff17e74ed469d9a515d25bfa59046f83a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12935,"rank":12935,"depth":0,"x":1123.236,"y":1686.331,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXM","tag":"0GXM","title":"Descending properties · Lemma 0GXM","summary":"In the situation above, if B is weakly pre-admissible, then A is weakly pre-admissible.","statement_latex":"In the situation above, if $B$ is weakly pre-admissible, then\n$A$ is weakly pre-admissible.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXM","source_file":"formal-spaces.tex","source_line":1603,"source_end_line":1607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1603-L1607","statement_sha256":"0f4b4f82c166019daf73131b2b4088d210c3dbfd2608a5b82c642ffac883cdd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12936,"rank":12936,"depth":0,"x":1065.367,"y":1532.298,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXN","tag":"0GXN","title":"Descending properties · Lemma 0GXN","summary":"In the situation above, if B is pre-admissible, then A is pre-admissible.","statement_latex":"In the situation above, if $B$ is pre-admissible, then $A$\nis pre-admissible.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXN","source_file":"formal-spaces.tex","source_line":1621,"source_end_line":1625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1621-L1625","statement_sha256":"68fa1412d210c2e77eaffe29f96662d585d27ccf37ef047f8cab82bab8a82c9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12937,"rank":12937,"depth":0,"x":1232.411,"y":1613.288,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXP","tag":"0GXP","title":"Descending properties · Lemma 0GXP","summary":"In the situation above, if B is weakly pre-adic, then A is weakly pre-adic.","statement_latex":"In the situation above, if $B$ is weakly pre-adic, then $A$\nis weakly pre-adic.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXP","source_file":"formal-spaces.tex","source_line":1639,"source_end_line":1643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1639-L1643","statement_sha256":"c9724fc0c9e4a040296852e1c3738bb01ec4eb8e1c385cd9d75a5b293f5425e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12938,"rank":12938,"depth":4,"x":1043.54,"y":1648.456,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXQ","tag":"0GXQ","title":"Descending properties · Lemma 0GXQ","summary":"In the situation above, if B is adic and has a finitely generated ideal of definition and A is complete, then A is adic and has a finitely generated ideal of definition.","statement_latex":"In the situation above, if $B$ is adic and has a finitely generated\nideal of definition and $A$ is complete, then $A$ is adic and\nhas a finitely generated ideal of definition.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXQ","source_file":"formal-spaces.tex","source_line":1675,"source_end_line":1680,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1675-L1680","statement_sha256":"00e9239547d4c0ddd307fbd7aed650ae88af5804e4efc8ce63a5b203e739a3bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12939,"rank":12939,"depth":48,"x":1154.861,"y":1514.957,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AI7","tag":"0AI7","title":"Affine formal algebraic spaces · Definition 0AI7","summary":"Let S be a scheme. We say a sheaf X on (Sch/S)_fppf is an affine formal algebraic space if there exist • a directed set Lambda, • a system (X_λ, f_λ μ) over Lambda in (Sch/S)_fppf where • each X_λ is affine, • each f_λ μ : X_λ → X_μ is a thickening, such that X ≅ colim_λ ∈ Lambda X_λ as fppf sheaves and X satisfies a set theoretic condition (see Remark [Tag 0AIS]). A morphism of affine formal algebraic spaces over S is a map of sheaves.","statement_latex":"Let $S$ be a scheme. We say a sheaf $X$ on $(\\Sch/S)_{fppf}$ is an\n{\\it affine formal algebraic space} if there exist\n\\begin{enumerate}\n\\item a directed set $\\Lambda$,\n\\item a system $(X_\\lambda, f_{\\lambda \\mu})$ over $\\Lambda$\nin $(\\Sch/S)_{fppf}$ where\n\\begin{enumerate}\n\\item each $X_\\lambda$ is affine,\n\\item each $f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a thickening,\n\\end{enumerate}\n\\end{enumerate}\nsuch that\n$$\nX \\cong \\colim_{\\lambda \\in \\Lambda} X_\\lambda\n$$\nas fppf sheaves and $X$ satisfies a set theoretic condition\n(see Remark \\ref{remark-set-theoretic}). A\n{\\it morphism of affine formal algebraic spaces}\nover $S$ is a map of sheaves.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AI7","source_file":"formal-spaces.tex","source_line":1757,"source_end_line":1778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1757-L1778","statement_sha256":"61545ace98f6e6fb78e76097c1731d2638c4aa1f3d97e8625249929d33c422e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12940,"rank":12940,"depth":0,"x":1180.205,"y":1677.044,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AI8","tag":"0AI8","title":"Affine formal algebraic spaces · Lemma 0AI8","summary":"Let S be a scheme. If X is an affine formal algebraic space over S, then the diagonal morphism Δ : X → X ×_S X is representable and a closed immersion.","statement_latex":"Let $S$ be a scheme. If $X$ is an affine formal algebraic space over\n$S$, then the diagonal morphism $\\Delta : X \\to X \\times_S X$\nis representable and a closed immersion.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AI8","source_file":"formal-spaces.tex","source_line":1792,"source_end_line":1797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1792-L1797","statement_sha256":"d46e065149df2e1cc1942bcbae4ed30f97ecc9b6f3b7fb05d740e36cde7ba9bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":12941,"rank":12941,"depth":5,"x":1030.732,"y":1571.594,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AI9","tag":"0AI9","title":"Affine formal algebraic spaces · Lemma 0AI9","summary":"Let X_λ, λ ∈ Lambda and X = colim X_λ be as in Definition [Tag 0AI7]. Then X_λ → X is representable and a thickening.","statement_latex":"Let $X_\\lambda, \\lambda \\in \\Lambda$ and $X = \\colim X_\\lambda$\nbe as in Definition \\ref{definition-affine-formal-algebraic-space}.\nThen $X_\\lambda \\to X$ is representable and a thickening.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AI9","source_file":"formal-spaces.tex","source_line":1839,"source_end_line":1844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1839-L1844","statement_sha256":"c93ec0bc08417c6c50678a79e165fe5903783dc041d204755a1c4a8f9b3ff3d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":12942,"rank":12942,"depth":6,"x":1226.323,"y":1564.516,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIA","tag":"0AIA","title":"Affine formal algebraic spaces · Lemma 0AIA","summary":"Let X_λ, λ ∈ Lambda and X = colim X_λ be as in Definition [Tag 0AI7]. If Y is a quasi-compact algebraic space over S, then any morphism Y → X factors through an X_λ.","statement_latex":"Let $X_\\lambda, \\lambda \\in \\Lambda$ and $X = \\colim X_\\lambda$\nbe as in Definition \\ref{definition-affine-formal-algebraic-space}.\nIf $Y$ is a quasi-compact algebraic space over $S$, then any\nmorphism $Y \\to X$ factors through an $X_\\lambda$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIA","source_file":"formal-spaces.tex","source_line":1862,"source_end_line":1868,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1862-L1868","statement_sha256":"19cbebffec7bb98abefebda3c44687f03dabddf6a226e059f0c1420de8d3bab0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12943,"rank":12943,"depth":1,"x":1087.385,"y":1681.056,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIB","tag":"0AIB","title":"Affine formal algebraic spaces · Lemma 0AIB","summary":"Let S be a scheme. Let X be a sheaf on (Sch/S)_fppf. Then X is an affine formal algebraic space if and only if the following hold • any morphism U → X where U is an affine scheme over S factors through a morphism T → X which is representable and a thickening with T an affine scheme over S, and • a set theoretic condition as in Remark [Tag 0AIS].","statement_latex":"Let $S$ be a scheme. Let $X$ be a sheaf on $(\\Sch/S)_{fppf}$.\nThen $X$ is an affine formal algebraic space if and only if\nthe following hold\n\\begin{enumerate}\n\\item any morphism $U \\to X$ where $U$ is an affine scheme over $S$\nfactors through a morphism $T \\to X$ which is representable and a\nthickening with $T$ an affine scheme over $S$, and\n\\item a set theoretic condition as in Remark \\ref{remark-set-theoretic}.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIB","source_file":"formal-spaces.tex","source_line":1878,"source_end_line":1889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1878-L1889","statement_sha256":"0af05fe17567d41cbc45bb7776498cd4d15806221e18feb68026981706135b4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12944,"rank":12944,"depth":7,"x":1096.143,"y":1515.807,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIC","tag":"0AIC","title":"Affine formal algebraic spaces · Lemma 0AIC","summary":"Let S be a scheme. Let X be an fppf sheaf on (Sch/S)_fppf which satisfies the set theoretic condition of Remark [Tag 0AIS]. The following are equivalent: • there exists a weakly admissible topological ring A over S (see Remark [Tag 0AI3]) such that X = colim_I ⊂ A weak ideal of definition Spec(A/I), • X is an affine formal algebraic space and there exists an S-algebra A and a map X → Spec(A) such that for a closed immersion T → X with T an affine scheme the composition T…","statement_latex":"Let $S$ be a scheme. Let $X$ be an fppf sheaf on $(\\Sch/S)_{fppf}$\nwhich satisfies the set theoretic condition of\nRemark \\ref{remark-set-theoretic}.\nThe following are equivalent:\n\\begin{enumerate}\n\\item there exists a weakly admissible topological ring $A$ over $S$\n(see Remark \\ref{remark-mcquillan}) such that\n$X = \\colim_{I \\subset A\\text{ weak ideal of definition}} \\Spec(A/I)$,\n\\item $X$ is an affine formal algebraic space and\nthere exists an $S$-algebra $A$ and a map $X \\to \\Spec(A)$\nsuch that for a closed immersion $T \\to X$ with $T$ an affine scheme\nthe composition $T \\to \\Spec(A)$ is a closed immersion,\n\\item $X$ is an affine formal algebraic space and\nthere exists an $S$-algebra $A$ and a map $X \\to \\Spec(A)$\nsuch that for a closed immersion $T \\to X$ with $T$ a scheme\nthe composition $T \\to \\Spec(A)$ is a closed immersion,\n\\item $X$ is an affine formal algebraic space and\nfor some choice of $X = \\colim X_\\lambda$ as in\nDefinition \\ref{definition-affine-formal-algebraic-space}\nthe projections $\\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})\n\\to \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$ are surjective,\n\\item $X$ is an affine formal algebraic space and for any choice\nof $X = \\colim X_\\lambda$ as in\nDefinition \\ref{definition-affine-formal-algebraic-space}\nthe projections $\\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})\n\\to \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$ are surjective.\n\\end{enumerate}\nMoreover, the weakly admissible topological ring is\n$A = \\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$\nendowed with its limit topology and the weak ideals of definition\nclassify exactly the morphisms $T \\to X$ which are representable\nand thickenings.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIC","source_file":"formal-spaces.tex","source_line":1917,"source_end_line":1951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L1917-L1951","statement_sha256":"6de45592a1c2a7ff9104e5860fc17b3fdb87a2249e07805287d9c0a76125c1d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":12945,"rank":12945,"depth":31,"x":1222.938,"y":1642.995,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AID","tag":"0AID","title":"Affine formal algebraic spaces · Definition 0AID","summary":"Let S be a scheme. Let X be an affine formal algebraic space over S. We say X is McQuillan if X satisfies the equivalent conditions of Lemma [Tag 0AIC]. Let A be the weakly admissible topological ring associated to X. We say • X is classical if X is McQuillan and A is admissible (More on Algebra, Definition [Tag 07E8]), • X is weakly adic if X is McQuillan and A is weakly adic (Definition [Tag 0GXE]), • X is adic if X is McQuillan and A is adic (More on Algebra,…","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.\nWe say $X$ is {\\it McQuillan} if $X$ satisfies the equivalent conditions\nof Lemma \\ref{lemma-mcquillan-affine-formal-algebraic-space}. Let $A$\nbe the weakly admissible topological ring associated to $X$. We say\n\\begin{enumerate}\n\\item $X$ is {\\it classical} if $X$ is McQuillan and $A$ is admissible\n(More on Algebra, Definition \\ref{more-algebra-definition-topological-ring}),\n\\item $X$ is {\\it weakly adic} if $X$ is McQuillan and $A$ is weakly adic\n(Definition \\ref{definition-weakly-adic}),\n\\item $X$ is {\\it adic} if $X$ is McQuillan and $A$ is adic\n(More on Algebra, Definition \\ref{more-algebra-definition-topological-ring}),\n\\item $X$ is {\\it adic*} if $X$ is McQuillan, $A$ is adic, and $A$\nhas a finitely generated ideal of definition, and\n\\item $X$ is {\\it Noetherian} if $X$ is McQuillan and $A$ is\nboth Noetherian and adic.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AID","source_file":"formal-spaces.tex","source_line":2021,"source_end_line":2039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2021-L2039","statement_sha256":"a64ca91844d14679893f094a8fffc97011fa8efd9b9c04357c2a75a81637346b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12946,"rank":12946,"depth":32,"x":1026.599,"y":1621.092,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIF","tag":"0AIF","title":"Affine formal algebraic spaces · Definition 0AIF","summary":"Let S be a scheme. Let A be a weakly admissible topological ring over S, see Definition [Tag 0AMV]. The formal spectrum of A is the affine formal algebraic space Spf(A) = colim Spec(A/I) where the colimit is over the set of weak ideals of definition of A and taken in the category Sh((Sch/S)_fppf).","statement_latex":"Let $S$ be a scheme. Let $A$ be a weakly admissible topological ring over\n$S$, see Definition \\ref{definition-weakly-admissible}\\footnote{See\nMore on Algebra, Definition\n\\ref{more-algebra-definition-topological-ring}\nfor the classical case and see Remark \\ref{remark-mcquillan}\nfor a discussion of differences.}.\nThe {\\it formal spectrum} of $A$ is the affine formal algebraic space\n$$\n\\text{Spf}(A) = \\colim \\Spec(A/I)\n$$\nwhere the colimit is over the set of weak ideals of definition of $A$\nand taken in the category $\\Sh((\\Sch/S)_{fppf})$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIF","source_file":"formal-spaces.tex","source_line":2086,"source_end_line":2100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2086-L2100","statement_sha256":"c9381319630180a6bb98ffb34c08b74662927fdb9c8dadeb1a46cf57c3eb1ce0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12947,"rank":12947,"depth":1,"x":1189.453,"y":1525.565,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AN0","tag":"0AN0","title":"Affine formal algebraic spaces · Lemma 0AN0","summary":"Let S be a scheme. Let A, B be weakly admissible topological rings over S. Any morphism f : Spf(B) → Spf(A) of affine formal algebraic spaces over S is equal to Spf(f^sharp) for a unique continuous S-algebra map f^sharp : A → B.","statement_latex":"Let $S$ be a scheme. Let $A$, $B$ be weakly admissible\ntopological rings over $S$. Any morphism $f : \\text{Spf}(B) \\to \\text{Spf}(A)$\nof affine formal algebraic spaces over $S$\nis equal to $\\text{Spf}(f^\\sharp)$ for a unique continuous\n$S$-algebra map $f^\\sharp : A \\to B$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AN0","source_file":"formal-spaces.tex","source_line":2137,"source_end_line":2144,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2137-L2144","statement_sha256":"2d92192bd92b1374e91062ba7cc8554a528752112756abae6eb222c250e1242d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12948,"rank":12948,"depth":11,"x":1146.064,"y":1688.873,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIG","tag":"0AIG","title":"Affine formal algebraic spaces · Lemma 0AIG","summary":"Let S be a scheme. Let f : X → Y be a map of presheaves on (Sch/S)_fppf. If X is an affine formal algebraic space and f is representable by algebraic spaces and locally quasi-finite, then f is representable (by schemes).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a map\nof presheaves on $(\\Sch/S)_{fppf}$. If $X$ is an affine formal algebraic\nspace and $f$ is representable by algebraic spaces and locally quasi-finite,\nthen $f$ is representable (by schemes).","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIG","source_file":"formal-spaces.tex","source_line":2163,"source_end_line":2169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2163-L2169","statement_sha256":"c7e3c0ba742bb84a064a33bcaf77ab9b924d7d6d63c0b8bcd53ce17ef98ded7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12949,"rank":12949,"depth":55,"x":1046.451,"y":1543.423,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AII","tag":"0AII","title":"Countably indexed affine formal algebraic spaces · Lemma 0AII","summary":"Let S be a scheme. Let X be an affine formal algebraic space over S. The following are equivalent • there exists a system X_1 → X_2 → X_3 → … of thickenings of affine schemes over S such that X = colim X_n, • there exists a choice X = colim X_λ as in Definition [Tag 0AI7] such that Lambda is countable.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists a system $X_1 \\to X_2 \\to X_3 \\to \\ldots$\nof thickenings of affine schemes over $S$ such that $X = \\colim X_n$,\n\\item there exists a choice $X = \\colim X_\\lambda$ as in\nDefinition \\ref{definition-affine-formal-algebraic-space}\nsuch that $\\Lambda$ is countable.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Countably indexed affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AII","source_file":"formal-spaces.tex","source_line":2205,"source_end_line":2216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2205-L2216","statement_sha256":"a279379e4306ae6c613df120f0c342033ee5e4aa397adf88a818a29af824deff","origin":"The Stacks Project","memory_eligible":false,"source_rank":12950,"rank":12950,"depth":1,"x":1237.404,"y":1594.295,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIJ","tag":"0AIJ","title":"Countably indexed affine formal algebraic spaces · Definition 0AIJ","summary":"Let S be a scheme. Let X be an affine formal algebraic space over S. We say X is countably indexed if the equivalent conditions of Lemma [Tag 0AII] are satisfied.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.\nWe say $X$ is {\\it countably indexed} if the equivalent conditions of\nLemma \\ref{lemma-countable-affine-formal-algebraic-space} are satisfied.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Countably indexed affine formal algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIJ","source_file":"formal-spaces.tex","source_line":2224,"source_end_line":2229,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2224-L2229","statement_sha256":"fab6f2773e8618749715f9df747395d2d983d2eb74a13d20e416a6587d917d4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12951,"rank":12951,"depth":2,"x":1055.182,"y":1665.331,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIK","tag":"0AIK","title":"Countably indexed affine formal algebraic spaces · Lemma 0AIK","summary":"Let X be an affine formal algebraic space over a scheme S. • If X is Noetherian, then X is adic*. • If X is adic*, then X is adic. • If X is adic, then X is weakly adic. • If X is weakly adic, then X is classical. • If X is weakly adic, then X is countably indexed. • If X is countably indexed, then X is McQuillan.","statement_latex":"Let $X$ be an affine formal algebraic space over a scheme $S$.\n\\begin{enumerate}\n\\item If $X$ is Noetherian, then $X$ is adic*.\n\\item If $X$ is adic*, then $X$ is adic.\n\\item If $X$ is adic, then $X$ is weakly adic.\n\\item If $X$ is weakly adic, then $X$ is classical.\n\\item If $X$ is weakly adic, then $X$ is countably indexed.\n\\item If $X$ is countably indexed, then $X$ is McQuillan.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Countably indexed affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIK","source_file":"formal-spaces.tex","source_line":2235,"source_end_line":2246,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2235-L2246","statement_sha256":"e197647092b1f426d142ee5a8ccc5db1f09b22ef657adb9430867d28bec6111a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12952,"rank":12952,"depth":1,"x":1132.639,"y":1509.123,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AN1","tag":"0AN1","title":"Countably indexed affine formal algebraic spaces · Lemma 0AN1","summary":"Let S be a scheme. Let X be a presheaf on (Sch/S)_fppf. The following are equivalent • X is a countably indexed affine formal algebraic space, • X = Spf(A) where A is a weakly admissible topological S-algebra which has a countable fundamental system of neighbourhoods of 0, • X = Spf(A) where A is a weakly admissible topological S-algebra which has a fundamental system A ⊃ I_1 ⊃ I_2 ⊃ I_3 ⊃ … of weak ideals of definition, • X = Spf(A) where A is a complete topological…","statement_latex":"Let $S$ be a scheme. Let $X$ be a presheaf on $(\\Sch/S)_{fppf}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is a countably indexed affine formal algebraic space,\n\\item $X = \\text{Spf}(A)$ where $A$ is a weakly admissible topological\n$S$-algebra which has a countable fundamental system of neighbourhoods of $0$,\n\\item $X = \\text{Spf}(A)$ where $A$ is a weakly admissible topological\n$S$-algebra which has a fundamental system\n$A \\supset I_1 \\supset I_2 \\supset I_3 \\supset \\ldots$\nof weak ideals of definition,\n\\item $X = \\text{Spf}(A)$ where $A$ is a complete topological $S$-algebra\nwith a fundamental system of open neighbourhoods of $0$ given by a\ncountable sequence $A \\supset I_1 \\supset I_2 \\supset I_3 \\supset \\ldots$\nof ideals such that $I_n/I_{n + 1}$ is locally nilpotent, and\n\\item $X = \\text{Spf}(A)$ where $A = \\lim B/J_n$ with the limit topology\nwhere $B \\supset J_1 \\supset J_2 \\supset J_3 \\supset \\ldots$ is a\nsequence of ideals in an $S$-algebra $B$ with $J_n/J_{n + 1}$\nlocally nilpotent.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Countably indexed affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AN1","source_file":"formal-spaces.tex","source_line":2275,"source_end_line":2296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2275-L2296","statement_sha256":"7b1b0785cf0403e3728d523873d150135a41faf8783f9a65755588611eda5062","origin":"The Stacks Project","memory_eligible":false,"source_rank":12953,"rank":12953,"depth":32,"x":1201.33,"y":1668.697,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKM","tag":"0AKM","title":"Countably indexed affine formal algebraic spaces · Lemma 0AKM","summary":"Let S be a scheme. Let X be an affine formal algebraic space. The following are equivalent • X is Noetherian, • X is adic* and for every closed immersion T → X with T a scheme, T is Noetherian, • X is adic* and for some choice of X = colim X_λ as in Definition [Tag 0AI7] the schemes X_λ are Noetherian, and • X is weakly adic and for some choice X = colim X_λ as in Definition [Tag 0AI7] the schemes X_λ are Noetherian.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine formal algebraic space.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X$ is Noetherian,\n\\item $X$ is adic* and for every closed immersion $T \\to X$ with $T$ a scheme,\n$T$ is Noetherian,\n\\item $X$ is adic* and for some choice of $X = \\colim X_\\lambda$ as in\nDefinition \\ref{definition-affine-formal-algebraic-space}\nthe schemes $X_\\lambda$ are Noetherian, and\n\\item $X$ is weakly adic and for some choice $X = \\colim X_\\lambda$\nas in Definition \\ref{definition-affine-formal-algebraic-space}\nthe schemes $X_\\lambda$ are Noetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Countably indexed affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKM","source_file":"formal-spaces.tex","source_line":2340,"source_end_line":2355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2340-L2355","statement_sha256":"291c9fcf601fde2af9abab139997b2391c6efb1dec6f7a62651b5e643fe13724","origin":"The Stacks Project","memory_eligible":false,"source_rank":12954,"rank":12954,"depth":32,"x":1021.863,"y":1589.79,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIM","tag":"0AIM","title":"Formal algebraic spaces · Definition 0AIM","summary":"Let S be a scheme. We say a sheaf X on (Sch/S)_fppf is a formal algebraic space if there exist a family of maps (X_i → X)_i ∈ I of sheaves such that • X_i is an affine formal algebraic space, • X_i → X is representable by algebraic spaces and étale, • coprod X_i → X is surjective as a map of sheaves and X satisfies a set theoretic condition (see Remark [Tag 0AIS]). A morphism of formal algebraic spaces over S is a map of sheaves.","statement_latex":"Let $S$ be a scheme. We say a sheaf $X$ on $(\\Sch/S)_{fppf}$ is a\n{\\it formal algebraic space} if there exist a family of maps\n$\\{X_i \\to X\\}_{i \\in I}$ of sheaves such that\n\\begin{enumerate}\n\\item $X_i$ is an affine formal algebraic space,\n\\item $X_i \\to X$ is representable by algebraic spaces and \\'etale,\n\\item $\\coprod X_i \\to X$ is surjective as a map of sheaves\n\\end{enumerate}\nand $X$ satisfies a set theoretic condition\n(see Remark \\ref{remark-set-theoretic}). A\n{\\it morphism of formal algebraic spaces}\nover $S$ is a map of sheaves.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Formal algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIM","source_file":"formal-spaces.tex","source_line":2417,"source_end_line":2431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2417-L2431","statement_sha256":"4c71cb7eb885fa5ba77d48b0ccf56379935d12771d396aa291d48595ca08262c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12955,"rank":12955,"depth":0,"x":1218.187,"y":1546.023,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIP","tag":"0AIP","title":"Formal algebraic spaces · Lemma 0AIP","summary":"Let S be a scheme. If X is a formal algebraic space over S, then the diagonal morphism Δ : X → X ×_S X is representable, a monomorphism, locally quasi-finite, locally of finite type, and separated.","statement_latex":"Let $S$ be a scheme. If $X$ is a formal algebraic space over\n$S$, then the diagonal morphism $\\Delta : X \\to X \\times_S X$\nis representable, a monomorphism, locally quasi-finite,\nlocally of finite type, and separated.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIP","source_file":"formal-spaces.tex","source_line":2449,"source_end_line":2455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2449-L2455","statement_sha256":"765ad2378d20428b57d1ac8b2ddaa39cfeb4b9151da57f328808744997630b63","origin":"The Stacks Project","memory_eligible":false,"source_rank":12956,"rank":12956,"depth":67,"x":1108.32,"y":1690.084,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIQ","tag":"0AIQ","title":"Formal algebraic spaces · Lemma 0AIQ","summary":"Let S be a scheme. Let f : X → Y be a morphism from an algebraic space over S to a formal algebraic space over S. Then f is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism from an\nalgebraic space over $S$ to a formal algebraic space over $S$.\nThen $f$ is representable by algebraic spaces.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIQ","source_file":"formal-spaces.tex","source_line":2509,"source_end_line":2514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2509-L2514","statement_sha256":"bea69cda359994b6c132327da96565fff0eaff56c8b052a98ce6f5d4dc87e473","origin":"The Stacks Project","memory_eligible":false,"source_rank":12957,"rank":12957,"depth":68,"x":1073.392,"y":1521.061,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIN","tag":"0AIN","title":"The reduction · Lemma 0AIN","summary":"Let S be a scheme. Let X be a formal algebraic space over S. There exists a reduced algebraic space X_red and a representable morphism X_red → X which is a thickening. A morphism U → X with U a reduced algebraic space factors uniquely through X_red.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nThere exists a reduced algebraic space $X_{red}$ and a representable\nmorphism $X_{red} \\to X$ which is a thickening. A morphism $U \\to X$\nwith $U$ a reduced algebraic space factors uniquely through $X_{red}$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The reduction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIN","source_file":"formal-spaces.tex","source_line":2615,"source_end_line":2621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2615-L2621","statement_sha256":"de8f4b84f106e10b965eff09ab39c4eca809fb03cbbf8b6e16c57f520b8b577a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12958,"rank":12958,"depth":56,"x":1235.504,"y":1626.157,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GB7","tag":"0GB7","title":"The reduction · Lemma 0GB7","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S which is representable by algebraic spaces and smooth (for example étale). Then X_red = X ×_Y Y_red.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nformal algebraic spaces over $S$ which is representable by\nalgebraic spaces and smooth (for example \\'etale).\nThen $X_{red} = X \\times_Y Y_{red}$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The reduction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GB7","source_file":"formal-spaces.tex","source_line":2705,"source_end_line":2711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2705-L2711","statement_sha256":"e29802814a8aa3ce48fbaa2058bb4cf2b6b710671bcad996915da2a598a417a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12959,"rank":12959,"depth":57,"x":1030.905,"y":1640.685,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GB8","tag":"0GB8","title":"The reduction · Lemma 0GB8","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S which is representable by algebraic spaces. Then f is surjective in the sense of Bootstrap, Definition [Tag 03XZ] if and only if f_red : X_red → Y_red is a surjective morphism of algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nformal algebraic spaces over $S$ which is representable by\nalgebraic spaces. Then $f$ is surjective in the sense of\nBootstrap, Definition \\ref{bootstrap-definition-property-transformation}\nif and only if $f_{red} : X_{red} \\to Y_{red}$ is a\nsurjective morphism of algebraic spaces.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The reduction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GB8","source_file":"formal-spaces.tex","source_line":2725,"source_end_line":2733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2725-L2733","statement_sha256":"5cb6f1614bca0bbf750e44ee455e91bacd4cb50cb6b2b9f196638fe6402aae69","origin":"The Stacks Project","memory_eligible":false,"source_rank":12960,"rank":12960,"depth":2,"x":1170.46,"y":1513.543,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIU","tag":"0AIU","title":"Colimits of algebraic spaces along thickenings · Lemma 0AIU","summary":"Let S be a scheme. Suppose given a directed set Lambda and a system of algebraic spaces (X_λ, f_λ μ) over Lambda where each f_λ μ : X_λ → X_μ is a thickening. Then X = colim_λ ∈ Lambda X_λ is a formal algebraic space over S.","statement_latex":"Let $S$ be a scheme. Suppose given a directed set\n$\\Lambda$ and a system of algebraic spaces $(X_\\lambda, f_{\\lambda \\mu})$\nover $\\Lambda$ where each $f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a\nthickening. Then $X = \\colim_{\\lambda \\in \\Lambda} X_\\lambda$\nis a formal algebraic space over $S$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Colimits of algebraic spaces along thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIU","source_file":"formal-spaces.tex","source_line":2755,"source_end_line":2762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2755-L2762","statement_sha256":"4e13e2f017299ed2a3da2334dbb2eca43f7ebb709cbcecba02e6b584977f5f2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12961,"rank":12961,"depth":67,"x":1169.796,"y":1686.938,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIY","tag":"0AIY","title":"Completion along a closed subset · Lemma 0AIY","summary":"Let S be a scheme. Let X be an affine scheme over S. Let T ⊂ |X| be a closed subset. Then the functor (Sch/S)_fppf → Sets, U ↦ (f : U → X mid f(|U|) ⊂ T) is a McQuillan affine formal algebraic space.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine scheme over $S$.\nLet $T \\subset |X|$ be a closed subset. Then the functor\n$$\n(\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad\nU \\longmapsto \\{f : U \\to X \\mid f(|U|) \\subset T\\}\n$$\nis a McQuillan affine formal algebraic space.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIY","source_file":"formal-spaces.tex","source_line":2931,"source_end_line":2940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2931-L2940","statement_sha256":"07f8d63d523ca0d99bb08fc7f571ae805b24ea62ebd5c7ca844d6d5c635f9fa6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12962,"rank":12962,"depth":33,"x":1030.482,"y":1558.366,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AIZ","tag":"0AIZ","title":"Completion along a closed subset · Lemma 0AIZ","summary":"Let S be a scheme. Let X be an algebraic space over S. Let T ⊂ |X| be a closed subset. Then the functor (Sch/S)_fppf → Sets, U ↦ (f : U → X mid f(|U|) ⊂ T) is a formal algebraic space.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset. Then the functor\n$$\n(\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad\nU \\longmapsto \\{f : U \\to X \\mid f(|U|) \\subset T\\}\n$$\nis a formal algebraic space.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AIZ","source_file":"formal-spaces.tex","source_line":2961,"source_end_line":2970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2961-L2970","statement_sha256":"998eff8c0ebd813c49d482f08f97c071c8b096b5c6aa26094bd6a220f0f01ed0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12963,"rank":12963,"depth":41,"x":1237.141,"y":1574.164,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AMC","tag":"0AMC","title":"Completion along a closed subset · Definition 0AMC","summary":"Let S be a scheme. Let X be an algebraic space over S. Let T ⊂ |X| be a closed subset. The formal algebraic space of Lemma [Tag 0AIZ] is called the completion of X along T.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset. The formal algebraic space\nof Lemma \\ref{lemma-completion-is-formal-algebraic-space}\nis called the {\\it completion of $X$ along $T$}.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subset","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AMC","source_file":"formal-spaces.tex","source_line":2992,"source_end_line":2998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L2992-L2998","statement_sha256":"d23f26e56c4cd93925f4231c7742be9cf69860e3510c6d03171e1397dc9700be","origin":"The Stacks Project","memory_eligible":false,"source_rank":12964,"rank":12964,"depth":42,"x":1071.622,"y":1680.052,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0APV","tag":"0APV","title":"Completion along a closed subset · Lemma 0APV","summary":"Let S be a scheme. Let f : X' → X be a morphism of algebraic spaces over S. Let T ⊂ |X| be a closed subset and let T' = |f|^-1(T) ⊂ |X'|. Then xymatrix X'_/T' ar[r] ar[d] & X' ar[d]^f X_/T ar[r] & X is a cartesian diagram of sheaves. In particular, the morphism X'_/T' → X_/T is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $f : X' \\to X$ be a morphism\nof algebraic spaces over $S$. Let $T \\subset |X|$\nbe a closed subset and let $T' = |f|^{-1}(T) \\subset |X'|$.\nThen\n$$\n\\xymatrix{\nX'_{/T'} \\ar[r] \\ar[d] & X' \\ar[d]^f \\\\\nX_{/T} \\ar[r] & X\n}\n$$\nis a cartesian diagram of sheaves. In particular, the morphism\n$X'_{/T'} \\to X_{/T}$ is representable by algebraic spaces.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APV","source_file":"formal-spaces.tex","source_line":3013,"source_end_line":3027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3013-L3027","statement_sha256":"d59a2021cd0cc3dc9f8708269cb008575f991a2e33fb993a04d81127347256fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":12965,"rank":12965,"depth":0,"x":1108.617,"y":1507.609,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GB9","tag":"0GB9","title":"Completion along a closed subset · Lemma 0GB9","summary":"Let S be a scheme. Let X be an algebraic space over S. Let T ⊂ |X| be a closed subset. The reduction (X_/T)_red of the completion X_/T of X along T is the reduced induced closed subspace Z of X corresponding to T.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset. The reduction $(X_{/T})_{red}$\nof the completion $X_{/T}$ of $X$ along $T$ is\nthe reduced induced closed subspace $Z$ of $X$ corresponding to $T$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GB9","source_file":"formal-spaces.tex","source_line":3037,"source_end_line":3043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3037-L3043","statement_sha256":"60ac2fcc9970542408a4fd93b0f4b313cdb81cc1aec0ef3c50daa0de08a6c2e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":12966,"rank":12966,"depth":57,"x":1220.296,"y":1656.138,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBA","tag":"0GBA","title":"Completion along a closed subset · Lemma 0GBA","summary":"Let S be a scheme. Let X = Spec(A) be an affine scheme over S. Let T ⊂ X be a closed subset. Let X_/T be the formal completion of X along T. • If X setminus T is quasi-compact, i.e., T is constructible, then X_/T is adic*. • If T = V(I) for some finitely generated ideal I ⊂ A, then X_/T = Spf(A^wedge) where A^wedge is the I-adic completion of A. • If X is Noetherian, then X_/T is Noetherian.","statement_latex":"Let $S$ be a scheme. Let $X = \\Spec(A)$ be an affine scheme over $S$.\nLet $T \\subset X$ be a closed subset. Let $X_{/T}$ be the\nformal completion of $X$ along $T$.\n\\begin{enumerate}\n\\item If $X \\setminus T$ is quasi-compact, i.e., $T$ is constructible,\nthen $X_{/T}$ is adic*.\n\\item If $T = V(I)$ for some finitely generated ideal $I \\subset A$,\nthen $X_{/T} = \\text{Spf}(A^\\wedge)$ where $A^\\wedge$ is the\n$I$-adic completion of $A$.\n\\item If $X$ is Noetherian, then $X_{/T}$ is Noetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBA","source_file":"formal-spaces.tex","source_line":3058,"source_end_line":3071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3058-L3071","statement_sha256":"fc6a22923042966c0e5aa062e64a768793569de537c16b2a8ad0a893568046ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":12967,"rank":12967,"depth":7,"x":1017.988,"y":1609.866,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0APW","tag":"0APW","title":"Completion along a closed subset · Lemma 0APW","summary":"Email by Ofer Gabber of September 11, 2014. Let S be a scheme. Let X = Spec(A) be an affine scheme over S. Let T ⊂ X be a closed subscheme. • If the formal completion X_/T is countably indexed and there exist countably many f_1, f_2, f_3, … ∈ A such that T = V(f_1, f_2, f_3, …), then X_/T is adic*. • The conclusion of (1) is wrong if we omit the assumption that T can be cut out by countably many functions in X.","statement_latex":"\\begin{reference}\nEmail by Ofer Gabber of September 11, 2014.\n\\end{reference}\nLet $S$ be a scheme. Let $X = \\Spec(A)$ be an affine scheme over $S$.\nLet $T \\subset X$ be a closed subscheme.\n\\begin{enumerate}\n\\item If the formal completion $X_{/T}$ is countably indexed\nand there exist countably many $f_1, f_2, f_3, \\ldots \\in A$ such that\n$T = V(f_1, f_2, f_3, \\ldots)$, then $X_{/T}$ is adic*.\n\\item The conclusion of (1) is wrong if we omit the assumption that\n$T$ can be cut out by countably many functions in $X$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subset","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APW","source_file":"formal-spaces.tex","source_line":3094,"source_end_line":3108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3094-L3108","statement_sha256":"38c83ce3fc137b43a68b468179a1ed7691187c7128495260d6b257b875e339e0","origin":"The Stacks Project","memory_eligible":false,"source_rank":12968,"rank":12968,"depth":4,"x":1204.851,"y":1528.987,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJ1","tag":"0AJ1","title":"Fibre products · Lemma 0AJ1","summary":"Let S be a scheme. Let (X_i → X)_i ∈ I be a family of maps of sheaves on (Sch/S)_fppf. Assume (a) X_i is a formal algebraic space over S, (b) X_i → X is representable by algebraic spaces and étale, and (c) coprod X_i → X is a surjection of sheaves. Then X is a formal algebraic space over S.","statement_latex":"Let $S$ be a scheme. Let $\\{X_i \\to X\\}_{i \\in I}$ be a family of maps\nof sheaves on $(\\Sch/S)_{fppf}$. Assume (a) $X_i$ is a\nformal algebraic space over $S$, (b) $X_i \\to X$ is representable\nby algebraic spaces and \\'etale, and (c) $\\coprod X_i \\to X$\nis a surjection of sheaves. Then $X$ is a formal algebraic space\nover $S$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJ1","source_file":"formal-spaces.tex","source_line":3252,"source_end_line":3260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3252-L3260","statement_sha256":"13fff3f2db2a6ef2de0f66318cf5d2509e4ea8a149d03d216ab12308545d0a0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":12969,"rank":12969,"depth":1,"x":1131.917,"y":1695.075,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJ2","tag":"0AJ2","title":"Fibre products · Lemma 0AJ2","summary":"Let S be a scheme. Let X, Y be formal algebraic spaces over S and let Z be a sheaf whose diagonal is representable by algebraic spaces. Let X → Z and Y → Z be maps of sheaves. Then X ×_Z Y is a formal algebraic space.","statement_latex":"Let $S$ be a scheme. Let $X, Y$ be formal algebraic spaces over $S$\nand let $Z$ be a sheaf whose diagonal is representable by\nalgebraic spaces. Let $X \\to Z$ and $Y \\to Z$ be maps of sheaves.\nThen $X \\times_Z Y$ is a formal algebraic space.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJ2","source_file":"formal-spaces.tex","source_line":3270,"source_end_line":3276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3270-L3276","statement_sha256":"08f1330dc9ef1e8fab8bb1db7e70a27609742d058d4f8195ebc83e7c26507801","origin":"The Stacks Project","memory_eligible":false,"source_rank":12970,"rank":12970,"depth":68,"x":1051.934,"y":1530.808,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJ3","tag":"0AJ3","title":"Fibre products · Lemma 0AJ3","summary":"Let S be a scheme. The category of formal algebraic spaces over S has fibre products.","statement_latex":"Let $S$ be a scheme. The category of formal algebraic spaces over $S$\nhas fibre products.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJ3","source_file":"formal-spaces.tex","source_line":3303,"source_end_line":3307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3303-L3307","statement_sha256":"eabff8cd5607ff31f7739f92aef263a18410edeef5fe9c031ac452acda1ea5e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":12971,"rank":12971,"depth":69,"x":1243.489,"y":1606.742,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0CB9","tag":"0CB9","title":"Fibre products · Lemma 0CB9","summary":"Let S be a scheme. Let X → Z and Y → Z be morphisms of formal algebraic spaces over S. Then (X ×_Z Y)_red = (X_red ×_Z_red Y_red)_red.","statement_latex":"Let $S$ be a scheme. Let $X \\to Z$ and $Y \\to Z$ be morphisms of\nformal algebraic spaces over $S$. Then\n$(X \\times_Z Y)_{red} = (X_{red} \\times_{Z_{red}} Y_{red})_{red}$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CB9","source_file":"formal-spaces.tex","source_line":3315,"source_end_line":3320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3315-L3320","statement_sha256":"c39209531e833573f64615d2b1304d7c65f1eec090d29cfc9c99db966f33f542","origin":"The Stacks Project","memory_eligible":false,"source_rank":12972,"rank":12972,"depth":57,"x":1040.672,"y":1659.573,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AN2","tag":"0AN2","title":"Fibre products · Lemma 0AN2","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. The diagonal morphism Δ : X → X ×_Y X is representable (by schemes), a monomorphism, locally quasi-finite, locally of finite type, and separated.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces\nover $S$. The diagonal morphism $\\Delta : X \\to X \\times_Y X$\nis representable (by schemes), a monomorphism, locally quasi-finite,\nlocally of finite type, and separated.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AN2","source_file":"formal-spaces.tex","source_line":3331,"source_end_line":3337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3331-L3337","statement_sha256":"ed58bc8c35b6ff4fa8c29a78fe80e97cdf54cb0067e29a981666634722299398","origin":"The Stacks Project","memory_eligible":false,"source_rank":12973,"rank":12973,"depth":68,"x":1148.006,"y":1505.151,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJ5","tag":"0AJ5","title":"Separation axioms for formal algebraic spaces · Lemma 0AJ5","summary":"Let S be a scheme. Let X be a formal algebraic space over S. The following are equivalent • the reduction of X (Lemma [Tag 0AIN]) is a quasi-separated algebraic space, • for U → X, V → X with U, V quasi-compact schemes the fibre product U ×_X V is quasi-compact, • for U → X, V → X with U, V affine the fibre product U ×_X V is quasi-compact.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item the reduction of $X$\n(Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a\nquasi-separated algebraic space,\n\\item for $U \\to X$, $V \\to X$ with $U$, $V$ quasi-compact schemes\nthe fibre product $U \\times_X V$ is quasi-compact,\n\\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine\nthe fibre product $U \\times_X V$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJ5","source_file":"formal-spaces.tex","source_line":3362,"source_end_line":3375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3362-L3375","statement_sha256":"cea071f190c3ad2b91e7b4ddfc332b209dbfcfca0ec745bf3ff881aacac00389","origin":"The Stacks Project","memory_eligible":false,"source_rank":12974,"rank":12974,"depth":68,"x":1193.152,"y":1680.354,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJ6","tag":"0AJ6","title":"Separation axioms for formal algebraic spaces · Lemma 0AJ6","summary":"Let S be a scheme. Let X be a formal algebraic space over S. The following are equivalent • the reduction of X (Lemma [Tag 0AIN]) is a separated algebraic space, • for U → X, V → X with U, V affine the fibre product U ×_X V is affine and O(U) ⊗_Z O(V) → O(U ×_X V) is surjective.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item the reduction of $X$\n(Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a separated\nalgebraic space,\n\\item for $U \\to X$, $V \\to X$ with $U$, $V$ affine\nthe fibre product $U \\times_X V$ is affine and\n$$\n\\mathcal{O}(U) \\otimes_\\mathbf{Z} \\mathcal{O}(V)\n\\longrightarrow\n\\mathcal{O}(U \\times_X V)\n$$\nis surjective.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJ6","source_file":"formal-spaces.tex","source_line":3392,"source_end_line":3409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3392-L3409","statement_sha256":"02f83f5dea7ccc43ee78af8546b24320ba2a5675a2431f272ec80e4bfe6bacfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12975,"rank":12975,"depth":68,"x":1018.543,"y":1576.525,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJ7","tag":"0AJ7","title":"Separation axioms for formal algebraic spaces · Definition 0AJ7","summary":"Let S be a scheme. Let X be a formal algebraic space over S. We say • X is quasi-separated if the equivalent conditions of Lemma [Tag 0AJ5] are satisfied. • X is separated if the equivalent conditions of Lemma [Tag 0AJ6] are satisfied.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nWe say\n\\begin{enumerate}\n\\item $X$ is {\\it quasi-separated} if the equivalent conditions of\nLemma \\ref{lemma-characterize-quasi-separated} are satisfied.\n\\item $X$ is {\\it separated} if the equivalent conditions of\nLemma \\ref{lemma-characterize-separated} are satisfied.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for formal algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJ7","source_file":"formal-spaces.tex","source_line":3468,"source_end_line":3478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3468-L3478","statement_sha256":"aa11ce6586a956f1f40832acdab0ef25d7d7d548268a004c90221ffadcf6bf62","origin":"The Stacks Project","memory_eligible":false,"source_rank":12976,"rank":12976,"depth":69,"x":1231.311,"y":1553.955,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AN3","tag":"0AN3","title":"Separation axioms for formal algebraic spaces · Lemma 0AN3","summary":"Let S be a scheme. Let X → Z and Y → Z be morphisms of formal algebraic spaces over S. Assume Z separated. • If X and Y are affine formal algebraic spaces, then so is X ×_Z Y. • If X and Y are McQuillan affine formal algebraic spaces, then so is X ×_Z Y. • If X, Y, and Z are McQuillan affine formal algebraic spaces corresponding to the weakly admissible topological S-algebras A, B, and C, then X ×_Z Y corresponds to A widehat⊗_C B.","statement_latex":"Let $S$ be a scheme. Let $X \\to Z$ and $Y \\to Z$ be morphisms\nof formal algebraic spaces over $S$. Assume $Z$ separated.\n\\begin{enumerate}\n\\item If $X$ and $Y$ are affine formal algebraic spaces, then\nso is $X \\times_Z Y$.\n\\item If $X$ and $Y$ are McQuillan affine formal algebraic spaces, then\nso is $X \\times_Z Y$.\n\\item If $X$, $Y$, and $Z$ are McQuillan affine formal algebraic spaces\ncorresponding to the weakly admissible topological $S$-algebras\n$A$, $B$, and $C$, then $X \\times_Z Y$ corresponds to\n$A \\widehat{\\otimes}_C B$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AN3","source_file":"formal-spaces.tex","source_line":3485,"source_end_line":3499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3485-L3499","statement_sha256":"bd7d27056f2062c668e5276db6128e3eb084edbbcbcb51fb2e0743cbd9ffb47c","origin":"The Stacks Project","memory_eligible":false,"source_rank":12977,"rank":12977,"depth":32,"x":1092.232,"y":1691.661,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0APX","tag":"0APX","title":"Separation axioms for formal algebraic spaces · Lemma 0APX","summary":"Let S be a scheme. Let X be a formal algebraic space over S. Let U → X be a morphism where U is a separated algebraic space over S. Then U → X is separated.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nLet $U \\to X$ be a morphism where $U$ is a separated algebraic\nspace over $S$. Then $U \\to X$ is separated.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APX","source_file":"formal-spaces.tex","source_line":3526,"source_end_line":3531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3526-L3531","statement_sha256":"3ffc48a78c50d24b2880695f00ed9ee4efc69b46c0c387fa4e0589a2b8c56daa","origin":"The Stacks Project","memory_eligible":false,"source_rank":12978,"rank":12978,"depth":69,"x":1084.028,"y":1510.766,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJ9","tag":"0AJ9","title":"Quasi-compact formal algebraic spaces · Lemma 0AJ9","summary":"Let S be a scheme. Let X be a formal algebraic space over S. The following are equivalent • the reduction of X (Lemma [Tag 0AIN]) is a quasi-compact algebraic space, • we can find (X_i → X)_i ∈ I as in Definition [Tag 0AIM] with I finite, • there exists a morphism Y → X representable by algebraic spaces which is étale and surjective and where Y is an affine formal algebraic space.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item the reduction of $X$\n(Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact\nalgebraic space,\n\\item we can find $\\{X_i \\to X\\}_{i \\in I}$ as in\nDefinition \\ref{definition-formal-algebraic-space} with $I$ finite,\n\\item there exists a morphism $Y \\to X$ representable by algebraic\nspaces which is \\'etale and surjective and where\n$Y$ is an affine formal algebraic space.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Quasi-compact formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJ9","source_file":"formal-spaces.tex","source_line":3558,"source_end_line":3572,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3558-L3572","statement_sha256":"003acd431439d46f103febfcb4ec3bb17e6103541e5a5c47226707d878b4e018","origin":"The Stacks Project","memory_eligible":false,"source_rank":12979,"rank":12979,"depth":57,"x":1235.912,"y":1639.81,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJA","tag":"0AJA","title":"Quasi-compact formal algebraic spaces · Definition 0AJA","summary":"Let S be a scheme. Let X be a formal algebraic space over S. We say X is quasi-compact if the equivalent conditions of Lemma [Tag 0AJ9] are satisfied.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nWe say $X$ is {\\it quasi-compact} if the equivalent conditions of\nLemma \\ref{lemma-characterize-quasi-compact} are satisfied.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Quasi-compact formal algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJA","source_file":"formal-spaces.tex","source_line":3578,"source_end_line":3583,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3578-L3583","statement_sha256":"4cab79d19f9d484754cbac08f1f1d18caf5ebd5093f3b13ef5a3a713557e9cab","origin":"The Stacks Project","memory_eligible":false,"source_rank":12980,"rank":12980,"depth":58,"x":1019.626,"y":1630.815,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJB","tag":"0AJB","title":"Quasi-compact formal algebraic spaces · Lemma 0AJB","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. The following are equivalent • the induced map f_red : X_red → Y_red between reductions (Lemma [Tag 0AIN]) is a quasi-compact morphism of algebraic spaces, • for every quasi-compact scheme T and morphism T → Y the fibre product X ×_Y T is a quasi-compact formal algebraic space, • for every affine scheme T and morphism T → Y the fibre product X ×_Y T is a quasi-compact formal algebraic space,…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces over $S$. The following are equivalent\n\\begin{enumerate}\n\\item the induced map $f_{red} : X_{red} \\to Y_{red}$ between reductions\n(Lemma \\ref{lemma-reduction-formal-algebraic-space}) is a quasi-compact\nmorphism of algebraic spaces,\n\\item for every quasi-compact scheme $T$ and morphism $T \\to Y$\nthe fibre product $X \\times_Y T$ is a quasi-compact formal\nalgebraic space,\n\\item for every affine scheme $T$ and morphism $T \\to Y$\nthe fibre product $X \\times_Y T$ is a quasi-compact formal\nalgebraic space, and\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nsuch that each $X \\times_Y Y_j$ is a quasi-compact formal algebraic space.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Quasi-compact formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJB","source_file":"formal-spaces.tex","source_line":3585,"source_end_line":3603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3585-L3603","statement_sha256":"0c7c0b10d43eb4df79e8d89516bf89bbf1a21684fcb9f5253f8e0a566c053af6","origin":"The Stacks Project","memory_eligible":false,"source_rank":12981,"rank":12981,"depth":57,"x":1186.742,"y":1514.446,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJC","tag":"0AJC","title":"Quasi-compact formal algebraic spaces · Definition 0AJC","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. We say f is quasi-compact if the equivalent conditions of Lemma [Tag 0AJB] are satisfied.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nformal algebraic spaces over $S$.\nWe say $f$ is {\\it quasi-compact} if the equivalent conditions of\nLemma \\ref{lemma-characterize-quasi-compact-morphism} are satisfied.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Quasi-compact formal algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJC","source_file":"formal-spaces.tex","source_line":3609,"source_end_line":3615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3609-L3615","statement_sha256":"47bb2d1a245afb4d34ea31a7db5d6d4db02ae2f1706d543e89596771c9fe5655","origin":"The Stacks Project","memory_eligible":false,"source_rank":12982,"rank":12982,"depth":58,"x":1157.023,"y":1695.508,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AM2","tag":"0AM2","title":"Quasi-compact formal algebraic spaces · Lemma 0AM2","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S which is representable by algebraic spaces. Then f is quasi-compact in the sense of Definition [Tag 0AJC] if and only if f is quasi-compact in the sense of Bootstrap, Definition [Tag 03XZ].","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces over $S$ which is representable by algebraic spaces.\nThen $f$ is quasi-compact in the sense of\nDefinition \\ref{definition-quasi-compact-morphism}\nif and only if $f$ is quasi-compact in the sense of\nBootstrap, Definition \\ref{bootstrap-definition-property-transformation}.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Quasi-compact formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AM2","source_file":"formal-spaces.tex","source_line":3622,"source_end_line":3630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3622-L3630","statement_sha256":"6dffb70d182c99da95d7cf73b5af28c6b8f9085948790b0d5c8de6bfac1feb00","origin":"The Stacks Project","memory_eligible":false,"source_rank":12983,"rank":12983,"depth":59,"x":1033.041,"y":1544.778,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJE","tag":"0AJE","title":"Quasi-compact and quasi-separated formal algebraic spaces · Lemma 0AJE","summary":"[Yasuda] Let S be a scheme. Let X be a quasi-compact and quasi-separated formal algebraic space over S. Then X = colim X_λ for a system of algebraic spaces (X_λ, f_λ μ) over a directed set Lambda where each f_λ μ : X_λ → X_μ is a thickening.","statement_latex":"\\begin{reference}\n\\cite[Proposition 3.32]{Yasuda}\n\\end{reference}\nLet $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated\nformal algebraic space over $S$. Then $X = \\colim X_\\lambda$\nfor a system of algebraic spaces $(X_\\lambda, f_{\\lambda \\mu})$\nover a directed set $\\Lambda$ where each\n$f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a thickening.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Quasi-compact and quasi-separated formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJE","source_file":"formal-spaces.tex","source_line":3648,"source_end_line":3658,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3648-L3658","statement_sha256":"e2ded71222c4b44609f0bb69a3ea2c8c055c6adbadce93f049c9345d13b27d1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12984,"rank":12984,"depth":58,"x":1246.175,"y":1585.67,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DE8","tag":"0DE8","title":"Quasi-compact and quasi-separated formal algebraic spaces · Lemma 0DE8","summary":"Let S be a scheme. Let X be a formal algebraic space over S. Then X is an affine formal algebraic space if and only if its reduction X_red (Lemma [Tag 0AIN]) is affine.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nThen $X$ is an affine formal algebraic space if and only if\nits reduction $X_{red}$ (Lemma \\ref{lemma-reduction-formal-algebraic-space})\nis affine.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Quasi-compact and quasi-separated formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DE8","source_file":"formal-spaces.tex","source_line":3800,"source_end_line":3806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3800-L3806","statement_sha256":"182b078b56620da88151cc2bfb3072b08cb2365f1e179289aeacbb532503e8cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12985,"rank":12985,"depth":70,"x":1055.686,"y":1676.666,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0APY","tag":"0APY","title":"Morphisms representable by algebraic spaces · Lemma 0APY","summary":"The composition of morphisms representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes).","statement_latex":"The composition of morphisms representable by algebraic spaces is\nrepresentable by algebraic spaces. The same holds for representable\n(by schemes).","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APY","source_file":"formal-spaces.tex","source_line":3836,"source_end_line":3841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3836-L3841","statement_sha256":"b47638f862df3146977d780742fee0b4f26fc685b1d3e82307c65b39fc7502ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":12986,"rank":12986,"depth":55,"x":1123.13,"y":1501.071,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0APZ","tag":"0APZ","title":"Morphisms representable by algebraic spaces · Lemma 0APZ","summary":"A base change of a morphism representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes).","statement_latex":"A base change of a morphism representable by algebraic spaces is\nrepresentable by algebraic spaces. The same holds for representable\n(by schemes).","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0APZ","source_file":"formal-spaces.tex","source_line":3847,"source_end_line":3852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3847-L3852","statement_sha256":"350d1870b26fa3fd154fa773cbe5a72d1ca98bffbf818cc9afec78da5ccb5dce","origin":"The Stacks Project","memory_eligible":false,"source_rank":12987,"rank":12987,"depth":1,"x":1214.817,"y":1669.21,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQ0","tag":"0AQ0","title":"Morphisms representable by algebraic spaces · Lemma 0AQ0","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of formal algebraic spaces over S. If g ∘ f : X → Z is representable by algebraic spaces, then f : X → Y is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of\nformal algebraic spaces over $S$. If $g \\circ f : X \\to Z$ is representable\nby algebraic spaces, then $f : X \\to Y$ is representable by algebraic spaces.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQ0","source_file":"formal-spaces.tex","source_line":3858,"source_end_line":3863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3858-L3863","statement_sha256":"5eb24adfbcd7953c4893c52ce69c7d68dab985fe60403bf574ff55e28a2c61e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":12988,"rank":12988,"depth":69,"x":1011.53,"y":1597.086,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AN4","tag":"0AN4","title":"Morphisms representable by algebraic spaces · Lemma 0AN4","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. The following are equivalent: • the morphism f is representable by algebraic spaces, • there exists a commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y where U, V are formal algebraic spaces, the vertical arrows are representable by algebraic spaces, U → X is surjective étale, and U → V is representable by algebraic spaces, • for any commutative diagram xymatrix U ar[d] ar[r]…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces over $S$. The following are equivalent:\n\\begin{enumerate}\n\\item the morphism $f$ is representable by algebraic spaces,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are formal algebraic spaces, the vertical arrows are\nrepresentable by algebraic spaces, $U \\to X$\nis surjective \\'etale, and $U \\to V$ is representable by algebraic spaces,\n\\item for any commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are formal algebraic spaces and the vertical arrows are\nrepresentable by algebraic spaces, the morphism $U \\to V$ is\nrepresentable by algebraic spaces,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nand for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space} such that\n$X_{ji} \\to Y_j$ is representable by algebraic spaces for each $j$ and $i$,\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces, such that $X_i \\to Y_i$ is representable by algebraic\nspaces, and\n\\item add more here.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AN4","source_file":"formal-spaces.tex","source_line":3876,"source_end_line":3915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3876-L3915","statement_sha256":"e585b57eb93cd921ed5b4478ae344abc2053df721edc0be780b860d0478831fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12989,"rank":12989,"depth":70,"x":1219.906,"y":1534.777,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJH","tag":"0AJH","title":"Morphisms representable by algebraic spaces · Lemma 0AJH","summary":"Let S be a scheme. Let Y be an affine formal algebraic space over S. Let f : X → Y be a map of sheaves on (Sch/S)_fppf which is representable by algebraic spaces. Then X is a formal algebraic space.","statement_latex":"Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space over $S$.\nLet $f : X \\to Y$ be a map of sheaves on $(\\Sch/S)_{fppf}$ which is\nrepresentable by algebraic spaces. Then $X$ is a formal\nalgebraic space.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJH","source_file":"formal-spaces.tex","source_line":3955,"source_end_line":3961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3955-L3961","statement_sha256":"0b2a5a9bc25d6eb61040c6cab751ac624a9327caacd381d624dca990b0ad59ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":12990,"rank":12990,"depth":68,"x":1116.124,"y":1699.335,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJI","tag":"0AJI","title":"Morphisms representable by algebraic spaces · Lemma 0AJI","summary":"Let S be a scheme. Let Y be a formal algebraic space over S. Let f : X → Y be a map of sheaves on (Sch/S)_fppf which is representable by algebraic spaces. Then X is a formal algebraic space.","statement_latex":"Let $S$ be a scheme. Let $Y$ be a formal algebraic space over $S$.\nLet $f : X \\to Y$ be a map of sheaves on $(\\Sch/S)_{fppf}$ which is\nrepresentable by algebraic spaces. Then $X$ is a formal\nalgebraic space.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJI","source_file":"formal-spaces.tex","source_line":3972,"source_end_line":3978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3972-L3978","statement_sha256":"5c0113be6afceb8bbe54c9ae1e0addb8746d58bf38ab39a6282fc843fc038fcb","origin":"The Stacks Project","memory_eligible":false,"source_rank":12991,"rank":12991,"depth":69,"x":1060.192,"y":1518.694,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKN","tag":"0AKN","title":"Morphisms representable by algebraic spaces · Lemma 0AKN","summary":"Let S be a scheme. Let f : X → Y be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then f is representable (by schemes) and affine.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\naffine formal algebraic spaces which is representable by\nalgebraic spaces. Then $f$ is representable (by schemes) and affine.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKN","source_file":"formal-spaces.tex","source_line":3991,"source_end_line":3996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L3991-L3996","statement_sha256":"c43f2ea501b122616d29fda659bf30c4df234e2a9c2fd734af604851effa892d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12992,"rank":12992,"depth":58,"x":1247.122,"y":1620.389,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AN5","tag":"0AN5","title":"Morphisms representable by algebraic spaces · Lemma 0AN5","summary":"Let S be a scheme. Let φ : A → B be a continuous map of weakly admissible topological rings over S. The following are equivalent • Spf(φ) : Spf(B) → Spf(A) is representable by algebraic spaces, • Spf(φ) : Spf(B) → Spf(A) is representable (by schemes), • φ is taut, see Definition [Tag 0AMX].","statement_latex":"Let $S$ be a scheme. Let $\\varphi : A \\to B$ be a continuous map of\nweakly admissible topological rings over $S$. The following\nare equivalent\n\\begin{enumerate}\n\\item $\\text{Spf}(\\varphi) : \\text{Spf}(B) \\to \\text{Spf}(A)$\nis representable by algebraic spaces,\n\\item $\\text{Spf}(\\varphi) : \\text{Spf}(B) \\to \\text{Spf}(A)$\nis representable (by schemes),\n\\item $\\varphi$ is taut, see Definition \\ref{definition-taut}.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AN5","source_file":"formal-spaces.tex","source_line":4021,"source_end_line":4033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4021-L4033","statement_sha256":"ac5aaea7ac84eafa949e5c35eee1908a74890a8e9e3440b5c8453c24408cc72a","origin":"The Stacks Project","memory_eligible":false,"source_rank":12993,"rank":12993,"depth":59,"x":1027.009,"y":1651.539,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKP","tag":"0AKP","title":"Morphisms representable by algebraic spaces · Lemma 0AKP","summary":"Let S be a scheme. Let Y be an affine formal algebraic space. Let f : X → Y be a map of sheaves on (Sch/S)_fppf which is representable and affine. Then • X is an affine formal algebraic space, • if Y is countably indexed, then X is countably indexed, • if Y is countably indexed and classical, then X is countably indexed and classical, • if Y is weakly adic, then X is weakly adic, • if Y is adic*, then X is adic*, and • if Y is Noetherian and f is (locally) of finite type,…","statement_latex":"Let $S$ be a scheme. Let $Y$ be an affine formal algebraic space.\nLet $f : X \\to Y$ be a map of sheaves on $(\\Sch/S)_{fppf}$ which\nis representable and affine. Then\n\\begin{enumerate}\n\\item $X$ is an affine formal algebraic space,\n\\item if $Y$ is countably indexed, then $X$ is countably indexed,\n\\item if $Y$ is countably indexed and classical, then $X$ is\ncountably indexed and classical,\n\\item if $Y$ is weakly adic, then $X$ is weakly adic,\n\\item if $Y$ is adic*, then $X$ is adic*, and\n\\item if $Y$ is Noetherian and $f$ is (locally) of finite type, then\n$X$ is Noetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKP","source_file":"formal-spaces.tex","source_line":4098,"source_end_line":4113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4098-L4113","statement_sha256":"8e68ba77a781c79b789e694e67e849136cc2f6f9e1f6637f2d98400373728138","origin":"The Stacks Project","memory_eligible":false,"source_rank":12994,"rank":12994,"depth":60,"x":1164.574,"y":1503.341,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKQ","tag":"0AKQ","title":"Morphisms representable by algebraic spaces · Lemma 0AKQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then • if Y is countably indexed, then X is countably indexed, • if Y is countably indexed and classical, then X is countably indexed and classical, • if Y is weakly adic, then X is weakly adic, • if Y is adic*, then X is adic*, and • if Y is Noetherian and f is (locally) of finite type, then X is Noetherian.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of affine formal\nalgebraic spaces which is representable by algebraic spaces. Then\n\\begin{enumerate}\n\\item if $Y$ is countably indexed, then $X$ is countably indexed,\n\\item if $Y$ is countably indexed and classical, then $X$ is countably\nindexed and classical,\n\\item if $Y$ is weakly adic, then $X$ is weakly adic,\n\\item if $Y$ is adic*, then $X$ is adic*, and\n\\item if $Y$ is Noetherian and $f$ is (locally) of finite type, then\n$X$ is Noetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKQ","source_file":"formal-spaces.tex","source_line":4155,"source_end_line":4168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4155-L4168","statement_sha256":"fe757eafe1213c66af1320357cd186ce1d205a0fefba64766ca0863f78d87bfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":12995,"rank":12995,"depth":61,"x":1182.352,"y":1691.095,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AN8","tag":"0AN8","title":"Morphisms representable by algebraic spaces · Lemma 0AN8","summary":"Let S be a scheme. Let Y be a McQuillan affine formal algebraic space over S, i.e., Y = Spf(B) for some weakly admissible topological S-algebra B. Then there is an equivalence of categories between • the category of morphisms f : X → Y of affine formal algebraic spaces which are representable by algebraic spaces and étale, and • the category of topological B-algebras of the form A^wedge where A is an étale B-algebra and A^wedge = lim A/JA with J ⊂ B running over the weak…","statement_latex":"Let $S$ be a scheme. Let $Y$ be a McQuillan affine formal algebraic space\nover $S$, i.e., $Y = \\text{Spf}(B)$ for some weakly admissible topological\n$S$-algebra $B$. Then there is an equivalence of categories between\n\\begin{enumerate}\n\\item the category of morphisms $f : X \\to Y$\nof affine formal algebraic spaces which are representable\nby algebraic spaces and \\'etale, and\n\\item the category of topological $B$-algebras of the form\n$A^\\wedge$ where $A$ is an \\'etale $B$-algebra and\n$A^\\wedge = \\lim A/JA$ with $J \\subset B$ running over the\nweak ideals of definition of $B$.\n\\end{enumerate}\nThe equivalence is given by sending $A^\\wedge$ to  $X = \\text{Spf}(A^\\wedge)$.\nIn particular, any $X$ as in (1) is McQuillan.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AN8","source_file":"formal-spaces.tex","source_line":4224,"source_end_line":4240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4224-L4240","statement_sha256":"2d4cbf77a40d72bfb96be9df7703ebe2042630adc5b590d5e9204f32ecb13f1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":12996,"rank":12996,"depth":59,"x":1017.892,"y":1562.451,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AN9","tag":"0AN9","title":"Morphisms representable by algebraic spaces · Lemma 0AN9","summary":"With notation and assumptions as in Lemma [Tag 0AN8] let f : X → Y correspond to B → A^wedge. The following are equivalent • f : X → Y is surjective, • B → A is faithfully flat, • for every weak ideal of definition J ⊂ B the ring map B/J → A/JA is faithfully flat, and • for some weak ideal of definition J ⊂ B the ring map B/J → A/JA is faithfully flat.","statement_latex":"With notation and assumptions as in Lemma \\ref{lemma-etale} let\n$f : X \\to Y$ correspond to $B \\to A^\\wedge$. The following are equivalent\n\\begin{enumerate}\n\\item $f : X \\to Y$ is surjective,\n\\item $B \\to A$ is faithfully flat,\n\\item for every weak ideal of definition $J \\subset B$\nthe ring map $B/J \\to A/JA$ is faithfully flat, and\n\\item for some weak ideal of definition $J \\subset B$\nthe ring map $B/J \\to A/JA$ is faithfully flat.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AN9","source_file":"formal-spaces.tex","source_line":4274,"source_end_line":4286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4274-L4286","statement_sha256":"17de22de1df2c4f167c87a1ccb68ff6eccd60769d500e3c35af220d79391fa0d","origin":"The Stacks Project","memory_eligible":false,"source_rank":12997,"rank":12997,"depth":60,"x":1243.111,"y":1563.997,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKS","tag":"0AKS","title":"Types of formal algebraic spaces · Lemma 0AKS","summary":"Let S be a scheme. Let X → Y be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then X is countably indexed if and only if Y is countably indexed.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine\nformal algebraic spaces which is representable by algebraic spaces,\nsurjective, and flat. Then $X$ is countably indexed if and only\nif $Y$ is countably indexed.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKS","source_file":"formal-spaces.tex","source_line":4331,"source_end_line":4337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4331-L4337","statement_sha256":"8e6af068d3c205837963225212b033c53fade3125dca1e16a026b8160976a9a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":12998,"rank":12998,"depth":62,"x":1075.427,"y":1690.929,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXR","tag":"0GXR","title":"Types of formal algebraic spaces · Lemma 0GXR","summary":"Let S be a scheme. Let X → Y be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then X is countably indexed and classical if and only if Y is countably indexed and classical.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine\nformal algebraic spaces which is representable by algebraic spaces,\nsurjective, and flat. Then $X$ is countably indexed and classical\nif and only if $Y$ is countably indexed and classical.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXR","source_file":"formal-spaces.tex","source_line":4379,"source_end_line":4385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4379-L4385","statement_sha256":"98f1d9a7b82cde7d1a3bd4e2c8e60c175d456e83af77f6a8675f2acde5f40c46","origin":"The Stacks Project","memory_eligible":false,"source_rank":12999,"rank":12999,"depth":63,"x":1097.047,"y":1501.77,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXS","tag":"0GXS","title":"Types of formal algebraic spaces · Lemma 0GXS","summary":"Let S be a scheme. Let X → Y be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then X is weakly adic if and only if Y is weakly adic.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine\nformal algebraic spaces which is representable by algebraic spaces,\nsurjective, and flat. Then $X$ is weakly adic\nif and only if $Y$ is weakly adic.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXS","source_file":"formal-spaces.tex","source_line":4410,"source_end_line":4416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4410-L4416","statement_sha256":"a2196aefe2d539a04bc24c351fc80f07aca2f2d9c5bfd7e350b78c1b04ae202f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13000,"rank":13000,"depth":64,"x":1233.518,"y":1653.852,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKT","tag":"0AKT","title":"Types of formal algebraic spaces · Lemma 0AKT","summary":"Let S be a scheme. Let X → Y be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then X is adic* if and only if Y is adic*.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine\nformal algebraic spaces which is representable by algebraic spaces,\nsurjective, and flat. Then $X$ is adic* if and only if $Y$ is adic*.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKT","source_file":"formal-spaces.tex","source_line":4424,"source_end_line":4429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4424-L4429","statement_sha256":"b6c14550aa5541e2d9f9d84517aff3076107b96a8b243e69e7e065ea0331d61a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13001,"rank":13001,"depth":64,"x":1010.103,"y":1619.07,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKW","tag":"0AKW","title":"Types of formal algebraic spaces · Lemma 0AKW","summary":"Let S be a scheme. Let X → Y be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and (locally) of finite type. Then X is Noetherian if and only if Y is Noetherian.","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ be a morphism of affine\nformal algebraic spaces which is representable by algebraic spaces,\nsurjective, flat, and (locally) of finite type. Then $X$ is Noetherian\nif and only if $Y$ is Noetherian.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKW","source_file":"formal-spaces.tex","source_line":4437,"source_end_line":4443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4437-L4443","statement_sha256":"3e26fd26fcaae04be79db34485248b2a218f6c2ebb6f6b7dedbfde2864329ff1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13002,"rank":13002,"depth":65,"x":1203.232,"y":1517.728,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKX","tag":"0AKX","title":"Types of formal algebraic spaces · Lemma 0AKX","summary":"Let S be a scheme. Let P ∈ ( countably indexed, countably indexed and classical, weakly adic, adic*, Noetherian ) Let X be a formal algebraic space over S. The following are equivalent • if Y is an affine formal algebraic space and f : Y → X is representable by algebraic spaces and étale, then Y has property P, • for some (X_i → X)_i ∈ I as in Definition [Tag 0AIM] each X_i has property P.","statement_latex":"Let $S$ be a scheme. Let\n$$\nP \\in\n\\left\\{\n\\begin{matrix}\ncountably\\ indexed,\\\\\ncountably\\ indexed\\ and\\ classical,\\\\\nweakly\\ adic,\\ adic*,\\ Noetherian\n\\end{matrix}\n\\right\\}\n$$\nLet $X$ be a formal algebraic space over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item if $Y$ is an affine formal algebraic space and\n$f : Y \\to X$ is representable by algebraic spaces and \\'etale,\nthen $Y$ has property $P$,\n\\item for some $\\{X_i \\to X\\}_{i \\in I}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\neach $X_i$ has property $P$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKX","source_file":"formal-spaces.tex","source_line":4464,"source_end_line":4487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4464-L4487","statement_sha256":"4c8e89eb96d411493508347d8295483ba717a18e717900e3748c1745d8bdba65","origin":"The Stacks Project","memory_eligible":false,"source_rank":13003,"rank":13003,"depth":69,"x":1142.186,"y":1702.439,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKY","tag":"0AKY","title":"Types of formal algebraic spaces · Definition 0AKY","summary":"Let S be a scheme. Let X be a formal algebraic space over S. We say X is locally countably indexed, locally countably indexed and classical, locally weakly adic, locally adic*, or locally Noetherian if the equivalent conditions of Lemma [Tag 0AKX] hold for the corresponding property.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nWe say $X$ is\n{\\it locally countably indexed},\n{\\it locally countably indexed and classical},\n{\\it locally weakly adic},\n{\\it locally adic*}, or\n{\\it locally Noetherian}\nif the equivalent conditions of Lemma \\ref{lemma-type-local}\nhold for the corresponding property.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKY","source_file":"formal-spaces.tex","source_line":4517,"source_end_line":4528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4517-L4528","statement_sha256":"0fef941ee4a551cbe2d3f1a458317f50a40388a61c05cb0f3a1bcc1f23e4cad3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13004,"rank":13004,"depth":70,"x":1038.44,"y":1531.231,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQ1","tag":"0AQ1","title":"Types of formal algebraic spaces · Lemma 0AQ1","summary":"Let S be a scheme. Let X be an algebraic space over S. Let T ⊂ |X| be a closed subset. Let X_/T be the formal completion of X along T. • If X setminus T → X is quasi-compact, then X_/T is locally adic*. • If X is locally Noetherian, then X_/T is locally Noetherian.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset. Let $X_{/T}$ be the\nformal completion of $X$ along $T$.\n\\begin{enumerate}\n\\item If $X \\setminus T \\to X$ is quasi-compact,\nthen $X_{/T}$ is locally adic*.\n\\item If $X$ is locally Noetherian, then $X_{/T}$ is locally\nNoetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQ1","source_file":"formal-spaces.tex","source_line":4534,"source_end_line":4545,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4534-L4545","statement_sha256":"4ea2857603cf68de7a9956036b10b50cdf0467d768fbd3bdc2bd8dbb83639634","origin":"The Stacks Project","memory_eligible":false,"source_rank":13005,"rank":13005,"depth":41,"x":1253.078,"y":1598.753,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBB","tag":"0GBB","title":"Types of formal algebraic spaces · Lemma 0GBB","summary":"Let S be a scheme. Let X → Y and Z → Y be morphisms of formal algebraic space over S. Then • If X and Z are locally countably indexed, then X ×_Y Z is locally countably indexed. • If X and Z are locally countably indexed and classical, then X ×_Y Z is locally countably indexed and classical. • If X and Z are weakly adic, then X ×_Y Z is weakly adic. • If X and Z are locally adic*, then X ×_Y Z is locally adic*. • If X and Z are locally Noetherian and X_red → Y_red is…","statement_latex":"Let $S$ be a scheme. Let $X \\to Y$ and $Z \\to Y$ be\nmorphisms of formal algebraic space over $S$. Then\n\\begin{enumerate}\n\\item If $X$ and $Z$ are locally countably indexed, then $X \\times_Y Z$\nis locally countably indexed.\n\\item If $X$ and $Z$ are locally countably indexed and classical,\nthen $X \\times_Y Z$ is locally countably indexed and classical.\n\\item If $X$ and $Z$ are weakly adic, then $X \\times_Y Z$\nis weakly adic.\n\\item If $X$ and $Z$ are locally adic*, then $X \\times_Y Z$ is\nlocally adic*.\n\\item If $X$ and $Z$ are locally Noetherian and $X_{red} \\to Y_{red}$\nis locally of finite type, then $X \\times_Y Z$ is locally Noetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBB","source_file":"formal-spaces.tex","source_line":4582,"source_end_line":4598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4582-L4598","statement_sha256":"3d6c44bf8b4ab758e59a0f34e0661099386819012559b2d28255853ed83e8b8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13006,"rank":13006,"depth":58,"x":1040.057,"y":1670.906,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHM","tag":"0GHM","title":"Types of formal algebraic spaces · Lemma 0GHM","summary":"Let S be a scheme. Let X be a locally Noetherian formal algebraic space over S. Then X = colim X_n for a system X_1 → X_2 → X_3 → … of finite order thickenings of locally Noetherian algebraic spaces over S where X_1 = X_red and X_n is the nth infinitesimal neighbourhood of X_1 in X_m for all m ≥ n.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian formal algebraic space\nover $S$. Then $X = \\colim X_n$ for a system $X_1 \\to X_2 \\to X_3 \\to \\ldots$\nof finite order thickenings of locally Noetherian algebraic spaces over $S$\nwhere $X_1 = X_{red}$ and $X_n$ is the $n$th infinitesimal neighbourhood of\n$X_1$ in $X_m$ for all $m \\geq n$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Types of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHM","source_file":"formal-spaces.tex","source_line":4701,"source_end_line":4708,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4701-L4708","statement_sha256":"ce6065fe883bce7b5326bf55d559416bf9f9f2b8a3631b836b4e1c6adca9b812","origin":"The Stacks Project","memory_eligible":false,"source_rank":13007,"rank":13007,"depth":0,"x":1139.321,"y":1496.461,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANB","tag":"0ANB","title":"Morphisms and continuous ring maps · Lemma 0ANB","summary":"Let A ∈ Ob(WAdm). Let A → A' be a ring map (no topology). Let (A')^wedge = lim_I ⊂ A w.i.d A'/IA' be the object of WAdm constructed in Example [Tag 0AN6]. • If A is in WAdm^count, so is (A')^wedge. • If A is in WAdm^cic, so is (A')^wedge. • If A is in WAdm^weakly adic, so is (A')^wedge. • If A is in WAdm^adic*, so is (A')^wedge. • If A is in WAdm^Noeth and A' is Noetherian, then (A')^wedge is in WAdm^Noeth.","statement_latex":"Let $A \\in \\Ob(\\textit{WAdm})$. Let $A \\to A'$ be a ring\nmap (no topology). Let $(A')^\\wedge = \\lim_{I \\subset A\\text{ w.i.d}} A'/IA'$\nbe the object of $\\textit{WAdm}$ constructed in\nExample \\ref{example-representable-morphism-from-completion}.\n\\begin{enumerate}\n\\item If $A$ is in $\\textit{WAdm}^{count}$, so is $(A')^\\wedge$.\n\\item If $A$ is in $\\textit{WAdm}^{cic}$, so is $(A')^\\wedge$.\n\\item If $A$ is in $\\textit{WAdm}^{weakly\\ adic}$, so is $(A')^\\wedge$.\n\\item If $A$ is in $\\textit{WAdm}^{adic*}$, so is $(A')^\\wedge$.\n\\item If $A$ is in $\\textit{WAdm}^{Noeth}$ and $A'$ is Noetherian, then\n$(A')^\\wedge$ is in $\\textit{WAdm}^{Noeth}$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms and continuous ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANB","source_file":"formal-spaces.tex","source_line":4811,"source_end_line":4825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4811-L4825","statement_sha256":"2b8f47b2673e720b38075dfb1b0c18bd6b188affe6b77b5b576af434ef8f18ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":13008,"rank":13008,"depth":7,"x":1206.55,"y":1681.808,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANG","tag":"0ANG","title":"Morphisms and continuous ring maps · Lemma 0ANG","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally countably indexed formal algebraic spaces over S. Let P be a local property of morphisms of WAdm^count. The following are equivalent • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a morphism of WAdm^count with property P, • there exists a covering…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally countably indexed formal algebraic spaces over $S$.\nLet $P$ be a local property of morphisms of $\\textit{WAdm}^{count}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a morphism of $\\textit{WAdm}^{count}$ with property $P$,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space} and for each $j$\na covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nsuch that each $X_{ji} \\to Y_j$  corresponds\nto a morphism of $\\textit{WAdm}^{count}$ with property $P$, and\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds\nto a morphism of $\\textit{WAdm}^{count}$ with property $P$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms and continuous ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANG","source_file":"formal-spaces.tex","source_line":4886,"source_end_line":4916,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L4886-L4916","statement_sha256":"805ebb504f4b2458495633f5be87a792b9f9c2e2062e829aedce629426beaa4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13009,"rank":13009,"depth":69,"x":1007.509,"y":1583.077,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBD","tag":"0GBD","title":"Morphisms and continuous ring maps · Lemma 0GBD","summary":"Let S be a scheme. Let P be a local property of morphisms of WAdm^count which is stable under base change. Let f : X → Y and g : Z → Y be morphisms of locally countably indexed formal algebraic spaces over S. If f satisfies the equivalent conditions of Lemma [Tag 0ANG] then so does pr_2 : X ×_Y Z → Z.","statement_latex":"Let $S$ be a scheme. Let $P$ be a local property of morphisms of\n$\\textit{WAdm}^{count}$ which is stable under base change.\nLet $f : X \\to Y$ and $g : Z \\to Y$ be morphisms of locally countably indexed\nformal algebraic spaces over $S$. If $f$ satisfies the equivalent conditions of\nLemma \\ref{lemma-property-defines-property-morphisms}\nthen so does $\\text{pr}_2 : X \\times_Y Z \\to Z$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms and continuous ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBD","source_file":"formal-spaces.tex","source_line":5008,"source_end_line":5016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5008-L5016","statement_sha256":"49c99e1b82de1d8097ed2743f211c597cc719c09931550aab80f9b6ae15d92bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13010,"rank":13010,"depth":70,"x":1234.148,"y":1542.856,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBI","tag":"0GBI","title":"Morphisms and continuous ring maps · Lemma 0GBI","summary":"Let S be a scheme. Let P be a local property of morphisms of WAdm^count which is stable under composition. Let f : X → Y and g : Y → Z be morphisms of locally countably indexed formal algebraic spaces over S. If f and g satisfies the equivalent conditions of Lemma [Tag 0ANG] then so does g ∘ f : X → Z.","statement_latex":"Let $S$ be a scheme. Let $P$ be a local property of morphisms of\n$\\textit{WAdm}^{count}$ which is stable under composition.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of locally countably indexed\nformal algebraic spaces over $S$. If $f$ and $g$\nsatisfies the equivalent conditions of\nLemma \\ref{lemma-property-defines-property-morphisms}\nthen so does $g \\circ f : X \\to Z$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms and continuous ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBI","source_file":"formal-spaces.tex","source_line":5112,"source_end_line":5121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5112-L5121","statement_sha256":"3700453f4b3aae63504131317271061062b53a889cb8960e6ed190363d652446","origin":"The Stacks Project","memory_eligible":false,"source_rank":13011,"rank":13011,"depth":70,"x":1099.092,"y":1701.443,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBM","tag":"0GBM","title":"Morphisms and continuous ring maps · Lemma 0GBM","summary":"Let S be a scheme. Let P be a local property of morphisms of WAdm^count which has the cancellation property. Let f : X → Y and g : Y → Z be morphisms of locally countably indexed formal algebraic spaces over S. If g ∘ f and g satisfies the equivalent conditions of Lemma [Tag 0ANG] then so does f : X → Y.","statement_latex":"Let $S$ be a scheme. Let $P$ be a local property of morphisms of\n$\\textit{WAdm}^{count}$ which has the cancellation property.\nLet $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of locally countably indexed\nformal algebraic spaces over $S$. If $g \\circ f$ and $g$\nsatisfies the equivalent conditions of\nLemma \\ref{lemma-property-defines-property-morphisms}\nthen so does $f : X \\to Y$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms and continuous ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBM","source_file":"formal-spaces.tex","source_line":5172,"source_end_line":5181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5172-L5181","statement_sha256":"7090c94c1f645c20d146732574e8c12c5a48bef509af936ca1007ad406acea75","origin":"The Stacks Project","memory_eligible":false,"source_rank":13012,"rank":13012,"depth":70,"x":1071.092,"y":1507.469,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANJ","tag":"0ANJ","title":"Taut ring maps and representability by algebraic spaces · Lemma 0ANJ","summary":"Let B → A be an arrow of WAdm^count. The following are equivalent • [(a)] B → A is taut (Definition [Tag 0AMX]), • [(b)] for B ⊃ J_1 ⊃ J_2 ⊃ J_3 ⊃ … a fundamental system of weak ideals of definitions there exist a commutative diagram xymatrix A ar[r] & … ar[r] & A_3 ar[r] & A_2 ar[r] & A_1 B ar[r] ar[u] & … ar[r] & B/J_3 ar[r] ar[u] & B/J_2 ar[r] ar[u] & B/J_1 ar[u] such that A_n + 1/J_nA_n + 1 = A_n and A = lim A_n as topological ring. Moreover, these equivalent…","statement_latex":"Let $B \\to A$ be an arrow of $\\textit{WAdm}^{count}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item[(a)] $B \\to A$ is taut (Definition \\ref{definition-taut}),\n\\item[(b)] for $B \\supset J_1 \\supset J_2 \\supset J_3 \\supset \\ldots$\na fundamental system of weak ideals of definitions there exist\na commutative diagram\n$$\n\\xymatrix{\nA \\ar[r] & \\ldots \\ar[r] & A_3 \\ar[r] & A_2 \\ar[r] & A_1 \\\\\nB \\ar[r] \\ar[u] & \\ldots \\ar[r] & B/J_3 \\ar[r] \\ar[u] &\nB/J_2 \\ar[r] \\ar[u] & B/J_1 \\ar[u]\n}\n$$\nsuch that $A_{n + 1}/J_nA_{n + 1} = A_n$ and $A = \\lim A_n$\nas topological ring.\n\\end{enumerate}\nMoreover, these equivalent conditions define a local property,\ni.e., they satisfy axioms (\\ref{item-axiom-1}), (\\ref{item-axiom-2}),\n(\\ref{item-axiom-3}).","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps and representability by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANJ","source_file":"formal-spaces.tex","source_line":5248,"source_end_line":5270,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5248-L5270","statement_sha256":"1c1e44dcaf49e42fc308b25afd92db25971efb25ff7e15d45c800fd716a117ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":13013,"rank":13013,"depth":71,"x":1248.091,"y":1634.876,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANK","tag":"0ANK","title":"Taut ring maps and representability by algebraic spaces · Lemma 0ANK","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally countably indexed formal algebraic spaces over S. The following are equivalent • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a taut map B → A of WAdm^count, • there exists a covering (Y_j → Y) as in Definition [Tag 0AIM] and for each j a covering…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally countably indexed formal algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a taut map $B \\to A$ of $\\textit{WAdm}^{count}$,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space} and for each $j$\na covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nsuch that each $X_{ji} \\to Y_j$  corresponds\nto a taut ring map in $\\textit{WAdm}^{count}$,\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds\nto a taut ring map in $\\textit{WAdm}^{count}$, and\n\\item $f$ is representable by algebraic spaces.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Taut ring maps and representability by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANK","source_file":"formal-spaces.tex","source_line":5337,"source_end_line":5367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5337-L5367","statement_sha256":"b4c80c74c76211561d1230e117bd4c946d57189e3ecc0456a11ff5dd2cc6e562","origin":"The Stacks Project","memory_eligible":false,"source_rank":13014,"rank":13014,"depth":72,"x":1014.639,"y":1641.375,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBS","tag":"0GBS","title":"Adic morphisms · Lemma 0GBS","summary":"Let A and B be pre-adic topological rings. Let φ : A → B be a continuous ring homomorphism. • If φ is adic, then φ is taut. • If B is complete, A has a finitely generated ideal of definition, and φ is taut, then φ is adic. In particular the conditions \"φ is adic\" and \"φ is taut\" are equivalent on the category WAdm^adic*.","statement_latex":"Let $A$ and $B$ be pre-adic topological rings. Let\n$\\varphi : A \\to B$ be a continuous ring homomorphism.\n\\begin{enumerate}\n\\item If $\\varphi$ is adic, then $\\varphi$ is taut.\n\\item If $B$ is complete, $A$ has a finitely generated\nideal of definition, and $\\varphi$ is taut, then $\\varphi$ is adic.\n\\end{enumerate}\nIn particular the conditions ``$\\varphi$ is adic'' and ``$\\varphi$ is taut''\nare equivalent on the category $\\textit{WAdm}^{adic*}$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Adic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBS","source_file":"formal-spaces.tex","source_line":5407,"source_end_line":5418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5407-L5418","statement_sha256":"4a640a2ca16ca7149bb74fb99c49b0a8ba471192a17e64cb38565c07c33e884f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13015,"rank":13015,"depth":6,"x":1181.898,"y":1503.837,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQ3","tag":"0AQ3","title":"Adic morphisms · Definition 0AQ3","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. Assume X and Y are locally adic*. We say f is an adic morphism if f is representable by algebraic spaces. See discussion above.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces over $S$. Assume $X$ and $Y$ are locally adic*. We say $f$ is\nan {\\it adic morphism} if $f$ is representable by algebraic spaces.\nSee discussion above.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Adic morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQ3","source_file":"formal-spaces.tex","source_line":5452,"source_end_line":5458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5452-L5458","statement_sha256":"e4c36cc6c2ccc6939034713dba8118394c77d29b46e8a050c232e2488807873e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13016,"rank":13016,"depth":0,"x":1169.142,"y":1700.561,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AM4","tag":"0AM4","title":"Morphisms of finite type · Definition 0AM4","summary":"Let S be a scheme. Let f : Y → X be a morphism of formal algebraic spaces over S. • We say f is locally of finite type if f is representable by algebraic spaces and is locally of finite type in the sense of Bootstrap, Definition [Tag 03XZ]. • We say f is of finite type if f is locally of finite type and quasi-compact (Definition [Tag 0AJC]).","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of formal algebraic\nspaces over $S$.\n\\begin{enumerate}\n\\item We say $f$ is {\\it locally of finite type}\nif $f$ is representable by algebraic spaces and is locally\nof finite type in the sense of\nBootstrap, Definition \\ref{bootstrap-definition-property-transformation}.\n\\item We say $f$ is of {\\it finite type} if $f$ is locally of finite type and\nquasi-compact (Definition \\ref{definition-quasi-compact-morphism}).\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms of finite type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AM4","source_file":"formal-spaces.tex","source_line":5478,"source_end_line":5490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5478-L5490","statement_sha256":"569066057665a5769a08c60dad53dd5b936acaeb9676dc64129c926d6940b7ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":13017,"rank":13017,"depth":59,"x":1020.047,"y":1547.953,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJJ","tag":"0AJJ","title":"Morphisms of finite type · Lemma 0AJJ","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. The following are equivalent • f is of finite type, • f is representable by algebraic spaces and is of finite type in the sense of Bootstrap, Definition [Tag 03XZ].","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is of finite type,\n\\item $f$ is representable by algebraic spaces and is of finite type in\nthe sense of\nBootstrap, Definition \\ref{bootstrap-definition-property-transformation}.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJJ","source_file":"formal-spaces.tex","source_line":5498,"source_end_line":5508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5498-L5508","statement_sha256":"baf2d78b11a2e4cbac6b5f1fac3cfd78ccdf344bf4abf8b3673aac924320d0e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13018,"rank":13018,"depth":60,"x":1253.18,"y":1575.94,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQ4","tag":"0AQ4","title":"Morphisms of finite type · Lemma 0AQ4","summary":"The composition of finite type morphisms is of finite type. The same holds for locally of finite type.","statement_latex":"The composition of finite type morphisms is of finite type.\nThe same holds for locally of finite type.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQ4","source_file":"formal-spaces.tex","source_line":5518,"source_end_line":5522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5518-L5522","statement_sha256":"29114cbfbfd7bef270f56fe8296d511b38800454e0768aee916b29c579651f51","origin":"The Stacks Project","memory_eligible":false,"source_rank":13019,"rank":13019,"depth":56,"x":1058.373,"y":1687.812,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQ5","tag":"0AQ5","title":"Morphisms of finite type · Lemma 0AQ5","summary":"A base change of a finite type morphism is finite type. The same holds for locally of finite type.","statement_latex":"A base change of a finite type morphism is finite type.\nThe same holds for locally of finite type.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQ5","source_file":"formal-spaces.tex","source_line":5530,"source_end_line":5534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5530-L5534","statement_sha256":"d2cce72dc29f174023a6766bc2b0fb301a292ad246c29acd7a1c207369745713","origin":"The Stacks Project","memory_eligible":false,"source_rank":13020,"rank":13020,"depth":7,"x":1112.164,"y":1494.395,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQ6","tag":"0AQ6","title":"Morphisms of finite type · Lemma 0AQ6","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of formal algebraic spaces over S. If g ∘ f : X → Z is locally of finite type, then f : X → Y is locally of finite type.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of\nformal algebraic spaces over $S$. If $g \\circ f : X \\to Z$ is locally of\nfinite type, then $f : X \\to Y$ is locally of finite type.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQ6","source_file":"formal-spaces.tex","source_line":5542,"source_end_line":5547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5542-L5547","statement_sha256":"13b59e73581037d3a088035cdcc640f1bcf33596df5b164788a477b12b797eba","origin":"The Stacks Project","memory_eligible":false,"source_rank":13021,"rank":13021,"depth":70,"x":1228.271,"y":1667.886,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANL","tag":"0ANL","title":"Morphisms of finite type · Lemma 0ANL","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. The following are equivalent: • the morphism f is locally of finite type, • there exists a commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y where U, V are formal algebraic spaces, the vertical arrows are representable by algebraic spaces and étale, U → X is surjective, and U → V is locally of finite type, • for any commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r]…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces over $S$. The following are equivalent:\n\\begin{enumerate}\n\\item the morphism $f$ is locally of finite type,\n\\item there exists a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are formal algebraic spaces, the vertical arrows are\nrepresentable by algebraic spaces and \\'etale, $U \\to X$\nis surjective, and $U \\to V$ is locally of finite type,\n\\item for any commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwhere $U$, $V$ are formal algebraic spaces and vertical arrows\nrepresentable by algebraic spaces and \\'etale, the morphism\n$U \\to V$ is locally of finite type,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nand for each $j$ a covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space} such that\n$X_{ji} \\to Y_j$ is locally of finite type for each $j$ and $i$,\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, such that $X_i \\to Y_i$ is\nlocally of finite type, and\n\\item add more here.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANL","source_file":"formal-spaces.tex","source_line":5561,"source_end_line":5600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5561-L5600","statement_sha256":"65fc565556e1326e9568c1401c90125a176744dbded81d3bd58f9c94b6f2faea","origin":"The Stacks Project","memory_eligible":false,"source_rank":13022,"rank":13022,"depth":71,"x":1002.693,"y":1605.715,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQ7","tag":"0AQ7","title":"Morphisms of finite type · Lemma 0AQ7","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. If Y is locally Noetherian and f locally of finite type, then X is locally Noetherian.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nformal algebraic spaces over $S$. If $Y$ is locally Noetherian and\n$f$ locally of finite type, then $X$ is locally Noetherian.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQ7","source_file":"formal-spaces.tex","source_line":5685,"source_end_line":5690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5685-L5690","statement_sha256":"3bcd1ec29ad972604c495402c5e6368b33ac639db8ce374f60d4df32d25a1f54","origin":"The Stacks Project","memory_eligible":false,"source_rank":13023,"rank":13023,"depth":72,"x":1219.455,"y":1523.398,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQ8","tag":"0AQ8","title":"Morphisms of finite type · Lemma 0AQ8","summary":"Let S be a scheme. Let f : X → Y and Z → Y be morphisms of formal algebraic spaces over S. If Z is locally Noetherian and f locally of finite type, then Z ×_Y X is locally Noetherian.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $Z \\to Y$ be morphisms\nof formal algebraic spaces over $S$. If $Z$ is locally\nNoetherian and $f$ locally of finite type, then\n$Z \\times_Y X$ is locally Noetherian.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQ8","source_file":"formal-spaces.tex","source_line":5699,"source_end_line":5705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5699-L5705","statement_sha256":"71d168180676eb87b44f8e148ef88b2fced6aba2e014cc1be26a92193567ef53","origin":"The Stacks Project","memory_eligible":false,"source_rank":13024,"rank":13024,"depth":73,"x":1125.63,"y":1707.454,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHP","tag":"0GHP","title":"Surjective morphisms · Definition 0GHP","summary":"Let S be a scheme. A morphism f : X → Y of formal algebraic spaces over S is said to be surjective if it induces a surjective morphism X_red → Y_red on underlying reduced algebraic spaces.","statement_latex":"Let $S$ be a scheme. A morphism $f : X \\to Y$ of formal algebraic spaces\nover $S$ is said to be {\\it surjective} if it induces a surjective morphism\n$X_{red} \\to Y_{red}$ on underlying reduced algebraic spaces.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Surjective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHP","source_file":"formal-spaces.tex","source_line":5729,"source_end_line":5734,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5729-L5734","statement_sha256":"4a763cb3f076c7ef1f04c3f95ef1107073215ff3f5d834592d7c897a3eecfaf1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13025,"rank":13025,"depth":0,"x":1046.648,"y":1518.125,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHQ","tag":"0GHQ","title":"Surjective morphisms · Lemma 0GHQ","summary":"The composition of two surjective morphisms is a surjective morphism.","statement_latex":"The composition of two surjective morphisms is a surjective morphism.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHQ","source_file":"formal-spaces.tex","source_line":5736,"source_end_line":5739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5736-L5739","statement_sha256":"7ba600d809f24c6c614c6563850d828f1d3ecc76ecf8ebac0716e7dcc2c72917","origin":"The Stacks Project","memory_eligible":false,"source_rank":13026,"rank":13026,"depth":0,"x":1257.552,"y":1613.103,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHR","tag":"0GHR","title":"Surjective morphisms · Lemma 0GHR","summary":"A base change of a surjective morphism is a surjective morphism.","statement_latex":"A base change of a surjective morphism is a surjective morphism.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHR","source_file":"formal-spaces.tex","source_line":5745,"source_end_line":5748,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5745-L5748","statement_sha256":"1bc5b96e9b431c592d083b272ab35e008b5488cd91f68f443241941fd620fd84","origin":"The Stacks Project","memory_eligible":false,"source_rank":13027,"rank":13027,"depth":0,"x":1025.205,"y":1662.836,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHS","tag":"0GHS","title":"Surjective morphisms · Lemma 0GHS","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. The following are equivalent • f is surjective, • for every scheme T and morphism T → Y the projection X ×_Y T → T is a surjective morphism of formal algebraic spaces, • for every affine scheme T and morphism T → Y the projection X ×_Y T → T is a surjective morphism of formal algebraic spaces, • there exists a covering (Y_j → Y) as in Definition [Tag 0AIM] such that each X ×_Y Y_j → Y_j is a…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is surjective,\n\\item for every scheme $T$ and morphism $T \\to Y$\nthe projection $X \\times_Y T \\to T$ is a surjective morphism\nof formal algebraic spaces,\n\\item for every affine scheme $T$ and morphism $T \\to Y$\nthe projection $X \\times_Y T \\to T$ is a surjective morphism of formal\nalgebraic spaces,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nsuch that each $X \\times_Y Y_j \\to Y_j$ is a surjective morphism of\nformal algebraic spaces,\n\\item there exists a surjective morphism $Z \\to Y$\nof formal algebraic spaces such that $X \\times_Y Z \\to Z$ is surjective, and\n\\item add more here.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHS","source_file":"formal-spaces.tex","source_line":5754,"source_end_line":5774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5754-L5774","statement_sha256":"bbcbb5f712f685ca028501dab44a0db44fee0d838fe40a38bc4f227264d7307a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13028,"rank":13028,"depth":1,"x":1156.795,"y":1493.998,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQB","tag":"0AQB","title":"Monomorphisms · Definition 0AQB","summary":"Let S be a scheme. A morphism of formal algebraic spaces over S is called a monomorphism if it is an injective map of sheaves.","statement_latex":"Let $S$ be a scheme.\nA morphism of formal algebraic spaces over $S$ is called a\n{\\it monomorphism} if it is an injective map of sheaves.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Monomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQB","source_file":"formal-spaces.tex","source_line":5792,"source_end_line":5797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5792-L5797","statement_sha256":"88c8da06f90a4f4d55b9e4b31a2ed0ea894add5e299ebbc8b2e89298f614dc65","origin":"The Stacks Project","memory_eligible":false,"source_rank":13029,"rank":13029,"depth":0,"x":1195.612,"y":1693.548,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHT","tag":"0GHT","title":"Monomorphisms · Lemma 0GHT","summary":"The composition of two monomorphisms is a monomorphism.","statement_latex":"The composition of two monomorphisms is a monomorphism.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHT","source_file":"formal-spaces.tex","source_line":5806,"source_end_line":5809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5806-L5809","statement_sha256":"f2dce142673770bbd0861a0e7c3ef814d6fcbe25f8819b0d1e8b64ae3483c1e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13030,"rank":13030,"depth":0,"x":1006.153,"y":1568.188,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHU","tag":"0GHU","title":"Monomorphisms · Lemma 0GHU","summary":"A base change of a monomorphism is a monomorphism.","statement_latex":"A base change of a monomorphism is a monomorphism.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHU","source_file":"formal-spaces.tex","source_line":5815,"source_end_line":5818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5815-L5818","statement_sha256":"fcdc85736eacb80295ca21d823f2e72d2905d2d9a3a153090412ccc5c3b61bbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13031,"rank":13031,"depth":0,"x":1247.129,"y":1553.094,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHV","tag":"0GHV","title":"Monomorphisms · Lemma 0GHV","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. The following are equivalent • f is a monomorphism, • for every scheme T and morphism T → Y the projection X ×_Y T → T is a monomorphism of formal algebraic spaces, • for every affine scheme T and morphism T → Y the projection X ×_Y T → T is a monomorphism of formal algebraic spaces, • there exists a covering (Y_j → Y) as in Definition [Tag 0AIM] such that each X ×_Y Y_j → Y_j is a…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is a monomorphism,\n\\item for every scheme $T$ and morphism $T \\to Y$\nthe projection $X \\times_Y T \\to T$ is a monomorphism\nof formal algebraic spaces,\n\\item for every affine scheme $T$ and morphism $T \\to Y$\nthe projection $X \\times_Y T \\to T$ is a monomorphism of formal\nalgebraic spaces,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nsuch that each $X \\times_Y Y_j \\to Y_j$ is a monomorphism of\nformal algebraic spaces, and\n\\item there exists a family of morphisms $\\{Y_j \\to Y\\}$ such\nthat $\\coprod Y_j \\to Y$ is a surjection of sheaves on\n$(\\Sch/S)_{fppf}$ such that each $X \\times_Y Y_j \\to Y_j$ is a\nmonomorphism for all $j$,\n\\item there exists a morphism $Z \\to Y$ of formal algebraic spaces\nwhich is representable by algebraic spaces, surjective, flat, and locally\nof finite presentation such that $X \\times_Y Z \\to X$ is a monomorphism, and\n\\item add more here.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Monomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHV","source_file":"formal-spaces.tex","source_line":5824,"source_end_line":5849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5824-L5849","statement_sha256":"8d27a612874392991a91973fbf24e97229d354a280a3ebd46d2c4bc438302d5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13032,"rank":13032,"depth":1,"x":1081.259,"y":1701.242,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANP","tag":"0ANP","title":"Closed immersions · Definition 0ANP","summary":"Let S be a scheme. Let f : Y → X be a morphism of formal algebraic spaces over S. We say f is a closed immersion if f is representable by algebraic spaces and a closed immersion in the sense of Bootstrap, Definition [Tag 03XZ].","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of formal algebraic\nspaces over $S$. We say $f$ is a {\\it closed immersion}\nif $f$ is representable by algebraic spaces and a closed immersion\nin the sense of\nBootstrap, Definition \\ref{bootstrap-definition-property-transformation}.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Closed immersions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANP","source_file":"formal-spaces.tex","source_line":5865,"source_end_line":5872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5865-L5872","statement_sha256":"ac5111b7321504f4f080cb9ed6671f3721c549508ae88ba95ce282b7ff1c8d69","origin":"The Stacks Project","memory_eligible":false,"source_rank":13033,"rank":13033,"depth":2,"x":1084.439,"y":1497.495,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHW","tag":"0GHW","title":"Closed immersions · Lemma 0GHW","summary":"The composition of two closed immersions is a closed immersion.","statement_latex":"The composition of two closed immersions is a closed immersion.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHW","source_file":"formal-spaces.tex","source_line":5877,"source_end_line":5880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5877-L5880","statement_sha256":"8c58898272aedf0b5abc96027888a2e9a777dc3f6d42a9d82550105781839ab9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13034,"rank":13034,"depth":0,"x":1246.247,"y":1649.827,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHX","tag":"0GHX","title":"Closed immersions · Lemma 0GHX","summary":"A base change of a closed immersion is a closed immersion.","statement_latex":"A base change of a closed immersion is a closed immersion.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHX","source_file":"formal-spaces.tex","source_line":5886,"source_end_line":5889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5886-L5889","statement_sha256":"568195b75fc585d3008b9ddeb3f096dc9df38b21252af1ede633fcef23fbef87","origin":"The Stacks Project","memory_eligible":false,"source_rank":13035,"rank":13035,"depth":0,"x":1003.975,"y":1629.274,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHY","tag":"0GHY","title":"Closed immersions · Lemma 0GHY","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. The following are equivalent • f is a closed immersion, • for every scheme T and morphism T → Y the projection X ×_Y T → T is a closed immersion, • for every affine scheme T and morphism T → Y the projection X ×_Y T → T is a closed immersion, • there exists a covering (Y_j → Y) as in Definition [Tag 0AIM] such that each X ×_Y Y_j → Y_j is a closed immersion, and • there exists a morphism Z →…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces over $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is a closed immersion,\n\\item for every scheme $T$ and morphism $T \\to Y$ the projection\n$X \\times_Y T \\to T$ is a closed immersion,\n\\item for every affine scheme $T$ and morphism $T \\to Y$\nthe projection $X \\times_Y T \\to T$ is a closed immersion,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nsuch that each $X \\times_Y Y_j \\to Y_j$ is a closed immersion, and\n\\item there exists a morphism $Z \\to Y$ of formal algebraic spaces\nwhich is representable by algebraic spaces, surjective, flat, and locally\nof finite presentation such that $X \\times_Y Z \\to X$ is a\nclosed immersion, and\n\\item add more here.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHY","source_file":"formal-spaces.tex","source_line":5895,"source_end_line":5914,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5895-L5914","statement_sha256":"7a140c02b47e65d76c18949b0f42ea90c9fcd9a1c0daf170a3a81f03736b939c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13036,"rank":13036,"depth":1,"x":1199.517,"y":1506.73,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANQ","tag":"0ANQ","title":"Closed immersions · Lemma 0ANQ","summary":"Let S be a scheme. Let X be a McQuillan affine formal algebraic space over S. Let f : Y → X be a closed immersion of formal algebraic spaces over S. Then Y is a McQuillan affine formal algebraic space and f corresponds to a continuous homomorphism A → B of weakly admissible topological S-algebras which is taut, has closed kernel, and has dense image.","statement_latex":"Let $S$ be a scheme. Let $X$ be a McQuillan affine formal algebraic space\nover $S$. Let $f : Y \\to X$ be a closed immersion of formal algebraic spaces\nover $S$. Then $Y$ is a McQuillan affine formal algebraic space and\n$f$ corresponds to a continuous homomorphism $A \\to B$ of weakly admissible\ntopological $S$-algebras which is taut, has closed kernel, and has\ndense image.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANQ","source_file":"formal-spaces.tex","source_line":5920,"source_end_line":5928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5920-L5928","statement_sha256":"75ae48120ed454649989f20883662674ff975022b2d2f83db4ab6b263eaa8288","origin":"The Stacks Project","memory_eligible":false,"source_rank":13037,"rank":13037,"depth":60,"x":1153.789,"y":1708.424,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GHZ","tag":"0GHZ","title":"Closed immersions · Lemma 0GHZ","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces. Assume • f is representable by algebraic spaces, • f is a monomorphism, • the inclusion Y_red → Y factors through f, and • f is locally of finite type or Y is locally Noetherian. Then f is a closed immersion.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces. Assume\n\\begin{enumerate}\n\\item $f$ is representable by algebraic spaces,\n\\item $f$ is a monomorphism,\n\\item the inclusion $Y_{red} \\to Y$ factors through $f$, and\n\\item $f$ is locally of finite type or $Y$ is locally Noetherian.\n\\end{enumerate}\nThen $f$ is a closed immersion.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Closed immersions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHZ","source_file":"formal-spaces.tex","source_line":5976,"source_end_line":5987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L5976-L5987","statement_sha256":"883c9bb469982dafe4ccc72bfdfa419d35878664bb007a8ca3f758c78e0c3742","origin":"The Stacks Project","memory_eligible":false,"source_rank":13038,"rank":13038,"depth":67,"x":1025.072,"y":1533.425,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANS","tag":"0ANS","title":"Algebras topologically of finite type · Definition 0ANS","summary":"Let A → B be a continuous map of topological rings (More on Algebra, Definition [Tag 07E8]). We say B is topologically of finite type over A if there exists an A-algebra map A[x_1, …, x_n] → B whose image is dense in B.","statement_latex":"Let $A \\to B$ be a continuous map of topological rings\n(More on Algebra, Definition \\ref{more-algebra-definition-topological-ring}).\nWe say $B$ is {\\it topologically of finite type over} $A$ if\nthere exists an $A$-algebra map $A[x_1, \\ldots, x_n] \\to B$ whose\nimage is dense in $B$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Algebras topologically of finite type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANS","source_file":"formal-spaces.tex","source_line":6105,"source_end_line":6112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6105-L6112","statement_sha256":"8816bce80db115f56a31efddccd789452ed7dcfa33bdfdf0972b0e7e2663bac5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13039,"rank":13039,"depth":1,"x":1261.153,"y":1589.534,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANT","tag":"0ANT","title":"Algebras topologically of finite type · Lemma 0ANT","summary":"Let S be a scheme. Let φ : A → B be a continuous map of weakly admissible topological rings over S. The following are equivalent • Spf(φ) : Y = Spf(B) → Spf(A) = X is of finite type, • φ is taut and B is topologically of finite type over A.","statement_latex":"Let $S$ be a scheme. Let $\\varphi : A \\to B$ be a continuous map of\nweakly admissible topological rings over $S$. The following\nare equivalent\n\\begin{enumerate}\n\\item $\\text{Spf}(\\varphi) : Y = \\text{Spf}(B) \\to \\text{Spf}(A) = X$\nis of finite type,\n\\item $\\varphi$ is taut and $B$ is topologically of finite type over $A$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Algebras topologically of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANT","source_file":"formal-spaces.tex","source_line":6125,"source_end_line":6135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6125-L6135","statement_sha256":"bbecbc5806f95c11b9c007197d33521ea4e5ad4c844e84ee1d83f4f704e6f9c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13040,"rank":13040,"depth":60,"x":1041.544,"y":1682.289,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AL1","tag":"0AL1","title":"Algebras topologically of finite type · Lemma 0AL1","summary":"Let S be a scheme. Let X be an affine formal algebraic space over S. Assume X is McQuillan and let A be the weakly admissible topological ring associated to X. Then there is an anti-equivalence of categories between • the category C introduced above, and • the category of maps Y → X of finite type of affine formal algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.\nAssume $X$ is McQuillan and let $A$ be the weakly admissible topological\nring associated to $X$. Then there is an anti-equivalence of categories\nbetween\n\\begin{enumerate}\n\\item the category $\\mathcal{C}$ introduced above, and\n\\item the category of maps $Y \\to X$ of finite type of\naffine formal algebraic spaces.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Algebras topologically of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AL1","source_file":"formal-spaces.tex","source_line":6187,"source_end_line":6198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6187-L6198","statement_sha256":"d251c05430c4e425f295bf1281b1d4d98461448af0dd78b4d84b33d4ee477128","origin":"The Stacks Project","memory_eligible":false,"source_rank":13041,"rank":13041,"depth":60,"x":1129.05,"y":1488.925,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQI","tag":"0AQI","title":"Algebras topologically of finite type · Lemma 0AQI","summary":"Let S be a scheme. Let X be a countably indexed affine formal algebraic space over S. Let f : Y → X be a closed immersion of formal algebraic spaces over S. Then Y is a countably indexed affine formal algebraic space and f corresponds to A → A/K where A is an object of WAdm^count (Section [Tag 0ANA]) and K ⊂ A is a closed ideal.","statement_latex":"Let $S$ be a scheme. Let $X$ be a countably indexed affine formal algebraic\nspace over $S$. Let $f : Y \\to X$ be a closed immersion of formal algebraic\nspaces over $S$. Then $Y$ is a countably indexed affine formal algebraic space\nand $f$ corresponds to $A \\to A/K$ where $A$ is an object of\n$\\textit{WAdm}^{count}$\n(Section \\ref{section-morphisms-rings})\nand $K \\subset A$ is a closed ideal.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Algebras topologically of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQI","source_file":"formal-spaces.tex","source_line":6316,"source_end_line":6325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6316-L6325","statement_sha256":"d94337c6642a1f7058a86c112395316e6976e4c6ca6c0ca587a283221e54303f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13042,"rank":13042,"depth":61,"x":1220.188,"y":1681.515,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANU","tag":"0ANU","title":"Algebras topologically of finite type · Lemma 0ANU","summary":"Let B → A be an arrow of WAdm^count, see Section [Tag 0ANA]. The following are equivalent • [(a)] B → A is taut and B/J → A/I is of finite type for every weak ideal of definition J ⊂ B where I ⊂ A is the closure of JA, • [(b)] B → A is taut and B/J_λ → A/I_λ is of finite type for a cofinal system (J_λ) of weak ideals of definition of B where I_λ ⊂ A is the closure of J_λ A, • [(c)] B → A is taut and A is topologically of finite type over B, • [(d)] A is isomorphic as a…","statement_latex":"Let $B \\to A$ be an arrow of $\\textit{WAdm}^{count}$, see\nSection \\ref{section-morphisms-rings}.\nThe following are equivalent\n\\begin{enumerate}\n\\item[(a)] $B \\to A$ is taut and $B/J \\to A/I$ is of finite type for\nevery weak ideal of definition $J \\subset B$ where $I \\subset A$ is the\nclosure of $JA$,\n\\item[(b)] $B \\to A$ is taut and $B/J_\\lambda \\to A/I_\\lambda$\nis of finite type for a cofinal system $(J_\\lambda)$\nof weak ideals of definition of $B$ where\n$I_\\lambda \\subset A$ is the closure of $J_\\lambda A$,\n\\item[(c)] $B \\to A$ is taut and $A$ is topologically of finite\ntype over $B$,\n\\item[(d)] $A$ is isomorphic as a topological $B$-algebra to a quotient of\n$B\\{x_1, \\ldots, x_n\\}$ by a closed ideal.\n\\end{enumerate}\nMoreover, these equivalent conditions define a local property,\ni.e., they satisfy\nAxioms (\\ref{item-axiom-1}), (\\ref{item-axiom-2}), (\\ref{item-axiom-3}).","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Algebras topologically of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANU","source_file":"formal-spaces.tex","source_line":6342,"source_end_line":6363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6342-L6363","statement_sha256":"4d3aa314591996d7c7f0449f7058896633a07a9a05c44efa9f9780ece01d1bd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13043,"rank":13043,"depth":72,"x":997.704,"y":1591.051,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0CB6","tag":"0CB6","title":"Algebras topologically of finite type · Lemma 0CB6","summary":"In Lemma [Tag 0ANU] if B is admissible (for example adic), then the equivalent conditions (a) -- (d) are also equivalent to • [(e)] B → A is taut and B/J → A/I is of finite type for some ideal of definition J ⊂ B where I ⊂ A is the closure of JA.","statement_latex":"In Lemma \\ref{lemma-quotient-restricted-power-series}\nif $B$ is admissible (for example adic), then the equivalent conditions\n(a) -- (d) are also equivalent to\n\\begin{enumerate}\n\\item[(e)] $B \\to A$ is taut and $B/J \\to A/I$ is of finite type for\nsome ideal of definition $J \\subset B$ where $I \\subset A$ is\nthe closure of $JA$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Algebras topologically of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CB6","source_file":"formal-spaces.tex","source_line":6443,"source_end_line":6453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6443-L6453","statement_sha256":"2ce56bd5f88669a4260dbdeec9ecb9283f4207f06ca592e57c716293170fbc49","origin":"The Stacks Project","memory_eligible":false,"source_rank":13044,"rank":13044,"depth":73,"x":1234.941,"y":1531.406,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANV","tag":"0ANV","title":"Algebras topologically of finite type · Lemma 0ANV","summary":"Let S be a scheme. Let f : X → Y be a morphism of affine formal algebraic spaces. Assume Y countably indexed. The following are equivalent • f is locally of finite type, • f is of finite type, • f corresponds to a morphism B → A of WAdm^count (Section [Tag 0ANA]) satisfying the equivalent conditions of Lemma [Tag 0ANU].","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\naffine formal algebraic spaces. Assume $Y$ countably indexed.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is locally of finite type,\n\\item $f$ is of finite type,\n\\item $f$ corresponds to a morphism $B \\to A$ of $\\textit{WAdm}^{count}$\n(Section \\ref{section-morphisms-rings})\nsatisfying the equivalent conditions of\nLemma \\ref{lemma-quotient-restricted-power-series}.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Algebras topologically of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANV","source_file":"formal-spaces.tex","source_line":6477,"source_end_line":6490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6477-L6490","statement_sha256":"a569944832750805b17e1cc8d8f2239d710a0d76036358dfe46b0ebcd52a1137","origin":"The Stacks Project","memory_eligible":false,"source_rank":13045,"rank":13045,"depth":73,"x":1107.739,"y":1710.325,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ANW","tag":"0ANW","title":"Algebras topologically of finite type · Lemma 0ANW","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally countably indexed formal algebraic spaces over S. The following are equivalent • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a morphism of WAdm^count which is taut and topologically of finite type, • there exists a covering (Y_j → Y) as in…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally countably indexed formal algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a morphism of $\\textit{WAdm}^{count}$ which is\ntaut and topologically of finite type,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space} and for each $j$\na covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nsuch that each $X_{ji} \\to Y_j$  corresponds\nto a morphism of $\\textit{WAdm}^{count}$ which is\ntaut and topologically of finite type,\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds\nto a morphism of $\\textit{WAdm}^{count}$ which is,\ntaut and topologically of finite type, and\n\\item $f$ is locally of finite type.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Algebras topologically of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ANW","source_file":"formal-spaces.tex","source_line":6509,"source_end_line":6542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6509-L6542","statement_sha256":"e19ba135793fbc120de0d9974769e7c70892585fbdb7157b82151d464a824e7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13046,"rank":13046,"depth":74,"x":1057.564,"y":1505.847,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARN","tag":"0ARN","title":"Separation axioms for morphisms · Definition 0ARN","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. Let Δ_X/Y : X → X ×_Y X be the diagonal morphism. • We say f is separated if Δ_X/Y is a closed immersion. • We say f is quasi-separated if Δ_X/Y is quasi-compact.","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$.\nLet $\\Delta_{X/Y} : X \\to X \\times_Y X$ be the diagonal morphism.\n\\begin{enumerate}\n\\item We say $f$ is {\\it separated} if $\\Delta_{X/Y}$ is a closed immersion.\n\\item We say $f$ is {\\it quasi-separated} if $\\Delta_{X/Y}$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARN","source_file":"formal-spaces.tex","source_line":6577,"source_end_line":6586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6577-L6586","statement_sha256":"16ef38a6a69ddd7f8fba1532261f38c28cd83d4d34867fc12887c66d1626dfae","origin":"The Stacks Project","memory_eligible":false,"source_rank":13047,"rank":13047,"depth":0,"x":1259.361,"y":1628.376,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARP","tag":"0ARP","title":"Separation axioms for morphisms · Lemma 0ARP","summary":"All of the separation axioms listed in Definition [Tag 0ARN] are stable under base change.","statement_latex":"All of the separation axioms listed in\nDefinition \\ref{definition-separated-morphism}\nare stable under base change.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARP","source_file":"formal-spaces.tex","source_line":6603,"source_end_line":6608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6603-L6608","statement_sha256":"a92d7233dbbaa4930bdb13655d02cb692c5315a87ca1799b726ecedbcec6f9bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13048,"rank":13048,"depth":1,"x":1011.58,"y":1652.573,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARQ","tag":"0ARQ","title":"Separation axioms for morphisms · Lemma 0ARQ","summary":"Let S be a scheme. Let f : X → Z, g : Y → Z and Z → T be morphisms of formal algebraic spaces over S. Consider the induced morphism i : X ×_Z Y → X ×_T Y. Then • i is representable (by schemes), locally of finite type, locally quasi-finite, separated, and a monomorphism, • if Z → T is separated, then i is a closed immersion, and • if Z → T is quasi-separated, then i is quasi-compact.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Z$, $g : Y \\to Z$ and $Z \\to T$\nbe morphisms of formal algebraic spaces over $S$. Consider the induced\nmorphism $i : X \\times_Z Y \\to X \\times_T Y$. Then\n\\begin{enumerate}\n\\item $i$ is representable (by schemes), locally of finite type,\nlocally quasi-finite, separated, and a monomorphism,\n\\item if $Z \\to T$ is separated, then $i$ is a closed immersion, and\n\\item if $Z \\to T$ is quasi-separated, then $i$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARQ","source_file":"formal-spaces.tex","source_line":6619,"source_end_line":6630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6619-L6630","statement_sha256":"a64bccc0f05845ec24f879834a3ba4a9af93756acdf5de70f2a59d8ce9cff7dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13049,"rank":13049,"depth":69,"x":1175.124,"y":1493.85,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARR","tag":"0ARR","title":"Separation axioms for morphisms · Lemma 0ARR","summary":"All of the separation axioms listed in Definition [Tag 0ARN] are stable under composition of morphisms.","statement_latex":"All of the separation axioms listed in\nDefinition \\ref{definition-separated-morphism}\nare stable under composition of morphisms.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARR","source_file":"formal-spaces.tex","source_line":6645,"source_end_line":6650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6645-L6650","statement_sha256":"886abef5a4bc68cdbb3e6668ab6e633763bfe0db63be78c70cf77a68a0b49355","origin":"The Stacks Project","memory_eligible":false,"source_rank":13050,"rank":13050,"depth":70,"x":1182.182,"y":1704.062,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARS","tag":"0ARS","title":"Separation axioms for morphisms · Lemma 0ARS","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. Let P be any of the separation axioms of Definition [Tag 0ARN]. The following are equivalent • f is P, • for every scheme Z and morphism Z → Y the base change Z ×_Y X → Z of f is P, • for every affine scheme Z and every morphism Z → Y the base change Z ×_Y X → Z of f is P, • for every affine scheme Z and every morphism Z → Y the formal algebraic space Z ×_Y X is P (see Definition [Tag…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces\nover $S$. Let $\\mathcal{P}$ be any of the separation axioms of\nDefinition \\ref{definition-separated-morphism}.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is $\\mathcal{P}$,\n\\item for every scheme $Z$ and morphism $Z \\to Y$ the\nbase change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the\nbase change $Z \\times_Y X \\to Z$ of $f$ is $\\mathcal{P}$,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the\nformal algebraic space $Z \\times_Y X$ is $\\mathcal{P}$ (see\nDefinition \\ref{definition-separated}),\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nsuch that the base change $Y_j \\times_Y X \\to Y_j$ has\n$\\mathcal{P}$ for all $j$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Separation axioms for morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARS","source_file":"formal-spaces.tex","source_line":6667,"source_end_line":6687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6667-L6687","statement_sha256":"4ed8514066be9c6265632d85e08e65912eaef5514fd599f5b8e4cd403ea0066e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13051,"rank":13051,"depth":70,"x":1007.622,"y":1552.792,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AM6","tag":"0AM6","title":"Proper morphisms · Definition 0AM6","summary":"Let S be a scheme. Let f : Y → X be a morphism of formal algebraic spaces over S. We say f is proper if f is representable by algebraic spaces and is proper in the sense of Bootstrap, Definition [Tag 03XZ].","statement_latex":"Let $S$ be a scheme. Let $f : Y \\to X$ be a morphism of formal algebraic\nspaces over $S$. We say $f$ is {\\it proper}\nif $f$ is representable by algebraic spaces and is proper in the sense of\nBootstrap, Definition \\ref{bootstrap-definition-property-transformation}.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Proper morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AM6","source_file":"formal-spaces.tex","source_line":6778,"source_end_line":6784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6778-L6784","statement_sha256":"389f6e14cd767d7bb5da80fd4ed2c9c2c2bb0e98e8db741c04a57c2957238963","origin":"The Stacks Project","memory_eligible":false,"source_rank":13052,"rank":13052,"depth":2,"x":1258.43,"y":1565.309,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ART","tag":"0ART","title":"Proper morphisms · Lemma 0ART","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. The following are equivalent • f is proper, • for every scheme Z and morphism Z → Y the base change Z ×_Y X → Z of f is proper, • for every affine scheme Z and every morphism Z → Y the base change Z ×_Y X → Z of f is proper, • for every affine scheme Z and every morphism Z → Y the formal algebraic space Z ×_Y X is an algebraic space proper over Z, • there exists a covering (Y_j → Y) as in…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic spaces\nover $S$. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is proper,\n\\item for every scheme $Z$ and morphism $Z \\to Y$ the\nbase change $Z \\times_Y X \\to Z$ of $f$ is proper,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the\nbase change $Z \\times_Y X \\to Z$ of $f$ is proper,\n\\item for every affine scheme $Z$ and every morphism $Z \\to Y$ the\nformal algebraic space $Z \\times_Y X$ is an algebraic space proper over $Z$,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nDefinition \\ref{definition-formal-algebraic-space}\nsuch that the base change $Y_j \\times_Y X \\to Y_j$ is proper for all $j$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ART","source_file":"formal-spaces.tex","source_line":6789,"source_end_line":6805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6789-L6805","statement_sha256":"6456222f38fba73aa0926ae87afabd4ac7a9e2be71131b07aaf97ad6296e4601","origin":"The Stacks Project","memory_eligible":false,"source_rank":13053,"rank":13053,"depth":1,"x":1063.078,"y":1698.628,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBT","tag":"0GBT","title":"Proper morphisms · Lemma 0GBT","summary":"Proper morphisms of formal algebraic spaces are preserved by base change.","statement_latex":"Proper morphisms of formal algebraic spaces are preserved by base change.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBT","source_file":"formal-spaces.tex","source_line":6811,"source_end_line":6814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6811-L6814","statement_sha256":"b92e060f2aa33376d92a108675d066a982cf4c0a2dee7ab985ece7fc1ffd9227","origin":"The Stacks Project","memory_eligible":false,"source_rank":13054,"rank":13054,"depth":2,"x":1099.981,"y":1489.106,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQD","tag":"0AQD","title":"Formal algebraic spaces and fpqc coverings · Lemma 0AQD","summary":"Formal algebraic spaces are fpqc sheaves Let S be a scheme. Let X be a formal algebraic space over S. Then X satisfies the sheaf property for the fpqc topology.","statement_latex":"\\begin{slogan}\nFormal algebraic spaces are fpqc sheaves\n\\end{slogan}\nLet $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then\n$X$ satisfies the sheaf property for the fpqc topology.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Formal algebraic spaces and fpqc coverings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQD","source_file":"formal-spaces.tex","source_line":6835,"source_end_line":6842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6835-L6842","statement_sha256":"348ecd3d46ba0d9a81c89d02185e8623c3168e01369cfa298295bf5eb905ab89","origin":"The Stacks Project","memory_eligible":false,"source_rank":13055,"rank":13055,"depth":70,"x":1241.508,"y":1664.851,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQF","tag":"0AQF","title":"Maps out of affine formal schemes · Lemma 0AQF","summary":"Let S be a scheme. Let A be a weakly admissible topological S-algebra. Let X be an affine scheme over S. Then the natural map Mor_S(Spec(A), X) → Mor_S(Spf(A), X) is bijective.","statement_latex":"Let $S$ be a scheme. Let $A$ be a weakly admissible topological\n$S$-algebra. Let $X$ be an affine scheme over $S$. Then\nthe natural map\n$$\n\\Mor_S(\\Spec(A), X)\n\\longrightarrow\n\\Mor_S(\\text{Spf}(A), X)\n$$\nis bijective.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Maps out of affine formal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQF","source_file":"formal-spaces.tex","source_line":6963,"source_end_line":6974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6963-L6974","statement_sha256":"e413945c2ae1eba99e527ce54c3cfe0000968ff6f6987b21a16e1f8805e2b65c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13056,"rank":13056,"depth":12,"x":995.39,"y":1615.479,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQG","tag":"0AQG","title":"Maps out of affine formal schemes · Lemma 0AQG","summary":"Let S be a scheme. Let A be a weakly admissible topological S-algebra such that A/I is a local ring for some weak ideal of definition I ⊂ A. Let X be a scheme over S. Then the natural map Mor_S(Spec(A), X) → Mor_S(Spf(A), X) is bijective.","statement_latex":"Let $S$ be a scheme. Let $A$ be a weakly admissible topological\n$S$-algebra such that $A/I$ is a local ring for some weak ideal\nof definition $I \\subset A$. Let $X$ be a scheme over $S$. Then\nthe natural map\n$$\n\\Mor_S(\\Spec(A), X)\n\\longrightarrow\n\\Mor_S(\\text{Spf}(A), X)\n$$\nis bijective.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Maps out of affine formal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQG","source_file":"formal-spaces.tex","source_line":6985,"source_end_line":6997,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L6985-L6997","statement_sha256":"9501c75703fac7fa2cb16421f3858ce24975b985f6beccb303f3e8ec99b641e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13057,"rank":13057,"depth":13,"x":1216.962,"y":1512.053,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQH","tag":"0AQH","title":"Maps out of affine formal schemes · Lemma 0AQH","summary":"Let S be a scheme. Let R be a complete local Noetherian S-algebra. Let X be an algebraic space over S. Then the natural map Mor_S(Spec(R), X) → Mor_S(Spf(R), X) is bijective.","statement_latex":"Let $S$ be a scheme. Let $R$ be a complete local Noetherian $S$-algebra.\nLet $X$ be an algebraic space over $S$. Then the natural map\n$$\n\\Mor_S(\\Spec(R), X)\n\\longrightarrow\n\\Mor_S(\\text{Spf}(R), X)\n$$\nis bijective.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Maps out of affine formal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQH","source_file":"formal-spaces.tex","source_line":7009,"source_end_line":7019,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7009-L7019","statement_sha256":"3afdcf9798738842f1de495c126ceea5d5c53c6ffb2b798c75af123d8fc50066","origin":"The Stacks Project","memory_eligible":false,"source_rank":13058,"rank":13058,"depth":53,"x":1136.612,"y":1714.392,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBU","tag":"0GBU","title":"Maps out of affine formal schemes · Lemma 0GBU","summary":"Let S be a scheme. Let X be an algebraic space over S. Let T ⊂ |X| be a closed subset such that X setminus T → X is quasi-compact. Let R be a complete local Noetherian S-algebra. Then an adic morphism p : Spf(R) → X_/T corresponds to a unique morphism g : Spec(R) → X such that g^-1(T) = ( m_R).","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset such that\n$X \\setminus T \\to X$ is quasi-compact. Let $R$ be a complete local\nNoetherian $S$-algebra. Then an adic morphism $p : \\text{Spf}(R) \\to X_{/T}$\ncorresponds to a unique morphism $g : \\Spec(R) \\to X$ such\nthat $g^{-1}(T) = \\{\\mathfrak m_R\\}$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Maps out of affine formal schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBU","source_file":"formal-spaces.tex","source_line":7097,"source_end_line":7105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7097-L7105","statement_sha256":"3505d0ae483484fcc1e6e81b6bf048427b324e81c76fccf161b478cf43f76a3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13059,"rank":13059,"depth":54,"x":1032.967,"y":1519.261,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DEA","tag":"0DEA","title":"The small étale site of a formal algebraic space · Definition 0DEA","summary":"Let S be a scheme. Let X be a formal algebraic space with reduction X_red (Lemma [Tag 0AIN]). • The small étale site X_etale of X is the site X_red, etale of Properties of Spaces, Definition [Tag 03ED]. • The site X_spaces, etale is the site X_red, spaces, etale of Properties of Spaces, Definition [Tag 03G0]. • The site X_affine, etale is the site X_red, affine, etale of Properties of Spaces, Lemma [Tag 04JS].","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space with\nreduction $X_{red}$ (Lemma \\ref{lemma-reduction-formal-algebraic-space}).\n\\begin{enumerate}\n\\item The {\\it small \\'etale site} $X_\\etale$ of $X$ is\nthe site $X_{red, \\etale}$ of Properties of Spaces, Definition\n\\ref{spaces-properties-definition-etale-site}.\n\\item The site $X_{spaces, \\etale}$ is the site\n$X_{red, spaces, \\etale}$ of Properties of Spaces, Definition\n\\ref{spaces-properties-definition-spaces-etale-site}.\n\\item The site $X_{affine, \\etale}$ is the site\n$X_{red, affine, \\etale}$ of Properties of Spaces, Lemma\n\\ref{spaces-properties-lemma-alternative}.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The small étale site of a formal algebraic space","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEA","source_file":"formal-spaces.tex","source_line":7169,"source_end_line":7184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7169-L7184","statement_sha256":"4bcd838c2d70ce497106ac5c43a8c884607ea5ca0ea8063a925b6f0213e5da13","origin":"The Stacks Project","memory_eligible":false,"source_rank":13060,"rank":13060,"depth":57,"x":1266.713,"y":1604.484,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DEC","tag":"0DEC","title":"The small étale site of a formal algebraic space · Lemma 0DEC","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. • There is a continuous functor Y_spaces, etale → X_spaces, etale which induces a morphism of sites f_spaces, etale : X_spaces, etale → Y_spaces, etale. • The rule f ↦ f_spaces, etale is compatible with compositions, in other words (f ∘ g)_spaces, etale = f_spaces, etale ∘ g_spaces, etale (see Sites, Definition [Tag 03CC]). • The morphism of topoi associated to f_spaces, etale induces, via…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of formal algebraic spaces over $S$.\n\\begin{enumerate}\n\\item There is a continuous functor\n$Y_{spaces, \\etale} \\to X_{spaces, \\etale}$\nwhich induces a morphism of sites\n$$\nf_{spaces, \\etale} : X_{spaces, \\etale} \\to Y_{spaces, \\etale}.\n$$\n\\item The rule $f \\mapsto f_{spaces, \\etale}$ is compatible with\ncompositions, in other words $(f \\circ g)_{spaces, \\etale}\n= f_{spaces, \\etale} \\circ g_{spaces, \\etale}$ (see\nSites, Definition \\ref{sites-definition-composition-morphisms-sites}).\n\\item The morphism of topoi associated to $f_{spaces, \\etale}$\ninduces, via (\\ref{equation-etale-topos}), a morphism of topoi\n$f_{small} : \\Sh(X_\\etale) \\to \\Sh(Y_\\etale)$\nwhose construction is compatible with compositions.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The small étale site of a formal algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEC","source_file":"formal-spaces.tex","source_line":7201,"source_end_line":7221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7201-L7221","statement_sha256":"9f03f10874192107b2e94e91ee039a083e2c959300502f50938f6bc361a058c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13061,"rank":13061,"depth":57,"x":1025.406,"y":1674.395,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DED","tag":"0DED","title":"The small étale site of a formal algebraic space · Lemma 0DED","summary":"Let S be a scheme, and let f : X → Y be a morphism of formal algebraic spaces over S. Assume f is representable by algebraic spaces and étale. In this case there is a cocontinuous functor j : X_etale → Y_etale. The morphism of topoi f_small is the morphism of topoi associated to j, see Sites, Lemma [Tag 00XO]. Moreover, j is continuous as well, hence Sites, Lemma [Tag 00XR] applies.","statement_latex":"Let $S$ be a scheme, and let $f : X \\to Y$ be a morphism of\nformal algebraic spaces over $S$. Assume $f$ is representable\nby algebraic spaces and \\'etale. In this case there is a\ncocontinuous functor $j : X_\\etale \\to Y_\\etale$.\nThe morphism of topoi $f_{small}$ is the\nmorphism of topoi associated to $j$, see\nSites, Lemma \\ref{sites-lemma-cocontinuous-morphism-topoi}.\nMoreover, $j$ is continuous as well, hence\nSites, Lemma \\ref{sites-lemma-when-shriek} applies.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The small étale site of a formal algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DED","source_file":"formal-spaces.tex","source_line":7236,"source_end_line":7247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7236-L7247","statement_sha256":"e9c03196be8a8958baf92d74d5a7b311e0d3bca517c81e909b3c1d2a6b4eaae7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13062,"rank":13062,"depth":48,"x":1147.33,"y":1485.598,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DEE","tag":"0DEE","title":"The small étale site of a formal algebraic space · Lemma 0DEE","summary":"Let S be a scheme. Let X be an affine formal algebraic space over S. Then X_affine, etale is equivalent to the category whose objects are morphisms φ : U → X of formal algebraic spaces such that • U is an affine formal algebraic space, • φ is representable by algebraic spaces and étale.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.\nThen $X_{affine, \\etale}$ is equivalent to the category whose objects\nare morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that\n\\begin{enumerate}\n\\item $U$ is an affine formal algebraic space,\n\\item $\\varphi$ is representable by algebraic spaces and \\'etale.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The small étale site of a formal algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEE","source_file":"formal-spaces.tex","source_line":7262,"source_end_line":7271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7262-L7271","statement_sha256":"bb125eeda64807060238cd4234acdcd3ab38dc105983990e854be24a25b9f861","origin":"The Stacks Project","memory_eligible":false,"source_rank":13063,"rank":13063,"depth":59,"x":1209.354,"y":1694.351,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DEF","tag":"0DEF","title":"The small étale site of a formal algebraic space · Lemma 0DEF","summary":"Let S be a scheme. Let X be an affine formal algebraic space over S. Assume X is McQuillan, i.e., equal to Spf(A) for some weakly admissible topological S-algebra A. Then (X_affine, etale)^opp is equivalent to the category whose • objects are A-algebras of the form B^wedge = lim B/JB where A → B is an étale ring map and J runs over the weak ideals of definition of A, and • morphisms are continuous A-algebra homomorphisms.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine formal\nalgebraic space over $S$. Assume $X$ is McQuillan, i.e.,\nequal to $\\text{Spf}(A)$ for some\nweakly admissible topological $S$-algebra $A$.\nThen $(X_{affine, \\etale})^{opp}$ is equivalent to\nthe category whose\n\\begin{enumerate}\n\\item objects are $A$-algebras of the form\n$B^\\wedge = \\lim B/JB$ where $A \\to B$ is an \\'etale ring map\nand $J$ runs over the weak ideals of definition of $A$, and\n\\item morphisms are continuous $A$-algebra homomorphisms.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The small étale site of a formal algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEF","source_file":"formal-spaces.tex","source_line":7311,"source_end_line":7325,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7311-L7325","statement_sha256":"0659819662f322e529f36de25292c7fafe316a25556e0c59e9b0eac44c4d449d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13064,"rank":13064,"depth":60,"x":995.383,"y":1575.413,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DEG","tag":"0DEG","title":"The small étale site of a formal algebraic space · Lemma 0DEG","summary":"Let S be a scheme. Let X be a formal algebraic space over S. Then X_spaces, etale is equivalent to the category whose objects are morphisms φ : U → X of formal algebraic spaces such that φ is representable by algebraic spaces and étale.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nThen $X_{spaces, \\etale}$ is equivalent to the category whose objects\nare morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that\n$\\varphi$ is representable by algebraic spaces and \\'etale.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The small étale site of a formal algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEG","source_file":"formal-spaces.tex","source_line":7331,"source_end_line":7337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7331-L7337","statement_sha256":"fd0d4353bc3781f53a73d2c68f7102443f29b53063eb9f5c44c2aa087d7e236f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13065,"rank":13065,"depth":49,"x":1249.237,"y":1541.647,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DEH","tag":"0DEH","title":"The small étale site of a formal algebraic space · Lemma 0DEH","summary":"Let S be a scheme. Let X be a formal algebraic space over S. Then X_affine, etale is equivalent to the category whose objects are morphisms φ : U → X of formal algebraic spaces such that • U is an affine formal algebraic space, • φ is representable by algebraic spaces and étale.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nThen $X_{affine, \\etale}$ is equivalent to the category whose objects\nare morphisms $\\varphi : U \\to X$ of formal algebraic spaces such that\n\\begin{enumerate}\n\\item $U$ is an affine formal algebraic space,\n\\item $\\varphi$ is representable by algebraic spaces and \\'etale.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The small étale site of a formal algebraic space","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEH","source_file":"formal-spaces.tex","source_line":7396,"source_end_line":7405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7396-L7405","statement_sha256":"24de5aca7db5dbed6b5da853df8b5ff33e5abd25d68bdcacce20772219ea0ce2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13066,"rank":13066,"depth":71,"x":1088.933,"y":1710.873,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DEJ","tag":"0DEJ","title":"The structure sheaf · Lemma 0DEJ","summary":"Every formal algebraic space has a structure sheaf.","statement_latex":"Every formal algebraic space has a structure sheaf.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEJ","source_file":"formal-spaces.tex","source_line":7447,"source_end_line":7450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7447-L7450","statement_sha256":"d8fafe4d9ada30dd85dd0eecfcc8bce5dd12c0d57ce68ba858c7cf48ba9c8602","origin":"The Stacks Project","memory_eligible":false,"source_rank":13067,"rank":13067,"depth":72,"x":1071.023,"y":1494.766,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0DEL","tag":"0DEL","title":"The structure sheaf · Lemma 0DEL","summary":"If X is a countably indexed affine formal algebraic space, then we have H^n(X_etale, O_X) = 0 for n > 0.","statement_latex":"If $X$ is a countably indexed affine formal algebraic space, then\nwe have $H^n(X_\\etale, \\mathcal{O}_X) = 0$ for $n > 0$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"The structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DEL","source_file":"formal-spaces.tex","source_line":7547,"source_end_line":7551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7547-L7551","statement_sha256":"17474430a9905146b87c5d581d2959c1e6367b5df1502776303f976438f1b256","origin":"The Stacks Project","memory_eligible":false,"source_rank":13068,"rank":13068,"depth":61,"x":1258.329,"y":1644.207,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GVM","tag":"0GVM","title":"Colimits of formal algebraic spaces · Lemma 0GVM","summary":"Let S be a scheme. Suppose given a directed set Lambda and a system of affine formal algebraic spaces (X_λ, f_λ μ) over Lambda where each f_λ μ : X_λ → X_μ is a closed immersion inducing an isomorphism X_λ, red → X_μ, red. Then X = colim_λ ∈ Lambda X_λ is an affine formal algebraic space over S.","statement_latex":"Let $S$ be a scheme. Suppose given a directed set\n$\\Lambda$ and a system of affine formal algebraic spaces\n$(X_\\lambda, f_{\\lambda \\mu})$ over $\\Lambda$ where each\n$f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a closed immersion\ninducing an isomorphism $X_{\\lambda, red} \\to X_{\\mu, red}$.\nThen $X = \\colim_{\\lambda \\in \\Lambda} X_\\lambda$\nis an affine formal algebraic space over $S$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Colimits of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVM","source_file":"formal-spaces.tex","source_line":7650,"source_end_line":7659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7650-L7659","statement_sha256":"05b9d8eae5a01b693ab84a318a784dcc62c81523007b9591ec9970a1131b4fda","origin":"The Stacks Project","memory_eligible":false,"source_rank":13069,"rank":13069,"depth":2,"x":999.606,"y":1640.287,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GVN","tag":"0GVN","title":"Colimits of formal algebraic spaces · Lemma 0GVN","summary":"Let S be a scheme. Suppose given a directed set Lambda and a system of formal algebraic spaces (X_λ, f_λ μ) over Lambda where each f_λ μ : X_λ → X_μ is a closed immersion inducing an isomorphism X_λ, red → X_μ, red. Then X = colim_λ ∈ Lambda X_λ is a formal algebraic space over S.","statement_latex":"Let $S$ be a scheme. Suppose given a directed set\n$\\Lambda$ and a system of formal algebraic spaces\n$(X_\\lambda, f_{\\lambda \\mu})$ over $\\Lambda$ where each\n$f_{\\lambda \\mu} : X_\\lambda \\to X_\\mu$ is a closed immersion\ninducing an isomorphism $X_{\\lambda, red} \\to X_{\\mu, red}$.\nThen $X = \\colim_{\\lambda \\in \\Lambda} X_\\lambda$\nis a formal algebraic space over $S$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Colimits of formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVN","source_file":"formal-spaces.tex","source_line":7685,"source_end_line":7694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7685-L7694","statement_sha256":"d9c158a75cd26be9452e84978a80b6717f76597e8d4f7543efa1117a542cec2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13070,"rank":13070,"depth":60,"x":1193.859,"y":1496.133,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GVQ","tag":"0GVQ","title":"Recompletion · Lemma 0GVQ","summary":"Let S be a scheme. Let X be an affine formal algebraic space over S. Let T ⊂ |X_red| be a closed subset. Then the functor X_/T : (Sch/S)_fppf → Sets, U ↦ (f : U → X : f(|U|) ⊂ T) is an affine formal algebraic space.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.\nLet $T \\subset |X_{red}|$ be a closed subset. Then the functor\n$$\nX_{/T} : (\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad\nU \\longmapsto \\{f : U \\to X : f(|U|) \\subset T\\}\n$$\nis an affine formal algebraic space.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Recompletion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVQ","source_file":"formal-spaces.tex","source_line":7751,"source_end_line":7760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7751-L7760","statement_sha256":"3fc0857046c16c6f81d7c712f6a1ac3b54bb6614d291ce6fed38b957efa4853f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13071,"rank":13071,"depth":58,"x":1166.499,"y":1713.012,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GVR","tag":"0GVR","title":"Recompletion · Lemma 0GVR","summary":"Let S be a scheme. Let X be a formal algebraic space over S. Let T ⊂ |X_red| be a closed subset. Then the functor X_/T : (Sch/S)_fppf → Sets, U ↦ (f : U → X mid f(|U|) ⊂ T) is a formal algebraic space.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nLet $T \\subset |X_{red}|$ be a closed subset. Then the functor\n$$\nX_{/T} : (\\Sch/S)_{fppf} \\longrightarrow \\textit{Sets},\\quad\nU \\longmapsto \\{f : U \\to X \\mid f(|U|) \\subset T\\}\n$$\nis a formal algebraic space.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Recompletion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVR","source_file":"formal-spaces.tex","source_line":7779,"source_end_line":7788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7779-L7788","statement_sha256":"0e7694dede288d9aa28625265ab8f3075d2f659d13164a56e3a9939f961a87f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13072,"rank":13072,"depth":59,"x":1012.011,"y":1537.274,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GVS","tag":"0GVS","title":"Recompletion · Definition 0GVS","summary":"Let S be a scheme. Let X be a formal algebraic space over S. Let T ⊂ |X_red| be a closed subset. The formal algebraic space X_/T of Lemma [Tag 0AIZ] is called the completion of X along T.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nLet $T \\subset |X_{red}|$ be a closed subset. The formal algebraic space\n$X_{/T}$ of Lemma \\ref{lemma-completion-is-formal-algebraic-space}\nis called the {\\it completion of $X$ along $T$}.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Recompletion","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVS","source_file":"formal-spaces.tex","source_line":7810,"source_end_line":7816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7810-L7816","statement_sha256":"fd4bdc00e0c6ee8b823f3bfc53ced3b0a7c0c752619a51778e0c479a3d560c3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13073,"rank":13073,"depth":42,"x":1267.671,"y":1579.269,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GVT","tag":"0GVT","title":"Recompletion · Lemma 0GVT","summary":"Let S be a scheme. Let f : X' → X be a morphism of formal algebraic spaces over S. Let T ⊂ |X_red| be a closed subset and let T' = |f_red|^-1(T) ⊂ |X'_red|. Then xymatrix X'_/T' ar[r] ar[d] & X' ar[d]^f X_/T ar[r] & X is a cartesian diagram of formal algebraic spaces over S.","statement_latex":"Let $S$ be a scheme. Let $f : X' \\to X$ be a morphism\nof formal algebraic spaces over $S$. Let $T \\subset |X_{red}|$\nbe a closed subset and let $T' = |f_{red}|^{-1}(T) \\subset |X'_{red}|$.\nThen\n$$\n\\xymatrix{\nX'_{/T'} \\ar[r] \\ar[d] & X' \\ar[d]^f \\\\\nX_{/T} \\ar[r] & X\n}\n$$\nis a cartesian diagram of formal algebraic spaces over $S$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Recompletion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVT","source_file":"formal-spaces.tex","source_line":7828,"source_end_line":7841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7828-L7841","statement_sha256":"2abad47c27413409a35dea29c0fddbe8c941b9399763b8a3304678c92b5eae92","origin":"The Stacks Project","memory_eligible":false,"source_rank":13074,"rank":13074,"depth":0,"x":1045.016,"y":1693.557,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GVU","tag":"0GVU","title":"Recompletion · Lemma 0GVU","summary":"Let S be a scheme. Let X be a formal algebraic space over S. Let T ⊂ |X_red| be a closed subset. The reduction (X_/T)_red of the completion X_/T of X along T is the reduced induced closed subspace Z of X_red corresponding to T.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nLet $T \\subset |X_{red}|$ be a closed subset. The reduction\n$(X_{/T})_{red}$ of the completion $X_{/T}$ of $X$ along $T$ is\nthe reduced induced closed subspace $Z$ of $X_{red}$ corresponding to $T$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Recompletion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVU","source_file":"formal-spaces.tex","source_line":7855,"source_end_line":7861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7855-L7861","statement_sha256":"18d1adf3df1b4c588a6d6f99eea210e192fb9bba09e9e4d0e2886362f28fcefe","origin":"The Stacks Project","memory_eligible":false,"source_rank":13075,"rank":13075,"depth":57,"x":1117.41,"y":1482.599,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GVV","tag":"0GVV","title":"Recompletion · Lemma 0GVV","summary":"Let S be a scheme. Let X be an affine formal algebraic space over S. Let T ⊂ X_red be a closed subset and let X_/T be the formal completion of X along T. Then • X_/T is an affine formal algebraic space, • if X is McQuillan, then X_/T is McQuillan, • if |X_red| setminus T is quasi-compact and X is countably indexed, then X_/T is countably indexed, • if |X_red| setminus T is quasi-compact and X is adic*, then X_/T is adic*, • if X is Noetherian, then X_/T is Noetherian.","statement_latex":"Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$.\nLet $T \\subset X_{red}$ be a closed subset and let $X_{/T}$\nbe the formal completion of $X$ along $T$. Then\n\\begin{enumerate}\n\\item $X_{/T}$ is an affine formal algebraic space,\n\\item if $X$ is McQuillan, then $X_{/T}$ is McQuillan,\n\\item if $|X_{red}| \\setminus T$ is quasi-compact and $X$\nis countably indexed, then $X_{/T}$ is countably indexed,\n\\item if $|X_{red}| \\setminus T$ is quasi-compact and $X$\nis adic*, then $X_{/T}$ is adic*,\n\\item if $X$ is Noetherian, then $X_{/T}$ is Noetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Recompletion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVV","source_file":"formal-spaces.tex","source_line":7876,"source_end_line":7890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7876-L7890","statement_sha256":"8aec163f40646b4ad0da86717caf4e74d3b4203f445e14164b2fddba4af6bcaf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13076,"rank":13076,"depth":59,"x":1233.863,"y":1679.554,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GVW","tag":"0GVW","title":"Recompletion · Lemma 0GVW","summary":"Let S be a scheme. Let X be a formal algebraic space over S. Let T ⊂ X_red be a closed subset and let X_/T be the formal completion of X along T. Then • if X_red setminus T → X_red is quasi-compact and X is locally countably indexed, then X_/T is locally countably indexed, • if X_red setminus T → X_red is quasi-compact and X is locally adic*, then X_/T is locally adic*, and • if X is locally Noetherian, then X_/T is locally Noetherian.","statement_latex":"Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$.\nLet $T \\subset X_{red}$ be a closed subset and let $X_{/T}$\nbe the formal completion of $X$ along $T$. Then\n\\begin{enumerate}\n\\item if $X_{red} \\setminus T \\to X_{red}$ is quasi-compact and $X$\nis locally countably indexed, then $X_{/T}$ is locally countably indexed,\n\\item if $X_{red} \\setminus T \\to X_{red}$ is quasi-compact and $X$\nis locally adic*, then $X_{/T}$ is locally adic*, and\n\\item if $X$ is locally Noetherian, then $X_{/T}$ is locally Noetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Recompletion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVW","source_file":"formal-spaces.tex","source_line":7934,"source_end_line":7946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7934-L7946","statement_sha256":"80746f3f3042f39df448897faa1573e3515202759982a80bd12f615e4ef6dacd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13077,"rank":13077,"depth":60,"x":989.21,"y":1600.272,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXU","tag":"0GXU","title":"Completion along a closed subspace · Definition 0GXU","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z ⊂ X be a closed subspace and denote Z_n ⊂ X the nth order infinitesimal neighbourhood. The formal algebraic space X^wedge_Z = colim Z_n (see Lemma [Tag 0GVN]) is called the completion of X along Z.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z \\subset X$ be a closed subspace and denote $Z_n \\subset X$\nthe $n$th order infinitesimal neighbourhood. The formal algebraic space\n$$\nX^\\wedge_Z = \\colim Z_n\n$$\n(see Lemma \\ref{lemma-colimit-formal-spaces-is-formal-space})\nis called the {\\it completion of $X$ along $Z$}.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subspace","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXU","source_file":"formal-spaces.tex","source_line":7978,"source_end_line":7988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L7978-L7988","statement_sha256":"974cbad1df7bb78a2d9e72e12d1572b62586a8262e56fa9c0601b1567d2753fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13078,"rank":13078,"depth":61,"x":1233.764,"y":1519.782,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXV","tag":"0GXV","title":"Completion along a closed subspace · Lemma 0GXV","summary":"Let S be a scheme. Let f : X' → X be a morphism of algebraic spaces over S. Let Z ⊂ X be a closed subspace and let Z' = f^-1(Z) = X' ×_X Z. Then xymatrix (X')^wedge_Z' ar[r] ar[d] & X' ar[d]^f X^wedge_Z ar[r] & X is a cartesian diagram of sheaves. In particular, the morphism (X')^wedge_Z' → X^wedge_Z is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $f : X' \\to X$ be a morphism\nof algebraic spaces over $S$. Let $Z \\subset X$\nbe a closed subspace and let $Z' = f^{-1}(Z) = X' \\times_X Z$.\nThen\n$$\n\\xymatrix{\n(X')^\\wedge_{Z'} \\ar[r] \\ar[d] & X' \\ar[d]^f \\\\\nX^\\wedge_Z \\ar[r] & X\n}\n$$\nis a cartesian diagram of sheaves. In particular, the morphism\n$(X')^\\wedge_{Z'} \\to X^\\wedge_Z$ is representable by algebraic spaces.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subspace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXV","source_file":"formal-spaces.tex","source_line":8015,"source_end_line":8029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L8015-L8029","statement_sha256":"dbc5ea94997a4cacedb496924019530886ebf721af67d08ee65122f475466a67","origin":"The Stacks Project","memory_eligible":false,"source_rank":13079,"rank":13079,"depth":0,"x":1117.975,"y":1718.22,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXW","tag":"0GXW","title":"Completion along a closed subspace · Lemma 0GXW","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z ⊂ X be a closed subspace. The reduction (X^wedge_Z)_red of the completion X^wedge_Z of X along Z is Z_red.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z \\subset X$ be a closed subspace. The reduction $(X^\\wedge_Z)_{red}$\nof the completion $X^\\wedge_Z$ of $X$ along $Z$ is $Z_{red}$.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subspace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXW","source_file":"formal-spaces.tex","source_line":8042,"source_end_line":8047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L8042-L8047","statement_sha256":"0664ae80b93c24b467bb07ec3e168f25bb1cb9659ebcf02fc7e54952788cc352","origin":"The Stacks Project","memory_eligible":false,"source_rank":13080,"rank":13080,"depth":0,"x":1043.66,"y":1505.853,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXX","tag":"0GXX","title":"Completion along a closed subspace · Lemma 0GXX","summary":"Let S be a scheme. Let X = Spec(A) be an affine scheme over S. Let Z ⊂ X be a closed subscheme corresponding to the ideal I ⊂ A. Then • The affine formal algebraic space X^wedge_Z is weakly adic. • If I is finitely generated, then X^wedge_Z = Spf(A^wedge) where A^wedge is the I-adic completion of A. • If Z → X is of finite presentation, i.e., I is finitely generated, then X^wedge_Z is adic*. • If X is Noetherian, then X^wedge_Z is Noetherian.","statement_latex":"Let $S$ be a scheme. Let $X = \\Spec(A)$ be an affine scheme over $S$.\nLet $Z \\subset X$ be a closed subscheme corresponding to the ideal\n$I \\subset A$. Then\n\\begin{enumerate}\n\\item The affine formal algebraic space $X^\\wedge_Z$ is weakly adic.\n\\item If $I$ is finitely generated, then\n$X^\\wedge_Z = \\text{Spf}(A^\\wedge)$ where $A^\\wedge$ is the\n$I$-adic completion of $A$.\n\\item If $Z \\to X$ is of finite presentation, i.e., $I$ is\nfinitely generated, then $X^\\wedge_Z$ is adic*.\n\\item If $X$ is Noetherian, then $X^\\wedge_Z$ is Noetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subspace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXX","source_file":"formal-spaces.tex","source_line":8053,"source_end_line":8067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L8053-L8067","statement_sha256":"1ee0db96206bccd6ee0078f151c36331f85dc6ecae62d623b2b777b81cd5346f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13081,"rank":13081,"depth":0,"x":1269.599,"y":1620.464,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GXY","tag":"0GXY","title":"Completion along a closed subspace · Lemma 0GXY","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z ⊂ X be a closed subspace. Let X^wedge_Z be the formal completion of X along Z. • The formal algebraic space X^wedge_Z is locally weakly adic. • If Z → X is of finite presentation, then X^wedge_Z is locally adic*. • If X is locally Noetherian, then X_Z is locally Noetherian.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z \\subset X$ be a closed subspace. Let $X^\\wedge_Z$ be the\nformal completion of $X$ along $Z$.\n\\begin{enumerate}\n\\item The formal algebraic space $X^\\wedge_Z$ is locally weakly adic.\n\\item If $Z \\to X$ is of finite presentation,\nthen $X^\\wedge_Z$ is locally adic*.\n\\item If $X$ is locally Noetherian, then $X_Z$ is locally\nNoetherian.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subspace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GXY","source_file":"formal-spaces.tex","source_line":8073,"source_end_line":8085,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L8073-L8085","statement_sha256":"6324fe68d3b1303d329fa067ef6851b87b2ae33159f2b0fbbed19b278d20aed7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13082,"rank":13082,"depth":0,"x":1010.415,"y":1664.224,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0H8N","tag":"0H8N","title":"Completion along a closed subspace · Lemma 0H8N","summary":"Let S be a scheme. Let X be an algebraic space over S. Let Z ⊂ X be a closed subspace and set T = |Z|. The canonical morphism c : X^wedge_Z → X_/T is an isomorphism if Z → X is of finite presentation, but not in general.","statement_latex":"Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.\nLet $Z \\subset X$ be a closed subspace and set $T = |Z|$. The canonical\nmorphism $c : X^\\wedge_Z \\to X_{/T}$ is an isomorphism if $Z \\to X$\nis of finite presentation, but not in general.","area":"Algebraic & Formal Geometry","chapter":"Formal Algebraic Spaces","chapter_id":"formal-spaces","section":"Completion along a closed subspace","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H8N","source_file":"formal-spaces.tex","source_line":8091,"source_end_line":8097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-spaces.tex#L8091-L8097","statement_sha256":"f7d773cb0396816178163057a5c775395773fe649d4a17b8c3fec68a8af4c51d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13083,"rank":13083,"depth":8,"x":1166.591,"y":1484.604,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJP","tag":"0AJP","title":"Two categories · Lemma 0AJP","summary":"Let A be a ring and let I ⊂ A be a finitely generated ideal. The functor C → C', (B_n) ↦ B = lim B_n is a quasi-inverse to ([Tag 0AJN]). The completions A[x_1, …, x_r]^wedge are in C' and any object of C' is of the form B = A[x_1, …, x_r]^wedge / J for some ideal J ⊂ A[x_1, …, x_r]^wedge.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be a finitely generated ideal.\nThe functor\n$$\n\\mathcal{C} \\longrightarrow \\mathcal{C}',\\quad\n(B_n) \\longmapsto B = \\lim B_n\n$$\nis a quasi-inverse to (\\ref{equation-from-complete-to-systems}).\nThe completions $A[x_1, \\ldots, x_r]^\\wedge$ are in $\\mathcal{C}'$ and\nany object of $\\mathcal{C}'$ is of the form\n$$\nB = A[x_1, \\ldots, x_r]^\\wedge / J\n$$\nfor some ideal $J \\subset A[x_1, \\ldots, x_r]^\\wedge$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Two categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJP","source_file":"restricted.tex","source_line":128,"source_end_line":143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L128-L143","statement_sha256":"606b8cf21e2e6b6dad0d983e2825d20c9b58fda9ca722c8cc337cc2efac49522","origin":"The Stacks Project","memory_eligible":false,"source_rank":13084,"rank":13084,"depth":7,"x":1195.921,"y":1706.021,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJQ","tag":"0AJQ","title":"Two categories · Lemma 0AJQ","summary":"[EGA1] Let A be a Noetherian ring and let I ⊂ A be an ideal. Then • every object of the category C' ([Tag 0AL4]) is Noetherian, • if B ∈ Ob(C') and J ⊂ B is an ideal, then B/J is an object of C', • for a finite type A-algebra C the I-adic completion C^wedge is in C', • in particular the completion A[x_1, …, x_r]^wedge is in C'.","statement_latex":"\\begin{reference}\n\\cite[Proposition 7.5.5]{EGA1}\n\\end{reference}\nLet $A$ be a Noetherian ring and let $I \\subset A$ be an ideal. Then\n\\begin{enumerate}\n\\item every object of the category $\\mathcal{C}'$\n(\\ref{equation-C-prime}) is Noetherian,\n\\item if $B \\in \\Ob(\\mathcal{C}')$ and $J \\subset B$ is an ideal,\nthen $B/J$ is an object of $\\mathcal{C}'$,\n\\item for a finite type $A$-algebra $C$ the $I$-adic completion\n$C^\\wedge$ is in $\\mathcal{C}'$,\n\\item in particular the completion $A[x_1, \\ldots, x_r]^\\wedge$\nis in $\\mathcal{C}'$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Two categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJQ","source_file":"restricted.tex","source_line":177,"source_end_line":193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L177-L193","statement_sha256":"14c76717a78675e102815f5b8c17b49b541ddeb55b1932a8be4e3b75e14d2d25","origin":"The Stacks Project","memory_eligible":false,"source_rank":13085,"rank":13085,"depth":8,"x":995.92,"y":1559.162,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAE","tag":"0GAE","title":"A naive cotangent complex · Lemma 0GAE","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let B be an object of ([Tag 0AL4]). The naive cotangent complex NL_B/A^wedge is well defined in K(B).","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nLet $B$ be an object of (\\ref{equation-C-prime}). The naive\ncotangent complex $\\NL_{B/A}^\\wedge$ is well defined in $K(B)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"A naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAE","source_file":"restricted.tex","source_line":288,"source_end_line":293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L288-L293","statement_sha256":"e8baba476169cdd9b7b4b0b84ec3360c2977706a83bc90e8f68255bb12efc142","origin":"The Stacks Project","memory_eligible":false,"source_rank":13086,"rank":13086,"depth":1,"x":1261.916,"y":1553.96,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAF","tag":"0GAF","title":"A naive cotangent complex · Lemma 0GAF","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let A → B be a finite type ring map. Choose a presentation α : A[x_1, …, x_n] → B. Then NL_B^wedge/A^wedge = lim NL(α) ⊗_B B^wedge as complexes and NL_B^wedge/A^wedge = NL_B/A ⊗_B^L B^wedge in D(B^wedge).","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nLet $A \\to B$ be a finite type ring map. Choose a presentation\n$\\alpha : A[x_1, \\ldots, x_n] \\to B$. Then\n$\\NL_{B^\\wedge/A}^\\wedge = \\lim \\NL(\\alpha) \\otimes_B B^\\wedge$\nas complexes and\n$\\NL_{B^\\wedge/A}^\\wedge = \\NL_{B/A} \\otimes_B^\\mathbf{L} B^\\wedge$\nin $D(B^\\wedge)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"A naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAF","source_file":"restricted.tex","source_line":342,"source_end_line":351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L342-L351","statement_sha256":"bebd63565b63adcfa5c69a8fd9518f01e05bff11b1e87ad736ab8cdb8454095a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13087,"rank":13087,"depth":9,"x":1069.656,"y":1708.972,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJS","tag":"0AJS","title":"A naive cotangent complex · Lemma 0AJS","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let B be an object of ([Tag 0AL4]). Then • the pro-objects (NL_B/A^wedge ⊗_B B/I^nB) and (NL_B_n/A_n) of D(B) are strictly isomorphic (see proof for elucidation), • NL_B/A^wedge = Rlim NL_B_n/A_n in D(B). Here B_n and A_n are as in Section [Tag 0AL2].","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nLet $B$ be an object of (\\ref{equation-C-prime}). Then\n\\begin{enumerate}\n\\item the pro-objects\n$\\{\\NL_{B/A}^\\wedge \\otimes_B B/I^nB\\}$ and $\\{\\NL_{B_n/A_n}\\}$\nof $D(B)$ are strictly isomorphic (see proof for elucidation),\n\\item $\\NL_{B/A}^\\wedge = R\\lim \\NL_{B_n/A_n}$ in $D(B)$.\n\\end{enumerate}\nHere $B_n$ and $A_n$ are as in Section \\ref{section-two-categories}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"A naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJS","source_file":"restricted.tex","source_line":378,"source_end_line":389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L378-L389","statement_sha256":"15c929da8e0461f54fdd95448708224878a4740bcdcdefbe1a78e7be1d2b0acf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13088,"rank":13088,"depth":18,"x":1086.797,"y":1485.227,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAG","tag":"0GAG","title":"A naive cotangent complex · Lemma 0GAG","summary":"Let (A_1, I_1) → (A_2, I_2) be as in Remark [Tag 0AL5] with A_1 and A_2 Noetherian. Let B_1 be in ([Tag 0AL4]) for (A_1, I_1). Let B_2 be the base change of B_1. Then there is a canonical map NL_B_1/A_1 ⊗_B_2 B_1 → NL_B_2/A_2 which induces and isomorphism on H^0 and a surjection on H^-1.","statement_latex":"Let $(A_1, I_1) \\to (A_2, I_2)$ be as in\nRemark \\ref{remark-base-change} with $A_1$ and $A_2$ Noetherian.\nLet $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$.\nLet $B_2$ be the base change of $B_1$. Then there is a canonical map\n$$\n\\NL_{B_1/A_1} \\otimes_{B_2} B_1 \\to \\NL_{B_2/A_2}\n$$\nwhich induces and isomorphism on $H^0$ and a surjection on $H^{-1}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"A naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAG","source_file":"restricted.tex","source_line":451,"source_end_line":461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L451-L461","statement_sha256":"942a3af5c2982428bf691f34d04c08a982280ff4d77c184baf27954c21855339","origin":"The Stacks Project","memory_eligible":false,"source_rank":13089,"rank":13089,"depth":0,"x":1254.348,"y":1660.211,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ALM","tag":"0ALM","title":"A naive cotangent complex · Lemma 0ALM","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let B → C be morphism of ([Tag 0AL4]). Then there is an exact sequence xymatrix C ⊗_B H^0(NL_B/A^wedge) ar[r] & H^0(NL_C/A^wedge) ar[r] & H^0(NL_C/B^wedge) ar[r] & 0 H^-1(NL_B/A^wedge ⊗_B C) ar[r] & H^-1(NL_C/A^wedge) ar[r] & H^-1(NL_C/B^wedge) ar[llu] See proof for elucidation.","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nLet $B \\to C$ be morphism of (\\ref{equation-C-prime}). Then\nthere is an exact sequence\n$$\n\\xymatrix{\nC \\otimes_B H^0(\\NL_{B/A}^\\wedge) \\ar[r] &\nH^0(\\NL_{C/A}^\\wedge) \\ar[r] &\nH^0(\\NL_{C/B}^\\wedge) \\ar[r] & 0 \\\\\nH^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\ar[r] &\nH^{-1}(\\NL_{C/A}^\\wedge) \\ar[r] &\nH^{-1}(\\NL_{C/B}^\\wedge) \\ar[llu]\n}\n$$\nSee proof for elucidation.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"A naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALM","source_file":"restricted.tex","source_line":507,"source_end_line":523,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L507-L523","statement_sha256":"0c3bec6c948c5fcda9ca8cacececcc64f334b29e2100b241f8176dadf6058d78","origin":"The Stacks Project","memory_eligible":false,"source_rank":13090,"rank":13090,"depth":9,"x":989.67,"y":1626.2,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQJ","tag":"0AQJ","title":"A naive cotangent complex · Lemma 0AQJ","summary":"With assumptions as in Lemma [Tag 0ALM] assume that B/I^nB → C/I^nC is a local complete intersection homomorphism for all n. Then H^-1(NL_B/A^wedge ⊗_B C) → H^-1(NL_C/A^wedge) is injective.","statement_latex":"With assumptions as in Lemma \\ref{lemma-exact-sequence-NL}\nassume that $B/I^nB \\to C/I^nC$ is a local complete intersection\nhomomorphism for all $n$. Then\n$H^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\to H^{-1}(\\NL_{C/A}^\\wedge)$\nis injective.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"A naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQJ","source_file":"restricted.tex","source_line":562,"source_end_line":569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L562-L569","statement_sha256":"fec5ff981f400aaf810d0ac2bc868bc3080552d40a13d79eb0f35dd876cea1cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13091,"rank":13091,"depth":19,"x":1212.536,"y":1500.901,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAI","tag":"0GAI","title":"Rig-smooth algebras · Definition 0GAI","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let B be an object of ([Tag 0AL4]). We say B is rig-smooth over (A, I) if there exists an integer c ≥ 0 such that I^c annihilates Ext^1_B(NL_B/A^wedge, N) for every B-module N.","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nLet $B$ be an object of (\\ref{equation-C-prime}). We say\n$B$ is {\\it rig-smooth over $(A, I)$} if there exists an integer $c \\geq 0$\nsuch that $I^c$ annihilates $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$ for every\n$B$-module $N$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAI","source_file":"restricted.tex","source_line":613,"source_end_line":620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L613-L620","statement_sha256":"6b178dae2a5329fff6eab72dfbd21f554280a1fd33f5d73d36ab11ef838c736a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13092,"rank":13092,"depth":0,"x":1148.861,"y":1720.092,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAJ","tag":"0GAJ","title":"Rig-smooth algebras · Lemma 0GAJ","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let B be an object of ([Tag 0AL4]). Write B = A[x_1, …, x_r]^wedge/J (Lemma [Tag 0AJQ]) and let NL_B/A^wedge = (J/J^2 → bigoplus Bdx_i) be its naive cotangent complex ([Tag 0AJR]). The following are equivalent • B is rig-smooth over (A, I), • the object NL_B/A^wedge of D(B) satisfies the equivalent conditions (1) -- (6) of More on Algebra, Lemma [Tag 0G9K] with respect to the ideal IB, • there exists a c ≥ 0 such that…","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nLet $B$ be an object of (\\ref{equation-C-prime}). Write\n$B = A[x_1, \\ldots, x_r]^\\wedge/J$\n(Lemma \\ref{lemma-topologically-finite-type-Noetherian})\nand let $\\NL_{B/A}^\\wedge = (J/J^2 \\to \\bigoplus B\\text{d}x_i)$\nbe its naive cotangent complex (\\ref{equation-NL}).\nThe following are equivalent\n\\begin{enumerate}\n\\item $B$ is rig-smooth over $(A, I)$,\n\\item the object $\\NL_{B/A}^\\wedge$ of $D(B)$ satisfies the equivalent\nconditions (1) -- (6) of More on Algebra, Lemma\n\\ref{more-algebra-lemma-ext-1-annihilated} with respect to the ideal $IB$,\n\\item there exists a $c \\geq 0$ such that for all $a \\in I^c$\nthere is a map $h : \\bigoplus B\\text{d}x_i \\to J/J^2$ such that\n$a : J/J^2 \\to J/J^2$ is equal to $h \\circ \\text{d}$,\n\\item there exist $b_1, \\ldots, b_s \\in B$ such that\n$V(b_1, \\ldots, b_s) \\subset V(IB)$ and such that for every\n$l = 1, \\ldots, s$ there exist $m \\geq 0$, $f_1, \\ldots, f_m \\in J$,\nand subset $T \\subset \\{1, \\ldots, n\\}$ with $|T| = m$ such that\n\\begin{enumerate}\n\\item $\\det_{i \\in T, j \\leq m}(\\partial f_j/ \\partial x_i)$\ndivides $b_l$ in $B$, and\n\\item $b_l J \\subset (f_1, \\ldots, f_m) + J^2$.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAJ","source_file":"restricted.tex","source_line":625,"source_end_line":652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L625-L652","statement_sha256":"b7aa15af3644d86339f2e655b1966e5a47ca52d6758abdb91595e7b8b8a6d035","origin":"The Stacks Project","memory_eligible":false,"source_rank":13093,"rank":13093,"depth":15,"x":1019.348,"y":1522.029,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAK","tag":"0GAK","title":"Rig-smooth algebras · Lemma 0GAK","summary":"Let A be a Noetherian ring and let I be an ideal. Let B be a finite type A-algebra. • If Spec(B) → Spec(A) is smooth over Spec(A) setminus V(I), then B^wedge is rig-smooth over (A, I). • If B^wedge is rig-smooth over (A, I), then there exists g ∈ 1 + IB such that Spec(B_g) is smooth over Spec(A) setminus V(I).","statement_latex":"Let $A$ be a Noetherian ring and let $I$ be an ideal.\nLet $B$ be a finite type $A$-algebra.\n\\begin{enumerate}\n\\item If $\\Spec(B) \\to \\Spec(A)$ is smooth over $\\Spec(A) \\setminus V(I)$,\nthen $B^\\wedge$ is rig-smooth over $(A, I)$.\n\\item If $B^\\wedge$ is rig-smooth over $(A, I)$,\nthen there exists $g \\in 1 + IB$ such that $\\Spec(B_g)$ is smooth\nover $\\Spec(A) \\setminus V(I)$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAK","source_file":"restricted.tex","source_line":725,"source_end_line":736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L725-L736","statement_sha256":"2e03719affc4027487bad5efa067b833c886d8c95fe9f2f92644077e6438ddc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13094,"rank":13094,"depth":20,"x":1274.517,"y":1594.7,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAL","tag":"0GAL","title":"Rig-smooth algebras · Lemma 0GAL","summary":"Let (A_1, I_1) → (A_2, I_2) be as in Remark [Tag 0AL5] with A_1 and A_2 Noetherian. Let B_1 be in ([Tag 0AL4]) for (A_1, I_1). Let B_2 be the base change of B_1. Let f_1 ∈ B_1 with image f_2 ∈ B_2. If Ext^1_B_1(NL_B_1/A_1^wedge, N_1) is annihilated by f_1 for every B_1-module N_1, then Ext^1_B_2(NL_B_2/A_2^wedge, N_2) is annihilated by f_2 for every B_2-module N_2.","statement_latex":"Let $(A_1, I_1) \\to (A_2, I_2)$ be as in\nRemark \\ref{remark-base-change} with $A_1$ and $A_2$ Noetherian.\nLet $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$.\nLet $B_2$ be the base change of $B_1$. Let $f_1 \\in B_1$\nwith image $f_2 \\in B_2$.\nIf $\\Ext^1_{B_1}(\\NL_{B_1/A_1}^\\wedge, N_1)$ is annihilated\nby $f_1$ for every $B_1$-module $N_1$, then\n$\\Ext^1_{B_2}(\\NL_{B_2/A_2}^\\wedge, N_2)$ is annihilated\nby $f_2$ for every $B_2$-module $N_2$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAL","source_file":"restricted.tex","source_line":798,"source_end_line":809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L798-L809","statement_sha256":"92374839737d32dadbb21aa3b2b856372be00ea31189aa3af9e6b1e25f5a6939","origin":"The Stacks Project","memory_eligible":false,"source_rank":13095,"rank":13095,"depth":13,"x":1027.541,"y":1686.042,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAM","tag":"0GAM","title":"Rig-smooth algebras · Lemma 0GAM","summary":"Let A_1 → A_2 be a map of Noetherian rings. Let I_i ⊂ A_i be an ideal such that V(I_1A_2) = V(I_2). Let B_1 be in ([Tag 0AL4]) for (A_1, I_1). Let B_2 be the base change of B_1 as in Remark [Tag 0AL5]. If B_1 is rig-smooth over (A_1, I_1), then B_2 is rig-smooth over (A_2, I_2).","statement_latex":"Let $A_1 \\to A_2$ be a map of Noetherian rings. Let $I_i \\subset A_i$\nbe an ideal such that $V(I_1A_2) = V(I_2)$. Let $B_1$ be in\n(\\ref{equation-C-prime}) for $(A_1, I_1)$.\nLet $B_2$ be the base change of $B_1$ as in\nRemark \\ref{remark-base-change}.\nIf $B_1$ is rig-smooth over $(A_1, I_1)$,\nthen $B_2$ is rig-smooth over $(A_2, I_2)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAM","source_file":"restricted.tex","source_line":825,"source_end_line":834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L825-L834","statement_sha256":"a75653876ba2e2fc53b217d8edd19962420c4ab320fbe8efad36c35c0671cbcc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13096,"rank":13096,"depth":14,"x":1136.369,"y":1478.231,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAQ","tag":"0GAQ","title":"Deformations of ring homomorphisms · Lemma 0GAQ","summary":"Assume given the following data • an integer c ≥ 0, • an ideal I of a Noetherian ring A, • B in ([Tag 0AL4]) for (A, I) such that I^c annihilates Ext^1_B(NL_B/A^wedge, N) for any B-module N, • a Noetherian I-adically complete A-algebra C; denote d = d(Gr_I(C)) and q_0 = q(Gr_I(C)) the integers found in Local Cohomology, Section [Tag 0GA6], • an integer n ≥ max(q_0 + (d + 1)c, 2(d + 1)c + 1), and • an A-algebra homomorphism ψ_n : B → C/I^nC. Then there exists a map φ : B →…","statement_latex":"Assume given the following data\n\\begin{enumerate}\n\\item an integer $c \\geq 0$,\n\\item an ideal $I$ of a Noetherian ring $A$,\n\\item $B$ in (\\ref{equation-C-prime}) for $(A, I)$ such that\n$I^c$ annihilates $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$\nfor any $B$-module $N$,\n\\item a Noetherian $I$-adically complete $A$-algebra $C$; denote\n$d = d(\\text{Gr}_I(C))$ and $q_0 = q(\\text{Gr}_I(C))$ the integers found in\nLocal Cohomology, Section \\ref{local-cohomology-section-uniform},\n\\item an integer $n \\geq \\max(q_0 + (d + 1)c, 2(d + 1)c + 1)$, and\n\\item an $A$-algebra homomorphism $\\psi_n : B \\to C/I^nC$.\n\\end{enumerate}\nThen there exists a map $\\varphi : B \\to C$ of $A$-algebras such\nthat $\\psi_n \\bmod I^{n - (d + 1)c} = \\varphi \\bmod I^{n - (d + 1)c}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Deformations of ring homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAQ","source_file":"restricted.tex","source_line":949,"source_end_line":966,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L949-L966","statement_sha256":"f05b556a0c538864af67f043b4ef260a1e8e567820b15bc7d75207d88adc9c29","origin":"The Stacks Project","memory_eligible":false,"source_rank":13097,"rank":13097,"depth":19,"x":1223.368,"y":1693.546,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AK6","tag":"0AK6","title":"Deformations of ring homomorphisms · Lemma 0AK6","summary":"Let I = (a) be a principal ideal of a Noetherian ring A. Let B be an object of ([Tag 0AL4]). Assume given an integer c ≥ 0 such that Ext^1_B(NL_B/A^wedge, N) is annihilated by a^c for all B-modules N. Let C be an I-adically complete A-algebra such that a is a nonzerodivisor on C. Let n > 2c. For any A-algebra map ψ_n : B → C/a^nC there exists an A-algebra map φ : B → C such that ψ_n bmod a^n - cC = φ bmod a^n - cC.","statement_latex":"Let $I = (a)$ be a principal ideal of a Noetherian ring $A$.\nLet $B$ be an object of (\\ref{equation-C-prime}).\nAssume given an integer $c \\geq 0$ such that $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$\nis annihilated by $a^c$ for all $B$-modules $N$.\nLet $C$ be an $I$-adically complete $A$-algebra such that\n$a$ is a nonzerodivisor on $C$. Let $n > 2c$. For any $A$-algebra\nmap $\\psi_n : B \\to C/a^nC$ there exists an $A$-algebra\nmap $\\varphi : B \\to C$ such that\n$\\psi_n \\bmod a^{n - c}C = \\varphi \\bmod a^{n - c}C$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Deformations of ring homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AK6","source_file":"restricted.tex","source_line":1024,"source_end_line":1035,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1024-L1035","statement_sha256":"d0e52a5fbc04dc1d11cd8f51d79865dd6293ab92b4afbd5c8c1e18393ea972a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13098,"rank":13098,"depth":0,"x":985.707,"y":1583.976,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AK7","tag":"0AK7","title":"Deformations of ring homomorphisms · Lemma 0AK7","summary":"Let I = (a) be a principal ideal of a Noetherian ring A. Let B be an object of ([Tag 0AL4]). Assume given an integer c ≥ 0 such that Ext^1_B(NL_B/A^wedge, N) is annihilated by a^c for all B-modules N. Let C be an I-adically complete A-algebra. Assume given an integer d ≥ 0 such that C[a^∞] ∩ a^dC = 0. Let n > max(2c, c + d). For any A-algebra map ψ_n : B → C/a^nC there exists an A-algebra map φ : B → C such that ψ_n bmod a^n - c = φ bmod a^n - c.","statement_latex":"Let $I = (a)$ be a principal ideal of a Noetherian ring $A$.\nLet $B$ be an object of (\\ref{equation-C-prime}).\nAssume given an integer $c \\geq 0$ such that $\\Ext^1_B(\\NL_{B/A}^\\wedge, N)$\nis annihilated by $a^c$ for all $B$-modules $N$.\nLet $C$ be an $I$-adically complete $A$-algebra.\nAssume given an integer $d \\geq 0$ such that $C[a^\\infty] \\cap a^dC = 0$.\nLet $n > \\max(2c, c + d)$. For any $A$-algebra map\n$\\psi_n : B \\to C/a^nC$ there exists an $A$-algebra map\n$\\varphi : B \\to C$ such\nthat $\\psi_n \\bmod a^{n - c} = \\varphi \\bmod a^{n - c}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Deformations of ring homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AK7","source_file":"restricted.tex","source_line":1076,"source_end_line":1088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1076-L1088","statement_sha256":"40de7ccfa353b8d282505c29066ed98b615e9811e6193434719f658e8c0ae924","origin":"The Stacks Project","memory_eligible":false,"source_rank":13099,"rank":13099,"depth":0,"x":1249.466,"y":1529.835,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAR","tag":"0GAR","title":"Algebraization of rig-smooth algebras over G-rings · Lemma 0GAR","summary":"Let I be an ideal of a Noetherian ring A. Let r ≥ 0 and write P = A[x_1, …, x_r] the I-adic completion. Consider a resolution P^⊕ t xrightarrowK P^⊕ m xrightarrowg_1, …, g_m P → B → 0 of a quotient of P. Assume B is rig-smooth over (A, I). Then there exists an integer n such that for any complex P^⊕ t xrightarrowK' P^⊕ m xrightarrowg'_1, …, g'_m P with g_i - g'_i ∈ I^nP and K - K' ∈ I^nMat(m × t, P) there exists an isomorphism B → B' of A-algebras where B' = P/(g'_1, …,…","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let $r \\geq 0$\nand write $P = A[x_1, \\ldots, x_r]$ the $I$-adic completion.\nConsider a resolution\n$$\nP^{\\oplus t} \\xrightarrow{K} P^{\\oplus m}\n\\xrightarrow{g_1, \\ldots, g_m} P \\to B \\to 0\n$$\nof a quotient of $P$. Assume $B$ is rig-smooth over $(A, I)$.\nThen there exists an integer $n$ such that for any complex\n$$\nP^{\\oplus t} \\xrightarrow{K'} P^{\\oplus m}\n\\xrightarrow{g'_1, \\ldots, g'_m} P\n$$\nwith $g_i - g'_i \\in I^nP$ and $K - K' \\in I^n\\text{Mat}(m \\times t, P)$\nthere exists an isomorphism $B \\to B'$ of $A$-algebras where\n$B' = P/(g'_1, \\ldots, g'_m)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-smooth algebras over G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAR","source_file":"restricted.tex","source_line":1145,"source_end_line":1163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1145-L1163","statement_sha256":"7d838a9cf6eaab6797911b8601c00b09f4872a5e86d7c4308c08c278e7135ac9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13100,"rank":13100,"depth":20,"x":1098.285,"y":1719.706,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAS","tag":"0GAS","title":"Algebraization of rig-smooth algebras over G-rings · Lemma 0GAS","summary":"Let I be an ideal of a Noetherian ring A. Let C^h be the henselization of a finite type A-algebra C with respect to the ideal IC. Let J ⊂ C^h be an ideal. Then there exists a finite type A-algebra B such that B^wedge ≅ (C^h/J)^wedge.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let $C^h$ be the henselization\nof a finite type $A$-algebra $C$ with respect to the ideal $IC$. Let\n$J \\subset C^h$ be an ideal. Then there exists a finite type $A$-algebra\n$B$ such that $B^\\wedge \\cong (C^h/J)^\\wedge$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-smooth algebras over G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAS","source_file":"restricted.tex","source_line":1234,"source_end_line":1240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1234-L1240","statement_sha256":"1645b84bd54067109a76e50aa08761c64cde02ab772754ebebd39094eaca10c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13101,"rank":13101,"depth":50,"x":1057.012,"y":1493.575,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAT","tag":"0GAT","title":"Algebraization of rig-smooth algebras over G-rings · Proposition 0GAT","summary":"Let I be an ideal of a Noetherian G-ring A. Let B be an object of ([Tag 0AL4]). If B is rig-smooth over (A, I), then there exists a finite type A-algebra C and an isomorphism B ≅ C^wedge of A-algebras.","statement_latex":"Let $I$ be an ideal of a Noetherian G-ring $A$. Let $B$ be an\nobject of (\\ref{equation-C-prime}). If $B$ is rig-smooth\nover $(A, I)$, then there exists a finite type $A$-algebra\n$C$ and an isomorphism $B \\cong C^\\wedge$ of $A$-algebras.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-smooth algebras over G-rings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAT","source_file":"restricted.tex","source_line":1257,"source_end_line":1263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1257-L1263","statement_sha256":"7b71bcfe99113400cddf1620448722a32d105b84c26021f2a0b5ae589f243eb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13102,"rank":13102,"depth":52,"x":1269.612,"y":1637.116,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AK4","tag":"0AK4","title":"Algebraization of rig-smooth algebras over G-rings · Lemma 0AK4","summary":"Let A be a Noetherian G-ring. Let I ⊂ A be an ideal. Let B, C be finite type A-algebras. For any A-algebra map φ : B^wedge → C^wedge of I-adic completions and any N ≥ 1 there exist • an étale ring map C → C' which induces an isomorphism C/IC → C'/IC', • an A-algebra map φ : B → C' such that φ and ψ agree modulo I^N into C^wedge = (C')^wedge.","statement_latex":"Let $A$ be a Noetherian G-ring. Let $I \\subset A$ be an ideal.\nLet $B, C$ be finite type $A$-algebras. For any $A$-algebra map\n$\\varphi : B^\\wedge \\to C^\\wedge$ of $I$-adic completions and any\n$N \\geq 1$ there exist\n\\begin{enumerate}\n\\item an \\'etale ring map $C \\to C'$ which induces\nan isomorphism $C/IC \\to C'/IC'$,\n\\item an $A$-algebra map $\\varphi : B \\to C'$\n\\end{enumerate}\nsuch that $\\varphi$ and $\\psi$ agree modulo $I^N$\ninto $C^\\wedge = (C')^\\wedge$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-smooth algebras over G-rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AK4","source_file":"restricted.tex","source_line":1322,"source_end_line":1335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1322-L1335","statement_sha256":"22f22c22ddb4bd90c67abf97ea8cf00c566cc157d51f1ef5ae2385362856c731","origin":"The Stacks Project","memory_eligible":false,"source_rank":13103,"rank":13103,"depth":52,"x":997.006,"y":1651.928,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAV","tag":"0GAV","title":"Algebraization of rig-smooth algebras · Lemma 0GAV","summary":"Let A be a ring. Let f_1, …, f_m ∈ A[x_1, …, x_n] and set B = A[x_1, …, x_n]/(f_1, …, f_m). Assume m ≤ n and set g = det_1 ≤ i, j ≤ m(∂ f_j/∂ x_i). Then • g annihilates Ext^1_B(NL_B/A, N) for every B-module N, • if n = m, then multiplication by g on NL_B/A is 0 in D(B).","statement_latex":"Let $A$ be a ring. Let $f_1, \\ldots, f_m \\in A[x_1, \\ldots, x_n]$\nand set $B = A[x_1, \\ldots, x_n]/(f_1, \\ldots, f_m)$. Assume $m \\leq n$\nand set $g = \\det_{1 \\leq i, j \\leq m}(\\partial f_j/\\partial x_i)$.\nThen\n\\begin{enumerate}\n\\item $g$ annihilates $\\Ext^1_B(\\NL_{B/A}, N)$ for every $B$-module $N$,\n\\item if $n = m$, then multiplication by $g$ on $\\NL_{B/A}$ is $0$ in $D(B)$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-smooth algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAV","source_file":"restricted.tex","source_line":1380,"source_end_line":1390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1380-L1390","statement_sha256":"219394d87be8dacfba40c995c7c16b57fa026072c2a161ad7e7e4a75f9adef98","origin":"The Stacks Project","memory_eligible":false,"source_rank":13104,"rank":13104,"depth":2,"x":1186.392,"y":1486.077,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAW","tag":"0GAW","title":"Algebraization of rig-smooth algebras · Lemma 0GAW","summary":"Let I be an ideal of a Noetherian ring A. Let B be an object of ([Tag 0AL4]). Let B = A[x_1, …, x_r]^wedge/J be a presentation. Assume there exists an element b ∈ B, 0 ≤ m ≤ r, and f_1, …, f_m ∈ J such that • V(b) ⊂ V(IB) in Spec(B), • the image of Δ = det_1 ≤ i, j ≤ m(∂ f_j/∂ x_i) in B divides b, and • b J ⊂ (f_1, …, f_m) + J^2. Then there exists a finite type A-algebra C and an A-algebra isomorphism B ≅ C^wedge.","statement_latex":"Let $I$ be an ideal of a Noetherian ring $A$. Let $B$ be an object\nof (\\ref{equation-C-prime}). Let $B = A[x_1, \\ldots, x_r]^\\wedge/J$\nbe a presentation. Assume there exists an element\n$b \\in B$, $0 \\leq m \\leq r$, and $f_1, \\ldots, f_m \\in J$\nsuch that\n\\begin{enumerate}\n\\item $V(b) \\subset V(IB)$ in $\\Spec(B)$,\n\\item the image of\n$\\Delta = \\det_{1 \\leq i, j \\leq m}(\\partial f_j/\\partial x_i)$\nin $B$ divides $b$, and\n\\item $b J \\subset (f_1, \\ldots, f_m) + J^2$.\n\\end{enumerate}\nThen there exists a finite type $A$-algebra $C$ and an $A$-algebra\nisomorphism $B \\cong C^\\wedge$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-smooth algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAW","source_file":"restricted.tex","source_line":1420,"source_end_line":1436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1420-L1436","statement_sha256":"2c266ba89f2bd8f589e94135fe97f9e9302602b44c40a3d0839053cf6f370d4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13105,"rank":13105,"depth":48,"x":1180.105,"y":1716.176,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GAY","tag":"0GAY","title":"Rig-étale algebras · Definition 0GAY","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let B be an object of ([Tag 0AL4]). We say B is rig-étale over (A, I) if there exists an integer c ≥ 0 such that for all a ∈ I^c multiplication by a on NL_B/A^wedge is zero in D(B).","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nLet $B$ be an object of (\\ref{equation-C-prime}). We say\n$B$ is {\\it rig-\\'etale over $(A, I)$} if there exists an integer\n$c \\geq 0$ such that for all $a \\in I^c$\nmultiplication by $a$ on $\\NL_{B/A}^\\wedge$\nis zero in $D(B)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale algebras","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GAY","source_file":"restricted.tex","source_line":1576,"source_end_line":1584,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1576-L1584","statement_sha256":"a4d71029ce43e908fa93ddfc6eb32d9bfc8fd7bc1f0c3f0a50ec0669ccf40244","origin":"The Stacks Project","memory_eligible":false,"source_rank":13106,"rank":13106,"depth":0,"x":999.435,"y":1542.68,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AJU","tag":"0AJU","title":"Rig-étale algebras · Lemma 0AJU","summary":"Let A be a Noetherian ring and let I ⊂ A be an ideal. Let B be an object of ([Tag 0AL4]). Write B = A[x_1, …, x_r]^wedge/J (Lemma [Tag 0AJQ]) and let NL_B/A^wedge = (J/J^2 → bigoplus Bdx_i) be its naive cotangent complex ([Tag 0AJR]). The following are equivalent • B is rig-étale over (A, I), • there exists a c ≥ 0 such that for all a ∈ I^c multiplication by a on NL_B/A^wedge is zero in D(B), • there exits a c ≥ 0 such that H^i(NL_B/A^wedge), i = -1, 0 is annihilated by…","statement_latex":"Let $A$ be a Noetherian ring and let $I \\subset A$ be an ideal.\nLet $B$ be an object of (\\ref{equation-C-prime}). Write\n$B = A[x_1, \\ldots, x_r]^\\wedge/J$\n(Lemma \\ref{lemma-topologically-finite-type-Noetherian})\nand let $\\NL_{B/A}^\\wedge = (J/J^2 \\to \\bigoplus B\\text{d}x_i)$\nbe its naive cotangent complex (\\ref{equation-NL}).\nThe following are equivalent\n\\begin{enumerate}\n\\item $B$ is rig-\\'etale over $(A, I)$,\n\\item\n\nthere exists a $c \\geq 0$ such that for all $a \\in I^c$ multiplication by $a$\non $\\NL_{B/A}^\\wedge$ is zero in $D(B)$,\n\\item\n\nthere exits a $c \\geq 0$ such that $H^i(\\NL_{B/A}^\\wedge)$, $i = -1, 0$ is\nannihilated by $I^c$,\n\\item\n\nthere exists a $c \\geq 0$ such that $H^i(\\NL_{B_n/A_n})$, $i = -1, 0$ is\nannihilated by $I^c$ for all $n \\geq 1$ where $A_n = A/I^n$ and $B_n = B/I^nB$,\n\\item\n\nfor every $a \\in I$ there exists a $c \\geq 0$ such that\n\\begin{enumerate}\n\\item $a^c$ annihilates $H^0(\\NL_{B/A}^\\wedge)$, and\n\\item there exist $f_1, \\ldots, f_r \\in J$ such that\n$a^c J \\subset (f_1, \\ldots, f_r) + J^2$.\n\\end{enumerate}\n\\item\n\nfor every $a \\in I$ there exist $f_1, \\ldots, f_r \\in J$ and $c \\geq 0$\nsuch that\n\\begin{enumerate}\n\\item $\\det_{1 \\leq i, j \\leq r}(\\partial f_j/\\partial x_i)$ divides\n$a^c$ in $B$, and\n\\item $a^c J \\subset (f_1, \\ldots, f_r) + J^2$.\n\\end{enumerate}\n\\item\n\nchoosing generators $f_1, \\ldots, f_t$ for $J$ we have\n\\begin{enumerate}\n\\item the Jacobian ideal of $B$ over $A$, namely the ideal in $B$\ngenerated by the $r \\times r$ minors of the matrx\n$(\\partial f_j/\\partial x_i)_{1 \\leq i \\leq r, 1 \\leq j \\leq t}$,\ncontains the ideal $I^cB$ for some $c$, and\n\\item the Cramer ideal of $B$ over $A$, namely the ideal in $B$\ngenerated by the image in $B$ of the $r$th Fitting ideal of $J$\nas an $A[x_1, \\ldots, x_r]^\\wedge$-module, contains $I^cB$ for some $c$.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AJU","source_file":"restricted.tex","source_line":1592,"source_end_line":1645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1592-L1645","statement_sha256":"40ebc754e6465979948d074d32f05a7f72e5b9e13b733a27db9f288f7ec3130a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13107,"rank":13107,"depth":19,"x":1272.581,"y":1568.134,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GB1","tag":"0GB1","title":"Rig-étale algebras · Lemma 0GB1","summary":"Let A be a Noetherian ring and let I be an ideal. Let B be an object of ([Tag 0AL4]). If B is rig-étale over (A, I), then B is rig-smooth over (A, I).","statement_latex":"Let $A$ be a Noetherian ring and let $I$ be an ideal.\nLet $B$ be an object of (\\ref{equation-C-prime}).\nIf $B$ is rig-\\'etale over $(A, I)$, then $B$ is rig-smooth over $(A, I)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GB1","source_file":"restricted.tex","source_line":1751,"source_end_line":1756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1751-L1756","statement_sha256":"28bfbae956b4e5d3bfdaea4505f5ec0293135220319da4561de941cf51793919","origin":"The Stacks Project","memory_eligible":false,"source_rank":13108,"rank":13108,"depth":1,"x":1050.372,"y":1704.554,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ALQ","tag":"0ALQ","title":"Rig-étale algebras · Lemma 0ALQ","summary":"Let A be a Noetherian ring and let I be an ideal. Let B be a finite type A-algebra. • If Spec(B) → Spec(A) is étale over Spec(A) setminus V(I), then B^wedge satisfies the equivalent conditions of Lemma [Tag 0AJU]. • If B^wedge satisfies the equivalent conditions of Lemma [Tag 0AJU], then there exists g ∈ 1 + IB such that Spec(B_g) is étale over Spec(A) setminus V(I).","statement_latex":"Let $A$ be a Noetherian ring and let $I$ be an ideal.\nLet $B$ be a finite type $A$-algebra.\n\\begin{enumerate}\n\\item If $\\Spec(B) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$,\nthen $B^\\wedge$ satisfies the equivalent conditions of\nLemma \\ref{lemma-equivalent-with-artin}.\n\\item If $B^\\wedge$ satisfies the equivalent conditions of\nLemma \\ref{lemma-equivalent-with-artin},\nthen there exists $g \\in 1 + IB$ such that $\\Spec(B_g)$ is \\'etale\nover $\\Spec(A) \\setminus V(I)$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALQ","source_file":"restricted.tex","source_line":1763,"source_end_line":1776,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1763-L1776","statement_sha256":"b60179bf01ca763ee27eb4534e2d52e9dd9c51b73235908787be2be1f29045c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13109,"rank":13109,"depth":20,"x":1104.6,"y":1477.541,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AK2","tag":"0AK2","title":"Rig-étale algebras · Lemma 0AK2","summary":"Let (A_1, I_1) → (A_2, I_2) be as in Remark [Tag 0AL5] with A_1 and A_2 Noetherian. Let B_1 be in ([Tag 0AL4]) for (A_1, I_1). Let B_2 be the base change of B_1. If multiplication by f_1 ∈ B_1 on NL^wedge_B_1/A_1 is zero in D(B_1), then multiplication by the image f_2 ∈ B_2 on NL^wedge_B_2/A_2 is zero in D(B_2).","statement_latex":"Let $(A_1, I_1) \\to (A_2, I_2)$ be as in\nRemark \\ref{remark-base-change} with $A_1$ and $A_2$ Noetherian.\nLet $B_1$ be in (\\ref{equation-C-prime}) for $(A_1, I_1)$.\nLet $B_2$ be the base change of $B_1$.\nIf multiplication by $f_1 \\in B_1$ on $\\NL^\\wedge_{B_1/A_1}$\nis zero in $D(B_1)$, then multiplication by\nthe image $f_2 \\in B_2$ on $\\NL^\\wedge_{B_2/A_2}$ is zero\nin $D(B_2)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AK2","source_file":"restricted.tex","source_line":1808,"source_end_line":1818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1808-L1818","statement_sha256":"92fec35bafa71c2ea05eb5687720baaf3d5964ec55f6127143f61a5a9305a97a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13110,"rank":13110,"depth":2,"x":1247.379,"y":1675.997,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GB2","tag":"0GB2","title":"Rig-étale algebras · Lemma 0GB2","summary":"Let A_1 → A_2 be a map of Noetherian rings. Let I_i ⊂ A_i be an ideal such that V(I_1A_2) = V(I_2). Let B_1 be in ([Tag 0AL4]) for (A_1, I_1). Let B_2 be the base change of B_1 as in Remark [Tag 0AL5]. If B_1 is rig-étale over (A_1, I_1), then B_2 is rig-étale over (A_2, I_2).","statement_latex":"Let $A_1 \\to A_2$ be a map of Noetherian rings. Let $I_i \\subset A_i$\nbe an ideal such that $V(I_1A_2) = V(I_2)$. Let $B_1$ be in\n(\\ref{equation-C-prime}) for $(A_1, I_1)$.\nLet $B_2$ be the base change of $B_1$ as in\nRemark \\ref{remark-base-change}.\nIf $B_1$ is rig-\\'etale over $(A_1, I_1)$,\nthen $B_2$ is rig-\\'etale over $(A_2, I_2)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GB2","source_file":"restricted.tex","source_line":1830,"source_end_line":1839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1830-L1839","statement_sha256":"e60ce0b7e1fd1189bf8c4e8363411f1eca377c1c220ceb11da02cd3194a99b49","origin":"The Stacks Project","memory_eligible":false,"source_rank":13111,"rank":13111,"depth":3,"x":982.117,"y":1610.58,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKJ","tag":"0AKJ","title":"Rig-étale algebras · Lemma 0AKJ","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let B be a finite type A-algebra such that Spec(B) → Spec(A) is étale over Spec(A) setminus V(I). Let C be a Noetherian A-algebra. Then any A-algebra map B^wedge → C^wedge of I-adic completions comes from a unique A-algebra map B → C^h where C^h is the henselization of the pair (C, IC) as in More on Algebra, Lemma [Tag 0A02]. Moreover, any A-algebra homomorphism B → C^h factors through some étale C-algebra C' such that…","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nLet $B$ be a finite type $A$-algebra such that\n$\\Spec(B) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$.\nLet $C$ be a Noetherian $A$-algebra. Then any $A$-algebra\nmap $B^\\wedge \\to C^\\wedge$ of $I$-adic completions\ncomes from a unique $A$-algebra map\n$$\nB \\longrightarrow C^h\n$$\nwhere $C^h$ is the henselization of the pair $(C, IC)$ as\nin More on Algebra, Lemma \\ref{more-algebra-lemma-henselization}.\nMoreover, any $A$-algebra homomorphism $B \\to C^h$ factors through\nsome \\'etale $C$-algebra $C'$ such that $C/IC \\to C'/IC'$ is an isomorphism.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKJ","source_file":"restricted.tex","source_line":1847,"source_end_line":1862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1847-L1862","statement_sha256":"1a96b3a7462ecc5248fb5406754849db86a6aefa5c12468ab2f75c8875b2488d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13112,"rank":13112,"depth":50,"x":1230.684,"y":1508.154,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ALT","tag":"0ALT","title":"A pushout argument · Lemma 0ALT","summary":"Let A be a Noetherian ring and I ⊂ A an ideal. Let J ⊂ A be a nilpotent ideal. Consider a commutative diagram xymatrix C ar[r] & C_0 ar@=[r] & C/JC & B_0 ar[u] A ar[r] ar[uu] & A_0 ar[u] ar@=[r] & A/J whose vertical arrows are of finite type such that • Spec(C) → Spec(A) is étale over Spec(A) setminus V(I), • Spec(B_0) → Spec(A_0) is étale over Spec(A_0) setminus V(IA_0), and • B_0 → C_0 is étale and induces an isomorphism B_0/IB_0 = C_0/IC_0. Then we can fill in the…","statement_latex":"Let $A$ be a Noetherian ring and $I \\subset A$ an ideal.\nLet $J \\subset A$ be a nilpotent ideal. Consider a commutative diagram\n$$\n\\xymatrix{\nC \\ar[r] & C_0 \\ar@{=}[r] & C/JC \\\\\n& B_0 \\ar[u] \\\\\nA \\ar[r] \\ar[uu] & A_0 \\ar[u] \\ar@{=}[r] & A/J\n}\n$$\nwhose vertical arrows are of finite type such that\n\\begin{enumerate}\n\\item $\\Spec(C) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$,\n\\item $\\Spec(B_0) \\to \\Spec(A_0)$ is \\'etale over\n$\\Spec(A_0) \\setminus V(IA_0)$, and\n\\item $B_0 \\to C_0$ is \\'etale and induces an isomorphism\n$B_0/IB_0 = C_0/IC_0$.\n\\end{enumerate}\nThen we can fill in the diagram above to a commutative diagram\n$$\n\\xymatrix{\nC \\ar[r] & C/JC \\\\\nB \\ar[u] \\ar[r] & B_0 \\ar[u] \\\\\nA \\ar[r] \\ar[u] & A/J \\ar[u]\n}\n$$\nwith $A \\to B$ of finite type, $B/JB = B_0$, $B \\to C$ \\'etale, and\n$\\Spec(B) \\to \\Spec(A)$ \\'etale over $\\Spec(A) \\setminus V(I)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"A pushout argument","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALT","source_file":"restricted.tex","source_line":1933,"source_end_line":1962,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L1933-L1962","statement_sha256":"b1aaf374f975b089091738aeedef14b98133c5e20a3efb1e5c56717b8da83f1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13113,"rank":13113,"depth":68,"x":1129.617,"y":1725.037,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ALS","tag":"0ALS","title":"Algebraization of rig-étale algebras · Lemma 0ALS","summary":"The rig-étale case of [Elkik] Let A be a Noetherian ring and I = (a) a principal ideal. Let B be an object of ([Tag 0AL4]) which is rig-étale over (A, I). Then there exists a finite type A-algebra C and an isomorphism B ≅ C^wedge.","statement_latex":"\\begin{reference}\nThe rig-\\'etale case of \\cite[III Theorem 7]{Elkik}\n\\end{reference}\nLet $A$ be a Noetherian ring and $I = (a)$ a principal ideal.\nLet $B$ be an object of (\\ref{equation-C-prime}) which is\nrig-\\'etale over $(A, I)$.\nThen there exists a finite type $A$-algebra $C$ and an\nisomorphism $B \\cong C^\\wedge$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALS","source_file":"restricted.tex","source_line":2063,"source_end_line":2073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2063-L2073","statement_sha256":"7766a469724efddb05bdd52be40e9aeeab18aa24404d1aa263e0f52cef999e60","origin":"The Stacks Project","memory_eligible":false,"source_rank":13114,"rank":13114,"depth":49,"x":1029.586,"y":1507.449,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKA","tag":"0AKA","title":"Algebraization of rig-étale algebras · Lemma 0AKA","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let B be an object of ([Tag 0AL4]) which is rig-étale over (A, I). Then there exists a finite type A-algebra C and an isomorphism B ≅ C^wedge.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nLet $B$ be an object of (\\ref{equation-C-prime}) which is\nrig-\\'etale over $(A, I)$.\nThen there exists a finite type $A$-algebra $C$ and an\nisomorphism $B \\cong C^\\wedge$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKA","source_file":"restricted.tex","source_line":2090,"source_end_line":2097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2090-L2097","statement_sha256":"250c115ad696040c20471da2f41a22df2b734c7b7e20d0a6969f10f48c60acb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13115,"rank":13115,"depth":69,"x":1278.684,"y":1611.285,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AKG","tag":"0AKG","title":"Algebraization of rig-étale algebras · Lemma 0AKG","summary":"Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let B be an I-adically complete A-algebra with A/I → B/IB of finite type. The equivalent conditions of Lemma [Tag 0AJU] are also equivalent to • [(8)] there exists a finite type A-algebra C such that Spec(C) → Spec(A) is étale over Spec(A) setminus V(I) and such that B ≅ C^wedge.","statement_latex":"Let $A$ be a Noetherian ring. Let $I \\subset A$ be an ideal.\nLet $B$ be an $I$-adically complete $A$-algebra with $A/I \\to B/IB$\nof finite type. The equivalent conditions of\nLemma \\ref{lemma-equivalent-with-artin} are also equivalent to\n\\begin{enumerate}\n\\item[(8)]\n\nthere exists a finite type $A$-algebra $C$ such that\n$\\Spec(C) \\to \\Spec(A)$ is \\'etale over $\\Spec(A) \\setminus V(I)$\nand such that $B \\cong C^\\wedge$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-étale algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AKG","source_file":"restricted.tex","source_line":2188,"source_end_line":2201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2188-L2201","statement_sha256":"497c7726775bad4b5c0cd7f5442e570bb1ed5d740233520868812ede9035f3a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13116,"rank":13116,"depth":70,"x":1011.117,"y":1676.154,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBW","tag":"0GBW","title":"Finite type morphisms · Lemma 0GBW","summary":"Let A and B be adic topological rings which have a finitely generated ideal of definition. Let φ : A → B be a continuous ring homomorphism. The following are equivalent: • φ is adic and B is topologically of finite type over A, • φ is taut and B is topologically of finite type over A, • there exists an ideal of definition I ⊂ A such that the topology on B is the I-adic topology and there exist an ideal of definition I' ⊂ A such that A/I' → B/I'B is of finite type, • for…","statement_latex":"Let $A$ and $B$ be adic topological rings which have a finitely generated\nideal of definition. Let $\\varphi : A \\to B$ be a continuous ring homomorphism.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\varphi$ is adic and $B$ is topologically of finite type over $A$,\n\\item $\\varphi$ is taut and $B$ is topologically of finite type over $A$,\n\\item there exists an ideal of definition $I \\subset A$ such that\nthe topology on $B$ is the $I$-adic topology and there exist an ideal\nof definition $I' \\subset A$ such that $A/I' \\to B/I'B$ is of finite type,\n\\item for all ideals of definition $I \\subset A$ the topology on $B$\nis the $I$-adic topology and $A/I \\to B/IB$ is of finite type,\n\\item there exists an ideal of definition $I \\subset A$ such that\nthe topology on $B$ is the $I$-adic topology and $B$ is in the category\n(\\ref{equation-C-prime}),\n\\item for all ideals of definition $I \\subset A$ the topology on $B$\nis the $I$-adic topology and $B$ is in the category (\\ref{equation-C-prime}),\n\\item $B$ as a topological $A$-algebra is the quotient of\n$A\\{x_1, \\ldots, x_r\\}$ by a closed ideal,\n\\item $B$ as a topological $A$-algebra is the quotient of\n$A[x_1, \\ldots, x_r]^\\wedge$ by a closed ideal where\n$A[x_1, \\ldots, x_r]^\\wedge$ is the completion of $A[x_1, \\ldots, x_r]$\nwith respect to some ideal of definition of $A$, and\n\\item add more here.\n\\end{enumerate}\nMoreover, these equivalent conditions define\na local property of morphisms of $\\text{WAdm}^{adic*}$ as defined in\nFormal Spaces, Remark \\ref{formal-spaces-remark-variant-adic-star}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBW","source_file":"restricted.tex","source_line":2221,"source_end_line":2250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2221-L2250","statement_sha256":"da8a4f41e74d128215fe846bf277e551937b0597a23a27d78962a942cc5e4a7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13117,"rank":13117,"depth":74,"x":1156.459,"y":1476.211,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBY","tag":"0GBY","title":"Finite type morphisms · Lemma 0GBY","summary":"Consider the property P on arrows of WAdm^adic* defined in Lemma [Tag 0GBW]. Then P is stable under base change as defined in Formal Spaces, Remark [Tag 0GBE].","statement_latex":"Consider the property $P$ on arrows of $\\textit{WAdm}^{adic*}$ defined in\nLemma \\ref{lemma-finite-type}. Then $P$ is stable under base change as\ndefined in Formal Spaces, Remark\n\\ref{formal-spaces-remark-base-change-variant-adic-star}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBY","source_file":"restricted.tex","source_line":2295,"source_end_line":2301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2295-L2301","statement_sha256":"a3222f1b3087024a803574cfffdedf07119b52712747f5d7c7b276b0f3f550ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":13118,"rank":13118,"depth":75,"x":1210.151,"y":1706.445,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GBZ","tag":"0GBZ","title":"Finite type morphisms · Lemma 0GBZ","summary":"Consider the property P on arrows of WAdm^adic* defined in Lemma [Tag 0GBW]. Then P is stable under composition as defined in Formal Spaces, Remark [Tag 0GBJ].","statement_latex":"Consider the property $P$ on arrows of $\\textit{WAdm}^{adic*}$ defined in\nLemma \\ref{lemma-finite-type}. Then $P$ is stable under composition as\ndefined in Formal Spaces, Remark\n\\ref{formal-spaces-remark-composition-variant-adic-star}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GBZ","source_file":"restricted.tex","source_line":2315,"source_end_line":2321,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2315-L2321","statement_sha256":"6c76cdb38e260a67dfec47d7b8c94df15a1e934d38e33bf946550f672abf454c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13119,"rank":13119,"depth":75,"x":985.092,"y":1566.942,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GC0","tag":"0GC0","title":"Finite type morphisms · Lemma 0GC0","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally adic* formal algebraic spaces over S. The following are equivalent • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to an arrow of WAdm^adic* which is adic and topologically of finite type, • there exists a covering (Y_j → Y) as in Formal Spaces,…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally adic* formal algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to an arrow of $\\textit{WAdm}^{adic*}$ which is\nadic and topologically of finite type,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $j$\na covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nsuch that each $X_{ji} \\to Y_j$  corresponds\nto an arrow of $\\textit{WAdm}^{adic*}$ which is adic and\ntopologically of finite type,\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds\nto an arrow of $\\textit{WAdm}^{adic*}$ which is adic and topologically\nof finite type, and\n\\item $f$ is locally of finite type.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GC0","source_file":"restricted.tex","source_line":2338,"source_end_line":2375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2338-L2375","statement_sha256":"64564cec7ddf70af749a0ec540c479298359e856efbd7a8f4bcefe9fd3f6f73b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13120,"rank":13120,"depth":75,"x":1263.626,"y":1542.07,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GC2","tag":"0GC2","title":"Finite type on reductions · Lemma 0GC2","summary":"For an arrow φ : A → B in WAdm^count consider the property P(φ)=\"the induced ring homomorphism A/ a → B/ b is of finite type\" where a ⊂ A and b ⊂ B are the ideals of topologically nilpotent elements. Then P is a local property as defined in Formal Spaces, Situation [Tag 0CBA].","statement_latex":"For an arrow $\\varphi : A \\to B$ in $\\text{WAdm}^{count}$ consider\nthe property $P(\\varphi)=$``the induced ring homomorphism\n$A/\\mathfrak a \\to B/\\mathfrak b$ is of finite type''\nwhere $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$ are the ideals\nof topologically nilpotent elements. Then $P$ is a local property\nas defined in\nFormal Spaces, Situation\n\\ref{formal-spaces-situation-local-property}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type on reductions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GC2","source_file":"restricted.tex","source_line":2413,"source_end_line":2423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2413-L2423","statement_sha256":"5c2aae18eca1f8b636aaf3684c6328413d569dfd2c71cb4290691630a66865aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13121,"rank":13121,"depth":61,"x":1077.978,"y":1718.706,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GC3","tag":"0GC3","title":"Finite type on reductions · Lemma 0GC3","summary":"Consider the property P on arrows of WAdm^count defined in Lemma [Tag 0GC2]. Then P is stable under base change (Formal Spaces, Situation [Tag 0GBC]).","statement_latex":"Consider the property $P$ on arrows of $\\textit{WAdm}^{count}$ defined in\nLemma \\ref{lemma-finite-type-red}. Then $P$ is stable under base change\n(Formal Spaces, Situation\n\\ref{formal-spaces-situation-base-change-local-property}).","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type on reductions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GC3","source_file":"restricted.tex","source_line":2479,"source_end_line":2485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2479-L2485","statement_sha256":"6ba1dc97eb18574c547d48cb639deb94daee6bcc348d7840744c14421803fb2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13122,"rank":13122,"depth":62,"x":1072.819,"y":1482.785,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GC4","tag":"0GC4","title":"Finite type on reductions · Lemma 0GC4","summary":"Consider the property P on arrows of WAdm^count defined in Lemma [Tag 0GC2]. Then P is stable under composition (Formal Spaces, Situation [Tag 0GBH]).","statement_latex":"Consider the property $P$ on arrows of $\\textit{WAdm}^{count}$ defined in\nLemma \\ref{lemma-finite-type-red}. Then $P$ is stable under composition\n(Formal Spaces, Situation\n\\ref{formal-spaces-situation-composition-local-property}).","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type on reductions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GC4","source_file":"restricted.tex","source_line":2508,"source_end_line":2514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2508-L2514","statement_sha256":"c7d7915cdae97610d61fd0edd287f9730eb4644062e7127aefb61ec5675ba896","origin":"The Stacks Project","memory_eligible":false,"source_rank":13123,"rank":13123,"depth":62,"x":1266.617,"y":1654.063,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GC5","tag":"0GC5","title":"Finite type on reductions · Lemma 0GC5","summary":"Let φ : A → B be an arrow of WAdm^count. If φ is taut and topologically of finite type, then φ satisfies the condition defined in Lemma [Tag 0GC2].","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{count}$.\nIf $\\varphi$ is taut and topologically of finite type, then $\\varphi$\nsatisfies the condition defined in Lemma \\ref{lemma-finite-type-red}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type on reductions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GC5","source_file":"restricted.tex","source_line":2520,"source_end_line":2525,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2520-L2525","statement_sha256":"97c502b76012e8fd29b3c3599d07f4928f26197756e01a6dd55a18872e5f1fc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13124,"rank":13124,"depth":62,"x":985.583,"y":1637.707,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GC6","tag":"0GC6","title":"Finite type on reductions · Lemma 0GC6","summary":"Let φ : A → B be an arrow of WAdm^Noeth satisfying the condition defined in Lemma [Tag 0GC2]. Then A → B is topologically of finite type.","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$\nsatisfying the condition defined in Lemma \\ref{lemma-finite-type-red}.\nThen $A \\to B$ is topologically of finite type.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type on reductions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GC6","source_file":"restricted.tex","source_line":2531,"source_end_line":2536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2531-L2536","statement_sha256":"6133c672a323fce186e16af1cc1cfc79b8f30225af63e8ea4077201a6cad83c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13125,"rank":13125,"depth":62,"x":1206.274,"y":1490.096,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GG6","tag":"0GG6","title":"Finite type on reductions · Lemma 0GG6","summary":"Let φ : A → B be an arrow of WAdm^Noeth. If φ is adic the following are equivalent • φ satisfies the condition defined in Lemma [Tag 0GC2] and • φ satisfies the condition defined in Lemma [Tag 0GBW].","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$.\nIf $\\varphi$ is adic the following are equivalent\n\\begin{enumerate}\n\\item $\\varphi$ satisfies the condition defined in\nLemma \\ref{lemma-finite-type-red} and\n\\item $\\varphi$ satisfies the condition defined in\nLemma \\ref{lemma-finite-type}.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type on reductions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GG6","source_file":"restricted.tex","source_line":2560,"source_end_line":2570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2560-L2570","statement_sha256":"44edccb1d872c7260986b52ad3a436f34441957e64485ea96a32d50a6864d692","origin":"The Stacks Project","memory_eligible":false,"source_rank":13126,"rank":13126,"depth":75,"x":1162.183,"y":1724.494,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GC7","tag":"0GC7","title":"Finite type on reductions · Lemma 0GC7","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally countably indexed formal algebraic spaces over S. The following are equivalent • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to an arrow of WAdm^count satisfying the property defined in Lemma [Tag 0GC2], • there exists a covering (Y_j → Y) as in…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally countably indexed formal algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to an arrow of $\\textit{WAdm}^{count}$ satisfying the\nproperty defined in Lemma \\ref{lemma-finite-type-red},\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $j$\na covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nsuch that each $X_{ji} \\to Y_j$  corresponds\nto an arrow of $\\textit{WAdm}^{count}$ satisfying the\nproperty defined in Lemma \\ref{lemma-finite-type-red},\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds\nto an arrow of $\\textit{WAdm}^{count}$ satisfying the\nproperty defined in Lemma \\ref{lemma-finite-type-red}, and\n\\item the morphism $f_{red} : X_{red} \\to Y_{red}$ is locally of finite type.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Finite type on reductions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GC7","source_file":"restricted.tex","source_line":2577,"source_end_line":2614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2577-L2614","statement_sha256":"f73b5e09ce8dd7c6103c3ca6f18a9ae1bd81e89e3faf8849e400fca54d9402a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13127,"rank":13127,"depth":70,"x":1005.982,"y":1526.359,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GC9","tag":"0GC9","title":"Flat morphisms · Lemma 0GC9","summary":"The property P(φ)=\"φ is flat\" on arrows of WAdm^Noeth is a local property as defined in Formal Spaces, Remark [Tag 0ANI].","statement_latex":"The property $P(\\varphi)=$``$\\varphi$ is flat'' on arrows\nof $\\textit{WAdm}^{Noeth}$ is a local property as defined in\nFormal Spaces, Remark \\ref{formal-spaces-remark-variant-Noetherian}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GC9","source_file":"restricted.tex","source_line":2672,"source_end_line":2677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2672-L2677","statement_sha256":"bca22e383f3340573b2763cf63ccf79d8fb51511d163c34088f9cc89861c115b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13128,"rank":13128,"depth":61,"x":1280.877,"y":1583.908,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCA","tag":"0GCA","title":"Flat morphisms · Lemma 0GCA","summary":"Denote P the property of arrows of WAdm^Noeth defined in Lemma [Tag 0GC9]. Denote Q the property defined in Lemma [Tag 0GC2] viewed as a property of arrows of WAdm^Noeth. Denote R the property defined in Lemma [Tag 0GBW] viewed as a property of arrows of WAdm^Noeth. Then • P is stable under base change by Q (Formal Spaces, Remark [Tag 0GBG]), and • P + R is stable under base change (Formal Spaces, Remark [Tag 0GBF]).","statement_latex":"Denote $P$ the property of arrows of $\\textit{WAdm}^{Noeth}$\ndefined in Lemma \\ref{lemma-flat-axioms}.\nDenote $Q$ the property defined in Lemma \\ref{lemma-finite-type-red}\nviewed as a property of arrows of $\\textit{WAdm}^{Noeth}$.\nDenote $R$ the property defined in Lemma \\ref{lemma-finite-type}\nviewed as a property of arrows of $\\textit{WAdm}^{Noeth}$. Then\n\\begin{enumerate}\n\\item $P$ is stable under base change by $Q$\n(Formal Spaces, Remark\n\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}), and\n\\item $P + R$ is stable under base change\n(Formal Spaces, Remark\n\\ref{formal-spaces-remark-base-change-variant-Noetherian}).\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCA","source_file":"restricted.tex","source_line":2733,"source_end_line":2749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2733-L2749","statement_sha256":"4d2eb119b2303f83aa42ee6b71205f7ecea4c920bf126cc6df3415eb6150afdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13129,"rank":13129,"depth":75,"x":1031.55,"y":1697.612,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCB","tag":"0GCB","title":"Flat morphisms · Lemma 0GCB","summary":"Denote P the property of arrows of WAdm^Noeth defined in Lemma [Tag 0GC9]. Then P is stable under composition (Formal Spaces, Remark [Tag 0GBK]).","statement_latex":"Denote $P$ the property of arrows of $\\textit{WAdm}^{Noeth}$\ndefined in Lemma \\ref{lemma-flat-axioms}.\nThen $P$ is stable under composition (Formal Spaces, Remark\n\\ref{formal-spaces-remark-composition-variant-Noetherian}).","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCB","source_file":"restricted.tex","source_line":2793,"source_end_line":2799,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2793-L2799","statement_sha256":"4712fc3922d2bbd59742e5ca055d2a5d36e39c969b73835a1795631a23db33b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13130,"rank":13130,"depth":62,"x":1124.091,"y":1471.984,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCC","tag":"0GCC","title":"Flat morphisms · Definition 0GCC","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. We say f is flat if for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a flat map of adic Noetherian topological rings.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally\nNoetherian formal algebraic spaces over $S$. We say $f$ is\n{\\it flat} if for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a flat map of adic Noetherian topological rings.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCC","source_file":"restricted.tex","source_line":2805,"source_end_line":2819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2805-L2819","statement_sha256":"8cf79e83c90296f1969e9543f85dfabe1289971057134722aa8d53ed59b4583c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13131,"rank":13131,"depth":0,"x":1237.452,"y":1691.169,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCD","tag":"0GCD","title":"Flat morphisms · Lemma 0GCD","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. The following are equivalent • f is flat, • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a flat map in WAdm^Noeth, • there exists a covering (Y_j → Y) as in Formal Spaces, Definition [Tag 0AIM] and for…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally Noetherian formal algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is flat,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a flat map in $\\textit{WAdm}^{Noeth}$,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $j$\na covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nsuch that each $X_{ji} \\to Y_j$  corresponds\nto a flat map in $\\textit{WAdm}^{Noeth}$, and\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds\nto a flat map in $\\textit{WAdm}^{Noeth}$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCD","source_file":"restricted.tex","source_line":2825,"source_end_line":2859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2825-L2859","statement_sha256":"3e7b05bd0494579532319149de066ccfbe801c64ab96d57cfef235748235486b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13132,"rank":13132,"depth":70,"x":977.242,"y":1593.735,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCE","tag":"0GCE","title":"Flat morphisms · Lemma 0GCE","summary":"Let S be a scheme. Let f : X → Y and g : Z → Y be morphisms of locally Noetherian formal algebraic spaces over S. • If f is flat and g_red : Z_red → Y_red is locally of finite type, then the base change X ×_Y Z → Z is flat. • If f is flat and locally of finite type, then the base change X ×_Y Z → Z is flat and locally of finite type.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\n\\begin{enumerate}\n\\item If $f$ is flat and $g_{red} : Z_{red} \\to Y_{red}$ is\nlocally of finite type, then the base change\n$X \\times_Y Z \\to Z$ is flat.\n\\item If $f$ is flat and locally of finite type, then\nthe base change $X \\times_Y Z \\to Z$ is flat and locally of finite type.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCE","source_file":"restricted.tex","source_line":2875,"source_end_line":2886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2875-L2886","statement_sha256":"dd1db0c625601615834566eaaac137c53f531321e5a718f16a470eb41fe887fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13133,"rank":13133,"depth":76,"x":1247.839,"y":1517.829,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCF","tag":"0GCF","title":"Flat morphisms · Lemma 0GCF","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of locally Noetherian formal algebraic spaces over S. If f and g are flat, then so is g ∘ f.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\nIf $f$ and $g$ are flat, then so is $g \\circ f$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCF","source_file":"restricted.tex","source_line":2905,"source_end_line":2910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2905-L2910","statement_sha256":"4c17f2d044066d3c9c71cd3c39305ff35056da61e75413963c47f85d756c8db9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13134,"rank":13134,"depth":63,"x":1109.158,"y":1727.63,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCG","tag":"0GCG","title":"Flat morphisms · Lemma 0GCG","summary":"Let S be a scheme. Let f : X → Y be a morphisms of locally Noetherian formal algebraic spaces over S. If f is representable by algebraic spaces and flat in the sense of Bootstrap, Definition [Tag 03XZ], then f is flat in the sense of Definition [Tag 0GCC].","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphisms of\nlocally Noetherian formal algebraic spaces over $S$.\nIf $f$ is representable by algebraic spaces and\nflat in the sense of Bootstrap, Definition\n\\ref{bootstrap-definition-property-transformation},\nthen $f$ is flat in the sense of Definition \\ref{definition-flat}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCG","source_file":"restricted.tex","source_line":2918,"source_end_line":2926,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L2918-L2926","statement_sha256":"3ebeeec368c2acb6179eae6d4a7530250c6536712a7481985f3814af441184d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13135,"rank":13135,"depth":73,"x":1042.614,"y":1493.918,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GG8","tag":"0GG8","title":"Rig-closed points · Lemma 0GG8","summary":"Let A be a Noetherian adic topological ring. Let q ⊂ A be a prime ideal. The following are equivalent • for some ideal of definition I ⊂ A we have I not ⊂ q and q is maximal with respect to this property, • for some ideal of definition I ⊂ A the prime q defines a closed point of Spec(A) setminus V(I), • for any ideal of definition I ⊂ A we have I not ⊂ q and q is maximal with respect to this property, • for any ideal of definition I ⊂ A the prime q defines a closed point…","statement_latex":"Let $A$ be a Noetherian adic topological ring. Let\n$\\mathfrak q \\subset A$ be a prime ideal. The following are\nequivalent\n\\begin{enumerate}\n\\item for some ideal of definition $I \\subset A$ we have\n$I \\not \\subset \\mathfrak q$ and $\\mathfrak q$ is maximal\nwith respect to this property,\n\\item for some ideal of definition $I \\subset A$ the prime\n$\\mathfrak q$ defines a closed point of $\\Spec(A) \\setminus V(I)$,\n\\item for any ideal of definition $I \\subset A$ we have\n$I \\not \\subset \\mathfrak q$ and $\\mathfrak q$ is maximal\nwith respect to this property,\n\\item for any ideal of definition $I \\subset A$ the prime\n$\\mathfrak q$ defines a closed point of $\\Spec(A) \\setminus V(I)$,\n\\item $\\dim(A/\\mathfrak q) = 1$ and for some ideal of definition\n$I \\subset A$ we have $I \\not \\subset \\mathfrak q$,\n\\item $\\dim(A/\\mathfrak q) = 1$ and for any ideal of definition\n$I \\subset A$ we have $I \\not \\subset \\mathfrak q$,\n\\item $\\dim(A/\\mathfrak q) = 1$ and the induced topology\non $A/\\mathfrak q$ is nontrivial,\n\\item $A/\\mathfrak q$ is a $1$-dimensional Noetherian complete local domain\nwhose maximal ideal is the radical of the image of any ideal of\ndefinition of $A$, and\n\\item add more here.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GG8","source_file":"restricted.tex","source_line":3004,"source_end_line":3031,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3004-L3031","statement_sha256":"d5e16f9e302b21bb8a2f6c6b2d4f42dd80f30ee3e3713b68bcb5852e90080c71","origin":"The Stacks Project","memory_eligible":false,"source_rank":13136,"rank":13136,"depth":49,"x":1279.948,"y":1628.677,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GG9","tag":"0GG9","title":"Rig-closed points · Definition 0GG9","summary":"Let A be a Noetherian adic topological ring. Let q ⊂ A be a prime ideal. We say q is rig-closed if the equivalent conditions of Lemma [Tag 0GG8] are satisfied.","statement_latex":"Let $A$ be a Noetherian adic topological ring. Let\n$\\mathfrak q \\subset A$ be a prime ideal. We say\n$\\mathfrak q$ is {\\it rig-closed} if the equivalent\nconditions of Lemma \\ref{lemma-rig-point} are satisfied.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GG9","source_file":"restricted.tex","source_line":3084,"source_end_line":3090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3084-L3090","statement_sha256":"2314d1a1ddf56bb525241e937e032b837d611d41b6d041bcfffdef18f905a0b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13137,"rank":13137,"depth":50,"x":996.189,"y":1664.025,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGA","tag":"0GGA","title":"Rig-closed points · Lemma 0GGA","summary":"Let φ : A → B in WAdm^Noeth. Denote a ⊂ A and b ⊂ B the ideals of topologically nilpotent elements. Assume A/ a → B/ b is of finite type. Let q ⊂ B be rig-closed. The residue field kappa of the local ring B/ q is a finite type A/ a-algebra.","statement_latex":"Let $\\varphi : A \\to B$ in $\\textit{WAdm}^{Noeth}$.\nDenote $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$\nthe ideals of topologically nilpotent elements. Assume\n$A/\\mathfrak a \\to B/\\mathfrak b$ is of finite type.\nLet $\\mathfrak q \\subset B$ be rig-closed.\nThe residue field $\\kappa$ of the local ring $B/\\mathfrak q$\nis a finite type $A/\\mathfrak a$-algebra.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGA","source_file":"restricted.tex","source_line":3096,"source_end_line":3105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3096-L3105","statement_sha256":"5ae352cec168b6e93379435c26b4fd8425653a66f98901012b04a2806b550910","origin":"The Stacks Project","memory_eligible":false,"source_rank":13138,"rank":13138,"depth":0,"x":1177.247,"y":1476.696,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGB","tag":"0GGB","title":"Rig-closed points · Lemma 0GGB","summary":"Let φ : A → B be an arrow of WAdm^Noeth which is adic and topologically of finite type. Let q ⊂ B be rig-closed. Let p = φ^-1( q) ⊂ A. Let a ⊂ A be the ideal of topologically nilpotent elements. The following are equivalent • the residue field kappa of B/ q is finite over A/ a, • p ⊂ A is rig-closed, • A/ p ⊂ B/ q is a finite extension of rings.","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$\nwhich is adic and topologically of finite type.\nLet $\\mathfrak q \\subset B$ be rig-closed.\nLet $\\mathfrak p = \\varphi^{-1}(\\mathfrak q) \\subset A$.\nLet $\\mathfrak a \\subset A$ be the ideal of topologically nilpotent\nelements.\nThe following are equivalent\n\\begin{enumerate}\n\\item the residue field $\\kappa$ of $B/\\mathfrak q$ is finite\nover $A/\\mathfrak a$,\n\\item $\\mathfrak p \\subset A$ is rig-closed,\n\\item $A/\\mathfrak p \\subset B/\\mathfrak q$ is a finite extension\nof rings.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGB","source_file":"restricted.tex","source_line":3115,"source_end_line":3131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3115-L3131","statement_sha256":"e2d2f96ca520ac4d2d4bd5ab4aa02605fabb180aa3c68c7069118885f94afd83","origin":"The Stacks Project","memory_eligible":false,"source_rank":13139,"rank":13139,"depth":63,"x":1194.405,"y":1717.89,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGC","tag":"0GGC","title":"Rig-closed points · Lemma 0GGC","summary":"Let φ : A → B be an arrow of WAdm^Noeth which is adic and topologically of finite type. Let q ⊂ B be rig-closed. If A/I is Jacobson for some ideal of definition I ⊂ A, then p = φ^-1( q) ⊂ A is rig-closed.","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$\nwhich is adic and topologically of finite type.\nLet $\\mathfrak q \\subset B$ be rig-closed.\nIf $A/I$ is Jacobson for some ideal of definition $I \\subset A$, then\n$\\mathfrak p = \\varphi^{-1}(\\mathfrak q) \\subset A$\nis rig-closed.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGC","source_file":"restricted.tex","source_line":3163,"source_end_line":3171,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3163-L3171","statement_sha256":"37c573475d23637d859628eb9a0b96229ca199cd4e458ecc1098ed5c6fe7c620","origin":"The Stacks Project","memory_eligible":false,"source_rank":13140,"rank":13140,"depth":64,"x":987.515,"y":1549.547,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGD","tag":"0GGD","title":"Rig-closed points · Lemma 0GGD","summary":"Let φ : A → B be an arrow of WAdm^Noeth which is adic and topologically of finite type. Let p ⊂ A be rig-closed. Let a ⊂ A and b ⊂ B be the ideals of topologically nilpotent elements. If φ is flat, then the following are equivalent • the maximal ideal of A/ p is in the image of Spec(B/ b) → Spec(A/ a), • there exists a rig-closed prime ideal q ⊂ B such that p = φ^-1( q). and if so then φ, p, and q satisfy the conclusions of Lemma [Tag 0GGB].","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$\nwhich is adic and topologically of finite type.\nLet $\\mathfrak p \\subset A$ be rig-closed.\nLet $\\mathfrak a \\subset A$ and $\\mathfrak b \\subset B$\nbe the ideals of topologically nilpotent elements. If $\\varphi$\nis flat, then the following are equivalent\n\\begin{enumerate}\n\\item the maximal ideal of $A/\\mathfrak p$ is in the image of\n$\\Spec(B/\\mathfrak b) \\to \\Spec(A/\\mathfrak a)$,\n\\item there exists a rig-closed prime ideal $\\mathfrak q \\subset B$\nsuch that $\\mathfrak p = \\varphi^{-1}(\\mathfrak q)$.\n\\end{enumerate}\nand if so then $\\varphi$, $\\mathfrak p$, and $\\mathfrak q$\nsatisfy the conclusions of Lemma \\ref{lemma-rig-closed-point-relative}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGD","source_file":"restricted.tex","source_line":3183,"source_end_line":3199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3183-L3199","statement_sha256":"c90779815c7e48eed8131ecccbdd9e76d4438d22afa93dd1a24a3608186019fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13141,"rank":13141,"depth":64,"x":1275.833,"y":1556.295,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGE","tag":"0GGE","title":"Rig-closed points · Definition 0GGE","summary":"Let A be an adic topological ring which has a finitely generated ideal of definition. Let f ∈ A. The completed principal localization A_(f) of A is the completion of A_f = A[1/f] of the principal localization of A at f with respect to any ideal of definition of A.","statement_latex":"Let $A$ be an adic topological ring which has a finitely generated ideal\nof definition. Let $f \\in A$. The {\\it completed principal localization}\n$A_{\\{f\\}}$ of $A$ is the completion of $A_f = A[1/f]$\nof the principal localization of $A$ at $f$ with respect to any\nideal of definition of $A$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGE","source_file":"restricted.tex","source_line":3229,"source_end_line":3236,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3229-L3236","statement_sha256":"47b9ae3db2b599dec7b7942b01696ea269d7e8b0d861d191ececf10f2d3ace81","origin":"The Stacks Project","memory_eligible":false,"source_rank":13142,"rank":13142,"depth":0,"x":1057.515,"y":1715.13,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGF","tag":"0GGF","title":"Rig-closed points · Lemma 0GGF","summary":"Let A be an adic Noetherian topological ring. Let p ⊂ A be a prime ideal. Let f ∈ A be an element mapping to a unit in A/ p. Then p A_(f) = p(A_f)^wedge = p ⊗_A (A_f)^wedge = ( p_f)^wedge is a prime ideal with quotient A/ p = (A/ p) ⊗_A (A_f)^wedge = (A_f)^wedge / p (A_f)^wedge = A_(f)/ p A_(f)","statement_latex":"Let $A$ be an adic Noetherian topological ring.\nLet $\\mathfrak p \\subset A$ be a prime ideal.\nLet $f \\in A$ be an element mapping to a unit in $A/\\mathfrak p$.\nThen\n$$\n\\mathfrak p A_{\\{f\\}} =\n\\mathfrak p(A_f)^\\wedge =\n\\mathfrak p \\otimes_A (A_f)^\\wedge =\n(\\mathfrak p_f)^\\wedge\n$$\nis a prime ideal with quotient\n$$\nA/\\mathfrak p = (A/\\mathfrak p) \\otimes_A (A_f)^\\wedge =\n(A_f)^\\wedge / \\mathfrak p (A_f)^\\wedge = A_{\\{f\\}}/\\mathfrak p A_{\\{f\\}}\n$$","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGF","source_file":"restricted.tex","source_line":3242,"source_end_line":3259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3242-L3259","statement_sha256":"f6f39589a30cb0f12fab0fb52a26a9f9c646c94adf0d44d53b46286fbe85c933","origin":"The Stacks Project","memory_eligible":false,"source_rank":13143,"rank":13143,"depth":0,"x":1090.814,"y":1473.808,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGG","tag":"0GGG","title":"Rig-closed points · Lemma 0GGG","summary":"Let φ : A → B be an arrow of WAdm^Noeth which is adic and topologically of finite type. Let q ⊂ B be rig-closed. There exists an f ∈ A which maps to a unit in B/ q such that we obtain a diagram vcenter xymatrix B ar[r] & B_(f) A ar[r] ar[u]_φ & A_(f) ar[u]_φ_(f) with primes vcenter xymatrix q ar@-[r] ar@-[d] & q' ar@-[d] ar@=[r] & q B_(f) p ar@-[r] & p' such that p' is rig-closed, i.e., the map A_(f) → B_(f) and the prime ideals q' and p' satisfy the equivalent conditions…","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$\nwhich is adic and topologically of finite type.\nLet $\\mathfrak q \\subset B$ be rig-closed.\nThere exists an $f \\in A$ which maps to a unit in\n$B/\\mathfrak q$ such that we obtain a diagram\n$$\n\\vcenter{\n\\xymatrix{\nB \\ar[r] &\nB_{\\{f\\}} \\\\\nA \\ar[r] \\ar[u]_\\varphi &\nA_{\\{f\\}} \\ar[u]_{\\varphi_{\\{f\\}}}\n}\n}\n\\quad\\text{with primes}\\quad\n\\vcenter{\n\\xymatrix{\n\\mathfrak q \\ar@{-}[r] \\ar@{-}[d] &\n\\mathfrak q' \\ar@{-}[d] \\ar@{=}[r] &\n\\mathfrak q B_{\\{f\\}} \\\\\n\\mathfrak p \\ar@{-}[r] &\n\\mathfrak p'\n}\n}\n$$\nsuch that $\\mathfrak p'$ is rig-closed, i.e.,\nthe map $A_{\\{f\\}} \\to B_{\\{f\\}}$ and the prime ideals\n$\\mathfrak q'$ and $\\mathfrak p'$ satisfy\nthe equivalent conditions of Lemma \\ref{lemma-rig-closed-point-relative}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGG","source_file":"restricted.tex","source_line":3270,"source_end_line":3301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3270-L3301","statement_sha256":"990a4af99863e52956a7585f84f79c4709790c0fb5fc6e2a337e2869ea4d9c35","origin":"The Stacks Project","memory_eligible":false,"source_rank":13144,"rank":13144,"depth":64,"x":1260.548,"y":1670.911,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGH","tag":"0GGH","title":"Rig-closed points · Lemma 0GGH","summary":"Let A be a Noetherian adic topological ring. Denote A(x_1, …, x_n) the restricted power series over A. Let q ⊂ A(x_1, …, x_n) be a prime ideal. Set q' = A[x_1, …, x_n] ∩ q and p = A ∩ q. If q and p are rig-closed, then the map A[x_1, …, x_n]_ q' → A(x_1, …, x_n)_ q defines an isomorphism on completions with respect to their maximal ideals.","statement_latex":"Let $A$ be a Noetherian adic topological ring. Denote $A\\{x_1, \\ldots, x_n\\}$\nthe restricted power series over $A$. Let\n$\\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}$ be a prime ideal.\nSet $\\mathfrak q' = A[x_1, \\ldots, x_n] \\cap \\mathfrak q$ and\n$\\mathfrak p = A \\cap \\mathfrak q$. If $\\mathfrak q$ and $\\mathfrak p$\nare rig-closed, then the map\n$$\nA[x_1, \\ldots, x_n]_{\\mathfrak q'}\n\\to\nA\\{x_1, \\ldots, x_n\\}_\\mathfrak q\n$$\ndefines an isomorphism on completions with respect to their maximal ideals.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGH","source_file":"restricted.tex","source_line":3320,"source_end_line":3334,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3320-L3334","statement_sha256":"baea0700432664aa9dc25ae5e60e0bd8fffb2501f64c3102c345add30245f519","origin":"The Stacks Project","memory_eligible":false,"source_rank":13145,"rank":13145,"depth":64,"x":976.509,"y":1621.817,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGI","tag":"0GGI","title":"Rig-closed points · Lemma 0GGI","summary":"Let φ : A → B be an arrow of WAdm^Noeth. Assume φ is adic, topologically of finite type, flat, and A/I → B/IB is étale for some (resp. any) ideal of definition I ⊂ A. Let q ⊂ B be rig-closed such that p = A ∩ q is rig-closed as well. Then p B_ q = q B_ q.","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$.\nAssume $\\varphi$ is adic, topologically of finite type, flat,\nand $A/I \\to B/IB$ is \\'etale for some (resp.\\ any)\nideal of definition $I \\subset A$. Let $\\mathfrak q \\subset B$\nbe rig-closed such that $\\mathfrak p = A \\cap \\mathfrak q$\nis rig-closed as well. Then\n$\\mathfrak p B_\\mathfrak q = \\mathfrak q B_\\mathfrak q$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGI","source_file":"restricted.tex","source_line":3358,"source_end_line":3367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3358-L3367","statement_sha256":"682bed2056de7711c65a5cb86873e96f12033e7a6d40287d33ab623842e03939","origin":"The Stacks Project","memory_eligible":false,"source_rank":13146,"rank":13146,"depth":40,"x":1225.763,"y":1496.683,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGJ","tag":"0GGJ","title":"Rig-closed points · Lemma 0GGJ","summary":"Let A be an adic Noetherian topological ring. Let p ⊂ A be a rig-closed prime. For any n ≥ 1 the ring map A/ p → A(x_1, …, x_n) ⊗_A A/ p = A/ p(x_1, …, x_n) is regular. In particular, the algebra A(x_1, …, x_n) ⊗_A kappa( p) is geometrically regular over kappa( p).","statement_latex":"Let $A$ be an adic Noetherian topological ring.\nLet $\\mathfrak p \\subset A$ be a rig-closed prime.\nFor any $n \\geq 1$ the ring map\n$$\nA/\\mathfrak p\n\\longrightarrow\nA\\{x_1, \\ldots, x_n\\} \\otimes_A A/\\mathfrak p =\nA/\\mathfrak p\\{x_1, \\ldots, x_n\\}\n$$\nis regular. In particular, the algebra\n$A\\{x_1, \\ldots, x_n\\} \\otimes_A \\kappa(\\mathfrak p)$\nis geometrically regular over $\\kappa(\\mathfrak p)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-closed points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGJ","source_file":"restricted.tex","source_line":3386,"source_end_line":3400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3386-L3400","statement_sha256":"095774b07e96f3e9ddcfa8063ad2aca21401c4c68dd92663f504f0e8ed9c7ac7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13147,"rank":13147,"depth":51,"x":1142.487,"y":1730.694,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGL","tag":"0GGL","title":"Rig-flat homomorphisms · Lemma 0GGL","summary":"Let φ : A → B be a morphism in WAdm^adic* (Formal Spaces, Section [Tag 0ANA]). Assume φ is adic. The following are equivalent: • B_f is flat over A for all topologically nilpotent f ∈ A, • B_g is flat over A for all topologically nilpotent g ∈ B, • B_ q is flat over A for all primes q ⊂ B which do not contain an ideal of definition, • B_ q is flat over A for every rig-closed prime q ⊂ B, and • add more here.","statement_latex":"Let $\\varphi : A \\to B$ be a morphism in $\\textit{WAdm}^{adic*}$\n(Formal Spaces, Section \\ref{formal-spaces-section-morphisms-rings}).\nAssume $\\varphi$ is adic. The following are equivalent:\n\\begin{enumerate}\n\\item $B_f$ is flat over $A$ for all\ntopologically nilpotent $f \\in A$,\n\\item $B_g$ is flat over $A$ for all\ntopologically nilpotent $g \\in B$,\n\\item $B_\\mathfrak q$ is flat over $A$\nfor all primes $\\mathfrak q \\subset B$ which do not contain\nan ideal of definition,\n\\item $B_\\mathfrak q$ is flat over $A$ for every rig-closed\nprime $\\mathfrak q \\subset B$, and\n\\item add more here.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGL","source_file":"restricted.tex","source_line":3432,"source_end_line":3449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3432-L3449","statement_sha256":"2d1e30e5ffb2384f3922c9454d08384fab83f862b2d22e2f3dcaffbf7c8e0787","origin":"The Stacks Project","memory_eligible":false,"source_rank":13148,"rank":13148,"depth":3,"x":1015.542,"y":1510.596,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGM","tag":"0GGM","title":"Rig-flat homomorphisms · Definition 0GGM","summary":"Let φ : A → B be a continuous ring homomorphism between adic Noetherian topological rings, i.e., φ is an arrow of WAdm^Noeth. We say φ is naively rig-flat if φ is adic, topologically of finite type, and satisfies the equivalent conditions of Lemma [Tag 0GGL].","statement_latex":"Let $\\varphi : A \\to B$ be a continuous ring homomorphism\nbetween adic Noetherian topological rings, i.e., $\\varphi$\nis an arrow of $\\textit{WAdm}^{Noeth}$. We say $\\varphi$ is\n{\\it naively rig-flat} if $\\varphi$ is adic, topologically\nof finite type, and satisfies the equivalent conditions of\nLemma \\ref{lemma-naively-rig-flat-continuous}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat homomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGM","source_file":"restricted.tex","source_line":3456,"source_end_line":3464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3456-L3464","statement_sha256":"770dab406dc66830fdc2d64889ae78a5adf4efe9de837bc4fa107e64545f1cba","origin":"The Stacks Project","memory_eligible":false,"source_rank":13149,"rank":13149,"depth":4,"x":1286.5,"y":1600.98,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGP","tag":"0GGP","title":"Rig-flat homomorphisms · Definition 0GGP","summary":"Let φ : A → B be a continuous ring homomorphism between adic Noetherian topological rings, i.e., φ is an arrow of WAdm^Noeth. We say φ is rig-flat if φ is adic, topologically of finite type, and for all f ∈ A the induced map A_(f) → B_(f) is naively rig-flat (Definition [Tag 0GGM]).","statement_latex":"Let $\\varphi : A \\to B$ be a continuous ring homomorphism between\nadic Noetherian topological rings, i.e., $\\varphi$ is an arrow of\n$\\textit{WAdm}^{Noeth}$. We say $\\varphi$ is {\\it rig-flat} if $\\varphi$\nis adic, topologically of finite type, and for all $f \\in A$ the induced map\n$$\nA_{\\{f\\}} \\longrightarrow B_{\\{f\\}}\n$$\nis naively rig-flat (Definition \\ref{definition-naively-rig-flat}).","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat homomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGP","source_file":"restricted.tex","source_line":3508,"source_end_line":3518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3508-L3518","statement_sha256":"9b3caa9e9981095b770349d30528c2a875d2d7df7f977ecc9efe8579a240f412","origin":"The Stacks Project","memory_eligible":false,"source_rank":13150,"rank":13150,"depth":5,"x":1013.66,"y":1688.194,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGQ","tag":"0GGQ","title":"Rig-flat homomorphisms · Lemma 0GGQ","summary":"Let φ : A → B be an arrow of WAdm^Noeth. If A/I is Jacobson for some (equivalently any) ideal of definition I ⊂ A and φ is naively rig-flat, then φ is rig-flat.","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$.\nIf $A/I$ is Jacobson for some (equivalently any) ideal of definition\n$I \\subset A$ and $\\varphi$ is naively rig-flat, then $\\varphi$ is\nrig-flat.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGQ","source_file":"restricted.tex","source_line":3526,"source_end_line":3532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3526-L3532","statement_sha256":"305e77bb8097907f493b969017a2681bbb34b3fe522eeeb4df171cdd86676110","origin":"The Stacks Project","memory_eligible":false,"source_rank":13151,"rank":13151,"depth":65,"x":1144.883,"y":1468.783,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGR","tag":"0GGR","title":"Rig-flat homomorphisms · Lemma 0GGR","summary":"Let φ : A → B and A → C be arrows of WAdm^Noeth. Assume φ is rig-flat and A → C adic and topologically of finite type. Then C → B widehat⊗_A C is rig-flat.","statement_latex":"Let $\\varphi : A \\to B$ and $A \\to C$ be arrows of $\\textit{WAdm}^{Noeth}$.\nAssume $\\varphi$ is rig-flat and $A \\to C$ adic and topologically\nof finite type. Then $C \\to B \\widehat{\\otimes}_A C$ is rig-flat.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGR","source_file":"restricted.tex","source_line":3583,"source_end_line":3588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3583-L3588","statement_sha256":"d1355cd6d58f80b1b18c673e6725f8b06ac191060a0635436d8ba7b8a24a851f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13152,"rank":13152,"depth":65,"x":1224.669,"y":1705.339,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGS","tag":"0GGS","title":"Rig-flat homomorphisms · Lemma 0GGS","summary":"Consider a commutative diagram xymatrix B ar[r] & B' A ar[r] ar[u]^φ & A' ar[u]_φ' in WAdm^Noeth with all arrows adic and topologically of finite type. Assume A → A' and B → B' are flat. Let I ⊂ A be an ideal of definition. If φ is rig-flat and A/I → A'/IA' is étale, then φ' is rig-flat.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nB \\ar[r] & B' \\\\\nA \\ar[r] \\ar[u]^\\varphi & A' \\ar[u]_{\\varphi'}\n}\n$$\nin $\\textit{WAdm}^{Noeth}$ with all arrows adic and topologically\nof finite type. Assume $A \\to A'$ and $B \\to B'$ are flat.\nLet $I \\subset A$ be an ideal of definition.\nIf $\\varphi$ is rig-flat and $A/I \\to A'/IA'$\nis \\'etale, then $\\varphi'$ is rig-flat.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGS","source_file":"restricted.tex","source_line":3674,"source_end_line":3688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3674-L3688","statement_sha256":"32449b7fe23da1e98abcee7047b88465a813597127876475f7197a5c8c671459","origin":"The Stacks Project","memory_eligible":false,"source_rank":13153,"rank":13153,"depth":65,"x":975.283,"y":1576.011,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGT","tag":"0GGT","title":"Rig-flat homomorphisms · Lemma 0GGT","summary":"Consider a commutative diagram xymatrix B ar[r] & B' A ar[r] ar[u]^φ & A' ar[u]_φ' in WAdm^Noeth with all arrows adic and topologically of finite type. Assume A → A' flat and B → B' faithfully flat. If φ' is rig-flat, then φ is rig-flat.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\nB \\ar[r] & B' \\\\\nA \\ar[r] \\ar[u]^\\varphi & A' \\ar[u]_{\\varphi'}\n}\n$$\nin $\\textit{WAdm}^{Noeth}$ with all arrows adic and topologically\nof finite type. Assume $A \\to A'$ flat and $B \\to B'$ faithfully flat.\nIf $\\varphi'$ is rig-flat, then $\\varphi$ is rig-flat.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGT","source_file":"restricted.tex","source_line":3757,"source_end_line":3769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3757-L3769","statement_sha256":"035b62c844e9665749136d412e9b7f95dd3b989a7a5d71a0668505cc6f0b475a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13154,"rank":13154,"depth":0,"x":1263.549,"y":1529.808,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGU","tag":"0GGU","title":"Rig-flat homomorphisms · Lemma 0GGU","summary":"The property P(φ)=\"φ is rig-flat\" on arrows of WAdm^Noeth is a local property as defined in Formal Spaces, Remark [Tag 0ANH].","statement_latex":"The property $P(\\varphi)=$``$\\varphi$ is rig-flat'' on arrows\nof $\\textit{WAdm}^{Noeth}$ is a local property as defined in\nFormal Spaces, Remark \\ref{formal-spaces-remark-variant-adic-star}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGU","source_file":"restricted.tex","source_line":3792,"source_end_line":3797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3792-L3797","statement_sha256":"0fc6c846d4111ea138a877d401286ed656d18f49f421dfbea5e1c97a48b52c42","origin":"The Stacks Project","memory_eligible":false,"source_rank":13155,"rank":13155,"depth":75,"x":1087.915,"y":1727.701,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGV","tag":"0GGV","title":"Rig-flat homomorphisms · Lemma 0GGV","summary":"The property P(φ)=\"φ is rig-flat\" on arrows of WAdm^Noeth is stable under composition as defined in Formal Spaces, Remark [Tag 0GBK].","statement_latex":"The property $P(\\varphi)=$``$\\varphi$ is rig-flat''\non arrows of $\\textit{WAdm}^{Noeth}$ is stable under composition\nas defined in Formal Spaces, Remark\n\\ref{formal-spaces-remark-composition-variant-Noetherian}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGV","source_file":"restricted.tex","source_line":3837,"source_end_line":3843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3837-L3843","statement_sha256":"38c58736be72290b4412185bf9f856a3f9b1f99d5b4456ff14a90143bb74c169","origin":"The Stacks Project","memory_eligible":false,"source_rank":13156,"rank":13156,"depth":76,"x":1058.247,"y":1481.804,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGX","tag":"0GGX","title":"Rig-flat morphisms · Definition 0GGX","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. We say f is rig-flat if for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a rig-flat map of adic Noetherian topological rings.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally\nNoetherian formal algebraic spaces over $S$. We say $f$ is\n{\\it rig-flat} if for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a rig-flat map of adic Noetherian topological rings.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGX","source_file":"restricted.tex","source_line":3877,"source_end_line":3891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3877-L3891","statement_sha256":"a99e1792b3a18ba9d5f3469300dc97e9da5f8f6cae8035584c7db829e3075901","origin":"The Stacks Project","memory_eligible":false,"source_rank":13157,"rank":13157,"depth":0,"x":1278.149,"y":1646.501,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGY","tag":"0GGY","title":"Rig-flat morphisms · Lemma 0GGY","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. The following are equivalent • f is rig-flat, • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a rig-flat map in WAdm^Noeth, • there exists a covering (Y_j → Y) as in Formal Spaces, Definition [Tag 0AIM] and…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally Noetherian formal algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is rig-flat,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a rig-flat map in $\\textit{WAdm}^{Noeth}$,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $j$\na covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nsuch that each $X_{ji} \\to Y_j$  corresponds\nto a rig-flat map in $\\textit{WAdm}^{Noeth}$, and\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds\nto a rig-flat map in $\\textit{WAdm}^{Noeth}$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGY","source_file":"restricted.tex","source_line":3897,"source_end_line":3931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3897-L3931","statement_sha256":"f6690620f631b07ffc19e4fdc60dba9acd434de7465d73d1f93da67aadcbb043","origin":"The Stacks Project","memory_eligible":false,"source_rank":13158,"rank":13158,"depth":76,"x":983.176,"y":1649.837,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GGZ","tag":"0GGZ","title":"Rig-flat morphisms · Lemma 0GGZ","summary":"Let S be a scheme. Let f : X → Y and g : Z → Y be morphisms of locally Noetherian formal algebraic spaces over S. If f is rig-flat and g is locally of finite type, then the base change X ×_Y Z → Z is rig-flat.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\nIf $f$ is rig-flat and $g$ is locally of finite type, then the base change\n$X \\times_Y Z \\to Z$ is rig-flat.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GGZ","source_file":"restricted.tex","source_line":3947,"source_end_line":3953,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3947-L3953","statement_sha256":"10e59314304b209ef7eb1b732b0564bf672646504e3884c176b2e44a21ea95b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13159,"rank":13159,"depth":66,"x":1198.275,"y":1479.786,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GH0","tag":"0GH0","title":"Rig-flat morphisms · Lemma 0GH0","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of locally Noetherian formal algebraic spaces over S. If f and g are rig-flat, then so is g ∘ f.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\nIf $f$ and $g$ are rig-flat, then so is $g \\circ f$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GH0","source_file":"restricted.tex","source_line":3964,"source_end_line":3969,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3964-L3969","statement_sha256":"768bdd1b77b9e3f2fe98c93d1ef4541ef00092dbbd0df3019b95e787e0234875","origin":"The Stacks Project","memory_eligible":false,"source_rank":13160,"rank":13160,"depth":77,"x":1176.386,"y":1727.546,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCI","tag":"0GCI","title":"Rig-smooth homomorphisms · Lemma 0GCI","summary":"Let A → B be a morphism in WAdm^Noeth (Formal Spaces, Section [Tag 0ANA]). The following are equivalent: • [(a)] A → B satisfies the equivalent conditions of Lemma [Tag 0GBW] and there exists an ideal of definition I ⊂ B such that B is rig-smooth over (A, I), and • [(b)] A → B satisfies the equivalent conditions of Lemma [Tag 0GBW] and for all ideals of definition I ⊂ A the algebra B is rig-smooth over (A, I).","statement_latex":"Let $A \\to B$ be a morphism in $\\textit{WAdm}^{Noeth}$\n(Formal Spaces, Section \\ref{formal-spaces-section-morphisms-rings}).\nThe following are equivalent:\n\\begin{enumerate}\n\\item[(a)] $A \\to B$ satisfies the equivalent conditions of\nLemma \\ref{lemma-finite-type} and there exists an ideal of definition\n$I \\subset B$ such that $B$ is rig-smooth over $(A, I)$, and\n\\item[(b)] $A \\to B$ satisfies the equivalent conditions of\nLemma \\ref{lemma-finite-type} and for all ideals of definition\n$I \\subset A$ the algebra $B$ is rig-smooth over $(A, I)$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCI","source_file":"restricted.tex","source_line":3996,"source_end_line":4009,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L3996-L4009","statement_sha256":"b2e4bdff426787892c9b1dbb50a8bd61c335574ae20e34273474273db3ee05d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13161,"rank":13161,"depth":75,"x":993.054,"y":1532.184,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCJ","tag":"0GCJ","title":"Rig-smooth homomorphisms · Definition 0GCJ","summary":"Let φ : A → B be a continuous ring homomorphism between adic Noetherian topological rings, i.e., φ is an arrow of WAdm^Noeth. We say φ is rig-smooth if the equivalent conditions of Lemma [Tag 0GCI] hold.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous ring homomorphism\nbetween adic Noetherian topological rings, i.e., $\\varphi$\nis an arrow of $\\textit{WAdm}^{Noeth}$. We say\n$\\varphi$ is {\\it rig-smooth} if the equivalent conditions\nof Lemma \\ref{lemma-rig-smooth-continuous} hold.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth homomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCJ","source_file":"restricted.tex","source_line":4023,"source_end_line":4030,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4023-L4030","statement_sha256":"595f446ee53dd574d0d19af960ac70dbb5ad732ef43a0b404a8afdabe3214743","origin":"The Stacks Project","memory_eligible":false,"source_rank":13162,"rank":13162,"depth":76,"x":1285.714,"y":1572.264,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCK","tag":"0GCK","title":"Rig-smooth homomorphisms · Lemma 0GCK","summary":"The property P(φ)=\"φ is rig-smooth\" on arrows of WAdm^Noeth is a local property as defined in Formal Spaces, Remark [Tag 0ANI].","statement_latex":"The property $P(\\varphi)=$``$\\varphi$ is rig-smooth'' on arrows\nof $\\textit{WAdm}^{Noeth}$ is a local property as defined in\nFormal Spaces, Remark \\ref{formal-spaces-remark-variant-Noetherian}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCK","source_file":"restricted.tex","source_line":4035,"source_end_line":4040,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4035-L4040","statement_sha256":"dfcf91377fa3d29afe99f6ea09d31dbb4c2b8dbe135d6736e69ba7be004327af","origin":"The Stacks Project","memory_eligible":false,"source_rank":13163,"rank":13163,"depth":75,"x":1037.367,"y":1708.944,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCL","tag":"0GCL","title":"Rig-smooth homomorphisms · Lemma 0GCL","summary":"Consider the properties P(φ)=\"φ is rig-smooth\" and Q(φ)=\"φ is adic\" on arrows of WAdm^Noeth. Then P is stable under base change by Q as defined in Formal Spaces, Remark [Tag 0GBG].","statement_latex":"Consider the properties $P(\\varphi)=$``$\\varphi$ is rig-smooth''\nand $Q(\\varphi)$=``$\\varphi$ is adic'' on arrows of $\\textit{WAdm}^{Noeth}$.\nThen $P$ is stable under base change by $Q$ as defined in\nFormal Spaces, Remark\n\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCL","source_file":"restricted.tex","source_line":4153,"source_end_line":4160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4153-L4160","statement_sha256":"b71b9bbe118b25829d61f2ad6bcb197abc6f3c0fb5af365bb81316ad4169b942","origin":"The Stacks Project","memory_eligible":false,"source_rank":13164,"rank":13164,"depth":76,"x":1110.672,"y":1466.937,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCM","tag":"0GCM","title":"Rig-smooth homomorphisms · Lemma 0GCM","summary":"The property P(φ)=\"φ is rig-smooth\" on arrows of WAdm^Noeth is stable under composition as defined in Formal Spaces, Remark [Tag 0GBK].","statement_latex":"The property $P(\\varphi)=$``$\\varphi$ is rig-smooth''\non arrows of $\\textit{WAdm}^{Noeth}$ is stable under composition\nas defined in Formal Spaces, Remark\n\\ref{formal-spaces-remark-composition-variant-Noetherian}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCM","source_file":"restricted.tex","source_line":4176,"source_end_line":4182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4176-L4182","statement_sha256":"51d38f627ed67af1c9ac313c9e2aa57bac989e773177b7d2813c558caf9093f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13165,"rank":13165,"depth":76,"x":1251.41,"y":1687.261,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GH1","tag":"0GH1","title":"Rig-smooth homomorphisms · Lemma 0GH1","summary":"Let φ : A → B be an arrow of WAdm^Noeth. If φ is rig-smooth, then φ is rig-flat, and for any presentation B = A(x_1, …, x_n)/J and prime J ⊂ q ⊂ A(x_1, …, x_n) not containing an ideal of definition the ideal J_ q ⊂ A(x_1, …, x_n)_ q is generated by a regular sequence.","statement_latex":"Let $\\varphi : A \\to B$ be an arrow of $\\textit{WAdm}^{Noeth}$.\nIf $\\varphi$ is rig-smooth, then $\\varphi$ is rig-flat, and\nfor any presentation $B = A\\{x_1, \\ldots, x_n\\}/J$\nand prime $J \\subset \\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}$\nnot containing an ideal of definition the ideal\n$J_\\mathfrak q \\subset A\\{x_1, \\ldots, x_n\\}_\\mathfrak q$\nis generated by a regular sequence.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GH1","source_file":"restricted.tex","source_line":4242,"source_end_line":4251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4242-L4251","statement_sha256":"6de18c6a8e713009134f8d6d8649a6b97c3120023e638c9bfa931d8e257bd95f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13166,"rank":13166,"depth":77,"x":970.103,"y":1604.552,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GH2","tag":"0GH2","title":"Rig-smooth homomorphisms · Lemma 0GH2","summary":"Let A → B → C be arrows in WAdm^Noeth which are adic and topologically of finite type. If B → C is rig-smooth, then the kernel of the map H^-1(NL_B/A^wedge ⊗_B C) → H^-1(NL_C/A^wedge) (see Lemma [Tag 0ALM]) is annihilated by an ideal of definition.","statement_latex":"Let $A \\to B \\to C$ be arrows in $\\textit{WAdm}^{Noeth}$\nwhich are adic and topologically of finite type. If $B \\to C$\nis rig-smooth, then the kernel of the map\n$$\nH^{-1}(\\NL_{B/A}^\\wedge \\otimes_B C) \\to H^{-1}(\\NL_{C/A}^\\wedge)\n$$\n(see Lemma \\ref{lemma-exact-sequence-NL})\nis annihilated by an ideal of definition.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GH2","source_file":"restricted.tex","source_line":4343,"source_end_line":4353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4343-L4353","statement_sha256":"65d1dda0590226a313c632d40e3f33f97404d9fa3f1533a0d4784ce67e554ecc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13167,"rank":13167,"depth":78,"x":1244.388,"y":1505.797,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCP","tag":"0GCP","title":"Rig-smooth morphisms · Definition 0GCP","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. We say f is rig-smooth if for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a rig-smooth map of adic Noetherian topological rings.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally\nNoetherian formal algebraic spaces over $S$. We say $f$ is\n{\\it rig-smooth} if for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a rig-smooth map of adic Noetherian topological rings.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCP","source_file":"restricted.tex","source_line":4404,"source_end_line":4418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4404-L4418","statement_sha256":"aa9917b0186a0c03f60ec92beff880c8cee6d96acbb7090262a97d5afa64fa8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13168,"rank":13168,"depth":0,"x":1121.397,"y":1734.537,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCQ","tag":"0GCQ","title":"Rig-smooth morphisms · Lemma 0GCQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. The following are equivalent • f is rig-smooth, • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a rig-smooth map in WAdm^Noeth, • there exists a covering (Y_j → Y) as in Formal Spaces, Definition [Tag 0AIM]…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally Noetherian formal algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is rig-smooth,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a rig-smooth map in $\\textit{WAdm}^{Noeth}$,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $j$\na covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nsuch that each $X_{ji} \\to Y_j$  corresponds\nto a rig-smooth map in $\\textit{WAdm}^{Noeth}$, and\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds\nto a rig-smooth map in $\\textit{WAdm}^{Noeth}$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCQ","source_file":"restricted.tex","source_line":4424,"source_end_line":4458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4424-L4458","statement_sha256":"98145224daa44716148ceafd2446573215005ad4116200d2d4d31efe8cfa2e1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13169,"rank":13169,"depth":76,"x":1028.026,"y":1495.784,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCR","tag":"0GCR","title":"Rig-smooth morphisms · Lemma 0GCR","summary":"Let S be a scheme. Let f : X → Y and g : Z → Y be morphisms of locally Noetherian formal algebraic spaces over S. If f is rig-smooth and g is adic, then the base change X ×_Y Z → Z is rig-smooth.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\nIf $f$ is rig-smooth and $g$ is adic, then the base change\n$X \\times_Y Z \\to Z$ is rig-smooth.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCR","source_file":"restricted.tex","source_line":4474,"source_end_line":4480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4474-L4480","statement_sha256":"290608a0c2c418088460d3940c54e855576254a803416d9b7c1105db9f4237d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13170,"rank":13170,"depth":77,"x":1289.198,"y":1619.008,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCS","tag":"0GCS","title":"Rig-smooth morphisms · Lemma 0GCS","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of locally Noetherian formal algebraic spaces over S. If f and g are rig-smooth, then so is g ∘ f.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\nIf $f$ and $g$ are rig-smooth, then so is $g \\circ f$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCS","source_file":"restricted.tex","source_line":4490,"source_end_line":4495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4490-L4495","statement_sha256":"1791fc20a96a0dc18e8bf813535e0310126506cc77c24ce6a822b141ebfb279a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13171,"rank":13171,"depth":77,"x":997.16,"y":1676.41,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GH3","tag":"0GH3","title":"Rig-smooth morphisms · Lemma 0GH3","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. If f is rig-smooth, then f is rig-flat.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally Noetherian formal algebraic spaces over $S$.\nIf $f$ is rig-smooth, then $f$ is rig-flat.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GH3","source_file":"restricted.tex","source_line":4503,"source_end_line":4508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4503-L4508","statement_sha256":"6c42f99adf95355778052f91b707f0dbad44dafc074670c4e23227b59cbd75c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13172,"rank":13172,"depth":78,"x":1166.552,"y":1468.118,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCU","tag":"0GCU","title":"Rig-étale homomorphisms · Lemma 0GCU","summary":"Let A → B be a morphism in WAdm^Noeth (Formal Spaces, Section [Tag 0ANA]). The following are equivalent: • [(a)] A → B satisfies the equivalent conditions of Lemma [Tag 0GBW] and there exists an ideal of definition I ⊂ B such that B is rig-étale over (A, I), and • [(b)] A → B satisfies the equivalent conditions of Lemma [Tag 0GBW] and for all ideals of definition I ⊂ A the algebra B is rig-étale over (A, I).","statement_latex":"Let $A \\to B$ be a morphism in $\\textit{WAdm}^{Noeth}$\n(Formal Spaces, Section \\ref{formal-spaces-section-morphisms-rings}).\nThe following are equivalent:\n\\begin{enumerate}\n\\item[(a)] $A \\to B$ satisfies the equivalent conditions of\nLemma \\ref{lemma-finite-type} and there exists an ideal of definition\n$I \\subset B$ such that $B$ is rig-\\'etale over $(A, I)$, and\n\\item[(b)] $A \\to B$ satisfies the equivalent conditions of\nLemma \\ref{lemma-finite-type} and for all ideals of definition\n$I \\subset A$ the algebra $B$ is rig-\\'etale over $(A, I)$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCU","source_file":"restricted.tex","source_line":4531,"source_end_line":4544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4531-L4544","statement_sha256":"0309840c579c51b71f2e6a614cda7dfdb06db1561cb9c5413aac4f8df2a0a351","origin":"The Stacks Project","memory_eligible":false,"source_rank":13173,"rank":13173,"depth":75,"x":1209.2,"y":1718.133,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCV","tag":"0GCV","title":"Rig-étale homomorphisms · Definition 0GCV","summary":"Let φ : A → B be a continuous ring homomorphism between adic Noetherian topological rings, i.e., φ is an arrow of WAdm^Noeth. We say φ is rig-etale if the equivalent conditions of Lemma [Tag 0GCU] hold.","statement_latex":"Let $\\varphi : A \\to B$ be a continuous ring homomorphism\nbetween adic Noetherian topological rings, i.e., $\\varphi$\nis an arrow of $\\textit{WAdm}^{Noeth}$. We say\n$\\varphi$ is {\\it rig-etale} if the equivalent conditions\nof Lemma \\ref{lemma-rig-etale-continuous} hold.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale homomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCV","source_file":"restricted.tex","source_line":4558,"source_end_line":4565,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4558-L4565","statement_sha256":"3bad5143dfb1486e4da3297d7873aaa9972e35d92a406bddc47b8529c6395d13","origin":"The Stacks Project","memory_eligible":false,"source_rank":13174,"rank":13174,"depth":76,"x":976.412,"y":1557.779,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQL","tag":"0AQL","title":"Rig-étale homomorphisms · Lemma 0AQL","summary":"The property P(φ)=\"φ is rig-étale\" on arrows of WAdm^Noeth is a local property as defined in Formal Spaces, Remark [Tag 0ANI].","statement_latex":"The property $P(\\varphi)=$``$\\varphi$ is rig-\\'etale'' on arrows\nof $\\textit{WAdm}^{Noeth}$ is a local property as defined in\nFormal Spaces, Remark \\ref{formal-spaces-remark-variant-Noetherian}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQL","source_file":"restricted.tex","source_line":4570,"source_end_line":4575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4570-L4575","statement_sha256":"922f849d8515b19a34b3b4cd3647a0de16856ab7eeb1e1f0bcd9401464342359","origin":"The Stacks Project","memory_eligible":false,"source_rank":13175,"rank":13175,"depth":76,"x":1277.386,"y":1543.915,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCW","tag":"0GCW","title":"Rig-étale homomorphisms · Lemma 0GCW","summary":"Consider the properties P(φ)=\"φ is rig-étale\" and Q(φ)=\"φ is adic\" on arrows of WAdm^Noeth. Then P is stable under base change by Q as defined in Formal Spaces, Remark [Tag 0GBG].","statement_latex":"Consider the properties $P(\\varphi)=$``$\\varphi$ is rig-\\'etale''\nand $Q(\\varphi)$=``$\\varphi$ is adic'' on arrows of $\\textit{WAdm}^{Noeth}$.\nThen $P$ is stable under base change by $Q$ as defined in\nFormal Spaces, Remark\n\\ref{formal-spaces-remark-base-change-variant-variant-Noetherian}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCW","source_file":"restricted.tex","source_line":4682,"source_end_line":4689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4682-L4689","statement_sha256":"27ec4794053516cdd5e3684ea9f29095b95445caf24a692beea2a8242cd8f3d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13176,"rank":13176,"depth":76,"x":1066.344,"y":1725.138,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCX","tag":"0GCX","title":"Rig-étale homomorphisms · Lemma 0GCX","summary":"The property P(φ)=\"φ is rig-étale\" on arrows of WAdm^Noeth is stable under composition as defined in Formal Spaces, Remark [Tag 0GBK].","statement_latex":"The property $P(\\varphi)=$``$\\varphi$ is rig-\\'etale''\non arrows of $\\textit{WAdm}^{Noeth}$ is stable under composition\nas defined in Formal Spaces, Remark\n\\ref{formal-spaces-remark-composition-variant-Noetherian}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCX","source_file":"restricted.tex","source_line":4705,"source_end_line":4711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4705-L4711","statement_sha256":"30a487e1010ee0c78a8bf939fb80309f8132fd319cc1f3f826b319ce6cbe6f72","origin":"The Stacks Project","memory_eligible":false,"source_rank":13177,"rank":13177,"depth":76,"x":1076.241,"y":1471.449,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCY","tag":"0GCY","title":"Rig-étale homomorphisms · Lemma 0GCY","summary":"The property P(φ)=\"φ is rig-étale\" on arrows of WAdm^Noeth has the cancellation property as defined in Formal Spaces, Remark [Tag 0GBP].","statement_latex":"The property $P(\\varphi)=$``$\\varphi$ is rig-\\'etale''\non arrows of $\\textit{WAdm}^{Noeth}$ has the cancellation property\nas defined in Formal Spaces, Remark\n\\ref{formal-spaces-remark-permanence-variant-Noetherian}.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale homomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCY","source_file":"restricted.tex","source_line":4747,"source_end_line":4753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4747-L4753","statement_sha256":"1a3370a5b4dde94e29a9d7e77be0b4b8e71bbae232a69bb00d860ab51e858736","origin":"The Stacks Project","memory_eligible":false,"source_rank":13178,"rank":13178,"depth":76,"x":1273.191,"y":1664.365,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQM","tag":"0AQM","title":"Rig-étale morphisms · Definition 0AQM","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. We say f is rig-étale if for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a rig-étale map of adic Noetherian topological rings.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally\nNoetherian formal algebraic spaces over $S$. We say $f$ is\n{\\it rig-\\'etale} if for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a rig-\\'etale map of adic Noetherian topological rings.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQM","source_file":"restricted.tex","source_line":4809,"source_end_line":4823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4809-L4823","statement_sha256":"0d6381ded9aa0b846132112bb0f60e0893953fd15d38140732ae0451f48104c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13179,"rank":13179,"depth":0,"x":972.462,"y":1633.829,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GCZ","tag":"0GCZ","title":"Rig-étale morphisms · Lemma 0GCZ","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. The following are equivalent • f is rig-étale, • for every commutative diagram xymatrix U ar[d] ar[r] & V ar[d] X ar[r] & Y with U and V affine formal algebraic spaces, U → X and V → Y representable by algebraic spaces and étale, the morphism U → V corresponds to a rig-étale map in WAdm^Noeth, • there exists a covering (Y_j → Y) as in Formal Spaces, Definition [Tag 0AIM]…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally Noetherian formal algebraic spaces over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is rig-\\'etale,\n\\item for every commutative diagram\n$$\n\\xymatrix{\nU \\ar[d] \\ar[r] & V \\ar[d] \\\\\nX \\ar[r] & Y\n}\n$$\nwith $U$ and $V$ affine formal algebraic spaces, $U \\to X$ and $V \\to Y$\nrepresentable by algebraic spaces and \\'etale, the morphism $U \\to V$\ncorresponds to a rig-\\'etale map in $\\textit{WAdm}^{Noeth}$,\n\\item there exists a covering $\\{Y_j \\to Y\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $j$\na covering $\\{X_{ji} \\to Y_j \\times_Y X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nsuch that each $X_{ji} \\to Y_j$  corresponds\nto a rig-\\'etale map in $\\textit{WAdm}^{Noeth}$, and\n\\item there exist a covering $\\{X_i \\to X\\}$ as in\nFormal Spaces,\nDefinition \\ref{formal-spaces-definition-formal-algebraic-space}\nand for each $i$ a factorization $X_i \\to Y_i \\to Y$ where $Y_i$\nis an affine formal algebraic space, $Y_i \\to Y$ is representable\nby algebraic spaces and \\'etale, and $X_i \\to Y_i$ corresponds\nto a rig-\\'etale map in $\\textit{WAdm}^{Noeth}$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GCZ","source_file":"restricted.tex","source_line":4829,"source_end_line":4863,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4829-L4863","statement_sha256":"ccafc487218d556d962735231bf30f5a0d70e61c4aa9f5e44549e41d0e09a3b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13180,"rank":13180,"depth":77,"x":1219.068,"y":1485.526,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQN","tag":"0AQN","title":"Rig-étale morphisms · Lemma 0AQN","summary":"A rig-étale morphism of locally Noetherian formal algebraic spaces is locally of finite type.","statement_latex":"A rig-\\'etale morphism of locally Noetherian formal algebraic spaces\nis locally of finite type.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQN","source_file":"restricted.tex","source_line":4881,"source_end_line":4885,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4881-L4885","statement_sha256":"46a951dbfa0cb5fd80afec867855525c22c1c32dfc25c0eacac3b4e30491d5a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13181,"rank":13181,"depth":77,"x":1156.411,"y":1735.113,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GD0","tag":"0GD0","title":"Rig-étale morphisms · Lemma 0GD0","summary":"A rig-étale morphism of locally Noetherian formal algebraic spaces is rig-smooth.","statement_latex":"A rig-\\'etale morphism of locally Noetherian formal algebraic spaces\nis rig-smooth.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GD0","source_file":"restricted.tex","source_line":4896,"source_end_line":4900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4896-L4900","statement_sha256":"b1c9b565e3fe8f287fa6ff57f875d48a6c1e981bd2570e64fdeb431f47485c56","origin":"The Stacks Project","memory_eligible":false,"source_rank":13182,"rank":13182,"depth":2,"x":1001.718,"y":1515.254,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GD1","tag":"0GD1","title":"Rig-étale morphisms · Lemma 0GD1","summary":"Let S be a scheme. Let f : X → Y and g : Z → Y be morphisms of locally Noetherian formal algebraic spaces over S. If f is rig-étale and g is adic, then the base change X ×_Y Z → Z is rig-étale.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\nIf $f$ is rig-\\'etale and $g$ is adic, then the base change\n$X \\times_Y Z \\to Z$ is rig-\\'etale.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GD1","source_file":"restricted.tex","source_line":4907,"source_end_line":4913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4907-L4913","statement_sha256":"684c4b99844c3ed3d4a9b9237678990ee809e24d1c3130eb53aff5bb919c3705","origin":"The Stacks Project","memory_eligible":false,"source_rank":13183,"rank":13183,"depth":77,"x":1292.938,"y":1589.687,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GD2","tag":"0GD2","title":"Rig-étale morphisms · Lemma 0GD2","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of locally Noetherian formal algebraic spaces over S. If f and g are rig-étale, then so is g ∘ f.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\nIf $f$ and $g$ are rig-\\'etale, then so is $g \\circ f$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GD2","source_file":"restricted.tex","source_line":4923,"source_end_line":4928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4923-L4928","statement_sha256":"ea055c04b4c262916230346357b7000dceaf35f45790860054799e2296f2119a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13184,"rank":13184,"depth":77,"x":1018.012,"y":1700.179,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GD3","tag":"0GD3","title":"Rig-étale morphisms · Lemma 0GD3","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be a morphism of locally Noetherian formal algebraic spaces over S. If g ∘ f and g are rig-étale, then so is f.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$\nbe a morphism of locally Noetherian formal algebraic spaces over $S$.\nIf $g \\circ f$ and $g$ are rig-\\'etale, then so is $f$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GD3","source_file":"restricted.tex","source_line":4936,"source_end_line":4941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4936-L4941","statement_sha256":"56a124b6cf5ac67fb7d4f715699aa3e99ddcaab23d5fd7d385e243aed930adbd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13185,"rank":13185,"depth":77,"x":1132.02,"y":1462.421,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GH4","tag":"0GH4","title":"Rig-étale morphisms · Lemma 0GH4","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of locally Noetherian formal algebraic spaces over S. If g ∘ f is rig-étale and g is an adic monomorphism, then f is rig-étale.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\nIf $g \\circ f$ is rig-\\'etale and $g$ is an adic monomorphism, then\n$f$ is rig-\\'etale.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GH4","source_file":"restricted.tex","source_line":4949,"source_end_line":4955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4949-L4955","statement_sha256":"2d80142d25ee75a75bb44514275c6e674ad793ed5d4eb3f17a200edd54e6ddd1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13186,"rank":13186,"depth":78,"x":1239.277,"y":1702.716,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GD4","tag":"0GD4","title":"Rig-étale morphisms · Lemma 0GD4","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces. Assume that X and Y are locally Noetherian and f is a closed immersion. The following are equivalent • f is rig-smooth, • f is rig-étale, • for every affine formal algebraic space V and every morphism V → Y which is representable by algebraic spaces and étale the morphism X ×_Y V → V corresponds to a surjective morphism B → A in WAdm^Noeth whose kernel J has the following property: I(J/J^2) = 0 for…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a\nclosed immersion. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is rig-smooth,\n\\item $f$ is rig-\\'etale,\n\\item for every affine formal algebraic space $V$ and every morphism\n$V \\to Y$ which is representable by algebraic spaces and \\'etale\nthe morphism $X \\times_Y V \\to V$ corresponds to a surjective morphism\n$B \\to A$ in $\\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following\nproperty: $I(J/J^2) = 0$ for some ideal of definition $I$ of $B$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GD4","source_file":"restricted.tex","source_line":4962,"source_end_line":4976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L4962-L4976","statement_sha256":"3bb81df5786c9cf6fd5cdf089a4bdfb4d9c0855e3f7478cd253d70cff799941d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13187,"rank":13187,"depth":78,"x":966.626,"y":1586.249,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQQ","tag":"0AQQ","title":"Rig-surjective morphisms · Definition 0AQQ","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces over S. Assume that X and Y are locally Noetherian and that f is locally of finite type. We say f is rig-surjective if for every solid diagram xymatrix Spf(R') ar@..>[r] ar@..>[d] & X ar[d]^f Spf(R) ar[r]^-p & Y where R is a complete discrete valuation ring and where p is an adic morphism there exists an extension of complete discrete valuation rings R ⊂ R' and a morphism Spf(R') → X making the…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal\nalgebraic spaces over $S$. Assume that $X$ and $Y$ are locally\nNoetherian and that $f$ is locally of finite type. We say\n$f$ is {\\it rig-surjective} if for every solid diagram\n$$\n\\xymatrix{\n\\text{Spf}(R') \\ar@{..>}[r] \\ar@{..>}[d] & X \\ar[d]^f \\\\\n\\text{Spf}(R) \\ar[r]^-p & Y\n}\n$$\nwhere $R$ is a complete discrete valuation ring and where\n$p$ is an adic morphism there exists an\nextension of complete discrete valuation rings $R \\subset R'$\nand a morphism $\\text{Spf}(R') \\to X$ making the displayed diagram commute.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQQ","source_file":"restricted.tex","source_line":5037,"source_end_line":5053,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5037-L5053","statement_sha256":"cb7754a29cf6c6e2d4e593b0ece29ae02b1bd1056b14eea02c5324798e5ab903","origin":"The Stacks Project","memory_eligible":false,"source_rank":13188,"rank":13188,"depth":0,"x":1261.683,"y":1517.34,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQR","tag":"0AQR","title":"Rig-surjective morphisms · Lemma 0AQR","summary":"Rig-surjectivity of locally finite type morphisms is preserved under composition Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of formal algebraic spaces over S. Assume X, Y, Z are locally Noetherian and f and g locally of finite type. Then if f and g are rig-surjective, so is g ∘ f.","statement_latex":"\\begin{slogan}\nRig-surjectivity of locally finite type morphisms is preserved under\ncomposition\n\\end{slogan}\nLet $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of formal\nalgebraic spaces over $S$. Assume $X$, $Y$, $Z$ are locally Noetherian and\n$f$ and $g$ locally of finite type. Then if $f$ and $g$ are rig-surjective,\nso is $g \\circ f$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQR","source_file":"restricted.tex","source_line":5089,"source_end_line":5099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5089-L5099","statement_sha256":"15d62fee1dd0404cb33267d652a5c1f9ff27bc39059f6472b04596bf63ad1875","origin":"The Stacks Project","memory_eligible":false,"source_rank":13189,"rank":13189,"depth":57,"x":1099.336,"y":1735.832,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQS","tag":"0AQS","title":"Rig-surjective morphisms · Lemma 0AQS","summary":"Let S be a scheme. Let f : X → Y and Z → Y be morphisms of formal algebraic spaces over S. Assume X, Y, Z are locally Noetherian and f and g locally of finite type. If f is rig-surjective, then the base change Z ×_Y X → Z is too.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $Z \\to Y$ be morphisms\nof formal algebraic spaces over $S$. Assume $X$, $Y$, $Z$ are locally\nNoetherian and $f$ and $g$ locally of finite type. If $f$ is\nrig-surjective, then the base change $Z \\times_Y X \\to Z$ is too.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQS","source_file":"restricted.tex","source_line":5106,"source_end_line":5112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5106-L5112","statement_sha256":"2fce76e57e35ebfaf09b48825b8cb14f2490e16e78648dfec6ab65eb4c31b801","origin":"The Stacks Project","memory_eligible":false,"source_rank":13190,"rank":13190,"depth":74,"x":1043.277,"y":1482.301,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GH5","tag":"0GH5","title":"Rig-surjective morphisms · Lemma 0GH5","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms locally of finite type of locally Noetherian formal algebraic spaces over S. If g ∘ f is rig-surjective and g is a monomorphism, then f is rig-surjective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$\nbe morphisms locally of finite type of locally Noetherian\nformal algebraic spaces over $S$. If $g \\circ f$ is rig-surjective\nand $g$ is a monomorphism, then $f$ is rig-surjective.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GH5","source_file":"restricted.tex","source_line":5120,"source_end_line":5126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5120-L5126","statement_sha256":"a667510ee7595a657d8b3d954f0e117b9e77b541464066a1add2647a93453324","origin":"The Stacks Project","memory_eligible":false,"source_rank":13191,"rank":13191,"depth":75,"x":1288.784,"y":1637.625,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQT","tag":"0AQT","title":"Rig-surjective morphisms · Lemma 0AQT","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of formal algebraic spaces over S. Assume X, Y, Z locally Noetherian and f and g locally of finite type. If g ∘ f : X → Z is rig-surjective, so is g : Y → Z.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of\nformal algebraic spaces over $S$. Assume $X$, $Y$, $Z$ locally Noetherian\nand $f$ and $g$ locally of finite type. If $g \\circ f : X \\to Z$\nis rig-surjective, so is $g : Y \\to Z$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQT","source_file":"restricted.tex","source_line":5133,"source_end_line":5139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5133-L5139","statement_sha256":"ccec140a0824466a2e123f9395f6adbaf5bb018dc7669df62f7d4938a0f35c23","origin":"The Stacks Project","memory_eligible":false,"source_rank":13192,"rank":13192,"depth":0,"x":982.486,"y":1662.427,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQU","tag":"0AQU","title":"Rig-surjective morphisms · Lemma 0AQU","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces which is representable by algebraic spaces, étale, and surjective. Then f is rig-surjective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian\nformal algebraic spaces which is representable by algebraic spaces, \\'etale,\nand surjective. Then $f$ is rig-surjective.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQU","source_file":"restricted.tex","source_line":5145,"source_end_line":5150,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5145-L5150","statement_sha256":"ce97e4963e93615682f4ee3b8b695f2a8ecae9e0b37c52d8f34e14edf5130e55","origin":"The Stacks Project","memory_eligible":false,"source_rank":13193,"rank":13193,"depth":60,"x":1188.641,"y":1470.113,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQV","tag":"0AQV","title":"Rig-surjective morphisms · Lemma 0AQV","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces which is locally of finite type. Let (g_i : Y_i → Y) be a family of morphisms of formal algebraic spaces which are representable by algebraic spaces and étale such that coprod g_i is surjective. Then f is rig-surjective if and only if each f_i : X ×_Y Y_i → Y_i is rig-surjective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally\nNoetherian formal algebraic spaces which is locally of finite type.\nLet $\\{g_i : Y_i \\to Y\\}$ be a family of morphisms of formal\nalgebraic spaces which are representable by algebraic spaces and\n\\'etale such that $\\coprod g_i$ is surjective.\nThen $f$ is rig-surjective if and only if each\n$f_i : X \\times_Y Y_i \\to Y_i$ is rig-surjective.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQV","source_file":"restricted.tex","source_line":5176,"source_end_line":5185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5176-L5185","statement_sha256":"31efae7457680d9fdf7b6b3b4d0a8f05befb0d32d6531013cbb55e22ce94880a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13194,"rank":13194,"depth":75,"x":1191.282,"y":1729.201,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQX","tag":"0AQX","title":"Rig-surjective morphisms · Lemma 0AQX","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Let B be an I-adically complete A-algebra. If A/I^n → B/I^nB is of finite type and flat for all n and faithfully flat for n = 1, then Spf(B) → Spf(A) is rig-surjective.","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nLet $B$ be an $I$-adically complete $A$-algebra.\nIf $A/I^n \\to B/I^nB$ is of finite type and flat for all $n$ and\nfaithfully flat for $n = 1$, then $\\text{Spf}(B) \\to \\text{Spf}(A)$\nis rig-surjective.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQX","source_file":"restricted.tex","source_line":5200,"source_end_line":5207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5200-L5207","statement_sha256":"683ead33f6a0452ee9c5f70e4ea798835fb2c298b71c292de333383dd7ed8336","origin":"The Stacks Project","memory_eligible":false,"source_rank":13195,"rank":13195,"depth":17,"x":980.738,"y":1539.432,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQY","tag":"0AQY","title":"Rig-surjective morphisms · Lemma 0AQY","summary":"Let A be a Noetherian ring complete with respect to an ideal I. Let B be an I-adically complete A-algebra. Assume that • the I-torsion in A is 0, • A/I^n → B/I^nB is flat and of finite type for all n. Then Spf(B) → Spf(A) is rig-surjective if and only if A/I → B/IB is faithfully flat.","statement_latex":"Let $A$ be a Noetherian ring complete with respect to an ideal $I$.\nLet $B$ be an $I$-adically complete $A$-algebra. Assume that\n\\begin{enumerate}\n\\item the $I$-torsion in $A$ is $0$,\n\\item $A/I^n \\to B/I^nB$ is flat and of finite type for all $n$.\n\\end{enumerate}\nThen $\\text{Spf}(B) \\to \\text{Spf}(A)$ is rig-surjective if and only\nif $A/I \\to B/IB$ is faithfully flat.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQY","source_file":"restricted.tex","source_line":5251,"source_end_line":5261,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5251-L5261","statement_sha256":"d60a535c8ad6b51c29628807d76647daf69288dfd1a5aeda3bfd96669eb9e50a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13196,"rank":13196,"depth":18,"x":1288.954,"y":1559.921,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AR0","tag":"0AR0","title":"Rig-surjective morphisms · Lemma 0AR0","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces. Assume X and Y are locally Noetherian, f locally of finite type, and f a monomorphism. Then f is rig surjective if and only if every adic morphism Spf(R) → Y where R is a complete discrete valuation ring factors through X.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces. Assume $X$ and $Y$ are locally Noetherian, $f$ locally of finite\ntype, and $f$ a monomorphism. Then $f$ is rig surjective if and only if\nevery adic morphism $\\text{Spf}(R) \\to Y$ where $R$ is a complete discrete\nvaluation ring factors through $X$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AR0","source_file":"restricted.tex","source_line":5307,"source_end_line":5314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5307-L5314","statement_sha256":"e4077d9988249b76045728729a40a9d2ac04edb0209ead803d91c14404b43618","origin":"The Stacks Project","memory_eligible":false,"source_rank":13197,"rank":13197,"depth":71,"x":1044.922,"y":1719.886,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GD5","tag":"0GD5","title":"Rig-surjective morphisms · Lemma 0GD5","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces. Assume that X and Y are locally Noetherian and f is a closed immersion. The following are equivalent • f is rig-surjective, and • for every affine formal algebraic space V and every morphism V → Y which is representable by algebraic spaces and étale the morphism X ×_Y V → V corresponds to a surjective morphism B → A in WAdm^Noeth whose kernel J has the following property: IJ^n = 0 for some ideal of…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a\nclosed immersion. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is rig-surjective, and\n\\item for every affine formal algebraic space $V$ and every morphism\n$V \\to Y$ which is representable by algebraic spaces and \\'etale\nthe morphism $X \\times_Y V \\to V$ corresponds to a surjective morphism\n$B \\to A$ in $\\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following\nproperty: $IJ^n = 0$ for some ideal of definition $I$ of $B$\nand some $n \\geq 1$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GD5","source_file":"restricted.tex","source_line":5328,"source_end_line":5342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5328-L5342","statement_sha256":"7fa4d6977599c90094559d737c6edd282bc1cd3e90396c8c4c92e8613825811d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13198,"rank":13198,"depth":76,"x":1096.287,"y":1463.164,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GD6","tag":"0GD6","title":"Rig-surjective morphisms · Lemma 0GD6","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces. Assume that X and Y are locally Noetherian and f is a closed immersion. The following are equivalent • f is rig-smooth and rig-surjective, • f is rig-étale and rig-surjective, and • for every affine formal algebraic space V and every morphism V → Y which is representable by algebraic spaces and étale the morphism X ×_Y V → V corresponds to a surjective morphism B → A in WAdm^Noeth whose kernel J…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces. Assume that $X$ and $Y$ are locally Noetherian and $f$ is a\nclosed immersion. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is rig-smooth and rig-surjective,\n\\item $f$ is rig-\\'etale and rig-surjective, and\n\\item for every affine formal algebraic space $V$ and every morphism\n$V \\to Y$ which is representable by algebraic spaces and \\'etale\nthe morphism $X \\times_Y V \\to V$ corresponds to a surjective morphism\n$B \\to A$ in $\\textit{WAdm}^{Noeth}$ whose kernel $J$ has the following\nproperty: $IJ = 0$ for some ideal of definition $I$ of $B$.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GD6","source_file":"restricted.tex","source_line":5384,"source_end_line":5398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5384-L5398","statement_sha256":"71dfd9bc5d677a32afc0830779d4601c4c0f30df379bb6aed852e3669a938277","origin":"The Stacks Project","memory_eligible":false,"source_rank":13199,"rank":13199,"depth":79,"x":1265.054,"y":1681.865,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GH6","tag":"0GH6","title":"Rig-surjective morphisms · Lemma 0GH6","summary":"Let S be a scheme. Let f : X → Y and g : Y → Z be morphisms of locally Noetherian formal algebraic spaces over S. Assume • g is locally of finite type, • f is rig-smooth (resp. rig-étale) and rig-surjective, • g ∘ f is rig-smooth (resp. rig-étale) then g is rig-smooth (resp. rig-étale).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Y \\to Z$\nbe morphisms of locally Noetherian formal algebraic spaces over $S$.\nAssume\n\\begin{enumerate}\n\\item $g$ is locally of finite type,\n\\item $f$ is rig-smooth (resp.\\ rig-\\'etale) and rig-surjective,\n\\item $g \\circ f$ is rig-smooth (resp.\\ rig-\\'etale)\n\\end{enumerate}\nthen $g$ is rig-smooth (resp.\\ rig-\\'etale).","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig-surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GH6","source_file":"restricted.tex","source_line":5408,"source_end_line":5419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5408-L5419","statement_sha256":"0cdc482cfa3af5bdb49bd9fd18e224dbf716ac43d4d6afa91345d086a16c1b88","origin":"The Stacks Project","memory_eligible":false,"source_rank":13200,"rank":13200,"depth":79,"x":964.39,"y":1616.285,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GD8","tag":"0GD8","title":"Formal algebraic spaces over cdvrs · Lemma 0GD8","summary":"Let X be a locally Noetherian formal algebraic space over a complete discrete valuation ring A. Then there exists a closed immersion X' → X of formal algebraic spaces such that X' is flat over A and such that any morphism Y → X of locally Noetherian formal algebraic spaces with Y flat over A factors through X'.","statement_latex":"Let $X$ be a locally Noetherian formal algebraic space over\na complete discrete valuation ring $A$.\nThen there exists a closed immersion $X' \\to X$\nof formal algebraic spaces such that $X'$ is flat over $A$\nand such that any morphism $Y \\to X$ of locally Noetherian formal algebraic\nspaces with $Y$ flat over $A$ factors through $X'$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Formal algebraic spaces over cdvrs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GD8","source_file":"restricted.tex","source_line":5534,"source_end_line":5542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5534-L5542","statement_sha256":"3d7c16e54976a4506b744274d41610de482bdf91e7efa23e6403e846a67fd4b9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13201,"rank":13201,"depth":74,"x":1239.147,"y":1493.899,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GD9","tag":"0GD9","title":"Formal algebraic spaces over cdvrs · Lemma 0GD9","summary":"Let X be a locally Noetherian formal algebraic space which is locally of finite type over a complete discrete valuation ring A. Let X' ⊂ X be as in Lemma [Tag 0GD8]. If X → X ×_Spf(A) X is rig-étale and rig-surjective, then X' = Spf(A) or X' = ∅.","statement_latex":"Let $X$ be a locally Noetherian formal algebraic space which is\nlocally of finite type over a complete discrete valuation ring $A$.\nLet $X' \\subset X$ be as in Lemma \\ref{lemma-flat-locus}.\nIf $X \\to X \\times_{\\text{Spf}(A)} X$ is rig-\\'etale and rig-surjective,\nthen $X' = \\text{Spf}(A)$ or $X' = \\emptyset$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Formal algebraic spaces over cdvrs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GD9","source_file":"restricted.tex","source_line":5614,"source_end_line":5621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5614-L5621","statement_sha256":"cf084db6627ae24da7a9b4a11c64c365a1e576e1355544c4be495b177ff98dc4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13202,"rank":13202,"depth":80,"x":1134.846,"y":1740.33,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDA","tag":"0GDA","title":"Formal algebraic spaces over cdvrs · Lemma 0GDA","summary":"Let S be a scheme. Let f : X → Y be a morphism of formal algebraic spaces. Assume • X and Y are locally Noetherian, • f locally of finite type, • Δ_f : X → X ×_Y X is rig-étale and rig-surjective. Then f is rig surjective if and only if every adic morphism Spf(R) → Y where R is a complete discrete valuation ring lifts to a morphism Spf(R) → X.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of formal algebraic\nspaces. Assume\n\\begin{enumerate}\n\\item $X$ and $Y$ are locally Noetherian,\n\\item $f$ locally of finite type,\n\\item $\\Delta_f : X \\to X \\times_Y X$ is rig-\\'etale and rig-surjective.\n\\end{enumerate}\nThen $f$ is rig surjective if and only if every adic morphism\n$\\text{Spf}(R) \\to Y$ where $R$ is a complete discrete\nvaluation ring lifts to a morphism $\\text{Spf}(R) \\to X$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Formal algebraic spaces over cdvrs","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDA","source_file":"restricted.tex","source_line":5682,"source_end_line":5694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5682-L5694","statement_sha256":"d207522fc92acd70512d0dca31c6cfa987e1ea69d9829385a29a5652ccb33005","origin":"The Stacks Project","memory_eligible":false,"source_rank":13203,"rank":13203,"depth":81,"x":1013.445,"y":1499.158,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQ9","tag":"0AQ9","title":"The completion functor · Lemma 0AQ9","summary":"In the situation above. If f is locally of finite type, then f_/T is locally of finite type.","statement_latex":"In the situation above. If $f$ is locally of finite type, then\n$f_{/T}$ is locally of finite type.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQ9","source_file":"restricted.tex","source_line":5791,"source_end_line":5795,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5791-L5795","statement_sha256":"9b5141bf9a4b319b0f6ded91f969da2bef4c543c6cb8f32997dbab7affe1a427","origin":"The Stacks Project","memory_eligible":false,"source_rank":13204,"rank":13204,"depth":7,"x":1297.231,"y":1608.233,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GI0","tag":"0GI0","title":"The completion functor · Lemma 0GI0","summary":"In the situation above. If f is étale, then f_/T is étale.","statement_latex":"In the situation above. If $f$ is \\'etale, then $f_{/T}$ is \\'etale.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GI0","source_file":"restricted.tex","source_line":5808,"source_end_line":5811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5808-L5811","statement_sha256":"72e221c99b49eeeca9cb918cf2f0e29bb7ee8863f67bd5196c33cb613ac69587","origin":"The Stacks Project","memory_eligible":false,"source_rank":13205,"rank":13205,"depth":47,"x":999.919,"y":1688.921,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDC","tag":"0GDC","title":"The completion functor · Lemma 0GDC","summary":"In the situation above. If f is a closed immersion, then f_/T is a closed immersion.","statement_latex":"In the situation above. If $f$ is a closed immersion, then\n$f_{/T}$ is a closed immersion.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDC","source_file":"restricted.tex","source_line":5819,"source_end_line":5823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5819-L5823","statement_sha256":"6a90a1dbfb190a345b67cae9e6055dcaecd40ec7bd26dcff910278f30c468768","origin":"The Stacks Project","memory_eligible":false,"source_rank":13206,"rank":13206,"depth":8,"x":1154.44,"y":1460.462,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDD","tag":"0GDD","title":"The completion functor · Lemma 0GDD","summary":"In the situation above. If f is proper, then f_/T is proper.","statement_latex":"In the situation above. If $f$ is proper, then $f_{/T}$ is proper.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDD","source_file":"restricted.tex","source_line":5835,"source_end_line":5838,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5835-L5838","statement_sha256":"9cb644e98ca5c1ccc16c61659ae905ea64a6fb973e412c9bc1313edd44c4e1c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13207,"rank":13207,"depth":8,"x":1224.297,"y":1716.892,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDE","tag":"0GDE","title":"The completion functor · Lemma 0GDE","summary":"In the situation above. If f is quasi-compact, then f_/T is quasi-compact.","statement_latex":"In the situation above. If $f$ is quasi-compact, then\n$f_{/T}$ is quasi-compact.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDE","source_file":"restricted.tex","source_line":5849,"source_end_line":5853,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5849-L5853","statement_sha256":"d47875721341031b6d7d802626df39d5d92b2179a463682c6a13347e4b0766c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13208,"rank":13208,"depth":70,"x":966.281,"y":1567.275,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDG","tag":"0GDG","title":"The completion functor · Lemma 0GDG","summary":"In the situation above. If f is (quasi-)separated, then f_/T is too.","statement_latex":"In the situation above. If $f$ is (quasi-)separated, then\n$f_{/T}$ is too.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDG","source_file":"restricted.tex","source_line":5888,"source_end_line":5892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5888-L5892","statement_sha256":"75868ff53f1a7757bfe636b4a501a1b8b67d0a969d6098b8f3e1fad357495bc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13209,"rank":13209,"depth":71,"x":1277.206,"y":1531.156,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDH","tag":"0GDH","title":"The completion functor · Lemma 0GDH","summary":"In the situation above. If X is locally Noetherian, f is locally of finite type, and U' → U is smooth, then f_/T is rig-smooth.","statement_latex":"In the situation above. If $X$ is locally Noetherian,\n$f$ is locally of finite type, and $U' \\to U$ is smooth, then\n$f_{/T}$ is rig-smooth.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDH","source_file":"restricted.tex","source_line":5905,"source_end_line":5910,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5905-L5910","statement_sha256":"5f23e2cb2125ca93c9961bfd0265d80e2254194d63354aef8a5586c3ecbbe328","origin":"The Stacks Project","memory_eligible":false,"source_rank":13210,"rank":13210,"depth":77,"x":1076.756,"y":1734.443,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AR2","tag":"0AR2","title":"The completion functor · Lemma 0AR2","summary":"In the situation above. If X is locally Noetherian, f is locally of finite type, and U' → U is étale, then f_/T is rig-étale.","statement_latex":"In the situation above. If $X$ is locally Noetherian,\n$f$ is locally of finite type, and $U' \\to U$ is \\'etale, then\n$f_{/T}$ is rig-\\'etale.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AR2","source_file":"restricted.tex","source_line":5968,"source_end_line":5973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5968-L5973","statement_sha256":"e45773092057cdaae59516fbf28d5477bb77f91953afc201f1066e7a70911ff2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13211,"rank":13211,"depth":78,"x":1061.069,"y":1470.507,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AQW","tag":"0AQW","title":"The completion functor · Lemma 0AQW","summary":"In the situation above. If X is locally Noetherian, f is proper, and U' → U is surjective, then f_/T is rig-surjective.","statement_latex":"In the situation above. If $X$ is locally Noetherian,\n$f$ is proper, and $U' \\to U$ is surjective, then $f_{/T}$ is rig-surjective.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AQW","source_file":"restricted.tex","source_line":5983,"source_end_line":5987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L5983-L5987","statement_sha256":"5692d3f844867c3e4c09da4e71032b364cb8637f3c9b9f06be1299fec5de4eba","origin":"The Stacks Project","memory_eligible":false,"source_rank":13212,"rank":13212,"depth":62,"x":1285.136,"y":1656.436,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDI","tag":"0GDI","title":"The completion functor · Lemma 0GDI","summary":"In the situation above. If X is locally Noetherian, f is separated and locally of finite type, and U' → U is a monomorphism, then Δ_f_/T is rig-surjective.","statement_latex":"In the situation above. If $X$ is locally Noetherian,\n$f$ is separated and locally of finite type, and $U' \\to U$ is\na monomorphism, then $\\Delta_{f_{/T}}$ is rig-surjective.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"The completion functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDI","source_file":"restricted.tex","source_line":6023,"source_end_line":6028,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6023-L6028","statement_sha256":"a05ad1d0572779386cfabbed6ccf9fa52dbde41da4bd64172709cd3fcc4f0740","origin":"The Stacks Project","memory_eligible":false,"source_rank":13213,"rank":13213,"depth":63,"x":970.043,"y":1646.465,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDK","tag":"0GDK","title":"Formal modifications · Definition 0GDK","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S. We say f is a formal modification if • f is a proper morphism (Formal Spaces, Definition [Tag 0AM6]), • f is rig-étale, • f is rig-surjective, • Δ_f : X → X ×_Y X is rig-surjective.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of\nlocally Noetherian formal algebraic spaces over $S$. We say $f$ is a\n{\\it formal modification} if\n\\begin{enumerate}\n\\item $f$ is a proper morphism (Formal Spaces, Definition\n\\ref{formal-spaces-definition-proper}),\n\\item $f$ is rig-\\'etale,\n\\item $f$ is rig-surjective,\n\\item $\\Delta_f : X \\to X \\times_Y X$ is rig-surjective.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Formal modifications","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDK","source_file":"restricted.tex","source_line":6057,"source_end_line":6069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6057-L6069","statement_sha256":"8eec258c6dd0570a5dc8e21b710f6b277a6a79f73b6513e0d7490d30a1eb4662","origin":"The Stacks Project","memory_eligible":false,"source_rank":13214,"rank":13214,"depth":3,"x":1210.674,"y":1474.835,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDM","tag":"0GDM","title":"Formal modifications · Lemma 0GDM","summary":"Let S, f : X' → X, T ⊂ |X|, U ⊂ X, T' ⊂ |X'|, and U' ⊂ X' be as in Section [Tag 0GDB]. If X is locally Noetherian, f is proper, and U' → U is an isomorphism, then f_/T : X'_/T' → X_/T is a formal modification.","statement_latex":"Let $S$, $f : X' \\to X$, $T \\subset |X|$, $U \\subset X$,\n$T' \\subset |X'|$, and $U' \\subset X'$ be as in\nSection \\ref{section-completion-functor}.\nIf $X$ is locally Noetherian, $f$ is proper, and $U' \\to U$ is an isomorphism,\nthen $f_{/T} : X'_{/T'} \\to X_{/T}$ is a formal modification.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Formal modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDM","source_file":"restricted.tex","source_line":6128,"source_end_line":6135,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6128-L6135","statement_sha256":"3724c06ca87b7711220ce5879638e9cc04896512a047c874052213145b9208c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13215,"rank":13215,"depth":79,"x":1171.212,"y":1738.224,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDN","tag":"0GDN","title":"Formal modifications · Lemma 0GDN","summary":"Let S be a scheme. Let f : X → Y be a morphism of locally Noetherian formal algebraic spaces over S which is a formal modification. Then for any adic morphism Y' → Y of locally Noetherian formal algebraic spaces, the base change f' : X ×_Y Y' → Y' is a formal modification.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of locally Noetherian\nformal algebraic spaces over $S$ which is a formal modification.\nThen for any adic morphism $Y' \\to Y$ of locally Noetherian formal\nalgebraic spaces, the base change $f' : X \\times_Y Y' \\to Y'$ is\na formal modification.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Formal modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDN","source_file":"restricted.tex","source_line":6148,"source_end_line":6155,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6148-L6155","statement_sha256":"acb05615857847e31b46e6d03fab4b9dc1c4f08cf4595238900bd137cd6b69a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13216,"rank":13216,"depth":78,"x":988.299,"y":1521.374,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AR4","tag":"0AR4","title":"Completions and morphisms, I · Lemma 0AR4","summary":"Let T ⊂ X be a closed subset of a Noetherian affine scheme X. Let W be a Noetherian affine formal algebraic space. Let g : W → X_/T be a rig-étale morphism. Then there exists an affine scheme X' and a finite type morphism f : X' → X étale over X setminus T such that there is an isomorphism X'_/f^-1T ≅ W compatible with f_/T and g. Moreover, if W → X_/T is étale, then X' → X is étale.","statement_latex":"Let $T \\subset X$ be a closed subset of a Noetherian affine scheme $X$.\nLet $W$ be a Noetherian affine formal algebraic space.\nLet $g : W \\to X_{/T}$ be a rig-\\'etale morphism. Then there exists\nan affine scheme $X'$ and a finite type morphism $f : X' \\to X$\n\\'etale over $X \\setminus T$ such that there is an isomorphism\n$X'_{/f^{-1}T} \\cong W$ compatible with $f_{/T}$ and $g$.\nMoreover, if $W \\to X_{/T}$ is \\'etale, then $X' \\to X$ is \\'etale.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AR4","source_file":"restricted.tex","source_line":6188,"source_end_line":6197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6188-L6197","statement_sha256":"7887c287b7caa11618f4ed8c9453080213775aae1acfe15e32fe600efc8ee18d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13217,"rank":13217,"depth":71,"x":1297.902,"y":1577.548,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AR3","tag":"0AR3","title":"Completions and morphisms, I · Lemma 0AR3","summary":"Assume we have • Noetherian affine schemes X, X', and Y, • a closed subset T ⊂ |X|, • a morphism f : X' → X locally of finite type and étale over X setminus T, • a morphism h : Y → X, • a morphism α : Y_/T → X'_/T over X_/T (see proof for notation). Then there exists an étale morphism b : Y' → Y of affine schemes which induces an isomorphism b_/T : Y'_/T → Y_/T and a morphism a : Y' → X' over X such that α = a_/T ∘ b_/T^-1.","statement_latex":"Assume we have\n\\begin{enumerate}\n\\item Noetherian affine schemes $X$, $X'$, and $Y$,\n\\item a closed subset $T \\subset |X|$,\n\\item a morphism $f : X' \\to X$ locally of finite type\nand \\'etale over $X \\setminus T$,\n\\item a morphism $h : Y \\to X$,\n\\item a morphism $\\alpha : Y_{/T} \\to X'_{/T}$ over $X_{/T}$\n(see proof for notation).\n\\end{enumerate}\nThen there exists an \\'etale morphism $b : Y' \\to Y$ of affine schemes\nwhich induces an isomorphism $b_{/T} : Y'_{/T} \\to Y_{/T}$\nand a morphism $a : Y' \\to X'$ over $X$\nsuch that $\\alpha = a_{/T} \\circ b_{/T}^{-1}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AR3","source_file":"restricted.tex","source_line":6207,"source_end_line":6223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6207-L6223","statement_sha256":"ee532e62eb35285aed1f8f9889fa81d4dd16478ef06ed0e8976c1ede869c8eb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13218,"rank":13218,"depth":51,"x":1024.13,"y":1711.95,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AR6","tag":"0AR6","title":"Completions and morphisms, I · Lemma 0AR6","summary":"Let S be a scheme. Let f : X → Y and g : Z → Y be morphisms of algebraic spaces. Let T ⊂ |X| be closed. Assume that • X is locally Noetherian, • g is a monomorphism and locally of finite type, • f|_X setminus T : X setminus T → Y factors through g, and • f_/T : X_/T → Y factors through g, then f factors through g.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ and $g : Z \\to Y$ be morphisms\nof algebraic spaces. Let $T \\subset |X|$ be closed.\nAssume that\n\\begin{enumerate}\n\\item $X$ is locally Noetherian,\n\\item $g$ is a monomorphism and locally of finite type,\n\\item $f|_{X \\setminus T} : X \\setminus T \\to Y$ factors through $g$, and\n\\item $f_{/T} : X_{/T} \\to Y$ factors through $g$,\n\\end{enumerate}\nthen $f$ factors through $g$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AR6","source_file":"restricted.tex","source_line":6236,"source_end_line":6248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6236-L6248","statement_sha256":"d0d2ee4e954690c71cf3d15ea6d3b46d464dbae076217dd655014604cbc35a85","origin":"The Stacks Project","memory_eligible":false,"source_rank":13219,"rank":13219,"depth":61,"x":1118.026,"y":1457.22,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GI1","tag":"0GI1","title":"Completions and morphisms, I · Lemma 0GI1","summary":"Let S be a scheme. Let X, W be algebraic spaces over S with X locally Noetherian. Let T ⊂ |X| be a closed subset. Let a, b : X → W be morphisms of algebraic spaces over S such that a|_X setminus T = b|_X setminus T and such that a_/T = b_/T as morphisms X_/T → W. Then a = b.","statement_latex":"Let $S$ be a scheme. Let $X$, $W$ be algebraic spaces over $S$ with\n$X$ locally Noetherian. Let $T \\subset |X|$ be a closed subset.\nLet $a, b : X \\to W$ be morphisms of algebraic spaces over $S$ such\nthat $a|_{X \\setminus T} = b|_{X \\setminus T}$ and such that\n$a_{/T} = b_{/T}$ as morphisms $X_{/T} \\to W$. Then $a = b$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GI1","source_file":"restricted.tex","source_line":6266,"source_end_line":6273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6266-L6273","statement_sha256":"f5c7db38d8a8ea4bc3067d40247846fb0ee1086307a47e4740e79d423f1c76de","origin":"The Stacks Project","memory_eligible":false,"source_rank":13220,"rank":13220,"depth":62,"x":1253.785,"y":1698.598,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AR7","tag":"0AR7","title":"Completions and morphisms, I · Lemma 0AR7","summary":"Let S be a scheme. Let X, Y be locally Noetherian algebraic spaces over S. Let T ⊂ |X| and T' ⊂ |Y| be closed subsets. Let a, b : X → Y be morphisms of algebraic spaces over S such that a|_X setminus T = b|_X setminus T, such that |a|(T) ⊂ T' and |b|(T) ⊂ T', and such that a_/T = b_/T as morphisms X_/T → Y_/T'. Then a = b.","statement_latex":"Let $S$ be a scheme. Let $X$, $Y$ be locally Noetherian algebraic spaces\nover $S$. Let $T \\subset |X|$ and $T' \\subset |Y|$ be closed subsets.\nLet $a, b : X \\to Y$ be morphisms of algebraic spaces over $S$ such\nthat $a|_{X \\setminus T} = b|_{X \\setminus T}$, such that\n$|a|(T) \\subset T'$ and $|b|(T) \\subset T'$, and such that\n$a_{/T} = b_{/T}$ as morphisms $X_{/T} \\to Y_{/T'}$.\nThen $a = b$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AR7","source_file":"restricted.tex","source_line":6284,"source_end_line":6293,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6284-L6293","statement_sha256":"0f93bfe36689918d5cf844997d9002380791473477a50021cc68590e5324ebfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13221,"rank":13221,"depth":63,"x":959.247,"y":1597.531,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AR8","tag":"0AR8","title":"Completions and morphisms, I · Lemma 0AR8","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let T ⊂ |X| be a closed subset. Let s, t : R → U be two morphisms of algebraic spaces over X. Assume • R, U are locally of finite type over X, • the base change of s and t to X setminus T is an étale equivalence relation, and • the formal completion (t_/T, s_/T) : R_/T → U_/T ×_X_/T U_/T is an equivalence relation too (see proof for notation). Then (t, s) : R → U ×_X U is an étale equivalence relation.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space\nover $S$. Let $T \\subset |X|$ be a closed subset.\nLet $s, t : R \\to U$ be two morphisms of algebraic spaces over $X$.\nAssume\n\\begin{enumerate}\n\\item $R$, $U$ are locally of finite type over $X$,\n\\item the base change of $s$ and $t$ to $X \\setminus T$\nis an \\'etale equivalence relation, and\n\\item the formal completion\n$(t_{/T}, s_{/T}) : R_{/T} \\to U_{/T} \\times_{X_{/T}} U_{/T}$\nis an equivalence relation too (see proof for notation).\n\\end{enumerate}\nThen $(t, s) : R \\to U \\times_X U$ is an \\'etale equivalence relation.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AR8","source_file":"restricted.tex","source_line":6299,"source_end_line":6314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6299-L6314","statement_sha256":"7839ee98e829a9acce7148ffaf019abd09a04061fa4deec1739d9a32c72cbdca","origin":"The Stacks Project","memory_eligible":false,"source_rank":13222,"rank":13222,"depth":62,"x":1258.035,"y":1504.828,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AR9","tag":"0AR9","title":"Completions and morphisms, I · Lemma 0AR9","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S and let T ⊂ |X| be a closed subset. Let f : X' → X be a morphism of algebraic spaces which is locally of finite type and étale outside of T. There exists a factorization X' → X\" → X of f with the following properties: X\" → X is locally of finite type, X\" → X is an isomorphism over X setminus T, and X'_/T → X\"_/T is an isomorphism (see proof for notation).","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$\nand let $T \\subset |X|$ be a closed subset. Let $f : X' \\to X$ be a morphism\nof algebraic spaces which is locally of finite type and \\'etale outside of $T$.\nThere exists a factorization\n$$\nX' \\longrightarrow X'' \\longrightarrow X\n$$\nof $f$ with the following properties:\n$X'' \\to X$ is locally of finite type,\n$X'' \\to X$ is an isomorphism over $X \\setminus T$, and\n$X'_{/T} \\to X''_{/T}$ is an isomorphism (see proof for notation).","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, I","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AR9","source_file":"restricted.tex","source_line":6336,"source_end_line":6349,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6336-L6349","statement_sha256":"e11a8d5823516ac2df03d87d136b065fbad2d5f54259f662708bfe313b7ef8f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13223,"rank":13223,"depth":67,"x":1112.104,"y":1742.984,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GI3","tag":"0GI3","title":"Rig glueing of morphisms · Proposition 0GI3","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let T ⊂ |X| be a closed subset with complementary open subspace U ⊂ X. Let f : X' → X be a proper morphism of algebraic spaces such that f^-1(U) → U is an isomorphism. For any algebraic space W over S the map Mor_S(X, W) → Mor_S(X', W) ×_Mor_S(X'_/T, W) Mor_S(X_/T, W) is bijective.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset with complementary open subspace\n$U \\subset X$. Let $f : X' \\to X$ be a proper morphism\nof algebraic spaces such that $f^{-1}(U) \\to U$ is an isomorphism.\nFor any algebraic space $W$ over $S$ the map\n$$\n\\Mor_S(X, W) \\longrightarrow\n\\Mor_S(X', W) \\times_{\\Mor_S(X'_{/T}, W)} \\Mor_S(X_{/T}, W)\n$$\nis bijective.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Rig glueing of morphisms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GI3","source_file":"restricted.tex","source_line":6420,"source_end_line":6432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6420-L6432","statement_sha256":"7eb5cff3d3569696aef7fd9e8eeaa22ea3231d27b75cebcd17ebd3cef56d655a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13224,"rank":13224,"depth":72,"x":1028.101,"y":1484.286,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDQ","tag":"0GDQ","title":"Algebraization of rig-étale morphisms · Lemma 0GDQ","summary":"In the situation above, let X_1 → X be a morphism of algebraic spaces with X_1 locally Noetherian. Denote T_1 ⊂ |X_1| the inverse image of T and U_1 ⊂ X_1 the inverse image of U. We denote • C_X, T the category whose objects are morphisms of algebraic spaces f : X' → X which are locally of finite type and such that U' = f^-1U → U is an isomorphism, • C_X_1, T_1 the category whose objects are morphisms of algebraic spaces f_1 : X_1' → X_1 which are locally of finite type…","statement_latex":"In the situation above, let $X_1 \\to X$ be a morphism of algebraic\nspaces with $X_1$ locally Noetherian. Denote $T_1 \\subset |X_1|$\nthe inverse image of $T$ and $U_1 \\subset X_1$ the inverse image of $U$.\nWe denote\n\\begin{enumerate}\n\\item $\\mathcal{C}_{X, T}$ the category whose objects are\nmorphisms of algebraic spaces $f : X' \\to X$ which are locally\nof finite type and such that $U' = f^{-1}U \\to U$ is an isomorphism,\n\\item $\\mathcal{C}_{X_1, T_1}$ the category whose objects are\nmorphisms of algebraic spaces $f_1 : X_1' \\to X_1$ which are locally\nof finite type and such that $f_1^{-1}U_1 \\to U_1$ is an isomorphism,\n\\item $\\mathcal{C}_{X_{/T}}$ the category whose objects are\nmorphisms $g : W \\to X_{/T}$ of formal algebraic spaces\nwith $W$ locally Noetherian and $g$ rig-\\'etale,\n\\item $\\mathcal{C}_{X_{1, /T_1}}$ the category whose objects are\nmorphisms $g_1 : W_1 \\to X_{1, /T_1}$ of formal algebraic spaces\nwith $W_1$ locally Noetherian and $g_1$ rig-\\'etale.\n\\end{enumerate}\nThen the diagram\n$$\n\\xymatrix{\n\\mathcal{C}_{X, T} \\ar[d] \\ar[r] &\n\\mathcal{C}_{X_{/T}} \\ar[d] \\\\\n\\mathcal{C}_{X_1, T_1} \\ar[r] &\n\\mathcal{C}_{X_{1, /T_1}}\n}\n$$\nis commutative where the horizontal arrows are given by\n(\\ref{equation-completion-functor})\nand the vertical arrows by base change along\n$X_1 \\to X$ and along $X_{1, /T_1} \\to X_{/T}$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDQ","source_file":"restricted.tex","source_line":6572,"source_end_line":6605,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6572-L6605","statement_sha256":"79d9522b581063e3dde92aa634dc3964f10f394881a1c498737251be22c9e41c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13225,"rank":13225,"depth":0,"x":1298.375,"y":1627.536,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDR","tag":"0GDR","title":"Algebraization of rig-étale morphisms · Lemma 0GDR","summary":"In the situation above. Let f : X' → X be a morphism of algebraic spaces which is locally of finite type and an isomorphism over U. Let g : Y → X be a morphism with Y locally Noetherian. Then completion defines a bijection Mor_X(Y, X') → Mor_X_/T(Y_/T, X'_/T) In particular, the functor ([Tag 0AR5]) is fully faithful.","statement_latex":"In the situation above. Let $f : X' \\to X$ be a morphism of algebraic spaces\nwhich is locally of finite type and an isomorphism over $U$. Let\n$g : Y \\to X$ be a morphism with $Y$ locally Noetherian. Then completion\ndefines a bijection\n$$\n\\Mor_X(Y, X') \\longrightarrow \\Mor_{X_{/T}}(Y_{/T}, X'_{/T})\n$$\nIn particular, the functor (\\ref{equation-completion-functor}) is\nfully faithful.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDR","source_file":"restricted.tex","source_line":6614,"source_end_line":6625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6614-L6625","statement_sha256":"ab5f5bdeebb2cc98b4400721da5ffbcbe7f135de02fe3e257688e05908993eb7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13226,"rank":13226,"depth":72,"x":983.541,"y":1675.317,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARA","tag":"0ARA","title":"Algebraization of rig-étale morphisms · Lemma 0ARA","summary":"In the situation above. Assume X is affine. Then the functor ([Tag 0AR5]) is an equivalence.","statement_latex":"In the situation above. Assume $X$ is affine. Then the functor\n(\\ref{equation-completion-functor}) is an equivalence.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARA","source_file":"restricted.tex","source_line":6728,"source_end_line":6732,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6728-L6732","statement_sha256":"2cdef5f9a8e6eec33fd7a402eaf4048d258b4bfa534d57a35879f05512857070","origin":"The Stacks Project","memory_eligible":false,"source_rank":13227,"rank":13227,"depth":0,"x":1177.48,"y":1461.209,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARB","tag":"0ARB","title":"Algebraization of rig-étale morphisms · Theorem 0ARB","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let T ⊂ |X| be a closed subset. Let U ⊂ X be the open subspace with |U| = |X| setminus T. The completion functor ([Tag 0AR5]) ( morphisms of algebraic spaces f : X' → X which are locally of finite type and such that f^-1U → U is an isomorphism ) → ( morphisms g : W → X_/T of formal algebraic spaces with W locally Noetherian and g rig-étale ) sending f : X' → X to f_/T : X'_/T' → X_/T is an equivalence.","statement_latex":"Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset. Let $U \\subset X$ be the open subspace\nwith $|U| = |X| \\setminus T$. The completion functor\n(\\ref{equation-completion-functor})\n$$\n\\left\\{\n\\begin{matrix}\n\\text{morphisms of algebraic spaces}\\\\\nf : X' \\to X\\text{ which are locally}\\\\\n\\text{of finite type and such that}\\\\\nf^{-1}U \\to U\\text{ is an isomorphism}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\left\\{\n\\begin{matrix}\n\\text{morphisms }g : W \\to X_{/T}\\\\\n\\text{of formal algebraic spaces}\\\\\n\\text{with }W\\text{ locally Noetherian}\\\\\n\\text{and }g\\text{ rig-\\'etale}\n\\end{matrix}\n\\right\\}\n$$\nsending $f : X' \\to X$ to $f_{/T} : X'_{/T'} \\to X_{/T}$ is an equivalence.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Algebraization of rig-étale morphisms","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARB","source_file":"restricted.tex","source_line":6797,"source_end_line":6823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6797-L6823","statement_sha256":"caa6766e83e9b77809952c01846860005587482b8d1bace4337855c7d5984bd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13228,"rank":13228,"depth":73,"x":1206.683,"y":1729.423,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARU","tag":"0ARU","title":"Completions and morphisms, II · Lemma 0ARU","summary":"With assumptions and notation as in Theorem [Tag 0ARB] let f : X' → X correspond to g : W → X_/T. Then f is quasi-compact if and only if g is quasi-compact.","statement_latex":"With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}\nlet $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.\nThen $f$ is quasi-compact if and only if $g$ is quasi-compact.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARU","source_file":"restricted.tex","source_line":6899,"source_end_line":6904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6899-L6904","statement_sha256":"3f507d20198630a027b4f97e69bb2dba8378011981fb085627ae8e7feb113025","origin":"The Stacks Project","memory_eligible":false,"source_rank":13229,"rank":13229,"depth":74,"x":969.204,"y":1548.022,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARV","tag":"0ARV","title":"Completions and morphisms, II · Lemma 0ARV","summary":"With assumptions and notation as in Theorem [Tag 0ARB] let f : X' → X correspond to g : W → X_/T. Then f is quasi-separated if and only if g is so.","statement_latex":"With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}\nlet $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.\nThen $f$ is quasi-separated if and only if $g$ is so.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARV","source_file":"restricted.tex","source_line":6937,"source_end_line":6942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6937-L6942","statement_sha256":"b840caad72b9b0784a211839c8d84bd98f60832b4b958b1121398319d5dfba3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13230,"rank":13230,"depth":75,"x":1290.539,"y":1547.032,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARW","tag":"0ARW","title":"Completions and morphisms, II · Lemma 0ARW","summary":"With assumptions and notation as in Theorem [Tag 0ARB] let f : X' → X correspond to g : W → X_/T. Then f is separated ⇔ g is separated and Δ_g : W → W ×_X_/T W is rig-surjective.","statement_latex":"With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}\nlet $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.\nThen $f$ is separated $\\Leftrightarrow$ $g$ is\nseparated and $\\Delta_g : W \\to W \\times_{X_{/T}} W$ is rig-surjective.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARW","source_file":"restricted.tex","source_line":6980,"source_end_line":6986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L6980-L6986","statement_sha256":"35b65335dad06438651f205c6bf96c92996e0055d467b8beedadfe50074efcc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13231,"rank":13231,"depth":76,"x":1054.135,"y":1730.291,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ARX","tag":"0ARX","title":"Completions and morphisms, II · Lemma 0ARX","summary":"With assumptions and notation as in Theorem [Tag 0ARB] let f : X' → X correspond to g : W → X_/T. Then f is proper if and only if g is a formal modification (Definition [Tag 0GDK]).","statement_latex":"With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}\nlet $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.\nThen $f$ is proper if and only if $g$ is a formal modification\n(Definition \\ref{definition-formal-modification}).","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARX","source_file":"restricted.tex","source_line":7061,"source_end_line":7067,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L7061-L7067","statement_sha256":"ce96e7fec4318d7cdfdec398bbb59da4f225676fd643eb59865d6e18b2c06469","origin":"The Stacks Project","memory_eligible":false,"source_rank":13232,"rank":13232,"depth":80,"x":1081.113,"y":1460.73,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GI4","tag":"0GI4","title":"Completions and morphisms, II · Lemma 0GI4","summary":"With assumptions and notation as in Theorem [Tag 0ARB] let f : X' → X correspond to g : W → X_/T. Then f is étale if and only if g is étale.","statement_latex":"With assumptions and notation as in Theorem \\ref{theorem-dilatations-general}\nlet $f : X' \\to X$ correspond to $g : W \\to X_{/T}$.\nThen $f$ is \\'etale if and only if $g$ is \\'etale.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Completions and morphisms, II","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GI4","source_file":"restricted.tex","source_line":7119,"source_end_line":7124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L7119-L7124","statement_sha256":"3048a0ea721a76aaa4637c9bfcee0579513e229d4c6006587832bfcbb9beb0c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13233,"rank":13233,"depth":74,"x":1278.203,"y":1675.036,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDU","tag":"0GDU","title":"Artin's theorem on dilatations · Theorem 0GDU","summary":"[ArtinII] Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let T ⊂ |X| be a closed subset. Let X = X_/T be the formal completion of X along T. Let f : X' → X be a formal modification (Definition [Tag 0GDK]). Then there exists a unique proper morphism f : X' → X which is an isomorphism over the complement of T in X whose completion f_/T recovers f.","statement_latex":"\\begin{reference}\n\\cite[Theorem 3.2]{ArtinII}\n\\end{reference}\nLet $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.\nLet $T \\subset |X|$ be a closed subset. Let\n$\\mathfrak X = X_{/T}$\nbe the formal completion of $X$ along $T$. Let\n$$\n\\mathfrak f : \\mathfrak X' \\to \\mathfrak X\n$$\nbe a formal modification (Definition \\ref{definition-formal-modification}).\nThen there exists a unique proper morphism $f : X' \\to X$ which is an\nisomorphism over the complement of $T$ in $X$ whose completion $f_{/T}$\nrecovers $\\mathfrak f$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Artin's theorem on dilatations","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDU","source_file":"restricted.tex","source_line":7158,"source_end_line":7174,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L7158-L7174","statement_sha256":"84335352b43b5f30135c747fac1e7e47f46d0df87b2c1407b78ef61cb2972b57","origin":"The Stacks Project","memory_eligible":false,"source_rank":13234,"rank":13234,"depth":81,"x":960.195,"y":1628.794,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0GDV","tag":"0GDV","title":"Artin's theorem on dilatations · Lemma 0GDV","summary":"Let S be a scheme. Let X' → X be a formal modification (Definition [Tag 0GDK]) of locally Noetherian formal algebraic spaces over S. Given • any adic Noetherian topological ring A, • any adic morphism Spf(A) → X there exists a proper morphism X → Spec(A) of algebraic spaces and an isomorphism Spf(A) ×_ X X' → X_/Z over Spf(A) of the base change of X with the formal completion of X along the \"closed fibre\" Z = X ×_Spec(A) Spf(A)_red of X over A.","statement_latex":"Let $S$ be a scheme. Let $\\mathfrak X' \\to \\mathfrak X$\nbe a formal modification (Definition \\ref{definition-formal-modification})\nof locally Noetherian formal algebraic spaces over $S$. Given\n\\begin{enumerate}\n\\item any adic Noetherian topological ring $A$,\n\\item any adic morphism $\\text{Spf}(A) \\longrightarrow \\mathfrak X$\n\\end{enumerate}\nthere exists a proper morphism $X \\to \\Spec(A)$ of algebraic spaces\nand an isomorphism\n$$\n\\text{Spf}(A) \\times_{\\mathfrak X} \\mathfrak X'\n\\longrightarrow\nX_{/Z}\n$$\nover $\\text{Spf}(A)$ of the base change of $\\mathfrak X$\nwith the formal completion of $X$ along the ``closed fibre''\n$Z = X \\times_{\\Spec(A)} \\text{Spf}(A)_{red}$ of $X$ over $A$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Artin's theorem on dilatations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GDV","source_file":"restricted.tex","source_line":7185,"source_end_line":7204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L7185-L7204","statement_sha256":"9ec64061d74e59b43f3eba2c02ca2d77676cab57f59842ad668adc5aa585cd48","origin":"The Stacks Project","memory_eligible":false,"source_rank":13235,"rank":13235,"depth":82,"x":1232.164,"y":1482.292,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AE5","tag":"0AE5","title":"Application to modifications · Lemma 0AE5","summary":"Let A → B be a ring homomorphism of Noetherian rings inducing an isomorphism on I-adic completions for some ideal I ⊂ A (for example if B is the I-adic completion of A). Then base change defines an equivalence of categories between the category ([Tag 0AS2]) for (A, I) with the category ([Tag 0AS2]) for (B, IB).","statement_latex":"Let $A \\to B$ be a ring homomorphism of Noetherian rings inducing an\nisomorphism on $I$-adic completions for some ideal $I \\subset A$\n(for example if $B$ is the $I$-adic completion of $A$).\nThen base change defines an equivalence of categories between the\ncategory (\\ref{equation-modification}) for $(A, I)$\nwith the category (\\ref{equation-modification}) for $(B, IB)$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Application to modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AE5","source_file":"restricted.tex","source_line":7248,"source_end_line":7256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L7248-L7256","statement_sha256":"5bcd4d0739606bf4b37c3255ebabd171a5efc0f887344066b11df860817f2158","origin":"The Stacks Project","memory_eligible":false,"source_rank":13236,"rank":13236,"depth":74,"x":1149.344,"y":1744.919,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0BH5","tag":"0BH5","title":"Application to modifications · Lemma 0BH5","summary":"Notation and assumptions as in Lemma [Tag 0AE5]. Let f : X' → Spec(A) correspond to g : Y' → Spec(B) via the equivalence. Then f is quasi-compact, quasi-separated, separated, proper, finite, and add more here if and only if g is so.","statement_latex":"Notation and assumptions as in Lemma \\ref{lemma-Noetherian-local-ring}.\nLet $f : X' \\to \\Spec(A)$ correspond to $g : Y' \\to \\Spec(B)$\nvia the equivalence. Then $f$ is quasi-compact, quasi-separated, separated,\nproper, finite, and add more here if and only if $g$ is so.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Application to modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BH5","source_file":"restricted.tex","source_line":7277,"source_end_line":7283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L7277-L7283","statement_sha256":"5a8233d3eca8d82d13d556b06c368c71fb492ba967ceeebfe306a01d6eb82cee","origin":"The Stacks Project","memory_eligible":false,"source_rank":13237,"rank":13237,"depth":81,"x":999.058,"y":1504.014,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AF7","tag":"0AF7","title":"Application to modifications · Lemma 0AF7","summary":"Let A → B be a local map of local Noetherian rings such that • A → B is flat, • m_B = m_A B, and • kappa( m_A) = kappa( m_B) Then the base change functor from the category ([Tag 0AS2]) for (A, m_A) to the category ([Tag 0AS2]) for (B, m_B) is an equivalence.","statement_latex":"Let $A \\to B$ be a local map of local Noetherian rings such that\n\\begin{enumerate}\n\\item $A \\to B$ is flat,\n\\item $\\mathfrak m_B = \\mathfrak m_A B$, and\n\\item $\\kappa(\\mathfrak m_A) = \\kappa(\\mathfrak m_B)$\n\\end{enumerate}\nThen the base change functor from the category\n(\\ref{equation-modification}) for $(A, \\mathfrak m_A)$ to the category\n(\\ref{equation-modification}) for $(B, \\mathfrak m_B)$\nis an equivalence.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Application to modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AF7","source_file":"restricted.tex","source_line":7319,"source_end_line":7331,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L7319-L7331","statement_sha256":"a0dbcf1db9b9c1153b7b07b7d76fc61217e24db9467bcf50410f97a15963bb3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13238,"rank":13238,"depth":75,"x":1303.927,"y":1596.475,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AE6","tag":"0AE6","title":"Application to modifications · Lemma 0AE6","summary":"Let (A, m, kappa) be a Noetherian local ring. Let f : X → S be an object of ([Tag 0AS2]) such that f is proper. Then there exists a U-admissible blowup S' → S which dominates X.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $f : X \\to S$ be an object of (\\ref{equation-modification})\nsuch that $f$ is proper.\nThen there exists a $U$-admissible blowup $S' \\to S$\nwhich dominates $X$.","area":"Algebraic & Formal Geometry","chapter":"Algebraization of Formal Spaces","chapter_id":"restricted","section":"Application to modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AE6","source_file":"restricted.tex","source_line":7341,"source_end_line":7348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/restricted.tex#L7341-L7348","statement_sha256":"20acb77be53757f51cb17f8ab5d8134a388ee05147145b1fc3c2000317738b6b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13239,"rank":13239,"depth":72,"x":1004.452,"y":1701.396,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0AE3","tag":"0AE3","title":"Modifications · Lemma 0AE3","summary":"Let (A, m, kappa) be a 2-dimensional Noetherian local domain such that U = Spec(A) setminus ( m) is a normal scheme. Then any modification f : X → Spec(A) is a morphism as in ([Tag 0AE2]).","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a $2$-dimensional Noetherian\nlocal domain such that $U = \\Spec(A) \\setminus \\{\\mathfrak m\\}$\nis a normal scheme. Then any modification $f : X \\to \\Spec(A)$\nis a morphism as in (\\ref{equation-modification}).","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AE3","source_file":"spaces-resolve.tex","source_line":74,"source_end_line":80,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L74-L80","statement_sha256":"2ede9f423d70ba908577e861e0eab62e203b8c8658e823b75f18708d9b362e40","origin":"The Stacks Project","memory_eligible":false,"source_rank":13240,"rank":13240,"depth":75,"x":765.627,"y":1726.354,"cluster":"geometry-of-spaces"},{"id":"stacks:0AGM","tag":"0AGM","title":"Modifications · Lemma 0AGM","summary":"Let (A, m, kappa) be a Noetherian local ring. Let g : X → Y be a morphism in the category ([Tag 0AE2]). If the induced morphism X_kappa → Y_kappa of special fibres is a closed immersion, then g is a closed immersion.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $g : X \\to Y$ be a morphism in the category (\\ref{equation-modification}).\nIf the induced morphism $X_\\kappa \\to Y_\\kappa$ of special fibres is\na closed immersion, then $g$ is a closed immersion.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGM","source_file":"spaces-resolve.tex","source_line":90,"source_end_line":96,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L90-L96","statement_sha256":"cdd74e07d5187ba4cd4dc27670570e10d63544f9165e7fcb5ca91e78119e2137","origin":"The Stacks Project","memory_eligible":false,"source_rank":13241,"rank":13241,"depth":59,"x":515.172,"y":1555.394,"cluster":"geometry-of-spaces"},{"id":"stacks:0AYJ","tag":"0AYJ","title":"Modifications · Lemma 0AYJ","summary":"Let (A, m, kappa) be a Noetherian local domain of dimension ≥ 1. Let f : X → Spec(A) be a morphism of algebraic spaces. Assume at least one of the following conditions is satisfied • f is a modification (Spaces over Fields, Definition [Tag 0AD8]), • f is an alteration (Spaces over Fields, Definition [Tag 0ADA]), • f is locally of finite type, quasi-separated, X is integral, and there is exactly one point of |X| mapping to the generic point of Spec(A), • f is locally of…","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local domain\nof dimension $\\geq 1$.\nLet $f : X \\to \\Spec(A)$ be a morphism of algebraic spaces.\nAssume at least one of the following conditions is satisfied\n\\begin{enumerate}\n\\item $f$ is a modification (Spaces over Fields, Definition\n\\ref{spaces-over-fields-definition-modification}),\n\\item $f$ is an alteration (Spaces over Fields, Definition\n\\ref{spaces-over-fields-definition-alteration}),\n\\item $f$ is locally of finite type, quasi-separated, $X$ is integral,\nand there is exactly one point of $|X|$ mapping to the generic point\nof $\\Spec(A)$,\n\\item $f$ is locally of finite type, $X$ is decent, and the points\nof $|X|$ mapping to the generic point of $\\Spec(A)$ are\nthe generic points of irreducible components of $|X|$,\n\\item add more here.\n\\end{enumerate}\nThen $\\dim(X_\\kappa) \\leq \\dim(A) - 1$.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AYJ","source_file":"spaces-resolve.tex","source_line":104,"source_end_line":124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L104-L124","statement_sha256":"89a2cd0f9f41d0a92251b3db10f737cbdae3cc08725cf678659a9a4a2abd3e92","origin":"The Stacks Project","memory_eligible":false,"source_rank":13242,"rank":13242,"depth":61,"x":837.487,"y":1539.336,"cluster":"geometry-of-spaces"},{"id":"stacks:0AGN","tag":"0AGN","title":"Modifications · Lemma 0AGN","summary":"If (A, m, kappa) is a complete Noetherian local domain of dimension 2, then every modification of Spec(A) is projective over A.","statement_latex":"If $(A, \\mathfrak m, \\kappa)$ is a complete Noetherian local domain\nof dimension $2$, then every modification of $\\Spec(A)$ is projective over $A$.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Modifications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AGN","source_file":"spaces-resolve.tex","source_line":154,"source_end_line":158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L154-L158","statement_sha256":"8ca4a3f606ad6386843a2c237c6997ec4511b50404160e2f26632f9130a52c24","origin":"The Stacks Project","memory_eligible":false,"source_rank":13243,"rank":13243,"depth":79,"x":612.624,"y":1734.158,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHB","tag":"0BHB","title":"Strategy · Lemma 0BHB","summary":"The functor F ([Tag 0BHA]) is an equivalence.","statement_latex":"The functor $F$ (\\ref{equation-equivalence}) is an equivalence.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Strategy","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHB","source_file":"spaces-resolve.tex","source_line":207,"source_end_line":210,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L207-L210","statement_sha256":"26ad1c9a1dca18d034b2442fb2c5705c17c7795e80ace0c822336457f5908b10","origin":"The Stacks Project","memory_eligible":false,"source_rank":13244,"rank":13244,"depth":57,"x":621.768,"y":1462.779,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHC","tag":"0BHC","title":"Strategy · Lemma 0BHC","summary":"Let X, x_i, U_i → X, u_i be as in ([Tag 0BHA]). If f : Y → X corresponds to g_i : Y_i → U_i under F, then f is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, if and only if g_i is so for i = 1, …, n.","statement_latex":"Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence}).\nIf $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$,\nthen $f$ is quasi-compact, quasi-separated, separated, locally of finite\npresentation, of finite presentation, locally of finite type, of finite type,\nproper, integral, finite, if and only if $g_i$ is so\nfor $i = 1, \\ldots, n$.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Strategy","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHC","source_file":"spaces-resolve.tex","source_line":230,"source_end_line":238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L230-L238","statement_sha256":"b9044ab3e84c1363eba4a92afaea9470ac259c949de2a7f9bef39a86e4ab6867","origin":"The Stacks Project","memory_eligible":false,"source_rank":13245,"rank":13245,"depth":58,"x":833.361,"y":1668.175,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHD","tag":"0BHD","title":"Strategy · Lemma 0BHD","summary":"Let X, x_i, U_i → X, u_i be as in ([Tag 0BHA]). If f : Y → X corresponds to g_i : Y_i → U_i under F, then Y_x_i ≅ (Y_i)_u_i as algebraic spaces.","statement_latex":"Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence}).\nIf $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$,\nthen $Y_{x_i} \\cong (Y_i)_{u_i}$ as algebraic spaces.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Strategy","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHD","source_file":"spaces-resolve.tex","source_line":245,"source_end_line":250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L245-L250","statement_sha256":"01cbcbbec1de2d017a49bb76cb91cd7ef88e520b94e8be5d21c355b9dd89a662","origin":"The Stacks Project","memory_eligible":false,"source_rank":13246,"rank":13246,"depth":0,"x":512.011,"y":1636.767,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHE","tag":"0BHE","title":"Dominating by quadratic transformations · Definition 0BHE","summary":"Let S be a scheme. Let X be a decent algebraic space over S. Let x ∈ |X| be a closed point. By Decent Spaces, Lemma [Tag 0AHB] we can represent x by a closed immersion i : Spec(k) → X. The blowing up X' → X of X at x means the blowing up of X in the closed subspace Z = i(Spec(k)) ⊂ X.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.\nLet $x \\in |X|$ be a closed point. By\nDecent Spaces, Lemma \\ref{decent-spaces-lemma-decent-space-closed-point}\nwe can represent $x$ by a closed immersion $i : \\Spec(k) \\to X$.\nThe {\\it blowing up $X' \\to X$ of $X$ at $x$} means the blowing up of $X$\nin the closed subspace $Z = i(\\Spec(k)) \\subset X$.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Dominating by quadratic transformations","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHE","source_file":"spaces-resolve.tex","source_line":268,"source_end_line":276,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L268-L276","statement_sha256":"277c469fe6ed65f4c06a02292279cd826e99796639c48e43b5abe7778f690457","origin":"The Stacks Project","memory_eligible":false,"source_rank":13247,"rank":13247,"depth":58,"x":774.35,"y":1477.51,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHF","tag":"0BHF","title":"Dominating by quadratic transformations · Lemma 0BHF","summary":"Let X, x_i, U_i → X, u_i be as in ([Tag 0BHA]) and assume f : Y → X corresponds to g_i : Y_i → U_i under F. Then there exists a factorization Y = Z_m → Z_m - 1 → … → Z_1 → Z_0 = X of f where Z_j + 1 → Z_j is the blowing up of Z_j at a closed point z_j lying over (x_1, …, x_n) if and only if for each i there exists a factorization Y_i = Z_i, m_i → Z_i, m_i - 1 → … → Z_i, 1 → Z_i, 0 = U_i of g_i where Z_i, j + 1 → Z_i, j is the blowing up of Z_i, j at a closed point z_i, j…","statement_latex":"Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence})\nand assume $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$.\nThen there exists a factorization\n$$\nY = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = X\n$$\nof $f$ where $Z_{j + 1} \\to Z_j$ is the blowing up of $Z_j$ at a closed\npoint $z_j$ lying over $\\{x_1, \\ldots, x_n\\}$ if and only if for each\n$i$ there exists a factorization\n$$\nY_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = U_i\n$$\nof $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the blowing up of $Z_{i, j}$\nat a closed point $z_{i, j}$ lying over $u_i$.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Dominating by quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHF","source_file":"spaces-resolve.tex","source_line":289,"source_end_line":305,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L289-L305","statement_sha256":"8034e6b20f4b4d9d093c9cd7718fe623952b2a9977de556d2bb9872363162483","origin":"The Stacks Project","memory_eligible":false,"source_rank":13248,"rank":13248,"depth":58,"x":708.945,"y":1743.926,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHG","tag":"0BHG","title":"Dominating by quadratic transformations · Lemma 0BHG","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let T ⊂ |X| be a finite set of closed points x such that (1) X is regular at x and (2) the local ring of X at x has dimension 2. Let I ⊂ O_X be a quasi-coherent sheaf of ideals such that O_X/I is supported on T. Then there exists a sequence X_m → X_m - 1 → … → X_1 → X_0 = X where X_j + 1 → X_j is the blowing up of X_j at a closed point x_j lying above a point of T such that IO_X_n is an invertible ideal sheaf.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $T \\subset |X|$ be a finite set of closed points $x$ such that\n(1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has\ndimension $2$. Let $\\mathcal{I} \\subset \\mathcal{O}_X$ be a quasi-coherent\nsheaf of ideals such that $\\mathcal{O}_X/\\mathcal{I}$ is supported on $T$.\nThen there exists a sequence\n$$\nX_m \\to X_{m - 1} \\to \\ldots \\to X_1 \\to X_0 = X\n$$\nwhere $X_{j + 1} \\to X_j$ is the blowing up of $X_j$ at a closed\npoint $x_j$ lying above a point of $T$ such that\n$\\mathcal{I}\\mathcal{O}_{X_n}$ is an invertible ideal sheaf.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Dominating by quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHG","source_file":"spaces-resolve.tex","source_line":337,"source_end_line":351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L337-L351","statement_sha256":"1e20efb76cafd60cf2fc92fed165ed3baf82128df90e0ee08d4113d3f6651a90","origin":"The Stacks Project","memory_eligible":false,"source_rank":13249,"rank":13249,"depth":59,"x":542.851,"y":1510.253,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHH","tag":"0BHH","title":"Dominating by quadratic transformations · Lemma 0BHH","summary":"Let S be a scheme. Let X be a Noetherian algebraic space over S. Let T ⊂ |X| be a finite set of closed points x such that (1) X is regular at x and (2) the local ring of X at x has dimension 2. Let f : Y → X be a proper morphism of algebraic spaces which is an isomorphism over U = X setminus T. Then there exists a sequence X_n → X_n - 1 → … → X_1 → X_0 = X where X_i + 1 → X_i is the blowing up of X_i at a closed point x_i lying above a point of T and a factorization X_n →…","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.\nLet $T \\subset |X|$ be a finite set of closed points $x$ such that\n(1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has\ndimension $2$. Let $f : Y \\to X$ be a proper morphism of\nalgebraic spaces which is an isomorphism over $U = X \\setminus T$.\nThen there exists a sequence\n$$\nX_n \\to X_{n - 1} \\to \\ldots \\to X_1 \\to X_0 = X\n$$\nwhere $X_{i + 1} \\to X_i$ is the blowing up of $X_i$ at a closed\npoint $x_i$ lying above a point of $T$ and a factorization $X_n \\to Y \\to X$\nof the composition.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Dominating by quadratic transformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHH","source_file":"spaces-resolve.tex","source_line":381,"source_end_line":395,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L381-L395","statement_sha256":"198a5b4a85e6a0936cfb8955d444f78cf2ab68182e4d5a0e139ccbf10554fa90","origin":"The Stacks Project","memory_eligible":false,"source_rank":13250,"rank":13250,"depth":72,"x":853.385,"y":1588.351,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHJ","tag":"0BHJ","title":"Dominating by normalized blowups · Definition 0BHJ","summary":"Let S be a scheme. Let X be a decent algebraic space over S satisfying the equivalent conditions of Morphisms of Spaces, Lemma [Tag 0BB1]. Let x ∈ |X| be a closed point. The normalized blowup of X at x is the composition X\" → X' → X where X' → X is the blowup of X at x (Definition [Tag 0BHE]) and X\" → X' is the normalization of X'.","statement_latex":"Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$ satisfying\nthe equivalent conditions of\nMorphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}.\nLet $x \\in |X|$ be a closed point. The {\\it normalized blowup of $X$ at $x$}\nis the composition $X'' \\to X' \\to X$ where $X' \\to X$ is the blowup\nof $X$ at $x$ (Definition \\ref{definition-blowup-at-point})\nand $X'' \\to X'$ is the normalization of $X'$.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Dominating by normalized blowups","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHJ","source_file":"spaces-resolve.tex","source_line":430,"source_end_line":439,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L430-L439","statement_sha256":"ac011e463dfdcf0c5109ab7d2012ece4539ce5bcbe4e4dbbd446de3c7dec65bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13251,"rank":13251,"depth":59,"x":561.461,"y":1707.023,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHK","tag":"0BHK","title":"Dominating by normalized blowups · Lemma 0BHK","summary":"In Definition [Tag 0BHJ] if X is Nagata, then the normalized blowing up of X at x is a normal Nagata algebraic space proper over X.","statement_latex":"In Definition \\ref{definition-normalized-blowup} if $X$ is Nagata,\nthen the normalized blowing up of $X$ at $x$ is a\nnormal Nagata algebraic space proper over $X$.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Dominating by normalized blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHK","source_file":"spaces-resolve.tex","source_line":461,"source_end_line":466,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L461-L466","statement_sha256":"e07972c5d1b4ee8b390266fe9373fc1cccdcd2414d3676c86bb67f3587a02ace","origin":"The Stacks Project","memory_eligible":false,"source_rank":13252,"rank":13252,"depth":60,"x":681.345,"y":1453.753,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHL","tag":"0BHL","title":"Dominating by normalized blowups · Lemma 0BHL","summary":"Let X, x_i, U_i → X, u_i be as in ([Tag 0BHA]) and assume f : Y → X corresponds to g_i : Y_i → U_i under F. Assume X satisfies the equivalent conditions of Morphisms of Spaces, Lemma [Tag 0BB1]. Then there exists a factorization Y = Z_m → Z_m - 1 → … → Z_1 → Z_0 = X of f where Z_j + 1 → Z_j is the normalized blowing up of Z_j at a closed point z_j lying over (x_1, …, x_n) if and only if for each i there exists a factorization Y_i = Z_i, m_i → Z_i, m_i - 1 → … → Z_i, 1 →…","statement_latex":"Let $X, x_i, U_i \\to X, u_i$ be as in (\\ref{equation-equivalence})\nand assume $f : Y \\to X$ corresponds to $g_i : Y_i \\to U_i$ under $F$.\nAssume $X$ satisfies the equivalent conditions of\nMorphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization}.\nThen there exists a factorization\n$$\nY = Z_m \\to Z_{m - 1} \\to \\ldots \\to Z_1 \\to Z_0 = X\n$$\nof $f$ where $Z_{j + 1} \\to Z_j$ is the normalized blowing up of $Z_j$\nat a closed point $z_j$ lying over $\\{x_1, \\ldots, x_n\\}$ if and only if\nfor each $i$ there exists a factorization\n$$\nY_i = Z_{i, m_i} \\to Z_{i, m_i - 1} \\to \\ldots \\to Z_{i, 1} \\to Z_{i, 0} = U_i\n$$\nof $g_i$ where $Z_{i, j + 1} \\to Z_{i, j}$ is the normalized blowing up of\n$Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_i$.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Dominating by normalized blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHL","source_file":"spaces-resolve.tex","source_line":490,"source_end_line":508,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L490-L508","statement_sha256":"aa23cdf351f6a8930b4a0ec33b93253d61813095702803c1852991f7f04e6a4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13253,"rank":13253,"depth":59,"x":796.671,"y":1708.654,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHM","tag":"0BHM","title":"Dominating by normalized blowups · Lemma 0BHM","summary":"Let S be a scheme. Let X be a Noetherian Nagata algebraic space over S with dim(X) = 2. Let f : Y → X be a proper birational morphism. Then there exists a commutative diagram xymatrix X_n ar[r] ar[d] & X_n - 1 ar[r] & … ar[r] & X_1 ar[r] & X_0 ar[d] Y ar[rrrr] & & & & X where X_0 → X is the normalization and where X_i + 1 → X_i is the normalized blowing up of X_i at a closed point.","statement_latex":"Let $S$ be a scheme. Let $X$ be a Noetherian Nagata algebraic space over $S$\nwith $\\dim(X) = 2$. Let $f : Y \\to X$ be a proper birational morphism.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\nX_n \\ar[r] \\ar[d] &\nX_{n - 1} \\ar[r] &\n\\ldots \\ar[r] &\nX_1 \\ar[r] &\nX_0 \\ar[d] \\\\\nY \\ar[rrrr]  & & & & X\n}\n$$\nwhere $X_0 \\to X$ is the normalization and\nwhere $X_{i + 1} \\to X_i$ is the normalized blowing up of $X_i$ at a closed\npoint.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Dominating by normalized blowups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHM","source_file":"spaces-resolve.tex","source_line":518,"source_end_line":536,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L518-L536","statement_sha256":"cf527e3b00d00ab74fd59c0cec0db1a1ee965e3344ab2243202a91d68d6d699e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13254,"rank":13254,"depth":75,"x":506.511,"y":1586.075,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHP","tag":"0BHP","title":"Base change to the completion · Lemma 0BHP","summary":"Let (A, m, kappa) be a local ring with finitely generated maximal ideal m. Let X be a decent algebraic space over A. Let Y = X ×_Spec(A) Spec(A^wedge) where A^wedge is the m-adic completion of A. For a point q ∈ |Y| with image p ∈ |X| lying over the closed point of Spec(A) the map O_X, p^h → O_Y, q^h of henselian local rings induces an isomorphism on completions.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local ring with finitely generated\nmaximal ideal $\\mathfrak m$. Let $X$ be a decent algebraic\nspace over $A$. Let $Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$ where\n$A^\\wedge$ is the $\\mathfrak m$-adic completion of $A$.\nFor a point $q \\in |Y|$ with image $p \\in |X|$ lying\nover the closed point of $\\Spec(A)$ the map\n$\\mathcal{O}_{X, p}^h \\to \\mathcal{O}_{Y, q}^h$\nof henselian local rings induces an isomorphism on completions.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHP","source_file":"spaces-resolve.tex","source_line":615,"source_end_line":625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L615-L625","statement_sha256":"8d9842fbc6fff5c4be92f9b57106d2ba564fa389f02068a8336fa392aacab6cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13255,"rank":13255,"depth":58,"x":819.191,"y":1511.785,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHQ","tag":"0BHQ","title":"Base change to the completion · Lemma 0BHQ","summary":"Let (A, m, kappa) be a Noetherian local ring. Let X → Spec(A) be a morphism which is locally of finite type with X a decent algebraic space. Set Y = X ×_Spec(A) Spec(A^wedge). Let y ∈ |Y| with image x ∈ |X|. Then • if O_Y, y^h is regular, then O_X, x^h is regular, • if y is in the closed fibre, then O_Y, y^h is regular ⇔ O_X, x^h is regular, and • If X is proper over A, then X is regular if and only if Y is regular.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring.\nLet $X \\to \\Spec(A)$ be a morphism which is locally of finite type\nwith $X$ a decent algebraic space. Set\n$Y = X \\times_{\\Spec(A)} \\Spec(A^\\wedge)$. Let $y \\in |Y|$\nwith image $x \\in |X|$. Then\n\\begin{enumerate}\n\\item if $\\mathcal{O}_{Y, y}^h$ is regular, then\n$\\mathcal{O}_{X, x}^h$ is regular,\n\\item if $y$ is in the closed fibre, then $\\mathcal{O}_{Y, y}^h$ is regular\n$\\Leftrightarrow \\mathcal{O}_{X, x}^h$ is regular, and\n\\item If $X$ is proper over $A$, then $X$ is regular\nif and only if $Y$ is regular.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHQ","source_file":"spaces-resolve.tex","source_line":638,"source_end_line":653,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L638-L653","statement_sha256":"990916229a59b1fe7f442ca875f070664ad37f62d2aa5d4d985079581caf9b80","origin":"The Stacks Project","memory_eligible":false,"source_rank":13256,"rank":13256,"depth":58,"x":648.288,"y":1744.096,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHR","tag":"0BHR","title":"Base change to the completion · Lemma 0BHR","summary":"Let (A, m) be a local Noetherian ring. Let X be an algebraic space over A. Assume • A is analytically unramified (Algebra, Definition [Tag 032X]), • X is locally of finite type over A, • X → Spec(A) is étale at every point of codimension 0 in X. Then the normalization of X is finite over X.","statement_latex":"Let $(A, \\mathfrak m)$ be a local Noetherian ring. Let\n$X$ be an algebraic space over $A$. Assume\n\\begin{enumerate}\n\\item $A$ is analytically unramified\n(Algebra, Definition \\ref{algebra-definition-analytically-unramified}),\n\\item $X$ is locally of finite type over $A$,\n\\item $X \\to \\Spec(A)$ is \\'etale at every point of codimension $0$ in $X$.\n\\end{enumerate}\nThen the normalization of $X$ is finite over $X$.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Base change to the completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHR","source_file":"spaces-resolve.tex","source_line":678,"source_end_line":689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L678-L689","statement_sha256":"3b9dc8352efc1adeb296deffab101e8ddd0f4f372e87d2df7f2646a90fba28e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13257,"rank":13257,"depth":39,"x":587.463,"y":1475.694,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHT","tag":"0BHT","title":"Implied properties · Lemma 0BHT","summary":"Let S be a scheme. Let Y be a Noetherian integral algebraic space over S. Assume there exists an alteration f : X → Y with X regular. Then the normalization Y^ν → Y is finite and Y has a dense open which is regular.","statement_latex":"Let $S$ be a scheme. Let $Y$ be a Noetherian integral algebraic space\nover $S$. Assume there exists an alteration\n$f : X \\to Y$ with $X$ regular. Then the normalization $Y^\\nu \\to Y$\nis finite and $Y$ has a dense open which is regular.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Implied properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHT","source_file":"spaces-resolve.tex","source_line":717,"source_end_line":723,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L717-L723","statement_sha256":"093d62e0239a09aa114b623f7d7dca3e3b87a15f63f50079f33a99eb8eb15149","origin":"The Stacks Project","memory_eligible":false,"source_rank":13258,"rank":13258,"depth":63,"x":848.277,"y":1639.172,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHU","tag":"0BHU","title":"Implied properties · Lemma 0BHU","summary":"Let (A, m, kappa) be a local Noetherian domain. Assume there exists an alteration f : X → Spec(A) with X regular. Then • there exists a nonzero f ∈ A such that A_f is regular, • the integral closure B of A in its fraction field is finite over A, • the m-adic completion of B is a normal ring, i.e., the completions of B at its maximal ideals are normal domains, and • the generic formal fibre of A is regular.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a local Noetherian domain.\nAssume there exists an alteration $f : X \\to \\Spec(A)$\nwith $X$ regular. Then\n\\begin{enumerate}\n\\item there exists a nonzero $f \\in A$ such that $A_f$ is regular,\n\\item the integral closure $B$ of $A$ in its fraction field is finite over $A$,\n\\item the $\\mathfrak m$-adic completion of $B$ is a normal ring, i.e., the\ncompletions of $B$ at its maximal ideals are normal domains, and\n\\item the generic formal fibre of $A$ is regular.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Implied properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHU","source_file":"spaces-resolve.tex","source_line":753,"source_end_line":765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L753-L765","statement_sha256":"acc412734b740ec7d51049faa54a32c34e32578eff1ab2cd99ab2fc9af0eabd6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13259,"rank":13259,"depth":64,"x":524.342,"y":1666.63,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHW","tag":"0BHW","title":"Resolution · Definition 0BHW","summary":"Let S be a scheme. Let Y be a Noetherian integral algebraic space over S. A resolution of singularities of X is a modification f : X → Y such that X is regular.","statement_latex":"Let $S$ be a scheme. Let $Y$ be a Noetherian integral algebraic space over\n$S$. A {\\it resolution of singularities} of $X$ is a modification\n$f : X \\to Y$ such that $X$ is regular.","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Resolution","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHW","source_file":"spaces-resolve.tex","source_line":849,"source_end_line":854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L849-L854","statement_sha256":"b3aed0600ccfff2a07fc146043e21e6287660807752bf3687e9e8751413f23ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":13260,"rank":13260,"depth":0,"x":741.226,"y":1462.48,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHX","tag":"0BHX","title":"Resolution · Definition 0BHX","summary":"Let S be a scheme. Let Y be a 2-dimensional Noetherian integral algebraic space over S. We say Y has a resolution of singularities by normalized blowups if there exists a sequence Y_n → X_n - 1 → … → Y_1 → Y_0 → Y where • Y_i is proper over Y for i = 0, …, n, • Y_0 → Y is the normalization, • Y_i → Y_i - 1 is a normalized blowup for i = 1, …, n, and • Y_n is regular.","statement_latex":"Let $S$ be a scheme. Let $Y$ be a $2$-dimensional Noetherian integral\nalgebraic space over $S$. We say $Y$ has a\n{\\it resolution of singularities by normalized blowups}\nif there exists a sequence\n$$\nY_n \\to X_{n - 1} \\to \\ldots \\to Y_1 \\to Y_0 \\to Y\n$$\nwhere\n\\begin{enumerate}\n\\item $Y_i$ is proper over $Y$ for $i = 0, \\ldots, n$,\n\\item $Y_0 \\to Y$ is the normalization,\n\\item $Y_i \\to Y_{i - 1}$ is a normalized blowup for $i = 1, \\ldots, n$, and\n\\item $Y_n$ is regular.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Resolution","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHX","source_file":"spaces-resolve.tex","source_line":859,"source_end_line":875,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L859-L875","statement_sha256":"4784a35559c71072d59c92875094b6befd23f722a5c469634838f44af5bf8272","origin":"The Stacks Project","memory_eligible":false,"source_rank":13261,"rank":13261,"depth":0,"x":745.473,"y":1736.209,"cluster":"geometry-of-spaces"},{"id":"stacks:0BHY","tag":"0BHY","title":"Resolution · Theorem 0BHY","summary":"Let S be a scheme. Let Y be a two dimensional integral Noetherian algebraic space over S. The following are equivalent • there exists an alteration X → Y with X regular, • there exists a resolution of singularities of Y, • Y has a resolution of singularities by normalized blowups, • the normalization Y^ν → Y is finite, Y^ν has finitely many singular points y_1, …, y_m ∈ |Y|, and for each i the completion of the henselian local ring O_Y^ν, y_i^h is normal.","statement_latex":"Let $S$ be a scheme. Let $Y$ be a two dimensional integral\nNoetherian algebraic space over $S$. The following are equivalent\n\\begin{enumerate}\n\\item there exists an alteration $X \\to Y$ with $X$ regular,\n\\item there exists a resolution of singularities of $Y$,\n\\item $Y$ has a resolution of singularities by normalized blowups,\n\\item the normalization $Y^\\nu \\to Y$ is finite, $Y^\\nu$ has\nfinitely many singular points $y_1, \\ldots, y_m \\in |Y|$, and\nfor each $i$ the completion of the henselian local ring\n$\\mathcal{O}_{Y^\\nu, y_i}^h$ is normal.\n\\end{enumerate}","area":"Geometry of Spaces","chapter":"Resolution of Surfaces Revisited","chapter_id":"spaces-resolve","section":"Resolution","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BHY","source_file":"spaces-resolve.tex","source_line":883,"source_end_line":896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/spaces-resolve.tex#L883-L896","statement_sha256":"546c0e02525d7a0c1ba3c66c20796b98087b641adbdbcbbe1bec5df80ac9e0e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13262,"rank":13262,"depth":74,"x":522.112,"y":1536.683,"cluster":"geometry-of-spaces"},{"id":"stacks:06GC","tag":"06GC","title":"The base category · Definition 06GC","summary":"Let Lambda be a Noetherian ring and let Lambda → k be a finite ring map where k is a field. We define C_Lambda to be the category with • objects are pairs (A, φ) where A is an Artinian local Lambda-algebra and where φ : A/ m_A → k is a Lambda-algebra isomorphism, and • morphisms f : (B, ψ) → (A, φ) are local Lambda-algebra homomorphisms such that φ ∘ (f bmod m) = ψ. We say we are in the classical case if Lambda is a Noetherian complete local ring and k is its residue field.","statement_latex":"Let $\\Lambda$ be a Noetherian ring and let $\\Lambda \\to k$ be a finite\nring map where $k$ is a field. We define {\\it $\\mathcal{C}_\\Lambda$} to be\nthe category with\n\\begin{enumerate}\n\\item objects are pairs $(A, \\varphi)$ where $A$ is an Artinian local\n$\\Lambda$-algebra and where $\\varphi : A/\\mathfrak m_A \\to k$ is a\n$\\Lambda$-algebra isomorphism, and\n\\item morphisms $f : (B, \\psi) \\to (A, \\varphi)$ are local $\\Lambda$-algebra\nhomomorphisms such that $\\varphi \\circ (f \\bmod \\mathfrak m) = \\psi$.\n\\end{enumerate}\nWe say we are in the {\\it classical case} if $\\Lambda$ is a Noetherian\ncomplete local ring and $k$ is its residue field.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GC","source_file":"formal-defos.tex","source_line":236,"source_end_line":250,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L236-L250","statement_sha256":"35d72341caf6dfaa1442102e8b33697ba505032150a7b5859684727e5fdd0d5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13263,"rank":13263,"depth":0,"x":1586.682,"y":1600.0,"cluster":"deformation-theory"},{"id":"stacks:06GD","tag":"06GD","title":"The base category · Definition 06GD","summary":"Let f: B → A be a ring map in C_Lambda. We say f is a small extension if it is surjective and Ker(f) is a nonzero principal ideal which is annihilated by m_B.","statement_latex":"Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$.  We say $f$\nis a {\\it small extension} if it is surjective and $\\Ker(f)$ is a nonzero\nprincipal ideal which is annihilated by $\\mathfrak{m}_B$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GD","source_file":"formal-defos.tex","source_line":268,"source_end_line":273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L268-L273","statement_sha256":"2584cd56830bf90254ed9aa7381ec45465dfa6eda80cad0b20c9229f9d24f37c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13264,"rank":13264,"depth":0,"x":1571.467,"y":1606.567,"cluster":"deformation-theory"},{"id":"stacks:06GE","tag":"06GE","title":"The base category · Lemma 06GE","summary":"Let f: B → A be a surjective ring map in C_Lambda. Then f can be factored as a composition of small extensions.","statement_latex":"Let $f: B \\to A$ be a surjective ring map in $\\mathcal{C}_\\Lambda$.\nThen $f$ can be factored as a composition of small extensions.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GE","source_file":"formal-defos.tex","source_line":279,"source_end_line":283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L279-L283","statement_sha256":"6f413d3b0d52314badb6e820c3a406219ba5685779f1042cf8fa11be44230816","origin":"The Stacks Project","memory_eligible":false,"source_rank":13265,"rank":13265,"depth":7,"x":1581.306,"y":1587.498,"cluster":"deformation-theory"},{"id":"stacks:06GG","tag":"06GG","title":"The base category · Lemma 06GG","summary":"Let A be a local Lambda-algebra with residue field k. Let M be an A-module. Then [k : k'] length_A(M) = length_Lambda(M). In the classical case we have length_A(M) = length_Lambda(M).","statement_latex":"Let $A$ be a local $\\Lambda$-algebra with residue field $k$.\nLet $M$ be an $A$-module. Then\n$[k : k'] \\text{length}_A(M) = \\text{length}_\\Lambda(M)$.\nIn the classical case we have\n$\\text{length}_A(M) = \\text{length}_\\Lambda(M)$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GG","source_file":"formal-defos.tex","source_line":323,"source_end_line":330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L323-L330","statement_sha256":"e47a467c0628f9941ad229a7a79d0523d2debbafc063b27a4158f69108889262","origin":"The Stacks Project","memory_eligible":false,"source_rank":13266,"rank":13266,"depth":3,"x":1590.756,"y":1611.784,"cluster":"deformation-theory"},{"id":"stacks:06S3","tag":"06S3","title":"The base category · Lemma 06S3","summary":"Let A → B be a ring map in C_Lambda. The following are equivalent • f is surjective, • m_A/ m_A^2 → m_B/ m_B^2 is surjective, and • m_A/( m_Lambda A + m_A^2) → m_B/( m_Lambda B + m_B^2) is surjective.","statement_latex":"Let $A \\to B$ be a ring map in $\\mathcal{C}_\\Lambda$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is surjective,\n\\item $\\mathfrak m_A/\\mathfrak m_A^2 \\to \\mathfrak m_B/\\mathfrak m_B^2$\nis surjective, and\n\\item $\\mathfrak m_A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)\n\\to \\mathfrak m_B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2)$ is surjective.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06S3","source_file":"formal-defos.tex","source_line":346,"source_end_line":357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L346-L357","statement_sha256":"5ac8c643be95d0e4b09e6f8f5918ce6b991495d55c243247095151e25949a296","origin":"The Stacks Project","memory_eligible":false,"source_rank":13267,"rank":13267,"depth":3,"x":1560.262,"y":1597.067,"cluster":"deformation-theory"},{"id":"stacks:06GY","tag":"06GY","title":"The base category · Definition 06GY","summary":"Let R → S be a local homomorphism of local rings. The relative cotangent space_Lambda over Lambda should not be defined simply as the k-linear dual of the relative cotangent space. In fact, the correct definition of the relative cotangent space is Ω_S/R ⊗_S S/ m_S. of R over S is the S/ m_S-vector space m_S/( m_R S + m_S^2).","statement_latex":"Let $R \\to S$ be a local homomorphism of local rings. The\n{\\it relative cotangent space}\\footnote{Caution: We will see later\nthat in our general setting the tangent\nspace of an object $A \\in \\mathcal{C}_\\Lambda$ over $\\Lambda$ should\nnot be defined simply as the $k$-linear dual of the relative\ncotangent space. In fact, the correct definition of the relative\ncotangent space is\n$\\Omega_{S/R} \\otimes_S S/\\mathfrak m_S$.} of $R$ over $S$ is the\n$S/\\mathfrak m_S$-vector space\n$\\mathfrak m_S/(\\mathfrak m_R S + \\mathfrak m_S^2)$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GY","source_file":"formal-defos.tex","source_line":403,"source_end_line":415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L403-L415","statement_sha256":"60bdc7b6d81a8593385692f62cd29f3deec94e68e454533714ca44178a484131","origin":"The Stacks Project","memory_eligible":false,"source_rank":13268,"rank":13268,"depth":0,"x":1598.698,"y":1590.009,"cluster":"deformation-theory"},{"id":"stacks:06GH","tag":"06GH","title":"The base category · Lemma 06GH","summary":"Let f_1 : A_1 → A and f_2 : A_2 → A be ring maps in C_Lambda. Then: • If f_1 or f_2 is surjective, then A_1 ×_A A_2 is in C_Lambda. • If f_2 is a small extension, then so is A_1 ×_A A_2 → A_1. • If the field extension k/k' is separable, then A_1 ×_A A_2 is in C_Lambda.","statement_latex":"Let $f_1 : A_1 \\to A$ and $f_2 : A_2 \\to A$ be ring maps in\n$\\mathcal{C}_\\Lambda$. Then:\n\\begin{enumerate}\n\\item If $f_1$ or $f_2$ is surjective, then\n$A_1 \\times_A A_2$ is in $\\mathcal{C}_\\Lambda$.\n\\item If $f_2$ is a small extension, then so is\n$A_1 \\times_A A_2 \\to A_1$.\n\\item If the field extension $k/k'$ is separable, then\n$A_1 \\times_A A_2$ is in $\\mathcal{C}_\\Lambda$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GH","source_file":"formal-defos.tex","source_line":450,"source_end_line":462,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L450-L462","statement_sha256":"2b040cb20131f4213010fe08b622f6141d22e3cdb398e72971ca503a35604503","origin":"The Stacks Project","memory_eligible":false,"source_rank":13269,"rank":13269,"depth":45,"x":1573.746,"y":1619.542,"cluster":"deformation-theory"},{"id":"stacks:06GF","tag":"06GF","title":"The base category · Definition 06GF","summary":"Let f: B → A be a ring map in C_Lambda. We say f is an essential surjection if it has the following properties: • f is surjective. • If g: C → B is a ring map in C_Lambda such that f ∘ g is surjective, then g is surjective.","statement_latex":"Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$.  We say $f$\nis an {\\it essential surjection} if it has the following properties:\n\\begin{enumerate}\n\\item $f$ is surjective.\n\\item If $g: C \\to B$ is a ring map in $\\mathcal{C}_\\Lambda$ such that\n$f \\circ g$ is surjective, then $g$ is surjective.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GF","source_file":"formal-defos.tex","source_line":525,"source_end_line":534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L525-L534","statement_sha256":"acb4d4a389fec156b296690b4b140823b8d347d47025db65490bf23413f73724","origin":"The Stacks Project","memory_eligible":false,"source_rank":13270,"rank":13270,"depth":0,"x":1568.073,"y":1580.709,"cluster":"deformation-theory"},{"id":"stacks:06S5","tag":"06S5","title":"The base category · Lemma 06S5","summary":"Let f: B → A be a ring map in C_Lambda. The following are equivalent • f is an essential surjection, • the map B/ m_B^2 → A/ m_A^2 is an essential surjection, and • the map B/( m_Lambda B + m_B^2) → A/( m_Lambda A + m_A^2) is an essential surjection.","statement_latex":"Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is an essential surjection,\n\\item the map $B/\\mathfrak m_B^2 \\to A/\\mathfrak m_A^2$ is an essential\nsurjection, and\n\\item the map\n$B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2) \\to\nA/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$ is an essential surjection.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06S5","source_file":"formal-defos.tex","source_line":540,"source_end_line":552,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L540-L552","statement_sha256":"250bbc5c20d6c9c2082a94cbfe5d84011742ad0a68af5427536ef4f89a72d319","origin":"The Stacks Project","memory_eligible":false,"source_rank":13271,"rank":13271,"depth":46,"x":1605.877,"y":1607.938,"cluster":"deformation-theory"},{"id":"stacks:06S9","tag":"06S9","title":"The base category · Lemma 06S9","summary":"There is a canonical map m_Lambda/ m_Lambda^2 → H_1(L_k/Lambda). If k' ⊂ k is separable (for example if the characteristic of k is zero), then this map induces an isomorphism m_Lambda/ m_Lambda^2 ⊗_k' k = H_1(L_k/Lambda). If k = k' (for example in the classical case), then m_Lambda/ m_Lambda^2 = H_1(L_k/Lambda). The composition m_Lambda/ m_Lambda^2 → H_1(L_k/Lambda) → m_A/ m_A^2 comes from the canonical map m_Lambda → m_A.","statement_latex":"There is a canonical map\n$$\n\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\longrightarrow H_1(L_{k/\\Lambda}).\n$$\nIf $k' \\subset k$ is separable (for example if the characteristic\nof $k$ is zero), then this map induces an isomorphism\n$\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\otimes_{k'} k = H_1(L_{k/\\Lambda})$.\nIf $k = k'$ (for example in the classical case), then\n$\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 = H_1(L_{k/\\Lambda})$.\nThe composition\n$$\n\\mathfrak m_\\Lambda/\\mathfrak m_\\Lambda^2 \\longrightarrow\nH_1(L_{k/\\Lambda}) \\longrightarrow \\mathfrak m_A/\\mathfrak m_A^2\n$$\ncomes from the canonical map $\\mathfrak m_\\Lambda \\to \\mathfrak m_A$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06S9","source_file":"formal-defos.tex","source_line":635,"source_end_line":652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L635-L652","statement_sha256":"d6b435ace8f26df9f76dece23760b9c933d2f64a6448245c5248ad1592d4aa62","origin":"The Stacks Project","memory_eligible":false,"source_rank":13272,"rank":13272,"depth":40,"x":1553.079,"y":1609.334,"cluster":"deformation-theory"},{"id":"stacks:06H0","tag":"06H0","title":"The base category · Lemma 06H0","summary":"Let f: B → A be a ring map in C_Lambda. Notation as in ([Tag 06S8]). • The equivalent conditions of Lemma [Tag 06S3] characterizing when f is surjective are also equivalent to • Im(d_B) → Im(d_A) is surjective, and • the map Ω_B/Lambda ⊗_B k → Ω_A/Lambda ⊗_A k is surjective. • The following are equivalent • f is an essential surjection (see Lemma [Tag 06S5]), • the map Im(d_B) → Im(d_A) is an isomorphism, and • the map Ω_B/Lambda ⊗_B k → Ω_A/Lambda ⊗_A k is an…","statement_latex":"Let $f: B \\to A$ be a ring map in $\\mathcal{C}_\\Lambda$.\nNotation as in (\\ref{equation-sequence-functorial}).\n\\begin{enumerate}\n\\item The equivalent conditions of\nLemma \\ref{lemma-surjective}\ncharacterizing when $f$ is surjective are also equivalent to\n\\begin{enumerate}\n\\item $\\Im(\\text{d}_B) \\to \\Im(\\text{d}_A)$ is surjective, and\n\\item the map $\\Omega_{B/\\Lambda} \\otimes_B k \\to\n\\Omega_{A/\\Lambda} \\otimes_A k$ is surjective.\n\\end{enumerate}\n\\item The following are equivalent\n\\begin{enumerate}\n\\item $f$ is an essential surjection\n(see Lemma \\ref{lemma-essential-surjection-mod-squares}),\n\\item the map $\\Im(\\text{d}_B) \\to \\Im(\\text{d}_A)$ is an\nisomorphism, and\n\\item the map $\\Omega_{B/\\Lambda} \\otimes_B k \\to\n\\Omega_{A/\\Lambda} \\otimes_A k$ is an isomorphism.\n\\end{enumerate}\n\\item If $k/k'$ is separable, then $f$ is an essential surjection if\nand only if the map\n$\\mathfrak m_B/(\\mathfrak m_\\Lambda B + \\mathfrak m_B^2) \\to\n\\mathfrak m_A/(\\mathfrak m_\\Lambda A + \\mathfrak m_A^2)$\nis an isomorphism.\n\\item If $f$ is a small extension, then $f$ is not essential if and only if\n$f$ has a section $s: A \\to B$ in $\\mathcal{C}_\\Lambda$\nwith $f \\circ s = \\text{id}_A$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06H0","source_file":"formal-defos.tex","source_line":669,"source_end_line":700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L669-L700","statement_sha256":"b41b94c3a57dc8f7bb224774ea5a66910176c697eef78c15988ca006a3e422aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13273,"rank":13273,"depth":47,"x":1592.978,"y":1576.705,"cluster":"deformation-theory"},{"id":"stacks:06GW","tag":"06GW","title":"The completed base category · Definition 06GW","summary":"Let Lambda be a Noetherian ring and let Lambda → k be a finite ring map where k is a field. We define widehatC_Lambda to be the category with • objects are pairs (R, φ) where R is a Noetherian complete local Lambda-algebra and where φ : R/ m_R → k is a Lambda-algebra isomorphism, and • morphisms f : (S, ψ) → (R, φ) are local Lambda-algebra homomorphisms such that φ ∘ (f bmod m) = ψ.","statement_latex":"Let $\\Lambda$ be a Noetherian ring and let $\\Lambda \\to k$ be a finite\nring map where $k$ is a field. We define {\\it $\\widehat{\\mathcal{C}}_\\Lambda$}\nto be the category with\n\\begin{enumerate}\n\\item objects are pairs $(R, \\varphi)$ where $R$ is a Noetherian complete\nlocal $\\Lambda$-algebra and where $\\varphi : R/\\mathfrak m_R \\to k$ is a\n$\\Lambda$-algebra isomorphism, and\n\\item morphisms $f : (S, \\psi) \\to (R, \\varphi)$ are local $\\Lambda$-algebra\nhomomorphisms such that $\\varphi \\circ (f \\bmod \\mathfrak m) = \\psi$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The completed base category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GW","source_file":"formal-defos.tex","source_line":818,"source_end_line":830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L818-L830","statement_sha256":"0c68ab7a5c7e4f463dd928a9b3cbb99dc1fc3f3b912f2f0f197f31fc7847d824","origin":"The Stacks Project","memory_eligible":false,"source_rank":13274,"rank":13274,"depth":0,"x":1589.59,"y":1625.683,"cluster":"deformation-theory"},{"id":"stacks:06GZ","tag":"06GZ","title":"The completed base category · Lemma 06GZ","summary":"Let f: R → S be a ring map in widehatC_Lambda. The following are equivalent • f is surjective, • the map m_R/ m_R^2 → m_S/ m_S^2 is surjective, and • the map m_R/( m_Lambda R + m_R^2) → m_S/( m_Lambda S + m_S^2) is surjective.","statement_latex":"Let $f: R \\to S$ be a ring map in $\\widehat{\\mathcal{C}}_\\Lambda$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is surjective,\n\\item the map\n$\\mathfrak m_R/\\mathfrak m_R^2 \\to \\mathfrak m_S/\\mathfrak m_S^2$\nis surjective, and\n\\item the map\n$\\mathfrak m_R/(\\mathfrak m_\\Lambda R + \\mathfrak m_R^2) \\to\n\\mathfrak m_S/(\\mathfrak m_\\Lambda S + \\mathfrak m_S^2)$\nis surjective.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The completed base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GZ","source_file":"formal-defos.tex","source_line":868,"source_end_line":882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L868-L882","statement_sha256":"0c9edaea79b942a6e3baff7decaeeba48c119fcd491cad3dfdfc2e782555f23a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13275,"rank":13275,"depth":4,"x":1551.095,"y":1585.929,"cluster":"deformation-theory"},{"id":"stacks:06SB","tag":"06SB","title":"The completed base category · Lemma 06SB","summary":"The category widehatC_Lambda admits pushouts.","statement_latex":"The category $\\widehat{\\mathcal{C}}_\\Lambda$ admits pushouts.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The completed base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SB","source_file":"formal-defos.tex","source_line":905,"source_end_line":908,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L905-L908","statement_sha256":"dc917131802184d50b6e8f9facfba30dd3458f60562eafec184412210903c758","origin":"The Stacks Project","memory_eligible":false,"source_rank":13276,"rank":13276,"depth":6,"x":1613.908,"y":1593.738,"cluster":"deformation-theory"},{"id":"stacks:06H1","tag":"06H1","title":"The completed base category · Lemma 06H1","summary":"The category widehatC_Lambda admits coproducts of pairs of objects.","statement_latex":"The category $\\widehat{\\mathcal{C}}_\\Lambda$ admits coproducts\nof pairs of objects.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The completed base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06H1","source_file":"formal-defos.tex","source_line":930,"source_end_line":934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L930-L934","statement_sha256":"cd71c99f1c7790d97e6c6ed84cecfeea4b0aee0545e30a6d30bf6be63442d80f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13277,"rank":13277,"depth":6,"x":1559.306,"y":1624.725,"cluster":"deformation-theory"},{"id":"stacks:06SC","tag":"06SC","title":"The completed base category · Lemma 06SC","summary":"Let S be an object of widehatC_Lambda. Then dim_k Der_Lambda(S, k) < ∞.","statement_latex":"Let $S$ be an object of $\\widehat{\\mathcal{C}}_\\Lambda$.\nThen $\\dim_k \\text{Der}_\\Lambda(S, k) < \\infty$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The completed base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SC","source_file":"formal-defos.tex","source_line":976,"source_end_line":980,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L976-L980","statement_sha256":"407b5b4cc479884794d4a7a03e632127712782bf5156613dc8a3fe719b13c3c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13278,"rank":13278,"depth":0,"x":1575.219,"y":1569.01,"cluster":"deformation-theory"},{"id":"stacks:06SD","tag":"06SD","title":"The completed base category · Lemma 06SD","summary":"Let f : R → S be a morphism of widehatC_Lambda. If Der_Lambda(S, k) → Der_Lambda(R, k) is injective, then f is surjective.","statement_latex":"Let $f : R \\to S$ be a morphism of $\\widehat{\\mathcal{C}}_\\Lambda$.\nIf $\\text{Der}_\\Lambda(S, k) \\to \\text{Der}_\\Lambda(R, k)$ is injective,\nthen $f$ is surjective.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The completed base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SD","source_file":"formal-defos.tex","source_line":1002,"source_end_line":1007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1002-L1007","statement_sha256":"eccf35a3391b6f190601b30e65b639e83003d169ac118604b4b0ce204bef00c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13279,"rank":13279,"depth":5,"x":1609.349,"y":1620.778,"cluster":"deformation-theory"},{"id":"stacks:06SE","tag":"06SE","title":"The completed base category · Lemma 06SE","summary":"Let R be an object of widehatC_Lambda. Let (J_n) be a decreasing sequence of ideals such that m_R^n ⊂ J_n. Set J = ⋂ J_n. Then the sequence (J_n/J) defines the m_R/J-adic topology on R/J.","statement_latex":"Let $R$ be an object of $\\widehat{\\mathcal{C}}_\\Lambda$. Let $(J_n)$ be a\ndecreasing sequence of ideals such that $\\mathfrak m_R^n \\subset J_n$.\nSet $J = \\bigcap J_n$. Then the sequence $(J_n/J)$ defines the\n$\\mathfrak m_{R/J}$-adic topology on $R/J$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The completed base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SE","source_file":"formal-defos.tex","source_line":1033,"source_end_line":1039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1033-L1039","statement_sha256":"335d8e81a83816dd9f57fa7bb2970a306c21cd7287ed5bc9b611d7c54e3d4d16","origin":"The Stacks Project","memory_eligible":false,"source_rank":13280,"rank":13280,"depth":4,"x":1540.505,"y":1601.372,"cluster":"deformation-theory"},{"id":"stacks:06SF","tag":"06SF","title":"The completed base category · Lemma 06SF","summary":"Let … → A_3 → A_2 → A_1 be a sequence of surjective ring maps in C_Lambda. If dim_k ( m_A_n/ m_A_n^2) is bounded, then S = lim A_n is an object in widehatC_Lambda and the ideals I_n = Ker(S → A_n) define the m_S-adic topology on S.","statement_latex":"Let $\\ldots \\to A_3 \\to A_2 \\to A_1$ be a sequence of surjective\nring maps in $\\mathcal{C}_\\Lambda$. If\n$\\dim_k (\\mathfrak m_{A_n}/\\mathfrak m_{A_n}^2)$ is bounded, then\n$S = \\lim A_n$ is an object in $\\widehat{\\mathcal{C}}_\\Lambda$\nand the ideals $I_n = \\Ker(S \\to A_n)$ define the\n$\\mathfrak m_S$-adic topology on $S$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The completed base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SF","source_file":"formal-defos.tex","source_line":1060,"source_end_line":1068,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1060-L1068","statement_sha256":"f6299a4b63a8d05f3b325fe1fedca171a4015f05762f84b7fe412be571f564c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13281,"rank":13281,"depth":5,"x":1608.808,"y":1575.919,"cluster":"deformation-theory"},{"id":"stacks:06SG","tag":"06SG","title":"The completed base category · Lemma 06SG","summary":"Let R', R ∈ Ob(widehatC_Lambda). Suppose that R = R' ⊕ I for some ideal I of R. Let x_1, …, x_r ∈ I map to a basis of I/ m_R I. Set S = R'[[X_1, …, X_r]] and consider the R'-algebra map S → R mapping X_i to x_i. Assume that for every n gg 0 the map S/ m_S^n → R/ m_R^n has a left inverse in C_Lambda. Then S → R is an isomorphism.","statement_latex":"Let $R', R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. Suppose that\n$R = R' \\oplus I$ for some ideal $I$ of $R$. Let $x_1, \\ldots, x_r \\in I$\nmap to a basis of $I/\\mathfrak m_R I$. Set $S = R'[[X_1, \\ldots, X_r]]$\nand consider the $R'$-algebra map $S \\to R$ mapping $X_i$ to $x_i$.\nAssume that for every $n \\gg 0$ the map\n$S/\\mathfrak m_S^n \\to R/\\mathfrak m_R^n$ has a left inverse in\n$\\mathcal{C}_\\Lambda$. Then $S \\to R$ is an isomorphism.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"The completed base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SG","source_file":"formal-defos.tex","source_line":1098,"source_end_line":1107,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1098-L1107","statement_sha256":"ddbe12ab2df5b66b411782d82382353850846f1208fb7a9be826494e37e18150","origin":"The Stacks Project","memory_eligible":false,"source_rank":13282,"rank":13282,"depth":5,"x":1578.073,"y":1635.013,"cluster":"deformation-theory"},{"id":"stacks:06GJ","tag":"06GJ","title":"Categories cofibered in groupoids · Definition 06GJ","summary":"Let C be a category. A category cofibered in groupoids over C is a category F equipped with a functor p: F → C such that F^opp is a category fibered in groupoids over C^opp via p^opp: F^opp → C^opp.","statement_latex":"Let $\\mathcal{C}$ be a category.  A {\\it category cofibered in groupoids over\n$\\mathcal{C}$} is a category $\\mathcal{F}$ equipped with a functor\n$p: \\mathcal{F} \\to \\mathcal{C}$ such that $\\mathcal{F}^{opp}$ is a category\nfibered in groupoids over $\\mathcal{C}^{opp}$ via\n$p^{opp}: \\mathcal{F}^{opp} \\to \\mathcal{C}^{opp}$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Categories cofibered in groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GJ","source_file":"formal-defos.tex","source_line":1141,"source_end_line":1148,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1141-L1148","statement_sha256":"2f52e527cd3f448eddfa7054e86965f1dd237767a3fc1a89283a1860a3d13b40","origin":"The Stacks Project","memory_eligible":false,"source_rank":13283,"rank":13283,"depth":0,"x":1552.589,"y":1572.408,"cluster":"deformation-theory"},{"id":"stacks:06GX","tag":"06GX","title":"Prorepresentable functors and predeformation categories · Definition 06GX","summary":"Let F : C_Lambda → Sets be a functor. We say F is prorepresentable if there exists an isomorphism F ≅ underlineR|_C_Lambda of functors for some R ∈ Ob(widehatC_Lambda).","statement_latex":"Let $F : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a functor.\nWe say $F$ is {\\it prorepresentable} if there exists an isomorphism\n$F \\cong \\underline{R}|_{\\mathcal{C}_\\Lambda}$ of functors for some\n$R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Prorepresentable functors and predeformation categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GX","source_file":"formal-defos.tex","source_line":1314,"source_end_line":1320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1314-L1320","statement_sha256":"19ef2ec435d9e4dc7badf532c2244edd6db8c1dd1d5b8903607bb8d785ec2061","origin":"The Stacks Project","memory_eligible":false,"source_rank":13284,"rank":13284,"depth":0,"x":1623.422,"y":1604.908,"cluster":"deformation-theory"},{"id":"stacks:06GS","tag":"06GS","title":"Prorepresentable functors and predeformation categories · Definition 06GS","summary":"A predeformation category F is a category cofibered in groupoids over C_Lambda such that F(k) is equivalent to a category with a single object and a single morphism, i.e., F(k) contains at least one object and there is a unique morphism between any two objects. A morphism of predeformation categories is a morphism of categories cofibered in groupoids over C_Lambda.","statement_latex":"A {\\it predeformation category} $\\mathcal{F}$ is a category cofibered\nin groupoids over $\\mathcal{C}_\\Lambda$ such that $\\mathcal{F}(k)$ is\nequivalent to a category with a single object and a single morphism,\ni.e., $\\mathcal{F}(k)$ contains at least one object and there is a\nunique morphism between any two objects. A {\\it morphism of predeformation\ncategories} is a morphism of categories cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Prorepresentable functors and predeformation categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06GS","source_file":"formal-defos.tex","source_line":1332,"source_end_line":1341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1332-L1341","statement_sha256":"f119c02d56170afb966367b472e92880dc2a8c790fbe549cb3a9cb28c7ab5cb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13285,"rank":13285,"depth":0,"x":1543.208,"y":1621.503,"cluster":"deformation-theory"},{"id":"stacks:06H3","tag":"06H3","title":"Formal objects and completion categories · Definition 06H3","summary":"Let F be a category cofibered in groupoids over C_Lambda. The category widehatF of formal objects of F is the category with the following objects and morphisms. • A formal object xi = (R, xi_n, f_n) of F consists of an object R of widehatC_Lambda, and a collection indexed by n ∈ N of objects xi_n of F(R/ m_R^n) and morphisms f_n : xi_n + 1 → xi_n lying over the projection R/ m_R^n + 1 → R/ m_R^n. • Let xi = (R, xi_n, f_n) and eta = (S, eta_n, g_n) be formal objects of F.…","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. The {\\it category $\\widehat{\\mathcal{F}}$ of formal\nobjects of  $\\mathcal{F}$} is the category with the following objects and\nmorphisms.\n\\begin{enumerate}\n\\item A {\\it formal object $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{F}$}\nconsists of an object $R$ of $\\widehat{\\mathcal{C}}_\\Lambda$, and a collection\nindexed by $n \\in \\mathbf{N}$ of objects $\\xi_n$ of\n$\\mathcal{F}(R/\\mathfrak m_R^n)$ and morphisms\n$f_n : \\xi_{n + 1} \\to \\xi_n$ lying over the projection\n$R/\\mathfrak m_R^{n + 1} \\to R/\\mathfrak m_R^n$.\n\\item Let $\\xi = (R, \\xi_n, f_n)$ and $\\eta = (S, \\eta_n, g_n)$ be\nformal objects of $\\mathcal{F}$.  A {\\it morphism $a : \\xi \\to \\eta$ of\nformal objects} consists of a map $a_0 : R \\to S$ in\n$\\widehat{\\mathcal{C}}_\\Lambda$ and a collection $a_n : \\xi_n \\to \\eta_n$\nof morphisms of $\\mathcal{F}$ lying over\n$R/\\mathfrak m_R^n \\to S/\\mathfrak m_S^n$,\nsuch that for every $n$ the diagram\n$$\n\\xymatrix{\n\\xi_{n + 1} \\ar[r]_{f_n} \\ar[d]_{a_{n + 1}} & \\xi_n \\ar[d]^{a_n} \\\\\n\\eta_{n + 1} \\ar[r]^{g_n} & \\eta_n\n}\n$$\ncommutes.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Formal objects and completion categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06H3","source_file":"formal-defos.tex","source_line":1398,"source_end_line":1426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1398-L1426","statement_sha256":"0f4db837eec4d9f0d150023d301a4b430d5caac7a0d7eae7177e38f8a2e05933","origin":"The Stacks Project","memory_eligible":false,"source_rank":13286,"rank":13286,"depth":0,"x":1590.054,"y":1562.461,"cluster":"deformation-theory"},{"id":"stacks:06H4","tag":"06H4","title":"Formal objects and completion categories · Lemma 06H4","summary":"Let p : F → C_Lambda be a category cofibered in groupoids. Then widehatp : widehatF → widehatC_Lambda is a category cofibered in groupoids.","statement_latex":"Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in\ngroupoids. Then\n$\\widehat{p} : \\widehat{\\mathcal{F}} \\to \\widehat{\\mathcal{C}}_\\Lambda$\nis a category cofibered in groupoids.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Formal objects and completion categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06H4","source_file":"formal-defos.tex","source_line":1434,"source_end_line":1440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1434-L1440","statement_sha256":"f07282a735ef04804532ec8b9c00d6731c2f4ee9aa3f536556e785dbf0326970","origin":"The Stacks Project","memory_eligible":false,"source_rank":13287,"rank":13287,"depth":0,"x":1603.254,"y":1634.088,"cluster":"deformation-theory"},{"id":"stacks:06H5","tag":"06H5","title":"Formal objects and completion categories · Definition 06H5","summary":"Let p : F → C_Lambda be a category cofibered in groupoids. The category cofibered in groupoids widehatp : widehat F → widehatC_Lambda is called the completion of F.","statement_latex":"Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in\ngroupoids. The category cofibered in groupoids\n$\\widehat{p} : \\widehat{\\mathcal  F} \\to \\widehat{\\mathcal{C}}_\\Lambda$\nis called the {\\it completion of $\\mathcal{F}$}.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Formal objects and completion categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06H5","source_file":"formal-defos.tex","source_line":1472,"source_end_line":1478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1472-L1478","statement_sha256":"9880e18be60c29c2df0893a8361adcbd252787813dccc7c60fca9d720aba2ddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13288,"rank":13288,"depth":0,"x":1534.542,"y":1587.818,"cluster":"deformation-theory"},{"id":"stacks:06H6","tag":"06H6","title":"Formal objects and completion categories · Lemma 06H6","summary":"In the situation above, widehatF_I(R) is equivalent to the category widehatF(R).","statement_latex":"In the situation above, $\\widehat{\\mathcal{F}}_\\mathcal{I}(R)$ is equivalent\nto the category $\\widehat{\\mathcal{F}}(R)$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Formal objects and completion categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06H6","source_file":"formal-defos.tex","source_line":1505,"source_end_line":1509,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1505-L1509","statement_sha256":"6ae4697bfc98805492f465d7d60d942e17a1c39b458f87ee63c70ecfe6d5b32c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13289,"rank":13289,"depth":0,"x":1624.157,"y":1582.863,"cluster":"deformation-theory"},{"id":"stacks:06HG","tag":"06HG","title":"Smooth morphisms · Definition 06HG","summary":"Let φ : F → G be a morphism of categories cofibered in groupoids over C_Lambda. We say φ is smooth if it satisfies the following condition: Let B → A be a surjective ring map in C_Lambda. Let y ∈ Ob(G(B)), x ∈ Ob(F(A)), and y → φ(x) be a morphism lying over B → A. Then there exists x' ∈ Ob(F(B)), a morphism x' → x lying over B → A, and a morphism φ(x') → y lying over id: B → B, such that the diagram xymatrix φ(x') ar[r] ar[dr] & y ar[d] & φ(x) commutes.","statement_latex":"Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories\ncofibered in groupoids over $\\mathcal{C}_\\Lambda$.  We say  $\\varphi$ is\n{\\it smooth} if it satisfies the following condition: Let $B \\to A$ be\na surjective ring map in $\\mathcal{C}_\\Lambda$.  Let $y \\in\n\\Ob(\\mathcal{G}(B)), x \\in \\Ob(\\mathcal{F}(A))$, and $y\n\\to \\varphi(x)$ be a morphism lying over $B \\to A$.  Then there\nexists $x' \\in \\Ob(\\mathcal{F}(B))$, a morphism $x' \\to x$\nlying over $B \\to A$, and a morphism $\\varphi(x') \\to y$ lying\nover $\\text{id}: B \\to B$, such that the diagram\n$$\n\\xymatrix{\n\\varphi(x') \\ar[r] \\ar[dr] & y \\ar[d] \\\\\n& \\varphi(x)\n}\n$$\ncommutes.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HG","source_file":"formal-defos.tex","source_line":1895,"source_end_line":1913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1895-L1913","statement_sha256":"12acbf7e32b13ba76af47646e6f9d69602befd2f484041b74265da087ec6178f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13290,"rank":13290,"depth":0,"x":1560.87,"y":1638.396,"cluster":"deformation-theory"},{"id":"stacks:06HH","tag":"06HH","title":"Smooth morphisms · Lemma 06HH","summary":"Let φ : F → G be a morphism of categories cofibered in groupoids over C_Lambda. Then φ is smooth if the condition in Definition [Tag 06HG] is assumed to hold only for small extensions B → A.","statement_latex":"Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories\ncofibered in groupoids over $\\mathcal{C}_\\Lambda$.  Then $\\varphi$ is smooth\nif the condition in Definition \\ref{definition-smooth-morphism} is assumed to\nhold only for small extensions $B \\to A$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HH","source_file":"formal-defos.tex","source_line":1915,"source_end_line":1921,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L1915-L1921","statement_sha256":"08b130672f8a59e88304f24b3480a85622b45b15ac3d87edf9ff65189f0e2d6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13291,"rank":13291,"depth":8,"x":1562.927,"y":1560.127,"cluster":"deformation-theory"},{"id":"stacks:06HL","tag":"06HL","title":"Smooth morphisms · Lemma 06HL","summary":"Let R → S be a ring map in widehatC_Lambda. Then the induced morphism underlineS|_C_Lambda → underlineR|_C_Lambda is smooth if and only if S is a power series ring over R.","statement_latex":"Let $R \\to S$ be a ring map in $\\widehat{\\mathcal{C}}_\\Lambda$. Then\nthe induced morphism\n$\\underline{S}|_{\\mathcal{C}_\\Lambda} \\to \\underline{R}|_{\\mathcal{C}_\\Lambda}$\nis smooth if and only if $S$ is a power series ring over $R$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HL","source_file":"formal-defos.tex","source_line":2024,"source_end_line":2030,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2024-L2030","statement_sha256":"cc8cff907d289c0b143b3b901d015c1ab4cb4680a4096461fc3f77b7ae97b746","origin":"The Stacks Project","memory_eligible":false,"source_rank":13292,"rank":13292,"depth":5,"x":1625.43,"y":1620.056,"cluster":"deformation-theory"},{"id":"stacks:06HM","tag":"06HM","title":"Smooth morphisms · Lemma 06HM","summary":"Let φ : F → G and ψ : G → H be morphisms of categories cofibered in groupoids over C_Lambda. • If φ and ψ are smooth, then ψ ∘ φ is smooth. • If φ is essentially surjective and ψ ∘ φ is smooth, then ψ is smooth. • If G' → G is a morphism of categories cofibered in groupoids and φ is smooth, then F ×_G G' → G' is smooth.","statement_latex":"Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and $\\psi : \\mathcal{G}\n\\to \\mathcal{H}$ be morphisms of categories cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$.\n\\begin{enumerate}\n\\item If $\\varphi$ and $\\psi$ are smooth, then $\\psi \\circ \\varphi$ is smooth.\n\\item If $\\varphi$ is essentially surjective and $\\psi \\circ \\varphi$ is\nsmooth, then $\\psi$ is smooth.\n\\item If $\\mathcal{G}' \\to \\mathcal{G}$ is a morphism of categories\ncofibered in groupoids and $\\varphi$ is smooth, then\n$\\mathcal{F} \\times_\\mathcal{G} \\mathcal{G}' \\to \\mathcal{G}'$ is smooth.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HM","source_file":"formal-defos.tex","source_line":2107,"source_end_line":2120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2107-L2120","statement_sha256":"99f6f4e354a627467e13a246e4c7214bf042c83246e33215c4d5d0d826feb535","origin":"The Stacks Project","memory_eligible":false,"source_rank":13293,"rank":13293,"depth":0,"x":1529.539,"y":1611.173,"cluster":"deformation-theory"},{"id":"stacks:06HN","tag":"06HN","title":"Smooth morphisms · Lemma 06HN","summary":"Let φ : F → G be a smooth morphism of categories cofibered in groupoids over C_Lambda. Assume φ : F(k) → G(k) is essentially surjective. Then φ : F → G and widehatφ : widehatF → widehatG are essentially surjective.","statement_latex":"Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a smooth morphism of\ncategories cofibered in groupoids over $\\mathcal{C}_\\Lambda$.  Assume\n$\\varphi : \\mathcal{F}(k) \\to \\mathcal{G}(k)$ is essentially surjective.\nThen $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and\n$\\widehat{\\varphi} : \\widehat{\\mathcal{F}} \\to \\widehat{\\mathcal{G}}$\nare essentially surjective.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HN","source_file":"formal-defos.tex","source_line":2131,"source_end_line":2139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2131-L2139","statement_sha256":"e9b2756c87c065538ba4d3263ea8eef3977555304cbebf6ffd0b4ed72bd164ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":13294,"rank":13294,"depth":0,"x":1608.682,"y":1562.53,"cluster":"deformation-theory"},{"id":"stacks:06HR","tag":"06HR","title":"Smooth morphisms · Definition 06HR","summary":"Let F be a category cofibered in groupoids. Let xi be a formal object of F lying over R ∈ Ob(widehatC_Lambda). We say xi is versal if the corresponding morphism underlinexi: underlineR|_C_Lambda → F of Remark [Tag 06HC] is smooth.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids.  Let $\\xi$ be a formal\nobject of $\\mathcal{F}$ lying over $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$.\nWe say $\\xi$ is {\\it versal} if the corresponding morphism\n$\\underline{\\xi}: \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$\nof Remark \\ref{remark-formal-objects-yoneda} is smooth.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HR","source_file":"formal-defos.tex","source_line":2186,"source_end_line":2193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2186-L2193","statement_sha256":"14237f535a163967430334c4bff3b3e3f1b18641b31d51e8356635bffe85c7a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13295,"rank":13295,"depth":0,"x":1589.124,"y":1644.596,"cluster":"deformation-theory"},{"id":"stacks:06HT","tag":"06HT","title":"Smooth morphisms · Lemma 06HT","summary":"Let F be a predeformation category. Let xi be a versal formal object of F. For any formal object eta of widehatF, there exists a morphism xi → eta.","statement_latex":"Let $\\mathcal{F}$ be a predeformation category.\nLet $\\xi$ be a versal formal object of $\\mathcal{F}$.\nFor any formal object $\\eta$ of $\\widehat{\\mathcal{F}}$,\nthere exists a morphism $\\xi \\to \\eta$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HT","source_file":"formal-defos.tex","source_line":2222,"source_end_line":2228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2222-L2228","statement_sha256":"fd57a722032b8b12a170f2a3ea9c30c3b523d9bcf74ad8ac534f144fe1c9ae2a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13296,"rank":13296,"depth":1,"x":1536.76,"y":1571.871,"cluster":"deformation-theory"},{"id":"stacks:06HP","tag":"06HP","title":"Smooth or unobstructed categories · Definition 06HP","summary":"Let p : F → C_Lambda be a category cofibered in groupoids. We say F is smooth or unobstructed if its structure morphism p is smooth in the sense of Definition [Tag 06HG].","statement_latex":"Let $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in\ngroupoids. We say $\\mathcal{F}$ is {\\it smooth} or {\\it unobstructed}\nif its structure morphism $p$ is smooth\nin the sense of Definition \\ref{definition-smooth-morphism}.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth or unobstructed categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HP","source_file":"formal-defos.tex","source_line":2263,"source_end_line":2269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2263-L2269","statement_sha256":"0769d683ba3fafb75df821a6d75092db2913a92288e6e3e857175148a36dae16","origin":"The Stacks Project","memory_eligible":false,"source_rank":13297,"rank":13297,"depth":1,"x":1635.311,"y":1596.151,"cluster":"deformation-theory"},{"id":"stacks:0DYL","tag":"0DYL","title":"Smooth or unobstructed categories · Lemma 0DYL","summary":"Let R ∈ Ob(widehatC_Lambda). The following are equivalent • underlineR|_C_Lambda is smooth, • Lambda → R is formally smooth in the m_R-adic topology, • Lambda → R is flat and R ⊗_Lambda k' is geometrically regular over k', and • Lambda → R is flat and k' → R ⊗_Lambda k' is formally smooth in the m_R-adic topology. In the classical case, these are also equivalent to • [(5)] R is isomorphic to Lambda[[x_1, …, x_n]] for some n.","statement_latex":"Let $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. The following are\nequivalent\n\\begin{enumerate}\n\\item $\\underline{R}|_{\\mathcal{C}_\\Lambda}$ is smooth,\n\\item $\\Lambda \\to R$ is formally smooth in the $\\mathfrak m_R$-adic topology,\n\\item $\\Lambda \\to R$ is flat and $R \\otimes_\\Lambda k'$ is\ngeometrically regular over $k'$, and\n\\item $\\Lambda \\to R$ is flat and $k' \\to R \\otimes_\\Lambda k'$ is\nformally smooth in the $\\mathfrak m_R$-adic topology.\n\\end{enumerate}\nIn the classical case, these are also equivalent to\n\\begin{enumerate}\n\\item[(5)] $R$ is isomorphic to $\\Lambda[[x_1, \\ldots, x_n]]$\nfor some $n$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth or unobstructed categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYL","source_file":"formal-defos.tex","source_line":2294,"source_end_line":2311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2294-L2311","statement_sha256":"c6b1e93152d8dc80bcc5cef204d68d40efd7d9200e1b88a3834ba6d47a8f9bc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13298,"rank":13298,"depth":46,"x":1541.768,"y":1634.715,"cluster":"deformation-theory"},{"id":"stacks:0DZK","tag":"0DZK","title":"Smooth or unobstructed categories · Lemma 0DZK","summary":"Let F be a predeformation category. Let xi be a versal formal object of F lying over R ∈ Ob(widehatC_Lambda). The following are equivalent • F is unobstructed, and • Lambda → R is formally smooth in the m_R-adic topology. In the classical case these are also equivalent to • [(3)] R ≅ Lambda[[x_1, …, x_n]] for some n.\\","statement_latex":"Let $\\mathcal{F}$ be a predeformation category.\nLet $\\xi$ be a versal formal object of $\\mathcal{F}$ lying over\n$R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$. The following are\nequivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is unobstructed, and\n\\item $\\Lambda \\to R$ is formally smooth in the $\\mathfrak m_R$-adic topology.\n\\end{enumerate}\nIn the classical case these are also equivalent to\n\\begin{enumerate}\n\\item[(3)] $R \\cong \\Lambda[[x_1, \\ldots, x_n]]$ for some $n$.\\\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth or unobstructed categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZK","source_file":"formal-defos.tex","source_line":2344,"source_end_line":2358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2344-L2358","statement_sha256":"b700ada90e9d0f9bba8f2084237d584217597309037b70fbf709b0ed32505882","origin":"The Stacks Project","memory_eligible":false,"source_rank":13299,"rank":13299,"depth":47,"x":1580.278,"y":1552.047,"cluster":"deformation-theory"},{"id":"stacks:06SM","tag":"06SM","title":"Smooth or unobstructed categories · Lemma 06SM","summary":"There exists an R ∈ Ob(widehatC_Lambda) such that the equivalent conditions of Lemma [Tag 0DYL] hold and moreover H_1(L_k/Lambda) = m_R/ m_R^2 and Ω_R/Lambda ⊗_R k = Ω_k/Lambda.","statement_latex":"There exists an $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$\nsuch that the equivalent conditions of Lemma \\ref{lemma-smooth}\nhold and moreover $H_1(L_{k/\\Lambda}) = \\mathfrak m_R/\\mathfrak m_R^2$\nand $\\Omega_{R/\\Lambda} \\otimes_R k = \\Omega_{k/\\Lambda}$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth or unobstructed categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SM","source_file":"formal-defos.tex","source_line":2380,"source_end_line":2386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2380-L2386","statement_sha256":"5fc87e5954ce34a26f1c5b866573db86f511ea4ffa5b28fa302b4a455abc4404","origin":"The Stacks Project","memory_eligible":false,"source_rank":13300,"rank":13300,"depth":47,"x":1618.878,"y":1636.0,"cluster":"deformation-theory"},{"id":"stacks:06HW","tag":"06HW","title":"Schlessinger's conditions · Definition 06HW","summary":"Let F be a category cofibered in groupoids over C_Lambda. We define conditions (S1) and (S2) on F as follows: • [(S1)] Every diagram in F vcenter xymatrix & x_2 ar[d] x_1 ar[r] & x lying over vcenter xymatrix & A_2 ar[d] A_1 ar[r] & A in C_Lambda with A_2 → A surjective can be completed to a commutative diagram vcenter xymatrix y ar[r] ar[d] & x_2 ar[d] x_1 ar[r] & x lying over vcenter xymatrix A_1 ×_A A_2 ar[r] ar[d] & A_2 ar[d] A_1 ar[r] & A. • [(S2)] The condition of…","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal\nC_\\Lambda$. We define {\\it conditions (S1) and (S2)}\non $\\mathcal{F}$ as follows:\n\\begin{enumerate}\n\\item[(S1)] Every diagram in $\\mathcal{F}$\n$$\n\\vcenter{\n\\xymatrix{\n           & x_2 \\ar[d] \\\\\nx_1 \\ar[r] & x\n}\n}\n\\quad\\text{lying over}\\quad\n\\vcenter{\n\\xymatrix{\n           & A_2 \\ar[d] \\\\\nA_1 \\ar[r] & A\n}\n}\n$$\nin $\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective can be completed\nto a commutative diagram\n$$\n\\vcenter{\n\\xymatrix{\ny \\ar[r] \\ar[d] & x_2 \\ar[d] \\\\\nx_1 \\ar[r]      & x\n}\n}\n\\quad\\text{lying over}\\quad\n\\vcenter{\n\\xymatrix{\nA_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\\nA_1 \\ar[r]      & A.\n}\n}\n$$\n\\item[(S2)]\nThe condition of (S1) holds for diagrams in $\\mathcal{F}$ lying over\na diagram in $\\mathcal{C}_\\Lambda$ of the form\n$$\n\\xymatrix{\n          & k[\\epsilon] \\ar[d] \\\\\nA  \\ar[r] & k.\n}\n$$\nMoreover, if we have two commutative diagrams in $\\mathcal{F}$\n$$\n\\vcenter{\n\\xymatrix{\ny \\ar[r]_c \\ar[d]_a & x_\\epsilon \\ar[d]^e \\\\\nx \\ar[r]^d          & x_0\n}\n}\n\\quad\\text{and}\\quad\n\\vcenter{\n\\xymatrix{\ny' \\ar[r]_{c'} \\ar[d]_{a'} & x_\\epsilon \\ar[d]^e \\\\\nx \\ar[r]^d                 & x_0\n}\n}\n\\quad\\text{lying over}\\quad\n\\vcenter{\n\\xymatrix{\nA \\times_k k[\\epsilon] \\ar[r] \\ar[d] & k[\\epsilon] \\ar[d] \\\\\nA  \\ar[r] & k\n}\n}\n$$\nthen there exists a morphism $b : y \\to y'$ in\n$\\mathcal{F}(A \\times_k k[\\epsilon])$ such that $a = a' \\circ b$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Schlessinger's conditions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HW","source_file":"formal-defos.tex","source_line":2513,"source_end_line":2587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2513-L2587","statement_sha256":"6f7681c7d01e041e2640e65f4a7ab83711d1843b12606ca7099565887f126b3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13301,"rank":13301,"depth":0,"x":1521.62,"y":1595.455,"cluster":"deformation-theory"},{"id":"stacks:06HX","tag":"06HX","title":"Schlessinger's conditions · Lemma 06HX","summary":"Let F be a category cofibered in groupoids over C_Lambda. Then F satisfies (S1) if the condition of (S1) is assumed to hold only when A_2 → A is a small extension.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal\nC_\\Lambda$. Then $\\mathcal{F}$ satisfies (S1) if the condition of (S1)\nis assumed to hold only when $A_2 \\to A$ is a small extension.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Schlessinger's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HX","source_file":"formal-defos.tex","source_line":2619,"source_end_line":2624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2619-L2624","statement_sha256":"1e50422aa71d055047eacb31c428a5dbf6e35c746df23970e44146e077ee53b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13302,"rank":13302,"depth":9,"x":1627.305,"y":1569.842,"cluster":"deformation-theory"},{"id":"stacks:06HZ","tag":"06HZ","title":"Schlessinger's conditions · Lemma 06HZ","summary":"Let F be a category cofibred in groupoids over C_Lambda. If F satisfies (S2), then the condition of (S2) also holds when k[ε] is replaced by k[V] for any finite dimensional k-vector space V.","statement_latex":"Let $\\mathcal{F}$ be a category cofibred in groupoids over\n$\\mathcal{C}_\\Lambda$. If $\\mathcal{F}$ satisfies (S2), then the\ncondition of (S2) also holds when $k[\\epsilon]$ is replaced by $k[V]$\nfor any finite dimensional $k$-vector space $V$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Schlessinger's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06HZ","source_file":"formal-defos.tex","source_line":2654,"source_end_line":2660,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2654-L2660","statement_sha256":"70f2a5e1849a9f89384e4c5a65e3e3c621996b25a6fdf533648214d0d42a3d5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13303,"rank":13303,"depth":0,"x":1569.237,"y":1649.697,"cluster":"deformation-theory"},{"id":"stacks:06I0","tag":"06I0","title":"Schlessinger's conditions · Lemma 06I0","summary":"Let F be a category cofibered in groupoids over C_Lambda. • If F satisfies (S1), then so does overlineF. • If F satisfies (S2), then so does overlineF provided at least one of the following conditions is satisfied • F is a predeformation category, • the category F(k) is a set or a setoid, or • for any morphism x_ε → x_0 of F lying over k[ε] → k the pushforward map Aut_k[ε](x_ε) → Aut_k(x_0) is surjective.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ satisfies (S1), then so does\n$\\overline{\\mathcal{F}}$.\n\\item If $\\mathcal{F}$ satisfies (S2), then so does\n$\\overline{\\mathcal{F}}$ provided at least one of the following conditions is\nsatisfied\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a predeformation category,\n\\item the category $\\mathcal{F}(k)$ is a set or a setoid, or\n\\item for any morphism $x_\\epsilon \\to x_0$ of $\\mathcal{F}$\nlying over $k[\\epsilon] \\to k$ the pushforward map\n$\\text{Aut}_{k[\\epsilon]}(x_\\epsilon) \\to \\text{Aut}_k(x_0)$\nis surjective.\n\\end{enumerate}\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Schlessinger's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06I0","source_file":"formal-defos.tex","source_line":2877,"source_end_line":2896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2877-L2896","statement_sha256":"bbf25c30628ff9c6e721a3c8284156f59fac4a486974ad38d7567a558f9ccc10","origin":"The Stacks Project","memory_eligible":false,"source_rank":13304,"rank":13304,"depth":0,"x":1547.579,"y":1556.724,"cluster":"deformation-theory"},{"id":"stacks:06SQ","tag":"06SQ","title":"Schlessinger's conditions · Lemma 06SQ","summary":"Let F be a category cofibered in groupoids over C_Lambda. Let x_0 ∈ Ob(F(k)). Let F_x_0 be the category cofibred in groupoids over C_Lambda constructed in Remark [Tag 06GU]. • If F satisfies (S1), then so does F_x_0. • If F satisfies (S2), then so does F_x_0.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$.\nLet $\\mathcal{F}_{x_0}$ be the category cofibred in groupoids over\n$\\mathcal{C}_\\Lambda$ constructed in\nRemark \\ref{remark-localize-cofibered-groupoid}.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ satisfies (S1), then so does $\\mathcal{F}_{x_0}$.\n\\item If $\\mathcal{F}$ satisfies (S2), then so does $\\mathcal{F}_{x_0}$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Schlessinger's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SQ","source_file":"formal-defos.tex","source_line":2973,"source_end_line":2984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2973-L2984","statement_sha256":"c7ef7c90c4bc89a376547016206ed5f55aac38201df70a1250deead5e169f0de","origin":"The Stacks Project","memory_eligible":false,"source_rank":13305,"rank":13305,"depth":1,"x":1639.41,"y":1613.677,"cluster":"deformation-theory"},{"id":"stacks:06IS","tag":"06IS","title":"Schlessinger's conditions · Lemma 06IS","summary":"Let p: F → C_Lambda be a category cofibered in groupoids. Consider a diagram of F vcenter xymatrix y ar[r] ar[d]_a & x_ε ar[d]_e x ar[r]^d & x_0 lying over vcenter xymatrix A ×_k k[ε] ar[r] ar[d] & k[ε] ar[d] A ar[r] & k. in C_Lambda. Assume F satisfies (S2). Then there exists a morphism s : x → y with a ∘ s = id_x if and only if there exists a morphism s_ε : x → x_ε with e ∘ s_ε = d.","statement_latex":"Let $p: \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ be a category cofibered in\ngroupoids. Consider a diagram of $\\mathcal{F}$\n$$\n\\vcenter{\n\\xymatrix{\ny \\ar[r] \\ar[d]_a & x_\\epsilon \\ar[d]_e \\\\\nx \\ar[r]^d        & x_0\n}\n}\n\\quad\\text{lying over}\\quad\n\\vcenter{\n\\xymatrix{\nA \\times_k k[\\epsilon] \\ar[r] \\ar[d] & k[\\epsilon] \\ar[d] \\\\\nA \\ar[r] & k.\n}\n}\n$$\nin $\\mathcal{C}_\\Lambda$. Assume $\\mathcal{F}$ satisfies (S2).\nThen there exists a morphism $s : x \\to y$ with $a \\circ s = \\text{id}_x$\nif and only if there exists a morphism $s_\\epsilon : x \\to x_\\epsilon$\nwith $e \\circ s_\\epsilon = d$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Schlessinger's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IS","source_file":"formal-defos.tex","source_line":2994,"source_end_line":3017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L2994-L3017","statement_sha256":"59e96388647375e202d5b45b82132479902cb6cb7234e6f11a304aadae38d0e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13306,"rank":13306,"depth":0,"x":1524.554,"y":1623.901,"cluster":"deformation-theory"},{"id":"stacks:06IT","tag":"06IT","title":"Schlessinger's conditions · Lemma 06IT","summary":"Consider a commutative diagram in a predeformation category F vcenter xymatrix y ar[r] ar[d] & x_2 ar[d]^a_2 x_1 ar[r]^a_1 & x lying over vcenter xymatrix A_1 ×_A A_2 ar[r] ar[d] & A_2 ar[d]^f_2 A_1 ar[r]^f_1 & A in C_Lambda where f_2 : A_2 → A is a small extension. Assume there is a map h : A_1 → A_2 such that f_2 = f_1 ∘ h. Let I = Ker(f_2). Consider the ring map g : A_1 ×_A A_2 → k[I] = k ⊕ I, (u, v) ↦ overlineu ⊕ (v - h(u)) Choose a pushforward y → g_*y. Assume F…","statement_latex":"Consider a commutative diagram in a predeformation category $\\mathcal{F}$\n$$\n\\vcenter{\n\\xymatrix{\ny \\ar[r] \\ar[d] & x_2 \\ar[d]^{a_2} \\\\\nx_1 \\ar[r]^{a_1}        & x\n}\n}\n\\quad\\text{lying over}\n\\vcenter{\n\\xymatrix{\nA_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d]^{f_2} \\\\\nA_1 \\ar[r]^{f_1} & A\n}\n}\n$$\nin $\\mathcal{C}_\\Lambda$ where\n$f_2 : A_2 \\to A$ is a small extension.\nAssume there is a map $h : A_1 \\to A_2$ such that $f_2 = f_1 \\circ h$.\nLet $I = \\Ker(f_2)$. Consider the ring map\n$$\ng : A_1 \\times_A A_2 \\longrightarrow k[I] = k \\oplus I, \\quad\n(u, v) \\longmapsto \\overline{u} \\oplus (v - h(u))\n$$\nChoose a pushforward $y \\to g_*y$. Assume $\\mathcal{F}$ satisfies (S2).\nIf there exists a morphism $x_1 \\to g_*y$, then there exists a\nmorphism $b: x_1 \\to x_2$ such that $a_1 =  a_2 \\circ b$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Schlessinger's conditions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IT","source_file":"formal-defos.tex","source_line":3044,"source_end_line":3073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3044-L3073","statement_sha256":"35b5e523f9f9b1ba50a7b9deba7dd444cd3720d565eee28079562342045c09a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13307,"rank":13307,"depth":1,"x":1601.911,"y":1550.354,"cluster":"deformation-theory"},{"id":"stacks:06I3","tag":"06I3","title":"Tangent spaces of functors · Definition 06I3","summary":"Let L: Mod^fg_R → Mod_R, resp. L: Mod_R → Mod_R be a functor. We say that L is R-linear if for every pair of objects M, N of Mod^fg_R, resp. Mod_R the map L : Hom_R(M, N) → Hom_R(L(M), L(N)) is a map of R-modules.","statement_latex":"Let $L: \\text{Mod}^{fg}_R \\to \\text{Mod}_R$,\nresp.\\ $L: \\text{Mod}_R \\to \\text{Mod}_R$\nbe a functor.  We say that $L$ is {\\it $R$-linear} if for every\npair of objects $M, N$ of $\\text{Mod}^{fg}_R$, resp.\\ $\\text{Mod}_R$\nthe map\n$$\nL : \\Hom_R(M, N) \\longrightarrow \\Hom_R(L(M), L(N))\n$$\nis a map of $R$-modules.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06I3","source_file":"formal-defos.tex","source_line":3117,"source_end_line":3128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3117-L3128","statement_sha256":"1e17667786211f6e2d08f346ede8097f99a19de21e7aa1a086cf91cd61943a1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13308,"rank":13308,"depth":0,"x":1604.032,"y":1649.588,"cluster":"deformation-theory"},{"id":"stacks:06I6","tag":"06I6","title":"Tangent spaces of functors · Lemma 06I6","summary":"Let L: Mod^fg_R → Sets, resp. L: Mod_R → Sets be a functor. Suppose L(0) is a one element set and L preserves finite products. Then there exists a unique R-linear functor widetildeL : Mod^fg_R → Mod_R, resp. widetildeL : Mod_R → Mod_R, such that vcenter xymatrix & Mod_R ar[dr]^forget & Mod^fg_R ar[ur]^widetildeL ar[rr]^L & & Sets resp. vcenter xymatrix & Mod_R ar[dr]^forget & Mod_R ar[ur]^widetildeL ar[rr]^L & & Sets commutes.","statement_latex":"Let $L: \\text{Mod}^{fg}_R \\to \\textit{Sets}$,\nresp.\\ $L: \\text{Mod}_R \\to \\textit{Sets}$ be a\nfunctor.  Suppose $L(0)$ is a one element set and $L$ preserves finite\nproducts.  Then there exists a unique $R$-linear functor\n$\\widetilde{L} : \\text{Mod}^{fg}_R \\to \\text{Mod}_R$,\nresp.\\  $\\widetilde{L} : \\text{Mod}_R \\to \\text{Mod}_R$,\nsuch that\n$$\n\\vcenter{\n\\xymatrix{\n& \\text{Mod}_R \\ar[dr]^{\\text{forget}} &   \\\\\n\\text{Mod}^{fg}_R  \\ar[ur]^{\\widetilde{L}} \\ar[rr]^{L} &  &\n\\textit{Sets}\n}\n}\n\\quad\\text{resp.}\\quad\n\\vcenter{\n\\xymatrix{\n& \\text{Mod}_R \\ar[dr]^{\\text{forget}} &   \\\\\n\\text{Mod}_R  \\ar[ur]^{\\widetilde{L}} \\ar[rr]^{L} &  &\n\\textit{Sets}\n}\n}\n$$\ncommutes.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06I6","source_file":"formal-defos.tex","source_line":3151,"source_end_line":3178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3151-L3178","statement_sha256":"a8b230be74cb579b1c60c8ffdc271674b1d2871d315ac2b250413c6ebf73b3b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13309,"rank":13309,"depth":0,"x":1521.774,"y":1576.821,"cluster":"deformation-theory"},{"id":"stacks:06I7","tag":"06I7","title":"Tangent spaces of functors · Lemma 06I7","summary":"Let L_1, L_2: Mod^fg_R → Sets be functors that take 0 to a one element set and preserve finite products. Let t : L_1 → L_2 be a morphism of functors. Then t induces a morphism widetildet : widetildeL_1 → widetildeL_2 between the functors guaranteed by Lemma [Tag 06I6], which is given simply by widetildet_M = t_M: widetildeL_1(M) → widetildeL_2(M) for each M ∈ Ob(Mod^fg_R). In other words, t_M: widetildeL_1(M) → widetildeL_2(M) is a map of R-modules.","statement_latex":"Let $L_1, L_2: \\text{Mod}^{fg}_R \\to \\textit{Sets}$ be\nfunctors that take $0$ to a one element set and preserve finite products.\nLet $t : L_1 \\to L_2$ be a morphism of functors. Then $t$ induces a morphism\n$\\widetilde{t} : \\widetilde{L}_1 \\to \\widetilde{L}_2$ between the\nfunctors guaranteed by Lemma \\ref{lemma-linear-functor}, which is given simply\nby $\\widetilde{t}_M = t_M: \\widetilde{L}_1(M) \\to \\widetilde{L}_2(M)$\nfor each $M \\in \\Ob(\\text{Mod}^{fg}_R)$. In other words,\n$t_M: \\widetilde{L}_1(M) \\to \\widetilde{L}_2(M)$ is a map of $R$-modules.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06I7","source_file":"formal-defos.tex","source_line":3205,"source_end_line":3215,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3205-L3215","statement_sha256":"ffaa9d78db067a2669db7c804aab6ac1773460dc2038cfe592b2aa59f0c6f025","origin":"The Stacks Project","memory_eligible":false,"source_rank":13310,"rank":13310,"depth":1,"x":1642.233,"y":1583.883,"cluster":"deformation-theory"},{"id":"stacks:06I8","tag":"06I8","title":"Tangent spaces of functors · Lemma 06I8","summary":"Let K be a field. Let L: Mod^fg_K → Mod_K be a K-linear functor. Then L is isomorphic to the functor L(K) ⊗_K - : Mod^fg_K → Mod_K.","statement_latex":"Let $K$ be a field. Let $L: \\text{Mod}^{fg}_K \\to\n\\text{Mod}_K$ be a $K$-linear functor.  Then $L$ is isomorphic to the\nfunctor $L(K) \\otimes_K - : \\text{Mod}^{fg}_K \\to\n\\text{Mod}_K$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06I8","source_file":"formal-defos.tex","source_line":3225,"source_end_line":3231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3225-L3231","statement_sha256":"97bb2fad97799a1c396e2e0be354fcb8a6af99652b88d3b893cc580233715cf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13311,"rank":13311,"depth":0,"x":1546.727,"y":1647.69,"cluster":"deformation-theory"},{"id":"stacks:06I9","tag":"06I9","title":"Tangent spaces of functors · Lemma 06I9","summary":"Let R be an S-algebra. Then the functor Mod_R → S-Alg/R described above preserves finite products.","statement_latex":"Let $R$ be an $S$-algebra. Then the functor\n$\\text{Mod}_R \\to S\\text{-Alg}/R$ described above preserves finite products.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06I9","source_file":"formal-defos.tex","source_line":3258,"source_end_line":3262,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3258-L3262","statement_sha256":"a2be278eb5c3ba457a80a8fcaf8ddece1c1147bde444a6d40f27d5243a4512aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13312,"rank":13312,"depth":0,"x":1566.042,"y":1545.401,"cluster":"deformation-theory"},{"id":"stacks:06IA","tag":"06IA","title":"Tangent spaces of functors · Lemma 06IA","summary":"Let R be an S-algebra, and let C be a strictly full subcategory of S-Alg/R containing R[M] for all M ∈ Ob(Mod^fg_R). Let F: C → Sets be a functor. Suppose that F(R) is a one element set and that for any M, N ∈ Ob(Mod^fg_R), the induced map F(R[M] ×_R R[N]) → F(R[M]) × F(R[N]) is a bijection. Then F(R[M]) has a natural R-module structure for any M ∈ Ob(Mod^fg_R).","statement_latex":"Let $R$ be an $S$-algebra, and let $\\mathcal{C}$ be a strictly full\nsubcategory of $S\\text{-Alg}/R$ containing $R[M]$ for all\n$M \\in \\Ob(\\text{Mod}^{fg}_R)$.\nLet $F: \\mathcal{C} \\to \\textit{Sets}$ be a functor. Suppose that\n$F(R)$ is a one element set and that for any $M, N \\in\n\\Ob(\\text{Mod}^{fg}_R)$, the induced map\n$$\nF(R[M] \\times_R R[N]) \\to F(R[M]) \\times F(R[N])\n$$\nis a bijection. Then $F(R[M])$ has a natural $R$-module structure for any $M\n\\in \\Ob(\\text{Mod}^{fg}_R)$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IA","source_file":"formal-defos.tex","source_line":3270,"source_end_line":3283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3270-L3283","statement_sha256":"7088f00286bb100cbef0ea0a09c346856ce5a8401f2edb4af268d4474beee96b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13313,"rank":13313,"depth":1,"x":1634.743,"y":1632.665,"cluster":"deformation-theory"},{"id":"stacks:06IB","tag":"06IB","title":"Tangent spaces of functors · Definition 06IB","summary":"Let C be a category as in Lemma [Tag 06IA]. Let F : C → Sets be a functor such that F(R) is a one element set. The tangent space TF of F is F(R[ε]).","statement_latex":"Let $\\mathcal{C}$ be a category as in\nLemma \\ref{lemma-tangent-space-functor}.\nLet $F : \\mathcal{C} \\to \\textit{Sets}$ be a functor such that\n$F(R)$ is a one element set. The {\\it tangent space $TF$ of $F$} is\n$F(R[\\epsilon])$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of functors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IB","source_file":"formal-defos.tex","source_line":3303,"source_end_line":3310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3303-L3310","statement_sha256":"3038140845c0de6c0905200780daf0d47aeac042ac6ad936eb82977b4f7fbafa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13314,"rank":13314,"depth":2,"x":1512.71,"y":1607.044,"cluster":"deformation-theory"},{"id":"stacks:06ID","tag":"06ID","title":"Tangent spaces of functors · Lemma 06ID","summary":"Let F, G: C → Sets be functors satisfying the hypotheses of Lemma [Tag 06IA]. Let t : F → G be a morphism of functors. For any M ∈ Ob(Mod^fg_R), the map t_R[M]: F(R[M]) → G(R[M]) is a map of R-modules, where F(R[M]) and G(R[M]) are given the R-module structure from Lemma [Tag 06IA]. In particular, t_R[ε] : TF → TG is a map of R-modules.","statement_latex":"Let $F, G: \\mathcal{C} \\to \\textit{Sets}$ be functors satisfying\nthe hypotheses of\nLemma \\ref{lemma-tangent-space-functor}.\nLet $t : F \\to G$ be a morphism of functors. For any\n$M \\in \\Ob(\\text{Mod}^{fg}_R)$, the map\n$t_{R[M]}: F(R[M]) \\to G(R[M])$ is a map of $R$-modules, where\n$F(R[M])$ and $G(R[M])$ are given the $R$-module structure from\nLemma \\ref{lemma-tangent-space-functor}.\nIn particular, $t_{R[\\epsilon]} : TF \\to TG$ is a map of $R$-modules.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ID","source_file":"formal-defos.tex","source_line":3370,"source_end_line":3381,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3370-L3381","statement_sha256":"7502ce8bf4b4ceff2c95b9dda91aa951d34132b9df47b794f99c1128162cfbab","origin":"The Stacks Project","memory_eligible":false,"source_rank":13315,"rank":13315,"depth":2,"x":1624.377,"y":1556.206,"cluster":"deformation-theory"},{"id":"stacks:06IE","tag":"06IE","title":"Tangent spaces of functors · Lemma 06IE","summary":"Let F: C → Sets be a functor satisfying the hypotheses of Lemma [Tag 06IA]. Assume R = K is a field. Then F(K[V]) ≅ TF ⊗_K V for any finite dimensional K-vector space V.","statement_latex":"Let $F: \\mathcal{C} \\to \\textit{Sets}$ be a functor satisfying the\nhypotheses of\nLemma \\ref{lemma-tangent-space-functor}.\nAssume $R = K$ is a field. Then $F(K[V]) \\cong TF \\otimes_K V$\nfor any finite dimensional $K$-vector space $V$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IE","source_file":"formal-defos.tex","source_line":3406,"source_end_line":3413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3406-L3413","statement_sha256":"b3fe725eecb32979d5c57970385ffd126eb18ad78aef239e08eb049c4cc7430a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13316,"rank":13316,"depth":2,"x":1582.519,"y":1658.017,"cluster":"deformation-theory"},{"id":"stacks:06IG","tag":"06IG","title":"Tangent spaces of predeformation categories · Definition 06IG","summary":"Let F be a predeformation category. The tangent space T F of F is the set overlineF(k[ε]) of isomorphism classes of objects in the fiber category F(k[ε]).","statement_latex":"Let $\\mathcal{F}$ be a predeformation category.\nThe {\\it tangent space $T \\mathcal{F}$ of $\\mathcal{F}$}\nis the set $\\overline{\\mathcal{F}}(k[\\epsilon])$\nof isomorphism classes of objects in the fiber category $\\mathcal\nF(k[\\epsilon])$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of predeformation categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IG","source_file":"formal-defos.tex","source_line":3436,"source_end_line":3443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3436-L3443","statement_sha256":"5e2a53ff24c00d4e28969d3c971c64952424be6a522b998349c57f4d2bbc8149","origin":"The Stacks Project","memory_eligible":false,"source_rank":13317,"rank":13317,"depth":0,"x":1531.036,"y":1558.264,"cluster":"deformation-theory"},{"id":"stacks:06IH","tag":"06IH","title":"Tangent spaces of predeformation categories · Lemma 06IH","summary":"Let F be a predeformation category such that overlineF satisfies (S2) satisfies (S2), see Lemma [Tag 06I0].. Then T F has a natural k-vector space structure. For any finite dimensional vector space V we have overlineF(k[V]) = TF ⊗_k V functorially in V.","statement_latex":"Let $\\mathcal{F}$ be a predeformation category such that\n$\\overline{\\mathcal{F}}$ satisfies (S2)\\footnote{For example\nif $\\mathcal{F}$ satisfies (S2), see\nLemma \\ref{lemma-S1-S2-associated-functor}.}. Then $T \\mathcal{F}$ has a\nnatural $k$-vector space structure. For any finite dimensional\nvector space $V$ we have\n$\\overline{\\mathcal{F}}(k[V]) = T\\mathcal{F} \\otimes_k V$\nfunctorially in $V$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of predeformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IH","source_file":"formal-defos.tex","source_line":3451,"source_end_line":3461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3451-L3461","statement_sha256":"b84d6ce8db3e7a10b4259b522ee8ff49ca3194bbc9043e2a0c633cb4fb02e17b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13318,"rank":13318,"depth":3,"x":1650.302,"y":1603.019,"cluster":"deformation-theory"},{"id":"stacks:06II","tag":"06II","title":"Tangent spaces of predeformation categories · Definition 06II","summary":"Let φ : F → G be a morphism predeformation categories. The differential d φ : T F → T G of φ is the map obtained by evaluating the morphism of functors overlineφ: overlineF → overlineG at A = k[ε].","statement_latex":"Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism predeformation\ncategories. The\n{\\it differential $d \\varphi : T \\mathcal{F} \\to T \\mathcal{G}$ of $\\varphi$}\nis the map obtained by evaluating the morphism of functors\n$\\overline{\\varphi}: \\overline{\\mathcal{F}} \\to \\overline{\\mathcal{G}}$\nat $A = k[\\epsilon]$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of predeformation categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06II","source_file":"formal-defos.tex","source_line":3488,"source_end_line":3496,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3488-L3496","statement_sha256":"eba15a0e7be9040139c5812a08824433795e5b9bad99cdcc220b3bdab29417e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13319,"rank":13319,"depth":0,"x":1525.246,"y":1638.002,"cluster":"deformation-theory"},{"id":"stacks:06IJ","tag":"06IJ","title":"Tangent spaces of predeformation categories · Lemma 06IJ","summary":"Let φ : F → G be a morphism of predeformation categories. Assume overlineF and overlineG both satisfy (S2). Then d φ : T F → T G is k-linear.","statement_latex":"Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of predeformation\ncategories. Assume $\\overline{\\mathcal{F}}$ and $\\overline{\\mathcal{G}}$ both\nsatisfy (S2). Then $d \\varphi : T \\mathcal{F} \\to T \\mathcal{G}$ is $k$-linear.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of predeformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IJ","source_file":"formal-defos.tex","source_line":3498,"source_end_line":3503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3498-L3503","statement_sha256":"9ea5e799aaf64315d69dfb065b8749034b8802baff02ec9737bc35267982a6dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13320,"rank":13320,"depth":4,"x":1589.901,"y":1540.39,"cluster":"deformation-theory"},{"id":"stacks:06SU","tag":"06SU","title":"Tangent spaces of predeformation categories · Lemma 06SU","summary":"Let F be a predeformation category over C_Lambda. If overlineF has (S2) then the maps γ_V are k-linear and we have a_V(D, x) = x + γ_V(D).","statement_latex":"Let $\\mathcal{F}$ be a predeformation category over $\\mathcal{C}_\\Lambda$.\nIf $\\overline{\\mathcal{F}}$ has (S2) then the maps $\\gamma_V$ are\n$k$-linear and we have $a_V(D, x) = x + \\gamma_V(D)$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Tangent spaces of predeformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SU","source_file":"formal-defos.tex","source_line":3593,"source_end_line":3598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3593-L3598","statement_sha256":"8a1b2849b354daf9204841056bb9bcf0f9247b5fe2ad7e67c3c243c6eed65c75","origin":"The Stacks Project","memory_eligible":false,"source_rank":13321,"rank":13321,"depth":4,"x":1620.987,"y":1650.001,"cluster":"deformation-theory"},{"id":"stacks:06SY","tag":"06SY","title":"Versal formal objects · Lemma 06SY","summary":"Let F be a predeformation category. Assume F has a versal formal object. Then F satisfies (S1).","statement_latex":"Let $\\mathcal{F}$ be a predeformation category.\nAssume $\\mathcal{F}$ has a versal formal object.\nThen $\\mathcal{F}$ satisfies (S1).","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Versal formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SY","source_file":"formal-defos.tex","source_line":3676,"source_end_line":3681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3676-L3681","statement_sha256":"10881a18c18f4dfa0096808d70f58b6d8a2dc06519f1b44dcd09b2440c59f794","origin":"The Stacks Project","memory_eligible":false,"source_rank":13322,"rank":13322,"depth":2,"x":1508.969,"y":1586.271,"cluster":"deformation-theory"},{"id":"stacks:06IU","tag":"06IU","title":"Versal formal objects · Lemma 06IU","summary":"Let F be a predeformation category satisfying (S1) and (S2). Let xi be a formal object of F corresponding to underlinexi : underlineR|_C_Lambda → F, see Remark [Tag 06HC]. Then xi is versal if and only if the following two conditions hold: • the map dunderlinexi : TunderlineR|_C_Lambda → TF on tangent spaces is surjective, and • given a diagram in widehatF vcenter xymatrix & y ar[d] xi ar[r] & x lying over vcenter xymatrix & B ar[d]^f R ar[r] & A in widehatC_Lambda with B…","statement_latex":"Let $\\mathcal{F}$ be a predeformation category satisfying (S1) and\n(S2). Let $\\xi$ be a formal object of $\\mathcal{F}$ corresponding to\n$\\underline{\\xi} : \\underline{R}|_{\\mathcal{C}_\\Lambda} \\to \\mathcal{F}$, see\nRemark \\ref{remark-formal-objects-yoneda}.\nThen $\\xi$ is versal if and only if the following two conditions hold:\n\\begin{enumerate}\n\\item the map\n$d\\underline{\\xi} : T\\underline{R}|_{\\mathcal{C}_\\Lambda} \\to T\\mathcal{F}$\non tangent spaces is surjective, and\n\\item given a diagram in $\\widehat{\\mathcal{F}}$\n$$\n\\vcenter{\n\\xymatrix{\n            &  y \\ar[d] \\\\\n\\xi \\ar[r]  &  x\n}\n}\n\\quad\\text{lying over}\\quad\n\\vcenter{\n\\xymatrix{\n         &   B  \\ar[d]^{f} \\\\\nR \\ar[r] &   A\n}\n}\n$$\nin $\\widehat{\\mathcal{C}}_\\Lambda$ with $B \\to A$ a small extension of\nArtinian rings, then there exists a ring map $R \\to B$ such that\n$$\n\\xymatrix{\n         &   B  \\ar[d]^{f} \\\\\nR \\ar[ur] \\ar[r] &   A\n}\n$$\ncommutes.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Versal formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IU","source_file":"formal-defos.tex","source_line":3716,"source_end_line":3753,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3716-L3753","statement_sha256":"3dd02cdd2f103f4a51b54c30df901e17810f1797fe045e404a7fa73fc4397180","origin":"The Stacks Project","memory_eligible":false,"source_rank":13323,"rank":13323,"depth":2,"x":1643.947,"y":1569.567,"cluster":"deformation-theory"},{"id":"stacks:06SZ","tag":"06SZ","title":"Versal formal objects · Lemma 06SZ","summary":"Let F be a category cofibred in groupoids over C_Lambda which has (S1). Let B → A be a surjection in C_Lambda with kernel I annihilated by m_B. Let x ∈ F(A). The set of ideals J = ( J ⊂ I mid there exists an y → x lying over B/J → A) has a smallest element.","statement_latex":"Let $\\mathcal{F}$ be a category cofibred in groupoids over\n$\\mathcal{C}_\\Lambda$ which has (S1). Let $B \\to A$ be a surjection\nin $\\mathcal{C}_\\Lambda$ with kernel $I$ annihilated by $\\mathfrak m_B$.\nLet $x \\in \\mathcal{F}(A)$. The set of ideals\n$$\n\\mathcal{J} = \\{ J \\subset I \\mid\n\\text{there exists an }y \\to x\\text{ lying over }B/J \\to A\\}\n$$\nhas a smallest element.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Versal formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06SZ","source_file":"formal-defos.tex","source_line":3833,"source_end_line":3844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3833-L3844","statement_sha256":"ef60c26092d2b4f02ef421d86a317efa3aa194c67a0994db28801379aef84038","origin":"The Stacks Project","memory_eligible":false,"source_rank":13324,"rank":13324,"depth":0,"x":1557.134,"y":1659.208,"cluster":"deformation-theory"},{"id":"stacks:06IW","tag":"06IW","title":"Versal formal objects · Lemma 06IW","summary":"Let F be a category cofibred in groupoids over C_Lambda. Assume the following conditions hold: • F is a predeformation category. • F satisfies (S1). • F satisfies (S2). • dim_k TF is finite. Then F has a versal formal object.","statement_latex":"Let $\\mathcal{F}$ be a category cofibred in groupoids over\n$\\mathcal{C}_\\Lambda$. Assume the following conditions hold:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a predeformation category.\n\\item $\\mathcal{F}$ satisfies (S1).\n\\item $\\mathcal{F}$ satisfies (S2).\n\\item $\\dim_k T\\mathcal{F}$ is finite.\n\\end{enumerate}\nThen $\\mathcal{F}$ has a versal formal object.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Versal formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IW","source_file":"formal-defos.tex","source_line":3866,"source_end_line":3877,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L3866-L3877","statement_sha256":"7b96677e179f6ba32c1e5a7c2686a3f736007183d9680ed74751d63c18bf2b58","origin":"The Stacks Project","memory_eligible":false,"source_rank":13325,"rank":13325,"depth":48,"x":1549.0,"y":1542.909,"cluster":"deformation-theory"},{"id":"stacks:06T1","tag":"06T1","title":"Minimal versal formal objects · Lemma 06T1","summary":"Let F be a category cofibred in groupoids over C_Lambda which has (S1). • For y → x in F a minimal object in S_y maps to a minimal object of S_x. • For y → x in F lying over a surjection f : B → A in C_Lambda every minimal object of S_x is the image of a minimal object of S_y.","statement_latex":"Let $\\mathcal{F}$ be a category cofibred in groupoids over\n$\\mathcal{C}_\\Lambda$ which has (S1).\n\\begin{enumerate}\n\\item For $y \\to x$ in $\\mathcal{F}$ a minimal\nobject in $\\mathcal{S}_y$ maps to a minimal object of $\\mathcal{S}_x$.\n\\item For $y \\to x$ in $\\mathcal{F}$ lying over a surjection\n$f : B \\to A$ in $\\mathcal{C}_\\Lambda$ every minimal object\nof $\\mathcal{S}_x$ is the image of a minimal object of\n$\\mathcal{S}_y$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Minimal versal formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06T1","source_file":"formal-defos.tex","source_line":4104,"source_end_line":4116,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4104-L4116","statement_sha256":"269e57d2c64332f59b03acc276b7b6adf19b41e2bf9f337ed7e9bc32eca02e16","origin":"The Stacks Project","memory_eligible":false,"source_rank":13326,"rank":13326,"depth":0,"x":1649.317,"y":1624.703,"cluster":"deformation-theory"},{"id":"stacks:06T2","tag":"06T2","title":"Minimal versal formal objects · Lemma 06T2","summary":"Let F be a category cofibred in groupoids over C_Lambda which has (S1). Let xi be a versal formal object of F lying over R. There exists a morphism xi' → xi lying over R' ⊂ R with the following minimality properties • for every f : R → A with A ∈ Ob(C_Lambda) the pushforwards vcenter xymatrix xi' ar[d] ar[r] & x' ar[d] xi ar[r] & x lying over vcenter xymatrix R' ar[d] ar[r] & f(R') ar[d] R ar[r] & A produce a minimal object x' → x of S_x, and • for any morphism of formal…","statement_latex":"Let $\\mathcal{F}$ be a category cofibred in groupoids over\n$\\mathcal{C}_\\Lambda$ which has (S1). Let $\\xi$ be a versal formal object\nof $\\mathcal{F}$ lying over $R$. There exists a morphism $\\xi' \\to \\xi$\nlying over $R' \\subset R$ with the following minimality properties\n\\begin{enumerate}\n\\item for every $f : R \\to A$ with $A \\in \\Ob(\\mathcal{C}_\\Lambda)$\nthe pushforwards\n$$\n\\vcenter{\n\\xymatrix{\n\\xi' \\ar[d] \\ar[r] & x' \\ar[d] \\\\\n\\xi \\ar[r] & x\n}\n}\n\\quad\\text{lying over}\\quad\n\\vcenter{\n\\xymatrix{\nR' \\ar[d] \\ar[r] & f(R') \\ar[d] \\\\\nR \\ar[r] & A\n}\n}\n$$\nproduce a minimal object $x' \\to x$ of $\\mathcal{S}_x$, and\n\\item for any morphism of formal objects $\\xi'' \\to \\xi'$\nthe corresponding morphism $R'' \\to R'$ is surjective.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Minimal versal formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06T2","source_file":"formal-defos.tex","source_line":4157,"source_end_line":4185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4157-L4185","statement_sha256":"5f3762f490b636178eec75f8b5ecdfd0d0d279b63324fb712d1e91d2b063e9f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13327,"rank":13327,"depth":6,"x":1508.467,"y":1621.282,"cluster":"deformation-theory"},{"id":"stacks:06T3","tag":"06T3","title":"Minimal versal formal objects · Lemma 06T3","summary":"Let F be a category cofibred in groupoids over C_Lambda which has (S1). Let xi be a versal formal object of F lying over R. Let xi' → xi be a morphism of formal objects lying over R' ⊂ R as constructed in Lemma [Tag 06T2]. Then R ≅ R'[[x_1, …, x_r]] is a power series ring over R'. Moreover, xi' is a versal formal object too.","statement_latex":"Let $\\mathcal{F}$ be a category cofibred in groupoids over\n$\\mathcal{C}_\\Lambda$ which has (S1). Let $\\xi$ be a versal formal object\nof $\\mathcal{F}$ lying over $R$. Let $\\xi' \\to \\xi$ be a morphism\nof formal objects lying over $R' \\subset R$ as constructed in\nLemma \\ref{lemma-smallest-where-descends-versal}.\nThen\n$$\nR \\cong R'[[x_1, \\ldots, x_r]]\n$$\nis a power series ring over $R'$.\nMoreover, $\\xi'$ is a versal formal object too.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Minimal versal formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06T3","source_file":"formal-defos.tex","source_line":4238,"source_end_line":4251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4238-L4251","statement_sha256":"504d097d9f66ebcf47c5745586de61c0419c89f89c32a05285025bf624595ad7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13328,"rank":13328,"depth":7,"x":1615.908,"y":1543.284,"cluster":"deformation-theory"},{"id":"stacks:06T4","tag":"06T4","title":"Minimal versal formal objects · Definition 06T4","summary":"Let F be a predeformation category. We say a versal formal object xi of F is minimal if for any morphism of formal objects xi' → xi the underlying map on rings is surjective. Sometimes a minimal versal formal object is called miniversal.","statement_latex":"Let $\\mathcal{F}$ be a predeformation category.\nWe say a versal formal object $\\xi$ of $\\mathcal{F}$ is\n{\\it minimal}\\footnote{This may be nonstandard terminology. Many\nauthors tie this notion in with properties of tangent spaces.\nWe will make the link in\nSection \\ref{section-miniversal-objects-existence}.}\nif for any morphism of formal objects\n$\\xi' \\to \\xi$ the underlying map on rings is surjective.\nSometimes a minimal versal formal object is called {\\it miniversal}.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Minimal versal formal objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06T4","source_file":"formal-defos.tex","source_line":4319,"source_end_line":4330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4319-L4330","statement_sha256":"ce536c7d2bbb861daf99bcc17f32b5dc8274449f86d247ae8acb35bdb6a3732c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13329,"rank":13329,"depth":0,"x":1599.277,"y":1662.668,"cluster":"deformation-theory"},{"id":"stacks:06T5","tag":"06T5","title":"Minimal versal formal objects · Lemma 06T5","summary":"Let F be a predeformation category which has a versal formal object. Then • F has a minimal versal formal object, • minimal versal objects are unique up to isomorphism, and • any versal object is the pushforward of a minimal versal object along a power series ring extension.","statement_latex":"Let $\\mathcal{F}$ be a predeformation category which\nhas a versal formal object. Then\n\\begin{enumerate}\n\\item $\\mathcal{F}$ has a minimal versal formal object,\n\\item minimal versal objects are unique up to isomorphism, and\n\\item any versal object is the pushforward of a minimal versal\nobject along a power series ring extension.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Minimal versal formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06T5","source_file":"formal-defos.tex","source_line":4340,"source_end_line":4350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4340-L4350","statement_sha256":"ef4bf3f03bf3a409308f713ba6dde0b084f11850db461606b1ad0ef1a95d3e80","origin":"The Stacks Project","memory_eligible":false,"source_rank":13330,"rank":13330,"depth":8,"x":1514.907,"y":1564.464,"cluster":"deformation-theory"},{"id":"stacks:06IR","tag":"06IR","title":"Miniversal formal objects and tangent spaces · Lemma 06IR","summary":"Let F be a predeformation category. Let xi be a versal formal object of F such that ([Tag 06T6]) holds. Then xi is a minimal versal formal object. In particular, such xi are unique up to isomorphism.","statement_latex":"Let $\\mathcal{F}$ be a predeformation category.\nLet $\\xi$ be a versal formal object of $\\mathcal{F}$ such that\n(\\ref{equation-bijective-orbits}) holds.\nThen $\\xi$ is a minimal versal formal object.\nIn particular, such $\\xi$ are unique up to isomorphism.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Miniversal formal objects and tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IR","source_file":"formal-defos.tex","source_line":4422,"source_end_line":4429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4422-L4429","statement_sha256":"fff57da77e1029d57cc7c39f64f625413c1ef95035e6edca21f2cc8a6a5ed9f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13331,"rank":13331,"depth":9,"x":1657.139,"y":1589.189,"cluster":"deformation-theory"},{"id":"stacks:06IV","tag":"06IV","title":"Miniversal formal objects and tangent spaces · Lemma 06IV","summary":"Let F be a predeformation category. Let xi be a versal formal object of F such that ([Tag 06IM]) holds. Then • F satisfies (S1). • F satisfies (S2). • dim_k TF is finite.","statement_latex":"Let $\\mathcal{F}$ be a predeformation category.\nLet $\\xi$ be a versal formal object of $\\mathcal{F}$ such that\n(\\ref{equation-bijective}) holds. Then\n\\begin{enumerate}\n\\item $\\mathcal{F}$ satisfies (S1).\n\\item $\\mathcal{F}$ satisfies (S2).\n\\item $\\dim_k T\\mathcal{F}$ is finite.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Miniversal formal objects and tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IV","source_file":"formal-defos.tex","source_line":4447,"source_end_line":4457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4447-L4457","statement_sha256":"661a3e2659db016e199ee37b1a3279a070ce2a1555ca08c79a1248630c58c141","origin":"The Stacks Project","memory_eligible":false,"source_rank":13332,"rank":13332,"depth":3,"x":1531.463,"y":1652.117,"cluster":"deformation-theory"},{"id":"stacks:06T8","tag":"06T8","title":"Miniversal formal objects and tangent spaces · Lemma 06T8","summary":"Let F be a predeformation category satisfying (S2) which has a versal formal object. Then its minimal versal formal object satisfies ([Tag 06T6]).","statement_latex":"Let $\\mathcal{F}$ be a predeformation category satisfying\n(S2) which has a versal formal object. Then its minimal versal\nformal object satisfies (\\ref{equation-bijective-orbits}).","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Miniversal formal objects and tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06T8","source_file":"formal-defos.tex","source_line":4526,"source_end_line":4531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4526-L4531","statement_sha256":"d370ca6b91c04841d7932628fafe6ebb1b45b048fce8fe61554e3da2b7377821","origin":"The Stacks Project","memory_eligible":false,"source_rank":13333,"rank":13333,"depth":48,"x":1573.835,"y":1533.557,"cluster":"deformation-theory"},{"id":"stacks:06IX","tag":"06IX","title":"Miniversal formal objects and tangent spaces · Theorem 06IX","summary":"Let F be a predeformation category. Consider the following conditions • F has a minimal versal formal object satisfying ([Tag 06IM]), • F has a minimal versal formal object satisfying ([Tag 06T6]), • the following conditions hold: • F satisfies (S1). • F satisfies (S2). • dim_k TF is finite. We always have (1) ⇒ (3) ⇒ (2). If k' ⊂ k is separable, then all three are equivalent.","statement_latex":"Let $\\mathcal{F}$ be a predeformation category.\nConsider the following conditions\n\\begin{enumerate}\n\\item $\\mathcal{F}$ has a minimal versal formal object satisfying\n(\\ref{equation-bijective}),\n\\item $\\mathcal{F}$ has a minimal versal formal object satisfying\n(\\ref{equation-bijective-orbits}),\n\\item the following conditions hold:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ satisfies (S1).\n\\item $\\mathcal{F}$ satisfies (S2).\n\\item $\\dim_k T\\mathcal{F}$ is finite.\n\\end{enumerate}\n\\end{enumerate}\nWe always have\n$$\n(1) \\Rightarrow (3) \\Rightarrow (2).\n$$\nIf $k' \\subset k$ is separable, then all three are equivalent.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Miniversal formal objects and tangent spaces","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06IX","source_file":"formal-defos.tex","source_line":4619,"source_end_line":4640,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4619-L4640","statement_sha256":"76fbacb06a4b969197543d2883910370bcad1223f8d15db618b3b19c9ca53e18","origin":"The Stacks Project","memory_eligible":false,"source_rank":13334,"rank":13334,"depth":49,"x":1638.386,"y":1645.817,"cluster":"deformation-theory"},{"id":"stacks:06J2","tag":"06J2","title":"Rim-Schlessinger conditions and deformation categories · Definition 06J2","summary":"Let F be a category cofibered in groupoids over C_Lambda. We say that F satisfies condition (RS) if for every diagram in F vcenter xymatrix & x_2 ar[d] x_1 ar[r] & x lying over vcenter xymatrix & A_2 ar[d] A_1 ar[r] & A in C_Lambda with A_2 → A surjective, there exists a fiber product x_1 ×_x x_2 in F such that the diagram vcenter xymatrix x_1 ×_x x_2 ar[r] ar[d] & x_2 ar[d] x_1 ar[r] & x lies over vcenter xymatrix A_1 ×_A A_2 ar[r] ar[d] & A_2 ar[d] A_1 ar[r] & A.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal\nC_\\Lambda$.  We say that $\\mathcal{F}$ satisfies {\\it condition (RS)}\nif for every diagram in $\\mathcal{F}$\n$$\n\\vcenter{\n\\xymatrix{\n           & x_2 \\ar[d] \\\\\nx_1 \\ar[r] & x\n}\n}\n\\quad\\text{lying over}\\quad\n\\vcenter{\n\\xymatrix{\n           & A_2 \\ar[d] \\\\\nA_1 \\ar[r] & A\n}\n}\n$$\nin $\\mathcal{C}_\\Lambda$ with $A_2 \\to A$ surjective, there exists a\nfiber product $x_1 \\times_x x_2$ in $\\mathcal{F}$ such that the diagram\n$$\n\\vcenter{\n\\xymatrix{\nx_1 \\times_x x_2 \\ar[r] \\ar[d] & x_2 \\ar[d] \\\\\nx_1 \\ar[r]      & x\n}\n}\n\\quad\\text{lies over}\\quad\n\\vcenter{\n\\xymatrix{\nA_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\\nA_1 \\ar[r]      & A.\n}\n}\n$$","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Rim-Schlessinger conditions and deformation categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06J2","source_file":"formal-defos.tex","source_line":4779,"source_end_line":4816,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4779-L4816","statement_sha256":"7d2bcfc3f56e2e5ed3aef223aadf205055569ff02c356dda242929b6571f0579","origin":"The Stacks Project","memory_eligible":false,"source_rank":13335,"rank":13335,"depth":0,"x":1499.548,"y":1599.341,"cluster":"deformation-theory"},{"id":"stacks:06J3","tag":"06J3","title":"Rim-Schlessinger conditions and deformation categories · Lemma 06J3","summary":"Let F be a category cofibered in groupoids over C_Lambda satisfying (RS). Given a commutative diagram in F vcenter xymatrix y ar[r] ar[d] & x_2 ar[d] x_1 ar[r] & x lying over vcenter xymatrix A_1 ×_A A_2 ar[r] ar[d] & A_2 ar[d] A_1 ar[r] & A. with A_2 → A surjective, then it is a fiber square.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$ satisfying (RS). Given a commutative diagram\nin $\\mathcal{F}$\n$$\n\\vcenter{\n\\xymatrix{\ny \\ar[r] \\ar[d] & x_2 \\ar[d]   \\\\\nx_1 \\ar[r]      & x\n}\n}\n\\quad\\text{lying over}\\quad\n\\vcenter{\n\\xymatrix{\nA_1 \\times_A A_2 \\ar[r] \\ar[d] & A_2 \\ar[d] \\\\\nA_1 \\ar[r]      & A.\n}\n}\n$$\nwith $A_2 \\to A$ surjective, then it is a fiber square.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Rim-Schlessinger conditions and deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06J3","source_file":"formal-defos.tex","source_line":4818,"source_end_line":4839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4818-L4839","statement_sha256":"78c8bf56a59b32aaa729ea94f3fc95847ac726a00f8efd7ba7da12c99efb3ee9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13336,"rank":13336,"depth":0,"x":1640.265,"y":1554.526,"cluster":"deformation-theory"},{"id":"stacks:06J4","tag":"06J4","title":"Rim-Schlessinger conditions and deformation categories · Lemma 06J4","summary":"Let F be a category cofibered in groupoids over C_Lambda. Then F satisfies (RS) if the condition in Definition [Tag 06J2] is assumed to hold only when A_2 → A is a small extension.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal\nC_\\Lambda$. Then $\\mathcal{F}$ satisfies (RS) if the condition in\nDefinition \\ref{definition-RS} is assumed to hold only when $A_2 \\to A$\nis a small extension.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Rim-Schlessinger conditions and deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06J4","source_file":"formal-defos.tex","source_line":4863,"source_end_line":4869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4863-L4869","statement_sha256":"15741599ccaa36f4c5f659d7914e7eaafe169043aa4346acf812e0c2fb95da38","origin":"The Stacks Project","memory_eligible":false,"source_rank":13337,"rank":13337,"depth":9,"x":1572.077,"y":1668.185,"cluster":"deformation-theory"},{"id":"stacks:06J5","tag":"06J5","title":"Rim-Schlessinger conditions and deformation categories · Lemma 06J5","summary":"Let F be a category cofibered in groupoids over C_Lambda. The following are equivalent • F satisfies (RS), • the functor F(A_1 ×_A A_2) → F(A_1) ×_F(A) F(A_2) see ([Tag 06SP]) is an equivalence of categories whenever A_2 → A is surjective, and • same as in (2) whenever A_2 → A is a small extension.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ satisfies (RS),\n\\item the functor\n$\\mathcal{F}(A_1 \\times_A A_2) \\to\n\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)$\nsee (\\ref{equation-compare}) is an equivalence of\ncategories whenever $A_2 \\to A$ is surjective, and\n\\item same as in (2) whenever $A_2 \\to A$ is a small extension.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Rim-Schlessinger conditions and deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06J5","source_file":"formal-defos.tex","source_line":4876,"source_end_line":4889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4876-L4889","statement_sha256":"ec822966b4a589940de1355442479864b35fd5f3d3ca8997b42a6b9e2da1e8eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13338,"rank":13338,"depth":10,"x":1530.683,"y":1544.861,"cluster":"deformation-theory"},{"id":"stacks:06J7","tag":"06J7","title":"Rim-Schlessinger conditions and deformation categories · Lemma 06J7","summary":"Let F be a category cofibered in groupoids over C_Lambda. The condition (RS) for F implies both (S1) and (S2) for F.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. The condition (RS) for $\\mathcal{F}$\nimplies both (S1) and (S2) for $\\mathcal{F}$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Rim-Schlessinger conditions and deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06J7","source_file":"formal-defos.tex","source_line":4946,"source_end_line":4951,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4946-L4951","statement_sha256":"1965c0c75afeef07caed2fd7b766153be6ca706332ef013651b8b88a2a525135","origin":"The Stacks Project","memory_eligible":false,"source_rank":13339,"rank":13339,"depth":11,"x":1661.238,"y":1612.759,"cluster":"deformation-theory"},{"id":"stacks:06J8","tag":"06J8","title":"Rim-Schlessinger conditions and deformation categories · Lemma 06J8","summary":"Let F be a category cofibered in groupoids over C_Lambda satisfying (RS). The following conditions are equivalent: • overlineF satisfies (RS). • Let f_1: A_1 → A and f_2: A_2 → A be ring maps in C_Lambda with f_2 surjective. The induced map of sets of isomorphism classes overlineF(A_1) ×_F(A) F(A_2) → overlineF(A_1) ×_overlineF(A) overlineF(A_2) is injective. • For every morphism x' → x in F lying over a surjective ring map A' → A, the map Aut_A'(x') → Aut_A(x) is…","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$ satisfying (RS).\nThe following conditions are equivalent:\n\\begin{enumerate}\n\\item $\\overline{\\mathcal{F}}$ satisfies (RS).\n\\item Let $f_1: A_1 \\to A$ and $f_2: A_2 \\to A$ be ring maps in\n$\\mathcal{C}_\\Lambda$ with $f_2$ surjective. The induced map\nof sets of isomorphism classes\n$$\n\\overline{\\mathcal{F}(A_1) \\times_{\\mathcal{F}(A)} \\mathcal{F}(A_2)}\n\\to \\overline{\\mathcal{F}}(A_1) \\times_{\\overline{\\mathcal{F}}(A)}\n\\overline{\\mathcal{F}}(A_2)\n$$\nis injective.\n\\item For every morphism $x' \\to x$ in $\\mathcal{F}$ lying over a\nsurjective ring map $A' \\to A$, the map\n$\\text{Aut}_{A'}(x') \\to \\text{Aut}_A(x)$ is surjective.\n\\item For every morphism $x' \\to x$ in $\\mathcal{F}$ lying over a small\nextension $A' \\to A$, the map\n$\\text{Aut}_{A'}(x') \\to \\text{Aut}_A(x)$ is surjective.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Rim-Schlessinger conditions and deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06J8","source_file":"formal-defos.tex","source_line":4979,"source_end_line":5002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L4979-L5002","statement_sha256":"4fcf5e540c6989ebf8238daeee32e2e472b9409de79f5a0f01b22b19bb96ace9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13340,"rank":13340,"depth":12,"x":1509.38,"y":1636.927,"cluster":"deformation-theory"},{"id":"stacks:06J9","tag":"06J9","title":"Rim-Schlessinger conditions and deformation categories · Definition 06J9","summary":"A deformation category is a predeformation category F satisfying (RS). A morphism of deformation categories is a morphism of categories over C_Lambda.","statement_latex":"A {\\it deformation category} is a predeformation category $\\mathcal{F}$\nsatisfying (RS). A morphism of deformation categories is a morphism of\ncategories over $\\mathcal{C}_\\Lambda$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Rim-Schlessinger conditions and deformation categories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06J9","source_file":"formal-defos.tex","source_line":5092,"source_end_line":5097,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5092-L5097","statement_sha256":"5e841db7abbb5d85e3d8ea2d1492c2cf43db7759ac80377a69f3909738e91be3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13341,"rank":13341,"depth":0,"x":1602.522,"y":1532.268,"cluster":"deformation-theory"},{"id":"stacks:06JC","tag":"06JC","title":"Rim-Schlessinger conditions and deformation categories · Lemma 06JC","summary":"Let F be a category cofibered in groupoids over C_Lambda. Let x_0 ∈ Ob(F(k)). Let F_x_0 be the category cofibred in groupoids over C_Lambda constructed in Remark [Tag 06GU]. If F satisfies (RS), then so does F_x_0. In particular, F_x_0 is a deformation category.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$.\nLet $\\mathcal{F}_{x_0}$ be the category cofibred in groupoids over\n$\\mathcal{C}_\\Lambda$ constructed in\nRemark \\ref{remark-localize-cofibered-groupoid}.\nIf $\\mathcal{F}$ satisfies (RS), then so does $\\mathcal{F}_{x_0}$.\nIn particular, $\\mathcal{F}_{x_0}$ is a deformation category.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Rim-Schlessinger conditions and deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JC","source_file":"formal-defos.tex","source_line":5131,"source_end_line":5140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5131-L5140","statement_sha256":"bccad156fef70711ff832436adcaf12fabd22f3380d8f8b688e22536574a4041","origin":"The Stacks Project","memory_eligible":false,"source_rank":13342,"rank":13342,"depth":1,"x":1618.101,"y":1663.121,"cluster":"deformation-theory"},{"id":"stacks:06L4","tag":"06L4","title":"Rim-Schlessinger conditions and deformation categories · Lemma 06L4","summary":"Let xymatrix H ×_F G ar[r] ar[d] & G ar[d]^g H ar[r]^f & F be 2-fibre product of categories cofibered in groupoids over C_Lambda. If F, G, H all satisfy (RS), then H ×_F G satisfies (RS).","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G} \\ar[r] \\ar[d] &\n\\mathcal{G} \\ar[d]^g \\\\\n\\mathcal{H} \\ar[r]^f & \\mathcal{F}\n}\n$$\nbe $2$-fibre product of categories cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. If $\\mathcal{F}, \\mathcal{G}, \\mathcal{H}$\nall satisfy (RS), then $\\mathcal{H} \\times_\\mathcal{F} \\mathcal{G}$\nsatisfies (RS).","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Rim-Schlessinger conditions and deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06L4","source_file":"formal-defos.tex","source_line":5154,"source_end_line":5168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5154-L5168","statement_sha256":"9b53358e7523143b08fc1821ff06edab26a47b9dba31e1030de341006426faf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13343,"rank":13343,"depth":1,"x":1500.653,"y":1574.919,"cluster":"deformation-theory"},{"id":"stacks:06JE","tag":"06JE","title":"Lifts of objects · Definition 06JE","summary":"Let F be a category cofibered in groupoids over C_Lambda. Let f: A' → A be a map in C_Lambda. Let x ∈ F(A). The category Lift(x, f) of lifts of x along f is the category with the following objects and morphisms. • Objects: A lift of x along f is a morphism x' → x lying over f. • Morphisms: A morphism of lifts from a_1 : x'_1 → x to a_2 : x'_2 → x is a morphism b : x'_1 → x'_2 in F(A') such that a_2 = a_1 ∘ b. The set Lift(x, f) of lifts of x along f is the set of…","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. Let $f: A' \\to A$ be a map in $\\mathcal{C}_\\Lambda$.\nLet $x \\in \\mathcal{F}(A)$. The category $\\textit{Lift}(x, f)$ of lifts of $x$\nalong $f$ is the category with the following objects and\nmorphisms.\n\\begin{enumerate}\n\\item Objects: A {\\it lift of $x$ along $f$} is a morphism $x' \\to x$\nlying over $f$.\n\\item Morphisms: A {\\it morphism of lifts} from $a_1 : x'_1 \\to x$ to\n$a_2 : x'_2 \\to x$ is a morphism $b : x'_1 \\to x'_2$ in\n$\\mathcal{F}(A')$ such that $a_2 = a_1 \\circ b$.\n\\end{enumerate}\nThe set $\\text{Lift}(x, f)$ of lifts of $x$ along $f$ is the set of\nisomorphism classes of $\\textit{Lift}(x, f)$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Lifts of objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JE","source_file":"formal-defos.tex","source_line":5251,"source_end_line":5267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5251-L5267","statement_sha256":"7b336253907e154265941abb0fddd305b30a2a9564e1ce4719b3a90ece2d5316","origin":"The Stacks Project","memory_eligible":false,"source_rank":13344,"rank":13344,"depth":0,"x":1659.164,"y":1573.307,"cluster":"deformation-theory"},{"id":"stacks:06JI","tag":"06JI","title":"Lifts of objects · Lemma 06JI","summary":"Let F be a deformation category. Let A' → A be a surjective ring map in C_Lambda whose kernel I is annihilated by m_A'. Let x ∈ Ob(F(A)). If Lift(x, A') is nonempty, then there is a free and transitive action of TF ⊗_k I on Lift(x, A').","statement_latex":"Let $\\mathcal{F}$ be a deformation category.\nLet $A' \\to A$ be a surjective ring map in\n$\\mathcal{C}_\\Lambda$ whose kernel $I$ is annihilated\nby $\\mathfrak m_{A'}$. Let $x \\in \\Ob(\\mathcal{F}(A))$.\nIf $\\text{Lift}(x, A')$ is nonempty,\nthen there is a free and transitive action of\n$T\\mathcal{F} \\otimes_k I$ on $\\text{Lift}(x, A')$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Lifts of objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JI","source_file":"formal-defos.tex","source_line":5347,"source_end_line":5356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5347-L5356","statement_sha256":"db8fb8d2a90d0d098308e55665caa6835b2de6e13b139bf95b3c6683a6b46241","origin":"The Stacks Project","memory_eligible":false,"source_rank":13345,"rank":13345,"depth":2,"x":1542.866,"y":1664.996,"cluster":"deformation-theory"},{"id":"stacks:06JL","tag":"06JL","title":"Schlessinger's theorem on prorepresentable functors · Lemma 06JL","summary":"Let F, G: C_Lambda → Sets be deformation functors. Let φ : F → G be a smooth morphism which induces an isomorphism dφ : TF → TG of tangent spaces. Then φ is an isomorphism.","statement_latex":"Let $F, G: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be deformation\nfunctors. Let $\\varphi : F \\to G$ be a smooth morphism which induces\nan isomorphism $d\\varphi : TF \\to TG$ of tangent\nspaces. Then $\\varphi$ is an isomorphism.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Schlessinger's theorem on prorepresentable functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JL","source_file":"formal-defos.tex","source_line":5477,"source_end_line":5483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5477-L5483","statement_sha256":"332bdd14a1eb35dc2300d04c65d00c779a8edc0c0fb4aec13caff2cd5a6f2ca8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13346,"rank":13346,"depth":3,"x":1554.964,"y":1530.587,"cluster":"deformation-theory"},{"id":"stacks:06JM","tag":"06JM","title":"Schlessinger's theorem on prorepresentable functors · Theorem 06JM","summary":"Let F: C_Lambda → Sets be a functor. Then F is prorepresentable if and only if (a) F is a deformation functor, (b) dim_k TF is finite, and (c) γ : Der_Lambda(k, k) → TF is injective.","statement_latex":"Let $F: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a functor.\nThen $F$ is prorepresentable if and only if\n(a) $F$ is a deformation functor,\n(b) $\\dim_k TF$ is finite, and (c) $\\gamma : \\text{Der}_\\Lambda(k, k) \\to TF$\nis injective.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Schlessinger's theorem on prorepresentable functors","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JM","source_file":"formal-defos.tex","source_line":5521,"source_end_line":5528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5521-L5528","statement_sha256":"202386b1cc5ce6eba5ca9aa4960e4446850dc4f8d58190c3938fef754f7db511","origin":"The Stacks Project","memory_eligible":false,"source_rank":13347,"rank":13347,"depth":50,"x":1654.724,"y":1637.198,"cluster":"deformation-theory"},{"id":"stacks:06JP","tag":"06JP","title":"Infinitesimal automorphisms · Definition 06JP","summary":"Let F be a category cofibered in groupoids over C_Lambda. Let x' → x be a morphism in F lying over A' → A. The kernel Inf(x'/x) = Ker(Aut_A'(x') → Aut_A(x)) is the group of infinitesimal automorphisms of x' over x.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal\nC_\\Lambda$. Let $x' \\to x$ be a morphism in $\\mathcal{F}$ lying over\n$A' \\to A$. The kernel\n$$\n\\text{Inf}(x'/x) = \\Ker(\\text{Aut}_{A'}(x') \\to \\text{Aut}_A(x))\n$$\nis the {\\it group of infinitesimal automorphisms of $x'$ over $x$}.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JP","source_file":"formal-defos.tex","source_line":5580,"source_end_line":5589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5580-L5589","statement_sha256":"1f8e26fc8747c7e62eb1661e9b3ccfb08ac8d3aa89afc424933e5485aea8cd76","origin":"The Stacks Project","memory_eligible":false,"source_rank":13348,"rank":13348,"depth":0,"x":1494.487,"y":1615.059,"cluster":"deformation-theory"},{"id":"stacks:06JQ","tag":"06JQ","title":"Infinitesimal automorphisms · Definition 06JQ","summary":"Let F be a category cofibered in groupoids over C_Lambda. Let x_0 ∈ Ob(F(k)). Assume a choice of pushforward x_0 → x_0' of x_0 along the map k → k[ε], a ↦ a has been made. Then there is a unique map x'_0 → x_0 such that x_0 → x_0' → x_0 is the identity on x_0. Then Inf_x_0( F) = Inf(x'_0/x_0) is the group of infinitesimal automorphisms of x_0","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over $\\mathcal\nC_\\Lambda$. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$. Assume a choice of\npushforward $x_0 \\to x_0'$ of $x_0$ along the map\n$k \\to k[\\epsilon], a \\mapsto a$ has been made.\nThen there is a unique map $x'_0 \\to x_0$ such that\n$x_0 \\to x_0' \\to x_0$ is the identity on $x_0$.\nThen\n$$\n\\text{Inf}_{x_0}(\\mathcal F) = \\text{Inf}(x'_0/x_0)\n$$\nis the {\\it group of infinitesimal automorphisms of $x_0$}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JQ","source_file":"formal-defos.tex","source_line":5591,"source_end_line":5604,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5591-L5604","statement_sha256":"5842276dbfe96be10f125b00ab8ac87b2070ca6b3f9e669c28f9889c2e5f9a1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13349,"rank":13349,"depth":0,"x":1631.242,"y":1540.027,"cluster":"deformation-theory"},{"id":"stacks:06JT","tag":"06JT","title":"Infinitesimal automorphisms · Definition 06JT","summary":"Let p : F → C be a category cofibered in groupoids over an arbitrary base category C. Assume a choice of pushforwards has been made. Let x ∈ Ob(F) and let U = p(x). Let U/C denote the category of objects under U. The automorphism functor of x is the functor mathitAut(x) : U/C → Sets sending an object f : U → V to Aut_V(f_*x) and sending a morphism xymatrix V' ar[rr] & & V & U ar[ul]^f' ar[ur]_f & to the homomorphism Aut_V'(f'_*x) → Aut_V(f_*x) coming from the unique…","statement_latex":"Let $p : \\mathcal{F} \\to \\mathcal{C}$ be a category cofibered in groupoids\nover an arbitrary base category $\\mathcal{C}$. Assume a choice of pushforwards\nhas been made. Let $x \\in \\Ob(\\mathcal{F})$ and let $U = p(x)$.\nLet $U/\\mathcal{C}$ denote the category of objects under $U$. The\n{\\it automorphism functor of $x$} is the functor\n$\\mathit{Aut}(x) : U/\\mathcal{C} \\to \\textit{Sets}$ sending an object\n$f : U \\to V$ to $\\text{Aut}_V(f_*x)$ and sending a morphism\n$$\n\\xymatrix{\nV' \\ar[rr] &                    & V\\\\\n          & U \\ar[ul]^{f'}  \\ar[ur]_f &\n}\n$$\nto the homomorphism\n$\\text{Aut}_{V'}(f'_*x) \\to \\text{Aut}_V(f_*x)$\ncoming from the unique morphism $f'_*x \\to f_*x$ lying over\n$V' \\to V$ and compatible with $x \\to f'_*x$ and $x \\to f_*x$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JT","source_file":"formal-defos.tex","source_line":5653,"source_end_line":5672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5653-L5672","statement_sha256":"c3da6b7d66b3644293529a5ed442616782a36d3057dfcaa4c17e4247be4ed211","origin":"The Stacks Project","memory_eligible":false,"source_rank":13350,"rank":13350,"depth":0,"x":1590.503,"y":1673.72,"cluster":"deformation-theory"},{"id":"stacks:06JU","tag":"06JU","title":"Infinitesimal automorphisms · Lemma 06JU","summary":"Let F be a category cofibered in groupoids over C_Lambda satisfying (RS). Let x ∈ Ob(F(A)). Then mathitAut(x): C_A → Sets satisfies (RS).","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$ satisfying (RS). Let\n$x \\in \\Ob(\\mathcal{F}(A))$. Then\n$\\mathit{Aut}(x): \\mathcal{C}_A \\to \\textit{Sets}$ satisfies (RS).","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JU","source_file":"formal-defos.tex","source_line":5695,"source_end_line":5701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5695-L5701","statement_sha256":"ec69b6e550215ab7a4bf7d4c798ebeb3a9ef6d9f7521c6bc8e00a927a121b019","origin":"The Stacks Project","memory_eligible":false,"source_rank":13351,"rank":13351,"depth":11,"x":1512.591,"y":1551.325,"cluster":"deformation-theory"},{"id":"stacks:06JV","tag":"06JV","title":"Infinitesimal automorphisms · Lemma 06JV","summary":"Let F be a category cofibered in groupoids over C_Lambda satisfying (RS). Let x ∈ Ob(F(A)). Let x_0 be a pushforward of x to F(k). • T_id_x_0 mathitAut(x) has a natural k-vector space structure such that addition agrees with composition in T_id_x_0 mathitAut(x). In particular, composition in T_id_x_0 mathitAut(x) is commutative. • There is a canonical isomorphism T_id_x_0 mathitAut(x) → T_id_x_0 mathitAut(x_0) of k-vector spaces.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$ satisfying (RS). Let\n$x \\in \\Ob(\\mathcal{F}(A))$. Let $x_0$ be a pushforward of $x$ to\n$\\mathcal{F}(k)$.\n\\begin{enumerate}\n\\item $T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$ has a natural $k$-vector\nspace structure such that addition agrees with composition in\n$T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$. In particular, composition in\n$T_{\\text{id}_{x_0}} \\mathit{Aut}(x)$ is commutative.\n\\item There is a canonical isomorphism\n$T_{\\text{id}_{x_0}} \\mathit{Aut}(x) \\to\nT_{\\text{id}_{x_0}} \\mathit{Aut}(x_0)$\nof $k$-vector spaces.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JV","source_file":"formal-defos.tex","source_line":5711,"source_end_line":5727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5711-L5727","statement_sha256":"2d1a6c9761c57ed5f2e395f494ad9e325bc6479b381faee8c9f45d03b0c95bed","origin":"The Stacks Project","memory_eligible":false,"source_rank":13352,"rank":13352,"depth":12,"x":1669.348,"y":1597.63,"cluster":"deformation-theory"},{"id":"stacks:06JX","tag":"06JX","title":"Infinitesimal automorphisms · Lemma 06JX","summary":"Let F be a category cofibered in groupoids over C_Lambda satisfying (RS). Let x_0 ∈ Ob(F(k)). Then Inf_x_0(F) is equal as a set to T_id_x_0 mathitAut(x_0), and so has a natural k-vector space structure such that addition agrees with composition of automorphisms.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x_0 \\in \\Ob(\\mathcal{F}(k))$.\nThen $\\text{Inf}_{x_0}(\\mathcal{F})$ is equal as a set to\n$T_{\\text{id}_{x_0}} \\mathit{Aut}(x_0)$, and so has a natural $k$-vector\nspace structure such that addition agrees with composition of automorphisms.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JX","source_file":"formal-defos.tex","source_line":5805,"source_end_line":5812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5805-L5812","statement_sha256":"ce37dc8c83baf2cbd61d82d0e1138c8f94dc8f4bbb453ed32ed050f9eccc4c8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13353,"rank":13353,"depth":13,"x":1515.667,"y":1652.737,"cluster":"deformation-theory"},{"id":"stacks:07W6","tag":"07W6","title":"Infinitesimal automorphisms · Lemma 07W6","summary":"Let φ : F → G be a morphism of categories cofibred in groupoids over C_Lambda satisfying (RS). Let x_0 ∈ Ob(F(k)). Then φ induces a k-linear map Inf_x_0(F) → Inf_φ(x_0)(G).","statement_latex":"Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be a morphism of categories\ncofibred in groupoids over $\\mathcal{C}_\\Lambda$ satisfying (RS).\nLet $x_0 \\in \\Ob(\\mathcal{F}(k))$. Then $\\varphi$ induces a $k$-linear\nmap $\\text{Inf}_{x_0}(\\mathcal{F}) \\to \\text{Inf}_{\\varphi(x_0)}(\\mathcal{G})$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07W6","source_file":"formal-defos.tex","source_line":5821,"source_end_line":5827,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5821-L5827","statement_sha256":"72029a8827e891b2b4ffd3a7b633f3341681da971e142d1b8d6bec52f6550863","origin":"The Stacks Project","memory_eligible":false,"source_rank":13354,"rank":13354,"depth":5,"x":1585.056,"y":1524.195,"cluster":"deformation-theory"},{"id":"stacks:06JY","tag":"06JY","title":"Infinitesimal automorphisms · Lemma 06JY","summary":"Let F be a category cofibered in groupoids over C_Lambda satisfying (RS). Let x' → x be a morphism lying over a surjective ring map A' → A with kernel I annihilated by m_A'. Let x_0 be a pushforward of x to F(k). Then Inf(x'/x) has a free and transitive action by T_id_x_0 mathitAut(x') ⊗_k I = Inf_x_0(F) ⊗_k I.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x' \\to x$ be a\nmorphism lying over a surjective ring map $A' \\to A$ with kernel $I$\nannihilated by $\\mathfrak m_{A'}$. Let $x_0$ be a pushforward of $x$ to\n$\\mathcal{F}(k)$. Then $\\text{Inf}(x'/x)$ has a free and transitive action by\n$T_{\\text{id}_{x_0}} \\mathit{Aut}(x') \\otimes_k I\n= \\text{Inf}_{x_0}(\\mathcal{F}) \\otimes_k I$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JY","source_file":"formal-defos.tex","source_line":5837,"source_end_line":5846,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5837-L5846","statement_sha256":"8bc56258b6d44518e110b2425bf8a5ce7a915e7dede69d4b186b93f179464b43","origin":"The Stacks Project","memory_eligible":false,"source_rank":13355,"rank":13355,"depth":13,"x":1637.543,"y":1659.086,"cluster":"deformation-theory"},{"id":"stacks:06JZ","tag":"06JZ","title":"Infinitesimal automorphisms · Lemma 06JZ","summary":"Let F be a category cofibered in groupoids over C_Lambda satisfying (RS). Let x' → x be a morphism in F lying over a surjective ring map. Let x_0 be a pushforward of x to F(k). If Inf_x_0(F) = 0 then Inf(x'/x) = 0.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$ satisfying (RS). Let $x' \\to x$ be a morphism\nin $\\mathcal{F}$ lying over a surjective ring map. Let $x_0$ be a pushforward\nof $x$ to $\\mathcal{F}(k)$. If $\\text{Inf}_{x_0}(\\mathcal{F}) = 0$ then\n$\\text{Inf}(x'/x) = 0$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06JZ","source_file":"formal-defos.tex","source_line":5875,"source_end_line":5882,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5875-L5882","statement_sha256":"5b0ce1824455f9b4acb81332495262cc57f0727a5d550ddd636a81a23199ba62","origin":"The Stacks Project","memory_eligible":false,"source_rank":13356,"rank":13356,"depth":14,"x":1489.571,"y":1589.024,"cluster":"deformation-theory"},{"id":"stacks:06K0","tag":"06K0","title":"Infinitesimal automorphisms · Lemma 06K0","summary":"Let F be a category cofibered in groupoids over C_Lambda satisfying (RS). Let x_0 ∈ Ob(F(k)). Then Inf_x_0(F) = 0 if and only if the natural morphism F_x_0 → overlineF_x_0 of categories cofibered in groupoids is an equivalence.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$ satisfying (RS). Let\n$x_0 \\in \\Ob(\\mathcal{F}(k))$. Then $\\text{Inf}_{x_0}(\\mathcal{F}) = 0$\nif and only if the natural morphism\n$\\mathcal{F}_{x_0} \\to \\overline{\\mathcal{F}_{x_0}}$ of\ncategories cofibered in groupoids is an equivalence.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Infinitesimal automorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06K0","source_file":"formal-defos.tex","source_line":5890,"source_end_line":5898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5890-L5898","statement_sha256":"f924db2b0ed53809c0cb124fafe2755fc9ec23325fde09efa4be9d988fa1a163","origin":"The Stacks Project","memory_eligible":false,"source_rank":13357,"rank":13357,"depth":15,"x":1655.909,"y":1556.552,"cluster":"deformation-theory"},{"id":"stacks:06L5","tag":"06L5","title":"Applications · Lemma 06L5","summary":"Let f : H → F and g : G → F be 1-morphisms of deformation categories. Then • W = H ×_F G is a deformation category, and • we have a 6-term exact sequence of vector spaces 0 → Inf(W) → Inf(H) ⊕ Inf(G) → Inf(F) → TW → TH ⊕ TG → TF","statement_latex":"Let $f : \\mathcal{H} \\to \\mathcal{F}$ and $g : \\mathcal{G} \\to \\mathcal{F}$\nbe $1$-morphisms of deformation categories. Then\n\\begin{enumerate}\n\\item $\\mathcal{W} = \\mathcal{H} \\times_\\mathcal{F} \\mathcal{G}$ is a\ndeformation category, and\n\\item we have a $6$-term exact sequence of vector spaces\n$$\n0 \\to \\text{Inf}(\\mathcal{W})\n\\to \\text{Inf}(\\mathcal{H}) \\oplus \\text{Inf}(\\mathcal{G})\n\\to \\text{Inf}(\\mathcal{F}) \\to\nT\\mathcal{W} \\to T\\mathcal{H} \\oplus T\\mathcal{G} \\to T\\mathcal{F}\n$$\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06L5","source_file":"formal-defos.tex","source_line":5931,"source_end_line":5946,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L5931-L5946","statement_sha256":"43332146175e2d87c60a803ee6500581f56b56ad24dc63780a4e9e844a5aca94","origin":"The Stacks Project","memory_eligible":false,"source_rank":13358,"rank":13358,"depth":6,"x":1558.855,"y":1675.505,"cluster":"deformation-theory"},{"id":"stacks:0DYN","tag":"0DYN","title":"Applications · Lemma 0DYN","summary":"Let H_1 → G, H_2 → G, and G → F be maps of categories cofibred in groupoids over C_Lambda. Assume • F and G are deformation categories, • TG → TF is injective, and • Inf(G) → Inf(F) is surjective. Then H_1 ×_G H_2 → H_1 ×_F H_2 is smooth.","statement_latex":"Let $\\mathcal{H}_1 \\to \\mathcal{G}$, $\\mathcal{H}_2 \\to \\mathcal{G}$, and\n$\\mathcal{G} \\to \\mathcal{F}$ be maps of categories cofibred in groupoids\nover $\\mathcal{C}_\\Lambda$. Assume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ and $\\mathcal{G}$ are deformation categories,\n\\item $T\\mathcal{G} \\to T\\mathcal{F}$ is injective, and\n\\item $\\text{Inf}(\\mathcal{G}) \\to \\text{Inf}(\\mathcal{F})$ is surjective.\n\\end{enumerate}\nThen $\\mathcal{H}_1 \\times_\\mathcal{G} \\mathcal{H}_2 \\to\n\\mathcal{H}_1 \\times_\\mathcal{F} \\mathcal{H}_2$ is smooth.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYN","source_file":"formal-defos.tex","source_line":6022,"source_end_line":6034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6022-L6034","statement_sha256":"80453dac1870d48116670b7915c397a57f3d1f97afadaca6aaab088ab36f82f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13359,"rank":13359,"depth":14,"x":1534.638,"y":1531.974,"cluster":"deformation-theory"},{"id":"stacks:0DYP","tag":"0DYP","title":"Applications · Lemma 0DYP","summary":"Let f : F → G be a map of deformation categories. Let x_0 ∈ Ob(F(k)) with image y_0 ∈ Ob(G(k)). If • the map TF → TG is surjective, and • for every small extension A' → A in C_Lambda and x ∈ F(A) with image y ∈ G(A) if there is a lift of y to A', then there is a lift of x to A', then F → G is smooth (and vice versa).","statement_latex":"Let $f : \\mathcal{F} \\to \\mathcal{G}$ be a map of deformation\ncategories. Let $x_0 \\in \\Ob(\\mathcal{F}(k))$ with image\n$y_0 \\in \\Ob(\\mathcal{G}(k))$. If\n\\begin{enumerate}\n\\item the map $T\\mathcal{F} \\to T\\mathcal{G}$ is surjective, and\n\\item for every small extension $A' \\to A$ in $\\mathcal{C}_\\Lambda$\nand $x \\in \\mathcal{F}(A)$ with image $y \\in \\mathcal{G}(A)$\nif there is a lift of $y$ to $A'$, then there is a lift\nof $x$ to $A'$,\n\\end{enumerate}\nthen $\\mathcal{F} \\to \\mathcal{G}$ is smooth (and vice versa).","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYP","source_file":"formal-defos.tex","source_line":6085,"source_end_line":6098,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6085-L6098","statement_sha256":"2c6d4c527a2ababa53829779a77a4b47c12fc4466db28fa6b7c4db9e8edff55d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13360,"rank":13360,"depth":9,"x":1668.608,"y":1624.548,"cluster":"deformation-theory"},{"id":"stacks:0E3R","tag":"0E3R","title":"Applications · Lemma 0E3R","summary":"Let F → G → H be maps of categories cofibred in groupoids over C_Lambda. If • F, G are deformation categories • the map TF → TG is surjective, and • F → H is smooth. Then F → G is smooth.","statement_latex":"Let $\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}$ be maps of categories\ncofibred in groupoids over $\\mathcal{C}_\\Lambda$. If\n\\begin{enumerate}\n\\item $\\mathcal{F}$, $\\mathcal{G}$ are deformation categories\n\\item the map $T\\mathcal{F} \\to T\\mathcal{G}$ is surjective, and\n\\item $\\mathcal{F} \\to \\mathcal{H}$ is smooth.\n\\end{enumerate}\nThen $\\mathcal{F} \\to \\mathcal{G}$ is smooth.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3R","source_file":"formal-defos.tex","source_line":6116,"source_end_line":6126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6116-L6126","statement_sha256":"6db77bcd4e4354e3183e884f808ebbd35e85655b60c6245872072119e11491ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":13361,"rank":13361,"depth":10,"x":1494.488,"y":1632.341,"cluster":"deformation-theory"},{"id":"stacks:06K3","tag":"06K3","title":"Groupoids in functors on an arbitrary category · Definition 06K3","summary":"Let C be a category. The category of groupoids in functors on C is the category with the following objects and morphisms. • Objects: A groupoid in functors on C is a quintuple (U, R, s, t, c) where U, R : C → Sets are functors and s, t : R → U and c : R ×_s, U, t R → R are morphisms with the following property: For any object T of C, the quintuple (U(T), R(T), s, t, c) is a groupoid category. • Morphisms: A morphism (U, R, s, t, c) → (U', R', s', t', c') of groupoids in…","statement_latex":"Let $\\mathcal{C}$ be a category. The\n{\\it category of groupoids in functors on $\\mathcal{C}$}\nis the category with the following objects and morphisms.\n\\begin{enumerate}\n\\item Objects: A {\\it groupoid in functors on $\\mathcal{C}$} is a quintuple\n$(U, R, s, t, c)$ where $U, R : \\mathcal{C} \\to \\textit{Sets}$ are\nfunctors and $s, t : R \\to U$ and $c : R \\times_{s, U, t} R \\to R$\nare morphisms with the following property: For any object $T$ of $\\mathcal{C}$,\nthe quintuple\n$$\n(U(T), R(T), s, t, c)\n$$\nis a groupoid category.\n\\item Morphisms: A {\\it morphism $(U, R, s, t, c) \\to (U', R', s', t', c')$ of\ngroupoids in functors on $\\mathcal{C}$} consists of morphisms $U \\to U'$\nand $R \\to R'$ with the following property: For any object $T$ of\n$\\mathcal{C}$, the induced maps $U(T) \\to U'(T)$ and\n$R(T) \\to R'(T)$ define a functor between groupoid categories\n$$\n(U(T), R(T), s, t, c) \\to (U'(T), R'(T), s', t', c').\n$$\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Groupoids in functors on an arbitrary category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06K3","source_file":"formal-defos.tex","source_line":6154,"source_end_line":6178,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6154-L6178","statement_sha256":"49ac588b55bb7e32e1db623968327a6c852d44dac905d70d821ede3d04637a4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13362,"rank":13362,"depth":0,"x":1617.235,"y":1527.266,"cluster":"deformation-theory"},{"id":"stacks:06K6","tag":"06K6","title":"Groupoids in functors on an arbitrary category · Definition 06K6","summary":"Let C be a category. A groupoid in functors on C is representable if it is isomorphic to one of the form (underlineU, underlineR, s, t, c) where U and R are objects of C and the pushout R amalg_s, U, t R exists.","statement_latex":"Let $\\mathcal{C}$ be a category. A groupoid in functors on $\\mathcal{C}$ is\n{\\it representable} if it is isomorphic to one of the form\n$(\\underline{U}, \\underline{R}, s, t, c)$ where $U$ and $R$ are objects of\n$\\mathcal{C}$ and the pushout $R \\amalg_{s, U, t} R$ exists.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Groupoids in functors on an arbitrary category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06K6","source_file":"formal-defos.tex","source_line":6207,"source_end_line":6213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6207-L6213","statement_sha256":"10ca156062e32abdbf33c88dad2edd7a35066e15f5cdf0263267998f9eb467c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13363,"rank":13363,"depth":0,"x":1611.189,"y":1675.134,"cluster":"deformation-theory"},{"id":"stacks:06K9","tag":"06K9","title":"Groupoids in functors on an arbitrary category · Definition 06K9","summary":"Let (U, R, s, t, c) be a groupoid in functors on a category C. Let C' be a subcategory of C. The restriction (U, R, s, t, c)|_C' of (U, R, s, t, c) to C' is the groupoid in functors on C' given by (U|_C', R|_ C', s|_C', t|_C', c|_C').","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\\mathcal{C}$.\nLet $\\mathcal{C}'$ be a subcategory of $\\mathcal{C}$. The\n{\\it restriction $(U, R, s, t, c)|_{\\mathcal{C}'}$ of $(U, R, s, t, c)$\nto $\\mathcal{C}'$} is the groupoid\nin functors on $\\mathcal{C}'$ given by $(U|_{\\mathcal{C}'}, R|_{\\mathcal\nC'}, s|_{\\mathcal{C}'}, t|_{\\mathcal{C}'}, c|_{\\mathcal{C}'})$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Groupoids in functors on an arbitrary category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06K9","source_file":"formal-defos.tex","source_line":6248,"source_end_line":6256,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6248-L6256","statement_sha256":"63a6d88a9cbd7e7dd592226c96f9c92243037258d164ffccfc3b1a24c528e7cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13364,"rank":13364,"depth":0,"x":1496.169,"y":1562.109,"cluster":"deformation-theory"},{"id":"stacks:06KB","tag":"06KB","title":"Groupoids in functors on an arbitrary category · Definition 06KB","summary":"Let (U, R, s, t, c) be a groupoid in functors on a category C. • The assignment T ↦ (U(T), R(T), s, t, c) determines a functor C → Groupoids. The quotient category cofibered in groupoids [U/R] → C is the category cofibered in groupoids over C associated to this functor (as in Remarks [Tag 06GK] ([Tag 06GN])). • The quotient morphism U → [U/R] is the morphism of categories cofibered in groupoids over C defined by the rules • x ∈ U(T) maps to the object (T, x) ∈…","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in functors on a category $\\mathcal{C}$.\n\\begin{enumerate}\n\\item The assignment $T \\mapsto  (U(T), R(T), s, t, c)$ determines a functor\n$\\mathcal{C} \\to \\textit{Groupoids}$. The {\\it quotient category\ncofibered in groupoids $[U/R] \\to \\mathcal{C}$} is the category\ncofibered in groupoids over $\\mathcal{C}$ associated to this functor (as in\nRemarks \\ref{remarks-cofibered-groupoids}\n(\\ref{item-construction-associated-cofibered-groupoid})).\n\\item The {\\it quotient morphism $U \\to [U/R]$} is the morphism of\ncategories cofibered in groupoids over $\\mathcal{C}$ defined by the\nrules\n\\begin{enumerate}\n\\item $x \\in U(T)$ maps to the object $(T, x) \\in \\Ob([U/R](T))$, and\n\\item $x \\in U(T)$ and $f : T \\to T'$ give rise to the morphism\n$(f, \\text{id}_{U(f)(x)}): (T, x) \\to (T, U(f)(x))$ lying over\n$f : T \\to T'$.\n\\end{enumerate}\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Groupoids in functors on an arbitrary category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KB","source_file":"formal-defos.tex","source_line":6265,"source_end_line":6285,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6265-L6285","statement_sha256":"500fe6cf594683092f6bc038d2a155f2a1e81009d8e346c5da2d2baa6fabc38b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13365,"rank":13365,"depth":0,"x":1672.739,"y":1580.276,"cluster":"deformation-theory"},{"id":"stacks:06KD","tag":"06KD","title":"Groupoids in functors on the base category · Definition 06KD","summary":"A groupoid in functors on C_Lambda is prorepresentable if it is isomorphic to (underlineR_0, underlineR_1, s, t, c)|_C_Lambda for some representable groupoid in functors (underlineR_0, underlineR_1, s, t, c) on the category widehatC_Lambda.","statement_latex":"A groupoid in functors on $\\mathcal{C}_\\Lambda$ is {\\it prorepresentable}\nif it is isomorphic to\n$(\\underline{R_0}, \\underline{R_1}, s, t, c)|_{\\mathcal{C}_\\Lambda}$\nfor some representable groupoid in functors\n$(\\underline{R_0}, \\underline{R_1}, s, t, c)$ on the category\n$\\widehat{\\mathcal{C}}_\\Lambda$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Groupoids in functors on the base category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KD","source_file":"formal-defos.tex","source_line":6303,"source_end_line":6311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6303-L6311","statement_sha256":"8852a0346eeab9436748437faf4b6a7667a929fa2e36023a806fc6936b2671a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13366,"rank":13366,"depth":0,"x":1527.223,"y":1667.491,"cluster":"deformation-theory"},{"id":"stacks:06KE","tag":"06KE","title":"Groupoids in functors on the base category · Definition 06KE","summary":"Let (U, R, s, t, c) be a groupoid in functors on C_Lambda. The completion (U, R, s, t, c)^wedge of (U, R, s, t, c) is the groupoid in functors (widehatU, widehatR, widehats, widehatt, widehatc) on widehatC_Lambda described above.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$.\nThe {\\it completion $(U, R, s, t, c)^{\\wedge}$ of $(U, R, s, t, c)$} is the\ngroupoid in functors\n$(\\widehat{U}, \\widehat{R}, \\widehat{s}, \\widehat{t}, \\widehat{c})$\non $\\widehat{\\mathcal{C}}_\\Lambda$ described above.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Groupoids in functors on the base category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KE","source_file":"formal-defos.tex","source_line":6334,"source_end_line":6341,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6334-L6341","statement_sha256":"a7d53de8f108ee0236677acaac7665dbc1e55fe80de6eec5acca1e9bbc45c178","origin":"The Stacks Project","memory_eligible":false,"source_rank":13367,"rank":13367,"depth":0,"x":1564.569,"y":1519.903,"cluster":"deformation-theory"},{"id":"stacks:06KG","tag":"06KG","title":"Groupoids in functors on the base category · Lemma 06KG","summary":"Let (U, R, s, t, c) be a groupoid in functors on C_Lambda. • (U, R, s, t, c) is prorepresentable if and only if its completion is representable as a groupoid in functors on widehatC_Lambda. • (U, R, s, t, c) is prorepresentable if and only if U and R are prorepresentable.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$.\n\\begin{enumerate}\n\\item $(U, R, s, t, c)$ is prorepresentable if and only if its completion is\nrepresentable as a groupoid in functors on $\\widehat{\\mathcal{C}}_\\Lambda$.\n\\item $(U, R, s, t, c)$ is prorepresentable if and only if $U$ and $R$ are\nprorepresentable.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Groupoids in functors on the base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KG","source_file":"formal-defos.tex","source_line":6359,"source_end_line":6368,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6359-L6368","statement_sha256":"37aa65ab674df6b3909965b49bbb4d80dc0918026f4ad58d184b3fc45dd317ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":13368,"rank":13368,"depth":7,"x":1656.15,"y":1650.545,"cluster":"deformation-theory"},{"id":"stacks:06KI","tag":"06KI","title":"Smooth groupoids in functors on the base category · Definition 06KI","summary":"Let (U, R, s, t, c) be a groupoid in functors on C_Lambda. We say (U, R, s, t, c) is smooth if s, t: R → U are smooth.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$. We\nsay $(U, R, s, t, c)$ is {\\it smooth} if $s, t: R \\to U$ are smooth.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth groupoids in functors on the base category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KI","source_file":"formal-defos.tex","source_line":6411,"source_end_line":6415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6411-L6415","statement_sha256":"3d2d619ff9d820733721e40cedcdb8621936251fc7a12816ff63302d50848aae","origin":"The Stacks Project","memory_eligible":false,"source_rank":13369,"rank":13369,"depth":0,"x":1482.745,"y":1605.966,"cluster":"deformation-theory"},{"id":"stacks:06KL","tag":"06KL","title":"Smooth groupoids in functors on the base category · Lemma 06KL","summary":"Let (U, R, s, t, c) be a groupoid in functors on C_Lambda. The following are equivalent: • The groupoid in functors (U, R, s, t, c) is smooth. • The morphism s : R → U is smooth. • The morphism t : R → U is smooth. • The quotient morphism U → [U/R] is smooth.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in functors on $\\mathcal{C}_\\Lambda$.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The groupoid in functors $(U, R, s, t, c)$ is smooth.\n\\item The morphism $s : R \\to U$ is smooth.\n\\item The morphism $t : R \\to U$ is smooth.\n\\item The quotient morphism $U \\to [U/R]$ is smooth.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Smooth groupoids in functors on the base category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KL","source_file":"formal-defos.tex","source_line":6439,"source_end_line":6449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6439-L6449","statement_sha256":"e24e9fad5dc26c4ad7d7d6dc3198e7831e32c90152a60b75b9ce9567da1957e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13370,"rank":13370,"depth":0,"x":1647.228,"y":1540.138,"cluster":"deformation-theory"},{"id":"stacks:06KT","tag":"06KT","title":"Deformation categories as quotients of groupoids in functors · Lemma 06KT","summary":"Let (U, R, s, t, c) be a smooth groupoid in functors on C_Lambda. Assume U and R satisfy (RS). Then [U/R] satisfies (RS).","statement_latex":"Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\\mathcal{C}_\\Lambda$.\nAssume $U$ and $R$ satisfy (RS). Then $[U/R]$ satisfies (RS).","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Deformation categories as quotients of groupoids in functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KT","source_file":"formal-defos.tex","source_line":6496,"source_end_line":6500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6496-L6500","statement_sha256":"dcf1dce8777444ce884bf289d8d211cc8caa0cea58c2e8fc4a082914d8649232","origin":"The Stacks Project","memory_eligible":false,"source_rank":13371,"rank":13371,"depth":0,"x":1578.56,"y":1682.668,"cluster":"deformation-theory"},{"id":"stacks:06KU","tag":"06KU","title":"Deformation categories as quotients of groupoids in functors · Lemma 06KU","summary":"Let (U, R, s, t, c) be a smooth groupoid in functors on C_Lambda. Assume U and R are deformation functors. Then: • The quotient [U/R] is a deformation category. • The tangent space of [U/R] is T[U/R] = Coker(ds-dt: TR → TU). • The space of infinitesimal automorphisms of [U/R] is Inf([U/R]) = Ker(ds ⊕ dt : TR → TU ⊕ TU).","statement_latex":"Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\\mathcal{C}_\\Lambda$.\nAssume $U$ and $R$ are deformation functors. Then:\n\\begin{enumerate}\n\\item The quotient $[U/R]$ is a deformation category.\n\\item The tangent space of $[U/R]$ is\n$$\nT[U/R] = \\Coker(ds-dt: TR \\to TU).\n$$\n\\item The space of infinitesimal automorphisms of $[U/R]$ is\n$$\n\\text{Inf}([U/R]) =\n\\Ker(ds \\oplus dt : TR \\to TU \\oplus TU).\n$$\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Deformation categories as quotients of groupoids in functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KU","source_file":"formal-defos.tex","source_line":6574,"source_end_line":6590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6574-L6590","statement_sha256":"5c3b898d353d9ce1bf8450473300716204976b87829009feb9a8eccb5742ec46","origin":"The Stacks Project","memory_eligible":false,"source_rank":13372,"rank":13372,"depth":1,"x":1514.283,"y":1537.942,"cluster":"deformation-theory"},{"id":"stacks:06KX","tag":"06KX","title":"Presentations of categories cofibered in groupoids · Definition 06KX","summary":"Let F be a category cofibered in groupoids over a category C. Let (U, R, s, t, c) be a groupoid in functors on C. A presentation of F by (U, R, s, t, c) is an equivalence φ : [U/R] → F of categories cofibered in groupoids over C.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over a category\n$\\mathcal{C}$. Let $(U, R, s, t, c)$ be a groupoid in functors on\n$\\mathcal{C}$. A\n{\\it presentation of $\\mathcal{F}$ by $(U, R, s, t, c)$} is an equivalence\n$\\varphi : [U/R] \\to \\mathcal{F}$ of categories cofibered in groupoids\nover $\\mathcal{C}$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Presentations of categories cofibered in groupoids","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KX","source_file":"formal-defos.tex","source_line":6613,"source_end_line":6621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6613-L6621","statement_sha256":"cfa0613890be50464a50e5a33817d3c45faf72c76de3f082d97b66f0846778bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13373,"rank":13373,"depth":0,"x":1678.81,"y":1608.51,"cluster":"deformation-theory"},{"id":"stacks:06KY","tag":"06KY","title":"Presentations of categories cofibered in groupoids · Lemma 06KY","summary":"Let F be category cofibered in groupoids over a category C. Let U : C → Sets be a functor. Let f : U → F be a morphism of categories cofibered in groupoids over C. Define R, s, t, c as follows: • R : C → Sets is the functor U ×_f, F, f U. • t, s : R → U are the first and second projections, respectively. • c : R ×_s, U, t R → R is the morphism given by projection onto the first and last factors of U ×_f, F, f U ×_f, F, f U under the canonical isomorphism R ×_s, U, t R → U…","statement_latex":"Let $\\mathcal{F}$ be category cofibered in groupoids over a category\n$\\mathcal{C}$. Let $U : \\mathcal{C} \\to \\textit{Sets}$ be a functor.\nLet $f : U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids\nover $\\mathcal{C}$. Define $R, s, t, c$ as follows:\n\\begin{enumerate}\n\\item $R : \\mathcal{C} \\to \\textit{Sets}$ is the functor\n$U \\times_{f, \\mathcal{F}, f} U$.\n\\item $t, s : R \\to U$ are the first and second projections,\nrespectively.\n\\item $c : R \\times_{s, U, t} R \\to R$ is the morphism given by projection\nonto the first and last factors of\n$U \\times_{f, \\mathcal{F}, f} U \\times_{f, \\mathcal{F}, f} U$\nunder the canonical isomorphism\n$R \\times_{s, U, t} R \\to\nU \\times_{f, \\mathcal{F}, f} U \\times_{f, \\mathcal{F}, f} U$.\n\\end{enumerate}\nThen $(U, R, s, t, c)$ is a groupoid in functors on $\\mathcal{C}$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Presentations of categories cofibered in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KY","source_file":"formal-defos.tex","source_line":6626,"source_end_line":6645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6626-L6645","statement_sha256":"e333fc8449b9679420cd9574213b4db9158197d8920347481a01e6e190958b1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13374,"rank":13374,"depth":0,"x":1499.937,"y":1650.016,"cluster":"deformation-theory"},{"id":"stacks:06KZ","tag":"06KZ","title":"Presentations of categories cofibered in groupoids · Lemma 06KZ","summary":"Let F be category cofibered in groupoids over a category C. Let U : C → Sets be a functor. Let f : U → F be a morphism of categories cofibered in groupoids over C. Let (U, R, s, t, c) be the groupoid in functors on C constructed from f : U → F in Lemma [Tag 06KY]. Then there is a natural morphism [f] : [U/R] → F such that: • [f]: [U/R] → F is fully faithful. • [f]: [U/R] → F is an equivalence if and only if f : U → F is essentially surjective.","statement_latex":"Let $\\mathcal{F}$ be category cofibered in groupoids over a category\n$\\mathcal{C}$. Let $U : \\mathcal{C} \\to \\textit{Sets}$ be a functor.\nLet $f : U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids\nover $\\mathcal{C}$. Let $(U, R, s, t, c)$ be the groupoid in functors on\n$\\mathcal{C}$ constructed from $f : U \\to \\mathcal{F}$ in\nLemma \\ref{lemma-presentation-construction}.\nThen there is a natural morphism $[f] : [U/R] \\to \\mathcal{F}$ such that:\n\\begin{enumerate}\n\\item $[f]: [U/R] \\to \\mathcal{F}$ is fully faithful.\n\\item $[f]: [U/R] \\to \\mathcal{F}$ is an equivalence if and only if\n$f : U \\to \\mathcal{F}$ is essentially surjective.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Presentations of categories cofibered in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KZ","source_file":"formal-defos.tex","source_line":6651,"source_end_line":6665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6651-L6665","statement_sha256":"1a2c3c7ad88b5b4c78cbae8931bffa2eff4c5bf97aa42e3b704be239a5e1fd88","origin":"The Stacks Project","memory_eligible":false,"source_rank":13375,"rank":13375,"depth":1,"x":1598.899,"y":1517.323,"cluster":"deformation-theory"},{"id":"stacks:06L1","tag":"06L1","title":"Presentations of deformation categories · Lemma 06L1","summary":"Let F be a category cofibered in groupoids over C_Lambda. Let U : C_Lambda → Sets be a functor. Let f : U → F be a smooth morphism of categories cofibered in groupoids. Then: • If (U, R, s, t, c) is the groupoid in functors on C_Lambda constructed from f : U → F in Lemma [Tag 06KY], then (U, R, s, t, c) is smooth. • If f : U(k) → F(k) is essentially surjective, then the morphism [f] : [U/R] → F of Lemma [Tag 06KZ] is an equivalence.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. Let $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$\nbe a functor. Let $f : U \\to \\mathcal{F}$ be a smooth morphism of\ncategories cofibered in groupoids. Then:\n\\begin{enumerate}\n\\item If $(U, R, s, t, c)$ is the groupoid in functors on\n$\\mathcal{C}_\\Lambda$ constructed from $f : U \\to \\mathcal{F}$ in\nLemma \\ref{lemma-presentation-construction}, then $(U, R, s, t, c)$\nis smooth.\n\\item If $f : U(k) \\to \\mathcal{F}(k)$ is essentially surjective,\nthen the morphism $[f] : [U/R] \\to \\mathcal{F}$ of\nLemma \\ref{lemma-presentation-morphism}\nis an equivalence.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Presentations of deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06L1","source_file":"formal-defos.tex","source_line":6685,"source_end_line":6701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6685-L6701","statement_sha256":"9ea0dfcc956e2a889b7bc096ffc8f6e88922d58cd10a6bb9a0e3ee018fd3cfe2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13376,"rank":13376,"depth":2,"x":1632.782,"y":1672.005,"cluster":"deformation-theory"},{"id":"stacks:06L6","tag":"06L6","title":"Presentations of deformation categories · Lemma 06L6","summary":"Let F be a deformation category. Let U : C_Lambda → Sets be a deformation functor. Let f: U → F be a morphism of categories cofibered in groupoids. Then U ×_f, F, f U is a deformation functor with tangent space fitting into an exact sequence of k-vector spaces 0 → Inf(F) → T(U ×_f, F, f U) → TU ⊕ TU","statement_latex":"Let $\\mathcal{F}$ be a deformation category.\nLet $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a deformation functor.\nLet $f: U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids.\nThen $U \\times_{f, \\mathcal{F}, f} U$ is a deformation functor\nwith tangent space fitting into an exact sequence of $k$-vector spaces\n$$\n0 \\to \\text{Inf}(\\mathcal{F}) \\to\nT(U \\times_{f, \\mathcal{F}, f} U) \\to TU \\oplus TU\n$$","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Presentations of deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06L6","source_file":"formal-defos.tex","source_line":6725,"source_end_line":6736,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6725-L6736","statement_sha256":"5c1949a6b332b872fceb387aa4a598b3b64d7102f8d91cb38d5880ecd8c61254","origin":"The Stacks Project","memory_eligible":false,"source_rank":13377,"rank":13377,"depth":7,"x":1482.751,"y":1576.753,"cluster":"deformation-theory"},{"id":"stacks:06L7","tag":"06L7","title":"Presentations of deformation categories · Lemma 06L7","summary":"Let F be a deformation category. Let U : C_Lambda → Sets be a prorepresentable functor. Let f : U → F be a morphism of categories cofibered in groupoids. Let (U, R, s, t, c) be the groupoid in functors on C_Lambda constructed from f : U → F in Lemma [Tag 06KY]. If dim_k Inf(F) < ∞, then (U, R, s, t, c) is prorepresentable.","statement_latex":"Let $\\mathcal{F}$ be a deformation category.\nLet $U : \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a prorepresentable functor.\nLet $f : U \\to \\mathcal{F}$ be a morphism of categories cofibered in groupoids.\nLet $(U, R, s, t, c)$ be the groupoid in functors on $\\mathcal{C}_\\Lambda$\nconstructed from $f : U \\to \\mathcal{F}$ in\nLemma \\ref{lemma-presentation-construction}. If\n$\\dim_k \\text{Inf}(\\mathcal{F}) < \\infty$, then\n$(U, R, s, t, c)$ is prorepresentable.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Presentations of deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06L7","source_file":"formal-defos.tex","source_line":6744,"source_end_line":6754,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6744-L6754","statement_sha256":"8fdafc3796c115fa68e10e047e939314c02e099fea152b1a168933d41bf189f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13378,"rank":13378,"depth":51,"x":1670.796,"y":1561.796,"cluster":"deformation-theory"},{"id":"stacks:06L8","tag":"06L8","title":"Presentations of deformation categories · Theorem 06L8","summary":"Let F be a category cofibered in groupoids over C_Lambda. Then F admits a presentation by a smooth prorepresentable groupoid in functors on C_Lambda if and only if the following conditions hold: • F is a deformation category. • dim_k TF is finite. • dim_k Inf(F) is finite.","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. Then $\\mathcal{F}$ admits a presentation by a\nsmooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$\nif and only if the following conditions hold:\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a deformation category.\n\\item $\\dim_k T\\mathcal{F}$ is finite.\n\\item $\\dim_k \\text{Inf}(\\mathcal{F})$ is finite.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Presentations of deformation categories","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06L8","source_file":"formal-defos.tex","source_line":6779,"source_end_line":6790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6779-L6790","statement_sha256":"ee2857a5836d4d49d18b33ead29a316c8db9c862321a05a63d7431cb53a36e9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13379,"rank":13379,"depth":52,"x":1543.616,"y":1680.034,"cluster":"deformation-theory"},{"id":"stacks:06KM","tag":"06KM","title":"Remarks regarding minimality · Definition 06KM","summary":"Let (U, R, s, t, c) be a smooth prorepresentable groupoid in functors on C_Lambda. • We say (U, R, s, t, c) is normalized if the groupoid (U(k[ε]), R(k[ε]), s, t, c) is totally disconnected, i.e., there are no morphisms between distinct objects. • We say (U, R, s, t, c) is minimal if the U → [U/R] is given by a minimal versal formal object of [U/R].","statement_latex":"Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors\non $\\mathcal{C}_\\Lambda$.\n\\begin{enumerate}\n\\item We say $(U, R, s, t, c)$ is {\\it normalized} if the groupoid\n$(U(k[\\epsilon]), R(k[\\epsilon]), s, t, c)$ is totally disconnected,\ni.e., there are no morphisms between distinct objects.\n\\item We say $(U, R, s, t, c)$ is {\\it minimal} if the $U \\to [U/R]$\nis given by a minimal versal formal object of $[U/R]$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Remarks regarding minimality","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KM","source_file":"formal-defos.tex","source_line":6841,"source_end_line":6852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6841-L6852","statement_sha256":"a18ce5d7dabcffd12552971296d6aa75dc68ccf5595f0c12c638e5f6f57ddbe3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13380,"rank":13380,"depth":0,"x":1542.308,"y":1519.999,"cluster":"deformation-theory"},{"id":"stacks:06KN","tag":"06KN","title":"Remarks regarding minimality · Lemma 06KN","summary":"Let (U, R, s, t, c) be a smooth prorepresentable groupoid in functors on C_Lambda. • (U, R, s, t, c) is normalized if and only if the morphism U → [U/R] induces an isomorphism on tangent spaces, and • (U, R, s, t, c) is minimal if and only if the kernel of TU → T[U/R] is contained in the image of Der_Lambda(k, k) → TU.","statement_latex":"Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in\nfunctors on $\\mathcal{C}_\\Lambda$.\n\\begin{enumerate}\n\\item $(U, R, s, t, c)$ is normalized if and only if the morphism\n$U \\to [U/R]$ induces an isomorphism on tangent spaces, and\n\\item $(U, R, s, t, c)$ is minimal if and only if the kernel of\n$TU \\to T[U/R]$ is contained in the image of\n$\\text{Der}_\\Lambda(k, k) \\to TU$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Remarks regarding minimality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KN","source_file":"formal-defos.tex","source_line":6861,"source_end_line":6872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6861-L6872","statement_sha256":"cf659185931987f39ee8df40761dfa528973db26f830ccf79f3905610172029d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13381,"rank":13381,"depth":53,"x":1672.517,"y":1637.763,"cluster":"deformation-theory"},{"id":"stacks:06KP","tag":"06KP","title":"Remarks regarding minimality · Lemma 06KP","summary":"Let U: C_Lambda → Sets be a prorepresentable functor. Let φ : U → U be a morphism such that dφ : TU → TU is an isomorphism. Then φ is an isomorphism.","statement_latex":"Let $U: \\mathcal{C}_\\Lambda \\to \\textit{Sets}$ be a\nprorepresentable functor. Let $\\varphi : U \\to U$ be a morphism such\nthat $d\\varphi : TU \\to TU$ is an isomorphism.  Then $\\varphi$ is an\nisomorphism.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Remarks regarding minimality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KP","source_file":"formal-defos.tex","source_line":6898,"source_end_line":6904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6898-L6904","statement_sha256":"5e9e0afb5a5fb0c0f055cdb86a65388b582e0428c705b3e53136660d82225310","origin":"The Stacks Project","memory_eligible":false,"source_rank":13382,"rank":13382,"depth":6,"x":1480.999,"y":1624.754,"cluster":"deformation-theory"},{"id":"stacks:06KQ","tag":"06KQ","title":"Remarks regarding minimality · Lemma 06KQ","summary":"Let (U, R, s, t, c) be a minimal smooth prorepresentable groupoid in functors on C_Lambda. If φ : [U/R] → [U/R] is an equivalence of categories cofibered in groupoids, then φ is an isomorphism.","statement_latex":"Let $(U, R, s, t, c)$ be a minimal smooth prorepresentable groupoid in\nfunctors on $\\mathcal{C}_\\Lambda$. If $\\varphi : [U/R] \\to [U/R]$ is an\nequivalence of categories cofibered in groupoids, then $\\varphi$ is an\nisomorphism.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Remarks regarding minimality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KQ","source_file":"formal-defos.tex","source_line":6923,"source_end_line":6929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6923-L6929","statement_sha256":"2ae0326fd197a2dcc5ddcdbb9035b805991fb092425efdac0e9fec324e3f4d5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13383,"rank":13383,"depth":54,"x":1633.316,"y":1525.262,"cluster":"deformation-theory"},{"id":"stacks:06KR","tag":"06KR","title":"Remarks regarding minimality · Lemma 06KR","summary":"Let (U, R, s, t, c) and (U', R', s', t', c') be minimal smooth prorepresentable groupoids in functors on C_Lambda. If φ : [U/R] → [U'/R'] is an equivalence of categories cofibered in groupoids, then φ is an isomorphism.","statement_latex":"Let $(U, R, s, t, c)$ and $(U', R', s', t', c')$ be minimal smooth\nprorepresentable groupoids in functors on $\\mathcal{C}_\\Lambda$. If\n$\\varphi : [U/R] \\to [U'/R']$ is an equivalence of categories cofibered\nin groupoids, then $\\varphi$ is an isomorphism.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Remarks regarding minimality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06KR","source_file":"formal-defos.tex","source_line":6962,"source_end_line":6968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6962-L6968","statement_sha256":"2e68c8b2a63fd57ead6bea75df4ab736360ce363e523ee45d285193432ca8a9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13384,"rank":13384,"depth":55,"x":1600.873,"y":1685.715,"cluster":"deformation-theory"},{"id":"stacks:06L2","tag":"06L2","title":"Remarks regarding minimality · Lemma 06L2","summary":"Let F be a deformation category such that dim_k TF <∞ and dim_k Inf(F) < ∞. Then there exists a minimal versal formal object xi of F. Say xi lies over R ∈ Ob(widehatC_Lambda). Let U = underlineR|_C_Lambda. Let f = underlinexi : U → F be the associated morphism. Let (U, R, s, t, c) be the groupoid in functors on C_Lambda constructed from f : U → F in Lemma [Tag 06KY]. Then (U, R, s, t, c) is a minimal smooth prorepresentable groupoid in functors on C_Lambda and there is an…","statement_latex":"Let $\\mathcal{F}$ be a deformation category such that\n$\\dim_k T\\mathcal{F} <\\infty$ and\n$\\dim_k \\text{Inf}(\\mathcal{F}) < \\infty$.\nThen there exists a minimal versal formal object $\\xi$ of $\\mathcal{F}$.\nSay $\\xi$ lies over $R \\in \\Ob(\\widehat{\\mathcal{C}}_\\Lambda)$.\nLet $U = \\underline{R}|_{\\mathcal{C}_\\Lambda}$.\nLet $f = \\underline{\\xi} : U \\to \\mathcal{F}$ be the associated\nmorphism. Let $(U, R, s, t, c)$ be the groupoid in functors on\n$\\mathcal{C}_\\Lambda$ constructed from $f : U \\to \\mathcal{F}$ in\nLemma \\ref{lemma-presentation-construction}.\nThen $(U, R, s, t, c)$ is a minimal smooth prorepresentable\ngroupoid in functors on $\\mathcal{C}_\\Lambda$ and there\nis an equivalence $[U/R] \\to \\mathcal{F}$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Remarks regarding minimality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06L2","source_file":"formal-defos.tex","source_line":6981,"source_end_line":6996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L6981-L6996","statement_sha256":"3bb671e1826c40df3d04217eda4743611a66eeaabc3d0e9e6f242e974f3bb3ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":13385,"rank":13385,"depth":52,"x":1495.334,"y":1548.429,"cluster":"deformation-theory"},{"id":"stacks:06L3","tag":"06L3","title":"Remarks regarding minimality · Lemma 06L3","summary":"Let F be category cofibered in groupoids over C_Lambda. Assume there exist presentations of F by minimal smooth prorepresentable groupoids in functors (U, R, s, t, c) and (U', R', s', t', c'). Then (U, R, s, t, c) and (U', R', s', t', c') are isomorphic.","statement_latex":"Let $\\mathcal{F}$ be category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. Assume there exist presentations of\n$\\mathcal{F}$ by minimal smooth prorepresentable groupoids\nin functors $(U, R, s, t, c)$ and $(U', R', s', t', c')$.\nThen $(U, R, s, t, c)$ and $(U', R', s', t', c')$ are isomorphic.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Remarks regarding minimality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06L3","source_file":"formal-defos.tex","source_line":7026,"source_end_line":7033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7026-L7033","statement_sha256":"9c32702adf01b33d26a15964e3882ed6d238ea9e1d8da56cfa129d5ead30194f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13386,"rank":13386,"depth":56,"x":1684.324,"y":1589.946,"cluster":"deformation-theory"},{"id":"stacks:06TE","tag":"06TE","title":"Remarks regarding minimality · Theorem 06TE","summary":"Let F be a category cofibered in groupoids over C_Lambda. Consider the following conditions • F admits a presentation by a normalized smooth prorepresentable groupoid in functors on C_Lambda, • F admits a presentation by a smooth prorepresentable groupoid in functors on C_Lambda, • F admits a presentation by a minimal smooth prorepresentable groupoid in functors on C_Lambda, and • F satisfies the following conditions • F is a deformation category. • dim_k TF is finite. •…","statement_latex":"Let $\\mathcal{F}$ be a category cofibered in groupoids over\n$\\mathcal{C}_\\Lambda$. Consider the following conditions\n\\begin{enumerate}\n\\item $\\mathcal{F}$ admits a presentation by a normalized\nsmooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$,\n\\item $\\mathcal{F}$ admits a presentation by a\nsmooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$,\n\\item $\\mathcal{F}$ admits a presentation by a minimal\nsmooth prorepresentable groupoid in functors on $\\mathcal{C}_\\Lambda$, and\n\\item $\\mathcal{F}$ satisfies the following conditions\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a deformation category.\n\\item $\\dim_k T\\mathcal{F}$ is finite.\n\\item $\\dim_k \\text{Inf}(\\mathcal{F})$ is finite.\n\\end{enumerate}\n\\end{enumerate}\nThen (2), (3), (4) are equivalent and are implied by (1).\nIf $k' \\subset k$ is separable, then (1), (2), (3), (4) are all equivalent.\nFurthermore, the minimal smooth prorepresentable groupoids in functors\nwhich provide a presentation of $\\mathcal{F}$ are unique up to isomorphism.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Remarks regarding minimality","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TE","source_file":"formal-defos.tex","source_line":7047,"source_end_line":7069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7047-L7069","statement_sha256":"afabfacfcd82be57471269f1b628dea5600287b91f1445f9db6b87bdcbbbe367","origin":"The Stacks Project","memory_eligible":false,"source_rank":13387,"rank":13387,"depth":57,"x":1510.882,"y":1666.878,"cluster":"deformation-theory"},{"id":"stacks:0DQB","tag":"0DQB","title":"Uniqueness of versal rings · Lemma 0DQB","summary":"Being formally homotopic is an equivalence relation on sets of morphisms in widehatC_Lambda.","statement_latex":"Being formally homotopic is an equivalence relation on\nsets of morphisms in $\\widehat{\\mathcal{C}}_\\Lambda$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Uniqueness of versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQB","source_file":"formal-defos.tex","source_line":7107,"source_end_line":7111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7107-L7111","statement_sha256":"9b158ad84103b9a35d9ec7753a581ca5ae475c2238fa4232a24739e1c2b54088","origin":"The Stacks Project","memory_eligible":false,"source_rank":13388,"rank":13388,"depth":7,"x":1577.175,"y":1511.113,"cluster":"deformation-theory"},{"id":"stacks:0DQC","tag":"0DQC","title":"Uniqueness of versal rings · Lemma 0DQC","summary":"In the category widehatC_Lambda, if f_1, f_2 : R → S are formally homotopic and g : S → S' is a morphism, then g ∘ f_1 and g ∘ f_2 are formally homotopic.","statement_latex":"In the category $\\widehat{\\mathcal{C}}_\\Lambda$, if $f_1, f_2 : R \\to S$\nare formally homotopic and $g : S \\to S'$ is a morphism, then\n$g \\circ f_1$ and $g \\circ f_2$ are formally homotopic.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Uniqueness of versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQC","source_file":"formal-defos.tex","source_line":7179,"source_end_line":7184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7179-L7184","statement_sha256":"f37538b60357ac572a4ffbf499d60180517e2d517d0ccdd9f696c271eda0a12b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13389,"rank":13389,"depth":0,"x":1653.855,"y":1664.193,"cluster":"deformation-theory"},{"id":"stacks:0DQD","tag":"0DQD","title":"Uniqueness of versal rings · Lemma 0DQD","summary":"Let F be a deformation category over C_Lambda with dim_k TF < ∞ and dim_k Inf(F) < ∞. Let xi be a versal formal object lying over R. Let eta be a formal object lying over S. Then any two maps f, g : R → S such that f_*xi ≅ eta ≅ g_*xi are formally homotopic.","statement_latex":"Let $\\mathcal{F}$ be a deformation category over $\\mathcal{C}_\\Lambda$\nwith $\\dim_k T\\mathcal{F} < \\infty$ and\n$\\dim_k \\text{Inf}(\\mathcal{F}) < \\infty$. Let $\\xi$ be a versal formal\nobject lying over $R$. Let $\\eta$ be a formal object lying over $S$.\nThen any two maps\n$$\nf, g : R \\to S\n$$\nsuch that $f_*\\xi \\cong \\eta \\cong g_*\\xi$ are formally homotopic.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Uniqueness of versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQD","source_file":"formal-defos.tex","source_line":7192,"source_end_line":7203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7192-L7203","statement_sha256":"a9b44417a5332d9fbcce04e36693a578f0071ccda435c355d475bbcf459a3bc0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13390,"rank":13390,"depth":53,"x":1473.502,"y":1594.55,"cluster":"deformation-theory"},{"id":"stacks:0DQE","tag":"0DQE","title":"Uniqueness of versal rings · Lemma 0DQE","summary":"In the category widehatC_Lambda, if f_1, f_2 : R → S are formally homotopic and p ⊂ R is a minimal prime ideal, then f_1( p)S = f_2( p)S as ideals.","statement_latex":"In the category $\\widehat{\\mathcal{C}}_\\Lambda$, if $f_1, f_2 : R \\to S$\nare formally homotopic and $\\mathfrak p \\subset R$ is a minimal\nprime ideal, then $f_1(\\mathfrak p)S = f_2(\\mathfrak p)S$ as ideals.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Uniqueness of versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQE","source_file":"formal-defos.tex","source_line":7227,"source_end_line":7232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7227-L7232","statement_sha256":"5ca37f56fb261494e2d36c506bd77f311ac4ec3763adf5f39bb9b86d1047a4d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13391,"rank":13391,"depth":0,"x":1663.235,"y":1543.369,"cluster":"deformation-theory"},{"id":"stacks:07WA","tag":"07WA","title":"Change of residue field · Lemma 07WA","summary":"With notation and assumptions as in Situation [Tag 07W8]. • We have overlineF_l/k = (overlineF)_l/k. • If F is a predeformation category, then F_l/k is a predeformation category. • If F satisfies (S1), then F_l/k satisfies (S1). • If F satisfies (S2), then F_l/k satisfies (S2). • If F satisfies (RS), then F_l/k satisfies (RS).","statement_latex":"With notation and assumptions as in Situation \\ref{situation-change-of-fields}.\n\\begin{enumerate}\n\\item We have $\\overline{\\mathcal{F}_{l/k}} = (\\overline{\\mathcal{F}})_{l/k}$.\n\\item If $\\mathcal{F}$ is a predeformation category, then $\\mathcal{F}_{l/k}$\nis a predeformation category.\n\\item If $\\mathcal{F}$ satisfies (S1), then $\\mathcal{F}_{l/k}$\nsatisfies (S1).\n\\item If $\\mathcal{F}$ satisfies (S2), then $\\mathcal{F}_{l/k}$\nsatisfies (S2).\n\\item If $\\mathcal{F}$ satisfies (RS), then $\\mathcal{F}_{l/k}$\nsatisfies (RS).\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Change of residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WA","source_file":"formal-defos.tex","source_line":7321,"source_end_line":7335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7321-L7335","statement_sha256":"83c646629feea0b98cfed89484799a51cc316fd8afa9096d5bbf9c1cc97fe6d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13392,"rank":13392,"depth":11,"x":1564.103,"y":1689.332,"cluster":"deformation-theory"},{"id":"stacks:07WB","tag":"07WB","title":"Change of residue field · Lemma 07WB","summary":"With notation and assumptions as in Situation [Tag 07W8]. Assume F is a predeformation category and overlineF satisfies (S2). Then there is a canonical l-vector space isomorphism TF ⊗_k l → TF_l/k of tangent spaces.","statement_latex":"With notation and assumptions as in Situation \\ref{situation-change-of-fields}.\nAssume $\\mathcal{F}$ is a predeformation category and\n$\\overline{\\mathcal{F}}$ satisfies (S2). Then there is a\ncanonical $l$-vector space isomorphism\n$$\nT\\mathcal{F} \\otimes_k l \\longrightarrow T\\mathcal{F}_{l/k}\n$$\nof tangent spaces.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Change of residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WB","source_file":"formal-defos.tex","source_line":7381,"source_end_line":7391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7381-L7391","statement_sha256":"b533ec73854aa5c880ba121f00c7791adab4e7cbc87fb2e45533a14c628c928e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13393,"rank":13393,"depth":12,"x":1519.653,"y":1524.821,"cluster":"deformation-theory"},{"id":"stacks:07WC","tag":"07WC","title":"Change of residue field · Lemma 07WC","summary":"With notation and assumptions as in Situation [Tag 07W8]. Assume F is a deformation category. Then there is a canonical l-vector space isomorphism Inf(F) ⊗_k l → Inf(F_l/k) of infinitesimal automorphism spaces.","statement_latex":"With notation and assumptions as in Situation \\ref{situation-change-of-fields}.\nAssume $\\mathcal{F}$ is a deformation category.\nThen there is a\ncanonical $l$-vector space isomorphism\n$$\n\\text{Inf}(\\mathcal{F}) \\otimes_k l\n\\longrightarrow\n\\text{Inf}(\\mathcal{F}_{l/k})\n$$\nof infinitesimal automorphism spaces.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Change of residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WC","source_file":"formal-defos.tex","source_line":7417,"source_end_line":7429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7417-L7429","statement_sha256":"b59b1dbb672f0921f096a2ac27c7a8549ebc8fd7aea9c727f7c58c366be02f95","origin":"The Stacks Project","memory_eligible":false,"source_rank":13394,"rank":13394,"depth":13,"x":1685.355,"y":1621.274,"cluster":"deformation-theory"},{"id":"stacks:07WD","tag":"07WD","title":"Change of residue field · Lemma 07WD","summary":"With notation and assumptions as in Situation [Tag 07W8]. If F → G is a smooth morphism of categories cofibred in groupoids over C_Lambda, k, then F_l/k → G_l/k is a smooth morphism of categories cofibred in groupoids over C_Lambda, l.","statement_latex":"With notation and assumptions as in Situation \\ref{situation-change-of-fields}.\nIf $\\mathcal{F} \\to \\mathcal{G}$ is a smooth morphism of categories cofibred\nin groupoids over $\\mathcal{C}_{\\Lambda, k}$, then\n$\\mathcal{F}_{l/k} \\to \\mathcal{G}_{l/k}$ is a smooth morphism of categories\ncofibred in groupoids over $\\mathcal{C}_{\\Lambda, l}$.","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Change of residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WD","source_file":"formal-defos.tex","source_line":7456,"source_end_line":7463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7456-L7463","statement_sha256":"6c61259f4d361da5b8657c87ab59fab5a04078e00abcef6759afdfc3208a3649","origin":"The Stacks Project","memory_eligible":false,"source_rank":13395,"rank":13395,"depth":0,"x":1484.847,"y":1644.26,"cluster":"deformation-theory"},{"id":"stacks:0DQF","tag":"0DQF","title":"Change of residue field · Lemma 0DQF","summary":"With notation and assumptions as in Situation [Tag 07W8]. Let xi be a versal formal object for F lying over R ∈ Ob(widehatC_Lambda, k). Then there exist • an S ∈ Ob(widehatC_Lambda, l) and a local Lambda-algebra homomorphism R → S which is formally smooth in the m_S-adic topology and induces the given field extension l/k on residue fields, and • a versal formal object of F_l/k lying over S.","statement_latex":"With notation and assumptions as in Situation \\ref{situation-change-of-fields}.\nLet $\\xi$ be a versal formal object for $\\mathcal{F}$ lying over\n$R \\in \\Ob(\\widehat{\\mathcal{C}}_{\\Lambda, k})$. Then there exist\n\\begin{enumerate}\n\\item an $S \\in \\Ob(\\widehat{\\mathcal{C}}_{\\Lambda, l})$\nand a local $\\Lambda$-algebra homomorphism $R \\to S$ which is\nformally smooth in the $\\mathfrak m_S$-adic topology and induces\nthe given field extension $l/k$ on residue fields, and\n\\item a versal formal object of $\\mathcal{F}_{l/k}$ lying over $S$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Formal Deformation Theory","chapter_id":"formal-defos","section":"Change of residue field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQF","source_file":"formal-defos.tex","source_line":7475,"source_end_line":7487,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/formal-defos.tex#L7475-L7487","statement_sha256":"ad4479b9b0297b60876b8ca6b9d5459ed6a300b8e5a489bd0299e276d26ef0ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":13396,"rank":13396,"depth":48,"x":1614.701,"y":1513.047,"cluster":"deformation-theory"},{"id":"stacks:08S5","tag":"08S5","title":"Deformations of rings and the naive cotangent complex · Lemma 08S5","summary":"Given a commutative diagram xymatrix & 0 ar[r] & N_2 ar[r] & B'_2 ar[r] & B_2 ar[r] & 0 & 0 ar[r]|hole & I_2 ar[u]_c_2 ar[r] & A'_2 ar[u] ar[r]|hole & A_2 ar[u] ar[r] & 0 0 ar[r] & N_1 ar[ruu] ar[r] & B'_1 ar[r] & B_1 ar[ruu] ar[r] & 0 0 ar[r] & I_1 ar[ruu]|hole ar[u]^c_1 ar[r] & A'_1 ar[ruu]|hole ar[u] ar[r] & A_1 ar[ruu]|hole ar[u] ar[r] & 0 with front and back solutions to ([Tag 08S4]) we have • There exist a canonical element in Ext^1_B_1(NL_B_1/A_1, N_2) whose…","statement_latex":"Given a commutative diagram\n$$\n\\xymatrix{\n& 0 \\ar[r] & N_2 \\ar[r] & B'_2 \\ar[r] & B_2 \\ar[r] & 0 \\\\\n& 0 \\ar[r]|\\hole & I_2 \\ar[u]_{c_2} \\ar[r] &\nA'_2 \\ar[u] \\ar[r]|\\hole & A_2 \\ar[u] \\ar[r] & 0 \\\\\n0 \\ar[r] & N_1 \\ar[ruu] \\ar[r] & B'_1 \\ar[r] & B_1 \\ar[ruu] \\ar[r] & 0 \\\\\n0 \\ar[r] & I_1 \\ar[ruu]|\\hole \\ar[u]^{c_1} \\ar[r] &\nA'_1 \\ar[ruu]|\\hole \\ar[u] \\ar[r] & A_1 \\ar[ruu]|\\hole \\ar[u] \\ar[r] & 0\n}\n$$\nwith front and back solutions to (\\ref{equation-to-solve}) we have\n\\begin{enumerate}\n\\item There exist a canonical element in\n$\\Ext^1_{B_1}(\\NL_{B_1/A_1}, N_2)$\nwhose vanishing is a necessary and sufficient condition for the existence\nof a ring map $B'_1 \\to B'_2$ fitting into the diagram.\n\\item If there exists a map $B'_1 \\to B'_2$ fitting into the diagram\nthe set of all such maps is a principal homogeneous space under\n$\\Hom_{B_1}(\\Omega_{B_1/A_1}, N_2)$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of rings and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08S5","source_file":"defos.tex","source_line":57,"source_end_line":80,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L57-L80","statement_sha256":"3803cf5601493e57da18331850d7855d60e644ff3ff1eb2c6488605ccef780f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13397,"rank":13397,"depth":0,"x":1624.502,"y":1684.12,"cluster":"deformation-theory"},{"id":"stacks:08S7","tag":"08S7","title":"Deformations of rings and the naive cotangent complex · Lemma 08S7","summary":"If there exists a solution to ([Tag 08S4]), then the set of isomorphism classes of solutions is principal homogeneous under Ext^1_B(NL_B/A, N).","statement_latex":"If there exists a solution to (\\ref{equation-to-solve}), then the set of\nisomorphism classes of solutions is principal homogeneous under\n$\\Ext^1_B(\\NL_{B/A}, N)$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of rings and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08S7","source_file":"defos.tex","source_line":158,"source_end_line":163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L158-L163","statement_sha256":"2f1ed1acb2d96a9c2aea9e22f001de7cdcbb6c0d56e07ac3240920884c146195","origin":"The Stacks Project","memory_eligible":false,"source_rank":13398,"rank":13398,"depth":1,"x":1479.168,"y":1563.087,"cluster":"deformation-theory"},{"id":"stacks:0GPT","tag":"0GPT","title":"Deformations of rings and the naive cotangent complex · Lemma 0GPT","summary":"Let A be a ring. Let B be an A-algebra. Let N be a B-module. The set of isomorphism classes of extensions of A-algebras 0 → N → B' → B → 0 where N is an ideal of square zero is canonically bijective to Ext^1_B(NL_B/A, N).","statement_latex":"Let $A$ be a ring. Let $B$ be an $A$-algebra. Let $N$ be a $B$-module.\nThe set of isomorphism classes of extensions of $A$-algebras\n$$\n0 \\to N \\to B' \\to B \\to 0\n$$\nwhere $N$ is an ideal of square zero is canonically bijective to\n$\\Ext^1_B(\\NL_{B/A}, N)$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of rings and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPT","source_file":"defos.tex","source_line":216,"source_end_line":225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L216-L225","statement_sha256":"73f13bb36734c7e0c381e19a4e48aa79e2e6d104fda23210ba593f0e88b1667a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13399,"rank":13399,"depth":2,"x":1684.417,"y":1569.895,"cluster":"deformation-theory"},{"id":"stacks:0GPV","tag":"0GPV","title":"Deformations of rings and the naive cotangent complex · Lemma 0GPV","summary":"Given ring maps A → B → C, a B-module M, a C-module N, a B-linear map c : M → N, and extensions of A-algebras with square zero kernels • [(a)] 0 → M → B' → B → 0 corresponding to xi ∈ Ext^1_B(NL_B/A, M), and • [(b)] 0 → N → C' → C → 0 corresponding to zeta ∈ Ext^1_C(NL_C/A, N). See Lemma [Tag 0GPT]. Then there is an A-algebra map B' → C' compatible with B → C and c if and only if xi and zeta map to the same element of Ext^1_B(NL_B/A, N).","statement_latex":"Given ring maps $A \\to B \\to C$, a $B$-module $M$, a $C$-module $N$,\na $B$-linear map $c : M \\to N$, and extensions of\n$A$-algebras with square zero kernels\n\\begin{enumerate}\n\\item[(a)] $0 \\to M \\to B' \\to B \\to 0$ corresponding to\n$\\xi \\in \\Ext^1_B(\\NL_{B/A}, M)$, and\n\\item[(b)] $0 \\to N \\to C' \\to C \\to 0$ corresponding to\n$\\zeta \\in \\Ext^1_C(\\NL_{C/A}, N)$.\n\\end{enumerate}\nSee Lemma \\ref{lemma-extensions-of-algebras}.\nThen there is an $A$-algebra map $B' \\to C'$ compatible with\n$B \\to C$ and $c$ if and only if $\\xi$ and $\\zeta$\nmap to the same element of\n$\\Ext^1_B(\\NL_{B/A}, N)$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of rings and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPV","source_file":"defos.tex","source_line":262,"source_end_line":278,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L262-L278","statement_sha256":"8487e811a403dc5cd8ae6c7efc15785a267df7465432f2c0eb95c5a5ae92b8c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13400,"rank":13400,"depth":3,"x":1527.022,"y":1681.744,"cluster":"deformation-theory"},{"id":"stacks:0GPX","tag":"0GPX","title":"Deformations of rings and the naive cotangent complex · Lemma 0GPX","summary":"Let 0 → I → A' → A → 0, A → B, and c : I → N be as in ([Tag 08S4]). Denote xi ∈ Ext^1_A(NL_A/A', I) the element corresponding to the extension A' of A by I via Lemma [Tag 0GPT]. The set of isomorphism classes of solutions is canonically bijective to the fibre of Ext^1_B(NL_B/A', N) → Ext^1_A(NL_A/A', N) over the image of xi.","statement_latex":"Let $0 \\to I \\to A' \\to A \\to 0$, $A \\to B$, and $c : I \\to N$ be as in\n(\\ref{equation-to-solve}). Denote $\\xi \\in \\Ext^1_A(\\NL_{A/A'}, I)$\nthe element corresponding to the extension $A'$ of $A$ by $I$ via\nLemma \\ref{lemma-extensions-of-algebras}. The set of isomorphism\nclasses of solutions is canonically bijective to the fibre of\n$$\n\\Ext^1_B(\\NL_{B/A'}, N) \\to \\Ext^1_A(\\NL_{A/A'}, N)\n$$\nover the image of $\\xi$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of rings and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPX","source_file":"defos.tex","source_line":368,"source_end_line":379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L368-L379","statement_sha256":"dacbb94e0df64bcc0c11953011f3d5863eb78ecabd2e186a54df65b816060e5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13401,"rank":13401,"depth":4,"x":1553.232,"y":1509.336,"cluster":"deformation-theory"},{"id":"stacks:08S6","tag":"08S6","title":"Deformations of rings and the naive cotangent complex · Lemma 08S6","summary":"If A → B is a local complete intersection ring map, then there exists a solution to ([Tag 08S4]).","statement_latex":"If $A \\to B$ is a local complete intersection ring map, then\nthere exists a solution to (\\ref{equation-to-solve}).","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of rings and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08S6","source_file":"defos.tex","source_line":423,"source_end_line":427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L423-L427","statement_sha256":"22c2460da06184083de4b6f305dcacf4e9ba53fed68ff3eaecd47d3ee6103295","origin":"The Stacks Project","memory_eligible":false,"source_rank":13402,"rank":13402,"depth":6,"x":1672.979,"y":1651.85,"cluster":"deformation-theory"},{"id":"stacks:08L1","tag":"08L1","title":"Thickenings of ringed spaces · Definition 08L1","summary":"In Situation [Tag 08KZ] we say that (f, f') is a strict morphism of thickenings if the map (f')^*J → I is surjective.","statement_latex":"In Situation \\ref{situation-morphism-thickenings} we say that $(f, f')$ is a\n{\\it strict morphism of thickenings}\nif the map $(f')^*\\mathcal{J} \\longrightarrow \\mathcal{I}$ is surjective.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Thickenings of ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08L1","source_file":"defos.tex","source_line":548,"source_end_line":553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L548-L553","statement_sha256":"9f7b3cd4c82a7ed814d2f7f5a47a3ed5dc0531ded4df6b815f4b1168c565b044","origin":"The Stacks Project","memory_eligible":false,"source_rank":13403,"rank":13403,"depth":0,"x":1469.35,"y":1614.577,"cluster":"deformation-theory"},{"id":"stacks:08L2","tag":"08L2","title":"Thickenings of ringed spaces · Lemma 08L2","summary":"In Situation [Tag 08KZ] the morphism (f, f') is a strict morphism of thickenings if and only if ([Tag 08L0]) is cartesian in the category of ringed spaces.","statement_latex":"In Situation \\ref{situation-morphism-thickenings} the morphism $(f, f')$\nis a strict morphism of thickenings if and only if\n(\\ref{equation-morphism-thickenings}) is cartesian in the category\nof ringed spaces.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Thickenings of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08L2","source_file":"defos.tex","source_line":560,"source_end_line":566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L560-L566","statement_sha256":"8059c68be71b8c58a6ff6241f9c214285e5dae4a1e873199452ce32bcb0a1d91","origin":"The Stacks Project","memory_eligible":false,"source_rank":13404,"rank":13404,"depth":0,"x":1650.116,"y":1526.206,"cluster":"deformation-theory"},{"id":"stacks:08L5","tag":"08L5","title":"Modules on first order thickenings of ringed spaces · Lemma 08L5","summary":"Let i : (X, O_X) → (X', O_X') be a first order thickening of ringed spaces. Assume given extensions 0 → K → F' → F → 0 and 0 → L → G' → G → 0 as in ([Tag 08L4]) and maps φ : F → G and ψ : K → L. • If there exists an O_X'-module map φ' : F' → G' compatible with φ and ψ, then the diagram xymatrix I ⊗_O_X F ar[r]_-c_F' ar[d]_1 ⊗ φ & K ar[d]^ψ I ⊗_O_X G ar[r]^-c_G' & L is commutative. • The set of O_X'-module maps φ' : F' → G' compatible with φ and ψ is, if nonempty, a…","statement_latex":"Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$\nbe a first order thickening of ringed spaces. Assume given\nextensions\n$$\n0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0\n\\quad\\text{and}\\quad\n0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0\n$$\nas in (\\ref{equation-extension})\nand maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and\n$\\psi : \\mathcal{K} \\to \\mathcal{L}$.\n\\begin{enumerate}\n\\item If there exists an $\\mathcal{O}_{X'}$-module\nmap $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$\nand $\\psi$, then the diagram\n$$\n\\xymatrix{\n\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}\n\\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} &\n\\mathcal{K} \\ar[d]^\\psi \\\\\n\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{G}\n\\ar[r]^-{c_{\\mathcal{G}'}} &\n\\mathcal{L}\n}\n$$\nis commutative.\n\\item The set of $\\mathcal{O}_{X'}$-module\nmaps $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$\nand $\\psi$ is, if nonempty, a principal homogeneous space under\n$\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{L})$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Modules on first order thickenings of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08L5","source_file":"defos.tex","source_line":603,"source_end_line":636,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L603-L636","statement_sha256":"7f3aa462d3eb420db3a6271b38d30f0ebfa0a0c1a66ae224006b5394741c9dc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13405,"rank":13405,"depth":4,"x":1587.667,"y":1694.531,"cluster":"deformation-theory"},{"id":"stacks:08L6","tag":"08L6","title":"Modules on first order thickenings of ringed spaces · Lemma 08L6","summary":"Let i : (X, O_X) → (X', O_X') be a first order thickening of ringed spaces. Assume given extensions 0 → K → F' → F → 0 and 0 → L → G' → G → 0 as in ([Tag 08L4]) and maps φ : F → G and ψ : K → L. Assume the diagram xymatrix I ⊗_O_X F ar[r]_-c_F' ar[d]_1 ⊗ φ & K ar[d]^ψ I ⊗_O_X G ar[r]^-c_G' & L is commutative. Then there exists an element o(φ, ψ) ∈ Ext^1_O_X(F, L) whose vanishing is a necessary and sufficient condition for the existence of a map φ' : F' → G' compatible…","statement_latex":"Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$\nbe a first order thickening of ringed spaces. Assume given\nextensions\n$$\n0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0\n\\quad\\text{and}\\quad\n0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0\n$$\nas in (\\ref{equation-extension})\nand maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and\n$\\psi : \\mathcal{K} \\to \\mathcal{L}$. Assume the diagram\n$$\n\\xymatrix{\n\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}\n\\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} &\n\\mathcal{K} \\ar[d]^\\psi \\\\\n\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{G}\n\\ar[r]^-{c_{\\mathcal{G}'}} &\n\\mathcal{L}\n}\n$$\nis commutative. Then there exists an element\n$$\no(\\varphi, \\psi) \\in\n\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{L})\n$$\nwhose vanishing is a necessary and sufficient condition for the existence\nof a map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with\n$\\varphi$ and $\\psi$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Modules on first order thickenings of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08L6","source_file":"defos.tex","source_line":654,"source_end_line":685,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L654-L685","statement_sha256":"a5a7a56d0bff452bfb5a57afd649215afcac722279f7147d651181d011c0687a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13406,"rank":13406,"depth":0,"x":1498.043,"y":1534.418,"cluster":"deformation-theory"},{"id":"stacks:08L7","tag":"08L7","title":"Modules on first order thickenings of ringed spaces · Lemma 08L7","summary":"Let i : (X, O_X) → (X', O_X') be a first order thickening of ringed spaces. Assume given O_X-modules F, K and an O_X-linear map c : I ⊗_O_X F → K. If there exists a sequence ([Tag 08L4]) with c_F' = c then the set of isomorphism classes of these extensions is principal homogeneous under Ext^1_O_X(F, K).","statement_latex":"Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$ be a first order\nthickening of ringed spaces.\nAssume given $\\mathcal{O}_X$-modules $\\mathcal{F}$, $\\mathcal{K}$\nand an $\\mathcal{O}_X$-linear map\n$c : \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}$.\nIf there exists a sequence (\\ref{equation-extension}) with\n$c_{\\mathcal{F}'} = c$ then the set of isomorphism classes of these\nextensions is principal homogeneous under\n$\\Ext^1_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Modules on first order thickenings of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08L7","source_file":"defos.tex","source_line":728,"source_end_line":739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L728-L739","statement_sha256":"fe22a85e9dcc71820f53ed95319ae58aca52c022dbb7fd31a0225ada9ac73a11","origin":"The Stacks Project","memory_eligible":false,"source_rank":13407,"rank":13407,"depth":4,"x":1693.564,"y":1601.862,"cluster":"deformation-theory"},{"id":"stacks:08L8","tag":"08L8","title":"Modules on first order thickenings of ringed spaces · Lemma 08L8","summary":"Let i : (X, O_X) → (X', O_X') be a first order thickening of ringed spaces. Assume given O_X-modules F, K and an O_X-linear map c : I ⊗_O_X F → K. Then there exists an element o(F, K, c) ∈ Ext^2_O_X(F, K) whose vanishing is a necessary and sufficient condition for the existence of a sequence ([Tag 08L4]) with c_F' = c.","statement_latex":"Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$\nbe a first order thickening of ringed spaces. Assume given\n$\\mathcal{O}_X$-modules $\\mathcal{F}$, $\\mathcal{K}$\nand an $\\mathcal{O}_X$-linear map\n$c : \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to \\mathcal{K}$.\nThen there exists an element\n$$\no(\\mathcal{F}, \\mathcal{K}, c) \\in\n\\Ext^2_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{K})\n$$\nwhose vanishing is a necessary and sufficient condition for the existence\nof a sequence (\\ref{equation-extension}) with $c_{\\mathcal{F}'} = c$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Modules on first order thickenings of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08L8","source_file":"defos.tex","source_line":763,"source_end_line":777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L763-L777","statement_sha256":"234170eac33f1861f1d3581b3cf98630b386314814b0c2846f87d44c9e8244ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":13408,"rank":13408,"depth":7,"x":1494.47,"y":1663.283,"cluster":"deformation-theory"},{"id":"stacks:08LH","tag":"08LH","title":"Infinitesimal deformations of modules on ringed spaces · Lemma 08LH","summary":"Let i : (X, O_X) → (X', O_X') be a first order thickening of ringed spaces. Let F', G' be O_X'-modules. Set F = i^*F' and G = i^*G'. Let φ : F → G be an O_X-linear map. The set of lifts of φ to an O_X'-linear map φ' : F' → G' is, if nonempty, a principal homogeneous space under Hom_O_X(F, IG').","statement_latex":"Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$\nbe a first order thickening of ringed spaces.\nLet $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules.\nSet $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map.\nThe set of lifts of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map\n$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal\nhomogeneous space under\n$\\Hom_{\\mathcal{O}_X}(\\mathcal{F}, \\mathcal{I}\\mathcal{G}')$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LH","source_file":"defos.tex","source_line":1277,"source_end_line":1288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1277-L1288","statement_sha256":"4c9997466cc501669e917213b7e99bd2de53227ada3da3e4750119cbfeb60526","origin":"The Stacks Project","memory_eligible":false,"source_rank":13409,"rank":13409,"depth":5,"x":1592.22,"y":1504.48,"cluster":"deformation-theory"},{"id":"stacks:08LI","tag":"08LI","title":"Infinitesimal deformations of modules on ringed spaces · Lemma 08LI","summary":"Let (f, f') be a morphism of first order thickenings of ringed spaces as in Situation [Tag 08KZ]. Let F' be an O_X'-module and set F = i^*F'. Assume that F is flat over S and that (f, f') is a strict morphism of thickenings (Definition [Tag 08L1]). Then the following are equivalent • F' is flat over S', and • the canonical map f^*J ⊗_O_X F → IF' is an isomorphism. Moreover, in this case the maps f^*J ⊗_O_X F → I ⊗_O_X F → IF' are isomorphisms.","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings of ringed spaces\nas in Situation \\ref{situation-morphism-thickenings}.\nLet $\\mathcal{F}'$ be an $\\mathcal{O}_{X'}$-module\nand set $\\mathcal{F} = i^*\\mathcal{F}'$.\nAssume that $\\mathcal{F}$ is flat over $S$\nand that $(f, f')$ is a strict morphism of thickenings\n(Definition \\ref{definition-strict-morphism-thickenings}).\nThen the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}'$ is flat over $S'$, and\n\\item the canonical map\n$f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to\n\\mathcal{I}\\mathcal{F}'$\nis an isomorphism.\n\\end{enumerate}\nMoreover, in this case the maps\n$$\nf^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to\n\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to\n\\mathcal{I}\\mathcal{F}'\n$$\nare isomorphisms.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LI","source_file":"defos.tex","source_line":1318,"source_end_line":1342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1318-L1342","statement_sha256":"ccdf5b8b0cf86307d4c0e111e9ed56f9864fd1505ca81cf8106a379ba8528db2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13410,"rank":13410,"depth":4,"x":1648.034,"y":1677.631,"cluster":"deformation-theory"},{"id":"stacks:08LJ","tag":"08LJ","title":"Infinitesimal deformations of modules on ringed spaces · Lemma 08LJ","summary":"Let (f, f') be a morphism of first order thickenings as in Situation [Tag 08KZ]. Let F', G' be O_X'-modules and set F = i^*F' and G = i^*G'. Let φ : F → G be an O_X-linear map. Assume that G' is flat over S' and that (f, f') is a strict morphism of thickenings. The set of lifts of φ to an O_X'-linear map φ' : F' → G' is, if nonempty, a principal homogeneous space under Hom_O_X(F, G ⊗_O_X f^*J)","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings as in\nSituation \\ref{situation-morphism-thickenings}.\nLet $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules and set\n$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map.\nAssume that $\\mathcal{G}'$ is flat over $S'$ and that\n$(f, f')$ is a strict morphism of thickenings.\nThe set of lifts of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map\n$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal\nhomogeneous space under\n$$\n\\Hom_{\\mathcal{O}_X}(\\mathcal{F},\n\\mathcal{G} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{J})\n$$","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LJ","source_file":"defos.tex","source_line":1393,"source_end_line":1409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1393-L1409","statement_sha256":"95b90db7e23bde1a270b34ec1abdd9d750466c1f0d97586c803668d1f331907a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13411,"rank":13411,"depth":6,"x":1467.026,"y":1581.297,"cluster":"deformation-theory"},{"id":"stacks:08LK","tag":"08LK","title":"Infinitesimal deformations of modules on ringed spaces · Lemma 08LK","summary":"Let i : (X, O_X) → (X', O_X') be a first order thickening of ringed spaces. Let F', G' be O_X'-modules and set F = i^*F' and G = i^*G'. Let φ : F → G be an O_X-linear map. There exists an element o(φ) ∈ Ext^1_O_X(Li^*F', IG') whose vanishing is a necessary and sufficient condition for the existence of a lift of φ to an O_X'-linear map φ' : F' → G'.","statement_latex":"Let $i : (X, \\mathcal{O}_X) \\to (X', \\mathcal{O}_{X'})$\nbe a first order thickening of ringed spaces.\nLet $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules and set\n$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map.\nThere exists an element\n$$\no(\\varphi) \\in\n\\Ext^1_{\\mathcal{O}_X}(Li^*\\mathcal{F}',\n\\mathcal{I}\\mathcal{G}')\n$$\nwhose vanishing is a necessary and sufficient condition for the\nexistence of a lift of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map\n$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LK","source_file":"defos.tex","source_line":1415,"source_end_line":1431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1415-L1431","statement_sha256":"5e1d063b3dd7e3eea4eac00f3943e1c6ab5f7930822e57a6ea296db566789049","origin":"The Stacks Project","memory_eligible":false,"source_rank":13412,"rank":13412,"depth":19,"x":1678.674,"y":1549.519,"cluster":"deformation-theory"},{"id":"stacks:08LL","tag":"08LL","title":"Infinitesimal deformations of modules on ringed spaces · Lemma 08LL","summary":"Let (f, f') be a morphism of first order thickenings as in Situation [Tag 08KZ]. Let F', G' be O_X'-modules and set F = i^*F' and G = i^*G'. Let φ : F → G be an O_X-linear map. Assume that F' and G' are flat over S' and that (f, f') is a strict morphism of thickenings. There exists an element o(φ) ∈ Ext^1_O_X(F, G ⊗_O_X f^*J) whose vanishing is a necessary and sufficient condition for the existence of a lift of φ to an O_X'-linear map φ' : F' → G'.","statement_latex":"Let $(f, f')$ be a morphism of first\norder thickenings as in Situation \\ref{situation-morphism-thickenings}.\nLet $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}_{X'}$-modules and set\n$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}_X$-linear map.\nAssume that $\\mathcal{F}'$ and $\\mathcal{G}'$ are flat over $S'$ and\nthat $(f, f')$ is a strict morphism of thickenings. There exists an element\n$$\no(\\varphi) \\in  \\Ext^1_{\\mathcal{O}_X}(\\mathcal{F},\n\\mathcal{G} \\otimes_{\\mathcal{O}_X} f^*\\mathcal{J})\n$$\nwhose vanishing is a necessary and sufficient condition for the\nexistence of a lift of $\\varphi$ to an $\\mathcal{O}_{X'}$-linear map\n$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LL","source_file":"defos.tex","source_line":1451,"source_end_line":1467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1451-L1467","statement_sha256":"45a6464c802500a151d89a4d10f8599b8cceb758ab3eaf3379b7db2d4a22353f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13413,"rank":13413,"depth":20,"x":1547.728,"y":1693.523,"cluster":"deformation-theory"},{"id":"stacks:08LM","tag":"08LM","title":"Infinitesimal deformations of modules on ringed spaces · Lemma 08LM","summary":"Let (f, f') be a morphism of first order thickenings as in Situation [Tag 08KZ]. Let F be an O_X-module. Assume (f, f') is a strict morphism of thickenings and F flat over S. If there exists a pair (F', α) consisting of an O_X'-module F' flat over S' and an isomorphism α : i^*F' → F, then the set of isomorphism classes of such pairs is principal homogeneous under Ext^1_O_X( F, I ⊗_O_X F).","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings as in\nSituation \\ref{situation-morphism-thickenings}.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module.\nAssume $(f, f')$ is a strict morphism of thickenings and\n$\\mathcal{F}$ flat over $S$. If there exists a pair\n$(\\mathcal{F}', \\alpha)$ consisting of an\n$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ flat over $S'$ and an isomorphism\n$\\alpha : i^*\\mathcal{F}' \\to \\mathcal{F}$, then the set of\nisomorphism classes of such pairs is principal homogeneous\nunder\n$\\Ext^1_{\\mathcal{O}_X}(\n\\mathcal{F}, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LM","source_file":"defos.tex","source_line":1517,"source_end_line":1531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1517-L1531","statement_sha256":"5f35252f05e7e5237c617a6fb9550558a82cfc9dab3953cfd91aff7b0eccb964","origin":"The Stacks Project","memory_eligible":false,"source_rank":13414,"rank":13414,"depth":5,"x":1528.418,"y":1512.438,"cluster":"deformation-theory"},{"id":"stacks:08LN","tag":"08LN","title":"Infinitesimal deformations of modules on ringed spaces · Lemma 08LN","summary":"Let (f, f') be a morphism of first order thickenings as in Situation [Tag 08KZ]. Let F be an O_X-module. Assume (f, f') is a strict morphism of thickenings and F flat over S. There exists an O_X'-module F' flat over S' with i^*F' ≅ F, if and only if • the canonical map f^*J ⊗_O_X F → I ⊗_O_X F is an isomorphism, and • the class o(F, I ⊗_O_X F, 1) ∈ Ext^2_O_X( F, I ⊗_O_X F) of Lemma [Tag 08L8] is zero.","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings as in\nSituation \\ref{situation-morphism-thickenings}.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module. Assume\n$(f, f')$ is a strict morphism of thickenings\nand $\\mathcal{F}$ flat over $S$. There exists an\n$\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ flat over $S'$ with\n$i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if\n\\begin{enumerate}\n\\item the canonical map $\nf^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F} \\to\n\\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}$\nis an isomorphism, and\n\\item the class\n$o(\\mathcal{F}, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F}, 1)\n\\in \\Ext^2_{\\mathcal{O}_X}(\n\\mathcal{F}, \\mathcal{I} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$\nof Lemma \\ref{lemma-inf-obs-ext} is zero.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LN","source_file":"defos.tex","source_line":1546,"source_end_line":1566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1546-L1566","statement_sha256":"cdd312684fecb24ecb51662c92c1beea3bcc97b6e4cf66c8d4ee4b9912f2af53","origin":"The Stacks Project","memory_eligible":false,"source_rank":13415,"rank":13415,"depth":8,"x":1688.806,"y":1635.414,"cluster":"deformation-theory"},{"id":"stacks:08VR","tag":"08VR","title":"Application to flat modules on flat thickenings of ringed spaces · Lemma 08VR","summary":"In the situation above. • There exists an O_X'-module F' flat over S' with i^*F' ≅ F, if and only if the class o(F, f^*J ⊗_O_X F, 1) ∈ Ext^2_O_X( F, f^*J ⊗_O_X F) of Lemma [Tag 08L8] is zero. • If such a module exists, then the set of isomorphism classes of lifts is principal homogeneous under Ext^1_O_X( F, f^*J ⊗_O_X F). • Given a lift F', the set of automorphisms of F' which pull back to id_F is canonically isomorphic to Ext^0_O_X( F, f^*J ⊗_O_X F).","statement_latex":"In the situation above.\n\\begin{enumerate}\n\\item There exists an $\\mathcal{O}_{X'}$-module $\\mathcal{F}'$ flat over\n$S'$ with $i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if\nthe class\n$o(\\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F}, 1)\n\\in \\Ext^2_{\\mathcal{O}_X}(\n\\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$\nof Lemma \\ref{lemma-inf-obs-ext} is zero.\n\\item If such a module exists, then the set of isomorphism classes\nof lifts is principal homogeneous under\n$\\Ext^1_{\\mathcal{O}_X}(\n\\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$.\n\\item Given a lift $\\mathcal{F}'$, the set of automorphisms of\n$\\mathcal{F}'$ which pull back to $\\text{id}_\\mathcal{F}$ is canonically\nisomorphic to $\\Ext^0_{\\mathcal{O}_X}(\n\\mathcal{F}, f^*\\mathcal{J} \\otimes_{\\mathcal{O}_X} \\mathcal{F})$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Application to flat modules on flat thickenings of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VR","source_file":"defos.tex","source_line":1607,"source_end_line":1627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1607-L1627","statement_sha256":"3dbd53919c7084db4f510db9814b3f123c22c1489bd1ea2a05bde1b3414081e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13416,"rank":13416,"depth":9,"x":1470.936,"y":1635.741,"cluster":"deformation-theory"},{"id":"stacks:08VT","tag":"08VT","title":"Application to flat modules on flat thickenings of ringed spaces · Lemma 08VT","summary":"In Situation [Tag 08VS] the modules π^*F and h^*F'_2 are O'_1-modules flat over S'_1 restricting to F on X. Their difference (Lemma [Tag 08VR]) is an element theta of Ext^1_O_X( F, f^*J_1 ⊗_O_X F) whose boundary in Ext^2_O_X( F, f^*J_3 ⊗_O_X F) equals the obstruction (Lemma [Tag 08VR]) to lifting F to an O'_3-module flat over S'_3.","statement_latex":"In Situation \\ref{situation-ses-flat-thickenings} the modules\n$\\pi^*\\mathcal{F}$ and $h^*\\mathcal{F}'_2$ are $\\mathcal{O}'_1$-modules\nflat over $S'_1$ restricting to $\\mathcal{F}$ on $X$.\nTheir difference (Lemma \\ref{lemma-flat}) is an element\n$\\theta$ of $\\Ext^1_{\\mathcal{O}_X}(\n\\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_{\\mathcal{O}_X} \\mathcal{F})$\nwhose boundary in\n$\\Ext^2_{\\mathcal{O}_X}(\n\\mathcal{F}, f^*\\mathcal{J}_3 \\otimes_{\\mathcal{O}_X} \\mathcal{F})$\nequals the obstruction (Lemma \\ref{lemma-flat})\nto lifting $\\mathcal{F}$ to an $\\mathcal{O}'_3$-module flat over $S'_3$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Application to flat modules on flat thickenings of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VT","source_file":"defos.tex","source_line":1660,"source_end_line":1673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1660-L1673","statement_sha256":"b87243ebd0353aded8ceecdc1bc86ff6d568a0c1db4f43c459b63db45616ac5c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13417,"rank":13417,"depth":10,"x":1631.847,"y":1511.475,"cluster":"deformation-theory"},{"id":"stacks:08U8","tag":"08U8","title":"Deformations of ringed spaces and the naive cotangent complex · Lemma 08U8","summary":"Assume given a commutative diagram of morphisms of ringed spaces vcenter xymatrix & (X_2, O_X_2) ar[r]_i_2 ar[d]_f_2 ar[ddl]_g & (X'_2, O_X'_2) ar[d]^f'_2 & (S_2, O_S_2) ar[r]^t_2 ar[ddl]|hole & (S'_2, O_S'_2) ar[ddl] (X_1, O_X_1) ar[r]_i_1 ar[d]_f_1 & (X'_1, O_X'_1) ar[d]^f'_1 (S_1, O_S_1) ar[r]^t_1 & (S'_1, O_S'_1) whose horizontal arrows are first order thickenings. Set G_j = Ker(i_j^sharp) and assume given a g-map ν : G_1 → G_2 of modules giving rise to the…","statement_latex":"Assume given a commutative diagram of morphisms of ringed spaces\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\n& (X_2, \\mathcal{O}_{X_2}) \\ar[r]_{i_2} \\ar[d]_{f_2} \\ar[ddl]_g &\n(X'_2, \\mathcal{O}_{X'_2}) \\ar[d]^{f'_2} \\\\\n& (S_2, \\mathcal{O}_{S_2}) \\ar[r]^{t_2} \\ar[ddl]|\\hole &\n(S'_2, \\mathcal{O}_{S'_2}) \\ar[ddl] \\\\\n(X_1, \\mathcal{O}_{X_1}) \\ar[r]_{i_1} \\ar[d]_{f_1} &\n(X'_1, \\mathcal{O}_{X'_1}) \\ar[d]^{f'_1} \\\\\n(S_1, \\mathcal{O}_{S_1}) \\ar[r]^{t_1} &\n(S'_1, \\mathcal{O}_{S'_1})\n}\n}\n\\end{equation}\nwhose horizontal arrows are first order thickenings. Set\n$\\mathcal{G}_j = \\Ker(i_j^\\sharp)$ and assume given\na $g$-map $\\nu : \\mathcal{G}_1 \\to \\mathcal{G}_2$ of modules\ngiving rise to the commutative diagram\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\n& 0 \\ar[r] & \\mathcal{G}_2 \\ar[r] &\n\\mathcal{O}_{X'_2} \\ar[r] &\n\\mathcal{O}_{X_2} \\ar[r] & 0 \\\\\n& 0 \\ar[r]|\\hole &\n\\mathcal{J}_2 \\ar[u]_{c_2} \\ar[r] &\n\\mathcal{O}_{S'_2} \\ar[u] \\ar[r]|\\hole &\n\\mathcal{O}_{S_2} \\ar[u] \\ar[r] & 0 \\\\\n0 \\ar[r] & \\mathcal{G}_1 \\ar[ruu] \\ar[r] &\n\\mathcal{O}_{X'_1} \\ar[r] &\n\\mathcal{O}_{X_1} \\ar[ruu] \\ar[r] & 0 \\\\\n0 \\ar[r] & \\mathcal{J}_1 \\ar[ruu]|\\hole \\ar[u]^{c_1} \\ar[r] &\n\\mathcal{O}_{S'_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] &\n\\mathcal{O}_{S_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] & 0\n}\n}\n\\end{equation}\nwith front and back solutions to (\\ref{equation-to-solve-ringed-spaces}).\n\\begin{enumerate}\n\\item There exist a canonical element in\n$\\Ext^1_{\\mathcal{O}_{X_2}}(Lg^*\\NL_{X_1/S_1}, \\mathcal{G}_2)$\nwhose vanishing is a necessary and sufficient condition for the existence\nof a morphism of ringed spaces $X'_2 \\to X'_1$ fitting into\n(\\ref{equation-huge-1}) compatibly with $\\nu$.\n\\item If there exists a morphism $X'_2 \\to X'_1$ fitting into\n(\\ref{equation-huge-1}) compatibly with $\\nu$ the set of all such morphisms\nis a principal homogeneous space under\n$$\n\\Hom_{\\mathcal{O}_{X_1}}(\\Omega_{X_1/S_1}, g_*\\mathcal{G}_2) =\n\\Hom_{\\mathcal{O}_{X_2}}(g^*\\Omega_{X_1/S_1}, \\mathcal{G}_2) =\n\\Ext^0_{\\mathcal{O}_{X_2}}(Lg^*\\NL_{X_1/S_1}, \\mathcal{G}_2).\n$$\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed spaces and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08U8","source_file":"defos.tex","source_line":1738,"source_end_line":1796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1738-L1796","statement_sha256":"d0ea84b4be320b268f6ce895e78eb3c21f722a08da85c14a9ac558087e10a687","origin":"The Stacks Project","memory_eligible":false,"source_rank":13418,"rank":13418,"depth":4,"x":1613.064,"y":1695.001,"cluster":"deformation-theory"},{"id":"stacks:08UB","tag":"08UB","title":"Deformations of ringed spaces and the naive cotangent complex · Lemma 08UB","summary":"Let X be a topological space. Let A → B be a homomorphism of sheaves of rings. Let G be a B-module. Let xi ∈ Ext^1_B(NL_B/A, G). There exists a map of sheaves of sets α : E → B such that xi ∈ Ext^1_B(NL(α), G) is the class of a map I/I^2 → G (see proof for notation).","statement_latex":"Let $X$ be a topological space. Let $\\mathcal{A} \\to \\mathcal{B}$ be a\nhomomorphism of sheaves of rings. Let $\\mathcal{G}$ be a $\\mathcal{B}$-module.\nLet\n$\\xi \\in \\Ext^1_\\mathcal{B}(\\NL_{\\mathcal{B}/\\mathcal{A}}, \\mathcal{G})$. \nThere exists a map of sheaves of sets $\\alpha : \\mathcal{E} \\to \\mathcal{B}$\nsuch that $\\xi \\in \\Ext^1_\\mathcal{B}(\\NL(\\alpha), \\mathcal{G})$\nis the class of a map $\\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$\n(see proof for notation).","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed spaces and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UB","source_file":"defos.tex","source_line":1940,"source_end_line":1950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1940-L1950","statement_sha256":"641683c9df1a86b391d4bdb6decfbe4e28fbae1385299ad5f1884f3ee2ea759b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13419,"rank":13419,"depth":4,"x":1478.901,"y":1548.546,"cluster":"deformation-theory"},{"id":"stacks:08UC","tag":"08UC","title":"Deformations of ringed spaces and the naive cotangent complex · Lemma 08UC","summary":"If there exists a solution to ([Tag 08U7]), then the set of isomorphism classes of solutions is principal homogeneous under Ext^1_O_X(NL_X/S, G).","statement_latex":"If there exists a solution to (\\ref{equation-to-solve-ringed-spaces}),\nthen the set of isomorphism classes of solutions is principal homogeneous\nunder $\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, \\mathcal{G})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed spaces and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UC","source_file":"defos.tex","source_line":1999,"source_end_line":2004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L1999-L2004","statement_sha256":"001afbde4015ffb4e3a795fee3c939d299f0eba083c07a69a66bdc623320323d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13420,"rank":13420,"depth":5,"x":1696.294,"y":1580.514,"cluster":"deformation-theory"},{"id":"stacks:0GPZ","tag":"0GPZ","title":"Deformations of ringed spaces and the naive cotangent complex · Lemma 0GPZ","summary":"Let f : (X, O_X) → (S, O_S) be a morphism of ringed spaces. Let G be a O_X-module. The set of isomorphism classes of extensions of f^-1O_S-algebras 0 → G → O_X' → O_X → 0 where G is an ideal of square zero → Ker(i^sharp) of O_X-modules. is canonically bijective to Ext^1_O_X(NL_X/S, G).","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of\nringed spaces. Let $\\mathcal{G}$ be a $\\mathcal{O}_X$-module.\nThe set of isomorphism classes of extensions of\n$f^{-1}\\mathcal{O}_S$-algebras\n$$\n0 \\to \\mathcal{G} \\to \\mathcal{O}_{X'} \\to \\mathcal{O}_X \\to 0\n$$\nwhere $\\mathcal{G}$ is an ideal of square zero\\footnote{In other words,\nthe set of isomorphism classes of first order thickenings\n$i : X \\to X'$ over $S$ endowed with an isomorphism\n$\\mathcal{G} \\to \\Ker(i^\\sharp)$ of $\\mathcal{O}_X$-modules.} \nis canonically bijective to\n$\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, \\mathcal{G})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed spaces and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GPZ","source_file":"defos.tex","source_line":2063,"source_end_line":2078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L2063-L2078","statement_sha256":"113027647d021985aed34538be59e7437d0044affe0913c07c2e0788c68186f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13421,"rank":13421,"depth":6,"x":1509.696,"y":1680.61,"cluster":"deformation-theory"},{"id":"stacks:0GQ1","tag":"0GQ1","title":"Deformations of ringed spaces and the naive cotangent complex · Lemma 0GQ1","summary":"Let f : (X, O_X) → (S, O_S) and g : (Y, O_Y) → (X, O_X) be morphisms of ringed spaces. Let F be a O_X-module. Let G be a O_Y-module. Let c : F → G be a g-map. Finally, consider • [(a)] 0 → F → O_X' → O_X → 0 an extension of f^-1O_S-algebras corresponding to xi ∈ Ext^1_O_X(NL_X/S, F), and • [(b)] 0 → G → O_Y' → O_Y → 0 an extension of g^-1f^-1O_S-algebras corresponding to zeta ∈ Ext^1_O_Y(NL_Y/S, G). See Lemma [Tag 0GPZ]. Then there is an S-morphism g' : Y' → X' compatible…","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ and\n$g : (Y, \\mathcal{O}_Y) \\to (X, \\mathcal{O}_X)$ be morphisms\nof ringed spaces. Let $\\mathcal{F}$ be a $\\mathcal{O}_X$-module.\nLet $\\mathcal{G}$ be a $\\mathcal{O}_Y$-module. Let\n$c : \\mathcal{F} \\to \\mathcal{G}$ be a $g$-map. Finally, consider\n\\begin{enumerate}\n\\item[(a)] $0 \\to \\mathcal{F} \\to \\mathcal{O}_{X'} \\to \\mathcal{O}_X \\to 0$\nan extension of $f^{-1}\\mathcal{O}_S$-algebras\ncorresponding to $\\xi \\in \\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, \\mathcal{F})$, and\n\\item[(b)] $0 \\to \\mathcal{G} \\to \\mathcal{O}_{Y'} \\to \\mathcal{O}_Y \\to 0$\nan extension of $g^{-1}f^{-1}\\mathcal{O}_S$-algebras\ncorresponding to $\\zeta \\in \\Ext^1_{\\mathcal{O}_Y}(\\NL_{Y/S}, \\mathcal{G})$.\n\\end{enumerate}\nSee Lemma \\ref{lemma-extensions-of-relative-ringed-spaces}.\nThen there is an $S$-morphism $g' : Y' \\to X'$\ncompatible with $g$ and $c$ if and only if $\\xi$ and $\\zeta$\nmap to the same element of\n$\\Ext^1_{\\mathcal{O}_Y}(Lg^*\\NL_{X/S}, \\mathcal{G})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed spaces and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQ1","source_file":"defos.tex","source_line":2140,"source_end_line":2160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L2140-L2160","statement_sha256":"525eca1b76fd16fcefe7be098ff20d62577d6405f155471bed04c2f1c6419dea","origin":"The Stacks Project","memory_eligible":false,"source_rank":13422,"rank":13422,"depth":7,"x":1566.976,"y":1500.357,"cluster":"deformation-theory"},{"id":"stacks:0GQ3","tag":"0GQ3","title":"Deformations of ringed spaces and the naive cotangent complex · Lemma 0GQ3","summary":"Let t : (S, O_S) → (S', O_S'), J = Ker(t^sharp), f : (X, O_X) → (S, O_S), G, and c : J → G be as in ([Tag 08U7]). Denote xi ∈ Ext^1_O_S(NL_S/S', J) the element corresponding to the extension O_S' of O_S by J via Lemma [Tag 0GPZ]. The set of isomorphism classes of solutions is canonically bijective to the fibre of Ext^1_O_X(NL_X/S', G) → Ext^1_O_X(Lf^*NL_S/S', G) over the image of xi.","statement_latex":"Let $t : (S, \\mathcal{O}_S) \\to (S', \\mathcal{O}_{S'})$,\n$\\mathcal{J} = \\Ker(t^\\sharp)$,\n$f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$, $\\mathcal{G}$, and\n$c : \\mathcal{J} \\to \\mathcal{G}$ be as in\n(\\ref{equation-to-solve-ringed-spaces}).\nDenote $\\xi \\in \\Ext^1_{\\mathcal{O}_S}(\\NL_{S/S'}, \\mathcal{J})$\nthe element corresponding to the extension $\\mathcal{O}_{S'}$\nof $\\mathcal{O}_S$ by $\\mathcal{J}$ via\nLemma \\ref{lemma-extensions-of-relative-ringed-spaces}.\nThe set of isomorphism classes of solutions is canonically bijective\nto the fibre of\n$$\n\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S'}, \\mathcal{G}) \\to\n\\Ext^1_{\\mathcal{O}_X}(Lf^*\\NL_{S/S'}, \\mathcal{G})\n$$\nover the image of $\\xi$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed spaces and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQ3","source_file":"defos.tex","source_line":2318,"source_end_line":2336,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L2318-L2336","statement_sha256":"4bc7220a350b14afbaf0b2f1f7f2518c22519d6e5375ded58683c0b461dd3546","origin":"The Stacks Project","memory_eligible":false,"source_rank":13423,"rank":13423,"depth":8,"x":1670.013,"y":1666.291,"cluster":"deformation-theory"},{"id":"stacks:0D14","tag":"0D14","title":"Deformations of schemes · Lemma 0D14","summary":"Let S ⊂ S' be a first order thickening of schemes. Let f : X → S be a flat morphism of schemes. If there exists a flat morphism f' : X' → S' of schemes and an isomorphism a : X → X' ×_S' S over S, then • the set of isomorphism classes of pairs (f' : X' → S', a) is principal homogeneous under Ext^1_O_X(NL_X/S, f^*C_S/S'), and • the set of automorphisms of φ : X' → X' over S' which reduce to the identity on X' ×_S' S is Ext^0_O_X(NL_X/S, f^*C_S/S').","statement_latex":"Let $S \\subset S'$ be a first order thickening of schemes.\nLet $f : X \\to S$ be a flat morphism of schemes.\nIf there exists a flat morphism $f' : X' \\to S'$ of schemes\nand an isomorphism $a : X \\to X' \\times_{S'} S$ over $S$, then\n\\begin{enumerate}\n\\item the set of isomorphism classes of pairs $(f' : X' \\to S', a)$ is\nprincipal homogeneous under\n$\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/S}, f^*\\mathcal{C}_{S/S'})$, and\n\\item the set of automorphisms of $\\varphi : X' \\to X'$\nover $S'$ which reduce to the identity on $X' \\times_{S'} S$\nis $\\Ext^0_{\\mathcal{O}_X}(\\NL_{X/S}, f^*\\mathcal{C}_{S/S'})$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D14","source_file":"defos.tex","source_line":2391,"source_end_line":2405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L2391-L2405","statement_sha256":"388f7d1e839f0fa6a194cabfa930af9a699a7548cc46e744a690c0ff13d09898","origin":"The Stacks Project","memory_eligible":false,"source_rank":13424,"rank":13424,"depth":6,"x":1459.947,"y":1602.2,"cluster":"deformation-theory"},{"id":"stacks:08M9","tag":"08M9","title":"Thickenings of ringed topoi · Definition 08M9","summary":"In Situation [Tag 08M7] we say that (f, f') is a strict morphism of thickenings if the map (f')^*J → I is surjective.","statement_latex":"In Situation \\ref{situation-morphism-thickenings-ringed-topoi}\nwe say that $(f, f')$ is a {\\it strict morphism of thickenings}\nif the map $(f')^*\\mathcal{J} \\longrightarrow \\mathcal{I}$ is surjective.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Thickenings of ringed topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08M9","source_file":"defos.tex","source_line":2527,"source_end_line":2532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L2527-L2532","statement_sha256":"7c007b0ebc25f7434ec8b791993340ab1d3057b7a91ca7807a475ec83e315d4f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13425,"rank":13425,"depth":0,"x":1667.022,"y":1530.042,"cluster":"deformation-theory"},{"id":"stacks:08MC","tag":"08MC","title":"Modules on first order thickenings of ringed topoi · Lemma 08MC","summary":"Let i : (Sh(C), O) → (Sh(D), O') be a first order thickening of ringed topoi. Assume given extensions 0 → K → F' → F → 0 and 0 → L → G' → G → 0 as in ([Tag 08MB]) and maps φ : F → G and ψ : K → L. • If there exists an O'-module map φ' : F' → G' compatible with φ and ψ, then the diagram xymatrix I ⊗_O F ar[r]_-c_F' ar[d]_1 ⊗ φ & K ar[d]^ψ I ⊗_O G ar[r]^-c_G' & L is commutative. • The set of O'-module maps φ' : F' → G' compatible with φ and ψ is, if nonempty, a principal…","statement_latex":"Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a first order thickening of ringed topoi. Assume given\nextensions\n$$\n0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0\n\\quad\\text{and}\\quad\n0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0\n$$\nas in (\\ref{equation-extension-ringed-topoi})\nand maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and\n$\\psi : \\mathcal{K} \\to \\mathcal{L}$.\n\\begin{enumerate}\n\\item If there exists an $\\mathcal{O}'$-module\nmap $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$\nand $\\psi$, then the diagram\n$$\n\\xymatrix{\n\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}\n\\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} &\n\\mathcal{K} \\ar[d]^\\psi \\\\\n\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{G}\n\\ar[r]^-{c_{\\mathcal{G}'}} &\n\\mathcal{L}\n}\n$$\nis commutative.\n\\item The set of $\\mathcal{O}'$-module\nmaps $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with $\\varphi$\nand $\\psi$ is, if nonempty, a principal homogeneous space under\n$\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{L})$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Modules on first order thickenings of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MC","source_file":"defos.tex","source_line":2570,"source_end_line":2603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L2570-L2603","statement_sha256":"1ee63c1b9eead83c9d9982ca6751b4607af200a6458a426420cb456b9e55d891","origin":"The Stacks Project","memory_eligible":false,"source_rank":13426,"rank":13426,"depth":12,"x":1572.065,"y":1701.272,"cluster":"deformation-theory"},{"id":"stacks:08MD","tag":"08MD","title":"Modules on first order thickenings of ringed topoi · Lemma 08MD","summary":"Let i : (Sh(C), O) → (Sh(D), O') be a first order thickening of ringed topoi. Assume given extensions 0 → K → F' → F → 0 and 0 → L → G' → G → 0 as in ([Tag 08MB]) and maps φ : F → G and ψ : K → L. Assume the diagram xymatrix I ⊗_O F ar[r]_-c_F' ar[d]_1 ⊗ φ & K ar[d]^ψ I ⊗_O G ar[r]^-c_G' & L is commutative. Then there exists an element o(φ, ψ) ∈ Ext^1_O(F, L) whose vanishing is a necessary and sufficient condition for the existence of a map φ' : F' → G' compatible with φ…","statement_latex":"Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a first order thickening of ringed topoi. Assume given extensions\n$$\n0 \\to \\mathcal{K} \\to \\mathcal{F}' \\to \\mathcal{F} \\to 0\n\\quad\\text{and}\\quad\n0 \\to \\mathcal{L} \\to \\mathcal{G}' \\to \\mathcal{G} \\to 0\n$$\nas in (\\ref{equation-extension-ringed-topoi})\nand maps $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ and\n$\\psi : \\mathcal{K} \\to \\mathcal{L}$. Assume the diagram\n$$\n\\xymatrix{\n\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}\n\\ar[r]_-{c_{\\mathcal{F}'}} \\ar[d]_{1 \\otimes \\varphi} &\n\\mathcal{K} \\ar[d]^\\psi \\\\\n\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{G}\n\\ar[r]^-{c_{\\mathcal{G}'}} &\n\\mathcal{L}\n}\n$$\nis commutative. Then there exists an element\n$$\no(\\varphi, \\psi) \\in\n\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{L})\n$$\nwhose vanishing is a necessary and sufficient condition for the existence\nof a map $\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ compatible with\n$\\varphi$ and $\\psi$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Modules on first order thickenings of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MD","source_file":"defos.tex","source_line":2621,"source_end_line":2651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L2621-L2651","statement_sha256":"05252323aab38bd9bf4d4cd13ed5698867c208324b84f17e1438366729cc28e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13427,"rank":13427,"depth":0,"x":1504.182,"y":1520.581,"cluster":"deformation-theory"},{"id":"stacks:08ME","tag":"08ME","title":"Modules on first order thickenings of ringed topoi · Lemma 08ME","summary":"Let i : (Sh(C), O) → (Sh(D), O') be a first order thickening of ringed topoi. Assume given O-modules F, K and an O-linear map c : I ⊗_O F → K. If there exists a sequence ([Tag 08MB]) with c_F' = c then the set of isomorphism classes of these extensions is principal homogeneous under Ext^1_O(F, K).","statement_latex":"Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a first order thickening of ringed topoi. Assume given\n$\\mathcal{O}$-modules $\\mathcal{F}$, $\\mathcal{K}$\nand an $\\mathcal{O}$-linear map\n$c : \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}$.\nIf there exists a sequence (\\ref{equation-extension-ringed-topoi}) with\n$c_{\\mathcal{F}'} = c$ then the set of isomorphism classes of these\nextensions is principal homogeneous under\n$\\Ext^1_\\mathcal{O}(\\mathcal{F}, \\mathcal{K})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Modules on first order thickenings of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08ME","source_file":"defos.tex","source_line":2694,"source_end_line":2705,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L2694-L2705","statement_sha256":"28ff526ddf5749f8569d7db4586c23841d89aa9f79ab44483eb3dfa5c2b9a64c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13428,"rank":13428,"depth":12,"x":1700.135,"y":1615.588,"cluster":"deformation-theory"},{"id":"stacks:08MF","tag":"08MF","title":"Modules on first order thickenings of ringed topoi · Lemma 08MF","summary":"Let i : (Sh(C), O) → (Sh(D), O') be a first order thickening of ringed topoi. Assume given O-modules F, K and an O-linear map c : I ⊗_O F → K. Then there exists an element o(F, K, c) ∈ Ext^2_O(F, K) whose vanishing is a necessary and sufficient condition for the existence of a sequence ([Tag 08MB]) with c_F' = c.","statement_latex":"Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a first order thickening of ringed topoi. Assume given\n$\\mathcal{O}$-modules $\\mathcal{F}$, $\\mathcal{K}$\nand an $\\mathcal{O}$-linear map\n$c : \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to \\mathcal{K}$.\nThen there exists an element\n$$\no(\\mathcal{F}, \\mathcal{K}, c) \\in\n\\Ext^2_\\mathcal{O}(\\mathcal{F}, \\mathcal{K})\n$$\nwhose vanishing is a necessary and sufficient condition for the existence\nof a sequence (\\ref{equation-extension-ringed-topoi})\nwith $c_{\\mathcal{F}'} = c$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Modules on first order thickenings of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MF","source_file":"defos.tex","source_line":2729,"source_end_line":2744,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L2729-L2744","statement_sha256":"b1342e185a762ca526c8d280a11b4f467622fd9be1630a997a611f0cc79cdc7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13429,"rank":13429,"depth":13,"x":1478.576,"y":1656.843,"cluster":"deformation-theory"},{"id":"stacks:08MP","tag":"08MP","title":"Infinitesimal deformations of modules on ringed topoi · Lemma 08MP","summary":"Let i : (Sh(C), O) → (Sh(D), O') be a first order thickening of ringed topoi. Let F', G' be O'-modules. Set F = i^*F' and G = i^*G'. Let φ : F → G be an O-linear map. The set of lifts of φ to an O'-linear map φ' : F' → G' is, if nonempty, a principal homogeneous space under Hom_O(F, IG').","statement_latex":"Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a first order thickening of ringed topoi.\nLet $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules.\nSet $\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map.\nThe set of lifts of $\\varphi$ to an $\\mathcal{O}'$-linear map\n$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal\nhomogeneous space under\n$\\Hom_\\mathcal{O}(\\mathcal{F}, \\mathcal{I}\\mathcal{G}')$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MP","source_file":"defos.tex","source_line":3366,"source_end_line":3377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3366-L3377","statement_sha256":"62825b8d8ba2f81cce4181ad8cc953c252e3b72fa11efb843a49bd7ebbaf2b51","origin":"The Stacks Project","memory_eligible":false,"source_rank":13430,"rank":13430,"depth":13,"x":1609.164,"y":1500.238,"cluster":"deformation-theory"},{"id":"stacks:08MQ","tag":"08MQ","title":"Infinitesimal deformations of modules on ringed topoi · Lemma 08MQ","summary":"Let (f, f') be a morphism of first order thickenings of ringed topoi as in Situation [Tag 08M7]. Let F' be an O'-module and set F = i^*F'. Assume that F is flat over O_B and that (f, f') is a strict morphism of thickenings (Definition [Tag 08M9]). Then the following are equivalent • F' is flat over O_B', and • the canonical map f^*J ⊗_O F → IF' is an isomorphism. Moreover, in this case the maps f^*J ⊗_O F → I ⊗_O F → IF' are isomorphisms.","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings of ringed topoi\nas in Situation \\ref{situation-morphism-thickenings-ringed-topoi}.\nLet $\\mathcal{F}'$ be an $\\mathcal{O}'$-module\nand set $\\mathcal{F} = i^*\\mathcal{F}'$.\nAssume that $\\mathcal{F}$ is flat over $\\mathcal{O}_\\mathcal{B}$\nand that $(f, f')$ is a strict morphism of thickenings\n(Definition \\ref{definition-strict-morphism-thickenings-ringed-topoi}).\nThen the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}'$ is flat over $\\mathcal{O}_{\\mathcal{B}'}$, and\n\\item the canonical map\n$f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to\n\\mathcal{I}\\mathcal{F}'$\nis an isomorphism.\n\\end{enumerate}\nMoreover, in this case the maps\n$$\nf^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to\n\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F} \\to\n\\mathcal{I}\\mathcal{F}'\n$$\nare isomorphisms.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MQ","source_file":"defos.tex","source_line":3407,"source_end_line":3431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3407-L3431","statement_sha256":"a895f9a9bf47fdc401505c7a86c5e41f6d8fc7ef995ca896094eaaa6355be540","origin":"The Stacks Project","memory_eligible":false,"source_rank":13431,"rank":13431,"depth":7,"x":1638.894,"y":1690.377,"cluster":"deformation-theory"},{"id":"stacks:08VU","tag":"08VU","title":"Infinitesimal deformations of modules on ringed topoi · Lemma 08VU","summary":"Let (f, f') be a morphism of first order thickenings of ringed topoi as in Situation [Tag 08M7]. Let F' be an O'-module and set F = i^*F'. Assume that F' is flat over O_B' and that (f, f') is a strict morphism of thickenings. Then the following are equivalent • F' is an O'-module of finite presentation, and • F is an O-module of finite presentation.","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings of ringed topoi\nas in Situation \\ref{situation-morphism-thickenings-ringed-topoi}.\nLet $\\mathcal{F}'$ be an $\\mathcal{O}'$-module\nand set $\\mathcal{F} = i^*\\mathcal{F}'$.\nAssume that $\\mathcal{F}'$ is flat over $\\mathcal{O}_{\\mathcal{B}'}$\nand that $(f, f')$ is a strict morphism of thickenings.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}'$ is an $\\mathcal{O}'$-module of finite presentation, and\n\\item $\\mathcal{F}$ is an $\\mathcal{O}$-module of finite presentation.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VU","source_file":"defos.tex","source_line":3454,"source_end_line":3467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3454-L3467","statement_sha256":"e61269b495ed9b2b5863f14abdee8c78cdc11ba307cc5a23b29f5927248a1e68","origin":"The Stacks Project","memory_eligible":false,"source_rank":13432,"rank":13432,"depth":11,"x":1463.552,"y":1566.677,"cluster":"deformation-theory"},{"id":"stacks:08MR","tag":"08MR","title":"Infinitesimal deformations of modules on ringed topoi · Lemma 08MR","summary":"Let (f, f') be a morphism of first order thickenings as in Situation [Tag 08M7]. Let F', G' be O'-modules and set F = i^*F' and G = i^*G'. Let φ : F → G be an O-linear map. Assume that G' is flat over O_B' and that (f, f') is a strict morphism of thickenings. The set of lifts of φ to an O'-linear map φ' : F' → G' is, if nonempty, a principal homogeneous space under Hom_O(F, G ⊗_O f^*J)","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings as in\nSituation \\ref{situation-morphism-thickenings-ringed-topoi}.\nLet $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules and set\n$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map.\nAssume that $\\mathcal{G}'$ is flat over $\\mathcal{O}_{\\mathcal{B}'}$ and that\n$(f, f')$ is a strict morphism of thickenings.\nThe set of lifts of $\\varphi$ to an $\\mathcal{O}'$-linear map\n$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$ is, if nonempty, a principal\nhomogeneous space under\n$$\n\\Hom_\\mathcal{O}(\\mathcal{F},\n\\mathcal{G} \\otimes_\\mathcal{O} f^*\\mathcal{J})\n$$","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MR","source_file":"defos.tex","source_line":3501,"source_end_line":3517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3501-L3517","statement_sha256":"6057316486828afaad35406450d1163bfe6b5a8d1a740b83549f9cb1b49a97e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13433,"rank":13433,"depth":14,"x":1692.994,"y":1558.375,"cluster":"deformation-theory"},{"id":"stacks:08MS","tag":"08MS","title":"Infinitesimal deformations of modules on ringed topoi · Lemma 08MS","summary":"Let i : (Sh(C), O) → (Sh(D), O') be a first order thickening of ringed topoi. Let F', G' be O'-modules and set F = i^*F' and G = i^*G'. Let φ : F → G be an O-linear map. There exists an element o(φ) ∈ Ext^1_O(Li^*F', IG') whose vanishing is a necessary and sufficient condition for the existence of a lift of φ to an O'-linear map φ' : F' → G'.","statement_latex":"Let $i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{D}), \\mathcal{O}')$\nbe a first order thickening of ringed topoi.\nLet $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules and set\n$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map.\nThere exists an element\n$$\no(\\varphi) \\in\n\\Ext^1_\\mathcal{O}(Li^*\\mathcal{F}', \\mathcal{I}\\mathcal{G}')\n$$\nwhose vanishing is a necessary and sufficient condition for the\nexistence of a lift of $\\varphi$ to an $\\mathcal{O}'$-linear map\n$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MS","source_file":"defos.tex","source_line":3524,"source_end_line":3539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3524-L3539","statement_sha256":"db8234ad85d2565fa3c0619090d05a6f768ca72ab72bfb5a161f0f20c44a43fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13434,"rank":13434,"depth":19,"x":1530.009,"y":1695.085,"cluster":"deformation-theory"},{"id":"stacks:08MT","tag":"08MT","title":"Infinitesimal deformations of modules on ringed topoi · Lemma 08MT","summary":"Let (f, f') be a morphism of first order thickenings as in Situation [Tag 08M7]. Let F', G' be O'-modules and set F = i^*F' and G = i^*G'. Let φ : F → G be an O-linear map. Assume that F' and G' are flat over O_B' and that (f, f') is a strict morphism of thickenings. There exists an element o(φ) ∈ Ext^1_O(F, G ⊗_O f^*J) whose vanishing is a necessary and sufficient condition for the existence of a lift of φ to an O'-linear map φ' : F' → G'.","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings as in\nSituation \\ref{situation-morphism-thickenings-ringed-topoi}.\nLet $\\mathcal{F}'$, $\\mathcal{G}'$ be $\\mathcal{O}'$-modules and set\n$\\mathcal{F} = i^*\\mathcal{F}'$ and $\\mathcal{G} = i^*\\mathcal{G}'$.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an $\\mathcal{O}$-linear map.\nAssume that $\\mathcal{F}'$ and $\\mathcal{G}'$ are flat over\n$\\mathcal{O}_{\\mathcal{B}'}$ and\nthat $(f, f')$ is a strict morphism of thickenings. There exists an element\n$$\no(\\varphi) \\in\n\\Ext^1_\\mathcal{O}(\\mathcal{F},\n\\mathcal{G} \\otimes_\\mathcal{O} f^*\\mathcal{J})\n$$\nwhose vanishing is a necessary and sufficient condition for the\nexistence of a lift of $\\varphi$ to an $\\mathcal{O}'$-linear map\n$\\varphi' : \\mathcal{F}' \\to \\mathcal{G}'$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MT","source_file":"defos.tex","source_line":3559,"source_end_line":3577,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3559-L3577","statement_sha256":"d2a9d484ed8069f6adee351a863907d0e6a28dc9425e73c6d1af79d5d45cc074","origin":"The Stacks Project","memory_eligible":false,"source_rank":13435,"rank":13435,"depth":20,"x":1540.283,"y":1501.235,"cluster":"deformation-theory"},{"id":"stacks:08MU","tag":"08MU","title":"Infinitesimal deformations of modules on ringed topoi · Lemma 08MU","summary":"Let (f, f') be a morphism of first order thickenings as in Situation [Tag 08M7]. Let F be an O-module. Assume (f, f') is a strict morphism of thickenings and F flat over O_B. If there exists a pair (F', α) consisting of an O'-module F' flat over O_B' and an isomorphism α : i^*F' → F, then the set of isomorphism classes of such pairs is principal homogeneous under Ext^1_O( F, I ⊗_O F).","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings as in\nSituation \\ref{situation-morphism-thickenings-ringed-topoi}.\nLet $\\mathcal{F}$ be an $\\mathcal{O}$-module.\nAssume $(f, f')$ is a strict morphism of thickenings and\n$\\mathcal{F}$ flat over $\\mathcal{O}_\\mathcal{B}$. If there exists a pair\n$(\\mathcal{F}', \\alpha)$ consisting of an\n$\\mathcal{O}'$-module $\\mathcal{F}'$ flat over $\\mathcal{O}_{\\mathcal{B}'}$\nand an isomorphism\n$\\alpha : i^*\\mathcal{F}' \\to \\mathcal{F}$, then the set of\nisomorphism classes of such pairs is principal homogeneous\nunder\n$\\Ext^1_\\mathcal{O}(\n\\mathcal{F}, \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MU","source_file":"defos.tex","source_line":3627,"source_end_line":3642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3627-L3642","statement_sha256":"ef78c051d64cfa46826b1df2bad7ce99178634de168558e865e7629e6682c320","origin":"The Stacks Project","memory_eligible":false,"source_rank":13436,"rank":13436,"depth":13,"x":1689.023,"y":1650.436,"cluster":"deformation-theory"},{"id":"stacks:08MV","tag":"08MV","title":"Infinitesimal deformations of modules on ringed topoi · Lemma 08MV","summary":"Let (f, f') be a morphism of first order thickenings as in Situation [Tag 08M7]. Let F be an O-module. Assume (f, f') is a strict morphism of thickenings and F flat over O_B. There exists an O'-module F' flat over O_B' with i^*F' ≅ F, if and only if • the canonical map f^*J ⊗_O F → I ⊗_O F is an isomorphism, and • the class o(F, I ⊗_O F, 1) ∈ Ext^2_O( F, I ⊗_O F) of Lemma [Tag 08MF] is zero.","statement_latex":"Let $(f, f')$ be a morphism of first order thickenings as in\nSituation \\ref{situation-morphism-thickenings-ringed-topoi}.\nLet $\\mathcal{F}$ be an $\\mathcal{O}$-module. Assume\n$(f, f')$ is a strict morphism of thickenings\nand $\\mathcal{F}$ flat over $\\mathcal{O}_\\mathcal{B}$. There exists an\n$\\mathcal{O}'$-module $\\mathcal{F}'$ flat over $\\mathcal{O}_{\\mathcal{B}'}$\nwith $i^*\\mathcal{F}' \\cong \\mathcal{F}$, if and only if\n\\begin{enumerate}\n\\item the canonical map\n$f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F} \\to\n\\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}$\nis an isomorphism, and\n\\item the class\n$o(\\mathcal{F}, \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F}, 1)\n\\in \\Ext^2_\\mathcal{O}(\n\\mathcal{F}, \\mathcal{I} \\otimes_\\mathcal{O} \\mathcal{F})$\nof Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} is zero.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Infinitesimal deformations of modules on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08MV","source_file":"defos.tex","source_line":3658,"source_end_line":3678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3658-L3678","statement_sha256":"624d78e662d10ab2ab687b7496b7498b8ef7b3cfdec99974200eb6fd799a93a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13437,"rank":13437,"depth":14,"x":1458.704,"y":1624.742,"cluster":"deformation-theory"},{"id":"stacks:08VW","tag":"08VW","title":"Application to flat modules on flat thickenings of ringed topoi · Lemma 08VW","summary":"In the situation above. • There exists an O'-module F' flat over O_B' with i^*F' ≅ F, if and only if the class o(F, f^*J ⊗_O F, 1) ∈ Ext^2_O( F, f^*J ⊗_O F) of Lemma [Tag 08MF] is zero. • If such a module exists, then the set of isomorphism classes of lifts is principal homogeneous under Ext^1_O( F, f^*J ⊗_O F). • Given a lift F', the set of automorphisms of F' which pull back to id_F is canonically isomorphic to Ext^0_O( F, f^*J ⊗_O F).","statement_latex":"In the situation above.\n\\begin{enumerate}\n\\item There exists an $\\mathcal{O}'$-module $\\mathcal{F}'$ flat over\n$\\mathcal{O}_{\\mathcal{B}'}$ with $i^*\\mathcal{F}' \\cong \\mathcal{F}$,\nif and only if\nthe class $o(\\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F}, 1)\n\\in \\Ext^2_\\mathcal{O}(\n\\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F})$\nof Lemma \\ref{lemma-inf-obs-ext-ringed-topoi} is zero.\n\\item If such a module exists, then the set of isomorphism classes\nof lifts is principal homogeneous under\n$\\Ext^1_\\mathcal{O}(\n\\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F})$.\n\\item Given a lift $\\mathcal{F}'$, the set of automorphisms of\n$\\mathcal{F}'$ which pull back to $\\text{id}_\\mathcal{F}$ is canonically\nisomorphic to $\\Ext^0_\\mathcal{O}(\n\\mathcal{F}, f^*\\mathcal{J} \\otimes_\\mathcal{O} \\mathcal{F})$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Application to flat modules on flat thickenings of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VW","source_file":"defos.tex","source_line":3721,"source_end_line":3741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3721-L3741","statement_sha256":"dfe0113848289129cc6f2dd8bea536f87043c8506feadb40ed5e4ebd6dc24eb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13438,"rank":13438,"depth":15,"x":1649.743,"y":1512.682,"cluster":"deformation-theory"},{"id":"stacks:0CYE","tag":"0CYE","title":"Application to flat modules on flat thickenings of ringed topoi · Lemma 0CYE","summary":"In Situation [Tag 0CYD] the obstruction class o(F, f^*J_2 ⊗_O F, 1) maps to the obstruction class o(F, f^*J_1 ⊗_O F, 1) under the canonical map Ext^2_O( F, f^*J_2 ⊗_O F) → Ext^2_O( F, f^*J_1 ⊗_O F)","statement_latex":"In Situation \\ref{situation-morphism-flat-thickenings-ringed-topoi}\nthe obstruction class\n$o(\\mathcal{F}, f^*\\mathcal{J}_2 \\otimes_\\mathcal{O} \\mathcal{F}, 1)$\nmaps to the obstruction class\n$o(\\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F}, 1)$\nunder the canonical map\n$$\n\\Ext^2_\\mathcal{O}(\n\\mathcal{F}, f^*\\mathcal{J}_2 \\otimes_\\mathcal{O} \\mathcal{F})\n\\to \\Ext^2_\\mathcal{O}(\n\\mathcal{F}, f^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F})\n$$","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Application to flat modules on flat thickenings of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYE","source_file":"defos.tex","source_line":3772,"source_end_line":3786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3772-L3786","statement_sha256":"e7f00e3e426ec1870b464c581269c38b3e0a2eda0b9cb373c3347c8406efa2ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":13439,"rank":13439,"depth":0,"x":1598.844,"y":1704.254,"cluster":"deformation-theory"},{"id":"stacks:08VY","tag":"08VY","title":"Application to flat modules on flat thickenings of ringed topoi · Lemma 08VY","summary":"In Situation [Tag 08VX] the modules π^*F and h^*F'_2 are O'_1-modules flat over O_B'_1 restricting to F on (Sh(C), O). Their difference (Lemma [Tag 08VW]) is an element theta of Ext^1_O(F, f^*J_1 ⊗_O F) whose boundary in Ext^2_O(F, f^*J_3 ⊗_O F) equals the obstruction (Lemma [Tag 08VW]) to lifting F to an O'_3-module flat over O_B'_3.","statement_latex":"In Situation \\ref{situation-ses-flat-thickenings-ringed-topoi} the modules\n$\\pi^*\\mathcal{F}$ and $h^*\\mathcal{F}'_2$ are $\\mathcal{O}'_1$-modules\nflat over $\\mathcal{O}_{\\mathcal{B}'_1}$ restricting to $\\mathcal{F}$ on\n$(\\Sh(\\mathcal{C}), \\mathcal{O})$.\nTheir difference (Lemma \\ref{lemma-flat-ringed-topoi}) is an element\n$\\theta$ of\n$\\Ext^1_\\mathcal{O}(\\mathcal{F},\nf^*\\mathcal{J}_1 \\otimes_\\mathcal{O} \\mathcal{F})$\nwhose boundary in\n$\\Ext^2_\\mathcal{O}(\\mathcal{F},\nf^*\\mathcal{J}_3 \\otimes_\\mathcal{O} \\mathcal{F})$\nequals the obstruction (Lemma \\ref{lemma-flat-ringed-topoi})\nto lifting $\\mathcal{F}$ to an $\\mathcal{O}'_3$-module flat over\n$\\mathcal{O}_{\\mathcal{B}'_3}$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Application to flat modules on flat thickenings of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VY","source_file":"defos.tex","source_line":3820,"source_end_line":3836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3820-L3836","statement_sha256":"b2eb888686e43ceb5a45487d440d09b0d4a0b6ceb6e4e3f24356d6400fde5361","origin":"The Stacks Project","memory_eligible":false,"source_rank":13440,"rank":13440,"depth":16,"x":1481.992,"y":1533.632,"cluster":"deformation-theory"},{"id":"stacks:08UG","tag":"08UG","title":"Deformations of ringed topoi and the naive cotangent complex · Lemma 08UG","summary":"Assume given a commutative diagram of morphisms ringed topoi vcenter xymatrix & (Sh(C_2), O_2) ar[r]_i_2 ar[d]_f_2 ar[ddl]_g & (Sh(C'_2), O'_2) ar[d]^f'_2 & (Sh(B_2), O_B_2) ar[r]^t_2 ar[ddl]|hole & (Sh(B'_2), O_B'_2) ar[ddl] (Sh(C_1), O_1) ar[r]_i_1 ar[d]_f_1 & (Sh(C'_1), O'_1) ar[d]^f'_1 (Sh(B_1), O_B_1) ar[r]^t_1 & (Sh(B'_1), O_B'_1) whose horizontal arrows are first order thickenings. Set G_j = Ker(i_j^sharp) and assume given a map of g^-1O_1-modules ν : g^-1G_1 → G_2…","statement_latex":"Assume given a commutative diagram of morphisms ringed topoi\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\n& (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\ar[r]_{i_2} \\ar[d]_{f_2} \\ar[ddl]_g &\n(\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\ar[d]^{f'_2} \\\\\n&\n(\\Sh(\\mathcal{B}_2), \\mathcal{O}_{\\mathcal{B}_2}) \\ar[r]^{t_2} \\ar[ddl]|\\hole &\n(\\Sh(\\mathcal{B}'_2), \\mathcal{O}_{\\mathcal{B}'_2}) \\ar[ddl] \\\\\n(\\Sh(\\mathcal{C}_1), \\mathcal{O}_1) \\ar[r]_{i_1} \\ar[d]_{f_1} &\n(\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1) \\ar[d]^{f'_1} \\\\\n(\\Sh(\\mathcal{B}_1), \\mathcal{O}_{\\mathcal{B}_1}) \\ar[r]^{t_1} &\n(\\Sh(\\mathcal{B}'_1), \\mathcal{O}_{\\mathcal{B}'_1})\n}\n}\n\\end{equation}\nwhose horizontal arrows are first order thickenings. Set\n$\\mathcal{G}_j = \\Ker(i_j^\\sharp)$ and assume given a\nmap of $g^{-1}\\mathcal{O}_1$-modules\n$\\nu : g^{-1}\\mathcal{G}_1 \\to \\mathcal{G}_2$\ngiving rise to the commutative diagram\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\n& 0 \\ar[r] & \\mathcal{G}_2 \\ar[r] &\n\\mathcal{O}'_2 \\ar[r] &\n\\mathcal{O}_2 \\ar[r] & 0 \\\\\n& 0 \\ar[r]|\\hole &\nf_2^{-1}\\mathcal{J}_2 \\ar[u]_{c_2} \\ar[r] &\nf_2^{-1}\\mathcal{O}_{\\mathcal{B}'_2} \\ar[u] \\ar[r]|\\hole &\nf_2^{-1}\\mathcal{O}_{\\mathcal{B}_2} \\ar[u] \\ar[r] & 0 \\\\\n0 \\ar[r] &\n\\mathcal{G}_1 \\ar[ruu] \\ar[r] &\n\\mathcal{O}'_1 \\ar[r] &\n\\mathcal{O}_1 \\ar[ruu] \\ar[r] & 0 \\\\\n0 \\ar[r] &\nf_1^{-1}\\mathcal{J}_1 \\ar[ruu]|\\hole \\ar[u]^{c_1} \\ar[r] &\nf_1^{-1}\\mathcal{O}_{\\mathcal{B}'_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] &\nf_1^{-1}\\mathcal{O}_{\\mathcal{B}_1} \\ar[ruu]|\\hole \\ar[u] \\ar[r] & 0\n}\n}\n\\end{equation}\nwith front and back solutions to (\\ref{equation-to-solve-ringed-topoi}).\n(The north-north-west arrows are maps on $\\mathcal{C}_2$ after applying\n$g^{-1}$ to the source.)\n\\begin{enumerate}\n\\item There exist a canonical element in\n$\\Ext^1_{\\mathcal{O}_2}(\nLg^*\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2)$\nwhose vanishing is a necessary and sufficient condition for the existence\nof a morphism of ringed topoi\n$(\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\to\n(\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1)$ fitting into\n(\\ref{equation-huge-1-ringed-topoi}) compatibly with $\\nu$.\n\\item If there exists a morphism\n$(\\Sh(\\mathcal{C}'_2), \\mathcal{O}'_2) \\to\n(\\Sh(\\mathcal{C}'_1), \\mathcal{O}'_1)$\nfitting into\n(\\ref{equation-huge-1-ringed-topoi}) compatibly with $\\nu$ the set\nof all such morphisms is a principal homogeneous space under\n$$\n\\Hom_{\\mathcal{O}_1}(\n\\Omega_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, g_*\\mathcal{G}_2) =\n\\Hom_{\\mathcal{O}_2}(\ng^*\\Omega_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2) =\n\\Ext^0_{\\mathcal{O}_2}(\nLg^*\\NL_{\\mathcal{O}_1/\\mathcal{O}_{\\mathcal{B}_1}}, \\mathcal{G}_2).\n$$\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed topoi and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UG","source_file":"defos.tex","source_line":3905,"source_end_line":3978,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L3905-L3978","statement_sha256":"d57fe0cbd7d7d84060ee51b2b31ca6178a2889bfd53665ba83bb3ad1cdb8825c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13441,"rank":13441,"depth":5,"x":1705.992,"y":1593.308,"cluster":"deformation-theory"},{"id":"stacks:08UJ","tag":"08UJ","title":"Deformations of ringed topoi and the naive cotangent complex · Lemma 08UJ","summary":"Let C be a site. Let A → B be a homomorphism of sheaves of rings on C. Let G be a B-module. Let xi ∈ Ext^1_B(NL_B/A, G). There exists a map of sheaves of sets α : E → B such that xi ∈ Ext^1_B(NL(α), G) is the class of a map I/I^2 → G (see proof for notation).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a\nhomomorphism of sheaves of rings on $\\mathcal{C}$.\nLet $\\mathcal{G}$ be a $\\mathcal{B}$-module.\nLet\n$\\xi \\in \\Ext^1_\\mathcal{B}(\\NL_{\\mathcal{B}/\\mathcal{A}}, \\mathcal{G})$. \nThere exists a map of sheaves of sets $\\alpha : \\mathcal{E} \\to \\mathcal{B}$\nsuch that $\\xi \\in \\Ext^1_\\mathcal{B}(\\NL(\\alpha), \\mathcal{G})$\nis the class of a map $\\mathcal{I}/\\mathcal{I}^2 \\to \\mathcal{G}$\n(see proof for notation).","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed topoi and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UJ","source_file":"defos.tex","source_line":4135,"source_end_line":4146,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4135-L4146","statement_sha256":"5708dd84cbc5c5d1c3f7d553500e8c88060c073e2bdfd1fee678d3a6f8849f28","origin":"The Stacks Project","memory_eligible":false,"source_rank":13442,"rank":13442,"depth":4,"x":1492.234,"y":1676.638,"cluster":"deformation-theory"},{"id":"stacks:08UK","tag":"08UK","title":"Deformations of ringed topoi and the naive cotangent complex · Lemma 08UK","summary":"If there exists a solution to ([Tag 08UF]), then the set of isomorphism classes of solutions is principal homogeneous under Ext^1_O( NL_O/O_B, G).","statement_latex":"If there exists a solution to (\\ref{equation-to-solve-ringed-topoi}),\nthen the set of isomorphism classes of solutions is principal homogeneous\nunder $\\Ext^1_\\mathcal{O}(\n\\NL_{\\mathcal{O}/\\mathcal{O}_\\mathcal{B}}, \\mathcal{G})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed topoi and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UK","source_file":"defos.tex","source_line":4195,"source_end_line":4201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4195-L4201","statement_sha256":"82c0ff2d393030007bbc2c627de66c1a28db71a04f93146583948da4f6505ac5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13443,"rank":13443,"depth":6,"x":1583.096,"y":1493.394,"cluster":"deformation-theory"},{"id":"stacks:0GQ5","tag":"0GQ5","title":"Deformations of ringed topoi and the naive cotangent complex · Lemma 0GQ5","summary":"Let f : (Sh(C), O) → (Sh(B), O_B) be a morphism of ringed topoi. Let G be an O-module. The set of isomorphism classes of extensions of f^-1O_B-algebras 0 → G → O' → O → 0 where G is an ideal of square zero), O) → (Sh(C), O') over (Sh(B), O_B) endowed with an isomorphism G → Ker(i^sharp) of O-modules. is canonically bijective to Ext^1_O(NL_O/O_B, G).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to\n(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ be a morphism of\nringed topoi. Let $\\mathcal{G}$ be an $\\mathcal{O}$-module.\nThe set of isomorphism classes of extensions of\n$f^{-1}\\mathcal{O}_\\mathcal{B}$-algebras\n$$\n0 \\to \\mathcal{G} \\to \\mathcal{O}' \\to \\mathcal{O} \\to 0\n$$\nwhere $\\mathcal{G}$ is an ideal of square zero\\footnote{In other words,\nthe set of isomorphism classes of first order thickenings\n$i : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to (\\Sh(\\mathcal{C}), \\mathcal{O}')$\nover $(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ endowed with an isomorphism\n$\\mathcal{G} \\to \\Ker(i^\\sharp)$ of $\\mathcal{O}$-modules.} \nis canonically bijective to\n$\\Ext^1_\\mathcal{O}(\\NL_{\\mathcal{O}/\\mathcal{O}_\\mathcal{B}}, \\mathcal{G})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed topoi and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQ5","source_file":"defos.tex","source_line":4265,"source_end_line":4282,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4265-L4282","statement_sha256":"28e081612fe4974774bd7f194e1fa79014ffa9b19a67c13361c0f65b5d2bea95","origin":"The Stacks Project","memory_eligible":false,"source_rank":13444,"rank":13444,"depth":7,"x":1663.675,"y":1680.587,"cluster":"deformation-theory"},{"id":"stacks:0GQ7","tag":"0GQ7","title":"Deformations of ringed topoi and the naive cotangent complex · Lemma 0GQ7","summary":"Let f : (Sh(C), O_C) → (Sh(B), O_B) and g : (Sh(D), O_D) → (Sh(C), O_C) be morphisms of ringed topoi. Let F be a O_C-module. Let G be a O_D-module. Let c : g^*F → G be a O_D-linear map. Finally, consider • [(a)] 0 → F → O_C' → O_C → 0 an extension of f^-1O_B-algebras corresponding to xi ∈ Ext^1_O_C( NL_O_C/O_B, F), and • [(b)] 0 → G → O_D' → O_D → 0 an extension of g^-1f^-1O_B-algebras corresponding to zeta ∈ Ext^1_O_D( NL_O_D/O_B, G). See Lemma [Tag 0GQ5]. Then there is…","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ and\n$g : (\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D}) \\to\n(\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C})$ be morphisms\nof ringed topoi. Let $\\mathcal{F}$ be a $\\mathcal{O}_\\mathcal{C}$-module.\nLet $\\mathcal{G}$ be a $\\mathcal{O}_\\mathcal{D}$-module. Let\n$c : g^*\\mathcal{F} \\to \\mathcal{G}$ be a $\\mathcal{O}_\\mathcal{D}$-linear\nmap. Finally, consider\n\\begin{enumerate}\n\\item[(a)]\n$0 \\to \\mathcal{F} \\to \\mathcal{O}_{\\mathcal{C}'} \\to\n\\mathcal{O}_\\mathcal{C} \\to 0$\nan extension of $f^{-1}\\mathcal{O}_\\mathcal{B}$-algebras\ncorresponding to\n$\\xi \\in \\Ext^1_{\\mathcal{O}_\\mathcal{C}}(\n\\NL_{\\mathcal{O}_\\mathcal{C}/\\mathcal{O}_\\mathcal{B}}, \\mathcal{F})$, and\n\\item[(b)]\n$0 \\to \\mathcal{G} \\to \\mathcal{O}_{\\mathcal{D}'} \\to\n\\mathcal{O}_\\mathcal{D} \\to 0$\nan extension of $g^{-1}f^{-1}\\mathcal{O}_\\mathcal{B}$-algebras\ncorresponding to\n$\\zeta \\in \\Ext^1_{\\mathcal{O}_\\mathcal{D}}(\n\\NL_{\\mathcal{O}_\\mathcal{D}/\\mathcal{O}_\\mathcal{B}}, \\mathcal{G})$.\n\\end{enumerate}\nSee Lemma \\ref{lemma-extensions-of-relative-ringed-topoi}.\nThen there is a morphism\n$$\ng' :\n(\\Sh(\\mathcal{D}), \\mathcal{O}_{\\mathcal{D}'})\n\\longrightarrow\n(\\Sh(\\mathcal{C}), \\mathcal{O}_{\\mathcal{C}'})\n$$\nof ringed topoi over $(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$\ncompatible with $g$ and $c$ if and only if $\\xi$ and $\\zeta$\nmap to the same element of\n$\\Ext^1_{\\mathcal{O}_\\mathcal{D}}(\nLg^*\\NL_{\\mathcal{O}_\\mathcal{C}/\\mathcal{O}_\\mathcal{B}}, \\mathcal{G})$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed topoi and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQ7","source_file":"defos.tex","source_line":4347,"source_end_line":4386,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4347-L4386","statement_sha256":"1cd98c4d06834f314304fe3351606f014601d7bd5b4764531a88606f42b8a8a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13445,"rank":13445,"depth":8,"x":1453.148,"y":1588.023,"cluster":"deformation-theory"},{"id":"stacks:0GQ9","tag":"0GQ9","title":"Deformations of ringed topoi and the naive cotangent complex · Lemma 0GQ9","summary":"Let t : (Sh(B), O_B) → (Sh(B'), O_B'), J = Ker(t^sharp), f : (Sh(C), O) → (Sh(B), O_B), G, and c : J → G be as in ([Tag 08UF]). Denote xi ∈ Ext^1_O_B( NL_O_B/O_B', J) the element corresponding to the extension O_B' of O_B by J via Lemma [Tag 0GQ5]. The set of isomorphism classes of solutions is canonically bijective to the fibre of Ext^1_O(NL_O/O_B', G)→ Ext^1_O( Lf^*NL_O_B/O_B', G) over the image of xi.","statement_latex":"Let $t : (\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B}) \\to\n(\\Sh(\\mathcal{B}'), \\mathcal{O}_{\\mathcal{B}'})$,\n$\\mathcal{J} = \\Ker(t^\\sharp)$,\n$f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to\n(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$, $\\mathcal{G}$, and\n$c : \\mathcal{J} \\to \\mathcal{G}$ be as in\n(\\ref{equation-to-solve-ringed-topoi}).\nDenote $\\xi \\in \\Ext^1_{\\mathcal{O}_\\mathcal{B}}(\n\\NL_{\\mathcal{O}_\\mathcal{B}/\\mathcal{O}_{\\mathcal{B}'}}, \\mathcal{J})$\nthe element corresponding to the extension $\\mathcal{O}_{\\mathcal{B}'}$\nof $\\mathcal{O}_\\mathcal{B}$ by $\\mathcal{J}$ via\nLemma \\ref{lemma-extensions-of-relative-ringed-topoi}.\nThe set of isomorphism classes of solutions is canonically bijective\nto the fibre of\n$$\n\\Ext^1_\\mathcal{O}(\\NL_{\\mathcal{O}/\\mathcal{O}_{\\mathcal{B}'}},\n\\mathcal{G})\\to\n\\Ext^1_\\mathcal{O}(\nLf^*\\NL_{\\mathcal{O}_\\mathcal{B}/\\mathcal{O}_{\\mathcal{B}'}}, \\mathcal{G})\n$$\nover the image of $\\xi$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of ringed topoi and the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQ9","source_file":"defos.tex","source_line":4570,"source_end_line":4593,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4570-L4593","statement_sha256":"a25354003c569df87c700e0a00d724a3dbee287a957f9019d99b54dd3b8d0807","origin":"The Stacks Project","memory_eligible":false,"source_rank":13446,"rank":13446,"depth":9,"x":1683.45,"y":1536.681,"cluster":"deformation-theory"},{"id":"stacks:0D16","tag":"0D16","title":"Deformations of algebraic spaces · Lemma 0D16","summary":"Let S be a scheme. Let i : Z → Z' be a morphism of algebraic spaces over S. The following are equivalent • i is a thickening of algebraic spaces as defined in More on Morphisms of Spaces, Section [Tag 05ZJ], and • the associated morphism i_small : (Sh(Z_etale), O_Z) → (Sh(Z'_etale), O_Z') of ringed topoi (Properties of Spaces, Lemma [Tag 03G8]) is a thickening in the sense of Section [Tag 08M6].","statement_latex":"Let $S$ be a scheme. Let $i : Z \\to Z'$ be a morphism of algebraic spaces\nover $S$. The following are equivalent\n\\begin{enumerate}\n\\item $i$ is a thickening of algebraic spaces as defined\nin More on Morphisms of Spaces, Section\n\\ref{spaces-more-morphisms-section-thickenings}, and\n\\item the associated morphism\n$i_{small} : (\\Sh(Z_\\etale), \\mathcal{O}_Z) \\to\n(\\Sh(Z'_\\etale), \\mathcal{O}_{Z'})$\nof ringed topoi (Properties of Spaces, Lemma\n\\ref{spaces-properties-lemma-morphism-ringed-topoi})\nis a thickening in the sense of\nSection \\ref{section-thickenings-ringed-topoi}.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D16","source_file":"defos.tex","source_line":4656,"source_end_line":4672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4656-L4672","statement_sha256":"e9c8b570d30c87ff9772f6c2e4d0857f4739d3476ffdc90217311c68fabc8969","origin":"The Stacks Project","memory_eligible":false,"source_rank":13447,"rank":13447,"depth":70,"x":1554.568,"y":1705.675,"cluster":"deformation-theory"},{"id":"stacks:0D17","tag":"0D17","title":"Deformations of algebraic spaces · Lemma 0D17","summary":"Let S be a scheme. Let Y ⊂ Y' be a first order thickening of algebraic spaces over S. Let f : X → Y be a flat morphism of algebraic spaces over S. If there exists a flat morphism f' : X' → Y' of algebraic spaces over S and an isomorphsm a : X → X' ×_Y' Y over Y, then • the set of isomorphism classes of pairs (f' : X' → Y', a) is principal homogeneous under Ext^1_O_X(NL_X/Y, f^*C_Y/Y'), and • the set of automorphisms of φ : X' → X' over Y' which reduce to the identity on…","statement_latex":"Let $S$ be a scheme.\nLet $Y \\subset Y'$ be a first order thickening of algebraic spaces\nover $S$.\nLet $f : X \\to Y$ be a flat morphism of algebraic spaces over $S$.\nIf there exists a flat morphism $f' : X' \\to Y'$ of algebraic spaces over $S$\nand an isomorphsm $a : X \\to X' \\times_{Y'} Y$ over $Y$, then\n\\begin{enumerate}\n\\item the set of isomorphism classes of pairs $(f' : X' \\to Y', a)$ is\nprincipal homogeneous under\n$\\Ext^1_{\\mathcal{O}_X}(\\NL_{X/Y}, f^*\\mathcal{C}_{Y/Y'})$, and\n\\item the set of automorphisms of $\\varphi : X' \\to X'$\nover $Y'$ which reduce to the identity on $X' \\times_{Y'} Y$\nis $\\Ext^0_{\\mathcal{O}_X}(\\NL_{X/Y}, f^*\\mathcal{C}_{Y/Y'})$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D17","source_file":"defos.tex","source_line":4716,"source_end_line":4732,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4716-L4732","statement_sha256":"d7c90f5afaead75770ebc839f1a21ff3926c545af68e3ce243d98dde4b52d2be","origin":"The Stacks Project","memory_eligible":false,"source_rank":13448,"rank":13448,"depth":71,"x":1513.594,"y":1507.399,"cluster":"deformation-theory"},{"id":"stacks:0D3P","tag":"0D3P","title":"Deformations of algebraic spaces · Lemma 0D3P","summary":"In the situation above assume that X is quasi-compact and quasi-separated and that DQ_X(F) → DQ_X(G) (Derived Categories of Spaces, Section [Tag 0CR3]) is an isomorphism. Then the functor F is an equivalence of categories.","statement_latex":"In the situation above assume that $X$ is quasi-compact and quasi-separated\nand that $DQ_X(\\mathcal{F}) \\to DQ_X(\\mathcal{G})$\n(Derived Categories of Spaces, Section\n\\ref{spaces-perfect-section-better-coherator})\nis an isomorphism. Then the functor $F$ is an equivalence of categories.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3P","source_file":"defos.tex","source_line":4827,"source_end_line":4834,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4827-L4834","statement_sha256":"ed3aaa0c6e75e6bcd413e4118df10ad1ee8d69e9ce3bb19b64773de66985d64a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13449,"rank":13449,"depth":55,"x":1703.764,"y":1630.684,"cluster":"deformation-theory"},{"id":"stacks:0D3Q","tag":"0D3Q","title":"Deformations of algebraic spaces · Lemma 0D3Q","summary":"In the situation above assume that X is quasi-compact and quasi-separated and that DQ_X(F) → DQ_X(G) (Derived Categories of Spaces, Section [Tag 0CR3]) is an isomorphism. Then the functor FT is an equivalence of categories.","statement_latex":"In the situation above assume that $X$ is quasi-compact and quasi-separated\nand that $DQ_X(\\mathcal{F}) \\to DQ_X(\\mathcal{G})$\n(Derived Categories of Spaces, Section\n\\ref{spaces-perfect-section-better-coherator})\nis an isomorphism. Then the functor $FT$ is an equivalence of categories.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3Q","source_file":"defos.tex","source_line":4877,"source_end_line":4884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4877-L4884","statement_sha256":"ec2c066efd54fa6a480bc1060f3c0df5519c288b7b5b58b8e6c870e1beae52a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13450,"rank":13450,"depth":56,"x":1463.755,"y":1647.727,"cluster":"deformation-theory"},{"id":"stacks:0DYR","tag":"0DYR","title":"Deformations of complexes · Lemma 0DYR","summary":"Let R' → R be a surjection of rings whose kernel is an ideal I of square zero. For every K ∈ D^-(R) there is a canonical map ω(K) : K → K ⊗_R^L I[2] in D(R) with the following properties • ω(K) = 0 if and only if there exists K' ∈ D(R') with K' ⊗_R'^L R = K, • given K → L in D^-(R) the diagram xymatrix K ar[d] ar[rr]_-ω(K) & & K ⊗^L_R I[2] ar[d] L ar[rr]^-ω(L) & & L ⊗^L_R I[2] commutes, and • formation of ω(K) is compatible with ring maps R' → S' (see proof for a precise…","statement_latex":"Let $R' \\to R$ be a surjection of rings whose kernel is an ideal\n$I$ of square zero. For every $K \\in D^-(R)$ there is a canonical\nmap\n$$\n\\omega(K) : K \\longrightarrow K \\otimes_R^\\mathbf{L} I[2]\n$$\nin $D(R)$ with the following properties\n\\begin{enumerate}\n\\item $\\omega(K) = 0$ if and only if there exists\n$K' \\in D(R')$ with $K' \\otimes_{R'}^\\mathbf{L} R = K$,\n\\item given $K \\to L$ in $D^-(R)$ the diagram\n$$\n\\xymatrix{\nK \\ar[d] \\ar[rr]_-{\\omega(K)} & &\nK \\otimes^\\mathbf{L}_R I[2] \\ar[d] \\\\\nL \\ar[rr]^-{\\omega(L)} & &\nL \\otimes^\\mathbf{L}_R I[2]\n}\n$$\ncommutes, and\n\\item formation of $\\omega(K)$ is compatible with ring maps $R' \\to S'$\n(see proof for a precise statement).\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of complexes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYR","source_file":"defos.tex","source_line":4917,"source_end_line":4942,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L4917-L4942","statement_sha256":"a01e4fb1db687061f18646d05f7d18210ed8404e111c25682c8fb1ac194160c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13451,"rank":13451,"depth":0,"x":1627.462,"y":1498.58,"cluster":"deformation-theory"},{"id":"stacks:0DIT","tag":"0DIT","title":"Deformations of complexes on ringed topoi · Lemma 0DIT","summary":"Let C be a site. Let O → O_0 be a surjection of sheaves of rings. Assume given the following data • flat O-modules G^n, • maps of O-modules G^n → G^n + 1, • a complex K_0^bullet of O_0-modules, • maps of O-modules G^n → K_0^n such that • [(a)] H^n(K_0^bullet) = 0 for n gg 0, • [(b)] G^n = 0 for n gg 0, • [(c)] with G^n_0 = G^n ⊗_O O_0 the induced maps determine a complex G_0^bullet and a map of complexes G_0^bullet → K_0^bullet. Then there exist • [(romannumeral1)] flat…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$\nbe a surjection of sheaves of rings. Assume given the following data\n\\begin{enumerate}\n\\item flat $\\mathcal{O}$-modules $\\mathcal{G}^n$,\n\\item maps of $\\mathcal{O}$-modules $\\mathcal{G}^n \\to \\mathcal{G}^{n + 1}$,\n\\item a complex $\\mathcal{K}_0^\\bullet$ of $\\mathcal{O}_0$-modules,\n\\item maps of $\\mathcal{O}$-modules $\\mathcal{G}^n \\to \\mathcal{K}_0^n$\n\\end{enumerate}\nsuch that\n\\begin{enumerate}\n\\item[(a)] $H^n(\\mathcal{K}_0^\\bullet) = 0$ for $n \\gg 0$,\n\\item[(b)] $\\mathcal{G}^n = 0$ for $n \\gg 0$,\n\\item[(c)] with\n$\\mathcal{G}^n_0 = \\mathcal{G}^n \\otimes_\\mathcal{O} \\mathcal{O}_0$\nthe induced maps determine a complex $\\mathcal{G}_0^\\bullet$ and a map\nof complexes $\\mathcal{G}_0^\\bullet \\to \\mathcal{K}_0^\\bullet$.\n\\end{enumerate}\nThen there exist\n\\begin{enumerate}\n\\item[(\\romannumeral1)]\nflat $\\mathcal{O}$-modules $\\mathcal{F}^n$,\n\\item[(\\romannumeral2)]\nmaps of $\\mathcal{O}$-modules $\\mathcal{F}^n \\to \\mathcal{F}^{n + 1}$,\n\\item[(\\romannumeral3)]\nmaps of $\\mathcal{O}$-modules $\\mathcal{F}^n \\to \\mathcal{K}_0^n$,\n\\item[(\\romannumeral4)]\nmaps of $\\mathcal{O}$-modules $\\mathcal{G}^n \\to \\mathcal{F}^n$,\n\\end{enumerate}\nsuch that $\\mathcal{F}^n = 0$ for $n \\gg 0$, such that the diagrams\n$$\n\\xymatrix{\n\\mathcal{G}^n \\ar[r] \\ar[d] & \\mathcal{G}^{n + 1} \\ar[d] \\\\\n\\mathcal{F}^n \\ar[r] & \\mathcal{F}^{n + 1}\n}\n$$\ncommute for all $n$, such that the composition\n$\\mathcal{G}^n \\to \\mathcal{F}^n \\to \\mathcal{K}_0^n$\nis the given map $\\mathcal{G}^n \\to \\mathcal{K}_0^n$, and such that with\n$\\mathcal{F}^n_0 = \\mathcal{F}^n \\otimes_\\mathcal{O} \\mathcal{O}_0$\nwe obtain a complex $\\mathcal{F}_0^\\bullet$ and map of complexes\n$\\mathcal{F}_0^\\bullet \\to \\mathcal{K}_0^\\bullet$ which is a\nquasi-isomorphism.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of complexes on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIT","source_file":"defos.tex","source_line":5050,"source_end_line":5094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L5050-L5094","statement_sha256":"6278edfc5b1e53cfec34eb7b20851aecc3349edef5efa9f0104ba00b9acafb06","origin":"The Stacks Project","memory_eligible":false,"source_rank":13452,"rank":13452,"depth":0,"x":1626.684,"y":1701.984,"cluster":"deformation-theory"},{"id":"stacks:0DIU","tag":"0DIU","title":"Deformations of complexes on ringed topoi · Lemma 0DIU","summary":"Let C be a site. Let O → O_0 be a surjection of sheaves of rings whose kernel is an ideal sheaf I of square zero. For every object K_0 in D^-(O_0) there is a canonical map ω(K_0) : K_0 → K_0 ⊗_O_0^L I[2] in D(O_0) such that for any map K_0 → L_0 in D^-(O_0) the diagram xymatrix K_0 ar[d] ar[rr]_-ω(K_0) & & (K_0 ⊗^L_O_0 I)[2] ar[d] L_0 ar[rr]^-ω(L_0) & & (L_0 ⊗^L_O_0 I)[2] commutes.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$\nbe a surjection of sheaves of rings whose kernel is an ideal sheaf\n$\\mathcal{I}$ of square zero. For every object\n$K_0$ in $D^-(\\mathcal{O}_0)$ there is a canonical map\n$$\n\\omega(K_0) :\nK_0 \\longrightarrow\nK_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I}[2]\n$$\nin $D(\\mathcal{O}_0)$ such that for any map\n$K_0 \\to L_0$ in $D^-(\\mathcal{O}_0)$ the diagram\n$$\n\\xymatrix{\nK_0 \\ar[d] \\ar[rr]_-{\\omega(K_0)} & &\n(K_0 \\otimes^\\mathbf{L}_{\\mathcal{O}_0} \\mathcal{I})[2] \\ar[d] \\\\\nL_0 \\ar[rr]^-{\\omega(L_0)} & &\n(L_0 \\otimes^\\mathbf{L}_{\\mathcal{O}_0} \\mathcal{I})[2]\n}\n$$\ncommutes.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of complexes on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIU","source_file":"defos.tex","source_line":5190,"source_end_line":5212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L5190-L5212","statement_sha256":"6a2049c3537b2dba4bea1c57b2aa07909b510583ea39ec523d6f91f975867490","origin":"The Stacks Project","memory_eligible":false,"source_rank":13453,"rank":13453,"depth":1,"x":1463.259,"y":1551.161,"cluster":"deformation-theory"},{"id":"stacks:0DIV","tag":"0DIV","title":"Deformations of complexes on ringed topoi · Lemma 0DIV","summary":"Let (C, O) be a ringed site. Let α : K → L be a map of D^-(O). Let F be a sheaf of O-modules. Let n ∈ Z. • If H^i(α) is an isomorphism for i ≥ n, then H^i(α ⊗_O^L id_F) is an isomorphism for i ≥ n. • If H^i(α) is an isomorphism for i > n and surjective for i = n, then H^i(α ⊗_O^L id_F) is an isomorphism for i > n and surjective for i = n.","statement_latex":"Let $(\\mathcal{C}, \\mathcal{O})$ be a ringed site.\nLet $\\alpha : K \\to L$ be a map of $D^-(\\mathcal{O})$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}$-modules.\nLet $n \\in \\mathbf{Z}$.\n\\begin{enumerate}\n\\item If $H^i(\\alpha)$ is an isomorphism for $i \\geq n$,\nthen $H^i(\\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}_\\mathcal{F})$\nis an isomorphism for $i \\geq n$.\n\\item If $H^i(\\alpha)$ is an isomorphism for $i > n$ \nand surjective for $i = n$,\nthen $H^i(\\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}_\\mathcal{F})$\nis an isomorphism for $i > n$ and surjective for $i = n$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of complexes on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIV","source_file":"defos.tex","source_line":5312,"source_end_line":5327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L5312-L5327","statement_sha256":"5bead54dcf0f2d5706cac457de5dcfddbb90eb2a822e46293cf200e40af2132e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13454,"rank":13454,"depth":10,"x":1705.684,"y":1569.693,"cluster":"deformation-theory"},{"id":"stacks:0DIW","tag":"0DIW","title":"Deformations of complexes on ringed topoi · Lemma 0DIW","summary":"Let C be a site. Let O → O_0 be a surjection of sheaves of rings whose kernel is an ideal sheaf I of square zero. For every object K_0 in D^-(O_0) the following are equivalent • the class ω(K_0) ∈ Ext^2_O_0(K_0, K_0 ⊗_O_0 I) constructed in Lemma [Tag 0DIU] is zero, • there exists K ∈ D^-(O) with K ⊗_O^L O_0 = K_0 in D(O_0).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$\nbe a surjection of sheaves of rings whose kernel is an ideal sheaf\n$\\mathcal{I}$ of square zero. For every object\n$K_0$ in $D^-(\\mathcal{O}_0)$ the following are equivalent\n\\begin{enumerate}\n\\item  the class\n$\\omega(K_0) \\in\n\\Ext^2_{\\mathcal{O}_0}(K_0, K_0 \\otimes_{\\mathcal{O}_0} \\mathcal{I})$\nconstructed in Lemma \\ref{lemma-canonical-class} is zero,\n\\item there exists $K \\in D^-(\\mathcal{O})$ with\n$K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0 = K_0$\nin $D(\\mathcal{O}_0)$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of complexes on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIW","source_file":"defos.tex","source_line":5357,"source_end_line":5372,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L5357-L5372","statement_sha256":"bf6805c4a200f65664d0216e9bfe08ef994e2874b6fc68540e0801edb75efb76","origin":"The Stacks Project","memory_eligible":false,"source_rank":13455,"rank":13455,"depth":11,"x":1511.518,"y":1693.906,"cluster":"deformation-theory"},{"id":"stacks:0DIX","tag":"0DIX","title":"Deformations of complexes on ringed topoi · Lemma 0DIX","summary":"Let C be a site. Let O → O_0 be a surjection of sheaves of rings. Assume given the following data • a complex of O-modules F^bullet, • a complex K_0^bullet of O_0-modules, • a quasi-isomorphism K_0^bullet → F^bullet ⊗_O O_0, Then there exist a quasi-isomorphism G^bullet → F^bullet such that the map of complexes G^bullet ⊗_O O_0 → F^bullet ⊗_O O_0 factors through K_0^bullet in the homotopy category of complexes of O_0-modules.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$\nbe a surjection of sheaves of rings. Assume given the following data\n\\begin{enumerate}\n\\item a complex of $\\mathcal{O}$-modules $\\mathcal{F}^\\bullet$,\n\\item a complex $\\mathcal{K}_0^\\bullet$ of $\\mathcal{O}_0$-modules,\n\\item a quasi-isomorphism $\\mathcal{K}_0^\\bullet \\to\n\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{O}_0$,\n\\end{enumerate}\nThen there exist a quasi-isomorphism\n$\\mathcal{G}^\\bullet \\to \\mathcal{F}^\\bullet$ such that the map\nof complexes\n$\\mathcal{G}^\\bullet  \\otimes_\\mathcal{O} \\mathcal{O}_0 \\to\n\\mathcal{F}^\\bullet \\otimes_\\mathcal{O} \\mathcal{O}_0$ factors\nthrough $\\mathcal{K}_0^\\bullet$ in the homotopy category\nof complexes of $\\mathcal{O}_0$-modules.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of complexes on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIX","source_file":"defos.tex","source_line":5497,"source_end_line":5514,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L5497-L5514","statement_sha256":"66129533dfd5cdcd10b74d33cf8d2ba4ce8a94584d0c489608d88032923a19f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13456,"rank":13456,"depth":3,"x":1554.916,"y":1491.619,"cluster":"deformation-theory"},{"id":"stacks:0DIY","tag":"0DIY","title":"Deformations of complexes on ringed topoi · Lemma 0DIY","summary":"Let C be a site. Let O → O_0 be a surjection of sheaves of rings whose kernel is an ideal sheaf I of square zero. Let K, L ∈ D^-(O). Set K_0 = K ⊗_O^L O_0 and L_0 = L ⊗_O^L O_0 in D^-(O_0). Given α_0 : K_0 → L_0 in D(O_0) there is a canonical element o(α_0) ∈ Ext^1_O_0(K_0, L_0 ⊗_O_0^L I) whose vanishing is necessary and sufficient for the existence of a map α : K → L in D(O) with α_0 = α ⊗_O^L id.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$\nbe a surjection of sheaves of rings whose kernel is an ideal sheaf\n$\\mathcal{I}$ of square zero. Let $K, L \\in D^-(\\mathcal{O})$.\nSet $K_0 = K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0$\nand $L_0 = L \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0$\nin $D^-(\\mathcal{O}_0)$. Given $\\alpha_0 : K_0 \\to L_0$ in $D(\\mathcal{O}_0)$\nthere is a canonical element\n$$\no(\\alpha_0) \\in \\Ext^1_{\\mathcal{O}_0}(K_0,\nL_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})\n$$\nwhose vanishing is necessary and sufficient for the\nexistence of a map $\\alpha : K \\to L$ in $D(\\mathcal{O})$\nwith $\\alpha_0 = \\alpha \\otimes_\\mathcal{O}^\\mathbf{L} \\text{id}$.","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of complexes on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIY","source_file":"defos.tex","source_line":5535,"source_end_line":5551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L5535-L5551","statement_sha256":"c69edde8f10b4178e0accdc00e77025ccfbd5bd220583751a7c9eb89305d4058","origin":"The Stacks Project","memory_eligible":false,"source_rank":13457,"rank":13457,"depth":2,"x":1685.924,"y":1665.854,"cluster":"deformation-theory"},{"id":"stacks:0DIZ","tag":"0DIZ","title":"Deformations of complexes on ringed topoi · Lemma 0DIZ","summary":"Let C be a site. Let O → O_0 be a surjection of sheaves of rings whose kernel is an ideal sheaf I of square zero. Let K_0 ∈ D^-(O). A lift of K_0 is a pair (K, α_0) consisting of an object K in D^-(O) and an isomorphism α_0 : K ⊗_O^L O_0 → K_0 in D(O_0). • Given a lift (K, α) the group of automorphism of the pair is canonically the cokernel of a map Ext^-1_O_0(K_0, K_0) → Hom_O_0(K_0, K_0 ⊗_O_0^L I) • If there is a lift, then the set of isomorphism classes of lifts is…","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{O} \\to \\mathcal{O}_0$\nbe a surjection of sheaves of rings whose kernel is an ideal sheaf\n$\\mathcal{I}$ of square zero. Let $K_0 \\in D^-(\\mathcal{O})$.\nA lift of $K_0$ is a pair $(K, \\alpha_0)$ consisting of an object\n$K$ in $D^-(\\mathcal{O})$ and an isomorphism\n$\\alpha_0 : K \\otimes_\\mathcal{O}^\\mathbf{L} \\mathcal{O}_0 \\to K_0$\nin $D(\\mathcal{O}_0)$.\n\\begin{enumerate}\n\\item Given a lift $(K, \\alpha)$ the group of automorphism of the pair\nis canonically the cokernel of a map\n$$\n\\Ext^{-1}_{\\mathcal{O}_0}(K_0, K_0)\n\\longrightarrow\n\\Hom_{\\mathcal{O}_0}(K_0, K_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})\n$$\n\\item If there is a lift, then the set of isomorphism classes of lifts\nis principal homogenenous under\n$\\Ext^1_{\\mathcal{O}_0}(K_0,\nK_0 \\otimes_{\\mathcal{O}_0}^\\mathbf{L} \\mathcal{I})$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Theory","chapter_id":"defos","section":"Deformations of complexes on ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DIZ","source_file":"defos.tex","source_line":5585,"source_end_line":5607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/defos.tex#L5585-L5607","statement_sha256":"4615655ef7fe99defab46471c20abcdfe8214529a1304d88a0ba68d3967e9937","origin":"The Stacks Project","memory_eligible":false,"source_rank":13458,"rank":13458,"depth":12,"x":1448.602,"y":1611.574,"cluster":"deformation-theory"},{"id":"stacks:08PM","tag":"08PM","title":"The cotangent complex of a ring map · Definition 08PM","summary":"Let A → B be a ring map. The standard resolution of B over A is the augmentation ε : P_bullet → B with terms P_0 = A[B], P_1 = A[A[B]], … and maps as constructed in Simplicial, Example [Tag 09CB].","statement_latex":"Let $A \\to B$ be a ring map. The {\\it standard resolution of $B$ over $A$}\nis the augmentation $\\epsilon : P_\\bullet \\to B$ with terms\n$$\nP_0 = A[B],\\quad P_1 = A[A[B]],\\quad \\ldots\n$$\nand maps as constructed in\nSimplicial, Example \\ref{simplicial-example-polynomial-algebra-maps}.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a ring map","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PM","source_file":"cotangent.tex","source_line":102,"source_end_line":111,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L102-L111","statement_sha256":"01c284ba1c345ec2b96e4868cbbb49b0806272a5b6a750bc0359873d29709d34","origin":"The Stacks Project","memory_eligible":false,"source_rank":13459,"rank":13459,"depth":0,"x":1667.806,"y":1516.697,"cluster":"deformation-theory"},{"id":"stacks:08PN","tag":"08PN","title":"The cotangent complex of a ring map · Definition 08PN","summary":"The cotangent complex L_B/A of a ring map A → B is the complex of B-modules associated to the simplicial B-module Ω_P_bullet/A ⊗_P_bullet, ε B where ε : P_bullet → B is the standard resolution of B over A.","statement_latex":"The {\\it cotangent complex} $L_{B/A}$ of a ring map $A \\to B$\nis the complex of $B$-modules associated to the simplicial $B$-module\n$$\n\\Omega_{P_\\bullet/A} \\otimes_{P_\\bullet, \\epsilon} B\n$$\nwhere $\\epsilon : P_\\bullet \\to B$ is the standard resolution\nof $B$ over $A$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a ring map","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PN","source_file":"cotangent.tex","source_line":117,"source_end_line":126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L117-L126","statement_sha256":"65dce15b1d5e967afe96e5a3916f1312640196bef75993401d26205ec4b5d2b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13460,"rank":13460,"depth":0,"x":1582.249,"y":1711.53,"cluster":"deformation-theory"},{"id":"stacks:08S9","tag":"08S9","title":"The cotangent complex of a ring map · Lemma 08S9","summary":"Let A_i → B_i be a system of ring maps over a directed index set I. Then colim L_B_i/A_i = L_colim B_i/colim A_i.","statement_latex":"Let $A_i \\to B_i$ be a system of ring maps over a directed index\nset $I$. Then $\\colim L_{B_i/A_i} = L_{\\colim B_i/\\colim A_i}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08S9","source_file":"cotangent.tex","source_line":156,"source_end_line":160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L156-L160","statement_sha256":"c31a28de21d0a1027170f6e88bd0ea211b75f6f806e64134f7105a4c96834818","origin":"The Stacks Project","memory_eligible":false,"source_rank":13461,"rank":13461,"depth":1,"x":1488.423,"y":1518.833,"cluster":"deformation-theory"},{"id":"stacks:08PS","tag":"08PS","title":"Simplicial resolutions and derived lower shriek · Lemma 08PS","summary":"With notation as above let P_bullet be a simplicial A-algebra endowed with an augmentation ε : P_bullet → B. Assume each P_n is a polynomial algebra over A and ε is a trivial Kan fibration on underlying simplicial sets. Then Lπ_!(F) = F(P_bullet, ε) in D(Ab), resp. D(B) functorially in F in Ab(C), resp. Mod(underlineB).","statement_latex":"With notation as above let $P_\\bullet$ be a simplicial $A$-algebra\nendowed with an augmentation $\\epsilon : P_\\bullet \\to B$.\nAssume each $P_n$ is a polynomial algebra over $A$ and $\\epsilon$\nis a trivial Kan fibration on underlying simplicial sets. Then\n$$\nL\\pi_!(\\mathcal{F}) = \\mathcal{F}(P_\\bullet, \\epsilon)\n$$\nin $D(\\textit{Ab})$, resp.\\ $D(B)$ functorially in $\\mathcal{F}$ in\n$\\textit{Ab}(\\mathcal{C})$, resp.\\ $\\textit{Mod}(\\underline{B})$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Simplicial resolutions and derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PS","source_file":"cotangent.tex","source_line":228,"source_end_line":239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L228-L239","statement_sha256":"90670cb4efa052b6e2274b4c2389badfe0067bc991b94a683c79cce72af9e330","origin":"The Stacks Project","memory_eligible":false,"source_rank":13462,"rank":13462,"depth":12,"x":1713.131,"y":1607.908,"cluster":"deformation-theory"},{"id":"stacks:08PT","tag":"08PT","title":"Simplicial resolutions and derived lower shriek · Lemma 08PT","summary":"Let A → B be a ring map. Let ε : P_bullet → B be the standard resolution of B over A. Let π be as in ([Tag 08PR]). Then Lπ_!(F) = F(P_bullet, ε) in D(Ab), resp. D(B) functorially in F in Ab(C), resp. Mod(underlineB).","statement_latex":"Let $A \\to B$ be a ring map. Let $\\epsilon : P_\\bullet \\to B$\nbe the standard resolution of $B$ over $A$. Let $\\pi$ be as in\n(\\ref{equation-pi}). Then\n$$\nL\\pi_!(\\mathcal{F}) = \\mathcal{F}(P_\\bullet, \\epsilon)\n$$\nin $D(\\textit{Ab})$, resp.\\ $D(B)$ functorially in $\\mathcal{F}$ in\n$\\textit{Ab}(\\mathcal{C})$, resp.\\ $\\textit{Mod}(\\underline{B})$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Simplicial resolutions and derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PT","source_file":"cotangent.tex","source_line":277,"source_end_line":287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L277-L287","statement_sha256":"84969d49221777d9b65078c79d0969539a0e099c0d1f76a4cbdc523ee71081a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13463,"rank":13463,"depth":13,"x":1475.212,"y":1669.883,"cluster":"deformation-theory"},{"id":"stacks:08PU","tag":"08PU","title":"Simplicial resolutions and derived lower shriek · Lemma 08PU","summary":"Let A → B be a ring map. Let π and i be as in ([Tag 08PR]). There is a canonical isomorphism L_B/A = Lπ_!(Li^*Ω_O/A) = Lπ_!(i^*Ω_O/A) = Lπ_!(Ω_O/A ⊗_O underlineB) in D(B).","statement_latex":"Let $A \\to B$ be a ring map.  Let $\\pi$ and $i$ be as in (\\ref{equation-pi}).\nThere is a canonical isomorphism\n$$\nL_{B/A} = L\\pi_!(Li^*\\Omega_{\\mathcal{O}/A}) =\nL\\pi_!(i^*\\Omega_{\\mathcal{O}/A}) =\nL\\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B})\n$$\nin $D(B)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Simplicial resolutions and derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PU","source_file":"cotangent.tex","source_line":328,"source_end_line":338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L328-L338","statement_sha256":"a5751dc3085d231d7e9de7566d604081fd49ce1ec3288cb14063453c439e57a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13464,"rank":13464,"depth":14,"x":1601.123,"y":1488.736,"cluster":"deformation-theory"},{"id":"stacks:08QE","tag":"08QE","title":"Simplicial resolutions and derived lower shriek · Lemma 08QE","summary":"If A → B is a ring map, then Lπ_!(π^-1M) = M with π as in ([Tag 08PR]).","statement_latex":"If $A \\to B$ is a ring map, then $L\\pi_!(\\pi^{-1}M) = M$\nwith $\\pi$ as in (\\ref{equation-pi}).","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Simplicial resolutions and derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QE","source_file":"cotangent.tex","source_line":354,"source_end_line":358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L354-L358","statement_sha256":"60a465796c17b3a2d910e44a4cb448cc6534736c865ca8bad555bb4cfc0df389","origin":"The Stacks Project","memory_eligible":false,"source_rank":13465,"rank":13465,"depth":13,"x":1654.081,"y":1694.261,"cluster":"deformation-theory"},{"id":"stacks:08QF","tag":"08QF","title":"Simplicial resolutions and derived lower shriek · Lemma 08QF","summary":"If A → B is a ring map, then H^0(L_B/A) = Ω_B/A.","statement_latex":"If $A \\to B$ is a ring map, then $H^0(L_{B/A}) = \\Omega_{B/A}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Simplicial resolutions and derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QF","source_file":"cotangent.tex","source_line":366,"source_end_line":369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L366-L369","statement_sha256":"c4bbc41eb2c3bcace5737a568a6338987e42a5b9a757dba7cc784cb9bdaa1314","origin":"The Stacks Project","memory_eligible":false,"source_rank":13466,"rank":13466,"depth":15,"x":1449.253,"y":1572.462,"cluster":"deformation-theory"},{"id":"stacks:08QG","tag":"08QG","title":"Simplicial resolutions and derived lower shriek · Lemma 08QG","summary":"If B is a polynomial algebra over the ring A, then with π as in ([Tag 08PR]) we have that π_! is exact and π_!F = F(B → B).","statement_latex":"If $B$ is a polynomial algebra over the ring $A$, then\nwith $\\pi$ as in (\\ref{equation-pi}) we have that\n$\\pi_!$ is exact and $\\pi_!\\mathcal{F} = \\mathcal{F}(B \\to B)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Simplicial resolutions and derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QG","source_file":"cotangent.tex","source_line":402,"source_end_line":407,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L402-L407","statement_sha256":"c3f603220874fb620f3f811a4b0250eebd5b46c391dc5e2f31e3775429d2f0b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13467,"rank":13467,"depth":13,"x":1698.845,"y":1545.986,"cluster":"deformation-theory"},{"id":"stacks:08QH","tag":"08QH","title":"Simplicial resolutions and derived lower shriek · Lemma 08QH","summary":"If B is a polynomial algebra over the ring A, then L_B/A is quasi-isomorphic to Ω_B/A[0].","statement_latex":"If $B$ is a polynomial algebra over the ring $A$, then\n$L_{B/A}$ is quasi-isomorphic to $\\Omega_{B/A}[0]$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Simplicial resolutions and derived lower shriek","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QH","source_file":"cotangent.tex","source_line":414,"source_end_line":418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L414-L418","statement_sha256":"19ad48bee4156bb5941d8542650c93825d72f9f417a9437d5ff07a6e11147cd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13468,"rank":13468,"depth":15,"x":1535.695,"y":1707.524,"cluster":"deformation-theory"},{"id":"stacks:08PW","tag":"08PW","title":"Constructing a resolution · Lemma 08PW","summary":"Let A be a Noetherian ring. Let A → B be a finite type ring map. Let A be the category of A-algebra maps C → B. Let n ≥ 0 and let P_bullet be a simplicial object of A such that • P_bullet → B is a trivial Kan fibration of simplicial sets, • P_k is finite type over A for k ≤ n, • P_bullet = cosk_n sk_n P_bullet as simplicial objects of A. Then P_n + 1 is a finite type A-algebra.","statement_latex":"Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map.\nLet $\\mathcal{A}$ be the category of $A$-algebra maps $C \\to B$. Let\n$n \\geq 0$ and let $P_\\bullet$ be a simplicial object of $\\mathcal{A}$\nsuch that\n\\begin{enumerate}\n\\item $P_\\bullet \\to B$ is a trivial Kan fibration of simplicial sets,\n\\item $P_k$ is finite type over $A$ for $k \\leq n$,\n\\item $P_\\bullet = \\text{cosk}_n \\text{sk}_n P_\\bullet$ as simplicial\nobjects of $\\mathcal{A}$.\n\\end{enumerate}\nThen $P_{n + 1}$ is a finite type $A$-algebra.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Constructing a resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PW","source_file":"cotangent.tex","source_line":437,"source_end_line":450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L437-L450","statement_sha256":"fb41aacadec96882c6c88058412f426dccdfa3edecc10275ff7e57a6fce2bf3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13469,"rank":13469,"depth":4,"x":1526.072,"y":1495.322,"cluster":"deformation-theory"},{"id":"stacks:08PX","tag":"08PX","title":"Constructing a resolution · Proposition 08PX","summary":"Let A be a Noetherian ring. Let A → B be a finite type ring map. There exists a simplicial A-algebra P_bullet with an augmentation ε : P_bullet → B such that each P_n is a polynomial algebra of finite type over A and such that ε is a trivial Kan fibration of simplicial sets.","statement_latex":"Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map.\nThere exists a simplicial $A$-algebra $P_\\bullet$ with an augmentation\n$\\epsilon : P_\\bullet \\to B$ such that each $P_n$ is a polynomial algebra\nof finite type over $A$ and such that $\\epsilon$ is a trivial\nKan fibration of simplicial sets.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Constructing a resolution","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PX","source_file":"cotangent.tex","source_line":554,"source_end_line":561,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L554-L561","statement_sha256":"d2c90a8a1d776c0195e65139ed4435071e27e9256daca5e0efdb5e4cab9329eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13470,"rank":13470,"depth":5,"x":1704.242,"y":1646.7,"cluster":"deformation-theory"},{"id":"stacks:08PY","tag":"08PY","title":"Constructing a resolution · Lemma 08PY","summary":"Let A be a Noetherian ring. Let A → B be a finite type ring map. Let π, underlineB be as in ([Tag 08PR]). If F is an underlineB-module such that F(P, α) is a finite B-module for all α : P = A[x_1, …, x_n] → B, then the cohomology modules of Lπ_!(F) are finite B-modules.","statement_latex":"Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map.\nLet $\\pi$, $\\underline{B}$ be as in (\\ref{equation-pi}).\nIf $\\mathcal{F}$ is an $\\underline{B}$-module such that\n$\\mathcal{F}(P, \\alpha)$ is a finite $B$-module for all\n$\\alpha : P = A[x_1, \\ldots, x_n] \\to B$, then the cohomology modules\nof $L\\pi_!(\\mathcal{F})$ are finite $B$-modules.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Constructing a resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PY","source_file":"cotangent.tex","source_line":637,"source_end_line":645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L637-L645","statement_sha256":"8f84598f7d31e76ff0c7989b4a2bf8c040906eda01623bc850abf90c2c4dd38b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13471,"rank":13471,"depth":13,"x":1450.523,"y":1636.148,"cluster":"deformation-theory"},{"id":"stacks:08PZ","tag":"08PZ","title":"Constructing a resolution · Lemma 08PZ","summary":"Let A be a Noetherian ring. Let A → B be a finite type ring map. Then H^n(L_B/A) is a finite B-module for all n ∈ Z.","statement_latex":"Let $A$ be a Noetherian ring. Let $A \\to B$ be a finite type ring map.\nThen $H^n(L_{B/A})$ is a finite $B$-module for all $n \\in \\mathbf{Z}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Constructing a resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08PZ","source_file":"cotangent.tex","source_line":655,"source_end_line":659,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L655-L659","statement_sha256":"1298f22a7c9c2fd4fb245c18f3482a3554c6f886f038d785984cc5924fe33b16","origin":"The Stacks Project","memory_eligible":false,"source_rank":13472,"rank":13472,"depth":15,"x":1646.562,"y":1499.639,"cluster":"deformation-theory"},{"id":"stacks:08QJ","tag":"08QJ","title":"Constructing a resolution · Lemma 08QJ","summary":"Let A → B be a ring map. Let π, O, underlineB be as in ([Tag 08PR]). For any O-module F we have Lπ_!(F) = Lπ_!(Li^*F) = Lπ_!(F ⊗_O^L underlineB) in D(Ab).","statement_latex":"Let $A \\to B$ be a ring map. Let $\\pi$, $\\mathcal{O}$, $\\underline{B}$\nbe as in (\\ref{equation-pi}). For any $\\mathcal{O}$-module $\\mathcal{F}$\nwe have\n$$\nL\\pi_!(\\mathcal{F}) = L\\pi_!(Li^*\\mathcal{F}) =\nL\\pi_!(\\mathcal{F} \\otimes_\\mathcal{O}^\\mathbf{L} \\underline{B})\n$$\nin $D(\\textit{Ab})$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Constructing a resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QJ","source_file":"cotangent.tex","source_line":708,"source_end_line":718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L708-L718","statement_sha256":"4ff2e3ce79a57fa78678743758f0ed22dddc60cc1b62fe7f3ae0e9326751caf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13473,"rank":13473,"depth":18,"x":1611.7,"y":1712.038,"cluster":"deformation-theory"},{"id":"stacks:08QK","tag":"08QK","title":"Constructing a resolution · Lemma 08QK","summary":"Let A → B be a ring map. Let π, O, underlineB be as in ([Tag 08PR]). We have Lπ_!(O) = Lπ_!(underlineB) = B and L_B/A = Lπ_!(Ω_O/A ⊗_O underlineB) = Lπ_!(Ω_O/A) in D(Ab).","statement_latex":"Let $A \\to B$ be a ring map. Let $\\pi$, $\\mathcal{O}$, $\\underline{B}$\nbe as in (\\ref{equation-pi}). We have\n$$\nL\\pi_!(\\mathcal{O}) = L\\pi_!(\\underline{B}) = B\n\\quad\\text{and}\\quad\nL_{B/A} = L\\pi_!(\\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}) =\nL\\pi_!(\\Omega_{\\mathcal{O}/A})\n$$\nin $D(\\textit{Ab})$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Constructing a resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QK","source_file":"cotangent.tex","source_line":736,"source_end_line":747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L736-L747","statement_sha256":"0fd963573a7343f4d779992739a77c786678ba88418fcec6f845e39b60b709da","origin":"The Stacks Project","memory_eligible":false,"source_rank":13474,"rank":13474,"depth":19,"x":1466.26,"y":1535.221,"cluster":"deformation-theory"},{"id":"stacks:08SA","tag":"08SA","title":"Constructing a resolution · Lemma 08SA","summary":"Let A → B → C be ring maps. If B is a polynomial algebra over A, then there is a distinguished triangle L_B/A ⊗_B^L C → L_C/A → L_C/B → L_B/A ⊗_B^L C[1] in D(C).","statement_latex":"Let $A \\to B \\to C$ be ring maps. If $B$ is a polynomial algebra over\n$A$, then there is a distinguished triangle \n$L_{B/A} \\otimes_B^\\mathbf{L} C \\to L_{C/A} \\to L_{C/B} \\to\nL_{B/A} \\otimes_B^\\mathbf{L} C[1]$ in $D(C)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Constructing a resolution","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SA","source_file":"cotangent.tex","source_line":758,"source_end_line":764,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L758-L764","statement_sha256":"1756eee79b051b8b117d65042612729be41f8d52b76413ebf6acb50021c7d0aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13475,"rank":13475,"depth":16,"x":1716.282,"y":1583.189,"cluster":"deformation-theory"},{"id":"stacks:08QP","tag":"08QP","title":"Functoriality · Lemma 08QP","summary":"Assume ([Tag 08QM]) induces a quasi-isomorphism B ⊗_A^L A' = B'. Then, with notation as in ([Tag 08QN]) and F' ∈ Ab(C'), we have Lπ_!(g^-1F') = Lπ'_!(F').","statement_latex":"Assume (\\ref{equation-commutative-square}) induces a quasi-isomorphism\n$B \\otimes_A^\\mathbf{L} A' = B'$. Then, with notation as in\n(\\ref{equation-double-square}) and\n$\\mathcal{F}' \\in \\textit{Ab}(\\mathcal{C}')$,\nwe have $L\\pi_!(g^{-1}\\mathcal{F}') = L\\pi'_!(\\mathcal{F}')$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Functoriality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QP","source_file":"cotangent.tex","source_line":957,"source_end_line":964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L957-L964","statement_sha256":"4e1dd638f0913915727e910a748cb3968e4d8adc31adb6e07d925f9106c9fc99","origin":"The Stacks Project","memory_eligible":false,"source_rank":13476,"rank":13476,"depth":13,"x":1492.824,"y":1689.934,"cluster":"deformation-theory"},{"id":"stacks:08QQ","tag":"08QQ","title":"Functoriality · Lemma 08QQ","summary":"If ([Tag 08QM]) induces a quasi-isomorphism B ⊗_A^L A' = B', then the functoriality map induces an isomorphism L_B/A ⊗_B^L B' → L_B'/A'","statement_latex":"If (\\ref{equation-commutative-square}) induces a quasi-isomorphism\n$B \\otimes_A^\\mathbf{L} A' = B'$, then the functoriality map\ninduces an isomorphism\n$$\nL_{B/A} \\otimes_B^\\mathbf{L} B' \\longrightarrow L_{B'/A'}\n$$","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Functoriality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QQ","source_file":"cotangent.tex","source_line":984,"source_end_line":992,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L984-L992","statement_sha256":"e1b50aa1ad31f6089f2d78f41117e78c8c261969da52743675de2a58b270bda6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13477,"rank":13477,"depth":15,"x":1571.941,"y":1483.95,"cluster":"deformation-theory"},{"id":"stacks:08SC","tag":"08SC","title":"Functoriality · Lemma 08SC","summary":"Let A → B and A → C be ring maps. Then the map L_B × C/A → L_B/A ⊕ L_C/A is an isomorphism in D(B × C).","statement_latex":"Let $A \\to B$ and $A \\to C$ be ring maps.\nThen the map $L_{B \\times C/A} \\to L_{B/A} \\oplus L_{C/A}$ is\nan isomorphism in $D(B \\times C)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Functoriality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SC","source_file":"cotangent.tex","source_line":1061,"source_end_line":1066,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1061-L1066","statement_sha256":"aa0432e0792098ad4e62eb25e2170b5b781bf7584400e1c1ebce58d08c9d9f64","origin":"The Stacks Project","memory_eligible":false,"source_rank":13478,"rank":13478,"depth":17,"x":1679.496,"y":1681.188,"cluster":"deformation-theory"},{"id":"stacks:08QU","tag":"08QU","title":"The fundamental triangle · Lemma 08QU","summary":"With notation as in ([Tag 08QT]) set Ω_1 = Ω_O/A ⊗_O underlineB on C_B/A Ω_2 = Ω_O/A ⊗_O underlineC on C_C/A Ω_3 = Ω_O/B ⊗_O underlineC on C_C/B Then we have a canonical short exact sequence of sheaves of underlineC-modules 0 → g_1^-1Ω_1 ⊗_underlineB underlineC → g_2^-1Ω_2 → g_3^-1Ω_3 → 0 on C_C/B/A.","statement_latex":"With notation as in (\\ref{equation-three-maps}) set\n$$\n\\begin{matrix}\n\\Omega_1 = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}\n\\text{ on }\\mathcal{C}_{B/A} \\\\\n\\Omega_2 = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{C}\n\\text{ on }\\mathcal{C}_{C/A} \\\\\n\\Omega_3 = \\Omega_{\\mathcal{O}/B} \\otimes_\\mathcal{O} \\underline{C}\n\\text{ on }\\mathcal{C}_{C/B}\n\\end{matrix}\n$$\nThen we have a canonical short exact sequence of sheaves\nof $\\underline{C}$-modules\n$$\n0 \\to g_1^{-1}\\Omega_1 \\otimes_{\\underline{B}} \\underline{C} \\to\ng_2^{-1}\\Omega_2 \\to\ng_3^{-1}\\Omega_3 \\to 0\n$$\non $\\mathcal{C}_{C/B/A}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The fundamental triangle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QU","source_file":"cotangent.tex","source_line":1172,"source_end_line":1193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1172-L1193","statement_sha256":"12950311f236f7bcd510c998d48f0d1937eb6467825754a06ab1b8559517e5f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13479,"rank":13479,"depth":6,"x":1441.026,"y":1596.582,"cluster":"deformation-theory"},{"id":"stacks:08QV","tag":"08QV","title":"The fundamental triangle · Lemma 08QV","summary":"With notation as in ([Tag 08QT]) suppose that C is a polynomial algebra over B. Then Lπ_!(g_3^-1F) = Lπ_3, !F = π_3, !F for any abelian sheaf F on C_C/B","statement_latex":"With notation as in (\\ref{equation-three-maps})\nsuppose that $C$ is a polynomial algebra over $B$. Then\n$L\\pi_!(g_3^{-1}\\mathcal{F}) = L\\pi_{3, !}\\mathcal{F} = \\pi_{3, !}\\mathcal{F}$\nfor any abelian sheaf $\\mathcal{F}$ on $\\mathcal{C}_{C/B}$","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The fundamental triangle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QV","source_file":"cotangent.tex","source_line":1221,"source_end_line":1227,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1221-L1227","statement_sha256":"15c83692afb3983dbaca4a9d449d116a1a79a127b9a54a5636066ac25f1aa42f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13480,"rank":13480,"depth":14,"x":1685.466,"y":1523.489,"cluster":"deformation-theory"},{"id":"stacks:08QW","tag":"08QW","title":"The fundamental triangle · Lemma 08QW","summary":"With notation as in ([Tag 08QT]) we have Lg_i, ! ∘ g_i^-1 = id for i = 1, 2, 3 and hence also Lπ_! ∘ g_i^-1 = Lπ_i, ! for i = 1, 2, 3.","statement_latex":"With notation as in (\\ref{equation-three-maps}) we have\n$Lg_{i, !} \\circ g_i^{-1} = \\text{id}$ for $i = 1, 2, 3$\nand hence also $L\\pi_! \\circ g_i^{-1} = L\\pi_{i, !}$ for\n$i = 1, 2, 3$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The fundamental triangle","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QW","source_file":"cotangent.tex","source_line":1289,"source_end_line":1295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1289-L1295","statement_sha256":"8a0478c9de27f659d9422a5bedc4107f378b2f8ffc9994281e88c0c951dbc5cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13481,"rank":13481,"depth":20,"x":1563.722,"y":1716.527,"cluster":"deformation-theory"},{"id":"stacks:08QX","tag":"08QX","title":"The fundamental triangle · Proposition 08QX","summary":"Let A → B → C be ring maps. There is a canonical distinguished triangle L_B/A ⊗_B^L C → L_C/A → L_C/B → L_B/A ⊗_B^L C[1] in D(C).","statement_latex":"Let $A \\to B \\to C$ be ring maps. There is a canonical distinguished\ntriangle\n$$\nL_{B/A} \\otimes_B^\\mathbf{L} C \\to L_{C/A} \\to L_{C/B} \\to\nL_{B/A} \\otimes_B^\\mathbf{L} C[1]\n$$\nin $D(C)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The fundamental triangle","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QX","source_file":"cotangent.tex","source_line":1369,"source_end_line":1378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1369-L1378","statement_sha256":"911a4233d6a2fd6b2ec6d559149488beaed5d382e91b98aecfcc1d606943c1e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13482,"rank":13482,"depth":21,"x":1498.11,"y":1504.623,"cluster":"deformation-theory"},{"id":"stacks:08QZ","tag":"08QZ","title":"Localization and étale ring maps · Lemma 08QZ","summary":"Let A → A' → B be ring maps such that B = B ⊗_A^L A'. Then L_B/A = L_B/A' in D(B).","statement_latex":"Let $A \\to A' \\to B$ be ring maps such that $B = B \\otimes_A^\\mathbf{L} A'$.\nThen $L_{B/A} = L_{B/A'}$ in $D(B)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Localization and étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08QZ","source_file":"cotangent.tex","source_line":1544,"source_end_line":1548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1544-L1548","statement_sha256":"e2e3f14a7649a647577a4fdb0a5506811be67aff40622383c86865c1bc22fa65","origin":"The Stacks Project","memory_eligible":false,"source_rank":13483,"rank":13483,"depth":15,"x":1717.393,"y":1623.917,"cluster":"deformation-theory"},{"id":"stacks:08R0","tag":"08R0","title":"Localization and étale ring maps · Lemma 08R0","summary":"Let A → B be a ring map such that B = B ⊗_A^L B. Then L_B/A = 0 in D(B).","statement_latex":"Let $A \\to B$ be a ring map such that $B = B \\otimes_A^\\mathbf{L} B$.\nThen $L_{B/A} = 0$ in $D(B)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Localization and étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08R0","source_file":"cotangent.tex","source_line":1575,"source_end_line":1579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1575-L1579","statement_sha256":"28e1f739310fc9877d413e4936a6447b732bf04351b4a788ce0ed3b44143093b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13484,"rank":13484,"depth":16,"x":1459.185,"y":1660.459,"cluster":"deformation-theory"},{"id":"stacks:08R1","tag":"08R1","title":"Localization and étale ring maps · Lemma 08R1","summary":"Let A → B be a ring map such that Tor^A_i(B, B) = 0 for i > 0 and such that L_B/B ⊗_A B = 0. Then L_B/A = 0 in D(B).","statement_latex":"Let $A \\to B$ be a ring map such that $\\text{Tor}^A_i(B, B) = 0$ for $i > 0$\nand such that $L_{B/B \\otimes_A B} = 0$.\nThen $L_{B/A} = 0$ in $D(B)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Localization and étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08R1","source_file":"cotangent.tex","source_line":1587,"source_end_line":1592,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1587-L1592","statement_sha256":"c1f6162bdf6e17b5b993e46535150644ff9202f3465c244fedd97405c2719271","origin":"The Stacks Project","memory_eligible":false,"source_rank":13485,"rank":13485,"depth":16,"x":1620.558,"y":1486.612,"cluster":"deformation-theory"},{"id":"stacks:08R2","tag":"08R2","title":"Localization and étale ring maps · Lemma 08R2","summary":"The cotangent complex L_B/A is zero in each of the following cases: • A → B and B ⊗_A B → B are flat, i.e., A → B is weakly étale (More on Algebra, Definition [Tag 092B]), • A → B is a flat epimorphism of rings, • B = S^-1A for some multiplicative subset S ⊂ A, • A → B is unramified and flat, • A → B is étale, • A → B is a filtered colimit of ring maps for which the cotangent complex vanishes, • B is a henselization of a local ring of A, • B is a strict henselization of a…","statement_latex":"The cotangent complex $L_{B/A}$ is zero in each of the following cases:\n\\begin{enumerate}\n\\item $A \\to B$ and $B \\otimes_A B \\to B$ are flat, i.e., $A \\to B$\nis weakly \\'etale\n(More on Algebra, Definition \\ref{more-algebra-definition-weakly-etale}),\n\\item $A \\to B$ is a flat epimorphism of rings,\n\\item $B = S^{-1}A$ for some multiplicative subset $S \\subset A$,\n\\item $A \\to B$ is unramified and flat,\n\\item $A \\to B$ is \\'etale,\n\\item $A \\to B$ is a filtered colimit of ring maps for which\nthe cotangent complex vanishes,\n\\item $B$ is a henselization of a local ring of $A$,\n\\item $B$ is a strict henselization of a local ring of $A$, and\n\\item add more here.\n\\end{enumerate}","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Localization and étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08R2","source_file":"cotangent.tex","source_line":1615,"source_end_line":1632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1615-L1632","statement_sha256":"76d4f2ef8fa7edb9b23b3581f69add8249b1892c85191298e4e92819eb2b8782","origin":"The Stacks Project","memory_eligible":false,"source_rank":13486,"rank":13486,"depth":46,"x":1641.413,"y":1706.861,"cluster":"deformation-theory"},{"id":"stacks:08R3","tag":"08R3","title":"Localization and étale ring maps · Lemma 08R3","summary":"Let A → B → C be ring maps such that L_C/B = 0. Then L_C/A = L_B/A ⊗_B^L C.","statement_latex":"Let $A \\to B \\to C$ be ring maps such that $L_{C/B} = 0$.\nThen $L_{C/A} = L_{B/A} \\otimes_B^\\mathbf{L} C$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Localization and étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08R3","source_file":"cotangent.tex","source_line":1648,"source_end_line":1652,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1648-L1652","statement_sha256":"44fffbc8a12f2bb08bd5658c27aa8dd687c9e2c296b6ca9736767fac10ebd08d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13487,"rank":13487,"depth":0,"x":1448.49,"y":1555.952,"cluster":"deformation-theory"},{"id":"stacks:08SF","tag":"08SF","title":"Localization and étale ring maps · Lemma 08SF","summary":"Let A → B be ring maps and S ⊂ A, T ⊂ B multiplicative subsets such that S maps into T. Then L_T^-1B/S^-1A = L_B/A ⊗_B T^-1B in D(T^-1B).","statement_latex":"Let $A \\to B$ be ring maps and $S \\subset A$, $T \\subset B$ multiplicative\nsubsets such that $S$ maps into $T$.\nThen $L_{T^{-1}B/S^{-1}A} = L_{B/A} \\otimes_B T^{-1}B$\nin $D(T^{-1}B)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Localization and étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SF","source_file":"cotangent.tex","source_line":1659,"source_end_line":1665,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1659-L1665","statement_sha256":"1cb24a7be820c5dd40670f978e92ea3072cc1cb494981b1bde1c1488e63bf01a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13488,"rank":13488,"depth":16,"x":1712.687,"y":1557.766,"cluster":"deformation-theory"},{"id":"stacks:08UN","tag":"08UN","title":"Localization and étale ring maps · Lemma 08UN","summary":"Let A → B be a local ring homomorphism of local rings. Let A^h → B^h, resp. A^sh → B^sh be the induced maps of henselizations, resp. strict henselizations. Then L_B^h/A^h = L_B^h/A = L_B/A ⊗_B^L B^h resp. L_B^sh/A^sh = L_B^sh/A = L_B/A ⊗_B^L B^sh in D(B^h), resp. D(B^sh).","statement_latex":"Let $A \\to B$ be a local ring homomorphism of local rings.\nLet $A^h \\to B^h$, resp.\\ $A^{sh} \\to B^{sh}$ be the induced\nmaps of henselizations, resp.\\ strict henselizations.\nThen\n$$\nL_{B^h/A^h} = L_{B^h/A} = L_{B/A} \\otimes_B^\\mathbf{L} B^h\n\\quad\\text{resp.}\\quad\nL_{B^{sh}/A^{sh}} = L_{B^{sh}/A} = L_{B/A} \\otimes_B^\\mathbf{L} B^{sh}\n$$\nin $D(B^h)$, resp.\\ $D(B^{sh})$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Localization and étale ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UN","source_file":"cotangent.tex","source_line":1674,"source_end_line":1686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1674-L1686","statement_sha256":"c4592dd10d701218aad4b247712ed80faf2d5a6da5ae266ab70bc3ef1f29f8f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13489,"rank":13489,"depth":47,"x":1515.982,"y":1706.666,"cluster":"deformation-theory"},{"id":"stacks:08R5","tag":"08R5","title":"Smooth ring maps · Lemma 08R5","summary":"If A → B is a smooth ring map, then L_B/A = Ω_B/A[0].","statement_latex":"If $A \\to B$ is a smooth ring map, then $L_{B/A} = \\Omega_{B/A}[0]$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Smooth ring maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08R5","source_file":"cotangent.tex","source_line":1715,"source_end_line":1718,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1715-L1718","statement_sha256":"4b40b808f342222ec4ce087c30a694b4da2db3db3aaf0163745402357dd62241","origin":"The Stacks Project","memory_eligible":false,"source_rank":13490,"rank":13490,"depth":47,"x":1541.344,"y":1484.769,"cluster":"deformation-theory"},{"id":"stacks:0G5Y","tag":"0G5Y","title":"Positive characteristic · Lemma 0G5Y","summary":"Let A → B be a ring map with p = 0 in A. Let P_bullet be the standard resolution of B over A. The map P_bullet → P_bullet induced by the diagram xymatrix B ar[r]_F_B & B A ar[u] ar[r]^F_A & A ar[u] discussed in Section [Tag 08QL] is homotopic to the Frobenius endomorphism P_bullet → P_bullet given by Frobenius on each P_n.","statement_latex":"Let $A \\to B$ be a ring map with $p = 0$ in $A$. Let $P_\\bullet$ be the\nstandard resolution of $B$ over $A$. The map $P_\\bullet \\to P_\\bullet$\ninduced by the diagram\n$$\n\\xymatrix{\nB \\ar[r]_{F_B} & B \\\\\nA \\ar[u] \\ar[r]^{F_A} & A \\ar[u]\n}\n$$\ndiscussed in Section \\ref{section-functoriality} is homotopic to the Frobenius\nendomorphism $P_\\bullet \\to P_\\bullet$ given by Frobenius on each $P_n$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5Y","source_file":"cotangent.tex","source_line":1749,"source_end_line":1762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1749-L1762","statement_sha256":"1f7df635cddcf4f023696ea70fed2b4695d31f057cf3363ac7e2f4a7cfa75f15","origin":"The Stacks Project","memory_eligible":false,"source_rank":13491,"rank":13491,"depth":6,"x":1701.433,"y":1663.172,"cluster":"deformation-theory"},{"id":"stacks:0G5Z","tag":"0G5Z","title":"Positive characteristic · Lemma 0G5Z","summary":"Let p be a prime number. Let A → B be a ring homomorphism and assume that p = 0 in A. The map L_B/A → L_B/A of Section [Tag 08QL] induced by the Frobenius maps F_A and F_B is homotopic to zero.","statement_latex":"Let $p$ be a prime number. Let $A \\to B$ be a ring homomorphism\nand assume that $p = 0$ in $A$. The map $L_{B/A} \\to L_{B/A}$\nof Section \\ref{section-functoriality} induced by the\nFrobenius maps $F_A$ and $F_B$ is homotopic to zero.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G5Z","source_file":"cotangent.tex","source_line":1800,"source_end_line":1806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1800-L1806","statement_sha256":"7a5920d2447b711c44e5727eaebd65231b9079aee328e880295b69a055bbb8d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13492,"rank":13492,"depth":7,"x":1439.352,"y":1622.371,"cluster":"deformation-theory"},{"id":"stacks:0G60","tag":"0G60","title":"Positive characteristic · Lemma 0G60","summary":"Let p be a prime number. Let A → B be a ring homomorphism and assume that p = 0 in A. If A and B are perfect, then L_B/A is zero in D(B).","statement_latex":"Let $p$ be a prime number. Let $A \\to B$ be a ring homomorphism\nand assume that $p = 0$ in $A$. If $A$ and $B$ are perfect, then\n$L_{B/A}$ is zero in $D(B)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Positive characteristic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G60","source_file":"cotangent.tex","source_line":1818,"source_end_line":1823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1818-L1823","statement_sha256":"fd508029ac0e0d861988db9f8ba8edfb0e1ac365f149f2d0bf71cc24bbda0258","origin":"The Stacks Project","memory_eligible":false,"source_rank":13493,"rank":13493,"depth":8,"x":1665.906,"y":1503.49,"cluster":"deformation-theory"},{"id":"stacks:08RA","tag":"08RA","title":"Comparison with the naive cotangent complex · Lemma 08RA","summary":"The cohomology of the cotangent complex of a surjective ring map is trivial in degree zero; it is the kernel modulo its square in degree -1. Let A → B be a surjective ring map with kernel I. Then H^0(L_B/A) = 0 and H^-1(L_B/A) = I/I^2. This isomorphism comes from the map ([Tag 08R9]) for the object (A → B) of C_B/A.","statement_latex":"\\begin{slogan}\nThe cohomology of the cotangent complex of a surjective ring map is trivial in\ndegree zero; it is the kernel modulo its square in degree $-1$.\n\\end{slogan}\nLet $A \\to B$ be a surjective ring map with kernel $I$.\nThen $H^0(L_{B/A}) = 0$ and $H^{-1}(L_{B/A}) = I/I^2$.\nThis isomorphism comes from the map (\\ref{equation-comparison-map})\nfor the object $(A \\to B)$ of $\\mathcal{C}_{B/A}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Comparison with the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RA","source_file":"cotangent.tex","source_line":1896,"source_end_line":1906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1896-L1906","statement_sha256":"18b2d3bdea92d4a611a3e01eae5b5d2cf15b20ffdb6930c5651c5780690c649b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13494,"rank":13494,"depth":14,"x":1594.295,"y":1720.168,"cluster":"deformation-theory"},{"id":"stacks:08RB","tag":"08RB","title":"Comparison with the naive cotangent complex · Lemma 08RB","summary":"Let A → B be a ring map. Then τ_≥ -1L_B/A is canonically quasi-isomorphic to the naive cotangent complex.","statement_latex":"Let $A \\to B$ be a ring map. Then $\\tau_{\\geq -1}L_{B/A}$\nis canonically quasi-isomorphic to the naive cotangent complex.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Comparison with the naive cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RB","source_file":"cotangent.tex","source_line":1937,"source_end_line":1941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L1937-L1941","statement_sha256":"4d19874c9064fb1b48284b12cbaf55347ced2db4beb00a210a18a4ebf62a269a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13495,"rank":13495,"depth":16,"x":1472.594,"y":1519.33,"cluster":"deformation-theory"},{"id":"stacks:08RD","tag":"08RD","title":"A spectral sequence of Quillen · Lemma 08RD","summary":"Notation and assumptions as in Cohomology on Sites, Example [Tag 08PF]. Assume C has a cosimplicial object as in Cohomology on Sites, Lemma [Tag 08Q9]. Let F be a flat underlineB-module such that H_0(C, F) = 0. Then H_l(C, Sym_underlineB^k(F)) = 0 for l < k.","statement_latex":"Notation and assumptions as in\nCohomology on Sites, Example \\ref{sites-cohomology-example-category-to-point}.\nAssume $\\mathcal{C}$ has a cosimplicial object as in\nCohomology on Sites, Lemma\n\\ref{sites-cohomology-lemma-compute-by-cosimplicial-resolution}.\nLet $\\mathcal{F}$ be a flat $\\underline{B}$-module such that\n$H_0(\\mathcal{C}, \\mathcal{F}) = 0$.\nThen $H_l(\\mathcal{C}, \\text{Sym}_{\\underline{B}}^k(\\mathcal{F})) = 0$\nfor $l < k$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"A spectral sequence of Quillen","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RD","source_file":"cotangent.tex","source_line":2064,"source_end_line":2075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2064-L2075","statement_sha256":"51678ffbd36d827a30285da534ef0a9f9e1516edce625f03c77dd89db8b5e64e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13496,"rank":13496,"depth":27,"x":1724.379,"y":1598.537,"cluster":"deformation-theory"},{"id":"stacks:08RE","tag":"08RE","title":"A spectral sequence of Quillen · Lemma 08RE","summary":"Let A be a ring. Let P = A[E] be a polynomial ring. Set I = (e; e ∈ E) ⊂ P. The maps Tor_i^P(A, I^n + 1) → Tor_i^P(A, I^n) are zero for all i and n.","statement_latex":"Let $A$ be a ring. Let $P = A[E]$ be a polynomial ring.\nSet $I = (e; e \\in E) \\subset P$. The maps\n$\\text{Tor}_i^P(A, I^{n + 1}) \\to \\text{Tor}_i^P(A, I^n)$\nare zero for all $i$ and $n$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"A spectral sequence of Quillen","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RE","source_file":"cotangent.tex","source_line":2137,"source_end_line":2143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2137-L2143","statement_sha256":"db7a94288aaf164806d0907e5bfefb86a35ff64f4ff528c8a0b5eab5bb0c823d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13497,"rank":13497,"depth":0,"x":1474.491,"y":1683.178,"cluster":"deformation-theory"},{"id":"stacks:08RF","tag":"08RF","title":"Quillen spectral sequence · Theorem 08RF","summary":"Let A → B be a surjective ring map. Consider the sheaf Ω = Ω_O/A ⊗_O underlineB of underlineB-modules on C_B/A, see Section [Tag 08PQ]. Then there is a spectral sequence with E_1-page E_1^p, q = H_- p - q(C_B/A, Sym^p_underlineB(Ω)) ⇒ Tor^A_- p - q(B, B) with d_r of bidegree (r, -r + 1). Moreover, H_i(C_B/A, Sym^k_underlineB(Ω)) = 0 for i < k.","statement_latex":"Let $A \\to B$ be a surjective ring map. Consider the sheaf\n$\\Omega = \\Omega_{\\mathcal{O}/A} \\otimes_\\mathcal{O} \\underline{B}$ of\n$\\underline{B}$-modules on $\\mathcal{C}_{B/A}$, see\nSection \\ref{section-compute-L-pi-shriek}.\nThen there is a spectral sequence with $E_1$-page\n$$\nE_1^{p, q} =\nH_{- p - q}(\\mathcal{C}_{B/A}, \\text{Sym}^p_{\\underline{B}}(\\Omega))\n\\Rightarrow \\text{Tor}^A_{- p - q}(B, B)\n$$\nwith $d_r$ of bidegree $(r, -r + 1)$.\nMoreover, $H_i(\\mathcal{C}_{B/A}, \\text{Sym}^k_{\\underline{B}}(\\Omega)) = 0$\nfor $i < k$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"A spectral sequence of Quillen","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08RF","source_file":"cotangent.tex","source_line":2162,"source_end_line":2177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2162-L2177","statement_sha256":"78c5cdf62b462e4e29dbd298c37a0b7e24d09763fd28fcf5a23c7bf522691a70","origin":"The Stacks Project","memory_eligible":false,"source_rank":13498,"rank":13498,"depth":28,"x":1590.934,"y":1478.542,"cluster":"deformation-theory"},{"id":"stacks:09CE","tag":"09CE","title":"Comparison with Lichtenbaum-Schlessinger · Definition 09CE","summary":"Let A → B be a ring map. Let M be a (B, B)-bimodule over A. An A-biderivation is an A-linear map λ : B → M such that λ(xy) = xλ(y) + λ(x)y.","statement_latex":"Let $A \\to B$ be a ring map. Let $M$ be a $(B, B)$-bimodule\nover $A$. An {\\it $A$-biderivation} is an $A$-linear map $\\lambda : B \\to M$\nsuch that $\\lambda(xy) = x\\lambda(y) + \\lambda(x)y$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Comparison with Lichtenbaum-Schlessinger","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CE","source_file":"cotangent.tex","source_line":2419,"source_end_line":2424,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2419-L2424","statement_sha256":"dd386ea041cff375b64ea26ca79e1e4cfc6038f745b6f119a5266356e25c61d7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13499,"rank":13499,"depth":0,"x":1669.799,"y":1695.967,"cluster":"deformation-theory"},{"id":"stacks:09CF","tag":"09CF","title":"Comparison with Lichtenbaum-Schlessinger · Lemma 09CF","summary":"Let P = A[S] be a polynomial ring over A. Let M be a (P, P)-bimodule over A. Given m_s ∈ M for s ∈ S, there exists a unique A-biderivation λ : P → M mapping s to m_s for s ∈ S.","statement_latex":"Let $P = A[S]$ be a polynomial ring over $A$. Let $M$ be a $(P, P)$-bimodule\nover $A$. Given $m_s \\in M$ for $s \\in S$, there exists a unique\n$A$-biderivation $\\lambda : P \\to M$ mapping $s$ to $m_s$ for $s \\in S$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Comparison with Lichtenbaum-Schlessinger","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CF","source_file":"cotangent.tex","source_line":2429,"source_end_line":2434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2429-L2434","statement_sha256":"c36d562d2af66d8385ec890845b7116fd1bdb42e9e96e993aa1ea843afb130c6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13500,"rank":13500,"depth":0,"x":1436.31,"y":1580.148,"cluster":"deformation-theory"},{"id":"stacks:09CG","tag":"09CG","title":"Comparison with Lichtenbaum-Schlessinger · Lemma 09CG","summary":"In the situation above denote L the complex ([Tag 09CD]). There is a canonical map L_B/A → L in D(B) which induces an isomorphism τ_≥ -2L_B/A → L in D(B).","statement_latex":"In the situation above denote $L$ the complex\n(\\ref{equation-lichtenbaum-schlessinger}).\nThere is a canonical map $L_{B/A} \\to L$ in $D(B)$ which\ninduces an isomorphism $\\tau_{\\geq -2}L_{B/A} \\to L$ in $D(B)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Comparison with Lichtenbaum-Schlessinger","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09CG","source_file":"cotangent.tex","source_line":2452,"source_end_line":2458,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2452-L2458","statement_sha256":"11168498b7950231623aad743792115886127c369345610dfc97560e5271bec6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13501,"rank":13501,"depth":17,"x":1702.173,"y":1532.966,"cluster":"deformation-theory"},{"id":"stacks:08SI","tag":"08SI","title":"The cotangent complex of a local complete intersection · Lemma 08SI","summary":"Let A = Z[x_1, …, x_n] → B = Z be the ring map which sends x_i to 0 for i = 1, …, n. Let I = (x_1, …, x_n) ⊂ A. Then L_B/A is quasi-isomorphic to I/I^2[1].","statement_latex":"Let $A = \\mathbf{Z}[x_1, \\ldots, x_n] \\to B = \\mathbf{Z}$\nbe the ring map which sends $x_i$ to $0$ for $i = 1, \\ldots, n$.\nLet $I = (x_1, \\ldots, x_n) \\subset A$. Then $L_{B/A}$ is quasi-isomorphic to\n$I/I^2[1]$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a local complete intersection","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SI","source_file":"cotangent.tex","source_line":2672,"source_end_line":2678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2672-L2678","statement_sha256":"7571115f16bf073bb912fc5b5d5ed3c4fdf58b0b68db0755895b3900d06ada99","origin":"The Stacks Project","memory_eligible":false,"source_rank":13502,"rank":13502,"depth":47,"x":1543.743,"y":1719.0,"cluster":"deformation-theory"},{"id":"stacks:08SJ","tag":"08SJ","title":"The cotangent complex of a local complete intersection · Lemma 08SJ","summary":"Let A → B be a surjective ring map whose kernel I is generated by a Koszul-regular sequence (for example a regular sequence). Then L_B/A is quasi-isomorphic to I/I^2[1].","statement_latex":"Let $A \\to B$ be a surjective ring map whose kernel $I$ is generated\nby a Koszul-regular sequence (for example a regular sequence).\nThen $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a local complete intersection","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SJ","source_file":"cotangent.tex","source_line":2701,"source_end_line":2706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2701-L2706","statement_sha256":"5e6d9ce92be46f059d53558e0ec7a19f541a7a5f337140f87c9f57e94c3262fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":13503,"rank":13503,"depth":48,"x":1510.897,"y":1491.455,"cluster":"deformation-theory"},{"id":"stacks:08SK","tag":"08SK","title":"The cotangent complex of a local complete intersection · Lemma 08SK","summary":"Let A → B be a surjective ring map whose kernel I is Koszul. Then L_B/A is quasi-isomorphic to I/I^2[1].","statement_latex":"Let $A \\to B$ be a surjective ring map whose kernel $I$ is Koszul.\nThen $L_{B/A}$ is quasi-isomorphic to $I/I^2[1]$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a local complete intersection","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SK","source_file":"cotangent.tex","source_line":2727,"source_end_line":2731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2727-L2731","statement_sha256":"0ab39c7ef66fcf59a364dbc6ce1db0127d5a8ac40f414ce4d66f33b22f47b8aa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13504,"rank":13504,"depth":16,"x":1718.529,"y":1640.912,"cluster":"deformation-theory"},{"id":"stacks:08SL","tag":"08SL","title":"The cotangent complex of a local complete intersection · Proposition 08SL","summary":"Let A → B be a local complete intersection map. Then L_B/A is a perfect complex with tor amplitude in [-1, 0].","statement_latex":"Let $A \\to B$ be a local complete intersection map.\nThen $L_{B/A}$ is a perfect complex with tor amplitude in $[-1, 0]$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a local complete intersection","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SL","source_file":"cotangent.tex","source_line":2741,"source_end_line":2745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2741-L2745","statement_sha256":"a94d2424b7be238fbb19602528f65e3f10ad331e0be37e17e172bb1a854cae5f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13505,"rank":13505,"depth":17,"x":1444.674,"y":1648.536,"cluster":"deformation-theory"},{"id":"stacks:09DA","tag":"09DA","title":"Tensor products and the cotangent complex · Lemma 09DA","summary":"If A and B are Tor independent R-algebras, then the object E in ([Tag 09D9]) is zero. In this case we have L_A ⊗_R B/R = L_A/R ⊗_A^L (A ⊗_R B) ⊕ L_B/R ⊗_B^L (A ⊗_R B) which is represented by the complex L_A/R ⊗_R B ⊕ L_B/R ⊗_R A of A ⊗_R B-modules.","statement_latex":"If $A$ and $B$ are Tor independent $R$-algebras, then the object $E$\nin (\\ref{equation-tensor-product}) is zero. In this case we have\n$$\nL_{A \\otimes_R B/R} =\nL_{A/R} \\otimes_A^\\mathbf{L} (A \\otimes_R B) \\oplus\nL_{B/R} \\otimes_B^\\mathbf{L} (A \\otimes_R B)\n$$\nwhich is represented by the complex\n$L_{A/R} \\otimes_R B \\oplus L_{B/R} \\otimes_R A $\nof $A \\otimes_R B$-modules.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Tensor products and the cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DA","source_file":"cotangent.tex","source_line":2820,"source_end_line":2832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2820-L2832","statement_sha256":"1b3f0afb0308308f001210da350b7f09eb247093790432c59e2db80d2dd783ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":13506,"rank":13506,"depth":16,"x":1640.88,"y":1487.194,"cluster":"deformation-theory"},{"id":"stacks:09DB","tag":"09DB","title":"Tensor products and the cotangent complex · Lemma 09DB","summary":"Let R be a ring and let A, B be R-algebras. The object E in ([Tag 09D9]) satisfies H^i(E) = ( 0 & if & i ≥ -1 Tor_1^R(A, B) & if & i = -2 .","statement_latex":"Let $R$ be a ring and let $A$, $B$ be $R$-algebras. The object $E$\nin (\\ref{equation-tensor-product}) satisfies\n$$\nH^i(E) =\n\\left\\{\n\\begin{matrix}\n0 & \\text{if} & i \\geq -1 \\\\\n\\text{Tor}_1^R(A, B) & \\text{if} & i = -2\n\\end{matrix}\n\\right.\n$$","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Tensor products and the cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DB","source_file":"cotangent.tex","source_line":2846,"source_end_line":2859,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2846-L2859","statement_sha256":"c3227d6e06dbee8cf27369c612ab47682292d80a70638c561cde5dbca5c97d91","origin":"The Stacks Project","memory_eligible":false,"source_rank":13507,"rank":13507,"depth":18,"x":1625.916,"y":1717.966,"cluster":"deformation-theory"},{"id":"stacks:08SP","tag":"08SP","title":"Deformations of ring maps and the cotangent complex · Lemma 08SP","summary":"In the situation above we have • There is a canonical element xi ∈ Ext^2_B(L_B/A, N) whose vanishing is a sufficient and necessary condition for the existence of a solution to ([Tag 08SN]). • If there exists a solution, then the set of isomorphism classes of solutions is principal homogeneous under Ext^1_B(L_B/A, N). • Given a solution B', the set of automorphisms of B' fitting into ([Tag 08SN]) is canonically isomorphic to Ext^0_B(L_B/A, N).","statement_latex":"In the situation above we have\n\\begin{enumerate}\n\\item There is a canonical element $\\xi \\in \\Ext^2_B(L_{B/A}, N)$\nwhose vanishing is a sufficient and necessary condition for the existence\nof a solution to (\\ref{equation-to-solve}).\n\\item If there exists a solution, then the set of\nisomorphism classes of solutions is principal homogeneous under\n$\\Ext^1_B(L_{B/A}, N)$.\n\\item Given a solution $B'$, the set of automorphisms of $B'$\nfitting into (\\ref{equation-to-solve}) is canonically isomorphic\nto $\\Ext^0_B(L_{B/A}, N)$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Deformations of ring maps and the cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SP","source_file":"cotangent.tex","source_line":2951,"source_end_line":2965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L2951-L2965","statement_sha256":"326dffda028ce9a8db18d8b42360bea5608e5d0bab1ef6bee85242847f78ea06","origin":"The Stacks Project","memory_eligible":false,"source_rank":13508,"rank":13508,"depth":17,"x":1451.017,"y":1538.945,"cluster":"deformation-theory"},{"id":"stacks:09DE","tag":"09DE","title":"The Atiyah class of a module · Definition 09DE","summary":"Let A → B be a ring map. Let M be a B-module. The map M → L_B/A ⊗_B^L M[1] in ([Tag 09DD]) is called the Atiyah class of M.","statement_latex":"Let $A \\to B$ be a ring map. Let $M$ be a $B$-module.\nThe map $M \\to L_{B/A} \\otimes_B^\\mathbf{L} M[1]$\nin (\\ref{equation-atiyah}) is called the {\\it Atiyah class} of $M$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The Atiyah class of a module","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DE","source_file":"cotangent.tex","source_line":3051,"source_end_line":3056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3051-L3056","statement_sha256":"28091c50ce485f70ecbc7352534fee53417cee8adbf2c3372bd6d9148fc2803d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13509,"rank":13509,"depth":0,"x":1724.499,"y":1571.776,"cluster":"deformation-theory"},{"id":"stacks:08SR","tag":"08SR","title":"The cotangent complex · Definition 08SR","summary":"Let C be a site. Let A → B be a homomorphism of sheaves of rings on C. The standard resolution of B over A is the augmentation ε : P_bullet → B with terms P_0 = A[B], P_1 = A[A[B]], … and maps as constructed above.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings\non $\\mathcal{C}$. The {\\it standard resolution of $\\mathcal{B}$ over\n$\\mathcal{A}$} is the augmentation\n$\\epsilon : \\mathcal{P}_\\bullet \\to \\mathcal{B}$\nwith terms\n$$\n\\mathcal{P}_0 = \\mathcal{A}[\\mathcal{B}],\\quad\n\\mathcal{P}_1 = \\mathcal{A}[\\mathcal{A}[\\mathcal{B}]],\\quad \\ldots\n$$\nand maps as constructed above.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SR","source_file":"cotangent.tex","source_line":3099,"source_end_line":3112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3099-L3112","statement_sha256":"73c64fe2a880c2419cc00ae2758daae1dc622e4cfdd0d50ab60aa6d7f57c4da5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13510,"rank":13510,"depth":0,"x":1495.977,"y":1703.01,"cluster":"deformation-theory"},{"id":"stacks:08SS","tag":"08SS","title":"The cotangent complex · Definition 08SS","summary":"Let C be a site. Let A → B be a homomorphism of sheaves of rings on C. The cotangent complex L_B/A is the complex of B-modules associated to the simplicial module Ω_P_bullet/A ⊗_P_bullet, ε B where ε : P_bullet → B is the standard resolution of B over A. We usually think of L_B/A as an object of D(B).","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings\non $\\mathcal{C}$.\nThe {\\it cotangent complex} $L_{\\mathcal{B}/\\mathcal{A}}$\nis the complex of $\\mathcal{B}$-modules associated to the\nsimplicial module\n$$\n\\Omega_{\\mathcal{P}_\\bullet/\\mathcal{A}}\n\\otimes_{\\mathcal{P}_\\bullet, \\epsilon} \\mathcal{B}\n$$\nwhere $\\epsilon : \\mathcal{P}_\\bullet \\to \\mathcal{B}$\nis the standard resolution of $\\mathcal{B}$ over\n$\\mathcal{A}$. We usually think of $L_{\\mathcal{B}/\\mathcal{A}}$\nas an object of $D(\\mathcal{B})$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SS","source_file":"cotangent.tex","source_line":3120,"source_end_line":3136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3120-L3136","statement_sha256":"d7fedb7d1ab93659922e42da8677485151a0d7e6b497c6834b011210ed9b4de9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13511,"rank":13511,"depth":0,"x":1559.078,"y":1476.118,"cluster":"deformation-theory"},{"id":"stacks:08SV","tag":"08SV","title":"The cotangent complex · Lemma 08SV","summary":"Let f : Sh(D) → Sh(C) be a morphism of topoi. Let A → B be a homomorphism of sheaves of rings on C. Then f^-1L_B/A = L_f^-1B/f^-1A.","statement_latex":"Let $f : \\Sh(\\mathcal{D}) \\to \\Sh(\\mathcal{C})$ be a morphism of topoi.\nLet $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings\non $\\mathcal{C}$. Then\n$f^{-1}L_{\\mathcal{B}/\\mathcal{A}} = L_{f^{-1}\\mathcal{B}/f^{-1}\\mathcal{A}}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SV","source_file":"cotangent.tex","source_line":3177,"source_end_line":3183,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3177-L3183","statement_sha256":"216235324539fe4d2356ca4a8948734b2402e525e1b4464c7c7e50723b570edd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13512,"rank":13512,"depth":0,"x":1695.279,"y":1679.635,"cluster":"deformation-theory"},{"id":"stacks:08SW","tag":"08SW","title":"The cotangent complex · Lemma 08SW","summary":"Let C be a site. Let A → B be a homomorphism of sheaves of rings on C. Then H^i(L_B/A) is the sheaf associated to the presheaf U ↦ H^i(L_B(U)/A(U)).","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a\nhomomorphism of sheaves of rings on $\\mathcal{C}$. Then\n$H^i(L_{\\mathcal{B}/\\mathcal{A}})$ is the sheaf associated to the\npresheaf $U \\mapsto H^i(L_{\\mathcal{B}(U)/\\mathcal{A}(U)})$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SW","source_file":"cotangent.tex","source_line":3197,"source_end_line":3203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3197-L3203","statement_sha256":"34205201321235540fa73d6edf52a6905581e2fc883512b58d3d797360262398","origin":"The Stacks Project","memory_eligible":false,"source_rank":13513,"rank":13513,"depth":1,"x":1430.66,"y":1606.704,"cluster":"deformation-theory"},{"id":"stacks:08UR","tag":"08UR","title":"The cotangent complex · Lemma 08UR","summary":"Let C be a site. Let A → B be a homomorphism of sheaves of rings on C. Then H^0(L_B/A) = Ω_B/A.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a\nhomomorphism of sheaves of rings on $\\mathcal{C}$. Then\n$H^0(L_{\\mathcal{B}/\\mathcal{A}}) = \\Omega_{\\mathcal{B}/\\mathcal{A}}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UR","source_file":"cotangent.tex","source_line":3240,"source_end_line":3245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3240-L3245","statement_sha256":"51cb3c09872cc4bd9d56a89503807646f9234400c005a8326fbba1813976888c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13514,"rank":13514,"depth":16,"x":1684.936,"y":1510.141,"cluster":"deformation-theory"},{"id":"stacks:08SY","tag":"08SY","title":"The cotangent complex · Lemma 08SY","summary":"Let C be a site. Let A → B and A → B' be homomorphisms of sheaves of rings on C. Then L_B × B'/A → L_B/A ⊕ L_B'/A is an isomorphism in D(B × B').","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$\nand $\\mathcal{A} \\to \\mathcal{B}'$ be homomorphisms of sheaves of rings\non $\\mathcal{C}$. Then\n$$\nL_{\\mathcal{B} \\times \\mathcal{B}'/\\mathcal{A}}\n\\longrightarrow\nL_{\\mathcal{B}/\\mathcal{A}} \\oplus L_{\\mathcal{B}'/\\mathcal{A}}\n$$\nis an isomorphism in $D(\\mathcal{B} \\times \\mathcal{B}')$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SY","source_file":"cotangent.tex","source_line":3253,"source_end_line":3264,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3253-L3264","statement_sha256":"b08bfc0bff721e7c40b9423089098359d27b1ed6a6da96b8349357aa913f0300","origin":"The Stacks Project","memory_eligible":false,"source_rank":13515,"rank":13515,"depth":18,"x":1574.874,"y":1726.051,"cluster":"deformation-theory"},{"id":"stacks:08SZ","tag":"08SZ","title":"The cotangent complex · Lemma 08SZ","summary":"Let D be a site. Let A → B → C be homomorphisms of sheaves of rings on D. There is a canonical distinguished triangle L_B/A ⊗_B^L C → L_C/A → L_C/B → L_B/A ⊗_B^L C[1] in D(C).","statement_latex":"Let $\\mathcal{D}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B} \\to \\mathcal{C}$\nbe homomorphisms of sheaves of rings on $\\mathcal{D}$.\nThere is a canonical distinguished triangle\n$$\nL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B}^\\mathbf{L} \\mathcal{C}\n\\to L_{\\mathcal{C}/\\mathcal{A}} \\to L_{\\mathcal{C}/\\mathcal{B}} \\to\nL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B}^\\mathbf{L} \\mathcal{C}[1]\n$$\nin $D(\\mathcal{C})$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SZ","source_file":"cotangent.tex","source_line":3277,"source_end_line":3288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3277-L3288","statement_sha256":"e29533c4825cb99a0d99af62544ad98572ed0041a91c8e0553359171f5744972","origin":"The Stacks Project","memory_eligible":false,"source_rank":13516,"rank":13516,"depth":19,"x":1482.222,"y":1503.955,"cluster":"deformation-theory"},{"id":"stacks:08T0","tag":"08T0","title":"The cotangent complex · Lemma 08T0","summary":"Let C be a site. Let A → B be a homomorphism of sheaves of rings on C. If p is a point of C, then (L_B/A)_p = L_B_p/A_p.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a\nhomomorphism of sheaves of rings on $\\mathcal{C}$. If $p$ is a point\nof $\\mathcal{C}$, then\n$(L_{\\mathcal{B}/\\mathcal{A}})_p = L_{\\mathcal{B}_p/\\mathcal{A}_p}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08T0","source_file":"cotangent.tex","source_line":3353,"source_end_line":3359,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3353-L3359","statement_sha256":"932076f98d462c995efb3290ade88ec1bc0854f7ce28c644ae4922277e58cbdf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13517,"rank":13517,"depth":1,"x":1729.628,"y":1615.37,"cluster":"deformation-theory"},{"id":"stacks:08US","tag":"08US","title":"The cotangent complex · Lemma 08US","summary":"Let C be a site. Let A → B be a homomorphism of sheaves of rings on C. There is a canonical map L_B/A → NL_B/A which identifies the naive cotangent complex with the truncation τ_≥ -1L_B/A.","statement_latex":"Let $\\mathcal{C}$ be a site. Let $\\mathcal{A} \\to \\mathcal{B}$ be a\nhomomorphism of sheaves of rings on $\\mathcal{C}$.\nThere is a canonical map\n$L_{\\mathcal{B}/\\mathcal{A}} \\to \\NL_{\\mathcal{B}/\\mathcal{A}}$\nwhich identifies the naive cotangent complex with the truncation\n$\\tau_{\\geq -1}L_{\\mathcal{B}/\\mathcal{A}}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08US","source_file":"cotangent.tex","source_line":3370,"source_end_line":3378,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3370-L3378","statement_sha256":"789e0eb9c29ae10de447347b046273179e061fa99e7deb626eb6234b7afbd33f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13518,"rank":13518,"depth":3,"x":1457.068,"y":1673.711,"cluster":"deformation-theory"},{"id":"stacks:09DI","tag":"09DI","title":"The Atiyah class of a sheaf of modules · Definition 09DI","summary":"Let C be a site. Let A → B be a homomorphism of sheaves of rings. Let F be a sheaf of B-modules. The map F → L_B/A ⊗_B^L F[1] in ([Tag 09DH]) is called the Atiyah class of F.","statement_latex":"Let $\\mathcal{C}$ be a site.\nLet $\\mathcal{A} \\to \\mathcal{B}$ be a homomorphism of sheaves of rings.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{B}$-modules.\nThe map $\\mathcal{F} \\to\nL_{\\mathcal{B}/\\mathcal{A}} \\otimes_\\mathcal{B}^\\mathbf{L} \\mathcal{F}[1]$\nin (\\ref{equation-atiyah-general}) is called the {\\it Atiyah class} of\n$\\mathcal{F}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The Atiyah class of a sheaf of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DI","source_file":"cotangent.tex","source_line":3494,"source_end_line":3503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3494-L3503","statement_sha256":"2ed3e7f9c9574907a7a81f52e217d67ea2743d7114908660f51237410febd4f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13519,"rank":13519,"depth":0,"x":1611.432,"y":1475.652,"cluster":"deformation-theory"},{"id":"stacks:08UU","tag":"08UU","title":"The cotangent complex of a morphism of ringed spaces · Definition 08UU","summary":"Let f : (X, O_X) → (S, O_S) be a morphism of ringed spaces. The cotangent complex L_f of f is L_f = L_O_X/f^-1O_S. We will also use the notation L_f = L_X/S = L_O_X/O_S.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of\nringed spaces. The {\\it cotangent complex} $L_f$ of $f$ is\n$L_f = L_{\\mathcal{O}_X/f^{-1}\\mathcal{O}_S}$.\nWe will also use the notation\n$L_f = L_{X/S} = L_{\\mathcal{O}_X/\\mathcal{O}_S}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of ringed spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UU","source_file":"cotangent.tex","source_line":3520,"source_end_line":3527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3520-L3527","statement_sha256":"31126a6ccd95855a36b29db90bb439dae0c981dd9f21bbe425d548951f55140b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13520,"rank":13520,"depth":0,"x":1656.968,"y":1709.738,"cluster":"deformation-theory"},{"id":"stacks:08UV","tag":"08UV","title":"The cotangent complex of a morphism of ringed spaces · Lemma 08UV","summary":"Let f : (X, O_X) → (S, O_S) be a morphism of ringed spaces. Then H^0(L_X/S) = Ω_X/S.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (S, \\mathcal{O}_S)$ be a morphism of\nringed spaces. Then $H^0(L_{X/S}) = \\Omega_{X/S}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UV","source_file":"cotangent.tex","source_line":3536,"source_end_line":3540,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3536-L3540","statement_sha256":"65fde2430a72d60ca92dcce6c9557875810a1489f3bfe7d8954530064a54577c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13521,"rank":13521,"depth":17,"x":1434.718,"y":1562.682,"cluster":"deformation-theory"},{"id":"stacks:08T4","tag":"08T4","title":"The cotangent complex of a morphism of ringed spaces · Lemma 08T4","summary":"Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Then there is a canonical distinguished triangle Lf^* L_Y/Z → L_X/Z → L_X/Y → Lf^*L_Y/Z[1] in D(O_X).","statement_latex":"Let $f : X \\to Y$ and $g : Y \\to Z$ be morphisms of ringed spaces.\nThen there is a canonical distinguished triangle\n$$\nLf^* L_{Y/Z} \\to L_{X/Z} \\to L_{X/Y} \\to Lf^*L_{Y/Z}[1]\n$$\nin $D(\\mathcal{O}_X)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08T4","source_file":"cotangent.tex","source_line":3546,"source_end_line":3554,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3546-L3554","statement_sha256":"ba1b2485ad59a15b9984fe1b3174b8e7bcefd0ab71d81752cfd70c28e5608f89","origin":"The Stacks Project","memory_eligible":false,"source_rank":13522,"rank":13522,"depth":20,"x":1717.4,"y":1544.976,"cluster":"deformation-theory"},{"id":"stacks:08UW","tag":"08UW","title":"The cotangent complex of a morphism of ringed spaces · Lemma 08UW","summary":"Let f : (X, O_X) → (Y, O_Y) be a morphism of ringed spaces. There is a canonical map L_X/Y → NL_X/Y which identifies the naive cotangent complex with the truncation τ_≥ -1L_X/Y.","statement_latex":"Let $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of\nringed spaces. There is a canonical map $L_{X/Y} \\to \\NL_{X/Y}$ which\nidentifies the naive cotangent complex with the truncation\n$\\tau_{\\geq -1}L_{X/Y}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of ringed spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UW","source_file":"cotangent.tex","source_line":3568,"source_end_line":3574,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3568-L3574","statement_sha256":"4a498c1348670d84e15f9a193290b7ace977f8a51bc0553a2b91d0c0f9341a1e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13523,"rank":13523,"depth":4,"x":1522.823,"y":1718.763,"cluster":"deformation-theory"},{"id":"stacks:08UZ","tag":"08UZ","title":"Deformations of ringed spaces and the cotangent complex · Lemma 08UZ","summary":"In the situation above we have • There is a canonical element xi ∈ Ext^2_O_X(L_X/S, G) whose vanishing is a sufficient and necessary condition for the existence of a solution to ([Tag 08UY]). • If there exists a solution, then the set of isomorphism classes of solutions is principal homogeneous under Ext^1_O_X(L_X/S, G). • Given a solution X', the set of automorphisms of X' fitting into ([Tag 08UY]) is canonically isomorphic to Ext^0_O_X(L_X/S, G).","statement_latex":"In the situation above we have\n\\begin{enumerate}\n\\item There is a canonical element\n$\\xi \\in \\Ext^2_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G})$\nwhose vanishing is a sufficient and necessary condition for the existence\nof a solution to (\\ref{equation-to-solve-ringed-spaces}).\n\\item If there exists a solution, then the set of\nisomorphism classes of solutions is principal homogeneous under\n$\\Ext^1_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G})$.\n\\item Given a solution $X'$, the set of automorphisms of $X'$\nfitting into (\\ref{equation-to-solve-ringed-spaces}) is canonically isomorphic\nto $\\Ext^0_{\\mathcal{O}_X}(L_{X/S}, \\mathcal{G})$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Deformations of ringed spaces and the cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08UZ","source_file":"cotangent.tex","source_line":3619,"source_end_line":3634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3619-L3634","statement_sha256":"9dd3400748737c61e6d28a0511733b30b239d17707a4d2bd58fc49f32a851ae2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13524,"rank":13524,"depth":21,"x":1526.554,"y":1479.755,"cluster":"deformation-theory"},{"id":"stacks:08SU","tag":"08SU","title":"The cotangent complex of a morphism of ringed topoi · Definition 08SU","summary":"Let (f, f^sharp) : (Sh(C), O_C) → (Sh(D), O_D) be a morphism of ringed topoi. The cotangent complex L_f of f is L_f = L_O_C/f^-1O_D. We sometimes write L_f = L_O_C/O_D.","statement_latex":"Let $(f, f^\\sharp) : (\\Sh(\\mathcal{C}), \\mathcal{O}_\\mathcal{C}) \\to\n(\\Sh(\\mathcal{D}), \\mathcal{O}_\\mathcal{D})$ be a morphism of ringed topoi.\nThe {\\it cotangent complex} $L_f$ of $f$ is\n$L_f = L_{\\mathcal{O}_\\mathcal{C}/f^{-1}\\mathcal{O}_\\mathcal{D}}$.\nWe sometimes write $L_f = L_{\\mathcal{O}_\\mathcal{C}/\\mathcal{O}_\\mathcal{D}}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of ringed topoi","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08SU","source_file":"cotangent.tex","source_line":3694,"source_end_line":3701,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3694-L3701","statement_sha256":"5119f32d830ffafb0ec8fefd874a68653eda6357a1d7c838f073d006c76098bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13525,"rank":13525,"depth":0,"x":1716.365,"y":1658.451,"cluster":"deformation-theory"},{"id":"stacks:08V0","tag":"08V0","title":"The cotangent complex of a morphism of ringed topoi · Lemma 08V0","summary":"Let f : (Sh(C), O) → (Sh(B), O_B) be a morphism of ringed topoi. Then H^0(L_f) = Ω_f.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to\n(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ be a morphism of\nringed topoi. Then $H^0(L_f) = \\Omega_f$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08V0","source_file":"cotangent.tex","source_line":3716,"source_end_line":3721,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3716-L3721","statement_sha256":"dccfce94d3d08e93119400392b5795de83d739ea9914819678fed453abab9845","origin":"The Stacks Project","memory_eligible":false,"source_rank":13526,"rank":13526,"depth":17,"x":1432.164,"y":1634.341,"cluster":"deformation-theory"},{"id":"stacks:08V1","tag":"08V1","title":"The cotangent complex of a morphism of ringed topoi · Lemma 08V1","summary":"Let f : (Sh(C_1), O_1) → (Sh(C_2), O_2) and g : (Sh(C_2), O_2) → (Sh(C_3), O_3) be morphisms of ringed topoi. Then there is a canonical distinguished triangle Lf^* L_g → L_g ∘ f → L_f → Lf^*L_g[1] in D(O_1).","statement_latex":"Let $f : (\\Sh(\\mathcal{C}_1), \\mathcal{O}_1) \\to\n(\\Sh(\\mathcal{C}_2), \\mathcal{O}_2)$ and\n$g : (\\Sh(\\mathcal{C}_2), \\mathcal{O}_2) \\to\n(\\Sh(\\mathcal{C}_3), \\mathcal{O}_3)$ be morphisms of ringed topoi.\nThen there is a canonical distinguished triangle\n$$\nLf^* L_g \\to L_{g \\circ f} \\to L_f \\to Lf^*L_g[1]\n$$\nin $D(\\mathcal{O}_1)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08V1","source_file":"cotangent.tex","source_line":3727,"source_end_line":3738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3727-L3738","statement_sha256":"4a09da47f499597504bee551e3220de46328e573f80dbdd36c256d60b17612df","origin":"The Stacks Project","memory_eligible":false,"source_rank":13527,"rank":13527,"depth":20,"x":1661.548,"y":1490.587,"cluster":"deformation-theory"},{"id":"stacks:08V2","tag":"08V2","title":"The cotangent complex of a morphism of ringed topoi · Lemma 08V2","summary":"Let f : (Sh(C), O) → (Sh(B), O_B) be a morphism of ringed topoi. There is a canonical map L_f → NL_f which identifies the naive cotangent complex with the truncation τ_≥ -1L_f.","statement_latex":"Let $f : (\\Sh(\\mathcal{C}), \\mathcal{O}) \\to\n(\\Sh(\\mathcal{B}), \\mathcal{O}_\\mathcal{B})$ be a morphism of\nringed topoi. There is a canonical map $L_f \\to \\NL_f$ which\nidentifies the naive cotangent complex with the truncation\n$\\tau_{\\geq -1}L_f$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of ringed topoi","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08V2","source_file":"cotangent.tex","source_line":3752,"source_end_line":3759,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3752-L3759","statement_sha256":"21fc8bc3e5eacf3b560aae4ac57c7f721afc682617999a6b293b04819ecf6604","origin":"The Stacks Project","memory_eligible":false,"source_rank":13528,"rank":13528,"depth":4,"x":1607.906,"y":1727.189,"cluster":"deformation-theory"},{"id":"stacks:08V5","tag":"08V5","title":"Deformations of ringed topoi and the cotangent complex · Lemma 08V5","summary":"In the situation above we have • There is a canonical element xi ∈ Ext^2_O(L_f, G) whose vanishing is a sufficient and necessary condition for the existence of a solution to ([Tag 08V4]). • If there exists a solution, then the set of isomorphism classes of solutions is principal homogeneous under Ext^1_O(L_f, G). • Given a solution X', the set of automorphisms of X' fitting into ([Tag 08V4]) is canonically isomorphic to Ext^0_O(L_f, G).","statement_latex":"In the situation above we have\n\\begin{enumerate}\n\\item There is a canonical element\n$\\xi \\in \\Ext^2_\\mathcal{O}(L_f, \\mathcal{G})$\nwhose vanishing is a sufficient and necessary condition for the existence\nof a solution to (\\ref{equation-to-solve-ringed-topoi}).\n\\item If there exists a solution, then the set of\nisomorphism classes of solutions is principal homogeneous under\n$\\Ext^1_\\mathcal{O}(L_f, \\mathcal{G})$.\n\\item Given a solution $X'$, the set of automorphisms of $X'$\nfitting into (\\ref{equation-to-solve-ringed-topoi}) is canonically isomorphic\nto $\\Ext^0_\\mathcal{O}(L_f, \\mathcal{G})$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Deformations of ringed topoi and the cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08V5","source_file":"cotangent.tex","source_line":3811,"source_end_line":3826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3811-L3826","statement_sha256":"170e9ad44c9548d87dc19e36d26d2a1d597a8ad98059745bfd1b0b2a12a55cf6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13529,"rank":13529,"depth":21,"x":1456.912,"y":1521.902,"cluster":"deformation-theory"},{"id":"stacks:08T2","tag":"08T2","title":"The cotangent complex of a morphism of schemes · Definition 08T2","summary":"Let f : X → Y be a morphism of schemes. The cotangent complex L_X/Y of X over Y is the cotangent complex of f as a morphism of ringed spaces (Definition [Tag 08UU]).","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. The {\\it cotangent complex\n$L_{X/Y}$ of $X$ over $Y$} is the cotangent complex of $f$ as a\nmorphism of ringed spaces\n(Definition \\ref{definition-cotangent-complex-morphism-ringed-spaces}).","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08T2","source_file":"cotangent.tex","source_line":3896,"source_end_line":3902,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3896-L3902","statement_sha256":"29b892872cc9e729005ed009c4e4e29481d1c8834da8ab418cdb008f9e5de2ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":13530,"rank":13530,"depth":1,"x":1733.852,"y":1587.722,"cluster":"deformation-theory"},{"id":"stacks:08T3","tag":"08T3","title":"The cotangent complex of a morphism of schemes · Lemma 08T3","summary":"Let f : X → Y be a morphism of schemes. Let U = Spec(B) ⊂ X and V = Spec(A) ⊂ Y be affine opens such that f(U) ⊂ V. There is a canonical map widetildeL_B/A → L_X/Y|_U of complexes which is an isomorphism in D(O_U). This map is compatible with restricting to smaller affine opens of X and Y.","statement_latex":"Let $f : X \\to Y$ be a morphism of schemes. Let $U = \\Spec(B) \\subset X$\nand $V = \\Spec(A) \\subset Y$ be affine opens such that $f(U) \\subset V$.\nThere is a canonical map\n$$\n\\widetilde{L_{B/A}} \\longrightarrow L_{X/Y}|_U\n$$\nof complexes which is an isomorphism in $D(\\mathcal{O}_U)$.\nThis map is compatible with restricting to smaller affine opens\nof $X$ and $Y$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08T3","source_file":"cotangent.tex","source_line":3912,"source_end_line":3923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3912-L3923","statement_sha256":"98fad92c1168317d8b1d5af48993d42fbd59e6fea337319fd53ce3453e7aaa2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13531,"rank":13531,"depth":17,"x":1476.235,"y":1696.53,"cluster":"deformation-theory"},{"id":"stacks:08V6","tag":"08V6","title":"The cotangent complex of a morphism of schemes · Lemma 08V6","summary":"Let Lambda be a ring. Let X be a scheme over Lambda. Then L_X/Spec(Lambda) = L_O_X/underlineLambda where underlineLambda is the constant sheaf with value Lambda on X.","statement_latex":"Let $\\Lambda$ be a ring. Let $X$ be a scheme over $\\Lambda$.\nThen\n$$\nL_{X/\\Spec(\\Lambda)} = L_{\\mathcal{O}_X/\\underline{\\Lambda}}\n$$\nwhere $\\underline{\\Lambda}$ is the constant sheaf with value\n$\\Lambda$ on $X$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08V6","source_file":"cotangent.tex","source_line":3950,"source_end_line":3959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L3950-L3959","statement_sha256":"79d810fe60b7ccac8713d4c5c10fb9570990a0620e04c12f0e6d675bb1301c9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13532,"rank":13532,"depth":47,"x":1578.886,"y":1469.702,"cluster":"deformation-theory"},{"id":"stacks:08V9","tag":"08V9","title":"The cotangent complex of a scheme over a ring · Lemma 08V9","summary":"In the situation above the category C_X/Lambda is fibred over X_Zar.","statement_latex":"In the situation above the category\n$\\mathcal{C}_{X/\\Lambda}$ is fibred over $X_{Zar}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a scheme over a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08V9","source_file":"cotangent.tex","source_line":4032,"source_end_line":4036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4032-L4036","statement_sha256":"3c830b80ad47937e0ed1d49fcb148b9a8c250af54546c1b378ed82fa9a2acb76","origin":"The Stacks Project","memory_eligible":false,"source_rank":13533,"rank":13533,"depth":0,"x":1685.797,"y":1695.623,"cluster":"deformation-theory"},{"id":"stacks:08T9","tag":"08T9","title":"The cotangent complex of a scheme over a ring · Lemma 08T9","summary":"In the situation above there is a canonical isomorphism L_X/Lambda = Lπ_!(Li^*Ω_O/underlineLambda) = Lπ_!(i^*Ω_O/underlineLambda) = Lπ_!(Ω_O/underlineLambda ⊗_O underlineO_X) in D(O_X).","statement_latex":"In the situation above there is a canonical isomorphism\n$$\nL_{X/\\Lambda} = \nL\\pi_!(Li^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) =\nL\\pi_!(i^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) =\nL\\pi_!(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}\n\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X)\n$$\nin $D(\\mathcal{O}_X)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a scheme over a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08T9","source_file":"cotangent.tex","source_line":4123,"source_end_line":4134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4123-L4134","statement_sha256":"f4f6c37fc29561cdfcca5b1b2f7d2c51765be60ecc8e355b7eb1d44e047fc604","origin":"The Stacks Project","memory_eligible":false,"source_rank":13534,"rank":13534,"depth":48,"x":1424.807,"y":1589.502,"cluster":"deformation-theory"},{"id":"stacks:08VD","tag":"08VD","title":"The cotangent complex of a morphism of algebraic spaces · Definition 08VD","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. The cotangent complex L_X/Y of X over Y is the cotangent complex of the morphism of ringed topoi f_small between the small étale sites of X and Y (see Properties of Spaces, Lemma [Tag 03G8] and Definition [Tag 08SU]).","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. The {\\it cotangent complex $L_{X/Y}$ of $X$ over $Y$} is the\ncotangent complex of the morphism of ringed topoi $f_{small}$\nbetween the small \\'etale sites of $X$ and $Y$\n(see\nProperties of Spaces, Lemma\n\\ref{spaces-properties-lemma-morphism-ringed-topoi}\nand\nDefinition \\ref{definition-cotangent-complex-morphism-ringed-topoi}).","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VD","source_file":"cotangent.tex","source_line":4180,"source_end_line":4191,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4180-L4191","statement_sha256":"b37d92ab08fe52cd93ebffb75e886c70b49a6d743e14cc74cc691834ea63f5e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13535,"rank":13535,"depth":11,"x":1703.103,"y":1519.535,"cluster":"deformation-theory"},{"id":"stacks:08VE","tag":"08VE","title":"The cotangent complex of a morphism of algebraic spaces · Lemma 08VE","summary":"Let S be a scheme. Consider a commutative diagram xymatrix U ar[d]_p ar[r]_g & V ar[d]^q X ar[r]^f & Y of algebraic spaces over S with p and q étale. Then there is a canonical identification L_X/Y|_U_etale = L_U/V in D(O_U).","statement_latex":"Let $S$ be a scheme. Consider a commutative diagram\n$$\n\\xymatrix{\nU \\ar[d]_p \\ar[r]_g & V \\ar[d]^q \\\\\nX \\ar[r]^f & Y\n}\n$$\nof algebraic spaces over $S$ with $p$ and $q$ \\'etale.\nThen there is a canonical identification\n$L_{X/Y}|_{U_\\etale} = L_{U/V}$ in $D(\\mathcal{O}_U)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VE","source_file":"cotangent.tex","source_line":4201,"source_end_line":4213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4201-L4213","statement_sha256":"72e4d1c8c3612cf70dcd33143389d87ad5135b39e8e62fcc17308b7f6e7a26b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13536,"rank":13536,"depth":49,"x":1553.887,"y":1729.419,"cluster":"deformation-theory"},{"id":"stacks:08VF","tag":"08VF","title":"The cotangent complex of a morphism of algebraic spaces · Lemma 08VF","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume X and Y representable by schemes X_0 and Y_0. Then there is a canonical identification L_X/Y = ε^*L_X_0/Y_0 in D(O_X) where ε is as in Derived Categories of Spaces, Section [Tag 071P] and L_X_0/Y_0 is as in Definition [Tag 08T2].","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces\nover $S$. Assume $X$ and $Y$ representable by schemes $X_0$ and $Y_0$.\nThen there is a canonical identification\n$L_{X/Y} = \\epsilon^*L_{X_0/Y_0}$ in $D(\\mathcal{O}_X)$\nwhere $\\epsilon$ is as in Derived Categories of Spaces, Section\n\\ref{spaces-perfect-section-derived-quasi-coherent-etale}\nand $L_{X_0/Y_0}$ is as in\n Definition \\ref{definition-cotangent-morphism-schemes}.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VF","source_file":"cotangent.tex","source_line":4229,"source_end_line":4239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4229-L4239","statement_sha256":"08ade8d189da790b67392445c7f4fd381a863a0286faa96953f7fda3049b3bd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13537,"rank":13537,"depth":53,"x":1495.027,"y":1489.552,"cluster":"deformation-theory"},{"id":"stacks:08VG","tag":"08VG","title":"The cotangent complex of a morphism of algebraic spaces · Lemma 08VG","summary":"Let Lambda be a ring. Let X be an algebraic space over Lambda. Then L_X/Spec(Lambda) = L_O_X/underlineLambda where underlineLambda is the constant sheaf with value Lambda on X_etale.","statement_latex":"Let $\\Lambda$ be a ring. Let $X$ be an algebraic space over $\\Lambda$.\nThen\n$$\nL_{X/\\Spec(\\Lambda)} = L_{\\mathcal{O}_X/\\underline{\\Lambda}}\n$$\nwhere $\\underline{\\Lambda}$ is the constant sheaf with value\n$\\Lambda$ on $X_\\etale$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of a morphism of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VG","source_file":"cotangent.tex","source_line":4300,"source_end_line":4309,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4300-L4309","statement_sha256":"9eaed502012eed170be8792be681e2fd8fe2ea39179118da833e50cfb8517a19","origin":"The Stacks Project","memory_eligible":false,"source_rank":13538,"rank":13538,"depth":53,"x":1731.749,"y":1633.287,"cluster":"deformation-theory"},{"id":"stacks:08VJ","tag":"08VJ","title":"The cotangent complex of an algebraic space over a ring · Lemma 08VJ","summary":"In the situation above the category C_X/Lambda is fibred over X_etale.","statement_latex":"In the situation above the category\n$\\mathcal{C}_{X/\\Lambda}$ is fibred over $X_\\etale$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of an algebraic space over a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VJ","source_file":"cotangent.tex","source_line":4397,"source_end_line":4401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4397-L4401","statement_sha256":"2f961bb712e9dce0daa50132b3cfa6af66a256f38081b5694fe69908501fac04","origin":"The Stacks Project","memory_eligible":false,"source_rank":13539,"rank":13539,"depth":0,"x":1441.085,"y":1661.671,"cluster":"deformation-theory"},{"id":"stacks:08VM","tag":"08VM","title":"The cotangent complex of an algebraic space over a ring · Lemma 08VM","summary":"In the situation above there is a canonical isomorphism L_X/Lambda = Lπ_!(Li^*Ω_O/underlineLambda) = Lπ_!(i^*Ω_O/underlineLambda) = Lπ_!(Ω_O/underlineLambda ⊗_O underlineO_X) in D(O_X).","statement_latex":"In the situation above there is a canonical isomorphism\n$$\nL_{X/\\Lambda} = \nL\\pi_!(Li^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) =\nL\\pi_!(i^*\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}) =\nL\\pi_!(\\Omega_{\\mathcal{O}/\\underline{\\Lambda}}\n\\otimes_\\mathcal{O} \\underline{\\mathcal{O}}_X)\n$$\nin $D(\\mathcal{O}_X)$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"The cotangent complex of an algebraic space over a ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08VM","source_file":"cotangent.tex","source_line":4488,"source_end_line":4499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4488-L4499","statement_sha256":"f4f6c37fc29561cdfcca5b1b2f7d2c51765be60ecc8e355b7eb1d44e047fc604","origin":"The Stacks Project","memory_eligible":false,"source_rank":13540,"rank":13540,"depth":54,"x":1632.933,"y":1475.479,"cluster":"deformation-theory"},{"id":"stacks:09DL","tag":"09DL","title":"Fibre products of algebraic spaces and the cotangent complex · Lemma 09DL","summary":"In the situation above, if X and Y are Tor independent over B, then the object E in ([Tag 09DK]) is zero. In this case we have L_X ×_B Y/B = Lp^*L_X/B ⊕ Lq^*L_Y/B","statement_latex":"In the situation above, if $X$ and $Y$ are Tor independent over $B$, then\nthe object $E$ in (\\ref{equation-fibre-product}) is zero. In this case we\nhave\n$$\nL_{X \\times_B Y/B} = Lp^*L_{X/B} \\oplus Lq^*L_{Y/B}\n$$","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Fibre products of algebraic spaces and the cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DL","source_file":"cotangent.tex","source_line":4579,"source_end_line":4587,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4579-L4587","statement_sha256":"900118ce8fcd59a9135b5d864075282521ba6996e3b2df7f7a67aa4c95b7163f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13541,"rank":13541,"depth":54,"x":1641.213,"y":1722.072,"cluster":"deformation-theory"},{"id":"stacks:09DM","tag":"09DM","title":"Fibre products of algebraic spaces and the cotangent complex · Lemma 09DM","summary":"Let S be a scheme. Let X → B and Y → B be morphisms of algebraic spaces over S. The object E in ([Tag 09DK]) satisfies H^i(E) = 0 for i = 0, -1 and for a geometric point (overlinex, overliney) : Spec(k) → X ×_B Y we have H^-2(E)_(overlinex, overliney) = Tor_1^R(A, B) ⊗_A ⊗_R B C where R = O_B, overlineb, A = O_X, overlinex, B = O_Y, overliney, and C = O_X ×_B Y, (overlinex, overliney).","statement_latex":"Let $S$ be a scheme. Let $X \\to B$ and $Y \\to B$ be morphisms of algebraic\nspaces over $S$. The object $E$ in (\\ref{equation-fibre-product}) satisfies\n$H^i(E) = 0$ for $i = 0, -1$ and for a geometric point\n$(\\overline{x}, \\overline{y}) : \\Spec(k) \\to X \\times_B Y$ we have\n$$\nH^{-2}(E)_{(\\overline{x}, \\overline{y})} =\n\\text{Tor}_1^R(A, B) \\otimes_{A \\otimes_R B} C\n$$\nwhere $R = \\mathcal{O}_{B, \\overline{b}}$, $A = \\mathcal{O}_{X, \\overline{x}}$,\n$B = \\mathcal{O}_{Y, \\overline{y}}$, and\n$C = \\mathcal{O}_{X \\times_B Y, (\\overline{x}, \\overline{y})}$.","area":"Deformation Theory","chapter":"The Cotangent Complex","chapter_id":"cotangent","section":"Fibre products of algebraic spaces and the cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DM","source_file":"cotangent.tex","source_line":4608,"source_end_line":4621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/cotangent.tex#L4608-L4621","statement_sha256":"f609ddffc05f5e51ca914a0c576205c007331a42e08eeb2de16edd6bda44ccdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13542,"rank":13542,"depth":19,"x":1436.442,"y":1544.621,"cluster":"deformation-theory"},{"id":"stacks:0DVP","tag":"0DVP","title":"Finite projective modules · Lemma 0DVP","summary":"Example [Tag 0D3I] satisfies the Rim-Schlessinger condition (RS). In particular, Deformationcategory_V is a deformation category for any finite dimensional vector space V over k.","statement_latex":"Example \\ref{example-finite-projective-modules}\nsatisfies the Rim-Schlessinger condition (RS).\nIn particular, $\\Deformationcategory_V$ is a deformation category\nfor any finite dimensional vector space $V$ over $k$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVP","source_file":"examples-defos.tex","source_line":178,"source_end_line":184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L178-L184","statement_sha256":"63b78ef657f1debb4cc3addf4714072c5e920ea7e0b968d75c5b145fc4d1c975","origin":"The Stacks Project","memory_eligible":false,"source_rank":13543,"rank":13543,"depth":11,"x":1730.657,"y":1559.305,"cluster":"deformation-theory"},{"id":"stacks:0DVQ","tag":"0DVQ","title":"Finite projective modules · Lemma 0DVQ","summary":"In Example [Tag 0D3I] let V be a finite dimensional k-vector space. Then TDeformationcategory_V = (0) and Inf(Deformationcategory_V) = End_k(V) are finite dimensional.","statement_latex":"In Example \\ref{example-finite-projective-modules}\nlet $V$ be a finite dimensional $k$-vector space. Then\n$$\nT\\Deformationcategory_V = (0)\n\\quad\\text{and}\\quad\n\\text{Inf}(\\Deformationcategory_V) = \\text{End}_k(V)\n$$\nare finite dimensional.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Finite projective modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVQ","source_file":"examples-defos.tex","source_line":212,"source_end_line":222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L212-L222","statement_sha256":"0ae949d72c0b947e43eb4b2dca5b724ccf3095325ce38b6c1f82282f77c57b37","origin":"The Stacks Project","memory_eligible":false,"source_rank":13544,"rank":13544,"depth":1,"x":1501.496,"y":1715.698,"cluster":"deformation-theory"},{"id":"stacks:0DVS","tag":"0DVS","title":"Representations of a group · Lemma 0DVS","summary":"Example [Tag 0D3J] satisfies the Rim-Schlessinger condition (RS). In particular, Deformationcategory_V, ρ_0 is a deformation category for any finite dimensional representation ρ_0 : Γ → GL_k(V).","statement_latex":"Example \\ref{example-representations}\nsatisfies the Rim-Schlessinger condition (RS).\nIn particular, $\\Deformationcategory_{V, \\rho_0}$ is a deformation category\nfor any finite dimensional representation\n$\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Representations of a group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVS","source_file":"examples-defos.tex","source_line":308,"source_end_line":315,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L308-L315","statement_sha256":"caf6eec964cc5f86a2a6764ec4ea47420ebba179ba25764959e3c6a1915f2a12","origin":"The Stacks Project","memory_eligible":false,"source_rank":13545,"rank":13545,"depth":12,"x":1544.786,"y":1469.913,"cluster":"deformation-theory"},{"id":"stacks:0DVT","tag":"0DVT","title":"Representations of a group · Lemma 0DVT","summary":"In Example [Tag 0D3J] let ρ_0 : Γ → GL_k(V) be a finite dimensional representation. Then TDeformationcategory_V, ρ_0 = Ext^1_k[Γ](V, V) = H^1(Γ, End_k(V)) and Inf(Deformationcategory_V, ρ_0) = H^0(Γ, End_k(V)) Thus Inf(Deformationcategory_V, ρ_0) is always finite dimensional and TDeformationcategory_V, ρ_0 is finite dimensional if Γ is finitely generated.","statement_latex":"In Example \\ref{example-representations} let \n$\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$\nbe a finite dimensional representation. Then\n$$\nT\\Deformationcategory_{V, \\rho_0} = \\Ext^1_{k[\\Gamma]}(V, V) =\nH^1(\\Gamma, \\text{End}_k(V))\n\\quad\\text{and}\\quad\n\\text{Inf}(\\Deformationcategory_{V, \\rho_0}) = H^0(\\Gamma, \\text{End}_k(V))\n$$\nThus $\\text{Inf}(\\Deformationcategory_{V, \\rho_0})$\nis always finite dimensional\nand $T\\Deformationcategory_{V, \\rho_0}$ is finite dimensional\nif $\\Gamma$ is finitely generated.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Representations of a group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVT","source_file":"examples-defos.tex","source_line":346,"source_end_line":361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L346-L361","statement_sha256":"5b3900f6dc7bb763e972bb286d377dd5a542bdcaf1e0ae7d57976bf8756eb7a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13546,"rank":13546,"depth":8,"x":1710.807,"y":1676.075,"cluster":"deformation-theory"},{"id":"stacks:0ET1","tag":"0ET1","title":"Representations of a group · Lemma 0ET1","summary":"In Example [Tag 0D3J] assume Γ finitely generated. Let ρ_0 : Γ → GL_k(V) be a finite dimensional representation. Assume Lambda is a complete local ring with residue field k (the classical case). Then the functor F : C_Lambda → Sets, A ↦ Ob(Deformationcategory_V, ρ_0(A))/≅ of isomorphism classes of objects has a hull. If H^0(Γ, End_k(V)) = k, then F is prorepresentable.","statement_latex":"In Example \\ref{example-representations} assume $\\Gamma$ finitely generated.\nLet $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional representation.\nAssume $\\Lambda$ is a complete local ring with residue field $k$\n(the classical case). Then the functor\n$$\nF : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad\nA \\longmapsto \\Ob(\\Deformationcategory_{V, \\rho_0}(A))/\\cong\n$$\nof isomorphism classes of objects has a hull. If\n$H^0(\\Gamma, \\text{End}_k(V)) = k$, then $F$ is\nprorepresentable.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Representations of a group","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET1","source_file":"examples-defos.tex","source_line":418,"source_end_line":431,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L418-L431","statement_sha256":"05b2a5198791439a265d897e0b103b4922451d30fc5b3eb1a811d70e9a141df0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13547,"rank":13547,"depth":51,"x":1422.093,"y":1618.158,"cluster":"deformation-theory"},{"id":"stacks:0DVV","tag":"0DVV","title":"Continuous representations · Lemma 0DVV","summary":"Example [Tag 0D3K] satisfies the Rim-Schlessinger condition (RS). In particular, Deformationcategory_V, ρ_0 is a deformation category for any finite dimensional continuous representation ρ_0 : Γ → GL_k(V).","statement_latex":"Example \\ref{example-continuous-representations}\nsatisfies the Rim-Schlessinger condition (RS).\nIn particular, $\\Deformationcategory_{V, \\rho_0}$ is a deformation category\nfor any finite dimensional continuous representation\n$\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Continuous representations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVV","source_file":"examples-defos.tex","source_line":528,"source_end_line":535,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L528-L535","statement_sha256":"c062f4d71668bb35bfc25c6635ca353722ca946dbe7db9bf63f7d05d479e53d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13548,"rank":13548,"depth":13,"x":1682.013,"y":1496.832,"cluster":"deformation-theory"},{"id":"stacks:0DVW","tag":"0DVW","title":"Continuous representations · Lemma 0DVW","summary":"In Example [Tag 0D3K] let ρ_0 : Γ → GL_k(V) be a finite dimensional continuous representation. Then TDeformationcategory_V, ρ_0 = H^1(Γ, End_k(V)) and Inf(Deformationcategory_V, ρ_0) = H^0(Γ, End_k(V)) Thus Inf(Deformationcategory_V, ρ_0) is always finite dimensional and TDeformationcategory_V, ρ_0 is finite dimensional if Γ is topologically finitely generated.","statement_latex":"In Example \\ref{example-continuous-representations} let\n$\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional\ncontinuous representation. Then\n$$\nT\\Deformationcategory_{V, \\rho_0} = H^1(\\Gamma, \\text{End}_k(V))\n\\quad\\text{and}\\quad\n\\text{Inf}(\\Deformationcategory_{V, \\rho_0}) = H^0(\\Gamma, \\text{End}_k(V))\n$$\nThus $\\text{Inf}(\\Deformationcategory_{V, \\rho_0})$\nis always finite dimensional\nand $T\\Deformationcategory_{V, \\rho_0}$ is finite dimensional\nif $\\Gamma$ is topologically finitely generated.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Continuous representations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVW","source_file":"examples-defos.tex","source_line":542,"source_end_line":556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L542-L556","statement_sha256":"b4ce55e8ffa563db179a9e499c90bc8260836264b00855936ca644a0093923b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13549,"rank":13549,"depth":9,"x":1587.756,"y":1734.19,"cluster":"deformation-theory"},{"id":"stacks:0ET2","tag":"0ET2","title":"Continuous representations · Lemma 0ET2","summary":"In Example [Tag 0D3K] assume Γ is topologically finitely generated. Let ρ_0 : Γ → GL_k(V) be a finite dimensional representation. Assume Lambda is a complete local ring with residue field k (the classical case). Then the functor F : C_Lambda → Sets, A ↦ Ob(Deformationcategory_V, ρ_0(A))/≅ of isomorphism classes of objects has a hull. If H^0(Γ, End_k(V)) = k, then F is prorepresentable.","statement_latex":"In Example \\ref{example-continuous-representations} assume $\\Gamma$\nis topologically finitely generated.\nLet $\\rho_0 : \\Gamma \\to \\text{GL}_k(V)$ be a finite dimensional representation.\nAssume $\\Lambda$ is a complete local ring with residue field $k$\n(the classical case). Then the functor\n$$\nF : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad\nA \\longmapsto \\Ob(\\Deformationcategory_{V, \\rho_0}(A))/\\cong\n$$\nof isomorphism classes of objects has a hull. If\n$H^0(\\Gamma, \\text{End}_k(V)) = k$, then $F$ is\nprorepresentable.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Continuous representations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET2","source_file":"examples-defos.tex","source_line":575,"source_end_line":589,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L575-L589","statement_sha256":"b3570c3a448b9f1fdc82e3d47c2f82ad2c95229fd658414eab8c3ae16dc93020","origin":"The Stacks Project","memory_eligible":false,"source_rank":13550,"rank":13550,"depth":52,"x":1466.173,"y":1505.288,"cluster":"deformation-theory"},{"id":"stacks:0DVY","tag":"0DVY","title":"Graded algebras · Lemma 0DVY","summary":"Example [Tag 0D3L] satisfies the Rim-Schlessinger condition (RS). In particular, Deformationcategory_P is a deformation category for any graded k-algebra P.","statement_latex":"Example \\ref{example-graded-algebras}\nsatisfies the Rim-Schlessinger condition (RS).\nIn particular, $\\Deformationcategory_P$ is a deformation category\nfor any graded $k$-algebra $P$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVY","source_file":"examples-defos.tex","source_line":629,"source_end_line":635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L629-L635","statement_sha256":"11566e2deb4b15d096de49bb1babfbb35ab54d26e8b0cb0e88785a78c3465fc6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13551,"rank":13551,"depth":12,"x":1740.374,"y":1605.26,"cluster":"deformation-theory"},{"id":"stacks:0DVZ","tag":"0DVZ","title":"Graded algebras · Lemma 0DVZ","summary":"In Example [Tag 0D3L] let P be a graded k-algebra. Then TDeformationcategory_P and Inf(Deformationcategory_P) = Der_k(P, P) are finite dimensional if P is finitely generated over k.","statement_latex":"In Example \\ref{example-graded-algebras} let $P$ be a graded $k$-algebra.\nThen\n$$\nT\\Deformationcategory_P\n\\quad\\text{and}\\quad\n\\text{Inf}(\\Deformationcategory_P) = \\text{Der}_k(P, P)\n$$\nare finite dimensional if $P$ is finitely generated over $k$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DVZ","source_file":"examples-defos.tex","source_line":663,"source_end_line":673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L663-L673","statement_sha256":"26f832a9a7afdb271ff285e711ed08a083958b62bbf7cd130d92b0d4346dbc89","origin":"The Stacks Project","memory_eligible":false,"source_rank":13552,"rank":13552,"depth":0,"x":1457.303,"y":1687.27,"cluster":"deformation-theory"},{"id":"stacks:0ET3","tag":"0ET3","title":"Graded algebras · Lemma 0ET3","summary":"In Example [Tag 0D3L] assume P is a finitely generated graded k-algebra. Assume Lambda is a complete local ring with residue field k (the classical case). Then the functor F : C_Lambda → Sets, A ↦ Ob(Deformationcategory_P(A))/≅ of isomorphism classes of objects has a hull.","statement_latex":"In Example \\ref{example-graded-algebras} assume $P$ is a finitely generated\ngraded $k$-algebra. Assume $\\Lambda$ is a complete local ring\nwith residue field $k$\n(the classical case). Then the functor\n$$\nF : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad\nA \\longmapsto \\Ob(\\Deformationcategory_P(A))/\\cong\n$$\nof isomorphism classes of objects has a hull.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Graded algebras","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET3","source_file":"examples-defos.tex","source_line":779,"source_end_line":790,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L779-L790","statement_sha256":"893c840feca4070fb1cc5e0920dcab7020cd20bb75a6415ca99627722cc9b078","origin":"The Stacks Project","memory_eligible":false,"source_rank":13553,"rank":13553,"depth":13,"x":1600.329,"y":1465.799,"cluster":"deformation-theory"},{"id":"stacks:0DY2","tag":"0DY2","title":"Rings · Lemma 0DY2","summary":"Example [Tag 0DY1] satisfies the Rim-Schlessinger condition (RS). In particular, Deformationcategory_P is a deformation category for any k-algebra P.","statement_latex":"Example \\ref{example-rings}\nsatisfies the Rim-Schlessinger condition (RS).\nIn particular, $\\Deformationcategory_P$ is a deformation category\nfor any $k$-algebra $P$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DY2","source_file":"examples-defos.tex","source_line":833,"source_end_line":839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L833-L839","statement_sha256":"25e519f95188524341c0b9394d288f754ecc4eba31a11892e9076b15c4c6d8ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":13554,"rank":13554,"depth":39,"x":1673.088,"y":1710.68,"cluster":"deformation-theory"},{"id":"stacks:0DY3","tag":"0DY3","title":"Rings · Lemma 0DY3","summary":"In Example [Tag 0DY1] let P be a k-algebra. Then TDeformationcategory_P = Ext^1_P(NL_P/k, P) and Inf(Deformationcategory_P) = Der_k(P, P)","statement_latex":"In Example \\ref{example-rings} let $P$ be a $k$-algebra. Then\n$$\nT\\Deformationcategory_P = \\text{Ext}^1_P(\\NL_{P/k}, P)\n\\quad\\text{and}\\quad\n\\text{Inf}(\\Deformationcategory_P) = \\text{Der}_k(P, P)\n$$","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DY3","source_file":"examples-defos.tex","source_line":854,"source_end_line":862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L854-L862","statement_sha256":"9b2b22b3a4e55eaeaba20cfb1389519ab07143c32073820255f4db36bc7695b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13555,"rank":13555,"depth":5,"x":1422.085,"y":1571.158,"cluster":"deformation-theory"},{"id":"stacks:0DZL","tag":"0DZL","title":"Rings · Lemma 0DZL","summary":"In Example [Tag 0DY1] let P be a smooth k-algebra. Then TDeformationcategory_P = (0).","statement_latex":"In Example \\ref{example-rings} let $P$ be a smooth $k$-algebra. Then\n$T\\Deformationcategory_P = (0)$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZL","source_file":"examples-defos.tex","source_line":892,"source_end_line":896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L892-L896","statement_sha256":"362eac2137d86770a809e6c0b80986c8e9a98d279b7aac2424f36400f17f875a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13556,"rank":13556,"depth":6,"x":1719.874,"y":1531.548,"cluster":"deformation-theory"},{"id":"stacks:0DY4","tag":"0DY4","title":"Rings · Lemma 0DY4","summary":"In Lemma [Tag 0DY3] if P is a finite type k-algebra, then • Inf(Deformationcategory_P) is finite dimensional if and only if dim(P) = 0, and • TDeformationcategory_P is finite dimensional if Spec(P) → Spec(k) is smooth except at a finite number of points.","statement_latex":"In Lemma \\ref{lemma-rings-TI} if $P$ is a finite type $k$-algebra, then\n\\begin{enumerate}\n\\item $\\text{Inf}(\\Deformationcategory_P)$ is finite dimensional if and only if\n$\\dim(P) = 0$, and\n\\item $T\\Deformationcategory_P$ is finite dimensional if\n$\\Spec(P) \\to \\Spec(k)$ is smooth except at a finite number of points.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DY4","source_file":"examples-defos.tex","source_line":906,"source_end_line":915,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L906-L915","statement_sha256":"37321a65986cfe32300d32ef95bbb9bf17ad0722056e5d53e56c2fb2cba5e5f4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13557,"rank":13557,"depth":37,"x":1531.826,"y":1730.061,"cluster":"deformation-theory"},{"id":"stacks:0ET4","tag":"0ET4","title":"Rings · Lemma 0ET4","summary":"In Example [Tag 0DY1] assume P is a finite type k-algebra such that Spec(P) → Spec(k) is smooth except at a finite number of points. Assume Lambda is a complete local ring with residue field k (the classical case). Then the functor F : C_Lambda → Sets, A ↦ Ob(Deformationcategory_P(A))/≅ of isomorphism classes of objects has a hull.","statement_latex":"In Example \\ref{example-rings} assume $P$ is a finite type\n$k$-algebra such that $\\Spec(P) \\to \\Spec(k)$ is smooth except\nat a finite number of points.\nAssume $\\Lambda$ is a complete local ring with residue field $k$\n(the classical case). Then the functor\n$$\nF : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad\nA \\longmapsto \\Ob(\\Deformationcategory_P(A))/\\cong\n$$\nof isomorphism classes of objects has a hull.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET4","source_file":"examples-defos.tex","source_line":991,"source_end_line":1003,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L991-L1003","statement_sha256":"66939f25c7ec1cba32a0b2aa859e29314278aa017ef9c054b6b66dc188a62ec1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13558,"rank":13558,"depth":40,"x":1510.815,"y":1476.553,"cluster":"deformation-theory"},{"id":"stacks:0DYS","tag":"0DYS","title":"Rings · Lemma 0DYS","summary":"In Example [Tag 0DY1] let P be a k-algebra. Let S ⊂ P be a multiplicative subset. There is a natural functor Deformationcategory_P → Deformationcategory_S^-1P of deformation categories.","statement_latex":"In Example \\ref{example-rings} let $P$ be a $k$-algebra.\nLet $S \\subset P$ be a multiplicative subset. There is a natural functor\n$$\n\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{S^{-1}P}\n$$\nof deformation categories.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYS","source_file":"examples-defos.tex","source_line":1012,"source_end_line":1020,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1012-L1020","statement_sha256":"113f6faa41a836ddca48699497b0bc27c37f85d4955e41e3bb44f91e9b59555e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13559,"rank":13559,"depth":0,"x":1730.539,"y":1651.857,"cluster":"deformation-theory"},{"id":"stacks:0DYT","tag":"0DYT","title":"Rings · Lemma 0DYT","summary":"In Example [Tag 0DY1] let P be a k-algebra. Let J ⊂ P be an ideal. Denote (P^h, J^h) the henselization of the pair (P, J). There is a natural functor Deformationcategory_P → Deformationcategory_P^h of deformation categories.","statement_latex":"In Example \\ref{example-rings} let $P$ be a $k$-algebra.\nLet $J \\subset P$ be an ideal.\nDenote $(P^h, J^h)$ the henselization of the pair $(P, J)$.\nThere is a natural functor\n$$\n\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^h}\n$$\nof deformation categories.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYT","source_file":"examples-defos.tex","source_line":1033,"source_end_line":1043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1033-L1043","statement_sha256":"14191ba8896829e84225064308b12312faffdca71df9532e51185b9ad05ae5ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":13560,"rank":13560,"depth":51,"x":1427.039,"y":1647.259,"cluster":"deformation-theory"},{"id":"stacks:0DYU","tag":"0DYU","title":"Rings · Lemma 0DYU","summary":"In Example [Tag 0DY1] let P be a k-algebra. Assume P is a local ring and let P^sh be a strict henselization of P. There is a natural functor Deformationcategory_P → Deformationcategory_P^sh of deformation categories.","statement_latex":"In Example \\ref{example-rings} let $P$ be a $k$-algebra.\nAssume $P$ is a local ring and let $P^{sh}$ be a strict henselization of $P$.\nThere is a natural functor\n$$\n\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^{sh}}\n$$\nof deformation categories.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYU","source_file":"examples-defos.tex","source_line":1062,"source_end_line":1071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1062-L1071","statement_sha256":"f0aa800455017e3a4646cb5ee32625bc4a973f34da05aa1541f47d82c61dbd01","origin":"The Stacks Project","memory_eligible":false,"source_rank":13561,"rank":13561,"depth":51,"x":1654.911,"y":1478.156,"cluster":"deformation-theory"},{"id":"stacks:0DYV","tag":"0DYV","title":"Rings · Lemma 0DYV","summary":"In Example [Tag 0DY1] let P be a k-algebra. Assume P Noetherian and let J ⊂ P be an ideal. Denote P^wedge the J-adic completion. There is a natural functor Deformationcategory_P → Deformationcategory_P^wedge of deformation categories.","statement_latex":"In Example \\ref{example-rings} let $P$ be a $k$-algebra.\nAssume $P$ Noetherian and let $J \\subset P$ be an ideal.\nDenote $P^\\wedge$ the $J$-adic completion.\nThere is a natural functor\n$$\n\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^\\wedge}\n$$\nof deformation categories.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYV","source_file":"examples-defos.tex","source_line":1091,"source_end_line":1101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1091-L1101","statement_sha256":"a09afd0661cb711c1b0c89ead950fc831cc9bfbc8376d5e6a54c357e03bf1bbf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13562,"rank":13562,"depth":6,"x":1622.816,"y":1732.571,"cluster":"deformation-theory"},{"id":"stacks:0DY5","tag":"0DY5","title":"Rings · Lemma 0DY5","summary":"In Lemma [Tag 0DY3] if P = k[[x_1, …, x_n]]/(f) for some nonzero f ∈ (x_1, …, x_n)^2, then • Inf(Deformationcategory_P) is finite dimensional if and only if n = 1, and • TDeformationcategory_P is finite dimensional if sqrt(f, ∂ f/∂ x_1, …, ∂ f/∂ x_n) = (x_1, …, x_n)","statement_latex":"In Lemma \\ref{lemma-rings-TI} if $P = k[[x_1, \\ldots, x_n]]/(f)$\nfor some nonzero $f \\in (x_1, \\ldots, x_n)^2$, then\n\\begin{enumerate}\n\\item $\\text{Inf}(\\Deformationcategory_P)$ is finite dimensional\nif and only if $n = 1$, and\n\\item $T\\Deformationcategory_P$ is finite dimensional if\n$$\n\\sqrt{(f, \\partial f/\\partial x_1, \\ldots,  \\partial f/\\partial x_n)} =\n(x_1, \\ldots, x_n)\n$$\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DY5","source_file":"examples-defos.tex","source_line":1124,"source_end_line":1137,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1124-L1137","statement_sha256":"492dced90675d31b857c6e7b5b797f8373f5a923591a39349160b83983710a46","origin":"The Stacks Project","memory_eligible":false,"source_rank":13563,"rank":13563,"depth":6,"x":1441.591,"y":1526.418,"cluster":"deformation-theory"},{"id":"stacks:0DY8","tag":"0DY8","title":"Schemes · Lemma 0DY8","summary":"Example [Tag 0DY7] satisfies the Rim-Schlessinger condition (RS). In particular, Deformationcategory_X is a deformation category for any scheme X over k.","statement_latex":"Example \\ref{example-schemes}\nsatisfies the Rim-Schlessinger condition (RS).\nIn particular, $\\Deformationcategory_X$ is a deformation category\nfor any scheme $X$ over $k$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DY8","source_file":"examples-defos.tex","source_line":1233,"source_end_line":1239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1233-L1239","statement_sha256":"1a3c8ac977c985af622424f4cc994aa297dd62ace5f5a20b9bb849786eddea6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13564,"rank":13564,"depth":45,"x":1741.498,"y":1575.682,"cluster":"deformation-theory"},{"id":"stacks:0DY9","tag":"0DY9","title":"Schemes · Lemma 0DY9","summary":"In Example [Tag 0DY7] let X be a scheme over k. Then Inf(Deformationcategory_X) = Ext^0_O_X(NL_X/k, O_X) = Hom_O_X(Ω_X/k, O_X) = Der_k(O_X, O_X) and TDeformationcategory_X = Ext^1_O_X(NL_X/k, O_X)","statement_latex":"In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Then\n$$\n\\text{Inf}(\\Deformationcategory_X) =\n\\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) =\n\\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X) =\n\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)\n$$\nand\n$$\nT\\Deformationcategory_X =\n\\text{Ext}^1_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)\n$$","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DY9","source_file":"examples-defos.tex","source_line":1264,"source_end_line":1278,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1264-L1278","statement_sha256":"026636235ebbd9e9b1772bcfd4c9c655b33ff3f324bee57748b2ce985b4aded9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13565,"rank":13565,"depth":16,"x":1480.307,"y":1709.749,"cluster":"deformation-theory"},{"id":"stacks:0DYA","tag":"0DYA","title":"Schemes · Lemma 0DYA","summary":"In Lemma [Tag 0DY9] if X is proper over k, then Inf(Deformationcategory_X) and TDeformationcategory_X are finite dimensional.","statement_latex":"In Lemma \\ref{lemma-schemes-TI} if $X$ is proper over $k$, then\n$\\text{Inf}(\\Deformationcategory_X)$ and $T\\Deformationcategory_X$ are\nfinite dimensional.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYA","source_file":"examples-defos.tex","source_line":1305,"source_end_line":1310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1305-L1310","statement_sha256":"fb169cb85f1cd473d730cff3210584d3c85965fc07d50f5fb0e4b9c80c4bfcb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13566,"rank":13566,"depth":35,"x":1565.231,"y":1462.281,"cluster":"deformation-theory"},{"id":"stacks:0ET5","tag":"0ET5","title":"Schemes · Lemma 0ET5","summary":"In Example [Tag 0DY7] assume X is a proper k-scheme. Assume Lambda is a complete local ring with residue field k (the classical case). Then the functor F : C_Lambda → Sets, A ↦ Ob(Deformationcategory_X(A))/≅ of isomorphism classes of objects has a hull. If Der_k(O_X, O_X) = 0, then F is prorepresentable.","statement_latex":"In Example \\ref{example-schemes} assume $X$ is a proper $k$-scheme.\nAssume $\\Lambda$ is a complete local ring with residue field $k$\n(the classical case). Then the functor\n$$\nF : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad\nA \\longmapsto \\Ob(\\Deformationcategory_X(A))/\\cong\n$$\nof isomorphism classes of objects has a hull. If\n$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$, then\n$F$ is prorepresentable.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET5","source_file":"examples-defos.tex","source_line":1336,"source_end_line":1348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1336-L1348","statement_sha256":"1d76b96c80c467123af98907ec46db30a5792dc9c89d17691e9f124b9ec8d736","origin":"The Stacks Project","memory_eligible":false,"source_rank":13567,"rank":13567,"depth":51,"x":1701.838,"y":1693.323,"cluster":"deformation-theory"},{"id":"stacks:0DYW","tag":"0DYW","title":"Schemes · Lemma 0DYW","summary":"In Example [Tag 0DY7] let X be a scheme over k. Let U ⊂ X be an open subscheme. There is a natural functor Deformationcategory_X → Deformationcategory_U of deformation categories.","statement_latex":"In Example \\ref{example-schemes} let $X$ be a scheme over $k$.\nLet $U \\subset X$ be an open subscheme.\nThere is a natural functor\n$$\n\\Deformationcategory_X \\longrightarrow \\Deformationcategory_U\n$$\nof deformation categories.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYW","source_file":"examples-defos.tex","source_line":1366,"source_end_line":1375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1366-L1375","statement_sha256":"7c0e67470db0974d72e34f8f660e66f7fa83b460ba637bbd6343a9c2d58d3306","origin":"The Stacks Project","memory_eligible":false,"source_rank":13568,"rank":13568,"depth":0,"x":1414.844,"y":1600.32,"cluster":"deformation-theory"},{"id":"stacks:0DYX","tag":"0DYX","title":"Schemes · Lemma 0DYX","summary":"In Example [Tag 0DY7] let X = Spec(P) be an affine scheme over k. With Deformationcategory_P as in Example [Tag 0DY1] there is a natural equivalence Deformationcategory_X → Deformationcategory_P of deformation categories.","statement_latex":"In Example \\ref{example-schemes} let $X = \\Spec(P)$ be an\naffine scheme over $k$. With $\\Deformationcategory_P$ as in\nExample \\ref{example-rings} there is a natural equivalence\n$$\n\\Deformationcategory_X \\longrightarrow \\Deformationcategory_P\n$$\nof deformation categories.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYX","source_file":"examples-defos.tex","source_line":1382,"source_end_line":1391,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1382-L1391","statement_sha256":"275a814b8c5d493df68696175d802d44b829c9988332698dbcb4b8fbc34632f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13569,"rank":13569,"depth":32,"x":1701.723,"y":1505.899,"cluster":"deformation-theory"},{"id":"stacks:0DZM","tag":"0DZM","title":"Schemes · Lemma 0DZM","summary":"In Example [Tag 0DY7] let X be a scheme over k Let p ∈ X be a point. With Deformationcategory_O_X, p as in Example [Tag 0DY1] there is a natural functor Deformationcategory_X → Deformationcategory_O_X, p of deformation categories.","statement_latex":"In Example \\ref{example-schemes} let $X$ be a scheme over $k$\nLet $p \\in X$ be a point. With $\\Deformationcategory_{\\mathcal{O}_{X, p}}$\nas in Example \\ref{example-rings} there is a natural functor\n$$\n\\Deformationcategory_X\n\\longrightarrow\n\\Deformationcategory_{\\mathcal{O}_{X, p}}\n$$\nof deformation categories.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZM","source_file":"examples-defos.tex","source_line":1400,"source_end_line":1411,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1400-L1411","statement_sha256":"8a829dcf2c0e4b79017c3379cb820d695d6cac96f5bb75d2dd37a0a72312c249","origin":"The Stacks Project","memory_eligible":false,"source_rank":13570,"rank":13570,"depth":33,"x":1565.894,"y":1738.68,"cluster":"deformation-theory"},{"id":"stacks:0DYZ","tag":"0DYZ","title":"Schemes · Lemma 0DYZ","summary":"In Situation [Tag 0DYY] there is an equivalence Deformationcategory_X = Deformationcategory_P_1 ×_Deformationcategory_P_12 Deformationcategory_P_2 of deformation categories, see Examples [Tag 0DY7] and [Tag 0DY1].","statement_latex":"In Situation \\ref{situation-glueing}\nthere is an equivalence\n$$\n\\Deformationcategory_X =\n\\Deformationcategory_{P_1}\n\\times_{\\Deformationcategory_{P_{12}}}\n\\Deformationcategory_{P_2}\n$$\nof deformation categories, see Examples \\ref{example-schemes} and\n\\ref{example-rings}.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DYZ","source_file":"examples-defos.tex","source_line":1433,"source_end_line":1445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1433-L1445","statement_sha256":"f5912ecbb8c25cf22b84bb0a655ddacfd736ab06040f46e4afb5dc4553f85f01","origin":"The Stacks Project","memory_eligible":false,"source_rank":13571,"rank":13571,"depth":33,"x":1478.717,"y":1489.559,"cluster":"deformation-theory"},{"id":"stacks:0E3U","tag":"0E3U","title":"Morphisms of Schemes · Lemma 0E3U","summary":"Example [Tag 0E3T] satisfies the Rim-Schlessinger condition (RS). In particular, Deformationcategory_X → Y is a deformation category for any morphism of schemes X → Y over k.","statement_latex":"Example \\ref{example-schemes-morphisms}\nsatisfies the Rim-Schlessinger condition (RS).\nIn particular, $\\Deformationcategory_{X \\to Y}$ is a deformation category\nfor any morphism of schemes $X \\to Y$ over $k$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Morphisms of Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3U","source_file":"examples-defos.tex","source_line":1524,"source_end_line":1530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1524-L1530","statement_sha256":"b2fa47b684aa685b811557474d042ca877a8433eb199c280dc1bb78d857c3629","origin":"The Stacks Project","memory_eligible":false,"source_rank":13572,"rank":13572,"depth":45,"x":1743.76,"y":1624.005,"cluster":"deformation-theory"},{"id":"stacks:0E3V","tag":"0E3V","title":"Morphisms of Schemes · Lemma 0E3V","summary":"In Example [Tag 0DY7] let f : X → Y be a morphism of schemes over k. There is a canonical exact sequence of k-vector spaces xymatrix 0 ar[r] & Inf(Deformationcategory_X → Y) ar[r] & Inf(Deformationcategory_X × Deformationcategory_Y) ar[r] & Der_k(O_Y, f_*O_X) ar[lld] & TDeformationcategory_X → Y ar[r] & T(Deformationcategory_X × Deformationcategory_Y) ar[r] & Ext^1_O_X(Lf^*NL_Y/k, O_X)","statement_latex":"In Example \\ref{example-schemes} let $f : X \\to Y$ be a morphism of schemes\nover $k$. There is a canonical exact sequence of $k$-vector spaces\n$$\n\\xymatrix{\n0 \\ar[r] &\n\\text{Inf}(\\Deformationcategory_{X \\to Y}) \\ar[r] &\n\\text{Inf}(\\Deformationcategory_X \\times \\Deformationcategory_Y) \\ar[r] &\n\\text{Der}_k(\\mathcal{O}_Y, f_*\\mathcal{O}_X) \\ar[lld] \\\\\n& T\\Deformationcategory_{X \\to Y} \\ar[r] &\nT(\\Deformationcategory_X \\times \\Deformationcategory_Y) \\ar[r] &\n\\text{Ext}^1_{\\mathcal{O}_X}(Lf^*\\NL_{Y/k}, \\mathcal{O}_X)\n}\n$$","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Morphisms of Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3V","source_file":"examples-defos.tex","source_line":1560,"source_end_line":1575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1560-L1575","statement_sha256":"9f3d9efa59d88d7c99aae8613d602a15dde982b2f8500e7aa114b174265b79c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13573,"rank":13573,"depth":16,"x":1439.719,"y":1675.34,"cluster":"deformation-theory"},{"id":"stacks:0E3W","tag":"0E3W","title":"Morphisms of Schemes · Lemma 0E3W","summary":"In Lemma [Tag 0E3V] if X and Y are both proper over k, then Inf(Deformationcategory_X → Y) and TDeformationcategory_X → Y are finite dimensional.","statement_latex":"In Lemma \\ref{lemma-schemes-morphisms-TI} if $X$ and $Y$ are both\nproper over $k$, then\n$\\text{Inf}(\\Deformationcategory_{X \\to Y})$ and\n$T\\Deformationcategory_{X \\to Y}$ are finite dimensional.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Morphisms of Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3W","source_file":"examples-defos.tex","source_line":1641,"source_end_line":1647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1641-L1647","statement_sha256":"8a4e4d31a76c1fdb2721af026159f4abb6d1b6fc14e1f850a91f3160b1865230","origin":"The Stacks Project","memory_eligible":false,"source_rank":13574,"rank":13574,"depth":36,"x":1622.923,"y":1464.632,"cluster":"deformation-theory"},{"id":"stacks:0ET6","tag":"0ET6","title":"Morphisms of Schemes · Lemma 0ET6","summary":"In Example [Tag 0E3T] assume X → Y is a morphism of proper k-schemes. Assume Lambda is a complete local ring with residue field k (the classical case). Then the functor F : C_Lambda → Sets, A ↦ Ob(Deformationcategory_X → Y(A))/≅ of isomorphism classes of objects has a hull. If Der_k(O_X, O_X) = Der_k(O_Y, O_Y) = 0, then F is prorepresentable.","statement_latex":"In Example \\ref{example-schemes-morphisms} assume $X \\to Y$\nis a morphism of proper $k$-schemes.\nAssume $\\Lambda$ is a complete local ring with residue field $k$\n(the classical case). Then the functor\n$$\nF : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad\nA \\longmapsto \\Ob(\\Deformationcategory_{X \\to Y}(A))/\\cong\n$$\nof isomorphism classes of objects has a hull. If\n$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) =\n\\text{Der}_k(\\mathcal{O}_Y, \\mathcal{O}_Y) = 0$, then\n$F$ is prorepresentable.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Morphisms of Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET6","source_file":"examples-defos.tex","source_line":1665,"source_end_line":1679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1665-L1679","statement_sha256":"25f98571130e234f681d1953e318c462e89d8da674ba48d39b6c360211e9da60","origin":"The Stacks Project","memory_eligible":false,"source_rank":13575,"rank":13575,"depth":51,"x":1657.33,"y":1724.371,"cluster":"deformation-theory"},{"id":"stacks:0E3X","tag":"0E3X","title":"Morphisms of Schemes · Lemma 0E3X","summary":"This is discussed in [Ravi-Murphys-Law] and [Ran-deformations]. In Example [Tag 0DY7] let f : X → Y be a morphism of schemes over k. If f_*O_X = O_Y and R^1f_*O_X = 0, then the morphism of deformation categories Deformationcategory_X → Y → Deformationcategory_X is an equivalence.","statement_latex":"\\begin{reference}\nThis is discussed in \\cite[Section 5.3]{Ravi-Murphys-Law} and\n\\cite[Theorem 3.3]{Ran-deformations}.\n\\end{reference}\nIn Example \\ref{example-schemes} let $f : X \\to Y$ be a morphism of schemes\nover $k$. If $f_*\\mathcal{O}_X = \\mathcal{O}_Y$ and $R^1f_*\\mathcal{O}_X = 0$,\nthen the morphism of deformation categories\n$$\n\\Deformationcategory_{X \\to Y} \\to \\Deformationcategory_X\n$$\nis an equivalence.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Morphisms of Schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E3X","source_file":"examples-defos.tex","source_line":1703,"source_end_line":1716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1703-L1716","statement_sha256":"a0b9f746b8bde0631e66fadc75fac77761574ca59cb09c700565fb0d866e1dbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13576,"rank":13576,"depth":25,"x":1422.715,"y":1552.095,"cluster":"deformation-theory"},{"id":"stacks:0E40","tag":"0E40","title":"Algebraic spaces · Lemma 0E40","summary":"Example [Tag 0E3Z] satisfies the Rim-Schlessinger condition (RS). In particular, Deformationcategory_X is a deformation category for any algebraic space X over k.","statement_latex":"Example \\ref{example-spaces}\nsatisfies the Rim-Schlessinger condition (RS).\nIn particular, $\\Deformationcategory_X$ is a deformation category\nfor any algebraic space $X$ over $k$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E40","source_file":"examples-defos.tex","source_line":1805,"source_end_line":1811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1805-L1811","statement_sha256":"8b9c3467823c591df50987555579c6ec00d0d180744dccb067a26ca2a9fb1104","origin":"The Stacks Project","memory_eligible":false,"source_rank":13577,"rank":13577,"depth":74,"x":1734.747,"y":1545.992,"cluster":"deformation-theory"},{"id":"stacks:0E41","tag":"0E41","title":"Algebraic spaces · Lemma 0E41","summary":"In Example [Tag 0E3Z] let X be an algebraic space over k. Then Inf(Deformationcategory_X) = Ext^0_O_X(NL_X/k, O_X) = Hom_O_X(Ω_X/k, O_X) = Der_k(O_X, O_X) and TDeformationcategory_X = Ext^1_O_X(NL_X/k, O_X)","statement_latex":"In Example \\ref{example-spaces} let $X$ be an algebraic space over $k$. Then\n$$\n\\text{Inf}(\\Deformationcategory_X) =\n\\text{Ext}^0_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X) =\n\\Hom_{\\mathcal{O}_X}(\\Omega_{X/k}, \\mathcal{O}_X) =\n\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X)\n$$\nand\n$$\nT\\Deformationcategory_X =\n\\text{Ext}^1_{\\mathcal{O}_X}(\\NL_{X/k}, \\mathcal{O}_X)\n$$","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E41","source_file":"examples-defos.tex","source_line":1836,"source_end_line":1850,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1836-L1850","statement_sha256":"dc76c0f1411fad854e14ac33ad0ea366763b9a681eb38b7704372d774dae9d92","origin":"The Stacks Project","memory_eligible":false,"source_rank":13578,"rank":13578,"depth":72,"x":1509.213,"y":1727.831,"cluster":"deformation-theory"},{"id":"stacks:0E42","tag":"0E42","title":"Algebraic spaces · Lemma 0E42","summary":"In Lemma [Tag 0E41] if X is proper over k, then Inf(Deformationcategory_X) and TDeformationcategory_X are finite dimensional.","statement_latex":"In Lemma \\ref{lemma-spaces-TI} if $X$ is proper over $k$, then\n$\\text{Inf}(\\Deformationcategory_X)$ and $T\\Deformationcategory_X$ are\nfinite dimensional.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E42","source_file":"examples-defos.tex","source_line":1879,"source_end_line":1884,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1879-L1884","statement_sha256":"1713f25d806de995fa6d5cc3c4caba9b99a257f7ab2f29edb2e6a715f592591e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13579,"rank":13579,"depth":73,"x":1529.32,"y":1465.363,"cluster":"deformation-theory"},{"id":"stacks:0ET7","tag":"0ET7","title":"Algebraic spaces · Lemma 0ET7","summary":"In Example [Tag 0E3Z] assume X is a proper algebraic space over k. Assume Lambda is a complete local ring with residue field k (the classical case). Then the functor F : C_Lambda → Sets, A ↦ Ob(Deformationcategory_X(A))/≅ of isomorphism classes of objects has a hull. If Der_k(O_X, O_X) = 0, then F is prorepresentable.","statement_latex":"In Example \\ref{example-spaces} assume $X$ is a proper algebraic space over $k$.\nAssume $\\Lambda$ is a complete local ring with residue field $k$\n(the classical case). Then the functor\n$$\nF : \\mathcal{C}_\\Lambda \\longrightarrow \\textit{Sets},\\quad\nA \\longmapsto \\Ob(\\Deformationcategory_X(A))/\\cong\n$$\nof isomorphism classes of objects has a hull. If\n$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$, then\n$F$ is prorepresentable.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ET7","source_file":"examples-defos.tex","source_line":1910,"source_end_line":1922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1910-L1922","statement_sha256":"75f67b8f60fb3305f663296d63c496c0280ee29661b0e7751421f1b7ddc9383b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13580,"rank":13580,"depth":75,"x":1725.869,"y":1670.632,"cluster":"deformation-theory"},{"id":"stacks:0DZ1","tag":"0DZ1","title":"Deformations of completions · Lemma 0DZ1","summary":"Let A' → A be a surjection of rings with nilpotent kernel. Let A' → P' be a flat ring map. Set P = P' ⊗_A' A. Let M be an A-flat P-module. Then the following are equivalent • there is an A'-flat P'-module M' with M' ⊗_P' P = M, and • there is an object K' ∈ D^-(P') with K' ⊗_P'^L P = M.","statement_latex":"Let $A' \\to A$ be a surjection of rings with nilpotent kernel.\nLet $A' \\to P'$ be a flat ring map.\nSet $P = P' \\otimes_{A'} A$.\nLet $M$ be an $A$-flat $P$-module.\nThen the following are equivalent\n\\begin{enumerate}\n\\item there is an $A'$-flat $P'$-module $M'$ with\n$M' \\otimes_{P'} P = M$, and\n\\item there is an object $K' \\in D^-(P')$ with\n$K' \\otimes_{P'}^\\mathbf{L} P = M$.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZ1","source_file":"examples-defos.tex","source_line":1952,"source_end_line":1965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1952-L1965","statement_sha256":"72b0062ff8572deb79fb7d6f6813a857bc5abe522d84784ff56ad69c0b0e65e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13581,"rank":13581,"depth":1,"x":1415.383,"y":1630.734,"cluster":"deformation-theory"},{"id":"stacks:0DZ2","tag":"0DZ2","title":"Deformations of completions · Lemma 0DZ2","summary":"Consider a commutative diagram of Noetherian rings xymatrix A' ar[d] ar[r] & P' ar[d] ar[r] & Q' ar[d] A ar[r] & P ar[r] & Q with cartesian squares, with flat horizontal arrows, and with surjective vertical arrows whose kernels are nilpotent. Let J' ⊂ P' be an ideal such that P'/J' = Q'/J'Q'. Let M be an A-flat P-module. Assume for all g ∈ J' there exists an A'-flat (P')_g-module lifting M_g. Then the following are equivalent • M has an A'-flat lift to a P'-module, and •…","statement_latex":"Consider a commutative diagram of Noetherian rings\n$$\n\\xymatrix{\nA' \\ar[d] \\ar[r] &\nP' \\ar[d] \\ar[r] &\nQ' \\ar[d] \\\\\nA \\ar[r] &\nP \\ar[r] &\nQ\n}\n$$\nwith cartesian squares, with flat horizontal arrows, and with\nsurjective vertical arrows whose kernels are nilpotent.\nLet $J' \\subset P'$ be an ideal such that $P'/J' = Q'/J'Q'$.\nLet $M$ be an $A$-flat $P$-module.\nAssume for all $g \\in J'$ there exists an $A'$-flat $(P')_g$-module\nlifting $M_g$. Then the following are equivalent\n\\begin{enumerate}\n\\item $M$ has an $A'$-flat lift to a $P'$-module, and\n\\item $M \\otimes_P Q$ has an $A'$-flat lift to a $Q'$-module.\n\\end{enumerate}","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZ2","source_file":"examples-defos.tex","source_line":1995,"source_end_line":2018,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L1995-L2018","statement_sha256":"bcec9820e6c7a7a828a0327d4faf8d84b52177f6f2d9e38730883730dfb70133","origin":"The Stacks Project","memory_eligible":false,"source_rank":13582,"rank":13582,"depth":20,"x":1676.82,"y":1483.75,"cluster":"deformation-theory"},{"id":"stacks:0DZ3","tag":"0DZ3","title":"Deformations of completions · Lemma 0DZ3","summary":"Let A' → A be a surjective map of Noetherian rings with nilpotent kernel. Let A → B be a finite type flat ring map. Let b ⊂ B be an ideal such that Spec(B) → Spec(A) is syntomic on the complement of V( b). Then B has a flat lift to A' if and only if the b-adic completion B^wedge has a flat lift to A'.","statement_latex":"Let $A' \\to A$ be a surjective map of Noetherian rings with nilpotent kernel.\nLet $A \\to B$ be a finite type flat ring map.\nLet $\\mathfrak b \\subset B$ be an ideal such that\n$\\Spec(B) \\to \\Spec(A)$ is syntomic on the complement of $V(\\mathfrak b)$.\nThen $B$ has a flat lift to $A'$ if and only if the $\\mathfrak b$-adic\ncompletion $B^\\wedge$ has a flat lift to $A'$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZ3","source_file":"examples-defos.tex","source_line":2075,"source_end_line":2083,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2075-L2083","statement_sha256":"5bd377813a6e5efb880539c7dd3e1d1c4c80fd66077af276558bd0ca3bf17315","origin":"The Stacks Project","memory_eligible":false,"source_rank":13583,"rank":13583,"depth":47,"x":1602.126,"y":1740.876,"cluster":"deformation-theory"},{"id":"stacks:0DZ4","tag":"0DZ4","title":"Deformations of completions · Lemma 0DZ4","summary":"Let k be a field. Let B be a finite type k-algebra. Let J ⊂ B be an ideal such that Spec(B) → Spec(k) is smooth on the complement of V(J). Let N be a finite B-module. Then there is a canonical bijection Exal_k(B, N) → Exal_k(B^wedge, N^wedge) Here B^wedge and N^wedge are the J-adic completions.","statement_latex":"Let $k$ be a field. Let $B$ be a finite type $k$-algebra.\nLet $J \\subset B$ be an ideal such that\n$\\Spec(B) \\to \\Spec(k)$ is smooth on the complement of $V(J)$.\nLet $N$ be a finite $B$-module.\nThen there is a canonical bijection\n$$\n\\text{Exal}_k(B, N) \\to \\text{Exal}_k(B^\\wedge, N^\\wedge)\n$$\nHere $B^\\wedge$ and $N^\\wedge$ are the $J$-adic completions.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZ4","source_file":"examples-defos.tex","source_line":2165,"source_end_line":2176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2165-L2176","statement_sha256":"75b65667aa23c47172440944e26ce0b0cea3b951ecbef66e4f11c7dd8895c577","origin":"The Stacks Project","memory_eligible":false,"source_rank":13584,"rank":13584,"depth":11,"x":1450.197,"y":1508.534,"cluster":"deformation-theory"},{"id":"stacks:0DZ5","tag":"0DZ5","title":"Deformations of completions · Lemma 0DZ5","summary":"In Example [Tag 0DY1] let P be a k-algebra. Let J ⊂ P be an ideal. Denote P^wedge the J-adic completion. If • k → P is of finite type, and • Spec(P) → Spec(k) is smooth on the complement of V(J). then the functor between deformation categories of Lemma [Tag 0DYV] Deformationcategory_P → Deformationcategory_P^wedge is smooth and induces an isomorphism on tangent spaces.","statement_latex":"In Example \\ref{example-rings} let $P$ be a $k$-algebra.\nLet $J \\subset P$ be an ideal.\nDenote $P^\\wedge$ the $J$-adic completion. If\n\\begin{enumerate}\n\\item $k \\to P$ is of finite type, and\n\\item $\\Spec(P) \\to \\Spec(k)$ is smooth on the complement of $V(J)$.\n\\end{enumerate}\nthen the functor between deformation categories of\nLemma \\ref{lemma-completion}\n$$\n\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^\\wedge}\n$$\nis smooth and induces an isomorphism on tangent spaces.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZ5","source_file":"examples-defos.tex","source_line":2274,"source_end_line":2289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2274-L2289","statement_sha256":"e7f6cad908411fcb4dd05e7d2ef0a45223a44b0fbbc58dd28a94d22d810f92e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13585,"rank":13585,"depth":48,"x":1749.528,"y":1593.783,"cluster":"deformation-theory"},{"id":"stacks:0DZ7","tag":"0DZ7","title":"Deformations of localizations · Lemma 0DZ7","summary":"Let A' → A be a surjective map of Noetherian rings with nilpotent kernel. Let A → B be a finite type flat ring map. Let S ⊂ B be a multiplicative subset such that if Spec(B) → Spec(A) is not syntomic at q, then S ∩ q = ∅. Then B has a flat lift to A' if and only if S^-1B has a flat lift to A'.","statement_latex":"Let $A' \\to A$ be a surjective map of Noetherian rings with nilpotent kernel.\nLet $A \\to B$ be a finite type flat ring map.\nLet $S \\subset B$ be a multiplicative subset such that\nif $\\Spec(B) \\to \\Spec(A)$ is not syntomic at $\\mathfrak q$,\nthen $S \\cap \\mathfrak q = \\emptyset$.\nThen $B$ has a flat lift to $A'$ if and only if\n$S^{-1}B$ has a flat lift to $A'$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of localizations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZ7","source_file":"examples-defos.tex","source_line":2314,"source_end_line":2323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2314-L2323","statement_sha256":"d6306ad4a7ccb20986e7a867711b33074ea29e5f0bc842b4f15e5957edc2becb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13586,"rank":13586,"depth":48,"x":1459.809,"y":1700.933,"cluster":"deformation-theory"},{"id":"stacks:0DZ8","tag":"0DZ8","title":"Deformations of localizations · Lemma 0DZ8","summary":"Let k be a field. Let B be a finite type k-algebra. Let S ⊂ B be a multiplicative subset ideal such that if Spec(B) → Spec(k) is not smooth at q then S ∩ q = ∅. Let N be a finite B-module. Then there is a canonical bijection Exal_k(B, N) → Exal_k(S^-1B, S^-1N)","statement_latex":"Let $k$ be a field. Let $B$ be a finite type $k$-algebra.\nLet $S \\subset B$ be a multiplicative subset ideal such that\nif $\\Spec(B) \\to \\Spec(k)$ is not smooth at $\\mathfrak q$\nthen $S \\cap \\mathfrak q = \\emptyset$.\nLet $N$ be a finite $B$-module.\nThen there is a canonical bijection\n$$\n\\text{Exal}_k(B, N) \\to \\text{Exal}_k(S^{-1}B, S^{-1}N)\n$$","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of localizations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZ8","source_file":"examples-defos.tex","source_line":2382,"source_end_line":2393,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2382-L2393","statement_sha256":"25764cbd285d7e903ae9fb610ad84095143530d4b2fd2224db2f168d1c1b8b73","origin":"The Stacks Project","memory_eligible":false,"source_rank":13587,"rank":13587,"depth":20,"x":1587.471,"y":1457.157,"cluster":"deformation-theory"},{"id":"stacks:0DZ9","tag":"0DZ9","title":"Deformations of localizations · Lemma 0DZ9","summary":"In Example [Tag 0DY1] let P be a k-algebra. Let S ⊂ P be a multiplicative subset. If • k → P is of finite type, and • Spec(P) → Spec(k) is smooth at all points of V(g) for all g ∈ S. then the functor between deformation categories of Lemma [Tag 0DYS] Deformationcategory_P → Deformationcategory_S^-1P is smooth and induces an isomorphism on tangent spaces.","statement_latex":"In Example \\ref{example-rings} let $P$ be a $k$-algebra.\nLet $S \\subset P$ be a multiplicative subset. If\n\\begin{enumerate}\n\\item $k \\to P$ is of finite type, and\n\\item $\\Spec(P) \\to \\Spec(k)$ is smooth at all points of\n$V(g)$ for all $g \\in S$.\n\\end{enumerate}\nthen the functor between deformation categories of\nLemma \\ref{lemma-localization}\n$$\n\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{S^{-1}P}\n$$\nis smooth and induces an isomorphism on tangent spaces.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of localizations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZ9","source_file":"examples-defos.tex","source_line":2435,"source_end_line":2450,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2435-L2450","statement_sha256":"36c5a356fe3141049b73d9cedfeeaa28c622bf26835c622e9207f76830457807","origin":"The Stacks Project","memory_eligible":false,"source_rank":13588,"rank":13588,"depth":49,"x":1689.528,"y":1709.736,"cluster":"deformation-theory"},{"id":"stacks:0DZB","tag":"0DZB","title":"Deformations of henselizations · Lemma 0DZB","summary":"Let A' → A be a surjective map of Noetherian rings with nilpotent kernel. Let A → B be a finite type flat ring map. Let b ⊂ B be an ideal such that Spec(B) → Spec(A) is syntomic on the complement of V( b). Let (B^h, b^h) be the henselization of the pair (B, b). Then B has a flat lift to A' if and only if B^h has a flat lift to A'.","statement_latex":"Let $A' \\to A$ be a surjective map of Noetherian rings with nilpotent kernel.\nLet $A \\to B$ be a finite type flat ring map.\nLet $\\mathfrak b \\subset B$ be an ideal such that\n$\\Spec(B) \\to \\Spec(A)$ is syntomic on the complement of $V(\\mathfrak b)$.\nLet $(B^h, \\mathfrak b^h)$ be the henselization of the pair $(B, \\mathfrak b)$.\nThen $B$ has a flat lift to $A'$ if and only if $B^h$ has a flat lift to $A'$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of henselizations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZB","source_file":"examples-defos.tex","source_line":2475,"source_end_line":2483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2475-L2483","statement_sha256":"9836c741100a49ddc54ac97fb41721cb666d31f7cc39ee5f8398ae8a01eeb554","origin":"The Stacks Project","memory_eligible":false,"source_rank":13589,"rank":13589,"depth":52,"x":1410.734,"y":1581.203,"cluster":"deformation-theory"},{"id":"stacks:0DZC","tag":"0DZC","title":"Deformations of henselizations · Lemma 0DZC","summary":"Let k be a field. Let B be a finite type k-algebra. Let J ⊂ B be an ideal such that Spec(B) → Spec(k) is smooth on the complement of V(J). Let N be a finite B-module. Then there is a canonical bijection Exal_k(B, N) → Exal_k(B^h, N^h) Here (B^h, J^h) is the henselization of (B, J) and N^h = N ⊗_B B^h.","statement_latex":"Let $k$ be a field. Let $B$ be a finite type $k$-algebra.\nLet $J \\subset B$ be an ideal such that\n$\\Spec(B) \\to \\Spec(k)$ is smooth on the complement of $V(J)$.\nLet $N$ be a finite $B$-module.\nThen there is a canonical bijection\n$$\n\\text{Exal}_k(B, N) \\to \\text{Exal}_k(B^h, N^h)\n$$\nHere $(B^h, J^h)$ is the henselization of $(B, J)$\nand $N^h = N \\otimes_B B^h$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of henselizations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZC","source_file":"examples-defos.tex","source_line":2556,"source_end_line":2568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2556-L2568","statement_sha256":"4326c813235ea16c7d8a5519cd135aa019284dd119835fe866baddbbea6f2aed","origin":"The Stacks Project","memory_eligible":false,"source_rank":13590,"rank":13590,"depth":52,"x":1720.142,"y":1517.691,"cluster":"deformation-theory"},{"id":"stacks:0DZD","tag":"0DZD","title":"Deformations of henselizations · Lemma 0DZD","summary":"In Example [Tag 0DY1] let P be a k-algebra. Let J ⊂ P be an ideal. Denote (P^h, J^h) the henselization of the pair (P, J). If • k → P is of finite type, and • Spec(P) → Spec(k) is smooth on the complement of V(J), then the functor between deformation categories of Lemma [Tag 0DYT] Deformationcategory_P → Deformationcategory_P^h is smooth and induces an isomorphism on tangent spaces.","statement_latex":"In Example \\ref{example-rings} let $P$ be a $k$-algebra.\nLet $J \\subset P$ be an ideal.\nDenote $(P^h, J^h)$ the henselization of the pair $(P, J)$. If\n\\begin{enumerate}\n\\item $k \\to P$ is of finite type, and\n\\item $\\Spec(P) \\to \\Spec(k)$ is smooth on the complement of $V(J)$,\n\\end{enumerate}\nthen the functor between deformation categories of\nLemma \\ref{lemma-henselization}\n$$\n\\Deformationcategory_P \\longrightarrow \\Deformationcategory_{P^h}\n$$\nis smooth and induces an isomorphism on tangent spaces.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Deformations of henselizations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZD","source_file":"examples-defos.tex","source_line":2617,"source_end_line":2632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2617-L2632","statement_sha256":"7e312af59408fa8c7502ffa561aea5b4d8cada91a8b7bde756bd4cd141da94e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13591,"rank":13591,"depth":53,"x":1542.796,"y":1740.424,"cluster":"deformation-theory"},{"id":"stacks:0DZF","tag":"0DZF","title":"Application to isolated singularities · Lemma 0DZF","summary":"In Example [Tag 0DY1] let P be a k-algebra. Assume that k → P is of finite type and that Spec(P) → Spec(k) is smooth except at the maximal ideals m_1, …, m_n of P. Let P_ m_i, P_ m_i^h, P_ m_i^wedge be the local ring, henselization, completion. Then the maps of deformation categories Deformationcategory_P → ∏ Deformationcategory_P_ m_i → ∏ Deformationcategory_P_ m_i^h → ∏ Deformationcategory_P_ m_i^wedge are smooth and induce isomorphisms on their finite dimensional…","statement_latex":"In Example \\ref{example-rings} let $P$ be a $k$-algebra.\nAssume that $k \\to P$ is of finite type and that $\\Spec(P) \\to \\Spec(k)$\nis smooth except at the maximal ideals\n$\\mathfrak m_1, \\ldots, \\mathfrak m_n$ of $P$.\nLet $P_{\\mathfrak m_i}$, $P_{\\mathfrak m_i}^h$, $P_{\\mathfrak m_i}^\\wedge$\nbe the local ring, henselization, completion.\nThen the maps of deformation categories\n$$\n\\Deformationcategory_P \\to\n\\prod \\Deformationcategory_{P_{\\mathfrak m_i}} \\to\n\\prod \\Deformationcategory_{P_{\\mathfrak m_i}^h} \\to\n\\prod \\Deformationcategory_{P_{\\mathfrak m_i}^\\wedge}\n$$\nare smooth and induce isomorphisms on their finite dimensional\ntangent spaces.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Application to isolated singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZF","source_file":"examples-defos.tex","source_line":2656,"source_end_line":2673,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2656-L2673","statement_sha256":"6580a36ab384fb9db23e6470d34be40a90dbbf239ff1000d28d2a959d4c8876f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13592,"rank":13592,"depth":54,"x":1494.38,"y":1475.156,"cluster":"deformation-theory"},{"id":"stacks:0DZH","tag":"0DZH","title":"Unobstructed deformation problems · Lemma 0DZH","summary":"In Example [Tag 0DY1] let P be a local complete intersection over k (Algebra, Definition [Tag 00S9]). Then Deformationcategory_P is unobstructed.","statement_latex":"In Example \\ref{example-rings} let $P$ be a local complete\nintersection over $k$ (Algebra, Definition \\ref{algebra-definition-lci-field}).\nThen $\\Deformationcategory_P$ is unobstructed.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Unobstructed deformation problems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZH","source_file":"examples-defos.tex","source_line":2735,"source_end_line":2740,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2735-L2740","statement_sha256":"e347cdecd110381f7c2393aaad931cf426645ccbce1e8b7caef85e3a2bd0849a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13593,"rank":13593,"depth":47,"x":1743.775,"y":1643.54,"cluster":"deformation-theory"},{"id":"stacks:0DZN","tag":"0DZN","title":"Unobstructed deformation problems · Lemma 0DZN","summary":"In Situation [Tag 0DYY] if U_12 → Spec(k) is smooth, then the morphism Deformationcategory_X → Deformationcategory_U_1 × Deformationcategory_U_2 = Deformationcategory_P_1 × Deformationcategory_P_2 is smooth. If in addition U_1 is a local complete intersection over k, then Deformationcategory_X → Deformationcategory_U_2 = Deformationcategory_P_2 is smooth.","statement_latex":"In Situation \\ref{situation-glueing} if $U_{12} \\to \\Spec(k)$ is smooth,\nthen the morphism\n$$\n\\Deformationcategory_X\n\\longrightarrow\n\\Deformationcategory_{U_1} \\times \\Deformationcategory_{U_2} =\n\\Deformationcategory_{P_1} \\times \\Deformationcategory_{P_2}\n$$\nis smooth. If in addition\n$U_1$ is a local complete intersection over $k$, then\n$$\n\\Deformationcategory_X\n\\longrightarrow\n\\Deformationcategory_{U_2} = \\Deformationcategory_{P_2}\n$$\nis smooth.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Unobstructed deformation problems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZN","source_file":"examples-defos.tex","source_line":2753,"source_end_line":2771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2753-L2771","statement_sha256":"c17e4032d51dd6a7ede196d5c0486d204c962458aa40bf7fb6f80e7540d3e133","origin":"The Stacks Project","memory_eligible":false,"source_rank":13594,"rank":13594,"depth":48,"x":1423.989,"y":1660.915,"cluster":"deformation-theory"},{"id":"stacks:0DZP","tag":"0DZP","title":"Unobstructed deformation problems · Lemma 0DZP","summary":"In Example [Tag 0DY7] let X be a scheme over k. Assume • X is separated, finite type over k and dim(X) ≤ 1, • X → Spec(k) is smooth except at the closed points p_1, …, p_n ∈ X. Let O_X, p_1, O_X, p_1^h, O_X, p_1^wedge be the local ring, henselization, completion. Consider the maps of deformation categories Deformationcategory_X → ∏ Deformationcategory_O_X, p_i → ∏ Deformationcategory_O_X, p_i^h → ∏ Deformationcategory_O_X, p_i^wedge The first arrow is smooth and the…","statement_latex":"In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Assume\n\\begin{enumerate}\n\\item $X$ is separated, finite type over $k$ and $\\dim(X) \\leq 1$,\n\\item $X \\to \\Spec(k)$ is smooth except at the closed\npoints $p_1, \\ldots, p_n \\in X$.\n\\end{enumerate}\nLet $\\mathcal{O}_{X, p_1}$, $\\mathcal{O}_{X, p_1}^h$,\n$\\mathcal{O}_{X, p_1}^\\wedge$ be the local ring, henselization, completion.\nConsider the maps of deformation categories\n$$\n\\Deformationcategory_X\n\\longrightarrow\n\\prod \\Deformationcategory_{\\mathcal{O}_{X, p_i}}\n\\longrightarrow\n\\prod \\Deformationcategory_{\\mathcal{O}_{X, p_i}^h}\n\\longrightarrow\n\\prod \\Deformationcategory_{\\mathcal{O}_{X, p_i}^\\wedge}\n$$\nThe first arrow is smooth and the second and third arrows\nare smooth and induce isomorphisms on tangent spaces.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Unobstructed deformation problems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZP","source_file":"examples-defos.tex","source_line":2813,"source_end_line":2835,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2813-L2835","statement_sha256":"a14b694a46e9c8557b47b8cb457061cf9dc0ebb8511d561348b5010a6e3b5d5a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13595,"rank":13595,"depth":55,"x":1646.152,"y":1466.36,"cluster":"deformation-theory"},{"id":"stacks:0DZQ","tag":"0DZQ","title":"Unobstructed deformation problems · Lemma 0DZQ","summary":"In Example [Tag 0DY7] let X be a scheme over k. Assume • X is separated, finite type over k and dim(X) ≤ 1, • X is a local complete intersection over k, and • X → Spec(k) is smooth except at finitely many points. Then Deformationcategory_X is unobstructed.","statement_latex":"In Example \\ref{example-schemes} let $X$ be a scheme over $k$. Assume\n\\begin{enumerate}\n\\item $X$ is separated, finite type over $k$ and $\\dim(X) \\leq 1$,\n\\item $X$ is a local complete intersection over $k$, and\n\\item $X \\to \\Spec(k)$ is smooth except at finitely many points.\n\\end{enumerate}\nThen $\\Deformationcategory_X$ is unobstructed.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Unobstructed deformation problems","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZQ","source_file":"examples-defos.tex","source_line":2855,"source_end_line":2864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2855-L2864","statement_sha256":"c3c66ece337799765f559a3a6f940bf890d395e3530a810d1f43e6989b6c99e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13596,"rank":13596,"depth":49,"x":1638.777,"y":1736.282,"cluster":"deformation-theory"},{"id":"stacks:0E7T","tag":"0E7T","title":"Smoothings · Lemma 0E7T","summary":"Let k be a field. Set S = Spec(k[[t]]) and S_n = Spec(k[t]/(t^n)). Let Y → S be a proper, flat morphism of schemes whose special fibre X is Cohen-Macaulay and equidimensional of dimension d. Denote X_n = Y ×_S S_n. If for some n ≥ 1 the dth Fitting ideal of Ω_X_n/S_n contains t^n - 1, then the generic fibre of Y → S is smooth.","statement_latex":"Let $k$ be a field. Set $S = \\Spec(k[[t]])$ and\n$S_n = \\Spec(k[t]/(t^n))$. Let $Y \\to S$ be a proper, flat morphism\nof schemes whose special fibre $X$ is Cohen-Macaulay and\nequidimensional of dimension $d$. Denote $X_n = Y \\times_S S_n$.\nIf for some $n \\geq 1$ the $d$th Fitting ideal of $\\Omega_{X_n/S_n}$\ncontains $t^{n - 1}$, then the generic fibre of $Y \\to S$ is smooth.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Smoothings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7T","source_file":"examples-defos.tex","source_line":2899,"source_end_line":2907,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2899-L2907","statement_sha256":"f55fbc852ba4acf9afc10888c301bdb9ad7f1426b36642cde4c55173c9778f09","origin":"The Stacks Project","memory_eligible":false,"source_rank":13597,"rank":13597,"depth":40,"x":1426.839,"y":1532.76,"cluster":"deformation-theory"},{"id":"stacks:0E7U","tag":"0E7U","title":"Smoothings · Lemma 0E7U","summary":"Let k be a field. Let 1 ≤ c ≤ n be integers. Let f_1, …, f_c ∈ k[x_1, … x_n] be elements. Let a_ij, 0 ≤ i ≤ n, 1 ≤ j ≤ c be variables. Consider g_j = f_j + a_0j + a_1jx_1 + … + a_njx_n ∈ k[a_ij][x_1, …, x_n] Denote Y ⊂ A^n + c(n + 1)_k the closed subscheme cut out by g_1, …, g_c. Denote π : Y → A^c(n + 1)_k the projection onto the affine space with variables a_ij. Then there is a nonempty Zariski open of A^c(n + 1)_k over which π is smooth.","statement_latex":"Let $k$ be a field. Let $1 \\leq c \\leq n$ be integers.\nLet $f_1, \\ldots, f_c \\in k[x_1, \\ldots x_n]$ be elements.\nLet $a_{ij}$, $0 \\leq i \\leq n$, $1 \\leq j \\leq c$ be\nvariables. Consider\n$$\ng_j = f_j + a_{0j} + a_{1j}x_1 + \\ldots + a_{nj}x_n \\in\nk[a_{ij}][x_1, \\ldots, x_n]\n$$\nDenote $Y \\subset \\mathbf{A}^{n + c(n + 1)}_k$\nthe closed subscheme cut out by $g_1, \\ldots, g_c$.\nDenote $\\pi : Y \\to \\mathbf{A}^{c(n + 1)}_k$ the projection\nonto the affine space with variables $a_{ij}$.\nThen there is a nonempty Zariski open \nof $\\mathbf{A}^{c(n + 1)}_k$ over which $\\pi$ is smooth.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Smoothings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7U","source_file":"examples-defos.tex","source_line":2933,"source_end_line":2949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2933-L2949","statement_sha256":"d713c18b99176536bdc400b4648cb571f64fb33b7a4bacb71d65e1818a80b62d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13598,"rank":13598,"depth":7,"x":1747.257,"y":1562.619,"cluster":"deformation-theory"},{"id":"stacks:0E7V","tag":"0E7V","title":"Smoothings · Lemma 0E7V","summary":"Let k be a field. Let A be a global complete intersection over k. There exists a flat finite type ring map k[[t]] → B with B/tB ≅ A such that B[1/t] is smooth over k((t)).","statement_latex":"Let $k$ be a field. Let $A$ be a global complete intersection\nover $k$. There exists a flat finite type ring map\n$k[[t]] \\to B$ with $B/tB \\cong A$ such that\n$B[1/t]$ is smooth over $k((t))$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Smoothings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7V","source_file":"examples-defos.tex","source_line":2998,"source_end_line":3004,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L2998-L3004","statement_sha256":"2610f69d37cb2efd0a21c4eb44f916edfba75e4dcdc1ebc488cd871c7adfa08e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13599,"rank":13599,"depth":25,"x":1486.591,"y":1722.65,"cluster":"deformation-theory"},{"id":"stacks:0E7W","tag":"0E7W","title":"Smoothings · Lemma 0E7W","summary":"Let k be a field. Let A be a finite dimensional k-algebra which is a local complete intersection over k. Then there is a finite flat k[[t]]-algebra B with B/tB ≅ A and B[1/t] étale over k((t)).","statement_latex":"Let $k$ be a field. Let $A$ be a finite dimensional $k$-algebra\nwhich is a local complete intersection over $k$. Then there is\na finite flat $k[[t]]$-algebra $B$ with $B/tB \\cong A$\nand $B[1/t]$ \\'etale over $k((t))$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Smoothings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7W","source_file":"examples-defos.tex","source_line":3043,"source_end_line":3049,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L3043-L3049","statement_sha256":"a69f74327d67980f206c40084da27727db6811340ec1bc39f2f49d319d86b188","origin":"The Stacks Project","memory_eligible":false,"source_rank":13600,"rank":13600,"depth":44,"x":1550.204,"y":1456.347,"cluster":"deformation-theory"},{"id":"stacks:0E7X","tag":"0E7X","title":"Smoothings · Lemma 0E7X","summary":"Let k be a field. Let A be a k-algebra. Assume • A is a local ring essentially of finite type over k, • A is a complete intersection over k (Algebra, Definition [Tag 00SD]). Set d = dim(A) + trdeg_k(kappa) where kappa is the residue field of A. Then there exists an integer n and a flat, essentially of finite type ring map k[[t]] → B with B/tB ≅ A such that t^n is in the dth Fitting ideal of Ω_B/k[[t]].","statement_latex":"Let $k$ be a field. Let $A$ be a $k$-algebra. Assume\n\\begin{enumerate}\n\\item $A$ is a local ring essentially of finite type over $k$,\n\\item $A$ is a complete intersection over $k$\n(Algebra, Definition \\ref{algebra-definition-lci-local-ring}).\n\\end{enumerate}\nSet $d = \\dim(A) + \\text{trdeg}_k(\\kappa)$ where $\\kappa$\nis the residue field of $A$. Then there exists an integer $n$\nand a flat, essentially of finite type ring map\n$k[[t]] \\to B$ with $B/tB \\cong A$ such that $t^n$ is in the\n$d$th Fitting ideal of $\\Omega_{B/k[[t]]}$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Smoothings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7X","source_file":"examples-defos.tex","source_line":3075,"source_end_line":3088,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L3075-L3088","statement_sha256":"1c197952474766f63e607c644318dd630851039efac4ad0713fd7afc3425b085","origin":"The Stacks Project","memory_eligible":false,"source_rank":13601,"rank":13601,"depth":40,"x":1717.693,"y":1689.15,"cluster":"deformation-theory"},{"id":"stacks:0E7Y","tag":"0E7Y","title":"Smoothings · Lemma 0E7Y","summary":"Let X be a scheme over a field k. Assume • X is proper over k, • X is a local complete intersection over k, • X has dimension ≤ 1, and • X → Spec(k) is smooth except at finitely many points. Then there exists a flat projective morphism Y → Spec(k[[t]]) whose generic fibre is smooth and whose special fibre is isomorphic to X.","statement_latex":"Let $X$ be a scheme over a field $k$. Assume\n\\begin{enumerate}\n\\item $X$ is proper over $k$,\n\\item $X$ is a local complete intersection over $k$,\n\\item $X$ has dimension $\\leq 1$, and\n\\item $X \\to \\Spec(k)$ is smooth except at finitely many points.\n\\end{enumerate}\nThen there exists a flat projective morphism $Y \\to \\Spec(k[[t]])$\nwhose generic fibre is smooth and whose special fibre is\nisomorphic to $X$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Smoothings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7Y","source_file":"examples-defos.tex","source_line":3111,"source_end_line":3123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L3111-L3123","statement_sha256":"e20cf41518459d06a09059241b2305259bb8711a2de7711e112c1e553895c4bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13602,"rank":13602,"depth":70,"x":1406.523,"y":1612.411,"cluster":"deformation-theory"},{"id":"stacks:0E7Z","tag":"0E7Z","title":"Smoothings · Lemma 0E7Z","summary":"Let k be a field and let X be a scheme over k. Assume • X is separated, finite type over k and dim(X) ≤ 1, • X is a local complete intersection over k, and • X → Spec(k) is smooth except at finitely many points. Then there exists a flat, separated, finite type morphism Y → Spec(k[[t]]) whose generic fibre is smooth and whose special fibre is isomorphic to X.","statement_latex":"Let $k$ be a field and let $X$ be a scheme over $k$. Assume\n\\begin{enumerate}\n\\item $X$ is separated, finite type over $k$ and $\\dim(X) \\leq 1$,\n\\item $X$ is a local complete intersection over $k$, and\n\\item $X \\to \\Spec(k)$ is smooth except at finitely many points.\n\\end{enumerate}\nThen there exists a flat, separated, finite type morphism $Y \\to \\Spec(k[[t]])$\nwhose generic fibre is smooth and whose special fibre is\nisomorphic to $X$.","area":"Deformation Theory","chapter":"Deformation Problems","chapter_id":"examples-defos","section":"Smoothings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7Z","source_file":"examples-defos.tex","source_line":3198,"source_end_line":3209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-defos.tex#L3198-L3209","statement_sha256":"ad2518f7a947fd30a8672c88194549b2cd61c8e91986c0071025d5e90c8978e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13603,"rank":13603,"depth":71,"x":1698.11,"y":1492.258,"cluster":"deformation-theory"},{"id":"stacks:02ZR","tag":"02ZR","title":"Representable morphisms of categories fibred in groupoids · Lemma 02ZR","summary":"Let f : X → Y be a morphism of (Sch/S)_fppf. Then the 1-morphism induced by f (Sch/X)_fppf → (Sch/Y)_fppf is a representable 1-morphism.","statement_latex":"Let $f : X \\to Y$ be a morphism of $(\\Sch/S)_{fppf}$.\nThen the $1$-morphism induced by $f$\n$$\n(\\Sch/X)_{fppf} \\longrightarrow (\\Sch/Y)_{fppf}\n$$\nis a representable $1$-morphism.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Representable morphisms of categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZR","source_file":"algebraic.tex","source_line":255,"source_end_line":263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L255-L263","statement_sha256":"341243d1ade3bdb929f5f5c372ee36c93fc01cc77b551f5a8c287f3a08e99ca7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13604,"rank":13604,"depth":0,"x":2034.343,"y":1600.0,"cluster":"algebraic-stacks"},{"id":"stacks:0456","tag":"0456","title":"Representable morphisms of categories fibred in groupoids · Lemma 0456","summary":"Let S be an object of Sch_fppf. Consider a 2-commutative diagram xymatrix X' ar[r] ar[d]_f' & X ar[d]^f Y' ar[r] & Y of 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. Assume the horizontal arrows are equivalences. Then f is representable if and only if f' is representable.","statement_latex":"Let $S$ be an object of $\\Sch_{fppf}$.\nConsider a $2$-commutative diagram\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r] & \\mathcal{Y}\n}\n$$\nof $1$-morphisms of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$.\nAssume the horizontal arrows are equivalences.\nThen $f$ is representable if and only if $f'$ is representable.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Representable morphisms of categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0456","source_file":"algebraic.tex","source_line":270,"source_end_line":284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L270-L284","statement_sha256":"2cf9fc16e7ec26371573fa4d3af4674f2a155934ae05349e2d530efc60d6b295","origin":"The Stacks Project","memory_eligible":false,"source_rank":13605,"rank":13605,"depth":0,"x":2024.454,"y":1604.268,"cluster":"algebraic-stacks"},{"id":"stacks:02ZS","tag":"02ZS","title":"Representable morphisms of categories fibred in groupoids · Lemma 02ZS","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y, Z be categories fibred in groupoids over (Sch/S)_fppf Let f : X → Y, g : Y → Z be representable 1-morphisms. Then g ∘ f : X → Z is a representable 1-morphism.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$\nbe categories fibred in groupoids over $(\\Sch/S)_{fppf}$\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$\nbe representable $1$-morphisms. Then\n$$\ng \\circ f : \\mathcal{X} \\longrightarrow \\mathcal{Z}\n$$\nis a representable $1$-morphism.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Representable morphisms of categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZS","source_file":"algebraic.tex","source_line":290,"source_end_line":301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L290-L301","statement_sha256":"3785028c4659be3b613c527908b1d2ada8e1050a8fd8fca21f67440e9726a91b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13606,"rank":13606,"depth":0,"x":2030.849,"y":1591.875,"cluster":"algebraic-stacks"},{"id":"stacks:02ZT","tag":"02ZT","title":"Representable morphisms of categories fibred in groupoids · Lemma 02ZT","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y, Z be categories fibred in groupoids over (Sch/S)_fppf Let f : X → Y be a representable 1-morphism. Let g : Z → Y be any 1-morphism. Consider the fibre product diagram xymatrix Z ×_g, Y, f X ar[r]_-g' ar[d]_f' & X ar[d]^f Z ar[r]^g & Y Then the base change f' is a representable 1-morphism.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$\nbe categories fibred in groupoids over $(\\Sch/S)_{fppf}$\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a representable $1$-morphism.\nLet $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism.\nConsider the fibre product diagram\n$$\n\\xymatrix{\n\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} &\n\\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Z} \\ar[r]^g & \\mathcal{Y}\n}\n$$\nThen the base change $f'$ is a representable $1$-morphism.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Representable morphisms of categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZT","source_file":"algebraic.tex","source_line":307,"source_end_line":323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L307-L323","statement_sha256":"5dd8fcfcecc5dd06ec149225a0a669f817323d55b1c10fa7cc00e5ce245416b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13607,"rank":13607,"depth":0,"x":2036.991,"y":1607.659,"cluster":"algebraic-stacks"},{"id":"stacks:02ZU","tag":"02ZU","title":"Representable morphisms of categories fibred in groupoids · Lemma 02ZU","summary":"Let S be a scheme contained in Sch_fppf. Let X_i, Y_i be categories fibred in groupoids over (Sch/S)_fppf, i = 1, 2. Let f_i : X_i → Y_i, i = 1, 2 be representable 1-morphisms. Then f_1 × f_2 : X_1 × X_2 → Y_1 × Y_2 is a representable 1-morphism.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}_i, \\mathcal{Y}_i$ be categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$, $i = 1, 2$.\nLet $f_i : \\mathcal{X}_i \\to \\mathcal{Y}_i$, $i = 1, 2$\nbe representable $1$-morphisms.\nThen\n$$\nf_1 \\times f_2 :\n\\mathcal{X}_1 \\times \\mathcal{X}_2\n\\longrightarrow\n\\mathcal{Y}_1 \\times \\mathcal{Y}_2\n$$\nis a representable $1$-morphism.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Representable morphisms of categories fibred in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZU","source_file":"algebraic.tex","source_line":329,"source_end_line":344,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L329-L344","statement_sha256":"64e040b5e227453168ed4db50d23f227eaf64ab124e0f2f27dad6415aeaef5e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13608,"rank":13608,"depth":1,"x":2017.171,"y":1598.094,"cluster":"algebraic-stacks"},{"id":"stacks:04SV","tag":"04SV","title":"Categories fibred in groupoids representable by algebraic spaces · Definition 04SV","summary":"Let S be a scheme contained in Sch_fppf. A category fibred in groupoids p : X → (Sch/S)_fppf is called representable by an algebraic space over S if there exists an algebraic space F over S and an equivalence j : X → S_F of categories over (Sch/S)_fppf.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nA category fibred in groupoids $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$\nis called {\\it representable by an algebraic space over $S$}\nif there exists an algebraic space $F$ over $S$ and an equivalence\n$j : \\mathcal{X} \\to \\mathcal{S}_F$\nof categories over $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Categories fibred in groupoids representable by algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SV","source_file":"algebraic.tex","source_line":434,"source_end_line":442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L434-L442","statement_sha256":"1c9b364364f5b7cb0b8aba861d88266fb0482ab40019018679ef91c3ae574546","origin":"The Stacks Project","memory_eligible":false,"source_rank":13609,"rank":13609,"depth":0,"x":2042.152,"y":1593.507,"cluster":"algebraic-stacks"},{"id":"stacks:02ZX","tag":"02ZX","title":"Categories fibred in groupoids representable by algebraic spaces · Lemma 02ZX","summary":"Let S be a scheme contained in Sch_fppf. Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Then X is representable by an algebraic space over S if and only if the following conditions are satisfied: • X is fibred in setoids, and • the presheaf U ↦ Ob(X_U)/ ≅ is an algebraic space.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$\nbe a category fibred in groupoids.\nThen $\\mathcal{X}$ is representable by an algebraic space over $S$\nif and only if the following conditions are satisfied:\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is fibred in setoids\\footnote{This means that\nit is fibred in groupoids and objects in the fibre categories\nhave no nontrivial automorphisms, see Categories,\nDefinition \\ref{categories-definition-category-fibred-setoids}.}, and\n\\item the presheaf $U \\mapsto \\Ob(\\mathcal{X}_U)/\\!\\!\\cong$ is\nan algebraic space.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Categories fibred in groupoids representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZX","source_file":"algebraic.tex","source_line":453,"source_end_line":468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L453-L468","statement_sha256":"3e2384c699b6514a6330f2b9993c29fcca9c54329d7f7f3c5f1214efcf1c64ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":13610,"rank":13610,"depth":6,"x":2025.935,"y":1612.701,"cluster":"algebraic-stacks"},{"id":"stacks:02ZW","tag":"02ZW","title":"Morphisms representable by algebraic spaces · Definition 02ZW","summary":"Let S be a scheme contained in Sch_fppf. A 1-morphism f : X → Y of categories fibred in groupoids over (Sch/S)_fppf is called representable by algebraic spaces if for any U ∈ Ob((Sch/S)_fppf) and any y : (Sch/U)_fppf → Y the category fibred in groupoids (Sch/U)_fppf ×_y, Y X over (Sch/U)_fppf is representable by an algebraic space over U.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nA $1$-morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$\nis called {\\it representable by algebraic spaces} if\nfor any $U \\in \\Ob((\\Sch/S)_{fppf})$\nand any $y : (\\Sch/U)_{fppf} \\to \\mathcal{Y}$\nthe category fibred in groupoids\n$$\n(\\Sch/U)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}\n$$\nover $(\\Sch/U)_{fppf}$\nis representable by an algebraic space over $U$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZW","source_file":"algebraic.tex","source_line":504,"source_end_line":518,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L504-L518","statement_sha256":"df1b00a271ebf7de8ed22569274563e9b6d91d5d2587cba22feb058c6d3803d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13611,"rank":13611,"depth":0,"x":2022.248,"y":1587.462,"cluster":"algebraic-stacks"},{"id":"stacks:02ZY","tag":"02ZY","title":"Morphisms representable by algebraic spaces · Lemma 02ZY","summary":"Let S be a scheme contained in Sch_fppf. Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. The following are necessary and sufficient conditions for f to be representable by algebraic spaces: • for each scheme U/S the functor f_U : X_U → Y_U between fibre categories is faithful, and • for each U and each y ∈ Ob(Y_U) the presheaf (h : V → U) ↦ ((x, φ) mid x ∈ Ob(X_V), φ : h^*y → f(x))/≅ is an algebraic space over U. Here we have made a…","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nof categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nThe following are necessary and sufficient conditions for\n$f$ to be representable by algebraic spaces:\n\\begin{enumerate}\n\\item for each scheme $U/S$ the\nfunctor $f_U : \\mathcal{X}_U \\longrightarrow \\mathcal{Y}_U$\nbetween fibre categories is faithful, and\n\\item for each $U$ and each $y \\in \\Ob(\\mathcal{Y}_U)$ the presheaf\n$$\n(h : V \\to U)\n\\longmapsto\n\\{(x, \\phi) \\mid x \\in \\Ob(\\mathcal{X}_V), \\phi : h^*y \\to f(x)\\}/\\cong\n$$\nis an algebraic space over $U$.\n\\end{enumerate}\nHere we have made a choice of pullbacks for $\\mathcal{Y}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZY","source_file":"algebraic.tex","source_line":551,"source_end_line":571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L551-L571","statement_sha256":"bc705ab4b6e469d71b4aa272f963a866e67ccbd0e913e0278234ba4245ee9708","origin":"The Stacks Project","memory_eligible":false,"source_rank":13612,"rank":13612,"depth":7,"x":2046.818,"y":1605.159,"cluster":"algebraic-stacks"},{"id":"stacks:0457","tag":"0457","title":"Morphisms representable by algebraic spaces · Lemma 0457","summary":"Let S be an object of Sch_fppf. Consider a 2-commutative diagram xymatrix X' ar[r] ar[d]_f' & X ar[d]^f Y' ar[r] & Y of 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. Assume the horizontal arrows are equivalences. Then f is representable by algebraic spaces if and only if f' is representable by algebraic spaces.","statement_latex":"Let $S$ be an object of $\\Sch_{fppf}$.\nConsider a $2$-commutative diagram\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r] & \\mathcal{Y}\n}\n$$\nof $1$-morphisms of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$.\nAssume the horizontal arrows are equivalences.\nThen $f$ is representable by algebraic spaces\nif and only if $f'$ is representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0457","source_file":"algebraic.tex","source_line":584,"source_end_line":599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L584-L599","statement_sha256":"62511fe02d1681e8c5c37ee80a961d88faa8558c2156e6de443edf3fd0b49bb6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13613,"rank":13613,"depth":0,"x":2012.503,"y":1606.067,"cluster":"algebraic-stacks"},{"id":"stacks:02ZZ","tag":"02ZZ","title":"Morphisms representable by algebraic spaces · Lemma 02ZZ","summary":"Let S be an object of Sch_fppf. Let f : X → Y be a 1-morphism of categories fibred in groupoids over S. If X and Y are representable by algebraic spaces over S, then the 1-morphism f is representable by algebraic spaces.","statement_latex":"Let $S$ be an object of $\\Sch_{fppf}$.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of categories fibred in groupoids over $S$.\nIf $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by\nalgebraic spaces over $S$, then the $1$-morphism $f$\nis representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02ZZ","source_file":"algebraic.tex","source_line":605,"source_end_line":613,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L605-L613","statement_sha256":"c84a2a6441698c4b7918879e9e57bb39c6fc50f5a5efa508f63598d73087db8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13614,"rank":13614,"depth":8,"x":2038.435,"y":1584.86,"cluster":"algebraic-stacks"},{"id":"stacks:0458","tag":"0458","title":"Morphisms representable by algebraic spaces · Lemma 0458","summary":"Let S be an object of Sch_fppf. Let a : F → G be a map of presheaves of sets on (Sch/S)_fppf. Denote a' : S_F → S_G the associated map of categories fibred in sets. Then a is representable by algebraic spaces (see Bootstrap, Definition [Tag 02YQ]) if and only if a' is representable by algebraic spaces.","statement_latex":"Let $S$ be an object of $\\Sch_{fppf}$.\nLet $a : F \\to G$ be a map of presheaves of sets on $(\\Sch/S)_{fppf}$.\nDenote $a' : \\mathcal{S}_F  \\to \\mathcal{S}_G$ the associated\nmap of categories fibred in sets.\nThen $a$ is representable by algebraic spaces (see\nBootstrap,\nDefinition \\ref{bootstrap-definition-morphism-representable-by-spaces})\nif and only if $a'$ is representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0458","source_file":"algebraic.tex","source_line":621,"source_end_line":631,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L621-L631","statement_sha256":"1c6e99c35e7675b84c83a42aff55f1c6840d3ae66d10905f5fb69eebd9a244d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13615,"rank":13615,"depth":1,"x":2036.233,"y":1616.692,"cluster":"algebraic-stacks"},{"id":"stacks:04SY","tag":"04SY","title":"Morphisms representable by algebraic spaces · Lemma 04SY","summary":"Let S be an object of Sch_fppf. Let f : X → Y be a 1-morphism of categories fibred in setoids over (Sch/S)_fppf. Let F, resp. G be the presheaf which to T associates the set of isomorphism classes of objects of X_T, resp. Y_T. Let a : F → G be the map of presheaves corresponding to f. Then a is representable by algebraic spaces (see Bootstrap, Definition [Tag 02YQ]) if and only if f is representable by algebraic spaces.","statement_latex":"Let $S$ be an object of $\\Sch_{fppf}$.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of\ncategories fibred in setoids over $(\\Sch/S)_{fppf}$.\nLet $F$, resp.\\ $G$ be the presheaf which to $T$ associates\nthe set of isomorphism classes of objects of\n$\\mathcal{X}_T$, resp.\\ $\\mathcal{Y}_T$.\nLet $a : F \\to G$ be the map of presheaves corresponding to $f$.\nThen $a$ is representable by algebraic spaces (see\nBootstrap,\nDefinition \\ref{bootstrap-definition-morphism-representable-by-spaces})\nif and only if $f$ is representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SY","source_file":"algebraic.tex","source_line":637,"source_end_line":650,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L637-L650","statement_sha256":"482e6b1d60250e5690ab19e83c6cd637c12ce42a5aa8b78a3a3bc37d3bc4a70e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13616,"rank":13616,"depth":2,"x":2011.214,"y":1590.855,"cluster":"algebraic-stacks"},{"id":"stacks:0302","tag":"0302","title":"Morphisms representable by algebraic spaces · Lemma 0302","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y, Z be categories fibred in groupoids over (Sch/S)_fppf. Let f : X → Y be a 1-morphism representable by algebraic spaces. Let g : Z → Y be any 1-morphism. Consider the fibre product diagram xymatrix Z ×_g, Y, f X ar[r]_-g' ar[d]_f' & X ar[d]^f Z ar[r]^g & Y Then the base change f' is a 1-morphism representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$\nbe categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nrepresentable by algebraic spaces.\nLet $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism.\nConsider the fibre product diagram\n$$\n\\xymatrix{\n\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} &\n\\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Z} \\ar[r]^g & \\mathcal{Y}\n}\n$$\nThen the base change $f'$ is a $1$-morphism representable by\nalgebraic spaces.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0302","source_file":"algebraic.tex","source_line":658,"source_end_line":676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L658-L676","statement_sha256":"9aaf7178141359a17da601d1350049f9e045edd2d28bddd17375065466420eee","origin":"The Stacks Project","memory_eligible":false,"source_rank":13617,"rank":13617,"depth":0,"x":2052.038,"y":1595.93,"cluster":"algebraic-stacks"},{"id":"stacks:0300","tag":"0300","title":"Morphisms representable by algebraic spaces · Lemma 0300","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y, Z be categories fibred in groupoids over (Sch/S)_fppf Let f : X → Y, g : Z → Y be 1-morphisms. Assume • f is representable by algebraic spaces, and • Z is representable by an algebraic space over S. Then the 2-fibre product Z ×_g, Y, f X is representable by an algebraic space.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$\nbe categories fibred in groupoids over $(\\Sch/S)_{fppf}$\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$,\n$g : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms.\nAssume\n\\begin{enumerate}\n\\item $f$ is representable by algebraic spaces, and\n\\item $\\mathcal{Z}$ is representable by an algebraic space over $S$.\n\\end{enumerate}\nThen the $2$-fibre product\n$\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X}$\nis representable by an algebraic space.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0300","source_file":"algebraic.tex","source_line":682,"source_end_line":697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L682-L697","statement_sha256":"514ed1501fa5223b958d81a3d14004d02f450aa2a5919bff7bc0967d2c2791f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13618,"rank":13618,"depth":53,"x":2016.55,"y":1616.07,"cluster":"algebraic-stacks"},{"id":"stacks:0301","tag":"0301","title":"Morphisms representable by algebraic spaces · Lemma 0301","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y, Z be categories fibred in groupoids over (Sch/S)_fppf. If f : X → Y, g : Y → Z are 1-morphisms representable by algebraic spaces, then g ∘ f : X → Z is a 1-morphism representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$\nbe categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$\nare $1$-morphisms representable by algebraic spaces, then\n$$\ng \\circ f : \\mathcal{X} \\longrightarrow \\mathcal{Z}\n$$\nis a $1$-morphism representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0301","source_file":"algebraic.tex","source_line":751,"source_end_line":762,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L751-L762","statement_sha256":"b769b0aaf3a9c4fa1478f8b25bf737350f640d8c7cd8c96ba4d5018660b60855","origin":"The Stacks Project","memory_eligible":false,"source_rank":13619,"rank":13619,"depth":54,"x":2026.893,"y":1579.859,"cluster":"algebraic-stacks"},{"id":"stacks:0303","tag":"0303","title":"Morphisms representable by algebraic spaces · Lemma 0303","summary":"Let S be a scheme contained in Sch_fppf. Let X_i, Y_i be categories fibred in groupoids over (Sch/S)_fppf, i = 1, 2. Let f_i : X_i → Y_i, i = 1, 2 be 1-morphisms representable by algebraic spaces. Then f_1 × f_2 : X_1 × X_2 → Y_1 × Y_2 is a 1-morphism representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}_i, \\mathcal{Y}_i$ be categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$, $i = 1, 2$.\nLet $f_i : \\mathcal{X}_i \\to \\mathcal{Y}_i$, $i = 1, 2$\nbe $1$-morphisms representable by algebraic spaces.\nThen\n$$\nf_1 \\times f_2 :\n\\mathcal{X}_1 \\times \\mathcal{X}_2\n\\longrightarrow\n\\mathcal{Y}_1 \\times \\mathcal{Y}_2\n$$\nis a $1$-morphism representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0303","source_file":"algebraic.tex","source_line":770,"source_end_line":785,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L770-L785","statement_sha256":"ead7db4d976ac30ce131e497d8be802ca422c2a277a60dacd82c4f10c2d2dfea","origin":"The Stacks Project","memory_eligible":false,"source_rank":13620,"rank":13620,"depth":55,"x":2049.075,"y":1613.504,"cluster":"algebraic-stacks"},{"id":"stacks:0CKY","tag":"0CKY","title":"Morphisms representable by algebraic spaces · Lemma 0CKY","summary":"Lemma in an email of Matthew Emerton dated June 15, 2016 Let S be a scheme contained in Sch_fppf. Let X → Z and Y → Z be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If X → Z is representable by algebraic spaces and Y is a stack in groupoids, then X ×_Z Y is a stack in groupoids.","statement_latex":"\\begin{reference}\nLemma in an email of Matthew Emerton dated June 15, 2016\n\\end{reference}\nLet $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X} \\to \\mathcal{Z}$ and $\\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $\\mathcal{X} \\to \\mathcal{Z}$ is representable by algebraic spaces\nand $\\mathcal{Y}$ is a stack in groupoids, then\n$\\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y}$ is a stack in groupoids.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKY","source_file":"algebraic.tex","source_line":801,"source_end_line":812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L801-L812","statement_sha256":"395dbe142d1d874185341bf9b9f9f50ab7cd23154cf10f6cd4b3b99dd0abbaee","origin":"The Stacks Project","memory_eligible":false,"source_rank":13621,"rank":13621,"depth":54,"x":2004.331,"y":1600.892,"cluster":"algebraic-stacks"},{"id":"stacks:03YK","tag":"03YK","title":"Properties of morphisms representable by algebraic spaces · Definition 03YK","summary":"Let S be a scheme contained in Sch_fppf. Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Assume f is representable by algebraic spaces. Let P be a property of morphisms of algebraic spaces which • is preserved under any base change, and • is fppf local on the base, see Descent on Spaces, Definition [Tag 03YH]. In this case we say that f has property P if for every U ∈ Ob((Sch/S)_fppf) and any y ∈ Y_U the resulting morphism of algebraic…","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nof categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nAssume $f$ is representable by algebraic spaces.\nLet $\\mathcal{P}$ be a property of morphisms of algebraic spaces which\n\\begin{enumerate}\n\\item is preserved under any base change, and\n\\item is fppf local on the base, see\nDescent on Spaces,\nDefinition \\ref{spaces-descent-definition-property-morphisms-local}.\n\\end{enumerate}\nIn this case we say that $f$ has {\\it property $\\mathcal{P}$} if for every\n$U \\in \\Ob((\\Sch/S)_{fppf})$ and\nany $y \\in \\mathcal{Y}_U$ the resulting morphism of algebraic spaces\n$f_y : F_y \\to U$, see\ndiagram (\\ref{equation-representable-by-algebraic-spaces}),\nhas property $\\mathcal{P}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YK","source_file":"algebraic.tex","source_line":843,"source_end_line":862,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L843-L862","statement_sha256":"82a67fe07eed2219b0d4472abbbbf34eda3e2a2938d04e4716015e72d5c614b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13622,"rank":13622,"depth":1,"x":2048.724,"y":1584.349,"cluster":"algebraic-stacks"},{"id":"stacks:0459","tag":"0459","title":"Properties of morphisms representable by algebraic spaces · Lemma 0459","summary":"Let S be an object of Sch_fppf. Let P be as in Definition [Tag 03YK]. Consider a 2-commutative diagram xymatrix X' ar[r] ar[d]_f' & X ar[d]^f Y' ar[r] & Y of 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. Assume the horizontal arrows are equivalences and f (or equivalently f') is representable by algebraic spaces. Then f has P if and only if f' has P.","statement_latex":"Let $S$ be an object of $\\Sch_{fppf}$.\nLet $\\mathcal{P}$ be as in\nDefinition \\ref{definition-relative-representable-property}.\nConsider a $2$-commutative diagram\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r] & \\mathcal{Y}\n}\n$$\nof $1$-morphisms of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$.\nAssume the horizontal arrows are equivalences and $f$ (or equivalently $f'$)\nis representable by algebraic spaces.\nThen $f$ has $\\mathcal{P}$ if and only if $f'$ has $\\mathcal{P}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0459","source_file":"algebraic.tex","source_line":872,"source_end_line":889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L872-L889","statement_sha256":"b8a54aa1dea5f0c346a99d0ead43ec0119e4fc095bd5255073bbc14a33f48248","origin":"The Stacks Project","memory_eligible":false,"source_rank":13623,"rank":13623,"depth":2,"x":2028.747,"y":1622.756,"cluster":"algebraic-stacks"},{"id":"stacks:045A","tag":"045A","title":"Properties of morphisms representable by algebraic spaces · Lemma 045A","summary":"Let S be a scheme contained in Sch_fppf. Let a : F → G be a map of presheaves on (Sch/S)_fppf. Let P be as in Definition [Tag 03YK]. Assume a is representable by algebraic spaces. Then a : F → G has property P (see Bootstrap, Definition [Tag 03XZ]) if and only if the corresponding morphism S_F → S_G of categories fibred in groupoids has property P.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $a : F \\to G$ be a map of presheaves on $(\\Sch/S)_{fppf}$.\nLet $\\mathcal{P}$ be as in\nDefinition \\ref{definition-relative-representable-property}.\nAssume $a$ is representable by algebraic spaces.\nThen $a : F \\to G$ has property $\\mathcal{P}$ (see\nBootstrap, Definition \\ref{bootstrap-definition-property-transformation})\nif and only if the corresponding morphism\n$\\mathcal{S}_F \\to \\mathcal{S}_G$ of categories fibred in groupoids\nhas property $\\mathcal{P}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045A","source_file":"algebraic.tex","source_line":900,"source_end_line":912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L900-L912","statement_sha256":"90b21e5b0662fe4dc6648fffacdb019e77871c9b584e30e4d3917a586a9745eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13624,"rank":13624,"depth":2,"x":2012.185,"y":1582.067,"cluster":"algebraic-stacks"},{"id":"stacks:04TC","tag":"04TC","title":"Properties of morphisms representable by algebraic spaces · Lemma 04TC","summary":"Let S be an object of Sch_fppf. Let P be as in Definition [Tag 03YK]. Let f : X → Y be a 1-morphism of categories fibred in setoids over (Sch/S)_fppf. Let F, resp. G be the presheaf which to T associates the set of isomorphism classes of objects of X_T, resp. Y_T. Let a : F → G be the map of presheaves corresponding to f. Then a has P if and only if f has P.","statement_latex":"Let $S$ be an object of $\\Sch_{fppf}$. Let $\\mathcal{P}$ be as in\nDefinition \\ref{definition-relative-representable-property}.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of\ncategories fibred in setoids over $(\\Sch/S)_{fppf}$.\nLet $F$, resp.\\ $G$ be the presheaf which to $T$ associates\nthe set of isomorphism classes of objects of\n$\\mathcal{X}_T$, resp.\\ $\\mathcal{Y}_T$.\nLet $a : F \\to G$ be the map of presheaves corresponding to $f$.\nThen $a$ has $\\mathcal{P}$ if and only if $f$ has $\\mathcal{P}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TC","source_file":"algebraic.tex","source_line":920,"source_end_line":931,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L920-L931","statement_sha256":"365847a1598d0c087ba9272ee146d0f6bebc80b0f2c30a7487fe3657b4803d16","origin":"The Stacks Project","memory_eligible":false,"source_rank":13625,"rank":13625,"depth":3,"x":2058.222,"y":1603.19,"cluster":"algebraic-stacks"},{"id":"stacks:045B","tag":"045B","title":"Properties of morphisms representable by algebraic spaces · Lemma 045B","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y, Z be categories fibred in groupoids over (Sch/S)_fppf. Let P be a property as in Definition [Tag 03YK] which is stable under composition. Let f : X → Y, g : Y → Z be 1-morphisms which are representable by algebraic spaces. If f and g have property P so does g ∘ f : X → Z.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$, $\\mathcal{Y}$, $\\mathcal{Z}$ be categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-relative-representable-property}\nwhich is stable under composition.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$,\n$g : \\mathcal{Y} \\to \\mathcal{Z}$ be $1$-morphisms which\nare representable by algebraic spaces.\nIf $f$ and $g$ have property $\\mathcal{P}$ so does\n$g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045B","source_file":"algebraic.tex","source_line":941,"source_end_line":954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L941-L954","statement_sha256":"b6f09dd67fd3350ae4c29a05780417f4b7e8b68796bbba0f4c67f5f93c42f1d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13626,"rank":13626,"depth":55,"x":2006.088,"y":1613.975,"cluster":"algebraic-stacks"},{"id":"stacks:045C","tag":"045C","title":"Properties of morphisms representable by algebraic spaces · Lemma 045C","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y, Z be categories fibred in groupoids over (Sch/S)_fppf. Let P be a property as in Definition [Tag 03YK]. Let f : X → Y be a 1-morphism representable by algebraic spaces. Let g : Z → Y be any 1-morphism. Consider the 2-fibre product diagram xymatrix Z ×_g, Y, f X ar[r]_-g' ar[d]_f' & X ar[d]^f Z ar[r]^g & Y If f has P, then the base change f' has P.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$\nbe categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-relative-representable-property}.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nrepresentable by algebraic spaces.\nLet $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism.\nConsider the $2$-fibre product diagram\n$$\n\\xymatrix{\n\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} &\n\\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Z} \\ar[r]^g & \\mathcal{Y}\n}\n$$\nIf $f$ has $\\mathcal{P}$, then the base change $f'$\nhas $\\mathcal{P}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045C","source_file":"algebraic.tex","source_line":962,"source_end_line":982,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L962-L982","statement_sha256":"039d0a967bd137714d70bec378c0cf71abacc4149de4cb06f0c553517fdf0749","origin":"The Stacks Project","memory_eligible":false,"source_rank":13627,"rank":13627,"depth":2,"x":2036.534,"y":1575.602,"cluster":"algebraic-stacks"},{"id":"stacks:045D","tag":"045D","title":"Properties of morphisms representable by algebraic spaces · Lemma 045D","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y, Z be categories fibred in groupoids over (Sch/S)_fppf. Let P be a property as in Definition [Tag 03YK]. Let f : X → Y be a 1-morphism representable by algebraic spaces. Let g : Z → Y be any 1-morphism. Consider the fibre product diagram xymatrix Z ×_g, Y, f X ar[r]_-g' ar[d]_f' & X ar[d]^f Z ar[r]^g & Y Assume that for every scheme U and object x of Y_U, there exists an fppf covering (U_i → U) such that x|_U_i is in the…","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$\nbe categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-relative-representable-property}.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nrepresentable by algebraic spaces.\nLet $g : \\mathcal{Z} \\to \\mathcal{Y}$ be any $1$-morphism.\nConsider the fibre product diagram\n$$\n\\xymatrix{\n\\mathcal{Z} \\times_{g, \\mathcal{Y}, f} \\mathcal{X} \\ar[r]_-{g'} \\ar[d]_{f'} &\n\\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Z} \\ar[r]^g & \\mathcal{Y}\n}\n$$\nAssume that for every scheme $U$ and object $x$ of $\\mathcal{Y}_U$,\nthere exists an fppf covering $\\{U_i \\to U\\}$ such that $x|_{U_i}$\nis in the essential image of the functor\n$g : \\mathcal{Z}_{U_i} \\to \\mathcal{Y}_{U_i}$.\nIn this case, if $f'$ has $\\mathcal{P}$, then $f$ has $\\mathcal{P}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045D","source_file":"algebraic.tex","source_line":990,"source_end_line":1013,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L990-L1013","statement_sha256":"578d2a0d9c04387d2b3b51c9766044c1a07ec5b8f20717e6647e1a4273919e75","origin":"The Stacks Project","memory_eligible":false,"source_rank":13628,"rank":13628,"depth":3,"x":2045.113,"y":1622.155,"cluster":"algebraic-stacks"},{"id":"stacks:045E","tag":"045E","title":"Properties of morphisms representable by algebraic spaces · Lemma 045E","summary":"Let S be a scheme contained in Sch_fppf. Let P be a property as in Definition [Tag 03YK] which is stable under composition. Let X_i, Y_i be categories fibred in groupoids over (Sch/S)_fppf, i = 1, 2. Let f_i : X_i → Y_i, i = 1, 2 be 1-morphisms representable by algebraic spaces. If f_1 and f_2 have property P so does f_1 × f_2 : X_1 × X_2 → Y_1 × Y_2 .","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{P}$ be a property as in\nDefinition \\ref{definition-relative-representable-property}\nwhich is stable under composition.\nLet $\\mathcal{X}_i, \\mathcal{Y}_i$ be categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$, $i = 1, 2$.\nLet $f_i : \\mathcal{X}_i \\to \\mathcal{Y}_i$, $i = 1, 2$\nbe $1$-morphisms representable by algebraic spaces.\nIf $f_1$ and $f_2$ have property $\\mathcal{P}$ so does\n$\nf_1 \\times f_2 :\n\\mathcal{X}_1 \\times \\mathcal{X}_2\n\\to\n\\mathcal{Y}_1 \\times \\mathcal{Y}_2\n$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045E","source_file":"algebraic.tex","source_line":1021,"source_end_line":1038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1021-L1038","statement_sha256":"f45b1098f959efc1f40367eb4930f08a0c44865b3e9caee846cbc10d52cd8e39","origin":"The Stacks Project","memory_eligible":false,"source_rank":13629,"rank":13629,"depth":56,"x":2000.455,"y":1592.082,"cluster":"algebraic-stacks"},{"id":"stacks:045F","tag":"045F","title":"Properties of morphisms representable by algebraic spaces · Lemma 045F","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y be categories fibred in groupoids over (Sch/S)_fppf. Let f : X → Y be a 1-morphism representable by algebraic spaces. Let P, P' be properties as in Definition [Tag 03YK]. Suppose that for any morphism of algebraic spaces a : F → G we have P(a) ⇒ P'(a). If f has property P then f has property P'.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$, $\\mathcal{Y}$ be categories fibred in groupoids\nover $(\\Sch/S)_{fppf}$.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism representable\nby algebraic spaces.\nLet $\\mathcal{P}$, $\\mathcal{P}'$ be properties as in\nDefinition \\ref{definition-relative-representable-property}.\nSuppose that for any morphism of algebraic spaces $a : F \\to G$\nwe have $\\mathcal{P}(a) \\Rightarrow \\mathcal{P}'(a)$.\nIf $f$ has property $\\mathcal{P}$ then\n$f$ has property $\\mathcal{P}'$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045F","source_file":"algebraic.tex","source_line":1046,"source_end_line":1059,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1046-L1059","statement_sha256":"f0b57d9a3b9f1725c743619f0d95cf4b3cf4fcb0897c6adb913b50d964779350","origin":"The Stacks Project","memory_eligible":false,"source_rank":13630,"rank":13630,"depth":2,"x":2058.699,"y":1588.862,"cluster":"algebraic-stacks"},{"id":"stacks:05UK","tag":"05UK","title":"Properties of morphisms representable by algebraic spaces · Lemma 05UK","summary":"Let S be a scheme contained in Sch_fppf. Let j : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Assume j is representable by algebraic spaces and a monomorphism (see Definition [Tag 03YK] and Descent on Spaces, Lemma [Tag 042D]). Then j is fully faithful on fibre categories.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $j : \\mathcal X \\to \\mathcal Y$ be a $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nAssume $j$ is representable by algebraic spaces and a monomorphism\n(see\nDefinition \\ref{definition-relative-representable-property}\nand\nDescent on Spaces, Lemma\n\\ref{spaces-descent-lemma-descending-property-monomorphism}).\nThen $j$ is fully faithful on fibre categories.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UK","source_file":"algebraic.tex","source_line":1065,"source_end_line":1077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1065-L1077","statement_sha256":"2f2cc637987ec85a233428eefe6131dfa4566e6f49c45c25fb29a57e3a930a45","origin":"The Stacks Project","memory_eligible":false,"source_rank":13631,"rank":13631,"depth":57,"x":2017.567,"y":1624.955,"cluster":"algebraic-stacks"},{"id":"stacks:045G","tag":"045G","title":"Properties of morphisms representable by algebraic spaces · Lemma 045G","summary":"Let S be a scheme contained in Sch_fppf. Let X be a category fibred in groupoids over (Sch/S)_fppf. The following are equivalent: • the diagonal X → X × X is representable by algebraic spaces, • for every scheme U over S, and any x, y ∈ Ob(X_U) the sheaf mathitIsom(x, y) is an algebraic space over U, • for every scheme U over S, and any x ∈ Ob(X_U) the associated 1-morphism x : (Sch/U)_fppf → X is representable by algebraic spaces, • for every pair of schemes T_1, T_2…","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$ be a category fibred in groupoids over\n$(\\Sch/S)_{fppf}$. The following are equivalent:\n\\begin{enumerate}\n\\item the diagonal $\\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nis representable by algebraic spaces,\n\\item for every scheme $U$ over $S$, and any\n$x, y \\in \\Ob(\\mathcal{X}_U)$ the sheaf\n$\\mathit{Isom}(x, y)$ is an algebraic space over $U$,\n\\item for every scheme $U$ over $S$, and any $x \\in \\Ob(\\mathcal{X}_U)$\nthe associated $1$-morphism $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$\nis representable by algebraic spaces,\n\\item for every pair of schemes $T_1, T_2$ over $S$, and any\n$x_i \\in \\Ob(\\mathcal{X}_{T_i})$, $i = 1, 2$ the $2$-fibre product\n$(\\Sch/T_1)_{fppf} \\times_{x_1, \\mathcal{X}, x_2}\n(\\Sch/T_2)_{fppf}$\nis representable by an algebraic space,\n\\item for every representable category fibred in groupoids $\\mathcal{U}$\nover $(\\Sch/S)_{fppf}$ every $1$-morphism\n$\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces,\n\\item for every pair $\\mathcal{T}_1, \\mathcal{T}_2$ of representable\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$ and any\n$1$-morphisms $x_i : \\mathcal{T}_i \\to \\mathcal{X}$, $i = 1, 2$ the\n$2$-fibre product $\\mathcal{T}_1 \\times_{x_1, \\mathcal{X}, x_2} \\mathcal{T}_2$\nis representable by an algebraic space,\n\\item for every category fibred in groupoids $\\mathcal{U}$\nover $(\\Sch/S)_{fppf}$ which is\nrepresentable by an algebraic space every $1$-morphism\n$\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces,\n\\item for every pair $\\mathcal{T}_1, \\mathcal{T}_2$ of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$ which are representable\nby algebraic spaces, and any $1$-morphisms\n$x_i : \\mathcal{T}_i \\to \\mathcal{X}$ the\n$2$-fibre product $\\mathcal{T}_1 \\times_{x_1, \\mathcal{X}, x_2} \\mathcal{T}_2$\nis representable by an algebraic space.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045G","source_file":"algebraic.tex","source_line":1119,"source_end_line":1157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1119-L1157","statement_sha256":"3afeb0d71a282a9d1e357be87f1e673aec3998b0560bd375c72a948598e38828","origin":"The Stacks Project","memory_eligible":false,"source_rank":13632,"rank":13632,"depth":54,"x":2018.904,"y":1574.085,"cluster":"algebraic-stacks"},{"id":"stacks:026O","tag":"026O","title":"Algebraic stacks · Definition 026O","summary":"Let S be a base scheme contained in Sch_fppf. An algebraic stack over S is a category p : X → (Sch/S)_fppf over (Sch/S)_fppf with the following properties: • The category X is a stack in groupoids over (Sch/S)_fppf. • The diagonal Δ : X → X × X is representable by algebraic spaces. • There exists a scheme U ∈ Ob((Sch/S)_fppf) and a 1-morphism (Sch/U)_fppf → X which is surjective and smooth as is customary in the literature. Another good alternative would be to formulate…","statement_latex":"Let $S$ be a base scheme contained in $\\Sch_{fppf}$.\nAn {\\it algebraic stack over $S$} is a category\n$$\np : \\mathcal{X} \\to (\\Sch/S)_{fppf}\n$$\nover $(\\Sch/S)_{fppf}$ with the following properties:\n\\begin{enumerate}\n\\item The category $\\mathcal{X}$ is a stack in groupoids over\n$(\\Sch/S)_{fppf}$.\n\\item The diagonal\n$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nis representable by algebraic spaces.\n\\item There exists a scheme $U \\in \\Ob((\\Sch/S)_{fppf})$\nand a $1$-morphism $(\\Sch/U)_{fppf} \\to \\mathcal{X}$\nwhich is surjective and smooth\\footnote{In future chapters we will denote\nthis simply $U \\to \\mathcal{X}$ as is customary in the literature. Another\ngood alternative would be to formulate this condition as the existence of a\nrepresentable category fibred in groupoids $\\mathcal{U}$ and a surjective\nsmooth $1$-morphism $\\mathcal{U} \\to \\mathcal{X}$.}.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/026O","source_file":"algebraic.tex","source_line":1288,"source_end_line":1310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1288-L1310","statement_sha256":"6007ded895cc919de87271128cf6a6aca3dd85efbebdf83638258b301d3761ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":13633,"rank":13633,"depth":0,"x":2059.526,"y":1613.035,"cluster":"algebraic-stacks"},{"id":"stacks:03YO","tag":"03YO","title":"Algebraic stacks · Definition 03YO","summary":"Let S be a scheme contained in Sch_fppf. Let X be an algebraic stack over S. We say X is a Deligne-Mumford stack if there exists a scheme U and a surjective étale morphism (Sch/U)_fppf → X.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$ be an algebraic stack over $S$.\nWe say $\\mathcal{X}$ is a {\\it Deligne-Mumford stack} if there exists\na scheme $U$ and a surjective \\'etale morphism\n$(\\Sch/U)_{fppf} \\to \\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YO","source_file":"algebraic.tex","source_line":1351,"source_end_line":1358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1351-L1358","statement_sha256":"d9af0a082813cbf0f3ec0007b51e11a97a1dfff23a6842bb4b74b217ca290eac","origin":"The Stacks Project","memory_eligible":false,"source_rank":13634,"rank":13634,"depth":0,"x":1997.204,"y":1607.262,"cluster":"algebraic-stacks"},{"id":"stacks:03YP","tag":"03YP","title":"Algebraic stacks · Definition 03YP","summary":"Let S be a scheme contained in Sch_fppf. The 2-category of algebraic stacks over S is the sub 2-category of the 2-category of categories fibred in groupoids over (Sch/S)_fppf (see Categories, Definition [Tag 02XS]) defined as follows: • Its objects are those categories fibred in groupoids over (Sch/S)_fppf which are algebraic stacks over S. • Its 1-morphisms f : X → Y are any functors of categories over (Sch/S)_fppf, as in Categories, Definition [Tag 003Y]. • Its…","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nThe {\\it $2$-category of algebraic stacks over $S$} is the\nsub $2$-category of the $2$-category of categories fibred in\ngroupoids over $(\\Sch/S)_{fppf}$ (see\nCategories,\nDefinition \\ref{categories-definition-categories-fibred-in-groupoids-over-C})\ndefined as follows:\n\\begin{enumerate}\n\\item Its objects are those categories fibred in groupoids\nover $(\\Sch/S)_{fppf}$ which are algebraic stacks over $S$.\n\\item Its $1$-morphisms $f : \\mathcal{X} \\to \\mathcal{Y}$ are\nany functors of categories over $(\\Sch/S)_{fppf}$, as in\nCategories, Definition \\ref{categories-definition-categories-over-C}.\n\\item Its $2$-morphisms are transformations between functors\nover $(\\Sch/S)_{fppf}$, as in\nCategories, Definition \\ref{categories-definition-categories-over-C}.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YP","source_file":"algebraic.tex","source_line":1369,"source_end_line":1388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1369-L1388","statement_sha256":"bfa13f3a29cdf03459e3c6dfd99aea455079899950d74862a8ffa90fff4a7d09","origin":"The Stacks Project","memory_eligible":false,"source_rank":13635,"rank":13635,"depth":3,"x":2048.642,"y":1575.647,"cluster":"algebraic-stacks"},{"id":"stacks:03YQ","tag":"03YQ","title":"Algebraic stacks · Lemma 03YQ","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y be categories over (Sch/S)_fppf. Assume X, Y are equivalent as categories over (Sch/S)_fppf. Then X is an algebraic stack if and only if Y is an algebraic stack. Similarly, X is a Deligne-Mumford stack if and only if Y is a Deligne-Mumford stack.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$, $\\mathcal{Y}$ be categories over $(\\Sch/S)_{fppf}$.\nAssume $\\mathcal{X}$, $\\mathcal{Y}$ are equivalent as categories over\n$(\\Sch/S)_{fppf}$. Then $\\mathcal{X}$ is an algebraic stack if and\nonly if $\\mathcal{Y}$ is an algebraic stack. Similarly, $\\mathcal{X}$\nis a Deligne-Mumford stack if and only if $\\mathcal{Y}$ is a Deligne-Mumford\nstack.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YQ","source_file":"algebraic.tex","source_line":1406,"source_end_line":1415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1406-L1415","statement_sha256":"a29dd31e42b280c0a8f9d56eae89816c856d036a5ae447d8dc21a6dd69671d53","origin":"The Stacks Project","memory_eligible":false,"source_rank":13636,"rank":13636,"depth":6,"x":2035.93,"y":1628.984,"cluster":"algebraic-stacks"},{"id":"stacks:03YS","tag":"03YS","title":"Algebraic stacks and algebraic spaces · Lemma 03YS","summary":"Let S be a scheme contained in Sch_fppf. • A category fibred in groupoids p : X → (Sch/S)_fppf which is representable by an algebraic space is a Deligne-Mumford stack. • If F is an algebraic space over S, then the associated category fibred in groupoids p : S_F → (Sch/S)_fppf is a Deligne-Mumford stack. • If X ∈ Ob((Sch/S)_fppf), then (Sch/X)_fppf → (Sch/S)_fppf is a Deligne-Mumford stack.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\n\\begin{enumerate}\n\\item A category fibred in groupoids\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$\nwhich is representable by an algebraic space is a Deligne-Mumford stack.\n\\item If $F$ is an algebraic space over $S$, then the associated\ncategory fibred in groupoids\n$p : \\mathcal{S}_F \\to (\\Sch/S)_{fppf}$\nis a Deligne-Mumford stack.\n\\item If $X \\in \\Ob((\\Sch/S)_{fppf})$, then\n$(\\Sch/X)_{fppf} \\to (\\Sch/S)_{fppf}$ is\na Deligne-Mumford stack.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks and algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YS","source_file":"algebraic.tex","source_line":1464,"source_end_line":1479,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1464-L1479","statement_sha256":"c8b69a86a8a8179e3ef8c0bb317ca6bd4387f78b6c5109fd49326b27c90b571b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13637,"rank":13637,"depth":9,"x":2001.897,"y":1581.718,"cluster":"algebraic-stacks"},{"id":"stacks:045H","tag":"045H","title":"Algebraic stacks and algebraic spaces · Lemma 045H","summary":"Let S be a scheme contained in Sch_fppf. Let X be an algebraic stack over S. The following are equivalent • X is a Deligne-Mumford stack and is a stack in setoids, • X is a Deligne-Mumford stack such that the canonical 1-morphism I_X → X is an equivalence, and • X is representable by an algebraic space.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$ be an algebraic stack over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is a Deligne-Mumford stack and is a stack in setoids,\n\\item $\\mathcal{X}$ is a Deligne-Mumford stack such that the\ncanonical $1$-morphism $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$\nis an equivalence, and\n\\item $\\mathcal{X}$ is representable by an algebraic space.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks and algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/045H","source_file":"algebraic.tex","source_line":1512,"source_end_line":1524,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1512-L1524","statement_sha256":"efd2c690b3c318f95c5c84b8866e03a0910b853abd1755afde75d4c1a638e185","origin":"The Stacks Project","memory_eligible":false,"source_rank":13638,"rank":13638,"depth":59,"x":2065.949,"y":1597.498,"cluster":"algebraic-stacks"},{"id":"stacks:04SZ","tag":"04SZ","title":"Algebraic stacks and algebraic spaces · Proposition 04SZ","summary":"Let S be a scheme contained in Sch_fppf. Let X be an algebraic stack over S. The following are equivalent • X is a stack in setoids, • the canonical 1-morphism I_X → X is an equivalence, and • X is representable by an algebraic space.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$ be an algebraic stack over $S$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is a stack in setoids,\n\\item the canonical $1$-morphism $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$\nis an equivalence, and\n\\item $\\mathcal{X}$ is representable by an algebraic space.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks and algebraic spaces","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04SZ","source_file":"algebraic.tex","source_line":1553,"source_end_line":1564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1553-L1564","statement_sha256":"2265beaba1e14a64edac850a2e0756bc595fc144b391d020a7f3c92f42928454","origin":"The Stacks Project","memory_eligible":false,"source_rank":13639,"rank":13639,"depth":67,"x":2005.152,"y":1622.563,"cluster":"algebraic-stacks"},{"id":"stacks:04TE","tag":"04TE","title":"2-Fibre products of algebraic stacks · Lemma 04TE","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y be algebraic stacks over S. Then X ×_(Sch/S)_fppf Y is an algebraic stack, and is a product in the 2-category of algebraic stacks over S.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$, $\\mathcal{Y}$ be algebraic stacks over $S$.\nThen $\\mathcal{X} \\times_{(\\Sch/S)_{fppf}} \\mathcal{Y}$\nis an algebraic stack, and is a product in the $2$-category of\nalgebraic stacks over $S$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"2-Fibre products of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TE","source_file":"algebraic.tex","source_line":1615,"source_end_line":1622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1615-L1622","statement_sha256":"93bd6dd60efade7bd60748589f7399fdbb0adb919012fd4717782e4e78b13468","origin":"The Stacks Project","memory_eligible":false,"source_rank":13640,"rank":13640,"depth":56,"x":2030.181,"y":1568.834,"cluster":"algebraic-stacks"},{"id":"stacks:04TF","tag":"04TF","title":"2-Fibre products of algebraic stacks · Lemma 04TF","summary":"Let S be a scheme contained in Sch_fppf. Let Z be a stack in groupoids over (Sch/S)_fppf whose diagonal is representable by algebraic spaces. Let X, Y be algebraic stacks over S. Let f : X → Z, g : Y → Z be 1-morphisms of stacks in groupoids. Then the 2-fibre product X ×_f, Z, g Y is an algebraic stack.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{Z}$ be a stack in groupoids over $(\\Sch/S)_{fppf}$\nwhose diagonal is representable by algebraic spaces.\nLet $\\mathcal{X}$, $\\mathcal{Y}$ be algebraic stacks over $S$.\nLet $f : \\mathcal{X} \\to \\mathcal{Z}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of stacks in groupoids. Then the $2$-fibre product\n$\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y}$ is an algebraic stack.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"2-Fibre products of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TF","source_file":"algebraic.tex","source_line":1669,"source_end_line":1678,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1669-L1678","statement_sha256":"d21c09ee3d9cdb08cda72ed81ad67c5b241731ac1113dbffd89d9a9d5179e3fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13641,"rank":13641,"depth":56,"x":2055.268,"y":1623.398,"cluster":"algebraic-stacks"},{"id":"stacks:04T2","tag":"04T2","title":"2-Fibre products of algebraic stacks · Lemma 04T2","summary":"Let S be a scheme contained in Sch_fppf. Let X, Y, Z be algebraic stacks over S. Let f : X → Z, g : Y → Z be 1-morphisms of algebraic stacks. Then the 2-fibre product X ×_f, Z, g Y is an algebraic stack. It is also the 2-fibre product in the 2-category of algebraic stacks over (Sch/S)_fppf.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be algebraic stacks over $S$.\nLet $f : \\mathcal{X} \\to \\mathcal{Z}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of algebraic stacks. Then the $2$-fibre product\n$\\mathcal{X} \\times_{f, \\mathcal{Z}, g} \\mathcal{Y}$ is an algebraic stack.\nIt is also the $2$-fibre product in the $2$-category of algebraic stacks\nover $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"2-Fibre products of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04T2","source_file":"algebraic.tex","source_line":1746,"source_end_line":1755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1746-L1755","statement_sha256":"07445d814d3552bec9da202a9e16f4372ec45c5a56d0fe9293ea83f58ad732d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13642,"rank":13642,"depth":57,"x":1992.057,"y":1597.046,"cluster":"algebraic-stacks"},{"id":"stacks:04T1","tag":"04T1","title":"Algebraic stacks, overhauled · Lemma 04T1","summary":"Let S be a scheme contained in Sch_fppf. Let f : X → Y be a 1-morphism of algebraic stacks over S. Let V ∈ Ob((Sch/S)_fppf). Let y : (Sch/V)_fppf → Y be surjective and smooth. Then there exists an object U ∈ Ob((Sch/S)_fppf) and a 2-commutative diagram xymatrix (Sch/U)_fppf ar[r]_a ar[d]_x & (Sch/V)_fppf ar[d]^y X ar[r]^f & Y with x surjective and smooth.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of algebraic\nstacks over $S$.\nLet $V \\in \\Ob((\\Sch/S)_{fppf})$.\nLet $y : (\\Sch/V)_{fppf} \\to \\mathcal{Y}$ be surjective and smooth.\nThen there exists an object $U \\in \\Ob((\\Sch/S)_{fppf})$\nand a $2$-commutative diagram\n$$\n\\xymatrix{\n(\\Sch/U)_{fppf} \\ar[r]_a \\ar[d]_x &\n(\\Sch/V)_{fppf} \\ar[d]^y \\\\\n\\mathcal{X} \\ar[r]^f & \\mathcal{Y}\n}\n$$\nwith $x$ surjective and smooth.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks, overhauled","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04T1","source_file":"algebraic.tex","source_line":1780,"source_end_line":1797,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1780-L1797","statement_sha256":"ca8616d9f2a2ef848e043e34d741ebf96a570982d63cf0b4e3d48c8b4a8b906d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13643,"rank":13643,"depth":56,"x":2060.745,"y":1580.399,"cluster":"algebraic-stacks"},{"id":"stacks:04Y5","tag":"04Y5","title":"Algebraic stacks, overhauled · Lemma 04Y5","summary":"Let S be a scheme contained in Sch_fppf. Let f : X → Y be a 1-morphism of algebraic stacks over S. The following are equivalent: • for U ∈ Ob((Sch/S)_fppf) the functor f : X_U → Y_U is faithful, • the functor f is faithful, and • f is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of algebraic\nstacks over $S$. The following are equivalent:\n\\begin{enumerate}\n\\item for $U \\in \\Ob((\\Sch/S)_{fppf})$\nthe functor $f : \\mathcal{X}_U \\to \\mathcal{Y}_U$ is faithful,\n\\item the functor $f$ is faithful, and\n\\item $f$ is representable by algebraic spaces.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks, overhauled","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Y5","source_file":"algebraic.tex","source_line":1828,"source_end_line":1839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1828-L1839","statement_sha256":"140c4cca3cb438688162c2decdba6bda7138a8e27aa447629719962cbf91ad87","origin":"The Stacks Project","memory_eligible":false,"source_rank":13644,"rank":13644,"depth":68,"x":2023.005,"y":1632.3,"cluster":"algebraic-stacks"},{"id":"stacks:05UL","tag":"05UL","title":"Algebraic stacks, overhauled · Lemma 05UL","summary":"Let S be a scheme contained in Sch_fppf. Let u : U → X be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. If • U is representable by an algebraic space, and • u is representable by algebraic spaces, surjective and smooth, then X is an algebraic stack over S.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $u : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of\nstacks in groupoids over $(\\Sch/S)_{fppf}$. If\n\\begin{enumerate}\n\\item $\\mathcal{U}$ is representable by an algebraic space, and\n\\item $u$ is representable by algebraic spaces, surjective and smooth,\n\\end{enumerate}\nthen $\\mathcal X$ is an algebraic stack over $S$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks, overhauled","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UL","source_file":"algebraic.tex","source_line":1864,"source_end_line":1874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1864-L1874","statement_sha256":"1ef81ff438fecc67959113766e1a6d69a45127521c81e9faa5ee276391b0542b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13645,"rank":13645,"depth":67,"x":2008.929,"y":1571.873,"cluster":"algebraic-stacks"},{"id":"stacks:05UM","tag":"05UM","title":"Algebraic stacks, overhauled · Lemma 05UM","summary":"Let S be a scheme contained in Sch_fppf. Let X → Y be a morphism of stacks in groupoids over (Sch/S)_fppf. Assume that • X → Y is representable by algebraic spaces, and • Y is an algebraic stack over S. Then X is an algebraic stack over S.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of stacks in groupoids\nover $(\\Sch/S)_{fppf}$. Assume that\n\\begin{enumerate}\n\\item $\\mathcal{X} \\to \\mathcal{Y}$ is representable by algebraic spaces, and\n\\item $\\mathcal{Y}$ is an algebraic stack over $S$.\n\\end{enumerate}\nThen $\\mathcal{X}$ is an algebraic stack over $S$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks, overhauled","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UM","source_file":"algebraic.tex","source_line":1954,"source_end_line":1964,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1954-L1964","statement_sha256":"ef8749d91514730efd0dde0828429c13e2e009424c9f5bd88e384596ce4c3d54","origin":"The Stacks Project","memory_eligible":false,"source_rank":13646,"rank":13646,"depth":68,"x":2068.613,"y":1608.889,"cluster":"algebraic-stacks"},{"id":"stacks:05UN","tag":"05UN","title":"Algebraic stacks, overhauled · Lemma 05UN","summary":"Removing the hypothesis that j is a monomorphism was observed in an email from Matthew Emerton dates June 15, 2016 Let S be a scheme contained in Sch_fppf. Let j : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Assume j is representable by algebraic spaces. Then, if Y is a stack in groupoids (resp. an algebraic stack), so is X.","statement_latex":"\\begin{reference}\nRemoving the hypothesis that $j$ is a monomorphism was observed\nin an email from Matthew Emerton dates June 15, 2016\n\\end{reference}\nLet $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $j : \\mathcal X \\to \\mathcal Y$ be a $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nAssume $j$ is representable by algebraic spaces.\nThen, if $\\mathcal{Y}$ is a stack in groupoids\n(resp.\\ an algebraic stack), so is $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Algebraic stacks, overhauled","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UN","source_file":"algebraic.tex","source_line":1983,"source_end_line":1995,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L1983-L1995","statement_sha256":"63faf4d3470f4582f731eaad91e656520f12ef845d9eef1de6fb7a95ea473f26","origin":"The Stacks Project","memory_eligible":false,"source_rank":13647,"rank":13647,"depth":69,"x":1993.964,"y":1615.534,"cluster":"algebraic-stacks"},{"id":"stacks:04T4","tag":"04T4","title":"From an algebraic stack to a presentation · Lemma 04T4","summary":"Let S be a scheme contained in Sch_fppf. Let X be an algebraic stack over S. Let U be an algebraic stack over S which is representable by an algebraic space. Let f : U → X be a 1-morphism. Then • the 2-fibre product R = U ×_f, X, f U is representable by an algebraic space, • there is a canonical equivalence U ×_f, X, f U ×_f, X, f U = R ×_pr_1, U, pr_0 R, • the projection pr_02 induces via (2) a 1-morphism pr_02 : R ×_pr_1, U, pr_0 R → R • let U, R be the algebraic spaces…","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$ be an algebraic stack over $S$.\nLet $\\mathcal{U}$ be an algebraic stack over $S$ which\nis representable by an algebraic space.\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ be a 1-morphism. Then\n\\begin{enumerate}\n\\item the $2$-fibre product\n$\\mathcal{R} = \\mathcal{U} \\times_{f, \\mathcal{X}, f} \\mathcal{U}$\nis representable by an algebraic space,\n\\item there is a canonical equivalence\n$$\n\\mathcal{U} \\times_{f, \\mathcal{X}, f} \\mathcal{U}\n\\times_{f, \\mathcal{X}, f} \\mathcal{U} =\n\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R},\n$$\n\\item the projection $\\text{pr}_{02}$ induces via (2) a $1$-morphism\n$$\n\\text{pr}_{02} :\n\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R}\n\\longrightarrow\n\\mathcal{R}\n$$\n\\item let $U$, $R$ be the algebraic spaces representing\n$\\mathcal{U}, \\mathcal{R}$ and $t, s : R \\to U$ and\n$c : R \\times_{s, U, t} R \\to R$ are the morphisms corresponding\nto the $1$-morphisms\n$\\text{pr}_0, \\text{pr}_1 : \\mathcal{R} \\to \\mathcal{U}$\nand\n$\\text{pr}_{02} :\n\\mathcal{R} \\times_{\\text{pr}_1, \\mathcal{U}, \\text{pr}_0} \\mathcal{R} \\to\n\\mathcal{R}$ above, then the quintuple $(U, R, s, t, c)$ is a groupoid in\nalgebraic spaces over $S$,\n\\item the morphism $f$ induces a canonical $1$-morphism\n$f_{can} : [U/R] \\to \\mathcal{X}$\nof stacks in groupoids over $(\\Sch/S)_{fppf}$, and\n\\item the $1$-morphism $f_{can} : [U/R] \\to \\mathcal{X}$ is fully faithful.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"From an algebraic stack to a presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04T4","source_file":"algebraic.tex","source_line":2032,"source_end_line":2071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2032-L2071","statement_sha256":"0eea910e009d666cf4bf5a2357a55331a6998325511a78a93b4c5ed2e59feaad","origin":"The Stacks Project","memory_eligible":false,"source_rank":13648,"rank":13648,"depth":55,"x":2044.241,"y":1567.733,"cluster":"algebraic-stacks"},{"id":"stacks:04T5","tag":"04T5","title":"From an algebraic stack to a presentation · Lemma 04T5","summary":"Let S be a scheme contained in Sch_fppf. Let X be an algebraic stack over S. Let U be an algebraic space over S. Let f : S_U → X be a surjective smooth morphism. Let (U, R, s, t, c) be the groupoid in algebraic spaces and f_can : [U/R] → X be the result of applying Lemma [Tag 04T4] to U and f. Then • the morphisms s, t are smooth, and • the 1-morphism f_can : [U/R] → X is an equivalence.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $\\mathcal{X}$ be an algebraic stack over $S$.\nLet $U$ be an algebraic space over $S$.\nLet $f : \\mathcal{S}_U \\to \\mathcal{X}$ be a surjective smooth morphism.\nLet $(U, R, s, t, c)$ be the groupoid in algebraic spaces\nand $f_{can} : [U/R] \\to \\mathcal{X}$ be the result of applying\nLemma \\ref{lemma-map-space-into-stack}\nto $U$ and $f$. Then\n\\begin{enumerate}\n\\item the morphisms $s$, $t$ are smooth, and\n\\item the $1$-morphism $f_{can} : [U/R] \\to \\mathcal{X}$\nis an equivalence.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"From an algebraic stack to a presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04T5","source_file":"algebraic.tex","source_line":2276,"source_end_line":2291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2276-L2291","statement_sha256":"d5892502fae771e0a500e05b85c7ed4fc65b9473cb9dcd590e901f1a9cbae6d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13649,"rank":13649,"depth":56,"x":2045.619,"y":1632.229,"cluster":"algebraic-stacks"},{"id":"stacks:04TH","tag":"04TH","title":"From an algebraic stack to a presentation · Definition 04TH","summary":"Let S be a scheme. Let B be an algebraic space over S. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. We say (U, R, s, t, c) is a smooth groupoid if s, t : R → U are smooth morphisms of algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nWe say $(U, R, s, t, c)$ is a {\\it smooth groupoid}\\footnote{This terminology\nmight be a bit confusing: it does not imply that $[U/R]$ is smooth\nover anything.}\nif $s, t : R \\to U$ are smooth morphisms of algebraic spaces.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"From an algebraic stack to a presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TH","source_file":"algebraic.tex","source_line":2322,"source_end_line":2330,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2322-L2330","statement_sha256":"613702bb38870f083673ecd29e839936d958f83f406b02a4c8f68de4e33cf407","origin":"The Stacks Project","memory_eligible":false,"source_rank":13650,"rank":13650,"depth":0,"x":1992.157,"y":1584.935,"cluster":"algebraic-stacks"},{"id":"stacks:04TI","tag":"04TI","title":"From an algebraic stack to a presentation · Definition 04TI","summary":"Let X be an algebraic stack over S. A presentation of X is given by a smooth groupoid (U, R, s, t, c) in algebraic spaces over S, and an equivalence f : [U/R] → X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over $S$.\nA {\\it presentation} of $\\mathcal{X}$ is given by a smooth groupoid\n$(U, R, s, t, c)$ in algebraic spaces over $S$, and an\nequivalence $f : [U/R] \\to \\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"From an algebraic stack to a presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TI","source_file":"algebraic.tex","source_line":2332,"source_end_line":2338,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2332-L2338","statement_sha256":"34661f30de55b212638c57289f6999077ab73e2cd0591d579174b25ddf960769","origin":"The Stacks Project","memory_eligible":false,"source_rank":13651,"rank":13651,"depth":0,"x":2070.447,"y":1589.525,"cluster":"algebraic-stacks"},{"id":"stacks:04WZ","tag":"04WZ","title":"The algebraic stack associated to a smooth groupoid · Lemma 04WZ","summary":"Let S be a scheme contained in Sch_fppf. Let (U, R, s, t, c) be a groupoid in algebraic spaces over S. Then the diagonal of [U/R] is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.\nThen the diagonal of $[U/R]$ is representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"The algebraic stack associated to a smooth groupoid","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WZ","source_file":"algebraic.tex","source_line":2353,"source_end_line":2358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2353-L2358","statement_sha256":"ac34cb311fef92739c99f0467dd39fba35ee1ca0a8da5125c99034d3f83f5853","origin":"The Stacks Project","memory_eligible":false,"source_rank":13652,"rank":13652,"depth":69,"x":2008.375,"y":1630.995,"cluster":"algebraic-stacks"},{"id":"stacks:04X0","tag":"04X0","title":"The algebraic stack associated to a smooth groupoid · Lemma 04X0","summary":"Let S be a scheme contained in Sch_fppf. Let (U, R, s, t, c) be a smooth groupoid in algebraic spaces over S. Then the morphism S_U → [U/R] is smooth and surjective.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$.\nThen the morphism $\\mathcal{S}_U \\to [U/R]$ is smooth and surjective.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"The algebraic stack associated to a smooth groupoid","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04X0","source_file":"algebraic.tex","source_line":2368,"source_end_line":2373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2368-L2373","statement_sha256":"42e42339372b9e153c5440e010ad4c9bd91504c054f1979d04c7be5af27f0773","origin":"The Stacks Project","memory_eligible":false,"source_rank":13653,"rank":13653,"depth":70,"x":2020.928,"y":1564.514,"cluster":"algebraic-stacks"},{"id":"stacks:04TK","tag":"04TK","title":"The algebraic stack associated to a smooth groupoid · Theorem 04TK","summary":"Let S be a scheme contained in Sch_fppf. Let (U, R, s, t, c) be a smooth groupoid in algebraic spaces over S. Then the quotient stack [U/R] is an algebraic stack over S.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$.\nThen the quotient stack $[U/R]$ is an algebraic stack over $S$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"The algebraic stack associated to a smooth groupoid","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04TK","source_file":"algebraic.tex","source_line":2424,"source_end_line":2429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2424-L2429","statement_sha256":"2822e88f29e83e3d172b64eb303192f8a2fd8ccfe447c2d87d2ba2811552a968","origin":"The Stacks Project","memory_eligible":false,"source_rank":13654,"rank":13654,"depth":71,"x":2065.579,"y":1621.23,"cluster":"algebraic-stacks"},{"id":"stacks:04X2","tag":"04X2","title":"Change of big site · Lemma 04X2","summary":"Suppose given big sites Sch_fppf and Sch'_fppf. Assume that Sch_fppf is contained in Sch'_fppf, see Topologies, Section [Tag 022I]. Let S be an object of Sch_fppf. Let f : (Sch'/S)_fppf → (Sch/S)_fppf the morphism of sites corresponding to the inclusion functor u : (Sch/S)_fppf → (Sch'/S)_fppf. Let X be a stack in groupoids over (Sch/S)_fppf. • if X is representable by some X ∈ Ob((Sch/S)_fppf), then f^-1X is representable too, in fact it is representable by the same…","statement_latex":"Suppose given big sites $\\Sch_{fppf}$ and $\\Sch'_{fppf}$.\nAssume that $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$,\nsee Topologies, Section \\ref{topologies-section-change-alpha}.\nLet $S$ be an object of $\\Sch_{fppf}$.\nLet $f : (\\Sch'/S)_{fppf} \\to (\\Sch/S)_{fppf}$ the morphism\nof sites corresponding to the inclusion functor\n$u : (\\Sch/S)_{fppf} \\to (\\Sch'/S)_{fppf}$.\nLet $\\mathcal{X}$ be a stack in groupoids over $(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item if $\\mathcal{X}$ is representable by some\n$X \\in \\Ob((\\Sch/S)_{fppf})$, then\n$f^{-1}\\mathcal{X}$ is representable too, in fact it is representable by the\nsame scheme $X$, now viewed as an object of $(\\Sch'/S)_{fppf}$,\n\\item if $\\mathcal{X}$ is representable by\n$F \\in \\Sh((\\Sch/S)_{fppf})$ which is\nan algebraic space, then $f^{-1}\\mathcal{X}$ is representable\nby the algebraic space $f^{-1}F$,\n\\item if $\\mathcal{X}$ is an algebraic stack, then $f^{-1}\\mathcal{X}$\nis an algebraic stack, and\n\\item if $\\mathcal{X}$ is a Deligne-Mumford stack, then $f^{-1}\\mathcal{X}$\nis a Deligne-Mumford stack too.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Change of big site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04X2","source_file":"algebraic.tex","source_line":2478,"source_end_line":2502,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2478-L2502","statement_sha256":"753601b010f044e642b983438526f2de531fb0fd35e36b104fd321418f7e684f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13655,"rank":13655,"depth":57,"x":1986.266,"y":1604.578,"cluster":"algebraic-stacks"},{"id":"stacks:04X3","tag":"04X3","title":"Change of big site · Lemma 04X3","summary":"Suppose Sch_fppf is contained in Sch'_fppf. Let S be an object of Sch_fppf. Denote Algebraic-Stacks/S the 2-category of algebraic stacks over S defined using Sch_fppf. Similarly, denote Algebraic-Stacks'/S the 2-category of algebraic stacks over S defined using Sch'_fppf. The rule X ↦ f^-1X of Lemma [Tag 04X2] defines a functor of 2-categories Algebraic-Stacks/S → Algebraic-Stacks'/S which defines equivalences of morphism categories Mor_Algebraic-Stacks/S(X, Y) →…","statement_latex":"Suppose $\\Sch_{fppf}$ is contained in $\\Sch'_{fppf}$.\nLet $S$ be an object of $\\Sch_{fppf}$. Denote\n$\\textit{Algebraic-Stacks}/S$ the $2$-category of algebraic stacks over $S$\ndefined using $\\Sch_{fppf}$. Similarly, denote\n$\\textit{Algebraic-Stacks}'/S$ the $2$-category of algebraic stacks over $S$\ndefined using $\\Sch'_{fppf}$. The rule\n$\\mathcal{X} \\mapsto f^{-1}\\mathcal{X}$ of\nLemma \\ref{lemma-change-big-site}\ndefines a functor of $2$-categories\n$$\n\\textit{Algebraic-Stacks}/S \\longrightarrow \\textit{Algebraic-Stacks}'/S\n$$\nwhich defines equivalences of morphism categories\n$$\n\\Mor_{\\textit{Algebraic-Stacks}/S}(\\mathcal{X}, \\mathcal{Y})\n\\longrightarrow\n\\Mor_{\\textit{Algebraic-Stacks}'/S}(f^{-1}\\mathcal{X}, f^{-1}\\mathcal{Y})\n$$\nfor every objects $\\mathcal{X}, \\mathcal{Y}$ of\n$\\textit{Algebraic-Stacks}/S$. An object\n$\\mathcal{X}'$ of $\\textit{Algebraic-Stacks}'/S$\nis equivalence to $f^{-1}\\mathcal{X}$ for some\n$\\mathcal{X}$ in $\\textit{Algebraic-Stacks}/S$\nif and only if it has a presentation $\\mathcal{X} = [U'/R']$\nwith $U', R'$ isomorphic to $f^{-1}U$, $f^{-1}R$ for some\n$U, R \\in \\textit{Spaces}/S$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Change of big site","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04X3","source_file":"algebraic.tex","source_line":2528,"source_end_line":2556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2528-L2556","statement_sha256":"fab1ed6f2d8ff07546d8e2223ec9e0787088de2328c54d2226d43c2ccc1ba5ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":13656,"rank":13656,"depth":58,"x":2058.842,"y":1571.537,"cluster":"algebraic-stacks"},{"id":"stacks:04X5","tag":"04X5","title":"Change of base scheme · Lemma 04X5","summary":"Let Sch_fppf be a big fppf site. Let S → S' be a morphism of this site. The constructions A and B of Stacks, Section [Tag 04WT] above give isomorphisms of 2-categories ( 2-category of algebraic stacks X over S ) ↔ ( 2-category of pairs (X', f) consisting of an algebraic stack X' over S' and a morphism f : X' → (Sch/S)_fppf of algebraic stacks over S' )","statement_latex":"Let $\\Sch_{fppf}$ be a big fppf site.\nLet $S \\to S'$ be a morphism of this site.\nThe constructions A and B of\nStacks, Section \\ref{stacks-section-localize}\nabove give isomorphisms of $2$-categories\n$$\n\\left\\{\n\\begin{matrix}\n2\\text{-category of algebraic}\\\\\n\\text{stacks }\\mathcal{X}\\text{ over }S\n\\end{matrix}\n\\right\\}\n\\leftrightarrow\n\\left\\{\n\\begin{matrix}\n2\\text{-category of pairs }(\\mathcal{X}', f)\\text{ consisting of an}\\\\\n\\text{algebraic stack }\\mathcal{X}'\\text{ over }S'\\text{ and a morphism}\\\\\nf : \\mathcal{X}' \\to (\\Sch/S)_{fppf}\\text{ of algebraic stacks over }S'\n\\end{matrix}\n\\right\\}\n$$","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Change of base scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04X5","source_file":"algebraic.tex","source_line":2575,"source_end_line":2598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2575-L2598","statement_sha256":"6c57e5109464fff0d7b07d877f01da1714729f7016f96dccb5e770dfb5ebc27c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13657,"rank":13657,"depth":1,"x":2031.637,"y":1637.707,"cluster":"algebraic-stacks"},{"id":"stacks:04X6","tag":"04X6","title":"Change of base scheme · Definition 04X6","summary":"Let Sch_fppf be a big fppf site. Let S → S' be a morphism of this site. If p : X → (Sch/S)_fppf is an algebraic stack over S, then X viewed as an algebraic stack over S' is the algebraic stack X → (Sch/S')_fppf gotten by applying construction A of Lemma [Tag 04X5] to X.","statement_latex":"Let $\\Sch_{fppf}$ be a big fppf site.\nLet $S \\to S'$ be a morphism of this site.\nIf $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$\nis an algebraic stack over $S$, then\n$\\mathcal{X}$ {\\it viewed as an algebraic stack over $S'$}\nis the algebraic stack\n$$\n\\mathcal{X} \\longrightarrow (\\Sch/S')_{fppf}\n$$\ngotten by applying construction A of\nLemma \\ref{lemma-category-of-spaces-over-smaller-base-scheme}\nto $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Change of base scheme","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04X6","source_file":"algebraic.tex","source_line":2616,"source_end_line":2630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2616-L2630","statement_sha256":"638fdd214b14e36067c6ccda35bfb3083a3597922c64f7a5d7ca2bfee6603749","origin":"The Stacks Project","memory_eligible":false,"source_rank":13658,"rank":13658,"depth":2,"x":1998.177,"y":1572.875,"cluster":"algebraic-stacks"},{"id":"stacks:04X7","tag":"04X7","title":"Change of base scheme · Definition 04X7","summary":"Let Sch_fppf be a big fppf site. Let S → S' be a morphism of this site. Let X' be an algebraic stack over S'. The change of base of X' is the algebraic stack X'_S over S described above.","statement_latex":"Let $\\Sch_{fppf}$ be a big fppf site.\nLet $S \\to S'$ be a morphism of this site.\nLet $\\mathcal{X}'$ be an algebraic stack over $S'$.\nThe {\\it change of base of $\\mathcal{X}'$} is the\nalgebraic stack $\\mathcal{X}'_S$ over $S$ described above.","area":"Algebraic Stacks","chapter":"Algebraic Stacks","chapter_id":"algebraic","section":"Change of base scheme","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04X7","source_file":"algebraic.tex","source_line":2648,"source_end_line":2655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/algebraic.tex#L2648-L2655","statement_sha256":"25daf1efa327b5f5ce62f9efd39b69a7ab5ed251ee233295017567ac0b6c9d28","origin":"The Stacks Project","memory_eligible":false,"source_rank":13659,"rank":13659,"depth":0,"x":2075.692,"y":1601.962,"cluster":"algebraic-stacks"},{"id":"stacks:04U3","tag":"04U3","title":"Quasi-coherent sheaves · Lemma 04U3","summary":"A morphism (f, φ) : (Y, G) → (X, F) of QCohstack is strongly cartesian if and only if the map φ induces an isomorphism f^*F → G.","statement_latex":"A morphism $(f, \\varphi) : (Y, \\mathcal{G}) \\to (X, \\mathcal{F})$\nof $\\QCohstack$ is strongly cartesian if and only if the\nmap $\\varphi$ induces an isomorphism $f^*\\mathcal{F} \\to \\mathcal{G}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04U3","source_file":"examples-stacks.tex","source_line":115,"source_end_line":120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L115-L120","statement_sha256":"966d287cd70c6404aad4d9415981e4bf4569c2cd68a406a95a231aa8daabb6e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13660,"rank":13660,"depth":2,"x":1994.414,"y":1624.699,"cluster":"algebraic-stacks"},{"id":"stacks:03YM","tag":"03YM","title":"Quasi-coherent sheaves · Lemma 03YM","summary":"The functor p : QCohstack → (Sch/S)_fppf satisfies conditions (1), (2) and (3) of Stacks, Definition [Tag 026F].","statement_latex":"The functor $p : \\QCohstack \\to (\\Sch/S)_{fppf}$\nsatisfies conditions (1), (2) and (3) of\nStacks, Definition \\ref{stacks-definition-stack}.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/03YM","source_file":"examples-stacks.tex","source_line":157,"source_end_line":162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L157-L162","statement_sha256":"c66f8c80ed249c117491d6a6afeb3e5cbe3744b72699d0ac9992aa42ac21ba4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13661,"rank":13661,"depth":21,"x":2036.435,"y":1561.258,"cluster":"algebraic-stacks"},{"id":"stacks:04U4","tag":"04U4","title":"The stack of finitely generated quasi-coherent sheaves · Lemma 04U4","summary":"The functor p_fg : QCohstack_fg → (Sch/S)_fppf satisfies conditions (1), (2) and (3) of Stacks, Definition [Tag 026F].","statement_latex":"The functor $p_{fg} : \\QCohstack_{fg} \\to (\\Sch/S)_{fppf}$\nsatisfies conditions (1), (2) and (3) of\nStacks, Definition \\ref{stacks-definition-stack}.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"The stack of finitely generated quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04U4","source_file":"examples-stacks.tex","source_line":199,"source_end_line":204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L199-L204","statement_sha256":"c29742d3997c8fce8d757c17e61c6eabeba8fa1074bfd798175a464f45ec7854","origin":"The Stacks Project","memory_eligible":false,"source_rank":13662,"rank":13662,"depth":8,"x":2056.639,"y":1632.498,"cluster":"algebraic-stacks"},{"id":"stacks:04U5","tag":"04U5","title":"The stack of finitely generated quasi-coherent sheaves · Lemma 04U5","summary":"Let (X, O_X) be a ringed space. • The category of finite type O_X-modules has a set of isomorphism classes. • The category of finite type quasi-coherent O_X-modules has a set of isomorphism classes.","statement_latex":"Let $(X, \\mathcal{O}_X)$ be a ringed space.\n\\begin{enumerate}\n\\item The category of finite type $\\mathcal{O}_X$-modules has a\nset of isomorphism classes.\n\\item The category of finite type quasi-coherent\n$\\mathcal{O}_X$-modules has a set of isomorphism classes.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"The stack of finitely generated quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04U5","source_file":"examples-stacks.tex","source_line":232,"source_end_line":241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L232-L241","statement_sha256":"94f6d7e82b22d987844fb6e47034908ca389b9bfc4fb4239941c7d8457f60cf2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13663,"rank":13663,"depth":0,"x":1983.835,"y":1591.077,"cluster":"algebraic-stacks"},{"id":"stacks:04U6","tag":"04U6","title":"The stack of finitely generated quasi-coherent sheaves · Lemma 04U6","summary":"There exists a subcategory QCohstack_fg, small ⊂ QCohstack_fg with the following properties: • the inclusion functor QCohstack_fg, small → QCohstack_fg is fully faithful and essentially surjective, and • the functor p_fg, small : QCohstack_fg, small → (Sch/S)_fppf turns QCohstack_fg, small into a stack over (Sch/S)_fppf.","statement_latex":"There exists a subcategory\n$\\QCohstack_{fg, small} \\subset \\QCohstack_{fg}$\nwith the following properties:\n\\begin{enumerate}\n\\item the inclusion functor\n$\\QCohstack_{fg, small} \\to \\QCohstack_{fg}$ is\nfully faithful and essentially surjective, and\n\\item the functor\n$p_{fg, small} : \\QCohstack_{fg, small} \\to (\\Sch/S)_{fppf}$\nturns $\\QCohstack_{fg, small}$ into a stack over $(\\Sch/S)_{fppf}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"The stack of finitely generated quasi-coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04U6","source_file":"examples-stacks.tex","source_line":271,"source_end_line":284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L271-L284","statement_sha256":"c62cb8acfea2d074949591628f2e50492d20fb17c1cff7c6f93ce065561cf03a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13664,"rank":13664,"depth":9,"x":2071.561,"y":1580.221,"cluster":"algebraic-stacks"},{"id":"stacks:0BLZ","tag":"0BLZ","title":"Finite étale covers · Lemma 0BLZ","summary":"The functor p : FÉt → (Sch/S)_fppf defines a stack over (Sch/S)_fppf.","statement_latex":"The functor\n$$\np : \\textit{F\\'Et} \\longrightarrow (\\Sch/S)_{fppf}\n$$\ndefines a stack over $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Finite étale covers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BLZ","source_file":"examples-stacks.tex","source_line":348,"source_end_line":355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L348-L355","statement_sha256":"b88e465190226a8fe860960365b21dde47f2e82b0a18ccc27407ccc1129aaf61","origin":"The Stacks Project","memory_eligible":false,"source_rank":13665,"rank":13665,"depth":47,"x":2015.138,"y":1638.481,"cluster":"algebraic-stacks"},{"id":"stacks:04U9","tag":"04U9","title":"Algebraic spaces · Lemma 04U9","summary":"A morphism (f, g) : X/U → Y/V of Spacesstack is strongly cartesian if and only if the map f induces an isomorphism X → U ×_g, V Y.","statement_latex":"A morphism $(f, g) : X/U \\to Y/V$\nof $\\Spacesstack$ is strongly cartesian if and only if the\nmap $f$ induces an isomorphism $X \\to U \\times_{g, V} Y$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04U9","source_file":"examples-stacks.tex","source_line":415,"source_end_line":420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L415-L420","statement_sha256":"e0ec69a3791e54387629df0e3fdfc78683340b487cd10d4dfeeb60903b62ab6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13666,"rank":13666,"depth":2,"x":2009.852,"y":1562.895,"cluster":"algebraic-stacks"},{"id":"stacks:04UA","tag":"04UA","title":"Algebraic spaces · Lemma 04UA","summary":"The functor p : Spacesstack → (Sch/S)_fppf satisfies conditions (1) and (2) of Stacks, Definition [Tag 026F].","statement_latex":"The functor $p : \\Spacesstack \\to (\\Sch/S)_{fppf}$\nsatisfies conditions (1) and (2) of\nStacks, Definition \\ref{stacks-definition-stack}.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04UA","source_file":"examples-stacks.tex","source_line":457,"source_end_line":462,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L457-L462","statement_sha256":"5c98fa99ef20d12bff7573ace7fddc51f15f10d6b2320839cb3863f5a17e18bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":13667,"rank":13667,"depth":7,"x":2075.051,"y":1616.055,"cluster":"algebraic-stacks"},{"id":"stacks:04UD","tag":"04UD","title":"The stack of finite type algebraic spaces · Lemma 04UD","summary":"The functor p_ft : Spacesstack_ft → (Sch/S)_fppf satisfies the conditions (1), (2) and (3) of Stacks, Definition [Tag 026F].","statement_latex":"The functor\n$p_{ft} : \\Spacesstack_{ft} \\to (\\Sch/S)_{fppf}$\nsatisfies the conditions (1), (2) and (3) of\nStacks, Definition \\ref{stacks-definition-stack}.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"The stack of finite type algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04UD","source_file":"examples-stacks.tex","source_line":535,"source_end_line":541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L535-L541","statement_sha256":"3c94ca2f88f2036468f344a01701d0d63e5257584a2d73825eb835959303f986","origin":"The Stacks Project","memory_eligible":false,"source_rank":13668,"rank":13668,"depth":70,"x":1983.508,"y":1613.832,"cluster":"algebraic-stacks"},{"id":"stacks:04UE","tag":"04UE","title":"The stack of finite type algebraic spaces · Lemma 04UE","summary":"There exists a subcategory Spacesstack_ft, small ⊂ Spacesstack_ft with the following properties: • the inclusion functor Spacesstack_ft, small → Spacesstack_ft is fully faithful and essentially surjective, and • the functor p_ft, small : Spacesstack_ft, small → (Sch/S)_fppf turns Spacesstack_ft, small into a stack over (Sch/S)_fppf.","statement_latex":"There exists a subcategory\n$\\Spacesstack_{ft, small} \\subset \\Spacesstack_{ft}$\nwith the following properties:\n\\begin{enumerate}\n\\item the inclusion functor\n$\\Spacesstack_{ft, small} \\to \\Spacesstack_{ft}$ is\nfully faithful and essentially surjective, and\n\\item the functor\n$p_{ft, small} : \\Spacesstack_{ft, small} \\to (\\Sch/S)_{fppf}$\nturns $\\Spacesstack_{ft, small}$ into a stack over\n$(\\Sch/S)_{fppf}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"The stack of finite type algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04UE","source_file":"examples-stacks.tex","source_line":608,"source_end_line":622,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L608-L622","statement_sha256":"56a9147dfe69e958f734ec3e04794ea03a9c4bf1e4e179d416ec6e393348835c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13669,"rank":13669,"depth":71,"x":2053.338,"y":1563.138,"cluster":"algebraic-stacks"},{"id":"stacks:04UK","tag":"04UK","title":"Classifying torsors · Lemma 04UK","summary":"Up to a replacement as in Stacks, Remark [Tag 03ZZ] the functor p : G-Torsors → (Sch/S)_fppf defines a stack in groupoids over (Sch/S)_fppf.","statement_latex":"Up to a replacement as in\nStacks, Remark \\ref{stacks-remark-stack-make-small}\nthe functor\n$$\np : \\mathcal{G}\\textit{-Torsors} \\longrightarrow (\\Sch/S)_{fppf}\n$$\ndefines a stack in groupoids over $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Classifying torsors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04UK","source_file":"examples-stacks.tex","source_line":822,"source_end_line":831,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L822-L831","statement_sha256":"a42ffd041be335b61cb3f2b24389fb070546bb5692aa4c542aea9a7dc93c57c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13670,"rank":13670,"depth":8,"x":2042.529,"y":1640.73,"cluster":"algebraic-stacks"},{"id":"stacks:04UM","tag":"04UM","title":"Classifying torsors · Lemma 04UM","summary":"Up to a replacement as in Stacks, Remark [Tag 03ZZ] the functor p : G/B-Torsors → (Sch/S)_fppf defines a stack in groupoids over (Sch/S)_fppf.","statement_latex":"Up to a replacement as in\nStacks, Remark \\ref{stacks-remark-stack-make-small}\nthe functor\n$$\np :\n\\mathcal{G}/\\mathcal{B}\\textit{-Torsors}\n\\longrightarrow\n(\\Sch/S)_{fppf}\n$$\ndefines a stack in groupoids over $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Classifying torsors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04UM","source_file":"examples-stacks.tex","source_line":920,"source_end_line":932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L920-L932","statement_sha256":"064a775a3bb804302c82315c7e78e449275b6bd13b7270bcb2a88e7afe5b7351","origin":"The Stacks Project","memory_eligible":false,"source_rank":13671,"rank":13671,"depth":9,"x":1987.694,"y":1576.904,"cluster":"algebraic-stacks"},{"id":"stacks:04US","tag":"04US","title":"Classifying torsors · Lemma 04US","summary":"Up to a replacement as in Stacks, Remark [Tag 03ZZ] the functor p : G-Torsors → (Sch/S)_fppf defines a stack in groupoids over (Sch/S)_fppf.","statement_latex":"Up to a replacement as in\nStacks, Remark \\ref{stacks-remark-stack-make-small}\nthe functor\n$$\np : G\\textit{-Torsors} \\longrightarrow (\\Sch/S)_{fppf}\n$$\ndefines a stack in groupoids over $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Classifying torsors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04US","source_file":"examples-stacks.tex","source_line":1121,"source_end_line":1130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L1121-L1130","statement_sha256":"b0d1dfa4804ad2b7546f0ed5b7a2ffadb5ba0a2da828527a1ba367907c74f345","origin":"The Stacks Project","memory_eligible":false,"source_rank":13672,"rank":13672,"depth":69,"x":2080.135,"y":1592.974,"cluster":"algebraic-stacks"},{"id":"stacks:04UT","tag":"04UT","title":"Classifying torsors · Lemma 04UT","summary":"Let B be an algebraic space over S. Let G be a group algebraic space over B. Denote G, resp. B the algebraic space G, resp. B seen as a sheaf on (Sch/S)_fppf. The functor G-Torsors → G/B-Torsors which associates to a triple (U, b, X) the triple (U, b, X) where X is X viewed as a sheaf is an equivalence of stacks in groupoids over (Sch/S)_fppf.","statement_latex":"Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic\nspace over $B$. Denote $\\mathcal{G}$, resp.\\ $\\mathcal{B}$ the algebraic\nspace $G$, resp.\\ $B$ seen as a sheaf on $(\\Sch/S)_{fppf}$.\nThe functor\n$$\nG\\textit{-Torsors} \\longrightarrow \\mathcal{G}/\\mathcal{B}\\textit{-Torsors}\n$$\nwhich associates to a triple $(U, b, X)$ the triple\n$(U, b, \\mathcal{X})$ where $\\mathcal{X}$ is $X$ viewed as a sheaf\nis an equivalence of stacks in groupoids over $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Classifying torsors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04UT","source_file":"examples-stacks.tex","source_line":1140,"source_end_line":1152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L1140-L1152","statement_sha256":"9c6576b0674f2084a846c307003d6d8669d27630bbd1212ead9eab8d09887572","origin":"The Stacks Project","memory_eligible":false,"source_rank":13673,"rank":13673,"depth":1,"x":1998.454,"y":1633.873,"cluster":"algebraic-stacks"},{"id":"stacks:0370","tag":"0370","title":"Quotients by group actions · Lemma 0370","summary":"Up to a replacement as in Stacks, Remark [Tag 03ZZ] the functor p : [[X/G]] → (Sch/S)_fppf defines a stack in groupoids over (Sch/S)_fppf.","statement_latex":"Up to a replacement as in\nStacks, Remark \\ref{stacks-remark-stack-make-small}\nthe functor\n$$\np : [[X/G]] \\longrightarrow (\\Sch/S)_{fppf}\n$$\ndefines a stack in groupoids over $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Quotients by group actions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0370","source_file":"examples-stacks.tex","source_line":1271,"source_end_line":1280,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L1271-L1280","statement_sha256":"b263b2538ea580c49538d6823756c97422e209c0d28c2be7dd363090cadd23d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13674,"rank":13674,"depth":70,"x":2025.993,"y":1556.816,"cluster":"algebraic-stacks"},{"id":"stacks:04WM","tag":"04WM","title":"Quotients by group actions · Proposition 04WM","summary":"In Situation [Tag 04WL] there exists a canonical equivalence [X/G] → [[X/G]] of stacks in groupoids over (Sch/S)_fppf.","statement_latex":"In\nSituation \\ref{situation-quotient-stack}\nthere exists a canonical equivalence\n$$\n[X/G] \\longrightarrow [[X/G]]\n$$\nof stacks in groupoids over $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Quotients by group actions","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WM","source_file":"examples-stacks.tex","source_line":1310,"source_end_line":1319,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L1310-L1319","statement_sha256":"2118d2eaa076b16aa77bb0372af4f8da8ea54d4f66fc85c37e220327f4998dba","origin":"The Stacks Project","memory_eligible":false,"source_rank":13675,"rank":13675,"depth":9,"x":2067.947,"y":1629.778,"cluster":"algebraic-stacks"},{"id":"stacks:0CQJ","tag":"0CQJ","title":"Quotients by group actions · Lemma 0CQJ","summary":"The classifying stack of a group scheme or group algebraic space. Let S be a scheme. Let B be an algebraic space over S. Let G be a group algebraic space over B. Then the stacks in groupoids [B/G], [[B/G]], G-Torsors, G/B-Torsors are all canonically equivalent. If G → B is flat and locally of finite presentation, then these are also equivalent to G-Principal.","statement_latex":"\\begin{slogan}\nThe classifying stack of a group scheme or group algebraic space.\n\\end{slogan}\nLet $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $G$ be a group algebraic space over $B$. Then the stacks\nin groupoids\n$$\n[B/G],\\quad\n[[B/G]],\\quad\nG\\textit{-Torsors},\\quad\n\\mathcal{G}/\\mathcal{B}\\textit{-Torsors}\n$$\nare all canonically equivalent.\nIf $G \\to B$ is flat and locally\nof finite presentation, then these are also equivalent to\n$G\\textit{-Principal}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Quotients by group actions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQJ","source_file":"examples-stacks.tex","source_line":1458,"source_end_line":1476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L1458-L1476","statement_sha256":"a7370003e6f933d1eb638708705b1c4e7620237b7e672bc77741a63d10c7d415","origin":"The Stacks Project","memory_eligible":false,"source_rank":13676,"rank":13676,"depth":39,"x":1977.711,"y":1599.571,"cluster":"algebraic-stacks"},{"id":"stacks:04WN","tag":"04WN","title":"The Picard stack · Lemma 04WN","summary":"Up to a replacement as in Stacks, Remark [Tag 03ZZ] the functor Picardstack_X/B → (Sch/S)_fppf defines a stack in groupoids over (Sch/S)_fppf.","statement_latex":"Up to a replacement as in\nStacks, Remark \\ref{stacks-remark-stack-make-small}\nthe functor\n$$\n\\Picardstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}\n$$\ndefines a stack in groupoids over $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"The Picard stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04WN","source_file":"examples-stacks.tex","source_line":1591,"source_end_line":1600,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L1591-L1600","statement_sha256":"f6c8ddb13123a1a675eec9802654011b2acd466010e3810cc657257f8d9f8a17","origin":"The Stacks Project","memory_eligible":false,"source_rank":13677,"rank":13677,"depth":52,"x":2069.168,"y":1570.445,"cluster":"algebraic-stacks"},{"id":"stacks:05WB","tag":"05WB","title":"Finite Hilbert stacks · Lemma 05WB","summary":"The category H_d(X/Y) endowed with the functor p above defines a stack in groupoids over (Sch/S)_fppf.","statement_latex":"The category $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ endowed with\nthe functor $p$ above defines a stack in groupoids over\n$(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Finite Hilbert stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WB","source_file":"examples-stacks.tex","source_line":1769,"source_end_line":1774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L1769-L1774","statement_sha256":"380e7eab87310a22fa15be0d68ef31cccee447d41e8334b742f78db670c6ec8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13678,"rank":13678,"depth":46,"x":2024.851,"y":1644.316,"cluster":"algebraic-stacks"},{"id":"stacks:05WC","tag":"05WC","title":"Finite Hilbert stacks · Definition 05WC","summary":"We will denote H_d(X/Y) the degree d finite Hilbert stack of X over Y constructed above. If Y = S we write H_d(X) = H_d(X/Y). If X = Y = S we denote it H_d.","statement_latex":"We will denote $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$\nthe {\\it degree $d$ finite Hilbert stack of $\\mathcal{X}$ over $\\mathcal{Y}$}\nconstructed above. If $\\mathcal{Y} = S$ we write\n$\\mathcal{H}_d(\\mathcal{X}) = \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$.\nIf $\\mathcal{X} = \\mathcal{Y} = S$ we denote it $\\mathcal{H}_d$.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Finite Hilbert stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WC","source_file":"examples-stacks.tex","source_line":1805,"source_end_line":1812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L1805-L1812","statement_sha256":"f43dd3a15de32266ac9733dffa327d9847dc32b16570d9a778de94acf6a27a4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13679,"rank":13679,"depth":0,"x":1997.947,"y":1564.163,"cluster":"algebraic-stacks"},{"id":"stacks:05XV","tag":"05XV","title":"Finite Hilbert stacks · Lemma 05XV","summary":"The 1-morphism H_d(X/Y) → H_d(X) is faithful.","statement_latex":"The $1$-morphism\n$\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{H}_d(\\mathcal{X})$\nis faithful.","area":"Algebraic Stacks","chapter":"Examples of Stacks","chapter_id":"examples-stacks","section":"Finite Hilbert stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XV","source_file":"examples-stacks.tex","source_line":1831,"source_end_line":1836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples-stacks.tex#L1831-L1836","statement_sha256":"e77a2dd3f19dce9e75c3f48fc0b8212554aee09a2915a93595430b6082167858","origin":"The Stacks Project","memory_eligible":false,"source_rank":13680,"rank":13680,"depth":0,"x":2082.799,"y":1608.292,"cluster":"algebraic-stacks"},{"id":"stacks:06TJ","tag":"06TJ","title":"Presheaves · Definition 06TJ","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. • A presheaf on X is a presheaf on the underlying category of X. • A morphism of presheaves on X is a morphism of presheaves on the underlying category of X. We denote PSh(X) the category of presheaves on X.","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in\ngroupoids.\n\\begin{enumerate}\n\\item A {\\it presheaf on $\\mathcal{X}$} is a presheaf on the\nunderlying category of $\\mathcal{X}$.\n\\item A {\\it morphism of presheaves on $\\mathcal{X}$} is a morphism of\npresheaves on the underlying category of $\\mathcal{X}$.\n\\end{enumerate}\nWe denote $\\textit{PSh}(\\mathcal{X})$ the category of presheaves on\n$\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Presheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TJ","source_file":"stacks-sheaves.tex","source_line":93,"source_end_line":105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L93-L105","statement_sha256":"128c92a8e8286956f5d6071029708c296f120a2cec276dbc7beef3f487eba9e9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13681,"rank":13681,"depth":0,"x":1984.102,"y":1624.0,"cluster":"algebraic-stacks"},{"id":"stacks:06TL","tag":"06TL","title":"Presheaves · Lemma 06TL","summary":"Let f : X → Y and g : Y → Z be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. Then (g ∘ f)^p = f^p ∘ g^p and there is a canonical isomorphism _p(g ∘ f) → _pg ∘ _pf compatible with adjointness of (f^p, _pf), (g^p, _pg), and ((g ∘ f)^p, _p(g ∘ f)).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$. Then $(g \\circ f)^p = f^p \\circ g^p$ and\nthere is a canonical isomorphism\n${}_p(g \\circ f) \\to {}_pg \\circ {}_pf$\ncompatible with adjointness of $(f^p, {}_pf)$, $(g^p, {}_pg)$, and\n$((g \\circ f)^p, {}_p(g \\circ f))$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TL","source_file":"stacks-sheaves.tex","source_line":152,"source_end_line":161,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L152-L161","statement_sha256":"41c18b4202ed91fdb8e71630fa908064382e0aa3533e433e9fd64bedc5318ff2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13682,"rank":13682,"depth":1,"x":2044.638,"y":1555.979,"cluster":"algebraic-stacks"},{"id":"stacks:06TM","tag":"06TM","title":"Presheaves · Lemma 06TM","summary":"Let f, g : X → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. Let t : f → g be a 2-morphism of categories fibred in groupoids over (Sch/S)_fppf. Assigned to t there are canonical isomorphisms of functors t^p : g^p → f^p and _pt : _pf → _pg which compatible with adjointness of (f^p, _pf) and (g^p, _pg) and with vertical and horizontal composition of 2-morphisms.","statement_latex":"Let $f, g : \\mathcal{X} \\to \\mathcal{Y}$ be $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let $t : f \\to g$\nbe a $2$-morphism of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$. Assigned to $t$ there are canonical\nisomorphisms of functors\n$$\nt^p : g^p \\longrightarrow f^p\n\\quad\\text{and}\\quad\n{}_pt : {}_pf \\longrightarrow {}_pg\n$$\nwhich compatible with adjointness of $(f^p, {}_pf)$ and\n$(g^p, {}_pg)$ and with\nvertical and horizontal composition of $2$-morphisms.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Presheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TM","source_file":"stacks-sheaves.tex","source_line":222,"source_end_line":237,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L222-L237","statement_sha256":"7fbea5303e2c18f571a53dc4b5e02ed0d956c7e78483e06318cc93d15772d764","origin":"The Stacks Project","memory_eligible":false,"source_rank":13683,"rank":13683,"depth":3,"x":2054.763,"y":1641.024,"cluster":"algebraic-stacks"},{"id":"stacks:06TP","tag":"06TP","title":"Sheaves · Definition 06TP","summary":"Let X be a category fibred in groupoids over (Sch/S)_fppf. • The associated Zariski site, denoted X_Zar, is the structure of site on X inherited from (Sch/S)_Zar. • The associated étale site, denoted X_etale, is the structure of site on X inherited from (Sch/S)_etale. • The associated smooth site, denoted X_smooth, is the structure of site on X inherited from (Sch/S)_smooth. • The associated syntomic site, denoted X_syntomic, is the structure of site on X inherited from…","statement_latex":"Let $\\mathcal{X}$ be a category fibred in groupoids over\n$(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item The {\\it associated Zariski site}, denoted $\\mathcal{X}_{Zar}$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_{Zar}$.\n\\item The {\\it associated \\'etale site}, denoted $\\mathcal{X}_\\etale$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_\\etale$.\n\\item The {\\it associated smooth site}, denoted $\\mathcal{X}_{smooth}$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_{smooth}$.\n\\item The {\\it associated syntomic site}, denoted $\\mathcal{X}_{syntomic}$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_{syntomic}$.\n\\item The {\\it associated fppf site}, denoted $\\mathcal{X}_{fppf}$,\nis the structure of site on $\\mathcal{X}$ inherited from\n$(\\Sch/S)_{fppf}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TP","source_file":"stacks-sheaves.tex","source_line":354,"source_end_line":375,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L354-L375","statement_sha256":"b6d7a5a1878d39fb168843e345319b7e2770919e7eaf9acfa05de684e4264c13","origin":"The Stacks Project","memory_eligible":false,"source_rank":13684,"rank":13684,"depth":0,"x":1978.43,"y":1583.699,"cluster":"algebraic-stacks"},{"id":"stacks:06TR","tag":"06TR","title":"Sheaves · Definition 06TR","summary":"Let X be a category fibred in groupoids over (Sch/S)_fppf. Let F be a presheaf on X. • We say F is a Zariski sheaf, or a sheaf for the Zariski topology if F is a sheaf on the associated Zariski site X_Zar. • We say F is an étale sheaf, or a sheaf for the étale topology if F is a sheaf on the associated étale site X_etale. • We say F is a smooth sheaf, or a sheaf for the smooth topology if F is a sheaf on the associated smooth site X_smooth. • We say F is a syntomic sheaf,…","statement_latex":"Let $\\mathcal{X}$ be a category fibred in groupoids over\n$(\\Sch/S)_{fppf}$. Let $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$.\n\\begin{enumerate}\n\\item We say $\\mathcal{F}$ is a {\\it Zariski sheaf}, or a\n{\\it sheaf for the Zariski topology} if $\\mathcal{F}$\nis a sheaf on the associated Zariski site $\\mathcal{X}_{Zar}$.\n\\item We say $\\mathcal{F}$ is an {\\it \\'etale sheaf}, or a\n{\\it sheaf for the \\'etale topology} if $\\mathcal{F}$\nis a sheaf on the associated \\'etale site $\\mathcal{X}_\\etale$.\n\\item We say $\\mathcal{F}$ is a {\\it smooth sheaf}, or a\n{\\it sheaf for the smooth topology} if $\\mathcal{F}$\nis a sheaf on the associated smooth site $\\mathcal{X}_{smooth}$.\n\\item We say $\\mathcal{F}$ is a {\\it syntomic sheaf}, or a\n{\\it sheaf for the syntomic topology} if $\\mathcal{F}$\nis a sheaf on the associated syntomic site $\\mathcal{X}_{syntomic}$.\n\\item We say $\\mathcal{F}$ is an {\\it fppf sheaf}, or a {\\it sheaf},\nor a {\\it sheaf for the fppf topology} if $\\mathcal{F}$\nis a sheaf on the associated fppf site $\\mathcal{X}_{fppf}$.\n\\end{enumerate}\nA morphism of sheaves is just a morphism of presheaves. We denote\nthese categories of sheaves\n$\\Sh(\\mathcal{X}_{Zar})$,\n$\\Sh(\\mathcal{X}_\\etale)$,\n$\\Sh(\\mathcal{X}_{smooth})$,\n$\\Sh(\\mathcal{X}_{syntomic})$, and\n$\\Sh(\\mathcal{X}_{fppf})$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TR","source_file":"stacks-sheaves.tex","source_line":397,"source_end_line":425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L397-L425","statement_sha256":"747e5f84d008d2174e3a35435e46e4ae8c93e24ac739258cbe935de1d43e0d9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13685,"rank":13685,"depth":0,"x":2081.452,"y":1582.651,"cluster":"algebraic-stacks"},{"id":"stacks:06TS","tag":"06TS","title":"Sheaves · Lemma 06TS","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). The functors _pf and f^p of ([Tag 06TK]) transform τ sheaves into τ sheaves and define a morphism of topoi f : Sh(X_τ) → Sh(Y_τ).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThe functors ${}_pf$ and $f^p$ of (\\ref{equation-pushforward-pullback})\ntransform $\\tau$ sheaves into $\\tau$ sheaves and define a morphism\nof topoi\n$f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TS","source_file":"stacks-sheaves.tex","source_line":460,"source_end_line":469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L460-L469","statement_sha256":"8d2611e925be4c8063d3212f710def49101a819da01ff8a9ab1315e423c79fff","origin":"The Stacks Project","memory_eligible":false,"source_rank":13686,"rank":13686,"depth":7,"x":2005.866,"y":1642.243,"cluster":"algebraic-stacks"},{"id":"stacks:06TT","tag":"06TT","title":"Sheaves · Definition 06TT","summary":"Let f : X → Y be a morphism of categories fibred in groupoids over (Sch/S)_fppf. We denote f = (f^-1, f_*) : Sh(X_fppf) → Sh(Y_fppf) the associated morphism of fppf topoi constructed above. Similarly for the associated Zariski, étale, smooth, and syntomic topoi.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. We denote\n$$\nf = (f^{-1}, f_*) :\n\\Sh(\\mathcal{X}_{fppf})\n\\longrightarrow\n\\Sh(\\mathcal{Y}_{fppf})\n$$\nthe {\\it associated morphism of fppf topoi} constructed above.\nSimilarly for the associated Zariski, \\'etale, smooth, and syntomic topoi.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TT","source_file":"stacks-sheaves.tex","source_line":483,"source_end_line":495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L483-L495","statement_sha256":"37dc7fe42bf50919573e52fc8c7ded583f376f2a1e8ebf96a76ee59af24963fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":13687,"rank":13687,"depth":0,"x":2013.728,"y":1554.886,"cluster":"algebraic-stacks"},{"id":"stacks:075B","tag":"075B","title":"Computing pushforward · Lemma 075B","summary":"Let S be a scheme. Let xymatrix Y' ×_Y X ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a 2-cartesian diagram of categories fibred in groupoids over S. Then we have a canonical isomorphism g^-1f_*F → f'_*(g')^-1F functorial in the presheaf F on X.","statement_latex":"Let $S$ be a scheme. Let\n$$\n\\xymatrix{\n\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r]_{g'} \\ar[d]_{f'} &\n\\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}\n}\n$$\nbe a $2$-cartesian diagram of categories fibred in groupoids over $S$.\nThen we have a canonical isomorphism\n$$\ng^{-1}f_*\\mathcal{F} \\longrightarrow f'_*(g')^{-1}\\mathcal{F}\n$$\nfunctorial in the presheaf $\\mathcal{F}$ on $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Computing pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075B","source_file":"stacks-sheaves.tex","source_line":547,"source_end_line":563,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L547-L563","statement_sha256":"723519b9c15daef33a1874bf8623b171e8df9d9ba9f13e04b7dc47bea84f0385","origin":"The Stacks Project","memory_eligible":false,"source_rank":13688,"rank":13688,"depth":0,"x":2078.565,"y":1624.176,"cluster":"algebraic-stacks"},{"id":"stacks:06W7","tag":"06W7","title":"Computing pushforward · Lemma 06W7","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. The following are equivalent • f is representable, and • for every y ∈ Ob(Y) the functor X^opp → Sets, x ↦ Mor_Y(f(x), y) is representable.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. The following are\nequivalent\n\\begin{enumerate}\n\\item $f$ is representable, and\n\\item for every $y \\in \\Ob(\\mathcal{Y})$ the functor\n$\\mathcal{X}^{opp} \\to \\textit{Sets}$,\n$x \\mapsto \\Mor_\\mathcal{Y}(f(x), y)$\nis representable.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Computing pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06W7","source_file":"stacks-sheaves.tex","source_line":585,"source_end_line":597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L585-L597","statement_sha256":"e6ad15562e17432eba1e76cb38d562c2bf90f936e17d83d41bab959eec125801","origin":"The Stacks Project","memory_eligible":false,"source_rank":13689,"rank":13689,"depth":0,"x":1974.422,"y":1609.787,"cluster":"algebraic-stacks"},{"id":"stacks:06W8","tag":"06W8","title":"Computing pushforward · Lemma 06W8","summary":"Let f : X → Y be a representable 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). Then the functor u : Y_τ → X_τ is continuous and defines a morphism of sites X_τ → Y_τ which induces the same morphism of topoi Sh(X_τ) → Sh(Y_τ) as the morphism f constructed in Lemma [Tag 06TS]. Moreover, f_*F(y) = F(u(y)) for any presheaf F on X.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a representable $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThen the functor $u : \\mathcal{Y}_\\tau \\to \\mathcal{X}_\\tau$ is continuous\nand defines a morphism of sites $\\mathcal{X}_\\tau \\to \\mathcal{Y}_\\tau$\nwhich induces the same morphism of topoi\n$\\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$\nas the morphism $f$ constructed in\nLemma \\ref{lemma-functoriality-sheaves}.\nMoreover, $f_*\\mathcal{F}(y) = \\mathcal{F}(u(y))$ for any presheaf\n$\\mathcal{F}$ on $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Computing pushforward","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06W8","source_file":"stacks-sheaves.tex","source_line":656,"source_end_line":669,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L656-L669","statement_sha256":"a1f6af4f7a8849aec44f599de507a5d9f7218f01f4c462e7e4ab5482b93e1144","origin":"The Stacks Project","memory_eligible":false,"source_rank":13690,"rank":13690,"depth":8,"x":2063.304,"y":1561.022,"cluster":"algebraic-stacks"},{"id":"stacks:06TV","tag":"06TV","title":"The structure sheaf · Definition 06TV","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. The structure sheaf of X is the sheaf of rings O_X = p^-1O.","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. The\n{\\it structure sheaf of $\\mathcal{X}$} is the sheaf of rings\n$\\mathcal{O}_\\mathcal{X} = p^{-1}\\mathcal{O}$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The structure sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TV","source_file":"stacks-sheaves.tex","source_line":731,"source_end_line":737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L731-L737","statement_sha256":"418a998ea88b0ba779d0546c58bec09a289dc40ea0743fe6cb2f9f3a1ca79a39","origin":"The Stacks Project","memory_eligible":false,"source_rank":13691,"rank":13691,"depth":0,"x":2036.826,"y":1647.913,"cluster":"algebraic-stacks"},{"id":"stacks:06TW","tag":"06TW","title":"The structure sheaf · Lemma 06TW","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). There is a canonical identification f^-1O_Y = O_X which turns f : Sh(X_τ) → Sh(Y_τ) into a morphism of ringed topoi.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThere is a canonical identification\n$f^{-1}\\mathcal{O}_\\mathcal{Y} = \\mathcal{O}_\\mathcal{X}$\nwhich turns\n$f : \\Sh(\\mathcal{X}_\\tau) \\to \\Sh(\\mathcal{Y}_\\tau)$\ninto a morphism of ringed topoi.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The structure sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TW","source_file":"stacks-sheaves.tex","source_line":748,"source_end_line":758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L748-L758","statement_sha256":"836297ab343b3429ad763adc10bfd1d0cc9786a9793955b3760f5de451b0733d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13692,"rank":13692,"depth":2,"x":1986.189,"y":1568.364,"cluster":"algebraic-stacks"},{"id":"stacks:06WB","tag":"06WB","title":"Sheaves of modules · Definition 06WB","summary":"Let X be a category fibred in groupoids over (Sch/S)_fppf. • A presheaf of modules on X is a presheaf of O_X-modules. The category of presheaves of modules is denoted PMod(O_X). • We say a presheaf of modules F is an O_X-module, or more precisely a sheaf of O_X-modules if F is an fppf sheaf. The category of O_X-modules is denoted Mod(O_X).","statement_latex":"Let $\\mathcal{X}$ be a category fibred in groupoids over\n$(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item A {\\it presheaf of modules on $\\mathcal{X}$} is a\npresheaf of $\\mathcal{O}_\\mathcal{X}$-modules. The category of\npresheaves of modules is denoted $\\textit{PMod}(\\mathcal{O}_\\mathcal{X})$.\n\\item We say a presheaf of modules $\\mathcal{F}$ is an\n{\\it $\\mathcal{O}_\\mathcal{X}$-module}, or more precisely a\n{\\it sheaf of $\\mathcal{O}_\\mathcal{X}$-modules} if $\\mathcal{F}$\nis an fppf sheaf. The category of $\\mathcal{O}_\\mathcal{X}$-modules\nis denoted $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Sheaves of modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WB","source_file":"stacks-sheaves.tex","source_line":797,"source_end_line":811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L797-L811","statement_sha256":"1633d5e941663db92ea6e38bc218744c58e9e53c0a335a7f4b96b39195a04ca5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13693,"rank":13693,"depth":0,"x":2088.07,"y":1598.459,"cluster":"algebraic-stacks"},{"id":"stacks:075I","tag":"075I","title":"Representable categories · Lemma 075I","summary":"Let S be a scheme. Let X be a category fibred in groupoids over (Sch/S). Assume X is representable by a scheme X. For τ ∈ (Zar,linebreak[0] etale,linebreak[0] smooth,linebreak[0] syntomic,linebreak[0] fppf) there is a canonical equivalence (X_τ, O_X) = ((Sch/X)_τ, O_X) of ringed sites.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred\nin groupoids over $(\\Sch/S)$. Assume $\\mathcal{X}$ is representable\nby a scheme $X$. For $\\tau \\in \\{Zar,\\linebreak[0] \\etale,\\linebreak[0]\nsmooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$\nthere is a canonical equivalence\n$$\n(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X}) =\n((\\Sch/X)_\\tau, \\mathcal{O}_X)\n$$\nof ringed sites.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Representable categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075I","source_file":"stacks-sheaves.tex","source_line":922,"source_end_line":934,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L922-L934","statement_sha256":"ab83a6ffdb0e7fd30baee3f7bf9c723489aba7e5a0f629ea2fd1d7dd104fe578","origin":"The Stacks Project","memory_eligible":false,"source_rank":13694,"rank":13694,"depth":0,"x":1988.188,"y":1634.276,"cluster":"algebraic-stacks"},{"id":"stacks:075J","tag":"075J","title":"Representable categories · Lemma 075J","summary":"Let S be a scheme. Let f : X → Y be a morphism of categories fibred in groupoids over S. Assume X, Y are representable by schemes X, Y. Let f : X → Y be the morphism of schemes corresponding to f. For τ ∈ (Zar,linebreak[0] etale,linebreak[0] smooth,linebreak[0] syntomic,linebreak[0] fppf) the morphism of ringed topoi f : (Sh(X_τ), O_X) → (Sh(Y_τ), O_Y) agrees with the morphism of ringed topoi f : (Sh((Sch/X)_τ), O_X) → (Sh((Sch/Y)_τ), O_Y) via the identifications of Lemma…","statement_latex":"Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism\nof categories fibred in groupoids over $S$.\nAssume $\\mathcal{X}$, $\\mathcal{Y}$ are representable by schemes\n$X$, $Y$. Let $f : X \\to Y$ be the morphism of schemes corresponding\nto $f$. For $\\tau \\in \\{Zar,\\linebreak[0] \\etale,\\linebreak[0]\nsmooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$\nthe morphism of ringed topoi\n$f : (\\Sh(\\mathcal{X}_\\tau), \\mathcal{O}_\\mathcal{X}) \\to\n(\\Sh(\\mathcal{Y}_\\tau), \\mathcal{O}_\\mathcal{Y})$\nagrees with the morphism of ringed topoi\n$f : (\\Sh((\\Sch/X)_\\tau), \\mathcal{O}_X) \\to \n(\\Sh((\\Sch/Y)_\\tau), \\mathcal{O}_Y)$ via the identifications of\nLemma \\ref{lemma-compare-with-scheme}.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Representable categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075J","source_file":"stacks-sheaves.tex","source_line":943,"source_end_line":958,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L943-L958","statement_sha256":"23ed59ec30c2d117af9d91e426e31895f28fb511c9978236c280f4fa9bc6fdd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13695,"rank":13695,"depth":1,"x":2033.286,"y":1550.731,"cluster":"algebraic-stacks"},{"id":"stacks:06W0","tag":"06W0","title":"Restriction · Lemma 06W0","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). Let x ∈ Ob(X) lying over U = p(x). The functor p induces an equivalence of sites X_τ/x → (Sch/U)_τ.","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nLet $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$.\nThe functor $p$ induces an equivalence of sites\n$\\mathcal{X}_\\tau/x \\to (\\Sch/U)_\\tau$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restriction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06W0","source_file":"stacks-sheaves.tex","source_line":976,"source_end_line":983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L976-L983","statement_sha256":"6b6a7bdd8173b50540189740628e2fef356f5343d44dbaf5e83d3a9bfa26d2dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13696,"rank":13696,"depth":5,"x":2067.399,"y":1638.402,"cluster":"algebraic-stacks"},{"id":"stacks:06W1","tag":"06W1","title":"Restriction · Definition 06W1","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let x ∈ Ob(X) lying over U = p(x). Let F be a presheaf on X. • The pullback x^-1F of F is the restriction F|_(X/x) viewed as a presheaf on (Sch/U)_fppf via the equivalence X/x → (Sch/U)_fppf of Lemma [Tag 06W0]. • The restriction of F to U_etale is x^-1F|_U_etale, abusively written F|_U_etale.","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. Let $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$.\nLet $\\mathcal{F}$ be a presheaf on $\\mathcal{X}$.\n\\begin{enumerate}\n\\item The {\\it pullback $x^{-1}\\mathcal{F}$ of $\\mathcal{F}$} is the\nrestriction $\\mathcal{F}|_{(\\mathcal{X}/x)}$ viewed as a presheaf on\n$(\\Sch/U)_{fppf}$ via the equivalence\n$\\mathcal{X}/x \\to (\\Sch/U)_{fppf}$ of\nLemma \\ref{lemma-localizing}.\n\\item The {\\it restriction of $\\mathcal{F}$ to $U_\\etale$}\nis $x^{-1}\\mathcal{F}|_{U_\\etale}$, abusively written\n$\\mathcal{F}|_{U_\\etale}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restriction","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06W1","source_file":"stacks-sheaves.tex","source_line":993,"source_end_line":1008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L993-L1008","statement_sha256":"746db5d907554afc5aeeae489f62ee2219b4270ce6fe6aa2dd0206e728b22437","origin":"The Stacks Project","memory_eligible":false,"source_rank":13697,"rank":13697,"depth":6,"x":1971.227,"y":1592.866,"cluster":"algebraic-stacks"},{"id":"stacks:075D","tag":"075D","title":"Restriction · Lemma 075D","summary":"Let F be an étale sheaf on X → (Sch/S)_fppf. • If φ : x → y and ψ : y → z are morphisms of X lying over a : U → V and b : V → W, then the composition a_small^-1(b_small^-1 (F|_W_etale)) xrightarrowa_small^-1c_ψ a_small^-1(F|_V_etale) xrightarrowc_φ F|_U_etale is equal to c_ψ ∘ φ via the identification (b ∘ a)_small^-1(F|_W_etale) = a_small^-1(b_small^-1 (F|_W_etale)). • If φ : x → y lies over an étale morphism of schemes a : U → V, then ([Tag 06W3]) is an isomorphism. •…","statement_latex":"Let $\\mathcal{F}$ be an \\'etale sheaf on $\\mathcal{X} \\to (\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item If $\\varphi : x \\to y$ and $\\psi : y \\to z$\nare morphisms of $\\mathcal{X}$ lying over $a : U \\to V$ and\n$b : V \\to W$, then the composition\n$$\na_{small}^{-1}(b_{small}^{-1} (\\mathcal{F}|_{W_\\etale}))\n\\xrightarrow{a_{small}^{-1}c_\\psi}\na_{small}^{-1}(\\mathcal{F}|_{V_\\etale})\n\\xrightarrow{c_\\varphi}\n\\mathcal{F}|_{U_\\etale}\n$$\nis equal to $c_{\\psi \\circ \\varphi}$ via the identification\n$$\n(b \\circ a)_{small}^{-1}(\\mathcal{F}|_{W_\\etale}) =\na_{small}^{-1}(b_{small}^{-1} (\\mathcal{F}|_{W_\\etale})).\n$$\n\\item If $\\varphi : x \\to y$ lies over an \\'etale morphism of schemes\n$a : U \\to V$, then (\\ref{equation-comparison}) is an isomorphism.\n\\item Suppose $f : \\mathcal{Y} \\to \\mathcal{X}$ is a $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$ and $y$ is\nan object of $\\mathcal{Y}$ lying over the scheme $U$ with image\n$x = f(y)$. Then there is a canonical identification\n$f^{-1}\\mathcal{F}|_{U_\\etale} = \\mathcal{F}|_{U_\\etale}$.\n\\item Moreover, given $\\psi : y' \\to y$ in $\\mathcal{Y}$ lying over\n$a : U' \\to U$ the comparison map\n$c_\\psi : a_{small}^{-1}(f^{-1}\\mathcal{F}|_{U_\\etale}) \\to\nf^{-1}\\mathcal{F}|_{U'_\\etale}$ is equal to the\ncomparison map $c_{f(\\psi)} : a_{small}^{-1}\\mathcal{F}|_{U_\\etale}\n\\to \\mathcal{F}|_{U'_\\etale}$ via the identifications in (3).\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restriction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075D","source_file":"stacks-sheaves.tex","source_line":1085,"source_end_line":1118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1085-L1118","statement_sha256":"1fe1dd6d250defc0de0a419fa84bf354c9a946605de6bc0e8c9b081eb6e4dc22","origin":"The Stacks Project","memory_eligible":false,"source_rank":13698,"rank":13698,"depth":0,"x":2079.336,"y":1571.762,"cluster":"algebraic-stacks"},{"id":"stacks:06W9","tag":"06W9","title":"Restriction · Lemma 06W9","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). Let x ∈ Ob(X) lying over U = p(x). The equivalence of Lemma [Tag 06W0] extends to an equivalence of ringed sites (X_τ/x, O_X|_x) → ((Sch/U)_τ, O).","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nLet $x \\in \\Ob(\\mathcal{X})$ lying over $U = p(x)$.\nThe equivalence of\nLemma \\ref{lemma-localizing}\nextends to an equivalence of ringed sites\n$(\\mathcal{X}_\\tau/x, \\mathcal{O}_\\mathcal{X}|_x) \\to\n((\\Sch/U)_\\tau, \\mathcal{O})$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restriction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06W9","source_file":"stacks-sheaves.tex","source_line":1127,"source_end_line":1137,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1127-L1137","statement_sha256":"ff6e34264cc75ca72a9c55ea2022b5ba216f83a80cd132057391e5c3c2320e91","origin":"The Stacks Project","memory_eligible":false,"source_rank":13699,"rank":13699,"depth":6,"x":2016.257,"y":1649.073,"cluster":"algebraic-stacks"},{"id":"stacks:06W4","tag":"06W4","title":"Restriction · Lemma 06W4","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). The site X_τ has enough points.","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. Let $\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThe site $\\mathcal{X}_\\tau$ has enough points.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restriction","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06W4","source_file":"stacks-sheaves.tex","source_line":1176,"source_end_line":1181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1176-L1181","statement_sha256":"f3b0d3623e6e463e885c05622e8995aca8cf065628713dfd9a4aa41fc306b471","origin":"The Stacks Project","memory_eligible":false,"source_rank":13700,"rank":13700,"depth":52,"x":2000.518,"y":1555.787,"cluster":"algebraic-stacks"},{"id":"stacks:073M","tag":"073M","title":"Restriction to algebraic spaces · Lemma 073M","summary":"Let S be a scheme. Let X → (Sch/S)_fppf be a category fibred in groupoids. Assume X is representable by an algebraic space F. Then there exists a continuous and cocontinuous functor F_etale → X_etale which induces a morphism of ringed sites π_F : (X_etale, O_X) → (F_etale, O_F) and a morphism of ringed topoi i_F : (Sh(F_etale), O_F) → (Sh(X_etale), O_X) such that π_F ∘ i_F = id. Moreover π_F, * = i_F^-1.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Assume $\\mathcal{X}$ is representable by an algebraic\nspace $F$. Then there exists a continuous and cocontinuous functor\n$\nF_\\etale \\to \\mathcal{X}_\\etale\n$\nwhich induces a morphism of ringed sites\n$$\n\\pi_F :\n(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})\n\\longrightarrow\n(F_\\etale, \\mathcal{O}_F)\n$$\nand a morphism of ringed topoi\n$$\ni_F :\n(\\Sh(F_\\etale), \\mathcal{O}_F)\n\\longrightarrow\n(\\Sh(\\mathcal{X}_\\etale), \\mathcal{O}_\\mathcal{X})\n$$\nsuch that $\\pi_F \\circ i_F = \\text{id}$. Moreover $\\pi_{F, *} = i_F^{-1}$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restriction to algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/073M","source_file":"stacks-sheaves.tex","source_line":1212,"source_end_line":1235,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1212-L1235","statement_sha256":"1019d145569c244cceb3a8450cd0444eb4188bb90f36eb59c9105bad058b5473","origin":"The Stacks Project","memory_eligible":false,"source_rank":13701,"rank":13701,"depth":7,"x":2087.589,"y":1615.955,"cluster":"algebraic-stacks"},{"id":"stacks:073N","tag":"073N","title":"Restriction to algebraic spaces · Lemma 073N","summary":"Let S be a scheme. Let f : X → Y be a morphism of categories fibred in groupoids over (Sch/S)_fppf. Assume X, Y are representable by algebraic spaces F, G. Denote f : F → G the induced morphism of algebraic spaces, and f_small : F_etale → G_etale the corresponding morphism of ringed topoi. Then xymatrix (Sh(X_etale), O_X) ar[d]_π_F ar[rr]_f & & (Sh(Y_etale), O_Y) ar[d]^π_G (Sh(F_etale), O_F) ar[rr]^f_small & & (Sh(G_etale), O_G) is a commutative diagram of ringed topoi.","statement_latex":"Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism\nof categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Assume\n$\\mathcal{X}$, $\\mathcal{Y}$ are representable by algebraic spaces $F$, $G$.\nDenote $f : F \\to G$ the induced morphism of algebraic spaces, and\n$f_{small} : F_\\etale \\to G_\\etale$\nthe corresponding morphism of ringed topoi. Then\n$$\n\\xymatrix{\n(\\Sh(\\mathcal{X}_\\etale), \\mathcal{O}_\\mathcal{X})\n\\ar[d]_{\\pi_F} \\ar[rr]_f & &\n(\\Sh(\\mathcal{Y}_\\etale), \\mathcal{O}_\\mathcal{Y}) \\ar[d]^{\\pi_G} \\\\\n(\\Sh(F_\\etale), \\mathcal{O}_F) \\ar[rr]^{f_{small}} & &\n(\\Sh(G_\\etale), \\mathcal{O}_G)\n}\n$$\nis a commutative diagram of ringed topoi.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restriction to algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/073N","source_file":"stacks-sheaves.tex","source_line":1335,"source_end_line":1353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1335-L1353","statement_sha256":"c2a734547c53d97ee3245abc38d68f7dcdea27c279cd3f9a919b602704ba9b97","origin":"The Stacks Project","memory_eligible":false,"source_rank":13702,"rank":13702,"depth":47,"x":1974.423,"y":1621.019,"cluster":"algebraic-stacks"},{"id":"stacks:06WG","tag":"06WG","title":"Quasi-coherent modules · Definition 06WG","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. A quasi-coherent module on X, or a quasi-coherent O_X-module is a quasi-coherent module on the ringed site (X_fppf, O_X) as in Modules on Sites, Definition [Tag 03DL]. The category of quasi-coherent sheaves on X is denoted QCoh(O_X).","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred\nin groupoids. A {\\it quasi-coherent module on $\\mathcal{X}$}, or a\n{\\it quasi-coherent $\\mathcal{O}_\\mathcal{X}$-module} is a\nquasi-coherent module on the ringed site\n$(\\mathcal{X}_{fppf}, \\mathcal{O}_\\mathcal{X})$ as in\nModules on Sites, Definition \\ref{sites-modules-definition-site-local}.\nThe category of quasi-coherent sheaves on $\\mathcal{X}$\nis denoted $\\QCoh(\\mathcal{O}_\\mathcal{X})$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WG","source_file":"stacks-sheaves.tex","source_line":1533,"source_end_line":1543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1533-L1543","statement_sha256":"348417eb71fad42cdb0eca073833afa4b4d785a19456979ff3748d7ce1f41a29","origin":"The Stacks Project","memory_eligible":false,"source_rank":13703,"rank":13703,"depth":2,"x":2054.2,"y":1552.728,"cluster":"algebraic-stacks"},{"id":"stacks:06WH","tag":"06WH","title":"Quasi-coherent modules · Lemma 06WH","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. The pullback functor f^* = f^-1 : Mod(O_Y) → Mod(O_X) preserves quasi-coherent sheaves.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$.\nThe pullback functor\n$f^* = f^{-1} : \\textit{Mod}(\\mathcal{O}_\\mathcal{Y}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\npreserves quasi-coherent sheaves.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WH","source_file":"stacks-sheaves.tex","source_line":1559,"source_end_line":1567,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1559-L1567","statement_sha256":"da5bdacad7f361bab742d62e64641c993a903c7e3ec85df4f00beb2d6662887f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13704,"rank":13704,"depth":11,"x":2050.271,"y":1648.832,"cluster":"algebraic-stacks"},{"id":"stacks:06WI","tag":"06WI","title":"Quasi-coherent modules · Lemma 06WI","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let F be a sheaf of O_X-modules. Then F is quasi-coherent if and only if x^*F is a quasi-coherent sheaf on (Sch/U)_fppf for every object x of X with U = p(x).","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Let $\\mathcal{F}$\nbe a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules. Then $\\mathcal{F}$\nis quasi-coherent if and only if $x^*\\mathcal{F}$ is a quasi-coherent\nsheaf on $(\\Sch/U)_{fppf}$ for every object $x$ of\n$\\mathcal{X}$ with $U = p(x)$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WI","source_file":"stacks-sheaves.tex","source_line":1580,"source_end_line":1588,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1580-L1588","statement_sha256":"5d74201ccad480c5aa179d5ede03b26ed637b751ad0d88db4114b1b1afdb77a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13705,"rank":13705,"depth":12,"x":1975.515,"y":1575.373,"cluster":"algebraic-stacks"},{"id":"stacks:0EM8","tag":"0EM8","title":"Quasi-coherent modules · Lemma 0EM8","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let F be a presheaf of modules on X. The following are equivalent • F is an object of Mod(X_Zar, O_X) and F is a quasi-coherent module on (X_Zar, O_X) in the sense of Modules on Sites, Definition [Tag 03DL], • F is an object of Mod(X_etale, O_X) and F is a quasi-coherent module on (X_etale, O_X) in the sense of Modules on Sites, Definition [Tag 03DL], and • F is a quasi-coherent module on X in the sense of…","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Let $\\mathcal{F}$ be a presheaf of\nmodules on $\\mathcal{X}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an object of\n$\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$\nand $\\mathcal{F}$ is a quasi-coherent module on\n$(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$ in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local},\n\\item $\\mathcal{F}$ is an object of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nand $\\mathcal{F}$ is a quasi-coherent module on\n$(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local}, and\n\\item $\\mathcal{F}$ is a quasi-coherent module on $\\mathcal{X}$\nin the sense of Definition \\ref{definition-quasi-coherent}.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EM8","source_file":"stacks-sheaves.tex","source_line":1601,"source_end_line":1620,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1601-L1620","statement_sha256":"6942c219f80c53898130fdf1014bdb9f2e93ae3749ad2c1bbf15001e820dd280","origin":"The Stacks Project","memory_eligible":false,"source_rank":13706,"rank":13706,"depth":40,"x":2090.274,"y":1587.181,"cluster":"algebraic-stacks"},{"id":"stacks:06WJ","tag":"06WJ","title":"Locally quasi-coherent modules · Definition 06WJ","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let F be a presheaf of O_X-modules. We say F is locally quasi-coherent if F is a sheaf for the étale topology and for every object x of X the restriction x^*F|_U_etale is a quasi-coherent sheaf. Here U = p(x).","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Let $\\mathcal{F}$\nbe a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules.\nWe say $\\mathcal{F}$ is {\\it locally quasi-coherent}\\footnote{This is\nnonstandard notation.} if\n$\\mathcal{F}$ is a sheaf for the \\'etale topology and\nfor every object $x$ of $\\mathcal{X}$ the restriction\n$x^*\\mathcal{F}|_{U_\\etale}$ is a quasi-coherent\nsheaf. Here $U = p(x)$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Locally quasi-coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WJ","source_file":"stacks-sheaves.tex","source_line":1659,"source_end_line":1670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1659-L1670","statement_sha256":"008824c8035688a70602d73715fa3ff386b78ff5cec4d0061bfb94ea7767899c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13707,"rank":13707,"depth":0,"x":1995.698,"y":1643.865,"cluster":"algebraic-stacks"},{"id":"stacks:06WK","tag":"06WK","title":"Locally quasi-coherent modules · Lemma 06WK","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let F be a presheaf of O_X-modules. Then F is quasi-coherent if and only if the following two conditions hold • F is locally quasi-coherent, and • for any morphism φ : x → y of X lying over f : U → V the comparison map c_φ : f_small^*F|_V_etale → F|_U_etale of ([Tag 06WC]) is an isomorphism.","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Let $\\mathcal{F}$\nbe a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules. Then $\\mathcal{F}$\nis quasi-coherent if and only if the following two conditions hold\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is locally quasi-coherent, and\n\\item for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ lying over\n$f : U \\to V$ the comparison map\n$c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to\n\\mathcal{F}|_{U_\\etale}$ of\n(\\ref{equation-comparison-modules}) is an isomorphism.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Locally quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WK","source_file":"stacks-sheaves.tex","source_line":1692,"source_end_line":1706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1692-L1706","statement_sha256":"e4d599bfb8271d3cc88d094c6218b6ad87a64f40e63fa9f871a848af925ad888","origin":"The Stacks Project","memory_eligible":false,"source_rank":13708,"rank":13708,"depth":22,"x":2019.971,"y":1547.942,"cluster":"algebraic-stacks"},{"id":"stacks:06WL","tag":"06WL","title":"Locally quasi-coherent modules · Lemma 06WL","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. The pullback functor f^* = f^-1 : Mod(Y_etale, O_Y) → Mod(X_etale, O_X) preserves locally quasi-coherent sheaves.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. The pullback functor\n$f^* = f^{-1} :\n\\textit{Mod}(\\mathcal{Y}_\\etale, \\mathcal{O}_\\mathcal{Y})\n\\to\n\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\npreserves locally quasi-coherent sheaves.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Locally quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WL","source_file":"stacks-sheaves.tex","source_line":1741,"source_end_line":1750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1741-L1750","statement_sha256":"bb461a503461a393245165ddab57cfaadbb8d3cb97e2385adf89576e5b057d8e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13709,"rank":13709,"depth":0,"x":2079.493,"y":1632.851,"cluster":"algebraic-stacks"},{"id":"stacks:06WM","tag":"06WM","title":"Locally quasi-coherent modules · Lemma 06WM","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. • The category LQCoh(O_X) has colimits and they agree with colimits in the category Mod(X_etale, O_X). • The category LQCoh(O_X) is abelian with kernels and cokernels computed in Mod(X_etale, O_X), in other words the inclusion functor is exact. • Given a short exact sequence 0 → F_1 → F_2 → F_3 → 0 of Mod(X_etale, O_X) if two out of three are locally quasi-coherent so is the third. • Given F, G in LQCoh(O_X) the…","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in\ngroupoids.\n\\begin{enumerate}\n\\item The category $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$\nhas colimits and they agree with colimits in the category\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.\n\\item The category $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$\nis abelian with kernels and cokernels computed in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$,\nin other words the inclusion functor is exact.\n\\item Given a short exact sequence\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$ of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nif two out of three are locally quasi-coherent so is the third.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$\nthe tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$\nin $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nis an object of $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$\nwith $\\mathcal{F}$ of finite presentation on\n$\\mathcal{X}_\\etale$ the sheaf\n$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$\nin $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nis an object of $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Locally quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WM","source_file":"stacks-sheaves.tex","source_line":1760,"source_end_line":1789,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1760-L1789","statement_sha256":"8931e22b26f32201adb0ea598ffe64043b188995fe309f661e1b0337c4d34fec","origin":"The Stacks Project","memory_eligible":false,"source_rank":13710,"rank":13710,"depth":41,"x":1966.791,"y":1603.878,"cluster":"algebraic-stacks"},{"id":"stacks:06WN","tag":"06WN","title":"Locally quasi-coherent modules · Lemma 06WN","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. • The category QCoh(O_X) has colimits and they agree with colimits in the categories Mod(X_Zar, O_X), Mod(X_etale, O_X), Mod(O_X), and LQCoh(O_X). • Given F, G in QCoh(O_X) the tensor products F ⊗_O_X G computed in Mod(X_Zar, O_X), Mod(X_etale, O_X), or Mod(O_X) agree and the common value is an object of QCoh(O_X). • Given F, G in QCoh(O_X) with F finite locally free (in fppf, or equivalently étale, or…","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids.\n\\begin{enumerate}\n\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nhas colimits and they agree with colimits in the categories\n$\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$,\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$,\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$, and\n$\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nthe tensor products $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$\ncomputed in $\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$,\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$, or\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ agree and the common value\nis an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nwith $\\mathcal{F}$ finite locally free (in fppf, or equivalently \\'etale, or\nequivalently Zariski topology) the internal homs\n$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$\ncomputed in $\\textit{Mod}(\\mathcal{X}_{Zar}, \\mathcal{O}_\\mathcal{X})$,\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$, or\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ agree and the common value\nis an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Locally quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WN","source_file":"stacks-sheaves.tex","source_line":1896,"source_end_line":1922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1896-L1922","statement_sha256":"5d43387aaa0359b940820785dd61f356deb2b60877d80ffbd23aa09e43f7939d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13711,"rank":13711,"depth":42,"x":2073.694,"y":1561.094,"cluster":"algebraic-stacks"},{"id":"stacks:06WQ","tag":"06WQ","title":"Stackification and sheaves · Lemma 06WQ","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. If f induces an equivalence of stackifications, then the morphism of topoi f : Sh(X_fppf) → Sh(Y_fppf) is an equivalence.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If\n$f$ induces an equivalence of stackifications, then the morphism\nof topoi\n$f : \\Sh(\\mathcal{X}_{fppf}) \\to \\Sh(\\mathcal{Y}_{fppf})$\nis an equivalence.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Stackification and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WQ","source_file":"stacks-sheaves.tex","source_line":1955,"source_end_line":1963,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1955-L1963","statement_sha256":"ec2cd6cefade0a21331582006ce3d8277b128f9859ee20ad7d1fc460ed51fc80","origin":"The Stacks Project","memory_eligible":false,"source_rank":13712,"rank":13712,"depth":8,"x":2029.064,"y":1653.729,"cluster":"algebraic-stacks"},{"id":"stacks:06WR","tag":"06WR","title":"Stackification and sheaves · Lemma 06WR","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. If f induces an equivalence of stackifications, then f^* induces equivalences Mod(O_X) → Mod(O_Y) and QCoh(O_X) → QCoh(O_Y).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If\n$f$ induces an equivalence of stackifications, then $f^*$\ninduces equivalences\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})$\nand\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\QCoh(\\mathcal{O}_\\mathcal{Y})$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Stackification and sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WR","source_file":"stacks-sheaves.tex","source_line":1979,"source_end_line":1990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L1979-L1990","statement_sha256":"136f610489fa80012d1b5b9de672c2a136ea975624e31934a14b60533bb3f205","origin":"The Stacks Project","memory_eligible":false,"source_rank":13713,"rank":13713,"depth":13,"x":1987.288,"y":1559.666,"cluster":"algebraic-stacks"},{"id":"stacks:0GQC","tag":"0GQC","title":"Quasi-coherent sheaves and presentations · Lemma 0GQC","summary":"Let S be a scheme. Let X → (Sch/S)_fppf be a category fibred in groupoids which is representable by an algebraic space F. If F is in LQCoh(O_X) then the restriction F|_F_etale ([Tag 075K]) is quasi-coherent.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids which is representable by an algebraic space $F$.\nIf $\\mathcal{F}$ is in $\\textit{LQCoh}(\\mathcal{O}_\\mathcal{X})$\nthen the restriction $\\mathcal{F}|_{F_\\etale}$ (\\ref{equation-restrict}) \nis quasi-coherent.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent sheaves and presentations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQC","source_file":"stacks-sheaves.tex","source_line":2029,"source_end_line":2036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2029-L2036","statement_sha256":"058da85c5e3295a30ea335fdc08dae3a34dae4714e20c28e386228559f85b19b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13714,"rank":13714,"depth":0,"x":2094.22,"y":1605.531,"cluster":"algebraic-stacks"},{"id":"stacks:0GQD","tag":"0GQD","title":"Quasi-coherent sheaves and presentations · Lemma 0GQD","summary":"Let S be a scheme. Let X → (Sch/S)_fppf be a category fibred in groupoids which is representable by an algebraic space F. The functor ([Tag 075K]) defines an equivalence QCoh(O_X) → QCoh(O_F), F ↦ F|_F_etale with quasi-inverse given by G ↦ π_F^*G. This equivalence is compatible with pullback for morphisms between categories fibred in groupoids representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids which is representable by an algebraic space $F$.\nThe functor (\\ref{equation-restrict}) defines an equivalence\n$$\n\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\QCoh(\\mathcal{O}_F),\\quad\n\\mathcal{F} \\longmapsto \\mathcal{F}|_{F_\\etale}\n$$\nwith quasi-inverse given by $\\mathcal{G} \\mapsto \\pi_F^*\\mathcal{G}$.\nThis equivalence is compatible with pullback for morphisms between\ncategories fibred in groupoids representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent sheaves and presentations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQD","source_file":"stacks-sheaves.tex","source_line":2045,"source_end_line":2057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2045-L2057","statement_sha256":"96c22c265e904775455c430cc3936d536f223059b7b63b8a8ec6c1a80a7a904c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13715,"rank":13715,"depth":48,"x":1977.965,"y":1632.507,"cluster":"algebraic-stacks"},{"id":"stacks:06WT","tag":"06WT","title":"Quasi-coherent sheaves and presentations · Proposition 06WT","summary":"Let (U, R, s, t, c) be a groupoid in algebraic spaces over S. Let X = [U/R] be the quotient stack. The category of quasi-coherent modules on X is equivalent to the category of quasi-coherent modules on (U, R, s, t, c).","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.\nLet $\\mathcal{X} = [U/R]$ be the quotient stack.\nThe category of quasi-coherent modules on $\\mathcal{X}$\nis equivalent to the category of quasi-coherent modules\non $(U, R, s, t, c)$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent sheaves and presentations","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WT","source_file":"stacks-sheaves.tex","source_line":2133,"source_end_line":2140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2133-L2140","statement_sha256":"59bc7017408d6452db030a73663aa5fd0302778ab5c1acdf800b8a31fc2fcb27","origin":"The Stacks Project","memory_eligible":false,"source_rank":13716,"rank":13716,"depth":49,"x":2042.283,"y":1546.266,"cluster":"algebraic-stacks"},{"id":"stacks:076S","tag":"076S","title":"Quasi-coherent sheaves and presentations · Lemma 076S","summary":"Let (U, R, s, t, c) be a groupoid in algebraic spaces over S. Assume s, t are flat and locally of finite presentation. Let X = [U/R] be the quotient stack. Denote x the object of X over U. Let F be a quasi-coherent O_X-module, and let H be any object of Mod(O_X). The map Hom_O_X(F, H) → Hom_O_U(x^*F|_U_etale, x^*H|_U_etale), φ ↦ x^*φ|_U_etale is injective and its image consists of exactly those φ : x^*F|_U_etale → x^*H|_U_etale which give rise to a commutative diagram…","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.\nAssume $s, t$ are flat and locally of finite presentation.\nLet $\\mathcal{X} = [U/R]$ be the quotient stack. Denote\n$x$ the object of $\\mathcal{X}$ over $U$.\nLet $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_\\mathcal{X}$-module, and let $\\mathcal{H}$ be any object\nof $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.\nThe map\n$$\n\\Hom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{H})\n\\longrightarrow\n\\Hom_{\\mathcal{O}_U}(x^*\\mathcal{F}|_{U_\\etale},\nx^*\\mathcal{H}|_{U_\\etale}),\n\\quad\n\\phi \\longmapsto x^*\\phi|_{U_\\etale}\n$$\nis injective and its image consists of exactly those\n$\\varphi : x^*\\mathcal{F}|_{U_\\etale} \\to\nx^*\\mathcal{H}|_{U_\\etale}$ which give rise to a commutative\ndiagram\n$$\n\\xymatrix{\ns_{small}^*(x^*\\mathcal{F}|_{U_\\etale})\n\\ar[r] \\ar[d]^{s_{small}^*\\varphi} &\n(x \\circ s)^*\\mathcal{F}|_{R_\\etale} =\n(x \\circ t)^*\\mathcal{F}|_{R_\\etale} &\nt_{small}^*(x^*\\mathcal{F}|_{U_\\etale})\n\\ar[l] \\ar[d]_{t_{small}^*\\varphi} \\\\\ns_{small}^*(x^*\\mathcal{H}|_{U_\\etale})\n\\ar[r] &\n(x \\circ s)^*\\mathcal{H}|_{R_\\etale} =\n(x \\circ t)^*\\mathcal{H}|_{R_\\etale} &\nt_{small}^*(x^*\\mathcal{H}|_{U_\\etale})\n\\ar[l]\n}\n$$\nof modules on $R_\\etale$\nwhere the horizontal arrows are the comparison maps\n(\\ref{equation-comparison-algebraic-spaces-modules}).","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent sheaves and presentations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/076S","source_file":"stacks-sheaves.tex","source_line":2246,"source_end_line":2287,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2246-L2287","statement_sha256":"0e59b5f79a2e055f76fec9299a33b68ac4e3a0747b2f35a05176352950371660","origin":"The Stacks Project","memory_eligible":false,"source_rank":13717,"rank":13717,"depth":50,"x":2064.305,"y":1646.798,"cluster":"algebraic-stacks"},{"id":"stacks:06WV","tag":"06WV","title":"Quasi-coherent sheaves on algebraic stacks · Lemma 06WV","summary":"Let X be an algebraic stack over S. • If [U/R] → X is a presentation of X then there is a canonical equivalence QCoh(O_X) ≅ QCoh(U, R, s, t, c). • The category QCoh(O_X) is abelian. • The inclusion functor QCoh(O_X) → Mod(O_X) is right exact but not exact in general. • The category QCoh(O_X) has colimits and they agree with colimits in the category Mod(O_X). • Given F, G in QCoh(O_X) the tensor product F ⊗_O_X G in Mod(O_X) is an object of QCoh(O_X). • Given F, G in…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over $S$.\n\\begin{enumerate}\n\\item If $[U/R] \\to \\mathcal{X}$ is a presentation of $\\mathcal{X}$\nthen there is a canonical equivalence\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\cong\n\\QCoh(U, R, s, t, c)$.\n\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ is abelian.\n\\item The inclusion functor $\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ is right exact but\n{\\bf not} exact in general.\n\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nhas colimits and they agree with colimits in the category\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\QCoh(\\mathcal{O}_\\mathcal{X})$\nthe tensor product $\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nis an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}, \\mathcal{G}$ in\n$\\QCoh(\\mathcal{O}_\\mathcal{X})$\nwith $\\mathcal{F}$ finite locally free the sheaf\n$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nis an object of $\\QCoh(\\mathcal{O}_\\mathcal{X})$.\n\\item Given a short exact sequence\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nwith $\\mathcal{F}_1$ and $\\mathcal{F}_3$ quasi-coherent, then\n$\\mathcal{F}_2$ is quasi-coherent.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent sheaves on algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WV","source_file":"stacks-sheaves.tex","source_line":2383,"source_end_line":2415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2383-L2415","statement_sha256":"417f6335d21945ee36dbc0521a81f69c8f9a91f3a1cc2f6ee32980f95e82a38b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13718,"rank":13718,"depth":50,"x":1966.795,"y":1584.891,"cluster":"algebraic-stacks"},{"id":"stacks:0781","tag":"0781","title":"Quasi-coherent sheaves on algebraic stacks · Proposition 0781","summary":"Let X be an algebraic stack over S. • The category QCoh(O_X) is a Grothendieck abelian category. Consequently, QCoh(O_X) has enough injectives and all limits. • The inclusion functor QCoh(O_X) → Mod(O_X) has a right adjoint. Q : Mod(O_X) → QCoh(O_X) such that for every quasi-coherent sheaf F the adjunction mapping Q(F) → F is an isomorphism.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over $S$.\n\\begin{enumerate}\n\\item The category $\\QCoh(\\mathcal{O}_\\mathcal{X})$ is a Grothendieck\nabelian category. Consequently, $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nhas enough injectives and all limits.\n\\item The inclusion functor\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ has a right adjoint\\footnote{This\nfunctor is sometimes called the {\\it coherator}.}\n$$\nQ :\n\\textit{Mod}(\\mathcal{O}_\\mathcal{X})\n\\longrightarrow\n\\QCoh(\\mathcal{O}_\\mathcal{X})\n$$\nsuch that for every quasi-coherent sheaf $\\mathcal{F}$ the adjunction mapping\n$Q(\\mathcal{F}) \\to \\mathcal{F}$ is an isomorphism.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent sheaves on algebraic stacks","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0781","source_file":"stacks-sheaves.tex","source_line":2439,"source_end_line":2459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2439-L2459","statement_sha256":"cbab64d1fc77580e4c48d3b8b89dd641cdffd16f12244d111ebeb790bc870875","origin":"The Stacks Project","memory_eligible":false,"source_rank":13719,"rank":13719,"depth":53,"x":2089.011,"y":1575.17,"cluster":"algebraic-stacks"},{"id":"stacks:075F","tag":"075F","title":"Cohomology · Lemma 075F","summary":"Let S be a scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Let x ∈ Ob(X) be an object lying over the scheme U. Let F be an object of Ab(X_τ) or Mod(X_τ, O_X). Then H^p_τ(x, F) = H^p((Sch/U)_τ, x^-1F) and if τ = etale, then we also have H^p_etale(x, F) = H^p(U_etale, F|_U_etale).","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids\nover $(\\Sch/S)_{fppf}$. Let $\\tau \\in \\{Zariski, \\etale, smooth,\nsyntomic, fppf\\}$. Let $x \\in \\Ob(\\mathcal{X})$ be an object lying\nover the scheme $U$. Let $\\mathcal{F}$ be\nan object of $\\textit{Ab}(\\mathcal{X}_\\tau)$ or\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$. Then\n$$\nH^p_\\tau(x, \\mathcal{F}) = H^p((\\Sch/U)_\\tau, x^{-1}\\mathcal{F})\n$$\nand if $\\tau = \\etale$, then we also have\n$$\nH^p_\\etale(x, \\mathcal{F}) =\nH^p(U_\\etale, \\mathcal{F}|_{U_\\etale}).\n$$","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075F","source_file":"stacks-sheaves.tex","source_line":2570,"source_end_line":2586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2570-L2586","statement_sha256":"5ce0c299b59914e25ecfd31f09136e74373d6f335c2c3a019730c7a87859cfd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13720,"rank":13720,"depth":10,"x":2006.353,"y":1652.017,"cluster":"algebraic-stacks"},{"id":"stacks:06WX","tag":"06WX","title":"Injective sheaves · Lemma 06WX","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). • f_*I is injective in Ab(Y_τ) for I injective in Ab(X_τ), and • f_*I is injective in Mod(Y_τ, O_Y) for I injective in Mod(X_τ, O_X).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\n\\begin{enumerate}\n\\item $f_*\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{Y}_\\tau)$\nfor $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{X}_\\tau)$, and\n\\item $f_*\\mathcal{I}$ is injective in\n$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$\nfor $\\mathcal{I}$ injective in\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Injective sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WX","source_file":"stacks-sheaves.tex","source_line":2617,"source_end_line":2630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2617-L2630","statement_sha256":"7b634ed2b07a7f7575c6bcd497245f0b87d4ca7f476ff544ad7ee71e9c06a8c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13721,"rank":13721,"depth":8,"x":2005.503,"y":1548.005,"cluster":"algebraic-stacks"},{"id":"stacks:06WY","tag":"06WY","title":"Injective sheaves · Lemma 06WY","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. • The category X has fibre products. • If the mathitIsom-presheaves of X are representable by algebraic spaces, then X has equalizers. • If X is an algebraic stack (or more generally a quotient stack), then X has equalizers.","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$\nbe a category fibred in groupoids.\n\\begin{enumerate}\n\\item The category $\\mathcal{X}$ has fibre products.\n\\item If the $\\mathit{Isom}$-presheaves of $\\mathcal{X}$\nare representable by algebraic spaces, then $\\mathcal{X}$ has equalizers.\n\\item If $\\mathcal{X}$ is an algebraic stack (or more generally\na quotient stack), then $\\mathcal{X}$ has equalizers.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Injective sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WY","source_file":"stacks-sheaves.tex","source_line":2646,"source_end_line":2657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2646-L2657","statement_sha256":"b73795fb9b90a88eb0b1c8400ae79dd9f21af0a646195c294ef6b3d5d2ac3a24","origin":"The Stacks Project","memory_eligible":false,"source_rank":13722,"rank":13722,"depth":69,"x":2090.13,"y":1624.543,"cluster":"algebraic-stacks"},{"id":"stacks:06WZ","tag":"06WZ","title":"Injective sheaves · Lemma 06WZ","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. • The functor f transforms fibre products into fibre products. • If f is faithful, then f transforms equalizers into equalizers.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item The functor $f$ transforms fibre products into fibre products.\n\\item If $f$ is faithful, then $f$ transforms equalizers into equalizers.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Injective sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06WZ","source_file":"stacks-sheaves.tex","source_line":2696,"source_end_line":2704,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2696-L2704","statement_sha256":"939c131a0dc385a0e9a64fd36b9a7c4101c992527ff3060792582d0bd2495c3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13723,"rank":13723,"depth":3,"x":1965.656,"y":1616.088,"cluster":"algebraic-stacks"},{"id":"stacks:06X0","tag":"06X0","title":"Injective sheaves · Lemma 06X0","summary":"Let f : X → Y, g : Z → Y be faithful 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. • the functor X ×_Y Z → Y is faithful, and • if X, Z have equalizers, so does X ×_Y Z.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Z} \\to \\mathcal{Y}$\nbe faithful $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item the functor $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Y}$\nis faithful, and\n\\item if $\\mathcal{X}, \\mathcal{Z}$ have equalizers, so does\n$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Injective sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06X0","source_file":"stacks-sheaves.tex","source_line":2727,"source_end_line":2738,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2727-L2738","statement_sha256":"feb24c1c1a2f208e3ed15e7bd99cc6732e5ed0d5ed2a3c5a9e2321f14044497e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13724,"rank":13724,"depth":4,"x":2064.652,"y":1551.425,"cluster":"algebraic-stacks"},{"id":"stacks:06X1","tag":"06X1","title":"Injective sheaves · Lemma 06X1","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). The functor f^-1 : Ab(Y_τ) → Ab(X_τ) has a left adjoint f_! : Ab(X_τ) → Ab(Y_τ). If f is faithful and X has equalizers, then • f_! is exact, and • f^-1I is injective in Ab(X_τ) for I injective in Ab(Y_τ).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThe functor\n$f^{-1} : \\textit{Ab}(\\mathcal{Y}_\\tau) \\to \\textit{Ab}(\\mathcal{X}_\\tau)$\nhas a left adjoint\n$f_! : \\textit{Ab}(\\mathcal{X}_\\tau) \\to \\textit{Ab}(\\mathcal{Y}_\\tau)$.\nIf $f$ is faithful and $\\mathcal{X}$ has equalizers, then\n\\begin{enumerate}\n\\item $f_!$ is exact, and\n\\item $f^{-1}\\mathcal{I}$ is injective in $\\textit{Ab}(\\mathcal{X}_\\tau)$\nfor $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{Y}_\\tau)$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Injective sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06X1","source_file":"stacks-sheaves.tex","source_line":2791,"source_end_line":2806,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2791-L2806","statement_sha256":"91bc2ffca7762053ef300bd325a5d4b47a2b41ce56b2008bab8419c72a7772fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13725,"rank":13725,"depth":70,"x":2043.566,"y":1655.709,"cluster":"algebraic-stacks"},{"id":"stacks:06X2","tag":"06X2","title":"Injective sheaves · Lemma 06X2","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). The functor f^* : Mod(Y_τ, O_Y) → Mod(X_τ, O_X) has a left adjoint f_! : Mod(X_τ, O_X) → Mod(Y_τ, O_Y) which agrees with the functor f_! of Lemma [Tag 06X1] on underlying abelian sheaves. If f is faithful and X has equalizers, then • f_! is exact, and • f^-1I is injective in Mod(X_τ, O_X) for I injective in Mod(Y_τ, O_X).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nThe functor\n$f^* : \\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y}) \\to\n\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$\nhas a left adjoint\n$f_! : \\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X}) \\to\n\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$ which\nagrees with the functor $f_!$ of Lemma \\ref{lemma-pullback-injective}\non underlying abelian sheaves.\nIf $f$ is faithful and $\\mathcal{X}$ has equalizers, then\n\\begin{enumerate}\n\\item $f_!$ is exact, and\n\\item $f^{-1}\\mathcal{I}$ is injective in\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$\nfor $\\mathcal{I}$ injective in\n$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Injective sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06X2","source_file":"stacks-sheaves.tex","source_line":2826,"source_end_line":2847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2826-L2847","statement_sha256":"1994efd5423c7853469ebceb276b8a5ee9e40e8896dce5a94f2a55b8077375b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13726,"rank":13726,"depth":71,"x":1974.973,"y":1566.482,"cluster":"algebraic-stacks"},{"id":"stacks:06X5","tag":"06X5","title":"The v Cech complex · Lemma 06X5","summary":"Generalities on v Cech complexes. • If xymatrix V ar[d]_g ar[r]_h & U ar[d]^f Y ar[r]^e & X is 2-commutative diagram of categories fibred in groupoids over (Sch/S)_fppf, then there is a morphism of v Cech complexes checkC^bullet(U → X, F) → checkC^bullet(V → Y, e^-1F) • if h and e are equivalences, then the map of (1) is an isomorphism, • if f, f' : U → X are 2-isomorphic, then the associated v Cech complexes are isomorphic.","statement_latex":"Generalities on {\\v C}ech complexes.\n\\begin{enumerate}\n\\item If\n$$\n\\xymatrix{\n\\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\\n\\mathcal{Y} \\ar[r]^e & \\mathcal{X}\n}\n$$\nis $2$-commutative diagram of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$, then there is a morphism of {\\v C}ech complexes\n$$\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{U} \\to \\mathcal{X}, \\mathcal{F})\n\\longrightarrow\n\\check{\\mathcal{C}}^\\bullet(\\mathcal{V} \\to \\mathcal{Y}, e^{-1}\\mathcal{F})\n$$\n\\item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,\n\\item if $f, f' : \\mathcal{U} \\to \\mathcal{X}$ are $2$-isomorphic, then\nthe associated {\\v C}ech complexes are isomorphic.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06X5","source_file":"stacks-sheaves.tex","source_line":2994,"source_end_line":3016,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L2994-L3016","statement_sha256":"673f096ba7bd784f480e44cf3388600c71a5c0c8094c2582565c97deec0d2638","origin":"The Stacks Project","memory_eligible":false,"source_rank":13727,"rank":13727,"depth":0,"x":2097.804,"y":1593.465,"cluster":"algebraic-stacks"},{"id":"stacks:06X6","tag":"06X6","title":"The v Cech complex · Lemma 06X6","summary":"If there exists a 1-morphism s : X → U such that f ∘ s is 2-isomorphic to id_X then the extended v Cech complex is homotopic to zero.","statement_latex":"If there exists a $1$-morphism $s : \\mathcal{X} \\to \\mathcal{U}$\nsuch that $f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{X}$\nthen the extended {\\v C}ech complex is homotopic to zero.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06X6","source_file":"stacks-sheaves.tex","source_line":3034,"source_end_line":3039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3034-L3039","statement_sha256":"6d8a73e5b69dcd33278f9029066397ef0f7051d5594dbd3d1f0c5efb437c39bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13728,"rank":13728,"depth":2,"x":1985.078,"y":1643.466,"cluster":"algebraic-stacks"},{"id":"stacks:06X9","tag":"06X9","title":"The relative v Cech complex · Lemma 06X9","summary":"Generalities on relative v Cech complexes. • If xymatrix V ar[d]_g ar[r]_h & U ar[d]^f Y ar[r]^e & X is 2-commutative diagram of categories fibred in groupoids over (Sch/S)_fppf, then there is a morphism e^-1K^bullet(f, F) → K^bullet(g, e^-1F). • if h and e are equivalences, then the map of (1) is an isomorphism, • if f, f' : U → X are 2-isomorphic, then the associated relative v Cech complexes are isomorphic,","statement_latex":"Generalities on relative {\\v C}ech complexes.\n\\begin{enumerate}\n\\item If\n$$\n\\xymatrix{\n\\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\\n\\mathcal{Y} \\ar[r]^e & \\mathcal{X}\n}\n$$\nis $2$-commutative diagram of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$, then there is a morphism\n$e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F}) \\to\n\\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})$.\n\\item if $h$ and $e$ are equivalences, then the map of (1) is an isomorphism,\n\\item if $f, f' : \\mathcal{U} \\to \\mathcal{X}$ are $2$-isomorphic, then\nthe associated relative {\\v C}ech complexes are isomorphic,\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06X9","source_file":"stacks-sheaves.tex","source_line":3171,"source_end_line":3190,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3171-L3190","statement_sha256":"7d98abdecddfdf05f3de8876d60768abc55bacf127f54e26fdfab232217856d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13729,"rank":13729,"depth":1,"x":2028.164,"y":1542.229,"cluster":"algebraic-stacks"},{"id":"stacks:06XA","tag":"06XA","title":"The relative v Cech complex · Lemma 06XA","summary":"If there exists a 1-morphism s : X → U such that f ∘ s is 2-isomorphic to id_X then the extended relative v Cech complex is homotopic to zero.","statement_latex":"If there exists a $1$-morphism $s : \\mathcal{X} \\to \\mathcal{U}$\nsuch that $f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{X}$\nthen the extended relative {\\v C}ech complex is homotopic to zero.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XA","source_file":"stacks-sheaves.tex","source_line":3199,"source_end_line":3204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3199-L3204","statement_sha256":"008c79ee59649a59d0fdec6419c53fad3ec83dad3874fc09123b13bc537dff9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13730,"rank":13730,"depth":3,"x":2078.001,"y":1641.721,"cluster":"algebraic-stacks"},{"id":"stacks:06XC","tag":"06XC","title":"The relative v Cech complex · Lemma 06XC","summary":"Let xymatrix V ar[d]_g ar[r]_h & U ar[d]^f Y ar[r]^e & X be a 2-fibre product of categories fibred in groupoids over (Sch/S)_fppf and let F be an abelian presheaf on X. Then the map e^-1K^bullet(f, F) → K^bullet(g, e^-1F) of Lemma [Tag 06X9] is an isomorphism of complexes of abelian presheaves.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{V} \\ar[d]_g \\ar[r]_h & \\mathcal{U} \\ar[d]^f \\\\\n\\mathcal{Y} \\ar[r]^e & \\mathcal{X}\n}\n$$\nbe a $2$-fibre product of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$ and let $\\mathcal{F}$ be an abelian presheaf\non $\\mathcal{X}$. Then the map\n$e^{-1}\\mathcal{K}^\\bullet(f, \\mathcal{F}) \\to\n\\mathcal{K}^\\bullet(g, e^{-1}\\mathcal{F})$\nof\nLemma \\ref{lemma-generalities-sheafified}\nis an isomorphism of complexes of abelian presheaves.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XC","source_file":"stacks-sheaves.tex","source_line":3248,"source_end_line":3265,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3248-L3265","statement_sha256":"c4fa3af78745747529bda49c7fe8ab83d81858ab8223e110182e8251a5f54b08","origin":"The Stacks Project","memory_eligible":false,"source_rank":13731,"rank":13731,"depth":2,"x":1960.783,"y":1596.458,"cluster":"algebraic-stacks"},{"id":"stacks:06XD","tag":"06XD","title":"The relative v Cech complex · Lemma 06XD","summary":"Let f : U → X be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). Let F → G → H be a complex in Ab(X_τ). Assume that • for every object x of X there exists a covering (x_i → x) in X_τ such that each x_i is isomorphic to f(u_i) for some object u_i of U, and • f^-1F → f^-1G → f^-1H is exact. Then the sequence F → G → H is exact.","statement_latex":"Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nLet\n$$\n\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}\n$$\nbe a complex in $\\textit{Ab}(\\mathcal{X}_\\tau)$. Assume that\n\\begin{enumerate}\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$, and\n\\item $f^{-1}\\mathcal{F} \\to f^{-1}\\mathcal{G} \\to f^{-1}\\mathcal{H}$ is exact.\n\\end{enumerate}\nThen the sequence $\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}$\nis exact.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XD","source_file":"stacks-sheaves.tex","source_line":3302,"source_end_line":3320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3302-L3320","statement_sha256":"cfedf37a7f198a656ac899cd104bf29292a928b617963cf8b2cda98ddc2b0eae","origin":"The Stacks Project","memory_eligible":false,"source_rank":13732,"rank":13732,"depth":0,"x":2084.097,"y":1563.194,"cluster":"algebraic-stacks"},{"id":"stacks:06XE","tag":"06XE","title":"The relative v Cech complex · Proposition 06XE","summary":"Let f : U → X be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). If • F is an abelian sheaf on X_τ, and • for every object x of X there exists a covering (x_i → x) in X_τ such that each x_i is isomorphic to f(u_i) for some object u_i of U, then the extended relative v Cech complex … → 0 → F → f_0, *f_0^-1F → f_1, *f_1^-1F → f_2, *f_2^-1F → … is exact in Ab(X_τ).","statement_latex":"Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nIf\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$, and\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\end{enumerate}\nthen the extended relative {\\v C}ech complex\n$$\n\\ldots \\to 0 \\to\n\\mathcal{F} \\to\nf_{0, *}f_0^{-1}\\mathcal{F} \\to\nf_{1, *}f_1^{-1}\\mathcal{F} \\to\nf_{2, *}f_2^{-1}\\mathcal{F} \\to \\ldots\n$$\nis exact in $\\textit{Ab}(\\mathcal{X}_\\tau)$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XE","source_file":"stacks-sheaves.tex","source_line":3340,"source_end_line":3361,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3340-L3361","statement_sha256":"43ad1cdc3c800ac9ce68cb77ea75d1e6d32b41c43d632ee9ae5596382b8f056e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13733,"rank":13733,"depth":4,"x":2019.668,"y":1658.06,"cluster":"algebraic-stacks"},{"id":"stacks:06XF","tag":"06XF","title":"The relative v Cech complex · Lemma 06XF","summary":"Let f : U → X be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). Assume • F is an abelian sheaf on X_τ, • for every object x of X there exists a covering (x_i → x) in X_τ such that each x_i is isomorphic to f(u_i) for some object u_i of U, • the category U has equalizers, and • the functor f is faithful. Then there is a first quadrant spectral sequence of abelian groups E_1^p, q = H^q((U_p)_τ, f_p^-1F) ⇒ H^p…","statement_latex":"Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$,\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\item the category $\\mathcal{U}$ has equalizers, and\n\\item the functor $f$ is faithful.\n\\end{enumerate}\nThen there is a first quadrant spectral sequence of abelian groups\n$$\nE_1^{p, q} = H^q((\\mathcal{U}_p)_\\tau, f_p^{-1}\\mathcal{F})\n\\Rightarrow\nH^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F})\n$$\nconverging to the cohomology of $\\mathcal{F}$ in the $\\tau$-topology.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XF","source_file":"stacks-sheaves.tex","source_line":3383,"source_end_line":3404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3383-L3404","statement_sha256":"8d711f5bac3b11ebaeaf2a1e40849853388b8b657e44b91933d80dcbc3250f1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13734,"rank":13734,"depth":71,"x":1990.779,"y":1551.139,"cluster":"algebraic-stacks"},{"id":"stacks:06XG","tag":"06XG","title":"The relative v Cech complex · Lemma 06XG","summary":"Let f : U → X be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, fppf). Assume • F is an object of Mod(X_τ, O_X), • for every object x of X there exists a covering (x_i → x) in X_τ such that each x_i is isomorphic to f(u_i) for some object u_i of U, • the category U has equalizers, and • the functor f is faithful. Then there is a first quadrant spectral sequence of Γ(O_X)-modules E_1^p, q = H^q((U_p)_τ, f_p^*F) ⇒…","statement_latex":"Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, fppf\\}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an object of\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$,\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\item the category $\\mathcal{U}$ has equalizers, and\n\\item the functor $f$ is faithful.\n\\end{enumerate}\nThen there is a first quadrant spectral sequence of\n$\\Gamma(\\mathcal{O}_\\mathcal{X})$-modules\n$$\nE_1^{p, q} = H^q((\\mathcal{U}_p)_\\tau, f_p^*\\mathcal{F})\n\\Rightarrow\nH^{p + q}(\\mathcal{X}_\\tau, \\mathcal{F})\n$$\nconverging to the cohomology of $\\mathcal{F}$ in the $\\tau$-topology.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XG","source_file":"stacks-sheaves.tex","source_line":3457,"source_end_line":3480,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3457-L3480","statement_sha256":"ca4d09c4adc1549cbeaec90b7d81f9ec5b26416b52ceed855b3cb934b68235b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13735,"rank":13735,"depth":72,"x":2098.474,"y":1613.827,"cluster":"algebraic-stacks"},{"id":"stacks:06XH","tag":"06XH","title":"The relative v Cech complex · Lemma 06XH","summary":"Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. • Assume that f is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then for any object y of Y there exists an fppf covering (y_i → y) and objects x_i of X such that f(x_i) ≅ y_i in Y. • Assume that f is representable by algebraic spaces, surjective, and smooth. Then for any object y of Y there exists an étale covering (y_i → y) and objects x_i of X…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item Assume that $f$ is representable by algebraic spaces, surjective,\nflat, and locally of finite presentation. Then for any object $y$ of\n$\\mathcal{Y}$ there exists an fppf covering $\\{y_i \\to y\\}$ and objects\n$x_i$ of $\\mathcal{X}$ such that $f(x_i) \\cong y_i$ in $\\mathcal{Y}$.\n\\item Assume that $f$ is representable by algebraic spaces, surjective,\nand smooth. Then for any object $y$ of\n$\\mathcal{Y}$ there exists an \\'etale covering $\\{y_i \\to y\\}$ and objects\n$x_i$ of $\\mathcal{X}$ such that $f(x_i) \\cong y_i$ in $\\mathcal{Y}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XH","source_file":"stacks-sheaves.tex","source_line":3497,"source_end_line":3511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3497-L3511","statement_sha256":"f0cd12f61f394146efdec8686c953e219d66332ea90cb18817bf7ca46c6df70a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13736,"rank":13736,"depth":56,"x":1968.157,"y":1628.766,"cluster":"algebraic-stacks"},{"id":"stacks:072D","tag":"072D","title":"The relative v Cech complex · Lemma 072D","summary":"Let f : U → X and g : X → Y be composable 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, linebreak[0] fppf). Assume • F is an abelian sheaf on X_τ, • for every object x of X there exists a covering (x_i → x) in X_τ such that each x_i is isomorphic to f(u_i) for some object u_i of U, • the category U has equalizers, and • the functor f is faithful. Then there is a first quadrant spectral sequence of abelian sheaves…","statement_latex":"Let $f : \\mathcal{U} \\to \\mathcal{X}$ and\n$g : \\mathcal{X} \\to \\mathcal{Y}$\nbe composable $1$-morphisms of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, \\linebreak[0] fppf\\}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an abelian sheaf on $\\mathcal{X}_\\tau$,\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\item the category $\\mathcal{U}$ has equalizers, and\n\\item the functor $f$ is faithful.\n\\end{enumerate}\nThen there is a first quadrant spectral sequence of abelian sheaves\non $\\mathcal{Y}_\\tau$\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nwhere all higher direct images are computed in the $\\tau$-topology.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072D","source_file":"stacks-sheaves.tex","source_line":3558,"source_end_line":3582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3558-L3582","statement_sha256":"d0b2f048c537fba4f55a0a27793e8905256a0c20e29fb6c1e4e834a65048e907","origin":"The Stacks Project","memory_eligible":false,"source_rank":13737,"rank":13737,"depth":72,"x":2052.554,"y":1543.486,"cluster":"algebraic-stacks"},{"id":"stacks:072E","tag":"072E","title":"The relative v Cech complex · Lemma 072E","summary":"Let f : U → X and g : X → Y be composable 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. Let τ ∈ (Zar, etale, smooth, syntomic, linebreak[0] fppf). Assume • F is an object of Mod(X_τ, O_X), • for every object x of X there exists a covering (x_i → x) in X_τ such that each x_i is isomorphic to f(u_i) for some object u_i of U, • the category U has equalizers, and • the functor f is faithful. Then there is a first quadrant spectral sequence in Mod(Y_τ, O_Y)…","statement_latex":"Let $f : \\mathcal{U} \\to \\mathcal{X}$ and\n$g : \\mathcal{X} \\to \\mathcal{Y}$\nbe composable $1$-morphisms of categories fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Let\n$\\tau \\in \\{Zar, \\etale, smooth, syntomic, \\linebreak[0] fppf\\}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is an object of\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$,\n\\item for every object $x$ of $\\mathcal{X}$ there exists a covering\n$\\{x_i \\to x\\}$ in $\\mathcal{X}_\\tau$ such that each $x_i$ is isomorphic\nto $f(u_i)$ for some object $u_i$ of $\\mathcal{U}$,\n\\item the category $\\mathcal{U}$ has equalizers, and\n\\item the functor $f$ is faithful.\n\\end{enumerate}\nThen there is a first quadrant spectral sequence in\n$\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nwhere all higher direct images are computed in the $\\tau$-topology.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"The relative v Cech complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072E","source_file":"stacks-sheaves.tex","source_line":3616,"source_end_line":3641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3616-L3641","statement_sha256":"7731629b33077e795272333946d77b2b8f6176d6ef497ddcca9d885f746316fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13738,"rank":13738,"depth":73,"x":2058.923,"y":1654.672,"cluster":"algebraic-stacks"},{"id":"stacks:06XJ","tag":"06XJ","title":"Cohomology on algebraic stacks · Proposition 06XJ","summary":"Cohomology of a stack can be computed on the Cech nerve of a presentation Let f : U → X be a 1-morphism of algebraic stacks. • Let F be an abelian étale sheaf on X. Assume that f is representable by algebraic spaces, surjective, and smooth. Then there is a spectral sequence E_1^p, q = H^q_etale(U_p, f_p^-1F) ⇒ H^p + q_etale(X, F) • Let F be an abelian sheaf on X. Assume that f is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then…","statement_latex":"\\begin{slogan}\nCohomology of a stack can be computed on the Cech nerve of a presentation\n\\end{slogan}\nLet $f : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of algebraic stacks.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be an abelian \\'etale sheaf on $\\mathcal{X}$.\nAssume that $f$ is representable by algebraic spaces, surjective, and smooth.\nThen there is a spectral sequence\n$$\nE_1^{p, q} = H^q_\\etale(\\mathcal{U}_p, f_p^{-1}\\mathcal{F})\n\\Rightarrow\nH^{p + q}_\\etale(\\mathcal{X}, \\mathcal{F})\n$$\n\\item Let $\\mathcal{F}$ be an abelian sheaf on $\\mathcal{X}$.\nAssume that $f$ is representable by algebraic spaces, surjective, flat,\nand locally of finite presentation. Then there is\na spectral sequence\n$$\nE_1^{p, q} = H^q_{fppf}(\\mathcal{U}_p, f_p^{-1}\\mathcal{F})\n\\Rightarrow\nH^{p + q}_{fppf}(\\mathcal{X}, \\mathcal{F})\n$$\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Cohomology on algebraic stacks","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06XJ","source_file":"stacks-sheaves.tex","source_line":3720,"source_end_line":3745,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3720-L3745","statement_sha256":"58a0d65174d81a9f094cad6fc7f6c00066b714c82c7fe60398d967b263f298f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13739,"rank":13739,"depth":72,"x":1964.466,"y":1576.009,"cluster":"algebraic-stacks"},{"id":"stacks:072G","tag":"072G","title":"Higher direct images and algebraic stacks · Proposition 072G","summary":"Let f : U → X and g : X → Y be composable 1-morphisms of algebraic stacks. • Assume that f is representable by algebraic spaces, surjective and smooth. • If F is in Ab(X_etale) then there is a spectral sequence E_1^p, q = R^q(g ∘ f_p)_*f_p^-1F ⇒ R^p + qg_*F in Ab(Y_etale) with higher direct images computed in the étale topology. • If F is in Mod(X_etale, O_X) then there is a spectral sequence E_1^p, q = R^q(g ∘ f_p)_*f_p^-1F ⇒ R^p + qg_*F in Mod(Y_etale, O_Y). • Assume…","statement_latex":"Let $f : \\mathcal{U} \\to \\mathcal{X}$ and $g : \\mathcal{X} \\to \\mathcal{Y}$\nbe composable $1$-morphisms of algebraic stacks.\n\\begin{enumerate}\n\\item Assume that $f$ is representable by algebraic spaces, surjective and\nsmooth.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is in $\\textit{Ab}(\\mathcal{X}_\\etale)$\nthen there is a spectral sequence\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nin $\\textit{Ab}(\\mathcal{Y}_\\etale)$ with higher direct images\ncomputed in the \\'etale topology.\n\\item If $\\mathcal{F}$ is in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ then\nthere is a spectral sequence\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nin $\\textit{Mod}(\\mathcal{Y}_\\etale, \\mathcal{O}_\\mathcal{Y})$.\n\\end{enumerate}\n\\item Assume that $f$ is representable by algebraic spaces, surjective,\nflat, and locally of finite presentation.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is in $\\textit{Ab}(\\mathcal{X})$ then there is\na spectral sequence\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nin $\\textit{Ab}(\\mathcal{Y})$ with higher direct images\ncomputed in the fppf topology.\n\\item If $\\mathcal{F}$ is in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ then\nthere is a spectral sequence\n$$\nE_1^{p, q} = R^q(g \\circ f_p)_*f_p^{-1}\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_*\\mathcal{F}\n$$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{Y})$.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Higher direct images and algebraic stacks","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072G","source_file":"stacks-sheaves.tex","source_line":3803,"source_end_line":3852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3803-L3852","statement_sha256":"c0ccfa1a22218659c249e37e4a8ac67eb190ffa3f89252e68551317a4511fbc7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13740,"rank":13740,"depth":74,"x":2097.864,"y":1580.433,"cluster":"algebraic-stacks"},{"id":"stacks:075G","tag":"075G","title":"Higher direct images and algebraic stacks · Lemma 075G","summary":"Let S be a scheme. Let f : X → Y be a 1-morphism of algebraic stacks. over S. Let τ ∈ (Zariski,linebreak[0] etale,linebreak[0] smooth,linebreak[0] syntomic,linebreak[0] fppf). Let F be an object of Ab(X_τ) or Mod(X_τ, O_X). Then the sheaf R^if_*F is the sheaf associated to the presheaf y ↦ H^i_τBig((Sch/V)_fppf ×_y, Y X, pr^-1FBig) Here y is an object of Y lying over the scheme V.","statement_latex":"Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a\n$1$-morphism of algebraic stacks\\footnote{This result should hold\nfor any $1$-morphism of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$.} over $S$.\nLet $\\tau \\in \\{Zariski,\\linebreak[0] \\etale,\\linebreak[0]\nsmooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$.\nLet $\\mathcal{F}$ be\nan object of $\\textit{Ab}(\\mathcal{X}_\\tau)$ or\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.\nThen the sheaf $R^if_*\\mathcal{F}$ is the sheaf associated to the\npresheaf\n$$\ny \\longmapsto\nH^i_\\tau\\Big((\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X},\n\\ \\text{pr}^{-1}\\mathcal{F}\\Big)\n$$\nHere $y$ is an object of $\\mathcal{Y}$ lying over the scheme $V$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Higher direct images and algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075G","source_file":"stacks-sheaves.tex","source_line":3874,"source_end_line":3893,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3874-L3893","statement_sha256":"d6fb98ada60f382925a41565f9f422bff181dcd6e8686fb1a7663c0dab12928a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13741,"rank":13741,"depth":72,"x":1995.568,"y":1653.128,"cluster":"algebraic-stacks"},{"id":"stacks:075H","tag":"075H","title":"Higher direct images and algebraic stacks · Lemma 075H","summary":"Let S be a scheme. Let τ ∈ (Zariski,linebreak[0] etale,linebreak[0] smooth,linebreak[0] syntomic,linebreak[0] fppf). Let xymatrix Y' ×_Y X ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a 2-cartesian diagram of algebraic stacks over S. Then the base change map is an isomorphism g^-1Rf_*F → Rf'_*(g')^-1F functorial for F in Ab(X_τ) or F in Mod(X_τ, O_X).","statement_latex":"Let $S$ be a scheme. Let\n$\\tau \\in \\{Zariski,\\linebreak[0] \\etale,\\linebreak[0]\nsmooth,\\linebreak[0] syntomic,\\linebreak[0] fppf\\}$. Let\n$$\n\\xymatrix{\n\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\ar[r]_{g'} \\ar[d]_{f'} &\n\\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}\n}\n$$\nbe a $2$-cartesian diagram of algebraic stacks over $S$. Then the base change\nmap is an isomorphism\n$$\ng^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F}\n$$\nfunctorial for $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{X}_\\tau)$\nor $\\mathcal{F}$ in $\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Higher direct images and algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075H","source_file":"stacks-sheaves.tex","source_line":3929,"source_end_line":3948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3929-L3948","statement_sha256":"9e293868e1470a85a2916ed3d2a3afdd249833bfd5aec6b1bd9a5cb27097a2f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13742,"rank":13742,"depth":73,"x":2012.603,"y":1541.075,"cluster":"algebraic-stacks"},{"id":"stacks:075L","tag":"075L","title":"Comparison · Lemma 075L","summary":"Let S be a scheme. Let X be an algebraic stack over S representable by the algebraic space F. • If I injective in Ab(X_etale), then I|_F_etale is injective in Ab(F_etale), • If I^bullet is a K-injective complex in Ab(X_etale), then I^bullet|_F_etale is a K-injective complex in Ab(F_etale). The same does not hold for modules.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$\nrepresentable by the algebraic space $F$.\n\\begin{enumerate}\n\\item If $\\mathcal{I}$ injective in $\\textit{Ab}(\\mathcal{X}_\\etale)$, then\n$\\mathcal{I}|_{F_\\etale}$ is injective in $\\textit{Ab}(F_\\etale)$,\n\\item If $\\mathcal{I}^\\bullet$ is a K-injective complex in\n$\\textit{Ab}(\\mathcal{X}_\\etale)$, then $\\mathcal{I}^\\bullet|_{F_\\etale}$\nis a K-injective complex in $\\textit{Ab}(F_\\etale)$.\n\\end{enumerate}\nThe same does not hold for modules.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Comparison","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075L","source_file":"stacks-sheaves.tex","source_line":3988,"source_end_line":4000,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L3988-L4000","statement_sha256":"69c67669b2e738531c46318b0f69aa605282536c76912f6f24521d470f37c602","origin":"The Stacks Project","memory_eligible":false,"source_rank":13743,"rank":13743,"depth":11,"x":2090.43,"y":1633.699,"cluster":"algebraic-stacks"},{"id":"stacks:075N","tag":"075N","title":"Comparison · Lemma 075N","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic stacks over S. Assume X, Y are representable by algebraic spaces F, G. Denote f : F → G the induced morphism of algebraic spaces. • For any F ∈ Ab(X_etale) we have (Rf_*F)|_G_etale = Rf_small, *(F|_F_etale) in D(G_etale). • For any object F of Mod(X_etale, O_X) we have (Rf_*F)|_G_etale = Rf_small, *(F|_F_etale) in D(O_G).","statement_latex":"Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism\nof algebraic stacks over $S$. Assume $\\mathcal{X}$, $\\mathcal{Y}$ are\nrepresentable by algebraic spaces $F$, $G$. Denote $f : F \\to G$ the\ninduced morphism of algebraic spaces.\n\\begin{enumerate}\n\\item For any $\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X}_\\etale)$\nwe have\n$$\n(Rf_*\\mathcal{F})|_{G_\\etale} =\nRf_{small, *}(\\mathcal{F}|_{F_\\etale})\n$$\nin $D(G_\\etale)$.\n\\item For any object $\\mathcal{F}$ of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nwe have\n$$\n(Rf_*\\mathcal{F})|_{G_\\etale} =\nRf_{small, *}(\\mathcal{F}|_{F_\\etale})\n$$\nin $D(\\mathcal{O}_G)$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Comparison","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075N","source_file":"stacks-sheaves.tex","source_line":4013,"source_end_line":4036,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4013-L4036","statement_sha256":"622409c6d0069faf834c59ae56e5f3c193f2e3ac0119c485e84c588e01eabb92","origin":"The Stacks Project","memory_eligible":false,"source_rank":13744,"rank":13744,"depth":48,"x":1958.085,"y":1609.474,"cluster":"algebraic-stacks"},{"id":"stacks:075P","tag":"075P","title":"Comparison · Lemma 075P","summary":"Let S be a scheme. Consider a 2-fibre product square xymatrix X' ar[r]_g' ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y of algebraic stacks over S. Assume that f is representable by algebraic spaces and that Y' is representable by an algebraic space G'. Then X' is representable by an algebraic space F' and denoting f' : F' → G' the induced morphism of algebraic spaces we have g^-1(Rf_*F)|_G'_etale = Rf'_small, *((g')^-1F|_F'_etale) for any F in Ab(X_etale) or in Mod(X_etale, O_X)","statement_latex":"Let $S$ be a scheme. Consider a $2$-fibre product square\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r]_{g'} \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}\n}\n$$\nof algebraic stacks over $S$. Assume that $f$ is representable by algebraic\nspaces and that $\\mathcal{Y}'$ is representable by an algebraic space $G'$.\nThen $\\mathcal{X}'$ is representable by an algebraic space $F'$ and\ndenoting $f' : F' \\to G'$ the induced morphism of algebraic spaces\nwe have\n$$\ng^{-1}(Rf_*\\mathcal{F})|_{G'_\\etale} =\nRf'_{small, *}((g')^{-1}\\mathcal{F}|_{F'_\\etale})\n$$\nfor any $\\mathcal{F}$ in $\\textit{Ab}(\\mathcal{X}_\\etale)$\nor in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Comparison","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075P","source_file":"stacks-sheaves.tex","source_line":4055,"source_end_line":4076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4055-L4076","statement_sha256":"65988506bd51fa71d6cfe0b38b53187d375ab08fb8af75773b5ae6a6b8307d05","origin":"The Stacks Project","memory_eligible":false,"source_rank":13745,"rank":13745,"depth":74,"x":2075.571,"y":1552.039,"cluster":"algebraic-stacks"},{"id":"stacks:076T","tag":"076T","title":"Change of topology · Lemma 076T","summary":"Let S be a scheme. Let X be an algebraic stack over S. Let F be a presheaf of O_X-modules. Assume • [(a)] F is locally quasi-coherent, and • [(b)] for any morphism φ : x → y of X which lies over a morphism of schemes f : U → V which is flat and locally of finite presentation the comparison map c_φ : f_small^*F|_V_etale → F|_U_etale of ([Tag 06WC]) is an isomorphism. Then F is a sheaf for the fppf topology.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.\nLet $\\mathcal{F}$ be a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules.\nAssume\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{F}$ is locally quasi-coherent, and\n\\item[(b)] for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ which lies\nover a morphism of schemes $f : U \\to V$ which is flat and\nlocally of finite presentation the comparison map\n$c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to\n\\mathcal{F}|_{U_\\etale}$ of\n(\\ref{equation-comparison-modules}) is an isomorphism.\n\\end{enumerate}\nThen $\\mathcal{F}$ is a sheaf for the fppf topology.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Change of topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/076T","source_file":"stacks-sheaves.tex","source_line":4103,"source_end_line":4118,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4103-L4118","statement_sha256":"51d14b58f5822a77871503c415df0b117dda39532ec88d0c1efb27b9ac1cdb10","origin":"The Stacks Project","memory_eligible":false,"source_rank":13746,"rank":13746,"depth":40,"x":2034.983,"y":1661.439,"cluster":"algebraic-stacks"},{"id":"stacks:075R","tag":"075R","title":"Change of topology · Lemma 075R","summary":"Let S be a scheme. Let X be an algebraic stack over S. Let F be a presheaf O_X-module such that • [(a)] F is locally quasi-coherent, and • [(b)] for any morphism φ : x → y of X which lies over a morphism of schemes f : U → V which is flat and locally of finite presentation, the comparison map c_φ : f_small^*F|_V_etale → F|_U_etale of ([Tag 06WC]) is an isomorphism. Then F is an O_X-module and we have the following • If ε : X_fppf → X_etale is the comparison morphism, then…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.\nLet $\\mathcal{F}$ be a presheaf $\\mathcal{O}_\\mathcal{X}$-module such that\n\\begin{enumerate}\n\\item[(a)] $\\mathcal{F}$ is locally quasi-coherent, and\n\\item[(b)] for any morphism $\\varphi : x \\to y$ of $\\mathcal{X}$ which lies\nover a morphism of schemes $f : U \\to V$ which is flat and\nlocally of finite presentation, the comparison map\n$c_\\varphi : f_{small}^*\\mathcal{F}|_{V_\\etale} \\to\n\\mathcal{F}|_{U_\\etale}$ of\n(\\ref{equation-comparison-modules}) is an isomorphism.\n\\end{enumerate}\nThen $\\mathcal{F}$ is an $\\mathcal{O}_\\mathcal{X}$-module and\nwe have the following\n\\begin{enumerate}\n\\item If $\\epsilon : \\mathcal{X}_{fppf} \\to \\mathcal{X}_\\etale$\nis the comparison morphism, then\n$R\\epsilon_*\\mathcal{F} = \\epsilon_*\\mathcal{F}$.\n\\item The cohomology groups $H^p_{fppf}(\\mathcal{X}, \\mathcal{F})$ are equal\nto the cohomology groups computed in the \\'etale topology on $\\mathcal{X}$.\nSimilarly for the cohomology groups $H^p_{fppf}(x, \\mathcal{F})$ and the\nderived versions $R\\Gamma(\\mathcal{X}, \\mathcal{F})$ and\n$R\\Gamma(x, \\mathcal{F})$.\n\\item If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a $1$-morphism of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$ then\n$R^if_*\\mathcal{F}$ is equal to the fppf-sheafification of the\nhigher direct image computed in the \\'etale cohomology.\nSimilarly for derived pullback.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Change of topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075R","source_file":"stacks-sheaves.tex","source_line":4133,"source_end_line":4163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4133-L4163","statement_sha256":"a1ae72656fe5d117306fc52b1358ef3e3b5ff1af7e800d3fbb594053755a59c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13747,"rank":13747,"depth":73,"x":1976.734,"y":1557.376,"cluster":"algebraic-stacks"},{"id":"stacks:07AK","tag":"07AK","title":"Change of topology · Lemma 07AK","summary":"Let S be a scheme. Let X be an algebraic stack over S. Let τ = etale (resp. τ = fppf). Let X' ⊂ X be a full subcategory with the following properties • if x → x' is a morphism of X which lies over a smooth (resp. flat and locally finitely presented) morphism of schemes and x' ∈ Ob(X'), then x ∈ Ob(X'), and • there exists an object x ∈ Ob(X') lying over a scheme U such that the associated 1-morphism x : (Sch/U)_fppf → X is smooth and surjective. We get a site X'_τ by…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.\nLet $\\tau = \\etale$ (resp.\\ $\\tau = fppf$). Let\n$\\mathcal{X}' \\subset \\mathcal{X}$ be a full subcategory with the\nfollowing properties\n\\begin{enumerate}\n\\item if $x \\to x'$ is a morphism of $\\mathcal{X}$ which lies over a\nsmooth (resp.\\ flat and locally finitely presented) morphism of\nschemes and $x' \\in \\Ob(\\mathcal{X}')$, then $x \\in \\Ob(\\mathcal{X}')$, and\n\\item there exists an object $x \\in \\Ob(\\mathcal{X}')$ lying over\na scheme $U$ such that the associated $1$-morphism\n$x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$ is smooth and surjective.\n\\end{enumerate}\nWe get a site $\\mathcal{X}'_\\tau$ by declaring a covering of $\\mathcal{X}'$\nto be any family of morphisms $\\{x_i \\to x\\}$ in $\\mathcal{X}'$ which is a\ncovering in $\\mathcal{X}_\\tau$. Then the inclusion functor\n$\\mathcal{X}' \\to \\mathcal{X}_\\tau$ is fully faithful, cocontinuous, and\ncontinuous, whence defines a morphism of topoi\n$$\ng : \\Sh(\\mathcal{X}'_\\tau) \\longrightarrow \\Sh(\\mathcal{X}_\\tau)\n$$\nand $H^p(\\mathcal{X}'_\\tau, g^{-1}\\mathcal{F}) =\nH^p(\\mathcal{X}_\\tau, \\mathcal{F})$ for all $p \\geq 0$ and all\n$\\mathcal{F} \\in \\textit{Ab}(\\mathcal{X}_\\tau)$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Change of topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AK","source_file":"stacks-sheaves.tex","source_line":4218,"source_end_line":4243,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4218-L4243","statement_sha256":"a775c995e2af6d6a3655eb5e04ab4f8da46c14faa3dd14a06bfbedee2fa1b1e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13748,"rank":13748,"depth":57,"x":2103.809,"y":1601.21,"cluster":"algebraic-stacks"},{"id":"stacks:0H09","tag":"0H09","title":"Restricting to affines · Definition 0H09","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. The associated affine site is the full subcategory X_affine of X whose objects are those x ∈ Ob(X) lying over a scheme U such that U is affine. The topology on X_affine will be the chaotic one, i.e., such that sheaves on X_affine are the same as presheaves.","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nThe {\\it associated affine site} is the full subcategory\n$\\mathcal{X}_{affine}$ of $\\mathcal{X}$ whose objects are those\n$x \\in \\Ob(\\mathcal{X})$ lying over a scheme $U$ such that $U$\nis affine. The topology on $\\mathcal{X}_{affine}$ will be the chaotic one,\ni.e., such that sheaves on $\\mathcal{X}_{affine}$ are the same as presheaves.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restricting to affines","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H09","source_file":"stacks-sheaves.tex","source_line":4421,"source_end_line":4429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4421-L4429","statement_sha256":"70acbe0adaacffabf3174a38862534d8b92dce2557dd64f7c0608e6cf8eb130a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13749,"rank":13749,"depth":0,"x":1974.411,"y":1641.13,"cluster":"algebraic-stacks"},{"id":"stacks:0H0A","tag":"0H0A","title":"Restricting to affines · Definition 0H0A","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. • The associated affine Zariski site X_affine, Zar is the structure of site on X_affine inherited from (Aff/S)_Zar. • The associated affine étale site X_affine, etale is the structure of site on X_affine inherited from (Aff/S)_etale. • The associated affine smooth site X_affine, smooth is the structure of site on X_affine inherited from (Aff/S)_smooth. • The associated affine syntomic site X_affine, syntomic is…","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\n\\begin{enumerate}\n\\item The {\\it associated affine Zariski site} $\\mathcal{X}_{affine, Zar}$\nis the structure of site on $\\mathcal{X}_{affine}$ inherited from\n$(\\textit{Aff}/S)_{Zar}$.\n\\item The {\\it associated affine \\'etale site} $\\mathcal{X}_{affine, \\etale}$\nis the structure of site on $\\mathcal{X}_{affine}$ inherited from\n$(\\textit{Aff}/S)_\\etale$.\n\\item The {\\it associated affine smooth site}\n$\\mathcal{X}_{affine, smooth}$\nis the structure of site on $\\mathcal{X}_{affine}$ inherited from\n$(\\textit{Aff}/S)_{smooth}$.\n\\item The {\\it associated affine syntomic site} $\\mathcal{X}_{affine, syntomic}$\nis the structure of site on $\\mathcal{X}_{affine}$ inherited from\n$(\\textit{Aff}/S)_{syntomic}$.\n\\item The {\\it associated affine fppf site} $\\mathcal{X}_{affine, fppf}$\nis the structure of site on $\\mathcal{X}_{affine}$ inherited from\n$(\\textit{Aff}/S)_{fppf}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restricting to affines","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0A","source_file":"stacks-sheaves.tex","source_line":4446,"source_end_line":4467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4446-L4467","statement_sha256":"e98bfadf201a91ed43541a6f8cefe327b10e8144d002dea4240be642fe1ffefa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13750,"rank":13750,"depth":0,"x":2037.942,"y":1537.918,"cluster":"algebraic-stacks"},{"id":"stacks:0H0B","tag":"0H0B","title":"Restricting to affines · Lemma 0H0B","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). The functor X_affine, τ → X_τ is a special cocontinuous functor. Hence it induces an equivalence of topoi from Sh(X_affine, τ) to Sh(X_τ).","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nLet $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$. The functor\n$\\mathcal{X}_{affine, \\tau} \\to \\mathcal{X}_\\tau$ is a special\ncocontinuous functor. Hence it induces an equivalence of topoi\nfrom $\\Sh(\\mathcal{X}_{affine, \\tau})$ to $\\Sh(\\mathcal{X}_\\tau)$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Restricting to affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0B","source_file":"stacks-sheaves.tex","source_line":4483,"source_end_line":4490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4483-L4490","statement_sha256":"d62ac7bee29dfc0ec39d488bd092a3b9bc94303cec97453bca6b7f2d7ff8eadc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13751,"rank":13751,"depth":37,"x":2074.217,"y":1650.455,"cluster":"algebraic-stacks"},{"id":"stacks:0H0E","tag":"0H0E","title":"Quasi-coherent modules and affines · Lemma 0H0E","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let F be an O-module on X_affine. The following are equivalent • for every morphism x → x' of X_affine the map F(x') ⊗_O(x') O(x) → F(x) is an isomorphism, • F is a quasi-coherent module on (X_affine, O) in the sense of Modules on Sites, Definition [Tag 03DL], • F is a sheaf for the Zariski topology on X_affine and a quasi-coherent module on (X_affine, Zar, O) in the sense of Modules on Sites, Definition [Tag…","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nLet $\\mathcal{F}$ be an $\\mathcal{O}$-module on $\\mathcal{X}_{affine}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every morphism $x \\to x'$ of $\\mathcal{X}_{affine}$ the map\n$\\mathcal{F}(x') \\otimes_{\\mathcal{O}(x')} \\mathcal{O}(x) \\to \\mathcal{F}(x)$\nis an isomorphism,\n\\item $\\mathcal{F}$ is a quasi-coherent module on\n$(\\mathcal{X}_{affine}, \\mathcal{O})$ in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local},\n\\item $\\mathcal{F}$ is a sheaf for the Zariski topology on\n$\\mathcal{X}_{affine}$ and a quasi-coherent module on\n$(\\mathcal{X}_{affine, Zar}, \\mathcal{O})$ in the sense of\nModules on Sites, Definition \\ref{sites-modules-definition-site-local},\n\\item same as in (3) for the \\'etale topology,\n\\item same as in (3) for the smooth topology,\n\\item same as in (3) for the syntomic topology,\n\\item same as in (3) for the fppf topology, and\n\\item $\\mathcal{F}$ corresponds to a quasi-coherent module\non $\\mathcal{X}$ via the equivalence (\\ref{equation-alternative-ringed}).\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent modules and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0E","source_file":"stacks-sheaves.tex","source_line":4533,"source_end_line":4556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4533-L4556","statement_sha256":"662df60f0012d4641aa82c811dd2f0dbc7e8aa215d6170df254b0e321a0b154f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13752,"rank":13752,"depth":23,"x":1956.574,"y":1587.844,"cluster":"algebraic-stacks"},{"id":"stacks:0H0F","tag":"0H0F","title":"Quasi-coherent modules and affines · Lemma 0H0F","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let F be an O-module on X_affine. The following are equivalent • for every morphism x → x' of X_affine such that p(x) → p(x') is an étale morphism (of affine schemes), the map F(x') ⊗_O(x') O(x) → F(x) is an isomorphism, • F is a sheaf for the étale topology on X_affine and for every object x of X_affine the restriction x^*F|_U_affine, etale is quasi-coherent where U = p(x), • F corresponds to a locally…","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nLet $\\mathcal{F}$ be an $\\mathcal{O}$-module on $\\mathcal{X}_{affine}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for every morphism $x \\to x'$ of $\\mathcal{X}_{affine}$ such that\n$p(x) \\to p(x')$ is an \\'etale morphism (of affine schemes), the map\n$\\mathcal{F}(x') \\otimes_{\\mathcal{O}(x')} \\mathcal{O}(x) \\to \\mathcal{F}(x)$\nis an isomorphism,\n\\item $\\mathcal{F}$ is a sheaf for the \\'etale topology on\n$\\mathcal{X}_{affine}$ and for every object $x$ of $\\mathcal{X}_{affine}$\nthe restriction $x^*\\mathcal{F}|_{U_{affine, \\etale}}$ is quasi-coherent\nwhere $U = p(x)$,\n\\item $\\mathcal{F}$ corresponds to a locally quasi-coherent module\non $\\mathcal{X}$ via the equivalence (\\ref{equation-alternative-ringed})\nfor the \\'etale topology.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent modules and affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0F","source_file":"stacks-sheaves.tex","source_line":4581,"source_end_line":4599,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4581-L4599","statement_sha256":"c3f3eb5e7703573861426cf86c03bc21c04a7b70072d54fadccbb0315907ce9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13753,"rank":13753,"depth":23,"x":2094.132,"y":1567.191,"cluster":"algebraic-stacks"},{"id":"stacks:0H0H","tag":"0H0H","title":"Quasi-coherent objects in the derived category · Definition 0H0H","summary":"Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let O be the sheaf of rings on X_affine introduced in Section [Tag 0H08]. We define the triangulated category of quasi-coherent objects in the derived category by the formula mathitQC(X) = mathitQC(X_affine, O) where the right hand side is as defined in Cohomology on Sites, Definition [Tag 0GYV].","statement_latex":"Let $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nLet $\\mathcal{O}$ be the sheaf of rings on $\\mathcal{X}_{affine}$ introduced\nin Section \\ref{section-alternative}. We define the\n{\\it triangulated category of quasi-coherent objects in the derived category}\nby the formula\n$$\n\\mathit{QC}(\\mathcal{X}) = \\mathit{QC}(\\mathcal{X}_{affine}, \\mathcal{O})\n$$\nwhere the right hand side is as defined in\nCohomology on Sites, Definition \\ref{sites-cohomology-definition-cartesian}.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent objects in the derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0H","source_file":"stacks-sheaves.tex","source_line":4701,"source_end_line":4713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4701-L4713","statement_sha256":"13534c2ead98a7c540300d5ca28b6162ed8c68c4e956191651b6e668de4450ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":13754,"rank":13754,"depth":1,"x":2009.025,"y":1660.784,"cluster":"algebraic-stacks"},{"id":"stacks:0H0I","tag":"0H0I","title":"Quasi-coherent objects in the derived category · Lemma 0H0I","summary":"In the situation of Definition [Tag 0H0H] suppose that M is an object of mathitQC(X) and b ∈ Z such that H^i(M) = 0 for all i > b. Then H^b(M) is a quasi-coherent module on (X_affine, O), see Lemma [Tag 0H0E].","statement_latex":"In the situation of Definition \\ref{definition-QC} suppose that\n$M$ is an object of $\\mathit{QC}(\\mathcal{X})$ and $b \\in \\mathbf{Z}$\nsuch that $H^i(M) = 0$ for all $i > b$.\nThen $H^b(M)$ is a quasi-coherent module on\n$(\\mathcal{X}_{affine}, \\mathcal{O})$, see\nLemma \\ref{lemma-quasi-coherent-alternative}.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0I","source_file":"stacks-sheaves.tex","source_line":4729,"source_end_line":4737,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4729-L4737","statement_sha256":"08d55dea88a9cd68ca81c3173d56707447b5e29f9e742081cc0ef1087c5e048a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13755,"rank":13755,"depth":24,"x":1996.475,"y":1543.09,"cluster":"algebraic-stacks"},{"id":"stacks:0H0J","tag":"0H0J","title":"Quasi-coherent objects in the derived category · Lemma 0H0J","summary":"Let S be a scheme. Let X → (Sch/S)_fppf be a category fibred in groupoids. The comparison morphism ε : X_affine, etale → X_affine satisfies the assumptions and conclusions of Cohomology on Sites, Lemma [Tag 0GZS].","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. The comparison morphism\n$\\epsilon : \\mathcal{X}_{affine, \\etale} \\to \\mathcal{X}_{affine}$\nsatisfies the assumptions and conclusions of Cohomology on Sites, Lemma\n\\ref{sites-cohomology-lemma-cartesion-plus-topology}.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0J","source_file":"stacks-sheaves.tex","source_line":4744,"source_end_line":4751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4744-L4751","statement_sha256":"28ce689ea826e1ebd2528c361bf4f88cb1d034d1d71ff9b15b2bcfe8fe73e0db","origin":"The Stacks Project","memory_eligible":false,"source_rank":13756,"rank":13756,"depth":27,"x":2100.716,"y":1623.017,"cluster":"algebraic-stacks"},{"id":"stacks:0H0K","tag":"0H0K","title":"Quasi-coherent objects in the derived category · Proposition 0H0K","summary":"Let S be a scheme. Let X → (Sch/S)_fppf be a category fibred in groupoids. Assume X is representable by an algebraic space X. Then mathitQC(X) is canonically equivalent to D_QCoh(O_X).","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Assume $\\mathcal{X}$ is representable by an algebraic\nspace $X$. Then $\\mathit{QC}(\\mathcal{X})$ is canonically equivalent to\n$D_\\QCoh(\\mathcal{O}_X)$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent objects in the derived category","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0K","source_file":"stacks-sheaves.tex","source_line":4785,"source_end_line":4791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4785-L4791","statement_sha256":"09047fa9d7db889f4420d8391c27fac470a0aee2aa6a6f858731904df68dbe97","origin":"The Stacks Project","memory_eligible":false,"source_rank":13757,"rank":13757,"depth":43,"x":1959.116,"y":1623.229,"cluster":"algebraic-stacks"},{"id":"stacks:0H0L","tag":"0H0L","title":"Quasi-coherent objects in the derived category · Proposition 0H0L","summary":"Let S be a scheme. Let X = Spf(A) where A is an an adic Noetherian topological S-algebra with ideal of definition I, see More on Algebra, Definition [Tag 07E8] and Formal Spaces, Definition [Tag 0AIF]. Let p : X → (Sch/S)_fppf the be category fibred in sets associated to the functor X, see Categories, Example [Tag 04TM]. Then mathitQC(X) is canonically equivalent to the category D_comp(A, I) of objects of D(A) which are derived complete with respect to I.","statement_latex":"Let $S$ be a scheme. Let $X = \\text{Spf}(A)$ where $A$ is an\nan adic Noetherian topological $S$-algebra with ideal of definition $I$, see\nMore on Algebra, Definition \\ref{more-algebra-definition-topological-ring}\nand Formal Spaces, Definition\n\\ref{formal-spaces-definition-affine-formal-spectrum}.\nLet $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$\nthe be category fibred in sets associated to the functor $X$, see\nCategories, Example \\ref{categories-example-presheaf}.\nThen $\\mathit{QC}(\\mathcal{X})$ is canonically equivalent to the\ncategory $D_{comp}(A, I)$ of objects of $D(A)$ which are\nderived complete with respect to $I$.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent objects in the derived category","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0L","source_file":"stacks-sheaves.tex","source_line":4919,"source_end_line":4932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4919-L4932","statement_sha256":"63b9428d5dce1003f2cd67ea20a69f83b376500c7566d689d83554021f7e1bfa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13758,"rank":13758,"depth":23,"x":2063.697,"y":1542.464,"cluster":"algebraic-stacks"},{"id":"stacks:0H0X","tag":"0H0X","title":"Quasi-coherent objects in the derived category · Lemma 0H0X","summary":"Let S be a scheme. Let X → (Sch/S)_fppf be a category fibred in groupoids. The comparison morphism ε : X_affine, fppf → X_affine satisfies the assumptions and conclusions of Cohomology on Sites, Lemma [Tag 0GZS].","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. The comparison morphism\n$\\epsilon : \\mathcal{X}_{affine, fppf} \\to \\mathcal{X}_{affine}$\nsatisfies the assumptions and conclusions of Cohomology on Sites, Lemma\n\\ref{sites-cohomology-lemma-cartesion-plus-topology}.","area":"Algebraic Stacks","chapter":"Sheaves on Algebraic Stacks","chapter_id":"stacks-sheaves","section":"Quasi-coherent objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0X","source_file":"stacks-sheaves.tex","source_line":4963,"source_end_line":4970,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-sheaves.tex#L4963-L4970","statement_sha256":"bd2e3eae15667a18b89f79c96f5451a5abd5f4033d6de75b665368b5041cc1b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13759,"rank":13759,"depth":44,"x":2051.49,"y":1661.744,"cluster":"algebraic-stacks"},{"id":"stacks:05XK","tag":"05XK","title":"Morphisms of stacks in groupoids · Lemma 05XK","summary":"Let X → Y → Z be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If X → Z and Y → Z are representable by algebraic spaces and étale so is X → Y.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$.\nIf $\\mathcal{X} \\to \\mathcal{Z}$ and $\\mathcal{Y} \\to \\mathcal{Z}$ are\nrepresentable by algebraic spaces and \\'etale so is\n$\\mathcal{X} \\to \\mathcal{Y}$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Morphisms of stacks in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XK","source_file":"criteria.tex","source_line":128,"source_end_line":136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L128-L136","statement_sha256":"8fc6150c95215c3101947c4a4b4d343a3028d38fd4212eec46bea21675abb7ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":13760,"rank":13760,"depth":45,"x":1964.292,"y":1566.558,"cluster":"algebraic-stacks"},{"id":"stacks:05XL","tag":"05XL","title":"Morphisms of stacks in groupoids · Lemma 05XL","summary":"Let X, Y, Z be stacks in groupoids over (Sch/S)_fppf. Suppose that X → Y and Z → Y are 1-morphisms. If • Y, Z are representable by algebraic spaces Y, Z over S, • the associated morphism of algebraic spaces Y → Z is surjective, flat and locally of finite presentation, and • Y ×_Z X is a stack in setoids, then X is a stack in setoids.","statement_latex":"Let $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ be stacks in groupoids\nover $(\\Sch/S)_{fppf}$. Suppose that $\\mathcal{X} \\to \\mathcal{Y}$\nand $\\mathcal{Z} \\to \\mathcal{Y}$ are $1$-morphisms.\nIf\n\\begin{enumerate}\n\\item $\\mathcal{Y}$, $\\mathcal{Z}$ are representable by algebraic spaces\n$Y$, $Z$ over $S$,\n\\item the associated morphism of algebraic spaces $Y \\to Z$ is surjective,\nflat and locally of finite presentation, and\n\\item $\\mathcal{Y} \\times_\\mathcal{Z} \\mathcal{X}$ is a stack in\nsetoids,\n\\end{enumerate}\nthen $\\mathcal{X}$ is a stack in setoids.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Morphisms of stacks in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XL","source_file":"criteria.tex","source_line":165,"source_end_line":180,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L165-L180","statement_sha256":"595bcaa16d8034e97fac92985a95a4a56aa02b7c3567931b029f54dda1eb1e7a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13761,"rank":13761,"depth":3,"x":2105.584,"y":1587.335,"cluster":"algebraic-stacks"},{"id":"stacks:05XW","tag":"05XW","title":"Morphisms of stacks in groupoids · Lemma 05XW","summary":"Let S be a scheme. Let u : U → X be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. If • U is representable by an algebraic space, and • u is representable by algebraic spaces, surjective, flat and locally of finite presentation, then Δ : X → X × X representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme.\nLet $u : \\mathcal{U} \\to \\mathcal{X}$ be a $1$-morphism of\nstacks in groupoids over $(\\Sch/S)_{fppf}$. If\n\\begin{enumerate}\n\\item $\\mathcal{U}$ is representable by an algebraic space, and\n\\item $u$ is representable by algebraic spaces, surjective, flat and\nlocally of finite presentation,\n\\end{enumerate}\nthen\n$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nrepresentable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Morphisms of stacks in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XW","source_file":"criteria.tex","source_line":194,"source_end_line":207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L194-L207","statement_sha256":"540748911dd4d337ffdea148fe4bc87c98c0c9ef23edd1ea8b1c0dcc606aaed4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13762,"rank":13762,"depth":67,"x":1984.307,"y":1652.391,"cluster":"algebraic-stacks"},{"id":"stacks:07WG","tag":"07WG","title":"Morphisms of stacks in groupoids · Lemma 07WG","summary":"Let X be a category fibred in groupoids over (Sch/S)_fppf. The following are equivalent • Δ_Δ : X → X ×_X × X X is representable by algebraic spaces, • for every 1-morphism V → X × X with V representable (by a scheme) the fibre product Y = X ×_Δ, X × X V has diagonal representable by algebraic spaces.","statement_latex":"Let $\\mathcal{X}$ be a category fibred in groupoids over $(\\Sch/S)_{fppf}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\Delta_\\Delta : \\mathcal{X} \\to\n\\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} \\mathcal{X}$\nis representable by algebraic spaces,\n\\item for every $1$-morphism $\\mathcal{V} \\to \\mathcal{X} \\times \\mathcal{X}$\nwith $\\mathcal{V}$ representable (by a scheme) the fibre product\n$\\mathcal{Y} =\n\\mathcal{X} \\times_{\\Delta, \\mathcal{X} \\times \\mathcal{X}} \\mathcal{V}$\nhas diagonal representable by algebraic spaces.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Morphisms of stacks in groupoids","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WG","source_file":"criteria.tex","source_line":274,"source_end_line":288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L274-L288","statement_sha256":"f9126023d70f64710ad73ac11c31286a541892ea60360faaea085df2e9884855","origin":"The Stacks Project","memory_eligible":false,"source_rank":13763,"rank":13763,"depth":0,"x":2021.535,"y":1535.238,"cluster":"algebraic-stacks"},{"id":"stacks:06CV","tag":"06CV","title":"Limit preserving on objects · Lemma 06CV","summary":"Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If p : X → Y is limit preserving on objects, then so is the base change p' : X ×_Y Z → Z of p by q.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$\nbe $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $p : \\mathcal{X} \\to \\mathcal{Y}$ is limit preserving on objects, then so\nis the base change\n$p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$\nof $p$ by $q$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Limit preserving on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CV","source_file":"criteria.tex","source_line":368,"source_end_line":376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L368-L376","statement_sha256":"0efb0b4c336ff861a87bacc7ca908183783c1478963cb7aaa281738d474d22a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13764,"rank":13764,"depth":0,"x":2088.502,"y":1643.085,"cluster":"algebraic-stacks"},{"id":"stacks:06CW","tag":"06CW","title":"Limit preserving on objects · Lemma 06CW","summary":"Let p : X → Y and q : Y → Z be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If p and q are limit preserving on objects, then so is the composition q ∘ p.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $p$ and $q$ are limit preserving on objects, then so is the composition\n$q \\circ p$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Limit preserving on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CW","source_file":"criteria.tex","source_line":410,"source_end_line":416,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L410-L416","statement_sha256":"76b3f8ae6046757a72956870766be7165145c5bf98152f9eba0e631bf991be2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13765,"rank":13765,"depth":0,"x":1951.973,"y":1601.43,"cluster":"algebraic-stacks"},{"id":"stacks:06CX","tag":"06CX","title":"Limit preserving on objects · Lemma 06CX","summary":"Let p : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. If p is representable by algebraic spaces, then the following are equivalent: • p is limit preserving on objects, and • p is locally of finite presentation (see Algebraic Stacks, Definition [Tag 03YK]).","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is\nrepresentable by algebraic spaces, then the following are equivalent:\n\\begin{enumerate}\n\\item $p$ is limit preserving on objects, and\n\\item $p$ is locally of finite presentation (see\nAlgebraic Stacks,\nDefinition \\ref{algebraic-definition-relative-representable-property}).\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Limit preserving on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CX","source_file":"criteria.tex","source_line":443,"source_end_line":454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L443-L454","statement_sha256":"35466a48bcac3ed87910c8a8ac14d3c15c8d09caed3feb6b66e56ee88f5b5a4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13766,"rank":13766,"depth":50,"x":2086.559,"y":1554.532,"cluster":"algebraic-stacks"},{"id":"stacks:06CY","tag":"06CY","title":"Limit preserving on objects · Lemma 06CY","summary":"Let p : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Assume p is representable by algebraic spaces and an open immersion. Then p is limit preserving on objects.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $p$ is representable\nby algebraic spaces and an open immersion. Then $p$ is limit preserving\non objects.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Limit preserving on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CY","source_file":"criteria.tex","source_line":511,"source_end_line":517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L511-L517","statement_sha256":"9257f70448a49a272cf6c9244b82ae68931e502eeb504c8206e548c331ba236d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13767,"rank":13767,"depth":51,"x":2024.843,"y":1665.82,"cluster":"algebraic-stacks"},{"id":"stacks:07WH","tag":"07WH","title":"Limit preserving on objects · Lemma 07WH","summary":"Let S be a scheme. Let kappa = size(T) for some T ∈ Ob((Sch/S)_fppf). Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf such that • Y → (Sch/S)_fppf is limit preserving on objects, • for an affine scheme V locally of finite presentation over S and y ∈ Ob(Y_V) the fibre product (Sch/V)_fppf ×_y, Y X is representable by an algebraic space of size ≤ kappa, • X and Y are stacks for the Zariski topology. Then f is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme.\nLet $\\kappa = \\text{size}(T)$ for some $T \\in \\Ob((\\Sch/S)_{fppf})$.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nof categories fibred in groupoids over $(\\Sch/S)_{fppf}$\nsuch that\n\\begin{enumerate}\n\\item $\\mathcal{Y} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects,\n\\item for an affine scheme $V$ locally of finite presentation over $S$ and\n$y \\in \\Ob(\\mathcal{Y}_V)$ the fibre product\n$(\\Sch/V)_{fppf} \\times_{y, \\mathcal{Y}} \\mathcal{X}$ is representable\nby an algebraic space of size $\\leq \\kappa$\\footnote{The condition on\nsize can be dropped by those ignoring set theoretic issues.},\n\\item $\\mathcal{X}$ and $\\mathcal{Y}$ are stacks for the Zariski topology.\n\\end{enumerate}\nThen $f$ is representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Limit preserving on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WH","source_file":"criteria.tex","source_line":539,"source_end_line":556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L539-L556","statement_sha256":"9977beec2ed10947d725e789bd7e483218323d13dd48256a6cab59ace57de23c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13768,"rank":13768,"depth":19,"x":1980.723,"y":1548.383,"cluster":"algebraic-stacks"},{"id":"stacks:07WI","tag":"07WI","title":"Limit preserving on objects · Lemma 07WI","summary":"Let S be a scheme. Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let P be a property of morphisms of algebraic spaces as in Algebraic Stacks, Definition [Tag 03YK]. If • f is representable by algebraic spaces, • Y → (Sch/S)_fppf is limit preserving on objects, • for an affine scheme V locally of finite presentation over S and y ∈ Y_V the resulting morphism of algebraic spaces f_y : F_y → V, see Algebraic Stacks, Equation ([Tag 0402]),…","statement_latex":"Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nof categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let $\\mathcal{P}$\nbe a property of morphisms of algebraic spaces as in\nAlgebraic Stacks, Definition\n\\ref{algebraic-definition-relative-representable-property}. If\n\\begin{enumerate}\n\\item $f$ is representable by algebraic spaces,\n\\item $\\mathcal{Y} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects,\n\\item for an affine scheme $V$ locally of finite presentation over $S$ and\n$y \\in \\mathcal{Y}_V$ the resulting morphism of algebraic spaces\n$f_y : F_y \\to V$, see Algebraic Stacks, Equation\n(\\ref{algebraic-equation-representable-by-algebraic-spaces}),\nhas property $\\mathcal{P}$.\n\\end{enumerate}\nThen $f$ has property $\\mathcal{P}$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Limit preserving on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WI","source_file":"criteria.tex","source_line":615,"source_end_line":632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L615-L632","statement_sha256":"85caccf2858d4a6077e52e6915cbaee363aa7dec2d4eacbe29ba31bf51a8701b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13769,"rank":13769,"depth":5,"x":2108.08,"y":1610.131,"cluster":"algebraic-stacks"},{"id":"stacks:06D1","tag":"06D1","title":"Formally smooth on objects · Lemma 06D1","summary":"Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If p : X → Y is formally smooth on objects, then so is the base change p' : X ×_Y Z → Z of p by q.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$\nbe $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $p : \\mathcal{X} \\to \\mathcal{Y}$ is formally smooth on objects, then so\nis the base change\n$p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$\nof $p$ by $q$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Formally smooth on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06D1","source_file":"criteria.tex","source_line":681,"source_end_line":689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L681-L689","statement_sha256":"2263c92c98fe3af26b28deb9467c1d904215a5d288307ff828e2d7c27469475b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13770,"rank":13770,"depth":0,"x":1964.081,"y":1636.944,"cluster":"algebraic-stacks"},{"id":"stacks:06D2","tag":"06D2","title":"Formally smooth on objects · Lemma 06D2","summary":"Let p : X → Y and q : Y → Z be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If p and q are formally smooth on objects, then so is the composition q ∘ p.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $p$ and $q$ are formally smooth on objects, then so is the composition\n$q \\circ p$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Formally smooth on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06D2","source_file":"criteria.tex","source_line":721,"source_end_line":727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L721-L727","statement_sha256":"895a65bc51acb5f4984953cc883288df0c093a7347b9572223dcd4c26539da54","origin":"The Stacks Project","memory_eligible":false,"source_rank":13771,"rank":13771,"depth":0,"x":2048.954,"y":1535.162,"cluster":"algebraic-stacks"},{"id":"stacks:06D3","tag":"06D3","title":"Formally smooth on objects · Lemma 06D3","summary":"Let p : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. If p is representable by algebraic spaces, then the following are equivalent: • p is formally smooth on objects, and • p is formally smooth (see Algebraic Stacks, Definition [Tag 03YK]).","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is\nrepresentable by algebraic spaces, then the following are equivalent:\n\\begin{enumerate}\n\\item $p$ is formally smooth on objects, and\n\\item $p$ is formally smooth (see\nAlgebraic Stacks,\nDefinition \\ref{algebraic-definition-relative-representable-property}).\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Formally smooth on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06D3","source_file":"criteria.tex","source_line":759,"source_end_line":770,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L759-L770","statement_sha256":"68c98b248a56b4ab09bbfec871cd9b44285445f9bf992e460789a4e062c60ac5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13772,"rank":13772,"depth":2,"x":2068.277,"y":1658.739,"cluster":"algebraic-stacks"},{"id":"stacks:06D5","tag":"06D5","title":"Surjective on objects · Lemma 06D5","summary":"Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If p : X → Y is surjective on objects, then so is the base change p' : X ×_Y Z → Z of p by q.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$\nbe $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $p : \\mathcal{X} \\to \\mathcal{Y}$ is surjective on objects, then so\nis the base change\n$p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$\nof $p$ by $q$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Surjective on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06D5","source_file":"criteria.tex","source_line":839,"source_end_line":847,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L839-L847","statement_sha256":"6383b9f96a38f86ec9054a0d7a69548264b37323f9f4e2b83c178cb2e7e7d67e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13773,"rank":13773,"depth":0,"x":1954.316,"y":1578.343,"cluster":"algebraic-stacks"},{"id":"stacks:06D6","tag":"06D6","title":"Surjective on objects · Lemma 06D6","summary":"Let p : X → Y and q : Y → Z be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If p and q are surjective on objects, then so is the composition q ∘ p.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $p$ and $q$ are surjective on objects, then so is the composition\n$q \\circ p$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Surjective on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06D6","source_file":"criteria.tex","source_line":859,"source_end_line":865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L859-L865","statement_sha256":"c1231555b14d420cace9d95fb032687a080ec6b4c0f5673df976a5fbcf8d2b01","origin":"The Stacks Project","memory_eligible":false,"source_rank":13774,"rank":13774,"depth":0,"x":2103.439,"y":1572.946,"cluster":"algebraic-stacks"},{"id":"stacks:06D7","tag":"06D7","title":"Surjective on objects · Lemma 06D7","summary":"Let p : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. If p is representable by algebraic spaces, then the following are equivalent: • p is surjective on objects, and • p is surjective (see Algebraic Stacks, Definition [Tag 03YK]).","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is\nrepresentable by algebraic spaces, then the following are equivalent:\n\\begin{enumerate}\n\\item $p$ is surjective on objects, and\n\\item $p$ is surjective (see\nAlgebraic Stacks,\nDefinition \\ref{algebraic-definition-relative-representable-property}).\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Surjective on objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06D7","source_file":"criteria.tex","source_line":878,"source_end_line":889,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L878-L889","statement_sha256":"3415b27a0403eac95ff178605125999ab22689badeee1d69b17956e7ac3fbb3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13775,"rank":13775,"depth":2,"x":1997.509,"y":1661.799,"cluster":"algebraic-stacks"},{"id":"stacks:06CF","tag":"06CF","title":"Algebraic morphisms · Definition 06CF","summary":"Let S be a scheme. Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. We say that F is algebraic if for every scheme T and every object xi of Y over T the 2-fibre product (Sch/T)_fppf ×_xi, Y X is an algebraic stack over S.","statement_latex":"Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a\n$1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$.\nWe say that $F$ is {\\it algebraic} if for every scheme $T$ and every\nobject $\\xi$ of $\\mathcal{Y}$ over $T$ the $2$-fibre product\n$$\n(\\Sch/T)_{fppf} \\times_{\\xi, \\mathcal{Y}} \\mathcal{X}\n$$\nis an algebraic stack over $S$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Algebraic morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CF","source_file":"criteria.tex","source_line":944,"source_end_line":954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L944-L954","statement_sha256":"f501bb287132766ba50246e9d88f44652cdc1afece356c6b0b05fa1e752b0e53","origin":"The Stacks Project","memory_eligible":false,"source_rank":13776,"rank":13776,"depth":0,"x":2004.187,"y":1535.809,"cluster":"algebraic-stacks"},{"id":"stacks:05XY","tag":"05XY","title":"Algebraic morphisms · Lemma 05XY","summary":"Let S be a scheme. Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. If • Y is an algebraic stack, and • F is algebraic (see above), then X is an algebraic stack.","statement_latex":"Let $S$ be a scheme.\nLet $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of\nstacks in groupoids over $(\\Sch/S)_{fppf}$. If\n\\begin{enumerate}\n\\item $\\mathcal{Y}$ is an algebraic stack, and\n\\item $F$ is algebraic (see above),\n\\end{enumerate}\nthen $\\mathcal{X}$ is an algebraic stack.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Algebraic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XY","source_file":"criteria.tex","source_line":961,"source_end_line":971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L961-L971","statement_sha256":"aaab795e816a5a6fb47942a8e39c3cbe351693ac3663a719b13fbb8c6ca313f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13777,"rank":13777,"depth":68,"x":2100.857,"y":1632.78,"cluster":"algebraic-stacks"},{"id":"stacks:06CG","tag":"06CG","title":"Algebraic morphisms · Lemma 06CG","summary":"Let S be a scheme. Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. If X is an algebraic stack and Δ : Y → Y × Y is representable by algebraic spaces, then F is algebraic.","statement_latex":"Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nof stacks in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X}$ is an\nalgebraic stack and $\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$\nis representable by algebraic spaces, then $F$ is algebraic.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Algebraic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CG","source_file":"criteria.tex","source_line":999,"source_end_line":1005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L999-L1005","statement_sha256":"9c6790937b34fa368a26c6b995ddafd62dd69c0ea167c31fe69b58fc4159b8f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13778,"rank":13778,"depth":68,"x":1951.165,"y":1616.081,"cluster":"algebraic-stacks"},{"id":"stacks:0D3R","tag":"0D3R","title":"Algebraic morphisms · Lemma 0D3R","summary":"Let S be a scheme. Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. If F is algebraic and Δ : Y → Y × Y is representable by algebraic spaces, then Δ : X → X × X is representable by algebraic spaces.","statement_latex":"Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nof stacks in groupoids over $(\\Sch/S)_{fppf}$.\nIf $F$ is algebraic and\n$\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$\nis representable by algebraic spaces, then\n$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nis representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Algebraic morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3R","source_file":"criteria.tex","source_line":1030,"source_end_line":1039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1030-L1039","statement_sha256":"4ab57e67dd6bbde18d19d9f3c5484b5dfa697bbe898929552d5d6bdba83c1b37","origin":"The Stacks Project","memory_eligible":false,"source_rank":13779,"rank":13779,"depth":55,"x":2075.328,"y":1543.249,"cluster":"algebraic-stacks"},{"id":"stacks:05XQ","tag":"05XQ","title":"Spaces of sections · Lemma 05XQ","summary":"Let Z → U be a finite morphism of schemes. Let W be an algebraic space and let W → Z be a surjective étale morphism. Then there exists a surjective étale morphism U' → U and a section σ : Z_U' → W_U' of the morphism W_U' → Z_U'.","statement_latex":"Let $Z \\to U$ be a finite morphism of schemes.\nLet $W$ be an algebraic space and let $W \\to Z$ be a\nsurjective \\'etale morphism. Then there exists a surjective\n\\'etale morphism $U' \\to U$ and a section\n$$\n\\sigma : Z_{U'} \\to W_{U'}\n$$\nof the morphism $W_{U'} \\to Z_{U'}$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Spaces of sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XQ","source_file":"criteria.tex","source_line":1103,"source_end_line":1113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1103-L1113","statement_sha256":"7605365bfc2928e66d717369e012e0ca6bb59bafc4242710b1ca8fe6e961936e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13780,"rank":13780,"depth":45,"x":2042.248,"y":1667.758,"cluster":"algebraic-stacks"},{"id":"stacks:05XR","tag":"05XR","title":"Spaces of sections · Lemma 05XR","summary":"Let Z → U be a finite locally free morphism of schemes. Let W be an algebraic space and let W → Z be an étale morphism. Then the functor F : (Sch/U)_fppf^opp → Sets, defined by the rule U' ↦ F(U') = (σ : Z_U' → W_U' section of W_U' → Z_U') is an algebraic space and the morphism F → U is étale.","statement_latex":"Let $Z \\to U$ be a finite locally free morphism of schemes.\nLet $W$ be an algebraic space and let $W \\to Z$ be an \\'etale morphism.\nThen the functor\n$$\nF : (\\Sch/U)_{fppf}^{opp} \\longrightarrow \\textit{Sets},\n$$\ndefined by the rule\n$$\nU' \\longmapsto\nF(U') =\n\\{\\sigma : Z_{U'} \\to W_{U'}\\text{ section of }W_{U'} \\to Z_{U'}\\}\n$$\nis an algebraic space and the morphism $F \\to U$ is \\'etale.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Spaces of sections","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XR","source_file":"criteria.tex","source_line":1161,"source_end_line":1176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1161-L1176","statement_sha256":"10b8f7255afd6bb768eaa14a566d2551606cf5208b8321373456f8bdd70cd1dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13781,"rank":13781,"depth":67,"x":1966.301,"y":1556.865,"cluster":"algebraic-stacks"},{"id":"stacks:05Y3","tag":"05Y3","title":"Relative morphisms · Lemma 05Y3","summary":"Let S be a scheme. Let Z → B and X → B be morphisms of algebraic spaces over S. Then • mathitMor_B(Z, X) is a sheaf on (Sch/S)_fppf. • If T is an algebraic space over S, then there is a canonical bijection Mor_Sh((Sch/S)_fppf)(T, mathitMor_B(Z, X)) = ((a, b) as in ([Tag 05Y1]))","statement_latex":"Let $S$ be a scheme. Let $Z \\to B$ and $X \\to B$ be morphisms of\nalgebraic spaces over $S$. Then\n\\begin{enumerate}\n\\item $\\mathit{Mor}_B(Z, X)$ is a sheaf on\n$(\\Sch/S)_{fppf}$.\n\\item If $T$ is an algebraic space over $S$, then there is a\ncanonical bijection\n$$\n\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\mathit{Mor}_B(Z, X))\n=\n\\{(a, b)\\text{ as in }(\\ref{equation-hom})\\}\n$$\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Y3","source_file":"criteria.tex","source_line":1299,"source_end_line":1314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1299-L1314","statement_sha256":"a394624f2d949a83eef2f3fca2d8823e9b89a8b2bcbd3f9c1ca2fb9bc67ac2a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13782,"rank":13782,"depth":55,"x":2111.887,"y":1595.651,"cluster":"algebraic-stacks"},{"id":"stacks:05Y4","tag":"05Y4","title":"Relative morphisms · Lemma 05Y4","summary":"Let S be a scheme. Let Z → B, X → B, and B' → B be morphisms of algebraic spaces over S. Set Z' = B' ×_B Z and X' = B' ×_B X. Then mathitMor_B'(Z', X') = B' ×_B mathitMor_B(Z, X) in Sh((Sch/S)_fppf).","statement_latex":"Let $S$ be a scheme. Let $Z \\to B$, $X \\to B$, and $B' \\to B$\nbe morphisms of algebraic spaces over $S$. Set $Z' = B' \\times_B Z$\nand $X' = B' \\times_B X$. Then\n$$\n\\mathit{Mor}_{B'}(Z', X')\n=\nB' \\times_B \\mathit{Mor}_B(Z, X)\n$$\nin $\\Sh((\\Sch/S)_{fppf})$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Y4","source_file":"criteria.tex","source_line":1372,"source_end_line":1383,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1372-L1383","statement_sha256":"37711f958609011ac8af9578698f13c42a3713230605a4353f4e1e0fdf375dd7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13783,"rank":13783,"depth":56,"x":1972.958,"y":1649.809,"cluster":"algebraic-stacks"},{"id":"stacks:05Y5","tag":"05Y5","title":"Relative morphisms · Lemma 05Y5","summary":"Let S be a scheme. Let Z → B and X' → X → B be morphisms of algebraic spaces over S. Assume • X' → X is étale, and • Z → B is finite locally free. Then mathitMor_B(Z, X') → mathitMor_B(Z, X) is representable by algebraic spaces and étale. If X' → X is also surjective, then mathitMor_B(Z, X') → mathitMor_B(Z, X) is surjective.","statement_latex":"Let $S$ be a scheme. Let $Z \\to B$ and $X' \\to X \\to B$ be morphisms of\nalgebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item $X' \\to X$ is \\'etale, and\n\\item $Z \\to B$ is finite locally free.\n\\end{enumerate}\nThen $\\mathit{Mor}_B(Z, X') \\to \\mathit{Mor}_B(Z, X)$ is representable\nby algebraic spaces and \\'etale. If $X' \\to X$ is also surjective,\nthen $\\mathit{Mor}_B(Z, X') \\to \\mathit{Mor}_B(Z, X)$ is surjective.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Y5","source_file":"criteria.tex","source_line":1394,"source_end_line":1405,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1394-L1405","statement_sha256":"4c24cdd0a6d69d7192cf9c759a4ea1cbde09c0499d683e1fbbee80115f4f98f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13784,"rank":13784,"depth":68,"x":2032.012,"y":1530.713,"cluster":"algebraic-stacks"},{"id":"stacks:05Y7","tag":"05Y7","title":"Relative morphisms · Proposition 05Y7","summary":"Let S be a scheme. Let Z → B and X → B be morphisms of algebraic spaces over S. If Z → B is finite locally free then mathitMor_B(Z, X) is an algebraic space.","statement_latex":"Let $S$ be a scheme. Let $Z \\to B$ and $X \\to B$ be morphisms of\nalgebraic spaces over $S$. If $Z \\to B$ is finite locally free\nthen $\\mathit{Mor}_B(Z, X)$ is an algebraic space.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Relative morphisms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05Y7","source_file":"criteria.tex","source_line":1424,"source_end_line":1429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1424-L1429","statement_sha256":"048bde7b987afc8790cdcfa204d26509523ae789fe822ec240c4899f94aef1cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13785,"rank":13785,"depth":69,"x":2084.383,"y":1652.376,"cluster":"algebraic-stacks"},{"id":"stacks:05YB","tag":"05YB","title":"Restriction of scalars · Lemma 05YB","summary":"Let S be a scheme. Let X → Z → B be morphisms of algebraic spaces over S. Then • Res_Z/B(X) is a sheaf on (Sch/S)_fppf. • If T is an algebraic space over S, then there is a canonical bijection Mor_Sh((Sch/S)_fppf)(T, Res_Z/B(X)) = ((a, b) as in ([Tag 05Y9]))","statement_latex":"Let $S$ be a scheme. Let $X \\to Z \\to B$ be morphisms of\nalgebraic spaces over $S$. Then\n\\begin{enumerate}\n\\item $\\text{Res}_{Z/B}(X)$ is a sheaf on\n$(\\Sch/S)_{fppf}$.\n\\item If $T$ is an algebraic space over $S$, then there is a\ncanonical bijection\n$$\n\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\text{Res}_{Z/B}(X))\n=\n\\{(a, b)\\text{ as in }(\\ref{equation-pairs})\\}\n$$\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Restriction of scalars","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YB","source_file":"criteria.tex","source_line":1534,"source_end_line":1549,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1534-L1549","statement_sha256":"89fabe925fccdd79c25181862fbed74f59e1899faf23891a93ba3afbf3bc524f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13786,"rank":13786,"depth":55,"x":1947.555,"y":1592.216,"cluster":"algebraic-stacks"},{"id":"stacks:05YC","tag":"05YC","title":"Restriction of scalars · Lemma 05YC","summary":"Let S be a scheme. Let X → Z → B and B' → B be morphisms of algebraic spaces over S. Set Z' = B' ×_B Z and X' = B' ×_B X. Then Res_Z'/B'(X') = B' ×_B Res_Z/B(X) in Sh((Sch/S)_fppf).","statement_latex":"Let $S$ be a scheme. Let $X \\to Z \\to B$ and $B' \\to B$\nbe morphisms of algebraic spaces over $S$.\nSet $Z' = B' \\times_B Z$ and $X' = B' \\times_B X$. Then\n$$\n\\text{Res}_{Z'/B'}(X')\n=\nB' \\times_B \\text{Res}_{Z/B}(X)\n$$\nin $\\Sh((\\Sch/S)_{fppf})$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Restriction of scalars","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YC","source_file":"criteria.tex","source_line":1612,"source_end_line":1623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1612-L1623","statement_sha256":"c20c47ff30f9ebfd2d0ae40280514e72fea5f0c6b57e7550cdb0a69013782056","origin":"The Stacks Project","memory_eligible":false,"source_rank":13787,"rank":13787,"depth":56,"x":2097.236,"y":1558.847,"cluster":"algebraic-stacks"},{"id":"stacks:05YD","tag":"05YD","title":"Restriction of scalars · Lemma 05YD","summary":"Let S be a scheme. Let X' → X → Z → B be morphisms of algebraic spaces over S. Assume • X' → X is étale, and • Z → B is finite locally free. Then Res_Z/B(X') → Res_Z/B(X) is representable by algebraic spaces and étale. If X' → X is also surjective, then Res_Z/B(X') → Res_Z/B(X) is surjective.","statement_latex":"Let $S$ be a scheme. Let $X' \\to X \\to Z \\to B$ be morphisms of\nalgebraic spaces over $S$. Assume\n\\begin{enumerate}\n\\item $X' \\to X$ is \\'etale, and\n\\item $Z \\to B$ is finite locally free.\n\\end{enumerate}\nThen $\\text{Res}_{Z/B}(X') \\to \\text{Res}_{Z/B}(X)$ is representable\nby algebraic spaces and \\'etale. If $X' \\to X$ is also surjective,\nthen $\\text{Res}_{Z/B}(X') \\to \\text{Res}_{Z/B}(X)$ is surjective.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Restriction of scalars","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YD","source_file":"criteria.tex","source_line":1634,"source_end_line":1645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1634-L1645","statement_sha256":"ed30326b0d707310b50d79658634c92c2ce52854eb7f54fb574666334dc72a87","origin":"The Stacks Project","memory_eligible":false,"source_rank":13788,"rank":13788,"depth":68,"x":2013.471,"y":1668.682,"cluster":"algebraic-stacks"},{"id":"stacks:05YE","tag":"05YE","title":"Restriction of scalars · Lemma 05YE","summary":"Let S be a scheme. Let X → Z → B be morphisms of algebraic spaces over S. The following diagram xymatrix mathitMor_B(Z, X) ar[r] & mathitMor_B(Z, Z) Res_Z/B(X) ar[r] ar[u] & B ar[u]_id_Z is a cartesian diagram of sheaves on (Sch/S)_fppf.","statement_latex":"Let $S$ be a scheme. Let $X \\to Z \\to B$ be morphisms of\nalgebraic spaces over $S$. The following diagram\n$$\n\\xymatrix{\n\\mathit{Mor}_B(Z, X) \\ar[r] & \\mathit{Mor}_B(Z, Z) \\\\\n\\text{Res}_{Z/B}(X) \\ar[r] \\ar[u] & B \\ar[u]_{\\text{id}_Z}\n}\n$$\nis a cartesian diagram of sheaves on $(\\Sch/S)_{fppf}$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Restriction of scalars","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YE","source_file":"criteria.tex","source_line":1673,"source_end_line":1684,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1673-L1684","statement_sha256":"06d8ac7a8bc0785b44399d44c9dd7b858015d870de72da9f3ba18d47ca0977f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13789,"rank":13789,"depth":0,"x":1986.841,"y":1539.815,"cluster":"algebraic-stacks"},{"id":"stacks:05YF","tag":"05YF","title":"Restriction of scalars · Proposition 05YF","summary":"Let S be a scheme. Let X → Z → B be morphisms of algebraic spaces over S. If Z → B is finite locally free then Res_Z/B(X) is an algebraic space.","statement_latex":"Let $S$ be a scheme. Let $X \\to Z \\to B$ be morphisms of\nalgebraic spaces over $S$. If $Z \\to B$ is finite locally free\nthen $\\text{Res}_{Z/B}(X)$ is an algebraic space.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Restriction of scalars","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YF","source_file":"criteria.tex","source_line":1691,"source_end_line":1696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1691-L1696","statement_sha256":"986d5812d5ab92c80684a1027fd0768154ea66dd03ab30d0f06c4ba30e03c264","origin":"The Stacks Project","memory_eligible":false,"source_rank":13790,"rank":13790,"depth":70,"x":2110.438,"y":1619.943,"cluster":"algebraic-stacks"},{"id":"stacks:05XN","tag":"05XN","title":"Finite Hilbert stacks · Lemma 05XN","summary":"Consider a 2-commutative diagram xymatrix X' ar[r]_G ar[d]_F' & X ar[d]^F Y' ar[r]^H & Y of stacks in groupoids over (Sch/S)_fppf with a given 2-isomorphism γ : H ∘ F' → F ∘ G. In this situation we obtain a canonical 1-morphism H_d(X'/Y') → H_d(X/Y). This morphism is compatible with the forgetful 1-morphisms of Examples of Stacks, Equation ([Tag 05WD]).","statement_latex":"Consider a $2$-commutative diagram\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r]_G \\ar[d]_{F'} & \\mathcal{X} \\ar[d]^F \\\\\n\\mathcal{Y}' \\ar[r]^H & \\mathcal{Y}\n}\n$$\nof stacks in groupoids over $(\\Sch/S)_{fppf}$ with a given\n$2$-isomorphism $\\gamma : H \\circ F' \\to F \\circ G$. In this situation we\nobtain a canonical $1$-morphism\n$\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\to\n\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$.\nThis morphism is compatible with the forgetful $1$-morphisms of\nExamples of Stacks,\nEquation (\\ref{examples-stacks-equation-diagram-hilbert-d-stack}).","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Finite Hilbert stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XN","source_file":"criteria.tex","source_line":1726,"source_end_line":1743,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1726-L1743","statement_sha256":"42db51dd75a11c922ed0a0dbc91c14684fe8c1abd6c9394d3ea4ce8065d1eaaa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13791,"rank":13791,"depth":1,"x":1954.449,"y":1631.019,"cluster":"algebraic-stacks"},{"id":"stacks:05XP","tag":"05XP","title":"Finite Hilbert stacks · Lemma 05XP","summary":"In the situation of Lemma [Tag 05XN] assume that the given square is 2-cartesian. Then the diagram xymatrix H_d(X'/Y') ar[r] ar[d] & H_d(X/Y) ar[d] Y' ar[r] & Y is 2-cartesian.","statement_latex":"In the situation of\nLemma \\ref{lemma-map-hilbert}\nassume that the given square is $2$-cartesian. Then the diagram\n$$\n\\xymatrix{\n\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\ar[r] \\ar[d] &\n\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\ar[d] \\\\\n\\mathcal{Y}' \\ar[r] &\n\\mathcal{Y}\n}\n$$\nis $2$-cartesian.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Finite Hilbert stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XP","source_file":"criteria.tex","source_line":1759,"source_end_line":1773,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1759-L1773","statement_sha256":"9c600d5b56de633373ca358aa2697d600cff825b05d17bddf684684892b714f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13792,"rank":13792,"depth":2,"x":2060.847,"y":1534.084,"cluster":"algebraic-stacks"},{"id":"stacks:05YG","tag":"05YG","title":"Finite Hilbert stacks · Lemma 05YG","summary":"In the situation of Lemma [Tag 05XN] assume • Y' = Y and H = id_Y, • G is representable by algebraic spaces and étale. Then H_d(X'/Y) → H_d(X/Y) is representable by algebraic spaces and étale. If G is also surjective, then H_d(X'/Y) → H_d(X/Y) is surjective.","statement_latex":"In the situation of\nLemma \\ref{lemma-map-hilbert}\nassume\n\\begin{enumerate}\n\\item $\\mathcal{Y}' = \\mathcal{Y}$ and $H = \\text{id}_\\mathcal{Y}$,\n\\item $G$ is representable by algebraic spaces and \\'etale.\n\\end{enumerate}\nThen $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}) \\to\n\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is representable by\nalgebraic spaces and \\'etale.\nIf $G$ is also surjective, then\n$\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}) \\to\n\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is surjective.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Finite Hilbert stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YG","source_file":"criteria.tex","source_line":1796,"source_end_line":1811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1796-L1811","statement_sha256":"76270308d3390287a38a537fa6801c20f51a4c622f3b5682cd00fcc91f5b8650","origin":"The Stacks Project","memory_eligible":false,"source_rank":13793,"rank":13793,"depth":68,"x":2060.341,"y":1666.283,"cluster":"algebraic-stacks"},{"id":"stacks:05XS","tag":"05XS","title":"Finite Hilbert stacks · Lemma 05XS","summary":"In the situation of Lemma [Tag 05XN]. Assume that G, H are representable by algebraic spaces and étale. Then H_d(X'/Y') → H_d(X/Y) is representable by algebraic spaces and étale. If also H is surjective and the induced functor X' → Y' ×_Y X is surjective, then H_d(X'/Y') → H_d(X/Y) is surjective.","statement_latex":"In the situation of\nLemma \\ref{lemma-map-hilbert}.\nAssume that $G$, $H$ are representable by algebraic spaces and \\'etale.\nThen $\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\to\n\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is representable by\nalgebraic spaces and \\'etale.\nIf also $H$ is surjective and the induced functor\n$\\mathcal{X}' \\to \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}$\nis surjective, then\n$\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y}') \\to\n\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is surjective.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Finite Hilbert stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05XS","source_file":"criteria.tex","source_line":1848,"source_end_line":1861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1848-L1861","statement_sha256":"033c2b419e058e8b86fa8ba98f4a5c0b884eaa936565c84dc3c4d3ebed46886a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13794,"rank":13794,"depth":69,"x":1954.126,"y":1568.258,"cluster":"algebraic-stacks"},{"id":"stacks:05YH","tag":"05YH","title":"Finite Hilbert stacks · Lemma 05YH","summary":"Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. Assume that Δ : Y → Y × Y is representable by algebraic spaces. Then H_d(X/Y) → H_d(X) × Y see Examples of Stacks, Equation ([Tag 05WD]) is representable by algebraic spaces.","statement_latex":"Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids\nover $(\\Sch/S)_{fppf}$. Assume that\n$\\Delta : \\mathcal{Y} \\to \\mathcal{Y} \\times \\mathcal{Y}$\nis representable by algebraic spaces. Then\n$$\n\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})\n\\longrightarrow\n\\mathcal{H}_d(\\mathcal{X}) \\times \\mathcal{Y}\n$$\nsee\nExamples of Stacks, Equation\n(\\ref{examples-stacks-equation-diagram-hilbert-d-stack})\nis representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Finite Hilbert stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YH","source_file":"criteria.tex","source_line":1898,"source_end_line":1913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1898-L1913","statement_sha256":"dcd5d52e0e9b07085704fdf133f6780ded8e2230174d7c2cd23ac57182876ff0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13795,"rank":13795,"depth":71,"x":2111.686,"y":1580.302,"cluster":"algebraic-stacks"},{"id":"stacks:05YJ","tag":"05YJ","title":"Finite Hilbert stacks · Lemma 05YJ","summary":"Let F : X → Y and G : X' → X be 1-morphisms of stacks in groupoids over (Sch/S)_fppf. If G is representable by algebraic spaces, then the 1-morphism H_d(X'/Y) → H_d(X/Y) is representable by algebraic spaces.","statement_latex":"Let $F : \\mathcal{X} \\to \\mathcal{Y}$ and $G : \\mathcal{X}' \\to \\mathcal{X}$\nbe $1$-morphisms of stacks in groupoids over $(\\Sch/S)_{fppf}$.\nIf $G$ is representable by algebraic spaces, then the $1$-morphism\n$$\n\\mathcal{H}_d(\\mathcal{X}'/\\mathcal{Y})\n\\longrightarrow\n\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})\n$$\nis representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Finite Hilbert stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YJ","source_file":"criteria.tex","source_line":1960,"source_end_line":1971,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1960-L1971","statement_sha256":"19dbbdacf5e94d9e86b6e0fa8579e574cf6a063fde2c85344aee808df11f94d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13796,"rank":13796,"depth":71,"x":1985.491,"y":1661.033,"cluster":"algebraic-stacks"},{"id":"stacks:06CH","tag":"06CH","title":"Finite Hilbert stacks · Lemma 06CH","summary":"Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. Assume F is representable by algebraic spaces and locally of finite presentation. Then p : H_d(X/Y) → Y is limit preserving on objects.","statement_latex":"Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids\nover $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic\nspaces and locally of finite presentation. Then\n$$\np : \\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{Y}\n$$\nis limit preserving on objects.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Finite Hilbert stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CH","source_file":"criteria.tex","source_line":1996,"source_end_line":2005,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L1996-L2005","statement_sha256":"0649525f363cb8bfcdd70537e64bc71d595eb2d09942c8005db19f537316fc19","origin":"The Stacks Project","memory_eligible":false,"source_rank":13797,"rank":13797,"depth":49,"x":2013.697,"y":1529.559,"cluster":"algebraic-stacks"},{"id":"stacks:05YP","tag":"05YP","title":"The finite Hilbert stack of a point · Lemma 05YP","summary":"The functor in groupoids FA_d defined in ([Tag 05YN]) is isomorphic (!) to the functor in groupoids which associates to a scheme T the category with • set of objects is X(T), • set of morphisms is G(T) × X(T), • s : G(T) × X(T) → X(T) is the projection map, • t : G(T) × X(T) → X(T) is a(T), and • composition G(T) × X(T) ×_s, X(T), t G(T) × X(T) → G(T) × X(T) is given by ((g, m), (g', m')) ↦ (gg', m').","statement_latex":"The functor in groupoids $FA_d$ defined in (\\ref{equation-FAd})\nis isomorphic (!) to the functor in groupoids which associates\nto a scheme $T$ the category with\n\\begin{enumerate}\n\\item set of objects is $X(T)$,\n\\item set of morphisms is $G(T) \\times X(T)$,\n\\item $s : G(T) \\times X(T) \\to X(T)$ is the projection map,\n\\item $t : G(T) \\times X(T) \\to X(T)$ is $a(T)$, and\n\\item composition $G(T) \\times X(T) \\times_{s, X(T), t} G(T) \\times X(T)\n\\to G(T) \\times X(T)$ is given by $((g, m), (g', m')) \\mapsto (gg', m')$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"The finite Hilbert stack of a point","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YP","source_file":"criteria.tex","source_line":2229,"source_end_line":2242,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2229-L2242","statement_sha256":"54b7aa61026b34b888fe92f24d5e298b64b349409c2abe70301a1e612a437ffa","origin":"The Stacks Project","memory_eligible":false,"source_rank":13798,"rank":13798,"depth":0,"x":2098.844,"y":1642.801,"cluster":"algebraic-stacks"},{"id":"stacks:05YQ","tag":"05YQ","title":"The finite Hilbert stack of a point · Proposition 05YQ","summary":"The stack H_d is equivalent to the quotient stack [X/G] described above. In particular H_d is an algebraic stack.","statement_latex":"The stack $\\mathcal{H}_d$ is equivalent to the quotient stack\n$[X/G]$ described above. In particular $\\mathcal{H}_d$ is an\nalgebraic stack.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"The finite Hilbert stack of a point","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YQ","source_file":"criteria.tex","source_line":2267,"source_end_line":2272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2267-L2272","statement_sha256":"f8d4250fff7303dce558140e28e9ee67448b7be8ff47ff62d5311af53f6be228","origin":"The Stacks Project","memory_eligible":false,"source_rank":13799,"rank":13799,"depth":7,"x":1944.6,"y":1607.522,"cluster":"algebraic-stacks"},{"id":"stacks:05YS","tag":"05YS","title":"Finite Hilbert stacks of spaces · Lemma 05YS","summary":"Let S be a scheme. Let X be an algebraic space over S. Then H_d(X) is an algebraic stack.","statement_latex":"Let $S$ be a scheme.\nLet $X$ be an algebraic space over $S$.\nThen $\\mathcal{H}_d(X)$ is an algebraic stack.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Finite Hilbert stacks of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05YS","source_file":"criteria.tex","source_line":2374,"source_end_line":2379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2374-L2379","statement_sha256":"835f7e037b8a2baaa191273de378c4fb3fb07231609e72a8f2713b3d488616a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13800,"rank":13800,"depth":72,"x":2087.068,"y":1545.858,"cluster":"algebraic-stacks"},{"id":"stacks:06CI","tag":"06CI","title":"Finite Hilbert stacks of spaces · Lemma 06CI","summary":"Let S be a scheme. Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf such that • X is representable by an algebraic space, and • F is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then H_d(X/Y) is an algebraic stack.","statement_latex":"Let $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nof stacks in groupoids over $(\\Sch/S)_{fppf}$ such that\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is representable by an algebraic space, and\n\\item $F$ is representable by algebraic spaces, surjective, flat, and\nlocally of finite presentation.\n\\end{enumerate}\nThen $\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})$ is an algebraic stack.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Finite Hilbert stacks of spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CI","source_file":"criteria.tex","source_line":2398,"source_end_line":2408,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2398-L2408","statement_sha256":"6225f93c41a4c11c0490d6ad17fb114d430cd8ccc8286fbcffc5956fb2b39d0b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13801,"rank":13801,"depth":73,"x":2031.462,"y":1672.487,"cluster":"algebraic-stacks"},{"id":"stacks:06CL","tag":"06CL","title":"LCI locus in the Hilbert stack · Lemma 06CL","summary":"Let S be a scheme. Fix a 1-morphism F : X → Y of stacks in groupoids over (Sch/S)_fppf. Assume F is representable by algebraic spaces, flat, and locally of finite presentation. Then H_d, lci(X/Y) is a stack in groupoids and the inclusion functor H_d, lci(X/Y) → H_d(X/Y) is representable and an open immersion.","statement_latex":"Let $S$ be a scheme. Fix a $1$-morphism\n$F : \\mathcal{X} \\longrightarrow \\mathcal{Y}$\nof stacks in groupoids over $(\\Sch/S)_{fppf}$.\nAssume $F$ is representable by algebraic spaces, flat, and locally\nof finite presentation. Then $\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})$\nis a stack in groupoids and the inclusion functor\n$$\n\\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})\n\\longrightarrow\n\\mathcal{H}_d(\\mathcal{X}/\\mathcal{Y})\n$$\nis representable and an open immersion.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"LCI locus in the Hilbert stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06CL","source_file":"criteria.tex","source_line":2567,"source_end_line":2581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2567-L2581","statement_sha256":"2cf5c3322d0b8ae8b2c0e1f941a9390946e71bd20ed9674c4575218cfe3c8f66","origin":"The Stacks Project","memory_eligible":false,"source_rank":13802,"rank":13802,"depth":70,"x":1970.481,"y":1547.247,"cluster":"algebraic-stacks"},{"id":"stacks:06D8","tag":"06D8","title":"LCI locus in the Hilbert stack · Lemma 06D8","summary":"Let U ⊂ U' be a first order thickening of affine schemes. Let X' be an algebraic space flat over U'. Set X = U ×_U' X'. Let Z → U be finite locally free of degree d. Finally, let f : Z → X be unramified and a local complete intersection morphism. Then there exists a commutative diagram xymatrix (Z ⊂ Z') ar[rd] ar[rr]_(f, f') & & (X ⊂ X') ar[ld] & (U ⊂ U') of algebraic spaces over U' such that Z' → U' is finite locally free of degree d and Z = U ×_U' Z'.","statement_latex":"Let $U \\subset U'$ be a first order thickening of affine schemes.\nLet $X'$ be an algebraic space flat over $U'$. Set $X = U \\times_{U'} X'$.\nLet $Z \\to U$ be finite locally free of degree $d$. Finally, let\n$f : Z \\to X$ be unramified and a local complete intersection morphism.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n(Z \\subset Z') \\ar[rd] \\ar[rr]_{(f, f')} & & (X \\subset X') \\ar[ld] \\\\\n& (U \\subset U')\n}\n$$\nof algebraic spaces over $U'$ such that $Z' \\to U'$ is finite locally free\nof degree $d$ and $Z = U \\times_{U'} Z'$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"LCI locus in the Hilbert stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06D8","source_file":"criteria.tex","source_line":2618,"source_end_line":2633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2618-L2633","statement_sha256":"6dd4d0c46be04e6a971ad9656afdb2a572016887593c49559308317d7cc60805","origin":"The Stacks Project","memory_eligible":false,"source_rank":13803,"rank":13803,"depth":60,"x":2116.526,"y":1605.14,"cluster":"algebraic-stacks"},{"id":"stacks:06D9","tag":"06D9","title":"LCI locus in the Hilbert stack · Lemma 06D9","summary":"Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. Assume F is representable by algebraic spaces, flat, and locally of finite presentation. Then p : H_d, lci(X/Y) → Y is formally smooth on objects.","statement_latex":"Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids\nover $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic\nspaces, flat, and locally of finite presentation. Then\n$$\np : \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y}) \\to \\mathcal{Y}\n$$\nis formally smooth on objects.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"LCI locus in the Hilbert stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06D9","source_file":"criteria.tex","source_line":2687,"source_end_line":2696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2687-L2696","statement_sha256":"4b9df0115026376dfcc751f9845ff902be476d2e57b6e2d12578b706c7628823","origin":"The Stacks Project","memory_eligible":false,"source_rank":13804,"rank":13804,"depth":71,"x":1961.895,"y":1645.42,"cluster":"algebraic-stacks"},{"id":"stacks:06DA","tag":"06DA","title":"LCI locus in the Hilbert stack · Lemma 06DA","summary":"Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. Assume F is representable by algebraic spaces, flat, surjective, and locally of finite presentation. Then coprod_d ≥ 1 H_d, lci(X/Y) → Y is surjective on objects.","statement_latex":"Let $F : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of stacks in groupoids\nover $(\\Sch/S)_{fppf}$. Assume $F$ is representable by algebraic\nspaces, flat, surjective, and locally of finite presentation. Then\n$$\n\\coprod\\nolimits_{d \\geq 1} \\mathcal{H}_{d, lci}(\\mathcal{X}/\\mathcal{Y})\n\\longrightarrow\n\\mathcal{Y}\n$$\nis surjective on objects.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"LCI locus in the Hilbert stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DA","source_file":"criteria.tex","source_line":2735,"source_end_line":2746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2735-L2746","statement_sha256":"4aecf236b77a0d7c78c8433094fb722fedf40afc71c86767fc274ed70d96876f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13805,"rank":13805,"depth":48,"x":2043.728,"y":1527.686,"cluster":"algebraic-stacks"},{"id":"stacks:06DC","tag":"06DC","title":"Bootstrapping algebraic stacks · Theorem 06DC","summary":"Artin's theorem on representability of flat groupoids Let S be a scheme. Let F : X → Y be a 1-morphism of stacks in groupoids over (Sch/S)_fppf. If • X is representable by an algebraic space, and • F is representable by algebraic spaces, surjective, flat and locally of finite presentation, then Y is an algebraic stack.","statement_latex":"\\begin{slogan}\nArtin's theorem on representability of flat groupoids\n\\end{slogan}\nLet $S$ be a scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of stacks in groupoids over $(\\Sch/S)_{fppf}$. If\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is representable by an algebraic space, and\n\\item $F$ is representable by algebraic spaces, surjective, flat and\nlocally of finite presentation,\n\\end{enumerate}\nthen $\\mathcal{Y}$ is an algebraic stack.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Bootstrapping algebraic stacks","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06DC","source_file":"criteria.tex","source_line":2828,"source_end_line":2841,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2828-L2841","statement_sha256":"c8e2a42efc50a49ec7eda32235bfba4839673e3012e3a84bb13e35bd84d31e99","origin":"The Stacks Project","memory_eligible":false,"source_rank":13806,"rank":13806,"depth":74,"x":2078.148,"y":1661.263,"cluster":"algebraic-stacks"},{"id":"stacks:06FH","tag":"06FH","title":"Applications · Lemma 06FH","summary":"Let S be a scheme contained in Sch_fppf. Let (U, R, s, t, c) be a groupoid in algebraic spaces over S. Assume s, t are flat and locally of finite presentation. Then the morphism S_U → [U/R] is flat, locally of finite presentation, and surjective.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.\nAssume $s, t$ are flat and locally of finite presentation.\nThen the morphism $\\mathcal{S}_U \\to [U/R]$ is flat, locally of\nfinite presentation, and surjective.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Applications","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FH","source_file":"criteria.tex","source_line":2967,"source_end_line":2974,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L2967-L2974","statement_sha256":"94bbb0200e6998702cdcb916aeef416c135ac2ad2ce239fc5af9e57db832cf12","origin":"The Stacks Project","memory_eligible":false,"source_rank":13807,"rank":13807,"depth":70,"x":1945.023,"y":1582.102,"cluster":"algebraic-stacks"},{"id":"stacks:06FI","tag":"06FI","title":"Applications · Theorem 06FI","summary":"Let S be a scheme contained in Sch_fppf. Let (U, R, s, t, c) be a groupoid in algebraic spaces over S. Assume s, t are flat and locally of finite presentation. Then the quotient stack [U/R] is an algebraic stack over S.","statement_latex":"Let $S$ be a scheme contained in $\\Sch_{fppf}$.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.\nAssume $s, t$ are flat and locally of finite presentation.\nThen the quotient stack $[U/R]$ is an algebraic stack over $S$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Applications","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FI","source_file":"criteria.tex","source_line":3027,"source_end_line":3033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L3027-L3033","statement_sha256":"e4f9c793eb3379c2ae3947423bc09db98d435cbd5658d81184c6fc4edf5a92a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13808,"rank":13808,"depth":75,"x":2107.241,"y":1564.894,"cluster":"algebraic-stacks"},{"id":"stacks:06PJ","tag":"06PJ","title":"When is a quotient stack algebraic? · Lemma 06PJ","summary":"Let S be a scheme and let B be an algebraic space over S. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. The quotient stack [U/R] is an algebraic stack if and only if there exists a morphism of algebraic spaces g : U' → U such that • the composition U' ×_g, U, t R → R xrightarrows U is a surjection of sheaves, and • the morphisms s', t' : R' → U' are flat and locally of finite presentation where (U', R', s', t', c') is the restriction of (U, R, s, t, c) via g.","statement_latex":"Let $S$ be a scheme and let $B$ be an algebraic space over $S$.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$.\nThe quotient stack $[U/R]$ is an algebraic stack if and only if\nthere exists a morphism of algebraic spaces $g : U' \\to U$ such that\n\\begin{enumerate}\n\\item the composition\n$U' \\times_{g, U, t} R \\to R \\xrightarrow{s} U$ is a surjection of\nsheaves, and\n\\item the morphisms $s', t' : R' \\to U'$ are flat and locally of finite\npresentation where $(U', R', s', t', c')$ is the restriction of\n$(U, R, s, t, c)$ via $g$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"When is a quotient stack algebraic?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PJ","source_file":"criteria.tex","source_line":3087,"source_end_line":3101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L3087-L3101","statement_sha256":"0324ceec6a6befc55f859fcc3d9fbe5550a8d93f22c94ba500f0777de10295f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13809,"rank":13809,"depth":76,"x":2001.205,"y":1669.884,"cluster":"algebraic-stacks"},{"id":"stacks:06PK","tag":"06PK","title":"When is a quotient stack algebraic? · Lemma 06PK","summary":"Let S be a scheme and let B be an algebraic space over S. Let G be a group algebraic space over B. Let X be an algebraic space over B and let a : G ×_B X → X be an action of G on X over B. The quotient stack [X/G] is an algebraic stack if and only if there exists a morphism of algebraic spaces φ : X' → X such that • G ×_B X' → X, (g, x') ↦ a(g, φ(x')) is a surjection of sheaves, and • the two projections X\" → X' of the algebraic space X\" given by the rule T ↦ ((x'_1, g,…","statement_latex":"Let $S$ be a scheme and let $B$ be an algebraic space over $S$.\nLet $G$ be a group algebraic space over $B$.\nLet $X$ be an algebraic space over $B$ and let $a : G \\times_B X \\to X$\nbe an action of $G$ on $X$ over $B$.\nThe quotient stack $[X/G]$ is an algebraic stack if and only if\nthere exists a morphism of algebraic spaces $\\varphi : X' \\to X$ such that\n\\begin{enumerate}\n\\item $G \\times_B X' \\to X$, $(g, x') \\mapsto a(g, \\varphi(x'))$ is a\nsurjection of sheaves, and\n\\item the two projections $X'' \\to X'$ of the algebraic space $X''$\ngiven by the rule\n$$\nT \\longmapsto \\{(x'_1, g, x'_2) \\in (X' \\times_B G \\times_B X')(T)\n\\mid \\varphi(x'_1) = a(g, \\varphi(x'_2))\\}\n$$\nare flat and locally of finite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"When is a quotient stack algebraic?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PK","source_file":"criteria.tex","source_line":3167,"source_end_line":3186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L3167-L3186","statement_sha256":"749f65637694b908e9f0b3f0505949a01291a6b2aac0eb20caf5312df0db473f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13810,"rank":13810,"depth":77,"x":1994.95,"y":1531.966,"cluster":"algebraic-stacks"},{"id":"stacks:06PL","tag":"06PL","title":"When is a quotient stack algebraic? · Lemma 06PL","summary":"Gerbes are algebraic if and only if the associated groups are flat and locally of finite presentation Let S be a scheme and let B be an algebraic space over S. Let G be a group algebraic space over B. Endow B with the trivial action of G. Then the quotient stack [B/G] is an algebraic stack if and only if G is flat and locally of finite presentation over B.","statement_latex":"\\begin{slogan}\nGerbes are algebraic if and only if the associated groups are flat\nand locally of finite presentation\n\\end{slogan}\nLet $S$ be a scheme and let $B$ be an algebraic space over $S$.\nLet $G$ be a group algebraic space over $B$.\nEndow $B$ with the trivial action of $G$.\nThen the quotient stack $[B/G]$ is an algebraic stack\nif and only if $G$ is flat and locally of finite presentation over $B$.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"When is a quotient stack algebraic?","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PL","source_file":"criteria.tex","source_line":3214,"source_end_line":3225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L3214-L3225","statement_sha256":"6b104d055f215af8f7b77ae46ee6dc05380ee1de0429c33ec3b4fc1e3b4807f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13811,"rank":13811,"depth":78,"x":2110.749,"y":1630.352,"cluster":"algebraic-stacks"},{"id":"stacks:076V","tag":"076V","title":"Algebraic stacks in the étale topology · Lemma 076V","summary":"Denote the common underlying category of Sch_fppf and Sch_etale by Sch_α (see Sheaves on Stacks, Section [Tag 06TN] and Topologies, Remark [Tag 03FF]). Let S be an object of Sch_α. Let p : X → Sch_α/S be a category fibred in groupoids with the following properties: • X is a stack in groupoids over (Sch/S)_etale, • the diagonal Δ : X → X × X is representable by algebraic spaces, and • there exists U ∈ Ob(Sch_α/S) and a 1-morphism (Sch/U)_etale → X which is surjective and…","statement_latex":"Denote the common underlying category of $\\Sch_{fppf}$\nand $\\Sch_\\etale$ by $\\Sch_\\alpha$ (see\nSheaves on Stacks, Section \\ref{stacks-sheaves-section-sheaves} and\nTopologies, Remark \\ref{topologies-remark-choice-sites}). Let $S$ be an object\nof $\\Sch_\\alpha$. Let\n$$\np : \\mathcal{X} \\to \\Sch_\\alpha/S\n$$\nbe a category fibred in groupoids with the following properties:\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is a stack in groupoids over $(\\Sch/S)_\\etale$,\n\\item the diagonal $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nis representable by algebraic spaces\\footnote{Here we can either mean\nsheaves in the \\'etale topology whose diagonal is representable and which\nhave an \\'etale surjective covering by a scheme or algebraic spaces as\ndefined in\nAlgebraic Spaces, Definition \\ref{spaces-definition-algebraic-space}.\nNamely, by Bootstrap, Lemma \\ref{bootstrap-lemma-spaces-etale}\nthere is no difference.}, and\n\\item there exists $U \\in \\Ob(\\Sch_\\alpha/S)$\nand a $1$-morphism $(\\Sch/U)_\\etale \\to \\mathcal{X}$\nwhich is surjective and smooth.\n\\end{enumerate}\nThen $\\mathcal{X}$ is an algebraic stack in the sense of\nAlgebraic Stacks, Definition \\ref{algebraic-definition-algebraic-stack}.","area":"Algebraic Stacks","chapter":"Criteria for Representability","chapter_id":"criteria","section":"Algebraic stacks in the étale topology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/076V","source_file":"criteria.tex","source_line":3290,"source_end_line":3317,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/criteria.tex#L3290-L3317","statement_sha256":"e41a95d3e6a812345c9b18855be45f85abb88de94c6b3c22d4a3587f422d9475","origin":"The Stacks Project","memory_eligible":false,"source_rank":13812,"rank":13812,"depth":72,"x":1945.849,"y":1623.494,"cluster":"algebraic-stacks"},{"id":"stacks:07T5","tag":"07T5","title":"Predeformation categories · Lemma 07T5","summary":"The functor p : F → C_Lambda defined above is a predeformation category.","statement_latex":"The functor $p : \\mathcal{F} \\to \\mathcal{C}_\\Lambda$ defined above\nis a predeformation category.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Predeformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07T5","source_file":"artin.tex","source_line":268,"source_end_line":272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L268-L272","statement_sha256":"20804af44c2d941156263761060bde0aba299549a40f1ee79720dbe1df1f0076","origin":"The Stacks Project","memory_eligible":false,"source_rank":13813,"rank":13813,"depth":4,"x":2073.261,"y":1534.772,"cluster":"algebraic-stacks"},{"id":"stacks:07WK","tag":"07WK","title":"Predeformation categories · Lemma 07WK","summary":"Let S be a locally Noetherian scheme. Let F : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Assume either • F is formally smooth on objects (Criteria for Representability, Section [Tag 06CZ]), • F is representable by algebraic spaces and formally smooth, or • F is representable by algebraic spaces and smooth. Then for every finite type field k over S and object x_0 of X over k the functor ([Tag 07WJ]) is smooth in the sense of Formal…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nAssume either\n\\begin{enumerate}\n\\item $F$ is formally smooth on objects (Criteria for Representability,\nSection \\ref{criteria-section-formally-smooth}),\n\\item $F$ is representable by algebraic spaces and formally smooth, or\n\\item $F$ is representable by algebraic spaces and smooth.\n\\end{enumerate}\nThen for every finite type field $k$ over $S$ and object\n$x_0$ of $\\mathcal{X}$ over $k$ the functor (\\ref{equation-functoriality})\nis smooth in the sense of\nFormal Deformation Theory, Definition\n\\ref{formal-defos-definition-smooth-morphism}.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Predeformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WK","source_file":"artin.tex","source_line":321,"source_end_line":337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L321-L337","statement_sha256":"3449309ca1070a8a522956122b49527ffe6ee579e705807ed5664907124a3c74","origin":"The Stacks Project","memory_eligible":false,"source_rank":13814,"rank":13814,"depth":59,"x":2050.603,"y":1672.817,"cluster":"algebraic-stacks"},{"id":"stacks:07WL","tag":"07WL","title":"Predeformation categories · Lemma 07WL","summary":"Let S be a locally Noetherian scheme. Let xymatrix W ar[d] ar[r] & Z ar[d] X ar[r] & Y be a 2-fibre product of categories fibred in groupoids over (Sch/S)_fppf. Let k be a finite type field over S and w_0 an object of W over k. Let x_0, z_0, y_0 be the images of w_0 under the morphisms in the diagram. Then xymatrix F_W, k, w_0 ar[d] ar[r] & F_Z, k, z_0 ar[d] F_X, k, x_0 ar[r] & F_Y, k, y_0 is a fibre product of predeformation categories.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$$\n\\xymatrix{\n\\mathcal{W} \\ar[d] \\ar[r] & \\mathcal{Z} \\ar[d] \\\\\n\\mathcal{X} \\ar[r] & \\mathcal{Y}\n}\n$$\nbe a $2$-fibre product of categories fibred in groupoids over\n$(\\Sch/S)_{fppf}$. Let $k$ be a finite type field over $S$ and\n$w_0$ an object of $\\mathcal{W}$ over $k$. Let $x_0, z_0, y_0$ be\nthe images of $w_0$ under the morphisms in the diagram. Then\n$$\n\\xymatrix{\n\\mathcal{F}_{\\mathcal{W}, k, w_0} \\ar[d] \\ar[r] &\n\\mathcal{F}_{\\mathcal{Z}, k, z_0} \\ar[d] \\\\\n\\mathcal{F}_{\\mathcal{X}, k, x_0} \\ar[r] & \\mathcal{F}_{\\mathcal{Y}, k, y_0}\n}\n$$\nis a fibre product of predeformation categories.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Predeformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WL","source_file":"artin.tex","source_line":352,"source_end_line":373,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L352-L373","statement_sha256":"ea9a59cf70843a3f1532e1eb69313de63d7eb1bcc36a519f99e39820ba700383","origin":"The Stacks Project","memory_eligible":false,"source_rank":13815,"rank":13815,"depth":0,"x":1956.077,"y":1557.898,"cluster":"algebraic-stacks"},{"id":"stacks:07WN","tag":"07WN","title":"Pushouts and stacks · Lemma 07WN","summary":"Algebraic stacks satisfy the (strong) Rim-Schlessinger condition Let S be a scheme. Let xymatrix X ar[r] ar[d] & X' ar[d] Y ar[r] & Y' be a pushout in the category of schemes over S where X → X' is a thickening and X → Y is affine, see More on Morphisms, Lemma [Tag 07RT]. Let Z be an algebraic stack over S. Then the functor of fibre categories Z_Y' → Z_Y ×_Z_X Z_X' is an equivalence of categories.","statement_latex":"\\begin{slogan}\nAlgebraic stacks satisfy the (strong) Rim-Schlessinger condition\n\\end{slogan}\nLet $S$ be a scheme. Let\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d] & X' \\ar[d] \\\\\nY \\ar[r] & Y'\n}\n$$\nbe a pushout in the category of schemes over $S$ where $X \\to X'$\nis a thickening and $X \\to Y$ is affine, see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.\nLet $\\mathcal{Z}$ be an algebraic stack over $S$.\nThen the functor of fibre categories\n$$\n\\mathcal{Z}_{Y'}\n\\longrightarrow\n\\mathcal{Z}_Y \\times_{\\mathcal{Z}_X} \\mathcal{Z}_{X'}\n$$\nis an equivalence of categories.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Pushouts and stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WN","source_file":"artin.tex","source_line":396,"source_end_line":419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L396-L419","statement_sha256":"dce4a1274de77f3ff4756a09afc2dfdd6cc635d824a737cb53d9aa100c51318f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13816,"rank":13816,"depth":74,"x":2118.574,"y":1589.074,"cluster":"algebraic-stacks"},{"id":"stacks:07WP","tag":"07WP","title":"The Rim-Schlessinger condition · Definition 07WP","summary":"Let S be a locally Noetherian scheme. Let Z be a category fibred in groupoids over (Sch/S)_fppf. We say Z satisfies condition (RS) if for every pushout xymatrix X ar[r] ar[d] & X' ar[d] Y ar[r] & Y' = Y amalg_X X' in the category of schemes over S where • X, X', Y, Y' are spectra of local Artinian rings, • X, X', Y, Y' are of finite type over S, and • X → X' (and hence Y → Y') is a closed immersion the functor of fibre categories Z_Y' → Z_Y ×_Z_X Z_X' is an equivalence of…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{Z}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{Z}$\nsatisfies {\\it condition (RS)} if for every pushout\n$$\n\\xymatrix{\nX \\ar[r] \\ar[d] & X' \\ar[d] \\\\\nY \\ar[r] & Y' = Y \\amalg_X X'\n}\n$$\nin the category of schemes over $S$ where\n\\begin{enumerate}\n\\item $X$, $X'$, $Y$, $Y'$ are spectra of local Artinian rings,\n\\item $X$, $X'$, $Y$, $Y'$ are of finite type over $S$, and\n\\item $X \\to X'$ (and hence $Y \\to Y'$) is a closed immersion\n\\end{enumerate}\nthe functor of fibre categories\n$$\n\\mathcal{Z}_{Y'}\n\\longrightarrow\n\\mathcal{Z}_Y \\times_{\\mathcal{Z}_X} \\mathcal{Z}_{X'}\n$$\nis an equivalence of categories.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"The Rim-Schlessinger condition","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WP","source_file":"artin.tex","source_line":492,"source_end_line":516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L492-L516","statement_sha256":"382b3b0e533ee02c1fb5f0c2a6eb955c83d6e68007bad61169794016b0ef4e23","origin":"The Stacks Project","memory_eligible":false,"source_rank":13817,"rank":13817,"depth":0,"x":1973.341,"y":1658.451,"cluster":"algebraic-stacks"},{"id":"stacks:07WQ","tag":"07WQ","title":"The Rim-Schlessinger condition · Lemma 07WQ","summary":"Let X be an algebraic stack over a locally Noetherian base S. Then X satisfies (RS).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over a locally Noetherian base\n$S$. Then $\\mathcal{X}$ satisfies (RS).","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"The Rim-Schlessinger condition","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WQ","source_file":"artin.tex","source_line":525,"source_end_line":529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L525-L529","statement_sha256":"aa987b559225ac2f8e055cda7f449e34d0999debee26cfbf0f223667622977e4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13818,"rank":13818,"depth":75,"x":2024.762,"y":1524.575,"cluster":"algebraic-stacks"},{"id":"stacks:07WR","tag":"07WR","title":"The Rim-Schlessinger condition · Lemma 07WR","summary":"Let S be a scheme. Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If X, Y, and Z satisfy (RS), then so does X ×_Y Z.","statement_latex":"Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and\n$q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X}$, $\\mathcal{Y}$,\nand $\\mathcal{Z}$ satisfy (RS), then so\ndoes $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"The Rim-Schlessinger condition","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WR","source_file":"artin.tex","source_line":535,"source_end_line":542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L535-L542","statement_sha256":"3314ad844a1f6b1449246177d7dcd4c294df56ebe6b8d71e87034b066da5f281","origin":"The Stacks Project","memory_eligible":false,"source_rank":13819,"rank":13819,"depth":1,"x":2094.666,"y":1652.767,"cluster":"algebraic-stacks"},{"id":"stacks:07WU","tag":"07WU","title":"Deformation categories · Lemma 07WU","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf satisfying (RS). For any field k of finite type over S and any object x_0 of X lying over k the predeformation category p : F_X, k, x_0 → C_Lambda ([Tag 07T4]) is a deformation category, see Formal Deformation Theory, Definition [Tag 06J9].","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$ satisfying (RS). For any field\n$k$ of finite type over $S$ and any object $x_0$ of $\\mathcal{X}$ lying\nover $k$ the predeformation category\n$p : \\mathcal{F}_{\\mathcal{X}, k, x_0} \\to \\mathcal{C}_\\Lambda$\n(\\ref{equation-predeformation-category}) is a deformation category, see\nFormal Deformation Theory, Definition\n\\ref{formal-defos-definition-deformation-category}.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Deformation categories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WU","source_file":"artin.tex","source_line":606,"source_end_line":616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L606-L616","statement_sha256":"d7ba07635567be6506ef7b5465590490c0a6c637491a7f550bfbebf6f8e943b4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13820,"rank":13820,"depth":43,"x":1939.676,"y":1597.779,"cluster":"algebraic-stacks"},{"id":"stacks:07WX","tag":"07WX","title":"Change of field · Lemma 07WX","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. Let k be a field of finite type over S and let l/k be a finite extension. Let x_0 be an object of F lying over Spec(k). Denote x_l, 0 the restriction of x_0 to Spec(l). Then there is a canonical functor (F_X, k , x_0)_l/k → F_X, l, x_l, 0 of categories cofibred in groupoids over C_Lambda, l. If X satisfies (RS), then this functor is an equivalence.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let $k$ be a\nfield of finite type over $S$ and let $l/k$ be a finite extension.\nLet $x_0$ be an object of $\\mathcal{F}$ lying over $\\Spec(k)$.\nDenote $x_{l, 0}$ the restriction of $x_0$ to $\\Spec(l)$.\nThen there is a canonical functor\n$$\n(\\mathcal{F}_{\\mathcal{X}, k , x_0})_{l/k}\n\\longrightarrow\n\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}\n$$\nof categories cofibred in groupoids over $\\mathcal{C}_{\\Lambda, l}$.\nIf $\\mathcal{X}$ satisfies (RS), then this functor is an equivalence.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Change of field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07WX","source_file":"artin.tex","source_line":685,"source_end_line":700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L685-L700","statement_sha256":"095de80c1b5cdfb84e41d29264464006c01beaa459a6c496f9df0b99abb71db8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13821,"rank":13821,"depth":43,"x":2098.546,"y":1550.273,"cluster":"algebraic-stacks"},{"id":"stacks:07X1","tag":"07X1","title":"Tangent spaces · Lemma 07X1","summary":"Let S be a locally Noetherian scheme. Assume • X is an algebraic stack, • U is a scheme locally of finite type over S, and • (Sch/U)_fppf → X is a smooth surjective morphism. Then, for any F = F_X, k, x_0 as in Section [Tag 07T2] the tangent space TF and infinitesimal automorphism space Inf(F) have finite dimension over k.","statement_latex":"Let $S$ be a locally Noetherian scheme. Assume\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is an algebraic stack,\n\\item $U$ is a scheme locally of finite type over $S$, and\n\\item $(\\Sch/U)_{fppf} \\to \\mathcal{X}$ is a smooth surjective\nmorphism.\n\\end{enumerate}\nThen, for any $\\mathcal{F} = \\mathcal{F}_{\\mathcal{X}, k, x_0}$ as in\nSection \\ref{section-predeformation-categories}\nthe tangent space $T\\mathcal{F}$ and infinitesimal automorphism space\n$\\text{Inf}(\\mathcal{F})$ have finite dimension over $k$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07X1","source_file":"artin.tex","source_line":812,"source_end_line":825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L812-L825","statement_sha256":"0079dc95209a02534ebd33706e51a106c6bbc32da432b97a416fdee4e9c4074b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13822,"rank":13822,"depth":76,"x":2019.42,"y":1675.735,"cluster":"algebraic-stacks"},{"id":"stacks:07X2","tag":"07X2","title":"Tangent spaces · Lemma 07X2","summary":"Let S be a locally Noetherian scheme. Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. Assume X, Y, Z satisfy (RS). Let k be a field of finite type over S and let w_0 be an object of W = X ×_Y Z over k. Denote x_0, y_0, z_0 the objects of X, Y, Z you get from w_0. Then there is a 6-term exact sequence xymatrix 0 ar[r] & Inf(F_W, k, w_0) ar[r] & Inf(F_X, k, x_0) ⊕ Inf(F_Z, k, z_0) ar[r] & Inf(F_Y, k, y_0) ar[lld] & TF_W, k,…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$\nand $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$,\n$\\mathcal{Y}$, $\\mathcal{Z}$ satisfy (RS).\nLet $k$ be a field of finite type over $S$ and let $w_0$ be an object of\n$\\mathcal{W} = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $k$.\nDenote $x_0, y_0, z_0$ the objects of $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$\nyou get from $w_0$. Then there is a $6$-term exact sequence\n$$\n\\xymatrix{\n0 \\ar[r] &\n\\text{Inf}(\\mathcal{F}_{\\mathcal{W}, k, w_0}) \\ar[r] &\n\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0}) \\oplus\n\\text{Inf}(\\mathcal{F}_{\\mathcal{Z}, k, z_0}) \\ar[r] &\n\\text{Inf}(\\mathcal{F}_{\\mathcal{Y}, k, y_0}) \\ar[lld] \\\\\n &\nT\\mathcal{F}_{\\mathcal{W}, k, w_0} \\ar[r] &\nT\\mathcal{F}_{\\mathcal{X}, k, x_0} \\oplus\nT\\mathcal{F}_{\\mathcal{Z}, k, z_0} \\ar[r] &\nT\\mathcal{F}_{\\mathcal{Y}, k, y_0}\n}\n$$\nof $k$-vector spaces.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Tangent spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07X2","source_file":"artin.tex","source_line":936,"source_end_line":961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L936-L961","statement_sha256":"f916f350eb58cb59918bbc8f5563585f8e92c28cf7fdeb0b4f65c077389c7439","origin":"The Stacks Project","memory_eligible":false,"source_rank":13823,"rank":13823,"depth":44,"x":1976.777,"y":1538.011,"cluster":"algebraic-stacks"},{"id":"stacks:07X4","tag":"07X4","title":"Formal objects · Definition 07X4","summary":"Let S be a locally Noetherian scheme. Let p : X → (Sch/S)_fppf be a category fibred in groupoids. • A formal object xi = (R, xi_n, f_n) of X consists of a Noetherian complete local S-algebra R, objects xi_n of X lying over Spec(R/ m_R^n), and morphisms f_n : xi_n → xi_n + 1 of X lying over Spec(R/ m^n) → Spec(R/ m^n + 1) such that R/ m is a field of finite type over S. • A morphism of formal objects a : xi = (R, xi_n, f_n) → eta = (T, eta_n, g_n) is given by morphisms a_n…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\n\\begin{enumerate}\n\\item A {\\it formal object} $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{X}$ consists\nof a Noetherian complete local $S$-algebra $R$, objects $\\xi_n$ of\n$\\mathcal{X}$ lying over $\\Spec(R/\\mathfrak m_R^n)$, and morphisms\n$f_n : \\xi_n \\to \\xi_{n + 1}$ of $\\mathcal{X}$ lying over\n$\\Spec(R/\\mathfrak m^n) \\to \\Spec(R/\\mathfrak m^{n + 1})$\nsuch that $R/\\mathfrak m$ is a field of finite type over $S$.\n\\item A {\\it morphism of formal objects}\n$a : \\xi = (R, \\xi_n, f_n) \\to \\eta = (T, \\eta_n, g_n)$\nis given by morphisms $a_n : \\xi_n \\to \\eta_n$ such that for every $n$\nthe diagram\n$$\n\\xymatrix{\n\\xi_n \\ar[r]_{f_n} \\ar[d]_{a_n} & \\xi_{n + 1} \\ar[d]^{a_{n + 1}} \\\\\n\\eta_n \\ar[r]^{g_n} & \\eta_{n + 1}\n}\n$$\nis commutative. Applying the functor $p$ we obtain a compatible collection\nof morphisms $\\Spec(R/\\mathfrak m_R^n) \\to \\Spec(T/\\mathfrak m_T^n)$ and\nhence a morphism $a_0 : \\Spec(R) \\to \\Spec(T)$ over $S$. We say that\n$a$ {\\it lies over} $a_0$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Formal objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07X4","source_file":"artin.tex","source_line":986,"source_end_line":1012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L986-L1012","statement_sha256":"b11311e6e6e88d3b732571f0036642b31a040e696c4407436e17509e76515298","origin":"The Stacks Project","memory_eligible":false,"source_rank":13824,"rank":13824,"depth":0,"x":2119.296,"y":1615.545,"cluster":"algebraic-stacks"},{"id":"stacks:07X5","tag":"07X5","title":"Formal objects · Lemma 07X5","summary":"Let S be a locally Noetherian scheme. Let F : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let eta = (R, eta_n, g_n) be a formal object of Y and let xi_1 be an object of X with F(xi_1) ≅ eta_1. If F is formally smooth on objects (see Criteria for Representability, Section [Tag 06CZ]), then there exists a formal object xi = (R, xi_n, f_n) of X such that F(xi) ≅ eta.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $F : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nLet $\\eta = (R, \\eta_n, g_n)$ be a formal object of $\\mathcal{Y}$\nand let $\\xi_1$ be an object of $\\mathcal{X}$ with $F(\\xi_1) \\cong \\eta_1$.\nIf $F$ is formally smooth on objects (see\nCriteria for Representability, Section \\ref{criteria-section-formally-smooth}),\nthen there exists a formal object $\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{X}$\nsuch that $F(\\xi) \\cong \\eta$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07X5","source_file":"artin.tex","source_line":1038,"source_end_line":1048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1038-L1048","statement_sha256":"96cfcff7b2521104bb012fbb9cfea36d358d81d7572d52f1ff03fe13c51fcba9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13825,"rank":13825,"depth":0,"x":1951.478,"y":1639.294,"cluster":"algebraic-stacks"},{"id":"stacks:07X7","tag":"07X7","title":"Formal objects · Definition 07X7","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. A formal object xi = (R, xi_n, f_n) of X is called effective if it is in the essential image of the functor ([Tag 07X6]).","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$. A formal object\n$\\xi = (R, \\xi_n, f_n)$ of $\\mathcal{X}$ is called {\\it effective}\nif it is in the essential image of the functor\n(\\ref{equation-approximation}).","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Formal objects","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07X7","source_file":"artin.tex","source_line":1092,"source_end_line":1099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1092-L1099","statement_sha256":"aa79fa4dccf99b7778f94c78895feded4bf31048fb21e1be8a87fd12d59c4e24","origin":"The Stacks Project","memory_eligible":false,"source_rank":13826,"rank":13826,"depth":0,"x":2056.36,"y":1526.305,"cluster":"algebraic-stacks"},{"id":"stacks:07X8","tag":"07X8","title":"Formal objects · Lemma 07X8","summary":"Let S be a locally Noetherian scheme. Let X be an algebraic stack over S. The functor ([Tag 07X6]) is an equivalence.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be an algebraic\nstack over $S$. The functor (\\ref{equation-approximation}) is an equivalence.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07X8","source_file":"artin.tex","source_line":1105,"source_end_line":1109,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1105-L1109","statement_sha256":"397fe2fdaf9e2b2e95643dbd55eddb81eb933d52c6d8eb96b8172e12dcf2ad9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13827,"rank":13827,"depth":59,"x":2069.914,"y":1669.453,"cluster":"algebraic-stacks"},{"id":"stacks:07X9","tag":"07X9","title":"Formal objects · Lemma 07X9","summary":"Let S be a scheme. Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If the functor ([Tag 07X6]) is an equivalence for X, Y, and Z, then it is an equivalence for X ×_Y Z.","statement_latex":"Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and\n$q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If the functor\n(\\ref{equation-approximation}) is an equivalence for \n$\\mathcal{X}$, $\\mathcal{Y}$, and $\\mathcal{Z}$, then it is \nan equivalence for $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Formal objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07X9","source_file":"artin.tex","source_line":1220,"source_end_line":1228,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1220-L1228","statement_sha256":"58ddc936786c8dc64523c0cf3ffea5a00f53e1e1f525e3959d763dba5d8dc359","origin":"The Stacks Project","memory_eligible":false,"source_rank":13828,"rank":13828,"depth":2,"x":1944.527,"y":1571.372,"cluster":"algebraic-stacks"},{"id":"stacks:07XB","tag":"07XB","title":"Approximation · Lemma 07XB","summary":"Let S be a locally Noetherian scheme. Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let x be an object of X lying over Spec(R) where R is a Noetherian complete local ring with residue field k of finite type over S. Let s ∈ S be the image of Spec(k) → S. Assume that (a) O_S, s is a G-ring and (b) p is limit preserving on objects. Then for every integer N ≥ 1 there exist • a finite type S-algebra A, • a maximal ideal m_A ⊂ A, • an object x_A of X over Spec(A),…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category\nfibred in groupoids. Let $x$ be an object of\n$\\mathcal{X}$ lying over $\\Spec(R)$ where $R$ is a Noetherian complete\nlocal ring with residue field $k$ of finite type over $S$. Let $s \\in S$\nbe the image of $\\Spec(k) \\to S$. Assume that (a) $\\mathcal{O}_{S, s}$ is\na G-ring and (b) $p$ is limit preserving on objects. Then for every\ninteger $N \\geq 1$ there exist\n\\begin{enumerate}\n\\item a finite type $S$-algebra $A$,\n\\item a maximal ideal $\\mathfrak m_A \\subset A$,\n\\item an object $x_A$ of $\\mathcal{X}$ over $\\Spec(A)$,\n\\item an $S$-isomorphism $R/\\mathfrak m_R^N \\cong A/\\mathfrak m_A^N$,\n\\item an isomorphism\n$x|_{\\Spec(R/\\mathfrak m_R^N)} \\cong x_A|_{\\Spec(A/\\mathfrak m_A^N)}$\ncompatible with (4), and\n\\item an isomorphism\n$\\text{Gr}_{\\mathfrak m_R}(R) \\cong \\text{Gr}_{\\mathfrak m_A}(A)$\nof graded $k$-algebras.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Approximation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XB","source_file":"artin.tex","source_line":1270,"source_end_line":1292,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1270-L1292","statement_sha256":"ba606807e1bdd518259a1feb819de4f4b12a5eb197555eccc2eb779fe1b6425c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13829,"rank":13829,"depth":54,"x":2116.238,"y":1572.55,"cluster":"algebraic-stacks"},{"id":"stacks:07XL","tag":"07XL","title":"Limit preserving · Definition 07XL","summary":"Let S be a scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. We say X is limit preserving if for every affine scheme T over S which is a limit T = lim T_i of a directed inverse system of affine schemes T_i over S, we have an equivalence colim X_T_i → X_T of fibre categories.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred in groupoids\nover $(\\Sch/S)_{fppf}$. We say $\\mathcal{X}$ is {\\it limit preserving}\nif for every affine scheme $T$ over $S$ which is a limit $T = \\lim T_i$\nof a directed inverse system of affine schemes $T_i$ over $S$, we have\nan equivalence\n$$\n\\colim \\mathcal{X}_{T_i} \\longrightarrow \\mathcal{X}_T\n$$\nof fibre categories.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Limit preserving","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XL","source_file":"artin.tex","source_line":1449,"source_end_line":1460,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1449-L1460","statement_sha256":"6dde3d42001d8ba3ec7497185df65fb957e85777daffc60a192b2ad27a9a839c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13830,"rank":13830,"depth":0,"x":1988.392,"y":1669.326,"cluster":"algebraic-stacks"},{"id":"stacks:07XM","tag":"07XM","title":"Limit preserving · Lemma 07XM","summary":"Let S be a scheme. Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. • If X → (Sch/S)_fppf and Z → (Sch/S)_fppf are limit preserving on objects and Y is limit preserving, then X ×_Y Z → (Sch/S)_fppf is limit preserving on objects. • If X, Y, and Z are limit preserving, then so is X ×_Y Z.","statement_latex":"Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and\n$q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$.\n\\begin{enumerate}\n\\item If $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ and\n$\\mathcal{Z} \\to (\\Sch/S)_{fppf}$ are limit preserving on objects and\n$\\mathcal{Y}$ is limit preserving, then\n$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to (\\Sch/S)_{fppf}$ is\nlimit preserving on objects.\n\\item If $\\mathcal{X}$, $\\mathcal{Y}$,\nand $\\mathcal{Z}$ are limit preserving, then so\nis $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Limit preserving","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XM","source_file":"artin.tex","source_line":1475,"source_end_line":1490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1475-L1490","statement_sha256":"09dc641edd5b2b6c62d1fbf1acefd76d9e59ebcc9c6cd7e5027c56455f9da5dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13831,"rank":13831,"depth":0,"x":2004.876,"y":1525.108,"cluster":"algebraic-stacks"},{"id":"stacks:07XN","tag":"07XN","title":"Limit preserving · Lemma 07XN","summary":"Let S be a scheme. Let X be an algebraic stack over S. Then the following are equivalent • X is a stack in setoids and X → (Sch/S)_fppf is limit preserving on objects, • X is a stack in setoids and limit preserving, • X is representable by an algebraic space locally of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack over $S$.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is a stack in setoids and\n$\\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects,\n\\item $\\mathcal{X}$ is a stack in setoids and limit preserving,\n\\item $\\mathcal{X}$ is representable by an algebraic space\nlocally of finite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Limit preserving","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XN","source_file":"artin.tex","source_line":1515,"source_end_line":1526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1515-L1526","statement_sha256":"e775003536c54fc64d7747287e4bb9e36fb9f504674314f89b5b710aa1ab2442","origin":"The Stacks Project","memory_eligible":false,"source_rank":13832,"rank":13832,"depth":68,"x":2108.924,"y":1641.057,"cluster":"algebraic-stacks"},{"id":"stacks:0CXI","tag":"0CXI","title":"Limit preserving · Lemma 0CXI","summary":"Let S be a scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. Assume Δ : X → X × X is representable by algebraic spaces and X is limit preserving. Then Δ is locally of finite type.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be a category fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Assume\n$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is\nrepresentable by algebraic spaces and $\\mathcal{X}$ is limit preserving.\nThen $\\Delta$ is locally of finite type.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Limit preserving","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXI","source_file":"artin.tex","source_line":1539,"source_end_line":1546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1539-L1546","statement_sha256":"843aa135a0058db258b5d82735132cc7b7beb9fc9babffd234c5afceb3a08526","origin":"The Stacks Project","memory_eligible":false,"source_rank":13833,"rank":13833,"depth":69,"x":1938.588,"y":1614.539,"cluster":"algebraic-stacks"},{"id":"stacks:0CXJ","tag":"0CXJ","title":"Versality · Definition 0CXJ","summary":"Let S be a locally Noetherian scheme. Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let xi = (R, xi_n, f_n) be a formal object. Set k = R/ m and x_0 = xi_1. We will say that xi is versal if xi as a formal object of F_X, k, x_0 (Remark [Tag 0CXH]) is versal in the sense of Formal Deformation Theory, Definition [Tag 06HR].","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nLet $\\xi = (R, \\xi_n, f_n)$ be a formal object. Set $k = R/\\mathfrak m$ and\n$x_0 = \\xi_1$. We will say that $\\xi$ is {\\it versal} if $\\xi$\nas a formal object of $\\mathcal{F}_{\\mathcal{X}, k, x_0}$\n(Remark \\ref{remark-formal-objects-match}) is versal in the sense\nof Formal Deformation Theory, Definition \\ref{formal-defos-definition-versal}.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Versality","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXJ","source_file":"artin.tex","source_line":1586,"source_end_line":1595,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1586-L1595","statement_sha256":"3c8d8d16360609c63862cadefef862b96d73e6d876d0bf32e779cf976c5cfe17","origin":"The Stacks Project","memory_eligible":false,"source_rank":13834,"rank":13834,"depth":1,"x":2085.833,"y":1537.275,"cluster":"algebraic-stacks"},{"id":"stacks:07XF","tag":"07XF","title":"Versality · Definition 07XF","summary":"Let S be a locally Noetherian scheme. Let X be fibred in groupoids over (Sch/S)_fppf. Let U be a scheme locally of finite type over S. Let x be an object of X lying over U. Let u_0 be a finite type point of U. We say x is versal at u_0 if the morphism hat x ([Tag 07XE]) is smooth, see Formal Deformation Theory, Definition [Tag 06HG].","statement_latex":"Let $S$ be a locally Noetherian scheme.\nLet $\\mathcal{X}$ be fibred in groupoids over $(\\Sch/S)_{fppf}$.\nLet $U$ be a scheme locally of finite type over $S$.\nLet $x$ be an object of $\\mathcal{X}$ lying over $U$.\nLet $u_0$ be a finite type point of $U$.\nWe say $x$ is {\\it versal} at $u_0$ if the morphism $\\hat x$\n(\\ref{equation-hat-x}) is smooth, see Formal Deformation Theory, Definition\n\\ref{formal-defos-definition-smooth-morphism}.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Versality","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XF","source_file":"artin.tex","source_line":1669,"source_end_line":1679,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1669-L1679","statement_sha256":"5f8e2cb980421f4ca924e16215cd51d213d788a6c589619b462617af6b536585","origin":"The Stacks Project","memory_eligible":false,"source_rank":13835,"rank":13835,"depth":1,"x":2039.291,"y":1678.101,"cluster":"algebraic-stacks"},{"id":"stacks:0CXK","tag":"0CXK","title":"Versality · Lemma 0CXK","summary":"With notation as in Definition [Tag 07XF]. Let R = O_U, u_0^wedge. Let xi be the formal object of X over R associated to x|_Spec(R), see ([Tag 07X6]). Then x is versal at u_0 ⇔ xi is versal","statement_latex":"With notation as in Definition \\ref{definition-versal}.\nLet $R = \\mathcal{O}_{U, u_0}^\\wedge$.\nLet $\\xi$ be the formal object of $\\mathcal{X}$\nover $R$ associated to $x|_{\\Spec(R)}$, see (\\ref{equation-approximation}).\nThen\n$$\nx\\text{ is versal at }u_0\n\\Leftrightarrow\n\\xi\\text{ is versal}\n$$","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXK","source_file":"artin.tex","source_line":1685,"source_end_line":1697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1685-L1697","statement_sha256":"cf0aa8f05304751beab8d9046fb5e152191788090ff4dd0d2a1b4350bc6d3386","origin":"The Stacks Project","memory_eligible":false,"source_rank":13836,"rank":13836,"depth":2,"x":1960.193,"y":1547.57,"cluster":"algebraic-stacks"},{"id":"stacks:0CXL","tag":"0CXL","title":"Versality · Lemma 0CXL","summary":"Let S be a locally Noetherian scheme. Let f : U → V be a morphism of schemes locally of finite type over S. Let u_0 ∈ U be a finite type point. The following are equivalent • f is smooth at u_0, • f viewed as an object of (Sch/V)_fppf over U is versal at u_0.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $f : U \\to V$\nbe a morphism of schemes locally of finite type over $S$.\nLet $u_0 \\in U$ be a finite type point. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is smooth at $u_0$,\n\\item $f$ viewed as an object of $(\\Sch/V)_{fppf}$ over $U$ is\nversal at $u_0$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXL","source_file":"artin.tex","source_line":1737,"source_end_line":1747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1737-L1747","statement_sha256":"4f47b69b867ab9e5e2266106ef88c16c8b92aaa46091d1ccc13766ab7e15ae4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13837,"rank":13837,"depth":8,"x":2123.837,"y":1599.049,"cluster":"algebraic-stacks"},{"id":"stacks:07XG","tag":"07XG","title":"Versality · Lemma 07XG","summary":"Let S, X, U, x, u_0 be as in Definition [Tag 07XF]. Let l be a field and let u_l, 0 : Spec(l) → U be a morphism with image u_0 such that l/k = kappa(u_0) is finite. Set x_l, 0 = x_0|_Spec(l). If X satisfies (RS) and x is versal at u_0, then F_(Sch/U)_fppf, l, u_l, 0 → F_X, l, x_l, 0 is smooth.","statement_latex":"Let $S$, $\\mathcal{X}$, $U$, $x$, $u_0$ be as in\nDefinition \\ref{definition-versal}. Let $l$ be a field and let\n$u_{l, 0} : \\Spec(l) \\to U$ be a morphism with image $u_0$ such that\n$l/k = \\kappa(u_0)$ is finite. Set $x_{l, 0} = x_0|_{\\Spec(l)}$.\nIf $\\mathcal{X}$ satisfies (RS) and $x$ is versal at $u_0$, then\n$$\n\\mathcal{F}_{(\\Sch/U)_{fppf}, l, u_{l, 0}}\n\\longrightarrow\n\\mathcal{F}_{\\mathcal{X}, l, x_{l, 0}}\n$$\nis smooth.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XG","source_file":"artin.tex","source_line":1758,"source_end_line":1771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1758-L1771","statement_sha256":"25540a2c0269cede7f3157b687a43958d6dd0b548a822d7af8fdfa7a0269af43","origin":"The Stacks Project","memory_eligible":false,"source_rank":13838,"rank":13838,"depth":76,"x":1961.426,"y":1654.06,"cluster":"algebraic-stacks"},{"id":"stacks:0CXM","tag":"0CXM","title":"Versality · Lemma 0CXM","summary":"Let S, X, U, x, u_0 be as in Definition [Tag 07XF]. Assume • Δ : X → X × X is representable by algebraic spaces, • Δ is locally of finite type (for example if X is limit preserving), and • X has (RS). Let V be a scheme locally of finite type over S and let y be an object of X over V. Form the 2-fibre product xymatrix Z ar[r] ar[d] & (Sch/U)_fppf ar[d]^x (Sch/V)_fppf ar[r]^y & X Let Z be the algebraic space representing Z and let z_0 ∈ |Z| be a finite type point lying over…","statement_latex":"Let $S$, $\\mathcal{X}$, $U$, $x$, $u_0$ be as in\nDefinition \\ref{definition-versal}. Assume\n\\begin{enumerate}\n\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nis representable by algebraic spaces,\n\\item $\\Delta$ is locally of finite type\n(for example if $\\mathcal{X}$ is limit preserving), and\n\\item $\\mathcal{X}$ has (RS).\n\\end{enumerate}\nLet $V$ be a scheme locally of finite type over $S$\nand let $y$ be an object of $\\mathcal{X}$ over $V$.\nForm the $2$-fibre product\n$$\n\\xymatrix{\n\\mathcal{Z} \\ar[r] \\ar[d] & (\\Sch/U)_{fppf} \\ar[d]^x \\\\\n(\\Sch/V)_{fppf} \\ar[r]^y & \\mathcal{X}\n}\n$$\nLet $Z$ be the algebraic space representing $\\mathcal{Z}$\nand let $z_0 \\in |Z|$ be a finite type point lying over $u_0$.\nIf $x$ is versal at $u_0$, then\nthe morphism $Z \\to V$ is smooth at $z_0$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXM","source_file":"artin.tex","source_line":1795,"source_end_line":1819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1795-L1819","statement_sha256":"07eee0501a227dc6262a5ed99fbd187fd3f709edfbaf151750b156e570299ffb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13839,"rank":13839,"depth":77,"x":2037.107,"y":1521.06,"cluster":"algebraic-stacks"},{"id":"stacks:07XH","tag":"07XH","title":"Versality · Lemma 07XH","summary":"Let S be a locally Noetherian scheme. Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let xi = (R, xi_n, f_n) be a formal object of X with xi_1 lying over Spec(k) → S with image s ∈ S. Assume • xi is versal, • xi is effective, • O_S, s is a G-ring, and • p : X → (Sch/S)_fppf is limit preserving on objects. Then there exist a morphism of finite type U → S, a finite type point u_0 ∈ U with residue field k, and an object x of X over U such that x is versal at u_0…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nLet $\\xi = (R, \\xi_n, f_n)$ be a formal object of $\\mathcal{X}$ with\n$\\xi_1$ lying over $\\Spec(k) \\to S$ with image $s \\in S$. Assume\n\\begin{enumerate}\n\\item $\\xi$ is versal,\n\\item $\\xi$ is effective,\n\\item $\\mathcal{O}_{S, s}$ is a G-ring, and\n\\item $p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects.\n\\end{enumerate}\nThen there exist a morphism of finite type $U \\to S$, a finite type\npoint $u_0 \\in U$ with residue field $k$, and an object $x$ of $\\mathcal{X}$\nover $U$ such that $x$ is versal at $u_0$ and such that\n$x|_{\\Spec(\\mathcal{O}_{U, u_0}/\\mathfrak m_{u_0}^n)} \\cong \\xi_n$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XH","source_file":"artin.tex","source_line":1874,"source_end_line":1890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1874-L1890","statement_sha256":"1a24a3260246fe12a4a522558764d28248a5c4f587f7c7a358a13f6b79ab99c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13840,"rank":13840,"depth":55,"x":2088.363,"y":1662.372,"cluster":"algebraic-stacks"},{"id":"stacks:07XQ","tag":"07XQ","title":"Openness of versality · Definition 07XQ","summary":"Let S be a locally Noetherian scheme. • Let X be a category fibred in groupoids over (Sch/S)_fppf. We say X satisfies openness of versality if given a scheme U locally of finite type over S, an object x of X over U, and a finite type point u_0 ∈ U such that x is versal at u_0, then there exists an open neighbourhood u_0 ∈ U' ⊂ U such that x is versal at every finite type point of U'. • Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. We…","statement_latex":"Let $S$ be a locally Noetherian scheme.\n\\begin{enumerate}\n\\item Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{X}$ satisfies\n{\\it openness of versality} if given a scheme $U$ locally of finite type\nover $S$, an object $x$ of $\\mathcal{X}$ over $U$, and a finite type point\n$u_0 \\in U$ such that $x$ is versal at $u_0$, then there exists an open\nneighbourhood $u_0 \\in U' \\subset U$ such that $x$ is versal at every finite\ntype point of $U'$.\n\\item Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. We say $f$ satisfies\n{\\it openness of versality} if given a scheme $U$ locally of finite type\nover $S$, an object $y$ of $\\mathcal{Y}$ over $U$, openness\nof versality holds for\n$(\\Sch/U)_{fppf} \\times_\\mathcal{Y} \\mathcal{X}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Openness of versality","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XQ","source_file":"artin.tex","source_line":1996,"source_end_line":2014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L1996-L2014","statement_sha256":"ffa668402edef52865fef2bab09dfc2602d0b354f78649dd122dcaf1d64d25eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13841,"rank":13841,"depth":0,"x":1936.611,"y":1587.097,"cluster":"algebraic-stacks"},{"id":"stacks:07XT","tag":"07XT","title":"Openness of versality · Lemma 07XT","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. Let U be a scheme locally of finite type over S. Let x be an object of X over U. Assume that x is versal at every finite type point of U and that X satisfies (RS). Then x : (Sch/U)_fppf → X satisfies ([Tag 07XS]).","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Let $U$ be a scheme locally\nof finite type over $S$. Let $x$ be an object of $\\mathcal{X}$ over $U$.\nAssume that $x$ is versal at every finite type point of $U$ and that\n$\\mathcal{X}$ satisfies (RS). Then $x : (\\Sch/U)_{fppf} \\to \\mathcal{X}$\nsatisfies (\\ref{equation-smooth}).","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Openness of versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XT","source_file":"artin.tex","source_line":2063,"source_end_line":2071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2063-L2071","statement_sha256":"f99957c45603b43120d3bfafc4bfc1c74e1d1d7de161a9695d592b53b5ce1a9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13842,"rank":13842,"depth":77,"x":2109.404,"y":1556.432,"cluster":"algebraic-stacks"},{"id":"stacks:07XU","tag":"07XU","title":"Openness of versality · Lemma 07XU","summary":"Let S be a locally Noetherian scheme. Let f : X → Y and g : Y → Z be composable 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If f and g satisfy ([Tag 07XS]) so does g ∘ f.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$\nand $g : \\mathcal{Y} \\to \\mathcal{Z}$ be composable $1$-morphisms of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$ and $g$\nsatisfy (\\ref{equation-smooth}) so does $g \\circ f$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Openness of versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XU","source_file":"artin.tex","source_line":2084,"source_end_line":2090,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2084-L2090","statement_sha256":"9b896152363801eefb6db97bce9cb2d6d5950f1910a0d2558dc45c15d6db6572","origin":"The Stacks Project","memory_eligible":false,"source_rank":13843,"rank":13843,"depth":1,"x":2006.435,"y":1677.342,"cluster":"algebraic-stacks"},{"id":"stacks:07XV","tag":"07XV","title":"Openness of versality · Lemma 07XV","summary":"Let S be a locally Noetherian scheme. Let f : X → Y and Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If f satisfies ([Tag 07XS]) so does the projection X ×_Y Z → Z.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$\nand $\\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of\ncategories fibred in groupoids over $(\\Sch/S)_{fppf}$. If $f$\nsatisfies (\\ref{equation-smooth}) so does the projection\n$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Openness of versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XV","source_file":"artin.tex","source_line":2097,"source_end_line":2104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2097-L2104","statement_sha256":"ba36f58d610b2704d8b1fa93088a35c6619dcb28401cd51020c5a340e836edb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13844,"rank":13844,"depth":1,"x":1985.087,"y":1529.453,"cluster":"algebraic-stacks"},{"id":"stacks:07XW","tag":"07XW","title":"Openness of versality · Lemma 07XW","summary":"Let S be a locally Noetherian scheme. Let f : X → Y be a 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If f is formally smooth on objects, then f satisfies ([Tag 07XS]). If f is representable by algebraic spaces and smooth, then f satisfies ([Tag 07XS]).","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $f$ is formally smooth on objects, then $f$ satisfies\n(\\ref{equation-smooth}). If $f$ is representable by algebraic spaces\nand smooth, then $f$ satisfies (\\ref{equation-smooth}).","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Openness of versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XW","source_file":"artin.tex","source_line":2114,"source_end_line":2121,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2114-L2121","statement_sha256":"f783a64c82cdac3d48d8589c59458eb7479f24e1c61201ca75d4115d1615a7d0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13845,"rank":13845,"depth":60,"x":2120.035,"y":1626.59,"cluster":"algebraic-stacks"},{"id":"stacks:07XX","tag":"07XX","title":"Openness of versality · Lemma 07XX","summary":"Let S be a locally Noetherian scheme. Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Assume • f is representable by algebraic spaces, • f satisfies ([Tag 07XS]), • X → (Sch/S)_fppf is limit preserving on objects, and • Y is limit preserving. Then f is smooth.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$\nbe a $1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nAssume\n\\begin{enumerate}\n\\item $f$ is representable by algebraic spaces,\n\\item $f$ satisfies (\\ref{equation-smooth}),\n\\item $\\mathcal{X} \\to (\\Sch/S)_{fppf}$ is limit preserving on objects, and\n\\item $\\mathcal{Y}$ is limit preserving.\n\\end{enumerate}\nThen $f$ is smooth.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Openness of versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XX","source_file":"artin.tex","source_line":2127,"source_end_line":2139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2127-L2139","statement_sha256":"9194e435e6643332c4bdc70ce3c0a421848e1f12043cb5bfa0ef8553189cca28","origin":"The Stacks Project","memory_eligible":false,"source_rank":13846,"rank":13846,"depth":69,"x":1942.047,"y":1631.545,"cluster":"algebraic-stacks"},{"id":"stacks:07XY","tag":"07XY","title":"Openness of versality · Lemma 07XY","summary":"Let S be a locally Noetherian scheme. Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Let k be a finite type field over S and let x_0 be an object of X over Spec(k) with image s ∈ S. Assume • Δ : X → X × X is representable by algebraic spaces, • X satisfies axioms [1], [2], [3] (see Section [Tag 07XJ]), • every formal object of X is effective, • openness of versality holds for X, and • O_S, s is a G-ring. Then there exist a morphism of finite type U → S and an…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nLet $k$ be a finite type field over $S$ and let $x_0$ be an object of\n$\\mathcal{X}$ over $\\Spec(k)$ with image $s \\in S$. Assume\n\\begin{enumerate}\n\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is\nrepresentable by algebraic spaces,\n\\item $\\mathcal{X}$ satisfies axioms [1], [2], [3] (see\nSection \\ref{section-axioms}),\n\\item every formal object of $\\mathcal{X}$ is effective,\n\\item openness of versality holds for $\\mathcal{X}$, and\n\\item $\\mathcal{O}_{S, s}$ is a G-ring.\n\\end{enumerate}\nThen there exist a morphism of finite type $U \\to S$ and an object\n$x$ of $\\mathcal{X}$ over $U$ such that\n$$\nx : (\\Sch/U)_{fppf} \\longrightarrow \\mathcal{X}\n$$\nis smooth and such that there exists a finite type point $u_0 \\in U$\nwhose residue field is $k$ and such that $x|_{u_0} \\cong x_0$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Openness of versality","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07XY","source_file":"artin.tex","source_line":2190,"source_end_line":2212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2190-L2212","statement_sha256":"bf06dfcb0578d2f60d38abaf93f9a80a77028850b02aac7c1e343a08da7b46af","origin":"The Stacks Project","memory_eligible":false,"source_rank":13847,"rank":13847,"depth":78,"x":2069.568,"y":1526.683,"cluster":"algebraic-stacks"},{"id":"stacks:07Y1","tag":"07Y1","title":"Algebraic spaces · Proposition 07Y1","summary":"Let S be a locally Noetherian scheme. Let F : (Sch/S)_fppf^opp → Sets be a functor. Assume that • Δ : F → F × F is representable by algebraic spaces, • F satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see Section [Tag 07XZ]), and • O_S, s is a G-ring for all finite type points s of S. Then F is an algebraic space.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$F : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$ be a functor. Assume that\n\\begin{enumerate}\n\\item $\\Delta : F \\to F \\times F$ is representable by algebraic spaces,\n\\item $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5]\n(see Section \\ref{section-axioms-functors}), and\n\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.\n\\end{enumerate}\nThen $F$ is an algebraic space.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Algebraic spaces","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Y1","source_file":"artin.tex","source_line":2423,"source_end_line":2434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2423-L2434","statement_sha256":"ed00ad2207097324d3cf86ad0f7f49bbe45f3273760d0554b2ffc034430d1cec","origin":"The Stacks Project","memory_eligible":false,"source_rank":13848,"rank":13848,"depth":79,"x":2059.843,"y":1676.67,"cluster":"algebraic-stacks"},{"id":"stacks:07Y2","tag":"07Y2","title":"Algebraic spaces · Lemma 07Y2","summary":"Let S be a locally Noetherian scheme. Let a : F → G be a transformation of functors (Sch/S)_fppf^opp → Sets. Assume that • a is injective, • F satisfies axioms [0], [1], [2], [4], and [5], • O_S, s is a G-ring for all finite type points s of S, • G is an algebraic space locally of finite type over S, Then F is an algebraic space.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $a : F \\to G$ be a transformation\nof functors $(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$.\nAssume that\n\\begin{enumerate}\n\\item $a$ is injective,\n\\item $F$ satisfies axioms [0], [1], [2], [4], and [5],\n\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$,\n\\item $G$ is an algebraic space locally of finite type over $S$,\n\\end{enumerate}\nThen $F$ is an algebraic space.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Y2","source_file":"artin.tex","source_line":2466,"source_end_line":2478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2466-L2478","statement_sha256":"6fec67478f6e48fbd79e2c67e22bedce281cecef6c4e2ff13d11a48fbe68f1d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13849,"rank":13849,"depth":80,"x":1946.17,"y":1560.318,"cluster":"algebraic-stacks"},{"id":"stacks:07Y4","tag":"07Y4","title":"Algebraic stacks · Lemma 07Y4","summary":"Let S be a locally Noetherian scheme. Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Assume that • Δ : X → X × X is representable by algebraic spaces, • X satisfies axioms [-1], [0], [1], [2], [3] (see Section [Tag 07XJ]), • every formal object of X is effective, • X satisfies openness of versality, and • O_S, s is a G-ring for all finite type points s of S. Then X is an algebraic stack.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nAssume that\n\\begin{enumerate}\n\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nis representable by algebraic spaces,\n\\item $\\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3] (see\nSection \\ref{section-axioms}),\n\\item every formal object of $\\mathcal{X}$ is effective,\n\\item $\\mathcal{X}$ satisfies openness of versality, and\n\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.\n\\end{enumerate}\nThen $\\mathcal{X}$ is an algebraic stack.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Y4","source_file":"artin.tex","source_line":2511,"source_end_line":2526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2511-L2526","statement_sha256":"e3a0cfa74f26d8164c2a2cfb8e7b0f67e0656fc58ba751d1a177f1f95009fb60","origin":"The Stacks Project","memory_eligible":false,"source_rank":13850,"rank":13850,"depth":79,"x":2123.915,"y":1581.657,"cluster":"algebraic-stacks"},{"id":"stacks:07Y5","tag":"07Y5","title":"Algebraic stacks · Proposition 07Y5","summary":"Let S be a locally Noetherian scheme. Let p : X → (Sch/S)_fppf be a category fibred in groupoids. Assume that • Δ_Δ : X → X ×_X × X X is representable by algebraic spaces, • X satisfies axioms [-1], [0], [1], [2], [3], [4], and [5] (see Section [Tag 07XJ]), • O_S, s is a G-ring for all finite type points s of S. Then X is an algebraic stack.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$ be a category fibred in groupoids.\nAssume that\n\\begin{enumerate}\n\\item $\\Delta_\\Delta : \\mathcal{X} \\to\n\\mathcal{X} \\times_{\\mathcal{X} \\times \\mathcal{X}} \\mathcal{X}$\nis representable by algebraic spaces,\n\\item $\\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3], [4], and [5]\n(see Section \\ref{section-axioms}),\n\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.\n\\end{enumerate}\nThen $\\mathcal{X}$ is an algebraic stack.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Algebraic stacks","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Y5","source_file":"artin.tex","source_line":2565,"source_end_line":2579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2565-L2579","statement_sha256":"eba093e1b7387208c60a3eb6bc02ea2382fb4f2073d133e3afce6f1df7a0c4b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13851,"rank":13851,"depth":80,"x":1975.391,"y":1666.95,"cluster":"algebraic-stacks"},{"id":"stacks:07Y8","tag":"07Y8","title":"Strong Rim-Schlessinger · Definition 07Y8","summary":"Let S be a scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. We say X satisfies condition (RS*) if given a fibre product diagram xymatrix B' ar[r] & B A' = A ×_B B' ar[u] ar[r] & A ar[u] of S-algebras, with B' → B surjective with square zero kernel, the functor of fibre categories X_Spec(A') → X_Spec(A) ×_X_Spec(B) X_Spec(B') is an equivalence of categories.","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$. We say $\\mathcal{X}$\nsatisfies {\\it condition (RS*)} if given a fibre product diagram\n$$\n\\xymatrix{\nB' \\ar[r] & B \\\\\nA' = A \\times_B B' \\ar[u] \\ar[r] & A \\ar[u]\n}\n$$\nof $S$-algebras, with $B' \\to B$ surjective with square zero kernel,\nthe functor of fibre categories\n$$\n\\mathcal{X}_{\\Spec(A')}\n\\longrightarrow\n\\mathcal{X}_{\\Spec(A)} \\times_{\\mathcal{X}_{\\Spec(B)}} \\mathcal{X}_{\\Spec(B')}\n$$\nis an equivalence of categories.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Strong Rim-Schlessinger","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Y8","source_file":"artin.tex","source_line":2647,"source_end_line":2666,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2647-L2666","statement_sha256":"82769af0930da76b80351ff30d17ff78e885e59e18003b60e9d748d2c595f1af","origin":"The Stacks Project","memory_eligible":false,"source_rank":13852,"rank":13852,"depth":0,"x":2016.402,"y":1519.485,"cluster":"algebraic-stacks"},{"id":"stacks:0CXP","tag":"0CXP","title":"Strong Rim-Schlessinger · Lemma 0CXP","summary":"Let X be an algebraic stack over a base S. Then X satisfies (RS*).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over a base $S$.\nThen $\\mathcal{X}$ satisfies (RS*).","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Strong Rim-Schlessinger","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXP","source_file":"artin.tex","source_line":2693,"source_end_line":2697,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2693-L2697","statement_sha256":"6ce22c59887ea2b00761d2a82cc1c13f56ca75b9291692690010c33a1d386f57","origin":"The Stacks Project","memory_eligible":false,"source_rank":13853,"rank":13853,"depth":75,"x":2104.923,"y":1651.757,"cluster":"algebraic-stacks"},{"id":"stacks:0CXQ","tag":"0CXQ","title":"Strong Rim-Schlessinger · Lemma 0CXQ","summary":"Let S be a scheme. Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If X, Y, and Z satisfy (RS*), then so does X ×_Y Z.","statement_latex":"Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and\n$q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If $\\mathcal{X}$, $\\mathcal{Y}$,\nand $\\mathcal{Z}$ satisfy (RS*), then so\ndoes $\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Strong Rim-Schlessinger","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXQ","source_file":"artin.tex","source_line":2704,"source_end_line":2711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2704-L2711","statement_sha256":"20d592a9b6cf0ee578d8575e2dd042a5e48ecdb622e67f4d066f5166fa86140d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13854,"rank":13854,"depth":2,"x":1932.939,"y":1604.357,"cluster":"algebraic-stacks"},{"id":"stacks:0G2J","tag":"0G2J","title":"Versality and generalizations · Lemma 0G2J","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf having (RS*). Let x be an object of X over an affine scheme U of finite type over S. Let u ∈ U be a finite type point such that x is not versal at u. Then there exists a morphism x → y of X lying over U → T satisfying • the morphism U → T is a first order thickening, • we have a short exact sequence 0 → kappa(u) → O_T → O_U → 0 • there does not exist a pair (W, α) consisting of…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$ having (RS*).\nLet $x$ be an object of $\\mathcal{X}$ over an affine scheme $U$\nof finite type over $S$. Let $u \\in U$ be a finite type point such that\n$x$ is not versal at $u$. Then there exists a morphism $x \\to y$\nof $\\mathcal{X}$ lying over $U \\to T$ satisfying\n\\begin{enumerate}\n\\item the morphism $U \\to T$ is a first order thickening,\n\\item we have a short exact sequence\n$$\n0 \\to \\kappa(u) \\to \\mathcal{O}_T \\to \\mathcal{O}_U \\to 0\n$$\n\\item there does {\\bf not} exist a pair $(W, \\alpha)$\nconsisting of an open neighbourhood $W \\subset T$ of $u$\nand a morphism $\\beta : y|_W \\to x$ such that the composition\n$$\nx|_{U \\cap W} \\xrightarrow{\\text{restriction of }x \\to y}\ny|_W \\xrightarrow{\\beta} x\n$$\nis the canonical morphism $x|_{U \\cap W} \\to x$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Versality and generalizations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2J","source_file":"artin.tex","source_line":2733,"source_end_line":2756,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2733-L2756","statement_sha256":"080d8822b566c1cf88d54214c7124f847d317ca4445ef859f4d1f55daab064f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13855,"rank":13855,"depth":3,"x":2098.202,"y":1541.597,"cluster":"algebraic-stacks"},{"id":"stacks:0G2K","tag":"0G2K","title":"Versality and generalizations · Lemma 0G2K","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. Assume • Δ : X → X × X is representable by algebraic spaces, • X has (RS*), • X is limit preserving. Let x be an object of X over a scheme U of finite type over S. Let u leadsto u_0 be a specialization of finite type points of U such that x is versal at u_0. Then x is versal at u.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Assume\n\\begin{enumerate}\n\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is\nrepresentable by algebraic spaces,\n\\item $\\mathcal{X}$ has (RS*),\n\\item $\\mathcal{X}$ is limit preserving.\n\\end{enumerate}\nLet $x$ be an object of $\\mathcal{X}$ over a scheme $U$ of finite type over\n$S$. Let $u \\leadsto u_0$ be a specialization of finite type points of $U$\nsuch that $x$ is versal at $u_0$. Then $x$ is versal at $u$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Versality and generalizations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2K","source_file":"artin.tex","source_line":2858,"source_end_line":2871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2858-L2871","statement_sha256":"cc1ff5322c5d4643d09e8095ec5ef138c384eae5a36593f500cdb88a733183cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13856,"rank":13856,"depth":78,"x":2026.668,"y":1681.925,"cluster":"algebraic-stacks"},{"id":"stacks:0G2S","tag":"0G2S","title":"Strong formal effectiveness · Lemma 0G2S","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf having (RS*). Let x be an object of X over an affine scheme U of finite type over S. Let u_n ∈ U, n ≥ 1 be a finite type points such that (a) there are no specializations u_n leadsto u_m for n not = m, and (b) x is not versal at u_n for all n. Then there exist morphisms x → x_1 → x_2 → … in X lying over U → U_1 → U_2 → … over S such that • for each n the morphism U → U_n is a…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$ having (RS*).\nLet $x$ be an object of\n$\\mathcal{X}$ over an affine scheme $U$ of finite type over $S$.\nLet $u_n \\in U$, $n \\geq 1$ be a finite type points such that\n(a) there are no specializations $u_n \\leadsto u_m$ for $n \\not = m$, and\n(b) $x$ is not versal at $u_n$ for all $n$. Then there exist morphisms\n$$\nx \\to x_1 \\to x_2 \\to \\ldots\n\\quad\\text{in }\\mathcal{X}\\text{ lying over }\\quad\nU \\to U_1 \\to U_2 \\to \\ldots\n$$\nover $S$ such that\n\\begin{enumerate}\n\\item for each $n$ the morphism $U \\to U_n$ is a first order\nthickening,\n\\item for each $n$ we have a short exact sequence\n$$\n0 \\to \\kappa(u_n) \\to \\mathcal{O}_{U_n} \\to \\mathcal{O}_{U_{n - 1}} \\to 0\n$$\nwith $U_0 = U$ for $n = 1$,\n\\item for each $n$ there does {\\bf not} exist a pair $(W, \\alpha)$\nconsisting of an open neighbourhood $W \\subset U_n$ of $u_n$\nand a morphism $\\alpha : x_n|_W \\to x$\nsuch that the composition\n$$\nx|_{U \\cap W} \\xrightarrow{\\text{restriction of }x \\to x_n}\nx_n|_W \\xrightarrow{\\alpha} x\n$$\nis the canonical morphism $x|_{U \\cap W} \\to x$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Strong formal effectiveness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2S","source_file":"artin.tex","source_line":2958,"source_end_line":2991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L2958-L2991","statement_sha256":"123944b83d4a84d5e1b5efe889cbef66e57f22ecfaacd3583476656bff86a5f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13857,"rank":13857,"depth":43,"x":1966.451,"y":1537.577,"cluster":"algebraic-stacks"},{"id":"stacks:0CXU","tag":"0CXU","title":"Strong formal effectiveness · Lemma 0CXU","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. Assume • Δ : X → X × X is representable by algebraic spaces, • X has (RS*), • X is limit preserving, • systems (xi_n) as in Remark [Tag 0CXT] where Ker(R_m → R_n) is an ideal of square zero for all m ≥ n are effective. Then X satisfies openness of versality.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Assume\n\\begin{enumerate}\n\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is\nrepresentable by algebraic spaces,\n\\item $\\mathcal{X}$ has (RS*),\n\\item $\\mathcal{X}$ is limit preserving,\n\\item systems $(\\xi_n)$ as in Remark \\ref{remark-strong-effectiveness}\nwhere $\\Ker(R_m \\to R_n)$ is an ideal of square zero for all $m \\geq n$\nare effective.\n\\end{enumerate}\nThen $\\mathcal{X}$ satisfies openness of versality.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Strong formal effectiveness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXU","source_file":"artin.tex","source_line":3064,"source_end_line":3078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L3064-L3078","statement_sha256":"fcaf1b435be746a29a8d8764a648d15a66c6ebee56ec0a72afe31d446d3809d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13858,"rank":13858,"depth":79,"x":2127.248,"y":1609.99,"cluster":"algebraic-stacks"},{"id":"stacks:07Y7","tag":"07Y7","title":"Infinitesimal deformations · Lemma 07Y7","summary":"Let S be a scheme. Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. Let xymatrix B' ar[r] & B A' ar[u] ar[r] & A ar[u] be a commutative diagram of S-algebras. Let x be an object of X over Spec(A), let y be an object of Y over Spec(B), and let φ : f(x)|_Spec(B) → y be a morphism of Y over Spec(B). Then there is a canonical functor Lift(x, A') → Lift(y, B') of categories of lifts induced by f and φ. The construction is compatible with…","statement_latex":"Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism\nof categories fibred in groupoids over $(\\Sch/S)_{fppf}$. Let\n$$\n\\xymatrix{\nB' \\ar[r] & B \\\\\nA' \\ar[u] \\ar[r] & A \\ar[u]\n}\n$$\nbe a commutative diagram of $S$-algebras. Let $x$ be an object of $\\mathcal{X}$\nover $\\Spec(A)$, let $y$ be an object of $\\mathcal{Y}$ over $\\Spec(B)$,\nand let $\\phi : f(x)|_{\\Spec(B)} \\to y$ be a morphism of $\\mathcal{Y}$\nover $\\Spec(B)$. Then there is a canonical functor\n$$\n\\textit{Lift}(x, A') \\longrightarrow \\textit{Lift}(y, B')\n$$\nof categories of lifts induced by $f$ and $\\phi$. The construction is\ncompatible with compositions of $1$-morphisms of categories fibred in\ngroupoids in an obvious manner.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Infinitesimal deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Y7","source_file":"artin.tex","source_line":3263,"source_end_line":3283,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L3263-L3283","statement_sha256":"383af630693c37de8398dda31e0cfc3372cff072d1a7d6208ff1a0bfac07ee5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13859,"rank":13859,"depth":0,"x":1950.103,"y":1647.907,"cluster":"algebraic-stacks"},{"id":"stacks:07Y9","tag":"07Y9","title":"Infinitesimal deformations · Lemma 07Y9","summary":"Let S be a scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. Assume X satisfies condition (RS*). Let A be an S-algebra and let x be an object of X over Spec(A). • There exists an A-linear functor Inf_x : Mod_A → Mod_A such that given a deformation situation (x, A' → A) and a lift x' there is an isomorphism Inf_x(I) → Inf(x'/x) where I = Ker(A' → A). • There exists an A-linear functor T_x : Mod_A → Mod_A such that • given M in Mod_A there is a bijection…","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$ satisfies\ncondition (RS*). Let $A$ be an $S$-algebra and let $x$ be an object of\n$\\mathcal{X}$ over $\\Spec(A)$.\n\\begin{enumerate}\n\\item There exists an $A$-linear functor\n$\\text{Inf}_x : \\text{Mod}_A \\to \\text{Mod}_A$\nsuch that given a deformation situation $(x, A' \\to A)$ and a lift $x'$\nthere is an isomorphism $\\text{Inf}_x(I) \\to \\text{Inf}(x'/x)$ where\n$I = \\Ker(A' \\to A)$.\n\\item There exists an $A$-linear functor\n$T_x : \\text{Mod}_A \\to \\text{Mod}_A$\nsuch that\n\\begin{enumerate}\n\\item given $M$ in $\\text{Mod}_A$ there is a bijection\n$T_x(M) \\to \\text{Lift}(x, A[M])$,\n\\item given a deformation situation $(x, A' \\to A)$ there is an action\n$$\nT_x(I) \\times \\text{Lift}(x, A') \\to \\text{Lift}(x, A')\n$$\nwhere $I = \\Ker(A' \\to A)$. It is simply transitive if\n$\\text{Lift}(x, A') \\not = \\emptyset$.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Infinitesimal deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Y9","source_file":"artin.tex","source_line":3309,"source_end_line":3335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L3309-L3335","statement_sha256":"9ee960a5081d5b8bdbbe4437afa1f91ebcbbfc12d0858ad803f8944e294c7814","origin":"The Stacks Project","memory_eligible":false,"source_rank":13860,"rank":13860,"depth":1,"x":2050.429,"y":1519.182,"cluster":"algebraic-stacks"},{"id":"stacks:0DNN","tag":"0DNN","title":"Infinitesimal deformations · Lemma 0DNN","summary":"Let S be a scheme. Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. Assume X, Y, Z satisfy (RS*). Let A be an S-algebra and let w be an object of W = X ×_Y Z over A. Denote x, y, z the objects of X, Y, Z you get from w. For any A-module M there is a 6-term exact sequence xymatrix 0 ar[r] & Inf_w(M) ar[r] & Inf_x(M) ⊕ Inf_z(M) ar[r] & Inf_y(M) ar[lld] & T_w(M) ar[r] & T_x(M) ⊕ T_z(M) ar[r] & T_y(M) of A-modules.","statement_latex":"Let $S$ be a scheme. Let $p : \\mathcal{X} \\to \\mathcal{Y}$\nand $q : \\mathcal{Z} \\to \\mathcal{Y}$ be $1$-morphisms of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. Assume $\\mathcal{X}$,\n$\\mathcal{Y}$, $\\mathcal{Z}$ satisfy (RS*).\nLet $A$ be an $S$-algebra and let $w$ be an object of\n$\\mathcal{W} = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$ over $A$.\nDenote $x, y, z$ the objects of $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$\nyou get from $w$. For any $A$-module $M$ there is a $6$-term exact sequence\n$$\n\\xymatrix{\n0 \\ar[r] &\n\\text{Inf}_w(M) \\ar[r] &\n\\text{Inf}_x(M) \\oplus \\text{Inf}_z(M) \\ar[r] &\n\\text{Inf}_y(M) \\ar[lld] \\\\\n &\nT_w(M) \\ar[r] &\nT_x(M) \\oplus T_z(M) \\ar[r] &\nT_y(M)\n}\n$$\nof $A$-modules.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Infinitesimal deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNN","source_file":"artin.tex","source_line":3509,"source_end_line":3532,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L3509-L3532","statement_sha256":"d71919664411b904ae983a91dbf7575a0024675bef60bea4719fccd18204ee7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13861,"rank":13861,"depth":7,"x":2080.024,"y":1671.323,"cluster":"algebraic-stacks"},{"id":"stacks:07YG","tag":"07YG","title":"Obstruction theories · Definition 07YG","summary":"Let S be a locally Noetherian base. Let X be a category fibred in groupoids over (Sch/S)_fppf. An obstruction theory is given by the following data • for every S-algebra A such that Spec(A) → S maps into an affine open and every object x of X over Spec(A) an A-linear functor O_x : Mod_A → Mod_A of obstruction modules, • for (x, A) as in (1), a ring map A → B, M ∈ Mod_A, N ∈ Mod_B, and an A-linear map M → N an induced A-linear map O_x(M) → O_y(N) where y = x|_Spec(B), and…","statement_latex":"Let $S$ be a locally Noetherian base. Let $\\mathcal{X}$ be a category fibred\nin groupoids over $(\\Sch/S)_{fppf}$. An {\\it obstruction theory} is \ngiven by the following data\n\\begin{enumerate}\n\\item for every $S$-algebra $A$ such that $\\Spec(A) \\to S$\nmaps into an affine open and every object $x$ of $\\mathcal{X}$ over\n$\\Spec(A)$ an $A$-linear functor\n$$\n\\mathcal{O}_x : \\text{Mod}_A \\to \\text{Mod}_A\n$$\nof {\\it obstruction modules},\n\\item for $(x, A)$ as in (1), a ring map $A \\to B$,\n$M \\in \\text{Mod}_A$, $N \\in \\text{Mod}_B$, and an $A$-linear\nmap $M \\to N$ an induced $A$-linear map $\\mathcal{O}_x(M) \\to \\mathcal{O}_y(N)$\nwhere $y = x|_{\\Spec(B)}$, and\n\\item for every deformation situation $(x, A' \\to A)$ an\n{\\it obstruction} element\n$o_x(A') \\in \\mathcal{O}_x(I)$ where $I = \\Ker(A' \\to A)$.\n\\end{enumerate}\nThese data are subject to the following conditions\n\\begin{enumerate}\n\\item[(i)] the functoriality maps turn the obstruction modules into a functor\nfrom the category of triples $(x, A, M)$ to sets,\n\\item[(ii)] for every morphism of deformation situations\n$(y, B' \\to B) \\to (x, A' \\to A)$ the element $o_x(A')$ maps\nto $o_y(B')$, and\n\\item[(iii)] we have\n$$\n\\text{Lift}(x, A') \\not = \\emptyset\n\\Leftrightarrow\no_x(A') = 0\n$$\nfor every deformation situation $(x, A' \\to A)$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Obstruction theories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YG","source_file":"artin.tex","source_line":3668,"source_end_line":3704,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L3668-L3704","statement_sha256":"5ae127cdaac47d9397702c096d8fd57f365648a0941dadfae9f511a67a3e2bf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13862,"rank":13862,"depth":0,"x":1935.577,"y":1575.746,"cluster":"algebraic-stacks"},{"id":"stacks:0CYF","tag":"0CYF","title":"Obstruction theories · Lemma 0CYF","summary":"This is [Hall-coherent] Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf. Assume • Δ : X → X × X is representable by algebraic spaces, • X has (RS*), • X is limit preserving, • there exists an obstruction theory, • for an object x of X over Spec(A) and A-modules M_n, n ≥ 1 we have • T_x(∏ M_n) = ∏ T_x(M_n), • O_x(∏ M_n) → ∏ O_x(M_n) is injective. Then X satisfies openness of versality.","statement_latex":"\\begin{reference}\nThis is \\cite[Theorem 4.4]{Hall-coherent}\n\\end{reference}\nLet $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Assume\n\\begin{enumerate}\n\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is\nrepresentable by algebraic spaces,\n\\item $\\mathcal{X}$ has (RS*),\n\\item $\\mathcal{X}$ is limit preserving,\n\\item there exists an obstruction theory\\footnote{Analyzing the proof\nthe reader sees that in fact it suffices to check\nthe functoriality (ii) of obstruction classes in\nDefinition \\ref{definition-obstruction-theory}\nfor maps $(y, B' \\to B) \\to (x, A' \\to A)$\nwith $B = A$ and $y = x$.},\n\\item for an object $x$ of $\\mathcal{X}$ over $\\Spec(A)$\nand $A$-modules $M_n$, $n \\geq 1$ we have\n\\begin{enumerate}\n\\item $T_x(\\prod M_n) = \\prod T_x(M_n)$,\n\\item $\\mathcal{O}_x(\\prod M_n) \\to \\prod \\mathcal{O}_x(M_n)$\nis injective.\n\\end{enumerate}\n\\end{enumerate}\nThen $\\mathcal{X}$ satisfies openness of versality.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Obstruction theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CYF","source_file":"artin.tex","source_line":3731,"source_end_line":3758,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L3731-L3758","statement_sha256":"d40f4cbba4e22291140393d86062a151349c5b9987f4182c2dcdb8d1d9e72394","origin":"The Stacks Project","memory_eligible":false,"source_rank":13863,"rank":13863,"depth":80,"x":2119.301,"y":1564.238,"cluster":"algebraic-stacks"},{"id":"stacks:07YK","tag":"07YK","title":"Naive obstruction theories · Lemma 07YK","summary":"Let A → k be a ring map with k a field. Let E ∈ D^-(A). Then Ext^i_A(E, k) = Hom_k(H^-i(E ⊗^L k), k).","statement_latex":"Let $A \\to k$ be a ring map with $k$ a field. Let $E \\in D^-(A)$.\nThen $\\Ext^i_A(E, k) = \\Hom_k(H^{-i}(E \\otimes^\\mathbf{L} k), k)$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Naive obstruction theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YK","source_file":"artin.tex","source_line":3932,"source_end_line":3936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L3932-L3936","statement_sha256":"4154c94138af866f38551dcf7896f4f40e57ff3f37699baf7f658699d77b0c76","origin":"The Stacks Project","memory_eligible":false,"source_rank":13864,"rank":13864,"depth":0,"x":1992.839,"y":1677.188,"cluster":"algebraic-stacks"},{"id":"stacks:07YL","tag":"07YL","title":"Naive obstruction theories · Lemma 07YL","summary":"Let Lambda → A → k be finite type ring maps of Noetherian rings with k = kappa( p) for some prime p of A. Let xi : E → NL_A/Lambda be morphism of D^-(A) such that H^-1(xi ⊗^L k) is not surjective. Then there exists a surjection A' → A of Lambda-algebras such that • [(a)] I = Ker(A' → A) has square zero and is isomorphic to k as an A-module, • [(b)] Ω_A'/Lambda ⊗ k = Ω_A/Lambda ⊗ k, and • [(c)] E → NL_A/A' is zero.","statement_latex":"Let $\\Lambda \\to A \\to k$ be finite type ring maps of Noetherian rings with\n$k = \\kappa(\\mathfrak p)$ for some prime $\\mathfrak p$ of $A$. Let\n$\\xi : E \\to \\NL_{A/\\Lambda}$ be morphism of $D^{-}(A)$ such that\n$H^{-1}(\\xi \\otimes^{\\mathbf{L}} k)$ is not surjective.\nThen there exists a surjection $A' \\to A$ of $\\Lambda$-algebras\nsuch that\n\\begin{enumerate}\n\\item[(a)] $I = \\Ker(A' \\to A)$ has square zero and is isomorphic to $k$\nas an $A$-module,\n\\item[(b)] $\\Omega_{A'/\\Lambda} \\otimes k = \\Omega_{A/\\Lambda} \\otimes k$, and\n\\item[(c)] $E \\to \\NL_{A/A'}$ is zero.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Naive obstruction theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YL","source_file":"artin.tex","source_line":3943,"source_end_line":3957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L3943-L3957","statement_sha256":"d6a03d6dc733b940be649a3ec00df32f3163a9b51a9001ede69fd8485eaa186d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13865,"rank":13865,"depth":9,"x":1995.264,"y":1521.848,"cluster":"algebraic-stacks"},{"id":"stacks:07YM","tag":"07YM","title":"Naive obstruction theories · Lemma 07YM","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf satisfying (RS*). Let U = Spec(A) be an affine scheme of finite type over S which maps into an affine open Spec(Lambda). Let x be an object of X over U. Let xi : E → NL_A/Lambda be a morphism of D^-(A). Assume • [(i)] for every deformation situation (x, A' → A) we have: x lifts to Spec(A') if and only if E → NL_A/Lambda → NL_A/A' is zero, and • [(ii)] there is an isomorphism of…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$ satisfying (RS*).\nLet $U = \\Spec(A)$ be an\naffine scheme of finite type over $S$ which maps into an affine open\n$\\Spec(\\Lambda)$. Let $x$ be an object of $\\mathcal{X}$ over $U$.\nLet $\\xi : E \\to \\NL_{A/\\Lambda}$ be a morphism of $D^{-}(A)$. Assume\n\\begin{enumerate}\n\\item[(i)] for every deformation situation $(x, A' \\to A)$ we have:\n$x$ lifts to $\\Spec(A')$ if and only if\n$E \\to \\NL_{A/\\Lambda} \\to \\NL_{A/A'}$ is zero, and\n\\item[(ii)] there is an isomorphism of functors\n$T_x(-) \\to \\Ext^0_A(E, -)$\nsuch that $E \\to \\NL_{A/\\Lambda} \\to \\Omega^1_{A/\\Lambda}$\ncorresponds to the canonical element (see\nRemark \\ref{remark-canonical-element}).\n\\end{enumerate}\nLet $u_0 \\in U$ be a finite type point with residue field\n$k = \\kappa(u_0)$. Consider the following statements\n\\begin{enumerate}\n\\item $x$ is versal at $u_0$, and\n\\item $\\xi : E \\to \\NL_{A/\\Lambda}$ induces a surjection\n$H^{-1}(E \\otimes_A^{\\mathbf{L}} k) \\to\nH^{-1}(\\NL_{A/\\Lambda} \\otimes_A^{\\mathbf{L}} k)$\nand an injection\n$H^0(E \\otimes_A^{\\mathbf{L}} k) \\to\nH^0(\\NL_{A/\\Lambda} \\otimes_A^{\\mathbf{L}} k)$.\n\\end{enumerate}\nThen we always have (2) $\\Rightarrow$ (1) and we have (1) $\\Rightarrow$ (2)\nif $u_0$ is a closed point.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Naive obstruction theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YM","source_file":"artin.tex","source_line":4017,"source_end_line":4048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4017-L4048","statement_sha256":"2a660547ddfb4d7f682eb73c2e390aae1bc6a71c2d7f32cca9a8fc8a9764379b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13866,"rank":13866,"depth":48,"x":2118.629,"y":1637.989,"cluster":"algebraic-stacks"},{"id":"stacks:07YN","tag":"07YN","title":"Naive obstruction theories · Lemma 07YN","summary":"Let S be a locally Noetherian scheme. Let X be a category fibred in groupoids over (Sch/S)_fppf satisfying (RS*). Let U = Spec(A) be an affine scheme of finite type over S which maps into an affine open Spec(Lambda). Let x be an object of X over U. Let xi : E → NL_A/Lambda be a morphism of D^-(A). Assume • [(i)] for every deformation situation (x, A' → A) we have: x lifts to Spec(A') if and only if E → NL_A/Lambda → NL_A/A' is zero, • [(ii)] there is an isomorphism of…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $\\mathcal{X}$ be a category\nfibred in groupoids over $(\\Sch/S)_{fppf}$ satisfying (RS*).\nLet $U = \\Spec(A)$ be an affine scheme of finite type over $S$ which maps\ninto an affine open $\\Spec(\\Lambda)$. Let $x$ be an object of $\\mathcal{X}$\nover $U$. Let $\\xi : E \\to \\NL_{A/\\Lambda}$ be a morphism of $D^{-}(A)$.\nAssume\n\\begin{enumerate}\n\\item[(i)] for every deformation situation $(x, A' \\to A)$ we have:\n$x$ lifts to $\\Spec(A')$ if and only if\n$E \\to \\NL_{A/\\Lambda} \\to \\NL_{A/A'}$ is zero,\n\\item[(ii)] there is an isomorphism of functors\n$T_x(-) \\to \\Ext^0_A(E, -)$\nsuch that $E \\to \\NL_{A/\\Lambda} \\to \\Omega^1_{A/\\Lambda}$\ncorresponds to the canonical element (see\nRemark \\ref{remark-canonical-element}),\n\\item[(iii)] the cohomology groups of $E$ are finite $A$-modules.\n\\end{enumerate}\nIf $x$ is versal at a closed point $u_0 \\in U$,\nthen there exists an open neighbourhood $u_0 \\in U' \\subset U$\nsuch that $x$ is versal at every finite type point of $U'$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Naive obstruction theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YN","source_file":"artin.tex","source_line":4146,"source_end_line":4168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4146-L4168","statement_sha256":"ea57fdd0609f1ac9d55afb0b4fb337aa9f41d5873e3c00d5186b90d65844198e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13867,"rank":13867,"depth":49,"x":1933.916,"y":1622.319,"cluster":"algebraic-stacks"},{"id":"stacks:07YP","tag":"07YP","title":"Naive obstruction theories · Definition 07YP","summary":"Let S be a locally Noetherian base. Let X be a category fibred in groupoids over (Sch/S)_fppf. Assume that X satisfies (RS*). A naive obstruction theory is given by the following data • for every S-algebra A such that Spec(A) → S maps into an affine open Spec(Lambda) ⊂ S and every object x of X over Spec(A) we are given an object E_x ∈ D^-(A) and a map xi_x : E → NL_A/Lambda, • given (x, A) as in ([Tag 07YQ]) there are transformations of functors Inf_x( - ) →…","statement_latex":"Let $S$ be a locally Noetherian base. Let $\\mathcal{X}$ be a category fibred\nin groupoids over $(\\Sch/S)_{fppf}$. Assume that $\\mathcal{X}$\nsatisfies (RS*). A {\\it naive obstruction theory} is \ngiven by the following data\n\\begin{enumerate}\n\\item\n\nfor every $S$-algebra $A$ such that $\\Spec(A) \\to S$\nmaps into an affine open $\\Spec(\\Lambda) \\subset S$ and every object $x$\nof $\\mathcal{X}$ over $\\Spec(A)$ we are given an object $E_x \\in D^-(A)$\nand a map $\\xi_x : E \\to \\NL_{A/\\Lambda}$,\n\\item\n\ngiven $(x, A)$ as in (\\ref{item-map}) there are transformations of\nfunctors\n$$\n\\text{Inf}_x( - ) \\to \\Ext^{-1}_A(E_x, -)\n\\quad\\text{and}\\quad\nT_x(-) \\to \\Ext^0_A(E_x, -)\n$$\n\\item\n\nfor $(x, A)$ as in (\\ref{item-map}) and a ring map $A \\to B$\nsetting $y = x|_{\\Spec(B)}$ there is a functoriality map\n$E_x \\to E_y$ in $D(A)$.\n\\end{enumerate}\nThese data are subject to the following conditions\n\\begin{enumerate}\n\\item[(i)]\nin the situation of (\\ref{item-functoriality}) the diagram\n$$\n\\xymatrix{\nE_y \\ar[r]_{\\xi_y} & \\NL_{B/\\Lambda} \\\\\nE_x \\ar[u] \\ar[r]^{\\xi_x} & \\NL_{A/\\Lambda} \\ar[u]\n}\n$$\nis commutative in $D(A)$,\n\\item[(ii)]\ngiven $(x, A)$ as in (\\ref{item-map}) and $A \\to B \\to C$\nsetting $y = x|_{\\Spec(B)}$ and $z = x|_{\\Spec(C)}$ the\ncomposition of the functoriality maps $E_x \\to E_y$ and $E_y \\to E_z$ is\nthe functoriality map $E_x \\to E_z$,\n\\item[(iii)]\nthe maps of (\\ref{item-inf}) are isomorphisms\ncompatible with the functoriality\nmaps and the maps of Remark \\ref{remark-functoriality},\n\\item[(iv)]\nthe composition $E_x \\to \\NL_{A/\\Lambda} \\to \\Omega_{A/\\Lambda}$\ncorresponds to the canonical element of\n$T_x(\\Omega_{A/\\Lambda}) = \\Ext^0(E_x, \\Omega_{A/\\Lambda})$, see\nRemark \\ref{remark-canonical-element},\n\\item[(v)]\ngiven a deformation situation $(x, A' \\to A)$ with $I = \\Ker(A' \\to A)$\nthe composition $E_x \\to \\NL_{A/\\Lambda} \\to \\NL_{A/A'}$ is zero in\n$$\n\\Hom_A(E_x, \\NL_{A/\\Lambda}) = \\Ext^0_A(E_x, \\NL_{A/A'}) =\n\\Ext^1_A(E_x, I)\n$$\nif and only if $x$ lifts to $A'$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Naive obstruction theories","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YP","source_file":"artin.tex","source_line":4190,"source_end_line":4252,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4190-L4252","statement_sha256":"816a33a8ff81a699f6b586a9dcc639ab0d7d3791f68cdaeccf6f33166b768d7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13868,"rank":13868,"depth":0,"x":2083.001,"y":1528.889,"cluster":"algebraic-stacks"},{"id":"stacks:07YT","tag":"07YT","title":"Naive obstruction theories · Lemma 07YT","summary":"Let S and X be as in Definition [Tag 07YP] and let X be endowed with a naive obstruction theory. Let A → B and y → x be as in ([Tag 07YS]). Let k be a B-algebra which is a field. Then the functoriality map E_x → E_y induces bijections H^i(E_x ⊗_A^L k) → H^i(E_y ⊗_B^L k) for i = 0, 1.","statement_latex":"Let $S$ and $\\mathcal{X}$ be as in\nDefinition \\ref{definition-naive-obstruction-theory}\nand let $\\mathcal{X}$ be endowed with a naive obstruction theory.\nLet $A \\to B$ and $y \\to x$ be as in (\\ref{item-functoriality}).\nLet $k$ be a $B$-algebra which is a field. Then the functoriality\nmap $E_x \\to E_y$ induces bijections\n$$\nH^i(E_x \\otimes_A^{\\mathbf{L}} k) \\to H^i(E_y \\otimes_B^{\\mathbf{L}} k)\n$$\nfor $i = 0, 1$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Naive obstruction theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YT","source_file":"artin.tex","source_line":4259,"source_end_line":4271,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4259-L4271","statement_sha256":"6d27036caf7258de1b7550a1609be17aed385f6c47ac8e13c5b7e3cb1c7ce397","origin":"The Stacks Project","memory_eligible":false,"source_rank":13869,"rank":13869,"depth":1,"x":2048.137,"y":1682.664,"cluster":"algebraic-stacks"},{"id":"stacks:07YU","tag":"07YU","title":"Naive obstruction theories · Lemma 07YU","summary":"Let S be a locally Noetherian scheme. Let p : X → (Sch/S)_fppf^opp be a category fibred in groupoids. Assume that X satisfies (RS*) and that X has a naive obstruction theory. Then openness of versality holds for X provided the complexes E_x of Definition [Tag 07YP] have finitely generated cohomology groups for pairs (A, x) where A is of finite type over S.","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}^{opp}$ be a category fibred in groupoids.\nAssume that $\\mathcal{X}$ satisfies (RS*)\nand that $\\mathcal{X}$ has a naive obstruction theory.\nThen openness of versality holds for $\\mathcal{X}$ provided the\ncomplexes $E_x$ of Definition \\ref{definition-naive-obstruction-theory}\nhave finitely generated cohomology groups for pairs $(A, x)$ where\n$A$ is of finite type over $S$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Naive obstruction theories","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YU","source_file":"artin.tex","source_line":4289,"source_end_line":4299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4289-L4299","statement_sha256":"e430d33d20d944e92760badd6116d67c2e8fd320a8bb9e81737a5e112cb4e9d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13870,"rank":13870,"depth":50,"x":1950.001,"y":1549.242,"cluster":"algebraic-stacks"},{"id":"stacks:07YY","tag":"07YY","title":"A dual notion · Lemma 07YY","summary":"In Situation [Tag 07YX]. Assume furthermore that • [(iv)] given a short exact sequence of deformation situations as in Remark [Tag 07YE] and a lift x'_2 ∈ Lift(x, A_2') then o_x(A_3') ∈ H^2(K^bullet ⊗_A^L I_3) equals ∂theta where theta ∈ H^1(K^bullet ⊗_A^L I_1) is the element corresponding to x'_2|_Spec(A_1') via A_1' = A[I_1] and the given map T_x(-) → H^1(K^bullet ⊗_A^L -). In this case there exists an element xi ∈ H^1(K^bullet ⊗_A^L NL_A/Lambda) such that • for every…","statement_latex":"In Situation \\ref{situation-dual}. Assume furthermore that\n\\begin{enumerate}\n\\item[(iv)] given a short exact sequence of deformation situations\nas in Remark \\ref{remark-short-exact-sequence-thickenings} and\na lift $x'_2 \\in \\text{Lift}(x, A_2')$ then\n$o_x(A_3') \\in H^2(K^\\bullet \\otimes_A^\\mathbf{L} I_3)$\nequals $\\partial\\theta$ where\n$\\theta \\in H^1(K^\\bullet \\otimes_A^\\mathbf{L} I_1)$\nis the element corresponding to $x'_2|_{\\Spec(A_1')}$ via\n$A_1' = A[I_1]$ and the given map\n$T_x(-) \\to H^1(K^\\bullet \\otimes_A^\\mathbf{L} -)$.\n\\end{enumerate}\nIn this case there exists an element\n$\\xi \\in H^1(K^\\bullet \\otimes_A^\\mathbf{L} \\NL_{A/\\Lambda})$\nsuch that\n\\begin{enumerate}\n\\item for every deformation situation $(x, A' \\to A)$ we have\n$\\xi_{A'} = o_x(A')$, and\n\\item $\\xi_{can}$ matches the canonical element of\nRemark \\ref{remark-canonical-element} via the given transformation\n$T_x(-) \\to H^1(K^\\bullet \\otimes_A^\\mathbf{L} -)$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"A dual notion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YY","source_file":"artin.tex","source_line":4398,"source_end_line":4422,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4398-L4422","statement_sha256":"8dc008da43eda215ccb6f254f7062a0ed5b9747a8a5b21b85177f67871fc3d39","origin":"The Stacks Project","memory_eligible":false,"source_rank":13871,"rank":13871,"depth":1,"x":2129.993,"y":1592.02,"cluster":"algebraic-stacks"},{"id":"stacks:07YZ","tag":"07YZ","title":"A dual notion · Lemma 07YZ","summary":"In Situation [Tag 07YX] assume that (iv) of Lemma [Tag 07YY] holds and that K^bullet is a perfect object of D(A). In this case, if x is versal at a closed point u_0 ∈ U then there exists an open neighbourhood u_0 ∈ U' ⊂ U such that x is versal at every finite type point of U'.","statement_latex":"In Situation \\ref{situation-dual} assume that (iv) of\nLemma \\ref{lemma-dual-obstruction} holds and that $K^\\bullet$ is a\nperfect object of $D(A)$. In this case, if $x$ is versal at a closed\npoint $u_0 \\in U$ then there exists an open neighbourhood\n$u_0 \\in U' \\subset U$ such that $x$ is versal at every finite type\npoint of $U'$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"A dual notion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07YZ","source_file":"artin.tex","source_line":4497,"source_end_line":4505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4497-L4505","statement_sha256":"971a8995703dce53b06fc68f9f71d7b898f8679faa025c9d55fe3f1c86a3aa17","origin":"The Stacks Project","memory_eligible":false,"source_rank":13872,"rank":13872,"depth":50,"x":1962.559,"y":1662.738,"cluster":"algebraic-stacks"},{"id":"stacks:0GE3","tag":"0GE3","title":"Limit preserving functors on Noetherian schemes · Lemma 0GE3","summary":"Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Restricting along the inclusion functor (Noetherian/S)_τ → (Sch/S)_τ defines an equivalence of categories between • the category of limit preserving sheaves on (Sch/S)_τ and • the category of limit preserving sheaves on (Noetherian/S)_τ","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nRestricting along the inclusion functor\n$(\\textit{Noetherian}/S)_\\tau \\to (\\Sch/S)_\\tau$\ndefines an equivalence of categories between\n\\begin{enumerate}\n\\item the category of limit preserving sheaves on\n$(\\Sch/S)_\\tau$ and\n\\item the category of limit preserving sheaves on\n$(\\textit{Noetherian}/S)_\\tau$\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Limit preserving functors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GE3","source_file":"artin.tex","source_line":4612,"source_end_line":4624,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4612-L4624","statement_sha256":"2f125f5f822325720c26067a4a427b3a0fbca01faf31784d73f615848bc84a4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13873,"rank":13873,"depth":44,"x":2029.276,"y":1515.315,"cluster":"algebraic-stacks"},{"id":"stacks:0GE4","tag":"0GE4","title":"Limit preserving functors on Noetherian schemes · Lemma 0GE4","summary":"Let X be an object of (Noetherian/S)_τ. If the functor of points h_X : (Noetherian/S)_τ^opp → Sets is limit preserving, then X is locally of finite presentation over S.","statement_latex":"Let $X$ be an object of $(\\textit{Noetherian}/S)_\\tau$. If the functor\nof points $h_X : (\\textit{Noetherian}/S)_\\tau^{opp} \\to \\textit{Sets}$\nis limit preserving, then $X$ is locally of finite presentation over $S$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Limit preserving functors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GE4","source_file":"artin.tex","source_line":4640,"source_end_line":4645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4640-L4645","statement_sha256":"54cfd66e428301b20431c4fe59a908f3c36d5317dc3b4e6c81afe2a998370310","origin":"The Stacks Project","memory_eligible":false,"source_rank":13874,"rank":13874,"depth":5,"x":2098.761,"y":1662.148,"cluster":"algebraic-stacks"},{"id":"stacks:0GE5","tag":"0GE5","title":"Limit preserving functors on Noetherian schemes · Lemma 0GE5","summary":"Let τ ∈ (Zariski, etale, smooth, syntomic, fppf). Let F', G' : (Sch/S)_τ^opp → Sets be limit preserving and sheaves. Let a' : F' → G' be a transformation of functors. Denote a : F → G the restriction of a' : F' → G' to (Noetherian/S)_τ. The following are equivalent • a' is representable (as a transformation of functors, see Categories, Definition [Tag 001X]), and • for every object V of (Noetherian/S)_τ and every map V → G the fibre product F ×_G V : (Noetherian/S)_τ^opp…","statement_latex":"Let $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$.\nLet $F', G' : (\\Sch/S)_\\tau^{opp} \\to \\textit{Sets}$ be limit preserving\nand sheaves. Let $a' : F' \\to G'$ be a transformation of functors.\nDenote $a : F \\to G$ the restriction of $a' : F' \\to G'$ to\n$(\\textit{Noetherian}/S)_\\tau$. The following are equivalent\n\\begin{enumerate}\n\\item $a'$ is representable (as a transformation of functors, see\nCategories, Definition \\ref{categories-definition-representable-morphism}), and\n\\item for every object $V$ of $(\\textit{Noetherian}/S)_\\tau$\nand every map $V \\to G$ the fibre product\n$F \\times_G V : (\\textit{Noetherian}/S)_\\tau^{opp} \\to \\textit{Sets}$\nis a representable functor, and\n\\item same as in (2) but only for $V$ affine finite type over $S$\nmapping into an affine open of $S$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Limit preserving functors on Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GE5","source_file":"artin.tex","source_line":4669,"source_end_line":4686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4669-L4686","statement_sha256":"ab8dd11ab8ed2a1442b5c99f0e67e0b0e80801f0d8db29f3ad1576be3b40a1fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13875,"rank":13875,"depth":45,"x":1929.135,"y":1593.177,"cluster":"algebraic-stacks"},{"id":"stacks:0GE7","tag":"0GE7","title":"Algebraic spaces in the Noetherian setting · Proposition 0GE7","summary":"Let S be a locally Noetherian scheme. Let F : (Noetherian/S)_etale^opp → Sets be a functor. Assume that • Δ : F → F × F is representable (as a transformation of functors, see Categories, Definition [Tag 001X]), • F satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see above), and • O_S, s is a G-ring for all finite type points s of S. Then there exists a unique algebraic space F' : (Sch/S)_fppf^opp → Sets whose restriction to (Noetherian/S)_etale is F (see proof for…","statement_latex":"Let $S$ be a locally Noetherian scheme. Let\n$F : (\\textit{Noetherian}/S)_\\etale^{opp} \\to \\textit{Sets}$\nbe a functor. Assume that\n\\begin{enumerate}\n\\item $\\Delta : F \\to F \\times F$ is representable\n(as a transformation of functors, see\nCategories, Definition \\ref{categories-definition-representable-morphism}),\n\\item $F$ satisfies axioms [-1], [0], [1], [2], [3], [4], [5]\n(see above), and\n\\item $\\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.\n\\end{enumerate}\nThen there exists a unique algebraic space\n$F' : (\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}$\nwhose restriction to $(\\textit{Noetherian}/S)_\\etale$ is $F$\n(see proof for elucidation).","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Algebraic spaces in the Noetherian setting","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GE7","source_file":"artin.tex","source_line":4854,"source_end_line":4871,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L4854-L4871","statement_sha256":"8d565dd9dd9bb5154306b90ef4745dae6e5f4469899e3438020199f138a5f949","origin":"The Stacks Project","memory_eligible":false,"source_rank":13876,"rank":13876,"depth":80,"x":2110.009,"y":1547.703,"cluster":"algebraic-stacks"},{"id":"stacks:0GHA","tag":"0GHA","title":"Artin's theorem on contractions · Lemma 0GHA","summary":"In Situation [Tag 0GH8] the rule F that sends a locally Noetherian scheme V over S to the set of triples (Z, u', hat x) satisfying the compatibility condition and which sends a a morphism φ : V_2 → V_1 of locally Noetherian schemes over S to the map F(φ) : F(V_1) → F(V_2) sending an element (Z_1, u'_1, hat x_1) of F(V_1) to (Z_2, u'_2, hat x_2) in F(V_2) given by • Z_2 ⊂ V_2 is the inverse image of Z_1 by φ, • u'_2 is the composition of u'_1 and φ|_V_2 setminus Z_2 : V_2…","statement_latex":"In Situation \\ref{situation-contractions} the rule $F$ that sends\na locally Noetherian scheme $V$ over $S$ to the set of triples\n$(Z, u', \\hat x)$ satisfying the compatibility condition and which sends a\na morphism $\\varphi : V_2 \\to V_1$ of locally Noetherian schemes over $S$\nto the map\n$$\nF(\\varphi) : F(V_1) \\longrightarrow F(V_2)\n$$\nsending an element $(Z_1, u'_1, \\hat x_1)$ of $F(V_1)$ to\n$(Z_2, u'_2, \\hat x_2)$ in $F(V_2)$ given by\n\\begin{enumerate}\n\\item $Z_2 \\subset V_2$ is the inverse image of $Z_1$ by $\\varphi$,\n\\item $u'_2$ is the composition of $u'_1$ and\n$\\varphi|_{V_2 \\setminus Z_2} : V_2 \\setminus Z_2 \\to V_1 \\setminus Z_1$,\n\\item $\\hat x_2$ is the composition of $\\hat x_1$ and\n$\\varphi_{/Z_2} : V_{2, /Z_2} \\to V_{1, /Z_1}$\n\\end{enumerate}\nis a contravariant functor.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHA","source_file":"artin.tex","source_line":5081,"source_end_line":5101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L5081-L5101","statement_sha256":"2132f4b9d2aff29341333d884ffc426c32291e02e7b5749f25a3819c260a7210","origin":"The Stacks Project","memory_eligible":false,"source_rank":13877,"rank":13877,"depth":0,"x":2013.028,"y":1684.114,"cluster":"algebraic-stacks"},{"id":"stacks:0GHB","tag":"0GHB","title":"Artin's theorem on contractions · Lemma 0GHB","summary":"In Situation [Tag 0GH8] if there exists a solution (f : X' → X, T, a) then there is a functorial bijection F(V) = Mor_S(V, X) on the category of locally Noetherian schemes V over S.","statement_latex":"In Situation \\ref{situation-contractions} if there exists a solution\n$(f : X' \\to X, T, a)$ then there is a functorial bijection\n$F(V) = \\Mor_S(V, X)$ on the category of\nlocally Noetherian schemes $V$ over $S$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHB","source_file":"artin.tex","source_line":5114,"source_end_line":5120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L5114-L5120","statement_sha256":"6f3c073012e34395222490bc6b5b426e9db725234dc588893e987ef15d1e20a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13878,"rank":13878,"depth":73,"x":1974.773,"y":1528.216,"cluster":"algebraic-stacks"},{"id":"stacks:0GHC","tag":"0GHC","title":"Artin's theorem on contractions · Lemma 0GHC","summary":"In Situation [Tag 0GH8] if there exists an algebraic space X locally of finite type over S and a functorial bijection F(V) = Mor_S(V, X) on the category of locally Noetherian schemes V over S, then X is a solution.","statement_latex":"In Situation \\ref{situation-contractions} if there exists an\nalgebraic space $X$ locally of finite type over $S$ and a\nfunctorial bijection $F(V) = \\Mor_S(V, X)$ on the category of\nlocally Noetherian schemes $V$ over $S$, then $X$ is a solution.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHC","source_file":"artin.tex","source_line":5154,"source_end_line":5160,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L5154-L5160","statement_sha256":"26c5209a0b7c57458d4679a5864b68bd0e7df146f9a9045c4c8cac2a57454f0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13879,"rank":13879,"depth":45,"x":2128.627,"y":1621.634,"cluster":"algebraic-stacks"},{"id":"stacks:0GHE","tag":"0GHE","title":"Artin's theorem on contractions · Lemma 0GHE","summary":"In Situation [Tag 0GH8] assume given a closed subset Z ⊂ S such that • the inverse image of Z in X' is T', • U' → S setminus Z is a closed immersion, • W → S_/Z is a closed immersion. Then there exists a solution (f : X' → X, T, a) and moreover X → S is a closed immersion.","statement_latex":"In Situation \\ref{situation-contractions} assume given a closed\nsubset $Z \\subset S$ such that\n\\begin{enumerate}\n\\item the inverse image of $Z$ in $X'$ is $T'$,\n\\item $U' \\to S \\setminus Z$ is a closed immersion,\n\\item $W \\to S_{/Z}$ is a closed immersion.\n\\end{enumerate}\nThen there exists a solution $(f : X' \\to X, T, a)$\nand moreover $X \\to S$ is a closed immersion.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHE","source_file":"artin.tex","source_line":5299,"source_end_line":5310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L5299-L5310","statement_sha256":"f1b613ac1fa1e7118fac69079646e818873d905591e98cf741ba0eb36372124c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13880,"rank":13880,"depth":80,"x":1939.715,"y":1640.082,"cluster":"algebraic-stacks"},{"id":"stacks:0GHF","tag":"0GHF","title":"Artin's theorem on contractions · Lemma 0GHF","summary":"In Situation [Tag 0GH8] assume X' → S and W → S are separated. Then the diagonal Δ : F → F × F is representable by closed immersions.","statement_latex":"In Situation \\ref{situation-contractions} assume $X' \\to S$\nand $W \\to S$ are separated. Then the diagonal $\\Delta : F \\to F \\times F$\nis representable by closed immersions.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHF","source_file":"artin.tex","source_line":5395,"source_end_line":5400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L5395-L5400","statement_sha256":"02b3252d64ab8d54852a16e4bbd92192a765023361a7956ed5d330e9556898dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13881,"rank":13881,"depth":81,"x":2064.403,"y":1519.07,"cluster":"algebraic-stacks"},{"id":"stacks:0GHG","tag":"0GHG","title":"Artin's theorem on contractions · Lemma 0GHG","summary":"In Situation [Tag 0GH8] the functor F satisfies the sheaf property for all étale coverings of locally Noetherian schemes over S.","statement_latex":"In Situation \\ref{situation-contractions} the functor\n$F$ satisfies the sheaf property for all \\'etale coverings\nof locally Noetherian schemes over $S$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHG","source_file":"artin.tex","source_line":5407,"source_end_line":5412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L5407-L5412","statement_sha256":"5fe40da9203f8dd56e537665ef2f9a19eb50a044f8f7da2077db1ed34dcbfa23","origin":"The Stacks Project","memory_eligible":false,"source_rank":13882,"rank":13882,"depth":0,"x":2069.784,"y":1679.339,"cluster":"algebraic-stacks"},{"id":"stacks:0GI5","tag":"0GI5","title":"Artin's theorem on contractions · Lemma 0GI5","summary":"In Situation [Tag 0GH8] the functor F is limit preserving: for any directed limit V = lim V_λ of Noetherian affine schemes over S we have F(V) = colim F(V_λ).","statement_latex":"In Situation \\ref{situation-contractions} the functor $F$ is limit preserving:\nfor any directed limit $V = \\lim V_\\lambda$ of Noetherian affine schemes\nover $S$ we have $F(V) = \\colim F(V_\\lambda)$.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GI5","source_file":"artin.tex","source_line":5418,"source_end_line":5423,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L5418-L5423","statement_sha256":"6f3236eb4578bcb482c08af14aebc5d52085944397bb6be0b5bbb6b34115e839","origin":"The Stacks Project","memory_eligible":false,"source_rank":13883,"rank":13883,"depth":81,"x":1936.697,"y":1564.008,"cluster":"algebraic-stacks"},{"id":"stacks:0GI6","tag":"0GI6","title":"Artin's theorem on contractions · Lemma 0GI6","summary":"In Situation [Tag 0GH8] the functor F satisfies the Rim-Schlessinger condition (RS).","statement_latex":"In Situation \\ref{situation-contractions} the functor $F$ satisfies\nthe Rim-Schlessinger condition (RS).","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GI6","source_file":"artin.tex","source_line":5957,"source_end_line":5961,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L5957-L5961","statement_sha256":"f19275870e8a8c69fa054a5eef3c47493310a76fe2c962f44f713ba95dd1ae53","origin":"The Stacks Project","memory_eligible":false,"source_rank":13884,"rank":13884,"depth":76,"x":2127.917,"y":1573.551,"cluster":"algebraic-stacks"},{"id":"stacks:0GI7","tag":"0GI7","title":"Artin's theorem on contractions · Lemma 0GI7","summary":"In Situation [Tag 0GH8] the tangent spaces of the functor F are finite dimensional.","statement_latex":"In Situation \\ref{situation-contractions} the tangent spaces of\nthe functor $F$ are finite dimensional.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GI7","source_file":"artin.tex","source_line":5994,"source_end_line":5998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L5994-L5998","statement_sha256":"caa0dfe17f021875ddc30aae76d02c7e1909555533f39cdc5eface996dab3449","origin":"The Stacks Project","memory_eligible":false,"source_rank":13885,"rank":13885,"depth":77,"x":1978.977,"y":1675.196,"cluster":"algebraic-stacks"},{"id":"stacks:0GI8","tag":"0GI8","title":"Artin's theorem on contractions · Lemma 0GI8","summary":"In Situation [Tag 0GH8] assume X' → S is separated. Then every formal object for F is effective.","statement_latex":"In Situation \\ref{situation-contractions} assume $X' \\to S$ is separated.\nThen every formal object for $F$ is effective.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GI8","source_file":"artin.tex","source_line":6019,"source_end_line":6023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L6019-L6023","statement_sha256":"0ffff3efa7f83a5a21ff484c9ce515985519787b05a0f7de5b0cbb2e529aec66","origin":"The Stacks Project","memory_eligible":false,"source_rank":13886,"rank":13886,"depth":82,"x":2007.113,"y":1515.452,"cluster":"algebraic-stacks"},{"id":"stacks:0GI9","tag":"0GI9","title":"Artin's theorem on contractions · Lemma 0GI9","summary":"Let S be a locally Noetherian scheme. Let V be a scheme locally of finite type over S. Let Z ⊂ V be closed. Let W be a locally Noetherian formal algebraic space over S such that W_red is locally of finite type over S. Let g : V_/Z → W be an adic morphism of formal algebraic spaces over S. Let v ∈ V be a closed point such that g is versal at v (as in Section [Tag 07XZ]). Then after replacing V by an open neighbourhood of v the morphism g is smooth (see proof).","statement_latex":"Let $S$ be a locally Noetherian scheme. Let $V$ be a scheme locally\nof finite type over $S$. Let $Z \\subset V$ be closed. Let $W$ be\na locally Noetherian formal algebraic space over $S$ such that\n$W_{red}$ is locally of finite type over $S$. Let $g : V_{/Z} \\to W$\nbe an adic morphism of formal algebraic spaces over $S$. Let $v \\in V$\nbe a closed point such that $g$ is versal at $v$ (as in\nSection \\ref{section-axioms-functors}).\nThen after replacing $V$ by an open neighbourhood of $v$ the\nmorphism $g$ is smooth (see proof).","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GI9","source_file":"artin.tex","source_line":6112,"source_end_line":6123,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L6112-L6123","statement_sha256":"3dd66925fb8e6ff2e9906334d71bd9a44197b4dd5680c5e2074f58e12d324439","origin":"The Stacks Project","memory_eligible":false,"source_rank":13887,"rank":13887,"depth":78,"x":2115.016,"y":1649.444,"cluster":"algebraic-stacks"},{"id":"stacks:0GIA","tag":"0GIA","title":"Artin's theorem on contractions · Lemma 0GIA","summary":"In Situation [Tag 0GH8] the functor F satisfies openness of versality.","statement_latex":"In Situation \\ref{situation-contractions} the functor\n$F$ satisfies openness of versality.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIA","source_file":"artin.tex","source_line":6234,"source_end_line":6238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L6234-L6238","statement_sha256":"4fd42cde7511eb9708bae4649d873c42bdc56fccc2a63cc85c1e7589db064081","origin":"The Stacks Project","memory_eligible":false,"source_rank":13888,"rank":13888,"depth":79,"x":1927.371,"y":1611.801,"cluster":"algebraic-stacks"},{"id":"stacks:0GIB","tag":"0GIB","title":"Artin's theorem on contractions · Theorem 0GIB","summary":"[ArtinII] Let S be a locally Noetherian scheme such that O_S, s is a G-ring for all finite type points s ∈ S. Let X' be an algebraic space locally of finite type over S. Let T' ⊂ |X'| be a closed subset. Let W be a locally Noetherian formal algebraic space over S with W_red locally of finite type over S. Finally, we let g : X'_/T' → W be a formal modification, see Algebraization of Formal Spaces, Definition [Tag 0GDK]. If X' and W are separated over S, then there exists a…","statement_latex":"\\begin{reference}\n\\cite[Theorem 3.1]{ArtinII}\n\\end{reference}\nLet $S$ be a locally Noetherian scheme such that $\\mathcal{O}_{S, s}$\nis a G-ring for all finite type points $s \\in S$. Let $X'$ be an algebraic\nspace locally of finite type over $S$. Let $T' \\subset |X'|$ be a closed\nsubset. Let $W$ be a locally Noetherian formal algebraic space over $S$\nwith $W_{red}$ locally of finite type over $S$. Finally, we let\n$$\ng : X'_{/T'} \\longrightarrow W\n$$\nbe a formal modification, see Algebraization of Formal Spaces, Definition\n\\ref{restricted-definition-formal-modification}. If $X'$ and $W$ are\nseparated\\footnote{See Remark \\ref{remark-separated-needed}.} over $S$, then\nthere exists a proper morphism $f : X' \\to X$ of algebraic spaces over $S$,\na closed subset $T \\subset |X|$, and an isomorphism $a : X_{/T} \\to W$\nof formal algebraic spaces such that\n\\begin{enumerate}\n\\item $T'$ is the inverse image of $T$ by $|f| : |X'| \\to |X|$,\n\\item $f : X' \\to X$ maps $X' \\setminus T'$ isomorphically to\n$X \\setminus T$, and\n\\item $g = a \\circ f_{/T}$ where $f_{/T} : X'_{/T'} \\to X_{/T}$\nis the induced morphism.\n\\end{enumerate}\nIn other words, $(f : X' \\to X, T, a)$ is a solution as defined earlier in\nthis section.","area":"Algebraic Stacks","chapter":"Artin's Axioms","chapter_id":"artin","section":"Artin's theorem on contractions","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GIB","source_file":"artin.tex","source_line":6285,"source_end_line":6313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/artin.tex#L6285-L6313","statement_sha256":"ad9d20eaf81bfcbc8915d37139b8df92e4ce6f9b9c7a5d14ecc8a97acd377e2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13889,"rank":13889,"depth":83,"x":2096.301,"y":1532.948,"cluster":"algebraic-stacks"},{"id":"stacks:08JV","tag":"08JV","title":"The Hom functor · Lemma 08JV","summary":"In Situation [Tag 08JT] the functor mathitHom(F, G) satisfies the sheaf property for the fpqc topology.","statement_latex":"In Situation \\ref{situation-hom} the functor\n$\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ \nsatisfies the sheaf property for the fpqc topology.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hom functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JV","source_file":"quot.tex","source_line":219,"source_end_line":224,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L219-L224","statement_sha256":"0367b974b8851cd9058460c3ed35189fbacc38f9917ec78eb58913a8f59da93c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13890,"rank":13890,"depth":52,"x":2553.59,"y":1564.978,"cluster":"moduli-theory"},{"id":"stacks:0D3S","tag":"0D3S","title":"The Hom functor · Lemma 0D3S","summary":"In Situation [Tag 08JT]. Let T be an algebraic space over S. We have Mor_Sh((Sch/S)_fppf)(T, mathitHom(F, G)) = ((h, u) mid h : T → B, u : F_T → G_T) where F_T, G_T denote the pullbacks of F and G to the algebraic space X ×_B, h T.","statement_latex":"In Situation \\ref{situation-hom}. Let $T$ be an algebraic space over $S$.\nWe have\n$$\n\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\mathit{Hom}(\\mathcal{F}, \\mathcal{G})) =\n\\{(h, u) \\mid h : T \\to B, u : \\mathcal{F}_T \\to \\mathcal{G}_T\\}\n$$\nwhere $\\mathcal{F}_T, \\mathcal{G}_T$ denote the pullbacks of $\\mathcal{F}$\nand $\\mathcal{G}$ to the algebraic space $X \\times_{B, h} T$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hom functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3S","source_file":"quot.tex","source_line":244,"source_end_line":254,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L244-L254","statement_sha256":"176664a3b29871d12e08af303b2b24b4a2733480d132538454a13f1e03336b0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13891,"rank":13891,"depth":52,"x":2453.686,"y":1668.136,"cluster":"moduli-theory"},{"id":"stacks:08K3","tag":"08K3","title":"The Hom functor · Lemma 08K3","summary":"In Situation [Tag 08JT] let (X_i → X)_i ∈ I be an fppf covering and for each i, j ∈ I let (X_ijk → X_i ×_X X_j) be an fppf covering. Denote F_i, resp. F_ijk the pullback of F to X_i, resp. X_ijk. Similarly define G_i and G_ijk. For every scheme T over B the diagram xymatrix mathitHom(F, G)(T) ar[r] & ∏_i mathitHom(F_i, G_i)(T) ar@<1ex>[r]^-pr_0^* ar@<-1ex>[r]_-pr_1^* & ∏_i, j, k mathitHom(F_ijk, G_ijk)(T) presents the first arrow as the equalizer of the other two.","statement_latex":"In Situation \\ref{situation-hom} let $\\{X_i \\to X\\}_{i \\in I}$ be an fppf\ncovering and for each $i, j \\in I$ let $\\{X_{ijk} \\to X_i \\times_X X_j\\}$\nbe an fppf covering. Denote $\\mathcal{F}_i$, resp.\\ $\\mathcal{F}_{ijk}$\nthe pullback of $\\mathcal{F}$ to $X_i$, resp.\\ $X_{ijk}$. Similarly\ndefine $\\mathcal{G}_i$ and $\\mathcal{G}_{ijk}$. For every scheme\n$T$ over $B$ the diagram\n$$\n\\xymatrix{\n\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})(T) \\ar[r] &\n\\prod\\nolimits_i\n\\mathit{Hom}(\\mathcal{F}_i, \\mathcal{G}_i)(T)\n\\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*}\n&\n\\prod\\nolimits_{i, j, k}\n\\mathit{Hom}(\\mathcal{F}_{ijk}, \\mathcal{G}_{ijk})(T)\n}\n$$\npresents the first arrow as the equalizer of the other two.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hom functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08K3","source_file":"quot.tex","source_line":300,"source_end_line":320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L300-L320","statement_sha256":"7bd68d8c3b645e84533b86972639f914eaeebc73bdadd55754a5902236c9540b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13892,"rank":13892,"depth":52,"x":2444.325,"y":1534.3,"cluster":"moduli-theory"},{"id":"stacks:08K4","tag":"08K4","title":"The Hom functor · Lemma 08K4","summary":"In Situation [Tag 08JT]. If F is of finite presentation and f is quasi-compact and quasi-separated, then mathitHom(F, G) is limit preserving.","statement_latex":"In Situation \\ref{situation-hom}. If $\\mathcal{F}$ is of finite presentation\nand $f$ is quasi-compact and quasi-separated, then\n$\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is limit preserving.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hom functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08K4","source_file":"quot.tex","source_line":337,"source_end_line":342,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L337-L342","statement_sha256":"efb29ad684c23ff5d645a419fa128c869930676f3a5ea4570c44955c9b233cc6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13893,"rank":13893,"depth":53,"x":2559.769,"y":1628.427,"cluster":"moduli-theory"},{"id":"stacks:08K5","tag":"08K5","title":"The Hom functor · Lemma 08K5","summary":"Let S be a scheme. Let B be an algebraic space over S. Let i : X' → X be a closed immersion of algebraic spaces over B. Let F be a quasi-coherent O_X-module and let G' be a quasi-coherent O_X'-module. Then mathitHom(F, i_*G') = mathitHom(i^*F, G') as functors on (Sch/B).","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $i : X' \\to X$ be a closed immersion of algebraic spaces\nover $B$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module\nand let $\\mathcal{G}'$ be a quasi-coherent $\\mathcal{O}_{X'}$-module.\nThen\n$$\n\\mathit{Hom}(\\mathcal{F}, i_*\\mathcal{G}') =\n\\mathit{Hom}(i^*\\mathcal{F}, \\mathcal{G}')\n$$\nas functors on $(\\Sch/B)$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hom functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08K5","source_file":"quot.tex","source_line":390,"source_end_line":402,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L390-L402","statement_sha256":"d213736e504fb29a93053fa9c81335c4781a15a6ef4daa9051f95bc1362f9697","origin":"The Stacks Project","memory_eligible":false,"source_rank":13894,"rank":13894,"depth":27,"x":2397.68,"y":1624.491,"cluster":"moduli-theory"},{"id":"stacks:08JX","tag":"08JX","title":"The Hom functor · Lemma 08JX","summary":"Let S be a scheme. Let B be an algebraic space over S. Let K be a pseudo-coherent object of D(O_B). • If for all g : T → B in (Sch/B) the cohomology sheaf H^-1(Lg^*K) is zero, then the functor (Sch/B)^opp → Sets, (g : T → B) ↦ H^0(T, H^0(Lg^*K)) is an algebraic space affine and of finite presentation over B. • If for all g : T → B in (Sch/B) the cohomology sheaves H^i(Lg^*K) are zero for i < 0, then K is perfect, K locally has tor amplitude in [0, b], and the functor…","statement_latex":"Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.\nLet $K$ be a pseudo-coherent object of $D(\\mathcal{O}_B)$.\n\\begin{enumerate}\n\\item If for all $g : T \\to B$ in $(\\Sch/B)$ the cohomology sheaf\n$H^{-1}(Lg^*K)$ is zero, then the functor\n$$\n(\\Sch/B)^{opp} \\longrightarrow \\textit{Sets},\\quad\n(g : T \\to B) \\longmapsto H^0(T, H^0(Lg^*K))\n$$\nis an algebraic space affine and of finite presentation over $B$.\n\\item If for all $g : T \\to B$ in $(\\Sch/B)$ the cohomology sheaves\n$H^i(Lg^*K)$ are zero for $i < 0$, then $K$ is perfect,\n$K$ locally has tor amplitude in $[0, b]$, and the functor\n$$\n(\\Sch/B)^{opp} \\longrightarrow \\textit{Sets},\\quad\n(g : T \\to B) \\longmapsto H^0(T, Lg^*K)\n$$\nis an algebraic space affine and of finite presentation over $B$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hom functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JX","source_file":"quot.tex","source_line":432,"source_end_line":453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L432-L453","statement_sha256":"d04005e6f40e3be3546fe70f2ca10489fcf20726db3b8639429c9bf1d74cf538","origin":"The Stacks Project","memory_eligible":false,"source_rank":13895,"rank":13895,"depth":69,"x":2521.322,"y":1534.731,"cluster":"moduli-theory"},{"id":"stacks:08JY","tag":"08JY","title":"The Hom functor · Lemma 08JY","summary":"In Situation [Tag 08JT] assume that • B is a Noetherian algebraic space, • f is locally of finite type and quasi-separated, • F is a finite type O_X-module, and • G is a finite type O_X-module, flat over B, with support proper over B. Then the functor mathitHom(F, G) is an algebraic space affine and of finite presentation over B.","statement_latex":"In Situation \\ref{situation-hom} assume that\n\\begin{enumerate}\n\\item $B$ is a Noetherian algebraic space,\n\\item $f$ is locally of finite type and quasi-separated,\n\\item $\\mathcal{F}$ is a finite type $\\mathcal{O}_X$-module, and\n\\item $\\mathcal{G}$ is a finite type $\\mathcal{O}_X$-module, flat over $B$,\nwith support proper over $B$.\n\\end{enumerate}\nThen the functor $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is\nan algebraic space affine and of finite presentation over $B$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hom functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08JY","source_file":"quot.tex","source_line":589,"source_end_line":601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L589-L601","statement_sha256":"073176213f07f12929564f7fd71df7c78b3ba245e9c0e8aab774cb0b3d0a561c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13896,"rank":13896,"depth":70,"x":2502.184,"y":1672.118,"cluster":"moduli-theory"},{"id":"stacks:08K6","tag":"08K6","title":"The Hom functor · Proposition 08K6","summary":"In Situation [Tag 08JT] assume that • f is of finite presentation, and • G is a finitely presented O_X-module, flat over B, with support proper over B. Then the functor mathitHom(F, G) is an algebraic space affine over B. If F is of finite presentation, then mathitHom(F, G) is of finite presentation over B.","statement_latex":"In Situation \\ref{situation-hom} assume that\n\\begin{enumerate}\n\\item $f$ is of finite presentation, and\n\\item $\\mathcal{G}$ is a finitely presented $\\mathcal{O}_X$-module,\nflat over $B$, with support proper over $B$.\n\\end{enumerate}\nThen the functor $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$ is\nan algebraic space affine over $B$. If $\\mathcal{F}$\nis of finite presentation, then $\\mathit{Hom}(\\mathcal{F}, \\mathcal{G})$\nis of finite presentation over $B$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hom functor","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08K6","source_file":"quot.tex","source_line":669,"source_end_line":681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L669-L681","statement_sha256":"bab2c9ab7b162856272e88137d8332b6de626daaf285a4b016e94fd85e0fd0d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13897,"rank":13897,"depth":71,"x":2405.091,"y":1559.106,"cluster":"moduli-theory"},{"id":"stacks:08K8","tag":"08K8","title":"The Isom functor · Lemma 08K8","summary":"In Situation [Tag 08JT] the functor mathitIsom(F, G) satisfies the sheaf property for the fpqc topology.","statement_latex":"In Situation \\ref{situation-hom} the functor\n$\\mathit{Isom}(\\mathcal{F}, \\mathcal{G})$ \nsatisfies the sheaf property for the fpqc topology.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Isom functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08K8","source_file":"quot.tex","source_line":774,"source_end_line":779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L774-L779","statement_sha256":"6b7e7ba1a9d78ddef703ee18cbed6559b6a1e00fbb04178fc45c658e044336ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":13898,"rank":13898,"depth":52,"x":2568.771,"y":1587.559,"cluster":"moduli-theory"},{"id":"stacks:0D3T","tag":"0D3T","title":"The Isom functor · Lemma 0D3T","summary":"In Situation [Tag 08JT]. Let T be an algebraic space over S. We have Mor_Sh((Sch/S)_fppf)(T, mathitIsom(F, G)) = ((h, u) mid h : T → B, u : F_T → G_T isomorphism) where F_T, G_T denote the pullbacks of F and G to the algebraic space X ×_B, h T.","statement_latex":"In Situation \\ref{situation-hom}. Let $T$ be an algebraic space over $S$.\nWe have\n$$\n\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\mathit{Isom}(\\mathcal{F}, \\mathcal{G})) =\n\\{(h, u) \\mid\nh : T \\to B, u : \\mathcal{F}_T \\to \\mathcal{G}_T\\text{ isomorphism}\\}\n$$\nwhere $\\mathcal{F}_T, \\mathcal{G}_T$ denote the pullbacks of $\\mathcal{F}$\nand $\\mathcal{G}$ to the algebraic space $X \\times_{B, h} T$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Isom functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3T","source_file":"quot.tex","source_line":799,"source_end_line":810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L799-L810","statement_sha256":"9bba62ed7eaac1e0d214c0261bb43a9cb0deb2a9ea462d0d0a22446cdd7b8ed2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13899,"rank":13899,"depth":53,"x":2424.144,"y":1659.976,"cluster":"moduli-theory"},{"id":"stacks:08K9","tag":"08K9","title":"The Isom functor · Proposition 08K9","summary":"In Situation [Tag 08JT] assume that • f is of finite presentation, and • F and G are finitely presented O_X-modules, flat over B, with support proper over B. Then the functor mathitIsom(F, G) is an algebraic space affine of finite presentation over B.","statement_latex":"In Situation \\ref{situation-hom} assume that\n\\begin{enumerate}\n\\item $f$ is of finite presentation, and\n\\item $\\mathcal{F}$ and $\\mathcal{G}$ are finitely presented\n$\\mathcal{O}_X$-modules, flat over $B$, with support proper over $B$.\n\\end{enumerate}\nThen the functor $\\mathit{Isom}(\\mathcal{F}, \\mathcal{G})$ is\nan algebraic space affine of finite presentation over $B$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Isom functor","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08K9","source_file":"quot.tex","source_line":820,"source_end_line":830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L820-L830","statement_sha256":"0a99e9201a22e6eb8d07db7f1b1ffd62ad10089aae9c9645f0082ea837e9ede1","origin":"The Stacks Project","memory_eligible":false,"source_rank":13900,"rank":13900,"depth":72,"x":2472.906,"y":1523.538,"cluster":"moduli-theory"},{"id":"stacks:08W5","tag":"08W5","title":"The stack of coherent sheaves · Lemma 08W5","summary":"In Situation [Tag 08KB] the functor p : Cohstack_X/B → (Sch/S)_fppf is fibred in groupoids.","statement_latex":"In Situation \\ref{situation-coherent} the functor\n$p : \\Cohstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$\nis fibred in groupoids.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08W5","source_file":"quot.tex","source_line":913,"source_end_line":918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L913-L918","statement_sha256":"44c35e3f03054639ef76e4a3d6ceb2139340c108c53b0b7f9a767738216f71e7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13901,"rank":13901,"depth":1,"x":2547.19,"y":1652.725,"cluster":"moduli-theory"},{"id":"stacks:08W6","tag":"08W6","title":"The stack of coherent sheaves · Lemma 08W6","summary":"In Situation [Tag 08KB]. Denote X = Cohstack_X/B. Then Δ : X → X × X is representable by algebraic spaces.","statement_latex":"In Situation \\ref{situation-coherent}. Denote\n$\\mathcal{X} = \\Cohstack_{X/B}$. Then\n$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is\nrepresentable by algebraic spaces.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08W6","source_file":"quot.tex","source_line":951,"source_end_line":957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L951-L957","statement_sha256":"9f63d0cf0c7d336e2afb9ccec1a12604bbf6322c2bd3b42b861938a5029685ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":13902,"rank":13902,"depth":73,"x":2387.416,"y":1599.241,"cluster":"moduli-theory"},{"id":"stacks:08KC","tag":"08KC","title":"The stack of coherent sheaves · Lemma 08KC","summary":"In Situation [Tag 08KB] the functor p : Cohstack_X/B → (Sch/S)_fppf is a stack in groupoids.","statement_latex":"In Situation \\ref{situation-coherent} the functor\n$p : \\Cohstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$\nis a stack in groupoids.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KC","source_file":"quot.tex","source_line":978,"source_end_line":983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L978-L983","statement_sha256":"beb12a56c1a0d0ab1f1e690551b40850faecc0c798a6eed267a273272acb1d93","origin":"The Stacks Project","memory_eligible":false,"source_rank":13903,"rank":13903,"depth":74,"x":2549.352,"y":1547.669,"cluster":"moduli-theory"},{"id":"stacks:08KD","tag":"08KD","title":"The stack of coherent sheaves · Lemma 08KD","summary":"In Situation [Tag 08KB] assume that B → S is locally of finite presentation. Then p : Cohstack_X/B → (Sch/S)_fppf is limit preserving (Artin's Axioms, Definition [Tag 07XL]).","statement_latex":"In Situation \\ref{situation-coherent} assume that $B \\to S$\nis locally of finite presentation. Then\n$p : \\Cohstack_{X/B} \\to (\\Sch/S)_{fppf}$ is limit preserving\n(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving}).","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KD","source_file":"quot.tex","source_line":1089,"source_end_line":1095,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1089-L1095","statement_sha256":"0857dc6830d0f12df4f204446fe6a54079ae7508499094c1885816c8ea165576","origin":"The Stacks Project","memory_eligible":false,"source_rank":13904,"rank":13904,"depth":60,"x":2470.883,"y":1678.467,"cluster":"moduli-theory"},{"id":"stacks:08LQ","tag":"08LQ","title":"The stack of coherent sheaves · Lemma 08LQ","summary":"In Situation [Tag 08KB]. Let xymatrix Z ar[r] ar[d] & Z' ar[d] Y ar[r] & Y' be a pushout in the category of schemes over S where Z → Z' is a thickening and Z → Y is affine, see More on Morphisms, Lemma [Tag 07RT]. Then the functor on fibre categories Cohstack_X/B, Y' → Cohstack_X/B, Y ×_Cohstack_X/B, Z Cohstack_X/B, Z' is an equivalence.","statement_latex":"In Situation \\ref{situation-coherent}. Let\n$$\n\\xymatrix{\nZ \\ar[r] \\ar[d] & Z' \\ar[d] \\\\\nY \\ar[r] & Y'\n}\n$$\nbe a pushout in the category of schemes over $S$ where\n$Z \\to Z'$ is a thickening and $Z \\to Y$ is affine, see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.\nThen the functor on fibre categories\n$$\n\\Cohstack_{X/B, Y'}\n\\longrightarrow\n\\Cohstack_{X/B, Y} \\times_{\\Cohstack_{X/B, Z}} \\Cohstack_{X/B, Z'}\n$$\nis an equivalence.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08LQ","source_file":"quot.tex","source_line":1137,"source_end_line":1156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1137-L1156","statement_sha256":"5123c0c7de67d99a5d19a1a8203fdeb6081aa3d703935cc0ad4e9fba7c8e8a14","origin":"The Stacks Project","memory_eligible":false,"source_rank":13905,"rank":13905,"depth":72,"x":2423.246,"y":1536.546,"cluster":"moduli-theory"},{"id":"stacks:08W7","tag":"08W7","title":"The stack of coherent sheaves · Lemma 08W7","summary":"Let xymatrix X ar[d] ar[r]_i & X' ar[d] T ar[r] & T' be a cartesian square of algebraic spaces where T → T' is a first order thickening. Let F' be an O_X'-module flat over T'. Set F = i^*F'. The following are equivalent • F' is a quasi-coherent O_X'-module of finite presentation, • F' is an O_X'-module of finite presentation, • F is a quasi-coherent O_X-module of finite presentation, • F is an O_X-module of finite presentation,","statement_latex":"Let\n$$\n\\xymatrix{\nX \\ar[d] \\ar[r]_i & X' \\ar[d] \\\\\nT \\ar[r] & T'\n}\n$$\nbe a cartesian square of algebraic spaces where $T \\to T'$ is a first\norder thickening. Let $\\mathcal{F}'$ be an $\\mathcal{O}_{X'}$-module\nflat over $T'$. Set $\\mathcal{F} = i^*\\mathcal{F}'$. The following\nare equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}'$ is a quasi-coherent $\\mathcal{O}_{X'}$-module\nof finite presentation,\n\\item $\\mathcal{F}'$ is an $\\mathcal{O}_{X'}$-module of finite presentation,\n\\item $\\mathcal{F}$ is a quasi-coherent $\\mathcal{O}_X$-module\nof finite presentation,\n\\item $\\mathcal{F}$ is an $\\mathcal{O}_X$-module of finite presentation,\n\\end{enumerate}","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08W7","source_file":"quot.tex","source_line":1182,"source_end_line":1203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1182-L1203","statement_sha256":"014de35d197f5195a2204847dbdbf82ce692469fdd1e10dbcedd4e9ab3af8b1d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13906,"rank":13906,"depth":12,"x":2573.488,"y":1614.682,"cluster":"moduli-theory"},{"id":"stacks:08W8","tag":"08W8","title":"The stack of coherent sheaves · Lemma 08W8","summary":"In Situation [Tag 08KB] assume that S is a locally Noetherian scheme and B → S is locally of finite presentation. Let k be a finite type field over S and let x_0 = (Spec(k), g_0, G_0) be an object of X = Cohstack_X/B over k. Then the spaces TF_X, k, x_0 and Inf(F_X, k, x_0) (Artin's Axioms, Section [Tag 07WY]) are finite dimensional.","statement_latex":"In Situation \\ref{situation-coherent} assume that $S$ is a locally Noetherian\nscheme and $B \\to S$ is locally of finite presentation.\nLet $k$ be a finite type field over $S$ and let\n$x_0 = (\\Spec(k), g_0, \\mathcal{G}_0)$\nbe an object of $\\mathcal{X} = \\Cohstack_{X/B}$ over $k$. Then\nthe spaces $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$ and\n$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$\n(Artin's Axioms, Section \\ref{artin-section-tangent-spaces})\nare finite dimensional.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08W8","source_file":"quot.tex","source_line":1212,"source_end_line":1223,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1212-L1223","statement_sha256":"4b717d7a7438373126c04999ed20108384e29903d861ad1dd1f7c3ce313e95a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13907,"rank":13907,"depth":77,"x":2398.732,"y":1642.495,"cluster":"moduli-theory"},{"id":"stacks:08W9","tag":"08W9","title":"The stack of coherent sheaves · Lemma 08W9","summary":"In Situation [Tag 08KB] assume that S is a locally Noetherian scheme and that f : X → B is separated. Let X = Cohstack_X/B. Then the functor Artin's Axioms, Equation ([Tag 07X6]) is an equivalence.","statement_latex":"In Situation \\ref{situation-coherent} assume that $S$ is a locally Noetherian\nscheme and that $f : X \\to B$ is separated.\nLet $\\mathcal{X} = \\Cohstack_{X/B}$. Then the functor\nArtin's Axioms, Equation (\\ref{artin-equation-approximation})\nis an equivalence.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08W9","source_file":"quot.tex","source_line":1287,"source_end_line":1294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1287-L1294","statement_sha256":"66c5d6959bb7a9c8b2df76815e9bf6a8eccf9ffcd1e42a99e63fcc19da8c3f46","origin":"The Stacks Project","memory_eligible":false,"source_rank":13908,"rank":13908,"depth":77,"x":2505.918,"y":1522.055,"cluster":"moduli-theory"},{"id":"stacks:08WA","tag":"08WA","title":"The stack of coherent sheaves · Lemma 08WA","summary":"In Situation [Tag 08KB] assume that S is a locally Noetherian scheme, S = B, and f : X → B is flat. Let X = Cohstack_X/B. Then we have openness of versality for X (see Artin's Axioms, Definition [Tag 07XQ]).","statement_latex":"In Situation \\ref{situation-coherent} assume that\n$S$ is a locally Noetherian scheme, $S = B$, and $f : X \\to B$ is flat.\nLet $\\mathcal{X} = \\Cohstack_{X/B}$. Then we have openness of\nversality for $\\mathcal{X}$ (see\nArtin's Axioms, Definition \\ref{artin-definition-openness-versality}).","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WA","source_file":"quot.tex","source_line":1328,"source_end_line":1335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1328-L1335","statement_sha256":"b4b1f5d8d99917b47eeda40b889e6d10d014673f3591e7ddc028854b6df9b5a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13909,"rank":13909,"depth":70,"x":2523.846,"y":1672.639,"cluster":"moduli-theory"},{"id":"stacks:08WC","tag":"08WC","title":"Algebraicity of the stack of coherent sheaves; flat case · Theorem 08WC","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. Assume that f is of finite presentation, separated, and flat. Then Cohstack_X/B is an algebraic stack over S.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces\nover $S$. Assume that $f$ is of finite presentation, separated, and\nflat\\footnote{This assumption is not necessary. See\nSection \\ref{section-not-flat}.}. Then $\\Cohstack_{X/B}$ is\nan algebraic stack over $S$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08WC","source_file":"quot.tex","source_line":1448,"source_end_line":1455,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1448-L1455","statement_sha256":"dc04bb8e4de31b9a181fd2bede3f58bb94ddb4907eb2238fa8fe9eb015d5a305","origin":"The Stacks Project","memory_eligible":false,"source_rank":13910,"rank":13910,"depth":80,"x":2388.688,"y":1571.137,"cluster":"moduli-theory"},{"id":"stacks:09DS","tag":"09DS","title":"Algebraicity of the stack of coherent sheaves; general case · Theorem 09DS","summary":"Let S be a scheme. Let f : X → B be morphism of algebraic spaces over S. Assume that f is of finite presentation and separated. Then Cohstack_X/B is an algebraic stack over S.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be morphism of algebraic spaces\nover $S$. Assume that $f$ is of finite presentation and separated. Then\n$\\Cohstack_{X/B}$ is an algebraic stack over $S$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of coherent sheaves in the non-flat case","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09DS","source_file":"quot.tex","source_line":1561,"source_end_line":1566,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1561-L1566","statement_sha256":"986f6cca217a7959fd0c1ff4b571669723d47aef9a4bdf62baf0b61f1c179c13","origin":"The Stacks Project","memory_eligible":false,"source_rank":13911,"rank":13911,"depth":80,"x":2571.102,"y":1569.282,"cluster":"moduli-theory"},{"id":"stacks:082P","tag":"082P","title":"The functor of quotients · Lemma 082P","summary":"In Situation [Tag 082M]. The functors Q_F/X/B and Q^fp_F/X/B satisfy the sheaf property for the fpqc topology.","statement_latex":"In Situation \\ref{situation-q}. The functors\n$\\text{Q}_{\\mathcal{F}/X/B}$ and\n$\\text{Q}^{fp}_{\\mathcal{F}/X/B}$\nsatisfy the sheaf property for the fpqc topology.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The functor of quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082P","source_file":"quot.tex","source_line":1801,"source_end_line":1807,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1801-L1807","statement_sha256":"7fe11387d471e6fc42f62e76a53e7b5f30305f9e5d4c4464d7a69f23210dda36","origin":"The Stacks Project","memory_eligible":false,"source_rank":13912,"rank":13912,"depth":56,"x":2437.267,"y":1674.798,"cluster":"moduli-theory"},{"id":"stacks:0D3U","tag":"0D3U","title":"The functor of quotients · Lemma 0D3U","summary":"In Situation [Tag 082M]. Let T be an algebraic space over S. We have Mor_Sh((Sch/S)_fppf)(T, Q_F/X/B) = ( (h, F_T → Q) where h : T → B and Q is quasi-coherent and flat over T ) where F_T denotes the pullback of F to the algebraic space X ×_B, h T. Similarly, we have Mor_Sh((Sch/S)_fppf)(T, Q^fp_F/X/B) = ( (h, F_T → Q) where h : T → B and Q is of finite presentation and flat over T )","statement_latex":"In Situation \\ref{situation-q}. Let $T$ be an algebraic space over $S$.\nWe have\n$$\n\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T,  \\text{Q}_{\\mathcal{F}/X/B}) =\n\\left\\{\n\\begin{matrix}\n(h, \\mathcal{F}_T \\to \\mathcal{Q}) \\text{ where }\nh : T \\to B \\text{ and}\\\\\n\\mathcal{Q}\\text{ is quasi-coherent and flat over }T\n\\end{matrix}\n\\right\\}\n$$\nwhere $\\mathcal{F}_T$ denotes the pullback of $\\mathcal{F}$\nto the algebraic space $X \\times_{B, h} T$. Similarly, we have\n$$\n\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T,  \\text{Q}^{fp}_{\\mathcal{F}/X/B}) =\n\\left\\{\n\\begin{matrix}\n(h, \\mathcal{F}_T \\to \\mathcal{Q}) \\text{ where }\nh : T \\to B \\text{ and}\\\\\n\\mathcal{Q}\\text{ is of finite presentation and flat over }T\n\\end{matrix}\n\\right\\}\n$$","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The functor of quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3U","source_file":"quot.tex","source_line":1839,"source_end_line":1865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1839-L1865","statement_sha256":"fc98288deb54afe75041dcc1dce66cadbaec6ba06bcd1da1e59f9d3b90adc904","origin":"The Stacks Project","memory_eligible":false,"source_rank":13913,"rank":13913,"depth":56,"x":2451.188,"y":1520.12,"cluster":"moduli-theory"},{"id":"stacks:08IV","tag":"08IV","title":"The functor of quotients · Lemma 08IV","summary":"In Situation [Tag 082M] let (X_i → X)_i ∈ I be an fpqc covering and for each i, j ∈ I let (X_ijk → X_i ×_X X_j) be an fpqc covering. Denote F_i, resp. F_ijk the pullback of F to X_i, resp. X_ijk. For every scheme T over B the diagram xymatrix Q_F/X/B(T) ar[r] & ∏_i Q_F_i/X_i/B(T) ar@<1ex>[r]^-pr_0^* ar@<-1ex>[r]_-pr_1^* & ∏_i, j, k Q_F_ijk/X_ijk/B(T) presents the first arrow as the equalizer of the other two. The same is true for the functor Q^fp_F/X/B.","statement_latex":"In Situation \\ref{situation-q} let $\\{X_i \\to X\\}_{i \\in I}$ be an fpqc\ncovering and for each $i, j \\in I$ let $\\{X_{ijk} \\to X_i \\times_X X_j\\}$\nbe an fpqc covering. Denote $\\mathcal{F}_i$, resp.\\ $\\mathcal{F}_{ijk}$\nthe pullback of $\\mathcal{F}$ to $X_i$, resp.\\ $X_{ijk}$. For every scheme\n$T$ over $B$ the diagram\n$$\n\\xymatrix{\nQ_{\\mathcal{F}/X/B}(T) \\ar[r] &\n\\prod\\nolimits_i\nQ_{\\mathcal{F}_i/X_i/B}(T)\n\\ar@<1ex>[r]^-{\\text{pr}_0^*} \\ar@<-1ex>[r]_-{\\text{pr}_1^*}\n&\n\\prod\\nolimits_{i, j, k}\nQ_{\\mathcal{F}_{ijk}/X_{ijk}/B}(T)\n}\n$$\npresents the first arrow as the equalizer of the other two.\nThe same is true for the functor $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The functor of quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IV","source_file":"quot.tex","source_line":1908,"source_end_line":1928,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1908-L1928","statement_sha256":"83e6a9b8002b103414bd13795a35dbfd3943d9be854fb11ef33f8cfc94cd01be","origin":"The Stacks Project","memory_eligible":false,"source_rank":13914,"rank":13914,"depth":56,"x":2565.991,"y":1642.807,"cluster":"moduli-theory"},{"id":"stacks:082Q","tag":"082Q","title":"The functor of quotients · Lemma 082Q","summary":"In Situation [Tag 082M] assume also that (a) f is quasi-compact and quasi-separated and (b) F is of finite presentation. Then the functor Q^fp_F/X/B is limit preserving in the following sense: If T = lim T_i is a directed limit of affine schemes over B, then Q^fp_F/X/B(T) = colim Q^fp_F/X/B(T_i).","statement_latex":"In Situation \\ref{situation-q} assume also that\n(a) $f$ is quasi-compact and quasi-separated and\n(b) $\\mathcal{F}$ is of finite presentation.\nThen the functor $\\text{Q}^{fp}_{\\mathcal{F}/X/B}$\nis limit preserving in the following sense: If $T = \\lim T_i$ is a\ndirected limit of affine schemes over $B$, then\n$\\text{Q}^{fp}_{\\mathcal{F}/X/B}(T) =\n\\colim \\text{Q}^{fp}_{\\mathcal{F}/X/B}(T_i)$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The functor of quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/082Q","source_file":"quot.tex","source_line":1945,"source_end_line":1955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1945-L1955","statement_sha256":"47b38e8b612f51429b1e232fd0a45252aa7cc7ab0365c4ab03b1103b3fd53d94","origin":"The Stacks Project","memory_eligible":false,"source_rank":13915,"rank":13915,"depth":55,"x":2381.592,"y":1617.33,"cluster":"moduli-theory"},{"id":"stacks:08IW","tag":"08IW","title":"The functor of quotients · Lemma 08IW","summary":"In Situation [Tag 082M]. Let xymatrix Z ar[r] ar[d] & Z' ar[d] Y ar[r] & Y' be a pushout in the category of schemes over B where Z → Z' is a thickening and Z → Y is affine, see More on Morphisms, Lemma [Tag 07RT]. Then the natural map Q_F/X/B(Y') → Q_F/X/B(Y) ×_Q_F/X/B(Z) Q_F/X/B(Z') is bijective. If X → B is locally of finite presentation, then the same thing is true for Q^fp_F/X/B.","statement_latex":"In Situation \\ref{situation-q}. Let\n$$\n\\xymatrix{\nZ \\ar[r] \\ar[d] & Z' \\ar[d] \\\\\nY \\ar[r] & Y'\n}\n$$\nbe a pushout in the category of schemes over $B$ where\n$Z \\to Z'$ is a thickening and $Z \\to Y$ is affine, see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.\nThen the natural map\n$$\nQ_{\\mathcal{F}/X/B}(Y') \\longrightarrow\nQ_{\\mathcal{F}/X/B}(Y) \\times_{Q_{\\mathcal{F}/X/B}(Z)} Q_{\\mathcal{F}/X/B}(Z')\n$$\nis bijective. If $X \\to B$ is locally of finite presentation, then\nthe same thing is true for $Q^{fp}_{\\mathcal{F}/X/B}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The functor of quotients","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IW","source_file":"quot.tex","source_line":1996,"source_end_line":2015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L1996-L2015","statement_sha256":"6f9bf24eaaebd1f95a144fb27a93cf5873d2c15cead1be67c0b067370f347249","origin":"The Stacks Project","memory_eligible":false,"source_rank":13916,"rank":13916,"depth":72,"x":2538.969,"y":1530.984,"cluster":"moduli-theory"},{"id":"stacks:09TT","tag":"09TT","title":"The Quot functor · Lemma 09TT","summary":"In Situation [Tag 09TR]. The functor Quotfunctor_F/X/B satisfies the sheaf property for the fpqc topology.","statement_latex":"In Situation \\ref{situation-quot}. The functor $\\Quotfunctor_{\\mathcal{F}/X/B}$\nsatisfies the sheaf property for the fpqc topology.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Quot functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TT","source_file":"quot.tex","source_line":2244,"source_end_line":2248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2244-L2248","statement_sha256":"b56806215e26982dc98b58836864139e6bcb86d27ecfb6a5eb5230220e2953f7","origin":"The Stacks Project","memory_eligible":false,"source_rank":13917,"rank":13917,"depth":60,"x":2492.087,"y":1684.837,"cluster":"moduli-theory"},{"id":"stacks:0D3V","tag":"0D3V","title":"The Quot functor · Lemma 0D3V","summary":"In Situation [Tag 09TR]. Let T be an algebraic space over S. We have Mor_Sh((Sch/S)_fppf)(T, Quotfunctor_F/X/B) = ( (h, F_T → Q) where h : T → B and Q is of finite presentation and flat over T with support proper over T ) where F_T denotes the pullback of F to the algebraic space X ×_B, h T.","statement_latex":"In Situation \\ref{situation-quot}. Let $T$ be an algebraic space over $S$.\nWe have\n$$\n\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T,  \\Quotfunctor_{\\mathcal{F}/X/B}) =\n\\left\\{\n\\begin{matrix}\n(h, \\mathcal{F}_T \\to \\mathcal{Q}) \\text{ where }\nh : T \\to B \\text{ and}\\\\\n\\mathcal{Q}\\text{ is of finite presentation and}\\\\\n\\text{flat over }T\\text{ with support proper over }T\n\\end{matrix}\n\\right\\}\n$$\nwhere $\\mathcal{F}_T$ denotes the pullback of $\\mathcal{F}$\nto the algebraic space $X \\times_{B, h} T$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Quot functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3V","source_file":"quot.tex","source_line":2269,"source_end_line":2286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2269-L2286","statement_sha256":"194dd001ff6cfc7b9da2c8b5d08d3dcd14e77d800ab31d5ce15c340610ceee2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13918,"rank":13918,"depth":60,"x":2402.426,"y":1543.985,"cluster":"moduli-theory"},{"id":"stacks:09TU","tag":"09TU","title":"The Quot functor · Proposition 09TU","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. Let F be a quasi-coherent sheaf on X. If f is of finite presentation and separated, then Quotfunctor_F/X/B is an algebraic space. If F is of finite presentation, then Quotfunctor_F/X/B → B is locally of finite presentation.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic\nspaces over $S$. Let $\\mathcal{F}$ be a quasi-coherent sheaf\non $X$. If $f$ is of finite presentation and separated, then\n$\\Quotfunctor_{\\mathcal{F}/X/B}$\nis an algebraic space. If $\\mathcal{F}$ is of finite presentation,\nthen $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is locally of finite presentation.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Quot functor","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09TU","source_file":"quot.tex","source_line":2312,"source_end_line":2320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2312-L2320","statement_sha256":"80e827e495dfdb9daed648bb52be54c86bb7ec470f49443ce5ef527badefaab0","origin":"The Stacks Project","memory_eligible":false,"source_rank":13919,"rank":13919,"depth":81,"x":2582.821,"y":1597.272,"cluster":"moduli-theory"},{"id":"stacks:0D00","tag":"0D00","title":"The Hilbert functor · Lemma 0D00","summary":"In Situation [Tag 0CZY] we have Hilbfunctor_X/B = Quotfunctor_O_X/X/B.","statement_latex":"In Situation \\ref{situation-hilb} we have\n$\\Hilbfunctor_{X/B} = \\Quotfunctor_{\\mathcal{O}_X/X/B}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hilbert functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D00","source_file":"quot.tex","source_line":2480,"source_end_line":2484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2480-L2484","statement_sha256":"aeb97b28096638d6730f91b63b95b37af675156924ddf6841c22cebf75c33dec","origin":"The Stacks Project","memory_eligible":false,"source_rank":13920,"rank":13920,"depth":56,"x":2405.966,"y":1660.69,"cluster":"moduli-theory"},{"id":"stacks:0D3W","tag":"0D3W","title":"The Hilbert functor · Lemma 0D3W","summary":"In Situation [Tag 0CZY]. Let T be an algebraic space over S. We have Mor_Sh((Sch/S)_fppf)(T, Hilbfunctor_X/B) = ( (h, Z) where h : T → B, Z ⊂ X_T finite presentation, flat, proper over T ) where X_T = X ×_B, h T.","statement_latex":"In Situation \\ref{situation-hilb}. Let $T$ be an algebraic space over $S$.\nWe have\n$$\n\\Mor_{\\Sh((\\Sch/S)_{fppf})}(T, \\Hilbfunctor_{X/B}) =\n\\left\\{\n\\begin{matrix}\n(h, Z)\\text{ where }h : T \\to B,\\ Z \\subset X_T \\\\\n\\text{finite presentation, flat, proper over }T\n\\end{matrix}\n\\right\\}\n$$\nwhere $X_T = X \\times_{B, h} T$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hilbert functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3W","source_file":"quot.tex","source_line":2542,"source_end_line":2556,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2542-L2556","statement_sha256":"79640f5702a6400dc3d9ad4e0d3a12409b2cbe734e49a51db1b7261c2df208ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":13921,"rank":13921,"depth":61,"x":2485.818,"y":1512.764,"cluster":"moduli-theory"},{"id":"stacks:0D01","tag":"0D01","title":"The Hilbert functor · Proposition 0D01","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. If f is of finite presentation and separated, then Hilbfunctor_X/B is an algebraic space locally of finite presentation over B.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic\nspaces over $S$. If $f$ is of finite presentation and separated, then\n$\\Hilbfunctor_{X/B}$ is an algebraic space locally of finite\npresentation over $B$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Hilbert functor","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D01","source_file":"quot.tex","source_line":2572,"source_end_line":2578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2572-L2578","statement_sha256":"ab9675a54da8f77a2f5c610afd5be9830c8291253a9a7940df5a4f1e8e88ce4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13922,"rank":13922,"depth":82,"x":2546.22,"y":1667.995,"cluster":"moduli-theory"},{"id":"stacks:0D03","tag":"0D03","title":"The Picard stack · Lemma 0D03","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S which is flat, of finite presentation, and proper. The natural map Picardstack_X/B → Cohstack_X/B is representable by open immersions.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic spaces\nover $S$ which is flat, of finite presentation, and proper.\nThe natural map\n$$\n\\Picardstack_{X/B} \\longrightarrow \\Cohstack_{X/B}\n$$\nis representable by open immersions.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D03","source_file":"quot.tex","source_line":2600,"source_end_line":2609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2600-L2609","statement_sha256":"fb71aa90043093d6852fce2296f71b9193b1f4d997515e2cf51a8179f0535081","origin":"The Stacks Project","memory_eligible":false,"source_rank":13923,"rank":13923,"depth":59,"x":2375.934,"y":1587.369,"cluster":"moduli-theory"},{"id":"stacks:0D04","tag":"0D04","title":"The Picard stack · Proposition 0D04","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. If f is flat, of finite presentation, and proper, then Picardstack_X/B is an algebraic stack.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic\nspaces over $S$. If $f$ is flat, of finite presentation, and proper, then\n$\\Picardstack_{X/B}$ is an algebraic stack.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard stack","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D04","source_file":"quot.tex","source_line":2659,"source_end_line":2664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2659-L2664","statement_sha256":"93611d9e90eb511b639de3f4dc5a24cacd12ef6acb491424adbc25d0cb974a34","origin":"The Stacks Project","memory_eligible":false,"source_rank":13924,"rank":13924,"depth":81,"x":2567.355,"y":1550.0,"cluster":"moduli-theory"},{"id":"stacks:0D26","tag":"0D26","title":"The Picard functor · Lemma 0D26","summary":"In Situation [Tag 0D25] the functor Picardfunctor_X/B is the sheafification of the functor T ↦ Ob(Picardstack_X/B, T)/≅.","statement_latex":"In Situation \\ref{situation-pic}\nthe functor $\\Picardfunctor_{X/B}$ is the sheafification of\nthe functor $T \\mapsto \\Ob(\\Picardstack_{X/B, T})/\\cong$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D26","source_file":"quot.tex","source_line":2741,"source_end_line":2746,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2741-L2746","statement_sha256":"8ea66dae5feb511c75b7574e3bb35efa7688100335d56b70ffeba18c08ed04e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13925,"rank":13925,"depth":0,"x":2455.666,"y":1686.891,"cluster":"moduli-theory"},{"id":"stacks:0D27","tag":"0D27","title":"The Picard functor · Lemma 0D27","summary":"In Situation [Tag 0D25]. If O_T → f_T, *O_X_T is an isomorphism for all schemes T over B, then 0 → Pic(T) → Pic(X_T) → Picardfunctor_X/B(T) is an exact sequence for all T.","statement_latex":"In Situation \\ref{situation-pic}.\nIf $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism\nfor all schemes $T$ over $B$, then\n$$\n0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/B}(T)\n$$\nis an exact sequence for all $T$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D27","source_file":"quot.tex","source_line":2760,"source_end_line":2769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2760-L2769","statement_sha256":"d13fbf7d35339aaa8ab59f64a20bb0c0890e946e95576d95f32ba770a61035d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13926,"rank":13926,"depth":21,"x":2427.798,"y":1521.716,"cluster":"moduli-theory"},{"id":"stacks:0D28","tag":"0D28","title":"The Picard functor · Lemma 0D28","summary":"In Situation [Tag 0D25] let σ : B → X be a section. Assume that O_T → f_T, *O_X_T is an isomorphism for all T over B. Then 0 → Pic(T) → Pic(X_T) → Picardfunctor_X/B(T) → 0 is a split exact sequence with splitting given by σ_T^* : Pic(X_T) → Pic(T).","statement_latex":"In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section.\nAssume that $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism\nfor all $T$ over $B$. Then\n$$\n0 \\to \\Pic(T) \\to \\Pic(X_T) \\to \\Picardfunctor_{X/B}(T) \\to 0\n$$\nis a split exact sequence with splitting given by\n$\\sigma_T^* : \\Pic(X_T) \\to \\Pic(T)$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D28","source_file":"quot.tex","source_line":2810,"source_end_line":2820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2810-L2820","statement_sha256":"b9e848f3b253ea2af7680faff247806a5a8682d20750bfe5f7dceb4d5dbb5a1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13927,"rank":13927,"depth":52,"x":2581.969,"y":1628.25,"cluster":"moduli-theory"},{"id":"stacks:0D29","tag":"0D29","title":"The Picard functor · Lemma 0D29","summary":"In Situation [Tag 0D25] let σ : B → X be a section. Then Picardstack_X/B, σ as defined above is a stack in groupoids over (Sch/S)_fppf.","statement_latex":"In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section.\nThen $\\Picardstack_{X/B, \\sigma}$ as defined above is a stack in\ngroupoids over $(\\Sch/S)_{fppf}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D29","source_file":"quot.tex","source_line":2901,"source_end_line":2906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2901-L2906","statement_sha256":"44eb115e7feb2ccadd660e1f66adc97bbc6354a7c6187ee9b7b4d52118b1075d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13928,"rank":13928,"depth":53,"x":2381.593,"y":1637.217,"cluster":"moduli-theory"},{"id":"stacks:0D2A","tag":"0D2A","title":"The Picard functor · Lemma 0D2A","summary":"In Situation [Tag 0D25] let σ : B → X be a section. The morphism Picardstack_X/B, σ → Picardstack_X/B is representable, surjective, and smooth.","statement_latex":"In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section.\nThe morphism $\\Picardstack_{X/B, \\sigma} \\to \\Picardstack_{X/B}$\nis representable, surjective, and smooth.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2A","source_file":"quot.tex","source_line":2934,"source_end_line":2939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2934-L2939","statement_sha256":"ed6c41b7399578bffabc79804775d0b19c0b60a188f141b211889dc088dfacbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13929,"rank":13929,"depth":73,"x":2522.849,"y":1516.298,"cluster":"moduli-theory"},{"id":"stacks:0D2B","tag":"0D2B","title":"The Picard functor · Lemma 0D2B","summary":"In Situation [Tag 0D25] let σ : B → X be a section. If O_T → f_T, *O_X_T is an isomorphism for all T over B, then Picardstack_X/B, σ → (Sch/S)_fppf is fibred in setoids with set of isomorphism classes over T given by coprod_h : T → B Ker(σ_T^* : Pic(X ×_B, h T) → Pic(T))","statement_latex":"In Situation \\ref{situation-pic} let $\\sigma : B \\to X$ be a section.\nIf $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism\nfor all $T$ over $B$, then\n$\\Picardstack_{X/B, \\sigma} \\to (\\Sch/S)_{fppf}$\nis fibred in setoids with set of isomorphism classes over $T$ given by\n$$\n\\coprod\\nolimits_{h : T \\to B}\n\\Ker(\\sigma_T^* : \\Pic(X \\times_{B, h} T) \\to \\Pic(T))\n$$","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2B","source_file":"quot.tex","source_line":2973,"source_end_line":2984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L2973-L2984","statement_sha256":"58dc77a8e6eba3d38a8b548e2829c439d088dacdbc5ccefabec7c413a84fcb06","origin":"The Stacks Project","memory_eligible":false,"source_rank":13930,"rank":13930,"depth":0,"x":2515.893,"y":1686.464,"cluster":"moduli-theory"},{"id":"stacks:0D2C","tag":"0D2C","title":"The Picard functor · Proposition 0D2C","summary":"Let S be a scheme. Let f : X → B be a morphism of algebraic spaces over S. Assume that • f is flat, of finite presentation, and proper, and • O_T → f_T, *O_X_T is an isomorphism for all schemes T over B. Then Picardfunctor_X/B is an algebraic space.","statement_latex":"Let $S$ be a scheme. Let $f : X \\to B$ be a morphism of algebraic\nspaces over $S$. Assume that\n\\begin{enumerate}\n\\item $f$ is flat, of finite presentation, and proper, and\n\\item $\\mathcal{O}_T \\to f_{T, *}\\mathcal{O}_{X_T}$ is an isomorphism\nfor all schemes $T$ over $B$.\n\\end{enumerate}\nThen $\\Picardfunctor_{X/B}$ is an algebraic space.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard functor","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2C","source_file":"quot.tex","source_line":3005,"source_end_line":3015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3005-L3015","statement_sha256":"7e2e0dea87ee5af7160645569b88387ef89676e665236abb307dbbcd5c6b4712","origin":"The Stacks Project","memory_eligible":false,"source_rank":13931,"rank":13931,"depth":0,"x":2383.527,"y":1556.395,"cluster":"moduli-theory"},{"id":"stacks:0D2D","tag":"0D2D","title":"The Picard functor · Lemma 0D2D","summary":"With assumptions and notation as in Proposition [Tag 0D2C]. Then the diagonal Picardfunctor_X/B → Picardfunctor_X/B ×_B Picardfunctor_X/B is representable by immersions. In other words, Picardfunctor_X/B → B is locally separated.","statement_latex":"With assumptions and notation as in Proposition \\ref{proposition-pic-functor}.\nThen the diagonal\n$\\Picardfunctor_{X/B} \\to \\Picardfunctor_{X/B} \\times_B \\Picardfunctor_{X/B}$\nis representable by immersions. In other words, $\\Picardfunctor_{X/B} \\to B$\nis locally separated.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D2D","source_file":"quot.tex","source_line":3056,"source_end_line":3063,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3056-L3063","statement_sha256":"f43d242fa44bc62fb7fc00564a36da75ab21bb73d16def56fb983c6a4df60b1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13932,"rank":13932,"depth":70,"x":2586.723,"y":1577.302,"cluster":"moduli-theory"},{"id":"stacks:0D1A","tag":"0D1A","title":"Relative morphisms · Lemma 0D1A","summary":"Let S be a scheme. Consider morphisms of algebraic spaces Z → B and X → B over S. If X → B is separated and Z → B is of finite presentation, flat, and proper, then there is a natural injective transformation of functors mathitMor_B(Z, X) → Hilbfunctor_Z ×_B X/B which maps a morphism f : Z_T → X_T to its graph.","statement_latex":"Let $S$ be a scheme. Consider morphisms\nof algebraic spaces $Z \\to B$ and $X \\to B$ over $S$.\nIf $X \\to B$ is separated and $Z \\to B$ is\nof finite presentation, flat, and proper,\nthen there is a natural\ninjective transformation of functors\n$$\n\\mathit{Mor}_B(Z, X) \\longrightarrow \\Hilbfunctor_{Z \\times_B X/B}\n$$\nwhich maps a morphism $f : Z_T \\to X_T$ to its graph.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1A","source_file":"quot.tex","source_line":3139,"source_end_line":3151,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3139-L3151","statement_sha256":"364349b62a146129bc394140459792690b4bcb024fd0988a250481924b014a33","origin":"The Stacks Project","memory_eligible":false,"source_rank":13933,"rank":13933,"depth":55,"x":2419.264,"y":1677.669,"cluster":"moduli-theory"},{"id":"stacks:0D1B","tag":"0D1B","title":"Relative morphisms · Lemma 0D1B","summary":"Assumption and notation as in Lemma [Tag 0D1A]. The transformation mathitMor_B(Z, X) → Hilbfunctor_Z ×_B X/B is representable by open immersions.","statement_latex":"Assumption and notation as in Lemma \\ref{lemma-Mor-into-Hilb}.\nThe transformation\n$\\mathit{Mor}_B(Z, X) \\longrightarrow \\Hilbfunctor_{Z \\times_B X/B}$\nis representable by open immersions.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1B","source_file":"quot.tex","source_line":3175,"source_end_line":3181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3175-L3181","statement_sha256":"5c423e4b4a8b2695bd1d51d8a62edba59199204fe825bba8dd202c7997476091","origin":"The Stacks Project","memory_eligible":false,"source_rank":13934,"rank":13934,"depth":62,"x":2462.242,"y":1507.825,"cluster":"moduli-theory"},{"id":"stacks:0D1C","tag":"0D1C","title":"Relative morphisms · Proposition 0D1C","summary":"Let S be a scheme. Let Z → B and X → B be morphisms of algebraic spaces over S. Assume X → B is of finite presentation and separated and Z → B is of finite presentation, flat, and proper. Then mathitMor_B(Z, X) is an algebraic space locally of finite presentation over B.","statement_latex":"Let $S$ be a scheme. Let $Z \\to B$ and $X \\to B$ be morphisms of algebraic\nspaces over $S$. Assume $X \\to B$ is of finite presentation and separated and\n$Z \\to B$ is of finite presentation, flat, and proper. Then\n$\\mathit{Mor}_B(Z, X)$ is an algebraic space locally of finite\npresentation over $B$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Relative morphisms","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1C","source_file":"quot.tex","source_line":3197,"source_end_line":3204,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3197-L3204","statement_sha256":"8ead67a38bd717f7abc2ee120c044ab8b5034f42e5b242139d21d3a74996f3bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13935,"rank":13935,"depth":83,"x":2567.633,"y":1658.167,"cluster":"moduli-theory"},{"id":"stacks:0D1E","tag":"0D1E","title":"The stack of algebraic spaces · Lemma 0D1E","summary":"The category Spacesstack'_ft is fibred in groupoids over Sch_fppf. The same is true for Spacesstack'_fp, flat, proper.","statement_latex":"The category $\\Spacesstack'_{ft}$ is fibred in groupoids\nover $\\Sch_{fppf}$. The same is true for\n$\\Spacesstack'_{fp, flat, proper}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1E","source_file":"quot.tex","source_line":3269,"source_end_line":3274,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3269-L3274","statement_sha256":"ac765124bc4ebc6677f732d41db2c5d333b5c9cb7e98a720de4eeba8dc786516","origin":"The Stacks Project","memory_eligible":false,"source_rank":13936,"rank":13936,"depth":18,"x":2368.08,"y":1606.866,"cluster":"moduli-theory"},{"id":"stacks:0D1F","tag":"0D1F","title":"The stack of algebraic spaces · Lemma 0D1F","summary":"The diagonal Δ : Spacesstack'_fp, flat, proper → Spacesstack'_fp, flat, proper × Spacesstack'_fp, flat, proper is representable by algebraic spaces.","statement_latex":"The diagonal\n$$\n\\Delta : \\Spacesstack'_{fp, flat, proper} \\longrightarrow\n\\Spacesstack'_{fp, flat, proper} \\times \\Spacesstack'_{fp, flat, proper}\n$$\nis representable by algebraic spaces.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1F","source_file":"quot.tex","source_line":3312,"source_end_line":3320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3312-L3320","statement_sha256":"b6982d038d1ae8400d84e89c46fa3a7dc66ba72371843bec475b039f18772c17","origin":"The Stacks Project","memory_eligible":false,"source_rank":13937,"rank":13937,"depth":84,"x":2557.366,"y":1531.111,"cluster":"moduli-theory"},{"id":"stacks:0D1G","tag":"0D1G","title":"The stack of algebraic spaces · Lemma 0D1G","summary":"The category Spacesstack'_ft is a stack in groupoids over Sch_fppf. The same is true for Spacesstack'_fp, flat, proper.","statement_latex":"The category $\\Spacesstack'_{ft}$ is a stack in groupoids\nover $\\Sch_{fppf}$. The same is true for\n$\\Spacesstack'_{fp, flat, proper}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1G","source_file":"quot.tex","source_line":3349,"source_end_line":3354,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3349-L3354","statement_sha256":"574f3bc269a766a1197897c62c147607b676136be1cb775a214908acee7634ea","origin":"The Stacks Project","memory_eligible":false,"source_rank":13938,"rank":13938,"depth":85,"x":2478.342,"y":1695.134,"cluster":"moduli-theory"},{"id":"stacks:0E93","tag":"0E93","title":"The stack of algebraic spaces · Lemma 0E93","summary":"Let T be an algebraic space over Z. Let S_T denote the corresponding algebraic stack (Algebraic Stacks, Sections [Tag 04SU], [Tag 02ZV], and [Tag 03YR]). We have an equivalence of categories ( morphisms of algebraic spaces X → T of finite type ) → Mor_Cat/Sch_fppf(S_T, Spacesstack'_ft) and an equivalence of categories ( morphisms of algebraic spaces X → T of finite presentation, flat, and proper ) → Mor_Cat/Sch_fppf(S_T, Spacesstack'_fp, flat, proper)","statement_latex":"Let $T$ be an algebraic space over $\\mathbf{Z}$. Let $\\mathcal{S}_T$\ndenote the corresponding algebraic stack (Algebraic Stacks, Sections\n\\ref{algebraic-section-split},\n\\ref{algebraic-section-representable-by-algebraic-spaces}, and\n\\ref{algebraic-section-stacks-spaces}).\nWe have an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{morphisms of algebraic spaces }\\\\\nX \\to T\\text{ of finite type}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\Mor_{\\textit{Cat}/\\Sch_{fppf}}(\\mathcal{S}_T, \\Spacesstack'_{ft})\n$$\nand an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n\\text{morphisms of algebraic spaces }X \\to T\\\\\n\\text{of finite presentation, flat, and proper}\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\Mor_{\\textit{Cat}/\\Sch_{fppf}}(\\mathcal{S}_T,\n\\Spacesstack'_{fp, flat, proper})\n$$","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E93","source_file":"quot.tex","source_line":3413,"source_end_line":3443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3413-L3443","statement_sha256":"64230bac39e210bfeb28e5dab49f145763930b29cdcb09ec5deb0324158d582c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13939,"rank":13939,"depth":70,"x":2404.374,"y":1528.584,"cluster":"moduli-theory"},{"id":"stacks:0D1I","tag":"0D1I","title":"The stack of algebraic spaces · Lemma 0D1I","summary":"The stack p'_fp, flat, proper : Spacesstack'_fp, flat, proper → Sch_fppf is limit preserving (Artin's Axioms, Definition [Tag 07XL]).","statement_latex":"The stack\n$p'_{fp, flat, proper} :\n\\Spacesstack'_{fp, flat, proper} \\to \\Sch_{fppf}$ is limit preserving\n(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving}).","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1I","source_file":"quot.tex","source_line":3522,"source_end_line":3528,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3522-L3528","statement_sha256":"1847e1c691a5dc66aad14a87f442cba50affce58e403058b064600508d6977dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13940,"rank":13940,"depth":59,"x":2593.71,"y":1609.793,"cluster":"moduli-theory"},{"id":"stacks:0D1J","tag":"0D1J","title":"The stack of algebraic spaces · Lemma 0D1J","summary":"Let xymatrix T ar[r] ar[d] & T' ar[d] S ar[r] & S' be a pushout in the category of schemes where T → T' is a thickening and T → S is affine, see More on Morphisms, Lemma [Tag 07RT]. Then the functor on fibre categories Spacesstack'_fp, flat, proper, S' downarrow Spacesstack'_fp, flat, proper, S ×_Spacesstack'_fp, flat, proper, T Spacesstack'_fp, flat, proper, T' is an equivalence.","statement_latex":"Let\n$$\n\\xymatrix{\nT \\ar[r] \\ar[d] & T' \\ar[d] \\\\\nS \\ar[r] & S'\n}\n$$\nbe a pushout in the category of schemes where\n$T \\to T'$ is a thickening and $T \\to S$ is affine, see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.\nThen the functor on fibre categories\n$$\n\\begin{matrix}\n\\Spacesstack'_{fp, flat, proper, S'} \\\\\n\\downarrow \\\\\n\\Spacesstack'_{fp, flat, proper, S}\n\\times_{\\Spacesstack'_{fp, flat, proper, T}}\n\\Spacesstack'_{fp, flat, proper, T'}\n\\end{matrix}\n$$\nis an equivalence.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1J","source_file":"quot.tex","source_line":3545,"source_end_line":3568,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3545-L3568","statement_sha256":"f6a1b75fd83e7a0304d14afe9cef90537c9fe82a26455a84558884460e0c2539","origin":"The Stacks Project","memory_eligible":false,"source_rank":13941,"rank":13941,"depth":74,"x":2387.864,"y":1657.558,"cluster":"moduli-theory"},{"id":"stacks:0D1K","tag":"0D1K","title":"The stack of algebraic spaces · Lemma 0D1K","summary":"Let k be a field and let x = (X → Spec(k)) be an object of X = Spacesstack'_fp, flat, proper over Spec(k). • If k is of finite type over Z, then the vector spaces TF_X, k, x and Inf(F_X, k, x) (see Artin's Axioms, Section [Tag 07WY]) are finite dimensional, and • in general the vector spaces T_x(k) and Inf_x(k) (see Artin's Axioms, Section [Tag 07Y6]) are finite dimensional.","statement_latex":"Let $k$ be a field and let $x = (X \\to \\Spec(k))$ be an object of\n$\\mathcal{X} = \\Spacesstack'_{fp, flat, proper}$ over $\\Spec(k)$.\n\\begin{enumerate}\n\\item If $k$ is of finite type over $\\mathbf{Z}$, then\nthe vector spaces $T\\mathcal{F}_{\\mathcal{X}, k, x}$ and\n$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x})$\n(see Artin's Axioms, Section \\ref{artin-section-tangent-spaces})\nare finite dimensional, and\n\\item in general the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$\n(see Artin's Axioms, Section \\ref{artin-section-inf})\nare finite dimensional.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1K","source_file":"quot.tex","source_line":3589,"source_end_line":3603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3589-L3603","statement_sha256":"31c47a61676b7ac90c16065e2ec68672443e853451c2ba2cff1fefe24cc5612a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13942,"rank":13942,"depth":75,"x":2501.749,"y":1504.856,"cluster":"moduli-theory"},{"id":"stacks:0D3X","tag":"0D3X","title":"The stack of algebraic spaces · Lemma 0D3X","summary":"The stack in groupoids X = Spacesstack'_fp, flat, proper satisfies openness of versality over Spec(Z). Similarly, after base change (Remark [Tag 0D1H]) openness of versality holds over any Noetherian base scheme S.","statement_latex":"The stack in groupoids $\\mathcal{X} = \\Spacesstack'_{fp, flat, proper}$\nsatisfies openness of versality over $\\Spec(\\mathbf{Z})$.\nSimilarly, after base change (Remark \\ref{remark-spaces-base-change})\nopenness of versality holds over any Noetherian base scheme $S$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3X","source_file":"quot.tex","source_line":3645,"source_end_line":3651,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3645-L3651","statement_sha256":"6871d39d5a1d894a01c79ac69f651f0f0ddae1b45ed7a626ef8601c156e71a76","origin":"The Stacks Project","memory_eligible":false,"source_rank":13943,"rank":13943,"depth":85,"x":2540.741,"y":1682.863,"cluster":"moduli-theory"},{"id":"stacks:0D3Z","tag":"0D3Z","title":"The stack of polarized proper schemes · Lemma 0D3Z","summary":"The category Polarizedstack is fibred in groupoids over Spacesstack'_fp, flat, proper. The category Polarizedstack is fibred in groupoids over Sch_fppf.","statement_latex":"The category $\\Polarizedstack$ is fibred in groupoids over\n$\\Spacesstack'_{fp, flat, proper}$.\nThe category $\\Polarizedstack$ is fibred in groupoids over $\\Sch_{fppf}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D3Z","source_file":"quot.tex","source_line":3818,"source_end_line":3823,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3818-L3823","statement_sha256":"16a2662f1ab6392bb35ddf9a133eee6e3581be1170720c264e803f42bd93f7d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":13944,"rank":13944,"depth":24,"x":2368.087,"y":1573.248,"cluster":"moduli-theory"},{"id":"stacks:0D40","tag":"0D40","title":"The stack of polarized proper schemes · Lemma 0D40","summary":"The category Polarizedstack is a stack in groupoids over Spacesstack'_fp, flat, proper (endowed with the inherited topology, see Stacks, Definition [Tag 06NV]). The category Polarizedstack is a stack in groupoids over Sch_fppf.","statement_latex":"The category $\\Polarizedstack$ is a stack in groupoids over\n$\\Spacesstack'_{fp, flat, proper}$ (endowed with the inherited topology,\nsee Stacks, Definition \\ref{stacks-definition-topology-inherited}).\nThe category $\\Polarizedstack$ is a stack in groupoids over $\\Sch_{fppf}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D40","source_file":"quot.tex","source_line":3866,"source_end_line":3872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3866-L3872","statement_sha256":"62847053bcb338c468b66e5496f058f1fbc64bd4a8bcbaf287655d8f54983c7d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13945,"rank":13945,"depth":86,"x":2584.488,"y":1556.035,"cluster":"moduli-theory"},{"id":"stacks:0E94","tag":"0E94","title":"The stack of polarized proper schemes · Lemma 0E94","summary":"Let T be an algebraic space over Z. Let S_T denote the corresponding algebraic stack (Algebraic Stacks, Sections [Tag 04SU], [Tag 02ZV], and [Tag 03YR]). We have an equivalence of categories ( (X → T, L) where X → T is a morphism of algebraic spaces, is proper, flat, and of finite presentation and L ample on X/T ) → Mor_Cat/Sch_fppf(S_T, Polarizedstack)","statement_latex":"Let $T$ be an algebraic space over $\\mathbf{Z}$. Let $\\mathcal{S}_T$\ndenote the corresponding algebraic stack (Algebraic Stacks, Sections\n\\ref{algebraic-section-split},\n\\ref{algebraic-section-representable-by-algebraic-spaces}, and\n\\ref{algebraic-section-stacks-spaces}).\nWe have an equivalence of categories\n$$\n\\left\\{\n\\begin{matrix}\n(X \\to T, \\mathcal{L})\\text{ where }X \\to T\\text{ is a morphism}\\\\\n\\text{of algebraic spaces, is proper, flat, and of}\\\\\n\\text{finite presentation and }\\mathcal{L}\\text{ ample on }X/T\n\\end{matrix}\n\\right\\}\n\\longrightarrow\n\\Mor_{\\textit{Cat}/\\Sch_{fppf}}(\\mathcal{S}_T, \\Polarizedstack)\n$$","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E94","source_file":"quot.tex","source_line":3957,"source_end_line":3976,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L3957-L3976","statement_sha256":"b30ed73d395ca57ab065c85f35bb70e8bfaa1e80cf9871f3129a3cbf16282c66","origin":"The Stacks Project","memory_eligible":false,"source_rank":13946,"rank":13946,"depth":71,"x":2438.129,"y":1692.102,"cluster":"moduli-theory"},{"id":"stacks:0D41","tag":"0D41","title":"The stack of polarized proper schemes · Lemma 0D41","summary":"The functor ([Tag 0D3Y]) defines a 1-morphism Polarizedstack → Spacesstack'_fp, flat, proper of stacks in groupoids over Sch_fppf which is algebraic in the sense of Criteria for Representability, Definition [Tag 06CF].","statement_latex":"The functor (\\ref{equation-over-proper-spaces}) defines a $1$-morphism\n$$\n\\Polarizedstack \\to \\Spacesstack'_{fp, flat, proper}\n$$\nof stacks in groupoids over $\\Sch_{fppf}$\nwhich is algebraic in the sense of\nCriteria for Representability, Definition\n\\ref{criteria-definition-algebraic}.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D41","source_file":"quot.tex","source_line":4012,"source_end_line":4022,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4012-L4022","statement_sha256":"9ddb654f01812ab792c2ffb3d7c3050504e5a6d3ab7c92c7b385f4c8886f10a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13947,"rank":13947,"depth":87,"x":2436.625,"y":1507.936,"cluster":"moduli-theory"},{"id":"stacks:0D42","tag":"0D42","title":"The stack of polarized proper schemes · Lemma 0D42","summary":"The diagonal Δ : Polarizedstack → Polarizedstack × Polarizedstack is representable by algebraic spaces.","statement_latex":"The diagonal\n$$\n\\Delta : \\Polarizedstack \\longrightarrow\n\\Polarizedstack \\times \\Polarizedstack\n$$\nis representable by algebraic spaces.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D42","source_file":"quot.tex","source_line":4049,"source_end_line":4057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4049-L4057","statement_sha256":"286d6e996c1e3f4840e6626a442c490ef76e6e9eac8eefa752f6ac1fdbe324fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13948,"rank":13948,"depth":88,"x":2586.467,"y":1643.457,"cluster":"moduli-theory"},{"id":"stacks:0D43","tag":"0D43","title":"The stack of polarized proper schemes · Lemma 0D43","summary":"The stack in groupoids Polarizedstack is limit preserving (Artin's Axioms, Definition [Tag 07XL]).","statement_latex":"The stack in groupoids $\\Polarizedstack$ is limit preserving\n(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving}).","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D43","source_file":"quot.tex","source_line":4067,"source_end_line":4071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4067-L4071","statement_sha256":"5c785179e4bc2a45e60d4b17332dcee4a054ece3a26bd7286624106c8b4ed145","origin":"The Stacks Project","memory_eligible":false,"source_rank":13949,"rank":13949,"depth":36,"x":2366.07,"y":1628.486,"cluster":"moduli-theory"},{"id":"stacks:0D44","tag":"0D44","title":"The stack of polarized proper schemes · Lemma 0D44","summary":"In Situation [Tag 08KB]. Let xymatrix T ar[r] ar[d] & T' ar[d] S ar[r] & S' be a pushout in the category of schemes where T → T' is a thickening and T → S is affine, see More on Morphisms, Lemma [Tag 07RT]. Then the functor on fibre categories Polarizedstack_S' → Polarizedstack_S ×_Polarizedstack_T Polarizedstack_T' is an equivalence.","statement_latex":"In Situation \\ref{situation-coherent}. Let\n$$\n\\xymatrix{\nT \\ar[r] \\ar[d] & T' \\ar[d] \\\\\nS \\ar[r] & S'\n}\n$$\nbe a pushout in the category of schemes where\n$T \\to T'$ is a thickening and $T \\to S$ is affine, see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.\nThen the functor on fibre categories\n$$\n\\Polarizedstack_{S'}\n\\longrightarrow\n\\Polarizedstack_S \\times_{\\Polarizedstack_T} \\Polarizedstack_{T'}\n$$\nis an equivalence.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D44","source_file":"quot.tex","source_line":4101,"source_end_line":4120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4101-L4120","statement_sha256":"faf1307e3c3ccc904994b7ab57d3ac304b4dc37c7cfeab063e6ab06b652535db","origin":"The Stacks Project","memory_eligible":false,"source_rank":13950,"rank":13950,"depth":45,"x":2541.356,"y":1513.992,"cluster":"moduli-theory"},{"id":"stacks:0D4S","tag":"0D4S","title":"The stack of polarized proper schemes · Lemma 0D4S","summary":"Let k be a field and let x = (X → Spec(k), L) be an object of X = Polarizedstack over Spec(k). • If k is of finite type over Z, then the vector spaces TF_X, k, x and Inf(F_X, k, x) (see Artin's Axioms, Section [Tag 07WY]) are finite dimensional, and • in general the vector spaces T_x(k) and Inf_x(k) (see Artin's Axioms, Section [Tag 07Y6]) are finite dimensional.","statement_latex":"Let $k$ be a field and let $x = (X \\to \\Spec(k), \\mathcal{L})$\nbe an object of $\\mathcal{X} = \\Polarizedstack$ over $\\Spec(k)$.\n\\begin{enumerate}\n\\item If $k$ is of finite type over $\\mathbf{Z}$, then\nthe vector spaces $T\\mathcal{F}_{\\mathcal{X}, k, x}$ and\n$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x})$\n(see Artin's Axioms, Section \\ref{artin-section-tangent-spaces})\nare finite dimensional, and\n\\item in general the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$\n(see Artin's Axioms, Section \\ref{artin-section-inf})\nare finite dimensional.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4S","source_file":"quot.tex","source_line":4192,"source_end_line":4206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4192-L4206","statement_sha256":"8991f2f446c9317bd77ac11e4919633425e0991c13e413c7353019b26c26151d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13951,"rank":13951,"depth":76,"x":2504.02,"y":1698.64,"cluster":"moduli-theory"},{"id":"stacks:0D4T","tag":"0D4T","title":"Strong formal effectiveness for polarized schemes · Lemma 0D4T","summary":"Grothendieck's algebraization theorem continues to hold in the non-Noetherian setting if one assumes flatness and finite presentation. Let (R_n) be an inverse system of rings with surjective transition maps whose kernels are locally nilpotent. Set R = lim R_n. Set S_n = Spec(R_n) and S = Spec(R). Consider a commutative diagram xymatrix X_1 ar[r]_i_1 ar[d] & X_2 ar[r]_i_2 ar[d] & X_3 ar[r] ar[d] & … S_1 ar[r] & S_2 ar[r] & S_3 ar[r] & … of schemes with cartesian squares.…","statement_latex":"\\begin{slogan}\nGrothendieck's algebraization theorem continues to hold in\nthe non-Noetherian setting if one assumes flatness and\nfinite presentation.\n\\end{slogan}\nLet $(R_n)$ be an inverse system of rings with surjective transition maps\nwhose kernels are locally nilpotent. Set $R = \\lim R_n$.\nSet $S_n = \\Spec(R_n)$ and $S = \\Spec(R)$. Consider a commutative diagram\n$$\n\\xymatrix{\nX_1 \\ar[r]_{i_1} \\ar[d] & X_2 \\ar[r]_{i_2} \\ar[d] & X_3 \\ar[r] \\ar[d] &\n\\ldots \\\\\nS_1 \\ar[r] & S_2 \\ar[r] & S_3 \\ar[r] & \\ldots\n}\n$$\nof schemes with cartesian squares. Suppose given $(\\mathcal{L}_n, \\varphi_n)$\nwhere each $\\mathcal{L}_n$ is an invertible sheaf on $X_n$ and\n$\\varphi_n : i_n^*\\mathcal{L}_{n + 1} \\to \\mathcal{L}_n$ is an isomorphism.\nIf\n\\begin{enumerate}\n\\item $X_n \\to S_n$ is proper, flat, of finite presentation, and\n\\item $\\mathcal{L}_1$ is ample on $X_1$\n\\end{enumerate}\nthen there exists a morphism of schemes $X \\to S$\nproper, flat, and of finite presentation\nand an ample invertible $\\mathcal{O}_X$-module $\\mathcal{L}$\nand isomorphisms $X_n \\cong X \\times_S S_n$ and\n$\\mathcal{L}_n \\cong \\mathcal{L}|_{X_n}$ compatible with\nthe morphisms $i_n$ and $\\varphi_n$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4T","source_file":"quot.tex","source_line":4292,"source_end_line":4323,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4292-L4323","statement_sha256":"ed26e9dea16c166306bd5909f93427b9154e4481197140485e9f3b0130f0ebc9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13952,"rank":13952,"depth":49,"x":2382.567,"y":1540.652,"cluster":"moduli-theory"},{"id":"stacks:0D4U","tag":"0D4U","title":"The stack of polarized proper schemes · Lemma 0D4U","summary":"Consider the stack Polarizedstack over the base scheme Spec(Z). Then every formal object is effective.","statement_latex":"Consider the stack $\\Polarizedstack$ over the base\nscheme $\\Spec(\\mathbf{Z})$. Then every formal object is effective.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4U","source_file":"quot.tex","source_line":4413,"source_end_line":4417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4413-L4417","statement_sha256":"375e4fb06a83d1775e0d77b8923eda0430ebe5fd959aa5e0e9e7aacd71d33d53","origin":"The Stacks Project","memory_eligible":false,"source_rank":13953,"rank":13953,"depth":50,"x":2600.055,"y":1588.43,"cluster":"moduli-theory"},{"id":"stacks:0D4V","tag":"0D4V","title":"The stack of polarized proper schemes · Lemma 0D4V","summary":"The stack in groupoids Polarizedstack satisfies openness of versality over Spec(Z). Similarly, after base change (Remark [Tag 0D1N]) openness of versality holds over any Noetherian base scheme S.","statement_latex":"The stack in groupoids $\\Polarizedstack$\nsatisfies openness of versality over $\\Spec(\\mathbf{Z})$.\nSimilarly, after base change (Remark \\ref{remark-polarized-base-change})\nopenness of versality holds over any Noetherian base scheme $S$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4V","source_file":"quot.tex","source_line":4426,"source_end_line":4432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4426-L4432","statement_sha256":"ebbcb3dbd5aa0fbc40be590e26025b568ae3cfc6401d4ba8431f98fe0fa8aed6","origin":"The Stacks Project","memory_eligible":false,"source_rank":13954,"rank":13954,"depth":89,"x":2400.459,"y":1676.962,"cluster":"moduli-theory"},{"id":"stacks:0D4X","tag":"0D4X","title":"Algebraicity of the stack of polarized schemes · Theorem 0D4X","summary":"The stack Polarizedstack (Situation [Tag 0D1M]) is algebraic. In fact, for any algebraic space B the stack B-Polarized (Remark [Tag 0D1N]) is algebraic.","statement_latex":"The stack $\\Polarizedstack$ (Situation \\ref{situation-polarized})\nis algebraic. In fact, for any algebraic space $B$ the stack\n$B\\textit{-Polarized}$ (Remark \\ref{remark-polarized-base-change})\nis algebraic.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of polarized proper schemes","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D4X","source_file":"quot.tex","source_line":4463,"source_end_line":4469,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4463-L4469","statement_sha256":"c6d270f804d0b2f149c2cb99534e9e8d43ad042537ae80972116101eadfd584e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13955,"rank":13955,"depth":90,"x":2476.749,"y":1497.71,"cluster":"moduli-theory"},{"id":"stacks:0D51","tag":"0D51","title":"The stack of curves · Lemma 0D51","summary":"The category Curvesstack is fibred in groupoids over Sch_fppf.","statement_latex":"The category $\\Curvesstack$ is fibred in groupoids over $\\Sch_{fppf}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D51","source_file":"quot.tex","source_line":4570,"source_end_line":4573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4570-L4573","statement_sha256":"4a54b0a219ccc17a22f79e00ae75695aa7aa86c5b87fbbd388b39290543b0218","origin":"The Stacks Project","memory_eligible":false,"source_rank":13956,"rank":13956,"depth":28,"x":2564.991,"y":1673.873,"cluster":"moduli-theory"},{"id":"stacks:0D52","tag":"0D52","title":"The stack of curves · Lemma 0D52","summary":"The category Curvesstack is a stack in groupoids over Sch_fppf.","statement_latex":"The category $\\Curvesstack$ is a stack in groupoids over $\\Sch_{fppf}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D52","source_file":"quot.tex","source_line":4595,"source_end_line":4598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4595-L4598","statement_sha256":"9d5a86739bc8035116de6a931e300d67e1109e8d0882bf9159b45495faaae89f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13957,"rank":13957,"depth":86,"x":2357.443,"y":1593.729,"cluster":"moduli-theory"},{"id":"stacks:0D53","tag":"0D53","title":"The stack of curves · Lemma 0D53","summary":"The diagonal Δ : Curvesstack → Curvesstack × Curvesstack is representable by algebraic spaces.","statement_latex":"The diagonal\n$$\n\\Delta : \\Curvesstack \\longrightarrow \\Curvesstack \\times \\Curvesstack\n$$\nis representable by algebraic spaces.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D53","source_file":"quot.tex","source_line":4618,"source_end_line":4625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4618-L4625","statement_sha256":"71638407160e71c7c5a6cbf7e8d3ccb5b1011bd6ce8ab6047f5ef4ac3adb57ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":13958,"rank":13958,"depth":85,"x":2575.786,"y":1534.83,"cluster":"moduli-theory"},{"id":"stacks:0D55","tag":"0D55","title":"The stack of curves · Lemma 0D55","summary":"The stack Curvesstack → Sch_fppf is limit preserving (Artin's Axioms, Definition [Tag 07XL]).","statement_latex":"The stack $\\Curvesstack \\to \\Sch_{fppf}$ is limit preserving\n(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving}).","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D55","source_file":"quot.tex","source_line":4657,"source_end_line":4661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4657-L4661","statement_sha256":"5c1f7fcc6609086e638c44fcd6fc3482e1da059b7a4f967ecfa850e2ad7946ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":13959,"rank":13959,"depth":60,"x":2461.706,"y":1702.803,"cluster":"moduli-theory"},{"id":"stacks:0D56","tag":"0D56","title":"The stack of curves · Lemma 0D56","summary":"Let xymatrix T ar[r] ar[d] & T' ar[d] S ar[r] & S' be a pushout in the category of schemes where T → T' is a thickening and T → S is affine, see More on Morphisms, Lemma [Tag 07RT]. Then the functor on fibre categories Curvesstack_S' → Curvesstack_S ×_Curvesstack_T Curvesstack_T' is an equivalence.","statement_latex":"Let\n$$\n\\xymatrix{\nT \\ar[r] \\ar[d] & T' \\ar[d] \\\\\nS \\ar[r] & S'\n}\n$$\nbe a pushout in the category of schemes where\n$T \\to T'$ is a thickening and $T \\to S$ is affine, see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.\nThen the functor on fibre categories\n$$\n\\Curvesstack_{S'}\n\\longrightarrow\n\\Curvesstack_S\n\\times_{\\Curvesstack_T}\n\\Curvesstack_{T'}\n$$\nis an equivalence.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D56","source_file":"quot.tex","source_line":4682,"source_end_line":4703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4682-L4703","statement_sha256":"08155f063ea502efc43a144c2750be92371efeb36bd4a7a08ae871cc7a805db3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13960,"rank":13960,"depth":75,"x":2410.554,"y":1513.484,"cluster":"moduli-theory"},{"id":"stacks:0D57","tag":"0D57","title":"The stack of curves · Lemma 0D57","summary":"Let k be a field and let x = (X → Spec(k)) be an object of X = Curvesstack over Spec(k). • If k is of finite type over Z, then the vector spaces TF_X, k, x and Inf(F_X, k, x) (see Artin's Axioms, Section [Tag 07WY]) are finite dimensional, and • in general the vector spaces T_x(k) and Inf_x(k) (see Artin's Axioms, Section [Tag 07Y6]) are finite dimensional.","statement_latex":"Let $k$ be a field and let $x = (X \\to \\Spec(k))$ be an object of\n$\\mathcal{X} = \\Curvesstack$ over $\\Spec(k)$.\n\\begin{enumerate}\n\\item If $k$ is of finite type over $\\mathbf{Z}$, then\nthe vector spaces $T\\mathcal{F}_{\\mathcal{X}, k, x}$ and\n$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x})$\n(see Artin's Axioms, Section \\ref{artin-section-tangent-spaces})\nare finite dimensional, and\n\\item in general the vector spaces $T_x(k)$ and $\\text{Inf}_x(k)$\n(see Artin's Axioms, Section \\ref{artin-section-inf})\nare finite dimensional.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D57","source_file":"quot.tex","source_line":4728,"source_end_line":4742,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4728-L4742","statement_sha256":"a847d31e23f70e15e41a17a4a5aac2811bee2af630082dabbf5713bff24578a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13961,"rank":13961,"depth":76,"x":2601.242,"y":1624.482,"cluster":"moduli-theory"},{"id":"stacks:0D58","tag":"0D58","title":"The stack of curves · Lemma 0D58","summary":"Consider the stack Curvesstack over the base scheme Spec(Z). Then every formal object is effective.","statement_latex":"Consider the stack $\\Curvesstack$ over the base\nscheme $\\Spec(\\mathbf{Z})$. Then every formal object is effective.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D58","source_file":"quot.tex","source_line":4751,"source_end_line":4755,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4751-L4755","statement_sha256":"ef3f09aa891d6d9081d69fef9be4d3896232a8af0c337e6678879dc4fac6fc99","origin":"The Stacks Project","memory_eligible":false,"source_rank":13962,"rank":13962,"depth":70,"x":2370.499,"y":1650.934,"cluster":"moduli-theory"},{"id":"stacks:0D59","tag":"0D59","title":"The stack of curves · Lemma 0D59","summary":"The stack in groupoids X = Curvesstack satisfies openness of versality over Spec(Z). Similarly, after base change (Remark [Tag 0D54]) openness of versality holds over any Noetherian base scheme S.","statement_latex":"The stack in groupoids $\\mathcal{X} = \\Curvesstack$\nsatisfies openness of versality over $\\Spec(\\mathbf{Z})$.\nSimilarly, after base change (Remark \\ref{remark-curves-base-change})\nopenness of versality holds over any Noetherian base scheme $S$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D59","source_file":"quot.tex","source_line":4776,"source_end_line":4782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4776-L4782","statement_sha256":"2b3a256db84acab3f60eeded64d35e3bd1fcac95192383026306cae73656dc17","origin":"The Stacks Project","memory_eligible":false,"source_rank":13963,"rank":13963,"depth":86,"x":2519.934,"y":1499.935,"cluster":"moduli-theory"},{"id":"stacks:0D5A","tag":"0D5A","title":"Algebraicity of the stack of curves · Theorem 0D5A","summary":"See [dJHS] and [Smyth]. The stack Curvesstack (Situation [Tag 0D4Z]) is algebraic. In fact, for any algebraic space B the stack B-Curvesstack (Remark [Tag 0D54]) is algebraic.","statement_latex":"\\begin{reference}\nSee \\cite[Proposition 3.3, page 8]{dJHS} and\n\\cite[Appendix B by Jack Hall, Theorem B.1]{Smyth}.\n\\end{reference}\nThe stack $\\Curvesstack$ (Situation \\ref{situation-curves})\nis algebraic. In fact, for any algebraic space $B$ the stack\n$B\\text{-}\\Curvesstack$ (Remark \\ref{remark-curves-base-change})\nis algebraic.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5A","source_file":"quot.tex","source_line":4791,"source_end_line":4801,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4791-L4801","statement_sha256":"6c59ec23bec88559adae988932970657f411190391c63fbd9ddb2ed343c674ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":13964,"rank":13964,"depth":87,"x":2531.212,"y":1696.804,"cluster":"moduli-theory"},{"id":"stacks:0D5B","tag":"0D5B","title":"The stack of curves · Lemma 0D5B","summary":"The 1-morphism ([Tag 0D50]) Curvesstack → Spacesstack'_fp, flat, proper is representable by open and closed immersions.","statement_latex":"The $1$-morphism (\\ref{equation-curves-over-proper-spaces})\n$$\n\\Curvesstack \\longrightarrow \\Spacesstack'_{fp, flat, proper}\n$$\nis representable by open and closed immersions.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5B","source_file":"quot.tex","source_line":4818,"source_end_line":4825,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4818-L4825","statement_sha256":"d4e6a1b4c228bb224b78698acb6fc8db64978d2ab78c5787560418aabeba5de5","origin":"The Stacks Project","memory_eligible":false,"source_rank":13965,"rank":13965,"depth":52,"x":2363.963,"y":1557.521,"cluster":"moduli-theory"},{"id":"stacks:0DLC","tag":"0DLC","title":"Moduli of complexes on a proper morphism · Lemma 0DLC","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is proper, flat, and of finite presentation. Let K, E ∈ D(O_X). Assume K is pseudo-coherent and E is Y-perfect (More on Morphisms of Spaces, Definition [Tag 0DKN]). For a field k and a morphism y : Spec(k) → Y denote K_y, E_y the pullback to the fibre X_y. • There is an open W ⊂ Y characterized by the property y ∈ |W| ⇔ Ext^i_O_X_y(K_y, E_y) = 0 for i < 0. • For any morphism V → Y…","statement_latex":"Let $S$ be a scheme.\nLet $f : X \\to Y$ be a morphism of algebraic spaces over $S$.\nAssume $f$ is proper, flat, and of finite presentation.\nLet $K, E \\in D(\\mathcal{O}_X)$. Assume $K$ is pseudo-coherent\nand $E$ is $Y$-perfect (More on Morphisms of Spaces, Definition\n\\ref{spaces-more-morphisms-definition-relatively-perfect}).\nFor a field $k$ and a morphism $y : \\Spec(k) \\to Y$ denote $K_y$, $E_y$\nthe pullback to the fibre $X_y$.\n\\begin{enumerate}\n\\item There is an open $W \\subset Y$ characterized by the property\n$$\ny \\in |W|\n\\Leftrightarrow\n\\Ext^i_{\\mathcal{O}_{X_y}}(K_y, E_y) = 0\n\\text{ for }i < 0.\n$$\n\\item For any morphism $V \\to Y$ factoring through $W$ we have\n$$\n\\Ext^i_{\\mathcal{O}_{X_V}}(K_V, E_V) = 0\n\\quad\\text{for}\\quad i < 0\n$$\nwhere $X_V$ is the base change of $X$ and $K_V$ and $E_V$\nare the derived pullbacks of $K$ and $E$ to $X_V$.\n\\item The functor $V \\mapsto \\Hom_{\\mathcal{O}_{X_V}}(K_V, E_V)$\nis a sheaf on $(\\textit{Spaces}/W)_{fppf}$ representable by an\nalgebraic space affine and of finite presentation over $W$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLC","source_file":"quot.tex","source_line":4936,"source_end_line":4965,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L4936-L4965","statement_sha256":"06fda97768e9040ebaf66a6411381883447a688f087135813b4cbc8e98e3c49c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13966,"rank":13966,"depth":69,"x":2600.163,"y":1565.355,"cluster":"moduli-theory"},{"id":"stacks:0DLD","tag":"0DLD","title":"Moduli of complexes on a proper morphism · Lemma 0DLD","summary":"Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. Assume f is proper, flat, and of finite presentation. Let E ∈ D(O_X). Assume • E is S-perfect (More on Morphisms of Spaces, Definition [Tag 0DKN]), and • for every point s ∈ S we have Ext^i_O_X_s(E_s, E_s) = 0 for i < 0 where E_s is the pullback to the fibre X_s. Then • [(a)] (1) and (2) are preserved by arbitrary base change V → Y, • [(b)] Ext^i_O_X_V(E_V, E_V) = 0 for i < 0 and all V over Y, •…","statement_latex":"Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic\nspaces over $S$. Assume $f$ is proper, flat, and of finite presentation.\nLet $E \\in D(\\mathcal{O}_X)$.\nAssume\n\\begin{enumerate}\n\\item $E$ is $S$-perfect (More on Morphisms of Spaces, Definition\n\\ref{spaces-more-morphisms-definition-relatively-perfect}), and\n\\item for every point $s \\in S$ we have\n$$\n\\Ext^i_{\\mathcal{O}_{X_s}}(E_s, E_s) = 0\n\\quad\\text{for}\\quad i < 0\n$$\nwhere $E_s$ is the pullback to the fibre $X_s$.\n\\end{enumerate}\nThen\n\\begin{enumerate}\n\\item[(a)] (1) and (2) are preserved by arbitrary base change $V \\to Y$,\n\\item[(b)] $\\Ext^i_{\\mathcal{O}_{X_V}}(E_V, E_V) = 0$ for $i < 0$\nand all $V$ over $Y$,\n\\item[(c)] $V \\mapsto \\Hom_{\\mathcal{O}_{X_V}}(E_V, E_V)$ is representable\nby an algebraic space affine and of finite presentation over $Y$.\n\\end{enumerate}\nHere $X_V$ is the base change of $X$ and $E_V$ is the derived pullback\nof $E$ to $X_V$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLD","source_file":"quot.tex","source_line":5104,"source_end_line":5130,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L5104-L5130","statement_sha256":"2c5920d669dc5658b42b533dff394071522f7e2675aaa339457d9e4966c93754","origin":"The Stacks Project","memory_eligible":false,"source_rank":13967,"rank":13967,"depth":70,"x":2419.034,"y":1694.071,"cluster":"moduli-theory"},{"id":"stacks:0DLF","tag":"0DLF","title":"Moduli of complexes on a proper morphism · Lemma 0DLF","summary":"In Situation [Tag 0DLE] the functor p : Complexesstack_X/B → (Sch/S)_fppf is fibred in groupoids.","statement_latex":"In Situation \\ref{situation-complexes} the functor\n$p : \\Complexesstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$\nis fibred in groupoids.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLF","source_file":"quot.tex","source_line":5170,"source_end_line":5175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L5170-L5175","statement_sha256":"cdb7c8dd9ca64ea9432ff977394b624e1faec68d57dff47968a66450e4715f17","origin":"The Stacks Project","memory_eligible":false,"source_rank":13968,"rank":13968,"depth":1,"x":2449.196,"y":1495.665,"cluster":"moduli-theory"},{"id":"stacks:0DLG","tag":"0DLG","title":"Moduli of complexes on a proper morphism · Lemma 0DLG","summary":"In Situation [Tag 0DLE]. Denote X = Complexesstack_X/B. Then Δ : X → X × X is representable by algebraic spaces.","statement_latex":"In Situation \\ref{situation-complexes}. Denote\n$\\mathcal{X} = \\Complexesstack_{X/B}$. Then\n$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$ is\nrepresentable by algebraic spaces.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLG","source_file":"quot.tex","source_line":5208,"source_end_line":5214,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L5208-L5214","statement_sha256":"6ae9b970ad93131d6fbb7020897edb187bfaada4378edea452247255356197fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13969,"rank":13969,"depth":73,"x":2587.0,"y":1659.669,"cluster":"moduli-theory"},{"id":"stacks:0DLH","tag":"0DLH","title":"Moduli of complexes on a proper morphism · Lemma 0DLH","summary":"In Situation [Tag 0DLE] the functor p : Complexesstack_X/B → (Sch/S)_fppf is a stack in groupoids.","statement_latex":"In Situation \\ref{situation-complexes} the functor\n$p : \\Complexesstack_{X/B} \\longrightarrow (\\Sch/S)_{fppf}$\nis a stack in groupoids.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLH","source_file":"quot.tex","source_line":5274,"source_end_line":5279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L5274-L5279","statement_sha256":"98ce7916d9f0b236eb6ff58a55907ee609ea9a50a859eb09ecec11c381599461","origin":"The Stacks Project","memory_eligible":false,"source_rank":13970,"rank":13970,"depth":74,"x":2352.664,"y":1616.775,"cluster":"moduli-theory"},{"id":"stacks:0DLJ","tag":"0DLJ","title":"Moduli of complexes on a proper morphism · Lemma 0DLJ","summary":"In Situation [Tag 0DLE] assume that B → S is locally of finite presentation. Then p : Complexesstack_X/B → (Sch/S)_fppf is limit preserving (Artin's Axioms, Definition [Tag 07XL]).","statement_latex":"In Situation \\ref{situation-complexes} assume that $B \\to S$\nis locally of finite presentation. Then\n$p : \\Complexesstack_{X/B} \\to (\\Sch/S)_{fppf}$ is limit preserving\n(Artin's Axioms, Definition \\ref{artin-definition-limit-preserving}).","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLJ","source_file":"quot.tex","source_line":5395,"source_end_line":5401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L5395-L5401","statement_sha256":"47211a45743e04a8eab2ee358c097d9c754712629babb843365b28311d641abb","origin":"The Stacks Project","memory_eligible":false,"source_rank":13971,"rank":13971,"depth":71,"x":2560.69,"y":1515.078,"cluster":"moduli-theory"},{"id":"stacks:0DLK","tag":"0DLK","title":"Moduli of complexes on a proper morphism · Lemma 0DLK","summary":"In Situation [Tag 0DLE]. Let xymatrix Z ar[r] ar[d] & Z' ar[d] Y ar[r] & Y' be a pushout in the category of schemes over S where Z → Z' is a finite order thickening and Z → Y is affine, see More on Morphisms, Lemma [Tag 07RT]. Then the functor on fibre categories Complexesstack_X/B, Y' → Complexesstack_X/B, Y ×_Complexesstack_X/B, Z Complexesstack_X/B, Z' is an equivalence.","statement_latex":"In Situation \\ref{situation-complexes}. Let\n$$\n\\xymatrix{\nZ \\ar[r] \\ar[d] & Z' \\ar[d] \\\\\nY \\ar[r] & Y'\n}\n$$\nbe a pushout in the category of schemes over $S$ where\n$Z \\to Z'$ is a finite order thickening and $Z \\to Y$ is affine, see\nMore on Morphisms, Lemma \\ref{more-morphisms-lemma-pushout-along-thickening}.\nThen the functor on fibre categories\n$$\n\\Complexesstack_{X/B, Y'}\n\\longrightarrow\n\\Complexesstack_{X/B, Y}\n\\times_{\\Complexesstack_{X/B, Z}}\n\\Complexesstack_{X/B, Z'}\n$$\nis an equivalence.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLK","source_file":"quot.tex","source_line":5447,"source_end_line":5468,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L5447-L5468","statement_sha256":"29d2b8ac14e13c9bc0918adf8ca753206c555380f2f4e83155362d2fb226f1ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":13972,"rank":13972,"depth":75,"x":2488.823,"y":1708.785,"cluster":"moduli-theory"},{"id":"stacks:0DLL","tag":"0DLL","title":"Moduli of complexes on a proper morphism · Lemma 0DLL","summary":"In Situation [Tag 0DLE] assume that S is a locally Noetherian scheme and B → S is locally of finite presentation. Let k be a finite type field over S and let x_0 = (Spec(k), g_0, E_0) be an object of X = Complexesstack_X/B over k. Then the spaces TF_X, k, x_0 and Inf(F_X, k, x_0) (Artin's Axioms, Section [Tag 07WY]) are finite dimensional.","statement_latex":"In Situation \\ref{situation-complexes} assume that $S$ is a locally Noetherian\nscheme and $B \\to S$ is locally of finite presentation.\nLet $k$ be a finite type field over $S$ and let\n$x_0 = (\\Spec(k), g_0, E_0)$\nbe an object of $\\mathcal{X} = \\Complexesstack_{X/B}$ over $k$.\nThen the spaces $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$ and\n$\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$\n(Artin's Axioms, Section \\ref{artin-section-tangent-spaces})\nare finite dimensional.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLL","source_file":"quot.tex","source_line":5546,"source_end_line":5557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L5546-L5557","statement_sha256":"55ed0f780e8065c8655f5abf8f78617b9bf65c3d2d293d1ad6e55e08ccf4e513","origin":"The Stacks Project","memory_eligible":false,"source_rank":13973,"rank":13973,"depth":77,"x":2385.685,"y":1524.528,"cluster":"moduli-theory"},{"id":"stacks:0DLM","tag":"0DLM","title":"Moduli of complexes on a proper morphism · Lemma 0DLM","summary":"In Situation [Tag 0DLE] assume B = S is locally Noetherian. Then strong formal effectiveness in the sense of Artin's Axioms, Remark [Tag 0CXT] holds for p : Complexesstack_X/S → (Sch/S)_fppf.","statement_latex":"In Situation \\ref{situation-complexes} assume $B = S$ is locally Noetherian.\nThen strong formal effectiveness in the sense of\nArtin's Axioms, Remark \\ref{artin-remark-strong-effectiveness}\nholds for $p : \\Complexesstack_{X/S} \\to (\\Sch/S)_{fppf}$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLM","source_file":"quot.tex","source_line":5628,"source_end_line":5634,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L5628-L5634","statement_sha256":"6905271375678bd4caadd8364f2578c0e918b1f4b74f8bb1e7269b9b6b517c3d","origin":"The Stacks Project","memory_eligible":false,"source_rank":13974,"rank":13974,"depth":71,"x":2610.689,"y":1602.142,"cluster":"moduli-theory"},{"id":"stacks:0DLN","tag":"0DLN","title":"Algebraicity of moduli of complexes on a proper morphism · Theorem 0DLN","summary":"[lieblich-complexes] Let S be a scheme. Let f : X → B be morphism of algebraic spaces over S. Assume that f is proper, flat, and of finite presentation. Then Complexesstack_X/B is an algebraic stack over S.","statement_latex":"\\begin{reference}\n\\cite{lieblich-complexes}\n\\end{reference}\nLet $S$ be a scheme. Let $f : X \\to B$ be morphism of algebraic spaces\nover $S$. Assume that $f$ is proper, flat, and of finite presentation.\nThen $\\Complexesstack_{X/B}$ is an algebraic stack over $S$.","area":"Moduli Theory","chapter":"Quot and Hilbert Spaces","chapter_id":"quot","section":"Moduli of complexes on a proper morphism","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLN","source_file":"quot.tex","source_line":5675,"source_end_line":5683,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/quot.tex#L5675-L5683","statement_sha256":"0a1de9207678cae7833b2a267566507b1b4815ed8a6c0cbe0552ac179a3d27a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13975,"rank":13975,"depth":80,"x":2381.572,"y":1672.825,"cluster":"moduli-theory"},{"id":"stacks:04ZP","tag":"04ZP","title":"Properties of morphisms representable by algebraic spaces · Lemma 04ZP","summary":"Let f : X → Y be a morphism of algebraic stacks. Let W be an algebraic space and let W → Y be surjective, locally of finite presentation, and flat. The following are equivalent • f is representable by algebraic spaces, and • W ×_Y X is an algebraic space.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $W$ be an algebraic space and let $W \\to \\mathcal{Y}$ be surjective,\nlocally of finite presentation, and flat. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is representable by algebraic spaces, and\n\\item $W \\times_\\mathcal{Y} \\mathcal{X}$ is an algebraic space.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZP","source_file":"stacks-properties.tex","source_line":176,"source_end_line":185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L176-L185","statement_sha256":"63a1c1103b4265c4998fe84efb05f00da6a93ea22a2b80f94808dd3d4ac4a83a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13976,"rank":13976,"depth":69,"x":2035.041,"y":1687.215,"cluster":"algebraic-stacks"},{"id":"stacks:04XC","tag":"04XC","title":"Properties of morphisms representable by algebraic spaces · Lemma 04XC","summary":"Let P be a property of morphisms of algebraic spaces as above. Let f : X → Y be a morphism of algebraic stacks representable by algebraic spaces. The following are equivalent: • f has P, • for every algebraic space Z and morphism Z → Y the morphism Z ×_Y X → Z has P.","statement_latex":"Let $P$ be a property of morphisms of algebraic spaces as above.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nrepresentable by algebraic spaces. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ has $P$,\n\\item for every algebraic space $Z$ and morphism $Z \\to \\mathcal{Y}$\nthe morphism $Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$ has $P$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XC","source_file":"stacks-properties.tex","source_line":432,"source_end_line":442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L432-L442","statement_sha256":"e3a131f66bcf1243aff7b1e328512448a86e10e333764e971d60fb1eaaf16ddd","origin":"The Stacks Project","memory_eligible":false,"source_rank":13977,"rank":13977,"depth":0,"x":1956.02,"y":1538.444,"cluster":"algebraic-stacks"},{"id":"stacks:04XD","tag":"04XD","title":"Properties of morphisms representable by algebraic spaces · Lemma 04XD","summary":"Let P be a property of morphisms of algebraic spaces as above. Let f : X → Y be a morphism of algebraic stacks representable by algebraic spaces. Let W be an algebraic space and let W → Y be surjective, locally of finite presentation, and flat. Set V = W ×_Y X. Then (f has P) ⇔ (the projection V → W has P).","statement_latex":"Let $P$ be a property of morphisms of algebraic spaces as above.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nrepresentable by algebraic spaces.\nLet $W$ be an algebraic space and let $W \\to \\mathcal{Y}$ be surjective,\nlocally of finite presentation, and flat.\nSet $V = W \\times_\\mathcal{Y} \\mathcal{X}$. Then\n$$\n(f\\text{ has }P) \\Leftrightarrow (\\text{the projection }V \\to W\\text{ has }P).\n$$","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XD","source_file":"stacks-properties.tex","source_line":465,"source_end_line":476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L465-L476","statement_sha256":"673d24ea8b48eecae25a9516de5ac8d0b01a79b53068cbc76f7143505041a874","origin":"The Stacks Project","memory_eligible":false,"source_rank":13978,"rank":13978,"depth":1,"x":2134.232,"y":1603.419,"cluster":"algebraic-stacks"},{"id":"stacks:06TY","tag":"06TY","title":"Properties of morphisms representable by algebraic spaces · Lemma 06TY","summary":"Let P be a property of morphisms of algebraic spaces as above. Let f : X → Y be a morphism of algebraic stacks representable by algebraic spaces. Let Z → Y be a morphism of algebraic stacks which is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Set W = Z ×_Y X. Then (f has P) ⇔ (the projection W → Z has P).","statement_latex":"Let $P$ be a property of morphisms of algebraic spaces as above.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nrepresentable by algebraic spaces.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which\nis representable by algebraic spaces, surjective, flat, and\nlocally of finite presentation.\nSet $\\mathcal{W} = \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$. Then\n$$\n(f\\text{ has }P) \\Leftrightarrow\n(\\text{the projection }\\mathcal{W} \\to \\mathcal{Z}\\text{ has }P).\n$$","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TY","source_file":"stacks-properties.tex","source_line":498,"source_end_line":511,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L498-L511","statement_sha256":"7b661e3559fc4e6cb2e17e6d55e5b40eb611c47f937624188c87a63962fc6661","origin":"The Stacks Project","memory_eligible":false,"source_rank":13979,"rank":13979,"depth":2,"x":1950.255,"y":1656.72,"cluster":"algebraic-stacks"},{"id":"stacks:06M2","tag":"06M2","title":"Properties of morphisms representable by algebraic spaces · Lemma 06M2","summary":"Let P be a property of morphisms of algebraic spaces as above. Let τ ∈ (etale, smooth, syntomic, fppf). Let X → Y and Y → Z be morphisms of algebraic stacks representable by algebraic spaces. Assume • X → Y is surjective and étale, smooth, syntomic, or flat and locally of finite presentation, • the composition has P, and • P is local on the source in the τ topology. Then Y → Z has property P.","statement_latex":"Let $P$ be a property of morphisms of algebraic spaces as above.\nLet $\\tau \\in \\{\\etale, smooth, syntomic, fppf\\}$.\nLet $\\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Y} \\to \\mathcal{Z}$\nbe morphisms of algebraic stacks representable by algebraic spaces.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{X} \\to \\mathcal{Y}$ is surjective and\n\\'etale, smooth, syntomic, or flat and locally of finite presentation,\n\\item the composition has $P$, and\n\\item $P$ is local on the source in the $\\tau$ topology.\n\\end{enumerate}\nThen $\\mathcal{Y} \\to \\mathcal{Z}$ has property $P$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06M2","source_file":"stacks-properties.tex","source_line":527,"source_end_line":541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L527-L541","statement_sha256":"f60681cb252b1c8233825914d81918f6a7f3340034600b1a319cec436581518b","origin":"The Stacks Project","memory_eligible":false,"source_rank":13980,"rank":13980,"depth":0,"x":2043.212,"y":1512.778,"cluster":"algebraic-stacks"},{"id":"stacks:04Y6","tag":"04Y6","title":"Properties of morphisms representable by algebraic spaces · Lemma 04Y6","summary":"Let g : X' → X be a morphism of algebraic stacks which is representable by algebraic spaces. Let [U/R] → X be a presentation. Set U' = U ×_X X', and R' = R ×_X X'. Then there exists a groupoid in algebraic spaces of the form (U', R', s', t', c'), a presentation [U'/R'] → X', and the diagram xymatrix [U'/R'] ar[d]_[pr] ar[r] & X' ar[d]^g [U/R] ar[r] & X is 2-commutative where the morphism [pr] comes from a morphism of groupoids pr : (U', R', s', t', c') → (U, R, s, t, c).","statement_latex":"Let $g : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks\nwhich is representable by algebraic spaces. Let $[U/R] \\to \\mathcal{X}$\nbe a presentation. Set $U' = U \\times_\\mathcal{X} \\mathcal{X}'$,\nand $R' = R \\times_\\mathcal{X} \\mathcal{X}'$.\nThen there exists a groupoid in algebraic spaces of the form\n$(U', R', s', t', c')$, a presentation $[U'/R'] \\to \\mathcal{X}'$,\nand the diagram\n$$\n\\xymatrix{\n[U'/R'] \\ar[d]_{[\\text{pr}]} \\ar[r] & \\mathcal{X}' \\ar[d]^g \\\\\n[U/R] \\ar[r] & \\mathcal{X}\n}\n$$\nis $2$-commutative where the morphism $[\\text{pr}]$ comes from a\nmorphism of groupoids\n$\\text{pr} : (U', R', s', t', c') \\to (U, R, s, t, c)$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of morphisms representable by algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Y6","source_file":"stacks-properties.tex","source_line":552,"source_end_line":570,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L552-L570","statement_sha256":"f840e38070e41b177d5a80665ce73dc5ece5410816818bbc7f9c104efd2729c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13981,"rank":13981,"depth":71,"x":2090.501,"y":1671.935,"cluster":"algebraic-stacks"},{"id":"stacks:04XF","tag":"04XF","title":"Points of algebraic stacks · Lemma 04XF","summary":"The notion above does indeed define an equivalence relation on morphisms from spectra of fields into the algebraic stack X.","statement_latex":"The notion above does indeed define an equivalence relation on\nmorphisms from spectra of fields into the algebraic stack $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Points of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XF","source_file":"stacks-properties.tex","source_line":709,"source_end_line":713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L709-L713","statement_sha256":"cc87e8743b83ba161f84695d7f240e14bf8b8c411e263f037615f8fbc355115e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13982,"rank":13982,"depth":0,"x":1927.366,"y":1581.254,"cluster":"algebraic-stacks"},{"id":"stacks:04XG","tag":"04XG","title":"Points of algebraic stacks · Definition 04XG","summary":"Let X be an algebraic stack. A point of X is an equivalence class of morphisms from spectra of fields into X. The set of points of X is denoted |X|.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nA {\\it point} of $\\mathcal{X}$ is an equivalence class of morphisms\nfrom spectra of fields into $\\mathcal{X}$.\nThe set of points of $\\mathcal{X}$ is denoted $|\\mathcal{X}|$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Points of algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XG","source_file":"stacks-properties.tex","source_line":762,"source_end_line":768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L762-L768","statement_sha256":"5fcd3cc9b2e62643453092fb92ef6e284493be4e4b57a23b2cfc31ffadb6c61e","origin":"The Stacks Project","memory_eligible":false,"source_rank":13983,"rank":13983,"depth":0,"x":2120.909,"y":1555.511,"cluster":"algebraic-stacks"},{"id":"stacks:04XH","tag":"04XH","title":"Points of algebraic stacks · Lemma 04XH","summary":"Let xymatrix Z ×_Y X ar[r] ar[d] & X ar[d] Z ar[r] & Y be a fibre product of algebraic stacks. Then the map of sets of points |Z ×_Y X| → |Z| ×_|Y| |X| is surjective.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\ar[r] \\ar[d] &\n\\mathcal{X} \\ar[d] \\\\\n\\mathcal{Z} \\ar[r] & \\mathcal{Y}\n}\n$$\nbe a fibre product of algebraic stacks. Then the map of sets\nof points\n$$\n|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}|\n\\longrightarrow\n|\\mathcal{Z}| \\times_{|\\mathcal{Y}|} |\\mathcal{X}|\n$$\nis surjective.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Points of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XH","source_file":"stacks-properties.tex","source_line":801,"source_end_line":819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L801-L819","statement_sha256":"db572933920bc353a050199babb2e0c256d7fee43a8436afdea7987543374ce3","origin":"The Stacks Project","memory_eligible":false,"source_rank":13984,"rank":13984,"depth":0,"x":1998.69,"y":1684.531,"cluster":"algebraic-stacks"},{"id":"stacks:04XI","tag":"04XI","title":"Points of algebraic stacks · Lemma 04XI","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent: • |f| : |X| → |Y| is surjective, and • f is surjective (in the sense of Section [Tag 04XB]).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich is representable by algebraic spaces. The following are equivalent:\n\\begin{enumerate}\n\\item $|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ is surjective, and\n\\item $f$ is surjective\n(in the sense of Section \\ref{section-properties-morphisms}).\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Points of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XI","source_file":"stacks-properties.tex","source_line":833,"source_end_line":842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L833-L842","statement_sha256":"de30c9fda4306e1333d9cddb7f4370204f789217089ff2015cc5cfd5d2d88711","origin":"The Stacks Project","memory_eligible":false,"source_rank":13985,"rank":13985,"depth":1,"x":1985.035,"y":1519.768,"cluster":"algebraic-stacks"},{"id":"stacks:04XJ","tag":"04XJ","title":"Points of algebraic stacks · Lemma 04XJ","summary":"Let X be an algebraic stack. Let X = [U/R] be a presentation of X, see Algebraic Stacks, Definition [Tag 04TI]. Then the image of |R| → |U| × |U| is an equivalence relation and |X| is the quotient of |U| by this equivalence relation.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $\\mathcal{X} = [U/R]$ be a presentation of $\\mathcal{X}$, see\nAlgebraic Stacks, Definition \\ref{algebraic-definition-presentation}.\nThen the image of $|R| \\to |U| \\times |U|$ is an equivalence relation\nand $|\\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Points of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XJ","source_file":"stacks-properties.tex","source_line":884,"source_end_line":891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L884-L891","statement_sha256":"1731f70a442886d5852ee4f03750eb64f5ba08d150fd5ae288fce975c459cad4","origin":"The Stacks Project","memory_eligible":false,"source_rank":13986,"rank":13986,"depth":71,"x":2127.84,"y":1633.704,"cluster":"algebraic-stacks"},{"id":"stacks:04XL","tag":"04XL","title":"Points of algebraic stacks · Lemma 04XL","summary":"There exists a unique topology on the sets of points of algebraic stacks with the following properties: • for every morphism of algebraic stacks X → Y the map |X| → |Y| is continuous, and • for every morphism U → X which is flat and locally of finite presentation with U an algebraic space the map of topological spaces |U| → |X| is continuous and open.","statement_latex":"There exists a unique topology on the sets of points\nof algebraic stacks with the following properties:\n\\begin{enumerate}\n\\item for every morphism of algebraic stacks $\\mathcal{X} \\to \\mathcal{Y}$\nthe map $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is continuous, and\n\\item for every morphism $U \\to \\mathcal{X}$ which is flat and locally\nof finite presentation with $U$ an algebraic space\nthe map of topological spaces $|U| \\to |\\mathcal{X}|$ is continuous and open.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Points of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XL","source_file":"stacks-properties.tex","source_line":937,"source_end_line":948,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L937-L948","statement_sha256":"883265057b68362d0d02dba82149b0464b20fafaa92108172fd2bbf02c58cb90","origin":"The Stacks Project","memory_eligible":false,"source_rank":13987,"rank":13987,"depth":57,"x":1930.585,"y":1630.715,"cluster":"algebraic-stacks"},{"id":"stacks:04Y8","tag":"04Y8","title":"Points of algebraic stacks · Definition 04Y8","summary":"Let X be an algebraic stack. The underlying topological space of X is the set of points |X| endowed with the topology constructed in Lemma [Tag 04XL].","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nThe underlying {\\it topological space} of $\\mathcal{X}$ is the set of points\n$|\\mathcal{X}|$ endowed with the topology constructed in\nLemma \\ref{lemma-topology-points}.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Points of algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Y8","source_file":"stacks-properties.tex","source_line":1037,"source_end_line":1043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1037-L1043","statement_sha256":"8a63644992758caea56f804359e7e57838b62f4697b6c3d05f49e60c212d6a0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":13988,"rank":13988,"depth":58,"x":2078.687,"y":1520.81,"cluster":"algebraic-stacks"},{"id":"stacks:04Y9","tag":"04Y9","title":"Points of algebraic stacks · Lemma 04Y9","summary":"Let X be an algebraic stack. Every point of |X| has a fundamental system of quasi-compact open neighbourhoods. In particular |X| is locally quasi-compact in the sense of Topology, Definition [Tag 0068].","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nEvery point of $|\\mathcal{X}|$ has a fundamental system of\nquasi-compact open neighbourhoods.\nIn particular $|\\mathcal{X}|$ is locally quasi-compact in the sense of\nTopology, Definition \\ref{topology-definition-locally-quasi-compact}.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Points of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Y9","source_file":"stacks-properties.tex","source_line":1049,"source_end_line":1056,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1049-L1056","statement_sha256":"1b1d43a710476da4676871fadd8586bf222dd854de89e53f8dec0c5b366d5fd9","origin":"The Stacks Project","memory_eligible":false,"source_rank":13989,"rank":13989,"depth":58,"x":2057.828,"y":1686.162,"cluster":"algebraic-stacks"},{"id":"stacks:04ZS","tag":"04ZS","title":"Surjective morphisms · Definition 04ZS","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is surjective if the map |f| : |X| → |Y| of associated topological spaces is surjective.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is {\\it surjective} if the map\n$|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ of associated topological spaces\nis surjective.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Surjective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZS","source_file":"stacks-properties.tex","source_line":1090,"source_end_line":1096,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1090-L1096","statement_sha256":"19fddabadcafb938e6e8599175d82a872bd52c39b98dc8a12fedcecdb6eef687","origin":"The Stacks Project","memory_eligible":false,"source_rank":13990,"rank":13990,"depth":0,"x":1940.043,"y":1552.177,"cluster":"algebraic-stacks"},{"id":"stacks:04ZT","tag":"04ZT","title":"Surjective morphisms · Lemma 04ZT","summary":"The composition of surjective morphisms is surjective.","statement_latex":"The composition of surjective morphisms is surjective.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZT","source_file":"stacks-properties.tex","source_line":1101,"source_end_line":1104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1101-L1104","statement_sha256":"4dd4fe35ca16f2c73e3e366b1f2912c07a5b5b2a6386e2f44bd252c2338c7567","origin":"The Stacks Project","memory_eligible":false,"source_rank":13991,"rank":13991,"depth":0,"x":2134.963,"y":1584.195,"cluster":"algebraic-stacks"},{"id":"stacks:04ZU","tag":"04ZU","title":"Surjective morphisms · Lemma 04ZU","summary":"The base change of a surjective morphism is surjective.","statement_latex":"The base change of a surjective morphism is surjective.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZU","source_file":"stacks-properties.tex","source_line":1110,"source_end_line":1113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1110-L1113","statement_sha256":"0d5614dbadbf8741512daa6ee61694cffd32e3cab3ae71b919ab1c129a04b3fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":13992,"rank":13992,"depth":1,"x":1965.206,"y":1671.329,"cluster":"algebraic-stacks"},{"id":"stacks:06PM","tag":"06PM","title":"Surjective morphisms · Lemma 06PM","summary":"Let f : X → Y be a morphism of algebraic stacks. Let Y' → Y be a surjective morphism of algebraic stacks. If the base change f' : Y' ×_Y X → Y' of f is surjective, then f is surjective.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Y}' \\to \\mathcal{Y}$ be a surjective morphism of algebraic\nstacks. If the base change $f' : \\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X}\n\\to \\mathcal{Y}'$ of $f$ is surjective, then $f$ is surjective.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PM","source_file":"stacks-properties.tex","source_line":1120,"source_end_line":1126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1120-L1126","statement_sha256":"a7e16253cdb190b12ac3f81884c56d4d50d33f53f5a067bc670ff6cff882766c","origin":"The Stacks Project","memory_eligible":false,"source_rank":13993,"rank":13993,"depth":1,"x":2020.401,"y":1510.492,"cluster":"algebraic-stacks"},{"id":"stacks:06PN","tag":"06PN","title":"Surjective morphisms · Lemma 06PN","summary":"Let X → Y → Z be morphisms of algebraic stacks. If X → Z is surjective so is Y → Z.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of\nalgebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is surjective\nso is $\\mathcal{Y} \\to \\mathcal{Z}$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Surjective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PN","source_file":"stacks-properties.tex","source_line":1133,"source_end_line":1138,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1133-L1138","statement_sha256":"f1d0f78df949e64187cad0ac037bd757bfd354a409d25eab01eed99c634f0577","origin":"The Stacks Project","memory_eligible":false,"source_rank":13994,"rank":13994,"depth":0,"x":2109.187,"y":1660.654,"cluster":"algebraic-stacks"},{"id":"stacks:04YB","tag":"04YB","title":"Quasi-compact algebraic stacks · Definition 04YB","summary":"Let X be an algebraic stack. We say X is quasi-compact if and only if |X| is quasi-compact.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nWe say $\\mathcal{X}$ is {\\it quasi-compact}\nif and only if $|\\mathcal{X}|$ is quasi-compact.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Quasi-compact algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YB","source_file":"stacks-properties.tex","source_line":1163,"source_end_line":1168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1163-L1168","statement_sha256":"5c6fe23cdbebd2055b664d1a8a9c123fcf8331651b748e77e4010b5027c1dfbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":13995,"rank":13995,"depth":0,"x":1922.659,"y":1600.208,"cluster":"algebraic-stacks"},{"id":"stacks:04YC","tag":"04YC","title":"Quasi-compact algebraic stacks · Lemma 04YC","summary":"Let X be an algebraic stack. The following are equivalent: • X is quasi-compact, • there exists a surjective smooth morphism U → X with U an affine scheme, • there exists a surjective smooth morphism U → X with U a quasi-compact scheme, • there exists a surjective smooth morphism U → X with U a quasi-compact algebraic space, and • there exists a surjective morphism U → X of algebraic stacks such that U is quasi-compact.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is quasi-compact,\n\\item there exists a surjective smooth morphism $U \\to \\mathcal{X}$\nwith $U$ an affine scheme,\n\\item there exists a surjective smooth morphism $U \\to \\mathcal{X}$\nwith $U$ a quasi-compact scheme,\n\\item there exists a surjective smooth morphism $U \\to \\mathcal{X}$\nwith $U$ a quasi-compact algebraic space, and\n\\item there exists a surjective morphism $\\mathcal{U} \\to \\mathcal{X}$\nof algebraic stacks such that $\\mathcal{U}$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Quasi-compact algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YC","source_file":"stacks-properties.tex","source_line":1170,"source_end_line":1185,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1170-L1185","statement_sha256":"7b6b199e617eb6a44e90533b8bf4f6fd980461f6a74cbbf5f6beaedd1a71b044","origin":"The Stacks Project","memory_eligible":false,"source_rank":13996,"rank":13996,"depth":12,"x":2109.112,"y":1538.841,"cluster":"algebraic-stacks"},{"id":"stacks:04YD","tag":"04YD","title":"Quasi-compact algebraic stacks · Lemma 04YD","summary":"A finite disjoint union of quasi-compact algebraic stacks is a quasi-compact algebraic stack.","statement_latex":"A finite disjoint union of quasi-compact algebraic stacks is\na quasi-compact algebraic stack.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Quasi-compact algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YD","source_file":"stacks-properties.tex","source_line":1207,"source_end_line":1211,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1207-L1211","statement_sha256":"360d4cf2c9091f70dceb2daa703a8e07270ead57c97b53f49d87b5ce480cc6a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":13997,"rank":13997,"depth":0,"x":2020.832,"y":1690.133,"cluster":"algebraic-stacks"},{"id":"stacks:04YF","tag":"04YF","title":"Properties of algebraic stacks defined by properties of schemes · Lemma 04YF","summary":"Let P be a property of schemes which is local in the smooth topology, see Descent, Definition [Tag 0348]. Let X be an algebraic stack. The following are equivalent • for some scheme U and some surjective smooth morphism U → X the scheme U has property P, • for every scheme U and every smooth morphism U → X the scheme U has property P, • for some algebraic space U and some surjective smooth morphism U → X the algebraic space U has property P, and • for every algebraic…","statement_latex":"Let $\\mathcal{P}$ be a property of schemes which is local in the smooth\ntopology, see\nDescent, Definition \\ref{descent-definition-property-local}.\nLet $\\mathcal{X}$ be an algebraic stack. The following are equivalent\n\\begin{enumerate}\n\\item for some scheme $U$ and some surjective smooth morphism\n$U \\to \\mathcal{X}$ the scheme $U$ has property $\\mathcal{P}$,\n\\item for every scheme $U$ and every smooth morphism $U \\to \\mathcal{X}$\nthe scheme $U$ has property $\\mathcal{P}$,\n\\item for some algebraic space $U$ and some surjective smooth morphism\n$U \\to \\mathcal{X}$ the algebraic space $U$ has property $\\mathcal{P}$, and\n\\item for every algebraic space $U$ and every smooth morphism\n$U \\to \\mathcal{X}$ the algebraic space $U$ has property $\\mathcal{P}$.\n\\end{enumerate}\nIf $\\mathcal{X}$ is a scheme $U$ this is equivalent to $\\mathcal{P}(U)$.\nIf $\\mathcal{X}$ is an algebraic space $X$ this is equivalent to\n$X$ having property $\\mathcal{P}$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of algebraic stacks defined by properties of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YF","source_file":"stacks-properties.tex","source_line":1230,"source_end_line":1249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1230-L1249","statement_sha256":"340833df628ca8420473f9747fe8964004b58f366ef75cde8f1f99bc3b14e50a","origin":"The Stacks Project","memory_eligible":false,"source_rank":13998,"rank":13998,"depth":48,"x":1964.173,"y":1528.221,"cluster":"algebraic-stacks"},{"id":"stacks:04YG","tag":"04YG","title":"Properties of algebraic stacks defined by properties of schemes · Definition 04YG","summary":"Let X be an algebraic stack. Let P be a property of schemes which is local in the smooth topology. We say X has property P if any of the equivalent conditions of Lemma [Tag 04YF] hold.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $\\mathcal{P}$ be a property of schemes which is\nlocal in the smooth topology.\nWe say $\\mathcal{X}$ {\\it has property $\\mathcal{P}$}\nif any of the equivalent conditions of\nLemma \\ref{lemma-type-property}\nhold.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of algebraic stacks defined by properties of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YG","source_file":"stacks-properties.tex","source_line":1279,"source_end_line":1288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1279-L1288","statement_sha256":"ca5f82790365c4db12b7495972270f57aa9e240748f65aa489ee5e1530639c07","origin":"The Stacks Project","memory_eligible":false,"source_rank":13999,"rank":13999,"depth":49,"x":2136.433,"y":1615.602,"cluster":"algebraic-stacks"},{"id":"stacks:04YI","tag":"04YI","title":"Properties of algebraic stacks defined by properties of schemes · Lemma 04YI","summary":"Let X be an algebraic stack. Let x ∈ |X| be a point of X. Let P be a property of germs of schemes which is smooth local, see Descent, Definition [Tag 04N1]. The following are equivalent • for any smooth morphism U → X with U a scheme and u ∈ U with a(u) = x we have P(U, u), • for some smooth morphism U → X with U a scheme and some u ∈ U with a(u) = x we have P(U, u), • for any smooth morphism U → X with U an algebraic space and u ∈ |U| with a(u) = x the algebraic space U…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $x \\in |\\mathcal{X}|$ be a point of $\\mathcal{X}$.\nLet $\\mathcal{P}$ be a property of germs of schemes which is smooth local, see\nDescent, Definition \\ref{descent-definition-local-at-point}.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any smooth morphism $U \\to \\mathcal{X}$ with $U$ a scheme\nand $u \\in U$ with $a(u) = x$ we have $\\mathcal{P}(U, u)$,\n\\item for some smooth morphism $U \\to \\mathcal{X}$ with $U$ a scheme\nand some $u \\in U$ with $a(u) = x$ we have $\\mathcal{P}(U, u)$,\n\\item for any smooth morphism $U \\to \\mathcal{X}$ with $U$ an algebraic space\nand $u \\in |U|$ with $a(u) = x$ the algebraic space $U$ has property\n$\\mathcal{P}$ at $u$, and\n\\item for some smooth morphism $U \\to \\mathcal{X}$ with $U$ an\nalgebraic space and some $u \\in |U|$ with $a(u) = x$ the algebraic space\n$U$ has property $\\mathcal{P}$ at $u$.\n\\end{enumerate}\nIf $\\mathcal{X}$ is representable, then this is equivalent to\n$\\mathcal{P}(\\mathcal{X}, x)$. If $\\mathcal{X}$ is an algebraic space then\nthis is equivalent to $\\mathcal{X}$ having property $\\mathcal{P}$ at $x$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of algebraic stacks defined by properties of schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YI","source_file":"stacks-properties.tex","source_line":1325,"source_end_line":1347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1325-L1347","statement_sha256":"6c7331e1eb7c7812ee8bf82ee8d7a238a0bd8733221c68029549afeb0ae68049","origin":"The Stacks Project","memory_eligible":false,"source_rank":14000,"rank":14000,"depth":48,"x":1938.826,"y":1648.966,"cluster":"algebraic-stacks"},{"id":"stacks:04YJ","tag":"04YJ","title":"Properties of algebraic stacks defined by properties of schemes · Definition 04YJ","summary":"Let P be a property of germs of schemes which is smooth local. Let X be an algebraic stack. Let x ∈ |X|. We say X has property P at x if any of the equivalent conditions of Lemma [Tag 04YI] holds.","statement_latex":"Let $\\mathcal{P}$ be a property of germs of schemes which is\nsmooth local. Let $\\mathcal{X}$ be an algebraic stack.\nLet $x \\in |\\mathcal{X}|$.\nWe say $\\mathcal{X}$ {\\it has property $\\mathcal{P}$ at $x$}\nif any of the equivalent conditions of\nLemma \\ref{lemma-local-source-target-at-point}\nholds.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Properties of algebraic stacks defined by properties of schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YJ","source_file":"stacks-properties.tex","source_line":1390,"source_end_line":1399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1390-L1399","statement_sha256":"4ad20c9aa57842a8aafd20577531728d0c7a8c4a0135ddd1a3858fdc799e9e99","origin":"The Stacks Project","memory_eligible":false,"source_rank":14001,"rank":14001,"depth":49,"x":2057.897,"y":1512.02,"cluster":"algebraic-stacks"},{"id":"stacks:04ZW","tag":"04ZW","title":"Monomorphisms of algebraic stacks · Definition 04ZW","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is a monomorphism if it is representable by algebraic spaces and a monomorphism in the sense of Section [Tag 04XB].","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is a {\\it monomorphism}\nif it is representable by algebraic spaces and a monomorphism in the sense of\nSection \\ref{section-properties-morphisms}.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Monomorphisms of algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZW","source_file":"stacks-properties.tex","source_line":1419,"source_end_line":1425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1419-L1425","statement_sha256":"7a0ca5b8eb3f08af9db6d11e80d1df8fd4139b5f77618172872f46340ca2ab08","origin":"The Stacks Project","memory_eligible":false,"source_rank":14002,"rank":14002,"depth":0,"x":2080.26,"y":1680.833,"cluster":"algebraic-stacks"},{"id":"stacks:04ZX","tag":"04ZX","title":"Monomorphisms of algebraic stacks · Lemma 04ZX","summary":"Let X → Y be a morphism of algebraic stacks. Let Z → Y be a monomorphism. Then Z ×_Y X → X is a monomorphism.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be a monomorphism.\nThen $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$\nis a monomorphism.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Monomorphisms of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZX","source_file":"stacks-properties.tex","source_line":1430,"source_end_line":1436,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1430-L1436","statement_sha256":"f687f8eea8101ea75cf8a3b6c2fa04baec035c95f22719d641ef880a22179204","origin":"The Stacks Project","memory_eligible":false,"source_rank":14003,"rank":14003,"depth":0,"x":1927.775,"y":1568.865,"cluster":"algebraic-stacks"},{"id":"stacks:04ZY","tag":"04ZY","title":"Monomorphisms of algebraic stacks · Lemma 04ZY","summary":"Compositions of monomorphisms of algebraic stacks are monomorphisms.","statement_latex":"Compositions of monomorphisms of algebraic stacks are monomorphisms.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Monomorphisms of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZY","source_file":"stacks-properties.tex","source_line":1443,"source_end_line":1446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1443-L1446","statement_sha256":"ac2acd81172c7abba8caba57b58ce1dab8bfef3de8ac8573c6275fc0ae58f140","origin":"The Stacks Project","memory_eligible":false,"source_rank":14004,"rank":14004,"depth":1,"x":2130.575,"y":1564.899,"cluster":"algebraic-stacks"},{"id":"stacks:04ZZ","tag":"04ZZ","title":"Monomorphisms of algebraic stacks · Lemma 04ZZ","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent: • f is a monomorphism, • f is fully faithful, • the diagonal Δ_f : X → X ×_Y X is an equivalence, and • there exists an algebraic space W and a surjective, flat morphism W → Y which is locally of finite presentation such that V = X ×_Y W is an algebraic space, and the morphism V → W is a monomorphism of algebraic spaces.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is a monomorphism,\n\\item $f$ is fully faithful,\n\\item the diagonal\n$\\Delta_f : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$\nis an equivalence, and\n\\item there exists an algebraic space $W$ and a surjective, flat morphism\n$W \\to \\mathcal{Y}$ which is locally of finite presentation such that\n$V = \\mathcal{X} \\times_\\mathcal{Y} W$ is an algebraic space, and the\nmorphism $V \\to W$ is a monomorphism of algebraic spaces.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Monomorphisms of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04ZZ","source_file":"stacks-properties.tex","source_line":1456,"source_end_line":1471,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1456-L1471","statement_sha256":"de2b1b430ebc8ac3fc5aebdbd43deae815a9ffb9b6448629d1662361e7f7cab6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14005,"rank":14005,"depth":70,"x":1983.993,"y":1683.082,"cluster":"algebraic-stacks"},{"id":"stacks:0500","tag":"0500","title":"Monomorphisms of algebraic stacks · Lemma 0500","summary":"Monomorphisms of stacks are injective on points. A monomorphism of algebraic stacks induces an injective map of sets of points.","statement_latex":"\\begin{slogan}\nMonomorphisms of stacks are injective on points.\n\\end{slogan}\nA monomorphism of algebraic stacks induces an injective map of\nsets of points.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Monomorphisms of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0500","source_file":"stacks-properties.tex","source_line":1515,"source_end_line":1522,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1515-L1522","statement_sha256":"fe6d664a7e50db7f4978e49c1b37e8381f8b04f993c2af5bf8638dad6cf9003b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14006,"rank":14006,"depth":71,"x":1997.061,"y":1512.495,"cluster":"algebraic-stacks"},{"id":"stacks:0CBB","tag":"0CBB","title":"Monomorphisms of algebraic stacks · Lemma 0CBB","summary":"Let X → X' → Y be morphisms of algebraic stacks. If X → X' is a monomorphism then the canonical diagram xymatrix X ar[r] ar[d] & X ×_Y X ar[d] X' ar[r] & X' ×_Y X' is a fibre product square.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{X}' \\to \\mathcal{Y}$ be morphisms\nof algebraic stacks. If $\\mathcal{X} \\to \\mathcal{X}'$ is a monomorphism\nthen the canonical diagram\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[r] \\ar[d] &\n\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\\\\n\\mathcal{X}' \\ar[r] &\n\\mathcal{X}' \\times_\\mathcal{Y} \\mathcal{X}'\n}\n$$\nis a fibre product square.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Monomorphisms of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBB","source_file":"stacks-properties.tex","source_line":1539,"source_end_line":1553,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1539-L1553","statement_sha256":"f3a484c79c2b3730566ace12fca4212ed5c3bf18bee5be57a82299f23e44ea2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14007,"rank":14007,"depth":71,"x":2124.805,"y":1645.906,"cluster":"algebraic-stacks"},{"id":"stacks:04YL","tag":"04YL","title":"Immersions of algebraic stacks · Definition 04YL","summary":"Immersions. • A morphism of algebraic stacks is called an open immersion if it is representable, and an open immersion in the sense of Section [Tag 04XB]. • A morphism of algebraic stacks is called a closed immersion if it is representable, and a closed immersion in the sense of Section [Tag 04XB]. • A morphism of algebraic stacks is called an immersion if it is representable, and an immersion in the sense of Section [Tag 04XB].","statement_latex":"Immersions.\n\\begin{enumerate}\n\\item A morphism of algebraic stacks is called an {\\it open immersion}\nif it is representable, and an open immersion\nin the sense of\nSection \\ref{section-properties-morphisms}.\n\\item A morphism of algebraic stacks is called a {\\it closed immersion}\nif it is representable, and a closed immersion\nin the sense of\nSection \\ref{section-properties-morphisms}.\n\\item A morphism of algebraic stacks is called an {\\it immersion}\nif it is representable, and an immersion\nin the sense of\nSection \\ref{section-properties-morphisms}.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YL","source_file":"stacks-properties.tex","source_line":1573,"source_end_line":1590,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1573-L1590","statement_sha256":"c143145671e1820d45daed0ea8f5e326e31f771b6504484c53dc360d47fe72d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14008,"rank":14008,"depth":0,"x":1923.01,"y":1619.975,"cluster":"algebraic-stacks"},{"id":"stacks:0501","tag":"0501","title":"Immersions of algebraic stacks · Lemma 0501","summary":"Let X → Y be a morphism of algebraic stacks. Let Z → Y be a (closed, resp. open) immersion. Then Z ×_Y X → X is a (closed, resp. open) immersion.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be a\n(closed, resp.\\ open) immersion.\nThen $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$\nis a (closed, resp.\\ open) immersion.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0501","source_file":"stacks-properties.tex","source_line":1603,"source_end_line":1610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1603-L1610","statement_sha256":"3cf3151a59ccef0be78aeb4643ae7d0a0ad206bbc3c69be6f2d439471fefa771","origin":"The Stacks Project","memory_eligible":false,"source_rank":14009,"rank":14009,"depth":0,"x":2092.927,"y":1524.445,"cluster":"algebraic-stacks"},{"id":"stacks:0502","tag":"0502","title":"Immersions of algebraic stacks · Lemma 0502","summary":"Compositions of immersions of algebraic stacks are immersions. Similarly for closed immersions and open immersions.","statement_latex":"Compositions of immersions of algebraic stacks are immersions.\nSimilarly for closed immersions and open immersions.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0502","source_file":"stacks-properties.tex","source_line":1617,"source_end_line":1621,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1617-L1621","statement_sha256":"ab9624375e16f0ee8040ee9a2602dcd93952a1eff56a2a9b944334d667dd03f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14010,"rank":14010,"depth":4,"x":2044.38,"y":1691.56,"cluster":"algebraic-stacks"},{"id":"stacks:0503","tag":"0503","title":"Immersions of algebraic stacks · Lemma 0503","summary":"Let f : X → Y be a morphism of algebraic stacks. Let W be an algebraic space and let W → Y be a surjective, flat morphism which is locally of finite presentation. The following are equivalent: • f is an (open, resp. closed) immersion, and • V = W ×_Y X is an algebraic space, and V → W is an (open, resp. closed) immersion.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $W$ be an algebraic space and let $W \\to \\mathcal{Y}$ be a surjective,\nflat morphism which is locally of finite presentation. The following\nare equivalent:\n\\begin{enumerate}\n\\item $f$ is an (open, resp.\\ closed) immersion, and\n\\item $V = W \\times_\\mathcal{Y} \\mathcal{X}$ is an algebraic space, and\n$V \\to W$ is an (open, resp.\\ closed) immersion.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0503","source_file":"stacks-properties.tex","source_line":1630,"source_end_line":1641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1630-L1641","statement_sha256":"ea8a3c77c178e0d3b3a313434aa1c24cf6920a66b9d531d4c4963db7537f4c27","origin":"The Stacks Project","memory_eligible":false,"source_rank":14011,"rank":14011,"depth":70,"x":1945.637,"y":1540.553,"cluster":"algebraic-stacks"},{"id":"stacks:0504","tag":"0504","title":"Immersions of algebraic stacks · Lemma 0504","summary":"An immersion is a monomorphism.","statement_latex":"An immersion is a monomorphism.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0504","source_file":"stacks-properties.tex","source_line":1651,"source_end_line":1654,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1651-L1654","statement_sha256":"85d2079fc532d859b335e401083da3e0bea9268f6e7472ab91a1c6d16adef484","origin":"The Stacks Project","memory_eligible":false,"source_rank":14012,"rank":14012,"depth":4,"x":2140.182,"y":1595.959,"cluster":"algebraic-stacks"},{"id":"stacks:0H20","tag":"0H20","title":"Immersions of algebraic stacks · Lemma 0H20","summary":"If f : X → Y is an immersion, then |f| : |X| → |Y| is a homeomorphism onto a locally closed subset. If f is a closed, resp. open immersion, then |f| is closed, resp. open.","statement_latex":"If $f : \\mathcal{X} \\to \\mathcal{Y}$ is an immersion, then\n$|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ is a homeomorphism\nonto a locally closed subset. If $f$ is a closed, resp.\\ open\nimmersion, then $|f|$ is closed, resp.\\ open.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H20","source_file":"stacks-properties.tex","source_line":1662,"source_end_line":1668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1662-L1668","statement_sha256":"08940a9b8bfec873a78616229b945775c521e13998a8bd11c0e8158ddd4dc13b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14013,"rank":14013,"depth":0,"x":1951.884,"y":1665.6,"cluster":"algebraic-stacks"},{"id":"stacks:0505","tag":"0505","title":"Immersions of algebraic stacks · Lemma 0505","summary":"Let (U, R, s, t, c) be a smooth groupoid in algebraic spaces. Let i : Z → [U/R] be an immersion. Then there exists an R-invariant locally closed subspace Z ⊂ U and a presentation [Z/R_Z] → Z where R_Z is the restriction of R to Z such that xymatrix [Z/R_Z] ar[dr] ar[rr] & & Z ar[ld]^i & [U/R] is 2-commutative. If i is a closed (resp. open) immersion then Z is a closed (resp. open) subspace of U.","statement_latex":"Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces.\nLet $i : \\mathcal{Z} \\to [U/R]$ be an immersion.\nThen there exists an $R$-invariant locally closed subspace\n$Z \\subset U$ and a presentation $[Z/R_Z] \\to \\mathcal{Z}$\nwhere $R_Z$ is the restriction of $R$ to $Z$ such that\n$$\n\\xymatrix{\n[Z/R_Z] \\ar[dr] \\ar[rr] & & \\mathcal{Z} \\ar[ld]^i \\\\\n& [U/R]\n}\n$$\nis $2$-commutative. If $i$ is a closed (resp.\\ open) immersion\nthen $Z$ is a closed (resp.\\ open) subspace of $U$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0505","source_file":"stacks-properties.tex","source_line":1678,"source_end_line":1693,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1678-L1693","statement_sha256":"11844a32be597902f3b693ab4d63101ac913b2065139b3c3eaa865490cc65022","origin":"The Stacks Project","memory_eligible":false,"source_rank":14014,"rank":14014,"depth":72,"x":2034.856,"y":1507.162,"cluster":"algebraic-stacks"},{"id":"stacks:04YN","tag":"04YN","title":"Immersions of algebraic stacks · Lemma 04YN","summary":"Let (U, R, s, t, c) be a smooth groupoid in algebraic spaces. Let X = [U/R] be the associated algebraic stack, see Algebraic Stacks, Theorem [Tag 04TK]. Let Z ⊂ U be an R-invariant locally closed subspace. Then [Z/R_Z] → [U/R] is an immersion of algebraic stacks, where R_Z is the restriction of R to Z. If Z ⊂ U is open (resp. closed) then the morphism is an open (resp. closed) immersion of algebraic stacks.","statement_latex":"Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces.\nLet $\\mathcal{X} = [U/R]$ be the associated algebraic stack, see\nAlgebraic Stacks,\nTheorem \\ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}.\nLet $Z \\subset U$ be an $R$-invariant locally closed subspace. Then\n$$\n[Z/R_Z] \\longrightarrow [U/R]\n$$\nis an immersion of algebraic stacks, where $R_Z$ is the restriction\nof $R$ to $Z$. If $Z \\subset U$ is open (resp.\\ closed) then the morphism\nis an open (resp.\\ closed) immersion of algebraic stacks.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YN","source_file":"stacks-properties.tex","source_line":1718,"source_end_line":1731,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1718-L1731","statement_sha256":"0a07d7c9a91ef555fd0a15743894c81c015d21c23d6ea5df5aa5ff89e24c7a7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14015,"rank":14015,"depth":72,"x":2101.186,"y":1671.321,"cluster":"algebraic-stacks"},{"id":"stacks:04YM","tag":"04YM","title":"Immersions of algebraic stacks · Definition 04YM","summary":"Let X be an algebraic stack. • An open substack of X is a strictly full subcategory X' ⊂ X such that X' is an algebraic stack and X' → X is an open immersion. • A closed substack of X is a strictly full subcategory X' ⊂ X such that X' is an algebraic stack and X' → X is a closed immersion. • A locally closed substack of X is a strictly full subcategory X' ⊂ X such that X' is an algebraic stack and X' → X is an immersion.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\n\\begin{enumerate}\n\\item An {\\it open substack} of $\\mathcal{X}$ is a strictly full subcategory\n$\\mathcal{X}' \\subset \\mathcal{X}$ such that $\\mathcal{X}'$ is an algebraic\nstack and $\\mathcal{X}' \\to \\mathcal{X}$ is an open immersion.\n\\item A {\\it closed substack} of $\\mathcal{X}$ is a strictly full subcategory\n$\\mathcal{X}' \\subset \\mathcal{X}$ such that $\\mathcal{X}'$ is an algebraic\nstack and $\\mathcal{X}' \\to \\mathcal{X}$ is a closed immersion.\n\\item A {\\it locally closed substack} of $\\mathcal{X}$ is a strictly full\nsubcategory $\\mathcal{X}' \\subset \\mathcal{X}$ such that $\\mathcal{X}'$\nis an algebraic stack and $\\mathcal{X}' \\to \\mathcal{X}$ is an immersion.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YM","source_file":"stacks-properties.tex","source_line":1768,"source_end_line":1782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1768-L1782","statement_sha256":"23087f2a4160ca7e576b4c54fb43ad7856224d88f601c9e6a4c8df702dd8311d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14016,"rank":14016,"depth":0,"x":1919.988,"y":1587.783,"cluster":"algebraic-stacks"},{"id":"stacks:0506","tag":"0506","title":"Immersions of algebraic stacks · Lemma 0506","summary":"For any immersion i : Z → X there exists a unique locally closed substack X' ⊂ X such that i factors as the composition of an equivalence i' : Z → X' followed by the inclusion morphism X' → X. If i is a closed (resp. open) immersion, then X' is a closed (resp. open) substack of X.","statement_latex":"For any immersion $i : \\mathcal{Z} \\to \\mathcal{X}$ there exists a\nunique locally closed substack $\\mathcal{X}' \\subset \\mathcal{X}$\nsuch that $i$ factors as the composition of\nan equivalence $i' : \\mathcal{Z} \\to \\mathcal{X}'$\nfollowed by the inclusion morphism $\\mathcal{X}' \\to \\mathcal{X}$.\nIf $i$ is a closed (resp.\\ open) immersion, then $\\mathcal{X}'$\nis a closed (resp.\\ open) substack of $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0506","source_file":"stacks-properties.tex","source_line":1793,"source_end_line":1802,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1793-L1802","statement_sha256":"0bfa0f1c2a1c44407bbe1e5df6ecffc39aae63fb8ef789f9fb62978c174b81d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14017,"rank":14017,"depth":0,"x":2121.083,"y":1546.505,"cluster":"algebraic-stacks"},{"id":"stacks:0507","tag":"0507","title":"Immersions of algebraic stacks · Lemma 0507","summary":"Let [U/R] → X be a presentation of an algebraic stack. There is a canonical bijection locally closed substacks Z of X → R-invariant locally closed subspaces Z of U which sends Z to U ×_X Z. Moreover, a morphism of algebraic stacks f : Y → X factors through Z if and only if Y ×_X U → U factors through Z. Similarly for closed substacks and open substacks.","statement_latex":"Let $[U/R] \\to \\mathcal{X}$ be a presentation of an algebraic stack.\nThere is a canonical bijection\n$$\n\\text{locally closed substacks }\\mathcal{Z}\\text{ of }\\mathcal{X}\n\\longrightarrow\nR\\text{-invariant locally closed subspaces }Z\\text{ of }U\n$$\nwhich sends $\\mathcal{Z}$ to $U \\times_\\mathcal{X} \\mathcal{Z}$.\nMoreover, a morphism of algebraic stacks $f : \\mathcal{Y} \\to \\mathcal{X}$\nfactors through $\\mathcal{Z}$ if and only if\n$\\mathcal{Y} \\times_\\mathcal{X} U \\to U$ factors through $Z$.\nSimilarly for closed substacks and open substacks.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0507","source_file":"stacks-properties.tex","source_line":1808,"source_end_line":1822,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1808-L1822","statement_sha256":"1c5dc5bb712b1b47550b963f14f004df1c2196a96c23ebcf3d79dc92b6a2f3d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14018,"rank":14018,"depth":73,"x":2005.82,"y":1691.266,"cluster":"algebraic-stacks"},{"id":"stacks:06FJ","tag":"06FJ","title":"Immersions of algebraic stacks · Lemma 06FJ","summary":"Let X be an algebraic stack. The rule U ↦ |U| defines an inclusion preserving bijection between open substacks of X and open subsets of |X|.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. The rule\n$\\mathcal{U} \\mapsto |\\mathcal{U}|$ defines an inclusion preserving\nbijection between open substacks of $\\mathcal{X}$ and open subsets\nof $|\\mathcal{X}|$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FJ","source_file":"stacks-properties.tex","source_line":1848,"source_end_line":1854,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1848-L1854","statement_sha256":"7edca859686c01c32fb81bd12089df39eb2558868a60e5042fdf669dc55d4eb3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14019,"rank":14019,"depth":74,"x":1974.353,"y":1518.86,"cluster":"algebraic-stacks"},{"id":"stacks:05UP","tag":"05UP","title":"Immersions of algebraic stacks · Lemma 05UP","summary":"Let X be an algebraic stack. Let U be an algebraic space and U → X a surjective smooth morphism. For an open immersion V hookrightarrow U, there exists an algebraic stack Y, an open immersion Y → X, and a surjective smooth morphism V → Y.","statement_latex":"Let $\\mathcal X$ be an algebraic stack. Let $U$ be an algebraic space and\n$U \\to \\mathcal X$ a surjective smooth morphism. For an open immersion\n$V \\hookrightarrow U$, there exists an algebraic stack $\\mathcal Y$, an\nopen immersion $\\mathcal Y \\to \\mathcal X$, and a surjective smooth\nmorphism $V \\to \\mathcal Y$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UP","source_file":"stacks-properties.tex","source_line":1867,"source_end_line":1874,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1867-L1874","statement_sha256":"fb36f471604b41637c05a16c6c792e27346a1987672b6b01127c6be649bd2e61","origin":"The Stacks Project","memory_eligible":false,"source_rank":14020,"rank":14020,"depth":70,"x":2136.443,"y":1628.298,"cluster":"algebraic-stacks"},{"id":"stacks:05UQ","tag":"05UQ","title":"Immersions of algebraic stacks · Lemma 05UQ","summary":"Let X be an algebraic stack and X_i ⊂ X a collection of open substacks indexed by i ∈ I. Then there exists an open substack, which we denote ⋃_i∈ I X_i ⊂ X, such that the X_i are open substacks covering it.","statement_latex":"Let $\\mathcal X$ be an algebraic stack and $\\mathcal{X}_i \\subset \\mathcal X$\na collection of open substacks indexed by $i \\in I$. Then there exists an\nopen substack, which we denote\n$\\bigcup_{i\\in I} \\mathcal{X}_i \\subset \\mathcal X$, such that\nthe $\\mathcal{X}_i$ are open substacks covering it.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UQ","source_file":"stacks-properties.tex","source_line":1928,"source_end_line":1935,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1928-L1935","statement_sha256":"2c70c86d521c1851b30466b89eeb1ce8570f66ef87e3a578618f2aee449d06af","origin":"The Stacks Project","memory_eligible":false,"source_rank":14021,"rank":14021,"depth":70,"x":1928.603,"y":1639.591,"cluster":"algebraic-stacks"},{"id":"stacks:05UR","tag":"05UR","title":"Immersions of algebraic stacks · Lemma 05UR","summary":"Let X be an algebraic stack and X' ⊂ X a quasi-compact open substack. Suppose that we have a collection of open substacks X_i ⊂ X indexed by i ∈ I such that X' ⊂ ⋃_i ∈ I X_i, where we define the union as in Lemma [Tag 05UQ]. Then there exists a finite subset I' ⊂ I such that X' ⊂ ⋃_i ∈ I' X_i.","statement_latex":"Let $\\mathcal X$ be an algebraic stack and $\\mathcal X' \\subset \\mathcal X$\na quasi-compact open substack. Suppose that we have a collection of open\nsubstacks $\\mathcal{X}_i \\subset \\mathcal X$  indexed by $i \\in I$ such\nthat $\\mathcal{X}' \\subset \\bigcup_{i \\in I} \\mathcal{X}_i$,\nwhere we define the union as in Lemma \\ref{lemma-union-open-substacks}.\nThen there exists a finite subset $I' \\subset I$ such that\n$\\mathcal{X}' \\subset \\bigcup_{i \\in I'} \\mathcal{X}_i$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05UR","source_file":"stacks-properties.tex","source_line":1976,"source_end_line":1985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L1976-L1985","statement_sha256":"123065da504f901f76ca30fa5a80dfea9135a0bcc149e99fd90b4ea67097ad8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14022,"rank":14022,"depth":71,"x":2072.995,"y":1513.143,"cluster":"algebraic-stacks"},{"id":"stacks:05WE","tag":"05WE","title":"Immersions of algebraic stacks · Lemma 05WE","summary":"Let X be an algebraic stack. Let X_i, i ∈ I be a set of open substacks of X. Assume • X = ⋃_i ∈ I X_i, and • each X_i is an algebraic space. Then X is an algebraic space.","statement_latex":"Let $\\mathcal X$ be an algebraic stack.\nLet $\\mathcal{X}_i$, $i \\in I$ be a set of open substacks of $\\mathcal{X}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{X} = \\bigcup_{i \\in I} \\mathcal{X}_i$, and\n\\item each $\\mathcal{X}_i$ is an algebraic space.\n\\end{enumerate}\nThen $\\mathcal{X}$ is an algebraic space.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WE","source_file":"stacks-properties.tex","source_line":2033,"source_end_line":2043,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2033-L2043","statement_sha256":"c069e183aa85d945c228d4eaa9f2f8594293eaf3e303a57ca1c8275c7aafc5d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14023,"rank":14023,"depth":68,"x":2068.201,"y":1688.574,"cluster":"algebraic-stacks"},{"id":"stacks:05WF","tag":"05WF","title":"Immersions of algebraic stacks · Lemma 05WF","summary":"Let X be an algebraic stack. Let X_i, i ∈ I be a set of open substacks of X. Assume • X = ⋃_i ∈ I X_i, and • each X_i is a scheme Then X is a scheme.","statement_latex":"Let $\\mathcal X$ be an algebraic stack.\nLet $\\mathcal{X}_i$, $i \\in I$ be a set of open substacks of $\\mathcal{X}$.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{X} = \\bigcup_{i \\in I} \\mathcal{X}_i$, and\n\\item each $\\mathcal{X}_i$ is a scheme\n\\end{enumerate}\nThen $\\mathcal{X}$ is a scheme.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WF","source_file":"stacks-properties.tex","source_line":2056,"source_end_line":2066,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2056-L2066","statement_sha256":"1797e4e44132db033afbcb503967ecb553ccf2818851965be9394ede8d052bee","origin":"The Stacks Project","memory_eligible":false,"source_rank":14024,"rank":14024,"depth":69,"x":1930.455,"y":1556.298,"cluster":"algebraic-stacks"},{"id":"stacks:06M3","tag":"06M3","title":"Immersions of algebraic stacks · Lemma 06M3","summary":"Let P, Q, R be properties of morphisms of algebraic spaces. Assume • P, Q, R are fppf local on the target and stable under arbitrary base change, • smooth ⇒ R, • for any morphism f : X → Y which has Q there exists a largest open subspace W(P, f) ⊂ X such that f|_W(P, f) has P, and • for any morphism f : X → Y which has Q, and any morphism Y' → Y which has R we have Y' ×_Y W(P, f) = W(P, f'), where f' : X_Y' → Y' is the base change of f. Let f : X → Y be a morphism of…","statement_latex":"Let $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ be properties of morphisms\nof algebraic spaces. Assume\n\\begin{enumerate}\n\\item $\\mathcal{P}, \\mathcal{Q}, \\mathcal{R}$ are fppf local on the target\nand stable under arbitrary base change,\n\\item $\\text{smooth} \\Rightarrow \\mathcal{R}$,\n\\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$ there exists a\nlargest open subspace $W(\\mathcal{P}, f) \\subset X$ such that\n$f|_{W(\\mathcal{P}, f)}$ has $\\mathcal{P}$, and\n\\item for any morphism $f : X \\to Y$ which has $\\mathcal{Q}$,\nand any morphism $Y' \\to Y$ which has $\\mathcal{R}$ we have\n$Y' \\times_Y W(\\mathcal{P}, f) = W(\\mathcal{P}, f')$, where\n$f' : X_{Y'} \\to Y'$ is the base change of $f$.\n\\end{enumerate}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nrepresentable by algebraic spaces. Assume $f$ has $\\mathcal{Q}$. Then\n\\begin{enumerate}\n\\item[(A)] there exists a largest open substack\n$\\mathcal{X}' \\subset \\mathcal{X}$ such that $f|_{\\mathcal{X}'}$ has\n$\\mathcal{P}$, and\n\\item[(B)] if $\\mathcal{Z} \\to \\mathcal{Y}$ is a morphism of algebraic\nstacks representable by algebraic spaces which has $\\mathcal{R}$\nthen $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}'$ is the largest open\nsubstack of $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$ over which\nthe base change $\\text{id}_\\mathcal{Z} \\times f$ has property $\\mathcal{P}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Immersions of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06M3","source_file":"stacks-properties.tex","source_line":2082,"source_end_line":2110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2082-L2110","statement_sha256":"4258e4513a72d67378207b806102a014b84e676db1cf33925c9094ca9fe52bbc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14025,"rank":14025,"depth":73,"x":2138.706,"y":1575.705,"cluster":"algebraic-stacks"},{"id":"stacks:0509","tag":"0509","title":"Reduced algebraic stacks · Lemma 0509","summary":"Let X be an algebraic stack. Let T ⊂ |X| be a closed subset. There exists a unique closed substack Z ⊂ X with the following properties: (a) we have |Z| = T, and (b) Z is reduced.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $T \\subset |\\mathcal{X}|$ be a closed subset.\nThere exists a unique closed substack $\\mathcal{Z} \\subset \\mathcal{X}$\nwith the following properties:\n(a) we have $|\\mathcal{Z}| = T$, and (b) $\\mathcal{Z}$ is reduced.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Reduced algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0509","source_file":"stacks-properties.tex","source_line":2239,"source_end_line":2246,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2239-L2246","statement_sha256":"18c221ea60baba50abc8f14b60e154b6a11906006fb0d8fa705357769a3ad4c7","origin":"The Stacks Project","memory_eligible":false,"source_rank":14026,"rank":14026,"depth":74,"x":1969.29,"y":1679.714,"cluster":"algebraic-stacks"},{"id":"stacks:050A","tag":"050A","title":"Reduced algebraic stacks · Lemma 050A","summary":"Let X be an algebraic stack. If X' ⊂ X is a closed substack, X is reduced and |X'| = |X|, then X' = X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nIf $\\mathcal{X}' \\subset \\mathcal{X}$\nis a closed substack, $\\mathcal{X}$ is reduced and\n$|\\mathcal{X}'| = |\\mathcal{X}|$, then $\\mathcal{X}' = \\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Reduced algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050A","source_file":"stacks-properties.tex","source_line":2290,"source_end_line":2296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2290-L2296","statement_sha256":"5d694dcf18c721ebbe7603f5d894d109b58c169e30178e29d918f9f835a7d142","origin":"The Stacks Project","memory_eligible":false,"source_rank":14027,"rank":14027,"depth":74,"x":2010.634,"y":1506.635,"cluster":"algebraic-stacks"},{"id":"stacks:050B","tag":"050B","title":"Reduced algebraic stacks · Lemma 050B","summary":"Let X, Y be algebraic stacks. Let Z ⊂ X be a closed substack Assume Y is reduced. A morphism f : Y → X factors through Z if and only if f(|Y|) ⊂ |Z|.","statement_latex":"Let $\\mathcal{X}$, $\\mathcal{Y}$ be algebraic stacks.\nLet $\\mathcal{Z} \\subset \\mathcal{X}$ be a closed substack\nAssume $\\mathcal{Y}$ is reduced.\nA morphism $f : \\mathcal{Y} \\to \\mathcal{X}$ factors through\n$\\mathcal{Z}$ if and only if\n$f(|\\mathcal{Y}|) \\subset |\\mathcal{Z}|$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Reduced algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050B","source_file":"stacks-properties.tex","source_line":2312,"source_end_line":2320,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2312-L2320","statement_sha256":"98c880a45120074b3923e2ea011db3fff90cc2829528302bc4dbce57547da544","origin":"The Stacks Project","memory_eligible":false,"source_rank":14028,"rank":14028,"depth":75,"x":2119.491,"y":1657.942,"cluster":"algebraic-stacks"},{"id":"stacks:050C","tag":"050C","title":"Reduced algebraic stacks · Definition 050C","summary":"Let X be an algebraic stack. Let Z ⊂ |X| be a closed subset. An algebraic stack structure on Z is given by a closed substack Z of X with |Z| equal to Z. The reduced induced algebraic stack structure on Z is the one constructed in Lemma [Tag 0509]. The reduction X_red of X is the reduced induced algebraic stack structure on |X|.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $Z \\subset |\\mathcal{X}|$ be a closed subset.\nAn {\\it algebraic stack structure on $Z$} is given by a closed substack\n$\\mathcal{Z}$ of $\\mathcal{X}$ with $|\\mathcal{Z}|$ equal to $Z$.\nThe {\\it reduced induced algebraic stack structure}\non $Z$ is the one constructed in\nLemma \\ref{lemma-reduced-closed-substack}.\nThe {\\it reduction $\\mathcal{X}_{red}$ of $\\mathcal{X}$}\nis the reduced induced algebraic stack structure on $|\\mathcal{X}|$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Reduced algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050C","source_file":"stacks-properties.tex","source_line":2339,"source_end_line":2350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2339-L2350","statement_sha256":"80ce09d6f92820fa46b5c74f97bfd01864171a37235c4a787836014ae354640e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14029,"rank":14029,"depth":75,"x":1917.252,"y":1608.066,"cluster":"algebraic-stacks"},{"id":"stacks:06MM","tag":"06MM","title":"Residual gerbes · Lemma 06MM","summary":"Let Z be an algebraic stack. Let k be a field and let Spec(k) → Z be surjective and flat. Then any morphism Spec(k') → Z where k' is a field is surjective and flat.","statement_latex":"Let $\\mathcal{Z}$ be an algebraic stack. Let $k$ be a field and let\n$\\Spec(k) \\to \\mathcal{Z}$ be surjective and flat. Then any\nmorphism $\\Spec(k') \\to \\mathcal{Z}$ where $k'$ is a field is\nsurjective and flat.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MM","source_file":"stacks-properties.tex","source_line":2409,"source_end_line":2415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2409-L2415","statement_sha256":"8255c4a8738e73e304a94939c88aae2c49736ffdd079ba0dab7889913381d1b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14030,"rank":14030,"depth":56,"x":2106.764,"y":1529.975,"cluster":"algebraic-stacks"},{"id":"stacks:06MN","tag":"06MN","title":"Residual gerbes · Lemma 06MN","summary":"Let Z be an algebraic stack. The following are equivalent • Z is reduced and |Z| is a singleton, • there exists a surjective flat morphism Spec(k) → Z where k is a field, and • there exists a locally of finite type, surjective, flat morphism Spec(k) → Z where k is a field.","statement_latex":"Let $\\mathcal{Z}$ be an algebraic stack. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{Z}$ is reduced and $|\\mathcal{Z}|$ is a singleton,\n\\item there exists a surjective flat morphism $\\Spec(k) \\to \\mathcal{Z}$\nwhere $k$ is a field, and\n\\item there exists a locally of finite type, surjective, flat morphism\n$\\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MN","source_file":"stacks-properties.tex","source_line":2436,"source_end_line":2446,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2436-L2446","statement_sha256":"f250e5a8d972f3fd3747a37c88199b77a49f50a849d91624c254a8741587b770","origin":"The Stacks Project","memory_eligible":false,"source_rank":14031,"rank":14031,"depth":57,"x":2029.708,"y":1695.331,"cluster":"algebraic-stacks"},{"id":"stacks:06MP","tag":"06MP","title":"Residual gerbes · Lemma 06MP","summary":"Let Z be an algebraic stack. The following are equivalent • Z is reduced, locally Noetherian, and |Z| is a singleton, and • there exists a locally finitely presented, surjective, flat morphism Spec(k) → Z where k is a field.","statement_latex":"Let $\\mathcal{Z}$ be an algebraic stack. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{Z}$ is reduced, locally Noetherian, and $|\\mathcal{Z}|$\nis a singleton, and\n\\item there exists a locally finitely presented, surjective, flat morphism\n$\\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MP","source_file":"stacks-properties.tex","source_line":2492,"source_end_line":2501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2492-L2501","statement_sha256":"75d98d81c321135f5caab87a440d6341b2273c20ab08fc5980ced0bc91661ddc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14032,"rank":14032,"depth":58,"x":1953.443,"y":1529.437,"cluster":"algebraic-stacks"},{"id":"stacks:06MQ","tag":"06MQ","title":"Residual gerbes · Lemma 06MQ","summary":"Let Z' → Z be a monomorphism of algebraic stacks. Assume there exists a field k and a locally finitely presented, surjective, flat morphism Spec(k) → Z. Then either Z' is empty or Z' → Z is an equivalence.","statement_latex":"Let $\\mathcal{Z}' \\to \\mathcal{Z}$ be a monomorphism of algebraic stacks.\nAssume there exists a field $k$ and a locally finitely presented, surjective,\nflat morphism $\\Spec(k) \\to \\mathcal{Z}$. Then either $\\mathcal{Z}'$\nis empty or $\\mathcal{Z}' \\to \\mathcal{Z}$ is an equivalence.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MQ","source_file":"stacks-properties.tex","source_line":2541,"source_end_line":2547,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2541-L2547","statement_sha256":"b8cd78d4b060d59186abdc6798dbb543a43f5775cafe02ddf5985bfc1e651e92","origin":"The Stacks Project","memory_eligible":false,"source_rank":14033,"rank":14033,"depth":71,"x":2143.359,"y":1608.604,"cluster":"algebraic-stacks"},{"id":"stacks:06MR","tag":"06MR","title":"Residual gerbes · Lemma 06MR","summary":"Let Z be an algebraic stack. Assume Z satisfies the equivalent conditions of Lemma [Tag 06MN]. Then there exists a unique strictly full subcategory Z' ⊂ Z such that Z' is an algebraic stack which satisfies the equivalent conditions of Lemma [Tag 06MP]. The inclusion morphism Z' → Z is a monomorphism of algebraic stacks.","statement_latex":"Let $\\mathcal{Z}$ be an algebraic stack. Assume $\\mathcal{Z}$ satisfies\nthe equivalent conditions of\nLemma \\ref{lemma-unique-point}.\nThen there exists a unique strictly full subcategory\n$\\mathcal{Z}' \\subset \\mathcal{Z}$ such that\n$\\mathcal{Z}'$ is an algebraic stack which satisfies the equivalent\nconditions of\nLemma \\ref{lemma-unique-point-better}.\nThe inclusion morphism $\\mathcal{Z}' \\to \\mathcal{Z}$ is a monomorphism\nof algebraic stacks.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MR","source_file":"stacks-properties.tex","source_line":2573,"source_end_line":2585,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2573-L2585","statement_sha256":"71b08efe74971b2dcd5ebbde9cdae33f85e910001a976ee751983e5f52cd4e8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14034,"rank":14034,"depth":76,"x":1939.362,"y":1658.062,"cluster":"algebraic-stacks"},{"id":"stacks:06MT","tag":"06MT","title":"Residual gerbes · Lemma 06MT","summary":"Let X be an algebraic stack. Let x ∈ |X| be a point. The following are equivalent • there exists an algebraic stack Z and a monomorphism Z → X such that |Z| is a singleton and such that the image of |Z| in |X| is x, • there exists a reduced algebraic stack Z and a monomorphism Z → X such that |Z| is a singleton and such that the image of |Z| in |X| is x, • there exists an algebraic stack Z, a monomorphism f : Z → X, and a surjective flat morphism z : Spec(k) → Z where k…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$ be a point.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists an algebraic stack $\\mathcal{Z}$ and a monomorphism\n$\\mathcal{Z} \\to \\mathcal{X}$ such that $|\\mathcal{Z}|$ is a singleton\nand such that the image of $|\\mathcal{Z}|$ in $|\\mathcal{X}|$ is $x$,\n\\item there exists a reduced algebraic stack $\\mathcal{Z}$ and a monomorphism\n$\\mathcal{Z} \\to \\mathcal{X}$ such that $|\\mathcal{Z}|$ is a singleton\nand such that the image of $|\\mathcal{Z}|$ in $|\\mathcal{X}|$ is $x$,\n\\item there exists an algebraic stack $\\mathcal{Z}$, a monomorphism\n$f : \\mathcal{Z} \\to \\mathcal{X}$, and a surjective flat morphism\n$z : \\Spec(k) \\to \\mathcal{Z}$ where $k$ is a field such that\n$x = f(z)$.\n\\end{enumerate}\nMoreover, if these conditions hold, then there exists a unique\nstrictly full subcategory $\\mathcal{Z}_x \\subset \\mathcal{X}$\nsuch that $\\mathcal{Z}_x$ is a reduced, locally Noetherian algebraic\nstack and $|\\mathcal{Z}_x|$ is a singleton which maps to $x$\nvia the map $|\\mathcal{Z}_x| \\to |\\mathcal{X}|$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MT","source_file":"stacks-properties.tex","source_line":2691,"source_end_line":2712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2691-L2712","statement_sha256":"5e5102da5b8e85457e229b818a57dc152c11d446425eefe1eeceb830bd25e133","origin":"The Stacks Project","memory_eligible":false,"source_rank":14035,"rank":14035,"depth":77,"x":2050.173,"y":1505.621,"cluster":"algebraic-stacks"},{"id":"stacks:06MU","tag":"06MU","title":"Residual gerbes · Definition 06MU","summary":"Let X be an algebraic stack. Let x ∈ |X|. • We say the residual gerbe of X at x exists if the equivalent conditions (1), (2), and (3) of Lemma [Tag 06MT] hold. • If the residual gerbe of X at x exists, then the residual gerbe of X at x is the strictly full subcategory Z_x ⊂ X constructed in Lemma [Tag 06MT].","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.\n\\begin{enumerate}\n\\item We say the {\\it residual gerbe of $\\mathcal{X}$ at $x$ exists}\nif the equivalent conditions (1), (2), and (3) of\nLemma \\ref{lemma-residual-gerbe}\nhold.\n\\item If the residual gerbe of $\\mathcal{X}$ at $x$ exists, then the\n{\\it residual gerbe of $\\mathcal{X}$ at $x$}\\footnote{This clashes with\n\\cite{LM-B} in spirit, but not in fact. Namely, in Chapter 11 they associate\nto any point on any quasi-separated algebraic stack a gerbe (not necessarily\nalgebraic) which they call the residual gerbe. We will see in\nMorphisms of Stacks, Lemma\n\\ref{stacks-morphisms-lemma-every-point-residual-gerbe}\nthat on a quasi-separated algebraic stack every point\nhas a residual gerbe in our sense which is then equivalent to theirs. For\nmore information on this topic see\n\\cite[Appendix B]{rydh_etale_devissage}.}\nis the strictly full\nsubcategory $\\mathcal{Z}_x \\subset \\mathcal{X}$ constructed in\nLemma \\ref{lemma-residual-gerbe}.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MU","source_file":"stacks-properties.tex","source_line":2763,"source_end_line":2786,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2763-L2786","statement_sha256":"e7027fef3a4a76641848f0f7239d5f4e0a43fa062151e062b081b5ea07ffdacb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14036,"rank":14036,"depth":78,"x":2091.109,"y":1681.155,"cluster":"algebraic-stacks"},{"id":"stacks:06MV","tag":"06MV","title":"Residual gerbes · Lemma 06MV","summary":"A reduced, locally Noetherian algebraic stack Z such that |Z| is a singleton is regular.","statement_latex":"A reduced, locally Noetherian algebraic stack $\\mathcal{Z}$ such that\n$|\\mathcal{Z}|$ is a singleton is regular.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MV","source_file":"stacks-properties.tex","source_line":2816,"source_end_line":2820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2816-L2820","statement_sha256":"c65f71c401a281f4233ecdb0959b4ba74390f0d22468740bb5faa8fdb035f0f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14037,"rank":14037,"depth":59,"x":1919.52,"y":1574.796,"cluster":"algebraic-stacks"},{"id":"stacks:06MW","tag":"06MW","title":"Residual gerbes · Lemma 06MW","summary":"Let X be an algebraic stack. Let x ∈ |X|. Assume that the residual gerbe Z_x of X exists. Let f : Spec(K) → X be a morphism where K is a field in the equivalence class of x. Then f factors through the inclusion morphism Z_x → X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.\nAssume that the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ exists.\nLet $f : \\Spec(K) \\to \\mathcal{X}$ be a morphism where $K$ is a field\nin the equivalence class of $x$. Then $f$ factors through the inclusion\nmorphism $\\mathcal{Z}_x \\to \\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MW","source_file":"stacks-properties.tex","source_line":2836,"source_end_line":2843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2836-L2843","statement_sha256":"bc5af137c0258ca364ba6b1ee98964b36a3a19cc06ab42950ae839daaede4656","origin":"The Stacks Project","memory_eligible":false,"source_rank":14038,"rank":14038,"depth":57,"x":2131.879,"y":1555.833,"cluster":"algebraic-stacks"},{"id":"stacks:06MX","tag":"06MX","title":"Residual gerbes · Lemma 06MX","summary":"Let X be an algebraic stack. Let x ∈ |X|. Let Z be an algebraic stack satisfying the equivalent conditions of Lemma [Tag 06MP] and let Z → X be a monomorphism such that the image of |Z| → |X| is x. Then the residual gerbe Z_x of X at x exists and Z → X factors as Z → Z_x → X where the first arrow is an equivalence.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.\nLet $\\mathcal{Z}$ be an algebraic stack satisfying the equivalent conditions of\nLemma \\ref{lemma-unique-point-better}\nand let $\\mathcal{Z} \\to \\mathcal{X}$ be a monomorphism such that the image\nof $|\\mathcal{Z}| \\to |\\mathcal{X}|$ is $x$. Then the residual gerbe\n$\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists and\n$\\mathcal{Z} \\to \\mathcal{X}$ factors as\n$\\mathcal{Z} \\to \\mathcal{Z}_x \\to \\mathcal{X}$ where the first arrow\nis an equivalence.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MX","source_file":"stacks-properties.tex","source_line":2858,"source_end_line":2869,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2858-L2869","statement_sha256":"b311235064f1f51edb2468ae7e256ef0d80591a6985b115557345e303a0a1066","origin":"The Stacks Project","memory_eligible":false,"source_rank":14039,"rank":14039,"depth":78,"x":1990.338,"y":1690.504,"cluster":"algebraic-stacks"},{"id":"stacks:0DTH","tag":"0DTH","title":"Residual gerbes · Lemma 0DTH","summary":"Let f : X → Y be a morphism of algebraic stacks. Let x ∈ |X| with image y ∈ |Y|. If the residual gerbes Z_x ⊂ X and Z_y ⊂ Y of x and y exist, then f induces a commutative diagram xymatrix X ar[d]_f & Z_x ar[l] ar[d] Y & Z_y ar[l]","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $x \\in |\\mathcal{X}|$ with image $y \\in |\\mathcal{Y}|$.\nIf the residual gerbes $\\mathcal{Z}_x \\subset \\mathcal{X}$\nand $\\mathcal{Z}_y \\subset \\mathcal{Y}$ of $x$ and $y$ exist,\nthen $f$ induces a commutative diagram\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[d]_f & \\mathcal{Z}_x \\ar[l] \\ar[d] \\\\\n\\mathcal{Y} & \\mathcal{Z}_y \\ar[l]\n}\n$$","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTH","source_file":"stacks-properties.tex","source_line":2883,"source_end_line":2896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2883-L2896","statement_sha256":"fdc241de5ff71653da4b11281b25cf4d5e3861474913508dbfdd3994012bbe2d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14040,"rank":14040,"depth":72,"x":1986.404,"y":1510.633,"cluster":"algebraic-stacks"},{"id":"stacks:0DTI","tag":"0DTI","title":"Residual gerbes · Lemma 0DTI","summary":"Let f : X → Y be a morphism of algebraic stacks. Let x ∈ |X| with image y ∈ |Y|. Assume the residual gerbes Z_x ⊂ X and Z_y ⊂ Y of x and y exist and that there exists a morphism Spec(k) → X in the equivalence class of x such that Spec(k) ×_X Spec(k) → Spec(k) ×_Y Spec(k) is an isomorphism. Then Z_x → Z_y is an isomorphism.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $x \\in |\\mathcal{X}|$ with image $y \\in |\\mathcal{Y}|$.\nAssume the residual gerbes $\\mathcal{Z}_x \\subset \\mathcal{X}$\nand $\\mathcal{Z}_y \\subset \\mathcal{Y}$ of $x$ and $y$ exist\nand that there exists a morphism $\\Spec(k) \\to \\mathcal{X}$\nin the equivalence class of $x$ such that\n$$\n\\Spec(k) \\times_\\mathcal{X} \\Spec(k)\n\\longrightarrow\n\\Spec(k) \\times_\\mathcal{Y} \\Spec(k)\n$$\nis an isomorphism. Then $\\mathcal{Z}_x \\to \\mathcal{Z}_y$\nis an isomorphism.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTI","source_file":"stacks-properties.tex","source_line":2908,"source_end_line":2923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2908-L2923","statement_sha256":"567936369a0929f80876d0f1280cc7ca2a389bf335ac24f8245ab897dccf8c3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14041,"rank":14041,"depth":79,"x":2134.159,"y":1641.218,"cluster":"algebraic-stacks"},{"id":"stacks:0AFM","tag":"0AFM","title":"Dimension of a stack · Lemma 0AFM","summary":"Let X be a locally Noetherian algebraic stack over a scheme S. Let x ∈ |X| be a point of X. Let [U/R] → X be a presentation (Algebraic Stacks, Definition [Tag 04TI]) where U is a scheme. Let u ∈ U be a point that maps to x. Let e : U → R be the \"identity\" map and let s : R → U be the \"source\" map, which is a smooth morphism of algebraic spaces. Let R_u be the fiber of s : R → U over u. The element dim_x(X) = dim_u(U) - dim_e(u)(R_u) ∈ Z ∪ ∞ is independent of the choice of…","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack over a scheme $S$.\nLet $x \\in |\\mathcal{X}|$ be a point of $\\mathcal{X}$.\nLet $[U/R] \\to \\mathcal{X}$ be a presentation\n(Algebraic Stacks, Definition \\ref{algebraic-definition-presentation})\nwhere $U$ is a scheme. Let $u \\in U$ be a point that maps to $x$.\nLet $e : U \\to R$ be the ``identity'' map and let $s : R \\to U$ be the\n``source'' map, which is a smooth morphism of algebraic spaces. Let $R_u$\nbe the fiber of $s : R \\to U$ over $u$. The element\n$$\n\\dim_x(\\mathcal{X}) = \\dim_u(U) - \\dim_{e(u)}(R_u) \\in \\mathbf{Z} \\cup \\infty\n$$\nis independent of the choice of presentation and the point $u$ over $x$.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Dimension of a stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFM","source_file":"stacks-properties.tex","source_line":2972,"source_end_line":2986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L2972-L2986","statement_sha256":"ca956fe9f1b2d6b063ac002710669110ee713a1831164758d45a1c9a9992ad2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14042,"rank":14042,"depth":49,"x":1919.894,"y":1628.749,"cluster":"algebraic-stacks"},{"id":"stacks:0AFN","tag":"0AFN","title":"Dimension of a stack · Definition 0AFN","summary":"Let X be a locally Noetherian algebraic stack over a scheme S. Let x ∈ |X| be a point of X. Let [U/R] → X be a presentation (Algebraic Stacks, Definition [Tag 04TI]) where U is a scheme and let u ∈ U be a point that maps to x. We define the dimension of X at x to be the element dim_x(X) ∈ Z ∪ ∞ such that dim_x(X) = dim_u(U)-dim_e(u)(R_u). with notation as in Lemma [Tag 0AFM].","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack over a scheme $S$.\nLet $x \\in |\\mathcal{X}|$ be a point of $\\mathcal{X}$.\nLet $[U/R] \\to \\mathcal{X}$ be a presentation\n(Algebraic Stacks, Definition \\ref{algebraic-definition-presentation})\nwhere $U$ is a scheme\nand let $u \\in U$ be a point that maps to $x$.\nWe define the {\\it dimension of $\\mathcal{X}$ at $x$} to be\nthe element $\\dim_x(\\mathcal{X}) \\in \\mathbf{Z} \\cup \\infty$\nsuch that \n$$\n\\dim_x(\\mathcal{X}) = \\dim_u(U)-\\dim_{e(u)}(R_u).\n$$\nwith notation as in Lemma \\ref{lemma-dimension-at-point-well-defined}.","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Dimension of a stack","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFN","source_file":"stacks-properties.tex","source_line":3037,"source_end_line":3052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L3037-L3052","statement_sha256":"0a83b53472e912f8f1d30c2215c7f98cbcee20e936ae1826a6fdfcbb92153357","origin":"The Stacks Project","memory_eligible":false,"source_rank":14043,"rank":14043,"depth":50,"x":2088.153,"y":1516.207,"cluster":"algebraic-stacks"},{"id":"stacks:0AFP","tag":"0AFP","title":"Dimension of a stack · Definition 0AFP","summary":"Let S be a scheme. Let X be a locally Noetherian algebraic stack over S. The dimension dim(X) of X is defined to be dim(X) = sup_x ∈ |X| dim_x(X)","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be\na locally Noetherian algebraic stack over $S$.\nThe {\\it dimension} $\\dim(\\mathcal{X})$ of $\\mathcal{X}$ is defined to be\n$$\n\\dim(\\mathcal{X}) = \\sup\\nolimits_{x \\in |\\mathcal{X}|} \\dim_x(\\mathcal{X})\n$$","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Dimension of a stack","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFP","source_file":"stacks-properties.tex","source_line":3062,"source_end_line":3070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L3062-L3070","statement_sha256":"14a3d6dae2e29130687edb9ab782a8ddd25f91e54477bd56e821cac4ed3eea2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14044,"rank":14044,"depth":0,"x":2054.537,"y":1694.917,"cluster":"algebraic-stacks"},{"id":"stacks:0DQH","tag":"0DQH","title":"Local irreducibility · Definition 0DQH","summary":"Let X be an algebraic stack. Let x ∈ |X|. • The number of geometric branches of X at x is either n ∈ N if the equivalent conditions of Lemma [Tag 04YI] hold for P_n defined above, or else ∞. • We say X is geometrically unibranch at x if the number of geometric branches of X at x is 1.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.\n\\begin{enumerate}\n\\item The {\\it number of geometric branches of $\\mathcal{X}$ at $x$}\nis either $n \\in \\mathbf{N}$ if the equivalent conditions of\nLemma \\ref{lemma-local-source-target-at-point} hold for\n$\\mathcal{P}_n$ defined above, or else $\\infty$.\n\\item We say $\\mathcal{X}$ is {\\it geometrically unibranch at $x$}\nif the number of geometric branches of $\\mathcal{X}$ at $x$ is $1$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Local irreducibility","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQH","source_file":"stacks-properties.tex","source_line":3145,"source_end_line":3156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L3145-L3156","statement_sha256":"4e0a01e401f7c0f4df84a123a93ad068eb5de6e81d4ae88cbd9acc76606741c2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14045,"rank":14045,"depth":49,"x":1935.446,"y":1543.855,"cluster":"algebraic-stacks"},{"id":"stacks:0DTK","tag":"0DTK","title":"Finiteness conditions and points · Lemma 0DTK","summary":"Let X be an algebraic stack. Let x ∈ |X| be a point. The following are equivalent • some morphism Spec(k) → X in the equivalence class of x is quasi-compact, and • any morphism Spec(k) → X in the equivalence class of x is quasi-compact.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$\nbe a point. The following are equivalent\n\\begin{enumerate}\n\\item some morphism $\\Spec(k) \\to \\mathcal{X}$ in the equivalence\nclass of $x$ is quasi-compact, and\n\\item any morphism $\\Spec(k) \\to \\mathcal{X}$ in the equivalence\nclass of $x$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Properties of Algebraic Stacks","chapter_id":"stacks-properties","section":"Finiteness conditions and points","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTK","source_file":"stacks-properties.tex","source_line":3171,"source_end_line":3181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-properties.tex#L3171-L3181","statement_sha256":"8077cfa9ad4e98f7ac9c6e7d96f318003845a8f309fd6af81cd86c81c6babd3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14046,"rank":14046,"depth":5,"x":2145.032,"y":1587.732,"cluster":"algebraic-stacks"},{"id":"stacks:04XR","tag":"04XR","title":"Properties of diagonals · Lemma 04XR","summary":"Let X be an algebraic stack. Let T be a scheme and let x, y be objects of the fibre category of X over T. Then the morphism mathitIsom_X(x, y) → T is locally of finite type.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $T$ be a scheme and let $x, y$ be objects of the fibre category of\n$\\mathcal{X}$ over $T$. Then the morphism\n$\\mathit{Isom}_\\mathcal{X}(x, y) \\to T$ is locally of finite type.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Properties of diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XR","source_file":"stacks-morphisms.tex","source_line":64,"source_end_line":70,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L64-L70","statement_sha256":"26f0262a5cd50306d8ad16739547dfec78c7d3a045fe8904a9922c6c42c0c482","origin":"The Stacks Project","memory_eligible":false,"source_rank":14047,"rank":14047,"depth":57,"x":1954.94,"y":1674.421,"cluster":"algebraic-stacks"},{"id":"stacks:04YP","tag":"04YP","title":"Properties of diagonals · Lemma 04YP","summary":"Let X be an algebraic stack. Let T be a scheme and let x, y be objects of the fibre category of X over T. Then • mathitIsom_X(y, y) is a group algebraic space over T, and • mathitIsom_X(x, y) is a pseudo torsor for mathitIsom_X(y, y) over T.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $T$ be a scheme and let $x, y$ be objects of the fibre category of\n$\\mathcal{X}$ over $T$. Then\n\\begin{enumerate}\n\\item $\\mathit{Isom}_\\mathcal{X}(y, y)$ is a group algebraic space\nover $T$, and\n\\item $\\mathit{Isom}_\\mathcal{X}(x, y)$ is a pseudo torsor for\n$\\mathit{Isom}_\\mathcal{X}(y, y)$ over $T$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Properties of diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YP","source_file":"stacks-morphisms.tex","source_line":97,"source_end_line":108,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L97-L108","statement_sha256":"39e9bb847cf32a2d2f74ee145363877079353e2dc26d4accc366354dc5bda6ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":14048,"rank":14048,"depth":1,"x":2025.495,"y":1502.398,"cluster":"algebraic-stacks"},{"id":"stacks:04XS","tag":"04XS","title":"Properties of diagonals · Lemma 04XS","summary":"Diagonals of morphisms of algebraic stacks are representable by algebraic spaces and locally of finite type. Let f : X → Y be a morphism of algebraic stacks. Then • Δ_f is representable by algebraic spaces, and • Δ_f is locally of finite type.","statement_latex":"\\begin{slogan}\nDiagonals of morphisms of algebraic stacks are representable by\nalgebraic spaces and locally of finite type.\n\\end{slogan}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThen\n\\begin{enumerate}\n\\item $\\Delta_f$ is representable by algebraic spaces,\nand\n\\item $\\Delta_f$ is locally of finite type.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Properties of diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04XS","source_file":"stacks-morphisms.tex","source_line":130,"source_end_line":143,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L130-L143","statement_sha256":"643da5c28c5e485edec54e7e0f941bed9f6ea822e960fd96f26dfed645bd32f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14049,"rank":14049,"depth":58,"x":2111.923,"y":1669.509,"cluster":"algebraic-stacks"},{"id":"stacks:04YQ","tag":"04YQ","title":"Properties of diagonals · Lemma 04YQ","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. Then • Δ_f is representable (by schemes), • Δ_f is locally of finite type, • Δ_f is a monomorphism, • Δ_f is separated, and • Δ_f is locally quasi-finite.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich is representable by algebraic spaces. Then\n\\begin{enumerate}\n\\item $\\Delta_f$ is representable\n(by schemes),\n\\item $\\Delta_f$ is locally of finite type,\n\\item $\\Delta_f$ is a monomorphism,\n\\item $\\Delta_f$ is separated, and\n\\item $\\Delta_f$ is locally quasi-finite.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Properties of diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YQ","source_file":"stacks-morphisms.tex","source_line":191,"source_end_line":203,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L191-L203","statement_sha256":"fdc5913204a17494b5d5c6ca52680c5b77f17e3612f2cc1285679add970bab7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14050,"rank":14050,"depth":59,"x":1913.535,"y":1595.224,"cluster":"algebraic-stacks"},{"id":"stacks:04YS","tag":"04YS","title":"Properties of diagonals · Lemma 04YS","summary":"Let f : X → Y be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent • f is separated, • Δ_f is a closed immersion, • Δ_f is proper, or • Δ_f is universally closed.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nrepresentable by algebraic spaces. Then the following are equivalent\n\\begin{enumerate}\n\\item $f$ is separated,\n\\item $\\Delta_f$ is a closed immersion,\n\\item $\\Delta_f$ is proper, or\n\\item $\\Delta_f$ is universally closed.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Properties of diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YS","source_file":"stacks-morphisms.tex","source_line":238,"source_end_line":248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L238-L248","statement_sha256":"cf56f450cd6406eb1a5b576f68158289dbe11f4c4f399d71a43bbe5f34ddb50b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14051,"rank":14051,"depth":54,"x":2119.843,"y":1537.351,"cluster":"algebraic-stacks"},{"id":"stacks:04YT","tag":"04YT","title":"Properties of diagonals · Lemma 04YT","summary":"Let f : X → Y be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent • f is quasi-separated, • Δ_f is quasi-compact, or • Δ_f is of finite type.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nrepresentable by algebraic spaces. Then the following are equivalent\n\\begin{enumerate}\n\\item $f$ is quasi-separated,\n\\item $\\Delta_f$ is quasi-compact, or\n\\item $\\Delta_f$ is of finite type.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Properties of diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YT","source_file":"stacks-morphisms.tex","source_line":285,"source_end_line":294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L285-L294","statement_sha256":"5f8124b4b856de8bc001408d548cb60d9593a25f0b5e3239901429db6fa5a2ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":14052,"rank":14052,"depth":60,"x":2014.11,"y":1697.307,"cluster":"algebraic-stacks"},{"id":"stacks:04YU","tag":"04YU","title":"Properties of diagonals · Lemma 04YU","summary":"Let f : X → Y be a morphism of algebraic stacks representable by algebraic spaces. Then the following are equivalent • f is locally separated, and • Δ_f is an immersion.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nrepresentable by algebraic spaces. Then the following are equivalent\n\\begin{enumerate}\n\\item $f$ is locally separated, and\n\\item $\\Delta_f$ is an immersion.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Properties of diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YU","source_file":"stacks-morphisms.tex","source_line":332,"source_end_line":340,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L332-L340","statement_sha256":"f393a57bd95fb0dafd1ab26693cc9d2d13fcbefb72e1a7a1ae0eb083f48a50d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14053,"rank":14053,"depth":61,"x":1963.375,"y":1519.121,"cluster":"algebraic-stacks"},{"id":"stacks:04YW","tag":"04YW","title":"Separation axioms · Definition 04YW","summary":"Let f : X → Y be a morphism of algebraic stacks. • We say f is DM if Δ_f is unramified the fibre product X_T = X ×_Y T is an algebraic stack over T whose diagonal is unramified, i.e., X_T is DM. This implies X_T is a Deligne-Mumford stack, see Theorem [Tag 06N3]. In other words a DM morphism is one whose \"fibres\" are Deligne-Mumford stacks. This hopefully at least motivates the terminology.. • We say f is quasi-DM if Δ_f is locally quasi-finite_T of X → Y are quasi-DM. An…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item We say $f$ is {\\it DM} if $\\Delta_f$ is unramified\\footnote{The\nletters DM stand for Deligne-Mumford. If $f$ is DM then given any scheme\n$T$ and any morphism $T \\to \\mathcal{Y}$ the fibre product\n$\\mathcal{X}_T = \\mathcal{X} \\times_\\mathcal{Y} T$\nis an algebraic stack over $T$ whose diagonal is unramified, i.e.,\n$\\mathcal{X}_T$ is DM. This implies $\\mathcal{X}_T$\nis a Deligne-Mumford stack, see Theorem \\ref{theorem-DM}.\nIn other words a DM morphism is one whose ``fibres'' are Deligne-Mumford\nstacks. This hopefully at least motivates the terminology.}.\n\\item We say $f$ is {\\it quasi-DM} if $\\Delta_f$ is\nlocally quasi-finite\\footnote{If $f$ is quasi-DM, then the\n``fibres'' $\\mathcal{X}_T$ of $\\mathcal{X} \\to \\mathcal{Y}$ are quasi-DM. An\nalgebraic stack $\\mathcal{X}$ is quasi-DM exactly if there exists a\nscheme $U$ and a surjective flat morphism $U \\to \\mathcal{X}$ of finite\npresentation which is locally quasi-finite, see\nTheorem \\ref{theorem-quasi-DM}.\nNote the similarity to being Deligne-Mumford, which\nis defined in terms of having an \\'etale covering by a scheme.}.\n\\item We say $f$ is {\\it separated} if $\\Delta_f$ is proper.\n\\item We say $f$ is {\\it quasi-separated} if $\\Delta_f$\nis quasi-compact and quasi-separated.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YW","source_file":"stacks-morphisms.tex","source_line":387,"source_end_line":413,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L387-L413","statement_sha256":"ecceeb57d7ac5247ce34c3f005b14b96e52c93c6dd4e54760235becf0c39a71c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14054,"rank":14054,"depth":0,"x":2144.323,"y":1621.864,"cluster":"algebraic-stacks"},{"id":"stacks:050D","tag":"050D","title":"Separation axioms · Definition 050D","summary":"Let X be an algebraic stack over the base scheme S. Denote p : X → S the structure morphism. • We say X is DM over S if p : X → S is DM. • We say X is quasi-DM over S if p : X → S is quasi-DM. • We say X is separated over S if p : X → S is separated. • We say X is quasi-separated over S if p : X → S is quasi-separated. • We say X is DM if X is DM being a Deligne-Mumford stack. over Spec(Z). • We say X is quasi-DM if X is quasi-DM over Spec(Z). • We say X is separated if X…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over the base scheme $S$.\nDenote $p : \\mathcal{X} \\to S$ the structure morphism.\n\\begin{enumerate}\n\\item We say $\\mathcal{X}$ is {\\it DM over $S$}\nif $p : \\mathcal{X} \\to S$ is DM.\n\\item We say $\\mathcal{X}$ is {\\it quasi-DM over $S$}\nif $p : \\mathcal{X} \\to S$ is quasi-DM.\n\\item We say $\\mathcal{X}$ is {\\it separated over $S$}\nif $p : \\mathcal{X} \\to S$ is separated.\n\\item We say $\\mathcal{X}$ is {\\it quasi-separated over $S$} if\n$p : \\mathcal{X} \\to S$ is quasi-separated.\n\\item We say $\\mathcal{X}$ is {\\it DM}\nif $\\mathcal{X}$ is DM\\footnote{Theorem \\ref{theorem-DM} shows\nthat this is equivalent to $\\mathcal{X}$ being a Deligne-Mumford stack.}\nover $\\Spec(\\mathbf{Z})$.\n\\item We say $\\mathcal{X}$ is {\\it quasi-DM}\nif $\\mathcal{X}$ is quasi-DM over $\\Spec(\\mathbf{Z})$.\n\\item We say $\\mathcal{X}$ is {\\it separated} if $\\mathcal{X}$\nis separated over $\\Spec(\\mathbf{Z})$.\n\\item We say $\\mathcal{X}$ is {\\it quasi-separated} if $\\mathcal{X}$\nis quasi-separated over $\\Spec(\\mathbf{Z})$.\n\\end{enumerate}\nIn the last 4 definitions we view $\\mathcal{X}$\nas an algebraic stack over $\\Spec(\\mathbf{Z})$\nvia\nAlgebraic Stacks, Definition \\ref{algebraic-definition-viewed-as}.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050D","source_file":"stacks-morphisms.tex","source_line":429,"source_end_line":457,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L429-L457","statement_sha256":"f67435dcb26691eaa11b7992e6fc9639aa330f62be4c43f9f475bedcfff1f508","origin":"The Stacks Project","memory_eligible":false,"source_rank":14055,"rank":14055,"depth":3,"x":1927.98,"y":1648.813,"cluster":"algebraic-stacks"},{"id":"stacks:050E","tag":"050E","title":"Separation axioms · Lemma 050E","summary":"Let f : X → Y be a morphism of algebraic stacks. • If f is separated, then f is quasi-separated. • If f is DM, then f is quasi-DM. • If f is representable by algebraic spaces, then f is DM.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item If $f$ is separated, then $f$ is quasi-separated.\n\\item If $f$ is DM, then $f$ is quasi-DM.\n\\item If $f$ is representable by algebraic spaces, then $f$ is DM.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050E","source_file":"stacks-morphisms.tex","source_line":468,"source_end_line":476,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L468-L476","statement_sha256":"39f0376ff74edfa5a1467d48d3e06e8cbfbf2ec99bb128217b9d1aa167bd06d5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14056,"rank":14056,"depth":60,"x":2066.022,"y":1505.991,"cluster":"algebraic-stacks"},{"id":"stacks:050F","tag":"050F","title":"Separation axioms · Lemma 050F","summary":"All of the separation axioms listed in Definition [Tag 04YW] are stable under base change.","statement_latex":"All of the separation axioms listed in\nDefinition \\ref{definition-separated}\nare stable under base change.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050F","source_file":"stacks-morphisms.tex","source_line":489,"source_end_line":494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L489-L494","statement_sha256":"4924bca4676ed9fd5e4240d42c0ec31cc4083d455e1e357e2adecd6ed72996c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14057,"rank":14057,"depth":2,"x":2079.103,"y":1689.881,"cluster":"algebraic-stacks"},{"id":"stacks:06TZ","tag":"06TZ","title":"Separation axioms · Lemma 06TZ","summary":"Let f : X → Y be a morphism of algebraic stacks. Let W → Y be a surjective, flat, and locally of finite presentation where W is an algebraic space. If the base change W ×_Y X → W has one of the separation properties of Definition [Tag 04YW] then so does f.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $W \\to \\mathcal{Y}$ be a surjective, flat, and locally of finite\npresentation where $W$ is an algebraic space. If the base change\n$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ has one of the separation properties\nof Definition \\ref{definition-separated}\nthen so does $f$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06TZ","source_file":"stacks-morphisms.tex","source_line":513,"source_end_line":521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L513-L521","statement_sha256":"dcc1e17850ac7ea689531a91c7566c06f13bd679932c0046ba7d2674996067c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14058,"rank":14058,"depth":3,"x":1921.367,"y":1561.534,"cluster":"algebraic-stacks"},{"id":"stacks:050G","tag":"050G","title":"Separation axioms · Lemma 050G","summary":"Let S be a scheme. The property of being quasi-DM over S, quasi-separated over S, or separated over S (see Definition [Tag 050D]) is stable under change of base scheme, see Algebraic Stacks, Definition [Tag 04X7].","statement_latex":"Let $S$ be a scheme. The property of being\nquasi-DM over $S$, quasi-separated over $S$, or separated over $S$ (see\nDefinition \\ref{definition-absolute-separated})\nis stable under change of base scheme, see\nAlgebraic Stacks, Definition \\ref{algebraic-definition-change-of-base}.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050G","source_file":"stacks-morphisms.tex","source_line":535,"source_end_line":542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L535-L542","statement_sha256":"a282428c29e9c08ac2f5229bca6c2eb4f84a101097f110174adf407a2fca3405","origin":"The Stacks Project","memory_eligible":false,"source_rank":14059,"rank":14059,"depth":4,"x":2141.186,"y":1566.679,"cluster":"algebraic-stacks"},{"id":"stacks:050H","tag":"050H","title":"Separation axioms · Lemma 050H","summary":"Let f : X → Z, g : Y → Z and Z → T be morphisms of algebraic stacks. Consider the induced morphism i : X ×_Z Y → X ×_T Y. Then • i is representable by algebraic spaces and locally of finite type, • if Δ_Z/T is quasi-separated, then i is quasi-separated, • if Δ_Z/T is separated, then i is separated, • if Z → T is DM, then i is unramified, • if Z → T is quasi-DM, then i is locally quasi-finite, • if Z → T is separated, then i is proper, and • if Z → T is quasi-separated,…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Z}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$\nand $\\mathcal{Z} \\to \\mathcal{T}$ be morphisms of algebraic stacks.\nConsider the induced morphism\n$i : \\mathcal{X} \\times_\\mathcal{Z} \\mathcal{Y} \\to\n\\mathcal{X} \\times_\\mathcal{T} \\mathcal{Y}$.\nThen\n\\begin{enumerate}\n\\item $i$ is representable by algebraic spaces and locally of finite type,\n\\item if $\\Delta_{\\mathcal{Z}/\\mathcal{T}}$ is quasi-separated, then\n$i$ is quasi-separated,\n\\item if $\\Delta_{\\mathcal{Z}/\\mathcal{T}}$ is separated, then\n$i$ is separated,\n\\item if $\\mathcal{Z} \\to \\mathcal{T}$ is DM,\nthen $i$ is unramified,\n\\item if $\\mathcal{Z} \\to \\mathcal{T}$ is quasi-DM,\nthen $i$ is locally quasi-finite,\n\\item if $\\mathcal{Z} \\to \\mathcal{T}$ is separated, then $i$ is proper, and\n\\item if $\\mathcal{Z} \\to \\mathcal{T}$ is quasi-separated, then\n$i$ is quasi-compact and quasi-separated.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050H","source_file":"stacks-morphisms.tex","source_line":549,"source_end_line":571,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L549-L571","statement_sha256":"117496299382eee63deca3df34692d0f32c69b1558aef10fd08e245bbd7a361a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14060,"rank":14060,"depth":59,"x":1974.736,"y":1687.777,"cluster":"algebraic-stacks"},{"id":"stacks:050I","tag":"050I","title":"Separation axioms · Lemma 050I","summary":"Let T be an algebraic stack. Let g : X → Y be a morphism of algebraic stacks over T. Consider the graph i : X → X ×_T Y of g. Then • i is representable by algebraic spaces and locally of finite type, • if Y → T is DM, then i is unramified, • if Y → T is quasi-DM, then i is locally quasi-finite, • if Y → T is separated, then i is proper, and • if Y → T is quasi-separated, then i is quasi-compact and quasi-separated.","statement_latex":"Let $\\mathcal{T}$ be an algebraic stack. Let $g : \\mathcal{X} \\to \\mathcal{Y}$\nbe a morphism of algebraic stacks over $\\mathcal{T}$. Consider the graph\n$i : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{T} \\mathcal{Y}$ of $g$. Then\n\\begin{enumerate}\n\\item $i$ is representable by algebraic spaces and locally of finite type,\n\\item if $\\mathcal{Y} \\to \\mathcal{T}$ is DM, then $i$ is unramified,\n\\item if $\\mathcal{Y} \\to \\mathcal{T}$ is quasi-DM, then $i$ is locally\nquasi-finite,\n\\item if $\\mathcal{Y} \\to \\mathcal{T}$ is separated, then $i$ is proper, and\n\\item if $\\mathcal{Y} \\to \\mathcal{T}$ is quasi-separated, then $i$ is\nquasi-compact and quasi-separated.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050I","source_file":"stacks-morphisms.tex","source_line":594,"source_end_line":608,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L594-L608","statement_sha256":"f5dbfc1a6691e26e83c687d0e9eb056f41a1c9da889a4ed995e02dafbbc286ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":14061,"rank":14061,"depth":60,"x":2000.124,"y":1503.789,"cluster":"algebraic-stacks"},{"id":"stacks:050J","tag":"050J","title":"Separation axioms · Lemma 050J","summary":"Let f : X → T be a morphism of algebraic stacks. Let s : T → X be a morphism such that f ∘ s is 2-isomorphic to id_T. Then • s is representable by algebraic spaces and locally of finite type, • if f is DM, then s is unramified, • if f is quasi-DM, then s is locally quasi-finite, • if f is separated, then s is proper, and • if f is quasi-separated, then s is quasi-compact and quasi-separated.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{T}$ be a morphism of algebraic stacks.\nLet $s : \\mathcal{T} \\to \\mathcal{X}$ be a morphism such that\n$f \\circ s$ is $2$-isomorphic to $\\text{id}_\\mathcal{T}$. Then\n\\begin{enumerate}\n\\item $s$ is representable by algebraic spaces and locally of finite type,\n\\item if $f$ is DM, then $s$ is unramified,\n\\item if $f$ is quasi-DM, then $s$ is locally quasi-finite,\n\\item if $f$ is separated, then $s$ is proper, and\n\\item if $f$ is quasi-separated, then $s$ is quasi-compact and quasi-separated.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050J","source_file":"stacks-morphisms.tex","source_line":617,"source_end_line":629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L617-L629","statement_sha256":"6e89002a17efdf09601870939ca4c77d36e9e8c12dbe1b606722eecb21e09f3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14062,"rank":14062,"depth":61,"x":2129.532,"y":1654.063,"cluster":"algebraic-stacks"},{"id":"stacks:050K","tag":"050K","title":"Separation axioms · Lemma 050K","summary":"All of the separation axioms listed in Definition [Tag 04YW] are stable under composition of morphisms.","statement_latex":"All of the separation axioms listed in\nDefinition \\ref{definition-separated}\nare stable under composition of morphisms.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050K","source_file":"stacks-morphisms.tex","source_line":638,"source_end_line":643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L638-L643","statement_sha256":"84571a41faba2c60773edcc144bebc0e3efad5b8215f8b2ffcdfbb6377ff2e7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14063,"rank":14063,"depth":60,"x":1912.976,"y":1616.633,"cluster":"algebraic-stacks"},{"id":"stacks:050L","tag":"050L","title":"Separation axioms · Lemma 050L","summary":"Let f : X → Y be a morphism of algebraic stacks over the base scheme S. • If Y is DM over S and f is DM, then X is DM over S. • If Y is quasi-DM over S and f is quasi-DM, then X is quasi-DM over S. • If Y is separated over S and f is separated, then X is separated over S. • If Y is quasi-separated over S and f is quasi-separated, then X is quasi-separated over S. • If Y is DM and f is DM, then X is DM. • If Y is quasi-DM and f is quasi-DM, then X is quasi-DM. • If Y is…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nover the base scheme $S$.\n\\begin{enumerate}\n\\item If $\\mathcal{Y}$ is DM over $S$ and $f$ is DM,\nthen $\\mathcal{X}$ is DM over $S$.\n\\item If $\\mathcal{Y}$ is quasi-DM over $S$ and $f$ is quasi-DM,\nthen $\\mathcal{X}$ is quasi-DM over $S$.\n\\item If $\\mathcal{Y}$ is separated over $S$ and $f$ is separated,\nthen $\\mathcal{X}$ is separated over $S$.\n\\item If $\\mathcal{Y}$ is quasi-separated over $S$ and $f$ is quasi-separated,\nthen $\\mathcal{X}$ is quasi-separated over $S$.\n\\item If $\\mathcal{Y}$ is DM and $f$ is DM,\nthen $\\mathcal{X}$ is DM.\n\\item If $\\mathcal{Y}$ is quasi-DM and $f$ is quasi-DM,\nthen $\\mathcal{X}$ is quasi-DM.\n\\item If $\\mathcal{Y}$ is separated and $f$ is separated,\nthen $\\mathcal{X}$ is separated.\n\\item If $\\mathcal{Y}$ is quasi-separated and $f$ is quasi-separated,\nthen $\\mathcal{X}$ is quasi-separated.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050L","source_file":"stacks-morphisms.tex","source_line":674,"source_end_line":696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L674-L696","statement_sha256":"3af7d9d5d604a3c669977c5c30f9f0bfbad813bcabe8c4d03a0c2c98a7c64a2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14064,"rank":14064,"depth":61,"x":2103.012,"y":1521.229,"cluster":"algebraic-stacks"},{"id":"stacks:050M","tag":"050M","title":"Separation axioms · Lemma 050M","summary":"Let f : X → Y and g : Y → Z be morphisms of algebraic stacks. • If g ∘ f is DM then so is f. • If g ∘ f is quasi-DM then so is f. • If g ∘ f is separated and Δ_g is separated, then f is separated. • If g ∘ f is quasi-separated and Δ_g is quasi-separated, then f is quasi-separated.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and\n$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks.\n\\begin{enumerate}\n\\item If $g \\circ f$ is DM then so is $f$.\n\\item If $g \\circ f$ is quasi-DM then so is $f$.\n\\item If $g \\circ f$ is separated and $\\Delta_g$ is separated, then\n$f$ is separated.\n\\item If $g \\circ f$ is quasi-separated and\n$\\Delta_g$ is quasi-separated, then $f$ is quasi-separated.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050M","source_file":"stacks-morphisms.tex","source_line":715,"source_end_line":727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L715-L727","statement_sha256":"81637df761b1cb630f525cc4de1616f33213c6e2b6099080f1a1f767ba49ecc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14065,"rank":14065,"depth":60,"x":2039.52,"y":1699.644,"cluster":"algebraic-stacks"},{"id":"stacks:050N","tag":"050N","title":"Separation axioms · Lemma 050N","summary":"Let X be an algebraic stack over the base scheme S. • X is DM ⇔ X is DM over S. • X is quasi-DM ⇔ X is quasi-DM over S. • If X is separated, then X is separated over S. • If X is quasi-separated, then X is quasi-separated over S. Let f : X → Y be a morphism of algebraic stacks over the base scheme S. • [(5)] If X is DM over S, then f is DM. • [(6)] If X is quasi-DM over S, then f is quasi-DM. • [(7)] If X is separated over S and Δ_Y/S is separated, then f is separated. •…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over the base scheme $S$.\n\\begin{enumerate}\n\\item\n$\\mathcal{X}$ is DM $\\Leftrightarrow$\n$\\mathcal{X}$ is DM over $S$.\n\\item\n$\\mathcal{X}$ is quasi-DM $\\Leftrightarrow$\n$\\mathcal{X}$ is quasi-DM over $S$.\n\\item If $\\mathcal{X}$ is separated, then\n$\\mathcal{X}$ is separated over $S$.\n\\item If $\\mathcal{X}$ is quasi-separated, then\n$\\mathcal{X}$ is quasi-separated over $S$.\n\\end{enumerate}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nover the base scheme $S$.\n\\begin{enumerate}\n\\item[(5)] If $\\mathcal{X}$ is DM over $S$, then $f$ is DM.\n\\item[(6)] If $\\mathcal{X}$ is quasi-DM over $S$, then $f$ is quasi-DM.\n\\item[(7)] If $\\mathcal{X}$ is separated over $S$ and\n$\\Delta_{\\mathcal{Y}/S}$ is separated, then $f$ is separated.\n\\item[(8)] If $\\mathcal{X}$ is quasi-separated over $S$ and\n$\\Delta_{\\mathcal{Y}/S}$ is quasi-separated, then $f$ is quasi-separated.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050N","source_file":"stacks-morphisms.tex","source_line":847,"source_end_line":872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L847-L872","statement_sha256":"38b20e33156dbebb371d902fb7b7e2c03ce4273f5f629a0f616fecdc117c03b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14066,"rank":14066,"depth":61,"x":1942.735,"y":1531.837,"cluster":"algebraic-stacks"},{"id":"stacks:06MB","tag":"06MB","title":"Separation axioms · Lemma 06MB","summary":"Let X be an algebraic stack. Let W be an algebraic space, and let f : W → X be a surjective, flat, locally finitely presented morphism. • If f is unramified (i.e., étale, i.e., X is Deligne-Mumford), then X is DM. • If f is locally quasi-finite, then X is quasi-DM.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $W$ be an algebraic space, and let $f : W \\to \\mathcal{X}$\nbe a surjective, flat, locally finitely presented morphism.\n\\begin{enumerate}\n\\item If $f$ is unramified (i.e., \\'etale, i.e., $\\mathcal{X}$\nis Deligne-Mumford), then $\\mathcal{X}$ is DM.\n\\item If $f$ is locally quasi-finite, then $\\mathcal{X}$ is quasi-DM.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MB","source_file":"stacks-morphisms.tex","source_line":903,"source_end_line":913,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L903-L913","statement_sha256":"596b2dbb56e6e4f4b958747d76d8b0cb688402cebe451e7c4932ca9366afc639","origin":"The Stacks Project","memory_eligible":false,"source_rank":14067,"rank":14067,"depth":47,"x":2149.318,"y":1600.747,"cluster":"algebraic-stacks"},{"id":"stacks:06MY","tag":"06MY","title":"Separation axioms · Lemma 06MY","summary":"A monomorphism of algebraic stacks is separated and DM. The same is true for immersions of algebraic stacks.","statement_latex":"A monomorphism of algebraic stacks is separated and DM.\nThe same is true for immersions of algebraic stacks.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MY","source_file":"stacks-morphisms.tex","source_line":945,"source_end_line":949,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L945-L949","statement_sha256":"d7833938869b272c98620e5deae099670410674c2cdd7847f4039893b9f91b0f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14068,"rank":14068,"depth":71,"x":1941.301,"y":1667.241,"cluster":"algebraic-stacks"},{"id":"stacks:06MZ","tag":"06MZ","title":"Separation axioms · Lemma 06MZ","summary":"Let X be an algebraic stack. Let x ∈ |X|. Assume the residual gerbe Z_x of X at x exists. If X is DM, resp. quasi-DM, resp. separated, resp. quasi-separated, then so is Z_x.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$.\nAssume the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists.\nIf $\\mathcal{X}$ is DM, resp.\\ quasi-DM, resp.\\ separated,\nresp.\\ quasi-separated, then so is $\\mathcal{Z}_x$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Separation axioms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MZ","source_file":"stacks-morphisms.tex","source_line":962,"source_end_line":968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L962-L968","statement_sha256":"c851d6a43e4a6401687bc4236be6f3ed93b557be5cca1fdd045eda09496a1451","origin":"The Stacks Project","memory_eligible":false,"source_rank":14069,"rank":14069,"depth":72,"x":2041.347,"y":1499.958,"cluster":"algebraic-stacks"},{"id":"stacks:050Q","tag":"050Q","title":"Inertia stacks · Lemma 050Q","summary":"Let X be an algebraic stack. Then the inertia stack I_X is an algebraic stack as well. The morphism I_X → X is representable by algebraic spaces and locally of finite type. More generally, let f : X → Y be a morphism of algebraic stacks. Then the relative inertia I_X/Y is an algebraic stack and the morphism I_X/Y → X is representable by algebraic spaces and locally of finite type.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Then the inertia stack\n$\\mathcal{I}_\\mathcal{X}$ is an algebraic stack as well.\nThe morphism\n$$\n\\mathcal{I}_\\mathcal{X} \\longrightarrow \\mathcal{X}\n$$\nis representable by algebraic spaces and locally of finite type.\nMore generally, let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism\nof algebraic stacks. Then the relative inertia\n$\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$ is an algebraic stack and the\nmorphism\n$$\n\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\longrightarrow \\mathcal{X}\n$$\nis representable by algebraic spaces and locally of finite type.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Inertia stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050Q","source_file":"stacks-morphisms.tex","source_line":1006,"source_end_line":1023,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1006-L1023","statement_sha256":"bb8561f118cad59f14ab050aa149e7bb4dc96e35f3553ebb1ff45e5e49d75210","origin":"The Stacks Project","memory_eligible":false,"source_rank":14070,"rank":14070,"depth":58,"x":2102.177,"y":1680.312,"cluster":"algebraic-stacks"},{"id":"stacks:06PP","tag":"06PP","title":"Inertia stacks · Definition 06PP","summary":"Let f : X → Y be a morphism of algebraic stacks. Let Z be an algebraic space. • Let x : Z → X be a morphism. We set mathitIsom_X/Y(x, x) = Z ×_x, X I_X/Y We endow it with the structure of a group algebraic space over Z by pulling back the composition law discussed in Remark [Tag 050R]. We will sometimes refer to mathitIsom_X/Y(x, x) as the relative sheaf of automorphisms of x. • Let x_1, x_2 : Z → X be morphisms. Set y_i = f ∘ x_i. Let α : y_1 → y_2 be a 2-morphism. Then…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $Z$ be an algebraic space.\n\\begin{enumerate}\n\\item Let $x : Z \\to \\mathcal{X}$ be a morphism. We set\n$$\n\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x, x) =\nZ \\times_{x, \\mathcal{X}} \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}\n$$\nWe endow it with the structure of a group algebraic space over $Z$\nby pulling back the composition law discussed in\nRemark \\ref{remark-inertia-is-group-in-spaces}.\nWe will sometimes refer to $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x, x)$\nas the {\\it relative sheaf of automorphisms of $x$}.\n\\item Let $x_1, x_2 : Z \\to \\mathcal{X}$ be morphisms. Set\n$y_i = f \\circ x_i$. Let $\\alpha : y_1 \\to y_2$ be a $2$-morphism.\nThen $\\alpha$ determines a morphism\n$\\Delta^\\alpha : Z \\to Z \\times_{y_1, \\mathcal{Y}, y_2} Z$ and we set\n$$\n\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2) =\n(Z \\times_{x_1, \\mathcal{X}, x_2} Z)\n\\times_{Z \\times_{y_1, \\mathcal{Y}, y_2} Z, \\Delta^\\alpha} Z.\n$$\nWe will sometimes refer to\n$\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2)$\nas the {\\it relative sheaf of isomorphisms from $x_1$ to $x_2$}.\n\\end{enumerate}\nIf $\\mathcal{Y} = \\Spec(\\mathbf{Z})$ or more generally when $\\mathcal{Y}$\nis an algebraic space, then we use the notation\n$\\mathit{Isom}_\\mathcal{X}(x, x)$ and $\\mathit{Isom}_\\mathcal{X}(x_1, x_2)$\nand we use the terminology {\\it sheaf of automorphisms of $x$}\nand {\\it sheaf of isomorphisms from $x_1$ to $x_2$}.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Inertia stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PP","source_file":"stacks-morphisms.tex","source_line":1143,"source_end_line":1176,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1143-L1176","statement_sha256":"1d26ca07fc2b35cba25f2c480f6ae4956b18ced955c33c822d571a88e2e8d8d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14071,"rank":14071,"depth":0,"x":1912.041,"y":1581.71,"cluster":"algebraic-stacks"},{"id":"stacks:0CPK","tag":"0CPK","title":"Inertia stacks · Lemma 0CPK","summary":"Let f : X → Y be a morphism of algebraic stacks. Let Z be an algebraic space and let x_i : Z → X, i = 1, 2 be morphisms. Then • mathitIsom_X/Y(x_2, x_2) is a group algebraic space over Z, • there is an exact sequence of groups 0 → mathitIsom_X/Y(x_2, x_2) → mathitIsom_X(x_2, x_2) → mathitIsom_Y(f ∘ x_2, f ∘ x_2) • there is a map of algebraic spaces mathitIsom_X(x_1, x_2) → mathitIsom_Y(f ∘ x_1, f ∘ x_2) such that for any 2-morphism α : f ∘ x_1 → f ∘ x_2 we obtain a…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $Z$ be an algebraic space and let $x_i : Z \\to \\mathcal{X}$, $i = 1, 2$\nbe morphisms. Then\n\\begin{enumerate}\n\\item $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)$\nis a group algebraic space over $Z$,\n\\item there is an exact sequence of groups\n$$\n0 \\to \\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)\n\\to \\mathit{Isom}_\\mathcal{X}(x_2, x_2)\n\\to \\mathit{Isom}_\\mathcal{Y}(f \\circ x_2, f \\circ x_2)\n$$\n\\item there is a map of algebraic spaces\n$\n\\mathit{Isom}_\\mathcal{X}(x_1, x_2)\n\\to \\mathit{Isom}_\\mathcal{Y}(f \\circ x_1, f \\circ x_2)\n$\nsuch that for any $2$-morphism $\\alpha : f \\circ x_1 \\to f \\circ x_2$\nwe obtain a cartesian diagram\n$$\n\\xymatrix{\n\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2) \\ar[d] \\ar[r] &\nZ \\ar[d]^\\alpha \\\\\n\\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\ar[r] &\n\\mathit{Isom}_\\mathcal{Y}(f \\circ x_1, f \\circ x_2)\n}\n$$\n\\item for any $2$-morphism $\\alpha : f \\circ x_1 \\to f \\circ x_2$ the\nalgebraic space $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}^\\alpha(x_1, x_2)$\nis a pseudo torsor for $\\mathit{Isom}_{\\mathcal{X}/\\mathcal{Y}}(x_2, x_2)$\nover $Z$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Inertia stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPK","source_file":"stacks-morphisms.tex","source_line":1178,"source_end_line":1212,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1178-L1212","statement_sha256":"371cd87ba96b9a27d767448bcc7cecd94831f7f3202e35a86463b97c592bd195","origin":"The Stacks Project","memory_eligible":false,"source_rank":14072,"rank":14072,"depth":2,"x":2131.82,"y":1546.485,"cluster":"algebraic-stacks"},{"id":"stacks:06PQ","tag":"06PQ","title":"Inertia stacks · Lemma 06PQ","summary":"Let π : X → Y and f : Y' → Y be morphisms of algebraic stacks. Set X' = X ×_Y Y'. Then both squares in the diagram xymatrix I_X'/Y' ar[r] ar[d]_ Categories, Equation ([Tag 04Z4]) & X' ar[r]_π' ar[d] & Y' ar[d]^f I_X/Y ar[r] & X ar[r]^π & Y are fibre product squares.","statement_latex":"Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ and\n$f : \\mathcal{Y}' \\to \\mathcal{Y}$ be morphisms of algebraic stacks.\nSet $\\mathcal{X}' = \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Y}'$.\nThen both squares in the diagram\n$$\n\\xymatrix{\n\\mathcal{I}_{\\mathcal{X}'/\\mathcal{Y}'} \\ar[r]\n\\ar[d]_{\n\\text{Categories, Equation}\\ (\\ref{categories-equation-functorial})\n} &\n\\mathcal{X}' \\ar[r]_{\\pi'} \\ar[d] & \\mathcal{Y}' \\ar[d]^f \\\\\n\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\ar[r] &\n\\mathcal{X} \\ar[r]^\\pi & \\mathcal{Y}\n}\n$$\nare fibre product squares.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Inertia stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PQ","source_file":"stacks-morphisms.tex","source_line":1222,"source_end_line":1240,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1222-L1240","statement_sha256":"c64c19b0c5715c807563e9516908b119cc8ce2057368a33f7206bf65a6061a17","origin":"The Stacks Project","memory_eligible":false,"source_rank":14073,"rank":14073,"depth":2,"x":1997.914,"y":1697.362,"cluster":"algebraic-stacks"},{"id":"stacks:06R5","tag":"06R5","title":"Inertia stacks · Lemma 06R5","summary":"Let f : X → Y be a monomorphism of algebraic stacks. Then the diagram xymatrix I_X ar[r] ar[d] & X ar[d] I_Y ar[r] & Y is a fibre product square.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a monomorphism of algebraic stacks.\nThen the diagram\n$$\n\\xymatrix{\n\\mathcal{I}_\\mathcal{X} \\ar[r] \\ar[d] &\n\\mathcal{X} \\ar[d] \\\\\n\\mathcal{I}_\\mathcal{Y} \\ar[r] &\n\\mathcal{Y}\n}\n$$\nis a fibre product square.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Inertia stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R5","source_file":"stacks-morphisms.tex","source_line":1266,"source_end_line":1279,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1266-L1279","statement_sha256":"c44fa1d1b8495349187e7ac2a3ec566a205b998ff92e6a15c0131b9514241a9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14074,"rank":14074,"depth":71,"x":1975.294,"y":1509.885,"cluster":"algebraic-stacks"},{"id":"stacks:06PR","tag":"06PR","title":"Inertia stacks · Lemma 06PR","summary":"Let X be an algebraic stack. Let [U/R] → X be a presentation. Let G/U be the stabilizer group algebraic space associated to the groupoid (U, R, s, t, c). Then xymatrix G ar[d] ar[r] & U ar[d] I_X ar[r] & X is a fibre product diagram.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $[U/R] \\to \\mathcal{X}$\nbe a presentation. Let $G/U$ be the stabilizer group algebraic space\nassociated to the groupoid $(U, R, s, t, c)$. Then\n$$\n\\xymatrix{\nG \\ar[d] \\ar[r] & U \\ar[d] \\\\\n\\mathcal{I}_\\mathcal{X} \\ar[r] & \\mathcal{X}\n}\n$$\nis a fibre product diagram.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Inertia stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PR","source_file":"stacks-morphisms.tex","source_line":1299,"source_end_line":1311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1299-L1311","statement_sha256":"7db562e51b50b0256560a81c4857f221bc9020da485080e207f9c7d1772e4402","origin":"The Stacks Project","memory_eligible":false,"source_rank":14075,"rank":14075,"depth":14,"x":2142.951,"y":1635.453,"cluster":"algebraic-stacks"},{"id":"stacks:0CL0","tag":"0CL0","title":"Higher diagonals · Lemma 0CL0","summary":"Let f : X → Y be a morphism of algebraic stacks. • The following are equivalent • I_X/Y → X is separated, • Δ_f, 1 = Δ_f : X → X ×_Y X is separated, and • Δ_f, 2 = e : X → I_X/Y is a closed immersion. • The following are equivalent • I_X/Y → X is quasi-separated, • Δ_f, 1 = Δ_f : X → X ×_Y X is quasi-separated, and • Δ_f, 2 = e : X → I_X/Y is quasi-compact. • The following are equivalent • I_X/Y → X is locally separated, • Δ_f, 1 = Δ_f : X → X ×_Y X is locally separated,…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$\nis separated,\n\\item $\\Delta_{f, 1} = \\Delta_f :\n\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$\nis separated, and\n\\item $\\Delta_{f, 2} = e :\n\\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$\nis a closed immersion.\n\\end{enumerate}\n\\item\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$\nis quasi-separated,\n\\item $\\Delta_{f, 1} = \\Delta_f :\n\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$\nis quasi-separated, and\n\\item $\\Delta_{f, 2} = e :\n\\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$\nis quasi-compact.\n\\end{enumerate}\n\\item\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$\nis locally separated,\n\\item $\\Delta_{f, 1} = \\Delta_f :\n\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$\nis locally separated, and\n\\item $\\Delta_{f, 2} = e :\n\\mathcal{X} \\to \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$\nis an immersion.\n\\end{enumerate}\n\\item\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$\nis unramified,\n\\item $f$ is DM.\n\\end{enumerate}\n\\item\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$\nis locally quasi-finite,\n\\item $f$ is quasi-DM.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Higher diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CL0","source_file":"stacks-morphisms.tex","source_line":1371,"source_end_line":1426,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1371-L1426","statement_sha256":"e79041f5440db8a261b305ec22b375ecc0fa43159080d7c0a1e9cbd70c527551","origin":"The Stacks Project","memory_eligible":false,"source_rank":14076,"rank":14076,"depth":59,"x":1918.058,"y":1637.997,"cluster":"algebraic-stacks"},{"id":"stacks:04YY","tag":"04YY","title":"Higher diagonals · Lemma 04YY","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent: • the morphism f is representable by algebraic spaces, • the second diagonal of f is an isomorphism, • the group stack I_X/Y is trivial over X, and • for a scheme T and a morphism x : T → X the kernel of mathitIsom_X(x, x) → mathitIsom_Y(f(x), f(x)) is trivial.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent:\n\\begin{enumerate}\n\\item the morphism $f$ is representable by algebraic spaces,\n\\item the second diagonal of $f$ is an isomorphism,\n\\item the group stack $ \\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}}$\nis trivial over $\\mathcal X$, and\n\\item for a scheme $T$ and a morphism $x : T \\to \\mathcal{X}$\nthe kernel of $\\mathit{Isom}_\\mathcal{X}(x, x) \\to\n\\mathit{Isom}_\\mathcal{Y}(f(x), f(x))$ is trivial.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Higher diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YY","source_file":"stacks-morphisms.tex","source_line":1487,"source_end_line":1500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1487-L1500","statement_sha256":"116a75a05ff153a7283d77c20bbc6583c69c0fedf9f4d43ef750b77301655769","origin":"The Stacks Project","memory_eligible":false,"source_rank":14077,"rank":14077,"depth":69,"x":2082.054,"y":1508.347,"cluster":"algebraic-stacks"},{"id":"stacks:0AHJ","tag":"0AHJ","title":"Higher diagonals · Lemma 0AHJ","summary":"A morphism f : X → Y of algebraic stacks is • a monomorphism if and only if Δ_f, 1 is an isomorphism, and • representable by algebraic spaces if and only if Δ_f, 1 is a monomorphism. Moreover, the second diagonal Δ_f, 2 is always a monomorphism.","statement_latex":"A morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks is\n\\begin{enumerate}\n\\item a monomorphism if and only if $\\Delta_{f, 1}$ is an isomorphism, and\n\\item representable by algebraic spaces if and only if $\\Delta_{f, 1}$\nis a monomorphism.\n\\end{enumerate}\nMoreover, the second diagonal $\\Delta_{f, 2}$ is always a monomorphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Higher diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AHJ","source_file":"stacks-morphisms.tex","source_line":1528,"source_end_line":1537,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1528-L1537","statement_sha256":"162a32882105e4ffe155538372a264aa5794188c061674e2227b8b3d2282780e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14078,"rank":14078,"depth":71,"x":2065.366,"y":1697.243,"cluster":"algebraic-stacks"},{"id":"stacks:04YZ","tag":"04YZ","title":"Higher diagonals · Lemma 04YZ","summary":"Let f : X → Y be a morphism of algebraic stacks. Then • Δ_f, 1 separated ⇔ Δ_f, 2 closed immersion ⇔ Δ_f, 2 proper ⇔ Δ_f, 2 universally closed, • Δ_f, 1 quasi-separated ⇔ Δ_f, 2 finite type ⇔ Δ_f, 2 quasi-compact, and • Δ_f, 1 locally separated ⇔ Δ_f, 2 immersion.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThen\n\\begin{enumerate}\n\\item $\\Delta_{f, 1}$ separated $\\Leftrightarrow$\n$\\Delta_{f, 2}$ closed immersion $\\Leftrightarrow$\n$\\Delta_{f, 2}$ proper $\\Leftrightarrow$\n$\\Delta_{f, 2}$ universally closed,\n\\item $\\Delta_{f, 1}$ quasi-separated $\\Leftrightarrow$\n$\\Delta_{f, 2}$ finite type $\\Leftrightarrow$ $\\Delta_{f, 2}$ quasi-compact,\nand\n\\item $\\Delta_{f, 1}$ locally separated $\\Leftrightarrow$\n$\\Delta_{f, 2}$ immersion.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Higher diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04YZ","source_file":"stacks-morphisms.tex","source_line":1554,"source_end_line":1569,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1554-L1569","statement_sha256":"b01d2e7032bf13ea9e5e2b91881a8292f2f2638938c5fd5c41264f2fecb706fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14079,"rank":14079,"depth":62,"x":1925.59,"y":1548.296,"cluster":"algebraic-stacks"},{"id":"stacks:04Z0","tag":"04Z0","title":"Higher diagonals · Lemma 04Z0","summary":"Let f : X → Y be a morphism of algebraic stacks. Then • f is separated if and only if Δ_f, 1 and Δ_f, 2 are universally closed, and • f is quasi-separated if and only if Δ_f, 1 and Δ_f, 2 are quasi-compact. • f is quasi-DM if and only if Δ_f, 1 and Δ_f, 2 are locally quasi-finite. • f is DM if and only if Δ_f, 1 and Δ_f, 2 are unramified.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThen\n\\begin{enumerate}\n\\item $f$ is separated if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$\nare universally closed, and\n\\item $f$ is quasi-separated if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$\nare quasi-compact.\n\\item $f$ is quasi-DM if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$\nare locally quasi-finite.\n\\item $f$ is DM if and only if $\\Delta_{f, 1}$ and $\\Delta_{f, 2}$\nare unramified.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Higher diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04Z0","source_file":"stacks-morphisms.tex","source_line":1583,"source_end_line":1597,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1583-L1597","statement_sha256":"56c9de5ce004d1df9db43075a2999bb975d00001426e44e5e3052c2a112dfaee","origin":"The Stacks Project","memory_eligible":false,"source_rank":14080,"rank":14080,"depth":63,"x":2148.719,"y":1578.854,"cluster":"algebraic-stacks"},{"id":"stacks:0CPL","tag":"0CPL","title":"Higher diagonals · Lemma 0CPL","summary":"Let f : X → Y be a separated (resp. quasi-separated, resp. quasi-DM, resp. DM) morphism of algebraic stacks. Then • given algebraic spaces T_i, i = 1, 2 and morphisms x_i : T_i → X, with y_i = f ∘ x_i the morphism T_1 ×_x_1, X, x_2 T_2 → T_1 ×_y_1, Y, y_2 T_2 is proper (resp. quasi-compact and quasi-separated, resp. locally quasi-finite, resp. unramified), • given an algebraic space T and morphisms x_i : T → X, i = 1, 2, with y_i = f ∘ x_i the morphism mathitIsom_X(x_1,…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a separated\n(resp.\\ quasi-separated, resp.\\ quasi-DM, resp.\\ DM)\nmorphism of algebraic stacks. Then\n\\begin{enumerate}\n\\item given algebraic spaces $T_i$, $i = 1, 2$ and morphisms\n$x_i : T_i \\to \\mathcal{X}$, with $y_i = f \\circ x_i$ the morphism\n$$\nT_1 \\times_{x_1, \\mathcal{X}, x_2} T_2 \\longrightarrow\nT_1 \\times_{y_1, \\mathcal{Y}, y_2} T_2\n$$\nis proper (resp.\\ quasi-compact and quasi-separated,\nresp.\\ locally quasi-finite, resp.\\ unramified),\n\\item given an algebraic space $T$ and morphisms\n$x_i : T \\to \\mathcal{X}$, $i = 1, 2$, with $y_i = f \\circ x_i$ the morphism\n$$\n\\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\longrightarrow\n\\mathit{Isom}_\\mathcal{Y}(y_1, y_2)\n$$\nis proper (resp.\\ quasi-compact and quasi-separated,\nresp.\\ locally quasi-finite, resp.\\ unramified).\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Higher diagonals","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPL","source_file":"stacks-morphisms.tex","source_line":1638,"source_end_line":1661,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1638-L1661","statement_sha256":"69dab462c04b53420ff48ce6788c18d15a29a6b5a7618b1589fcb12336d1c69b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14081,"rank":14081,"depth":0,"x":1959.375,"y":1683.061,"cluster":"algebraic-stacks"},{"id":"stacks:050T","tag":"050T","title":"Quasi-compact morphisms · Lemma 050T","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent: • f is quasi-compact (as in Properties of Stacks, Section [Tag 04XB]), and • for every quasi-compact algebraic stack Z and any morphism Z → Y the algebraic stack Z ×_Y X is quasi-compact.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich is representable by algebraic spaces. The following are equivalent:\n\\begin{enumerate}\n\\item $f$ is quasi-compact (as in Properties of Stacks,\nSection \\ref{stacks-properties-section-properties-morphisms}), and\n\\item for every quasi-compact algebraic stack $\\mathcal{Z}$\nand any morphism $\\mathcal{Z} \\to \\mathcal{Y}$ the algebraic stack\n$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$ is quasi-compact.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050T","source_file":"stacks-morphisms.tex","source_line":1705,"source_end_line":1716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1705-L1716","statement_sha256":"174512024e803b8cc6edba50dc669f78e5491e67903747520ad500005de953e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14082,"rank":14082,"depth":43,"x":2015.264,"y":1498.55,"cluster":"algebraic-stacks"},{"id":"stacks:050U","tag":"050U","title":"Quasi-compact morphisms · Definition 050U","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is quasi-compact if for every quasi-compact algebraic stack Z and morphism Z → Y the fibre product Z ×_Y X is quasi-compact.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is {\\it quasi-compact} if for every quasi-compact\nalgebraic stack $\\mathcal{Z}$ and morphism $\\mathcal{Z} \\to \\mathcal{Y}$\nthe fibre product $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$\nis quasi-compact.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050U","source_file":"stacks-morphisms.tex","source_line":1753,"source_end_line":1760,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1753-L1760","statement_sha256":"230d3aeedc216cee7f46d7e14164c003dbc8b75bc2312dba13f9901ceb5c4ccd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14083,"rank":14083,"depth":0,"x":2122.565,"y":1666.529,"cluster":"algebraic-stacks"},{"id":"stacks:050V","tag":"050V","title":"Quasi-compact morphisms · Lemma 050V","summary":"The base change of a quasi-compact morphism of algebraic stacks by any morphism of algebraic stacks is quasi-compact.","statement_latex":"The base change of a quasi-compact morphism of algebraic stacks\nby any morphism of algebraic stacks is quasi-compact.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050V","source_file":"stacks-morphisms.tex","source_line":1770,"source_end_line":1774,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1770-L1774","statement_sha256":"77e3e27a7d746afb5fbf632a2766eb6afc068494b2e6e0ccfb25ec940f4c3365","origin":"The Stacks Project","memory_eligible":false,"source_rank":14084,"rank":14084,"depth":0,"x":1908.091,"y":1603.47,"cluster":"algebraic-stacks"},{"id":"stacks:050W","tag":"050W","title":"Quasi-compact morphisms · Lemma 050W","summary":"The composition of a pair of quasi-compact morphisms of algebraic stacks is quasi-compact.","statement_latex":"The composition of a pair of quasi-compact morphisms of algebraic stacks\nis quasi-compact.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050W","source_file":"stacks-morphisms.tex","source_line":1780,"source_end_line":1784,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1780-L1784","statement_sha256":"f8cb7a10e6dbaef0fbf357f2e0dc4e262f67d4716e7eba2714525b7f656b5b45","origin":"The Stacks Project","memory_eligible":false,"source_rank":14085,"rank":14085,"depth":0,"x":2117.211,"y":1528.178,"cluster":"algebraic-stacks"},{"id":"stacks:0CL1","tag":"0CL1","title":"Quasi-compact morphisms · Lemma 0CL1","summary":"A closed immersion of algebraic stacks is quasi-compact.","statement_latex":"A closed immersion of algebraic stacks is quasi-compact.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CL1","source_file":"stacks-morphisms.tex","source_line":1790,"source_end_line":1793,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1790-L1793","statement_sha256":"d0a112eec8414e85ed07a643a2d9849c1f84bdd6709a8679c7e9e3e754cb5f26","origin":"The Stacks Project","memory_eligible":false,"source_rank":14086,"rank":14086,"depth":0,"x":2023.441,"y":1702.574,"cluster":"algebraic-stacks"},{"id":"stacks:050X","tag":"050X","title":"Quasi-compact morphisms · Lemma 050X","summary":"Let xymatrix X ar[rr]_f ar[rd]_p & & Y ar[dl]^q & Z be a 2-commutative diagram of morphisms of algebraic stacks. If f is surjective and p is quasi-compact, then q is quasi-compact.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[rr]_f \\ar[rd]_p & &\n\\mathcal{Y} \\ar[dl]^q \\\\\n& \\mathcal{Z}\n}\n$$\nbe a $2$-commutative diagram of morphisms of algebraic stacks.\nIf $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050X","source_file":"stacks-morphisms.tex","source_line":1800,"source_end_line":1812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1800-L1812","statement_sha256":"327452709b521e38807f7268c20653387e88c0a009bdd46c529bea65d1641099","origin":"The Stacks Project","memory_eligible":false,"source_rank":14087,"rank":14087,"depth":13,"x":1952.253,"y":1520.544,"cluster":"algebraic-stacks"},{"id":"stacks:050Y","tag":"050Y","title":"Quasi-compact morphisms · Lemma 050Y","summary":"Let f : X → Y and g : Y → Z be morphisms of algebraic stacks. If g ∘ f is quasi-compact and g is quasi-separated then f is quasi-compact.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and\n$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks.\nIf $g \\circ f$ is quasi-compact and $g$ is quasi-separated\nthen $f$ is quasi-compact.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/050Y","source_file":"stacks-morphisms.tex","source_line":1830,"source_end_line":1836,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1830-L1836","statement_sha256":"f6b46228fdf551b414c045e119ef2bfcc36fef617f540dfe7af758db0c4e1dd3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14088,"rank":14088,"depth":62,"x":2151.375,"y":1614.492,"cluster":"algebraic-stacks"},{"id":"stacks:075S","tag":"075S","title":"Quasi-compact morphisms · Lemma 075S","summary":"Let f : X → Y be a morphism of algebraic stacks. • If X is quasi-compact and Y is quasi-separated, then f is quasi-compact. • If X is quasi-compact and quasi-separated and Y is quasi-separated, then f is quasi-compact and quasi-separated. • A fibre product of quasi-compact and quasi-separated algebraic stacks is quasi-compact and quasi-separated.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item If $\\mathcal{X}$ is quasi-compact and $\\mathcal{Y}$ is\nquasi-separated, then $f$ is quasi-compact.\n\\item If $\\mathcal{X}$ is quasi-compact and quasi-separated and $\\mathcal{Y}$\nis quasi-separated, then $f$ is quasi-compact and quasi-separated.\n\\item A fibre product of quasi-compact and quasi-separated algebraic stacks\nis quasi-compact and quasi-separated.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075S","source_file":"stacks-morphisms.tex","source_line":1854,"source_end_line":1865,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1854-L1865","statement_sha256":"22023a75594cbca5a7009e34df2a07bb489d0ca8a99b9ca4654c5e954cd77842","origin":"The Stacks Project","memory_eligible":false,"source_rank":14089,"rank":14089,"depth":63,"x":1928.721,"y":1658.257,"cluster":"algebraic-stacks"},{"id":"stacks:0CL2","tag":"0CL2","title":"Quasi-compact morphisms · Lemma 0CL2","summary":"Let f : X → Y be a quasi-compact morphism of algebraic stacks. Let y ∈ |Y| be a point in the closure of the image of |f|. There exists a valuation ring A with fraction field K and a commutative diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d] Spec(A) ar[r] & Y such that the closed point of Spec(A) maps to y.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact morphism of\nalgebraic stacks. Let $y \\in |\\mathcal{Y}|$ be a point in the closure\nof the image of $|f|$. There exists a valuation ring $A$ with\nfraction field $K$ and a commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\\n\\Spec(A) \\ar[r] & \\mathcal{Y}\n}\n$$\nsuch that the closed point of $\\Spec(A)$ maps to $y$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CL2","source_file":"stacks-morphisms.tex","source_line":1887,"source_end_line":1900,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1887-L1900","statement_sha256":"c44c7c243c44566f57bd93b5ae2dd01612e2cf80addf78c17735e023973c8803","origin":"The Stacks Project","memory_eligible":false,"source_rank":14090,"rank":14090,"depth":58,"x":2057.868,"y":1499.451,"cluster":"algebraic-stacks"},{"id":"stacks:0DTL","tag":"0DTL","title":"Quasi-compact morphisms · Lemma 0DTL","summary":"Let f : X → Y be a morphism of algebraic stacks. Let W → Y be surjective, flat, and locally of finite presentation where W is an algebraic space. If the base change W ×_Y X → W is quasi-compact, then f is quasi-compact.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $W \\to \\mathcal{Y}$ be surjective, flat, and locally of finite\npresentation where $W$ is an algebraic space. If the base change\n$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is quasi-compact, then\n$f$ is quasi-compact.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-compact morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTL","source_file":"stacks-morphisms.tex","source_line":1940,"source_end_line":1947,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1940-L1947","statement_sha256":"401fed218b52de1cdc6c476257647101a1c7afc16c6d2c0979a7c67efe459a50","origin":"The Stacks Project","memory_eligible":false,"source_rank":14091,"rank":14091,"depth":0,"x":2090.384,"y":1690.067,"cluster":"algebraic-stacks"},{"id":"stacks:0510","tag":"0510","title":"Noetherian algebraic stacks · Definition 0510","summary":"Let X be an algebraic stack. We say X is Noetherian if X is quasi-compact, quasi-separated and locally Noetherian.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. We say $\\mathcal{X}$ is\n{\\it Noetherian} if $\\mathcal{X}$ is quasi-compact, quasi-separated\nand locally Noetherian.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Noetherian algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0510","source_file":"stacks-morphisms.tex","source_line":1985,"source_end_line":1990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L1985-L1990","statement_sha256":"f003b4bf326faa037cd2bdc0682a361337067b8fe493a166553960ce878c3637","origin":"The Stacks Project","memory_eligible":false,"source_rank":14092,"rank":14092,"depth":0,"x":1912.902,"y":1567.81,"cluster":"algebraic-stacks"},{"id":"stacks:0CPM","tag":"0CPM","title":"Noetherian algebraic stacks · Lemma 0CPM","summary":"Let j : X → Y be an immersion of algebraic stacks. • If Y is locally Noetherian, then X is locally Noetherian and j is quasi-compact. • If Y is Noetherian, then X is Noetherian.","statement_latex":"Let $j : \\mathcal{X} \\to \\mathcal{Y}$ be an immersion of algebraic stacks.\n\\begin{enumerate}\n\\item If $\\mathcal{Y}$ is locally Noetherian, then\n$\\mathcal{X}$ is locally Noetherian and $j$ is quasi-compact.\n\\item If $\\mathcal{Y}$ is Noetherian, then $\\mathcal{X}$ is Noetherian.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Noetherian algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPM","source_file":"stacks-morphisms.tex","source_line":2009,"source_end_line":2017,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2009-L2017","statement_sha256":"f4606970dd428e507b10aaa6e8206970bd9cdeac66641700f42b1ea6fe4e0865","origin":"The Stacks Project","memory_eligible":false,"source_rank":14093,"rank":14093,"depth":62,"x":2142.37,"y":1557.24,"cluster":"algebraic-stacks"},{"id":"stacks:0DQI","tag":"0DQI","title":"Noetherian algebraic stacks · Lemma 0DQI","summary":"Let X be an algebraic stack. • If X is locally Noetherian then |X| is a locally Noetherian topological space. • If X is quasi-compact and locally Noetherian, then |X| is a Noetherian topological space.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\n\\begin{enumerate}\n\\item If $\\mathcal{X}$ is locally Noetherian then $|\\mathcal{X}|$\nis a locally Noetherian topological space.\n\\item If $\\mathcal{X}$ is quasi-compact and locally Noetherian, then\n$|\\mathcal{X}|$ is a Noetherian topological space.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Noetherian algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQI","source_file":"stacks-morphisms.tex","source_line":2039,"source_end_line":2048,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2039-L2048","statement_sha256":"fc4714177bddddce8863834d85d8645eb3ecb8173d3c5785c73cb1ec1b7c8587","origin":"The Stacks Project","memory_eligible":false,"source_rank":14094,"rank":14094,"depth":3,"x":1981.468,"y":1695.408,"cluster":"algebraic-stacks"},{"id":"stacks:0GVX","tag":"0GVX","title":"Noetherian algebraic stacks · Lemma 0GVX","summary":"Let X be a locally Noetherian algebraic stack. Then |X| is quasi-sober (Topology, Definition [Tag 004X]).","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nThen $|\\mathcal{X}|$ is quasi-sober (Topology, Definition\n\\ref{topology-definition-generic-point}).","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Noetherian algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVX","source_file":"stacks-morphisms.tex","source_line":2066,"source_end_line":2071,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2066-L2071","statement_sha256":"a9626fe81005cb4fdf42ad92c8c0c3b3095404bd484a21b076ee74b29834178e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14095,"rank":14095,"depth":3,"x":1989.013,"y":1501.991,"cluster":"algebraic-stacks"},{"id":"stacks:0CHQ","tag":"0CHQ","title":"Affine morphisms · Definition 0CHQ","summary":"A morphism of algebraic stacks is said to be affine if it is representable and affine in the sense of Properties of Stacks, Section [Tag 04XB].","statement_latex":"A morphism of algebraic stacks is said to be {\\it affine}\nif it is representable and affine in the sense of\nProperties of Stacks, Section\n\\ref{stacks-properties-section-properties-morphisms}.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Affine morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHQ","source_file":"stacks-morphisms.tex","source_line":2099,"source_end_line":2105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2099-L2105","statement_sha256":"16b871cfd86996f58d708bf006cdcad8586a1ce98b80241b53beab3f120d4a01","origin":"The Stacks Project","memory_eligible":false,"source_rank":14096,"rank":14096,"depth":0,"x":2139.172,"y":1649.073,"cluster":"algebraic-stacks"},{"id":"stacks:0CHR","tag":"0CHR","title":"Affine morphisms · Lemma 0CHR","summary":"Let X → Y be a morphism of algebraic stacks. Let Z → Y be an affine morphism of algebraic stacks. Then Z ×_Y X → X is an affine morphism of algebraic stacks.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be an affine morphism of algebraic\nstacks. Then $\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$\nis an affine morphism of algebraic stacks.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHR","source_file":"stacks-morphisms.tex","source_line":2119,"source_end_line":2125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2119-L2125","statement_sha256":"e708c946fb5a76f48e5ae2e4e9706ea67f572b3d00ff130496d00043a8de3ff2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14097,"rank":14097,"depth":0,"x":1909.891,"y":1625.792,"cluster":"algebraic-stacks"},{"id":"stacks:0CHS","tag":"0CHS","title":"Affine morphisms · Lemma 0CHS","summary":"Compositions of affine morphisms of algebraic stacks are affine.","statement_latex":"Compositions of affine morphisms of algebraic stacks are affine.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHS","source_file":"stacks-morphisms.tex","source_line":2133,"source_end_line":2136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2133-L2136","statement_sha256":"24cd3755a811a6d6472b2f3cab486908e343a1c0ae57cca94433c105ee3841cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14098,"rank":14098,"depth":1,"x":2097.907,"y":1512.723,"cluster":"algebraic-stacks"},{"id":"stacks:0GQE","tag":"0GQE","title":"Affine morphisms · Lemma 0GQE","summary":"Let xymatrix X ar[rr]_f ar[rd]_a & & Y ar[dl]^b & Z be a commutative diagram of morphisms of algebraic stacks. If a is affine and Δ_b is affine, then f is affine.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[rr]_f \\ar[rd]_a & & \\mathcal{Y} \\ar[dl]^b \\\\\n& \\mathcal{Z}\n}\n$$\nbe a commutative diagram of morphisms of algebraic stacks.\nIf $a$ is affine and $\\Delta_b$ is affine, then\n$f$ is affine.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Affine morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQE","source_file":"stacks-morphisms.tex","source_line":2146,"source_end_line":2158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2146-L2158","statement_sha256":"9abc9817c68382e7bb74c668b84184a4284b63870042be96d23408f29f012a31","origin":"The Stacks Project","memory_eligible":false,"source_rank":14099,"rank":14099,"depth":2,"x":2050.136,"y":1703.012,"cluster":"algebraic-stacks"},{"id":"stacks:0CHU","tag":"0CHU","title":"Integral and finite morphisms · Definition 0CHU","summary":"Let f : X → Y be a morphism of algebraic stacks. • We say f is integral if f is representable and integral in the sense of Properties of Stacks, Section [Tag 04XB]. • We say f is finite if f is representable and finite in the sense of Properties of Stacks, Section [Tag 04XB].","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item We say $f$ is {\\it integral} if $f$ is representable and integral\nin the sense of Properties of Stacks, Section\n\\ref{stacks-properties-section-properties-morphisms}.\n\\item We say $f$ is {\\it finite} if $f$ is representable and finite\nin the sense of Properties of Stacks, Section\n\\ref{stacks-properties-section-properties-morphisms}.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Integral and finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHU","source_file":"stacks-morphisms.tex","source_line":2186,"source_end_line":2197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2186-L2197","statement_sha256":"5995c25a34909f48e619797496406d37d67996d97500c74acf489ac3cc86665b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14100,"rank":14100,"depth":0,"x":1932.195,"y":1535.388,"cluster":"algebraic-stacks"},{"id":"stacks:0CHV","tag":"0CHV","title":"Integral and finite morphisms · Lemma 0CHV","summary":"Let X → Y be a morphism of algebraic stacks. Let Z → Y be an integral (or finite) morphism of algebraic stacks. Then Z ×_Y X → X is an integral (or finite) morphism of algebraic stacks.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be an integral (or finite)\nmorphism of algebraic stacks. Then\n$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{X}$\nis an integral (or finite) morphism of algebraic stacks.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHV","source_file":"stacks-morphisms.tex","source_line":2212,"source_end_line":2219,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2212-L2219","statement_sha256":"390c6c415a534d1b28807da4a7be86dc16c6e865efcafae3fac236d206a6a339","origin":"The Stacks Project","memory_eligible":false,"source_rank":14101,"rank":14101,"depth":0,"x":2154.227,"y":1592.137,"cluster":"algebraic-stacks"},{"id":"stacks:0CHW","tag":"0CHW","title":"Integral and finite morphisms · Lemma 0CHW","summary":"Compositions of integral, resp. finite morphisms of algebraic stacks are integral, resp. finite.","statement_latex":"Compositions of integral, resp.\\ finite morphisms of algebraic stacks\nare integral, resp.\\ finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Integral and finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHW","source_file":"stacks-morphisms.tex","source_line":2227,"source_end_line":2231,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2227-L2231","statement_sha256":"048929e3ceee5ba51d999cf1d82421cb6737ec6d505a50ae23fb401b1a69c704","origin":"The Stacks Project","memory_eligible":false,"source_rank":14102,"rank":14102,"depth":1,"x":1944.618,"y":1676.378,"cluster":"algebraic-stacks"},{"id":"stacks:06U1","tag":"06U1","title":"Open morphisms · Lemma 06U1","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent • f is universally open (as in Properties of Stacks, Section [Tag 04XB]), and • for every morphism of algebraic stacks Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is open.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of\nalgebraic stacks which is representable by algebraic spaces.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally open (as in Properties of Stacks,\nSection \\ref{stacks-properties-section-properties-morphisms}), and\n\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$\nthe morphism of topological spaces\n$|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is open.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06U1","source_file":"stacks-morphisms.tex","source_line":2255,"source_end_line":2267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2255-L2267","statement_sha256":"51656ffb1be1cc27d802a6ca3fcb957c59b35492c8df5e748242e0da3bbac0ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":14103,"rank":14103,"depth":0,"x":2031.54,"y":1495.108,"cluster":"algebraic-stacks"},{"id":"stacks:06U2","tag":"06U2","title":"Open morphisms · Definition 06U2","summary":"Let f : X → Y be a morphism of algebraic stacks. • We say f is open if the map of topological spaces |X| → |Y| is open. • We say f is universally open if for every morphism of algebraic stacks Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is open, i.e., the base change Z ×_Y X → Z is open.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item We say $f$ is {\\it open} if the map of topological\nspaces $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is open.\n\\item We say $f$ is {\\it universally open} if for every morphism\nof algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$\nthe morphism of topological spaces\n$$\n|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|\n$$\nis open, i.e., the base change\n$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is open.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Open morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06U2","source_file":"stacks-morphisms.tex","source_line":2297,"source_end_line":2312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2297-L2312","statement_sha256":"36ddd57c9ea1163bbe7e8535a519265c699b82dc48237552ff11f7f509ac0a5e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14104,"rank":14104,"depth":0,"x":2113.315,"y":1678.313,"cluster":"algebraic-stacks"},{"id":"stacks:06U3","tag":"06U3","title":"Open morphisms · Lemma 06U3","summary":"The base change of a universally open morphism of algebraic stacks by any morphism of algebraic stacks is universally open.","statement_latex":"The base change of a universally open morphism of algebraic stacks\nby any morphism of algebraic stacks is universally open.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06U3","source_file":"stacks-morphisms.tex","source_line":2314,"source_end_line":2318,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2314-L2318","statement_sha256":"c95cdaa99ecd5180e31476c418af033ad47a75c2100fd61fda85f056665f6cd5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14105,"rank":14105,"depth":0,"x":1905.441,"y":1589.516,"cluster":"algebraic-stacks"},{"id":"stacks:06U4","tag":"06U4","title":"Open morphisms · Lemma 06U4","summary":"The composition of a pair of (universally) open morphisms of algebraic stacks is (universally) open.","statement_latex":"The composition of a pair of (universally) open morphisms of\nalgebraic stacks is (universally) open.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Open morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06U4","source_file":"stacks-morphisms.tex","source_line":2324,"source_end_line":2328,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2324-L2328","statement_sha256":"15c6423f7644e2e81ab6d458bd3aff7b1ab9c0315f27c968fa560432f6eabff4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14106,"rank":14106,"depth":0,"x":2130.398,"y":1536.979,"cluster":"algebraic-stacks"},{"id":"stacks:0CHX","tag":"0CHX","title":"Submersive morphisms · Lemma 0CHX","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent • f is universally submersive (as in Properties of Stacks, Section [Tag 04XB]), and • for every morphism of algebraic stacks Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is submersive.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of\nalgebraic stacks which is representable by algebraic spaces.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally submersive (as in Properties of Stacks,\nSection \\ref{stacks-properties-section-properties-morphisms}), and\n\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$\nthe morphism of topological spaces\n$|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is submersive.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Submersive morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHX","source_file":"stacks-morphisms.tex","source_line":2351,"source_end_line":2363,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2351-L2363","statement_sha256":"4d19ae3ee0bfa4796d0c3cc2148e27107a2e36d3a272391d66fcc2dce775ea8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14107,"rank":14107,"depth":0,"x":2006.621,"y":1703.561,"cluster":"algebraic-stacks"},{"id":"stacks:06U6","tag":"06U6","title":"Submersive morphisms · Definition 06U6","summary":"Let f : X → Y be a morphism of algebraic stacks. • We say f is submersive if the continuous map |X| → |Y| is submersive, see Topology, Definition [Tag 0406]. • We say f is universally submersive if for every morphism of algebraic stacks Y' → Y the base change Y' ×_Y X → Y' is submersive.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item We say $f$ is {\\it submersive}\\footnote{This is very different\nfrom the notion of a submersion of differential manifolds.}\nif the continuous map $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is submersive, see\nTopology, Definition \\ref{topology-definition-submersive}.\n\\item We say $f$ is {\\it universally submersive} if for every\nmorphism of algebraic stacks $\\mathcal{Y}' \\to \\mathcal{Y}$\nthe base change $\\mathcal{Y}' \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Y}'$\nis submersive.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Submersive morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06U6","source_file":"stacks-morphisms.tex","source_line":2393,"source_end_line":2406,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2393-L2406","statement_sha256":"041cda49bb80619bf15b59697ebde434679c450ed13f7c333ca99b4e4a963a24","origin":"The Stacks Project","memory_eligible":false,"source_rank":14108,"rank":14108,"depth":1,"x":1963.881,"y":1510.265,"cluster":"algebraic-stacks"},{"id":"stacks:0CHY","tag":"0CHY","title":"Submersive morphisms · Lemma 0CHY","summary":"The base change of a universally submersive morphism of algebraic stacks by any morphism of algebraic stacks is universally submersive.","statement_latex":"The base change of a universally submersive morphism of algebraic stacks\nby any morphism of algebraic stacks is universally submersive.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Submersive morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHY","source_file":"stacks-morphisms.tex","source_line":2411,"source_end_line":2415,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2411-L2415","statement_sha256":"46f0897ff2842099463d0eaf1286a21c20e81dea6561d1858cbc5012ce517a86","origin":"The Stacks Project","memory_eligible":false,"source_rank":14109,"rank":14109,"depth":0,"x":2151.06,"y":1628.686,"cluster":"algebraic-stacks"},{"id":"stacks:0CHZ","tag":"0CHZ","title":"Submersive morphisms · Lemma 0CHZ","summary":"The composition of a pair of (universally) submersive morphisms of algebraic stacks is (universally) submersive.","statement_latex":"The composition of a pair of (universally) submersive morphisms of\nalgebraic stacks is (universally) submersive.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Submersive morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CHZ","source_file":"stacks-morphisms.tex","source_line":2421,"source_end_line":2425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2421-L2425","statement_sha256":"867cc12be9debc39e6444ceff9ed9b58d932b7bd458e3752344a98a3075d3b33","origin":"The Stacks Project","memory_eligible":false,"source_rank":14110,"rank":14110,"depth":0,"x":1917.533,"y":1647.595,"cluster":"algebraic-stacks"},{"id":"stacks:0512","tag":"0512","title":"Universally closed morphisms · Lemma 0512","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent • f is universally closed (as in Properties of Stacks, Section [Tag 04XB]), and • for every morphism of algebraic stacks Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is closed.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of\nalgebraic stacks which is representable by algebraic spaces.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed (as in Properties of Stacks,\nSection \\ref{stacks-properties-section-properties-morphisms}), and\n\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$\nthe morphism of topological spaces\n$|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is closed.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0512","source_file":"stacks-morphisms.tex","source_line":2453,"source_end_line":2465,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2453-L2465","statement_sha256":"c22273062b20d54e1d27116d8172904b51916acc883bdd2289fffb15cdb7b4c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14111,"rank":14111,"depth":0,"x":2074.709,"y":1500.972,"cluster":"algebraic-stacks"},{"id":"stacks:0513","tag":"0513","title":"Universally closed morphisms · Definition 0513","summary":"Let f : X → Y be a morphism of algebraic stacks. • We say f is closed if the map of topological spaces |X| → |Y| is closed. • We say f is universally closed if for every morphism of algebraic stacks Z → Y the morphism of topological spaces |Z ×_Y X| → |Z| is closed, i.e., the base change Z ×_Y X → Z is closed.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item We say $f$ is {\\it closed} if the map of topological\nspaces $|\\mathcal{X}| \\to |\\mathcal{Y}|$ is closed.\n\\item We say $f$ is {\\it universally closed} if for every morphism\nof algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$\nthe morphism of topological spaces\n$$\n|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|\n$$\nis closed, i.e., the base change\n$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$ is closed.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally closed morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0513","source_file":"stacks-morphisms.tex","source_line":2495,"source_end_line":2510,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2495-L2510","statement_sha256":"df46b8edb986cb493efb0aeea6761bf086e4f16375630b8c119a196fe5b1f12f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14112,"rank":14112,"depth":0,"x":2076.723,"y":1698.505,"cluster":"algebraic-stacks"},{"id":"stacks:0514","tag":"0514","title":"Universally closed morphisms · Lemma 0514","summary":"The base change of a universally closed morphism of algebraic stacks by any morphism of algebraic stacks is universally closed.","statement_latex":"The base change of a universally closed morphism of algebraic stacks\nby any morphism of algebraic stacks is universally closed.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0514","source_file":"stacks-morphisms.tex","source_line":2512,"source_end_line":2516,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2512-L2516","statement_sha256":"34f8e80fbc01995671058049927034551d3aa1f037a23f2ada290586430f7d34","origin":"The Stacks Project","memory_eligible":false,"source_rank":14113,"rank":14113,"depth":0,"x":1916.2,"y":1553.822,"cluster":"algebraic-stacks"},{"id":"stacks:0515","tag":"0515","title":"Universally closed morphisms · Lemma 0515","summary":"The composition of a pair of (universally) closed morphisms of algebraic stacks is (universally) closed.","statement_latex":"The composition of a pair of (universally) closed morphisms of\nalgebraic stacks is (universally) closed.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0515","source_file":"stacks-morphisms.tex","source_line":2522,"source_end_line":2526,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2522-L2526","statement_sha256":"6c6d9326c80e370aa6c09c37ecf9952a32e365d6b6406f07c23362130ef63ad5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14114,"rank":14114,"depth":0,"x":2151.19,"y":1569.443,"cluster":"algebraic-stacks"},{"id":"stacks:0CL3","tag":"0CL3","title":"Universally closed morphisms · Lemma 0CL3","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • f is universally closed, • for every scheme Z and every morphism Z → Y the projection |Z ×_Y X| → |Z| is closed, • for every affine scheme Z and every morphism Z → Y the projection |Z ×_Y X| → |Z| is closed, and • there exists an algebraic space V and a surjective smooth morphism V → Y such that V ×_Y X → V is a universally closed morphism of algebraic stacks.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed,\n\\item for every scheme $Z$ and every morphism $Z \\to \\mathcal{Y}$\nthe projection $|Z \\times_\\mathcal{Y} \\mathcal{X}| \\to |Z|$\nis closed,\n\\item for every affine scheme $Z$ and every morphism $Z \\to \\mathcal{Y}$\nthe projection $|Z \\times_\\mathcal{Y} \\mathcal{X}| \\to |Z|$ is\nclosed, and\n\\item there exists an algebraic space $V$ and a surjective smooth morphism\n$V \\to \\mathcal{Y}$ such that $V \\times_\\mathcal{Y} \\mathcal{X} \\to V$\nis a universally closed morphism of algebraic stacks.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CL3","source_file":"stacks-morphisms.tex","source_line":2532,"source_end_line":2548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2532-L2548","statement_sha256":"2004cecfaa34b0ac0d7e2435d3ab5716a8fdb27e64b763a5ec5cdb19e4526adc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14115,"rank":14115,"depth":58,"x":1965.135,"y":1691.404,"cluster":"algebraic-stacks"},{"id":"stacks:0CI1","tag":"0CI1","title":"Universally injective morphisms · Lemma 0CI1","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent • f is universally injective (as in Properties of Stacks, Section [Tag 04XB]), and • for every morphism of algebraic stacks Z → Y the map |Z ×_Y X| → |Z| is injective.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of\nalgebraic stacks which is representable by algebraic spaces.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally injective (as in Properties of Stacks,\nSection \\ref{stacks-properties-section-properties-morphisms}), and\n\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$\nthe map $|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$\nis injective.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CI1","source_file":"stacks-morphisms.tex","source_line":2620,"source_end_line":2632,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2620-L2632","statement_sha256":"248344ebabaf353ba09e8e64f06c24e6ad151476263e9c4489e76d7a19b1c8b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14116,"rank":14116,"depth":0,"x":2004.297,"y":1495.674,"cluster":"algebraic-stacks"},{"id":"stacks:0CI2","tag":"0CI2","title":"Universally injective morphisms · Definition 0CI2","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is universally injective if for every morphism of algebraic stacks Z → Y the map |Z ×_Y X| → |Z| is injective.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is {\\it universally injective} if for every morphism\nof algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$ the map\n$$\n|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|\n$$\nis injective.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally injective morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CI2","source_file":"stacks-morphisms.tex","source_line":2662,"source_end_line":2671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2662-L2671","statement_sha256":"101882148443a852118f7e8b709a87dfc2507549a74e98729e2433420a3fa7bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14117,"rank":14117,"depth":0,"x":2132.967,"y":1662.415,"cluster":"algebraic-stacks"},{"id":"stacks:0CI3","tag":"0CI3","title":"Universally injective morphisms · Lemma 0CI3","summary":"The base change of a universally injective morphism of algebraic stacks by any morphism of algebraic stacks is universally injective.","statement_latex":"The base change of a universally injective morphism of algebraic stacks\nby any morphism of algebraic stacks is universally injective.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CI3","source_file":"stacks-morphisms.tex","source_line":2673,"source_end_line":2677,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2673-L2677","statement_sha256":"3bac8df01e542e9dc99e1788ae1b4307590cd13d1bf2c99b9f4b405f0719908d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14118,"rank":14118,"depth":0,"x":1903.736,"y":1612.416,"cluster":"algebraic-stacks"},{"id":"stacks:0CI4","tag":"0CI4","title":"Universally injective morphisms · Lemma 0CI4","summary":"The composition of a pair of universally injective morphisms of algebraic stacks is universally injective.","statement_latex":"The composition of a pair of universally injective morphisms of\nalgebraic stacks is universally injective.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CI4","source_file":"stacks-morphisms.tex","source_line":2683,"source_end_line":2687,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2683-L2687","statement_sha256":"5a65640c76bd0846ecfbaadd62131220ef866d3d7322549618eff35ca83a694c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14119,"rank":14119,"depth":0,"x":2113.216,"y":1519.107,"cluster":"algebraic-stacks"},{"id":"stacks:0CPN","tag":"0CPN","title":"Universally injective morphisms · Lemma 0CPN","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • f is universally injective, • Δ : X → X ×_Y X is surjective, and • for an algebraically closed field, for x_1, x_2 : Spec(k) → X, and for a 2-arrow β : f ∘ x_1 → f ∘ x_2 there is a 2-arrow α : x_1 → x_2 with β = id_f star α.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally injective,\n\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$\nis surjective, and\n\\item for an algebraically closed field, for\n$x_1, x_2 : \\Spec(k) \\to \\mathcal{X}$, and for a $2$-arrow\n$\\beta : f \\circ x_1 \\to f \\circ x_2$ there is a\n$2$-arrow $\\alpha : x_1 \\to x_2$ with\n$\\beta = \\text{id}_f \\star \\alpha$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPN","source_file":"stacks-morphisms.tex","source_line":2693,"source_end_line":2707,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2693-L2707","statement_sha256":"a244dc93136400ae84d9dce86cf96a482102df1f59dc05bb17635047e831f789","origin":"The Stacks Project","memory_eligible":false,"source_rank":14120,"rank":14120,"depth":59,"x":2033.694,"y":1706.99,"cluster":"algebraic-stacks"},{"id":"stacks:0DTM","tag":"0DTM","title":"Universally injective morphisms · Lemma 0DTM","summary":"Let f : X → Y be a universally injective morphism of algebraic stacks. Let y : Spec(k) → Y be a morphism where k is an algebraically closed field. If y is in the image of |X| → |Y|, then there is a morphism x : Spec(k) → X with y = f ∘ x.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a universally injective\nmorphism of algebraic stacks. Let $y : \\Spec(k) \\to \\mathcal{Y}$\nbe a morphism where $k$ is an algebraically closed field.\nIf $y$ is in the image of $|\\mathcal{X}| \\to |\\mathcal{Y}|$,\nthen there is a morphism $x : \\Spec(k) \\to \\mathcal{X}$\nwith $y = f \\circ x$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTM","source_file":"stacks-morphisms.tex","source_line":2763,"source_end_line":2771,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2763-L2771","statement_sha256":"40fa3fcbcd00e078cbd46e927c00ff2ab989834fcef286faf3e6d63517109e83","origin":"The Stacks Project","memory_eligible":false,"source_rank":14121,"rank":14121,"depth":60,"x":1941.136,"y":1523.116,"cluster":"algebraic-stacks"},{"id":"stacks:0DTN","tag":"0DTN","title":"Universally injective morphisms · Lemma 0DTN","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent: • f is universally injective, • for every affine scheme Z and any morphism Z → Y the morphism Z ×_Y X → Z is universally injective, and • add more here.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent:\n\\begin{enumerate}\n\\item $f$ is universally injective,\n\\item for every affine scheme $Z$ and any morphism\n$Z \\to \\mathcal{Y}$ the morphism $Z \\times_\\mathcal{Y} \\mathcal{X} \\to Z$\nis universally injective, and\n\\item add more here.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTN","source_file":"stacks-morphisms.tex","source_line":2819,"source_end_line":2830,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2819-L2830","statement_sha256":"1a4d0b5d55470c6bc06f9fd3b5ef6556d5c4b4a5f8abfcad1c0000b941c6b97b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14122,"rank":14122,"depth":60,"x":2157.5,"y":1606.277,"cluster":"algebraic-stacks"},{"id":"stacks:0DTP","tag":"0DTP","title":"Universally injective morphisms · Lemma 0DTP","summary":"Let f : X → Y be a morphism of algebraic stacks. Let W → Y be surjective, flat, and locally of finite presentation where W is an algebraic space. If the base change W ×_Y X → W is universally injective, then f is universally injective.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $W \\to \\mathcal{Y}$ be surjective, flat, and locally of finite\npresentation where $W$ is an algebraic space. If the base change\n$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is universally injective,\nthen $f$ is universally injective.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universally injective morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTP","source_file":"stacks-morphisms.tex","source_line":2848,"source_end_line":2855,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2848-L2855","statement_sha256":"2dc1dd516fb1c1ed37930792cf2a66c1961fc379da8303feec2d9d265529a275","origin":"The Stacks Project","memory_eligible":false,"source_rank":14123,"rank":14123,"depth":60,"x":1930.824,"y":1667.795,"cluster":"algebraic-stacks"},{"id":"stacks:0CI6","tag":"0CI6","title":"Universal homeomorphisms · Lemma 0CI6","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent • f is a universal homeomorphism (Properties of Stacks, Section [Tag 04XB]), and • for every morphism of algebraic stacks Z → Y the map of topological spaces |Z ×_Y X| → |Z| is a homeomorphism.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of\nalgebraic stacks which is representable by algebraic spaces.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is a universal homeomorphism (Properties of Stacks,\nSection \\ref{stacks-properties-section-properties-morphisms}), and\n\\item for every morphism of algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$\nthe map of topological spaces\n$|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|$ is\na homeomorphism.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CI6","source_file":"stacks-morphisms.tex","source_line":2886,"source_end_line":2899,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2886-L2899","statement_sha256":"9ebfa116a2a6d22746de3cb1f2212591384ac092969fb3461b00f5817d9071f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14124,"rank":14124,"depth":0,"x":2048.633,"y":1493.614,"cluster":"algebraic-stacks"},{"id":"stacks:0CI7","tag":"0CI7","title":"Universal homeomorphisms · Definition 0CI7","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is a universal homeomorphism if for every morphism of algebraic stacks Z → Y the map of topological spaces |Z ×_Y X| → |Z| is a homeomorphism.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is a {\\it universal homeomorphism} if for every morphism\nof algebraic stacks $\\mathcal{Z} \\to \\mathcal{Y}$\nthe map of topological spaces\n$$\n|\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}| \\to |\\mathcal{Z}|\n$$\nis a homeomorphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universal homeomorphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CI7","source_file":"stacks-morphisms.tex","source_line":2929,"source_end_line":2939,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2929-L2939","statement_sha256":"ce02ab755565e2716abc839fcd805a388f4a33a8e32992bdb952e65a9fd55547","origin":"The Stacks Project","memory_eligible":false,"source_rank":14125,"rank":14125,"depth":0,"x":2101.894,"y":1689.121,"cluster":"algebraic-stacks"},{"id":"stacks:0CI8","tag":"0CI8","title":"Universal homeomorphisms · Lemma 0CI8","summary":"The base change of a universal homeomorphism of algebraic stacks by any morphism of algebraic stacks is a universal homeomorphism.","statement_latex":"The base change of a universal homeomorphism of algebraic stacks\nby any morphism of algebraic stacks is a universal homeomorphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CI8","source_file":"stacks-morphisms.tex","source_line":2941,"source_end_line":2945,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2941-L2945","statement_sha256":"9c441178f5a6335d1bdaf71826748e6532f9766efde4de1461d74868dde5b4e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14126,"rank":14126,"depth":0,"x":1905.178,"y":1575.05,"cluster":"algebraic-stacks"},{"id":"stacks:0CI9","tag":"0CI9","title":"Universal homeomorphisms · Lemma 0CI9","summary":"The composition of a pair of universal homeomorphisms of algebraic stacks is a universal homeomorphism.","statement_latex":"The composition of a pair of universal homeomorphisms of\nalgebraic stacks is a universal homeomorphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CI9","source_file":"stacks-morphisms.tex","source_line":2951,"source_end_line":2955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2951-L2955","statement_sha256":"5cfb91bf38b33684ccb2743a9ab67cc80089a462b7638ff3c815f0ce1c0a43e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14127,"rank":14127,"depth":0,"x":2142.232,"y":1547.512,"cluster":"algebraic-stacks"},{"id":"stacks:0DTQ","tag":"0DTQ","title":"Universal homeomorphisms · Lemma 0DTQ","summary":"Let f : X → Y be a morphism of algebraic stacks. Let W → Y be surjective, flat, and locally of finite presentation where W is an algebraic space. If the base change W ×_Y X → W is a universal homeomorphism, then f is a universal homeomorphism.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $W \\to \\mathcal{Y}$ be surjective, flat, and locally of finite\npresentation where $W$ is an algebraic space. If the base change\n$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is a universal homeomorphism,\nthen $f$ is a universal homeomorphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Universal homeomorphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTQ","source_file":"stacks-morphisms.tex","source_line":2961,"source_end_line":2968,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L2961-L2968","statement_sha256":"731410599839b28321df74d0b4cd6661d8df692f2eab1f020a12188b9165af81","origin":"The Stacks Project","memory_eligible":false,"source_rank":14128,"rank":14128,"depth":61,"x":1989.406,"y":1702.502,"cluster":"algebraic-stacks"},{"id":"stacks:06FM","tag":"06FM","title":"Types of morphisms smooth local on source-and-target · Lemma 06FM","summary":"Let P be a property of morphisms of algebraic spaces which is smooth local on the source-and-target. Let f : X → Y be a morphism of algebraic stacks. Consider commutative diagrams xymatrix U ar[d]_a ar[r]_h & V ar[d]^b X ar[r]^f & Y where U and V are algebraic spaces and the vertical arrows are smooth. The following are equivalent • for any diagram as above such that in addition U → X ×_Y V is smooth the morphism h has property P, and • for some diagram as above with a :…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces\nwhich is smooth local on the source-and-target.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nConsider commutative diagrams\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\n\\mathcal{X} \\ar[r]^f & \\mathcal{Y}\n}\n$$\nwhere $U$ and $V$ are algebraic spaces and the vertical arrows are smooth.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any diagram as above such that in addition\n$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is smooth\nthe morphism $h$ has property $\\mathcal{P}$, and\n\\item for some diagram as above with $a : U \\to \\mathcal{X}$ surjective\nthe morphism $h$ has property $\\mathcal{P}$.\n\\end{enumerate}\nIf $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by algebraic spaces,\nthen this is also equivalent to $f$ (as a morphism of algebraic spaces)\nhaving property $\\mathcal{P}$. If $\\mathcal{P}$ is also preserved under\nany base change, and fppf local on the base, then for morphisms $f$\nwhich are representable by algebraic spaces this\nis also equivalent to $f$ having property $\\mathcal{P}$ in the sense\nof\nProperties of Stacks,\nSection \\ref{stacks-properties-section-properties-morphisms}.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Types of morphisms smooth local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FM","source_file":"stacks-morphisms.tex","source_line":3023,"source_end_line":3053,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3023-L3053","statement_sha256":"8e8f386982d2e538b94003795fa28f3d7f0508df96f8c5febffb479dd827d830","origin":"The Stacks Project","memory_eligible":false,"source_rank":14129,"rank":14129,"depth":3,"x":1977.446,"y":1501.272,"cluster":"algebraic-stacks"},{"id":"stacks:06FN","tag":"06FN","title":"Types of morphisms smooth local on source-and-target · Definition 06FN","summary":"Let P be a property of morphisms of algebraic spaces which is smooth local on the source-and-target. We say a morphism f : X → Y of algebraic stacks has property P if the equivalent conditions of Lemma [Tag 06FM] hold.","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces\nwhich is smooth local on the source-and-target.\nWe say a morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks\n{\\it has property $\\mathcal{P}$} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Types of morphisms smooth local on source-and-target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FN","source_file":"stacks-morphisms.tex","source_line":3158,"source_end_line":3166,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3158-L3166","statement_sha256":"985d008eaac979bfc3419637352affd042c1754aae8a321bdd63aa5db3613054","origin":"The Stacks Project","memory_eligible":false,"source_rank":14130,"rank":14130,"depth":4,"x":2148.279,"y":1643.028,"cluster":"algebraic-stacks"},{"id":"stacks:06FS","tag":"06FS","title":"Morphisms of finite type · Definition 06FS","summary":"Let f : X → Y be a morphism of algebraic stacks. • We say f locally of finite type if the equivalent conditions of Lemma [Tag 06FM] hold with P = locally of finite type. • We say f is of finite type if it is locally of finite type and quasi-compact.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item We say $f$\n{\\it locally of finite type} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with\n$\\mathcal{P} = \\text{locally of finite type}$.\n\\item We say $f$ is\n{\\it of finite type} if it is locally of finite type and quasi-compact.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FS","source_file":"stacks-morphisms.tex","source_line":3260,"source_end_line":3272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3260-L3272","statement_sha256":"07033a3281c6f31749260013750669b3c81fb02fd3cc1560f0244c4d4dae7926","origin":"The Stacks Project","memory_eligible":false,"source_rank":14131,"rank":14131,"depth":4,"x":1908.046,"y":1635.426,"cluster":"algebraic-stacks"},{"id":"stacks:06FT","tag":"06FT","title":"Morphisms of finite type · Lemma 06FT","summary":"The composition of finite type morphisms is of finite type. The same holds for locally of finite type.","statement_latex":"The composition of finite type morphisms is of finite type.\nThe same holds for locally of finite type.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FT","source_file":"stacks-morphisms.tex","source_line":3274,"source_end_line":3278,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3274-L3278","statement_sha256":"29114cbfbfd7bef270f56fe8296d511b38800454e0768aee916b29c579651f51","origin":"The Stacks Project","memory_eligible":false,"source_rank":14132,"rank":14132,"depth":7,"x":2091.507,"y":1504.572,"cluster":"algebraic-stacks"},{"id":"stacks:06FU","tag":"06FU","title":"Morphisms of finite type · Lemma 06FU","summary":"A base change of a finite type morphism is finite type. The same holds for locally of finite type.","statement_latex":"A base change of a finite type morphism is finite type.\nThe same holds for locally of finite type.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FU","source_file":"stacks-morphisms.tex","source_line":3287,"source_end_line":3291,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3287-L3291","statement_sha256":"d2cce72dc29f174023a6766bc2b0fb301a292ad246c29acd7a1c207369745713","origin":"The Stacks Project","memory_eligible":false,"source_rank":14133,"rank":14133,"depth":7,"x":2061.421,"y":1705.384,"cluster":"algebraic-stacks"},{"id":"stacks:06FV","tag":"06FV","title":"Morphisms of finite type · Lemma 06FV","summary":"An immersion is locally of finite type.","statement_latex":"An immersion is locally of finite type.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FV","source_file":"stacks-morphisms.tex","source_line":3300,"source_end_line":3303,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3300-L3303","statement_sha256":"f76ed2ac68b4b6bf58906b1272ce8dbcebfcac4153d67ac089e49a5a6c9a3600","origin":"The Stacks Project","memory_eligible":false,"source_rank":14134,"rank":14134,"depth":4,"x":1921.964,"y":1540.054,"cluster":"algebraic-stacks"},{"id":"stacks:06R6","tag":"06R6","title":"Morphisms of finite type · Lemma 06R6","summary":"Let f : X → Y be a morphism of algebraic stacks. If f is locally of finite type and Y is locally Noetherian, then X is locally Noetherian.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $f$ is locally of finite type and $\\mathcal{Y}$ is locally Noetherian,\nthen $\\mathcal{X}$ is locally Noetherian.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R6","source_file":"stacks-morphisms.tex","source_line":3311,"source_end_line":3316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3311-L3316","statement_sha256":"4a4cc2dbe37aa58cd1ced9703736547377ee190c46dd48be8faf1edb4998b4cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14135,"rank":14135,"depth":2,"x":2158.012,"y":1582.882,"cluster":"algebraic-stacks"},{"id":"stacks:06U7","tag":"06U7","title":"Morphisms of finite type · Lemma 06U7","summary":"Let f : X → Y be a morphism of algebraic stacks. Let W → Y be a surjective, flat, and locally of finite presentation where W is an algebraic space. If the base change W ×_Y X → W is locally of finite type, then f is locally of finite type.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $W \\to \\mathcal{Y}$ be a surjective, flat, and locally of finite\npresentation where $W$ is an algebraic space. If the base change\n$W \\times_\\mathcal{Y} \\mathcal{X} \\to W$ is\nlocally of finite type, then $f$ is locally of finite type.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06U7","source_file":"stacks-morphisms.tex","source_line":3340,"source_end_line":3347,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3340-L3347","statement_sha256":"9885ad98042750aa1c7557f607fdfcb9732cfac5bcfc98dc079618061033f7a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14136,"rank":14136,"depth":43,"x":1949.283,"y":1685.353,"cluster":"algebraic-stacks"},{"id":"stacks:06U8","tag":"06U8","title":"Morphisms of finite type · Lemma 06U8","summary":"Let X → Y → Z be morphisms of algebraic stacks. Assume X → Z is locally of finite type and that X → Y is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Then Y → Z is locally of finite type.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of\nalgebraic stacks. Assume $\\mathcal{X} \\to \\mathcal{Z}$ is locally of finite\ntype and that $\\mathcal{X} \\to \\mathcal{Y}$ is representable by algebraic\nspaces, surjective, flat, and locally of finite presentation.\nThen $\\mathcal{Y} \\to \\mathcal{Z}$ is locally of finite type.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06U8","source_file":"stacks-morphisms.tex","source_line":3372,"source_end_line":3379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3372-L3379","statement_sha256":"f0dac6c35606221de4e8f5e54e80c90d66ac117d2f0a1e72357fee20f09df517","origin":"The Stacks Project","memory_eligible":false,"source_rank":14137,"rank":14137,"depth":49,"x":2020.871,"y":1491.142,"cluster":"algebraic-stacks"},{"id":"stacks:06U9","tag":"06U9","title":"Morphisms of finite type · Lemma 06U9","summary":"Let f : X → Y, g : Y → Z be morphisms of algebraic stacks. If g ∘ f : X → Z is locally of finite type, then f : X → Y is locally of finite type.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$,\n$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks.\nIf $g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$ is locally of finite type,\nthen $f : \\mathcal{X} \\to \\mathcal{Y}$ is locally of finite type.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06U9","source_file":"stacks-morphisms.tex","source_line":3394,"source_end_line":3400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3394-L3400","statement_sha256":"dfc1bc5f16456b8da12ff13ab69f2faf0e67ae10fa90fad80120aeab3005a79c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14138,"rank":14138,"depth":6,"x":2124.376,"y":1675.173,"cluster":"algebraic-stacks"},{"id":"stacks:06FX","tag":"06FX","title":"Points of finite type · Lemma 06FX","summary":"Let X be an algebraic stack. Let x ∈ |X|. The following are equivalent: • There exists a morphism Spec(k) → X which is locally of finite type and represents x. • There exists a scheme U, a closed point u ∈ U, and a smooth morphism φ : U → X such that φ(u) = x.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $x \\in |\\mathcal{X}|$. The following are equivalent:\n\\begin{enumerate}\n\\item There exists a morphism $\\Spec(k) \\to \\mathcal{X}$\nwhich is locally of finite type and represents $x$.\n\\item There exists a scheme $U$, a closed point $u \\in U$, and a smooth\nmorphism $\\varphi : U \\to \\mathcal{X}$ such that $\\varphi(u) = x$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FX","source_file":"stacks-morphisms.tex","source_line":3442,"source_end_line":3452,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3442-L3452","statement_sha256":"15464418ad802114836cd2de455885c61bd35a1aacafdd4126d3d40768f497fe","origin":"The Stacks Project","memory_eligible":false,"source_rank":14139,"rank":14139,"depth":48,"x":1899.815,"y":1598.118,"cluster":"algebraic-stacks"},{"id":"stacks:06FY","tag":"06FY","title":"Points of finite type · Definition 06FY","summary":"Let X be an algebraic stack. We say a point x ∈ |X| is a finite type point if the equivalent conditions of Lemma [Tag 06FX] are satisfied. We denote X_ft-pts the set of finite type points of X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. We say a point $x \\in |\\mathcal{X}|$\nis a {\\it finite type point}\\footnote{This is a\nslight abuse of language as it would perhaps be more correct to say\n``locally finite type point''.} if the equivalent conditions of\nLemma \\ref{lemma-point-finite-type}\nare satisfied. We denote $\\mathcal{X}_{\\text{ft-pts}}$\nthe set of finite type points of $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points of finite type","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FY","source_file":"stacks-morphisms.tex","source_line":3483,"source_end_line":3492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3483-L3492","statement_sha256":"581e0e5a3588bd28b3d53792d9ca70aadfc15144749f026520065181e7857868","origin":"The Stacks Project","memory_eligible":false,"source_rank":14140,"rank":14140,"depth":49,"x":2127.616,"y":1527.439,"cluster":"algebraic-stacks"},{"id":"stacks:06FZ","tag":"06FZ","title":"Points of finite type · Lemma 06FZ","summary":"Let X be an algebraic stack. We have X_ft-pts = ⋃_φ : U → X smooth |φ|(U_0) where U_0 is the set of closed points of U. Here we may let U range over all schemes smooth over X or over all affine schemes smooth over X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. We have\n$$\n\\mathcal{X}_{\\text{ft-pts}} =\n\\bigcup\\nolimits_{\\varphi : U \\to \\mathcal{X}\\text{ smooth}} |\\varphi|(U_0)\n$$\nwhere $U_0$ is the set of closed points of $U$.\nHere we may let $U$ range over all schemes smooth over $\\mathcal{X}$\nor over all affine schemes smooth over $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06FZ","source_file":"stacks-morphisms.tex","source_line":3497,"source_end_line":3507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3497-L3507","statement_sha256":"00d5d5f2f195db6d40e2470a710c2cd9c49e874143146976cccf071436fffa9d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14141,"rank":14141,"depth":49,"x":2016.356,"y":1709.014,"cluster":"algebraic-stacks"},{"id":"stacks:06G0","tag":"06G0","title":"Points of finite type · Lemma 06G0","summary":"Let f : X → Y be a morphism of algebraic stacks. If f is locally of finite type, then f(X_ft-pts) ⊂ Y_ft-pts.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $f$ is locally of finite type, then\n$f(\\mathcal{X}_{\\text{ft-pts}}) \\subset \\mathcal{Y}_{\\text{ft-pts}}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06G0","source_file":"stacks-morphisms.tex","source_line":3514,"source_end_line":3519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3514-L3519","statement_sha256":"4c28be3c0fd9ea36c1bbe2436f46888e3d902f8c343ad44e071e3f026e7e00da","origin":"The Stacks Project","memory_eligible":false,"source_rank":14142,"rank":14142,"depth":8,"x":1952.311,"y":1511.777,"cluster":"algebraic-stacks"},{"id":"stacks:06G1","tag":"06G1","title":"Points of finite type · Lemma 06G1","summary":"Let f : X → Y be a morphism of algebraic stacks. If f is locally of finite type and surjective, then f(X_ft-pts) = Y_ft-pts.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $f$ is locally of finite type and surjective, then\n$f(\\mathcal{X}_{\\text{ft-pts}}) = \\mathcal{Y}_{\\text{ft-pts}}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06G1","source_file":"stacks-morphisms.tex","source_line":3529,"source_end_line":3534,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3529-L3534","statement_sha256":"f89ca057878132203b8a10242b612867910a73001b31a02a8dac33314bf1178a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14143,"rank":14143,"depth":50,"x":2158.372,"y":1620.994,"cluster":"algebraic-stacks"},{"id":"stacks:06G2","tag":"06G2","title":"Points of finite type · Lemma 06G2","summary":"Let X be an algebraic stack. For any locally closed subset T ⊂ |X| we have T not = ∅ ⇒ T ∩ X_ft-pts not = ∅. In particular, for any closed subset T ⊂ |X| we see that T ∩ X_ft-pts is dense in T.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nFor any locally closed subset $T \\subset |\\mathcal{X}|$ we have\n$$\nT \\not = \\emptyset\n\\Rightarrow\nT \\cap \\mathcal{X}_{\\text{ft-pts}} \\not = \\emptyset.\n$$\nIn particular, for any closed subset $T \\subset |\\mathcal{X}|$ we\nsee that $T \\cap \\mathcal{X}_{\\text{ft-pts}}$ is dense in $T$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06G2","source_file":"stacks-morphisms.tex","source_line":3553,"source_end_line":3564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3553-L3564","statement_sha256":"eea5b676274263d4acb0482bf8c7c70f98b7cc8f832b99dd373facb941fe00be","origin":"The Stacks Project","memory_eligible":false,"source_rank":14144,"rank":14144,"depth":50,"x":1918.337,"y":1657.423,"cluster":"algebraic-stacks"},{"id":"stacks:06G3","tag":"06G3","title":"Points of finite type · Lemma 06G3","summary":"Let X be an algebraic stack. Let x ∈ |X|. The following are equivalent: • x is a finite type point, • there exists an algebraic stack Z whose underlying topological space |Z| is a singleton, and a morphism f : Z → X which is locally of finite type such that (x) = |f|(|Z|), and • the residual gerbe Z_x of X at x exists and the inclusion morphism Z_x → X is locally of finite type.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $x \\in |\\mathcal{X}|$. The following are equivalent:\n\\begin{enumerate}\n\\item $x$ is a finite type point,\n\\item there exists an algebraic stack $\\mathcal{Z}$\nwhose underlying topological space $|\\mathcal{Z}|$ is a singleton,\nand a morphism $f : \\mathcal{Z} \\to \\mathcal{X}$ which is\nlocally of finite type such that $\\{x\\} = |f|(|\\mathcal{Z}|)$, and\n\\item the residual gerbe $\\mathcal{Z}_x$ of $\\mathcal{X}$ at $x$ exists\nand the inclusion morphism $\\mathcal{Z}_x \\to \\mathcal{X}$ is locally of\nfinite type.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points of finite type","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06G3","source_file":"stacks-morphisms.tex","source_line":3589,"source_end_line":3603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3589-L3603","statement_sha256":"40d67ae43b9ec07b4873a6ef2ba57f1b112043026d9936e73572f696e1fb8793","origin":"The Stacks Project","memory_eligible":false,"source_rank":14145,"rank":14145,"depth":79,"x":2066.2,"y":1494.183,"cluster":"algebraic-stacks"},{"id":"stacks:0DTS","tag":"0DTS","title":"Automorphism groups · Lemma 0DTS","summary":"In the situation above G_x is a scheme if one of the following holds • Δ : X → X × X is quasi-separated • Δ : X → X × X is locally separated, • X is quasi-DM, • I_X → X is quasi-separated, • I_X → X is locally separated, or • I_X → X is locally quasi-finite.","statement_latex":"In the situation above $G_x$ is a scheme if one of the following\nholds\n\\begin{enumerate}\n\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nis quasi-separated\n\\item $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nis locally separated,\n\\item $\\mathcal{X}$ is quasi-DM,\n\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$\nis quasi-separated,\n\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$\nis locally separated, or\n\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$\nis locally quasi-finite.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Automorphism groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTS","source_file":"stacks-morphisms.tex","source_line":3724,"source_end_line":3741,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3724-L3741","statement_sha256":"c88641fdbe9e27c899971a412da567854e96c27d3b6b2d37117ce3d1dae1eacc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14146,"rank":14146,"depth":73,"x":2088.465,"y":1698.674,"cluster":"algebraic-stacks"},{"id":"stacks:0DTT","tag":"0DTT","title":"Automorphism groups · Lemma 0DTT","summary":"Let X be an algebraic stack. Let x ∈ |X| be a point. Let P be a property of algebraic spaces over fields which is invariant under ground field extensions; for example P(X/k) = X → Spec(k) is finite. The following are equivalent • for some morphism x : Spec(k) → X in the class of x the automorphism group algebraic space G_x/k has P, and • for any morphism x : Spec(k) → X in the class of x the automorphism group algebraic space G_x/k has P.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$ be a point.\nLet $P$ be a property of algebraic spaces over fields which is invariant\nunder ground field extensions; for example\n$P(X/k) = X \\to \\Spec(k)\\text{ is finite}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item for some morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the\nclass of $x$ the automorphism group algebraic space $G_x/k$\nhas $P$, and\n\\item for any morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the\nclass of $x$ the automorphism group algebraic space $G_x/k$\nhas $P$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Automorphism groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTT","source_file":"stacks-morphisms.tex","source_line":3764,"source_end_line":3779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3764-L3779","statement_sha256":"9126879ddb35bee54f67036ebb728b761319f3b1d095505fe80bc97468af59e6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14147,"rank":14147,"depth":0,"x":1907.407,"y":1560.371,"cluster":"algebraic-stacks"},{"id":"stacks:0DTV","tag":"0DTV","title":"Automorphism groups · Lemma 0DTV","summary":"Let f : X → Y be a morphism of algebraic stacks. Let x ∈ |X| be a point. The following are equivalent • for some morphism x : Spec(k) → X in the class of x setting y = f ∘ x the map G_x → G_y of automorphism group algebraic spaces is an isomorphism, and • for any morphism x : Spec(k) → X in the class of x setting y = f ∘ x the map G_x → G_y of automorphism group algebraic spaces is an isomorphism.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $x \\in |\\mathcal{X}|$ be a point. The following are equivalent\n\\begin{enumerate}\n\\item for some morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the\nclass of $x$ setting $y = f \\circ x$ the map\n$G_x \\to G_y$ of automorphism group algebraic spaces\nis an isomorphism, and\n\\item for any morphism $x : \\Spec(k) \\to \\mathcal{X}$ in the\nclass of $x$ setting $y = f \\circ x$ the map\n$G_x \\to G_y$ of automorphism group algebraic spaces\nis an isomorphism.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Automorphism groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTV","source_file":"stacks-morphisms.tex","source_line":3800,"source_end_line":3814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3800-L3814","statement_sha256":"64a177aa4143a11f4695fabcba4ab5822cdbb7f4a076cfac467294730c8aa508","origin":"The Stacks Project","memory_eligible":false,"source_rank":14148,"rank":14148,"depth":56,"x":2152.398,"y":1559.616,"cluster":"algebraic-stacks"},{"id":"stacks:0DTY","tag":"0DTY","title":"Presentations and properties of algebraic stacks · Lemma 0DTY","summary":"Let (U, R, s, t, c) be a groupoid in algebraic spaces such that s, t : R → U are flat and locally of finite presentation. Consider the algebraic stack X = [U/R] (see above). • If R → U × U is separated, then Δ_X is separated. • If U, R are separated, then Δ_X is separated. • If R → U × U is locally quasi-finite, then X is quasi-DM. • If s, t : R → U are locally quasi-finite, then X is quasi-DM. • If R → U × U is proper, then X is separated. • If s, t : R → U are proper…","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces such that\n$s, t : R \\to U$ are flat and locally of finite presentation.\nConsider the algebraic stack $\\mathcal{X} = [U/R]$ (see above).\n\\begin{enumerate}\n\\item If $R \\to U \\times U$ is separated, then\n$\\Delta_\\mathcal{X}$ is separated.\n\\item If $U$, $R$ are separated, then $\\Delta_\\mathcal{X}$ is separated.\n\\item If $R \\to U \\times U$ is locally quasi-finite, then $\\mathcal{X}$\nis quasi-DM.\n\\item If $s, t : R \\to U$ are locally quasi-finite, then\n$\\mathcal{X}$ is quasi-DM.\n\\item If $R \\to U \\times U$ is proper, then $\\mathcal{X}$ is separated.\n\\item If $s, t : R \\to U$ are proper and $U$ is separated, then\n$\\mathcal{X}$ is separated.\n\\item Add more here.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Presentations and properties of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTY","source_file":"stacks-morphisms.tex","source_line":3860,"source_end_line":3878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3860-L3878","statement_sha256":"486bed0287ecce6c0b19780b45781860e557f114bfd8cebab022f8da0dfd6b29","origin":"The Stacks Project","memory_eligible":false,"source_rank":14149,"rank":14149,"depth":71,"x":1972.159,"y":1699.336,"cluster":"algebraic-stacks"},{"id":"stacks:0DTZ","tag":"0DTZ","title":"Presentations and properties of algebraic stacks · Lemma 0DTZ","summary":"Let (U, R, s, t, c) be a groupoid in algebraic spaces such that s, t : R → U are flat and locally of finite presentation. Consider the algebraic stack X = [U/R] (see above). Then the image of |R| → |U| × |U| is an equivalence relation and |X| is the quotient of |U| by this equivalence relation.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces such that\n$s, t : R \\to U$ are flat and locally of finite presentation.\nConsider the algebraic stack $\\mathcal{X} = [U/R]$ (see above).\nThen the image of $|R| \\to |U| \\times |U|$ is an equivalence relation\nand $|\\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Presentations and properties of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DTZ","source_file":"stacks-morphisms.tex","source_line":3912,"source_end_line":3919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3912-L3919","statement_sha256":"cd6ed056ce3068f0fa1a94e3c042837eb6755934908124e7809e0a3a118bf59f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14150,"rank":14150,"depth":71,"x":1992.728,"y":1493.818,"cluster":"algebraic-stacks"},{"id":"stacks:06MD","tag":"06MD","title":"Special presentations of algebraic stacks · Lemma 06MD","summary":"Let X be an algebraic stack. Consider a cartesian diagram xymatrix U ar[d] & F ar[l]^p ar[d] X & Spec(k) ar[l] where U is an algebraic space, k is a field, and U → X is flat and locally of finite presentation. Let f_1, …, f_r ∈ Γ(U, O_U) and z ∈ |F| such that f_1, …, f_r map to a regular sequence in the local ring O_F, overlinez. Then, after replacing U by an open subspace containing p(z), the morphism V(f_1, …, f_r) → X is flat and locally of finite presentation.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nConsider a cartesian diagram\n$$\n\\xymatrix{\nU \\ar[d] & F \\ar[l]^p \\ar[d] \\\\\n\\mathcal{X} & \\Spec(k) \\ar[l]\n}\n$$\nwhere $U$ is an algebraic space, $k$ is a field, and $U \\to \\mathcal{X}$\nis flat and locally of finite presentation. Let\n$f_1, \\ldots, f_r \\in \\Gamma(U, \\mathcal{O}_U)$\nand $z \\in |F|$ such that $f_1, \\ldots, f_r$ map to a regular sequence\nin the local ring $\\mathcal{O}_{F, \\overline{z}}$.\nThen, after replacing $U$ by an open subspace containing $p(z)$, the morphism\n$$\nV(f_1, \\ldots, f_r) \\longrightarrow \\mathcal{X}\n$$\nis flat and locally of finite presentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Special presentations of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MD","source_file":"stacks-morphisms.tex","source_line":3963,"source_end_line":3983,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L3963-L3983","statement_sha256":"dc055ac1b17a2a4d034a7ccc0aaeedddfaff9211471a1fec1363e83f3614a46d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14151,"rank":14151,"depth":57,"x":2142.993,"y":1657.209,"cluster":"algebraic-stacks"},{"id":"stacks:06ME","tag":"06ME","title":"Special presentations of algebraic stacks · Lemma 06ME","summary":"Let X be an algebraic stack. Consider a cartesian diagram xymatrix U ar[d] & F ar[l]^p ar[d] X & Spec(k) ar[l] where U is an algebraic space, k is a field, and U → X is locally of finite type. Let z ∈ |F| be such that dim_z(F) = 0. Then, after replacing U by an open subspace containing p(z), the morphism U → X is locally quasi-finite.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram\n$$\n\\xymatrix{\nU \\ar[d] & F \\ar[l]^p \\ar[d] \\\\\n\\mathcal{X} & \\Spec(k) \\ar[l]\n}\n$$\nwhere $U$ is an algebraic space, $k$ is a field, and $U \\to \\mathcal{X}$\nis locally of finite type. Let $z \\in |F|$ be such that $\\dim_z(F) = 0$.\nThen, after replacing $U$ by an open subspace containing $p(z)$, the morphism\n$$\nU \\longrightarrow \\mathcal{X}\n$$\nis locally quasi-finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Special presentations of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06ME","source_file":"stacks-morphisms.tex","source_line":4034,"source_end_line":4050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4034-L4050","statement_sha256":"47a5539cb7285492128cf773db46374a0cab90e94e0dc7e47ca554ec14cf5b29","origin":"The Stacks Project","memory_eligible":false,"source_rank":14152,"rank":14152,"depth":29,"x":1900.538,"y":1621.953,"cluster":"algebraic-stacks"},{"id":"stacks:06MF","tag":"06MF","title":"Special presentations of algebraic stacks · Theorem 06MF","summary":"Let X be an algebraic stack. The following are equivalent • X is quasi-DM, and • there exists a scheme W and a surjective, flat, locally finitely presented, locally quasi-finite morphism W → X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is quasi-DM, and\n\\item there exists a scheme $W$ and a surjective, flat, locally finitely\npresented, locally quasi-finite morphism $W \\to \\mathcal{X}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Special presentations of algebraic stacks","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06MF","source_file":"stacks-morphisms.tex","source_line":4072,"source_end_line":4080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4072-L4080","statement_sha256":"1bcb1583ba86e41577c7a24ec55b54842279a765f4e91a0f12d315715b469799","origin":"The Stacks Project","memory_eligible":false,"source_rank":14153,"rank":14153,"depth":80,"x":2107.892,"y":1510.257,"cluster":"algebraic-stacks"},{"id":"stacks:06N0","tag":"06N0","title":"Special presentations of algebraic stacks · Lemma 06N0","summary":"Let Z be a DM, locally Noetherian, reduced algebraic stack with |Z| a singleton. Then there exists a field k and a surjective étale morphism Spec(k) → Z.","statement_latex":"Let $\\mathcal{Z}$ be a DM, locally Noetherian, reduced algebraic stack\nwith $|\\mathcal{Z}|$ a singleton. Then there exists a field $k$ and\na surjective \\'etale morphism $\\Spec(k) \\to \\mathcal{Z}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Special presentations of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06N0","source_file":"stacks-morphisms.tex","source_line":4192,"source_end_line":4197,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4192-L4197","statement_sha256":"c8f861e87c52e1d6b30183b15d8a1ec429d67a6bbcec72426feb5755c4d6c831","origin":"The Stacks Project","memory_eligible":false,"source_rank":14154,"rank":14154,"depth":69,"x":2044.748,"y":1710.489,"cluster":"algebraic-stacks"},{"id":"stacks:06N2","tag":"06N2","title":"Special presentations of algebraic stacks · Lemma 06N2","summary":"Let X be an algebraic stack. Consider a cartesian diagram xymatrix U ar[d] & F ar[l]^p ar[d] X & Spec(k) ar[l] where U is an algebraic space, k is a field, and U → X is flat and locally of finite presentation. Let z ∈ |F| be such that F → Spec(k) is unramified at z. Then, after replacing U by an open subspace containing p(z), the morphism U → X is étale.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram\n$$\n\\xymatrix{\nU \\ar[d] & F \\ar[l]^p \\ar[d] \\\\\n\\mathcal{X} & \\Spec(k) \\ar[l]\n}\n$$\nwhere $U$ is an algebraic space, $k$ is a field, and $U \\to \\mathcal{X}$\nis flat and locally of finite presentation. Let $z \\in |F|$ be such that\n$F \\to \\Spec(k)$ is unramified at $z$. Then, after replacing $U$ by\nan open subspace containing $p(z)$, the morphism\n$$\nU \\longrightarrow \\mathcal{X}\n$$\nis \\'etale.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Special presentations of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06N2","source_file":"stacks-morphisms.tex","source_line":4334,"source_end_line":4351,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4334-L4351","statement_sha256":"870ab1c20c436a91ac4b3de6c4e829da0ce13e84a0505091c5841e4fa893a308","origin":"The Stacks Project","memory_eligible":false,"source_rank":14155,"rank":14155,"depth":47,"x":1930.168,"y":1526.818,"cluster":"algebraic-stacks"},{"id":"stacks:06N3","tag":"06N3","title":"Special presentations of algebraic stacks · Theorem 06N3","summary":"Let X be an algebraic stack. The following are equivalent • X is DM, • X is Deligne-Mumford, and • there exists a scheme W and a surjective étale morphism W → X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is DM,\n\\item $\\mathcal{X}$ is Deligne-Mumford, and\n\\item there exists a scheme $W$ and a surjective \\'etale\nmorphism $W \\to \\mathcal{X}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Special presentations of algebraic stacks","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06N3","source_file":"stacks-morphisms.tex","source_line":4375,"source_end_line":4384,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4375-L4384","statement_sha256":"9df2c4571f3d0fe8d7ab5b6e815350909eb6e2e1663d4b6563f1002416062838","origin":"The Stacks Project","memory_eligible":false,"source_rank":14156,"rank":14156,"depth":80,"x":2162.605,"y":1597.313,"cluster":"algebraic-stacks"},{"id":"stacks:0CIA","tag":"0CIA","title":"Special presentations of algebraic stacks · Lemma 0CIA","summary":"Let f : X → Y be a DM morphism of algebraic stacks. Then • For every DM algebraic stack Z and morphism Z → Y there exists a scheme and a surjective étale morphism U → X ×_Y Z. • For every algebraic space Z and morphism Z → Y there exists a scheme and a surjective étale morphism U → X ×_Y Z.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a DM morphism of algebraic\nstacks. Then\n\\begin{enumerate}\n\\item For every DM algebraic stack $\\mathcal{Z}$ and morphism\n$\\mathcal{Z} \\to \\mathcal{Y}$ there exists a scheme and\na surjective \\'etale morphism\n$U \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z}$.\n\\item For every algebraic space $Z$ and morphism\n$Z \\to \\mathcal{Y}$ there exists a scheme and\na surjective \\'etale morphism\n$U \\to \\mathcal{X} \\times_\\mathcal{Y} Z$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Special presentations of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIA","source_file":"stacks-morphisms.tex","source_line":4532,"source_end_line":4546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4532-L4546","statement_sha256":"f926f3e5cd12fb3733695f4fdfaadd53e8be14477d31420d21101fd22b6c9a9a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14157,"rank":14157,"depth":81,"x":1934.28,"y":1677.306,"cluster":"algebraic-stacks"},{"id":"stacks:0DSM","tag":"0DSM","title":"The Deligne-Mumford locus · Lemma 0DSM","summary":"Let X be an algebraic stack. There exist open substacks X\" ⊂ X' ⊂ X such that X\" is DM, X' is quasi-DM, and such that these are the largest open substacks with these properties.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. There exist open substacks\n$$\n\\mathcal{X}'' \\subset \\mathcal{X}' \\subset \\mathcal{X}\n$$\nsuch that $\\mathcal{X}''$ is DM, $\\mathcal{X}'$ is quasi-DM, and\nsuch that these are the largest open substacks with these properties.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"The Deligne-Mumford locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSM","source_file":"stacks-morphisms.tex","source_line":4578,"source_end_line":4586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4578-L4586","statement_sha256":"81fa752386b3884ffe34e89726619d624ce74266db7e12e835a621482cdd97ae","origin":"The Stacks Project","memory_eligible":false,"source_rank":14158,"rank":14158,"depth":81,"x":2038.424,"y":1488.565,"cluster":"algebraic-stacks"},{"id":"stacks:0DSN","tag":"0DSN","title":"The Deligne-Mumford locus · Lemma 0DSN","summary":"Let X be an algebraic stack. Let x ∈ |X| correspond to x : Spec(k) → X. Let G_x/k be the automorphism group algebraic space of x. Then • x is in the DM locus of X if and only if G_x → Spec(k) is unramified, and • x is in the quasi-DM locus of X if and only if G_x → Spec(k) is locally quasi-finite.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x \\in |\\mathcal{X}|$\ncorrespond to $x : \\Spec(k) \\to \\mathcal{X}$. Let $G_x/k$\nbe the automorphism group algebraic space of $x$. Then\n\\begin{enumerate}\n\\item  $x$ is in the DM locus of $\\mathcal{X}$\nif and only if $G_x \\to \\Spec(k)$ is unramified, and\n\\item $x$ is in the quasi-DM locus of $\\mathcal{X}$\nif and only if $G_x \\to \\Spec(k)$ is locally quasi-finite.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"The Deligne-Mumford locus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSN","source_file":"stacks-morphisms.tex","source_line":4603,"source_end_line":4614,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4603-L4614","statement_sha256":"8460b26f9ca5d6cf16838ab78aa8a38ca4b9ff7e2c4ec71c3cd60048ec5a335b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14159,"rank":14159,"depth":74,"x":2113.488,"y":1687.041,"cluster":"algebraic-stacks"},{"id":"stacks:06UA","tag":"06UA","title":"Locally quasi-finite morphisms · Lemma 06UA","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume f is representable by algebraic spaces. The following are equivalent • f is locally quasi-finite (as in Properties of Stacks, Section [Tag 04XB]), and • f is locally of finite type and for every morphism Spec(k) → Y where k is a field the space |Spec(k) ×_Y X| is discrete.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nAssume $f$ is representable by algebraic spaces.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite (as in Properties of Stacks,\nSection \\ref{stacks-properties-section-properties-morphisms}), and\n\\item $f$ is locally of finite type and for every morphism\n$\\Spec(k) \\to \\mathcal{Y}$ where $k$ is a field the\nspace $|\\Spec(k) \\times_\\mathcal{Y} \\mathcal{X}|$ is discrete.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UA","source_file":"stacks-morphisms.tex","source_line":4700,"source_end_line":4712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4700-L4712","statement_sha256":"50e122f89f50649cf6b9866bc97220ee9cb79aeb60b1bed9b1a282f3a54f81c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14160,"rank":14160,"depth":51,"x":1898.304,"y":1583.173,"cluster":"algebraic-stacks"},{"id":"stacks:06PU","tag":"06PU","title":"Locally quasi-finite morphisms · Definition 06PU","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is locally quasi-finite if f is quasi-DM, locally of finite type, and for every morphism Spec(k) → Y where k is a field the space |X_k| is discrete.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is {\\it locally quasi-finite} if $f$ is quasi-DM, locally of\nfinite type, and for every morphism $\\Spec(k) \\to \\mathcal{Y}$\nwhere $k$ is a field the space $|\\mathcal{X}_k|$ is discrete.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Locally quasi-finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PU","source_file":"stacks-morphisms.tex","source_line":4745,"source_end_line":4751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4745-L4751","statement_sha256":"dba627ac1c4ff577687ddf6d2fa98906d2134914b518a6b7850b9a3bd37c7a62","origin":"The Stacks Project","memory_eligible":false,"source_rank":14161,"rank":14161,"depth":0,"x":2140.758,"y":1537.616,"cluster":"algebraic-stacks"},{"id":"stacks:06UB","tag":"06UB","title":"Locally quasi-finite morphisms · Lemma 06UB","summary":"A base change of a locally quasi-finite morphism is locally quasi-finite.","statement_latex":"A base change of a locally quasi-finite morphism is locally quasi-finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UB","source_file":"stacks-morphisms.tex","source_line":4775,"source_end_line":4778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4775-L4778","statement_sha256":"e2e078cdbd0e9f252ce7b001d2d58bf2194b4c7e3793e7e7cdb42bfc5dbc3297","origin":"The Stacks Project","memory_eligible":false,"source_rank":14162,"rank":14162,"depth":8,"x":1998.464,"y":1708.961,"cluster":"algebraic-stacks"},{"id":"stacks:06UC","tag":"06UC","title":"Locally quasi-finite morphisms · Lemma 06UC","summary":"Let X → Spec(k) be a locally quasi-finite morphism where X is an algebraic stack and k is a field. Let f : V → X be a locally quasi-finite morphism where V is a scheme. Then V → Spec(k) is locally quasi-finite.","statement_latex":"Let $\\mathcal{X} \\to \\Spec(k)$ be a locally quasi-finite morphism\nwhere $\\mathcal{X}$ is an algebraic stack and $k$ is a field.\nLet $f : V \\to \\mathcal{X}$ be a locally quasi-finite morphism where\n$V$ is a scheme. Then $V \\to \\Spec(k)$ is locally quasi-finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UC","source_file":"stacks-morphisms.tex","source_line":4788,"source_end_line":4794,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4788-L4794","statement_sha256":"13affbde6ea887f360ed79fbadc120acedb869f737f74e0e63cfdca7c187425c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14163,"rank":14163,"depth":28,"x":1965.564,"y":1501.658,"cluster":"algebraic-stacks"},{"id":"stacks:06UD","tag":"06UD","title":"Locally quasi-finite morphisms · Lemma 06UD","summary":"A composition of a locally quasi-finite morphisms is locally quasi-finite.","statement_latex":"A composition of a locally quasi-finite morphisms is locally quasi-finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UD","source_file":"stacks-morphisms.tex","source_line":4823,"source_end_line":4826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4823-L4826","statement_sha256":"240a496234584d90dce2e57f4b2f74f7a26c49489c59e2159342681640332616","origin":"The Stacks Project","memory_eligible":false,"source_rank":14164,"rank":14164,"depth":81,"x":2156.729,"y":1635.991,"cluster":"algebraic-stacks"},{"id":"stacks:06UE","tag":"06UE","title":"Locally quasi-finite morphisms · Lemma 06UE","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • f is quasi-DM, • for any morphism V → Y with V an algebraic space there exists a surjective, flat, locally finitely presented, locally quasi-finite morphism U → X ×_Y V where U is an algebraic space, and • there exist algebraic spaces U, V and a morphism V → Y which is surjective, flat, and locally of finite presentation, and a morphism U → X ×_Y V which is surjective, flat, locally of finite…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is quasi-DM,\n\\item for any morphism $V \\to \\mathcal{Y}$ with $V$ an algebraic space\nthere exists a surjective, flat, locally finitely presented, locally\nquasi-finite morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ where\n$U$ is an algebraic space, and\n\\item there exist algebraic spaces $U$, $V$ and a morphism\n$V \\to \\mathcal{Y}$ which is surjective, flat, and\nlocally of finite presentation, and a morphism\n$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ which is surjective, flat,\nlocally of finite presentation, and locally quasi-finite.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UE","source_file":"stacks-morphisms.tex","source_line":4872,"source_end_line":4888,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4872-L4888","statement_sha256":"ddcb238a1545018c94d0dc3b7c59c51764db92a8515eb6872c282c684196210f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14165,"rank":14165,"depth":81,"x":1907.483,"y":1645.417,"cluster":"algebraic-stacks"},{"id":"stacks:06UF","tag":"06UF","title":"Locally quasi-finite morphisms · Lemma 06UF","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • f is locally quasi-finite, • f is quasi-DM and for any morphism V → Y with V an algebraic space and any locally quasi-finite morphism U → X ×_Y V where U is an algebraic space the morphism U → V is locally quasi-finite, • for any morphism V → Y from an algebraic space V there exists a surjective, flat, locally finitely presented, and locally quasi-finite morphism U → X ×_Y V where U is an…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is locally quasi-finite,\n\\item $f$ is quasi-DM and for any morphism $V \\to \\mathcal{Y}$ with $V$\nan algebraic space and any locally quasi-finite morphism\n$U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ where $U$ is an algebraic space\nthe morphism $U \\to V$ is locally quasi-finite,\n\\item for any morphism $V \\to \\mathcal{Y}$ from an algebraic space $V$\nthere exists a surjective, flat, locally finitely presented, and locally\nquasi-finite morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ where\n$U$ is an algebraic space such that $U \\to V$ is locally quasi-finite,\n\\item there exists algebraic spaces $U$, $V$, a surjective, flat,\nand locally of finite presentation morphism $V \\to \\mathcal{Y}$,\nand a morphism $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ which\nis surjective, flat, locally of finite presentation, and\nlocally quasi-finite such that $U \\to V$ is locally quasi-finite.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UF","source_file":"stacks-morphisms.tex","source_line":4921,"source_end_line":4941,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L4921-L4941","statement_sha256":"6fff31401282eeed8811f6f2e2f2c8fefdac1a2eed077f681b8a5dfa34c76b8a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14166,"rank":14166,"depth":82,"x":2083.875,"y":1496.885,"cluster":"algebraic-stacks"},{"id":"stacks:06UG","tag":"06UG","title":"Locally quasi-finite morphisms · Lemma 06UG","summary":"Let X → Y → Z be morphisms of algebraic stacks. Assume that X → Z is locally quasi-finite and Y → Z is quasi-DM. Then X → Y is locally quasi-finite.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms\nof algebraic stacks. Assume that $\\mathcal{X} \\to \\mathcal{Z}$\nis locally quasi-finite and $\\mathcal{Y} \\to \\mathcal{Z}$ is quasi-DM.\nThen $\\mathcal{X} \\to \\mathcal{Y}$ is locally quasi-finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Locally quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UG","source_file":"stacks-morphisms.tex","source_line":5004,"source_end_line":5010,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5004-L5010","statement_sha256":"70095e636a8132160b8634f78b69e143f7aa32d26bf2ac7516e7f0cb67dd0f70","origin":"The Stacks Project","memory_eligible":false,"source_rank":14167,"rank":14167,"depth":82,"x":2073.24,"y":1706.715,"cluster":"algebraic-stacks"},{"id":"stacks:0G2M","tag":"0G2M","title":"Quasi-finite morphisms · Definition 0G2M","summary":"[rydh_approx] Let f : X → Y be a morphism of algebraic stacks. We say f is quasi-finite if f is locally quasi-finite (Definition [Tag 06PU]) and quasi-compact (Definition [Tag 050U]).","statement_latex":"\\begin{reference}\n\\cite{rydh_approx}\n\\end{reference}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is {\\it quasi-finite} if $f$ is locally quasi-finite\n(Definition \\ref{definition-locally-quasi-finite})\nand quasi-compact (Definition \\ref{definition-quasi-compact}).","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-finite morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2M","source_file":"stacks-morphisms.tex","source_line":5061,"source_end_line":5070,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5061-L5070","statement_sha256":"a7f5cb2b7957d91607dca3da5dc058438e5f879157aac1863cb37386e65c639d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14168,"rank":14168,"depth":1,"x":1912.178,"y":1545.79,"cluster":"algebraic-stacks"},{"id":"stacks:0G2N","tag":"0G2N","title":"Quasi-finite morphisms · Lemma 0G2N","summary":"The composition of quasi-finite morphisms is quasi-finite.","statement_latex":"The composition of quasi-finite morphisms is quasi-finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2N","source_file":"stacks-morphisms.tex","source_line":5072,"source_end_line":5075,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5072-L5075","statement_sha256":"59c761aa65b73cb755dd4506a16045af9b39595224e1aa176e42d94424f7282f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14169,"rank":14169,"depth":82,"x":2160.608,"y":1573.091,"cluster":"algebraic-stacks"},{"id":"stacks:0G2P","tag":"0G2P","title":"Quasi-finite morphisms · Lemma 0G2P","summary":"A base change of a quasi-finite morphism is quasi-finite.","statement_latex":"A base change of a quasi-finite morphism is quasi-finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2P","source_file":"stacks-morphisms.tex","source_line":5083,"source_end_line":5086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5083-L5086","statement_sha256":"cbe06abeaccf277516806dd80fe8e9f579891b50215653c5b5e06d10c24952c4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14170,"rank":14170,"depth":9,"x":1955.255,"y":1694.048,"cluster":"algebraic-stacks"},{"id":"stacks:0G2Q","tag":"0G2Q","title":"Quasi-finite morphisms · Lemma 0G2Q","summary":"Let f : X → Y and g : Y → Z be morphisms of algebraic stacks. If g ∘ f is quasi-finite and g is quasi-separated and quasi-DM then f is quasi-finite.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and\n$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks.\nIf $g \\circ f$ is quasi-finite and $g$ is quasi-separated and quasi-DM\nthen $f$ is quasi-finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Quasi-finite morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G2Q","source_file":"stacks-morphisms.tex","source_line":5094,"source_end_line":5100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5094-L5100","statement_sha256":"37f59baa4c0fcd47c31cad2557e4067ea6a82d1a737c9d46b5c798e2807ed31c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14171,"rank":14171,"depth":83,"x":2009.464,"y":1488.124,"cluster":"algebraic-stacks"},{"id":"stacks:06PW","tag":"06PW","title":"Flat morphisms · Definition 06PW","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is flat if the equivalent conditions of Lemma [Tag 06FM] hold with P = flat.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is {\\it flat} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with $\\mathcal{P} = \\text{flat}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Flat morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PW","source_file":"stacks-morphisms.tex","source_line":5141,"source_end_line":5147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5141-L5147","statement_sha256":"c33266bbb746a9362f9765f88e6791207d6577b7d75792267d7c71d5a9641842","origin":"The Stacks Project","memory_eligible":false,"source_rank":14172,"rank":14172,"depth":4,"x":2135.217,"y":1670.915,"cluster":"algebraic-stacks"},{"id":"stacks:06PX","tag":"06PX","title":"Flat morphisms · Lemma 06PX","summary":"The composition of flat morphisms is flat.","statement_latex":"The composition of flat morphisms is flat.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PX","source_file":"stacks-morphisms.tex","source_line":5149,"source_end_line":5152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5149-L5152","statement_sha256":"1cef9b5636167763759bac6e58fd08ed733a3d6d6e3faac26fee58c85f9d5eef","origin":"The Stacks Project","memory_eligible":false,"source_rank":14173,"rank":14173,"depth":6,"x":1895.25,"y":1607.419,"cluster":"algebraic-stacks"},{"id":"stacks:06PY","tag":"06PY","title":"Flat morphisms · Lemma 06PY","summary":"A base change of a flat morphism is flat.","statement_latex":"A base change of a flat morphism is flat.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PY","source_file":"stacks-morphisms.tex","source_line":5162,"source_end_line":5165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5162-L5165","statement_sha256":"7bacb2984b5c2494eadec6a87d71bf6de5b19afe4dd875a1160c0aebe20339af","origin":"The Stacks Project","memory_eligible":false,"source_rank":14174,"rank":14174,"depth":4,"x":2123.491,"y":1517.986,"cluster":"algebraic-stacks"},{"id":"stacks:06PZ","tag":"06PZ","title":"Flat morphisms · Lemma 06PZ","summary":"Let f : X → Y be a morphism of algebraic stacks. Let Z → Y be a surjective flat morphism of algebraic stacks. If the base change Z ×_Y X → Z is flat, then f is flat.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective flat morphism of algebraic\nstacks. If the base change\n$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$\nis flat, then $f$ is flat.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06PZ","source_file":"stacks-morphisms.tex","source_line":5175,"source_end_line":5182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5175-L5182","statement_sha256":"fcc973a8cffcd7ebba8c41910e652ff259e0d0348edabf5801ce51a871395301","origin":"The Stacks Project","memory_eligible":false,"source_rank":14175,"rank":14175,"depth":47,"x":2027.011,"y":1713.639,"cluster":"algebraic-stacks"},{"id":"stacks:06Q0","tag":"06Q0","title":"Flat morphisms · Lemma 06Q0","summary":"Let X → Y → Z be morphisms of algebraic stacks. If X → Z is flat and X → Y is surjective and flat, then Y → Z is flat.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of\nalgebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is flat\nand $\\mathcal{X} \\to \\mathcal{Y}$ is surjective and flat, then\n$\\mathcal{Y} \\to \\mathcal{Z}$ is flat.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Q0","source_file":"stacks-morphisms.tex","source_line":5219,"source_end_line":5225,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5219-L5225","statement_sha256":"8a00b8cf99a67ff00ec17b6aa094c9d8ed69e7543ae0f9d27359cddf649dba46","origin":"The Stacks Project","memory_eligible":false,"source_rank":14176,"rank":14176,"depth":56,"x":1940.728,"y":1514.422,"cluster":"algebraic-stacks"},{"id":"stacks:0DN5","tag":"0DN5","title":"Flat morphisms · Lemma 0DN5","summary":"Let f : X → Y be a flat morphism of algebraic stacks. Let Spec(A) → Y be a morphism where A is a valuation ring. If the closed point of Spec(A) maps to a point of |Y| in the image of |X| → |Y|, then there exists a commutative diagram xymatrix Spec(A') ar[r] ar[d] & X ar[d] Spec(A) ar[r] & Y where A → A' is an extension of valuation rings (More on Algebra, Definition [Tag 0ASG]).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a flat morphism\nof algebraic stacks. Let $\\Spec(A) \\to \\mathcal{Y}$ be a morphism\nwhere $A$ is a valuation ring. If the closed point of $\\Spec(A)$ maps to a\npoint of $|\\mathcal{Y}|$ in the image of $|\\mathcal{X|} \\to |\\mathcal{Y}|$,\nthen there exists a commutative diagram\n$$\n\\xymatrix{\n\\Spec(A') \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\\n\\Spec(A) \\ar[r] & \\mathcal{Y}\n}\n$$\nwhere $A \\to A'$ is an extension of valuation rings\n(More on Algebra, Definition\n\\ref{more-algebra-definition-extension-valuation-rings}).","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Flat morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DN5","source_file":"stacks-morphisms.tex","source_line":5241,"source_end_line":5257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5241-L5257","statement_sha256":"8c830811184e0331b5dc075443103ce72de4b43968c8254b1626d37e48c1e651","origin":"The Stacks Project","memory_eligible":false,"source_rank":14177,"rank":14177,"depth":8,"x":2164.782,"y":1612.461,"cluster":"algebraic-stacks"},{"id":"stacks:0CIC","tag":"0CIC","title":"Flat at a point · Lemma 0CIC","summary":"Let f : X → Y be a morphism of algebraic stacks. Let x ∈ |X|. Consider commutative diagrams vcenter xymatrix U ar[d]_a ar[r]_h & V ar[d]^b X ar[r]^f & Y with points vcenter xymatrix u ∈ |U| ar[d] x ∈ |X| where U and V are algebraic spaces, b is flat, and (a, h) : U → X ×_Y V is flat. The following are equivalent • h is flat at u for one diagram as above, • h is flat at u for every diagram as above.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $x \\in |\\mathcal{X}|$. Consider commutative diagrams\n$$\n\\vcenter{\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\n\\mathcal{X} \\ar[r]^f & \\mathcal{Y}\n}\n}\n\\quad\\text{with points}\n\\vcenter{\n\\xymatrix{\nu \\in |U| \\ar[d] \\\\\nx \\in |\\mathcal{X}|\n}\n}\n$$\nwhere $U$ and $V$ are algebraic spaces, $b$ is flat, and\n$(a, h) : U \\to \\mathcal{X} \\times_\\mathcal{Y} V$\nis flat. The following are equivalent\n\\begin{enumerate}\n\\item $h$ is flat at $u$ for one diagram as above,\n\\item $h$ is flat at $u$ for every diagram as above.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Flat at a point","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIC","source_file":"stacks-morphisms.tex","source_line":5285,"source_end_line":5311,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5285-L5311","statement_sha256":"58ac1a6b3c4df079db881b70297da84b7023a534dbd70d881b487eb79960a275","origin":"The Stacks Project","memory_eligible":false,"source_rank":14178,"rank":14178,"depth":56,"x":1920.483,"y":1667.357,"cluster":"algebraic-stacks"},{"id":"stacks:0CID","tag":"0CID","title":"Flat at a point · Definition 0CID","summary":"Let f : X → Y be a morphism of algebraic stacks. Let x ∈ |X|. We say f is flat at x if the equivalent conditions of Lemma [Tag 0CIC] hold.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $x \\in |\\mathcal{X}|$. We say $f$ is {\\it flat at $x$} if the\nequivalent conditions of Lemma \\ref{lemma-flat-at-point} hold.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Flat at a point","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CID","source_file":"stacks-morphisms.tex","source_line":5348,"source_end_line":5353,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5348-L5353","statement_sha256":"e22e35a4dcdf5171f3604c971f2445db3384672399c2f95ae5b091c458695c72","origin":"The Stacks Project","memory_eligible":false,"source_rank":14179,"rank":14179,"depth":57,"x":2056.616,"y":1488.077,"cluster":"algebraic-stacks"},{"id":"stacks:06Q2","tag":"06Q2","title":"Morphisms of finite presentation · Definition 06Q2","summary":"Let f : X → Y be a morphism of algebraic stacks. • We say f locally of finite presentation if the equivalent conditions of Lemma [Tag 06FM] hold with P = locally of finite presentation. • We say f is of finite presentation if it is locally of finite presentation, quasi-compact, and quasi-separated.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item We say $f$\n{\\it locally of finite presentation} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with\n$\\mathcal{P} = \\text{locally of finite presentation}$.\n\\item We say $f$ is\n{\\it of finite presentation} if it is locally of finite presentation,\nquasi-compact, and quasi-separated.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Q2","source_file":"stacks-morphisms.tex","source_line":5381,"source_end_line":5394,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5381-L5394","statement_sha256":"21495631312a6dae2ac92dde701f2e0edcfe164903d10232731c77fc92c4311e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14180,"rank":14180,"depth":4,"x":2100.449,"y":1697.73,"cluster":"algebraic-stacks"},{"id":"stacks:06Q3","tag":"06Q3","title":"Morphisms of finite presentation · Lemma 06Q3","summary":"The composition of finitely presented morphisms is of finite presentation. The same holds for morphisms which are locally of finite presentation.","statement_latex":"The composition of finitely presented morphisms is of finite presentation.\nThe same holds for morphisms which are locally of finite presentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Q3","source_file":"stacks-morphisms.tex","source_line":5400,"source_end_line":5404,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5400-L5404","statement_sha256":"dc2dbc34d07bbdff7c34373aeb0dedbdfb375ad4ecd5f116633e908fc71af5f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14181,"rank":14181,"depth":56,"x":1899.329,"y":1567.878,"cluster":"algebraic-stacks"},{"id":"stacks:06Q4","tag":"06Q4","title":"Morphisms of finite presentation · Lemma 06Q4","summary":"A base change of a finitely presented morphism is of finite presentation. The same holds for morphisms which are locally of finite presentation.","statement_latex":"A base change of a finitely presented morphism is of finite presentation.\nThe same holds for morphisms which are locally of finite presentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Q4","source_file":"stacks-morphisms.tex","source_line":5414,"source_end_line":5418,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5414-L5418","statement_sha256":"0dcda06aca7f83ddf032ab538adf105202922c5fd94af068a703557e4ad8bc54","origin":"The Stacks Project","memory_eligible":false,"source_rank":14182,"rank":14182,"depth":18,"x":2152.308,"y":1549.491,"cluster":"algebraic-stacks"},{"id":"stacks:06Q5","tag":"06Q5","title":"Morphisms of finite presentation · Lemma 06Q5","summary":"A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type.","statement_latex":"A morphism which is locally of finite presentation is locally of finite type.\nA morphism of finite presentation is of finite type.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Q5","source_file":"stacks-morphisms.tex","source_line":5428,"source_end_line":5432,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5428-L5432","statement_sha256":"92df17b1f929423d789954b1427ef46ce17eb80205bc17747f44a7084dc1c0e5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14183,"rank":14183,"depth":47,"x":1980.381,"y":1706.751,"cluster":"algebraic-stacks"},{"id":"stacks:0DQJ","tag":"0DQJ","title":"Morphisms of finite presentation · Lemma 0DQJ","summary":"Let f : X → Y be a morphism of algebraic stacks. • If Y is locally Noetherian and f locally of finite type then f is locally of finite presentation. • If Y is locally Noetherian and f of finite type and quasi-separated then f is of finite presentation.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item If $\\mathcal{Y}$ is locally Noetherian and $f$ locally of finite type\nthen $f$ is locally of finite presentation.\n\\item If $\\mathcal{Y}$ is locally Noetherian and $f$ of finite type and\nquasi-separated then $f$ is of finite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQJ","source_file":"stacks-morphisms.tex","source_line":5442,"source_end_line":5451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5442-L5451","statement_sha256":"06aea90ee590aaad88ab6e99303b3bb681b0ed1419d216146468038c1540489b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14184,"rank":14184,"depth":47,"x":1980.693,"y":1493.022,"cluster":"algebraic-stacks"},{"id":"stacks:06Q6","tag":"06Q6","title":"Morphisms of finite presentation · Lemma 06Q6","summary":"Let f : X → Y and g : Y → Z be morphisms of algebraic stacks If g ∘ f is locally of finite presentation and g is locally of finite type, then f is locally of finite presentation.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and\n$g : \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of algebraic stacks\nIf $g \\circ f$ is locally of finite presentation and $g$ is locally of\nfinite type, then $f$ is locally of finite presentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Q6","source_file":"stacks-morphisms.tex","source_line":5469,"source_end_line":5475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5469-L5475","statement_sha256":"a895569b8b553c88bc421e9d00e086e5fd2e1655a1b86229dc7818b12498fac5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14185,"rank":14185,"depth":20,"x":2152.508,"y":1650.956,"cluster":"algebraic-stacks"},{"id":"stacks:0CMG","tag":"0CMG","title":"Morphisms of finite presentation · Lemma 0CMG","summary":"Let f : X → Y be a morphism of algebraic stacks with diagonal Δ : X → X ×_Y X. If f is locally of finite type then Δ is locally of finite presentation. If f is quasi-separated and locally of finite type, then Δ is of finite presentation.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwith diagonal\n$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$.\nIf $f$ is locally of finite type then $\\Delta$ is\nlocally of finite presentation. If $f$ is\nquasi-separated and locally of finite type, then $\\Delta$ is of finite\npresentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMG","source_file":"stacks-morphisms.tex","source_line":5490,"source_end_line":5499,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5490-L5499","statement_sha256":"2c45f87026c62bf25593e6eac310cd993a02ab9d9afcd0c1c228d191bc390490","origin":"The Stacks Project","memory_eligible":false,"source_rank":14186,"rank":14186,"depth":21,"x":1898.558,"y":1631.971,"cluster":"algebraic-stacks"},{"id":"stacks:06Q7","tag":"06Q7","title":"Morphisms of finite presentation · Lemma 06Q7","summary":"An open immersion is locally of finite presentation.","statement_latex":"An open immersion is locally of finite presentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Q7","source_file":"stacks-morphisms.tex","source_line":5514,"source_end_line":5517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5514-L5517","statement_sha256":"d440827d1bad79ba869ec9151cb790ee8feef936e28ada3ae58ca42053b62108","origin":"The Stacks Project","memory_eligible":false,"source_rank":14187,"rank":14187,"depth":4,"x":2101.284,"y":1501.745,"cluster":"algebraic-stacks"},{"id":"stacks:0CPP","tag":"0CPP","title":"Morphisms of finite presentation · Lemma 0CPP","summary":"Let P be a property of morphisms of algebraic spaces which is fppf local on the target and preserved by arbitrary base change. Let f : X → Y be a morphism of algebraic stacks representable by algebraic spaces. Let Z → Y be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Set W = Z ×_Y X. Then (f has P) ⇔ (the projection W → Z has P). For the meaning of this statement see Properties of Stacks, Section [Tag 04XB].","statement_latex":"Let $P$ be a property of morphisms of algebraic spaces which is\nfppf local on the target and preserved by arbitrary base change.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nrepresentable by algebraic spaces.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be a morphism of algebraic stacks which\nis surjective, flat, and locally of finite presentation.\nSet $\\mathcal{W} = \\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X}$. Then\n$$\n(f\\text{ has }P) \\Leftrightarrow\n(\\text{the projection }\\mathcal{W} \\to \\mathcal{Z}\\text{ has }P).\n$$\nFor the meaning of this statement see\nProperties of Stacks, Section\n\\ref{stacks-properties-section-properties-morphisms}.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPP","source_file":"stacks-morphisms.tex","source_line":5527,"source_end_line":5543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5527-L5543","statement_sha256":"7b95632c781fb11223b6f48ec298c5c9299e1156119a9487e4aaf33bd06ca1af","origin":"The Stacks Project","memory_eligible":false,"source_rank":14188,"rank":14188,"depth":57,"x":2056.477,"y":1713.011,"cluster":"algebraic-stacks"},{"id":"stacks:0DN6","tag":"0DN6","title":"Morphisms of finite presentation · Lemma 0DN6","summary":"Let P be a property of morphisms of algebraic spaces which is smooth local on the source-and-target and fppf local on the target. Let f : X → Y be a morphism of algebraic stacks. Let Z → Y be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change Z ×_Y X → Z has P, then f has P.","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces\nwhich is smooth local on the source-and-target and fppf local\non the target.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely\npresented morphism of algebraic stacks. If the base change\n$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$\nhas $\\mathcal{P}$, then $f$ has $\\mathcal{P}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DN6","source_file":"stacks-morphisms.tex","source_line":5563,"source_end_line":5573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5563-L5573","statement_sha256":"714a639d975b16550b31d5b1a7afd3d1679b2ba16e4b3d5f1e80e08b6960bf98","origin":"The Stacks Project","memory_eligible":false,"source_rank":14189,"rank":14189,"depth":57,"x":1919.488,"y":1531.624,"cluster":"algebraic-stacks"},{"id":"stacks:06Q8","tag":"06Q8","title":"Morphisms of finite presentation · Lemma 06Q8","summary":"Let f : X → Y be a morphism of algebraic stacks. Let Z → Y be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change Z ×_Y X → Z is locally of finite presentation, then f is locally of finite presentation.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely\npresented morphism of algebraic stacks. If the base change\n$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$\nis locally of finite presentation, then $f$ is locally of finite\npresentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Q8","source_file":"stacks-morphisms.tex","source_line":5619,"source_end_line":5627,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5619-L5627","statement_sha256":"b5fa526668623da449e0f62f9260147540dfdded0ed5b8508310bac503938885","origin":"The Stacks Project","memory_eligible":false,"source_rank":14190,"rank":14190,"depth":58,"x":2166.61,"y":1587.7,"cluster":"algebraic-stacks"},{"id":"stacks:06Q9","tag":"06Q9","title":"Morphisms of finite presentation · Lemma 06Q9","summary":"Let X → Y → Z be morphisms of algebraic stacks. If X → Z is locally of finite presentation and X → Y is surjective, flat, and locally of finite presentation, then Y → Z is locally of finite presentation.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be morphisms of\nalgebraic stacks. If $\\mathcal{X} \\to \\mathcal{Z}$ is locally of finite\npresentation and $\\mathcal{X} \\to \\mathcal{Y}$ is surjective, flat, and\nlocally of finite presentation, then $\\mathcal{Y} \\to \\mathcal{Z}$\nis locally of finite presentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06Q9","source_file":"stacks-morphisms.tex","source_line":5639,"source_end_line":5646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5639-L5646","statement_sha256":"d0bb3da2c6434848dabd0dd0f2f8d6c97af7931735c7d7ffabd0ba6d395e2024","origin":"The Stacks Project","memory_eligible":false,"source_rank":14191,"rank":14191,"depth":49,"x":1939.069,"y":1686.67,"cluster":"algebraic-stacks"},{"id":"stacks:06QA","tag":"06QA","title":"Morphisms of finite presentation · Lemma 06QA","summary":"Let f : X → Y be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Then for every scheme U and object y of Y over U there exists an fppf covering (U_i → U) and objects x_i of X over U_i such that f(x_i) ≅ y|_U_i in Y_U_i.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich is surjective, flat, and locally of finite presentation.\nThen for every scheme $U$ and object $y$ of $\\mathcal{Y}$ over $U$\nthere exists an fppf covering $\\{U_i \\to U\\}$ and objects $x_i$\nof $\\mathcal{X}$ over $U_i$ such that $f(x_i) \\cong y|_{U_i}$ in\n$\\mathcal{Y}_{U_i}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QA","source_file":"stacks-morphisms.tex","source_line":5663,"source_end_line":5671,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5663-L5671","statement_sha256":"94b8e39aa6f04220c13b82cf6d90f0f0bcee60869b6007e70e3d515b0708113e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14192,"rank":14192,"depth":57,"x":2027.351,"y":1484.382,"cluster":"algebraic-stacks"},{"id":"stacks:07AN","tag":"07AN","title":"Morphisms of finite presentation · Lemma 07AN","summary":"Let f_j : X_j → X, j ∈ J be a family of morphisms of algebraic stacks which are each flat and locally of finite presentation and which are jointly surjective, i.e., |X| = ⋃ |f_j|(|X_j|). Then for every scheme U and object x of X over U there exists an fppf covering (U_i → U)_i ∈ I, a map a : I → J, and objects x_i of X_a(i) over U_i such that f_a(i)(x_i) ≅ y|_U_i in X_U_i.","statement_latex":"Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$, $j \\in J$ be a family of morphisms\nof algebraic stacks which are each flat and locally of finite presentation\nand which are jointly surjective, i.e.,\n$|\\mathcal{X}| = \\bigcup |f_j|(|\\mathcal{X}_j|)$.\nThen for every scheme $U$ and object $x$ of $\\mathcal{X}$ over $U$\nthere exists an fppf covering $\\{U_i \\to U\\}_{i \\in I}$, a map\n$a : I \\to J$, and objects $x_i$ of $\\mathcal{X}_{a(i)}$ over $U_i$\nsuch that $f_{a(i)}(x_i) \\cong y|_{U_i}$ in $\\mathcal{X}_{U_i}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AN","source_file":"stacks-morphisms.tex","source_line":5699,"source_end_line":5709,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5699-L5709","statement_sha256":"ee7b560d04446f2b18930d8eae89bddde0d590e1c3447f28b0e126823b8da6c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14193,"rank":14193,"depth":58,"x":2125.023,"y":1683.834,"cluster":"algebraic-stacks"},{"id":"stacks:06R7","tag":"06R7","title":"Morphisms of finite presentation · Lemma 06R7","summary":"Let f : X → Y be flat and locally of finite presentation. Then |f| : |X| → |Y| is open.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be flat and locally of finite\npresentation. Then $|f| : |\\mathcal{X}| \\to |\\mathcal{Y}|$ is open.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R7","source_file":"stacks-morphisms.tex","source_line":5723,"source_end_line":5727,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5723-L5727","statement_sha256":"8c0bb1859c98292ef4f9cdbfd335e28ef27899ebb8dce8590f18eb5460188108","origin":"The Stacks Project","memory_eligible":false,"source_rank":14194,"rank":14194,"depth":18,"x":1892.381,"y":1592.093,"cluster":"algebraic-stacks"},{"id":"stacks:0DQK","tag":"0DQK","title":"Morphisms of finite presentation · Lemma 0DQK","summary":"Let f : X → Y be a morphism of algebraic stacks. Let Z → Y be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change Z ×_Y X → Z is quasi-compact, then f is quasi-compact.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely\npresented morphism of algebraic stacks. If the base change\n$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$\nis quasi-compact, then $f$ is quasi-compact.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQK","source_file":"stacks-morphisms.tex","source_line":5742,"source_end_line":5749,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5742-L5749","statement_sha256":"e49ab002f69cfc3f195337b44a7ececd6e3d5d56d7ee05544aa6acb85a365cce","origin":"The Stacks Project","memory_eligible":false,"source_rank":14195,"rank":14195,"depth":19,"x":2137.941,"y":1527.673,"cluster":"algebraic-stacks"},{"id":"stacks:0CPQ","tag":"0CPQ","title":"Morphisms of finite presentation · Lemma 0CPQ","summary":"Let f : X → Y, g : Y → Z be composable morphisms of algebraic stacks with composition h = g ∘ f : X → Z. If f is surjective, flat, locally of finite presentation, and universally injective and if h is separated, then g is separated.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$, $g : \\mathcal{Y} \\to \\mathcal{Z}$\nbe composable morphisms of algebraic stacks with composition\n$h = g \\circ f : \\mathcal{X} \\to \\mathcal{Z}$.\nIf $f$ is surjective, flat, locally of finite presentation,\nand universally injective and if $h$ is separated, then\n$g$ is separated.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPQ","source_file":"stacks-morphisms.tex","source_line":5772,"source_end_line":5780,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5772-L5780","statement_sha256":"8281b13814b6ff6fb4e2712b098adc3bd2731abf95da5938d740ee97f505ee22","origin":"The Stacks Project","memory_eligible":false,"source_rank":14196,"rank":14196,"depth":60,"x":2008.549,"y":1714.692,"cluster":"algebraic-stacks"},{"id":"stacks:06QC","tag":"06QC","title":"Gerbes · Definition 06QC","summary":"Let f : X → Y be a morphism of algebraic stacks. We say X is a gerbe over Y if X is a gerbe over Y as stacks in groupoids over (Sch/S)_fppf, see Stacks, Definition [Tag 06P2]. We say an algebraic stack X is a gerbe if there exists a morphism X → X where X is an algebraic space which turns X into a gerbe over X.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $\\mathcal{X}$ is a {\\it gerbe over} $\\mathcal{Y}$ if\n$\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ as stacks\nin groupoids over $(\\Sch/S)_{fppf}$, see\nStacks, Definition \\ref{stacks-definition-gerbe-over-stack-in-groupoids}.\nWe say an algebraic stack $\\mathcal{X}$ is a {\\it gerbe} if there exists\na morphism $\\mathcal{X} \\to X$ where $X$ is an algebraic space which\nturns $\\mathcal{X}$ into a gerbe over $X$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QC","source_file":"stacks-morphisms.tex","source_line":5854,"source_end_line":5864,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5854-L5864","statement_sha256":"b35fb4cb34eed742714a1e7ce65a864e3dce8065c8d6ab6f9e033639dc75c941","origin":"The Stacks Project","memory_eligible":false,"source_rank":14197,"rank":14197,"depth":9,"x":1953.512,"y":1503.162,"cluster":"algebraic-stacks"},{"id":"stacks:06QD","tag":"06QD","title":"Gerbes · Lemma 06QD","summary":"Let X be an algebraic stack. If X is a gerbe, then the sheafification of the presheaf (Sch/S)_fppf^opp → Sets, U ↦ Ob(X_U)/ ≅ is an algebraic space and X is a gerbe over it.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{X}$ is a gerbe, then\nthe sheafification of the presheaf\n$$\n(\\Sch/S)_{fppf}^{opp} \\to \\textit{Sets}, \\quad\nU \\mapsto \\Ob(\\mathcal{X}_U)/\\!\\!\\cong\n$$\nis an algebraic space and $\\mathcal{X}$ is a gerbe over it.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QD","source_file":"stacks-morphisms.tex","source_line":5877,"source_end_line":5886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5877-L5886","statement_sha256":"03a6ee7f23831733fcb18934afb56eb886238f167e430de83b5f7cb6e80fbc4d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14198,"rank":14198,"depth":8,"x":2164.405,"y":1628.032,"cluster":"algebraic-stacks"},{"id":"stacks:06QE","tag":"06QE","title":"Gerbes · Lemma 06QE","summary":"Let xymatrix X' ar[r] ar[d] & X ar[d] Y' ar[r] & Y be a fibre product of algebraic stacks. If X is a gerbe over Y, then X' is a gerbe over Y'.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\\n\\mathcal{Y}' \\ar[r] & \\mathcal{Y}\n}\n$$\nbe a fibre product of algebraic stacks.\nIf $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then\n$\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QE","source_file":"stacks-morphisms.tex","source_line":5915,"source_end_line":5927,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5915-L5927","statement_sha256":"eb201eb5f17c53ee6c39134ecab2b0d899bbd532b8fe0686033bee090f398c09","origin":"The Stacks Project","memory_eligible":false,"source_rank":14199,"rank":14199,"depth":9,"x":1908.232,"y":1655.647,"cluster":"algebraic-stacks"},{"id":"stacks:06R8","tag":"06R8","title":"Gerbes · Lemma 06R8","summary":"Let X → Y and Y → Z be morphisms of algebraic stacks. If X is a gerbe over Y and Y is a gerbe over Z, then X is a gerbe over Z.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y}$ and $\\mathcal{Y} \\to \\mathcal{Z}$\nbe morphisms of algebraic stacks. If $\\mathcal{X}$ is a gerbe over\n$\\mathcal{Y}$ and $\\mathcal{Y}$ is a gerbe over $\\mathcal{Z}$, then\n$\\mathcal{X}$ is a gerbe over $\\mathcal{Z}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R8","source_file":"stacks-morphisms.tex","source_line":5934,"source_end_line":5940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5934-L5940","statement_sha256":"29e444aefd1c1ca15be7115dd5514f846ba9bd53d43ae8670cf3441c47949220","origin":"The Stacks Project","memory_eligible":false,"source_rank":14200,"rank":14200,"depth":9,"x":2075.083,"y":1489.767,"cluster":"algebraic-stacks"},{"id":"stacks:06QF","tag":"06QF","title":"Gerbes · Lemma 06QF","summary":"Let xymatrix X' ar[r] ar[d] & X ar[d] Y' ar[r] & Y be a fibre product of algebraic stacks. If Y' → Y is surjective, flat, and locally of finite presentation and X' is a gerbe over Y', then X is a gerbe over Y.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d] \\\\\n\\mathcal{Y}' \\ar[r] & \\mathcal{Y}\n}\n$$\nbe a fibre product of algebraic stacks.\nIf $\\mathcal{Y}' \\to \\mathcal{Y}$ is surjective, flat, and locally\nof finite presentation and $\\mathcal{X}'$ is a gerbe over $\\mathcal{Y}'$,\nthen $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QF","source_file":"stacks-morphisms.tex","source_line":5947,"source_end_line":5960,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5947-L5960","statement_sha256":"0e9d761c680edc65607e334b7f5a39110d9416e49fefb4684956874f3fb93932","origin":"The Stacks Project","memory_eligible":false,"source_rank":14201,"rank":14201,"depth":58,"x":2085.456,"y":1706.968,"cluster":"algebraic-stacks"},{"id":"stacks:06QG","tag":"06QG","title":"Gerbes · Lemma 06QG","summary":"Let π : X → U be a morphism from an algebraic stack to an algebraic space and let x : U → X be a section of π. Set G = mathitIsom_X(x, x), see Definition [Tag 06PP]. If X is a gerbe over U, then • there is a canonical equivalence of stacks in groupoids x_can : [U/G] → X. where [U/G] is the quotient stack for the trivial action of G on U, • G → U is flat and locally of finite presentation, and • U → X is surjective, flat, and locally of finite presentation.","statement_latex":"Let $\\pi : \\mathcal{X} \\to U$ be a morphism from an algebraic stack to\nan algebraic space and let $x : U \\to \\mathcal{X}$ be a section of $\\pi$.\nSet $G = \\mathit{Isom}_\\mathcal{X}(x, x)$, see\nDefinition \\ref{definition-isom}.\nIf $\\mathcal{X}$ is a gerbe over $U$, then\n\\begin{enumerate}\n\\item there is a canonical equivalence of stacks in groupoids\n$$\nx_{can} : [U/G] \\longrightarrow \\mathcal{X}.\n$$\nwhere $[U/G]$ is the quotient stack for the trivial\naction of $G$ on $U$,\n\\item $G \\to U$ is flat and locally of finite presentation, and\n\\item $U \\to \\mathcal{X}$ is surjective, flat, and locally of finite\npresentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QG","source_file":"stacks-morphisms.tex","source_line":5969,"source_end_line":5987,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L5969-L5987","statement_sha256":"d6748538d4e1899d4ae7fe42ad5c82a087c7f4f5b4f9520d1a07b9c8c99baef4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14202,"rank":14202,"depth":79,"x":1902.966,"y":1552.546,"cluster":"algebraic-stacks"},{"id":"stacks:06QH","tag":"06QH","title":"Gerbes · Lemma 06QH","summary":"Let π : X → Y be a morphism of algebraic stacks. The following are equivalent • X is a gerbe over Y, and • there exists an algebraic space U, a group algebraic space G flat and locally of finite presentation over U, and a surjective, flat, and locally finitely presented morphism U → Y such that X ×_Y U ≅ [U/G] over U.","statement_latex":"Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, and\n\\item there exists an algebraic space $U$, a group algebraic space $G$\nflat and locally of finite presentation over $U$, and a\nsurjective, flat, and locally finitely presented\nmorphism $U \\to \\mathcal{Y}$ such that\n$\\mathcal{X} \\times_\\mathcal{Y} U \\cong [U/G]$ over $U$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QH","source_file":"stacks-morphisms.tex","source_line":6034,"source_end_line":6046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6034-L6046","statement_sha256":"c6bff2a67eeb97bc97c1f81c305d8e2d0fb6932280046db95b1481a7b9f8d8a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14203,"rank":14203,"depth":80,"x":2161.96,"y":1562.875,"cluster":"algebraic-stacks"},{"id":"stacks:06QI","tag":"06QI","title":"Gerbes · Lemma 06QI","summary":"Let π : X → Y be a morphism of algebraic stacks. If X is a gerbe over Y, then π is surjective, flat, and locally of finite presentation.","statement_latex":"Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then $\\pi$ is surjective,\nflat, and locally of finite presentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QI","source_file":"stacks-morphisms.tex","source_line":6073,"source_end_line":6078,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6073-L6078","statement_sha256":"e58795bbb063b2a3a88f614e85a91b501fb8b97288e3a7ca153bae7942aa1f27","origin":"The Stacks Project","memory_eligible":false,"source_rank":14204,"rank":14204,"depth":81,"x":1962.486,"y":1702.351,"cluster":"algebraic-stacks"},{"id":"stacks:06QJ","tag":"06QJ","title":"Gerbes · Proposition 06QJ","summary":"Let X be an algebraic stack. The following are equivalent • X is a gerbe, and • I_X → X is flat and locally of finite presentation.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is a gerbe, and\n\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is flat and locally of\nfinite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QJ","source_file":"stacks-morphisms.tex","source_line":6105,"source_end_line":6113,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6105-L6113","statement_sha256":"eeb3ac7212f0801d7fe381c8bb9e62207ab7f96daf7ffaa889e5150a06bcb179","origin":"The Stacks Project","memory_eligible":false,"source_rank":14205,"rank":14205,"depth":81,"x":1997.446,"y":1486.111,"cluster":"algebraic-stacks"},{"id":"stacks:0CPR","tag":"0CPR","title":"Gerbes · Lemma 0CPR","summary":"Let f : X → Y be a morphism of algebraic stacks which makes X a gerbe over Y. Then • I_X/Y → X is flat and locally of finite presentation, • X → X ×_Y X is surjective, flat, and locally of finite presentation, • given algebraic spaces T_i, i = 1, 2 and morphisms x_i : T_i → X, with y_i = f ∘ x_i the morphism T_1 ×_x_1, X, x_2 T_2 → T_1 ×_y_1, Y, y_2 T_2 is surjective, flat, and locally of finite presentation, • given an algebraic space T and morphisms x_i : T → X, i = 1,…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich makes $\\mathcal{X}$ a gerbe over $\\mathcal{Y}$. Then\n\\begin{enumerate}\n\\item $\\mathcal{I}_{\\mathcal{X}/\\mathcal{Y}} \\to \\mathcal{X}$\nis flat and locally of finite presentation,\n\\item $\\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$\nis surjective, flat, and locally of finite presentation,\n\\item given algebraic spaces $T_i$, $i = 1, 2$ and morphisms\n$x_i : T_i \\to \\mathcal{X}$, with $y_i = f \\circ x_i$ the morphism\n$$\nT_1 \\times_{x_1, \\mathcal{X}, x_2} T_2 \\longrightarrow\nT_1 \\times_{y_1, \\mathcal{Y}, y_2} T_2\n$$\nis surjective, flat, and locally of finite presentation,\n\\item given an algebraic space $T$ and morphisms\n$x_i : T \\to \\mathcal{X}$, $i = 1, 2$, with $y_i = f \\circ x_i$ the morphism\n$$\n\\mathit{Isom}_\\mathcal{X}(x_1, x_2) \\longrightarrow\n\\mathit{Isom}_\\mathcal{Y}(y_1, y_2)\n$$\nis surjective, flat, and locally of finite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPR","source_file":"stacks-morphisms.tex","source_line":6215,"source_end_line":6239,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6215-L6239","statement_sha256":"46503d0fd1a83994653c1637d71b61a305e9131435906abd8c23b3759fcf9c96","origin":"The Stacks Project","memory_eligible":false,"source_rank":14206,"rank":14206,"depth":82,"x":2145.701,"y":1665.57,"cluster":"algebraic-stacks"},{"id":"stacks:0CPS","tag":"0CPS","title":"Gerbes · Proposition 0CPS","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • X is a gerbe over Y, and • f : X → Y and Δ : X → X ×_Y X are surjective, flat, and locally of finite presentation.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, and\n\\item $f : \\mathcal{X} \\to \\mathcal{Y}$ and\n$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$\nare surjective, flat, and locally of finite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPS","source_file":"stacks-morphisms.tex","source_line":6306,"source_end_line":6316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6306-L6316","statement_sha256":"b608faf32ec7618940864b1871ed1d7b8cd5a2f01d4f887985ab7f4e241d81b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14207,"rank":14207,"depth":83,"x":1891.823,"y":1617.318,"cluster":"algebraic-stacks"},{"id":"stacks:06QK","tag":"06QK","title":"Gerbes · Lemma 06QK","summary":"Let Z be a reduced, locally Noetherian algebraic stack such that |Z| is a singleton. Then Z is a gerbe over a reduced, locally Noetherian algebraic space Z with |Z| a singleton.","statement_latex":"Let $\\mathcal{Z}$ be a reduced, locally Noetherian algebraic stack\nsuch that $|\\mathcal{Z}|$ is a singleton. Then $\\mathcal{Z}$ is a gerbe\nover a reduced, locally Noetherian algebraic space $Z$ with $|Z|$ a\nsingleton.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06QK","source_file":"stacks-morphisms.tex","source_line":6359,"source_end_line":6365,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6359-L6365","statement_sha256":"15fbf7690c7f8291313bf3ecaa6d50e24888fba5a4d1cfd3f430b588046e3a56","origin":"The Stacks Project","memory_eligible":false,"source_rank":14208,"rank":14208,"depth":82,"x":2118.046,"y":1508.74,"cluster":"algebraic-stacks"},{"id":"stacks:06R9","tag":"06R9","title":"Gerbes · Lemma 06R9","summary":"Let f : X → Y be a morphism of algebraic stacks. If X is a gerbe over Y then f is a universal homeomorphism.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ then $f$ is a\nuniversal homeomorphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06R9","source_file":"stacks-morphisms.tex","source_line":6404,"source_end_line":6409,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6404-L6409","statement_sha256":"af2b8d5e3e83fa04853e25890772096dc9eadf2871b893dd05316316238d7519","origin":"The Stacks Project","memory_eligible":false,"source_rank":14209,"rank":14209,"depth":82,"x":2038.473,"y":1717.364,"cluster":"algebraic-stacks"},{"id":"stacks:0DQL","tag":"0DQL","title":"Gerbes · Lemma 0DQL","summary":"Let f : X → Y be a morphism of algebraic stacks such that X is a gerbe over Y. If Δ_X is quasi-compact, so is Δ_Y.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nsuch that $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$.\nIf $\\Delta_\\mathcal{X}$ is quasi-compact, so is $\\Delta_\\mathcal{Y}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQL","source_file":"stacks-morphisms.tex","source_line":6437,"source_end_line":6442,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6437-L6442","statement_sha256":"a1c17c1dc7d0dcdeffc2658a3ce690c426e3c25c68b3c9f9cf2929990c7c9fa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14210,"rank":14210,"depth":84,"x":1929.276,"y":1518.189,"cluster":"algebraic-stacks"},{"id":"stacks:06RA","tag":"06RA","title":"Gerbes · Lemma 06RA","summary":"Let X be an algebraic stack. If X is a gerbe then for every x ∈ |X| the residual gerbe of X at x exists.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{X}$ is a gerbe\nthen for every $x \\in |\\mathcal{X}|$ the residual gerbe of $\\mathcal{X}$\nat $x$ exists.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RA","source_file":"stacks-morphisms.tex","source_line":6470,"source_end_line":6475,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6470-L6475","statement_sha256":"5c8ec21efabd777889fabc422d3d0564b2cfdcd92580f7c93218d190db48d14b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14211,"rank":14211,"depth":83,"x":2170.194,"y":1603.176,"cluster":"algebraic-stacks"},{"id":"stacks:06RC","tag":"06RC","title":"Stratification by gerbes · Proposition 06RC","summary":"Let X be a reduced algebraic stack such that I_X → X is quasi-compact. Then there exists a dense open substack U ⊂ X which is a gerbe.","statement_latex":"Let $\\mathcal{X}$ be a reduced algebraic stack such that\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact.\nThen there exists a dense open substack $\\mathcal{U} \\subset \\mathcal{X}$\nwhich is a gerbe.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Stratification by gerbes","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RC","source_file":"stacks-morphisms.tex","source_line":6532,"source_end_line":6538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6532-L6538","statement_sha256":"74d915faf6da195ffcddc80115c92c871c24a621e0e5f0453d8b97767e71e92e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14212,"rank":14212,"depth":82,"x":1923.969,"y":1677.279,"cluster":"algebraic-stacks"},{"id":"stacks:06RF","tag":"06RF","title":"Stratification by gerbes · Lemma 06RF","summary":"Let X be an algebraic stack such that I_X → X is quasi-compact. Then there exists a well-ordered index set I and for every i ∈ I a reduced locally closed substack U_i ⊂ X such that • each U_i is a gerbe, • we have |X| = ⋃_i ∈ I |U_i|, • T_i = |X| setminus ⋃_i' < i |U_i'| is closed in |X| for all i ∈ I, and • |U_i| is open in T_i. We can moreover arrange it so that either (a) |U_i| ⊂ T_i is dense, or (b) U_i is quasi-compact. In case (a), if we choose U_i as large as…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack such that\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact.\nThen there exists a well-ordered index set $I$ and for every $i \\in I$\na reduced locally closed substack $\\mathcal{U}_i \\subset \\mathcal{X}$ such that\n\\begin{enumerate}\n\\item each $\\mathcal{U}_i$ is a gerbe,\n\\item we have $|\\mathcal{X}| = \\bigcup_{i \\in I} |\\mathcal{U}_i|$,\n\\item $T_i = |\\mathcal{X}| \\setminus \\bigcup_{i' < i} |\\mathcal{U}_{i'}|$\nis closed in $|\\mathcal{X}|$ for all $i \\in I$, and\n\\item $|\\mathcal{U}_i|$ is open in $T_i$.\n\\end{enumerate}\nWe can moreover arrange it so that either (a) $|\\mathcal{U}_i| \\subset T_i$\nis dense, or (b) $\\mathcal{U}_i$ is quasi-compact. In case (a), if\nwe choose $\\mathcal{U}_i$ as large as possible (see proof for details), then\nthe stratification is canonical.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Stratification by gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RF","source_file":"stacks-morphisms.tex","source_line":6616,"source_end_line":6633,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6616-L6633","statement_sha256":"90644ad130de082dc02a56d06ff924a9ebe556c6e6908dc094112007a9b0e388","origin":"The Stacks Project","memory_eligible":false,"source_rank":14213,"rank":14213,"depth":83,"x":2046.054,"y":1482.741,"cluster":"algebraic-stacks"},{"id":"stacks:0DQN","tag":"0DQN","title":"The topological space of an algebraic stack · Lemma 0DQN","summary":"Let X be a quasi-compact algebraic stack whose diagonal Δ is quasi-compact. Then |X| is a spectral topological space.","statement_latex":"Let $\\mathcal{X}$ be a quasi-compact algebraic stack\nwhose diagonal $\\Delta$ is quasi-compact.\nThen $|\\mathcal{X}|$ is a spectral topological space.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"The topological space of an algebraic stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQN","source_file":"stacks-morphisms.tex","source_line":6720,"source_end_line":6725,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6720-L6725","statement_sha256":"a7e5749bd4dbb99ecc8950b27c785d553f3d88a6bbf2cc35bed6d1cc9d593a61","origin":"The Stacks Project","memory_eligible":false,"source_rank":14214,"rank":14214,"depth":85,"x":2112.535,"y":1695.663,"cluster":"algebraic-stacks"},{"id":"stacks:0DQP","tag":"0DQP","title":"The topological space of an algebraic stack · Lemma 0DQP","summary":"Let X be a quasi-compact and quasi-separated algebraic stack. Then |X| is a spectral topological space.","statement_latex":"Let $\\mathcal{X}$ be a quasi-compact and quasi-separated algebraic stack.\nThen $|\\mathcal{X}|$ is a spectral topological space.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"The topological space of an algebraic stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQP","source_file":"stacks-morphisms.tex","source_line":6794,"source_end_line":6798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6794-L6798","statement_sha256":"ef75ab285e249b8cd08fb9ae086319e8788f5023174448c78ea92640433556dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14215,"rank":14215,"depth":86,"x":1892.082,"y":1576.269,"cluster":"algebraic-stacks"},{"id":"stacks:0DQQ","tag":"0DQQ","title":"The topological space of an algebraic stack · Lemma 0DQQ","summary":"Let X be an algebraic stack whose diagonal is quasi-compact (for example if X is quasi-separated). Then there is an open covering |X| = ⋃ U_i with U_i spectral. In particular |X| is a sober topological space.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack whose diagonal is quasi-compact\n(for example if $\\mathcal{X}$ is quasi-separated).\nThen there is an open covering $|\\mathcal{X}| = \\bigcup U_i$\nwith $U_i$ spectral. In particular $|\\mathcal{X}|$ is\na sober topological space.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"The topological space of an algebraic stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQQ","source_file":"stacks-morphisms.tex","source_line":6804,"source_end_line":6811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6804-L6811","statement_sha256":"3af1027a43230914671f0c9efd74e16b6f98115bf0fd6e2f081292bbb98f79ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":14216,"rank":14216,"depth":86,"x":2150.894,"y":1539.184,"cluster":"algebraic-stacks"},{"id":"stacks:06RD","tag":"06RD","title":"Existence of residual gerbes · Lemma 06RD","summary":"Let X be an algebraic stack such that I_X → X is quasi-compact. Then the residual gerbe of X at x exists for every x ∈ |X|.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack such that\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact.\nThen the residual gerbe of $\\mathcal{X}$ at $x$ exists for\nevery $x \\in |\\mathcal{X}|$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Existence of residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RD","source_file":"stacks-morphisms.tex","source_line":6837,"source_end_line":6843,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6837-L6843","statement_sha256":"4a817566eab172000f1a30922ba55f9c4374f9dc48d4e9de88b115ed4d684f70","origin":"The Stacks Project","memory_eligible":false,"source_rank":14217,"rank":14217,"depth":84,"x":1989.724,"y":1713.548,"cluster":"algebraic-stacks"},{"id":"stacks:06UI","tag":"06UI","title":"Existence of residual gerbes · Lemma 06UI","summary":"Let X be a quasi-DM algebraic stack. Then the residual gerbe of X at x exists for every x ∈ |X|.","statement_latex":"Let $\\mathcal{X}$ be a quasi-DM algebraic stack.\nThen the residual gerbe of $\\mathcal{X}$ at $x$ exists for\nevery $x \\in |\\mathcal{X}|$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Existence of residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UI","source_file":"stacks-morphisms.tex","source_line":6874,"source_end_line":6879,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6874-L6879","statement_sha256":"32fd7161f6a4058d3e8139419715167d0ac5fa0b53de07541f02a37f78ce829e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14218,"rank":14218,"depth":81,"x":1968.33,"y":1493.319,"cluster":"algebraic-stacks"},{"id":"stacks:0H22","tag":"0H22","title":"Existence of residual gerbes · Lemma 0H22","summary":"Let X be a locally Noetherian algebraic stack. Then the residual gerbe of X at x exists for every x ∈ |X|.","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nThen the residual gerbe of $\\mathcal{X}$ at $x$ exists for\nevery $x \\in |\\mathcal{X}|$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Existence of residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H22","source_file":"stacks-morphisms.tex","source_line":6951,"source_end_line":6956,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L6951-L6956","statement_sha256":"a48f4cdf37e1072c7bdf0595fdee3ff79d32219816937bdb640cdd133ad1cd95","origin":"The Stacks Project","memory_eligible":false,"source_rank":14219,"rank":14219,"depth":76,"x":2161.386,"y":1643.712,"cluster":"algebraic-stacks"},{"id":"stacks:0DU1","tag":"0DU1","title":"Étale local structure · Lemma 0DU1","summary":"Let Y be an algebraic space. Let (U, R, s, t, c) be a groupoid in algebraic spaces over Y. Assume U → Y is flat and locally of finite presentation and R → U ×_Y U an open immersion. Then X = [U/R] = U/R is an algebraic space and X → Y is étale.","statement_latex":"Let $Y$ be an algebraic space.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $Y$.\nAssume $U \\to Y$ is flat and locally of finite presentation\nand $R \\to U \\times_Y U$ an open immersion.\nThen $X = [U/R] = U/R$ is an algebraic space and $X \\to Y$\nis \\'etale.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale local structure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DU1","source_file":"stacks-morphisms.tex","source_line":7038,"source_end_line":7046,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7038-L7046","statement_sha256":"3fd81b04675ffec4ae66c3a1c0cf000a323b27da042360f0d4acaf577d4be193","origin":"The Stacks Project","memory_eligible":false,"source_rank":14220,"rank":14220,"depth":76,"x":1897.844,"y":1642.358,"cluster":"algebraic-stacks"},{"id":"stacks:0DU2","tag":"0DU2","title":"Étale local structure · Lemma 0DU2","summary":"Let S be a scheme. Let (U, R, s, t, c) be a groupoid in algebraic spaces over S. Assume s, t are flat and locally of finite presentation. Let P ⊂ R be an open subspace such that (U, P, s|_P, t|_P, c|_P ×_s, U, t P) is a groupoid in algebraic spaces over S. Then [U/P] → [U/R] is a morphism of algebraic stacks which is representable by algebraic spaces, surjective, and étale.","statement_latex":"Let $S$ be a scheme.\nLet $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$.\nAssume $s, t$ are flat and locally of finite presentation.\nLet $P \\subset R$ be an open subspace such that\n$(U, P, s|_P, t|_P, c|_{P \\times_{s, U, t} P})$ is a\ngroupoid in algebraic spaces over $S$. Then\n$$\n[U/P] \\longrightarrow [U/R]\n$$\nis a morphism of algebraic stacks which\nis representable by algebraic spaces, surjective, and \\'etale.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale local structure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DU2","source_file":"stacks-morphisms.tex","source_line":7086,"source_end_line":7099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7086-L7099","statement_sha256":"eafcec890e8c70422eeb0ffd2ead44cbd73766567be31a73f034bd5e1887f985","origin":"The Stacks Project","memory_eligible":false,"source_rank":14221,"rank":14221,"depth":77,"x":2093.445,"y":1493.679,"cluster":"algebraic-stacks"},{"id":"stacks:0DU3","tag":"0DU3","title":"Étale local structure · Lemma 0DU3","summary":"Let X be an algebraic stack. Assume X is quasi-DM with separated diagonal (equivalently I_X → X is locally quasi-finite and separated). Let x ∈ |X|. Then there exists a morphism of algebraic stacks U → X with the following properties • there exists a point u ∈ |U| mapping to x, • U → X is representable by algebraic spaces and étale, • U = [U/R] where (U, R, s, t, c) is a groupoid scheme with U, R affine, and s, t finite, flat, and locally of finite presentation.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{X}$ is\nquasi-DM with separated diagonal (equivalently\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite and\nseparated). Let $x \\in |\\mathcal{X}|$. Then there exists a\nmorphism of algebraic stacks\n$$\n\\mathcal{U} \\longrightarrow \\mathcal{X}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item there exists a point $u \\in |\\mathcal{U}|$ mapping to $x$,\n\\item $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces and\n\\'etale,\n\\item $\\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid\nscheme with $U$, $R$ affine, and $s, t$ finite, flat, and\nlocally of finite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale local structure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DU3","source_file":"stacks-morphisms.tex","source_line":7144,"source_end_line":7163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7144-L7163","statement_sha256":"9a70e3f480a642f02d6bd939dd6250aea1282be4e72377c4ce3fe4b4c57d511e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14222,"rank":14222,"depth":81,"x":2068.752,"y":1714.505,"cluster":"algebraic-stacks"},{"id":"stacks:0DU4","tag":"0DU4","title":"Étale local structure · Lemma 0DU4","summary":"Let X be an algebraic stack. Assume X is quasi-DM with separated diagonal (equivalently I_X → X is locally quasi-finite and separated). Let x ∈ |X|. Assume the automorphism group of X at x is finite (Remark [Tag 0DTU]). Then there exists a morphism of algebraic stacks g : U → X with the following properties • there exists a point u ∈ |U| mapping to x and g induces an isomorphism between automorphism groups at u and x (Remark [Tag 0DTW]), • U → X is representable by…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{X}$ is\nquasi-DM with separated diagonal (equivalently\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite and\nseparated). Let $x \\in |\\mathcal{X}|$. Assume the\nautomorphism group of $\\mathcal{X}$ at $x$ is finite\n(Remark \\ref{remark-property-automorphism-groups}).\nThen there exists a morphism of algebraic stacks\n$$\ng : \\mathcal{U} \\longrightarrow \\mathcal{X}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item there exists a point $u \\in |\\mathcal{U}|$ mapping to $x$ and\n$g$ induces an isomorphism between automorphism groups at $u$ and $x$\n(Remark \\ref{remark-identify-automorphism-groups}),\n\\item $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces and\n\\'etale,\n\\item $\\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid\nscheme with $U$, $R$ affine, and $s, t$ finite, flat, and\nlocally of finite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale local structure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DU4","source_file":"stacks-morphisms.tex","source_line":7218,"source_end_line":7241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7218-L7241","statement_sha256":"de5447884d8e21a64d3f09d2335e7add22c695a6df7f4214b8f081a222221515","origin":"The Stacks Project","memory_eligible":false,"source_rank":14223,"rank":14223,"depth":82,"x":1909.233,"y":1537.498,"cluster":"algebraic-stacks"},{"id":"stacks:0DU5","tag":"0DU5","title":"Étale local structure · Lemma 0DU5","summary":"Let X be an algebraic stack. Assume X is quasi-DM with separated diagonal (equivalently I_X → X is locally quasi-finite and separated). Let x ∈ |X|. Assume x can be represented by a quasi-compact morphism Spec(k) → X. Then there exists a morphism of algebraic stacks g : U → X with the following properties • there exists a point u ∈ |U| mapping to x and g induces an isomorphism between the residual gerbes at u and x, • U → X is representable by algebraic spaces and étale,…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Assume $\\mathcal{X}$ is\nquasi-DM with separated diagonal (equivalently\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally quasi-finite and\nseparated). Let $x \\in |\\mathcal{X}|$. Assume $x$ can be represented\nby a quasi-compact morphism $\\Spec(k) \\to \\mathcal{X}$.\nThen there exists a morphism of algebraic stacks\n$$\ng : \\mathcal{U} \\longrightarrow \\mathcal{X}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item there exists a point $u \\in |\\mathcal{U}|$ mapping to $x$ and\n$g$ induces an isomorphism between the residual gerbes at $u$ and $x$,\n\\item $\\mathcal{U} \\to \\mathcal{X}$ is representable by algebraic spaces and\n\\'etale,\n\\item $\\mathcal{U} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid\nscheme with $U$, $R$ affine, and $s, t$ finite, flat, and\nlocally of finite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale local structure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DU5","source_file":"stacks-morphisms.tex","source_line":7273,"source_end_line":7294,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7273-L7294","statement_sha256":"60eced22a1cb6e3e033aec5503b8a0399e1095561dc1997d8d8a823d5e5d27cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":14224,"rank":14224,"depth":82,"x":2169.442,"y":1577.542,"cluster":"algebraic-stacks"},{"id":"stacks:075U","tag":"075U","title":"Smooth morphisms · Definition 075U","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is smooth if the equivalent conditions of Lemma [Tag 06FM] hold with P = smooth.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is {\\it smooth} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with $\\mathcal{P} = \\text{smooth}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075U","source_file":"stacks-morphisms.tex","source_line":7379,"source_end_line":7385,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7379-L7385","statement_sha256":"4c4ed40b25e82d29fff696cbc061e05dafdcaccadf73659648b99dbfd114ae10","origin":"The Stacks Project","memory_eligible":false,"source_rank":14225,"rank":14225,"depth":4,"x":1945.161,"y":1695.771,"cluster":"algebraic-stacks"},{"id":"stacks:075V","tag":"075V","title":"Smooth morphisms · Lemma 075V","summary":"The composition of smooth morphisms is smooth.","statement_latex":"The composition of smooth morphisms is smooth.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075V","source_file":"stacks-morphisms.tex","source_line":7387,"source_end_line":7390,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7387-L7390","statement_sha256":"bd19c8deb448decac76bd06f1603021371747cddd17692e4c3999c0f730881af","origin":"The Stacks Project","memory_eligible":false,"source_rank":14226,"rank":14226,"depth":39,"x":2015.53,"y":1481.132,"cluster":"algebraic-stacks"},{"id":"stacks:075W","tag":"075W","title":"Smooth morphisms · Lemma 075W","summary":"A base change of a smooth morphism is smooth.","statement_latex":"A base change of a smooth morphism is smooth.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075W","source_file":"stacks-morphisms.tex","source_line":7400,"source_end_line":7403,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7400-L7403","statement_sha256":"86373d7914e682d43d67f43c5091147bb0d53d4dc70448c29da8a58bc0a060ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":14227,"rank":14227,"depth":39,"x":2136.357,"y":1679.513,"cluster":"algebraic-stacks"},{"id":"stacks:0DN7","tag":"0DN7","title":"Smooth morphisms · Lemma 0DN7","summary":"Let f : X → Y be a morphism of algebraic stacks. Let Z → Y be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change Z ×_Y X → Z is smooth, then f is smooth.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $\\mathcal{Z} \\to \\mathcal{Y}$ be a surjective, flat, locally finitely\npresented morphism of algebraic stacks. If the base change\n$\\mathcal{Z} \\times_\\mathcal{Y} \\mathcal{X} \\to \\mathcal{Z}$\nis smooth, then $f$ is smooth.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DN7","source_file":"stacks-morphisms.tex","source_line":7413,"source_end_line":7420,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7413-L7420","statement_sha256":"8f1c1a439caf59fdeb5a2aeba0d86539d8d2210295b11756d65cee8b9870ae4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14228,"rank":14228,"depth":58,"x":1887.502,"y":1601.719,"cluster":"algebraic-stacks"},{"id":"stacks:0DNP","tag":"0DNP","title":"Smooth morphisms · Lemma 0DNP","summary":"A smooth morphism of algebraic stacks is locally of finite presentation.","statement_latex":"A smooth morphism of algebraic stacks is locally of finite presentation.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNP","source_file":"stacks-morphisms.tex","source_line":7431,"source_end_line":7434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7431-L7434","statement_sha256":"2c5d76228754e8a8b8d5af09903c964d143f0a580d5dfc9b9f2a31f3f2d60162","origin":"The Stacks Project","memory_eligible":false,"source_rank":14229,"rank":14229,"depth":0,"x":2133.787,"y":1517.801,"cluster":"algebraic-stacks"},{"id":"stacks:0DZR","tag":"0DZR","title":"Smooth morphisms · Lemma 0DZR","summary":"Let f : X → Y be a morphism of algebraic stacks. There is a largest open substack U ⊂ X such that f|_U : U → Y is smooth. Moreover, formation of this open commutes with • precomposing by smooth morphisms, • base change by morphisms which are flat and locally of finite presentation, • base change by flat morphisms provided f is locally of finite presentation.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThere is a largest open substack $\\mathcal{U} \\subset \\mathcal{X}$\nsuch that $f|_\\mathcal{U} : \\mathcal{U} \\to \\mathcal{Y}$ is smooth.\nMoreover, formation of this open commutes with\n\\begin{enumerate}\n\\item precomposing by smooth morphisms,\n\\item base change by morphisms which are flat and locally of\nfinite presentation,\n\\item base change by flat morphisms provided $f$ is locally of\nfinite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZR","source_file":"stacks-morphisms.tex","source_line":7440,"source_end_line":7453,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7440-L7453","statement_sha256":"24ae5317b66e1fa5af7082bb29cc2949b7b168da34247cf0de1df8f1912fd400","origin":"The Stacks Project","memory_eligible":false,"source_rank":14230,"rank":14230,"depth":75,"x":2019.562,"y":1719.612,"cluster":"algebraic-stacks"},{"id":"stacks:0DLS","tag":"0DLS","title":"Smooth morphisms · Lemma 0DLS","summary":"Let X → Y be a smooth morphism of algebraic spaces. Let G be a group algebraic space over Y which is flat and locally of finite presentation over Y. Let G act on X over Y. Then the quotient stack [X/G] is smooth over Y.","statement_latex":"Let $X \\to Y$ be a smooth morphism of algebraic spaces.\nLet $G$ be a group algebraic space over $Y$ which is flat\nand locally of finite presentation over $Y$. Let $G$ act on $X$ over $Y$.\nThen the quotient stack $[X/G]$ is smooth over $Y$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLS","source_file":"stacks-morphisms.tex","source_line":7484,"source_end_line":7490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7484-L7490","statement_sha256":"18b1cb029b6ef3fadfed92b90f1bb43dcb6f1e01948d1d9e3dddc282cb0109d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14231,"rank":14231,"depth":0,"x":1941.429,"y":1505.792,"cluster":"algebraic-stacks"},{"id":"stacks:0DN8","tag":"0DN8","title":"Smooth morphisms · Lemma 0DN8","summary":"Let π : X → Y be a morphism of algebraic stacks. If X is a gerbe over Y, then π is surjective and smooth.","statement_latex":"Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$, then $\\pi$ is surjective\nand smooth.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DN8","source_file":"stacks-morphisms.tex","source_line":7514,"source_end_line":7519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7514-L7519","statement_sha256":"97fb8ade6d3dabd6b5e055e709237ec335ef31d79eaf64ef65ef0a6a6596e952","origin":"The Stacks Project","memory_eligible":false,"source_rank":14232,"rank":14232,"depth":82,"x":2171.198,"y":1619.227,"cluster":"algebraic-stacks"},{"id":"stacks:0CIF","tag":"0CIF","title":"Types of morphisms étale-smooth local on source-and-target · Lemma 0CIF","summary":"Let P be a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target. Let f : X → Y be a DM morphism of algebraic stacks. Consider commutative diagrams xymatrix U ar[d]_a ar[r]_h & V ar[d]^b X ar[r]^f & Y where U and V are algebraic spaces, V → Y is smooth, and U → X ×_Y V is étale. The following are equivalent • for any diagram as above the morphism h has property P, and • for some diagram as above with a : U → X surjective the…","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces\nwhich is \\'etale-smooth local on the source-and-target.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a DM morphism of algebraic stacks.\nConsider commutative diagrams\n$$\n\\xymatrix{\nU \\ar[d]_a \\ar[r]_h & V \\ar[d]^b \\\\\n\\mathcal{X} \\ar[r]^f & \\mathcal{Y}\n}\n$$\nwhere $U$ and $V$ are algebraic spaces, $V \\to \\mathcal{Y}$ is smooth,\nand $U \\to \\mathcal{X} \\times_\\mathcal{Y} V$ is \\'etale.\nThe following are equivalent\n\\begin{enumerate}\n\\item for any diagram as above the morphism $h$ has property $\\mathcal{P}$, and\n\\item for some diagram as above with $a : U \\to \\mathcal{X}$ surjective\nthe morphism $h$ has property $\\mathcal{P}$.\n\\end{enumerate}\nIf $\\mathcal{X}$ and $\\mathcal{Y}$ are representable by algebraic spaces,\nthen this is also equivalent to $f$ (as a morphism of algebraic spaces)\nhaving property $\\mathcal{P}$. If $\\mathcal{P}$ is also preserved under\nany base change, and fppf local on the base, then for morphisms $f$\nwhich are representable by algebraic spaces this\nis also equivalent to $f$ having property $\\mathcal{P}$ in the sense\nof\nProperties of Stacks,\nSection \\ref{stacks-properties-section-properties-morphisms}.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Types of morphisms étale-smooth local on source-and-target","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIF","source_file":"stacks-morphisms.tex","source_line":7551,"source_end_line":7580,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7551-L7580","statement_sha256":"ab5c274f45863d34d2546034b82abdbd46292249412cb8f217df8735673abf5b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14233,"rank":14233,"depth":82,"x":1910.314,"y":1666.0,"cluster":"algebraic-stacks"},{"id":"stacks:0CIG","tag":"0CIG","title":"Types of morphisms étale-smooth local on source-and-target · Definition 0CIG","summary":"Let P be a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target. We say a DM morphism f : X → Y of algebraic stacks has property P if the equivalent conditions of Lemma [Tag 06FM] hold.","statement_latex":"Let $\\mathcal{P}$ be a property of morphisms of algebraic spaces\nwhich is \\'etale-smooth local on the source-and-target.\nWe say a DM morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks\n{\\it has property $\\mathcal{P}$} if the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Types of morphisms étale-smooth local on source-and-target","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIG","source_file":"stacks-morphisms.tex","source_line":7705,"source_end_line":7713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7705-L7713","statement_sha256":"ffc4f139dd14ce5e6ec69ef162826b3f3e5e372c0e17e88131c2a5634101d0ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":14234,"rank":14234,"depth":4,"x":2065.211,"y":1483.315,"cluster":"algebraic-stacks"},{"id":"stacks:0CIL","tag":"0CIL","title":"Étale morphisms · Definition 0CIL","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is étale if f is DM and the equivalent conditions of Lemma [Tag 0CIF] hold with P = etale.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is {\\it \\'etale} if $f$ is DM and the equivalent conditions of\nLemma \\ref{lemma-etale-smooth-local-source-target}\nhold with $\\mathcal{P} = \\etale$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIL","source_file":"stacks-morphisms.tex","source_line":7864,"source_end_line":7870,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7864-L7870","statement_sha256":"b05d5c2bdb7b48cae5c5ad58f34c823d4773b0c4c185767c15fbd1c6d3985ad4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14235,"rank":14235,"depth":83,"x":2097.932,"y":1706.117,"cluster":"algebraic-stacks"},{"id":"stacks:0CIM","tag":"0CIM","title":"Étale morphisms · Lemma 0CIM","summary":"The composition of étale morphisms is étale.","statement_latex":"The composition of \\'etale morphisms is \\'etale.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIM","source_file":"stacks-morphisms.tex","source_line":7878,"source_end_line":7881,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7878-L7881","statement_sha256":"e1ff0e898b5e672a5d7ccfa49954239c207749e1f8efe23b1bb98b84b2d1c602","origin":"The Stacks Project","memory_eligible":false,"source_rank":14236,"rank":14236,"depth":2,"x":1894.451,"y":1560.262,"cluster":"algebraic-stacks"},{"id":"stacks:0CIN","tag":"0CIN","title":"Étale morphisms · Lemma 0CIN","summary":"A base change of an étale morphism is étale.","statement_latex":"A base change of an \\'etale morphism is \\'etale.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIN","source_file":"stacks-morphisms.tex","source_line":7889,"source_end_line":7892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7889-L7892","statement_sha256":"fbfd19c0ed546e400f97007350d47ab9d4211cb04bc86c8361f69bbf8dd5df97","origin":"The Stacks Project","memory_eligible":false,"source_rank":14237,"rank":14237,"depth":47,"x":2162.026,"y":1552.346,"cluster":"algebraic-stacks"},{"id":"stacks:0CIP","tag":"0CIP","title":"Étale morphisms · Lemma 0CIP","summary":"An open immersion is étale.","statement_latex":"An open immersion is \\'etale.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIP","source_file":"stacks-morphisms.tex","source_line":7901,"source_end_line":7904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7901-L7904","statement_sha256":"6c1a5661c1605cfe241811e2dd8acc2563d4410a28afedeafa55e7bdd1066d9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14238,"rank":14238,"depth":61,"x":1970.916,"y":1710.152,"cluster":"algebraic-stacks"},{"id":"stacks:0CIQ","tag":"0CIQ","title":"Étale morphisms · Lemma 0CIQ","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • f is étale, • f is DM and for any morphism V → Y where V is an algebraic space and any étale morphism U → V ×_Y X where U is an algebraic space, the morphism U → V is étale, • there exists some surjective, locally of finite presentation, and flat morphism W → Y where W is an algebraic space and some surjective étale morphism T → W ×_Y X where T is an algebraic space such that the morphism T →…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is \\'etale,\n\\item $f$ is DM and for any morphism $V \\to \\mathcal{Y}$\nwhere $V$ is an algebraic space and any \\'etale morphism\n$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ where $U$ is an algebraic space,\nthe morphism $U \\to V$ is \\'etale,\n\\item there exists some surjective, locally of finite presentation, and flat\nmorphism $W \\to \\mathcal{Y}$ where $W$ is an algebraic space and some\nsurjective \\'etale morphism $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$\nwhere $T$ is an algebraic space such that the morphism $T \\to W$ is \\'etale.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIQ","source_file":"stacks-morphisms.tex","source_line":7917,"source_end_line":7932,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7917-L7932","statement_sha256":"0b6c1784d2569463049480b31982eb78f0a23b9ab573a3f2b6241d528ea82de7","origin":"The Stacks Project","memory_eligible":false,"source_rank":14239,"rank":14239,"depth":84,"x":1984.946,"y":1485.148,"cluster":"algebraic-stacks"},{"id":"stacks:0CIR","tag":"0CIR","title":"Étale morphisms · Lemma 0CIR","summary":"Let X, Y be algebraic stacks étale over an algebraic stack Z. Any morphism X → Y over Z is étale.","statement_latex":"Let $\\mathcal{X}, \\mathcal{Y}$ be algebraic stacks \\'etale over\nan algebraic stack $\\mathcal{Z}$. Any morphism\n$\\mathcal{X} \\to \\mathcal{Y}$ over $\\mathcal{Z}$ is \\'etale.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Étale morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIR","source_file":"stacks-morphisms.tex","source_line":7984,"source_end_line":7989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L7984-L7989","statement_sha256":"465542699d43068755730e0a97df18edc434cf114d1dd61e18982c3715a4ed24","origin":"The Stacks Project","memory_eligible":false,"source_rank":14240,"rank":14240,"depth":82,"x":2155.694,"y":1659.179,"cluster":"algebraic-stacks"},{"id":"stacks:0CIT","tag":"0CIT","title":"Unramified morphisms · Definition 0CIT","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is unramified if f is DM and the equivalent conditions of Lemma [Tag 0CIF] hold with P =\"unramified\".","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is {\\it unramified} if $f$ is DM and the equivalent conditions of\nLemma \\ref{lemma-etale-smooth-local-source-target}\nhold with $\\mathcal{P} =$``unramified''.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIT","source_file":"stacks-morphisms.tex","source_line":8059,"source_end_line":8065,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8059-L8065","statement_sha256":"f78242ffdb6316f74a7fab72c6311062bd5eec11bc71383b822e2d58c6f1cfa7","origin":"The Stacks Project","memory_eligible":false,"source_rank":14241,"rank":14241,"depth":83,"x":1889.601,"y":1627.709,"cluster":"algebraic-stacks"},{"id":"stacks:0CIU","tag":"0CIU","title":"Unramified morphisms · Lemma 0CIU","summary":"The composition of unramified morphisms is unramified.","statement_latex":"The composition of unramified morphisms is unramified.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIU","source_file":"stacks-morphisms.tex","source_line":8073,"source_end_line":8076,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8073-L8076","statement_sha256":"48d704be9aec3ae374190c7fb628f18f90d875388af9d0dc01808bb62aa080cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14242,"rank":14242,"depth":8,"x":2111.317,"y":1499.813,"cluster":"algebraic-stacks"},{"id":"stacks:0CIV","tag":"0CIV","title":"Unramified morphisms · Lemma 0CIV","summary":"A base change of an unramified morphism is unramified.","statement_latex":"A base change of an unramified morphism is unramified.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIV","source_file":"stacks-morphisms.tex","source_line":8084,"source_end_line":8087,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8084-L8087","statement_sha256":"05ea2a4842108d4080f35a5553e828f126aa47b501c7979f0b11aef5744e2f31","origin":"The Stacks Project","memory_eligible":false,"source_rank":14243,"rank":14243,"depth":8,"x":2050.624,"y":1720.124,"cluster":"algebraic-stacks"},{"id":"stacks:0CIW","tag":"0CIW","title":"Unramified morphisms · Lemma 0CIW","summary":"An étale morphism is unramified.","statement_latex":"An \\'etale morphism is unramified.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIW","source_file":"stacks-morphisms.tex","source_line":8096,"source_end_line":8099,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8096-L8099","statement_sha256":"2ef1ba20ed831d88e6c511f18a2a3ae3b12524d41556fe128cffd80898230d4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14244,"rank":14244,"depth":48,"x":1918.094,"y":1523.057,"cluster":"algebraic-stacks"},{"id":"stacks:0CIX","tag":"0CIX","title":"Unramified morphisms · Lemma 0CIX","summary":"An immersion is unramified.","statement_latex":"An immersion is unramified.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIX","source_file":"stacks-morphisms.tex","source_line":8107,"source_end_line":8110,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8107-L8110","statement_sha256":"bd086e63d472c5edeaf0c5e5013f4db672b59de7f1d2f3e403bade37e05ec495","origin":"The Stacks Project","memory_eligible":false,"source_rank":14245,"rank":14245,"depth":61,"x":2174.52,"y":1593.232,"cluster":"algebraic-stacks"},{"id":"stacks:0CIY","tag":"0CIY","title":"Unramified morphisms · Lemma 0CIY","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • f is unramified, • f is DM and for any morphism V → Y where V is an algebraic space and any étale morphism U → V ×_Y X where U is an algebraic space, the morphism U → V is unramified, • there exists some surjective, locally of finite presentation, and flat morphism W → Y where W is an algebraic space and some surjective étale morphism T → W ×_Y X where T is an algebraic space such that the…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is unramified,\n\\item $f$ is DM and for any morphism $V \\to \\mathcal{Y}$\nwhere $V$ is an algebraic space and any \\'etale morphism\n$U \\to V \\times_\\mathcal{Y} \\mathcal{X}$ where $U$ is an algebraic space,\nthe morphism $U \\to V$ is unramified,\n\\item there exists some surjective, locally of finite presentation, and flat\nmorphism $W \\to \\mathcal{Y}$ where $W$ is an algebraic space and some\nsurjective \\'etale morphism $T \\to W \\times_\\mathcal{Y} \\mathcal{X}$\nwhere $T$ is an algebraic space such that the morphism $T \\to W$ is unramified.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIY","source_file":"stacks-morphisms.tex","source_line":8124,"source_end_line":8139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8124-L8139","statement_sha256":"25cd35939a179eaa568bbabcd91f410caa74bd759490a472b2b06204a668007d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14246,"rank":14246,"depth":84,"x":1928.787,"y":1687.071,"cluster":"algebraic-stacks"},{"id":"stacks:0H2Z","tag":"0H2Z","title":"Unramified morphisms · Lemma 0H2Z","summary":"An unramified morphism of algebraic stacks is locally quasi-finite.","statement_latex":"An unramified morphism of algebraic stacks is locally quasi-finite.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2Z","source_file":"stacks-morphisms.tex","source_line":8191,"source_end_line":8194,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8191-L8194","statement_sha256":"097163d3bb775d012ca3c2eef52df0bfd1200d945cb59b4d27c7dd7fbbc7a711","origin":"The Stacks Project","memory_eligible":false,"source_rank":14247,"rank":14247,"depth":85,"x":2034.617,"y":1478.258,"cluster":"algebraic-stacks"},{"id":"stacks:0CIZ","tag":"0CIZ","title":"Unramified morphisms · Lemma 0CIZ","summary":"Let X → Y → Z be morphisms of algebraic stacks. If X → Z is unramified and Y → Z is DM, then X → Y is unramified.","statement_latex":"Let $\\mathcal{X} \\to \\mathcal{Y} \\to \\mathcal{Z}$ be\nmorphisms of algebraic stacks.\nIf $\\mathcal{X} \\to \\mathcal{Z}$ is unramified and\n$\\mathcal{Y} \\to \\mathcal{Z}$ is DM, then\n$\\mathcal{X} \\to \\mathcal{Y}$ is unramified.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CIZ","source_file":"stacks-morphisms.tex","source_line":8206,"source_end_line":8213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8206-L8213","statement_sha256":"e295912927249ec94e90a68a1d45abbe982cc361d0d5e6b34a0d9dc03327bd55","origin":"The Stacks Project","memory_eligible":false,"source_rank":14248,"rank":14248,"depth":82,"x":2124.58,"y":1692.471,"cluster":"algebraic-stacks"},{"id":"stacks:0CJ0","tag":"0CJ0","title":"Unramified morphisms · Lemma 0CJ0","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • f is unramified, and • f is locally of finite type and its diagonal is étale.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is unramified, and\n\\item $f$ is locally of finite type and its diagonal is \\'etale.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJ0","source_file":"stacks-morphisms.tex","source_line":8236,"source_end_line":8244,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8236-L8244","statement_sha256":"ff41a36b82d6d486e18f1846ffa821fcba44481fd033e9469fb7a600b2205fc6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14249,"rank":14249,"depth":83,"x":1885.769,"y":1585.467,"cluster":"algebraic-stacks"},{"id":"stacks:0CJ1","tag":"0CJ1","title":"Unramified morphisms · Lemma 0CJ1","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • f is étale, and • f is locally of finite presentation, flat, and unramified, • f is locally of finite presentation, flat, and its diagonal is étale.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is \\'etale, and\n\\item $f$ is locally of finite presentation, flat, and unramified,\n\\item $f$ is locally of finite presentation, flat, and its diagonal\nis \\'etale.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Unramified morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJ1","source_file":"stacks-morphisms.tex","source_line":8325,"source_end_line":8335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8325-L8335","statement_sha256":"056f90a193610330c2ab03bc34f887112238a3c35d0a80a0abb86cf71d3dae59","origin":"The Stacks Project","memory_eligible":false,"source_rank":14250,"rank":14250,"depth":84,"x":2148.143,"y":1528.814,"cluster":"algebraic-stacks"},{"id":"stacks:0CL5","tag":"0CL5","title":"Proper morphisms · Definition 0CL5","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is proper if f is separated, finite type, and universally closed.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$\nbe a morphism of algebraic stacks.\nWe say $f$ is {\\it proper} if $f$ is separated, finite type, and\nuniversally closed.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Proper morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CL5","source_file":"stacks-morphisms.tex","source_line":8370,"source_end_line":8376,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8370-L8376","statement_sha256":"931e5f7fd8552c25267cf17b69310b2d43c4a2c6d4efdd11e8311f47cba41bd0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14251,"rank":14251,"depth":0,"x":2000.103,"y":1719.633,"cluster":"algebraic-stacks"},{"id":"stacks:0CL6","tag":"0CL6","title":"Proper morphisms · Lemma 0CL6","summary":"A base change of a proper morphism is proper.","statement_latex":"A base change of a proper morphism is proper.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CL6","source_file":"stacks-morphisms.tex","source_line":8397,"source_end_line":8400,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8397-L8400","statement_sha256":"d30cd8ba74796aa24cebe9df0b79419746fe7185db40fc848efc7e9bf749daa1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14252,"rank":14252,"depth":8,"x":1955.776,"y":1494.729,"cluster":"algebraic-stacks"},{"id":"stacks:0CL7","tag":"0CL7","title":"Proper morphisms · Lemma 0CL7","summary":"A composition of proper morphisms is proper.","statement_latex":"A composition of proper morphisms is proper.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CL7","source_file":"stacks-morphisms.tex","source_line":8409,"source_end_line":8412,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8409-L8412","statement_sha256":"a2215eeaf72a90343a8435f29bbba8328292bda0f1848d568ab048e6c11c78d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14253,"rank":14253,"depth":61,"x":2169.509,"y":1635.54,"cluster":"algebraic-stacks"},{"id":"stacks:0CL8","tag":"0CL8","title":"Proper morphisms · Lemma 0CL8","summary":"A closed immersion of algebraic stacks is a proper morphism of algebraic stacks.","statement_latex":"A closed immersion of algebraic stacks is a proper morphism of\nalgebraic stacks.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CL8","source_file":"stacks-morphisms.tex","source_line":8421,"source_end_line":8425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8421-L8425","statement_sha256":"548d57ade0f4e3f5026b6d3c3589bddc1cb95b430bafeb3dfd15e3922761758c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14254,"rank":14254,"depth":16,"x":1898.435,"y":1653.0,"cluster":"algebraic-stacks"},{"id":"stacks:0CPT","tag":"0CPT","title":"Proper morphisms · Lemma 0CPT","summary":"Consider a commutative diagram xymatrix X ar[rr] ar[rd] & & Y ar[ld] & Z & of algebraic stacks. • If X → Z is universally closed and Y → Z is separated, then the morphism X → Y is universally closed. In particular, the image of |X| in |Y| is closed. • If X → Z is proper and Y → Z is separated, then the morphism X → Y is proper.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[rr] \\ar[rd] & &\n\\mathcal{Y} \\ar[ld] \\\\\n& \\mathcal{Z} &\n}\n$$\nof algebraic stacks.\n\\begin{enumerate}\n\\item If $\\mathcal{X} \\to \\mathcal{Z}$ is universally closed and\n$\\mathcal{Y} \\to \\mathcal{Z}$ is separated,\nthen the morphism $\\mathcal{X} \\to \\mathcal{Y}$ is universally closed.\nIn particular, the image of $|\\mathcal{X}|$ in $|\\mathcal{Y}|$ is closed.\n\\item If $\\mathcal{X} \\to \\mathcal{Z}$ is proper and\n$\\mathcal{Y} \\to \\mathcal{Z}$ is separated, then\nthe morphism $\\mathcal{X} \\to \\mathcal{Y}$ is proper.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPT","source_file":"stacks-morphisms.tex","source_line":8439,"source_end_line":8459,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8439-L8459","statement_sha256":"69ba3bab1a96d567aba15d8896a1d732b93e2ed091e0c6ffa39d93ceca1d80c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14255,"rank":14255,"depth":63,"x":2084.44,"y":1486.167,"cluster":"algebraic-stacks"},{"id":"stacks:0CQK","tag":"0CQK","title":"Proper morphisms · Lemma 0CQK","summary":"Let Z be an algebraic stack. Let f : X → Y be a morphism of algebraic stacks over Z. If X is universally closed over Z and f is surjective then Y is universally closed over Z. In particular, if also Y is separated and of finite type over Z, then Y is proper over Z.","statement_latex":"Let $\\mathcal{Z}$ be an algebraic stack.\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$\nbe a morphism of algebraic stacks over $\\mathcal{Z}$.\nIf $\\mathcal{X}$ is universally closed over $\\mathcal{Z}$\nand $f$ is surjective then $\\mathcal{Y}$\nis universally closed over $\\mathcal{Z}$.\nIn particular, if also $\\mathcal{Y}$ is\nseparated and of finite type over $\\mathcal{Z}$,\nthen $\\mathcal{Y}$ is proper over $\\mathcal{Z}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Proper morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQK","source_file":"stacks-morphisms.tex","source_line":8481,"source_end_line":8492,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8481-L8492","statement_sha256":"98dea7ca72329a9f41bd5f3856ac30e551be081171103703c8f8e5a662f1524d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14256,"rank":14256,"depth":2,"x":2081.442,"y":1714.928,"cluster":"algebraic-stacks"},{"id":"stacks:0CMI","tag":"0CMI","title":"Scheme theoretic image · Definition 0CMI","summary":"Let f : X → Y be a morphism of algebraic stacks. The scheme theoretic image of f is the smallest closed substack Z ⊂ Y through which f factors.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe {\\it scheme theoretic image} of $f$ is the smallest closed substack\n$\\mathcal{Z} \\subset \\mathcal{Y}$ through which $f$\nfactors\\footnote{We will see in\nLemma \\ref{lemma-scheme-theoretic-image-existence}\nthat the scheme theoretic image always exists.}.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Scheme theoretic image","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMI","source_file":"stacks-morphisms.tex","source_line":8522,"source_end_line":8530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8522-L8530","statement_sha256":"5a0fcb96d10acc149ba639d45657f71b9160ea06efca0799a8d918453b6a3d98","origin":"The Stacks Project","memory_eligible":false,"source_rank":14257,"rank":14257,"depth":0,"x":1899.533,"y":1544.395,"cluster":"algebraic-stacks"},{"id":"stacks:0CMJ","tag":"0CMJ","title":"Scheme theoretic image · Lemma 0CMJ","summary":"Let f : X → Y be a morphism of algebraic stacks. Let g : W → X be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Then the scheme theoretic image of f exists if and only if the scheme theoretic image of f ∘ g exists and if so then these scheme theoretic images are the same.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $g : \\mathcal{W} \\to \\mathcal{X}$ be a morphism of algebraic stacks\nwhich is surjective, flat, and locally of finite presentation.\nThen the scheme theoretic image of $f$ exists if and only if the\nscheme theoretic image of $f \\circ g$ exists and if so then these\nscheme theoretic images are the same.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMJ","source_file":"stacks-morphisms.tex","source_line":8540,"source_end_line":8548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8540-L8548","statement_sha256":"10ae9566d271b49c493a051d86f47bf470e60c60f175781c4a5fdfcf6c67c633","origin":"The Stacks Project","memory_eligible":false,"source_rank":14258,"rank":14258,"depth":0,"x":2171.041,"y":1566.944,"cluster":"algebraic-stacks"},{"id":"stacks:0CPU","tag":"0CPU","title":"Scheme theoretic image · Lemma 0CPU","summary":"Let f : Y → X be a morphism of algebraic stacks. Then the scheme theoretic image of f exists.","statement_latex":"Let $f : \\mathcal{Y} \\to \\mathcal{X}$ be a morphism of algebraic stacks.\nThen the scheme theoretic image of $f$ exists.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPU","source_file":"stacks-morphisms.tex","source_line":8569,"source_end_line":8573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8569-L8573","statement_sha256":"16668b205ef4a7eb690cecef809f34855f1a4c2d2b5f23a1a172cabf7d944722","origin":"The Stacks Project","memory_eligible":false,"source_rank":14259,"rank":14259,"depth":74,"x":1952.515,"y":1704.495,"cluster":"algebraic-stacks"},{"id":"stacks:0CPV","tag":"0CPV","title":"Scheme theoretic image · Lemma 0CPV","summary":"Let xymatrix X_1 ar[d] ar[r]_f_1 & Y_1 ar[d] X_2 ar[r]^f_2 & Y_2 be a commutative diagram of algebraic stacks. Let Z_i ⊂ Y_i, i = 1, 2 be the scheme theoretic image of f_i. Then the morphism Y_1 → Y_2 induces a morphism Z_1 → Z_2 and a commutative diagram xymatrix X_1 ar[r] ar[d] & Z_1 ar[d] ar[r] & Y_1 ar[d] X_2 ar[r] & Z_2 ar[r] & Y_2","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{X}_1 \\ar[d] \\ar[r]_{f_1} & \\mathcal{Y}_1 \\ar[d] \\\\\n\\mathcal{X}_2 \\ar[r]^{f_2} & \\mathcal{Y}_2\n}\n$$\nbe a commutative diagram of algebraic stacks.\nLet $\\mathcal{Z}_i \\subset \\mathcal{Y}_i$, $i = 1, 2$ be\nthe scheme theoretic image of $f_i$. Then the morphism\n$\\mathcal{Y}_1 \\to \\mathcal{Y}_2$ induces a morphism\n$\\mathcal{Z}_1 \\to \\mathcal{Z}_2$ and a\ncommutative diagram\n$$\n\\xymatrix{\n\\mathcal{X}_1 \\ar[r] \\ar[d] &\n\\mathcal{Z}_1 \\ar[d] \\ar[r] &\n\\mathcal{Y}_1 \\ar[d] \\\\\n\\mathcal{X}_2 \\ar[r] &\n\\mathcal{Z}_2 \\ar[r] &\n\\mathcal{Y}_2\n}\n$$","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPV","source_file":"stacks-morphisms.tex","source_line":8637,"source_end_line":8662,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8637-L8662","statement_sha256":"2d627e606aaa64967f0b77503dabcae92f06f3d8023fae9dfd105845dc4077e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14260,"rank":14260,"depth":0,"x":2003.081,"y":1478.877,"cluster":"algebraic-stacks"},{"id":"stacks:0CMK","tag":"0CMK","title":"Scheme theoretic image · Lemma 0CMK","summary":"Let f : X → Y be a quasi-compact morphism of algebraic stacks. Then formation of the scheme theoretic image commutes with flat base change.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact\nmorphism of algebraic stacks. Then formation of the scheme theoretic image\ncommutes with flat base change.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMK","source_file":"stacks-morphisms.tex","source_line":8671,"source_end_line":8676,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8671-L8676","statement_sha256":"c9c96a329f8176d030881bc5008392c21ff4c5ad5897171874d7846460da7139","origin":"The Stacks Project","memory_eligible":false,"source_rank":14261,"rank":14261,"depth":74,"x":2147.354,"y":1674.103,"cluster":"algebraic-stacks"},{"id":"stacks:0CML","tag":"0CML","title":"Scheme theoretic image · Lemma 0CML","summary":"Let f : X → Y be a quasi-compact morphism of algebraic stacks. Let Z ⊂ Y be the scheme theoretic image of f. Then |Z| is the closure of the image of |f|.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact\nmorphism of algebraic stacks. Let $\\mathcal{Z} \\subset \\mathcal{Y}$\nbe the scheme theoretic image of $f$. Then $|\\mathcal{Z}|$\nis the closure of the image of $|f|$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CML","source_file":"stacks-morphisms.tex","source_line":8773,"source_end_line":8779,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8773-L8779","statement_sha256":"08c023aebffc9d16ba1294921890dbe2947c1b64d88d9cce8497da5bb37af365","origin":"The Stacks Project","memory_eligible":false,"source_rank":14262,"rank":14262,"depth":75,"x":1883.748,"y":1611.957,"cluster":"algebraic-stacks"},{"id":"stacks:0CPW","tag":"0CPW","title":"Scheme theoretic image · Lemma 0CPW","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces and separated. Let V ⊂ Y be an open substack such that V → Y is quasi-compact. Let s : V → X be a morphism such that f ∘ s = id_V. Let Y' be the scheme theoretic image of s. Then Y' → Y is an isomorphism over V.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich is representable by algebraic spaces and separated.\nLet $\\mathcal{V} \\subset \\mathcal{Y}$ be an open substack such that\n$\\mathcal{V} \\to \\mathcal{Y}$ is quasi-compact.\nLet $s : \\mathcal{V} \\to \\mathcal{X}$ be a morphism such that\n$f \\circ s = \\text{id}_\\mathcal{V}$.\nLet $\\mathcal{Y}'$ be the scheme theoretic image of $s$.\nThen $\\mathcal{Y}' \\to \\mathcal{Y}$ is an isomorphism over $\\mathcal{V}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Scheme theoretic image","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPW","source_file":"stacks-morphisms.tex","source_line":8805,"source_end_line":8815,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8805-L8815","statement_sha256":"cea61faad5914c3d6b654ca28270ca558a6589a580e137f27906539d52908415","origin":"The Stacks Project","memory_eligible":false,"source_rank":14263,"rank":14263,"depth":75,"x":2128.312,"y":1508.118,"cluster":"algebraic-stacks"},{"id":"stacks:0CLA","tag":"0CLA","title":"Valuative criteria · Definition 0CLA","summary":"Let f : X → Y be a morphism of algebraic stacks. Consider a 2-commutative solid diagram vcenter xymatrix Spec(K) ar[r]_-x ar[d]_j & X ar[d]^f Spec(A) ar[r]^-y ar@..>[ru] & Y where A is a valuation ring with field of fractions K. Let γ : y ∘ j → f ∘ x be a 2-morphism witnessing the 2-commutativity of the diagram. (Notation as in Categories, Sections [Tag 003D] and [Tag 003G].) Given ([Tag 0CLB]) and γ a dotted arrow is a triple (a, α, β) consisting of a morphism a :…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nConsider a $2$-commutative solid diagram\n\\begin{equation}\n\n\\vcenter{\n\\xymatrix{\n\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[r]^-y \\ar@{..>}[ru] & \\mathcal{Y}\n}\n}\n\\end{equation}\nwhere $A$ is a valuation ring with field of fractions $K$. Let\n$$\n\\gamma : y \\circ j \\longrightarrow f \\circ x\n$$\nbe a $2$-morphism witnessing the $2$-commutativity of the diagram.\n(Notation as in Categories, Sections \\ref{categories-section-formal-cat-cat}\nand \\ref{categories-section-2-categories}.)\nGiven (\\ref{equation-diagram}) and $\\gamma$\na {\\it dotted arrow} is a triple $(a, \\alpha, \\beta)$ consisting of a\nmorphism $a : \\Spec(A) \\to \\mathcal{X}$ and $2$-arrows\n$\\alpha : a \\circ j \\to x$, $\\beta : y \\to f \\circ a$\nsuch that\n$\\gamma = (\\text{id}_f \\star \\alpha) \\circ (\\beta \\star \\text{id}_j)$,\nin other words such that\n$$\n\\xymatrix{\n& f \\circ a \\circ j \\ar[rd]^{\\text{id}_f \\star \\alpha} \\\\\ny \\circ j \\ar[ru]^{\\beta \\star \\text{id}_j} \\ar[rr]^\\gamma & &\nf \\circ x\n}\n$$\nis commutative. A {\\it morphism of dotted arrows}\n$(a, \\alpha, \\beta) \\to (a', \\alpha', \\beta')$ is a\n$2$-arrow $\\theta : a \\to a'$ such that\n$\\alpha = \\alpha' \\circ (\\theta \\star \\text{id}_j)$ and\n$\\beta' = (\\text{id}_f \\star \\theta) \\circ \\beta$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLA","source_file":"stacks-morphisms.tex","source_line":8847,"source_end_line":8886,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8847-L8886","statement_sha256":"1aeeab629f6481a66e86634bbb3a01cb727c7562042f116ab6a55d9ff98173ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":14264,"rank":14264,"depth":0,"x":2031.396,"y":1723.642,"cluster":"algebraic-stacks"},{"id":"stacks:0CLC","tag":"0CLC","title":"Valuative criteria · Lemma 0CLC","summary":"In the situation of Definition [Tag 0CLA] the category of dotted arrows is a groupoid. If Δ_f is separated, then it is a setoid.","statement_latex":"In the situation of Definition \\ref{definition-fill-in-diagram}\nthe category of dotted arrows is a groupoid. If $\\Delta_f$\nis separated, then it is a setoid.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLC","source_file":"stacks-morphisms.tex","source_line":8901,"source_end_line":8906,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8901-L8906","statement_sha256":"7be6c40287fd00fd5c37cd9281bb6597f951dc4bb1cea8558aea1c686aaf9010","origin":"The Stacks Project","memory_eligible":false,"source_rank":14265,"rank":14265,"depth":60,"x":1929.456,"y":1509.544,"cluster":"algebraic-stacks"},{"id":"stacks:0CLD","tag":"0CLD","title":"Valuative criteria · Lemma 0CLD","summary":"In Definition [Tag 0CLA] assume I_Y → Y is proper (for example if Y is separated or if Y is separated over an algebraic space). Then the category of dotted arrows is independent (up to noncanonical equivalence) of the choice of γ and the existence of a dotted arrow (for some and hence equivalently all γ) is equivalent to the existence of a diagram xymatrix Spec(K) ar[r]_-x ar[d]_j & X ar[d]^f Spec(A) ar[r]^-y ar[ru]_a & Y with 2-commutative triangles (without checking the…","statement_latex":"In Definition \\ref{definition-fill-in-diagram}\nassume $\\mathcal{I}_\\mathcal{Y} \\to \\mathcal{Y}$ is proper\n(for example if $\\mathcal{Y}$ is separated or if $\\mathcal{Y}$\nis separated over an algebraic space). Then the category of dotted arrows\nis independent (up to noncanonical equivalence) of the choice of $\\gamma$\nand the existence of a dotted arrow\n(for some and hence equivalently all $\\gamma$)\nis equivalent to the existence of a diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[r]^-y \\ar[ru]_a & \\mathcal{Y}\n}\n$$\nwith $2$-commutative triangles\n(without checking the $2$-morphisms compose correctly).","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLD","source_file":"stacks-morphisms.tex","source_line":8937,"source_end_line":8955,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8937-L8955","statement_sha256":"19d47fe9166a665c4ff74aa585604ee8dad7b5db5411dd630322d0e62b58da7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14266,"rank":14266,"depth":62,"x":2177.006,"y":1609.658,"cluster":"algebraic-stacks"},{"id":"stacks:0CLE","tag":"0CLE","title":"Valuative criteria · Lemma 0CLE","summary":"Assume given a 2-commutative diagram xymatrix Spec(K) ar[r]_-x' ar[d]_j & X' ar[d]^p ar[r]_q & X ar[d]^f Spec(A) ar[r]^-y' & Y' ar[r]^g & Y with the right square 2-cartesian. Choose a 2-arrow γ' : y' ∘ j → p ∘ x'. Set x = q ∘ x', y = g ∘ y' and let γ : y ∘ j → f ∘ x be the composition of γ' with the 2-arrow implicit in the 2-commutativity of the right square. Then the category of dotted arrows for the left square and γ' is equivalent to the category of dotted arrows for…","statement_latex":"Assume given a $2$-commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_-{x'} \\ar[d]_j &\n\\mathcal{X}' \\ar[d]^p \\ar[r]_q &\n\\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[r]^-{y'} &\n\\mathcal{Y}' \\ar[r]^g &\n\\mathcal{Y}\n}\n$$\nwith the right square $2$-cartesian. Choose a $2$-arrow\n$\\gamma' : y' \\circ j \\to p \\circ x'$. Set\n$x = q \\circ x'$, $y = g \\circ y'$ and let\n$\\gamma : y \\circ j \\to f \\circ x$ be the composition of\n$\\gamma'$ with the $2$-arrow implicit in the $2$-commutativity\nof the right square. Then the category of dotted arrows\nfor the left square and $\\gamma'$ is equivalent to the category of dotted\narrows for the outer rectangle and $\\gamma$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLE","source_file":"stacks-morphisms.tex","source_line":8973,"source_end_line":8994,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L8973-L8994","statement_sha256":"18a0fa57a7afc68c5a60cdacadcd95f000cb9c09f0fa469425b404a351f1d39a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14267,"rank":14267,"depth":63,"x":1913.735,"y":1676.358,"cluster":"algebraic-stacks"},{"id":"stacks:0CLF","tag":"0CLF","title":"Valuative criteria · Lemma 0CLF","summary":"Assume given a 2-commutative diagram xymatrix Spec(K) ar[r]_-x ar[dd]_j & X ar[d]^f & Y ar[d]^g Spec(A) ar[r]^-z & Z Choose a 2-arrow γ : z ∘ j → g ∘ f ∘ x. Let C be the category of dotted arrows for the outer rectangle and γ. Let C' be the category of dotted arrows for the square xymatrix Spec(K) ar[r]_-f ∘ x ar[d]_j & Y ar[d]^g Spec(A) ar[r]^-z & Z and γ. Then C is equivalent to a category C\" which has the following property: there is a functor C\" → C' which turns C\"…","statement_latex":"Assume given a $2$-commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_-x \\ar[dd]_j & \\mathcal{X} \\ar[d]^f \\\\\n& \\mathcal{Y} \\ar[d]^g \\\\\n\\Spec(A) \\ar[r]^-z & \\mathcal{Z}\n}\n$$\nChoose a $2$-arrow $\\gamma : z \\circ j \\to g \\circ f \\circ x$.\nLet $\\mathcal{C}$ be the category of dotted arrows for\nthe outer rectangle and $\\gamma$. Let $\\mathcal{C}'$ be the\ncategory of dotted arrows for the square\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_-{f \\circ x} \\ar[d]_j & \\mathcal{Y} \\ar[d]^g \\\\\n\\Spec(A) \\ar[r]^-z & \\mathcal{Z}\n}\n$$\nand $\\gamma$. Then $\\mathcal{C}$ is equivalent to a category $\\mathcal{C}''$ \nwhich has the following property: there is \na functor $\\mathcal{C}'' \\to \\mathcal{C}'$\nwhich turns $\\mathcal{C}''$ into a category fibred in groupoids over\n$\\mathcal{C}'$ and whose fibre categories are categories of dotted arrows\nfor certain squares of the form\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[r]^-y & \\mathcal{Y}\n}\n$$\nand some choices of $y \\circ j \\to f \\circ x$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLF","source_file":"stacks-morphisms.tex","source_line":9017,"source_end_line":9050,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9017-L9050","statement_sha256":"1480a2b25bc38ea5e4cb191b6a5eddf0602b9d73da8d1175f32ec5b3dbad5cf4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14268,"rank":14268,"depth":1,"x":2054.348,"y":1477.62,"cluster":"algebraic-stacks"},{"id":"stacks:0CLG","tag":"0CLG","title":"Valuative criteria · Definition 0CLG","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f satisfies the uniqueness part of the valuative criterion if for every diagram ([Tag 0CLB]) and γ as in Definition [Tag 0CLA] the category of dotted arrows is either empty or a setoid with exactly one isomorphism class.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ satisfies the {\\it uniqueness part of the valuative criterion}\nif for every diagram (\\ref{equation-diagram}) and $\\gamma$\nas in Definition \\ref{definition-fill-in-diagram}\nthe category of dotted arrows is either empty or\na setoid with exactly one isomorphism class.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLG","source_file":"stacks-morphisms.tex","source_line":9057,"source_end_line":9065,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9057-L9065","statement_sha256":"57c4ad6ea20826ff24f57523088a85d884a49e9c219603c49924881f9cd33b0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14269,"rank":14269,"depth":1,"x":2110.528,"y":1704.144,"cluster":"algebraic-stacks"},{"id":"stacks:0CLH","tag":"0CLH","title":"Valuative criteria · Lemma 0CLH","summary":"The base change of a morphism of algebraic stacks which satisfies the uniqueness part of the valuative criterion by any morphism of algebraic stacks is a morphism of algebraic stacks which satisfies the uniqueness part of the valuative criterion.","statement_latex":"The base change of a morphism of algebraic stacks which satisfies the\nuniqueness part of the valuative criterion by any morphism of\nalgebraic stacks is a morphism of algebraic stacks which satisfies the\nuniqueness part of the valuative criterion.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLH","source_file":"stacks-morphisms.tex","source_line":9067,"source_end_line":9073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9067-L9073","statement_sha256":"ca73e63c37898fb5dcbbb06c2f4b7885288f1596e07ac6ee303a47403a30c261","origin":"The Stacks Project","memory_eligible":false,"source_rank":14270,"rank":14270,"depth":64,"x":1886.749,"y":1568.873,"cluster":"algebraic-stacks"},{"id":"stacks:0CLI","tag":"0CLI","title":"Valuative criteria · Lemma 0CLI","summary":"The composition of morphisms of algebraic stacks which satisfy the uniqueness part of the valuative criterion is another morphism of algebraic stacks which satisfies the uniqueness part of the valuative criterion.","statement_latex":"The composition of morphisms of algebraic stacks which satisfy the\nuniqueness part of the valuative criterion is another\nmorphism of algebraic stacks which satisfies the\nuniqueness part of the valuative criterion.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLI","source_file":"stacks-morphisms.tex","source_line":9080,"source_end_line":9086,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9080-L9086","statement_sha256":"2f0ace1f40b2f87fc6e2272cb0fa6edaccabffe8364d69db8eec8de461587203","origin":"The Stacks Project","memory_eligible":false,"source_rank":14271,"rank":14271,"depth":2,"x":2160.772,"y":1541.619,"cluster":"algebraic-stacks"},{"id":"stacks:0CLJ","tag":"0CLJ","title":"Valuative criteria · Lemma 0CLJ","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent • f satisfies the uniqueness part of the valuative criterion, • for every scheme T and morphism T → Y the morphism X ×_Y T → T satisfies the uniqueness part of the valuative criterion as a morphism of algebraic spaces.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich is representable by algebraic spaces. Then the following are equivalent\n\\begin{enumerate}\n\\item $f$ satisfies the uniqueness part of the valuative criterion,\n\\item for every scheme $T$ and morphism $T \\to \\mathcal{Y}$\nthe morphism $\\mathcal{X} \\times_\\mathcal{Y} T \\to T$ satisfies\nthe uniqueness part of the valuative criterion as a morphism\nof algebraic spaces.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLJ","source_file":"stacks-morphisms.tex","source_line":9093,"source_end_line":9104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9093-L9104","statement_sha256":"e3db5294060e37d432ff15b51ced35b17274b29737fb58d789ded4d54ad01c24","origin":"The Stacks Project","memory_eligible":false,"source_rank":14272,"rank":14272,"depth":64,"x":1980.477,"y":1717.35,"cluster":"algebraic-stacks"},{"id":"stacks:0CLK","tag":"0CLK","title":"Valuative criteria · Definition 0CLK","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f satisfies the existence part of the valuative criterion if for every diagram ([Tag 0CLB]) and γ as in Definition [Tag 0CLA] there exists an extension K'/K of fields, a valuation ring A' ⊂ K' dominating A such that the category of dotted arrows for the outer rectangle of the diagram xymatrix Spec(K') ar[r] ar@/^2em/[rr]_x' ar[d]_j' & Spec(K) ar[d]_j ar[r]_-x & X ar[d]^f Spec(A') ar[r] ar@/_2em/[rr]^y' & Spec(A)…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ satisfies the {\\it existence part of the valuative criterion}\nif for every diagram (\\ref{equation-diagram}) and $\\gamma$\nas in Definition \\ref{definition-fill-in-diagram}\nthere exists an extension $K'/K$ of fields, a valuation ring $A' \\subset K'$\ndominating $A$ such that the category of dotted arrows for the\nouter rectangle of the diagram\n$$\n\\xymatrix{\n\\Spec(K') \\ar[r] \\ar@/^2em/[rr]_{x'} \\ar[d]_{j'} &\n\\Spec(K) \\ar[d]_j \\ar[r]_-x &\n\\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A') \\ar[r] \\ar@/_2em/[rr]^{y'} &\n\\Spec(A) \\ar[r]^-y &\n\\mathcal{Y}\n}\n$$\nwith induced $2$-arrow $\\gamma' : y' \\circ j' \\to f \\circ x'$ is nonempty.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLK","source_file":"stacks-morphisms.tex","source_line":9111,"source_end_line":9131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9111-L9131","statement_sha256":"1b274939f2d6919bb78a567bc1824b220dc7be3bdd87f7aaba31714363271790","origin":"The Stacks Project","memory_eligible":false,"source_rank":14273,"rank":14273,"depth":1,"x":1972.099,"y":1485.272,"cluster":"algebraic-stacks"},{"id":"stacks:0CLL","tag":"0CLL","title":"Valuative criteria · Lemma 0CLL","summary":"The base change of a morphism of algebraic stacks which satisfies the existence part of the valuative criterion by any morphism of algebraic stacks is a morphism of algebraic stacks which satisfies the existence part of the valuative criterion.","statement_latex":"The base change of a morphism of algebraic stacks which satisfies the\nexistence part of the valuative criterion by any morphism of\nalgebraic stacks is a morphism of algebraic stacks which satisfies the\nexistence part of the valuative criterion.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLL","source_file":"stacks-morphisms.tex","source_line":9169,"source_end_line":9175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9169-L9175","statement_sha256":"d72dae2f1757a85c821394556482cc1048fa89c6636774b6a53c43c99c75325d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14274,"rank":14274,"depth":64,"x":2165.069,"y":1651.788,"cluster":"algebraic-stacks"},{"id":"stacks:0CLM","tag":"0CLM","title":"Valuative criteria · Lemma 0CLM","summary":"The composition of morphisms of algebraic stacks which satisfy the existence part of the valuative criterion is another morphism of algebraic stacks which satisfies the existence part of the valuative criterion.","statement_latex":"The composition of morphisms of algebraic stacks which satisfy the\nexistence part of the valuative criterion is another\nmorphism of algebraic stacks which satisfies the\nexistence part of the valuative criterion.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLM","source_file":"stacks-morphisms.tex","source_line":9182,"source_end_line":9188,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9182-L9188","statement_sha256":"337fe63e8edcc0f6cdd9d1842360035bbf9c115f0461f2c6175fadf3c2b004a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14275,"rank":14275,"depth":2,"x":1888.638,"y":1638.486,"cluster":"algebraic-stacks"},{"id":"stacks:0CLN","tag":"0CLN","title":"Valuative criteria · Lemma 0CLN","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent • f satisfies the existence part of the valuative criterion, • for every scheme T and morphism T → Y the morphism X ×_Y T → T satisfies the existence part of the valuative criterion as a morphism of algebraic spaces.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich is representable by algebraic spaces. Then the following are equivalent\n\\begin{enumerate}\n\\item $f$ satisfies the existence part of the valuative criterion,\n\\item for every scheme $T$ and morphism $T \\to \\mathcal{Y}$\nthe morphism $\\mathcal{X} \\times_\\mathcal{Y} T \\to T$ satisfies\nthe existence part of the valuative criterion as a morphism\nof algebraic spaces.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLN","source_file":"stacks-morphisms.tex","source_line":9195,"source_end_line":9206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9195-L9206","statement_sha256":"0d2beb986abf3e99633f477d68cfd44c5c511c10e30283c337712c790521131d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14276,"rank":14276,"depth":64,"x":2103.35,"y":1491.318,"cluster":"algebraic-stacks"},{"id":"stacks:0CLP","tag":"0CLP","title":"Valuative criteria · Lemma 0CLP","summary":"A closed immersion of algebraic stacks satisfies both the existence and uniqueness part of the valuative criterion.","statement_latex":"A closed immersion of algebraic stacks satisfies both\nthe existence and uniqueness part of the valuative criterion.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criteria","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLP","source_file":"stacks-morphisms.tex","source_line":9213,"source_end_line":9217,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9213-L9217","statement_sha256":"622024c7a557ab1df8caf86159c8ebb631a2197d2b8436302dc08abff4b51fb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14277,"rank":14277,"depth":65,"x":2063.34,"y":1721.862,"cluster":"algebraic-stacks"},{"id":"stacks:0CLR","tag":"0CLR","title":"Valuative criterion for second diagonal · Lemma 0CLR","summary":"Let f : X → Y be a morphism of algebraic stacks. If Δ_f is quasi-separated and if for every diagram ([Tag 0CLB]) and choice of γ as in Definition [Tag 0CLA] the category of dotted arrows is a setoid, then Δ_f is separated.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $\\Delta_f$ is quasi-separated and if for every diagram\n(\\ref{equation-diagram}) and choice of $\\gamma$ as in\nDefinition \\ref{definition-fill-in-diagram}\nthe category of dotted arrows\nis a setoid, then $\\Delta_f$ is separated.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criterion for second diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLR","source_file":"stacks-morphisms.tex","source_line":9237,"source_end_line":9245,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9237-L9245","statement_sha256":"c7e98bb7b99b39183a9dba7dbfd25058361a9e55f4c20f26a6f43155d5e250cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14278,"rank":14278,"depth":65,"x":1907.316,"y":1529.0,"cluster":"algebraic-stacks"},{"id":"stacks:0E8L","tag":"0E8L","title":"Valuative criterion for the diagonal · Lemma 0E8L","summary":"Let f : X → Y be a morphism of algebraic stacks. Consider a 2-commutative solid diagram xymatrix Spec(K) ar[rr]_-x ar[d]_j & & X ar[d]^Δ_f Spec(A) ar[rr]^(a_1, a_2, φ) ar@..>[rru] & & X ×_Y X where A is a valuation ring with field of fractions K. Let γ : (a_1, a_2, φ) ∘ j → Δ_f ∘ x be a 2-morphism witnessing the 2-commutativity of the diagram. Then • Writing γ = (α_1, α_2) with α_i : a_i ∘ j → x we obtain two dotted arrows (a_1, α_1, id) and (a_2, α_2, φ) in the diagram…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nConsider a $2$-commutative solid diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[rr]_-x \\ar[d]_j & &\n\\mathcal{X} \\ar[d]^{\\Delta_f} \\\\\n\\Spec(A) \\ar[rr]^{(a_1, a_2, \\varphi)} \\ar@{..>}[rru] & &\n\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}\n}\n$$\nwhere $A$ is a valuation ring with field of fractions $K$. Let\n$\\gamma : (a_1, a_2, \\varphi) \\circ j \\longrightarrow \\Delta_f \\circ x$\nbe a $2$-morphism witnessing the $2$-commutativity of the diagram.\nThen\n\\begin{enumerate}\n\\item Writing $\\gamma = (\\alpha_1, \\alpha_2)$ with\n$\\alpha_i : a_i \\circ j \\to x$ we obtain two dotted arrows\n$(a_1, \\alpha_1, \\text{id})$ and $(a_2, \\alpha_2, \\varphi)$ in\nthe diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[r]^-{f \\circ a_1} \\ar@{..>}[ru] & \\mathcal{Y}\n}\n$$\n\\item The category of dotted arrows for the original diagram\nand $\\gamma$ is a setoid whose set of isomorphism\nclasses of objects equal to the set of morphisms\n$(a_1, \\alpha_1, \\text{id}) \\to (a_2, \\alpha_2, \\varphi)$ in\nthe category of dotted arrows.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criterion for the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8L","source_file":"stacks-morphisms.tex","source_line":9329,"source_end_line":9362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9329-L9362","statement_sha256":"159da4d879dc020b3e98c72947cf3bfa83b1baf036a5788419bf4f9e73ea6568","origin":"The Stacks Project","memory_eligible":false,"source_rank":14279,"rank":14279,"depth":61,"x":2177.684,"y":1582.727,"cluster":"algebraic-stacks"},{"id":"stacks:0CLT","tag":"0CLT","title":"Valuative criterion for the diagonal · Lemma 0CLT","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume f is quasi-separated. If f satisfies the uniqueness part of the valuative criterion, then f is separated.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nAssume $f$ is quasi-separated.\nIf $f$ satisfies the uniqueness part of the valuative criterion,\nthen $f$ is separated.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criterion for the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLT","source_file":"stacks-morphisms.tex","source_line":9373,"source_end_line":9379,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9373-L9379","statement_sha256":"40835c0f36bedd5579e1716ff1b21e1f47bed3d5c1fa495aed0999b309eda46d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14280,"rank":14280,"depth":66,"x":1934.913,"y":1696.616,"cluster":"algebraic-stacks"},{"id":"stacks:0CLU","tag":"0CLU","title":"Valuative criterion for the diagonal · Lemma 0CLU","summary":"Let f : X → Y be a morphism of algebraic stacks. If f is separated, then f satisfies the uniqueness part of the valuative criterion.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $f$ is separated, then $f$ satisfies the\nuniqueness part of the valuative criterion.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criterion for the diagonal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLU","source_file":"stacks-morphisms.tex","source_line":9412,"source_end_line":9417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9412-L9417","statement_sha256":"eb9ad502c6857225e61f87afa5083d7903d1dbe5b1b0a1ac8488a2c0b134248e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14281,"rank":14281,"depth":65,"x":2022.414,"y":1474.699,"cluster":"algebraic-stacks"},{"id":"stacks:0CLW","tag":"0CLW","title":"Valuative criterion for universal closedness · Lemma 0CLW","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume • f is quasi-compact, and • f satisfies the existence part of the valuative criterion. Then f is universally closed.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nAssume\n\\begin{enumerate}\n\\item $f$ is quasi-compact, and\n\\item $f$ satisfies the existence part of the valuative criterion.\n\\end{enumerate}\nThen $f$ is universally closed.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLW","source_file":"stacks-morphisms.tex","source_line":9481,"source_end_line":9490,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9481-L9490","statement_sha256":"55cfe1197bb3abf71431ce60b9a8c2e8f3d1975b03113aaa0f83a0285c552632","origin":"The Stacks Project","memory_eligible":false,"source_rank":14282,"rank":14282,"depth":76,"x":2136.446,"y":1688.163,"cluster":"algebraic-stacks"},{"id":"stacks:0CLX","tag":"0CLX","title":"Valuative criterion for universal closedness · Lemma 0CLX","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume • f is quasi-separated, and • f is universally closed. Then f satisfies the existence part of the valuative criterion.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nAssume\n\\begin{enumerate}\n\\item $f$ is quasi-separated, and\n\\item $f$ is universally closed.\n\\end{enumerate}\nThen $f$ satisfies the existence part of the valuative criterion.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criterion for universal closedness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLX","source_file":"stacks-morphisms.tex","source_line":9564,"source_end_line":9573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9564-L9573","statement_sha256":"0949d399e5b51a56b305d1fa0a41337a02bc44779494b62bc95171081ad9d997","origin":"The Stacks Project","memory_eligible":false,"source_rank":14283,"rank":14283,"depth":64,"x":1880.488,"y":1595.386,"cluster":"algebraic-stacks"},{"id":"stacks:0CLZ","tag":"0CLZ","title":"Valuative criterion for properness · Lemma 0CLZ","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume f is of finite type and quasi-separated. Then the following are equivalent • f is proper, and • f satisfies both the uniqueness and existence parts of the valuative criterion.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nAssume $f$ is of finite type and quasi-separated.\nThen the following are equivalent\n\\begin{enumerate}\n\\item $f$ is proper, and\n\\item $f$ satisfies both the uniqueness and existence parts\nof the valuative criterion.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Valuative criterion for properness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CLZ","source_file":"stacks-morphisms.tex","source_line":9636,"source_end_line":9646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9636-L9646","statement_sha256":"0681551f87cd2d4284f0d462b0381006f40dec3cf07568db18477d6e354d70bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14284,"rank":14284,"depth":77,"x":2144.052,"y":1518.499,"cluster":"algebraic-stacks"},{"id":"stacks:0CJ3","tag":"0CJ3","title":"Local complete intersection morphisms · Definition 0CJ3","summary":"Let f : X → Y be a morphism of algebraic stacks. We say f is a local complete intersection morphism or Koszul if the equivalent conditions of Lemma [Tag 06FM] hold with P = local complete intersection.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nWe say $f$ is a {\\it local complete intersection morphism} or {\\it Koszul}\nif the equivalent conditions of\nLemma \\ref{lemma-local-source-target}\nhold with $\\mathcal{P} = \\text{local complete intersection}$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Local complete intersection morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJ3","source_file":"stacks-morphisms.tex","source_line":9683,"source_end_line":9690,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9683-L9690","statement_sha256":"cde28ab9ad92b1ab791f58fadacfd2a8bc40b6943a41052706d763eb31ad368e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14285,"rank":14285,"depth":4,"x":2011.426,"y":1724.916,"cluster":"algebraic-stacks"},{"id":"stacks:0CJ4","tag":"0CJ4","title":"Local complete intersection morphisms · Lemma 0CJ4","summary":"The composition of local complete intersection morphisms is a local complete intersection.","statement_latex":"The composition of local complete intersection morphisms is\na local complete intersection.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJ4","source_file":"stacks-morphisms.tex","source_line":9692,"source_end_line":9696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9692-L9696","statement_sha256":"b400c1f1a44ef0beab8d6e80f6ca54430a57d34ab814be0f0d8982abc2736b6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14286,"rank":14286,"depth":40,"x":1943.171,"y":1497.265,"cluster":"algebraic-stacks"},{"id":"stacks:0CJ5","tag":"0CJ5","title":"Local complete intersection morphisms · Lemma 0CJ5","summary":"A flat base change of a local complete intersection morphism is a local complete intersection morphism.","statement_latex":"A flat base change of a local complete intersection morphism is\na local complete intersection morphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJ5","source_file":"stacks-morphisms.tex","source_line":9706,"source_end_line":9710,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9706-L9710","statement_sha256":"f7c0c73d1f47d28ef06db2fcaaf8ae077b18879931a44b2f6c44d5458229db72","origin":"The Stacks Project","memory_eligible":false,"source_rank":14287,"rank":14287,"depth":18,"x":2176.762,"y":1626.508,"cluster":"algebraic-stacks"},{"id":"stacks:0CJ6","tag":"0CJ6","title":"Local complete intersection morphisms · Lemma 0CJ6","summary":"Let xymatrix X ar[rr]_f ar[rd] & & Y ar[ld] & Z be a commutative diagram of morphisms of algebraic stacks. Assume Y → Z is smooth and X → Z is a local complete intersection morphism. Then f : X → Y is a local complete intersection morphism.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[rr]_f \\ar[rd] & & \\mathcal{Y} \\ar[ld] \\\\\n& \\mathcal{Z}\n}\n$$\nbe a commutative diagram of morphisms of algebraic stacks.\nAssume $\\mathcal{Y} \\to \\mathcal{Z}$ is smooth and\n$\\mathcal{X} \\to \\mathcal{Z}$ is a local complete intersection morphism.\nThen $f : \\mathcal{X} \\to \\mathcal{Y}$ is a\nlocal complete intersection morphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Local complete intersection morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJ6","source_file":"stacks-morphisms.tex","source_line":9719,"source_end_line":9733,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9719-L9733","statement_sha256":"19e54040f83223e4346c8e4261dbb564ab9a0abd5b54f0c01c740849b4b083a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14288,"rank":14288,"depth":1,"x":1900.357,"y":1663.782,"cluster":"algebraic-stacks"},{"id":"stacks:0DU7","tag":"0DU7","title":"Stabilizer preserving morphisms · Lemma 0DU7","summary":"Let f : X → Y be a morphism of algebraic stacks. If I_X → X ×_Y I_Y is an isomorphism, then f is representable by algebraic spaces.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $\\mathcal{I}_\\mathcal{X} \\to\n\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$ is an isomorphism,\nthen $f$ is representable by algebraic spaces.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Stabilizer preserving morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DU7","source_file":"stacks-morphisms.tex","source_line":9766,"source_end_line":9772,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9766-L9772","statement_sha256":"edfc4aee8b89ea761fe3b45f81d20a346803d9c343bf8655b2aae2a430d502a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14289,"rank":14289,"depth":70,"x":2074.341,"y":1479.308,"cluster":"algebraic-stacks"},{"id":"stacks:0DU9","tag":"0DU9","title":"Stabilizer preserving morphisms · Lemma 0DU9","summary":"Let f : X → Y be an unramified morphism of algebraic stacks. The following are equivalent • I_X → X ×_Y I_Y is an isomorphism, and • f induces an isomorphism between automorphism groups at x and f(x) (Remark [Tag 0DTW]) for all x ∈ |X|.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be an unramified\nmorphism of algebraic stacks. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{I}_\\mathcal{X} \\to\n\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$\nis an isomorphism, and\n\\item $f$ induces an isomorphism between automorphism groups at $x$ and $f(x)$\n(Remark \\ref{remark-identify-automorphism-groups}) for all\n$x \\in |\\mathcal{X}|$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Stabilizer preserving morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DU9","source_file":"stacks-morphisms.tex","source_line":9798,"source_end_line":9810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9798-L9810","statement_sha256":"154ac998c5721b774d409de8f0f825770320a45d0d603e91f3f833bdd3f7122a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14290,"rank":14290,"depth":84,"x":2094.413,"y":1714.25,"cluster":"algebraic-stacks"},{"id":"stacks:0DUA","tag":"0DUA","title":"Stabilizer preserving morphisms · Lemma 0DUA","summary":"[rydh_quotients] and [alper_quotient] Let f : X → Y be a morphism of algebraic stacks. Assume • f is representable by algebraic spaces and unramified, and • I_Y → Y is proper. Then the set of x ∈ |X| such that f induces an isomorphism between automorphism groups at x and f(x) (Remark [Tag 0DTW]) is open. Letting U ⊂ X be the corresponding open substack, the morphism I_U → U ×_Y I_Y is an isomorphism.","statement_latex":"\\begin{reference}\n\\cite[Proposition 3.5]{rydh_quotients} and\n\\cite[Proposition 2.5]{alper_quotient}\n\\end{reference}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nAssume\n\\begin{enumerate}\n\\item $f$ is representable by algebraic spaces and unramified, and\n\\item $\\mathcal{I}_\\mathcal{Y} \\to \\mathcal{Y}$ is proper.\n\\end{enumerate}\nThen the set of $x \\in |\\mathcal{X}|$ such that $f$ induces an\nisomorphism between automorphism groups at $x$ and $f(x)$\n(Remark \\ref{remark-identify-automorphism-groups}) is open.\nLetting $\\mathcal{U} \\subset \\mathcal{X}$ be the corresponding open substack,\nthe morphism\n$\\mathcal{I}_\\mathcal{U} \\to\n\\mathcal{U} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$\nis an isomorphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Stabilizer preserving morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUA","source_file":"stacks-morphisms.tex","source_line":9832,"source_end_line":9852,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9832-L9852","statement_sha256":"5eb1c77abc018295ad8da13a61877b57f90670b474e2ac0bf7b63b54ff3a3b02","origin":"The Stacks Project","memory_eligible":false,"source_rank":14291,"rank":14291,"depth":85,"x":1890.513,"y":1552.265,"cluster":"algebraic-stacks"},{"id":"stacks:0DUB","tag":"0DUB","title":"Stabilizer preserving morphisms · Lemma 0DUB","summary":"Let xymatrix X' ar[r] ar[d]_f' & X ar[d]^f Y' ar[r] & Y be a cartesian diagram of algebraic stacks. • Let x' ∈ |X'| with image x ∈ |X|. If f induces an isomorphism between automorphism groups at x and f(x) (Remark [Tag 0DTW]), then f' induces an isomorphism between automorphism groups at x' and f(x'). • If I_X → X ×_Y I_Y is an isomorphism, then I_X' → X' ×_Y' I_Y' is an isomorphism.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r] & \\mathcal{Y}\n}\n$$\nbe a cartesian diagram of algebraic stacks.\n\\begin{enumerate}\n\\item Let $x' \\in |\\mathcal{X}'|$ with image $x \\in |\\mathcal{X}|$.\nIf $f$ induces an isomorphism between automorphism groups at\n$x$ and $f(x)$ (Remark \\ref{remark-identify-automorphism-groups}), then\n$f'$ induces an isomorphism between automorphism groups at $x'$ and $f(x')$.\n\\item If $\\mathcal{I}_\\mathcal{X} \\to\n\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{I}_\\mathcal{Y}$ is an isomorphism,\nthen $\\mathcal{I}_{\\mathcal{X}'} \\to\n\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{I}_{\\mathcal{Y}'}$\nis an isomorphism.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Stabilizer preserving morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUB","source_file":"stacks-morphisms.tex","source_line":9883,"source_end_line":9904,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9883-L9904","statement_sha256":"f3ab849dc6ae927da86e04d41a2ffe00d0b37731a8eeb12aba26ba270f02ed28","origin":"The Stacks Project","memory_eligible":false,"source_rank":14292,"rank":14292,"depth":0,"x":2171.357,"y":1556.015,"cluster":"algebraic-stacks"},{"id":"stacks:0DUC","tag":"0DUC","title":"Stabilizer preserving morphisms · Lemma 0DUC","summary":"Let xymatrix X' ar[r] ar[d]_f' & X ar[d]^f Y' ar[r]^g & Y be a cartesian diagram of algebraic stacks. If f induces an isomorphism between automorphism groups at points (Remark [Tag 0DTW]), then Mor(Spec(k), X') → Mor(Spec(k), Y') × Mor(Spec(k), X) is injective on isomorphism classes for any field k.","statement_latex":"Let\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[r] \\ar[d]_{f'} & \\mathcal{X} \\ar[d]^f \\\\\n\\mathcal{Y}' \\ar[r]^g & \\mathcal{Y}\n}\n$$\nbe a cartesian diagram of algebraic stacks. If $f$ induces an isomorphism\nbetween automorphism groups at points\n(Remark \\ref{remark-identify-automorphism-groups}),\nthen\n$$\n\\Mor(\\Spec(k), \\mathcal{X}')\n\\longrightarrow\n\\Mor(\\Spec(k), \\mathcal{Y}') \\times \\Mor(\\Spec(k), \\mathcal{X})\n$$\nis injective on isomorphism classes for any field $k$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Stabilizer preserving morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUC","source_file":"stacks-morphisms.tex","source_line":9910,"source_end_line":9929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9910-L9929","statement_sha256":"d3358cb0b65e4c2a1a541588925975e813532f2e98ed041a23749373f24e1782","origin":"The Stacks Project","memory_eligible":false,"source_rank":14293,"rank":14293,"depth":0,"x":1961.081,"y":1712.734,"cluster":"algebraic-stacks"},{"id":"stacks:0DUD","tag":"0DUD","title":"Stabilizer preserving morphisms · Lemma 0DUD","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume f is étale, f induces an isomorphism between automorphism groups at points (Remark [Tag 0DTW]), and for every algebraically closed field k the functor f : Mor(Spec(k), X) → Mor(Spec(k), Y) is an equivalence. Then f is an isomorphism.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nAssume $f$ is \\'etale, $f$ induces an isomorphism\nbetween automorphism groups at points\n(Remark \\ref{remark-identify-automorphism-groups}),\nand for every algebraically closed field $k$ the functor\n$$\nf : \\Mor(\\Spec(k), \\mathcal{X}) \\longrightarrow \\Mor(\\Spec(k), \\mathcal{Y})\n$$\nis an equivalence. Then $f$ is an isomorphism.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Stabilizer preserving morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUD","source_file":"stacks-morphisms.tex","source_line":9946,"source_end_line":9957,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9946-L9957","statement_sha256":"cf89494900040ce1db2fdf777bba254e8e38ca7fa939a805bb3f6db2b6260425","origin":"The Stacks Project","memory_eligible":false,"source_rank":14294,"rank":14294,"depth":85,"x":1990.13,"y":1477.667,"cluster":"algebraic-stacks"},{"id":"stacks:0GMI","tag":"0GMI","title":"Normalization · Lemma 0GMI","summary":"Let X be an algebraic stack. The following are equivalent • there is a surjective smooth morphism U → X where U is a scheme such that every quasi-compact open of U has finitely many irreducible components, • for every scheme U and every smooth morphism U → X every quasi-compact open of U has finitely many irreducible components, • for every algebraic space Y and smooth morphism Y → X the space Y satisfies the equivalent conditions of Morphisms of Spaces, Lemma [Tag 0BB1],…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent\n\\begin{enumerate}\n\\item there is a surjective smooth morphism $U \\to \\mathcal{X}$ where $U$\nis a scheme such that every quasi-compact open of $U$ has\nfinitely many irreducible components,\n\\item for every scheme $U$ and every smooth morphism\n$U \\to \\mathcal{X}$ every quasi-compact open of $U$ has finitely many\nirreducible components,\n\\item for every algebraic space $Y$ and smooth morphism $Y \\to \\mathcal{X}$\nthe space $Y$ satisfies the equivalent conditions of\nMorphisms of Spaces, Lemma \\ref{spaces-morphisms-lemma-prepare-normalization},\nand\n\\item for every quasi-compact algebraic stack $\\mathcal{Y}$ smooth over\n$\\mathcal{X}$ the space $|\\mathcal{Y}|$ has finitely many irreducible\ncomponents.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMI","source_file":"stacks-morphisms.tex","source_line":9984,"source_end_line":10002,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L9984-L10002","statement_sha256":"90128483cb325778235b3b4dcebb3f6d43e49e9ce140aa392a8b33d2155fc358","origin":"The Stacks Project","memory_eligible":false,"source_rank":14295,"rank":14295,"depth":49,"x":2157.879,"y":1667.638,"cluster":"algebraic-stacks"},{"id":"stacks:0GMJ","tag":"0GMJ","title":"Normalization · Lemma 0GMJ","summary":"Let X be an algebraic stack satisfying the equivalent conditions of Lemma [Tag 0GMI]. Then there exists an integral morphism of algebraic stacks X^ν → X such that for every scheme U and smooth morphism U → X the fibre product X^ν ×_X U is the normalization of U.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack satisfying\nthe equivalent conditions of Lemma \\ref{lemma-prepare-normalization}.\nThen there exists an integral morphism of algebraic stacks\n$$\n\\mathcal{X}^\\nu \\longrightarrow \\mathcal{X}\n$$\nsuch that for every scheme $U$ and smooth morphism $U \\to \\mathcal{X}$\nthe fibre product $\\mathcal{X}^\\nu \\times_\\mathcal{X} U$\nis the normalization of $U$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMJ","source_file":"stacks-morphisms.tex","source_line":10022,"source_end_line":10033,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10022-L10033","statement_sha256":"024224d37792021b1c49b7bfba567d7d54b9258e725759c30be01fbaecfabc71","origin":"The Stacks Project","memory_eligible":false,"source_rank":14296,"rank":14296,"depth":72,"x":1881.192,"y":1622.705,"cluster":"algebraic-stacks"},{"id":"stacks:0GMK","tag":"0GMK","title":"Normalization · Definition 0GMK","summary":"Let X be an algebraic stack satisfying the equivalent conditions of Lemma [Tag 0GMI]. We define the normalization of X as the morphism ν : X^ν → X constructed in Lemma [Tag 0GMJ].","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack satisfying the\nequivalent conditions of Lemma \\ref{lemma-prepare-normalization}.\nWe define the {\\it normalization} of $\\mathcal{X}$ as the morphism\n$$\n\\nu : \\mathcal{X}^\\nu \\longrightarrow \\mathcal{X}\n$$\nconstructed in Lemma \\ref{lemma-normalization}.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Normalization","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GMK","source_file":"stacks-morphisms.tex","source_line":10096,"source_end_line":10105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10096-L10105","statement_sha256":"0703594b32747bf2b9bc78cd148c5277a2690b5a8925f0f543da6272f01fd0f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14297,"rank":14297,"depth":73,"x":2121.544,"y":1498.739,"cluster":"algebraic-stacks"},{"id":"stacks:0GVZ","tag":"0GVZ","title":"Points and specializations · Lemma 0GVZ","summary":"Let X be an algebraic stack. Let f : U → X be a smooth morphism where U is an algebraic space. Let x' leadsto x be a specialization of points of |X|. Let u ∈ |U| with f(u) = x. If (X, x') satisfy the equivalent conditions of Properties of Stacks, Lemma [Tag 0DTK], then there exists a specialization u' leadsto u in |U| with f(u') = x'.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $f : U \\to \\mathcal{X}$\nbe a smooth morphism where $U$ is an algebraic space. Let\n$x' \\leadsto x$ be a specialization of points of $|\\mathcal{X}|$.\nLet $u \\in |U|$ with $f(u) = x$. If $(\\mathcal{X}, x')$ satisfy the\nequivalent conditions of Properties of Stacks,\nLemma \\ref{stacks-properties-lemma-UR-quasi-compact-above-x},\nthen there exists a specialization\n$u' \\leadsto u$ in $|U|$ with $f(u') = x'$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points and specializations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GVZ","source_file":"stacks-morphisms.tex","source_line":10123,"source_end_line":10133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10123-L10133","statement_sha256":"1a1b3ac7c3400621ca2afb1652ad8c71f3f00763842d13a04a9bcad74b184018","origin":"The Stacks Project","memory_eligible":false,"source_rank":14298,"rank":14298,"depth":20,"x":2043.939,"y":1726.714,"cluster":"algebraic-stacks"},{"id":"stacks:0GW1","tag":"0GW1","title":"Decent algebraic stacks · Definition 0GW1","summary":"Let X be an algebraic stack. We say X is decent if for every x ∈ |X| the equivalent conditions of Properties of Stacks, Lemma [Tag 0DTK] are satisfied.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. We say $\\mathcal{X}$ is {\\it decent}\nif for every $x \\in |\\mathcal{X}|$ the equivalent conditions of\nProperties of Stacks, Lemma\n\\ref{stacks-properties-lemma-UR-quasi-compact-above-x} are satisfied.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Decent algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GW1","source_file":"stacks-morphisms.tex","source_line":10173,"source_end_line":10179,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10173-L10179","statement_sha256":"ff792fbc41e731a1618ca9793ecb387dbda176fadfc737f5b00d070e97c6e770","origin":"The Stacks Project","memory_eligible":false,"source_rank":14299,"rank":14299,"depth":6,"x":1917.732,"y":1514.404,"cluster":"algebraic-stacks"},{"id":"stacks:0GW2","tag":"0GW2","title":"Decent algebraic stacks · Lemma 0GW2","summary":"A quasi-separated algebraic stack X is decent. More generally, if Δ : X → X × X is quasi-compact, then X is decent.","statement_latex":"A quasi-separated algebraic stack $\\mathcal{X}$ is decent.\nMore generally, if $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times \\mathcal{X}$\nis quasi-compact, then $\\mathcal{X}$ is decent.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Decent algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GW2","source_file":"stacks-morphisms.tex","source_line":10188,"source_end_line":10193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10188-L10193","statement_sha256":"4becbbbf6421352fc43c2172b81450bfc0a3aeee6d14c30df984ec3089d2a4bb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14300,"rank":14300,"depth":63,"x":2181.739,"y":1599.413,"cluster":"algebraic-stacks"},{"id":"stacks:0GW3","tag":"0GW3","title":"Decent algebraic stacks · Lemma 0GW3","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume Y is decent and f is representable (by schemes) or f is representable by algebraic spaces and quasi-separated. Then X is decent.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism\nof algebraic stacks. Assume $Y$ is decent\nand $f$ is representable (by schemes) or $f$ is representable\nby algebraic spaces and quasi-separated.\nThen $\\mathcal{X}$ is decent.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Decent algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GW3","source_file":"stacks-morphisms.tex","source_line":10211,"source_end_line":10218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10211-L10218","statement_sha256":"ef006c50172506f41d06ae2a26e1fe411ad4fa192ad17056c21009d25527fc2e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14301,"rank":14301,"depth":57,"x":1918.493,"y":1686.602,"cluster":"algebraic-stacks"},{"id":"stacks:0GW4","tag":"0GW4","title":"Decent algebraic stacks · Lemma 0GW4","summary":"Let f : X → Y be a morphism of algebraic stacks. If f is quasi-compact and surjective and X is decent, then Y is decent.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $f$ is quasi-compact and surjective and $\\mathcal{X}$ is decent,\nthen $\\mathcal{Y}$ is decent.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Decent algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GW4","source_file":"stacks-morphisms.tex","source_line":10252,"source_end_line":10257,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10252-L10257","statement_sha256":"a7bdd7e577fdc2a8a1d6bdd8888c5e0c75bd5f069d502fd1fbb3ed44a44bae08","origin":"The Stacks Project","memory_eligible":false,"source_rank":14302,"rank":14302,"depth":2,"x":2042.59,"y":1472.768,"cluster":"algebraic-stacks"},{"id":"stacks:0GW5","tag":"0GW5","title":"Decent algebraic stacks · Lemma 0GW5","summary":"Let f : X → Y be a morphism of algebraic stacks. If X is a gerbe over Y and X is decent, then Y is decent.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nIf $\\mathcal{X}$ is a gerbe over $\\mathcal{Y}$ and $\\mathcal{X}$ is decent,\nthen $\\mathcal{Y}$ is decent.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Decent algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GW5","source_file":"stacks-morphisms.tex","source_line":10279,"source_end_line":10284,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10279-L10284","statement_sha256":"bccdd924830ee91f1253cf33bb5ab1c47f8ded187e5b58710b1a53fe3d72fb30","origin":"The Stacks Project","memory_eligible":false,"source_rank":14303,"rank":14303,"depth":83,"x":2123.107,"y":1701.043,"cluster":"algebraic-stacks"},{"id":"stacks:0GW7","tag":"0GW7","title":"Points on decent stacks · Lemma 0GW7","summary":"Let X be a decent algebraic stack. Then |X| is Kolmogorov (see Topology, Definition [Tag 004X]).","statement_latex":"Let $\\mathcal{X}$ be a decent algebraic stack.\nThen $|\\mathcal{X}|$ is Kolmogorov (see\nTopology, Definition \\ref{topology-definition-generic-point}).","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points on decent stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GW7","source_file":"stacks-morphisms.tex","source_line":10305,"source_end_line":10310,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10305-L10310","statement_sha256":"e1118a6048e45cc8d8d1702f96e005079985d4957203b86705d413db15ded095","origin":"The Stacks Project","memory_eligible":false,"source_rank":14304,"rank":14304,"depth":58,"x":1879.969,"y":1578.306,"cluster":"algebraic-stacks"},{"id":"stacks:0GW8","tag":"0GW8","title":"Points on decent stacks · Lemma 0GW8","summary":"Let X be a decent, locally Noetherian algebraic stack. Then |X| is a sober locally Noetherian topological space.","statement_latex":"Let $\\mathcal{X}$ be a decent, locally Noetherian algebraic stack.\nThen $|\\mathcal{X}|$ is a sober locally Noetherian topological space.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points on decent stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GW8","source_file":"stacks-morphisms.tex","source_line":10346,"source_end_line":10350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10346-L10350","statement_sha256":"988e8d09f86c1afe4a6dcd6d877fcee852bcc600d435f677238df4ebefcd581f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14305,"rank":14305,"depth":59,"x":2158.177,"y":1530.811,"cluster":"algebraic-stacks"},{"id":"stacks:0GW9","tag":"0GW9","title":"Points on decent stacks · Proposition 0GW9","summary":"Let X be a decent algebraic stack such that I_X → X is quasi-compact. Then |X| is sober.","statement_latex":"Let $\\mathcal{X}$ be a decent algebraic stack such that\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact.\nThen $|\\mathcal{X}|$ is sober.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Points on decent stacks","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GW9","source_file":"stacks-morphisms.tex","source_line":10361,"source_end_line":10366,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10361-L10366","statement_sha256":"0bf4858a5f34b81e67ca5aeec502b2f664f3a8fb364dbdcade57276e4c01ea3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14306,"rank":14306,"depth":84,"x":1991.093,"y":1723.848,"cluster":"algebraic-stacks"},{"id":"stacks:0GWB","tag":"0GWB","title":"Integral algebraic stacks · Definition 0GWB","summary":"We say an algebraic stack X is integral if it is reduced, decent, I_X → X is quasi-compact, and |X| is irreducible.","statement_latex":"We say an algebraic stack $\\mathcal{X}$ is\n{\\it integral} if it is reduced, decent,\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact,\nand $|\\mathcal{X}|$ is irreducible.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Integral algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWB","source_file":"stacks-morphisms.tex","source_line":10424,"source_end_line":10430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10424-L10430","statement_sha256":"cc827af5810ce4c5b580321447e65f90ab6b41fb648f8192ecdb831f800572f9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14307,"rank":14307,"depth":0,"x":1959.039,"y":1486.51,"cluster":"algebraic-stacks"},{"id":"stacks:0GWC","tag":"0GWC","title":"Integral algebraic stacks · Lemma 0GWC","summary":"Let X be an integral algebraic stack. Then • |X| is sober, irreducible, and has a unique generic point, • there exists an open substack U ⊂ X which is a gerbe over an integral scheme U.","statement_latex":"Let $\\mathcal{X}$ be an integral algebraic stack. Then\n\\begin{enumerate}\n\\item $|\\mathcal{X}|$ is sober, irreducible, and has a unique generic point,\n\\item there exists an open substack $\\mathcal{U} \\subset \\mathcal{X}$ \nwhich is a gerbe over an integral scheme $U$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Integral algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWC","source_file":"stacks-morphisms.tex","source_line":10437,"source_end_line":10445,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10437-L10445","statement_sha256":"19aab4ad1d6efe9389b1051d98fdf5d87db6448fb8c0910c208da1ac527d40ab","origin":"The Stacks Project","memory_eligible":false,"source_rank":14308,"rank":14308,"depth":85,"x":2173.704,"y":1643.454,"cluster":"algebraic-stacks"},{"id":"stacks:0GWD","tag":"0GWD","title":"Integral algebraic stacks · Lemma 0GWD","summary":"Let X be an algebraic stack which is reduced and quasi-separated and whose associated topological space |X| is irreducible. Then X is integral.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack which is reduced and quasi-separated\nand whose associated topological space $|\\mathcal{X}|$ is irreducible.\nThen $\\mathcal{X}$ is integral.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Integral algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWD","source_file":"stacks-morphisms.tex","source_line":10473,"source_end_line":10478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10473-L10478","statement_sha256":"f3b49c3a187fc734c036f3c118d042c5ee24ede6d734adfdc7d423018d861939","origin":"The Stacks Project","memory_eligible":false,"source_rank":14309,"rank":14309,"depth":64,"x":1888.979,"y":1649.537,"cluster":"algebraic-stacks"},{"id":"stacks:0GWE","tag":"0GWE","title":"Integral algebraic stacks · Lemma 0GWE","summary":"Let X be a decent algebraic stack such that I_X → X is quasi-compact. There are canonical bijections between the following sets: • the set of points of X, i.e., |X|, • the set of irreducible closed subsets of |X|, • the set of integral closed substacks of X. The bijection from (1) to (2) sends x to overline(x). The bijection from (3) to (2) sends Z to |Z|.","statement_latex":"Let $\\mathcal{X}$ be a decent algebraic stack such that\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact.\nThere are canonical bijections between the following sets:\n\\begin{enumerate}\n\\item the set of points of $\\mathcal{X}$, i.e., $|\\mathcal{X}|$,\n\\item the set of irreducible closed subsets of $|\\mathcal{X}|$,\n\\item the set of integral closed substacks of $\\mathcal{X}$.\n\\end{enumerate}\nThe bijection from (1) to (2) sends $x$ to $\\overline{\\{x\\}}$.\nThe bijection from (3) to (2) sends $\\mathcal{Z}$ to $|\\mathcal{Z}|$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Integral algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GWE","source_file":"stacks-morphisms.tex","source_line":10494,"source_end_line":10506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10494-L10506","statement_sha256":"e62bad401ec7f3d5d56d3bf9a3dcca3802046895bf0f62964acdf6be6fd6b608","origin":"The Stacks Project","memory_eligible":false,"source_rank":14310,"rank":14310,"depth":85,"x":2094.2,"y":1483.362,"cluster":"algebraic-stacks"},{"id":"stacks:0H24","tag":"0H24","title":"Residual gerbes · Lemma 0H24","summary":"Let π : X → Y be a morphism of algebraic stacks. Let x ∈ |X| with image y ∈ |Y|. Assume the residual gerbe Z_y ⊂ Y of Y at y exists and that X is a gerbe over Y. Then Z_x = Z_y ×_Y X is the residual gerbe of X at x.","statement_latex":"Let $\\pi : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nLet $x \\in |\\mathcal{X}|$ with image $y \\in |\\mathcal{Y}|$.\nAssume the residual gerbe $\\mathcal{Z}_y \\subset \\mathcal{Y}$\nof $\\mathcal{Y}$ at $y$ exists and that $\\mathcal{X}$ is a gerbe\nover $\\mathcal{Y}$. Then\n$\\mathcal{Z}_x = \\mathcal{Z}_y \\times_\\mathcal{Y} \\mathcal{X}$ is\nthe residual gerbe of $\\mathcal{X}$ at $x$.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H24","source_file":"stacks-morphisms.tex","source_line":10541,"source_end_line":10550,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10541-L10550","statement_sha256":"6272cebdad3515166d3cdd0dfa42b152a5cadec646e1f2153ff9bd5d42ba20d2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14311,"rank":14311,"depth":83,"x":2076.493,"y":1722.532,"cluster":"algebraic-stacks"},{"id":"stacks:0H25","tag":"0H25","title":"Residual gerbes · Lemma 0H25","summary":"Let f : Y → X be a morphism of algebraic stacks. Let x ∈ |X| be a point. Assume • X is decent or locally Noetherian (or both), • I_X → X is quasi-compact, • |f|(|Y|) is contained in (x) ⊂ |X|, and • Y is reduced. Then f factors through the residual gerbe Z_x of X at x (whose existence is guaranteed by Lemma [Tag 06RD] or [Tag 0H22]).","statement_latex":"Let $f : \\mathcal{Y} \\to \\mathcal{X}$ be a morphism of algebraic stacks.\nLet $x \\in |\\mathcal{X}|$ be a point. Assume\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is decent or locally Noetherian (or both),\n\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is quasi-compact,\n\\item $|f|(|\\mathcal{Y}|)$ is contained in $\\{x\\} \\subset |\\mathcal{X}|$, and\n\\item $\\mathcal{Y}$ is reduced.\n\\end{enumerate}\nThen $f$ factors through the residual gerbe $\\mathcal{Z}_x$\nof $\\mathcal{X}$ at $x$ (whose existence is guaranteed by\nLemma \\ref{lemma-every-point-residual-gerbe} or\n\\ref{lemma-every-point-residual-gerbe-locally-Noetherian}).","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H25","source_file":"stacks-morphisms.tex","source_line":10567,"source_end_line":10581,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10567-L10581","statement_sha256":"647ca867e2f473708c6bd81af9334410f15d4e318b999823918f8fd5fbf631fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14312,"rank":14312,"depth":85,"x":1897.075,"y":1535.978,"cluster":"algebraic-stacks"},{"id":"stacks:0H27","tag":"0H27","title":"Residual gerbes · Lemma 0H27","summary":"Let X be a locally Noetherian algebraic stack. Let x ∈ |X| with residual gerbe Z_x ⊂ X (Lemma [Tag 0H22]). Then x is a closed point of |X| if and only if the morphism Z_x → X is a closed immersion.","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nLet $x \\in |\\mathcal{X}|$ with residual gerbe\n$\\mathcal{Z}_x \\subset \\mathcal{X}$\n(Lemma \\ref{lemma-every-point-residual-gerbe-locally-Noetherian}).\nThen $x$ is a closed point of $|\\mathcal{X}|$ if and only if\nthe morphism $\\mathcal{Z}_x \\to \\mathcal{X}$ is a closed immersion.","area":"Algebraic Stacks","chapter":"Morphisms of Algebraic Stacks","chapter_id":"stacks-morphisms","section":"Residual gerbes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H27","source_file":"stacks-morphisms.tex","source_line":10633,"source_end_line":10641,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-morphisms.tex#L10633-L10641","statement_sha256":"9290eda25f1529a61927d569a76150c6dbe419de2d3ae37a7378a0248928a469","origin":"The Stacks Project","memory_eligible":false,"source_rank":14313,"rank":14313,"depth":77,"x":2179.618,"y":1571.762,"cluster":"algebraic-stacks"},{"id":"stacks:0CMR","tag":"0CMR","title":"Morphisms of finite presentation · Definition 0CMR","summary":"Let S be a scheme. Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. We say f is limit preserving if for every directed limit U = lim U_i of affine schemes over S the diagram xymatrix colim X_U_i ar[r] ar[d]_f & X_U ar[d]^f colim Y_U_i ar[r] & Y_U of fibre categories is 2-cartesian.","statement_latex":"Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a\n$1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nWe say $f$ is {\\it limit preserving} if for every directed limit\n$U = \\lim U_i$ of affine schemes over $S$ the diagram\n$$\n\\xymatrix{\n\\colim \\mathcal{X}_{U_i} \\ar[r] \\ar[d]_f & \\mathcal{X}_U \\ar[d]^f \\\\\n\\colim \\mathcal{Y}_{U_i} \\ar[r] & \\mathcal{Y}_U\n}\n$$\nof fibre categories is $2$-cartesian.","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Morphisms of finite presentation","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMR","source_file":"stacks-limits.tex","source_line":58,"source_end_line":71,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L58-L71","statement_sha256":"de9809ec593acff816c04ce45130f3ed006d024877ec2d9496cd91feab350604","origin":"The Stacks Project","memory_eligible":false,"source_rank":14314,"rank":14314,"depth":0,"x":1942.314,"y":1705.801,"cluster":"algebraic-stacks"},{"id":"stacks:0CMS","tag":"0CMS","title":"Morphisms of finite presentation · Lemma 0CMS","summary":"Let S be a scheme. Let f : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. If f is limit preserving (Definition [Tag 0CMR]), then f is limit preserving on objects (Criteria for Representability, Section [Tag 06CT]).","statement_latex":"Let $S$ be a scheme. Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a\n$1$-morphism of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $f$ is limit preserving (Definition \\ref{definition-limit-preserving}),\nthen $f$ is limit preserving on objects (Criteria for Representability, Section\n\\ref{criteria-section-limit-preserving}).","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMS","source_file":"stacks-limits.tex","source_line":73,"source_end_line":80,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L73-L80","statement_sha256":"9aee0af74931d88c0c1dedf7d59c91bbc486ebcab3a6f7f88b63e20bb4d335db","origin":"The Stacks Project","memory_eligible":false,"source_rank":14315,"rank":14315,"depth":1,"x":2009.56,"y":1472.129,"cluster":"algebraic-stacks"},{"id":"stacks:0CMT","tag":"0CMT","title":"Morphisms of finite presentation · Lemma 0CMT","summary":"Let p : X → Y and q : Z → Y be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If p : X → Y is limit preserving, then so is the base change p' : X ×_Y Z → Z of p by q.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Z} \\to \\mathcal{Y}$\nbe $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $p : \\mathcal{X} \\to \\mathcal{Y}$ is limit preserving, then so\nis the base change\n$p' : \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{Z} \\to \\mathcal{Z}$\nof $p$ by $q$.","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMT","source_file":"stacks-limits.tex","source_line":92,"source_end_line":100,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L92-L100","statement_sha256":"42b5695ab18304068140a7ddd60530b11a309b2610207f212362c88eac03b7bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14316,"rank":14316,"depth":0,"x":2147.994,"y":1682.757,"cluster":"algebraic-stacks"},{"id":"stacks:0CMU","tag":"0CMU","title":"Morphisms of finite presentation · Lemma 0CMU","summary":"Let p : X → Y and q : Y → Z be 1-morphisms of categories fibred in groupoids over (Sch/S)_fppf. If p and q are limit preserving, then so is the composition q ∘ p.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ and $q : \\mathcal{Y} \\to \\mathcal{Z}$\nbe $1$-morphisms of categories fibred in groupoids over $(\\Sch/S)_{fppf}$.\nIf $p$ and $q$ are limit preserving, then so is the composition $q \\circ p$.","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMU","source_file":"stacks-limits.tex","source_line":132,"source_end_line":137,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L132-L137","statement_sha256":"5767e8a9a940538e1b0017594b0996eaf7c1bfcfe7da3237d937500dbc8ed58f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14317,"rank":14317,"depth":0,"x":1876.323,"y":1605.934,"cluster":"algebraic-stacks"},{"id":"stacks:0CMV","tag":"0CMV","title":"Morphisms of finite presentation · Lemma 0CMV","summary":"Let p : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. If p is representable by algebraic spaces, then the following are equivalent: • p is limit preserving, • p is limit preserving on objects, and • p is locally of finite presentation (see Algebraic Stacks, Definition [Tag 03YK]).","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. If $p$ is\nrepresentable by algebraic spaces, then the following are equivalent:\n\\begin{enumerate}\n\\item $p$ is limit preserving,\n\\item $p$ is limit preserving on objects, and\n\\item $p$ is locally of finite presentation (see\nAlgebraic Stacks,\nDefinition \\ref{algebraic-definition-relative-representable-property}).\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMV","source_file":"stacks-limits.tex","source_line":155,"source_end_line":167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L155-L167","statement_sha256":"fa5d75094fbeaf1f01fa707e6af3d14da90b70e67952b42f523a169cea4ab9bf","origin":"The Stacks Project","memory_eligible":false,"source_rank":14318,"rank":14318,"depth":54,"x":2138.631,"y":1508.353,"cluster":"algebraic-stacks"},{"id":"stacks:0CMW","tag":"0CMW","title":"Morphisms of finite presentation · Lemma 0CMW","summary":"Let p : X → Y be a 1-morphism of categories fibred in groupoids over (Sch/S)_fppf. The following are equivalent • the diagonal Δ : X → X ×_Y X is limit preserving, and • for every directed limit U = lim U_i of affine schemes over S the functor colim X_U_i → X_U ×_Y_U colim Y_U_i is fully faithful. In particular, if p is limit preserving, then Δ is too.","statement_latex":"Let $p : \\mathcal{X} \\to \\mathcal{Y}$ be a $1$-morphism of categories\nfibred in groupoids over $(\\Sch/S)_{fppf}$. The following are equivalent\n\\begin{enumerate}\n\\item the diagonal\n$\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$\nis limit preserving, and\n\\item for every directed limit $U = \\lim U_i$ of affine schemes over $S$\nthe functor\n$$\n\\colim \\mathcal{X}_{U_i} \\longrightarrow\n\\mathcal{X}_U \\times_{\\mathcal{Y}_U} \\colim \\mathcal{Y}_{U_i}\n$$\nis fully faithful.\n\\end{enumerate}\nIn particular, if $p$ is limit preserving, then $\\Delta$ is too.","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMW","source_file":"stacks-limits.tex","source_line":255,"source_end_line":272,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L255-L272","statement_sha256":"26f043d8fe82b9a5856cc7d345761393d49546a8da11e232242d7632bc135180","origin":"The Stacks Project","memory_eligible":false,"source_rank":14319,"rank":14319,"depth":4,"x":2023.592,"y":1729.319,"cluster":"algebraic-stacks"},{"id":"stacks:0CMX","tag":"0CMX","title":"Morphisms of finite presentation · Lemma 0CMX","summary":"Let S be a scheme. Let X be an algebraic stack over S. If X → S is locally of finite presentation, then X is limit preserving in the sense of Artin's Axioms, Definition [Tag 07XL] (equivalently: the morphism X → S is limit preserving).","statement_latex":"Let $S$ be a scheme. Let $\\mathcal{X}$ be an algebraic stack\nover $S$. If $\\mathcal{X} \\to S$ is locally of finite presentation,\nthen $\\mathcal{X}$ is limit preserving in the sense of\nArtin's Axioms, Definition \\ref{artin-definition-limit-preserving}\n(equivalently: the morphism $\\mathcal{X} \\to S$ is limit preserving).","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Morphisms of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMX","source_file":"stacks-limits.tex","source_line":360,"source_end_line":367,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L360-L367","statement_sha256":"e2fcee365fc95c6165d30d3b7c5b7baf6990c0e8169bf08378dd59b76edf510e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14320,"rank":14320,"depth":57,"x":1930.654,"y":1500.93,"cluster":"algebraic-stacks"},{"id":"stacks:0CMY","tag":"0CMY","title":"Morphisms of finite presentation · Proposition 0CMY","summary":"This is a special case of [Emerton-Gee] Let f : X → Y be a morphism of algebraic stacks. The following are equivalent • f is limit preserving, • f is limit preserving on objects, and • f is locally of finite presentation.","statement_latex":"\\begin{reference}\nThis is a special case of \\cite[Lemma 2.3.15]{Emerton-Gee}\n\\end{reference}\nLet $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $f$ is limit preserving,\n\\item $f$ is limit preserving on objects, and\n\\item $f$ is locally of finite presentation.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Morphisms of finite presentation","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CMY","source_file":"stacks-limits.tex","source_line":444,"source_end_line":456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L444-L456","statement_sha256":"779185075a17fc1174ad11ba50662e8605b2cb626297f818eaf604084d494b03","origin":"The Stacks Project","memory_eligible":false,"source_rank":14321,"rank":14321,"depth":58,"x":2183.044,"y":1616.695,"cluster":"algebraic-stacks"},{"id":"stacks:0CPZ","tag":"0CPZ","title":"Descending properties · Lemma 0CPZ","summary":"In Situation [Tag 0CPY] assume that X_0 → Y_0 is a morphism from algebraic stack to Y_0. Assume X_0 is quasi-compact and quasi-separated. If Y ×_Y_0 X_0 → Y is separated, then Y_i ×_Y_0 X_0 → Y_i is separated for all sufficiently large i ∈ I.","statement_latex":"In Situation \\ref{situation-descent} assume that $\\mathcal{X}_0 \\to Y_0$\nis a morphism from algebraic stack to $Y_0$. Assume $\\mathcal{X}_0$\nis quasi-compact and quasi-separated.\nIf $Y \\times_{Y_0} \\mathcal{X}_0 \\to Y$ is separated, then\n$Y_i \\times_{Y_0} \\mathcal{X}_0 \\to Y_i$ is separated for all\nsufficiently large $i \\in I$.","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Descending properties","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CPZ","source_file":"stacks-limits.tex","source_line":531,"source_end_line":539,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L531-L539","statement_sha256":"65aede3178b54d5bf8d6788bed1eadddb6cb3960021d3834eb46619a7c8a6c54","origin":"The Stacks Project","memory_eligible":false,"source_rank":14322,"rank":14322,"depth":59,"x":1903.625,"y":1674.588,"cluster":"algebraic-stacks"},{"id":"stacks:0CN4","tag":"0CN4","title":"Descending relative objects · Lemma 0CN4","summary":"Let I be a directed set. Let (X_i, f_ii') be an inverse system of algebraic spaces over I. Assume • the morphisms f_ii' : X_i → X_i' are affine, • the spaces X_i are quasi-compact and quasi-separated. Let X = lim X_i. If X is an algebraic stack of finite presentation over X, then there exists an i ∈ I and an algebraic stack X_i of finite presentation over X_i with X ≅ X_i ×_X_i X as algebraic stacks over X.","statement_latex":"Let $I$ be a directed set. Let $(X_i, f_{ii'})$ be an inverse system\nof algebraic spaces over $I$. Assume\n\\begin{enumerate}\n\\item the morphisms $f_{ii'} : X_i \\to X_{i'}$ are affine,\n\\item the spaces $X_i$ are quasi-compact and quasi-separated.\n\\end{enumerate}\nLet $X = \\lim X_i$.\nIf $\\mathcal{X}$ is an algebraic stack of finite presentation over $X$,\nthen there exists an $i \\in I$ and an algebraic stack $\\mathcal{X}_i$\nof finite presentation over $X_i$ with\n$\\mathcal{X} \\cong \\mathcal{X}_i \\times_{X_i} X$ as\nalgebraic stacks over $X$.","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Descending relative objects","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CN4","source_file":"stacks-limits.tex","source_line":584,"source_end_line":598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L584-L598","statement_sha256":"fa0834198f3c86a350503806e69c8003cc44b27f892847957c3194c37d146032","origin":"The Stacks Project","memory_eligible":false,"source_rank":14323,"rank":14323,"depth":72,"x":2063.231,"y":1473.195,"cluster":"algebraic-stacks"},{"id":"stacks:0CQ1","tag":"0CQ1","title":"Finite type closed in finite presentation · Lemma 0CQ1","summary":"Let f : X → Y be a morphism from an algebraic stack to an algebraic space. Assume: • f is of finite type and quasi-separated, • Y is quasi-compact and quasi-separated. Then there exists a morphism of finite presentation f' : X' → Y and a closed immersion X → X' of algebraic stacks over Y.","statement_latex":"Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack\nto an algebraic space. Assume:\n\\begin{enumerate}\n\\item $f$ is of finite type and quasi-separated,\n\\item $Y$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nThen there exists a morphism of finite presentation\n$f' : \\mathcal{X}' \\to Y$ and a closed immersion\n$\\mathcal{X} \\to \\mathcal{X}'$ of\nalgebraic stacks over $Y$.","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQ1","source_file":"stacks-limits.tex","source_line":674,"source_end_line":686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L674-L686","statement_sha256":"129516460520be5a6375c78eb5f89224c69745ef9ba61da7f4a62423ab2110a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14324,"rank":14324,"depth":72,"x":2107.529,"y":1712.446,"cluster":"algebraic-stacks"},{"id":"stacks:0CQ2","tag":"0CQ2","title":"Finite type closed in finite presentation · Lemma 0CQ2","summary":"Let f : X → Y be a morphism from an algebraic stack to an algebraic space. Assume: • f is of finite type and separated, • Y is quasi-compact and quasi-separated. Then there exists a separated morphism of finite presentation f' : X' → Y and a closed immersion X → X' of algebraic stacks over Y.","statement_latex":"Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack\nto an algebraic space. Assume:\n\\begin{enumerate}\n\\item $f$ is of finite type and separated,\n\\item $Y$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nThen there exists a separated morphism of finite presentation\n$f' : \\mathcal{X}' \\to Y$ and a closed immersion\n$\\mathcal{X} \\to \\mathcal{X}'$ of\nalgebraic stacks over $Y$.","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Finite type closed in finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQ2","source_file":"stacks-limits.tex","source_line":816,"source_end_line":828,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L816-L828","statement_sha256":"cec7be09fa1d4ba8ee51598bd284a841e2665d46ad3fbc97aabb5d041d0a238f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14325,"rank":14325,"depth":73,"x":1882.292,"y":1561.046,"cluster":"algebraic-stacks"},{"id":"stacks:0H29","tag":"0H29","title":"Universally closed morphisms · Lemma 0H29","summary":"Let g : Z → Y be a morphism of affine schemes. Let f : X → Y be a quasi-compact morphism of algebraic stacks. Let z ∈ Z and let T ⊂ |X ×_Y Z| be a closed subset with z not ∈ Im(T → |Z|). If X is quasi-compact, then there exist an open neighbourhood V ⊂ Z of z, a commutative diagram xymatrix V ar[d] ar[r]_a & Z' ar[d]^b Z ar[r]^g & Y, and a closed subset T' ⊂ |X ×_Y Z'| such that • Z' is an affine scheme of finite presentation over Y, • with z' = a(z) we have z' not ∈…","statement_latex":"Let $g : Z \\to Y$ be a morphism of affine schemes.\nLet $f : \\mathcal{X} \\to Y$ be a quasi-compact morphism of algebraic stacks.\nLet $z \\in Z$ and let $T \\subset |\\mathcal{X} \\times_Y Z|$\nbe a closed subset with $z \\not \\in \\Im(T \\to |Z|)$.\nIf $\\mathcal{X}$ is quasi-compact, then there exist\nan open neighbourhood $V \\subset Z$ of $z$,\na commutative diagram\n$$\n\\xymatrix{\nV \\ar[d] \\ar[r]_a & Z' \\ar[d]^b \\\\\nZ \\ar[r]^g & Y,\n}\n$$\nand a closed subset $T' \\subset |X \\times_Y Z'|$ such that\n\\begin{enumerate}\n\\item $Z'$ is an affine scheme of finite presentation over $Y$,\n\\item with $z' = a(z)$ we have $z' \\not \\in \\Im(T' \\to |Z'|)$, and\n\\item the inverse image of $T$ in $|\\mathcal{X} \\times_Y V|$\nmaps into $T'$ via $|\\mathcal{X} \\times_Y V| \\to |\\mathcal{X} \\times_Y Z'|$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H29","source_file":"stacks-limits.tex","source_line":897,"source_end_line":919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L897-L919","statement_sha256":"15eba3332547fbd2732965ae19abcf9170b37997de1eb8090cbcaea60edb6d40","origin":"The Stacks Project","memory_eligible":false,"source_rank":14326,"rank":14326,"depth":1,"x":2170.351,"y":1544.869,"cluster":"algebraic-stacks"},{"id":"stacks:0H2A","tag":"0H2A","title":"Universally closed morphisms · Lemma 0H2A","summary":"Let f : X → Y be a quasi-compact morphism of algebraic stacks. The following are equivalent • f is universally closed, • for every morphism Z → Y which is locally of finite presentation and where Z is an affine scheme the map |X ×_Y Z| → |Z| is closed, and • there exists a scheme V and a surjective smooth morphism V → Y such that |A^n × (X ×_Y V)| → |A^n × V| is closed for all n ≥ 0.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact morphism of\nalgebraic stacks. The following are equivalent\n\\begin{enumerate}\n\\item $f$ is universally closed,\n\\item for every morphism $Z \\to \\mathcal{Y}$ which is locally of finite\npresentation and where $Z$ is an affine scheme\nthe map $|\\mathcal{X} \\times_Y Z| \\to |Z|$ is closed, and\n\\item there exists a scheme $V$ and a surjective smooth morphism\n$V \\to \\mathcal{Y}$ such that\n$|\\mathbf{A}^n \\times (\\mathcal{X} \\times_\\mathcal{Y} V)|\n\\to |\\mathbf{A}^n \\times V|$ is closed for all $n \\geq 0$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Limits of Algebraic Stacks","chapter_id":"stacks-limits","section":"Universally closed morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2A","source_file":"stacks-limits.tex","source_line":974,"source_end_line":988,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-limits.tex#L974-L988","statement_sha256":"17376d09272b1906c14da24b55c6a2c974d6047e8e6aa7eb8106304ab33bf56a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14327,"rank":14327,"depth":59,"x":1970.797,"y":1720.383,"cluster":"algebraic-stacks"},{"id":"stacks:076X","tag":"076X","title":"Pullback of quasi-coherent modules · Lemma 076X","summary":"If f : X → Y is a flat morphism of algebraic stacks then f^* : QCoh(O_Y) → QCoh(O_X) is an exact functor.","statement_latex":"If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a flat morphism of algebraic stacks\nthen $f^* : \\QCoh(\\mathcal{O}_\\mathcal{Y}) \\to\n\\QCoh(\\mathcal{O}_\\mathcal{X})$ is an exact functor.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Pullback of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/076X","source_file":"stacks-cohomology.tex","source_line":129,"source_end_line":134,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L129-L134","statement_sha256":"0e514b18febd962662d7f0f2003bec87de08e121bbf184055475d8cbbbf1044a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14328,"rank":14328,"depth":51,"x":1976.806,"y":1477.545,"cluster":"algebraic-stacks"},{"id":"stacks:0GQF","tag":"0GQF","title":"Pullback of quasi-coherent modules · Lemma 0GQF","summary":"Let X be an algebraic stack. Let I be a set and for i ∈ I let x_i : U_i → X be an object of X. Assume that x_i is flat and coprod x_i : coprod U_i → X is surjective. Let φ : F → G be an arrow of QCoh(O_X). Denote φ_i the restriction of φ to (U_i)_etale. Then φ is injective, resp. surjective, resp. an isomorphism if and only if each φ_i is so.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $I$ be a set and\nfor $i \\in I$ let $x_i : U_i \\to \\mathcal{X}$ be an object\nof $\\mathcal{X}$. Assume that $x_i$ is flat and\n$\\coprod x_i : \\coprod U_i \\to \\mathcal{X}$ is surjective.\nLet $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ be an arrow of\n$\\QCoh(\\mathcal{O}_\\mathcal{X})$. Denote $\\varphi_i$\nthe restriction of $\\varphi$ to $(U_i)_\\etale$.\nThen $\\varphi$ is injective, resp.\\ surjective, resp.\\ an isomorphism\nif and only if each $\\varphi_i$ is so.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Pullback of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQF","source_file":"stacks-cohomology.tex","source_line":166,"source_end_line":177,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L166-L177","statement_sha256":"32859d8c4148cc72d1f7451075759f8c9cf09b8242842b9ff79d7055f2d629a7","origin":"The Stacks Project","memory_eligible":false,"source_rank":14329,"rank":14329,"depth":23,"x":2167.804,"y":1660.159,"cluster":"algebraic-stacks"},{"id":"stacks:076Z","tag":"076Z","title":"Higher direct images of types of modules · Lemma 076Z","summary":"Let M be a rule which associates to every algebraic stack X a subcategory M_X of Mod(X_etale, O_X) such that • M_X is a weak Serre subcategory of Mod(X_etale, O_X) (see Homology, Definition [Tag 02MO]) for all algebraic stacks X, • for a smooth morphism of algebraic stacks f : Y → X the functor f^* maps M_X into M_Y, • if f_i : X_i → X is a family of smooth morphisms of algebraic stacks with |X| = ⋃ |f_i|(|X_i|), then an object F of Mod(X_etale, O_X) is in M_X if and only…","statement_latex":"Let $\\mathcal{M}$ be a rule which associates to every algebraic stack\n$\\mathcal{X}$ a subcategory $\\mathcal{M}_\\mathcal{X}$ of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nsuch that\n\\begin{enumerate}\n\\item $\\mathcal{M}_\\mathcal{X}$ is a weak Serre subcategory\nof $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\n(see Homology, Definition \\ref{homology-definition-serre-subcategory})\nfor all algebraic stacks $\\mathcal{X}$,\n\\item for a smooth morphism of algebraic stacks\n$f : \\mathcal{Y} \\to \\mathcal{X}$ the functor $f^*$ maps\n$\\mathcal{M}_\\mathcal{X}$ into $\\mathcal{M}_\\mathcal{Y}$,\n\\item if $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ is a family of smooth\nmorphisms of algebraic stacks with\n$|\\mathcal{X}| = \\bigcup |f_i|(|\\mathcal{X}_i|)$, then an object\n$\\mathcal{F}$ of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nis in $\\mathcal{M}_\\mathcal{X}$ if and only if\n$f_i^*\\mathcal{F}$ is in $\\mathcal{M}_{\\mathcal{X}_i}$ for all $i$, and\n\\item if $f : \\mathcal{Y} \\to \\mathcal{X}$ is a morphism of algebraic\nstacks such that $\\mathcal{X}$ and $\\mathcal{Y}$ are representable\nby affine schemes, then $R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$\ninto $\\mathcal{M}_\\mathcal{X}$.\n\\end{enumerate}\nThen for any quasi-compact and quasi-separated morphism \n$f : \\mathcal{Y} \\to \\mathcal{X}$ of algebraic stacks\n$R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$\ninto $\\mathcal{M}_\\mathcal{X}$. (Higher direct images computed in \\'etale\ntopology.)","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Higher direct images of types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/076Z","source_file":"stacks-cohomology.tex","source_line":236,"source_end_line":267,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L236-L267","statement_sha256":"fe2e05d6652680728c7cb8643e402e8c3092a7fcbb2c39f0a86cae12d37832a1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14330,"rank":14330,"depth":75,"x":1879.893,"y":1633.859,"cluster":"algebraic-stacks"},{"id":"stacks:0770","tag":"0770","title":"Higher direct images of types of modules · Lemma 0770","summary":"Let M be a rule which associates to every algebraic stack X a subcategory M_X of Mod(O_X) such that • O_X is a weak Serre subcategory of Mod(O_X) for all algebraic stacks X, • for a smooth morphism of algebraic stacks f : Y → X the functor f^* maps M_X into M_Y, • if f_i : X_i → X is a family of smooth morphisms of algebraic stacks with |X| = ⋃ |f_i|(|X_i|), then an object F of Mod(O_X) is in M_X if and only if f_i^*F is in M_X_i for all i, and • if f : Y → X is a…","statement_latex":"Let $\\mathcal{M}$ be a rule which associates to every algebraic stack\n$\\mathcal{X}$ a subcategory $\\mathcal{M}_\\mathcal{X}$ of\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nsuch that\n\\begin{enumerate}\n\\item $\\mathcal{O}_\\mathcal{X}$ is a weak Serre subcategory\nof $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nfor all algebraic stacks $\\mathcal{X}$,\n\\item for a smooth morphism of algebraic stacks\n$f : \\mathcal{Y} \\to \\mathcal{X}$ the functor $f^*$ maps\n$\\mathcal{M}_\\mathcal{X}$ into $\\mathcal{M}_\\mathcal{Y}$,\n\\item if $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ is a family of smooth\nmorphisms of algebraic stacks with\n$|\\mathcal{X}| = \\bigcup |f_i|(|\\mathcal{X}_i|)$, then an object\n$\\mathcal{F}$ of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nis in $\\mathcal{M}_\\mathcal{X}$ if and only if\n$f_i^*\\mathcal{F}$ is in $\\mathcal{M}_{\\mathcal{X}_i}$ for all $i$, and\n\\item if $f : \\mathcal{Y} \\to \\mathcal{X}$ is a morphism of algebraic\nstacks and $\\mathcal{X}$ and $\\mathcal{Y}$ are representable\nby affine schemes, then $R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$\ninto $\\mathcal{M}_\\mathcal{X}$.\n\\end{enumerate}\nThen for any quasi-compact and quasi-separated morphism \n$f : \\mathcal{Y} \\to \\mathcal{X}$ of algebraic stacks\n$R^if_*$ maps $\\mathcal{M}_\\mathcal{Y}$\ninto $\\mathcal{M}_\\mathcal{X}$. (Higher direct images computed in fppf\ntopology.)","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Higher direct images of types of modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0770","source_file":"stacks-cohomology.tex","source_line":368,"source_end_line":397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L368-L397","statement_sha256":"ff4389f82e7bd6d087f83484a6e6eb214e66fb614b157d86cf4f3132bdc044ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":14331,"rank":14331,"depth":76,"x":2113.521,"y":1489.775,"cluster":"algebraic-stacks"},{"id":"stacks:075Y","tag":"075Y","title":"Locally quasi-coherent modules · Lemma 075Y","summary":"Let X be an algebraic stack. Let f_j : X_j → X be a family of smooth morphisms of algebraic stacks with |X| =⋃ |f_j|(|X_j|). Let F be a sheaf of O_X-modules on X_etale. If each f_j^-1F is locally quasi-coherent, then so is F.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let\n$f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of smooth\nmorphisms of algebraic stacks with\n$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules\non $\\mathcal{X}_\\etale$. If each $f_j^{-1}\\mathcal{F}$\nis locally quasi-coherent, then so is $\\mathcal{F}$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Locally quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075Y","source_file":"stacks-cohomology.tex","source_line":425,"source_end_line":434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L425-L434","statement_sha256":"40acafb557a311811503f9d2130b4d2738184b8e19b02a0571168728306c6ebc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14332,"rank":14332,"depth":57,"x":2057.073,"y":1728.768,"cluster":"algebraic-stacks"},{"id":"stacks:075Z","tag":"075Z","title":"Locally quasi-coherent modules · Lemma 075Z","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic stacks. Let F be a locally quasi-coherent O_X-module on X_etale. Then R^if_*F (computed in the étale topology) is locally quasi-coherent on Y_etale.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and\nquasi-separated morphism of algebraic stacks. Let \n$\\mathcal{F}$ be a locally quasi-coherent\n$\\mathcal{O}_\\mathcal{X}$-module on $\\mathcal{X}_\\etale$.\nThen $R^if_*\\mathcal{F}$ (computed in the \\'etale topology) is\nlocally quasi-coherent on $\\mathcal{Y}_\\etale$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Locally quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/075Z","source_file":"stacks-cohomology.tex","source_line":464,"source_end_line":472,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L464-L472","statement_sha256":"dc958e40a4e5571171128f85c81074a77ada580886c9df0342eba5e835c3caa4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14333,"rank":14333,"depth":76,"x":1906.392,"y":1520.35,"cluster":"algebraic-stacks"},{"id":"stacks:07AP","tag":"07AP","title":"Locally quasi-coherent modules · Lemma 07AP","summary":"Let X be an algebraic stack. Let f_j : X_j → X be a family of flat and locally finitely presented morphisms of algebraic stacks with |X| =⋃ |f_j|(|X_j|). Let F be a sheaf of O_X-modules on X_fppf. If each f_j^-1F is locally quasi-coherent, then so is F.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let\n$f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of flat\nand locally finitely presented morphisms of algebraic stacks with\n$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules\non $\\mathcal{X}_{fppf}$. If each $f_j^{-1}\\mathcal{F}$\nis locally quasi-coherent, then so is $\\mathcal{F}$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Locally quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AP","source_file":"stacks-cohomology.tex","source_line":512,"source_end_line":521,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L512-L521","statement_sha256":"7ba75023fd8312ee111bd680748bb6aafbb52f4f0310900e057ad304ec0ba818","origin":"The Stacks Project","memory_eligible":false,"source_rank":14334,"rank":14334,"depth":77,"x":2185.316,"y":1588.586,"cluster":"algebraic-stacks"},{"id":"stacks:0762","tag":"0762","title":"Flat comparison maps · Definition 0762","summary":"Let X be an algebraic stack and let F in Mod(X_etale, O_X). We say F has the flat base change property if and only if c_φ is an isomorphism whenever f is flat.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack and let $\\mathcal{F}$ in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.\nWe say $\\mathcal{F}$ has the {\\it flat base change property}\\footnote{This\nmay be nonstandard notation.}\nif and only if $c_\\varphi$ is an isomorphism whenever $f$ is flat.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Flat comparison maps","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0762","source_file":"stacks-cohomology.tex","source_line":603,"source_end_line":610,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L603-L610","statement_sha256":"aeec0b23253e83f42d6946ee2db0e409b34ff357c2b253a7492b49a3588326f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14335,"rank":14335,"depth":0,"x":1924.572,"y":1696.619,"cluster":"algebraic-stacks"},{"id":"stacks:0764","tag":"0764","title":"Flat comparison maps · Lemma 0764","summary":"Let X be an algebraic stack. Let F be an O_X-module on X_etale. • If F has the flat base change property then for any morphism g : Y → X of algebraic stacks, the pullback g^*F does too. • The full subcategory of Mod(X_etale, O_X) consisting of modules with the flat base change property is a weak Serre subcategory. • Let f_i : X_i → X be a family of smooth morphisms of algebraic stacks such that |X| = ⋃_i |f_i|(|X_i|). If each f_i^*F has the flat base change property then…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{F}$\nbe an $\\mathcal{O}_\\mathcal{X}$-module on $\\mathcal{X}_\\etale$.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ has the flat base change property then for any morphism\n$g : \\mathcal{Y} \\to \\mathcal{X}$ of algebraic stacks, the\npullback $g^*\\mathcal{F}$ does too.\n\\item The full subcategory of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nconsisting of modules with the flat base change property\nis a weak Serre subcategory.\n\\item  Let $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a family of\nsmooth morphisms of algebraic stacks such that\n$|\\mathcal{X}| = \\bigcup_i |f_i|(|\\mathcal{X}_i|)$. If each\n$f_i^*\\mathcal{F}$ has the flat base change property then so does\n$\\mathcal{F}$.\n\\item The category of $\\mathcal{O}_\\mathcal{X}$-modules\non $\\mathcal{X}_\\etale$ with the flat base change property\nhas colimits and they agree with colimits in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}$ and $\\mathcal{G}$ in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nwith the flat base change property then the tensor product\n$\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$\nhas the flat base change property.\n\\item Given $\\mathcal{F}$ and $\\mathcal{G}$ in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nwith $\\mathcal{F}$ of finite presentation and $\\mathcal{G}$ having\nthe flat base change property then the sheaf\n$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$\nhas the flat base change property.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Flat comparison maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0764","source_file":"stacks-cohomology.tex","source_line":615,"source_end_line":648,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L615-L648","statement_sha256":"23043b951da50aa98945d3b6e1d1fb9f8efdc512ff9fc20d8e0712ab7831875b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14336,"rank":14336,"depth":57,"x":2030.042,"y":1468.833,"cluster":"algebraic-stacks"},{"id":"stacks:0765","tag":"0765","title":"Flat comparison maps · Lemma 0765","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic stacks. Let F be an object of Mod(X_etale, O_X) which is locally quasi-coherent and has the flat base change property. Then each R^if_*F (computed in the étale topology) has the flat base change property.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and\nquasi-separated morphism of algebraic stacks. Let \n$\\mathcal{F}$ be an object of\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nwhich is locally quasi-coherent and has the flat base change property.\nThen each $R^if_*\\mathcal{F}$ (computed in the \\'etale topology)\nhas the flat base change property.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Flat comparison maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0765","source_file":"stacks-cohomology.tex","source_line":789,"source_end_line":798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L789-L798","statement_sha256":"b8873cdc526d1b1b4ee5d70250c2c02f83d9edf37871b379a426e5f217e63550","origin":"The Stacks Project","memory_eligible":false,"source_rank":14337,"rank":14337,"depth":77,"x":2135.529,"y":1696.817,"cluster":"algebraic-stacks"},{"id":"stacks:0771","tag":"0771","title":"Locally quasi-coherent modules with the flat base change property · Proposition 0771","summary":"Summary of results on locally quasi-coherent modules having the flat base change property. • Let X be an algebraic stack. If F is in LQCoh^fbc(O_X), then F is a sheaf for the fppf topology, i.e., it is an object of Mod(O_X). • The category LQCoh^fbc(O_X) is a weak Serre subcategory of both Mod(O_X) and Mod(X_etale, O_X). • Pullback f^* along any morphism of algebraic stacks f : X → Y induces a functor f^* : LQCoh^fbc(O_Y) → LQCoh^fbc(O_X). • If f : X → Y is a…","statement_latex":"Summary of results on locally quasi-coherent modules having the flat\nbase change property.\n\\begin{enumerate}\n\\item Let $\\mathcal{X}$ be an algebraic stack.\nIf $\\mathcal{F}$ is in $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$,\nthen $\\mathcal{F}$ is a sheaf for the fppf topology, i.e., it is\nan object of $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.\n\\item The category $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$\nis a weak Serre subcategory of both $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nand $\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.\n\\item Pullback $f^*$ along any morphism of algebraic stacks\n$f : \\mathcal{X} \\to \\mathcal{Y}$ induces a functor\n$f^* : \\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{Y}) \\to\n\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\n\\item If $f : \\mathcal{X} \\to \\mathcal{Y}$ is a\nquasi-compact and quasi-separated morphism of algebraic stacks\nand $\\mathcal{F}$ is an object of\n$\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$, then\n\\begin{enumerate}\n\\item the total direct image $Rf_*\\mathcal{F}$ and the higher direct\nimages $R^if_*\\mathcal{F}$ can be computed in either the \\'etale or the\nfppf topology with the same result, and\n\\item each $R^if_*\\mathcal{F}$ is an object of\n$\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{Y})$.\n\\end{enumerate}\n\\item The category $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$ has\ncolimits and they agree with colimits in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nas well as in $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}$ and $\\mathcal{G}$ in\n$\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$ then the tensor product\n$\\mathcal{F} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{G}$\nis in $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\n\\item Given $\\mathcal{F}$ of finite presentation and $\\mathcal{G}$ in\n$\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$ then\n$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$\nis in $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Locally quasi-coherent modules with the flat base change property","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0771","source_file":"stacks-cohomology.tex","source_line":900,"source_end_line":940,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L900-L940","statement_sha256":"35d3a5f66d2a9b6296dd72d8cbd69f821ce583533f1d8bcddbca6db94301f0eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14338,"rank":14338,"depth":78,"x":1874.21,"y":1588.479,"cluster":"algebraic-stacks"},{"id":"stacks:07AQ","tag":"07AQ","title":"Locally quasi-coherent modules with the flat base change property · Lemma 07AQ","summary":"Let X be an algebraic stack. • Let f_j : X_j → X be a family of smooth morphisms of algebraic stacks with |X| =⋃ |f_j|(|X_j|). Let F be a sheaf of O_X-modules on X_etale. If each f_j^-1F is in LQCoh^fpc(O_X_i), then F is in LQCoh^fbc(O_X). • Let f_j : X_j → X be a family of flat and locally finitely presented morphisms of algebraic stacks with |X| =⋃ |f_j|(|X_j|). Let F be a sheaf of O_X-modules on X_fppf. If each f_j^-1F is in LQCoh^fbc(O_X_i), then F is in LQCoh^fbc(O_X).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\n\\begin{enumerate}\n\\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of smooth\nmorphisms of algebraic stacks with\n$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules\non $\\mathcal{X}_\\etale$. If each $f_j^{-1}\\mathcal{F}$\nis in $\\textit{LQCoh}^{fpc}(\\mathcal{O}_{\\mathcal{X}_i})$, then\n$\\mathcal{F}$ is in\n$\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\n\\item  Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of flat\nand locally finitely presented morphisms of algebraic stacks with\n$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules\non $\\mathcal{X}_{fppf}$. If each $f_j^{-1}\\mathcal{F}$\nis in $\\textit{LQCoh}^{fbc}(\\mathcal{O}_{\\mathcal{X}_i})$, then\n$\\mathcal{F}$ is in $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Locally quasi-coherent modules with the flat base change property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AQ","source_file":"stacks-cohomology.tex","source_line":1015,"source_end_line":1035,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1015-L1035","statement_sha256":"7dca4204e529f3d4095e51394b740c945b9de7bc27e064da8d9408c60f4b957b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14339,"rank":14339,"depth":79,"x":2154.234,"y":1520.036,"cluster":"algebraic-stacks"},{"id":"stacks:0GQH","tag":"0GQH","title":"Locally quasi-coherent modules with the flat base change property · Lemma 0GQH","summary":"Let f : X → Y be a morphism of algebraic stacks which is quasi-compact, quasi-separated, and representable by algebraic spaces. Let F be in LQCoh^fbc(O_X). Then for an object y : V → Y of Y we have (R^if_*F)|_V_etale = R^if'_small, *(F|_U_etale) where f' : U = V ×_Y X → V is the base change of f.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich is quasi-compact, quasi-separated, and\nrepresentable by algebraic spaces. Let $\\mathcal{F}$ be in\n$\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\nThen for an object $y : V \\to \\mathcal{Y}$ of $\\mathcal{Y}$ we have\n$$\n(R^if_*\\mathcal{F})|_{V_\\etale} = R^if'_{small, *}(\\mathcal{F}|_{U_\\etale})\n$$\nwhere $f' : U = V \\times_\\mathcal{Y} \\mathcal{X} \\to V$ is the base\nchange of $f$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Locally quasi-coherent modules with the flat base change property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQH","source_file":"stacks-cohomology.tex","source_line":1092,"source_end_line":1104,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1092-L1104","statement_sha256":"b240cafbfc397dc6a95960837b2576ae2c553bc77f876164480e9d5208c3b85f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14340,"rank":14340,"depth":79,"x":2002.675,"y":1729.554,"cluster":"algebraic-stacks"},{"id":"stacks:0GQI","tag":"0GQI","title":"Locally quasi-coherent modules with the flat base change property · Lemma 0GQI","summary":"Let f : X → Y be an affine morphism of algebraic stacks. The functor f_* : LQCoh^fbc(O_X) → LQCoh^fbc(O_Y) is exact and commutes with direct sums. The functors R^if_* for i > 0 vanish on LQCoh^fbc(O_X).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be an affine morphism of algebraic\nstacks. The functor $f_* : \\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{Y})$ is exact and commutes\nwith direct sums. The functors $R^if_*$ for $i > 0$ vanish on\n$\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Locally quasi-coherent modules with the flat base change property","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQI","source_file":"stacks-cohomology.tex","source_line":1122,"source_end_line":1129,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1122-L1129","statement_sha256":"4bd46454b785cb12d47558987cc3ea6fdd1b63421f67e76586a274c884ba5b67","origin":"The Stacks Project","memory_eligible":false,"source_rank":14341,"rank":14341,"depth":80,"x":1945.902,"y":1488.881,"cluster":"algebraic-stacks"},{"id":"stacks:0773","tag":"0773","title":"Parasitic modules · Definition 0773","summary":"Let X be an algebraic stack. A presheaf of O_X-modules F is parasitic if we have F(x) = 0 for any object x of X which lies over a scheme U such that the corresponding morphism x : U → X is flat.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nA presheaf of $\\mathcal{O}_\\mathcal{X}$-modules $\\mathcal{F}$ is\n{\\it parasitic} if we have $\\mathcal{F}(x) = 0$ for any object $x$\nof $\\mathcal{X}$ which lies over a scheme $U$ such that the corresponding\nmorphism $x : U \\to \\mathcal{X}$ is flat.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Parasitic modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0773","source_file":"stacks-cohomology.tex","source_line":1165,"source_end_line":1172,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1165-L1172","statement_sha256":"7f3174ca885aae5c3525a294f70144b4ad3c7a90075d696c8e2962431b9c198c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14342,"rank":14342,"depth":0,"x":2181.484,"y":1634.244,"cluster":"algebraic-stacks"},{"id":"stacks:0774","tag":"0774","title":"Parasitic modules · Lemma 0774","summary":"Let X be an algebraic stack. Let F be a presheaf of O_X-modules. • If F is parasitic and g : Y → X is a flat morphism of algebraic stacks, then g^*F is parasitic. • For τ ∈ (Zariski, etale, smooth, syntomic, fppf) we have • the τ sheafification of a parasitic presheaf of modules is parasitic, and • the full subcategory of Mod(X_τ, O_X) consisting of parasitic modules is a Serre subcategory. • Suppose F is a sheaf for the étale topology. Let f_i : X_i → X be a family of…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{F}$\nbe a presheaf of $\\mathcal{O}_\\mathcal{X}$-modules.\n\\begin{enumerate}\n\\item If $\\mathcal{F}$ is parasitic and\n$g : \\mathcal{Y} \\to \\mathcal{X}$ is a flat morphism of algebraic stacks,\nthen $g^*\\mathcal{F}$ is parasitic.\n\\item For $\\tau \\in \\{Zariski, \\etale, smooth, syntomic, fppf\\}$\nwe have\n\\begin{enumerate}\n\\item the $\\tau$ sheafification of a parasitic presheaf of modules is\nparasitic, and\n\\item the full subcategory of\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$\nconsisting of parasitic modules is a Serre subcategory.\n\\end{enumerate}\n\\item Suppose $\\mathcal{F}$ is a sheaf for the \\'etale topology.\nLet $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a family of\nsmooth morphisms of algebraic stacks such that\n$|\\mathcal{X}| = \\bigcup_i |f_i|(|\\mathcal{X}_i|)$. If each\n$f_i^*\\mathcal{F}$ is parasitic then so is $\\mathcal{F}$.\n\\item Suppose $\\mathcal{F}$ is a sheaf for the fppf topology.\nLet $f_i : \\mathcal{X}_i \\to \\mathcal{X}$ be a family of\nflat and locally finitely presented morphisms of algebraic stacks such that\n$|\\mathcal{X}| = \\bigcup_i |f_i|(|\\mathcal{X}_i|)$. If each\n$f_i^*\\mathcal{F}$ is parasitic then so is $\\mathcal{F}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Parasitic modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0774","source_file":"stacks-cohomology.tex","source_line":1177,"source_end_line":1205,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1177-L1205","statement_sha256":"2ad0450f75d2960a2769becbc57f463a1f9b470cdf9fdb809bdacb93aa1bfe1a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14343,"rank":14343,"depth":48,"x":1890.656,"y":1660.75,"cluster":"algebraic-stacks"},{"id":"stacks:0775","tag":"0775","title":"Parasitic modules · Lemma 0775","summary":"Let τ ∈ (etale, fppf). Let X be an algebraic stack. Let F be a parasitic object of Mod(X_τ, O_X). • H^i_τ(X, F) = 0 for all i. • Let f : X → Y be a morphism of algebraic stacks. Then R^if_*F (computed in τ-topology) is a parasitic object of Mod(Y_τ, O_Y).","statement_latex":"Let $\\tau \\in \\{\\etale, fppf\\}$.\nLet $\\mathcal{X}$ be an algebraic stack.\nLet $\\mathcal{F}$ be a parasitic object of\n$\\textit{Mod}(\\mathcal{X}_\\tau, \\mathcal{O}_\\mathcal{X})$.\n\\begin{enumerate}\n\\item $H^i_\\tau(\\mathcal{X}, \\mathcal{F}) = 0$ for all $i$.\n\\item Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThen $R^if_*\\mathcal{F}$ (computed in $\\tau$-topology) is a\nparasitic object of $\\textit{Mod}(\\mathcal{Y}_\\tau, \\mathcal{O}_\\mathcal{Y})$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Parasitic modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0775","source_file":"stacks-cohomology.tex","source_line":1276,"source_end_line":1288,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1276-L1288","statement_sha256":"051c4f4a807de29cbd2403bbe33538ea88de106e4d77074195bca4c34d37a2e8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14344,"rank":14344,"depth":76,"x":2083.936,"y":1476.045,"cluster":"algebraic-stacks"},{"id":"stacks:0776","tag":"0776","title":"Parasitic modules · Lemma 0776","summary":"Let X be an algebraic stack. Let α : F → G and β : G → H be maps in QCoh(O_X) with β ∘ α = 0. The following are equivalent: • in the abelian category QCoh(O_X) the complex F → G → H is exact at G, • Ker(β)/Im(α) computed in either Mod(X_etale, O_X) or Mod(X_fppf, O_X) is parasitic.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let\n$\\alpha : \\mathcal{F} \\to \\mathcal{G}$ and\n$\\beta : \\mathcal{G} \\to \\mathcal{H}$\nbe maps in $\\QCoh(\\mathcal{O}_\\mathcal{X})$ with\n$\\beta \\circ \\alpha = 0$. The following are equivalent:\n\\begin{enumerate}\n\\item in the abelian category $\\QCoh(\\mathcal{O}_\\mathcal{X})$\nthe complex $\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}$\nis exact at $\\mathcal{G}$,\n\\item $\\Ker(\\beta)/\\Im(\\alpha)$ computed in either\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ or\n$\\textit{Mod}(\\mathcal{X}_{fppf}, \\mathcal{O}_\\mathcal{X})$\nis parasitic.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Parasitic modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0776","source_file":"stacks-cohomology.tex","source_line":1340,"source_end_line":1356,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1340-L1356","statement_sha256":"e0f25868404e9a617c232d5631ccd7c760aa8bd00f3cf32de3b284be40806f91","origin":"The Stacks Project","memory_eligible":false,"source_rank":14345,"rank":14345,"depth":79,"x":2089.954,"y":1722.098,"cluster":"algebraic-stacks"},{"id":"stacks:0778","tag":"0778","title":"Quasi-coherent modules · Lemma 0778","summary":"Let X be an algebraic stack. Let LQCoh^fbc(O_X) be the category of locally quasi-coherent modules with the flat base change property, see Section [Tag 0GQG]. The inclusion functor i : QCoh(O_X) → LQCoh^fbc(O_X) has a right adjoint Q : LQCoh^fbc(O_X) → QCoh(O_X) such that Q ∘ i is the identity functor.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let\n$\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$\nbe the category of locally quasi-coherent modules with the\nflat base change property, see\nSection \\ref{section-loc-qcoh-flat-base-change}.\nThe inclusion functor\n$i : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$\nhas a right adjoint\n$$\nQ : \\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X}) \\to\n\\QCoh(\\mathcal{O}_\\mathcal{X})\n$$\nsuch that $Q \\circ i$ is the identity functor.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0778","source_file":"stacks-cohomology.tex","source_line":1414,"source_end_line":1430,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1414-L1430","statement_sha256":"acc319aa0def3297c20183587c1f7bc48e8d73bc56ec9a120edf7c82b5dd1d97","origin":"The Stacks Project","memory_eligible":false,"source_rank":14346,"rank":14346,"depth":57,"x":1887.497,"y":1543.945,"cluster":"algebraic-stacks"},{"id":"stacks:0779","tag":"0779","title":"Quasi-coherent modules · Lemma 0779","summary":"Let X be an algebraic stack. Let Q : LQCoh^fbc(O_X) → QCoh(O_X) be the functor constructed in Lemma [Tag 0778]. • The kernel of Q is exactly the collection of parasitic objects of LQCoh^fbc(O_X). • For any object F of LQCoh^fbc(O_X) both the kernel and the cokernel of the adjunction map Q(F) → F are parasitic. • The functor Q is exact and commutes with all limits and colimits.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $Q : \\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X}) \\to\n\\QCoh(\\mathcal{O}_\\mathcal{X})$\nbe the functor constructed in Lemma \\ref{lemma-adjoint}.\n\\begin{enumerate}\n\\item The kernel of $Q$ is exactly the collection of parasitic objects\nof $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\n\\item For any object $\\mathcal{F}$\nof $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$ both the kernel and the\ncokernel of the\nadjunction map $Q(\\mathcal{F}) \\to \\mathcal{F}$ are parasitic.\n\\item The functor $Q$ is exact and commutes with all limits and colimits.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0779","source_file":"stacks-cohomology.tex","source_line":1498,"source_end_line":1513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1498-L1513","statement_sha256":"a2a293e78adda0d086fbbd5b1b660b2d09580852462f9f3508cb3c568b294d2c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14347,"rank":14347,"depth":58,"x":2180.269,"y":1560.446,"cluster":"algebraic-stacks"},{"id":"stacks:0GQJ","tag":"0GQJ","title":"Quasi-coherent modules · Lemma 0GQJ","summary":"Let f : X → Y be a flat morphism of algebraic stacks. Then Q_X ∘ f^* = f^* ∘ Q_Y where Q_X and Q_Y are as in Lemma [Tag 0778].","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a flat morphism of algebraic stacks.\nThen $Q_\\mathcal{X} \\circ f^* = f^* \\circ Q_\\mathcal{Y}$ where\n$Q_\\mathcal{X}$ and $Q_\\mathcal{Y}$ are as in Lemma \\ref{lemma-adjoint}.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQJ","source_file":"stacks-cohomology.tex","source_line":1598,"source_end_line":1603,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1598-L1603","statement_sha256":"f4b0ee1bce8ee8be518abff67a2ff43de9333bb9ca4b196e855a27eaccd5c439","origin":"The Stacks Project","memory_eligible":false,"source_rank":14348,"rank":14348,"depth":80,"x":1950.944,"y":1714.517,"cluster":"algebraic-stacks"},{"id":"stacks:0GQK","tag":"0GQK","title":"Quasi-coherent modules · Lemma 0GQK","summary":"Let X be an algebraic stack. Let x be an object of X lying over the scheme U such that x : U → X is flat. Then for F in QCoh^fbc(O_X) we have Q(F)|_U_etale = F|_U_etale.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $x$ be an object of $\\mathcal{X}$\nlying over the scheme $U$ such that $x : U \\to \\mathcal{X}$ is flat.\nThen for $\\mathcal{F}$ in $\\QCoh^{fbc}(\\mathcal{O}_\\mathcal{X})$\nwe have $Q(\\mathcal{F})|_{U_\\etale} = \\mathcal{F}|_{U_\\etale}$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQK","source_file":"stacks-cohomology.tex","source_line":1624,"source_end_line":1630,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1624-L1630","statement_sha256":"3e1f58559aab3ef3f235d88dddd3d2c63882cd901b5e495f185485af53184f6f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14349,"rank":14349,"depth":59,"x":1996.178,"y":1470.603,"cluster":"algebraic-stacks"},{"id":"stacks:0GQN","tag":"0GQN","title":"Quasi-coherent modules · Lemma 0GQN","summary":"Let X be an algebraic stack. Let F be an O_X-module of finite presentation and let G be a quasi-coherent O_X-module. The internal homs SheafHom_O_X(F, G) computed in Mod(X_etale, O_X) or Mod(O_X) agree and the common value is an object of LQCoh^fbc(O_X). The quasi-coherent module hom(F, G) = Q(SheafHom_O_X(F, G)) has the following universal property Hom_X(H, hom(F, G)) = Hom_X(H ⊗_O_X F, G) for H in QCoh(O_X).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $\\mathcal{F}$ be an\n$\\mathcal{O}_\\mathcal{X}$-module of finite presentation and let\n$\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_\\mathcal{X}$-module.\nThe internal homs\n$\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G})$\ncomputed in\n$\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$ or\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ agree and the common value\nis an object of $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\nThe quasi-coherent module\n$\nhom(\\mathcal{F}, \\mathcal{G}) =\nQ(\\SheafHom_{\\mathcal{O}_\\mathcal{X}}(\\mathcal{F}, \\mathcal{G}))\n$\nhas the following universal property\n$$\n\\Hom_\\mathcal{X}(\\mathcal{H}, hom(\\mathcal{F}, \\mathcal{G})) =\n\\Hom_\\mathcal{X}(\\mathcal{H} \\otimes_{\\mathcal{O}_\\mathcal{X}} \\mathcal{F},\n\\mathcal{G})\n$$\nfor $\\mathcal{H}$ in $\\QCoh(\\mathcal{O}_\\mathcal{X})$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQN","source_file":"stacks-cohomology.tex","source_line":1746,"source_end_line":1769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1746-L1769","statement_sha256":"aadaf04331f79136978c9c5abe21f7127a624c852d4f1e021b7607b9cb2b7bd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14350,"rank":14350,"depth":58,"x":2159.092,"y":1676.28,"cluster":"algebraic-stacks"},{"id":"stacks:0GQP","tag":"0GQP","title":"Quasi-coherent modules · Lemma 0GQP","summary":"Let f : X → Y be a flat morphism of algebraic stacks. Let F be an O_Y-module of finite presentation and let G be a quasi-coherent O_Y-module. Then f^*hom(F, G) = hom(f^*F, f^*G) with notation as in Lemma [Tag 0GQN].","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a flat morphism of algebraic stacks.\nLet $\\mathcal{F}$ be an\n$\\mathcal{O}_\\mathcal{Y}$-module of finite presentation and let\n$\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_\\mathcal{Y}$-module.\nThen $f^*hom(\\mathcal{F}, \\mathcal{G}) = hom(f^*\\mathcal{F}, f^*\\mathcal{G})$\nwith notation as in Lemma \\ref{lemma-internal-hom-fp-into-qcoh}.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQP","source_file":"stacks-cohomology.tex","source_line":1801,"source_end_line":1809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1801-L1809","statement_sha256":"05a4bbfc5157611638a5216bcc85d865600ab0486a803ede141d31a0c7a3e00d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14351,"rank":14351,"depth":81,"x":1873.352,"y":1617.015,"cluster":"algebraic-stacks"},{"id":"stacks:077A","tag":"077A","title":"Pushforward of quasi-coherent modules · Proposition 077A","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor f^* : QCoh(O_Y) → QCoh(O_X) has a right adjoint f_QCoh, * : QCoh(O_X) → QCoh(O_Y) which can be defined as the composition QCoh(O_X) → LQCoh^fbc(O_X) xrightarrowf_* LQCoh^fbc(O_Y) xrightarrowQ QCoh(O_Y) where the functors f_* and Q are as in Proposition [Tag 0771] and Lemma [Tag 0778]. Moreover, if we define R^if_QCoh, * as the composition QCoh(O_X) → LQCoh^fbc(O_X)…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated\nmorphism of algebraic stacks. The functor\n$f^* : \\QCoh(\\mathcal{O}_\\mathcal{Y}) \\to\n\\QCoh(\\mathcal{O}_\\mathcal{X})$\nhas a right adjoint\n$$\nf_{\\QCoh, *} :\n\\QCoh(\\mathcal{O}_\\mathcal{X})\n\\longrightarrow\n\\QCoh(\\mathcal{O}_\\mathcal{Y})\n$$\nwhich can be defined as the composition\n$$\n\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})\n\\xrightarrow{f_*} \\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{Y})\n\\xrightarrow{Q} \\QCoh(\\mathcal{O}_\\mathcal{Y})\n$$\nwhere the functors $f_*$ and $Q$ are as in\nProposition \\ref{proposition-loc-qcoh-flat-base-change}\nand\nLemma \\ref{lemma-adjoint}.\nMoreover, if we define $R^if_{\\QCoh, *}$ as the composition\n$$\n\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})\n\\xrightarrow{R^if_*} \\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{Y})\n\\xrightarrow{Q} \\QCoh(\\mathcal{O}_\\mathcal{Y})\n$$\nthen the sequence of functors $\\{R^if_{\\QCoh, *}\\}_{i \\geq 0}$\nforms a cohomological $\\delta$-functor.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Pushforward of quasi-coherent modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077A","source_file":"stacks-cohomology.tex","source_line":1904,"source_end_line":1937,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L1904-L1937","statement_sha256":"30b07e6e1990b93eed404589408dff01efa4c5b62306f2da878aeea678f0797a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14352,"rank":14352,"depth":80,"x":2131.901,"y":1498.493,"cluster":"algebraic-stacks"},{"id":"stacks:0GQQ","tag":"0GQQ","title":"Pushforward of quasi-coherent modules · Lemma 0GQQ","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic stacks. Let y : V → Y in Ob(Y) with y a flat morphism. Let F be in QCoh(O_X). Then (f_*F)(y) = (f_QCoh, *F)(y) and (R^if_*F)(y) = (R^if_QCoh, *F)(y) for all i ∈ Z.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated\nmorphism of algebraic stacks. Let $y : V \\to \\mathcal{Y}$ in $\\Ob(\\mathcal{Y})$\nwith $y$ a flat morphism. Let $\\mathcal{F}$ be in\n$\\QCoh(\\mathcal{O}_\\mathcal{X})$.\nThen $(f_*\\mathcal{F})(y) = (f_{\\QCoh, *}\\mathcal{F})(y)$\nand $(R^if_*\\mathcal{F})(y) = (R^if_{\\QCoh, *}\\mathcal{F})(y)$\nfor all $i \\in \\mathbf{Z}$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Pushforward of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQQ","source_file":"stacks-cohomology.tex","source_line":2031,"source_end_line":2040,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2031-L2040","statement_sha256":"d95200d24477f955e865bc45bde07665d1fc209c786cb00e9cb7838eb9f68fa5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14353,"rank":14353,"depth":81,"x":2036.493,"y":1732.768,"cluster":"algebraic-stacks"},{"id":"stacks:0782","tag":"0782","title":"Pushforward of quasi-coherent modules · Lemma 0782","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic stacks. Let F be a quasi-coherent sheaf on X. Then there exists a spectral sequence with E_2-page E_2^p, q = H^p(Y, R^qf_QCoh, *F) converging to H^p + q(X, F).","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$\nbe a quasi-compact and quasi-separated morphism of algebraic stacks.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathcal{X}$. Then\nthere exists a spectral sequence with $E_2$-page\n$$\nE_2^{p, q} = H^p(\\mathcal{Y}, R^qf_{\\QCoh, *}\\mathcal{F})\n$$\nconverging to $H^{p + q}(\\mathcal{X}, \\mathcal{F})$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Pushforward of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0782","source_file":"stacks-cohomology.tex","source_line":2083,"source_end_line":2093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2083-L2093","statement_sha256":"c6fed6713f3ff35e37cca0468b83297913b12f3a7dca784b8e6332194acfe204","origin":"The Stacks Project","memory_eligible":false,"source_rank":14354,"rank":14354,"depth":77,"x":1918.362,"y":1505.714,"cluster":"algebraic-stacks"},{"id":"stacks:0783","tag":"0783","title":"Pushforward of quasi-coherent modules · Lemma 0783","summary":"Let f : X → Y and g : Y → Z be quasi-compact and quasi-separated morphisms of algebraic stacks. Let F be a quasi-coherent sheaf on X. Then there exists a spectral sequence with E_2-page E_2^p, q = R^pg_QCoh, *(R^qf_QCoh, *F) converging to R^p + q(g ∘ f)_QCoh, *F.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $g : \\mathcal{Y} \\to \\mathcal{Z}$\nbe quasi-compact and quasi-separated morphisms of algebraic stacks.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf on $\\mathcal{X}$. Then\nthere exists a spectral sequence with $E_2$-page\n$$\nE_2^{p, q} = R^pg_{\\QCoh, *}(R^qf_{\\QCoh, *}\\mathcal{F})\n$$\nconverging to $R^{p + q}(g \\circ f)_{\\QCoh, *}\\mathcal{F}$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Pushforward of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0783","source_file":"stacks-cohomology.tex","source_line":2115,"source_end_line":2125,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2115-L2125","statement_sha256":"a6d94c5dbf83652728b24216f3cc40b7b29c48f7e435341933820f72217ba42d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14355,"rank":14355,"depth":79,"x":2188.258,"y":1606.183,"cluster":"algebraic-stacks"},{"id":"stacks:0784","tag":"0784","title":"Pushforward of quasi-coherent modules · Proposition 0784","summary":"Let f : U → X be a morphism of algebraic stacks. Assume f is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Let F be a quasi-coherent O_X-module. Then there is a spectral sequence E_2^p, q = H^q(U_p, f_p^*F) ⇒ H^p + q(X, F) where f_p is the morphism U ×_X … ×_X U → X (p + 1 factors).","statement_latex":"Let $f : \\mathcal{U} \\to \\mathcal{X}$ be a morphism of algebraic stacks.\nAssume $f$ is representable by algebraic spaces, surjective, flat, and\nlocally of finite presentation. Let $\\mathcal{F}$ be a quasi-coherent\n$\\mathcal{O}_\\mathcal{X}$-module. Then there is a spectral sequence\n$$\nE_2^{p, q} = H^q(\\mathcal{U}_p, f_p^*\\mathcal{F})\n\\Rightarrow\nH^{p + q}(\\mathcal{X}, \\mathcal{F})\n$$\nwhere $f_p$ is the morphism\n$\\mathcal{U} \\times_\\mathcal{X} \\ldots \\times_\\mathcal{X} \\mathcal{U} \\to\n\\mathcal{X}$ ($p + 1$ factors).","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Pushforward of quasi-coherent modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0784","source_file":"stacks-cohomology.tex","source_line":2161,"source_end_line":2175,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2161-L2175","statement_sha256":"ab90bef05aec177a32acb2551ccccf334f170dd56997c888a15358a1dee2b14b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14356,"rank":14356,"depth":73,"x":1908.242,"y":1685.302,"cluster":"algebraic-stacks"},{"id":"stacks:0785","tag":"0785","title":"Pushforward of quasi-coherent modules · Proposition 0785","summary":"Let f : U → X and g : X → Y be composable morphisms of algebraic stacks. Assume that • f is representable by algebraic spaces, surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated, and • g is quasi-compact and quasi-separated. If F is in QCoh(O_X) then there is a spectral sequence E_2^p, q = R^q(g ∘ f_p)_QCoh, *f_p^*F ⇒ R^p + qg_QCoh, *F in QCoh(O_Y).","statement_latex":"Let $f : \\mathcal{U} \\to \\mathcal{X}$ and $g : \\mathcal{X} \\to \\mathcal{Y}$\nbe composable morphisms of algebraic stacks.\nAssume that\n\\begin{enumerate}\n\\item $f$ is representable by algebraic spaces, surjective,\nflat, locally of finite presentation, quasi-compact, and quasi-separated, and\n\\item $g$ is quasi-compact and quasi-separated.\n\\end{enumerate}\nIf $\\mathcal{F}$ is in $\\QCoh(\\mathcal{O}_\\mathcal{X})$ then\nthere is a spectral sequence\n$$\nE_2^{p, q} = R^q(g \\circ f_p)_{\\QCoh, *}f_p^*\\mathcal{F}\n\\Rightarrow\nR^{p + q}g_{\\QCoh, *}\\mathcal{F}\n$$\nin $\\QCoh(\\mathcal{O}_\\mathcal{Y})$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Pushforward of quasi-coherent modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0785","source_file":"stacks-cohomology.tex","source_line":2183,"source_end_line":2201,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2183-L2201","statement_sha256":"07629e45ee676f156e2f94cbacfc74a185cc0b3fcdc578b89b9200e33abd45f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14357,"rank":14357,"depth":75,"x":2051.2,"y":1467.915,"cluster":"algebraic-stacks"},{"id":"stacks:0GQV","tag":"0GQV","title":"Colimits and cohomology · Lemma 0GQV","summary":"Let X be a quasi-compact and quasi-separated algebraic stack. Then colim_i H^p(X, F_i) → H^p(X, colim_i F_i) is an isomorphism for every filtered diagram of abelian sheaves on X. The same is true for abelian sheaves on X_etale taking cohomology in the étale topology.","statement_latex":"Let $\\mathcal{X}$ be a quasi-compact and quasi-separated algebraic stack.\nThen\n$$\n\\colim_i H^p(\\mathcal{X}, \\mathcal{F}_i)\n\\longrightarrow\nH^p(\\mathcal{X}, \\colim_i \\mathcal{F}_i)\n$$\nis an isomorphism for every filtered diagram of abelian sheaves on\n$\\mathcal{X}$. The same is true for abelian sheaves on $\\mathcal{X}_\\etale$\ntaking cohomology in the \\'etale topology.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Colimits and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQV","source_file":"stacks-cohomology.tex","source_line":2386,"source_end_line":2398,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2386-L2398","statement_sha256":"fdcadab8c86e81362543fd8319b80a03dfa855d50ed83e898dd186fba2ff0841","origin":"The Stacks Project","memory_eligible":false,"source_rank":14358,"rank":14358,"depth":64,"x":2120.652,"y":1709.507,"cluster":"algebraic-stacks"},{"id":"stacks:0GQW","tag":"0GQW","title":"Colimits and cohomology · Lemma 0GQW","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic stacks. Let F = colim F_i be a filtered colimit of abelian sheaves on X. Then for any p ≥ 0 we have R^pf_*F = colim R^pf_*F_i. The same is true for abelian sheaves on X_etale taking higher direct images in the étale topology.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated\nmorphism of algebraic stacks. Let $\\mathcal{F} = \\colim \\mathcal{F}_i$\nbe a filtered colimit of abelian sheaves on $\\mathcal{X}$.\nThen for any $p \\geq 0$ we have\n$$\nR^pf_*\\mathcal{F} = \\colim R^pf_*\\mathcal{F}_i.\n$$\nThe same is true for abelian sheaves on $\\mathcal{X}_\\etale$\ntaking higher direct images in the \\'etale topology.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Colimits and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQW","source_file":"stacks-cohomology.tex","source_line":2445,"source_end_line":2456,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2445-L2456","statement_sha256":"16265028cfbfe19283fcb350aefb88f6c40826abecf35eb01af5420b06d48ca5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14359,"rank":14359,"depth":73,"x":1874.98,"y":1570.668,"cluster":"algebraic-stacks"},{"id":"stacks:0GQX","tag":"0GQX","title":"Colimits and cohomology · Lemma 0GQX","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor f_QCoh, * and the functors R^if_QCoh, * commute with direct sums and filtered colimits.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and quasi-separated\nmorphism of algebraic stacks. The functor $f_{\\QCoh, *}$\nand the functors $R^if_{\\QCoh, *}$ commute with direct sums\nand filtered colimits.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Colimits and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQX","source_file":"stacks-cohomology.tex","source_line":2477,"source_end_line":2483,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2477-L2483","statement_sha256":"e31769b5c27d8f3e1395136c0f92453cde3e0fb8db7b1533e838a21fa85c9506","origin":"The Stacks Project","memory_eligible":false,"source_rank":14360,"rank":14360,"depth":74,"x":2167.997,"y":1533.619,"cluster":"algebraic-stacks"},{"id":"stacks:0GQY","tag":"0GQY","title":"Colimits and cohomology · Lemma 0GQY","summary":"Let f : X → Y be an affine morphism of algebraic stacks. The functors R^if_QCoh, *, i > 0 vanish and the functor f_QCoh, * is exact and commutes with direct sums and all colimits.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be an affine morphism of algebraic\nstacks. The functors $R^if_{\\QCoh, *}$, $i > 0$ vanish and the functor\n$f_{\\QCoh, *}$ is exact and commutes with direct sums and all colimits.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Colimits and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQY","source_file":"stacks-cohomology.tex","source_line":2493,"source_end_line":2498,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2493-L2498","statement_sha256":"5c5a1b86abde13bdc0eb01444aeec7ac58fe4d0f1e8d7beaeb692752465a09b7","origin":"The Stacks Project","memory_eligible":false,"source_rank":14361,"rank":14361,"depth":81,"x":1981.59,"y":1727.344,"cluster":"algebraic-stacks"},{"id":"stacks:0GQZ","tag":"0GQZ","title":"Colimits and cohomology · Lemma 0GQZ","summary":"Let X be a quasi-compact and quasi-separated algebraic stack. Let I be a directed set and let (F_i, φ_ii') be a system over I of O_X-modules. Let G be an O_X-module of finite presentation. Then we have colim_i Hom_X(G, F_i) = Hom_X(G, colim_i F_i). In particular, Hom_X(G, -) commutes with filtered colimits in QCoh(O_X).","statement_latex":"Let $\\mathcal{X}$ be a quasi-compact and quasi-separated algebraic stack.\nLet $I$ be a directed set and let $(\\mathcal{F}_i, \\varphi_{ii'})$ be a\nsystem over $I$ of $\\mathcal{O}_\\mathcal{X}$-modules. Let $\\mathcal{G}$ be an\n$\\mathcal{O}_\\mathcal{X}$-module of finite presentation. Then we have\n$$\n\\colim_i \\Hom_\\mathcal{X}(\\mathcal{G}, \\mathcal{F}_i)\n=\n\\Hom_\\mathcal{X}(\\mathcal{G}, \\colim_i \\mathcal{F}_i).\n$$\nIn particular, $\\Hom_\\mathcal{X}(\\mathcal{G}, -)$ commutes with filtered\ncolimits in $\\QCoh(\\mathcal{O}_\\mathcal{X})$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Colimits and cohomology","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GQZ","source_file":"stacks-cohomology.tex","source_line":2516,"source_end_line":2529,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2516-L2529","statement_sha256":"392f0c04d7233b2ef98bf6ee89583e8dd490eb93d40cb5c49524631a16f6ea83","origin":"The Stacks Project","memory_eligible":false,"source_rank":14362,"rank":14362,"depth":64,"x":1963.242,"y":1478.542,"cluster":"algebraic-stacks"},{"id":"stacks:0787","tag":"0787","title":"The lisse-étale and the flat-fppf sites · Definition 0787","summary":"Let X be an algebraic stack. • The lisse-étale site of X is the full subcategory X_lisse,etale(X) or Lis-Et(X) and the associated topos is denoted X_lis-ét or X_lis-et. In the Stacks project our convention is to name the site and denote the corresponding topos by Sh(C). of X whose objects are those x ∈ Ob(X) lying over a scheme U such that x : U → X is smooth. A covering of X_lisse,etale is a family of morphisms (x_i → x)_i ∈ I of X_lisse,etale which forms a covering of…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\n\\begin{enumerate}\n\\item The {\\it lisse-\\'etale site} of $\\mathcal{X}$ is the full subcategory\n$\\mathcal{X}_{lisse,\\etale}$\\footnote{In the literature the\nsite is denoted $\\text{Lis-\\'et}(\\mathcal{X})$ or\n$\\text{Lis-Et}(\\mathcal{X})$ and the associated topos is denoted\n$\\mathcal{X}_{\\text{lis-\\'e}t}$ or $\\mathcal{X}_{\\text{lis-et}}$.\nIn the Stacks project our convention is to name the site and\ndenote the corresponding topos by $\\Sh(\\mathcal{C})$.} of $\\mathcal{X}$\nwhose objects are those $x \\in \\Ob(\\mathcal{X})$ lying over a scheme $U$\nsuch that $x : U \\to \\mathcal{X}$ is smooth. A covering of\n$\\mathcal{X}_{lisse,\\etale}$ is a family of morphisms\n$\\{x_i \\to x\\}_{i \\in I}$ of $\\mathcal{X}_{lisse,\\etale}$\nwhich forms a covering of $\\mathcal{X}_\\etale$.\n\\item The {\\it flat-fppf site} of $\\mathcal{X}$ is the full subcategory\n$\\mathcal{X}_{flat,fppf}$ of $\\mathcal{X}$\nwhose objects are those $x \\in \\Ob(\\mathcal{X})$ lying over a scheme $U$\nsuch that $x : U \\to \\mathcal{X}$ is flat. A covering of\n$\\mathcal{X}_{flat,fppf}$ is a family of morphisms\n$\\{x_i \\to x\\}_{i \\in I}$ of $\\mathcal{X}_{flat,fppf}$\nwhich forms a covering of $\\mathcal{X}_{fppf}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"The lisse-étale and the flat-fppf sites","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0787","source_file":"stacks-cohomology.tex","source_line":2592,"source_end_line":2616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2592-L2616","statement_sha256":"326d531700d6b0a4379a4b24e5db5b58a513563a27aa539265af79a0d9c5f63a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14363,"rank":14363,"depth":0,"x":2177.005,"y":1651.719,"cluster":"algebraic-stacks"},{"id":"stacks:0788","tag":"0788","title":"The lisse-étale and the flat-fppf sites · Lemma 0788","summary":"Let X be an algebraic stack. • The inclusion functor X_lisse,etale → X_etale is fully faithful, continuous and cocontinuous. It follows that • there is a morphism of topoi g : Sh(X_lisse,etale) → Sh(X_etale) with g^-1 given by restriction, • the functor g^-1 has a left adjoint g_!^Sh on sheaves of sets, • the adjunction maps g^-1g_* → id and id → g^-1g_!^Sh are isomorphisms, • the functor g^-1 has a left adjoint g_! on abelian sheaves, • the adjunction map id → g^-1g_! is…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\n\\begin{enumerate}\n\\item The inclusion functor\n$\\mathcal{X}_{lisse,\\etale} \\to \\mathcal{X}_\\etale$\nis fully faithful, continuous and cocontinuous. It follows that\n\\begin{enumerate}\n\\item there is a morphism of topoi\n$$\ng :\n\\Sh(\\mathcal{X}_{lisse,\\etale})\n\\longrightarrow\n\\Sh(\\mathcal{X}_\\etale)\n$$\nwith $g^{-1}$ given by restriction,\n\\item the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets,\n\\item the adjunction maps $g^{-1}g_* \\to \\text{id}$ and\n$\\text{id} \\to g^{-1}g_!^{Sh}$ are isomorphisms,\n\\item the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves,\n\\item the adjunction map $\\text{id} \\to g^{-1}g_!$ is an isomorphism, and\n\\item we have $g^{-1}\\mathcal{O}_\\mathcal{X} =\n\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$ hence $g$ induces a flat\nmorphism of ringed topoi such that $g^{-1} = g^*$.\n\\end{enumerate}\n\\item The inclusion functor\n$\\mathcal{X}_{flat,fppf} \\to \\mathcal{X}_{fppf}$\nis fully faithful, continuous and cocontinuous. It follows that\n\\begin{enumerate}\n\\item there is a morphism of topoi\n$$\ng :\n\\Sh(\\mathcal{X}_{flat,fppf})\n\\longrightarrow\n\\Sh(\\mathcal{X}_{fppf})\n$$\nwith $g^{-1}$ given by restriction,\n\\item the functor $g^{-1}$ has a left adjoint $g_!^{Sh}$ on sheaves of sets,\n\\item the adjunction maps $g^{-1}g_* \\to \\text{id}$ and\n$\\text{id} \\to g^{-1}g_!^{Sh}$ are isomorphisms,\n\\item the functor $g^{-1}$ has a left adjoint $g_!$ on abelian sheaves,\n\\item the adjunction map $\\text{id} \\to g^{-1}g_!$ is an isomorphism, and\n\\item we have $g^{-1}\\mathcal{O}_\\mathcal{X} =\n\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$ hence $g$ induces a flat\nmorphism of ringed topoi such that $g^{-1} = g^*$.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"The lisse-étale and the flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0788","source_file":"stacks-cohomology.tex","source_line":2625,"source_end_line":2672,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2625-L2672","statement_sha256":"ff2adfddf9fb2621f0a4ed1357e85acf2553eeb8aaeee7ad26f35a43902d9972","origin":"The Stacks Project","memory_eligible":false,"source_rank":14364,"rank":14364,"depth":8,"x":1879.902,"y":1645.311,"cluster":"algebraic-stacks"},{"id":"stacks:0GR0","tag":"0GR0","title":"The lisse-étale and the flat-fppf sites · Lemma 0GR0","summary":"Let X be an algebraic stack. Notation as in Lemma [Tag 0788]. • For an abelian sheaf F on X_etale we have • H^p(X_etale, F) = H^p(X_lisse,etale, g^-1F), and • H^p(x, F) = H^p(X_lisse,etale/x, g^-1F) for any object x of X_lisse,etale. The same holds for sheaves of modules. • For an abelian sheaf F on X_fppf we have • H^p(X_fppf, F) = H^p(X_flat,fppf, g^-1F), and • H^p(x, F) = H^p(X_flat,fppf/x, g^-1F) for any object x of X_flat,fppf. The same holds for sheaves of modules.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Notation as in\nLemma \\ref{lemma-lisse-etale}.\n\\begin{enumerate}\n\\item For an abelian sheaf $\\mathcal{F}$ on $\\mathcal{X}_\\etale$ we have\n\\begin{enumerate}\n\\item $H^p(\\mathcal{X}_\\etale, \\mathcal{F}) =\nH^p(\\mathcal{X}_{lisse,\\etale}, g^{-1}\\mathcal{F})$, and\n\\item $H^p(x, \\mathcal{F}) =\nH^p(\\mathcal{X}_{lisse,\\etale}/x, g^{-1}\\mathcal{F})$\nfor any object $x$ of $\\mathcal{X}_{lisse,\\etale}$.\n\\end{enumerate}\nThe same holds for sheaves of modules.\n\\item For an abelian sheaf $\\mathcal{F}$ on $\\mathcal{X}_{fppf}$ we have\n\\begin{enumerate}\n\\item $H^p(\\mathcal{X}_{fppf}, \\mathcal{F}) =\nH^p(\\mathcal{X}_{flat,fppf}, g^{-1}\\mathcal{F})$, and\n\\item $H^p(x, \\mathcal{F}) =\nH^p(\\mathcal{X}_{flat,fppf}/x, g^{-1}\\mathcal{F})$\nfor any object $x$ of $\\mathcal{X}_{flat,fppf}$.\n\\end{enumerate}\nThe same holds for sheaves of modules.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"The lisse-étale and the flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GR0","source_file":"stacks-cohomology.tex","source_line":2688,"source_end_line":2712,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2688-L2712","statement_sha256":"7ae9662ee579e8dcfcef1a85732f5f44a5d9d8debc6706af2bfa7e52dbfa0cad","origin":"The Stacks Project","memory_eligible":false,"source_rank":14365,"rank":14365,"depth":58,"x":2104.296,"y":1481.334,"cluster":"algebraic-stacks"},{"id":"stacks:0789","tag":"0789","title":"The lisse-étale and the flat-fppf sites · Lemma 0789","summary":"Let X be an algebraic stack. Notation as in Lemma [Tag 0788]. • There exists a functor g_! : Mod(X_lisse,etale, O_X_lisse,etale) → Mod(X_etale, O_X) which is left adjoint to g^*. Moreover it agrees with the functor g_! on abelian sheaves and g^*g_! = id. • There exists a functor g_! : Mod(X_flat,fppf, O_X_flat,fppf) → Mod(X_fppf, O_X) which is left adjoint to g^*. Moreover it agrees with the functor g_! on abelian sheaves and g^*g_! = id.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Notation as in\nLemma \\ref{lemma-lisse-etale}.\n\\begin{enumerate}\n\\item There exists a functor\n$$\ng_! :\n\\textit{Mod}(\\mathcal{X}_{lisse,\\etale},\n\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_{\\mathcal{X}})\n$$\nwhich is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$\non abelian sheaves and $g^*g_! = \\text{id}$.\n\\item There exists a functor\n$$\ng_! :\n\\textit{Mod}(\\mathcal{X}_{flat,fppf},\n\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})\n\\longrightarrow\n\\textit{Mod}(\\mathcal{X}_{fppf}, \\mathcal{O}_{\\mathcal{X}})\n$$\nwhich is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$\non abelian sheaves and $g^*g_! = \\text{id}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"The lisse-étale and the flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0789","source_file":"stacks-cohomology.tex","source_line":2724,"source_end_line":2750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2724-L2750","statement_sha256":"148aecc077112fd2348ba4ad5f8844e941dc8ccd050145055516b9a806a3cbe5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14366,"rank":14366,"depth":9,"x":2070.672,"y":1729.753,"cluster":"algebraic-stacks"},{"id":"stacks:078A","tag":"078A","title":"The lisse-étale and the flat-fppf sites · Lemma 078A","summary":"Let X be an algebraic stack. Notation as in Lemmas [Tag 0788] and [Tag 0789]. • We have g_!O_X_lisse,etale = O_X. • We have g_!O_X_flat, fppf = O_X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Notation as in\nLemmas \\ref{lemma-lisse-etale} and \\ref{lemma-lisse-etale-modules}.\n\\begin{enumerate}\n\\item We have $g_!\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}} =\n\\mathcal{O}_\\mathcal{X}$.\n\\item We have $g_!\\mathcal{O}_{\\mathcal{X}_{flat, fppf}} =\n\\mathcal{O}_\\mathcal{X}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"The lisse-étale and the flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/078A","source_file":"stacks-cohomology.tex","source_line":2800,"source_end_line":2810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2800-L2810","statement_sha256":"5f7c99eba2027ac4fdbde57ac2a93cd86261220e90a0dc0450ecac464cfae319","origin":"The Stacks Project","memory_eligible":false,"source_rank":14367,"rank":14367,"depth":49,"x":1895.569,"y":1527.349,"cluster":"algebraic-stacks"},{"id":"stacks:07AR","tag":"07AR","title":"The lisse-étale and the flat-fppf sites · Lemma 07AR","summary":"Let X be an algebraic stack. • Let F be an O_X-module with the flat base change property on X_etale. The following are equivalent • F is parasitic, and • g^*F = 0 where g : Sh(X_lisse,etale) → Sh(X_etale) is as in Lemma [Tag 0788]. • Let F be an O_X-module on X_fppf. The following are equivalent • F is parasitic, and • g^*F = 0 where g : Sh(X_flat,fppf) → Sh(X_fppf) is as in Lemma [Tag 0788].","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\n\\begin{enumerate}\n\\item Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{X}$-module\nwith the flat base change property on $\\mathcal{X}_\\etale$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is parasitic, and\n\\item $g^*\\mathcal{F} = 0$ where\n$g : \\Sh(\\mathcal{X}_{lisse,\\etale}) \\to\n\\Sh(\\mathcal{X}_\\etale)$ is as in Lemma \\ref{lemma-lisse-etale}.\n\\end{enumerate}\n\\item Let $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{X}$-module on\n$\\mathcal{X}_{fppf}$. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is parasitic, and\n\\item $g^*\\mathcal{F} = 0$ where\n$g :  \\Sh(\\mathcal{X}_{flat,fppf}) \\to \\Sh(\\mathcal{X}_{fppf})$\nis as in Lemma \\ref{lemma-lisse-etale}.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"The lisse-étale and the flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AR","source_file":"stacks-cohomology.tex","source_line":2914,"source_end_line":2936,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2914-L2936","statement_sha256":"b722fa45ca229e886d08991d151c9cb07f8736b11974e23dc7186375e70271a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14368,"rank":14368,"depth":9,"x":2187.665,"y":1577.276,"cluster":"algebraic-stacks"},{"id":"stacks:07AT","tag":"07AT","title":"Functoriality of the lisse-étale and flat-fppf sites · Lemma 07AT","summary":"Let f : X → Y be a morphism of algebraic stacks. • If f is smooth, then f restricts to a continuous and cocontinuous functor X_lisse,etale → Y_lisse,etale which gives a morphism of ringed topoi fitting into the following commutative diagram xymatrix Sh(X_lisse,etale) ar[r]_g' ar[d]_f' & Sh(X_etale) ar[d]^f Sh(Y_lisse,etale) ar[r]^g & Sh(Y_etale) We have f'_*(g')^-1 = g^-1f_* and g'_!(f')^-1 = f^-1g_!. • If f is flat, then f restricts to a continuous and cocontinuous…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\n\\begin{enumerate}\n\\item If $f$ is smooth, then $f$ restricts to a continuous and cocontinuous\nfunctor\n$\\mathcal{X}_{lisse,\\etale} \\to \\mathcal{Y}_{lisse,\\etale}$\nwhich gives a morphism of ringed topoi fitting into the following\ncommutative diagram\n$$\n\\xymatrix{\n\\Sh(\\mathcal{X}_{lisse,\\etale}) \\ar[r]_{g'} \\ar[d]_{f'} &\n\\Sh(\\mathcal{X}_\\etale) \\ar[d]^f \\\\\n\\Sh(\\mathcal{Y}_{lisse,\\etale}) \\ar[r]^g &\n\\Sh(\\mathcal{Y}_\\etale)\n}\n$$\nWe have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$.\n\\item If $f$ is flat, then $f$ restricts to a continuous and cocontinuous\nfunctor\n$\\mathcal{X}_{flat,fppf} \\to \\mathcal{Y}_{flat,fppf}$\nwhich gives a morphism of ringed topoi fitting into the following\ncommutative diagram\n$$\n\\xymatrix{\n\\Sh(\\mathcal{X}_{flat,fppf}) \\ar[r]_{g'} \\ar[d]_{f'} &\n\\Sh(\\mathcal{X}_{fppf}) \\ar[d]^f \\\\\n\\Sh(\\mathcal{Y}_{flat,fppf}) \\ar[r]^g &\n\\Sh(\\mathcal{Y}_{fppf})\n}\n$$\nWe have $f'_*(g')^{-1} = g^{-1}f_*$ and $g'_!(f')^{-1} = f^{-1}g_!$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Functoriality of the lisse-étale and flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AT","source_file":"stacks-cohomology.tex","source_line":2979,"source_end_line":3012,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L2979-L3012","statement_sha256":"10f9cd65cb74f7035e0fa23d09f5e9e183b141722ca0a5c6d6e2630c3215c5af","origin":"The Stacks Project","memory_eligible":false,"source_rank":14369,"rank":14369,"depth":58,"x":1931.944,"y":1706.295,"cluster":"algebraic-stacks"},{"id":"stacks:0GR2","tag":"0GR2","title":"Functoriality of the lisse-étale and flat-fppf sites · Lemma 0GR2","summary":"With assumptions and notation as in Lemma [Tag 07AT]. Let H be an abelian sheaf on X_lisse,etale (resp. X_flat,fppf). Then R^pf'_*H = sheaf associated to y ↦ H^p((V ×_y, Y X)', (pr')^-1H) Here y is an object of Y_lisse,etale (resp. Y_flat,fppf) lying over the scheme V and the notation (V ×_y, Y X)' and pr' are explained in the proof.","statement_latex":"With assumptions and notation as in Lemma \\ref{lemma-lisse-etale-functorial}.\nLet $\\mathcal{H}$  be an abelian sheaf on $\\mathcal{X}_{lisse,\\etale}$\n(resp.\\ $\\mathcal{X}_{flat,fppf}$). Then\n\\begin{equation}\n\nR^pf'_*\\mathcal{H} =\n\\text{sheaf associated to }y \\longmapsto\nH^p((V \\times_{y, \\mathcal{Y}} \\mathcal{X})', (\\text{pr}')^{-1}\\mathcal{H})\n\\end{equation}\nHere $y$ is an object of $\\mathcal{Y}_{lisse,\\etale}$\n(resp.\\ $\\mathcal{Y}_{flat,fppf}$) lying over the scheme $V$\nand the notation $(V \\times_{y, \\mathcal{Y}} \\mathcal{X})'$\nand $\\text{pr}'$ are explained in the proof.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Functoriality of the lisse-étale and flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GR2","source_file":"stacks-cohomology.tex","source_line":3097,"source_end_line":3112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3097-L3112","statement_sha256":"af924b37e1fd518d248eb4ce2f99c0cf8c6ae3f9508d841f5ff9f07b1a655b61","origin":"The Stacks Project","memory_eligible":false,"source_rank":14370,"rank":14370,"depth":59,"x":2016.816,"y":1465.885,"cluster":"algebraic-stacks"},{"id":"stacks:0GR3","tag":"0GR3","title":"Functoriality of the lisse-étale and flat-fppf sites · Lemma 0GR3","summary":"With assumptions and notation as in Lemma [Tag 07AT] the canonical (base change) map g^-1Rf_*F → Rf'_*(g')^-1F is an isomorphism for any abelian sheaf F on X_etale (resp. X_fppf).","statement_latex":"With assumptions and notation as in Lemma \\ref{lemma-lisse-etale-functorial}\nthe canonical (base change) map\n$$\ng^{-1}Rf_*\\mathcal{F} \\longrightarrow Rf'_*(g')^{-1}\\mathcal{F}\n$$\nis an isomorphism for any abelian sheaf $\\mathcal{F}$\non $\\mathcal{X}_\\etale$ (resp.\\ $\\mathcal{X}_{fppf}$).","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Functoriality of the lisse-étale and flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GR3","source_file":"stacks-cohomology.tex","source_line":3172,"source_end_line":3181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3172-L3181","statement_sha256":"28ee327f260d77befedc803f7da8e79261fb31c2002780b6ac0d2bca1fbe575b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14371,"rank":14371,"depth":73,"x":2147.657,"y":1691.479,"cluster":"algebraic-stacks"},{"id":"stacks:07AZ","tag":"07AZ","title":"Quasi-coherent modules and the lisse-étale and flat-fppf sites · Lemma 07AZ","summary":"Let X be an algebraic stack. • Let f_j : X_j → X be a family of smooth morphisms of algebraic stacks with |X| =⋃ |f_j|(|X_j|). Let F be a sheaf of O_X-modules on X_etale. If each f_j^-1F is quasi-coherent, then so is F. • Let f_j : X_j → X be a family of flat and locally finitely presented morphisms of algebraic stacks with |X| =⋃ |f_j|(|X_j|). Let F be a sheaf of O_X-modules on X_fppf. If each f_j^-1F is quasi-coherent, then so is F.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\n\\begin{enumerate}\n\\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of smooth\nmorphisms of algebraic stacks with\n$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules\non $\\mathcal{X}_\\etale$. If each $f_j^{-1}\\mathcal{F}$\nis quasi-coherent, then so is $\\mathcal{F}$.\n\\item Let $f_j : \\mathcal{X}_j \\to \\mathcal{X}$ be a family of flat and\nlocally finitely presented morphisms of algebraic stacks with\n$|\\mathcal{X}| =\\bigcup |f_j|(|\\mathcal{X}_j|)$.\nLet $\\mathcal{F}$ be a sheaf of $\\mathcal{O}_\\mathcal{X}$-modules\non $\\mathcal{X}_{fppf}$. If each $f_j^{-1}\\mathcal{F}$\nis quasi-coherent, then so is $\\mathcal{F}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules and the lisse-étale and flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AZ","source_file":"stacks-cohomology.tex","source_line":3217,"source_end_line":3234,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3217-L3234","statement_sha256":"760ae3a403ccb5d96b616a799900d7b29f9900b5a75630f386e3bbd2181d9c6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14372,"rank":14372,"depth":57,"x":1869.563,"y":1599.306,"cluster":"algebraic-stacks"},{"id":"stacks:07B0","tag":"07B0","title":"Quasi-coherent modules and the lisse-étale and flat-fppf sites · Lemma 07B0","summary":"Let X be an algebraic stack. Notation as in Lemma [Tag 0788]. • Let H be a quasi-coherent O_X_lisse,etale-module on the lisse-étale site of X. Then g_!H is a quasi-coherent module on X. • Let H be a quasi-coherent O_X_flat,fppf-module on the flat-fppf site of X. Then g_!H is a quasi-coherent module on X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Notation as in\nLemma \\ref{lemma-lisse-etale}.\n\\begin{enumerate}\n\\item Let $\\mathcal{H}$ be a quasi-coherent\n$\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$-module \non the lisse-\\'etale site of $\\mathcal{X}$. Then $g_!\\mathcal{H}$ is a\nquasi-coherent module on $\\mathcal{X}$.\n\\item Let $\\mathcal{H}$ be a quasi-coherent\n$\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$-module \non the flat-fppf site of $\\mathcal{X}$. Then $g_!\\mathcal{H}$ is a\nquasi-coherent module on $\\mathcal{X}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules and the lisse-étale and flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07B0","source_file":"stacks-cohomology.tex","source_line":3300,"source_end_line":3314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3300-L3314","statement_sha256":"a1f25f15c875e5da10cc3ad7fa4a012f6b728eeebe2fcb32ac5528e3e2f7aacf","origin":"The Stacks Project","memory_eligible":false,"source_rank":14373,"rank":14373,"depth":59,"x":2148.947,"y":1509.411,"cluster":"algebraic-stacks"},{"id":"stacks:07B1","tag":"07B1","title":"Quasi-coherent modules and the lisse-étale and flat-fppf sites · Lemma 07B1","summary":"Let X be an algebraic stack. • With g as in Lemma [Tag 0788] for the lisse-étale site we have • the functors g^-1 and g_! define mutually inverse functors xymatrix QCoh(O_X) ar@<1ex>[r]^-g^-1 & QCoh(X_lisse,etale, O_X_lisse,etale) ar@<1ex>[l]^-g_! • if F is in LQCoh^fbc(O_X) then g^-1F is in QCoh(O_X_lisse,etale) and • Q(F) = g_!g^-1F where Q is as in Lemma [Tag 0778]. • With g as in Lemma [Tag 0788] for the flat-fppf site we have • the functors g^-1 and g_! define…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\n\\begin{enumerate}\n\\item With $g$ as in Lemma \\ref{lemma-lisse-etale}\nfor the lisse-\\'etale site we have\n\\begin{enumerate}\n\\item the functors $g^{-1}$ and $g_!$ define mutually inverse functors\n$$\n\\xymatrix{\n\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} &\n\\QCoh(\\mathcal{X}_{lisse,\\etale},\n\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\ar@<1ex>[l]^-{g_!}\n}\n$$\n\\item if $\\mathcal{F}$ is in $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$\nthen $g^{-1}\\mathcal{F}$ is in\n$\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$ and\n\\item $Q(\\mathcal{F}) = g_!g^{-1}\\mathcal{F}$ where $Q$ is as in\nLemma \\ref{lemma-adjoint}.\n\\end{enumerate}\n\\item With $g$ as in Lemma \\ref{lemma-lisse-etale}\nfor the flat-fppf site we have\n\\begin{enumerate}\n\\item the functors $g^{-1}$ and $g_!$ define mutually inverse functors\n$$\n\\xymatrix{\n\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^{-1}} &\n\\QCoh(\\mathcal{X}_{flat,fppf},\n\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\ar@<1ex>[l]^-{g_!}\n}\n$$\n\\item if $\\mathcal{F}$ is in $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$\nthen $g^{-1}\\mathcal{F}$ is in\n$\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$\nand\n\\item $Q(\\mathcal{F}) = g_!g^{-1}\\mathcal{F}$ where $Q$ is as in\nLemma \\ref{lemma-adjoint}.\n\\end{enumerate}\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules and the lisse-étale and flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07B1","source_file":"stacks-cohomology.tex","source_line":3359,"source_end_line":3399,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3359-L3399","statement_sha256":"b6882708e02c23db66b0bdf6acf3d715031607d73dfa1e1c408503df9152f0ff","origin":"The Stacks Project","memory_eligible":false,"source_rank":14374,"rank":14374,"depth":60,"x":2015.129,"y":1734.387,"cluster":"algebraic-stacks"},{"id":"stacks:07B4","tag":"07B4","title":"Quasi-coherent modules and the lisse-étale and flat-fppf sites · Lemma 07B4","summary":"Let X be an algebraic stack. • QCoh(O_X_lisse,etale) is a weak Serre subcategory of Mod(O_X_lisse,etale). • QCoh(O_X_flat,fppf) is a weak Serre subcategory of Mod(O_X_flat,fppf).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\n\\begin{enumerate}\n\\item $\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$\nis a weak Serre subcategory of\n$\\textit{Mod}(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$.\n\\item $\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$\nis a weak Serre subcategory of\n$\\textit{Mod}(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Quasi-coherent modules and the lisse-étale and flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07B4","source_file":"stacks-cohomology.tex","source_line":3440,"source_end_line":3451,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3440-L3451","statement_sha256":"4a02bb406fd4f11dd6e37848fa226b87392313c537012a20927d875d46c6bc01","origin":"The Stacks Project","memory_eligible":false,"source_rank":14375,"rank":14375,"depth":79,"x":1932.827,"y":1492.39,"cluster":"algebraic-stacks"},{"id":"stacks:0GR5","tag":"0GR5","title":"Coherent sheaves on locally Noetherian stacks · Lemma 0GR5","summary":"Let X be a locally Noetherian algebraic stack. Let F be an O_X-module. The following are equivalent • F is a quasi-coherent, finite type O_X-module, • F is an O_X-module of finite presentation, • F is quasi-coherent and for any morphism f : U → X where U is a locally Noetherian algebraic space, the pullback f^*F|_U_etale is coherent, and • F is quasi-coherent and there exists an algebraic space U and a morphism f : U → X which is locally of finite type, flat, and…","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_\\mathcal{X}$-module.\nThe following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{F}$ is a quasi-coherent, finite type\n$\\mathcal{O}_\\mathcal{X}$-module,\n\\item $\\mathcal{F}$ is an $\\mathcal{O}_\\mathcal{X}$-module\nof finite presentation,\n\\item $\\mathcal{F}$ is quasi-coherent and for any morphism\n$f : U \\to \\mathcal{X}$ where $U$ is a locally Noetherian algebraic space,\nthe pullback $f^*\\mathcal{F}|_{U_\\etale}$ is coherent, and\n\\item $\\mathcal{F}$ is quasi-coherent and there exists an algebraic space\n$U$ and a morphism $f : U \\to \\mathcal{X}$ which is locally of finite type,\nflat, and surjective, such that the pullback $f^*\\mathcal{F}|_{U_\\etale}$\nis coherent.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on locally Noetherian stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GR5","source_file":"stacks-cohomology.tex","source_line":3573,"source_end_line":3591,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3573-L3591","statement_sha256":"ad8d25a3e88d182e34d502c03851e287349bfd8350911b8d731fe48cda5bc893","origin":"The Stacks Project","memory_eligible":false,"source_rank":14376,"rank":14376,"depth":49,"x":2188.303,"y":1624.228,"cluster":"algebraic-stacks"},{"id":"stacks:0GR6","tag":"0GR6","title":"Coherent sheaves on locally Noetherian stacks · Definition 0GR6","summary":"Let X be a locally Noetherian algebraic stack. An O_X-module F is called coherent if F satisfies one (and hence all) of the equivalent conditions of Lemma [Tag 0GR5]. The category of coherent O_X-modules is denote Coh(O_X).","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nAn $\\mathcal{O}_\\mathcal{X}$-module $\\mathcal{F}$ is called {\\it coherent}\nif $\\mathcal{F}$ satisfies one (and hence all) of the equivalent\nconditions of Lemma \\ref{lemma-coherent-Noetherian}.\nThe category of coherent $\\mathcal{O}_\\mathcal{X}$-modules is\ndenote $\\textit{Coh}(\\mathcal{O}_\\mathcal{X})$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on locally Noetherian stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GR6","source_file":"stacks-cohomology.tex","source_line":3627,"source_end_line":3635,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3627-L3635","statement_sha256":"f185014740461b06dbdc6f6892f004a0087d134fe58cc6162ed1231794796c15","origin":"The Stacks Project","memory_eligible":false,"source_rank":14377,"rank":14377,"depth":50,"x":1893.69,"y":1672.01,"cluster":"algebraic-stacks"},{"id":"stacks:0GR7","tag":"0GR7","title":"Coherent sheaves on locally Noetherian stacks · Lemma 0GR7","summary":"Let X be a locally Noetherian algebraic stack. The module O_X is coherent, any invertible O_X-module is coherent, and more generally any finite locally free O_X-module is coherent.","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nThe module $\\mathcal{O}_\\mathcal{X}$ is coherent, any invertible\n$\\mathcal{O}_\\mathcal{X}$-module is coherent, and more generally any\nfinite locally free $\\mathcal{O}_\\mathcal{X}$-module is coherent.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on locally Noetherian stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GR7","source_file":"stacks-cohomology.tex","source_line":3637,"source_end_line":3643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3637-L3643","statement_sha256":"49c653352fe203f2bb1121483b50df05ebb482e37f8d9c8b1b609fb6441a857b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14378,"rank":14378,"depth":9,"x":2072.634,"y":1469.463,"cluster":"algebraic-stacks"},{"id":"stacks:0GR8","tag":"0GR8","title":"Coherent sheaves on locally Noetherian stacks · Lemma 0GR8","summary":"Let f : X → Y be a morphism of locally Noetherian algebraic stacks. Then f^* sends coherent modules on Y to coherent modules on X.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of locally\nNoetherian algebraic stacks. Then $f^*$ sends coherent modules\non $\\mathcal{Y}$ to coherent modules on $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on locally Noetherian stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GR8","source_file":"stacks-cohomology.tex","source_line":3651,"source_end_line":3656,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3651-L3656","statement_sha256":"d93e660024603522ab9d375e708f680e714d441edb57eee81a0cb5538d482154","origin":"The Stacks Project","memory_eligible":false,"source_rank":14379,"rank":14379,"depth":11,"x":2103.588,"y":1720.532,"cluster":"algebraic-stacks"},{"id":"stacks:0GR9","tag":"0GR9","title":"Coherent sheaves on locally Noetherian stacks · Lemma 0GR9","summary":"Let X be a locally Noetherian algebraic stack. The category of coherent O_X-modules is abelian. If φ : F → G is a map of coherent O_X-modules, then • the cokernel Coker(φ) computed in Mod(O_X) is a coherent O_X-module, • the image Im(φ) computed in Mod(O_X) is a coherent O_X-module, and • the kernel Ker(φ) computed in Mod(O_X) may not be coherent, but it is in LQCoh^fbc(O_X) and Q(Ker(φ)) is coherent and is the kernel of φ in Coh(O_X). The inclusion functor Coh(O_X) →…","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nThe category of coherent $\\mathcal{O}_\\mathcal{X}$-modules is abelian.\nIf $\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a map\nof coherent $\\mathcal{O}_\\mathcal{X}$-modules, then\n\\begin{enumerate}\n\\item the cokernel $\\Coker(\\varphi)$ computed in\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ is a coherent\n$\\mathcal{O}_\\mathcal{X}$-module,\n\\item the image $\\Im(\\varphi)$ computed in\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$ is a coherent\n$\\mathcal{O}_\\mathcal{X}$-module, and\n\\item the kernel $\\Ker(\\varphi)$ computed in\n$\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nmay not be coherent, but it is\nin $\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$ and $Q(\\Ker(\\varphi))$\nis coherent and is the kernel of $\\varphi$ in\n$\\textit{Coh}(\\mathcal{O}_\\mathcal{X})$.\n\\end{enumerate}\nThe inclusion functor $\\textit{Coh}(\\mathcal{O}_\\mathcal{X}) \\to\n\\QCoh(\\mathcal{O}_\\mathcal{X})$ is exact.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on locally Noetherian stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GR9","source_file":"stacks-cohomology.tex","source_line":3664,"source_end_line":3686,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3664-L3686","statement_sha256":"91399857137a80651c0cb4b978266e05c84ffa911bd2888c609f86b407d56fbb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14380,"rank":14380,"depth":59,"x":1878.702,"y":1552.844,"cluster":"algebraic-stacks"},{"id":"stacks:0GRA","tag":"0GRA","title":"Coherent sheaves on locally Noetherian stacks · Lemma 0GRA","summary":"Let X be a locally Noetherian algebraic stack. Given a short exact sequence 0 → F_1 → F_2 → F_3 → 0 in Mod(O_X) with F_1 and F_3 coherent, then F_2 is coherent.","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nGiven a short exact sequence\n$0 \\to \\mathcal{F}_1 \\to \\mathcal{F}_2 \\to \\mathcal{F}_3 \\to 0$\nin $\\textit{Mod}(\\mathcal{O}_\\mathcal{X})$\nwith $\\mathcal{F}_1$ and $\\mathcal{F}_3$ coherent, then\n$\\mathcal{F}_2$ is coherent.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on locally Noetherian stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRA","source_file":"stacks-cohomology.tex","source_line":3708,"source_end_line":3716,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3708-L3716","statement_sha256":"b85901c055dad6cf6c49567aea20a76ce2424b91185d65f400b38530e2c65c9e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14381,"rank":14381,"depth":51,"x":2179.592,"y":1548.886,"cluster":"algebraic-stacks"},{"id":"stacks:0GRB","tag":"0GRB","title":"Coherent sheaves on locally Noetherian stacks · Lemma 0GRB","summary":"Let X be a locally Noetherian algebraic stack. Then Coh(O_X) is a Serre subcategory of QCoh(O_X). Let φ : F → G be a map of quasi-coherent O_X-modules. We have • if F is coherent and φ surjective, then G is coherent, • if F is coherent, then Im(φ) is coherent, and • if G coherent and Ker(φ) parasitic, then F is coherent.","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nThen $\\textit{Coh}(\\mathcal{O}_\\mathcal{X})$ is a Serre subcategory of\n$\\QCoh(\\mathcal{O}_\\mathcal{X})$. Let $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nbe a map of quasi-coherent $\\mathcal{O}_\\mathcal{X}$-modules. We have\n\\begin{enumerate}\n\\item if $\\mathcal{F}$ is coherent and $\\varphi$ surjective,\nthen $\\mathcal{G}$ is coherent,\n\\item if $\\mathcal{F}$ is coherent, then $\\Im(\\varphi)$ is coherent, and\n\\item if $\\mathcal{G}$ coherent and $\\Ker(\\varphi)$ parasitic, then\n$\\mathcal{F}$ is coherent.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on locally Noetherian stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRB","source_file":"stacks-cohomology.tex","source_line":3734,"source_end_line":3747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3734-L3747","statement_sha256":"b2f9d11fc66b3938caa4884b865bfd6d4a6bd02eea9356cdcec9e04048234c3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14382,"rank":14382,"depth":60,"x":1960.749,"y":1722.658,"cluster":"algebraic-stacks"},{"id":"stacks:0GRC","tag":"0GRC","title":"Coherent sheaves on locally Noetherian stacks · Lemma 0GRC","summary":"In the situation discussed above, the equivalence QCoh(O_X) ≅ QCoh(U, R, s, t, c) sends coherent sheaves to coherent sheaves and vice versa, i.e., induces an equivalence Coh(O_X) ≅ Coh(U, R, s, t, c).","statement_latex":"In the situation discussed above, the equivalence\n$\\QCoh(\\mathcal{O}_\\mathcal{X}) \\cong \\QCoh(U, R, s, t, c)$\nsends coherent sheaves to coherent sheaves and vice versa, i.e.,\ninduces an equivalence\n$\\textit{Coh}(\\mathcal{O}_\\mathcal{X}) \\cong \\textit{Coh}(U, R, s, t, c)$.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on locally Noetherian stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRC","source_file":"stacks-cohomology.tex","source_line":3801,"source_end_line":3808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3801-L3808","statement_sha256":"6b9c566e91d565dee2a394a2b6e24249973254e048dd14d57a15b17256cd9260","origin":"The Stacks Project","memory_eligible":false,"source_rank":14383,"rank":14383,"depth":56,"x":1982.393,"y":1470.168,"cluster":"algebraic-stacks"},{"id":"stacks:0GRD","tag":"0GRD","title":"Coherent sheaves on locally Noetherian stacks · Lemma 0GRD","summary":"Let X be a locally Noetherian algebraic stack. Let F and G be coherent be O_X-modules. Then the internal hom hom(F, G) constructed in Lemma [Tag 0GQN] is a coherent O_X-module.","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack. Let $\\mathcal{F}$\nand $\\mathcal{G}$ be coherent be $\\mathcal{O}_\\mathcal{X}$-modules. Then\nthe internal hom $hom(\\mathcal{F}, \\mathcal{G})$\nconstructed in Lemma \\ref{lemma-internal-hom-fp-into-qcoh}\nis a coherent $\\mathcal{O}_\\mathcal{X}$-module.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on locally Noetherian stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRD","source_file":"stacks-cohomology.tex","source_line":3825,"source_end_line":3832,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3825-L3832","statement_sha256":"605fb9effa4ce95bd036d3cc65aa7e6d75aa4d02640ae805a4f8113992e7811c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14384,"rank":14384,"depth":59,"x":2169.609,"y":1668.771,"cluster":"algebraic-stacks"},{"id":"stacks:0GRF","tag":"0GRF","title":"Coherent sheaves on Noetherian stacks · Lemma 0GRF","summary":"Let X be a Noetherian algebraic stack. Every quasi-coherent O_X-module is the filtered colimit of its coherent submodules.","statement_latex":"Let $\\mathcal{X}$ be a Noetherian algebraic stack. Every quasi-coherent\n$\\mathcal{O}_\\mathcal{X}$-module is the filtered colimit of its coherent\nsubmodules.","area":"Algebraic Stacks","chapter":"Cohomology of Algebraic Stacks","chapter_id":"stacks-cohomology","section":"Coherent sheaves on Noetherian stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRF","source_file":"stacks-cohomology.tex","source_line":3856,"source_end_line":3861,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-cohomology.tex#L3856-L3861","statement_sha256":"dfe003d0ea647dbbfaa9f6f179a07aba7dd5052384965c601634f0fe51125b57","origin":"The Stacks Project","memory_eligible":false,"source_rank":14385,"rank":14385,"depth":64,"x":1871.64,"y":1628.527,"cluster":"algebraic-stacks"},{"id":"stacks:07AS","tag":"07AS","title":"The lisse-étale and the flat-fppf sites · Lemma 07AS","summary":"Let X be an algebraic stack. Notation as in Cohomology of Stacks, Lemmas [Tag 0788] and [Tag 0789]. • The functor g_! : Ab(X_lisse,etale) → Ab(X_etale) has a left derived functor Lg_! : D(X_lisse,etale) → D(X_etale) which is left adjoint to g^-1 and such that g^-1Lg_! = id. • The functor g_! : Mod(X_lisse,etale, O_X_lisse,etale) → Mod(X_etale, O_X) has a left derived functor Lg_! : D(O_X_lisse,etale) → D(X_etale, O_X) which is left adjoint to g^* and such that g^*Lg_! =…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nNotation as in\nCohomology of Stacks,\nLemmas \\ref{stacks-cohomology-lemma-lisse-etale} and\n\\ref{stacks-cohomology-lemma-lisse-etale-modules}.\n\\begin{enumerate}\n\\item The functor\n$g_! : \\textit{Ab}(\\mathcal{X}_{lisse,\\etale}) \\to\n\\textit{Ab}(\\mathcal{X}_\\etale)$\nhas a left derived functor\n$$\nLg_! :\nD(\\mathcal{X}_{lisse,\\etale})\n\\longrightarrow\nD(\\mathcal{X}_\\etale)\n$$\nwhich is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \\text{id}$.\n\\item The functor $g_! : \n\\textit{Mod}(\\mathcal{X}_{lisse,\\etale},\n\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}) \\to\n\\textit{Mod}(\\mathcal{X}_\\etale, \\mathcal{O}_{\\mathcal{X}})$\nhas a left derived functor\n$$\nLg_! :\nD(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})\n\\longrightarrow\nD(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})\n$$\nwhich is left adjoint to $g^*$ and such that $g^*Lg_! = \\text{id}$.\n\\item The functor $g_! : \\textit{Ab}(\\mathcal{X}_{flat,fppf}) \\to\n\\textit{Ab}(\\mathcal{X}_{fppf})$\nhas a left derived functor\n$$\nLg_! :\nD(\\mathcal{X}_{flat, fppf})\n\\longrightarrow\nD(\\mathcal{X}_{fppf})\n$$\nwhich is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \\text{id}$.\n\\item The functor $g_! :\n\\textit{Mod}(\\mathcal{X}_{flat,fppf},\n\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\to\n\\textit{Mod}(\\mathcal{X}_{fppf}, \\mathcal{O}_{\\mathcal{X}})$\nhas a left derived functor\n$$\nLg_! :\nD(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}})\n\\longrightarrow\nD(\\mathcal{O}_\\mathcal{X})\n$$\nwhich is left adjoint to $g^*$ and such that $g^*Lg_! = \\text{id}$.\n\\end{enumerate}\nWarning: It is not clear (a priori) that $Lg_!$ on modules agrees\nwith $Lg_!$ on abelian sheaves, see\nCohomology on Sites, Remark\n\\ref{sites-cohomology-remark-when-derived-shriek-equal}.","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"The lisse-étale and the flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AS","source_file":"stacks-perfect.tex","source_line":68,"source_end_line":126,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L68-L126","statement_sha256":"b4238c5ffc9b876a6248664374e0aef17d58fe71a1d5130a551787127c730e7c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14386,"rank":14386,"depth":25,"x":288.983,"y":544.447,"cluster":"derived-categories"},{"id":"stacks:07AV","tag":"07AV","title":"The lisse-étale and the flat-fppf sites · Lemma 07AV","summary":"With assumptions and notation as in Cohomology of Stacks, Lemma [Tag 07AT]. We have g^-1 ∘ Rf_* = Rf'_* ∘ (g')^-1 and L(g')_! ∘ (f')^-1 = f^-1 ∘ Lg_! on unbounded derived categories (both for the case of modules and for the case of abelian sheaves).","statement_latex":"With assumptions and notation as in\nCohomology of Stacks,\nLemma \\ref{stacks-cohomology-lemma-lisse-etale-functorial}.\nWe have\n$$\ng^{-1} \\circ Rf_* = Rf'_* \\circ (g')^{-1}\n\\quad\\text{and}\\quad\nL(g')_! \\circ (f')^{-1} = f^{-1} \\circ Lg_!\n$$\non unbounded derived categories\n(both for the case of modules and for the case of abelian sheaves).","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"The lisse-étale and the flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07AV","source_file":"stacks-perfect.tex","source_line":145,"source_end_line":158,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L145-L158","statement_sha256":"149fc5b6392b76b03453bb00d6949d57aa39b6554fe14b95ec7dda2d82798093","origin":"The Stacks Project","memory_eligible":false,"source_rank":14387,"rank":14387,"depth":74,"x":295.563,"y":813.522,"cluster":"derived-categories"},{"id":"stacks:07B3","tag":"07B3","title":"The lisse-étale and the flat-fppf sites · Lemma 07B3","summary":"Let X be an algebraic stack. Notation as in Cohomology of Stacks, Lemma [Tag 0788]. • Let H be a quasi-coherent O_X_lisse,etale-module on the lisse-étale site of X. For all p ∈ Z the sheaf H^p(Lg_!H) is a locally quasi-coherent module with the flat base change property on X. • Let H be a quasi-coherent O_X_flat,fppf-module on the flat-fppf site of X. For all p ∈ Z the sheaf H^p(Lg_!H) is a locally quasi-coherent module with the flat base change property on X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Notation as in\nCohomology of Stacks,\nLemma \\ref{stacks-cohomology-lemma-lisse-etale}.\n\\begin{enumerate}\n\\item Let $\\mathcal{H}$ be a quasi-coherent\n$\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}}$-module \non the lisse-\\'etale site of $\\mathcal{X}$. For all $p \\in \\mathbf{Z}$\nthe sheaf $H^p(Lg_!\\mathcal{H})$ is a locally quasi-coherent module with\nthe flat base change property on $\\mathcal{X}$.\n\\item Let $\\mathcal{H}$ be a quasi-coherent\n$\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}$-module \non the flat-fppf site of $\\mathcal{X}$. For all $p \\in \\mathbf{Z}$\nthe sheaf $H^p(Lg_!\\mathcal{H})$ is a locally quasi-coherent module with the\nflat base change property on $\\mathcal{X}$.\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"The lisse-étale and the flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07B3","source_file":"stacks-perfect.tex","source_line":243,"source_end_line":260,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L243-L260","statement_sha256":"4522b88bf6ff567ebe6b2addbb146cc82979ad96fc2ede3843c80e31fd474c56","origin":"The Stacks Project","memory_eligible":false,"source_rank":14388,"rank":14388,"depth":80,"x":74.165,"y":618.7,"cluster":"derived-categories"},{"id":"stacks:0H0Z","tag":"0H0Z","title":"Cohomology and the lisse-étale and flat-fppf sites · Lemma 0H0Z","summary":"Let X be an algebraic stack. We have Lg_!Z = Z for either Lg_! as in Lemma [Tag 07AS] part (1) or Lg_! as in Lemma [Tag 07AS] part (3).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. We have\n$Lg_!\\mathbf{Z} = \\mathbf{Z}$ for either $Lg_!$ as in\nLemma \\ref{lemma-shriek-derived} part (1) or $Lg_!$ as in\nLemma \\ref{lemma-shriek-derived} part (3).","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Cohomology and the lisse-étale and flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H0Z","source_file":"stacks-perfect.tex","source_line":351,"source_end_line":357,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L351-L357","statement_sha256":"ba6dadf2571d14ad66dbc5c525b59a5eb27a9df0a5aeaa702d87aa17f2bcedba","origin":"The Stacks Project","memory_eligible":false,"source_rank":14389,"rank":14389,"depth":75,"x":394.328,"y":636.745,"cluster":"derived-categories"},{"id":"stacks:0H10","tag":"0H10","title":"Cohomology and the lisse-étale and flat-fppf sites · Lemma 0H10","summary":"Let X be an algebraic stack. Notation as in Lemma [Tag 07AS]. • For K in D(X_etale) we have • RΓ(X_etale, K) = RΓ(X_lisse,etale, g^-1K), and • RΓ(x, K) = RΓ(X_lisse,etale/x, g^-1K) for any object x of X_lisse,etale. • For K in D(X_fppf) we have • RΓ(X_fppf, K) = RΓ(X_flat,fppf, g^-1K), and • H^p(x, K) = RΓ(X_flat,fppf/x, g^-1K) for any object x of X_flat,fppf. In both cases, the same holds for modules, since we have g^-1 = g^* and there is no difference in computing…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Notation as in\nLemma \\ref{lemma-shriek-derived}.\n\\begin{enumerate}\n\\item For $K$ in $D(\\mathcal{X}_\\etale)$ we have\n\\begin{enumerate}\n\\item $R\\Gamma(\\mathcal{X}_\\etale, K) =\nR\\Gamma(\\mathcal{X}_{lisse,\\etale}, g^{-1}K)$, and\n\\item $R\\Gamma(x, K) =\nR\\Gamma(\\mathcal{X}_{lisse,\\etale}/x, g^{-1}K)$\nfor any object $x$ of $\\mathcal{X}_{lisse,\\etale}$.\n\\end{enumerate}\n\\item For $K$ in $D(\\mathcal{X}_{fppf})$ we have\n\\begin{enumerate}\n\\item $R\\Gamma(\\mathcal{X}_{fppf}, K) =\nR\\Gamma(\\mathcal{X}_{flat,fppf}, g^{-1}K)$, and\n\\item $H^p(x, K) =\nR\\Gamma(\\mathcal{X}_{flat,fppf}/x, g^{-1}K)$\nfor any object $x$ of $\\mathcal{X}_{flat,fppf}$.\n\\end{enumerate}\n\\end{enumerate}\nIn both cases, the same holds for modules, since we have\n$g^{-1} = g^*$ and there is no difference in computing\ncohomology by Cohomology on Sites, Lemma\n\\ref{sites-cohomology-lemma-modules-abelian-unbounded}.","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Cohomology and the lisse-étale and flat-fppf sites","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H10","source_file":"stacks-perfect.tex","source_line":393,"source_end_line":419,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L393-L419","statement_sha256":"d16139796f549c14af7ba2194e79f980190dd8e7f3ee13295a91708417e0ad09","origin":"The Stacks Project","memory_eligible":false,"source_rank":14390,"rank":14390,"depth":76,"x":143.548,"y":805.232,"cluster":"derived-categories"},{"id":"stacks:07B6","tag":"07B6","title":"Derived categories of quasi-coherent modules · Definition 07B6","summary":"Let X be an algebraic stack. With notation as above we define the derived category of O_X-modules with quasi-coherent cohomology sheaves as the Verdier quotient D_QCoh(O_X) = D_LQCoh^fbc(O_X)/ D_Parasitic ∩ LQCoh^fbc(O_X)","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. With notation as above\nwe define the {\\it derived category of $\\mathcal{O}_\\mathcal{X}$-modules with\nquasi-coherent cohomology sheaves} as the Verdier quotient\\footnote{This\ndefinition is different from the one in the literature, see\n\\cite[6.3]{olsson_sheaves}, but it agrees with that definition\nby Lemma \\ref{lemma-derived-quasi-coherent}.}\n$$\nD_\\QCoh(\\mathcal{O}_\\mathcal{X}) =\nD_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})/\nD_{\\textit{Parasitic} \\cap \\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})\n$$","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Derived categories of quasi-coherent modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07B6","source_file":"stacks-perfect.tex","source_line":525,"source_end_line":538,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L525-L538","statement_sha256":"bd95192769b530ec9a5b45d31d29ac7ed49bea8f4311d9be56c0ac8d96b2f2a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14391,"rank":14391,"depth":0,"x":193.013,"y":538.497,"cluster":"derived-categories"},{"id":"stacks:07B8","tag":"07B8","title":"Derived categories of quasi-coherent modules · Lemma 07B8","summary":"Let X be an algebraic stack. Abbreviate P_X = Parasitic(O_X) ∩ LQCoh^fbc(O_X). The comparison morphism ε : X_fppf → X_etale induces a commutative diagram xymatrix D_Parasitic ∩ LQCoh^fbc(O_X) ar[r] & D_LQCoh^fbc(O_X) ar[r] & D(O_X) D_P_X(X_etale, O_X) ar[r] ar[u]^ε^* & D_LQCoh^fbc(O_X)( X_etale, O_X) ar[r] ar[u]^ε^* & D(X_etale, O_X) ar[u]^ε^* Moreover, the left two vertical arrows are equivalences of triangulated categories, hence we also obtain an equivalence…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Abbreviate\n$\\mathcal{P}_\\mathcal{X} = \\textit{Parasitic}(\\mathcal{O}_\\mathcal{X}) \\cap\n\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\nThe comparison morphism\n$\\epsilon : \\mathcal{X}_{fppf} \\to \\mathcal{X}_\\etale$\ninduces a commutative diagram\n$$\n\\xymatrix{\nD_{\\textit{Parasitic} \\cap \\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})\n\\ar[r] &\nD_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] &\nD(\\mathcal{O}_\\mathcal{X}) \\\\\nD_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})\n\\ar[r] \\ar[u]^{\\epsilon^*} &\nD_{\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})}(\n\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})\n\\ar[r] \\ar[u]^{\\epsilon^*} &\nD(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})\n\\ar[u]^{\\epsilon^*}\n}\n$$\nMoreover, the left two vertical arrows are equivalences of triangulated\ncategories, hence we also obtain an equivalence\n$$\nD_{\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})}\n(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})\n/\nD_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})\n\\longrightarrow\nD_\\QCoh(\\mathcal{O}_\\mathcal{X})\n$$","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Derived categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07B8","source_file":"stacks-perfect.tex","source_line":592,"source_end_line":625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L592-L625","statement_sha256":"8a1c60256251c89aecc5fc3fed0521fc0b0b8471d24b733876c2eb5ffe05df28","origin":"The Stacks Project","memory_eligible":false,"source_rank":14392,"rank":14392,"depth":79,"x":371.17,"y":763.417,"cluster":"derived-categories"},{"id":"stacks:07B9","tag":"07B9","title":"Derived categories of quasi-coherent modules · Lemma 07B9","summary":"Let X be an algebraic stack. Set P_X = Parasitic(O_X) ∩ LQCoh^fbc(O_X). • Let F^bullet be an object of D_LQCoh^fbc(O_X) (X_etale, O_X). With g as in Cohomology of Stacks, Lemma [Tag 0788] for the lisse-étale site we have • g^*F^bullet is in D_QCoh(O_X_lisse,etale), • g^*F^bullet = 0 if and only if F^bullet is in D_P_X(X_etale, O_X), • Lg_!H^bullet is in D_LQCoh^fbc(O_X)( X_etale, O_X) for H^bullet in D_QCoh(O_X_lisse,etale), and • the functors g^* and Lg_! define mutually…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Set\n$\\mathcal{P}_\\mathcal{X} = \\textit{Parasitic}(\\mathcal{O}_\\mathcal{X}) \\cap\n\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})$.\n\\begin{enumerate}\n\\item\nLet $\\mathcal{F}^\\bullet$ be an object of\n$D_{\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})}\n(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$.\nWith $g$ as in\nCohomology of Stacks,\nLemma \\ref{stacks-cohomology-lemma-lisse-etale}\nfor the lisse-\\'etale site we have\n\\begin{enumerate}\n\\item $g^*\\mathcal{F}^\\bullet$ is in\n$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$,\n\\item $g^*\\mathcal{F}^\\bullet = 0$ if and only if\n$\\mathcal{F}^\\bullet$ is in\n$D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$,\n\\item $Lg_!\\mathcal{H}^\\bullet$ is in\n$D_{\\textit{LQCoh}^{fbc}(\\mathcal{O}_\\mathcal{X})}(\n\\mathcal{X}_\\etale, \\mathcal{O}_\\mathcal{X})$\nfor $\\mathcal{H}^\\bullet$ in\n$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})$, and\n\\item the functors $g^*$ and $Lg_!$ define mutually inverse functors\n$$\n\\xymatrix{\nD_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^*} &\nD_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{lisse,\\etale}})\n\\ar@<1ex>[l]^-{Lg_!}\n}\n$$\n\\end{enumerate}\n\\item\nLet $\\mathcal{F}^\\bullet$ be an object of\n$D_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})$. With $g$ as in\nCohomology of Stacks,\nLemma \\ref{stacks-cohomology-lemma-lisse-etale}\nfor the flat-fppf site we have\n\\begin{enumerate}\n\\item $g^*\\mathcal{F}^\\bullet$ is in\n$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}})$,\n\\item $g^*\\mathcal{F}^\\bullet = 0$ if and only if\n$\\mathcal{F}^\\bullet$ is in\n$D_{\\mathcal{P}_\\mathcal{X}}(\\mathcal{O}_\\mathcal{X})$,\n\\item $Lg_!\\mathcal{H}^\\bullet$ is in\n$D_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})$\nfor $\\mathcal{H}^\\bullet$ in\n$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}})$, and\n\\item the functors $g^*$ and $Lg_!$ define mutually inverse functors\n$$\n\\xymatrix{\nD_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\ar@<1ex>[r]^-{g^*} &\nD_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat,fppf}}) \\ar@<1ex>[l]^-{Lg_!}\n}\n$$\n\\end{enumerate}\n\\end{enumerate}","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Derived categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07B9","source_file":"stacks-perfect.tex","source_line":692,"source_end_line":751,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L692-L751","statement_sha256":"70c4447ff555371511f501e82ffc313c9dc9fa74947b40c10b4056cf7d14ec81","origin":"The Stacks Project","memory_eligible":false,"source_rank":14393,"rank":14393,"depth":81,"x":58.696,"y":698.607,"cluster":"derived-categories"},{"id":"stacks:07BA","tag":"07BA","title":"Derived categories of quasi-coherent modules · Lemma 07BA","summary":"Let X be an algebraic stack. Let E be an object of D_LQCoh^fbc(O_X). There exists a canonical distinguished triangle E' → E → P → E'[1] in D_LQCoh^fbc(O_X) such that P is in D_Parasitic ∩ LQCoh^fbc (O_X) and Hom_D(O_X)(E', P') = 0 for all P' in D_Parasitic ∩ LQCoh^fbc(O_X).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack.\nLet $E$ be an object of $D_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})$.\nThere exists a canonical distinguished triangle\n$$\nE' \\to E \\to P \\to E'[1]\n$$\nin $D_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})$ such that\n$P$ is in $D_{\\textit{Parasitic} \\cap \\textit{LQCoh}^{fbc}}\n(\\mathcal{O}_\\mathcal{X})$\nand\n$$\n\\Hom_{D(\\mathcal{O}_\\mathcal{X})}(E', P') = 0\n$$\nfor all $P'$ in\n$D_{\\textit{Parasitic} \\cap \\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})$.","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Derived categories of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BA","source_file":"stacks-perfect.tex","source_line":861,"source_end_line":878,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L861-L878","statement_sha256":"c2db970115236e0de7c52cc9979ddcef02c001fa931bb8307fe708ecd5bf433a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14394,"rank":14394,"depth":82,"x":341.435,"y":568.996,"cluster":"derived-categories"},{"id":"stacks:07BC","tag":"07BC","title":"Derived pushforward of quasi-coherent modules · Proposition 07BC","summary":"Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor Rf_* induces a commutative diagram xymatrix D^+_Parasitic ∩ LQCoh^fbc(O_X) ar[r] ar[d]^Rf_* & D^+_LQCoh^fbc(O_X) ar[r] ar[d]^Rf_* & D(O_X) ar[d]^Rf_* D^+_Parasitic ∩ LQCoh^fbc(O_Y) ar[r] & D^+_LQCoh^fbc(O_Y) ar[r] & D(O_Y) and hence induces a functor Rf_QCoh, * : D^+_QCoh(O_X) → D^+_QCoh(O_Y) on quotient categories. Moreover, the functor R^if_QCoh of Cohomology of Stacks,…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a quasi-compact and\nquasi-separated morphism of algebraic stacks.\nThe functor $Rf_*$ induces a commutative diagram\n$$\n\\xymatrix{\nD^{+}_{\\textit{Parasitic} \\cap \\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})\n\\ar[r] \\ar[d]^{Rf_*} &\nD^{+}_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})\n\\ar[r] \\ar[d]^{Rf_*} &\nD(\\mathcal{O}_\\mathcal{X})\n\\ar[d]^{Rf_*} \\\\\nD^{+}_{\\textit{Parasitic} \\cap \\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{Y})\n\\ar[r] &\nD^{+}_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{Y}) \\ar[r] &\nD(\\mathcal{O}_\\mathcal{Y})\n}\n$$\nand hence induces a functor\n$$\nRf_{\\QCoh, *} :\nD^{+}_\\QCoh(\\mathcal{O}_\\mathcal{X})\n\\longrightarrow\nD^{+}_\\QCoh(\\mathcal{O}_\\mathcal{Y})\n$$\non quotient categories. Moreover, the functor $R^if_\\QCoh$\nof\nCohomology of Stacks,\nProposition \\ref{stacks-cohomology-proposition-direct-image-quasi-coherent}\nare equal to $H^i \\circ Rf_{\\QCoh, *}$ with $H^i$ as in\n(\\ref{equation-Hi-quasi-coherent}).","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Derived pushforward of quasi-coherent modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BC","source_file":"stacks-perfect.tex","source_line":954,"source_end_line":986,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L954-L986","statement_sha256":"88dae46c479944f60013bb2d9e70250abe46fca6c6059811da5df3513857c455","origin":"The Stacks Project","memory_eligible":false,"source_rank":14395,"rank":14395,"depth":81,"x":237.101,"y":825.19,"cluster":"derived-categories"},{"id":"stacks:07BE","tag":"07BE","title":"Derived pullback of quasi-coherent modules · Proposition 07BE","summary":"Let f : X → Y be a morphism of algebraic stacks. The exact functor f^* induces a commutative diagram xymatrix D_LQCoh^fbc(O_X) ar[r] & D(O_X) D_LQCoh^fbc(O_Y) ar[r] ar[u]^f^* & D(O_Y) ar[u]^f^* The composition D_LQCoh^fbc(O_Y) xrightarrowf^* D_LQCoh^fbc(O_X) xrightarrowq_X D_QCoh(O_X) is left derivable with respect to the localization D_LQCoh^fbc(O_Y) → D_QCoh(O_Y) and we may define Lf^*_QCoh as its left derived functor Lf_QCoh^* : D_QCoh(O_Y) → D_QCoh(O_X) (see Derived…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe exact functor $f^*$ induces a commutative diagram\n$$\n\\xymatrix{\nD_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X}) \\ar[r] &\nD(\\mathcal{O}_\\mathcal{X}) \\\\\nD_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{Y})\n\\ar[r] \\ar[u]^{f^*} &\nD(\\mathcal{O}_\\mathcal{Y}) \\ar[u]^{f^*}\n}\n$$\nThe composition\n$$\nD_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{Y})\n\\xrightarrow{f^*}\nD_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{X})\n\\xrightarrow{q_\\mathcal{X}}\nD_\\QCoh(\\mathcal{O}_\\mathcal{X})\n$$\nis left derivable with respect to the localization\n$D_{\\textit{LQCoh}^{fbc}}(\\mathcal{O}_\\mathcal{Y}) \\to\nD_\\QCoh(\\mathcal{O}_\\mathcal{Y})$\nand we may define $Lf^*_\\QCoh$ as its left derived functor\n$$\nLf_\\QCoh^* :\nD_\\QCoh(\\mathcal{O}_\\mathcal{Y})\n\\longrightarrow\nD_\\QCoh(\\mathcal{O}_\\mathcal{X})\n$$\n(see\nDerived Categories,\nDefinitions \\ref{derived-definition-right-derived-functor-defined} and\n\\ref{derived-definition-everywhere-defined}). If $f$ is quasi-compact\nand quasi-separated, then $Lf^*_\\QCoh$ and $Rf_{\\QCoh, *}$\nsatisfy the following adjointness:\n$$\n\\Hom_{D_\\QCoh(\\mathcal{O}_\\mathcal{X})}(Lf^*_\\QCoh A, B)\n=\n\\Hom_{D_\\QCoh(\\mathcal{O}_\\mathcal{Y})}(A, Rf_{\\QCoh, *}B)\n$$\nfor $A \\in D_\\QCoh(\\mathcal{O}_\\mathcal{Y})$ and\n$B \\in D^{+}_\\QCoh(\\mathcal{O}_\\mathcal{X})$.","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Derived pullback of quasi-coherent modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BE","source_file":"stacks-perfect.tex","source_line":1013,"source_end_line":1057,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L1013-L1057","statement_sha256":"befdcfb1cc117260832c2a90bff4caae0f821a341b888e885ef4ed566c42ab23","origin":"The Stacks Project","memory_eligible":false,"source_rank":14396,"rank":14396,"depth":83,"x":107.917,"y":576.893,"cluster":"derived-categories"},{"id":"stacks:0H13","tag":"0H13","title":"Quasi-coherent objects in the derived category · Lemma 0H13","summary":"Let X be an algebraic stack. Let K be an object of D(X_fppf) whose cohomology sheaves are parasitic. Then RΓ(x, K) = 0 for all objects x of X lying over a scheme U such that U → X is flat.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $K$ be an object of\n$D(\\mathcal{X}_{fppf})$ whose cohomology sheaves are parasitic. Then\n$R\\Gamma(x, K) = 0$ for all objects $x$ of $\\mathcal{X}$ lying\nover a scheme $U$ such that $U \\to \\mathcal{X}$ is flat.","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Quasi-coherent objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H13","source_file":"stacks-perfect.tex","source_line":1153,"source_end_line":1159,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L1153-L1159","statement_sha256":"20495d37a2ef42e7eb528bce91d06c6ea9d48a2586f5725fdf116180494c6888","origin":"The Stacks Project","memory_eligible":false,"source_rank":14397,"rank":14397,"depth":77,"x":403.064,"y":686.762,"cluster":"derived-categories"},{"id":"stacks:0H14","tag":"0H14","title":"Quasi-coherent objects in the derived category · Lemma 0H14","summary":"Let X be an algebraic stack. Let K be an object of D(X_fppf) such that RΓ(x, K) = 0 for all objects x of X lying over an affine scheme U such that U → X is flat. Then H^i(X, K) = 0 for all i.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $K$ be an object of\n$D(\\mathcal{X}_{fppf})$ such that $R\\Gamma(x, K) = 0$ for all objects\n$x$ of $\\mathcal{X}$ lying over an affine scheme $U$ such that\n$U \\to \\mathcal{X}$ is flat. Then $H^i(\\mathcal{X}, K) = 0$ for all $i$.","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Quasi-coherent objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H14","source_file":"stacks-perfect.tex","source_line":1175,"source_end_line":1181,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L1175-L1181","statement_sha256":"ce0cf05efbd96cf9f7313d237ff3e78930938ec99895a15a94155e894771613e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14398,"rank":14398,"depth":77,"x":96.85,"y":773.283,"cluster":"derived-categories"},{"id":"stacks:0H15","tag":"0H15","title":"Quasi-coherent objects in the derived category · Lemma 0H15","summary":"Let X be an algebraic stack. Let K be an object of D_QCoh(O_X_flat, fppf). Then Lg_!K satisfies the following property: for any morphism x → x' of X_affine the map RΓ(x', Lg_!K) ⊗_O(x')^L O(x) → RΓ(x, Lg_!K) is a quasi-isomorphism.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $K$ be an object of\n$D_\\QCoh(\\mathcal{O}_{\\mathcal{X}_{flat, fppf}})$. Then $Lg_!K$ satisfies\nthe following property: for any morphism $x \\to x'$ of\n$\\mathcal{X}_{affine}$ the map\n$$\nR\\Gamma(x', Lg_!K) \\otimes_{\\mathcal{O}(x')}^\\mathbf{L} \\mathcal{O}(x)\n\\longrightarrow\nR\\Gamma(x, Lg_!K)\n$$\nis a quasi-isomorphism.","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Quasi-coherent objects in the derived category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H15","source_file":"stacks-perfect.tex","source_line":1196,"source_end_line":1208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L1196-L1208","statement_sha256":"6417740cc6ff96915e5f17d458e83a8a81f4317a4d06c9bd13c058b5869403a0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14399,"rank":14399,"depth":82,"x":253.184,"y":535.557,"cluster":"derived-categories"},{"id":"stacks:0H16","tag":"0H16","title":"Quasi-coherent objects in the derived category · Proposition 0H16","summary":"Let X be an algebraic stack. Then mathitQC(X) is canonically equivalent to D_QCoh(O_X).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Then $\\mathit{QC}(\\mathcal{X})$\nis canonically equivalent to $D_\\QCoh(\\mathcal{O}_\\mathcal{X})$.","area":"Derived Categories","chapter":"Derived Categories of Stacks","chapter_id":"stacks-perfect","section":"Quasi-coherent objects in the derived category","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H16","source_file":"stacks-perfect.tex","source_line":1310,"source_end_line":1314,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-perfect.tex#L1310-L1314","statement_sha256":"70316dce55eb98855002100b96fd53896b50baeb4a5c70b48c013c010d814026","origin":"The Stacks Project","memory_eligible":false,"source_rank":14400,"rank":14400,"depth":83,"x":329.134,"y":799.752,"cluster":"derived-categories"},{"id":"stacks:072M","tag":"072M","title":"Key fact · Lemma 072M","summary":"The functor Sch^opp → Sets, T ↦ ((a, a', α) as above) is representable by a scheme S ×_M_1, 1 S'.","statement_latex":"The functor $\\Sch^{opp} \\to \\textit{Sets}$,\n$T \\mapsto \\{(a, a', \\alpha)\\text{ as above}\\}$\nis representable by a scheme $S \\times_{\\mathcal{M}_{1, 1}} S'$.","area":"Algebraic Stacks","chapter":"Introducing Algebraic Stacks","chapter_id":"stacks-introduction","section":"Fibre products","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072M","source_file":"stacks-introduction.tex","source_line":201,"source_end_line":206,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-introduction.tex#L201-L206","statement_sha256":"c598c296870e8339cd4cf31df23e96efd1e10e597618dbbaaf96ffba323604ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":14401,"rank":14401,"depth":0,"x":2123.897,"y":1489.03,"cluster":"algebraic-stacks"},{"id":"stacks:072P","tag":"072P","title":"Fibre products · Definition 072P","summary":"We say a morphism S → M_1, 1 is smooth if for every morphism S' → M_1, 1 the projection morphism S ×_M_1, 1 S' → S' is smooth.","statement_latex":"We say a morphism $S \\to \\mathcal{M}_{1, 1}$ is {\\it smooth} if for every\nmorphism $S' \\to \\mathcal{M}_{1, 1}$ the projection morphism\n$$\nS \\times_{\\mathcal{M}_{1, 1}} S' \\longrightarrow S'\n$$\nis smooth.","area":"Algebraic Stacks","chapter":"Introducing Algebraic Stacks","chapter_id":"stacks-introduction","section":"Fibre products","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072P","source_file":"stacks-introduction.tex","source_line":230,"source_end_line":238,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-introduction.tex#L230-L238","statement_sha256":"0286e6356e4c60785c089e776bdbb4a17fc8618656ffe7f7daf5d3db42dee1ce","origin":"The Stacks Project","memory_eligible":false,"source_rank":14402,"rank":14402,"depth":0,"x":2050.015,"y":1735.201,"cluster":"algebraic-stacks"},{"id":"stacks:072R","tag":"072R","title":"The definition · Definition 072R","summary":"We say M_1, 1 is an algebraic stack if and only if • We have descent for objects for the étale topology on Sch. • The key fact holds. • there exists a surjective and smooth morphism S → M_1, 1.","statement_latex":"We say $\\mathcal{M}_{1, 1}$ is an {\\it algebraic stack} if and only if\n\\begin{enumerate}\n\\item We have descent for objects for the \\'etale topology on $\\Sch$.\n\\item The key fact holds.\n\\item there exists a surjective and smooth morphism\n$S \\to \\mathcal{M}_{1, 1}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"Introducing Algebraic Stacks","chapter_id":"stacks-introduction","section":"The definition","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072R","source_file":"stacks-introduction.tex","source_line":260,"source_end_line":269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-introduction.tex#L260-L269","statement_sha256":"120738e5dcae3cc09087de8bd52b43d432a1418d1aeb29dd49a45fd68daf80b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14403,"rank":14403,"depth":0,"x":1906.431,"y":1511.601,"cluster":"algebraic-stacks"},{"id":"stacks:072T","tag":"072T","title":"A smooth cover · Lemma 072T","summary":"The morphism W xrightarrow(E_W, f_W, 0_W) M_1, 1 is smooth and surjective.","statement_latex":"The morphism $W \\xrightarrow{(E_W, f_W, 0_W)} \\mathcal{M}_{1, 1}$ is smooth\nand surjective.","area":"Algebraic Stacks","chapter":"Introducing Algebraic Stacks","chapter_id":"stacks-introduction","section":"A smooth cover","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/072T","source_file":"stacks-introduction.tex","source_line":385,"source_end_line":389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-introduction.tex#L385-L389","statement_sha256":"8558913c8d2d5f72e7f5bfc5fae59453e4a8c4d4ac94e7d987be295a3ad46161","origin":"The Stacks Project","memory_eligible":false,"source_rank":14404,"rank":14404,"depth":0,"x":2192.319,"y":1595.065,"cluster":"algebraic-stacks"},{"id":"stacks:0BPP","tag":"0BPP","title":"Thickenings · Definition 0BPP","summary":"Thickenings. • We say an algebraic stack X' is a thickening of an algebraic stack X if X is a closed substack of X' and the associated topological spaces are equal. • Given two thickenings X ⊂ X' and Y ⊂ Y' a morphism of thickenings is a morphism f' : X' → Y' of algebraic stacks such that f'|_X factors through the closed substack Y. In this situation we set f = f'|_X : X → Y and we say that (f, f') : (X ⊂ X') → (Y ⊂ Y') is a morphism of thickenings. • Let Z be an…","statement_latex":"Thickenings.\n\\begin{enumerate}\n\\item We say an algebraic stack $\\mathcal{X}'$ is a {\\it thickening}\nof an algebraic stack $\\mathcal{X}$ if $\\mathcal{X}$ is a closed substack\nof $\\mathcal{X}'$ and the associated topological spaces are equal.\n\\item Given two thickenings $\\mathcal{X} \\subset \\mathcal{X}'$ and\n$\\mathcal{Y} \\subset \\mathcal{Y}'$ a {\\it morphism of thickenings}\nis a morphism $f' : \\mathcal{X}' \\to \\mathcal{Y}'$ of algebraic stacks\nsuch that $f'|_\\mathcal{X}$ factors through the closed\nsubstack $\\mathcal{Y}$. In this situation we set\n$f = f'|_\\mathcal{X} : \\mathcal{X} \\to \\mathcal{Y}$ and we say that\n$(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to\n(\\mathcal{Y} \\subset \\mathcal{Y}')$ is a morphism of thickenings.\n\\item Let $\\mathcal{Z}$ be an algebraic stack. We similarly define\n{\\it thickenings over $\\mathcal{Z}$} and\n{\\it morphisms of thickenings over $\\mathcal{Z}$}.\nThis means that the algebraic stacks\n$\\mathcal{X}'$ and $\\mathcal{Y}'$\nare endowed with a structure\nmorphism to $\\mathcal{Z}$ and that $f'$ fits into a suitable\n$2$-commutative diagram of algebraic stacks.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Thickenings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPP","source_file":"stacks-more-morphisms.tex","source_line":49,"source_end_line":73,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L49-L73","statement_sha256":"7d7680d784e3d03e20b00bfb377e798b75b159ef56c504d68b107d9f3aeb181b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14405,"rank":14405,"depth":0,"x":1914.197,"y":1695.809,"cluster":"algebraic-stacks"},{"id":"stacks:0CJ7","tag":"0CJ7","title":"Thickenings · Lemma 0CJ7","summary":"Let i : X → X' be a morphism of algebraic stacks. The following are equivalent • i is a thickening of algebraic stacks (abuse of language as above), and • i is representable by algebraic spaces and is a thickening in the sense of Properties of Stacks, Section [Tag 04XB]. In this case i is a closed immersion and a universal homeomorphism.","statement_latex":"Let $i : \\mathcal{X} \\to \\mathcal{X}'$ be a morphism of algebraic stacks.\nThe following are equivalent\n\\begin{enumerate}\n\\item $i$ is a thickening of algebraic stacks (abuse of language as above), and\n\\item $i$ is representable by algebraic spaces and\nis a thickening in the sense of Properties of Stacks, Section\n\\ref{stacks-properties-section-properties-morphisms}.\n\\end{enumerate}\nIn this case $i$ is a closed immersion and a universal homeomorphism.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJ7","source_file":"stacks-more-morphisms.tex","source_line":90,"source_end_line":101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L90-L101","statement_sha256":"f7830c38cd51e726795e6ae36a892aca93b1882fc61c7bca96b9180cc0e8cdef","origin":"The Stacks Project","memory_eligible":false,"source_rank":14406,"rank":14406,"depth":59,"x":2038.35,"y":1463.548,"cluster":"algebraic-stacks"},{"id":"stacks:0BPQ","tag":"0BPQ","title":"Thickenings · Definition 0BPQ","summary":"We say an algebraic stack X' is a first order thickening of an algebraic stack X if X is a closed substack of X' and X → X' is a first order thickening in the sense of Properties of Stacks, Section [Tag 04XB].","statement_latex":"We say an algebraic stack $\\mathcal{X}'$ is a {\\it first order thickening}\nof an algebraic stack $\\mathcal{X}$ if $\\mathcal{X}$ is a closed substack\nof $\\mathcal{X}'$ and $\\mathcal{X} \\to \\mathcal{X}'$ is a first order\nthickening in the sense of Properties of Stacks, Section\n\\ref{stacks-properties-section-properties-morphisms}.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Thickenings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPQ","source_file":"stacks-more-morphisms.tex","source_line":132,"source_end_line":139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L132-L139","statement_sha256":"1f3aad0d1e6178d1400b4b36276a1035695ed6a5959216ed8ed2ba90e2c8a836","origin":"The Stacks Project","memory_eligible":false,"source_rank":14407,"rank":14407,"depth":0,"x":2133.645,"y":1705.429,"cluster":"algebraic-stacks"},{"id":"stacks:0BPR","tag":"0BPR","title":"Thickenings · Lemma 0BPR","summary":"Let Y ⊂ Y' be a thickening of algebraic stacks. Let X' → Y' be a morphism of algebraic stacks and set X = Y ×_Y' X'. Then (X ⊂ X') → (Y ⊂ Y') is a morphism of thickenings. If Y ⊂ Y' is a first order thickening, then X ⊂ X' is a first order thickening.","statement_latex":"Let $\\mathcal{Y} \\subset \\mathcal{Y}'$ be a thickening of algebraic stacks.\nLet $\\mathcal{X}' \\to \\mathcal{Y}'$ be a morphism of algebraic stacks\nand set $\\mathcal{X} = \\mathcal{Y} \\times_{\\mathcal{Y}'} \\mathcal{X}'$.\nThen\n$(\\mathcal{X} \\subset \\mathcal{X}') \\to (\\mathcal{Y} \\subset \\mathcal{Y}')$\nis a morphism of thickenings. If $\\mathcal{Y} \\subset \\mathcal{Y}'$ is a first\norder thickening, then $\\mathcal{X} \\subset \\mathcal{X}'$ is a first\norder thickening.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPR","source_file":"stacks-more-morphisms.tex","source_line":146,"source_end_line":156,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L146-L156","statement_sha256":"f44c90c3d8e648e0e87bad8e35864c29a50444b8d3f5477c6f94c26790344155","origin":"The Stacks Project","memory_eligible":false,"source_rank":14408,"rank":14408,"depth":1,"x":1868.68,"y":1581.056,"cluster":"algebraic-stacks"},{"id":"stacks:0BPS","tag":"0BPS","title":"Thickenings · Lemma 0BPS","summary":"If X ⊂ X' and X' ⊂ X\" are thickenings of algebraic stacks, then so is X ⊂ X\".","statement_latex":"If $\\mathcal{X} \\subset \\mathcal{X}'$ and $\\mathcal{X}' \\subset \\mathcal{X}''$\nare thickenings of algebraic stacks, then so is\n$\\mathcal{X} \\subset \\mathcal{X}''$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPS","source_file":"stacks-more-morphisms.tex","source_line":165,"source_end_line":170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L165-L170","statement_sha256":"5ca56103c3a582174aadc6361594481409192a4e1e7274daacea22a21fd91b72","origin":"The Stacks Project","memory_eligible":false,"source_rank":14409,"rank":14409,"depth":1,"x":2164.282,"y":1522.378,"cluster":"algebraic-stacks"},{"id":"stacks:0BPU","tag":"0BPU","title":"Thickenings · Lemma 0BPU","summary":"Let (f, f') : (X ⊂ X') → (Y ⊂ Y') be a morphism of thickenings of algebraic stacks. Then X ×_Y X → X' ×_Y' X' is a thickening and the canonical diagram xymatrix X ar[r]_-Δ ar[d] & X ×_Y X ar[d] X' ar[r]^-Δ' & X' ×_Y' X' is cartesian.","statement_latex":"Let $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to\n(\\mathcal{Y} \\subset \\mathcal{Y}')$ be a morphism of thickenings\nof algebraic stacks. Then\n$\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\to\n\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'$\nis a thickening and the canonical diagram\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[r]_-\\Delta \\ar[d] &\n\\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X} \\ar[d] \\\\\n\\mathcal{X}' \\ar[r]^-{\\Delta'} &\n\\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'\n}\n$$\nis cartesian.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPU","source_file":"stacks-more-morphisms.tex","source_line":192,"source_end_line":209,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L192-L209","statement_sha256":"9068cc7a26ffd6ae44e4a8d075f9ed2b49320ada0f7b2c81b7c849cfb8799f7e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14410,"rank":14410,"depth":72,"x":1993.379,"y":1733.524,"cluster":"algebraic-stacks"},{"id":"stacks:0BPV","tag":"0BPV","title":"Thickenings · Lemma 0BPV","summary":"Let (f, f') : (X ⊂ X') → (Y ⊂ Y') be a morphism of thickenings of algebraic stacks. Let Δ : X → X ×_Y X and Δ' : X' → X' ×_Y' X' be the corresponding diagonal morphisms. Then each property from the following list is satisfied by Δ if and only if it is satisfied by Δ': (a) representable by schemes, (b) affine, (c) surjective, (d) quasi-compact, (e) universally closed, (f) integral, (g) quasi-separated, (h) separated, (i) universally injective, (j) universally open, (k)…","statement_latex":"Let $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to\n(\\mathcal{Y} \\subset \\mathcal{Y}')$ be a morphism of thickenings\nof algebraic stacks.\nLet $\\Delta : \\mathcal{X} \\to \\mathcal{X} \\times_\\mathcal{Y} \\mathcal{X}$ and\n$\\Delta' : \\mathcal{X}' \\to \\mathcal{X}' \\times_{\\mathcal{Y}'} \\mathcal{X}'$\nbe the corresponding diagonal morphisms.\nThen each property from the following list is satisfied by $\\Delta$ if\nand only if it is satisfied by $\\Delta'$:\n(a) representable by schemes, (b) affine, (c) surjective, (d) quasi-compact,\n(e) universally closed, (f) integral, (g) quasi-separated, (h) separated,\n(i) universally injective, (j) universally open, (k) locally quasi-finite,\n(l) finite, (m) unramified, (n) monomorphism, (o) immersion,\n(p) closed immersion, and (q) proper.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPV","source_file":"stacks-more-morphisms.tex","source_line":236,"source_end_line":251,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L236-L251","statement_sha256":"1d117fd24bb5b30e6b9ed168b64fbeff8af49e02b9f33e4d07f7d945a3f3c28d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14411,"rank":14411,"depth":73,"x":1949.573,"y":1480.68,"cluster":"algebraic-stacks"},{"id":"stacks:0BPW","tag":"0BPW","title":"Thickenings · Lemma 0BPW","summary":"[Conrad-moduli] Let X ⊂ X' be a thickening of algebraic stacks. Then • X is an algebraic space if and only if X' is an algebraic space, • X is a scheme if and only if X' is a scheme, • X is DM if and only if X' is DM, • X is quasi-DM if and only if X' is quasi-DM, • X is separated if and only if X' is separated, • X is quasi-separated if and only if X' is quasi-separated, and • add more here.","statement_latex":"\\begin{reference}\n\\cite[Theorem 2.2.5]{Conrad-moduli}\n\\end{reference}\nLet $\\mathcal{X} \\subset \\mathcal{X}'$ be a thickening of algebraic\nstacks. Then\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is an algebraic space if and only if $\\mathcal{X}'$\nis an algebraic space,\n\\item $\\mathcal{X}$ is a scheme if and only if $\\mathcal{X}'$ is a scheme,\n\\item $\\mathcal{X}$ is DM if and only if $\\mathcal{X}'$ is DM,\n\\item $\\mathcal{X}$ is quasi-DM if and only if $\\mathcal{X}'$ is quasi-DM,\n\\item $\\mathcal{X}$ is separated if and only if $\\mathcal{X}'$ is separated,\n\\item $\\mathcal{X}$ is quasi-separated if and only if $\\mathcal{X}'$ is\nquasi-separated, and\n\\item add more here.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0BPW","source_file":"stacks-more-morphisms.tex","source_line":289,"source_end_line":307,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L289-L307","statement_sha256":"2f69e8d7926bbbece94d9b9aee9956ceed847dcc6ecebfd92ab02ed45018b5b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14412,"rank":14412,"depth":74,"x":2185.366,"y":1642.377,"cluster":"algebraic-stacks"},{"id":"stacks:0CJ9","tag":"0CJ9","title":"Morphisms of thickenings · Lemma 0CJ9","summary":"Let (f, f') : (X ⊂ X') → (Y ⊂ Y') be a morphism of thickenings of algebraic stacks. Then • f is an affine morphism if and only if f' is an affine morphism, • f is a surjective morphism if and only if f' is a surjective morphism, • f is quasi-compact if and only if f' quasi-compact, • f is universally closed if and only if f' is universally closed, • f is integral if and only if f' is integral, • f is universally injective if and only if f' is universally injective, • f is…","statement_latex":"Let $(f, f') : (\\mathcal{X} \\subset \\mathcal{X}') \\to\n(\\mathcal{Y} \\subset \\mathcal{Y}')$\nbe a morphism of thickenings of algebraic stacks. Then\n\\begin{enumerate}\n\\item $f$ is an affine morphism if and only if $f'$ is an affine morphism,\n\\item $f$ is a surjective morphism if and only if $f'$ is a surjective morphism,\n\\item $f$ is quasi-compact if and only if $f'$ quasi-compact,\n\\item $f$ is universally closed if and only if $f'$ is universally closed,\n\\item $f$ is integral if and only if $f'$ is integral,\n\\item $f$ is universally injective if and only if $f'$ is universally injective,\n\\item $f$ is universally open if and only if $f'$ is universally open,\n\\item $f$ is quasi-DM if and only if $f'$ is quasi-DM,\n\\item $f$ is DM if and only if $f'$ is DM,\n\\item $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,\n\\item $f$ is representable if and only if $f'$ is representable,\n\\item $f$ is representable by algebraic spaces if and only if $f'$ is\nrepresentable by algebraic spaces,\n\\item add more here.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Morphisms of thickenings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJ9","source_file":"stacks-more-morphisms.tex","source_line":359,"source_end_line":380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L359-L380","statement_sha256":"ba60d1559f2a742f26b141a46de69b11ce662b039af93018fcd24abf899cbd08","origin":"The Stacks Project","memory_eligible":false,"source_rank":14413,"rank":14413,"depth":75,"x":1881.255,"y":1656.949,"cluster":"algebraic-stacks"},{"id":"stacks:0CJB","tag":"0CJB","title":"Infinitesimal deformations of algebraic stacks · Lemma 0CJB","summary":"Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (B ⊂ B') of thickenings of algebraic stacks. Assume • Y' → B' is locally of finite type, • X' → B' is flat and locally of finite presentation, • f is flat, and • X = B ×_B' X' and Y = B ×_B' Y'. Then f' is flat and for all y' ∈ |Y'| in the image of |f'| the morphism Y' → B' is flat at y'.","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n(\\mathcal{X} \\subset \\mathcal{X}') \\ar[rr]_{(f, f')} \\ar[rd] & &\n(\\mathcal{Y} \\subset \\mathcal{Y}') \\ar[ld] \\\\\n& (\\mathcal{B} \\subset \\mathcal{B}')\n}\n$$\nof thickenings of algebraic stacks. Assume\n\\begin{enumerate}\n\\item $\\mathcal{Y}' \\to \\mathcal{B}'$ is locally of finite type,\n\\item $\\mathcal{X}' \\to \\mathcal{B}'$ is\nflat and locally of finite presentation,\n\\item $f$ is flat, and\n\\item $\\mathcal{X} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{X}'$ and\n$\\mathcal{Y} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{Y}'$.\n\\end{enumerate}\nThen $f'$ is flat and for all $y' \\in |\\mathcal{Y}'|$ in the image of $|f'|$\nthe morphism $\\mathcal{Y}' \\to \\mathcal{B}'$ is flat at $y'$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Infinitesimal deformations of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJB","source_file":"stacks-more-morphisms.tex","source_line":446,"source_end_line":467,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L446-L467","statement_sha256":"0304f7df4760a291f02ccb04090fa5a1fcc6607a9a2c611eaa1fc7f4f766c0fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14414,"rank":14414,"depth":16,"x":2093.929,"y":1473.519,"cluster":"algebraic-stacks"},{"id":"stacks:0CJC","tag":"0CJC","title":"Infinitesimal deformations of algebraic stacks · Lemma 0CJC","summary":"Consider a commutative diagram xymatrix (X ⊂ X') ar[rr]_(f, f') ar[rd] & & (Y ⊂ Y') ar[ld] & (B ⊂ B') of thickenings of algebraic stacks. Assume Y' → B' locally of finite type, X' → B' flat and locally of finite presentation, X = B ×_B' X', and Y = B ×_B' Y'. Then • f is flat if and only if f' is flat, • f is an isomorphism if and only if f' is an isomorphism, • f is an open immersion if and only if f' is an open immersion, • f is a monomorphism if and only if f' is a…","statement_latex":"Consider a commutative diagram\n$$\n\\xymatrix{\n(\\mathcal{X} \\subset \\mathcal{X}') \\ar[rr]_{(f, f')} \\ar[rd] & &\n(\\mathcal{Y} \\subset \\mathcal{Y}') \\ar[ld] \\\\\n& (\\mathcal{B} \\subset \\mathcal{B}')\n}\n$$\nof thickenings of algebraic stacks.\nAssume $\\mathcal{Y}' \\to \\mathcal{B}'$ locally of finite type,\n$\\mathcal{X}' \\to \\mathcal{B}'$ flat and locally of finite presentation,\n$\\mathcal{X} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{X}'$, and\n$\\mathcal{Y} = \\mathcal{B} \\times_{\\mathcal{B}'} \\mathcal{Y}'$. Then\n\\begin{enumerate}\n\\item $f$ is flat if and only if $f'$ is flat,\n\n\\item $f$ is an isomorphism if and only if $f'$ is an isomorphism,\n\n\\item $f$ is an open immersion if and only if $f'$ is an open immersion,\n\n\\item $f$ is a monomorphism if and only if $f'$ is a monomorphism,\n\n\\item $f$ is locally quasi-finite if and only if $f'$ is locally quasi-finite,\n\n\\item $f$ is syntomic if and only if $f'$ is syntomic,\n\n\\item $f$ is smooth if and only if $f'$ is smooth,\n\n\\item $f$ is unramified if and only if $f'$ is unramified,\n\n\\item $f$ is \\'etale if and only if $f'$ is \\'etale,\n\n\\item $f$ is finite if and only if $f'$ is finite, and\n\n\\item add more here.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Infinitesimal deformations of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJC","source_file":"stacks-more-morphisms.tex","source_line":493,"source_end_line":531,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L493-L531","statement_sha256":"1986bbedd69f2d76a57b240add78ea65e0ff73915e9ecc7139042e073b145b79","origin":"The Stacks Project","memory_eligible":false,"source_rank":14415,"rank":14415,"depth":76,"x":2084.61,"y":1729.629,"cluster":"algebraic-stacks"},{"id":"stacks:0CJS","tag":"0CJS","title":"Lifting affines · Lemma 0CJS","summary":"For any morphism ([Tag 0CJR]) the map f' : V' → U' is étale.","statement_latex":"For any morphism (\\ref{equation-morphism}) the map $f' : V' \\to U'$ is \\'etale.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Lifting affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJS","source_file":"stacks-more-morphisms.tex","source_line":732,"source_end_line":735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L732-L735","statement_sha256":"7de002ec94094cfccc09d000ace0ccddb5e873f4f74a69ec63140df786587e7b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14416,"rank":14416,"depth":77,"x":1885.389,"y":1535.357,"cluster":"algebraic-stacks"},{"id":"stacks:0CJT","tag":"0CJT","title":"Lifting affines · Lemma 0CJT","summary":"The category p : C → W_spaces, etale constructed in Remark [Tag 0CJP] is fibred in groupoids.","statement_latex":"The category $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed\nin Remark \\ref{remark-gerbe-of-lifts} is fibred in groupoids.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Lifting affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJT","source_file":"stacks-more-morphisms.tex","source_line":746,"source_end_line":750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L746-L750","statement_sha256":"29ca4af2f43b270b12823013b5ec3d5b660b7068204eece244f023e87af7bee0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14417,"rank":14417,"depth":77,"x":2188.726,"y":1565.588,"cluster":"algebraic-stacks"},{"id":"stacks:0CJU","tag":"0CJU","title":"Lifting affines · Lemma 0CJU","summary":"The category p : C → W_spaces, etale constructed in Remark [Tag 0CJP] is a stack in groupoids.","statement_latex":"The category $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed\nin Remark \\ref{remark-gerbe-of-lifts} is a stack in groupoids.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Lifting affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJU","source_file":"stacks-more-morphisms.tex","source_line":801,"source_end_line":805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L801-L805","statement_sha256":"477237e9b91250fd8e9966ccf4c74f9c06e4fc2beabc71ecbb8ae2e36d3051cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14418,"rank":14418,"depth":78,"x":1940.57,"y":1715.518,"cluster":"algebraic-stacks"},{"id":"stacks:0CJV","tag":"0CJV","title":"Lifting affines · Lemma 0CJV","summary":"Let X ⊂ X' be a thickening of algebraic stacks. Let W be an algebraic space and let W → X be a smooth morphism. There exists an étale covering (W_i → W)_i ∈ I and for each i a cartesian diagram xymatrix W_i ar[r] ar[d] & W_i' ar[d] X ar[r] & X' with W_i' → X' smooth.","statement_latex":"Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a thickening of algebraic stacks.\nLet $W$ be an algebraic space and let $W \\to \\mathcal{X}$ be a smooth morphism.\nThere exists an \\'etale covering $\\{W_i \\to W\\}_{i \\in I}$ and for each $i$\na cartesian diagram\n$$\n\\xymatrix{\nW_i \\ar[r] \\ar[d] & W_i' \\ar[d] \\\\\n\\mathcal{X} \\ar[r] & \\mathcal{X}'\n}\n$$\nwith $W_i' \\to \\mathcal{X}'$ smooth.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Lifting affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJV","source_file":"stacks-more-morphisms.tex","source_line":881,"source_end_line":894,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L881-L894","statement_sha256":"934628bf2577e146c5357f92fb2373fc46399832691c8eac912a4bc60b8c33f2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14419,"rank":14419,"depth":49,"x":2003.029,"y":1463.981,"cluster":"algebraic-stacks"},{"id":"stacks:0CJW","tag":"0CJW","title":"Lifting affines · Lemma 0CJW","summary":"Let X ⊂ X' be a thickening of algebraic stacks. Consider a commutative diagram xymatrix W\" ar[d]_x\" & W ar[l] ar[r] ar[d]_x & W' ar[d]^x' X' & X ar[l] ar[r] & X' with cartesian squares where W', W, W\" are algebraic spaces and the vertical arrows are smooth. Then there exist • an étale covering (f'_k : W'_k → W')_k ∈ K, • étale morphisms f\"_k : W'_k → W\", and • 2-morphisms γ_k : x\" ∘ f\"_k → x' ∘ f'_k such that (a) (f'_k)^-1(W) = (f\"_k)^-1(W), (b) f'_k|_(f'_k)^-1(W) =…","statement_latex":"Let $\\mathcal{X} \\subset \\mathcal{X}'$ be a thickening of algebraic stacks.\nConsider a commutative diagram\n$$\n\\xymatrix{\nW'' \\ar[d]_{x''} & W \\ar[l] \\ar[r] \\ar[d]_x & W' \\ar[d]^{x'} \\\\\n\\mathcal{X}' & \\mathcal{X} \\ar[l] \\ar[r] & \\mathcal{X}'\n}\n$$\nwith cartesian squares where $W', W, W''$ are algebraic spaces and\nthe vertical arrows are smooth. Then there exist\n\\begin{enumerate}\n\\item an \\'etale covering $\\{f'_k : W'_k \\to W'\\}_{k \\in K}$,\n\\item \\'etale morphisms $f''_k : W'_k \\to W''$, and\n\\item $2$-morphisms $\\gamma_k : x'' \\circ f''_k \\to x' \\circ f'_k$\n\\end{enumerate}\nsuch that (a) $(f'_k)^{-1}(W) = (f''_k)^{-1}(W)$, (b)\n$f'_k|_{(f'_k)^{-1}(W)} = f''_k|_{(f''_k)^{-1}(W)}$, and\n(c) pulling back $\\gamma_k$ to the closed subscheme of (a)\nagrees with the $2$-morphism given by the commutativity of\nthe initial diagram over $W$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Lifting affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJW","source_file":"stacks-more-morphisms.tex","source_line":993,"source_end_line":1015,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L993-L1015","statement_sha256":"52552b500fb817c0a848e815249d35e9b0b059c60026643bd5240ff7265e51fc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14420,"rank":14420,"depth":77,"x":2159.358,"y":1685.052,"cluster":"algebraic-stacks"},{"id":"stacks:0CJX","tag":"0CJX","title":"Lifting affines · Lemma 0CJX","summary":"The category p : C → W_spaces, etale constructed in Remark [Tag 0CJP] is a gerbe.","statement_latex":"The category $p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed\nin Remark \\ref{remark-gerbe-of-lifts} is a gerbe.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Lifting affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CJX","source_file":"stacks-more-morphisms.tex","source_line":1088,"source_end_line":1092,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1088-L1092","statement_sha256":"a72bae4c14a4e05d903e28ca631e29ab950eb05b6afcdc5436dafb559f480b8c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14421,"rank":14421,"depth":79,"x":1866.106,"y":1610.692,"cluster":"algebraic-stacks"},{"id":"stacks:0CKG","tag":"0CKG","title":"Lifting affines · Lemma 0CKG","summary":"In Remark [Tag 0CJP] assume X ⊂ X' is a first order thickening. Then • the automorphism sheaves of objects of the gerbe p : C → W_spaces, etale constructed in Remark [Tag 0CJP] are abelian, and • the sheaf of groups G constructed in Stacks, Lemma [Tag 0CJY] is a quasi-coherent O_W-module.","statement_latex":"In Remark \\ref{remark-gerbe-of-lifts} assume $\\mathcal{X} \\subset \\mathcal{X}'$\nis a first order thickening. Then\n\\begin{enumerate}\n\\item the automorphism sheaves of objects of the gerbe\n$p : \\mathcal{C} \\to W_{spaces, \\etale}$ constructed\nin Remark \\ref{remark-gerbe-of-lifts} are abelian, and\n\\item the sheaf of groups $\\mathcal{G}$ constructed in\nStacks, Lemma \\ref{stacks-lemma-gerbe-abelian-auts}\nis a quasi-coherent $\\mathcal{O}_W$-module.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Lifting affines","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKG","source_file":"stacks-more-morphisms.tex","source_line":1105,"source_end_line":1117,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1105-L1117","statement_sha256":"4dbe2bd2009777ed0ae5077cb66ea7c5df751ffdd5209facd1de3300f3a957b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14422,"rank":14422,"depth":78,"x":2142.33,"y":1499.05,"cluster":"algebraic-stacks"},{"id":"stacks:0CKI","tag":"0CKI","title":"Emerton · Proposition 0CKI","summary":"Email of Matthew Emerton dated April 27, 2016. Let X ⊂ X' be a first order thickening of algebraic stacks. Let W be an affine scheme and let W → X be a smooth morphism. Then there exists a cartesian diagram xymatrix W ar[d] ar[r] & W' ar[d] X ar[r] & X' with W' → X' smooth and W' affine.","statement_latex":"\\begin{reference}\nEmail of Matthew Emerton dated April 27, 2016.\n\\end{reference}\nLet $\\mathcal{X} \\subset \\mathcal{X}'$ be a first order thickening\nof algebraic stacks. Let $W$ be an affine scheme and let\n$W \\to \\mathcal{X}$ be a smooth morphism. Then there exists\na cartesian diagram\n$$\n\\xymatrix{\nW \\ar[d] \\ar[r] & W' \\ar[d] \\\\\n\\mathcal{X} \\ar[r] & \\mathcal{X}'\n}\n$$\nwith $W' \\to \\mathcal{X}'$ smooth and $W'$ affine.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Lifting affines","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CKI","source_file":"stacks-more-morphisms.tex","source_line":1280,"source_end_line":1296,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1280-L1296","statement_sha256":"2a0666e45abbb6a700d12f774fdc416a0183e6fe67f60878ffaceec9978456bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14423,"rank":14423,"depth":80,"x":2028.349,"y":1738.271,"cluster":"algebraic-stacks"},{"id":"stacks:0DNR","tag":"0DNR","title":"Infinitesimal deformations · Lemma 0DNR","summary":"Let X be an algebraic stack over a scheme S. Assume I_X → X is locally of finite presentation. Let A → B be a flat S-algebra homomorphism. Let x be an object of X over A and set y = x|_B. Then Inf_x(M) ⊗_A B = Inf_y(M ⊗_A B).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over a scheme $S$.\nAssume $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally\nof finite presentation. Let $A \\to B$ be a flat $S$-algebra homomorphism.\nLet $x$ be an object of $\\mathcal{X}$ over $A$ and set $y = x|_B$.\nThen $\\text{Inf}_x(M) \\otimes_A B = \\text{Inf}_y(M \\otimes_A B)$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Infinitesimal deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNR","source_file":"stacks-more-morphisms.tex","source_line":1348,"source_end_line":1355,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1348-L1355","statement_sha256":"86bb386e497d73a35f44c42a91e63bc2f343f0a25b12cd8498ca3dd5d4f90c82","origin":"The Stacks Project","memory_eligible":false,"source_rank":14424,"rank":14424,"depth":50,"x":1919.949,"y":1497.035,"cluster":"algebraic-stacks"},{"id":"stacks:0DNS","tag":"0DNS","title":"Infinitesimal deformations · Lemma 0DNS","summary":"Let X be an algebraic stack over a base scheme S. Assume I_X → X is locally of finite presentation. Let (A' → A, x) be a deformation situation. Then the functor F : B' ↦ (lifts of x|_B' ⊗_A' A to B')/isomorphisms is a sheaf on the site (Aff/Spec(A'))_fppf of Topologies, Definition [Tag 021S].","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over a base scheme $S$.\nAssume $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is locally\nof finite presentation. Let $(A' \\to A, x)$ be a deformation situation.\nThen the functor\n$$\nF : B' \\longmapsto\n\\{\\text{lifts of }x|_{B' \\otimes_{A'} A}\\text{ to } B'\\}/\\text{isomorphisms}\n$$\nis a sheaf on the site $(\\textit{Aff}/\\Spec(A'))_{fppf}$ of\nTopologies, Definition \\ref{topologies-definition-big-small-fppf}.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Infinitesimal deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNS","source_file":"stacks-more-morphisms.tex","source_line":1389,"source_end_line":1401,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1389-L1401","statement_sha256":"2969e8bc0b384bb0dc1e6d5719bc82bc99b4e5cc2ddc5ef9c0cd96159010e6a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14425,"rank":14425,"depth":51,"x":2194.062,"y":1613.489,"cluster":"algebraic-stacks"},{"id":"stacks:0DNT","tag":"0DNT","title":"Infinitesimal deformations · Lemma 0DNT","summary":"Let X be an algebraic stack over a scheme S whose structure morphism X → S is locally of finite presentation. Let A → B be a flat S-algebra homomorphism. Let x be an object of X over A. Then T_x(M) ⊗_A B = T_y(M ⊗_A B).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over a scheme $S$ whose\nstructure morphism $\\mathcal{X} \\to S$ is locally of finite presentation.\nLet $A \\to B$ be a flat $S$-algebra homomorphism.\nLet $x$ be an object of $\\mathcal{X}$ over $A$.\nThen $T_x(M) \\otimes_A B = T_y(M \\otimes_A B)$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Infinitesimal deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNT","source_file":"stacks-more-morphisms.tex","source_line":1481,"source_end_line":1488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1481-L1488","statement_sha256":"c1b65a7facfc497e0219db81dd092dc6825df62c94cdc8759d00baeda5f9f4d4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14426,"rank":14426,"depth":59,"x":1898.088,"y":1683.202,"cluster":"algebraic-stacks"},{"id":"stacks:0DNU","tag":"0DNU","title":"Infinitesimal deformations · Lemma 0DNU","summary":"Let X be an algebraic stack over a scheme S whose structure morphism X → S is locally of finite presentation. Let (A' → A, x) be a deformation situation. If there exists a faithfully flat finitely presented A'-algebra B' and an object y' of X over B' lifting x|_B' ⊗_A' A, then there exists an object x' over A' lifting x.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack over a scheme $S$ whose\nstructure morphism $\\mathcal{X} \\to S$ is locally of finite presentation.\nLet $(A' \\to A, x)$ be a deformation situation. If there exists a\nfaithfully flat finitely presented $A'$-algebra $B'$ and an\nobject $y'$ of $\\mathcal{X}$ over $B'$ lifting $x|_{B' \\otimes_{A'} A}$,\nthen there exists an object $x'$ over $A'$ lifting $x$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Infinitesimal deformations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNU","source_file":"stacks-more-morphisms.tex","source_line":1578,"source_end_line":1586,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1578-L1586","statement_sha256":"fa90551f50b211dfc6ebf3ad6cff278372eb1116b2067c1f14199ed334118185","origin":"The Stacks Project","memory_eligible":false,"source_rank":14427,"rank":14427,"depth":60,"x":2060.382,"y":1463.706,"cluster":"algebraic-stacks"},{"id":"stacks:0DNW","tag":"0DNW","title":"Formally smooth morphisms · Definition 0DNW","summary":"A morphism f : X → Y of algebraic stacks is said to be formally smooth if it is formally smooth on objects as a 1-morphism in categories fibred in groupoids as explained in Criteria for Representability, Section [Tag 06CZ].","statement_latex":"A morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks is said to be\n{\\it formally smooth} if it is formally smooth on objects as a\n$1$-morphism in categories fibred in groupoids as explained in\nCriteria for Representability, Section \\ref{criteria-section-formally-smooth}.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Formally smooth morphisms","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNW","source_file":"stacks-more-morphisms.tex","source_line":1664,"source_end_line":1670,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1664-L1670","statement_sha256":"ad248e3cca0c645126610458762dc509aee17a1f62c6b8a4dd81ba4979145e0c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14428,"rank":14428,"depth":0,"x":2117.259,"y":1717.819,"cluster":"algebraic-stacks"},{"id":"stacks:0DNY","tag":"0DNY","title":"Formally smooth morphisms · Lemma 0DNY","summary":"A morphism f : X → Y of algebraic stacks is formally smooth (Definition [Tag 0DNW]) if and only if for every diagram ([Tag 0DNX]) and γ the category of dotted arrows is nonempty.","statement_latex":"A morphism $f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks is\nformally smooth (Definition \\ref{definition-formally-smooth})\nif and only if for every diagram (\\ref{equation-diagram}) and $\\gamma$\nthe category of dotted arrows is nonempty.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNY","source_file":"stacks-more-morphisms.tex","source_line":1719,"source_end_line":1725,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1719-L1725","statement_sha256":"931ccf4153a6b2597c32b1ac7312956f5c829854d52964d1219756dd571125a2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14429,"rank":14429,"depth":1,"x":1870.803,"y":1562.61,"cluster":"algebraic-stacks"},{"id":"stacks:0H1I","tag":"0H1I","title":"Formally smooth morphisms · Lemma 0H1I","summary":"The base change of a formally smooth morphism of algebraic stacks by any morphism of algebraic stacks is formally smooth.","statement_latex":"The base change of a formally smooth morphism of algebraic stacks\nby any morphism of algebraic stacks is formally smooth.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1I","source_file":"stacks-more-morphisms.tex","source_line":1731,"source_end_line":1735,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1731-L1735","statement_sha256":"f821a237c82978b2366541b47dc23cf5da6a85eafb4634717db7644efc1ce2ed","origin":"The Stacks Project","memory_eligible":false,"source_rank":14430,"rank":14430,"depth":1,"x":2177.556,"y":1537.197,"cluster":"algebraic-stacks"},{"id":"stacks:0H1J","tag":"0H1J","title":"Formally smooth morphisms · Lemma 0H1J","summary":"The composition of formally smooth morphisms of algebraic stacks is formally smooth.","statement_latex":"The composition of formally smooth morphisms of algebraic stacks\nis formally smooth.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1J","source_file":"stacks-more-morphisms.tex","source_line":1743,"source_end_line":1747,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1743-L1747","statement_sha256":"16f6fc62bd8aba52b76b8db269e8e22f4209a0ddde36c1376a70cd54807b1522","origin":"The Stacks Project","memory_eligible":false,"source_rank":14431,"rank":14431,"depth":1,"x":1971.66,"y":1730.124,"cluster":"algebraic-stacks"},{"id":"stacks:0H1K","tag":"0H1K","title":"Formally smooth morphisms · Lemma 0H1K","summary":"Let f : X → Y be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent • f is formally smooth, • for every scheme T and morphism T → Y the morphism X ×_Y T → T is formally smooth as a morphism of algebraic spaces.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks\nwhich is representable by algebraic spaces. Then the following are equivalent\n\\begin{enumerate}\n\\item $f$ is formally smooth,\n\\item for every scheme $T$ and morphism $T \\to \\mathcal{Y}$\nthe morphism $\\mathcal{X} \\times_\\mathcal{Y} T \\to T$\nis formally smooth as a morphism of algebraic spaces.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H1K","source_file":"stacks-more-morphisms.tex","source_line":1755,"source_end_line":1765,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1755-L1765","statement_sha256":"6c3e45f131e1ad30ff45c7496938d96619b976549f6f40a98f8bff6e55b4ac9b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14432,"rank":14432,"depth":1,"x":1968.336,"y":1470.859,"cluster":"algebraic-stacks"},{"id":"stacks:0DNZ","tag":"0DNZ","title":"Formally smooth morphisms · Lemma 0DNZ","summary":"Let T → T' be a first order thickening of affine schemes. Let X' be an algebraic stack over T' whose structure morphism X' → T' is smooth. Let x : T → X' be a morphism over T'. Then there exists a morphsm x' : T' → X' over T' with x'|_T = x.","statement_latex":"Let $T \\to T'$ be a first order thickening of affine schemes.\nLet $\\mathcal{X}'$ be an algebraic stack over $T'$\nwhose structure morphism $\\mathcal{X}' \\to T'$ is smooth.\nLet $x : T \\to \\mathcal{X}'$ be a morphism\nover $T'$. Then there exists a morphsm $x' : T' \\to \\mathcal{X}'$\nover $T'$ with $x'|_T = x$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNZ","source_file":"stacks-more-morphisms.tex","source_line":1773,"source_end_line":1781,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1773-L1781","statement_sha256":"b32f8a06a2af42f3914aaa50fce0b3cb9f5186fd1dd8c689fdf7aa4d29c83688","origin":"The Stacks Project","memory_eligible":false,"source_rank":14433,"rank":14433,"depth":61,"x":2179.421,"y":1660.277,"cluster":"algebraic-stacks"},{"id":"stacks:0DP0","tag":"0DP0","title":"Infinitesimal lifting criterion · Lemma 0DP0","summary":"Let f : X → Y be a morphism of algebraic stacks. The following are equivalent: • The morphism f is smooth. • The morphism f is locally of finite presentation and formally smooth.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nThe following are equivalent:\n\\begin{enumerate}\n\\item The morphism $f$ is smooth.\n\\item The morphism $f$ is locally of finite presentation and\nformally smooth.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Formally smooth morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DP0","source_file":"stacks-more-morphisms.tex","source_line":1817,"source_end_line":1826,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1817-L1826","statement_sha256":"2ad2a38a7806ff52063768336f946219fcb7cb364b7283da94ae718dffbeb420","origin":"The Stacks Project","memory_eligible":false,"source_rank":14434,"rank":14434,"depth":62,"x":1871.241,"y":1640.365,"cluster":"algebraic-stacks"},{"id":"stacks:0CQ4","tag":"0CQ4","title":"Blowing up and flatness · Lemma 0CQ4","summary":"Let f : X → Y be a morphism from an algebraic stack to an algebraic space. Let V ⊂ Y be an open subspace. Assume • Y is quasi-compact and quasi-separated, • f is of finite type and quasi-separated, • V is quasi-compact, and • X_V is flat and locally of finite presentation over V. Then there exists a V-admissible blowup Y' → Y and a closed substack X' ⊂ X_Y' with X'_V = X_V such that X' → Y' is flat and of finite presentation.","statement_latex":"Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack\nto an algebraic space. Let $V \\subset Y$ be an open subspace. Assume\n\\begin{enumerate}\n\\item $Y$ is quasi-compact and quasi-separated,\n\\item $f$ is of finite type and quasi-separated,\n\\item $V$ is quasi-compact, and\n\\item $\\mathcal{X}_V$ is flat and locally of finite presentation over $V$.\n\\end{enumerate}\nThen there exists a $V$-admissible blowup $Y' \\to Y$\nand a closed substack $\\mathcal{X}' \\subset \\mathcal{X}_{Y'}$\nwith $\\mathcal{X}'_V = \\mathcal{X}_V$ such that\n$\\mathcal{X}' \\to Y'$ is flat and of finite presentation.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Blowing up and flatness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQ4","source_file":"stacks-more-morphisms.tex","source_line":1898,"source_end_line":1912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1898-L1912","statement_sha256":"ced9ba5a38ed76e7c2c72081d614d32903b6022d66b1975a9440af03ee467ab7","origin":"The Stacks Project","memory_eligible":false,"source_rank":14435,"rank":14435,"depth":74,"x":2114.662,"y":1480.073,"cluster":"algebraic-stacks"},{"id":"stacks:0CQ6","tag":"0CQ6","title":"Chow's lemma for algebraic stacks · Lemma 0CQ6","summary":"Let Y be a quasi-compact and quasi-separated algebraic space. Let V ⊂ Y be a quasi-compact open. Let f : X → V be surjective, flat, and locally of finite presentation. Then there exists a finite surjective morphism g : Y' → Y such that V' = g^-1(V) → Y factors Zariski locally through f.","statement_latex":"Let $Y$ be a quasi-compact and quasi-separated algebraic space.\nLet $V \\subset Y$ be a quasi-compact open. Let $f : \\mathcal{X} \\to V$\nbe surjective, flat, and locally of finite presentation.\nThen there exists a finite surjective morphism $g : Y' \\to Y$ such that\n$V' = g^{-1}(V) \\to Y$ factors Zariski locally through $f$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Chow's lemma for algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQ6","source_file":"stacks-more-morphisms.tex","source_line":1983,"source_end_line":1990,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L1983-L1990","statement_sha256":"1c63f28244308dd9a026998669ca3862c15a2395a412c79030c1fc037a0fa36d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14436,"rank":14436,"depth":67,"x":2064.036,"y":1736.563,"cluster":"algebraic-stacks"},{"id":"stacks:0CQ7","tag":"0CQ7","title":"Chow's lemma for algebraic stacks · Lemma 0CQ7","summary":"Let f : X → Y be a morphism from an algebraic stack to an algebraic space. Let V ⊂ Y be an open subspace. Assume • f is separated and of finite type, • Y is quasi-compact and quasi-separated, • V is quasi-compact, and • X_V is a gerbe over V. Then there exists a commutative diagram xymatrix overlineZ ar[rd]_overlineg & Z ar[l]^j ar[d]_g ar[r]_h & X ar[ld]^f & Y with j an open immersion, overlineg and h proper, and such that |V| is contained in the image of |g|.","statement_latex":"Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack\nto an algebraic space. Let $V \\subset Y$ be an open subspace.\nAssume\n\\begin{enumerate}\n\\item $f$ is separated and of finite type,\n\\item $Y$ is quasi-compact and quasi-separated,\n\\item $V$ is quasi-compact, and\n\\item $\\mathcal{X}_V$ is a gerbe over $V$.\n\\end{enumerate}\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n\\overline{Z} \\ar[rd]_{\\overline{g}} &\nZ \\ar[l]^j \\ar[d]_g \\ar[r]_h & \\mathcal{X} \\ar[ld]^f \\\\\n& Y\n}\n$$\nwith $j$ an open immersion, $\\overline{g}$ and $h$ proper,\nand such that $|V|$ is contained in the image of $|g|$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Chow's lemma for algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQ7","source_file":"stacks-more-morphisms.tex","source_line":2013,"source_end_line":2034,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L2013-L2034","statement_sha256":"040c9df5c03c8ad5cc73fd43c33fcae67d7e4f7766e19540553a6d5c7257ece5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14437,"rank":14437,"depth":83,"x":1894.993,"y":1518.56,"cluster":"algebraic-stacks"},{"id":"stacks:0CQ8","tag":"0CQ8","title":"Chow's lemma · Theorem 0CQ8","summary":"This is a result due to Ofer Gabber, see [olsson_proper] Let f : X → Y be a morphism from an algebraic stack to an algebraic space. Assume • Y is quasi-compact and quasi-separated, • f is separated of finite type. Then there exists a commutative diagram xymatrix X ar[rd] & X ar[l] ar[d] ar[r] & overlineX ar[ld] & Y where X → X is proper surjective, X → overlineX is an open immersion, and overlineX → Y is proper morphism of algebraic spaces.","statement_latex":"\\begin{reference}\nThis is a result due to Ofer Gabber, see\n\\cite[Theorem 1.1]{olsson_proper}\n\\end{reference}\nLet $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack\nto an algebraic space. Assume\n\\begin{enumerate}\n\\item $Y$ is quasi-compact and quasi-separated,\n\\item $f$ is separated of finite type.\n\\end{enumerate}\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n\\mathcal{X} \\ar[rd] & X \\ar[l] \\ar[d] \\ar[r] & \\overline{X} \\ar[ld] \\\\\n& Y\n}\n$$\nwhere $X \\to \\mathcal{X}$ is proper surjective,\n$X \\to \\overline{X}$ is an open immersion, and\n$\\overline{X} \\to Y$ is proper morphism of algebraic spaces.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Chow's lemma for algebraic stacks","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQ8","source_file":"stacks-more-morphisms.tex","source_line":2241,"source_end_line":2263,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L2241-L2263","statement_sha256":"636f8e857532aa742cfb67f101f026231b27fcdbf9a858d02597d2f87a62daf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14438,"rank":14438,"depth":84,"x":2195.153,"y":1583.436,"cluster":"algebraic-stacks"},{"id":"stacks:0H2B","tag":"0H2B","title":"Noetherian valuative criterion · Lemma 0H2B","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume f finite type and Y locally Noetherian. Let y ∈ |Y| be a point in the closure of the image of |f|. Then there exists a commutative diagram xymatrix Spec(K) ar[r] ar[d] & X ar[d]^f Spec(A) ar[r] & Y of algebraic stacks where A is a discrete valuation ring and K is its field of fractions mapping the closed point of Spec(A) to y.","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nAssume $f$ finite type and $\\mathcal{Y}$ locally Noetherian.\nLet $y \\in |\\mathcal{Y}|$ be a point in the closure of the image of $|f|$.\nThen there exists a commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r] \\ar[d] & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[r] & \\mathcal{Y}\n}\n$$\nof algebraic stacks where $A$ is a discrete valuation ring and $K$\nis its field of fractions mapping the closed point of $\\Spec(A)$ to $y$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2B","source_file":"stacks-more-morphisms.tex","source_line":2582,"source_end_line":2596,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L2582-L2596","statement_sha256":"9a79939352f952f28506fd27f238eb0b1a16caa9d9a78108656036848e993901","origin":"The Stacks Project","memory_eligible":false,"source_rank":14439,"rank":14439,"depth":21,"x":1921.468,"y":1705.996,"cluster":"algebraic-stacks"},{"id":"stacks:0E80","tag":"0E80","title":"Noetherian valuative criterion · Lemma 0E80","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume • Y is locally Noetherian, • f is locally of finite type and quasi-separated, • for every commutative diagram xymatrix Spec(K) ar[r]_x ar[d]_j & X ar[d]^f Spec(A) ar[r]^y ar@-->[ru] & Y where A is a discrete valuation ring and K its fraction field and any 2-arrow γ : y ∘ j → f ∘ x the category of dotted arrows (Morphisms of Stacks, Definition [Tag 0CLA]) is either empty or a setoid with exactly one isomorphism class.…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks. Assume\n\\begin{enumerate}\n\\item $\\mathcal{Y}$ is locally Noetherian,\n\\item $f$ is locally of finite type and quasi-separated,\n\\item for every commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[r]^y \\ar@{-->}[ru] & \\mathcal{Y}\n}\n$$\nwhere $A$ is a discrete valuation ring and $K$ its fraction field\nand any $2$-arrow $\\gamma : y \\circ j \\to f \\circ x$ the category\nof dotted arrows (Morphisms of Stacks, Definition\n\\ref{stacks-morphisms-definition-fill-in-diagram})\nis either empty or a setoid with exactly one isomorphism class.\n\\end{enumerate}\nThen $f$ is separated.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E80","source_file":"stacks-more-morphisms.tex","source_line":2623,"source_end_line":2643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L2623-L2643","statement_sha256":"595ac5a54847d586105aaaa24d08ac30a733adc33e88d7cbd374d7bcf73eec47","origin":"The Stacks Project","memory_eligible":false,"source_rank":14440,"rank":14440,"depth":64,"x":2024.788,"y":1460.164,"cluster":"algebraic-stacks"},{"id":"stacks:0CQM","tag":"0CQM","title":"Noetherian valuative criterion · Lemma 0CQM","summary":"Let f : X → Y and h : U → X be morphisms of algebraic stacks. Assume that Y is locally Noetherian, that f and h are of finite type, that f is separated, and that the image of |h| : |U| → |X| is dense in |X|. If given any 2-commutative diagram xymatrix Spec(K) ar[r]_-u ar[d]_j & U ar[r]_h & X ar[d]^f Spec(A) ar[rr]^-y & & Y where A is a discrete valuation ring with field of fractions K and γ : y ∘ j → f ∘ h ∘ u there exist an extension K'/K of fields, a valuation ring A' ⊂…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $h : \\mathcal{U} \\to \\mathcal{X}$\nbe morphisms of algebraic stacks. Assume that $\\mathcal{Y}$ is\nlocally Noetherian, that $f$ and $h$ are of finite type,\nthat $f$ is separated, and that the image of\n$|h| : |\\mathcal{U}| \\to |\\mathcal{X}|$ is dense in $|\\mathcal{X}|$.\nIf given any $2$-commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_-u \\ar[d]_j & \\mathcal{U} \\ar[r]_h & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[rr]^-y & & \\mathcal{Y}\n}\n$$\nwhere $A$ is a discrete valuation ring with field of fractions $K$\nand $\\gamma : y \\circ j \\to f \\circ h \\circ u$ there\nexist an extension $K'/K$ of fields, a valuation ring $A' \\subset K'$\ndominating $A$ such that the category of dotted arrows for the\ninduced diagram\n$$\n\\xymatrix{\n\\Spec(K') \\ar[r]_-{x'} \\ar[d]_{j'} & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A') \\ar[r]^-{y'} \\ar@{..>}[ru] & \\mathcal{Y}\n}\n$$\nwith induced $2$-arrow $\\gamma' : y' \\circ j' \\to f \\circ x'$\nis nonempty (Morphisms of Stacks, Definition\n\\ref{stacks-morphisms-definition-fill-in-diagram}), then $f$ is proper.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CQM","source_file":"stacks-more-morphisms.tex","source_line":2691,"source_end_line":2719,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L2691-L2719","statement_sha256":"328ff9c656870389cfb9e931b2d4397b555004bb5db92456a5237ac3d5927a85","origin":"The Stacks Project","memory_eligible":false,"source_rank":14441,"rank":14441,"depth":85,"x":2146.372,"y":1700.221,"cluster":"algebraic-stacks"},{"id":"stacks:0E95","tag":"0E95","title":"Noetherian valuative criterion · Lemma 0E95","summary":"Let f : X → Y and h : U → X be morphisms of algebraic stacks. Assume that Y is locally Noetherian, that f is locally of finite type and quasi-separated, that h is of finite type, and that the image of |h| : |U| → |X| is dense in |X|. If given any 2-commutative diagram xymatrix Spec(K) ar[r]_-u ar[d]_j & U ar[r]_h & X ar[d]^f Spec(A) ar[rr]^-y ar@..>[rru] & & Y where A is a discrete valuation ring with field of fractions K and γ : y ∘ j → f ∘ h ∘ u, the category of dotted…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ and $h : \\mathcal{U} \\to \\mathcal{X}$\nbe morphisms of algebraic stacks. Assume that $\\mathcal{Y}$ is\nlocally Noetherian, that $f$ is locally of finite type and quasi-separated,\nthat $h$ is of finite type, and that the image of\n$|h| : |\\mathcal{U}| \\to |\\mathcal{X}|$ is dense in $|\\mathcal{X}|$.\nIf given any $2$-commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_-u \\ar[d]_j & \\mathcal{U} \\ar[r]_h & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[rr]^-y \\ar@{..>}[rru] & & \\mathcal{Y}\n}\n$$\nwhere $A$ is a discrete valuation ring with field of fractions $K$\nand $\\gamma : y \\circ j \\to f \\circ h \\circ u$, the category\nof dotted arrows is either empty or a setoid with exactly\none isomorphism class, then $f$ is separated.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E95","source_file":"stacks-more-morphisms.tex","source_line":2802,"source_end_line":2820,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L2802-L2820","statement_sha256":"b5636dc062b6b364c785588725e5b3d1b257388688739c4ed2aaacd88b73e9fa","origin":"The Stacks Project","memory_eligible":false,"source_rank":14442,"rank":14442,"depth":86,"x":1863.485,"y":1592.126,"cluster":"algebraic-stacks"},{"id":"stacks:0H2C","tag":"0H2C","title":"Noetherian valuative criterion · Lemma 0H2C","summary":"Let f : X → Y be a morphism of algebraic stacks. Assume that Y is locally Noetherian and that f is of finite type. If given any 2-commutative diagram xymatrix Spec(K) ar[r]_-x ar[d]_j & X ar[d]^f Spec(A) ar[r]^-y & Y where A is a discrete valuation ring with field of fractions K and γ : y ∘ j → f ∘ x there exist an extension K'/K of fields, a valuation ring A' ⊂ K' dominating A such that the category of dotted arrows for the induced diagram xymatrix Spec(K') ar[r]_-x'…","statement_latex":"Let $f : \\mathcal{X} \\to \\mathcal{Y}$ be a morphism of algebraic stacks.\nAssume that $\\mathcal{Y}$ is locally Noetherian and that $f$ is of finite type.\nIf given any $2$-commutative diagram\n$$\n\\xymatrix{\n\\Spec(K) \\ar[r]_-x \\ar[d]_j & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A) \\ar[r]^-y & \\mathcal{Y}\n}\n$$\nwhere $A$ is a discrete valuation ring with field of fractions $K$\nand $\\gamma : y \\circ j \\to f \\circ x$ there exist an extension $K'/K$\nof fields, a valuation ring $A' \\subset K'$ dominating $A$ such that\nthe category of dotted arrows for the induced diagram\n$$\n\\xymatrix{\n\\Spec(K') \\ar[r]_-{x'} \\ar[d]_{j'} & \\mathcal{X} \\ar[d]^f \\\\\n\\Spec(A') \\ar[r]^-{y'} \\ar@{..>}[ru] & \\mathcal{Y}\n}\n$$\nwith induced $2$-arrow $\\gamma' : y' \\circ j' \\to f \\circ x'$\nis nonempty (Morphisms of Stacks, Definition\n\\ref{stacks-morphisms-definition-fill-in-diagram}), then $f$ is\nuniversally closed.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Noetherian valuative criterion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H2C","source_file":"stacks-more-morphisms.tex","source_line":2871,"source_end_line":2896,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L2871-L2896","statement_sha256":"38f1fa0c9d3371fbee7044011035dc7491012bdbaaf21e11302e4046d4a7e8de","origin":"The Stacks Project","memory_eligible":false,"source_rank":14443,"rank":14443,"depth":64,"x":2159.203,"y":1511.263,"cluster":"algebraic-stacks"},{"id":"stacks:0DUG","tag":"0DUG","title":"Moduli spaces · Definition 0DUG","summary":"Let X be an algebraic stack. Let f : X → Y be a morphism to an algebraic space Y. • We say f is a categorical moduli space if any morphism X → W to an algebraic space W factors uniquely through f. • We say f is a uniform categorical moduli space if for any flat morphism Y' → Y of algebraic spaces the base change f' : Y' ×_Y X → Y' is a categorical moduli space. Let C be a full subcategory of the category of algebraic spaces. • [(3)] We say f is a categorical moduli space…","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let\n$f : \\mathcal{X} \\to Y$ be a morphism to an algebraic space $Y$.\n\\begin{enumerate}\n\\item We say $f$ is a {\\it categorical moduli space} if any morphism\n$\\mathcal{X} \\to W$ to an algebraic space $W$ factors uniquely through $f$.\n\\item We say $f$ is a {\\it uniform categorical moduli space}\nif for any flat morphism $Y' \\to Y$ of algebraic spaces the base change\n$f' : Y' \\times_Y \\mathcal{X} \\to Y'$ is a categorical moduli space.\n\\end{enumerate}\nLet $\\mathcal{C}$ be a full subcategory of the category of algebraic\nspaces.\n\\begin{enumerate}\n\\item[(3)] We say $f$ is a {\\it categorical moduli space in $\\mathcal{C}$}\nif $Y \\in \\Ob(\\mathcal{C})$ and any morphism $\\mathcal{X} \\to W$ with\n$W \\in \\Ob(\\mathcal{C})$ factors uniquely through $f$.\n\\item[(4)] We say is a {\\it uniform categorical moduli space in $\\mathcal{C}$}\nif $Y \\in \\Ob(\\mathcal{C})$ and for every flat morphism $Y' \\to Y$ in\n$\\mathcal{C}$ the base change $f' : Y' \\times_Y \\mathcal{X} \\to Y'$ is a\ncategorical moduli space in $\\mathcal{C}$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Moduli spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUG","source_file":"stacks-more-morphisms.tex","source_line":2971,"source_end_line":2993,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L2971-L2993","statement_sha256":"8cfc03b8c33e4d3a0091cf5f198bba96affd08e4f9fad5e3d5f966068530df5d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14444,"rank":14444,"depth":0,"x":2006.072,"y":1738.837,"cluster":"algebraic-stacks"},{"id":"stacks:0DUH","tag":"0DUH","title":"Moduli spaces · Lemma 0DUH","summary":"Let (U, R, s, t, c) be a groupoid in algebraic spaces with s, t : R → U flat and locally of finite presentation. Consider the algebraic stack X = [U/R]. Given an algebraic space Y there is a 1-to-1 correspondence between morphisms f : X → Y and R-invariant morphisms φ : U → Y.","statement_latex":"Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces with\n$s, t : R \\to U$ flat and locally of finite presentation.\nConsider the algebraic stack $\\mathcal{X} = [U/R]$.\nGiven an algebraic space $Y$ there is a $1$-to-$1$ correspondence between\nmorphisms $f : \\mathcal{X} \\to Y$ and $R$-invariant morphisms\n$\\phi : U \\to Y$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Moduli spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUH","source_file":"stacks-more-morphisms.tex","source_line":2999,"source_end_line":3007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L2999-L3007","statement_sha256":"145ea1da4f9e1ce3e2f6c5e70247489dfb1a6e7b7437fa4e2fcff181f3dfb2e2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14445,"rank":14445,"depth":76,"x":1935.934,"y":1483.971,"cluster":"algebraic-stacks"},{"id":"stacks:0DUI","tag":"0DUI","title":"Moduli spaces · Lemma 0DUI","summary":"With assumption and notation as in Lemma [Tag 0DUH]. Then f is a (uniform) categorical moduli space if and only if φ is a (uniform) categorical quotient. Similarly for moduli spaces in a full subcategory.","statement_latex":"With assumption and notation as in Lemma \\ref{lemma-quotient-compare}.\nThen $f$ is a (uniform) categorical moduli space\nif and only if $\\phi$ is a (uniform) categorical quotient.\nSimilarly for moduli spaces in a full subcategory.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Moduli spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUI","source_file":"stacks-more-morphisms.tex","source_line":3029,"source_end_line":3035,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3029-L3035","statement_sha256":"d034d08e3ab9402d341ad26a38ca129ebf67c3b4be6ccfc0c83b0be29a582f9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14446,"rank":14446,"depth":77,"x":2192.777,"y":1632.204,"cluster":"algebraic-stacks"},{"id":"stacks:0DUJ","tag":"0DUJ","title":"Moduli spaces · Lemma 0DUJ","summary":"Let f : X → Y be a morphism from an algebraic stack to an algebraic space. If for every affine scheme Y' and flat morphism Y' → Y the base change f' : Y' ×_Y X → Y' is a categorical moduli space, then f is a uniform categorical moduli space.","statement_latex":"Let $f : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack\nto an algebraic space. If for every affine scheme $Y'$ and flat\nmorphism $Y' \\to Y$ the base change\n$f' : Y' \\times_Y \\mathcal{X} \\to Y'$ is a categorical moduli space,\nthen $f$ is a uniform categorical moduli space.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Moduli spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUJ","source_file":"stacks-more-morphisms.tex","source_line":3059,"source_end_line":3066,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3059-L3066","statement_sha256":"0813777d3aac0d1de10224433c191ced8d423592804ae24a8b44b42b9ae9bb63","origin":"The Stacks Project","memory_eligible":false,"source_rank":14447,"rank":14447,"depth":0,"x":1883.978,"y":1668.661,"cluster":"algebraic-stacks"},{"id":"stacks:0DUL","tag":"0DUL","title":"The Keel-Mori theorem · Definition 0DUL","summary":"Let X be an algebraic stack. We say X is well-nigh affine if there exists an affine scheme U and a surjective, flat, finite, and finitely presented morphism U → X.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. We say $\\mathcal{X}$\nis {\\it well-nigh affine} if there exists an affine scheme $U$\nand a surjective, flat, finite, and finitely presented morphism\n$U \\to \\mathcal{X}$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUL","source_file":"stacks-more-morphisms.tex","source_line":3109,"source_end_line":3115,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3109-L3115","statement_sha256":"124be1d470a6cfc2354ea3ba6296322781f529eeae7f3fa81a864581cedbf40d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14448,"rank":14448,"depth":0,"x":2082.493,"y":1466.427,"cluster":"algebraic-stacks"},{"id":"stacks:0DUM","tag":"0DUM","title":"The Keel-Mori theorem · Lemma 0DUM","summary":"Let X be an algebraic stack. The following are equivalent • X is well-nigh affine, and • there exists a groupoid scheme (U, R, s, t, c) with U and R affine and s, t : R → U finite locally free such that X = [U/R]. If true then X is quasi-compact, quasi-DM, and separated.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. The following are equivalent\n\\begin{enumerate}\n\\item $\\mathcal{X}$ is well-nigh affine, and\n\\item there exists a groupoid scheme $(U, R, s, t, c)$ with $U$ and\n$R$ affine and $s, t : R \\to U$ finite locally free such that\n$\\mathcal{X} = [U/R]$.\n\\end{enumerate}\nIf true then $\\mathcal{X}$ is quasi-compact, quasi-DM, and separated.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUM","source_file":"stacks-more-morphisms.tex","source_line":3121,"source_end_line":3131,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3121-L3131","statement_sha256":"0986ade7b0513a2b14de00e8736e16eebc5a4d7fc82d60b1715b2dba99e61878","origin":"The Stacks Project","memory_eligible":false,"source_rank":14449,"rank":14449,"depth":76,"x":2098.753,"y":1728.364,"cluster":"algebraic-stacks"},{"id":"stacks:0DUN","tag":"0DUN","title":"The Keel-Mori theorem · Lemma 0DUN","summary":"Let the algebraic stack X be well-nigh affine. • If X is an algebraic space, then it is affine. • If X' → X is an affine morphism of algebraic stacks, then X' is well-nigh affine.","statement_latex":"Let the algebraic stack $\\mathcal{X}$ be well-nigh affine.\n\\begin{enumerate}\n\\item If $\\mathcal{X}$ is an algebraic space, then it is affine.\n\\item If $\\mathcal{X}' \\to \\mathcal{X}$ is an affine morphism\nof algebraic stacks, then $\\mathcal{X}'$ is well-nigh affine.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUN","source_file":"stacks-more-morphisms.tex","source_line":3178,"source_end_line":3186,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3178-L3186","statement_sha256":"71ce4581eee6578ee8943d5ee1face87726225f22a66b651aa276198689765cf","origin":"The Stacks Project","memory_eligible":false,"source_rank":14450,"rank":14450,"depth":77,"x":1875.976,"y":1544.322,"cluster":"algebraic-stacks"},{"id":"stacks:0DUP","tag":"0DUP","title":"The Keel-Mori theorem · Lemma 0DUP","summary":"Let the algebraic stack X be well-nigh affine. There exists a uniform categorical moduli space f : X → M in the category of affine schemes. Moreover f is separated, quasi-compact, and a universal homeomorphism.","statement_latex":"Let the algebraic stack $\\mathcal{X}$ be well-nigh affine. There exists\na uniform categorical moduli space\n$$\nf : \\mathcal{X} \\longrightarrow M\n$$\nin the category of affine schemes. Moreover\n$f$ is separated, quasi-compact, and a universal homeomorphism.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUP","source_file":"stacks-more-morphisms.tex","source_line":3207,"source_end_line":3216,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3207-L3216","statement_sha256":"00d5d8fcd662655ee3878704478c6f6b4b7bfa5069f2c86e10306a5b70dceeb5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14451,"rank":14451,"depth":78,"x":2188.452,"y":1553.629,"cluster":"algebraic-stacks"},{"id":"stacks:0DUQ","tag":"0DUQ","title":"The Keel-Mori theorem · Lemma 0DUQ","summary":"Let h : X' → X be a morphism of algebraic stacks. Assume X' and X are well-nigh affine, h is étale, and h induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark [Tag 0DTW]). Then there exists a cartesian diagram xymatrix X' ar[d] ar[r] & X ar[d] M' ar[r] & M where M' → M is étale and the vertical arrows are the moduli spaces constructed in Lemma [Tag 0DUP].","statement_latex":"Let $h : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks.\nAssume $\\mathcal{X}'$ and $\\mathcal{X}$ are well-nigh affine,\n$h$ is \\'etale, and $h$ induces isomorphisms on automorphism groups\n(Morphisms of Stacks, Remark\n\\ref{stacks-morphisms-remark-identify-automorphism-groups}).\nThen there exists a cartesian diagram\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d] \\\\\nM' \\ar[r] & M\n}\n$$\nwhere $M' \\to M$ is \\'etale and\nthe vertical arrows are the moduli spaces constructed in\nLemma \\ref{lemma-well-nigh-affine-moduli-space}.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUQ","source_file":"stacks-more-morphisms.tex","source_line":3299,"source_end_line":3316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3299-L3316","statement_sha256":"c875e469f8d0ada27cdfe35043fd9dc69f56e066dfd348b91dc145962a4d5173","origin":"The Stacks Project","memory_eligible":false,"source_rank":14452,"rank":14452,"depth":85,"x":1950.399,"y":1724.184,"cluster":"algebraic-stacks"},{"id":"stacks:0DUR","tag":"0DUR","title":"The Keel-Mori theorem · Lemma 0DUR","summary":"Let the algebraic stack X be well-nigh affine. The morphism f : X → M of Lemma [Tag 0DUP] is a uniform categorical moduli space.","statement_latex":"Let the algebraic stack $\\mathcal{X}$ be well-nigh affine. The morphism\n$$\nf : \\mathcal{X} \\longrightarrow M\n$$\nof Lemma \\ref{lemma-well-nigh-affine-moduli-space}\nis a uniform categorical moduli space.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUR","source_file":"stacks-more-morphisms.tex","source_line":3374,"source_end_line":3382,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3374-L3382","statement_sha256":"e5a59a960b69ad26ccaa06d7b920fb702a995cb7380c5c53630887d8d7333fb8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14453,"rank":14453,"depth":86,"x":1988.805,"y":1463.173,"cluster":"algebraic-stacks"},{"id":"stacks:0DUS","tag":"0DUS","title":"The Keel-Mori theorem · Lemma 0DUS","summary":"Let h : X' → X be a morphism of algebraic stacks. Assume X is well-nigh affine, h is étale, h is separated, and h induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark [Tag 0DTW]). Then there exists a cartesian diagram xymatrix X' ar[d] ar[r] & X ar[d] M' ar[r] & M where M' → M is a separated étale morphism of schemes and X → M is the moduli space constructed in Lemma [Tag 0DUP].","statement_latex":"Let $h : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks.\nAssume $\\mathcal{X}$ is well-nigh affine, $h$ is \\'etale, $h$ is separated,\nand $h$ induces isomorphisms on automorphism groups\n(Morphisms of Stacks, Remark\n\\ref{stacks-morphisms-remark-identify-automorphism-groups}).\nThen there exists a cartesian diagram\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[d] \\ar[r] & \\mathcal{X} \\ar[d] \\\\\nM' \\ar[r] & M\n}\n$$\nwhere $M' \\to M$ is a separated \\'etale morphism of schemes and\n$\\mathcal{X} \\to M$ is the moduli space constructed in\nLemma \\ref{lemma-well-nigh-affine-moduli-space}.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUS","source_file":"stacks-more-morphisms.tex","source_line":3474,"source_end_line":3491,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3474-L3491","statement_sha256":"2e6a18b164874340fb2f5ee2dc25834e429aa417987b8a862d276efa71ee67b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14454,"rank":14454,"depth":87,"x":2170.5,"y":1677.57,"cluster":"algebraic-stacks"},{"id":"stacks:0DUE","tag":"0DUE","title":"The Keel-Mori theorem · Lemma 0DUE","summary":"Let X be an algebraic stack. Assume I_X → X is finite. Then there exist a set I and for i ∈ I a morphism of algebraic stacks g_i : X_i → X with the following properties • |X| = ⋃ |g_i|(|X_i|), • X_i is well-nigh affine, • I_X_i → X_i ×_X I_X is an isomorphism, and • g_i : X_i → X is representable by algebraic spaces, separated, and étale,","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Assume\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite.\nThen there exist a set $I$ and for $i \\in I$ a morphism of algebraic stacks\n$$\ng_i : \\mathcal{X}_i \\longrightarrow \\mathcal{X}\n$$\nwith the following properties\n\\begin{enumerate}\n\\item $|\\mathcal{X}| = \\bigcup |g_i|(|\\mathcal{X}_i|)$,\n\\item $\\mathcal{X}_i$ is well-nigh affine,\n\\item $\\mathcal{I}_{\\mathcal{X}_i} \\to\n\\mathcal{X}_i \\times_\\mathcal{X} \\mathcal{I}_\\mathcal{X}$\nis an isomorphism, and\n\\item $g_i : \\mathcal{X}_i \\to \\mathcal{X}$ is representable\nby algebraic spaces, separated, and \\'etale,\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUE","source_file":"stacks-more-morphisms.tex","source_line":3580,"source_end_line":3598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3580-L3598","statement_sha256":"2e9d00cfcf9fec1445e5f3422d205f6812d521ba625dfb2567f4e3ec5fbfce4a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14455,"rank":14455,"depth":86,"x":1863.911,"y":1622.539,"cluster":"algebraic-stacks"},{"id":"stacks:0DUT","tag":"0DUT","title":"Keel-Mori · Theorem 0DUT","summary":"Let X be an algebraic stack. Assume I_X → X is finite. Then there exists a uniform categorical moduli space f : X → M and f is separated, quasi-compact, and a universal homeomorphism.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Assume\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite.\nThen there exists a uniform categorical moduli space\n$$\nf : \\mathcal{X} \\longrightarrow M\n$$\nand $f$ is separated, quasi-compact, and a universal homeomorphism.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUT","source_file":"stacks-more-morphisms.tex","source_line":3636,"source_end_line":3645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3636-L3645","statement_sha256":"6dd87a7bf18fbe26346eb41e71ec1d5ae41c93c62e7cdb8a17a9a632a5fbc5df","origin":"The Stacks Project","memory_eligible":false,"source_rank":14456,"rank":14456,"depth":88,"x":2134.414,"y":1489.066,"cluster":"algebraic-stacks"},{"id":"stacks:0DUV","tag":"0DUV","title":"The Keel-Mori theorem · Lemma 0DUV","summary":"Let h : X' → X be a morphism of algebraic stacks. Assume • I_X → X is finite, • h is étale, separated, and induces isomorphisms on automorphism groups (Morphisms of Stacks, Remark [Tag 0DTW]). Then there exists a cartesian diagram xymatrix X' ar[d] ar[r] & X ar[d] M' ar[r] & M where M' → M is a separated étale morphism of algebraic spaces and the vertical arrows are the moduli spaces constructed in Theorem [Tag 0DUT].","statement_latex":"Let $h : \\mathcal{X}' \\to \\mathcal{X}$ be a morphism of algebraic stacks.\nAssume\n\\begin{enumerate}\n\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite,\n\\item $h$ is \\'etale, separated, and induces isomorphisms on\nautomorphism groups (Morphisms of Stacks, Remark\n\\ref{stacks-morphisms-remark-identify-automorphism-groups}).\n\\end{enumerate}\nThen there exists a cartesian diagram\n$$\n\\xymatrix{\n\\mathcal{X}' \\ar[d] \\ar[r] &\n\\mathcal{X} \\ar[d] \\\\\nM' \\ar[r] &\nM\n}\n$$\nwhere $M' \\to M$ is a separated \\'etale morphism of algebraic spaces and\nthe vertical arrows are the moduli spaces constructed in\nTheorem \\ref{theorem-keel-mori}.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"The Keel-Mori theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUV","source_file":"stacks-more-morphisms.tex","source_line":3890,"source_end_line":3912,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3890-L3912","statement_sha256":"ee5a5886f2b4880da6d82660383b9ba44e61ab2a4b717c1a1bc8fb6972c19d85","origin":"The Stacks Project","memory_eligible":false,"source_rank":14457,"rank":14457,"depth":89,"x":2042.225,"y":1741.139,"cluster":"algebraic-stacks"},{"id":"stacks:0DUX","tag":"0DUX","title":"Properties of moduli spaces · Lemma 0DUX","summary":"Let p : X → Y be a morphism of an algebraic stack to an algebraic space. Assume • I_X → X is finite, • Y is locally Noetherian, and • p is locally of finite type. Let f : X → M be the moduli space constructed in Theorem [Tag 0DUT]. Then M → Y is locally of finite type.","statement_latex":"Let $p : \\mathcal{X} \\to Y$ be a morphism of an algebraic stack to an\nalgebraic space. Assume\n\\begin{enumerate}\n\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite,\n\\item $Y$ is locally Noetherian, and\n\\item $p$ is locally of finite type.\n\\end{enumerate}\nLet $f : \\mathcal{X} \\to M$ be the moduli space constructed in\nTheorem \\ref{theorem-keel-mori}.\nThen $M \\to Y$ is locally of finite type.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Properties of moduli spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUX","source_file":"stacks-more-morphisms.tex","source_line":3986,"source_end_line":3998,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L3986-L3998","statement_sha256":"4881281f588df9c714b3419bd978377d91a34371bb491d1b2e9cd152ff3444c9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14458,"rank":14458,"depth":90,"x":1907.407,"y":1502.801,"cluster":"algebraic-stacks"},{"id":"stacks:0DUY","tag":"0DUY","title":"Properties of moduli spaces · Lemma 0DUY","summary":"Let X be an algebraic stack. Assume I_X → X is finite. Let f : X → M be the moduli space constructed in Theorem [Tag 0DUT]. • If X is quasi-separated, then M is quasi-separated. • If X is separated, then M is separated. • Add more here, for example relative versions of the above.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Assume\n$\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite.\nLet $f : \\mathcal{X} \\to M$ be the moduli space constructed in\nTheorem \\ref{theorem-keel-mori}.\n\\begin{enumerate}\n\\item If $\\mathcal{X}$ is quasi-separated, then $M$ is quasi-separated.\n\\item If $\\mathcal{X}$ is separated, then $M$ is separated.\n\\item Add more here, for example relative versions of the above.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Properties of moduli spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUY","source_file":"stacks-more-morphisms.tex","source_line":4027,"source_end_line":4038,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4027-L4038","statement_sha256":"92c68fc1efc6f79f61e276a015e14050231540c633c9549897f4b46a8c713633","origin":"The Stacks Project","memory_eligible":false,"source_rank":14459,"rank":14459,"depth":89,"x":2198.671,"y":1602.112,"cluster":"algebraic-stacks"},{"id":"stacks:0DUZ","tag":"0DUZ","title":"Properties of moduli spaces · Lemma 0DUZ","summary":"Let p : X → Y be a morphism from an algebraic stack to an algebraic space. Assume • I_X → X is finite, • p is proper, and • Y is locally Noetherian. Let f : X → M be the moduli space constructed in Theorem [Tag 0DUT]. Then M → Y is proper.","statement_latex":"Let $p : \\mathcal{X} \\to Y$ be a morphism from an algebraic stack\nto an algebraic space. Assume\n\\begin{enumerate}\n\\item $\\mathcal{I}_\\mathcal{X} \\to \\mathcal{X}$ is finite,\n\\item $p$ is proper, and\n\\item $Y$ is locally Noetherian.\n\\end{enumerate}\nLet $f : \\mathcal{X} \\to M$ be the moduli space constructed in\nTheorem \\ref{theorem-keel-mori}. Then $M \\to Y$ is proper.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Properties of moduli spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DUZ","source_file":"stacks-more-morphisms.tex","source_line":4066,"source_end_line":4077,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4066-L4077","statement_sha256":"be2ccc335cebefd4d5292338a286b8d37f344175c69631e0e9e6f71d0a1e1399","origin":"The Stacks Project","memory_eligible":false,"source_rank":14460,"rank":14460,"depth":91,"x":1903.845,"y":1694.211,"cluster":"algebraic-stacks"},{"id":"stacks:0GRH","tag":"0GRH","title":"Stacks and fpqc coverings · Proposition 0GRH","summary":"Proposition 3.3.6 of \"Intro to Algebraic Stacks\" by Anatoly Preygel. Let X be an algebraic stack with quasi-affine diagonal. Then X satisfies descent for fpqc coverings.","statement_latex":"\\begin{reference}\nProposition 3.3.6 of ``Intro to Algebraic Stacks'' by\nAnatoly Preygel.\n\\end{reference}\nLet $\\mathcal{X}$ be an algebraic stack with quasi-affine\\footnote{It suffices\nto assume ind-quasi-affine.} diagonal. Then\n$\\mathcal{X}$ satisfies descent for fpqc coverings.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Stacks and fpqc coverings","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRH","source_file":"stacks-more-morphisms.tex","source_line":4103,"source_end_line":4112,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4103-L4112","statement_sha256":"55cbf3f476cbbc2e220e2eb48b568999cd91e70eeec7b1cbdf36a2ff72e2e7ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":14461,"rank":14461,"depth":54,"x":2047.274,"y":1458.857,"cluster":"algebraic-stacks"},{"id":"stacks:0GRJ","tag":"0GRJ","title":"Tensor functors · Lemma 0GRJ","summary":"Let X and Y be Noetherian algebraic stacks. Any right exact tensor functor F : Coh(O_X) → Coh(O_Y) extends uniquely to a right exact tensor functor F : QCoh(O_X) → QCoh(O_Y) commuting with all colimits.","statement_latex":"Let $\\mathcal{X}$ and $\\mathcal{Y}$ be Noetherian algebraic stacks.\nAny right exact tensor functor $F : \\textit{Coh}(\\mathcal{O}_\\mathcal{X}) \\to\n\\textit{Coh}(\\mathcal{O}_\\mathcal{Y})$ extends uniquely to a\nright exact tensor functor\n$F : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\QCoh(\\mathcal{O}_\\mathcal{Y})$\ncommuting with all colimits.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Tensor functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRJ","source_file":"stacks-more-morphisms.tex","source_line":4278,"source_end_line":4286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4278-L4286","statement_sha256":"490feb2776c2269296c1df3267d3248736e4dcdf79c9d1f58df0428c42e309b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14462,"rank":14462,"depth":65,"x":2130.831,"y":1713.952,"cluster":"algebraic-stacks"},{"id":"stacks:0GRK","tag":"0GRK","title":"Tensor functors · Lemma 0GRK","summary":"Let X be an algebraic stack with affine diagonal. Let B be a ring. Let F : QCoh(O_X) → Mod_B be a right exact tensor functor which commutes with direct sums. Let g : U → X be a morphism with U = Spec(A) affine. Then • C = F(g_QCoh, *O_U) is a commutative B-algebra and • there is a ring map A → C such that F ∘ g_QCoh, * : Mod_A → Mod_B sends M to M ⊗_A C seen as B-module.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack with affine diagonal.\nLet $B$ be a ring. Let $F : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\text{Mod}_B$\nbe a right exact tensor functor which commutes with direct sums.\nLet $g : U \\to \\mathcal{X}$ be a morphism with $U = \\Spec(A)$ affine. Then\n\\begin{enumerate}\n\\item $C = F(g_{\\QCoh, *}\\mathcal{O}_U)$ is a commutative $B$-algebra and\n\\item there is a ring map $A \\to C$\n\\end{enumerate}\nsuch that $F \\circ g_{\\QCoh, *} : \\text{Mod}_A \\to \\text{Mod}_B$\nsends $M$ to $M \\otimes_A C$ seen as $B$-module.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Tensor functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRK","source_file":"stacks-more-morphisms.tex","source_line":4357,"source_end_line":4369,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4357-L4369","statement_sha256":"023463707b1bdda8c209646c6c5c5436da26c1ea242340c3d54879884549a58d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14463,"rank":14463,"depth":82,"x":1863.906,"y":1573.17,"cluster":"algebraic-stacks"},{"id":"stacks:0GRL","tag":"0GRL","title":"Tensor functors · Lemma 0GRL","summary":"Notation as in Lemma [Tag 0GRK]. Assume X is Noetherian and g is surjective and flat. Then B → C is universally injective.","statement_latex":"Notation as in Lemma \\ref{lemma-affine}. Assume $\\mathcal{X}$ is\nNoetherian and $g$ is surjective and flat.\nThen $B \\to C$ is universally injective.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Tensor functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRL","source_file":"stacks-more-morphisms.tex","source_line":4428,"source_end_line":4433,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4428-L4433","statement_sha256":"a2f6f8bb6d8061f258e888c523806880fb28512a65af3a712f58acf04afb4e89","origin":"The Stacks Project","memory_eligible":false,"source_rank":14464,"rank":14464,"depth":83,"x":2174.142,"y":1525.491,"cluster":"algebraic-stacks"},{"id":"stacks:0GRM","tag":"0GRM","title":"Tensor functors · Lemma 0GRM","summary":"Let B → C be a ring map. If • the coprojections C → C ⊗_B C are flat and • B → C is universally injective, then B → C is faithfully flat.","statement_latex":"Let $B \\to C$ be a ring map. If\n\\begin{enumerate}\n\\item the coprojections $C \\to C \\otimes_B C$ are flat and\n\\item $B \\to C$ is universally injective,\n\\end{enumerate}\nthen $B \\to C$ is faithfully flat.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Tensor functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRM","source_file":"stacks-more-morphisms.tex","source_line":4498,"source_end_line":4506,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4498-L4506","statement_sha256":"d881eb25696f561353215a8c542563c738c433505b61d5298e6cd1977705d345","origin":"The Stacks Project","memory_eligible":false,"source_rank":14465,"rank":14465,"depth":6,"x":1983.6,"y":1736.819,"cluster":"algebraic-stacks"},{"id":"stacks:0GRN","tag":"0GRN","title":"Tensor functors · Lemma 0GRN","summary":"Let a : Y → X and b : Z → X be representable by schemes, quasi-compact, quasi-separated, and flat. Then a_QCoh, *O_Y ⊗_O_X b_QCoh, *O_Z = f_QCoh, *O_Y ×_X Z where f : Y ×_X Z → X is the obvious morphism.","statement_latex":"Let $a : \\mathcal{Y} \\to \\mathcal{X}$ and $b : \\mathcal{Z} \\to \\mathcal{X}$\nbe representable by schemes, quasi-compact, quasi-separated, and flat.\nThen $a_{\\QCoh, *}\\mathcal{O}_\\mathcal{Y}\n\\otimes_{\\mathcal{O}_\\mathcal{X}}\nb_{\\QCoh, *}\\mathcal{O}_\\mathcal{Z} =\nf_{\\QCoh, *}\\mathcal{O}_{\\mathcal{Y} \\times_\\mathcal{X} \\mathcal{Z}}$\nwhere $f : \\mathcal{Y} \\times_\\mathcal{X} \\mathcal{Z} \\to \\mathcal{X}$\nis the obvious morphism.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Tensor functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRN","source_file":"stacks-more-morphisms.tex","source_line":4520,"source_end_line":4530,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4520-L4530","statement_sha256":"470ff1d5f5e9bb80fbd15cd307ee990cf4c419eb010bb9c4b3a16ea9dbc98272","origin":"The Stacks Project","memory_eligible":false,"source_rank":14466,"rank":14466,"depth":79,"x":1954.14,"y":1472.703,"cluster":"algebraic-stacks"},{"id":"stacks:0GRP","tag":"0GRP","title":"Tensor functors · Lemma 0GRP","summary":"Let X be an algebraic stack with affine diagonal. Let B be a ring. Let f_i : Spec(B) → X, i = 1, 2 be two morphisms. Let t : f_1^* → f_2^* be an isomorphism of the tensor functors f_i^* : QCoh(O_X) → Mod_B. Then there is a 2-arrow f_1 → f_2 inducing t.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack with affine diagonal.\nLet $B$ be a ring. Let $f_i : \\Spec(B) \\to \\mathcal{X}$, $i = 1, 2$\nbe two morphisms. Let $t : f_1^* \\to f_2^*$ be an isomorphism\nof the tensor functors\n$f_i^* : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\text{Mod}_B$.\nThen there is a $2$-arrow $f_1 \\to f_2$ inducing $t$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Tensor functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRP","source_file":"stacks-more-morphisms.tex","source_line":4574,"source_end_line":4582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4574-L4582","statement_sha256":"7f06b3db1654ec4e6a5690a2a02d6514d9cd46ebe5d925e855ef44bcbacd82d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14467,"rank":14467,"depth":83,"x":2188.407,"y":1650.855,"cluster":"algebraic-stacks"},{"id":"stacks:0GRQ","tag":"0GRQ","title":"Tensor functors · Lemma 0GRQ","summary":"Let X be a Noetherian algebraic stack with affine diagonal. Let B be a ring. Let F : QCoh(O_X) → Mod_B be a right exact tensor functor which commutes with direct sums. Then F comes from a unique morphism Spec(B) → X.","statement_latex":"Let $\\mathcal{X}$ be a Noetherian algebraic stack with affine diagonal.\nLet $B$ be a ring.\nLet $F : \\QCoh(\\mathcal{O}_\\mathcal{X}) \\to \\text{Mod}_B$\nbe a right exact tensor functor which commutes with direct sums.\nThen $F$ comes from a unique morphism $\\Spec(B) \\to \\mathcal{X}$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Tensor functors","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRQ","source_file":"stacks-more-morphisms.tex","source_line":4630,"source_end_line":4637,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4630-L4637","statement_sha256":"f2b04eea8e72e858f0bf704f957dd7bf7b4d3fe96c5b9959a81317cc4ee28fc8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14468,"rank":14468,"depth":84,"x":1872.197,"y":1652.418,"cluster":"algebraic-stacks"},{"id":"stacks:0GRR","tag":"0GRR","title":"Tensor functors · Theorem 0GRR","summary":"Let X be a Noetherian algebraic stack with affine diagonal. Let B be a Noetherian ring. Let F : Coh(O_X) → Mod^fg_B be a right exact tensor functor. Then F comes from a unique morphism Spec(B) → X.","statement_latex":"Let $\\mathcal{X}$ be a Noetherian algebraic stack with affine diagonal.\nLet $B$ be a Noetherian ring.\nLet $F : \\text{Coh}(\\mathcal{O}_\\mathcal{X}) \\to \\text{Mod}^{fg}_B$\nbe a right exact tensor functor.\nThen $F$ comes from a unique morphism $\\Spec(B) \\to \\mathcal{X}$.","area":"Algebraic Stacks","chapter":"More on Morphisms of Stacks","chapter_id":"stacks-more-morphisms","section":"Tensor functors","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GRR","source_file":"stacks-more-morphisms.tex","source_line":4719,"source_end_line":4726,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-more-morphisms.tex#L4719-L4726","statement_sha256":"296c95ee502c579ade1d26d8372f0b737d8adcece0c75c272ad8093decbaba92","origin":"The Stacks Project","memory_eligible":false,"source_rank":14469,"rank":14469,"depth":85,"x":2104.255,"y":1471.726,"cluster":"algebraic-stacks"},{"id":"stacks:0DQV","tag":"0DQV","title":"Versal rings · Definition 0DQV","summary":"In Situation [Tag 0DQU] let x_0 : Spec(k) → X be a morphism, where k is a finite type field over S. A versal ring to X at x_0 is a complete Noetherian local S-algebra A with residue field k such that there exists a versal formal object (A, xi_n, f_n) as in Artin's Axioms, Definition [Tag 0CXJ] with xi_1 ≅ x_0 (a 2-isomorphism).","statement_latex":"In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nbe a morphism, where $k$ is a finite type field over $S$.\nA {\\it versal ring to $\\mathcal{X}$ at $x_0$} is a complete\nNoetherian local $S$-algebra $A$ with residue field $k$\nsuch that there exists a versal formal object\n$(A, \\xi_n, f_n)$ as in Artin's Axioms, Definition\n\\ref{artin-definition-versal-formal-object}\nwith $\\xi_1 \\cong x_0$ (a $2$-isomorphism).","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Versal rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQV","source_file":"stacks-geometry.tex","source_line":50,"source_end_line":60,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L50-L60","statement_sha256":"77af5ea64efc9f4f9aff3e4a09b7549e3f683f3884b51e75ebc3aaa877493255","origin":"The Stacks Project","memory_eligible":false,"source_rank":14470,"rank":14470,"depth":2,"x":2078.431,"y":1736.808,"cluster":"algebraic-stacks"},{"id":"stacks:0DQW","tag":"0DQW","title":"Versal rings · Lemma 0DQW","summary":"In Situation [Tag 0DQU] let x_0 : Spec(k) → X be a morphism, where k is a finite type field over S. Then F_X, k, x_0 is a deformation category and TF_X, k, x_0 and Inf(F_X, k, x_0) are finite dimensional k-vector spaces.","statement_latex":"In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nbe a morphism, where $k$ is a finite type field over $S$.\nThen $\\mathcal{F}_{\\mathcal{X}, k, x_0}$\nis a deformation category and $T\\mathcal{F}_{\\mathcal{X}, k, x_0}$\nand $\\text{Inf}(\\mathcal{F}_{\\mathcal{X}, k, x_0})$\nare finite dimensional $k$-vector spaces.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQW","source_file":"stacks-geometry.tex","source_line":68,"source_end_line":76,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L68-L76","statement_sha256":"5b4fc88f2c7a3aa1485ee646af667f1a3a74a368f9f5f74f6583af053145e18c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14471,"rank":14471,"depth":77,"x":1884.178,"y":1526.554,"cluster":"algebraic-stacks"},{"id":"stacks:0DQX","tag":"0DQX","title":"Versal rings · Lemma 0DQX","summary":"In Situation [Tag 0DQU] let x_0 : Spec(k) → X be a morphism, where k is a finite type field over S. Then a versal ring to X at x_0 exists. Given a pair A, A' of these, then A ≅ A'[[t_1, …, t_r]] or A' ≅ A[[t_1, …, t_r]] as S-algebras for some r.","statement_latex":"In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nbe a morphism, where $k$ is a finite type field over $S$.\nThen a versal ring to $\\mathcal{X}$ at $x_0$ exists. Given a pair\n$A$, $A'$ of these, then $A \\cong A'[[t_1, \\ldots, t_r]]$\nor $A' \\cong A[[t_1, \\ldots, t_r]]$ as $S$-algebras\nfor some $r$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQX","source_file":"stacks-geometry.tex","source_line":97,"source_end_line":105,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L97-L105","statement_sha256":"343b587ddd628cecea47c7f619ce2d664b6d9ffbcc112fe92808d9a7ea76b018","origin":"The Stacks Project","memory_eligible":false,"source_rank":14472,"rank":14472,"depth":78,"x":2196.695,"y":1571.397,"cluster":"algebraic-stacks"},{"id":"stacks:0DQY","tag":"0DQY","title":"Versal rings · Lemma 0DQY","summary":"In Situation [Tag 0DQU] let x_0 : Spec(k) → X be a morphism, where k is a finite type field over S. Let l/k be a finite extension of fields and denote x_l, 0 : Spec(l) → X the induced morphism. Given a versal ring A to X at x_0 there exists a versal ring A' to X at x_l, 0 such that there is a S-algebra map A → A' which induces the given field extension l/k and is formally smooth in the m_A'-adic topology.","statement_latex":"In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nbe a morphism, where $k$ is a finite type field over $S$.\nLet $l/k$ be a finite extension of fields and denote\n$x_{l, 0} : \\Spec(l) \\to \\mathcal{X}$ the induced morphism.\nGiven a versal ring $A$ to $\\mathcal{X}$ at $x_0$ there exists\na versal ring $A'$ to $\\mathcal{X}$ at $x_{l, 0}$ such that\nthere is a $S$-algebra map $A \\to A'$ which induces the given\nfield extension $l/k$ and is formally smooth in the $\\mathfrak m_{A'}$-adic\ntopology.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQY","source_file":"stacks-geometry.tex","source_line":121,"source_end_line":132,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L121-L132","statement_sha256":"2bf0c9c829c03af5e7bb5e0a85bbc37886ff397e23dd5f6390b8ad7ba11e02bd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14473,"rank":14473,"depth":76,"x":1930.02,"y":1715.751,"cluster":"algebraic-stacks"},{"id":"stacks:0DQZ","tag":"0DQZ","title":"Versal rings · Lemma 0DQZ","summary":"In Situation [Tag 0DQU] let x : U → X be a morphism where U is a scheme locally of finite type over S. Let u_0 ∈ U be a finite type point. Set k = kappa(u_0) and denote x_0 : Spec(k) → X the induced map. The following are equivalent • x is versal at u_0 (Artin's Axioms, Definition [Tag 07XF]), • hat x : F_U, k, u_0 → F_X, k, x_0 is smooth, • the formal object associated to x|_Spec(O_U, u_0^wedge) is versal, and • there is an open neighbourhood U' ⊂ U of x such that x|_U'…","statement_latex":"In Situation \\ref{situation-versal} let $x : U \\to \\mathcal{X}$ be a\nmorphism where $U$ is a scheme locally of finite type over $S$.\nLet $u_0 \\in U$ be a finite type point.\nSet $k = \\kappa(u_0)$ and denote $x_0 : \\Spec(k) \\to \\mathcal{X}$\nthe induced map. The following are equivalent\n\\begin{enumerate}\n\\item $x$ is versal at $u_0$\n(Artin's Axioms, Definition \\ref{artin-definition-versal}),\n\\item $\\hat x : \\mathcal{F}_{U, k, u_0} \\to \\mathcal{F}_{\\mathcal{X}, k, x_0}$\nis smooth,\n\\item the formal object associated to\n$x|_{\\Spec(\\mathcal{O}_{U, u_0}^\\wedge)}$ is versal, and\n\\item there is an open neighbourhood $U' \\subset U$ of $x$ such that\n$x|_{U'} : U' \\to \\mathcal{X}$ is smooth.\n\\end{enumerate}\nMoreover, in this case the completion $\\mathcal{O}_{U, u_0}^\\wedge$\nis a versal ring to $\\mathcal{X}$ at $x_0$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQZ","source_file":"stacks-geometry.tex","source_line":144,"source_end_line":163,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L144-L163","statement_sha256":"c56f0d8932efc06d7d82b3ebc5496ba2abf971beb768c9fbc40d70d58480eb23","origin":"The Stacks Project","memory_eligible":false,"source_rank":14474,"rank":14474,"depth":78,"x":2010.628,"y":1457.827,"cluster":"algebraic-stacks"},{"id":"stacks:0DZS","tag":"0DZS","title":"Versal rings · Lemma 0DZS","summary":"In Situation [Tag 0DQU]. Let x_0 : Spec(k) → X be a morphism such that Spec(k) → S is of finite type with image s. Let A be a versal ring to X at x_0. The following are equivalent • x_0 is in the smooth locus of X → S (Morphisms of Stacks, Lemma [Tag 0DZR]), • O_S, s → A is formally smooth in the m_A-adic topology, and • F_X, k, x_0 is unobstructed.","statement_latex":"In Situation \\ref{situation-versal}. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nbe a morphism such that $\\Spec(k) \\to S$ is of finite type with image $s$.\nLet $A$ be a versal ring to $\\mathcal{X}$ at $x_0$. The following\nare equivalent\n\\begin{enumerate}\n\\item $x_0$ is in the smooth locus of $\\mathcal{X} \\to S$\n(Morphisms of Stacks, Lemma \\ref{stacks-morphisms-lemma-where-smooth}),\n\\item $\\mathcal{O}_{S, s} \\to A$ is formally smooth in the\n$\\mathfrak m_A$-adic topology, and\n\\item $\\mathcal{F}_{\\mathcal{X}, k, x_0}$ is unobstructed.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZS","source_file":"stacks-geometry.tex","source_line":200,"source_end_line":213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L200-L213","statement_sha256":"0b996da2af1837f568a7d844de4bf8b9a75fc4a7565ca86ca527a7ec2414d320","origin":"The Stacks Project","memory_eligible":false,"source_rank":14475,"rank":14475,"depth":79,"x":2158.697,"y":1693.903,"cluster":"algebraic-stacks"},{"id":"stacks:0DR0","tag":"0DR0","title":"Versal rings · Lemma 0DR0","summary":"In Situation [Tag 0DQU]. Let x_0 : Spec(k) → X be a morphism such that Spec(k) → S is of finite type with image s. Let A be a versal ring to X at x_0. If O_S, s is a G-ring, then we may find a smooth morphism U → X whose source is a scheme and a point u_0 ∈ U with residue field k, such that • Spec(k) → U → X coincides with the given morphism x_0, • there is an isomorphism O_U, u_0^wedge ≅ A.","statement_latex":"In Situation \\ref{situation-versal}. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nbe a morphism such that $\\Spec(k) \\to S$ is of finite type with image $s$.\nLet $A$ be a versal ring to $\\mathcal{X}$ at $x_0$.\nIf $\\mathcal{O}_{S, s}$ is a G-ring, then we may find a smooth morphism\n$U \\to \\mathcal{X}$ whose source is a scheme and a point\n$u_0 \\in U$ with residue field $k$, such that\n\\begin{enumerate}\n\\item $\\Spec(k) \\to U \\to \\mathcal{X}$ coincides with the given morphism $x_0$,\n\\item there is an isomorphism $\\mathcal{O}_{U, u_0}^\\wedge \\cong A$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DR0","source_file":"stacks-geometry.tex","source_line":263,"source_end_line":275,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L263-L275","statement_sha256":"54249c2b9c47c116d720b84afad514ac4f5e04a703090540ed6548a7558b138c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14476,"rank":14476,"depth":79,"x":1859.48,"y":1603.786,"cluster":"algebraic-stacks"},{"id":"stacks:0DR2","tag":"0DR2","title":"Versal rings · Lemma 0DR2","summary":"In Situation [Tag 0DQU] let x_0 : Spec(k) → X be a morphism, where k is a finite type field over S. Let A be a versal ring to X at x_0. Then the morphism Spec(A) → X of Remark [Tag 0DR1] is flat.","statement_latex":"In Situation \\ref{situation-versal} let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nbe a morphism, where $k$ is a finite type field over $S$.\nLet $A$ be a versal ring to $\\mathcal{X}$ at $x_0$.\nThen the morphism $\\Spec(A) \\to \\mathcal{X}$ of\nRemark \\ref{remark-upgrade} is flat.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Versal rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DR2","source_file":"stacks-geometry.tex","source_line":332,"source_end_line":339,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L332-L339","statement_sha256":"c8ea234225f213ec510929c9f8c50db4563c032b3b9ff8418fce583727c68bdc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14477,"rank":14477,"depth":80,"x":2152.772,"y":1500.389,"cluster":"algebraic-stacks"},{"id":"stacks:0DR5","tag":"0DR5","title":"Multiplicities of components of algebraic stacks · Lemma 0DR5","summary":"Let f : U → X be a smooth morphism from a scheme to a locally Noetherian algebraic stack. The closure of the image of any irreducible component of |U| is an irreducible component of |X|. If U → X is surjective, then all irreducible components of |X| are obtained in this way.","statement_latex":"Let $f : U \\to \\mathcal{X}$ be a smooth morphism from a scheme\nto a locally Noetherian algebraic stack. The closure of the image of any\nirreducible component of $|U|$ is an irreducible component of $|\\mathcal{X}|$.\nIf $U \\to \\mathcal{X}$ is surjective, then all irreducible components of\n$|\\mathcal{X}|$ are obtained in this way.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Multiplicities of components of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DR5","source_file":"stacks-geometry.tex","source_line":500,"source_end_line":507,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L500-L507","statement_sha256":"92404c14d9f6fc1005a83a3d0ac568e975dde3111ee3baecf9367c05ec634da1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14478,"rank":14478,"depth":58,"x":2019.568,"y":1743.205,"cluster":"algebraic-stacks"},{"id":"stacks:0DR6","tag":"0DR6","title":"Multiplicities of components of algebraic stacks · Lemma 0DR6","summary":"Let U → X be a smooth morphism of locally Noetherian schemes. Let T' is an irreducible component of U. Let T be the irreducible component of X obtained as the closure of the image of T'. Then m_T', U = m_T, X.","statement_latex":"Let $U \\to X$ be a smooth morphism of locally Noetherian schemes.\nLet $T'$ is an irreducible component of $U$. Let $T$ be the\nirreducible component of $X$ obtained as the closure of the\nimage of $T'$. Then $m_{T', U} = m_{T, X}$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Multiplicities of components of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DR6","source_file":"stacks-geometry.tex","source_line":535,"source_end_line":541,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L535-L541","statement_sha256":"5143a321c5569cfeb432816bde365364411ebdec894b852f838ed80f5de68b4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14479,"rank":14479,"depth":44,"x":1922.464,"y":1488.414,"cluster":"algebraic-stacks"},{"id":"stacks:0DR7","tag":"0DR7","title":"Multiplicities of components of algebraic stacks · Lemma 0DR7","summary":"Let U_1 → X and U_2 → X be two smooth morphisms from schemes to a locally Noetherian algebraic stack X. Let T_1' and T_2' be irreducible components of |U_1| and |U_2| respectively. Assume the closures of the images of T_1' and T_2' are the same irreducible component T of |X|. Then m_T_1', U_1 = m_T_2', U_2.","statement_latex":"Let $U_1 \\to \\mathcal{X}$ and $U_2 \\to \\mathcal{X}$ be two smooth\nmorphisms from schemes to a locally Noetherian algebraic stack $\\mathcal{X}$.\nLet $T_1'$ and $T_2'$ be irreducible components of $|U_1|$\nand $|U_2|$ respectively. Assume the closures of the images of\n$T_1'$ and $T_2'$ are the same irreducible component $T$ of $|\\mathcal{X}|$.\nThen $m_{T_1', U_1} = m_{T_2', U_2}$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Multiplicities of components of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DR7","source_file":"stacks-geometry.tex","source_line":574,"source_end_line":582,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L574-L582","statement_sha256":"3a253a000b675f55dec28d9e6a582b5f154d5b431222d9c52e9ca66e3d68c9d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14480,"rank":14480,"depth":59,"x":2199.135,"y":1621.277,"cluster":"algebraic-stacks"},{"id":"stacks:0DR8","tag":"0DR8","title":"Multiplicities of components of algebraic stacks · Definition 0DR8","summary":"Let X be a locally Noetherian algebraic stack. Let T ⊂ |X| be an irreducible component. The multiplicity of T in X is defined as m_T, X = m_T', U where f : U → X is a smooth morphism from a scheme and T' ⊂ |U| is an irreducible component with f(T') ⊂ T.","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack. Let\n$T \\subset |\\mathcal{X}|$ be an irreducible component.\nThe {\\it multiplicity} of $T$ in $\\mathcal{X}$ is defined as\n$m_{T, \\mathcal{X}} = m_{T', U}$ where $f : U \\to \\mathcal{X}$\nis a smooth morphism from a scheme and $T' \\subset |U|$\nis an irreducible component with $f(T') \\subset T$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Multiplicities of components of algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DR8","source_file":"stacks-geometry.tex","source_line":615,"source_end_line":623,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L615-L623","statement_sha256":"a82b3126a83da5e266cecae342f7d64a0e9d8c476a717579afb3048998f83fd4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14481,"rank":14481,"depth":0,"x":1888.084,"y":1680.333,"cluster":"algebraic-stacks"},{"id":"stacks:0DRA","tag":"0DRA","title":"Formal branches and multiplicities · Definition 0DRA","summary":"Let X be an algebraic stack locally of finite type over a locally Noetherian scheme S. Let x_0 : Spec(k) → X is a morphism where k is a field of finite type over S. The formal branches of X through x_0 is the set of irreducible components of Spec(A) for any choice of versal ring to X at x_0 identified for different choices of A by the procedure described above.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack locally of finite type\nover a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nis a morphism where $k$ is a field of finite type over $S$.\nThe {\\it formal branches of $\\mathcal{X}$ through $x_0$}\nis the set of irreducible components of $\\Spec(A)$\nfor any choice of versal ring to $\\mathcal{X}$ at $x_0$\nidentified for different choices of $A$ by the procedure described above.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Formal branches and multiplicities","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRA","source_file":"stacks-geometry.tex","source_line":679,"source_end_line":688,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L679-L688","statement_sha256":"e6ae5ae21e0e80972f6333bebdb02e3d6f7cd3f4278cee46cddca55dd6b31a38","origin":"The Stacks Project","memory_eligible":false,"source_rank":14482,"rank":14482,"depth":0,"x":2070.071,"y":1460.15,"cluster":"algebraic-stacks"},{"id":"stacks:0DRB","tag":"0DRB","title":"Formal branches and multiplicities · Lemma 0DRB","summary":"In the situation of Definition [Tag 0DRA] there is a canonical surjection from the set of formal branches of X through x_0 to the set of irreducible components of |X| containing x_0 in |X|.","statement_latex":"In the situation of Definition \\ref{definition-formal-branches}\nthere is a canonical surjection from the set of formal branches of\n$\\mathcal{X}$ through $x_0$ to the set of irreducible components of\n$|\\mathcal{X}|$ containing $x_0$ in $|\\mathcal{X}|$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Formal branches and multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRB","source_file":"stacks-geometry.tex","source_line":714,"source_end_line":720,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L714-L720","statement_sha256":"96835dd6a29e2a5bc0d496b0e6664b88d108f5fde66e78a4fbe24ec8d0b2de45","origin":"The Stacks Project","memory_eligible":false,"source_rank":14483,"rank":14483,"depth":81,"x":2112.967,"y":1725.938,"cluster":"algebraic-stacks"},{"id":"stacks:0DRC","tag":"0DRC","title":"Formal branches and multiplicities · Definition 0DRC","summary":"Let X be an algebraic stack locally of finite type over a locally Noetherian scheme S. Let x_0 : Spec(k) → X is a morphism where k is a field of finite type over S. The multiplicity of a formal branch of X through x_0 is the multiplicity of the corresponding irreducible component of Spec(A) for any choice of versal ring to X at x_0 (see discussion above).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack locally of finite type\nover a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nis a morphism where $k$ is a field of finite type over $S$.\nThe {\\it multiplicity of a formal branch of $\\mathcal{X}$ through $x_0$}\nis the multiplicity of the corresponding irreducible component of\n$\\Spec(A)$ for any choice of versal ring to $\\mathcal{X}$ at $x_0$\n(see discussion above).","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Formal branches and multiplicities","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRC","source_file":"stacks-geometry.tex","source_line":787,"source_end_line":796,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L787-L796","statement_sha256":"61e5dbae5e4ecc112417314a8a11711720d502e62b7adbfbc753fb45c91b2602","origin":"The Stacks Project","memory_eligible":false,"source_rank":14484,"rank":14484,"depth":0,"x":1867.444,"y":1554.184,"cluster":"algebraic-stacks"},{"id":"stacks:0DRD","tag":"0DRD","title":"Formal branches and multiplicities · Lemma 0DRD","summary":"Let X be an algebraic stack locally of finite type over a locally Noetherian scheme S. Let x_0 : Spec(k) → X is a morphism where k is a field of finite type over S with image s ∈ S. If O_S, s is a G-ring, then the map of Lemma [Tag 0DRB] preserves multiplicities.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack locally of finite type\nover a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nis a morphism where $k$ is a field of finite type over $S$ with\nimage $s \\in S$. If $\\mathcal{O}_{S, s}$ is a G-ring, then\nthe map of Lemma \\ref{lemma-branches} preserves multiplicities.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Formal branches and multiplicities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRD","source_file":"stacks-geometry.tex","source_line":798,"source_end_line":805,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L798-L805","statement_sha256":"ff5835127bd669cd55afc4000d1ad8ec8e5d2d5a79ea5bf8080b282bb84938cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14485,"rank":14485,"depth":82,"x":2186.807,"y":1541.51,"cluster":"algebraic-stacks"},{"id":"stacks:0DRG","tag":"0DRG","title":"Dimension theory of algebraic stacks · Definition 0DRG","summary":"If f : T → X is a locally of finite type morphism from an algebraic space to an algebraic stack, and if t ∈ |T| is a point with image x ∈ | X|, then we define the relative dimension of f at t, denoted dim_t(T_x), as follows: choose a morphism Spec k → X, with source the spectrum of a field, which represents x, and choose a point t' ∈ |T ×_X Spec k| mapping to t under the projection to |T| (such a point t' exists, by Properties of Stacks, Lemma [Tag 04XH]); then dim_t(T_x)…","statement_latex":"If $f : T \\to \\mathcal{X}$ is a locally of finite type morphism from an\nalgebraic space to an algebraic stack,\nand if $t \\in |T|$ is a point with image $x \\in | \\mathcal{X}|$, then we define\n{\\it the relative dimension} of $f$ at $t$, denoted\n$\\dim_t(T_x),$\nas follows:\nchoose a morphism $\\Spec k \\to \\mathcal{X}$, with source the spectrum of\na field, which represents $x$, and choose a point\n$t' \\in |T \\times_{\\mathcal{X}} \\Spec k|$\nmapping to $t$ under the projection to $|T|$\n(such a point $t'$ exists, by\nProperties of Stacks, Lemma \\ref{stacks-properties-lemma-points-cartesian});\nthen\n$$\n\\dim_t(T_x) = \\dim_{t'}(T \\times_{\\mathcal{X}} \\Spec k ).\n$$","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRG","source_file":"stacks-geometry.tex","source_line":955,"source_end_line":973,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L955-L973","statement_sha256":"3dc996d7764c3e354496d7486e3124d14222d6073bd5fba01b0599bf5ef8d101","origin":"The Stacks Project","memory_eligible":false,"source_rank":14486,"rank":14486,"depth":1,"x":1961.367,"y":1732.188,"cluster":"algebraic-stacks"},{"id":"stacks:0DRI","tag":"0DRI","title":"Dimension theory of algebraic stacks · Lemma 0DRI","summary":"If f: U → X is a smooth morphism of locally Noetherian algebraic spaces, and if u ∈ |U| with image x ∈ |X|, then dim_u (U) = dim_x(X) + dim_u (U_x) where dim_u (U_x) is defined via Definition [Tag 0DRG].","statement_latex":"If $f: U \\to X$ is a smooth morphism of locally Noetherian algebraic\nspaces, and\nif $u \\in |U|$ with image $x \\in |X|$, then\n$$\n\\dim_u (U) = \\dim_x(X) + \\dim_{u} (U_x)\n$$\nwhere $\\dim_u (U_x)$ is defined via\nDefinition \\ref{definition-relative-dimension}.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRI","source_file":"stacks-geometry.tex","source_line":1004,"source_end_line":1014,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1004-L1014","statement_sha256":"f153a53a05c320f23481f62eb4e715f0840d8b3c0e3c3594f3b07362176f2272","origin":"The Stacks Project","memory_eligible":false,"source_rank":14487,"rank":14487,"depth":49,"x":1974.273,"y":1463.498,"cluster":"algebraic-stacks"},{"id":"stacks:0DRJ","tag":"0DRJ","title":"Dimension theory of algebraic stacks · Lemma 0DRJ","summary":"If X is a locally Noetherian algebraic stack and x ∈ |X|. Let U → X be a smooth morphism from an algebraic space to X, let u be any point of |U| mapping to x. Then we have dim_x(X) = dim_u(U) - dim_u(U_x) where the relative dimension dim_u(U_x) is defined by Definition [Tag 0DRG] and the dimension of X at x is as in Properties of Stacks, Definition [Tag 0AFN].","statement_latex":"If $\\mathcal{X}$ is a locally Noetherian algebraic stack and\n$x \\in |\\mathcal{X}|$. Let $U \\to \\mathcal{X}$ be a smooth morphism\nfrom an algebraic space to $\\mathcal{X}$, let $u$ be any point of $|U|$\nmapping to $x$. Then we have\n$$\n\\dim_x(\\mathcal{X}) =  \\dim_u(U) - \\dim_{u}(U_x)\n$$\nwhere the relative dimension $\\dim_u(U_x)$ is defined\nby Definition \\ref{definition-relative-dimension} and the\ndimension of $\\mathcal{X}$ at $x$ is as in\nProperties of Stacks, Definition\n\\ref{stacks-properties-definition-dimension-at-point}.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRJ","source_file":"stacks-geometry.tex","source_line":1023,"source_end_line":1037,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1023-L1037","statement_sha256":"2af91e524ab626276d681e4e885754bea1a9a8e8cb5bb43332849544d56e25f8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14488,"rank":14488,"depth":51,"x":2180.956,"y":1669.076,"cluster":"algebraic-stacks"},{"id":"stacks:0DRL","tag":"0DRL","title":"Dimension theory of algebraic stacks · Definition 0DRL","summary":"If f : T → X is a locally of finite type morphism between locally Noetherian algebraic stacks, and if t ∈ |T| is a point with image x ∈ |X|, then we define the relative dimension of f at t, denoted dim_t(T_x), as follows: choose a morphism Spec k → X, with source the spectrum of a field, which represents x, and choose a point t' ∈ |T ×_X Spec k| mapping to t under the projection to |T| (such a point t' exists, by Properties of Stacks, Lemma [Tag 04XH]; then dim_t(T_x) =…","statement_latex":"If $f : \\mathcal{T} \\to \\mathcal{X}$\nis a locally of finite type morphism between\nlocally Noetherian algebraic stacks, and if\n$t \\in |\\mathcal{T}|$ is a point with image $x \\in |\\mathcal{X}|$, then\nwe define the {\\it relative dimension} of $f$ at $t$, denoted\n$\\dim_t(\\mathcal{T}_x),$ as follows:\nchoose a morphism $\\Spec k \\to \\mathcal{X}$, with source the spectrum of\na field, which represents $x$, and choose a point\n$t' \\in |\\mathcal{T} \\times_{\\mathcal{X}} \\Spec k|$\nmapping to $t$ under the projection to $|\\mathcal{T}|$\n(such a point $t'$ exists, by\nProperties of Stacks, Lemma\n\\ref{stacks-properties-lemma-points-cartesian}; then\n$$\n\\dim_t(\\mathcal{T}_x) = \\dim_{t'}(\\mathcal{T} \\times_{\\mathcal{X}} \\Spec k ).\n$$","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRL","source_file":"stacks-geometry.tex","source_line":1085,"source_end_line":1103,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1085-L1103","statement_sha256":"5ee44a9d2a5cde35a3b2906c31313e7726940f1880da5ddec1b3aa32d6afd97d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14489,"rank":14489,"depth":1,"x":1863.035,"y":1634.742,"cluster":"algebraic-stacks"},{"id":"stacks:0DRN","tag":"0DRN","title":"Dimension theory of algebraic stacks · Lemma 0DRN","summary":"Suppose given a Cartesian square of morphisms of locally Noetherian stacks xymatrix T' ar[d]ar[r] & T ar[d] X' ar[r] & X in which the vertical morphisms are locally of finite type. If t' ∈ |T'|, with images t, x', and x in |T|, |X'|, and |X| respectively, then dim_t'(T'_x') = dim_t(T_x).","statement_latex":"Suppose given\na Cartesian square of morphisms of locally Noetherian stacks\n$$\n\\xymatrix{\n\\mathcal{T}' \\ar[d]\\ar[r] & \\mathcal{T} \\ar[d] \\\\\n\\mathcal{X}' \\ar[r] & \\mathcal{X}\n}\n$$\nin which the vertical morphisms are locally of finite type.\nIf $t' \\in |\\mathcal{T}'|$,\nwith images $t$, $x'$, and $x$ in $|\\mathcal{T}|$, $|\\mathcal{X}'|$, and\n$|\\mathcal{X}|$\nrespectively, then $\\dim_{t'}(\\mathcal{T}'_{x'}) = \\dim_{t}(\\mathcal{T}_x).$","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRN","source_file":"stacks-geometry.tex","source_line":1126,"source_end_line":1141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1126-L1141","statement_sha256":"796c811bf290736b8472a56c7ff96250a7ecf52b6515a262291356b523a50979","origin":"The Stacks Project","memory_eligible":false,"source_rank":14490,"rank":14490,"depth":0,"x":2125.237,"y":1479.568,"cluster":"algebraic-stacks"},{"id":"stacks:0DRP","tag":"0DRP","title":"Dimension theory of algebraic stacks · Lemma 0DRP","summary":"If f: U → X is a smooth morphism of locally Noetherian algebraic stacks, and if u ∈ |U| with image x ∈ |X|, then dim_u (U) = dim_x(X) + dim_u (U_x).","statement_latex":"If $f: \\mathcal{U} \\to \\mathcal{X}$ is a smooth morphism of locally Noetherian\nalgebraic stacks, and\nif $u \\in |\\mathcal{U}|$ with image $x \\in |\\mathcal{X}|$,\nthen\n$$\n\\dim_u (\\mathcal{U}) = \\dim_x(\\mathcal{X}) + \\dim_{u} (\\mathcal{U}_x).\n$$","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRP","source_file":"stacks-geometry.tex","source_line":1148,"source_end_line":1157,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1148-L1157","statement_sha256":"366a304c48802e1437415f07ff3304660de6a05ea669f59a060d1cf099ab8274","origin":"The Stacks Project","memory_eligible":false,"source_rank":14491,"rank":14491,"depth":50,"x":2056.638,"y":1742.933,"cluster":"algebraic-stacks"},{"id":"stacks:0DRQ","tag":"0DRQ","title":"Dimension theory of algebraic stacks · Lemma 0DRQ","summary":"Let f: T → X be a locally of finite type morphism of algebraic stacks. • The function t ↦ dim_t(T_f(t)) is upper semi-continuous on |T|. • If f is smooth, then the function t ↦ dim_t(T_f(t)) is locally constant on |T|.","statement_latex":"Let $f: \\mathcal{T} \\to \\mathcal{X}$ be a locally of finite type morphism of\nalgebraic stacks.\n\\begin{enumerate}\n\\item\nThe function $t \\mapsto \\dim_t(\\mathcal{T}_{f(t)})$ is upper semi-continuous\non $|\\mathcal{T}|$.\n\\item If $f$ is smooth, then\nthe function $t \\mapsto \\dim_t(\\mathcal{T}_{f(t)})$ is locally constant\non $|\\mathcal{T}|$.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRQ","source_file":"stacks-geometry.tex","source_line":1209,"source_end_line":1221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1209-L1221","statement_sha256":"3dbbecd84bde25dc7043572204884155c38f16cd20a3ecc27c22be42f01a30d3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14492,"rank":14492,"depth":39,"x":1895.332,"y":1509.664,"cluster":"algebraic-stacks"},{"id":"stacks:0DRR","tag":"0DRR","title":"Dimension theory of algebraic stacks · Lemma 0DRR","summary":"If X is a finite dimensional scheme, then there exists a closed (and hence finite type) point x ∈ X such that dim_x X = dim X.","statement_latex":"If $X$ is a finite dimensional scheme,\nthen there exists a closed (and hence finite type) point $x \\in X$\nsuch that $\\dim_x X = \\dim X$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRR","source_file":"stacks-geometry.tex","source_line":1290,"source_end_line":1295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1290-L1295","statement_sha256":"52371a7c94b6fee03a08af232d9de882a9e33a72d7b06accabddd2fb34cf1aa0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14493,"rank":14493,"depth":0,"x":2202.053,"y":1590.193,"cluster":"algebraic-stacks"},{"id":"stacks:0DRT","tag":"0DRT","title":"Dimension theory of algebraic stacks · Lemma 0DRT","summary":"If X is an irreducible, Jacobson, catenary, and locally Noetherian scheme of finite dimension, then dim U = dim X for every non-empty open subset U of X. Equivalently, dim_x X is a constant function on X.","statement_latex":"If $X$ is an irreducible, Jacobson, catenary, and locally Noetherian\nscheme of finite dimension,\nthen $\\dim U = \\dim X$ for every\nnon-empty open subset $U$ of $X$.\nEquivalently, $\\dim_x X$ is a constant function on $X$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRT","source_file":"stacks-geometry.tex","source_line":1328,"source_end_line":1335,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1328-L1335","statement_sha256":"f85912c23ab7c83f3dab61f8ac714f406a43c21d711a8bd22428df4d3bb4550e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14494,"rank":14494,"depth":20,"x":1910.944,"y":1704.923,"cluster":"algebraic-stacks"},{"id":"stacks:0DRU","tag":"0DRU","title":"Dimension theory of algebraic stacks · Definition 0DRU","summary":"We say that a locally Noetherian algebraic stack X is pseudo-catenary if there exists a smooth and surjective morphism U → X whose source is a universally catenary scheme.","statement_latex":"We say that a locally Noetherian algebraic stack $\\mathcal{X}$\nis {\\it pseudo-catenary} if there exists a smooth\nand surjective morphism $U \\to \\mathcal{X}$ whose source is\na universally catenary scheme.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRU","source_file":"stacks-geometry.tex","source_line":1437,"source_end_line":1443,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1437-L1443","statement_sha256":"2598e57432cbd018ee706c39d90a3f03848545d0d1cd5a4daa270fdef3b8ac70","origin":"The Stacks Project","memory_eligible":false,"source_rank":14495,"rank":14495,"depth":0,"x":2033.416,"y":1454.988,"cluster":"algebraic-stacks"},{"id":"stacks:0DRW","tag":"0DRW","title":"Dimension theory of algebraic stacks · Lemma 0DRW","summary":"If X is a pseudo-catenary locally Noetherian algebraic stack, and if Y → X is a locally of finite type morphism, then there exists a smooth surjective morphism V → Y whose source is a universally catenary scheme; thus Y is again pseudo-catenary.","statement_latex":"If $\\mathcal{X}$ is a pseudo-catenary locally Noetherian algebraic\nstack, and if $\\mathcal{Y} \\to \\mathcal{X}$ is a locally of finite type\nmorphism,\nthen there exists a smooth surjective morphism $V \\to \\mathcal{Y}$\nwhose source is a universally catenary scheme; thus\n$\\mathcal{Y}$ is again pseudo-catenary.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRW","source_file":"stacks-geometry.tex","source_line":1462,"source_end_line":1470,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1462-L1470","statement_sha256":"d8ffaee7ac65ee14c8bcac3f6519e8bcd892f12723820ce6c90bc4ea176310fd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14496,"rank":14496,"depth":7,"x":2144.166,"y":1708.935,"cluster":"algebraic-stacks"},{"id":"stacks:0DRX","tag":"0DRX","title":"Dimension theory of algebraic stacks · Lemma 0DRX","summary":"If X is a Jacobson, pseudo-catenary, and locally Noetherian algebraic stack for which |X| is irreducible, then dim_x(X) is a constant function on |X|.","statement_latex":"If $\\mathcal{X}$ is\na Jacobson, pseudo-catenary, and locally Noetherian  algebraic stack\nfor which $|\\mathcal{X}|$ is irreducible,\nthen $\\dim_x(\\mathcal{X})$ is a constant function on $|\\mathcal{X}|$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRX","source_file":"stacks-geometry.tex","source_line":1497,"source_end_line":1503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1497-L1503","statement_sha256":"45ee2bb9ac23fb5c29d0a375b024e74afcaeef55fab9e9fce8d5438849d56bfd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14497,"rank":14497,"depth":50,"x":1858.109,"y":1584.444,"cluster":"algebraic-stacks"},{"id":"stacks:0DRY","tag":"0DRY","title":"Dimension theory of algebraic stacks · Lemma 0DRY","summary":"If Z hookrightarrow X is a closed immersion of locally Noetherian algebraic stacks, and if z ∈ |Z| has image x ∈ |X|, then dim_z (Z) ≤ dim_x(X).","statement_latex":"If $\\mathcal{Z} \\hookrightarrow \\mathcal{X}$ is a closed immersion\nof locally Noetherian algebraic stacks,\nand if $z \\in |\\mathcal{Z}|$ has image $x \\in |\\mathcal{X}|$,\nthen $\\dim_z (\\mathcal{Z}) \\leq \\dim_x(\\mathcal{X})$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRY","source_file":"stacks-geometry.tex","source_line":1569,"source_end_line":1575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1569-L1575","statement_sha256":"e8c84236aff6408f4cb4cd9818e68df11251021834b1fc16e4f29a5ee2afbe08","origin":"The Stacks Project","memory_eligible":false,"source_rank":14498,"rank":14498,"depth":1,"x":2169.344,"y":1513.883,"cluster":"algebraic-stacks"},{"id":"stacks:0DRZ","tag":"0DRZ","title":"Dimension theory of algebraic stacks · Lemma 0DRZ","summary":"If X is a locally Noetherian algebraic stack, and if x ∈ |X|, then dim_x(X) = sup_T ( dim_x(T) ) , where T runs over all the irreducible components of |X| passing through x (endowed with their induced reduced structure).","statement_latex":"If $\\mathcal{X}$ is a locally Noetherian algebraic stack, and if\n$x \\in |\\mathcal{X}|$,\nthen $\\dim_x(\\mathcal{X}) = \\sup_{\\mathcal{T}} \\{ \\dim_x(\\mathcal{T}) \\} $,\nwhere $\\mathcal{T}$ runs over all the irreducible components\nof $|\\mathcal{X}|$ passing through $x$ (endowed with their\ninduced reduced structure).","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DRZ","source_file":"stacks-geometry.tex","source_line":1599,"source_end_line":1607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1599-L1607","statement_sha256":"7d2f28831dcb3b2703ec8f14186f68007007c8191893271ae8ae51333d460d78","origin":"The Stacks Project","memory_eligible":false,"source_rank":14499,"rank":14499,"depth":2,"x":1996.482,"y":1742.655,"cluster":"algebraic-stacks"},{"id":"stacks:0DS1","tag":"0DS1","title":"Dimension theory of algebraic stacks · Lemma 0DS1","summary":"If X is a locally Noetherian algebraic stack, and if x ∈ |X|, then for any open substack V of X containing x, there is a finite type point x_0 ∈ |V| such that dim_x_0(X) = dim_x(V).","statement_latex":"If $\\mathcal{X}$ is a locally Noetherian algebraic stack, and if\n$x \\in |\\mathcal{X}|$, then\nfor any open substack $\\mathcal{V}$ of $\\mathcal{X}$ containing $x$,\nthere is a finite type point $x_0 \\in |\\mathcal{V}|$ such that\n$\\dim_{x_0}(\\mathcal{X}) = \\dim_x(\\mathcal{V})$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DS1","source_file":"stacks-geometry.tex","source_line":1639,"source_end_line":1646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1639-L1646","statement_sha256":"580968f60e83c009aab132d9602bba3ea3243478c30dd346b575e5f891fe4028","origin":"The Stacks Project","memory_eligible":false,"source_rank":14500,"rank":14500,"depth":50,"x":1939.941,"y":1475.714,"cluster":"algebraic-stacks"},{"id":"stacks:0DS2","tag":"0DS2","title":"Dimension theory of algebraic stacks · Lemma 0DS2","summary":"Let T hookrightarrow X be a locally of finite type monomorphism of algebraic stacks, with X (and thus also T) being Jacobson, pseudo-catenary, and locally Noetherian. Suppose further that T is irreducible of some (finite) dimension d, and that X is reduced and of dimension less than or equal to d. Then there is a non-empty open substack V of T such that the induced monomorphism V hookrightarrow X is an open immersion which identifies V with an open subset of an…","statement_latex":"Let $\\mathcal{T} \\hookrightarrow \\mathcal{X}$ be a locally\nof finite type monomorphism of algebraic stacks,\nwith $\\mathcal{X}$ (and thus also $\\mathcal{T}$)\nbeing Jacobson, pseudo-catenary, and locally Noetherian.\nSuppose further that $\\mathcal{T}$ is irreducible\nof some (finite) dimension $d$, and that $\\mathcal{X}$ is reduced\nand of dimension less\nthan or equal to $d$.\nThen there is a non-empty open substack $\\mathcal{V}$ of $\\mathcal{T}$ such\nthat the induced\nmonomorphism $\\mathcal{V} \\hookrightarrow \\mathcal{X}$ is an open immersion\nwhich identifies\n$\\mathcal{V}$ with an open subset of an irreducible component of $\\mathcal{X}$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DS2","source_file":"stacks-geometry.tex","source_line":1684,"source_end_line":1699,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1684-L1699","statement_sha256":"57916d564eadbfff684db4cfcbfca51925e5a3c1ca7550820c221bf63d8644e3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14501,"rank":14501,"depth":56,"x":2196.456,"y":1640.57,"cluster":"algebraic-stacks"},{"id":"stacks:0DS4","tag":"0DS4","title":"Dimension theory of algebraic stacks · Lemma 0DS4","summary":"Let f: T → X be a locally of finite type morphism of Jacobson, pseudo-catenary, and locally Noetherian algebraic stacks, whose source is irreducible and whose target is quasi-separated, and let Z hookrightarrow X denote the scheme-theoretic image of T. Then for all t ∈ |T|, we have that dim_t( T_f(t)) ≥ dim T - dim Z, and there is a non-empty (equivalently, dense) open subset of |T| over which equality holds.","statement_latex":"Let $f: \\mathcal{T} \\to \\mathcal{X}$ be a locally of finite type\nmorphism of Jacobson, pseudo-catenary, and locally Noetherian\nalgebraic stacks,\nwhose source is irreducible and whose target is quasi-separated,\nand let $\\mathcal{Z} \\hookrightarrow \\mathcal{X}$ denote the scheme-theoretic\nimage of $\\mathcal{T}$.\nThen for all $t \\in |T|$,\nwe have that\n$\\dim_t( \\mathcal{T}_{f(t)}) \\geq \\dim \\mathcal{T}  - \\dim \\mathcal{Z}$,\nand there is a non-empty (equivalently, dense)\nopen subset of $|\\mathcal{T}|$ over which equality holds.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DS4","source_file":"stacks-geometry.tex","source_line":1779,"source_end_line":1792,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1779-L1792","statement_sha256":"c5cf3a632a86d55bf7cc48faaa9933889cc9ff70dab011132c679624ee0b90ac","origin":"The Stacks Project","memory_eligible":false,"source_rank":14502,"rank":14502,"depth":63,"x":1874.538,"y":1664.575,"cluster":"algebraic-stacks"},{"id":"stacks:0DS6","tag":"0DS6","title":"Dimension theory of algebraic stacks · Lemma 0DS6","summary":"Let f: T → X be a locally of finite type morphism of Jacobson, pseudo-catenary, and locally Noetherian algebraic stacks which is quasi-DM, whose source is irreducible and whose target is quasi-separated, and let Z hookrightarrow X denote the scheme-theoretic image of T. Then dim Z ≤ dim T, and furthermore, exactly one of the following two conditions holds: • for every finite type point t ∈ |T|, we have dim_t(T_f(t)) > 0, in which case dim Z < dim T; or • T and Z are of…","statement_latex":"Let $f: \\mathcal{T} \\to \\mathcal{X}$ be a locally of finite type\nmorphism of Jacobson, pseudo-catenary, and locally Noetherian\nalgebraic stacks\nwhich is quasi-DM,\nwhose source is irreducible and whose target is quasi-separated,\nand let $\\mathcal{Z} \\hookrightarrow \\mathcal{X}$ denote the scheme-theoretic\nimage of $\\mathcal{T}$.\nThen $\\dim \\mathcal{Z} \\leq \\dim \\mathcal{T}$,\nand furthermore, exactly one of the following two conditions holds:\n\\begin{enumerate}\n\\item for every finite type point $t \\in |T|,$\nwe have\n$\\dim_t(\\mathcal{T}_{f(t)}) > 0,$ in which\ncase $\\dim \\mathcal{Z} < \\dim \\mathcal{T}$; or\n\\item   $\\mathcal{T}$ and $\\mathcal{Z}$\nare of the same dimension.\n\\end{enumerate}","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"Dimension theory of algebraic stacks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DS6","source_file":"stacks-geometry.tex","source_line":1988,"source_end_line":2007,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L1988-L2007","statement_sha256":"7fb33681e5e32db29105f946404989d935396e1ed11bd9f6b673d078b9ee736f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14503,"rank":14503,"depth":0,"x":2092.744,"y":1464.089,"cluster":"algebraic-stacks"},{"id":"stacks:0DS8","tag":"0DS8","title":"The dimension of the local ring · Lemma 0DS8","summary":"Let X be a locally Noetherian algebraic stack. Let U → X be a smooth morphism and let u ∈ U. Then dim(O_U, overlineu) - dim(O_R_u, e(overlineu)) = 2dim(O_U, overlineu) - dim(O_R, e(overlineu)) Here R = U ×_X U with projections s, t : R → U and diagonal e : U → R and R_u is the fibre of s : R → U over u.","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nLet $U \\to \\mathcal{X}$ be a smooth morphism and let $u \\in U$.\nThen\n$$\n\\dim(\\mathcal{O}_{U, \\overline{u}}) -\n\\dim(\\mathcal{O}_{R_u, e(\\overline{u})}) =\n2\\dim(\\mathcal{O}_{U, \\overline{u}}) -\n\\dim(\\mathcal{O}_{R, e(\\overline{u})})\n$$\nHere $R = U \\times_\\mathcal{X} U$ with projections $s, t : R \\to U$ and\ndiagonal $e : U \\to R$ and $R_u$ is the fibre of $s : R \\to U$ over $u$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"The dimension of the local ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DS8","source_file":"stacks-geometry.tex","source_line":2032,"source_end_line":2045,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L2032-L2045","statement_sha256":"c39db7548a2d303ddb7d7b9aaa6f539c2a6a79cf2adc0de70c19125ce3777e3a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14504,"rank":14504,"depth":12,"x":2093.068,"y":1735.904,"cluster":"algebraic-stacks"},{"id":"stacks:0DS9","tag":"0DS9","title":"The dimension of the local ring · Lemma 0DS9","summary":"Let X be a locally Noetherian algebraic stack. Let x ∈ |X| be a finite type point Morphisms of Stacks, Definition [Tag 06FY]). Let d ∈ Z. The following are equivalent • there exists a scheme U, a smooth morphism U → X, and a finite type point u ∈ U mapping to x such that 2dim(O_U, overlineu) - dim(O_R, e(overlineu)) = d, and • for any scheme U, a smooth morphism U → X, and finite type point u ∈ U mapping to x we have 2dim(O_U, overlineu) - dim(O_R, e(overlineu)) = d. Here…","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nLet $x \\in |\\mathcal{X}|$ be a finite type point \nMorphisms of Stacks, Definition\n\\ref{stacks-morphisms-definition-finite-type-point}).\nLet $d \\in \\mathbf{Z}$.\nThe following are equivalent\n\\begin{enumerate}\n\\item there exists a scheme $U$, a smooth morphism $U \\to \\mathcal{X}$,\nand a finite type point $u \\in U$ mapping to $x$ such that\n$2\\dim(\\mathcal{O}_{U, \\overline{u}}) -\n\\dim(\\mathcal{O}_{R, e(\\overline{u})}) = d$, and\n\\item for any scheme $U$, a smooth morphism $U \\to \\mathcal{X}$,\nand finite type point $u \\in U$ mapping to $x$ we have\n$2\\dim(\\mathcal{O}_{U, \\overline{u}}) -\n\\dim(\\mathcal{O}_{R, e(\\overline{u})}) = d$.\n\\end{enumerate}\nHere $R = U \\times_\\mathcal{X} U$ with projections $s, t : R \\to U$ and\ndiagonal $e : U \\to R$ and $R_u$ is the fibre of $s : R \\to U$ over $u$.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"The dimension of the local ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DS9","source_file":"stacks-geometry.tex","source_line":2060,"source_end_line":2080,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L2060-L2080","statement_sha256":"9dcf2a44839c4a4331afb3cbb1edaca49540f09119acd8986945587cc1e42e26","origin":"The Stacks Project","memory_eligible":false,"source_rank":14505,"rank":14505,"depth":50,"x":1874.111,"y":1535.534,"cluster":"algebraic-stacks"},{"id":"stacks:0DSA","tag":"0DSA","title":"The dimension of the local ring · Definition 0DSA","summary":"Let X be a locally Noetherian algebraic stack. Let x ∈ |X| be a finite type point. The dimension of the local ring of X at x is d ∈ Z if the equivalent conditions of Lemma [Tag 0DS9] are satisfied.","statement_latex":"Let $\\mathcal{X}$ be a locally Noetherian algebraic stack.\nLet $x \\in |\\mathcal{X}|$ be a finite type point.\nThe {\\it dimension of the local ring of $\\mathcal{X}$ at $x$}\nis $d \\in \\mathbf{Z}$ if the equivalent conditions of\nLemma \\ref{lemma-dimension-local-ring} are satisfied.","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"The dimension of the local ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSA","source_file":"stacks-geometry.tex","source_line":2140,"source_end_line":2147,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L2140-L2147","statement_sha256":"fa0a066aaed691c61f3e256aac982df5c8e97b9a6d8d5728b6963c7c42546fac","origin":"The Stacks Project","memory_eligible":false,"source_rank":14506,"rank":14506,"depth":51,"x":2196.893,"y":1559.056,"cluster":"algebraic-stacks"},{"id":"stacks:0DSB","tag":"0DSB","title":"The dimension of the local ring · Lemma 0DSB","summary":"Suppose that X is an algebraic stack, locally of finite type over a locally Noetherian scheme S. Let x_0 : Spec(k) → X be a morphism where k is a field of finite type over S. Represent F_X, k, x_0 as in Remark [Tag 0DR3] by a cogroupoid (A, B, s, t, c) of Noetherian complete local S-algebras with residue field k. Then the dimension of the local ring of X at x_0 = 2dim A - dim B","statement_latex":"Suppose that $\\mathcal{X}$ is an algebraic stack, locally of finite type\nover a locally Noetherian scheme $S$. Let $x_0 : \\Spec(k) \\to \\mathcal{X}$\nbe a morphism where $k$ is a field of finite type over $S$. Represent\n$\\mathcal{F}_{\\mathcal{X}, k, x_0}$ as in Remark \\ref{remark-groupoid-defo}\nby a cogroupoid $(A, B, s, t, c)$ of Noetherian complete local $S$-algebras\nwith residue field $k$. Then\n$$\n\\text{the dimension of the local ring of }\\mathcal{X}\\text{ at }x_0 =\n2\\dim A - \\dim B\n$$","area":"Algebraic Stacks","chapter":"The Geometry of Stacks","chapter_id":"stacks-geometry","section":"The dimension of the local ring","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSB","source_file":"stacks-geometry.tex","source_line":2157,"source_end_line":2169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/stacks-geometry.tex#L2157-L2169","statement_sha256":"3c69904d16ad42c3bc63c7e8d1c18bac1bba63409352fae2c4b8f902dc7e2cf0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14507,"rank":14507,"depth":81,"x":1939.808,"y":1724.965,"cluster":"algebraic-stacks"},{"id":"stacks:0DLY","tag":"0DLY","title":"Properties of the stack of coherent sheaves · Lemma 0DLY","summary":"The diagonal of Cohstack_X/B over B is affine and of finite presentation.","statement_latex":"The diagonal of $\\Cohstack_{X/B}$ over $B$ is affine\nand of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLY","source_file":"moduli.tex","source_line":120,"source_end_line":124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L120-L124","statement_sha256":"8a7bf6a85f73849a34e6102ac217262b124d92cd25e1538e2ca48646127b15a9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14508,"rank":14508,"depth":74,"x":2494.063,"y":1490.076,"cluster":"moduli-theory"},{"id":"stacks:0DLZ","tag":"0DLZ","title":"Properties of the stack of coherent sheaves · Lemma 0DLZ","summary":"The morphism Cohstack_X/B → B is quasi-separated and locally of finite presentation.","statement_latex":"The morphism $\\Cohstack_{X/B} \\to B$ is quasi-separated and\nlocally of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DLZ","source_file":"moduli.tex","source_line":137,"source_end_line":141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L137-L141","statement_sha256":"c651ac56d648eb3374d91680e1307c629cf45ee2cc27e617b0ddafaea27cb61e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14509,"rank":14509,"depth":75,"x":2558.293,"y":1689.337,"cluster":"moduli-theory"},{"id":"stacks:0DM0","tag":"0DM0","title":"Properties of the stack of coherent sheaves · Lemma 0DM0","summary":"Assume X → B is proper as well as of finite presentation. Then Cohstack_X/B → B satisfies the existence part of the valuative criterion (Morphisms of Stacks, Definition [Tag 0CLK]).","statement_latex":"Assume $X \\to B$ is proper as well as of finite presentation.\nThen $\\Cohstack_{X/B} \\to B$ satisfies the existence part\nof the valuative criterion (Morphisms of Stacks, Definition\n\\ref{stacks-morphisms-definition-existence}).","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DM0","source_file":"moduli.tex","source_line":156,"source_end_line":162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L156-L162","statement_sha256":"3b19b952eb6225a6bd6ba7a28005f15b17286cdf3097a790d4050a80cabbac1f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14510,"rank":14510,"depth":62,"x":2349.99,"y":1578.477,"cluster":"moduli-theory"},{"id":"stacks:0DN9","tag":"0DN9","title":"Properties of the stack of coherent sheaves · Lemma 0DN9","summary":"Let B be an algebraic space. Let π : X → Y be a quasi-finite morphism of algebraic spaces which are separated and of finite presentation over B. Then π_* induces a morphism Cohstack_X/B → Cohstack_Y/B.","statement_latex":"Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a quasi-finite\nmorphism of algebraic spaces which are separated and of finite presentation\nover $B$. Then $\\pi_*$ induces a morphism\n$\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DN9","source_file":"moduli.tex","source_line":207,"source_end_line":213,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L207-L213","statement_sha256":"ffaf1cc5ac47da3b826912267245ac83766ae7ad4a6eb5eb5934b54e11a8a99e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14511,"rank":14511,"depth":60,"x":2593.554,"y":1541.907,"cluster":"moduli-theory"},{"id":"stacks:0DNA","tag":"0DNA","title":"Properties of the stack of coherent sheaves · Lemma 0DNA","summary":"Let B be an algebraic space. Let π : X → Y be an open immersion of algebraic spaces which are separated and of finite presentation over B. Then the morphism Cohstack_X/B → Cohstack_Y/B of Lemma [Tag 0DN9] is an open immersion.","statement_latex":"Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an open immersion\nof algebraic spaces which are separated and of finite presentation over $B$.\nThen the morphism $\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$ of\nLemma \\ref{lemma-coherent-functorial} is an open immersion.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNA","source_file":"moduli.tex","source_line":280,"source_end_line":286,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L280-L286","statement_sha256":"bd9aaf39a156aaed8bb9d3d618e034835eb1e382536810b56cbe3a9149b39e10","origin":"The Stacks Project","memory_eligible":false,"source_rank":14512,"rank":14512,"depth":61,"x":2442.862,"y":1707.626,"cluster":"moduli-theory"},{"id":"stacks:0DNB","tag":"0DNB","title":"Properties of the stack of coherent sheaves · Lemma 0DNB","summary":"Let B be an algebraic space. Let π : X → Y be a closed immersion of algebraic spaces which are separated and of finite presentation over B. Then the morphism Cohstack_X/B → Cohstack_Y/B of Lemma [Tag 0DN9] is a closed immersion.","statement_latex":"Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a closed immersion\nof algebraic spaces which are separated and of finite presentation over $B$.\nThen the morphism $\\Cohstack_{X/B} \\to \\Cohstack_{Y/B}$ of\nLemma \\ref{lemma-coherent-functorial} is a closed immersion.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNB","source_file":"moduli.tex","source_line":294,"source_end_line":300,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L294-L300","statement_sha256":"50a6b6d6fb96ae6ec0987368fb0fc3a8bdd208dfab58e76867a41f27152a1625","origin":"The Stacks Project","memory_eligible":false,"source_rank":14513,"rank":14513,"depth":63,"x":2420.64,"y":1499.234,"cluster":"moduli-theory"},{"id":"stacks:0DND","tag":"0DND","title":"Properties of the stack of coherent sheaves · Lemma 0DND","summary":"In Situation [Tag 0DNC] the stack Cohstack^P_X/B is algebraic and Cohstack^P_X/B → Cohstack_X/B is a flat closed immersion. If I is finite or B is locally Noetherian, then Cohstack^P_X/B is an open and closed substack of Cohstack_X/B.","statement_latex":"In Situation \\ref{situation-numerical} the stack\n$\\Cohstack^P_{X/B}$ is algebraic and\n$$\n\\Cohstack^P_{X/B} \\longrightarrow \\Cohstack_{X/B}\n$$\nis a flat closed immersion. If $I$ is finite or $B$ is locally\nNoetherian, then $\\Cohstack^P_{X/B}$ is an open and closed substack of\n$\\Cohstack_{X/B}$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DND","source_file":"moduli.tex","source_line":370,"source_end_line":380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L370-L380","statement_sha256":"ad5344e805bc2e1e5363cb6c20ca25a22b509a66364de612f86a66d8d9ddce92","origin":"The Stacks Project","memory_eligible":false,"source_rank":14514,"rank":14514,"depth":76,"x":2605.213,"y":1640.754,"cluster":"moduli-theory"},{"id":"stacks:0DNE","tag":"0DNE","title":"Properties of the stack of coherent sheaves · Lemma 0DNE","summary":"Let f : X → B be as in the introduction to this section. Let E_1, …, E_r ∈ D(O_X) be perfect. Let I = Z^⊕ r and consider the map I → D(O_X), (n_1, …, n_r) ↦ E_1^⊗ n_1 ⊗ … ⊗ E_r^⊗ n_r Let P : I → Z be a map. Then Cohstack^P_X/B ⊂ Cohstack_X/B as defined in Situation [Tag 0DNC] is an open and closed substack.","statement_latex":"Let $f : X \\to B$ be as in the introduction to this section.\nLet $E_1, \\ldots, E_r \\in D(\\mathcal{O}_X)$ be perfect.\nLet $I = \\mathbf{Z}^{\\oplus r}$ and consider the map\n$$\nI \\longrightarrow D(\\mathcal{O}_X),\\quad\n(n_1, \\ldots, n_r) \\longmapsto\nE_1^{\\otimes n_1}\n\\otimes \\ldots \\otimes\nE_r^{\\otimes n_r}\n$$\nLet $P : I \\to \\mathbf{Z}$ be a map. Then\n$\\Cohstack^P_{X/B} \\subset \\Cohstack_{X/B}$\nas defined in Situation \\ref{situation-numerical}\nis an open and closed substack.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of coherent sheaves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNE","source_file":"moduli.tex","source_line":409,"source_end_line":425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L409-L425","statement_sha256":"5e2ef6905ce06101987b5e9ac9df48bea2307ebbc87ef9054525fa40132ddb54","origin":"The Stacks Project","memory_eligible":false,"source_rank":14515,"rank":14515,"depth":77,"x":2354.49,"y":1641.13,"cluster":"moduli-theory"},{"id":"stacks:0DM2","tag":"0DM2","title":"Properties of Quot · Lemma 0DM2","summary":"The diagonal of Quotfunctor_F/X/B → B is a closed immersion. If F is of finite type, then the diagonal is a closed immersion of finite presentation.","statement_latex":"The diagonal of $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is a closed immersion.\nIf $\\mathcal{F}$ is of finite type, then the diagonal is a closed\nimmersion of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DM2","source_file":"moduli.tex","source_line":512,"source_end_line":517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L512-L517","statement_sha256":"0d7fff312a6de0ef12375c4dfe9690e893416ebb9e5725f1700d6a8cbfc6f243","origin":"The Stacks Project","memory_eligible":false,"source_rank":14516,"rank":14516,"depth":63,"x":2539.665,"y":1498.126,"cluster":"moduli-theory"},{"id":"stacks:0DM3","tag":"0DM3","title":"Properties of Quot · Lemma 0DM3","summary":"The morphism Quotfunctor_F/X/B → B is separated. If F is of finite presentation, then it is also locally of finite presentation.","statement_latex":"The morphism $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ is separated.\nIf $\\mathcal{F}$ is of finite presentation, then it is also\nlocally of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DM3","source_file":"moduli.tex","source_line":537,"source_end_line":542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L537-L542","statement_sha256":"1ba38ecdf66ce87e9f89a0d8d9c9c49407c8a1bda96369c7a49210c40554c16a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14517,"rank":14517,"depth":82,"x":2518.05,"y":1709.326,"cluster":"moduli-theory"},{"id":"stacks:0DM4","tag":"0DM4","title":"Properties of Quot · Lemma 0DM4","summary":"Assume X → B is proper as well as of finite presentation and F quasi-coherent of finite type. Then Quotfunctor_F/X/B → B satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition [Tag 03IX]).","statement_latex":"Assume $X \\to B$ is proper as well as of finite presentation\nand $\\mathcal{F}$ quasi-coherent of finite type.\nThen $\\Quotfunctor_{\\mathcal{F}/X/B} \\to B$ satisfies the existence part\nof the valuative criterion (Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-valuative-criterion}).","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DM4","source_file":"moduli.tex","source_line":552,"source_end_line":559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L552-L559","statement_sha256":"90b2aede5cc63e1a1abee88806a049daaecb362fba15ef9d3bda609d8900c2d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14518,"rank":14518,"depth":58,"x":2363.656,"y":1540.787,"cluster":"moduli-theory"},{"id":"stacks:0DP1","tag":"0DP1","title":"Properties of Quot · Lemma 0DP1","summary":"Let B be an algebraic space. Let π : X → Y be an affine quasi-finite morphism of algebraic spaces which are separated and of finite presentation over B. Let F be a quasi-coherent O_X-module. Then π_* induces a morphism Quotfunctor_F/X/B → Quotfunctor_π_*F/Y/B.","statement_latex":"Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an affine quasi-finite\nmorphism of algebraic spaces which are separated and of finite presentation\nover $B$. Let $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen $\\pi_*$ induces a morphism\n$\\Quotfunctor_{\\mathcal{F}/X/B} \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DP1","source_file":"moduli.tex","source_line":610,"source_end_line":617,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L610-L617","statement_sha256":"57c7b15f7208c62e5f07a03db071a3e86a958454f6bc8a1998cf54a7abfaae6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14519,"rank":14519,"depth":61,"x":2613.831,"y":1577.575,"cluster":"moduli-theory"},{"id":"stacks:0DP2","tag":"0DP2","title":"Properties of Quot · Lemma 0DP2","summary":"Let B be an algebraic space. Let π : X → Y be an affine open immersion of algebraic spaces which are separated and of finite presentation over B. Let F be a quasi-coherent O_X-module. Then the morphism Quotfunctor_F/X/B → Quotfunctor_π_*F/Y/B of Lemma [Tag 0DP1] is an open immersion.","statement_latex":"Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an affine open immersion\nof algebraic spaces which are separated and of finite presentation over $B$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module. Then the morphism\n$\\Quotfunctor_{\\mathcal{F}/X/B} \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}$ of\nLemma \\ref{lemma-quot-functorial} is an open immersion.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DP2","source_file":"moduli.tex","source_line":636,"source_end_line":643,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L636-L643","statement_sha256":"8139cb85409cc81d20bdd603e6166e3a50a7de854fb69c06ab65b021f059613a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14520,"rank":14520,"depth":62,"x":2399.095,"y":1692.765,"cluster":"moduli-theory"},{"id":"stacks:0DP3","tag":"0DP3","title":"Properties of Quot · Lemma 0DP3","summary":"Let B be an algebraic space. Let j : X → Y be an open immersion of algebraic spaces which are separated and of finite presentation over B. Let G be a quasi-coherent O_Y-module and set F = j^*G. Then there is an open immersion Quotfunctor_F/X/B → Quotfunctor_G/Y/B of algebraic spaces over B.","statement_latex":"Let $B$ be an algebraic space. Let $j : X \\to Y$ be an open immersion\nof algebraic spaces which are separated and of finite presentation over $B$.\nLet $\\mathcal{G}$ be a quasi-coherent $\\mathcal{O}_Y$-module and set\n$\\mathcal{F} = j^*\\mathcal{G}$. Then there is an open immersion\n$$\n\\Quotfunctor_{\\mathcal{F}/X/B}\n\\longrightarrow\n\\Quotfunctor_{\\mathcal{G}/Y/B}\n$$\nof algebraic spaces over $B$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DP3","source_file":"moduli.tex","source_line":652,"source_end_line":664,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L652-L664","statement_sha256":"5201318424fae32ca5209ffb670478db181be3e618aa48e8405112451addddad","origin":"The Stacks Project","memory_eligible":false,"source_rank":14521,"rank":14521,"depth":61,"x":2465.012,"y":1485.331,"cluster":"moduli-theory"},{"id":"stacks:0DP4","tag":"0DP4","title":"Properties of Quot · Lemma 0DP4","summary":"Let B be an algebraic space. Let π : X → Y be a closed immersion of algebraic spaces which are separated and of finite presentation over B. Let F be a quasi-coherent O_X-module. Then the morphism Quotfunctor_F/X/B → Quotfunctor_π_*F/Y/B of Lemma [Tag 0DP1] is an isomorphism.","statement_latex":"Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a closed immersion\nof algebraic spaces which are separated and of finite presentation over $B$.\nLet $\\mathcal{F}$ be a quasi-coherent $\\mathcal{O}_X$-module.\nThen the morphism\n$\\Quotfunctor_{\\mathcal{F}/X/B} \\to \\Quotfunctor_{\\pi_*\\mathcal{F}/Y/B}$ of\nLemma \\ref{lemma-quot-functorial} is an isomorphism.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DP4","source_file":"moduli.tex","source_line":683,"source_end_line":691,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L683-L691","statement_sha256":"ce43ee7a0634abbba6fc69da0cffe3f066b1e59d2286f7ab339fb279b98459b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14522,"rank":14522,"depth":62,"x":2583.586,"y":1676.287,"cluster":"moduli-theory"},{"id":"stacks:0DP5","tag":"0DP5","title":"Properties of Quot · Lemma 0DP5","summary":"Let X → B be as in the introduction to this section. Let F → G be a surjection of quasi-coherent O_X-modules. Then there is a canonical closed immersion Quotfunctor_G/X/B → Quotfunctor_F/X/B.","statement_latex":"Let $X \\to B$ be as in the introduction to this section. Let\n$\\mathcal{F} \\to \\mathcal{G}$ be a surjection of quasi-coherent\n$\\mathcal{O}_X$-modules. Then there is a canonical closed immersion\n$\\Quotfunctor_{\\mathcal{G}/X/B} \\to \\Quotfunctor_{\\mathcal{F}/X/B}$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DP5","source_file":"moduli.tex","source_line":707,"source_end_line":713,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L707-L713","statement_sha256":"736d9c313ba05affd45cb59ab129f7c81d61a728bd4897d613a60586fff12008","origin":"The Stacks Project","memory_eligible":false,"source_rank":14523,"rank":14523,"depth":63,"x":2341.844,"y":1602.532,"cluster":"moduli-theory"},{"id":"stacks:0DP7","tag":"0DP7","title":"Properties of Quot · Lemma 0DP7","summary":"Let f : X → B and F be as in the introduction to this section. Let L be an invertible O_X-module. Then tensoring with L defines an isomorphism Quotfunctor_F/X/B → Quotfunctor_F ⊗_O_X L/X/B Given a numerical polynomial P(t), then setting P'(t) = P(t + 1) this map induces an isomorphism Quotfunctor^P_F/X/B → Quotfunctor^P'_F ⊗_O_X L/X/B of open and closed substacks.","statement_latex":"Let $f : X \\to B$ and $\\mathcal{F}$ be as in the introduction to this section.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen tensoring with $\\mathcal{L}$ defines an isomorphism\n$$\n\\Quotfunctor_{\\mathcal{F}/X/B}\n\\longrightarrow\n\\Quotfunctor_{\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}/X/B}\n$$\nGiven a numerical polynomial $P(t)$, then setting $P'(t) = P(t + 1)$\nthis map induces an isomorphism\n$\\Quotfunctor^P_{\\mathcal{F}/X/B}\n\\longrightarrow\n\\Quotfunctor^{P'}_{\\mathcal{F} \\otimes_{\\mathcal{O}_X} \\mathcal{L}/X/B}$\nof open and closed substacks.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DP7","source_file":"moduli.tex","source_line":766,"source_end_line":782,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L766-L782","statement_sha256":"98c91944caf2972e120bd22771605a60cb926c032b75cfb3da884cf37ca3b084","origin":"The Stacks Project","memory_eligible":false,"source_rank":14524,"rank":14524,"depth":0,"x":2580.145,"y":1519.493,"cluster":"moduli-theory"},{"id":"stacks:0DP8","tag":"0DP8","title":"Properties of Quot · Lemma 0DP8","summary":"Let f : X → B and F be as in the introduction to this section. Let L be an invertible O_X-module. Then Quotfunctor^P, L_F/X/B = Quotfunctor^P', L^⊗ n_F/X/B where P'(t) = P(nt).","statement_latex":"Let $f : X \\to B$ and $\\mathcal{F}$ be as in the introduction to this section.\nLet $\\mathcal{L}$ be an invertible $\\mathcal{O}_X$-module.\nThen\n$$\n\\Quotfunctor^{P, \\mathcal{L}}_{\\mathcal{F}/X/B} =\n\\Quotfunctor^{P', \\mathcal{L}^{\\otimes n}}_{\\mathcal{F}/X/B}\n$$\nwhere $P'(t) = P(nt)$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DP8","source_file":"moduli.tex","source_line":800,"source_end_line":810,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L800-L810","statement_sha256":"cb77b63a243d6315589622ee553e102d4c9d19f8862c4795b2814002f262f6f6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14525,"rank":14525,"depth":0,"x":2470.869,"y":1716.543,"cluster":"moduli-theory"},{"id":"stacks:0DPA","tag":"0DPA","title":"Boundedness for Quot · Lemma 0DPA","summary":"Let n ≥ 0, r ≥ 1, P ∈ Q[t]. The algebraic space X = Quotfunctor^P_O^⊕ r_P^n_Z/ P^n_Z/Z parametrizing quotients of O_P^n_Z^⊕ r with Hilbert polynomial P is proper over Spec(Z).","statement_latex":"Let $n \\geq 0$, $r \\geq 1$, $P \\in \\mathbf{Q}[t]$.\nThe algebraic space\n$$\nX = \\Quotfunctor^P_{\\mathcal{O}^{\\oplus r}_{\\mathbf{P}^n_\\mathbf{Z}}/\n\\mathbf{P}^n_\\mathbf{Z}/\\mathbf{Z}}\n$$\nparametrizing quotients of $\\mathcal{O}_{\\mathbf{P}^n_\\mathbf{Z}}^{\\oplus r}$\nwith Hilbert polynomial $P$ is proper over $\\Spec(\\mathbf{Z})$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Boundedness for Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPA","source_file":"moduli.tex","source_line":829,"source_end_line":839,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L829-L839","statement_sha256":"40aaaae22a33833785eeac1b2589cf81898a25708342611658b5907b78c81ee9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14526,"rank":14526,"depth":83,"x":2392.749,"y":1508.605,"cluster":"moduli-theory"},{"id":"stacks:0DPB","tag":"0DPB","title":"Boundedness for Quot · Lemma 0DPB","summary":"Let B be an algebraic space. Let X = B × P^n_Z. Let L be the pullback of O_P^n(1) to X. Let F be an O_X-module of finite presentation. The algebraic space Quotfunctor^P_F/X/B parametrizing quotients of F having Hilbert polynomial P with respect to L is proper over B.","statement_latex":"Let $B$ be an algebraic space. Let $X = B \\times \\mathbf{P}^n_\\mathbf{Z}$.\nLet $\\mathcal{L}$ be the pullback of $\\mathcal{O}_{\\mathbf{P}^n}(1)$ to $X$.\nLet $\\mathcal{F}$ be an $\\mathcal{O}_X$-module of finite\npresentation. The algebraic space $\\Quotfunctor^P_{\\mathcal{F}/X/B}$\nparametrizing quotients of $\\mathcal{F}$\nhaving Hilbert polynomial $P$ with respect to $\\mathcal{L}$\nis proper over $B$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Boundedness for Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPB","source_file":"moduli.tex","source_line":975,"source_end_line":984,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L975-L984","statement_sha256":"de061494aa3d341d09e9e151ff56c53ac28c6d599180203b32ecd6c6d549360d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14527,"rank":14527,"depth":84,"x":2618.25,"y":1617.939,"cluster":"moduli-theory"},{"id":"stacks:0DPC","tag":"0DPC","title":"Boundedness for Quot · Lemma 0DPC","summary":"Let f : X → B be a proper morphism of finite presentation of algebraic spaces. Let F be a finitely presented O_X-module. Let L be an invertible O_X-module ample on X/B, see Divisors on Spaces, Definition [Tag 0D31]. The algebraic space Quotfunctor^P_F/X/B parametrizing quotients of F having Hilbert polynomial P with respect to L is proper over B.","statement_latex":"Let $f : X \\to B$ be a proper morphism of finite presentation\nof algebraic spaces. Let $\\mathcal{F}$ be a finitely presented\n$\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module ample on $X/B$, see\nDivisors on Spaces, Definition\n\\ref{spaces-divisors-definition-relatively-ample}.\nThe algebraic space $\\Quotfunctor^P_{\\mathcal{F}/X/B}$\nparametrizing quotients of $\\mathcal{F}$\nhaving Hilbert polynomial $P$ with respect to $\\mathcal{L}$\nis proper over $B$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Boundedness for Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPC","source_file":"moduli.tex","source_line":1015,"source_end_line":1027,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1015-L1027","statement_sha256":"f8830e7b058d796404e466cb7ea4626c0483a71eac786f0f94cd32d04313feba","origin":"The Stacks Project","memory_eligible":false,"source_rank":14528,"rank":14528,"depth":85,"x":2363.282,"y":1665.414,"cluster":"moduli-theory"},{"id":"stacks:0DPD","tag":"0DPD","title":"Boundedness for Quot · Lemma 0DPD","summary":"Let f : X → B be a separated morphism of finite presentation of algebraic spaces. Let F be a finitely presented O_X-module. Let L be an invertible O_X-module ample on X/B, see Divisors on Spaces, Definition [Tag 0D31]. The algebraic space Quotfunctor^P_F/X/B parametrizing quotients of F having Hilbert polynomial P with respect to L is separated of finite presentation over B.","statement_latex":"Let $f : X \\to B$ be a separated morphism of finite presentation\nof algebraic spaces. Let $\\mathcal{F}$ be a finitely presented\n$\\mathcal{O}_X$-module. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module ample on $X/B$, see\nDivisors on Spaces, Definition\n\\ref{spaces-divisors-definition-relatively-ample}.\nThe algebraic space $\\Quotfunctor^P_{\\mathcal{F}/X/B}$\nparametrizing quotients of $\\mathcal{F}$\nhaving Hilbert polynomial $P$ with respect to $\\mathcal{L}$\nis separated of finite presentation over $B$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Boundedness for Quot","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPD","source_file":"moduli.tex","source_line":1052,"source_end_line":1064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1052-L1064","statement_sha256":"78301d5e6d40414e2777aee2551f48c6179cafebb357666748a39832290c06b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14529,"rank":14529,"depth":86,"x":2513.561,"y":1485.195,"cluster":"moduli-theory"},{"id":"stacks:0DM6","tag":"0DM6","title":"Properties of the Hilbert functor · Lemma 0DM6","summary":"The diagonal of Hilbfunctor_X/B → B is a closed immersion of finite presentation.","statement_latex":"The diagonal of $\\Hilbfunctor_{X/B} \\to B$ is a closed immersion\nof finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Hilbert functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DM6","source_file":"moduli.tex","source_line":1136,"source_end_line":1140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1136-L1140","statement_sha256":"47111456777aabf4b5e4f31be699815646c97cee946356f762f3ac848399e693","origin":"The Stacks Project","memory_eligible":false,"source_rank":14530,"rank":14530,"depth":64,"x":2547.775,"y":1704.006,"cluster":"moduli-theory"},{"id":"stacks:0DM7","tag":"0DM7","title":"Properties of the Hilbert functor · Lemma 0DM7","summary":"The morphism Hilbfunctor_X/B → B is separated and locally of finite presentation.","statement_latex":"The morphism $\\Hilbfunctor_{X/B} \\to B$ is separated\nand locally of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Hilbert functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DM7","source_file":"moduli.tex","source_line":1148,"source_end_line":1152,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1148-L1152","statement_sha256":"e0f10b425fd294bb482c4d19d68ddf24582f8f39c9dedd37300a951902eaa6f1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14531,"rank":14531,"depth":83,"x":2345.992,"y":1561.653,"cluster":"moduli-theory"},{"id":"stacks:0DM8","tag":"0DM8","title":"Properties of the Hilbert functor · Lemma 0DM8","summary":"Assume X → B is proper as well as of finite presentation. Then Hilbfunctor_X/B → B satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition [Tag 03IX]).","statement_latex":"Assume $X \\to B$ is proper as well as of finite presentation.\nThen $\\Hilbfunctor_{X/B} \\to B$ satisfies the existence part\nof the valuative criterion (Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-valuative-criterion}).","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Hilbert functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DM8","source_file":"moduli.tex","source_line":1162,"source_end_line":1168,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1162-L1168","statement_sha256":"710f98330b62e710333800713591d2bb878d58f9208e2b51e021e10a9dc0a9b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14532,"rank":14532,"depth":59,"x":2610.033,"y":1552.098,"cluster":"moduli-theory"},{"id":"stacks:0DPE","tag":"0DPE","title":"Properties of the Hilbert functor · Lemma 0DPE","summary":"Let B be an algebraic space. Let π : X → Y be an open immersion of algebraic spaces which are separated and of finite presentation over B. Then π induces an open immersion Hilbfunctor_X/B → Hilbfunctor_Y/B.","statement_latex":"Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be an open immersion\nof algebraic spaces which are separated and of finite presentation over $B$.\nThen $\\pi$ induces an open immersion\n$\\Hilbfunctor_{X/B} \\to \\Hilbfunctor_{Y/B}$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Hilbert functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPE","source_file":"moduli.tex","source_line":1176,"source_end_line":1182,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1176-L1182","statement_sha256":"a36be3928ec49883ca33e69ac49da2e961dec6d362e326a2525c3d226c37ca53","origin":"The Stacks Project","memory_eligible":false,"source_rank":14533,"rank":14533,"depth":0,"x":2422.471,"y":1709.423,"cluster":"moduli-theory"},{"id":"stacks:0DPF","tag":"0DPF","title":"Properties of the Hilbert functor · Lemma 0DPF","summary":"Let B be an algebraic space. Let π : X → Y be a closed immersion of algebraic spaces which are separated and of finite presentation over B. Then π induces a closed immersion Hilbfunctor_X/B → Hilbfunctor_Y/B.","statement_latex":"Let $B$ be an algebraic space. Let $\\pi : X \\to Y$ be a closed immersion\nof algebraic spaces which are separated and of finite presentation\nover $B$. Then $\\pi$ induces a closed immersion\n$\\Hilbfunctor_{X/B} \\to \\Hilbfunctor_{Y/B}$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Hilbert functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPF","source_file":"moduli.tex","source_line":1193,"source_end_line":1199,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1193-L1199","statement_sha256":"7a187275dbeb219b1d7c746f682c0c753c910a14b2e86148423659cdb78069ad","origin":"The Stacks Project","memory_eligible":false,"source_rank":14534,"rank":14534,"depth":64,"x":2434.294,"y":1486.342,"cluster":"moduli-theory"},{"id":"stacks:0DPH","tag":"0DPH","title":"Properties of the Hilbert functor · Lemma 0DPH","summary":"Let f : X → B be a proper morphism of finite presentation of algebraic spaces. Let L be an invertible O_X-module ample on X/B, see Divisors on Spaces, Definition [Tag 0D31]. The algebraic space Hilbfunctor^P_X/B parametrizing closed subschemes having Hilbert polynomial P with respect to L is proper over B.","statement_latex":"Let $f : X \\to B$ be a proper morphism of finite presentation\nof algebraic spaces. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module ample on $X/B$, see\nDivisors on Spaces, Definition\n\\ref{spaces-divisors-definition-relatively-ample}.\nThe algebraic space $\\Hilbfunctor^P_{X/B}$\nparametrizing closed subschemes\nhaving Hilbert polynomial $P$ with respect to $\\mathcal{L}$\nis proper over $B$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Hilbert functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPH","source_file":"moduli.tex","source_line":1238,"source_end_line":1249,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1238-L1249","statement_sha256":"33e59f80c57dd3c7c435f68828409c99229fb21251d427d38cbffff130279697","origin":"The Stacks Project","memory_eligible":false,"source_rank":14535,"rank":14535,"depth":86,"x":2605.462,"y":1658.041,"cluster":"moduli-theory"},{"id":"stacks:0DPI","tag":"0DPI","title":"Properties of the Hilbert functor · Lemma 0DPI","summary":"Let f : X → B be a separated morphism of finite presentation of algebraic spaces. Let L be an invertible O_X-module ample on X/B, see Divisors on Spaces, Definition [Tag 0D31]. The algebraic space Hilbfunctor^P_X/B parametrizing closed subschemes having Hilbert polynomial P with respect to L is separated of finite presentation over B.","statement_latex":"Let $f : X \\to B$ be a separated morphism of finite presentation\nof algebraic spaces. Let $\\mathcal{L}$ be an invertible\n$\\mathcal{O}_X$-module ample on $X/B$, see\nDivisors on Spaces, Definition\n\\ref{spaces-divisors-definition-relatively-ample}.\nThe algebraic space $\\Hilbfunctor^P_{X/B}$\nparametrizing closed subschemes\nhaving Hilbert polynomial $P$ with respect to $\\mathcal{L}$\nis separated of finite presentation over $B$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Hilbert functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPI","source_file":"moduli.tex","source_line":1258,"source_end_line":1269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1258-L1269","statement_sha256":"e66a0189c5da1f0bbff675cb2dedd596a6d2dd5cf5ca16b06769b173e2d992f0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14536,"rank":14536,"depth":87,"x":2340.413,"y":1628.473,"cluster":"moduli-theory"},{"id":"stacks:0DMA","tag":"0DMA","title":"Properties of the Picard stack · Lemma 0DMA","summary":"The diagonal of Picardstack_X/B over B is affine and of finite presentation.","statement_latex":"The diagonal of $\\Picardstack_{X/B}$ over $B$ is affine\nand of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMA","source_file":"moduli.tex","source_line":1295,"source_end_line":1299,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1295-L1299","statement_sha256":"12f70a90cd187f21c188ac003979bb80c2f002389c79f2c96685c2ec2c0c90d6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14537,"rank":14537,"depth":75,"x":2560.259,"y":1499.515,"cluster":"moduli-theory"},{"id":"stacks:0DMB","tag":"0DMB","title":"Properties of the Picard stack · Lemma 0DMB","summary":"The morphism Picardstack_X/B → B is quasi-separated and locally of finite presentation.","statement_latex":"The morphism $\\Picardstack_{X/B} \\to B$ is quasi-separated and\nlocally of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMB","source_file":"moduli.tex","source_line":1308,"source_end_line":1312,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1308-L1312","statement_sha256":"38e50f67ebd12e64b60d7bd839468c165b4eee70224e84eef17e76b867fd486b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14538,"rank":14538,"depth":76,"x":2501.686,"y":1719.975,"cluster":"moduli-theory"},{"id":"stacks:0DNG","tag":"0DNG","title":"Properties of the Picard stack · Lemma 0DNG","summary":"Assume X → B is smooth in addition to being proper. Then Picardstack_X/B → B satisfies the existence part of the valuative criterion (Morphisms of Stacks, Definition [Tag 0CLK]).","statement_latex":"Assume $X \\to B$ is smooth in addition to being proper.\nThen $\\Picardstack_{X/B} \\to B$ satisfies the existence part\nof the valuative criterion (Morphisms of Stacks, Definition\n\\ref{stacks-morphisms-definition-existence}).","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNG","source_file":"moduli.tex","source_line":1321,"source_end_line":1327,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1321-L1327","statement_sha256":"cdaf733afd15cec0e8945c3cc6af171ecea516253f8fe8710d5bcbcecf0601d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14539,"rank":14539,"depth":63,"x":2367.213,"y":1523.624,"cluster":"moduli-theory"},{"id":"stacks:0DNH","tag":"0DNH","title":"Properties of the Picard stack · Lemma 0DNH","summary":"Assume f_T, *O_X_T ≅ O_T for all schemes T over B. Then the inertia stack of Picardstack_X/B is equal to G_m × Picardstack_X/B.","statement_latex":"Assume $f_{T, *}\\mathcal{O}_{X_T} \\cong \\mathcal{O}_T$ for all\nschemes $T$ over $B$. Then the inertia stack of $\\Picardstack_{X/B}$\nis equal to $\\mathbf{G}_m \\times \\Picardstack_{X/B}$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNH","source_file":"moduli.tex","source_line":1353,"source_end_line":1358,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1353-L1358","statement_sha256":"25cb91db145e59ad048cc3d68e1f0847a8649b34eb9572110983a8e571ae0a3e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14540,"rank":14540,"depth":0,"x":2624.991,"y":1592.299,"cluster":"moduli-theory"},{"id":"stacks:0DPJ","tag":"0DPJ","title":"Properties of the Picard stack · Lemma 0DPJ","summary":"Assume f : X → B has relative dimension ≤ 1 in addition to the other assumptions in this section. Then Picardstack_X/B → B is smooth.","statement_latex":"Assume $f : X \\to B$ has relative dimension $\\leq 1$ in addition to\nthe other assumptions in this section. Then $\\Picardstack_{X/B} \\to B$\nis smooth.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPJ","source_file":"moduli.tex","source_line":1365,"source_end_line":1370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1365-L1370","statement_sha256":"5f63663695379888c78b7c483f253a311788dd1ca9c0f4228309d4c3ef7d434a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14541,"rank":14541,"depth":77,"x":2378.999,"y":1688.194,"cluster":"moduli-theory"},{"id":"stacks:0DME","tag":"0DME","title":"Properties of the Picard functor · Lemma 0DME","summary":"The morphism Picardstack_X/B → Picardfunctor_X/B turns the Picard stack into a gerbe over the Picard functor.","statement_latex":"The morphism $\\Picardstack_{X/B} \\to \\Picardfunctor_{X/B}$\nturns the Picard stack into a gerbe over the Picard functor.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DME","source_file":"moduli.tex","source_line":1421,"source_end_line":1425,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1421-L1425","statement_sha256":"45ac42ffdbc55a4b1bf6daadf925f108c3e63202f4233ffe8261b5ca7a697cde","origin":"The Stacks Project","memory_eligible":false,"source_rank":14542,"rank":14542,"depth":10,"x":2483.563,"y":1477.319,"cluster":"moduli-theory"},{"id":"stacks:0DMF","tag":"0DMF","title":"Properties of the Picard functor · Lemma 0DMF","summary":"The diagonal of Picardfunctor_X/B over B is a quasi-compact immersion.","statement_latex":"The diagonal of $\\Picardfunctor_{X/B}$ over $B$ is a quasi-compact immersion.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMF","source_file":"moduli.tex","source_line":1458,"source_end_line":1461,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1458-L1461","statement_sha256":"59c3bc7e109d95c1b06940bb86deb9c4b2ef153e15a6fa799d67d40460377516","origin":"The Stacks Project","memory_eligible":false,"source_rank":14543,"rank":14543,"depth":85,"x":2576.293,"y":1692.739,"cluster":"moduli-theory"},{"id":"stacks:0DNI","tag":"0DNI","title":"Properties of the Picard functor · Lemma 0DNI","summary":"The morphism Picardfunctor_X/B → B is quasi-separated and locally of finite presentation.","statement_latex":"The morphism $\\Picardfunctor_{X/B} \\to B$ is quasi-separated and\nlocally of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNI","source_file":"moduli.tex","source_line":1474,"source_end_line":1478,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1474-L1478","statement_sha256":"8ffee93e49f17ef4e1075ed18c6ce609b73bf992c619161f62064c1327bf23f5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14544,"rank":14544,"depth":86,"x":2334.02,"y":1586.217,"cluster":"moduli-theory"},{"id":"stacks:0DNJ","tag":"0DNJ","title":"Properties of the Picard functor · Lemma 0DNJ","summary":"Assume the geometric fibres of X → B are integral in addition to the other assumptions in this section. Then Picardfunctor_X/B → B is separated.","statement_latex":"Assume the geometric fibres of $X \\to B$ are integral\nin addition to the other assumptions in this section.\nThen $\\Picardfunctor_{X/B} \\to B$ is separated.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DNJ","source_file":"moduli.tex","source_line":1496,"source_end_line":1501,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1496-L1501","statement_sha256":"aeb81360a6986181839956f3de7b8d73c8e499a8e32d64eeabcf2159336906ec","origin":"The Stacks Project","memory_eligible":false,"source_rank":14545,"rank":14545,"depth":82,"x":2599.049,"y":1527.133,"cluster":"moduli-theory"},{"id":"stacks:0DPK","tag":"0DPK","title":"Properties of the Picard functor · Lemma 0DPK","summary":"Assume f : X → B has relative dimension ≤ 1 in addition to the other assumptions in this section. Then Picardfunctor_X/B → B is smooth.","statement_latex":"Assume $f : X \\to B$ has relative dimension $\\leq 1$ in addition to\nthe other assumptions in this section. Then $\\Picardfunctor_{X/B} \\to B$\nis smooth.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the Picard functor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPK","source_file":"moduli.tex","source_line":1573,"source_end_line":1578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1573-L1578","statement_sha256":"e4ed2e6165b60a7edb42f372c763eab0bdf6382f34a19e5598e4f4ade375f74a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14546,"rank":14546,"depth":83,"x":2450.733,"y":1721.61,"cluster":"moduli-theory"},{"id":"stacks:0DPM","tag":"0DPM","title":"Properties of relative morphisms · Lemma 0DPM","summary":"The diagonal of mathitMor_B(Y, X) → B is a closed immersion of finite presentation.","statement_latex":"The diagonal of $\\mathit{Mor}_B(Y, X) \\to B$ is a closed immersion\nof finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPM","source_file":"moduli.tex","source_line":1611,"source_end_line":1615,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1611-L1615","statement_sha256":"dd7df175c1a51614aad48459fe71f18d4e9c12289ab3a837656484502eec9dd2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14547,"rank":14547,"depth":65,"x":2403.581,"y":1493.435,"cluster":"moduli-theory"},{"id":"stacks:0DPN","tag":"0DPN","title":"Properties of relative morphisms · Lemma 0DPN","summary":"The morphism mathitMor_B(Y, X) → B is separated and locally of finite presentation.","statement_latex":"The morphism $\\mathit{Mor}_B(Y, X) \\to B$ is separated\nand locally of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPN","source_file":"moduli.tex","source_line":1625,"source_end_line":1629,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1625-L1629","statement_sha256":"195affbd64b54e61e980f68bb0e1036f3e3ed148518d7a53c52237c0eba7494f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14548,"rank":14548,"depth":84,"x":2622.426,"y":1635.311,"cluster":"moduli-theory"},{"id":"stacks:0DPP","tag":"0DPP","title":"Properties of relative morphisms · Lemma 0DPP","summary":"With B, X, Y as in the introduction of this section, in addition assume X → B is proper. Then the subfunctor mathitIsom_B(Y, X) ⊂ mathitMor_B(Y, X) of isomorphisms is an open subspace.","statement_latex":"With $B, X, Y$ as in the introduction of this section, in addition\nassume $X \\to B$ is proper. Then the\nsubfunctor $\\mathit{Isom}_B(Y, X) \\subset \\mathit{Mor}_B(Y, X)$\nof isomorphisms is an open subspace.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPP","source_file":"moduli.tex","source_line":1639,"source_end_line":1645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1639-L1645","statement_sha256":"73026136fdb149144e3bb4827a9daabf11bf10115088a1a8fd71eb2da7e76c33","origin":"The Stacks Project","memory_eligible":false,"source_rank":14549,"rank":14549,"depth":62,"x":2346.226,"y":1654.923,"cluster":"moduli-theory"},{"id":"stacks:0DPR","tag":"0DPR","title":"Properties of relative morphisms · Lemma 0DPR","summary":"With B, X, Y as in the introduction of this section, let L be ample on X/B and let N be ample on Y/B. See Divisors on Spaces, Definition [Tag 0D31]. Let P be a numerical polynomial. Then mathitMor^P, M_B(Y, X) → B is separated and of finite presentation where M = pr_1^*N ⊗_O_Y ×_B X pr_2^*L.","statement_latex":"With $B, X, Y$ as in the introduction of this section, let\n$\\mathcal{L}$ be ample on $X/B$ and let $\\mathcal{N}$ be ample on $Y/B$.\nSee Divisors on Spaces, Definition\n\\ref{spaces-divisors-definition-relatively-ample}.\nLet $P$ be a numerical polynomial. Then\n$$\n\\mathit{Mor}^{P, \\mathcal{M}}_B(Y, X) \\longrightarrow B\n$$\nis separated and of finite presentation where\n$\\mathcal{M} = \\text{pr}_1^*\\mathcal{N}\n\\otimes_{\\mathcal{O}_{Y \\times_B X}} \\text{pr}_2^*\\mathcal{L}$.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of relative morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPR","source_file":"moduli.tex","source_line":1681,"source_end_line":1694,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1681-L1694","statement_sha256":"a31de0b60667a4f4b83faa811319a9183920cb188250de1b90e1421019ae910b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14550,"rank":14550,"depth":88,"x":2534.619,"y":1483.286,"cluster":"moduli-theory"},{"id":"stacks:0DPT","tag":"0DPT","title":"Properties of the stack of polarized proper schemes · Lemma 0DPT","summary":"The diagonal of Polarizedstack is separated and of finite presentation.","statement_latex":"The diagonal of $\\Polarizedstack$ is separated\nand of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPT","source_file":"moduli.tex","source_line":1765,"source_end_line":1769,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1765-L1769","statement_sha256":"a80c4ccfc2616250fb0519525a8bdb009914bdd71e8aff322bd556c2cd6c6514","origin":"The Stacks Project","memory_eligible":false,"source_rank":14551,"rank":14551,"depth":89,"x":2533.724,"y":1717.362,"cluster":"moduli-theory"},{"id":"stacks:0DPU","tag":"0DPU","title":"Properties of the stack of polarized proper schemes · Lemma 0DPU","summary":"The morphism Polarizedstack → Spec(Z) is quasi-separated and locally of finite presentation.","statement_latex":"The morphism $\\Polarizedstack \\to \\Spec(\\mathbf{Z})$ is quasi-separated and\nlocally of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPU","source_file":"moduli.tex","source_line":1894,"source_end_line":1898,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1894-L1898","statement_sha256":"96af9800d1d26cf1b7913fecf3a2d4447c0b6ca8f63bfa7b8568b4a356e28ba5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14552,"rank":14552,"depth":90,"x":2345.655,"y":1543.796,"cluster":"moduli-theory"},{"id":"stacks:0E96","tag":"0E96","title":"Properties of the stack of polarized proper schemes · Lemma 0E96","summary":"Let n ≥ 1 be an integer and let P be a numerical polynomial. Let T ⊂ |Polarizedstack| be a subset with the following property: for every xi ∈ T there exists a field k and an object (X, L) of Polarizedstack over k representing xi such that • the Hilbert polynomial of L on X is P, and • there exists a closed immersion i : X → P^n_k such that i^*O_P^n(1) ≅ L. Then T is a Noetherian topological space, in particular quasi-compact.","statement_latex":"Let $n \\geq 1$ be an integer and let $P$ be a numerical polynomial.\nLet\n$$\nT \\subset |\\Polarizedstack|\n$$\nbe a subset with the following property: for every $\\xi \\in T$\nthere exists a field $k$ and an object $(X, \\mathcal{L})$\nof $\\Polarizedstack$ over $k$ representing $\\xi$ such that\n\\begin{enumerate}\n\\item the Hilbert polynomial of $\\mathcal{L}$ on $X$ is $P$, and\n\\item there exists a closed immersion $i : X \\to \\mathbf{P}^n_k$\nsuch that $i^*\\mathcal{O}_{\\mathbf{P}^n}(1) \\cong \\mathcal{L}$.\n\\end{enumerate}\nThen $T$ is a Noetherian topological space, in particular quasi-compact.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of the stack of polarized proper schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E96","source_file":"moduli.tex","source_line":1911,"source_end_line":1927,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L1911-L1927","statement_sha256":"eed59ef0289a5d96c649bc662d1d7d759e6c98a3afb31c775eca39d45e533bdd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14553,"rank":14553,"depth":91,"x":2624.636,"y":1565.123,"cluster":"moduli-theory"},{"id":"stacks:0DPW","tag":"0DPW","title":"Properties of moduli of complexes on a proper morphism · Lemma 0DPW","summary":"The diagonal of Complexesstack_X/B over B is affine and of finite presentation.","statement_latex":"The diagonal of $\\Complexesstack_{X/B}$ over $B$ is affine\nand of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPW","source_file":"moduli.tex","source_line":2004,"source_end_line":2008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L2004-L2008","statement_sha256":"4f93bb29fb9885c506d07324fee295eefa004caf415061bc9d30a3bc39c85668","origin":"The Stacks Project","memory_eligible":false,"source_rank":14554,"rank":14554,"depth":74,"x":2401.191,"y":1708.067,"cluster":"moduli-theory"},{"id":"stacks:0DPX","tag":"0DPX","title":"Properties of moduli of complexes on a proper morphism · Lemma 0DPX","summary":"The morphism Complexesstack_X/B → B is quasi-separated and locally of finite presentation.","statement_latex":"The morphism $\\Complexesstack_{X/B} \\to B$ is quasi-separated and\nlocally of finite presentation.","area":"Moduli Theory","chapter":"Moduli Stacks","chapter_id":"moduli","section":"Properties of moduli of complexes on a proper morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPX","source_file":"moduli.tex","source_line":2048,"source_end_line":2052,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli.tex#L2048-L2052","statement_sha256":"70e8148ae8f101f0d2d75562a2633720ce7a0b1a014feed33d13dac22820e87b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14555,"rank":14555,"depth":75,"x":2451.134,"y":1475.275,"cluster":"moduli-theory"},{"id":"stacks:0DMK","tag":"0DMK","title":"The stack of curves · Lemma 0DMK","summary":"Let T → B be a morphism of algebraic spaces. The category Mor_B(T, B-Curvesstack) = Mor(T, Curvesstack) is the category of families of curves over T.","statement_latex":"Let $T \\to B$ be a morphism of algebraic spaces. The category\n$$\n\\Mor_B(T, B\\text{-}\\Curvesstack) = \\Mor(T, \\Curvesstack)\n$$\nis the category of families of curves over $T$.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"The stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DMK","source_file":"moduli-curves.tex","source_line":81,"source_end_line":88,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L81-L88","statement_sha256":"2a4272c360fe5fe0d66ff37fb4cdc2a905c364ac5fa4723d68cdae696cc6e81c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14556,"rank":14556,"depth":70,"x":2601.897,"y":1675.784,"cluster":"moduli-theory"},{"id":"stacks:0DPZ","tag":"0DPZ","title":"The stack of polarized curves · Lemma 0DPZ","summary":"The morphism PolarizedCurves → Polarizedstack is an open and closed immersion.","statement_latex":"The morphism\n$\\textit{PolarizedCurves} \\to\n\\Polarizedstack$ is an open and closed immersion.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"The stack of polarized curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DPZ","source_file":"moduli-curves.tex","source_line":202,"source_end_line":207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L202-L207","statement_sha256":"f9c623d709db5dba48c9b5991afcf0b09eb8f5ad59de22ee5b409dd8dc53f6a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14557,"rank":14557,"depth":53,"x":2328.788,"y":1613.319,"cluster":"moduli-theory"},{"id":"stacks:0DQ0","tag":"0DQ0","title":"The stack of polarized curves · Lemma 0DQ0","summary":"The morphism PolarizedCurves → Curvesstack is smooth and surjective.","statement_latex":"The morphism\n$\\textit{PolarizedCurves} \\to \\Curvesstack$\nis smooth and surjective.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"The stack of polarized curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DQ0","source_file":"moduli-curves.tex","source_line":216,"source_end_line":221,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L216-L221","statement_sha256":"bd89cd9ab07d3c2c0bdf587a3859b5d43816e8262c7257c0f560cf6cbc2a878b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14558,"rank":14558,"depth":78,"x":2581.046,"y":1504.135,"cluster":"moduli-theory"},{"id":"stacks:0E6F","tag":"0E6F","title":"The stack of polarized curves · Lemma 0E6F","summary":"Let X → S be a family of curves. Then there exists an étale covering (S_i → S) such that X_i = X ×_S S_i is a scheme. We may even assume X_i is H-projective over S_i.","statement_latex":"Let $X \\to S$ be a family of curves.\nThen there exists an \\'etale covering $\\{S_i \\to S\\}$\nsuch that $X_i = X \\times_S S_i$ is a scheme. We may even\nassume $X_i$ is H-projective over $S_i$.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"The stack of polarized curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6F","source_file":"moduli-curves.tex","source_line":253,"source_end_line":259,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L253-L259","statement_sha256":"cb22b6519068b87f83df73c1d0948880b00a023127cd77326bd427980f9debe3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14559,"rank":14559,"depth":79,"x":2482.588,"y":1728.348,"cluster":"moduli-theory"},{"id":"stacks:0DSQ","tag":"0DSQ","title":"Properties of the stack of curves · Lemma 0DSQ","summary":"The diagonal of Curvesstack is separated and of finite presentation.","statement_latex":"The diagonal of $\\Curvesstack$ is separated\nand of finite presentation.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Properties of the stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSQ","source_file":"moduli-curves.tex","source_line":291,"source_end_line":295,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L291-L295","statement_sha256":"65c4e0ad495d3e59e071925fdd85a3b77dbce2c40d3414ae41a390336b600310","origin":"The Stacks Project","memory_eligible":false,"source_rank":14560,"rank":14560,"depth":89,"x":2374.614,"y":1506.593,"cluster":"moduli-theory"},{"id":"stacks:0DSS","tag":"0DSS","title":"Properties of the stack of curves · Lemma 0DSS","summary":"The morphism Curvesstack → Spec(Z) is quasi-separated and locally of finite presentation.","statement_latex":"The morphism $\\Curvesstack \\to \\Spec(\\mathbf{Z})$ is quasi-separated and\nlocally of finite presentation.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Properties of the stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSS","source_file":"moduli-curves.tex","source_line":437,"source_end_line":441,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L437-L441","statement_sha256":"b779508e8701d0031f642e14b1acaa7663b9b717212fcf3e2872310097d4bab4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14561,"rank":14561,"depth":90,"x":2633.206,"y":1609.101,"cluster":"moduli-theory"},{"id":"stacks:0DSW","tag":"0DSW","title":"Curves with finite reduced automorphism groups · Lemma 0DSW","summary":"There exist an open substack Curvesstack^DM ⊂ Curvesstack with the following properties • Curvesstack^DM ⊂ Curvesstack is the maximal open substack which is DM, • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^DM, • the group algebraic space mathitAut_S(X) is unramified over S, • given X a proper scheme over a field k of dimension ≤ 1 the following are equivalent • the classifying morphism…","statement_latex":"There exist an open substack $\\Curvesstack^{DM} \\subset \\Curvesstack$\nwith the following properties\n\\begin{enumerate}\n\\item $\\Curvesstack^{DM} \\subset \\Curvesstack$ is the maximal\nopen substack which is DM,\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors through\n$\\Curvesstack^{DM}$,\n\\item the group algebraic space $\\mathit{Aut}_S(X)$ is unramified over $S$,\n\\end{enumerate}\n\\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{DM}$,\n\\item $\\mathit{Aut}(X)$ is geometrically reduced over $k$ and\nhas dimension $0$,\n\\item $\\mathit{Aut}(X) \\to \\Spec(k)$ is unramified.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Curves with finite reduced automorphism groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSW","source_file":"moduli-curves.tex","source_line":640,"source_end_line":663,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L640-L663","statement_sha256":"6d32ea70dec998bd1ad92ca72a83d24d1c1ed38ff0872d2c988761d364ab6e3f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14562,"rank":14562,"depth":82,"x":2359.411,"y":1680.421,"cluster":"moduli-theory"},{"id":"stacks:0E6G","tag":"0E6G","title":"Curves with finite reduced automorphism groups · Lemma 0E6G","summary":"Let X be a proper scheme over a field k of dimension ≤ 1. Then properties (3)(a), (b), (c) are also equivalent to Der_k(O_X, O_X) = 0.","statement_latex":"Let $X$ be a proper scheme over a field $k$ of dimension $\\leq 1$.\nThen properties (3)(a), (b), (c) are also equivalent to\n$\\text{Der}_k(\\mathcal{O}_X, \\mathcal{O}_X) = 0$.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Curves with finite reduced automorphism groups","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6G","source_file":"moduli-curves.tex","source_line":691,"source_end_line":696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L691-L696","statement_sha256":"6ed998f1666c32f8672c977406547102a3f78489a8489e02723c3e2dcaa0fdfb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14563,"rank":14563,"depth":90,"x":2504.308,"y":1471.958,"cluster":"moduli-theory"},{"id":"stacks:0E0I","tag":"0E0I","title":"Cohen-Macaulay curves · Lemma 0E0I","summary":"There exist an open substack Curvesstack^CM ⊂ Curvesstack such that • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^CM, • the morphism X → S is Cohen-Macaulay, • given a scheme X proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^CM, • X is Cohen-Macaulay.","statement_latex":"There exist an open substack $\\Curvesstack^{CM} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{CM}$,\n\\item the morphism $X \\to S$ is Cohen-Macaulay,\n\\end{enumerate}\n\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{CM}$,\n\\item $X$ is Cohen-Macaulay.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Cohen-Macaulay curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0I","source_file":"moduli-curves.tex","source_line":749,"source_end_line":768,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L749-L768","statement_sha256":"a6b30c2bed01e71188ca6db5712d59fd5f8006b5f04625f4d2f8972a34a420cb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14564,"rank":14564,"depth":38,"x":2565.252,"y":1708.475,"cluster":"moduli-theory"},{"id":"stacks:0E1F","tag":"0E1F","title":"Cohen-Macaulay curves · Lemma 0E1F","summary":"There exist an open substack Curvesstack^CM, 1 ⊂ Curvesstack such that • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^CM, 1, • the morphism X → S is Cohen-Macaulay and has relative dimension 1 (Morphisms of Spaces, Definition [Tag 06LR]), • given a scheme X proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through…","statement_latex":"There exist an open substack $\\Curvesstack^{CM, 1} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{CM, 1}$,\n\\item the morphism $X \\to S$ is Cohen-Macaulay and has\nrelative dimension $1$ (Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-relative-dimension}),\n\\end{enumerate}\n\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{CM, 1}$,\n\\item $X$ is Cohen-Macaulay and $X$ is equidimensional of\ndimension $1$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Cohen-Macaulay curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1F","source_file":"moduli-curves.tex","source_line":790,"source_end_line":812,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L790-L812","statement_sha256":"1e3c39f4e49aa4f2558711d6b88d76eff11622c2ca62fce0f2113a4de504b643","origin":"The Stacks Project","memory_eligible":false,"source_rank":14565,"rank":14565,"depth":39,"x":2329.537,"y":1568.309,"cluster":"moduli-theory"},{"id":"stacks:0E6I","tag":"0E6I","title":"Curves of a given genus · Lemma 0E6I","summary":"There exist an open substack Curvesstack^h0, 1 ⊂ Curvesstack such that • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^h0, 1, • f_*O_X = O_S, this holds after arbitrary base change, and the fibres of f have dimension 1, • given a scheme X proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^h0,…","statement_latex":"There exist an open substack $\\Curvesstack^{h0, 1} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{h0, 1}$,\n\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds\nafter arbitrary base change, and the fibres of $f$ have dimension $1$,\n\\end{enumerate}\n\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{h0, 1}$,\n\\item $H^0(X, \\mathcal{O}_X) = k$ and $\\dim(X) = 1$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Curves of a given genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6I","source_file":"moduli-curves.tex","source_line":860,"source_end_line":880,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L860-L880","statement_sha256":"a85c2f3b6ba8d4ede720275fc761d3ff0a2f004f5ef5833a676851ecf55128a3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14566,"rank":14566,"depth":70,"x":2616.766,"y":1537.841,"cluster":"moduli-theory"},{"id":"stacks:0E6J","tag":"0E6J","title":"Curves of a given genus · Lemma 0E6J","summary":"We have Curvesstack^h0, 1 ⊂ Curvesstack^CM, 1 as open substacks of Curvesstack.","statement_latex":"We have $\\Curvesstack^{h0, 1} \\subset \\Curvesstack^{CM, 1}$\nas open substacks of $\\Curvesstack$.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Curves of a given genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6J","source_file":"moduli-curves.tex","source_line":899,"source_end_line":903,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L899-L903","statement_sha256":"fcc9c4c5b62aefc2fc13c0e5fc75d3cdc903635faabfa30b2cffaa3b1e8eaaa5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14567,"rank":14567,"depth":71,"x":2429.014,"y":1723.738,"cluster":"moduli-theory"},{"id":"stacks:0E1J","tag":"0E1J","title":"Curves of a given genus · Lemma 0E1J","summary":"Let f : X → S be a family of curves such that kappa(s) = H^0(X_s, O_X_s) for all s ∈ S, i.e., the classifying morphism S → Curvesstack factors through Curvesstack^h0, 1 (Lemma [Tag 0E6I]). Then • f_*O_X = O_S and this holds universally, • R^1f_*O_X is a finite locally free O_S-module, • for any morphism h : S' → S if f' : X' → S' is the base change, then h^*(R^1f_*O_X) = R^1f'_*O_X'.","statement_latex":"Let $f : X \\to S$ be a family of curves such that\n$\\kappa(s) = H^0(X_s, \\mathcal{O}_{X_s})$ for all $s \\in S$, i.e.,\nthe classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{h0, 1}$ (Lemma \\ref{lemma-pre-genus-curves}). Then\n\\begin{enumerate}\n\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds universally,\n\\item $R^1f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_S$-module,\n\\item for any morphism $h : S' \\to S$ if $f' : X' \\to S'$ is the base change,\nthen $h^*(R^1f_*\\mathcal{O}_X) = R^1f'_*\\mathcal{O}_{X'}$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Curves of a given genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1J","source_file":"moduli-curves.tex","source_line":910,"source_end_line":922,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L910-L922","statement_sha256":"724dc33e89c3850e5b0b779d96ef5ea4d57bc4494a6e0313cde309db403cd650","origin":"The Stacks Project","memory_eligible":false,"source_rank":14568,"rank":14568,"depth":71,"x":2417.94,"y":1479.537,"cluster":"moduli-theory"},{"id":"stacks:0E6K","tag":"0E6K","title":"Curves of a given genus · Lemma 0E6K","summary":"There is a decomposition into open and closed substacks Curvesstack^h0, 1 = coprod_g ≥ 0 Curvesstack_g where each Curvesstack_g is characterized as follows: • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack_g, • f_*O_X = O_S, this holds after arbitrary base change, the fibres of f have dimension 1, and R^1f_*O_X is a locally free O_S-module of rank g, • given a scheme X proper over a…","statement_latex":"There is a decomposition into open and closed substacks\n$$\n\\Curvesstack^{h0, 1} = \\coprod\\nolimits_{g \\geq 0} \\Curvesstack_g\n$$\nwhere each $\\Curvesstack_g$ is characterized as follows:\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack_g$,\n\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds after\narbitrary base change, the fibres of $f$ have dimension $1$, and\n$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$,\n\\end{enumerate}\n\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack_g$,\n\\item $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$, and\nthe genus of $X$ is $g$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Curves of a given genus","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6K","source_file":"moduli-curves.tex","source_line":950,"source_end_line":975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L950-L975","statement_sha256":"822b4bbfcc0764cc6e455281fb866cff97391ef4c76c277f76b5024d88d76694","origin":"The Stacks Project","memory_eligible":false,"source_rank":14569,"rank":14569,"depth":72,"x":2622.977,"y":1653.741,"cluster":"moduli-theory"},{"id":"stacks:0E0G","tag":"0E0G","title":"Geometrically reduced curves · Lemma 0E0G","summary":"There exist an open substack Curvesstack^geomred ⊂ Curvesstack such that • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^geomred, • the fibres of the morphism X → S are geometrically reduced (More on Morphisms of Spaces, Definition [Tag 0E08]), • given a scheme X proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors…","statement_latex":"There exist an open substack $\\Curvesstack^{geomred} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{geomred}$,\n\\item the fibres of the morphism $X \\to S$ are geometrically reduced\n(More on Morphisms of Spaces, Definition\n\\ref{spaces-more-morphisms-definition-geometrically-reduced-fibre}),\n\\end{enumerate}\n\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{geomred}$,\n\\item $X$ is geometrically reduced over $k$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Geometrically reduced curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E0G","source_file":"moduli-curves.tex","source_line":1003,"source_end_line":1024,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1003-L1024","statement_sha256":"fbd06cc125b047b67f705a61ab53427a1d746257b25122cd1e9178b9b74a4809","origin":"The Stacks Project","memory_eligible":false,"source_rank":14570,"rank":14570,"depth":60,"x":2330.999,"y":1641.599,"cluster":"moduli-theory"},{"id":"stacks:0E1G","tag":"0E1G","title":"Geometrically reduced curves · Lemma 0E1G","summary":"We have Curvesstack^geomred ⊂ Curvesstack^CM as open substacks of Curvesstack.","statement_latex":"We have $\\Curvesstack^{geomred} \\subset \\Curvesstack^{CM}$\nas open substacks of $\\Curvesstack$.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Geometrically reduced curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1G","source_file":"moduli-curves.tex","source_line":1043,"source_end_line":1047,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1043-L1047","statement_sha256":"c067257515075012a658ab4287fd7b277e15c9a6737dae7ee51a4c95038b1b0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14571,"rank":14571,"depth":11,"x":2556.599,"y":1484.505,"cluster":"moduli-theory"},{"id":"stacks:0E1I","tag":"0E1I","title":"Geometrically reduced and connected curves · Lemma 0E1I","summary":"There exist an open substack Curvesstack^grc, 1 ⊂ Curvesstack such that • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^grc, 1, • the geometric fibres of the morphism X → S are reduced, connected, and have dimension 1, • given a scheme X proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^grc, 1, •…","statement_latex":"There exist an open substack $\\Curvesstack^{grc, 1} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{grc, 1}$,\n\\item the geometric fibres of the morphism $X \\to S$ are\nreduced, connected, and have dimension $1$,\n\\end{enumerate}\n\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{grc, 1}$,\n\\item $X$ is geometrically reduced, geometrically connected,\nand has dimension $1$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Geometrically reduced and connected curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1I","source_file":"moduli-curves.tex","source_line":1068,"source_end_line":1089,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1068-L1089","statement_sha256":"1530e1e4ef6e92fb10c13fb6c62cced5139e0ad22b5a7d88b99eaa8861ac9736","origin":"The Stacks Project","memory_eligible":false,"source_rank":14572,"rank":14572,"depth":75,"x":2516.48,"y":1728.932,"cluster":"moduli-theory"},{"id":"stacks:0E6L","tag":"0E6L","title":"Geometrically reduced and connected curves · Lemma 0E6L","summary":"We have Curvesstack^grc, 1 ⊂ Curvesstack^h0, 1 as open substacks of Curvesstack. In particular, given a family of curves f : X → S whose geometric fibres are reduced, connected and of dimension 1, then R^1f_*O_X is a finite locally free O_S-module whose formation commutes with arbitrary base change.","statement_latex":"We have $\\Curvesstack^{grc, 1} \\subset \\Curvesstack^{h0, 1}$\nas open substacks of $\\Curvesstack$. In particular, given\na family of curves $f : X \\to S$\nwhose geometric fibres are reduced, connected and of dimension $1$, then\n$R^1f_*\\mathcal{O}_X$ is a finite locally free $\\mathcal{O}_S$-module\nwhose formation commutes with arbitrary base change.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Geometrically reduced and connected curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6L","source_file":"moduli-curves.tex","source_line":1125,"source_end_line":1133,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1125-L1133","statement_sha256":"dcf35fa4df500d9a2eeae015b5e6a7541818b7c388746c008ebf0441c5ece929","origin":"The Stacks Project","memory_eligible":false,"source_rank":14573,"rank":14573,"depth":76,"x":2349.109,"y":1525.452,"cluster":"moduli-theory"},{"id":"stacks:0E1K","tag":"0E1K","title":"Geometrically reduced and connected curves · Lemma 0E1K","summary":"There is a decomposition into open and closed substacks Curvesstack^grc, 1 = coprod_g ≥ 0 Curvesstack^grc, 1_g where each Curvesstack^grc, 1_g is characterized as follows: • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^grc, 1_g, • the geometric fibres of the morphism f : X → S are reduced, connected, of dimension 1 and R^1f_*O_X is a locally free O_S-module of rank g, • given a…","statement_latex":"There is a decomposition into open and closed substacks\n$$\n\\Curvesstack^{grc, 1} = \\coprod\\nolimits_{g \\geq 0} \\Curvesstack^{grc, 1}_g\n$$\nwhere each $\\Curvesstack^{grc, 1}_g$ is characterized as follows:\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{grc, 1}_g$,\n\\item the geometric fibres of the morphism $f : X \\to S$ are\nreduced, connected, of dimension $1$ and\n$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module\nof rank $g$,\n\\end{enumerate}\n\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{grc, 1}_g$,\n\\item $X$ is geometrically reduced, geometrically connected,\nhas dimension $1$, and has genus $g$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Geometrically reduced and connected curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1K","source_file":"moduli-curves.tex","source_line":1143,"source_end_line":1169,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1143-L1169","statement_sha256":"b3ca8d30e4034f8e4e075dc39b4b7b11ee121db7cb1b9c17432da4bcc61c7fad","origin":"The Stacks Project","memory_eligible":false,"source_rank":14574,"rank":14574,"depth":77,"x":2636.832,"y":1580.654,"cluster":"moduli-theory"},{"id":"stacks:0E1M","tag":"0E1M","title":"Gorenstein curves · Lemma 0E1M","summary":"There exist an open substack Curvesstack^Gorenstein ⊂ Curvesstack such that • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^Gorenstein, • the morphism X → S is Gorenstein, • given a scheme X proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^Gorenstein, • X is Gorenstein.","statement_latex":"There exist an open substack $\\Curvesstack^{Gorenstein} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{Gorenstein}$,\n\\item the morphism $X \\to S$ is Gorenstein,\n\\end{enumerate}\n\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{Gorenstein}$,\n\\item $X$ is Gorenstein.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Gorenstein curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1M","source_file":"moduli-curves.tex","source_line":1203,"source_end_line":1222,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1203-L1222","statement_sha256":"21e16cb95484ed514c85eeb56d1efaf0eb322da877e50f4bd722c222add40466","origin":"The Stacks Project","memory_eligible":false,"source_rank":14575,"rank":14575,"depth":69,"x":2379.678,"y":1703.496,"cluster":"moduli-theory"},{"id":"stacks:0E6M","tag":"0E6M","title":"Gorenstein curves · Lemma 0E6M","summary":"There exist an open substack Curvesstack^Gorenstein, 1 ⊂ Curvesstack such that • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^Gorenstein, 1, • the morphism X → S is Gorenstein and has relative dimension 1 (Morphisms of Spaces, Definition [Tag 06LR]), • given a scheme X proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack…","statement_latex":"There exist an open substack\n$\\Curvesstack^{Gorenstein, 1} \\subset \\Curvesstack$ such that\n\\begin{enumerate}\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{Gorenstein, 1}$,\n\\item the morphism $X \\to S$ is Gorenstein and has\nrelative dimension $1$ (Morphisms of Spaces, Definition\n\\ref{spaces-morphisms-definition-relative-dimension}),\n\\end{enumerate}\n\\item given a scheme $X$ proper over a field $k$ with $\\dim(X) \\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{Gorenstein, 1}$,\n\\item $X$ is Gorenstein and $X$ is equidimensional of\ndimension $1$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Gorenstein curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6M","source_file":"moduli-curves.tex","source_line":1244,"source_end_line":1266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1244-L1266","statement_sha256":"c4fed6adaf0fe173f9e5e60d6c23b631379cf72ee85a02af6f9935408121b9ba","origin":"The Stacks Project","memory_eligible":false,"source_rank":14576,"rank":14576,"depth":70,"x":2470.726,"y":1466.45,"cluster":"moduli-theory"},{"id":"stacks:0DZV","tag":"0DZV","title":"Local complete intersection curves · Lemma 0DZV","summary":"There exist an open substack Curvesstack^lci ⊂ Curvesstack such that • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^lci, • X → S is a local complete intersection morphism, and • X → S is a syntomic morphism. • given X a proper scheme over a field k of dimension ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^lci, • X is a…","statement_latex":"There exist an open substack $\\Curvesstack^{lci} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors through\n$\\Curvesstack^{lci}$,\n\\item $X \\to S$ is a local complete intersection morphism, and\n\\item $X \\to S$ is a syntomic morphism.\n\\end{enumerate}\n\\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{lci}$,\n\\item $X$ is a local complete intersection over $k$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Local complete intersection curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZV","source_file":"moduli-curves.tex","source_line":1293,"source_end_line":1313,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1293-L1313","statement_sha256":"45c03cecf7788e1e0eec21420c099008db9d885729bc231deac7109b708623ee","origin":"The Stacks Project","memory_eligible":false,"source_rank":14577,"rank":14577,"depth":49,"x":2594.499,"y":1693.43,"cluster":"moduli-theory"},{"id":"stacks:0DZW","tag":"0DZW","title":"Curves with isolated singularities · Lemma 0DZW","summary":"There exist an open substack Curvesstack^+ ⊂ Curvesstack such that • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^+, • the singular locus of X → S endowed with any/some closed subspace structure is finite over S. • given X a proper scheme over a field k of dimension ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^+, • X →…","statement_latex":"There exist an open substack\n$\\Curvesstack^{+} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors through\n$\\Curvesstack^{+}$,\n\\item the singular locus of $X \\to S$ endowed\nwith any/some closed subspace structure is finite over $S$.\n\\end{enumerate}\n\\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{+}$,\n\\item $X \\to \\Spec(k)$ is smooth except at finitely many points.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Curves with isolated singularities","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZW","source_file":"moduli-curves.tex","source_line":1341,"source_end_line":1362,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1341-L1362","statement_sha256":"1f0af6c7ccb35352426c1c7a31bb994e4934011823894b344708dcb97afa06dc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14578,"rank":14578,"depth":61,"x":2320.07,"y":1596.067,"cluster":"moduli-theory"},{"id":"stacks:0DZU","tag":"0DZU","title":"The smooth locus of the stack of curves · Lemma 0DZU","summary":"In the situation above the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through the open where Curvesstack → Spec(Z) is smooth, • the deformation category Deformationcategory_X is unobstructed.","statement_latex":"In the situation above the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough the open where $\\Curvesstack \\to \\Spec(\\mathbf{Z})$ is smooth,\n\\item the deformation category $\\Deformationcategory_X$ is unobstructed.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"The smooth locus of the stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZU","source_file":"moduli-curves.tex","source_line":1409,"source_end_line":1417,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1409-L1417","statement_sha256":"659ebfd2d165eec547ff7c3e898dbbac6a9abef8c9e07d4e5c6ee21ce5f915c5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14579,"rank":14579,"depth":91,"x":2601.37,"y":1511.951,"cluster":"moduli-theory"},{"id":"stacks:0DZX","tag":"0DZX","title":"The smooth locus of the stack of curves · Lemma 0DZX","summary":"The open substack Curvesstack^lci+ = Curvesstack^lci ∩ Curvesstack^+ ⊂ Curvesstack has the following properties • Curvesstack^lci+ → Spec(Z) is smooth, • given a family of curves X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^lci+, • X → S is a local complete intersection morphism and the singular locus of X → S endowed with any/some closed subspace structure is finite over S, • given X a proper scheme over a…","statement_latex":"The open substack\n$$\n\\Curvesstack^{lci+} =\n\\Curvesstack^{lci} \\cap \\Curvesstack^{+}\n\\subset \\Curvesstack\n$$\nhas the following properties\n\\begin{enumerate}\n\\item $\\Curvesstack^{lci+} \\to \\Spec(\\mathbf{Z})$ is smooth,\n\\item given a family of curves $X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors through\n$\\Curvesstack^{lci+}$,\n\\item $X \\to S$ is a local complete intersection morphism and\nthe singular locus of $X \\to S$ endowed with any/some closed subspace\nstructure is finite over $S$,\n\\end{enumerate}\n\\item given $X$ a proper scheme over a field $k$ of dimension $\\leq 1$\nthe following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{lci+}$,\n\\item $X$ is a local complete intersection over $k$ and\n$X \\to \\Spec(k)$ is smooth except at finitely many points.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"The smooth locus of the stack of curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZX","source_file":"moduli-curves.tex","source_line":1467,"source_end_line":1495,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1467-L1495","statement_sha256":"7b169cd91c04e39cee672f3291b43d13d3b0119be377600d0e05e05d4d467853","origin":"The Stacks Project","memory_eligible":false,"source_rank":14580,"rank":14580,"depth":92,"x":2461.268,"y":1734.098,"cluster":"moduli-theory"},{"id":"stacks:0DZZ","tag":"0DZZ","title":"Smooth curves · Lemma 0DZZ","summary":"There exist an open substacks Curvesstack^smooth, 1 ⊂ Curvesstack^smooth ⊂ Curvesstack such that • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^smooth, resp. Curvesstack^smooth, 1, • f is smooth, resp. smooth of relative dimension 1, • given X a scheme proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through…","statement_latex":"There exist an open substacks\n$$\n\\Curvesstack^{smooth, 1} \\subset \\Curvesstack^{smooth} \\subset \\Curvesstack\n$$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{smooth}$, resp.\\ $\\Curvesstack^{smooth, 1}$,\n\\item $f$ is smooth, resp.\\ smooth of relative dimension $1$,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$\nfactors through $\\Curvesstack^{smooth}$, resp.\\ $\\Curvesstack^{smooth, 1}$,\n\\item $X$ is smooth over $k$, resp.\\ $X$ is smooth over $k$ and\n$X$ is equidimensional of dimension $1$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DZZ","source_file":"moduli-curves.tex","source_line":1520,"source_end_line":1543,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1520-L1543","statement_sha256":"f1bfec6868d6808800c933565e318cecf9f099db7fead6e11fdd852e570035b6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14581,"rank":14581,"depth":41,"x":2385.762,"y":1490.241,"cluster":"moduli-theory"},{"id":"stacks:0E1N","tag":"0E1N","title":"Smooth curves · Lemma 0E1N","summary":"The morphism Curvesstack^smooth → Spec(Z) is smooth.","statement_latex":"The morphism $\\Curvesstack^{smooth} \\to \\Spec(\\mathbf{Z})$ is smooth.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E1N","source_file":"moduli-curves.tex","source_line":1572,"source_end_line":1575,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1572-L1575","statement_sha256":"8e27706318f35a2b1e4d4c3dd19ca0e5ad4f37745376d277672e660d10da7221","origin":"The Stacks Project","memory_eligible":false,"source_rank":14582,"rank":14582,"depth":93,"x":2638.111,"y":1627.524,"cluster":"moduli-theory"},{"id":"stacks:0E81","tag":"0E81","title":"Smooth curves · Lemma 0E81","summary":"There exist an open substack Curvesstack^smooth, h0 ⊂ Curvesstack such that • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^smooth, • f_*O_X = O_S, this holds after any base change, and f is smooth of relative dimension 1, • given X a scheme proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through…","statement_latex":"There exist an open substack\n$\\Curvesstack^{smooth, h0} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{smooth}$,\n\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$, this holds after any base change,\nand $f$ is smooth of relative dimension $1$,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$\nfactors through $\\Curvesstack^{smooth, h0}$,\n\\item $X$ is smooth, $\\dim(X) = 1$, and $k = H^0(X, \\mathcal{O}_X)$,\n\\item $X$ is smooth, $\\dim(X) = 1$, and $X$ is geometrically connected,\n\\item $X$ is smooth, $\\dim(X) = 1$, and $X$ is geometrically integral, and\n\\item $X_{\\overline{k}}$ is a smooth curve.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E81","source_file":"moduli-curves.tex","source_line":1583,"source_end_line":1607,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1583-L1607","statement_sha256":"1ea9869c1e5bed28e3da06815e37f246a957b2234ca5048aeb9e1b33effc1f4e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14583,"rank":14583,"depth":71,"x":2340.966,"y":1669.576,"cluster":"moduli-theory"},{"id":"stacks:0E82","tag":"0E82","title":"Smooth curves · Definition 0E82","summary":"[DM] We denote M and we name it the moduli stack of smooth proper curves the algebraic stack Curvesstack^smooth, h0 parametrizing families of curves introduced in Lemma [Tag 0E81]. For g ≥ 0 we denote M_g and we name it the moduli stack of smooth proper curves of genus g the algebraic stack introduced in Lemma [Tag 0E83].","statement_latex":"\\begin{reference}\n\\cite{DM}\n\\end{reference}\nWe denote $\\mathcal{M}$ and we name it the\n{\\it moduli stack of smooth proper curves}\nthe algebraic stack\n$\\Curvesstack^{smooth, h0}$ parametrizing families of curves\nintroduced in Lemma \\ref{lemma-smooth-curves-h0}.\nFor $g \\geq 0$ we denote $\\mathcal{M}_g$ and we name it the\n{\\it moduli stack of smooth proper curves of genus $g$}\nthe algebraic stack introduced in\nLemma \\ref{lemma-smooth-one-piece-per-genus}.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Smooth curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E82","source_file":"moduli-curves.tex","source_line":1632,"source_end_line":1646,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1632-L1646","statement_sha256":"411783904ba62a439c3f77911b720b327018a89df4eda5d2ae91db06f30d8b8d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14584,"rank":14584,"depth":72,"x":2526.674,"y":1469.514,"cluster":"moduli-theory"},{"id":"stacks:0E83","tag":"0E83","title":"Smooth curves · Lemma 0E83","summary":"There is a decomposition into open and closed substacks M = coprod_g ≥ 0 M_g where each M_g is characterized as follows: • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through M_g, • X → S is smooth, f_*O_X = O_S, this holds after any base change, and R^1f_*O_X is a locally free O_S-module of rank g, • given X a scheme proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying…","statement_latex":"There is a decomposition into open and closed substacks\n$$\n\\mathcal{M} = \\coprod\\nolimits_{g \\geq 0} \\mathcal{M}_g\n$$\nwhere each $\\mathcal{M}_g$ is characterized as follows:\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\mathcal{M}_g$,\n\\item $X \\to S$ is smooth, $f_*\\mathcal{O}_X = \\mathcal{O}_S$,\nthis holds after any base change, and $R^1f_*\\mathcal{O}_X$\nis a locally free $\\mathcal{O}_S$-module of rank $g$,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$\nfactors through $\\mathcal{M}_g$,\n\\item $X$ is smooth, $\\dim(X) = 1$, $k = H^0(X, \\mathcal{O}_X)$,\nand $X$ has genus $g$,\n\\item $X$ is smooth, $\\dim(X) = 1$, $X$ is geometrically connected, and\n$X$ has genus $g$,\n\\item $X$ is smooth, $\\dim(X) = 1$, $X$ is geometrically integral, and\n$X$ has genus $g$, and\n\\item $X_{\\overline{k}}$ is a smooth curve of genus $g$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E83","source_file":"moduli-curves.tex","source_line":1651,"source_end_line":1681,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1651-L1681","statement_sha256":"3d68e951da8aa1c6ca07b9dbca187c4d3c318e45042d6ee9b67bfbb2dd61d58d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14585,"rank":14585,"depth":78,"x":2550.673,"y":1722.975,"cluster":"moduli-theory"},{"id":"stacks:0E84","tag":"0E84","title":"Smooth curves · Lemma 0E84","summary":"The morphisms M → Spec(Z) and M_g → Spec(Z) are smooth.","statement_latex":"The morphisms $\\mathcal{M} \\to \\Spec(\\mathbf{Z})$ and\n$\\mathcal{M}_g \\to \\Spec(\\mathbf{Z})$\nare smooth.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E84","source_file":"moduli-curves.tex","source_line":1691,"source_end_line":1696,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1691-L1696","statement_sha256":"c4095c4dc4a48dfd9659c05755ae0b781147783867aa94e8cc44ab2b38bc134a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14586,"rank":14586,"depth":93,"x":2328.659,"y":1549.31,"cluster":"moduli-theory"},{"id":"stacks:0E86","tag":"0E86","title":"Density of smooth curves · Lemma 0E86","summary":"The inclusion |Curvesstack^smooth| ⊂ |Curvesstack^lci+| is that of an open dense subset.","statement_latex":"The inclusion\n$$\n|\\Curvesstack^{smooth}| \\subset |\\Curvesstack^{lci+}|\n$$\nis that of an open dense subset.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Density of smooth curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E86","source_file":"moduli-curves.tex","source_line":1718,"source_end_line":1725,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1718-L1725","statement_sha256":"a82266e635b24a55ce0be0a8e8437f28a16d165877a9338ba809aed6d7f271a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14587,"rank":14587,"depth":93,"x":2632.696,"y":1551.397,"cluster":"moduli-theory"},{"id":"stacks:0DSY","tag":"0DSY","title":"Nodal curves · Lemma 0DSY","summary":"There exist an open substack Curvesstack^nodal ⊂ Curvesstack such that • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^nodal, • f is at-worst-nodal of relative dimension 1, • given X a scheme proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^nodal, • the singularities of X are at-worst-nodal…","statement_latex":"There exist an open substack $\\Curvesstack^{nodal} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{nodal}$,\n\\item $f$ is at-worst-nodal of relative dimension $1$,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{nodal}$,\n\\item the singularities of $X$ are at-worst-nodal and $X$\nis equidimensional of dimension $1$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0DSY","source_file":"moduli-curves.tex","source_line":1771,"source_end_line":1791,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1771-L1791","statement_sha256":"fc7c62ee464de66a8566e3bea7cff4d516bb37e21991761ee3cb592f710320a5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14588,"rank":14588,"depth":46,"x":2406.328,"y":1722.751,"cluster":"moduli-theory"},{"id":"stacks:0E00","tag":"0E00","title":"Nodal curves · Lemma 0E00","summary":"The morphism Curvesstack^nodal → Spec(Z) is smooth.","statement_latex":"The morphism $\\Curvesstack^{nodal} \\to \\Spec(\\mathbf{Z})$ is smooth.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Nodal curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E00","source_file":"moduli-curves.tex","source_line":1811,"source_end_line":1814,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1811-L1814","statement_sha256":"9ec493d8258e8d7ed82512e4ae8f7bbaa5bfa316a736b702fe32ca8aaa2de04d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14589,"rank":14589,"depth":93,"x":2435.514,"y":1467.393,"cluster":"moduli-theory"},{"id":"stacks:0E6P","tag":"0E6P","title":"The relative dualizing sheaf · Lemma 0E6P","summary":"Let X → S be a family of curves with Cohen-Macaulay fibres equidimensional of dimension 1 (Lemma [Tag 0E1F]). Then ω_X/S^bullet = ω_X/S[1] where ω_X/S is a pseudo-coherent O_X-module flat over S whose formation commutes with arbitrary base change.","statement_latex":"Let $X \\to S$ be a family of curves with Cohen-Macaulay fibres\nequidimensional of dimension $1$ (Lemma \\ref{lemma-CM-1-curves}).\nThen $\\omega_{X/S}^\\bullet = \\omega_{X/S}[1]$ where $\\omega_{X/S}$\nis a pseudo-coherent $\\mathcal{O}_X$-module flat over $S$ whose\nformation commutes with arbitrary base change.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"The relative dualizing sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6P","source_file":"moduli-curves.tex","source_line":1902,"source_end_line":1909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1902-L1909","statement_sha256":"35900dd992eda613e1d846d5e508cc5db48c13bb6f410c0c75b439df2f756582","origin":"The Stacks Project","memory_eligible":false,"source_rank":14590,"rank":14590,"depth":91,"x":2619.745,"y":1672.698,"cluster":"moduli-theory"},{"id":"stacks:0E6Q","tag":"0E6Q","title":"The relative dualizing sheaf · Definition 0E6Q","summary":"Let f : X → S be a family of curves with Cohen-Macaulay fibres equidimensional of dimension 1 (Lemma [Tag 0E1F]). Then the O_X-module ω_X/S = H^-1(ω_X/S^bullet) studied in Lemma [Tag 0E6P] is called the relative dualizing sheaf of f.","statement_latex":"Let $f : X \\to S$ be a family of curves with Cohen-Macaulay fibres\nequidimensional of dimension $1$ (Lemma \\ref{lemma-CM-1-curves}).\nThen the $\\mathcal{O}_X$-module\n$$\n\\omega_{X/S} = H^{-1}(\\omega_{X/S}^\\bullet)\n$$\nstudied in Lemma \\ref{lemma-CM-dualizing}\nis called the {\\it relative dualizing sheaf} of $f$.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"The relative dualizing sheaf","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6Q","source_file":"moduli-curves.tex","source_line":1949,"source_end_line":1959,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1949-L1959","statement_sha256":"88d717a6625faa006d674acec73ed0b7e168603ea58204e890435f52fbbacc71","origin":"The Stacks Project","memory_eligible":false,"source_rank":14591,"rank":14591,"depth":92,"x":2318.143,"y":1625.744,"cluster":"moduli-theory"},{"id":"stacks:0E6R","tag":"0E6R","title":"The relative dualizing sheaf · Lemma 0E6R","summary":"Let X → S be a family of curves with Gorenstein fibres equidimensional of dimension 1 (Lemma [Tag 0E6M]). Then the relative dualizing sheaf ω_X/S is an invertible O_X-module whose formation commutes with arbitrary base change.","statement_latex":"Let $X \\to S$ be a family of curves with Gorenstein fibres\nequidimensional of dimension $1$ (Lemma \\ref{lemma-gorenstein-1-curves}).\nThen the relative dualizing sheaf $\\omega_{X/S}$ is an\ninvertible $\\mathcal{O}_X$-module whose\nformation commutes with arbitrary base change.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"The relative dualizing sheaf","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6R","source_file":"moduli-curves.tex","source_line":1984,"source_end_line":1991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L1984-L1991","statement_sha256":"06d818fc9884caf6722879dda30daf69e53d27d5f1fc6d9e252a95df8a1983a8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14592,"rank":14592,"depth":92,"x":2578.861,"y":1488.937,"cluster":"moduli-theory"},{"id":"stacks:0E6T","tag":"0E6T","title":"Prestable curves · Definition 0E6T","summary":"Let f : X → S be a family of curves. We say f is a prestable family of curves if • f is at-worst-nodal of relative dimension 1, and • f_*O_X = O_S and this holds after any base change_X = O_S because the Stein factorization of f is étale in this case, see More on Morphisms of Spaces, Lemma [Tag 0E0D]. The condition may also be replaced by asking the geometric fibres to be connected, see Lemma [Tag 0E6L]..","statement_latex":"Let $f : X \\to S$ be a family of curves. We say $f$ is a\n{\\it prestable family of curves} if\n\\begin{enumerate}\n\\item $f$ is at-worst-nodal of relative dimension $1$, and\n\\item $f_*\\mathcal{O}_X = \\mathcal{O}_S$ and this holds after\nany base change\\footnote{In fact, it suffices to require\n$f_*\\mathcal{O}_X = \\mathcal{O}_S$ because the Stein factorization\nof $f$ is \\'etale in this case, see\nMore on Morphisms of Spaces, Lemma\n\\ref{spaces-more-morphisms-lemma-stein-factorization-etale}.\nThe condition may also be replaced by asking the geometric\nfibres to be connected, see Lemma \\ref{lemma-geomredcon-in-h0-1}.}.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Prestable curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6T","source_file":"moduli-curves.tex","source_line":2017,"source_end_line":2032,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2017-L2032","statement_sha256":"15abfbe1af813c2d308d953af236e7bcbd88623f4f69ea75dd8243bc3b7e8115","origin":"The Stacks Project","memory_eligible":false,"source_rank":14593,"rank":14593,"depth":77,"x":2496.451,"y":1738.288,"cluster":"moduli-theory"},{"id":"stacks:0E6U","tag":"0E6U","title":"Prestable curves · Lemma 0E6U","summary":"There exist an open substack Curvesstack^prestable ⊂ Curvesstack such that • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^prestable, • X → S is a prestable family of curves, • given X a scheme proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^prestable, • the singularities of X are…","statement_latex":"There exist an open substack $\\Curvesstack^{prestable} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{prestable}$,\n\\item $X \\to S$ is a prestable family of curves,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$\nfactors through $\\Curvesstack^{prestable}$,\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\nand $k = H^0(X, \\mathcal{O}_X)$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Prestable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6U","source_file":"moduli-curves.tex","source_line":2053,"source_end_line":2073,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2053-L2073","statement_sha256":"41053ee6d42f905537af1ddd43eed4e99c270757c8afad11088a6a9eacdc8864","origin":"The Stacks Project","memory_eligible":false,"source_rank":14594,"rank":14594,"depth":71,"x":2356.398,"y":1507.165,"cluster":"moduli-theory"},{"id":"stacks:0E6V","tag":"0E6V","title":"Prestable curves · Lemma 0E6V","summary":"There is a decomposition into open and closed substacks Curvesstack^prestable = coprod_g ≥ 0 Curvesstack^prestable_g where each Curvesstack^prestable_g is characterized as follows: • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^prestable_g, • X → S is a prestable family of curves and R^1f_*O_X is a locally free O_S-module of rank g, • given X a scheme proper over a field k with…","statement_latex":"There is a decomposition into open and closed substacks\n$$\n\\Curvesstack^{prestable} = \\coprod\\nolimits_{g \\geq 0}\n\\Curvesstack^{prestable}_g\n$$\nwhere each $\\Curvesstack^{prestable}_g$ is characterized as follows:\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{prestable}_g$,\n\\item $X \\to S$ is a prestable family of curves and\n$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$\nfactors through $\\Curvesstack^{prestable}_g$,\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\n$k = H^0(X, \\mathcal{O}_X)$, and the genus of $X$ is $g$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Prestable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6V","source_file":"moduli-curves.tex","source_line":2099,"source_end_line":2124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2099-L2124","statement_sha256":"669ab3e109209ec52743e95733ba88743eb9994989b93f1f9590f24cddf483d8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14595,"rank":14595,"depth":73,"x":2646.15,"y":1598.317,"cluster":"moduli-theory"},{"id":"stacks:0E6W","tag":"0E6W","title":"Prestable curves · Lemma 0E6W","summary":"The morphisms Curvesstack^prestable → Spec(Z) and Curvesstack^prestable_g → Spec(Z) are smooth.","statement_latex":"The morphisms\n$\\Curvesstack^{prestable} \\to \\Spec(\\mathbf{Z})$ and\n$\\Curvesstack^{prestable}_g \\to \\Spec(\\mathbf{Z})$ are\nsmooth.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Prestable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6W","source_file":"moduli-curves.tex","source_line":2133,"source_end_line":2139,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2133-L2139","statement_sha256":"56f09a6b55b6c9f20bed5da499587ccf1854aac4f91cf4042f083b12d20142b0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14596,"rank":14596,"depth":94,"x":2358.581,"y":1695.721,"cluster":"moduli-theory"},{"id":"stacks:0E6Y","tag":"0E6Y","title":"Semistable curves · Lemma 0E6Y","summary":"Let f : X → S be a prestable family of curves of genus g ≥ 1. Let s ∈ S be a point of the base scheme. Let m ≥ 2. The following are equivalent • X_s does not have a rational tail (Algebraic Curves, Example [Tag 0E3H]), and • f^*f_*ω_X/S^⊗ m → ω_X/S^⊗ m, is surjective over f^-1(U) for some s ∈ U ⊂ S open.","statement_latex":"Let $f : X \\to S$ be a prestable family of curves of genus $g \\geq 1$.\nLet $s \\in S$ be a point of the base scheme. Let $m \\geq 2$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X_s$ does not have a rational tail\n(Algebraic Curves, Example \\ref{curves-example-rational-tail}), and\n\\item $f^*f_*\\omega_{X/S}^{\\otimes m} \\to \\omega_{X/S}^{\\otimes m}$,\nis surjective over $f^{-1}(U)$ for some $s \\in U \\subset S$ open.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Semistable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6Y","source_file":"moduli-curves.tex","source_line":2156,"source_end_line":2167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2156-L2167","statement_sha256":"5f438bd8dfe0db1b8bb72f30f6c175c525c8fafe5951889ebd4a141a8adcec6c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14597,"rank":14597,"depth":72,"x":2492.583,"y":1460.227,"cluster":"moduli-theory"},{"id":"stacks:0E6Z","tag":"0E6Z","title":"Semistable curves · Definition 0E6Z","summary":"Let f : X → S be a family of curves. We say f is a semistable family of curves if • X → S is a prestable family of curves, and • X_s has genus ≥ 1 and does not have a rational tail for all s ∈ S.","statement_latex":"Let $f : X \\to S$ be a family of curves.\nWe say $f$ is a {\\it semistable family of curves} if\n\\begin{enumerate}\n\\item $X \\to S$ is a prestable family of curves, and\n\\item $X_s$ has genus $\\geq 1$ and\ndoes not have a rational tail for all $s \\in S$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Semistable curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E6Z","source_file":"moduli-curves.tex","source_line":2260,"source_end_line":2269,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2260-L2269","statement_sha256":"eaa24baa1c85a1c8cbf35b9ad9e0d3c45054b2065607694cb1c4f514e1f62b01","origin":"The Stacks Project","memory_eligible":false,"source_rank":14598,"rank":14598,"depth":0,"x":2583.34,"y":1710.438,"cluster":"moduli-theory"},{"id":"stacks:0E70","tag":"0E70","title":"Semistable curves · Lemma 0E70","summary":"There exist an open substack Curvesstack^semistable ⊂ Curvesstack such that • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^semistable, • X → S is a semistable family of curves, • given X a scheme proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^semistable, • the singularities of X are…","statement_latex":"There exist an open substack $\\Curvesstack^{semistable} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{semistable}$,\n\\item $X \\to S$ is a semistable family of curves,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$\nfactors through $\\Curvesstack^{semistable}$,\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\n$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $\\geq 1$, and\n$X$ has no rational tails,\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\n$k = H^0(X, \\mathcal{O}_X)$, and $\\omega_{X_s}^{\\otimes m}$ is\nglobally generated for $m \\geq 2$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Semistable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E70","source_file":"moduli-curves.tex","source_line":2292,"source_end_line":2316,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2292-L2316","statement_sha256":"bd7e210c784acd5773b62eec46ed8ecc641fe9696a16a881d9d967c9d93b20b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14599,"rank":14599,"depth":73,"x":2314.643,"y":1577.154,"cluster":"moduli-theory"},{"id":"stacks:0E71","tag":"0E71","title":"Semistable curves · Lemma 0E71","summary":"There is a decomposition into open and closed substacks Curvesstack^semistable = coprod_g ≥ 1 Curvesstack^semistable_g where each Curvesstack^semistable_g is characterized as follows: • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^semistable_g, • X → S is a semistable family of curves and R^1f_*O_X is a locally free O_S-module of rank g, • given X a scheme proper over a field k with…","statement_latex":"There is a decomposition into open and closed substacks\n$$\n\\Curvesstack^{semistable} = \\coprod\\nolimits_{g \\geq 1}\n\\Curvesstack^{semistable}_g\n$$\nwhere each $\\Curvesstack^{semistable}_g$ is characterized as follows:\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{semistable}_g$,\n\\item $X \\to S$ is a semistable family of curves and\n$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$\nfactors through $\\Curvesstack^{semistable}_g$,\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\n$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and $X$\nhas no rational tail,\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\n$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and\n$\\omega_{X_s}^{\\otimes m}$ is globally generated for $m \\geq 2$.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Semistable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E71","source_file":"moduli-curves.tex","source_line":2335,"source_end_line":2364,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2335-L2364","statement_sha256":"45fa3f1e49767016a3a7de02dd32fdc7c92f0a5085b916858f4c3f1628556362","origin":"The Stacks Project","memory_eligible":false,"source_rank":14600,"rank":14600,"depth":74,"x":2620.596,"y":1522.858,"cluster":"moduli-theory"},{"id":"stacks:0E72","tag":"0E72","title":"Semistable curves · Lemma 0E72","summary":"The morphisms Curvesstack^semistable → Spec(Z) and Curvesstack^semistable_g → Spec(Z) are smooth.","statement_latex":"The morphisms\n$\\Curvesstack^{semistable} \\to \\Spec(\\mathbf{Z})$ and\n$\\Curvesstack^{semistable}_g \\to \\Spec(\\mathbf{Z})$\nare smooth.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Semistable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E72","source_file":"moduli-curves.tex","source_line":2371,"source_end_line":2377,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2371-L2377","statement_sha256":"fd89419040eb56442b1d41330ab24f99b4269e682394a2a5b720b73764e37306","origin":"The Stacks Project","memory_eligible":false,"source_rank":14601,"rank":14601,"depth":94,"x":2438.276,"y":1736.944,"cluster":"moduli-theory"},{"id":"stacks:0E74","tag":"0E74","title":"Stable curves · Lemma 0E74","summary":"Let f : X → S be a prestable family of curves of genus g ≥ 2. Let s ∈ S be a point of the base scheme. The following are equivalent • X_s does not have a rational tail and does not have a rational bridge (Algebraic Curves, Examples [Tag 0E3H] and [Tag 0E3M]), and • ω_X/S is ample on f^-1(U) for some s ∈ U ⊂ S open.","statement_latex":"Let $f : X \\to S$ be a prestable family of curves of genus $g \\geq 2$.\nLet $s \\in S$ be a point of the base scheme.\nThe following are equivalent\n\\begin{enumerate}\n\\item $X_s$ does not have a rational tail and does not have a\nrational bridge\n(Algebraic Curves, Examples\n\\ref{curves-example-rational-tail} and\n\\ref{curves-example-rational-bridge}), and\n\\item $\\omega_{X/S}$ is ample on $f^{-1}(U)$ for some $s \\in U \\subset S$ open.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E74","source_file":"moduli-curves.tex","source_line":2397,"source_end_line":2410,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2397-L2410","statement_sha256":"dfbbea810a8a362cebe03731b1f6d8b9ed769de2aa191bff99f92b1e519e4705","origin":"The Stacks Project","memory_eligible":false,"source_rank":14602,"rank":14602,"depth":75,"x":2400.476,"y":1475.087,"cluster":"moduli-theory"},{"id":"stacks:0E75","tag":"0E75","title":"Stable curves · Definition 0E75","summary":"Let f : X → S be a family of curves. We say f is a stable family of curves if • X → S is a prestable family of curves, and • X_s has genus ≥ 2 and does not have a rational tails or bridges for all s ∈ S.","statement_latex":"Let $f : X \\to S$ be a family of curves.\nWe say $f$ is a {\\it stable family of curves} if\n\\begin{enumerate}\n\\item $X \\to S$ is a prestable family of curves, and\n\\item $X_s$ has genus $\\geq 2$ and does not have a rational tails\nor bridges for all $s \\in S$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E75","source_file":"moduli-curves.tex","source_line":2431,"source_end_line":2440,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2431-L2440","statement_sha256":"e58f16de2561214756046c7d831f7dec354f25ea47d4dbb90a9ae8b25b8cd477","origin":"The Stacks Project","memory_eligible":false,"source_rank":14603,"rank":14603,"depth":0,"x":2639.418,"y":1647.082,"cluster":"moduli-theory"},{"id":"stacks:0E76","tag":"0E76","title":"Stable curves · Lemma 0E76","summary":"There exist an open substack Curvesstack^stable ⊂ Curvesstack such that • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through Curvesstack^stable, • X → S is a stable family of curves, • given X a scheme proper over a field k with dim(X) ≤ 1 the following are equivalent • the classifying morphism Spec(k) → Curvesstack factors through Curvesstack^stable, • the singularities of X are at-worst-nodal,…","statement_latex":"There exist an open substack $\\Curvesstack^{stable} \\subset \\Curvesstack$\nsuch that\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\Curvesstack^{stable}$,\n\\item $X \\to S$ is a stable family of curves,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$\nfactors through $\\Curvesstack^{stable}$,\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\n$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $\\geq 2$, and\n$X$ has no rational tails or bridges,\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\n$k = H^0(X, \\mathcal{O}_X)$, and $\\omega_{X_s}$ is ample.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E76","source_file":"moduli-curves.tex","source_line":2461,"source_end_line":2484,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2461-L2484","statement_sha256":"ecf5134f5c9ebf2564d4f1b30c38cf8ea40c1cabb7b8e3d3b24b61b0d825d0a6","origin":"The Stacks Project","memory_eligible":false,"source_rank":14604,"rank":14604,"depth":76,"x":2324.267,"y":1655.854,"cluster":"moduli-theory"},{"id":"stacks:0E77","tag":"0E77","title":"Stable curves · Definition 0E77","summary":"[DM] We denote overlineM and we name the moduli stack of stable curves the algebraic stack Curvesstack^stable parametrizing stable families of curves introduced in Lemma [Tag 0E76]. For g ≥ 2 we denote overlineM_g and we name the moduli stack of stable curves of genus g the algebraic stack introduced in Lemma [Tag 0E78].","statement_latex":"\\begin{reference}\n\\cite{DM}\n\\end{reference}\nWe denote $\\overline{\\mathcal{M}}$ and we name the\n{\\it moduli stack of stable curves} the algebraic stack\n$\\Curvesstack^{stable}$ parametrizing stable families of curves\nintroduced in Lemma \\ref{lemma-stable-curves}.\nFor $g \\geq 2$ we denote $\\overline{\\mathcal{M}}_g$ and we name the\n{\\it moduli stack of stable curves of genus $g$}\nthe algebraic stack introduced in Lemma \\ref{lemma-stable-one-piece-per-genus}.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable curves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E77","source_file":"moduli-curves.tex","source_line":2501,"source_end_line":2513,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2501-L2513","statement_sha256":"98bf50e539514001ab1fa860c32dc2dd83d5e490670883a7151a708734b5aea3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14605,"rank":14605,"depth":77,"x":2550.061,"y":1470.183,"cluster":"moduli-theory"},{"id":"stacks:0E78","tag":"0E78","title":"Stable curves · Lemma 0E78","summary":"There is a decomposition into open and closed substacks overlineM = coprod_g ≥ 2 overlineM_g where each overlineM_g is characterized as follows: • given a family of curves f : X → S the following are equivalent • the classifying morphism S → Curvesstack factors through overlineM_g, • X → S is a stable family of curves and R^1f_*O_X is a locally free O_S-module of rank g, • given X a scheme proper over a field k with dim(X) ≤ 1 the following are equivalent • the…","statement_latex":"There is a decomposition into open and closed substacks\n$$\n\\overline{\\mathcal{M}} = \\coprod\\nolimits_{g \\geq 2} \\overline{\\mathcal{M}}_g\n$$\nwhere each $\\overline{\\mathcal{M}}_g$ is characterized as follows:\n\\begin{enumerate}\n\\item given a family of curves $f : X \\to S$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $S \\to \\Curvesstack$ factors\nthrough $\\overline{\\mathcal{M}}_g$,\n\\item $X \\to S$ is a stable family of curves and\n$R^1f_*\\mathcal{O}_X$ is a locally free $\\mathcal{O}_S$-module of rank $g$,\n\\end{enumerate}\n\\item given $X$ a scheme proper over a field $k$ with\n$\\dim(X) \\leq 1$ the following are equivalent\n\\begin{enumerate}\n\\item the classifying morphism $\\Spec(k) \\to \\Curvesstack$\nfactors through $\\overline{\\mathcal{M}}_g$,\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\n$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and $X$\nhas no rational tails or bridges.\n\\item the singularities of $X$ are at-worst-nodal, $\\dim(X) = 1$,\n$k = H^0(X, \\mathcal{O}_X)$, the genus of $X$ is $g$, and\n$\\omega_{X_s}$ is ample.\n\\end{enumerate}\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E78","source_file":"moduli-curves.tex","source_line":2518,"source_end_line":2546,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2518-L2546","statement_sha256":"c2a89a80927b09a6a66c5723f73d2e25864ccdb11b8ee3fb4eef59394f0bb4ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":14606,"rank":14606,"depth":77,"x":2532.84,"y":1735.754,"cluster":"moduli-theory"},{"id":"stacks:0E79","tag":"0E79","title":"Stable curves · Lemma 0E79","summary":"The morphisms overlineM → Spec(Z) and overlineM_g → Spec(Z) are smooth.","statement_latex":"The morphisms\n$\\overline{\\mathcal{M}} \\to \\Spec(\\mathbf{Z})$ and\n$\\overline{\\mathcal{M}}_g \\to \\Spec(\\mathbf{Z})$\nare smooth.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E79","source_file":"moduli-curves.tex","source_line":2553,"source_end_line":2559,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2553-L2559","statement_sha256":"a47dd019e98d4a698ee2bc810db197dfdeae5763a4dbd97a15271ff686c1aa27","origin":"The Stacks Project","memory_eligible":false,"source_rank":14607,"rank":14607,"depth":94,"x":2331.567,"y":1529.738,"cluster":"moduli-theory"},{"id":"stacks:0E7A","tag":"0E7A","title":"Stable curves · Lemma 0E7A","summary":"The stacks overlineM and overlineM_g are open substacks of Curvesstack^DM. In particular, overlineM and overlineM_g are DM (Morphisms of Stacks, Definition [Tag 050D]) as well as Deligne-Mumford stacks (Algebraic Stacks, Definition [Tag 03YO]).","statement_latex":"The stacks $\\overline{\\mathcal{M}}$ and\n$\\overline{\\mathcal{M}}_g$\nare open substacks of $\\Curvesstack^{DM}$.\nIn particular, $\\overline{\\mathcal{M}}$ and\n$\\overline{\\mathcal{M}}_g$ are DM\n(Morphisms of Stacks, Definition\n\\ref{stacks-morphisms-definition-absolute-separated})\nas well as Deligne-Mumford stacks\n(Algebraic Stacks, Definition \\ref{algebraic-definition-deligne-mumford}).","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7A","source_file":"moduli-curves.tex","source_line":2567,"source_end_line":2578,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2567-L2578","statement_sha256":"2c91f4f665872d118764357f68978fe0b8701a5c122b97e56660b0cf00344d62","origin":"The Stacks Project","memory_eligible":false,"source_rank":14608,"rank":14608,"depth":91,"x":2646.289,"y":1567.52,"cluster":"moduli-theory"},{"id":"stacks:0E87","tag":"0E87","title":"Stable curves · Lemma 0E87","summary":"Let g ≥ 2. The inclusion |M_g| ⊂ |overlineM_g| is that of an open dense subset.","statement_latex":"Let $g \\geq 2$. The inclusion\n$$\n|\\mathcal{M}_g| \\subset |\\overline{\\mathcal{M}}_g|\n$$\nis that of an open dense subset.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E87","source_file":"moduli-curves.tex","source_line":2605,"source_end_line":2612,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2605-L2612","statement_sha256":"aeab6ad622783a1c47f0a1f98cad17e5710a6dcc672c61b15ac5a46b8144bb21","origin":"The Stacks Project","memory_eligible":false,"source_rank":14609,"rank":14609,"depth":94,"x":2383.307,"y":1718.543,"cluster":"moduli-theory"},{"id":"stacks:0E88","tag":"0E88","title":"Contraction morphisms · Lemma 0E88","summary":"Let S be a scheme and s ∈ S a point. Let f : X → S and g : Y → S be families of curves. Let c : X → Y be a morphism over S. If c_s, *O_X_s = O_Y_s and R^1c_s, *O_X_s = 0, then after replacing S by an open neighbourhood of s we have O_Y = c_*O_X and R^1c_*O_X = 0 and this remains true after base change by any morphism S' → S.","statement_latex":"Let $S$ be a scheme and $s \\in S$ a point.\nLet $f : X \\to S$ and $g : Y \\to S$ be families of curves.\nLet $c : X \\to Y$ be a morphism over $S$. If\n$c_{s, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_s}$ and\n$R^1c_{s, *}\\mathcal{O}_{X_s} = 0$, then\nafter replacing $S$ by an open neighbourhood of $s$\nwe have $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$\nand this remains true after base change by any morphism $S' \\to S$.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Contraction morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E88","source_file":"moduli-curves.tex","source_line":2647,"source_end_line":2657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2647-L2657","statement_sha256":"0f00a158aad6f7bc4181aa67f72c2c099d0cbe1189287b6da5b5b05356f30f9f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14610,"rank":14610,"depth":80,"x":2455.923,"y":1457.438,"cluster":"moduli-theory"},{"id":"stacks:0E89","tag":"0E89","title":"Contraction morphisms · Lemma 0E89","summary":"Let S be a scheme and s ∈ S a point. Let f : X → S and g_i : Y_i → S, i = 1, 2 be families of curves. Let c_i : X → Y_i be morphisms over S. Assume there is an isomorphism Y_1, s ≅ Y_2, s of fibres compatible with c_1, s and c_2, s. If c_1, s, *O_X_s = O_Y_1, s and R^1c_1, s, *O_X_s = 0, then there exist an open neighbourhood U of s and an isomorphism Y_1, U ≅ Y_2, U of families of curves over U compatible with the given isomorphism of fibres and with c_1 and c_2.","statement_latex":"Let $S$ be a scheme and $s \\in S$ a point.\nLet $f : X \\to S$ and $g_i : Y_i \\to S$, $i = 1, 2$ be families of curves.\nLet $c_i : X \\to Y_i$ be morphisms over $S$.\nAssume there is an isomorphism $Y_{1, s} \\cong Y_{2, s}$\nof fibres compatible with $c_{1, s}$ and $c_{2, s}$.\nIf $c_{1, s, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_{1, s}}$ and\n$R^1c_{1, s, *}\\mathcal{O}_{X_s} = 0$, then there exist an\nopen neighbourhood $U$ of $s$ and an isomorphism\n$Y_{1, U} \\cong Y_{2, U}$ of families of curves over $U$\ncompatible with the given isomorphism of fibres and with\n$c_1$ and $c_2$.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Contraction morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E89","source_file":"moduli-curves.tex","source_line":2682,"source_end_line":2695,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2682-L2695","statement_sha256":"a06d48fec0fb793198702cbeb0d7d8d6d10bee5f30f1306bffe9664d1b213578","origin":"The Stacks Project","memory_eligible":false,"source_rank":14611,"rank":14611,"depth":91,"x":2612.662,"y":1691.643,"cluster":"moduli-theory"},{"id":"stacks:0E7C","tag":"0E7C","title":"Contraction morphisms · Lemma 0E7C","summary":"Let f : X → S be a family of curves. Let s ∈ S be a point. Let h_0 : X_s → Y_0 be a morphism to a proper scheme Y_0 over kappa(s) such that h_0, *O_X_s = O_Y_0 and R^1h_0, *O_X_s = 0. Then there exist an elementary étale neighbourhood (U, u) → (S, s), a family of curves Y → U, and a morphism h : X_U → Y over U whose fibre in u is isomorphic to h_0.","statement_latex":"Let $f : X \\to S$ be a family of curves. Let $s \\in S$ be a point.\nLet $h_0 : X_s \\to Y_0$ be a morphism to a proper scheme $Y_0$ over $\\kappa(s)$\nsuch that $h_{0, *}\\mathcal{O}_{X_s} = \\mathcal{O}_{Y_0}$ and\n$R^1h_{0, *}\\mathcal{O}_{X_s} = 0$. Then there exist an elementary\n\\'etale neighbourhood $(U, u) \\to (S, s)$, a family of curves $Y \\to U$,\nand a morphism $h : X_U \\to Y$ over $U$ whose fibre in $u$\nis isomorphic to $h_0$.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Contraction morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E7C","source_file":"moduli-curves.tex","source_line":2769,"source_end_line":2778,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2769-L2778","statement_sha256":"c0f0e7143c06e5ef9b60b702c1c1251f226b78489af4de0cae78f486aaa72919","origin":"The Stacks Project","memory_eligible":false,"source_rank":14612,"rank":14612,"depth":91,"x":2308.141,"y":1607.715,"cluster":"moduli-theory"},{"id":"stacks:0E8A","tag":"0E8A","title":"Contraction morphisms · Lemma 0E8A","summary":"Let f : X → S be a prestable family of curves of genus g ≥ 2. There is a factorization X → Y → S of f where g : Y → S is a stable family of curves and c : X → Y has the following properties • O_Y = c_*O_X and R^1c_*O_X = 0 and this remains true after base change by any morphism S' → S, and • for any s ∈ S the morphism c_s : X_s → Y_s is the contraction of rational tails and bridges discussed in Algebraic Curves, Section [Tag 0E7N]. Moreover c : X → Y is unique up to…","statement_latex":"Let $f : X \\to S$ be a prestable family of curves of genus $g \\geq 2$.\nThere is a factorization $X \\to Y \\to S$ of $f$ where $g : Y \\to S$\nis a stable family of curves and $c : X \\to Y$ has the following\nproperties\n\\begin{enumerate}\n\\item $\\mathcal{O}_Y = c_*\\mathcal{O}_X$ and $R^1c_*\\mathcal{O}_X = 0$\nand this remains true after base change by any morphism $S' \\to S$, and\n\\item for any $s \\in S$ the morphism $c_s : X_s \\to Y_s$ is the\ncontraction of rational tails and bridges discussed in\nAlgebraic Curves, Section \\ref{curves-section-contracting-to-stable}.\n\\end{enumerate}\nMoreover $c : X \\to Y$ is unique up to unique isomorphism.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Contraction morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8A","source_file":"moduli-curves.tex","source_line":2909,"source_end_line":2923,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2909-L2923","statement_sha256":"a70227b561e83b27deca7a718e4c6fbf02468331f4eea1c5875c49d12c369a52","origin":"The Stacks Project","memory_eligible":false,"source_rank":14613,"rank":14613,"depth":92,"x":2600.76,"y":1496.591,"cluster":"moduli-theory"},{"id":"stacks:0E8B","tag":"0E8B","title":"Contraction morphisms · Lemma 0E8B","summary":"Let g ≥ 2. There is a morphism of algebraic stacks over Z stabilization : Curvesstack^prestable_g → overlineM_g which sends a prestable family of curves X → S of genus g to the stable family Y → S associated to it in Lemma [Tag 0E8A].","statement_latex":"Let $g \\geq 2$. There is a morphism of algebraic stacks over $\\mathbf{Z}$\n$$\nstabilization :\n\\Curvesstack^{prestable}_g\n\\longrightarrow\n\\overline{\\mathcal{M}}_g\n$$\nwhich sends a prestable family of curves $X \\to S$ of genus $g$\nto the stable family $Y \\to S$ associated to it in\nLemma \\ref{lemma-contract-prestable-to-stable}.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Contraction morphisms","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8B","source_file":"moduli-curves.tex","source_line":2977,"source_end_line":2989,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L2977-L2989","statement_sha256":"9d28c00716cbb9793d90419091480752d4b238701e257cf7c3e68933eb032914","origin":"The Stacks Project","memory_eligible":false,"source_rank":14614,"rank":14614,"depth":93,"x":2474.101,"y":1745.059,"cluster":"moduli-theory"},{"id":"stacks:0E8D","tag":"0E8D","title":"Stable reduction theorem · Lemma 0E8D","summary":"Let R be a discrete valuation ring with fraction field K. Let C be a smooth projective curve over K with K = H^0(C, O_C) having genus g ≥ 2. The following are equivalent • C has semistable reduction (Semistable Reduction, Definition [Tag 0CDH]), or • there is a stable family of curves over R with generic fibre C.","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$.\nLet $C$ be a smooth projective curve over $K$ with\n$K = H^0(C, \\mathcal{O}_C)$ having genus $g \\geq 2$.\nThe following are equivalent\n\\begin{enumerate}\n\\item $C$ has semistable reduction\n(Semistable Reduction, Definition \\ref{models-definition-semistable}), or\n\\item there is a stable family of curves over $R$ with generic fibre $C$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable reduction theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E8D","source_file":"moduli-curves.tex","source_line":3028,"source_end_line":3039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L3028-L3039","statement_sha256":"059178661af5bdd36810d0e8e3b1251fc135cd84bc8966a6ff50fdca5d59c5b2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14615,"rank":14615,"depth":93,"x":2367.478,"y":1489.472,"cluster":"moduli-theory"},{"id":"stacks:0E97","tag":"0E97","title":"Stable reduction theorem · Lemma 0E97","summary":"Let R be a discrete valuation ring with fraction field K. Let C be a smooth proper curve over K with K = H^0(C, O_C) and genus g. If X and X' are models of C (Semistable Reduction, Section [Tag 0C2R]) and X and X' are stable families of genus g curves over R, then there exists a unique isomorphism X → X' of models.","statement_latex":"Let $R$ be a discrete valuation ring with fraction field $K$.\nLet $C$ be a smooth proper curve over $K$\nwith $K = H^0(C, \\mathcal{O}_C)$ and genus $g$.\nIf $X$ and $X'$ are models of $C$\n(Semistable Reduction, Section \\ref{models-section-models})\nand $X$ and $X'$ are stable families of genus $g$ curves over $R$,\nthen there exists a unique isomorphism $X \\to X'$ of models.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable reduction theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E97","source_file":"moduli-curves.tex","source_line":3055,"source_end_line":3064,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L3055-L3064","statement_sha256":"c1dda84c8f8bc22069741e0c44a96d3d6197bb6e9432b79118700c99fa82dc12","origin":"The Stacks Project","memory_eligible":false,"source_rank":14616,"rank":14616,"depth":93,"x":2652.19,"y":1617.688,"cluster":"moduli-theory"},{"id":"stacks:0E98","tag":"0E98","title":"Stable reduction theorem · Theorem 0E98","summary":"[DM] Let R be a discrete valuation ring with fraction field K. Let C be a smooth projective curve over K with H^0(C, O_C) = K and genus g ≥ 2. Then • there exists an extension of discrete valuation rings R ⊂ R' inducing a finite separable extension of fraction fields K'/K and a stable family of curves Y → Spec(R') of genus g with Y_K' ≅ C_K' over K', and • there exists a finite separable extension L/K and a stable family of curves Y → Spec(A) of genus g where A ⊂ L is the…","statement_latex":"\\begin{reference}\n\\cite[Corollary 2.7]{DM}\n\\end{reference}\nLet $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a\nsmooth projective curve over $K$ with $H^0(C, \\mathcal{O}_C) = K$\nand genus $g \\geq 2$. Then\n\\begin{enumerate}\n\\item there exists an extension of discrete valuation rings $R \\subset R'$\ninducing a finite separable extension of fraction fields $K'/K$ and\na stable family of curves $Y \\to \\Spec(R')$ of genus $g$ with\n$Y_{K'} \\cong C_{K'}$ over $K'$, and\n\\item there exists a finite separable extension $L/K$ and a stable\nfamily of curves $Y \\to \\Spec(A)$ of genus $g$ where $A \\subset L$\nis the integral closure of $R$ in $L$ such that\n$Y_L \\cong C_L$ over $L$.\n\\end{enumerate}","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Stable reduction theorem","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E98","source_file":"moduli-curves.tex","source_line":3166,"source_end_line":3184,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L3166-L3184","statement_sha256":"b6d5e7ad36a885845df5db5ce89be87afcebd04ea7605dd040f038cf6fe8b7f3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14617,"rank":14617,"depth":94,"x":2338.531,"y":1684.826,"cluster":"moduli-theory"},{"id":"stacks:0E9A","tag":"0E9A","title":"Properties of the stack of stable curves · Lemma 0E9A","summary":"Let g ≥ 2. The stack overlineM_g is separated.","statement_latex":"Let $g \\geq 2$. The stack $\\overline{\\mathcal{M}}_g$ is separated.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Properties of the stack of stable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9A","source_file":"moduli-curves.tex","source_line":3229,"source_end_line":3232,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L3229-L3232","statement_sha256":"2f2f157a10d1941adc7eea4376e56027f9cda258d7004c2da3cfb35853e00b8b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14618,"rank":14618,"depth":95,"x":2516.172,"y":1456.901,"cluster":"moduli-theory"},{"id":"stacks:0E9B","tag":"0E9B","title":"Properties of the stack of stable curves · Lemma 0E9B","summary":"Let g ≥ 2. The stack overlineM_g is quasi-compact.","statement_latex":"Let $g \\geq 2$. The stack $\\overline{\\mathcal{M}}_g$ is quasi-compact.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Properties of the stack of stable curves","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9B","source_file":"moduli-curves.tex","source_line":3286,"source_end_line":3289,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L3286-L3289","statement_sha256":"de52c0e0f14c5303a982131d4cec4aaf0b0e633ce8ff4d56261ec32934ef3927","origin":"The Stacks Project","memory_eligible":false,"source_rank":14619,"rank":14619,"depth":92,"x":2568.574,"y":1726.286,"cluster":"moduli-theory"},{"id":"stacks:0E9C","tag":"0E9C","title":"Properties of the stack of stable curves · Theorem 0E9C","summary":"Let g ≥ 2. The algebraic stack overlineM_g is a Deligne-Mumford stack, proper and smooth over Spec(Z). Moreover, the locus M_g parametrizing smooth curves is a dense open substack.","statement_latex":"Let $g \\geq 2$. The algebraic stack $\\overline{\\mathcal{M}}_g$ is a\nDeligne-Mumford stack, proper and smooth over $\\Spec(\\mathbf{Z})$.\nMoreover, the locus $\\mathcal{M}_g$ parametrizing smooth curves\nis a dense open substack.","area":"Moduli Theory","chapter":"Moduli of Curves","chapter_id":"moduli-curves","section":"Properties of the stack of stable curves","kind":"theorem","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0E9C","source_file":"moduli-curves.tex","source_line":3342,"source_end_line":3348,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/moduli-curves.tex#L3342-L3348","statement_sha256":"680ad8aa915c6804c30566e3e5450fcfd8bf756dd5bf43ac793badc014553743","origin":"The Stacks Project","memory_eligible":false,"source_rank":14620,"rank":14620,"depth":96,"x":2312.811,"y":1557.055,"cluster":"moduli-theory"},{"id":"stacks:09ZK","tag":"09ZK","title":"Non-quasi-compact inverse limit of quasi-compact spaces · Lemma 09ZK","summary":"There exists an inverse system of quasi-compact topological spaces over N whose limit is not quasi-compact.","statement_latex":"There exists an inverse system of quasi-compact topological spaces\nover $\\mathbf{N}$ whose limit is not quasi-compact.","area":"Topology","chapter":"Examples","chapter_id":"examples","section":"Non-quasi-compact inverse limit of quasi-compact spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09ZK","source_file":"examples.tex","source_line":120,"source_end_line":124,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L120-L124","statement_sha256":"1c18b4ddab73e4bf14d62a8ef73f6274a35e26fe5890b888100c54d3b3da5ede","origin":"The Stacks Project","memory_eligible":false,"source_rank":14621,"rank":14621,"depth":0,"x":507.492,"y":235.195,"cluster":"topology"},{"id":"stacks:05JC","tag":"05JC","title":"Noncomplete completion · Lemma 05JC","summary":"There exists a local ring R and a maximal ideal m such that the completion R^wedge of R with respect to m has the following properties • R^wedge is local, but its maximal ideal is not equal to m R^wedge, • R^wedge is not a complete local ring, and • R^wedge is not m-adically complete as an R-module.","statement_latex":"There exists a local ring $R$ and a maximal ideal $\\mathfrak m$ such that\nthe completion $R^\\wedge$ of $R$ with respect to $\\mathfrak m$ has the\nfollowing properties\n\\begin{enumerate}\n\\item $R^\\wedge$ is local, but its maximal ideal is not equal to\n$\\mathfrak m R^\\wedge$,\n\\item $R^\\wedge$ is not a complete local ring, and\n\\item $R^\\wedge$ is not $\\mathfrak m$-adically complete as an $R$-module.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Noncomplete completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JC","source_file":"examples.tex","source_line":436,"source_end_line":447,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L436-L447","statement_sha256":"790f14767b90306458e3a119bc06fa462dd2ecb9a6a217f410c15728a24a4285","origin":"The Stacks Project","memory_eligible":false,"source_rank":14622,"rank":14622,"depth":0,"x":1874.014,"y":159.375,"cluster":"commutative-algebra"},{"id":"stacks:05JE","tag":"05JE","title":"Noncomplete quotient · Lemma 05JE","summary":"There exists a ring R complete with respect to a principal ideal I and a principal ideal J such that R/J is not I-adically complete.","statement_latex":"There exists a ring $R$ complete with respect to a principal ideal\n$I$ and a principal ideal $J$ such that $R/J$ is not $I$-adically\ncomplete.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Noncomplete quotient","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JE","source_file":"examples.tex","source_line":512,"source_end_line":517,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L512-L517","statement_sha256":"dc2ae92750875f3d37c60ccfe198d47445946e83f43336f7e98007813587d7cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14623,"rank":14623,"depth":0,"x":2193.835,"y":176.177,"cluster":"commutative-algebra"},{"id":"stacks:05JG","tag":"05JG","title":"Completion is not exact · Lemma 05JG","summary":"Completion is neither left nor right exact in general. Completion is not an exact functor in general; it is not even right exact in general. This holds even when I is finitely generated on the category of finitely presented modules.","statement_latex":"\\begin{slogan}\nCompletion is neither left nor right exact in general.\n\\end{slogan}\nCompletion is not an exact functor in general; it is not even\nright exact in general. This holds even when $I$ is finitely\ngenerated on the category of finitely presented modules.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Completion is not exact","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JG","source_file":"examples.tex","source_line":556,"source_end_line":564,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L556-L564","statement_sha256":"e8bff12b44612586459e13f1765544ace084c641646f0f602b4464af5d7dd6b1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14624,"rank":14624,"depth":0,"x":1944.399,"y":345.325,"cluster":"commutative-algebra"},{"id":"stacks:0HB3","tag":"0HB3","title":"The category of complete modules is not abelian · Lemma 0HB3","summary":"Let I ⊂ R be an ideal of a ring. Let M be an I-adically complete R-module and let N ⊂ M be a submodule. The following are equivalent • N is closed in M, • N = ⋂_n ≥ 1 (N + I^n M), • M/N is I-adically complete If I is finitely generated, these conditions imply that N is I-adically complete.","statement_latex":"Let $I \\subset R$ be an ideal of a ring. Let $M$ be an $I$-adically complete\n$R$-module and let $N \\subset M$ be a submodule. The following are equivalent\n\\begin{enumerate}\n\\item $N$ is closed in $M$,\n\\item $N = \\bigcap_{n \\geq 1} (N + I^n M)$,\n\\item $M/N$ is $I$-adically complete\n\\end{enumerate}\nIf $I$ is finitely generated, these conditions imply that $N$\nis $I$-adically complete.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"The category of complete modules is not abelian","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HB3","source_file":"examples.tex","source_line":590,"source_end_line":601,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L590-L601","statement_sha256":"0530f0021d6bc30cef714b5f4b58ea7918dcf118542385b71254d01575459101","origin":"The Stacks Project","memory_eligible":false,"source_rank":14625,"rank":14625,"depth":5,"x":1992.324,"y":78.963,"cluster":"commutative-algebra"},{"id":"stacks:07JR","tag":"07JR","title":"The category of complete modules is not abelian · Lemma 07JR","summary":"Let R be a ring and let I ⊂ R be a finitely generated ideal. The category of I-adically complete R-modules has kernels and cokernels but is not abelian in general even when R is Noetherian.","statement_latex":"Let $R$ be a ring and let $I \\subset R$ be a finitely generated ideal.\nThe category of $I$-adically complete $R$-modules has kernels and\ncokernels but is not abelian in general even when $R$ is Noetherian.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"The category of complete modules is not abelian","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JR","source_file":"examples.tex","source_line":644,"source_end_line":649,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L644-L649","statement_sha256":"591e654705affd77c6ac8ca562ad6526ea41395bc51e8f89910338fd9caf51db","origin":"The Stacks Project","memory_eligible":false,"source_rank":14626,"rank":14626,"depth":0,"x":2171.253,"y":302.651,"cluster":"commutative-algebra"},{"id":"stacks:0ARD","tag":"0ARD","title":"The category of derived complete modules · Lemma 0ARD","summary":"Let A be a ring and let I ⊂ A be an ideal. The category C of derived complete modules is abelian and the inclusion functor F : C → Mod_A is exact and commutes with arbitrary limits. If I is finitely generated, then C has arbitrary direct sums and colimits, but F does not commute with these in general. Finally, filtered colimits are not exact in C in general, hence C is not a Grothendieck abelian category.","statement_latex":"Let $A$ be a ring and let $I \\subset A$ be an ideal.\nThe category $\\mathcal{C}$ of derived complete modules\nis abelian and the inclusion functor $F : \\mathcal{C} \\to \\text{Mod}_A$\nis exact and commutes with arbitrary limits.\nIf $I$ is finitely generated, then $\\mathcal{C}$ has\narbitrary direct sums and colimits, but $F$ does not commute with these\nin general. Finally, filtered colimits are not exact in $\\mathcal{C}$\nin general, hence $\\mathcal{C}$ is not a Grothendieck abelian category.","area":"Derived Categories","chapter":"Examples","chapter_id":"examples","section":"The category of derived complete modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARD","source_file":"examples.tex","source_line":714,"source_end_line":724,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L714-L724","statement_sha256":"cd4ce257fbb2288589824e036bd24c1c6d22f7e132fefad8f72502f7a70251b5","origin":"The Stacks Project","memory_eligible":false,"source_rank":14627,"rank":14627,"depth":1,"x":60.476,"y":647.924,"cluster":"derived-categories"},{"id":"stacks:0AL9","tag":"0AL9","title":"Nonflat completions · Lemma 0AL9","summary":"Let R be a ring. Let M be an R-module which is countable. Then M is a finite R-module if and only if M ⊗_R R^N → M^N is surjective.","statement_latex":"Let $R$ be a ring. Let $M$ be an $R$-module which is countable.\nThen $M$ is a finite $R$-module if and only if\n$M \\otimes_R R^\\mathbf{N} \\to M^\\mathbf{N}$ is surjective.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Nonflat completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AL9","source_file":"examples.tex","source_line":745,"source_end_line":750,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L745-L750","statement_sha256":"e6bfdc98915064f21ec1262153bbe2901effa41b7ab78b46a3f63018ae85b09c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14628,"rank":14628,"depth":1,"x":1859.313,"y":239.212,"cluster":"commutative-algebra"},{"id":"stacks:0ALA","tag":"0ALA","title":"Nonflat completions · Lemma 0ALA","summary":"Let R be a countable ring. Let M be a countable R-module. Then M is finitely presented if and only if the canonical map M ⊗_R R^N → M^N is an isomorphism.","statement_latex":"Let $R$ be a countable ring. Let $M$ be a countable $R$-module. Then $M$\nis finitely presented if and only if the canonical map\n$M \\otimes_R R^\\mathbf{N} \\to M^\\mathbf{N}$ is an isomorphism.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Nonflat completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALA","source_file":"examples.tex","source_line":762,"source_end_line":767,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L762-L767","statement_sha256":"de28653f49b2c105f2deaa7b8911db53acd32c80078300fb0567e6a5891c1f07","origin":"The Stacks Project","memory_eligible":false,"source_rank":14629,"rank":14629,"depth":2,"x":2140.453,"y":108.941,"cluster":"commutative-algebra"},{"id":"stacks:0ALB","tag":"0ALB","title":"Nonflat completions · Lemma 0ALB","summary":"Let R be a countable ring. Then R is coherent if and only if R^N is a flat R-module.","statement_latex":"Let $R$ be a countable ring. Then $R$ is coherent if and only if\n$R^\\mathbf{N}$ is a flat $R$-module.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Nonflat completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALB","source_file":"examples.tex","source_line":783,"source_end_line":787,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L783-L787","statement_sha256":"4042b8950f4ffd0d17f6b037f4d83231a26c57913ea3aa333e381e6b846e108b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14630,"rank":14630,"depth":4,"x":2037.867,"y":364.621,"cluster":"commutative-algebra"},{"id":"stacks:0ALC","tag":"0ALC","title":"Nonflat completions · Lemma 0ALC","summary":"There exists a ring such that the completion R[[x]] of R[x] at (x) is not flat over R and a fortiori not flat over R[x].","statement_latex":"There exists a ring such that the completion $R[[x]]$ of $R[x]$\nat $(x)$ is not flat over $R$ and a fortiori not flat over $R[x]$.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Nonflat completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALC","source_file":"examples.tex","source_line":815,"source_end_line":819,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L815-L819","statement_sha256":"dbc56e4d3f571b8d73e56f18ae970b127b0d73b93b1f2eaa88ab1966f0210003","origin":"The Stacks Project","memory_eligible":false,"source_rank":14631,"rank":14631,"depth":0,"x":1907.853,"y":117.785,"cluster":"commutative-algebra"},{"id":"stacks:0F1Y","tag":"0F1Y","title":"Nonflat completions · Lemma 0F1Y","summary":"Let R be a domain with fraction field K. If R[[x]] is flat over R[x], then R is normal if and only if R is completely normal (Algebra, Definition [Tag 00GW]).","statement_latex":"Let $R$ be a domain with fraction field $K$.\nIf $R[[x]]$ is flat over $R[x]$, then $R$ is normal if and only\nif $R$ is completely normal\n(Algebra, Definition \\ref{algebra-definition-almost-integral}).","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Nonflat completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1Y","source_file":"examples.tex","source_line":836,"source_end_line":842,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L836-L842","statement_sha256":"dd701c9d0fbf19e61a4985f992c33de4750022233b0f6362056fd0480396a745","origin":"The Stacks Project","memory_eligible":false,"source_rank":14632,"rank":14632,"depth":3,"x":2202.332,"y":226.065,"cluster":"commutative-algebra"},{"id":"stacks:0F1Z","tag":"0F1Z","title":"Nonflat completions · Lemma 0F1Z","summary":"If R is a valuation ring of dimension > 1, then R[[x]] is flat over R but not flat over R[x].","statement_latex":"If $R$ is a valuation ring of dimension $> 1$, then $R[[x]]$\nis flat over $R$ but not flat over $R[x]$.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Nonflat completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0F1Z","source_file":"examples.tex","source_line":868,"source_end_line":872,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L868-L872","statement_sha256":"d32e8f7570681306cbb2e199d6efa9471870b7f7ecd216563fe644fd070da852","origin":"The Stacks Project","memory_eligible":false,"source_rank":14633,"rank":14633,"depth":7,"x":1897.999,"y":313.347,"cluster":"commutative-algebra"},{"id":"stacks:0ALE","tag":"0ALE","title":"Nonflat completions · Lemma 0ALE","summary":"There exists a ring A complete with respect to a principal ideal I and an element f ∈ A such that the I-adic completion A_f^wedge of A_f is not flat over A.","statement_latex":"There exists a ring $A$ complete with respect to a principal ideal $I$\nand an element $f \\in A$ such that the $I$-adic completion\n$A_f^\\wedge$ of $A_f$ is not flat over $A$.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Nonflat completions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALE","source_file":"examples.tex","source_line":924,"source_end_line":929,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L924-L929","statement_sha256":"8ea9e0a484153bd6512e42bdc20d82f75a93f6095f5bcbf3028eb23453ad7caf","origin":"The Stacks Project","memory_eligible":false,"source_rank":14634,"rank":14634,"depth":0,"x":2052.276,"y":76.213,"cluster":"commutative-algebra"},{"id":"stacks:0ALG","tag":"0ALG","title":"Nonabelian category of quasi-coherent modules · Lemma 0ALG","summary":"The category of quasi-coherent(A), namely complete A-modules. modules on a formal algebraic space X is not abelian in general, even if X is a Noetherian affine formal algebraic space.","statement_latex":"The category of quasi-coherent\\footnote{With quasi-coherent modules\nas defined above. Due to how things are setup in the Stacks project,\nthis is really the correct definition; as seen above our definition\nagrees with what one would naively have defined to be quasi-coherent modules\non $\\text{Spf}(A)$, namely complete $A$-modules.}\nmodules on a formal algebraic space\n$X$ is not abelian in general, even if $X$ is a Noetherian affine\nformal algebraic space.","area":"Sheaves & Sites","chapter":"Examples","chapter_id":"examples","section":"Nonabelian category of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ALG","source_file":"examples.tex","source_line":986,"source_end_line":996,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L986-L996","statement_sha256":"b9b819cf4e0b2e591c4ef8ec84d7bb29a4a8a2eb2ea62f3c1447ed6146dc7148","origin":"The Stacks Project","memory_eligible":false,"source_rank":14635,"rank":14635,"depth":0,"x":1302.681,"y":241.723,"cluster":"sheaves-sites"},{"id":"stacks:0H7G","tag":"0H7G","title":"Nonsplit locally split sequence · Lemma 0H7G","summary":"There exists a ring R and a nonsplit sequence of modules which becomes split Zariski locally.","statement_latex":"There exists a ring $R$ and a nonsplit sequence of modules\nwhich becomes split Zariski locally.","area":"Sheaf Cohomology","chapter":"Examples","chapter_id":"examples","section":"Nonsplit locally split sequence","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0H7G","source_file":"examples.tex","source_line":1021,"source_end_line":1025,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L1021-L1025","statement_sha256":"e12ddff83b41aaa648845662a38620836a1238c2715caa6cb7731bb679ee45b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14636,"rank":14636,"depth":0,"x":956.787,"y":662.547,"cluster":"sheaf-cohomology"},{"id":"stacks:0640","tag":"0640","title":"Regular sequences and base change · Lemma 0640","summary":"There exists a local ring R and a regular sequence x, y, z (in the maximal ideal) such that there exists a nonzero element δ ∈ R/zR with xδ = yδ = 0.","statement_latex":"There exists a local ring $R$ and a regular sequence $x, y, z$\n(in the maximal ideal) such that there exists a nonzero element\n$\\delta \\in R/zR$ with $x\\delta = y\\delta = 0$.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Regular sequences and base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0640","source_file":"examples.tex","source_line":1115,"source_end_line":1120,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L1115-L1120","statement_sha256":"7c4d52c914604d49df22b319e66e7ab2613b05254fbdc375ecfdc1e1ab9b25db","origin":"The Stacks Project","memory_eligible":false,"source_rank":14637,"rank":14637,"depth":0,"x":2129.24,"y":338.71,"cluster":"commutative-algebra"},{"id":"stacks:0641","tag":"0641","title":"Regular sequences and base change · Lemma 0641","summary":"There exists a local homomorphism of local rings A → B and a regular sequence x, y in the maximal ideal of B such that B/(x, y) is flat over A, but such that the images overlinex, overliney of x, y in B/ m_AB do not form a regular sequence, nor even a Koszul-regular sequence.","statement_latex":"There exists a local homomorphism of local rings $A \\to B$\nand a regular sequence $x, y$ in the maximal ideal of $B$ such that\n$B/(x, y)$ is flat over $A$, but such that the images\n$\\overline{x}, \\overline{y}$ of $x, y$ in $B/\\mathfrak m_AB$ do not\nform a regular sequence, nor even a Koszul-regular sequence.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Regular sequences and base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0641","source_file":"examples.tex","source_line":1133,"source_end_line":1140,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L1133-L1140","statement_sha256":"686548595fa97adef2a1c0fb59d773079463f7ed0a847bb73d8640f9f3ff349f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14638,"rank":14638,"depth":3,"x":1861.296,"y":188.764,"cluster":"commutative-algebra"},{"id":"stacks:0GHI","tag":"0GHI","title":"Another local ring with nonreduced completion · Lemma 0GHI","summary":"There exists a local Noetherian 2-dimensional domain (B, m) complete with respect to a principal ideal I = (b) and an element f ∈ m, f not ∈ I such that the I-adic completion C = (B_f)^wedge of the principal localization B_f is nonreduced and even such that C_b = C[1/b] = (B_f)^wedge[1/b] is nonreduced.","statement_latex":"There exists a local Noetherian $2$-dimensional domain $(B, \\mathfrak m)$\ncomplete with respect to a principal ideal $I = (b)$ and an\nelement $f \\in \\mathfrak m$, $f \\not \\in I$ such that\nthe $I$-adic completion $C = (B_f)^\\wedge$ of the principal\nlocalization $B_f$ is nonreduced and even such that\n$C_b = C[1/b] = (B_f)^\\wedge[1/b]$ is nonreduced.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Another local ring with nonreduced completion","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GHI","source_file":"examples.tex","source_line":1389,"source_end_line":1397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L1389-L1397","statement_sha256":"6610297994f6b9dbf23900fa54d47e84c3d4b4ced3609e0662db607c76d4a8c0","origin":"The Stacks Project","memory_eligible":false,"source_rank":14639,"rank":14639,"depth":0,"x":2179.573,"y":147.279,"cluster":"commutative-algebra"},{"id":"stacks:0EEI","tag":"0EEI","title":"Dimension in Noetherian Jacobson rings · Lemma 0EEI","summary":"There exists a Jacobson, universally catenary, Noetherian domain B with maximal ideals m_1, m_2 such that dim(B_ m_1) = 1 and dim(B_ m_2) = 2.","statement_latex":"There exists a Jacobson, universally catenary, Noetherian domain $B$\nwith maximal ideals $\\mathfrak m_1, \\mathfrak m_2$ such that\n$\\dim(B_{\\mathfrak m_1}) = 1$ and $\\dim(B_{\\mathfrak m_2}) = 2$.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Dimension in Noetherian Jacobson rings","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EEI","source_file":"examples.tex","source_line":1701,"source_end_line":1706,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L1701-L1706","statement_sha256":"f497049bc3521f2f5a7b19ae8097a70f66a0872a5f03b2778feae0067543ccb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14640,"rank":14640,"depth":20,"x":1978.168,"y":358.546,"cluster":"commutative-algebra"},{"id":"stacks:0272","tag":"0272","title":"Non-quasi-affine variety with quasi-affine normalization · Lemma 0272","summary":"Let k be a field. There exists a variety X whose normalization is quasi-affine but which is itself not quasi-affine.","statement_latex":"Let $k$ be a field.\nThere exists a variety $X$ whose normalization is quasi-affine but\nwhich is itself not quasi-affine.","area":"Varieties & Curves","chapter":"Examples","chapter_id":"examples","section":"Non-quasi-affine variety with quasi-affine normalization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0272","source_file":"examples.tex","source_line":1853,"source_end_line":1858,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L1853-L1858","statement_sha256":"5a5bdceda250285f6d0951dfa2eae958c0d2c43875afa5d137311376d3919539","origin":"The Stacks Project","memory_eligible":false,"source_rank":14641,"rank":14641,"depth":0,"x":398.488,"y":1100.511,"cluster":"varieties-curves"},{"id":"stacks:0GZM","tag":"0GZM","title":"Images of locally closed subsets · Lemma 0GZM","summary":"There exists a morphism f : X → Y of finite presentation between affine schemes and a locally closed subset T of X such that f(T) is not a finite union of locally closed subsets of Y.","statement_latex":"There exists a morphism $f : X \\to Y$ of finite presentation\nbetween affine schemes and a locally closed subset $T$ of $X$\nsuch that $f(T)$ is not a finite union of locally closed subsets of $Y$.","area":"Schemes","chapter":"Examples","chapter_id":"examples","section":"Images of locally closed subsets","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GZM","source_file":"examples.tex","source_line":1949,"source_end_line":1954,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L1949-L1954","statement_sha256":"e712e5d131cfde53238f51d31ec36ce4ab4b8a65b508b879cf7df2906c420214","origin":"The Stacks Project","memory_eligible":false,"source_rank":14642,"rank":14642,"depth":0,"x":1700.55,"y":782.283,"cluster":"schemes"},{"id":"stacks:086H","tag":"086H","title":"Nonexistence of suitable opens · Lemma 086H","summary":"Nonexistence quasi-compact opens of affines: • There exist an affine scheme S and affine open U ⊂ S such that there is no quasi-compact open V ⊂ S with U ∩ V = ∅ and U ∪ V dense in S. • There exists an affine scheme S and a closed point s ∈ S such that S setminus (s) does not contain a quasi-compact dense open.","statement_latex":"Nonexistence quasi-compact opens of affines:\n\\begin{enumerate}\n\\item There exist an affine scheme $S$ and affine open $U \\subset S$\nsuch that there is no quasi-compact open $V \\subset S$ with\n$U \\cap V = \\emptyset$ and $U \\cup V$ dense in $S$.\n\\item There exists an affine scheme $S$ and a closed point $s \\in S$ such that\n$S \\setminus \\{s\\}$ does not contain a quasi-compact dense open.\n\\end{enumerate}","area":"Schemes","chapter":"Examples","chapter_id":"examples","section":"Nonexistence of suitable opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086H","source_file":"examples.tex","source_line":2059,"source_end_line":2069,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2059-L2069","statement_sha256":"923525b38ec70efdd6f3126a913545da7a59d32edb9cc906869ac0c0375b7c1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14643,"rank":14643,"depth":0,"x":1408.621,"y":672.972,"cluster":"schemes"},{"id":"stacks:086I","tag":"086I","title":"Nonexistence of suitable opens · Lemma 086I","summary":"There exists a quasi-compact and quasi-separated scheme X which does not contain a separated quasi-compact dense open.","statement_latex":"There exists a quasi-compact and quasi-separated scheme $X$ which does\nnot contain a separated quasi-compact dense open.","area":"Schemes","chapter":"Examples","chapter_id":"examples","section":"Nonexistence of suitable opens","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/086I","source_file":"examples.tex","source_line":2097,"source_end_line":2101,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2097-L2101","statement_sha256":"8f171e8133cb68547e7f524d19771ca458d2b9f0ca080d3d8e4e3f033cf872b3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14644,"rank":14644,"depth":0,"x":1712.205,"y":587.812,"cluster":"schemes"},{"id":"stacks:087I","tag":"087I","title":"Nonexistence of quasi-compact dense open subscheme · Lemma 087I","summary":"There exists a quasi-compact and quasi-separated algebraic space which does not contain a quasi-compact dense open subscheme.","statement_latex":"There exists a quasi-compact and quasi-separated algebraic space\nwhich does not contain a quasi-compact dense open subscheme.","area":"Schemes","chapter":"Examples","chapter_id":"examples","section":"Nonexistence of quasi-compact dense open subscheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087I","source_file":"examples.tex","source_line":2161,"source_end_line":2165,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2161-L2165","statement_sha256":"c9e4c18c9efbc397e51e67f50f87ed33ad4680d4fba330221856cb5a7d2e8d63","origin":"The Stacks Project","memory_eligible":false,"source_rank":14645,"rank":14645,"depth":0,"x":1556.617,"y":823.188,"cluster":"schemes"},{"id":"stacks:088W","tag":"088W","title":"Affines over algebraic spaces · Lemma 088W","summary":"There exists a finite type morphism of algebraic spaces Y → X with Y affine and X quasi-separated, such that there does not exist an immersion Y → A^n_X over X.","statement_latex":"There exists a finite type morphism of algebraic spaces $Y \\to X$\nwith $Y$ affine and $X$ quasi-separated, such that there does not exist\nan immersion $Y \\to \\mathbf{A}^n_X$ over $X$.","area":"Algebraic Spaces","chapter":"Examples","chapter_id":"examples","section":"Affines over algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/088W","source_file":"examples.tex","source_line":2221,"source_end_line":2226,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2221-L2226","statement_sha256":"1aafab8c138ad73fcc6a76742b9e2165ad2d7004f469b8a8f1c41417bbd7065d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14646,"rank":14646,"depth":0,"x":331.55,"y":1719.569,"cluster":"algebraic-spaces"},{"id":"stacks:078D","tag":"078D","title":"Pushforward of quasi-coherent modules · Lemma 078D","summary":"Schemes, Lemma [Tag 01LC] is sharp in the sense that one can neither drop the assumption of quasi-compactness nor the assumption of quasi-separatedness.","statement_latex":"Schemes, Lemma \\ref{schemes-lemma-push-forward-quasi-coherent}\nis sharp in the sense that one can neither drop the assumption\nof quasi-compactness nor the assumption of quasi-separatedness.","area":"Scheme Morphisms","chapter":"Examples","chapter_id":"examples","section":"Pushforward of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/078D","source_file":"examples.tex","source_line":2297,"source_end_line":2302,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2297-L2302","statement_sha256":"943a3d6dc8b53a81de81a1ee8e67f94fda92d4ec2af1c3abb11fe86cad47e047","origin":"The Stacks Project","memory_eligible":false,"source_rank":14647,"rank":14647,"depth":14,"x":2193.665,"y":731.818,"cluster":"scheme-morphisms"},{"id":"stacks:0CC0","tag":"0CC0","title":"A noninvertible ideal invertible in stalks · Lemma 0CC0","summary":"There exists a domain A and a nonzero ideal I ⊂ A such that I_ q ⊂ A_ q is a principal ideal for all primes q ⊂ A but I is not an invertible A-module.","statement_latex":"There exists a domain $A$ and a nonzero ideal $I \\subset A$\nsuch that $I_\\mathfrak q \\subset A_\\mathfrak q$ is a principal\nideal for all primes $\\mathfrak q \\subset A$ but $I$ is not an invertible\n$A$-module.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"A noninvertible ideal invertible in stalks","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CC0","source_file":"examples.tex","source_line":2382,"source_end_line":2388,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2382-L2388","statement_sha256":"2d25538bfea60c15b5ce633070f25dbebf383d586bdcf2364ae65fff7b6cf3cc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14648,"rank":14648,"depth":0,"x":1956.779,"y":88.379,"cluster":"commutative-algebra"},{"id":"stacks:05FY","tag":"05FY","title":"A finite flat module which is not projective · Lemma 05FY","summary":"Strange flat modules. • There exists a ring R and a finite flat R-module M which is not projective. • There exists a closed immersion which is flat but not open.","statement_latex":"Strange flat modules.\n\\begin{enumerate}\n\\item There exists a ring $R$ and a finite flat $R$-module $M$ which is\nnot projective.\n\\item There exists a closed immersion which is flat but not open.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"A finite flat module which is not projective","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05FY","source_file":"examples.tex","source_line":2421,"source_end_line":2429,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2421-L2429","statement_sha256":"5915b75075265d677ad1bc26097379cc1e06cf85a810047e2ec5e9fdf4f3fbf7","origin":"The Stacks Project","memory_eligible":false,"source_rank":14649,"rank":14649,"depth":0,"x":2189.897,"y":275.528,"cluster":"commutative-algebra"},{"id":"stacks:05WH","tag":"05WH","title":"A projective module which is not locally free · Lemma 05WH","summary":"Let R be a ring. Let I ⊂ R be an ideal generated by a countable collection of idempotents. Then I is projective as an R-module.","statement_latex":"Let $R$ be a ring. Let $I \\subset R$ be an ideal generated by\na countable collection of idempotents. Then $I$ is projective\nas an $R$-module.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"A projective module which is not locally free","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WH","source_file":"examples.tex","source_line":2444,"source_end_line":2449,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2444-L2449","statement_sha256":"a62d4a48c993556950200bcf4cb890b1bd9b16b3376910832b9b35b13b77e44d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14650,"rank":14650,"depth":2,"x":1867.38,"y":269.802,"cluster":"commutative-algebra"},{"id":"stacks:05WJ","tag":"05WJ","title":"A projective module which is not locally free · Lemma 05WJ","summary":"There exists a ring R and an ideal I such that I is projective as an R-module but not locally free as an R-module.","statement_latex":"There exists a ring $R$ and an ideal $I$ such that $I$ is projective as\nan $R$-module but not locally free as an $R$-module.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"A projective module which is not locally free","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WJ","source_file":"examples.tex","source_line":2490,"source_end_line":2494,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2490-L2494","statement_sha256":"1f4b22066a8ccf39f65872e2ee687cb45b4713a4ecdc3c6c610b5ef4e968cab8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14651,"rank":14651,"depth":0,"x":2109.893,"y":90.955,"cluster":"commutative-algebra"},{"id":"stacks:05WK","tag":"05WK","title":"A projective module which is not locally free · Lemma 05WK","summary":"Let K be a field. Let C_i, i = 1, …, n be smooth, projective, geometrically irreducible curves over K. Let P_i ∈ C_i(K) be a rational point and let Q_i ∈ C_i be a point such that [kappa(Q_i) : K] = 2. Then [P_1 × … × P_n] is nonzero in CH_0(U_1 ×_K … ×_K U_n) where U_i = C_i setminus (Q_i).","statement_latex":"Let $K$ be a field.\nLet $C_i$, $i = 1, \\ldots, n$ be smooth, projective, geometrically irreducible\ncurves over $K$. Let $P_i \\in C_i(K)$ be a rational point and\nlet $Q_i \\in C_i$ be a point such that $[\\kappa(Q_i) : K] = 2$.\nThen $[P_1 \\times \\ldots \\times P_n]$ is nonzero in\n$\\CH_0(U_1 \\times_K \\ldots \\times_K U_n)$ where $U_i = C_i \\setminus \\{Q_i\\}$.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"A projective module which is not locally free","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WK","source_file":"examples.tex","source_line":2511,"source_end_line":2519,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2511-L2519","statement_sha256":"9d3234af48e803f92be294caefef1dd019b8ef3887314cdd2fc0cbd68fdad079","origin":"The Stacks Project","memory_eligible":false,"source_rank":14652,"rank":14652,"depth":0,"x":2074.879,"y":360.541,"cluster":"commutative-algebra"},{"id":"stacks:05WL","tag":"05WL","title":"A projective module which is not locally free · Lemma 05WL","summary":"There exists a countable ring R and a projective module M which is a direct sum of countably many locally free rank 1 modules such that M is not locally free.","statement_latex":"There exists a countable ring $R$ and a projective module $M$\nwhich is a direct sum of countably many locally free rank $1$\nmodules such that $M$ is not locally free.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"A projective module which is not locally free","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05WL","source_file":"examples.tex","source_line":2611,"source_end_line":2616,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2611-L2616","statement_sha256":"61321f8ebe9cbd689999e96d402d5d7a5b7c14088efb8eada6ea7ecec920966d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14653,"rank":14653,"depth":0,"x":1883.834,"y":141.804,"cluster":"commutative-algebra"},{"id":"stacks:05G0","tag":"05G0","title":"Zero dimensional local ring with nonzero flat ideal · Lemma 05G0","summary":"Zero dimensional ring with flat ideal. There exists a local ring R with a unique prime ideal and a nonzero ideal I ⊂ R which is a flat R-module","statement_latex":"\\begin{slogan}\nZero dimensional ring with flat ideal.\n\\end{slogan}\nThere exists a local ring $R$ with a unique prime ideal\nand a nonzero ideal $I \\subset R$ which is a flat $R$-module","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Zero dimensional local ring with nonzero flat ideal","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05G0","source_file":"examples.tex","source_line":2661,"source_end_line":2668,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2661-L2668","statement_sha256":"be2ec256d02f7e31ac6b590c88841f807ea3692ba533b48745de71180f11137c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14654,"rank":14654,"depth":0,"x":2200.726,"y":194.713,"cluster":"commutative-algebra"},{"id":"stacks:06RI","tag":"06RI","title":"An epimorphism of zero-dimensional rings which is not surjective · Lemma 06RI","summary":"There exists an epimorphism of local rings of dimension 0 which is not a surjection.","statement_latex":"There exists an epimorphism of local rings of dimension $0$\nwhich is not a surjection.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"An epimorphism of zero-dimensional rings which is not surjective","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RI","source_file":"examples.tex","source_line":2707,"source_end_line":2711,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2707-L2711","statement_sha256":"6a16133aa3505222a71c455b66d74e9e55c0c4eb26a877ed6430f530f08bb67f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14655,"rank":14655,"depth":0,"x":1924.406,"y":335.562,"cluster":"commutative-algebra"},{"id":"stacks:05G2","tag":"05G2","title":"Finite type, not finitely presented, flat at prime · Lemma 05G2","summary":"There exists a local ring A, a finite type ring map A → B and a prime q lying over m_A such that B_ q is flat over A, and for any element g ∈ B, g not ∈ q the ring B_g is neither finitely presented over A nor flat over A.","statement_latex":"There exists a local ring $A$, a finite type ring map $A \\to B$ and a prime\n$\\mathfrak q$ lying over $\\mathfrak m_A$ such that $B_{\\mathfrak q}$ is flat\nover $A$, and for any element $g \\in B$, $g \\not \\in \\mathfrak q$\nthe ring $B_g$ is neither finitely presented over $A$ nor flat over $A$.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Finite type, not finitely presented, flat at prime","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05G2","source_file":"examples.tex","source_line":2812,"source_end_line":2818,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2812-L2818","statement_sha256":"446fbda24f5cc2a9f7a51edfd9c051c99ae83c6fe8e46d18246b0c598e6229ca","origin":"The Stacks Project","memory_eligible":false,"source_rank":14656,"rank":14656,"depth":0,"x":2014.926,"y":74.816,"cluster":"commutative-algebra"},{"id":"stacks:05LC","tag":"05LC","title":"Finite type, flat and not of finite presentation · Lemma 05LC","summary":"There exist examples of • a flat finite type ring map with geometrically irreducible complete intersection fibre rings which is not of finite presentation, • a flat finite type ring map with geometrically connected, geometrically reduced, dimension 1, complete intersection fibre rings which is not of finite presentation, • a proper flat morphism of schemes X → S each of whose fibres is isomorphic to either P^1_s or to the vanishing locus of X_1X_2 in P^2_s which is not of…","statement_latex":"There exist examples of\n\\begin{enumerate}\n\\item a flat finite type ring map with geometrically irreducible\ncomplete intersection fibre rings which is not of finite presentation,\n\\item a flat finite type ring map with geometrically connected,\ngeometrically reduced, dimension 1, complete intersection fibre rings\nwhich is not of finite presentation,\n\\item a proper flat morphism of schemes $X \\to S$ each of whose fibres\nis isomorphic to either $\\mathbf{P}^1_s$ or to the vanishing locus of\n$X_1X_2$ in $\\mathbf{P}^2_s$ which is not of finite presentation, and\n\\item a proper flat morphism of schemes $X \\to S$ each of whose\nfibres is isomorphic to either $\\mathbf{P}^1_s$ or $\\mathbf{P}^2_s$\nwhich is not of finite presentation.\n\\end{enumerate}","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Finite type, flat and not of finite presentation","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LC","source_file":"examples.tex","source_line":2875,"source_end_line":2891,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2875-L2891","statement_sha256":"4ca6815420d47458f7ada5d8a4d68c1816231557ec0a9564446f59016674c0dd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14657,"rank":14657,"depth":0,"x":2157.915,"y":318.539,"cluster":"commutative-algebra"},{"id":"stacks:05JI","tag":"05JI","title":"Topology of a finite type ring map · Lemma 05JI","summary":"There exists a local homomorphism A → B of local domains which is essentially of finite type and such that A/ m_A → B/ m_B is finite such that for every prime q not = m_B of B the ring map A → B/ q is not the localization of a quasi-finite ring map.","statement_latex":"There exists a local homomorphism $A \\to B$ of local domains which is\nessentially of finite type and such that $A/\\mathfrak m_A \\to B/\\mathfrak m_B$\nis finite such that for every prime\n$\\mathfrak q \\not = \\mathfrak m_B$ of $B$ the ring map\n$A \\to B/\\mathfrak q$ is not the localization of a quasi-finite ring map.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Topology of a finite type ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JI","source_file":"examples.tex","source_line":2943,"source_end_line":2950,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L2943-L2950","statement_sha256":"394891549cfaa95df17b5ca90acc11c3a7eab1181d81cef1cd7230f4a9eb55d1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14658,"rank":14658,"depth":0,"x":1856.372,"y":219.921,"cluster":"commutative-algebra"},{"id":"stacks:05JK","tag":"05JK","title":"Pure not universally pure · Lemma 05JK","summary":"There exists a morphism of affine schemes of finite presentation X → S and an O_X-module F of finite presentation such that F is pure relative to S, but not universally pure relative to S.","statement_latex":"There exists a morphism of affine schemes of finite presentation\n$X \\to S$ and an $\\mathcal{O}_X$-module $\\mathcal{F}$ of finite presentation\nsuch that $\\mathcal{F}$ is pure relative to $S$, but not universally\npure relative to $S$.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Pure not universally pure","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05JK","source_file":"examples.tex","source_line":3023,"source_end_line":3029,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3023-L3029","statement_sha256":"9f488db30ea60b800274c0b7bd46e674086f04acb6161e490d1ca6502fcfff1b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14659,"rank":14659,"depth":0,"x":2158.142,"y":121.502,"cluster":"commutative-algebra"},{"id":"stacks:057W","tag":"057W","title":"A formally smooth non-flat ring map · Lemma 057W","summary":"There exists a formally smooth ring map which is not flat.","statement_latex":"There exists a formally smooth ring map which is not flat.","area":"Advanced Algebra","chapter":"Examples","chapter_id":"examples","section":"A formally smooth non-flat ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/057W","source_file":"examples.tex","source_line":3091,"source_end_line":3094,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3091-L3094","statement_sha256":"cba65bc35b56497995b12988900158ce696772a6355c6ab6dc7cfb5ab840f98e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14660,"rank":14660,"depth":0,"x":853.86,"y":692.518,"cluster":"advanced-algebra"},{"id":"stacks:060I","tag":"060I","title":"A formally étale non-flat ring map · Lemma 060I","summary":"There exist formally étale nonflat ring maps.","statement_latex":"There exist formally \\'etale nonflat ring maps.","area":"Advanced Algebra","chapter":"Examples","chapter_id":"examples","section":"A formally étale non-flat ring map","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/060I","source_file":"examples.tex","source_line":3133,"source_end_line":3136,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3133-L3136","statement_sha256":"2f9603e00b6d7ceb616fd649e5f074808b75031c3497ab13fd34706044e8e821","origin":"The Stacks Project","memory_eligible":false,"source_rank":14661,"rank":14661,"depth":0,"x":541.662,"y":769.467,"cluster":"advanced-algebra"},{"id":"stacks:06E6","tag":"06E6","title":"A formally étale ring map with nontrivial cotangent complex · Lemma 06E6","summary":"There exists a formally étale surjective ring map A → B with L_B/A not equal to zero.","statement_latex":"There exists a formally \\'etale surjective ring map $A \\to B$\nwith $L_{B/A}$ not equal to zero.","area":"Deformation Theory","chapter":"Examples","chapter_id":"examples","section":"A formally étale ring map with nontrivial cotangent complex","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06E6","source_file":"examples.tex","source_line":3185,"source_end_line":3189,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3185-L3189","statement_sha256":"6b7d9b878b8131f80f7a5bb606764a0e93b76c073c4b122717a90af90ef47013","origin":"The Stacks Project","memory_eligible":false,"source_rank":14662,"rank":14662,"depth":0,"x":1579.55,"y":1746.678,"cluster":"deformation-theory"},{"id":"stacks:0G65","tag":"0G65","title":"Flat and formally unramified is not formally étale · Lemma 0G65","summary":"Let A = F_p[T] be the polynomial ring in one variable over F_p. Let A_perf denote the perfect closure of A. Then A → A_perf is flat and formally unramified, but not formally étale.","statement_latex":"Let $A = \\mathbb{F}_p[T]$ be the polynomial ring in one variable over\n$\\mathbb{F}_p$. Let $A_{perf}$  denote the perfect closure of $A$.\nThen $A \\rightarrow A_{perf}$ is flat and formally unramified,\nbut not formally \\'etale.","area":"Advanced Algebra","chapter":"Examples","chapter_id":"examples","section":"Flat and formally unramified is not formally étale","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G65","source_file":"examples.tex","source_line":3212,"source_end_line":3218,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3212-L3218","statement_sha256":"e71482bcbb6d4c3898f56c2d7a0dfd744c2c514fa3e87f79d80876bc232ae499","origin":"The Stacks Project","memory_eligible":false,"source_rank":14663,"rank":14663,"depth":48,"x":710.077,"y":535.46,"cluster":"advanced-algebra"},{"id":"stacks:0G66","tag":"0G66","title":"Flat and formally unramified is not formally étale · Lemma 0G66","summary":"Let (A, m, kappa) be a Noetherian local ring of prime characteristic p > 0 such that [kappa : kappa^p] < ∞. Then the canonical map A → A^wedge to the completion of A is flat and formally unramified. However, if A is regular but not excellent, then this map is not formally étale.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a Noetherian local ring of prime\ncharacteristic $p > 0$ such that $[\\kappa : \\kappa^p] < \\infty$.\nThen the  canonical map  $A \\to A^\\wedge$ to the completion of $A$\nis flat and formally unramified. However, if $A$ is regular but not\nexcellent, then this map is not formally \\'etale.","area":"Advanced Algebra","chapter":"Examples","chapter_id":"examples","section":"Flat and formally unramified is not formally étale","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G66","source_file":"examples.tex","source_line":3266,"source_end_line":3273,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3266-L3273","statement_sha256":"df4dfe6e2949ca41d8fa15e68f8e2e4ec476b740be2048a10b8fa2f7e3899b9c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14664,"rank":14664,"depth":6,"x":774.104,"y":803.71,"cluster":"advanced-algebra"},{"id":"stacks:0G68","tag":"0G68","title":"Flat and formally unramified is not formally étale · Lemma 0G68","summary":"Let (A, m, kappa) be a regular local ring of characteristic p > 0. Suppose [kappa : kappa^p] < ∞. Then A is excellent if and only if A → A^wedge is formally étale.","statement_latex":"Let $(A, \\mathfrak m, \\kappa)$ be a regular local ring of characteristic\n$p > 0$. Suppose $[\\kappa : \\kappa^p] < \\infty$. Then $A$ is excellent\nif and only if $A \\to A^\\wedge$ is formally \\'etale.","area":"Advanced Algebra","chapter":"Examples","chapter_id":"examples","section":"Flat and formally unramified is not formally étale","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0G68","source_file":"examples.tex","source_line":3365,"source_end_line":3370,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3365-L3370","statement_sha256":"5480c207cee275a23f171f91a774f560cd2e36798e4eae94d0367c2f56abc966","origin":"The Stacks Project","memory_eligible":false,"source_rank":14665,"rank":14665,"depth":49,"x":511.04,"y":642.156,"cluster":"advanced-algebra"},{"id":"stacks:04QL","tag":"04QL","title":"Ideals generated by sets of idempotents and localization · Lemma 04QL","summary":"There exists an affine scheme X = Spec(A) and a closed subscheme T ⊂ X such that T is Zariski locally on X cut out by ideals generated by idempotents, but T is not cut out by an ideal generated by idempotents.","statement_latex":"There exists an affine scheme $X = \\Spec(A)$ and a\nclosed subscheme $T \\subset X$ such that $T$ is Zariski locally\non $X$ cut out by ideals generated by idempotents, but\n$T$ is not cut out by an ideal generated by idempotents.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Ideals generated by sets of idempotents and localization","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04QL","source_file":"examples.tex","source_line":3458,"source_end_line":3464,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3458-L3464","statement_sha256":"eca8eee86452fb6a094f80d9d42423c552eafafb30261e5e8b0e7fe0ad3610fb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14666,"rank":14666,"depth":0,"x":2014.715,"y":365.395,"cluster":"commutative-algebra"},{"id":"stacks:09AP","tag":"09AP","title":"A ring map which identifies local rings which is not ind-étale · Lemma 09AP","summary":"There is a ring map A → B which identifies local rings but which is not ind-étale. A fortiori it is not ind-Zariski.","statement_latex":"There is a ring map $A \\to B$ which identifies local rings but\nwhich is not ind-\\'etale. A fortiori it is not ind-Zariski.","area":"Advanced Algebra","chapter":"Examples","chapter_id":"examples","section":"A ring map which identifies local rings which is not ind-étale","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09AP","source_file":"examples.tex","source_line":3496,"source_end_line":3500,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3496-L3500","statement_sha256":"1ac575af332384ee47ca743d1bb054b9dbe6b2618942935a6bd19855f9b64cf9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14667,"rank":14667,"depth":0,"x":835.099,"y":612.0,"cluster":"advanced-algebra"},{"id":"stacks:0274","tag":"0274","title":"Non flasque quasi-coherent sheaf associated to injective module · Lemma 0274","summary":"There exists an affine scheme X = Spec(A) and an injective A-module J such that widetildeJ is not a flasque sheaf on X. Even the restriction Γ(X, widetildeJ) → Γ(U, widetildeJ) with U a standard open need not be surjective.","statement_latex":"There exists an affine scheme $X = \\Spec(A)$ and an injective\n$A$-module $J$ such that $\\widetilde{J}$ is not a flasque sheaf on $X$.\nEven the restriction $\\Gamma(X, \\widetilde{J}) \\to \\Gamma(U, \\widetilde{J})$\nwith $U$ a standard open need not be surjective.","area":"Sheaf Cohomology","chapter":"Examples","chapter_id":"examples","section":"Non flasque quasi-coherent sheaf associated to injective module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0274","source_file":"examples.tex","source_line":3551,"source_end_line":3557,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3551-L3557","statement_sha256":"817ca89b39e88e645b8bfd50dbb2000bf11995b2d47a0b5ab089a27dce55e342","origin":"The Stacks Project","memory_eligible":false,"source_rank":14668,"rank":14668,"depth":0,"x":1271.884,"y":594.51,"cluster":"sheaf-cohomology"},{"id":"stacks:0CRZ","tag":"0CRZ","title":"Non flasque quasi-coherent sheaf associated to injective module · Lemma 0CRZ","summary":"There exists an affine scheme X = Spec(A) whose underlying topological space is Noetherian and an injective A-module I such that widetildeI has nonvanishing H^1 on some quasi-compact open U of X.","statement_latex":"There exists an affine scheme $X = \\Spec(A)$ whose underlying\ntopological space is Noetherian and an injective\n$A$-module $I$ such that $\\widetilde{I}$ has nonvanishing $H^1$\non some quasi-compact open $U$ of $X$.","area":"Sheaf Cohomology","chapter":"Examples","chapter_id":"examples","section":"Non flasque quasi-coherent sheaf associated to injective module","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CRZ","source_file":"examples.tex","source_line":3592,"source_end_line":3598,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3592-L3598","statement_sha256":"916fa41540ff48f669b7a8fcc4d9bdc217b4500607e6c34b975c93b502d29654","origin":"The Stacks Project","memory_eligible":false,"source_rank":14669,"rank":14669,"depth":0,"x":1094.095,"y":823.672,"cluster":"sheaf-cohomology"},{"id":"stacks:06E8","tag":"06E8","title":"A non-separated flat group scheme · Lemma 06E8","summary":"There exists a flat group scheme of finite type over the affine line which is not separated.","statement_latex":"There exists a flat group scheme of finite type over the affine line\nwhich is not separated.","area":"Groupoids & Quotients","chapter":"Examples","chapter_id":"examples","section":"A non-separated flat group scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06E8","source_file":"examples.tex","source_line":3641,"source_end_line":3645,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3641-L3645","statement_sha256":"ae1e7d9cf7486a89cd8fdc742959bdda47773a4a39760fce2847ca2f28e1d774","origin":"The Stacks Project","memory_eligible":false,"source_rank":14670,"rank":14670,"depth":0,"x":966.649,"y":1090.247,"cluster":"groupoids-quotients"},{"id":"stacks:08IX","tag":"08IX","title":"A non-separated flat group scheme · Lemma 08IX","summary":"There exists a flat group scheme of finite type over the infinite dimensional affine space which is not quasi-separated.","statement_latex":"There exists a flat group scheme of finite type over the infinite\ndimensional affine space which is not quasi-separated.","area":"Groupoids & Quotients","chapter":"Examples","chapter_id":"examples","section":"A non-separated flat group scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IX","source_file":"examples.tex","source_line":3651,"source_end_line":3655,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3651-L3655","statement_sha256":"392b8b24c603d06af313b86bc0d6527d297d2e74f6862df9327ffdf1c99407b8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14671,"rank":14671,"depth":0,"x":1290.714,"y":1083.91,"cluster":"groupoids-quotients"},{"id":"stacks:06RK","tag":"06RK","title":"A non-flat group scheme with flat identity component · Lemma 06RK","summary":"There exists a group scheme G over a base S whose identity component is flat over S but which is not flat over S.","statement_latex":"There exists a group scheme $G$ over a base $S$ whose identity\ncomponent is flat over $S$ but which is not flat over $S$.","area":"Groupoids & Quotients","chapter":"Examples","chapter_id":"examples","section":"A non-flat group scheme with flat identity component","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06RK","source_file":"examples.tex","source_line":3699,"source_end_line":3703,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3699-L3703","statement_sha256":"5106846bc11a957c13e6ba262c3c8422be48fd438b36bcae3988ae008864224a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14672,"rank":14672,"depth":0,"x":1056.483,"y":1272.761,"cluster":"groupoids-quotients"},{"id":"stacks:06EA","tag":"06EA","title":"A non-separated group algebraic space over a field · Lemma 06EA","summary":"There exists a group algebraic space of finite type over a field which is not separated (and not even quasi-separated or locally separated).","statement_latex":"There exists a group algebraic space of finite type over a field\nwhich is not separated (and not even quasi-separated or locally separated).","area":"Groupoids & Quotients","chapter":"Examples","chapter_id":"examples","section":"A non-separated group algebraic space over a field","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06EA","source_file":"examples.tex","source_line":3735,"source_end_line":3739,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3735-L3739","statement_sha256":"2487b40d1f2e7c36eca79b99dea30447c67db75828d7c208dc2fbe48804037c1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14673,"rank":14673,"depth":0,"x":1077.365,"y":1000.171,"cluster":"groupoids-quotients"},{"id":"stacks:06UK","tag":"06UK","title":"Specializations between points in fibre étale morphism · Lemma 06UK","summary":"There exists an étale morphism of algebraic spaces f : X → Y and a nontrivial specialization of points x leadsto x' in |X| with f(x) = f(x') in |Y|.","statement_latex":"There exists an \\'etale morphism of algebraic spaces $f : X \\to Y$\nand a nontrivial specialization of points $x \\leadsto x'$ in $|X|$ with\n$f(x) = f(x')$ in $|Y|$.","area":"Étale Geometry","chapter":"Examples","chapter_id":"examples","section":"Specializations between points in fibre étale morphism","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/06UK","source_file":"examples.tex","source_line":3803,"source_end_line":3808,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3803-L3808","statement_sha256":"35335ac0347b88df87b390e03f18ddf083783c161b4c4042ff509b9344260d11","origin":"The Stacks Project","memory_eligible":false,"source_rank":14674,"rank":14674,"depth":0,"x":1474.572,"y":1257.178,"cluster":"tale-geometry"},{"id":"stacks:077B","tag":"077B","title":"A torsor which is not an fppf torsor · Lemma 077B","summary":"Let S be a scheme. Let G be a group scheme over S. The stack G-Principal classifying principal homogeneous G-spaces (see Examples of Stacks, Subsection [Tag 04UN]) and the stack G-Torsors classifying fppf G-torsors (see Examples of Stacks, Subsection [Tag 04UR]) are not equivalent in general.","statement_latex":"Let $S$ be a scheme. Let $G$ be a group scheme over $S$.\nThe stack $G\\textit{-Principal}$ classifying principal homogeneous $G$-spaces\n(see Examples of Stacks, Subsection\n\\ref{examples-stacks-subsection-principal-homogeneous-spaces})\nand the stack $G\\textit{-Torsors}$ classifying fppf $G$-torsors\n(see Examples of Stacks, Subsection\n\\ref{examples-stacks-subsection-fppf-torsors})\nare not equivalent in general.","area":"Groupoids & Quotients","chapter":"Examples","chapter_id":"examples","section":"A torsor which is not an fppf torsor","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/077B","source_file":"examples.tex","source_line":3880,"source_end_line":3890,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3880-L3890","statement_sha256":"065ef65f3baf2b6bb6b77c0eb81dee5f288ade8756ad38d452331ad4eb97440b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14675,"rank":14675,"depth":0,"x":1281.494,"y":1213.356,"cluster":"groupoids-quotients"},{"id":"stacks:04AH","tag":"04AH","title":"Stack with quasi-compact flat covering which is not algebraic · Lemma 04AH","summary":"Let k be a field. Let G be an affine group scheme over k. If the stack [Spec(k)/G] has a smooth covering by a scheme, then G is of finite type over k.","statement_latex":"Let $k$ be a field. Let $G$ be an affine group scheme over $k$.\nIf the stack $[\\Spec(k)/G]$ has a smooth covering by a\nscheme, then $G$ is of finite type over $k$.","area":"Algebraic Stacks","chapter":"Examples","chapter_id":"examples","section":"Stack with quasi-compact flat covering which is not algebraic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/04AH","source_file":"examples.tex","source_line":3986,"source_end_line":3991,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L3986-L3991","statement_sha256":"69a309c728f7f1addd5953ac9048fdf0fb2c3358b4c38401c32ee7a80da41982","origin":"The Stacks Project","memory_eligible":false,"source_rank":14676,"rank":14676,"depth":0,"x":1995.992,"y":1456.589,"cluster":"algebraic-stacks"},{"id":"stacks:07Z1","tag":"07Z1","title":"Limit preserving on objects, not limit preserving · Lemma 07Z1","summary":"Let S be a nonempty scheme. There exists a stack in groupoids p : X → (Sch/S)_fppf such that p is limit preserving on objects, but X is not limit preserving.","statement_latex":"Let $S$ be a nonempty scheme. There exists a stack in groupoids\n$p : \\mathcal{X} \\to (\\Sch/S)_{fppf}$\nsuch that $p$ is limit preserving on objects, but $\\mathcal{X}$ is not\nlimit preserving.","area":"Algebraic Stacks","chapter":"Examples","chapter_id":"examples","section":"Limit preserving on objects, not limit preserving","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07Z1","source_file":"examples.tex","source_line":4035,"source_end_line":4041,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4035-L4041","statement_sha256":"6471ca436d9c0b0306789841faf8348172e8a4751c21cf45bacbf83201281228","origin":"The Stacks Project","memory_eligible":false,"source_rank":14677,"rank":14677,"depth":0,"x":2170.488,"y":1686.504,"cluster":"algebraic-stacks"},{"id":"stacks:078F","tag":"078F","title":"Sheaf with quasi-compact flat covering which is not algebraic · Lemma 078F","summary":"There exists a functor F : Sch^opp → Sets which satisfies the sheaf condition for the fpqc topology, has representable diagonal Δ : F → F × F, and such that there exists a surjective, flat, universally open, quasi-compact morphism U → F where U is a scheme, but such that F is not an algebraic space.","statement_latex":"There exists a functor $F : \\Sch^{opp} \\to \\textit{Sets}$\nwhich satisfies the sheaf condition for the fpqc topology, has representable\ndiagonal $\\Delta : F \\to F \\times F$, and such that there exists a\nsurjective, flat, universally open, quasi-compact morphism\n$U \\to F$ where $U$ is a scheme, but such that $F$ is not an algebraic space.","area":"Algebraic Stacks","chapter":"Examples","chapter_id":"examples","section":"Sheaf with quasi-compact flat covering which is not algebraic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/078F","source_file":"examples.tex","source_line":4191,"source_end_line":4198,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4191-L4198","statement_sha256":"a600333b2f0ef2514773b6d40e76f3efc6285a0a8a5f21880887951d795cb382","origin":"The Stacks Project","memory_eligible":false,"source_rank":14678,"rank":14678,"depth":0,"x":1856.738,"y":1615.939,"cluster":"algebraic-stacks"},{"id":"stacks:0HAR","tag":"0HAR","title":"The étale topology vs Zariski and finite étale Covers · Lemma 0HAR","summary":"The étale cover U → X cannot be refined by any finite composition of open immersions and finite étale morphisms.","statement_latex":"The \\'etale cover $U \\to X$ cannot be refined by any finite composition\nof open immersions and finite \\'etale morphisms.","area":"Étale Geometry","chapter":"Examples","chapter_id":"examples","section":"The étale topology vs Zariski and finite étale Covers","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAR","source_file":"examples.tex","source_line":4237,"source_end_line":4241,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4237-L4241","statement_sha256":"5c14afbed1301c477b7060e8125d826ad3ea3edb343d6939a577296257f126ef","origin":"The Stacks Project","memory_eligible":false,"source_rank":14679,"rank":14679,"depth":0,"x":1563.501,"y":993.695,"cluster":"tale-geometry"},{"id":"stacks:05LE","tag":"05LE","title":"Sheaves and specializations · Lemma 05LE","summary":"There exists a sheaf of abelian groups G on Sch_etale with the following properties • G(X) = 0 whenever dim(X) < n, • G(X) is not zero if dim(X) ≥ n, and • if X ⊂ X' is a thickening, then G(X) = G(X').","statement_latex":"There exists a sheaf of abelian groups $G$ on\n$\\Sch_\\etale$ with the following properties\n\\begin{enumerate}\n\\item $G(X) = 0$ whenever $\\dim(X) < n$,\n\\item $G(X)$ is not zero if $\\dim(X) \\geq n$, and\n\\item if $X \\subset X'$ is a thickening, then $G(X) = G(X')$.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Examples","chapter_id":"examples","section":"Sheaves and specializations","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LE","source_file":"examples.tex","source_line":4388,"source_end_line":4397,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4388-L4397","statement_sha256":"9c41ace762fc40a0dd75e240082d793473ede5cdb709a7dd916c713824886e65","origin":"The Stacks Project","memory_eligible":false,"source_rank":14680,"rank":14680,"depth":0,"x":985.109,"y":302.016,"cluster":"sheaves-sites"},{"id":"stacks:05LH","tag":"05LH","title":"Sheaves and constructible functions · Lemma 05LH","summary":"There exists a sheaf of abelian groups G on Sch_etale with the following properties • G(Spec(k)) = 0 whenever k is a field, • G is limit preserving, • if X ⊂ X' is a thickening, then G(X) = G(X'), and • G is not zero.","statement_latex":"There exists a sheaf of abelian groups $G$ on\n$\\Sch_\\etale$ with the following properties\n\\begin{enumerate}\n\\item $G(\\Spec(k)) = 0$ whenever $k$ is a field,\n\\item $G$ is limit preserving,\n\\item if $X \\subset X'$ is a thickening, then $G(X) = G(X')$, and\n\\item $G$ is not zero.\n\\end{enumerate}","area":"Sheaves & Sites","chapter":"Examples","chapter_id":"examples","section":"Sheaves and constructible functions","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/05LH","source_file":"examples.tex","source_line":4495,"source_end_line":4505,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4495-L4505","statement_sha256":"37edb3cce0d76d1a7de1e6dd3f0a895747af1d7f40330637f062a18c763a7e4b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14681,"rank":14681,"depth":0,"x":1170.911,"y":77.219,"cluster":"sheaves-sites"},{"id":"stacks:07BG","tag":"07BG","title":"The lisse-étale site is not functorial · Lemma 07BG","summary":"The lisse-étale site is not functorial, even for morphisms of schemes.","statement_latex":"The lisse-\\'etale site is not functorial, even for morphisms of schemes.","area":"Algebraic Stacks","chapter":"Examples","chapter_id":"examples","section":"The lisse-étale site is not functorial","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07BG","source_file":"examples.tex","source_line":4570,"source_end_line":4573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4570-L4573","statement_sha256":"f7499bbcf12f8a6028a489734d52c93c0fac5534bfadb565f3c2e47fefce6a0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14682,"rank":14682,"depth":0,"x":2145.012,"y":1489.868,"cluster":"algebraic-stacks"},{"id":"stacks:0GE9","tag":"0GE9","title":"Sheaves on the category of Noetherian schemes · Lemma 0GE9","summary":"With S = Spec(F_p) the inclusion functor (Noetherian/S)_fppf → (Sch/S)_fppf does not define a morphism of sites.","statement_latex":"With $S = \\Spec(\\mathbf{F}_p)$ the inclusion functor\n$(\\textit{Noetherian}/S)_{fppf} \\to (\\Sch/S)_{fppf}$\ndoes not define a morphism of sites.","area":"Sheaves & Sites","chapter":"Examples","chapter_id":"examples","section":"Sheaves on the category of Noetherian schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0GE9","source_file":"examples.tex","source_line":4684,"source_end_line":4689,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4684-L4689","statement_sha256":"8d673221a8a3d63c0dd62ad298f175ba38b903bb77e7c1e918153bb3dbabf479","origin":"The Stacks Project","memory_eligible":false,"source_rank":14683,"rank":14683,"depth":0,"x":1214.706,"y":348.578,"cluster":"sheaves-sites"},{"id":"stacks:08IY","tag":"08IY","title":"Derived pushforward of quasi-coherent modules · Lemma 08IY","summary":"Let X be an algebraic stack. Let K be an object of D(O_X) whose cohomology sheaves are locally quasi-coherent (Sheaves on Stacks, Definition [Tag 06WJ]) and satisfy the flat base change property (Cohomology of Stacks, Definition [Tag 0762]). Then there exists a distinguished triangle K → ∏_n ≥ 0 τ_≥ -n K → ∏_n ≥ 0 τ_≥ -n K → K[1] in D(O_X). In other words, K is the derived limit of its canonical truncations.","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. Let $K$ be an object of\n$D(\\mathcal{O}_\\mathcal{X})$ whose cohomology sheaves are locally\nquasi-coherent (Sheaves on Stacks, Definition\n\\ref{stacks-sheaves-definition-locally-quasi-coherent})\nand satisfy the flat base change property (Cohomology of Stacks,\nDefinition \\ref{stacks-cohomology-definition-flat-base-change}).\nThen there exists a distinguished triangle\n$$\nK \\to\n\\prod\\nolimits_{n \\geq 0} \\tau_{\\geq -n} K \\to\n\\prod\\nolimits_{n \\geq 0} \\tau_{\\geq -n} K \\to K[1]\n$$\nin $D(\\mathcal{O}_\\mathcal{X})$. In other words, $K$ is the derived\nlimit of its canonical truncations.","area":"Derived Categories","chapter":"Examples","chapter_id":"examples","section":"Derived pushforward of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IY","source_file":"examples.tex","source_line":4772,"source_end_line":4788,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4772-L4788","statement_sha256":"e3df11ba4ba4168f2af2df86539fac73387cf7d97b3aba276991689caf394053","origin":"The Stacks Project","memory_eligible":false,"source_rank":14684,"rank":14684,"depth":74,"x":380.908,"y":607.408,"cluster":"derived-categories"},{"id":"stacks:08IZ","tag":"08IZ","title":"Derived pushforward of quasi-coherent modules · Lemma 08IZ","summary":"Let X be an algebraic stack. If F_n is a collection of locally quasi-coherent sheaves with the flat base change property on X, then ⊕_n F_n[n] → ∏_n F_n[n] is an isomorphism in D(O_X).","statement_latex":"Let $\\mathcal{X}$ be an algebraic stack. If $\\mathcal{F}_n$ is a collection\nof locally quasi-coherent sheaves with the flat base change property on\n$\\mathcal{X}$, then $\\oplus_n \\mathcal{F}_n[n] \\to \\prod_n \\mathcal{F}_n[n]$\nis an isomorphism in $D(\\mathcal{O}_\\mathcal{X})$.","area":"Derived Categories","chapter":"Examples","chapter_id":"examples","section":"Derived pushforward of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08IZ","source_file":"examples.tex","source_line":4811,"source_end_line":4817,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4811-L4817","statement_sha256":"2cbe116f62ab1774efe6fcc80dc85cd581ae50605e0ef37c0caf5067f1897b4c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14685,"rank":14685,"depth":75,"x":177.061,"y":819.258,"cluster":"derived-categories"},{"id":"stacks:07DD","tag":"07DD","title":"Derived pushforward of quasi-coherent modules · Lemma 07DD","summary":"A quasi-compact and quasi-separated morphism f : X → Y of algebraic stacks need not induce a functor Rf_* : D_QCoh(O_X) → D_QCoh(O_Y).","statement_latex":"A quasi-compact and quasi-separated morphism\n$f : \\mathcal{X} \\to \\mathcal{Y}$ of algebraic stacks\nneed not induce a functor\n$Rf_* : D_\\QCoh(\\mathcal{O}_\\mathcal{X}) \\to\nD_\\QCoh(\\mathcal{O}_\\mathcal{Y})$.","area":"Derived Categories","chapter":"Examples","chapter_id":"examples","section":"Derived pushforward of quasi-coherent modules","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07DD","source_file":"examples.tex","source_line":4842,"source_end_line":4849,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4842-L4849","statement_sha256":"13df48f728680a10f456006b0f663eb0d5d3c001435601df9fc65d02ba04202c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14686,"rank":14686,"depth":0,"x":156.996,"y":547.178,"cluster":"derived-categories"},{"id":"stacks:07JT","tag":"07JT","title":"A big abelian category · Lemma 07JT","summary":"There exists a \"big\" abelian category A whose Ext-groups are proper classes.","statement_latex":"There exists a ``big'' abelian category $\\mathcal{A}$ whose\n$\\Ext$-groups are proper classes.","area":"Categories & Foundations","chapter":"Examples","chapter_id":"examples","section":"A big abelian category","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/07JT","source_file":"examples.tex","source_line":4905,"source_end_line":4909,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4905-L4909","statement_sha256":"a79718d793700cc3507dae2064107dbba1417a6168afe841890e9eb86a00c16e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14687,"rank":14687,"depth":0,"x":374.263,"y":140.846,"cluster":"categories-foundations"},{"id":"stacks:084K","tag":"084K","title":"Weakly associated points and scheme theoretic density · Lemma 084K","summary":"There exists a reduced scheme X and a schematically dense open U ⊂ X such that some weakly associated point x ∈ X is not in U.","statement_latex":"There exists a reduced scheme $X$ and a schematically dense open\n$U \\subset X$ such that some weakly associated point $x \\in X$ is not in $U$.","area":"Schemes","chapter":"Examples","chapter_id":"examples","section":"Weakly associated points and scheme theoretic density","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/084K","source_file":"examples.tex","source_line":4981,"source_end_line":4985,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L4981-L4985","statement_sha256":"21ff2dee349deaa1fbac745d4ae0c696c00545e7929262e164bb3ad1fa7ca4c8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14688,"rank":14688,"depth":0,"x":1481.961,"y":560.986,"cluster":"schemes"},{"id":"stacks:087L","tag":"087L","title":"Example of non-additivity of traces · Lemma 087L","summary":"There exists a ring R, a distinguished triangle (K, L, M, α, β, γ) in the homotopy category K(R), and an endomorphism (a, b, c) of this distinguished triangle, such that K, L, M are perfect complexes and Tr_K(a) + Tr_M(c) not = Tr_L(b).","statement_latex":"There exists a ring $R$, a distinguished triangle\n$(K, L, M, \\alpha, \\beta, \\gamma)$ in the homotopy category $K(R)$,\nand an endomorphism $(a, b, c)$ of this distinguished triangle, such that\n$K$, $L$, $M$ are perfect complexes and\n$\\text{Tr}_K(a) + \\text{Tr}_M(c) \\not = \\text{Tr}_L(b)$.","area":"Duality & Cohomology","chapter":"Examples","chapter_id":"examples","section":"Example of non-additivity of traces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/087L","source_file":"examples.tex","source_line":5032,"source_end_line":5039,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L5032-L5039","statement_sha256":"81f0c91bcc11e34134aab1ca1977757ffd86657b30d5d2bd20933726ee63ef6e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14689,"rank":14689,"depth":3,"x":2112.767,"y":1269.291,"cluster":"duality-cohomology"},{"id":"stacks:0HAT","tag":"0HAT","title":"Nonfinite cohomology of the structure sheaf of a projective scheme · Lemma 0HAT","summary":"Non-finite H^0. • There exists a ring R and a projective scheme X over R such that H^0(X, O_X) is not a finite R-module. • There exists a ring R and a finite type quasi-coherent module F on P^1_R such that H^0(P^1_R, F) is not a finite R-module.","statement_latex":"Non-finite $H^0$.\n\\begin{enumerate}\n\\item There exists a ring $R$ and a projective scheme $X$\nover $R$ such that $H^0(X, \\mathcal{O}_X)$ is not\na finite $R$-module.\n\\item There exists a ring $R$ and a finite type quasi-coherent module\n$\\mathcal{F}$ on $\\mathbf{P}^1_R$ such that\n$H^0(\\mathbf{P}^1_R, \\mathcal{F})$ is not a finite $R$-module.\n\\end{enumerate}","area":"Duality & Cohomology","chapter":"Examples","chapter_id":"examples","section":"Nonfinite cohomology of the structure sheaf of a projective scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0HAT","source_file":"examples.tex","source_line":5117,"source_end_line":5128,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L5117-L5128","statement_sha256":"bdf7e26a949eeff98e8bd1280fa7ee96e2c4c23ca4032ac2f54401086aab3bf8","origin":"The Stacks Project","memory_eligible":false,"source_rank":14690,"rank":14690,"depth":0,"x":1864.849,"y":1091.584,"cluster":"duality-cohomology"},{"id":"stacks:08J1","tag":"08J1","title":"Being projective is not local on the base · Lemma 08J1","summary":"The properties • [] P(f) =\"f is projective\", and • [] P(f) =\"f is quasi-projective\" are not Zariski local on the base. A fortiori, they are not fpqc local on the base.","statement_latex":"The properties\n\\begin{enumerate}\n\\item[] $\\mathcal{P}(f) =$``$f$ is projective'', and\n\\item[] $\\mathcal{P}(f) =$``$f$ is quasi-projective''\n\\end{enumerate}\nare not Zariski local on the base. A fortiori, they are not fpqc local\non the base.","area":"Schemes","chapter":"Examples","chapter_id":"examples","section":"Being projective is not local on the base","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08J1","source_file":"examples.tex","source_line":5153,"source_end_line":5162,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L5153-L5162","statement_sha256":"8707a865fb44c1a42350d39dd604f40fa456e33972db5aca72d615dce1727618","origin":"The Stacks Project","memory_eligible":false,"source_rank":14691,"rank":14691,"depth":50,"x":1748.228,"y":712.173,"cluster":"schemes"},{"id":"stacks:08KF","tag":"08KF","title":"Non-effective descent data for projective schemes · Lemma 08KF","summary":"There is an etale covering X→ S of schemes and a descent datum (V/X,φ) relative to X→ S such that V→ X is projective, but the descent datum is not effective in the category of schemes.","statement_latex":"There is an etale covering $X\\to S$ of schemes and a descent datum\n$(V/X,\\varphi)$ relative to $X\\to S$ such that \n$V\\to X$ is projective,\nbut the descent datum is not effective in the category of schemes.","area":"Descent","chapter":"Examples","chapter_id":"examples","section":"Non-effective descent data for projective schemes","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08KF","source_file":"examples.tex","source_line":5241,"source_end_line":5247,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L5241-L5247","statement_sha256":"217ee18a0ccc146def5c7132d5ccfa3ec6747bb00171c595466966fd5596d388","origin":"The Stacks Project","memory_eligible":false,"source_rank":14692,"rank":14692,"depth":51,"x":823.31,"y":1055.83,"cluster":"descent"},{"id":"stacks:0D5E","tag":"0D5E","title":"A family of curves whose total space is not a scheme · Lemma 0D5E","summary":"There exists a field k and a family of curves X → A^1_k such that X is not a scheme.","statement_latex":"There exists a field $k$ and a family of curves\n$X \\to \\mathbf{A}^1_k$ such that $X$ is not a scheme.","area":"Algebraic Spaces","chapter":"Examples","chapter_id":"examples","section":"A family of curves whose total space is not a scheme","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D5E","source_file":"examples.tex","source_line":5481,"source_end_line":5485,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L5481-L5485","statement_sha256":"82ccc56683542e34845c5fc61841dd7b407716fb44e12607b2d3f6117bb4b22e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14693,"rank":14693,"depth":0,"x":58.874,"y":1569.437,"cluster":"algebraic-spaces"},{"id":"stacks:08J3","tag":"08J3","title":"Derived base change · Lemma 08J3","summary":"Let R → R' and R → A be ring maps. In general there does not exist a functor T : D(A) → D(A ⊗_R R') of triangulated categories such that an A-module M gives an object T(M) of D(A ⊗_R R') which maps to M ⊗_R^L R' under the map D(A ⊗_R R') → D(R').","statement_latex":"Let $R \\to R'$ and $R \\to A$ be ring maps. In general there does not\nexist a functor $T : D(A) \\to D(A \\otimes_R R')$\nof triangulated categories such that an $A$-module $M$ gives an\nobject $T(M)$ of $D(A \\otimes_R R')$ which maps to\n$M \\otimes_R^\\mathbf{L} R'$ under the map $D(A \\otimes_R R') \\to D(R')$.","area":"Derived Categories","chapter":"Examples","chapter_id":"examples","section":"Derived base change","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/08J3","source_file":"examples.tex","source_line":5544,"source_end_line":5551,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L5544-L5551","statement_sha256":"4e030abe66dccb513a1fa6018b096b18ee927c051ad2a5e26868516932e8272d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14694,"rank":14694,"depth":0,"x":390.759,"y":736.557,"cluster":"derived-categories"},{"id":"stacks:09R5","tag":"09R5","title":"An interesting compact object · Lemma 09R5","summary":"There exists a differential graded algebra (A, d) and a compact object E of D(A, d) such that E cannot be represented by a finite and graded projective differential graded A-module.","statement_latex":"There exists a differential graded algebra $(A, \\text{d})$ and\na compact object $E$ of $D(A, \\text{d})$ such that $E$ cannot\nbe represented by a finite and graded projective differential\ngraded $A$-module.","area":"Derived Categories","chapter":"Examples","chapter_id":"examples","section":"An interesting compact object","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/09R5","source_file":"examples.tex","source_line":5636,"source_end_line":5642,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L5636-L5642","statement_sha256":"351195b1debf64cf99a76b11f11c5354080f6eb87aef33deca402011dc1bcfb1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14695,"rank":14695,"depth":0,"x":65.858,"y":729.55,"cluster":"derived-categories"},{"id":"stacks:0D1R","tag":"0D1R","title":"The stack of proper algebraic spaces is not algebraic · Lemma 0D1R","summary":"The stack in groupoids p'_fp, flat, proper : Spacesstack'_fp, flat, proper → Sch_fppf whose category of sections over a scheme S is the category of flat, proper, finitely presented algebraic spaces over S (see Quot, Section [Tag 0D1D]) is not an algebraic stack.","statement_latex":"The stack in groupoids\n$$\np'_{fp, flat, proper} :\n\\Spacesstack'_{fp, flat, proper}\n\\longrightarrow\n\\Sch_{fppf}\n$$\nwhose category of sections over a scheme $S$ is the category of\nflat, proper, finitely presented algebraic spaces over $S$\n(see Quot, Section \\ref{quot-section-stack-of-spaces})\nis not an algebraic stack.","area":"Algebraic Stacks","chapter":"Examples","chapter_id":"examples","section":"The stack of proper algebraic spaces is not algebraic","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0D1R","source_file":"examples.tex","source_line":5906,"source_end_line":5919,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L5906-L5919","statement_sha256":"22ab8f6c4f4d8d9e9e94e6956d094430daa86f789d46d07d48952a87a4abf19c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14696,"rank":14696,"depth":60,"x":2033.76,"y":1746.558,"cluster":"algebraic-stacks"},{"id":"stacks:0AF9","tag":"0AF9","title":"An example of a non-algebraic Hom-stack · Lemma 0AF9","summary":"Let W be a two dimensional regular integral Noetherian scheme with function field K. Let G → W be an abelian scheme. Then the map H^1_fppf(W, G) → H^1_fppf(Spec(K), G) is injective.","statement_latex":"Let $W$ be a two dimensional regular integral Noetherian scheme\nwith function field $K$. Let $G \\to W$ be an abelian scheme.\nThen the map $H^1_{fppf}(W, G) \\to H^1_{fppf}(\\Spec(K), G)$\nis injective.","area":"Algebraic Stacks","chapter":"Examples","chapter_id":"examples","section":"An example of a non-algebraic Hom-stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AF9","source_file":"examples.tex","source_line":6002,"source_end_line":6008,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6002-L6008","statement_sha256":"bbaefd0907e2c27c8b9a11a35140156e9f54170304efbcc0fc99869b243b4ce4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14697,"rank":14697,"depth":74,"x":1909.298,"y":1494.0,"cluster":"algebraic-stacks"},{"id":"stacks:0AFA","tag":"0AFA","title":"An example of a non-algebraic Hom-stack · Lemma 0AFA","summary":"Let G be a smooth commutative group algebraic space over a field K. Then H^1_fppf(Spec(K), G) is torsion.","statement_latex":"Let $G$ be a smooth commutative group algebraic space over a field $K$.\nThen $H^1_{fppf}(\\Spec(K), G)$ is torsion.","area":"Algebraic Stacks","chapter":"Examples","chapter_id":"examples","section":"An example of a non-algebraic Hom-stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFA","source_file":"examples.tex","source_line":6054,"source_end_line":6058,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6054-L6058","statement_sha256":"f9512f8574d32c76c8516cc7d6cf4d2640dc252c26345f3dc9c1b5494aa3dd03","origin":"The Stacks Project","memory_eligible":false,"source_rank":14698,"rank":14698,"depth":0,"x":2204.35,"y":1609.679,"cluster":"algebraic-stacks"},{"id":"stacks:0AFB","tag":"0AFB","title":"An example of a non-algebraic Hom-stack · Lemma 0AFB","summary":"The canonical map X(S) → lim X(S_n) is not essentially surjective.","statement_latex":"The canonical map $\\mathcal{X}(S) \\to \\lim \\mathcal{X}(S_n)$\nis not essentially surjective.","area":"Algebraic Stacks","chapter":"Examples","chapter_id":"examples","section":"An example of a non-algebraic Hom-stack","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFB","source_file":"examples.tex","source_line":6089,"source_end_line":6093,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6089-L6093","statement_sha256":"e20d531bed63b49280c239d07c7afc44a39ee4338fea4a84eb40deed68e3f11f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14699,"rank":14699,"depth":75,"x":1893.572,"y":1691.848,"cluster":"algebraic-stacks"},{"id":"stacks:0AFC","tag":"0AFC","title":"An example of a non-algebraic Hom-stack · Proposition 0AFC","summary":"The stack X = underlineMor_S(X, [S/A]) is not algebraic.","statement_latex":"The stack $\\mathcal{X} = \\underline{\\Mor}_S(X, [S/A])$ is not algebraic.","area":"Algebraic Stacks","chapter":"Examples","chapter_id":"examples","section":"An example of a non-algebraic Hom-stack","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0AFC","source_file":"examples.tex","source_line":6138,"source_end_line":6141,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6138-L6141","statement_sha256":"6120dd6cd285cf5f5236d34f010fd2ceb760f916eac24df9a4b2f2f7aac79636","origin":"The Stacks Project","memory_eligible":false,"source_rank":14700,"rank":14700,"depth":0,"x":2056.754,"y":1454.774,"cluster":"algebraic-stacks"},{"id":"stacks:0CXX","tag":"0CXX","title":"An algebraic stack not satisfying strong formal effectiveness · Lemma 0CXX","summary":"Let k be an algebraically closed field which is not the closure of a finite field. Let A be an abelian variety over k. Let X = [Spec(k)/A]. There exists an inverse system of k-algebras R_n with surjective transition maps whose kernels are locally nilpotent and a system (xi_n) of X lying over the system (Spec(R_n)) such that this system is not effective in the sense of Artin's Axioms, Remark [Tag 0CXT].","statement_latex":"Let $k$ be an algebraically closed field which is not the closure\nof a finite field. Let $A$ be an abelian variety over $k$.\nLet $\\mathcal{X} = [\\Spec(k)/A]$.\nThere exists an inverse system of $k$-algebras $R_n$\nwith surjective transition maps whose kernels are locally nilpotent\nand a system $(\\xi_n)$ of $\\mathcal{X}$ lying over the system\n$(\\Spec(R_n))$ such that this system is not effective\nin the sense of Artin's Axioms, Remark \\ref{artin-remark-strong-effectiveness}.","area":"Deformation Theory","chapter":"Examples","chapter_id":"examples","section":"An algebraic stack not satisfying strong formal effectiveness","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CXX","source_file":"examples.tex","source_line":6198,"source_end_line":6208,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6198-L6208","statement_sha256":"936b2060cfdb3c693ec7cb543c0d5cfacd92c59a9bafee32ae4dbde1a18328eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14701,"rank":14701,"depth":0,"x":1462.208,"y":1491.431,"cluster":"deformation-theory"},{"id":"stacks:0ARF","tag":"0ARF","title":"A counter example to Grothendieck's existence theorem · Lemma 0ARF","summary":"Counter examples to algebraization of coherent sheaves. • Grothendieck's existence theorem as stated in Cohomology of Schemes, Theorem [Tag 088E] is false if we drop the assumption that X → Spec(A) is separated. • The stack of coherent sheaves Cohstack_X/B of Quot, Theorems [Tag 09DS] and [Tag 08WC] is in general not algebraic if we drop the assumption that X → S is separated • The functor Quotfunctor_F/X/B of Quot, Proposition [Tag 09TU] is not an algebraic space in…","statement_latex":"Counter examples to algebraization of coherent sheaves.\n\\begin{enumerate}\n\\item Grothendieck's existence theorem as stated in\nCohomology of Schemes, Theorem \\ref{coherent-theorem-grothendieck-existence}\nis false if we drop the assumption that $X \\to \\Spec(A)$ is separated.\n\\item The stack of coherent sheaves $\\Cohstack_{X/B}$\nof Quot, Theorems \\ref{quot-theorem-coherent-algebraic-general} and\n\\ref{quot-theorem-coherent-algebraic} is in general\nnot algebraic if we drop the assumption that $X \\to S$ is separated\n\\item The functor $\\Quotfunctor_{\\mathcal{F}/X/B}$ of\nQuot, Proposition \\ref{quot-proposition-quot}\nis not an algebraic space in general if we drop the assumption\nthat $X \\to B$ is separated.\n\\end{enumerate}","area":"Algebraic & Formal Geometry","chapter":"Examples","chapter_id":"examples","section":"A counter example to Grothendieck's existence theorem","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ARF","source_file":"examples.tex","source_line":6250,"source_end_line":6266,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6250-L6266","statement_sha256":"670f04c051f3832ddb486925b82de6dea840005528643df9d5d241ee1eb3921e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14702,"rank":14702,"depth":82,"x":1141.049,"y":1453.84,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0CBC","tag":"0CBC","title":"Affine formal algebraic spaces · Lemma 0CBC","summary":"There exists an affine formal algebraic space which is not McQuillan.","statement_latex":"There exists an affine formal algebraic space which is not McQuillan.","area":"Algebraic & Formal Geometry","chapter":"Examples","chapter_id":"examples","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBC","source_file":"examples.tex","source_line":6298,"source_end_line":6301,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6298-L6301","statement_sha256":"6925b3c8b2bf1fa9dad929d9f38cb4f1ae5191261cc7d987b94b742fa03e0a0e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14703,"rank":14703,"depth":0,"x":1239.504,"y":1714.165,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0CBD","tag":"0CBD","title":"Affine formal algebraic spaces · Lemma 0CBD","summary":"There exists an affine formal algebraic space X whose regular functions do not separate points, in the following sense: If we write X = colim X_λ as in Formal Spaces, Definition [Tag 0AI7] then lim Γ(X_λ, O_X_λ) is a field, but X_red has infinitely many points.","statement_latex":"There exists an affine formal algebraic space $X$\nwhose regular functions do not separate points, in the following sense:\nIf we write $X = \\colim X_\\lambda$ as in\nFormal Spaces, Definition\n\\ref{formal-spaces-definition-affine-formal-algebraic-space}\nthen $\\lim \\Gamma(X_\\lambda, \\mathcal{O}_{X_\\lambda})$\nis a field, but $X_{red}$ has infinitely many points.","area":"Algebraic & Formal Geometry","chapter":"Examples","chapter_id":"examples","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBD","source_file":"examples.tex","source_line":6324,"source_end_line":6333,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6324-L6333","statement_sha256":"16b698d949dfdbf4568036c3130018417682ba5c1113f351ae5957378fd2a748","origin":"The Stacks Project","memory_eligible":false,"source_rank":14704,"rank":14704,"depth":1,"x":957.265,"y":1577.929,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0CBE","tag":"0CBE","title":"Affine formal algebraic spaces · Lemma 0CBE","summary":"There exists a representable morphism f : X → Y of affine formal algebraic spaces with Y McQuillan, but X not McQuillan.","statement_latex":"There exists a representable morphism $f : X \\to Y$ of\naffine formal algebraic spaces with $Y$ McQuillan, but $X$ not\nMcQuillan.","area":"Algebraic & Formal Geometry","chapter":"Examples","chapter_id":"examples","section":"Affine formal algebraic spaces","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0CBE","source_file":"examples.tex","source_line":6422,"source_end_line":6427,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6422-L6427","statement_sha256":"050b0fa00308d8fb7927db22e7b640a0afc5fbd491a67ce2ddad7a11f155b05e","origin":"The Stacks Project","memory_eligible":false,"source_rank":14705,"rank":14705,"depth":0,"x":1275.272,"y":1518.176,"cluster":"algebraic-formal-geometry"},{"id":"stacks:0ATF","tag":"0ATF","title":"Flat maps are not directed limits of finitely presented flat maps · Lemma 0ATF","summary":"There exists a commutative ring A and a flat A-algebra B which cannot be written as a filtered colimit of finitely presented flat A-algebras. In fact, we may either choose A to be a finite type F_p-algebra or a 1-dimensional Noetherian local ring with residue field of characteristic 0.","statement_latex":"There exists a commutative ring $A$ and a flat $A$-algebra $B$\nwhich cannot be written as a filtered colimit of finitely\npresented flat $A$-algebras. In fact, we may either choose $A$ to\nbe a finite type $\\mathbf{F}_p$-algebra or a $1$-dimensional\nNoetherian local ring with residue field of characteristic $0$.","area":"Commutative Algebra","chapter":"Examples","chapter_id":"examples","section":"Flat maps are not directed limits of finitely presented flat maps","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0ATF","source_file":"examples.tex","source_line":6535,"source_end_line":6542,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6535-L6542","statement_sha256":"8d90a0b82ca406322e978c9e4b4ef88bf15c04be7cae01de3e9545fc3c7a0aec","origin":"The Stacks Project","memory_eligible":false,"source_rank":14706,"rank":14706,"depth":0,"x":1924.31,"y":104.072,"cluster":"commutative-algebra"},{"id":"stacks:0EA5","tag":"0EA5","title":"The category of modules modulo torsion modules · Proposition 0EA5","summary":"Let A be a ring. Let S be a multiplicative subset of A. Let Mod_A denote the category of A-modules and T its Serre subcategory of modules for which any element is annihilated by some element of S. Then there is a canonical equivalence Mod_A/T → Mod_S^-1A.","statement_latex":"Let $A$ be a ring. Let $S$ be a multiplicative subset of $A$.\nLet $\\text{Mod}_A$ denote the category of $A$-modules and $\\mathcal{T}$ its\nSerre subcategory of modules for which any element is annihilated by some\nelement of $S$. Then there is a canonical equivalence\n$\\text{Mod}_A/\\mathcal{T} \\rightarrow \\text{Mod}_{S^{-1}A}$.","area":"Categories & Foundations","chapter":"Examples","chapter_id":"examples","section":"The category of modules modulo torsion modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EA5","source_file":"examples.tex","source_line":6566,"source_end_line":6573,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6566-L6573","statement_sha256":"fbe1ddb65d15a0b7b42237c2504712cb9a8284a2b1aef9c6eda0846c0ef85d2b","origin":"The Stacks Project","memory_eligible":false,"source_rank":14707,"rank":14707,"depth":9,"x":187.188,"y":360.516,"cluster":"categories-foundations"},{"id":"stacks:0B0K","tag":"0B0K","title":"The category of modules modulo torsion modules · Proposition 0B0K","summary":"Let A be a ring. Let Q(A) denote its total quotient ring (as in Algebra, Example [Tag 02C5]). Let Mod_A denote the category of A-modules and T its Serre subcategory of torsion modules. Let Mod_Q(A) denote the category of Q(A)-modules. Then there is a canonical equivalence Mod_A/T → Mod_Q(A).","statement_latex":"Let $A$ be a ring. Let $Q(A)$ denote its total quotient ring\n(as in Algebra, Example \\ref{algebra-example-localize-at-prime}). Let\n$\\text{Mod}_A$ denote the category of $A$-modules and $\\mathcal{T}$ its\nSerre subcategory of torsion modules. Let $\\text{Mod}_{Q(A)}$\ndenote the category\nof $Q(A)$-modules. Then there is a canonical equivalence\n$\\text{Mod}_A/\\mathcal{T} \\rightarrow \\text{Mod}_{Q(A)}$.","area":"Categories & Foundations","chapter":"Examples","chapter_id":"examples","section":"The category of modules modulo torsion modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0K","source_file":"examples.tex","source_line":6600,"source_end_line":6609,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6600-L6609","statement_sha256":"abd7c2ac654a6e368cf0ef646a37bec45e0dd39b9705e585eb0db475a57a6488","origin":"The Stacks Project","memory_eligible":false,"source_rank":14708,"rank":14708,"depth":10,"x":148.402,"y":91.829,"cluster":"categories-foundations"},{"id":"stacks:0B0L","tag":"0B0L","title":"The category of modules modulo torsion modules · Proposition 0B0L","summary":"Let A be a Noetherian integral domain. Let K denote its field of fractions. Let Mod_A^fg denote the category of finitely generated A-modules and T^fg its Serre subcategory of finitely generated torsion modules. Then Mod_A^fg/T^fg is canonically equivalent to the category of finite dimensional K-vector spaces.","statement_latex":"Let $A$ be a Noetherian integral domain. Let $K$ denote its field of fractions.\nLet $\\text{Mod}_A^{fg}$ denote the category of finitely generated $A$-modules\nand $\\mathcal{T}^{fg}$ its Serre subcategory of finitely generated torsion\nmodules. Then $\\text{Mod}_A^{fg}/\\mathcal{T}^{fg}$ is canonically equivalent\nto the category of finite dimensional $K$-vector spaces.","area":"Categories & Foundations","chapter":"Examples","chapter_id":"examples","section":"The category of modules modulo torsion modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0L","source_file":"examples.tex","source_line":6619,"source_end_line":6626,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6619-L6626","statement_sha256":"2842bf0d2f5a0c17e11c8b54a91e4c7edbb5d8f05e16c2471b61dff4f3251709","origin":"The Stacks Project","memory_eligible":false,"source_rank":14709,"rank":14709,"depth":11,"x":393.576,"y":268.31,"cluster":"categories-foundations"},{"id":"stacks:0B0M","tag":"0B0M","title":"The category of modules modulo torsion modules · Proposition 0B0M","summary":"The quotient of the category of abelian groups modulo its Serre subcategory of torsion groups is the category of Q-vector spaces.","statement_latex":"The quotient of the category of abelian groups modulo its\nSerre subcategory of torsion groups is the category of\n$\\mathbf{Q}$-vector spaces.","area":"Categories & Foundations","chapter":"Examples","chapter_id":"examples","section":"The category of modules modulo torsion modules","kind":"proposition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B0M","source_file":"examples.tex","source_line":6642,"source_end_line":6647,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6642-L6647","statement_sha256":"703d35a501c6c863f3e3ef15be18c4b081e0bebe2b10becbeb00447b45afc27a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14710,"rank":14710,"depth":11,"x":70.206,"y":277.311,"cluster":"categories-foundations"},{"id":"stacks:0B2Z","tag":"0B2Z","title":"Different colimit topologies · Lemma 0B2Z","summary":"There exists a system G_1 → G_2 → G_3 → … of (abelian) topological groups such that colim G_n taken in the category of topological spaces is different from colim G_n taken in the category of topological groups.","statement_latex":"There exists a system $G_1 \\to G_2 \\to G_3 \\to \\ldots$ of (abelian)\ntopological groups such that $\\colim G_n$ taken in the category of\ntopological spaces is different from $\\colim G_n$ taken in the category\nof topological groups.","area":"Topology","chapter":"Examples","chapter_id":"examples","section":"Different colimit topologies","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0B2Z","source_file":"examples.tex","source_line":6694,"source_end_line":6700,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6694-L6700","statement_sha256":"ee007cb182a4c275ee338330f04fa9c33601ce884c8fb1efabd586e65d240977","origin":"The Stacks Project","memory_eligible":false,"source_rank":14711,"rank":14711,"depth":0,"x":795.277,"y":110.634,"cluster":"topology"},{"id":"stacks:0EU9","tag":"0EU9","title":"Universally submersive but not V covering · Lemma 0EU9","summary":"There exists a morphism X → Y of affine schemes which is universally submersive such that (X → Y) is not a V covering.","statement_latex":"There exists a morphism $X \\to Y$ of affine schemes\nwhich is universally submersive such that $\\{X \\to Y\\}$\nis not a V covering.","area":"Scheme Morphisms","chapter":"Examples","chapter_id":"examples","section":"Universally submersive but not V covering","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EU9","source_file":"examples.tex","source_line":6772,"source_end_line":6777,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6772-L6777","statement_sha256":"948259c9b305848312221736e0d288f2c64c23b6f19fa6f8d3583e34480e3ccd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14712,"rank":14712,"depth":0,"x":1867.59,"y":734.676,"cluster":"scheme-morphisms"},{"id":"stacks:0EUF","tag":"0EUF","title":"The spectrum of the integers is not quasi-compact · Lemma 0EUF","summary":"There exists a ring A and an infinite family of flat ring maps (A → A_i)_i ∈ I such that for every A-module M M = Equalizer( xymatrix ∏_i ∈ I M ⊗_A A_i ar@<1ex>[r] ar@<-1ex>[r] & ∏_i, j ∈ I M ⊗_A A_i ⊗_A A_j ) but there is no finite subfamily where the same thing is true.","statement_latex":"There exists a ring $A$ and an infinite family of flat ring maps\n$\\{A \\to A_i\\}_{i \\in I}$ such that for every $A$-module $M$ \n$$\nM =\n\\text{Equalizer}\\left(\n\\xymatrix{\n\\prod\\nolimits_{i \\in I} M \\otimes_A A_i \\ar@<1ex>[r] \\ar@<-1ex>[r] &\n\\prod\\nolimits_{i, j \\in I} M \\otimes_A A_i \\otimes_A A_j\n}\n\\right)\n$$\nbut there is no finite subfamily where the same thing is true.","area":"Topology","chapter":"Examples","chapter_id":"examples","section":"The spectrum of the integers is not quasi-compact","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUF","source_file":"examples.tex","source_line":6904,"source_end_line":6918,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6904-L6918","statement_sha256":"8b1bf6f486788294f37f4aafd8f43ad2277aacd749491000aa38775cbc387d23","origin":"The Stacks Project","memory_eligible":false,"source_rank":14713,"rank":14713,"depth":0,"x":682.953,"y":366.424,"cluster":"topology"},{"id":"stacks:0EUG","tag":"0EUG","title":"The spectrum of the integers is not quasi-compact · Lemma 0EUG","summary":"The scheme Spec(Z) is not quasi-compact in the canonical topology on the category of schemes.","statement_latex":"The scheme $\\Spec(\\mathbf{Z})$ is not quasi-compact\nin the canonical topology on the category of schemes.","area":"Topology","chapter":"Examples","chapter_id":"examples","section":"The spectrum of the integers is not quasi-compact","kind":"lemma","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0EUG","source_file":"examples.tex","source_line":6971,"source_end_line":6975,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/examples.tex#L6971-L6975","statement_sha256":"287ffe9c2a3748e455b6ca9514b32daaad019e0df5345299a26b3a43bb54d58d","origin":"The Stacks Project","memory_eligible":false,"source_rank":14714,"rank":14714,"depth":28,"x":559.772,"y":113.438,"cluster":"topology"},{"id":"stacks:078I","tag":"078I","title":"Colimits · Definition 078I","summary":"A directed set is a nonempty set I endowed with a preorder ≤ such that given any pair i, j ∈ I there exists a k ∈ I such that i ≤ k and j ≤ k. A system of rings over I is given by a ring A_i for each i ∈ I and a map of rings φ_ij : A_i → A_j whenever i ≤ j such that the composition A_i → A_j → A_k is equal to A_i → A_k whenever i ≤ j ≤ k.","statement_latex":"A {\\it directed set} is a nonempty set $I$ endowed with a preorder $\\leq$\nsuch that given any pair $i, j \\in I$ there exists a $k \\in I$ such that\n$i \\leq k$ and $j \\leq k$. A {\\it system of rings} over $I$ is given by a\nring $A_i$ for each $i \\in I$ and a map of rings $\\varphi_{ij} : A_i \\to A_j$\nwhenever $i \\leq j$ such that the composition $A_i \\to A_j \\to A_k$ is equal to\n$A_i \\to A_k$ whenever $i \\leq j \\leq k$.","area":"Categories & Foundations","chapter":"Exercises","chapter_id":"exercises","section":"Colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/078I","source_file":"exercises.tex","source_line":162,"source_end_line":170,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L162-L170","statement_sha256":"1ce7495834f4c1ae847210ba58ebb773e84ada67de5be228e017ed9fd768062c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14715,"rank":14715,"depth":0,"x":301.888,"y":86.797,"cluster":"categories-foundations"},{"id":"stacks:078K","tag":"078K","title":"Colimits · Definition 078K","summary":"The ring A constructed in Exercise [Tag 078J] is called the colimit of the system. Notation colim A_i.","statement_latex":"The ring $A$ constructed in Exercise \\ref{exercise-directed-colimit}\nis called the {\\it colimit} of the system. Notation $\\colim A_i$.","area":"Categories & Foundations","chapter":"Exercises","chapter_id":"exercises","section":"Colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/078K","source_file":"exercises.tex","source_line":189,"source_end_line":193,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L189-L193","statement_sha256":"69ae52fb6e642bad2a680cbd40dbb90d14053e6106a4cd248e3d7b34fdbc341f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14716,"rank":14716,"depth":0,"x":284.218,"y":359.295,"cluster":"categories-foundations"},{"id":"stacks:0278","tag":"0278","title":"Colimits · Definition 0278","summary":"A module M over R is said to be of finite presentation over R if it is isomorphic to the cokernel of a map of finite free modules R^⊕ n → R^⊕ m.","statement_latex":"A module $M$ over $R$ is said to be of {\\it finite presentation} over\n$R$ if it  is isomorphic to the cokernel of a map of finite free modules\n$ R^{\\oplus n} \\to R^{\\oplus m}$.","area":"Categories & Foundations","chapter":"Exercises","chapter_id":"exercises","section":"Colimits","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0278","source_file":"exercises.tex","source_line":243,"source_end_line":248,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L243-L248","statement_sha256":"3bb99276b860fb7e9b298042b6d525fc65cec25241a51d5dca17203a2b838c41","origin":"The Stacks Project","memory_eligible":false,"source_rank":14717,"rank":14717,"depth":0,"x":77.696,"y":147.906,"cluster":"categories-foundations"},{"id":"stacks:027B","tag":"027B","title":"The Spectrum of a ring · Definition 027B","summary":"A topological space X is called quasi-compact if for any open covering X = ⋃_i∈ I U_i there is a finite subset (i_1, …, i_n)⊂ I such that X = U_i_1∪… U_i_n.","statement_latex":"A topological space $X$ is called {\\it quasi-compact}\nif for any open covering $X = \\bigcup_{i\\in I} U_i$ there is a finite\nsubset $\\{i_1, \\ldots, i_n\\}\\subset I$ such that $X = U_{i_1}\\cup\\ldots\nU_{i_n}$.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"The Spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027B","source_file":"exercises.tex","source_line":482,"source_end_line":488,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L482-L488","statement_sha256":"af5d0dfce41f994c9549513a7da74ad154cfc1548fcb889f399ba6ed9fbbfda7","origin":"The Stacks Project","memory_eligible":false,"source_rank":14718,"rank":14718,"depth":0,"x":2201.222,"y":245.522,"cluster":"commutative-algebra"},{"id":"stacks:027C","tag":"027C","title":"The Spectrum of a ring · Definition 027C","summary":"A topological space X is said to verify the separation axiom T_0 if for any pair of points x, y∈ X, xnot = y there is an open subset of X containing one but not the other. We say that X is Hausdorff if for any pair x, y∈ X, xnot = y there are disjoint open subsets U, V such that x∈ U and y∈ V.","statement_latex":"A topological space $X$ is said to verify the separation axiom $T_0$\nif for any pair of points $x, y\\in X$, $x\\not = y$ there is an open\nsubset of $X$ containing one but not the other.\nWe say that $X$ is {\\it Hausdorff} if for any pair $x, y\\in X$, $x\\not = y$\nthere are disjoint open subsets $U, V$ such that $x\\in U$\nand $y\\in V$.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"The Spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027C","source_file":"exercises.tex","source_line":495,"source_end_line":503,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L495-L503","statement_sha256":"e07ced653ac8ca18fbe41c7bd37816731d20e4e61af1ad6426a9a0ab6a4f58c3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14719,"rank":14719,"depth":0,"x":1883.166,"y":298.365,"cluster":"commutative-algebra"},{"id":"stacks:027D","tag":"027D","title":"The Spectrum of a ring · Definition 027D","summary":"A topological space X is called irreducible if X is not empty and if X = Z_1∪ Z_2 with Z_1, Z_2⊂ X closed, then either Z_1 = X or Z_2 = X. A subset T⊂ X of a topological space is called irreducible if it is an irreducible topological space with the topology induced from X. This definition implies T is irreducible if and only if the closure bar T of T in X is irreducible.","statement_latex":"A topological space $X$ is called {\\it irreducible} if $X$ is not empty\nand if $X = Z_1\\cup Z_2$ with $Z_1, Z_2\\subset X$ closed, then either\n$Z_1 = X$ or $Z_2 = X$. A subset $T\\subset X$ of a topological space\nis called {\\it irreducible} if it is an irreducible\ntopological space with the topology induced from $X$.\nThis definition implies $T$ is irreducible if and only\nif the closure $\\bar T$ of $T$ in $X$ is irreducible.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"The Spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027D","source_file":"exercises.tex","source_line":518,"source_end_line":527,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L518-L527","statement_sha256":"2cd68079b54f588ba776a9015b26a7f0eeb4ebe1e73bb327bb9ae30f00d45cc3","origin":"The Stacks Project","memory_eligible":false,"source_rank":14720,"rank":14720,"depth":0,"x":2075.271,"y":78.847,"cluster":"commutative-algebra"},{"id":"stacks:027E","tag":"027E","title":"The Spectrum of a ring · Definition 027E","summary":"A point x of an irreducible topological space X is called a generic point of X if X is equal to the closure of the subset (x).","statement_latex":"A point $x$ of an irreducible topological space $X$ is called\na {\\it generic point} of $X$ if $X$ is equal to the closure of\nthe subset $\\{x\\}$.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"The Spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027E","source_file":"exercises.tex","source_line":543,"source_end_line":548,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L543-L548","statement_sha256":"5e987a3502e567e9e17802bcd1ca8523908b2ff27b7c0ac4b254d70c3630e5a4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14721,"rank":14721,"depth":0,"x":2110.159,"y":349.819,"cluster":"commutative-algebra"},{"id":"stacks:027F","tag":"027F","title":"The Spectrum of a ring · Definition 027F","summary":"A topological space X is called Noetherian if any decreasing sequence Z_1⊃ Z_2 ⊃ Z_3⊃ … of closed subsets of X stabilizes. (It is called Artinian if any increasing sequence of closed subsets stabilizes.)","statement_latex":"A topological space $X$ is called {\\it Noetherian} if any\ndecreasing sequence $Z_1\\supset Z_2 \\supset Z_3\\supset \\ldots$\nof closed subsets of $X$ stabilizes.\n(It is called {\\it Artinian} if any increasing sequence of closed\nsubsets stabilizes.)","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"The Spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027F","source_file":"exercises.tex","source_line":572,"source_end_line":579,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L572-L579","statement_sha256":"a5d78a843be5b4a24fe585735998bccda504be73321609337663b6c4c25bfe7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14722,"rank":14722,"depth":0,"x":1866.436,"y":169.739,"cluster":"commutative-algebra"},{"id":"stacks:027G","tag":"027G","title":"The Spectrum of a ring · Definition 027G","summary":"A maximal irreducible subset T⊂ X is called an irreducible component of the space X. Such an irreducible component of X is automatically a closed subset of X.","statement_latex":"A maximal irreducible subset $T\\subset X$ is called an\n{\\it irreducible component} of the space $X$. Such an irreducible\ncomponent of $X$ is automatically a closed subset of $X$.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"The Spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027G","source_file":"exercises.tex","source_line":589,"source_end_line":594,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L589-L594","statement_sha256":"fd7f5f9b9fa87692df11a8f254f6aedd005bcc8b8a455c027aff649a56752783","origin":"The Stacks Project","memory_eligible":false,"source_rank":14723,"rank":14723,"depth":0,"x":2191.087,"y":164.231,"cluster":"commutative-algebra"},{"id":"stacks:027H","tag":"027H","title":"The Spectrum of a ring · Definition 027H","summary":"A point x∈ X is called closed if overline(x) = ( x). Let x, y be points of X. We say that x is a specialization of y, or that y is a generalization of x if x∈ overline(y).","statement_latex":"A point $x\\in X$ is called {\\it closed} if $\\overline{\\{x\\}} = \\{ x\\}$.\nLet $x, y$ be points of $X$. We say that $x$ is a {\\it specialization}\nof $y$, or that $y$ is a {\\it generalization} of $x$ if\n$x\\in \\overline{\\{y\\}}$.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"The Spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027H","source_file":"exercises.tex","source_line":619,"source_end_line":625,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L619-L625","statement_sha256":"660521e908ab29878b8b160b376c7e113a69385f6848b2187d366e1e0ec98a07","origin":"The Stacks Project","memory_eligible":false,"source_rank":14724,"rank":14724,"depth":0,"x":1956.038,"y":352.576,"cluster":"commutative-algebra"},{"id":"stacks:027I","tag":"027I","title":"The Spectrum of a ring · Definition 027I","summary":"A topological space X is called connected if it is nonempty and not the union of two nonempty disjoint open subsets. A connected component of X is a maximal connected subset. Any point of X is contained in a connected component of X and any connected component of X is closed in X. (But in general a connected component need not be open in X.)","statement_latex":"A topological space $X$ is called {\\it connected} if it is nonempty and not the\nunion of two nonempty disjoint open subsets. A {\\it connected component}\nof $X$ is a maximal connected subset. Any point of $X$ is contained\nin a connected component of $X$ and any connected component of $X$ is\nclosed in $X$. (But in general a connected component need not be open in $X$.)","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"The Spectrum of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027I","source_file":"exercises.tex","source_line":650,"source_end_line":657,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L650-L657","statement_sha256":"c93c6f587652d21e00db6ef660a6a2710a4742c6f6e733ef32ef62729f96a3e1","origin":"The Stacks Project","memory_eligible":false,"source_rank":14725,"rank":14725,"depth":0,"x":1977.906,"y":80.222,"cluster":"commutative-algebra"},{"id":"stacks:076E","tag":"076E","title":"Length · Definition 076E","summary":"Let A be a ring. Let M be an A-module. The length of M as an R-module is length_A(M) = sup ( n mid ∃ 0 = M_0 ⊂ M_1 ⊂ … ⊂ M_n = M, M_i not = M_i + 1 ). In other words, the supremum of the lengths of chains of submodules.","statement_latex":"Let $A$ be a ring. Let $M$ be an $A$-module. The\n{\\it length} of $M$ as an $R$-module is\n$$\n\\text{length}_A(M)\n=\n\\sup\n\\{\nn\n\\mid\n\\exists\\ 0 = M_0 \\subset M_1 \\subset \\ldots \\subset M_n = M,\n\\text{ }M_i \\not = M_{i + 1}\n\\}.\n$$\nIn other words, the supremum of the lengths of chains of submodules.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"Length","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/076E","source_file":"exercises.tex","source_line":793,"source_end_line":809,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L793-L809","statement_sha256":"5535d852bf2a184ee442e7b888e49e9c0cf8c8a0df9480622002fdc94a57ed6a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14726,"rank":14726,"depth":0,"x":2180.873,"y":293.537,"cluster":"commutative-algebra"},{"id":"stacks:027O","tag":"027O","title":"Catenary rings · Definition 027O","summary":"A Noetherian ring A is said to be catenary if for any triple of prime ideals p_1 ⊂ p_2 ⊂ p_3 we have ht( p_3 / p_1) = ht( p_3/ p_2) + ht( p_2/ p_1). Here ht( p/ q) means the height of p/ q in the ring A/ q. In a formula ht( p/ q) = dim(A_ p/ qA_ p) = dim((A/ q)_ p) = dim((A/ q)_ p/ q) A topological space X is catenary, if given T ⊂ T' ⊂ X with T and T' closed and irreducible, then there exists a maximal chain of irreducible closed subsets T = T_0 ⊂ T_1 ⊂ … ⊂ T_n = T' and…","statement_latex":"A Noetherian ring $A$ is said to be {\\it catenary}\nif for any triple of prime ideals\n${\\mathfrak p}_1 \\subset {\\mathfrak p}_2 \\subset {\\mathfrak p}_3$\nwe have\n$$\nht({\\mathfrak p}_3 / {\\mathfrak p}_1) = ht({\\mathfrak p}_3/{\\mathfrak p}_2) +\nht({\\mathfrak p}_2/{\\mathfrak p}_1).\n$$\nHere $ht(\\mathfrak p/\\mathfrak q)$ means the height of\n$\\mathfrak p/\\mathfrak q$ in the ring $A/\\mathfrak q$.\nIn a formula\n$$\nht(\\mathfrak p/\\mathfrak q) =\n\\dim(A_\\mathfrak p/\\mathfrak qA_\\mathfrak p) =\n\\dim((A/\\mathfrak q)_\\mathfrak p) =\n\\dim((A/\\mathfrak q)_{\\mathfrak p/\\mathfrak q})\n$$\nA topological space $X$ is {\\it catenary}, if given $T \\subset T' \\subset X$\nwith $T$ and $T'$ closed and irreducible, then there exists a maximal chain\nof irreducible closed subsets\n$$\nT = T_0 \\subset T_1 \\subset \\ldots \\subset T_n = T'\n$$\nand every such chain has the same (finite) length.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"Catenary rings","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027O","source_file":"exercises.tex","source_line":1354,"source_end_line":1380,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1354-L1380","statement_sha256":"3d113ed408498e976ba7f65af2f5b066f1def59b151d6b2788bfb1208f3eba1c","origin":"The Stacks Project","memory_eligible":false,"source_rank":14727,"rank":14727,"depth":0,"x":1859.55,"y":251.396,"cluster":"commutative-algebra"},{"id":"stacks:027R","tag":"027R","title":"Finite locally free modules · Definition 027R","summary":"Let A be a ring. Recall that a finite locally free A-module M is a module such that for every p ∈ Spec(A) there exists an f∈ A, f not ∈ p such that M_f is a finite free A_f-module. We say M is an invertible module if M is finite locally free of rank 1, i.e., for every p ∈ Spec(A) there exists an f∈ A, f not ∈ p such that M_f ≅ A_f as an A_f-module.","statement_latex":"Let $A$ be a ring. Recall that a {\\it finite locally free} $A$-module\n$M$ is a module such that for every ${\\mathfrak p} \\in \\Spec(A)$\nthere exists an\n$f\\in A$, $f \\not \\in {\\mathfrak p}$ such that $M_f$ is a finite free\n$A_f$-module. We say $M$ is an {\\it invertible module} if\n$M$ is finite locally free of rank $1$, i.e., for every\n${\\mathfrak p} \\in \\Spec(A)$ there exists an\n$f\\in A$, $f \\not \\in \\mathfrak p$ such that $M_f \\cong A_f$\nas an $A_f$-module.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"Finite locally free modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027R","source_file":"exercises.tex","source_line":1533,"source_end_line":1544,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1533-L1544","statement_sha256":"7969d096d7d2f6e3f9d59fd69c7a31cf9e63460a056bafe0df534adf4b3d7554","origin":"The Stacks Project","memory_eligible":false,"source_rank":14728,"rank":14728,"depth":0,"x":2130.476,"y":100.089,"cluster":"commutative-algebra"},{"id":"stacks:078Q","tag":"078Q","title":"Finite locally free modules · Definition 078Q","summary":"Let A be a ring. The class group of A, sometimes called the Picard group of A is the set Pic(A) of isomorphism classes of invertible A-modules endowed with a group operation defined by tensor product (see Exercise [Tag 078P]).","statement_latex":"Let $A$ be a ring. The {\\it class group of $A$}, sometimes called\nthe {\\it Picard group of $A$} is the set $\\Pic(A)$\nof isomorphism classes of invertible $A$-modules endowed with\na group operation defined by tensor product (see\nExercise \\ref{exercise-tensor-finite-locally-free}).","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"Finite locally free modules","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/078Q","source_file":"exercises.tex","source_line":1553,"source_end_line":1560,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1553-L1560","statement_sha256":"0ae03793d5709ce4c7137b794550088cabc31ce5f76d158c3e180e0c686b1a7f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14729,"rank":14729,"depth":0,"x":2052.348,"y":365.486,"cluster":"commutative-algebra"},{"id":"stacks:027U","tag":"027U","title":"Going up and going down · Definition 027U","summary":"Let φ : A → B be a homomorphism of rings. We say that the going-up theorem holds for φ if the following condition is satisfied: • [(GU)] for any p, p' ∈ Spec(A) such that p ⊂ p', and for any P ∈ Spec(B) lying over p, there exists P'∈ Spec(B) lying over p' such that P ⊂ P'. Similarly, we say that the going-down theorem holds for φ if the following condition is satisfied: • [(GD)] for any p, p' ∈ Spec(A) such that p ⊂ p', and for any P' ∈ Spec(B) lying over p', there exists…","statement_latex":"Let $\\phi : A \\to B$ be a homomorphism of rings. We say\nthat the {\\it going-up theorem} holds for $\\phi$ if the\nfollowing condition is satisfied:\n\\begin{itemize}\n\\item[(GU)] for any ${\\mathfrak p}, {\\mathfrak p}' \\in \\Spec(A)$ such that\n${\\mathfrak p} \\subset {\\mathfrak p}'$, and for any $P \\in \\Spec(B)$ lying\nover ${\\mathfrak p}$, there exists $P'\\in \\Spec(B)$ lying\nover ${\\mathfrak p}'$ such that $P \\subset P'$.\n\\end{itemize}\nSimilarly, we say that the {\\it going-down theorem} holds for $\\phi$\nif the following condition is satisfied:\n\\begin{itemize}\n\\item[(GD)] for any ${\\mathfrak p}, {\\mathfrak p}' \\in \\Spec(A)$ such that\n${\\mathfrak p} \\subset {\\mathfrak p}'$, and for any\n$P' \\in \\Spec(B)$ lying\nover ${\\mathfrak p}'$, there exists $P\\in \\Spec(B)$ lying\nover ${\\mathfrak p}$ such that $P \\subset P'$.\n\\end{itemize}","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"Going up and going down","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027U","source_file":"exercises.tex","source_line":1678,"source_end_line":1698,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1678-L1698","statement_sha256":"b039cbbb2e1a197dc1d35c07fec296530cfc40f70ec066d804a14edaa9286b18","origin":"The Stacks Project","memory_eligible":false,"source_rank":14730,"rank":14730,"depth":0,"x":1896.477,"y":125.368,"cluster":"commutative-algebra"},{"id":"stacks:027X","tag":"027X","title":"Hilbert functions · Definition 027X","summary":"A numerical polynomial is a polynomial f(x) ∈ Q[x] such that f(n) ∈ Z for every integer n.","statement_latex":"A {\\it numerical polynomial} is a polynomial $f(x) \\in {\\mathbf Q}[x]$\nsuch that $f(n) \\in {\\mathbf Z}$ for every integer $n$.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"Hilbert functions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027X","source_file":"exercises.tex","source_line":1794,"source_end_line":1798,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1794-L1798","statement_sha256":"fb94534e378a10f6c1212acf71803f6bc886e6a460c7eccb1c656ba41053fe0a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14731,"rank":14731,"depth":0,"x":2204.621,"y":214.014,"cluster":"commutative-algebra"},{"id":"stacks:027Y","tag":"027Y","title":"Hilbert functions · Definition 027Y","summary":"A graded module M over a ring A is an A-module M endowed with a direct sum decomposition bigoplus_n ∈ Z M_n into A-submodules. We will say that M is locally finite if all of the M_n are finite A-modules. Suppose that A is a Noetherian ring and that φ is a Euler-Poincaré function on finite A-modules. This means that for every finitely generated A-module M we are given an integer φ(M) ∈ Z and for every short exact sequence 0 → M' → M → M\" → 0 we have φ(M) = φ(M') + φ(M\").…","statement_latex":"A {\\it graded module} $M$ over a ring $A$ is an $A$-module $M$\nendowed with a direct sum decomposition\n$\n\\bigoplus\\nolimits_{n \\in {\\mathbf Z}} M_n\n$\ninto $A$-submodules. We will say that $M$ is {\\it locally finite} if all of\nthe $M_n$ are finite $A$-modules. Suppose that $A$ is a Noetherian ring and\nthat $\\varphi$ is a {\\it Euler-Poincar\\'e function} on finite $A$-modules.\nThis means that for every finitely generated $A$-module $M$ we are given an\ninteger $\\varphi(M) \\in {\\mathbf Z}$ and for every short exact sequence\n$$\n0\n\\longrightarrow\nM'\n\\longrightarrow\nM\n\\longrightarrow\nM''\n\\longrightarrow\n0\n$$\nwe have $\\varphi(M) = \\varphi(M') + \\varphi(M'')$. The {\\it Hilbert function}\nof a locally finite graded module $M$ (with respect to $\\varphi$) is the\nfunction $\\chi_\\varphi(M, n) = \\varphi(M_n)$. We say that $M$ has a\n{\\it Hilbert polynomial} if there is some numerical polynomial\n$P_\\varphi$ such that $\\chi_\\varphi(M, n) = P_\\varphi(n)$ for all sufficiently\nlarge integers $n$.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"Hilbert functions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027Y","source_file":"exercises.tex","source_line":1800,"source_end_line":1829,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1800-L1829","statement_sha256":"923e38f5bb111e674fecc05ec70ecb504bf521fe993cebf3c9398ad95867c96f","origin":"The Stacks Project","memory_eligible":false,"source_rank":14732,"rank":14732,"depth":0,"x":1906.006,"y":323.535,"cluster":"commutative-algebra"},{"id":"stacks:027Z","tag":"027Z","title":"Hilbert functions · Definition 027Z","summary":"A graded A-algebra is a graded A-module B = bigoplus_n ≥ 0 B_n together with an A-bilinear map B × B → B, (b, b') ↦ bb' that turns B into an A-algebra so that B_n · B_m ⊂ B_n + m. Finally, a graded module M over a graded A-algebra B is given by a graded A-module M together with a (compatible) B-module structure such that B_n · M_d ⊂ M_n + d. Now you can define homomorphisms of graded modules/rings, graded submodules, graded ideals, exact sequences of graded modules, etc, etc.","statement_latex":"A {\\it graded $A$-algebra} is a graded $A$-module\n$B = \\bigoplus_{n \\geq 0} B_n$ together with an $A$-bilinear map\n$$\nB \\times B \\longrightarrow B, \\ (b, b') \\longmapsto bb'\n$$\nthat turns $B$ into an $A$-algebra so that $B_n \\cdot B_m \\subset B_{n + m}$.\nFinally, a {\\it graded module $M$ over a graded $A$-algebra $B$} is given\nby a graded $A$-module $M$ together with a (compatible) $B$-module structure\nsuch that $B_n \\cdot M_d \\subset M_{n + d}$. Now you can define\n{\\it homomorphisms of graded modules/rings}, {\\it graded submodules},\n{\\it graded ideals}, {\\it exact sequences of graded modules}, etc, etc.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"Hilbert functions","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/027Z","source_file":"exercises.tex","source_line":1831,"source_end_line":1844,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1831-L1844","statement_sha256":"496859f37d220504a23ab2d1a6f7762145fa2de7bd594c904b9394dc7a6ef328","origin":"The Stacks Project","memory_eligible":false,"source_rank":14733,"rank":14733,"depth":0,"x":2038.174,"y":73.245,"cluster":"commutative-algebra"},{"id":"stacks:0281","tag":"0281","title":"Proj of a ring · Definition 0281","summary":"Let R be a graded ring. A homogeneous ideal is simply an ideal I ⊂ R which is also a graded submodule of R. Equivalently, it is an ideal generated by homogeneous elements. Equivalently, if f ∈ I and f = f_0 + f_1 + … + f_n is the decomposition of f into homogeneous pieces in R then f_i ∈ I for each i.","statement_latex":"Let $R$ be a graded ring. A {\\it homogeneous} ideal is simply an ideal\n$I \\subset R$ which is also a graded submodule of $R$. Equivalently,\nit is an ideal generated by homogeneous elements. Equivalently, if\n$f \\in I$ and\n$$\nf = f_0 + f_1 + \\ldots + f_n\n$$\nis the decomposition of $f$ into homogeneous pieces in $R$ then $f_i \\in I$\nfor each $i$.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Proj of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0281","source_file":"exercises.tex","source_line":1900,"source_end_line":1911,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1900-L1911","statement_sha256":"042793c188bb3c9c3816b882ffd6fa35d69978360bfb04ebc6a919546c10a713","origin":"The Stacks Project","memory_eligible":false,"source_rank":14734,"rank":14734,"depth":0,"x":1429.877,"y":751.829,"cluster":"schemes"},{"id":"stacks:0282","tag":"0282","title":"Proj of a ring · Definition 0282","summary":"We define the homogeneous spectrum Proj(R) of the graded ring R to be the set of homogeneous, prime ideals p of R such that R_+ not ⊂ p. Note that Proj(R) is a subset of Spec(R) and hence has a natural induced topology.","statement_latex":"We define the {\\it homogeneous spectrum $\\text{Proj}(R)$}\nof the graded ring $R$ to be the set of homogeneous, prime ideals\n${\\mathfrak p}$ of $R$ such that $R_{+} \\not \\subset {\\mathfrak p}$.\nNote that $\\text{Proj}(R)$ is a subset of $\\Spec(R)$ and hence has a\nnatural induced topology.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Proj of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0282","source_file":"exercises.tex","source_line":1913,"source_end_line":1920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1913-L1920","statement_sha256":"85fcb132c8f46ce895e861351d653b22b7d7263f37c763513b6f5618d1601ce4","origin":"The Stacks Project","memory_eligible":false,"source_rank":14735,"rank":14735,"depth":0,"x":1633.007,"y":541.665,"cluster":"schemes"},{"id":"stacks:0283","tag":"0283","title":"Proj of a ring · Definition 0283","summary":"Let R = ⊕_d ≥ 0 R_d be a graded ring, let f∈ R_d and assume that d ≥ 1. We define R_(f) to be the subring of R_f consisting of elements of the form r/f^n with r homogeneous and deg(r) = nd. Furthermore, we define D_+(f) = ( p ∈ Proj(R) | f not∈ p ). Finally, for a homogeneous ideal I ⊂ R we define V_+(I) = V(I) ∩ Proj(R).","statement_latex":"Let $R = \\oplus_{d \\geq 0} R_d$ be a graded ring, let $f\\in R_d$ and\nassume that $d \\geq 1$. We define {\\it $R_{(f)}$} to be the subring of\n$R_f$ consisting of elements of the form $r/f^n$ with $r$ homogeneous and\n$\\deg(r) = nd$. Furthermore, we define\n$$\nD_{+}(f) = \\{ {\\mathfrak p} \\in \\text{Proj}(R) | f \\not\\in {\\mathfrak p} \\}.\n$$\nFinally, for a homogeneous ideal $I \\subset R$ we define\n$V_{+}(I) = V(I) \\cap \\text{Proj}(R)$.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Proj of a ring","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0283","source_file":"exercises.tex","source_line":1922,"source_end_line":1933,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L1922-L1933","statement_sha256":"3c258947d35bf5a533abd5d69130b009b780733a41a09d104934c298fa42f461","origin":"The Stacks Project","memory_eligible":false,"source_rank":14736,"rank":14736,"depth":0,"x":1652.256,"y":812.261,"cluster":"schemes"},{"id":"stacks:0285","tag":"0285","title":"Cohen-Macaulay rings of dimension 1 · Definition 0285","summary":"A Noetherian local ring A is said to be Cohen-Macaulay of dimension d if it has dimension d and there exists a system of parameters x_1, …, x_d for A such that x_i is a nonzerodivisor in A/(x_1, …, x_i-1) for i = 1, …, d.","statement_latex":"A Noetherian local ring $A$ is said to be {\\it Cohen-Macaulay}\nof dimension $d$ if it has dimension $d$ and there exists a system\nof parameters $x_1, \\ldots, x_d$ for $A$ such that $x_i$ is a nonzerodivisor\nin $A/(x_1, \\ldots, x_{i-1})$ for $i = 1, \\ldots, d$.","area":"Commutative Algebra","chapter":"Exercises","chapter_id":"exercises","section":"Cohen-Macaulay rings of dimension 1","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/0285","source_file":"exercises.tex","source_line":2056,"source_end_line":2062,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L2056-L2062","statement_sha256":"e85c6e357cd43dec092cbed80c4dff8d741ee53257cf3ac24ae4225a4427f1af","origin":"The Stacks Project","memory_eligible":false,"source_rank":14737,"rank":14737,"depth":0,"x":2142.03,"y":332.894,"cluster":"commutative-algebra"},{"id":"stacks:028A","tag":"028A","title":"Filtered derived category · Definition 028A","summary":"Let A be an abelian category. Let I be a filtered object of A. Assume the filtration on I is finite. We say I is filtered injective if each gr^p(I) is an injective object of A.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $I$ be a filtered object of $\\mathcal{A}$.\nAssume the filtration on $I$ is finite.\nWe say $I$ is {\\it filtered injective} if each $\\text{gr}^p(I)$ is\nan injective object of $\\mathcal{A}$.","area":"Derived Categories","chapter":"Exercises","chapter_id":"exercises","section":"Filtered derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/028A","source_file":"exercises.tex","source_line":2414,"source_end_line":2421,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L2414-L2421","statement_sha256":"8e25d517f534ad50d3506414d380410fc26926884b65911d12b23c85e725f4eb","origin":"The Stacks Project","memory_eligible":false,"source_rank":14738,"rank":14738,"depth":0,"x":311.247,"y":550.232,"cluster":"derived-categories"},{"id":"stacks:028B","tag":"028B","title":"Filtered derived category · Definition 028B","summary":"Let A be an abelian category. We denote Fil^f(A) the full subcategory of Fil(A) whose objects consist of those A ∈ Ob(Fil(A)) whose filtration is finite.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nWe denote {\\it $\\text{Fil}^f(\\mathcal{A})$} the full subcategory\nof $\\text{Fil}(\\mathcal{A})$ whose objects consist of\nthose $A \\in \\Ob(\\text{Fil}(\\mathcal{A}))$\nwhose filtration is finite.","area":"Derived Categories","chapter":"Exercises","chapter_id":"exercises","section":"Filtered derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/028B","source_file":"exercises.tex","source_line":2427,"source_end_line":2434,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L2427-L2434","statement_sha256":"1ec537b0d105fedbb45b124bb0bf8b5b0a12632694712e6f2173247ea7dd16cd","origin":"The Stacks Project","memory_eligible":false,"source_rank":14739,"rank":14739,"depth":0,"x":274.478,"y":821.893,"cluster":"derived-categories"},{"id":"stacks:028D","tag":"028D","title":"Filtered derived category · Definition 028D","summary":"Let A be an abelian category. Let α : K^bullet → L^bullet be a morphism of complexes of Fil(A). We say that α is a filtered quasi-isomorphism if for each p ∈ Z the morphism gr^p(K^bullet) → gr^p(L^bullet) is a quasi-isomorphism.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $\\alpha : K^\\bullet \\to L^\\bullet$ be a morphism of\ncomplexes of $\\text{Fil}(\\mathcal{A})$. We say that\n$\\alpha$ is a {\\it filtered quasi-isomorphism} if\nfor each $p \\in \\mathbf{Z}$ the morphism\n$\\text{gr}^p(K^\\bullet) \\to \\text{gr}^p(L^\\bullet)$ is\na quasi-isomorphism.","area":"Derived Categories","chapter":"Exercises","chapter_id":"exercises","section":"Filtered derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/028D","source_file":"exercises.tex","source_line":2445,"source_end_line":2454,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L2445-L2454","statement_sha256":"6a0c003b5ec025a8b4616f369b39396b180ec1f5ecee57821b571276d656fde2","origin":"The Stacks Project","memory_eligible":false,"source_rank":14740,"rank":14740,"depth":0,"x":82.991,"y":600.551,"cluster":"derived-categories"},{"id":"stacks:028E","tag":"028E","title":"Filtered derived category · Definition 028E","summary":"Let A be an abelian category. Let K^bullet be a complex of Fil^f(A). We say that K^bullet is filtered acyclic if for each p ∈ Z the complex gr^p(K^bullet) is acyclic.","statement_latex":"Let $\\mathcal{A}$ be an abelian category.\nLet $K^\\bullet$ be a complex of $\\text{Fil}^f(\\mathcal{A})$.\nWe say that $K^\\bullet$ is {\\it filtered acyclic} if\nfor each $p \\in \\mathbf{Z}$ the complex $\\text{gr}^p(K^\\bullet)$ is\nacyclic.","area":"Derived Categories","chapter":"Exercises","chapter_id":"exercises","section":"Filtered derived category","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/028E","source_file":"exercises.tex","source_line":2456,"source_end_line":2463,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L2456-L2463","statement_sha256":"d7657d87018dff9a619bdeee0f17b4f43421dc72f959fee53c65d49496f2b557","origin":"The Stacks Project","memory_eligible":false,"source_rank":14741,"rank":14741,"depth":0,"x":402.416,"y":655.15,"cluster":"derived-categories"},{"id":"stacks:028Y","tag":"028Y","title":"Schemes · Definition 028Y","summary":"A scheme X is called integral if X is nonempty and for every nonempty affine open U ⊂ X the ring Γ(U, O_X) = O_X(U) is a domain.","statement_latex":"A scheme $X$ is called {\\it integral} if $X$ is nonempty and\nfor every nonempty affine open $U \\subset X$ the ring\n$\\Gamma(U, \\mathcal{O}_X) = \\mathcal{O}_X(U)$ is a domain.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Schemes","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/028Y","source_file":"exercises.tex","source_line":2915,"source_end_line":2920,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L2915-L2920","statement_sha256":"d4c128a0c371a7efadac17868ebe2d00887b50147f101c93aca610e01c373736","origin":"The Stacks Project","memory_eligible":false,"source_rank":14742,"rank":14742,"depth":0,"x":1420.145,"y":623.397,"cluster":"schemes"},{"id":"stacks:029D","tag":"029D","title":"Tangent Spaces · Definition 029D","summary":"For any ring R we denote R[ε] the ring of dual numbers. As an R-module it is free with basis 1, ε. The ring structure comes from setting ε^2 = 0.","statement_latex":"For any ring $R$ we denote $R[\\epsilon]$ the ring\nof {\\it dual numbers}. As an $R$-module it is free with\nbasis $1$, $\\epsilon$. The ring structure comes from setting\n$\\epsilon^2 = 0$.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Tangent Spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/029D","source_file":"exercises.tex","source_line":3161,"source_end_line":3167,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3161-L3167","statement_sha256":"9674b98b4cb2b5ed6d6f2747bab8c30ece8cbd1532a3aaed2b9759d176bba48a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14743,"rank":14743,"depth":0,"x":1743.611,"y":630.967,"cluster":"schemes"},{"id":"stacks:029F","tag":"029F","title":"Tangent Spaces · Definition 029F","summary":"Let f : X → S be a morphism of schemes. Let x ∈ X. We dub the set of dotted arrows of Exercise [Tag 029E] the tangent space of X over S and we denote it T_X/S, x. An element of this space is called a tangent vector of X/S at x.","statement_latex":"Let $f : X \\to S$ be a morphism of schemes.\nLet $x \\in X$. We dub the set of dotted arrows\nof Exercise \\ref{exercise-tangent-space-Zariski}\nthe {\\it tangent space of $X$ over $S$}\nand we denote it $T_{X/S, x}$. An element of this\nspace is called a {\\it tangent vector} of $X/S$ at $x$.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Tangent Spaces","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/029F","source_file":"exercises.tex","source_line":3199,"source_end_line":3207,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3199-L3207","statement_sha256":"d059911bfd6f8f06f16d980653d52e5b8e993f8516fb771bc10fb5f308ed16bc","origin":"The Stacks Project","memory_eligible":false,"source_rank":14744,"rank":14744,"depth":0,"x":1498.678,"y":809.164,"cluster":"schemes"},{"id":"stacks:029O","tag":"029O","title":"Quasi-coherent Sheaves · Definition 029O","summary":"Let X be a scheme. A sheaf F of O_X-modules is quasi-coherent if for every affine open Spec(R) = U ⊂ X the restriction F|_U is of the form widetilde M for some R-module M.","statement_latex":"Let $X$ be a scheme.\nA sheaf $\\mathcal{F}$ of $\\mathcal{O}_X$-modules is {\\it quasi-coherent}\nif for every affine open $\\Spec(R) = U \\subset X$ the restriction\n$\\mathcal{F}|_U$ is of the form $\\widetilde M$ for some $R$-module\n$M$.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Quasi-coherent Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/029O","source_file":"exercises.tex","source_line":3330,"source_end_line":3337,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3330-L3337","statement_sha256":"4d8829f10ae1421f4b2192f2f289658afe1f6fe9ae6ca4c0054c6fbbafecad22","origin":"The Stacks Project","memory_eligible":false,"source_rank":14745,"rank":14745,"depth":0,"x":1536.037,"y":538.425,"cluster":"schemes"},{"id":"stacks:029P","tag":"029P","title":"Quasi-coherent Sheaves · Definition 029P","summary":"Let X be a topological space. Let x, x' ∈ X. We say x is a specialization of x' if and only if x ∈ overline(x').","statement_latex":"Let $X$ be a topological space. Let $x, x' \\in X$.\nWe say $x$ is a {\\it specialization} of $x'$\nif and only if $x \\in \\overline{\\{x'\\}}$.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Quasi-coherent Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/029P","source_file":"exercises.tex","source_line":3345,"source_end_line":3350,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3345-L3350","statement_sha256":"6678b882ab744dd337346626dc2eafaa7af958c3d1b8dd9bbed0b35041128b09","origin":"The Stacks Project","memory_eligible":false,"source_rank":14746,"rank":14746,"depth":0,"x":1726.462,"y":759.555,"cluster":"schemes"},{"id":"stacks:029S","tag":"029S","title":"Quasi-coherent Sheaves · Definition 029S","summary":"A scheme X is called locally Noetherian if and only if for every point x ∈ X there exists an affine open Spec(R) = U ⊂ X such that R is Noetherian. A scheme is Noetherian if it is locally Noetherian and quasi-compact.","statement_latex":"A scheme $X$ is called {\\it locally Noetherian} if and only if\nfor every point $x \\in X$ there exists an affine open\n$\\Spec(R) = U \\subset X$ such that $R$ is Noetherian.\nA scheme is {\\it Noetherian} if it is locally Noetherian and quasi-compact.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Quasi-coherent Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/029S","source_file":"exercises.tex","source_line":3368,"source_end_line":3374,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3368-L3374","statement_sha256":"2f9f13dec02aca831f656a858d99bba5d84034a04abf0039854d235a37a9f09a","origin":"The Stacks Project","memory_eligible":false,"source_rank":14747,"rank":14747,"depth":0,"x":1407.798,"y":704.476,"cluster":"schemes"},{"id":"stacks:029T","tag":"029T","title":"Quasi-coherent Sheaves · Definition 029T","summary":"Let X be a locally Noetherian scheme. Let F be a quasi-coherent sheaf of O_X-modules. We say F is coherent if for every point x ∈ X there exists an affine open Spec(R) = U ⊂ X such that F|_U is isomorphic to widetilde M for some finite R-module M.","statement_latex":"Let $X$ be a locally Noetherian scheme.\nLet $\\mathcal{F}$ be a quasi-coherent sheaf of\n$\\mathcal{O}_X$-modules. We say $\\mathcal{F}$ is {\\it coherent}\nif for every point $x \\in X$ there exists an affine open\n$\\Spec(R) = U \\subset X$ such that $\\mathcal{F}|_U$\nis isomorphic to $\\widetilde M$ for some finite $R$-module $M$.","area":"Schemes","chapter":"Exercises","chapter_id":"exercises","section":"Quasi-coherent Sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/029T","source_file":"exercises.tex","source_line":3381,"source_end_line":3389,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3381-L3389","statement_sha256":"1105c37ebaba142005c9a53ac6025af32da4deb809f9b1ae56274d44052bde50","origin":"The Stacks Project","memory_eligible":false,"source_rank":14748,"rank":14748,"depth":0,"x":1687.437,"y":564.088,"cluster":"schemes"},{"id":"stacks:02AD","tag":"02AD","title":"Invertible sheaves · Definition 02AD","summary":"Let X be a locally ringed space. An invertible O_X-module on X is a sheaf of O_X-modules L such that every point has an open neighbourhood U ⊂ X such that L|_U is isomorphic to O_U as O_U-module. We say that L is trivial if it is isomorphic to O_X as a O_X-module.","statement_latex":"Let $X$ be a locally ringed space.\nAn {\\it invertible ${\\mathcal O}_X$-module} on $X$\nis a sheaf of ${\\mathcal O}_X$-modules ${\\mathcal L}$ such that every point\nhas an open neighbourhood $U \\subset X$ such that ${\\mathcal L}|_U$\nis isomorphic to ${\\mathcal O}_U$ as ${\\mathcal O}_U$-module.\nWe say that ${\\mathcal L}$ is trivial if it is isomorphic to\n${\\mathcal O}_X$ as a ${\\mathcal O}_X$-module.","area":"Divisors & Intersection Theory","chapter":"Exercises","chapter_id":"exercises","section":"Invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02AD","source_file":"exercises.tex","source_line":3766,"source_end_line":3775,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3766-L3775","statement_sha256":"0bbdb868bbc5e4252e7bdfdff1965c307927947a9c997d5a1b17cd44f3d4e202","origin":"The Stacks Project","memory_eligible":false,"source_rank":14749,"rank":14749,"depth":0,"x":2561.247,"y":550.232,"cluster":"divisors-intersection-theory"},{"id":"stacks:02AG","tag":"02AG","title":"Invertible sheaves · Definition 02AG","summary":"Let R be a ring. An invertible module M is an R-module M such that widetilde M is an invertible sheaf on the spectrum of R. We say M is trivial if M ≅ R as an R-module.","statement_latex":"Let $R$ be a ring. An {\\it invertible module $M$} is an $R$-module\n$M$ such that $\\widetilde M$ is an invertible sheaf on the\nspectrum of $R$. We say $M$ is {\\it trivial} if $M \\cong R$ as\nan $R$-module.","area":"Divisors & Intersection Theory","chapter":"Exercises","chapter_id":"exercises","section":"Invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02AG","source_file":"exercises.tex","source_line":3805,"source_end_line":3811,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3805-L3811","statement_sha256":"0fe4707295445e14a91fff85363d5644538bc2fb51538d528e5efe463c83e0d9","origin":"The Stacks Project","memory_eligible":false,"source_rank":14750,"rank":14750,"depth":0,"x":2524.478,"y":821.893,"cluster":"divisors-intersection-theory"},{"id":"stacks:02AM","tag":"02AM","title":"Invertible sheaves · Definition 02AM","summary":"Let X be a locally ringed space. The Picard group of X is the set Pic(X) of isomorphism classes of invertible O_X-modules with addition given by tensor product. See Modules, Definition [Tag 01CX]. For a ring R we set Pic(R) = Pic(Spec(R)).","statement_latex":"Let $X$ be a locally ringed space.\nThe {\\it Picard group of $X$} is the set $\\Pic(X)$\nof isomorphism classes of invertible $\\mathcal{O}_X$-modules\nwith addition given by tensor product.\nSee Modules, Definition \\ref{modules-definition-pic}.\nFor a ring $R$ we set $\\Pic(R) = \\Pic(\\Spec(R))$.","area":"Divisors & Intersection Theory","chapter":"Exercises","chapter_id":"exercises","section":"Invertible sheaves","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02AM","source_file":"exercises.tex","source_line":3884,"source_end_line":3892,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3884-L3892","statement_sha256":"856414eef972991eb2b4d7ee697b39709f6bbfcd5cad059cc1bf11e29730a292","origin":"The Stacks Project","memory_eligible":false,"source_rank":14751,"rank":14751,"depth":1,"x":2332.991,"y":600.551,"cluster":"divisors-intersection-theory"},{"id":"stacks:02AP","tag":"02AP","title":"v Cech Cohomology · Definition 02AP","summary":"(Definition of delta.) Suppose that 0 → F_1 → F_2 → F_3 → 0 is a short exact sequence of abelian sheaves on any topological space X. The boundary map δ : H^0(X, F_3) → check H^1(X, F_1) is defined as follows. Take an element τ ∈ H^0(X, F_3). Choose an open covering U : X = ⋃_i∈ I U_i such that for each i there exists a section tilde τ_i ∈ F_2 lifting the restriction of τ to U_i. Then consider the assignment (i_0, i_1) ↦ tilde τ_i_0|_U_i_0i_1 - tilde τ_i_1|_U_i_0i_1. This…","statement_latex":"(Definition of delta.) Suppose that\n$$\n0 \\to {\\mathcal F}_1 \\to {\\mathcal F}_2 \\to {\\mathcal F}_3 \\to 0\n$$\nis a short exact sequence of abelian sheaves on any topological space $X$.\nThe boundary map\n$\\delta : H^0(X, {\\mathcal F}_3) \\to {\\check H}^1(X, {\\mathcal F}_1)$\nis defined as follows. Take an element $\\tau \\in H^0(X, {\\mathcal F}_3)$.\nChoose an open covering ${\\mathcal U} : X = \\bigcup_{i\\in I} U_i$ such\nthat for each $i$ there exists a section $\\tilde \\tau_i \\in {\\mathcal F}_2$\nlifting the restriction of $\\tau$ to $U_i$. Then consider the assignment\n$$\n(i_0, i_1) \\longmapsto\n\\tilde \\tau_{i_0}|_{U_{i_0i_1}} - \\tilde \\tau_{i_1}|_{U_{i_0i_1}}.\n$$\nThis is clearly a 1-coboundary in the {\\v C}ech complex\n${\\check C}^\\ast({\\mathcal U}, {\\mathcal F}_2)$. But we observe that\n(thinking of ${\\mathcal F}_1$ as a subsheaf of ${\\mathcal F}_2$) the RHS\nalways is a section of ${\\mathcal F}_1$ over $U_{i_0i_1}$. Hence we\nsee that the assignment defines a 1-cochain in the complex\n${\\check C}^\\ast({\\mathcal U}, {\\mathcal F}_2)$. The cohomology\nclass of this 1-cochain is by definition {\\it $\\delta(\\tau)$}.","area":"Sheaf Cohomology","chapter":"Exercises","chapter_id":"exercises","section":"v Cech Cohomology","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02AP","source_file":"exercises.tex","source_line":3957,"source_end_line":3981,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L3957-L3981","statement_sha256":"d6e2d6cfad0df961331cfdbe1daf277d90a8ecf204c208d5be8006009e008865","origin":"The Stacks Project","memory_eligible":false,"source_rank":14752,"rank":14752,"depth":0,"x":1040.861,"y":553.575,"cluster":"sheaf-cohomology"},{"id":"stacks:02AR","tag":"02AR","title":"Divisors · Definition 02AR","summary":"Throughout, let S be any scheme and let X be a Noetherian, integral scheme. • A Weil divisor on X is a formal linear combination Sigma n_i[Z_i] of prime divisors Z_i with integer coefficients. • A prime divisor is a closed subscheme Z ⊂ X, which is integral with generic point xi ∈ Z such that O_X, xi has dimension 1. We will use the notation O_X, Z = O_X, xi when xi ∈ Z ⊂ X is as above. Note that O_X, Z ⊂ K(X) is a subring of the function field of X. • The Weil divisor…","statement_latex":"Throughout, let $S$ be any scheme and let\n$X$ be a Noetherian, integral scheme.\n\\begin{enumerate}\n\\item A {\\it Weil divisor} on $X$ is a formal linear combination\n$\\Sigma n_i[Z_i]$ of prime divisors $Z_i$ with integer coefficients.\n\\item A {\\it prime divisor} is a closed subscheme $Z \\subset X$,\nwhich is integral with generic point $\\xi \\in Z$ such that\n${\\mathcal O}_{X, \\xi}$ has dimension $1$. We will use the notation\n${\\mathcal O}_{X, Z} = {\\mathcal O}_{X, \\xi}$\nwhen $\\xi \\in Z \\subset X$ is as above. Note that ${\\mathcal O}_{X, Z} \\subset\nK(X)$ is a subring of the function field of $X$.\n\\item The {\\it Weil divisor associated to a rational function\n$f \\in K(X)^\\ast$} is the sum $\\Sigma v_Z(f)[Z]$. Here $v_Z(f)$ is\ndefined as follows\n\\begin{enumerate}\n\\item If $f \\in {\\mathcal O}_{X, Z}^\\ast$ then $v_Z(f) = 0$.\n\\item If $f \\in {\\mathcal O}_{X, Z}$ then\n$$\nv_Z(f) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(f)).\n$$\n\\item If $f = \\frac{a}{b}$ with $a, b \\in {\\mathcal O}_{X, Z}$\nthen\n$$\nv_Z(f) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(a)) -\n\\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(b)).\n$$\n\\end{enumerate}\n\\item An {\\it effective Cartier divisor} on a scheme $S$\nis a closed subscheme $D \\subset S$ such that every point $d\\in D$\nhas an affine open neighbourhood $\\Spec(A) = U \\subset S$ in $S$\nso that $D \\cap U = \\Spec(A/(f))$ with $f \\in A$ a nonzerodivisor.\n\\item The {\\it Weil divisor $[D]$ associated to an effective\nCartier divisor $D \\subset X$} of our Noetherian integral\nscheme $X$ is defined as the sum $\\Sigma v_Z(D)[Z]$ where\n$v_Z(D)$ is defined as follows\n\\begin{enumerate}\n\\item If the generic point $\\xi$ of $Z$ is not in $D$\nthen $v_Z(D) = 0$.\n\\item If the generic point $\\xi$ of $Z$ is in $D$\nthen\n$$\nv_Z(D) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(f))\n$$\nwhere $f \\in {\\mathcal O}_{X, Z} = {\\mathcal O}_{X, \\xi}$ is the nonzerodivisor\nwhich defines $D$ in an affine neighbourhood of $\\xi$ (as in (4) above).\n\\end{enumerate}\n\\item Let $S$ be a scheme. The {\\it sheaf of total quotient\nrings ${\\mathcal K}_S$} is the sheaf of ${\\mathcal O}_S$-algebras which is\nthe sheafification of the pre-sheaf ${\\mathcal K}'$ defined as follows.\nFor $U \\subset S$ open we set ${\\mathcal K}'(U) = S_U^{-1}{\\mathcal O}_S(U)$\nwhere $S_U \\subset {\\mathcal O}_S(U)$ is the multiplicative subset\nconsisting of sections $f \\in {\\mathcal O}_S(U)$ such that the germ\nof $f$ in ${\\mathcal O}_{S, u}$ is a nonzerodivisor for every $u\\in U$.\nIn particular the elements of $S_U$ are all nonzerodivisors.\nThus ${\\mathcal O}_S$ is a subsheaf of ${\\mathcal K}_S$, and we get a\nshort exact sequence\n$$\n0 \\to {\\mathcal O}_S^\\ast \\to {\\mathcal K}_S^\\ast \\to\n{\\mathcal K}_S^\\ast/{\\mathcal O}_S^\\ast \\to 0.\n$$\n\\item A {\\it Cartier divisor} on a scheme $S$ is a global\nsection of the quotient sheaf ${\\mathcal K}_S^\\ast/{\\mathcal O}_S^\\ast$.\n\\item The {\\it Weil divisor associated to a Cartier divisor}\n$\\tau \\in \\Gamma(X, {\\mathcal K}_X^\\ast/{\\mathcal O}_X^\\ast)$ over our\nNoetherian integral scheme\n$X$ is the sum $\\Sigma v_Z(\\tau)[Z]$ where $v_Z(\\tau)$ is defined\nas by the following recipe\n\\begin{enumerate}\n\\item If the germ of $\\tau$ at the generic point $\\xi$\nof $Z$ is zero -- in other words the image of $\\tau$ in the stalk\n$({\\mathcal K}^\\ast/{\\mathcal O}^\\ast)_\\xi$ is ``zero'' -- then $v_Z(\\tau) = 0$.\n\\item Find an affine open neighbourhood $\\Spec(A) = U \\subset X$\nso that $\\tau|_U$ is the image of a section $f \\in {\\mathcal K}(U)$\nand moreover $f = a/b$ with $a, b \\in A$. Then we set\n$$\nv_Z(f) = \\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(a)) -\n\\text{length}_{{\\mathcal O}_{X, Z}}({\\mathcal O}_{X, Z}/(b)).\n$$\n\\end{enumerate}\n\\end{enumerate}","area":"Divisors & Intersection Theory","chapter":"Exercises","chapter_id":"exercises","section":"Divisors","kind":"definition","layer":"informal","status":"Published source","source_url":"https://stacks.math.columbia.edu/tag/02AR","source_file":"exercises.tex","source_line":4784,"source_end_line":4866,"source_git_url":"https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/exercises.tex#L4784-L4866","statement_sha256":"227ad853780f24200b5174308b40abe9e6b8bfe2c82d2eaa894e9c70e5d61c34","origin":"The Stacks 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